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Sheridan and Smith]Nick Sheridan and Ivan Smith Nick Sheridan, School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, U.K. Ivan Smith, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, U.K. Abstract: We study the symplectic topology of certain K3 surfaces (including the “mirror quartic” and “mirror double plane”),equipped with certain Kähler forms.In particular, we prove that the symplectic Torelli group may be infinitely generated, and derive new constraints on Lagrangian tori. The key input, via homological mirror symmetry, is a result of Bayer and Bridgeland on the autoequivalence group of the derived category of an algebraic K3 surface of Picard rank one.Symplectic topology of K3 surfaces via mirror symmetry [ Received September 15, 1996; accepted March 16, 1997 ========================================================§ INTRODUCTION Let (X,ω) be a simply-connected closed symplectic 4-manifold.Elucidating the structure of the symplectic mapping class group π_0(X,ω), and of its natural map to π_0(X), are central aims in four-dimensional topology.This paper brings to bear insight from homological mirror symmetry, in the specific case of (certain symplectic forms on) K3 surfaces. §.§ Context and questions Many aspects of π_0(X) seem a priori out of reach, since we have no knowledge of π_0_(^4) – there is therefore no smooth closed four-manifold for which the smooth mapping class group is known –but in the symplectic setting we have Gromov's theorem _(^2,ω_) ≃{1}, via his spectacular determination that (^2,ω_) ≃ U(3) and (^1 ×^1, ω⊕ω) ≃ (SO(3) × SO(3)) ⋊/2. Away from the setting of rational or ruled surfaces, much less is known.We restrict our attention to the symplectic mapping class group and its variants:G(X,ω)π_0(X,ω)(symplectic mapping class group)I(X,ω)[G(X,ω) →(H^*(X;ℤ))](symplectic Torelli group)K(X,ω)[G(X,ω) →π_0(X)] (smoothly trivial symplectic mapping class group).Results of Seidel <cit.> and Tonkonog <cit.> show that K(X,ω) is infinite for “most” even-dimensional hypersurfaces in projective space, and Seidel <cit.> has shown that for certain symplectic K3 surfaces K(X,ω) contains a subgroup isomorphic to the pure braid group PB_m for m ≤ 15.However, very little is known about the structure of G(X,ω) itself. For instance, the following questions[These attributions are folkloric.] remain open in almost all cases: () (Seidel) Is G(X,ω) finitely generated?() (Fukaya) Does G(X,ω) have finitely many orbits on the set of Lagrangian spheres in X?() (Donaldson) Is I(X,ω) generated by squared Dehn twists in Lagrangian spheres?Contrast with the (symplectic) mapping class group Γ_g of a closed oriented surface Σ_g of genus g, which is finitely presented, generated by Dehn twists, and acts with finitely many orbits on the infinite set of isotopy classes of simple closed curves on Σ_g, or the classical Torelli group, which is infinitely generated when g=2 and finitely generated when g>2.One also has long-standing questions concerning the structure of Lagrangian submanifolds in a complex projective manifold, for example: () (Donaldson <cit.>) Do all Lagrangian spheres in a complex projective manifold arise from the vanishing cycles of deformations to singular varieties? A negative answer for certain rigid Calabi-Yau threefolds was very recently given in <cit.>.More specific to a symplectic Calabi–Yau surface (X,ω), we have:()(Seidel) Does every Lagrangian torus T^2 ⊂ X of Maslov class zero represent a non-zero class in H_2(X,)? The corresponding result for the four-dimensional torus (with the standard flat Kähler form)has a positive answer, see <cit.>. §.§ Sample results We are able to give partial answers to some of these questions, in particular we prove: There is a symplectic K3 surface (X,ω) for which the symplectic Torelli group I(X,ω) surjects onto a free group of (countably) infinite rank, and in particular is infinitely generated. To be more explicit, the result holds for ambient-irrational Kähler forms on either of the following two K3 surfaces: * X is the “mirror quartic”, i.e. the crepant resolution of a quotient of the Fermat quartic hypersurface in ^3 by (/4)^2;* X is the “mirror double plane”, i.e. the crepant resolution of a quotient of the Fermat sextic hypersurface in (1,1,1,3) by the group /6 ×/2.In both cases, X ⊂ Y is the proper transform of an anticanonical hypersurface X̅⊂Y̅ in the toric variety Y̅ associated to an appropriate reflexive simplex , where Y →Y̅ is a toric resolution of singularities that is chosen in such a way that X is a crepant resolution of X̅.A Kähler form on X is “ambient” if it is restricted from Y, and “ambient-irrational” if it is ambient and [ω]^⊥∩ H^2(X,) is as small as possible amongst ambient Kähler forms.Although we only establish Theorem <ref> in these rather specific cases, the underlying strategy – which extracts information on symplectic mapping class groups from homological mirror symmetry – is of interest in itself, and has much broader potential.As we elaborate in Section <ref> below, this paper essentially shows that Theorem <ref> holds whenever one has proved homological mirror symmetry for (X,ω), and has proved Bridgeland's conjecture on the autoequivalences of the derived category of its mirror algebraic K3 surface X^∘.In particular, it seems likely that ongoing developments will give the same conclusion in much greater generality, cf. for instance Remark <ref>. In fact we prove a stronger result: the surjection from I(X,ω) onto an infinite-rank free group remains surjective when restricted to the subgroup K(X,ω) ≤ I(X,ω) (cf. Corollary <ref>).This has the following consequence, suggested to us by Dietmar Salamon.Following<cit.>, if _0(X) denotes the group of diffeomorphisms which are smoothly isotopic to the identity, then _0(X) acts on Ω = {a∈Ω^2(X) | ais symplectic and cohomologous to ω}via the map f ↦ (f^-1)^*ω. Moser's theorem shows the action is transitive on the connected component containing ω, with stabiliser _0(X) = (X,ω) ∩_0(X). The long exact sequence of homotopy groups for the associated Serre fibration yields a surjective homomorphism π_1(Ω,ω) ↠ K(X,ω).It follows that the fundamental group of the space of symplectic formson X isotopic to ω surjects onto an infinite-rank free group.See also Lemma <ref>. Seidel's question () above, in the case of a symplectic K3 surface (X,ω), was motivated by the following beautiful line of thought, explained to us by Seidel: suppose that L ⊂ (X,ω) is a Maslov-zero Lagrangian torus with vanishing homology class, and X^∘ is a K3 mirror to (X,ω).There exist non-commutative deformations of (X^∘) which destroy all point-like objects (i.e., for which all point-like objects become obstructed); therefore the corresponding symplectic deformations of (X,ω) should destroy the point-like object L.However if [L] = 0, then we can deform L to remain Lagrangian under any deformation of the symplectic form by a Moser-type argument: a contradiction, so such L could not have existed. We use a variation on this idea to prove:Let X be a mirror quartic or mirror double plane, and ω any ambient Kähler form on X. Then every Maslov-zero Lagrangian torus L⊂ (X,ω) has non-trivial homology class. The Maslov class hypothesis is obviously necessary, since there are Lagrangian tori in a Darboux chart. Under stronger hypotheses one can say more; in particular, if the ambient Kähler form is ambient-irrational, then Maslov-zero Lagrangian tori are homologically primitive.Some of our results do not concern the groups G(X,ω) and I(X,ω) themselves, but rather their homological algebraic cousin (X), where (X) is the split-closed derived Fukaya category.For instance, we show that for the particular symplectic K3 surfaces considered in Theorem <ref>,the corresponding Torelli-type group [(X) → HH_*((X))] is indeed generated by squared Dehn twists, whilst (X) itself is finitely presented, but not generated by Dehn twists, cf. Corollary <ref>.In the same vein, we give positive answers to weakened versions of questions () and () in certain circumstances:Let X be a mirror quartic or mirror double plane, and ω an ambient-irrational Kähler form on X. Then: * Every Lagrangian sphere is Fukaya-isomorphic to a vanishing cycle.* G(X,ω) acts transitively on the set of Fukaya-isomorphism classes of Lagrangian spheres. In the body of this paper we will only consider four-dimensional symplectic Calabi–Yau manifolds, for which a complete construction of the Fukaya category can be carried out using classical pseudoholomorphic curve theory. It is interesting to note that our results would have implications in higher dimensions, once the relevant foundational theory is established. For instance, let (X,ω) be a symplectic K3 surface as in Theorem <ref>. Consider the product Z = (X× S^2, ω⊕ω_FS). Conjecturally, (Z) ≃(X) ⊕(X)splits as a direct product of two copies of (X), compare to <cit.>.Given this, one infers that the symplectic Torelli group I(Z) surjects onto an infinite-rank free group. By contrast, for simply-connected manifolds of dimension ≥ 5, Sullivanshowed <cit.> that the differentiable Torelli group [π_0(Z) → H^*(Z)] is commensurable with an arithmetic group, in particular is finitely presented. Thus, such a stabilised version of Theorem <ref> would be intrinsically symplectic in nature. (Our lack of knowledge of four-dimensional smooth mapping class groups makes it hard to draw precisely the same contrast in the setting of Theorem <ref> itself.)§.§ Outlines The results above are obtained by combining two main ingredients: * the proof of homological mirror symmetry for Greene–Plesser mirrors, cf. <cit.>;* Bayer and Bridgeland's proof <cit.> of Bridgeland's conjecture <cit.> on the group of autoequivalences of the derived category (X^∘) of a K3 surface X^∘, in the case that X^∘ has Picard rank one.An important point is that Bayer and Bridgeland prove Bridgeland's conjecture for Picard rank one K3 surfaces, but the higher-rank case remains open.In particular, their proof does not apply to the mirror of the quartic surface (which has Picard rank 19).That is why we cannot use Seidel's proof of homological mirror symmetry in that case<cit.> as input for our results, but must instead use the more general results proved in <cit.>.Another slightdelicacy is that the mirror to a symplectic K3 surface (X,ω), in the sense of homological mirror symmetry, is an algebraic K3 surface over the Novikov field Λ, whilst the work of Bayer and Bridgeland, at least as written, is for K3 surfaces over . We circumvent this using standard ideas around the “Lefschetz principle”, i.e. the fact that any algebraic K3 surface over a field of characteristic zero is in fact defined over a finitely generated extension field of the rationals, and such fields admit embeddings in .We now explain how we combine these two ingredients to obtain information about symplectic mapping class groups.We study G(X,ω) via its action on the Fukaya category.It turns out always to act by “Calabi–Yau” autoequivalences, and the action is only well-defined up to the action of even shifts, so that we have a homomorphismG(X,ω) →_CY(X,ω)/[2].If X^∘ is homologically mirror to (X,ω), then the autoequivalence group of (X,ω) can be identified with the autoequivalence group of the derived category (X^∘).The Bayer–Bridgeland theorem identifies the subgroup of Calabi–Yau autoequivalences of (X^∘), modulo even shifts, with π_1(), where = _Käh(X^∘) is a version of the “Kähler moduli space” of X^∘ defined via Bridgeland's stability conditions.We identify _Käh(X^∘) with the “complex moduli space” _cpx(X,ω), and construct a symplectic monodromy homomorphism π_1(_cpx(X,ω)) → G(X,ω).We show that the compositionπ_1() → G(X,ω) →_CY(X,ω)/[2] ≅_CY(X^∘)/[2] ≅π_1()is an isomorphism. Considering the respective Torelli subgroups, we have a similar compositionπ_1() → I(X,ω) →^0 (X,ω)/[2] ≅^0 (X^∘)/[2] ≅π_1()which is also an isomorphism, whereis a certain cover of(denoted Ω^+_0(X,ω), or alternatively Ω^+_0(X^∘), in the body of the paper).It follows, in particular, that I(X,ω) surjects onto π_1().Theorem <ref> then follows from the fact thatis the complement of a countably infinite discrete set of points in the upper half plane, and in particular its fundamental group is a free group of countably infinite rank. §.§ Further questions The fact that the compositions (<ref>), (<ref>) are isomorphisms implies thatG(X,ω)≅Z(X,ω)⋊π_1()andI(X,ω)≅Z(X,ω) ⋊π_1() ,whereZ(X,ω)[G(X,ω) →(X,ω)/[2]]denotes the “Floer-theoretically trivial” subgroup of the symplectic mapping class group (it is contained in I(X,ω)).Thus, we have essentially reduced the problem of computing G(X,ω) and I(X,ω) to the problem of computing Z(X,ω).This raises the following:() Let (X,ω) be a symplectic K3 whose Fukaya category is non-degenerate (i.e., the open-closed map hits the unit). Can Z(X,ω) be non-trivial? If the answer to () is negative for ambient-irrational Kähler forms on the mirror quartic or mirror double plane, then our results show that the answers to () and () are positive in those cases.The caveat of non-degeneracy of the Fukaya category is added to () to avoid a negative answer due to the symplectic manifold “not having enough Lagrangians to detect symplectomorphisms”. In a similar vein, Theorem <ref> reduces questions () and () to the following general:() Can there exist Fukaya-isomorphic but non-Hamiltonian-isotopic Lagrangian spheres in a symplectic K3?AcknowledgementsI.S. is indebted to Daniel Huybrechts forpatient explanations of the material of Section <ref>.He is furthermore grateful to Arend Bayer, Tom Bridgeland,Aurel Page, Oscar Randal-Williams,Tony Scholl and Richard Thomas forhelpful conversations and correspondence.Both authors are grateful to the anonymous referees for their queries and suggestions, which have greatly improved the exposition.N.S. was partially supported by a Sloan Research Fellowship, a Royal Society University Research Fellowship, and by the National Science Foundation through Grant number DMS-1310604 and under agreement number DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.N.S. also acknowledges support from Princeton University and the Institute for Advanced Study. I.S. was partially supported by a Fellowship from EPSRC.§ PRELIMINARIES In this section we define some of the moduli spaces that will arise in this paper, following the notation of <cit.> closely. The moduli spaces will appear in different ways on the two sides of mirror symmetry, so here we simply give the abstract definitions. §.§ Lattices In this paper, a lattice is a free -module of finite rank equipped with a non-degenerate integral symmetric bilinearpairing (·,·).We will sometimes write x^2 for (x,x). We denote the set of (-2)-classes in a lattice L byΔ(L) {δ∈ L | (δ,δ) = -2}.If L is a lattice, we denote by L^- the lattice with the same underlying -module but the sign of the pairing reversed. If A is an abelian group, we denote L_AL ⊗_ A.If A is a -algebra, then the pairing on L induces a symmetric bilinear pairing on L_A.Given A ⊂ B and a subset M ⊂ L_A, we define M^⊥_B {n ∈ L_B |(n,m) = 0for all m ∈ M}.For m ∈ L_A we will denote m^⊥_B {m}^⊥_B (and if A=B we will drop the subscript `B').In particular, for δ∈Δ(L), we will write δ^⊥_ and δ^⊥_ for the orthogonals to δ in the corresponding real or complex vector spaces.For the rest of the paper, let L = U^⊕ 3⊕ E_8^⊕ 2 be the “K3 lattice” (isomorphic to H^2(X,) equipped with the cup product pairing, for any complex K3 surface X; here U is the hyperbolic lattice, with matrix [ 0 1; 1 0 ] in an appropriate basis).L has signature (3,19).§.§ Abstract moduli spaceThe period domain associated to a lattice N is[ Our convention is that if V is a complex vector space, then its projectivization (V) is the set of complex lines in V.]Ω(N) {Ω∈(N_) | (Ω,Ω) = 0, (Ω, Ω) > 0 }.Ω(N) is an open complex submanifold of a quadric hypersurface in (N_).The group of lattice automorphisms (N) acts on Ω(N) on the left. For the rest of this section, let N be a lattice of signature (2,t).Observe that Ω(N) has two connected components, distinguished by the orientation of the positive-definite two-plane spanned by the real and imaginary parts of Ω, and interchanged by complex conjugation.We denote them by Ω^±(N).We define Ω^±_(N) Ω^±(N) ∖⋃_δ∈Δ(N)(δ^⊥_). We define Γ(N) ⊂(N) to be the subgroup of isometries acting trivially on the discriminant group N^*/N.We define Γ^+(N) ⊂Γ(N) to be the subgroup preserving Ω^+(N). (It may have index one or two, cf. <cit.>.) The significance of Γ(N) is explained by the following, which is an immediate consequence of <cit.> (cf. <cit.>): Suppose that N is a primitive sublattice of the unimodular lattice M. Then the image of the natural embedding{σ∈(M): σ|_N^⊥ = 𝕀}↪(N)is Γ(N). We define the topological stack(N)[Ω^+(N)/Γ^+(N)], and similarly _(N). (The curious reader may consult <cit.> on the general theory of topological stacks and their fundamental groups – however the only topological stacks in this paper will be global quotient stacks, and they will only appear via their stacky fundamental groups, whose elementary definition we now give.)The stacky fundamental group π_1((N),[p]), by definition, consists of all pairs (γ,y) where γ∈Γ^+(N) and y is a homotopy class of paths from p to γ· p.The composition (γ_1,y_1) · (γ_2,y_2) is defined to be (γ_1 ·γ_2, (γ_2 · y_1) # y_2) where `#' denotes concatenation of paths.Thus there is a short exact sequence@R=0em1 [r]π_1(Ω^+_(N),p) [r]π_1(_(N),[p]) [r]Γ^+(N) [r] 1 y @|->[r](1,y)(γ,y) @|->[r]γ.If Γ^+(N) acts properly discontinuously then this stacky fundamental group coincides with the fundamental group of the manifold Ω^+(N)/Γ^+(N); more generally, if (N) is an orbifold (as will be the case in this paper) then the stacky fundamental group coincides with the orbifold fundamental group.§.§ Kähler moduli space of a complex K3 A complex K3 surface is a simply connected compact complex surface X with trivial canonical sheaf K_X ≅𝒪_X.General references on K3 surfaces include <cit.>. The Picard lattice of a complex K3 is (X) = (X) = H^2(X,)∩ H^1,1(X,), equipped with the cup-product pairing; this is a lattice of rank 0 ≤ρ(X) ≤ 20. If X is algebraic it has signature (1,ρ(X) - 1) (non-algebraic K3 surfaces may have Picard rank zero).The cohomology H^*(X,) carries a polarized weight-two Hodge structure, whose algebraic part is(X)H^0(X, ) ⊕(X) ⊕ H^4(X, ),and whose polarization is given by the Mukai symmetric form⟨ (r_1,D_1,s_1),(r_2,D_2,s_2)⟩ =D_1· D_2-r_1s_2 -r_2s_1. If X is algebraic, then (X) ≅ U ⊕(X) has signature (2,ρ(X)).We briefly explain how the lattice (X) is related to the bounded derived category (X). Any object E of (X) has a Mukai vector v(E) = ch(E) √(td (X))∈(X).Riemann–Roch takes the formχ(E,F).∑_i (-1)^i ^i(E,F)=-(v(E),v(F)).As a consequence, any spherical object[Recall that an object E is called spherical if ^i(E,E) is isomorphic to the ith cohomology of the 2-sphere: i.e., it has rank one if i=0,2 and zero otherwise.] E has Mukai vector v(E) ∈Δ((X^∘)).Let X be an algebraic K3 surface. DefineΩ^+_0(X)Ω^+_0((X)),Γ^+(X)Γ^+((X)),and _Käh(X)_((X)).We call the latter the Kähler moduli space of X.§.§ Complex moduli space of a symplectic K3 A symplectic K3 is a symplectic manifold (X,ω) which is symplectomorphic to a complex K3 surface equipped with a Kähler form.Given a symplectic K3, we define the lattice(X,ω)[ω]^⊥∩ H^2(X,),equipped with the intersection pairing.We briefly explain how the lattice (X,ω) is related to the Fukaya category.Any oriented Lagrangian submanifold of X (and in particular, any object of the Fukaya category) has a homology class [L] ∈(X,ω).If K and L are objects of the Fukaya category, we haveχ(K,L) ∑_i (-1)^i ^i(K,L) = -[K]· [L],because both sides can be identified as a signed count of intersection points between K and L.[More generally, we have χ(K,L) = (-1)^n(n+1)/2 [K]· [L] where n is half the real dimension on X.]As a consequence, any spherical object (in particular, any Lagrangian sphere) has homology class [L] ∈Δ((X,ω)).Suppose that (X,ω) has signature (2,t).DefineΩ^+_0(X,ω)Ω^+_0((X,ω)),Γ^+(X,ω)Γ^+((X,ω)),and _cpx(X,ω)_((X,ω)).We call the latter the complex moduli space of (X,ω). Remarks <ref> and <ref> give some evidence for the following conjecture, which is implicit in <cit.>: If a symplectic K3 (X,ω) is mirror to a complex K3 X^∘, then we have an isometry(X,ω) ≅(X^∘).As a consequence, when (X,ω) and X^∘ are mirror there is an isomorphism_cpx(X,ω) ≅_Käh(X^∘). It may seem peculiar that Definition <ref> is only given when (X,ω) has signature (2,t).However this condition is natural in light of Conjecture <ref>, which shows that it is expected to hold whenever the mirror X^∘ is algebraic. A version of the complex moduli space, which does not require the assumption on the signature of (X,ω), is defined in Section <ref>.(It is explained in Section <ref> why _cpx(X,ω) is superior when it is defined.)We briefly explain the connection with <cit.>.If M is a lattice of signature (1,t), Dolgachev defines an “M-polarized K3 surface” to be one equipped with an embedding M ↪(X).Given an embedding M ↪ L, he identifies Ω^+_0(M^⊥) as the “moduli space of ample marked M-polarized K3 surfaces”, and _0(M^⊥) as the “moduli space of ample M-polarized K3 surfaces”.He claims that an M-polarized K3 surface X, equipped with a Kähler form ω satisfying [ω] ∈ M_, should be mirror to an M^∘-polarized K3 surface X^∘, where M^⊥≅ U ⊕ M^∘.If [ω] ∈ M_ is “irrational”, then (X,ω) = M^⊥; and if X^∘ is very general, then (X^∘) = M^∘.This gives rise to Conjecture <ref>. §.§ Picard rank one case We will be interested in the case that (X,ω) is a Kähler K3 surface mirror to an algebraic K3 surface X^∘.According to Conjecture <ref>, this means that(X,ω) ≅(X^∘) ≅ U ⊕(X^∘)where (X^∘) has signature (1,ρ(X^∘) - 1) and is furthermore even (since the K3 lattice is even).The smallest such examples are (X^∘) = ⟨ 2n ⟩ for n ≥ 1.Dolgachev has computed the relevant moduli spaces explicitly in this case <cit.>.The space Ω^+(U ⊕⟨ 2n⟩) is isomorphic to the upper half plane 𝔥{z ∈:z > 0} via the map𝔥 →Ω^+(U ⊕⟨ 2n⟩) z↦ [-nz^2:1:z].The set Δ(U ⊕⟨ 2n ⟩) consists of all (a,b,c) ∈^3 satisfying ab+nc^2+1 = 0.Given such δ = (a,b,c), the intersection of (δ^⊥_) with Ω^+(U ⊕⟨ 2n ⟩) corresponds to the single point p_δ = c/b + i/b√(n)∈𝔥. Therefore, the subset Ω^+_0(U ⊕⟨ 2n ⟩) corresponds to the subset𝔥_0 𝔥∖{.c/b + i/b√(n)| c ∈,b ∈_>0,b|nc^2+1}.The group Γ^+(U ⊕⟨ 2n⟩) is isomorphic to the Fricke modular group Γ_0^+(n) ⊂PSL(2,), which is generated by the matrices ( [ a b; c d ]) ∈SL(2,) withn|c , and ( [ 0 -1/√(n);√(n) 0 ]).§.§ More moduli spaces Still assuming N to be a lattice of signature (2,t), we define(N) {Ω∈ N_| (Ω,Ω) = 0, (Ω,Ω)>0}.It is clearly a ^*-bundle over Ω(N). Now observe that (N) consists of classes Ω∈ N_ satisfying(Re Ω)^2 = (Im Ω)^2 > 0, (Re Ω, Im Ω) = 0.Thus it can be identified with the set of conformal bases for positive-definite two-planes in N_.We now define (N) ⊂ N_ to be the open subset of vectors whose real and imaginary parts span a positive-definite two-plane. The subspace (N) ⊂(N) is a deformation retract.By Gram–Schmidt. Furthermore, by taking conformal bases, one sees that (N) is a GL^+(2,)-bundle over Ω(N) and (N) ⊂(N) a CO^+(2,)-subbundle, where CO^+(2,) ≅^+ ×SO(2,) is the group of orientation-preserving conformal transformations of ^2.We denote the component of (N) lying over Ω^±(N) (respectively, Ω^±_(N)) by ^±(N) (respectively, ^±_(N)), and define ^±_(N) ⊂^±(N) similarly. We define_0(N) [_0^+(N)/Γ^+(N)], _Käh(X^∘)_0((X^∘)) if X^∘ is an algebraic K3, and _cpx(X,ω)_0((X,ω)) if (X,ω) has signature (2,t).§ SYMPLECTIC MAPPING CLASS GROUPS OF K3 SURFACESIn this section we explain how monodromy can be used to construct classes in the symplectic mapping class group of a K3 surface equipped with a Kähler form. §.§ Symplectic monodromy of Kähler manifolds Let p:→ B be a family of compact complex manifolds equipped with a locally-constant family κ of Kähler classes. I.e., p is a proper holomorphic submersion between complex manifolds, and κ_b ∈ H^2(X_b;) is locally constant and Kähler for each b ∈ B. Then there exists a smoothly-varying family of Kähler forms ω_b ∈Ω^1,1(X_b;) representing the Kähler classes κ_b, by <cit.>.Moser's theorem says that the family of symplectic manifolds (X_b,ω_b) is symplectically locally trivial, so there is a monodromy mapπ_1(B,b) → G(X_b,ω_b).Because the space of Kähler forms representing a given Kähler class is convex and therefore contractible, this does not depend on the choice of forms ω_b (again by Moser).We require a generalization of this construction: suppose that we have a discrete group Γ acting on the family p:→ B, respecting the classes κ_b (i.e., Γ acts onand B by biholomorphisms, so that p is equivariant, and γ^* κ_γ· b = κ_b).Then we obtain a monodromy mapπ_1([B/Γ],[b]) → G(X_b,ω_b),which sends a class (γ,y) to the composition (X_b,ω_b)(X_γ· b,ω_γ· b)(X_b,γ^* ω_γ· b)(X_b,ω_b).The final arrow is defined using Moser's theorem, and the fact that γ^* ω_γ· b and ω_b are Kähler forms representing the same Kähler class. By construction, the morphisms (<ref>) and (<ref>) fit into a commutative diagram1 [r]π_1(B,b) [d]_-(<ref>)[r]π_1([B/Γ],[b]) [d]_-(<ref>)[r]Γ[r] 1G(X_b,ω_b) @=[r] G(X_b,ω_b)where the top row is exact. §.§ Moduli space of K3 surfaces A marked K3 surface is a pair (X,ϕ) with X a K3 surface and ϕ: H^2(X,) → L a lattice isometry.A marked K3 surface (X,ϕ) has a period pointϕ_(H^2,0(X)) ∈Ω(L).There is a moduli space of marked K3 surfaces, and the period mapping (<ref>) defines a local isomorphism between this moduli space and the 20-dimensional complex manifold Ω(L).The period mapping is furthermore surjective, however it is not injective, and the moduli space is not Hausdorff (see <cit.>). We define a Kähler K3 surface to be a K3 surface X equipped with a Kähler class κ∈ H^1,1(X;); thus a marked Kähler K3 surface is a triple (X,ϕ,κ) where (X,ϕ) is a marked K3 surface and κ∈ H^1,1(X;) is a Kähler class. We define𝒦(L) := { (Ω,κ) ∈Ω(L) × L_: κ^2>0, (Ω,κ)=0}.A marked Kähler K3 surface (X,ϕ,κ) has a period point(ϕ_(H^2,0(X)),ϕ_(κ)) ∈𝒦(L).There is a moduli space of marked Kähler K3 surfaces, and the period mapping (<ref>) defines an isomorphism with the subspace 𝒦_(L) 𝒦(L) ∖⋃_δ∈Δ(L)(δ^⊥_) ×δ^⊥_(see <cit.>). Given κ∈ L_ satisfying κ^2>0, we defineΩ(κ){Ω∈Ω(L): (Ω,κ)=0}⊂Ω(L) Ω_(κ)Ω(κ) ∖⋃_δ∈Δ(L) ∩κ^⊥(δ^⊥_).We observe that κ^⊥ has signature (2,19), and therefore Ω(κ) ⊂((κ^⊥)_) has two connected components Ω^±(κ), distinguished by the orientation of the positive-definite two-plane in κ^⊥ spanned by the real and imaginary parts of Ω and interchanged by complex conjugation.We similarly denote the connected components of Ω_(κ) by Ω^±_(κ). Now the moduli space of marked Kähler K3 surfaces is fine, meaning it carries a universal family →𝒦_(L).It is clear from the definitions that Ω^+_(κ) ×{κ}⊂𝒦_(L), so we may restrict the universal family to this submanifold to obtain a family of K3 surfaces (κ) →Ω^+_(κ) equipped with a locally-constant family of Kähler classes ϕ_^-1(κ).By the previous section we obtain a mapπ_1(Ω^+_(κ)) → G(X,ω)where ω is a Kähler form with class κ.In fact, because the cohomology of the family is globally trivialized by the marking ϕ, the image of this map lies in I(X,ω) ⊂ G(X,ω). Let us consider a neighbourhood of a point on one of the hypersurfaces (δ^⊥_) ∩Ω^+(κ) that get removed to obtain Ω^+_0(κ), which does not intersect any of the other hypersurfaces.Inside this neighbourhood, the family (κ) can be extended across the hypersurface, at the expense that it develops a node along the hypersurface.The resulting vanishing cycle is a Lagrangian sphere L with homology class satisfying ϕ([L]) = ±δ, and the symplectic monodromy around the hypersurface is the squared Dehn twist τ_L^2.In particular, every class in Δ((X,ω)) is the homology class of a vanishing cycle.§.§ Getting rid of the marking The fibres of the family (κ) →Ω^+_0(κ) are marked Kähler K3 surfaces.We now determine which fibres are isomorphic as unmarked Kähler K3 surfaces.We do this by applying the Torelli theorem for K3 surfaces <cit.>, which states that two Kähler K3 surfaces (X_0,κ_0) and (X_1,κ_1) are isomorphic if and only if there is a Hodge isometry ψ: H^2(X_0,)H^2(X_1,) such that ψ_(κ_0) = κ_1 (and the isomorphism, if it exists, is unique). Let κ∈ L_ satisfy κ^2>0. DefineΓ(κ) {σ∈(L): σ_(κ) = κ}.It is clear that Γ(κ) acts on Ω(κ); we define Γ^+(κ) ⊂Γ(κ) to be the subgroup preserving Ω^+(κ). The fibres of the family (κ) →Ω^+_0(κ) over points b_0,b_1 ∈Ω^+_0(κ) are isomorphic, as unmarked Kähler K3 surfaces, if and only if γ· b_0 = b_1 for some γ∈Γ^+(κ). When the isomorphism exists, it is unique. Follows from the Torelli theorem: the required Hodge isometry is ψ = ϕ_1^-1∘σ∘ϕ_0.The action of Γ^+(κ) on Ω^+_(κ) extends uniquely to an action on the family (κ) →Ω^+_(κ), respecting Kähler classes. Given σ∈Γ^+(κ), we consider the family of marked Kähler K3 surfaces (κ)_σ→Ω^+_(κ) obtained from (κ) →Ω^+_(κ) by post-composing the marking with σ^-1.Because (κ) →Ω^+_(κ) is a universal family (see <cit.>), (κ)_σ is classified by a unique map Ω^+_0(κ) →Ω^+_0(κ).By Lemma <ref>, this map coincides with the action of σ on Ω^+_0(κ); and the resulting isomorphism (κ)_σ≅σ^* (κ) provides the desired lift of the action of σ to (κ). Now let (X,ϕ,κ) be a marked Kähler K3 surface, and ω a Kähler form representing ϕ_^-1(κ).It is clear that the image of the mapG(X,ω) → (H^2(X,)) ϕ≃ (L)lies in Γ(κ).We have the following result of Donaldson: The image of (<ref>) lies in Γ^+(κ). It is clear from the definition that the action of Γ^+(κ) respects the Kähler classes of the family (κ) →Ω^+_(κ), so the construction of Section <ref> gives a commutative diagram1 [r]π_1(Ω^+_(κ)) [r] [d]π_1([Ω^+_(κ)/Γ^+(κ)]) [r] [d]Γ^+(κ) [r] @=[d] 1 1 [r] I(X,ω) [r] G(X,ω) [r]Γ^+(κ) [r] 1.Proposition <ref> allows us to write the bottom Γ^+(κ) instead of (H^2(X,)); the bottom `→ 1' can then be added because the map G(X,ω) →Γ^+(κ) is surjective by commutativity of the diagram (cf. <cit.>).§.§ Shrinking the family The family (κ) →Ω^+_(κ) is the universal family of marked Kähler K3 surfaces with Kähler class κ (more precisely, the universal family is the disjoint union of this together with its complex conjugate), and the group Γ^+(κ) is as large as possible by Lemma <ref>, so the monodromy maps appearing in (<ref>) are “the best we can do using the construction of Section <ref>”.However it turns out that in many interesting situations, including those we study in this paper, the family (κ) →Ω^+_0(κ) is “bigger than it needs to be”: we can construct a submanifold Ω^+_0(N_κ) ⊂Ω^+_0(κ) which is a deformation retract.Thus for the purposes of studying the symplectic mapping class group, we might just as well consider the monodromy homomorphism arising from the restriction of the family (κ) →Ω^+_0(κ) to this submanifold.The benefit is twofold: firstly, it is easier to understand Ω^+_0(N_κ) (in the main examples considered in this paper, it is one-dimensional whereas Ω^+_0(κ) is nineteen-dimensional), and hence compute its fundamental group; secondly, it is directly comparable with the moduli space of stability conditions on the mirror. Suppose that κ∈ L_ satisfies κ^2>0, and N_κκ^⊥∩ L is a sublattice of signature (2,t).Then Ω^+_(N_κ) ⊂Ω^+_(κ) is a deformation retract. We first check that Ω^+_(N_κ) does lie inside Ω^+_(κ): this follows from the fact that Δ(N_κ) = Δ(L) ∩κ^⊥.Now let (κ) ⊂ L_ be the set of vectors which are orthogonal to κ and whose real and imaginary parts span a positive-definite two-plane.It has two connected components, and we define ^+(κ) to be the one containing ^+(N_κ).We define ^+_(κ) ^+(κ) ∖⋃_δ∈Δ(N_κ)δ^⊥_,and observe that it is a GL^+(2,)-bundle over Ω^+_(κ) as in Section <ref>.Now observe that we have an orthogonal direct sum decompositionκ^⊥ = (N_κ)_⊕((N_κ^⊥)_∩κ^⊥), and the second summand is negative definite because L has signature (3,19), N_κ has signature (2,t), and κ^2>0.This gives a direct sum decomposition of κ^⊥_, which we use to define a linear retraction onto (N_κ)_.Negative-definiteness of the second summand implies that this retraction respects ^+(κ), and the fact that the second summand is orthogonal to Δ(N_κ) implies that it respects ^+_(κ); hence it retracts the latter onto ^+_(N_κ).The retraction is clearly GL^+(2,)-equivariant, so descends to the required retraction of Ω^+_(κ) onto Ω^+_(N_κ). We define (N_κ,κ) →Ω^+_0(N_κ) to be the restriction of the family (κ) →Ω^+_0(κ) to Ω^+_0(N_κ). We remark that this family of marked K3 surfaces does depend on κ (cf. <cit.>). If N_κ⊂ L is a sublattice of signature (2,t), then Γ^+(N_κ) ≅Γ^+(κ). The isomorphism (<ref>) identifies Γ(N_κ) with{σ∈(L): σ|_N_κ^⊥ = 𝕀},and this is equal to Γ(κ).Indeed, if σ fixes N_κ^⊥ then it fixes κ∈ (N_κ^⊥)_; conversely, if σ∈(L) fixes κ, then (σ - 𝕀) is a primitive subgroup containing κ in its -span, and it is straightforward to show that N_κ^⊥ is the smallest such subgroup.It is clear that the subgroups Γ^+(N_κ) and Γ^+(κ) correspond under this identification. Putting Lemmas <ref> and <ref> together, we have shown: Let X be a K3 surface and ω a Kähler form, such that (X,ω) is a sublattice of signature (2,t). Then we have a morphism of short exact sequences1 [r]π_1(Ω^+_(X,ω)) [r] [d]π_1(_cpx(X,ω)) [r] [d]Γ^+(X,ω) [r] @=[d] 1 1 [r] I(X,ω) [r] G(X,ω) [r]Γ^+(X,ω) [r]1.This diagram is isomorphic to (<ref>).§.§ Examples We consider Kähler K3 surfaces (X,ω) which are mirror to an algebraic K3 surface X^∘ of the smallest possible Picard rank ρ(X^∘) = 1.According to Section <ref>, this means we have(X,ω) ≅(X^∘) ≅ U ⊕⟨ 2n ⟩for some positive integer n.We will focus on the cases n=1,2.The mirrors X^∘ arise geometrically as follows:* A very general quartic hypersurface X^∘⊂^3 has (X^∘) ≅⟨ 4⟩; * A very general sextic hypersurface X^∘⊂(1,1,1,3) (also called a “double plane”) has (X^∘) ≅⟨ 2⟩. Note that in the second case, (1,1,1,3) has an isolated singularity, and the generic hypersurface is disjoint from that point and hence smooth. We will now give explicit descriptions of _0(U ⊕⟨ 2n ⟩) ≅ [𝔥_0/Γ^+_0(n)] in the cases n=1,2.We will also give explicit descriptions of the universal families [(U ⊕⟨ 2n ⟩,κ)/Γ^+(U ⊕⟨ 2n ⟩)] →_0(U ⊕⟨ 2n ⟩) for certain κ, in these cases.The families will be constructed as families of hypersurfaces X⊂ Y, where Y is a certain toric variety.The intersections of X with the toric boundary divisors (which will always be transverse) span a sublattice _tor(X) ⊂(X).We say that a Kähler form ω on X is “ambient” if it is the restriction of a Kähler form on Y, and “ambient-irrational” if it is ambient and furthermore [ω] ∈_tor(X)_ is not contained in any proper rational subspace. In the latter case, we have(X,ω) = _tor(X)^⊥. [The case n=2] We consider the “Dwork family”Q_λ = {x_0^4 +x_1^4+x_2^4+x_3^4+ 4λ x_0x_1x_2x_3 = 0}⊂^3of quartics, parametrized by λ∈^1. These are all invariant under the group Π = /4×/4 of diagonal projective transformations [i^a_0,i^a_1,i^a_2,i^a_3] of ^3 with ∑ a_j = 0.If λ^4≠ 1 then the quotient Q_λ/Π has six A_3-singularities; the fibres over λ^4=1 have an additional nodal singularity. There is atoric resolution of the ambient toric variety ^3/Π, yielding a simultaneous resolution of the Dwork family, which defines a family of K3's over ^1 \{λ^4=1}. This is the Greene–Plesser mirror family to the family of smooth quartics in ^3.A member of this family is called a “mirror quartic”.Any mirror quartic X satisfies _tor(X)^⊥≅ U ⊕⟨ 4 ⟩ (see <cit.>, and also <cit.>).Therefore, if ω is an ambient-irrational Kähler form, we have (X,ω)≅ U ⊕⟨ 4 ⟩,and in particular, _cpx(X,ω)≅_0(U ⊕⟨ 4 ⟩).There is an action of /4 on the family by multiplying one coordinate by i, which covers the action λ↦ iλ on (^1 \{λ^4 = 1}). Dolgachev proves that the quotient [^1/(/4)] is in fact isomorphic to (U ⊕⟨ 4 ⟩), and the complement of {λ^4=1} corresponds to _0(U ⊕⟨ 4 ⟩).He furthermore identifies the quotient of the family by this /4-action, with a family [(U ⊕⟨ 4 ⟩,κ)/Γ^+(U ⊕⟨ 4 ⟩)] for appropriate κ.We can compactify [^1/(/4)] to an orbifold ^1, by adding a point at λ^4=∞.The orbifold has 3 special points: the “cusp” λ^4 = ∞ which must be removed to get (U ⊕⟨ 4 ⟩); the “nodal point” λ^4 = 1 which must further be removed to get _(U ⊕⟨ 4 ⟩); and the order-4 orbifold point λ^4 = 0.[The case n=1] The one-dimensional family of hypersurfacesP_λ = { x_0^6 + x_1^6 + x_2^6 + x_3^2 + λ x_0x_1x_2x_3 }⊂(1,1,1,3)are all invariant under a group Π' = /6 ×/2 of diagonal projective transformations. The simultaneous crepant resolution of the quotient defines the Greene–Plesser mirror family to the family of double planes, and its fibres are called “mirror double planes”.Any mirror double plane X satisfies _tor^⊥≅ U ⊕⟨ 2 ⟩ (see <cit.> and <cit.>).In particular, if ω is an ambient-irrational Kähler form then we have _cpx(X,ω) ≅_0(U ⊕⟨ 2 ⟩).Similarly to the previous example, (U ⊕⟨ 2 ⟩) compactifies to an orbifold ^1 with 3 special points: the cusp, a nodal point, and an order-3 orbifold point. §.§ Consequences We now record some consequences of the explicit descriptions of _(U ⊕⟨ 2n ⟩) that were given in Examples <ref> and <ref>.These can of course be proven more directly, by the techniques that Dolgachev uses to arrive at these explicit descriptions.Suppose N = U ⊕⟨ 2 ⟩ or U ⊕⟨ 4 ⟩.Then Γ^+(N) acts transitively on Δ(N)/±𝕀. There is a bijective correspondence between classes [δ] ∈Δ(N)/±𝕀 and nodal points p_δ∈Ω^+(N) that must be removed to obtain Ω^+_0(N): precisely, (δ^⊥_) ∩Ω^+(N)= {p_δ}.Therefore there is a bijective correspondence between orbits of the action of Γ^+(N) on Δ(N)/±𝕀 and nodal points in (N).We have seen in Examples <ref> and <ref> that (N) contains a unique nodal point when N = U ⊕⟨ 2 ⟩ or U ⊕⟨ 4 ⟩. Suppose N = U ⊕⟨ 2 ⟩ (respectively, N =U ⊕⟨ 4⟩).Then we have identifications π_1(_0(N)) @<->[d]^[origin=c]90∼[r]^-(<ref>) Γ^+(N) @<->[d]^[origin=c]90∼ ∗/p [r]/2 ∗/p,where p=3 (respectively, p=4) and the homomorphism on the bottom row is the obvious one.The identification π_1(_) ≅∗/p is immediate from the description of _ given in Examples <ref> and <ref>: the factorcorresponds to loops around the nodal point, and the factor /p to loops around the orbifold point.By the short exact sequence (<ref>), the kernel of the map π_1(_) →Γ^+ is equal to the image of π_1(Ω^+_).We recall that Ω^+_ is the complement of the infinite set of points p_δ∈𝔥 corresponding to [δ] ∈Δ(N)/±𝕀, so its fundamental group is generated by loops around these points.The covering group acts transitively on these points by Lemma <ref>, so the image of the fundamental group is generated by elements conjugate to the image of a loop around a single point p_δ.A loop around a single point p_δ∈Ω^+ maps to a loop going twice around the nodal point in , which can be chosen to correspond to the element 2 ∈.Therefore the kernel of the map ∗/p →Γ^+ is generated by the elements conjugate to 2 ∈, which means the map can be identified with the projection ∗/p ↠/2 ∗/p as required. If X is a mirror quartic or mirror double plane and ω is ambient-irrational, then G(X,ω) is not generated by Dehn twists, although X contains a Lagrangian sphere.Let N (X,ω). If X is a mirror quartic (respectively, a mirror double plane), we saw in Example <ref> (respectively, Example <ref>) that N = U ⊕⟨ 4 ⟩ (respectively, N = U ⊕⟨ 2 ⟩). Applying Proposition <ref> and Lemma <ref>, we have a surjective homomorphism G(X,ω) ↠Γ^+(N) ≅/2 ∗/p.The Dehn twist in a given vanishing cycle can be arranged to map to 1 ∈/2 (cf. Remark <ref>).We know that the action of a Dehn twist in L on homology is given by the Picard–Lefschetz reflection in the corresponding homology class up to sign, [L] ∈Δ(N)/±𝕀.It follows by Lemma <ref> that all Dehn twists map to elements conjugate to 1 ∈/2.Passing to abelianizations, it follows that the image of a Dehn twist under the mapG(X,ω) ↠ (/2 ∗/p)^ab = /2 ⊕/pis (1,0), so Dehn twists can not generate G(X,ω).§ DERIVED AUTOEQUIVALENCES OF COMPLEX K3S Let X be a complex algebraic K3 surface, and (X) its bounded derived category.In this section we briefly survey what is known about the group (X) of triangulated, -linear autoequivalences of (X), following <cit.> closely. There are no original results in this section. §.§ Action of autoequivalences on cohomology Let K((X)) denote the Grothendieck group of (X). It admits the Euler form (E,F) ↦χ(E,F), whose left- and right-kernels coincide by Serre duality:χ(E,F)=0 ∀ F χ(F,E)=0 ∀ F.The numerical Grothendieck group K_num((X)) is the quotient of the Grothendieck group by this kernel. The Euler pairing descends to it, by construction. The association [E] ↦ v(E) defines an isometryK_num((X)) ≃(X)^-,cf. Remark <ref> (the fact that this is an isomorphism is specific to K3 surfaces: see <cit.>).Triangulated autoequivalences clearly act on the numerical Grothendieck group by isometries, so we obtain a homomorphismϖ_K(X) ⟶(X). Recall from Section <ref> that H^*(X,) carries a polarized weight-two Hodge structure, of which (X) is the algebraic part. The action (<ref>) of derived autoequivalences on the algebraic part extends to an action on the whole polarized Hodge structure: i.e., there is a group homomorphismϖ(X)⟶ H^*(X,)to the group of Hodge isometries, such that ϖ_K = ϖ|_(X). Indeed, by a theorem of Orlov <cit.>, any autoequivalence is of Fourier–Mukai type, meaning it has the form π_2*(E ⊗π_1^*(-)) for some E ∈(X × X) called the Fourier–Mukai kernel. The Mukai vector v(E) ∈ H^*(X × X,) of the kernel induces a correspondence, whose action on cohomology preserves the Hodge filtration, the integral structure and the Mukai pairing.This defines the desired map ϖ. The image of ϖ is the index-2 subgroup ^+ H^*(X,) consisting of Hodge isometries which preserve orientations of positive-definite 4-planes, by work of Huybrechts, Macrì and Stellari:Let ^0 (X) denote the kernel of ϖ. Then there is a short exact sequence1 →^0 (X) →(X) ^+ H^*(X,) → 1. We now define ^+_CY H^*(X,) ⊂^+ H^*(X,) to be the subgroup of Hodge isometries acting by the identity on H^2,0(X,).An autoequivalence is called Calabi–Yau if its image under ϖ lies in this subgroup, and the subgroup of Calabi–Yau autoequivalences is denoted by _CY(X) ⊂(X).There is a short exact sequence1 →^0 (X) →_CY(X) Γ^+(X) → 1. Note that H^2,0(X,) ⊂(X)^⊥_ is not contained in any proper rational subspace, so any ϕ∈_CYH^*(X,) acts by the identity on (X)^⊥.Thus we have an identification _CY H^*(X,) ≃Γ((X)) by Lemma <ref>.An element of _CY H^*(X,) acts by the identity on the positive-definite 2-planes in ((X)^⊥)_, so it preserves orientations of positive-definite 4-planes if and only if the corresponding element of Γ((X)) preserves orientations of positive-definite 2-planes. Therefore the image of ϖ_K|__CY(X) is identified with Γ^+((X)), by Theorem <ref>.§.§ Stability conditions on K3 surfaces Fix a triangulated categorylinear over a field .Supposeis proper, i.e. that ⊕_i ∈_(E,F[i]) is finite-dimensional for all objects E, F.We let() denote the set of stability conditions onwhichare numerical, full and locally-finite in the terminology of <cit.>. Recall that a numerical stability condition σ = (Z,) oncomprises a group homomorphism Z: K_num() →, called the central charge, and a collection of full additive subcategories (ϕ) ⊂ for ϕ∈ which satisfy various axioms, see <cit.>. The main result of op. cit. asserts that the space () has the structure of a complex manifold, such that the forgetful mapπ()⟶_(K_num(),),taking a stability condition to its central charge, is a local isomorphism.[More precisely, <cit.> shows that for each connected component of the space of numerical, locally-finite stability conditions, the forgetful map defines a local isomorphism to a linear subspace V of _(K_num(),), and <cit.> defines such a stability condition to be full if the subspace V is all of _(K_num(),).] The group of triangulated autoequivalences () acts on (); an element Φ∈() acts byΦ (Z,)↦ (Z','),Z'(E)=Z(Φ^-1(E)), '=Φ().Suppose now that X is a complex algebraic K3 surface, and (X) its bounded derived category.We denote (X) := ((X)). We can identify _(K_num((X)),) ≅(X) ⊗ via (<ref>) together with the Mukai pairing, so (<ref>) becomes a local isomorphism π: (X) →(X) ⊗.This map is (X)-equivariant, where the action on (X) ⊗ is via ϖ_K.Recall that (X) has signature (2, ρ(X)).Define _0(X) _0((X)) and let ^+_0(X) be the component containing vectors of the form (1,iω, -ω^2) for an ample class ω∈(X)⊗.Let ^†(X)⊂(X) be the connected component containing the (non-empty) set of “geometric” stability conditions, for which all skyscraper sheaves𝒪_x are stable of the same phase. The map (<ref>) induces a normal covering mapπ: ^†(X) →_0^+(X),whose group of deck transformations is the subgroup of ^0 (X) which preserves ^†(X). §.§ Bridgeland's conjecture, and the Bayer–Bridgeland theorem Bridgeland has conjectured: The component ^†(X) is simply-connected, and preserved by the group (X).Bayer and Bridgeland have proved the conjecture in the Picard rank one case:Conjecture <ref> holds when ρ(X) = 1. We now state some Corollaries of Theorems <ref> and <ref>.The hypothesis ρ(X)=1 can be removed from each if Bridgeland's conjecture holds.If ρ(X) = 1, then there is an isomorphism of short exact sequences1 [r]π_1 (^+_0(X)) [r] @<->[d]^[origin=c]90∼ π_1([^+_0(X)/^+ H^*(X)]) [r] @<->[d]^[origin=c]90∼ ^+ H^*(X,) [r] @=[d] 1 1 [r]^0 (X) [r](X) [r]^-ϖ ^+ H^*(X,)[r] 1.Follows from the Galois correspondence, using the fact that π is (X)-equivariant.If ρ(X)=1, then there is an isomorphism of short exact sequences1 [r]π_1 (^+_0(X)) [r] @<->[d]^[origin=c]90∼ π_1(_Käh(X)) [r] @<->[d]^[origin=c]90∼ Γ^+(X) [r] @=[d] 1 1 [r]^0 (X) [r]_CY(X) [r]^-ϖ_K Γ^+(X) [r] 1.Follows from Corollary <ref>, together with the fact that ^+_0(X) ⊂^+_0(X) is a deformation retract (cf. Lemma <ref>). If ρ(X) = 1, then there is an isomorphism of short exact sequences1 [r]π_1(Ω^+_0(X)) [r] @<->[d]^[origin=c]90∼ π_1(_Käh(X)) [r] @<->[d]^[origin=c]90∼ Γ^+(X) [r] @=[d] 1 1 [r]^0 (X)/[2] [r]_CY(X)/[2] [r]^-ϖ_K Γ^+(X) [r] 1.Recall that ^+_0(X) is a ^*-bundle over Ω^+_0(X).The image of the composition = π_1(^*) →π_1(^+_0(X)) ≅^0 (X)consists of the even shifts (cf. <cit.>).The result now follows from Corollary <ref> by taking a quotient.§ HOMOLOGICAL MIRROR SYMMETRY A version of homological mirror symmetry was proved for certain K3 surfaces in <cit.>.In this section we recall the precise statement, and give some immediate formal consequences of it.These formal consequences are not specific to the setting of mirror symmetry for K3 surfaces: they should work the same for general compact Calabi–Yau mirror varieties.We will prove our main results in Section <ref> by combining these formal consequences with geometric input specific to K3 surfaces. §.§ Homological mirror symmetry statement Let Λ denote the universal Novikov field over :Λ{∑_j=0^∞ c_j · q^λ_j: c_j ∈, λ_j ∈, lim_j →∞λ_j = +∞}.It is an algebraically closed field extension of , with a non-Archimedean valuation: Λ →∪{∞} (∑_j=0^∞ c_j · q^λ_j)min_j{λ_j: c_j ≠ 0}.If c_1(X) = 0, then the Fukaya category (X,ω) is a Λ-linear -graded (non-curved) A_∞-category whose objects are Lagrangian branes: closed Lagrangian submanifolds L ⊂ X equipped with orientations, gradings and Pin structures.A strictly unobstructed Lagrangian brane is a pair (L,J_L) where L is a Lagrangian brane and J_L an ω-compatible almost-complex structure such that there are no non-constant J_L-holomorphic spheres intersecting L, nor non-constant J_L-holomorphic discs with boundary on L.For the purposes of this paper we define (X,ω) to be the Fukaya category of strictly unobstructed Lagrangian branes, since this can be constructed by classical means <cit.>.This restriction on the objects is harmless when X has real dimension ≤ 4 (as is the case for K3 surfaces), because any Lagrangian brane L admits a J_L turning it into a strictly unobstructed Lagrangian brane by <cit.>.[We recall the argument: somewhere-injective discs can be ruled out for generic J_L by standard regularity and dimension-counting, and arbitrary discs are then ruled out by the theorem of Kwon–Oh <cit.> or Lazzarini <cit.>, showing that any such disc contains a somewhere-injective disc in its image.]Let (X,ω) be a symplectic K3, and X^∘ a smooth Calabi–Yau algebraic surface over Λ.We say that (X,ω) is homologically mirror to X^∘ if there is a -graded Λ-linear A_∞ quasi-equivalence(X,ω) ≃(X^∘)where the left-hand side denotes the split-closure of the category of twisted complexes on the Fukaya category (i.e., Π(Tw) in the notation of <cit.>), and the right-hand side denotes a dg enhancement of the bounded derived category of coherent sheaves on X^∘. We recall Batyrev's construction of mirror Calabi–Yau hypersurfaces in toric varieties <cit.>, in the two-dimensional case.Fix polar dual reflexive three-dimensional lattice polytopesand ^∘.Let Ξ_0 denote the set of boundary lattice points of ^∘ which do not lie in the interior of a codimension-one facet. Fix a vector λ∈ (_>0)^Ξ_0.On the A-side, we will consider hypersurfaces in resolutions of the toric variety associated to . The elements of Ξ_0 index the toric divisors which intersect such a hypersurface, and the coefficients λ_κ of λ will determine the cohomology class of the Kähler form on the hypersurface.The vector λ determines a function ψ_λ: ^∘→ as the smallest convex function such that ψ_λ(0) = 0 and ψ_λ(κ) ≥ -λ_κ. We assume: 2em (∗) The decomposition of ^∘ into domains of linearity of ψ_λ is a simplicial refinement Σ of the normal fan Σ̅ to , whose rays are generated by the elements of Ξ_0.In the language of <cit.>, condition (∗) holds if and only if λ lies in the interior of the cone cpl(Σ) of the secondary fan (or Gelfand–Kapranov–Zelevinskij decomposition) of Ξ_0 ⊂^3, where Σ is a simplified projective subdivision of Σ̅.The morphism of fans Σ→Σ̅ induces a birational morphism of the corresponding toric varieties Y →Y̅, and Y is an orbifold.We will consider a smooth Calabi–Yau hypersurface X ⊂ Y which avoids the orbifold points, and which is the proper transform of X̅⊂Y̅, a hypersurface with Newton polytope .There is a toric -Cartier divisor D_λ=∑_κλ_κ· D_κ with support function ψ_λ, which determines a toric Kähler form on Y, which restricts to a Kähler form ω_λ on X. Its cohomology class [ω_λ] is the restriction of PD(D_λ) to X.The pair (X,ω_λ) depends up to symplectomorphism only on (,λ).On the B-side, the polytope ^∘ defines a line bundle over the toric variety Y̅^∘, whose global sections have a basis indexed by the lattice points of ^∘.[In general, this is a toric stack, and one should consider the stacky derived category of hypersurfaces in this stack; in the situations relevant to our applications, the hypersurfaces will be smooth as schemes, and the stacky derived category coincides with the derived category of the underlying scheme.] For any d ∈Λ^Ξ_0 we have a corresponding hypersurfaceX^∘_d { -χ^0 + ∑_κ∈Ξ_0 d_κ·χ^κ = 0 } ⊂ Y̅^∘. Letand ^∘ be polar dual reflexive three-dimensional simplices, and let λ∈ (_>0)^Ξ_0 satisfy (∗). Then there exists a d∈Λ^Ξ_0, with (d) = λ, such that (X,ω_λ) is homologically mirror to X^∘_d.This is <cit.>.We remark that the “embeddedness” and “no ” conditions of op. cit. are automatic in this case. The fact that (d) = λ, with λ satisfying (∗), implies that X^∘_d is smooth, see <cit.>.One can describe the “mirror map” λ↦ d(λ) more precisely, but we shall not need the more precise version in this paper.Theorem <ref> should continue to hold without the assumption that Δ and Δ^∘ are simplices, but it has not been proved.§.§ Examples revisitedWe want to use Theorem <ref> to find examples of K3s (X,ω) which are homologically mirror to an algebraic K3 X^∘ of Picard rank 1, so that we may apply the Bayer–Bridgeland theorem to compute the derived autoequivalences of X^∘.Recall that the moduli space of K3 surfaces is 20-dimensional, and the subspace of algebraic K3 surfaces forms a countable union of hypersurfaces. Recall that _tor(X) ⊂ H^2(X,) is the sublattice spanned by the intersections of X with the toric boundary divisors of Y.Theorem <ref> proves homological mirror symmetry for Kähler classes lying in an open subset of _tor(X)_, matching them up with a mirror family of algebraic K3 surfaces.According to Conjecture <ref>, we expect members of the mirror family to satisfy(X^∘) ≅(X,ω) ⊇_tor(X)^⊥,with equality holding if and only if ω is ambient-irrational.In particular their Picard rank is ≥ 20 -(_tor(X)), so can be equal to 1 only if (_tor(X)) is equal to 19 or 20. One might hope to find an example where _tor(X) has rank 20, and the mirror family is an open subset of the moduli space of K3 surfaces which could potentially intersect infinitely many of the codimension-1 families of K3 surfaces of Picard rank ≥ 1.However this can't happen, because by construction our mirror families consist entirely of algebraic K3s, so must be contained in one of the codimension-1 loci of such.[One might hope to prove a homological mirror symmetry statement involving the bounded derived category of analytic coherent sheaves on a non-algebraic K3 surface, but in general this is a poorly-behaved invariant (not saturated, doesn't admit a generator…). Thus, any meaningful statement of homological mirror symmetry for such K3 surfaces would require a significant revision of the definitions of the categories involved.]Thus the best one can hope for is an example where _tor(X) has rank 19, in which case we expect any ambient-irrational ω to be mirror to a K3 of Picard rank 1. There are precisely two such examples arising from the Greene–Plesser construction, namely the mirror quartics and mirror double planes considered in Section <ref>: Letbe the reflexive 3-simplex withvertices {(1,0,0), (0,1,0), (0,0,1), (-1,-1,-1)}, and ^∘ its polar dual, which is given by the convex hull Conv(0,4e_1,4e_2,4e_3)-(1,1,1), so with vertices {(3,-1,-1), (-1,3,-1), (-1,-1,3), (-1,-1,-1)}. Then Y̅^∘= ^3 is smooth, and the anticanonical hypersurface X^∘_d is a quartic surface over Λ.The toric variety associated tois Y̅ = [^3 / Π] ≅{x_0^4 - x_1x_2x_3x_4 = 0}⊂^4where Π = (/4)^2 is as in Example <ref>. Anticanonical hypersurfaces in Y̅ are hyperplane sections; these include the Π-invariant quarticsQ_λ⊂^3 considered in (<ref>), and in particular the Fermat quartic Q_0. X is the crepant resolution of Q_0 / Π, resolving the 6 A_3-singularities.It supports a 19-dimensional family of ambient Kähler forms.Theorem <ref> gives a quasi-equivalence(X,ω) ≃(X^∘_d);X^∘_d ⊂^3a quartic for ambient Kähler forms ω satisfying (∗).The valuations of the coefficients d_κ in 19 of the 20 quartic monomials defining X^∘_d are determined by the ω-areas of the exceptional resolution curves in X and by its total volume, and the coefficient of the 20th monomial (corresponding to the interior point of Δ) is -1.Letbe the reflexive 3-simplex with vertices {(1,0,0), (0,1,0), (0,0,1), (-1,-1,-3)}, and ^∘ its polar dual, which is given by the convex hull Conv(0,6e_1,6e_2,2e_3)-(1,1,1), so with vertices { (-1,-1,-1), (5,-1,-1), (-1,5,-1), (-1,-1,1)}.Then the toric variety Y̅^∘ = (1,1,1,3), and the toric variety Y̅ is an orbifold quotient [(1,1,1,3)/ Π'] with Π' as in Example <ref>. In the notation of that example, let X denote the crepant resolution of P_0 / Π', a hypersurface in the toric resolution of [(1,1,1,3)/ Π'].This supports a 19-dimensional family of ambient Kähler forms.Theorem <ref> gives a quasi-equivalence(X,ω)≃(X^∘_d);X^∘_d ⊂(1,1,1,3) a double planefor ambient Kähler forms ω satisfying (∗).Here X^∘_d →^2 ⊃Σ_d is the double branched cover of ^2 branched over a smooth sextic curve Σ_d whose defining equation depends via the mirror map on ω.In <cit.> we elaborate a version of homological mirror symmetry similar to Theorem <ref>, taking the form(X_20,ω)^≃_d(ω). It involves a K3 surface X_20 which is a complete intersection inside a toric variety, and has _tor(X) of rank 20.As we explained above, the mirror to the resulting 20-dimensional family of Kähler forms could not be a 20-dimensional family of algebraic K3 surfaces; rather, the mirror is Kuznetsov's K3-category _d of a cubic four-fold <cit.>.This is equivalent to the derived category of certain algebraic K3 surfaces along certain 19-dimensional loci <cit.> (see also <cit.>).These loci should be mirror to certain rational hyperplanes in _tor(X), so one should be able to prove that (X_20,ω) is mirror to a K3 of Picard rank 1, for [ω] an irrational point on such a hyperplane; these would then give many new examples of symplectic K3 surfaces to which our main results (e.g., Theorem <ref>) apply. Indeed, whereas Examples <ref> and <ref> have (X,ω) ≅ U ⊕⟨ d ⟩ for d=2,4, this construction would yield examples for all d in the (infinite) set of natural numbers satisfying <cit.>.§.§ A_∞ autoequivalences and Hochschild homologyThe results of Section <ref> concern autoequivalences of triangulated categories, however our homological mirror equivalence (<ref>) concerns A_∞ categories.Therefore we will develop the relevant theory for A_∞ categories, before giving the relationship with triangulated categories in Section <ref>. Letbe an A_∞ category. We define () to be the group of A_∞ functors F:, such that H^0(F) is an equivalence, considered up to isomorphism in H^0(nu-fun(,)), where nu-fun(,) denotes the category of non-unital A_∞-functors (see <cit.>). The Hochschild homology of an A_∞ categoryisa graded vector space HH_*() (see <cit.>).There is a natural action ofon HH_*() by graded automorphisms. Thus, a homological mirror equivalence (<ref>) induces an isomorphism of exact sequences1 [r]^0 (X) @<->[d]^[origin=c]90∼[r](X) @<->[d]^[origin=c]90∼[r] HH_*((X)) @<->[d]^[origin=c]90∼1 [r]^0 (X^∘) [r](X^∘) [r] HH_*((X^∘)),where ^0 is the subgroup consisting of autoequivalences acting trivially on HH_*.For the Fukaya category we have the open–closed map: HH_*((X)) → H^*+n(X,Λ),which under certain hypotheses is an isomorphism.For example, this is a formal consequence of the existence of a homological mirror X^∘ which is maximally unipotent (see <cit.>) or smooth (see <cit.>), and in particular holds in the context of Theorem <ref>. For the derived category, <cit.> provides an isomorphismHH_*((X^∘)) ≅ HH_*(X^∘)where the right-hand side denotes the Hochschild homology of the variety X^∘ (see <cit.>).On the other hand, we have the Hochschild–Kostant–Rosenberg isomorphismHH_*(X^∘) ≃ H^*(Ω^-*X^∘)(see, for instance, <cit.>).Thus HH_*((X^∘)) ≃ H^*(Ω^-*X^∘). §.§ Calabi–Yau autoequivalencesAn n-Calabi–Yau (n-CY) structure[There are various notions of Calabi–Yau structures, relevant in different contexts; the one used here was called a `weak proper Calabi–Yau structure of dimension n' in <cit.>.] on a -linear A_∞-category is a map ϕ: HH_n() → with the property that the pairing^*(K,L) ⊗^n-*(L,K) [r]^-[μ^2] ^n(L,L) → HH_n() [r]^-ϕis non-degenerate, for any objects K,L.Ifcomes equipped with an n-CY structure ϕ, we define _CY⊂ to be the subgroup fixing ϕ.In general it may depend on the choice of ϕ, but this is not the case for the categories involved in a homological mirror equivalence (<ref>).These categories admit 2-CY structures, and have HH_2 is of rank 1 (being isomorphic to H^4(X,Λ)≃ H^2(𝒪_X^∘)), so all 2-CY structures are proportional.Thus, _CY does not depend on the choice of 2-CY structure: if an autoequivalence preserves one it preserves all of them.It is clear that ^0 ⊂_CY, so a homological mirror equivalence (<ref>) induces an isomorphism of exact sequences1 [r]^0 (X) @<->[d]^[origin=c]90∼[r]_CY(X) @<->[d]^[origin=c]90∼[r] HH_*((X)) @<->[d]^[origin=c]90∼1 [r]^0 (X^∘) [r]_CY(X^∘) [r] HH_*(X^∘).§.§ Enhancements and autoequivalences We would now like to relate the A_∞ autoequivalence groups considered in the preceding sections with the triangulated autoequivalence groups considered by Bayer and Bridgeland.The relationship between A_∞ and triangulated categories is given in <cit.>, which implies that ifis a triangulated A_∞ category, then the cohomological category H^0() has a natural triangulated structure, and there is a homomorphism → H^0() to the triangulated autoequivalence group, sending F ↦ H^0(F).We say thatis an A_∞ enhancement of H^0(). In our setting, (X^∘) has a unique dg (in particular, A_∞) enhancement (X^∘) by <cit.>, and the results of <cit.>imply that (X^∘) →(X^∘) is in fact an isomorphism.The homomorphism → HH_*() does not have an analogue in the general setting of triangulated categories; in the particular geometric setting that we consider, however, there is a substitute. Any autoequivalence of (X^∘) is of Fourier–Mukai type, and the Fourier–Mukai kernel induces an action on HH_*(X^∘) (see <cit.>), so we obtain a map (X^∘) → HH_*(X^∘).This is identified with the map (X^∘) → HH_*((X^∘)) by <cit.>. Since _CY(X^∘) consists precisely of the autoequivalences acting trivially on HH_2(X^∘) by <cit.>, it gets identified with _CY(X^∘).The upshot is an isomorphism of exact sequences1 [r]^0 (X^∘) @<->[d]^[origin=c]90∼[r]_CY(X^∘) @<->[d]^[origin=c]90∼[r] HH_*((X^∘)) @<->[d]^[origin=c]90∼_(<ref>)1 [r]^0 (X^∘) [r]_CY(X^∘) [r] HH_*(X^∘).§.§ The symplectic mapping class group acts on the Fukaya category Now recall <cit.> that the group of graded symplectomorphisms ^gr(X) is a central extension byof the group of all symplectomorphisms (X) of X.The Hamiltonian subgroup Ham^gr(X) is the connected component of ^gr(X) containing the identity, where we use the “Hamiltonian topology” on (X) (see <cit.> for the definition: when H^1(X;) = 0, there is no distinction between symplectic and Hamiltonian isotopy, so the Hamiltonian topology coincides with the C^∞ topology).In particular we have an isomorphism ^gr(X) / Ham^gr(X) ≅π_0^gr(X). There is a natural homomorphismπ_0 ^gr(X) ⟶(X). The construction of a homomorphism ^gr/Ham^gr→ is explained in the setting of exact manifolds in <cit.>, and the proof carries over to the strictly unobstructed setting in which we work.We compose this homomorphism with the homomorphism→,which exists for any A_∞ category .We now recall that (X) admits a canonical n-CY structure ϕ (where n is half the real dimension of X).It is defined byϕ(α) ∫_X 𝒪𝒞(α)where 𝒪𝒞: HH_n((X)) → H^2n(X;Λ) is the open–closed string map(compare <cit.>).This induces an n-CY structure on (X), by Morita invariance of Hochschild homology. The map (<ref>) lands in _CY(X), the subgroup of Calabi–Yau autoequivalences.Let ψ be a graded symplectomorphism, inducing an autoequivalence Ψ on (X). We have∫_X 𝒪𝒞(Ψ_* α)= ∫_X (ψ^-1)^* 𝒪𝒞(α) (naturality of 𝒪𝒞)= ∫_X 𝒪𝒞(α) (ψ is orientation-preserving),so Ψ preserves ϕ as required. We denote the element of ^gr(X) corresponding to k ∈ by [k].That is because the homomorphism (<ref>) sends [k] to the shift functor [k].The central extension ^gr(X)/[2] of (X) by /2 is canonically split (see, e.g., <cit.>), so we obtain a homomorphism(X)/Ham(X) →_CY(X)/[2].If the open–closed map 𝒪𝒞: HH_*((X)) → H^*+n(X;Λ) is an isomorphism, then there is a natural homomorphism of exact sequences1 [r] I(X,ω) [d] [r] G(X,ω) [r] [d] H^*(X,) @^(->[d]1 [r]^0(X,ω)/[2] [r]_CY(X,ω)/[2] [r] HH_*((X,ω)).Here the rightmost vertical arrow is obtained by identifying HH_*() ≅ H^*+n(X,) ⊗Λ via 𝒪𝒞. Commutativity follows from the naturality of 𝒪𝒞.§.§ Noncommutative Chern characterIfis triangulated, we define its Grothendieck group K() to be equal to the Grothendieck group of the triangulated category H^0().The noncommutative Chern character <cit.> is a homomorphism: K() → HH_0().In the case of the Fukaya category, the compositionK((X)) [r]^- HH_0((X)) [r]^- H^n(X;Λ)takes the class of a Lagrangian [L] to PD([L]), the Poincaré dual of its homology class. This follows because we have arranged that L bounds no non-constant holomorphic discs, and the constant discs sweep out the fundamental cycle of L (cf. <cit.>, <cit.>).In the case of the derived category, we introduce the isomorphismHH_*((X^∘))HH_*(X^∘)H^*( Ω^-*X^∘)H^*(Ω^-*X^∘).Then the composition K((X^∘))HH_0((X^∘)) ⊕ H^p(Ω^p_X^∘)sends [E] to its Mukai vector v(E) <cit.>. (If we had omitted the final twist by the square root of the Todd class in (<ref>) it would have sent [E] to its Chern character.) If X^∘ is a complex K3 surface, then the diagram(X^∘) [rr] @<->[d]^[origin=c]90∼ HH_*((X^∘)) @<->[d]^[origin=c]90∼_-(<ref>) (X^∘) [r]^-ω_H^*(X,) @<->[r]^-∼H^*(Ω^-* X^∘)commutes, by <cit.>. This would not have been the case had we not twisted by the square root of the Todd class in (<ref>).§ K3 SURFACES OVER THE NOVIKOV FIELDA K3 surface over an arbitrary fieldis an algebraic surface X overwith H^1(X,𝒪_X) = 0 and K_X ≅𝒪_X. This again has a Picard group scheme (X); we let (X) denote the Neron–Severi group, which is the quotient (X) / ^0(X) of the Picard group by its identity component. The fact that H^1(X,𝒪_X) = 0 implies that (X) consists of rigid isolated points, so (X) = (X).§.§ Lefschetz principle We will use the “Lefschetz principle” (see e.g. <cit.>) to translate results about complex K3 surfaces into corresponding results about K3 surfaces over an arbitrary field of characteristic zero (of course, we have the Novikov field Λ in mind).The results we collect here are taken from <cit.>, see also <cit.>, and will be standard in the relevant community. We include a discussion since the results may be less familiar to symplectic topologists.Letbe a field of characteristic zero, and X a K3 surface over .We can define X using only a finite number of elements of , so there exists a finitely-generated field ⊂_0 ⊂ and a variety X_0 over _0 such that X = X_0 ×__0.We can embed _0 in , so we obtain a variety X_ = X_0 ×__0 (which depends on the choice of embedding).Using the fact that flat base change commutes with coherent cohomology, one can show that X_0, and therefore X_, are also K3 surfaces. We call X_ a “complex model” of X; the basic idea of the Lefschetz principle is to translate results about the complex K3 surfaceX_ into results about X.Let X be a K3 surface over an algebraically closed field , / a field extension, and X_ := X ×_. Then the pull-back map (X) →(X_) is a bijection.This is <cit.>.To prove injectivity of the pull-back map, we observe that a line bundle L ∈(X) is trivial if and only if the composition map(L,𝒪_X)⊗(𝒪_X, L) →(𝒪_X,𝒪_X) ≅is non-zero.Since coherent cohomology commutes with flat base change, this holds for L if and only if it holds for the pull-back of L to X_ (note that this is true even ifis not algebraically closed).Surjectivity relies on the fact thatis algebraically closed.Any line bundle on X_K can be defined using finitely many elements of , hence is defined oversome finitely-generated extension of , which is the quotient field of a finitely-generated -algebra A.Localizing A with respect to finitely many denominators, we may assume that the line bundle can in fact be defined over A, so it can be viewed as a family of line bundles on X parametrized by (A).This family is classified by a morphism (A) →(X).The Picard scheme (X) is reduced and has dimension equal to the rank of H^1(X,𝒪_X), which for a K3 surface is 0. Sinceis algebraically closed, it follows that (X) is simply a disjoint union of points: so the classifying morphism must be constant, which means that the line bundle is pulled back from X as required.A crucial point in the proof of surjectivity in Lemma <ref> was that line bundles on a K3 surface are rigid: they do not admit non-trivial deformations.Let X be a K3 surface over an algebraically closed field of characteristic zero, and X_ a complex model.Then (X) ≅(X_). We observe that the algebraic closure _0 of _0 embeds intoand , so the restriction maps (X) →(X_0 ×__0_0) ←(X_)are isomorphisms by Lemma <ref>.It follows immediately that any K3 surface over an algebraically closed field of characteristic zero has Picard rank ≤ 20, since this is true of complex K3 surfaces.This is not true for K3 surfaces in finite characteristic.If X_ and X'_ are different complex models for X, then Corollary <ref> shows that they have isomorphic Picard lattices. However the embeddings (X_) ↪ H^2(X_,) and (X'_) ↪ H^2(X'_,) need not be isomorphic. The following two results are discussed in <cit.>: Let X be a K3 surface over an algebraically closed field of characteristic zero, and X_ a complex model.Then the set of isomorphism classes of spherical objects of (X) is in bijection with the set of isomorphism classes of spherical objects of (X_). The proof follows that of Corollary <ref>, using the fact that spherical objectsare rigid because ^1(,) = 0 by definition. Let X be a K3 surface over an algebraically closed field of characteristic zero, and X_ a complex model.Then there is an isomorphism (X) ≅(X_), inducing isomorphisms:1 [r]^0(X^∘_)/[2] [r] @<->[d]^[origin=c]90∼ _CY(X^∘_)/[2] [r] @<->[d]^[origin=c]90∼HH_*(X^∘_) @<–>[d] 1 [r]^0(X^∘)/[2] [r]_CY(X^∘)/[2] [r] HH_*(X^∘).The dashed vertical arrow signifies an isomorphism between the images of the rightmost horizontal arrows.In other words, this diagram can be completed to an isomorphism of short exact sequences, by replacing the rightmost terms by the images of the rightmost horizontal arrows.The proof follows that of Corollary <ref>, using the fact that the Fourier–Mukai kernels defining autoequivalences are rigid (because H^1(X,𝒪_X) = 0).The action of autoequivalences on Hochschild homology is compatible with flat base change, so these isomorphisms preserve ^0 (the subgroup acting trivially on HH_*) and _CY (the subgroup acting trivially on HH_2).The dashed vertical arrow is induced by the inclusionsHH_*(X_) ↩ HH_*(X_0 ×__0_0) ↪ HH_*(X)which are induced by base change by the inclusions ↩_0 ↪.The previous result does not hold for general varieties, since if H^1(X;𝒪_X) is non-zero then one can tensor by (flat) line bundles with different structure group after extending scalars; the result relies on the fact that the Picard group is discrete.Now we will consider point-like objectsof (X), i.e. objects satisfying ^*(,) ≅∧^* (^⊕ 2).Given such an , we may choose a finitely-generated ⊂_0 ⊂ such that X and the objectare defined over _0 (indeed, for any finite set of objects we can choose a finitely-generated field _0 over which they are all defined).Thus we have an object _0 of (X_0 ×__0_0) which pulls back toon X and to _ on the complex model X_.Since coherent cohomology commutes with flat base change, _0 and _ are also point-like.Let X be a K3 surface over an algebraically closed field of characteristic zero, with ρ(X) = 1. Then any point-like objectof (X) has non-zero Chern character () ≠ 0 ∈ HH_0(X). First we prove the result assuming =.Bayer and Bridgeland show that any point-like (or spherical) objectof (X) is quasi-stable for some stability condition, meaning that it is semistable and all its stable factors have positively proportional Mukai vector. Indeed, for a stability condition σ, let theσ-width of a point-like object be the difference between the phases of the maximal and minimal semistable factors in its Harder–Narasimhan filtration. Then <cit.> shows that there exists a stability condition σ such thathas σ-width 0, and <cit.> shows that eitheris σ-quasi-stable, or it is σ'-quasi-stable for a stability condition σ' near σ. A quasi-stable object has non-trivial Mukai vector v() ≠ 0 ∈(X), and therefore non-trivial Chern character () ≠ 0 ∈ HH_0(X) as required.Now we address the general case.Letbe a point-like object of (X), and let us choose a complex model X_ for X over whichis defined.Taking Hochschild homology and Chern characters commute with flat base change, sohas non-zero Chern character if and only if _ does: since ρ(X_) = ρ(X) = 1 by Corollary <ref>, the result follows from the case = proved above. Let X be a K3 surface over an algebraically closed field of characteristic zero, with (X) ≅⟨ 2n ⟩ where n is square-free.Then for any point-like objectof (X), there exists a spherical object S with χ(,S) = 1.First we prove the result for =.Letbe a point-like object of (X): we will start by showing that there exists a K3 surface Y and an equivalence η: (X) (Y) takingto the skyscraper sheaf of a point.The set of stability conditions makingquasi-stable is open <cit.>, and non-empty sinceis point-like, cf. the proof of Lemma <ref>. Thus we can pick a stability condition σ which makesquasi-stable and which is generic in the sense of <cit.>.Suppose the Mukai vector ofis v = m · v_0 where v_0 is primitive and m ∈_+.The Mukai pairing (v_0,v_0)=0, since χ(,) = 0 using the fact thatis point-like.Now, in <cit.>, Bayer and Macrì show that there is a non-empty projective moduli stack of σ-semistable objects with the same Mukai vector as . Let (v) be its coarse moduli space; then the coarse moduli space of (v_0) is again a K3 surface, which has a distinguished Brauer class α.There is a Fourier–Mukai equivalenceη: (X) ((v_0), α)(where the right-hand side denotes the derived category of twisted sheaves), which takes any complex in (X) defining a point of (v) to a torsion sheaf on the twisted K3 surface (v_0) of dimension 0 and length m. <cit.> proves that η identifies (v) with the m-th symmetric product ^m((v_0)). It follows that the general point of (v) corresponds, under this identification, to the skyscraper sheaf of an m-tuple of pairwise distinct points, which has ^1 of rank 2m. Since ^1 varies upper semicontinuously,it follows that ^1(,) has rank at least 2m; sinceis point-like, m=1 and v=v_0.The derived equivalence η then takesto the skyscraper sheaf of a point on the twisted K3 surface ((v), α). Recall that a K3 surface Y equipped with a Brauer class β is called a twisted Fourier–Mukai partner of X if there is an equivalence (X) ≃(Y,β): so (Y,β)((v),α) is a twisted Fourier–Mukai partner of X.Ma gives a formula for the number of twisted Fourier–Mukai partners of X with Brauer class of a given order <cit.>.His result shows that if (X) ≅⟨ 2n ⟩ with n square-free, then any twisted Fourier–Mukai partner of X has trivial Brauer class; since this holds by hypothesis in our case, we must have β=0. Now 𝒪_Y is a spherical object of (Y) with χ(𝒪_y,𝒪_Y)=1 for any y ∈ Y, so if we setSη^-1(𝒪_Y) then χ(,S) = 1 as required.Now we address the general case.Letbe a point-like object, and let us choose a complex model X_ for X over whichis defined.Then _ is a point-like object of (X_), and (X_) ≅(X) by Corollary <ref>, so there exists a spherical object S_ of (X_) for which χ(_,S_)=1 by the previous argument.The spherical object descends to X_0 ×__0_0 by Lemma <ref>, so we obtain a spherical object S of (X) with χ(,S) = 1 as required.§.§ Obtaining a Picard rank one mirror Let K be an algebraically closed field of characteristic zero.Let p: → be a family of K3 surfaces over K, i.e. a proper smooth morphism of relative dimension 2 with both the relative dualising sheaf and R^1p_*𝒪_ being trivial. Let ρ_0 be the Picard rank of the generic fibre _η. The locus {t ∈|ρ(_t) > ρ_0 } (called the Noether–Lefschetz locus) is a countable union of positive-codimension algebraic subvarieties.This is classical; references which explicitly deal with general algebraically closed fields include <cit.> and <cit.>. Now we consider the family ^∘ = ^Ξ_0 of hypersurfaces in Y̅^∘ that is defined by (<ref>) (it is defined over , hence over any field).Suppose that Y̅^∘ is smooth as a scheme, away from its toric fixed points, so that the generic fibre of ^∘ is a smooth K3 surface.Suppose furthermore that, after base changing to , the generic fibre has Picard rank ρ_0. There exists a set Υ⊂^Ξ_0, a countable union of hyperplanes of rational slope passing through the origin, such that if d ∈Λ^Ξ_0 has valuation (d) ∉Υ, then X^∘_d is smooth with Picard rank ρ_0. Let Δ⊂ denote the discriminant locus of the family ^∘.By Lemma <ref>, the generic fibre of the family after base changing to Λ has Picard rank ρ_0. By Theorem <ref>, the hypersurfaces of higher Picard rank are contained in a countable family of algebraic hypersurfaces in Λ^Ξ_0∖Δ. We now restrict the family to (Λ^*)^Ξ_0∖Δ.The valuation image of any algebraic hypersurface in (Λ^*)^Ξ_0 is a polyhedral complex of dimension |Ξ_0|-1 called the tropical amoeba (see, e.g., <cit.>).In particular it is contained in a finite union of affine hyperplanes of rational slope. Now we observe that, for any a ∈_>0, the map q ↦ q^a extends to an automorphism ψ_a of Λ.We define a corresponding automorphism Ψ_a = (ψ_a,…,ψ_a) of Λ^Ξ_0, which satisfies (Ψ_a(d)) = a ·(d) and X^∘_Ψ_a(d)≅ψ_a^* X^∘_d.It follows that the discriminant and Noether–Lefschetz loci are invariant under Ψ_a, hence that their tropical amoebae are invariant under scaling by a.Since a ∈_>0 was arbitrary, it follows that the affine hyperplanes making up the tropical amoebae pass through the origin.Therefore we can take Υ to be the union of linear hyperplanes containing the tropical amoebae of the discriminant and Noether–Lefschetz loci. In the situation of Theorem <ref>, suppose that λ∈ (_>0)^Ξ_0 is irrational, i.e. does not lie on any rational hyperplane.ThenX^∘_d has minimal Picard rank ρ_0.This follows from Proposition <ref>:since the hyperplanes making up Υ have rational slope, they cannot contain λ = (d). § SYMPLECTIC CONSEQUENCESLet (X,ω) be a symplectic K3, and X^∘ an algebraic K3 surface over Λ.Our standing assumptions for this section are as follows.We assume that (X,ω) is homologically mirror to X^∘:(X,ω) ≃(X^∘);and furthermore that they satisfy the analogue of Conjecture <ref>:(X,ω) ≅(X^∘)U ⊕(X^∘). These assumptions are satisfied if X is a mirror quartic or mirror double plane, ω_λ is an ambient-irrational Kähler form, and X^∘ = X^∘_d is the Greene–Plesser mirror.Firstly, (<ref>) holds by Theorem <ref>.Ambient-irrationality of ω_λ allows us to verify (<ref>) as follows.Firstly, because ω_λ is ambient-irrational, we may choose λ = (d) to be irrational, so that (X^∘_d) = ⟨ 2n ⟩ by Proposition <ref> (where n=2 for the quartic, n=1 for the double plane).Secondly, ambient-irrationality implies that (X,ω_λ) ≅ U ⊕⟨ 2n ⟩ (see Examples <ref> and <ref>). Thus (<ref>) also holds. For the purposes of this section we will pick a complex model X^∘_ of X^∘.We will abbreviate (X^∘). §.§ Spherical objects The homological mirror equivalence (<ref>) gives us an isomorphism of numerical Grothendieck groupsK_num((X)) ≃ K_num((X^∘)).The Mukai vector defines an isomorphismK_num((X^∘))^- ≃(X^∘) = U ⊕,see <cit.> and Remark <ref> (recall the `-' denotes negation of the pairing).Composing, we obtain an isomorphismK_num((X))^- ≃ U ⊕. We now consider spherical objects in (X). Their classes in the numerical Grothendieck group correspond, under (<ref>), to elements of Δ(U ⊕). One example of a spherical object is a Lagrangian sphere, and one example of a Lagrangian sphere is a vanishing cycle.Our aim in this subsection is to prove the following: Any class in Δ(U ⊕) is represented by a vanishing cycle in X. Before proving Lemma <ref> we will prove some preliminary results, the first of which concerns the compositionK((X))HH_0((X))H^n(X,Λ).The map (<ref>) descends to an injective mapK_num((X)) ↪ H^n(X,Λ).Because the Chern character map : K((X^∘)) → HH_0((X^∘)) descends to an injection K_num((X^∘)) ↪ HH_0((X^∘)), the same holds true for (X) by our homological mirror symmetry assumption.Composing with 𝒪𝒞, which is an isomorphism in this case, gives the result. We now define 𝒮⊂ K_num((X)) to be the set of classes represented by vanishing cycles; 𝕊⊂ K_num((X)) the subgroup they generate; and 𝕋⊂ U ⊕ the subgroup generated by Δ(U ⊕).The map (<ref>) identifies 𝒮 with Δ((X,ω)), and 𝕊^- isometrically with the subgroup of (X,ω) ⊂ H^n(X,Λ)generated by Δ((X,ω)). The map (<ref>) sends a Lagrangian sphere to its homology class in (X,ω), by Lemma <ref>.This homology class must lie in Δ((X,ω)), and in fact all such classes are realized by vanishing cycles, by Remark <ref>.The identification respects pairings, by Remark <ref>. The isomorphism (<ref>) identifies 𝕊 with 𝕋. We start by observing that the subgroup 𝕋⊂ U ⊕ has full rank by an elementary argument <cit.>.In particular, the pairing restricted to it is nondegenerate, so 𝕋 is a sublattice.Now the isomorphism (<ref>) clearly identifies 𝕊 with a subgroup of 𝕋.On the other hand, Lemma <ref> identifies 𝕊^- with the subgroup of (X,ω) generated by Δ((X,ω)), which the isomorphism (<ref>) identifies with 𝕋^-.The result now follows because it is impossible to embed the lattice 𝕋 properly inside itself, by discriminant considerations.Follows by combining Lemmas <ref> and <ref>.Of course one expects (<ref>) to identify K_num((X))^- with (X,ω), however we have not proved this.The issue is that a general object of (X) is an idempotent summand of a twisted complex, and while Lemma <ref> tells us what the image under (<ref>) of the twisted complex is, it does not tell us anything about the image of the idempotent summand (e.g., whether it lies in H^n(X,)).§.§ Symplectic mapping class groups via HMS Suppose () = 1 (as in Example <ref>). We compose the various morphisms of short exact sequences of groups that we have constructed: the maps on the central terms areπ_1(_0(U ⊕)) G(X,ω) _CY(X,ω)/[2] (<ref>)_CY(X^∘)/[2] … …(<ref>)_CY(X^∘)/[2] (<ref>)_CY(X^∘_)/[2] (<ref>)π_1(_0(U ⊕)).Note that in some cases we wrote exact sequences without a `→ 1' on the right: we turn these into short exact sequences in the canonical way by replacing the rightmost group with the image of the rightmost homomorphism in the exact sequence, so that all of the morphisms in (<ref>) are morphisms of short exact sequences.We recall the meanings of these morphisms: * (<ref>) comes from symplectic monodromy.* (<ref>) comes from the action of the symplectic mapping class group on the Fukaya category; it relies on the open–closed map being an isomorphism, which is a consequence of homological mirror symmetry.* (<ref>) exists by our assumption that (X,ω) and X^∘ are homologically mirror. * (<ref>) maps Calabi–Yau autoequivalences of (X^∘) to Calabi–Yau autoequivalences of (X^∘) by taking H^0.* (<ref>) identifies the derived autoequivalence group of a K3 over Λ with that of a K3 over .* (<ref>) existsby the theorem of Bayer–Bridgeland, because (X^∘_) ≅ has rank 1. We have also replaced HH_*(X^∘_) with H^*(X^∘_,) via (<ref>), which is valid by Remark <ref>. Considering the right-most terms in our exact sequences, we observe that (<ref>) maps to H^2(X,) whereas (<ref>) maps from H^*(X,).In order to be able to turn the composition into a morphism of short exact sequences, we use the fact that any symplectomorphism acts trivially on H^0(X,) ⊕ H^4(X,). Suppose () = 1. Then the composition of morphisms of short exact sequences (<ref>) is an isomorphism. By the 5-lemma, it suffices to prove the result on the initial and final terms of the short exact sequences.The maps between final terms are all isomorphisms, so it remains to prove the result for the initial terms. The composition of morphisms between initial terms has the formπ_1(Ω^+_0(U ⊕)) → I(X,ω) →^0 (X,ω)/[2] ≅^0 (X^∘)/[2] … …≅^0(X^∘)/[2] ≅^0 (X^∘_)/[2] ≅π_1(Ω^+_0(U ⊕)).Because ()=1, Section <ref> gives an identificationΩ^+_0(U ⊕) ≅𝔥∖⋃_δ∈Δ(U ⊕)/±𝕀{p_δ}.Therefore π_1(Ω^+_0(U ⊕)) is a free group with generators indexed by Δ(U ⊕)/±𝕀. Choose paths from a basepoint in Ω^+_0(U ⊕) to each point p_δ: these specify generators for the free group.The monodromy around p_δ is τ_L^2 ∈ I(X,ω), a squared Dehn twist in the corresponding vanishing cycle L (cf. Remark <ref>). The corresponding autoequivalence of (X,ω) is Tw^2_L, the squared algebraic twist in the corresponding spherical object by <cit.>, which gets sent to the autoequivalence Tw^2_S for the corresponding spherical object S of (X^∘_). Corollary <ref> says that ^0 (X^∘_)/[2] ≅π_1(Ω^+_0(U ⊕)). By <cit.>, for every ϵ∈Δ(U ⊕) there exists a spherical vector bundle S_ϵ, so that the spherical twist Tw^2_S_ϵ corresponds to a loop enclosing the hole p_ϵ under this isomorphism. Furthermore, by <cit.>, ^0 (X^∘_) acts transitively on the set of spherical objects S with fixed Mukai vector v(S) = ϵ.It follows that every squared spherical twist in a spherical object S with Mukai vector v(S) = ϵ corresponds to a loop in π_1(Ω^+_0(U ⊕)) enclosing the hole p_ϵ.In particular, the composition (<ref>) sends a loop enclosing p_δ to a loop enclosing p_ϵ, where δ is the image of [L] under the map (<ref>) and ϵ is the Mukai vector of S, and the spherical objects L and S correspond under mirror symmetry.The map δ↦ϵ is bijective by Lemma <ref>.Therefore the composition (<ref>) sends one set of generators for the free group bijectively to another set of generators for the free group, so the map is an isomorphism as required.Let us briefly mention a minor generalization of Proposition <ref>: it continues to hold if we replace _0(U ⊕) with _0(U ⊕), the symplectic mapping class group G(X,ω) with the graded symplectic mapping class group π_0^gr(X,ω), and remove all of the quotients by even shifts in (<ref>). The only point that requires extra explanation is the construction of the symplectic monodromy map (<ref>). The crucial points are that a choice of fibrewise holomorphic volume form on the family → B induces a lift of the symplectic monodromy map to the graded symplectic mapping class group, and the fibre of the ^*-bundle _0(U ⊕) →_0(U ⊕) can be identified with the space of holomorphic volume forms on the corresponding fibre of the family (U ⊕,κ) →_0(U ⊕). The details are left to the reader. Now recall thatZ(X,ω)[G(X,ω) →(X,ω)/[2] ]denotes the “Floer-theoretically trivial” subgroup of the symplectic mapping class group (it is contained in I(X,ω), when the open–closed map is an isomorphism).We have the following immediate corollary of Proposition <ref>:Suppose () = 1. Then we haveG(X,ω)≅ Z(X,ω)⋊π_1(_cpx(X,ω))andI(X,ω)≅ Z(X,ω) ⋊π_1(Ω^+_0(X,ω)) .We also recall thatK(X,ω)[π_0(X,ω) →π_0(X)]denotes the smoothly trivial symplectic mapping class group.It is contained in I(X,ω).Suppose () = 1.Then we have homomorphismsπ_1(Ω^+_0(X,ω)) → K(X,ω) →π_1(Ω^+_0(X,ω)) whose composition is an isomorphism.In particular, K(X,ω) is infinitely generated. We consider the composition π_1(Ω^+_0(X,ω)) → I(X,ω) →π_1(Ω^+_0(X,ω)) appearing in (<ref>), which is shown to be an isomorphism in Proposition <ref>.Observe that the image of the map π_1(Ω^+_0(X,ω)) → I(X,ω) is generated by squared Dehn twists, which are known to be smoothly trivial: so we can replace I(X,ω) with the subgroup K(X,ω).Now we can refine Remark <ref>, which we recall concerned the space Ω of symplectic forms on X cohomologous to ω.It essentially consisted of the observation that the compositionπ_1(Ω) ↠ K(X,ω) ↠π_1(Ω^+_0(X,ω))is surjective, where the first map is the connecting map in the long exact sequence associated to a Serre fibration, and the second was constructed in the proof of Corollary <ref>; this shows that π_1(Ω) is infinitely generated.There exists a map i:Ω^+_0(X,ω) →Ω, such that the compositionπ_1(Ω^+_0(X,ω)) π_1(Ω) π_1(Ω^+_0(X,ω))is an isomorphism. Consider the family of marked Kähler K3 surfaces ((X,ω),[ω]) →Ω^+_0(X,ω).We saw in the proof of Corollary <ref> that the monodromy homomorphisms are smoothly trivial; since Ω^+_0(X,ω) is an Eilenberg–MacLane space, it follows that the family is smoothly trivial.Pulling back a choice of fibrewise Kähler form via a smooth trivialization induces a map Ω^+_0(X,ω) →Ω, and it is clear from the definitions that the composition π_1(Ω^+_0(X,ω)) π_1(Ω) → K(X,ω)is precisely the symplectic monodromy homomorphism.It follows that the composition (<ref>) is equal to the composition (<ref>), which is an isomorphism by Corollary <ref>.§.§ Lagrangian spheres We now consider Lagrangian spheres L ⊂ (X,ω).We declare two Lagrangian spheres to be “Fukaya-isomorphic” if they can be equipped with a grading and Pin structure such that the corresponding objects of (X,ω) are quasi-isomorphic. Suppose () = 1.Then any Lagrangian sphere L ⊂ (X,ω) is Fukaya-isomorphic to a vanishing cycle. Let L be a Lagrangian sphere, and _L the mirror spherical object of (X^∘_).By Lemma <ref>, there exists a vanishing cycle V with v(_V) = v(_L). We can apply <cit.> (because ρ(X^∘_)=1), which says that there exists Φ∈^0 (X^∘_) with Φ(_V) ≅_L. We have a homomorphism π_1(Ω^+_0(X,ω)) →^0 (X^∘_) from the composition of short exact sequences (<ref>), which is surjective by Proposition <ref>.Therefore, there exists a ∈π_1(Ω^+_0(X,ω)) such that a · V ≅ L.It is clear that a · V is the vanishing cycle whose vanishing path is the concatenation of the vanishing path of V with the loop a, so we are done.Suppose () = 1.Then the orbits of G(X,ω) on the set of Fukaya-isomorphism classes of Lagrangian spheres in (X,ω) are in bijection with the orbits of Γ^+(U ⊕) on Δ(U ⊕)/±𝕀. Recall that the image of G(X,ω) → H^2(X,) is precisely Γ^+(U ⊕) (cf. Proposition <ref>). Thus there is a well-defined map{Lagrangian spheres up to Fukaya-isomorphism}/G(X,ω) → (Δ(U ⊕)/±𝕀)/Γ^+(U ⊕)sending L to its homology class (the choice of orientation of L does not matter since we quotient by ±𝕀).This map is surjective by Lemma <ref>.Suppose that L_1 and L_2 have the same image under (<ref>).By Lemma <ref>, we may assume that L_1 and L_2 are vanishing cycles.Because G(X,ω)→Γ^+(U ⊕) is surjective, we may assume that [L_1] = [L_2] = δ in H_2(X,); so L_1 and L_2 are vanishing cycles from the same point p_δ.It follows that their vanishing paths differ by an element of π_1(Ω^+_0(U ⊕)), so the vanishing cycles differ by its image under π_1(Ω^+_0(U ⊕)) → I(X,ω) ⊂ G(X,ω).In particular, L_1 and L_2 represent the same class on the left-hand side of (<ref>), so the map is injective as required.If () = 1, then G(X,ω) has finitely many orbits on the set of Fukaya-isomorphism classes of Lagrangian spheres. By Lemma <ref>, it suffices to check that Γ^+(U ⊕) has finitely many orbits on Δ(U ⊕).This follows from the following general result of Borel and Harish-Chandra <cit.>.Let G be a reductive algebraic group over , V a rational finite-dimensional representation of G, Γ⊂ V_ a G_-invariant lattice, and Q a closed orbit of G. Then Q∩Γ consists of finitely many G_-orbits.If = ⟨ 2 ⟩ or ⟨ 4 ⟩, then G(X,ω) acts transitively on the set ofFukaya-isomorphism classes of Lagrangian spheres. Follows by Lemma <ref> and Lemma <ref>.Suppose = ⟨ 2 ⟩ or ⟨ 4 ⟩.If L and L' are Lagrangian spheres in X, then the Dehn twists τ_L and τ_L' are conjugate in _CY(X).It is sufficient to prove that the algebraic twist functors Tw_L and Tw_L' are conjugate in _CY(X).The twist functors are conjugate provided the underlying spherical objects lie in the same orbit of the autoequivalence group, by <cit.>. The Dehn twist in a simple non-separating curve on a closed surface of genus g≥ 2 has non-trivial roots in the mapping class group <cit.>.For g ≥ 2, the image of a Dehn twistin the symplectic group Sp(2g,) has a cube root (this fails when g=1).Suppose = ⟨ 2 ⟩ or ⟨ 4 ⟩. Let L⊂ X be a Lagrangian sphere.The Dehn twist τ_L admits no cube root in G(X,ω).By Proposition <ref> and Lemma <ref>, we have a surjection G(X,ω) ↠π_1(_0(U ⊕)) ≅∗/p.Furthermore the Dehn twist in a given vanishing cycle can be arranged to correspond to 1 ∈ (cf. Remark <ref>).By Corollary <ref>, it follows that all Dehn twists map to elements conjugate to 1 ∈.Now we pass to the abelianization (∗/p)^ab≅⊕/p: all Dehn twists map to (1,0), and therefore have no cube root. Note that since τ_L^2 is smoothly trivial, τ_L is its own cube root in π_0Diff(X), so this is a symplectic and not smooth phenomenon. §.§ Lagrangian tori We now consider embedded Lagrangian tori L ⊂ (X,ω) with Maslov class zero.Suppose that = ⟨ 2n ⟩ where n is square-free (observe that this is true in the situation of Example <ref>, where n = 1 or 2). Then a Maslov-zero Lagrangian torus L ⊂ (X,ω) represents a primitive homology class.The object L of (X,ω) corresponds to a point-like objectof (X^∘).Therefore there exists a spherical object S of (X^∘) with χ(,S) = 1, by Lemma <ref>.By Lemma <ref> there exists a vanishing cycle V whose class in the numerical Grothendieck group corresponds to that of S under mirror symmetry, so we have-[L] · [V] = χ(L,V) = χ(,S) = 1.It follows that [L] is primitive. It would be nice to prove Lemma <ref> by arguing that the Chern character of the mirror objectto L is primitive, and that therefore the Chern character [L] of L is primitive.However this would require us to prove that the map (<ref>) identifies K_num((X)) with a primitive sublattice of H^2(X,), which we have not done (cf. Remark <ref>): that is why we circumvent this issue by appealingto Lemma <ref>. For our final result, we change our assumptions on the symplectic K3 surface (X,ω).Rather than assuming that (X,ω) itself has a homological mirror satisfying certain hypotheses, we assume that ω is a limit of ρ^∘=1 classes.We define this to mean that there exists a smooth one-parameter family of Kähler forms ω(t) with ω(0) = ω, having the following property: there exist times t_1,t_2,… converging to 0 such that for all j, (X,ω(t_j)) admits a homological mirror X^∘_j of Picard rank ρ(X^∘_j) = 1. Suppose ω is a limit of ρ^∘=1 classes.If L ⊂ (X,ω) is a Maslov-zero Lagrangian torus, then [L] ≠ 0 ∈ H_2(X,).Suppose L⊂ (X,ω) is a Maslov-zero Lagrangian torus with vanishing homology class.A Moser-type argument shows that L deforms to give a Maslov-zero Lagrangian torus L(t) ⊂ (X,ω(t)) for t ∈ [0,ϵ) sufficiently small that L(t) remains embedded.Thus there exists τ∈ (0,ϵ) such that L(τ) ⊂ (X,ω(τ)) is embedded and (X,ω(τ)) admits a homological mirror X^∘ of Picard rank 1. L(τ) defines a point-like object of (X,ω(τ)), and we denote the mirror point-like object by ∈(X^∘).Under the isomorphismH^2(X;Λ) ≅ HH_0((X,ω(τ))) ≅ HH_0((X^∘)) ≅ HH_0(X^∘)we see thathas trivial Chern character.However X^∘ has Picard rank 1, so this contradicts Lemma <ref>. It is obvious that, if (X,ω) is mirror to a K3 surface of Picard rank 1, then ω is a limit of ρ^∘=1 classes.However there are more examples, as we now describe. Recall that we constructed the mirror quartic and mirror double plane as hypersurfaces X ⊂ Y depending on data (,λ).The vector λ∈ (_>0)^Ξ_0 determines a refinement Σ of Σ̅, and hence a toric resolution of singularities Y →Y̅; and X ⊂ Y is the proper transform of X̅⊂Y̅.The vector λ also determines a Kähler class [ω_λ] on X.The K3 surface X does not depend on the refinement Σ, so long as the rays of Σ are generated by Ξ_0: changing Σ corresponds to changing Y by a birational modification in a region disjoint from X (the same is not true in higher dimensions). A convenient way of organizing the different refinements is provided by the “secondary fan” or “Gelfand–Kapranov–Zelevinskij decomposition” associated to Ξ_0 ⊂^3, whose cones cpl(Σ) are indexed by fans Σ whose rays are generated by subsets of Ξ_0 (see <cit.> or <cit.>).Our prescription for determining Σ from λ is equivalent to defining Σ to be the unique fan such that λ is contained in the interior of cpl(Σ).For any λ lying in the interior of a cone cpl(Σ), where Σ is a refinement of Σ̅ whose rays are generated by Ξ_0, we obtain a Kähler class ω_λ on X.We call these Kähler classes the ambient Kähler classes on X. Theorem <ref> only provides a homological mirror to (X,ω_λ) if Σ is furthermore simplicial (the corresponding cones cpl(Σ) are the top-dimensional ones). Nevertheless we have:Let X be the mirror quartic or mirror double plane, and ω an ambient Käher class.Then ω is a limit of ρ^∘=1 classes.Suppose ω = ω_λ; then λ lies in the closure of a top-dimensional cone cpl(Σ'), where Σ' is a simplicial refinement of Σ̅ whose rays are generated by Ξ_0.Therefore we can choose a path λ(t) converging linearly to λ(0) = λ along a line of irrational slope, such that λ(t) lies in the interior of cpl(Σ') for t ≠ 0.Then Theorem <ref> provides a mirror X^∘_d(t) to (X,ω_λ(t)) for all t ≠ 0.Because λ(t) moves linearly along a line of irrational slope, there is a sequence of times t_j converging to 0 at which λ(t_j) is irrational, so that the mirror X^∘_d(t_j) has Picard rank 1 by Proposition <ref>. Lemmas <ref> and <ref> combine to prove Theorem <ref>.amsalpha | http://arxiv.org/abs/1709.09439v3 | {
"authors": [
"Nick Sheridan",
"Ivan Smith"
],
"categories": [
"math.SG"
],
"primary_category": "math.SG",
"published": "20170927104026",
"title": "Symplectic topology of K3 surfaces via mirror symmetry"
} |
A note on truncated long-range percolation with heavy tails on oriented graphs C.T.M. Alves[Departamento de Estatística, IMECC, Universidade Estadual de Campinas,rua Sérgio Buarque de Holanda 651, 13083–859, Campinas SP, Brazil], M. Hilário[Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627 C.P. 702 CEP 30123-970 Belo Horizonte-MG, Brazil], B.N.B. de Lima^†, D. Valesin[Johann Bernoulli Instituut, Rijksuniversiteit Groningen, Nijenborgh 9 9747 AG Groningen, The Netherlands]=========================================================================================================================================================================================================================================================================================================================================================================================================================================================== We consider oriented long-range percolation on a graph with vertex set ^d ×_+ and directed edges of the form ⟨ (x,t), (x+y,t+1)⟩, for x,y in ^d and t ∈_+. Any edge of this form is open with probability p_y, independently for all edges. Under the assumption that the values p_y do not vanish at infinity, we show that there is percolation even if all edges of length more than k are deleted, for k large enough. We also state the analogous result for a long-range contact process on ^d.Keywords: contact processes; oriented percolation; long-range percolation; truncation MSC numbers:60K35, 82B43§ INTRODUCTION Let G = (𝕍, 𝔼) be the graph with set of vertices 𝕍 = ^d ×_+ andset of (oriented) bonds 𝔼 ={⟨ (x,t), (x+y,t+1)⟩ :x,y ∈^d,t ∈_+ }.Let (p_y)_y ∈^d be a family of numbers in the interval [0,1]and consider a Bernoulli bond percolation model where each bond ⟨ (x,t), (x+y,t+1)⟩∈𝔼 is open with probability p_y, independently for all bonds.That is, take (Ω,𝒜, P), where Ω = {0,1}^𝔼, 𝒜 is the canonical product σ-algebra, and P = ∏_e ∈𝔼μ_e, where μ_e(ω_e = 1) = p_y = 1- μ_e(ω_e = 0) for e = ⟨ (x,t), (x+y,t+1)⟩∈𝔼. An element ω∈Ω is called a percolation configuration.A (finite or infinite) sequence (v_0, v_1, … ) with v_i ∈ G for each i is called an oriented path if, for each i, v_i-v_i-1=(y,1) for some y∈ℤ^d; the oriented path is open if each oriented edge ⟨ v_i, v_i+1⟩ is open. For (x,t), (x',t') ∈𝕍 with t < t', we denote by {(x,t) ⇝ (x',t')} the event that there is an open oriented path from (x,t) to (x',t'). If A ⊂𝕍, we denote by {(x,t) ⇝ A} the event that (x,t) is connected by an open oriented path to some vertex of A. Finally,we denote by {(x,t) ⇝∞} the event that there is an infinite open oriented path started from (x,t).We now consider a truncation of the family (p_y)_y ∈^d at some finite range k. More precisely, for each k∈ℕ consider the truncated family (p_y^k)_y ∈^d, defined byp_y^k={[ p_y,y_∞≤ k,;0, , ].and the measure P^k = ∏_e ∈𝔼μ^k_e, where μ^k_e(ω_e = 1) = p^k_y = 1- μ^k_e(ω_e = 0) for e = ⟨ (x,t), (x+y,t+1)⟩∈𝔼. Then, one can ask the truncation question: is it the case that, whenever percolation can occur for a sequence of connection probabilities, it can also occur for a sufficiently high truncation of the sequence? That is: in case P{0 ⇝∞} > 0, is there a large enough truncation constant k for which we still have P^k( 0 ⇝∞) >0 ?Numerous works (<cit.> in chronological order) addressed this question considering different models (such as: the Ising model, oriented and non-oriented percolation, the contact process) or different assumptions on the sequence (p_n) or on the graph. We direct the reader to the introductory sections of <cit.> and <cit.> for a more thorough discussion. Our main contribution is the following: If there exists ε > 0 such that p_y > ε for infinitely many vectors y, then the truncation question has an affirmative answer. Moreover,lim_k→∞P^k{(0,0) ⇝∞} = 1. This result generalizes the analogous result obtained in <cit.> for non-oriented percolation on the square lattice. In that paper, the authors were able to construct a proper subgraph of ^2 with long (but limited) range edges that was isomorphic to a slab with two “unbounded” directions and arbitrarily large number of “bounded” dimensions and thickness. This allowed them to apply <cit.> to obtain their result. In our case however, this approach is fruitless, since <cit.> is not applicable in the case of oriented percolation processes. Therefore, we need to devise a new strategy.In Section <ref>, we present two settings where a positive answer to the truncation question can be readily obtained from the above theorem: an anisotropic two-dimensional oriented percolation model and a long-range contact process on ^d. We prove Theorem <ref> in Section <ref>.§ PROOF OF THEOREM <REF> We first prove the theorem for the case where d = 1, so that the family (p_y) is given by a doubly-infinite sequence (…, p_-1, p_0,p_1,…) (we replace y by n in the notation). Moreover, we assume that p_n = 0 if n ≤ 0. In the end of this section, we will show howwe can obtain the general statement from this particular case.By assumption, we can take ϵ>0 such that lim sup_n→∞p_n>ϵ>0. Define the sequence (a_n)_n asa_1=inf{i:p_i>ϵ}, a_n=inf{i>a_n-1:p_i>ϵ},n >1.Fix δ∈ (0,1) to be chosen later. Define the integers L_0 and L_1P(Bin(L_0,ϵ)≥ 1)>1-δ/3, P(Bin(L_1,ϵ)≥ L_0)>1-δ/3(here Bin(n,p) denotes a Binomial distribution with parameters n and p). Next, define R such thatR=max{a_L_1, a_2L_1-a_L_1}.Finally, take L_2 large enough such that a_L_2>a_1+3R. Given a vertex (x,y)∈ℤ^2_+ and i∈ℕ, define the eventsR_i^(x,y)={[ ⟨ (x,y),(x+i,y+1)⟩; ⟨ (x+i,y+1),(x+i+a_1,y+2)⟩ ]},S_i^(x,y)={[ ⟨ (x,y),(x+i,y+1)⟩; ⟨ (x+i,y+1),(x+i+a_L_2,y+2)⟩ ]}.Also defineT_-^(x,y)=(∪_i=1^a_L_1R_i^(x,y))∩(∪_i=1^a_L_1S_i^(x,y)),T_+^(x,y)=(∪_i=a_L_1+1^a_2L_1R_i^(x,y))∩(∪_i=a_L_1+1^a_2L_1S_i^(x,y)). Observe that by (<ref>), P^a_L_2( T_-^(x,y))>1-δ, P^a_L_2( T_+^(x,y))>1-δ.Also,onT^(x,y)_-, (x,y) ⇝ [x+2a_1, x+a_1 +a_L_1] ×{y+2},(x,y) ⇝ [x+a_1+ a_L_2, x+a_L_1 +a_L_2] ×{y+2}andonT^(x,y)_+,(x,y) ⇝ [x+a_1 + a_L_1, x+a_1 +a_2L_1] ×{y+2}, (x,y) ⇝ [x+a_L_2 + a_L_1, x+a_L_2 +a_2L_1] ×{y+2}(note that, by (<ref>) and (<ref>), the two horizontal segments in (<ref>) are disjoint, and similarly in (<ref>)).Thus, if there is an antenna (left or right, it does not matter) at (x,y) this implies that there are at least one active vertex in the interval [x+a_L_1+a_1-R,x+a_L_1+a_1+R]×{y+2} and at least one active vertex in the interval [x+a_L_1+a_L_2-R,x+a_L_1+a_L_2+R]×{y+2}. The condition <ref> holds that these intervals are disjoint. The next step is to define a renormalized lattice G^* (also an oriented graph); vertices of G^* will correspond to certain horizontal line segments in the original graph G. An exploration of the points reachable from the origin in G under the measure P^a_L_2 will produce, as its `coarse-grained' counterpart, a site percolation configuration on G^*. As is usual, two properties will result from the coupling: first, percolation in G^* will occur with high probability, and second, percolation in G^* will imply percolation in G.We let G^*=(𝕍^*,𝔼^*), where 𝕍^*={(i,j)∈ℤ×ℤ_+; i+j} and𝔼^* is the set of oriented edges 𝔼^*={⟨(i,j),(i± 1,j+1)⟩;(i,j)∈𝕍^*}. Define the following order in 𝕍^*: given (i_1,j_1),(i_2,j_2)∈𝕍^* we say that (i_1,j_1)≺(i_2,j_2) if and only if j_1<j_2 or (j_1=j_2i_1<i_2). Given S⊂ℤ×ℤ_+, we define the exterior boundary of S as the set∂_e S={(i,j)∈𝕍^*\ S; (i-1,j-1)∈ S(i+1,j-1)∈ S}. For each (i,j) ∈𝕍^*, define z_i,j = j· a_L_1 + i+j/2· a_L_2 + j-i/2·a_1.Also letv_i,j = (z_i,j, 2j) ∈𝕍, I_i,j = [z_i,j-R, z_i,j+R] ×{2j}⊂𝕍.These vertices and intervals are depicted in Figure <ref>. Note that, for all (i,j),z_i+2,j - z_i,j = a_L_2 - a_1,so, by the choice of L_2 in (<ref>), the segments I_i,j are pairwise disjoint. Additionally,z_i-1,j+1 - z_i,j = a_1 + a_L_1,z_i+1,j+1 - z_i,j = a_L_1 + a_L_2. Let us now present our exploration algorithm. We will define inductively two increasing sequences (A_i)_i and (B_i)_i of subsets of 𝕍^*. Set A_0=B_0=∅ and x_0=(0,0). We declare the vertex x_0=(0,0) as good if the event T_-^(0,0) occurs. Then, we define:A_1 =A_0∪{x_0},x_0,A_0,,B_1= B_0,x_0,B_0∪{x_0},If x_0 is not good, then we stop our recursive procedure. Note that, if x_0 is good, then by (<ref>) and (<ref>),(0,0) ⇝ [2a_1, a_1 + a_L_1] ×{2}⊂ [a_1 + a_L_1 - R,a_1 + a_L_1 + R] ×{2} = I_-1,1,(0,0) ⇝ [a_1 + a_L_2, a_L_1 + a_L_2] ×{2}⊂ [a_L_1 + a_L_2 - R,a_L_1 + a_L_2 + R] ×{2} = I_1,1. Assume A_n, B_n have been defined for n ≥ 1, and the following conditions are satisfied: (a) A_n is connected,(b) B_n ⊂∂_e A_n, (c) bonds started from vertices outside ∪_(i,j) ∈ A_n ∪ B_n I_i,j are still unexplored, and(d) (0,0) ⇝ I_i,j for each (i,j) ∈ (∂_e A_n) \ B_n.Now, if (∂_e A_n)\ B_n = ∅ we stop our recursive definition. Otherwise we let x_n= (i,j) be the minimal point of (∂_e A_n)\ B_n with respect to the order ≺ defined above. By property (d) above, we can fix a vertex (u,2j) ∈ I_i,j such that (0,0) ⇝ (u,2j). In caseu ∈ [z_i,j-R, z_i,j] that is, (u,2j) belongs to the left half of I_i,j (including the midpoint), then we declare that x_n is good if the event T^(u,2j)_+ occurs. In case (u,2j) ∈ (z_i,j,z_i,j+R], then we declare the x_n is good if the event T^(u,2j)_- occurs.Then we defineA_n+1 =A_n∪{x_n},x_n,A_n,,B_n+1= B_n,x_n,B_n∪{x_n},It is clear that (a), (b), (c) listed above are satisfied with A_n+1, B_n+1 in the place of A_n, B_n. Let us now verify that our steering mechanism (that is, choosing T_+ or T_- according to the position of (u,2j)) guarantees property (d). Consider first the case where (u,2j) is in the left half of I_i,j, that is, u ∈ [z_i,j-R,z_i,j]; then,u+a_1+a_L_1≥ z_i,j - R +a_1 + a_L_1(<ref>)= z_i-1,j+1 -R,u+a_1+a_2L_1≤ z_i,j + a_1 + a_2L_1(<ref>)= z_i-1,j+1 + a_2L_1-a_L_1(<ref>)≤ z_i-1,j+1 + R,u+a_L_2 + a_L_1≥ z_i,j - R + a_L_2 + a_L_1(<ref>)= z_i+1,j+1 - R,u+a_L_2 + a_2L_1≤ z_i,j + a_L_2 + a_2L_1(<ref>)= z_i+1,j+1 + a_2L_1 - a_L_1(<ref>)= z_i+1,j+1+R,so (<ref>) implies that, if T^(u,2j)_+ occurs, we have(0,0) ⇝ (u,2j) ⇝ I_i-1,j+1,(0,0)⇝ (u,2j) ⇝ I_i+1,j+1.The case where (u,2j) is in the right half of I_i,j is treated similarly (using (<ref>)). This completes the proof that (d) remains satisfied after each recursion step.Regardless of whether or not the recursion ever ends, we letC be the union of all sets A_n that have been defined. By construction, it follows that {| C| =∞}⊆{(0,0) ⇝∞}.Now, observe thatP^a_L_2(x_n| (A_m,B_m): 0 ≤ m ≤ n)≥ 1-δ.This implies that C stochastically dominates the cluster of the origin in Bernoulli oriented site percolation on G^* with parameter 1-δ (see Lemma 1 of <cit.>). As δ can be taken arbitrarily small, this proves the desired result for d = 1.Now let us show how the statement of Theorem <ref> can be obtained from the case we have already treated. Take ϵ > 0 as in the assumption of the theorem; we can then take an infinite set S ⊂^d so that p_y > ϵ for all y ∈ S.Let Π^-_i(S) = {[x_i < 0:(y_1,…, y_i-1, x_i,y_i+1,…, y_d) ∈ S; for somey_1,…, y_i-1,y_i+1,…, y_d ∈ ]},Π^+_i(S) = {[x_i > 0:(y_1,…, y_i-1, x_i,y_i+1,…, y_d) ∈ S; for somey_1,…, y_i-1,y_i+1,…, y_d ∈ ]}(in words, these sets are given by the projection of S to the ith axis, intersected with (-∞,0) and (0,∞), respectively). Since S is infinite, there exists i ∈{1,…, d} and a∈{-,+} such that Π^a_i(S) is infinite; for simplicity, assume that this is the case for a = + and i = 1. It is then easy to see that the cluster of 0 for percolation on G, when projected on the first coordinate axis times _+, stochastically dominates a percolation configuration on ×_+ which belongs to the case we have already treated. Percolation of this configuration then implies percolation on G.§ TRUNCATION QUESTION FOR RELATED ORIENTED MODELS In this section, we consider different oriented percolation models in which the truncation question can be posed, and an affirmative answer follows almost directly from Theorem <ref>. §.§ Anisotropic oriented percolation on the square latticeFor the first model, let G=(ℤ^2, E), where E =E_v ∪ (∪_n=1^∞ E_h,n):E_v={⟨(x,y),(x,y+1)⟩: x,y∈ℤ_+}, E_h,n ={⟨(x,y),(x+n,y)⟩: x,y∈ℤ_+,n∈ℕ},that is, G is an oriented square lattice equipped with long-range horizontal bonds. Given σ > 0 and (q_n)_n with q_n ∈ [0,1] for each n, we define an oriented bond percolation model where each bond e is open, independently of each other,with probability σ or q_n, if e∈ E_v or e∈ E_h,n, respectively. Let P be a probability measure under which this model is defined.For the graph G, an oriented path is a sequence (v_1, v_2,…) such that, for each i, v_i+1-v_i=(0,1) or (n,0) for some n∈ℕ; the path is open if each oriented bond in it is open. We use also the notation {(0,0) ⇝∞} to denote the set of configurations such that the origin is connected to infinitely many vertices by oriented open paths on G.As in Section 1, we denote by (q_n^k)_n and P^k the truncated sequence and the truncated probability measure. Thus, for this graph we have a result analogous to Theorem <ref>: For the Bernoulli long-range oriented percolation model on G, if lim sup q_n>0,then the truncation problem has an affirmative answer. Moreover,lim_k→∞ P^k{(0,0) ⇝∞} = 1. From the percolation model on G, we define an induced bond percolation model on the graph G of the previous sections with d = 1, that is, G = (𝕍,𝔼) with 𝕍 = ×_+ and 𝔼 as in (<ref>). We declare each bond ⟨ (x,y);(x+n,y+1)⟩ in G as open if and only if both thebonds ⟨ (x,y);(x+n,y)⟩ and ⟨ (x+n,y);(x+n,y+1)⟩ in G are open. Observe that: * each bond ⟨ (x,y);(x+n,y+1)⟩∈𝔼 is open with probability p_n:=σ q_n and by hypothesis lim sup q_n>0;* if there is an infinite open path in the induced model on G then this implies the existence of an infinite oriented path in the original model on G;* the induced percolation model on G is not an independent model, because the open or closed statuses for bonds with the same end vertex are positively correlated. However, given any collection of bonds in which any two bonds have distinct end vertices, the statuses of all these bonds are independent. Therefore, there exist no problems regarding the definition of events analogous to T^(x,y)_- and T^(x,y)_+ and in showing lower bounds like (<ref>).The range of the dependence on the induced model on G goes to infinity as the truncation parameter k→∞. Hence, the conclusion of Theorem <ref> could not be derived from standard techniques of stochastic comparison with product measures (see for instance the main result in <cit.>).We can now prove that, for the induced model on G, percolation occurs with high probability if k is large by an argument that is almost identical to the one of the previous section. The only difference is that, in the renormalized site percolation configuration on G^* that results from the exploration algorithm, some dependence with range one now arises. This is because the probability that a vertex x_n = (i,j) ∈𝕍^* in our exploration is good will be affected by a previous query of the vertex (i-2,j). This issue is settled by choosing the constant δ so that, for one-dependent oriented percolation configurations on G^* with density of open bonds above 1-δ, percolation occurs with high probability, since we are now in the context of one-dependent percolation, where <cit.> applies. §.§ Long-range contact processes on ℤ^d The second model we consider is a contact process on ℤ^d with long-range interactions, such as the one considered in <cit.>. To define the model, we fix a family of non-negative real numbers (λ_y)_y ∈^d, and take a family of independent Poisson point processes on [0,∞): * a process D^x of rate 1 for each x ∈ℤ^d;* a process B^(x,y) of rate λ_x-y for each ordered pair (x, y) with x,y∈ℤ^d.We view each of these processes as a random discrete subset of [0,∞) and write, for 0≤ a < b, D^x_[a,b] = D^x∩ [a, b] and B^(x,y)_[a,b] = B^(x,y)∩ [a,b]. We let 𝒫 be a probability measure under which these processes are defined.Fix k ∈ℕ. Given x, y ∈ℤ^d and 0 ≤ s ≤ t, we say (x,s) and (y,t) are k-connected, and write (x,s) k⇝ (y,t), if there exists a function γ:[s,t] →ℤ^d that is right-continuous, constant between jumps and satisfies:γ(s) =x, γ(t) = yand, for all r ∈ [s,t], r ∉ D^γ(r),r ∈ B^(γ(r-),γ(r)) if γ(r) ≠γ(r-),γ(r) - γ(r-)_∞≤ k .This provides a continuous-time percolation structure for the lattice ℤ^d. From the point of view of interacting particle systems, one usually definesξ_t,k(x) = 1{(0,0) k⇝ (x,t)}, x ∈ℤ^d,t ≥ 0,where 1 denotes the indicator function, thus obtaining a Markov process (ξ_t,k)_t≥ 0 on the state space {0,1}^^d. For this process, the identicallyzeroconfiguration (denoted by 0) is absorbing.For the long-range contact process on ℤ^d, if there exists λ> 0 such that λ_y>λ for infinitely many y, thenlim_k →∞𝒫(ξ_t,k≠0 for allt ) = 1. Similarly to the proof in Section <ref>, we can easily reduce the proof to the case of d= 1 andλ_a_n > λ > 0 for an increasing sequence (a_n)_n∈ℕ. So we now turn to this case.Fix δ > 0. Choose τ > 0 such that𝒫(D^0_[0,τ]≠∅) < δ/4.Given the Poisson processes {(D^x)_x, (B^(x,y))_(x,y)}, we now define a percolation configuration on the graph G = (𝕍,𝔼). We declare a bond ⟨ (x,n),(x+y,n+1)⟩ of 𝔼 to be open if:D^x_[τ n, τ(n+1)] = ∅, D^x+y_[τ n, τ(n+1)] = ∅, B^(x,x+y)_[τ n, τ(n+1)]≠∅.Let P be the probability distribution of this induced percolation configuration, and P^k the corresponding truncation (that is, the induced configuration obtained from {(D^x)_x, (B^(x,y))_(x,y)} by suppressing the Poisson processes B^(x,y) with |y-x| > k). We observe that * each bond ⟨ (x,n),(x+y,n+1)⟩∈𝔼 is open with probability larger than(1-δ/4)^2 · (1-exp{-λ_y τ}); * if there is an infinite open path in the induced model on G for some k, then we can construct a function γ:[0,∞) →ℤ with γ(0) = 0 and satisfying the three last requirements of (<ref>) for r ∈ [0,∞), so that have ξ_k,t≠0 for all t;* given any collection of bonds in which any two bonds have distinct start vertices and distinct end vertices, the statuses of all these bonds are independent. Note that there is more dependence here than in the model of Section <ref> (since bonds with coinciding starting points are dependent here), so we have to be more careful in implementing the proof of Section <ref>. We let ϵ = (1-δ4)^2· (1-exp{-λτ}) and choose L_0 and L_1 such thatP(Bin(L_0,ϵ) > 0) > 1-δ/8, P(Bin(L_1,ϵ) > L_0) > 1-δ/8.We now choose R and L_2 and, for (x,y) ∈^2_+, we define events R^(x,y)_i, S^(x,y)_i, T^(x,y)_- and T^(x,y)_+ exactly as in Section <ref>. Note that the event ∪_i=1^a_L_1 R^(0,0)_i is guaranteed to occur if the following items are satisfied: (a) D^0_[0,τ] = ∅;(b) for at least L_0 indices i ∈{1,…, L_1}, we haveD^a_i_[0,τ] = ∅, B^(0,a_i)_[0,τ]≠∅. (c) out of the indices i satisfying the requirements of item (b), at least one also satisfiesD^a_i_[τ,2τ] = ∅, D^a_i + a_1_[τ,2τ] = ∅, B^(a_i,a_i + a_1)_[τ,2τ]≠∅. Hence, by (<ref>) and (<ref>), we have P(∪_i=1^a_L_1 R^(0,0)_i) > 1-δ/2, and, by translation invariance of the Poisson processes, P(∪_i=1^a_L_1 R^(x,y)_i) > 1-δ/2 for any (x,y) ∈^2_+. Similarly, we have P(∪_i=1^a_L_1 S^(x,y)_i) > 1-δ/2, so thatP(T^(x,y)_- ) > 1-δ,and the same argument shows thatP(T^(x,y)_+ ) > 1-δalso holds.From here onward, the proof proceeds as in Section <ref>, with the only difference that already appeared in the treatment of the model of Section <ref>: in the site percolation configurationin the lattice G^* that results from the exploration algorithm, dependence of range one arises. As in Section <ref>, this issue is resolved (and percolation is guaranteed) as soon as 1-δ is supercritical for one-dependent site percolation on G^*.§ ACKNOWLEDGEMENTSThe authors would like to thank Daniel Ungaretti and Rangel Baldasso for helpful discussions. The research of B.N.B.L. was supported in part by CNPq grant 309468/2014-0 and FAPEMIG (Programa Pesquisador Mineiro). C.A. was supported by FAPESP, grant 2013/24928-2, and is thankful for the hospitality of the UFMG Mathematics Department. The research of M.H. was partially supported by CNPq grant 406659/2016-1. 999 Be Berger, N., Transience, Recurrence and Critical Behavior for Long-Range Percolation, Commun. Math. Phys. 226, 531-558 (2002). ELV van Enter A.C.D., de Lima B.N.B., Valesin D. Truncated Long-Range Percolation on Oriented Graphs, Journal of Statistical Physics 164 1, 166-173 (2016).FL Friedli S., de Lima B.N.B., On the truncation of systems with non-summable Interactions, Journal of Statistical Physics 122 6, 1215-1236 (2006).FLS Friedli S., de Lima B.N.B., Sidoravicius V. On Long Range Percolation with Heavy Tails, Electronic Communications in Probability9, 175-177 (2004).GM Grimmett G., Marstrand J.M., The Supercritical Phase of Percolation is Well Behaved, Proc. Roy. Soc. London Ser A 430, 439-457 (1990).harris Harris, T. E.Contact interactions on a lattice, The Annals of Probability, 969-988 (1974). LS de Lima B.N.B., Sapozhnikov A. On the truncated long range percolation on ℤ^2, Journal of Applied Probability45, 287-291 (2008).LSS Liggett T.M., Schonmann R.H., Stacey A.M. Domination by product measures, Ann. Probab. 25, 71-95 (1997).MS Meester R., Steif J. On the continuity of the critical value for long range percolation in the exponential case, Communications in Mathematical Physics 180.2, 483-504 (1996).MSV Menshikov M., Sidoravicius V. and Vachkovskaia M.A note on two-dimensional truncated long-range percolation, Adv. Appl. Prob. 33, 912-929 (2001).SSV Sidoravicius V., Surgailis D., Vares M.E., On the Truncated Anisotropic Long-Range Percolation on 𝐙^2, Stoch. Proc. and Appl. 81, 337-349 (1999). | http://arxiv.org/abs/1709.09757v1 | {
"authors": [
"Caio T. M. Alves",
"Marcelo Hilário",
"Bernardo N. B. de Lima",
"Daniel Valesin"
],
"categories": [
"math.PR",
"60K35, 82B43"
],
"primary_category": "math.PR",
"published": "20170927230756",
"title": "A note on truncated long-range percolation with heavy tails on oriented graphs"
} |
Dynamic Label Graph Matching for Unsupervised Video Re-Identification Mang Ye^1, Andy J Ma^1, Liang Zheng^2, Jiawei Li^1, Pong C Yuen^1 ^1 Hong Kong Baptist University ^2 University of Technology Sydney {mangye,andyjhma,jwli,pcyuen}@comp.hkbu.edu.hk, [email protected] 30, 2023 ========================================================================================================================================================================================================================emptyLabel estimation is an important component in an unsupervised person re-identification (re-ID) system. This paper focuses on cross-camera label estimation, which can be subsequently used in feature learning to learn robust re-ID models. Specifically, we propose to construct a graph for samples in each camera, and then graph matching scheme is introduced for cross-camera labeling association. While labels directly output from existing graph matching methods may be noisy and inaccurate due to significant cross-camera variations, this paper propose a dynamic graph matching (DGM) method. DGM iteratively updates the image graph and the label estimation process by learning a better feature space with intermediate estimated labels. DGM is advantageous in two aspects: 1) the accuracy of estimated labels is improved significantly with the iterations; 2) DGM is robust to noisy initial training data. Extensive experiments conducted on three benchmarks including the large-scale MARS dataset show that DGM yields competitive performance to fully supervised baselines, and outperforms competing unsupervised learning methods.[Code is available at <www.comp.hkbu.edu.hk/%7e mangye/>]§ INTRODUCTIONPerson re-identification (re-ID), a retrieval problem in its essence <cit.>, aims to search for the queried person from a gallery of disjoint cameras. In recent years, impressive progress has been reported in video based re-ID <cit.>, because video sequences provide rich visual and temporal information and can be trivially obtained by tracking algorithms <cit.> in practical video surveillance applications. Nevertheless, the annotation difficulty limits the scalability of supervised methods in large-scale camera networks, which motivates us to investigate an unsupervised solution for video re-ID. The difference between unsupervised learning and supervised learning consists in the availability of labels. Considering the good performance of supervised methods, an intuitive idea for unsupervised learning is to estimate re-ID labels as accurately as possible. In previous works, part from directly using hand-crafted descriptors <cit.>, some other unsupervised re-ID methods focus on finding shared invariant information (saliency <cit.> or dictionary <cit.>) among cameras. Deviating from the idea of estimating labels, these methods <cit.> might be less competitive compared with the supervised counterparts. Meanwhile, these methods also suffer from large cross-camera variations. For example, salient features are not stable due to occlusions or viewpoint variations. Different from the existing unsupervised person re-ID methods, this paper is based on a more customized solution, , cross-camera label estimation. In other words, we aim to mine the labels (matched or unmatched video pairs) across cameras. With the estimated labels, the remaining steps are exactly the same with supervised learning. To mine labels across cameras, we leverage the graph matching technique (, <cit.>) by constructing a graph for samples in each camera for label estimation. Instead of estimating labels independently, the graph matching approach has shown good property in finding correspondences by minimize the globally matching cost with intra-graph relationship. Meanwhile, label estimation problem for re-ID task is to link the same person across different cameras, which perfectly matches the graph matching problem by treating each person as a graph node. However, labels directly estimated by existing graph matching are very likely to be inaccurate and noisy due to the significant appearance changes across cameras. So a fixed graph constructed in the original feature space usually does not produce satisfying results. Moreover, the assumption that the assignment cost or affinity matrix is fixed in most graph matching methods may be unsuitable for re-ID due to large cross-camera variations <cit.>.In light of the above discussions, this paper proposes a dynamic graph matching (DGM) method to improve the label estimation performance for unsupervised video re-ID (the main idea is shown in Fig. <ref>). Specifically, our pipeline is an iterative process. In each iteration, a bipartite graph is established, labels are then estimated, and then a discriminative metric is learnt. Throughout this procedure, labels gradually become more accurate, and the learnt metric more discriminative. Additionally, our method includes a label re-weighting strategy which provides soft labels instead of hard labels, a beneficial step against the noisy intermediate label estimation output from graph matching.The main contributions are summarized as follows: * We propose a dynamic graph matching (DGM) method to estimate cross-camera labels for unsupervised re-ID, which is robust to distractors and noisy initial training data. The estimated labels can be used for further discriminative re-ID models learning.* Our experiment confirms that DGM is only slightly inferior to its supervised baselines and yields competitive re-ID accuracy compared with existing unsupervised re-ID methods on three video benchmarks. § RELATED WORKUnsupervised Re-ID. Since unsupervised methods could alleviate the reliance on large-scale supervised data, a number of unsupervised methods have been developed. Some transfer learning based methods <cit.> are proposed. Andy et al. <cit.> present a multi-task learning method by aligning the positive mean on the target dataset to learn the re-ID models for the target dataset. Peng et al. <cit.> try to adopt the pre-trained models on the source datasets to estimate the labels on the target datasets. Besides that, Zhao et al. <cit.> present a patch based matching method with inconsistent salience for re-ID. An unsupervised cross dataset transfer learning method with graph Laplacian regularization terms is introduced in <cit.>, and a similar constraint with graph Laplacian regularization term for dictionary learning is proposed in <cit.> to address the unsupervised re-ID problem. Khan et al. <cit.> select multiple frames in a video sequence as positive samples for unsupervised metric learning, which has limited extendability to the cross-camera settings.Two main differences between the proposed method and previous unsupervised re-ID methods are summarized. Firstly, this paper estimates labels with graph matching to address the cross-camera variation problem instead of directly learning an invariant representation.Secondly, output estimated labels of dynamic graph matching can be easily expanded with other advanced supervised learning methods, which provides much flexibility for practical applications in large-scale camera network.Two contemporary methods exists <cit.> which also employ the idea of label estimation for unsupervised re-ID. Liu et al. <cit.> use a retrieval method for labeling, while Fan et al. <cit.> employ k-means for label clustering. Graph Matching for Re-ID. Graph matching has been widely studied in many computer vision tasks, such as object recognition and shape matching <cit.>. It has shown superiority in finding consistent correspondences in two sets of features in an unsupervised manner. The relationships between nodes and edges are usually represented by assignment cost matrix <cit.> or affinity matrix <cit.>. Currently graph matching mainly focuses on optimizing the matching procedure with two fixed graphs. That is to say, the affinity matrix is fixed first, and then graph matching is formulated as linear integer programs <cit.> or quadratic integer programs <cit.>. Different from the literature, the graph constructed based on the original feature space is sub-optimal for re-ID task, since we need to model the camera variations besides the intra-graph deformations. Therefore, we design a dynamic graph strategy to optimize matching. Specifically, partial reliable matched results are utilized to learn discriminative metrics for accurate graph matching in each iteration.Graph matching has been introduced in previous re-ID works which fall into two main categories. (1) Constructing a graph for each person by representing each node with body parts <cit.> or local regions <cit.>, and then a graph matching procedure is conducted to do re-identification. (2) Establishing a graph for each camera view, Hamid et al. <cit.> introduces a joint graph matching to refine final matching results. They assume that all the query and gallery persons are available for testing, and then the matching results can be optimized by considering their joint distribution. However, it is hard to list a practical application for this method, since only the query person is available during testing stage in most scenarios. Motivated by <cit.>, we construct a graph for each camera by considering each person as a node during the training procedure. Subsequently, we could mine the positive video pairs in two cameras with graph matching.§ GRAPH MATCHING FOR VIDEO RE-IDSuppose that unlabelled graph 𝒢_𝒜 contains m persons, which is represented by [𝒜] = {𝐱^i_a|i = 1,2,⋯,m} for camera A, and another graph 𝒢_ℬ consists of n persons denoted by [ℬ]_0 = {𝐱^j_b|j = 0,1,2,⋯,n} for camera B. Note that [ℬ]_0 contains another 0 element besides the n persons. The main purpose is to model the situation that more than one person in 𝒢_𝒜 cannot find its correspondences in 𝒢_ℬ, allowing person-to-dummy assignments. To mine the label information across cameras, we follow <cit.> to formulate it as a binary linear programming with linear constraints:G(𝐲) = min_Y C^T𝐲s.t. ∀ i ∈ [𝒜],∀ j ∈ [ℬ]_0:y^j_i∈{0,1},∀ j ∈ [ℬ]_0:∑_i ∈ [𝒜]y_i^j ≤ 1,∀ i ∈ [𝒜]:∑_j ∈ [ℬ]_0y_i^j = 1,where 𝐲= {y_i^j}∈ℝ^ m(n+1)× 1 is an assignment indicator of node i and j, representing whether i and j are the same person (y_i^j =1) or not (y_i^j =0). C = {C(i,j)} is the assignment cost matrix with each element illustrating the distance of node i to node j. The assignment cost is usually defined by node distance like C(i,j) = Dist(𝐱^i_a,𝐱^j_b), as done in <cit.>. Additionally, some geometry information is added in many feature point matching models <cit.>.For video re-ID, each node (person) is represented by a set of frames. Therefore, Sequence Cost (C_S) and Neighborhood Cost (C_N) are designed as the assignment cost in the graph matching model for video re-ID under a certain metric. The former cost penalizes matchings with mean set-to-set distance, while the latter one constrains the graph matching with within-graph data structure. The assignment cost between person i and j is then formulated as a combination of two costs with a weighting parameter λ in a log-logistic form:C = log(1+ e^ (C_S + λ C_N)).Sequence Cost. The sequence cost C_S penalizes the matched sequences with the sequence difference. Under a discriminative metric M learnt from frame-level features, the average set distance between video sequences {x_a^i} and {x_b^j} is defined as the sequence cost, , C_S(i,j) = 1/|{x_a^i}||{x_b^j}|∑_∑_D_M( x_a^i_m,x_b^j_n).Neighborhood Cost. The neighborhood cost C_N models the within camera data structure with neighborhood similarity constraints. Specifically, the correctly matched person pair's neighborhood under two cameras should be similar <cit.>. A primarily experiment on PRID2011 dataset with features in <cit.> is conducted to justify this point. Results shown in Fig. <ref> illustrates that the percentages of the same person having common neighbors are much larger than that of different persons. It means that the same person under two different cameras should share similar neighborhood <cit.>. Moreover, compared with image-based re-ID, the neighborhood similarity constraints for video-based re-ID are much more effective. It verifies our idea to integrate the neighborhood constraints for graph matching in video re-ID, which could help to address the camera camera variations. The neighborhood cost C_N penalizes the neighborhood difference between all matched sequences, which is formulated by,C_N(i,j) =1/|𝒩^i_a||𝒩^j_b|∑_x̅_a^i'∈𝒩^i_a∑_x̅_b^j'∈𝒩^j_bD_M( x̅_a^i', x̅_b^j') s.t. 𝒩^i_a(i,k) = {x̅_a^i' | D_M(x̅_a^i, x̅_a^i') < k }, 𝒩^j_b(j,k) = {x̅_b^j' | D_M(x̅_b^j, x̅_b^j') < k },where 𝒩^i_a and 𝒩^j_b denote the neighborhood of person i in camera A and person j in camera B, k is the neighborhood parameter. For simplicity, a general kNN method is adopted in our paper, and k is set as 5 for all experiments. Meanwhile, a theoretical analysis of the neighborhood constraints is presented. Let x̅_a^p be a neighbor of person i in camera A and x̅_b^q be its neighbor in camera B. From the geometry perspective, we haveD_M(x̅_a^p,x̅_b^q)≤ D_M(x̅_a^p,x̅_a^i) + D_M(x̅_b^i,x̅_b^q) +D_M(x̅_a^i,x̅_b^i).Since x̅_a^p and x̅_b^q are the neighbors of x̅_a^i and x̅_b^i, respectively, D_M(x̅_a^p,x̅_a^i) and D_M(x̅_b^i,x̅_b^q) are small positive numbers. On the other hand, D_M(x̅_a^i,x̅_b^i) is also a small positive under a discriminative metric D_M. Thus, the distance between two neighbors x̅_a^p and x̅_b^q is small enough, ,D_M(x̅_a^p,x̅_b^q)≤ε.§ DYNAMIC GRAPH MATCHING A number of effective graph matching optimization methods could be adopted to solve the matching problem. After that, an intuitive idea to solve unsupervised video re-ID is learning a re-identification model based on the output of graph matching. However, there still remains two obvious shortcomings:* Since existing graphs are usually constructed in the original feature space with fixed assignment cost, it is not good enough for re-ID problem due to the large cross camera variations. Therefore, we need to learn a discriminative feature space to optimize the graph matching results.* The estimated labels output by graph matching may bring in many false positives and negatives to the training process. Moreover, the imbalanced positive and negative video pairs would worsen this situation further. Therefore, it is reasonable to re-encode the weights of labels for overall learning, especially for the uncertain estimated positive video pairs. To address above two shortcomings, a dynamic graph matching method is proposed. It iteratively learns a discriminative metric with intermediate estimated labels to update the graph construction, and then the graph matching is improved. Specifically, a re-weighting scheme is introduced for the estimated positive and negative video pairs. Then, a discriminative metric learning method is introduced to update the graph matching. The block diagram of the proposed method is shown in Fig. <ref>.§.§ Label Re-weightingThis part introduces the designed label re-weighting scheme. Note that the following re-weighting scheme is based on the output (𝐲) of optimization problem Eq. <ref>. y_i^j ∈{0,1} is a binary indicator representing whether i and j are the same person (y_i^j =1) or not (y_i^j =0).Positive Re-weighting. All y_i^j=1 estimated by graph matching are positive video pairs. Since the labels are uncertain, it means that considering all y_i^j=1 equally is unreasonable. Therefore, we design a soft label l_+(i,j) encoded with a Gaussian kernel for y_i^j=1,l_+(i,j) = {[e^-C(i,j), if C(i,j) < λ_+;0,others ].where λ_+ is the pre-defined threshold. C means the assignment cost computed in Eq. <ref> in current iteration. In this manner, the positive labels (y =1) are converted into soft labels, with smaller distance assigned larger weights while larger distance with smaller weights. Meanwhile, the filtering strategy could reduce the impact of false positives.Negative Re-weighting. Since abundant negative video pairs exist in video re-ID task compared with positive video pairs, some hard negative are selected for efficient training, l_-(i,j) for all y_i^j =0 is defined asl_-(i,j) = {[-1 , if C(i,j) < λ_-;0, others, ].where λ_- is the pre-defined threshold. Considering both Eq. <ref> and Eq. <ref>, we define λ_+ = λ_- = c_m based on the observation shown in Fig <ref>. c_m denotes the mean of C, which would be quite efficient. Thus, the label re-weighting scheme is refined by l(i,j) = {[ e^-C(i,j) * y_i^j, if 0 < y_i^jC(i,j) < c_m; 0,if C(i,j) > c_m;-1,others.;]. The label re-weighting scheme has the following advantages: (1) for positive video pairs, it could filter some false positives and then assign different positive sample pairs different weights; (2) for negative video pairs, a number of easy negatives would be filtered. The re-weighing scheme is simple but effective as shown in the experiments. §.§ Metric Learning with Re-weighted LabelsWith the label re-weighting scheme, we could learn a discriminative metric similar to many previous supervised metric learning works. We define the loss function by log-logistic metric learning as done in <cit.>, ,f^*_M(x̅^i_a,x̅^j_b) = log (1 + e^l(i,j)(D_M(x̅^i_a,x̅^j_b)-c_0)),where c_0 is a positive constant bias to ensure D_M has a lower bound. It is usually defined by the average distance between two cameras. The function D_M denotes the distance of x̅^i_a and x̅^j_b under the distance metric M, which is defined by D_M(x̅^i_a,x̅^j_b) = (x̅^i_a - x̅^j_b)^TM(x̅^i_a - x̅^j_b). We choose the first-order statistics x̅^i_a and x̅^j_b to represent each person as done in <cit.>.By summing up all of sequence pairs, we obtain the probabilistic metric learning problem under an estimated 𝐲 formulated by,F(M;𝐲) = ∑ ^m_i=1∑ ^n_j=1ω _ijf^*_M (x̅^i_a,x̅^j_b),where ω _ij is a weighting parameter to deal with the imbalanced positive and negative pairs. The weights ω _ij are caculated by ω _ij = 1/|{l(i,j)|l(i,j)>0}| if l(i,j)>0, and ω _ij = 1/|{l(i,j)|l(i,j)=-1}| if l(i,j)=-1, where |·| denotes the number of candidates in the set. Note that some uncertain pairs are assigned with label l(i,j) = 0 without affecting the overall metric learning. The discriminative metric can be optimized by minimizing Eq. <ref> using existing accelerated proximal gradient algorithms (, <cit.>). §.§ Iterative Updating With the label information estimated by graph matching, we could learn an improved metric by selecting high-confident labeled video pairs. By utilizing the learnt metric, the assignment cost of Eq. <ref> and Eq. <ref> could be dynamically updated for better graph matching in a new iteration. After that, better graph matching could provide more reliable matching results, so as to improve the previous learnt metric. Iteratively, a stable graph matching result is finally achieved by a discriminative metric. The matched result could provide label data for further supervised learning methods. Meanwhile, a distance metric learnt in an unsupervised way could also be directly adopted for re-ID. The proposed approach is summarized in Algorithm <ref>. Convergence Analysis. Note that we have two objective functions F and G optimizing 𝐲 and M in each iteration. To ensure the overall convergence of the proposed dynamic graph matching, we design a similar strategy as discussed in <cit.>. Specifically, M can be easily optimized by choosing a suitable working step size η≤ L, where L is the Lipschitz constant of the gradient function ▽ F(M,𝐲). Thus, it could ensure F(M^t;𝐲^t-1)≤ F(M^t-1;𝐲^t-1), a detailed proof is shown in <cit.>. For 𝐲^t at iteration t, we constrain the updating procedure by keep on updating the assignment cost matrix C^t until getting a better 𝐲 which satisfies G(M^t;𝐲^t)≤ G(M^t;𝐲^t-1), similar proof can be derived from <cit.>. By constrain the updating procedure, it could satisfy the criteria G^t(𝐲;M) + F^t(M;𝐲) ≤ G^t-1(𝐲;M) + F^t-1(M;𝐲). This is validated in our experiments as discussed in Section <ref>. Particularly, the proposed method converges steadily.Complexity Analysis. In the proposed method, most computational costs focus on the iterative procedure, since we need to conduct the graph matching with Hungarian algorithm at each iteration. We need to compute the sequence cost O(n^2) and neighborhood cost O(kn+n^2) for each camera, and then graph matching time complexity is O(n^3). Updating M with accelerated proximal gradient is extremely fast as illustrated in <cit.>. However, the proposed method is conducted offline to estimate labels, which is suitable for practical applications. During the online testing procedure, we only need to compute the distance between the query person p and the gallery persons with the learnt re-identification model. The distance computation complexity is O(n) and ranking complexity is O(nlog n), which is the same as existing methods <cit.>.§ EXPERIMENTAL RESULTS §.§ Experimental Settings Datasets. Three publicly available video re-ID datasets are used for evaluation: PRID-2011 <cit.>, iLIDS-VID <cit.> and MARS <cit.> dataset. The PRID-2011 dataset is collected from two disjoint surveillance cameras with significant color inconsistency. It contains 385 person video tracks in camera A and 749 person tracks in camera B. Among all persons, 200 persons are recorded in both camera views. Following <cit.>, 178 person video pairs with no less than 27 frames are employed for evaluation. iLIDS-VID dataset is captured by two non-overlapping cameras located in an airport arrival hall, 300 person videos tracks are sampled in each camera, each person track contains 23 to 192 frames. MARS dataset is a large scale dataset, it contains 1,261 different persons whom are captured by at least 2 cameras, totally 20,715 image sequences achieved by DPM detector and GMCCP tracker automatically.Feature Extraction. The hand-craft feature LOMO <cit.> is selected as the frame feature on all three datasets. LOMO extracts the feature representation with the Local Maximal Occurrence rule. All the image frames are normalized to 128 × 64. The original 26960-dim features for each frame are then reduced to a 600-dim feature vector by a PCA method for efficiency considerations on all three datasets. Meanwhile, we conduct a max-pooling for every 10 frames to get more robust video feature representations.Settings. All the experiments are conducted following the evaluation protocol in existing works <cit.>. PRID-2011 and iLIDS-VID datasets are randomly split by half, one for training and the other for testing. In testing procedure, the regularized minimum set distance <cit.> of two persons is adopted. Standard cumulated matching characteristics (CMC) curve is adopted as our evaluation metric. The procedure are repeated for 10 trials to achieve statistically reliable results, the training/testing splits are originated from <cit.>. Since MARS dataset contains 6 cameras with imbalanced tracklets in different cameras, we initialize the tracklets in camera 1 as the base graph, the same number of tracklets from other five cameras are randomly selected to construct a graph for matching. The evaluation protocol on MARS dataset is the same as <cit.>, CMC curve and mAP (mean average precision) value are both reported.Implementation. Both the graph matching and metric learning optimization problems can be solved separately using existing methods. We adopt Hungarian algorithm to solve the graph matching problem for efficiency considerations, and metric learning method (MLAPG) in <cit.> as the baseline methods. Some advanced graph matching and metric learning methods may be adopted as alternatives to produce even better results as shown in Section <ref>. We report the results at 10th iteration, with λ =0.5 for all three datasets if without specification. §.§ Self EvaluationEvaluation of iterative updating. To demonstrate the effectiveness of the iterative updating strategy, the rank-1 matching rates of training and testing at each iteration on three datasets are reported in Fig. <ref>. Specifically, the rank-1 accuracy for testing is achieved with the learnt metric at each iteration, which could directly reflect the improvements for re-ID task. Meanwhile, the overall objective values on three datasets are reported.Fig. <ref>(a) shows that the performance is improved with iterative updating procedure. We could achieve 81.57% accuracy for PRID-2011, 49.33% for iLIDS-VID and 59.64% for MARS dataset. Compare with iteration 1, the improvement at each iteration is significant. After about 5 iterations, the testing performance fluctuates mildly. This fluctuation may be caused by the data difference of the training data and testing data. It should be pointed out that there is a huge gap on the MARS dataset, this is caused by the abundant distractors during the testing procedure, while there is no distractors for training <cit.>. Experimental results on the three datasets show that the proposed iterative updating algorithm improves the performance remarkably. Although without theoretical proof, it is shown in Fig. <ref>(b) that DGM converges to steady and satisfactory performance.Evaluation of label re-weighting. We also compare the performance without label re-weighting strategy. The intermediate labels output by graph matching are simply transformed to 1 for matched and -1 for unmatched pairs. The rank-1 matching rates on three datasets are shown Table <ref>. Consistent improvements on three datasets illustrate that the proposed label-re-weighting scheme could improve the re-ID model learning.Evaluation of label estimation. To illustrate the label estimation performance, we adopt the general precision, recall and F-score as the evaluation criteria. The results on three datasets are shown in Table <ref>. Since graph matching usually constrains full matching, the precision score is quite close to the recall on the PRID-2011 and iLIDS-VID datasets. Note that the precision score is slightly higher than recall is due to the proposed positive re-weighting strategy.Running time. The running times on three datasets with the settings described in Section 5.1 are evaluated. It is implemented with Matlab and executed on a desktop PC with i7-4790K @4.0 GHz CPU and 16GB RAM. The training and testing time are reported by the average running time in 10 trials. For training, since we adopt an efficient graph matching algorithm and accelerated metric learning <cit.>, the training time is acceptable. The training time for the PRID2011 dataset is about 13s, about 15s for iLIDS-VID dataset, about 2.5 hours for the MARS dataset due to the large amount of tracklets. For testing, the running time is fast for our method, since standard 1-vs-N matching scheme is employed. The testing times are less than 0.001s on PRID2011 and iLIDS-VID datasets for each query process, and around 0.01s on MARS with 636 gallery persons.§.§ Estimated Labels for Supervised LearningThis subsection evaluates the effectiveness of the output estimated labels for other supervised learning methods. Compared with the re-identification performances with groundtruth labels (GT), they provide upper bounds as references to illustrate the effectiveness of DGM. Specifically, two metric learning methods MLAPG <cit.> and XQDA <cit.>, and an ID-discriminative Embedding (IDE) deep model <cit.> are selected for evaluation as shown in Fig. <ref>. Configured with MLAPG and XQDA, the performances outperform the baseline l_2-norm on all three datasets, usually by a large margin. The results show that the estimated labels also match well with other supervised methods. Compared with the upper bounds provided by supervised metric learning methods with groundtruth labels, the results on PRID-2011 and MARS datasets are quite close to the upper bounds. Although the results on iLIDS-VID dataset are not that competitive, the main reason can be attributed to its complex environment with many background clutters, such as luggage, passengers and so on, which cannot be effectively solved by a global descriptor (LOMO) <cit.>.Another experiment with IDE deep model on the three datasets shows the expendability of the proposed method to deep learning methods. Specifically, about 441k out of 518k image frames are labelled for 625 identities on the large scale MARS dataset, while others are left with Eq. <ref>. The labelled images are then resized to 227 × 227 pixels as done in <cit.>, square regions 224 × 224 are randomly cropped from the resized images. Three fully convolutional layers with 1,024, 1,024 and N blobs are defined by using AlexNet <cit.>, where N denotes the labelled identities on three datasets. The FC-7 layer features (1,024-dim) are extracted from testing frames, maxpooling strategy is adopted for each sequence <cit.>. Our IDE model is implemented with MxNet. Fig. <ref> shows that the performance is improved with a huge gap to hand-craft features with deep learning technique on the large scale MARS dataset. Comparably, it does not perform well on two small scale datasets (PRID-2011 and iLIDS-VID dataset) compared to hand-craft features due to the limited training data. Meanwhile, the gap between the estimated labels to fully supervised deep learning methods is consistent to that of metric learning methods. Note that since one person may appear in more than one cameras on the MARS dataset, the rank-1 matching rates may be even higher than label estimation accuracy.§.§ Comparison with Unsupervised re-IDThis section compares the performances to existing unsupervised re-ID methods. Specifically, two image-based re-ID methods, Salience <cit.> results originated from <cit.>, and GRDL <cit.> is implemented by averaging multiple frame features in a video sequence to a single feature vector. Four state-of-the-art unsupervised video re-ID methods are included, including DVDL <cit.>, FV3D <cit.>, STFV3D <cit.> and UnKISS <cit.>. Meanwhile, our unsupervised estimated labels are configured with three supervised baselines MLAPG <cit.>, XQDA <cit.> and IDE <cit.> to learn the re-identification models as shown in Table <ref>.It is shown in Table <ref> that the proposed method outperforms other unsupervised re-ID methods on PRID-2011 and MARS dataset often by a large margin. Meanwhile, a comparable performance with other state-of-the-art performances is obtained on iLIDS-VID dataset even with a poor baseline input. In most cases, our re-ID performance could achieve the best performances on all three datasets with the learnt metric directly. We assume that the proposed method may yield better results by adopting better baseline descriptors, other advanced supervised learning methods would also boost the performance further. The advantages can be attributed to two folds: (1) unsupervised estimating cross cameras labels provides a good solution for unsupervised re-ID, since it is quite hard to learn invariant feature representations without cross-camera label information; (2) dynamic graph matching is a good solution to select matched video pairs with the intra-graph relationship to address the cross camera variations.§.§ Robustness in the WildThis subsection mainly discusses whether the proposed method still works under practical conditions.Distractors. In real applications, some persons may not appear in both cameras. To simulate this situation for training, we use the additional 158 person sequences in camera A and 549 persons in camera B of PRID-2011 dataset to conduct the experiments. d% * N distractor persons are randomly selected from these additional person sequences for each camera. They are added to the training set as distractors. N is the size of training set. We use these distractors to model the practical application, in which many persons cannot find their correspondences in another camera.Trajectory segments. One person may have multiple sequences in each camera due to tracking errors or reappear in the camera views. Therefore, multiple sequences of the same person may be unavoidable to be false treated as different persons. To test the performance, p% *N person sequences are randomly selected to be divided into two halves in each camera on PRID-2011 dataset. In this manner, about p% persons would be false matched since the p% are both randomly selected for two cameras.Table <ref> shows that the performance without one-to-one matching assumption is still stable, with only a little degradation in both situations, this is because: (1) Without one-to-one assumption, it will increase the number of negative matching pairs, but due to the abundant negatives pairs in re-ID task, the influence is not that much. (2) The label re-weighting strategy would reduce the effects of low-confidence matched positive pairs.§ CONCLUSIONThis paper proposes a dynamic graph matching method to estimate labels for unsupervised video re-ID. The graph is dynamically updated by learning a discriminative metric. Benefit from the two layer cost designed for graph matching, a discriminative metric and an accurate label graph are updated iteratively. The estimated labels match well with other advanced supervised learning methods, and superior performances are obtained in extensive experiments. The dynamic graph matching framework provides a good solution for unsupervised re-ID.Acknowledgement This work is partially supported by Hong Kong RGC General Research Fund HKBU (12202514), NSFC (61562048). Thanks Guangcan Mai for the IDE implementation.ieee | http://arxiv.org/abs/1709.09297v1 | {
"authors": [
"Mang Ye",
"Andy J Ma",
"Liang Zheng",
"Jiawei Li",
"P C Yuen"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170927010710",
"title": "Dynamic Label Graph Matching for Unsupervised Video Re-Identification"
} |
[email protected] Astronomical Institute of the Academy of Sciences of the Czech Republic, Boční II 1401/1a, CZ-141 31 Prague, Czech RepublicThe implications of two different time constraints on theMathisson-Papapetrou-Dixon (MPD) equations are discussed under three spinsupplementary conditions (SSC). For this reason the MPD equations are revisitedwithout specifying the affine parameter and several relations are reintroducedin their general form. The latter allows to investigate the consequences ofcombining the Mathisson-Pirani (MP) SSC, the Tulczyjew-Dixon (TD) SSC and theOhashi-Kyrian-Semerák (OKS) SSC with two affine parameter types:the proper time on one hand and the parameterizations introduced in[Gen. Rel. Grav. 8, 197 (1977)] on the other.For the MP SSC and the TD SSC it is shown that quantities that are constant ofmotion for the one affine parameter are not for the other, while for the OKSSSC it is shown that the two affine parameters are the same. To clarify the relation between the two affine parameters in the case of theTD SSC the MPD equations are evolved and discussed.Time parameterizations and spin supplementary conditions of theMathisson-Papapetrou-Dixon equations Georgios Lukes-Gerakopoulos December 30, 2023 =====================================================================================================§ INTRODUCTION The motion of a small mass body whose effect on the spacetime background isnegligible had been first studied in terms of the multipole moments of the bodyby Mathisson <cit.> and Papapetrou <cit.>. A covariant formalism was achieved by Dixon in <cit.>, who also reformulated therespective equations of motion. These equations of motion are known now asMathisson-Papapetrou-Dixon (MPD) equations.In the case of a solely gravitational interaction within the pole-dipoleapproximation the MPD equations readṗ^μ=-1/2 R^μ_νκλ v^ν S^κλ, Ṡ^μν=p^μ v^ν-v^μ p^ν≡ 2 p^[μv^ν],where p^μ is the four-momentum, v^μ=d x^μ/d χ is the tangentvector and S^μν is the spin tensor of the body. Moreover, R^μ_νκλ is the Riemann tensor and the dot denotes acovariant differentiation along the worldline x^μ(χ), where χ is anevolution parameter along the worldline not necessarily the proper time τ.Thus, it is not assumed that the tangent vector is the four-velocity, and thecontractionv^μ v_μ≡ -v^2 does not represent necessarily the four-velocity preservation v^2=1.The notion of mass can be defined either with respect to the momentum p^ν,i.e. m^2 ≡-p^ν p_ν,or with respect to the tangent vector v^ν, i.e.m_v≡-v^ν p_ν.*Units and notation: The unitsemployed in this work are geometric (G=c=1), and the signature ofthe metric g_μν is (-,+,+,+). Greek letters denote the indicescorresponding to spacetime (running from 0 to 3). The Riemann tensoris defined as R^α_βγδ=Γ^α_γλΓ^λ_δβ- ∂_δΓ^α_γβ- Γ^α_δλΓ^λ_γβ+ ∂_γΓ^α_δβ, where Γ are the Christoffel symbols. The Levi-Civita tensor is ϵ_κλμν=√(-g)ϵ̃_κλμν with the Levi-Civita symbol defined as ϵ̃_0123=1. § SOME USEFUL RELATIONSBy keeping in mind that v^2 is not necessarily constant, some useful consequences of MPD equations are presented below. These consequences coincidewith expressions presented in <cit.> when v^2=1. Contracting Eq. (<ref>) with v_μgives ṗ^μ v_μ=0 . Contraction of Eq. (<ref>) with v_μ givesp^μ=1/v^2(m_v v^μ-Ṡ^μν v_ν) , while contracting with v̇_μ and using relation (<ref>) gives v̇_μṠ^μν =v̇_μ/v^2 (v^μṠ^νρ-Ṡ^μρ v^ν) v_ρ. Contracting Eq. (<ref>) with p_μ givesp^μ=1/m_v(m^2 v^μ-Ṡ^μν p_ν) , while contracting with ṗ_ν and using relation (<ref>) gives ṗ_νṠ^μν =1/m_vṗ_νṠ^νρ p_ρ v^μ. Contracting Eq. (<ref>) with v^μ leads tom_v^2- m^2 v^2=v_μṠ^μν p_ν, which combined with Eq. (<ref>) givesm^2 v^4-m_v^2 v^2=v^κṠ_κμṠ^μν v_ν. Furthermore, one finds that the evolution equation of the mass m isṁ=-1/mṗ_μ p^μ=1/m m_vṗ_μṠ^μν p_ν,for which result Eqs. (<ref>), (<ref>) are used, while the evolution equation of the mass m_v is ṁ_v=-1/v^2(v_μṠ^μν+m_v v^ν)v̇_ν,for which result Eq. (<ref>) is used, and Eq. (<ref>) is taken into account. The square of the spin's measure is S^2=1/2 S_μν S^μν,and its evolution equation readsṠ^2 = 2 p_μ S^μν v_ν,in which calculation Eq (<ref>) is used.§ CHOOSING A WORLDLINE The MPD equation system, consisted of (<ref>), (<ref>) and d x^μ/d χ=v^μ,is under-defined. Namely, there are only 14 independent equations of motion for the 18 variables {x^μ,v^μ,p^μ,S^μν}[Because the spin tensor S^μν is antisymmetric, it contributes only 6 independent equations and variables.]. To define a worldline we have to supplement the system with 4 additional constraints.One of these constraints comes from choosing the evolution parameter χ. A common choice for the evolution parameter is to identify χ with the proper time τ, see, e.g., <cit.>. Then, v^2=1 and the tangent vector v^μ identifies with the four-velocity.Another interesting choice was introduced in <cit.>, according to which χ scales in such way that v_μ u^μ =-1 ,where u^μ=p^μ/m. An apparent consequence of this choice is that m_v=m. This affine parameter is denoted as σ.After having chosen the evolution parameter, the remaining necessary constraints are devoted to choose the center of the mass of the system. The center of mass is called often centroid. By choosing the centroid and the evolutionparameter one defines the evolution along the worldline that the body described by the MP equations follows. In particular, the centroid is fixed by choosing an observer through a time-like vector V^μ for which V_μ S^μν=0. This constraint is known as spin supplementary condition (SSC). In the bibliography there are five established choices of SSC:* the Mathisson-Pirani (MP) condition V^μ=v^μ <cit.>. * the Tulczyjew-Dixon (TD) conditionV^μ=u^μ <cit.>. * the Corinaldesi-Papapetrou conditionV^μ = v_lab <cit.>,where v_lab is a a congruence of “laboratory” observers. * the Newton-Wigner condition V^μ∝ v_lab+u^μ <cit.>. * the Ohashi-Kyrian-Semerák (OKS) condition<cit.>),for which the V^μ is chosen in such way that p^μ∥ v^μ.§ DISCUSSING THE CONSTRAINTS In this section are investigated the consequences of combining the timeconstraints and particularly the condition (<ref>) with the MP SSC,the TD SSC and the OKS SSC. For these three SSCs Eq. (<ref>)shows that the measure of the spin is conserved independently from the timeconstraint choice. §.§ The Mathisson-Pirani SSC Contracting the covariant derivative of MP SSC with v̇_ν resultsin v_μṠ^μνv̇_ν=0, taking this into accountEq. (<ref>) for the MP SSC gives ṁ_v/m_v=v̇^̇2̇/2 v^2⇒m_v^2/v^2=const. . For the condition v^2=1, Eq. (<ref>) gives that m_vis a constant of motion. For the condition (<ref>), Eq. (<ref>) gives thatm^2/v^2=const., since m=m_v. §.§ The Tulczyjew-Dixon SSCFor TD SSC Eq. (<ref>) shows that the mass m is a constant ofmotion. Since for the condition (<ref>) m=m_v, then m_v isconstant as well. Thus, Eq. (<ref>) gives(v_μṠ^μν+m_v v^ν)v̇_ν=0 . According to Eq. (<ref>), if v^νv̇_ν=0, then it holdsthat v_μṠ^μνv̇_ν=0 as well. The former impliesthat v^2 is a constant, while the latter implies that MP SSC holds alongwith the assumed TD SSC. The last implication is proven as follows:contracting Eq. (<ref>) with v_ν gives Ṡ^μρ v_ρ=0,because v_μṠ^μνv̇_ν=0 and it is reasonable to assume that v^2=0 is not the case. Since v^2 and m_v are constantsand it has been shown that Ṡ^μρ v_ρ=0, Eq. (<ref>)results in p^μ||v^μ. If p^μ||v^μ, then Eq. (<ref>) givesv^2=1. Therefore, when v^νv̇_ν=0 the affine parameterdefined by the condition (<ref>) is the proper time, i.e.σ=τ.The cases of TD SSC for which p^μ||v^μ holds are very special cases.Thus, it is reasonable to assume that for the condition (<ref>) in general v^νv̇_ν≠ 0 is true. Under this assumption,Eq. (<ref>) gives m_v= -v_μṠ^μνv̇_ν/v^νv̇_ν=const. . In this case Eq. (<ref>) implies that the variation of v^2 duringthe evolution is reflected on the v_μṠ^μν p_ν evolution.Actually, if one uses the v^2=1 condition instead of thecondition (<ref>), then the variation of m_v^2 duringthe evolution is reflected on the v_μṠ^μν p_ν evolution.Another interesting relation comes from eq. (<ref>), when one usesthe covariant derivative of the TD SSC and then applies eq. (<ref>),then we getv^2=1/m^2(m_v^2-1/2 v_σ S^σμ R_μνρα v^ν S^ρα) .§.§ The Ohashi-Kyrian-SemerákSSCSince for OKS SSC by definition p^μ||u^μ, then it holds thatv_μṠ^μν p_ν=0. Combing the latter with the fact thatm=m_v for the condition (<ref>),Eq. (<ref>) gives thatv^2=1. This means that OKS SSC is satisfying both time constraintssimultaneously; or in other words the affine parameter σ is identicalwith the proper time τ for OKS SSC. Note that the latter holds when(p^μ||u^μ) independently of the implemented SSC. § NUMERIC COMPARISON FOR TULCZYJEW-DIXON SSCThis section examines the evolution of the MPD equations under the two time choices τ and σ. In particular, we are going to examine numericallythe MPD under the TD SSC, since for OKS SSC the two evolution parameterchoices are equivalent and for MP SSC the helical motion introduces anunnecessary complication. §.§ Preliminary considerations To do a numerical comparison, the first issue is the initial condition setup,i.e. the initial position, momentum and spin tensor have to be properly chosen.Since we have the same SSC (TD SSC), we have two observers with initiallythe same position x^μ. The definitions of the momentum and the spintensor depend only on the position x^μ along the worldline<cit.>, hence the initial conditions x^μ, p_μ and S^μνfor two observers are the same.However, the two observers are equipped with clocks that do not tick the same,i.e. they follow different affine parameters. If themomentum and the spintensor are not affected by the different choices of the affine parameter,then a time reparametrization of the MPDequations (<ref>)-(<ref>) just means that the MPD equations will reproduce the same worldline under different affineparameter χ. The validity of the last statement is what is in thissection checked. To evolve the MPD with TD SSC, one needs the relation v^μ = m_v/m^2( p^μ + 2 S^μν R_νρκλ p^ρ S^κλ/4 m^2 + R_αβγδ S^αβ S^γδ) , which gives v^μ as function of x^μ, p_μ and S^μν. Aninteresting fact about relation (<ref>) is that its derivation doesnot depend on the time constraint (see, e.g., <cit.> for the derivation). A related fact is that the relation (<ref>)is invariant under affine parameter changes, since the scalar m_v contains the tangent vector v^μ (definition (<ref>)).The background, on which the MPD are to be evolved, is the Kerr spacetime.The metric tensor of Kerr in Boyer-Lindquist (BL) coordinates{t,r,θ,ϕ} reads g_tt =-1+2 M r/Σ ,g_tϕ = -2 a M r sin^2θ/Σ, g_ϕϕ = Λsin^2θ/Σ ,g_rr = Σ/Δ,g_θθ = Σ ,where Σ = r^2+ a^2 cos^2θ, Δ = ϖ^2-2 M r, ϖ^2= r^2+a^2, Λ = ϖ^4-a^2Δsin^2θ. and M defines the mass and a the Kerr parameter. The motion of a smallspinning body in the stationary and axisymmetric Kerr spacetime preserves,respectively, the energy E= -p_t+1/2g_tμ,νS^μν, and the component of the total angular momentum along the symmetry axis zJ_z= p_ϕ-1/2g_ϕμ,νS^μν.The MPD equations are valid when the size of the spinning body is much smallerthan the curvature, i.e. when λ=|R_μνκλ|/ρ^2≪ 1 , where |R_μνκλ| is the magnitude of the Riemann tensor andρ is the radius of the spinning body. If the radius ρ isapproximated by the Møller radius <cit.>, then ρ=S/m.Thus, since |R_μνκλ|∼ M/r^3, we get λ∼(S/m M)^2(M/r)^3. For the computations of the MPD equations, the dimensionless counterparts ofthe involved quantities are employed. For example, in theirs dimensionlessforms the spin of the small body reads S/(m M), the BL radius reads r/Mand for the momentum holds that p^μ=u^μ. Numerically the values ofdimensionless quantities are equal to the dimensionful by setting M=m=1.From this point on in the article there is no distinction between thedimensionful and the dimensionless quantities. Since our discussion istheoretical, the spin of the small body does not need to be verysmall[See, e.g., Ref. <cit.> for thorough discussion the astrophysically relevant spin values] as long as λ≪ 1. For a radius r ∼ 10 and λ∼ 10^-3, approximation (<ref>) gives that S∼ 1. It is advantageousto use large spin values, because the larger the value is, the greatermight be the difference between v^μ and u^μ, and consequently thedivergence between the two time constraints. Following the initial condition setup presented in <cit.>, instead ofthe spin tensor S^μν the spin four-vectorS_μ≡ -1/2ϵ_μνρσu^νS^ρσ is utilized. According to Ref. <cit.> setup, one can set t=ϕ=0and provide the initial values for r, θ, u^r, S^r, S^θ, while therest of the initial conditions u^t, u^θ, u^ϕ, S^t, and S^ϕ,are fixed by m (Eq. (<ref>)), S (Eq. (<ref>)),E (Eq. (<ref>)), J_z (Eq. (<ref>)) and theconstraint u_μ S^μ=0. The latter constraint is obtained from contractingEq. (<ref>) with u^μ, while inEqs. (<ref>), (<ref>), (<ref>)the inverse relation of Eq. (<ref>)S^ρσ=-η^ρσγδ S_γ u_δ is employed. §.§ Numerical resultsTo show whether the under discussion time constraints reproduce the sameworldline or not, initial conditions leading to a generic non-equatorial orbithas to be chosen. Such initial conditions produce the orbits shown inFig. <ref>. These orbits cover a non-zero width spheroidal shell aroundthe central Kerr black hole (left panel). The pseudocartesian coordinates(x, y, z) used in Fig. <ref> relate to the BL ordinates as followsx=r cosϕsinθ, y=r sinϕsinθ, z=r cosθ.The orbits evolve in a non-trivial manner; examples of trivial motion isa circular or a radial orbit. However, the orbits appear to follow exactlythe same paths until the end (right panel of Fig. <ref>). This impliesthat the two time constraints reproduce the same worldline.To ensure that what is shown in Fig. <ref> is not just an opticalartifact, in the top panel of Fig. <ref> is displayed the relativeradial difference between the two orbits Δ r=|1-r_σ(t)/r_τ(t)| as a function of the coordinate time t. r_τ denotes the radialcomponent of the orbit evolved using the proper time, while r_σ denotesthe radial component of the orbit using the affine parameter σ definedby the constraint (<ref>). The coordinate time introduces a thirdobserver at infinity with his own clock.This clock provides a common time by which the orbits can be compared.The top panel of Fig. <ref> shows that the discrepancies inthe radial component start being at the level of the computational accuracy, whichis double precision, and after t∼ 10^3 they appear to drift away on averagelinearly. This drift resembles Fig. 11 in Ref. <cit.>, where the integrationscheme of s-stage Gauss Runge-Kutta used in this work was tested, and a similardrift was assigned to the interpolation used in the scheme. Actually, in orderto produce the top panel of Fig. <ref> interpolation was employedto get from a two component functional {r_χ(χ),t(χ)} to the functionr_χ(t), since both orbits were computed using their respective affineparameters (χ=σ, τ). Moreover, the bottom panel ofFig. <ref> shows that the relative error of the spin measure Δ S^2=|1-S^2(t)/S^2| increases on average linearly after t∼ 10^3 as well. S^2(t) denotesthe numerically computed value of S^2 at time t, while S^2 denotes theinitial value of the spin. In few words, the drift between the two orbitsshown in the top panel of Fig. <ref> arises for numerical reasons, and the two orbits reproduce the same worldline up to numerical accuracy.The fact that we do not see this drift in Fig. <ref> for the four-velocity conservation (<ref>) in the case of the proper time (black dots, top panel) and for the mass m_v in the case of the affine parameter σ (gray dots, bottom panel) is that at each step quantities v^2 and m_v are normalized in order to compute the velocity through the relation (<ref>). Namely, for the proper time v^2 is kept equal to 1, while for σ m_v is kept equal to m. Thus, it is no wonder why the relative errors of four-velocity conservation v^2=1 in the first case and of the mass m_v=1 in the second case stay at the computational accuracy level for so long. An interesting aspect of Fig. <ref> is the evolution of the relative difference between the initial value of v^2 and the value of v^2 at time t for the affine parameter σ (gray dots, top panel), and of the relative difference between the initial value of m_v and the value of m_v at time t for the proper time (black dots, bottom panel). The respective curves of the above two relative differences are practically identical, even the oscillations during the evolution take place at the same time. These curves provide a numerical example of the analysis provided in Sec. <ref> and show that the orbit does not belong to the special case for which p^μ||v^μ.It is notable that the phase space of the system does not change its dimensionality for the two time parameterization choices, i.e. the number of the constants of motion the same for τ and σ. Namely, in the case of the proper time the four-velocity (<ref>) is preserved and the mass (<ref>) is not, and for the affine parameter the preservation is vice versa. If the number of constants was not the same, then this would imply that the two affine parameter choices alter the nature of the MPD equations and this choice is not just a gauge.§ CONCLUSIONS This article revisited relations derived from the Mathisson-Papapetrou-Dixon equations without specifying the affine parameter nor the spin supplementary condition. Next, the proper time choice versus the affine parameter choice introduced in <cit.> were discussed in the case of the Mathisson-Pirani SSC, the Tulczyjew-Dixon SSC and the Ohashi-Kyrian-Semerák SSC, and the implications of this choice were analyzed. In particular, it was found that under OKS SSC the affine parameters are identical, while for the MP SSC the choice of the affine parameter affects the preservation of the mass m_v. Namely, for the proper time choice m_v is aconstant of motion, while for the Ref. <cit.> choice the quantity m_v^2/v^2 is preserved instead.The TD SSC was not only approached analytically, but also numerically. The analytical approach focused on the implications brought by the fact that m=m_v. The numerical approach proved that the affine parameter choices τ and σ are just a gauge choice, since both reproduce the same worldline when the MPD equations are evolved from the same initial conditions.G.L.-G. acknowledges the support from Grant No. GACR-17-06962Y and thanksO. Semerák for useful discussions. 9 Mathisson37M. Mathisson, “Neue mechanik materieller systemes”,Acta Phys. Polonica 6, 163 (1937) Papapetrou51A. Papapetrou, “Spinning test particles in general relativity. 1.”, Proc. R. Soc. LondonSer. A 209, 248 (1951) Dixon64W. G. Dixon, “A covariant multipole formalism for extended test bodies ingeneral relativity” Nuovo Cim. 34, 317 (1964)Semerak99 O. Semerák, “Spinning test particles in a Kerr field - I”, Mon. Not. R. Astron. S. 308, 863 (1999) Ehlers77 J. Ehlers and E. Rudolph, “Dynamics of extended bodies in general relativity: centre-of-mass description and quasirigidity”, Gen. Rel. Grav. 8, 197 (1977)Tulczyjew59W. Tulczyjew, “Motion of multipole particles in general relativity theory”,Acta Phys. Polonica 18, 393 (1959) Dixon1970IW. G. Dixon, “Dynamics of extended bodies in general relativity I: Momentum and Angular Momentum”,Proc. Roy. Soc. Lond. A. 314, 499 (1970)Pirani56F. A. E. Pirani, “On the physical significance of the Riemann tensor”,Acta Phys. Polonica 15, 389 (1956) Corinaldesi51E. Corinaldesi and A. Papapetrou,“Spinning test particles in general relativity. 2”, Proc. R. Soc. LondonSer. A 209, 259 (1951) NewtonWigner49T. D. Newton andE. P. Wigner, “Localized States for Elementary Systems”,Rev. Mod. Phys. 21, 400 (1949) Ohashi03 A. Ohashi,“Multipole particle in relativity” Phys. Rev. D 68, 044009 (2003)Kyrian07 K. Kyrian and O. Semerák, “Spinning test particles in a Kerr field - II”, Mon. Not. R. Astron. S. 382, 1922 (2007) Moller49C. Møller,“Sur la dynamique des systemes ayant un moment angulaire interne”,Ann. Inst. Henri Poincaré 11, 251 (1949) Hartl03 M. D. Hartl, “Dynamics of spinning test particles in Kerr spacetime” Phys. Rev. D 67, 024005 (2003); “Survey of spinning test particle orbits in Kerr spacetime” Phys. Rev. D 67, 104023 (2003)LGSK G. Lukes-Gerakopoulos, J. Seyrich and D. Kunst, “Investigating spinning test particles: Spin supplementary conditions and the Hamiltonian formalism” Phys. Rev. D 90, 104019 (2014) | http://arxiv.org/abs/1709.08942v1 | {
"authors": [
"Georgios Lukes-Gerakopoulos"
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"published": "20170926111229",
"title": "Time parameterizations and spin supplementary conditions of the Mathisson-Papapetrou-Dixon equations"
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[email protected] Department of Physics and Astronomy, University of British Columbia, Vancouver BC V6T 1Z1 CanadaAxions and axion-like particles are a leading model for the dark matter in the Universe; therefore, dark matter halos may be boson stars in the process of collapsing.We examine a class of static boson stars with a non-minimal coupling to gravity.We modify the gravitational density of the boson field to be proportional to an arbitrary power of the modulus of the field, introducing a non-standard coupling.We find a class of solutions very similar to Newtonian polytropic stars that we denote “quantum polytropes.” These quantum polytropes are supported by a non-local quantum pressure and follow an equation very similar to the Lane-Emden equation for classical polytropes.Furthermore, we derive a simple condition on the exponent of the non-linear gravitational coupling, α>8/3, beyond which the equilibrium solutions are unstable. The Modified Schrodinger Poisson Equation — Quantum Polytropes David Shinkaruk December 30, 2023 ==============================================================§ INTRODUCTION Bosonic dark matter possibly in the form of low-mass axions is a leading contender to explain some inconsistencies in the standard cold dark matter model (CDM) <cit.>.It is inspired from both a theoretical point of view <cit.> as emerging from string theory and observationally where bosonic dark matter can address some potential discrepancies in the standard CDM model <cit.>.Because the bosons can collapse to form a star-like object <cit.>, small-scale structure would be different if the dark matter were dominated by light bosons.Furthermore the collisions of these dark matter cores or boson stars would result in potentially observable interference <cit.>.It is these boson stars that are the focus of this investigation. The Schrodinger-Poisson equation provides a model for a boson star <cit.> in the Newtonian limit.We will explore the solutions to the Schrodinger-Poisson equation with a small yet non-trivial modification. The modified Schrodinger-Poisson equation is given by the following two equationsi ∂ψ/∂ t = -1/2∇^2 ψ + V ψwhere∇^2 V = |ψ|^αwhere we have taken m=1 and 4π G=1.For α=2 this equation is the well-known non-relativistic limit of the Klein-Gordon equation coupled to gravity <cit.>. For α≠ 2, this is not the case.Although the Newtonian limit of a self-gravitating scalar field with a potential of the form |ψ|^α would yield Eq. <ref>, one would not get Eq. <ref>, the Schrodinger equation, as the non-relativistic limit for the dynamics of the scalar field. Instead Eqs. <ref> and <ref> result as the Newtonian limit of a relativistic scalar field with a non-minimal coupling to gravity such as the following scalar-tensor actionS = ∫ d^4 x √(-g) [ R+L_m/|ψ|^α-2 +∂^μψ̅∂_μψ -|ψ|^2 ]where R is the Ricci scalar, g is the determinant of the metric and L_m is the Lagrangian density of the matter.The small change in Eq. <ref> yields a new richness to the solutions for Newtonian boson stars that we will call “quantum polytropes” for reasons that will become obvious later.Although authors have considered other modifications to the Schrodinger-Poisson equation such as an electromagnetic field <cit.> or non-linear gravitational terms <cit.>, the non-linear coupling of the gravitational source proposed here is novel.§ HOMOLOGYWe can examine how the equations change under a homology or scale transformation.Let us replace the four variables with scaled versions asψ→ A ψ, V → A^a V, r → A^b r and t → A^c tand try to find the values of the exponents that result in the same equations again.i A^1-c∂ψ/∂ t = -1/2 A^1-2b∇^2 ψ + A^1+a V ψandA^a-2b∇^2 V = A^α |ψ|^α.This yields the following equations for the exponents1-c=1-2b=1+a, a-2b=αand the following scalingsψ→ A ψ, V →A^α/2 V, r → A^-α/4 r and t → A^-α/2 t.The total norm of a solution which is conserved is given byN = ∫_0^∞ 4π r^2 |ψ|^2 d rand scales under the homology transformation as N → A^(8-3α)/4.For a static solution the value of the energy eigenvalue (E) scales as A^α/2.Because the solution is not normalized, the total energy will scale as the product of the eigenvalue and the norm, yielding A^(8-α)/4.We see that for α=8/3, one can increase the central value of the wavefunction ψ(0) without changing the norm but increasing the magnitude of the energy resulting in a more bound configuration.For larger values of α the value of the norm decreases.We can argue that the this decrease in the norm results in an unstable configuration.Let us divide the configuration arbitrarily into a central region and an arbitrarily small envelope.If we let the central region collapse slightly, energy is released but according to the decrease in norm of this central region, we still have some material left to add to the diffuse envelope to carry the excess energy and the process can continue to release energy.The star is unstable. For α<8/3 the slight collapse results in an increase in the norm of the central region but there is no material to add except from the arbitrarily small envelope, so the collapse fails.If we let the star expand a bit in this case, the norm decreases.However, the expansion costs energy so the star is again stable to the radial perturbation. For α=8/3 the norm is independent of ψ(0) and only depends on the number of nodes of the solution; therefore, it is natural to compare solutions for different values of α by choosing to normalize them to the value of the norm for α=8/3 for the corresponding state. § REAL EQUATIONS OF MOTIONWe would like examine the static solutions of Eq. (<ref>) and Eq. (<ref>).We will make the following substitutionψ = a e^iSwhere the functions a=a( r,t) and S=S( r,t) are explicitly real.This results in the three equations∂ a^2/∂ t + ∇· ( a^2 ∇ S)=0,∂ S/∂ t + 1/2 ( ∇ S)^2 + V - 1/2 a∇^2 a=0,∇^2 V=|a|^αthat in analogy with fluid mechanics we can call the continuity equation, the Euler equation and the Poisson equation. We can develop this analogy further by defining ρ=a^2 and U=∇ S and taking the gradient of Eq. (<ref>) to yield∂ρ/∂ t + ∇· ( ρ U )=0,∂ U/∂ t +(U·∇ )U + ∇ ( V -1/2 a∇^2 a)=0.These are simply the Madelung equations <cit.>. If we had retained constants such as the Planck constant h in the Schrodinger equation, we would find the that final term in the Euler equation is proportional to h^2 and is a quantum mechanical specific enthalpy,w = - 1/2 a∇^2 a.Furthermore, because U=∇ S the vorticity of the flow must vanish.We can exploit the fluid analogy further to write the equations in a Lagrangian form usingd/dt = ∂/∂ t +( U·∇ )to yieldd ρ/d t + ρ∇· U =0,dU/d t + ∇ ( V -1/2 a∇^2 a)=0.A static solution to these equations will have S=-E t in analogy with the time-independent Schrodinger equation and a=a( r) where a satisfies-E a -1/2∇^2 a + V a = 0.An alternative treatment would exploit the fact that U must vanish for this static solution so1/2a∇^2 a = V + constantwhere we can identify the constant with the value of E in Eq. <ref>. Furthermore we have∇^2 V = |a|^α = ∇^2( 1/2a∇^2 a)so if we specialize to a spherically symmetric solution, we have- 1/rd^2/dr^2 [ 1/2ad^2/dr^2 ( r a)]+ |a|^α = 0This equation is reminiscent of the Lane-Emden equation for polytropes 1/rd^2/d r^2 ( r θ ) + θ^n = 0,so a natural designation for these objects is “quantum polytropes.”Our equation is of course fourth order with a negative sign.We must supply four boundary conditions.In principle these are a(0)=a_0,.da/dr|_r=0 = 0, . -1/2a rd^2 (ra)/dr^2 |_r=0 =w_0andd/dr [ 1/2a rd^2 (ra)/dr^2 ]_r=0 = 0.Of course not all values of a_0 and w_0 will yield physically reasonable configurations, so we must vary w_0 for example to find solutions such that lim_r→∞ a(r) = 0.However, using the scaling rules in <ref>, once the value of w_0 is determined, one can rescale the solution.In the case of the Lane-Emden equation for n>5 one can find solutions where θ=0 at a finite radius, i.e. a star with a surface.From Eq. <ref> we find thatE = -lim_r→∞1/2a rd^2 (ra)/dr^2 = lim_r→∞ w(r).Therefore, if E≠ 0, the quantum system must extend to an infinite radius.To examine the regularity conditions near the centre, let us expand the solution near the centre asa(r) = a_0 + a_2 r^2 + a_4 r^4where we have dropped the odd terms to ensure that the derivative of the density and the derivative of the enthalpy vanish at the centre. We find thatw_0 = -3 a_2/a_0anda_4 = a_0^α a_0^2 + 18 a_2^2/60 a_0 = a_0( |a_0|^α/60 + w_0^2/30 ).As we would like to focus on the ground state where the function a(r) has no nodes, we can also make the substitution that a(r)=e^b which yields a simpler differential equation for b(r),b^(4)(r) = 2[ e^α b - 2/r ( b' b”+ b”') - b' b”' -(b”)^2]andw = -b”+(b')^2/2-b'/r.An examination of Eq. <ref> and <ref> yields the boundary conditions at r=0,b'(0) = 0,b”(0) = -2/3 w_0,b”'(0) = 0so a series expansion about r=0 for b(r) yieldsb(r) = b_0 - w_0/3 r^2 + 3 e^α b_0 - 4 w_0^2/180 r^4 +O(r^5)Furthermore, we can examine the behavior at large distances from Eq. <ref> to find thatlim_r→∞ b(r) ≈ -r √(-2E) = -r√(-2w)Fig. <ref> depicts the ground state wavefunction b(r)=lnψ(r) for various values of α.The wavefunction is normalized such that N = ∫ dV |ψ|^2 is constant. Furthermore, we have verified that the scaling relations of <ref> hold for these solutions.At fixed total normalization the wavefunction is more spatial extended as α increases.The slope for large values of r decreases gradually with increasing α reflecting the modest decrease in the binding energy as α increases. § EXCITED STATESTo study the excited states <cit.> where a(r) may have nodes, we have a more complicated differential equation of the forma^(4)(r) = 2 a |a|^α - 4 a”'/r + N_1/a + N_2/a^2whereN_1 = 2 a' a”' +(a”)^2 + 8/r a' a”andN_2 = -2(a')^2 a” - 4/r (a')^3.Rather than deal with these singular points we can return to the coupled differential equations <ref> and <ref> to examine the excited states.We will make the substitutions that u=ψ(r) r e^-iEt and v=V(r) r to yield the following equationsE u = -1/2u” + v u/randv” = |u|^α r^1-α,where we have focused on spherically symmetric configurations. Because equations <ref> and <ref> are non-linear we cannot follow the strategy of expanding the solutions in terms of spherical harmonics to yield a simple solution beyond spherical symmetry.The general solution is beyond the scope of this paper.We must supply four boundary conditions for the functions u and v and these are u=0, u'=ψ(0), v=0 and v'=V(0) where we take V(0)=0 because we can shift both the value of E and V(r) by a constant and retain the same equations.We generally shift E and V(r) such that lim_r→∞ V(r)=0. We can also take ψ(0)=1 and scale the resulting solution using the scaling relations in <ref>.Finally only specific values of E will result in normalizable solutions, so we shoot from the origin to large radii and find the values of E that result in normalizable solutions.Fig. <ref> depicts the ground state and the excited states for α=2 and α=3 where the wavefunction has been normalized such that ψ(0)=1.It is important to note that the various states correspond to different total normalizations, i.e. different numbers of particles.Furthermore, we will call the ground state the state without any nodes and excited states states with nodes, so the quantum number n denotes the number of anti-nodes or extrema, starting with one; therefore, Fig. <ref> shows the wavefunctions for n=1 to n=8.The wavefunctions for α=2 and α=3 appear quite similar modulo a size scaling. The α=3 wavefunctions with this particular normalization extend over a larger range in radius than the α=2 wavefunctions. Of course, what is most interesting are the configurations for a fixed number of particles, so a particular value of N=∫ dV |ψ|^2. For α≠ 8/3 the total normalization, N, can take any value. However, for α=8/3 the normalization is fixed to the values of the ground and the various excited states.Fig. <ref> depicts the binding energy as a function of α for two particular choices of normalization.As both the logarithm of the normalization and the value of the energy E are smooth functions of α for ψ(0)=1, we calculate these values for α=2,7/3,8/3,3 and 10/3 and interpolate or extrapolate over the plotted range.We then use the scaling relations from <ref> to find the eignenvalues for a particular normalization. What is most striking about the energy levels is that for α<8/3 we have the normal ordering where states with more nodes are less bound.For α>8/3 as the number of nodes increases so does the binding energy of the state.The energy levels are not bounded from below in this case, a hallmark of instability.For the limiting case α=8/3 we see that at most one state is bound for a particular total normalization, N, but that its energy is arbitrary because we can scale the value of the wavefunction which changes the energy eigenvalue without changing the total normalization.§ PERTURBATIONSThe results from scaling in <ref> and from the examination of the excited states in <ref> give very strong hints that quantum polytropes with α>8/3 are unstable.We will prove that α>8/3 is a sufficient condition for instability for an arbitrary stationary configuration. Let us take a constant background and examine small perturbations of the forma = a_0 + a_1( r,t) and U =U_1( r,t)so we have2 a_0∂ a_1/∂ t + a_0^2 ∇· U_1=0, ∂ U_1/∂ t + ∇ ( V_1 -1/2 a_0∇^2 a_1)=0.Now if we take the time derivative of Eq. (<ref>) and the divergence of Eq. (<ref>), we can combine the equations to yield2 a_0 ∂^2 a_1/∂ t^2 - a_0^2 ∇^2 V_1 + a_0/2∇^4 a_1 = 0and∂^2 a_1/∂ t^2 - α/2 a_1 |a_0|^α + 1/4∇^4 a_1 = 0.If we expand the perturbations in Fourier components we obtain the following dispersion relationω^2 =k^4/4 - α/2 |a_0|^αwhere the first term is the standard result for the deBroglie wavelength of a particle and the second term is due to the self-gravity of the perturbation.We can be a bit more sophisticated now and assume that small perturbations lie near a static solution soa = a_0( r) + a_1( r,t) and U =U_1( r,t)thus we have2 a_0∂ a_1/∂ t + ∇· ( a_0^2U_1) =0, ∂ U_1/∂ t + ∇ ( a_1/a_0 V_0 + V_1 -1/2 a_0∇^2 a_1)=0.and if we take the time derivative of Eq. (<ref>), we can combine the equations to yield2 a_0 ∂^2 a_1/∂ t^2 = ∇· [ a_0^2∇ ( a_1/a_0 V_0 + V_1 -1/2 a_0∇^2 a_1)].Furthermore, the perturbation of the potential satisfies∇^2 V_1 = αa_1/a_0 |a_0|^α-1.These again yield a self-gravitating wave equation where the static background affects the propagation.To examine the question of stability we can return to the Lagrangian formulation of the equations of motion, Eq. <ref> and Eq. <ref>.We can take the time derivative of Eq. <ref> to getd^2 ρ/dt^2 + dρ/dt∇· U + ρd/dt∇· U = 0and the divergence of Eq. <ref> to yieldd /d t∇· U + ∇^2( V -1/2 a∇^2 a) = 0If we have a perturbation on a static solution we find a simpler equation for the perturbations in the Lagrangian formulationd^2 ρ_1/dt^2 = ∇^2( a_1/a_0 V_0 + V_1 -1/2 a_0∇^2 a) .We will examine a homologous transformation wherer =r_0(1 + ϵsinω t).From Eq. (<ref>) this givesρ = ρ_0 (1 - 3 ϵsinω t) and a = a_0 (1 - 3/2ϵsinω t). Of course this pertubation is not a solution of Eq. <ref>; however, we can use it to derive an upper bound on the squared frequency of the oscillation.From Eq. <ref> we obtain to order ϵ∫ dV 3 ϵω^2 sinω t a_0^2 < ∫ dV[ a_0^α ( 1 - 3/2αϵsinω t ) -( 1 - 4 ϵsinω t)∇^21/2 a_0∇^2 a_0],and we can use the zeroth-order solution to simplify this to yield∫ dV3 ϵω^2 sinω t a_0^2 < ∫ dV[ |a_0|^α ( 1 - 3/2αϵsinω t ) -( 1 - 4 ϵsinω t) |a_0|^α ] and3ω^2 ∫ dV a_0^2 < ∫ dV( 8-3α/2 ) |a_0|^αsoω^2 <( 8-3α/6 )∫ dV |a_0|^α [ ∫ dV a_0^2 ]^-1 = 8-3α/6M/N.where M is the gravitational mass of the system and N is the number of particles. Therefore, α>8/3 is a sufficient condition for ω^2<0 and instability for at least one perturbative mode regardless of the static configuration, as we argued from the homology transformations in <ref>.If we examine an initially stationary configuration where U≠ 0 but dρ/dt=0 so ∇· U=0, we find to first order in the perturbation that the same stability condition applies when one uses the homologous transformation and the variational principle, so we find that α>8/3 is a sufficient condition for instability in general.§ CONCLUSIONSWe examine a natural generalization of the Schrodinger-Poisson equation and develop the theory of the static solutions to this equation that we denote quantum polytropes and their stability.These solutions obey a natural fourth-order generalization of the Lane-Emden equation, the second order equation for classical polytropes. Furthermore, as for classical polytropes the question of the stability of the solutions comes down to the exponent of the coupling. In the classical case this is how the pressure depends on density with power-law indices greater than 4/3 indicating stability.In the quantum case , it is how the boson field generates the gravitational field that leads to instability with power-law indices greater than 8/3 indicating instability. We demonstrate the instability in three ways and the criteria all coincide.We employ two classical techniques, a homology scaling argument and perturbation analysis, and one quantum technique the observation that the states are not bounded from below for α>8/3.This is a sufficient condition for instability not a necessary one.In particular the excited states even for α=2 are unstable <cit.>. The modified Schrodinger-Poisson presented here allows for richer possibilities for the modeling of dark matter halos and structure formation, andcan naturally emerge as the Newtonian limit from an underlying relativistic field theory. In particular if α>8/3 the dark matter halos may develop a quasi-static core that ultimately collapses to form a cusp like standard cold dark matter <cit.> or disperses, providing for especially rich phenomenology. This work was supported by the Natural Sciences and Engineering Research Council of Canada.prsty | http://arxiv.org/abs/1709.09291v2 | {
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"David Shinkaruk"
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"title": "The Modified Schrodinger Poisson Equation -- Quantum Polytropes"
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Image Milnor number and _e-codimension]Image Milnor number and _e-codimension for maps between weighted homogeneous irreducible curvesDepartamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676, 13560-905, São Carlos, SP, [email protected] de Matemàtiques, Universitat de València, Campus de Burjassot, 46100 Burjassot [email protected] de Matemática, Universidade Federal de São Carlos, Caixa Postal 676, 13560-905, São Carlos, SP, [email protected] first author has been supported by CAPES. The second author has been partially supported by DGICYT Grant MTM2015–64013–P. The third author is partially supported by CNPq Grant 309626/2014-5 and FAPESP Grant 2016/04740-7.[2000]Primary 32S30; Secondary 58K60, 32S05Let (X,0)⊂ (^n,0) be an irreducible weighted homogeneous singularity curve and let f:(X,0)→(^2,0) be a map germ finite, one-to-one and weighted homogeneous with the same weights of (X,0). We show that _e-(X,f)=μ_I(f), where _e-(X,f) is the _e-codimension, i.e., the minimum number of parameters in a versal deformation and μ_I(f) is the image Milnor number, i.e., the number of vanishing cycles in the image of a stabilisation of f. [ D. A. H. Ament, J. J. Nuño-Ballesteros, J. N. Tomazella December 30, 2023 ===========================================================§ INTRODUCTION Let α:(,S) →(^2,0) be a map germ of finite _e-codimension. D. Mond shows in <cit.> that the image Milnor number of α, μ_I(α), is determined by the number of branches r(X,0) of the curve and the number δ(X,0) of nodes appearing in a stable perturbation of α:μ_I(α)=δ(X,0)-r(X,0)+1. D. Mond also proves in <cit.> a relation between the image Milnor number and the _e-codimension.[<cit.>] Let α: (,S) → (^2,0) be a map germ of finite _e-codimension. Then, _e(α) ≤μ_I(α),with equality if α is weighted homogeneous. Inspired by the previous inequality, the first and second authors consider in <cit.> map germs f:(X,0)→(^2,0), where (X,0) is a plane curve. They define the image Milnor number of f andobtain the similar inequality.[<cit.>] Let (X,0) be a plane curve and let f:(X,0)→(^2,0) be a finite map germ of degree 1 onto its image (Y,0).Then,_e-(X,f) ≤μ_I(f),with equality if and only if (Y,0) is weighted homogeneous. Furthermore, following the proof of this result, it is possible obtain an equality,_e(X,f) + μ(Y,0) - τ(Y,0) = μ_I(f). In this work, we consider map germs f:(X,0)→(^2,0), where (X,0) is an isolated complete intersection singularity (ICIS) of dimension one, then we can consider the _e-codimension, _e-(X,f), in the sense of <cit.>. On the other hand, we define the image Milnor number in this case as the number of vanishing cycles in the image of a stable perturbation f_s:X_s→ B_ϵ, where B_ϵ is a ball of radius ϵ centered at the origin in ^2. Here, stable means that X_s is smooth and f_s is stable in the usual sense.We show that _e-(X,f)= μ_I(f), if (X,0) is irreducible and both (X,0) and f are weighted homogeneous (with the same weights).Furthermore, if we consider (X,0) parametrized by a map α:(,0)→(^n,0), we have also the following equalities.μ_I(f)=δ(X,0)+μ_I(f∘α)and_e(f)=_e(f∘α)-1/n-1_e(α)§ MAPS ON SINGULAR VARIETIES In this section, we give the basic definitions we will use in the work.First, we introduce some notations. We denote 𝒪_n the local ring of the analytic functions germs f:(^n,0) →, (X,0)⊂ (^n,0) is a germ of analytic variety (possibly with singularities), I(X,0) is the ideal of 𝒪_n of functions vanishing on (X,0),Ø_X,0=Ø_n/I(X,0) is the local ring of (X,0), Θ_n is the 𝒪_n-module of the vector fields in (^n,0) and Θ_X,0 is the 𝒪_X,0-module of vector fields tangents on (X,0).We refer to <cit.> for the general definition of _e-codimension for analytic map germs f:(X,0)→(^p,0). The _e-codimension is equal to_e(f)=_Θ(f)/tf(Θ_X,0)+ ω f(Θ_p),where Θ(f) is the 𝒪_X,0-module of vector fields along f, that is, holomorphic germs ξ:(X,0)→ T^p such that π∘ξ=f (where π:T^p→^p is the canonical projection). The map tf:Θ_X,0→Θ(f) is the morphism of 𝒪_X,0-modules given by tf(ξ)=df∘ξ and ω f:Θ_p→Θ(f)is the morphism of 𝒪_p-modules given by ω f(η)=η∘ f (where Θ(f) is considered as an 𝒪_p-module via f^*:Ø_p→Ø_X,0). We can denote T_e f=tf(Θ_X,0)+ ω f(Θ_p).We say that f is 𝒜-finite if this codimension is finite. We say that f has finite singularity type if_Θ(f)/tf(Θ_X,0)+(f^*m_p)Θ_p<∞. In the case that (X,0) is an ICIS and f has finite singularity type, we have the following important result due to Mond and Montaldi <cit.>: Let (X,0) be an ICIS and assume f:(X,0)→(^p,0) has finite singularity type. The minimal number of parameters in a versal unfolding of f is equal to the number𝒜_e-(X,f):=𝒜_e-(f)+τ(X,0),where τ(X,0) is the Tjurina number of (X,0), that is, the minimal number of parameters in a versal deformation of (X,0). As a corollary of this theorem, we have the following consequences. The second one is a generalization of the Mather-Gaffney criterion (see <cit.>). Let (X,0) be an ICIS and assume f:(X,0)→(^p,0) has finite singularity type.* f is stable if and only if X is smooth and f is stable in the usual sense.* f is 𝒜-finite if and only if f has isolated instability (i.e., there is a representative f:X→ B_ϵ such that for any y∈ B_ϵ∖{0}, the multigerm of f at f^-1(y)∩ S is stable).Another consequence of the theorem is the existence of stabilizations, at least in the range of Mather's nice dimensions. Given an analytic map germ f:(X,0)→(^p,0), a stabilization is a 1-parameter unfolding F:(𝒳,0)→ (×^p) with the property that for all s0 small enough, the map f_s:X_s→ B_ϵ is stable, where B_ϵ is a ball of radius ϵ in ^p. By Theorem <ref>, if f:(X,0)→(^p,0) is 𝒜-finite, a stabilization of f exists if (r,p) are nice dimensions in the Mather's sense (r=(X,0)). We will consider f:(X,0)→(^2,0), where (X,0)⊂(^n,0) is a curve ICIS.Observe that by Corollary <ref>, f is stable if and only if X is smooth and f is an immersion with only transverse double points, called nodes.As a consequence, f is 𝒜-finite if and only if f is finite and has degree one onto its image. § WEIGHTED HOMOGENEOUS MAPS AND VARIETIES We introduce the definition of weighted homogeneous for variety germs and map germs.We fix positive integer numbers w_1,⋯,w_n such that (w_1,⋯,w_n)=1.Given h ∈𝒪_n, we say that h is weighted homogeneous of type (w_1,⋯,w_n;d) if it satisfiesh(t^w_1x_1,⋯,t^w_nx_n)=t^d h(x_1,⋯,x_n), ∀ x ∈^n, ∀ t ∈.In this case, we call (w_1,⋯,w_n) the weights of h and d the weighted degree of h. Let (X,0)⊂(^n,0) be a variety germ, which is the zero set of an ideal I ⊂𝒪_n, we say that (X,0) is weighted homogeneous of type (w_1,⋯,w_n;d_1,⋯,d_m)if I can be generated by weighted homogeneous map germs ϕ_1,⋯,ϕ_m where each ϕ_i is weighted homogeneous of type (w_1,⋯,w_n,d_i), i=1,⋯,n. Finally, let f:(X,0)→ (^p,0) be a map germ, with (X,0)⊂ (^n,0) a weighted homogeneous variety germ of type (w_1,⋯,w_n;d_1,⋯,d_m).We say that f is weighted homogeneous if fis the restriction of a map germ f̃=(f_1,⋯,f_p):(^n,0)→(^p,0) where each f_j ∈𝒪_n is weighted homogeneous and we say that f is consistent with (X,0) if each f_j is weighted homogeneous with the weights (w_1,⋯,w_n), we observe that the weighted degree may be different.It follows from <cit.> that if (X,0) is a curve ICIS weighted homogeneous of type (w_1,⋯,w_n;d_1,⋯,d_n-1), then the Milnor number satisfiesμ(X,0)=d_1⋯ d_n-1(d_1 + ⋯+ d_n-1 - w_1 - ⋯ - w_n)/w_1 ⋯ w_n+1.Moreover, if (X,0) is irreducible, then d_1⋯ d_n-1/w_1 ⋯ w_n=1, thus the above equality is given by μ(X,0)=d_1 + ⋯ + d_n-1 - w_1 - ⋯ - w_n + 1. Wahl <cit.> shows how to compute the generators of Θ_X,0 when (X,0) is a weighted homogeneous ICIS.In particular, if we suppose (X,0)=h^-1(0) a curve, where h=(h_1, ⋯, h_n-1), then Θ_X,0is generated by vector fields h_i ∂/∂ x_j with i=1,⋯,n-1 and j=1, ⋯, n, the Euler field ϵ= (w_1x_1,⋯,w_nx_n) and ℋ = (ξ_1,⋯,ξ_n), whereℋ = | [ ∂/∂ x_1 ⋯ ∂/∂ x_n; ∂ h_1/∂ x_1 ⋯ ∂ h_1/∂ x_n; ⋮ ⋱ ⋮; ∂ h_n-1/∂ x_1 ⋯ ∂ h_n-1/∂ x_n; ]|= (-1)^1+1| [∂h_1/∂ x_1 ∂ h_1/∂ x_2 ⋯ ∂ h_1/∂ x_n; ⋮ ⋮ ⋱ ⋮; ∂ h_n-1/∂ x_1 ∂ h_n-1/∂ x_2 ⋯ ∂ h_n-1/∂ x_n; ]| ∂/∂ x_1 + ⋯ +(-1)^n+1| [ ∂ h_1/∂ x_1 ⋯ ∂ h_1/∂ x_n-1∂h_1/∂ x_n; ⋮ ⋱ ⋮ ⋮; ∂ h_n-1/∂ x_1 ⋯ ∂ h_n-1/∂ x_n-1 ∂ h_n-1/∂ x_n; ]| ∂/∂ x_n=ξ_1 ∂/∂ x_1 + ⋯ + ξ_n ∂/∂ x_n,we useto indicate that we should exclude this column. Let J_1,J_2∈Ø_X,0 be defined asJ_i= | [ ∂ h_1/∂ x_1 ⋯ ∂ h_1/∂ x_n; ⋮ ⋱ ⋮; ∂ h_n-1/∂ x_1 ⋯ ∂ h_n-1/∂ x_n; ∂ f_i/∂ x_1 ⋯ ∂ f_i/∂ x_n; ]|=(-1)^n-1(∂ f_i/∂ x_1ξ_1+⋯+ ∂ f_i/∂ x_nξ_n),with i=1,2.We have the matrix df= ( [∂ f_1/∂x_1 ⋯∂ f_1/∂x_n; ∂ f_2/∂ x_1 ⋯ ∂ f_2/∂ x_n; ]). Thus, tf(Θ_X,0) is generated by vector fields(∂ f_1/∂x_1 h_1, ∂ f_2/∂x_1h_1),⋯,(∂ f_1/∂x_nh_1,∂ f_2/∂x_nh_1),⋯, (∂ f_1/∂x_1h_n-1,∂ f_2/∂x_1h_n-1),⋯,(∂ f_1/∂x_nh_n-1,∂ f_2/∂x_nh_n-1), df ∘ϵ= (w_1x_1∂ f_1/∂x_1+⋯+w_nx_n∂ f_1/∂x_n,w_1x_1∂ f_2/∂x_1+⋯+w_nx_n∂ f_2/∂x_n)=(l_1 β_ 1 f_1,l_2 β_2 f_2)anddf∘ℋ= (ξ_1 ∂ f_1/∂ x_1+ ⋯+ξ_n ∂ f_1/∂ x_n , ξ_1 ∂ f_2/∂ x_1+ ⋯+ξ_n ∂ f_2/∂ x_n)= (-1)^n+1(J_1,J_2). § MAPS BETWEEN CURVES Let (X,0)⊂(^n,0) be an irreducible curve ICIS, weighted homogeneous of type (w_1,⋯,w_n;d_1,⋯,d_n-1).Apart from the case that (X,0) is smooth (which is a trivial case), it follows that (X,0) can be parametrized by a monomial mapα:(,0)→(^n,0) of the formα(t)=(α_1 t^w_1,⋯,α_n t^w_n), (α_1,⋯,α_n)∈^n ∖{0}. The local ring of (X,0) is Ø_X,0={t^w_1,⋯,t^w_n}, which is a subring of Ø_1. We denote by Γ_X the associated semigroup. Since (X,0) is irreducible, Γ_X has a conductor c, which satisfies that c-1 ∉Γ_X and if n∈, with n ≥ c, then n∈Γ_X. We have that Γ_X is symmetric, because (X,0) is an ICIS, thus #∖Γ_X=c/2. The main invariant of (X,0) is the delta invariant, which is in this case δ_X=_{ t }/{ t^w_1,⋯,t^w_n}=#∖Γ_X.Since (X,0) is irreducible, the Milnor number μ(X,0), which to simplify the notation we will denote by μ_X, is μ_X=2δ_X=c, by Milnor's formula.For more details, see for instance <cit.>.Let f:(X,0)→(^2,0) be a finite map germ of degree 1 onto its image (Y,0) and f is consistent with (X,0). We denote by l_1,l_2 the weighted degrees of f_1,f_2, respectively.Then, (Y,0) is also a weighted homogeneous irreducible plane curve with weights (l_1, l_2) and it admits a parametrization of the formf(α(t))=(β_1 t^l_1,β_2 t^l_2),β_1,β_2∈∖{0}.We have Ø_Y,0={t^l_1,t^l_2}⊂Ø_X,0,Γ_Y=⟨ l_1,l_2⟩⊂Γ_X and δ_Y=(l_1-1)(l_2-1)/2. Finally, we assume that (Y,0) has defining equation g(u,v)=0, where g∈{u,v} is weighted homogeneous. Thus, g(u,v)=β_2^l_1u^l_2-β_1^l_2v^l_1 is weighted homogeneousof type (l_1, l_2;l_1l_2).We consider f=i∘f̅, where f̅:(X,0)→(Y,0) is the restriction of f and i:(Y,0)→(^2,0) is the inclusion map. We have,J_g𝒪_X,0 = { a (∂ g/∂ u∘ f )+ b (∂ g/∂ v∘ f ) ; a, b ∈𝒪_X,0},andJ_g𝒪_Y,0 = {(a ∘f̅) ( (∂ g/∂ u∘ i ) ∘f̅) + ( b ∘f̅) ( ( ∂ g/∂ v∘ i ) ∘f̅) ; a, b ∈𝒪_Y,0}, where J_g be the Jacobian ideal of g, thus J_g𝒪_Y,0⊂ J_g𝒪_X,0.We consider the evaluation mapev: Θ(f)/tf(Θ_X,0)+ wf(Θ_2)⟶J_g𝒪_X,0/J_g𝒪_Y,0given by ev([ξ]) = [ξ(g)].The evaluation map is well-defined and surjective, hence_e(f) = _(ev) + _J_g𝒪_X,0/J_g𝒪_Y,0.Let us see that ev is well-defined.If ξ∈ tf(Θ_X,0), then ξ = tf(η), for some η∈Θ_X,0, hence η(l)=0 in 𝒪_X,0, for all l∈ I(X,0). Since g ∘f̃ = λ_1 h_1 + ⋯λ_n-1 h_n-1, with λ_i ∈𝒪_n, i=1,⋯,n, we have ξ(g) =tf(η)(g)=(df∘η)(g) = η(g ∘f̃)= η(λ_1 h_1 + ⋯λ_n-1 h_n-1)=0. If ξ∈ω f(Θ_2), then ξ = ζ∘ f, for some ζ∈Θ_2. Writeζ = ã∂/∂ u + b̃∂/∂ v, with ã,b̃∈Ø_2. Thus, ξ=ζ∘ f =( ã∘ f ) ∂/∂ u + (b̃∘ f ) ∂/∂ v. Then, ξ(g)=( ã∘ f ) ∂ g/∂ u∘ f + (b̃∘ f ) ∂ g/∂ v∘ f= ( (ã∘ i ) ∘f̅) ( (∂ g/∂ u∘ i ) ∘f̅) + ( (b̃∘ i ) ∘f̅) ( ( ∂ g/∂ v∘ i ) ∘f̅),with ã∘ i ,b̃∘ i ∈𝒪_Y,0. Hence, ξ(g) ∈ J_g𝒪_Y,0.If ξ= a ∂/∂ u + b ∂/∂ v, with a, b ∈𝒪_X,0, then ξ(g) = a ∂ g/∂ u∘ f + b ∂ g/∂ v∘ f, hence ev is surjective. The evaluation map is not injective in general, so we need to know its kernel in order to compute the _e-codimension of f. To this we need the following result.With the above notation, we have∂ g/∂ u∘ f= C λ_f J_2∂ g/∂ v∘ f = - Cλ_fJ_1,when λ_f=t^μ_Y-μ_X∈Ø_X,0. By definitions, we obtain∂ g/∂ u∘ f=l_2 β_2^l_1β_1^l_2-1t^l_1(l_2-1)∂ g/∂ v∘ f=-l_1 β_1^l_2β_2^l_1-1 t^l_2(l_1-1). We also haveJ_j =| [t^d_1-w_1∂ h_1/∂ x_1(α_1,⋯,α_n)⋯t^d_1-w_n∂ h_1/∂ x_n(α_1,⋯,α_n);⋮⋱⋮; t^d_n-1-w_1∂ h_n-1/∂ x_1 (α_1,⋯,α_n)⋯t^d_n-1-w_n∂ h_n-1/∂ x_n(α_1,⋯,α_n); t^l_j-w_1∂ f_j/∂ x_1 (α_1,⋯,α_n)⋯t^l_j-w_n∂ f_j/∂ x_n(α_1,⋯,α_n);]|. We denote ∂ h_i/∂ x_k(α_1,⋯,α_n)=γ_i,k with i=1,⋯,n-1, k=1,⋯,n and ∂ f_j/∂ x_k (α_1,⋯,α_n)=ζ_j,k with j=1,2 and k=1,⋯,n.Furthermore, since h_1, ⋯, h_n-1, f_1, f_2 are weighted homogeneous with weights w_1, ⋯, w_n, they satisfies:d_ih_i=w_1 x_1∂ h_i/∂ x_1+⋯+w_n x_n∂ h_i/∂ x_n, l_jf_j=w_1 x_1∂ f_j/∂ x_1+⋯+w_n x_n∂ f_j/∂ x_n,with i=1,⋯,n-1 and j=1,2. Thus,w_1 α_1t^w_1t^d_i-w_1γ_i,1+⋯+w_n α_n t^w_n t^d_i-w_nγ_i,n=0, w_1 α_1t^w_1t^l_j-w_1ζ_j,1+⋯+w_n α_n t^w_n t^l_j-w_nζ_j,n=l_jβ_j t^l_j,with i=1,⋯,n-1 and j=1,2. Rewriting in matrix form( [ t^d_1-w_1γ_1,1⋯ t^d_1-w_nγ_1,n;⋮⋱⋮; t^d_n-1-w_1γ_n-1,1⋯ t^d_n-1-w_nγ_n-1,n; t^l_j-w_1ζ_j,1⋯ t^l_j-w_nζ_j,n;])( [w_1 α_1t^w_1; ⋮; w_n α_n t^w_n; ])= ( [0;⋮;0; l_jβ_j t^l_j;]). By Cramer's Rule,w_1 α_1 t^w_1 J_j= | [0 t^d_1-w_2γ_1,2⋯ t^d_1-w_nγ_1,n;⋮⋮⋱⋮;0 t^d_n-1-w_2γ_n-1,2⋯ t^d_n-1-w_nγ_n-1,n; l_jβ_j t^l_j t^l_j-w_2ζ_j,2⋯ t^l_j-w_nζ_j,n;]|. Thus,J_j =(-1)^1+nl_jβ_j/w_1α_1 t^l_j-w_1| [ t^d_1-w_2γ_1,2⋯ t^d_1-w_nγ_1,n;⋮⋱⋮; t^d_n-1-w_2γ_n-1,2⋯ t^d_n-1-w_nγ_n-1,n;]| = l_jβ_j t^l_j - w_1 + d_1+ ⋯ + d_n-1- w_2 - ⋯ - w_n A,with A≠ 0.We obtain,J_1 = l_1 β_1 At^l_1+μ_X-1 J_2 = l_2 β_2 A t^l_2+μ_X-1 Therefore,∂ g/∂ u∘ f =l_2 β_2^l_1β_1^l_2-1t^l_1(l_2-1)= β_2^l_1-1β_1^l_2-1/A t^μ_Y - μ_X l_2 β_2 At^l_2+μ_X-1= C λ_f J_2and∂ g/∂ v∘ f =- l_1 β_1^l_2β_2^l_1-1t^l_2(l_1-1)=- β_1^l_2-1β_2^l_1-1/A t^μ_Y - μ_X l_1 β_1 At^l_1+μ_X-1= -C λ_f J_1. We consider the map ev: Θ(f)/tf(Θ_X,0)⟶ J_g Ø_X,0,given by ev([ξ])=ξ(g). Then, (ev) ≅(ev). We observe that the map ev is well-defined and it is surjective. We define the morphism (ev)→(ev) given by [ξ] ↦ [ξ], where the classes have to be considered in the respective quotients.We claim that if [ξ] ∈(ev), there exists representative ξ such thatξ(g) = a ∂ g/∂ u∘ f + b ∂ g/∂ v∘ f = 0.In fact, if [ξ] ∈(ev), we choose a representative ξ = a ∂/∂ u + b ∂/∂ v, a, b ∈𝒪_X,0, thus ξ(g) = a ∂ g/∂ u∘ f + b ∂ g/∂ v∘ f. On the other hand, since [ξ] ∈(ev), we have ξ(g) ∈ J_g𝒪_Y,0, so,ξ(g) = ( ã∘ i ∘f̅) ( (∂ g/∂ u∘ i ) ∘f̅) + ( b̃∘ i ∘f̅) ( ( ∂ g/∂ v∘ i ) ∘f̅),with ã∘ i, b̃∘ i∈𝒪_Y,0. Therefore,(a - ã∘ i ∘f̅) ( ∂ g/∂ u∘f ) + (b - b̃∘ i ∘f̅)( ∂ g/∂ v∘ f ) = 0.We define a̅ = a - ã∘ i ∘f̅, b̅ = b - b̃∘ i ∘f̅ and ξ̅ = a̅∂/∂ u + b̅∂/∂ v, thus ξ - ξ̅=a ∂/∂ u + b ∂/∂ v -( a̅∂/∂ u + b̅∂/∂ v) =( ã∘ i ∘f̅) ∂/∂ u + (b̃∘ i ∘f̅) ∂/∂ v=(ã∘ f) ∂/∂ u +( b̃∘ f ) ∂/∂ v =ζ̃∘ f,where ζ̃=ã∂/∂ u + b̃∂/∂ v, with ã, b̃∈𝒪_2. Thus, ξ - ξ̅∈ wf(Θ_2), i.e., [ξ]=[ξ̅] with ξ̅(g) = 0.This prove the claim.Now, we will show that the morphism is well-defined. In fact, let [ξ] = [ξ̃] ∈(ev), i.e., ξ + T_e f = ξ̃+T_e f, then ξ - ξ̃∈ T_e f, with ξ(g)=ξ̃(g)=0. We will show that ξ - ξ̃∈ tf(Θ_X).Sinceξ - ξ̃∈ T_e f, thenξ - ξ̃ = df ∘η + ρ∘ f, η∈Θ_X, ρ∈Θ_2, with 0 = ξ(g) - ξ̃(g) = df ∘η(g) + ρ∘ f(g), moreover df ∘η(g) = 0, thus ρ∘ f(g)=0.We have ρ =(a,b), with a, b ∈𝒪_2, then, ρ∘ f = (a ∘ f,b ∘ f) such that (a ∘ f) ∂ g/∂ u∘ f + (b ∘ f) ∂ g/∂ v∘ f =0. Thus,( a ∂ g/∂ u + b ∂ g/∂ v)∘ f =0, 𝒪_Yi.e.,a ∂ g/∂ u + b ∂ g/∂ v = κ g,κ∈𝒪_2.Since g is weighted homogeneous,l_1 l_2 g = l_1 u ∂ g/∂ u+l_2 v ∂ g/∂ v,so,l_1 l_2 (a ∂ g/∂ u + b ∂ g/∂ v) = l_1 l_2 κg = κ(l_1 u ∂ g/∂ u+l_2 v ∂ g/∂ v),(l_1 l_2 a - κ l_1 u) ∂ g/∂ u + (l_1 l_2 b - κ l_2 v) ∂ g/∂ v = 0,thenl_1 l_2 a - κ l_1 u = - ∂ g/∂ v l_1 l_2 b - κ l_2 v = ∂ g/∂ u.Therefore,(l_1 l_2 (a ∘ f) - (κ∘ f)l_1 β_1 f_1) = - ∂ g/∂ v∘ f (l_1 l_2 (b∘ f) - (κ∘ f)l_2 β_2 f_2)= ∂ g/∂ u∘ f Thus,l_1 l_2 (ρ∘ f)= (l_1 l_2 (a ∘ f) ,l_1 l_2 (b ∘ f))= (κ∘ f )(l_1 β_1 f_1,l_2 β_2 f_2) + (- ∂ g/∂ v∘ f, ∂ g/∂ u∘ f). We have(κ∘ f) ( l_1 β_1 f_1, l_2 β_2 f_2) ∈ tf(Θ_X),we will show that (- ∂ g/∂ v∘ f,∂ g/∂ u∘ f) ∈ tf(Θ_X). Since ∂ g/∂ u∘ f= C λ_f J_2 and ∂ g/∂ v∘ f = - Cλ_fJ_1, then(- ∂ g/∂ v∘ f,∂ g/∂ u∘ f) = Cλ_f(J_1,J_2) ∈ tf(Θ_X). Therefore, ρ∘ f ∈ tf(Θ_X) and, consequently, ξ - ξ̃ = df ∘η + ρ∘ f ∈ tf(Θ_X). Since t f(Θ_X) ⊆ t f (Θ_X) + ω f(Θ_2), the map is injective and it is surjective. By using the coordinates in source and target, we can identify Θ(f) with Ø_X,0^2=Ø_X,0⊕Ø_X,0. The module Θ_X,0 can be seen as a submodule of Ø_X,0^n. With these identifications, the morphism tf:Θ_X,0→Θ(f) is the restriction of the map tf:Ø_X,0^n→Ø_X,0^2, whose matrix in the canonical basis is the Jacobian matrix of f. The following result describes the elements of (ev).Let k = l_2-l_1. Then,(ev) ≅{ (l_1 β_1 t^r,l_2 β_2 t^r + k): (t^r, t^r + k) ∈𝒪^2_X,0, r ∈ℕ}/⟨ (l_1 β_1 t^l_1,l_2 β_2 t^l_1+k),(l_1 β_1 A t^l_1+μ_X-1,l_2 β_2 A t^l_1+μ_X-1+k)⟩.For each (a,b)∈Ø_X,0^2, we have:ev([(a,b)])=a ∂ g/∂ u∘ f+b ∂ g/∂ v∘ f=a l_2 β_2^l_1β_1^l_2-1 t^l_1 l_2 - l_1 - b l_1 β_1^l_2β_2^l_1 -1 t^l_1 l_2 - l_2.Then, [(a,b)]∈(ev) if and only if (a,b)=c(l_1 β_1 t^s_1,l_2 β_2 t^s_2 ) such that l_1 l_2-l_1+s_1=l_1 l_2 - l_2 +s_2, for some c∈Ø_X,0. It follows that (ev) is generated by monomial pairs of the form (l_1 β_1 t^r,l_2 β_2 t^r+k), for some r∈ such that (t^r,t^r+k)∈Ø^2_X,0.We remember that tf(Θ_X) is generated on 𝒪_X,0 by vectors fields(∂ f_1/∂x_1 h_1, ∂ f_2/∂x_1h_1),⋯,(∂ f_1/∂x_nh_1,∂ f_2/∂x_nh_1),⋯, (∂ f_1/∂x_1h_n-1,∂ f_2/∂x_1h_n-1),⋯,(∂ f_1/∂x_nh_n-1,∂ f_2/∂x_nh_n-1), df ∘ϵ = (l_1 β_1 f_1,l_2 β_2 f_2)df∘ℋ= (-1)^n+1( J_1, J_2). Since Ø_X,0={t^w_1,⋯,t^w_n} and Ø_Y,0={t^l_1,t^l_2}⊂Ø_X,0, we have (l_1 β_1 f_1,l_2 β_2 f_2)=(l_1 β_1 t^l_1,l_2 β_2 t^l_1 +k). We remember thatJ_1 = l_1 β_1 At^l_1+μ_X-1 J_2 = l_2 β_2 A t^l_2+μ_X-1 Thus, df∘ℋ = (-1)^n+1(l_1 β_1 At^l_1+μ_X-1, l_2 β_2 A t^l_1+μ_X-1+k). We denote by s the number of elements of the form (t^r,t^r+k) in 𝒪^2_X,0 such that 0≤ r<μ_X. Then, _(ev)= l_1 - δ_X + s -1and_𝒪_X,0/⟨ J_1,J_2 ⟩ =l_1 + δ_X + s - 1By Lemma <ref> andLemma <ref>,(ev) ≅{ (l_1 β_1 t^r,l_2 β_2 t^r + k): (t^r, t^r + k) ∈𝒪^2_X,0, r ∈ℕ}/⟨ (l_1 β_1 t^l_1,l_2 β_2 t^l_1+k),(l_1 β_1 A t^l_1+μ_X-1,l_2 β_2 A t^l_1+μ_X-1+k)⟩. We consider the set Γ= { l_1+ i:i∈Γ_X}∪{l_1+μ_X-1} which is a subset ofΓ_X. We denoted by Γ_X ⊕Γ_X the associated semigroup to Ø^2_X,0 e we take the sets:Γ_k={ (r,r+k)∈Γ_ X ⊕Γ_X }Γ_Θ={ (l,l+k): l ∈Γ}⊂Γ_k We can identify the elements of (ev) with the elements of Γ_k∖Γ_Θ. Therefore, we compute the number of elements in Γ_k∖Γ_Θ.We observe that Γ_k is equal to{(r,r+k): 0 ≤ r < μ_X }_selements∪{(μ_X,μ_X+k),(μ_X+1,μ_X+1+k), ⋯}_all elementsand Γ_Θ is { (l_1+p,l_1+p+k): 0 ≤ p < μ_X, p∈Γ_X }^δ_X ∪ { (l_1+ μ_X - 1,l_1+ μ_X - 1 +k), (l_1 + μ_X,l_1+μ_X+k),⋯}_all elements. Then, we describe Γ_k∖Γ_Θ as {(r,r+k): 0 ≤ r < μ_X }^selements∪{(μ_X,μ_X+k),⋯, (l_1+μ_X-2,l_1+μ_X-2+k)}^l_1-1elements ∖{ (l_1+p,l_1+p+k): 0 ≤ p < μ_X,p∈Γ_X }_δ_X Therefore,_(ev) = s +l_1 - 1 - δ_X = l_1 - δ_X + s - 1. Now, we compute the dimension of Ø_X / ⟨ J_1,J_2 ⟩. We consider the sets Γ_J_1= { l_1+μ_X-1 + i:i∈Γ_X} andΓ_J_2= { l_1+μ_X-1 + k+ j:j ∈Γ_X}which are subset ofΓ_X.We can identify the elements of Ø_X / ⟨ J_1,J_2⟩ with the elements of Γ_X∖ (Γ_J_1∪Γ_J_2). Therefore, we compute the number of elements in Γ_X∖ (Γ_J_1∪Γ_J_2).We observe that there exists common elements in Γ_J_1 and Γ_J_2, we need to know the elements. We have that all the elements after l_1 + μ_X -1 + k+ μ_X are common elements in Γ_J_1 and Γ_J_2.Then, we consider the subsets A={ l_1 + μ_X - 1 + i : 0≤ i ≤μ_X + k, i ∈Γ_X } and B={ l_1 + μ_X -1 + k + j : 0≤ j < μ_X , j ∈Γ_X }.Given r and r+k in Γ_X such that 0 ≤ r < μ_X, the elements common in A and B are the elements of form:l_1 + μ_X -1 +(r+k) = l_1 + μ_X - 1 + k + r. Since s is the number of elements r ∈Γ_X, 0 ≤ r < μ_X, such that r+k ∈Γ_X, then A∩ B contained s common elements. We have that A contained δ_X +k + 1 elements and B contained δ_X elements, thus A∪ B contained μ_X + k + 1 - s elements.Thus, Γ_X∖ (Γ_J_1∪Γ_J_2) is equivalent to{j ∈Γ_X; 0≤ j < μ_X}^δ_X∪{μ_X,⋯,l_1+μ_X + μ_X + k - 1 }^l_1 + μ_X + k ∖{ l_1 + μ_X - 1 + i : 0≤ i ≤μ_X}∪{ l_1 + μ_X -1+ k + j : 0≤ j < μ_X }_μ_X + k + 1 - s .Therefore,_𝒪_X,0/⟨ J_1,J_2 ⟩ = δ_X + l_1 + μ_X + k - (μ_X+ k + 1 - s) = l_1 + δ_X + s -1. Let (X,0)⊂(^n,0) be an irreducible curve ICIS, weighted homogeneous and let f:(X,0)→(^2,0) be a finite map germ of degree 1 onto its image (Y,0) and f is consistent with (X,0). Then,_e(f) =_𝒪_X,0/⟨ J_1,J_2 ⟩ -μ_X + _J_g𝒪_X,0/J_g𝒪_Y,0.By Lemma <ref>,_e(f) =_(ev) + _J_g𝒪_X,0/J_g𝒪_Y,0. By Theorem <ref>,_𝒪_X,0/⟨ J_1,J_2 ⟩ - _(ev)= l_1 + δ_X + s - 1- ( l_1 - δ_X + s -1 )= μ_X. Therefore,_e(f) = _𝒪_X,0/⟨ J_1,J_2 ⟩ -μ_X+ _J_g𝒪_X,0/J_g𝒪_Y,0.§ THE IMAGE MILNOR NUMBER The first and second authors in <cit.>, define the image Milnor number μ_I(f) of f:(X,0)→(^2,0), where (X,0) is a plane curve and f is -finite. In this work, naturally, we generalize the concept of the image Milnor number to the case that (X,0) is a space curve ICIS. We consider (X,0)⊂(^n,0) a curve ICIS and f:(X,0)→(^2,0) be a finite map germ of degree 1 onto its image (Y,0). We suppose f has finite singularity type, it follows from Theorem <ref> that f always admits a stabilisation F: (𝒳,0) → (×^2,0), given by F(s,x)=(s,f_s(x)). This means that f_s:X_s→ B_ϵ is stable, for all s0 small enough, where B_ϵ is a small enough ball centered at the origin in ^2.Thus the image Y_s=f_s(X_s) has the homotopy type of a wedge of 1-spheres. The image Milnor number μ_I(f) isthe number of 1-spheres in Y_s. Analogous to case where (X,0) is a plane curve <cit.>, μ_I(f) is well-defined, that is, it is independent of the stabilisation and of the representative. The image Milnor number satisfies the equality.μ_I(f) = μ(Y,0) + μ(X,0)/2. For the next results, we use the definition of the delta invariant of f, δ(f), which was introduced in <cit.> for mapsof degree 1 between curves.[<cit.>] Let f:(X,0)→(Y,0) be a holomorphic map of degree 1 between curves (X,0) and (Y,0). The delta invariant of f isδ(f)=_Ø_X,0/f^* Ø_Y,0.This number also satisfies δ(Y,0)=δ(X,0)+δ(f) when (X,0) and (Y,0) are irreducible curves. Therefore, when (X,0) and (Y,0) are irreducible curvesμ(Y,0)-μ(X,0)=2δ(f) (see <cit.>), thusμ_I(f)=μ(Y,0) - δ(f)=μ(X,0)+δ(f). Now we consider again, (X,0)⊂(^n,0) an irreducible curve ICIS and weighted homogeneous of type (w_1,⋯,w_n;d_1,⋯,d_n-1), and f:(X,0)→(^2,0) a finite map germ of degree 1 onto its image (Y,0), such that f is consistent with (X,0). We take 𝒞 the ideal generated by λ_f in Ø_X,0. Then we have the following result. Let (X,0)⊂(^n,0) be an irreducible curve ICIS and weighted homogeneous of type (w_1,⋯,w_n;d_1,⋯,d_n-1) and let f:(X,0)→(^2,0) be a finite map germ of degree 1 onto its image (Y,0) and consistent with (X,0). Then, _Ø_X,0/𝒞 = 2δ(f) _Ø_Y,0/𝒞 = δ(f).Furthermore, we have that 𝒪_X,0/⟨ J_1,J_2 ⟩ is isomorphic to 𝒞/J_g 𝒪_X,0.By Lemma <ref>,λ_f=t^μ_Y-μ_X=t^2δ(f).Therefore, the weighted degree of λ_f with the weights w_1,⋯,w_n is 2δ(f).Denoted by (h_1,⋯,h_n-1):(^n,0)→ (^n-1,0) the map that generates (X,0),by Bezout theorem, we obtain_Ø_X,0/𝒞 = _Ø_X,0/⟨λ_f ⟩ = _Ø_2/⟨ h_1,⋯,h_n-1, λ_f ⟩=d_1⋯ d_n-12δ(f)/w_1 ⋯ w_n=2δ(f). Therefore,_Ø_X,0/𝒞 = 2δ(f). To obtain the second equality, we first show that 𝒞⊂Ø_Y,0. We recall that 𝒞 is the ideal generated by λ_f=t^2δ(f) in Ø_X,0.Then, we need to show that σ t^2δ(f)∈Ø_Y,0, for all σ∈Ø_X,0. To see this, we consider the numerical semigroups Γ_X and Γ_Y associated the curves (X,0) and (Y,0) respectively. We recall that Γ_X and Γ_Y are symmetric. We show that if 2δ(f) ∈Γ_Y, then t^2δ(f)∈Ø_Y,0.In fact,since Γ_Y is symmetric we have 2δ(f) ∈Γ_Y or μ_Y-1-2δ(f) ∈Γ_Y. We suppose μ_Y-1-2δ(f) ∈Γ_Y, thenμ_Y-1-2δ(f)= μ_Y - 1 - μ_Y + μ_X = μ_X-1 ∈Γ_Y ⊂Γ_X.But μ_X-1 ∉Γ_X, because μ_X is the conductor of Γ_X.Therefore, 2δ(f) ∈Γ_Y, thus t^2δ(f)∈Ø_Y,0.Now, let σ=t^a∈Ø_X,0 with a ∈Γ_X, then 2δ(f)+a ∈Γ_Y.In fact, we have 2δ(f)+a ∈Γ_Y or μ_Y-1-(2δ(f)+a) ∈Γ_Y. If μ_Y-1-(2δ(f)+a) ∈Γ_Y, thenμ_Y-1-μ_Y+μ_X- a= μ_X-1-a ∈Γ_Y ⊂Γ_X.Since Γ_X is symmetric, we have a ∉Γ_X, but σ=t^a∈Ø_X,0.Thus, μ_Y-1-(2δ(f)+a) ∉Γ_Y andconsequently 2δ(f)+a ∈Γ_Y. Therefore, 𝒞⊂Ø_Y,0. We obtain,_Ø_Y,0/𝒞=_Ø_X,0/𝒞 - _Ø_X,0/f^* Ø_Y,0 = 2δ(f) -δ(f) = δ(f). To complete the proof, we observe that multiplication by λ_f gives an isomorphism ϕ:Ø_X,0→𝒞. Moreover, ϕ(⟨ J_1,J_2 ⟩)=J_g 𝒪_X,0 and hence, it induces an isomorphism𝒪_X,0/⟨ J_1,J_2 ⟩⟶𝒞/J_g 𝒪_X,0. Let (X,0)⊂(^n,0) be an irreducible weighted homogeneous curve ICIS and let f:(X,0)→(^2,0) be a finite map germ of degree 1 onto its image (Y,0) and consistent with (X,0). Then, _e(X,f) = μ_I(f).Moreover, let α:(,0)→(X,0) be a parametrisation of (X,0). Then, μ_I(f)=δ(X,0)+μ_I(f∘α)and_e(f)=_e(f∘α)-1/n-1_e(α)By Corollary <ref> and by Proposition <ref>, we obtain_e(f)= _𝒪_X,0/⟨ J_1,J_2 ⟩-μ_X + _J_g𝒪_X,0/J_g𝒪_Y,0= 𝒞/J_g 𝒪_X,0 - μ_X + _J_g𝒪_X,0/J_g𝒪_Y,0. We have the exact sequence, 0 ⟶J_g𝒪_X,0/J_g𝒪_Y,0⟶𝒞/J_g 𝒪_Y,0⟶𝒞/J_g 𝒪_X,0⟶ 0,thus, by the exact sequence and by Proposition <ref>,_e(f)=_𝒞/J_g 𝒪_Y,0 - μ_X=_Ø_Y,0/J_g 𝒪_Y,0- _Ø_Y,0/𝒞 - μ_X = μ_Y - δ(f) - μ_X = δ(f). Since (X,0) is weighted homogeneous, we have μ_X=τ(X,0), hence_e(X,f) = _e(f) + μ_X =δ(f) + μ_X = μ_I(f). To prove the second part, we observe that f∘α:(,0)→(Y,0) is a parametrisation of (Y,0)⊂(^2,0), then by <cit.>, _e(f∘α)=μ_I(f∘α)=δ(Y,0).By <cit.>, δ(Y,0)=δ(X,0)+δ(f), and by definition of image Milnor number, we obtainμ_I(f) =μ(X,0)+δ(f)=μ(X,0)+δ(Y,0)-δ(X,0)=δ(X,0)+δ(Y,0)=δ(X,0)+μ_I(f∘α). We also have,_e(f) = δ(f)= δ(Y,0) - δ(X,0) =_e(f∘α)-δ(X,0).It follows from <cit.> that_e(α)=(n-1) δ(X,0), thus_e(f) = _e(f∘α)-1/n-1_e(α).We observe that, in particular, if (X,0) is a plane curve, by <cit.>, μ_I(α)=δ(X,0) and we obtain μ_I(f)=μ_I(α)+μ_I(f∘α).We consider (X,0) a weighted homogeneouscurve defined by h(x,y) = x^2-y^3and f:(X,0) → (^2,0) a map defined by f(x,y)=(x,y^2). Then, (Y,0) is weighted homogeneous plane curve defined by g(u,v) = u^4-v^3.We have, μ(X,0) =2, μ(Y,0)=6, then μ_I(f)=(μ(Y,0) + μ(X,0))/2 = (6 + 2)/2 = 4.By <cit.>, _e(X,f)=_Θ(γ)/tγ(Θ_2)+γ^*( D(G)),where G:^3→^4 defined by G(x,y,s)=(x,y^2,x^2-y^3+sy,s) is a stability of map (h,f̅): ^2 →^3, given by (h,f̅)(x,y)=(x^2-y^3,x,y^2). We observe that, D(G)=Im(G), because G is no surjective, then D(G)=(G). Hence, to compute _e(X,f), we find H: ^4 → such that (G)=H^-1(0). We do this using the Singular software,H(z,u,v,w)= z^2-v^3-2u^2z+2v^2w-vw^2+u^4.We use the map Hto compute (G) and then we can calculate_Θ(γ)/tγ(Θ_2)+γ^*((G)). Therefore,_e(X,f)=_Θ(γ)/tγ(Θ_2)+γ^*((G)) = 4 = μ_I(f).and_e(f)= _e(X,f)-μ(X,0)=4-2=2=δ(f). Now, we consider α:(,0)→(X,0), given by α(t)=(t^3,t^2), the parametrisation of (X,0). Then, _e(α)=μ_I(α)=δ(X,0)=1 and _e(f∘α)=μ_I(f∘α)=δ(Y,0)=3. Therefore,μ_I(f)=μ_I(α)+μ_I(f∘α).and_e(f)= _e(f∘α)- _e(α).We consider (X,0)⊂ (^3,0) a weighted homogeneous curve defined by h(x,y,z) = (x^3-y^2,xy-z) with parametrization α:(,0)→ (^3,0) defined by α(t)=(t^2,t^3,t^5)and f:(X,0) → (^2,0) a map defined by f(x,y,z)=(x,y^2z), thus f is consistent with (X,0). Then, (Y,0) is a weighted homogeneous plane curve defined by g(u,v) = u^11-v^2 with parametrization β:(,0)→ (^2,0) defined by β(t)=(t^2,t^11). We use the Singular software to calculate the Milnor number, we have μ(X,0)=2 and μ(Y,0)=10. Thus, μ_I(f)=(μ(Y,0) + μ(X,0)) / 2 = (10 + 2) / 2 = 6.By <cit.>, _e(X,f)=_Θ(γ)/tγ(Θ_2)+γ^*( D(G)),where G:^4→^5 defined by G(x,y,z,s)=(x^3-y^2,xy-z,x,y^2z+sy,s) is a stability of map (h,f̅): ^3 →^4, given by (h,f̅)(x,y,z)=(x^3-y^2,xy-z,x,y^2z). As in example <ref>,we use the Singular software, thus,_e(X,f)=_Θ(γ)/tγ(Θ_2)+γ^*((G)) = 6=μ_I(f). We consider (X,0)⊂ (^3,0) a weighted homogeneous curve defined by h(x,y,z) = (x^5-y^2,xy-z) with parametrization α:(,0)→ (^3,0) defined by α(t)=(t^2,t^5,t^7)and f:(X,0) → (^2,0) a map defined by f(x,y,z)=(x^2,y+z). We observe that f is not consistent with (X,0). Then, (Y,0) is a plane curve defined by g(u,v) = u^7-2u^6 +4u^3 v^2 + u^5-v^4 with parametrization β:(,0)→ (^2,0) defined by β(t)=(t^4,t^5 + t^7). We use the Singular software to calculate the Milnor number and the Tjurina number, we have μ(X,0)=4, τ(X,0)=4, μ(Y,0)=12 and τ(Y,0)=11. Thus, (Y,0) is not weighted homogeneous. We also have, μ_I(f)=(μ(Y,0) + μ(X,0)) / 2 = (12 + 4) / 2 = 8.By <cit.>, _e(X,f)=_Θ(γ)/tγ(Θ_2)+γ^*( D(G)),where G:^4→^5 defined by G(x,y,z,s)=(x^5-y^2+sx,xy-z,x^2,y+z,s) is a stability of map (h,f̅): ^3 →^4, given by (h,f̅)(x,y,z)=(x^5-y^2,xy-z,x^2,y+z). As in example <ref>,we use the Singular software, thus,_e(X,f)=_Θ(γ)/tγ(Θ_2)+γ^*((G)) = 7.Therefore,_e(X,f)<μ_I(f).We observe that the hypotheses f consistent with of (X,0) is necessary to obtain an equality. amsplain | http://arxiv.org/abs/1709.09504v1 | {
"authors": [
"Daiane Alice Henrique Ament",
"Juan Jose Nuño Ballesteros",
"João Nivaldo Tomazella"
],
"categories": [
"math.AG",
"math.CV",
"32S30 (Primary), 58K60, 32S05 (Secondary)"
],
"primary_category": "math.AG",
"published": "20170927133651",
"title": "Image Milnor number and $\\mathscr{A}_e$-codimension for maps between weighted homogeneous irreducible curves"
} |
The role of first neighbors geometry in the electronic and mechanical properties of atomic contacts M. J. Caturla December 30, 2023 ===================================================================================================empty empty Car sharing is one of the key elements of a Mobility-on-Demand system, but it still suffers from several shortcomings, the most significant of which is the fleet unbalance during the day. What is typically observed in car sharing systems, in fact, is a vehicle shortage in so-called hot spots (i.e., areas with high demand) and vehicle accumulation in cold spots, due to the patterns in people flows during the day. In this work, we overview the main approaches to vehicle redistribution based on the type of vehicles the car sharing fleet is composed of, and we evaluate their performance using a realistic car sharing demand derived for a suburban area around Lyon, France. The main result of this paper is that stackable vehicles can achieve a relocation performance close to that of autonomous vehicles, significantly improving over the no-relocation approach and over traditional relocation with standard cars.§ INTRODUCTIONMobility-on-Demand systems are a new intermodal mobility concept, also declined as Mobility-as-a-service. The key idea is that people will not own their private car (and be stuck with it) anymore. Instead they will have a fully-fledged choice of transportation modes, seamlessly integrated with each other and with the smart city thanks to ICT, automation, and big data. When it comes to motorised modes for personal mobility, car sharing is a staple of MoD systems. Car sharing's positive effects have already been measured: car sharing members use cars less, rely more on public transport, and in some cases they even shed their private car (or refrain from buying a second one for their family) <cit.>. Car sharing can also act as a last-kilometre solution for connecting people with public transport hubs, hence becoming a feeder to traditional public transit <cit.>.Initial car sharing policies offered limited flexibility to the customers, forcing them to bring the shared cars back to the starting point of their journey. This is the case of two-way car sharing. The recent growth in car sharing popularity and usage, though, has been the result of the availability of car sharing solutions more suitable to customers' needs in terms of flexibility. One-way car sharing, in fact, allows customers to pick up and drop off vehicles at any of the car sharing stations deployed in the city (for station-based systems like Autolib in Paris) or at any location with operator's service area (for free-floating systems like Car2go). Unfortunately, this freedom is often paid in terms of vehicle availability. In fact, cars will follow the natural flows of people in a city, hence accumulating in commercial/business areas in the morning and in residential areas at night <cit.>. As a result, the availability of cars can become extremely unbalanced during the day, and certain areas of high demand (hot spots) may end up being underserved while areas with low demand (cold spots) may have several cars idly parked and that nobody wants to pick up.Previous research has proposed several approaches to solve the vehicle unbalance problem, including: user-based relocation, i.e., price incentives for the users to relocate the vehicles themselves <cit.>; operator-based relocation, i.e., workforce that moves vehicles from where they are not needed to where there is a significant demand <cit.>; and optimal planning of station deployment to achieve better service accessibility and a more favourable distribution of vehicles <cit.>. It is important to point out that the relocation process is intrinsically inefficient: as one driver per car is needed, for relocating several cars a large workforce or many willing customers are necessary. Autonomous cars can be a breakthrough from the car sharing perspective: therelocation workforce can be shed, as vehicles will move proactively and autonomously where they are needed.Autonomous vehicles have been the subject of extensive research in the recent years. The first pilot programs have highlighted the great potential of the technology, while also causing some headaches <cit.>. While the fact that self-driving cars are going to replace traditional vehicles has never been in question, when exactly they are going to be in people's hands is not clear, with predictions ranging from a few years to a few decades. In the meantime, alternative, intermediate solutions should be sought after to address the problems of currently available transportation systems. As far as car sharing is concerned, new vehicle concepts with stackable capabilities have been recently released or are under development, which can be stacked into a train (through a mechanical and electric coupling) and/or folded together. Then, the train can be driven either by a car sharing worker or by a customer. An illustration of this type of vehicle prototyped within the ESPRIT project <cit.> is provided in Fig. <ref>.Such stackable cars come with the promises of significantly improving the system manageability of future car sharing services and of being market-ready in just a few years. As such, they could be an effective intermediate solutions before autonomous cars become a reality. It is therefore important to understand how they rankwith respect to future and currently available vehicles. For this reason, in this paper we identify and compare different strategies for vehicle redistribution based on the type of vehicle used. Relying on a realistic car sharing demand, we study how well each of them is faring and we highlight their limitations.First, we apply a theoretical upper bounds about relocation with self-driving cars developed in <cit.> to determine the minimum fleet size that would allow to perfectly rebalanced the considered demand. Second, we developed simple heuristics to quantify the fleet unbalance over time and to define periodic relocation tasks that could help in rebalancing the system. Then we discuss how to perform these relocation tasks with and without operators, using conventional cars or stackable cars. The results show that stackable vehicles can significantly improve the quality of service provided by the car sharing operator with respect traditional cars, and hence should be considered as a viable, intermediate option on the road to self-driving vehicles. § THE CAR SHARING DEMANDIn this work we consider the car sharing demand obtained from a population synthesiser and demand model developed within the ESPRIT project for the suburban area of Lyon, France <cit.>. This demand corresponds to the potential demand, i.e., the demand obtained from the mobility needs ofthe people in the study area and their propensity towards using car sharing. The population synthesiser takes into account a set of 37,402 facilities (i.e., sources of mobility requests, such as households, commercial buildings, PT stops, etc.) scattered in the area, while the demand model determines the pickup and drop-off locations for each car sharing trip request. In this work, we focus on a station-based system, thus we need to deploy stations in the study area in order to associate the car sharing requests to the closest station. The definition of an optimal deployment algorithm is out of the scope of the paper, and for this reason we consider a straightforward deployment whereby we divide the study area using a grid with cells of side length 1km and we place a station in the centroid of the cell if there is at least one facility in that cell. This implies that an hypothetical customer will find at least one station at a ∼500m distance. After this procedure is completed, we end up with 70 active station in the study area.The demand generated by the model will provide the basis for our evaluation in Section <ref>. As in all car sharing systems, stations are used very differently by the customers depending on the time of the day and where they are located <cit.>. This is confirmed in Figure <ref>, where we focus on individual stations and we compute the difference between their inflow and outflow, which correspond, respectively, to the number of incoming vs outgoing vehicles per day. We observe that there are station that see more drop-offs than pickups, and vice versa. § RELOCATION: CURRENT SOLUTIONS §.§ No relocation Relocation can be very costly for car sharing operators, and this is the reason why most of them do not implement redistribution policies in their systems. For example, Car2go has recently announced that only one year ago they have designed their relocation strategy, and that since then they have successively introduced it in all their locations <cit.>. Given the practical relevance, the no-relocation case will serve as the baseline for our discussion. In particular, it is interesting to consider how many cars one would need to satisfy the demand in Section <ref> without ever dropping one request. This number can be computed by simply deriving the number of cars needed in the system at midnight such that the availability at the station is never negative during the following day. Using this approach, we found that we would need 1542 shared vehicles in order to accept all pickup requests in the Lyon study area.§.§ Relocation with standard carsWhen the shared fleet is composed of standard cars, relocation can only be performed moving one vehicle at a time. The car sharing workers (the relocators) can either work in pairs or alone. When in pairs, one drives the other one (using a service car) to the location where there is shared car to be relocated, then drives again to the relocation endpoint to pick up the relocator. An alternative solution is for the relocator to use a folding bicycle to reach the car to be relocated, then to store the folded bike in the trunk, to drive the car where it is need, then cycle again to reach the car of the following relocation task <cit.>. This process is clearly not very efficient. In the first case, two workers are needed to relocate one car. In the second case, the relocation delays are long since it takes some time for the relocators to reach the next car by bike.§.§ Theoretical upper-bound As far as relocation is concerned, you can never do better than with autonomous cars. In fact, assuming that a demand prediction tool is in place, self-driving cars can autonomously dispatch themselves where they are needed. Pavone et al. have derived in <cit.> a theoretical results about relocation with self-driving cars. Their model is based on a representation of the trips in the car sharing network in terms of flows in a fluid model. Pavone et al. prove that, without vehicle redistribution, the car sharing system cannot reach equilibrium (i.e., customers waiting for cars will accumulate indefinitely at stations). Instead, with a redistribution policy in place and a sufficient number of cars, it is always possible to stabilise the car sharing system. Pavone et al. also define an optimisation problem to derive the optimal relocation flows. Clearly, the fluid model is just an approximation of a real car sharing system, but it is very useful and important in order to establish what, in practice, is a theoretical upper-bound on vehicle rebalancing.We apply the model presented in <cit.> in order to derive the optimal rebalancing flows with autonomous vehicles for the car sharing demand described in Section <ref>. The results are shown in Figure <ref>. The blue and orange bars show the inbound/outbound relocation flows from each station. According to the model in <cit.>, optimal rebalancing is possible using 292 vehicles. Please note that this is 5 times less than what it would be needed without relocation (Section <ref>). § RELOCATION WITH STACKABLE VEHICLESIn car sharing systems with stackable cars like ESPRIT <cit.>,relocation becomes more efficient, since a single relocator can redistribute several vehicles at a time (up to 7, in ESPRIT). But how much more efficient with respect to regular cars and autonomous cars? In order to evaluate the efficacy of redistributing stackable vehicles, we first need to define a redistribution strategy.Before discussing the selected approach, we first introduce some terminology. The balance of vehicles at stations can be positive (more available cars than needed with respect to the foreseen demand) or negative (fewer cars than needed). Stations with positive balance are called feeders, because they can provide cars to stations in need of cars (which, in turn, are called recipients).Also, we denote with v_T the maximum train size allowed. Similarly to the related literature <cit.>, we perform relocation periodically (every T minutes). The proposed algorithm follows three steps: i) identify the expected vehicle balance/unbalance, ii) match feeders and recipients, iii) match relocators and feeder-recipient pairs. These steps are discussed in detail below.§.§ Identify the expected vehicle balance/unbalance First, we want to understand the excess/deficiencies of vehicles in the network. We denote with b_i(kT) the overall balance of vehicles at station i during time interval [kT, (k+1)T) and we write it as follows: b_i(kT) = v_i(kT) + drop_i^[kT, (k+1)T) + - pick_i^[kT, (k+1)T) - v_i^control(kT), where v_i(kT) denotes the number of cars parked at station i at time kT, drop_i^[kT, (k+1)T) and pick_i^[kT, (k+1)T) are, respectively, the expected number of drop-offs and pickups in the next time interval according to the demand and the currently relocating vehicles. Quantity v_i^control(kT) is a control knob that denotes how many vehicles should be present at station i no matter what (e.g., it can be set to 1 only if the rest of the right-hand side of Equation <ref> is positive, in order to be conservative in the choice of feeders). Intuitively, the excess/deficiencies of vehicles are computed balancing the expected in/out flows at each station and taking into account the current situation. If b_i(kT)>0, station i is expected to have an abundance of vehicles in [kT, (k+1)T), hence we can use these vehicles for relocation. Vice versa, if b_i(kT)<0, station i is expected to have a deficiency of vehicles in [kT, (k+1)T). Thus, we identify as feeders all the stations for which b_i(kT)>0 (and we denote their set as ℱ). Analogously, we define the set of recipients ℛ = { i: b_i(kT)<0 }.More complex strategies for demand prediction could be developed, e.g., exploiting machine learning techniques. However, to the purpose of the paper, suffices to use the same approach for all the relocation strategies considered, in order to have a fair comparison.§.§ Match feeders and recipientsWe now want to assign to each recipient one or more feeders in order to address its unbalance. This can be achieved in several ways. The idea behind our approach is that, since relocators are costly, we want use as few of them as possible. We have the two sets of feeders (ℱ) and recipients (ℛ), each ordered in decreasing order of cars in excess/missing[In the following, we drop the reference to kT, as it is implicit that we are referring to the current relocation interval. ]. We define V_excess = ∑_j ∈ℱ b_j and V_deficit = -∑_i ∈ℛ b_i. Please note that if V_excess≥ V_deficit, then all deficits can be addressed, at least in principle. Vice versa, if V_excess≤ V_deficit, we know that we are not able to fulfil the requests. Then, we proceed according to the pseudo-code in Algorithm <ref> below, which must be run every T. The main idea behind it is to first address the unbalance of the most unbalanced recipients (those with the smaller b_i, which is negative for feeders) using the most rich feeders. If the needs of the recipient are not completely satisfied using one feeder, the recipient is put back in the list of recipients with unfulfilled requests, with an updated balance. Similarly, if the excess of cars at a feeder are not exhausted by a single recipient, the feeder is kept in ℱ but its balance is updated.§.§ Match relocators and feeder-recipient pairsThe output of the previous step provides us with a list of feeder-recipient pairs and the number of cars to be relocated between them. Now we have to assign one relocator to each of these pairs. In the following, we denote with 𝒫 the set of matched feeder-recipient pairs and with 𝒪 the set of relocators. Please note that if | 𝒫| > |𝒪| the relocation demand might not be satisfied (unless some operators finish their task quickly and carry out another relocation within the same T). The idea is to assign relocators in order to minimise the overall relocation time (composed of the time for the relocator to reach the feeder from its current position plus the time to go from feeder to recipient). Since relocators may not be enough to satisfy all relocation requests, we prioritise relocation tasks that move the most vehicles. This approach is summarised in the Algorithm <ref>. Please note that Algorithm <ref> can be run more frequently than every T (possibly, as soon as there is an idle relocator available).§ RELOCATION WITH AUTONOMOUS VEHICLES As discussed earlier, autonomous vehicles provide the maximum flexibility in terms of relocation, since they can proactively move towards a hot spot when they end up in a cold spot. A simple relocation strategy for autonomous vehicles can be obtained by modifying the algorithms for relocation with stackable vehicles described in Section <ref>. Basically, it is enough to drop step 3 (corresponding to Algorithm <ref>), and assume that, for each feeder-recipient pair p = (f_i, r_j), v_p vehicles will autonomously move from f_i to r_j at the time of relocation. § EVALUATION In order to evaluate the performance of relocation based on stackable vehicles, we have developed a custom C++ simulator that models how vehicles move across the stations in the system. The simulator takes as input the car sharing demand discussed in Section <ref>, so shared cars move between stations according to the pickup and drop-off requests in the demand[Please note that the goal of the paper is not to reproduce the daily evolution of mobility on the transport network of the study area but to study, in simplified settings, the potentialities of relocation with stackable vehicles. For this reason, the richness of transport simulators like MATSim <cit.>, is not needed here. We also neglect the effect of congestion in the road network of the study areas. While more detailed analyses are planned as future work, we argue here that congestion will affect all policies approximately in the same way, and hence this simplified analysis still capture the main aspects of relocation performance.]. The travel times between stations are estimated based on a map of the study area. On top of this mobility process we deploy the relocation strategies described in Sections <ref>-<ref>. We assume that the fleet size (i.e., the number of cars in the car sharing system) is 350, i.e., a bit more than 292, the number needed according to the theoretical model discussed in Section <ref> for perfect rebalancing. Parameter v_T is varied during simulations from 2 to 8.Please note that a relocator driving a train of 8 vehicles will be able to relocate at most 7, since he/she needs one of the cars in the train to reach the next feeder. Finally, the relocation interval T is varied in {5, 15, 30} minutes. The main performance metric considered is the percentage of total accepted requests with respect to the original demand. In Figure <ref> we plot it for different train sizes and relocation intervals, as well as a function of the number of relocators. It is interesting to note that, with a relocation window of 30 minutes and ∼30 relocators (Figure <ref>), stackable vehicles provide exactly the same level of service as autonomous cars when the maximum allowed train size is greater than or equal to 5 cars. In all cases, relocating provides better performances than no relocation, but standard relocation needs a lot of relocators before it catches up with autonomous cars. In all cases, even autonomous cars are not able to satisfy 100% of the demand. There may be several reasons for that. First, the number of vehicles may be too low for the demand. The number of vehicles was chosen so that it was above the threshold obtained from the theoretical model in <cit.>. However, the model was obtained under simplifying assumptions that may affect the results when cast onto real systems (e.g., real demands have a time-varying behaviour during the day, while the model does not). Second, the demand prediction tool may be too simplistic. In any case, this simple setup is enough to highlight striking differences between the relocation of different types of vehicles. A second important metric (especially from the car sharing business model standpoint) is the number of relocators employed to perform relocation, and how long they are busy relocating. We show these quantities in Figure <ref> for a relocation interval of 30 minutes. Peaks in relocation activities are clearly visible during the day. If we focus on the case of 30 relocators and maximum train size 8 (the best case scenario discussed above), we observe that only early in the morning the entire relocation workforce is needed. For the rest of the day, only a fraction of the workforce is busy. This is definitely helpful for organising work shifts and deriving the full-time equivalents in the business model.Finally, we plot the train size distribution as a function of the relocation interval and the number of relocators (Figure <ref>). There is a clear trade-off between the two. When relocators are a lot and relocation is performed frequently, relocators mostly use short trains (because the overall relocation burden is low). Vice versa, with just a few relocators and infrequent relocations, long trains are needed to meet all relocation requests. § CONCLUSIONS In this work we have presented a preliminary analysis, based on a realistic car sharing demand, of the relocation capabilities of stackable vehicles. Our results show that stackable vehicles are able to bridge the relocation gap between autonomous vehicles and standard cars, by exploiting trains of vehicles that can be driven by a single relocator. More specifically, with just 30 relocators, a car sharing system with stackable vehicle achieves the same performance in terms of accepted demand as a car sharing system with self-driving cars. These results show that stackable vehicles can be a cost-effective, efficient solutions while waiting for self-driving cars to become a mainstream reality. § ACKNOWLEDGEMENT We would like to thank Helen Porter and Peter Davidson for letting us use in this paper the car sharing demand that they have produced for the ESPRIT study area in Lyon. IEEEtran | http://arxiv.org/abs/1709.09553v1 | {
"authors": [
"Chiara Boldrini",
"Raffaele Bruno"
],
"categories": [
"cs.OH"
],
"primary_category": "cs.OH",
"published": "20170927143443",
"title": "Stackable vs Autonomous Cars for Shared Mobility Systems: a Preliminary Performance Evaluation"
} |
headingsPredicting Disease-Gene AssociationsHendrik ter Horst et al.Hendrik ter Horst, Matthias Hartung, Roman Klinger, Matthias Zwick and Philipp CimianoSemantic Computing Group, CITEC, Bielefeld University{hterhors,mhartung,cimiano}@techfak.uni-bielefeld.de Institute for Natural Language Processing, University of [email protected] Networking, Boehringer Ingelheim Pharma GmbH & Co. [email protected] Disease-Gene Associations using Cross-Document Graph-based Features Hendrik ter Horst1 Matthias Hartung1 Roman Klinger1,2 Matthias Zwick3 Philipp Cimiano1 December 30, 2023 ============================================================================================ In the context of personalized medicine, text mining methods pose an interesting option for identifying disease-gene associations, as they can be used to generate novel links between diseases and genes which may complement knowledge from structured databases. The most straightforward approach to extract such links from text is to rely on a simple assumption postulating an association between all genes and diseases that co-occur within the same document. However, this approach (i) tends to yield a number of spurious associations, (ii) does not capture different relevant types of associations, and (iii) is incapable of aggregating knowledge that is spread across documents. Thus, we propose an approach in which disease-gene co-occurrences and gene-gene interactions are represented in an RDF graph. A machine learning-based classifier is trained that incorporates features extracted from the graph to separate disease-gene pairs into valid disease-gene associations and spurious ones. On the manually curated Genetic Testing Registry, our approach yields a 30 points increase in F_1 score over a plain co-occurrence baseline.§ INTRODUCTION Most current approaches in personalized medicine, irrespective of particular treatment modalities (e.g., small molecules, biologics, novel approaches like gene therapy), are centered around modulating a gene in order to modulate aspects of a disease <cit.>. Therefore, the detection of disease-gene links is an important starting point in drug discovery.Text mining methods pose an interesting option for identifying disease-gene links, as they can be used to generate new target (and therefore treatment) hypotheses and, in combination with experimental data, support the prioritization of research aimed at the discovery of new drug targets. Until now,text mining methods for disease-gene associations mostly rely on the degree of textual co-occurrence <cit.>. While those approaches are largely reliable in detecting well-known links for well-known diseases <cit.>, such well-known links are of minor interest in drug research, as they do not support the discovery of new targets. Such novel targets are difficult to detect, as they often require the aggregation of evidence across individual documents; at the same time, they potentially shed light on yet unknown disease-gene links. In this paper, we propose a classification model for predicting noveldisease-gene associations from biomedical text. The model combines private (intra-document) as well as public (cross-document) knowledge as defined by Swanson et al.<cit.> in terms of features based on local co-occurrences within documents and relations between diseases and genes that have been aggregated across individual documents. In an evaluation against an existing database, we address the following research questions: (i) Can such a combined model outperform a purely co-occurrence-based approach? (ii) What is the impact of features measuring the connectivity of diseases and genes? (iii) What is the impact of graph-based features capturing interactions between genes across documents?§ SYSTEM ARCHITECTURE Our system architecture consists of the following components (cf. Fig. <ref>):* Medline Corpus: All analyses are done on Medline[<http://www.ncbi.nlm.nih.gov/pubmed>], comprising a total of 21.5M abstracts.* Information Extraction: We rely on existing information extraction systems to identify disease names, genes/proteins and interactions between them.For disease recognition, we use a state-of-the-art CRF tagger <cit.> that has been trained on the NCBI Disease Corpus <cit.>. Using the normalization procedure described in <cit.>, disease names are reduced to a vocabulary of approx. 15K unique identifiers extracted from MeSH[<https://www.nlm.nih.gov/mesh/>] and OMIM[<http://omim.org>].For the identification of genes/proteins and their interactions, we rely on TEES <cit.>, a state-of-the-art event extraction system that has been tailored to the detection of molecular interactions from biomedical text. All extracted genes are normalized using GeNo <cit.> and afterwards filtered by human genes using taxonomic information from EntrezGene <cit.>. * Postprocessing: The complexity of the interaction graphs produced by TEES (see Fig. <ref>) is incrementally reduced. Fig. <ref> gives a running example of the individual steps described in the following: * RDFication: TEES graphs are broken down into binary relations by selecting all shortest paths that connect two proteins with respect to their semantic relation. These are represented as one RDF triple connecting two proteins. The path between the two proteins is serialized as a string and used as the name of an RDF property connecting both proteins.* Simplification: Semantic role information (i.e., cause and theme) is omitted. The direction of interactions is still captured in the directed edges of the graph.* Generalization: The TEES extraction scheme, originally consisting of 9 relations, was reduced to five relations after discussion with a domain expert: Expression, Catabolism, Localization, Binding and Regulation.* Compression: Consecutive occurrences of identical relations within a path signature are compressed by reducing them to one relation.* Path joining: To extract longer dependencies between genes/proteins as well, we join paths connecting two genes up to a distance of two edges. The join of the paths is serialized again, the above post-processing steps are applied and the results are stored as RDF triples. We refer to such serialized paths as path signatures.* Database: The results of all the steps described above are stored in an RDF database, Blazegraph[<http://www.blazegraph.com>].* Gene Classification System: Given a disease as input, protein candidates are classified as to whether or not they interact with that disease. The classifier relies on features extracted for each pair of disease and protein. Being our main contribution, this component is described in the next section. § GENE CLASSIFICATION SYSTEM Our gene classification system takes a disease as input and predicts, for each gene in the database, whether it interacts with the given disease or not. The classifier is implemented as a Support Vector Machine relying on features that are extracted for each disease-gene pair. Seven feature groups are defined in total, which can be divided into co-occurrence-based (CBF) and graph-based (GBF) features, as described below.Our notation is as follows: Let D be the set of all diseases and G the set of all genes in the database[D and G comprise all diseases and genes recognized during preprocessing the Medline corpus (cf. Section <ref>). This amounts to 7.640 diseases and 11.201 genes, in total.] and P a vocabulary of predicates denoting semantic relations between them. Then, T denotes the set of all triples in the database, such that: T ⊂ (D × P × G) = {⟨ d,p,g⟩ | p=𝑐𝑜𝑜𝑐𝑐}∪{⟨ g,p,g'⟩ | p=𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡}.§.§ Co-occurrence-based Features CBF features are based on the co-occurrence between diseases and genes. We consider T_d ⊂ T = {⟨ d', 𝑐𝑜𝑜𝑐𝑐, g⟩ | d'=d}, the set of all genes co-occurring with a particular disease d, and analogously T_g ⊂ T = {⟨ d, 𝑐𝑜𝑜𝑐𝑐, g'⟩ | g'=g}. Moreover, T_dg⊂ T= {⟨ d', 𝑐𝑜𝑜𝑐𝑐, g'⟩ | d'=d, g'=g} denotes the set of all co-occurrences of a particular disease d and a particular gene g. Entropy. We compute the entropy H(g) of a gene g in order to measure the specificity of g in terms of the diseases it co-occurs with. If g co-occurs with only a few specific diseases, this results in low entropy and high specificity. Co-occurrence with many diseases yields high entropy and low specificity. We compute the entropy of g asH(g) = -∑_d^D p(d|g) ·log_2 p(d|g),where p(d|g) = |T_dg|/|T_g|.Analogously, we compute the entropy/specificity H(d) of a disease d in terms of the genes it co-occurs with. Co-occurrence Frequencies. This feature group combines relative co-occurrence frequencies of a disease-gene pair (d,g):Occ(d,g) = |T_dg|/d' ∈ Dmax |T_d'| Besides the normalization given in Equation (<ref>), two other alternatives are used. In all variants, Occ(d,g) measures the strength of the connection of d and g. Ranging from 0 to 1, small values indicate a weak connection, whereas larger values indicate a strong connection.Grades. This feature group consists of two features which capture a normalized frequency of triples that contain d or g, respectively:Grade(d) = |T_d| / d' ∈ Dmax |T_d'| and Grade(g) = |T_g| / g' ∈ Gmax |T_g'|. Odds Ratio is used to assess the degree of association between d and g: Odds(d,g) = |T_dg|·(|T|-|T_dg|)/(|T_dg|-|T_d|)·(|T_dg|-|T_g|) The higher 𝑂𝑑𝑑𝑠(d,g), the stronger the association between d and g. 𝑂𝑑𝑑𝑠(d,g)=0 can only be achieved if |T_dg|=0. TF-IDF. In order to assess the relevance of a gene g for a disease d, we apply the 𝑡𝑓𝑖𝑑𝑓 metric from information retrieval which takes term frequency (𝑡𝑓) and inverted document frequency (𝑖𝑑𝑓) into account <cit.>: 𝑡𝑓𝑖𝑑𝑓(d,g) = 𝑡𝑓(d,g)·𝑖𝑑𝑓(g). Considering a disease as a “bag of genes”, 𝑡𝑓(d,g) is equivalent to |T_dg|, while 𝑖𝑑𝑓(g) can be computed in terms of Equation (<ref>): idf(g)= log(|D|/∑_d ∈ D f(d,g)), wheref(d,g) ={[ 1 if |T_dg| > 0; 0else ]. High values of 𝑡𝑓𝑖𝑑𝑓(d,g) indicate that g is mentioned frequently in the context of d, but still sufficiently specific to be informative for d, which we expect to be indicative of a relevant association between d and g.§.§ Graph-based FeaturesIn contrast to the previously described feature groups which take a disease and a gene into account, GBF features are calculated independently of a particular disease in that they are entirely based on the gene interaction graph, i.e., the set of triples I ⊂ T = {⟨ g,𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡,g'⟩|g,g' ∈ G}.Path Signatures. Each gene g is described in terms of a “bag of (outgoing) path signatures”, S_out(g) ⊂ I = {⟨ g', 𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡, g”⟩|g=g'}, which have been constructed by joining individual edges in the gene interaction network (cf. Section <ref>, postprocessing step (3e)). We use interact as a placeholder for all predicates constructed in this process. The strength of interaction between a pair ⟨ g, g'⟩∈ S_out(g) is weighted by the 𝑡𝑓𝑖𝑑𝑓 metric, expressing 𝑡𝑓(g,g') and 𝑖𝑑𝑓(g') analogously to the definitions in Section <ref> (cf. Equation(<ref>)). Path signatures encoding an important relation between g and g' in terms of a high 𝑡𝑓𝑖𝑑𝑓 value are considered useful for predicting novel disease-gene associations in cases where no direct evidence from co-occurrence relations is available yet.tfidf_sig(s,g,G) = tf_sig(s,g) ·idf_sig(s,G)tf_sig(s,g) = ∑_s' ∈ S(g) f(s,s')/max{∑_s' ∈ S(g) f(s,s') : s∈ S(g)}, wheref(s,s') = {[ 1 if s=s'; 0else ].idf_sig(s,G) = log(|G|/∑_g ∈ G f(g,s)), wheref(g,s) ={[ 1 if s ∈ S(g); 0else ]. I propose to formulate the indicator functions with 1_s=s' and analogously for S(g).It is really not clear to me what an outgoing signature could be and how set relations are defined on them. I assume these should be sets of edges (or nodes?), but I am not sure.Gene Connectivity. This feature group describes the connectivity of a gene within the graph. Analogously to S_out(g) above, we define S_in(g,l) and S_out(g,l) as lists of all incoming and outgoing signatures of path length l, respectively. Further, L denotes the maximum path length.Based on these definitions, we count the number of incoming and outgoing signatures for each path length 1 ≤ l ≤ L separately, as given in (<ref>), and by accumulation over all path lengths. In these features, higher values indicate a higher connectivity of g in the network. Out_l(g) = |S_out(g,l)|/g' ∈ Gmax |S_out(g',l)|In_l(g) = |S_in(g,l)|/g' ∈ Gmax |S_in(g',l)|We also measure the ratio of outgoing and incoming signatures per gene in terms of 𝐼𝑂𝑅𝑎𝑡𝑖𝑜(g)=|S_out(g)|/|S_in(g)|. If 𝐼𝑂𝑅𝑎𝑡𝑖𝑜(g)>1, g has a manipulating role in the network; otherwise, g tends to be manipulated by other genes. § EXPERIMENTAL EVALUATION§.§ Experimental SettingsGold Standard. The Genetic Testing Registry (GTR; <cit.>) is a manually curated database for results from biomedical experiments, mostly at the intersection of Mendelian disorders and human genes. Our GTR dump contains 5,800 disease-gene associations built from 4,200 diseases and 2,800 genes. Training and Testing Data are created from GTR as follows: All disease-gene associations in GTR are considered as positive examples. For each disease in GTR, we additionally generate the same amount of negative training examples by pairing the disease with genes that co-occur in Medline but are not attested in GTR as a valid disease-gene association. The resulting data set is split into 80% used for training and 20% for testing. The training set contains 3,665 diseases with 1,781 negative and 1,884 positive examples (i.e., associated genes). The test set comprises 910 diseases with 440 positive and 470 negative examples. Experimental Procedure. We train an SVM classifier using an RBF kernel <cit.> and apply grid search for meta-parameter optimization based on the LibSVM[<http://www.csie.ntu.edu.tw/ cjlin/libsvm>] and WEKA[<http://www.cs.waikato.ac.nz/ml/weka/>] toolkits. The trained model is applied to the task of predicting genes that are associated with a given disease. We evaluate the model on the GTR test set described above, reporting precision, recall and F_1 score. §.§ Results and Discussion Evaluation results are reported in Table <ref>. The left part displays classification performance of individudal feature groups; in the right part, testing performance of feature combinations (as selected by cross-validation on the training data) are shown. The baseline refers to the performance of a single-feature classifier relying on co-occurrence counts as described in Equation (<ref>). The results clearly indicate a positive impact of both cooccurrence-based and graph-based features, as all feature combinations yield an increase over the baseline in both precision and recall. The CBF combination achieves highest overall precision, whereas connectivity and signature features improve recall (at the expense of slight losses in precision). As for path signatures, it is most effective to select only a small number of individual paths. We determine the best 50 path signature features based on information gain <cit.>. In the best-performing configuration (CBF+Conn+Best50Sig), our system outperforms the co-occurrence baseline by 1.7 points in precision and 43.4 points in recall.Increasing the recall relative to the co-occurrence baseline is a key prerequisite towards our goal of discovering novel disease-gene associations. Given that the GTR gold standard is relatively small and slightly biased towards Mendelian disorders, we subject the best-performing model to another evaluation in a practical use case, as described in the next section. §.§ Case Study: Pulmonary Fibrosis In this experiment, our classification model was applied to the entire Medline corpus in order to predict genes related to pulmonary fibrosis (PF). The resulting hits were sorted by their corpus frequency and the 200 most frequent candidates were manually evaluated by a biomedical expert (who is not a PF researcher, though). Table <ref> shows the preliminary results of this analysis: 38.5% of the predictions are unanimously correct, whereas only 22% are clear errors. The missing mass is due to candidates for which hints were found that the gene may be associated with PF through relevant mechanisms or pathways. Thus, these candidates constitute plausible hypotheses which need further investigation by a PF expert. Main sources of erroneous predictions are false co-occurrences (e.g., due to negation contexts) or false positives as produced by the gene recognition component. Some errors of the latter type may be eliminated by incorporating the filtering approach proposed by <cit.>. In sum, this analysis clearly shows that our system is capable of generating promising candidates worth further investigation.§ RELATED WORK Three types of approaches have been proposed to tackle the problem of extracting explicitly mentioned disease-associated genes (DAGs) but also generating novel hypotheses from scientific publications. First, several authors extract DAGs from existing biomedical databases such as GeneSeeker <cit.> or PolySearch <cit.>. Piñero et al. developed DisGeNET <cit.>, a database quantifying the degree of disease-gene associations by a combination of different sources of evidence, with textual co-occurrence being one of the main sources. Obviously, these approaches lack the ability to discover new target hypotheses. Second, text mining techniques have been considered as an alternative and are mostly based on textual co-occurrence (sentence or document-based). Such systems can be optimized on precision <cit.> or recall <cit.>. Al-Mubaid presents a technique using various information-theoretic concepts to support the co-occurrence-based extraction <cit.>. Third, a promising alternative to overcome mere textual co-occurrence is to aggregate knowledge across single publications (cf. <cit.>) into larger interaction graphs, as we also do in our approach. Nevertheless, the knowledge extraction to build those interaction graphs often relies on text mining techniques and natural language processing methods as in the BITOLA system <cit.> or in the approaches of Wren et al. <cit.> and Wilkinson et al. <cit.>. Closely related to our approach is the one by Özgür et al. <cit.> who extract interaction paths from dependency networks and rely on graph centrality measures to rank proteins for a given disease. Contrary to our model, they do not use complex features extracted from the graph and do not combine different types of features. § CONCLUSIONS AND OUTLOOK In this paper, we have presented a system and a model for predicting disease-gene associations from biomedical text, using a combination of features based on disease-gene co-occurrences and gene interactions that are represented in a graph database. In a classification experiment against a manually curated database used as gold standard, we were able to demonstrate the effectiveness of both types of features, outperforming a plain co-occurrence baseline by more than 30 points in F_1 score. Moreover, preliminary investigation of a practical use case from pharmaceutical industry suggests that almost 80% of the candidates predicted by our model are plausible and may support pharmaceutical researchers in hypothesis generation. In future work, we will carry out a more detailed evaluation of the case study and supplement our classification approach by a ranking model that not only separates positive and negative candidates but also reflects relative differences in these candidates' plausibility.splncs03 | http://arxiv.org/abs/1709.09239v1 | {
"authors": [
"Hendrik ter Horst",
"Matthias Hartung",
"Roman Klinger",
"Matthias Zwick",
"Philipp Cimiano"
],
"categories": [
"cs.CL"
],
"primary_category": "cs.CL",
"published": "20170926195916",
"title": "Predicting Disease-Gene Associations using Cross-Document Graph-based Features"
} |
Weighted Sum-Throughput Maximization for Energy Harvesting Powered MIMO Multi-Access Channels Zheng Nan^1, Wenming Li^2=============================================================================================We consider a problem we call StateIsomorphism: given two quantum states of n qubits, can one be obtained from the other by rearranging the qubit subsystems? Our main goal is to study the complexity of this problem, which is a natural quantum generalisation of the problem StringIsomorphism. We show that StateIsomorphism is at least as hard as GraphIsomorphism, and show that these problems have a similar structure by presenting evidence to suggest that StateIsomorphism is an intermediate problem for QCMA. In particular, we show that the complement of the problem, StateNonIsomorphism, has a two message quantum interactive proof system, and that this proof system can be made statistical zero-knowledge. We consider also StabilizerStateIsomorphism (SSI) and MixedStateIsomorphism (MSI), showing that the complement of SSI has a quantum interactive proof system that uses classical communication only, and that MSI is QSZK-hard.§ INTRODUCTION AND STATEMENT OF RESULTSLadner's theorem <cit.> states that if P≠NP then there exists NP-intermediate problems: NP problems that are neither NP-hard, nor in P. While of course the P vs. NP problem is unresolved, the problem of testing if two graphs are isomorphic (GraphIsomorphism) has the characteristics of such an intermediate problem. GraphIsomorphism is trivially in NP, since isomorphism of two graphs can be certified by describing the permutation that maps one to the other, but as Boppana and Håstad show <cit.>, if it is NP-complete then the polynomial hierarchy collapses to the second level. Furthermore, while many instances of the problem are solvable efficiently in practice <cit.>, it is still not known if there exists a polynomial time algorithm for the problem.Recall that Quantum Merlin Arthur (QMA) is considered to be the quantum analogue of NP: the certificate is a quantum state, and the verifier has the ability to perform quantum computation. The class QCMA is defined in the same way but with certificates restricted to be classical bitstrings. In this paper, we show that there are problems that exhibit similar hallmarks of being intermediate for QCMA <cit.>. Succinctly: we formulate problems in QCMA that are not obviously in BQP, and which are unlikely to be QCMA-complete. Babai's recent quasi-polynomial time algorithm for GraphIsomorphism <cit.> has revived a fruitful body of work that links the problem to algorithmic group theory <cit.>. This literature deals with a closely related problem called StringIsomorphism: given bitstrings x,y∈{0,1}^n and a permutation group G, is there σ∈ G such that σ(x)=y (where permutations act in the obvious way on the strings)? This problem has a number of similarities with GraphIsomorphism, and, as we show, can be recast in terms of quantum states.We study what is arguably the most direct quantum generalisation of this problem, a problem we call StateIsomorphism. Such a generalisation is obtained by replacing the strings x and y by n-qubit pure states, and by considering the permutations in the group G to act as “reshufflings” of the qubits. The problem is obviously in QCMA: if there is a permutation mapping one state to the other then its permutation matrix acts as the certificate. Equality of two quantum states can be verified via an efficient quantum procedure known as the SWAP test <cit.>. Also, if there is an efficient quantum algorithm then the same can be used as an algorithm for GraphIsomorphism: as we shall see later, there exists a polynomial time many-one reduction from GraphIsomorphism to StateIsomorphism.We first establish that in terms of interactive proof systems that solve the problem, StateIsomorphism has a number of similarities with its classical counterpart. A central part of the Boppana-Håstad collapse result is that GraphIsomorphism belongs in co-IP(2): that is, that GraphNonIsomorphism has a two round interactive proof system. We show that StateIsomorphism is in co-QIP(2): its complement has a two round quantum interactive proof system. GraphIsomorphism also admits a statistical zero knowledge proof system, and indeed, we prove that StateIsomorphism has an honest verifier quantum statistical zero knowledge proof system. These results are summarised in the following theorem, where QSZK is the class of problems with (honest verifier) quantum statistical zero knowledge proof systems, defined by Watrous in <cit.>. Note that since QIP(2)⊇QSZK = co-QSZK (see <cit.>), inclusion in co-QIP(2) follows as a corollary.StateIsomorphism is in QSZK.A corollary of this theorem provides evidence to suggest that StateIsomorphism is not QCMA-complete. If it were, then every problem in QCMA would have an honest verifier quantum statistical zero knowledge proof system.Furthermore, this result is evidence against the problem being NP-hard: it is unlikely that NP⊆QSZK. If StateIsomorphism is QCMA-complete then QCMA⊆QSZK. In pursuit of stronger evidence against QCMA-hardness of StateIsomorphism, we consider a quantum polynomial hierarchy in the same vein as those considered by Gharibian and Kempe <cit.>, and Yamakami <cit.>. This hierarchy is defined in terms of quantum ∃ and ∀ complexity class operators like those of <cit.>, but from our definitions it is easy to verify that lower levels correspond to well known complexity classes. In particular, Σ_0=Π_0=BQP, and Σ_1=QCMA or Σ_1=QMA depending on whether we take the certificates to be classical or quantum (see Section <ref>). Also, from the definition we provide, it is clear that the class cq-Σ_2 corresponds directly to the identically named class in <cit.>.We prove the following, where QPH=∪_i=1^∞Σ_i, and QCAM is the quantum generalisation of the class AM where all communication between Arthur and Merlin is restricted to be classical <cit.>. Let A be a promise problem in QCMA∩co-QCAM. If A is QCMA-complete, then QPH⊆Σ_2. While the relationship between the levels of this hierarchy and the levels of the classical hierarchy remains an open research question <cit.>, the fact that the lower levels of this quantum hierarchy coincide with well known classes gives weight to collapse results of this kind. We draw attention to the fact that the collapse implication in Theorem <ref> is for the classical certificate classes QCMA and QCAM, rather than for the more well known QMA and QAM <cit.>. While the problems we consider are in QCMA, meaning that the current statement of the theorem is all we need, already we have an interesting open question: is there a similar collapse theorem that relates QMA and QAM? The proof of Theorem <ref> relies on the fact that QCMAM=QCAM (proved by Kobayashi et al. in <cit.>), but it is unlikely that QMAM=QAM, since QMAM=QIP=PSPACE <cit.>. As we shall see in Section <ref>, there is a barrier that prevents us from applying Theorem <ref> to StateIsomorphism: our quantum interactive proof systems for StateNonIsomorphism require quantum communication between verifier and prover. This prevents us from proving inclusion in QCAM. It is not clear that the problem admits such a proof system. However, if it is possible to produce an efficient classical description of the quantum states in the problem instance that is independent from how they are specified in the input, then it is possible to prove inclusion in QCAM. We show that this is the case for a restricted family of quantum states called stabilizer states, a fact which allows us to prove the following. If StabilizerStateIsomorphism is QCMA-complete, then QPH⊆Σ_2. Furthermore, the fact that stabilizer states can be described classically also implies the following.StabilizerStateNonIsomorphism is in QCSZK. Finally, we consider the state isomorphism problem for mixed quantum states. We show that this problem is QSZK-hard by reduction from the QSZK-complete problem of determining if a mixed state is product or separable.(ϵ,1-ϵ)-MixedStateIsomorphism is QSZK-hard. While these state isomorphism problems all have classical certificates, we have been able to demonstrate that the complexity of each problem depends precisely on the inherent computational difficulty of working with the input states. Stabilizer states form one end of the spectrum: with a polynomial number of measurements a classical description can be produced. The other extreme is the mixed states, these are so computationally difficult to work with that it is not clear that MixedStateIsomorphism even belongs in QMA; even the problem of testing equivalence of two such states is QSZK-complete (see <cit.>). Between these two extremes we have StateIsomorphism. While such states can be efficiently processed by a quantum circuit, and isomorphism can be certified classically, the analysis in Section <ref> uncovers an interesting caveat. It seems that the ability to communicate quantum states is still required when we wish to check non-isomorphism by interacting with a prover, or perhaps even to certify isomorphism with statistical zero knowledge. We thus draw attention to the following open question: can our protocols be modified to use exclusively classical communication?The fact that an efficient quantum algorithm for StateIsomorphism would also yield one for GraphIsomorphism, combined with Corollary <ref>, gives weight to the idea that this problem can be thought of as a candidate for a QCMA-intermediate problem. The fact that there are problems “in between” BQP and QCMA, and furthermore, that such problems are obtained by generalising StringIsomorphism suggests an interesting parallel between the classical and quantum classes.In Section <ref> we give an overview of the tools and notation we will use for the rest of the paper. We also define the key problems and complexity classes we will be working with and prove some initial results that we build on later. In Section <ref> we demonstrate quantum interactive proof systems for the StateIsomorphism problems. In Section <ref> we define a notion of a quantum polynomial hierarchy, and prove the hierarchy collapse results.§ PRELIMINARIES AND DEFINITIONSRecall that quantum states are represented by unit trace positive semi-definite operators ρ on a Hilbert space ℋ called the state space of the system. A state is pure if ρ^2=ρ. Otherwise, we say that the state is mixed.By definition then, for any pure state ρ on ℋ we have that ρ=|ψ⟩⟨ψ| for some unit vector |ψ⟩∈ℋ, and we refer to pure states by their corresponding state vector |ψ⟩ (which is unique up to multiplication by a phase). Mixed states are convex combinations of the outer products of some set of state vectors ρ=∑_i p_i|ψ_i⟩⟨ψ_i|. In what follows we refer to the Hilbert space ℂ^2 by ℋ_2. Recall that an n-qubit pure state |ψ⟩∈ℋ_2^⊗ n is product if |ψ⟩=|ψ_1⟩⊗⋯⊗|ψ_n⟩ where ⊗ denotes tensor product and for all i, |ψ_i⟩∈ℋ_2. For any bitstring x_1… x_n∈{0,1}^n, we say that |x⟩=⊗_i=1^n|x_i⟩ is a computational basis state.A useful measure of the distinguishability of a pair of quantum states is the trace distance. Let ρ,σ be quantum states with the same state space. Their trace distance is the quantity D(ρ,σ)=1/2‖ρ-σ‖_1, where ‖ M ‖_1=tr[|M|] is the trace norm.We say that a quantum circuit Q accepts a state |ψ⟩ if measuring the first qubit of the state Q|ψ⟩ in the computational basis yields outcome 1. We say that the circuit rejects the state otherwise. Let X be an index set. We say that a uniform family of quantum circuits {Q_x : x∈ X} is polynomial-time generated if there exists a polynomial-time Turing machine that takes as input x∈ X and halts with an efficient description of the circuit Q_x on its tape. Such a definition neatly captures the notion of an efficient quantum computation <cit.>.We make use of a quantum circuit known as the SWAP test <cit.>, illustrated in Figure <ref>. This circuit takes as input pure states |ψ⟩,|ϕ⟩ and accepts (denoted T(|ψ⟩,|ϕ⟩)=1) with probability (1+|⟨ψ|ϕ⟩|^2)/2. Note that T(|ψ⟩,|ϕ⟩)=1 with probability 1 if |ψ⟩=e^iτ|ϕ⟩ for some τ∈[-2π,2π], but is equal to 1 with probability 1/2 if they are orthogonal. The SWAP test can be therefore be used as an efficient quantum algorithm for testing if two quantum states are equivalent. In what follows we use some notation from complexity theory and formal language theory. In particular, if a problem A is polynomial-time many-one reducible to a problem B we denote this by A ≤_p B. We denote by {0,1}^n the set of bitstrings of length n, furthermore, {0,1}^* denotes the set of all bitstrings. For a bitstring x, we denote by |x| the length of the bitstring. We say that a function f:ℕ→ [0,1] is negligible if for every constant c there exists n_c such that for all n≥ n_c, f(n)<1/n^c. We use the shorthand f(n)=poly(n) (resp. f(n)=exp(n)) to state that f scales as a polynomially bounded (exponentially bounded) function in n.A decision problem is a set of bitstrings A⊆{0,1}^*. An algorithm is said to decide A if for all x∈{0,1}^* it outputs YES if x∈ A and NO otherwise. In quantum computational complexity it is useful to use the less well known notion of a promise problem to allow for more control over problem instances. A promise problem is a pair of sets (A_YES,A_NO)⊆{0,1}^*×{0,1}^* such that A_YES∩ A_NO=∅. An algorithm is said to decide (A_YES,A_NO) if for all x∈ A_YES it outputs YES and for all x∈ A_NO it outputs NO. Note that the algorithm is not required to do anything in the case where an input x does not belong to A_YES or A_NO. §.§ Quantum Merlin-Arthur, Quantum Arthur-MerlinFor convenience, we give a number of definitions related to quantum generalisations of public coin proof systems. In particular, we focus on Quantum Arthur-Merlin (QAM) and Quantum Merlin-Arthur, the quantum versions of AM and MA respectively. We use the definitions in <cit.> as our guide. A promise problem A=A is in QMA(a,b) for functions a,b:ℕ→[0,1] if there exists a polynomial-time generated uniform family of quantum circuits {V_x : x∈{0,1}^*} and polynomially bounded p:ℕ→ℕ such that * for all x∈A there exists |ψ⟩∈ℋ_2^⊗ p(|x|) such that [V_xaccepts|ψ⟩] ≥ a(|x|); * for all x∈A and for all |ψ⟩∈ℋ_2^⊗ p(|x|), [V_xaccepts|ψ⟩] ≤ b(|x|). The class QCMA is defined in the same way, but with the restriction that the certificate |ψ⟩ must be a computational basis state |x⟩.A QAM verification procedure is a tuple (V,m,s) whereV={V_x,y : x∈{0,1}^*,y∈{0,1}^s(|x|)}is a uniform family of polynomial time generated quantum circuits, and m,s:ℕ→ℕ are polynomially bounded functions. Each circuit acts on m(|x|) qubits sent by Merlin, and k(|x|) qubits which correspond to Arthur's workspace. For all x,y, we say that V_x,y accepts (resp. rejects) a state |ψ⟩∈ℋ_2^⊗ m(|x|) if, upon measuring the first qubit of the stateV_x,y |ψ⟩|0⟩^⊗ k(|x|)in the standard basis, the outcome is `1' (resp. `0'). A promise problem A=A is in QAM(a,b) for functions a,b:ℕ→[0,1] if there exists a QAM verification procedure (V,m,s) such that * for all x∈A, there exists a collection of m(|x|)-qubit quantum states {|ψ_y⟩} such that 1/2^s(|x|)∑_y∈{0,1}^s(|x|)[V_x,y accepts|ψ_y⟩]≥ a(|x|); * for all x∈A, and for all collections of m(|x|)-qubit quantum states {|ψ_y⟩}, it holds that 1/2^s(|x|)∑_y∈{0,1}^s(|x|)[V_x,y accepts|ψ_y⟩]≤ b(|x|). The class QCAM is defined in the same way but with the states {|ψ_y⟩} restricted to computational basis states. The class QCMAM is similar, but has an extra round of interaction. A promise problem A=A is in QCMAM(a,b) for functions a,b:ℕ→[0,1] if there exists a QAM verification procedure (V,m,s) and a polynomially bounded function p:ℕ→ℕ such that * for all x∈A, there is a certificate bitstring c∈{0,1}^p(|x|) and a collection of length m(|x|) bitstrings {z^c_y} such that 1/2^s(|x|)∑_y∈{0,1}^s(|x|)[V_x,y accepts|c⟩⊗|z^c_y⟩]≥ a(|x|); * for all x∈A, all certificate bitstrings c∈{0,1}^p(|x|) and all collections of length m(|x|) bitstrings {z^c_y}, it holds that 1/2^s(|x|)∑_y∈{0,1}^s(|x|)[V_x,y accepts|c⟩⊗|z^c_y⟩]≤ b(|x|). §.§ Quantum interactive proofs and zero knowledgeAn interactive proof system consists of a verifier and a prover. The computationally unbounded prover attempts to convince the computationally limited verifier that a particular statement is true. A quantum interactive proof system is where the verifier is equipped with a quantum computer, and quantum information can be transferred between verifier and prover. Our formal definitions will follow those of Watrous <cit.>.A quantum verifier is a polynomial time computable function V, where for each x∈{0,1}^*, V(x) is an efficient classical description of a sequence of quantum circuits V(x)_1,…,V(x)_k(|x|). Each circuit in the sequence acts on v(|x|) qubits that make up the verifier's private workspace, and a buffer of c(|x|) communication qubits that both verifier and prover have read/write access to.A quantum prover is a function P where for each x∈{0,1}^*, P(x) is a sequence of quantum circuits P(x)_1,… P(x)_l(|x|). Each circuit in the sequence acts on p(|x|) qubits that make up the prover's private workspace, and the c(|x|) communication qubits that are shared with each verifier circuit. Note that no restrictions are placed on the circuits P(x), since we wish the prover to be computationally unbounded. We say that a verifier V and a prover P are compatible if all their circuits act on the same number of communication qubits, and if for all x∈{0,1}^*, k(|x|)=⌊ m(|x|)/2+1⌋ and l(|x|)=⌊ m(|x|)/2+1/2⌋, for some m(|x|) which is taken to be the number of messages exchanged between the prover and verifier. We say that (P,V) are a compatible m-message prover-verifier pair.Given some compatible m-message prover-verifier pair (P,V), we define the quantum circuit(P(x),V(x)):=V(x)_1· P(x)_1… P(x)_m(|x|)/2· V(x)_m(|x|)/2+1ifm(|x|)is even,P(x)_1· V(x)_1… P(x)_(m(|x|)+1)/2· V(x)_(m(|x|)+1)/2ifm(|x|)is odd.Let q(|x|)=p(|x|)+c(|x|)+v(|x|). We say that (P,V) accepts an input x∈{0,1}^* if the result of measuring the verifier's first workspace qubit of the state(P(x),V(x))|0^q(|x|)⟩in the computational basis is 1, and that it rejects the input if the measurement result is 0.Let M=M be a promise problem, let a,b:ℕ→[0,1]be functions and k∈ℕ. Then M∈QIP(k)(a,b) if and only if there exists a k-message verifier V such that * if x∈M then max_P([(P,V)acceptsx]) ≥ a(|x|), * if x∈M then max_P ([(P,V)acceptsx]) ≤ b(|x|),where the maximisation is performed over all compatible k-message provers. We say that the pair (P,V) is an interactive proof system for M. Let us now define what it means for a quantum interactive proof system to be statistical zero-knowledge. Define the function view_V,P(x,j)=tr_P[(P(x),V(x))_j|0^q(|x|)⟩⟨ 0^q(|x|)|(P(x),V(x))_j^†],where (P(x),V(x))_j is the circuit obtained from running (P(x),V(x)) up to the j^th message. For some index set X, we say that a set of density operators {ρ_x : x∈ X} is polynomial-time preparable if there exists a polynomial-time uniformly generated family of quantum circuits {Q_x : x∈ X}, each with a designated set of output qubits, such that for all x∈ X, the state of the output qubits after running Q_x on a canonical initial state |0⟩^⊗ n is equal to ρ_x.Let M=M be a promise problem, let a,b:ℕ→[0,1] and k:ℕ→ℕ be functions. Then M∈HVQSZK(k)(a,b) if and only if M∈QIP(k)(a,b) with quantum interactive proof system (P,V) such that there exists a polynomial-time preparable set of density operators {σ_x,i} such that for all x∈{0,1}^*, if x∈M thenD(σ_x,i,view_P,V(x,i))≤δ(|x|)for some negligible function δ. It is known that the class of problems that have quantum statistical zero knowledge proof systems (QSZK) is equivalent to the class of problems that have honest verifier quantum statistical zero knowledge proof systems (HVQSZK) <cit.>. Therefore, we refer to HVQSZK as QSZK, and only consider honest verifiers.In the next section we give a formal definition of StringIsomorphism. §.§ Permutations and StringIsomorphism Let Ω be a finite set. A bijection σ:Ω→Ω is called a permutation of the set Ω. The set of all permutations of a finite set Ω forms a group under composition. This group is called the symmetric group, and we denote it by 𝔖(Ω). For x∈Ω and σ∈𝔖(Ω), we denote the image of x under σ by σ(x).A string 𝔰:Ω→Σ is an assignment of letters from a finite set Σ called an alphabet to the elements of a finite index set Ω. Let 𝔰:Ω→Σ be a string. The letters of 𝔰 are indexed by elements of the index set Ω. The letter corresponding to i∈Ω is thus denoted by 𝔰_i. Let σ∈𝔖(Ω) be a permutation. Then the action of σ on 𝔰 is denoted by σ(𝔰), and is a string such that for all i∈Ω, σ(𝔰)_i=𝔰_σ(i). In this paper we often deal with permutations of strings indexed by natural numbers. Hence, we denote the symmetric group𝔖([n]) by 𝔖_n, where [n]:={1,…,n}. In what follows we denote the fact that a group G is a subgroup of a group H by G≤ H. The following decision problem is related to GraphIsomorphism<cit.>, and forms the basis of our work. StringIsomorphism Input: Finite sets Ω,Σ, a permutation group G≤𝔖(Ω) specified by a set of generators, and strings 𝔰,𝔱:Ω→Σ. Output: Yes if and only if there exists σ∈ G such that σ(𝔰)=𝔱. It is clear that StringIsomorphism is at least as hard as GraphIsomorphism: a polynomial time many-one reduction can be obtained from GraphIsomorphism by “flattening” the adjacency matrices of the graphs in question into bitstrings. The set of string permutations that correspond to graph isomorphisms form a proper subgroup of the full symmetric group. Indeed thealgorithm in <cit.> is actually an algorithm for StringIsomorphism, which solves GraphIsomorphism as a special case.§.§ Stabilizer statesThe Gottesmann-Knill theorem <cit.> states that any quantum circuit made up of CNOT, Hadamard and phase gates along with single qubit measurements can be simulated in polynomial time by a classical algorithm. Such circuits are called stabilizer circuits, and any n-qubit quantum state |ψ⟩ such that |ψ⟩=Q|0⟩^⊗ n for a stabilizer circuit Q is referred to as a stabilizer state. Let |ψ⟩ be an n-qubit state. A unitary U is said to be a stabilizer of |ψ⟩ if U|ψ⟩=± 1|ψ⟩. The set of stabilizers of a state |ψ⟩ forms a finite group under composition called the stabilizer group of |ψ⟩, denoted Stab(|ψ⟩).The Pauli matrices are the unitariesσ_00:=[ 1 0; 0 1 ], σ_01:=[ 0 1; 1 0 ], σ_10:=[10;0 -1 ], σ_11:=[0 -i;i0 ],which form a finite group 𝒫 under composition called the single qubit Pauli group. The n-qubit Pauli group 𝒫_n is the group with elements {(± 1)U_1⊗…⊗ (± 1 )U_n : U_j∈𝒫}∪{(± i)U_1⊗…⊗ (± i)U_n : U_j∈𝒫}.It is well known (c.f. <cit.> Theorem 1) that an n-qubit stabilizer state |ψ⟩ is uniquely determined by the finite group S(|ψ⟩):=Stab(|ψ⟩)∩𝒫_n, of size 2^n. Hence, |ψ⟩ is determined by the n=log(2^n) elements of 𝒫_n that generate S(|ψ⟩). These elements each take 2n bits to specify the Pauli matrices in the tensor product, and an extra bit to specify the overall ± 1 phase. This fact, along with the following theorem, means that given a polynomial number of copies of a stabilizer state |ψ⟩, we can produce an efficient classical description of that state by means of the generators of S(|ψ⟩). There exists a quantum algorithm with the following properties: * Given access to O(n) copies of an n-qubit stabilizer state |ψ⟩, the algorithm outputs a bitstring describing a set of n-qubit Pauli operators s_1,…, s_n∈𝒫_n such that ⟨ s_1,… s_n⟩ = S(|ψ⟩);* the algorithm halts after O(n^3) classical time steps;* all collective measurements are performed over at most two copies of the state |ψ⟩;* the algorithm succeeds with probability 1-1/exp(n). §.§ Permutations of quantum states and isomorphism Let σ∈𝔖_n be a permutation. Then the following is a unitary map acting on n-partite states that implements it as a permutation of the subsystems (see e.g. <cit.>)P_σ:=∑_i_1,…,i_n∈[d]|i_σ(1)… i_σ(n)⟩⟨ i_1,… i_n|.Note that P_σ depends on the dimensions of the subsystems of the n-partite states on which it acts. Nevertheless, here we will only consider quantum states where each subsystem is a qubit.The focus of this work is on a number of variations on the following promise problem, StateIsomorphism. In what follows, let 𝒬_m,n for m≥ n denote the set of all quantum circuits with m input qubits and n output qubits. In particular, Q_n,n is the set of all pure state quantum circuits on n qubits. Then, for m>n, Q_m,n is the set of all mixed state circuits that can be obtained by discarding the last m-n output qubits of the circuits in Q_m,m.When we specify a circuit with a subscript label, such asQ_ψ∈𝒬_m,n, we do so to easily refer to the state of the output qubits when the circuit is applied to the state |0⟩^m. In particular, when m=n this is the pure state |ψ⟩∈ℂ^2^n, and the mixed state ψ acting on ℂ^2^n otherwise. StateIsomorphism (SI)Input: Efficient descriptions of quantum circuits Q_ψ_0 and Q_ψ_1 in 𝒬_n,n, a set of permutations {τ_1,…τ_k} generating some permutation group ⟨τ_1…τ_k⟩=:G≤𝔖_n, and a function ϵ:ℕ→[0,1] such that ϵ(n)≥ 1/poly(n) for all n.YES: There exists a permutation σ∈ G such that |⟨ψ_1|P_σ |ψ_0⟩| = 1. NO: For all permutations σ∈ G,|⟨ψ_1|P_σ |ψ_0⟩| ≤ϵ(n). The next problem is a special case of the above, defined in terms of stabilizer states. StabilizerStateIsomorphism (SSI)Input: Efficient descriptions of quantum circuits Q_ψ_0 and Q_ψ_1 in 𝒬_n,n such that |ψ_0⟩ and |ψ_1⟩ are stabilizer states, a set of permutations {τ_1,…τ_k} generating some permutation group ⟨τ_1…τ_k⟩=:G≤𝔖_n, and ϵ:ℕ→[0,1] such that ϵ(n)≥ 1/poly(n) for all n.YES: There exists a permutation σ∈ G such that |⟨ψ_1|P_σ |ψ_0⟩| = 1. NO: For all permutations σ∈ G,|⟨ψ_1|P_σ |ψ_0⟩| ≤ϵ(n). Finally, we consider the state isomorphism problem for mixed states. (ϵ,1-ϵ)-MixedStateIsomorphism (MSI)Input: Efficient descriptions of quantum circuits Q_ρ_0 and Q_ρ_1 in 𝒬_2n,n, a set of permutations {τ_1,…τ_k} generating some permutation group ⟨τ_1…τ_k⟩=:G≤𝔖_n, and ϵ:ℕ→[0,1]. YES: There exists a permutation σ∈ G such that D(P_σρ_0 P_σ^†,ρ_1) ≤ϵ(n). NO: For all permutations σ∈ G,D(P_σρ_0 P_σ^†,ρ_1) ≥ 1-ϵ(n).We also consider the above problems where the permutation group specified is equal to the symmetric group G=𝔖_n. We denote these problems with the prefix 𝔖_n, for example, 𝔖_n-SI. It is clear that SSI≤_p SI≤_pMSI. We now show that SI is in QCMA. StateIsomorphism∈QCMA. In the case of a YES instance, there exists σ∈ G such that |⟨ψ_1|P_σ|ψ_0⟩|=1. The latter equality can be verified by means of a SWAP-test on the states P_σ|ψ_0⟩ and |ψ_1⟩, which by definition will accept with probability equal to 1. Since the states |ψ_0⟩ and |ψ_1⟩ are given as an efficient classical descriptions of quantum circuits that will prepare them, this verification can be performed in quantum polynomial time. Furthermore, there exists an efficient classical description of the permutation σ in terms of the generators of the group specified in the input, each of which can be described via their permutation matrices. The unitary P_σ can be implemented efficiently by Arthur given the description of σ. Determining membership/non-membership of some permutation σ∈𝔖_n in the permutation group G≤𝔖_n specified by the set of generators {τ_1,…τ_k} can be verified in classical polynomial time by utilizing standard techniques from computational group theory. In particular, since we are considering permutation groups we can use the Schreier-Sims algorithm to obtain a base and a strong generating set for G in polynomial time from {τ_1,…,τ_k}. These new objects can then be used to efficiently verify membership in G <cit.>.In the case that the states are not isomorphic, we have by definition that for all permutations σ∈ G, |⟨ψ_1|P_σ|ψ_0⟩|≤ϵ(n), which can again be verified by using the SWAP-test, which will accept the states with probability at most 1/2+ϵ(n).It is not clear if MSI is in QCMA, or even in QMA. While the isomorphism σ can still be specified efficiently classically, it is not known if there exists an efficient quantum circuit for testing if two mixed states are close in trace distance. In fact, this problem is known as the StateDistinguishability problem, and is QSZK-complete <cit.>. There exists a polynomial-time many-one reduction from GraphIsomorphism to SSI, indeed it is identical to the reduction from GraphIsomorphism to StringIsomorphism. SSI is in turn trivially reducible to the isomorphism problems for pure and mixed states respectively. These problems are therefore at least as hard as GraphIsomorphism. Interestingly however, there also exists a reduction from GraphIsomorphism to a restricted form of SI where the permutation group G is equal to the full symmetric group 𝔖_n (as stated earlier, we refer to this problem as 𝔖_n-StateIsomorphism). In order to demonstrate this, we require a family of quantum states referred to as graph states <cit.>. Let G=(V,E) be an n-vertex graph. For each vertex v∈ V, define the observable K^(v):=σ_x^(v)∏_w∈ N(v)σ_z^(w) where N(v) is the neighborhood of v, and σ_i^(j) denotes the n-qubit operator consisting of Pauli σ_i applied to the j^th qubit and identity on the rest. The graph state |G⟩ is defined to be the state stabilized by the set S_G:={K^(v) : v∈ V}, that is, K^(v)|G⟩=|G⟩ for all v∈ V. Since the stabilizers of a graph state |G⟩ are all elements of the |V| qubit Pauli group, graph states are stabilizer states, and the following theorem provides an upper bound on the overlap of non-equal graph states. Let |ψ⟩,|ϕ⟩ be non-orthogonal stabiliser states, and let s be the minimum, taken over all sets of generators {P_1,… P_n} for S(|ψ⟩) and {Q_1,… Q_n} for S(|ϕ⟩), of the number of i values such that P_i≠ Q_i. Then |⟨ψ|ϕ⟩|=2^-s/2. We can now describe the reduction.GraphIsomorphism≤_p 𝔖_n-StateIsomorphism. Consider two n-vertex graphs G and H. If G=H then clearly |⟨ G|H⟩|^2=1 since |G⟩ and |H⟩ are the same state up to a global phase. Suppose G≠ H. Then necessarily s>0, so by Theorem <ref> we have that |⟨ G|H⟩|^2≤1/2. Consider a permutation σ∈𝔖_n. Then for each v∈ V, K^(σ(v))=P_σ K^(v) P_σ^T, so |⟨σ(G)|P_σ|G⟩|^2=1. Explicitly, if G≅ H then there exists a permutation of the vertices σ such that σ(G)=H and so |⟨σ(G)|H⟩|^2=|⟨ G|P_σ^T |H⟩|^2=1. If G≇H then for all σ, ⟨ G|P_σ^T |H⟩|^2≤1/2.To complete the reduction we must show that for any graph G=(V,E), a description of a quantum circuit that prepares |G⟩ can be produced efficiently classically. This is trivial, an alternate definition of graph states <cit.> gives us that |G⟩=Π_{i,j}∈ ECZ_ij|+⟩^⊗ |V|, where CZ_ij is the controlled-σ_z operator with qubit i as control and j as output. Therefore, the StateIsomorphism problem where no restriction is placed on the permutations is at least as hard as GraphIsomorphism. This is in stark contrast to the complexity of the corresponding classical problem, which is trivially in P: two bitstrings are isomorphic under 𝔖_n if and only if they have the same Hamming weight, which is easily determined.§ INTERACTIVE PROOF SYSTEMSIn this section we will prove Theorem <ref>. To do so, we will first demonstrate a quantum interactive proof system for StateNonIsomorphism (SNI) with two messages. We then show that this quantum interactive proof system can be made statistical zero knowledge. In order to prove the former, we will require the following lemma. Given access to a sequence of unitaries U_1,…, U_n, along with their inverses U_1^†,…, U_n^† and controlled implementations c-U_1,…,c-U_n, as well as the ability to produce copies of a state |ψ⟩ promised that one of the following cases holds: * For some i, U_i|ψ⟩=|ψ⟩; * For all i, |⟨ψ|U_i|ψ⟩|≤ 1-δ. Then there exists a quantum algorithm which distinguishes between these cases using O(log n /δ) copies of |ψ⟩, succeeding with probability at least 2/3.We can now prove the following.StateNonIsomorphism is in QIP(2). We will prove that the following constitutes a two message quantum interactive proof system for SNI.* (V) Uniformly at random, select σ∈ G and j∈{0,1}. Send the state |Ψ⟩^⊗ k to the prover, where k=O(log(|G|)/(1-ϵ(n))) and |Ψ⟩=P_σ|ψ_j⟩.* (P) Send j'∈{0,1} to the verifier.* (V) Accept if and only if j'=j. Obtaining a uniformly random element from G as in step 1 can be achieved efficiently if the verifier is in possession of a base and a strong generating set for G. These can be obtained in polynomial time from any generating set of G by using Schreier-Sims algorithm <cit.>.For a permutation π∈ G, we define the 2n qubit circuit U^(j)_π=SWAP· (P_π^-1⊗ P_π), where the SWAP acts so as to swap the two n qubit states, that is, SWAP|ψ_0⟩|ψ_1⟩=|ψ_1⟩|ψ_0⟩. Now consider the sets of quantum circuits C^(j)_G={U^(j)_π : π∈ G} for j∈{0,1}, each of cardinality |G|. Since each circuit in C^(0)_G∪ C^(1)_G is made up two permutations and a SWAP gate, each of their inverses can easily be obtained. Additionally, the controlled versions of these gates can be implemented via standard techniques.Consider first the YES case. The k=O(log(|G|)/(1-ϵ(n))) copies of |Ψ⟩ enables the prover to determine j with success probability at least 2/3 in the following manner.* Uniformly at random, select j'∈{0,1}.* Prepare k copies of the state |Ψ⟩|ψ_j'⟩* Use the HLM algorithm with the state |Ψ⟩|ψ_j'⟩ and the set of circuits C^(j')_G as input. If the algorithm reports case 1 then output j', otherwise output j'⊕ 1. Let us check that the HLM algorithm will work for our purposes. In the case that the prover's guess is correct and j'=j, we have that |Ψ⟩|ψ_j'⟩ = (P_σ⊗ I)|ψ_j⟩|ψ_j⟩, and so U_σ(P_σ⊗ I)|ψ_j⟩|ψ_j⟩ = SWAP· (P_σ^-1⊗ P_σ)·(P_σ⊗ I)|ψ_j⟩|ψ_j⟩=SWAP·(I⊗ P_σ)|ψ_j⟩|ψ_j⟩=|Ψ⟩|ψ_j⟩.This corresponds to case 1 of Lemma <ref>. If the prover's guess is incorrect j'≠ j then for all π∈ G|⟨Ψ |⟨ψ_j' |U_π|Ψ⟩|ψ_j'⟩| =|⟨Ψ |⟨ψ_j' |SWAP·(P_π^-1⊗ P_π)(P_σ⊗ I)|ψ_j⟩|ψ_j'⟩|=|⟨Ψ|⟨ψ_j'|(P_π⊗ P_π^-1·σ)|ψ_j'⟩|ψ_j⟩|≤ |⟨ψ_j|P_σ^† P_π|ψ_j'⟩| · |⟨ψ_j'| P_π^-1·σ|ψ_j⟩|≤ϵ(n)^2,with the last inequality following from the fact that we are in the YES case: for all σ∈ G, we have that |⟨ψ_2|P_σ|ψ_1⟩|≤ a(n). This corresponds to case 2 of Lemma <ref>. Therefore, the HLM algorithm allows the prover to determine if her guess was correct or not, with success probability at least 2/3.Consider now the NO case, where we have that for some σ∈ G, |⟨ψ_1|P_σ|ψ_2⟩|=1. To determine j correctly, a cheating prover must be able to distinguish the mixed states ρ_j=1/|G|∑_π∈ G(P_π|ψ_j⟩⟨ψ_j|P_π^†)^⊗ k correctly for j∈{1,2}, when given k copies. However,‖ρ_1-ρ_2‖_1= 1/|G|‖∑_π∈ GP_π^⊗ k(|ψ_1⟩⟨ψ_1|)^⊗ kP_π^†⊗ k-∑_π∈ GP_π^⊗ k(|ψ_2⟩⟨ψ_2|)^⊗ kP_π^†⊗ k‖_1= 1/|G|‖∑_π∈ GP_π^⊗ kP_σ^⊗ k(|ψ_1⟩⟨ψ_1|)^⊗ kP_σ^†⊗ kP_π^†⊗ k-∑_π∈ GP_π^⊗ k(|ψ_2⟩⟨ψ_2|)^⊗ kP_π^†⊗ k‖_1= 1/|G|‖∑_π∈ GP_π^⊗ k(|ψ_2⟩⟨ψ_2|)^⊗ kP_π^†⊗ k-∑_π∈ GP_π^⊗ k(|ψ_2⟩⟨ψ_2|)^⊗ kP_π^†⊗ k‖_1=0,so they are indistinguishable. Note that the fact that the prover has been given k copies does not help, as the overlap is 0. In this case, the probability that the prover can guess j correctly is therefore equal to 1/2. We can use a standard amplification argument to modify the above protocol so that it has negligible completeness error, which means that it can be made statistical zero knowledge. We prove this now. StateNonIsomorphism is in QSZK. We first show that the protocol above can be modified to have exponentially small completeness error. This allows us to show that the protocol is quantum statistical zero knowledge. First, the verifier sends the prover k'=O(nlog(|G|)/(1-ϵ(n))) copies of the state |Ψ⟩. The prover can then use HLM n times to guess j, responding with the value of j that appears in n/2 or more of the trials. Let X_1… X_n∈{`T',`F'} be the set of independent random variables corresponding to whether or not the prover guessed correctly on the i^th repetition. By Lemma <ref>, we have that [X_i=`T']≥2/3 and so [Prover guesses correctly] =1-[1/n∑_i=1^n X_i < 1/2]=1-[1/n∑_i=1^n X_i-2/3<-1/6]≥ 1-2^-Ω(n)via the Chernoff bound (explicitly, for p,q∈[0,1], we have that [∑_i=1^n (X_i-p)/n<-q ]<e^-q^2n/2p(1-p)). Clearly, sending k' copies of |Ψ⟩ rather than k gives no advantage to the prover, the trace distance between the mixed states ρ_0 and ρ_1 is still 0 in the NO case.What remains is to show that the protocol is statistical zero knowledge. This is easily obtained, and follows by similar reasoning to the protocol in <cit.>: the view of the verifier after the first step can be obtained by the simulator by selecting σ and j then preparing k' copies of the state |Ψ⟩. The view of the verifier after the prover's response can be obtained by tracing out the message qubits and supplying the verifier with the value j. Since completeness error is exponentially small, the trace distance between the simulated view and the actual view is a negligible function. If we change (relax) the condition for the two states to be non isomorphic (NO instance) to: `There exists σ∈ G such that |⟨ψ_2|P_σ|ψ_1⟩|≥ b(n)' then the distance between the two states ρ_j=1/|G|∑_π∈ G(P_π|ψ_j⟩⟨ψ_j|P_π^†)^⊗ k for j∈{1,2} is upper bounded by‖ρ_1-ρ_2‖_1=1/|G|‖∑_π∈ G(P_π)^⊗ k(P_σ|ψ_1⟩⟨ψ_1|P_σ^† - |ψ_2⟩⟨ψ_2|)^⊗ k(P_π^†)^⊗ k‖_1≤1/|G|∑_π∈ G‖(P_π)^⊗ k(P_σ|ψ_1⟩⟨ψ_1|P_σ^† - |ψ_2⟩⟨ψ_2|)^⊗ k(P_π^†)^⊗ k‖_1= ‖(P_σ|ψ_1⟩⟨ψ_1|P_σ^† - |ψ_2⟩⟨ψ_2|)^⊗ k‖_1=2 √(1- |⟨ψ_1|P_σ^†|ψ_2⟩|^2 k)≤ 2 √(1- ϵ(n)^2 k),where first inequality is just triangular inequality, last inequality follows from the promise and last equality is just rewriting the trace distance for pure states in terms of their scalar product. Now, putting the value of k=log n/1-a(n), algebraic manipulations and using the fact that log(1-x)>-2x for all x∈ (0,1/2), we get, for any b(n)∈ (1/2,1),‖ρ_1-ρ_2‖_1 = 2 √(1- b(n)^2 log n/1-a(n))= 2 √(1- n^2logb(n)/1-a(n))≤2 √(1- n^-4(1-b(n))/1-a(n)).Then the maximal probability of distinguishing between these two states is upper bounded byp≤ 1/2+√(1- n^-4(1-b(n))/1-a(n)).We have thus proved Theorem <ref>. Corollary <ref> follows easily: if SI was QCMA-complete then all QCMA problems would be reducible to it, and would belong in QSZK.While SI belongs in QCMA, the above protocol requires quantum communication. It is not clear if a similar protocol exists that uses classical communication only. In the next theorem we show that such a protocol exists for StabilizerStateNonIsomorphism, since stabilizer states can be described efficiently classically.StabilizerStateNonIsomorphism is in QCSZK. It suffices to show that the state |Ψ⟩ in the protocol above can be communicated to the prover using classical communication only. We know from Theorem <ref> that a classical description can be obtained efficiently from O(n) copies of |Ψ⟩. These copies can be prepared efficiently, since they are specified in the problem instance by quantum circuits that prepare them.We now prove that MixedStateIsomorphism is QSZK-hard (Theorem <ref>). We actually prove the following stronger result. (ϵ,1-ϵ)-𝔖_n-MixedStateIsomorphism is QSZK-hard for all ϵ(n)=1/exp(n).We prove this by reduction from the following problem (α,β)-ProductState, which as shown in <cit.> is QSZK-hard even when α=ϵ,β=1-ϵ and ϵ goes exponentially small in n. (α,β)-ProductState Input: Efficient description of a quantum circuit Q_ρ in 𝒬_0,n. YES: There exists an n-partite product state σ_1⊗⋯⊗σ_n such that D(ρ,σ_1⊗⋯⊗σ_n)≤α NO: For all n-partite product states σ_1⊗⋯⊗σ_n, D(ρ,σ_1⊗⋯⊗σ_n)≥β.We make use of the following lemma.For an n-partite mixed state ρ, let ρ_i denote the state of the i^th subsystem, obtained by tracing out the other subsystems. Let ρ be an n qubit state. If there exists a product state σ_1⊗⋯⊗σ_n such that ‖ρ-σ_1⊗⋯⊗σ_n‖_1≤α, then ‖ρ-ρ_1⊗⋯⊗ρ_n‖_1≤ (n+1)α We now must show that every instance of (α,β)-ProductState can be converted to an instance of (α',β')-𝔖_n-MixedStateIsomorphism. In particular, consider an instance ρ of (α,β)-ProductState. Our reduction takes this to an instance (ρ,ρ') of ((n+1)α,β)-𝔖_n-MixedStateIsomorphism, where ρ'=ρ_1⊗…⊗ρ_n can be prepared in the following way from n copies of the state ρ. Denote these n copies as ρ^(1),…,ρ^(n). The i^th qubit line of ρ' is the i^th qubit line of ρ^(i), all unused qubit lines are discarded (illustrated in Figure <ref>). Let ρ be an n-partite state. If ρ is product thenD(ρ,ρ_1⊗⋯⊗ρ_n) ≤ (n+1)α/2 and so (ρ,ρ') correspond to a YES instance of ((n+1)α,β)-𝔖_n-MixedStateIsomorphism. If ρ is a NO instance of (α,β)-ProductState then D(ρ,θ)≥β for all product states θ. This means that D(ρ,P_σρ_1⊗⋯⊗ρ_n P_σ)≥β for all σ∈𝔖_n since all such states are product. In this section we have shown that StateIsomorphism is in QSZK, and so is unlikely to be QCMA-complete unless all problems in QCMA have quantum statistical zero knowledge proof systems. We have also shown that StabilizerStateIsomorphism has a quantum statistical zero knowledge proof system that uses classical communication only, and that MixedStateIsomorphism is QSZK-hard.In the next section, we show that the quantum polynomial hierarchy collapses if StabilizerStateIsomorphism is QCMA-complete. § A QUANTUM POLYNOMIAL HIERARCHY Yamakami <cit.> considers a more general framework of quantum complexity theory, where computational problems are specified with quantum states as inputs, rather than just classical bitstrings. We find that using this more general view of computational problems makes it easier to define a very general quantum polynomial-time hierarchy, which can then be “pulled back” to a hierarchy that has more conventional complexity classes (e.g. BQP, QMA) as its lowest levels.Following <cit.> we consider classes of quantum promise problems, where the YES and NO sets are made up of quantum states. We use the work's notion of quantum ∃ and ∀ complexity class operators in our definitions. These yield classes that are more general than we need, so we use restricted versions where all instances are computational basis states. Let |ψ⟩∈ℋ_2^⊗ n be an n-qubit state. Then in analogy to the length of a classical bitstring |x_1… x_n|=n, we define the length of the state |ψ⟩ as | |ψ⟩|=n. The set {0,1}^*:=∪_i=1^∞{0,1}^i is the set of all bitstrings. Analogously, the set ℋ_2^*:= ⋃_i=1^∞ℋ_2^⊗ i is the set of all qubit states. A quantum promise problem is therefore a pair of sets 𝒜_YES,𝒜_NO⊆ℋ_2^* with 𝒜_YES∩𝒜_NO=∅. Note that to differentiate quantum promise problems from the traditional definition with bitstrings, we use the calligraphic font. We make use of the following complexity class, made up of quantum promise problems. A quantum promise problem A is in the class BQP^q(a,b), for functions a,b:ℕ→ [0,1] if there exists a polynomial-time generated uniform family of quantum circuits {Q_n : n∈ℕ} such that for all |ψ⟩∈ℋ_2^* * if |ψ⟩∈A then [Q_l accepts|ψ⟩]≥ a(l); * if |ψ⟩∈A then [Q_l accepts|ψ⟩]≤ b(l), where l=||ψ⟩|.Classes made up of quantum promise problems will always be denoted with the `q' superscript. It is clear that BQP⊆BQP^q, because any classical promise problem can be converted to a quantum promise problem by considering bitstrings as computational basis states. There is nothing to be gained computationally by imposing that inputs are expressed as computational basis states rather than bitstrings, so we make no distinction between the “bitstring promise problems” and the “computational basis state” promise problems. Indeed let C^q be a quantum promise problem class. Then we defineC:={𝒜∈C^q : all states in A and A are computational basis states.}The classes BQP^q and BQP are related in this way.For the remainder of this work we will assume that all complexity classes are made up of quantum promise problems. It will be convenient for us to consider even conventional complexity classes such as QMA and QCMA to be defined with problem instances specified as computational basis states, rather than as bitstrings. Defining them in this way does not affect the classes in any meaningful way, but it is useful for our purposes. In particular, instead of referring to instances of a promise problem x∈ A_YES∪ A_NO, we will refer to computational basis states in a quantum promise problem |x⟩∈𝒜_YES∪𝒜_NO.The following operators are well known from classical complexity theory, and are adapted here for quantum promise problem classes. Let C be a complexity class. A promise problem A is in∃_sC for s∈{q,c} if there exists a promise problem B∈C and a polynomially bounded function p:ℕ→ℕ such that A={|ψ⟩∈ℋ_2^* : ∃ |y⟩∈ S |ψ⟩⊗ |y⟩∈B}, and A={|ψ⟩∈ℋ_2^* : ∀ |y⟩∈ S |ψ⟩⊗ |y⟩∈B}, where the set S is equal to {|x⟩ : x∈{0,1}^p(||ψ⟩|)} if s=c, and ℋ_2^⊗ p(||ψ⟩|) if s=q. The class ∀_s C is defined analogously, but with the quantifiers swapped. We can now define the quantum polynomial hierarchy. Let Σ^q_0=Π^q_0=BQP^q. For k≥ 1, let s_1… s_k∈{c,q}^k. Then s_1… s_k-Σ_k^q=∃_s_1s_2⋯ s_k-Π_k-1^q and s_1… s_k-Π_k^q= ∀_s_1s_2⋯ s_k-Σ_k-1^qThis definition leads to complexity classes that include promise problems with quantum inputs. Such classes are not well understood, so we do not use this hierarchy in its full generality. Instead we take each level Σ_i^q or Π_i^q, and strip out all problems except those defined in terms of computational basis states by using Σ_i or Π_i. Doing so makes familiar classes emerge, indeed it is clear that Σ_0=Π_0=BQP, c-Σ_1=QCMA and q-Σ_1=QMA. This provides a generalisation of the ideas of Gharibian and Kempe <cit.> into a full hierarchy: our definition of the class cq-Σ_2 corresponds directly to theirs. For our purposes we require the following technical lemma. For all k, let C_k=s-Σ_k^q or C_k=s-Π_k^q for any s∈{q,c}^k. Then *∃_c∃_c C_k=∃_c C_k * ∀_c∀_c C_k=∀_c C_k *∃_c C_k⊆∃_q C_k * ∀_c C_k⊆∀_q C_k * ∃_c∃_q C_k=∃_q∃_c C_k=∃_q C_k * ∀_c∀_q C_k=∀_q∀_c C_k=∀_q C_k (<ref>) and (<ref>) are trivial. (<ref>) follows because a BQP verifier circuit can force all certificates to be classical by measuring each qubit in the standard basis before processing. (<ref>) follows because this class is complementary. (<ref>) follows by a similar argument: take A∈∃_c∃_q C_k, where the classical certificate is of length p_1(|x|), and the quantum certificate is of length p_2(|x|). Clearly A is in ∃_q C_k with certificate length p_1(|x|)+p_2(|x|), since the first p_1(|x|) qubits can be measured before processing, so that they are forced to be computational basis states. The other direction, ∃_q C_k⊆∃_c∃_q C_k, follows trivially by setting the classical certificate length to 0. Then (<ref>) follows from (<ref>) because the classes are complementary. §.§ Quantum hierarchy collapse Our main focus in this paper is on problems in QCMA. Therefore, it is sufficient to adopt the definition of the hierarchy with all certificates classical. Let QPH^q:=⋃_i=0^∞cc⋯c-Σ_i^q. We consider the restricted hierarchy QPH, (N.B., without the `q' superscript). Since each certificate is classical, when we refer to classes at each level we omit the certificate specification, referring to each level as simply Σ_i or Π_i. Also, note that we are considering the computational basis state restriction of each level of the hierarchy so we omit the `q' superscript. We make use of the following lemmas.For all i≥ 1, ∃_c Σ_i=Σ_i and∀_cΠ_i=Π_i. Both follow as corollaries of Lemma <ref>, parts (<ref>) and (<ref>). For all i≥ 1, if Σ_i⊆Π_i or Π_i⊆Σ_i then QPH⊆Σ_i. We prove first that if the equality Σ_i=Π_i held for some i≥ 1 then for all j>i, Σ_j⊆Σ_i. We prove this by induction on j. Consider the base case j=i+1. By definition, if 𝒜∈Σ_i+1 then 𝒜∈∃_cΠ_i=∃_cΣ_i=Σ_i. Assume for the induction hypothesis that if Σ_i=Π_i then Σ_j⊆Σ_i. Let k=j-i+1. For k odd and 𝒜∈Σ_j+1 we have that 𝒜∈∃_c ∀_c⋯∃_c_kΠ_i=∃_c ∀_c⋯∃_c_kΣ_i=∃_c∀_c⋯∀_c_k-1Σ_i=Σ_j. By the induction hypothesis this is a subclass of Σ_i. The case for even k follows in thesame way. Since for all i≥ 0, Σ_i=co-Π_i, we have that if Σ_i⊆Π_i or Π_i⊆Σ_i then Σ_i=Π_i, and so the hierarchy collapses. The following two propositions are important for our purposes, and can be proved using similar techniques to those used in the proofs of AM=BP·NP and AM⊆Π_2^P. We emphasise that the latter is in terms of the quantum polynomial hierarchy, indeed it would be remarkable if a similar result held for in terms of the classical hierarchy. The proofs follow in Sections <ref> and <ref>.QCAM⊆BP·QCMA, and QAM⊆BP·QMA.A corollary of this is the following. QCAM⊆cc-Π_2, and QAM⊆cq-Π_2. In what follows, we will refer to the class QCMAM: a generalisation of QCAM which has an extra round of interaction between Arthur and Merlin. Kobayashi et al. <cit.> show that this class is equal to QCAM. QCMAM=QCAM.The next proposition uses this fact, and allows us to complete the proof of Theorem <ref>. If co-QCMA⊆QCAM thenQPH⊆QCAM⊆Π_2. Let 𝒜=A∈Σ_2. Then by definition there exists a promise problem ℬ=B∈Π_1=co-QCMA and a polynomially bounded function p such that for all |x⟩∈A,∃ y∈{0,1}^p(|x|)|x⟩⊗ |y⟩∈B,and for all |x⟩∈A,∀ y∈{0,1}^p(|x|)|x⟩⊗ |y⟩∈B. If co-QCMA⊆QCAM then ℬ∈QCAM. The existentially (Eq. <ref>) and universally (Eq. <ref>) quantified y's can be thought of as certificate strings, and so 𝒜∈QCMAM. By Theorem <ref>, QCMAM=QCAM, and so 𝒜∈Π_2. Hence, Σ_2⊆Π_2, and the hierarchy collapses to the second level by Lemma <ref>.We now have the tools we need to prove <ref>.Suppose 𝒜∈QCMA∩co-QCAM. If 𝒜 is QCMA-complete then this implies that QCMA⊆co-QCAM, equivalently co-QCMA⊆QCAM. The hierarchy then collapses to the second level via Proposition <ref>. We may now finish this section by providing evidence that StabilizerStateIsomorphism is not QCMA-complete, encapsulated in Corollary <ref>. We do this by proving the following.StabilizerStateNonIsomorphism is in QCAM. For a stabilizer state |ψ⟩, denote by s_ψ^(1),…, s_ψ^(n)∈{± I,± X,± Y,± Z}^n the classical strings that describe the stabilizer generators of |ψ⟩ that we can obtain efficiently using the algorithm of Theorem <ref>. We denote by s_ψ the length 2n string that is obtained by concatenating these stabilizer strings, that is s_ψ = s_ψ^(1)… s_ψ^(n). Then for any permutation σ∈𝔖_n, we take σ(s_ψ) = s_ψ^(σ(1)),…, s_ψ^(σ(n)). For a permutation group G≤𝔖_n, consider the setS_G := ⋃_j∈{0,1},σ∈ G{(σ(s_ψ_j),π) : π∈ Gσ(s_ψ_j) = σ(s_ψ_j)}.If there exists σ such that |⟨ψ_1|P_σ|ψ_0⟩| = 1 then σ(s_ψ_0) = s_ψ_1, and so in this case |S_G| = |G|. If for all σ∈ G we have that |⟨ψ_1|P_σ|ψ_0⟩|≤ 1-ϵ(n) then likewise for all σ∈ G, σ(s_ψ_0) ≠ s_ψ_1 and therefore |S_G| = 2|G|. If we can show that membership in S_G can be efficiently verified by Arthur then we can apply the Goldwasser-Sipser set lower bound protocol <cit.> to determine isomorphism of the states. To convince Arthur with high probability that (σ(s_ψ_j),π)∈ S_G, Merlin sends the permutation σ and the index j∈{0,1}. Arthur can then obtain the string s_ψ_j with probability greater than 1-1/exp(n) using Montanaro's algorithm of Theorem <ref> applied to U_ψ_j|0⟩. He can then verify in polynomial time that the string he received is equal to σ(s_ψ_j), that π is an automorphism of σ(s_ψ_j), and that the permutation σ is in the group G. We have provided evidence that SSI can be thought of as an intermediate problem for QCMA. In particular, we have shown that if it were in BQP, then GraphIsomorphism would also be in BQP, and furthermore, that its QCMA-completeness would collapse the quantum polynomial hierarchy. Such evidence is unfortunately currently out of reach for StateIsomorphism, because we have been unable to show that StateNonIsomorphism is in QCAM. Perhaps Arthur and Merlin must always use quantum communication if Arthur is to be convinced that two states are NOT isomorphic. This would be interesting, because he can be convinced that they are isomorphic using classical communication only (StateIsomorphism∈QCMA). §.§ Proof of Proposition <ref> We begin by giving a definition of the BP complexity class operator. Note that we are still working in terms of the quantum promise problems defined earlier, which is clear from the use of the calligraphic font 𝒜. In the following we take x∼ X to mean that x is an element drawn uniformly at random from a finite set X. Let C be a complexity class. A promise problem A is in BP(a,b)·C for functions a,b:ℕ→[0,1] if there exists B∈C and a polynomially bounded function p:ℕ→ℕ such that * For all |ψ⟩∈A, _y∼{0,1}^p(|x|)[|ψ⟩⊗ |y⟩∈B]≥ a(||ψ⟩|); * For all |ψ⟩∈A, _y∼{0,1}^p(|x|)[|ψ⟩⊗ |y⟩∉B]≤ b(||ψ⟩|). It is clear that the probabilities a,b can be amplified in the usual way by repeating the protocol a sufficient number of times and taking a majority vote. Let ({V_x,y},m,s) be a QAM verification procedure. In what follows we make use of the functionsμ(m,V_x,y):=max_|ψ⟩∈ℋ_2^⊗m(|x|)([V_x,y accepts|ψ⟩])andν(m,V_x,y):=min_|ψ⟩∈ℋ_2^⊗m(|x|)([V_x,y rejects|ψ⟩]).The following results of Marriott and Watrous <cit.> are useful for our purposes. Let a,b:ℕ→[0,1] and polynomially bounded q:ℕ→[0,1] satisfy a(n)-b(n)≥1/q(n) for all n∈ℕ. Then QAM(a,b) ⊆QAM(1-2^-r,2^-r), for all polynomially bounded r:ℕ→[0,1]. Let({V_x,y : x∈{0,1}^*,y∈{0,1}^s(|x|)},m:ℕ→ℕ,s:ℕ→ℕ) be a QAM verification procedure for a promise problem A with completeness and soundness errors bounded by 1/9. Then for any x∈{0,1}^* and for y∈{0,1}^s(|x|) chosen uniformly at random, * if |x⟩∈A then [μ(m,V_x,y)≥ 2/3]≥ 2/3; * if |x⟩∈A then [μ(m,V_x,y)≤ 1/3]≥ 2/3. We can use these tools to prove Proposition <ref>. We prove it for QAM, the result follows for QCAM by similar reasoning. Suppose 𝒜=A∈QAM(a,b). By Theorem <ref>, there exists a QAM verification procedure ({V_x,y},m,s) with completeness and soundness errors bounded by 1/9. Thus by Proposition <ref> we know that for all x∈{0,1}^*, if |x⟩∈A then _y∼{0,1}^s(|x|)[μ(m,V_x,y)≥ 2/3]≥ 2/3, which means that 1/2^s(|x|)|{y∈{0,1}^s(|x|) : ∃ |z⟩∈ℋ_2^⊗ m(|x|)[V_x,y accepts |z⟩]≥ 2/3}|≥ 2/3. By similar reasoning, if |x⟩∈A then 1/2^s(|x|)|{y∈{0,1}^s(|x|) : ∀ |z⟩∈ℋ_2^⊗ m(|x|)[V_x,y accepts |z⟩]≤ 1/3}|≥ 2/3. These conditions are precisely the conditions for a promise problem to belong in QMA. This means we can fix some promise problem B∈QMA(2/3,1/3) and re-express these statements in the following form: * if |x⟩∈A then _y∼{0,1}^s(|x|)[|x⟩⊗ |y⟩∈B]≥ 2/3 * if |x⟩∈A then _y∼{0,1}^s(|x|)[|x⟩⊗ |y⟩∈B]≥ 2/3, and so 𝒜∈BP(2/3,1/3)·QMA(2/3,1/3). §.§ Proof of Proposition <ref> The following well known lemmas allow us to put BP·QMA (resp. BP·QCMA), and thus QAM (resp. QCAM), in the second level of the quantum polynomial-time hierarchy. We follow <cit.> but recast them in a more helpful form for our purposes. For a set of bitstrings S⊆{0,1}^m and x∈{0,1}^m, we take S⊕ x={s⊕ x : s∈ S}. Let S⊆{0,1}^m for m≥ 1 such that |S|≥ (1-2^-k)· 2^m, for 2^k≥ m. Then there exists t_1,…, t_m∈{0,1}^m such that ⋃_i=1^m S⊕ t_i={0,1}^m. We prove this via the probabilistic method. Consider uniformly random t_1,…, t_m∈{0,1}^m. Then _r∼{0,1}^m[r∉⋃_i=1^m S⊕ t_i] =∏_i=1^m_r∼{0,1}^m[r∉ S⊕ t_i]≤ 2 ^-km. Consider the probability that there exists some v∈{0,1}^m such that v∉⋃_i=1^m S⊕ t_i, [∃ v∈{0,1}^m.v∉⋃_i=1^m S⊕ t_i] ≤∑_i=1^2^m 2^-km=2^m/2^km<1. Hence, [⋃_i=1^m S⊕ t_i={0,1}^m]>0, and so there must exist t_1,…, t_m as required.This yields the following corollary. Let S⊆{0,1}^m for m≥ 1 such that |S|≥ (1-2^-k)· 2^m, for 2^k≥ m. Then there exists t_1,…, t_m such that for all v∈{0,1}^m, there exists i∈[m] such that t_i⊕ v∈ S.We also require the following lemma, which comes from the opposite direction. Let S⊆{0,1}^m for m≥ 1 such that |S|≥ (1-2^-k)· 2^m, for 2^k≥ m. Then for all t_1,…, t_m∈{0,1}^m, there exists v∈{0,1}^m such that ⋀_i∈[m](u_i⊕ v∈ S). Assume that there exists t_1… t_m such that for all v∈{0,1}^m there exists i∈[m] with t_i⊕ v∉ S. This implies that there exists i∈{1,…,m} such that, for at least 2^m/m elements v∈{0,1}^m, we have that t_i⊕ v∉ S. Then |S|<2^m-2^m/m=2^m(1-1/m)≤ (1-2^-k)· 2^m, contradicting our assumption about the cardinality of S.We can now prove the Proposition <ref>. We prove it for BP·QMA; the result for BP·QCMA follows in the same way. Let A∈BP·QMA. Then by definition there exists B∈QMA and polynomially bounded p,r:ℕ→ℕ such that if |x⟩∈A,_y∼{0,1}^p(|x|)[|x⟩⊗ |y⟩∈B]≥ 1-2^-r(|x|). Set S_x={y∈{0,1}^p(|x|) : |x⟩∘ |y⟩∈B}. Then |x⟩∈B implies that |S_x|≥ (1-2^r(|x|))· 2^p(|x|). By amplification of BP, we can choose r to be whatever we want, so we choose it such that 2^r(|x|)≥ p(|x|). Then by Lemma <ref>, x∈A∀ t_1… t_p(|x|)∈{0,1}^p(|x|)∃ v∈{0,1}^p(|x|)∃ i∈{1… p(|x|)}|x⟩⊗ |t_i⊕ v⟩∈B. By definition of QMA, for any bitstring y such that |x⟩⊗ |y⟩∈B, ∃ |ψ⟩∈ℋ_2^s(|x|).[Q_x∘ y accepts|ψ⟩] ≥ 2/3. From Lemma <ref> we know we can collapse the classical ∃ quantifiers into the quantum one, obtaining∀_c∃_q. This means that Eq. (<ref>) is of the form required by a promise problem in cq-Π_2. Set S'_x={y∈{0,1}^p(|x|) : |x⟩⊗ |y⟩∈B}. For x∈A, |S'_x|≥ 1-2^-r(|x|)2^p(|x|), for any r via amplification. Then by Corollary <ref> we know that this can be written as a ∃_c ∀_c statement about belonging to B. By definition the membership condition for B is a ∀_q statement. Again, the classical and quantum ∀ statements can be collapsed so we obtain a ∃_c∀_q statement for the NO instances, meaning that A∈cq-Π_2. § ACKNOWLEDGEMENTSThe authors thank Scott Aaronson, László Babai, Toby Cubitt, Aram Harrow, Will Matthews, Ashley Montanaro, Andrea Rocchetto, and Simone Severini for very helpful discussions. JL acknowledges financial support by the Engineering and Physical Sciences Research Council [grant number EP/L015242/1]. CEGG thanks UCL CSQ group for hospitality during the first semester of 2017, and acknowledges financial support from the Spanish MINECO (project MTM2014-54240-P) and MECD “José Castillejo”program (CAS16/00339).9 ladnerR. 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Barak, “Computational Complexity: A Modern Approach” Cambridge University Press (2009). | http://arxiv.org/abs/1709.09622v1 | {
"authors": [
"Joshua Lockhart",
"Carlos E. González Guillén"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170927165452",
"title": "Quantum State Isomorphism"
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headingsLocal Structure Theorems for GraphsJan Dreier, Philipp Kuinke, Ba Le Xuan and Peter Rossmanith Theoretical Computer Science, Dept. of Computer Science, RWTH Aachen University, Aachen, Germany The Sirindhorn International Thai-German Graduate School of Engineering,King Mongkut’s University of Technology North Bangkok, ThailandLocal Structure Theorems for Graphs and their Algorithmic Applications Jan Dreier1 Philipp Kuinke1 Ba Le Xuan2 Peter Rossmanith1 December 30, 2023 ====================================================================== We analyze local properties of sparse graphs, where d(n)/n is the edge probability. In particular we study the behavior of very short paths. For d(n)=n^o(1) we show thathas asymptotically almost surely () bounded local treewidth and therefore is nowhere dense. We also discover a new and simpler proof thathas bounded expansion for constant d. The local structure of sparse is very special: The r-neighborhood of a vertex is a tree with some additional edges, where the probability that there are m additional edges decreases with m. This implies efficient algorithms for subgraph isomorphism, in particular for finding subgraphs with small diameter. Finally, experiments suggest that preferential attachment graphs might have similar properties after deleting a small number of vertices. § INTRODUCTION One of the earliest and most intensively studied random graph models is the model <cit.>. Graphs from this class are usually depicted as a random variable G(n,p), which is a graph consisting of n vertices where each pair of vertices is connected independently uniformly at random with probability p. The edge probability p may also depend on the size of the graph, , p=d/n. Many properties of are well studied including but not limited to, threshold phenomena, the sizes of components, diameter, and lengths of paths <cit.>. One particular impressive result is the 0-1 law: Let φ be a first-order formula.If we take a random graph G=G(n,1/2), then the probability of Gφ is either 0 or 1 as n→∞ <cit.>.“Instead of the worst case running time, it is also interesting to consider the average case. Here even the most basic questions are wide open.” as Grohe puts it <cit.>. One can find an optimal coloring of G(n,p) in expected linear time for p<1.01/n <cit.>. The 0-1 law on the other hand has (not yet) an efficient accompanying algorithm that can decide whether Gφ for G=G(n,1/2) and a fixed formula φ.One possibility to open up a whole graph class to efficient algorithms are algorithmic meta-theorems.Such meta-theorems were developed for more and more general graph classes: planar, bounded genus, bounded degree, H-minor free, H-topological minor free .In all these graph classes we can decide properties that are expressible in first-order logic in linear time for a fixed formula φ <cit.>. Unfortunately, random graph classes do not belong to any of these classes. For example G(n,1.1/n) has linear treewidth and does contain constant-size cliques of arbitrary size <cit.>. Recently, however, graph classes of bounded expansion were introduced by and Ossona de Mendez <cit.>.These classes also admit linear time FO-model checking and generalize the older meta-theorems <cit.>. The most general model checking algorithm runs in time O(n^1+ϵ) on nowhere-dense classes <cit.>. In , the value d is the expected density of a random graph. For constant d it was shown thathas bounded expansion <cit.>. Unfortunately, this does not automatically imply that one can test first-order properties onin linear (expected) time, but only that we can test such a property in linear time with a failure probability of o(1) while the expected runtime might be unbounded. This is for example the case if the runtime grows faster than the failure probability converges to zero. One example of an (expected-time) fpt-algorithm is one that finds a k-clique in G(n,p(n)) in time f(k)n^O(1), for many choices of p <cit.>.In Section <ref> we find an easier proof for the fact thathas bounded expansion for constant d and give concrete probability bounds, which were missing up to now. Then we investigate local properties of . The expected density of G(n,d(n)/n) is d(n) and therefore, if d(n) is not constant, unbounded. This implies that G(n,d(n)/n) does not have bounded expansion. Nevertheless, we show that subgraphs with small diameter are tree-like with only a few additional edges. From this it follows that G(n,n^o(1)/n) has locally bounded treewidth, which implies that they are nowhere dense. Locally bounded treewidth <cit.> and more generally, locally excluding a minor <cit.> are useful concepts for developing first-order model checking algorithms that run in time O(n^1+ϵ).We discussed that a random graph class that is nowhere dense or has bounded expansion may not directly admit efficient algorithms. It is known that one can check first-order properties inin time O(g(|φ|)n^1+o(1)) for d(n) = n^o(1) and some function g <cit.>. For constant d one can check first-order properties in time O(g(|φ|)n). In Section <ref> we use the locally tree-like structure of to construct an efficient algorithm for subgraph isomorphism. We show that one can find a subgraph H with h vertices and radius r inin time 2^O(h)(d(n)log n)^O(r)n, while a naive algorithm may need time O(d(n)^h n). Therefore, our method may be faster for finding large pattern graphs with small radius.It can be argued that are not a good model for real-world networks and therefore efficient algorithms for admit only limited practical applications. Recently, there were more and more efforts to model real world networks with random graph models. One candidate to meet this goal were the , which use a preferential attachment paradigm to produce graphs with a degree distribution that tries to mimic the heavy-tailed distribution observed in many real-world networks <cit.>.This model is particularly interesting from the point of mathematical analysis because of its simple formulation and interesting characteristics, which is why they have been widely studied in the literature <cit.>. It was also shown that this model does not have bounded expansion <cit.>.In Section <ref> we discuss experiments to see how similar the local structure of is to .Not surprisingly, it seems that they are quite different and the formercontain dense subgraphs and are likely to be somewhere dense. If we, however, remove the relatively small dense early part of these graphs, the local structure of the remaining part looks quite similar to and indicators hint that the remaining part is indeed nowhere dense. As the dense part is quite small it gives us hope that hybrid algorithms exist that combine different methods for the dense part and the structurally simple part.To search for a subgraph H, for example, could be done by guessing which vertices of H lie in the dense part and then using methods from Section <ref> to find the remaining vertices in the simple part.§ PRELIMINARIES In this work we will denote probabilities by […] and expectation by […]. We use common graph theory notation <cit.>. For a graph G let V(G) be its vertex set and E(G) its edge set. For v ∈ V(G) we denote the r-neighborhood of v by N_r(v). The degree of a vertex v in graph G is denoted by deg(v). We write G^'⊆ G if G^' is a subgraph of G. For X ⊆ V(G) we denote by G[X] the subgraph of G that is induced by the vertices in X. The graph G[V(G)-X] obtained from G by deleting the vertices in X and their incident edges, is denoted by G-X. The treewidth tw(G) of a graph is a measure how tree-like a graph is. We denote graphs by a random variableand distinguish between graphs with constant d and graphs , where we allow d to grow (slowly) with n. We will use various ways to measure the sparsity of a graph or graph class.A graph M is an r-shallow topological minor of G if M is isomorphic to a subgraph G' of G if we allow the edges of M to be paths of length up to 2r+1 in G'.We call G' a model of M in G. For simplicity we assume by default that V(M) ⊆ V(G') such that the isomorphism between M and G' is the identity when restricted to V(M). The vertices V(M) are called nails[also known as principal vertices] and the vertices V(G') ∖ V(M) subdivision vertices.The set of all r-shallow topological minors of a graph G is denoted by Gr. With that we can define the clique size over all topological minors of G asω(Gr) = max_H∈ Gr ω(H).For a graph G and an integer r ≥ 0, the topological grad at depth r is defined as_r(G) = max_H ∈ Gr|E(H)|/|V(H)|For a graph class 𝒢, define _r(𝒢) = sup_G∈𝒢_r(G).A graph class 𝒢 has bounded expansion if and only if there exists a function f such that _r(𝒢) < f(r) for all r≥ 0.A graph class 𝒢 has locally bounded treewidth if and only if there exists a function f, such that for all r≥ 0 every subgraph with radius r has treewidth at most f(r).A graph class 𝒢 is nowhere dense if there exists a function f such that ω( Gr ) < f(r) for all G∈𝒢 and all r ≥ 0. If a graph class has locally bounded treewidth it is also nowhere dense <cit.>.§ LOCAL STRUCTURE AND ALGORITHMIC APPLICATIONS In this section, we observe the local structure of and how to exploit it algorithmically. It is already known that have bounded expansion if the edge probability is d/n for constant d <cit.>. We present a simpler proof via a direct method, that also gives concrete probability bounds. The original proof did not give such concrete bounds so we feel that this new proof has applications in the design of efficient algorithms. To make our calculations easier we assume that d≥2, since are only sparser for smaller d, our techniques will also work in this case. §.§ Bounded Expansion The technique we use to bound the probability that certain shallow topological minors exists is to bound the probability that a path of length at most r exists between two arbitrary vertices. Let p_r be the probability that there is a path of length at most r between two arbitrary but fixed vertices in . It holds thatd/n≤ p_r ≤2d^r/n. Since all edges are independent, we do not need to identify the start and end vertices of the path. We prove by induction over r that the probability of the existence of a path of length exactly r is bounded by d^r/n. For r=1 the statement holds: p_1 ≤d/n.The probability of a path of length r is at most that of some path of length r-1 times the probability of a single edge:p_r ≤∑_k=0^n p_r-1 p_1 ≤∑_k=0^n d^r-1d/n^2≤d^r/nBy using the union bound and assuming that d≥ 2, the joint probability is bounded by 2d^r/n. Having this bound in place, we can show thathas no r-shallow topological minors of large density from which it follows that they are contained in a graph class of bounded expansion .is contained in a graph class of bounded expansion.In particular, for d≥16 the probability that such a random graph contains some r-shallow topological minor of size k and at least 8kd^2r+1 edges is at most max{n^-2k,2^-n^2/3}. For d<16 the same result holds for at least 8k16^2r+1 edges.We will now investigate the probability that a random graph G= contains some model of an r-shallow topological minor H with nail set v_1,…,v_k.Such a model consists of the nails themselves and vertex-disjoint paths of maximal length q=2r+1 between them.Each such path models one edge of H.Assume that V(H)={u_1,…,u_k} and that u_i is modeled by v_i.Then an edge u_iu_j∈ E(H) is modeled by a path from u_i to u_j in G and all these paths are vertex-disjoint.What is the probability that such a model exists? A first path exists with a probability of p_q. The probability that the second path exists under the condition that it does not cross all candidates for the first path is slightly less than p_q. Continuing this argument shows that the probability of finding such a model is at most p_q^|E(H)| and more specifically it is at most the probability of getting |E(H)| heads after |E(H)| independent coin tosses with a head probability of p_q.Moreover, this implies that the probability of finding a model for some H with m edges is at most the probability of getting at least m heads after tossing k2 such coins.Let X be the sum of k2 independent Bernoulli variables with [X=1]=p_q. Using the bounds of Lemma <ref> we have4k^2/n≤k^2d/n≤[X] ≤k22d^q/n≤k^2d^q/n. Using Chernoff bounds with δ=8n/k-1 we get[X>(1+δ)k^2d^q/n] ≤[X>(1+δ) [X]] ≤(e^δ/(1+δ)^1+δ)^[X]≤(ek/8n)^32k.This means for fixed k nails, with probability of at most (ek/8n)^32k the graph G contains a model for an r-shallow topological minor with these nails and at least (1+δ)k^2d^q/n = 8kd^q edges. The density of such a topological minor is therefore 8d^q. There are only n k≤(ne/k)^k possibilities to choose the nails, so an r-shallow topological minor with k nails and density at least 8d^q exist with a probability of at most (ne/k)^k(ek/8n)^32k, which is n^-2k if k≤ n^2/3.For bigger k it is bounded by 2^-n^2/3. Therefore, every r-shallow topological minor in G has a density of at most 8d^q.§.§ Locally Simple Structure It is known that even for constant d the treewidth ofgrows with Ω(n) <cit.>. Furthermore, G(n,d(n)/n) does a.a.s not have bounded expansion if d(n) is unbounded. We now show that G(n,n^o(1)/n) nevertheless has locally bounded treewidth and thus is nowhere dense. We start by counting the expected number of occurrences of a certain subgraph in . The expected number of induced subgraphs with k vertices and at least k+m edges inis at most k^2k+2md(n)^k+m/n^m. There are nk≤ n^k induced subgraphs H of size k in G. For each such H there are k2k+m≤ k^2k+2m ways to choose k+m edges.The probability that these k+m edges are present in H is then exactly (d(n)/n)^k+m and the probability that H has k+m edgesis at most k^2k+2m(d(n)/n)^k+m.Finally, the expected number of such induced subgraphs is at most k^2k+2md(n)^k+m/n^m.From Lemma <ref> we can conclude a well known property of : The expected number of cycles of fixed length r is O(d(n)^r) (which is a constant if d is constant) by setting k=r and m=0. We now use this Lemma to make statements about the density of neighborhoods.The probability that there is an r-neighborhood inwith m more edges than vertices is at most f(r,m)d(n)^2r(d(n)^2r+1/n)^m for some function f. Consider any r-neighborhood with ℓ vertices. Assume the neighborhood contains at least m more edges than vertices. Let T be a breadth-first search spanning tree of this neighborhood. Since T contains ℓ vertices and ℓ-1 edges, there are m+1 edges which are not contained in T. Each extra edge is incident to two vertices. Let U be the set of these vertices. Let H be the graph induced by the union of the m+1 extra edges and the unique paths in T from u to the root of T for each u∈ U. Since |U| ≤ 2(m+1) and each path to the root in the breadth-first-search tree T has length at most r, the number of vertices of H is bounded by 2r(m+1).In summary, if there exists an r-neighborhood with at least m more edges than vertices then there exists a subgraph with k ≤ 2r(m+1) vertices and m more edges than vertices. But according to Lemma <ref>, the expected number of such subgraphs is bounded by((2r(m+1))^2d(n))^2r(m+1)+m/n^m = f(r,m)d(n)^2r(d(n)^2r+1/n)^m.This also bounds the probability that such a subgraph exists.Let d(n) = n^o(1).Thenhas locally bounded treewidth. The show that a graph has locally bounded treewidth we have to show that the treewidth of every r-neighborhood is bounded by a function of r alone.Since d(n) = n^o(1), there exists a monotone decreasing function g(n) with d(n) ≤ n^g(n) and lim_n→∞ g(n) = 0. Let h(r) be the inverse function of 1/8g(r). Since g(n) is monotone decreasing, h(r) exists and is monotone increasing. We show that for all r≥0 every subgraph with radius r has treewidth at most h(r). We distinguish between two cases. The first case is r < 1/8g(n) and f(r,1)<n^1/4.According to Lemma <ref>, an r-neighborhood of G has more edges than vertices with probability at mostf(r,1)d(n)^4r+1/n≤ f(r,1)n^g(n) (4 1/8g(n)+1)-1≤ f(r,1)n^-1/2 + g(n) = o(1)We can conclude that every r-neighborhood has treewidth at most 2.The second case is r≥1/8g(n), which means h(r) ≥ n, so even the treewidth of the whole graph is bounded by h(r) and the third case is given by f(r,1)≥ n^1/4 and the (total) treewidth is bounded by f(r,1)^4. Altogether, the treewidth of an r-neighborhood is bounded by 2, by h(r), or by f(r,1)^4. § ALGORITHM FOR SUBGRAPH ISOMORPHISM In this section we solve Subgraph Isomorphism, which given a graph G and a graph H asks, whether G contains H as a subgraph. This is equivalent to FO-model checking restricted to only existential quantifiers.Let H be a connected graph with h vertices and radius r. In this section we discuss how fast it can be decided whethercontains H as a subgraph. We first discuss the runtime of simple branching algorithms on graphs and how exploiting local structure may lead to better run-times. We discovered that if the radius r of the pattern graph is small, an approach based on local structure is significantly faster.For low-degree graphs there exists a simple branching algorithm to decide whether a graph G contains H as a subgraph in time O(Δ^h n), where Δ is the maximal degree in G. Let us first assume that d(n)=d is constant. There is nevertheless a non-vanishing probability that the maximal degree ofis as large as √(log n). Therefore, the maximal degree cannot be bounded by any function of d. This implies that a naive, maximal degree based algorithm may have at least a quasi-linear dependence on n, while we present an algorithm which has only a linear dependence on n.Let us also assume that that d(n) is of order logn and even that the maximum degree is bounded by O(d(n)). A naive branching algorithm may therefore decide whethercontains H in expected time O(d(n))^h n. We improve this result, not making any assumption about the maximal degree, by replacing the factor O(d(n))^h in the runtime with 2^O(h)(d(n)log n)^O(r), where r is the radius of H. For graphs with small radius, the runtime is no longer dominated by a factor O(d(n))^h.The new algorithm may be significantly smaller when d(n) is, for example, of order log n.So far we only discussed connected subgraphs. Using color-coding techniques, the results in this section can easily be extended to disconnected subgraphs, where the radius of each component is bounded by r. Color-coding may, however, lead to an additional factor of c^h in the runtime: Assume H has c components where the size of H is h. We want to color each vertex of G uniformly at random. Assume G contains H, then the probability that every component of H can be embedded using vertices of a single color is at least 1/c^h. So if H can be embedded in G we will answer yes after an expected number of c^h runs.For the following result notice that if d(n) is poly-logarithmic in n the runtime is quasi-linear in n. For d(n)=n^o(1) the dependence on n is n^1+o(1). The algorithm is given in the proof for Theorem <ref>. Inholds with probability of at least 1-n^-1/4log(n) that every r-neighborhood has size at most log(n)^2r d(n)^r. The Chernoff Bound states for the degree D of an individual vertex that [D ≥x] ≤ e^-(1/3x/d(n)-1)d(n) and therefore [D ≥log(n)^2 d(n)] ≤ n^-1/4log(n). Let D̂ be the maximal degree of the graph. With the union bound we have a similar bound for D̂. Every r-neighborhood has size at most D̂^r.Let H be a connected graph with h vertices and radius r.There is a deterministic algorithm that can find out whether H occurs as a subgraph inin expected time2^O(h)(d(n)log n)^O(r)n. We sketch the algorithm briefly.The algorithm works on a graph G=.In the following we assume that every r-neighborhood in G has size at most d(n)^rlog(n)^2r. By Lemma <ref> this assumption holds with a probability of at least 1-n^log(n)/4 and we can easily check it within the stated time bounds. Should the assumption be wrong, we can use a brute force algorithm without affecting the average running time.In a preprocessing step we look at the connected graph H and construct a subgraph H' that is also connected, but consists only of a tree with two additional edges (if possible, otherwise we set H'=H).We enumerate all r-neighborhoods in G and try to find H in every one of them as follows:By using color-coding we enumerate all subgraphs in the r-neighborhood that are isomorphic to H'.This can be done by using the algorithm forfinding a graph of bounded treewidth <cit.> with the enumeration techniques in <cit.>. The expected time needed is 2^O(h)(d(n)log(n))^O(r) times the number of subgraphs that are found.However, by Lemma <ref> the latter number is bounded by a constant.After enumerating all subgraphs isomorphic to H' we have to find out whether G contains H as a subgraph.If this turns out to be true, then H can be found only somewhere where H' was found.Hence, it suffices to look at all found H' in G and see whether by adding a subset of the possible h2 edges we can find H.This can be done in time O(2^h^2d(n)^rlog(n)^2), which is asymptotically faster than the remaining part.§ EXPERIMENTAL EVALUATION OF -GRAPHS In the previous section, we showed that have bounded expansion for edge probability p=d/n (with constant d) and are nowhere dense with p = n^o(1)/n. In this section, we discuss the sparsity of the model. It is known that this model has not bounded expansion, because it contains an unbounded clique with non-vanishing probability <cit.>. It is not known, however, if it is (or is not) somewhere-dense. Our experiments seem to imply that on average seem to be dense but that this density is limited to early vertices: In the model, vertices with high degree tend to be preferred for new connections. This means that edge probabilities are not independent. Moreover, the expected degree d(i) = √(n/i) for a vertex i is less uniform than it is for , where d(i)=pn.To evaluate the expansion properties of the -model, we compute transitive fraternal augmentations and p-centered colorings. These have been introduced by and Ossona de Mendez, and are highly related to bounded expansion and a tool for developing new and faster algorithms. A graph class has bounded expansion if and only if the maximum in-degree of transitive fraternal augmentations is bounded, or the graph admits a p-centered coloring with bounded number of colors. Let G be a directed graph. A 1-transitive fraternal augmentation of G is a directed graph H with the same vertex set, including all the arcs of G and such that, for any vertices x, y, z, * if (x, z) and (z, y) are arcs of G then (x, y) is an arc of H (),* if (x, z) and (y, z) are arcs of G then (x, y) or (y, x) is an arc of H ().A transitive fraternal augmentation of a directed graph G is a sequence G_1 ⊆…⊆G_i ⊆…⊆G_n, such that G_i+1 is a 1-transitive fraternal augmentation of G_i. For an integer p, a p-centered coloring of G is a coloring of the vertices such that any connected subgraph H induced on the vertices of an arbitrary set of i colors (i ≤ p), H must have at least one color that appears exactly once. Showing that the maximum in-degree of a transitive fraternal augmentation or the number of colors needed for a p-centered coloring does not grow with the size of the graph is a way to prove that a graph has bounded expansion <cit.>. When designing algorithms, p-centered colorings can be used to solve hard problems efficiently. By using p-centered colorings, we can decompose a graph into small, well-structured subgraphs such that -hard problems can be solved easily on each subgraph before combining these small solutions to get a solution for the entire graph. It is important that the number of colors needed for a p-centered coloring for a fixed p is small, as the runtime usually is a function of the number of colors needed. If a graph class does not have bounded expansion; that is, the number of colors grows with n, but very slowly, such as loglogn, using these algorithms might still be practical.One example problem which can be solved directly using p-centered colorings is Subgraph Isomorphism, where one asks if a graph H is contained in a graph G as a subgraph. In general graphs, this problem is -hard when parameterizing by the size of H <cit.>. However, there exist an algorithm, whose runtime is a function of the number of colors needed for a p-centered coloring, where p depends on the size of H <cit.>. So, regardless of the fact whether are theoretically sparse or not, calculating the number of colors of a p-centered coloring for different graph sizes has direct impact on the feasibility of a whole class of algorithms on these graphs.§.§ Experiment OverviewWe analyze the expansion properties of by computing transitive fraternal augmentations and p-centered colorings. In the following, we describe the heuristics used to compute these. In order to compute the transitive fraternal augmentations of a graph G, the graph is oriented to a directed graph G_1 by using low-degree orientation, in which every edge (u, v) in G is transformed to an arc (u, v) in G_1 if the degree of u is greater than the degree of v. Then, transitive fraternal augmentations are applied to G_1, which yield a sequence G_1, G_2, …, G_i. The augmentation heuristic we used was proposed in earlier work <cit.>: To build graph G_i from G_i-1 we need to perform transitive fraternal augmentations. First we create the set F of fraternal edges of G_i-1. Let G_F be the graph induced by F. Now we can orient the edges of G_F by the same low-degree orientation performed earlier to get the directed fraternal edges F that are added to G_i-1 and result in G_i. Now we can color the undirected graph G_i of G_i by iterating through the vertices in a descending- degree order and assign each vertex the lowest color that does not appear in its neighborhood. We then check whether that coloring is a p-centered coloring of the input graph G. If this is not the case we repeat this procedure for G_i+1. §.§ Graphs are Empirically Dense First, we analyze the maximum in-degree of transitive fraternal augmentations. We ran the previously described algorithm on random with d = 2 for different sizes (500 ≤ n ≤ 3000) and calculated the maximum in-degree of up to five transitive fraternal augmentation steps. The results are shown in Figure <ref>. Each data point is an average over ten runs with the same n. For all graphs both the maximum in-degree grows with n, which would not be the case for graphs with bounded expansion.To evaluate how well the expansion properties of can be practically exploited, we analyzed the number of colors needed to construct p-centered colorings. We constructed 3- and 4-centered colorings. with the same graph parameters and sizes than before. The results are shown in Figure <ref>. For the analyzed range, the number of colors needed grows steadily. Furthermore, the number of colors needed to construct 4-centered colorings is substantially higher than the number of colors needed for 3-centered colorings. Computing higher order colorings or colorings for larger graphs was infeasible with the used algorithm. It seems practically impossible to use p-centered colorings algorithmically for . We have to note that the used algorithm is only a heuristic and the real values might be much better than what we have computed. But since these heuristics work well for graphs that have low treedepth colorings, it is unlikely that the graphs have bounded coloring number for p-centered colorings. §.§ Density Seems Limited to Early VerticesPreviously, we showed that the colors needed to construct p-centered colorings of small graphs can be very high. In this section we discover that the early vertices of the random process heavily affect these results. We remove the first 10% of the vertices added in the random process and analyze the maximum in-degree of transitive fraternal augmentations and number of colors needed to construct p-centered colorings. By removing those 10%, we can construct p-centered colorings for much larger graphs (5000 ≤ n ≤ 30000), see Fig. <ref>. The required number of colors for p-centered colorings and maximum in-degree of transitive fraternal augmentations remain stable and do not seem to depend on the number of vertices. This suggests that these 10% of the early vertices contain almost all of the density of . This is of course a linear factor and it remains to see if one can use much smaller functions of n, like for example logn. The sizes of the graphs at hand, however, were not large enough to investigate sub-linear functions of n with a meaningful result.§ CONCLUSION In this work we gave an alternative proof thathas bounded expansion and have shown thatwith d(n) = n^o(1) has locally bounded treewidth. Our results are based on the fact that local neighborhoods of are tree-like with high probability. It is known <cit.> that for a graph G = with d(n)=n^o(1) and a first-order formula φ one can decide whether G φ in expected time f(|φ|)n^1+o(1) for some functions f and g. This result can also be proven using our techniques. It remains to show whether it is possible to answer this question in linear expected fpt-time (where d(n)n is the expected number of edges), O(f(|φ|)d(n)n). In this paper, we also presented a more efficient algorithm for the subgraph isomorphism problem on if the pattern graph has small radius. It would be interesting to consider other measures for the pattern graph as well, such as treewidth or treedepth. Furthermore, we gathered empirical evidence which suggests that are somewhere dense. It would be interesting to prove this conjecture. splncs | http://arxiv.org/abs/1709.09152v2 | {
"authors": [
"Jan Dreier",
"Philipp Kuinke",
"Ba Le Xuan",
"Peter Rossmanith"
],
"categories": [
"cs.DM"
],
"primary_category": "cs.DM",
"published": "20170926174100",
"title": "Local Structure Theorems for Erdos Renyi Graphs and their Algorithmic Application"
} |
Physics and Astronomy Department, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 Physics and Astronomy Department, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 Space Telescope Science Institute, Baltimore, MD 21218 Physics and Astronomy Department, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 Department of Physics, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa Physics and Astronomy Department, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 Physics and Astronomy Department, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 We present a simple method for identifying candidate white dwarf systems with dusty exoplanetary debris based on a single temperature blackbody model fit to the infrared excess. We apply this technique to a sample of Southern Hemisphere white dwarfs from the recently completed Edinburgh-Cape Blue Object Survey and identify four new promising dusty debris disk candidates. We demonstrate the efficacy of our selection method by recovering three of the four Spitzer confirmed dusty debris disk systems in our sample. Further investigation using archival high resolution imaging shows Spitzer data of the un-recovered fourth object is likely contaminated by a line-of-sight object that either led to a mis-classification as a dusty disk in the literature or is confounding our method. Finally, in our diagnostic plot we show that dusty white dwarfs which also host gaseous debris lie along a boundary of our dusty debris disk region, providing clues to the origin and evolution of these especially interesting systems.§ INTRODUCTION The now firmly established link between remnant exoplanetary systems, compact circumstellar dust disks, and atmospheric heavy metal pollution in white dwarf stars has demonstrated that exoplanetary science has much to gain by continuing studies post-mortem. In contrast with their dead host stars, the surviving exoplanetary systems are dynamically active, with large outer planets scattering smaller rocky bodies into disruptively compact orbits, providing a source of rocky material for both the observed compact debris disks and the otherwise unexpected atmospheric metals <cit.>. These post main-sequence exoplanetary systems have revealed detailed rocky exoplanetary abundances <cit.>, complex debris disk dynamics <cit.>, and even transits of actively disrupting bodies <cit.>. Despite these exciting discoveries, statistical analyses of the white dwarfs with observable exoplanetary debris disks suggest our picture is far from complete. The frequency of white dwarfs with exoplanetary systems as observed via accretion of heavy metals in white dwarf atmospheres is highly discrepant from the frequency of observable debris disks around white dwarf stars (see <cit.> for a recent review), suggesting that majority of the sources of the accreting material remain undetected <cit.>. The continued discovery of new white dwarf debris disk systems will expand the range of exoplanetary accretion phenomena we observe, and once our understanding of their origin and evolution is mature, allow us to translate the observed sample properties through the initial-final mass relationship to improve our understanding of planetary formation around stars like our sun (e.g. <cit.>). The large number of new white dwarf stars expected to be discovered by the Gaia space telescope offers the chance to increase the sample of white dwarf exoplanetary systems by more than an order of magnitude, once the dusty debris disk systems can be identified and confirmed <cit.>.The process of identifying candidate dusty debris disk systems in the modern astronomical era of cross matching and database mining is simple: given a list of white dwarf stars, a spectral energy distribution can be constructed entirely from publicly available photometry, then compared against atmospheric models to reveal targets with excess infrared radiation. Unfortunately the infrared signature of dusty debris disks is only prominent beyond 2μm, forcing data miners to rely heavily on the all-sky survey of the Wide-field Infrared Survey Explorer (hereafter WISE, <cit.>). While the photometry from WISE is well calibrated, its large imaging beam leads to a high probability of un-resolved source contamination <cit.>, and the two longest wavelength survey passbands, W3 and W4 at 12 and 22μm, are not sufficiently deep for studies of white dwarf debris disks, leaving us with only one or two significant points to discern the excess. In the past astronomers have turned to space-based observatories such as the Spitzer Space Telescope to confirm white dwarf debris disk candidates at higher spatial resolution and longer wavelengths than can be observed from the ground, which has continued to provide observations well into its warm mission <cit.>. This has worked well for small samples of interesting targets, but is impractical for the hundreds of new dusty infrared excess candidates expected from Gaia. In this work, as an extension of the WISE InfraRed Excess around Degenerates Survey (hereafter WIRED, <cit.>), we present a simple but robust method for identifying the most promising dusty debris disk candidates based on the best-fitting effective temperature and radius of a single temperature blackbody. This technique is well suited to handle the sparse infrared excess points for studies that rely on WISE photometry. We apply this technique to the set of hydrogen atmosphere (DA) white dwarfs identified in the recently completed Edinburgh-Cape Blue Object Survey (hereafter EC Survey, <cit.>, <cit.>). This sample of Southern Hemisphere white dwarfs is relatively bright (V ≤ 17.5) and provides a good proxy for some of the issues that can be expected and will need to be overcome for the sample of bright white dwarfs from Gaia, which will initially have little spectroscopic data and varying photometric coverage. We present four new, promising dusty debris disk candidates, and identify a known dusty debris disk hosting white dwarf as an outlier with our technique, which we show to have a nearby contaminant that is unresolved in previous Spitzer studies. We also find that among the Spitzer-confirmed dusty debris disk hosting white dwarfs, those that also host an observable gaseous component lie along the boundary of our dusty debris disk selection region, confirming a relationship between the infrared disk luminosity and its propensity to host gas. § TARGET SELECTION AND COLLECTED PHOTOMETRY In this study we focus on hydrogen atmosphere (DA) white dwarfs identified in the EC Survey. The EC Survey utilized U-B colors to select candidate objects and relied on follow-up photometry and low-resolution spectroscopy to classify each blue object <cit.>. The authors follow the identification scheme described in <cit.> to identify common white dwarf types, and note that the broad spectral features of white dwarfs make them easy to classify. Of the 2,637 unique hot objects identified in the EC survey, we find that 489 have been designated as type DA or possible type DA (e.g. DA?, DAweak, etc.). Candidates that have an uncertain but possible DA spectral type are also included in this study and are discussed later in the context of possible contaminants. For each target, we extract additional photometry from the GALEX All-sky Imaging Survey GR5 (<cit.>, hereafter GALEX), AAVSO Photometric All-Sky Survey DR9 (<cit.>, hereafter APASS), 2MASS All-Sky Point-Source Catalog <cit.>, VISTA Hemisphere Survey (<cit.>, hereafter VHS), and the AllWISE Data Release of the Wide-Field Infrared Survey Explorer <cit.>. Data collection for the EC Survey began in the 1980s so in order to minimize source mis-identification while cross-matching our targets across nearly three decades worth of surveys, we also collected proper motions for our targets from the PPMXL catalog <cit.>. Using the J2000 epoch from PPMXL, we queried each photometric catalog for sources within 2.5of the proper motion corrected target position, corrected to the mid-point of each survey's data collection period. To ensure we have selected the correct PPMXL source, we use a method similar to that described in <cit.>, which essentially selects all nearby sources from the proper motion catalog of choice, then corrects their positions to the known epoch of the target before automatically selecting the nearest source. Unfortunately, the later releases of the EC Survey are increasingly lacking in epochs for each object coordinates, so the procedure was modified to enable user selected sources. To do this, we overlaid the J2000 corrected EC target coordinates and proper motion projections of all PPMXL sources within 15on POSS2 imaging plates <cit.>. This search returned 3 or less PPMXL sources for 468/489 candidates with many having only a single nearby source, leading to simple, unique source identifications based on proximity to the target coordinates. 4/489 targets had no PPMXL sources within 15. For the remaining 17/489 targets where multiple sources were found near the target coordinates, we selected the PPMXL source most consistent with both the EC Survey position and the measured EC Survey B magnitude. These results suggest that up 5% of our candidates could be mis-identified, potentially leading to spurious infrared excess selections or classifications, which we discuss in later sections.It is well known that the large PSF of the WISE beam (∼ 6.0 in W1) can lead to contamination from nearby sources, and care must be taken to ensure the measured AllWISE fluxes are consistent with a single source <cit.>. To identify targets with potentially contaminated WISE photometry, we collected cutouts of survey images from the VISTA-VHS and VST-ATLAS <cit.> catalogs in K_s and z bands from the VISTA Science Archive. The VISTA Data Flow System pipeline processing and science archive are described in <cit.>, <cit.>, and <cit.>. When both images were available, the VHS K_s images were preferred as the photometric band is much closer to the WISE photometry. Examples of images from each catalog are shown in Figure 1. Overplotted on each image is the position of the white dwarf target as identified in the EC Survey, and the position of the corresponding AllWISE detection, including a 7.8circle around the WISE position, which is the approximate limit of the automatic deblending routine used for the ALLWISE pipeline. The imaging circle allowed us to quickly identify and flag targets with potentially contaminated WISE photometry. Each target was assigned an image quality flag based on the results of studying the collected images by eye, which is included in the summary tables in the appendix. Targets that were identified as having potentially contaminated WISE photometry were not excluded and should not be ruled out without more careful analysis of the contaminating source, but their WISE excess should be given more scrutiny than those with clean images.§ WHITE DWARF MODEL FITTING AND INFRARED EXCESS IDENTIFICATION Our first step in identifying systems with an infrared excess is fitting the collected photometry of each target with a white dwarf model. We use a grid of hydrogen atmosphere white dwarf models, kindly extended to include GALEX and WISE photometry by P. Bergeron <cit.>. To ensure the model and collected photometry were on the same magnitude scale, we applied zero-point offsets to the GALEX, EC, APASS, 2MASS, and VHS magnitude[VHS magnitudes were transformed to the 2MASS system using the color-color equations described at http://casu.ast.cam.ac.uk/surveys-projects/vista/technical/photometric-properties] as defined in <cit.> and <cit.>. Each transformed magnitude was then converted to flux density units using published zero points <cit.>. Our photometric uncertainties are derived from reported catalog values, and we assume a 5% relative flux uncertainty floor. Compared with previous WIRED surveys, our sample has a few unique features; first, our targets do not have a consistent set of optical measurements which are needed to anchor the white dwarf model photometry. The U, B, and V band photometry from the EC Survey is incomplete and the APASS survey is ongoing, leading to sporadic coverage across the different photometric bands. The second is that fewer than half of our white dwarfs have a prior spectroscopic effective temperature and surface gravity determination, the latter of which is often necessary to split the degeneracy between the solid angle subtended by the white dwarf and its photometric distance. To address these issues, for our photometric white dwarf model fits we fixed the surface gravity of our model atmosphere grid to logg=8.0 (g measured in cm s^-2). For each model in our grid, we determine an initial flux scaling based only on the available optical data (0.4μm≤ λ ≤0.7μm). The photometric scale factors were then transformed to initial distance estimates, and following the prescription of <cit.>, we apply photometric reddening corrections to our photometry for all sources beyond 100pc. The white dwarf model was then re-fit to the corrected photometry. The best-fitted model was chosen by minimizing the chi-square metric as computed for each scaled model using all photometry at wavelengths below 1.0 μm. As discussed in Section 4.2, for white dwarfs identified with strong stellar excesses that obviously extended into the near-infrared and optical, we limited the photometry used to determine the best-fit to wavelengths below 0.5 μm. For those targets with prior spectroscopic logg and T_ eff solutions from either <cit.> or <cit.>, we assumed the spectroscopic logg and T_ eff for our white dwarf atmospheric parameters, and generated model photometry scaled to the observed, de-reddened photometry with the method described above.We compare our photometrically derived effective temperatures to the spectroscopic determinations for apparently single objects where we have both in Figure 2, with the spectroscopic surface gravity displayed as a colorscale. The most egregious outliers on Figure 2, shown as open symbols, are cases where fewer than 3 photometric points were available to constrain the photometric fit. We note that none of our new infrared excess candidates suffered from this severe lack of data. Using the scatter in the relationship, we can establish uncertainties for our white dwarfs which only have photometric fits. Below 15,000 K, the fits are generally good, with an uncertainty of ∼ 1000 K. Between 15,000–30,000 K, the scatter is greater, resulting in an uncertainty of ∼ 3000 K. Above 30,000 K, the photometric fits are generally unreliable which reflects a lack of short wavelength optical and ultra-violet photometry needed to constrain the bluer SEDs. Despite the agreement below 30,000 K, there are still a handful of 3σ outliers given the uncertainties above, all of which exhibit a bias toward lower photometric temperatures. We find the culprit to be sporadically poor U band photometry from the EC Survey, examples of which can be seen in the SEDs of EC 02566-1802 and EC 23379-3725 in Figures 4 and 9 respectively. Unfortunately, we found no way to determine a priori if the EC U band photometry was poor, and therefore cannot correct for it in this sample. Based on the number of 3σ outliers in Figure 2 below 30,000 K we estimate the poor U band photometry to be affecting less than 10% of our sample.Targets showing a 5σ excess in either the W1 or W2 bands or a 3σ excess in the W1 and W2 bands were flagged as infrared excess candidates. These criteria flagged 111 out of 378 white dwarfs with AllWISE detections as infrared excess candidates. These candidates comprise the sample discussed in the remainder of the paper. § INFRARED EXCESS CLASSIFICATION While the selection of infrared excess candidates is straightforward, classification of the infrared source without a clean separation of the SEDs can be misleading. For programmatic searches of infrared excesses like the prior WIRED studies <cit.>, the root of this problem is the shortage of infrared excess data points, which in our case is exacerbated by an incomplete near-infrared dataset. Techniques for classifying excesses as dusty debris disks are particularly lacking. Conventional color-color selection <cit.> can miss the subtle infrared excesses that likely comprise a majority of dusty debris disks <cit.>, and it does not make use of all of the available information gained from fitting a model white dwarf atmosphere to the observed photometry, namely the photometric distance. More complex models can in theory distinguish between stellar and dusty infrared excesses <cit.>, but reduced chi-square metrics are often degenerate between stellar and dusty classifications given the limited number of infrared excess points constraining the models. Because of these concerns, and our uniquely deficient photometry, we sought a simple technique that could distinguish between whether the excess is consistent with a dusty debris disk, an unresolved stellar or sub-stellar companion, or a background contaminant, and that would quickly highlight the best candidates for follow-up studies.The simplest model that describes the infrared excess is a single temperature blackbody assumed to be at the photometric distance of the white dwarf star. Assuming the white dwarf atmospheric parameters and photometric distances derived above, we fit a single temperature blackbody to the observed infrared excess for each infrared excess candidate in our sample, with only the blackbody effective temperature and radius as free parameters. Figure 3 shows the results of the single temperature blackbody fits for our entire sample, plotted as the effective temperature versus radius of the infrared source as scaled to the white dwarf radius. It is important to keep in mind that what is actually being fitted is a ratio of the solid angle subtended by the single temperature blackbody source to that of the white dwarf star, and so errors in the assumed white dwarf radius and distance, particularly for those with photometric atmospheric solutions where we have assumed a surface gravity, propagate into this measurement. Figure 3 should not be interpreted as giving an accurate description of the temperature and radius of the infrared source, particularly for dusty debris disk candidates which are neither perfectly circular nor at a single temperature, but rather it can be used as guidance for selecting targets of interest. To help guide the reader, we also plot in the background as light grey squares the effective temperature and radius for the stellar and sub-stellar models of <cit.> and <cit.>, which extend from early M dwarf stars down to sub-stellar and late type brown-dwarf stars, scaled to a typical logg=8.0 white dwarf radius. Finally, we also show the parameters for single temperature blackbody fits to all known dusty debris disks confirmed with Spitzer, independently fitted by <cit.>, in the background as grey diamonds. We have derived the blackbody radius used in the fits of <cit.> from the white dwarf effective temperature, blackbody temperature, and fractional infrared luminosity from their Table 3, assuming the radius of a logg=8.0 white dwarf, consistent with what the authors used when fitting their single temperature blackbodies.We also performed a literature search for all of the objects in our sample and found 6 with published infrared excesses identified as dusty debris disks, and 44 with published infrared excesses identified as stellar or sub-stellar companions. We indicate these on Figure 3 with blue hexagons (dusty disks) and red stars (stellar companions) and see that, in general, these groups occupy distinct regions of the plot. We define three regions of interest in this plot. Region I is defined as a region of low effective temperature (T < 2000K) andvarying radius. We see that the 5/6 known dusty debris disks in our sample cluster in this region in the lower left corner of our plot. The one identified dusty debris disk in our sample which does not follow this trend, PG 1457-086, is discussed in more detail in section 4.3. While the 5 literature-identified dusty debris disks in this region provide a nice set of boundaries for selecting new dusty debris disk candidates within our sample, the single temperature blackbody fits of the sample of 35 Spitzer confirmed dusty white dwarf systems from <cit.>, overplotted as grey diamonds, provide an independent view of the extent of the dusty debris disk region. In Region II, known stellar companions to white dwarfs congregate at higher temperatures and radii, in a locus around 3000K and 30R_ WD (0.4R_⊙), which are the temperature and radii expected for an unresolved M dwarf type companion. Region II extends down into the low temperature and small radius regime of Region I, where the overlap between dusty debris disks and late-type stellar and sub-stellar companions forces us to a less certain conclusion about the source of the infrared excess. Objects in Region III, which consists of infrared excess best reproduced by objects of higher temperature (T > 3500K) and small radius R < 10 R_ WD (0.1R_⊙), have no obvious source, but are likely the result of mis-classification or contamination. We discuss each region and the objects they contain below.§.§ Region I: Compact, Dusty Debris Disks In addition to the empirical boundaries given by the single-temperature blackbody fits to the known dusty debris disks, there is a corresponding theoretical expectation that dusty debris disks should congregate in this region. The formation of dusty debris disks via the tidal disruption of asteroids suggests they should not extend well beyond the asteroid tidal disruption radius at 1.0R_⊙, or 85R_ WD for typical white dwarf masses around 0.6 M_⊙ and asteroid densities ∼ 2 g/cm^-3 <cit.>. At their inner edge, the cm to micron sized dust is only expected to be able to survive at temperatures below 2000 K before sublimating into gas <cit.>. Since the dust within this region is expected be optically thick, the majority of it is shielded from direct radiation and it's temperature falls off rapidly with distance from the white dwarf <cit.>, with the outer dust near the tidal disruption radius radiating at only a few hundred degrees kelvin. The temperature of the single-temperature blackbody fit along the x-axis of Figure 3 can be thought of as an area weighted temperature average of the dust disk, which given the expected inner and outer boundaries should be between 500-1500K depending on the width of the disk. The radius of the single-temperature blackbody plotted along the y-axis is less straightforward to interpret as it is dominated by the inclination of the dust disk. There is some overlap with the stellar candidates defined by Region II and the bottom of Region I. We choose to list the objects in this overlapping region as dusty debris disk candidates. Because the new dusty debris disk candidates are of particular interest to this study, we provide the spectral energy distributions with the single temperature blackbody fits for each new dusty debris disk candidate in Figure 4 and in the appendix in Figure 9. Table 1 provides a summary of the propertiesof all objects in Region I, including new debris disk candidates, known debris disk systems recovered by our search, and candidates we have chosen to reject as dusty debris disks based on either an obviously poorly fit SED or an independent physical reason, detailed in the Rejected Candidates section of the appendix. We also performed extensive literature searches on each object in this region and provide notes, additional data, and recommendations for follow-up on each object. In the remainder of this section, we highlight our follow-up on four promising candidates which appear firmly inside of Region I, have high resolution spatial follow-up which suggests their WISE photometry is unlikely to be contaminated, and have optical spectroscopy to search for atmospheric metal pollution. The signature of recent accretion is not necessarily independent confirmation that an observed WISE infrared excess is consistent with a dusty debris disk. Recent studies have shown that the frequency of atmospheric pollution consistent with metal rich exoplanetary debris accretion could be as high as 25% <cit.>. However, to date every confirmed dusty debris disk hosting white dwarf has been demonstrated to be actively accreting metal rich debris <cit.> so limits on the atmospheric pollution from optical spectra can be used as an easy way to prioritize dusty debris disk candidates. The most commonly detected transition in the optical is the Ca2 K resonance line at 3934Å<cit.>. For each object we provide measurements or conservative upper limits to equivalent widths of the Ca2 K line based on archival or collected spectra. Ultimately, only higher spatial resolution, longer wavelength observations, or low-resolutionnear-infrared spectroscopy can confirm WISE excesses as necessarily due to dust. EC 01071-1917: Also known as GD 685 and WD 0107-192. The best-fitted single temperature blackbody parameters place it well away from the overlapping stellar models of Region II. The ATLAS z band image is free of nearby contaminants. This object was also targeted as part of the SPY survey <cit.>, from which we have adopted the spectroscopic effective temperature and surface gravity (T_ eff=14,304 K, logg=7.8) for our white dwarf model parameters in the SED fit. We collected and combined the pipeline reduced archival SPY spectra using the ESO Science Archive to search for atmospheric Ca. We detect a subtle Ca K absorption feature shown in Figure 5 with an equivalent width of 28.0± 9.0mÅat a heliocentric corrected velocity of 1.6±1.4km/s, which is consistent with the white dwarf photospheric velocity of 1.5±0.9km/s as measured with a gaussian fit to hydrogen alpha NLTE line core. EC 02566-1802: Also known as HE 0256-1802 and WD 0256-180. The best-fitted single temperature blackbody parameters place it well away from the overlapping stellar models. ATLAS z band imaging is free of nearby contaminants. We also obtained additional K_s follow-up with the SPARTAN infrared camera on the SOAR telescope which confirm the lack of nearby sources, but are unable to provide additional calibrated photometry due to poor observing conditions. We adopt a spectroscopic temperature of 26,120 K and surface gravity of logg=7.76 from the SPY survey <cit.> for our SED fits, which agrees with our independent photometric temperature. The white dwarf effective temperature is high, but not prohibitively so, for white dwarfs which host dusty debris disks. At this temperature, the optical spectra are less useful as probes for atmospheric pollution due to higher ionization states of atmospheric metals with transition wavelengths in the ultra-violet <cit.>. Nevertheless we place an upper limit to the eqw Ca2 K line of 17.5mÅ. A nearby absorption feature of ∼ 21.0mÅwas detected at 10.3±1.1km/s, but is inconsistent with the white dwarf photospheric velocity of 25.5±1.7km/s, and is likely interstellar. EC 03103-6226: Also known as WD 0310-624. The white dwarf model is well constrained and the best-fitted single temperature blackbody parameters place it within the region of known dusty debris disks, though there is some overlap with the stellar models of Region II. VHS K_s band imaging is free of nearby contaminants. This object was identified as a white dwarf candidate and followed up spectroscopically by <cit.>, where it is noted that the difference in spectroscopic and photometric temperatures are suggestive of it being an unresolved double-degenerate candidate. Our independent photometric temperature is slightly hotter than their photometric fit (15,250 K vs 13,900K) and in better agreement with their spectroscopic temperature of ∼ 17,000K. Considering that both estimates of the photometric temperature assume a surface gravity of logg=8.0, the remaining discrepancy could be within the photometric temperature error budget. We also obtained high signal-to-noise optical spectroscopy on 2015 November 18 with the Goodman Spectrograph on SOAR. We used the 0.46slit in combination with the 1800 l/mm grating to cover a wavelength range from 3740 Åto 4580 Åwith a resolving power of R ∼ 7000. No absorption features are detected near Ca K, resulting in an upper limit of 46.5mÅ. EC 21548-5908 The white dwarf model is well constrained by multiple near-infrared photometry points (2MASS and VHS). VHS K_s band imaging is free of nearby contaminants. To confirm the atmospheric parameters, we obtained low resolution optical spectroscopy with the Goodman Spectrograph on SOAR covering the hydrogen Balmer series from Hβ blueward. Using the techniques described in <cit.>, we fit the spectra to a grid of hydrogen atmosphere white dwarf models kindly provided by D. Koester <cit.>, and determined a spectroscopic effective temperature and surface gravity of 12,330 K and logg=8.04, which are consistent with the photometric fit. We also obtained high signal-to-noise optical spectroscopy on 2015 October 18 with the Goodman Spectrograph on SOAR, using the same instrument setup described for EC 03103-6226. No absorption features are detected near Ca K, resulting in an upper limit of 65.9mÅ.§.§ Region II: Unresolved Stellar/Sub-Stellar Companions The results of objects that fall within Region II are summarized in Table 2. We provide examples of SEDs of objects with the stellar classification in Figure 11 of the . We note that for the brighter stellar excesses which begin in the optical wavebands, our white dwarf models often attempted to fit some of the additional optical flux from the unresolved companion, resulting in white dwarf model fits that were systematically overluminous, and single temperature blackbody fits that were systematically underluminous, pulling the single temperature blackbody fits to lower temperatures. To account for this, we re-fit the white dwarf models restricting our photometry to wavelengths less than 0.5 μm. The positions of objects in Figure 3 and the fitted parameters given in Table 2 reflect our best-fitted values after this correction was applied. More detailed modeling is necessary to determine the stellar companion spectral type, and is beyond the scope of this paper.§.§ Region III: High Tempertaure, Small Radius The third region of Figure 3 consists of objects with an infrared excess that is best fitted by a single temperature blackbody with a high temperature (T>4000K) and small radius (R<10R_ WD). The results of objects that fall within this region are summarized in Table 3. Examples of objects in this region are shown in the appendix in Figure 12. The nature of the objects populating this region is less obvious than the other two regions. Furthermore, since Region III includes one of the known dusty debris disks in our sample, PG 1457-086, we took great care in investigating this region. We propose these objects are the result of some combination of the following four scenarios.1) Erroneous stellar classification/poor photometry/poor WD model fit: The first thing that stands out when looking through the results in Table 3 is that 6/15 of the objects have an uncertain EC spectral type and 5 of them are identified as potential hot sub-dwarfs. With uncertainty about the blue object classification, we can no longer rely on our DA white dwarf atmospheric models to accurately predict the photometric flux in the near-infrared. 5/9 remaining objects have high signal-to-noise spectral follow-up and atmospheric model parameter fits from either the Gianninas or Koester spectroscopic surveys so they cannot be accounted for by misclassification. Finally, if the white dwarf model is fitted at a higher temperature than the true white dwarf temperature, the difference in slope can result in the excesses observed in this section. This appears to be the case for at least one of the white dwarfs where we have assumed a spectroscopic temperature, EC 10188-1019, as evidenced by the highly discrepant GALEX photometry. Our fitted photometric temperature is much lower than the reported spectroscopic temperature from <cit.> (11,000 K vs 17,720 K), and the difference in temperature completely accounts for the observed infrared excess.We note that EC 10188-1019 is flagged by <cit.> as being magnetic, which is likely affecting the spectroscopic temperature as non-magnetic models were assumed for the fits. As discussed in both <cit.> and <cit.>, the Balmer features used to determine spectroscopic atmospheric parameters peak in strength around 13,000-14,000 K, and fits of white dwarfs near this temperature often suffer from a hot/cold solution degeneracy across this boundary. Photometric fits are one way to break this degeneracy. 3 white dwarfs in this region have spectroscopic temperatures near this boundary, the magnetic white dwarf EC 10188-1019 discussed above, EC02121-5743, and EC 22185-2706. Cooler white dwarf models could explain the observed excess.2) Irradiated sub-stellar/planetary mass companion: The best-fitted radii of objects in this region are consistent with Jupiter and brown-dwarf sized companions, but the temperatures require a substantial amount of additional heating. The post-main sequence evolution of the white dwarf progenitor is expected to result in planetary re-heating via accretion and irradiation <cit.>, but the temperatures needed to fit the infrared excesses in this region (T>4000K) are well beyond what is expected. Furthermore, the thermal relaxation timescale of the re-heated planets is on the order of hundreds of millions of years, meaning the re-heated planets would only be expected around the youngest white dwarfs in this sample <cit.>. There are a handful of confirmed white dwarf-brown dwarf binaries in compact orbits which suggest that, despite engulfment, the brown-dwarf survives post-main sequence evolution relatively unscathed <cit.>. The compact orbits lead to tidally synchronous orbits and significant differences between dayside and nightside brown-dwarf surface temperatures <cit.>. Despite the strong irradiation, the dayside temperatures (∼ 3000K) are still too cool to explain the excesses seen in this region <cit.>. Nonetheless, if these excesses are the result of irradiated brown-dwarf companions, there should be observable spectral features in the near-infrared and optical, and brightness modulations from tidal and reflection effects from the companion. The lack of these features could quickly rule out compact brown-dwarf companions. Finally, it is also worth pointing out that previous studies have found the brown dwarf companion fraction to white dwarfs to be low. In a search for binary companions that included near-infrared direct imaging and near-infrared excess techniques, <cit.> find the white dwarf brown dwarf companion fraction to be < 0.5%. In a previous WIRED study, <cit.> find the observed frequency to be between 1.3±0.6% after accounting for likely contaminants. Even in the most optimistic case, in our sample of 383 white dwarfs with WISE detections we should only expected 7-8 white dwarf brown dwarf systems, which is not sufficient to explain all of the observed excesses in this region.3) Unresolved contaminants: Another way to produce the subtle excess is with an unresolved, line-of-sight object which is at a different distance than the white dwarf. Even with high quality near-infrared follow-up from the VISTA-VHS survey, care must still be taken to confirm the excesses seen in this section are not the result of a contaminant. We find EC 14572-0837, also known as PG 1457-086 (hereafter EC 14572), to be an example of contamination by an unresolved background source.EC 14572 is identified in the literature as a dusty debris hosting white dwarf, with strong atmospheric metal pollution <cit.>. The infrared excess was confirmed by Spitzer and determined to be most consistent with a dusty debris disk. EC 14572 is included in the sample of Spitzer confirmed dusty debris disks identified by <cit.>EC 14572 is the only published dusty debris disk that we failed to correctly identify in our single temperature blackbody selection. Because this might indicate a short-coming in our dusty debris disk selection technique, we performed a thorough archival data search to determine if any additional data could help resolve the discrepancy between the infrared excess as seen when scaling to the optical versus near-infrared photometry. Our ESO archive search revealed that EC 14572 was a target in a multi-epoch, high-contrast and high-spatial resolution imaging search for Jupiter sized planets around dusty debris disk hosting white dwarfs with the NAOS+CONICA near-infrared imager on the VLT as a part of program 085.D-0673(A) led by M. Radiszcz. The high quality J band imaging reveals a close contaminant. Figure 6 presents two epochs of imaging from this study for EC 14572, taken 3 years apart, centered on the brighter object, with approximate separation measurements. The change in separation between the two epochs of 0.126is consistent with the direction and magnitude of the proper motion of EC 14572 as measured by the PPMXL survey (μ_α=2.4 mas yr^-1 and μ_δ=-38.8 mas yr^-1), suggesting the two objects are not in a common proper motion pair. We performed aperture photometry on the two sources and find the flux ratio between the white dwarf and the contaminant to be 3.2±0.6, which is agreement with the excess flux above the white dwarf model observed in the J band. The best-fitted blackbody that can explain the excess has a temperature of 4400 K.Any additional flux from the unforeseen companion is certainly contaminating our near-infrared and WISE data, and is very likely present in the Spitzer data of <cit.>. Though <cit.> scaled the white dwarf model photometry to the near-infrared data when modeling the excess as a dusty debris disk, effectively including the near-infrared flux of the unresolved contaminant in their stellar model, if the contaminant is significantly cooler than the white dwarf the difference in slope of the spectral energy distribution at these wavelengths could be solely responsible for the infrared excess measured for EC 14572. Given the difference in proper motion between the white dwarf and contaminant, increased separation should allow future follow-up to independently measure the near-infrared flux of the white dwarf and the contaminant, and definitively resolve the source of the excess infrared radiation.§ LOCUS OF GASEOUS DEBRIS HOSTING DISKSAs a by-product to our search for new dusty debris disks, we also found an interesting relation among dusty debris disks which are also known to host gaseous debris in emission. Circumstellar gaseous emission has been observed in the optical spectra of 8 white dwarfs which also dusty debris disks <cit.>. The gaseous debris is believed to be spatially coincident with the dusty debris <cit.>, and the interaction between the gas and dust is likely to play large role in the evolution and accretion of the dust disk <cit.>. The double-peaked emission calcium triplet emission features exhibited by these disks lend themselves to more detailed dynamical modeling than can be accomplished with the infrared excesses of dusty debris disks <cit.>, and several have been shown to be variable on timescales of decades <cit.>. Figure 7 shows an expanded region of our Figure 3 with the blackbody fits from <cit.>. It is worth reiterating here that the disks themselves are not spherical, and the best fitted “radius” is a proxy for apparent surface area, which is affected by inclination and the inner/outer disk radius. In red diamonds with black outlines we highlight all of the literature identified gaseous debris systems within the <cit.> sample. We have also been surveying known dusty debris disk hosting white dwarfs for calcium triplet emission, the most common tracer of gaseous debris in these systems, and we include our non-detections as filled grey diamonds. Our Ca triplet observations are being carried out with the Goodman Spectrograph <cit.> on the SOAR telescope, using the 1200 l/mm grating and the 1.07slit with a wavelength coverage of 7900 Åto 9000 Åand routinely reach a signal-to-noise of ∼ 40 per 1.2Åresolution element. Examples of non-detections are shown in Figure 8. Open grey diamonds are systems we have not yet surveyed. The white dwarfs with gaseous and dusty debris all lie along the terminus of the dusty white dwarf region of the blackbody fit plot. In other words, for any projected surface area, the dusty disks with gaseous debris congregate at the highest temperatures. This is not surprising as the high temperature side of the dust disk region should be defined by the dust sublimation temperature. It seems natural then that gas disks appear most frequently in systems with copious amounts of dust at the sublimation temperature.Another interpretation is that for a given temperature the white dwarfs with gaseous debris host the largest, and therefore most luminous dusty debris disks. The observation that white dwarfs with gaseous and dusty debris tend to have brighter dusty debris disks is not novel <cit.>, but we find it particularly interesting in the context of the results from the dynamical modeling that has been performed on the emission profiles. Typically, high inclinations (i > 60^∘) are needed to reproduce the large velocity dispersion and deep inner regions of the Ca emission profiles <cit.>. This is difficult to reconcile with the brightness of the infrared dusty components, as all other things considered equal, one would expect the low inclination, face-on dust disks to be the highest luminosity disks. The implication is that systems which host gaseous debris in combination with dusty debris may not display equivalent, flat geometry as those without gaseous debris. This could be expected if the gas was collisionally produced, perhaps during a recent disruption or collision with an existing disk as described in <cit.>. § CONCLUSIONSThe EC Survey has provided a number of new, bright, spectroscopically confirmed white dwarf stars in the Southern Hemisphere, which we have surveyed for infrared excesses. The challenges of extending the WIRED techniques to a survey with incomplete spectroscopic and photometry were discussed, and a new technique for separating dusty debris disk candidates from stellar companion candidates based on single temperature blackbody fits to the excess radiation, which yields four new promising dusty debris disk candidates. We emphasize however that all infrared excesses discussed in this paper should be considered as candidates until independently confirmed. The selection of dusty debris candidates via single temperature blackbody fits works in a uniform way with good to poor photometry, and should prove useful for Gaia white dwarf infrared excess studies. Gaia searches will benefit greatly from the independent distance estimates and the precise, space-based G band flux measurement, which can be used to anchor the white dwarf model photometry.Along the way, we identified EC 14572 as an outlier among the literature identified dusty debris disks, and an archival search reveals high-contrast, high-spatial resolution imaging that suggests the observed excess could be contaminated by an unresolved contaminant. It remains to be seen if the companion can account for all of the observed infrared excess, or if the system still requires a dusty debris disk. We also identified the gaseous debris disk hosting white dwarfs on the blackbody temperature and radius plane, and find that they form the terminus for dusty debris disks, providing clues to their origin and evolution.We would like to thank the anonymous referee for a detailed review which greatly improved this manuscript. E. Dennihy, J. C. Clemens, P. C. O'Brien, and J. T. Fuchs acknowledge the support of the National Science Foundation, under award AST-1413001. D. Kilkenny acknowledges financial support from the National Research Foundation of South Africa. This work is based on data obtained from (1) the Wide-Field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory (JPL), California Institute of Technology (Caltech), funded by the National Aeronautics and Space Administration (NASA); (2) the Two Micron All Sky Survey, a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center (IPAC)/Caltech, funded by NASA and the National Science Foundation (NSF); (3) the ESO Science Archive Facility (4) the Southern Astrophysical Research (SOAR) telescope, which is a joint project of the Ministério da Ciência, Tecnologia, e Inovação (MCTI) da República Federativa do Brasil, the U.S. National Optical Astronomy Observatory (NOAO), the University of North Carolina at Chapel Hill (UNC), and Michigan State University (MSU); (5) the VizieR catalog access tool, CDS, Strasbourg, France; (6) the NASA/IPAC Infrared Science Archive, which is operated by JPL, Caltech, under a contract with NASA; (7) the NASA Astrophysics Data System. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.§ NOTES ON REMAINING DUSTY DEBRIS DISK CANDIDATESEC 00169-2205: Also known as GD 597 and WD 0016-220. This object has a tenuous excess, which does not continue into the W2 band, although the error bars on W2 are large. It was included as part of a high resolution imaging survey by <cit.> to uncover low luminosity companions to white dwarfs and no companion was detected. EC 00169-2205 was also included in the <cit.> search for metals in white dwarfs via the ca2 K line. No Ca was detected, and <cit.> provides an upper limit to the Ca K equivalent width of <10 Å, which at this temperature, corresponds to an atmospheric abundance upper limits of [Ca/H]<-10. This would be an unusually low abundance for an object with an infrared bright dust disk <cit.> and we therefore believe the detected excess to be the result of poor WISE photometry.EC 01129-5223: Also known as JL 237. This object has a tenuous excess, which does not continue into the W2 band, although the error bars on W2 are quite large and the excess does begin in the Ks band which is high quality. The VHS K_s band image available is free of nearby contaminants. The spectral energy distribution is not very well constrained by the limited optical photometry, but the departure from blackbody in the K_s band and W1 band are statistically significant. EC 05276-4305: The excess is small but the white dwarf model is well constrained by the multiple near-infrared data points (2MASS and VHS). The VHS K_s band image is free of nearby contaminants. EC 13140-1520: Also known as LP 737-47 and WD 1314-153. This object has a tenuous excess, and no high spatial resolution imaging exists to search for nearby contaminants. EC 20036-6613: The white dwarf model is well constrained by the multiple optical photometry data points but the object is lacking in near-infrared photometry. We obtained follow-up K_s band imaging with the SPARTAN infrared camera on the SOAR telescope, which shows a nearby source likely contaminating the WISE photometry.EC 21010-1741: The white dwarf model is well constrained by the high quality VISTA infrared data points, but the infrared excess is not consistent with a single source between W1 and W2. The VHS K_s band image available shows a potential nearby contaminant, which is likely the source of the infrared excess.EC 21459-3548: The white dwarf model is not well constrained by the limited available photometry, particularly in the near-infrared. ATLAS z band imaging reveals a nearby contaminant that is too close to be resolved in WISE photometry.EC 23379-3725 The best-fitted single temperature blackbody parameters place it well away from the overlapping stellar models. The discrepant near-infrared data present some concern for contamination, but the VHS K_s band imaging is free of nearby contaminants.Rejected Candidates: We chose to reject 9 candidate dusty debris white dwarfs for a variety of reasons. Their SEDs are shown in Figure 10. EC 00323-3146 and EC 04552-2812 were designated spectral type 'DAwk' in the EC catalog, indicating narrow/weak Balmer lines preventing clear DA white dwarf classification. EC 19442-4207 is identified in the EC catalog as a CV/DAe, indicating the system is a cataclysmic variable <cit.>. Both EC 12303-3052 and EC 11023-1821 were followed up with Spitzer by <cit.> and found to have no infrared excess, indicating the WISE excesses are the result of contamination. EC 04114-1243 and EC05024-5705 both have white dwarf temperatures which are too high for optically thick dust to survive sublimation within their tidal disruption radius <cit.>. EC 04139-4029 and EC 04516-4428 are both are strong outliers in Figure 3. Their spectral energy distributions are suggestive of source confusion, particularly the highly discrepant near-infrared data from 2MASS and VHS. The higher spatial resolution images for both objects from VHS show nearby sources, indicating the WISE excesses are very likely the result of contamination.§ EXAMPLE SPECTRAL ENERGY DISTRIBUTIONS WITH SINGLE TEMPERATURE BLACKBODY FITS§ TABLES OF CANDIDATES FROM EACH REGIONlcccccccccr Region I: Dusty White Dwarf Candidates 11 EC Name Right Ascension Declination V EC SptypeWD T_ eff WD logg BB T_ eff BB Rad Im FlagRef(J2000) (J2000) (mag)(K) (cm^-2) (K) (R_ WD) New Candidates: 00169-2205 4.8676032 -21.817987 15.33 DA 13264.0a 7.78a 1675.0 2.0 20 1 01071-1917 17.3880821 -19.0216157 16.16 DA 14304.0a 7.79a 537.5 39.0 10 1 01129-5223 18.7553121 -52.1286677 16.47 DA 22000.0 8.0 1725.0 5.0 00 1 02566-1802 44.7483175 -17.8387548 16.51 DA 26212.0a 7.76a 575.0 49.0 10 1 03103-6226 47.8357894 -62.2545421 16.05 DA 16000.0 8.0 1050.0 11.0 00 1 05276-4305 82.3005907 -43.0595245 16.10 DA 13000.0 8.0 1150.0 5.0 00 1 13140-1520 199.1818104 -15.5976591 14.86 DA3 16152.0a 7.72a 1862.5 3.0 20 1 20036-6613 302.099844 -66.0769588 15.91 DA 24250.0 8.0 1500.0 8.0 20 1 21010-1741 315.9666489 -17.4904601 16.57 DA 18750.0 8.0 975.0 13.0 01 1 21548-5908 329.5997382 -58.8983987 15.75 DA 12000.0 8.0 1037.5 9.0 00 1 23379-3725 355.1531593 -37.1454489 16.18 DA 13250.0 8.0 837.5 18.0 00 1 Previously Identified: 04203-7310 64.9071296 -73.0622893 15.61 DA 19000.0 8.0 1125.0 19.0 20 2 05365-4759 84.473051 -47.9679045 15.63 DA 22250.0 8.0 1037.5 16.0 01 311507-1519 178.3133581 -15.6099161 16.00 DA5 12132.0a 8.03a 862.5 31.0 20 4 21159-5602 319.9006626 -55.8370306 14.27 DA 9625.0a 8.01a 900.0 10.0 00 5 22215-1631 336.0727067 -16.263386 15.45 DA 9937.0a 8.16a 925.0 6.0 00 6 Rejected Candidates: 00323-3146 8.7072847 -31.4978588 16.09 DAwk 36965.0a 7.19a 1225.0 11.0 10 1 04114-1243 63.4385557 -12.5944927 16.70 DA 50000.0 8.0 1500.0 16.0 20 1 04139-4029 63.9157837 -40.3757732 16.23 DA 43000.0 8.0 825.0 83.0 01 1 04516-4428 73.3031017 -44.394365 15.35 DA 16750.0 8.0 662.5 92.0 01 1 04552-2812 74.3051311 -28.1312704 13.98 DAwk 54386.0a 7.68a 1975.0 4.0 20 7 05024-5705 75.8461981 -57.0227396 16.22 DA 44000.0 8.0 1237.5 14.0 01 1 11023-1821 166.1945881 -18.6200168 15.99 DA5 8057.0a 7.85a 962.5 4.0 20 8 12303-3052 188.2520412 -31.1432902 15.81 DA2 22764.0a 8.28a 1150.0 6.0 20 1 19442-4207 296.9186198 -42.0074873 10.38 CV/DAe 9500.0 8.0 1325.0 5.0 01 9 aSpectroscopic parameters from <cit.> bSpectroscopic parameters from <cit.> References: (1) This Paper; (2) <cit.>; (3) <cit.>; (4) <cit.>; (5) <cit.>; (6) <cit.>; (7) <cit.>; (8) <cit.>; (9) <cit.> Under the column Im Flag, the first bit refers to the quality of imaging, with objects that have VHS K_s band-images receiving a 0, VST-ATLAS z band images a 1, and those without follow-up images a 2. The second bit refers to the potential for contamination. Objects which appeared as single stars were assigned a 0, those with one or more potential contaminants within the 7.8circle were assigned a 1. lcccccccccr Region II: White Dwarfs with Stellar/Sub-Stellar Excesses 11 EC Name Right Ascension Declination V EC SptypeWD T_ eff WD logg BB T_ eff BB Rad Im FlagRef(J2000) (J2000) (mag)(K) (cms^-2) (K) (R_ WD) New Candidates: 00050-1622 1.8951187 -16.0922234 16.29 DA 15141.0a 7.59a 2837.5 6.0 10 1 00166-4340 4.775084 -43.4051481 15.53 DA 7250.0a 8.0a 2812.5 6.0 00 1 00286-6338 7.7279283 -63.3624649 15.23 DA 19750.0 8.0 3225.0 10.0 00 1 00370-4201 9.8542309 -41.7470551 16.37 DA 10750.0 8.0 3875.0 1.0 00 1 00594-5701 15.3797183 -56.7644763 16.55 DA 13000.0 8.0 3187.5 9.0 00 1 01077-8047 17.0733956 -80.5236636 14.47 DA 5500.0 8.0 2912.5 4.0 20 1 01176-8233 19.3494814 -82.3011543 16.43 DA 17750.0 8.0 3737.5 17.0 20 1 01346-4042 24.2001527 -40.4593443 16.36 DA/DAB 13500.0 8.0 2950.0 17.0 00 1 02223-2630 36.1504952 -26.2812935 15.68 DA 23198.0a 7.91a 2087.5 6.0 10 2 02434-1254 41.4726889 -12.7056873 15.05 DA 29250.0 8.0 3275.0 11.0 10 1 03155-1747 49.4484126 -17.601521 16.48 DA 22750.0 8.0 2875.0 32.0 10 1 03378-8348 53.0676708 -83.6389586 16.24 DA 36750.0 8.0 3387.5 11.0 20 1 04094-3233 62.838107 -32.4373756 16.01 DA 18250.0a 8.0a 3037.5 16.0 20 1 04233-2822 66.3363893 -28.255434 16.48 DA 10907.0a 8.07a 2087.5 6.0 20 1 04310-3259 68.2266693 -32.8872894 17.6 DA 3750.0 8.0 2600.0 2.0 20 1 04365-1633 69.6966934 -16.4545871 16.03 DA 14092.0a 7.96a 2825.0 2.0 20 1 04567-2347 74.714622 -23.7150737 16.62 DA 23645.0a 7.79a 3175.0 5.0 20 1 05089-5933 77.4280658 -59.4939338 15.78 DA 28500.0 8.0 3750.0 61.0 00 1 05230-3821 81.1923171 -38.3099344 16.55 DA 18250.0 8.0 2712.5 11.0 20 1 05237-3856 81.3667125 -38.903283 16.17 DA 15750.0a 8.0a 2937.5 15.0 20 1 05387-3558 85.1301338 -35.9572189 13.97 DA 13250.0 8.0 3925.0 2.0 20 1 05430-4711 86.09625 -47.1715794 15.97 DA 8000.0 8.0 2650.0 9.0 00 1 12204-2915 185.7709471 -29.5410766 15.79 DA3 17702.0a 7.89a 2387.5 2.0 20 1 13123-2523 198.7660094 -25.6497229 15.69 DA1 75463.0a 7.68a 3425.0 17.0 20 1 13324-2255 203.7936553 -23.1771076 16.30 DA3 20264.0a 7.86a 2125.0 6.0 20 1 14265-2737 217.3638143 -27.8498806 15.92 DA3 18087.0a 7.66a 3062.5 2.0 20 1 14361-1832 219.744523 -18.7615606 16.56 DA? 29250.0 8.0 3737.5 8.0 20 1 19272-7152 293.2369964 -71.7669411 15.92 DA 20250.0 8.0 3525.0 12.0 20 1 20453-7549 312.7909386 -75.6400976 16.05 DA 25750.0 8.0 2500.0 3.0 20 1 20503-4650 313.4347278 -46.6575692 15.76 DAwk 19250.0 8.0 4100.0 10.0 01 1 21053-8201 318.3144577 -81.8191237 13.63 DA 10600.0a 8.24a 3587.5 1.0 20 1 21105-5128 318.4991085 -51.2753573 16.68 DA 16500.0 8.0 3262.5 4.0 01 1 21161-2610 319.7694033 -25.9702383 16.10 DA 24750.0 8.0 3400.0 2.0 01 1 21188-2715 320.4373582 -27.0364774 15.16 DA 5250.0 8.0 2625.0 6.0 20 1 21335-3637 324.162532 -36.4000738 15.73 DA 26940.0b 7.75b 3725.0 2.0 10 1 21459-3548 327.2267922 -35.5801944 16.35 DA 12000.0 8.0 2400.0 3.0 11 1 21470-5412 327.6000537 -53.9776573 15.26 DA 11500.0 8.0 2875.0 23.0 00 1 21473-1405 327.5153847 -13.8626911 15.75 DA 22250.0a 8.0a 3025.0 30.0 00 1 22016-3015 331.1442181 -30.0183606 15.58 DAe 12750.0 8.0 2762.5 31.0 10 1 22158-2027 334.6492133 -20.2113139 16.00 DA 15500.0 8.0 3750.0 2.0 20 1 23016-4857 346.12937 -48.6825705 15.58 DA 4750.0 8.0 2862.5 2.0 00 1 23227-6739 351.4332031 -67.3785049 16.71 DAwk 31500.0 8.0 2900.0 9.0 20 1 Previously Identified: 00370-6328 9.8125271 -63.2073005 15.82 *DAe sdB+G 50000.0 8.0 3175.0 41.0 00 3,4 01162-2310 19.6547306 -22.9156167 16.15 DA 31990.0b 7.63b 3225.0 20.0 11 5 01319-1622 23.6002247 -16.1189799 13.94 DA 50110.0a 7.87a 3337.5 18.0 10 5 01450-2211 26.8410187 -21.9475691 14.85 DA(Z) 11747.0a 8.07a 2387.5 5.0 10 5 01450-7035 26.5471352 -70.339152 15.77 DA 19000.0 8.0 3375.0 6.0 20 5 02083-1520 32.6787961 -15.1095691 15.19 DA+dM 22620.0b 7.92b 3250.0 28.0 11 5 03094-2730 47.888501 -27.3236914 15.70 DA ? 56610.0b 7.53b 3712.5 40.0 10 5,6 03319-3541 53.4679871 -35.5210623 14.38 DA+dMe 23000.0a 8.0a 3200.0 31.0 10 4,5 03338-6410 53.643096 -64.0156465 14.32 DA 50000.0 8.0 3050.0 34.0 20 7 03479-1344 57.5607475 -13.5872177 14.96 DA 14250.0a 7.76a 2862.5 23.0 11 5 03569-2320 59.7703036 -23.2070023 15.87 DAwk 74710.0b 7.86b 3625.0 17.0 11 5 10150-1722 154.3701471 -17.6187202 16.77 DA4 32370.0b 7.58b 4075.0 15.0 20 5,6 12477-1738 192.5921204 -17.9129074 16.20 DA+dMe 21620.0a 8.16a 3287.5 27.0 20 5,8 12540-1318 194.1649138 -13.5783955 16.04 DA2 23710.0b 7.92b 3325.0 14.0 20 5 13077-1411 197.5938438 -14.4525458 16.44 DA4 26400.0b 7.92b 3737.5 25.0 20 5,6 13198-2849 200.6679344 -29.0922276 15.99 DA+dM 16620.0a 7.75a 2925.0 34.0 20 5,8 13349-3237 204.4613066 -32.872784 16.34 hot DA? sd? 7000.0a 8.0a 3262.5 4.0 20 9 13471-1258 207.4669075 -13.2272276 14.80 DA+dM 3000.0a 8.0a 2612.5 2.0 20 8 14329-1625 218.9405799 -16.63817 14.89 DA+dMe 16500.0a 8.0a 2837.5 27.0 20 8,9 14363-2137 219.8024563 -21.8369585 15.94 DA6 23690.0a 7.91a 3225.0 31.0 20 5 20220-2243 306.2477455 -22.5559774 16.45 DAwk/cont 50000.0 8.0 2200.0 12.0 20 10 20246-4855 307.0651962 -48.7608551 15.69 DA+dM 16250.0 8.0 3175.0 31.0 00 11 21016-3627 316.1957085 -36.2570282 16.79 DA 50000.0 8.0 4137.5 26.0 20 5 21083-4310 317.9062002 -42.9697055 15.77 DA 7750.0 8.0 2787.5 10.0 00 5 21384-6423 325.593066 -64.1624192 15.87 DA+dMe 17750.0 8.0 2962.5 30.0 01 3,4 22049-5839 332.0912832 -58.4093375 14.22 DA+dM 19500.0 8.0 3475.0 26.0 00 3,4 23260-2226 352.1615109 -22.1721571 16.80 DA 19590.0b 8.01b 3212.5 32.0 11 5 aSpectroscopic parameters from <cit.> bSpectroscopic parameters from <cit.> References: (1) This Paper; (2) <cit.> (3) <cit.>; (4) <cit.>; (5) <cit.>; (6) <cit.>; (7) <cit.>; (8) <cit.>; (9) <cit.>; (10) <cit.>; (11) <cit.> lcccccccccr Region III: White Dwarfs with High Temperature/Low Radius Excesses 11 EC Name Right Ascension Declination V EC SptypeWD T_ eff WD logg BB T_ eff BB Rad Im FlagRef(J2000) (J2000) (mag)(K) (cm^-2) (K)(R_ WD) 00169-3216 4.8518404 -31.9982455 15.67 sdB/DA? 30750.0 8.0 4287.5 5.0 10 1 02121-5743 33.4391834 -57.4967862 14.34 DAwk 17000.0 8.0 4625.0 2.0 00 1 03120-6650 48.1736862 -66.6561967 16.73 DA/sdB 24250.0 8.0 5125.0 2.0 20 1 03372-5808 54.5997256 -57.9739414 16.45 DA 44250.0 8.0 5500.0 3.0 00 1 03572-5455 59.6228876 -54.7779686 16.09 sdB?/DA? 22250.0 8.0 5287.5 3.0 00 1 04536-2933 73.8992797 -29.4835602 14.97 DAB 20640.0b 7.61b 4925.0 1.0 20 1 10188-1019 155.3301617 -10.5804209 16.35 DA5 17720.0b 8.52b 4612.5 2.0 20 1 11437-3124 176.575735 -31.6839625 17.32 DA1 38810.0b 8.04b 5412.5 4.0 20 1 14572-0837 224.9707361 -8.8247843 15.77 DA2 21448.0a 7.92a 4500.0 2.0 20 2 19579-7344 300.9522623 -73.5956704 16.65 DA 25250.0 8.0 4975.0 3.0 20 1 20228-5030 306.6264874 -50.3451028 17.13 DA 21250.0 8.0 5500.0 3.0 20 1 21591-7353 330.8978119 -73.6455014 14.46 DA 19750.0 8.0 4812.5 3.0 20 1 22185-2706 335.3493648 -26.8484764 14.76 DA 15039.0a 7.8a 4825.0 1.0 10 1 23127-4239 348.8769107 -42.392647 16.24 sdB/DAwk 29750.0 8.0 5500.0 2.0 00 1 23513-5536 358.4795859 -55.3316608 16.23 sdB/DA 24000.0 8.0 5275.0 3.0 00 1 aSpectroscopic parameters from <cit.> bSpectroscopic parameters from <cit.> References: (1) This Paper; (2) <cit.>[Barber et al.(2012)]bar12 Barber, S. 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M., & Jura, M. 2007, , 671, 872 | http://arxiv.org/abs/1709.09675v1 | {
"authors": [
"E. Dennihy",
"J. C. Clemens",
"John H. Debes",
"B. H. Dunlap",
"D. Kilkenny",
"P. C. O'Brien",
"J. T. Fuchs"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170927180027",
"title": "WIRED for EC: New White Dwarfs with $\\textit{WISE}$ Infrared Excesses and New Classification Schemes from the Edinburgh-Cape Blue Object Survey"
} |
UUITP-32/17𝒩 = 2 supersymmetric AdS_4 solutions of type IIB supergravityAchilleas Passias^1, Gautier Solard^2 and Alessandro Tomasiello^2 ^1Department of Physics and Astronomy, Uppsala University,Box 516, SE-75120 Uppsala, Sweden ^2Dipartimento di Fisica, Università di Milano–Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy and INFN, sezione di Milano–Bicocca [email protected], [email protected], [email protected] Abstract.1in We analyze general 𝒩=2 supersymmetric AdS_4 solutions of type IIB supergravity. Utilizing a set of pure spinor equations directly adapted to 𝒩=2,the necessary and sufficient conditions for supersymmetry are reduced to aconcise system of partial differential equations for two functions which determine the solutions.We show that using this system analytic solutions can be generated, thus potentially expanding the rather limited set of known AdS_4 solutions in type IIB supergravity.§ INTRODUCTION Compactifications of string theories with a negative cosmological constant, although not realistic cosmologically, are more abundant than those with a positive one, and in the framework of the AdS/CFT correspondence serve as a description of strongly coupled conformal field theories (CFTs); vice versa, CFT intuition can often suggest new classes of anti-deSitter (AdS) solutions. In the context of AdS_4/CFT_3 correspondence, solutions with extended supersymmetry (N≥ 2) are far easier to deal with than N=1 solutions. The latter are dual to CFT_3's with only two Poincaré or Q supercharges, which usually do not provide much computational power.A prominent class of AdS_4 solutions with N= 2 supersymmetry is the one of compactifications of M-theory on Sasaki–Einstein seven-manifolds (Freund–Rubin class), arising as the near-horizon geometries of M2-branes probing Calabi-Yau four-fold singularities. They can be generalized by additional flux on the internal space (see <cit.> for a general analysis), or reduced to type IIA supergravity (see e.g. <cit.> for an explicit discussion) where can also be modified by adding fluxes <cit.>. In the first case, <cit.>, 𝒩=2 supersymmetry was imposed from the outset, whereas some solutions in the latter case relied on N=1 classifications <cit.>, ensuring enhanced supersymmetry byimposing the presence of a U(1) symmetry corresponding to the U(1) R-symmetry[Namely an isometry which does not leave the supercharges invariant, and can thus be used to generate more of them.] of the N = 2 superalgebra. This is in turn was achieved using an Ansatz inspired by the reduction of the Sasaki–Einstein solutions <cit.>. More recently, N = 2 AdS_4 solutions were found in massive type IIA supergravityby uplifting solutions of a four-dimensional gauged supergravity <cit.>.In this paper, we impose N=2 supersymmetry in full generality, using an extension of the pure spinor approach <cit.>. In the N=1 case, the internal part of the supercharges defines a pair of polyforms ϕ_± on the internal manifold M_6, called pure spinors, which satisfy a system of differential equations <cit.>. In the N=2 case, one can define a 2×2 matrix ϕ_±^IJ of such pure spinors. As described above, one can impose the differential equations on one entry of this matrix, say ϕ_±^11, and rely on R-symmetry to generate the others. Here we choose instead to derive a system of “extended pure spinor equations” on ϕ_±^IJ which ensures directly N=2 supersymmetry. R-symmetry is then obtained as a by-product. Deriving the extended pure spinor equations can be done relatively painlessly by using the ten-dimensional approach <cit.>. This consists of a system that can be applied to any supersymmetric solution (even with one supercharge), without an AdS_4× M_6 or any other factorization. It was checked in <cit.> that it reproduces quickly and correctly the pure spinor equations <cit.>; here we use it in a similar fashion to deal with extended supersymmetry. Some of the equations in the system we obtain are a natural extension to the whole matrix ϕ_±^IJ of the N=1 system that one would apply to one entry; others are new. Most notably, one of the equations gives an expression for the Ramond–Ramond (R–R) fluxes that does not involve the Hodge star. From this it follows that all the Bianchi identities for the R–R fluxes are automatically satisfied, which is typically not the case for systems with N=1 supersymmetry. We decided to apply this new system to type IIB supergravity, where supersymmetric AdS_4 solutions appear to be scarcer than in type IIA: with minimal supersymmetry there are a few isolated examples <cit.>, while with extended supersymmetry there exists a notable class <cit.>, based on work in <cit.>, which has N=4 supersymmetry and is the dual of the Hanany–Witten theories <cit.>. Other solutions with extended supersymmetry have been obtained by applying non-abelian T-duality transformations on type IIA solutions <cit.>, or uplifting lower-dimensional vacua <cit.>.After parameterizing the pure spinors in terms of an identity structure on the internal manifold M_6, and “running” the extended pure spinor equations, we obtain a set of differential equations for the identity structure. (This in turn gives rise to an SU(3) structure closely resembling the structure of the aforementioned Ansatz usually employed in type IIA <cit.>.) As it often happens, many of the equations are redundant, and in the end only a small set of rather simple equations survives, which can be interpreted as defining local coordinates. One in particular defines a transversely-holomorphic foliation.[A similar foliation also appeared for example in the study of supersymmetric theories on curved spaces <cit.>.] Using these local coordinates, we can finally reduce the entire system to three partial differential equations (PDEs) for two functions. They are relatively simple in form, and evoke results obtained for other similar problems. One of the equations, for example, is a version with a source term of the Toda equation that appeared in <cit.>. Exploring the space of solutions to this system is an elaborate task which we will not undertake here. We do however describe a couple of elementary Ansätze that simplify the system, so that we recover the maximally supersymmetric AdS_5× S^5 solution (considered as a warped AdS_4 solution) and generate a few new formal solutions. While we are not certain that there is a compact and physical M_6 among these, further study of the PDEs is likely to be rewarding.In section <ref> we will describe how to obtain our extended pure spinor equations from the ten-dimensional system of <cit.>. After introducing a parameterization for the pure spinors ϕ_± in section <ref>, we will analyze the equations in section <ref>, obtaining a relatively simple set of conditions summarized in section <ref>. As is often the case, these conditions will suggest a choice of local coordinates, which we will use in section <ref> to simplify the equations further, arriving at our final system in section <ref>. We will end in section <ref> by discussing a few solutions. § REDUCTION OF THE 10D SUPERSYMMETRY EQUATIONSIn <cit.> a system of equations was obtained, which constitute necessary and sufficient conditions for any ten-dimensional solution of type II supergravity to preserve superymmetry. We will specialize this system to the case of an AdS_4 background of type IIB supergravity, preserving 𝒩=2 supersymmetry.Let us review the system of equations of <cit.>, which are summarized in section 3.1 of that paper. Let us focus on the following subset of equations: d_H(e^-ϕΦ)= -(K∧ + ι_K) F_(10d) , d K = ι_K H.Here ϕ is the dilaton, H is the NS–NS three-form field strength, d_H ≡ d - H∧, and F_(10d) is the sum of the R–R field strengths. The latter sum, following the “democratic formulation” of type II supergravities, includes all the p-form field strengths, with p odd for type IIB, subject to the self-duality constraint F=*λ(F). λ is an operator acting on a p-form F_p as λ(F_p) = (-1)^[p/2] F_p where square brackets denote the integer part.Φ is a bispinor constructed out of the supersymmetry parameters ϵ_1 and ϵ_2:Φ≡ϵ_1 ϵ_2 . The latter are Majorana–Weyl spinors of positive chirality. K and K are respectively a vector and a 1-form bilinear:K ≡164(ϵ_1Γ^M ϵ_1 + ϵ_2Γ^M ϵ_2) ∂_M, K≡164(ϵ_1Γ_M ϵ_1 - ϵ_2Γ_M ϵ_2) dx^M,with K being a Killing vector, and more general the generator of a symmetry of the full solution.We now turn to applying these equations to the AdS_4 background of interest. To do so we will make a “4+6” split of the ten-dimensional fields and the supersymmetry parameters.We want to allow for the most general geometry with an AdS_4 factor, leaving the symmetries of the latter intact. This amounts to taking the 10d spacetime to be a warped product of AdS_4 and a six-dimensional manifold M_6, with the warp factor being a function only on M_6. The corresponding line element is:ds^2_10 = e^2A ds^2_AdS_4 + ds^2_M_6 , where A is the warp factor.Accordingly, the H field is a form only on M_6, while the R–R field strengths are decomposed asF_(10d) = e^4A_4 ∧ * λ(F) + F, F = F_1 + F_3 + F_5.Turning to the supersymmetry parameters, we will take them to be a product of Spin(1,3) and Spin(6) spinors. For an 𝒩=1 supersymmetric AdS_4 solution this decomposition is:ϵ_i = χ_+ ⊗η_i+ + χ_- ⊗η_i-,i=1,2,where the χ's are AdS_4 Killing spinors and the η's spinors on M_6. A plus or minus subscript denotes the chirality of the spinor.Since we are interested in 𝒩=2 supersymmetry we need to add a second pair of χ's and η's. The decomposition Ansatz thus becomesϵ_i = ∑_I = 1^2 χ_+^I ⊗η_i+^I + ∑_J = 1^2 χ_-^J ⊗η_i-^J,with I,J indices upon which an SO(2) R-symmetry acts. As noted, the χ's are AdS_4 Killing spinors, i.e. they satisfy[Using χ_-^I = B (χ_+^I)^*, _μ B =B ^*_μ , ^0 _μ = - _μ^†^0.See appendix <ref> for more details on Cliff(1,3) conventions. ]∇_μχ^I_± = 1/2γ_μχ^I_∓ , ∇_μχ^I_± = - 1/2χ^I_∓γ_μ .We will consider the case that χ^1_+ and χ^2_+ are linearly independent, since otherwise we would only have 𝒩 = 1 supersymmetry. To see this consider χ^2_+ = a χ^1_+; since both χ^1_+ and χ^2_+ satisfy the Killing spinor equation it is easy to see that in fact a is a constant. Thenϵ_i = χ_+^1 ⊗ (η^1_i+ + a η^2_i+) +c.c. , where c.c. denotes the complex conjugates. Since a is constant we can define η̃_i = η^1_i+ + a η^2_i+ and we end up with an 𝒩 = 1 decomposition.Finally, in reducing the 10d equations we will use the following decomposition of Cliff(1,9):Γ_μ = e^A γ^(4)_μ⊗𝕀 , Γ_m + 3 = _5^(4)⊗γ^(6)_m, Γ_11≡Γ^0 …Γ^9 = γ^(4)_5 ⊗γ^(6)_7,with μ = 0,1,2,3 and m=1,2,… 6. _5^(4) and γ^(6)_7 are the chirality operators in 1+3 and 6 dimensions respectively.We can now proceed with the reduction.We first look at (<ref>). K, K decompose as[Henceforth, we drop the (4) and (6) superscripts from the gamma matrices.]K _μ = 1/32∑_I,J=1^2χ^I_+γ_μχ^J_+ e^A (η^I_1+η^J_1+ - η^I_2+η^J_2+),K_m= -1/16 ( χ^1_+χ^2_- ξ̃_m ), K^μ = 1/32∑_I,J=1^2χ^I_+γ^μχ^J_+ e^-A (η^I_1+η^J_1+ + η^I_2+η^J_2+), K^m= -1/16 ( χ^1_+χ^2_- ξ^m ), whereξ̃_m ≡η^1_1+γ_m η^2_1- - η^1_2+γ_m η^2_2- , ξ^m ≡η^1_1+γ^m η^2_1- + η^1_2+γ^m η^2_2- .We thus findη_1+^(Iη_1+^J) =η_2+^(Iη_2+^J) ,d(e^A f)=-1/2(ξ̃) , d ξ̃= i_ξ H,where2 ϵ^IJ f ≡ -iη_1+^[Iη_1+^J] + iη_2+^[Iη_2+^J] ,ϵ^IJ being the Levi–Civita symbol with ϵ^12 = 1.Next, we impose the condition that K is a Killling vector i.e. ∇_(M K_N) = 0. Doing so we get η_1+^(Iη_1+^J) =η_2+^(Iη_2+^J)≡1/2 c^IJ e^A,-iη_1+^[Iη_1+^J] = iη_2+^[Iη_2+^J]≡ϵ^IJ f,where c^IJ are constants.In addition(ξ) = 0, ∇_(nξ_m) = 0;thus ξ is a Killing vector.Next comes equation (<ref>). In order to reduce (<ref>), we need to write Φ as a product of external and internal (poly)forms. To do so we decompose the Fierz expansion of Φ, utilizing (<ref>) and (<ref>). We find[Powers of e^A coming from (<ref>) have been suppressed.]Φ = ∑_IJ ( χ^I_+χ^J_+∧η^I_1+η^J_2+ +χ^I_+χ^J_-∧η^I_1+η^J_2- - χ^I_-χ^J_+∧η^I_1-η^J_2+ + χ^I_-χ^J_-∧η^I_1-η^J_2- ).From (<ref>) we want to obtain differential conditions for the “internal” bispinors and in order to do so we need the derivatives of the “external” ones. The latter can be derived from (<ref>): d(χ_±^Iχ_±^J) = 2 (1-14(-1)^k(4-2k))(χ_∓^Iχ_±^J), d(χ_±^Iχ_∓^J) = 2i(1+14(-1)^k(4-2k))(χ_∓^Iχ_∓^J), where k is the degree of the individual components of the bispinor, considered as a polyform.Schematically, (<ref>) then becomesext∧[int + d_H( int) ] = F,where “ext” represents collectively the external part of Φ, “int” the internal part, and F the term involving the R–R fluxes. Next, the external part is expanded in linearly independent p-form components. Each resulting term has to vanish separately, giving an equation for the internal part. More details can be found in appendix A of <cit.>. The calculation there is for AdS_6 × M_4 backgrounds but the procedure is essentially the same. In the end we obtain d_H(e^2A-ϕϕ^(IJ)_-) + 2 e^A-ϕϕ_+^(IJ) = 0,d_H(e^3A - ϕϕ_+^[IJ])= 0, d_H(e^A-ϕϕ^[IJ]_+) + e^-ϕϕ_-^[IJ] = -1/8 e^A f F ϵ^IJ ;andd_H(e^3A-ϕϕ_+^(IJ)) + 3 e^2A-ϕϕ_-^(IJ) = -1/16 c^IJ e^4A * λ(F), d_H(e^-ϕϕ_-^[IJ])= -1/16(ξ̅̃̅∧ + ι_ξ) F ϵ^IJ , d_H(e^4A-ϕϕ_-^[IJ]) + 4 e^3A-ϕϕ_+^[IJ] = -i/16(ξ̅̃̅∧ + ι_ξ) e^4A * λ(F)ϵ^IJ , whereϕ_+^IJ≡η^I_1+η^J_2+ , ϕ_-^IJ≡η^I_1+η^J_2- ,and ξ̅̃̅ is the complex conjugate of ξ̃.Although the system (<ref>) appears large, in fact it has a high degree of redundancy: for instance (and as we will see in the sections that follow) c^IJ can be set proportional to the identity, and following that, except for (<ref>), the equations that involve the R–R fields are redundant. (This is why we have separated the equations in two blocks.)The system (<ref>) is also redundant in another, more trivial way. Consider its diagonal components, I=J. Then only the two equations (<ref>), (<ref>) survive: they are two copies of the pure spinor equations <cit.> for N=1 AdS_4 solutions. Solving them gives by definition two solutions of the supersymmetry equations, with the same fluxes and geometry; in other words, it gives an N=2 solution. Thus the I≠ J equations are redundant. Even though (<ref>) is highly redundant, it will be more convenient for our analysis. For example, some of the information that would appear at high form order in the subsystem (<ref>), (<ref>), (<ref>) appears at lower form order in the full system (<ref>), and is easier to handle. We can now also comment about the remaining equations in <cit.>, called (3.1c) and (3.1d). Those “pairing equations” are in general needed, but for AdS_4 vacua they are redundant. Indeed, as we have remarked, the N=1 supersymmetry system is already reproduced by (<ref>), (<ref>) above. (How exactly they become redundant was shown in <cit.> for Minkowski_4; that logic can be adapted to AdS_4 once again following <cit.>.) Thus, the pairing equations are not needed for our N=2 classification; they would make our system (<ref>) even more redundant. We are free to ignore them, and in the following we have done so.§ PARAMETRIZATION OF THE PURE SPINORS The spinors η_i+^I define an identity structure in six dimensions; see appendix <ref>. In this section we will introduce a set of 1-forms parametrizing the latter and express the pure spinors ϕ_±^IJ in terms of these. Before doing so we will manipulate the results of the previous section in two ways.The first one is fixing the constants c^IJ of (<ref>) asc^IJ = 2 δ^IJ , where δ^IJ is the Kronecker delta. We can do so because the decomposition Ansatz (<ref>) doesn't fix the spinors η^I_i+ uniquely.Specifically, one is free to make a GL(2,ℝ) transformation that leaves the fixed (by (<ref>)) norms η_i+^I = e^A invariant, leading to real linear combinations of the external spinors χ_+^I. The details of this transformation can be found in appendix <ref>.Note that since c^12 = η_i+^(1η_i+^2) = 0, from η_i+^[1η_i+^2] = η_i+^1η_i+^2 and |η^1_i+η^2_i+| ≤√(η^1_i+η^2_i+) it follows that|f| ≤ e^A. The second one is that instead of η^I_i+ we will work withη^±_i+ = 1/√(2) (η^1_i+± i η^2_i+)which have charge ± 1 under the U(1)≃ SO(2) R-symmetry. The conditions (<ref>) (with c^IJ = 2 δ^IJ) becomeη^±_i+η^∓_i+ = 0, η^±_1+η^±_1+ = f_∓ , η^±_2+η^±_2+ = f_± ,where f_±≡ e^A ± f.Given a chiral spinor η_+ of positive chirality (and its complex conjugate η_- ≡ (η_+)^c), we can express η_i+^±, taking into account (<ref>), as follows: η^+_1+ = √(f_-)η_+,η^-_1+ = √(f_+)1/2 w_1 η_-,η^+_2+ = √(f_+)(a η_+ + 1/2 b w_3 η_-),η^-_2+ = √(f_-)1/2 c w_2 (a^* η_- - 1/2 b w_3η_+). Here a ∈ℂ and b, c ∈ℝ. They satisfy|a|^2 + b^2 = 1, c^-1 = (|z_1|^2b^2+|a|^2)^1/2 ,with z_1 defined below. The 1-forms { w_1, w_2, w_3 } parametrize the identity structure and are holomorphic with respect to the almost complex structure J defined by η_+; see appendix <ref>.We introducez_1 ≡1/2(w_2,w_3), z_2 ≡1/2(w_3,w_1), z_3 ≡1/2(w_1,w_2), where (·,·) denotes the inner product.We then have(w_a,w_b) = 2 Z_ab ,Z ≡[ 1 z_3 z^*_2; z_3^* 1 z_1; z_2 z_1^* 1 ] .The determinant of Z is Z = 1-|z_1|^2 - |z_2|^2 - |z_3|^2 - 2 (z_1z_2z_3). The pair (J, Ω) that characterize the SU(3) structure defined by η_+ are expressed in terms of { w_1, w_2, w_3 } asJ = i/2 (Z^-1)^ab w_a w_b , Ω = e^i ϑ/√( Z) w_1 ∧ w_2 ∧ w_3where ϑ∈ℝ.We can now express the pure spinors ϕ_+^±±≡η^±_1+η^±_2+ , ϕ_-^±±≡η^±_1+η^±_2- ,in terms of forms:ϕ^++_+= 1/8√(f_+f_-)[a^* e^-iJ + 1/2 b (w_3∧Ω-Ω_w_3)],ϕ^++_-=1/8√(f_+f_-)[- a Ω - b w_3 ∧ e^-iJ],ϕ^+-_+= 1/8f_- [1/2 ac (w_2∧Ω-Ω_w_2) -b c z_1e^-iJ],ϕ^+-_-= 1/8f_- [-a^* c w_2 ∧ e^-iJ +b c z_1^* Ω],ϕ^-+_+= 1/8f_+ [1/2a^*(w_1∧Ω+Ω_w_1) + b z_2 e^iJ+b w_1∧w_3∧ e^iJ],ϕ^-+_-= 1/8f_+ [a w_1 ∧ e^iJ - 1/4 b (w_1,w_3,Ω)],ϕ^–_+= 1/8√(f_+f_-)[a c z_3^* e^iJ +a c w_1∧w_2∧ e^iJ- 1/2 b c z_1 (w_1∧Ω+Ω_w_1)],ϕ^–_-= 1/8√(f_+f_-)[- 1/4 a^*c (w_1,w_2,Ω) - bcz_1^* w_1 ∧ e^iJ]. In the above(u,w,Ω) ≡ι_uι_w Ω+ u ∧ι_w Ω + w ∧ι_u Ω - u ∧ w ∧Ω . We also have (ξ)^♭ = i√(f_+f_-)[w_1+b^2c z_1w_3+ |a|^2cw_2-1/4abc ι_w_2ι_w_3Ω],ξ̃ = i√(f_+f_-)[w_1-b^2c z_1w_3- |a|^2cw_2+1/4abc ι_w_2ι_w_3Ω], where (ξ)^♭ is the 1-form dual to the ξ vector.§ ANALYSIS OF THE SUPERSYMMETRY EQUATIONSIn this section we initiate the analysis of the supersymmetry equations obtained in section <ref>.We will first analyze those which do not involve the R–R field strengths, leaving the analysis of the latter for the end. As we anticipated, not all the equations are independent, and we will be able to reduce them to a significantly smaller set.§.§ System of equationsAfter switching from the ϕ^IJ_± to the ϕ_±^±± pure spinors introduced in the previous section, the system of supersymmetry equations is as follows: d_H[e^2A-ϕϕ^++_-]+e^A-ϕ(ϕ^+-_+ + ϕ^-+_+)= 0, d_H[e^2A-ϕ()]+2e^A-ϕ()= 0, d_H[e^2A-ϕϕ^–_-]+e^A-ϕ(ϕ^+-_++ ϕ^-+_+) = 0, d_H[e^3A-ϕ()]= 0, d_H[e^3A-ϕ(ϕ^+-_+ -ϕ^-+_+)]+3e^2A-ϕ(ϕ^++_- - ϕ^–_-)= 0,d_H[e^A-ϕ()]+e^-ϕ()= - 1/4e^A f F, and d_H[e^3A-ϕ()]+3e^2A-ϕ()= -1/4e^4A∗λ(F), d_H[e^-ϕ()]= i/8(ξ̅̃̅∧+ι_ξ)F, d_H[e^4A-ϕ()]+4i e^3A-ϕ()= -1/8(ξ̅̃̅∧+ι_ξ) e^4A∗λ(F). The reason we have separated the last three equations is that they are in fact redundant given the ones above,[With the exception of the 0-form component of (<ref>) which does not involve the R–R fields due to ξ being real.] as we will see in section <ref>. We also have (ξ)= 0, d(e^A f) + 1/2(ξ̃) = 0, d ξ̃- i_ξ H= 0,∇_(nξ_m) = 0, which were obtained from (<ref>) and the condition that the ten-dimensional vector K is Killing.§.§ Scalar and 1-form equationsThe 0-form components of (<ref>) and (<ref>) give: a cz_3^*=-a, (e^A+f) z_2^*= (e^A-f) c z_1, where in (<ref>) we have used (<ref>) to “decouple” F. Moving on to the 1-form equations, imposing (<ref>) yields (1-|z_1|^2)V+1 =0, |a|^2c-V(z_3^*-z_1z_2) =0, (1-|a|^2)cz_1^*-V(z_2-z_1^*z_3^*) =0, whereV≡a b c e^iϑ/√((Z)) .Combining the above with (<ref>) we arrive atz_1=z_2=0,z_3=-1/c=-|a|,V=-1.The 1-form components of (<ref>) are then satisfied trivially and we are left with (<ref>).Henceforth, we will use the following parametrization for the scalars:a = cosβ e^iα,f = e^A cos(2θ),following the relations (<ref>) and (<ref>). (<ref>) now readsd(e^2Acos(2θ)) = - e^A sin(2θ) (w_1).§.§ Orthonormal frame and 1-form basisBefore proceeding, we will introduce an orthonormal frame constructed out of { w_1, w_2, w_3 } and a new (non-orthogonal) 1-form basis that will prove useful in analysing the remaining supersymmetry equations. The orthonormal frame is:^1= 1/sinβ((w_1) + cosβ(w_2) ), ^2 = (w_1),^3= (w_2), ^4 = 1/sinβ((w_2) + cosβ(w_1) ),^5= (w_3), ^6 = (w_3).In terms of these, the SU(3) structure in (<ref>) reads J = ^1 ∧ (-sinβ^2 - cosβ^4)+ ^3 ∧ (cosβ^2 - sinβ^4)+ ^5 ∧^6, Ω = e^-i α[ ^1 + i (-sinβ^2 - cosβ^4) ] ∧[ ^3 + i (cosβ^2 - sinβ^4) ] ∧ (^5 + i ^6). This structure is the same that appeared in several IIA solutions: see for example <cit.>. There, it was identified from existing solutions (obtained by reductionfrom eleven dimensions) and later imposed as an Ansatz. In our approach, it is coming out naturally.The new basis is: v^1= 2 (e^A sinβsin(2 θ))^-1 ^1,v^2= -e^A sin(2θ)^2,v^3= e^3A-ϕ(sinαcosβcos(2θ)^2-cosα ^3 + sinαsinβ ^4) , v^4= (e^A+ϕcos(2θ))^-1(cosαcosβ(cos(2θ))^-1 ^2 + sinα ^3 + cosαsinβ ^4 ), v^5=-2e^3A-ϕsinβsin(2θ)^5, v^6=-2e^3A-ϕsinβsin(2θ)^6. We will also usev = (e^Asin(2θ))^-1(cos(2θ)^2 - β ^4),or expressed in terms of the “v basis” (<ref>):v= - (cos(2θ)/e^2Asin^2(2θ) + cos^2βsin^2αcos^2(2θ)+cos^2βcos^2α/e^2Asin^2βsin^2(2θ)cos(2θ)) v^2- e^-4A+ϕsinαcosβ/sin^2βsin(2θ) v^3 - e^ϕcosβcosαcos(2θ)/sin^2βsin(2θ) v^4. §.§ 2-form equationsFrom the 2-form components of (<ref>) we get:d(v^5+iv^6) = (2 v -i v^1) ∧ (v^5+iv^6),and dv^3 = 0,dv^4 = 0. We are left with (<ref>) which we will rewrite asH = - 1/4 d(v^1 ∧(ξ̃)) + H_0,using the fact that ξ = 2 e^A sinβsin(2 θ) (^1)^♯, where (^1)^♯ denotes the vector dual to ^1. §.§ (p > 2)-form equationsThe 3-form components of (<ref>) are satisfied trivially given the results derived so far, whereas the 4-form components yield the conditions(d δ_1 + i H_0 ) ∧ (v^5 + i v^6) = 0,and( d δ_2 + 4 v ∧δ_2 ) ∧ v^56 = 2 e^4Acos^2(2θ) v^24∧ dv^1 + 4H_0 ∧ v^3 ,( d δ_3 + 4 v ∧δ_3 ) ∧ v^56 = 2 v^23∧ dv^1 - 4 e^4Acos^2(2θ)H_0 ∧ v^4 + 2e^-8A+2ϕ/sin^2βsin^2(2θ) v^23∧ v^56 .In the above δ_1≡1/sinβ^3 ∧^4,δ_2≡e^-3A+ϕ/sin^2βsin^2(2θ)(e^A+ϕcos(2θ) v^4+ e^-Acosαcosβtan(2θ) v^2 ),δ_3≡e^-3A+ϕ/sin^2βsin^2(2θ)(e^-3A+ϕcos(2θ) v^3- e^-Asinαcosβsin(2θ)v^2 ). Also v^56≡ v^5 ∧ v^6 etc.Finally, the 5-form and 6-form components of (<ref>), given the conditions derived so far, are trivially satisfied.§.§ Equations with R–R fields Out of the equations which involve the R–R fields, only (<ref>) is independent, with (<ref>), (<ref>) and (<ref>) following from it given the rest of the supersymmetry equations. Here is a sketch, for example, of how to show that (<ref>) is redundant. One can act with ξ̅̃̅∧ + ι_ξon (<ref>). The right-hand side of (<ref>) now becomes proportional to the right-hand side of (<ref>). For the left-hand side we can use{ξ̅̃̅∧ + ι_ξ , d_H} = (dξ̅̃̅ - ι_ξ H )∧ + L_ξ= L_ξ ,the Lie derivative under ξ, where the last equality follows from (<ref>). The action of L_ξ on the pure spinors is the one dictated by their total R-charge: L_ξϕ^±±_+=0, L_ξϕ^±∓_-=0, and ϕ^±∓_+, ϕ^±±_- have charges ± 2.Using several Fierz identities one can show (ξ̅̃̅∧ + ι_ξ) (ϕ^++_+ - ϕ^–_+)=(ξ̅̃̅∧ + ι_ξ) (ϕ^++_+ - ϕ^–_+)= -4i (e^A ϕ^(+-) + f ϕ^[+-]), (ξ̅̃̅∧ + ι_ξ) ϕ^[+-]_- = 0, (ξ̅̃̅∧ + ι_ξ) ϕ^[+-]_- = -2i e^A Re (ϕ^++_+ + ϕ^–_+) - 2 fIm (ϕ^++_+-ϕ^–_+) .Using also (<ref>), one can now massage the result to obtain (<ref>). A similar argument shows that (<ref>) follows from (<ref>). In spite of being redundant, (<ref>) and (<ref>) are useful for showing in a straightforward way that the equations of motion and the Bianchi identities of the R–R fields are automatically satisfied.Acting with d_H on (<ref>), and using the imaginary part of (<ref>) it follows thatd_H(e^4A*λ(F)) = 0,which are the equations of motion. Acting with d_H on (<ref>), using (<ref>), and subtracting the real part of (<ref>), it follows thatd_H F = 0,which are the Bianchi identities of the R–R fields.Finally, equation (<ref>) determines the R–R fields. We give their expressions in section <ref>. §.§ SummaryWe have formulated the supersymmetry equations as a set of differential constraints on an identity strucure parametrized by the set of functions {A, ϕ, θ, α, β} and the 1-forms (<ref>), which are subject tov^2 = d(e^2Acos(2θ)),dv^3 = 0,dv^4 = 0, d(v^5+iv^6) = (2 v -i v^1) ∧ (v^5+iv^6),(with v given by (<ref>)), as well as (<ref>), (<ref>) and (<ref>). Finally,ξ = 1/4||ξ||^2 (v^1)^♯ (where ^♯ denotes raising the index) is a Killing vector.In the next section we will refine the analysis of these constraints by introducing coordinates, thus reducing them to partial differential equations.The NS–NS field strength is given by (<ref>), with H_0 determined by (<ref>)–(<ref>); we will give its explicit expression in the next section. The R–R field strengths are given by (<ref>). Note that the Bianchi identities for the form fields need to be imposed on top of the supersymmetry equations. However, as we saw in section<ref>, the Bianchi identities for the R–R fields are already implied by the latter. The Bianchi identity for H still needs to be imposed and we will do so in the next section.§ LOCAL COORDINATES AND PARTIAL DIFFERENTIAL EQUATIONSIn this section we introduce local coordinates and a new set of functions that will allow us to solve some of the conditions derived in the previous section, and reduce the rest to a system of partial differential equations. §.§ Local coordinates and a new set of functions We start by introducing the coordinates { y ≡ e^2Acos(2θ), λ_1, λ_2 } so that (<ref>) are solved asv^2 = dy,v^3 = dλ_1,v^4 = dλ_2. Next, we introduce the coordinate ψ adapted to the Killing vector ξ:ξ = 4 ∂_ψ .It follows thatv^1 = dψ + ρ ,for a 1-form ρ. Finally, the differential equation (<ref>) can be solved byv^5 + i v^6 = e^-iψ e^2Σ(dx_1 + i dx_2),for a function Σ = Σ(y, λ_1, λ_2, x_1, x_2).We give a detailed explanation of this in appendix <ref>, but a summary is that one needs the “complex Frobenius theorem” by Nirenberg <cit.>, which is a mix between the real Frobenius theorem and the Newlander–Nirenberg theorem about integrability of complex structures. In general it says the following: let M be a manifold of dimension n. Given a subbundle Ω⊂ (T^*M)^ C of dimension k, and Λ≡Ω∩Ω̅ of dimension k', then there exist locally adapted coordinates such that Ω is spanned by dx_a+ i dx_a+l, a=1,…,l≡ k-k' and dx_σ, σ=n-k'+1,…,n, if and only if d Ω⊂ the ideal generated by Ω, and d Λ⊂ the ideal generated by Λ. It is used in the theory of transversely holomorphic foliations (THF); see for example <cit.>.[In physics, a THF appears for example as a condition on which three-manifolds preserve at least one supercharge of a supersymmetric field theory <cit.>, with the only difference that the leaves there are one-dimensional. Another physics application is to A-branes <cit.>. Finally, the logic explained here was also used (implicitly) in <cit.>.] In our case, we can take Ω to be the span of v^5+iv^6; Λ={0}. Then the conditiond Ω⊂ the ideal generated by Ω is simply (<ref>). This implies that there are adapted coordinates such that Ω is the span of dx_1+idx_2.With (<ref>), (<ref>) now yieldsv = Σ_,y dy + Σ_,λ_1 dλ_1 + Σ_,λ_2 dλ_2,andρ = - 2 Σ_,x_2 dx^1 + 2 Σ_,x_1 dx^2.Here Σ_,y≡∂_y Σ etc..Via the two expressions for v, (<ref>) and (<ref>), we can exhange some of the functions we have been using in the supersymmetry equations with derivatives of Σ.In particular Σ_,λ_1 = - e^-4A+ϕsinαcosβ/sin^2βsin(2θ) ,Σ_,λ_2 = - e^ϕcosβcosαcos(2θ)/sin^2βsin(2θ) , Σ_,y = -cos(2θ)/e^2Asin^2(2θ) - cos^2βsin^2αcos^2(2θ)+cos^2βcos^2α/e^2Asin^2βsin^2(2θ)cos(2θ) . By also introducingΛ = e^-2A+2ϕcos(2θ)/sin^2β ,we can express {A, ϕ, θ, α, β} in terms of{y, Σ_,y, Σ_,λ_1, Σ_,λ_2, Λ}, of which y is used as a coordinate, thus reducing the number of functions that characterize the solutions to two: Σ and Λ. Explicitly,e^4A = Λ/U + y^2, e^2ϕ = - (Λ + y^2 U)^2/(y^-1Σ_,λ_2)^2 + Σ_,y (Λ + y^2 U) ,cos(2θ)= y (Λ/U + y^2)^-1/2 , tanα = yΣ_,λ_1/Σ_,λ_2(Λ/U + y^2)^1/2 ,^2(β)= (Σ_,λ_2)^2/y(Λ + y^2 U) + y/U(Σ_,λ_1)^2 .whereU ≡ - y^-1(Σ_,yΛ +(yΣ_,λ_1)^2 + (y^-1Σ_,λ_2)^2)is not an independent function, but will be convenient to use.In the following section we will reduce the rest of the supersymmetry conditions to a set of partial differential equations for Σ and Λ. §.§ Partial differential equationsBefore moving on with the analysis of the supersymmetry equations, we define the Hodge star operators *_x:*_x dx_1 = dx_2,*_x dx_2 = - dx_1,and *_λ:*_λ dλ_1 = y^2 dλ_2,*_λ dλ_2 = - y^-2 dλ_1,and the corresponding LaplaciansΔ_x = ∂^2_x_1 + ∂^2_x_2 , Δ_λ = y^2 ∂^2_λ_1 + y^-2∂^2_λ_2 .We will also use d_λ≡ dλ_1 ∧∂_λ_1 + dλ_2 ∧∂_λ_2 and d_x ≡ dx_1 ∧∂_x_1 + dx_2 ∧∂_x_2.The supersymmetry conditions to analyze are (<ref>), (<ref>) and (<ref>); they will yield two partial differential equations for {Σ, Λ} and an expression for H_0. In terms of the coordinates and the new functions: δ_1= dy ∧ *_λ d_λΣ- Λdλ_1 ∧ dλ_2,δ_2= (y^2 U + Λ) dλ_2 - y^-2Σ_,λ_2 dy,δ_3= U dλ_1 + Σ_,λ_1 dy.Let us start with the differential equations. (<ref>) givesΔ_λΣ = - Λ_,y ,which combined with (<ref>) can be alternatively written as Δ_λ e^4Σ = - 4 (e^4ΣΛ)_,y - 16 e^4Σ y U.(<ref>) and (<ref>) give two expressions for the (x_1,x_2) components of dv^1 (dv^1)|_x_1x_2 = 1/2( e^4Σ U )_,y + 1/y e^4Σ U + 1/2y^2[ 1/4 y^-2 (e^4Σ)_,λ_2λ_2 + (e^4ΣΛ)_,y], (dv^1)|_x_1x_2 = 1/2( e^4Σ U )_,y - 1/y e^4Σ U- 1/8 (e^4Σ)_,λ_1λ_1 ,which given (<ref>) can be shown to be equivalent. Combining these with (<ref>) and (<ref>) we obtain the equationΔ_x Σ+ 1/16 (e^4Σ)_,λ_1λ_1 = 1/4 y^2 ( e^4Σ y^-2 U )_,y . Turning to H_0, (<ref>) determines its{(y,λ_1,2,x_1,2),(λ_1,λ_2,x_1,2)} components, while (<ref>) and (<ref>) its {(y,x_1,x_2),(λ_1,2,x_1,x_2)} components. In total we get:H_0= 1/2 dy ∧ *_λ d_λρ- dλ_1 ∧ dλ_2 ∧ *_x d_x Λ+ [ 1/16 y^-2 (e^4Σ)_,λ_2λ_1 dy- 1/4 *_λ d_λ (e^4Σ U) - 1/4(e^4ΣΛ)_,λ_1 dλ_2 ] ∧ dx_1 ∧ dx_2Having fully specified H, via (<ref>) and (<ref>) we can impose its Bianchi identity, dH=0. By doing so we getΔ_x Λ = - 1/4Δ_λ (e^4ΣU) - 1/4 (e^4ΣΛ)_,λ_1λ_1 . § SUMMARY OF FINAL RESULTSWe have reduced the proplem of finding supersymmetric AdS_4 solutions to solving three partial differential equations (PDEs) for two functions Σ and Λ of five variables {y, λ_1, λ_2, x_1, x_2}:[box=]align Δ_λΣ= - Λ_,y,Δ_x Σ+ 1/16 (e^4Σ)_,λ_1λ_1 = 1/4 y^2 ( e^4Σ y^-2 U )_,y,Δ_x Λ= - 1/4 Δ_λ(e^4ΣU) - 1/4 (e^4ΣΛ)_,λ_1λ_1,U ≡- y^-1(Σ_,yΛ+(yΣ_,λ_1)^2 + (y^-1Σ_,λ_2)^2),where Δ_x = ∂^2_x_1 + ∂^2_x_2,Δ_λ = y^2 ∂^2_λ_1 + y^-2∂^2_λ_2 . By inverting (<ref>) so that the orhonormal frame is expressed in terms of the v's, and eventually in terms of the coordinates introduced in the previous section, we can write down the metric for M_6:ds^2_6= e^-6A+2ϕ U^-1{y [1/4(dψ+ρ)^2 + (v)^2] + U dλ_1^2 +y^2 e^4A U dλ_2^2 -2Σ_,λ_2dy dλ_2 - Σ_,ydy^2 }+ 1/4 y^-1U e^4Σ+2A(dx_1^2 + dx_2^2),where the warp function A and the dilaton ϕ are given by e^4A = Λ/U + y^2, e^2ϕ = - (Λ + y^2 U)^2/(y^-1Σ_,λ_2)^2 + Σ_,y (Λ + y^2 U) ,while v = Σ_,y dy + Σ_,λ_1 dλ_1 + Σ_,λ_2 dλ_2 and ρ = - 2 Σ_,x_2 dx^1 + 2 Σ_,x_1 dx^2. M_6 has a transversely holomorphic foliation of codimension 1, with the coordinates on the leaves being {ψ, y, λ_1, λ_2}. There is a U(1) isometry acting on ψ, which is a symmetry of the full solution, and corresponds to the R-symmetry of the dual superconformal field theory. Moreover, the ψ circle is fibered over the surface parameterized by {x_1,x_2}. The NS–NS field readsH =- 1/4 d((dψ+ρ) ∧(ξ̃)) + 1/2 dy ∧ *_λ d_λρ- dλ_1 ∧ dλ_2 ∧ *_x d_x Λ+ [ 1/16 y^-2 (e^4Σ)_,λ_2λ_1 dy- 1/4 *_λ d_λ (e^4Σ U) - 1/4(e^4ΣΛ)_,λ_1 dλ_2 ] ∧ dx_1 ∧ dx_2,where(ξ̃) = -2 e^-8A+2ϕ/U^2(Σ_,λ_1Σ_,λ_2 dy - y^2 e^4A U Σ_,λ_1 dλ_2 + U Σ_,λ_2 dλ_1 ). The R–R fields read: F_1= df_0 + dλ_2, F_3= d_+ f_2 - H_+ f_0+ f_3, F_5= d_+ f_4 - H_+ ∧ f_2, whered_+ ≡ d + e^4AU/yΛ dy ∧ , H_+ ≡ H + 1/2e^-4A+2ϕy/Λ (dψ+ρ) ∧δ_1,with δ_1 given by (<ref>), and f_0≡Σ_,λ_2/ye^4AU , f_2≡ -1/2e^-8A+2ϕ/U^2 (dψ+ρ) ∧(U dλ_1 + yΣ_,λ_1v)-1/4yΣ_,λ_1_x, f_3≡1/4 U (Λ^-1Σ_,λ_1dy-y^-2dλ_1) ∧_x, f_4≡1/8e^-4A/Σ_,λ_2[e^4AU dy- e^2ϕΣ_,λ_1 dλ_1 + e^2ϕ(1 + (Σ_,λ_2)^2/ye^4AU) v ] ∧ dψ∧_x, where _x = e^4Σ dx_1 ∧ dx_2.§ SOLUTIONS §.§ AdS_5 × S^5In this section we recover the AdS_5 × S^5 solution from our system of equations, by imposing that the (ψ, x_1, x_2) subspace forms a round three-sphere, as well as constant axion (F_1 = 0) and dilaton. It will be convenient to work with the functions {Σ,A}. The first condition amounts to Σ = 1/2 A_0(x_1,x_2) + s(y,λ_1,λ_2), Δ_x A_0 = - e^2A_0ande^4A U = y g_s e^-2s ,g_s = e^ϕ =const. .Requiring that F_1 = 0 and constant dilaton gives respectively: e^4A U = s_,λ_2/y(C_0-λ_2) , g_s^2 = - e^8AU^2/(y^-1s_,λ_2)^2 + e^4 AU s_,y , where C_0 is constant. In what follows, by shifting λ_2, we will set it to zero. Combining (<ref>), (<ref>), and (<ref>) together with e^4A(U,Λ,y) from (<ref>) and U(Σ,Λ,y) from (<ref>) we arrive at e^2s = -y^2(g_s^-1+g_sλ_2^2) + h(λ_1),with h(λ_1) satisfying (dh/dλ_1)^2 = 4(1 - g_s e^-4A h).From the latter equation we conclude that A=A(λ_1).Finally we need to solve the PDEs that comprise our system of equations.Starting with (<ref>),we find that it givesd^2h/dλ_1^2 = 2 g_s e^-4A .Combining the above with (<ref>) we geth= c_0 e^2A ,c_0 =const. , dλ_1= ±c_0 e^2A/√(1-g_s c_0 e^-2A) dA.The rest of the PDEs, (<ref>) and (<ref>), are then automatically satisfied. Turning to the internal metric we write it asg_s^-1 ds^2_6 = e^2s-2A(ds^2_S^3 + ds^2) +g_s c^2_0 e^-2A/1-g_s c_0 e^-2A dA^2 + c_0-e^2s-2A/(g_s^-1+g_sλ_2^2)^2 dλ_2^2 + e^-2A (d√(c_0 e^2A-e^2s))^2 ,effectively switching coordinates from {y,λ_1} to {s,A}. Hereds^2_S^3 = 1/4[(dψ+ρ)^2 + e^2A_0(dx_1^2 + dx_2^2)],is the metric on the round three-sphere, of unit radius. Introducing new coordinates {x,ϕ_1,ϕ_2} viaA = log(√(g_s c_0)coshϱ),e^2s-2A = c_0 sin^2(ϕ_1), ϕ_2 = arctan(g_s λ_2),the ten-dimensional metric becomes the AdS_5 × S^5 metricds^2_10 = L^2 ( dϱ^2 + cosh^2(ϱ) ds^2_ AdS_4 + dϕ_1^2 + sin^2(ϕ_1) ds^2_S^3 + cos^2(ϕ_1) dϕ_2^2 ),with L^2 = g_s c_0 and ds^2_ AdS_5=dϱ^2 + cosh^2(ϱ) ds^2_ AdS_4.Finally, looking at the form fields, as expected F_3 and H are zero, whereasF_5 = 4 g_s c_0^2 _S^5 .Flux quantizationN ≡1/16π^4α'^2∫ F_5 = g_s c_0^2/4πα'^2givesc_0^2 = 4πα'^2 N g_s^-1 ,and hence, L^2=α' √(4 π g_s N) ,F_5 = 16 πα'^2 N _S^5 .§.§ Separation of variables Ansatz We will now discuss an Ansatz that allows for several classes of new solutions. It involves the natural assumption that the two-dimensional surface parameterized by {x_1, x_2} is a Riemann surface of constant curvature. As we warned in the introduction, we have not pursued a global analysis to the point of making sure there is a class for which the internal space is compact and physical. However, new solutions seem to be generated easily enough that this is likely to be achieved. We expect to report on this in the future.The Ansatz consists of Σ= 1/2 A_0 (x_1,x_2) + s(y, λ_1, λ_2),Λ = Λ(y, λ_1, λ_2),where A_0 is a solution of Liouville's equationΔ_x A_0 = - κ e^2A_0 ,κ∈{-1,0,1} ,and can be taken to be A_0=- log((1+κ(x_1^2 + x_2^2))/2).It proves useful to defineE≡ e^4s , V ≡ U e^4s , L ≡Λ e^4s .With these definitions the system (<ref>) becomes[The first equation is a modification of the corresponding one in (<ref>) using the rest.](L+y^2 V)_,y= -1/4 y^-2 E_,λ_2 λ_2 - 2 κ y^2 , -2κ= y^2 (y^-2V)_,y-1/4 E_,λ_1 λ_1 , (L+ y^2 V)_,λ_1 λ_1 = -y^-2V_,λ_2 λ_2 , -yVE =1/4 L E_,y + 1/16(y E_,λ_1)^2 + 1/16(y^-1 E_,λ_2)^2.Notice that three of the above equations are linear in E, V and L, and only one is quadratic. This feature makes it easier to find solutions.The dilaton and metric becomee^-2ϕ= -1/e^8A V^2( 1/16(y^-1 E_,λ_2)^2 + 1/4 e^4A V E_,y) ; ds^2_6= e^2AV/4y ds^2_𝒞 + e^-6A+2 ϕ/4V[y (E D ψ^2 + (d√(E))^2) + 4V(d λ_1^2 + y^2 e^4Ad λ_2^2) - 2E_,λ_2 d λ_2 d y - E_,y d y^2 ] , where e^4A= L/V + y^2, ds^2_𝒞 = e^2A_0(dx_1^2 + dx_2^2) is the line element of a Riemann surface of scalar curvature 2κ, and Dψ≡ dψ + ρ. Here the coordinates are { x_i, y, λ_i }, i=1,2, with dE = E_,y dy + E_,λ_id λ_i.As can be seen, at the locus where E goes to zero, the ψ circle shrinks regularly by fixing the period of ψ to be 2π.One can also eliminate y as a coordinate in favor of E. This leads to the alternative expression for the metric: ds^2_6= e^2AV/4y ds^2_𝒞 + e^-6A+2 ϕ/4V ds^2_4 , ds^2_4= yED ψ^2 + (y/4 E - y_,E)dE^2 - 2y_,λ_1 dE d λ_1 +(4V-(y_,λ_1)^2/y_,E) d λ_1^2 + (4 V y^2 e^4A+(y_,λ_2)^2/y_,E)dλ_2^2.Although this expression appears longer, it has the advantage of having fewer non-diagonal components.We will now explore two classes of sub-Ansätze. §.§.§ Compactification Ansatz The first class comes about by demanding that the line element of the Riemann surface 𝒞 has the same prefactor as that of AdS_4, so that the metric takes the form ds^2_10 = e^2A(ds^2_ AdS_4+ 14 ds^2_𝒞) + … . The holographic interpretation of this class of solutions is that of the dual of a five-dimensional field theory compactified on 𝒞. An analogous class was studied in <cit.>, where the so-called “compactification Ansatz” was applied to AdS_5 solutions. From (<ref>) we see that this Ansatz amounts to imposing V= r ywhere r is a constant proportional to the curvature radius of 𝒞. With (<ref>), two of the equations in (<ref>) determine E= 2(2κ-r) λ_1^2 + K_1 λ_1 + K_2 , L=L_1 λ_1 + L_2 ,where K_i and L_i are functions of λ_2 and y. In the remaining two equations, λ_1 only appears linearly or quadratically; thus one can expand in it, and obtain several PDEs in λ_2 and y only. Some of them are quadratic, but further assumptions make them manageable. For example one may impose that K_i and L_i do not depend on λ_2. The most “physically promising” solution one finds like this isE= -8 λ_1^2 + k_1 λ_1 - 1/32 k_1^2 , L=ℓ_1 λ_1 + ℓ_2-4/3 y^3 , r=-2κ=2 ,where k_i, ℓ_i are constant. More complicated solutions exist; for example: E= -8 λ_1^2 -4 ℓ_1 y λ_1 λ_2 - y^2(1/2(ℓ_2^2+ℓ_1^2) +4 ℓ_2 y +8 y^2) λ_2^2 ,L=ℓ_1 λ_1/λ_2 + 1/4(ℓ_2^2 + ℓ_1^2)y + ℓ_2 y^2 , r=-2κ=2 . Given the holographic interpretation of the present Ansatz, that we mentioned above, we expect that it contains solutions descending from theAdS_6 solutions of <cit.>. §.§.§ Another sub-Ansatz Another possibility we can explore is L= L(y),V= V(y) .The third equation in (<ref>) is then automatically satisfied. The first two imply that E is a polynomial of total degree 2 in λ_i: E= ∑_0≤ a+b≤ 2 L_abλ_1^a λ_2^b.Moreover, they determine L_20 and L_02 in terms of L and V. The fourth, quadratic equation in (<ref>) then gives a system of six ODEs in the y coordinate, one for each monomial λ_1^a λ_2^b, 0≤ a+ b ≤ 2. One observes that the system simplifies substantially by assuming L_11=0. Moreover, the ODE corresponding to the monomials of total degree <2 are linear in L_ab, a+b=0,1 once the ODEs corresponding to total degree 2 have been solved. The latter are now equations for L and V, and can be solved, for example, with a power-law assumption. This way we getκ=0 , V= r y^2 , L= ℓ - r y^4 , L_20=L_02=0 ,L_10= ℓ_1 L,L_01= ℓ_2 L , L_00= L (ℓ_2^2/4y + ℓ_0 - ℓ_1^2/12 y^3 ), where r, ℓ, ℓ_a, a=0,1,2 are constants. § ACKNOWLEDGEMENTSWe would like to thank John Estes for useful correspondence. AP and AT are grateful to the Mainz Institute for Theoretical Physics (MITP) for its hospitality during the completion of this work. Our research was supported in part by INFN and by the European Research Council under the European Union's Seventh Framework Program (FP/2007-2013) – ERC Grant Agreement n. 307286 (XD-STRING). AT was also supported by the MIUR-FIRB grant RBFR10QS5J “String Theory and Fundamental Interactions”. AP is also supported by the Knut and Alice Wallenberg Foundation under grant Dnr KAW 2015.0083.§ GL(2,ℝ) TRANSFORMATION In this appendix we show how c^IJ can be set equal to 2 δ^IJ, by a GL(2,ℝ) transformation of η^I_i+.First, by rescaling the χ's in (<ref>) we can setc^11 = c^22 = 2.Our analysis then splits into two cases: (a) |c^12| ≠ 2 and (b) |c^12| = 2.For case (a) we definex ≡1/2 c^12, so that x^2≠ 1. Then the GL(2,ℝ) map ([ η^1_i+; η^2_i+ ])→( [-1 0; -x/√(1-x^2)1/√(1-x^2) ]) ([ η^1_i+; η^2_i+ ])leaves the norms η_i+^I = e^A invariant and in the new basis η^(1_i+η^2)_i+ =(η^1_i+η^2_i+) =1/2 c^12 e^A = 0.In the second case η^(1_i+η^2)_i+ =(η^1_i+η^2_i+) = 1/2c^12e^A = ± e^A, and from the Cauchy–Schwarz inequality√((η^1_i+η^2_i+)^2 + (η^1_i+η^2_i+)^2 ) = |η^1_i+η^2_i+| ≤√(η^1_i+η^2_i+) = e^Ait follows that ( η^1_i+η^2_i+ ) =0; in addition since the inequality is saturated η^1_i and η_i^2 should be proportional. The factor of proportionality is fixed by their norms and inner product |η^1_i+η^2_i+| = e^A to be ± 1. But in this case there is only 𝒩 = 1 supersymmetry, as can be readily inferred from the 10d spinor decomposition Ansatz.§ SPINORS AND 𝒢-STRUCTURES We look at the 𝒢-structures defined by spinors in 1+3 and 6 dimensions. They are characterized by a set of tensors constructed as spinor bilinears which we will assemble into bispinors ϵϵ̅, since the latter, via the Fierz expansion (schematically)[_m_1 … m_p denotes the antisymmetric product of _m_1, …, _m_p.]ϵϵ̅∝∑_p 1/p!γ^m_p… m_1ϵ̅γ_m_1 … m_pϵ ,and the mapγ^m_p… m_1→ dx^m_p∧…∧ dx^m_1can be treated as polyforms.§.§ Dimension d=1+3 In this appendix we examine the identity structure defined by two spinors, ζ^1_+ and ζ^2_+, of positive chirality in 1+3 dimensions[One chiral spinor, ζ_+, defines an ℝ^2 structure; see for example<cit.>.]. The generators of Cliff(1, 3) satisfy {^α , ^β} = 2 η^αβ, α, β = 0,1,2,3, where η^αβ is the Minkowski metric of “mostly plus” signature; they are chosen so that (^α)^† = ^0 ^α^0. The chirality operator is _5 ≡-i ^0 ^1 ^2 ^3 and has the property (^5)^2 = 𝕀. We introduce the intertwiner B that relates ^α, α = 0,1,2,3 and its complex conjugate (^α)^* as ^α B =B (^α)^*. It satisfies B^* =B^-1 and B^† = B^-1. The complex conjugate of a spinor ζ is then ζ^c ≡ B ζ^*; note that ζ^cc =ζ. Complex conjugation changes chirality. We look at the case of a “strict” identity structure where ζ^1_+ and ζ_+^2 are orthogonal i.e. (ζ^1_+)^†ζ_+^2 = 0.Employing the Fierz identity and_α_1 …α_k =(-1)^k(k-1)/2-i/(4-k)!ϵ_α_1 …α_4^α_k+1…α_4_5,the following expansions for the bispinors can be obtained:[^i_1i_2 … i_n denotes the wedge product ^i_1∧^i_2∧…^i_n.]ζ^1_+ ζ^1_+ = 1/4(^+ + i * ^+), ζ^1_+ ζ^1_- = 1/4^+1 ,ζ^2_+ ζ^2_+ = 1/4(^- + i * ^-), ζ^2_+ ζ^2_- = -1/4^-1̅ ,Here ζ≡ζ^†γ^0 and ζ_- = (ζ_+)^c. The set of 1-forms {^+, ^-, ^1, ^1̅} make up a complex frame defining the identity structure. A real framecan be constructed as^0 = 1/2(^+ + ^-), ^3 = 1/2(^+ - ^-), ^1 = 1/2 (^1 + ^1̅),^2 = -i/2 (^1 - ^1̅).The volume element is _4 = ^0123 and the Hodge star is defined via a ∧ * b = (a,b) _4, where (.,.) is the inner product with respect to the Minkowski metric. Thus, for example,* 1 = ^0123 ,* ^0 = - ^123 ,* ^3 = - ^012.Furthermore, ζ^1_+ ζ^2_+ = 1/4(^1 + i * ^1), ζ^1_+ ζ^2_- = -1/4(1 + 12^-+ + 12^11̅ - i * 1). We record the following identitiesζ^I_- ζ^J_- = B (ζ^I_+ ζ^J_+)^* B^-1 , ζ^I_- ζ^J_+ = B (ζ^I_+ ζ^J_-)^* B^-1 .for generic spinors ζ's. Bearing in mind that B^-1^α B =(^α)^*, we conclude that,a plus to minus interchange is equivalent to complex conjugation. For exampleζ^1_- ζ^1_- = 1/4(^+ - i * ^+).Finally,ζ^I_+ ζ^J_∓ = - (-1)^k(k+1)/2ζ^J_±ζ^I_- . §.§ Dimension d=6 In this appendix we take a look at the 𝒢-structures defined by chiral spinors in six dimensions. Given a representation {_1, _2, …, _6 } of Cliff(6) we introduce𝔤_1 ≡1/2(_1 + i _2), 𝔤_2 ≡1/2(_3 + i _4), 𝔤_3 ≡1/2(_5 + i _6).The Cliffora algebra then takes the form{𝔤_ a, 𝔤_b̅} = δ_ ab̅ , {𝔤_ a, 𝔤_ b}= {𝔤_a̅, 𝔤_b̅} = 0,a,b = 1,2,3,where 𝔤_1̅ = 12(𝔤_1 - i 𝔤_2) etc.We take |↓↓↓⟩ as the state which is annihilated by all 𝔤_ a. Starting from |↓↓↓⟩ and acting with 𝔤_a̅ we can construct the 2^3-dimensional Dirac representation of Spin(6). We denote |↑↓↓⟩ = 𝔤_1̅|↓↓↓⟩ etc. Expanding γ_7 ≡ i _1 …_6 in terms of 𝔤_ a, 𝔤_a̅, we conclude that spinors with an even number of ↑ have positive chirality while spinors with an odd number of ↑ have negative chirality. The intertwiner B, which relates _a, a=1,2,…,6 and (_a)^* as _a B = - B (_a)^*,interchanges ↓ and ↑ and hence chirality. For example B |↓↓↓⟩ = |↑↑↑⟩.A chiral spinor η_+ ≡|↓↓↓⟩ defines an SU(3) structure, characterized by a real 2-form J and a decomposable complex 3-form Ω, asη_+η_+ = 1/8(1 - iJ - *J + i *1 ), η_+η_- = -1/8Ω ,where η_+≡η_+^†,-iJ = 1/2 (^1 1̅ + ^2 2̅ + ^3 3̅), Ω = ^1 2 3 ,and {^1, ^2, ^3} are a complex frame. J obeys J ∧ J ∧ J = 6 _6,*J = 1/2 J ∧ J.Accordingly,η_+η_+ = 1/8 e^-iJ . Two chiral spinors η^1_+ and η^2_+ define an SU(2) structure as follows: we take η^1_+ ≡|↓↓↓⟩ and η^2_+ to be orthogonal. The stabilizer group 𝒢 of η^1_+ in Spin(6) ≃ SU(4) is SU(3). We can thus perform an SU(3) transformation that leaves η^1_+ invariant and sets η^2_+ = |↑↑↓⟩ = 𝔤_3 |↑↑↑⟩. Then ^3, ω≡ι_^3̅Ω , -i j ≡ - i J - 1/2^3 3̅ ,define an SU(2) structure in six dimensions, where ^3 is the 1-form bilinear constructed out of η^1_+ and η^2_+.Along the same lines four chiral spinors |↓↓↓⟩, 𝔤_1 |↑↑↑⟩, 𝔤_2 |↑↑↑⟩ and 𝔤_3 |↑↑↑⟩ define a (strict) identity structure.We record the following identitiesη^I_- η^J_- =B (η^I_+ η^J_+)^* B^-1 , η^I_- η^J_+ = B (η^I_+ η^J_-)^* B^-1 .Bearing in mind that B^-1_a B =- ^*_a, we conclude that,in the first case a plus to minus interchange is equivalent to complex conjugation but in the second case minus complex conjugation.Finally,η^I_+ η^J_∓ =(-1)^k(k+1)/2η^J_±η^I_- .§ COORDINATES ON THE {V^5,V^6}-SUBSPACEWe want to show here that (<ref>) implies (<ref>). Let us callβ≡ v^5 + i v^6, α≡ 2v - i v^1 ,so that (<ref>) readsd β = α∧β .Separating (<ref>) in real and imaginary parts, we see that it reads dv^a = a^ab∧ v^b, a^56+ i a^66=i(a^55+ia^65)=i α .By the dual version of the Frobenius theorem (see for example <cit.>), it follows that the four-dimensional distribution D_4 ⊂ T orthogonal to v^5 and v^6, D_4 = { X ∈ T | ι_X v^a=0,a=5,6}, is integrable. This means that D_4 is a foliation: there exist (generically) four-dimensional leaves, such that the union of all of them is the whole manifold M_6. In other words, at every point there is a leaf that goes through that point. These can be parameterized by 6-4=2 real numbers, which we can call x_1, x_2, so that the leaves can be labeled as L_x_1,x_2. We can also use x_1, x_2 as coordinates on M_6. They are constant on each leaf, since they parameterize them. In other words dx_i ⊥ D_4, i=1,2. But by definition the v^a are also orthogonal to each leaf. So we conclude v^a= m^ai dx_i.We can also define the differential d_L such that d= d_L + dx_i ∧∂_x_i. Using the coordinates we have just introduced, we can write β= β^i dx_i; moreover, we can decompose α=α_L + α^i dx_i. From (<ref>) it now follows that d_L β^i = α_L β^i. From this we conclude that d_L(β^2/β^1)=0. So w ≡β^2/β^1 is a function of x_1 and x_2 only. Now β = β^1 (dx_1 + w dx_2); but dx_1 + w dx_2 defines an almost complex structure in two dimensions, which is always integrable. Thus there exists a complex coordinate z such that (dx_1 + w dx_2) is proportional to dz. Hence β = e^φ dz.Redefining x_1, x_2 so that z=x_1 + i x_2, we arrive at (<ref>) with φ = 2Σ - i ψ.utphys | http://arxiv.org/abs/1709.09669v1 | {
"authors": [
"Achilleas Passias",
"Gautier Solard",
"Alessandro Tomasiello"
],
"categories": [
"hep-th"
],
"primary_category": "hep-th",
"published": "20170927180002",
"title": "N = 2 supersymmetric AdS4 solutions of type IIB supergravity"
} |
This work was supported in part by the National Nature Science Foundation of China (NSFC) under Grant No. 61601290, Shanghai Sailing Program under Grant No. 16YF1407700, the Hong Kong Research Grant Council under Grant No. 16200214, the National Natural Science Foundation of China under Project Nos. 61671269 and 61621091, the Chinese National 973 Program under Project No. 2013CB336600, and the 10000-Talent Program of China.Y. Shi is with the School of Information Science and Technology, ShanghaiTech University, Shanghai, China (e-mail: [email protected]). J. Zhang is with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong (e-mail: [email protected]).W. Chen is with the Department of Electronic Engineering, Tsinghua University, Beijing, China (e-mail: [email protected]).K. B. Letaief is with Hamad bin Khalifa University, Ar-Rayyan, Qatar, and also with the Hong Kong University of Science and Technology, Hong Kong (e-mail: [email protected]; [email protected]).Generalized Sparse and Low-Rank Optimization for Ultra-Dense Networks Yuanming Shi, Jun Zhang, Wei Chen, and Khaled B. Letaief December 30, 2023 =====================================================================Generalized Sparse and Low-Rank Optimization for Ultra-Dense Networks Yuanming Shi, Jun Zhang, Wei Chen, and Khaled B. Letaief December 30, 2023 ===================================================================== Ultra-dense network (UDN) is a promising technology to further evolve wireless networks and meet the diverse performance requirements of 5G networks. With abundant access points, each with communication, computation and storage resources, UDN brings unprecedented benefits, including significant improvement innetwork spectral efficiency and energy efficiency, greatly reduced latency to enable novel mobile applications, and the capability of providing massive access for Internet of Things (IoT) devices. However, such great promises come with formidable research challenges. To design and operate such complex networks with various types of resources, efficient and innovative methodologies will be needed. This motivates the recent introduction of highly structured and generalizable models for network optimization. In this article, we present some recently proposedlarge-scale sparse and low-rank frameworks for optimizing UDNs, supported by various motivating applications. A special attention is paid on algorithmic approaches to deal with nonconvex objective functions and constraints, as well as computational scalability.§ INTRODUCTION As mobile data traffic keeps growing at an exponential rate, and mobile applications pose more and more stringent and diverse requirements, wireless networks are facing unprecedented pressures. To further evolve wireless networks and maintain their competitiveness, network infrastructure densification stands out as a promising approach. By deploying more radio access points, supplemented with storage and computational capabilities, we can not only increase network capacity, but also improve network energy efficiency, enable low-latency mobile applications, and provide access for massive mobile devices. Such ultra-dense network (UDN) provides an ideal platform to develop disruptive proposals to advance wireless information technologies, including cloud radio access networks (C-RANs), wireless edge caching, and mobile edge computing. These are achieved by leveraging innovative ideas in different areas, such as software-defined networking, network function virtualization, content-centric networking, cloud and fog computing. By enabling capabilities ofcloud computing and software-defined networking, UDNs can easily support C-RANas an effective network architecture to exploit the benefits ofnetwork densification via centralized signal processing and interference management <cit.>. This is achieved by moving the basebandprocessing functionality to the cloud data center via high-capacity fronthaul links, supported by massively deployed low-cost remote radio heads (RRHs). Meanwhile, the Internet is shifting from the “connection-centric" mode to the “content-centric" mode to support high-volume content delivery <cit.>. By enabling content caching at radio access points, i.e., wireless edge caching, UDNs can assist the Internet architecture evolution and achieve more efficient content delivery for mobile users <cit.>. Another trend is theincreasing computation intensity in mobile applications, which puts a heavy burden on resource-constrained mobile devices. Mobile edge computing was recently proposed as a promising solution, by offloading computation tasks of mobile applications to servers at nearby access points. It avoids excessive propagation delay in the backbone network, compared to mobile cloud computing, and thus enables latency-critical applications. All of these systems are built upon the UDN platform, which enables integration of the storage, computing, control and networking functionalities at the ubiquitous access points. In particular, C-RANs serve the purpose of providing higher data rates, while mobile edge caching and computing networks enable low-latency content delivery and mobile applications. However, all theemerging networking paradigms associated with UDNs bring formidable challenges to network optimization, signal processing and resource allocation, given the highly complex network topology, the massive amount of required side information, and the high computational requirement. Typical design problems are nonconvex in nature, and of enormously large scales, i.e., with large numbers of constraints and optimization variables. For examples, the uncertainty or estimation error in the available channel state information (CSI) yields nonconvex quality-of-service (QoS) constraints, while such network performance metrics as sum throughput and energy efficiency lead to nonconvex objective functions. Thus effective and scalable design methodologies, with the capability of handling nonconvex constraints and objectives, will be needed to fully exploit the benefits of UDNs. The aim of this article is to present recent advances in sparse and low-rank techniques for optimizing dense wireless networks <cit.>, with a comprehensive coverage including modeling, algorithm design, and theoretical analysis. We identify two representative classes of design problems in UDNs, i.e., large-scale network adaption and side information assisted network optimization.The first class of design problems are for the efficient network adaptation in UDNs, includingradio access point selection <cit.>, backhaul data assignment, user admission control, user association <cit.>, and active user detection <cit.>. Such large-scale network adaptation problems involve both discrete and continuous decision variables, which motivates us to enforce sparsity structures in the solutions. The success of the structured sparse optimization for network adaptation comes from the key observation that such adaptation can be achieved by enforcing structured sparsities in the solution, which will be presentedin Section <ref> in details. The second class of design problems involve how to effectively utilize the available side information for network optimization, including topological interference management <cit.>, wireless distributed computing <cit.>, and mobile edge caching <cit.>. Network side information is critical to design UDNs, and it can take various forms, such as the network connectivity information, cache content placement at access points, and locally computedintermediate values in wireless distributed computing. In Section <ref>, we will present a general incomplete matrix framework to model various network side information, which leads to a unified network performance metric via the rank of the modeling matrix for optimizing UDNs.Although the structured sparse and low-rank techniques enjoy the benefits of modeling flexibility, the sparse function and rank function are nonconvex, which brings computational challenges <cit.>. Furthermore, typical optimization problems in UDNs bear complicated structures, which make most of the existing algorithms and theoretical results inapplicable. To address these algorithmic challenges, we present various convexification procedures for both objectives and constraints throughout our discussion. Moreover, scalable convex optimization algorithms and nonconvex optimization techniques, such as Riemannian optimization, will be presented in Section <ref>. This article shall serve the purpose of providing network modeling methodologies and scalable computational tools for optimizing complex UDNs, as summarized in Table <ref>.§ STRUCTURED SPARSE OPTIMIZATION FOR LARGE-SCALE NETWORK ADAPTATION In UDNs, to effectively utilize densely deployed access pointsto support massive mobile devices, large-scale network adaptation will play a pivotal role. For various network adaptation problems in UDNs, the solution vector is expected to be sparse in a structured manner, e.g., radio access point selection results in a group sparsity structure. To illustrate the powerfulness of the generalized sparse representation and scalable optimizationparadigms, in this section, we present representative examples of group sparse beamforming for green C-RANs, and structured sparse optimization for active users detection and user admission control.§.§ Generalized Structured Sparse Models In this part, two motivating applications of generalized sparse models for large-scale network adaptation are presented. §.§.§ Large-Scale Structured Optimization We take green C-RAN as an example toillustrate structured optimization for network adaptation. In C-RANs, the network power consumption consists of the transmit power of active RRHs and the power of the corresponding active fronthaul links. By exploiting the spatial and temporal data traffic fluctuation, network adaption via dynamically switching off RRHs and the associated fronthaul links can significantly reduce the network power consumption. To minimize the network power of a C-RAN, we need to optimize over both the discrete variables (i.e., the selection of RRHs and fronthaul links) and continuous variables (i.e., downlink beamforming coefficients), yielding a mixed combinatorial optimization problems, which is highly-intractable. To support efficient algorithm design and analysis, a principled group sparse beamforming framework was proposed in <cit.> by enforcing the group sparsity structure in the solution vectors. This is achieved by a group sparsity representation of the discrete optimization variables for RRHs selection as shown in Fig. <ref>. Specifically, by regarding all the beamforming coefficients of one RRH as a group, switching off this RRH corresponds to setting all the associated beamforming coefficients in the same group to be zero simultaneously. We thus enforce the group sparsity structure in the aggregative beamforming vector to guide switching off the corresponding RRHs to minimize the network power consumption. Similar to group sparse beamforming for RRH selection, there is a corresponding user side node selection problem. Withcrowded mobile devices, it is critical to maximize the user capacity, i.e., the number of admitted users. This user admission problem is equivalent to minimizing the number of violated QoSconstraints (modeled as g_i(x)≤ 0 for the i-th user and may be infeasible), which can further be modeled as minimizing the individual sparsity of the auxiliary vector z=[z_i] with z_i≥ 0 indicating the violations of the QoS constraints. That is, for the constraint g_i(x)≤ z_i (always feasible as the auxiliary variable z_i≥ 0), z_i=0 indicates that the original QoS constraint g_i(x)≤ 0 is feasible, while z_i>0 indicates that the original QoS constraint g_i(x)≤ 0 is infeasible. Therefore, by enforcing this structured sparsity in the solution, user admission can be effectively handled. §.§.§ High-Dimensional Structured Estimation With limited radio resources, it is challenging to support massive device connectivityfor such applications as IoT. Fortunately,only part of the massive devices will be active at a time given the sporadic traffic for the emerging applications (e.g., machine-type communications, Internet-of-Things (IoT)) <cit.>. Active user detection is thus a key problem for providing massive connectivity in UDNs, which turns out to be a structured sparse estimation problem. Specifically, suppose we have N single-antenna mobile devices (K of which are active) and one M-antenna base station (BS). The received signalat the BS has the form Y=HΣQ+W, where Σ∈ℝ^N× N is the unknown diagonal activity matrix with K non-zero diagonals whose positions are to be estimated, H∈ℂ^M× N is the unknown channel matrix from all the devices to the BS, Q∈ℂ^N× L is the known pilot matrix with training length L, and W is the additive noise. We thus need to simultaneously estimate the channel matrix H and Σ, which poses a great challenge. We observe that detecting the active users is equivalent to estimating the group sparsity structure of the combined matrix Θ=HΣ∈ℂ^M× N, which has a group structured sparsity in columns of matrix Θ, induced by the structure of Σ. That is, when mobile device n is inactive, all the entries in the n-th column in matrix Θ become zeros simultaneously. Due to thelimited radio resources, the training length L will be much smaller than the channel dimension N, and thus, the estimation problem is ill-posed and yields a high-dimensional structured estimation problem. Fortunately, the embedded low-dimensional structure (i.e., the structured sparsity) can be algorithmically exploited to ensure the success for the high-dimensional structured estimation, as illustrated in Fig. <ref> for the behaviors of phase transitions and normalized mean square error (NMSE) . Phase transition defines a sharp change in the behavior of a computational problem as its parameters vary. Convex geometry and conic integral geometry provide principled ways to theoretical predicate the phase transitions precisely <cit.>. In particular, the phase transition phenomenon in Fig. <ref> (a) reveals the fundamental limits of sparsity recoveryin the best cases, i.e., without noise. Specifically, such study reveals that the required training length, or the number of measurements, depends on the sparsity level of Θ, and highly accurate user activity detection can be achieved with sufficient measurements. Fig. <ref> (b) further demonstrates that the low-dimensional structure can be exploited to significantly reduce the training length for active users detection even in the noisy scenarios. §.§ A Generalized Sparse Optimization ParadigmWe have demonstrated that effective network adaptation can be achieved by either inducing vector sparsity in the structured manner or estimating the structured sparsity pattern. In this part, we providea generalized sparse optimization framework to algorithmically exploit the low-dimensional structures in UDNs. This is achieved by optimizing a constrained composite combinatorial objective: minimize_z∈ℂ^n f(z):=f_1(Supp(z))+ f_2(z) subject to z∈𝒞,where Supp(z) is the index set of non-zero coefficients of a vector z, f_1 is a combinatorial positive-valued set-function to control the structured sparsity in z, f_2 is a continuous convex function in z to represent the system performance such as transmit power consumption, and the constraint set 𝒞 serves the purpose of modeling system constraints, e.g, transmit power constraints and QoS constraints. The most natural convex surrogate for a nonconvex function f is its convex envelope, i.e., its tightest convex lower bound. The main motivation for convexifying function f is that the convexified optimization problems make it possible to use the convex geometry theory <cit.> to reveal benign properties about the globally optimal solutions, which can be computed with efficient algorithms. For example, the individual sparsity function with ℓ_0-norm in z can be convexified to the ℓ_1-norm. The group sparsity function can be convexified by the mixed ℓ_1/ℓ_2-norm.More general convex relaxation results can be derived based on the principles ofconvex analysis <cit.>.Note that, it is critical to establish the optimality for various convex relaxation approaches in UDNs. Forexample, for the nonconvex active user detection problem in Section <ref>, the optimality condition can be established via the conic geometry approach in <cit.>. The constraint set 𝒞 serves the purpose of modeling various QoS constraints including unicast beamforming, multicast beamforming, and stochastic beamforming, just to name a few. For example, the nonconvex QoS constraints for unicast beamforming can be equivalently transformed into convex second-order cone constraints <cit.>. Furthermore, physical layer integration techniques can effectively improve the network performance via providing multicast services, which, however, yield nonconvex quadratic QoS constraints. Thesemidefinite relaxation (SDR)technique turns out to be effective to convexify the nonconvex quadratic constraints via lifting the original vector problem to higher matrix dimensions, followed by dropping the rank-one constraints. For stochastic beamforming with probabilistic QoS constraints due to CSI uncertainty, the probabilistic QoS constraints can be convexified based on the principles of the majorization-minimization procedure, yielding sequential convex approximations. In summary, the general formulation in (<ref>) enables efficient algorithm design and analysis for network adaptation in UDNs. § GENERALIZED LOW-RANK OPTIMIZATION WITH NETWORK SIDE INFORMATION UDNs are highly complex to optimize, for which it is critical to exploit the available network side information. For example, network connectivity information,cached content at the access points, and locally computed intermediate values, all serve as exploitable side information for efficiently designing coding and decoding in UDNs. In this section, we provide a generalized low-rank matrix modeling framework to exploit the network side information, which helps to efficiently optimize across the communication, computation, and storage resources. To demonstrate the powerfulness of this framework, we present topological interference alignment as a concrete example and then extend it tocache-aided interference channels and wireless distributed computing systems. A general low-rank optimization problem is then formulated by incorporating the network side information. §.§ Network Side Information Modeling via Incomplete MatrixTo exploit the full performance gains of network densification, recent years have seen progresses on interference management under various scenarios depending on the amount of shared CSI and user messages. Typical interference management strategies include interference alignment, interference coordination, coordinated multipoint transmission and reception, to name just a few. However, the significant overhead of acquiring global CSI motivates numerous research efforts on CSI overhead reduction strategies, e.g., delayed CSI, alternating CSI and mixed CSI. One of the most promising strategies is topological interference management (TIM) <cit.>, for which only network connectivity information is required. Thisis based on the fact that most of the wireless channel propagation links are weak enough to be ignored, thanks to pathloss and shadowing. However, the TIM problem turns out to be linear index coding problems <cit.>, which are in general highly intractable and only partial results exist for special cases.Recently, a new proposal was made for the TIM problem, which can greatly assist the algorithm design. The main innovation is to model the network connectivity pattern in UDNs as an incomplete matrix. Then the TIM problem can be formulated as a generalized matrix completion problem[B. Hassibi, “Topological interference alignment in wireless networks," Smart Antennas Workshop, Aug. 2014.], which helps to develop effective linear precoding and decoding strategies. Fig. <ref> demonstrates the modeling framework, with Fig. <ref> (a) showing a 5-user interference channel as an example and Fig. <ref> (c) showing the corresponding modeling matrix. The task of TIM is to complete the side information modeling matrix, which will then determine the precoder and decoder <cit.>.This modeling framework is very powerful, and can be adopted to consider other design problems in UDNs. By equipping the densely deployed radio access points and mobile devices with isolated cache storages, caching the content at the edge of the network provides a promising way to improve the throughput and reduce latency, as well as reducing the load of the core network and radio access networks <cit.>. In general, content-centric communications consist of two phases, a content placement phase followed by a content delivery phase. However, due to the coupled wireline and wireless communications in cache-aided UDNs, unique challenges arise in the edge caching problem. Fortunately, the incomplete matrix modeling framework cancapture the information of the content cached at different nodes. Fig. <ref> (b) shows an example for cache-aided 5-user interference channels, where the side information is represented in the side information modeling matrix in Fig. <ref> (c). Similarly, this modeling framework can also be extended to wireless distributed computing networks <cit.>. For the prevalent distributed computing structures like MapReduce and Spark, the basic idea is that intermediate values computed in the “map" phase based on the locally available dataset, can be regarded as the side information for the “reduce" phase to compute the output value for a given input. This thus can help reduce the communication overhead in the “shuffle" phase to obtain the intermediate values that are not computed locally in the “map" phase. The incomplete matrix modeling approach will help to formulate the design problems for wireless caching and distributed computing systems. §.§ A Generalized Low-Rank Optimization Paradigm We have presented an effective and general framework to model various network side information in UDNs. Next we present a low-rank optimization formulation to exploit the available network side information. The side information modeling matrix M as shown in Fig. <ref> (c) helps cancel interference over n channel uses, yielding an interference-free channel with 1/n degrees-of-freedom (DoF), i.e., the first-order data characterization. Observe that the rank of the side information modeling matrix M, denoted by rank (M), equals the number of channel uses n, which equals the inverse of the achievable DoF. To maximize the achievable degrees-of-freedom (DoF), we thus can minimize the rank of the side information modeling matrix,yielding thefollowing generalized low-rank optimization problem: minimize_M∈ℂ^p× qrank(M) subject to M∈𝒟, where the constraint set 𝒟 encodes the network side information. Low-rank optimization has been proved to be a key design tool in machine learning, high-dimensional statistics, signal processing and computational mathematics <cit.>. The rank function is nonconvex and thus is computationally difficult, but convexifying it leads to efficient algorithms. For example, the nuclear norm (i.e., the summation of singular valves of a matrix) provides a convex surrogate of the rank function that is analogous to the ℓ_1-norm relaxation of the cardinality of a vector.Given the special structure of the side information modeling matrix in UDNs, most existing algorithmic and theoretical results for low-rank optimization are inapplicable. The recent work <cit.> contributed a novel proposal of nonconvex paradigms for solving the generalized low-rank optimization problem (<ref>) by optimizing over the nonconvex rank constraints directly via Riemannian optimization and matrix factorization. Fig. <ref>illustrates the phase transition behavior for the generalized low-rank optimization in topological interference management, which characterizes the relationships between the achievable DoF and the number of connected interference links on average.Given the rank, representing the achievable DoF, with more connected interference links, the success probability for recovering the incomplete side information modeling matrix is lower. It thus provides the guidelines for network deployment in dense wireless networks, content placement in cache-aided interference channels, and dataset placement in wireless distributed computing systems. § OPTIMIZATION ALGORITHMS AND ANALYSISWe have seen quite a few algorithmic challenges for the sparse and low-rank modeling frameworks for UDNs. In this section, we present some new trends in optimizationalgorithms for solving the generalized sparse and low-rank optimization problems in the forms of (<ref>) and (<ref>), respectively. Basically, numerical optimization algorithms can be classified in terms of first versus second order methods, depending on whether they use only gradient-based information versus calculations of both the first and second derivatives. The convergence rates of second-order methods are usually faster with the caveat that each iteration is more expensive. In general, there is a trade-off between the per-iteration computation cost versus the total number of iterations, though first-order methods often scale better to large-scale high-dimensional statistics problems <cit.>. While optimization problems in communication systems are typically solved in the convex paradigm with the second-order methods, thanks to the ease of use of the CVX toolbox, we have observed the necessity of the first-order methods and the importance of the nonconvex paradigm, as will be elaborated in the following parts.§.§ Convex Optimization AlgorithmsWe have presented a variety of methodologies to convexify the nonconvex objective functions and nonconvex constraints for the generalized sparse optimization problem (<ref>). Newton iteration based interior-point methods supported by many user-friendly software packages (e.g., CVX) provide a general way to solve constrained convex optimization problems. However, the cubic computational complexity of each Newton step limits its capabilityto scale to large network sizes in UDNs. This motivates enormous research efforts to improve the computational efficiency for convex programs, including the techniques of first-order methods, randomization, parallel and distributed computing. Parallel and distributed optimization provides a principled way to exploit the distributed computing environments to increase the levels of scalability, while reducing the communication costs. To solve a general large-scale convex programs, a principled two-stage framework has recently been proposed in <cit.> with the capability of providing certificates of infeasibility, enabling parallel and scale computing. This is achieved, in the first stage, by the matrix stuffing technique to fast transform the original convex programs into the standard conic optimization problem form via updating the associated values in the pre-stored structure of the standard conic program. In the second stage, the ADMM based algorithm is adopted to solve the standard large-scale conic optimization problem via exploiting the problem structures <cit.> to enable parallel cone projection at each iterate. Other lines of works have focused on the use of first-order methods and randomization to solve large convex programs. In particular, for sparse convex optimization problems,Frank-Wolfe-type algorithms (a.k.a., conditional gradient) have recently gained enormous interests, fueled by the excellent scalability with projection-free operations via exploiting the well-structured sparsity constraints. The coordinate descent method has gained its popularity for scalability by choosing a single coordinate (or a block of coordinates) to be updated within each iteration, thereby reducing the iteration computing cost. Approximation techniques, including randomization methods and sketching methods, further provide algorithmic opportunities to enable scalability for, in particular, first-order methods, via speeding up numerical linear algebra or reducing problem dimensions. In particular, the stochastic gradient method provides a generic way to stochastically approximate the gradient descent method to solve large-scale machine learning problems. All the above presented algorithmic and theoretical results may be leveraged to solve large-scale convex optimization problems in UDNs. §.§ Nonconvex Optimization AlgorithmsRecently, a new line of work has attracted significant attentions, which focuses on solving the nonconvex optimization problems directly via developing efficientnonconvex procedures, sometimes with optimality guarantee. We have seen recent progress on nonconvex procedures based on various algorithms (e.g., projected/stochastic/conditional gradient methods, Riemannian manifold optimization algorithms) for a class of high-dimensional statistical problems and machine learning problems, including low-rank matrix completion, phase retrieval and blind deconvolution, to name just a few. In particular, optimization by directly exploiting problems' manifold structures is becoming a general and powerful approach to solve various nonconvex optimization problems. The structured constraints such as rank and orthogonality appear in many machine learning applications, including sensor network localization, dimensionality reduction, low-rank matrix recovery, phase synchronization, and community detection. At a high-level standpoint, Riemannian optimization is the extension of standard unconstrained optimizationsearching in the Euclidean space to optimizing in the Riemannian manifold spaceby generalizing the concepts such as the gradient and Hessian <cit.>. A graphic representation ofRiemannian optimization algorithms is illustrated in Fig. <ref>. Specifically, the Euclidean gradient ∇ f(X_k) needs to be projected to the tangent space T_X_kℳ of manifold ℳ to definea search direction ξ_X_k (which can be computed based on the principles of the conjugate gradient method or trust-region method), followed by the retraction operator ℛ_X_k to define a new iterate X_k+1=ℛ_X_k(αξ_k) (α is the step size) on the manifold ℳ. In particular, we exploit the manifold geometry of fixed-rank matrices to solve the low-rank optimization problem (<ref>) efficiently. Fig. <ref> (b) demonstrates the effectiveness of Riemannian optimization based methods. It shows that the Riemannian optimization enjoys fast convergence rates, e.g., compared with an existing approach based on alternating minimization.§ CONCLUSIONS AND FUTURE DIRECTIONSThis article presented generalized sparse and low-rank optimization techniques for optimizing across communication, computation and storage resources in UDNs by exploitingnetwork structures and side information. Illustrated by important application examples, various structured sparse modeling methods were introduced, and an incomplete matrix representation was presented to model different types of network side information. Methodologies of designing scalable algorithms were discussed, including both convex and nonconvex methods. The presented results and methodologies demonstrated the effectiveness of structured optimization techniques for designing UDNs.Despite the encouraging progress, there still remain a variety of interesting open questions. To date, generalized sparse and low-rank optimization techniques are mainly applied to improve the network energy efficiency and spectral efficiency in UDNs. However, emerging mobile applications have strong demands for user privacy and ultra-low latency communications, which call for more general mathematical models and formulations. Other interesting problems concern the theoretical analysis for the generalized sparse and low-rank optimization models and algorithms. Although we have seen significant progresses for theoretical understanding of sparse and low-rank optimization problems via convex relaxation approaches <cit.> and nonconvex procedures, it is challenging to apply existing results to the generalized sparse and low-rank optimization problems (<ref>) and (<ref>)due to the complicated structures. Finally, there are a variety of interesting research directions associated with improving the computational scaling behaviour of various algorithms via recent proposals, e.g., randomized algorithms based on sketching.ieeetr10Tony_CRAN17 T. Q. S. Quek, M. Peng, O. Simeone, and W. Yu, Cloud Radio Access Networks: Principles, Technologies, and Applications. Cambridge University Press, 2017.Haijun_WCmag16 H. Zhang, Y. Dong, J. Cheng, M. J. Hossain, and V. C. M. Leung, “Fronthauling for 5G LTE-U ultra dense cloud small cell networks,” IEEE Wireless Commun. Mag., vol. 23, pp. 48–53, Dec. 2016.Ververidis_CST14 G. Xylomenos, C. N. Ververidis, V. A. 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Letaief, “Low-rank matrix completion for topological interference management by Riemannian pursuit,” IEEE Trans. Wireless Commun., vol. 15, pp. 4703–4717, Jul. 2016.Thomas_IEEEAccess20155G G. Wunder, H. Boche, T. Strohmer, and P. Jung, “Sparse signal processing concepts for efficient 5G system design,” IEEE Access, vol. 3, pp. 195–208, 2015.Haijun_JSAC17 H. Zhang, S. Huang, C. Jiang, K. Long, V. C. M. Leung, and H. V. Poor, “Energy efficient user association and power allocation in millimeter-wave-based ultra dense networks with energy harvesting base stations,” IEEE J. Sel. Areas Commun., vol. 35, pp. 1936–1947, Sept. 2017.Jafar_TIT2013TIM S. Jafar, “Topological interference management through index coding,” IEEE Trans. Inf. Theory, vol. 60, pp. 529–568, Jan. 2014.Ali_ComMag17 S. Li, M. A. Maddah-Ali, and A. S. Avestimehr, “Coding for distributed fog computing,” IEEE Commun. Mag., vol. 55, pp. 34–40, Apr. 2017.Wainwright2014structured M. J. Wainwright, “Structured regularizers for high-dimensional problems: Statistical and computational issues,” Annu. Rev. Stat. Appl., vol. 1, pp. 233–253, 2014.Romberg_JSTSP16lrmc M. A. Davenport and J. Romberg, “An overview of low-rank matrix recovery from incomplete observations,” IEEE J. Sel. Topics Signal Process., vol. 10, pp. 608–622, Jun. 2016.Tropp_livingedge2014 D. Amelunxen, M. Lotz, M. B. McCoy, and J. A. Tropp, “Living on the edge: phase transitions in convex programs with random data,” Inf. Inference, vol. 3, pp. 224–294, Jun. 2014.§ BIOGRAPHIES Yuanming Shi [S'13-M'15] ([email protected]) received the B.S. degree from Tsinghua University in 2011, and the Ph.D. degree from The Hong Kong University of Science and Technology (HKUST) in 2015. He is currently an Assistant Professor at ShanghaiTech University. He received the 2016 IEEE Marconi Prize Paper Award and the 2016 Young Author Best Paper Award by the IEEE Signal Processing Society. His research interests include dense wireless networks, intelligent IoT, mobile AI,machine learning, statistics, and optimization. Jun Zhang [M'10-SM'15] ([email protected]) received the Ph.D. degree from the University of Texas at Austin. He is currently a Research Assistant Professor at Hong Kong University of Science and Technology. He received the 2016 Marconi Prize Paper Award in Wireless Communications, and the 2016 IEEE ComSoc Asia-Pacific Best Young Researcher Award. His research interests include dense wireless cooperative networks, mobile edge caching and computing, cloud computing, and big data analytics systems. Wei Chen [S'05-M'07-SM'13] ([email protected]) received his BS and Ph.D. degrees (Hons.) from Tsinghua University in 2002 and 2007, respectively. Since 2007, he has been on the faculty at Tsinghua University, where he is a tenured full Professor and a member of the University Council. He is a member of the National 10000-Talent Program and a Cheung Kong Young Scholar. He received the IEEE Marconi Prize Paper Award and the IEEE Comsoc Asia Pacific Board Best Young Researcher Award. Khaled B. Letaief [S'85-M'86-SM'97-F'03] ([email protected]) received Ph.D. Degree from Purdue University, USA. From 1990 to 1993, he was faculty member at University of Melbourne, Australia.He has been with HKUST since 1993 where he was Dean of Engineering.From September 2015, he joined HBKU in Qatar as Provost.He is Fellow of IEEE, ISI Highly Cited Researcher, and recipients of many distinguished awards.He served in many IEEE leadership positions including ComSoc Vice-President for Technical Activities and Vice-President for Conferences. | http://arxiv.org/abs/1709.09103v1 | {
"authors": [
"Yuanming Shi",
"Jun Zhang",
"Wei Chen",
"Khaled B. Letaief"
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"published": "20170926160053",
"title": "Generalized Sparse and Low-Rank Optimization for Ultra-Dense Networks"
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http://arxiv.org/abs/1709.09673v2 | {
"authors": [
"Abhinav Prem",
"Michael Pretko",
"Rahul Nandkishore"
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"title": "Emergent Phases of Fractonic Matter"
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acks § ACKNOWLEDGMENTStocsectionAcknowledgments 1]Charles Corbière 1,2]Hedi Ben-Younes 1]Alexandre Ramé 1]Charles Ollion [1]Heuritech, Paris, France [2]UPMC-LIP6, Paris, France [ ]{corbiere,hbenyounes,rame,ollion}@heuritech.comLeveraging Weakly Annotated Data for Fashion Image Retrieval and Label Prediction [===================================================================================empty In this paper, we present a method to learn a visual representation adapted for e-commerce products. Based on weakly supervised learning, our model learns from noisy datasets crawled on e-commerce website catalogs and does not require any manual labeling. We show that our representation can be used for downward classification tasks over clothing categories with different levels of granularity. We also demonstrate that the learnt representation is suitable for image retrieval. We achieve nearly state-of-art results on the DeepFashion In-Shop Clothes Retrieval and Categories Attributes Prediction <cit.> tasks, without using the provided training set. § INTRODUCTION While online shopping has been an exponentially growing market for the last two decades, finding exactly what you want from online shops is still not a solved problem. Traditional fashion search engines allow consumers to look for products based on well chosen keywords. Such engines match those textual queries with meta-data of products, such as a title, a description or a set of tags. In online luxury fashion for instance, they still play an important role to address this customer pain point: 46% of customers use a search engine to find a specific product; 31% use it to find the brand they're looking for [http://www.mckinsey.com/business-functions/marketing-and-sales/our-insights/the-opportunity-in-online-luxury-fashion]. However, those meta-data informations may be incomplete, or use a biased vocabulary. For instance, a description may denote as "marinière" a long sleeves shirt with blue/white stripes. It then appears crucial for online retailers to have a rich labeled catalog to ensure good search. Moreover, these search engines don't incorporate the visual information of the image associated to the product. Computer vision for fashion e-commerce images has drawn an increasing interest in the last decade. It has been used for similarity search <cit.>, automatic image tagging <cit.>, fine-grained classification <cit.> or N-shot learning <cit.>. In all of these tasks, a model's performance is highly dependent on a visual feature extractor. Using a Convolutional Neural Network (CNN) trained on ImageNet <cit.> provides a good baseline. However, there are two main problems with this representation. First, it has been trained on an image distribution that is very far from e-commerce, as it has never (or rarely) seen such pictures. Second, the set of classes it has been trained on is different from a set of classes that could be meaningful in e-commerce. A useful representation should separate different types of clothing (e.g. a skirt and a dress), but it should also discriminate between different lengths of sleeves for shirts, trouser cuts, types of handbags, textures, colors, shapes,... Our goal is to learn a visual feature extractor designed for e-commerce images. This representation should: * encode multiple levels of visual semantics: from low level signals (color, shapes, textures, fabric,...) to high level information (style, brand),* be separable over visual concepts, so we can train very simple classifiers over clothing types, colors, attributes, textures, etc., * provide a meaningful similarity between images, so we can use it in the context of image retrieval.To these ends, we train a visual feature extractor on a large set of weakly annotated images crawled from the Internet. These annotations correspond to the textual description associated to the image. The model is learned on a dataset at zero labeling cost, and is exclusively constituted of data points extracted from e-commerce websites. Our main contribution is an in-depth analysis of the model presented in <cit.>, through applications to fashion image recognition tasks such as image retrieval and attribute tagging. We also improved the method by upgrading the CNN architecture and dealt with multiple languages, mainly English and French. In Section <ref>, we explain the model, how we handle noise in the dataset, as well as some implementation details. In Section <ref>, we provide results given by our representation on image retrieval and classification, over multiple datasets. Finally in Section <ref>, we conclude and go over some possible improvement tracks. § LEARNING IMAGE AND TEXT EMBEDDINGS WITH WEAK SUPERVISIONOne major issue in applied machine learning for fashion is the lack of large clothing e-commerce datasets with a rich, unique and clean labeling. Some very interesting work has been done on collecting datasets for fashion <cit.>. However, we believe it is very hard to be exhaustive in describing every visual attribute (pieces of clothing, texture, color, shape, etc.) in an image. Moreover, even if this labeling work could be perfectly carried, it would come at very high cost, and should be manually done each time we wanted to add a new attribute. A possible source of annotated data is the e-commerce website catalogs. They provide a great amount of product images associated with descriptions, such as the one in Figure <ref>. While this description contains information about the visual concepts in the image, it also comes with a lot of noise that could harm the learning. We explain now the approach we used to train a visual feature extractor on noisy weakly annotated data.§.§ Weakly Supervised ApproachLearning with noisy labeled training data is not new to the machine learning and computer vision community <cit.>. Label noise in image classification <cit.> usually refers to errors in labeling, or to cases where the image does not belong to any of the classes, but mistakenly has one of the corresponding labels. In our setting, in addition to these types of noise, there are some labels in the classes vocabulary that are not relevant to any input. Text descriptions are noisy as they contain common words (e.g. 'we', 'present'), subjective words (e.g. 'wonderful') or non visual words (e.g. 'xl', 'cm'), which are not related to the input image. As we don't have any prior information on which labels are relevant and which are not, we keep the preprocessing of textual data as light as possible. §.§ ModelOur work builds upon the one presented in <cit.>, which we explain in this section. The model's training scheme is exposed in Figure <ref>Let x ∈ℐ be an image, and y ∈{0,1}^K the associated multi-label vector, such that ∀ k ∈ [1, K], y^k = 1 if the k-th label of the vocabulary is true for the image x.We use a CNN to compute a visual feature z = f(x, θ) ∈ℝ^I, where θ are the weights of our convolutional neural network. This image embedding is given to a classification layer:ŷ = softmax( W^T z )where W ∈ℝ^I × K. Note that for all k ∈ [1,K], the column vector in w_k = W[:,k] corresponds to the embedding of the k-th word in the vocabulary. §.§ Label imbalance management As seen on Figure <ref>, the distribution of words in our dataset is highly unbalanced.Due to our minimal preprocessing, we observe a high frequency for some non-visual words such as "xl", "cm" or "size" as they appear very frequently in descriptions. Many examples contain those non-visual words that our model would be asked to predict, which is likely to harm the training.To overcome this issue, and as it was done in <cit.>, we perform uniform sampling. Specifically, during training, we sample uniformly a word w from the vocabulary. We then randomly choose an image x whose bag-of-words contains w and we try to predict w given x.§.§ LossAs we want to predict one label among a vocabulary K for each image, we use the cross-entropy loss. It minimizes the negative sum of the log-probabilities over all positive labelsL(θ, W, 𝒟) = -1/N∑_n=1^N∑_k=1^K y^k_nlogexp(w_k^T f(x_n,θ))/∑_i=1^K exp(w_i^T f(x_n,θ))§.§ Implementation details§.§.§ Negative samplingWe operate in a context where the vocabulary can be of arbitrary size. Computing probabilities for all those classes for each sample can be very slow. Negative sampling <cit.> is one way of addressing this problem. After selecting a positive label for an image sample, we randomly draw N_neg negative words within the vocabulary. We compute the scores and the softmax only over those chosen words.§.§.§ Learning We trained our model with stochastic gradient descent (SGD) on batch of size 20. We consider that an epoch is achieved when the model saw a number of images equivalent to 1/10 of the dataset size, which is approximately 1.3M images in total. After each epoch, we compute a validation error based on a held-out validation set. The initial rate was set to 0.1 and divided by 10 after 10 epochs without improvement. We stop the training after 20 epochs without improvements on our validation dataset. We use the ResNet50 architecture <cit.> for the visual feature extractor f(x,θ), with pre-trained weights on ImageNet. Because the last layer has been initialized randomly, we start by learning only the last layer W for 20 epochs, and then we fine-tune the parameters θ in the CNN.§.§ Training datasetWe built a dataset of about 1,3M images and their associated labels from multiple e-commerce website sources, mostly French, English and Italian. We crawled most of the time one image per product, except when multiple images where available. In that case, we consider them as four different samples with same associated bag of labels.For each source, we select the relevant fields to keep (title, category name, description,...) and concatenate them. After lower-casing and removing punctuation, we use the RegexpTokenizer provided by NLTK <cit.> to get a list of words. We remove stopwords, frequent non-relevant words (name of the website, 'collection', 'buy',...) and non alphabetic words. Our final dataset is a list of product images, associated to their respective bag-of-words obtained by the previous preprocessing.We have deliberately kept preprocessing as minimal as possible, so it is easy to scale to many sources. Thus, we need our model to adapt to this noise in the data. After preprocessing and aggregating the multiple sources, our final vocabulary is composed of 218,536 words. We chose to restrain the vocabulary to the 30,000 most frequent words. The average number of labels per sample is 26,88. We split our dataset into a training and a validation dataset. The validation set is made of the same labels as the training dataset and represent 0.5% of the total size. § EXPERIMENTS AND EVALUATION After learning the representation on our large weakly annotated dataset, we want to evaluate this representation. To what extent is this representation useful for tasks such as garment classification, attribute tagging or image retrieval ?§.§ Evaluation datasets We evaluate our representation on 5 datasets: two public datasets (DeepFashion) used for tagging and image retrieval; three in-house datasets used respectively for category classification, fine-grained classification and image retrieval.DeepFashion Categories and Attributes Predictionevaluates the performance on clothing category classification, and on attribute prediction (multi-labelling). It contains 289,222 images for 50 clothing categories and 1,000 clothing attributes. While an image can only be affected to one class, it can be associated to multiple labels. The average number of labels for an image is 3.38. For each image in train and test sets, we select a crop available from a ground truth bounding box.DeepFashion In-Shop Retrievalcontains 7,982 clothing items with 52,712 images. 3,997 classes are for training (25,882 images) and 3,985 items are for testing (28,760 images). Thetestsetis composed of aquerysetand a gallery set, where query set contains 14,218 images of 3,985 items and database set contains 12,612 images of 3,985 items. As in the Categories and Attributes Prediction benchmark, we cropped each image using a ground truth bounding box.ClothingType We have labeled a dataset with 18 classes, each one corresponding to a garment type (e.g. bag, dress, pants, shoes, ...). This in-house dataset contains approximately 736,000 images.HandBag In addition to the previous dataset, this in-house dataset focuses on bags for fine-grained recognition. Here, the differences between classes are more subtle: bucket bag, doctor bag, duffel bag, etc... It contains 3,060 samples within 13 classes, each one corresponding to a specific type of handbag.Dress Retrieval This in-house similarity dataset was gathered by crawling an e-commerce website. We collected a list of sets of images, each corresponding to the same item. We used an image classifier to filter out all non-dress items. The final dataset contains 9,009 items for training (20,200 images) and 1,001 items for testing.On this dataset, we keep only images where clothing are worn on humans. §.§ Image retrievalIn this task, given a query image containing an item, we aim at retrieving images that contain the same item. To do so, we compute the score between two images using the cosine similarity between their representation. For a given query image, we sort all gallery images in decreasing order of similarity, and evaluate our retrieval performance using top-k retrieval accuracy, as in <cit.>. For a given test query image, we give the model a score of 1 if an image of the same item is within the k highest scoring gallery images, 0 else. We adopt this metric for both our image retrieval datasets (DeepFashion In-Shop Retrieval and Dress Retrieval).In Figure <ref>, we show the results on top-k retrieval accuracy on DeepFashion In-shop Retrieval dataset, for multiple values of k. FashionNet corresponds to the model presented in <cit.>, and HDC+Contrastive is the model in <cit.>. We denote by [F] (resp. [C]) models that use the full image (resp. an image cropped on the product) to compute retrieval scores. We provide the ImageNet baseline both [F] and [C] models, where we use as feature extractor the penultimate layer of a CNN trained on ImageNet. We would like to emphasize on the fact that our Weakly model, as well as the ImageNet baseline, do not use the training set of DeepFashion In-shop Retrieval, unlike HDC+Contrastive and FashionNet.First, we note that not using bounding boxes for our ImageNet baseline or our Weakly model considerably increase accuracy. Our intuition is that human models in the DeepFashion In-shop dataset often wear the same ensemble of items together, meaning for one shirt item considered for instance, the human model would be wearing the same pants and shoes on all item's image. As a consequence of this bias, it seems easier to evaluate similarity on a ensemble of clothings than on a single clothing on this dataset. Our Weakly model without crop performs as well as FashionNet, and even outperforms it when k ≥ 20: considering top-20 retrieval accuracy, it predicts the correct item 78,1% of the time, against 76,4% for FashionNet. Besides, in both the [F] and [C] setups, our Weakly model improves over the ImageNet baseline (from 48% to 78.1% for [F], and from 43.7% to 64.0% for [C]). This validates our hypothesis that our model has learned a specific e-commerce representation. In Figure <ref>, we show an example of a query image, its top-5 similar images according to our weakly learned visual features, and its top-5 similar images according to ImageNet based visual features. As we can see, the similarity encoded by network trained on ImageNet brings together products that are on a same coarse semantic concept, while our representation encodes a more precise and rich closeness, which is based not only the image type, but also on their shape, texture, and fabric. Plus, our representation seems less dependent to human model's pose. On our in-house dress retrieval dataset, we also observed that the Weakly model improved over ImageNet Model on retrieval accuracy. The Weakly model obtained a top-20 retrieval accuracy of 83,71%, against 65,65% for the ImageNet model. Once again, we point out that we do not perform any training on the retrieval task of this dataset. §.§ Tagging We conducted multi-class classification and multi-labelling experiments to assess the quality of our visual representation on transfer learning. On the public DeepFashion Categories dataset, we pre-computed images representation using our Weakly image feature extractor on image crops. Then, we train a simple classifier using a fully-connected layer followed by a softmax activation function. The results are shown on the Table <ref>, at the column Category. With this simple classifier, our results are on par with the state-of-art model by Lu et al. <cit.>. On DeepFashion Attributes, we train a fully-connected layer with a sigmoid output and a binary cross-entropy loss. As we can see in Table <ref>, our model significantly improves over previous state-of-the-art on textures and shape labels top-k recall. However, part and style attributes seem more difficult to separate for our Weakly representation. This might be due to the fact that texture-like and shape-like labels are more represented than part and style words in the large weak dataset. This would require further investigation.We carried out experiments on our in-house ClothingType dataset where images are annotated according to their clothing category (such as bags, shirt, dress, shoes, etc.). Table <ref> shows the improvement on AUC scores over the ImageNet model for each of the clothing categories using our new representation. This indicates that our training scheme was able to learn discriminative features for garment classification. Finally, we now focus on a fine-grained recognition task. The HandBag dataset contains images annotated with their specific type of bag. In this dataset, the differences between classes are more subtle than in the ClothingType dataset. The training and evaluation are the same as for the previous experiment. As in the previous experiment, we improved AUC scores for nearly each type of bags (see Table <ref>).§.§ Exploratory visualization using t-SNETo obtain some insight about our Weakly representation, we applied t-SNE <cit.> on features extracted using our Weakly feature extractor. We did this for 1,000 images from DeepFashion Categories test set. Figure <ref> shows full map and some interesting close-ups. On top left (a), we can see a cluster of dresses sub-divised into multiple sub-clusters corresponding to different colors. The cluster (b) shows a focus on black pants. In the zone (c), we can easily see that the model gathered images containing stripes, and it seems like it has separated tops from dresses inside this cluster (with large striped sweaters on top). Checked clothings are grouped in cluster (d), while printed t-shirts are represented in cluster (e). This plot shows that our representation is able to group together concepts that are close in terms of clothing type, texture, color and style. z § DISCUSSION AND FUTURE WORKWe presented in the future a method to learn a visual representation adapted to fashion. This method has the major advantage to overcome the issue of finding a large and clean e-commerce dataset. The results shows clear improvements compared to a visual representation trained on ImageNet, improving performance on multiple tasks such as image retrieval, classification and fine-grained recognition. In the future, we would like to investigate on the possibility to better train our visual feature extractor using an external knowledge base of textual concepts. The authors would like to thank all the Heuritech team for providing an efficient network infrastructure. We'd also like to thank Alexandre Ramé for his help on the transfer learning evaluation task. ieee | http://arxiv.org/abs/1709.09426v1 | {
"authors": [
"Charles Corbière",
"Hedi Ben-Younes",
"Alexandre Ramé",
"Charles Ollion"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170927100210",
"title": "Leveraging Weakly Annotated Data for Fashion Image Retrieval and Label Prediction"
} |
firstpage–lastpage 2017Validating neural-network refinements of nuclear mass models J. Piekarewicz Accepted 2017 September 15. Received 2017 September 6; in original form 2017 June 26. =========================================================================================High-resolution mid-infrared observations carried out by the Spitzer Space Telescope allowed one to resolve the fine structure of many astrospheres. In particular, they showed that the astrosphere around the B0.7 Ia star κ Cas (HD 2905) has a clear-cut arc structure with numerous cirrus-like filaments beyond it. Previously, we suggested a physical mechanism for the formation of such filamentary structures. Namely, we showed theoretically that they might represent the non-monotonic spatial distribution of the interstellar dust in astrospheres (viewed as filaments) caused by interaction of the dust grains with the interstellar magnetic field disturbed in the astrosphere due to colliding of the stellar and interstellar winds. In this paper, we invoke this mechanism to explain the structure of the astrosphere around κ Cas. We performed 3D magnetohydrodynamic modelling of the astrosphere for realistic parameters of the stellar wind and space velocity. The dust dynamics and the density distribution in the astrosphere were calculated in the framework of a kinetic model. It is found that the model results with the classical MRN size distribution of dust in the interstellar medium do not match the observations, and that the observed filamentary structure of the astrosphere can be reproduced only if the dust is composed mainly of big (μm-sized) grains. Comparison of the model results with observations allowed us to estimate parameters (number density and magnetic field strength) of the surrounding interstellar medium.shock waves – methods: numerical – stars: individual: κ Cas – (ISM:)dust, extinction. § INTRODUCTIONInteraction of the stellar wind with the circum- and interstellar medium (ISM) results in the formation of structures called astrospheres. Severe interstellar extinction at low Galactic latitudes, where the majority of (massive) wind-blowing stars is concentrated, makes the infrared (IR) observations the most effective way for detection and study of astrospheres (e.g. van Buren, Noriega-Crespo & Dgani 1995). Nowadays, with the advent of the Spitzer Space Telescope, Wide-field Infrared Survey Explorer (WISE) and Herschel Space Observatory, many hundreds of new astrospheres were revealed (Peri et al. 2012; Cox et al. 2012; Kobulnicky et al. 2016). Some of them have a distinct filamentary (cirrus-like) structure (e.g. Gvaramadze et al. 2011a, b). Although the origin of this structure is not well understood, it seems likely that the regular interstellar magnetic field might play an important role in its formation (Gvaramadze et al. 2011b).Recently, Katushkina et al. (2017; hereafter Paper I) presented a physical mechanism possibly responsible for the origin of the cirrus-like structure of astrospheres around runaway stars. It was shown that, under proper conditions, alternating minima and maxima of the dust density (seen like filaments) might appear between the astrospheric bow shock (BS) and the astropause (AP) because of periodical gyromotion of the dust grains around the interstellar magnetic field lines.In the present work, we apply this mechanism to explain the morphology of the astrosphere around the runaway blue supergiant κ Cas (HD 2905). We choose this particular astrosphere because of its distinct cirrus-like structure, as well as because the basic parameters of its associated star are known fairly well (see Section <ref>). In Section <ref>, we perform numerical modelling of interaction between the stellar wind and the magnetized ISM in the framework of a 3D magnetohydrodynamic (MHD) model. In Section <ref>, we present the kinetic modelling of the interstellar dust distribution in the wind-ISM interaction region. In Section <ref>, we post-process the simulations to make synthetic maps of infrared dust emission. In Section <ref>, we compare the model results with observations, estimate the interstellar plasma number density and magnetic field strength, and derive constraints on the dust parameters, required to better reproduce the observations. Summary and discussion are presented in Section <ref>.§ OBSERVATIONAL DATAThe astrosphere around the blue supergiant (B0.7 Ia; Walborn 1972) κ Cas was discovered by van Buren & McCray (1988) using the Infrared Astronomical Satellite (IRAS) all-sky survey and presented for the first time in van Buren, Noriega-Crespo & Dgani (1995; see their fig. 2c). In the IRAS 60 μm image, the astrosphere has an arc-like shape, typical of bow shocks. The low resolution of the IRAS data, however, did not allow to see fine details of the astrosphere, which were revealed only with the advent of the Spitzer Space Telescope and the WISE mission with their much better angular resolution.κ Cas was observed by Spitzer on 2007 September 18 (Program Id.: 30088, PI: A.Noriega-Crespo) using the Multiband Imaging Photometer for Spitzer (MIPS; Rieke et al. 2004). We retrieved the post-basic data calibrated MIPS 24 μm image of κ Cas (with units of MJy sr^-1) from the NASA/IPAC infrared science archive[http://irsa.ipac.caltech.edu/]. In this image (presented for the first time in Gvaramadze et al. 2011b) the astrosphere of κ Cas appears (see Fig. <ref>) as a clear arcuate structure with numerous cirrus-like filaments beyond it, some of which are apparently attached to the main (brightest) arc. The surface brightness of this arc is ≈35 MJy sr^-1, while that of the background is ≈20 MJy sr^-1. Fig. <ref> also shows several filaments intersecting the arc at almost right angle to its surface in the south part of the astrosphere. A possible origin of these filaments is discussed in Section <ref>.We define the linear characteristic scale of the astrosphere as the minimum projected distance between κ Cas and the brightest arc, R_0^ obs (see Fig. <ref>), which is related to the observed angular separation between the star and the apex of the arc, Ω, through the relationship R_0^ obs=Ω d, where d is the heliocentric distance to κ Cas. For Ω=2.6 arcmin and d=1 kpc (see below), one has R_0^ obs≈0.75 pc or 2.3×10^18 cm.It is believed that κ Cas belongs to the Cas OB14 association, which is located at a distance of d≈1.0±0.1 kpc (Humphreys 1978; Mel'nik & Dambis 2009). This distance is generally accepted in studies of κ Cas (e.g. Crowther, Lennon & Walborn 2006; Searle et al. 2008). The presence of the astrosphere around κ Cas, however, suggests that this star is a runaway and that it might be formed far away from its present position on the sky (cf. Gvaramadze, Pflamm-Altenburg & Kroupa 2011). Correspondingly, κ Cas is not necessary a member of Cas OB14, unless this star has obtained its peculiar space velocity because of dissolution of a binary system in a recent supernova explosion in the association. In this connection, we note that among four members of the association listed in Humphreys (1978) one more star, HD 2619 (B0.5 III), produces a bow shock as well. The orientation of this bow shock (visible in WISE 22 and 12 μm images) suggests that HD 2619 was injected in Cas OB14 from the open star cluster Berkeley 59, located at a distance of ≈1 kpc (Pandey et al. 2008) and at ≈35 (or ≈60pc in projection) to the northwest from the star. It is possible therefore that Cas OB14 is actually a spurious association.A somewhat larger distance to κ Cas follows from the Hipparcos parallax (van Leeuwen 2007) and the empirical relationship between the strength of the interstellar Ca ii lines and the distances to early-type stars (Megier et al. 2009), yielding respectively d=1.37^+0.42 _-0.25 and d=1.46±0.30 kpc. Taken at face value, these two distance estimates imply a too high bolometric luminosity of log(L_*/)≈5.8, which along with the effective temperature of κ Cas of T_*=23.5±1.5 kK (Searle et al. 2008) would place this star on the S Doradus instability strip (Wolf 1989) in the Hertzsprung-Russell diagram. Since κ Cas does not show variability typical of stars in this region of the Hertzsprung-Russell diagram, it is likely that it is located at a shorter distance. In what follows, we adopt the distance to κ Cas of d=1 kpc. The basic parameters of κ Cas (T_*, L_*, stellar wind velocity v_∞, mass loss rate Ṁ and radius R_*) are compiled in Table <ref>.In Table <ref> we provide astrometric and kinematic data on κ Cas. The proper motion measurements, μ _αcosδ and μ _δ, are based on the new reduction of the Hipparcos data by van Leeuwen (2007). The heliocentric radial velocity of the star, v_ r,hel, is taken from Gontcharov (2006). Using these data, the Solar galactocentric distance R_0 = 8.0 kpc and the circular Galactic rotation velocity Θ _0 =240 (Reid et al. 2009), and the solar peculiar motion (U_⊙,V_⊙,W_⊙)=(11.1,12.2,7.3) (Schönrich, Binney & Dehnen 2010), we calculated the peculiar transverse velocity V_ tr=(v_ l^2 +v_ b^2)^1/2, where v_ l and v_ b are, respectively, the velocity components along the Galactic longitude and latitude, the peculiar radial velocity v_ r, and the total space velocity V_* of the star. For the error calculation, only the errors of the proper motion and the radial velocity measurements were considered. The obtained space velocity of ≈30 implies that κ Cas is a classical runaway star (e.g. Blaauw 1961).The orientation of the symmetry axis of astrospheres around moving stars is determined by the orientation of the stellar velocity relative to the local ISM. In a static, homogeneous ISM and for a spherically-symmetric stellar wind, the symmetry axis of an astrosphere is aligned with the vector of stellar space motion. In this case, the geometry of detected astrospheres (bow shocks) can be used to infer the direction of stellar motion and thereby to determine possible parent clusters for the bow-shock-producing stars (e.g. Gvaramadze & Bomans 2008). In reality, however, the ISM might not necessary be at rest owing to the effects of nearby supernova explosions, expanding regions, or outflows from massive star clusters. Also, the shape of astrospheres of hot (runaway) stars might be affected by photoevaporation flows from nearby regions of enhanced density (cloudlets) caused by ultraviolet emission of these stars (e.g. Mackey et al. 2015; Gvaramadze et al. 2017).Fig. <ref> shows that the vector of the peculiar (transverse) velocity of κ Cas is misaligned with the symmetry axis (median line) of the astrosphere by an angle α≈35. This misalignment might be caused by inaccuracy of the space velocity calculation[Note that the misalignment would be even stronger if κ Cas is located at d>1 kpc.] or by the presence of a regular flow in the local ISM. We consider the latter possibility to be more likely because the Hipparcos proper motion measurement for κ Cas is very reliable (see Table <ref>). We suggest, therefore, that the orientation of the astrosphere around κ Cas is affected by a flow of the local ISM.To reconcile the orientation of the astrosphere around κ Cas with the orientation of the stellar transverse motion, one needs to assume that the ISM is moving in the north-south direction (i.e. almost along the Galactic longitude) with a transverse velocity of V_ ISM,tr≈15. In this case, the transverse component of the relative velocity between the star and the ISM is V_ rel,tr≈26. Allowing the possibility that the ISM could have a velocity component in the radial direction as large as in the transverse one, i.e. ±15, one finds that the total relative velocity, V_ rel, could range from 26 to 42. For the sake of certainty, in our calculations we adopt an intermediate value of V_ rel of 35, which corresponds to the angle between the vector of the total relative velocity and the line of sight of θ=48. Note that the orientation of the astrosphere would not change if the transverse velocity of the ISM has a component in the east-west direction as well (i.e. parallel to the Galactic plane). The actual value of V_ rel (or the sonic Mach number), however, is not critically important for our calculations because in the considered case (see below) the overall structure of the astrosphere is mostly determined by the interstellar magnetic field.§ NUMERICAL MODELTo compare the model astrospheres with observations one needs to produce synthetic maps of the thermal emission from the dust accumulated at the region of interaction between the stellar wind and the surrounding ISM. For this, the following steps should be performed: 1) MHD modelling of the plasma and magnetic field distributions in the astrosphere, 2) kinetic modelling of the interstellar dust distribution in the wind-ISM interaction region, and 3) determination of the dust temperature and calculation of the thermal emission intensity. These steps are presented in the next subsections. §.§ 3D MHD modelling of the astrosphereTo determine the distribution of plasma and magnetic field in and around the astrosphere we use the steady-state 3D MHD model described in Paper I. Here we also take into account the radiative cooling of plasma with the cooling function from Cowie et al. (1981). Note that the interstellar magnetic field is essential for this study due to two reasons. Firstly, the proposed physical mechanism for formation of filaments is based on gyrorotation of dust grains around magnetic field lines frozen into the interstellar plasma. Secondarily, in the absence of the magnetic field the radiative cooling strongly reduces the thickness of the layer between the AP and the BS because of loss of energy (see Fig. <ref>). Therefore, the filaments formed in this region would be unresolvable. In the presence of significant interstellar magnetic field the outer shock layer does not collapse (see Fig. <ref>) and the filaments might be observable.Note that hereafter all distances in plots are dimensionless, they are normalized to the stand-off distance, which is the distance from the star to the AP in the upwind direction in the unmagnetized medium:D^*=√(Ṁ v_∞/4πρ_ p,ISM V_ rel ^2),where ρ_ p,ISM= 1.4n_ p,ISM m_ p, n_ p,ISM is the ISM number density of protons and m_ p is the proton mass.In the dimensionless form the solution of the problem depends on the sonic (M_ ISM) and Alfvenic (M_ A,ISM) Mach numbers of the ISM. In general, there is one more dimensionless parameter characterizing the efficiency of the radiative cooling. However, in the case of relatively high ISM number density (n_ p,ISM≥2 cm^-3, see Section <ref>) and for the adopted cooling function the plasma temperature in the outer shock region (i.e. between the AP and the BS) remains almost constant (see the solid curve in Fig. <ref>) and, correspondingly, the results do not depend on the dimensionless parameter related to the radiative cooling.We perform calculations with the “perpendicular" interstellar magnetic field, i.e. with the magnetic field vector in the undisturbed ISM (B_ ISM) perpendicular to the relative ISM velocity vector (V_ ISM=- V_ rel). We choose this orientation of the magnetic field because the nose part of the astrosphere around κ Cas seems to be axisymmetric and because we know from Paper I that in the case of parallel magnetic field the dust is accumulated in filaments at flanks of the astrosphere and is absent in its nose part (see fig. 10 in Paper I). The effect of the stellar magnetic field is neglected in our calculations.We assume that the ISM temperature is ≈7000-8000 K, which along with V_ rel=35 (see Section <ref>) corresponds to the sonic Mach number of M_ ISM≈3. We performed calculations for several values of M_ A,ISM. The magnitude of the interstellar magnetic field (and Alfvenic Mach number) determines the thickness of the outer shock layer. It is found that the best qualitative agreement between the model results and the observations could be achieved for M_ A,ISM≈1.5-3 (see discussion in Section <ref>). The main part of the calculations presented in this work is performed for M_ A,ISM=1.77.Fig. <ref> plots 2D distributions of the plasma density and the magnetic field in the ( V_ ISM, B_ ISM)-plane. Note that the thickness of the outer shock layer between the AP and the BS is determined by the chosen Alfvenic Mach number, i.e. the weaker the magnetic field (or the larger M_ A,ISM) the thinner the layer. §.§ Kinetic modelling of the dust distribution in the astrosphereTo calculate the dust distribution in the astrosphere we use the kinetic model described in Alexashov et al. (2016) and Paper I. In general, the dynamics of charged interstellar dust grains in the astrosphere is determined by the following forces: the electromagnetic Lorentz force F_ L, the stellar gravitational force F_ G, the stellar radiation pressure F_ rad, and the drag force F_ drag due to interaction of the dust grains with protons and electrons through the direct and Coulomb collisions. Therefore, the motion equation of a charged dust grain is the following:d v_ d dt=q m_d c( v_ rel× B)+(-GM_* + σ_ dQ̅_ rpL_* 4π m_d c) e_ r r^2 + σ_ d n_ p kT_p Ĝ(v_ rel, T_p) v_ rel | v_ rel|,where q and m_d are the charge and mass of the grain, c is the speed of light, v_d=ẋ_ d, x_ d is the position of a dust grain, r=| x_ d| is the distance from the star, e_ r is the unit vector in the radial direction, v_ p is the local plasma velocity, v_ rel= v_ d- v_ p is the relative velocity between the plasma and the dust grain, B is the magnetic field, G is the gravitational constant, M_* and L_* are the stellar mass and luminosity, σ_ d=π r_ d ^2 is the geometrical cross section of the dust grain, r_ d is the radius of the grain, Q̅_ rp≈1 is the flux-weighted mean radiation pressure efficiency of the grain, n_ p and T_p are the plasma number density and temperature, k is the Boltzmann constant, and Ĝ(v_ rel, T) is the dimensionless function determining the drag force (see, e.g., Draine & Salpeter 1979; Ochsendorf et al. 2014). The plasma parameters (n_ p, v_ p and B) are taken from the MHD model described above.It is assumed that the dust grain charge q is constant along the trajectory and q=U_ d,ISM r_ d, where U_ d,ISM=+0.75 V is the dust surface potential commonly assumed for the local ISM around the Sun (Grün & Svestka 1996). The potential is positive due to an influence of accretion of protons, photoelectric emission of stellar and interstellar radiation, and secondary electron emission (see, e.g. Kimura & Mann 1998; Akimkin et al. 2015).We estimated the relative contribution of the listed forces to the dust dynamics in an astrosphere of κ Cas and found that the stellar gravitation attractive force and the drag force are negligible compared with others. Stellar radiation force is especially important for small dust grains with radii r_ d≤0.1μm. These dust grains are swept out far away by the stellar radiation and do not cross the BS. Therefore, in our calculations we consider the dust grains with r_ d>0.2μm.Classical power-law MRN size distribution (Mathis, Rumpl & Nordsieck, 1977) is assumed for the dust grains in the undisturbed ISM, n_ d,ISM(r_ d)∼ r_ d^-3.5, with the minimum and maximum dust grain radii of 0.2 and 3 μm, respectively. Corresponding mass of the dust grains ranges from 8.37×10^-14 to 2.83×10^-10 g (it is assumed that the dust grains are spherical and have a uniform density ρ_ d=2.5 gcm^-3). It is also assumed that in the undisturbed ISM all dust grains have the same velocity of V_ ISM. The kinetic equation (<ref>) is solved by the imitative Monte-Carlo method.Fig. <ref> plots the distribution of the dust number density in the (B_ ISM, V_ ISM)-plane for dust grains with r_ d=1 μm and 2 μm. In both cases, the filaments are clearly seen. Their formation is caused by periodical gyrorotation of charged dust grains around the magnetic field lines. This physical mechanism is extensively discussed in Paper I. The filaments produced by larger dust grains are sparser because of the larger gyroradius. It is shown in Paper I that the characteristic separation between filaments is:D_ gyr=v_ p,z2π m_d/B q=v_ p,z8 π^2 ρ_ d/3r_ d ^2/B U_d,ISM,where v_ p,z is the component of the plasma velocity along the Z-axis. For small D_ gyr the filaments merge with each other and cannot be resolved. For the chosen model parameters (M_ A,ISM, M_ ISM, ρ_ d and U_ ISM) the distinct filamentary structure is formed for r_ d≈1.5-3μm. From equation (<ref>) it follows that for smaller values of B and/or U_ ISM the filamentary structure of the astrosphere would be discernible if the size of the dust grains would be reduced accordingly. §.§ Synthetic maps of thermal dust emissionThe astrosphere around κ Cas, like the majority of other known astrospheres, is visible only via its infrared emission, whose origin can be attributed mostly to the thermal dust emission (e.g. van Buren & McCray 1988). To compare the model astrosphere with the Spitzer MIPS image we produce a synthetic map of thermal dust emission at 24 μm. For this, we integrate the local emissivity over the line of sight: I_ν(r_ d)=∫ j_ν(r_ d,s)ds, where s is the coordinate along this line.The local emissivity can be expressed as follows:j_ν(r_ d,s)=π r_ d^2 n_ d(s,r_ d) Q_ν (r_ d) B_ν (T_ d(s,r_ d)),where n_ d is the dust number density, Q_ν is the dimensionless efficiency absorption factor of dust, B_ν is the Planck function, T_ d is the dust temperature. Q_ν(r_ d) is calculated using the Mie theory (Bohren & Huffman, 1983) for the chosen dust material. In this work, we consider astronomical silicates (Draine 2003), graphite (Draine 2003) and pure carbon (Jäger, Mutschke & Henning 1998). Plots of Q_ν(r_d) for different materials and grain radii are presented in Fig. <ref>. The dust temperature can be calculated from local energy balance between dust heating by the stellar radiation and cooling due to thermal emission. Details of the temperature calculations are given in Appendix A. The obtained temperature depends on the grain material and radius, and the distance from the star (see Fig. <ref>). Dust number density can be represented as n_ d(s,r_ d)=n̂_d(s,r_ d)n_ d,ISM(r_ d), where n̂_ d is the dimensionless dust number density taken from the model results. In the case of the MRN size distribution, the ISM number density of dust grains with radii in the range [r_ d- dr_ d/2; r_ d+ dr_ d/2] is dn_ d,ISM(r_ d)=N_ ISMr_ d^-3.5 d r_ d. The dimensional coefficient N_ ISM can be found from the assumption that the typical gas to dust mass ratio is equal to 100 (see Appendix B). Thus, the local emissivity of the dust grains with radii in the above range isdj_ν(s,r_ d)=π r_ d^2 Q_ν(r_ d) B_ν(T_ d(s,r_ d))n̂_ d(s,r_ d)N_ ISMr_ d^-3.5dr_ dand the total intensity is:I_ν=πN_ ISM ∫_r_ d,min^r_ d,max dr_ d∫r_ d^-1.5Q_ν(r_ d) B_ν(T_ d(s,r_ d))×n̂_ d(s,r_ d) ds.We also performed calculations for certain dust grain radii with a uniform dust distribution in a narrow range [r_ d,0- dr_ d/2, r_ d,0+dr_ d/2]. In this case, in the formula for the total intensity r_ d^-1.5 should be replaced with r_ d,0^2.§ COMPARISON OF THE MODEL RESULTS WITH OBSERVATIONS AND EVALUATION OF THE ISM PARAMETERSTo compare the model results with the observational data the intensity of the thermal dust emission should be calculated in the plane perpendicular to the line of sight. The orientation of the stellar velocity with respect to the line of sight is determined by two spherical angles θ and ϕ (see Fig. <ref> for illustration). In our calculations the angle θ is fixed at 48 (see Section <ref>), while the angle ϕ is a free parameter of the model because it is determined by the unknown orientation of the ( B_ ISM, V_ ISM)-plane. Therefore, we performed calculations at different planes and found that ϕ≈110-150 provides the best agreement with the observations. The results are presented for intermediate ϕ≈135.Fig. <ref> plots the dust number density obtained for the MRN size distribution in the ISM with the range of grain sizes of r_ d=0.2-3 μm. It is seen that no filaments are visible. The reason for this is that the filaments formed by dust grain with continuous size distribution merge with each other in a wide arcuate structure between the BS and the AP.Before considering the intensity maps, we note that all our model calculations were performed in dimensionless form. In order to transform the dimensionless solution in the dimensional form and find absolute values of intensities one needs to specify the required dimensional parameters of the model (e.g. the characteristic distances and the dust number density in the ISM). This can be done in the following way. We assume that the brightest arc in the astrosphere around κ Cas coincides with the astropause (our numerical calculations below support this assumption). By comparison of the obtained dimensionless solution with the known distance to the brightest arc, R_0^ obs, it is possible to determine the characteristic distance D^*, the corresponding ISM number density and the magnetic field strength, which are consistent with the observations and the model results. Namely, from the numerical modelling we know the dimensionless distance from the star to the astropause in the nose part of the astrosphere, R̂_̂0̂, and:R_0^ obs=R̂_̂0̂D^*.Combining this equation with equation (<ref>) one has:n_ p,ISM=Ṁ v_∞/5.6π m_ pV_ rel^2(R̂_̂0̂/R_0^ obs)^2.Then we obtain B_ ISM from the Alfvenic Mach number:B_ ISM=(5.6π m_ pn_p,ISM)^1/2 V_ rel/M_ A,ISM.The following estimates are obtained: n_ p,ISM=3-11 cm^-3, B_ ISM=18-35μ G. The ranges of n_ p,ISM and B_ ISM are due to uncertainties in Ṁ and v_∞ of κ Cas (see Table <ref>). From the gas to dust mass ratio of 100, one obtains the dust number density and the constant N_ ISM (see Appendix B). Note that the intensity of the thermal dust emission is proportional to N_ISM, which in turn is proportional to n_ p,ISM. All intensity maps presented below are computed for the intermediate value of n_ p,ISM=5 cm^-3 and the corresponding uncertainties in intensity are a factor of few.Fig. <ref> shows the MIPS 24 μm image of the astrosphere around κ Cas along with synthetic maps of emission from silicate, graphite and pure carbon dust grains at the same wavelength. All intensities are given in units of MJy sr^-1. Note that we added a constant intensity of 20 MJy sr^-1 to all synthetic maps to mimic the background emission that is seen in the data. It is seen that for all types of dust grains there is a maximum of intensity at the nose part of the astrosphere close to the astropause. However, no separate filaments are visible. This is explained by two effects. The first one is the same as was discussed above for the distribution of the dust number density – no filaments can be distinguished for the mixture of dust grains with the MNR size distribution. For the intensity maps this effect is even more pronounced than for the number density. The reason is that small dust grains are more heated than the larger ones and therefore their contribution to the total emission intensity is larger. In our model the filaments are clearly seen for grains with radii r_ d=1-2 μm. But these large grains are too cool to contribute to the total intensity maps. Note that the temperature of carbon grains is much higher than the temperature of silicon and graphite ones (Fig. <ref>), but this is still not enough to make filaments visible because the efficiency absorption factor Q_ν at 24 μm for carbon is much smaller than that for graphite and silicon (see Fig. <ref>).The second effect is connected with the distribution of the dust temperature, which decreases with distance from the star because the dust grains are heated mostly by the stellar radiation. Correspondingly, the intensity of the thermal dust emission, which is proportional to the Planck function, is a strong function of temperature and hence of the distance from the star. As a result, the thermal dust emission near the AP is much stronger than near the BS.Fig. <ref> also shows that the emission intensity ratio of the brightest arc of the astrosphere to the background is about an order of magnitude smaller compared with the observations.Thus, we found that the observed filamentary structure cannot be explained in the framework of the model with the classical MRN size distribution. The actual dust size distribution however could differ from the MRN one. For example, Wang, Li & Jiang (2015) reported that the existence of the very large (0.5-6μm) dust grains in the ISM is confirmed by several independent observational evidences (see also Lehtinen & Mattila 1996; Pagani et al. 2010; Steinacker et al. 2015). Moreover, Wang et al. (2015) noted that “if a substantial fraction of interstellar dust is from supernova condensates, then μm-sized grains may be prevalent in the ISM”.Assuming that the large dust grains are indeed prevail in the local ISM, we examine the range of dust parameters for which one can reproduce the observed filamentary structure of the astrosphere around κ Cas. In particular, we assumed that the dust in the local ISM is composed only of big grains (with the gas to dust mass ratio of 100) and performed calculations for two narrow ranges of grain sizes with uniform distribution inside each range: r_ d∈[1.3,1.7] μm and r_ d∈[1.8,2.2] μm; hereafter ranges SO1 and SO2, respectively. Figs <ref> A and <ref> D plot the distributions of the dust number density in the observational plane for graphite and carbon grains with radii in the above two ranges. One can discern five and three filaments, respectively, for SO1 and SO2. The filaments are wider than in Fig. <ref> because now we consider a range of grain radii, while Fig. <ref> was obtained for grains of a particular radius. We also calculated corresponding intensity maps, but the filaments do not appear on them because of the low dust temperature, which rapidly decreases with distance from the star (we do not show these maps since they are very similar to those shown in Fig. <ref>). The temperature obtained as a solution of the local energy balance (see Appendix A) is high enough to produce filaments in the intensity maps at 24 μm only in a narrow region close to the astropause, while at larger distances the dust emission at this wavelength is rapidly deceases.We speculate that the dust grains might be hotter due to some additional heating processes. To check how this will affect the intensity maps, we artificially increased the dust temperature for both graphite and carbon grains by 20 K everywhere in the astrosphere. The resulting emission maps are presented in Figs <ref> B–C and <ref> E–F for both ranges of grain sizes SO1 and SO2. One can see that with the increase of the dust temperature the filaments become more pronounced. Qualitatively, these intensity maps are quite similar to the MIPS image of the astrosphere around κ Cas. The absolute values of the emission intensity are also similar to the observed ones (recall that the calculated intensity is accurate within a factor of few due to the uncertainty in the ISM plasma density estimate). We note also that the dust density maximum visible near the BS in the panels A and D of Fig. <ref> is absent in the intensity maps. This is again because of small dust temperature in this remote part of the astrosphere.Finally, we note that the smaller the dust grains the higher their temperature (see the panel A in Fig. <ref>), which implies that one can avoid the artificial increase of the dust temperature if one adopts smaller dust grain radii. On the other hand, to make the filaments observable, one needs to keep the same separation between them, which for the given dust grain radius is inversely proportional to the magnetic field strength and the dust surface potential (see equation (<ref>)). From this it follows that the decrease of the grain size should be compensated by decrease of B or U_ d,ISM (or both). A strong decrease of the magnetic field strength, however, is less appropriate because, as discussed above, this would lead to the collapse of the outer shock region. The surface potential of the dust grains is, in principle, a free parameter of the model and many different processes can affect its value. If one adopts a factor of 10 smaller potential, then toproduce the same number of filaments the radii of the dust grains could be a factor of ≈3 smaller than those adopted in our modelling. Such grains are hot enough to produce observable filaments in the intensity maps.§ SUMMARY AND DISCUSSIONIn this paper, we performed 3D MHD numerical modelling of the astrosphere around κ Cas in order to produce a synthetic map of its thermal dust emission at 24 μm and to explain its filamentary structure. We found that distinct filaments would appear in the emission map only if quite large (μm-sized) dust grains are prevalent in the local ISM. The filamentary structure is not seen for continuous power-law size distribution of dust because individual filaments merge with each other due to the influence of small grains. Our model with large (1.3-2.2 μm) graphite and pure carbon dust grains reproduces the observational data quite well, if the temperature of these grains in the region where the filaments are formed is about 40 and 75 K, respectively. Comparison of the observed distance from the star to the brightest arc in the astrosphere with the model results allows us to estimate the ISM number density to be 3-11 cm^-3. We also constrain the local interstellar magnetic field strength to be 18-35μG, which exceeds the typical field strength in the warm phase of the ISM (Troland & Heiles 1986; Harvey-Smith et al. 2011).We performed test calculations with larger Alfvenic Mach numbers M_ A,ISM=3 and 12 that corresponds to weaker interstellar magnetic field(10 and 2.7 μG, respectively, for n_ p,ISM=3 cm^-3). The synthetic maps of thermal dust emission for graphite dust grains with radii of 1.45-1.55μm are presented in Fig. <ref>. In this figure the filaments are more distinct compared to, e.g., Fig. <ref> because the more narrow range of grain radii is considered. For smaller B_ ISM the outer shock layer (confined between the BS and the AP) becomes thinner. The model with M_ A,ISM=3 is still appropriate: the filaments are seen and the emission intensity is close to the observed one. Note that the outermost filament in the panel A of Fig. <ref> is in fact located between the AP and the BS, and appears beyond the BS because of the projection effect. The model with M_ A,ISM=12 provides a too small separation between the AP and the BS, so that the filaments merge with each other and the model cannot reproduce the observations for any grain size distribution. It is also interesting to note that there is a small arc close to the star in the case of weak magnetic field (M_ A,ISM=12). This arc is formed by large dust grains penetrating inside the IS. Near the star these grains are swept out by the stellar radiation and appear as an arc.Our numerical calculations are performed under assumption of constant dust charge. In general, the grain charge is determined by the balance between three main processes: impinging of protons and electrons, secondary electron emission due to electron impacts (this is especially important for hot plasma with T10^5 K) and photoelectron emission caused by the external interstellar and stellar radiation. We performed estimations of the changes of the dust charge and found that they are not more than 30 per cent in the considered region between the BS and the AP. Therefore we can neglect them and assume that the dust grain charge does not vary along the trajectory.In our calculations, we neglected the drag forces caused by direct collisions of the dust grains with ions and electrons (direct drag force), and by the electromagnetic Coulomb interaction (Coulomb drag force), although our numerical model allows us to take them into account. Ochsendorf et al. (2014) found that the Coulomb drag force can affect the formation of so-called “dust waves” – arclike enhancements of dust density around weak-wind stars – created because of decoupling of the dust grains from the gas by stellar radiation force. Namely, they showed that inclusion of the Coulomb drag in the model leads to a strong dust-gas coupling, which prevents the formation of the dust waves. To clarify whether or not the drag forces could be important in our calculations, below we discuss the model parameters which determine their relative contributions to the dust motion in the astrosphere. Fig. <ref> shows accelerations (a=F/m_ d) of a dust grain (with radius r_ d=1μm) along its trajectory due to three forces: the Lorentz force (a_ L), the direct drag force (a_ R) and the Coulomb drag force (a_ C). Note that a_ L∼ q/m_ d∼ U_ d/r_ d^2, a_ R∼ n_ p/r_ d, and a_ C∼ n_ p U_ d^2/r_ d. These relations determine the balance between different forces for the chosen model parameters. Accelerations are calculated for the following four pairs of values of the proton number density and the initial dust surface potential: n_ p,ISM=3 cm^-3, U_ d,ISM=0.75 V (hereafter, case 1; see panel A in Fig. <ref>), n_ p,ISM=10 cm^-3, U_ d,ISM=0.75 V (case 2; panel B), n_ p,ISM=3 cm^-3, U_ d,ISM=-1.3 V (case 3; panel C), and n_ p,ISM=10 cm^-3, U_ d,ISM=-1.6 V (case 4; panel D). It is seen from Fig. <ref> that in cases 3 and 4 the Coulomb drag force is larger than the direct drag force because of the large potential of the dust grains. One can see also that the Lorentz force is by one-two orders of magnitude larger than the drag forces in cases 1–3 and becomes comparable to the Coulomb drag force in case 4. Thus, it is seen that for the parameters considered in our paper (n_ p,ISM=3-11 cm^-3 and U_ d,ISM=0.75 V) the influence of the drag forces can be safely neglected. However, for higher gas densities and/or dust potentials their effect could be significant.Our explanation of the cirrus-like structure of the astrosphere around κ Cas implies that the thermal mid-IR emission of the dust originates in regions spatially separated from the region of the bulk optical line emission. Also, we expect that in the optical wavelengths the astrosphere should have a smooth appearance, unless it is deformed by (magneto)hydrodynamic instabilities. But even in this case, the optical filaments should not correlate with the mid-IR cirrus-like ones. In principle, the difference between the mid-IR and optical appearances of an astrosphere could be detected, provided that it is nearby enough to allow us to resolve the layer between the astropause and the bow shock. Unfortunately, with the existing optical surveys we were not able to detect the optical counterpart of the astrosphere of κ Cas, particularly because this bright (V≈4 mag) star outshines all around it. Optical imaging with narrow-band filters could potentially be of value in detection of the astrosphere and in verifying our model.We realize that the mechanism for origin of the filamentary structure of astrospheres is not unique and that the observed filaments might be produced by various different processes. For example, they could arise because of the rippling effect in radiative shocks, i.e. due to variations in the projection of the shock velocity along the line of sight (Hester 1987). Also, the filaments could originate because of time-dependent variations of the wind velocity (Decin et al. 2006) or might be caused by instabilities in the bow shock and contact discontinuity (Dgani, Van Buren & Noriega-Crespo 1996). Numerical simulations by many authors (e.g., van Marle et al. 2011, 2014; Mackey et al. 2012; Meyer et al. 2014a,b; Acreman et al. 2016) indeed show that astrospheres may be subject to various types of instabilities. However, it is a challenge to separate the real instabilities from the numerical ones, and we refrain in this paper from discussing this problem in depth.To conclude, we note that our model does not explain the origin of the almost straight filaments in the south part of the astrosphere, which intersect the brightest arc at almost right angle (see Fig. <ref>). These filaments might be due to much more complex structure of the interstellar magnetic field than assumed in our modelling (cf. Gvaramadze et al. 2011b). Also, inspection of the WISE 22 μm image of the region around κ Cas shows that these filaments have the same orientation as an elongated pillar to the west of the star (see Fig. <ref>), which points to the possibility that their origin might be due to interaction of the astrosphere with the inhomogeneous local ISM. Modelling of this interaction is, however, beyond the scope of the present paper. § ACKNOWLEDGMENTSThis work is supported by the Russian Science Foundation grant No. 14-12-01096 and is based in part on archival data obtained with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA, and has made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration, the SIMBAD data base and the VizieR catalogue access tool, both operated at CDS, Strasbourg, France. Acreman D. M., Stevens I. R., Harries T. J., 2016, MNRAS, 456, 136Akimkin V. V., Kirsanova M. S., Pavlyuchenkov Ya. N., Wiebe D. S., 2015, MNRAS, 449, 440Alexashov D. B., Katushkina O. A., Izmodenov V. V., Akaev P. S., 2016, MNRAS, 458, 2553Blaauw A., 1961, Bull. Astron. Inst. 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We do not consider the radiative transfer and multiple scattering of photons because of small dust number density that allows us to use an optically thin approximation. The dust grains are heated due to absorption of stellar and interstellar radiation, while their cooling is caused mostly by thermal black body emission. The energy balance for a single dust grain can be stated as follows (see, e.g., Hocuk et al. 2017):4π r_ d^2 ∫_0^∞Q_νB_ν(T_ d) dν=r_ d^2∫_0^∞ Q_νF^*_ν(r,T_*) dν++2π r_ d^2 ∫_0^∞ Q_νJ_ν dν,where F^*_ν=π(R_*/r)^2 B_ν(T_*) is the stellar radiation at point r, J_ν is the intensity of the isotropic stellar radiation field (see fig. 2 in Hocuk et al. 2017). R_* and T_* for κ Cas are given in Table <ref>. Equation (<ref>) is solved numerically and T_ d is found for any grain radii r_ d and distances r.§ DUST NUMBER DENSITY IN THE ISM In the case of power-law size distribution, the ISM number density of dust grains with radius of [r_d-dr_d/2; r_d+dr_d/2] is dn_ ISM(r_ d)=N_ ISMr_ d^-3.5 d r_ d and their mass density is d ρ_ d,ISM=m_ ddn_ d,ISM(r_ d). m_ d=4/3πr_ d^3ρ_ d is the mass of a dust grain, ρ_ d=2.5 gcm^-3 is a typical mass density for the interstellar grain material. We assume that the gas to dust mass ratio in the ISM is about 100. The mass density of the dust in the ISM is:ρ_ d,ISM=4/3πρ_ d N_ ISM∫_r_ d,min^r_ d,maxr_ d^-0.5 dr_ d.For the gas to dust mass ratio of ρ_ p,ISM/ρ_ d,ISM=100, one hasN_ ISM=3/800 πρ_ p,ISM/ρ_ d1/r_ d,max^0.5-r_ d,min^0.5.Note that [N_ ISM]= cm^-0.5. | http://arxiv.org/abs/1709.09494v1 | {
"authors": [
"O. A. Katushkina",
"D. B. Alexashov",
"V. V. Gvaramadze",
"V. V. Izmodenov"
],
"categories": [
"astro-ph.SR",
"astro-ph.GA"
],
"primary_category": "astro-ph.SR",
"published": "20170927132532",
"title": "An astrosphere around the blue supergiant kappa Cas: possible explanation of its filamentary structure"
} |
𝐫̊ 𝐩 ξ ε ℰ 𝒦Tr Re Im arg sign erf sn cn arcsinharcsinh arccosharccosh ḍ ≳ ≲ł sgn #1#1 Email: [email protected] Institute for Nuclear Research, 03680 Kyiv, UkraineDepartment of Physics, Nagoya Institute ofTechnology, Nagoya 466-8555, JapanThe Fedoriuk-Maslov catastrophe theory of caustics and turning points is extendedto solve the bifurcation problems by the improved stationary phase method (ISPM). The trace formulas for the radial power-law (RPL) potentials are presented by the ISPM based on the second- and third-order expansion of the classical action near the stationary point.A considerable enhancement of contributions of the two orbits (pair of consisting of the parent and newbornorbits) at their bifurcation is shown.The ISPM trace formula is proposed for a simple bifurcation scenario of Hamiltonian systems with continuous symmetries,where the contributions of the bifurcating parent orbits vanish upon approaching the bifurcation point due to the reduction of the end-point manifold.This occurs since the contribution of the parent orbits is included in the term corresponding to the family of the newborn daughter orbits. Taking this feature into account, the ISPM level densities calculated for the RPL potential model are shown to be in good agreement with the quantum results at the bifurcations and asymptotically far from the bifurcation points. Semiclassical catastrophe theory of simple bifurcations K. Arita December 30, 2023 =======================================================§ INTRODUCTION Semiclassical periodic-orbit theory (POT) is a powerful tool for the study of shell structures in the single-particle level density of finite fermionic systems <cit.>. This theory relates the oscillating level density and shell-correction energy to the sum over contributions ofclassical periodic orbits. It thus gives the correspondence between the fluctuation properties of the quantum dynamics and the characteristics of the periodic motion embedded in the classical dynamics.Gutzwiller <cit.> suggested the semiclassical evaluation of the Green's function in the Feynman path integral representation to derive the POT trace formula for the level density if the single-particle Hamiltonian has no continuous symmetries other than the time-translational invariance.In this case, the energy E is the single integral of motion for a particle dynamics in the mean-field potential.For a given E, all the generic periodic orbits (POs) are isolated, i.e., any variation of the initial condition perpendicular to the PO will violate its periodicity. The original version of the POT was extended for Hamiltonians with continuous symmetries (extended Gutzwiller approach), in particular for the rotational [(n)] and oscillator-type [(n)] symmetries <cit.>. Berry and Tabor <cit.> derived the POT for integrablesystems by applying the Poisson-summation method through the semiclassical torus-quantization condition.It is also helpful in the case of a high classical degeneracy of POs.Here the classical degeneracy is defined by the number of independent parameters 𝒦 for a continuous family ofclassical periodic orbits at a given energy of the particle.Some applications of the POT to the deformations of nuclei and metallic clusters by usingphase-space variables were presented in Refs. <cit.>. The pronounced shell effects caused by the deformations have been discussed.Within the improved stationary-phase method <cit.> (ISPM), the divergences and discontinuities of the standard stationary-phase method(SSPM) <cit.> near the symmetry-breaking and bifurcation points were removed.Bifurcations of the isolated POs in non-integrable Hamiltonian systems are classified by the normal-form theory <cit.>, based on a pioneering work of Meyer on the differentiable symplectic mappings <cit.>.The change in the number of POs depends on the types ofbifurcations: from zero to two in the isochronous or saddle-node bifurcations, from one to three in the period-doubling or pitchfork bifurcations, and so on.In the integrablesystems, the classical phase space is entirely covered by the tori. The classical orbits on the rational tori (in which the frequencies of the independent motions are commensurable) form degenerate families of POs.In such systems, the bifurcations mostly occur at a surface of the physical phase-space volume occupied by the classical trajectories. Forbrevity, below, such a surface will be called the end point.Note that even if the commensurability of the frequencies is not fully satisfied at the end point, one has a PO there because the mode with incommensurable frequencies haszero amplitude.In the action-angle representation (I_n,Θ_n) for the nth mode, it corresponds to I_n=0, where the variation of Θ_n will not generate a family.Thus, the end-point POs have smaller degeneracies than those inside the physical region.With varying potential parameter, a new PO family appears in the transition from an unphysical to a physical region for a rational torus where the frequencies of the end-point PO become commensurable. It is considered as the bifurcation of the end-point PO generating a new orbit family.Such bifurcations take place repeatedly with varying potential parameter and a new family of orbits with a higher degeneracy and different frequency ratio is generated at each bifurcation point.Typical examples are the equatorial orbits with =1 in the spheroidal cavity where the generic family has =2, and the circular orbits with =2 in a spherical potential where the generic family appears with =3.Our ISPM is based on the catastrophe theory by Fedoriuk and Maslov for solving problems with caustic and turning points in calculations of the integrals by using the saddle-point method <cit.>.In the SSPM, the catastrophe integrals are evaluated by an expansion of the action integral in the exponent up tosecond-order terms and amplitude up to zeroth-order terms near the stationary point, and the integration limits are extended to an infinite interval.The trace formula based on the SSPM encounters a divergence or discontinuity at the bifurcation point.Such catastrophe problems are due to the zeros (caustics) or infinities (turning points) of the second-order derivatives of theaction integral in this expansion.Fedoriuk was the firstto prove <cit.> the so-called Maslov theorem <cit.>: Each simple caustic or turning point (having a finite nonzero third-order derivative of the action integral) leads to a shift of the phase in the exponent of the catastrophe integral by -π/2 along the classical trajectory.This providesan extension of the one-dimensional WKB formula to higher dimensions. Thus, in the asymptotic region far from the caustic and turning points, one can use second-order expansions of theaction integral (and zeroth-orderexpansions of the amplitude) in the semiclassical Green's function taking into account the shift of the phase according to the Maslov theorem.However, a proper foundation for an extension of the Fedoriuk-Maslov catastrophe theory (FMCT) to the derivations of the trace formulas near the PO bifurcations, where one has to treat the semiclassical propagators near the catastrophe points, is still an open question.Uniform approximations based on the normal-form theories give alternative ways of solving the bifurcation problems, see Refs. <cit.>.The trace formula valid near the bifurcation points is derived by calculating the catastrophe integral in local uniform approximations. It gives an indistinct combination ofcontributions of bifurcating POs, and the asymptotic regions (where each of the POs has aseparate contribution to the trace integral) are connected by a kind of the interpolation through the bifurcations in theglobal uniform approximations.The ISPM provides much simpler semiclassical formula in which one needs no artificial interpolation procedures over the bifurcations.Within the simplest ISPM, the contributions of the bifurcating orbits are given separately through the bifurcationand they are basically independentof the type of bifurcation. This allows one togive analytic expressions for the Gaussian-averaged level densities and the energy shell corrections <cit.>.In the present work we apply the FMCT to solve the bifurcation problemsthat arisefor some parameters of the mean-field potential for the particle motion in the end-point phase space.For the simplest but a rich exemplary casethat nevertheless includes all of the necessary points of the general behavior of the integrable systems, wewill consider the spherical radial power-law (RPL) potential V∝ r^α as a function of the radial power parameter α, which controls the surface diffuseness of the system <cit.>.Some of these results are general for any integrable and nonintegrable Hamiltonian systems, and can be applied to a more realistic nuclear mean-field potential having the deformation.The spherical RPL model has already been analyzed within the ISPM in Ref. <cit.>.Good agreement between the semiclassical and quantum shell structures was shown in the level-density and energy shell correctionsfor several values of the surface diffuseness parameter including its symmetry-breaking and bifurcation values. Quantum-classical correspondences in the deformed RPL models with and without spin-orbit coupling were also studied and various properties of the nuclear shape dynamics,such as the origins of exotic deformations and the prolate-oblate asymmetries, have been clarified <cit.>. Therefore, a proper study of the general aspects of the bifurcation problem within the ISPM, even taking the simplest spherical RPL potential as anexample for which one can achievemuch progress in analytical derivations, is expected to be helpful.This article is organized in the following way. In Sec. <ref> we present a general semiclassical phase-space trace formula for the level density as a typical catastrophe integral. Section <ref> shows the Fedoriuk-Maslov method for solving the simple caustic- and turning-point singularity problems. Section <ref> is devoted to the application of the FMCT to the local PO bifurcations formore general Hamiltonian systems with continuous symmetries. Section <ref> presents the specific application of the ISPM trace formula to the bifurcations in the spherical RPL potential model.Contributions of the bifurcating orbitsto the trace formula are discussed.In Sec. <ref>, we compare our semiclassical results, obtained at a bifurcation point and asymptotically far from bifurcations, with the quantum calculations.These results are summarized in Sec. <ref>.A moreprecise trace formula based on the third-order expansions of the action integral is presented in the Appendix. § TRACE FORMULA łsec2:traceThe general semiclassical expression for the level density, g(E)=∑_i δ(E-E_i), is determined by the energy levels E_i for the single-particle Hamiltonian Ĥ=T̂+V of 𝒟 degrees of freedom. The specific expression generic to integrable and nonintegrable systems can be obtained by the following trace formula in the 2𝒟-dimensional phase space <cit.>:łpstrace g_ scl(E)=1/(2πħ)^𝒟∑_ CT∫̣̊^''∫^' δ(E - H(^̊',^')) ×|𝒥_ CT(^'_⊥, ^''_⊥)|^1/2exp( i/ħ Φ_ CT- iπ/2μ^_ CT - iϕ^_𝒟).Here H(,̊) is the classical Hamiltonian in thephase-space variables ,̊ and Φ_ CT is the phase integralłlegendtransΦ_ CT ≡ S_ CT(^', ^'',t^_ CT) + (^''-^') ·^̊''=S_ CT(^̊', ^̊'',E) + ^'·(^̊' - ^̊'') (see the derivations in Ref. <cit.>).In Eq. (<ref>), the sum is taken over all discrete classical trajectories (CTs) for a particle motion from the initial point (^̊', p^') to the final point (^̊'', p^'') with a given energy E<cit.>.A CT can uniquely be specified by fixing, for instance, the final coordinate ^̊'' and the initial momentum p^' for a given time t^_ CT of the motion along a CT. Here S_ CT(^',^'',t^_ CT) is the classical action in the momentum representation,łactionp S_ CT(^',^'',t^_ CT) =-∫_^'^^''·(̊) . Integration by parts relates Eq. (<ref>) to the classical action in the coordinate space,łactionr S_ CT(^̊',^̊'',E) =∫_^̊'^^̊''̣̊·()̊ , by the Legendre transformation [Eq. (<ref>)]. The factor μ^_ CT is the number of conjugate points along a CT with respect to the initial phase-space point ('̊,').They are, e.g., the focal and caustic points where the main curvatures of the energy surface(second derivatives of the phase integral Φ_ CT) vanish.In addition, there arethe turning points where these curvatures become divergent. The number of conjugate points evaluated along a POis called the Maslov index <cit.>. An extra phase component ϕ^_𝒟, which is independent of the individual CT, is determined by the dimension of thesystem and the classical degeneracy 𝒦[ϕ_𝒟 is zero when all orbits are isolated (𝒦=0),as defined in Ref. <cit.>]. In Eq. (<ref>) we introduced the local phase-space variables thatconsist of the three-dimensional(𝒟=3) coordinate =̊{x,y,z} and momentum ={p_x,p_y,p_z}. It is determined locally along a reference CT so that the variables (r_∥=x, p_∥=p_x) are parallel and(_̊⊥={y,z},_⊥={p_y,p_z}) areperpendicular to the CT <cit.>.Here 𝒥_ CT(^'_⊥,^''_⊥)is the Jacobian for the transformation of the momentum component perpendicular to a CT from the initial value _⊥^' to the final value_⊥^''. For calculations of the trace integral by the stationary-phase method(SPM), one may write the stationary-phase conditions for both ^'and ^̊'' variables. According to the definitions (<ref>)and (<ref>), the stationary-phase conditions are given byłstatcond(∂Φ_ CT/∂^')^* ≡(^̊' - ^̊'')^*=0 , (∂Φ_ CT/∂^̊'')^* ≡(^'' - ^')^*=0 .The asterisk indicates that quantities in large parentheses are taken at the stationary point. Equations (<ref>) express that the stationary-phase conditions are equivalent to the PO equations (”̊,”)^*=('̊, p')^*.One of the SPM integrations in Eq. (<ref>), e.g., over the parallel momentum p_∥' in the local Cartesian coordinate system introduced above, is the identity because of the energy conservation E=H(”̊,”)=H('̊, p').Therefore, it can be taken exactly.If the system has continuous symmetries, the integrations with respect to the corresponding cyclic variables can be carried out exactly.Notice that the exact integration is performed finally also along a parallel spatial coordinate (along the PO).Applying the ISPM with the PO equations (<ref>), accounting for the bifurcations and the breaking of symmetries, one may arrive at the trace formula in terms of the sum over POs <cit.>.The total ISPM trace formula is the sum over all of POs [families with the classical degeneracy K≥ 1 and isolated orbits (K=0)],łdgscδ g(E) ≃δ g_ scl(E) =∑_ POδ g^_ PO(E) ,where δ g^_ PO(E)={A_ POexp[i/ħ S_ PO(E) -iπ/2μ^_ PO -iϕ^_𝒟]}.The amplitude A_ PO depends on the classical degeneracy 𝒦 and the stability of the PO. In the exponent phase,S_ PO(E)=∮· d$̊ is the action andμ^_ POis the Maslov index <cit.>.§ FEDORIUK-MASLOV CATASTROPHE THEORY łsec3:cat3In this section we present the essence of the ISPM, following basically the Fedoriuk-Maslov catastrophe theory(see Refs. <cit.>).§.§ Caustic and turning points łsec31:causticsLet us assume that the integration interval in one of the integrals of Eq. (<ref>) over a phase-space variablecontains a stationary catastrophe point where the second derivative of the phase integralΦis zero [see Eq. (<ref>)]. This catastrophe integralℐ(κ,ϵ)can be considered as a function of the two dimensionless parameters ℐ(κ,ϵ) = ∫__-^_+ A(,ϵ) exp[i κ Φ(,ϵ)],whereA(,ϵ)is the amplitude andΦ(,ϵ)thedimensionless phaseintegral, which is proportional toΦ_ CTgiven byEq. (<ref>). One of these parametersκis related to a large semiclassical parameterκ∝ 1/ħ→∞, whenħ→ 0, through the relationshipκΦ=Φ_ CT/ħ(see also the Appendixfor a clear example). Another critical parameterϵis a small dimensionlessperturbationof the phase integralΦ(,ϵ)[Eq. (<ref>)] and the amplitudeA(,ϵ)through the potentialV(,̊ϵ). For instance,ϵcan be a dimensionless distance from the catastrophepoint by perturbingthe parameter of a potentialV(,̊ϵ)<cit.>, e.g.,thedeformation and diffuseness parameters (see examples in Refs. <cit.>). In Eq. (<ref>), the integration limits_±crossing the catastrophe point^∗(0)are generally assumed to be finite.We assume also that the integral (<ref>) hasthe simplest (first-order) caustic-catastrophe point^*(ϵ)atϵ=0defined by[ In general, the caustic point of the nth order (n≥ 1) is defined as the point where the derivatives up to the (n+1)th order vanish but the (n+2)th derivatives remain finite.] łcaustpoint Φ'(ξ^∗) =0,Φ”(ξ^∗) = 0,Φ”'(ξ^∗) = O(ϵ^0) ^∗=_0=^∗(0),where the asterisk indicates that the derivatives with respect toξare taken at=^∗.The mixed derivative(∂^2 Φ/∂∂ϵ)^∗,A^∗=A(ξ^∗,ϵ), is assumed to be of the zeroth order inϵas well as the third derivative in Eq. (<ref>). In the limitϵ→ 0at largeκfor the caustics, the two simple stationary points^∗(ϵ)coincide and formone caustic point given by Eq. (<ref>).To remove the indetermination, let us consider a small perturbation of the catastrophe integralI(ϵ,κ)[Eq. (<ref>)]by changingϵthrough the phasefunctionΦ(,ϵ)and the amplitudeA(,ϵ)near the caustic pointϵ=0.For any small nonzeroϵwe first study theexpansion of the functionΦ(,ϵ)overϵina power series near the stationary point^*(ϵ)fora smallϵ,łSexpΦ(,ϵ)= Φ^∗+ 1/2 Φ”(ξ^∗) (-^∗)^2 + 1/6 Φ”'(ξ^∗) (-^∗)^3 +⋯ . Similarly, for the amplitude expansion, one hasłAexp A(,ϵ) = A^∗ + A'(^∗)(-^∗) + ⋯ .The asterisks in Eqs. (<ref>) and (<ref>)indicate thatthe derivatives with respect toare taken at=^*(ϵ)for a small but finiteϵ,Φ^* =Φ(^*,ϵ).Using a small perturbation of the actionΦ(,ϵ)and amplitudeA(,ϵ)byϵvariations,one finds the first derivative in Eq. (<ref>) and the second derivative in Eq. (<ref>) as small butnonzero quantities.For asymptotic values ofκ→∞, one maytruncate the series (<ref>) for the phaseΦ(,ϵ)in the exponent of the catastrophe integralℐ(κ,ϵ)[Eq. (<ref>)] and corresponding one (<ref>) for its amplitudeA(,ϵ)at the third andzeroth orders, respectively,keeping, however, a small nonzeroϵ.In the following, definingξas a dimensionless variable, one can simply take the second derivative of a phaseΦin Eq. (<ref>), divided by 2, as the parameterϵ, łepsϵ=1/2Φ”(^∗)=σ|ϵ| , σ=(ϵ)=[Φ”(^*)] .By definition of the simplest caustic point of the first order[Eq. (<ref>)], the third derivative of the phaseΦnear the caustic pointϵ=0is not zero at any smallϵ. Therefore, one can truncate the expansion of the phase integral (<ref>) up to the third order asłnorm3Φ=Φ^∗ + ϵ (-^*)^2+ a (-^*)^3 ,wherełaQ a=1/6Φ”'(ξ^∗) .Here we assume thatA(^*,ϵ)has a finite nonzero limit atϵ→ 0.Therefore, it can be cut at zeroth order in-^*, namely,A(ξ,ϵ) ≈A^*=A(ξ^*,0).Following Refs. <cit.>, one can consider the linear transformationof the coordinate fromtozas ξ-ξ^∗ =Υ + Λ z ,where łtransformcoeffΥ=-ϵ/3a ,Λ= 1/(3a κ)^1/3 .Notice that bothcoefficients of this linear transformation,Υ(ϵ)andΛ(κ), depend on different critical parametersϵandκ, respectively [Υ(0)=Λ(∞)=0].Substituting Eq. (<ref>) into Eq. (<ref>), one can express the catastrophe integral (<ref>) in an analytical form,łcatint2ai ℐ(κ,ϵ)= πΛ A^∗ exp(iκ Φ^* + 2iσ/3 w^3/2) ×[(-w,𝒵_-,𝒵_+) +i (-w,𝒵_-,𝒵_+)] . The generalized incomplete Airy and Gairy integrals are defined in a similar way as the standard ones but with the finite integration limits,łairynoncomp (-w,z_1,z_2)=1/π∫_z_1^z_2 ẓcos sin(-w z+z^3/3).The argumentwof these functions and finite integrationlimits𝒵_±inEqs. (<ref>) and (<ref>) are given by łww= κ^2/3 ϵ^2/(3a)^4/3 > 0,andłzpm𝒵_± = ξ_± -ξ^*/Λ +σ√(w) .As can be seen from a cubic form of the phase in parenthesis onthe right-hand side of Eq. (<ref>), the caustic catastrophe can be considered as a crossing point of the two simple close stationary-point curves for anysmall nonzeroϵ,łstatpoints3 z_±^*(ϵ)=±√(w(ϵ)) .They degenerate into one caustic pointz_±^*(ϵ) → 0(<ref>) in the limitϵ→ 0because ofw → 0at any finiteκ.The final result is a sum of the contributions of thesestationary points.Note that, according to Eq. (<ref>), the value ofwis large for any nonzeroϵwhenκbecomes large.Nevertheless, it becomessmall for a large fixedfinite nonzeroκwhenϵis small.However, wemay considerboth cases by using the same formula(<ref>) because the two parametersϵandκappear in(<ref>) through one parameterwfor a finite constanta.§.§ The Maslov theorem łsec32:maslovFor any small nonzeroϵ, one may find a value ofκfor whichwis so large that the two above-mentioned stationary pointsz_±^*(ϵ)can be treated separately in the SPM. At the same time,wcan be sufficientlysmallthat the stationary points|z_±^*|are much smaller than the integration limits|𝒵_±|. In practice, it is enough to consider𝒵_+>0and𝒵_-<0, andwe split the integration interval into two parts, i.e., from𝒵_-to0and from0to𝒵_+, in order to separate the contributions of negative and positive stationary points-√(w)and√(w). According to the phase-space flow around the PO (see Fig. <ref>), the curvatureΦ”∝∂ p_/∂(p_denotes the momentum conjugate to) will always change its sign from positive to negative at the caustic catastrophe point <cit.>. Let us consider such crossings with a catastrophe point. Outside the catastrophe point, whereϵ>0(σ=1),the stationary point^*corresponds toz_+^*and its contribution is given by the second integral over positivez. For this integral, one can extend the upper integration limit𝒵_+toinfinity, like in the SSPM, because𝒵_+∝κ^1/3≫ 1and𝒵_+≫ z_+^*=√(w)∝κ^1/3ϵfora small finiteϵ[see Eq. (<ref>)]. Within this approximation, one can use the standard complete Airy and Gairy functions (-w) =1/π∫_0^∞ẓcos sin(-w z+z^3/3).They correspond to the limitsz_1=𝒵_-=0andz_2=𝒵_+=∞in Eq. (<ref>).Using the asymptotic form of the Airy and Gairy functions forw →∞,łAiGiAsympt (-w)→1/√(π)w^1/4sin cos(2/3w^3/2+π/4) ,one can evaluate the contributionℐ_+of the positive stationary point to Eq. (<ref>), asymptoticallyfar from the caustic point (<ref>). Then one obtains the same resultas onewould get by the standard second-order expansion of the phaseΦ(and zeroth order of the amplitudeA) at a simple stationary point^*, ℐ_+(κ,ϵ) ≃πΛ A^*exp(iκΦ^*+ 2i/3 w^3/2) [(-w). +. i(-w)] →√(2π/κ|Φ”(^*)|)A^*exp( iκΦ^*+iπ/4) .On the other side of the crossing with the catastrophe point,ϵ<0(σ=-1), the stationary point^*corresponds toz_-^*.Therefore, one should consider theother part of the integralℐ_-over the negative values ofz. Obviously, considering in an analogous way with a change of the integration variablez → -z, one obtains ℐ_-(κ,ϵ) ≃πΛ A^* exp(iκΦ^*-2i/3w^3/2) [(-w). -.i(-w)] →√(2π/κ|Φ”(^*)|)A^* exp(iκΦ^*-iπ/4) .Comparing the rightmost expression in Eq. (<ref>) with that in Eq. (<ref>), one sees a shift of phase by-π/2. Thus, the famous Maslov theorem <cit.> on the shift of the phaseΦby-π/2at each simple caustic point(<ref>) of the CT(in particular, the PO) in the SSPM(i.e., the Maslov index is increased by one) has been proved by using the Fedoriukcatastrophe method <cit.>.For the case of a turning point, onehas the conditions (<ref>) with only thereplacement of zero by infinity in the second derivative. In this case, Fedoriuk used a linearcoordinate transformation fromztoz, which has the formłturningtrans z=Λ/ϵz + Υ,whereΛandΥare new constantsthat are not singular inϵ(Λis independent ofϵ).This transformation reduces theturning-point singularity to thecaustic-point one. Indeed, the divergent second derivative of the phaseΦover the variablezis transformed, in a new variable, to its zero valuełnewderPhi1/2(∂^2Φ/∂^2z)^*= Λ^2/ϵ(see the second paper in Ref. <cit.>).Therefore, one obtains the same shift by-π/2at each simplestturning point (per a sign change of one momentum component perpendicular to the boundary) along a CT.Finally,the next part of the Maslov theorem <cit.> concerning the Maslov index generated by such aturningpoint has been proved, too, within the same catastrophetheory of Fedoriuk <cit.>. § SYMMETRY-BREAKING AND BIFURCATIONS łsec4:bifurcationsAssuming a convergence of the expansion [Eq. (<ref>)] to the second order, as shown in thepreceding section, one can use the same approachwithin the simplest ISPM2 of second order[We call thenth-order ISPM the ISPMn, in which we use the expansion of the phaseintegral Φ up to the nth-order terms and the amplitude up tothe (n-2)th order near the stationary point.The simplest ISPM for n=2 is called the ISPM2.] to investigate the symmetry-breaking and bifurcation problems in the POT.Therefore, one arrives at a sum over the separate contributions of different kinds ofisolated and degenerated orbits, as in the derivation of the Maslov theorem but within the finite integration limits. The latter is important because the bifurcation point is located at a boundary of the classically accessible region. This section is devoted to the extension of the FMCT to the bifurcation catastrophe problems.In the presence of continuous symmetries, the stationary points form a family of POsthat cover a(𝒦 + 1)-dimensional submanifold𝒬_ POof phase space, whereby𝒦is the classical degeneracy of the PO family. The integration over𝒬_ POmust be performed exactly. In anysystems with continuous symmetries, it is an advantage to transform the phase-spacevariablesfrom the Cartesian to the corresponding action-angle variables (see, e.g.,Ref. <cit.>). Then the actionΦ_ CTin Eq. (<ref>) is independent of the angle variables conjugate to the conserving action variables and the integrations over these cyclic angle variables areexactly carried out. For the integrable case, for instance, integrating over theremaining action variables and using thestandard SPM, one obtains the so-called Berry-Tabor trace formula <cit.>. Under the existence of additional symmetries such as SU(3) or O(4),some of the integrations over the action variables can also be performed exactly because of a higher degeneracy. For partially integrable systems, the integrations over partial set ofcyclic variables also greatly simplify the ISPM derivations of the traceformula (<ref>) near the bifurcations.To solve the bifurcation problems, some of the SPM integrations have to be done in a more exact way. For definiteness, we will consider first a simple bifurcationdefined as a caustic point of the first order [Eqs. (<ref>) and (<ref>)] where the degeneracy parameter𝒦is locally increased by one. In the SPM, after performing exact integrations overa submanifold𝒬_ PO, one uses an expansion of the action phaseΦ_ CTin phase space variablesξ={”̊,'}_⊥, perpendicular to𝒬_ POin the integrand of Eq. (<ref>) overξnear the stationary pointξ^∗, łispmexp Φ_ CT(ξ)=Φ^_ PO + 1/2Φ_ PO”(ξ^∗)(ξ-ξ^∗)^2 + 1/6Φ_ PO”'(ξ^∗) (ξ-ξ^∗)^3+ ··· ,where łPhiPOΦ_ PO=Φ^∗_ CT=Φ_ CT(ξ^∗) ,ξ^∗is the stationary point,ξ^∗=ξ^_ PO, andΦ^∗_ CT=Φ_ CT(ξ^∗)=Φ_ PO. To demonstrate the key point of our derivations of the trace formula,we focus on one of the phase-space variables in Eq. (<ref>), denoted byξ, which is associated with a catastrophe behavior. In the standard SPM, the above expansion is truncated at the second-order term and the integration over the variableξis extended to±∞. The integration can be performed analytically and yields a Fresnel integral (see, e.g., Refs. <cit.>). However, one encounters a singularity in the SSPM thatis related to the zero or infinite value ofΦ_ PO”(ξ^∗),whileΦ_ PO”'(ξ^∗)remains finite in the simplest case under consideration.This singularity occurs when a PO (isolated or degenerated) undergoes a simple bifurcation atthe stationary pointξ^∗under thevariation of a parameter of the potential(e.g., energy, deformation, orsurface diffuseness). The SSPM approximation to the Fresnel (error) functions by the Gaussian integrals breaks down because one has a divergence.Notice that the bifurcation problem is similar to thecaustic singularity considered by Fedoriuk within the catastrophe theory(see Sec. <ref> and Refs. <cit.>). The FMCT is adopted, however,for the specific position of such a singularity at the end point in the phase-space volume accessible for a classical motion(see the Introduction and also Ref. <cit.>).In systems with continuous symmetries, theorbit at the end point causes the bifurcation where it coincides with one of the rational tori thatappears in the transition from the unphysical to the physical region.The contribution of this end-point orbit is derived using a local phase-space variableξalong it, independently of the torus orbits.Near the bifurcation point, the contribution of these end-point orbits is mostly includedto the newborn orbit term. Therefore, we should consider a kind of separation of the phase space occupied by the newborn andend-point orbits to evaluate their contributions to the trace integral near the bifurcation point. Below we will callthe latter an end-point manifold.By definition of the end-point manifold, its measure is zero at thebifurcation limit where the minimal_-and maximal_+coincide with the stationary point^∗, i.e.,łbifboundlim_-→_+→^∗ (Φ_ PO”→ 0).Thus, although the contributions of POs participating in the bifurcation are considered separately, the parent orbit contribution vanishes at the bifurcation point andthere no risk of double counting. To describe the transition from the bifurcation point to the asymptotic region, one should properly define the end-point manifold. We need it to extract the additional contribution by the end-point orbit that is not covered by the term for the newborn orbit. The detailed treatment of such a transitionis still open, but is beyond the scope of the present paper. We are ready now to employ what we call the improvedstationary-phase method <cit.> evaluating the trace integral for the semiclassical level density. Hereby, the integration overξin Eq. (<ref>) is restrictedto the finite limits defined by the classically allowed phase-spaceregion through the energy-conservingδfunction in the integrand of Eq. (<ref>). The phase and amplitude are expanded around the stationary point up to the second- and zeroth-order terms inξ -ξ^*, respectively, and to higher-order terms if necessary.In the simplest version of the ISPM (ISPM2), the expansion of the phase is truncated at second order, keeping the finite integration limitsξ_-andξ_+given by the accessible region of the classical motion inEq. (<ref>). It will lead to a factor like[ For the case of several variables ξ for which we find find zeros andinfinities in eigenvalues of the matrix with second-order derivatives of Φ_ PO(ξ) at =ξ^*, wediagonalize this matrix and reduce the Fresnel-like integrals to products oferror functions similar to Eq. (<ref>).] łerrfuns e^iΦ_ PO/ħ∫_ξ_-^ξ_+exp[i/2ħΦ_ PO” (ξ-ξ^∗)^2] ξ∝1/√(Φ_ PO”) e^iΦ_ PO/ħ [𝒵_-,𝒵_+],where(z_1,z_2)is the generalized error function,łispmlimits(z_1,z_2)=2/√(π)∫_z_1^z_2e^-z^2z =(z_2)-(z_1), with the complex argumentsłcomparg𝒵_±=(ξ_±-ξ^*) √(-i/2ħΦ_ PO”) .Note that the expression (<ref>) has no divergence at thebifurcation point whereΦ_ PO”(ξ^∗)=0, since the error function (<ref>) also goes to zero as(𝒵_-,𝒵_+) ∝𝒵_+-𝒵_- ∝(_+-_-)√(Φ”_ PO) .Thus, the factor√(Φ_ PO”)in the denominator of Eq. (<ref>) is canceled with the same in the numerator. In addition, also taking into account Eq. (<ref>) and the discussion around it, one finds the zero contribution of the end-point term inthe trace formula at the bifurcation point, which seems to be a consistent semiclassical picture.This procedure is proved to be valid in the semiclassical limitκ→∞by the FMCT <cit.>. In this way, we can derive the separate PO contributions thatare free of divergences, discontinuities, and double counting at any bifurcation point. The oscillating part of the level density can be approximated by the semiclassicaltrace formula (<ref>). In Eq. (<ref>), the sum runs over all periodic orbits(isolated or degenerated) in the classical system. The termS_ PO(E)is the action integral along a PO.The amplitudeA_ PO(E)(which, in general, is complex) isof the order of the phase-space volume occupied by CTs.The factor given in Eq. (<ref>) depends on the degeneracies and stabilities of the POs, respectively(see Sec. <ref>).Notice that any additional exact integration in Eq. (<ref>) with respect to a bifurcation (catastrophe) variableof the improved SPM can lead to an enhancement of the amplitudeA_ POin the transition from the bifurcation point to the asymptotic region. This enhancement is of the order1/ħ^1/2as compared to theresult of the standard SPM integration.In particular, for the newborn family with the extra degeneracyΔ𝒦higher thanthat of the parent PO, one has such enhancements of the order1/ħ^Δ𝒦/2near the bifurcation.The trace formula (<ref>) thus relates the quantum oscillationsin the leveldensity to quantities that are purely determined by the classical system. Therefore, one can understand the shell effects in terms of classical pictures.The sum over POs in Eq. (<ref>) is asymptoticallycorrect to the leading order in1/ħ^1/2andit is hampered by convergence problems <cit.>. However, one is free from those problems by taking the coarse-grained level densityłdeltadenavδ g^_Γ, scl(E)= ∑_ POδ g^ scl_ PO(E) exp{-(Γ t^_ PO/2ħ)^2},whereΓis an averaging width. The value ofΓis sufficiently smaller than the distance between the major shellsnear the Fermi surface (seeRefs. <cit.>). Heret^_ PO=∂ S_ PO(E)/∂ Eis the period of the particle motion along a PO taking into account its repetition number. We see that, depending on thesmoothing widthΓ, longer orbits are automaticallysuppressed in the above expressions and the PO sum converges, which itusually does not <cit.> for nonintegrable systems in the limitΓ→ 0. Thus, one can highlight the major-shell structure in the level density using a smoothing width that is much larger than the meansingle-particle level spacing butsmaller than the main shell spacing (the distance between major shells) nearthe Fermi surface. Alternatively, a finer shell structure can be considered by using essentially smaller smoothing widths, which is important for studying the symmetry-breaking (bifurcation) phenomenon associated with longer POs. It is an advantage of this approach that the major-shell effects inδ g(E)canoften be explained semiclassicallyin terms of only a few of the shortest POs in the system.Examples will be given in Sec. <ref>. However, if one wants to study a finer shell structure, specifically at large deformations, some longer orbits have to be included. Hereby, bifurcations of POs play a crucial role, as it will be exemplified in Sec. <ref>.§ ISPM FOR THE SPHERICAL RPL POTENTIAL MODEL łsec4:ispmLet us apply the general FMCT to the RPL potential as an analytically solvable example. §.§ Scaling property łsec5A:scalA realistic mean-field potential for nuclei and metallic clusters is given by the well-known Woods-Saxon (WS) potential V_ WS(r)=-W_0/1+expr-R/a ,whereW_0is the depth of the potential,Ris the nuclear radius, andais the surface diffuseness. As suggested in Refs. <cit.>, this potential can be approximated by the RPL potential for a wide range of mass numbers asłramod V_ WS(r) ≈ -W_0 + W_0/2(r/R)^α ,with an appropriate choice of the radial power parameterα. One finds good agreement of the quantum spectra for the approximation (<ref>) to the WS potential up to and around the Fermi energyE^_F. Eliminating the constant term on the right-hand side of (<ref>), we define the RPL model Hamiltonian asłpotenra H=p^2/2m+V_0(r/R_0)^α,V_0=ħ^2/mR_0^2 ,wheremis the mass of a particle andR_0is an arbitrary length parameter. In the spherical RPL model (<ref>), as well as in general spherical potential models, one has the diameter and circle POs that form the two-parameter (𝒦=2) families. The diameter and circle POs have minimum and maximum values of the angular momentumL=0andL_C, respectively. They correspond to the end points of the energy surfaceH(I_r,L)=Eimplicitly given by the relationship (see, e.g., Ref. <cit.>)łensurf I_r=1/π∫_r_ min^r_ max p_r ṛ= I_r(E,L), wherep_ris the radial momentumłpr p_r=√(p^2(E,L)-L^2/r^2), withpthe particle momentumłp p(E,L)=√(2m[E-V(r)]) .The integration limitsr_ minandr_ maxin Eq. (<ref>) are functions of the energyEand angular momentumLfor a given spherical potentialV(r).In the spherical RPL potential, they are defined by the two real roots of the transcendent equation for the variabler:łpr2eqp_r(L,α)≡√(2m(E-V_0(r/R_0)^α)-L^2/r^2)=0 .Another key quantity in the POT is the curvature of the energy surface (<ref>)łcurvature K=∂^2 I_r/∂ L^2 . Using the invariance of the equations of motion under the scale transformationłscaling→̊s^1/α,̊→ s^1/2 ,t → s^1/2-1/αtE→ sE , one may factorizethe action integralS_ PO(E)along the PO asłactionscS_ PO(E)=∮_ PO(E)·̣̊ =(EV_0)^1/2+1/α∮_ PO(E=V_0)·̣̊≡ħτ^_ PO . In Eq. (<ref>) we define the dimensionless variablesandτ^_ POas classical characteristics of the particle motionłeq:scaledentau1=(E/V_0)^1/2+1/α and łeq:scaledentau2τ^_ PO=1/ħ∮_ PO(E=V_0)·̣̊ .We call them the scaled energy and the scaled period, respectively. To realize the advantage of the scaling invariance under the transformation (<ref>), it is helpful to useandτ^_ POin place of the energyEand the periodt_ POfor the particle motion along a PO, respectively. In the harmonic-oscillator (HO) limit (α→ 2),andτ^_ POareproportional toEandt^_ PO, while in the cavity limit (α→∞), they are proportionalto the momentumpand the geometrical POlengthℒ_ PO, respectively.The PO condition (<ref>) determinesseveral PO families in the RPL potential, namely, thepolygonlike (𝒦=3), the circular, and the diametric (𝒦=2) POs.This condition for the integrable spherical Hamiltonian isidentical to a resonance condition, which is expressed in the spherical variablesr, θ, φasłrescondω^_r/ω^_θ =n^_r/n^_θ ,ω^_θ≡ω^_φ ,whereω_r,ω_θ, andω_φare frequencies in the radial and angular motion. Figure <ref>shows these POs in the RPL potential (<ref>) in the(τ,α)plane, whereτ^_ PO=τ(α,L_ PO)is the scaled period for the PO specified by(n_r,n_θ),which satisfies the resonance condition (<ref>). As shown in this figure, the polygonlike orbitM(n_r,n_θ)continues to existafter its emergence at the bifurcation pointα=α_ biffrom the parent circularorbitMC(Mth repetition of the primitive circle orbitC). The exceptions are the diameter orbitsM(2,1) that exist for all values ofαand form families with a higher degeneracy at the HO symmetry-breaking pointα=2. §.§ Three-parameter PO families łsec5B:PISPMFor the contribution of the three-parameter(𝒦=3)families into the densityshell correctionδ g(E) [Eq. (<ref>)],one obtains <cit.> łdeltag3ispδ g^_ scl, P(E) =∑_M P A_M P^(E)×exp[i/ħS_M P(E)-i π/2μ^_M P -iϕ^_𝒟] .The sum is taken overfamilies of the three-parameter (𝒦=3) polygonlike orbitsMP(Pstands for polygonlike, andMfor the repetition numberM=1,2,...). In Eq. (<ref>),S_M P(E)is the action along the PO, S_M P(E) = 2π M [n_r I_r(E,L^*)+n_θ L^*] ,whereI_ris the radial action variable in the sphericalphase-space coordinates [Eq. (<ref>)]. The angular momentumL^∗is given by the classical value for a particle motion along thePorbitL^∗=L_Pin an azimuthal plane. The numbersn_randn_θspecify the orbitPwithn_r >2 n_θ.For the amplitudeA_M P^[Eq. (<ref>)], within the ISPM2, one findsłamp3isp A_M P^= L_P T_P/πħ^5/2√(M n_rK_ P) (𝒵_M P^+, 𝒵_M P^-) e^i π/4 ,whereK_Prepresents the curvature (<ref>) atL=L^∗andT_P=T_n_r,n_θis the period of the primitive (M=1) polygonlike orbit,P(n_θ,n_r). For a three-parameter family at the stationary pointL=L^*determined bythe PO (stationary-phase) equation, one has łperiod3 T_P=2π n_r/ω_r=2π n_θ/ω_θ .The function(𝒵_M P^+, 𝒵_M P^-)in Eq. (<ref>) is given by the generalized error function(<ref>). Its complex arguments𝒵_M P^±are expressed in terms of the curvatureK_P,łargerrorpar𝒵_M P^- = √(-i πM n_rK_P/ħ)(L_--L_P) , Z_M P^+ = √(-iπ M n_rK_P/ħ) (L_+-L_P) . We used here the ISPM2for the finite integration limits within the tori, i.e., between the minimalL_-=0and maximalL_+=L_Cvalues of the angular-momentum integration variable for the K=3familycontribution. The phase factorμ^_M Pin Eq. (<ref>)isthe Maslov index (see Sec. <ref>) as in the asymptotic Berry-Tabor trace formula. The amplitude (<ref>)obtained for the formula (<ref>) is regular at the bifurcation points where the stationary point is located at the end pointL=L^*=L_Cof the action (L) part of a torus. For the case of the power parameterαsufficiently far from the bifurcation points, one arrives at the SSPM limit of the trace formula (<ref>) with the amplitudeA_M P, identical to theBerry-Tabor trace formula<cit.>A_M P^( SSP)=2 L_P T_P/πħ^5/2√(M n_rK_P)e^iπ/4.According to Sec. <ref>, the Maslov indexμ^_M Pof Eq. (<ref>)is determinedby the number of turning and caustic points within the FMCT(see Refs. <cit.>),łmasl3Draμ^_M P =3 M n_r + 4 M n_θ , ϕ^_𝒟=-π/2 .The total Maslov indexμ_M P^ (tot)is defined asthe sum of this asymptotic part (<ref>) and the argument of the complex density amplitude (<ref>) <cit.>. The total indexμ_M P^ (tot)behaves as a smooth function of the energyEand the power parameterαthrough the bifurcation point. §.§Two-parameter circle families in the ISPM2 łsec5CFor contributions of the circular PO families to the trace formula (<ref>), one obtainsłdengenCδ g_ scl,C^(E)= ∑_M A_M C^× exp[i/ħ S_M C(E) - i π/2μ^_M C-iϕ^_𝒟] . The sum is taken overthe repetition number for the circlePO andM=1,2,... . HereS_M C(E)is the action along the orbitMC, łactionC S_M C(E)=M ∫_0^2π Lθ = 2π M L_C ,whereL_Cis the angular momentum of the particle moving along the orbitC. For amplitudes of theMC-orbit contributions, one obtains łamp2ispCA_M C^(E) = i L_C T_C/π ħ^2√(F_M C)× (𝒵_p M C^+) (𝒵_r M C^-, 𝒵_r M C ^+) ,whereT_C=2 π/ω^_Cis the period of a particle motion along the primitive (M=1) orbit C.Here,ω^_Cis the azimuthal frequency,łomtcraω^_C =ω_θ(L=L_C)=L_C/(m r_C^2) ,r^_Cthe radius of theCorbit, andL_Cis the angular momentum for a particle motion along theCPO <cit.>,łrcLcd2Fcrar^_C= R_0(2E/(2+α) V_0)^1/α, L_C=p(r^_C)r^_C .In Eq. (<ref>),F_M Cis the stability factor (the trace of the monodromy matrix)łfgutz F_M C= 4 sin^2(π M√(α+2))and𝒥_M C^(p)is the JacobianłjacpCres𝒥_M C^(p)= 2 π (α+2)MK_C r_C^2 ,whereK_Cis the curvature forCorbits <cit.>,łcurvraC K_C = -(α+1)(α-2)/12(√(α+2))^3L_C.The finite limits in the error functionsof Eq. (<ref>),𝒵_p M C^±and𝒵_r M C^±, are given by łZplimCfin𝒵_p M C^- = 0, 𝒵_p M C^+=L_C√(-i π/ħ (α+2) M K_C) , 𝒵_r M C^- = (r_ min/r^_ C-1) √(i F_M C/4 π(α+2) ħ M K_C) , 𝒵_r M C^+ = (r_ max/r^_C-1) √(i F_M C/4 π(α+2) ħ M K_C) ,wherer_ minandr_ maxare the radial turning points specified below. The interval between them covers the CT manifoldincluding the stationary pointr^_C.Asymptotically farfrom the bifurcations(also far from the symmetrybreaking pointα=2),the amplitude (<ref>) approaches the SSPM limit A_M C^(E) →A_M C^ SSP(E) =2 L_C T_C/πħ^2 √(F_M C) .In these SSPM derivations, the radial integration limitsr_ minandr_ maxturn into the asymptotic valuesłrmimax r_ min=0, r_ max= R_0ℰ^2/(2+α) .They are given bythe two real solutions of Eq. (<ref>) atL=0. The upper limitr_ maxwas extended to infinityin the derivation of Eq. (<ref>) because the stationary pointr^_ Cis far away fromboth integration boundariesr_ minandr_ max. The Maslov indexμ^_M Cin Eq. (<ref>) is given by μ^_M C=2 M ,ϕ^_𝒟=π/2 .For the calculation of this asymptoticMaslovindexμ^_M Cthrough the turning and caustic points [see the trace formula(<ref>) with the ISPM2(<ref>) and the SSPM (<ref>) amplitude], one can use the FMCT(Sec. <ref>). The total Maslov indexμ_M C^ (tot)can be introduced as above (seeRefs. <cit.>). Taking the opposite limitα→α_ bif=n_r^2/n_θ^2-2to the bifurcationswhereF_M C→ 0, but far away fromthe HO limitα=2,one finds that the argument of the seconderror function in Eq. (<ref>), coming from the radial-coordinate integration,tends to zero proportional to√(|F_M C|)[Eq. (<ref>)].Thus, as in a general case (Sec. <ref>), the singular stabilityfactor√(F_M C)ofthe denominator in Eq. (<ref>) is exactly canceled bythe same from the numerator.At the bifurcationF_M C→ 0, one obtainsłamp2ispCbif A_M C^ ISP(E)→ 2L_C/ħ^3/2 ω^_C √(iπ (α+2) M K_C r^2_C) ×( Z_p M C^(+)) (r^_ min-r^_ max) .Therefore, Eq. (<ref>) gives the finite result through the bifurcation. Taking the reduction of the end-point manifold for the parent-orbit term, according to Eq. (<ref>),łrboundbif r^_ min→ r^_ max→ r^_ C ,the amplitude (<ref>) vanishes at the bifurcation point <cit.> [see Eq. (<ref>)]. This is in line with general arguments for the bifurcation limit [see Sec. <ref> around Eq. (<ref>)].As shown in the Appendix, following the FMCT(Sec. <ref>) one can derive the ISPM3 expression for the oscillating level density. One may note that the parameterwgiven by Eq. (<ref>) can be considered as the dimensionless semiclassical measure of the distance from a bifurcation. Similarly, as for the caustic catastrophe points, in the case of the application of the FMCT(Sec. <ref>) to the bifurcation of the circular orbits (see also Sec. <ref>), one can usesimply the ISPM2 (Sec. <ref>) as the simplest approximationnear the bifurcation, i.e.,w1 .However, working out properlythe transition itself from the asymptotic SSPM radial-integration limitsr^_±tothe same valuer^_Cat the bifurcation within its close vicinity (w1 )is left for future work.§.§ Two-parameter diameter families łsec5D:DSSPMFor the diameter-orbit (𝒦=2) family contribution to the trace formula (<ref>) for the RPL potential, the ISPM is needed only near the symmetry-breaking atα=2of the harmonic-oscillator limit <cit.>.For our purpose, we simply use the SSPM approximation for the diameter families, valid at the values of the power parameterαfar from the symmetry-breaking limit,łdengenDδ g_ scl D^(E)= ∑^_M A_M D^ ×exp[i/ħ S_M D(E) - iπ/2μ^_M D-iϕ_𝒟],wherełamp2sspD A_M D^=1/i π M K_Dω_r ħ^2 .The frequencyω_ris expressed through the radial period,łperiodfreqT_r=2 π/ω_r=∫_0^r_ max2mṛ/p(r) =√(2mπ/E)r_maxΓ(1+1/α)/Γ(1/2 + 1/α) ,whereΓ(x)is the Gamma function. In Eq. (<ref>),K_Dis the diameter curvature <cit.>łcurvraD K_D= Γ(1-1/α)/Γ(1/2-1/α) √(2 π m R_0^2 V_0) .For the Maslov index [Eq. (<ref>)], one obtainsłmaslDμ^_M D=2M , ϕ^_𝒟=-π/2 .§.§ TOTAL TRACE FORMULAS FOR THE SPHERICAL RPL POTENTIAL łsec5E:totalThe total ISPM trace formula for the RPL potential is the sum of the contribution of the𝒦=3polygonlike (P) familiesδ g_P^(E)[Eqs. (<ref>) with (<ref>)], the𝒦=2circular (C) familiesδ g_C^(E)[Eqs. (<ref>) and (<ref>) for the ISPM2and Eqs. (<ref>) and (<ref>) for the ISPM3],and the𝒦=2diameter (D) familiesδ g_D^(E)[Eqs. (<ref>) and (<ref>)], łdeltadenstotprlpδ g_ scl(E) =δ g_ scl,P^(E) + δ g_ scl,C^(E) + δ g_ scl,D^(E) .This trace formula has the correct finite asymptotic limits to the SSPM: The Berry-Tabor result [Eqs. (<ref>)and (<ref>)]for thePorbits (𝒦=3) and forCorbits (𝒦=2) [Eqs. (<ref>) and (<ref>)]; see the sameforDorbits [Eqs. (<ref>) and (<ref>)]. Transforming the variable from the ordinary energyEto the scaled energy(<ref>), one obtains the trace formula for the scaled-energy level density,łscldenstyra1δ𝒢()=δ g(E)dE/d =∑_ POδ𝒢_ PO(),withδ𝒢_ PO() = [𝒜_ PO(). ×.exp(iτ_ PO-iπ/2μ_ PO -iϕ_𝒟)]. łscldenspoThe Fourier transform of this scaled-energy level density, truncated by the Gaussian with the cutoffγ, is expressed as F_γ(τ) =∫𝒢()e^iτe^-(/γ)^2 d=∑_ PO𝒜̃_ PO(τ) e^-γ^2(τ-τ_ PO)^2/4. łftlsclThis gives a function with successive peaks at the scaled periods of the classical POsτ=τ_ POwith the height|𝒜̃_ PO|, which is proportional to the amplitude𝒜_ POof the contribution of the orbit PO to the semiclassical level density. Evaluating the same Fourier transform by the exact quantum level density, one hasłfourierpower1 F(τ) = ∫[∑_iδ(-_i)] e^iτe^-(/γ)^2= ∑_i e^i_iτe^-(_i/γ)^2,withłepsi_i=(E_i/V_0)^1/2+1/α .Thus, one can extract the contribution of classical periodic orbits to the level density from the Fourier transform of the quantum level density. In what follows we consider the classical-quantum correspondence using this Fourier transformation technique, in addition to the direct comparison of quantum and semiclassical level densities.§ COMPARISON WITH QUANTUM RESULTS Figure <ref>shows the Fourier transform of thequantum-mechanical level density𝒢()for the RPL potential [see Eq. (<ref>)]. At the HO limitα=2, all the classical orbits are periodic and form the four-parameter family for a given energy.The Fourier transform exhibits the equidistant identical peaks atτ_n=√(2)π n, corresponding to thenth repetitions of the primitive PO family. With increasingα, each peak is split into two peaks corresponding to the diameter (D) and circle (C) orbits and the amplitudes of the oscillating level density for these orbits are decreased. However, one finds a growth of the peak atτ∼ 6.2corresponding to theCorbit around the bifurcation pointα^_ bif=7.0. Note that, approaching the bifurcation,the contribution of theCorbit is strongly enhanced until it forms a local family of POs with a higher degeneracy at the bifurcation point.From this point a trianglelikeP(3,1) family bifurcates. This family has high degeneracy𝒦=3. It remains important, also for largerα >α^_ bif. The above enhancement in the Fourier peaksF(τ)is directly associated with the oscillating ISPM level-density amplitudeA^_ POof the bifurcating PO family having a high degeneracy. This family is a major term in theħexpansion in the comparison with the SSPM asymptotics(see Sec. <ref> and Ref. <cit.>).The Fourier peak atτ∼ 6.2in Fig. <ref> shows the enhancement of the amplitude of the newbornP(3,1) family contribution includingC(1,1) orbits as the end points (see the Introduction and Sec. <ref>).As the significance of bifurcations is confirmed through the Fourier analysis [Eqs. (<ref>) and (<ref>)],let us now investigate the oscillating part of the scaled-energy level densities averaged with the Gaussian averaging parameterγ,łavdentotδ𝒢_γ() =∫exp[-(-'/γ)^2] δ𝒢(')d'.The semiclassical shell-correction densityδ𝒢_γ,scl()is given by łscdentotδ𝒢_γ,scl() =∑_ POδ𝒢_ PO()exp(-τ_ PO^2γ^2/4) .[see Eq. (<ref>) forδ𝒢_ PO]. For the quantum density, one hasłdGQMδ𝒢_γ, QM=𝒢_γ, QM- 𝒢_ QM ,where łqmdentot𝒢_γ,QM() =∑_i exp[-(-_i/γ)^2] .The smooth level density𝒢_ QMis calculated for the scaled spectrum_i. For these calculations we employedthe standard Strutinsky averaging (over the scaled energy), finding a good plateau[ It may worth pointing out that thequality of the plateau in the SCM calculations of the level density is much better when using the scaled-energy variableratherthan the energy E itself <cit.>.] around theGaussian averaging widthγ=2-3and curvature-correctiondegree M=6.Figures <ref>–<ref> show good agreement of the coarse-grained (γ=0.6) and fine-resolved (γ=0.1-0.2) semiclassical and quantum results forδ𝒢^_γ()(divided by) as functionsof the scaled energyatα=6.0, 7.0, and8.0. Atα=6, as well asα=2and4, the analytic expressions for all of the classical PO characteristics are available, and can be used to check the precision of the numerical calculations <cit.>. The valuesα=6(Fig. <ref>) and8(Fig. <ref>) are taken as examples thatare sufficiently farfrom the bifurcation pointα=α_ bif=7(Fig. <ref>). The ISPM results at these values ofαshow good convergence to the SSPM results. TheCandDPOs with the shortest (scaled) periodsτare dominating the PO sum at a large averaging parameterγ=0.6(thecoarse-grained ormajor-shell structure). Manymore families with a relatively long periodτatγ=0.1-0.2(the fine-resolved shell structure) become significantin comparison with the quantum results<cit.>.Figure <ref> shows the results for the bifurcation pointα=7where the trianglelike (3,1) PO family emerges from the parentC(1,1) family in a typical bifurcation scenario.One also finds good agreement of the ISPM with quantum results here. As the SSPM approximation fails atthe bifurcation,it is not presented in Fig. <ref>. As discussed above, the SSPM at the bifurcation yields a sharp discontinuity of theP(3,1) amplitude and a divergent behavior of theC(1,1), in contrast to the continuous ISPM components. Our results in Fig. <ref> demonstrate that the ISPM successfully solves these catastrophe problems of the SSPM for all averaging parametersγ. In contrast to the results shown in Figs. <ref> and <ref>, the coarse-grained (γ=0.6) density oscillations at the bifurcation pointα=7(Fig. <ref>) do not contain any contributions from theC(1,1) end-point term but instead theP(3,1) term becomesdominant.Note that many more families with relatively long periodsτbecome necessary to account for the fine-resolved shell structures (γ=0.1-0.2) <cit.>.For the exemplary bifurcationα=7.0, at smaller averaging parameters (γ 0.2) the dominating orbitsbecome the bifurcating newbornP(3,1) of the highest degeneracy𝒦=3along with the leadingP(5,2),P(7,3), andP(8,3) POswhich are born at smallerα(see Fig. <ref>). They include the parentC-orbit end-point manifolds. As also shown in the quantum Fourier transforms in Fig. <ref>, these POs yield largercontributions at the bifurcation values ofαand are even more enhancedon their right in a wide region ofα.§ CONCLUSIONS łsec6:conclThe Fedoriuk-Maslov catastrophe theory is extended to simple bifurcation problems in the POT. Within the extended FMCT, we overcome the divergence and discontinuity of semiclassical amplitudes of thestandard stationary-phase method, in particular, in the Berry-Tabor formula near bifurcations.A fast convergence in the PO expansion of the averaged level densityfor a large Gaussian averaging parameter is shown too. This allows one often to express significantfeatures of the shell structure in terms of afew short periodic orbits. We have formulated our ISPM trace formula for a simple bifurcation scenario so that the parent orbits at the end points have vanishing contributions at the bifurcation point, which allows us to consider themeverywhere separately from the term for a newborn family of the periodic orbits.The extended FMCT is used for derivations of the trace formula in the case of the three-dimensional spherical RPL potential by employing the improved stationary phase method. We presented a class of the radial power-law potentials that,up to a constant, provides a good approximation to the WS potential in the spatial region where the particles are bound.The RPL potential is capable of controlling surface diffuseness and contains the popular harmonic-oscillator and cavity potentials in the two limiting cases of the power parameterα. Its advantage is the scaling invariance of the classical equations of motion. This invariance makes the POT calculations and the Fourier analysis of the level density very easy. The contribution of the POs to the semiclassical level density and shell energies is expressed analytically (and even all the PO characteristics are given explicitly, e.g., forα=6) in terms of the simple special functions. The quantum Fourier spectra yield directly the amplitudes of the quantum level density at the periods (actions) of the corresponding classical POs.We have derived the semiclassical trace formulas thatare also valid in the bifurcation region and examined them at the bifurcation catastrophe points and asymptotically far from them in the spherical RPL potential model. They are based onthe SPM improved to account for the effect of the bifurcations by using the extended FMCT. The ISPM overcomes the problems of singularities in the SSPM and provides the generic trace formula that relates the oscillating component of the level density for aquantum system to a sum over POs of the correspondingclassical system. We showed good convergence of this improved trace formula to the simplest ISPM based on the second-order expansion of the classical actionat several characteristic values ofthe power parameterαincluding the bifurcations and asymptotically far from them. We obtained good agreement between theISPM semiclassical and quantum results for thelevel-density shell correctionsat different values of the power parameterα, both at the bifurcations and far from them. Sufficiently far from the bifurcation of the leading short POs with a maximal degeneracy, one finds also good convergence of the ISPM trace formulas to the SSPMapproximation.Weemphasize thesignificant influenceof the bifurcations of short POs on the main characteristicsof oscillating components of the single-particle level density for a fermionicquantum system.They appear in the significant fluctuations of the energy spectrum (visualizedby its Fourier transform), namely, the shell structure. In line with the general arguments of the extended FMCT,the stationary points forming the circular-orbit families are located at the end point of the classically accessible region and they coincide with the newborn family of the polygonlike orbits at the bifurcation. Taking into account the reduction of the end-point manifold in the bifurcation limit, the parentC-family contribution is transformed into the newbornP-family term that presents now their common result. Thus, one has the separate contributions ofthe parentCand newbornPorbits through the bifurcation scenario, but with no concern about double counting.Future work should study in detail thetransition of the ISPM trace formula from the bifurcation pointsto its asymptotic SSPM region. This will enable us to understand more properly the shape dynamics of the finite fermion systems.In particular, the improved stationary phase method can be applied to describe the deformed shell structures where bifurcations play an essential role in formations of the superdeformed minima along a potential energy valley <cit.>. One of the remarkable tasks might be to clarify, in terms of the symmetry-breaking (restoration) and bifurcation phenomena,the reasons of the exotic deformations such asthe octupole and tetrahedral ones within the suggestedISPM.In this way, it would be worth extending our present local bifurcation FMCTto describe, e.g.,a bridge (non-local) bifurcation phenomenonfound in a more realistic mean field in the fermionic systems (see alsoRefs. <cit.>).Our semiclassical analysis may therefore leadto a deeper understanding of the shell effects in the finite fermionic systems such as atomic nuclei, metallic clusters,trapped fermionic atoms, andsemiconductor quantum dots <cit.>. Their level densities, conductance, and magneticsusceptibilities are significantly modified by shell effects.As a first step towards the collective dynamics, the oscillating parts of the nuclear moment of inertiashould be studied semiclassically in terms of POs taking into account the bifurcations <cit.>.§ ACKNOWLEDGMENTS The authors gratefully acknowledge M. Brack and K. Matsuyanagi for fruitful collaborations andmany useful discussions. One of us (A.G.M.) is also very grateful for hospitality during his working visits to theDepartment of Physical Science and Engineering of the Nagoya Institute ofTechnology and for financial support fromthe Japanese Society of Promotion of Sciencesthrough Grant No. S-14130.§ THE ISPM3 APPROXIMATION Following the FMCT (Sec. <ref>), one can derive the improved (ISPM3) contribution of theCorbits by taking into account the third-order terms of the action expansion [see Eq. (<ref>)] in the integrationover the catastrophe variabler. Within the ISPM3, one obtainsłdenC3 δ g_ scl,C^(3)(E)= ∑_M A_M C^(3)×exp[i/ħ S_M C(E) - i π/2μ^_M C+2i/3w^3/2-iϕ^_𝒟].The ISPM3 amplitudesA_M C^(3)are given by łampc3 A_M C^(3) = 2Λ √(L_C)/ħ^5/2ω^_C (𝒵^+_p,M C)/√(2 π i (α+2)M K_C)×[(-w,𝒵_M C^(-,3), 𝒵_M C^(+,3)) .+.i (-w,𝒵_M C^(-,3),𝒵_M C^(+,3))] ,wherełwC w=w^_M C=[κ^1/3 F_M C/ 4 π (α+2) M K_C L_C(3a)^2/3]^2 ,with łakS a=r^3_C/6L_C Φ_CT”'(r^_C) ,κ=L_C/ħ .The parameterϵof Eq. (<ref>) [Eq. (<ref>)] used in deriving the above expressionsis proportional to the stability factorF_M C[Eq. (<ref>)],łepsCϵ=F_M C/4 π (α+2) MK_CL_C .The incomplete Ai (Gi) integrals in Eq. (<ref>) are defined by Eq. (<ref>). The integration limits of thesefunctions are the same as those given byEq. (<ref>),łlimrC3𝒵_M C^(±,3)= r^_±-r^_C/r^_CΛ +σ√(w^_M C) , σ=(ϵ) ,wherer_±are the upper (r^_+>r^_C) and lower(r^_-<r^_C) limits for the radial integration. These integration limits are defined in Eq. (<ref>). In the bifurcation limitα→α^_ bif, both terms in square brackets in Eq. (<ref>) [see also Eq. (<ref>)] go to zero for the same reason as in the ISPM2 case.The incomplete Ai and Gi functions of theintegrand [Eq. (<ref>)] have no singularities in thebifurcation limitw → 0.In addition,the radial integration limitsr_ minandr_ maxapproach the stationary pointr^*=r^_C, which is theC-orbit radius [Eq. (<ref>)]. This ensures the disappearance of the end-point manifold in this limit and therefore, in line with the general arguments of Sec. <ref> (see also Sec. <ref>), one finds the zero contribution ofthe circular orbit term exactly at this bifurcation. In turn, the contribution of the circular orbit is includedin the newbornPorbit term.In the opposite limit, sufficiently far from the bifurcation points, where the stability factorF_M C(α)takes a finite nonzero value, the second term in Eq. (<ref>) changes with increasingϵmuch faster (proportional toκ^1/3ϵ) than the first component (proportional toκ^1/3). Thus, one has|𝒵_M C^(±,3)|≫ |z^∗,±_M C|in this limit. The ISPM3 expression[Eq. (<ref>) with Eq. (<ref>)] for the oscillating level density suggests that the parameterw(w ∝ F_M C^2 ∝ϵ^2) can be considered as a dimensionless measure of the distance from the bifurcation [see Eqs. (<ref>) and (<ref>)]. For a large distance from thebifurcationw ≫ 1[Eq. (<ref>)], one can extend the radial integration limits asr_-=0andr_+→∞. In this limit, the incomplete Airy and Gairy functions (<ref>) can be approximated by the complete ones(<ref>). Since the argumentwof these standard functions [Eq. (<ref>)] at a finite stability factorF_M C(α)becomes large in the semiclassicallimitκ≫ 1,one can use their asymptotic expressions (<ref>). Thus,we arrive at the same SSPM result[Eqs. (<ref>) and (<ref>)] for theCfamily contributions as obtained fromthe ISPM2C-trace formula[see Eq. (<ref>) for its amplitude]. For the simplest catastrophe problem, the ISPM3 might become important when the PO is distant from the bifurcation points to some extent but not asymptotically far from them.It is also necessary for the higher-order catastrophe problem, which is not found in the RPL model discussed in this paper. The definition of the end-point manifold might also beaffected by the consideration of higher expansion terms.This is also a problem to be solved in the future in order to describe the transition from a bifurcation vicinity to the asymptoticregion. 99GUTZpr M. C. Gutzwiller, J. Math. Phys.12, 343 (1971).GUTZbook90 M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).BBann72 R. B. Balian and C. Bloch, Ann. Phys. (N.Y.)69, 76(1972).sclbook M. Brack and R. K. Bhaduri,Semiclassical Physics(Westview, Boulder, 2003).SMepp76 V. M. Strutinsky, Nucleonica, 20, 679 (1975); V. M. Strutinskyand A. G. Magner, Sov. J. Part. 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Creagh, A. G. Magner,P. Meier, and S. M. Reimann,Z. Phys. D40, 276 (1997).FKMSprb08 S. Frauendorf, V. M. Kolomietz, A. G. Magner, andA. I. Sanzhur, Phys. Rev. B58, 5622 (1998).MSKBprc10 A.G. Magner, A.S. Sitdikov, A.A. Khamzin, andJ. Bartel, Phys. Rev. C81, 064302 (2010).MGBpan14 A. G. Magner, D. V. Gorpinchenko, and J. Bartel,Phys. Atom. Nucl.77, 1229 (2014). GMBBprc16D. V. Gorpinchenko, A. G. Magner, J. Bartel, and J. P. Blocki,Phys. Rev. C93, 024304 (2016).MGBpan17 A.G. Magner, D.V. Gorpinchenko, and J. Bartel,Phys. Atom. Nucl. ,80, 122 (2017). | http://arxiv.org/abs/1709.10403v2 | {
"authors": [
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"K. Arita"
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"categories": [
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"primary_category": "math.DS",
"published": "20170927163346",
"title": "Semiclassical catastrophe theory of simple bifurcations"
} |
On generalized G_2-structures and T-duality]On generalized G_2-structures and T-duality [email protected] (Argentina) and IMECC-UNICAMP (Brazil) [email protected] IMECC-UNICAMP This is a short note on generalized G_2-structures obtained as a consequence of a T-dual construction given in <cit.>. Given classical G_2-structure on certain seven dimensional manifolds, either closed orco-closed, we obtain integrable generalized G_2-structures which are no longer an usual one, and with non-zero three form in general. In particular we obtain manifolds admitting closed generalized G_2-structures not admitting closed (usual) G_2-structures.V. del Barco supported by FAPESP grant 2015/23896-5. L. Grama supported by FAPESP grant 2016/22755-1 and 2012/18780-0.53C30,22E25,17B01, 81T30,53D18[ Lino Grama December 30, 2023 =====================§ INTRODUCTION Generalized G_2-structures were introduced by Witt <cit.> with the intention of generalizing the classical concept of G_2-structure defined by a 3-form φ <cit.>. A generalized G_2-structure on a 7-dimensionaldifferentiable manifoldM is a reduction from the structure group ^*× Spin(7,7,) of the principal bundle TM⊕ T^*Mto G_2× G_2. This reduction determines a generalized metric and a 2-form on M, which split TM⊕ T^*M into submodules with positive and negative definite metrics. Therefore there is associated a pair of spinors Ψ_± in the irreducible spin representation Δ=^8 of Spin(7). TheG_2× G_2 invariant tensor Ψ_+⊗Ψ_-∈Δ⊗Δ can be consideredas a differentialform on M, so it induces elements (Ψ_+⊗Ψ_-)^even/odd corresponding to the even and odd degrees <cit.> . Up to a B-field transformation and a dilaton we have (see also <cit.>) ρ = Ψ_+⊗Ψ_-^even =c-c⋆φ+s ⋆( α∧⋆φ)-s α∧φ-s⋆α, ρ̂ = Ψ_+⊗Ψ_-^odd =sα-cφ-s ⋆( α∧φ)-s α∧⋆φ+c1/7φ∧⋆φ,where α is a unit 1-form,φ is a 3-form and the parameters s,c correspond, respectively, to the sine and cosine of the angle between the spinors Ψ_±; in particulars^2+c^2=1.Let H be a 3-form on M.A generalized G_2-structure defined by the spinors ρ, ρ̂ as above, is called strongly integrable with respect to H if d_Hρ=d_Hρ̂=0,where d_H·=d·+H∧ is the twisted operator of d. The generalized G_2-structure is called weakly integrable of odd (resp. even) type if d_Hρ̂=λρ,d_Hρ=λρ̂.for some non-zero constant λ. The real number λ is called the Killing number.An usual G_2-structure φ on a 7-manifold induces a generalized G_2-structure on M. In this case Ψ_+=Ψ_- and s=0, therefore the even and odd spinors are given byρ = 1-⋆φ,ρ̂= -φ+dV.When H is zero, we see that the generalized structure is strongly integrable (<ref>) if and only if the usual G_2 is parallel, that is, φ is closed and co-closed. Instead,weak integrability of odd type cannot occur for usual G_2-structures, independently of H (see <cit.>). If we let H to be any closed 3-form, the only compact strongly integrable generalized G_2 are the usual parallel G_2-structures <cit.>. Fino and Tomassinigave the first example of a compact strongly integrable generalized G_2-structure for H non-closed <cit.>.Making an analogy with the classical case we define a generalized G_2-structure with spinors ρ and ρ̂ to be a closed (resp. co-closed) structure if d_Hρ̂=0,d_Hρ=0.Given a closed G_2-structure, the generalized structure associated to φ is closed with respect to any 3-form H such that H∧φ=0. To the contrary, if φ is co-closed, then the generalized structure is co-closed only for H=0. In fact by (<ref>), d_Hρ=H-H∧⋆φ-d⋆φ=0 if and only if H=0.Witt himself was interested in the relation of generalized G_2-structures defined on T-dual manifolds. Recall that T-duality is a concept introduced by Bouwknegt, Evslin, Hannabuss and Mathai <cit.>, <cit.> for manifolds having the structures of principal torus bundles. According to <cit.>, if H and H^∨ are closed 3-forms on T-dual manifolds M and M^∨ are T-dual then there exists an isomorphism τ between the differential complexes (Ω_T^k^∙ (M),d_H) and (Ω_T^k^∙ (M^∨),d_H^∨). Such isomorphism τ satisfiesd_H^∨τ (ρ)=τ (d_H ρ).Therefore the integrability conditions are preserved by T-duality, thus generalized G_2-structures on M integrable with respect to H, induce generalized G_2-structures on M^∨, integrable with respect to H^∨. And vice-versa. The aim of this short note is to contribute with further examples of integrable generalized G_2-structures following the construction of T-duals the authors developed in <cit.>, together with Leonardo Soriani.We start with an usual G_2-structure on a seven dimensional manifold seen as a generalized G_2 withrespectto certain three forms H. In the dual manifold to the given one we obtain an integrable generalized G_2 which is no longer an usual one, and with non-zero three form in general. Ourprevious work focuses on invariant structures on compact quotients of solvable Lie groups by discrete subgroups. So this framework maintains here and we develop theexamples at the Lie algebra level. In <cit.> we indicated how duality would contribute with the study of symplectic structures by dualizing generalized complex structures. A similar spirit is pursued here, but in this case we work directly with the spinors defining the generalized structures. In this manner we are able to present manifolds admitting closed generalized G_2-structures not admitting closed (usual) G_2-structures. This note was motivated by the papers of Witt, Fino and Tomassini <cit.>.§ LIE ALGEBRAS AND INFINITESIMAL DUALITYWe briefly recall the concept of duality of Lie algebras <cit.>.Letbe a Lie algebra together with a closed 3-form H. Letbe an abelian ideal of, we say that the triple (,,H) is admissible if H(x,y,·)=0 for all x,y∈. Notice that when =1 then any closed 3-form gives an admissible triple. In this case, denotethe quotient Lie algebra and q:and q^∨:^∨ the quotient maps. The subspaceof ⊕^∨={(x,y)∈⊕^∨: q(x)=q^∨(y)}is a Lie subalgebra and thefollowing diagram iscommutative[ld]_p[rd]^p^∨[dr]_q^∨[ld]^q^∨.Here p and p^∨ are the projections over the first and second component, respectively. A 2-form F∈Λ^2^* is said to be non-degenerate in the fibers if for all x∈={(x,0)∈:x∈}, there exists some y∈^∨={(0,y)∈:y∈^∨} such that F(x,y)≠ 0. Such an F exists if and only if =^∨.Twoadmissible triples (,,H) and (^∨,^∨,H^∨) are said to be dual if / ≃≃^∨/^∨ and there exist a 2-form F inwhich is non-degenerate in the fibers such that p^*H-p^∨*H^∨=dF. The dual of a given admissible triple always exists and its construction is described in the following result.<cit.> Let (,,H) be an admissible triple witha central ideal and let {x_1,…,x_m} be a basisof. Define* Ψ^∨=(ι_x_1H,…,ι_x_mH), * ^∨=(/)_Ψ^∨ and* H^∨=∑_k=1^m z^k∧ dx^k+δ where {z_1,…,z_m} is a basis of ^∨ and δ is the basic component of H.Then (^∨,^∨,H^∨) is an admissible triple and is dual to (,,H). Conversely, if (^∨,^∨,H^∨) is dual to (,,H), then there exist a basis {x_1,…,x_m} of and a basis {z_1,…,z_m} of ^∨ such that the formulas above hold. Here (/)_Ψ^∨ denotes the central extension of / by the closed 2-form Ψ^∨ (see <cit.> for details).G_2 and SU(3)-structures on Lie algebras were treated by several authors. In our context, the main references are the work of Conti, Fernandez, Fino, Manero, Raffero and Tomassini (see <cit.> and references therein). A G_2-structure on a Lie algebrais a non-degenerate 3-form φ such that is some adapted basis {e^1,…, e^7} of the dual ^*of , it is written as φ=e^127+e^347+e^567+e^135-e^236-e^146-e^245.To continue, we consider Lie algebrasendowed with G_2-structures and with non-trivial center. We consider central ideals inand their dimensions is what we call the dimension of the fiber, having in mind a possible torus bundle structure on the Lie group associated to . After fixing a closed 3-form giving integrability of the usual G_2-structure, we compute the dual triple and the generalized G_2-structure arising on it. §.§ One dimensional fiber Letbe a Lie algebra with non-trivial center and letbe a one dimensional central ideal. Fixa generator x∈ and let =⊕ be an orthogonal decomposition with respect to the metric induced by φ. Letφ be a G_2-structure on , then if α is the dual element to x we haveφ=α∧ω+ψ_+,ι_xω=0ι_xψ_+=0.Up to a normalization of coefficients, the forms (ω,ψ) define an SU(3)-structure on =/ <cit.>.The spinors associated to the generalized G_2-structure induced by φ are given by (<ref>):ρ = 1-1/2ω^2-ψ_-∧α,ρ̂= -α∧ω-ψ_++dV. Any closed 3-form H inmakes (,,H) an admissible triple, since =1. According to Theorem <ref>, the dual triple is (^∨,^∨,H^∨) where ^∨ is the central extension ofby ι_xH. Explicitly, ^∨= z⊕ and the Lie bracket of ^∨ satisfies[u,v]=[u,v]_+(ι_xH)(u,v)z [z,u]=0, u,v∈.Letbe the 1-form such that (z)=1 and ()=0, then the closed 3-form on ^∨ is H^∨=∧ dα +δ where δ the basic part of H. Note that H≠ 0 if and only if dα≠ 0 which means that x is not a direct factor of . The 2-form F∈Λ^2^* giving the duality on the correspondencespace is given by F=α̃∧α.The dual spinors ρ^∨ and ρ̂^∨ are given by (see <cit.>)ρ^∨=ι_xe^Fρ,ρ̂^∨=ι_xe^Fρ̂.Explicitly, we obtainρ^∨= -α̃+ψ_-+1/2α̃∧ω^2,ρ̂^∨ =-ω+α̃∧ψ_++1/6ω^3.The spinors above correspond to a generalized G_2-structure associated to the SU(3)-structure (ω,ψ_+), when one imposes the angle of the spinors to be π/2. These structures where considered in <cit.>. Notice that the dual of an usual G_2-structure is never an usual G_2-structure, but a pure generalized G_2.Letbe the Lie algebra spanned by {e_1, …, e_7} and satisfying the bracket relations[e_1,e_7]=-e_3, [e_1,e_5]=-e_4, [e_2,e_7]=-e_4, [e_1,e_3]=-e_6,and zero in the other cases. The Lie differential in the dual basis is de^3=e^17, de^4=e^15+e^27 and de^6=e^13. Consider the central ideal =e_6.The 3-formφ=e^127+e^347+e^567+e^135-e^236-e^146-e^245 is a closed G_2-structure <cit.>. Notice that φ=e^6∧ω+ψ_+ with ω=-e^14-e^23-e^57 and ψ_+=e^127+e^347+e^135-e^245. The pair (ω,ψ_+) define an SU(3)-structure on / We consider all possible closed 3-forms H such that d_Hρ̂=0. Canonical computations give that H is of the formH = a_1(e^134+e^267+e^123+e^357) +a_3(e^136+e^137+e^145-e^247) +a_2(e^126-e^237)+a_4(e^156-e^357)+a_5(e^157+e^134+e^267+e^357) +a_6(e^167-e^257)+a_7(e^367-e^457)+a_8(e^146+e^236) +a_9(e^347+e^567)+a_10(e^124+e^257)+a_11(e^125+e^137) +a_12e^127+a_13e^135+a_14 e^245+a_15(e^145+e^235), a_i∈. For any H as in (<ref>), the triple (,,H) is a compatible triple and d_Hρ̂=0. Thus ρ,ρ̂ define a closed generalized G_2-structure with respect to H. Now we describe the dual triple.The Lie algebra ^∨ has a dual basis {f^1, …,f^7} such that the Lie algebra differential is[df^3 = f^17,;df^4 =f^15+f^27,;df^6 = ι_e_6H=-a_6 f^17-(a_1+a_5) f^27-a_7 f^37-a_9 f^57; +a_2 f^12+a_3f^13+a_4 f^15+a_8 (f^14+f^23). ]Here we identify e_i with f_i for i=1, …, 5, 7, taking into account that /≃^∨/^∨. The dual 3-form is H^∨ = f^136+a_1(f^134+f^123+f^357)-a_2 f^237+a_3(f^137-f^247+f^145) -a_4 f^357+a_5(f^157+f^134+f^357)-a_6 f^257-a_7 f^457+a_9 f^347 +a_10(f^124+f^257)+a_11(f^125+f^137)+a_15(f^145+f^235) +a_12f^127+a_13f^135+a_14 f^245.and the dual spinor ρ̂^∨ is, by (<ref>), ρ̂^∨=-ω+f^6∧ψ_+=f^14+f^23+f^57+f^2456+f^1267+f^3467-f^1356.One can check directly that d_H^∨ρ̂^∨=0.Depending on the coefficients a_i of H in (<ref>), we reach non-isomorphic Lie algebras. For instance, if H=e^136+e^137-e^247+e^145 (a_3=1 while the others are zero), then ^∨≃. Meanwhile ^∨has one dimensional center for H=e^146+e^236. If H=0 then the only non-trivial Lie brackets of ^∨ are [f_1,f_7]=-f_3 and [f_1,f_5]=-f_4=[f_2,f_7].When H is as in (<ref>) and satisfies a_4· a_7=0 and a_3^2+a_8^2>0then (ι_e_6σ)^3=0 for any closed 3-form in the dual Lie algebra ^∨, so these Lie algebras do not admit closed G_2-structures, but they admit closed generalized G_2, as we showed above. The Lie algebraof dimension 7 with non-zero Lie brackets [e_2,e_5]=-e_6, [e_4,e_5]=e_7 admits closed G_2-structures <cit.>, such as, for instance, φ=e^127+e^347+e^567+e^135-e^146-e^236-e^245. Consider the central idealspanned by e_7 and let H be the 3-formH = a_1(e^124-e^456)+a_2(e^125-e^345)-a_3(e^134-e^156)+a_4 e^135+a_5(e^145-e^235)+ a_6(e^145+e^246)+a_7(e^234-e^256)+a_8 e^245,where a_i are real coefficients. Thus H is closed and satisfies d_Hρ̂=d_H(-φ+dV)=-dφ-H∧φ=0. Therefore, we have a closed generalized G_2-structure with respect to H. Since ι_e_7H=0, the dual Lie algebra ^∨ is defined by the only non-zero bracket relation [f_2,f_5]=-f_6 (as before we identify e_i with f_i for i=1, …, 6). The dual 3-form isH^∨ = -f^457+a_1(f^124-f^456 )+a_2(f^125-f^345)-a_3(f^134-f^156)+a_4 f^135+a_5(f^145-f^235)+a_6(f^145+f^246)+a_7(f^234-f^256) +a_8 f^245.The duality preserving the integrability, implies thatd_H^∨ρ̂^∨=0. Thus we re-obtained Example 5.1. of Fino and Tomassini in <cit.>.§.§ Fiber of dimension 2Whenhas dimension greater than one, there is no global expressions as (<ref>) for the dual spinors, so their computations need to be done by hand in each case.Letbe the Lie algebra such that the Lie algebra differential on a dual basis {e^1,…,e^7} isde^1=e^35+e^46, de^3=e^67, de^4=e^57, de^5=e^47, de^6=e^37, de^2=de^7=0.This Lie algebra is solvableand unimodular, and it admits a co-closed G_2 form. Indeed, φ=e^127+e^347+e^567-e^136-e^145-e^235+e^246satisfies⋆φ= e^1234+e^1256+e^3456+e^1357-e^1467-e^2367-e^2457and d⋆φ=0 (see <cit.>).Denote the Lie algebra with basis {e_1, …,e_6} and unique non-zero differential de^1=e^35+e^46. Thenis theone dimensional extension ofby the derivation D defined as De_3+i=e_6-i, i=0,…,3.The spinors associated to this G_2-structure are given in (<ref>):ρ = 1- (e^1234+e^1256+e^3456+e^1357-e^1467-e^2367-e^2457) ρ̂ = -(e^127+e^347+e^567-e^136-e^145-e^235+e^246)+e^1234567. The triple (,,H=0) with=()={e_1,e_2} is a compatible triple. And the generalized G_2-structure is co-closed with respect to H=0 (recall that co-closed G_2-structures are co-closed generalized G_2-structures only for H=0). We shall compute the dual triple and structure.Since ι_e_2H=ι_e_1H=0, the dual Lie algebra ^∨ is the trivial extension of =/, thus it has a dual basis {f^1,… ,f^7} such that the differentials account todf^3=f^67, df^4=f^57, df^5=f^47, df^6=f^37, df^1=df^2=df^7=0. One has that ^∨ is isomorphic to ^2⊕ (⋉_D ^4), with Das above, induced to ^4. The dual 3-form is H^∨=f^135+f^146.The dual spinor ρ^∨ is ρ^∨ = -(f^34+f^56+f^12)+f^1(f^367+f^457)+ f^2(f^357-f^467)+f^123456= -(f^12+f^34+f^56)+(f^136+f^145+f^235 -f^246)f^7+f^123456One can verify that d_Hρ^∨=0. The Lie algebra ^∨ also admits a co-closed usual G_2-structure since it is of the form f_7 ⋉_D ^6 for the derivation given above and ω=f^12+f^34+f^56, ψ_+=f^135-f^146-f^236-f^246 define a half flat structure on ^6. SoProposition 2.1. in <cit.> give co-closed G_2-structures on ^∨. plain | http://arxiv.org/abs/1709.09154v1 | {
"authors": [
"Viviana del Barco",
"Lino Grama"
],
"categories": [
"math.DG",
"53C30, 22E25, 17B01, 81T30, 53D18"
],
"primary_category": "math.DG",
"published": "20170926174128",
"title": "On generalized $G_2$-structures and $T$-duality"
} |
[][email protected] Department of Physics, Dogus University, Acibadem-Kadikoy, 34722Istanbul, Turkey[][email protected] of Physics, Dogus University, Acibadem-Kadikoy, 34722Istanbul, Turkey School of Physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran The magnetic dipole moment of the exotic Z_b(10610) state is calculated within the light cone QCD sum rule methodusing the diquark-antidiquark and molecule interpolating currents.The magnetic dipole moment is obtainedas μ_Z_b=1.73± 0.63 μ_N in diquark-antidiquark picture and μ_Z_b=1.59± 0.58 μ_N in the molecular case.The obtained results in both pictures together with the results of other theoretical studies on the spectroscopic parameters of the Z_b(10610) state may be useful in determination of the nature and quark organization ofthis state. Magnetic dipole moment of Z_b(10610) in light-cone QCDK. Azizi December 30, 2023 ======================================================= § INTRODUCTIONAccording to QCD and the conventional quark model,not only the standard hadrons, but also exotic states such as meson-baryon molecules,tetraquarks, pentaquarks, glueball and hybrids can exist.The first theoretical prediction on the existence of the multiquarkstructures was made by Jaffe in 1976 <cit.>. Although it was predicted in the 1970's,there was not significant experimentalevidence of their existence until 2003.The first observation on the exotic stateswas discovery of X(3872) made by Belle Collaboration <cit.>in the decay B^+→ K^+X(3872) J/ψπ^+ π^-.Subsequently, it was confirmed by BABAR <cit.>, CDF II <cit.>, D0 <cit.>, LHCb <cit.>and CMS <cit.> Collaborations. The discovery of the X(3872) state turned out to be theforerunner of a new direction in hadron physics. So far, more than twenty exotic states have been observedexperimentally [for details, see <cit.>]. The failure of these states to fit the standard particles' structures and violation of some conservation laws such as isospin symmetry, make these states suitable tools for studying the nonperturbative nature of QCD. In 2011, Belle Collaborationdiscovered two chargedbottomonium-like states Z_b (10610) and Z_b(10650) (hereafter we will denote these states as Z_b and Z_b^',respectively) in theprocesses Υ(5S) →ππΥ(nS),andΥ(5S) →ππ h_b (kP ) <cit.>.Here, n = 1, 2, 3 and k = 1, 2. The masses and widths of the two states have been measured as M_Z_b=10607.2 ± 2 MeV, Γ_Z_b= 18.4 ± 2.4 MeV, M_Z_b^'=10652.2 ± 1.5 MeV, Γ_Z_b^'= 11.5 ± 2.2 MeV.The analysis of the angular distribution shows that the quantum numbers of both states are I^G(J^P) = 1^+ (1^+). Both Z_b and Z_b^'belong to the family of charged hidden-bottom states. Since they are the first observed charged bottomoniumlike statesand also very close to the thresholds of B B̅^*̄(10604.6 MeV) and B^* B̅^*(10650.2 MeV), Z_b and Z_b^' states have attracted attention of many theoretical groups. The spectroscopic parameters and decays of Z_b and Z_b^' states have been studiedwith different models and approaches. Most of these investigations are based ondiquark-antidiquark <cit.>and molecular interpretations <cit.>,using the analogy to the charm sector. Although the spectroscopic features of these states have been studied sufficiently, the inner structure of these states have not exactly enlightened. Different kinds of analyses, such as interaction with the photon can shed light onthe internal structure of these multiquark states.A comprehensive analysis of the electromagnetic properties of hadrons ensurescrucial information on the nonperturbative nature of QCD and their geometric shapes. The electromagnetic multipole moments contain the spatial distributions of thecharge and magnetization in the particle and therefore,these observables are directly related to the spatial distributions of quarks and gluons in hadrons.In this study, the magnetic dipole moment of theexotic state Z_b is extracted byusing the diquark-antidiquark and molecule interpolating currents in the framework of the light cone QCD sum rule (LCSR).This method has already been successfully applied to study the dynamical andstatical properties of hadrons for decades such as, form factors,coupling constants and multipole moments.In the LCSR, the properties of the particles are characterized in terms of the light-cone distribution amplitudes (DAs) and the vacuum condensates [for details, see for instance <cit.>].The rest of the paper is organized as follows: In Sec. II,the light-cone QCD sum rule for the electromagnetic form factors of Z_b is applied and its magnetic dipole moment is derived. Section III, encompasses our numerical analysis and discussion. The explicit expressions of the photon DAs are moved to theAppendix A.§ FORMALISM To obtain the magnetic dipole moment of the Z_b state by using the LCSR approach, we begin with the subsequent correlation function,Π _μνα(p,q)=i^2∫ d^4x ∫ d^4y e^ip· x+iq · y ⟨ 0|𝒯{J_μ^Z_b(x) J_α(y) J_ν^Z_b†(0)}|0⟩.Here, J_μ(ν) is the interpolating current of the Z_b state and the electromagnetic current J_α is given as,J_α =∑_q= u,d,b e_q q̅γ_α q,where e_q is the electric charge of the corresponding quark.From technical point of view, it is more convenient to rewrite the correlation function by using the external background electromagnetic (BGEM) field, Π _μν(p,q)=i∫ d^4x e^ip· x⟨ 0|𝒯{J_μ^Z_b(x) J_ν^Z_b†(0)}|0⟩_F,whereF is the external BGEM field andF_αβ= i (ε_α q_β-ε_β q_α) with q_α and ε_βbeing the four-momentum and polarization of the BGEM field. Since the external BGEM field can be made arbitrarily small, the correlation function in Eq. (<ref>) can be acquired by expandingin powers of the BGEM field,Π _μν(p,q) = Π _μν^(0)(p,q) + Π _μν^(1)(p,q)+.... ,and keeping only terms Π _μν^(1)(p,q), which correspondsto the single photon emission <cit.> (the technical details about the external BGEM field method can be found in <cit.>). The main advantage of using the BGEM field approach relies on the fact that it separates the soft and hard photon emissions in an explicitly gaugeinvariant way <cit.>.The Π _μν^(0)(p,q) is the correlation function in theabsence of the BGEM field, and gives rise to the mass sum rules of the hadrons, which is not relevant for our case.After these general remarks, we can now proceed deriving the LCSR for the magnetic dipole moment of the Z_b state. The correlation function given in Eq. (<ref>) can be obtainedin terms of hadronic parameters, known as hadronic representation.Additionally it can be calculated in terms of the quark and gluon parametersin the deep Euclidean region, known as QCD representation. We can insert a complete set of intermediate hadronic states with the same quantum numbers as the interpolating current of the Z_b into the correlation function to obtain the hadronic representation.Then, by isolating the ground state contributions,we obtain the following expression: Π_μν^Had (p,q) = ⟨ 0 | J_μ^Z_b| Z_b(p) ⟩/p^2 - m_Z_b^2⟨ Z_b(p) | Z_b(p+q) ⟩_F ⟨ Z_b(p+q) |J^†_ν^Z_b| 0 ⟩/(p+q)^2 - m_Z_b^2 + ⋯,where dots denote the contributions coming from the higher states and continuum.The matrix element appearing in Eq. (<ref>) can be written in terms of three invariant form factors as follows <cit.>: ⟨ Z_b(p,ε^θ) |Z_b (p+q,ε^δ)⟩_F = - ε^τ (ε^θ)^α (ε^δ)^β[ G_1(Q^2) (2p+q)_τ g_αβ+ G_2(Q^2) ( g_τβ q_α -g_τα q_β) - 1/2 m_Z_b^2 G_3(Q^2) (2p+q)_τ q_α q_β] ,where ε^τ is the polarization vector of the BGEM field; and ε^θand ε^δ are thepolarization vectors of the initial and final Z_b states. The remaining matrix element, that of the interpolating currentbetween the vacuum and particle state, ⟨ 0 | J_μ^Z_b| Z_b ⟩, is parametrized as ⟨ 0 | J_μ^Z_b| Z_b ⟩ = λ_Z_bε_μ^θ ,where λ_Z_b is residue of the Z_b state.The form factors G_1(Q^2), G_2(Q^2)and G_3(Q^2) can be defined in terms of the charge F_C(Q^2), magnetic F_M(Q^2) and quadrupole F_ D(Q^2) form factors as follows F_C(Q^2) = G_1(Q^2) + 2/3 (Q^2/4 m_Z_b^2) F_ D(Q^2) ,F_M(Q^2) = G_2(Q^2) ,F_ D(Q^2) = G_1(Q^2)-G_2(Q^2)+(1+Q^2/4 m_Z_b^2) G_3(Q^2) , At Q^2 = 0, the form factors F_C(Q^2=0), F_M(Q^2=0), and F_ D(Q^2=0) are related to the electric charge, magnetic moment μ and the quadrupole momentD ase F_C(0) = e, e F_M(0) = 2 m_Z_bμ , e F_ D(0) = m_Z_b^2D .Inserting the matrix elements in Eqs. (<ref>) and (<ref>) into the correlation function in Eq. (<ref>) and imposing the condition q·ε = 0, we obtain the correlation function in terms of the hadronic parameters as Π_μν^Had(p,q)= λ_Z_b^2ε^τ/ [m_Z_b^2 - (p+q)^2][m_Z_b^2 - p^2][2 p_τ F_C(0) (g_μν -p_μ q_ν-p_ν q_μ/ m_Z_b^2 ) + F_M (0) (q_μ g_ντ - q_ν g_μτ + 1/m_Z_b^2 p_τ (p_μ q_ν - p_ν q_μ ) ) - (F_C(0) + F_ D(0)) p_τ/m_Z_b^2 q_μ q_ν] . To obtain the expression of the correlation function in terms of the quark and gluon parameters, the explicit form for the interpolating current of the Z_bstate needs to be chosen. In this study, we consider the Zb statewith the quantum numbersJ^PC=1^+-. Then in the diquark-antidiquark model the interpolating current J_μ^Z_b isdefined by the following expressionin terms of quark fields:J_μ^Z_b(Di)(x)= iϵϵ̃/√(2){ [ u_a^T(x)Cγ _5b_b(x)] [ d _d(x)γ _μCb_e^T(x)]-[ u_a^T(x)Cγ _μb_b(x)] [d_d(x)γ _5Cb_e^T(x)] },where C isthe charge conjugation matrix, ϵ =ϵ _abc, ϵ̃=ϵ _dec; and a,b,... are color indices.One can also construct the interpolating currentby considering the Z_b as a molecular form of B B̅^* and B^* B̅ state, J_μ^Zb(Mol)(x) = 1/√(2){[ d̅_a(x) iγ_5 b_a(x)][b̅_b(x) γ_μ u_b(x)]+[d̅_a(x) γ_μ b_a(x)][b̅_b(x) iγ_5 u_b(x)]}.After contracting pairs of the light and heavy quark operators, the correlation function becomes: Π _μν^QCD(p,q) = -iϵϵ̃ϵ^'ϵ̃^'/2∫ d^4xe^ipx⟨ 0 | {Tr[γ _5S_u^aa^'(x)γ _5S_c^bb^'(x)] Tr[γ _μS_c^e^'e(-x)γ _νS_d^d^'d(-x)]-Tr[ γ _μS_c^e^'e(-x)γ _5S_d^d^'d(-x)]Tr[ γ_νS_u^aa^'(x)γ _5S_c^bb^'(x)]-Tr[γ _5S_u^a^'a(x)γ _μS_c^b^'b(x)] Tr[ γ _5S_c^e^'e(-x)γ _νS_d^d^'d(-x)]+Tr[γ _νS_u^aa^'(x)γ _μS_c^bb^'(x)]Tr[γ _5S_c^e^'e(-x)γ_5S_d^d^'d(-x)]}| 0 ⟩_F,in the diquark-antidiquark picture, and Π _μν^QCD(p,q) = -i/2∫ d^4xe^ipx⟨ 0 | {Tr[γ _5S_b^aa^'(x)γ _5S_d^a^'a(-x)] Tr[γ _μS_u^bb^'(x)γ _νS_b^b^'b(-x)] +Tr[ γ _5 S_b^aa^'(x)γ _νS_d^a^'a(-x)]Tr[ γ_μS_u^bb^'(x)γ _5S_b^b^'b(-x)] +Tr[γ _μS_b^aa^'(x)γ _5 S_d^a^'a(-x)] Tr[ γ _5S_u^bb^'(x)γ _νS_b^b^'b(-x)] +Tr[γ _μS_b^aa^'(x)γ _νS_d^a^'a(-x)]Tr[γ _5S_u^bb^'(x)γ_5S_b^b^'b(-x)]}| 0 ⟩_F,in the molecular picture, whereS_b(q)(x)=CS_b(q)^T(x)C,with S_q(x) and S_b(x) being the light and heavy quark propagators, respectively.To calculate the correlation functions in QCD representations,the light and heavy quark propagators are required. Theirexplicit expressionsin the x-space are given as S_q(x)=i /2π ^2x^4- ⟨q̅q ⟩/12( 1+m_0^2 x^2/16)-i g_s /32 π^2 x^2 G^μν(x) [/xσ_μν+σ_μν/x], andS_b(x)=m_b^2/4 π^2[ K_1(m_b√(-x^2)) /√(-x^2) +i K_2( m_b√(-x^2))/(√(-x^2))^2] -g_sm_b/16π ^2∫_0^1 dvG^μν(vx)[ (σ _μν +σ _μν)K_1( m_b√(-x^2)) /√(-x^2) +2σ_μνK_0( m_b√(-x^2))],where K_i are the second kind Bessel functions, v is line variable and G^μν is the gluon field strength tensor.The correlation function includesdifferent types of contributions. In first case, one of the free quarkpropagators in Eqs. (<ref>-<ref>) is replaced byS^free→∫ d^4yS^free (x-y) /A(y)S^free (y) ,where S^free is the first term of the light or heavy quark propagators and the remaining three propagators are replaced with the full quark propagators.The LCSR calculations are most conveniently done in the fixed-point gauge. For electromagnetic field, it is defined by x_μ A^μ =0.In this gauge, the electromagnetic potential is given byA_α = -1/2 F_αβy^β = -1/2 (ε_α q_β-ε_β q_α) y^β.The Eq. (<ref>) is plugged into Eq. (<ref>), as a result of which we obtainS^free→ -1/2 (ε_α q_β-ε_β q_α) ∫d^4yy^β S^free (x-y) γ_α S^free (y) , After some calculations for S_q^free and S_b^free we getS_q^free=e_q/32 π^2 x^2(ε_α q_β-ε_β q_α)(σ_αβ+σ_αβ),S_b^free=-ie_b m_b/32 π^2(ε_α q_β-ε_β q_α) [2σ_αβK_0( m_b√(-x^2))+K_1( m_b√(-x^2)) /√(-x^2)(σ_αβ+σ_αβ)].In second case one of the light quarkpropagators in Eqs. (<ref>-<ref>) are replaced byS_αβ^ab→ -1/4 (q̅^a Γ_i q^b)(Γ_i)_αβ,and the remaining propagators are full quark propagators includingthe perturbative as well as the nonperturbative contributions.Here as an example, we give a short detail of the calculations of the QCD representations.In second case for simplicity, we only consider the first term in Eq. (<ref>), Π _μν^QCD(p,q)=-iϵϵ̃ϵ^'ϵ̃^'/2∫ d^4xe^ipx⟨ 0 | Tr[γ _5S_u^aa^'(x)γ _5S_b^bb^'(x)] Tr[γ _μS_b^e^'e(-x)γ _νS_d^d^'d(-x)] |0⟩_F+... By replacing one of light propagators with the expressions in Eq. (<ref>)and making use ofq̅^a(x)Γ_i q^a'(0)→1/3δ^aa'q̅(x)Γ_i q(0),the Eq. (<ref>) takes the formΠ _μν^QCD(p,q)=-iϵϵ̃ϵ^'ϵ̃^'/2∫ d^4xe^ipx{Tr[γ _5Γ_i γ _5S_b^bb^'(x)] Tr[γ _μS_b^e^'e(-x)γ _νS_d^d^'d(-x)] 1/12δ^aa' +Tr[γ _5S_u^aa^'(x)γ _5S_b^bb^'(x)] Tr[γ _μS_b^e^'e(-x)γ _νΓ_i] 1/12δ^dd'}⟨γ(q) |q̅(x)Γ_i q(0)|0⟩+...,where Γ_i = I, γ_5, γ_μ, iγ_5 γ_μ, σ_μν/2. Similarly, when a light propagator interacts with the photon,a gluon may be released from one of the remaining three propagators.The expression obtained in this case is as follows:Π _μν^QCD(p,q)=-iϵϵ̃ϵ^'ϵ̃^'/2∫ d^4xe^ipx{Tr[γ _5Γ_i γ _5S_b^bb^'(x)] Tr[γ _μS_b^e^'e(-x)γ _νS_d^d^'d(-x)] [(δ^abδ^a'b'-1/3δ^aa'δ^bb')+(δ^aeδ^a'e'-1/3δ^aa'δ^ee')+(δ^adδ^a'd'-1/3δ^aa'δ^dd')]+Tr[γ _5S_u^aa^'(x)γ _5S_b^bb^'(x)] Tr[γ _μS_b^e^'e(-x)γ _νΓ_i] [(δ^dbδ^d'b'-1/3δ^dd'δ^bb')+(δ^deδ^d'e'-1/3δ^dd'δ^ee')+(δ^adδ^a'd'-1/3δ^aa'δ^dd')] }1/32⟨γ(q) |q̅(x)Γ_i G_μν(vx) q(0)|0⟩+...,where we insertedq̅^a(x)Γ_i G_μν^bb'(vx) q^a'(0)→1/8(δ^abδ^a'b'-1/3δ^aa'δ^bb')q̅(x)Γ_i G_μν(vx) q(0).As is seen, there appear matrix elements such as ⟨γ(q)q̅(x) Γ_i q(0)0⟩ and ⟨γ(q)q̅(x) Γ_i G_μν(vx)q(0)0⟩, representing the nonperturbative contributions.These matrix elements can be expressed in termsof photon DAs and wave functions with definite twists, whose expressions are given in Appendix A.The QCD representation of the correlation function is obtained by usingEqs. (<ref>-<ref>).Then, the Fourier transformation is applied to transfer expressions in x-space to the momentum space. The sum rule for themagnetic dipole moment are obtained bymatching the expressionsof the correlation function in terms of QCD parameters and its expression in terms of the hadronic parameters,using their spectral representation.To eliminate the contributions of the excited and continuum statesin the spectral representation of the correlation function, a double Borel transformation with respect to the variables p^2 and (p + q)^2 is applied.After the transformation, these contributions are exponentially suppressed.Eventually, we choose the structure (ε.p)( p_μ q_ν- q_μ p_ν) for the magnetic dipole moment and obtain μ^Di =e^m_Z_b^2/M^2/λ_Z_b^2m_Z_b^2[Π_1+Π_2],μ^Mol =e^m_Z_b^2/M^2/λ_Z_b^2 m_Z_b^2[Π_3+Π_4].The explicit forms of the functions that appear in the above sum rules are given as follows:Π_1 = 3m_b^4/256 π^6(e_u-e_d){32N[3,3,0]-2M^2 N[3,3,1] -16m_b N[3,4,1]+m_b M^2 N[3,4,2]} -m_b^2 ⟨ g_s^2 G^2⟩/9216 π^6(e_u-e_d)(-M^2 N[1,1,0]+2m_b N[1,2,0]) -m_b^2 ⟨ g_s^2 G^2⟩/147456 π^6(2m_b M^2 N[1,2,1]+π^2 ⟨q̅q ⟩(16 N[1,2,1]+5 N[1,2,2])) +m_b^4 ⟨ g_s^2 G^2⟩/294912 π^6(e_u-e_d)(16N[1,3,1]-M^2N[1,3,2]) -m_b^2 ⟨ g_s^2 G^2⟩/294912 π^6(e_u-e_d)(128 N[2,2,0]-8(2m_b^2+M^2)N[2,2,1]+m_b^2 M^2N[2,2,2]) -m_b^3/98304 π^6(e_u-e_d)( 16(⟨ g_s^2G^2 ⟩ -192 π^2 m_b ⟨q̅q ⟩)N[2,3,1] +M^2(13⟨ g_s^2G^2 ⟩ -960 π^2 m_b ⟨q̅q ⟩)N[2,3,2]-m_b^3 m_0^2 ⟨q̅q ⟩/384 M^8 π^4(e_u+e_d)(64m_b^6FlP[-3,4,0] -48m_b^4FlP[-2,4,0]+12m_b^2FlP[-1,4,0]-FlP[0,4,0])-m_b m_0^2 ⟨ g_s^2G^2 ⟩⟨q̅q ⟩/36864 M^8 π^4 (e_u-e_d)(16m_b^4FlP[-1,2,0]-8m_b^2FlP[0,2,0]+FlP[1,2,0]),Π_2=-m_b ⟨ g_s^2G^2 ⟩⟨q̅q ⟩^2 /2592 M^10π^2(e_u-e_d)(m_0^2-M^2)I_3[h_γ](4m_b^2FlNP[0,1,0]-FlNP[-1,1,0]) +m_b m_0^2 ⟨ g_s^2G^2 ⟩⟨q̅q ⟩^2/10368 M^10π^2(e_u-e_d)I_3[h_γ](4m_b^2FlNP[2,1,1]-FlNP[3,1,1]) -f_3γ m_0^2 ⟨ g_s^2G^2 ⟩⟨q̅q ⟩/110592 M^10π^2[-(4e_u-3e_d)ψ^a(u_0)+2e_d I_3[ψ^ν]](16m_b^4FlNP[1,2,1]-8m_b^2FlNP[2,2,1] +FlNP[3,2,1]) -m_b m_0^2 ⟨ g_s^2G^2 ⟩⟨q̅q ⟩/995328 M^12π^2(8e_u-5e_d)(A(u_0)+8I_3[h_γ])(16m_b^4FlNP[2,3,2]-8m_b^2FlNP[3,3,2]+FlNP[4,3,2]) +m_b ⟨q̅q ⟩^2 /497664 M^12π^2(8e_u-5e_d)[-⟨ g_s^2G^2 ⟩(-(5m_0^2-2M^2)A(u_0)-2 m_0^2 χ M^2φ_γ(u_0)) -8{-5m_0^2 ⟨ g_s^2G^2 ⟩+2(M^2 ⟨ g_s^2G^2 ⟩ +432 m_b^2 m_0^2 M^2 ) I_3[h_γ]}] (16m_b^4 FlNP[0,3,1]-8m_b^2FlNP[1,3,1]+FlNP[2,3,1]) +m_b/165888 M^10π^4[(e_u-e_d)f_3γ(7M^2 ⟨ g_s^2G^2 ⟩ -576π^2 m_b ⟨q̅q ⟩ (m_0^2-M^2))ψ^a(u_0)+72π^2 ⟨q̅q ⟩^2(m_0^2-M^2) (-2e_u I_1[S̃]+e_d(3I_2[𝒯_1]-3I_2[𝒯_2] -5I_2[S̃]))](-64m_b^6FlNP[3,4,0] +48m_b^4FlNP[2,4,0] -12m_b^2FlNP[1,4,0]+FlNP[0,4,0])-m_b m_0^2⟨q̅q ⟩/9216 M^10π^4[-8(e_u-e_d)m_b f_3γψ^a(u_0) - ⟨q̅q ⟩(-2e_u I_1[S̃]+e_d(3I_2[𝒯_1] -3I_2[𝒯_2]-5I_2[S̃]))] (64m_b^6FlNP[-1,4,1]-48m_b^4FlNP[0,4,1] +12m_b^2FlNP[1,4,1]-FlNP[2,4,1])+⟨q̅q ⟩/1990656 M^12π^4[ ⟨ g_s^2G^2 ⟩{2e_u(-3M^4(16m_b^4FlNP[1,3,0]-8m_b^2FlNP[0,3,0]+FlNP[-1,3,0]) -32π^2 m_b⟨q̅q ⟩ (5m_0^2-4M^2)(16m_b^4FlNP[2,3,0]-8m_b^2FlNP[1,3,0]+FlNP[0,3,0])) +e_d(40π^2 m_b ⟨q̅q ⟩(5m_0^2-4M^2)(16m_b^4FlNP[2,3,0]-8m_b^2FlNP[1,3,0]+FlNP[0,3,0]) +3M^4(-48m_b^6FlNP[2,3,0]+56m_b^4FlNP[1,3,0]-19m_b^2FlNP[0,3,0]+2FlNP[-1,3,0]))}A(u_0) +8m_b{-4(8e_u-5e_d)π^2 M^2 χ⟨ g_s^2G^2 ⟩⟨q̅q ⟩(m_0^2-M^2)φ_γ(u_0) -3(40e_u-43e_d)m_b M^4 ⟨ g_s^2G^2 ⟩ +8(8e_u-5e_d)π^2⟨ g_s^2G^2 ⟩⟨q̅q ⟩(5m_0^2-M^2) -432(e_u-e_d)π^2 m_b^2 M^2 ⟨q̅q⟩ (m_0^2-4M^2)}I_3[h_γ]] (16m_b^4FlNP[2,3,0]-8m_b^2FlNP[1,3,0]+FlNP[0,3,0]),Π_3 = 9 m_b^4/1024 π^6(e_u-e_d){32N[3,3,0]-2M^2N[3,3,1]-16m_bN[3,4,1]+m_b M^2N[3,4,2] } -m_b^3/32768 π^6(e_u-e_d)(⟨g_s^2 G^2 ⟩ +48 π^2 m_b ⟨q̅q⟩ )( 16N[2,3,1]+5M^2N[2,3,2]) -m_b^3 m_0^2 ⟨q̅q⟩/512 M^8 π^2(e_u-e_d)(64m_b^64FlP[-3,4,0]-48m_b^4FlP[-2,4,0]+12m_b^2FlP[-1,4,0]+FlP[0,4,0]), and Π_4 = m_b^2 ⟨ g_s^2 G^2⟩⟨q̅q⟩/294912 π^4[e_u(-3I_1[𝒮]-2I_1[S̃])+e_d(3I_2[𝒮]+2I_2[S̃])](M^2N[1,2,2]-8N[1,2,1])+3m_b^2 ⟨q̅q⟩/64 π^4(e_u-e_d)I_3[h_γ](M^2N[2,3,2]-8N[2,3,1])+3m_b^4 f_3γ/128 π^4(e_u+e_d)ψ^a(u_0)(M^2N[3,3,2]-8N[3,3,1]) -m_b^3 m_0^2 ⟨q̅q⟩^2 /96 M^10π^2(e_u-e_d)I_3[h_γ](64 m_b^6FlNP[0,3,1]-48m_b^4FlNP[1,3,1]+FlNP[2,3,1]) -m_b^3 m_0^2 f_3γ⟨q̅q⟩^2 /1536 M^10π^2(e_u-e_d)ψ^a(u_0)(64m_b^6FlNP[-1,4,1]-48FlNP[0,4,1]+12m_b^2FlNP[1,4,1] -FlNP[2,4,1]) +m_b f_3γ/18432 M^10π^4(e_u+e_d)[(-M^2 ⟨ g_s^2 G^2⟩-48π^2m_b⟨q̅q⟩(m_0^2-M^2))ψ^a(u_0)](-64m_b^6FlNP[3,4,0] +48m_b^4FlNP[2,4,0]-12m_b^2FlNP[1,4,0]+FlNP[0,4,0])-m_b ⟨q̅q ⟩/1152 M^10π^4(e_u-e_d)[(-M^2 ⟨ g_s^2 G^2⟩-48π^2m_b⟨q̅q⟩(m_0^2-M^2))I_3[h_γ]](-16m_b^4FlNP[2,3,0] -8m_b^2FlNP[1,3,0]+FlNP[0,3,0]).where, m_b is the mass of the b quark, e_q is the corresponding electric charge, χ is the magnetic susceptibility of the quark condensate, m_0^2 = ⟨q̅ g σ_αβ G^αβ q ⟩ /⟨q̅q ⟩,⟨q̅q ⟩ and ⟨ g_s^2 G^2⟩ are quark and gluon condensates, respectively. The functions N[n,m,k], FlP[n,m,k], FlNP[n,m,k], I_1[𝒜], I_2[𝒜] and I_3[𝒜] are defined as:N[n,m,k] =∫_0^∞ dt∫_0^∞ dt' e^-m_b/2(t+t')/t^n (m_b/t+m_b/t')^k t'^m , FlP[n,m,k] = ∫_4m_b^2^s_0 ds ∫_4m_b^2^s dl e^-l^2/ϕ l^n (l-s)^m/(4m_b^2-l)^2 ϕ^k, FlNP[n,m,k] =∫_4m_b^2^s_0 ds ∫_4m_b^2^s dl e^-l^2/β l^n (l-s)^m/(l-2m_b^2) β^k, I_1[𝒜] =∫ D_α_i∫_0^1 dv 𝒜(α_q̅,α_q,α_g)δ(α_ q +v̅α_g-u_0),I_2[𝒜] =∫ D_α_i∫_0^1 dv 𝒜(α_q̅,α_q,α_g)δ(α_q̅+ v α_g-u_0), I_3[𝒜] =∫_0^1 du A(u), whereβ=4 l M^2-16 m_b^2M^2,ϕ=8 l M^2-32 m_b^2M^2. The functions Π_1 and Π_3 indicate the case that one of thequark propagators enters the perturbative interaction with the photon andthe remaining three propagators are taken as full propagators. The functions Π_2 and Π_4 show the contributions that one of the light quark propagatorsenters the nonperturbative interaction with the photon andthe remaining three propagators are taken as full propagators.The reader canfind some details about the calculationssuch as Fourier and Borel transformations as well as continuumsubtraction in Appendix C of Ref. <cit.>.As we already mentioned, the calculations have been done in the fixed-point gauge, x_μ A^μ =0, for simplicity.In order to show whetherour results are gauge invariant or not we examine the Lorentz gauge, ∂_μ A^μ =0. In this gauge, the electromagnetic vector potential is written as A_μ(x) = ε_μ e^-iq.x,with ε_μ q^μ =0. In this gauge, the corresponding gauge invariant electromagnetic field strength tensor is writtenasF_μν=i(ε_μ q_ν-q_με_ν)e^-iq.x.We repeat all the calculations in this gauge and find the same resultsfor the magnetic dipole moment of the state under consideration.Therefore the results obtained in the present study are gauge invariant.§ NUMERICAL ANALYSIS AND CONCLUSIONIn this section, we numerically analyze the resultsof calculations for magnetic dipole moment of the Z_b state.We use m_Z_b= 10607.2 ±2 MeV, m_b(m_b) = (4.18_-0.03^+0.04) GeV <cit.>,f_3γ=-0.0039 GeV^2 <cit.>, ⟨q̅q⟩(1 GeV) =(-0.24±0.01)^3 GeV^3 <cit.>, m_0^2 = 0.8 ± 0.1 GeV^2, ⟨ g_s^2G^2⟩ = 0.88 GeV^4 <cit.> andχ(1 GeV)=-2.85 ± 0.5 GeV^-2 <cit.>.To evaluate a numerical prediction for the magnetic moment,we need also specify the values of the residue of the Z_b state. The residue is obtained from the mass sum rule asλ_Z_b= m_Z_b f_Z_b withf_Z_b=(2.79^+0.55_-0.65)×10^-2 GeV^4 <cit.>for diquark-antidiquark picture and λ_Z_b=0.27± 0.07 GeV^5 <cit.> for molecular picture.The parameters used in the photon DAs are givenin Appendix A, as well.The estimations for themagnetic dipole moment of the Z_b state depend on twoauxiliary parameters; the continuum threshold s_0 and Borel mass parameter M^2.The continuum threshold is not completely an arbitrary parameter, and there are some physical restrictions for it. The s_0 signals the scale at which, the excited states and continuum start to contribute to the correlation function. The working interval for this parameter is chosen such that the maximum pole contribution is acquired and the results relatively weakly depend on its choices. Our numerical calculations leadto the interval [119-128] GeV^2 for this parameter.The Borel parameter can vary in the interval that the results weakly depend on it according to the standard prescriptions.The upper bound of it is found demanding the maximum pole contributions and its lower bound is found the convergence of the operator product expansion and exceeding of the perturbativepart over nonperturbative contributions. Under these constraints, the working region of the Borel parameteris determined as 15 GeV^2 ≤ M ^ 2 ≤ 17 GeV^2. In Fig. 1, we plot the dependency of the magnetic dipole moment of the Z_b state on M^2 at different fixed values of the continuum threshold.From the figure we observe that the results considerably depend on the variations of the Borel parameter.Themagnetic dipole moment isstable under variation of s_0 in its working region. In Fig. 2, we show the contributions of Π_1, Π_2, Π_3 and Π_4 functionsto the results obtained at average value of s_0 with respect to the Borel mass parameter. It is clear that Π_1 is dominant in the resultsobtained when using diquark-antidiquark current butΠ_3 is dominant while using the molecular current.The contribution of the Π_2 and Π_4 functions seems to be almost zero.When the results are analyzed in detail, almost (95-97)% of the total contribution comes from the perturbative part and the remaining (3-5)% belongs to the nonperturbative contributions.Our predictions on the numerical value of the magnetic dipole moment in both pictures are presented in Table I. The errors in the results come from the variations in the calculations of the working regions ofM^2 and from the uncertainties in the values of the input parameters as well as the photon DAs.We shall remark that the main source of uncertainties is the variations with respect to variations of M^2. In conclusion, we have computed the magnetic dipole momentof the Z_b(10610) by modeling it as the diquark-antidiquark and molecule states. In our calculationswe have employed the light-cone QCD sum rule in electromagnetic background field. Although the central values of themagnetic dipole moment obtained via two pictures differ slightly from each other but they are consistent within the errors.In Ref. <cit.>, both the spectroscopic parametersand someof the strong decays of the Z_b state have been studied usingdiquark-antidiquark interpolating current. Although the obtained mass in <cit.> is in agreement withthe experimental data, the result obtained for the width of Z_b in the diquark-antidiquark picturein <cit.> differ considerably from the experimental data. They suggested, as a result, that the Z_b state may not have a pure diquark-antidiquark structure.When we combine the obtained results in the present study with those of the predictions on the mass obtainedvia both pictures in the literature and those result obtainedfor the width of Z_b in Ref. <cit.>we conclude that both pictures can be considered for the internal structure of Z_b.May be a mixed current will be a better choice for interpolating this particle.More theoretical and experimental studies are still needed to be performed in this respect.Finally, themagnetic dipole moment encodes important information aboutthe inner structure of particlesand their geometric shape.The results obtained for themagnetic dipole moment of Z_b state in both the diquark-antidiquark and molecule pictures, within a factor 2, are of the same order of magnitude as the proton's magnetic moment and not such small that it appears hopeless to try to measure the valueof the magnetic dipole moment of this state. By the recent progresses in the experimental side, we hope that we can measure the multipole moments of the newly founded exotic states, especially the Z_b particle in future.Comparison of any experimental data on themagnetic dipole moment ofZ_b will be useful to gainexact knowledge on its quark organizations and will help us in the course of undestanding the structures of the newly observed exotic states and theirquantum chromodynamics. § ACKNOWLEDGEMENTThis work has been supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) under the Grant No. 115F183.§ APPENDIX A: PHOTON DAS AND WAVE FUNCTIONS In this appendix, we present the definitions of the matrix elements of the forms ⟨γ(q)q̅(x) Γ_i q(0)0⟩ and ⟨γ(q)q̅(x) Γ_i G_μνq(0)0⟩ in terms of the photon DAs and wave functions <cit.>, ⟨γ(q) |q̅(x) γ_μ q(0) | 0 ⟩ = e_q f_3 γ(ε_μ - q_με x/q x) ∫_0^1 du e^i u̅ q xψ^v(u)⟨γ(q) |q̅(x) γ_μγ_5 q(0) | 0 ⟩= - 1/4 e_q f_3 γϵ_μναβε^ν q^α x^β∫_0^1 du e^i u̅ q xψ^a(u)⟨γ(q) |q̅(x) σ_μν q(0) |0 ⟩= -i e_q ⟨q̅ q ⟩ (ε_μ q_ν - ε_ν q_μ) ∫_0^1 du e^i u̅ qx(χφ_γ(u) + x^2/16𝔸(u) )-i/2(qx)e_q q̅q [x_ν(ε_μ - q_με x/qx) - x_μ(ε_ν - q_νε x/q x) ] ∫_0^1 du e^i u̅ q x h_γ(u)⟨γ(q) | q̅(x) g_s G_μν (v x) q(0) | 0 ⟩ = -i e_q ⟨q̅ q ⟩(ε_μ q_ν - ε_ν q_μ) ∫ Dα_i e^i (α_q̅ + v α_g) q x S(α_i)⟨γ(q) | q̅(x) g_s G̃_μν(v x) i γ_5q(0) | 0 ⟩ = -i e_q ⟨q̅ q ⟩(ε_μ q_ν - ε_ν q_μ) ∫ Dα_i e^i (α_q̅ + v α_g) q xS̃(α_i)⟨γ(q) |q̅(x) g_s G̃_μν(v x) γ_αγ_5 q(0) | 0 ⟩ = e_q f_3 γ q_α (ε_μ q_ν - ε_ν q_μ) ∫ Dα_i e^i (α_q̅ + v α_g) q x A(α_i)⟨γ(q) |q̅(x) g_s G_μν(v x) i γ_α q(0) | 0 ⟩ = e_q f_3 γ q_α (ε_μ q_ν - ε_ν q_μ) ∫ Dα_i e^i (α_q̅ + v α_g) q x V(α_i)⟨γ(q) |q̅(x) σ_αβ g_s G_μν(v x) q(0) | 0 ⟩= e_q ⟨q̅ q ⟩{[(ε_μ - q_με x/q x)(g_αν - 1/qx (q_α x_ν + q_ν x_α)) . . q_β-(ε_μ - q_με x/q x)(g_βν - 1/qx (q_β x_ν + q_ν x_β)) q_α -(ε_ν - q_νε x/q x)(g_αμ - 1/qx (q_α x_μ + q_μ x_α)) q_β +. (ε_ν - q_νε x/q.x)( g_βμ - 1/qx (q_β x_μ + q_μ x_β)) q_α]∫ Dα_i e^i (α_q̅ + v α_g) qx T_1(α_i)+ [(ε_α - q_αε x/qx) (g_μβ - 1/qx(q_μ x_β + q_β x_μ)) . q_ν -(ε_α - q_αε x/qx) (g_νβ - 1/qx(q_ν x_β + q_β x_ν))q_μ-(ε_β - q_βε x/qx) (g_μα - 1/qx(q_μ x_α + q_α x_μ)) q_ν +. (ε_β - q_βε x/qx) (g_να - 1/qx(q_ν x_α + q_α x_ν) ) q_μ] ∫ Dα_i e^i (α_q̅ + v α_g) qx T_2(α_i)+1/qx (q_μ x_ν - q_ν x_μ) (ε_α q_β - ε_β q_α) ∫ Dα_i e^i (α_q̅ + v α_g) qx T_3(α_i)+ . 1/qx (q_α x_β - q_β x_α) (ε_μ q_ν - ε_ν q_μ) ∫ Dα_i e^i (α_q̅ + v α_g) qx T_4(α_i) } ,where φ_γ(u) is the leading twist-2, ψ^v(u), ψ^a(u), A(α_i) and V(α_i), are the twist-3, and h_γ(u), 𝔸(u), S(α_i), S̃(α_i), T_1(α_i), T_2(α_i), T_3(α_i)and T_4(α_i) are the twist-4 photon DAs. The measure Dα_i is defined as∫ Dα_i = ∫_0^1 d α_q̅∫_0^1 d α_q ∫_0^1 d α_g δ(1-α_q̅-α_q-α_g) . The expressions of the DAs entering into the above matrix elements are defined as: φ_γ(u)=6 u u̅( 1 + φ_2(μ) C_2^3/2(u - u̅) ),ψ^v(u)=3 (3 (2 u - 1)^2 -1 )+3/64(15 w^V_γ - 5 w^A_γ) (3 - 30 (2 u - 1)^2 + 35 (2 u -1)^4 ),ψ^a(u)= (1- (2 u -1)^2)(5 (2 u -1)^2 -1) 5/2(1 + 9/16 w^V_γ - 3/16 w^A_γ), h_γ(u)=- 10 (1 + 2 κ^+) C_2^1/2(u - u̅),𝔸(u)=40 u^2 u̅^2 (3 κ - κ^+ +1)+ 8 (ζ_2^+ - 3 ζ_2) [u u̅ (2 + 13 u u̅) . + . 2 u^3 (10 -15 u + 6 u^2) ln(u) + 2 u̅^3 (10 - 15 u̅ + 6 u̅^2) ln(u̅) ], A(α_i)=360 α_q α_q̅α_g^2 (1 + w^A_γ1/2 (7 α_g - 3)), V(α_i)=540 w^V_γ (α_q - α_q̅) α_q α_q̅α_g^2, T_1(α_i)=-120 (3 ζ_2 + ζ_2^+)(α_q̅ - α_q) α_q̅α_q α_g, T_2(α_i)=30 α_g^2 (α_q̅ - α_q) ((κ - κ^+) + (ζ_1 - ζ_1^+)(1 - 2α_g) + ζ_2 (3 - 4 α_g)), T_3(α_i)=- 120 (3 ζ_2 - ζ_2^+)(α_q̅ -α_q) α_q̅α_q α_g, T_4(α_i)=30 α_g^2 (α_q̅ - α_q) ((κ + κ^+) + (ζ_1 + ζ_1^+)(1 - 2α_g) + ζ_2 (3 - 4 α_g)),S(α_i)=30α_g^2{(κ + κ^+)(1-α_g)+(ζ_1 + ζ_1^+)(1 - α_g)(1 - 2α_g) +ζ_2[3 (α_q̅ - α_q)^2-α_g(1 - α_g)]}, S̃(α_i)= -30α_g^2{(κ -κ^+)(1-α_g)+(ζ_1 - ζ_1^+)(1 - α_g)(1 - 2α_g) +ζ_2 [3 (α_q̅ -α_q)^2-α_g(1 - α_g)]}. 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"authors": [
"U. Ozdem",
"K. Azizi"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170927195331",
"title": "Magnetic dipole moment of $Z_b(10610)$ in light-cone QCD"
} |
§ INTRODUCTION Gamma rays are one of the best probes to study our surrounding Universe. Due to their neutral nature, gamma rays are not deflected by the interstellar magnetic field which means that they can be used to pin-point the emitting astrophysical sources. Very high-energy gamma rays, comprised in an energy range between ∼ 100GeV and ∼ 100TeV are particularly interesting as they allow us to investigate some of the most extreme phenomena.At energies up to ∼ 100GeV, this radiation can be detected directly using instruments placed in artificial satellites, as for instance Fermi. Above this energies the gamma ray flux becomes too small and indirect methods prevail. These methods take advantage of the interaction of the gamma-ray with Earth's atmosphere which produces a cascade of particles designated usually as Extensive Air Showers (EAS).At ground there are two main techniques: Imaging Atmospheric Cherenkov Telescope Arrays (IACTAs) and EAS arrays. The former technique measures the Cherenkov light produce by relativistic shower particles while the later measures the secondaries particles of the EAS that reach the ground.Both techniques are placed at high altitude to minimize the atmosphere's attenuation. There are advantages and disadvantages to both approaches. IACTs have a better energy and geometry reconstruction resolution but can only survey small portions of the sky. Moreover, they have a limited duty cycle. This makes them perfect to study astrophysical sources but not to look for transient phenomena. Presently there is no wide Field Of View (FoV) experiment operating in the Southern hemisphere, nor able to cover the gap between satellite and ground array gamma-ray experiments.A low energy threshold and a large duty cycle wide FoV experiment would be fully complementary to planned project such as the Cherenkov Telescope Array (CTA), as itcould issue alert of transient phenomena. Moreover, such an experiment is able to perform long term observations of variable sources and search for emissions from extended regions, arising from phenomena such as the Fermi bubbles or dark matter annihilations from the centre of our galaxy.Hence, we propose a novel hybrid detector to be installed at ∼ 5200m a.s.l. which has an improved sensitivity at the 100 GeV energy region. This manuscript is organised as follows: in section 2 we describe the detector and the layout of the experiment. In section 3 we discussed the its performance and the achieved preliminary sensitivity. Finally, we end with a summary.§ DETECTOR DESCRIPTION Lowering the energy threshold requires that one would be able to trigger on the shower secondary photons which are more numerous by a factor of 5-7 than secondary charged particles. As such, we propose to build a dense array with an area of 20 000m^2 constituted by modular hybrid detectors (see figure <ref> (left)). Each station is composed by two low-cost Resistive Plate Chambers (RPC) on top of a Water Cherenkov Detector (WCD), as shown in figure <ref> (right). Each RPC has 16 charge collecting pads covering a total area of 1.5 × 1.5m^2 . The WCD has a rectangular structure with dimensions 3×1.5× 0.5m^3. The signals are read by two photomultipliers (PMTs) at both ends of the smallest vertical face of the WCD. This detector concept ensures the ability to trigger at low energies while maintaining a reasonable energy and geometry reconstruction accuracy, which is essential to distinguish the gamma-ray showers from the ones initiated by cosmic rays.The geometry reconstruction can be further improved by adding on the top of the RPCs a thin lead plate (5.6 mm). This allows to convert secondary photons, which have a stronger correlation with the shower axis while removing low energy electrons who have poorer correlation due to multiple scattering in the atmosphere.Hence, the RPCs contributes, with its high segmentation and time resolution, which is essential for the shower geometric reconstruction, while the WCD provides a calorimetric measurement of the shower secondary particles lowering effectively the experiment energy threshold. § DETECTOR PERFORMANCE The performance of this detector has been assessed using an end-to-end realistic Monte Carlo simulation. Extensive Air Showers (EAS) have been simulated using CORSIKA (COsmic Ray Simulations for KAscade) <cit.>, while the detector response has been treated with Geant4 <cit.>. A total of 5×10^6 gamma and proton showers have been simulated with energies between 10GeV and 300TeV. The simulations were generated uniformly in logarithm of the energy and afterwards weighted accordingly to the corresponding particle fluxes. The zenith angle for gammas was fixed to 10^∘, while for protons the range was between 5 and 15 degrees. An altitude of 5200 a.s.l. was chosen to investigate the experiment performance. §.§ Trigger and effective area As stated before, the trigger is ensured by the WCD. As such a station is considered to be triggered if each PMT collects at least 5 photoelectrons. To have an event one should have at least 3 active WCD stations. The effective area has been computed using simulations and is shown in figure <ref>. From this figure it is possible to see that, even after quality cuts, one still has for gamma primaries with an energy of 100 GeV an effective area of about 10^4m^2. The applied quality cuts shall be described in the next sections.§.§ Energy reconstructionThe energy estimation has been obtained using the signal measured by the WCDs. A calibration was built relating the total signal, S_tot. recorded in all WCD stations for each event with the true energy of the primary gamma, E_0. From this curve it is possible to evaluate the reconstructed energy, E_rec, for each shower and assess the energy resolution for this detector. In figure <ref> (right) it can be seen that the energy resolution improves, as expected, with the increase of the shower energy, while at lower energy it degrades considerably. The latter is due to shower-to-shower fluctuations. A comparison with HAWC results, presented in <cit.>, is also shown in this figure. It can be seen that the LATTES energy resolution is better above 500GeV and is comparable below. The reason why LATTES performs so well at high energies is related to its higher segmentation and compactness.§.§ Geometry reconstruction The reconstruction of the shower core was done fitting a gamma average 2D LDF to the WCD station signal with a procedure similar to the one discussed in <cit.>. To require a clear maximum inside of the array, a cut on the topology of the event has been done. In figure <ref> (left) it can be seen that the shower core can be reconstructed with an accuracy better than 10m for gamma induced showers with energies above 300 GeV.The shower geometry reconstruction is being done taking advantage of the RPC segmentation and fast timing. It was considered a time resolution of 1ns. The primary direction is obtained using a shower front plane model that has as ingredients the position and time of the recorded hits in the RPC. In order to improve the quality of the reconstruction it is required that the event has at least 10 hits on the RPCs pads. Only RPCs on top of active WCDs are considered to apply this cut. Moreover, late arrival particles are also discarded. The reconstructed angle was compared to the simulated one, and we calculate the 68% containment angle, σ_θ,68. In figure <ref> (right) it is shown σ_θ,68 as a function of the shower reconstructed energy for two conditions: when the reconstructed core is contained in the array and when the reconstructed core is required to be at a distance smaller than 20m from the array center. From this figure it can be seen thatat energies around 100 GeV, a reasonable resolution, better that 1.5^∘, can be achieved. Moreover, it is noticeable from this plot thatat higher energies the events reconstructed near to the array center have a better resolution. This clearly shows that the front plane model is not good enough to deal with the border effects of the array. This simplistic approach should be substituted by a shower conic fit, which accounts for the shower front curvature. §.§ Gamma-hadron discrimination The shower characteristics may be used to distinguish a pure electromagnetic shower, induced by a gamma-ray, from a shower generated by a hadronic primary. Here, we attempt to identify observables able to distinguish the two kind of primaries based solely on the WCD signal. Two promising variables, inspired in the procedures followed in <cit.> were found: * Sum of the signal of all WCD stations above 40m of distance to the reconstructed shower core with a signal above the expected one for a muon. This variable is then normalised to the total amount of signal encountered in stations far away from the shower core (> 40m);* distance of the WCD station signal to the shape of the average gamma-ray lateral distribution function (LDF), also known as compactness.Both variables were combined using a Fisher linear discriminant and the gamma selection efficiency and hadron rejection results are presented in figure <ref>. As one can seen, this preliminary analysis gives results already as good as those reached by the HAWC collaboration. §.§ Sensitivity to steady sources Using the results described in the previous sections, one can now compute this detector sensitivity to steady sources. We compute the differential sensitivity as the flux of a source giving N_excess /√(N_bkg) = 5 after 1 year of effective observation time. It was assumed that the source is visible one fourth of the time. This is roughly the time that the galactic centre is visible in the Southern hemisphere. The results from this study are shown in figure <ref> and are compared with the 1 year sensitivities of FERMI and HAWC. One can clearly see that this detector would be able to cover the gap between the two of the most sensitive experiments in this energy range. § SUMMARY We have presented a novel hybrid detector which combines Resistive Plate Counters with a Water Cherenkov Detetor, able to extend the sensitivity of previous experiments down to the region of 100 GeV.This modular compact and low cost detector, has been assessed with a realistic simulation, giving encouraging results. However, its capabilities are far from being fully explored. More sophisticated analysis are expected to be achieved in order to take fully advantage of LATTES hybrid concept. Studies to include a sparse array to extend the LATTES energy range up to 100 TeV are also undergoing.Finally, with the advent of the Cherenkov Telescope Array, LATTES would be a fully complementary project that could provide not only triggers to transient phenomena but also contribute with long term observations of variable sources.jhep | http://arxiv.org/abs/1709.09624v2 | {
"authors": [
"P. Assis",
"U. Barres de Almeida",
"A. Blanco",
"R. Conceição",
"B. D'Ettore Piazzoli",
"A. De Angelis",
"M. Doro",
"P. Fonte",
"L. Lopes",
"G. Matthiae",
"M. Pimenta",
"R. Shellard",
"B. Tomé"
],
"categories": [
"astro-ph.IM",
"astro-ph.HE",
"hep-ex",
"physics.ins-det"
],
"primary_category": "astro-ph.IM",
"published": "20170927170123",
"title": "LATTES: a novel detector concept for a gamma-ray experiment in the Southern hemisphere"
} |
Diffuse Ionized Gas in the Milky Way Disk T. M. Bania December 30, 2023 ========================================= This paper examines the Evolutionary programming (EP) method for optimizing PID parameters. PID is the most common type of regulator within control theory, partly because it's relatively simple and yields stable results for most applications. The p, i and d parameters vary for each application; therefore, choosing the right parameters is crucial for obtaining good results but also somewhat difficult. EP is a derivative-free optimization algorithm which makes it suitable for PID optimization. The experiments in this paper demonstrate the power of EP to solve the problem of optimizing PID parameters without getting stuck in local minimums. § INTRODUCTION This paper will examine one approach to the optimization of Proportional-Integral-Derivative (PID) parameters. PID is the most common type of regulator within control theory. The mathematical expression of a PID regulator is relatively simple and yields stable results for most applications. The p, i and d parameters vary for each application; therefore, choosing the right parameters is crucial for obtaining good results but also somewhat difficult. According to K.J.Åström and T.Hägglund <cit.>, there are several methods used in the industry for tuning the p, i and d parameters. Some methods involve developing models while others involve manual tuning by trial and error.This paper examines the Evolutionary programming (EP) method for optimizing PID parameters. EP is an iterative algorithm that runs until some performance criteria are met. The basic steps are as follows: EP generates a population, evaluates every member in the population, selects the best member, creates a new population based on the best member, and then repeats the aforementioned process with an evaluation of the new population. The new population is generated by adding a Gaussian mutation to the parent. The results of this paper are compared with the results generated in R.Johns <cit.>, which uses Genetic algorithm (GA), and S.Hadenius <cit.>, which uses Particle Swarm optimization (PSO). The GA and PSO methods are further explained in <cit.>. All training and testing of the algorithm was done in a simulated environment called MORSE on a Robot Operating System (ROS). The algorithm and movements of the robot were developed as three ROS nodes. An ROS node is one subsystem; typically, a robotics environment is built with multiple ROS nodes. The actual simulation of the robot was done by MORSE which is a generic 3D simulator for robots. Nodes are only able to communicate with other nodes using streaming topics, Remote Procedure Calls (RPC) services, and the Parameter Server. One route was used for training and a separate route was used for evaluating the step response. The experiments in this paper tuned two PIDs simultaneously. The PIDs tuned were the ones controlling linear velocity and angular velocity on the robot. Only the Husky robot was used. EP can perform effectively in this type of domain due to its ability to optimize without getting stuck in local minimums and its efficiency over brute force. The resulting parameters were evaluated by looking at the step response. Important measurements are: rise time, overshoot, and the steady state error.§ METHOD §.§ Set up environmentThis paper describes three different experiments that all ran on the same environment. Everything required to run the experiments is listed in table <ref>.§.§ DefinitionsThe Determinator receives both the actual velocity and desired velocity of the robot 50 times per second. With that data, an error can be estimated by calculating abs(desired - actual). That calculation is executed for every sample, summed, and then divided by the total number of samples. Thus, the total error for one run of a route is the average error of that route. This average error (AE) is used by the algorithm to evolve the parameters. Fitness refers to lowest AE in this paper. A population, or generation, is a collection of individuals in which each individual contains values for k_pv, k_iv, k_dv, k_pa, k_ia, k_da. The values denoted k_pv, k_iv, k_dv controls the linear velocity PID and values denoted k_pa, k_ia, k_da controls the angular velocity PID. The velocity and angular parameters were evaluated separately from one another. Figure <ref> provides a visual illustration of the generations within the EP <ref> algorithm. Each box in figure <ref> illustrates an individual and each row illustrates a population. The green boxes indicate the fittest individual of each population (i.e., lowest AE). §.§ Evolutionary programming algorithmThe version of evolutionary programming used in this paper follows the algorithm described in <cit.>. * Generate an initial population of individuals. The number of individuals differs between the experiments.*Evaluate fitness as described in <ref>.* Select the fittest individual of the population by selecting the individual with lowest AE.* If the fittest individual has an average error better 0.01, return that individual and exit. 0.01 was chosen as a way to try to get the AE lower than 1 percent.* Else generate a new population by applying a mutation to the fittest individual. The mutation differs between the experiments but the general principle is to add a random number from a Gaussian distribution to each of the new offspring. The mutation used for the experiments is described in detail in section <ref>.* Go to step <ref>.§.§ ExperimentsTable <ref> describes all of the experimental configurations. All experiments ran for 100 generations and no experiment reached the set criteria of AE < 0.01. All experiments began with the same initial population, which looked like the following: k_pv∼𝒰(0, 1),k_iv∼𝒰(0, 0.1),k_dv∼𝒰(0, 0.01), k_pa∼𝒰(0, 1),k_ia∼𝒰(0, 0.1),k_da∼𝒰(0, 0.01).The mutate algorithm provided in <cit.> and used in experiment 1 did not perform well for this application because the parameters differed in size by more than an order of magnitude. Therefore, experiments 2 and 3 used a mutate algorithm that scales the mutation with the value being mutated.§ RESULTFor each experiment, this paper will present the AE value as well as a graph illustrating the step response of the PID for both train and test. Table <ref> lists every experiment's best parameters and lowest AE for train and test.In addition to the results presented in Table <ref>, Figure <ref> illustrates how the PID parameters evolve over every generation in experiment 2. For experiment 2, Figure <ref> shows the AE across all generations and Figure <ref> shows the step response for train and test for both linear and angular velocity. § DISCUSSIONOverall, we can conclude that using EP is successful in finding sufficient parameters given enough training time as described in <cit.>. The results presented in this paper are somewhat surprising. As we can see, the d part of PID becomes either 0 or negligible after running the algorithm for a few generations. The same thing happens to the i part, but to an lesser extent. An explanation for this behavior could be the way that the simulator is built. This begs the question of whether the results are useful in an environment outside of our simulation. Finally, we noticed that increasing the population size from 10 to 20 did not significantly improve the results. When comparing the results with the results from R.Johns <cit.> and S.Hadenius <cit.>, we can see that they got similar results, including very small ID-parameters. This adds confidence that the algorithm works but that the simulation is not optimal for tuning the PIDs. There are several variations to the experiment that could be done to improve the results. One way is to divide the route into different parts and then calculate the AE for each part. That would minimize the risk of a good integral value being punished by the algorithm for a slow rise time. Second, other fitness functions could be used instead of AE, including mean squared error. Third, training the parameters on a dynamic route could take the algorithm out of a local minima. Lastly, running the robot with the best known parameters in between every run would allow each individual to start with as low of an error as possible. In this paper, bad parameters from a previous individual could affect the AE of the next tested individual. 0.2in | http://arxiv.org/abs/1709.09227v1 | {
"authors": [
"Adam Nyberg"
],
"categories": [
"cs.NE"
],
"primary_category": "cs.NE",
"published": "20170926192031",
"title": "Optimizing PID parameters with machine learning"
} |
K. S. Alexander]Kenneth S. Alexander K. S. Alexander, Department of Mathematics, KAP 104University of Southern CaliforniaLos Angeles, CA90089-2532 USA [email protected]. Berger]Quentin Berger Q. Berger, Sorbonne Université, LPSMCampus Pierre et Marie Curie, case 1584 place Jussieu, 75252 Paris Cedex 5, France [email protected][2010]Primary: 60K35, 82B43 Geodesics Toward Corners in First Passage Percolation [ December 30, 2023 ===================================================== For stationary first passage percolation in two dimensions, the existence and uniqueness of semi-infinite geodesics directed in particular directions or sectors has been considered by Damron and Hanson <cit.>, Ahlberg and Hoffman <cit.>, and others. However the main results do not cover geodesics in the direction of corners of the limit shape ℬ, where two facets meet. We construct an example with the following properties: (i) the limiting shape is an octagon, (ii) semi-infinite geodesics exist only in the four axis directions, and (iii) in each axis direction there are multiple such geodesics.Consequently, the set of points of ∂ℬ which are in the direction of some geodesic does not have all of ℬ as its convex hull.§ INTRODUCTIONWe consider stationary first passage percolation (FPP) on a lattice 𝕃 with site set ^2, and with a set of bonds which we denote . We are mainly interested in the usual set of nearest-neighbor bonds = { (x,y) ;‖ x-y ‖_1 =1}, though in Section <ref> we considerwith added diagonal bonds to construct a simpler example. Each bond e ofis assigned a random passage time τ_e≥ 0, and the configuration τ is assumed stationary under lattice translations; the measure on the space Ω = [0,∞)^ of configurations τ is denoted , with corresponding expectation .For sites x,y of 𝕃, a path γ from x to y in 𝕃 is a sequence x=x_0,…,x_n=y with x_i,x_i+1 adjacent in 𝕃 for all i; we may equivalently view γ as a sequence of edges.The passage time T(γ) of γ is T(γ) = ∑_e∈γτ_e.For sites x,y we defineT(x,y) = inf{T(γ): γ is a path from x toy}.A geodesic from x to y is a path which achieves this infimum.A semi-infinite geodesic Γ from a site x is a path with (necessarily distinct) sites x=x_0,x_1,… for which every finite segment is a geodesic, and the direction of Γ, denoted (Γ), is the set of limit points of {x_n/|x_n|: n≥ 1}. It is of interest to understand semi-infinite geodesics, and in particular the set of directions in which they exist. It is standard to make the following assumptions, from <cit.>.Assumption A1.(i)is ergodic with respect to lattice translation;(ii) (τ_e^2+)<∞ for any e∈, for some >0.Under A1, an easy application of Kingman's sub-additive theorem gives that for each x∈^2 the limitμ(x) = lim_n→∞T(0,nx)/nexists.This μ extends to ^2 by restricting to n for which nx∈^2, and then to ^2 by continuity; the resulting function is a norm (provided the limit shape defined below is bounded). Its unit ball is a nonempty convex symmetric set which we denote . The wet region at time t is (t) = {x+[- 12, 12]^2: T(0,x)≤ t}. The shape theorem of Boivin <cit.> says that with probability one, given >0, for all sufficiently large t we have (1-)⊂(t)/t⊂ (1+), sois called the limit shape. Häggström and Meester <cit.> showed that every compact convex B with the symmetries of ℤ^2 arises as the limit shape for some stationary FPP process.We add the following assumptions, also used in <cit.>, <cit.> and outlined in <cit.> (see A2).Assumption A2. (iii)has all the symmetries of the lattice 𝕃;(iv) if α,γ are finite paths, with the same endpoints, differing in at least one edge then T(α) ≠ T(γ) a.s.;(v)has upward finite energy: for any bond e and any t such that (τ_e>t)>0, we have (τ_e>t|{τ_f:f≠ e}) >0 a.s.; (vi) the limit shapeis bounded (equivalently, μ is strictly positive except at the origin.) Thanks to (iv), the union of all geodesics from a fixed site x to sites y∈^2 is a tree, and we denote it _x=_x(τ).By <cit.>, _x contains at least 4 semi-infinite geodesics, andBrito and Hoffman <cit.> give an example in which there are only 4 geodesics, and the direction for each corresponds to an entire closed quadrant of the lattice. To describe the directions in which semi-infinite geodesics may exist, we introduce some terminology.A facet ofis a maximal closed line segment F of positive length contained in ∂; the unique linear functional equal to 1 on F is denoted ρ_F.For each angle from 0 there corresponds a unique point of ∂ in the ray from 0 at that angle; a facet thus corresponds to a sector of angles, or of unit vectors. We say a point v∈∂ is of type i (i=0,1,2) if v is an endpoint of i facets.We may divide points of ∂ (or equivalently, all angles) into 6 classes: (1)exposed points of differentiability, that is, exposed points of ∂ where ∂ is differentiable, necessarily type 0;(2) facet endpoints of differentiability, or equivalently, type-1 points where ∂ is differentiable;(3)facet interior points, necessarily type-0;(4) half rounded corners, that is, type-1 points where ∂ is not differentiable;(5) fully rounded corners, that is, type-0 points where ∂ is not differentiable;(6) true corners, meaning type-2 points.Associated to any semi-infinite geodesic Γ={x_0,x_1,…} is its Busemann function B_Γ:^2×^2→ given byB_Γ(x,y) = lim_n→∞ (T(x,x_n) - T(y,x_n)).From <cit.>, we know the following, under Assumptions A1 and A2. Almost surely, there exists for any semi-infinite geodesic Γ a linear functional ρ_Γ on ^2 with the property that B_Γ is linear to ρ_Γ, that is, lim_|x|→∞1/|x|| B_Γ(0,x) - ρ_Γ(x) | = 0.Still from <cit.>, the set {ρ_Γ=1} is always a supporting line of , so its intersection with ∂ is either an exposed point v or a facet F, and then (Γ) is equal to {v} or contained in F (modulo normalizing to unit vectors.)Thus (Γ) determines ρ_Γ, unless (Γ) consists of only a corner of some type.Furthermore, there is a closed set _* of linear functionals such that the set of functionals ρ_Γ which appear for some Γ is almost surely equal to _*.If v∈∂ is not a corner, there is a ρ such that {ρ=1} is the unique tangent line to ∂ at v, and we have ρ∈_*.In<cit.>, Ahlberg and Hoffman definea random coalescing geodesic (or RC geodesic), which is, in loose terms, a mapping which selects measurably for each τ a semi-infinite geodesic Γ_0=Γ_0(τ) in _0(τ), in such a way that when the mapping is applied via translation to obtain Γ_x∈_x, Γ_0 and Γ_x coalesce a.s. for all x.The following statements are valid under Assumptions A1 and A2: they are part of, or immediate consequences of, results of Ahlberg and Hoffman <cit.>, strengthening earlier results from <cit.> and <cit.>.(I) For each exposed point of differentiability v∈∂, there is a.s. a unique RC geodesic Γ with (Γ) = {v/|v|}. (II) For each half rounded corner v∈∂, there is a.s. at least one RC geodesic Γ with (Γ) = {v/|v|}; for one such Γ the linear functional ρ_Γ corresponds to a limit of supporting lines taken from the non-facet side of v.This uses the fact that _* is closed. (III) For each fully rounded corner v∈∂, there are a.s. at least two RC geodesics Γ with (Γ) = {v/|v|}, with distinct linear functionals ρ_Γ corresponding to limits of supporting lines from each side of v. (IV) Given a facet F with corresponding sector S_F of unit vectors, there is a.s. a unique RC geodesic Γ with (Γ)⊂ S_F and ρ_Γ=ρ_F.For any other RC geodesic Γ with (Γ)∩ S_F≠∅, this intersection is a single endpoint of S_F which must be a corner.But relatively little has been proved about geodesics, or RC geodesics, in the directions of true corners(where several supporting lines coexist), for instance when the limit shape is a polygon, see e.g. the discussion in Section 3.1 of <cit.>.One may ask, must every true-corner direction be in (Γ) for some geodesic Γ?Equivalently, must the convex hull of _ geo := {v∈∂: v/|v| ∈(Γ)for some semi-infinite geodesic Γ}be all of ? Further, we can consider the nonuniqueness set:= { u∈ S^1:there exist multiple semi-infinite geodesics Γ with (Γ) = {u}}.For each fully rounded corner v we have (v∈)=1. For each non-corner v∈∂ we have (v∈)=0, but in the case ofwith no corners this does not meanis empty.If every point of ∂ is an exposed point of differentiability then there is at least one geodesic in every direction; the union of all semi-infinite geodesics from 0 is therefore a tree with infinitely many branches, and each branching produces a point of , sois infinite a.s.In the example of Brito and Hoffman <cit.>, the limit shape is a diamond with true corners on the axes, and for each of these corners v there is a.s. no geodesic Γ with (Γ)= {v/|v|}, so (v∈)=0. This suggests the question, must (v∈)=0 for true corners? Our primary result is an example of FPP process, which we call fast diagonals FPP, in which some true corners have no geodesic, and others have multiple geodesics a.s.This means in particular that the convex hull of _ geo is not all of . The fast diagonals FPP process (defined in Section <ref>) satisfies Assumptions A1-A2, andhas the following properties: (i) The limit shape is an octagon, with corners on the axes and main diagonals.(ii) Almost surely, every semi-infinite geodesic Γ is directed in an axis direction (that is, (Γ) consists of a single axis direction.)(iii) Almost surely, for each axis direction there exist at least two semi-infinite geodesics directed in that direction.We first introduce a simpler example in Section <ref>, which does not satisfy Assumption A2 (in particular (iv) and (v)), but encapsulates the key ideas of our construction. The remaining main part of the paper is devoted to modifying this example in order to satisfy Assumption A2,which brings many complications, see Section <ref>. In Theorem <ref> the shape is an octagon, and therefore <cit.> tells that there must therefore be at least eight geodesics, one for each flat edge of the shape. In our example, there are no geodesics associated to the supporting lines that only touch the shape at the diagonal corners, and the geodesics associated to the flat pieces are asymptotically directed along the axes.However, questions remains regarding the geodesics directed along the axis—for example, are there only two geodesics in each axis direction? Put differently, are there geodesics associated to supporting lines that intersect the shape only at the corners on the axes? Our guess is that, in our example, there are indeed a.s. only two geodesics in the direction of the axis, but the question remains as to whether it is possible to build a model with more than two geodesicsin a corner direction.Our result can perhaps be adapted to produce more general polygons, with polygon vertex directions alternating between those having two or more geodesics and those having none, and with no other directions with geodesics. In higher dimensions, the possibility of analogous examples is unclear.We make some informal comments here, without full proof, about the linear functionals associated to the geodesics in Theorem <ref>, and about geodesics vs. RC geodesics.Let us consider the collection _E,x of geodesics directed eastward from x, and the union _E,x of all such geodesics.For x=0, each such eastward geodesic Γ has a height h(Γ) of its final point of intersection with the vertical axis.We claim that all semi-infinite geodesics contained in _E,0 are in _E,0, and h(Γ) is bounded over Γ∈_E,0, a.s.In fact there can be no westward geodesic contained in _E,0 because any eastward geodesic from 0 passing through any point near the negative horizontal axis sufficiently far west from 0 must cross a northward or southward geodesic on its way back eastward, contradicting uniqueness of point-to-point geodesics (i.e. Assumption A2(iv).)Regarding northward geodesics contained in _E,0, they are ruled out when we show in the proof of (ii) in Section <ref> that there is a.s. a random R such that, roughly speaking, no eastward geodesic “goes approximately northward for a distance greater than R before turning eastward,” and similarly for southward in place of northward. This also shows the boundedness of h(Γ).Now any limit of geodesics in _E,0 must be contained in _E,0, and it follows from our claim that any such limit is in _E,0.It follows that among all eastward geodesics Γ in _E,0 with a given value of h(Γ), there is a leftmost one, where “leftmost” is defined in terms of the path Γ in the right half plane after the point (0,h(Γ)). Hence boundedness of h shows there is a leftmost geodesic Γ_L overall in _E,0.It is then straightforward to show that ρ_Γ_L must be the linear functional equal to 1 on the side ofconnecting the positive horizontal axis to the main diagonal.By <cit.>, Γ_L is a.s. the unique geodesic with corresponding linear functional ρ_Γ_L, so by <cit.>,Γ_L, viewed now as a function of the configuration τ and initial point x, must be an RC geodesic.Similar considerations apply symmetrically to other directions and to rightmost geodesics.Therefore we can replace “geodesics” with “RC geodesics” in Theorem <ref>(iii).§ SIMPLE EXAMPLE: DIAGONAL HIGHWAYS ONLYFor this section we consider the lattice 𝕃 with site set ^2, with the set of bonds = { (x,y) ; ‖ x-y ‖_∞=1 }, which adds diagonal bonds to the usual square lattice. We frequently identify bonds and path steps by map directions: either SW/NE or SE/NW for diagonal bonds, and N, NE, etc. for steps.By axis directions we mean horizontal and vertical, or N, E, W, S, depending on the context.For a preceding b in a path γ, we write γ[a,b] for the segment of γ from a to b.We assign all horizontal and vertical bonds passage time 1.(This makes the model in a sense degenerate, which is one of the aspects we modify in Section <ref>.)Let 12 < θ < 1 and (2θ)^-1<η<1. For diagonal bonds, for k≥ 1, a highway of class k consists of 2^k-1 consecutive bonds, all oriented SW/NE or all SE/NW.The collection of all highways of all classes is denoted ℍ, and a highway configuration, denoted ω, is an element of {0,1}^ℍ. When a coordinate is 1 in ω we say the corresponding highway is present in ω.To obtain a random highway configuration, for each of the two orientations we let southernmost points of class-k highways occur at each x∈^2 independently with probability (θ/2)^k, for each k≥ 1.Every diagonal bond is a highway of class 0. For each diagonal bond e we have(eis in a present class-k highway) = 1 - ( 1 - θ^k/2^k)^2^k-1≤θ^k,k≥ 1,so with probability one, e is in only finitely many present highways.Thus for diagonal e we can define k(e)=max{k: e is in a present class-k highway} if this set is nonempty, and k(e)=0 otherwise, and then define its passage timeτ_e = √(2)(1+η^k(e))ifk(e)≥ 1;3ifk(e)=0.For all horizontal and vertical bonds we define τ_e=1. Note that the value 3 ensures non-highway diagonal bonds never appear in geodesics.Let A_1,A_2 denote the positive horizontal and vertical axes, respectively, each including 0.Letr_k = ∑_j= k^∞θ^j = θ^k/1-θ,so that ( e is in some present highway of class ≥ k)≤ r_k. We fix C and take k_0 large enough soη^k_0<1/32, r_k_0≤1/2, 0.1 · 2^k(η^k-1-η^k) > 4C/r_kfor allk≥ k_0,the last being possible by our choice of η. The stationary first passage percolation process defined as above has the following properties: (i) The limit shape is an octagon, with corners on the axes and main diagonals.(ii) The only infinite geodesics are vertical and horizontal lines (and the only geodesics starting at the origin are indeed the vertical and horizontal axes). Let us first show thatis an octagon with Euclidean distance 1 in the axis and diagonal directions, and a facet in each of the 8 sectors of angle π/4 between an axis and a diagonal. As a lower bound for the passage time from (0,0) to (a,b), we readily have that for 0≤ b≤ a(the other cases being treated symmetrically):τ( (0,0),(a,b) ) ≥√(2)b + (a-b).For an upper bound, it follows from θ>1/2 that the horizontal (or vertical) distance from the origin to the nearest diagonal highway H connecting the postive horizontal axis to height b is o(b) a.s. (see (<ref>) below), so as b→∞τ( (0,0),(a,b) ) ≤(√(2) +o(1)) b + (a-b).This reflects the fact that one route from (0,0) to (a,b) is to follow the axis horizontally to a diagonal highway H, then follow H to the top or right side of the rectangle [0,a]×[0,b], then follow that side of the rectangle to (a,b), provided H intersects the rectangle. The linearity of the asymptotic expression √(2) b + (a-b) means that the limit shape is flat between any diagonal and an adjacent axis, while the asymptotic speed is 1 out any axis or diagonal, so the limit shapeis an octogon.This proves item (i) and we focus on item (ii).Step 1. Construction of a “success” event. For x in the first quadrant Q, let Δ(x) denote the Euclidean distance in the SW direction from x to A_1∪ A_2. Let Ĝ_k={x∈ Q:Δ(x) = (2^k-1+1)√(2)}, which is a translate of A_1∪ A_2. For j> k define three random sets of highways:_k,j = {all present SW/NE highways of class j intersecting both A_1∪ A_2 and Ĝ_k}, _k,j' = {all present SW/NE highways of class j crossing A_1∪ A_2 but not Ĝ_k}, _k,j” = {all present SW/NE highways of class j crossing Ĝ_k but not A_1∪ A_2}.Note that these three sets are independent of each other, and intersections with each along any given line have density at most θ^j, by (<ref>).We also highlight that _k,j for j≤ k-1 is empty, since highways of class ≤ k-1 are too short to connect A_1∪ A_2 and Ĝ_k. Again, using (<ref>), intersections with _k,k have density (over sites) (0∈_k,k) ≥ 1 - ( 1 - θ^k/2^k)^2^k-1≥ 1 - e^-θ^k/2≥θ^k/2.Here 0∈_k,k is a shorthand notation for 0 being in a highway in _k,k, and the last inequality holds provided that k is sufficiently large. LetĤ_i,k = the highway intersecting A_i closest to 0 among all in ∪_j ≥ k_k,j,i=1,2,and let Û_k=(X̂_1,k,0) and V̂_k=(0,X̂_2,k) denote the corresponding intersection points in A_1 and A_2. Let Ω̂_k denote the open region bounded by A_1∪ A_2,Ĝ_k, Ĥ_1,k, and Ĥ_2,k, see Figure <ref>. We define the event F̂_k = Î_k∩M̂_k (success at stage k) where (i) Î_k: max(X̂_1,k,X̂_2,k) ≤ C/r_k, with C from (<ref>); (ii) M̂_k: every SW/NE highway intersecting Ω̂_k is in classes 1,…,k-1.Note that a highway intersecting Ω̂_k cannot intersect both A_1∪ A_2 and Ĝ_k, by definition of Ĥ_i,k.We claim there exists λ>0 such that (F̂_k) ≥λ for all k≥ k_0. Let us first prove that (Î_k) is bounded away from 0.Similarly to (<ref>), we have that( 0∈⋃_j≥ k_k,j)≥ r_k/3,so for i=1,2, by independence,(X̂_i,k > C/r_k) ≤exp( -C/3 ) =: ζ <1,for allk≥ k_0.By independence, we get that (Î_k)≥ (1-ζ)^2.We next prove that (M̂_k|Î_k) is bounded away from 0 for k≥ k_0.In fact, by the above-mentioned independence of the three sets of highways, by (<ref>) and (<ref>) we have(M̂_k|Î_k) ≥min_x_1,x_2≤ C/r_k( M̂_k|X̂_1,k=x_1,X̂_2,k=x_2) ≥min_x_1,x_2≤ C/r_k (1-r_k)^x_1+x_2-1≥ e^-2C. This completes the proof of (<ref>).A slight modification of this proof shows that for fixed ℓ, for k sufficiently large we have (F̂_k|σ(F̂_1,…,F̂_ℓ))>λ/2, and it follows that (F̂_k i.o.)=1. Step 2. Properties of geodesics in case of a success. We now show that when F̂_k occurs, for every x∉ A_1∪ A_2 ∪Ω̂_k in the first quadrant, every geodesic Γ̂_0x from 0 to x follows A_1 from 0 to Û_k, or A_2 from 0 to V̂_k.Since x is in the first quadrant, it is easily seen that any geodesic from 0 to x has only N, NE and E steps.Let p̂_x=(r,s) be the first site of Γ̂_0x not in A_1∪ A_2 ∪Ω̂_k.Besides the geodesic _0x[0, p̂_x], we define an alternate path ψ_x from 0 to p̂_x as follows; for this we assume p̂_x is in the first quadrant on or below the main diagonal, and make the definition symmetric for p̂_x elsewhere. (i) If p̂_x ∈Ĥ_1,k we let ψ_x follow A_1 east from 0 to Û_k, then NE from Û_k to p̂_x on Ĥ_1,k.(ii) If p̂_x is in the horizontal part of Ĝ_k we let Û_k' be the intersection of Ĥ_1,k with the vertical line through p̂_x, and let ψ_x be the path east from 0 to Û_k, then NE to Û_k', then north to p̂_x.(Here we assume k_0 is chosen large enough so that C/r_k < 2^k-1, ensuring that p̂_x is farther east than Û_k.) (iii) Otherwise p̂_x is adjacent to A_1 and the final step of _0x[0,p̂_x] is from some Û_k”∈ A_1 to p̂_x, N or NE.We let ψ_x go east from 0 to Û_k”, then take one step (N or NE) to p̂_x.In case (i), the path ψ_x has no N steps, and it is easily seen that any path in A_1∪ A_2∪Ω_k from 0 to p̂_x containing some N steps will be strictly slower than ψ_x, and hence is not a geodesic.Thus every geodesic from 0 to p̂_x has s NE and r-s E steps.Since success occurs at stage k, any diagonal bonds in _0x[0,p̂_x] \Ĥ_1,k have passage time strictly more than √(2)(1+η^k), making _0x[0,p̂_x] strictly slower than ψ_x, which contradicts the fact that _0x[0,p̂_x] is a geodesic.It follows that we must have _0x[0,p̂_x] = ψ_x, which means Γ̂_0x indeed follows A_1 from 0 to Û_k.In case (ii), we have s=2^k-1+1 and 0≤ r-s ≤X̂_1,k.Let q be the number of NE steps in _0x[0,p̂_x], so it must have s-q N and r-q E steps. Each of the diagonal bonds has passage time at least √(2)(1+η^k-1), so its passage time satisfiesT(_0x[0,p̂_x]) ≥√(2)(1+η^k-1)q + (s-q) + (r-q) ≥√(2)(1+η^k-1)q + 2(s-q) ≥√(2)(1+η^k-1)s.By contrast, the northward segment of ψ_x has length X̂_1,k-(r-s), soT(ψ_x)≤ 2X̂_1,k - (r-s) + √(2)(1+η^k)(r-X̂_1,k) ≤ 2 X̂_1,k + √(2)(1+η^k)s ≤√(2)(1+η^k)s + 2 C/r_k.Hence by (<ref>),T(_0x[0,p̂_x]) - T(ψ_x) ≥√(2)(η^k-1-η^k)s - 2C/r_k= (2^k-1+1)√(2)(η^k-1-η^k) - 2C/r_k > 0.But this contradicts the fact that Γ̂_0x isa geodesic.Thus we cannot have p̂_x in the horizontal part of Ĝ_k – and similarly not in the vertical part.In case (iii), since the unique geodesic between any two points of A_1 is a segment of A_1, it is straightforward that we must have _0x[0,p̂_x] = ψ_x. Again, Γ̂_0xfollows A_1from 0 to Û_k.Step 3. Conclusion.If Γ̂ is any semi-infinite geodesic from 0 which has infinitely many points in the (closed) first quadrant, then for each of the infinitely many k for which F̂_k occurs, the initial segment of Γ̂ must follow an axis from the origin to Û_k or V̂_k.But since X̂_1,k,X̂_2,k→∞, this means Γ̂ itself must be one of these axes. It follows that all horizontal and vertical lines are semi-infinite geodesics, and no other paths.§ MODIFICATION FOR SQUARE LATTICE, FINITE ENERGY, AND UNIQUE GEODESICS The preceding simpler example does not satisfy A2 (iv) or (v), and it does not allow the use of results known only for the usual square lattice.The first difficulty is to adapt our construction of Section 2 so that it does not have diagonal bonds: we remove the diagonal bonds, and we replace diagonal highways with zigzag highways (alternating horizontal and vertical steps) as done in <cit.>. Then, in order to verify the upward finite energy, we need to introduce horizontal/vertical highways, and make highways of class k not have a fixed length. In order to ensurethe unique geodesics condition, we add auxiliary randomization. All together, this adds significant complications.Primarily, since the graph is planar, there is sharing of bonds between, for example, horizontal and zigzag highways where they cross. Since the passage times are different in the two types of highways, each shared bond slows or speeds the total passage time along at least one of the highways, compared to what it would be without the other highway.We must ensure that the number of such crossings is not a primary determinant of which paths are geodesics. Once we have properly define a passage time configuration (the fast diagonals FPP), our strategy will be similar to that of the simple example of Section <ref>: we will define a “success” event, and show that when a success occurs geodesics approximately follow an axis for a long distance, at least until they reach a very fastzigzag highway. §.§ Definition of the fast diagonals FPP process We now select parameters η,,θ,,μ: θ^k (resp. θ̃^k) will roughly correspond to the densities of zigzag (resp. horizontal/vertical) highways of class k, defined below, and η^k (resp. η̃^k) will roughly correspond to the slowdown of a zigzag (resp. horizontal/vertical) highway — that is the higher the class, the faster the highway. We will choose the parameters so that the horizontal/vertical highways are not too infrequent relative to zigzag ones (θ̃^c_θ > θ for a certain power c_θ<1) and are less slowed down (η̃<η) than zigzag ones.Loosely speaking we want the passage times to be much more affected by the zigzag highways encountered than by the horizontal/vertical ones, but fast zigzag highways have to be easily reachable by nearby horizontal/vertical highways, so the choice of parameters must be precise.The actual choice of the parameters is as follows: we choose c_θ∈ (0.4,0.5) and c_θ̃ , >0 to verifyθ=2^-c_θ, = 2^-c_, θ^c_θ c_/(1-c_θ (c_-4δ)) < η< min(7/8,θ^2/3), < min( 1/2θ, η/2) ,andθ < μ < min(^c_θ,η,(θη)^1-c_θ (c_-4δ)θ^-4c_θδ).To see that this choice of parameters is possible, notice that c_θ /(1-c_θ )>2/3 and θ^c_θ /(1-c_θ ) < 7/8 so we can choose 0<4δ<c_<1 such that c_θ c_/(1-c_θ (c_-4δ))>2/3 and θ^c_θ c_/(1-c_θ (c_-4δ)) < 7/8. Note that since c_θ c_<0.5, the third condition in (<ref>) guarantees η>θ; since c_θ<0.5 this also means2ηθ >2θ^2>1, guaranteeing that one can make a modification of (<ref>) hold also here:0.1 · 2^k(η^k-1-η^k) > 4C/r_k + 0.2 for allk≥ k_0 by choosing k_0 large enough.Further, the first inequality in that third condition in (<ref>) is equivalent to θ < (θη)^1-c_θ (c_-4δ)θ^-4c_θδ.Together these show μ can be chosen to satisfy (<ref>).Highways and types of bonds. A zigzag highway is a set of (adjacent) bonds in any finite path which either (i) alternates between N and E steps, starting with either, called a SW/NE highway, or (ii) alternates between N and W steps, starting with either, called a SE/NW highway.If the first step in the path is N, we say the highway is V-start; if the first step is W or E we say it is H-start. A SW/NE highway is called upper if it is above the main diagonal, and lower if it is below, and analogously for SE/NW highways. Note a SW/NE highway is not oriented toward SW or NE, it is only a set of bonds, and similarly for SE/NW.The length |H| of a highway H is the number of bonds it contains.To each zigzag highway H we associate a random variable 𝒰_H uniformly distributed in [0,1] and independent from highway to highway. For each k≥ 1 we construct a random configuration ω^(k) of zigzag highways of class k: these highways can have any length 1,…,2^k+3. Formally we can view ω^(k) as an element of {0,1}^ℍ_k, where ℍ_k is the set of all possible class-k zigzag highways; when a coordinate is 1 in ω^(k) we say the corresponding highway is present in ω^(k). To specify the distribution, for each length j≤ 2^k+3 and each x∈^2, a SW-most endpoint of a present length-j H-start SW/NE highway of class k occurs at x with probability θ^k/2^2k+4, independently over sites x and classes k, with the same for V-start, and similarly for SE/NW highways.We write ω=(ω^(1),ω^(2),…) for the configuration of zigzag highways of all classes. Note that due to independence, for k< l, a given zigzag highway (of length at most 2^k+3) may be present when viewed as a class-k highway, and either present or not present when viewed as a class-l highway, and vice versa. Formally, then, a present highway in a configuration ω is an ordered pair (H,k), with k specifying a class in which H is present, but we simply refer to H when confusion is unlikely.A horizontal highway of length j is a collection of j consecutive horizontal bonds, and similarly for a vertical highway.Highways of both these types are called HV highways. For each k≥ 1 we construct a configuration ^(k) of HV highways of class k: these highways can have any length 1,…,2^k, and for each length j≤ 2^k and each x∈^2, a leftmost endpoint of a present length-j horizontal highway of class k occurs at x with probability ^k/2^2k, independently over sites x, and similarly for vertical highways.We write = (^(1),^(2),…) for the configuration of HV highways of all classes. We now combine the classes of zigzag highways and “thin” them into a single configuration by deletions —we stress that following our definitions, each site is a.s. in only finitely many highways.We do the thinning in two stages, first removing those which are too close to certain other zigzag highways, then those which are crossed by aHV highway with sufficiently high class.Specifically, for stage-1 deletions we define a linear ordering (a ranking) of the SW/NE highways present in ω, as follows. Highway (H',k) ranks above highway (H,l) if one of the following holds: (i) k>l; (ii) k=l and |H'|>|H|; (iii)k=l, |H'|=|H|, and 𝒰_H'>𝒰_H. Let d_1(A,B) denote the ℓ^1 distance between the sets A and B of sites or bonds.We then delete any SW/NE highway (H,k) from any ω^(k) if there exists another present SW/NE highway (H',l), with d_1(H,H')≤ 22 which ranks higher than (H,k). We then do the same for SE/NW highways. The configuration of highways that remain in some ω^(k) after stage-1 deletions is denoted ω^ zig,thin,1.Here the condition d_1(H,H')≤ 22 is chosen to follow from d_1(H,H')<3/(1-η)-1; we have chosen η<7/8 in (<ref>) so the value 22 works.For stage-2 deletions, we let ζ = δ/(c_-δ), and for each m≥ 1 we delete from ω^ zig,thin,1 each zigzag highway of class m which shares a bond with an HV highway of class (1+ζ)m/c_ or more (as is always the case when such highways intersect, unless the intersection consists of a single endpoint of one of the highways.)The configuration of highways that remain in some ω^(k) after both stage-1 and stage-2 deletions is denoted ω^ zig,thin,2.For a given zigzag highway (H,m) in ω^ zig,thin,1, for each bond e of H there are at most 2^ℓ possible lengths and 2^ℓ possible endpoint locations for a class-ℓ HV highway containing e, so the probability (H,m) is deleted in stage 2 is at most 2^m+3∑_ℓ≥ (1+ζ)m/c_ 2^2ℓ·^ℓ/2^2ℓ = 8/1- 2^-ζ m. Following stage-2 deletions we make one further modification, which we call stage-3 trimming. Suppose (H,k),(H',l) are zigzag highways of opposite orientation (SW/NE vs SE/NW) in ω^ zig,thin,2, and x is an endpoint of H.If d_1(x,H')≤ 1, then we delete from H the 4 final bonds of H, ending at x, creating a shortened highway Ĥ.(Formally this means we delete (H,k) from ω^ zig,thin,2 and make (Ĥ,k) present, if it isn't already.)The resulting configuration is denoted ω^ zig,thin. This ensures that for any present SW/NE zigzag highway H and SE/NW zigzag highway H', either H and H' fully cross (meaning they intersect, and there are at least 2 bonds of each highway on either side of the intersection bond) or they satisfy d_1(H,H')≥ 2. This construction creates several types of bonds, which will have different definitions for their passage times. A bond e which is in no highway in ω^ zig,thin but which has at least one endpoint in some highway in ω^ zig,thin is called a boundary bond.A bond e in any highway in ω^ zig,thin is called a zigzag bond. A HV bond is a bond in some HV highway in some ^(k).An HV-only bond is an HV bond which is not a zigzag bond. A backroad bond is a bond which is not a zigzag bond, HV-only bond, or boundary bond.Moreover, there are special types of boundary and zigzag bondsthat arise when a SW/NE zigzag highway crosses a SE/NW one, so we need the following definitions for bonds in ω^ zig,thin. For zigzag bonds, we distinguish: (i) The first and last bonds (or sites) of any zigzag highway are called terminal bonds (or terminal sites.)A bond which is by itself a length-1 highway is called a doubly terminal bond; other terminal bonds are singly terminal bonds.Bonds which are not the first or last bond of a specified path are called interior bonds.(ii) An adjacent pair of zigzag bonds in the same direction (both N/S or both E/W, which are necessarily from different highways, one SW/NE and one SE/NW) is called a meeting pair, and each bond in the pair is a meeting zigzag bond.(iii) A zigzag bond for which both endpoints are meeting-pair midpoints is called an intersection zigzag bond. Equivalently, when a SW/NE highway intersects a SE/NW one, the bond forming the intersection is an intersection zigzag bond.(v) A zigzag bond which is not a meeting, intersection, or terminal zigzag bond is called a normal zigzag bond.For boundary bonds, we distinguish the following: (vi) A boundary bond is called a semislow boundary bond if either (a) it is adjacent to two meeting bonds (and is necessarily parallel to the intersection bond, separated by distance 1), called an entry/exit bond, or (b) it is adjacent to an intersection bond. (vi) A boundary bond e is called a skimming boundary bond if one endpoint is a terminal site of a zigzag highway, and the corresponding terminal bond is perpendicular to e.(vii) A boundary bond which is not a semislow or skimming boundary bond is called a normal boundary bond. These special type of bonds are represented in Figure <ref>. Definition of the passage times. For each edge e we associate its zigzag class k(e)= max{k: eis in a class-k zigzag highway in ω^ zig, thin} if this set is nonempty, 0,otherwise,and its HV class(e)= max{k: eis in a class-k HV highway in ω̃^(k)} if this set is nonempty, 0,otherwise.We then define a bond e to be slow if ξ_e' ≤ 4^-k(e)∨(e),where ξ_e' is uniform in [0,1], independent from bond to bond. Slow bonds exist only to ensure that upward finite energy holds.We next define the raw core passage time α_e=α_e(,ω^ zig,thin) of each bond e, mimicking ideas of the simple example of Section <ref>.For non-slow e we setα_e = 0.7ife is a zigzag bond0.9ife is an HV-only bond1if e is a backroad bond or non-HV boundary bond,and for slow e, we set α_e=1.2. Then we define the compensated core passage time α_e^*=α_e^*(ω,ω^ zig,thin) for non-slow e byα_e^* = 0.5if e is an intersection zigzag bond0.7ife is a normal zigzag bond0.8if e is a meeting bond, or a singly terminal zigzag bond0.9ife isa doubly terminal zigzag bond, or an HV-only bond which is either non-boundary or a skimming boundary bond1if e is a backroad bond, ora normal boundary bond1.1if e is a semislow boundary bond,and for slow e, we setα_e^*=1.3. The term “compensated” refers mainly to the following:when an HV highway crosses a zigzag one, it typically intersects one normal zigzag bond and two normal boundary bonds.The sum of the raw core passage times for these 3 bonds is 0.9+0.7+0.9=2.5, whereas in the absence of the zigzag highway the sum would be 3×0.9=2.7.With the compensated times, the sum is restored to 2.7, and in that sense the HV highway does not “feel” the zigzag highway.The compensation picture is more complicated when the HV highway crosses near the intersection of a SW/NE highway and a SE/NW highway (which necessarily fully cross.) It must be done so that (<ref>) and (<ref>) below hold, whether the HV highway contains intersection bonds, meeting bonds or entry/exit bonds, see Figure <ref>.In a similar sense, a SW/NE highway does not “feel” a crossing by a SE/NW highway.The idea in the definition of α_e^* is that we compensate for the “too fast” zigzag bonds in an HV highway (0.7 vs 0.9) by extracting a toll of 0.1 for entering or exiting a zigzag highway.If the entry/exit is through a terminal or meeting zigzag bond (as when passing through a meeting block), then the toll is paid by increasing the time of that bond from 0.7 to 0.8.In the meeting case, to avoid increasing the total time along the zigzag highway, the core passage time of the adjacent intersection bond is reduced to 0.5.If the entrance/exit for the zigzag highway is made through any other type of zigzag bond, the toll is paid by increasing the adjacent boundary bond in the path from 0.9 (if it's an HV bond) to 1.There is an exception in an entry/exit bond, which may be both entrance and exit: the toll for such a bond is 0.2.Due to the stage-1 deletions, any two parallel zigzag highways H,H' in ω^ zig,thin satisfy d_1(H,H')≥ 23, and as a result, the compensation described by (<ref>) is sufficient for our purposes; without a lower bound like (<ref>), more complicated highway-crossing situations would be possible, producing for example meeting zigzag bonds which are also intersection zigzag bonds.We thus have the following property:suppose H is an HV highway for which the first and last bonds are HV-only, and Γ is a path which starts and ends with non-zigzag bonds.Then ∑_e∈ Hα_e^* = 0.9|H|, ∑_e∈Γα_e^* ≥ 0.7|Γ|.End effects may alter this for general HV highways and paths, but it is easily checked that every HV highway H and path Γ satisfies| ∑_e∈ Hα_e^* - 0.9|H| | ≤ 0.4, ∑_e∈Γα_e^* ≥ 0.7|Γ| - 0.2.We use another auxiliary randomization to ensure unique geodesics:for each bond e we let ξ_e be uniform in [0,1], independent from bond to bond (and independent of the ξ_e''s used above). We can now define the full passage times τ_e (based on the configurations ω^ zig,thin and ω̃) by τ_e= α_e^* + σ_e,where σ_e (the slowdown) is defined to be 0.1 ξ_e if e is a slow bond, and for non-slow e σ_e = 0.1 ×η^k(e) + ^k(e)ξ_eif e is a zigzag bond, ^(e) + ^(e)ξ_eif e is an HV-only bond, ^(e)ξ_eif e is either a backroad bond, or a boundary bond which is not HV. Hence, this corresponds to a slowdown of order η^k (per bond) in class-k zigzag higways, and of order ^k̃ (per bond) in class-k̃ HV-highways. We refer to the resulting stationary FPP process as the fast diagonals FPP process.We stress that the presence of the independent variables ξ_e ensures that Assumption A2(iv) is satisfied, and (since σ_e≤ 0.2 in all cases) the presence of slow bonds ensures the positive finite energy conditionA2(v); the rest of A1 and A2 are straightforward. First observations and notations. It is important that since η>, passage times along long zigzag highways are much more affected by the class of the highway than are times along HV highways. In fact, by increasing k_0 we may assume (2)^k_0<η^k_0<1/16, recalling we chose η̃<η/2. Then if H is an HV highway of class k≥ k_0(so of length at most 2^k+3), we have0.1 ∑_e∈ H (^k + ^kξ_e) ≤ 1.6 (2)^k < 0.1,so the maximum effect on ∑_e∈ Hτ_e of all the variables involvingis less than 0.1, hence is less than the effect of the α_e^* value for any single bond e. Henceforth we consider only k≥ k_0, n setting the scale of Ω̂_k. Let r_k = ∑_j=k^∞^j and _k = ∑_j=k^∞^j. Define q_k = log_2(2C/r_k) = c_θ k+b where b is a constant. The following subsections prove Theorem <ref>.The strategy is similar to that of Section <ref>: we first construct in Section <ref> an event F_k that a.s. occurs for a positive fraction of all k's, and then show in Section <ref> that when F_k occurs geodesics have to stay near the axis. We conclude the proof of Theorem <ref> in Section <ref>. §.§ Construction of a “success” event Analogously to Section <ref>, we construct a random region _k (which is an enlarged random version of Ω̂_k), and a deterministic region Θ_k which may contain _k, as follows.As before we write A_1,A_2 for the positive horizontal and vertical axes, Q for the first quadrant, and now also A_3, A_4 for the negative horizontal and vertical axes, respectively.We write G_k^1 for the set {x∈ Q:Δ(x) = 2^k√(2)} (formerly denoted Ĝ_k)and G̃_k^1 for {x∈ Q:Δ(x) = (2^k+4)√(2)}; successively rotating the lattice by 90 degrees yields corresponding sets G_k^2,G_k^3,G_k^4 and G̃_k^2,G̃_k^3,G̃_k^4 in the second, third and fourth quadrants, respectively. Let H_NE,L,k and H_NE,U,k be the lower and upper zigzag highways in ω^ zig,thin,2 of class k or more, intersecting both G̃_k^1 and G̃_k^3, which intersect A_1 and A_2, respectively, closest to 0.Let U_k^NE = (X_1,k^NE,0) be the leftmost point of A_1∩ H_NE,L,k, and V_k^NE=(0,X_2,k^NE) the lowest point of A_2∩ H_NE,U,k.Rotating the lattice 90 degrees yields analogous highways H_NW,L,k and H_NW,U,k each intersecting G̃_k^2 and G̃_k^4, and intersections points U_k^NW = (X_1,k^NW,0) and V_k^NW=(0,X_2,k^NW) with axes A_3 and A_2, respectively.Here we have used G̃_k^i and not G_k^i so that stage-3 trimming does not prevent H_*,·,k from reaching appropriate G_k^i. Let ℓ_1,k and ℓ_3,k be the vertical lines {± 2C/r_k}× crossing A_1 and A_3 respectively, and ℓ_2,k and ℓ_4,k the horizontal lines ×{± 2C/r_k} crossing A_2 and A_4. Let J_N,k (and J_S,k, respectively) denote the lowest (and highest) horizontal highway above (and below) the horizontal axis intersecting both ℓ_1,k and ℓ_3,k. Analogously, let J_E,k (and J_W,k) be the leftmost (rightmost) vertical highway to the right (left) of the vertical axisintersecting both ℓ_2,k and ℓ_4,k.Let (Y_E,k,0) be the intersection of J_E,k with A_1, and analogously for (0,Y_N,k),(Y_W,k,0) and (0,Y_S,k) in A_2,A_3, and A_4. Let _k^NE be the open region bounded by H_NE,L,k,H_NE,U,k,G_k^1, and G_k^3, and let _k^NW be the open region bounded by H_NW,L,k,H_NW,U,k,G_k^2, and G_k^4.Then let _k=_k^NW∪_k^NE (an X-shaped region), see Figure <ref> below.Let h_NE,L,k and h_NE,U,k denote the SW/NE diagonal lines through (C/r_k,0) and (0,C/r_k) respectively, and let h_NW,L,k and let h_NW,U,k denote the SE/NW diagonal lines through (-C/r_k,0) and (0,C/r_k), respectively. Let Θ_k^NE denote the closed region bounded by h_NE,L,k,h_NE,U,k,G_k^1 and G_k^3, and Θ_k^NW the closed region bounded by h_NW,L,k,h_NW,U,k,G_k^2 and G_k^4.Then let Θ_k=Θ_k^NW∪Θ_k^NE.In the event of interest to us, the nonrandom region Θ_k will contain the random region Ω_k.We now construct an event F_k, which we will show occurs for a positive fraction of all k, a.s.We show in Section <ref> that F_k ensures that all finite geodesics from 0 to points outside _k must stay near one of the axes until leaving _k^NW∩_k^NE.For m>c_q_k/(1+ζ) let R_k,m^* be the number of class-m zigzag highways in ω^(m) intersecting J_*,k in Θ_k, for *= N, E, S, W.Here ζ is from the definition of stage-2 deletions. Note that since the class of J_*,k is at least q_k, any intersecting highwaysof class m≤ c_q_k/(1+ζ) are removed in stage-2 deletions, so R_k,m^* counts those which might remain (depending on the class of J_*,k). Fix c∈ (0,C/2) and define the eventsI_k^*:={ c/r_k≤ X_i,k^* ≤C/r_k for i=1,2 },for *= NE, NW,I_k = I_k^NE∩ I_k^NW, M_k^NE:={8.5cm every SW/NE highway H∉{H_NE,L,k,H_NE,U,k} in ω intersecting Θ_k^NEis in classes 1,…,k-1}, and analogously forM_k^NW,and let M_k = M_k^NE∩ M_k^NW. Define also_k: |Y_*,k| ≤C/_q_k for*=E,N,W,S. E_1,k: ∑_m>c_q_k/(1+ζ) R_k,m^*η^m ≤ c_1(η^3θ^-2)^c_θδ k (μ^-1η)^k,for*=E,N,W,S.with c_1 to be specified and δ from (<ref>), noting that by the bound on η in (<ref>) we have η^3θ^-2<1, andE_2,k: there are no slow bonds in any J_*,k (*= N, E, S, W)nor in any H_*,·,k (*= NE, NW, ·= U, L).Finally, we set F_k = I_k ∩ M_k ∩_k ∩ E_1,k∩ E_2,k. Note that when F_k occurs we have _k⊂Θ_k.There exists some κ_1 such thatlim inf_n→∞1/n∑_k=1^n 1_F_k≥κ_1a.s. Moreover, letting n_1()<n_2() <⋯ be the indices for which ∈F_k, we have lim sup_j→∞n_j+1/n_j =1a.s. In the events I_k and M_k, the highways H_*,·,k and associated values X_i,k^* are taken from the configuration ω^ zig,thin,2.Using instead the configuration ω^ zig,thin,1 yields different events, which we denote I_k^1 and M_k^1 respectively.Let F_k^1 = I_k^1 ∩ M_k^1 ∩_k (noting we do not intersect with E_1,k, E_2,k here). First, we prove the following. There exists κ_1>0 such that lim inf_n→∞1/n∑_k=1^n 1_F_k^1≥κ_1a.s.As a first step, we show thatinf_k≥ k_0(I_k^1∩ M_k^1)>0.For a lower SW/NE zigzag highway H, let (x_0(H),0) be the intersection point in A_1 closest to (0,0) when one exists, and similarly for an upper SW/NE zigzag highway H let (0,y_0(H)) be the intersection point in A_2 closest to (0,0). We call H k-connecting if H intersects both G_k^1 and G_k^3. The event I_k^1∩ M_k^1 contains the eventA_k: 14.3cm there exists exactly one lower SW/NE highway H in ω of class k or more intersecting Θ_k^NE, and this H is k-connecting and satsifies cr_k≤ x_0(H)≤Cr_k-23; further, the analogous statement holds for upper SW/NE highways with y_0(·) in place of x_0(·), and for lower and upper SE/NW highways.The parts of A_k for the 4 types of zigzag highways (upper vs lower, SW/NE vs SE/NW) are independent, so to bound the probability of A_k we can consider just one of these parts and take the 4th power of the corresponding probability.In particular,considering lower SW/NE highways H of class ℓ≥ k intersecting Θ_k^NE, there are at most 2^ℓ+3 possible lengths for H, at most 2^ℓ+3C/r_k possible SW-most points, and the choice of H-start or V-start, so at most 2^2ℓ+7C/r_k possible highways.Also, considering lower SW/NE highways H of class m≥ k which are k-connecting and satisfy cr_k≤ x_0(H)≤Cr_k-23, there are at least 2^m+1(C-c)/r_k possible lower endpoints, and 2^m+1 possible lengths, for H. Therefore considering only lower highways, and since c<C/2 we have(I_k^1∩ M_k^1)^1/4≥(A_k)^1/4 ≥∑_m≥ k2^2m+2(C-c)/r_k θ^m/2^2m+4∏_ℓ≥ k( 1 - θ^ℓ/2^2ℓ+4)^2^2ℓ+7C/r_k≥( ∑_m≥ kCθ^m/8r_k) exp( -∑_ℓ≥ k16Cθ^ℓ/r_k) = C/8e^-16C,proving (<ref>).A similar but simpler proof also using a count of highways yields that inf_k≥ k_0(_k)>0, so by independence we haveinf_k≥ k_0(F_k^1)≥κ_0>0for some κ_0. But Claim <ref> is a stronger statement, and we now complete its proof, using (<ref>).Let _k denote the set of all SW/NE highways in ω which intersect Θ_k^NE, together with all SE/NW highways which intersect Θ_k^NW. Let _k denote the σ-field generated by _k, and let _k denote the largest j such that _k contains a highway of class j.For m≥ k,class-m SW/NE highways intersecting Θ_k^NE have at most (2C/r_k)(2^m+3+2^k+3) possible SW-most endpoints, and 2^m+3 possible lengths, and 2 directions for the initial step (V- or H-start), so the number of such highwaysis bounded by a sum of Bernoulli random variables, each with parameter (success probability) less than 1/2 and with total mean at most 2^m+42C/r_k(2^m+3+2^k+3)θ^m/2^2m+4≤32Cθ^m/r_k.The same is true for SE/NW highways. Hence for n≥ 0 the number of highways of class at least k+n is bounded by a similar sum of Bernoulli variables of total mean at most 32Cr_k+n/r_k.It follows that for any n≥ 0,(_k-k ≥ n)≤(some highway in ω of class k+n or more intersects Θ_k)≤32Cr_k+n/r_k = 32Cθ^n.Let n_0 be the least integer with θ^n_0<c/C. Define a random sequence of indices 1=K_1<K_2<… inductively as follows: having defined K_i, let K_i+1:=max{_K_i , (K_i+n_0) } +1. Here k≥ K_i+n_0 ensures C/r_K_i < c/r_k, and k>_K_i ensures that H_NE,L,k,H_NE,U,k do not intersect Θ_K_i^NE, and likewise for NW in place of NE. For any j,ℓ, and any A∈_j,the event A∩{K_i=j,_j=ℓ} only conditions zigzag highways in _j, and ensures that no SW/NE (or SE/NW) highways of class ≥ k = max(j+n_0,ℓ)+1 intersect Θ_j^NE (or Θ_j^NW, respectively); in particular it ensures that _i,k^*>C/r_j for *= NE, NW.Therefore since C/r_j < c/r_k this event increases the probability of I_k^1∩ M_k^1: for such j,ℓ,k,A,(I_k^1∩ M_k^1 | A∩{K_i=j,_j=ℓ} ) ≥( I_k^1∩ M_k^1).Similarly the bound in (<ref>) is still valid conditionally: for such j,ℓ,k,(_k-k ≥ n | A∩{K_i=j,_j=ℓ} ) ≤ 32Cθ^n for alln≥ 0.It follows that for some c_2>0,lim sup_i→∞K_i/i≤ c_2 a.s. On the other hand, it is straightforward that for some κ_2>0, for all j<k and all B∈σ(_1,…,_j),(_k | B ) ≥κ_2(_k)a.s.,so by independence of HV highways from zigzag ones, using (<ref>), (F_k^1 | A∩ B ∩{K_i=j,_j=ℓ} ) ≥κ_2(F_k^1) ≥κ_2κ_0.Since A,B are arbitrary, it follows that the variables 1_F_K_i dominate an independent Bernoulli sequence with parameter κ_2κ_0, solim inf_m→∞1/m∑_i=1^m 1_F_K_i^1≥κ_2κ_0 a.s.This, combined with (<ref>), proves Claim <ref> with κ_1=κ_2κ_0/c_2.Note that by (<ref>), the variables K_i+1-K_i, i≥ 1 are dominated by an i.i.d. sequence of the form “constant plus geometric random variable.”Let us go back to the proof of Lemma <ref>.Let B_k denote the event that none of the 4 highways H_*,·,k(ω^ zig,thin,1) are deleted in stage-2 deletions, then we haveI_k^1 ∩ M_k^1 ∩ B_k ⊂ I_k∩ M_k.We have∑_k (B_k^c∩ I_k^1) < ∞ ; ∑_k (E_2,k^c∩_k) < ∞ and ∑_k (E_1,k^c) < ∞. Let Z_*,·,k^H denote the class of the zigzag highway H_*,·,k(ω^ zig,thin,1), for *= NE, NW and ·= U, L, and let Z_*,k^J denote the class of the HV highway J_*,k for *= N, E, W, S.Then, by (<ref>), for n≥ 0(B_k^c | I_k^1∩{Z_NE,L,k^H = k+n} ) ≤8/1- 2^-ζ(k+n)≤8/1- 2^-ζ k.Since the upper bound is independent of n, summing over n we get that (B_k^c ∩ I_k^1)≤ 8 (1-)^-1 2^-ζ k, and the first item of Claim <ref> is proven.Similarly,(E_2,k^c |_k∩{Z_N,k^J = q_k+n} ) ≤ 2^q_k+n+34^-(q_k+n)≤ 8· 2^-q_k.so we get that ( E_2,k^c ∩_k ) ≤ 2^- q_k +3, and the second item of Claim <ref> is proven. For the last item, recall c_/(1+ζ) = c_-δ. We have from the upper bound for μ in (<ref>)∑_m>c_q_k/(1+ζ)θ^m-k-c_θδ mη^m ≤ c_1 (θ^1-c_θδη)^(c_θ (c_-δ)-1)kμ^k θ^-c_θδ k (μ^-1η)^k ≤ c_1 (η^3 θ^-2)^c_θδ k (μ^-1η)^k,so (E_1,k^c) ≤ 4∑_m>c_q_k/(1+ζ)( R_k,m^N > θ^m-k-c_θδ m) .Analogously to (<ref>) we have that (R_k,m^N) ≤ c_3θ^m /r_k and by Markov's inequality,(E_1,k^c) ≤ c_4 ∑_m>c_q_k/(1+ζ)θ^c_θδ m≤ c_5θ^c_6k.This proves the last item of Claim <ref>. Equation (<ref>) in Lemma <ref> now follows from (<ref>), Claims <ref> and <ref>, and the Borel-Cantelli lemma. Equation (<ref>) comes additionally from the remark made at the end of the proof of Claim <ref>. §.§ Properties of geodesics in case of a success For x∉_k, we let Γ_0x be the geodesic from 0 to x (unique since the ξ_e are continuous random variables.) For p,q∈Γ_0x we denote by Γ_0x[p,q] the segment of _0x from p to q. We let p_x be the first point of Γ_0x outside _k.We then define t_x in the boundary of the “near-rectangle” _k^NE∩_k^NW as below.Note that this boundary consists of 4 zigzag segments, one from each highway H_*,·,k (*= NE, NW; ·= U, L).Some boundary points (one bond or site at each “corner”) are contained in 2 such segments; we call these double points.Removing all double points leaves 4 connected components of the boundary, which we call disjoint sides of _k^NE∩_k^NW, each contained in a unique highway H_*,·,k. The set _k \ (_k^NE∩_k^NW) has 4 connected components, which we call arms, extending from _k^NE∩_k^NW in the directions NW, NE, SE, SW.Each arm includes one disjoint side of _k^NE∩_k^NW(recalling that the sets _k^* are open.)If p_x is not a double point then it is contained in the boundary of one of the arms, and we let t_x be the first point of _0x[0,p_x] in that arm (necessarily in a disjoint side; see Case 3 of Figure <ref>.) If instead p_x is a double point, then we pick arbitrarily one of the two highways H_*,·,k containing it, and let t_x be the first site of _0x[0,p_x] in that highway.For μ from (<ref>), let L_N,k denote the horizontal line ×{μ^-k}which, at least on the event F_k, lies above J_N,k and below G_k^1 (since 1/2 < μ < θ̃^c_θ); L_*,k is defined analogously for *= E, S, W.We define Λ_H,k to be the “horizontal axis corridor,” meaning the closure of the portion of _k^NE∩_k^NW strictly between L_S,k and L_N,k, and let Λ_V,k denote the similar “vertical axis corridor.”The primary part of establishing directedness in axis directions is showing that all semi-infinite geodesics remain in these corridors until they exit out the far end, at least for many k, via the following deterministic result.For sufficiently large k, when F_k occurs, for all x∉Ω_k we have either _0x[0, t_x]⊂Λ_H,k or _0x[0, t_x]⊂Λ_V,k.Let us start with a claim analogous to what we proved in Section <ref>. Recall the definitions of Γ_0x and p_x.When F_k occurs, for all x∉_k we have p_x ∉ G_k^1∪ G_k^2∪ G_k^3∪ G_k^4. Suppose p_x=(r,s) is in the horizontal part of G_k^1 (so r≥ s=2^k) and let U_k' be the upper endpoint of the bond which is the intersection of H_NE,L,k and the vertical line through p_x.Define the alternate path π_x from 0 east to U_k^NE, then NE along H_NE,L,k to U_k', then north to p_x.We now compare the passage times of _0x[0,p_x] versus π_x. We divide the bonds of the lattice into NW/SE diagonal rows: the jth diagonal row R_j consists of those bonds with one endpoint in {(x_1,x_2):x_1+x_2=j-1} and the other in {(x_1,x_2):x_1+x_2=j}. We call a bond e∈_0x[0,p_x] a first bond if for some j≥ 1, e is the first bond of _0x[0,p_x] in R_j, and we let N be the number of first bonds in _0x[0,p_x] which are in SE/NW highways.There are at least 2s first bonds, and any first bond e not in any SE/NW highway satisfies τ_e ≥ 0.7+0.1η^k-1, since from F_k⊂ M_k we have k(e)≤ k-1.From (<ref>), letting q=| _0x[0,p_x]|≥ 2s, at most q/24 SE/NW highways intersect _0x[0,p_x]. Since no two first bonds can be in the same SE/NW highway,we thus have N≤ q/24.Therefore using (<ref>),T( _0x[0,p_x])≥ (2s-N)(0.7+0.1η^k-1) + (q-2s+N)0.7 - 0.2 = 2s(0.7+0.1η^k) + 2s(0.1η^k-1-0.1η^k) + (q-2s)0.7 - 0.1Nη^k-1 - 0.2 ≥ 2s(0.7+0.1η^k) + 2s(0.1η^k-1-0.1η^k) + (q-2s)0.7 - q/240η^k-1 - 0.2.If q≥ 3s then q(0.7-η^k-1/240) ≥ 1.4s so from (<ref>) we get T( _0x[0,p_x]) ≥ 2s(0.7+0.1η^k) + 2s(0.1η^k-1-0.1η^k) - 0.2.If instead q<3s then since 1-η≥ 1/8 we have s(0.1η^k-1-0.1η^k) ≥ sη^k-1/80 ≥ qη^k-1/240, soT( _0x[0,p_x]) ≥ 2s(0.7+0.1η^k) + s(0.1η^k-1-0.1η^k) -0.2.By contrast, the NE segment of π_x has length 2(r-X_1,k^NE), the N segment has length X_1,k^NE-(r-s), and all bonds have passage times at most 1.4, so we have for large kT(π_x)≤ 1.4 X_1,k^NE + 1.4(X_1,k^NE - (r-s)) + 2(0.7+0.1η^k + 0.1^k)(r-X_1,k^NE) ≤ 2.8X_1,k^NE + 2s(0.7+0.1η^k +0.1 ^k) ≤ 2s(0.7+0.1η^k) + 4C/r_k, where we used that X_1,k^NE≤ C/r_k, and that ^k s = (2)^k ≤ C/r_k, using (<ref>). Therefore by (<ref>) we getT(_0x[0,p_x]) - T(π_x) ≥ 0.1· 2^k(η^k-1-η^k) - 4C/r_k - 0.2 > 0. This contradicts _0x[0,p_x] being a geodesic, so p_x cannot be in the horizontal part of G_k^1. All other cases are symmetric, so Claim <ref> is proved.We need to further restrict the location of _0x[0,p_x].When F_k occurs, we begin by dividing the sites of each H_*,·,k (with *= NW, NE and ·= L, U) into accessible and inaccessible sites In H_NE,L,k we define as inaccessible the sites strictly between its intersection with J_N,k and its intersection with J_W,k; the rest of the sites are accessible.Lattice symmetry yields the definition of accessible in the other 3 zigzag paths.This definition enables us to define canonical paths to reach accessible points, which we need below. Given a site a∈ J_N,k∩_k^NE and an accessible site b∈ H_NE,L,k in the first quadrant, there is a canonical path from a to b which follows J_N,k from a to H_NE,L,k, then (changing direction 45 degrees) follows H_NE,L,k to b.Similarly we can define canonical paths from sites a∈ J_S,k∩_k^NW to accessible b∈ H_NW,U,k in the 4th quadrant, with further extension by lattice symmetries. When two points a,b lie in the same horizontal or vertical line, we define the canonical path from a to b to be the one which follows that line. We say _0x[0,p_x] is returning if it contains a point of L_*,k, followed by a point of J_*,k, where both values *(N, E, W or S) are the same. We recall that by definition, from (<ref>) we have θ<μ<^c_θ which ensures that, when F_k occurs, the height of Λ_H,k is much less than its length, but much more than the height of J_N,k. Recall the definitions of p_x,t_x from the beginning of the section. If F_k occurs and Γ_0x[0,t_x]⊄Λ_H,k∪Λ_V,k, then either _0x[0,p_x] is returning, or at least one of t_x,p_x is accessible. Suppose F_k occurs, Γ_0x[0,t_x]⊄Λ_H,k∪Λ_V,k, and neither t_x nor p_x is accessible. We may assume p_x lies in H_NE,L,k on or above H_NW,U,k, as other cases are symmetric; then t_x∈ H_NW,U,k (or we may assume so, if p_x is a double point.)Since Γ_0x[0,t_x]⊄Λ_H,k∪Λ_V,k, Γ_0x[0,t_x] must intersect L_N,k∪ L_S,k;let q_x be the first such point of intersection.If q_x∈ L_S,k, then the fact that t_x is not accessible (it must lie above J_S,k) meansthat _0x[0,p_x] is returning. If q_x∈ L_N,k, then the fact that p_x is not accessible (hence lying below J_N,k)again means that _0x[0,p_x] is returning. This proves Claim <ref>.We now complete the proof of Lemma <ref>, by a contradiction argument. Assume that F_k occurs, but for some x∉_k we have Γ_0x[0,t_x] ⊄Λ_H,k∪Λ_V,k.As in the proof of Claim <ref>, we may assume p_x lies in H_NE,L,k on or above H_NW,U,k, and then that t_x∈ H_NW,U,k.Let q_x∈ L_N,k∪ L_S,k be as in the proof of Claim <ref>, and let a_x be the last point of Γ_0x[0,q_x] in J_*,k, with subscript *= N or S according as q_x∈ L_N,k or q_x∈ L_S,k. Thus _0x[0,p_x] follows a path 0→ a_x→ q_x→ t_x→ p_x. By Claim <ref>, we now have three cases:Case 1. _0x[0,p_x] is returning.In this case we define b_x to be the first point of Γ_0x[q_x,p_x] in the line J_*,k (*= N or S) containing a_x.Case 2. _0x[0,p_x] is not returning, and p_x is accessible (hence above J_N,k, so necessarily q_x ∈ L_N,k).In this case we define b_x=p_x.Case 3. _0x[0,p_x] is not returning, p_x is inaccessible (hence below J_N,k, so necessarily q_x ∈ L_S,k), and t_x is accessible. In this case we define b_x=t_x.In all three cases we compare passage times for _0x[a_x,b_x] to that of the canonical path from a_x to b_x, which we denote γ_x, and we obtain our contradiction by showing that T(γ_x) < T( _0x[a_x,b_x]),see Figure <ref>. Since E_2,k occurs, there are no slow bonds in γ_x, so to prove (<ref>) we may and do assume there are no slow bonds at all. Cases 2 and 3 are essentially symmetric, so we focus onCase 2 first, then the simpler Case 1.We write (in Case 2)a_x = (u,Y_N,k),q_x = (y,μ^-k),b_x = p_x= (r,s),and we let (d,Y_N,k) denote the left endpoint of the bond J_N,k∩ H_NE,L,k.The class of the highway J_N,k is at least q_k, so due to stage-2 deletions, the only zigzag highways intersecting J_N,k have class greater than c_q_k/(1+ζ). From (<ref>), (<ref>) and the definition of E_1,k we haveT(γ_x)≤ 0.9(d-u) + 0.4 + ∑_m>c_q_k/(1+ζ) R_k,mη^m + 2(s-Y_N,k)(0.7+0.1η^k) + 0.2 ≤ 0.9(d-u) + 2(s-Y_N,k)(0.7+0.1η^k) + 0.6 + c_1(η^3θ^-2)^c_θδ k (μ^-1η)^k.Our aim is to show that for some c_7,T( _0x[a_x,b_x]) ≥ 0.9(d-u) + 2(s-Y_N,k)(0.7+0.1η^k)+ c_7 (μ^-1η)^k,which with (<ref>) is sufficient to yield (<ref>) for all large k, since μ^-1η>1 by (<ref>) and η^3θ^-2<1 by (<ref>). The rest of the proof is devoted to showing (<ref>), by analyzing the type (and number of each type) of bonds that the path Γ_0x uses.We observe first that the intersection of a geodesic with any one zigzag highway is always connected, since we have assumed there are no slow bonds. A singleton bond in a path Γ is a zigzag bond in Γ which is preceded and followed in Γ by boundary bonds.We divide _0x[a_x,b_x] = (a_x=z_0,z_1,…,z_n=b_x) into the following types of segments:(i) zigzag segments: maximal subsegments which do not contain two consecutive non-zigzag bonds.A zigzag segment must start and end with a boundary bond, unless it starts at a_x or ends at b_x.(ii) intermediate segments: the segments in between two consecutive zigzag segments, the segment up to the first zigzag segment, and the segment after the final zigzag segment (any of which may be empty.)Within zigzag segments we find (iii) component segments: maximal subsegments contained in a single zigzag highway. For each component segment in _0x[a_x,b_x] that is not both an intersection bond and a singleton bond, there isa unique highway orientation (SW/NE or SE/NW) determined by the zigzag highway containing the segment.We define Φ(e) to be this orientation, for each bond e in the segment.For singleton bonds in Γ that are also intersection bonds, we assign Φ(e) arbitrarily. We say that a zigzag highway H intersects _0x[a_x,b_x] redundantly if the intersection is a single bond e and the orientation of H is not Φ(e).We let z_n_2j-2,z_n_2j-1 be the endpoints of the jth intermediate segment, 1≤ j≤ J+1, so β_j=_0x[z_n_2j-1,z_n_2j] is the jth zigzag segment, 1≤ j ≤ J. For any path Γ we defineT_α^*(Γ) = ∑_e∈Γα_e^*.For *= Z, B, H, V we define N_*(Γ) = | {e∈Γ: ehas property* }|, where subscripts and corresponding properties are as follows:Z:zigzag, B:not zigzag, H:horizontal,V:vertical,N:northward step in Γ (E, W, S similar).We may combine subscripts to require multiple properties, for example N_ZH(Γ) = | {e∈Γ: e is a horizontal zigzag bond}|, and we use superscripts NE or NW to restrict the count to zigzag bonds with Φ(e) = SW/NE or SE/NW, respectively. We also let N_Hi(Γ) be the number of zigzag highways intersecting Γ non-redundantly.Note that if for example a SW/NE highway segment is traversed by Γ in the NE direction, the number of N and E steps differs by at most 1. It follows that for every geodesic Γ we haveN_Hi(Γ) ≥ D(Γ) := |N_ZE^NE (Γ) - N_ZN^NE(Γ)|+ |N_ZW^NE(Γ) - N_ZS^NE(Γ)| + |N_ZE^NW(Γ) - N_ZS^NW(Γ)| + |N_ZW^NW(Γ) - N_ZN^NW(Γ)|.We now make some observations about geodesics and zigzag segments.We show that a zigzag segment β_j may contain multiple bonds of at most one zigzag highway (which we call primary, when it exists)—any zigzag bonds in β_j not in the primary highway are necessarily singleton bonds, and there are at most 2 of these; if β_j intersects 3 zigzag highways non-redundantly then the primary highway must lie between the two singletons. Moreover, any non-zigzag interior bond of some β_j must be an entry/exit bond. Indeed, due to the stage-3 trimming,in order for Γ to switch from one zigzag highway to another within β_j (with at least two bonds on each) there would need to be one of the following succession of steps (or some lattice rotation thereof): (i) N, E, E, E, S with the middle one being an exit/entry bond; (ii) N, E, E, S with the middle 2 being meeting zigzag bonds; (iii) N, E, S with the middle one being an intersection bond. We refer to Figure <ref> for a picture. But(since we are assuming no slow bonds) none of these patterns can occur in a geodesic, because omitting the N and S steps always produces a faster path. Then (<ref>) and the stage-3 trimming establish the remaining properties mentioned.For 2≤ j≤ J-1, if β_j contains a primary highway then the bonds of β_j, from the initial bond through the first bond of the primary highway, must follow one of the following patterns (we refer to figure <ref>): (i) skimming boundary, terminal(ii) boundary (not skimming), zigzag (not intersection)(iii) semislow boundary, intersection, meeting(iv) boundary (not skimming), meeting, meeting(v) boundary (not skimming), meeting, intersection (followed by meeting)(vi) boundary (not skimming), zigzag (not intersection), entry/exit, zigzag (not intersection).The same is true in reverse order at the opposite end of β_j. (Note that certain patterns cannot appear in a geodesic Γ, for example a semislow boundary bond with both endpoints in meeting bonds cannot be adjacent in Γ to either meeting bond, so these are not listed here.)If there is no primary highway then the full β_j follows one of the following patterns, or its reverse: (vii) boundary (not skimming), zigzag (not intersection), boundary (not skimming)(viii) skimming boundary, singly terminal, boundary (not skimming) (ix) skimming boundary, doubly terminal, skimming boundary (x) semislow boundary, intersection, semislow boundary(xi) normal boundary, zigzag (not intersection), entry/exit, zigzag (not intersection), normal boundary. It is readily checked from this that in all cases T_α^*(β_j)≥ 0.9N_B(β_j) + 0.7N_Z(β_j) + 0.2N_Hi(β_j),2≤ j≤ J-1.For j=1,J, β_j is a truncation of a path as in (i)–(xi), omitting a (possibly empty) segment of bonds at one end, and we similarly haveT_α^*(β_j) ≥ 0.9N_B(β_j) + 0.7N_Z(β_j) + 0.2N_Hi(β_j) - 0.2,j=1,J,and thereforeT_α^*( _0x[a_x,b_x])≥ 0.9N_B( _0x[a_x,b_x]) + 0.7N_Z( _0x[a_x,b_x]) + 0.2N_Hi( _0x[a_x,b_x]) - 0.4 .By (<ref>) and (<ref>) we haveT_α^*(_0x[a_x,b_x]) ≥ 0.9N_B(_0x[a_x,b_x]) + 0.7N_Z(_0x[a_x,b_x]) + 0.2D(_0x[a_x,b_x]) - 0.4,and from the definition of M_k^NE,T(_0x[a_x,b_x])≥ T_α^*(_0x[a_x,b_x]) + 0.1η^k-1( N_ZN^NE(_0x[a_x,b_x]) + N_ZE^NE(_0x[a_x,b_x]) ). In view of (<ref>) and (<ref>)-(<ref>), let us consider the question of minimizing0.9 (n_BE + n_BW + n_BN + n_BS)+ 0.7(n_ZN^NE + n_ZE^NE + n_ZW^NE + n_ZS^NE+ n_ZN^NW + n_ZE^NW + n_ZW^NW + n_ZS^NW)+ 0.2( |n_ZE^NE - n_ZN^NE|+ |n_ZW^NE - n_ZS^NE| + |n_ZE^NW - n_ZS^NW| + |n_ZW^NW - n_ZN^NW| )+ 0.1η^k-1(n_ZN^NE + n_ZE^NE) - 0.4subject to all 8 variables being nonnegative integers satisfyingn_ZE^NE + n_ZE^NW - n_ZW^NE - n_ZW^NW + n_BE - n_BW = (d-u)+(s-Y_N,k) = N_H(γ_x), n_ZN^NE - n_ZS^NW - n_ZS^NE + n_ZN^NW + n_BN - n_BS = s-Y_N,k = N_V(γ_x), n_ZN^NE + n_ZN^NW + n_BN = (s-Y_N,k) + g, n_ZS^NE + n_ZS^NW + n_BS = g, n_ZN^NW + n_BN = j,for some fixed g≥ 0 and 0≤ j≤ (s-Y_N,k)+g. Here (<ref>) is redundant but we include it for ready reference, and despite (<ref>) we formulate the problem with n_ZN^NW as a variable, to match the rest of the problem.Further, g may be viewed as an “overshoot”, the number of northward steps beyond the minimum needed to reach the height s of p_x, and j is the number of northward steps taken “inefficiently,” that is, not in NE/SW zigzag highways.Since q_x is at height μ^-k, we may restrict to s+g≥μ^-k, and thus from the definition of D̃_k, also tos-Y_N,k+g≥μ^-k - C/r̃_q_k≥μ^-k/2, the last inequality being valid for large k, following from the fact that r̃_q_k is a constant multiple of ^ -c_k while μ < ^ c_ by (<ref>).To study this we use the concept of shifting mass from one variable n_∙^* to a second one, by which we mean incrementing the second by 1 and the first by -1.We also use canceling mass between two variables in (<ref>) or two in (<ref>), one appearing with “+" and the other with “-", by which we mean decreasing each variable by 1.Shifting mass from n_BS to n_ZS^NE, or from n_BN to n_ZN^NW, does not increase (<ref>), so a minimum exists with n_BS=n_BN=0, so we may eliminate those two variables. Among the variables n_∙^* in (<ref>), if a variable with “+” and a variable with “-” are both nonzero, then canceling (unit) mass between them decreases (<ref>) by at least 1; this means that a minimum exists with all negative terms on the left in (<ref>) equal to 0.Then shifting mass in (<ref>) from n_ZS^NE to n_ZS^NW, or in (<ref>) from n_BE to n_ZE^NW, does not increase (<ref>) (since the preceding step has set n_ZW^NE and n_ZW^NW to 0), so there is a minimum with also n_ZS^NE=n_BE=0.With these variables set to 0, the problem becomes minimizing 0.7 (n_ZE^NE + n_ZE^NW + s-Y_N,k+2g ) + 0.2( |n_ZE^NE - (s-Y_N,k+g-j)| + |n_ZE^NW - g| + j)+ 0.1η^k-1(s-Y_N,k+g-j + n_ZE^NE) - 0.4subject to n_ZE^NE + n_ZE^NW = (d-u)+(s-Y_N,k).Setting n_ZE^NE=z, n_ZE^NW = (d-u)+(s-Y_N,k)-z and considering the effect of incrementing z by 1, we see that (<ref>) is minimized (not necessarily uniquely) at z= min((s-Y_N,k)+g-j,[(s-Y_N,k)+(d-u)-g]∨ 0).We now consider two cases.Case 2A. 2g-j≤ d-u.Here we have z=(s-Y_N,k)+g-j≥ 0 in (<ref>), and the corresponding minimum value of (<ref>) is 0.9 (d-u) + 0.7· 2(s-Y_N,k) + g+0.4j + 0.1η^k-1(2(s-Y_N,k)+2(g-j)) - 0.4≥ 0.9(d-u) + 0.7· 2(s-Y_N,k) +g+ 0.1η^k-1· 2(s-Y_N,k) - 0.4≥ 0.9(d-u) + (0.7 + 0.1 η^k-1) 2(s-Y_N,k)+ 0.1η^k(η^-1-1) · 2 (s-Y_N,k+g) - 0.4.For the first inequality, we used that 0.4 j≥0.2 η^k-1 j, and for the second one that g ≥ 0.2η^k(η^-1-1)g, for k sufficiently large.With (<ref>) and (<ref>) this shows thatT ( _0x[a_x,b_x]) ≥ 0.9(d-u) + (0.7 + 0.1η^k)· 2(s-Y_N,k) + 0.1 η^k(η^-1-1)μ^-k - 0.4.Since η>μ by (<ref>), this proves (<ref>).Case 2B. 2g-j>d-u, so that0.8g ≥ 0.4(d-u) + 0.1η^k-1 j.Here we have z=[(s-Y_N,k)+(d-u)-g]∨ 0 in (<ref>), and the corresponding minimum value of (<ref>) in the case z = (s-Y_N,k)+(d-u) -g>0 (the case z=0 being treated similarly) is 0.5 (d-u) + 0.7· 2(s-Y_N,k) + 1.8g + 0.1η^k-1((d-u) + 2(s-Y_N,k) - j) - 0.4 ≥ 0.9(d-u) + 0.7· 2(s-Y_N,k) + g+0.1η^k-1· 2(s-Y_N,k)- 0.4where the inequality follows from (<ref>). Then (<ref>) follows as in Case 2A. We now briefly explain the modifications of the above argument to treat Case 1. Analogously to (<ref>), we writea_x = (u,Y_N,k),q_x = (y,μ^-k),b_x = (v,Y_N,k),and we may assume v-u>0, since otherwise we can consider the path _0x[a_x,b_x] running backwards.Similarly to (<ref>), we haveT(γ_x)≤ 0.9(v-u) + 0.5 + ∑_m>c_q_k/(1+ζ) R_k,mη^m ≤ 0.9(v-u) + 0.5 + c_1(η^3θ^-2)^c_θδ k (μ^-1η)^k.and similarly to (<ref>) we want to showT(_0x[a_x,b_x]) ≥ 0.9(v-u) + c_7 (μ^-1η)^kin order to obtain (<ref>). Then, in the proof above, we replace d-u with v-u and s-Y_N,k with 0 in the constraints (<ref>)–(<ref>) and in (<ref>). Otherwise the proof remains the same, establishing (<ref>). In the end, in all the Cases 1–3 we have the contradiction (<ref>), and Lemma <ref> is proven.§.§ Conclusion of the proof of Theorem <ref>Item (ii). This follows from the combination of Lemma <ref> and Lemma <ref>. For a configuration ω let n_1(ω)<n_2(ω)<… be the indices k for which ω∈ F_k. Let Γ_0 be an infinite geodesic starting from the origin, with sites 0=x_0,x_1,….Then Lemma <ref> says that for each j≥ 1, Γ_0 is contained in either Λ_H,n_j or Λ_V,n_j until it leaves _n_j^NE∩_n_j^NW; accordingly, we say Γ_0 is horizontal at stage j or vertical at stage j.Suppose Γ_0 is horizontal at stage j, and vertical at stage j+1. The horizontal coordinate of the first point of Γ_0 outside _n_j^NE∩_n_j^NW then has magnitude at least c/r_n_j - μ^-n_j but at most μ^-n_j+1(since Γ_0 is vertical at stage j+1.)Since θ < μ by (<ref>), this means that for large j we have c/(2r_n_j) ≤μ^-n_j+1.Choosing >0 small enough so that μ^1+>θ, thanks to (<ref>) we get that for large j, n_j+1≤ (1+) n_j, so that(μ^1+/θ)^n_j≤μ^n_j+1/r_n_j≤2/c.But this can only be true for finitely many j. Hence there exists a random J_0 such that for j≥ J_0, either Γ_0 is horizontal at stage j for all j≥ J_0, or Γ_0 is vertical at stage j for all j≥ J_0. (We call Γ_0 horizontal or vertical, accordingly.) Using again that n_j+1/n_j → 1, we get thatμ^-n_j+1≪c/ r_n_jasj→∞,that is, the width of Λ_*,n_j+1 is much less than the length of Λ_*,n_j, for *= H, V. This guarantees that for such Γ_0, the angle to x_i from an axis approaches 0, that is, Γ_0 is directed in an axis direction, proving Theorem <ref>(ii). The above reasoning gives that geodesics reaching horizontal distance n= c/r_k= c' θ^-k deviate from the horizontal axis by at most μ^-k. In the other direction, heuristically, in order to reach horizontal distance n = c/r_k, the most efficient way should involve a route going as soon as possible (through a succession of horizontal and zigzag highways) to the closest horizontal highway that reaches at least distance c/r_k, and then following that highway. This suggests that in order to reach distance n=c/r_k, geodesics have a transversal fluctuation of order at least 1/r̃_q_k = c θ̃^- c_θk=cθ^-c_k, or equivalently order n^c_, as this is the typical vertical distance to the closest horizontal highway reaching n. Suppose c_>10/11.Then choosing c_θ slightly less than 0.5, then η slightly less than 2/3, and then δ sufficiently small, we satisfy (<ref>), (<ref>), and the conditions preceding them, and further, ^c_θ is the smallest of the 3 quantities on the right side of (<ref>).This means we can choose μ arbitrarily close to ^c_θ=θ^c_.Thus our upper bound of μ^-k becomes n^c_ +o(1), nearly matching the heuristic lower bound. Hence in this case we expect geodesics reaching horizontal distance n to have transversal fluctuations of order n^c_ +o(1), with at least all values c_∈ (10/11,1) being possible. Item (i).It follows readily from an upper bound in the same style as the lower bound (<ref>) that for ϵ>0, (X_i,k^∗≥ϵ 2^k) is summable over k, so that X_i,k^∗ =o(2^k) a.s. (meaning there are long diagonal highways close to the origin), and thereforethe asymptotic speed in any diagonal direction is √(2)/1.4, and similarly along an axis it is 1/0.9.Observe that (<ref>) in the proof of Lemma <ref> is valid for all geodesics, as that is the only property of Γ_0x(a_x,b_x) that is used.Consider then the geodesic from (0,0) (in place of a_x) to some point (r,s) with 0≤ s≤ r (in place of b_x.)It is easy to see that the right side of (<ref>) is not increased if we replace the geodesic with a path of 2s consecutive zigzag bonds (heading NE) and r-s horizontal bonds heading east—effectively this means consolidating all zigzag bonds into a single highway. It follows that the passage time is τ((0,0),(r,s)) ≥(.9(r-s) + 1.4s)(1+o(1)) asb→∞.From the fact that there are, with high probability, long HV and zigzag highways close to any given point (reflected in the fact that X_*,k=o(2^k) and Y_*,k=o(2^k)), we readily obtain the reverse inequality.The linearity of the asymptotic expression .9(r-s) + 1.4s means that the limit shape is flat between any diagonal and an adjacent axis. It follows that the limit shapeis an octogon, with vertex (1/.9,0) on the horizontal axis, (√(2)/1.4,√(2)/1.4) on the SW/NE diagonal, and symmetrically in other quadrants, proving Theorem <ref>(i).Item (iii). Let F be the facet of B in the first quadrant between the horizontal axis and the main diagonal,and let ρ_F be the linear functional equal to 1 on F.We have from Theorem 1.11 in <cit.> that there is a semi-infinite geodesic Γ_F with (Γ_F) ⊂{v/|v|:v∈ F}, and Theorem 4.3, Corollary 4.7 and Proposition 5.1 of <cit.> show that Γ_F has Busemann function linear to ρ_F.But all geodesics are directed in axis directions so we must have (Γ_F)={(1,0)}. Letbe the facet which is the mirror image of F across the horizontal axis.From lattice symmetry, we have (Γ_)={(1,0)} and its Busemann function is linear to ρ_.Since the Busemann functions differ, we must have Γ_F≠Γ_. This and lattice symmetry prove Theorem <ref>(iii).99AH16 Ahlberg, D. and Hoffman, C., Random coalescing geodesics in first-passage percolation, (2016), arXiv:1609.02447 [math.PR]Bo90 Boivin, D. First passage percolation:the stationary case, Probab. Theory Related Fields 86 (1990), no. 4, pp. 491–499.BH17 Brito, G. and Hoffman, C., personal communication.DH14 Damron, M. and Hanson, J., Busemann functions and infinite geodesics in two-dimensional first-passage percolation, Comm. Math. Phys. 325 (2014), no. 3, pp. 917–963.DH17 Damron, M. and Hanson, J., Bigeodesics in first-passage percolation, Comm. Math. Phys. 349 (2017), no. 2, pp. 753–756.HM95 Häggström, O. and Meester, R., Asymptotic shapes for stationary first passage percolation, Ann. Probab. 23 (1995), no. 4, pp. 1511–1522.Ho08 Hoffman, C., Geodesics in first passage percolation, Ann. Appl. Probab. 18, (2008), no. 5, pp. 1944–1969. | http://arxiv.org/abs/1709.09072v2 | {
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Quasi-random Monte Carlo application in CGE systematic sensitivity analysis Theodoros Chatzivasileiadis Institute for Environmental Studies, Vrije Universiteit Amsterdam, The NetherlandsContact T. Chatzivasileiadis, Email: [email protected], Address: De Boelelaan 1087, 1081 HV Amsterdam, The Netherlands.=================================================================================================================================================================================================================================================== The uncertainty and robustness of Computable General Equilibrium models can be assessed by conducting a Systematic Sensitivity Analysis. Different methods have been used in the literature for SSA of CGE models such as Gaussian Quadrature and Monte Carlo methods. This paper explores the use of Quasi-random Monte Carlo methods based on the Halton and Sobol' sequences as means to improve the efficiency over regular Monte Carlo SSA, thus reducing the computational requirements of the SSA. The findings suggest that by using low-discrepancy sequences, the number of simulations required by the regular MC SSA methods can be notably reduced, hence lowering the computational time required for SSA of CGE models.§ INTRODUCTIONThe use of Systematic Sensitivity Analysis (SSA) in the context of Computable General Equilibrium (CGE) models is gaining a new momentum within the CGE literature. SSA sought to address the input and calibration-data uncertainty of the models in order to capture the uncertainties surrounding CGE models and their results. This paper discusses the use of quasi-random Monte Carlo methods in SSA of CGE models.We begin by defining the general form of a computable general equilibrium model as:G(x,β)=0The most common approach of SSA in CGE models is based on Gaussian Quadrature (GQ) design. That choice is based on the fact that GQ requires only a few data points to approximate the central moments of stochastic variables, making this method very computational inexpensive. An alternative to the GQ is the regular Monte Carlo (rMC) SSA that requires a large amount of realisations such that: the mean (x̅) in the univariate case as defined by equation <ref>, is an unbiased estimator of K(β), I=E[K(β)]=∫_a^b K(β)p(β)dβ =x̅I_N=∑_n=1^Nw_nK(β_n) where w_n=1/N for each realization n, x (eq. <ref>) represents a vector of results or endogenous variables (such as prices, welfare etc.), β (eq. <ref>) is a vector of exogenous variables, x^⋆(β) ≡ K(β) is a vector of results for each given parameter β and p is the non-zero probability density function (pdf).The difference between QG and rMC is in the way w_n is defined. While in rMC all realizations have equal weight 1/N, in QG we choose the most appropriate points within the interval [a, b] and associated weights w_n, such that, the crude moments of the approximating distribution equals the moments of the true distribution from zero to some specified order <cit.>. Thus the GQ method is able to economise the computational requirements of the SSA (i.e small number of simulations are required).Economising the SSA through GQ is not without its drawbacks. For example, a significant amount of information is lost regarding the shape of the distribution, its higher-order moments, and its range. This information might be important when the shocks applied to the CGE model are expected to have asymmetric impacts[Additional information about the differences between GQ and rMC can be found in <cit.> and <cit.>.]. By using rMC SSA instead, we sample from the total distribution of inputs, thus avoiding the GQ drawbacks at the cost of computation time. Moreover, the number of points required by the GQ is based on the shocks evaluated and the degree of the quadratures. For example in the case of <cit.>, 4096 simulations were required by the Liu quadrature based on the GEMPACK software. Thus, in that case, the advantage of GQ that economises the SSA is questioned. '[...]An advantage of the rMC method, compared to GQ, lies on the estimation of the error. In the rMC method, the error is estimated from the generated data, whereas in the QG more global measures of error estimation are required such as the Chebyshev's inequality for the confidence bounds. The Chebyshev's inequality, will produce confidence bounds that are extremely conservative compared to the Central Limit Theorem which provides narrower confidence intervals if the available number of data points is sufficiently large[...]'<cit.>. From the above, the conundrum choosing between; economising the SSA with GQ and gaining the advantages of rMC SSA is apparent. This paper discusses a way to keep the advantages of the rMC SSA methodology while minimising the number of simulations required. We contribute to the existing literature by applying a Quasi-Monte Carlo (QMC) SSA on a static CGE model in order to economise the SSA without loosing important information as in the GQ SSA, in a computationally inexpensive way, in contrast to the rMC SSA.§ METHODS AND DATA §.§ rMC and QMCBased on the model of <cit.> and the input SSA discussed in <cit.>, we explore the use of quasi-random realisations in the MC SSA of CGE models. We begin with a brief description of the differences between rMC and QMC SSA.Going back to equation <ref>, according to the Strong Law of Large Numbers for the rMC method, the approximation is convergent with probability one, i.e: lim_N→∞ I_N→ I Then the error of the MC integration is:ϵ_N=I-I_N≈σ N^-1/2 xif N is sufficiently large based on the Central Limit Theorem, where σ is √(Var(x))=[∫_a^b (K(β)-E[K(β)])^2p(β)dβ]^1/2 Based on the above, the convergence rate of the rMC is O(N^-1/2) which is independent of dimensions.In the case of rMC, we randomly select points and proximate equation <ref> by the empirical average in equation <ref>. In QMC the points are selected semi-deterministically, such that the chosen points provide the best possible spread. The chosen points are highly equidistributed thus providing greater uniformity compared to the pseudo-random numbers. In QMC, the resulting convergence rate is O((log N)^k N^-1), which is dependent on the k dimensions[See <cit.> for more information.]. As a result, for the same number of evaluations, the QMC method, using low-discrepancy sequences, achieves higher accuracy thus faster convergence. This characteristic of the QMC method is very appealing in our case. We are interested in faster convergence, thus smaller number of simulations required to conduct a SSA of our CGE model.One issue with QMC is that since the convergence is dependent on the dimensions of the problem, improved accuracy is lost in problems of high dimension. Another problem is due to existing correlations between the points of the quasi-random sequence used. The latter problem can be resolved by skip, leap over, or scramble the values in the sequence. §.§ SSA input generationIn our analysis we have used two different low-discrepancy sequences: the Halton; and Sobol' sequences <cit.>. The Halton sequence is generated by using different prime bases to generate the sequence, where for the Sobol' a base of 2 is used with a reordering of the coordinates in each dimension. In the Halton sequence we skip 1,000 and leap 100 using RR2 scrambler. The Sobol' sequence is generated using leap 10,000 and leap 100 using the MatousekAffineOwen scrambler.Based on <cit.>, we have 13 regions and a total of 39 shocks applied to the CGE model. The process initially used for the rMC is the following: * Generate 39 pseudo-random samples (10,000 realisations) based on the uniform distribution; * Transform to the triangular distribution given the minimum, median and maximum for each regional changes of Land, Capital and Productivity; * Run 10,000 simulations based on these inputs. In the QMC method the only difference is in step 1. Instead of the 39 pseudo-random samples we generate 39 quasi-random sequences based on the Halton and Sobol' sequences. Steps 2 and 3 are followed as described above.§ RESULTSFor simplicity, we focus on one result of the CGE model; the Hicksian Equivalent Variation (HEV) for two of the regions only: Central-Asia and North-America. The choice of the regional results is arbitrary since all regions show similar differences between the rMC and the two QMC SSA methods. First we discuss the differences between the shocks' distributions. Here we only present the Halton sequence-based inputs, since we saw no substantial difference in the histogram with the Sobol' generated inputs. Looking at both figures <ref> and <ref> for Land, Capital and Productivity shocks, we see the expected; the QMC shocks are evenly distributed without any clustering (i.e spikes in the histogram as around -20 for Central Asia's productivity in figure <ref>) compared to the rMC shocks' histograms.In Central Asia (figure <ref>) the mean is stable after approximately 4,000 simulations which are already 96 less than the QG method required as discussed above. In the Halton-based SSA results, we find that the mean converges to the rMC 10,000 runs mean (black line) after only 500 simulations and the confidence interval (CI) is similar at 2,000 simulations with the CI of 4,000 simulations in the rMC. Looking at the Sobol' QMC results, even though the mean converges at around 800 simulations, the Sobol'-based QMC is outperformed by the Halton QMC method, but is preferred from the rMC SSA. The same conclusion can be derived by looking at figure <ref> of the results' standard deviation (SD). The SD is more stable in the Halton QMC method after 500 simulations compared to the 1,500 required in the Sobol' QMC. Clearly though, both quasi-random methods produce significantly lower SD estimates compared to the rMC results. In North America (figure <ref>), the superiority of the Halton-based method is more prominent. Clearly, the Halton-based results converge after 500 simulations, but the Sobol'-based results have not converged yet, before 2,000 simulations. Nevertheless, these results are superior to the 4,000 simulations required by the rMC method. The North American SD results (figure <ref>) indicate that the SD reaches a plateau after 500 simulations in the Halton and 1,000 in the Sobol'-based results and that plateau is smaller than the rMC one.§ CONCLUSIONIn this paper, we discuss the use of Quasi-Monte Carlo methods based on the Halton and Sobol' sequences for systematic sensitivity analysis of CGE model. Our main interest is to provide evidence of efficiency gains with using QMC in SSA compared to rMC. Based on our empirical results, both Halton and Sobol' based QMC methods are more efficient than the rMC method, but the Halton-based results are preferred. This is based mainly on the convergence of the results. Our results indicate that there is a clear computational-time advantage in the use of low-discrepancy sequence compared to pseudo-random numbers and this method should be considered when conducting a SSA of a CGE model. In this paper we have only used two quasi-random sequences, but it would be interesting in the future to compare these results with other quasi-random sequences such as: the Faure sequence; the Hammersley set; and the Niederreiter sequence.§ REFERENCES5 [Caflisch(1998)]caflisch Caflisch, Russel E. 1998. “Monte carlo and quasi-monte carlo methods.” Acta numerica 7: 1–49. [Chatzivasileiadis et al.(2017)]TC2 Chatzivasileiadis, Theodoros, Francisco Estrada, Marjan Hofkes, and Richard Tol. 2017. Systematic sensitivity analysis of the full economic impacts of sea level rise. Under review. [Chatzivasileiadis et al.(2016)]TC1 Chatzivasileiadis, Theodoros, Marjan Hofkes, Onno Kuik, and Richard Tol. 2016. Full economic impacts of sea level rise: loss of productive resources and transport disruptions. Working paper series. Department of Economics, University of Sussex. <http://EconPapers.repec.org/RePEc:sus:susewp:09916>. [Jank(2005)]jank Jank, Wolfgang. 2005. “Quasi-Monte Carlo sampling to improve the efficiency of Monte Carlo EM.” Computational statistics & data analysis 48 (4): 685–701. <http://www.sciencedirect.com/science/article/pii/S0167947304001033>. [Villoria, Preckel et al.(2017)]VP Villoria, Nelson B, Paul V Preckel, et al. 2017. “Gaussian Quadratures vs. Monte Carlo Experiments for Systematic Sensitivity Analysis of Computable General Equilibrium Model Results.” Economics Bulletin 37 (1): 480–487. § FIGURES | http://arxiv.org/abs/1709.09755v1 | {
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0cm 0cm 0cm 16cm defiDefinitionthm[defi]Theorem ex[defi]Example rem[defi]Remarkprop[defi]Proposition lemme[defi]Lemma cor[defi]Corollary | http://arxiv.org/abs/1709.09506v1 | {
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Quasi-random Monte Carlo application in CGE systematic sensitivity analysis Theodoros Chatzivasileiadis Institute for Environmental Studies, Vrije Universiteit Amsterdam, The NetherlandsContact T. Chatzivasileiadis, Email: [email protected], Address: De Boelelaan 1087, 1081 HV Amsterdam, The Netherlands.=================================================================================================================================================================================================================================================== Previous studies have demonstrated the empirical success of word embeddings in various applications. In this paper, we investigate the problem of learning distributed representations for text documents which many machine learning algorithms take as input for a number of NLP tasks.We propose a neural network model, /, which learns document representations with the goal of preserving key semantics of the input text. It enables the learned low-dimensional vectors to retain the topics and important information from the documents that will flow to downstream tasks. Our empirical evaluations show the superior quality of / representations in two different document understanding tasks. § INTRODUCTION In recent years, the use of word representations, such as word2vec <cit.> and GloVe <cit.>, has become a key “secret sauce” for the success of many natural language processing (NLP), information retrieval (IR) and machine learning (ML) tasks. The empirical success of word embeddings raises an interesting research question: Beyond words, can we learn fixed-length distributed representations for pieces of texts? The texts can be of variable-length, ranging from paragraphs to documents. Such document representations play a vital role in a large number of downstream NLP/IR/ML applications, such as text clustering, sentiment analysis, and document retrieval, which treat each piece of text as an instance. Learning a good representation that captures the semantics of each document is thus essential for the success of such applications.In this paper, we introduce /, a neural network model that learns densely distributed representations for documents of variable-length. In order to capture semantics, the document representations are trained and optimized in a way to recover key information of the documents. In particular, given a document, the / model constructs a fixed-length vector to be able to predict both salient sentences and key words in the document. In this way, / conquers the problem of prior embedding models which treat every word and every sentence equally, failing to identify the key information that a document conveys. As a result, the vectorial representations generated by / can naturally capture the topics of the documents, and thus should yield good performance in downstream tasks.We evaluate our / on two text understanding tasks: document retrieval and document clustering. As shown in the experimental section <ref>, / yields generic document representations that perform better than state-of-the-art embedding models.§ RELATED WORKLe et al. proposed a Paragraph Vector model, which extends word2vec to vectorial representations for text paragraphs <cit.>. It projects both words and paragraphs into a single vector space by appending paragraph-specific vectors to typical word2vec. Different from our /, Paragraph Vector does not specifically model key information of a given piece of text, while capturing its sequential information. In addition, Paragraph Vector requires extra iterative inference to generate embeddings for unseen paragraphs, whereas our / embeds new documents simply via a single feed-forward run.In another recent work <cit.>, Djuric et al. introduced a Hierarchical Document Vector (HDV) model to learn representations from a document stream. Our / differs from HDV in that we do not assume the existence of a document stream and HDV does not model sentences.§ / MODEL Given a document D consisting of N sentences { s_1, s_2, …, s_N }, our / model aims to learn a fixed-length vectorial representation of D, denoted as /. Figure <ref> illustrates an overview of the / model consisting of two cascaded neural network components: a Neural Reader and a Neural Encoder, as described below. §.§ Neural Reader The Neural Reader learns to understand the topics of every given input document with paying attention to the salient sentences. It computes a dense representation for each sentence in the given document, and derives its probability of being a salient sentence. The identified set of salient sentences, together with the derived probabilities, will be used by the Neural Encoder to generate a document-level embedding.Since the Reader operates in embedding space, we first represent discrete words in each sentence by their word embeddings. The sentence encoder in Reader then derives sentence embeddings from the word representations to capture the semantics of each sentence. After that, a Recurrent Neural Network (RNN) is employed to derive document-level semantics by consolidating constituent sentence embeddings. Finally, we identify key sentences in every document by computing the probability of each sentence being salient.§.§.§ Sentence EncoderSpecifically, for the i-th sentence s_i={ w_i1, w_i2, …, w_iM} with M words, Neural Reader maps each word w_im into a word embedding 𝐰_im∈ℝ^d_𝒲. Pre-trained word embeddings like word2vec or GloVe may be used to initialize the embedding table. In our experiments, we use domain-specific word embeddings trained by word2vec on our corpus.Given the set of word embeddings for each sentence, Neural Reader then derives sentence-level embeddings 𝐬_𝐢 using a sentence encoder g(·):𝐬_i = g(𝐰_i1, 𝐰_i2, …, 𝐰_iM),where g(·) is implemented by a Convolutional Neural Network (CNN) with a max-pooling operation, in a way similar to <cit.>. Note that other modeling choices, such as an RNN, are possible as well. We used a CNN here because of its simplicity and high efficiency when running on GPUs. The sentence encoder generates an embedding 𝐬_i of 150 dimensions for each sentence.§.§.§ Identifying Salient SentencesGiven the embeddings of sentences {𝐬_1, 𝐬_2, …, 𝐬_N } in a document d, Neural Reader computes the probability of each sentence s_i being a key sentence, denoted as p(s_i|d).We employ a Long Short-Term Memory (LSTM) <cit.> to compose constituent sentence embeddings into a document representation. At the i-th time step, LSTM takes as input the current sentence embedding 𝐬_i, and computes a hidden state 𝐡_i. We place an LSTM in both directions, and concatenate the outputs of the two LSTMs. For the i-th sentence, 𝐡_i is semantically richer than sentence embedding 𝐬_i, as 𝐡_i incorporates the context information from surrounding sentences to model the temporal interactions between sentences. The probability of sentence s_i being a key sentence then follows a logistic sigmoid of a linear function of 𝐡_i:p(s_i|d) = σ (𝐰_l^⊤𝐡_i+b_l),where 𝐰_l∈ℝ^|𝐡_i| is a trainable weight vector, and b_l∈ℝ is a trainable bias scalar. §.§ Neural EncoderThe Neural Encoder computes document-level embeddings based on the salient sentences identified by the Reader. In order to capture the topics of a document and the importance of its individual sentences, we perform a weighted pooling over the constituent sentences, with the weights specified by p(s_i|d), which gives the document-level embedding 𝐝 through a tanh transformation:𝐝 = tanh (𝐖_d(∑_i=1^N p(s_i|d)/∑_j=1^N p(s_j|d)·𝐡_i)+𝐛_d),where 𝐖_d∈ℝ^|𝐝|× |𝐡| is a trainable weight matrix, and 𝐛_d∈ℝ^|𝐝| is a trainable bias vector.Weighted pooling functions are commonly used as the attention mechanism <cit.> in neural sequence learning tasks. The “share” each sentence contributes to the final embedding is proportional to its probability of being a salient sentence. As a result, 𝐝 will be dominated by salient sentences with high p(s_i|d), which preserves the key information in a document, and thus allows long documents to be encoded and embedded semantically.§ MODEL LEARNINGIn this section, we describe the learning process of the parameters of /. Similarly to most neural network models, / can be trained using Stochastic Gradient Descent (SGD), where the Neural Reader and Neural Encoder are jointly optimized. In particular, the parameters of Reader and Encoder are learned simultaneously by maximizing the joint likelihood of the two components:ℒ = ℒ_ read + ℒ_ enc,where ℒ_ read and ℒ_ enc denotes the log likelihood functions of Reader and Encoder, respectively. §.§ Reader's Objective: ℒ_ readTo optimize Reader, we take a surrogate approach to heuristically generate a set of salient sentences from a document collection, which constitute a training dataset for learning the probabilities of salient sentences p(s_i|d, θ) parametrized by θ. More specifically, given a training set 𝒟 of documents (e.g., body-text of research papers) and their associated summaries (e.g., abstracts) {⟨ d_k, S_k ⟩}_k=1^|𝒟|, where S_k is a gold summary of document d_k, we employ a state-of-the-art sentence similarity model, DSSM <cit.>, to find the set of top-K[K=10 in our experiments] sentences S^*_k={ s'_i } in d_k, such that the similarity between s'_i ∈ S^*_k and any sentence in the gold summary S_k is above a pre-defined threshold. Note that here we assume each training document is associated with a gold summary composed of sentences that might not come from d_k. We make this assumption only for the sake of generating the set of salient sentences S^*_k which is usually not readily available.The log likelihood objective of the Neural Reader is then given by maximizing the probability of S^*_k being the set of key sentences, denoted as p(S^*_k|d_k):ℒ_ read= ∑_k=1^|𝒟|log p(S^*_k|d_k) = ∑_k=1^|𝒟|log( ∏_s_i ∈ S^*_k p(s_i|d_k) ·∏_s_i ∈ d_k\ S^*_k(1 - p(s_i|d_k) ) ) = ∑_k=1^|𝒟|( ∑_s_i ∈ S^*_klog p(s_i|d_k)+∑_s_i ∈ d_k\ S^*_klog(1 - p(s_i|d_k) ) ),where d_k\ S^*_k is the set of non-key sentences. Intuitively, this likelihood function gives the probability of each sentence in the generated key sentence set S^*_k being a key sentence, and the rest of sentences being non-key ones. §.§ Encoder's Objective: ℒ_ enc The final output of Encoder is a document embedding 𝐝, derived from LSTM's hidden states {𝐡} of Reader. Given our goal of developing a general-purpose model for embedding documents, we would like 𝐝 to be semantically rich to encode as much key information as possible. To this end, we impose an additional objective on Encoder: the final document embedding needs to be able to reproduce the key words in the document, as illustrated in Figure <ref>.In document d_k, the set of key words W_k is composed of top 30 words in S_k (i.e., the gold summary of d_k) with the highest TF-IDF scores. Encoder's objective is then formalized by maximizing the probability of predicting the key words in W_k using the document embedding 𝐝_k:ℒ_ enc = ∑_k=1^|𝒟|∑_w ∈ W_klog p(w ∈ W_k|𝐝_k),where p(w ∈ W_k|𝐝_k) is implemented as a softmax function with output dimensionality being the size of the vocabulary.Combining the objectives of Reader and Encoder yields the joint objective function in Eq (<ref>). By jointly optimizing the two objectives with SGD, the / model is capable of learning to identify salient sentences from input documents, and thus generating semantically rich document-level embeddings.§ EXPERIMENTS AND RESULTS To verify the effectiveness, we evaluate the / model on two text understanding tasks that take continuous distributed vectors as the representations for documents: document retrieval and document clustering. §.§ Document RetrievalThe goal of the document retrieval task is to decide if a document should be retrieved given a query. In the experiments, our document pool contained 669 academic papers published by IEEE, from which top-k relevant papers are retrieved. We created 70 search queries, each composed of the text in a Wikipedia page on a field of study (e.g., <https://en.wikipedia.org/wiki/Deep_learning>). We retrieved relevant papers based on cosine similarity between document embeddings of 100 dimensions for Wikipedia pages and academic papers. For each query, a good document-embedding model should lead to a list of academic papers in one of the 70 fields of study. Table <ref> presents P@10, MAP and MRR results of our / model and competing embedding methods in academic paper retrieval. word2vec averaging generates an embedding for a document by averaging the word2vec vectors of its constituent words. In the experiment, we used two different versions of word2vec: one from public release, and the other one trained specifically on our own academic corpus (113 GB). From Table <ref>, we observe that as a document-embedding model, Paragraph Vector gave better retrieval results than word2vec averagings did. In contrast, our / outperforms all the competitors given its unique capability of capturing and embedding the key information of documents. §.§ Document ClusteringIn the document clustering task, we aim to cluster the academic papers by the venues in which they are published. There are a total of 850 academic papers, and 186 associated venues which are used as ground-truth for evaluation. Each academic paper is represented as a vector of 100 dimensions.To compare embedding methods in academic paper clustering, we calculate F1, V-measure (a conditional entropy-based clustering measure <cit.>), and ARI (Adjusted Rand index <cit.>). As shown in Table <ref>, similarly to document retrieval, Paragraph Vector performed better than word2vec averagings in clustering documents, while our / consistently performed the best among all the compared methods. § CONCLUSIONS In this work, we present a neural network model, /, that learns continuous representations for text documents in which key semantic patterns are retained.In the future, we plan to employ the Minimum Risk Training scheme to train Neural Reader directly on original summary, without needing to resort to a sentence similarity model.emnlp_natbib | http://arxiv.org/abs/1709.09749v1 | {
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Department of Physics, McGill University, Montréal, Québec, Canada.Institute for Molecular Engineering, University of Chicago, Chicago, Illinois, USA.Institute for Molecular Engineering, University of Chicago, Chicago, Illinois, USA.We present and analyze a method where parametric (two-photon) driving of a cavity is used to exponentially enhance the light-matter coupling in a generic cavity QED setup, with time-dependent control.Our method allows one to enhance weak-coupling systems, such that they enter the strong coupling regime (where the coupling exceeds dissipative rates) and even the ultra-strong coupling regime (where the coupling is comparable to the cavity frequency).As an example, we show how the scheme allows one to use a weak-coupling system to adiabatically prepare the highly entangled ground state of the ultra-strong coupling system.The resulting state could be used for remote entanglement applications. Enhancing cavity QED via anti-squeezing:synthetic ultra-strong coupling A. A. Clerk========================================================================= Introduction– Cavity QED (CQED), the interaction between a single two-level system (qubit) and a quantized mode of a cavity, is a ubiquitous platform <cit.> that has widespread utility, ranging from the study of fundamental physics <cit.>, to the cutting edge of quantum information <cit.>.The most interesting regimes of CQED correspond to a qubit-cavity coupling that is strong enough to dominate dissipation rates.While this has been achieved in several architectures, e.g. <cit.>, in many others <cit.> it remains extremely challenging. Even more challenging is reaching the so called ultra-strong coupling (USC) regime, where the coupling strength is comparable to the qubit/cavity frequency. Here, counter-rotating terms cannot be ignored, and the system is best described by the quantum Rabi model <cit.>, which is known to exhibit a wide range of interesting phenomena, such as strongly entangled and nonclassical eigenstates <cit.>. To date, only specially designed architectures have reached USC experimentally <cit.>, though simulations of USC have also been considered <cit.>.In this paper, we show how simple detuned parametric driving of a cavity can be used to dramatically enhance the effective qubit-cavity coupling in a generic CQED system.This can be used to turn a weakly coupled system into a strongly coupled one, and even push one from strong coupling to the USC regime. After describing the general idea, we show it enables the direct study of Rabi-model physics in a system whose bare coupling is far from the USC regime.Further, the ability to turn on and off the coupling enhancement leads to new applications.We show how our approach allows one to leverage the entanglement of the USC ground state to generate remote entanglement between the qubit and a traveling waveguide mode.This is a key ingredient in quantum teleportation <cit.> and teleported gates <cit.>, quantum repeaters <cit.>, and quantum networks <cit.>. Note that related approaches for coupling enhancement have been considered in the context of cavity optomechanics <cit.>, a system with a markedly different kind of light-matter interaction. Model– We start by considering a qubit weakly coupled to a cavity, with the cavity subject to a two-photon (i.e. parametric) drive, see <ref>.Working in a frame rotating at half the parametric drive frequency ω_p/2, the Hamiltonian isĤ(t) = δ_c â^†â + δ_q/2σ̂_z - λ(t)/2 ( â^† 2+â^2) + g(â^†σ̂_-+σ̂_+â),where â is the cavity annihilation operator and σ̂_± are qubit raising/lowering operators. λ(t) is the time-dependent parametric drive amplitude, g is the qubit-cavity coupling strength, and δ_c/q = ω_c/q - ω_p/2, are the cavity and qubit detunings (with ω_c/q being the cavity/qubit frequencies).The weak value of g implies that the qubit-cavity interaction is well-described by the excitation-conserving Jaynes-Cummings coupling written above.Note that a parametric drive can be implemented in many different physical architectures.For example, in circuit QED, one can modulate the flux through a SQUID embedded in the cavity (see e.g. <cit.>). In what follows, we will be interested exclusively in detuned parametric drives with |δ_c| > λ, which ensures that <ref> is stable. The instantaneous cavity-only part of Ĥ(t) can be diagonalized by the unitary Û_ S[r(t)] = exp[r(t)(â^2-â^† 2)/2], where the squeeze parameter r(t) is defined viatanh 2r(t) = λ(t)/δ_c.The Hamiltonian in the time-dependent squeezed frame described by Û_ S[r(t)] isĤ^ S(t) ≡Û_ S[r(t)]ĤÛ_ S^†[r(t)] - i Û_ SÛ̇_ S^†Ĥ^ S(t) = Ĥ_ Rabi(t) + Ĥ_ Err(t) + Ĥ_ DA(t),whereĤ_ Rabi(t) = Ω_c[r(t)] â^†â + δ_q/2σ̂_z + g/2e^r(t) (â^†+â)(σ̂_++σ̂_-),Ĥ_ Err(t) = - g/2e^-r(t) (â^†-â)(σ̂_+-σ̂_-), Ĥ_ DA(t) = - iṙ(t)/2(â^† 2-â^2) .The Hamiltonian Ĥ_ Rabi(t) has the form of the usual Rabi Hamiltonian, with an enhanced coupling g̃ = g e^r(t)/2 and an effective cavity frequency Ω_c[r(t)] = δ_c 2r(t) (which decreases with increasing r). As e^r(t) becomes arbitrarily large as we approach the instability threshold λ = |δ_c|, the effective qubit-cavity coupling in Ĥ_ Rabi can be orders of magnitude larger than the original coupling g.The remaining terms in Eqs. (<ref>) describe undesired corrections to the ideal Rabi Hamiltonian.Ĥ_ Err(t) is explicitly suppressed by e^-r(t)/2, and thus is negligible in the large amplification limit e^r(t)→∞ (as long as the cavity population in the squeezed frame remains finite).Note that the cavity vacuum in the squeezed frame corresponds in the original lab frame to a squeezed vacuum state with squeeze parameter r(t). The last correction term Ĥ_ DA(t) vanishes explicitly for a time-independent drive amplitude (and only plays a limited role in the adiabatic preparation scheme discussed later).Thus, for large parametric drives, we have that our system is unitarily equivalent to the Rabi-model Hamiltonian Ĥ_ Rabi(t) with an exponentially enhanced coupling strength. This enhancement is a consequence of the coherent parametric drive modifying the eigenstates of the cavity Hamiltonian: these are now squeezed photons, whose amplified fluctuations directly yield a larger interaction with the qubit.We stress that this enhancement is not equivalent to simply injecting squeezed light into the cavity (as this does not change the Hamiltonian).It is also distinct from the usual √(n) enhancement associated with the Jaynes-Cumming interaction between a qubit and an n-photon Fock state, as in the squeezed frame, the interaction is enhanced for both small and large photon numbers.Weak to Strong Coupling– The simplest application of our approach is to enhance the coupling in a weak-coupling CQED system (where g is much smaller than the cavity damping rate κ).Even if the resulting enhanced coupling g̃ < κ, the increase could lead to dramatic enhancement of measurement sensitivity for spin or qubit detection, as the signal to noise ratio scales quadratically with g. This could be of particular utility in systems involving electronic or nuclear spins coupled to microwave cavities, where couplings are naturally weak <cit.>.The enhancement of a dispersive qubit measurement in this regime (where g̃ < κ) can in some cases equivalently be understood from the perspective of amplification, see Ref. Levitan:2016aa for a full discussion. Perhaps more interesting is the ability of our scheme to push a system with a weak bare coupling (g < κ) into the regime of strong effective coupling, g̃≳κ.In this regime, we expect that the parametrically driven system will exhibit features of a true strong coupling CQED system.A hallmark of strong coupling is vacuum Rabi splitting (VRS) <cit.>, where, e.g., the qubit absorption spectrum splits as a function of frequency (due to qubit-cavity hybridization).In <ref> we show the qubit absorption spectrum (obtained from a master equation simulation <cit.>) as a function of frequency for a bare coupling g = 0.20 κ, both with and without a parametric drive.As expected, the coupling enhancement due to the drive leads to a clear VRS. Note that to obtain a simple zero-temperature spectrum, we assume that the cavity is driven by squeezed vacuum noise in the lab frame, which corresponds to simple vacuum noise in the squeezed frame used in Eq. (<ref>).This ensures that the system starts in the ground state in the squeezed frame <cit.>.Strong coupling in CQED enables a number of applications, ranging from nonlinear quantum optics at the two-photon level to single-atom lasing <cit.>.Parametric driving makes these accessible even in systems with a weak bare coupling. Dynamical Simulation of USC Quench– Parametric driving can even be pushed further, allowing a weak or strong coupling CQED system to be enhanced into the USC regime.In Fig. (<ref>) , we show that our approach allows a faithful realization of the USC regime by comparing the dynamical evolution of the parametrically driven system (including all terms in<ref>) against a simulation of just the ideal Rabi Hamiltonian Ĥ_ Rabi.We start the system in the g=0 ground state of<ref>, and thus are simulating a quench-type protocol where the (ultra-strong) coupling is suddenly turned on.Fig. (<ref>) plots the time-dependent fidelity between the simulated state and the ideal Rabi-model state, for several values of parametric drive strength.The parametrically driven system faithfully reproduces the ideal Rabi-model evolution over long timescales.The fidelity is even better for larger coupling enhancements, as the larger the squeezing, the more the suppression of the unwanted terms in Ĥ_ Err (c.f. Eqs. (<ref>)).Adiabatic preparation of entangled USC ground states–In contrast to the above quench protocol, we can start with the trivial ground state of a weakly coupled CQED system (i.e. ground state of <ref> for λ(t)=0), and then adiabatically prepare the ground state of the ultra-strong coupling Rabi model by slowly ramping up the parametric drive amplitude λ(t).Further, once the desired state is achieved, the parametric drive can be turned off, returning the system to weak-coupling dynamics.The ability to prepare strong coupling ground states and then return to weak coupling allows a number of useful protocols.After preparation, one could turn off the coupling and allow the cavity state to leak into the waveguide or transmission line coupled to the cavity, implying that any cavity-qubit entanglement is now qubit-propagating photon entanglement; this enables remote entanglement protocols.Alternatively, as the protocol ends with the system in a weak coupling regime, the cavity state can be directly probed using standard weak-coupling techniques, for example by using the qubit<cit.>. This addresses the long-standing issue of how to observe the non-trivial aspects of the ground state of the quantum Rabi model.While one could use this approach to prepare the ground state of the Rabi Hamiltonian Ĥ_ Rabi(t) in any parameter regime, we focus on the case δ_q = 0, g̃≳Ω_c, where the ground state has the form of an entangled cat state:|Ψ_ Target(t)⟩ = |α(t)⟩|+x⟩ - |-α(t)⟩|-x⟩/√(2).Here |α⟩ denotes a coherent state in the cavity, and |± x⟩ denote σ̂_x qubit eigenstates; the displacement α∝g̃ (see EPAPS for full expression) <cit.>.Note that as this is the ground state in the squeezed frame, in the original lab frame the state will correspond to the qubit being entangled with squeezed, displaced cavity pointer states.As discussed above, preparing this state and then turning off the parametric drive allows one to create a nontrivial entangled state where the qubit is entangled with a propagating (squeezed, displaced) wavepacket.Unlike more standard approaches(e.g. <cit.>), this is accomplished without any controls or drives applied directly to the qubit.To consider the robustness of our approach, we simulate adiabatic state preparation in the presence of both cavity and qubit dissipation.These are treated via a standard Linblad master equation, which in the original lab frame takes the form:ρ̇̂̇ = i ρ̂Ĥ(t)+ γ𝒟[σ̂_-] ρ̂ + κ𝒟[â] ρ̂,where 𝒟[x̂] ρ̂ = x̂ρ̂x̂^†-1/2x̂^†x̂ρ̂, κ is the cavity damping rate, γ is the intrinsic qubit decay rate, and we have assumed zero temperature environments. Note that the simple form of this master equation is justified by the fact that we have a driven system with a large drive frequency ω_p (see EPAPS <cit.>); as a result, complications associated with strong-coupling master equations <cit.> do not apply. We parameterize the time-dependent parametric drive amplitude λ(t) via r(t) = r_ maxtanh(t/2τ) and Eq. (<ref>), where τ sets the effective protocol speed, and r_ maxis the final maximum value of the time-dependent squeeze parameter.The evolution runs from t=0 to t = t_f ≫τ.We quantify the success of the protocol using the fidelity F(t) between the dynamically generated state and the final desired target state in <ref>,F(t) = √(Ψ_ Target(t_f)ρ̂^ S(t)Ψ_ Target(t_f)),where ρ̂^ S(t) denotes the system density matrix in the squeezed frame. Achieving a good fidelity involves picking a value of τ that is large enough to ensure adiabaticity, but not so large that dissipative effects corrupt the evolution.<ref> summarizes the the results of a simulation of our scheme for a system with g=0.1 δ_c, where the system starts in the zero-coupling ground state |0⟩|-z⟩, and the protocol time scale is τ = 10δ_c^-1.Panel (b) shows the Wigner function of the cavity state obtained if the parametric drive is off during this evolution time:it corresponds to vacuum.Panel (a) shows instead the Wigner function obtained when the parametric drive is ramped such that e^2r_ max = 11 dB. One clearly sees the double-blob structure associated with the target state in Eq. (<ref>), and <ref>(c) shows that one indeed has good fidelity with this state, with the expected near-maximal amount of qubit-cavity entanglement (as characterized by the log negativity E_N).It is also interesting to consider the performance of the adiabatic state preparation as a function of the parametric gain e^2 r_ max.<ref>(a) shows the behaviour of the fidelity F(t_f) at the end of the protocol, as a function of the gain; different curves are for different ramp rates 1/τ.The fidelity is seen to generically drop off at high amplification factors.This is due to increased non-adiabatic errors (due to Ĥ_ DA in Eqs. (<ref>)), as well as due to dissipation.In the lab frame, the cavity is driven by vacuum noise associated with the loss κ.However, in the squeezed frame used to write Eq. (<ref>), this noise appears squeezed.This unwanted squeezing is oriented such that it enhances the spurious terms in Ĥ_ Err in Eq. (<ref>), and also has effects akin to heating; this all leads to errors in the adiabatic protocol.<ref>(b) shows the corresponding behaviour of the qubit-cavity entanglement (measured by the log negativity E_N).Surprisingly, the entanglement does not simply mirror the behaviour of the fidelity, and for rapid protocols, can even be larger than in the ideal target state. Conclusion–We have analyzed how parametric driving of a cavity can enable a strong coupling enhancement in cQED, even letting a weak coupling system reach the regime of ultra-strong coupling.The time-dependent control of the enhancement allows a variety of protocols, including the adiabatic preparation of strong-coupling highly entangled states. Our scheme is well-suited for contemporary circuit QED technology, where strong parametric interactions <cit.>, and high coherence times <cit.> are commonplace. It can also be implemented in other cavity QED architectures, including ones incorporating nitrogen-vacancy centers <cit.>, Rydberg atoms <cit.>, or quantum dots <cit.>. Additionally, our scheme can be generalized to realize regimes of ultra-strong coupling in lattices of cQED cavities <cit.>, by introducing local parametric driving at each site or a subset of sites, and can be applied to generate multi-qubit entanglement in a multi-qubit, single mode setup <cit.>. This work was supported by NSERC and the AFOSR MURI FA9550-15-1-0029. Supplemental Material for “Enhancing cavity QED via anti-squeezing:synthetic ultra-strong coupling”§ TIME EVOLUTION In this section, we calculate the explicit form of α(t) used in Eq. (5) of the main text. In the squeezed frame, the time-dependent coherent evolution of the system is described by the propagatorK̂^ S(t) = 𝒯exp(-i∫_0^t Ĥ^ S(t')dt'),where 𝒯 is the time-ordering operator. This can be expressed asK̂^ S(t) = K̂_ Rabi(t) 𝒯exp(-i∫_0^t (Ĥ_ Err^ Rabi(t')+Ĥ_ DA^ Rabi(t'))dt'),whereK̂_ Rabi(t) = 𝒯exp(-i∫_0^t Ĥ_ Rabi(t')dt'),is the propagator for the target Rabi Hamiltonian evolution andĤ_ Err/ DA^ Rabi(t)=K̂^†_ Rabi(t)(Ĥ_ Err/DA(t))K̂_ Rabi(t),lead to small corrections to the ideal evolution. For δ_q = 0 the Rabi propagator can be further simplified via a Magnus expansion,K̂_ Rabi(t) = exp((α(t)â^†-α^*(t)â)(σ̂_++σ̂_-)) exp(-iΛ_c(t,0)â^†â),where Λ_c(t,t') = ∫_t'^tΩ_c(t”)dt” is the integrated cavity frequency andα(t) = g/2i∫_0^t exp(r(t')-iΛ_c(t,t')) dt',is the cavity displacement in the squeezed frame. In the adiabatic limit ṙ(t) ≈ 0, <ref> reduces to the optimal displacement ge^r(t)/2Ω_c(t). <Ref> then propagates the state |0,-z⟩ to the entangled cat-state|Ψ_ Rabi(t)⟩ = |+α(t),+x⟩-|-α(t),-x⟩/√(2).Close to the adiabatic limit (i.e. α(t) ∼ g e^r(t)/2Ω_c(t)), Ĥ_ Err^ Rabi(t)∼g e^-r(t) Im[α(t)]/2 ≪α (t), and Ĥ_ Err^ Rabi(t) can be safely ignored for the entire protocol as long as Re[α(t)] ≫ Im[α(t)]. Ĥ_ DA^ Rabi(t)∼ṙ(t) α(t), and in general this is only constrained by the stability condition ṙ(t)≪Ω_c(t). This shows the importance of remaining adiabatic, ṙ(t)≈ 0, as Ĥ_ DA^ Rabi(t) contributes to the growth of Im[α(t)], and therefore of Ĥ_ Err^ Rabi(t). In the adiabatic limit ṙ(t)≈ 0 and <ref> is always valid.§ MASTER EQUATION DERIVATION In this section we outline the derivation of the master equation in the lab frame found in the main text. We start with the full Hamiltonian including the system-environment interactionĤ_ Lab = ω_c â^†â + ω_qσ̂_z/2 - λ/2 ( â^† 2e^-iω_pt+â^2e^iω_pt) + g(â^†σ̂_-+σ̂_+â) + Ĥ_ E + Ĥ_ SE,where we consider bosonic environments for the cavity and the qubit, such thatĤ_ E = ∑_νω^c_νb̂_ν^†b̂_ν +∑_νω^q_νĉ_ν^†ĉ_ν, Ĥ_ SE = ∑_νJ_ν^c(â + â^†)X̂_ν^c + ∑_νJ_ν^qσ̂_xX̂_ν^q = ∑_νJ_ν^c(â + â^†)(b̂_ν + b̂_ν^†) + ∑_νJ_ν^qσ̂_x(ĉ_ν + ĉ_ν^†),where J_ν^c/q are the system-environment coupling strengths. We focus only on transversal coupling between the qubit and the environment, as this is most detrimental to our scheme.Moving to a frame rotating at ω_p/2 for both the system and the environments, the system-environment interaction Hamiltonian becomes time-dependentĤ_ SE(t)= ∑_νJ_ν^c(âe^-iω_p/2t + â^† e^iω_p/2t)(b̂_νe^-iω_p/2t + b̂_ν^† e^iω_p/2t) + ∑_νJ_ν^q(σ̂_-e^-iω_p/2t + σ̂_+e^iω_p/2t)(ĉ_νe^-iω_p/2t + ĉ_ν^† e^iω_p/2t).However, as ω_p is much larger than any system frequency, we can make a rotating-wave approximation and drop counter-rotating terms to arrive atĤ_ SE(t) = ∑_νJ_ν^c(âb̂_ν^† + â^†b̂_ν) + ∑_νJ_ν^q(σ̂_-ĉ_ν^† + σ̂_+ĉ_ν).Crucially, in this frame the bath contains bosonic modes at negative frequencies down to -ω_p/2Ĥ_ E = ∑_ν(ω^c_ν-ω_p/2)b̂_ν^†b̂_ν +∑_ν(ω^q_ν-ω_p/2)ĉ_ν^†ĉ_ν.As a result, for this driven-dissipative system, the environment can mediate transitions from a lower energy system eigenstate to a higher energy eigenstate, even at zero temperature. This is in contrast to the undriven ultra-strong coupling situation, where such transitions are forbidden <cit.>.More formally, we consider diagonalizing the system self-Hamiltonian in the lab frame (rotating at ω_p/2), such thatĤ = ∑_jϵ_j|j⟩⟨j|,where ϵ_j are the eigenfrequencies, and {|j⟩}_j the eigenstates of the system. Moving to the interaction picture with respect to the system and environment self-Hamiltonians the system opertors can be written in the interaction picture asâ^ I(t) = ∑_j,k|j⟩⟨k|⟨j|â|k⟩e^iϵ_jkt, σ̂_-^ I(t) = ∑_j,k|j⟩⟨k|⟨j|σ̂_-|k⟩e^iϵ_jkt,where ϵ_jk = ϵ_j - ϵ_k is the transition frequency of the eigenstate transition |j⟩⟨k|. As the environment contains negative frequency modes down to -ω_p/2, and ω_p≫ϵ_jk, the environment can mediate all eigenstate transitions, regardless of whether ϵ_jk > 0 or ϵ_jk <0, as the environment spectral density will be nonzero at all ϵ_jk. In the standard case, where the system is not driven, only the transitions with ϵ_jk > 0 have nonzero spectral density and are possible.In addition, we consider Markovian environments with white spectrums, such that the transition rates for all eigenstate transitions will be the same. Combining this with the fact that all transitions are allowed, we can assign a single jump operator for the cavity and a single jump operator for the qubit, i.e. â^ I(t) and σ̂_-^ I(t), with decay rates κ and γ respectively. In the lab frame, these simply correspond to the jump operators â and σ̂_-. From here, a standard derivation of the Lindblad master equation will obtain Eq. (6) of the main text. § QUBIT ABSORPTION SPECTRUM The absorption spectrum S[ω] of the qubit is proportional to the rate at which the qubit would absorb power from a monochromatic qubit drive at frequency ω.As is standard, it is defined via the two-time correlation functionS(ω) = ∫_-∞^+∞σ̂_-(t)σ̂_+(t)_ss e^-iω t dt,where ∙_ss is the expectation value with respect to the steady state of the system ρ_ss. We use the quantum regression theorem to compute the spectrum,S(ω) = ∫_-∞^+∞ Tr[σ̂_-(t)[σ̂_+ ρ̂_ss]] e^-iω t dt= ∫_-∞^+∞ Tr[σ̂_- [σ̂_+ ρ̂_ss](t)] e^-iω t dt,which shows that the spectrum can be computed using the single-time correlation σ̂_-(t) with an initial pseudo-state σ̂_+ρ̂_ss. We calculate the absorption spectrum numerically, and plot S(ω) in the main text Fig. 2.The simulations for the spectrum shown in Fig. 2 assume that the qubit is driven by squeezed vacuum noise in the lab frame, with a squeeze parameter which matches the value of r determined by the parametric drive.As a result, in the squeezed frame, ρ_ss corresponds to vacuum, and the obtained spectrum corresponds to the zero temperature absorption spectrum of a strong-coupling cQED system.If no external squeezing was injected into the qubit, in the squeezed frame it would appear that the system is being driven by squeezed radiation.As a result, the absorption spectrum would correspond to a strong-coupling cQED system driven by squeezed light.Such spectra are characterized by peak asymmetries, as has been studied previously <cit.>. | http://arxiv.org/abs/1709.09091v1 | {
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"A. A. Clerk"
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http://arxiv.org/abs/1709.09704v2 | {
"authors": [
"Roberto Casadio",
"Piero Nicolini",
"Roldao da Rocha"
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"published": "20170927191347",
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emptyAmplitude Variations in Pulsating Red Giants. II. Some SystematicsJohn R. PercyDepartment of Astronomy and Astrophysics, and Dunlap Institute of Astronomy and Astrophysics, University of Toronto, Toronto ON Canada M5S 3H4, [email protected] LaingDepartment of Astronomy and Astrophysics, University of Toronto, Toronto ON Canada M5S 3H4, [email protected] order to extend our previous studies of the unexplained phenomenon of cyclic amplitude variations in pulsating red giants, we have used the AAVSO time-series analysis package VSTAR to analyze long-term AAVSO visual observations of 50 such stars, mostly Mira stars. The relative amount of the variation, typically a factor of 1.5, and the time scale of the variation, typically 20-35 pulsation periods, are not significantly different in longer-period, shorter-period, and carbon stars in our sample, and they also occur in stars whose period is changing secularly, perhaps due to a thermal pulse.The time scale of thevariations is similar to that in smaller-amplitude SR variables, but the relative amount of the variation appears to be larger in smaller-amplitude stars, and is therefore more conspicuous.The cause of the amplitude variations remains unknown.AAVSO keywords = AAVSO International Database; photometry, visual; pulsating variables; giants, red; period analysis; amplitude analysisADS keywords = stars; stars: late-type; techniques: photometric; methods: statistical; stars: variable; stars: oscillations1. IntroductionPercy and Abachi (2013) showed that, in almost all pulsating red giants (PRGs), the pulsation amplitude varied by a factor of up to 10, on a time scale of 20-40 pulsation periods.The authors were initially concerned that the variation might be an artifact of wavelet analysis, but it can be confirmed by Fourier analysis of individual sections of the dataset.Similar amplitude variations were found in pulsating red supergiants (Percy and Khatu 2014) and yellow supergiants (Percy and Kim 2014).There were already sporadic reports in the literature of amplitude variations in PRGs (e.g. Templeton et al. 2008, Price and Klingenberg 2005), but these stars tended to be the rare few which also showed large changes in period, and which may be undergoing thermal pulses (Uttenthaler et al. 2011).Furthermore: it is well known that stars such as Mira do not repeat exactly from cycle to cycle.Percy and Abachi (2013), however, was the first systematic study of this phenomenon.Since these amplitude variations remain unexplained, we have examined the behavior of more PRGs, to investigate some of the systematics of this phenomenon.We have analyzed samples of large-amplitude PRGs, mostly Mira stars, in each of four groups: A: 17 shorter-period stars; B: 20 longer-period stars; C: 15 carbon stars; D: 8 stars with significant secular period changes (Templeton et al. 2005).The stars in groups A, B, and C were drawn randomly from among the 547 studied by Templeton et al. (2005) and which did not show significant secular period changes.As did Templeton et al. (2005), we used visual observations from the American Association of Variable Star Observers (AAVSO) International Database.We did not analyze stars for which the data were sparse, or had significant gaps. Note that Templeton et al. (2005) specifically studied Mira variables, which, by definition, have full ranges greater than 2.5 in visual light – an arbitrary limit.The purposes of this paper are: (1) to present our analyses of these 50 PRGs, and (2) to remind the astronomical community, once again, that the amplitude variations in PRGs require an explanation.2. Data and AnalysisWe analyzed visual observations from the AAVSO International Database (AID: Kafka 2017) using the AAVSO's VSTAR software package (Benn 2013). It includes both a Fourier and wavelet analysis routine; we used primarily the latter.For each star, we noted the Modified Julian Date MJD(1) after which the data were suitable for analysis– not sparse, no significant gaps.From the WWZ wavelet plots, we determined the maximum (Amx), minimum (Amn), and average (Ā) amplitude, the number of cycles N of amplitude increase and decrease, and the average length L of these cycles.See Percy and Abachi (2013) for a discussion of these quantities and their uncertainties; N and therefore L can be quite uncertain because the cycles are irregular, and few in number, especially if they are long.This is doubly true for the few stars in which the length of the dataset is shorter than average. The maximum and minimum amplitudes are also uncertain since they are determined over a limited interval of time.We then calculated the ratio of L to the pulsation period P, the ratio of maximum to minimum amplitude, the difference ΔA between the maximum and minimum amplitude, and the ratio of this to the average amplitude Ā.The periods were taken from the VSX catalog, and rounded off; the periods of stars like these “wander" by several percent, due to random cycle-to-cycle fluctuations.All this information is listed in Tables 1-4.In the “Notes" column, the symbols are as follows: “s" – the data were sparse in places; “g" – there were one or more gaps in the data (but not enough to interfere with the analysis); “d" – the star is discordant in one or more graphs mentioned below, but there were no reasons to doubt the data or analysis; asterisk (*) – see Note in Section 3.2.Note that the amplitudes that we determine and list are “half-amplitudes" rather than the full ranges i.e. they are the coefficient of the sine function which fit to the data.3. ResultsWe plotted L/P, Amx/Amn, and ΔA/Ā against period for each of the four groups of stars A,B,C, and D.There was no substantial trend in any case, except as noted below (Figures 1-3). We therefore determined the mean M and standard error of the mean SEM, for each of the three quantities, for each of the four groups. These are given in Table 5.We also flagged any outliers in the graphs, and reexamined the data and analysis.If there was anything requiring comment, that comment is given in Section 3.2.In stars which are undergoing large, secular period changes, possibly as a result of a thermal pulse, the size and length of the amplitude variation cycles is marginally larger, but this may be partly due to the difficulty of separating the cyclic and secular variations.Note that cyclic variations in amplitude are present during the secular ones in these stars.We also found that, for the shorter-period stars, Ā increased with increasing period (Figure 4), but this is a well-known correlation.The very shortest-period PRGs have amplitudes of only hundredths of a magnitude. There was no trend in amplitude for the longer-period stars.The relative amount of variation in amplitude is slightly larger in shorter-period, smaller-amplitude stars (Figure 5).This is consistent with the results of Percy and Abachi (2013), as discussed in Section 4.The Ā for the carbon stars are systematically lower than for the oxygen stars (Figures 2 and 3).Again, this is well-known; in the oxygen stars, the visual amplitude is amplified by the temperature sensitivity of TiO bands, which are not present in carbon stars.Note also that the carbon stars have longer periods, since they are in a larger, cooler, and more highly evolved state. 3.1 Stars with Secular Amplitude VariationsAlthough our main interest was in the cyclic variations in pulsation amplitude, the secular variations in amplitude are also of interest, though they have already been studied and discussed by other authors, as mentioned in the Introduction.We performed a quick wavelet analysis of the 547 Miras in Templeton et al.'s (2005) paper, to identify stars in which secular amplitude variations might dominate the cyclic ones.Of the 21 stars whose period varied secularly at the three-sigma level or greater, four (T UMi, LX Cyg, R Cen, and RU Sco) seemed to show such secular amplitude variations. There were no other stars in Templeton et al.'s (2005) sample which showed strong secular variations.Note that, in each case, cyclic amplitude variations were superimposed on the secular ones.3.2 Notes on Individual StarsThis section includes notes on two kinds of stars: the ones for which the data or analysis required comments, and ones which appear to be outliers in some of the graphs that we have plotted.R Cen: this star has a secular decrease in amplitude, and period, so it is not surprising that the star is discordant in some of the relationships; see also Templeton et al. (2005).T Dra: this star has unusually large cyclic variations in amplitude.R Lep: this star has unusual large variations in mean magnitude.RZ Sco: this star, with a relatively short period, has a secular change in period, but only at the 3σ level (Templeton et al. 2005).Z Tau: this star is exceptional in that it is an S-type star.Also: its light curve shows non-sinusoidal variations, and flat minima suggestive that the variable may have a faint companion star.Indeed, SIMBAD lists two faint stars within 5 arc seconds of Z Tau.This star is discussed by Templeton et al. (2005).4. DiscussionPercy and Abachi (2013) obtained a median value of L/P = 44 for 28 monoperiodic smaller-amplitude PRGs.They calculated the median, in part because there were a few stars with very large values of L/P. We have reanalyzed those stars, and realized that Percy and Abachi (2013) adopted a more conservative definition for amplitude variations. Figure 6 shows an example of this: for the smaller-amplitude PRG RY Cam, Percy and Abachi (2013) estimated N = 1.5 whereas, based on our subsequent experience, we would estimate N = 6.7.Based on our reanalysis, the L/P values are now strongly clustered between 20 and 30, with a mean of 26.6.This is consistent with the values which we obtained for shorter- and longer-period PRGs.The values of ΔA/Ā, obtained by Percy and Abachi (2013), for smaller-amplitude (1.0 down to 0.1) variables, are typically about 0.5 to 2.0.This is consistent with the trend shown in Figure 5. The amplitude variations are relatively larger and more conspicuous in small-amplitude stars.Templeton et al. (2008) call attention to three other PRGs with variable amplitudes.The amplitude variations in RT Hya are the largest (0.1 to 1.0) and are cyclic (L/P = 40).The amplitude variations in W Tau are almost as large (0.1 to 0.6) and are also cyclic (L/P = 24). Those in Y Per are less extreme (0.3 to 0.9) and also cyclic (L/P = 29). These three stars therefore behave similarly to PRGs in our sample.There are therefore at least three unexplained phenomena in the pulsation of PRGs: (1) random, cycle-to-cycle fluctuations which cause the period to “wander"; (2) “long secondary periods", 5-10 times the pulsation period; and now (3) cyclic variations in pulsation amplitudes, on timescales of 20-30 pulsation periods. PRGs have large outer convective envelopes.Stothers and Leung (1971) proposed that the long secondary periods represented the overturning time of giant convective cells in the outer envelope, and Stothers (2010)amplified this conclusion.Random convective cells may well explain the random cycle-to-cycle period fluctuations, as well.The amplitude variations might then be due to rotational modulation, since the rotation periods of PRGs are significantly longer than the long secondary periods according to Olivier and Wood (2003).5. ConclusionsSignificant cyclic amplitude variations occurs in all of our sample of 50 mostly-Mira stars.The relative amount of the variation (typically Amx/Amn = 1.5) and the time scale of the variation (typically 20-35 times the pulsation period) are not significantly different in the shorter-period and longer-period stars, and in the carbon stars.The time scales are consistent with those found by Percy and Abachi (2013) in a sample of mostly smaller-amplitude SR variables, but therelative amplitude variations are larger in the smaller-amplitude stars.As was previously known: the average amplitudes increase with period for the shorter-period stars, and the carbon stars have smaller visual amplitudes than the oxygen stars.AcknowledgementsWe thank the AAVSO observers who made the observations on which this project is based, the AAVSO staff who archived them and made them publicly available, and the developers of the VSTAR package which we used for analysis.This paper is based, in part, on a short summer research project by undergraduate astronomy and physics student co-author JL. We acknowledge and thank the University of Toronto Work-Study Program for existing, and for financial support.This project made use of the SIMBAD database, maintained in Strasbourg, France.ReferencesBenn, D. 2013, VSTAR data analysis software (http://www.aavso.org/node/803)Kafka, S. 2017, observations from the AAVSO International Database (https://www.aavso.org/aavso-international-database)Olivier, E.A., and Wood, P.R. 2003, Astrophys. J., 584, 1035. Percy, J.R., and Abachi, R. 2013, J. Amer. Assoc. Var. Star Obs., 41, 193.Percy, J.R., and Khatu, V.C. 2014, J. Amer. Assoc. Var. Star Obs., 42, 1.Percy, J.R., and Kim, R.Y.H. 2014, J. Amer. Assoc. Var. Star Obs., 42, 267.Price, A., and Klingenberg, G. 2005, J. Amer. Assoc. Var. Star Obs., 34, 23.Stothers, R.B., and Leung, K.C. 1971, Astron. Astrophys., 10, 290.Stothers, R.B. 2010, Astrophys. J., 725, 1170.Templeton, M.R. et al. 2005, Astron. J., 130, 776.Templeton, M.R. et al. 2008, J. Amer. Assoc. Var. Star Obs., 36, 1.Uttenthaler, S. et al. 2011, Astron. Astrophys. 531, A88. | http://arxiv.org/abs/1709.09696v1 | {
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"title": "Amplitude Variations in Pulsating Red Giants. II. Some Systematics"
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#1 1 0 Better estimates from binned income data: Interpolated CDFs and mean-matching Paul T. von HippelLBJ School of Public Affairs, University of Texas at Austin [email protected] J. Hunter, McKalie Drown Drown is grateful for support from a Tensor Grant of the Mathematical Association of America. Department of Mathematics and Computer Science, Westmont College [email protected], [email protected] 30, 2023 ==================================================================================================================================================================================================================================================================================================================================================================================================================== 1 Better estimates from binned incomes:Interpolated CDFs and mean-matching Researchers often estimate income statistics from summaries that report the number of incomes in bins such as $0-10,000, $10,001-20,000,…,$200,000+. Some analysts assign incomes to bin midpoints, but this treats income as discrete. Other analysts fit a continuous parametric distribution, but the distribution may not fit well. We fit nonparametric continuous distributions that reproduce the bin counts perfectly by interpolating the cumulative distribution function (CDF). We also show how both midpoints and interpolated CDFs can be constrained to reproduce the mean of income when it is known.We evaluate the methods in estimating the Gini coefficients of all 3,221 US counties. Fitting parametric distributions is very slow. Fitting interpolated CDFs is much faster and slightly more accurate. Both interpolated CDFs and midpoints give dramatically better estimates if constrained to match a known mean.We have implemented interpolated CDFs in the binsmooth package for R. We have implemented the midpoint method in the rpme command for Stata. Both implementations can be constrained to match a known mean.Keywords:Gini, inequality, income brackets, grouped data 1.45§ INTRODUCTIONSurveys often ask respondents to report income in brackets or bins, such as $0-10,000, $10,000-20,000,…,$200,000+. Even in surveys where respondents report exact incomes, incomes may be binned before publication, either to protect privacy or to summarize the income distribution compactly with the number of incomes in each bin. Table <ref> gives a binned summary of household incomes in Nantucket, the richest county in the US. Binning presents challenges to investigators who want to estimate simple summary statistics such as the mean, median, or standard deviation, or inequality statistics such as the Gini coefficient, the Theil index, the coefficient of variation, or the mean log deviation. Researchers have implemented several methods for calculating estimates from binned incomes.The simplest and most popular approach is to assign each case to the midpoint of its bin—using a robust pseudo-midpoint for the top bin, whose upper bound is typically undefined(e.g., Table <ref>). The weakness of the midpoint approach is that it treats income as a discrete variable, but the method also has several strengths. The midpoint method is easy to implement and runs quickly. Midpoint estimates are also “bin-consistent” <cit.> in the sense that midpoint estimates will get arbitrarily close to their estimands if the bins are sufficiently numerous and narrow. Another approach is to fit the bin counts to a continuous parametric distribution. Popular distributions include 2-, 3-, and 4-parameter distributions from the generalized beta family, which includes the Pareto, lognormal, Weibull, Dagum, and other distributions <cit.>. One implementation fits up to 10 distributions and selects the one that fits best. An alternative is to use the AIC or BIC to calculated a weighted average of income statistics across several candidate distributions <cit.>.A strength of the parametric approach is that it treats income as continuous. A weakness is that even the best-fitting parametric distribution may not fit the bin counts particularly well. If the fit is poor, the parametric approach is not bin-consistent; that is, even with an infinite number of bins, each infinitesimally narrow, a parametric distribution may produce poor estimates if it is not a good fit to the underlying distribution of income. A practical weakness of the parametric approach is that it is typically implemented using iterative methods which can be slow. The speed of a parametric fit may be acceptable if you fit a single distribution to a single binned dataset, but runtimes of hours are possible if you fit several distributions to thousands of binned datasets—such as every county or school district in the US. Other computational issues include nonconvergence and undefined estimates. These issues are rare but inevitable when you run thousands of binned datasets <cit.>.Neither approach—midpoint or parametric—is uniformly more accurate. With many bins, midpoint estimates are better because of bin-consistency, while with fewer than 8 bins, parametric estimates can be better because of their smoothness <cit.>. Empirically, the parametric and midpoint approaches produce similarly accurate estimates from typical US income data with 15 to 25 bins. Both methods typically estimate the Gini within a few percentage points of its true value. This is accurate enough for many purposes, but can lead to errors when estimating small differences or changes such as the 5% increase in the Gini of US family income that occurred between 1970 and 1980 <cit.>. A potential improvement is to fit binned incomes to a flexible nonparametric continuous density. Like parametric densities, a nonparametric density treats income as continuous. Like the midpoint method, a nonparametric density can be bin-consistent and fit the bin counts as closely as we like.Unfortunately, past nonparametric approaches have been disappointing. A nonparametric approach using kernel density estimation had substantial bias under some circumstances <cit.>. A nonparametric approach using a spline to model the log of the density <cit.> had even greater bias <cit.>, though it was not clear whether the bias came from the method or its software implementation.In this paper, we implement and test a nonparametric continuous method that outperforms its predecessors in both speed and accuracy. The method, which we call CDF interpolation, simply connects points on the empirical cumulative distribution function (CDF). The method can connect the points using line segments or cubic splines. When cubic splines are used, the method is similar to “histospline” or “histopolation” methods which fit a spline to a histogram <cit.>. But histosplines are limited to histograms which have bins of equal width <cit.>. CDF interpolation is more general approach which can handle income data where the bins have unequal width and the top bins has no upper bound (e.g., Table <ref>).We have implemented CDF interpolation in our R package binsmooth <cit.>, which is available for download from the Comprehensive R Archive Network (CRAN).Our results will show that statistics estimated with CDF interpolation are slightly more accurate than estimates obtained using midpoints or parametric distributions. In addition, CDF interpolation is much faster than parametric estimation, though not as fast as the midpoint method.We also show that the differences between methods are dwarfed by the improvement we get if we constrain a method to match the grand mean of income, which the US Census Bureau often reports alongside the bin counts. If we constrain either the interpolated CDF or the midpoint method to match a known mean, we get dramatically better estimates of the Gini. Our binsmooth package can constrain an interpolated CDF to match a known mean, and our new version (2.0) of the rpme command for Stata can constrain the midpoint method to match a known mean as well.In the rest of this paper, we define the midpoint, parametric, and interpolated CDF methods more precisely, then compare the accuracy of estimates in binned data summarizing household incomes within US counties. We also show how much estimates improve if we have the mean as well as the bin counts. § METHODS §.§ Binned dataA binned data set, such as Table <ref>, consists of counts n_1, n_2, …, n_B specifying the number of cases in each of B bins. The total number of cases is T = n_1 + n_2 + ⋯ + n_B. Each bin b is defined by an interval [l_b,u_b), b=1, …, B, where l_b and u_b are the lower and upper bound of income for that bin. The bottom bin often starts at zero (l_1=0), and the top bin may have no upper bound (u_B=∞).§.§ A midpoint method The oldest and simplest way to analyze binned data is the midpoint method, which within each bin b assigns incomes to the bin midpoint m_b=(l_b+u_b)/2 (Heitjan1989 for a review). Then statistics such as the Gini can be calculated by applying sample formulas to the midpoints m_b weighted by the counts n_b.When the top bin has no upper bound, we must define a pseudo-midpoint for it. The traditional choice is μ_B = l_B α/(α-1), which would define the arithmetic mean of the top bin if top-bin incomes followed a Pareto distribution with shape parameter α > 1 <cit.>. The problem with this choice is that the arithmetic mean of a Pareto distribution is undefined if α≤ 1 and grows arbitrarily large as α approaches 1 from above. A more robust choice is the harmonic mean h_B = l_B (1+1/α), which is defined for all α>0 <cit.>. We use the harmonic mean in this article. We estimate α by fitting a Pareto distribution to the top two bins and calculating the maximum likelihood estimate from the following formula <cit.>:α̂= ln((n_B-1+n_B)/n_B)/ln(l_B/l_B-1) If the survey provides the grand mean μ, then we need not assume that top-bin incomes follow a Pareto distribution. Instead, we can calculate μ̂_B = 1/n (T μ - ∑_b=1^B-1 n_B m_B ) which would be the mean of the top bin if the means of the lower bins were the midpoints m_b. Then μ̂_B can serve as a pseudo-midpoint for the top bin. When estimated in this way, it occasionally happens (e.g., in 4 percent of US counties) that the top bin's mean μ̂_B is slightly less than its lower bound l_B. This is infelicitous, but μ̂_B can still be used, and the resulting Gini estimates are not necessarily bad.[It might be a little better to set μ̂_B=l_B and move the other midpoints slightly the the left, but this would affect only 4 percent of counties, and those only slightly since the affected counties typically have few cases in the top bin.]The midpoint methods described in this section are implemented by the rpme command for Stata <cit.>, whererpme stands for “robust Pareto midpoint estimator” <cit.>. Except for mean-matching, the approach is also implemented in the binequality package for R <cit.>. §.§ Fitting parametric distributions The weakness of the midpoint method is that it treats income as discrete. An alternative is to model income as a continuous variable X that fits some parametric CDF F(X|θ). Here θ is a vector of parameters, which can be estimated by iteratively maximizing the log likelihood:ℓ(θ|X) = ln ∏_b=1^B (P(l_b<X<u_b))^n_b where P(l_b<X<u_b)=F(u_b)-F(l_b) is the probability, according to the fitted distribution, that an income is in the bin [l_b,u_b).[This formula includes the top bin B, where u_B=∞ and F(u_B)=1. Some older articles (e.g., <cit.>) add the following constant to the log likelihood: ln(T!)-∑_b=1^Bln(n_b!). This is not wrong, but it is unnecessary since adding a constant does not change the parameter values at which the log likelihood is maximized <cit.>.]While any parametric distribution can be considered, in practice it is hard to fit a distribution unless the number of parameters is small compared to the number of well-populated bins. Most investigators favor 2-, 3-, and 4-parameter distributions from the generalized beta family <cit.>, which includes the following 10 distributions: the log normal, the log logistic, the Pareto (type 2), the gamma and generalized gamma, the beta 2 and generalized beta (type 2), the Dagum, the Singh-Maddala, and the Weibull. A priori it is hard to know which distribution, if any, will fit well. The fit of a distribution can be tested by the following goodness-of-fit likelihood ratio statistic <cit.>: G^2=-2 (ℓ̂ - ∑_b=1^Bn_b ln(n_b/T) where ℓ̂ is the maximized log likelihood. Under the null hypothesis that the fitted distribution is the true distribution of income, G^2 would follow a chi-square distribution with B^*-k degrees of freedom, where k is the number of parameters, B^*=min(B_>0,B-1), and B_>0 is the number of bins with nonzero counts. We reject the fit of a distribution if G^2 has p<.05 in the null distribution. In empirical data it is common to reject every distribution in the generalized beta family <cit.>. In addition, some distributions may fail to converge, or may converge on parameter values that imply that the mean or variance of the distribution is undefined <cit.>. A solution is to fit all 10 distributions, screen out any with undefined moments and, among the distributions remaining, select the one that fits best according to the Akaike or Bayes information criterion (AIC or BIC):AIC= 2k - 2 ℓ̂BIC= ln(T) k- 2 ℓ̂ Or, instead of selecting a single best-fitting distribution, one can average estimates across several candidate distributions weighted proportionately to a function of the AIC or BIC (specifically exp(-AIC/2) or exp(-BIC/2)) an approach known as model averaging<cit.>. In general, model averaging yields better estimates than model selection, but when modeling binned incomes the advantage of model averaging is negligible <cit.>. Although it sounds broad-minded to fit 10 different distributions, there is limited diversity in the generalized beta family. All the distributions in the generalized beta family are unimodal and skewed to the right. Some distributions are quite similar (e.g., Dagum and generalized beta), and others rarely fit well (e.g., log normal, log logistic, Pareto). So in practice the range of viable and contrasting distributions in the generalized beta family is small; you can fit just 3 well-chosen distributions (e.g., Dagum, gamma, and generalized gamma) and get estimates almost as good as those obtained from fitting all 10 <cit.>.We use the fitted distributios to estimate income statistics such as the mean, variance, Gini, or Theil. Sometimes the income statistic is a simple function of the distributional parameters θ, but other times the function is unknown, or hard to calculate. As a general solution, it is easier to calculate income statistics by applying numeric integration to appropriate functions of the fitted distribution <cit.>.When the grand mean is available, the distributional parameters could in theory be constrained to match the grand mean as well as approximate the bin counts. This would be difficult, though, since for distributions in the generalized beta family the mean is a complicated nonlinear function of the parameters. We have not attempted to constrain our parametric distributions to match a known mean.The parametric approaches described in this section are implemented in the binequality package for R <cit.> and the mgbe command for Stata <cit.>. Here mgbe stands for “multi-model generalized beta estimator” <cit.>. §.§ Interpolated CDFs Since parametric distributions may fit poorly, an alternative is to define a flexible nonparametric density that fits the bin counts exactly. Our nonparametric approach is called CDF interpolation.To understand CDF interpolation, consider that binned data define B discrete points on the empirical cumulative distribution function (CDF). The empirical CDF for Nantucket is given in the last column of Table <ref>, which shows that 5% of Nantucket households make less than $10,000, 8% make less than $15,000, and so on. Formally, at an income of 0 the empirical CDF is F̂(0)=0, and at each bin's upper bound u_b the empirical CDF is the observed fraction of incomes that are less than u_b—i.e., F̂(u_b) = (n_1 + n_2 + ⋯ + n_b)/T.[If the binned incomes come from a sample (as they do in Table <ref>), then the estimate F̂(u_b) may differ from the true CDF F(u_b) because of sampling error.]Now to estimate a continuous CDF F̂(x), x>0, we just connect the dots. That is, we define a continuous nondecreasing function that interpolates between the B discrete points of the empirical CDF. Then the estimated probability density function (PDF) is just the derivative of the interpolated CDF F̂(x). Note that the estimated PDF “preserves areas” <cit.>—i.e., preserves bin counts. That is, within in each bin (P̂(l_b<x<u_b)=F̂(u_b)-F̂(l_b)), the total of the estimated density is equal to the observed fraction of incomes that are in that bin (n_b/T). The shape of the PDF depends on the function that interpolates the CDF.* If the CDF is interpolated by line segments, then the CDF is polygonal, and the PDF is a step function that is discontinuous at the bin boundaries. * If the CDF is interpolated more smoothly, by a continuously differentiable monotone cubic spline, then the PDF is piecewise quadratic — i.e., continuous at the bin boundaries and quadratic between them.There remains a question of how to shape the CDF in the top bin, which typically has no upper bound. In our implementation, the CDF of the top bin can be rectangular, exponential, or Pareto.[In our implementation, the exponential and Pareto tails are approximated by a sequence of rectangles of decreasing heights.] Each of these distributions has one parameter, which is estimated as follows.* If the grand mean of income is known, then the parameter shaping the top bin is constrained so that the mean of the fitted distribution matches the grand mean. It occasionally happens (e.g., in 4 percent of US counties) that we cannot reproduce the known mean this way, because the known mean is already less than the mean of the lower B-1 bins without the tail. In that case, we make an ad hoc adjustment by shrinking the bin boundaries toward the origin – that is, by replacing (l_b,u_b) with (s l_b,s u_b), where the shrinkage factor s<1 is chosen so that a small tail can be added to reproduce the grand mean. The shrinkage factor is rarely less than .995. * If the mean income is not known, we substitute an ad hoc estimate. We obtain that estimate by temporarily setting the upper bound of the top bin to u_B=2l_B and calculating the mean of a step PDF fit to all B bins. Then we unbound the top bin and proceed as though the mean were known.Income statistics are estimated by applying numerical integration to functions of the fitted PDF or CDF.The methods in this section are implemented by our binsmooth package for R <cit.>. Within the binsmooth package, the stepbins function implements a step-function PDF (and polygonal CDF), while the splinebins function implements a cubic spline CDF (and piecewise quadratic PDF).§.§ Recursive subdivision Another way to obtain a smooth PDF estimate that preserves bin areas is to subdivide the bins into smaller bins, and then adjust the heights of the subdivided bins to shorten the jumps at the bin boundaries. This method, recursive subdivision, is implemented by the recbin command in the binsmooth package for R <cit.>. Recursive subdivision is slower and more computationally intensive than CDF interpolation and the resulting estimates are practically identical. We present the details of recursive subdivision in the Appendix.§ DATA AND RESULTS Between 2006 and 2010, the American Community Survey (ACS) took a 1-in-8 sample of US households . Household incomes were inflated to 2010 dollars and summarized in binned income tables foreach of the 3,221 US counties. The published bin counts are estimates of the population counts. We can approximate the sample counts by dividing the population counts by 8. Dividing counts by a constant makes no difference to any of our statistics, except for the BIC and G^2 statistics that are used when fitting parametric distributions.The Census also published means and Gini coefficients for each county <cit.>. These statistics were estimated from exact incomes before binning, and so are more accurate than any estimate that could be calculated from the binned data. They are sample estimates which may differ from population values, but they remain a useful standard of comparison for our binned-data estimates. §.§ Results for Nantucket Table <ref> summarized the binned incomes for Nantucket County. Figure <ref> fits several distributions to the Nantucket data. The midpoint method is illustrated by gray spikes at the bin midpoints; the spike heights are proportional to the bin counts. The black step function is the PDF implied by a linear interpolation of the CDF, and the blue curve is the piecewise quadratic PDF implied by a cubic spline of the CDF. Both CDF interpolations fit the bin counts perfectly; in fact, their jagged appearance suggests they may overfit the data—a concern that we will revisit in the Conclusion. The step PDF looks slightly less volatile than the piecewise quadratic PDF, suggesting that the step PDF may be less overfit.The purple curve is the Dagum distribution. The Dagum fits Nantucket better than other distributions from the generalized beta family, but it does not fit well. It fails the G^2 goodness-of-fit test, and visually it fits the bin counts poorly. For example, between $70,000 and $150,000 the Dagum curve suggests there should be substantially fewer households than there are, and above $150,000 it suggests that there should be more.Except for the Dagum distribution, all the methods in Figure <ref> are calibrated to reproduce the grand mean.Table <ref> summarizes the Nantucket estimates. The true mean is $137,000 and the true Gini is .547. All the methods underestimate these quantities. When fit without knowledge of the true mean, every method underestimates the mean by 12 to 20 percent and the Gini by 15 to 21 percent, with the simple midpoint method coming closer than its more sophisticated competitors.When given the true mean, the midpoint and CDF interpolation methods do much better. They still underestimate the Gini, but only by 2 to 7 percent. The closest estimate is obtained by linear interpolation of the CDF. A smoother cubic spline interpolation does a little worse, but still better than the midpoint method.Although the estimates for Nantucket are less accurate overall than the estimates for most other counties, the relative performance of different methods in Nantucket is similar to what we will see elsewhere. §.§ Results for all US countiesFigure <ref> evaluates all county Gini estimates graphically by plotting the estimated Gini θ̂_j of each county against the published Gini θ_j. In the bottom row, where the methods are constrained to match the published mean, the Gini estimates are close to a diagonal reference line (θ̂_j=θ) indicating nearly perfect estimation. In the top row, where the methods do not match the mean, the estimates are more scattered, indicating lower accuracy.We can summarize the accuracy of Gini estimates in several ways. For a single county j, the percent estimation error is e_j = 100×(θ̂_j - θ_j)/θ_j. Then across all counties, the percent relative bias is the mean of e_j, the percent root mean squared error (RMSE) is the square root of the mean of e_j^2, and the reliability is the squared correlation between θ_j and θ̂_j. Table <ref> summarizes our findings. If the estimators ignore the published county means, then estimated Ginis have biases between 0% and -3%, RMSEs between 3% and 4%, and reliabilities between 82% and 88%. The interpolated CDF estimates have the best bias, the best RMSE, and the second best reliability, and they are just as good with linear interpolation as with cubic spline interpolation. The parametric estimates have the best reliability, but the worst bias, the worst RMSE, and by far the worst runtime, at 4.5 hours. When the methods are constrained to match the published county means, the estimates improve dramatically. The bias shrinks to 0-1%, the RMSE shrinks to 1-2%, and the reliability grows to 98-99%. The midpoint estimates are excellent, and the interpolated CDF estimates are even better, and just as good with linear interpolation as with cubic spline interpolation.The differences among the methods are much smaller than the improvement that comes from constraining any method to match the mean. Of course, this observation is only helpful when the mean is known.§ CONCLUSIONCDF interpolation produces estimates that are at least a little better than midpoint or parametric estimates, whether the true mean is known or not. And CDF interpolation runs much faster than parametric estimation, thought not as fast as midpoint estimation.We initially suspected that cubic spline interpolation would improve on simple linear interpolation, but empirically this turns out to be false. In estimating county Ginis, linear CDF interpolation was at least as accurate as cubic spline interpolation. The accuracy of linear CDF interpolation is remarkable, since it implies a step function for the PDF. Step PDFs look clearly unrealistic, especially in the top and bottom bins where the step function is flat while the true distribution likely has an upward or downward slope <cit.>. Our step PDF permits a downward Pareto or exponential slope in the top bin, and our cubic spline CDF can fit an upward slope to the bottom bin. But neither of these refinements does much to improve the accuracy of Gini estimates.The differences in accuracy among the methods are small, and they are dwarfed by the improvement in accuracy that comes from knowing the grand mean. By constraining binned-data methods to match a known mean, we can typically get county Gini estimates that are within 1-2% of the estimates we would get if the data were not binned. Our binsmooth package for R can constrain interpolated CDFs to match a known mean, and our rpme command for Stata can constrain the top-bin midpoint to match a known mean as well. We have not constrained our parametric distributions to match a known mean; we believe it would be difficult to do so.While the mean-constrained estimates are very accurate, there may be room for improvement when the mean is unknown. Perhaps the most promising idea is additional smoothing. As we noticed in Figure fig:comparepdfs, interpolated CDFs can be a bit jagged and may “overfit” the sample in the sense that they find nooks and crannies that might not appear in another sample or in the population. Likewise interpolated CDFs may be overfit to a specific set of bin boundaries. If the fitted CDF were a little smoother and did not quite preserve the counts of the least populous bins, it might fit the population and other samples (perhaps with different bin boundaries) a little better. § APPENDIX: PDF SMOOTHING BY RECURSIVE SUBDIVISIONRecursive subdivision is another way to smooth the fitted PDF. Like CDF interpolation, recursive subdivision preserves bin areas, but recursive subdivision is more computationally intensive and produces very similar results. Recursive subdivision is implemented by the recbins function in our binsmooth package for R.A slight change of notation will be helpful. Since the upper bound u_b of each bin is equal to the lower bound l_b+1 of the next, we can think the bins as having a set of “edges” e_0,e_1,…,e_B, where e_0=0, and the other e_b=u_b are the upper bounds of bins 1,…,B. Start by fitting a step PDF. Let h_b be the height of the step PDF in the bin [e_b, e_b+1). Given parameters ε_1 ∈ (0,0.5) and ε_2 ∈ (0,1), the subdivision process begins by introducing new bin edges l and r between e_b and e_b+1 such that (l+r)/2 = (e_b+e_b+1)/2 and r-l = (e_b+1-e_b)ε_2. The height of the new bin on the left with edges e_b and l is then shifted horizontally by (h_b-1-h_b)ε_1, while the height of the new bin on the right with edges r and e_b+1 is shifted horizontally by (h_b+1-h_b)ε_1. Finally, the new middle bin with edges l and r is shifted horizontally so that the area of the three new bins equals the area of the original bin.[We do not subdivide the bin in those rare cases where subdivision would yield a middle bin with negative height.] See Figure <ref>.In order for the above formulas to apply to the top and bottom bins, we create pseudo-bins above and below them with heights of zero. This ensures that the subdivided PDF will tend toward a height of zero at the lower edge of the bottom bin and the upper edge of the top bin. The smoothed PDF is obtained from the step PDF by applying the subdivision process to each bin, then applying the process again to each subdivided bin, and so on, until the desired level of smoothness is reached. In practice, three rounds of subdivision are sufficient to produce a reasonably smooth PDF, and we found that choosing ε_1 = 0.25 and ε_2 = 0.75 produced nicely smoothed PDF's from most empirical data sets. Figure <ref> shows the result of recursive subdivision in Nantucket. Unfortunately, if the original step PDF was constrained to match a known mean, the subdivision process may cause the mean to deviate slightly. But the estimated Gini typically remains quite accurate. | http://arxiv.org/abs/1709.09705v3 | {
"authors": [
"Paul T. von Hippel",
"David J. Hunter",
"McKalie Drown"
],
"categories": [
"stat.ME"
],
"primary_category": "stat.ME",
"published": "20170927191350",
"title": "Better estimates from binned income data: Interpolated CDFs and mean-matching"
} |
Diplomarbeitzur Erlangung des akademischen Grades:Diplom-Informatiker0.5mm Integration of Japanese PapersInto the DBLP Data Set 0.5mm Version for arXiv[t]0.4 vorgelegt vonPaul Christian Sommerhoff Matrikelnummer: XXXXXXvorgelegt zumApril 2013[t]0.4 vorgelegt amLehrstuhl für Datenbanken und Informationssysteme der Abteilung Informatik der Universität TrierProf. Dr. Bernd WalterRomanis-alphafancy [b][C]If someone is looking for a certain publication in the field of computer science, the searching person is likely to use the DBLP to find the desired publication. The DBLP data set is continuously extended with new publications, or rather their metadata, for example the names of involved authors, the title and the publication date. While the size of the data set is already remarkable, specific areas can still be improved.The DBLP offers a huge collection of English papers because most papers concerning computer science are published in English. Nevertheless, there are official publications in other languages which are supposed to be added to the data set. One kind of these are Japanese papers. This diploma thesis will show a way to automatically process publication lists of Japanese papers and to make them ready for an import into the DBLP data set. Especially important are the problems along the way of processing, such as transcription handling and Personal Name Matching with Japanese names.UTF8CHAPTER: LIST OF ACRONYMStocchapterList of Acronyms[OAI-PMH]ACMAssociation for Computing MachineryASCIIAmerican Standard Code for Information InterchangeAPIApplication Programming InterfaceBHTBibliography HyperTextDBLPDigital Bibliography & Library Project (former meaning: DataBase systems and Logic Programming)FAQFrequently Asked QuestionsGBGigaByteHTMLHyperText Markup LanguageHTTPHyperText Transfer ProtocolIDIdentifierIEEEInstitute of Electrical and Electronics EngineersIFIPInternational Federation for Information ProcessingIPSJInformation Processing Society of JapanIPSJ DLDigital Library of the Information Processing Society of JapanISOInternational Organization for StandardizationJARJava ARchiveJDBCJava DataBase ConnectivityJDKJava Development KitOAIOpen Archives InitiativeOAI-PMHOpen Archives Initiative - Protocol for Metadata HarvestingPDFPortable Document FormatRAMRandom Access MemorySAXSimple API for XMLSQLStructured Query LanguageSPFSingle Publication FormatTOCTables Of ContentsURLUniform Resource LocatorXMLeXtensible Markup LanguagetocchapterList of Figuresarabic CHAPTER: ABOUT THIS DIPLOMA THESISThe idea for this work was born when the author was searching for a possibility to combine computer science with his minor subject Japan studies in his diploma thesis. After dismissing some ideas leaning towards Named Entity Recognition and computer linguistics the author chose “Integration of Japanese Papers Into the DBLP Data Set” as his subject. The DBLP is a well-known and useful tool for finding papers published in the context of computer science. The challenge to deal with such a huge database and the problems that occur when processing Japanese input data was the reason why this idea has been chosen. The hope is that, in the future, many Japanese papers can be added by the responsible people of the DBLP project. § MOTIVATIONComputer scientists are likely to use the DBLP to find information about certain papers or authors.Therefore, the DBLP is supposed to provide information about as many papers as possible. For example, one could be interested in the paper “Analysis of an Entry Term Set of a Civil Engineering Dictionary and Its Application to Information Retrieval Systems” by Akiko Aizawa et al. (2005) but DBLP does not include it yet. Japanese scientists might look for the original (Japanese) title “土木関連用語辞典の見出し語の分析と検索システムにおける活用に関する考察” or use Aizawa's name in Japanese characters (相澤彰子) for a search in DBLP. The DBLP contains the author “Akiko Aizawa” but does not contain this specific paper or the author's original name in Japanese characters. Our work is to implement a tool which addresses these questions, support the DBLP team in the integration of Japanese papers and reveal the difficulties of realizing the integration. § COMPOSITION OF THE DIPLOMA THESISDates are displayed in the ISO 8601 standard format YYYY-MM-DD, e.g. 2012-10-19.Although scientific works about the Japanese language often display the Sino-Japanese reading of kanji (a Japanese character set) with uppercase letters to distinguish them from the other “pure” Japanese reading, we will not use uppercase letters to distinguish them in this work.When a Japanese word is used in its plural form in this work, the word always stays unmodified. The reason is that in the Japanese language there is no differentiation between a singular and plural form.We use a macron instead of a circumflex to display a long vowel of a Japanese word in Latin transcription (see section <ref>).§ ACKNOWLEDGEMENTFirst I would like to thank Prof. Dr. Bernd Walter and Prof. Dr. Peter Sturm for making this diploma thesis possible. Special thanks go to Florian Reitz for the great support and the useful answers for the questions I had while I have been working on this diploma thesis. I also want to acknowledge the help of Peter Sommerhoff, Daniel Fett, David Christ and Kana Matsumoto for proofreading my work. I thank Dr. Michael Ley, Oliver Hoffmann, Peter Birke and the other members of the Chair of Database and Information Systems of the University of Trier. Last but not least I want to tell some personal words to my family in my and their native language German: Ich möchte nun noch meinen Eltern und meinem Bruder Peter dafür danken, dass sie mich in meiner Diplomarbeitsphase, meinem Studium und auch schon davor immer unterstützt habenund immer für mich da waren, wenn ich sie brauchte. Ich weiß es zu schätzen.CHAPTER: WRITING IN JAPANESE“My view is that if your philosophy is not unsettled dailythen you are blind to all the universe has to offer.”(Neil deGrasse Tyson)First we need to understand some aspects of the Japanese language and especially the different ways of writing Japanese because the peculiarities of the Japanese writing system are a crucial point of our work. It lays the foundation for all Japanese-related subjects such as the structure of Japanese names (discussed in section <ref>), a dictionary for Japanese names (discussed in section <ref>) or the publication metadata source for Japanese publications (discussed in section <ref>).Hadamitzky (<cit.>, p. 8-57) gives an overview about the basics of Japanese writing. The Japanese writing system includes kanji, hiragana, katakana and the possibility to use Latin characters.§ KANJIKanji is the Japanese script which consists of traditional Chinese characters. It came to Japan around the 4th century. Since the Japanese had not developed an own writing system yet they began to use the Chinese characters. At the beginning, the characters were linked phonetically with a certain sound, so that they could write down all existing words by their sound. Applying this principle the man'yōgana were created. Every character had one defined way to pronounce it. In addition to this, a second principle was introduced to write Japanese. This time the people orientated themselves on the meaning of the Chinese characters to choose a writing for a word. Applying the second principle, the kanji were created. While the man'yōgana were simplified to hiragana and katakana (see following sections <ref> and <ref>) the general usage of kanji did not change.Due to an increase in number and possible readings of characters, the government began to try to simplify the Japanese writing system after the Meiji Restoration at the end of the 19th century. The last important reform took place after World War II. Along with some other changes and regulations, the permitted characters in official documents (tōyō kanji) were limited to 1850 in 1946 and increased to 1900 in a draft from 1977. In 1981 they were replaced by the “List of Characters for General Use” (jōyō kanji[for further information about jōyō kanji (and modern Japanese writing) see <cit.>, p. 165-171]) containing 1945 characters. In 1951 the government published a list of additional 92 kanji permitted for personal names. The number of kanji permitted for personal names increased with time passing by. Eschbach-Szabo (<cit.>, p. 175) says the last change permitted 983 kanji for personal names in 2004.The press tries to abide by the jōyō kanji. Japanese literature (science, fiction, etc.) uses about 4000 characters (comprehensive Sino-Japanese kanji dictionaries contain ca. 10000 characters).Japanese people know approximately 3000 kanji on average.Due to their capability to give a word a meaning, kanji are used in substantives, verbs, adjectives and Japanese personal names.An important aspect is reading a kanji because there are several possibilities to read one. Saitō and Silberstein (<cit.>, p. 31-34) describe how to read a kanji. There is a Japanese reading kun and a Sino-Japanese reading on. Depending on the text and grammar context either the kun or on reading is required. For example the kanji 生 is read sei in 学生 (gakusei, meaning: student, on reading) but is read u in 生まれる (umareru, meaning: being born, kun reading). A single kanji can have several kun and several on readings.For our work it is important to know that one character can have several readings in names too. § HIRAGANAThe syllabary hiragana evolved from the man'yōgana by simplifying the characters. Every syllable is phonetically assigned to one sound of the spoken language (with two exceptions which can have two sounds each). The gojūon table shown in figure <ref> lists the 46 syllables used today in a certain way (it can be compared with the ABC for letters).[Koop and Inada (<cit.>, p. 20-29) provide information about hiragana] Another but obsolete way to order the syllables is iroha which is a poem containing all syllables. Although the name implies 50 sounds (gojū means “50”, on means “sound”) there are only 46 syllables left in modern Japanese. Actually, only 45 syllables belong to the gojūon table. The n counts as extra symbol (see gojūon tables in figures <ref> and <ref>).Other additional syllables are dakuon (e.g. だ/da, recognizable by two little strokes), handakuon (e.g. ぱ/pa, recognizable by a little circle) and yōon (e.g. しゃ/sha, recognizable by a normally sized character that is followed by a smaller character).You can write every Japanese word in hiragana but if possible, kanji are usually preferred to avoid problems with homonyms (we take a look at homonyms in chapter <ref>). Hiragana is mainly used to write words not covered by kanji and as inflected endings. Kanji and hiragana are often combined within one word. For example 読む (yomu) is the basic form of the verb “to read”. The kanji 読 means reading by itself and in combination with the hiragana syllable む it becomes the verb “to read” in a special grammatical form specifying tense, politeness level and other properties.[see also <cit.> to get a beginner's guide to the Japanese language] § KATAKANAThe syllabary katakana also evolved from the man'yōgana by simplifying the characters, consists of 46 characters nowadays (representing the same syllables as hiragana) and is usually ordered by the gojūon table. Figure <ref> presents the katakana in a gojūon table. Besides optical differences with hiragana, katakana are used in other contexts. Japanese mostly use them to write foreign words including foreign personal names.So foreigners often apply katakana for their names. For example, the author's name can be transcribed as パウル·ソマホフ. The dot · in the middle separates family and given name. Foreign names are often written with the given name preceding the family name.§ LATIN CHARACTERS/TRANSCRIPTIONTranscription systems which convert kanji, hiragana and katakana to Latin characters are usually called rōmaji. Japanese can be easily transcribed by 22 letters and two additional signs. Due to many words having the same pronunciation, the meaning of words is sometimes ambiguous if they are transcribed into Latin characters. In 1954 the government released recommendations for transcribing Japanese. It recommended following two transcription systems: Kunreishiki rōmaji The kunreishiki rōmaji assigns transcriptions according to the order in the gojūon table without regard to phonetic divergences of some consonants (we will discuss these divergences later). It has been introduced for official usage by the government only slightly different in 1937. It became the preferred transcription system in the standard ISO 3602 “Documentation - Romanization of Japanese (kana script)” <cit.>.Hebonshiki rōmaji The hebonshiki rōmaji was developed by a council of Japanese and foreign erudites in 1885 and spread by the American missionaryJames C. Hepburn (Hebon in Japanese), especially thanks to his Japanese-English dictionary published one year later. This work also employs hebonshiki. Kunreishiki would lead to transcriptions like kunreisiki, hebonsiki and kanzi.Although the kunreishiki became the preferred system of the government, the international community often prefers the Hepburn system because the written words suggest a more intuitive pronunciation than kunreishiki. There are also language-related transcription systems that are rarely used. Kaneko and Stickel (<cit.>, p. 53-55) mention them:The important aspect are the system differences because we need to know where they occur when we deal with Personal Name Matching problems later. Figure <ref> in the appendix reveals the differences between the transcription systems. It summarizes 18 differences in all syllables including dakuon, handakuon and yo̅on.Unfortunately, there can be even more transcription differences. ISO 3602 highlights some more special cases when it comes to transcribing Japanese. One is the question whether to put an apostrophe after an n. To avoid misunderstandings, one should put an apostrophe behind an n in certain cases. Otherwise, people could misinterpret the syllable n followed by a syllable composed of a vowel or “y” and a vowel as syllables na, ni, nu, ne, no, nya, nyu or nyo. We will outline a practical example of this case in section <ref>.A second irregularity occurs when the same vowel appears right after another. If there is a morpheme boundary between the vowels, they should be transcribed as “aa”, “ii”, etc. but should be transcribed by an additional circumflex otherwise. Koop and Inada <cit.> write about another difficulty called nigori. “The nigori (濁, literally, `turbidity', `impurity') ... [means] modifying the pronunciation of the consonant in certain of the kana sounds. It may be either (1) inherent, as in suge (`sedge'), suzu (`grelot'), go (`five'), or (2) applied incidentally to the initial consonant of a word or name-element following another in composition, e.g., Shimabara from shima and hara, nenjū from nen and chū, Harada from hara and ta.” (<cit.>, p. 34) So, if we want to derive a transcription from the family name 中田, we cannot tell whether to take Nakata or Nakada as the rightful transcription. CHAPTER: JAPANESE PERSONAL NAMES七転び、八起き。Nana korobi, ya oki.(Fall seven times, get up eight times.)Japanese sayingOne of the central problems in this work is to deal with Japanese personal names. We need to get a picture of Japanese personal names in general to deal with multiple data sources (like the introduced publication metadata sources in chapter <ref>) which may represent the same name with different scripts or transcription methods. The dictionary ENAMDICT[will be introduced in section <ref>] will be very helpful when it comes to extracting and verifying name information. § STRUCTURE OF JAPANESE NAMESHaving the urge to name things is part of the human nature. Names make it easy to refer to things, people or any other object in this world. When it comes to name giving, history shows a development in the Japanese society.Japanese names are divided into family and given name, similar to the system in the Western culture. When Japanese write their name in kanji they put the family name first, followed by the given name (usually without leaving spaces between them), for example 中村武志[this name can also be found in the ENAMDICT dictionary as particular person born in 1967] (Takeshi Nakamura).[We intentionally use the expressions “given name” and “family name” instead of “first name” and “last name” in this work because of these circumstances and to avoid confusion.] While introducing themselves, they often tell their family name and skip the given name.When Japanese refer to others, they have many name particles they put after a name to express the relationship to the other person. There is the neutral san, chan for children, kun particular for boys or sensei for teachers and doctors. (<cit.>, p. 18-19) Kagami (<cit.>, p. 913) writes about Japanese personal names. Only the samurai and nobility were allowed to carry family names before the Meiji Restoration in 1868. Merchants carried shop names instead (recognizable by the suffix -ya), for example Kinokuniya (shop name) Bunzaemon (given name). Then everybody had to pick a family name after the Meiji Restoration. Approximately 135000 family names are recognized now. The most common family names are Suzuki, Satō, Tanaka, Yamamoto, Watanabe, Takahashi, Kobayashi, Nakamura, Itō, Saitō and others.“In the feudal age, first and second given names were used as male names. The first name was Kemyoo which was the order of brothers, and the second name was the formal name given at the coming of age ceremony (genpuku), e.g. the name of a famous general in 12c.: Minamoto (family name) no (of) Kuroo (kemyoo) Yoshitune (formal given name), and before the genpuku ceremony, he was called by Yoomyoo (child name) Ushiwakamaru.” (<cit.>, p. 913)While there were no restrictions to the number of personal names visible until the Meiji Restoration, due to modernization, Japanese people got the restriction to carry only one given and one family name. (<cit.>, p. 167-169)Some indicators for assigning the gender to a name also exist. The suffixes -ko (e.g. Hanako), -mi (Natsumi) and -yo (Yachiyo) indicate a female name. Male names are harder to identify because they have no fixed pattern. The suffix -o (Kazuo) mostly belongs to a male name though.Family names often consist of two kanji characters, rarely of one or three characters. (<cit.>, p. 913) Eschbach-Szabo (<cit.>, p. 157-309) dedicates an elaborate chapter to Japanese personal names. Compared to the Chinese system, the Japanese naming system shows more tolerance. Several readings are left besides each other, formal rules are not always applied in practice. Japanese apprehend names mainly visually by the characters, secondarily by the reading and sound. This is why several readings for a written name are still acceptable in the modern Japanese world.In the feudal system, names were needed to determine the position and roles of a person in the family and the society rather than characterizing him or her as an individual. Japan has an open naming system which allows adding new names. This is a difference to the exclusive name lists in Germany or France. (<cit.>, p. 157-166)Even the apparently simple kanji 正 has a lot of possible readings: Akira, Kami, Sada, Taka, Tadashi, Tsura, Nao, Nobu, Masa. We can see the same phenomenon in recently approved kanji too. When we see 昴 we cannot be sure whether it is read Kō or Subaru. (<cit.>)“Conversely, it often happens that one does not know to write a name of given pronunciation. For example, Ogawa can be written 尾川 or 小川. In Japan, when two people meet for the first time, they exchange business cards. This custom often baffles foreigners, but for Japanese it is a ritual with practical purpose: Japanese do not feel at ease until they see how a name is spelled out in kanji.” (<cit.>)Figure <ref> illustrates the problem. The cashier tries to read the customer's name and cannot determine the right name. According to the customer's reaction, his first two trials Hiroko and Yūko seem to be wrong. Ogawa considers the name polygraphy as a reason why the creation of new name characters is still allowed. Some characteristics of the Japanese naming system are: * only little renaming of people* semantic variance (names indicate different meanings/attributes)* admission of foreign elements (foreign names get assimilated)* possibility of polygraphic writing* diversity of writing (many scripts usable, weak orthographic normalization)* number of personal names for one personIn academic circles a Sino-Japanese reading led to a more reputable name. So the famous linguist 上田万年 from the Meiji era became known as Kazutoshi Ueda AND Mannen Ueda (Mannen is the Sino-Japanese on reading, Kazutoshi is the Japanese kun reading). Modern guidebooks underline that maybe one has to take a loan word from another language to find the corresponding reading for a name in kanji. For example, 宇宙 could be read as Kosumo (from the Greek word for cosmos) instead of Uchū. Also ノイ (Noi), derived from the German word “neu” (new), became a Japanese given name. Another imaginable name is “Sky” written as 空海 (meanings: 空 Sky, 海 sea) and transcribed as Sukai (actually kūkai). This would finally show the impact of globalization also on the Japanese naming system. If one has lived in Japan for a while and wants to adapt or register his or her Western name, one can choose corresponding kanji either by meaning or reading of the original name. Another possibility is transcribing the name with katakana.(<cit.>, p. 170-171, 305-309)The name Anna exists in many cultures. The girls in figure <ref> are both called Anna. Both turn around when they hear their name and respond in their mother tongue (“Yes!” and “Hai!”, respectively). One principle of Japanese name giving is ateji. Ateji (当て字) means “appropriate characters”. It says Japanese try to find characters with good, positive meanings for their children's name. Examples are 愛子 (愛: ai, love; 子: ko, child), 夏美 (夏: natsu, summer; 美: mi, beauty) or 正 (Tadashi, correct, honest). There is also a list with characters that are allowed but should be avoided because of bad associations. Characters like 蟻 (ari, ant), 苺 (ichigo, strawberry), 陰 (kage, shadow), 悪 (aku, bad/evil) belong to this list. (<cit.>, p. 172-176)A particular case drew public attention from June 1993 to February 1994 when Shigeru Satō wanted to call his son Akuma, written as 悪魔 (devil/demon). The civil registry office declined the registration after some discussion because they were worried about other children teasing him. The father went to court but the judges also declined the wish. Although the father wanted to give his son a unique, rememberable name, the judges saw a possible problem in his individual identification process and also getting teased (ijime) by other children in school someday. Then Satō tried to choose other characters while keeping the reading Akuma. But also changing the name partly into man'yōgana (亜久魔[亜: a, asia; 久: ku, long; 魔: ma, ghost]) did not change anything about the declination because of the phonological equality implying the same negative associations. Thereupon the father picked the character 神 (god) and its unusual reading Jin. Even though Shintoistic gods can be good or evil, the civil registry office accepted the name. Satō announced his intention to keep calling his son Akuma anyway. So a new (yet unofficial) reading for a character might be established. (<cit.>, p. 271-278)An article of “Japan Today” from December 2012 shows that there is still a debate about this subject.“[...]Shinzo Abe[Shinzo Abe is currently Prime Minister of Japan (since 2012-12-26)], the leader of the Liberal Democratic Party made a stand against kirakira[literally “sparkling”, Japanese expression for flashy names] names last week when he stated that giving a child a name like Pikachu[Pikachu is one of the most famous Pokémon (Pokémon is a video game franchise owned by Nintendo)], which could be written something like 光宙 (`light' and `space'), is tantamount to child abuse, saying: `Children are not pets; we have to provide guidance for parents who would name their child in such a way.' ”(<cit.>)Despite regulations, the discussion about the culture of name giving does not seem to have ended yet. Japanese comics like the one in figure <ref> suggest a happy-go-lucky life if one has a common everyday name like Keiko. Today's registration of names allows 2983 kanji for given names, 4000 kanji for family names, 700 man'yōgana, 46 hiragana and 46 katakana. There are still people whose names are written with the obsolete kana syllabary hentaigana which has been prohibited in 1948 (<cit.>, p. 176-177; <cit.>). Regarding this variety of characters (and readings) it is not surprising that even well educated Japanese have problems reading certain names too, respectively they cannot be sure that the chosen reading is the correct reading in the current situation. Forbidden is the usage of geometrical and punctuation signs. The sign ◯ (maru) is an example of such a forbidden one. Also forbidden is the usage of Latin characters (rōmaji) at the registration of a name. Rōmaji can be used privately, though. (<cit.>, p. 176-177)Names can be changed by marriage, adoption or getting a pseudonym or special posthumous name. Titles can be acquired too. (<cit.>, p. 251)After disestablishing the patriarchal ie system in which a man (for example the husband) is the dominating householder of a family, the family name has not been focused on the affiliation to a family anymore but has been focused on the couple living together in joint lives. (<cit.>, p. 253-255) Writing a Japanese name can be ambiguous. While the name written in kanji is definite, displaying it in Latin characters leads to several possibilities. Japanese themselves usually write their name using kanji. To find matching authors in the DBLP[we will discuss the DBLP project more detailed in section <ref>], it will be crucial for us to have names in Latin characters later on (in chapter <ref>) because the standard encoding format of the file containing the main data of the DBLP project is ISO 8859-1 (Latin-1). We sometimes talk about “kanji names” or “names in kanji representation” in this work. Although the expression does not suggest it, they shall include all names in Japanese characters, ergo names in kanji, hiragana and katakana. § ENAMDICTTo automatically detect where a Japanese family name in kanji notation ends and the given name begins, we should factor a name dictionary into our work. It is important that this dictionary includes the names written in kanji and a clear transcription for them in Latin characters. A useful dictionary for our purposes is ENAMDICT.ENAMDICT <cit.> is a free dictionary for Japanese proper names, maintained by the Monash University in Victoria (Australia). The Electronic Dictionary Research and Development Group[<http://www.edrdg.org/>] owns the copyright. In 1995, ENAMDICT became an independent project by dividing the universal dictionary EDICT into two projects. ENAMDICT contains person names and non-person names like places and companies as well. Table <ref> shows the online statistics about the content of the ENAMDICT file. We will call the categories “name types” in subsequent chapters. “A proper name is a word or group of words which is recognized as having identification as its specific purpose, and which achieves, or tends to achieve that purpose by means of its distinctive sound alone, without regard to any meaning possessed by that sound from the start, or aquired by it through association with the object thereby identified.” (<cit.>, p. 73)these intern abbreviations occur again when we construct a database for Japanese names in chapter <ref>CHAPTER: PUBLICATION METADATA SOURCES百語より一笑 Hyaku go yori isshō(A smile is more worth than a hundred words.)Japanese sayingThis chapter gives an overview of the publication metadata sources that we will need later. We take a look at these sources because we will discuss a way to extract metadata information from one source containing Japanese papers and import them into another source in chapter <ref>. § DIGITAL LIBRARY OF THE IPSJThe IPSJ is a Japanese society in the area of information processing and computer science. It was founded in April 1960 and, by its own account, helps evolving computer science and technology and contributes new ideas in the digital age. It regularly publishes the magazine “Information Processing” (jōhō shori) and a journal, holds symposiums and seminars, Special Interest Groups issue technical reports and hold conferences. It is also the Japan representative member of the IFIP and established partnerships with the IEEE, ACM and other organizations. -2 IPSJ develops drafts of international standards and Japanese industrial standards as well. Eight regional research sections are widespread over Japan. IPSJ had over 17000 members in March 2011. (<cit.>; <cit.>) The IPSJ provides a Digital Library[<https://ipsj.ixsq.nii.ac.jp/ej/>, accessed at 2012-10-10] (referenced as IPSJ DL in this work) where everybody can search Japanese papers in the field of computer science. The search page can be displayed in Japanese and English, most papers are written in Japanese. Free papers are accessible in PDF format, non-free can be bought. A tree view provides the order structure of the papers and there is a keyword search available. We are especially interested in the metadata export functions, though. The online application offers following export formats: * OAI-PMH* BibTeX* OWL SWRC* WEKO ExportFor our purposes the OAI-PMH is the most suitable solution because we can send simple HTTP requests to the server and get publication metadata as a result. It “provides an application-independent interoperability framework based on metadata harvesting” (<cit.>) and consists of two groups of participants. Data Providers can be servers hosting and supplying the metadata. Service Providers take the harvester role and process the recieved metadata from the Data Provider. The application-independent interoperability is achieved by using XML as basic exchange format.[further information about XML can be found in <cit.>] Arbitrary programs can parse XML input data very easily, so can we.While accessing the server, the data can be extracted in several ways. We can either access an OAI-PMH repository by the repository name, the metadata format prefix of the record and a unique identifier[Figure <ref> shows a part of the result of such a request.] or get a list of records with only one request. A request for a list of records looks like this: []http http: //ipsj.ixsq.nii.ac.jp/ej/ ?action=repository_oaipmh verb=ListRecordsmetadataPrefix=oai_dc It may also contain a start date and an end date or a resumption token. The headers of records include a corresponding time stamp. The server's response to a request offers only 100 publications. We need this resumption token because it determines the point where we resume the harvest.In the beginning and for debugging, it was more comfortable to increment a counter that acts as the unique identifier and send requests for single entries with the respective ID multiple times. Fortunately, the entries can be addressed by such an integer ID (plus some constant name): []http2 http: //ipsj.ixsq.nii.ac.jp/ej/ ?action=repository_oaipmh verb=GetRecord metadataPrefix=oai_dc(*@identifier@*)=oai:ipsj.ixsq.nii.ac.jp:(*@27130 @*)The last entry containing real publication metadata has the suffix integer 87045 in its ID.[The maximum ID was determined at 2012-11-13. Due to new publications the value is presumably higher in the future.] After that some entries with status deleted follow. If we continue requesting even higher IDs, we soon get only a reply with the error code idDoesNotExist anymore, implying there are no publications with higher IDs. We will discuss the implementation of an OAI-PMH harvester for the IPSJ DL in section <ref>. § DBLP PROJECTThe DBLP[<http://dblp.uni-trier.de>, accessed at 2012-10-10] is a worldwide known database for publication metadata in the field of computer science. Ley <cit.> gives a brief explanation of the DBLP, additional information is extracted from the online DBLP FAQ <cit.>. It was started in 1993 as a test server for web technologies and named “Database systems and Logic Programming” in the beginning. But it grew and became a popular web application for computer scientists. The Computer Science department of the University of Trier founded the project, since summer 2011 it is a joint project of Schloss Dagstuhl - Leibniz Center for Informatics and the University of Trier. “For computer science researchers the DBLP web site is a popular tool to trace the work of colleagues and to retrieve bibliographic details when composing the lists of references for new papers. Ranking and profiling of persons, institutions, journals, or conferences is another sometimes controversial usage of DBLP.” (<cit.>)The publication metadata is stored in the XML file dblp.xml[available at <http://dblp.uni-trier.de/xml/>] containing more than 2 million publications and exceeding a size of 1 GB (state of October 2012). An excerpt of the beginning of dblp.xml can be found in the appendix section <ref>.The header dictates ISO-8859-1 (Latin-1) as encoding of the file. Considering that we want to import Japanese names in kanji (which are not included in Latin-1) we must handle that issue somehow. We will discuss the solution in section <ref>. The web front end of the DBLP provides an overview of coauthor relationships by a Coauthor Index (see figure <ref>). The Coauthor Index can be found at the author's page after the list of the author's publications itself. It shows all coauthors, common papers and categorizes the coauthors into groups that worked together by giving the author names corresponding background colors. In his diploma thesis Vollmer <cit.> gives useful hints in terms of converting the dblp.xml file to a relational database. He also compares the performance of several relational database management systems for this conversion. The DBLP team developed a special format for the integration of new publications. It is called Bibliography Hypertext (BHT), is based on HTML and similar to the HTML code of the tables of contents (TOCs) at the DBLP website. An example of a publication list in BHT format can be found in the appendix in section <ref>. A BHT file has the following structure. The header (text between h2 tags) contains the volume, the number/issue and the date of issue. A list of corresponding publications follows next. The list is surrounded by a beginning and a closing ul tag, single publication entries start with a li tag. A comma is used for the separation of authors while there should be a colon after the last author name. Then comes the title which has to end with a period, question mark or exclamation point. The next line provides the start and end page in the volume/issue. At last, an optional URL can be added by an ee element to specify an “electronic edition” for a paper. Some guidelines need to be considered, too:* there is no closing li tag * initials should be avoided (full name is preferred) * titles with only upper case letters should be avoided * “0-” is the default page number value if the page information is missingThe BHT file may contain additional information. For example, conference proceedings may have more headers to achieve a better clarity. But it should be as close to the proposed format as possible to guarantee an easy import without unnecessary burdens. (<cit.>; <cit.>, “What is the preferred format to enter publications into DBLP?”)We will extend the original format in section <ref> to satisfy our needs in the context of Japanese papers. CHAPTER: PERSONAL NAME MATCHING“The important thing is not to stop questioning;curiosity has its own reason for existing.”(Albert Einstein)After looking at transcription systems, Japanese personal names and publication metadata sources, we will now have to look at Personal Name Matching to enable us to deal with the Japanese names extracted from the metadata sources. First we will discuss Personal Name Matching in general and then problems of Personal Name Matching for Japanese names in particular.The expression Personal Name Matching comes from the work by Borgman and Siegfried <cit.> and is used here as in the extended definition from Reuther's work (<cit.>, p. 48-51). Borgman and Siegfried only talk about synonyms. Synonyms are possible names for the same person. Reuther extended the definition by also including homonyms. A name is a homonym if it can belong to several persons. Personal Name Matching is known by other titles in literature, too. Niu et al. <cit.> discuss Cross Document Name Disambiguation:“Cross document name disambiguation is required for various tasks of knowledge discovery from textual documents, such as entity tracking, link discovery, information fusion and event tracking. This task is part of the co-reference task: if two mentions of the same name refer to same (different) entities, by definition, they should (should not) be co-referenced. As far as names are concerned, co-reference consists of two sub-tasks: * name disambiguation to handle the problem of different entities happening to use the same name;* alias association to handle the problem of the same entity using multiple names (aliases).”(<cit.>) On et al. <cit.> formally express their Name Disambiguation problem as follows: “Given two long lists of author names, X and Y, for each author name x (∈ X), find a set of author names, y_1,y_2, ..., y_n (∈ Y) such that both x and y_i (1 ≤ i ≤ n) are name variants of the same author.” (<cit.>)In contrast to the previous definitions Han et al. <cit.> define Name Disambiguation like this:“Name disambiguation can have several causes. Because of name variations, identical names, name misspellings or pseudonyms, two types of name ambiguities in research papers and bibliographies (citations) can be observed. The first type is that an author has multiple name labels. For example, the author `David S. Johnson' may appear in multiple publications under different name abbreviations such as `David Johnson', `D. Johnson', or `D. S. Johnson', or a misspelled name such as `Davad Johnson'. The second type is that multiple authors may share the same name label.For example, 'D. Johnson' may refer to `David B. Johnson' from Rice University, `David S. Johnson' from AT&T research lab, or `David E. Johnson' from Utah University (assuming the authors still have these affiliations).”(<cit.>)The citations above show that there are many expressions for Personal Name Matching (or sub-categories) which are not equally used by different authors. Niu et al. and On et al. restrict Name Disambiguation to finding synonyms, Han et al. include homonyms in their definition.Even more related expressions can be found in literature. As mentioned, we will use Personal Name Matching in this work as Reuther uses it.The main aspect of Personal Name Matching is handling synonyms and homonyms. Trying to express the problems formally leads to the following description: Let X be a set of persons, especially characterized by their names, in a certain data set and P a set of all existing persons. We are also being given a function Label(x_i) → String and a relation Entity = ((x_i,p_j) | p_j is the real person concealed by x_i)_X × P. The actual problems can be described as* ∀ x_i,x_j ∈ X ∃ (x_i,p_m),(x_j,p_n) ∈ Entity | Label(x_i) ≠ Label(x_j) ∧ p_m = p_n * ∀ x_i,x_j ∈ X ∃ (x_i,p_m),(x_j,p_n) ∈ Entity | Label(x_i) = Label(x_j) ∧ p_m ≠ p_nwith 1 ≤ i ≤ |X|; i ≤ j ≤ |X|; 1 ≤ m,n ≤ |P|. Case <ref> checks for each person x_i from the person set X whether another person x_j from X exists, so that their name labels are different (Label(x_i) ≠ Label(x_j)) but the person is the same (p_m = p_n). So this case covers the synonym problem because the same person has several names here.Case <ref> checks for each person x_i from the person set X whether another person x_j exists in X, so that their name labels are equal (Label(x_i) = Label(x_j)) but the persons behind the names differ (p_m ≠ p_n). So this case covers the homonym problem because the same name is taken by several people.The problem Personal Name Matching arises because such a relation Entity usually does not exist and needs to be approximated as good as possible:Entity^* = ((x_i,p_j) | sim(x_i,p_j) ≥θ⇒ p_j is concealed by x_i)_X × P≈ EntityThanks to appropriate similarity measurements and a matching threshold θ, we can find such a relation Entity^* which is approximately equivalent to the original relation Entity. The main task in Personal Name Matching is finding a good similarity measure for the described problem. (<cit.>, p. 52) Let us have a look at a vivid example.The birth name of the famous actor Michael Keaton is Michael John Douglas[see entry about Michael Keaton at <http://www.imdb.com/name/nm0000474/bio>].Keaton took a pseudonym because he could have been confused with the more famous actor Michael Douglas. Synonyms for Keaton are “Michael Keaton”, “Michael Douglas”,“Michael John Douglas”, “Michael J. Douglas”, “M. Keaton” or “M. J. Douglas”. -1On the other hand, when we hear the name “Michael Douglas” we cannot be sure which famous actor is referred to, because Michael Douglas is a valid name for both of them. Figure <ref> illustrates this Personal Name Matching problem with Michael Keaton. The process of Personal Name Matching can be divided into the following steps (<cit.>, p. 56-87): Standardization First the personal name data have to get standardized not to compare apples and oranges. Especially if the data source has a low data quality, this step is quite important. Otherwise, false or bad similarity values would be computed and assigned by the similarity function because the names are displayed by a different usage of upper and lower case letters or use national characters that are not used in the other name.Blocking We create blocks by assigning people to these blocks in this step. The goal is reducing the number of name comparisons and thereby improving the runtime. Depending on whether we are looking for synonyms or homonyms, the blocking method differs. Known methods are Sorted Neighbourhood <cit.>, clustering <cit.>, phonetic hashing (Soundex <cit.>, Phonix <cit.>, Metaphone <cit.>) or token based blocking <cit.>.Analysis This phase analyzes the names and contains the name comparisons. The task is finding possible synonyms and homonyms which will be evaluated in the next phase.Decision Model After the detection of possible synonyms and homonyms we have to evaluate the similarity of the names and make the decision to handle them as matches or not.Performance Measurement At last, we can evaluate the performance and correctness of the whole process. The purpose is finding out how reliable our data are and thinking about improvements to get an approximation to our optimal relation Entity that is as close as possible.Criteria for the evaluation of such a process are Precision and Recall (<cit.>, p. 75-81; <cit.>, p. 83-85). Let U be a set of items, R be the set of relevant items (e.g. synonyms) with R ⊆ U and A be the answer of a request. In our scenario, the request is usually the question “Is the item u ∈ U a synonym, or accordingly u ∈ R?”. Then we can define:Precision = |R ∩ A|/|A| Recall = |R ∩ A|/|R|Precision testifies whether the reported synonyms during the Name Matching process are really synonyms, Recall allows us to say whether there are synonyms which have not been found.We use a combination of the Jaccard Similarity Coefficient and Levenshtein Distance in our tool. Bilenko et al. <cit.> explain these string matching methods isolated. Given two word sets S and T, the simple Jaccard Similarity Coefficient is:Jac(S,T) = |S ∩ T|/|S ∪ T| [4] The Levenshtein Distance uses the operations replacement, insertion and deletion of a character and is defined by a matrix. Let s ∈ S and t ∈ T be words, a = |s| and b = |t| their lengths. Then we can define:D_0,0 = 0D_i,0 = i,1 ≤ i ≤ aD_0,j = j,1 ≤ j ≤ bD_i,j = min {[ D_i-1,j-1 ifs_i = t_i;D_i-1,j-1 +1 (replacement);D_i,j-1 +1 (insertion);D_i-1,j +1(deletion) ]. 1 ≤ i ≤ a, 1 ≤ j ≤ b.We modify the Jaccard Similarity Coefficient in a way that it classifies two set items as intersected if their Levenshtein Distance is lower than a certain threshold. In addition to the general Personal Name Matching, we must take the characteristics of Japanese names into account. Particularly the usage of kanji and several possibilities to transcribe a name make it hard to compare Japanese names.[see chapter <ref>] For example, we cannot compare kanji names from the IPSJ DL with the author names in DBLP. Even though kanji are suited best for name comparison it does not work here because the standard encoding of names in DBLP is “Latin-1” which does not support kanji natively.A big problem for our work is revealed by looking at the given name Akiko with its kanji representation 章子. As we can see in table <ref> 章子 has several possible readings besides Akiko (left column) and Akiko written in Latin characters does not determine a nonambiguous match in kanji (right column). The same problem applies to Japanese family names. Table <ref> presents the problem with Kojima as a family name example.CHAPTER: PREPARATION OF JAPANESE PAPERS FOR THE IMPORT INTO THE DBLP DATA SET大事の前の小事 Daiji no mae no shōji(Who wants to achieve big things must do the little things first.)Japanese sayingThis chapter explains the approach to process and combine the various data sources so that we can import Japanese publications in the end. We will proceed step by step to make the ideas behind the solution as comprehensible as possible. § GENERAL APPROACHFirst we will construct a table in a relational database containing information about Japanese names and their transcriptions by converting the ENAMDICT name dictionary. Then we set up a data structure for Japanese names that handles the problem of assigning a given and a family name to a newly instantiated author during parsing the publications of IPSJ DL. At last, we will discuss the actual and titular integration of Japanese papers into the DBLP data set including an explanation that shows how to create a harvester for the OAI-PMH protocol. § CONVERTING AN ENAMDICT FILE TO A RELATIONAL DATABASEThe first step towards being able to handle Japanese names is distinguishing given and family name in the input text. A relational database containing information about Japanese names and their transcriptions is useful for this task. The database should contain names in kanji, their transcriptions in hiragana and Latin characters and the name type to have a good match with the data source ENAMDICT and to provide all necessary name information we need.To fill the empty database, the ENAMDICT file needs to be analyzed and its data needs to be extracted.The entries usually have the form KANJI [TRANSCRIPTION[the transcription in hiragana is located at this place (we usually mean Latin characters when we talk about transcriptions, but not this time)]] /LATIN (TYPE)/ . We can take the following line as an example of an existing entry: 森田 [もりだ] /Morida (s)/ A parser should export the single entries. First it saves the text between the slashes and searches for the type of the entry. It must be assured that all person name types and no undesired or alleged types will be stored. Types can consist of the characters “s” (surname), “g” (given name), “f” (female name), “m” (male name), “u” (unclassified name), “p” (place name), “h” (full name of a particular person), “pr” (product name), “co” (company name) or “st” (station name). But only the types “s”, “g”, “f” and “m” are important in this case because the parser should only store person names in the database.One exception are the unclassified names and they need to be stored too because they can also contain person names.Using unclassified names carelessly leads to problems, though. On the one hand it is useful if you find a match for the given name but not for the assumed family name. Then it helps to find an unclassified name matching the assumed family name. On the other hand some unclassified names in the ENAMDICT file decrease the data quality of the database. The entry スターウォーズ /(u) Star Wars (film)/ shows that there are undesired names like film titles in the category “unclassified”. The example also reveals that there is no overall standard for an entry format. Analyzing the file leads to following observations:* text in round brackets might be type or additional commentary (see entry example above) * when only hiragana or katakana are used instead of kanji to display the Japanese name the transcription part is missing because it is not required (see entry example above) * the type information in brackets might actually consist of several type declarations, separated by commas * the type information might be placed before or after the transcription in Latin characters * one entry line might contain several possibilities to interpret the name, the example イブ /(f) Eve/(u) Ib/Ibu (f)/(m) Yves/ clarifies this aspectWe must consider these observations when we implement the parser.To handle the problems in <ref> and <ref> we can filter the contents in round brackets. One possibility is using a regular expression liketo filter all valid types. Regular expressions are powerful and popular tools for pattern matching. In our case we are looking for valid type expressions including commas to get rid of commentaries. After eliminating commentaries we also want to get rid of unwanted types like place names. So we filter again and only process desired types this way. To handle <ref> we just ignore missing transcriptions in square brackets. Our parser also needs to be flexible enough to deal with observation <ref> which means that it must expect the type(s) at two possible places (before and after the transcription in Latin characters). We can handle the last observation <ref> by using recursive function calls. We call the function that exports one entry with a modified parameter value within the function itself when there is more than one entry in the input line (noticeable by additional slashes). Before parsing we need to change the original encoding of the ENAMDICT file from “EUC-JP” to “UTF-8” to make it compatible with our program.During parsing a few inconsistencies in the syntax of the ENAMDICT file occurred: * there were four times no slash in the end of the entry: 甲子太郎 [かしたろう] /Kashitarou (m)* there was once an unnecessary closing bracket without an opening bracket: 近松秋江 [ちかまつしゅうこう] /Chikamatsu Shuukou) (h)/* there was once a backslash where a square bracket was supposed to be put: キルギス共和国 [キルギスきょうわこく\ /(p) Kyrgyz Republic/Kirghiz Republic/Instead of constructing a workaround for these problems we should rather correct the only few inconsistencies manually.[The author decided to contact Jim Breen and told him about the found inconsistencies. Mr Breen corrected them in the ENAMDICT master file. So the inconsistencies should not exist anymore in the latest downloadable version.] § A DATA STRUCTURE FOR JAPANESE NAMESWe will construct a class which is responsible for handling Japanese names and representing them in a convenient way. Therefore, it must be able to save the name in kanji and in at least one Latin transcription. The transcription is necessary to compare found authors in IPSJ DL with authors in the DBLP. The kanji name can be stored as additional author metadata in the DBLP later. Our goal is a standardized representation of a Japanese person. So first we can construct a simple helper class for a single name containing given and family name as strings. This class can be applied to both kanji and Latin names. Our Japanese person usually has these two name representations.When getting an input name from the IPSJ DL we try to determine the separation point and categorize the tokens into given and family names. The separation point can mostly be identified by white space or a comma between the words. The categorization is done by including information from ENAMDICT. Thanks to ENAMDICT's classification into name types we can use this information to categorize our input name tokens into given and family names. However, we have to cover some unusual cases too because IPSJ DL has no standardized way to provide names. So we get names in various formats. For example, there are entries in which the family name follows the given name directly without any separation markers. Then we can try to take advantage of upper and lower case letters assuming that an uppercase letter means the beginning of a new name token. But we must also be aware of existing input names like “KenjiTODA”. If we get a longer sequence of uppercase letters, this sequence is probably a family name. We can filter these names with a regular expression like(first character is an uppercase letter, followed by at least one lowercase letter, followed by at least three uppercase letters). We also have to recognize abbreviated names and normalize Latin names.Let us have a look at what we can observe about necessary transcription customizations. One peculiarity is that Japanese like to transcribe their names with an h instead of a double vowel. An example is “Hitoshi Gotoh”. The h symbolizes the lengthening of a vowel and is a substitute for o or u in this case. To enable our class to find names like this in ENAMDICT, we have to replace the h's lengthening a vowel by the vowel itself because ENAMDICT entries contain double vowels instead of h's with this semantic function.Another observation is ENAMDICT's usage of the Hepburn transcription system throughout the entire dictionary. So we have to convert the name to match the Hepburn system and to check a name via ENAMDICT. The needed character replacements for a conversion into the Hepburn system are shown in table <ref> (see also figure <ref> in the appendix).In addition to the replacements from table <ref>, we must consider that names usually start with uppercase letters and replace “Tu”, “Ti”, “Sya” and so on by “Tsu”, “Chi”, “Sha”, etc. as well.The Japanese n is sometimes transcribed as m. If n is followed by b or p, this n is likely to be transcribed as m. The reason is a correlative modification in the pronunciation of n in these cases. For example, the family name Kanbe is often transcribed as Kambe in the IPSJ DL data set. -1Double vowels are sometimes completely dropped in some IPSJ DL author elements. While this might be okay for aesthetic reasons when transcribing the own name, it becomes a problem when we try to find a matching name in a dictionary like ENAMDICT. So we also have to check additional modified names. If there is a single vowel in the name, we must also check the same name whose vowel has become a double vowel. If several single vowels occur in a name, the number of names to be checked rapidly increases too.We have to pay special attention to the doubling of the vowel o because oo and ou are possible doublings for the single o. Doubling the vowel e leads either to ee or ei. All other double vowels are intuitive: a becomes aa, i becomes ii, u becomes uu. Taking “Gotoh” as an example we remove the h first and check a list of names via ENAMDICT. The list of names consists of “Goto”, “Gooto”, “Gouto”, “Gotoo”, “Gotou”, “Gootoo”, “Goutoo”, “Gootou” and “Goutou”. We can remove “Goto”, “Gooto” and “Gouto” from the list if we know that the h (representing a double vowel) has been removed before.If the input metadata contains a Latin and kanji representation of the author's name, we will try to find a match for these. Names in kanji usually do not have any separation mark, so we must distinguish given and family name by taking advantage of the ENAMDICT dictionary and checking the possible name combinations. Processing author names without kanji representation is okay but a missing Latin representation becomes a problem when it comes to actually integrating the publication into the DBLP data set because all DBLP data are supposed to have a Latin representation. The solution is a search for name candidates (we will discuss it more detailed in section <ref>).We cannot be sure that our name matching for Latin and kanji names always succeeds. Therefore, we add some status information to our Japanese name to get a chance to evaluate the outcome of the program. Possible status types are:okThe status “ok” means that given and family name have successfully been found in the name dictionary and (if available) the kanji names have successfully been assigned to their corresponding name in Latin characters. undefined/Latin name missingAn undefined status usually means that the Latin name is missing. A missing Latin name leads to a never changed name status. In these cases, the name in kanji usually exists anyway. abbreviatedThis is the status type for an abbreviated name like “T. Nakamura”. not found in name dictionaryIf this status occurs, the Latin name could not be found in the name dictionary. no kanji matching foundIf a kanji name has not been found in the name dictionary or could not be assigned to the Latin name, this status will occur. bad data quality in sourceAs the name suggests, this status means that the data quality of the publication metadata source is most likely bad. Our tool can handle some of these cases well by normalizing the name. possible name anomalyWe could have stumbled upon a name anomaly when we see this status type. During implementation this status was narrowed down to a possible name anomaly for abbreviated names. name anomalyThis status indicates a critical name anomaly. This is the only case in which the tool cannot even give a recommendation for given and family name. The output is the full name of the input data for both given and family name.In chapter <ref> we discussed synonyms and homonyms. With the strategies from above we can deal with synonyms pretty well. Yet, homonyms cannot be recognized this way and are not covered at all by our tool. § IMPORT INTO THE DBLP DATA SETTo be able to import the harvested data into the DBLP, we still need to make the existing publication data processable in an appropriate way for our program, construct a coauthor table for these data, compare publications from the Digital Library of the IPSJ with those available in the DBLP project and provide the new publication metadata for the DBLP adequately.§.§ Converting the DBLP XML File to a Relational DatabaseIt is important to convert the DBLP file dblp.xml to a relational database to gain an easier and more efficient access to the data while running our program. We are mainly interested in the basic publication metadata. So we will skip some non-publication records of the DBLP like www elements[see also section <ref> to get an example of a www record]. Our publication database table shall contain columns for an ID, the authors, title, publication year, journal title, journal pages and the volume. Whenever we come across the beginning of a publication type element (article, inproceedings, proceedings, book, incollection, phdthesis, mastersthesis) during parsing, we reinitialize the variables which store this metadata for the table columns. When we encounter the according XML end tag of the publication we add an SQL INSERT command to a batch of commands. This batch is regularly executed after processing a certain amount of publications. The regular execution of batches allows a better performance than sending single INSERT commands to the database server. There are some recommendations in the DBLP FAQ <cit.> for parsing the dblp.xml file. We use the Apache Xerces[<http://xerces.apache.org/>] parser instead of the standard Java SAX parser and need to increase the allocatable heap space for our parser.While parsing the DBLP file we can construct a table with coauthor relationships along with the DBLP publication table. This coauthor table stores two author names and a publication ID. The ID shows which publication has been written together by the authors and matches the ID in the DBLP publication table.New coauthor relationships will only be inserted if there are at least two authors mentioned in the metadata. If the metadata mentions more than two authors, every possible pair of authors will be inserted into the database. §.§ Implementation of an OAI-PMH HarvesterAs already explained in section <ref>, we access the OAI-PMH repository by the repository name and the metadata format prefix to get a list of publication metadata entries. The specification of OAI-PMH 2.0 <cit.> describes a possibility to retrieve a list of all metadata formats which a Data Provider has to offer. The HTTP request []http3 http: //ipsj.ixsq.nii.ac.jp/ej/?action=repository_oaipmhverb=ListMetadataFormatsinforms us that there are two metadata formats called oai_dc and junii2. oai_dc[<http://www.openarchives.org/OAI/2.0/oai_dc.xsd>, version from 19.12.2002] is the standard Dublin Core format all Data Providers provide, also traceable in the protocol specification. The “Implementation Guidelines for the Open Archives Initiative Protocol for Metadata Harvesting” <cit.> classify the metadata format oai_dc as mandatory. The name junii2[<http://irdb.nii.ac.jp/oai/junii2.xsd>, state of 2012-10-10] suggests that it is a self-developed format of the National Institute of Informatics (in Tokyo). Comparing these two in IPSJ DL, we notice that junii2 provides a more accurate description of the data, for example regarding additional XML attributes telling us whether the element value is English or Japanese. This additional information is helpful when we process the data in a later step and is missing in the oai_dc representation of the IPSJ server's data. So we will take the metadata prefix junii2 as initial point for harvesting the server's metadata. Figure <ref> shows an according metadata example (also compare figure <ref>). The harvesting includes the following steps: * we load the DBLP publication, coauthor relationship and the ENAMDICT data into the RAM* we access the IPSJ server to get publication metadata* we parse the accessed XML metadata (concerning the thoughts from section <ref>) and store the needed publication data temporarily in the RAM.* we add the parsed publication to an SQL command batch to insert the metadata into a relational database (the batch is regularly executed)* we create a BHT file for the parsed publication[see section <ref>]* at the end we go into all directories with BHT files and concatenate them to one bigger BHT fileDuring the implementation and testing, some exceptional incidents occurred. We try to cover them besides the expected difficulties like Personal Name Matching and transcriptions. For example, we get “NobukazuYOSHIOKA” as a full input name. Algorithm <ref> shows a way to handle these unusual input data. Japanese sometimes write their family names in upper case letters to distinguish given and family name. Another observation during testing the program and checking the data is the following. Searching the Japanese given name “Shin'ichi” in the DBLP we notice that there is no uniform way to store certain names in the database. We find “Shin'ichi Aihara” but also “Shin-ichi Adachi” along with other results indicating the same phenomenon. So we see the apostrophe and the hyphen are used equally as syllable separators (we discussed the syllable separation in chapter <ref>).Comparing the author “Shinichi Horiden” from the IPSJ data set and the one from the DBLP data set we can assume they are the same person because they have common coauthors (e.g. Kenji Taguchi and Kiyoshi Itoh) in both databases. The IPSJ data set tells us that the name written in kanji is 本位田真一. We are interested in the part 真一 (Shin'ichi) because we get to know that the separator symbol is sometimes missing. The kanji indicates the syllables shi-n-i-chi, especially focused on n and i instead of ni. We would expect an additional separator symbol for a clear (nonambiguous) transcription; but obviously, it has been dropped in this case.A separator symbol can also be found when some double vowels occur. For example, we find “Toru Moto'oka” (元岡達) instead of “Toru Motooka”. This makes it easier to identify the reading of a single kanji (元 moto, 岡 oka, 達 toru). When a separator symbol is needed for a clear transcription, an apostrophe is used as separator symbol in ENAMDICT. While ENAMDICT always uses an apostrophe as separator symbol, DBLP and IPSJ DL use an apostrophe, a hyphen or the separator symbol is missing. We must consider these differences in the data sources for a successful import. For an easier name matching between names in the ENAMDICT and IPSJ DL data set we can add names containing an apostrophe once as they are and once without apostrophes to the relational database when we parse the ENAMDICT file to store person names in a relational database.Our tool has a statistics class to get an overview over the parsed input data and the quality of the output data. We will have a look at these statistics created after the harvest. There are 81597 records with publication metadata and 8562 records which are marked as deleted in the parsed data. Figure <ref> shows a visualization in pie chart form.[4]The publication types are declared as “Technical Report”, “Conference Paper”, “Journal Article”, “Departmental Bulletin Paper” or “Article” (compare the table <ref> and figure <ref>).The statistics also reveal that 74971 publications are published in Japanese, only 4456 in English (compare the pie chart in figure <ref>). Our tool detects 1325 publications which are already included in DBLP. A publication is considered found in both databases if the title is the same and at least one author is the same.The most interesting statistics for our work are these about the evaluation of the quality of author name assignments (compare the bar chart in figure <ref>):Fortunately, 180221 of 231162 author names could be matched successfully. There are many reasons for the remaining uncovered cases.9073 Latin names could not be found in the name dictionary ENAMDICT and 14827 name matchings between the names' Latin and kanji representations did not succeed.These names might be missing at all in the dictionary, delivered in a very unusual format that the tool does not cover, or might not be Japanese or human names at all. Of course, Japanese computer scientists sometimes also cooperate with foreign colleagues but our tool expects Japanese names and is optimized for them.Both IPSJ DL and ENAMDICT provide katakana representations for some Western names. However, katakana representations for Western names are irrelevant for projects like DBLP. But for instance, Chinese names in Chinese characters are relevant. Understandably, our tool does not support any special Personal Name Matching for Chinese names yet because our work is focused on Japanese names. The tool does not take account of the unclassified[compare section <ref>] names of ENAMDICT by default. We can increase the general success rate of the Name Matching process by enabling the inclusion of unclassified names in the configuration file but the quality of the Name Matching process will decrease because the correct differentiation between given and family name cannot be guaranteed anymore. An unclassified name may substitute a given or a family name.There are 1203 entries that were qualified as “bad data quality in publication metadata source”. They might be handled alright but they are particularly marked to indicate that these cases should also be reviewed manually before any import action is performed.The numbers of abbreviated names, possible name anomalies and name anomalies are very low. While processing author names which will be later qualified as “possible name anomaly”, the tool cannot decide whether the assignment has been correct or the name is an anomaly. “Name anomalies” are critical anomalies that could not be categorized into any other status.There could be a few uncovered flaws, for example HTML orcode in titles. We must be aware of those when we do the actual import into the DBLP data set.§.§ Creation of BHT Files for Japanese PapersWe will discuss the creation of BHT files and important extensions for the BHT format that fit the requirements of Japanese papers well, based on our knowledge from section <ref>. As mentioned, the header dictates ISO-8859-1 (Latin-1) as encoding of the file dblp.xml. Ley's work <cit.> reveals that we can use XML/HTML entities to solve this problem. Authors have person records in the DBLP providing additional information. For example, we can find the following entry for Atsuyuki Morishima (森嶋厚行) in the XML file: [language=XML] <www mdate="2008-02-20" key="homepages/m/AtsuyukiMorishima"> <author>Atsuyuki Morishima</author> <title>Home Page</title> <url>http://www.kc.tsukuba.ac.jp/ mori/index.html</url> <note> #x68EE; #x5D8B; #x539A; #x884C;</note> </www>We must extend the BHT format to fulfill the requirements and add extra metadata for authors, title and relevant process information. The author talked to members of the DBLP team personally and got the permission to extend the original BHT format to enable us to adapt the format to Japanese papers. Our additions are well formed XML elements. We must substitute all non-ASCII[ASCII is a encoding scheme for characters of the alphabet and consists of 128 characters.] characters by escape characters (XML entities) to ensure the compatibility for DBLP. The additional elements are:* Every author that has a kanji representation in its metadata gets an originalname element: [language=XML] <originalname latin="Shinsuke Mori"> #x68EE;, #x4FE1; #x4ECB; </originalname>If available, the Latin representation is added as an attribute latin to avoid confusion on assigning the extra information to the right author later on. The element content has a fixed structure. The family name comes first, followed by a comma and the given name. * Every author gets a status information that evaluates the author name assignment. It is displayed by a status element: [language=XML] <status name="Shinsuke Mori">ok</status>The connected author is added as an attribute name. * If there is no Latin representation of the name of an author, we will add Latin name candidates to the BHT file: [language=XML] <namecandidates kanji=" #x83C5; #x8C37; #x6B63; #x5F18;">Shougu Sugatani, Seihiro Sugatani, Tadahiro Sugatani, Masahiro Sugatani, Shougu Suganoya, Seihiro Suganoya, Tadahiro Suganoya, Masahiro Suganoya, Shougu Sugaya, Seihiro Sugaya, Tadahiro Sugaya, Masahiro Sugaya, Shougu Sugetani, Seihiro Sugetani, Tadahiro Sugetani, Masahiro Sugetani, Shougu Sugenoya, Seihiro Sugenoya, Tadahiro Sugenoya, Masahiro Sugenoya</namecandidates>The connected kanji representation is added as an attribute kanji in the namecandidates element. We seek the kanji in ENAMDICT and output all possible name combinations in a comma separated list. * If the original language of the title is Japanese, we will add this title to the BHT file: [language=XML] <originaltitle lang="ja" type="Journal Article"> #x70B9; #x4E88; #x6E2C; #x306B; #x3088; #x308B; #x81EA; #x52D5; #x5358; #x8A9E; #x5206; #x5272;</originaltitle>The XML element originaltitle has the attributes lang (for the paper language) and type (for the publication type). * The tool searches the authors in DBLP and tries to find additional common coauthors in DBLP. If at least two of the main authors of the paper also worked with a certain other person (that is retrieved from DBLP), this person is added to the comma separated list. The Personal Name Matching of author names uses a combination of Levenshtein Distance and Jaccard Similarity Coefficient here. [language=XML] <commoncoauthors>Masato Mimura</commoncoauthors>* If the tool finds the paper in DBLP, we also add the DBLP key. Records, such as elements with publication metadata, have a unique key in DBLP. [language=XML] <dblpkey>conf/iscas/HiratsukaGI06</dblpkey> An example of a BHT file in SPF can be found in the appendix in section <ref> (also compare with the original BHT format in section <ref>). After we have finished parsing all Japanese papers, we concatenate the BHT files in SPF that belong together to one bigger BHT file all.bht. Publications, respectively BHT files, that belong together are recognizable by the directory structure. If they belong together, they will be in the same directory. We must simply go through the BHT root directory recursively. CHAPTER: CONCLUSION AND FUTURE WORK“Creativity is seeing what everyone else sees,but then thinking a new thought that has never beenthought before and expressing it somehow.”(Neil deGrasse Tyson) The integration of Japanese papers into the DBLP data set has revealed some major problems.The nonambiguous representation of Japanese names (and paper titles, etc.) is done by kanji while DBLP's standard encoding is Latin-1 and Japanese characters are only optionally added to the publications' metadata. This leads to the need of transcribing the Japanese names which in turn also evokes new problems because there is not the transcription but rather a lot of transcription possibilities.In addition to that, we must ensure a certain data quality even if one data source sometimes lacks this quality. Due to name matching with a name dictionary, format checking and conversions (if necessary), we can actually correct some flaws or at least assimilate the data into our project. The problem of synonyms is dealt with by transcription manipulations, homonyms could not be addressed in this work. Reuther (<cit.>, p. 159-164) describes an idea tohomonyms. We could extend our tool by a Coauthor Index[see section <ref>] as in DBLP for the publications of the IPSJ DL. The idea is based on the assumption that scientists often publish their papers with the same people as coauthors. If the coauthors match a certain coauthor group, the author is considered the same. -1 If the author's coauthors are not members of the expected coauthor groups, the author could be a different person than we expected and we might have a homonym here. The developed tool is usable and provides among relational databases customized Bibliography Hypertext (BHT) files as output data. Customizations were necessary to optimize the BHT files for Japanese papers and additional important metadata information. Desired but missing metadata like contributors or a short description of the content of a paper can be added without much effort because the relational database already contains these data, only the source code of Kankoukanyuu (our tool) needs to be extended by a few lines.Though having been created with care regarding correct and well-formed output data, it is not recommended to import the newly created BHT files unchecked. The DBLP team should check the files not to compromise the data quality of DBLP. There might still be undesired format anomalies in the BHT files. The DBLP team also needs to adapt their import system to the extended BHT format developed in this work for the actual import into DBLP.Titles might be in uppercase letters. This could be improved but we have to pay attention because a primitive solution will not work well. For example, we have to be aware of the popular usage of acronyms in computer science. So some words in uppercase letters can be correct. Our tool is optimized for the Digital Library of the IPSJ and their OAI-PMH metadata prefix junii2. It can easily be adapted to support the similar and commonly used metadata prefix oai_dc. So the tool would be able to handle other publication metadata sources that support OAI-PMH.The algorithm[algorithm mentioned in section <ref>] for detecting common papers in DBLP and IPSJ DL may be modified to achieve an even better comparison between the databases and detect more common papers.It would be useful to include a Chinese name dictionary in the future and extend the name search of our tool to cover Chinese names as well. -1One improvement in the future could be storing the most common names (for example, the 100 most common given and family names) in a separate data structure in the RAM. This way we can improve the runtime by often skipping the search in the huge name data.We can still increase the success rate of the Name Matching process too. One way is swapping kanji. A typical Japanese name has two kanji for the given name and two kanji for the family name. The family name shall precede the given name. However, this principle could be violated by the publication source. If the Name Matching is not successful, we may swap the first two for the last two characters and try to find a match again.A second advancement is the additional support of a special Latin character set that is used by Japanese. For instance, we can find the name “Ｋａｉ” instead of “Kai” in the metadata of IPSJ DL. They look very similar and both represent simple Latin letters but their character codes are different. So programs handle them differently. A simple (but yet unimplemented) substitution function can cover these rare and unusual cases.Another possibility to take advantage of this work is extracting the author names in kanji from the relational database. So the DBLP team can insert author metadata for already existing authors in DBLP. We can also have a look at what phases of the Personal Name Matching process[see chapter <ref>] have been implemented in this work and to which degree. There are actually different types of Personal Name Matching included in our tool:* we try to match transcribed Japanese names with their original kanji representation * we try to distinguish given and family name (name in kanji not necessarily needed) * if a paper is also found in DBLP, we will try to look for common coauthorsThe “Standardization” is accomplished by a normalization of the Latin input names at the beginning of the process. Kanji input names get trimmed by removing all whitespace. We do not have a “Blocking” phase as it is proposed by Reuther <cit.>. When searching a match between transcribed Japanese names with their original kanji representation we even go a contrary way and increase the number of comparisons by adding reasonable other transcriptions to the matching process. Due to efficient data structures and a comparatively small amount of Japanese papers (less than 100000), our tool has an acceptable runtime (the retrieval of the publication metadata from the IPSJ server takes much longer than processing it). In addition, the search for common coauthors will only be done if the author exists in DBLP. The phases “Analysis” and “Decision Model” are entangled in our tool. If we find a match between a (normalized or modified) input name and a name in the name dictionary, we will immediately consider them a successful match and continue parsing the metadata. When we try to find coauthors in DBLP, we take advantage of the combined Jaccard Levenshtein Distance as explained in chapter <ref>.Instead of checking the complete output data in the “Performance Measurement” phase, we could only take control samples while implementing, debugging, testing and improving our program. A broad manual check of approximately 90000 publications is not possible within the scope of a diploma thesis. The control samples had the expected and desired content but we cannot guarantee the correctness of the output. Under the assumption that ENAMDICT's entries are correct, the predicted Precision[see definition in chapter <ref>] should be about 1.0 because the tool probably does not produce many false positives. But we cannot say anything about the Recall because ENAMDICT does not cover all names that occur in IPSJ DL. All exceptions resulting from the limits of a name dictionary and a bad data quality are supposed to be handled by the status for author name assignments (described in section <ref>). This gives us the chance to manually handle the noted exceptions afterwards. All in all, this work is a first approach for an integration of Japanese papers into the DBLP data set and provides a not yet perfect but usable tool for this task. Some major obstacles are overcome.tocchapterReferences CHAPTER: THE TOOL§ ABOUT THE TOOLThe developed tool that is also part of this project is named Kankoukanyuu (刊行加入). Kankou means publication, kanyuu means admission. The whole name indicates the ability to import publications. The tool also allows the assimilation of imported publications, of course. The usable functionalities are:* Parsing the DBLP file dplb.xml and converting it to a MySQL database * Converting an ENAMDICT name dictionary file to a MySQL database * Harvesting the IPSJ server, processing the publication metadata and storing it in a MySQL database * Making the harvested publications ready for an import into the DBLP data set by making BHT files § USAGEThe tool has been developed and tested on a Linux system with Intel Core 2 Quad and 8 GB RAM in the local computer pool. It has to be executed by command line like this: []exec java -Xmx5400M -jar kankoukanyuu.jarThe parameter -Xmx5400M allows our program to allocate more than 5 GB RAM and store all necessary data in the RAM for an unproblematic execution. Possible command line arguments are: --parse-dblp, -d Parse dplb.xml and fill database tables--enamdict, -e Convert ENAMDICT dictionary file to a relational database--harvest, -h Harvest the IPSJ server, fill OAI-PMH data into databases and create BHT files (in SPF) - requires DBLP and ENAMDICT database tables from steps above--concatenate-bht, -b Concatenate BHT files in Single Publication Format to one bigger file (file all.bht will be created in every folder with BHT files) - requires BHT files in SPF from step above--all, -a Do all of the above--help, -help Show help text about usage of the toolThe configuration file config.ini allows us to change following parameters: * Database related parameters (in [db] section): URL (url), database name (db), user name (user) and password (password)* ENAMDICT related parameter (in [enamdict] section): location of ENAMDICT file (file)* ENAMDICT database related parameters (in [japnamesdb] section): database table name (table), decision whether to use unclassified names (useunclassifiednames)* DBLP related parameter (in [dblp] section): location of dblp.xml (xmlfile)* DBLP database related parameters (in [dblpdb] section): database table name for publications (dblptable), database table name for coauthor relationships (authorscounttable)* OAI-PMH database (contains output after harvest and parsing process) related parameters (in [oaidb] section): publication table (publicationtable), authors table (authorstable), titles table (titlestable), contributors table (contributorstable), descriptions table (descriptionstable)* Harvester related parameters (in [harvester] section): location for storing the harvest (filespath), start ID for harvester (minid), end ID for harvester (maxid), decision whether to use record lists (uselistrecords)* BHT export related parameters (in [bhtexport] section): location for BHT output files (path), decision whether to compute and show common coauthors (showcommoncoauthors)* Log related parameter (in [log] section): location of log files (path)A configuration example can be found in the appendix section <ref>. The system must support the Japanese language (meaning Japanese characters) to ensure a successful run. Kankoukanyuu does not use any Linux-only commands but has not been tested on Microsoft Windows yet. § USED TECHNOLOGIESThe tool itself has been written in Java, using the OpenJDK 6. The handling of databases is done by MySQL 5 and JDBC is used to provide MySQL functionalities within Java.External libraries are the Apache Xerces[<http://xerces.apache.org/>] parser and the MySQL Connector/J[<http://dev.mysql.com/downloads/connector/j/>]. The Fat Jar Eclipse Plug-In[<http://fjep.sourceforge.net/>] is used to deploy the complete project into one executable Java JAR[Archive which includes compiled Java code, project metadata, libraries and possibly more] file.The execution of Kankoukanyuu becomes more user-friendly this way because external libraries are already included and class paths for external libraries does not need to be specified anymore. § RUNTIMEMeasurement indicates the following approximated runtimes of Kankoukanyuu: We can make some observations. During the harvest, only ca. 30 minutes were spent on processing the harvested data, the rest is needed to retrieve the data from the Japanese server. Depending on whether the local file system or network file system was used, the runtime for the concatenation differs immensely. CHAPTER: ADDITIONAL MATERIAL§ DIFFERENCES BETWEEN THE TRANSCRIPTION SYSTEMS § OAI-PMH EXAMPLE: JUNII2 ELEMENT § BHT EXAMPLE PROPOSED BY DBLP[language=XML] Computer Languages, Systemsamp; Structures(journals/cl)<h2>Volume 34, Numbers 2-3, July-October 2008</h2> Best Papers 2006 International Smalltalk Conference <ul> <li>Wolfgang De Meuter: Preface. 45 <ee>http://dx.doi.org/10.1016/j.cl.2007.07.001</ee> <li>David R ouml;thlisberger, Marcus Denker,Eacute;ric Tanter: Unanticipated partial behavioral reflection: Adapting applications at runtime. 46-65 <ee>http://dx.doi.org/10.1016/j.cl.2007.05.001</ee> <li>Johan Brichau, Andy Kellens, Kris Gybels, Kim Mens, Robert Hirschfeld, Theo D'Hondt: Application-specific models and pointcuts using a logic metalanguage. 66-82 <ee>http://dx.doi.org/10.1016/j.cl.2007.05.004</ee> <li>Alexandre Bergel, St eacute;phane Ducasse, Oscar Nierstrasz, Roel Wuyts: Stateful traits and their formalization. 83-108 <ee>http://dx.doi.org/10.1016/j.cl.2007.05.003</ee> <li>Alexandre Bergel, St eacute;phane Ducasse, Colin Putney, Roel Wuyts: Creating sophisticated development tools with OmniBrowser. 109-129 <ee>http://dx.doi.org/10.1016/j.cl.2007.05.005</ee> <li>Luc Fabresse, Christophe Dony, Marianne Huchard: Foundations of a simple and unified component-oriented language. 130-149 <ee>http://dx.doi.org/10.1016/j.cl.2007.05.002</ee> </ul>This is a BHT example proposed by the DBLP team in the DBLP FAQ <cit.>.[<http://dblp.uni-trier.de/db/about/faqex1.txt>, accessed at 2013-01-15] § BHT EXAMPLE FILE CREATED BY KANKOUKANYUU[language=XML] <h2>Volume 52, Number 10, October 2011</h2> <ul> <li>Shinsuke Mori, Graham Neubig, Yuuta Tsuboi: A Pointwise Approach to Automatic Word Segmentation. 2944-2952 <ee>http://id.nii.ac.jp/1001/00078161/</ee> <originalname latin="Shinsuke Mori"> #x68EE;, #x4FE1; #x4ECB;</originalname> <status name="Shinsuke Mori">ok</status> <originalname latin="Graham Neubig"> #x30CB; #x30E5; #x30FC; #x30D3; #x30C3; #x30B0; #x30B0; #x30E9; #x30E0;,</originalname> <status name="Graham Neubig">no kanji matching found</status> <originalname latin="Yuuta Tsuboi"> #x576A; #x4E95;, #x7950; #x592A;</originalname> <status name="Yuuta Tsuboi">ok</status> <originaltitle lang="ja" type="Journal Article"> #x70B9; #x4E88; #x6E2C; #x306B; #x3088; #x308B; #x81EA; #x52D5; #x5358; #x8A9E; #x5206; #x5272;</originaltitle> <commoncoauthors>Masato Mimura</commoncoauthors> </ul>This is an output example of a BHT file in Single Publication Format (before the concatenation step), created by our tool. § EXCERPT FROM DBLP.XML[language=XML]dblp.xml <?xml version="1.0" encoding="ISO-8859-1"?> <!DOCTYPE dblp SYSTEM "dblp.dtd"> <dblp> <article mdate="2002-01-03" key="persons/Codd71a"> <author>E. F. Codd</author> <title>Further Normalization of the Data Base Relational Model.</title> <journal>IBM Research Report, San Jose, California</journal> <volume>RJ909</volume> <month>August</month> <year>1971</year> <cdrom>ibmTR/rj909.pdf</cdrom> <ee>db/labs/ibm/RJ909.html</ee> </article><article mdate="2002-01-03" key="persons/Hall74"> <author>Patrick A. V. Hall</author> <title>Common Subexpression Identification in General Algebraic Systems.</title> <journal>Technical Rep. UKSC 0060, IBM United Kingdom Scientific Centre</journal> <month>November</month> <year>1974</year> </article><article mdate="2002-01-03" key="persons/Tresch96"> <author>Markus Tresch</author> <title>Principles of Distributed Object Database Languages.</title> <journal>technical Report 248, ETH Z uuml;rich, Dept. of Computer Science</journal> <month>July</month> <year>1996</year> </article> ...§ CONFIGURATION FILE OF OUR TOOL[]config.ini [db] url=myserver db=mydbname user=myusername password=mypassword [japnamesdb] table=japnames useunclassifiednames=false [dblpdb] authorscounttable=dblpauthors dblptable=dblp [oaidb] publicationtable=oai_publications authorstable=oai_authors titlestable=oai_titles contributorstable=oai_contributors descriptionstable=oai_descriptions [enamdict] file=./enamdict [harvester] filespath=./files-harvester minid=1 maxid=100000 uselistrecords=true [dblp] xmlfile=/dblp/dblp.xml [bhtexport] path=./bht showcommoncoauthors=true [log] path=./log | http://arxiv.org/abs/1709.09119v1 | {
"authors": [
"Paul Christian Sommerhoff"
],
"categories": [
"cs.CL",
"cs.DL"
],
"primary_category": "cs.CL",
"published": "20170926163359",
"title": "Integration of Japanese Papers Into the DBLP Data Set"
} |
Exact path-integral evaluation of locally interacting systems: The subtlety of operator ordering Nobuhiko Taniguchi December 30, 2023 ================================================================================================ We propose a Bayesian regression method that accounts for multi-way interactions of arbitrary orders among the predictor variables.Our model makes use of a factorization mechanism for representing the regressioncoefficients of interactions among the predictors, while the interaction selection is guided bya prior distribution on random hypergraphs, a constructionwhich generalizes the Finite Feature Model. We present a posterior inferencealgorithm based on Gibbs sampling, and establish posterior consistencyof our regression model. Our method is evaluated with extensive experiments on simulated data anddemonstrated to be able to identify meaningful interactions in applications in genetics and retail demand forecasting.[Code is available at <https://github.com/moonfolk/MiFM>.]§ INTRODUCTIONA fundamental challenge in supervised learning, particularly in regression, is the need for learning functions which produce accurate prediction of the response, whileretaining the explanatory power for the role of the predictor variables in the model. The standard linear regression method is favored for the latter requirement, but it fails the former when there are complex interactions among the predictor variables in determining the response. The challenge becomes even more pronounced in a high-dimensional setting – there are exponentially many potential interactions among the predictors, for which it is simply not computationally feasible to resort to standard variable selection techniques (cf. <cit.>).There are numerous examples where accounting for the predictors' interactions isof interest, including problems of identifying epistasis (gene-gene) and gene-environment interactions in genetics<cit.>, modeling problems in political science <cit.> and economics <cit.>. In the business analytics of retail demand forecasting, a strong prediction model that also accurately accounts for the interactions ofrelevant predictors such as seasons, product types, geography, promotions, etc. plays a critical role in the decision making of marketing design. A simple way to address the aforementioned issue in the regression problem is to simply restrict our attentionto lower order interactions (i.e. 2- or 3-way) among predictor variables. This can be achieved, for instance, via a support vector machine (SVM) using polynomial kernels <cit.>, which pre-determine the maximum order of predictor interactions. In practice, for computational reasons the degree of the polynomial kernel tends to be small.Factorization machines <cit.> can be viewed as an extension of SVM to sparse settings where most interactions are observed only infrequently, subject toa constraint that the interaction order (a.k.a. interaction depth) is given.Neither SVM nor FM can perform any selection of predictor interactions, but several authors have extended the SVM by combining it with ℓ_1 penalty for the purpose of featureselection <cit.> and gradient boosting for FM <cit.> to select interacting features.It is also an option to perform linear regression on as many interactions as we can and combine it withregularization procedures for selection (e.g. LASSO <cit.> or Elastic net<cit.>). It is noted that such methods are still not computationally feasible for accounting for interactions that involve a large number of predictor variables.In this work we propose a regression method capable of adaptive selection of multi-wayinteractions of arbitrary order (MiFM for short),while avoiding the combinatorial complexity growth encountered by the methods described above.MiFM extends the basic factorization mechanism for representing the regression coefficients of interactions among the predictors, while the interaction selection is guided by a prior distribution on random hypergraphs. Theprior, which does not insist on the upper bound on the order of interactions among the predictor variables, is motivated from but also generalizes Finite Feature Model, a parametric form of the well-known Indian Buffet process (IBP) <cit.>. We introduce a notion of the hypergraph of interactions and show how a parametric distribution over binary matrices can be utilized to express interactions of unbounded order. In addition, our generalizedconstruction allows us to exert extra control on the tail behavior of the interaction order. IBP was initially used for infinite latent feature modeling and later utilized in the modeling of a variety of domains (see a review paper by <cit.>).In developing MiFM, our contributions are the following: (i) we introduce a Bayesian multi-linear regression model, which aims to account for the multi-way interactions among predictor variables;part of our model construction includes a prior specification on the hypergraph of interactions —in particular we show how our prior can be used to model theincidence matrix of interactions in several ways; (ii) we propose a procedure to estimate coefficients of arbitrary interactions structure; (iii) we establish posterior consistency of the resulting MiFM model, i.e., the property that the posterior distribution on the true regression function represented by the MiFM model contracts toward the truth under some conditions, without requiring an upper bound on the order of the predictor interactions; and (iv) we present a comprehensive simulation study of our model and analyze its performance forretail demand forecasting and case-control genetics datasets with epistasis.The unique strength of the MiFM method is the ability to recover meaningful interactions among the predictors while maintaining a competitive prediction quality compared to existing methods thattarget prediction only.The paper proceeds as follows. Section <ref> introduces the problem of modeling interactions in regression, and gives a brief background on the Factorization Machines.Sections <ref> and <ref> carry out the contributions outlined above.Section <ref> presents results of the experiments. We conclude with a discussion in Section <ref>.§ BACKGROUND AND RELATED WORK Our starting point is a model which regresses a response variable y∈ℝto observed covariates (predictor variables) x∈ℝ^D by a non-linear functional relationship. In particular, we consider a multi-linear structure to account for the interactions among the covariates in the model:(Y|x) = w_0 + ∑_i=1^Dw_ix_i + ∑_j=1^Jβ_j∏_i∈ Z_jx_i.Here, w_i for i=0,…,D are bias and linear weights as in the standard linear regression model, J is the number of multi-way interactions where Z_j, β_j for j=1,…,J representthe interactions, i.e., sets of indices ofinteracting covariates and the corresponding interaction weights, respectively. Fitting such a model is very challenging even if dimension D is of magnitude of a dozen,since there are 2^D - 1 possible interactions to choose from in addition to other parameters.The goal of our work is to perform interaction selection and estimate corresponding weights. Before doing so, let us first discuss a model that puts a priori assumptions on the number and the structure of interactions. §.§ Factorization MachinesFactorization Machines (FM) <cit.> is a special case of the general interactions model defined in Eq. (<ref>). Let J=∑_l=2^dDl and Z:=⋃_j=1^J Z_j=⋃_l=2^d {(i_1,…,i_l) | i_1<…<i_l; i_1,…,i_l∈{1,…,D}}. I.e., restrictingthe set of interactions to 2,…,d-way, so (<ref>) becomes:(Y|x) = w_0 + ∑_i=1^Dw_ix_i + ∑_l=2^d ∑ _i_1=1^D …∑ _i_l=i_l-1 + 1^D β_i_1,…,i_l∏_t=1^l x_i_t,where coefficients β_j := β_i_1,…,i_l quantify the interactions. In order to reduce model complexity and handle sparse data more effectively, <cit.> suggested to factorize interaction weights usingPARAFAC <cit.>: β_i_1,…,i_l:= ∑_f=1^k_l∏_t=1^l v_i_t, f^(l), where V^(l)∈ℝ^D× k_l, k_l ∈ℕ and k_l ≪ D for l=2,…,d.Advantages of the FM over SVM are discussed in details by <cit.>.FMs turn out to be successful in the recommendation systems setups, since they utilize various context information <cit.>. Parameter estimation is typically achieved via stochastic gradient descent technique, or in the case of Bayesian FM <cit.> via MCMC. In practice only d=2 or d=3 are typically used, since the number of interactions and hence the computational complexity grow exponentially. We are interested in methods that can adapt to fewer interactions but of arbitrarily varying orders.§ MIFM: MULTI-WAY FACTORIZATION MACHINE We start by defining a mathematical object that can encode sets of interacting variables Z_1,…,Z_J of Eq. (<ref>) and selecting an appropriate prior to model it.§.§ Modeling hypergraph of interactions Multi-way interactions are naturally represented by hypergraphs, which are defined as follows.Given D vertices indexed by S={1,…,D}, let Z={Z_1,…,Z_J} be the set of J subsets of S. Then we say that G=(S,Z) is a hypergraph with D vertices and J hyperedges.A hypergraph can be equivalently represented as an incidence binary matrix.Therefore, with a bit abuse of notation, we recast Z as the matrix of interactions, i.e.,Z∈{0,1}^D× J, where Z_i_1 j = Z_i_2 j = 1iff i_1 and i_2 are part of a hyperedge indexed by column/interaction j. Placing a prior on multi-way interactions is the same as specifying the prior distribution on the space of binary matrices.We will at first adopt the Finite Feature Model (FFM) prior <cit.>, which is based on the Beta-Bernoulli construction: π_j|γ_1, γ_2 iidBeta(γ_1,γ_2) and Z_ij|π_j iidBernoulli(π_j). This simple prior has the attractive feature of treating the variables involved in each interaction (hyperedge) in an symmetric fashion and admits exchangeabilility among the variables inside interactions. In Section <ref> we will present an extension of FFM which allows to incorporate extra information about the distribution of the interaction degrees and explain the choice of the parametric construction. §.§ Modeling regression with multi-way interactionsNow that we know how to model unknown interactions of arbitrary order, we combine it with the Bayesian FMto arrive at a complete specification of MiFM, the Multi-way interacting Factorization Machine. Starting with the specification for hyperparameters: σΓ(α_1/2,β_1/2), λΓ(α_0/2,β_0/2), μ𝒩(μ_0, 1/γ_0),λ_kΓ(α_0/2,β_0/2), μ_k𝒩(μ_0, 1/γ_0) for k=1,…,K.Interactions and their weights:w_i|μ, λ𝒩(μ, 1/λ) for i=0,…,D, ZFFM(γ_1,γ_2), v_ik|μ_k, λ_k 𝒩(μ_k, 1/λ_k) for i=1,…,D; k=1,…,K.Likelihood specification given data pairs (y_n,x_n=(x_n1,…,x_nD))_n=1^N:y_n|Θ𝒩(y(x_n, Θ),σ), where y(x, Θ):= w_0 + ∑_i=1^Dw_ix_i + ∑_j=1^J∑_k=1^K∏_i∈ Z_jx_i v_ik,for n=1,…,N, and Θ={Z,V,σ,w_0,…,D}. Note that while the specification above utilizes Gaussian distributions, the main innovation of MiFM is the idea to utilize incidence matrix of the hypergraph of interactions Z with a low rank matrix V to model the mean response as in Eq. <ref>. Therefore, within the MiFM framework, different distributional choices can be made according to the problem at hand — e.g. Poisson likelihood and Gamma priors for count data or logistic regression for classification. Additionally, if selection of linear terms is desired, ∑_i=1^Dw_ix_i can be removed from the model since FFM can select linear interactions besides higher order ones. §.§ MiFM for Categorical Variables In numerous real world scenarios such as retail demand forecasting, recommender systems, genotype structures,most predictor variables may be categorical (e.g. color, season).Categorical variables with multiple attributes are often handled by so-called “one-hot encoding”,via vectors of binary variables (e.g., IS_blue; IS_red), which must bemutually exclusive.The FFM cannot immediately be applied to such structures since it assigns positive probabilityto interactions between attributes of the same category.To this end, we model interactions between categories in Z, while with V wemodel coefficients of interactions between attributes. For example, for an interaction between “product type” and “season” in Z, V will have individual coefficients for “jacket-summer” and “jacket-winter” leading to a more refined predictive model of jackets sales (see examplesin Section <ref>).We proceed to describe MiFM for the case of categorical variables as follows. Let U be thenumber of categories and d_u be the set of attributes for the category u, for u=1,…, U. Then D=∑_u=1^U (d_u) is the number of binary variables in the one-hot encodingand _u=1^U d_u = {1,…,D}. In this representation the input data of predictors is X, a N× U matrix, where x_nu is an active attribute of category u of observation n.Coefficients matrix V∈ℝ^D× K and interactionsZ∈{0,1}^U× J. All priors and hyperpriors are as before, while the mean response (<ref>) is replacedby:y(x, Θ):= w_0 + ∑_u=1^Uw_x_u + ∑_k=1^K∑_j=1^J ∏_u ∈ Z_j v_x_uk.Note that this model specification is easy to combine withcontinuous variables, allowing MiFM to handle data with different variable types. §.§ Posterior Consistency of the MiFMIn this section we shall establish posterior consistency of MiFM model, namely:the posterior distribution Π of the conditional distribution P(Y|X), given the training N-data pairs, contracts in a weak sense toward the truth as sample size N increases.Suppose that the data pairs (x_n,y_n)_n=1^N∈ℝ^D×ℝare i.i.d. samples from the joint distribution P^*(X,Y),according to which the marginal distribution for X and the conditional distribution ofY given X admit density functions f^*(x) and f^*(y|x), respectively, with respect to Lebesgue measure. In particular, f^*(y|x) is defined by. Y=y_n|X=x_n,Θ^*𝒩(y(x_n, Θ^*),σ), where Θ^*={β^*_1,…,β^*_J,Z^*_1,…,Z^*_J},y(x, Θ^*):= ∑_j=1^Jβ^*_j∏_i∈ Z^*_jx_i, and x_n∈ℝ^D, y_n∈ℝ, β^*_j∈ℝ, Z^*_j⊂{1,…,D}.for n=1,…,N,j=1,…,J. In the above Θ^* represents the true parameter for the conditional density f^*(y|x) that generates data sample y_n given x_n, for n=1,…,N.A key step in establishing posterior consistency for the MiFM (here we omit linear terms since, as mentioned earlier, they can be absorbed into the interaction structure) is to show that our PARAFAC type structure can approximate arbitrarily well the true coefficients β^*_1,…,β^*_J for the model given by (<ref>). Given natural number J≥ 1, β_j ∈ℝ∖{0} and Z_j⊂{1,…,D} for j=1,… J, exists K_0 < J such that for all K≥ K_0 system of polynomial equations β_j = ∑_k=1^K ∏_i ∈ Z_j v_ik, j=1,…,m has at least one solution in terms of v_11,…,v_DK.The upper bound K_0=J-1 is only required when all interactions are of the depth D-1. This is typically not expected to be the case in practice, therefore smaller values of K are often sufficient.By conditioning on the training data pairs (x_n,y_n) to account for the likelihood induced by the PARAFAC representation, the statistician obtains the posterior distribution on the parameters of interest, namely, Θ := (Z,V), which in turn induces the posterior distribution on the conditional density, to be denoted by f(y|x), according to the MiFM model (<ref>) without linear terms. The main result of this section is to show that under some conditions this posterior distribution Π will place most ofits mass on the true conditional density f^*(y|x) as N→∞. To state the theorem precisely, we need to adopt a suitable notion of weak topology on the space of conditional densities, namely the set of f(y|x),which is induced by the weak topology on the space of joint densities on X,Y,that is the set of f(x,y) = f^*(x) f(y|x), where f^*(x) is the true (but unknown) marginal density on X (see <cit.>, Sec. 2 for a formal definition).Given any true conditional density f^*(y|x) given by (<ref>), and assuming that the support of f^*(x) is bounded, there is a constant K_0< J such that by setting K ≥ K_0, the following statement holds: for any weak neighborhood U of f^*(y|x), under the MiFM model, the posterior probabilityΠ(U|(X_n,Y_n)_n=1^N) → 1 with P^*-probability one, as N→∞.The proof's sketch for this theorem is given in the [supp_th]Supplement. § PRIOR CONSTRUCTIONS FOR INTERACTIONS: FFM REVISITED AND EXTENDED The adoption of the FFM prior on the hypergraph of interactions carries a distinct behavior in contrast to the typical Latent Feature modeling setting. In astandard Latent Feature modeling setting <cit.>,each row of Z describes one of the data points in terms of its featurerepresentation; controlling row sums is desired to induce sparsity of the features.By contrast, for us a column of Z is identified with an interaction; its sum represents the interaction depth, which we want to control a priori. Interaction selection using MCMC samplerOne interesting issue of practical consequence arises in the aggregation of the MCMC samples (details of the sampler are in the [supp_gibbs]Supplement). When aggregating MCMC samples in the context of latent feature modeling one would always obtain exactly J latent features. However, in interaction modeling, different samples might have no interactions in common (i.e. no exactly matching columns), meaning that support of the resulting posterior estimate can have up to min{2^D-1, IJ} unique interactions, where I is the number of MCMC samples. In practice, we can obtain marginal distributions of all interactions across MCMC samples and use those marginals for selection. One approach is to pick J interactions with highest marginals and another is to consider interactions with marginal above some threshold (e.g. 0.5). We will resort to the second approach in our experiments in Section <ref> as it seems to be in more agreement with the concept of "selection". Lastly, we note that while a data instance may a priori possess unbounded number of features, the number of possible interactions in the data is bounded by 2^D-1, therefore taking J→∞ might not be appropriate.In any case, we do not want to encourage the number of interactions to be too high for regression modeling,which would lead to overfitting. The above considerations led us to opt for a parametric prior such as the FFM for interactions structure Z,as opposed to going fully nonparametric. J can then be chosen using model selection procedures (e.g. cross validation), or simply taken as the model input parameter. Generalized construction and induced distribution of interactions depths We now proceed to introduce a richer family of prior distributions on hypergraphs of which the FFM is one instance.Our construction is motivated by the induced distribution on the column sums and the conditional probability updates that arise in the original FFM. Recall that under the FFM prior, interactions are a priori independent. Fix an interaction j, for the remainder of this section let Z_i denote the indicator of whether variable i is present in interaction j or not (subscript j is dropped from Z_ij to simplify notation). Let M_i = Z_1 + … + Z_i denote the number of variables among the first i present in the corresponding interaction.By the Beta-Bernoulli conjugacy, one obtains (Z_i=1|Z_1,…,Z_i-1) = M_i-1 + γ_1/i-1+γ_1+γ_2. This highlights the “rich-gets-richer” effect of the FFM prior, which encourages the existence of very deep interactions while most other interactions have very small depths. In some situations we may prefer a relatively larger number of interactions of depths in the medium range.An intuitive but somewhat naive alternative sampling process is to allow a variable to be included into an interaction according to its present "shallowness" quantified by (i-1-M_i-1) (instead of M_i-1 in the FFM).It can be verified that this construction will lead to a distribution of interactions whichconcentrates most its mass around D/2; moreover, exchangeability among Z_i would be lost.To maintain exchangeability, we define the sampling process for the sequence Z=(Z_1,…,Z_D) ∈{0,1}^D as follows:let σ(·) be a random uniform permutation of {1,…,D} and let σ_1 = σ^-1(1),…,σ_D = σ^-1(D). Note that σ_1,…,σ_D are discrete random variables and (σ_k = i)=1/D for any i,k = 1,…,D. For i=1,…,D, set(Z_σ_i=1|Z_σ_1,…,Z_σ_i-1) = α M_i-1 + (1-α)(i-1-M_i-1)+γ_1/i-1+γ_1+γ_2,(Z_σ_i=0|Z_σ_1,…,Z_σ_i-1) = (1-α)M_i-1 + α(i-1-M_i-1)+γ_2/i-1+γ_1 + γ_2,where γ_1>0,γ_2>0,α∈ [0,1] are given parameters and M_i = Z_σ_1 + … + Z_σ_i. The collection of Z generated by this process shall be called to follow FFM_α. When α=1 we recover the original FFM prior. When α=0, we get the other extremal behavior mentioned at the beginning of the paragraph. Allowing α∈ [0,1] yields a richer spectrumspanning the two distinct extremal behaviors.Details of the process and some of its properties are given in the [ffm_all]Supplement. Here we briefly describe how FFM_α a priori ensures "poor gets richer" behavior and offers extra flexibility in modeling interaction depths compared to the original FFM. The depth of an interaction of D variables is described by the distribution of M_D. Consider the conditionals obtained for a Gibbs sampler where index of a variable to be updated is random and based on (σ_D = i|Z) (it is simply 1/D for FFM_1). Suppose we want to assess howlikely it is to add a variable into an existing interaction via the expression∑_i:Z^(k)_i = 0(Z^(k+1)_i = 1, σ_D = i| Z^(k)), where k+1 is the next iteration of the Gibbs sampler's conditional update. This probability is a function of M_D^(k); for small values of M_D^(k) it quantifies the tendency for the "poor gets richer" behavior. For the FFM_1 it is given by D-M_D^(k)/DM_D^(k) + γ_1/D-1+γ_1+γ_2. In Fig. <ref>(a) we show that FFM_1's behavior is opposite of "poor gets richer", while α≤ 0.7 appears to ensure the desired property. Next, in Fig.<ref> (b-f) we show the distribution of M_D for various α, which exhibits a broader spectrum of behavior. § EXPERIMENTAL RESULTS §.§ Simulation Studies We shall compare MiFM methods against a variety of other regression techniques in theliterature, including Bayesian Factorization Machines (FM), lasso-type regression, Support Vector Regression (SVR), multilayer perceptron neural network (MLP).[Random Forest Regression and optimization based FM showed worse results than other methods.]The comparisons are done on the basis of prediction accuracy of responses (Root Mean Squared Error on the held out data), quality of regression coefficient estimates and the interactions recovered.§.§.§ Predictive PerformanceIn this set of experiments we demonstrate that MiFMs with either α=0.7 or α=1 have dominant predictive performance when high order interactions are in play.In Fig. <ref>(a) we analyzed 70 random interactions of varying orders. We see that MiFM can handle arbitrary complexity of the interactions, while other methods are comparative only when interaction structure is simple (i.e. linear or 2-way on the right of the Fig. <ref>(a)).Next, to assess the effectiveness of MiFM in handling categorical variables (cf. Section <ref>) we vary the number of continuous variables from 1 (and 29 attributes across categories) to 30 (no categorical variables). Results in Fig. <ref>(b) demonstrate that our models can handle both variable types in the data (including continuous-categorical interactions), and still exhibit competitive RMSE performance.§.§.§ Interactions QualityCoefficients of the interactions This experiment verifies the posterior consistency result of Theorem <ref> and validates our factorization model for coefficients approximation. In Fig. <ref>(c) we compare MiFMs versus OLS fitted with the corresponding sets of chosen interactions. Additionally we benchmark against Elastic net <cit.> based on the expanded data matrix with interactions of all depths included, that is 2^D - 1 columns, and a corresponding OLS with only selected interactions. Selection of the interactions In this experiments we assess how well MiFM can recover true interactions. We consider three interaction structures: a realistic one with five linear, five 2-way, three 3-way and one of each 4,…,8-way interactions, and two artificial ones with 15 either only 4- or only 6-way interactions to challenge our model. Both binary and continuous variables are explored.Fig. <ref>(d) shows that MiFM can exactly recover up to 83% of the interactions and with α=0.8 it recovers 75% of the interaction in 4 out of 6 scenarios. Situation with 6-way interactions is more challenging, where 36% for binary data is recovered and almost half for continuous. It is interesting to note that lower values of α handle binary data better, while higher values are more appropriate for continuous, which is especially noticeable on the "only 6-way" case. We think it might be related to the fact that high order interactions between binary variables are very rare in the data (i.e. product of 6 binary variables is equal to 0 most of the times) and we need a prior eager to explore (α=0) to find them. §.§ Real world applications §.§.§ Finding epistasisIdentifying epistasis (i.e. interactions between genes) is one of the major questions in the field of human genetics. Interactions between multiple genes and environmental factors can often tell a lot more about the presence of a certain disease than any of the genes individually <cit.>. Our analysis of the epistasis is based on the data from <cit.>. These authors show that interactions between single nucleotide polymorphisms (SNPs) are often powerful predictors of various diseases, while individually SNPs might not contain important information at all. They developed a model free approach to simulate data mimicking relationships between complex gene interactions and the presence of a disease.We used datasets with five SNPs and either 3-,4- and 5-way interactions or only 5-way interactions. For this experiment we compared MiFM_1, MiFM_0; refitted logistic regression for each of our models based on the selected interactions (LMiFM_1 and LMiFM_0), Multilayer Perceptron with 3 layers and Random Forest.[FM, SVM and logistic regression had low accuracy of around 50% and are not reported.] Results in Table <ref> demonstrate that MiFM produces competitive performance compared to the very best black-box techniques on this data set,while it also selects interacting genes (i.e. finds epistasis).We don't know which of the 3- and 4-way interactions are present in the data, but since there is only one possible 5-way interaction we can check if it was identified or not — both MiFM_1 and MiFM_0 had a 5-way interaction in at least 95% of the posterior samples. §.§.§ Understanding retail demandWe finally report the analysis of data obtained from a major retailer with stores in multiple locations all over the world. This dataset has 430k observations and 26 variables spanning over 1100 binary variables after the one-hot encoding. Sales of a variety of products on different days and in different stores are provided as response. We will compare MiFM_1 and MiFM_0, both fitted with K=12 and J=150, versus Factorization Machines in terms of adjusted mean absolute percent error AMAPE=100∑_n|ŷ_n-y_n|/∑_n y_n, a common metric for evaluating sales forecasts. FM is currently a method of choice by the company for this data set, partly because the data is sparse and is similar in nature to the recommender systems.AMAPE for MiFM_1 is 92.4; for MiFM_0 - 92.45; for FM - 92.0.Posterior analysis of predictor interactions The unique strength of MiFM is the ability to provide valuable insights about the data through its posterior analysis. MiFM_1 recovered 62 non-linear interactions among which there are five 3-way and three 4-way. MiFM_0 selected 63 non-linear interactions including nine 3-way and four 4-way. We note that choice α=0 was made to explore deeper interactions and as we see MiFM_0 has more deeper interactions than MiFM_1. Coefficients for a 3-way interaction of MiFM_1 for two stores in France across years and months are shown in Fig. <ref>(a,b). We observe different behavior, which would not be captured by a low order interaction. In Fig. <ref>(c,d) we plot coefficients of a 4-way MiFM_0 interaction for the same two stores in France. It is interesting to note negative correlation between Saturday and Sunday coefficients for the store in Merignac, while the store in Perols is not affected by this interaction - this is an example of how MiFM can select interactions between attributes across categories.§ DISCUSSION We have proposed a novel regression method which is capable of learning interactions of arbitrary orders among the regression predictors. Our model extends Finite Feature Model and utilizes the extension to specify a hypergraph of interactions, while adopting a factorization mechanism for representing the corresponding coefficients. We found that MiFM performs very well when there are some important interactions among a relatively high number (higher than two) of predictor variables. This is the situation where existing modeling techniques may be ill-equipped at describing and recovering. There are several future directions that we would like to pursue. A thorough understanding of the fully nonparametric version of the FFM_α is of interest, that is, when the number of columns is taken to infinity. Such understanding may lead to an extension of the IBP and new modeling approaches in various domains.§.§.§ AcknowledgmentsThis research is supported in part by grants NSF CAREER DMS-1351362, NSF CNS-1409303, a research gift from Adobe Research and a Margaret and Herman Sokol Faculty Award.§ SUPPLEMENTARY MATERIALIn the Supplementary material we will start by proving consistency of the MiFM theorem, then we will show several important results related to FFM_α: how exchangeability is achieved using uniform permutation prior on the order in which variables enter the process, how it leads to a Gibbs sampler using distribution of the index of the variable entering FFM_α last and how to obtain distribution of the interaction depths M_D and compute its expectation. Lastly we will present a Gibbs sampling algorithm for the MiFM under the FFM_α prior on interactions structure Z. §.§ Proof of the Consistency Theorem <ref> First let us remind the reader of the problem setup. Suppose that the data pairs (x_n,y_n)_n=1^N∈ℝ^D×ℝare i.i.d. samples from the joint distribution P^*(X,Y),according to which marginal distribution for X and the conditional distribution ofY given X admit density functions f^*(x) and f^*(y|x), respectively, with respect to Lebesgue measure. In particular, f^*(y|x) is defined as in Eq. (<ref>):. Y=y_n|X=x_n,Θ^*𝒩(y(x_n, Θ^*),σ), where Θ^*={β^*_1,…,β^*_J,Z^*_1,…,Z^*_J},y(x, Θ^*):= ∑_j=1^Jβ^*_j∏_i∈ Z^*_jx_i, and x_n∈ℝ^D, y_n∈ℝ, β^*_j∈ℝ, Z^*_j⊂{1,…,D},for n=1,…,N,j=1,…,J. .In the above Θ^* represents the true parameter for the conditional density f^*(y|x) that generates data sample y_n given x_n, for n=1,…,N.On the other hand, the statistical modeler has access only to the MiFM:.ZFFM_α(γ_1, γ_2),v_ik|μ_k, λ_k 𝒩(μ_k, 1/λ_k) for i=1,…,D; k=1,…,K, y_n|Θ𝒩(y(x_n, Θ),σ), where y(x, Θ):= ∑_j=1^J∑_k=1^K∏_i∈ Z_jx_i v_ik,for n=1,…,N, and Θ=(Z,V). .We omitted linear terms in the MiFM since they can naturally be parts of the interaction structure Z and discarded hyperpriors for the ease of representation. Now we show that under some conditions posterior distribution Π will place most of its mass on the true conditional density f^*(y|x) as N→∞. [<ref>] Given any true conditional density f^*(y|x) given by (<ref>), and assuming that the support of f^*(x) is bounded, there is a constant K_0< J such that by setting K ≥ K_0, the following statement holds: for any weak neighborhood U of f^*(y|x), under the MiFM model (<ref>), the posterior probabilityΠ(U|(X_n,Y_n)_n=1^N) → 1 with P^*-probability one, as N→∞.A key part in the proof of this theorem is to clarify the role of parameter K, and the fact that under model (<ref>), the regression coefficient β_j associated withinteraction j is parameterized by β_j := ∑_k=1^K ∏_i ∈ Z_j v_ik, for j=1,…,J, which for some suitable choice of Θ = (Z,V)can represent exactly the true parameters β_1^*,…,β^*_J, provided that K is sufficiently large. The following basic lemma is informative. Let m ∈ [1,J] be a natural number, β_j ∈ℝ∖{0} for j=1,…, m. Suppose that the m subsets Z_j⊂{1,…,D} forj=1,… m have non-empty intersection, then as long as K≥ m, the system of polynomial equations∑_k=1^K ∏_i ∈ Z_j v_ik = β_j, j=1,…,mhas at least one solution in terms of v_11,…,v_DK such that the following collection of K vectors in ℝ^m, namely, {(∏_i ∈ Z_1v_ik,…,∏_i ∈ Z_mv_ik), k=1,…,K} contains mlinearly independent vectors. Let i_0 be an element of the intersection of all Z_j, for j=1,…, m. We consider system (<ref>) as linear with respect to {v_i_01,…,v_i_0K}, where corresponding coefficients are given by ∏_i ∈ Z_j∖{i_0} v_i,k, which we can pick to form a matrix of nonzero determinant. Hence by Rouché–Capelli theorem the system has at least one solution if K≥ m and, since β_j≠ 0 for ∀ j, the resulting {(∏_i ∈ Z_1v_ik,…,∏_i ∈ Z_mv_ik), k=1,…,K} contains at least m linearly independent vectors.[<ref>] Given natural number J≥ 1, β_j ∈ℝ∖{0} and Z_j⊂{1,…,D} for j=1,… J, exists K_0 < J: ∀ K≥ K_0 system of polynomial equations (<ref>) has at least one solution in terms of v_11,…,v_DK. The proof proceeds by performing an elimination process on the collection of variables v_ik according toan ordering that we now define. Let J_i = ({Z_j|i∈ Z_j}) for i=1,…,D.Define J^0=min_i J_i and i_0=_i J_i.If K ≥ J^0 by Lemma <ref> we can find a solution of the reduced system of equations∑_k=1^K ∏_i ∈ Z_j v_i,k = β_j, j∈{j|i_0∈ Z_j},while maintaining the linear independence needed to apply Lemma <ref> again further. Now we know that we can find a solution for equations indexed by {j|i_0∈ Z_j}. We remove them from system (<ref>) and recompute J^1=min_i≠ i_0 J_i and i_1=_i≠ i_0 J_i to apply Lemma <ref> again. Iteratively we will remove all the equations, meaning that there is at least one solution. Note that J_i are decreasing since whenever we remove equations, number of Z_js containing certain i can only decrease. Therefore, we will need K≥ K_0 :=max(J^0, J^1, …, 0) in order to apply Lemma <ref> on every elimination step.From the proof of Lemma <ref>, it can be observed that K_0 = max(J^0,J^1,…) ≪ J when we anticipate only few interactionsper variable, whereas the upper bound K_0=J-1 is attained when there are only (D-1)-way interactions.Now we are ready to present a proof of the main theorem.(of main theorem). By Lemma <ref> and the fact that the probability of a finite number of independent continuous randomvectors being linearly dependent is 0 it follows that under the MiFM prior on V as in (<ref>)and ∀β_1,…,β_J∈ℝ∖{0}, distinct Z_1,…,Z_J and ϵ>0Π(∑_j=1^J(β_j - ∑_k ∏_i ∈ Z_j v_ik)^2 < ϵ | Z_1,…,Z_J) > 0.From Eq. (<ref>) it follows that for any Z_1,…,Z_J, the prior probability of the corresponding incidence matrix is bounded away from 0. Combining this with (<ref>), we now establish that the probability of the true model parameters to be arbitrary close to the MiFM parameters under the MiFM prior as in (<ref>):Π((∑_j=1^Jβ_j - ∑_j=1^J∑_k ∏_i ∈ Z_j v_ik)^2 < ϵ) > 0, ∀ϵ>0.We shall appeal to Schwartz's theorem (cf.<cit.>), which asserts that the desired posterior consistency holdsas soon as we can establish that the true joint distribution P^*(X,Y) lies in the Kullback-Leibler support of the prior Π on the joint distribution P(X,Y). That is, Π(KL(P^*||P)<ϵ)>0, for ∀ϵ>0.Since the KL divergence of the two Gaussian distributions is proportional to the mean difference, we have (𝔼_X^* denotes expectation with respect to the true marginal distribution of X). KL(P^*||P) ∝𝔼_X^* 1/2(y(X,Θ)-y(X,Θ^*))^2 ∝𝔼_X^* (∑_j=1^Jβ_j∏_i∈ Z_jx_i - ∑_j=1^J∑_k ∏_i ∈ Z_j v_ikx_i)^2 ≲ (∑_j=1^Jβ_j - ∑_j=1^J∑_k ∏_i ∈ Z_j v_ik)^2. .Due to (<ref>) this quantity can be made arbitrarily close to 0 with positive probability. Therefore (<ref>) and then Schwartz theorem hold, which concludes the proof.§.§ Analyzing FFM_α §.§.§ Model definition and exchangeabilityHere we remind the reader the construction of FFM_α — the distribution over finite collection of binary random variables that we used to model interactions. Let D be the number of variables in the data and Z∈{0,1}^D is j-th interaction (subscript j is dropped to simplify notation). Let σ(·) be a random uniform permutation of {1,…,D} and let σ_1 = σ^-1(1),…,σ_D = σ^-1(D). Note that σ_1,…,σ_D are discrete random variables and (σ_k = i)=1/D for any i,k = 1,…,D. Next recall FFM_α from Eq. (<ref>):(Z_σ_i=1|Z_σ_1,…,Z_σ_i-1) = α M_i-1 + (1-α)(i-1-M_i-1)+γ_1/i-1+γ_1+γ_2,(Z_σ_i=0|Z_σ_1,…,Z_σ_i-1) = (1-α)M_i-1 + α(i-1-M_i-1)+γ_2/i-1+γ_1 + γ_2,where γ_1>0,γ_2>0,α∈ [0,1] are given parameters and M_i = Z_σ_1 + … + Z_σ_i. Due to the random permutation of indices, distribution of Z_1,…,Z_D is exchangeable because any ordering of variables entering the process has same probability. Next, we need to integrate the permutation part out to obtain a tractable full conditional representation.§.§.§ Gibbs sampling for FFM_α and distribution of interaction depths M_D To construct a Gibbs sampler for the the FFM_α we will use an additional latent variable - index of the variable entering the process last, σ_D. Additionally observe that when permutation is integrated out (Z_1,…, Z_D)=(M_D=Z_1+… + Z_D) since (M_D=m) is precisely the summation over all possible orderings of Z_1,…,Z_D such that Z_1 + … + Z_D = m.. (σ_D = i|Z_1,…, Z_D)∝ Z_i(σ_D = i|Z_σ_D=1,Z)(Z_σ_D=1|M_D-1=∑_k=1^D Z_k - 1)(M_D-1=∑_k=1^D Z_k - 1) ++ (1-Z_i)(σ_D = i|Z_σ_D=0,Z)(Z_σ_D=0|M_D-1=∑_k=1^D Z_k)(M_D-1=∑_k=1^D Z_k), .then if Z_i=1 and ∑_k=1^D Z_k = m we obtain .(σ_D = i|Z_-i,Z_i=1) = (σ_D = i| M_D=m, Z_i=1) = = (M_D-1 = m - 1)(Z_σ_D=1|M_D-1=m-1)/m(M_D = m), .where (Z_σ_D=1|M_D-1=m-1) and (Z_σ_D=0|M_D-1=m) can be computed as in Eq. <ref>. Our next step is to analyze probability (M_D = m). Indeed it is easy to obtain this distribution recursively:.(M_D = m) = (M_D-1=m)(Z_σ_D=0|M_D-1=m) + + (M_D-1=m-1)(Z_σ_D=1|M_D-1=m-1). .The base of recursion is given by the following identities:. (M_0 = 0) = 1, (M_i = 0) = ∏_k=0^i-1α(i-1-k) + γ_2/k+γ_1+γ_2 = ∏_k=0^i-1α k + γ_2/k+γ_1+γ_2,(M_i = i) = ∏_k=0^i-1α k + γ_1/k+γ_1+γ_2. . The above formulation allows us compute (M_i=k), D≥ i≥ k dynamically (computations are very fast since we only need to perform (D+1)(D+2)/2-1 calculations) before running MiFM inference and utilize the table of probabilities during it. The last step of the Gibbs sampler is clearly the update of the Z_i|σ_D=i,Z_-i which is done simply using the FFM_α definition <ref>. Recall Figure 1 (a) of the main text which illustrates the behavior of.∑_i:Z^(k)_i = 0(Z^(k+1)_i = 1, σ_D = i| Z^(k)) = (Z_σ_D=0|Z)(Z_i=1|σ_D=i,Z_-i), .and since we choose index of a variable to update based on the probability of it being last, the expression above reads as the probability that we choose to update a variable not present in the interaction and then add it to the interaction, therefore increasing the depth of the interaction.§.§.§ Mean Behavior of the FFM_αFrom Eq. (<ref>) it follows that .M_D = ∑_m=0^Dm(M_D = m) == 1/D-1+γ_1+γ_2{ (1-2α) M_D-1^2+ (α(D-1)+ γ_2)M_D-1 ++ (2α-1)(M_D-1+1)^2+ ((1-α)D - α + γ_1)(M_D-1+1) } = 1/D-1+γ_1+γ_2{ M_D-1(D+2α+γ_1+γ_2-2) +D(1-α)+α + γ_1-1 }. .For α = 0, this relation is simplified to be. (D-1+γ_1+γ_2)M_D=M_D-1(D+γ_1+γ_2-2)+ (D+γ_1 -1) == (D+γ_1-1) + … + γ_1 = 1/2D(D+2γ_1-1). .§.§ Gibbs Sampler for the MiFM Our Gibbs sampling algorithm consists of two parts — updating factorization coefficients V (based on the results from <cit.>) and then updating interactions Z based on the analysis of Section <ref>. Recall the MiFM model construction. First we have a layer of hyperpriors:σΓ(α_1/2,β_1/2), λΓ(α_0/2,β_0/2), μ𝒩(μ_0, 1/γ_0), λ_kΓ(α_0/2,β_0/2),μ_k𝒩(μ_0, 1/γ_0) for k=1,…,K,Then interactions and their weights:w_i|μ, λ𝒩(μ, 1/λ) for i=0,…,D,ZFFM_α(γ_1,γ_2), v_ik|μ_k, λ_k 𝒩(μ_k, 1/λ_k) for i=1,…,D; k=1,…,K,And finally the model's likelihood from Eq. (<ref>).y_n|Θ𝒩(y(x_n, Θ),1/σ), where y(x, Θ):= w_0 + ∑_i=1^Dw_ix_i + ∑_j=1^J∑_k=1^K∏_i∈ Z_jx_i v_ik,for n=1,…,N, and Θ={Z,V,σ,w_0,…,D}. .Inference in the context of Bayesian modeling is often related to learning the posterior distribution (Θ|X,Y). Then, if one wants point estimates, certain statistics of the posterior can be used, i.e. mean or median. In most situations (including MiFM) analytical form of the posterior is intractable, but with the help of Bayes rule it is often possible to compute it up to a proportionality constant:. (Θ,μ,γ,μ_1,…μ_K,λ_1,…,λ_K|Y) ∝∏_n=1^N (y_n|Z,V,σ,w_0,…,D)··(Z)(V|μ_1,…,μ_K, λ_1,…,λ_K)(σ,μ,γ,μ_1,…μ_K,λ_1,…,λ_K). .One can maximize this quantity to obtain MAP estimate, but this is very complicated due to the combinatorial complexity of interactions in Z and, additionally, often leads to overfitting. We use Gibbs sampling procedure for learning the posterior of our model. Due to normal-normal conjugacy and a priori independence of Z and other latent variables, we can derive closed form full conditional (i.e. variable given all the rest and the data) distributions for each of the latent variables in the model. Updating hyperprior parametersσΓ(α_1 + N/2; ∑_n=1^N(y_n - y(x_n,Θ))^2 + β_1)/2),λΓ(α_0 + D + 1/2; ∑_i=0^D(w_i - μ)^2 + β_0/2),μ𝒩(∑_i=0^D w_i + γ_0μ_0/D+1+γ_0; 1/λ(D+1+γ_0)),λ_k Γ(α_0 + D/2; ∑_i=1^D(v_ik - μ_k)^2 + β_0/2),μ_k 𝒩(∑_i=1^D v_ik + γ_0μ_0/D+γ_0; 1/λ_k(D+γ_0)),for k=1,… ,K. Updating factorization coefficients V For updating coefficients of the model we can utilize the multi-linear property also used for the Factorization Machines MCMC updates <cit.>. Note that for any θ∈{w_0,…, w_D, v_11,…, v_DK} we can write y(x,Θ) = l_θ(x) + θ m_θ(x), where l_θ(·) are all the terms independent of θ and m_θ(·) are the terms multiplied by θ. For example, if θ=w_0, then m_θ(x)=1 and l_θ(x)=∑_i=1^Dw_ix_i + ∑_j=1^J∑_k=1^K∏_i∈ Z_jx_i v_ik. Next we give updating distribution that can be used for any θ∈{w_0,…, w_D, v_11,…, v_DK}.. θ𝒩(μ^*_θ, σ_θ^2),where σ_θ^2 = (σ∑_n=1^N m_θ(x_n)^2 + λ_θ)^-1,μ^*_θ = σ_θ^2(σ∑_n=1^N (y_n - l_θ(x_n))m_θ(x_n) + μ_θλ_θ), .and μ_θ,λ_θ are the corresponding hyperprior parameters.Updating interactions Z Posterior updates of Z can be decomposed into prior times the likelihood:(Z_i|Z_-i,V,Y) ∝(Z_i|Z_-i)(Y|V,Z),where second part is the Gaussian likelihood as in Eq. (<ref>). To sample Z_i|Z_-i we use the construction from Section <ref>, where we first sample the value of Z_σ_D for fixed j:.(Z_σ_D=1|Z) = (σ_D = i| M_D=m, Z_i=1) = = (M_D-1 = m - 1)(Z_σ_D=1|M_D-1=m-1)/(M_D = m), .and then uniformly choose and index i to update among {i:Z_i=Z_σ_D}. Next Z_i can simply be updated using the process construction <ref> assuming it to be last. Recall that (M_D = m) should be computed beforehand using Eq. (<ref>). icml2017 | http://arxiv.org/abs/1709.09301v1 | {
"authors": [
"Mikhail Yurochkin",
"XuanLong Nguyen",
"Nikolaos Vasiloglou"
],
"categories": [
"stat.ML"
],
"primary_category": "stat.ML",
"published": "20170927015119",
"title": "Multi-way Interacting Regression via Factorization Machines"
} |
Quasi-random Monte Carlo application in CGE systematic sensitivity analysis Theodoros Chatzivasileiadis Institute for Environmental Studies, Vrije Universiteit Amsterdam, The NetherlandsContact T. Chatzivasileiadis, Email: [email protected], Address: De Boelelaan 1087, 1081 HV Amsterdam, The Netherlands.=================================================================================================================================================================================================================================================== Many automatic skin lesion diagnosis systems use segmentation as a preprocessing step to diagnose skin conditions because skin lesion shape, border irregularity, and size can influence the likelihood of malignancy. This paper presents, examines and compares two different approaches to skin lesion segmentation. The first approach uses U-Nets and introduces a histogram equalization based preprocessing step. The second approach is a C-Means clustering based approach that is much simpler to implement and faster to execute. The Jaccard Index between the algorithm output and hand segmented images by dermatologists is used to evaluate the proposed algorithms. While many recently proposed deep neural networks to segment skin lesions require a significant amount of computational power for training (i.e., computer with GPUs), the main objective of this paper is to present methods that can be used with only a CPU. This severely limits, for example, the number of training instances that can be presented to the U-Net. Comparing the two proposed algorithms, U-Nets achieved a significantly higher Jaccard Index compared to the clustering approach. Moreover, using the histogram equalization for preprocessing step significantly improved the U-Net segmentation results.Skin lesion Segmentation; U-Nets; C-Means Clustering; Melanoma; Histogram Equalization; Color Space § INTRODUCTIONWith between 2 and 3 million cases occurring globally each year, skin cancer is the most common cancer worldwide <cit.>. Since skin cancer occurs at the surface of the skin, melanomas, one of the world's most deadly cancer, can be diagnosed with visual inspection by a dermatologist <cit.>. To improve the accuracy, dermatologists use a dermatoscope, which eliminates some of the surface reflection and enhances the deeper layers of skin <cit.>. Dermatologists often look for five specific signs when classifying melanomas. These include skin lesion asymmetry, irregular borders, uneven distribution of color, diameter, and evolution of moles over time <cit.>. Standard skin cancer computer-aided diagnosis systems include five important steps: acquisition, preprocessing, segmentation, feature extraction, and finally, classification <cit.>. Since the shape and border of the skin lesion is extremely important in the diagnosis, automatic diagnosis systems must be able to accurately identify and segment the skin lesion in the image.Automatic skin lesion segmentation contains many different challenges. The lesion color, texture, and size can all vary considerably for different patients. On top of this, medical gauzes, hair, veins, and light reflections in the image make it even more difficult to identify the skin lesion. Finally, in the areas surrounding a skin lesion, there are often sub-regions that are darker than others. This makes it difficult to consistently identify which sub-regions are part of the lesion and which sub-regions are just part of the surrounding skin. Finally, lack of large labeled datasets (segmented by dermatologists) makes training sophisticated networks rather difficult.This paper presents and compares multiple approaches to automatic segmentation of dermoscopic skin lesion images. Firstly, we present some different segmentation approaches using U-nets including one with a novel histogram-based preprocessing technique. After this, the U-net approach is compared with a much simpler clustering algorithm. While U-nets and other deep networks often require a lot of training and computational power, the approaches presented in this paper are designed such that they do not require GPU power. All training and testing in this paper have been done with a single CPU.The paper is organized as follows: Section <ref> briefly reviews the literature of existing techniques for skin lesion segmentation. Section <ref> describes the proposed dataset used to benchmark proposed methods. Section <ref> explains the proposed methods; results and conclusion are discussed in Section <ref> and Section <ref>, respectively.§ BACKGROUND REVIEW Machine learning schemes like reinforcement learning and neural networks have been used for segmentation of medical images <cit.>. Since skin lesion segmentation is frequently used as an important first step in many automated skin lesion recognition systems, there have been many different approaches. A summary of several skin lesion segmentation methods by Celebi et al. <cit.> reviews many recent techniques proposed in literature. Frequently, proposed lesion segmentation techniques include a preprocessing step which removes unwanted artifacts such as hair or light reflections. Another very common preprocessing step involves performing color space transformation in order to make segmentation easier. Common techniques that have recently been proposed in the literature include, among others, histogram thresholding, clustering and active contours. Most recently, deep neural networks have led to many breakthroughs, producing excellent results in image segmentation <cit.>.Color space transformations can help facilitate skin lesion detection. For instance, Garnavi et al. <cit.>, take advantage of RGB, HSV, HSI, CIE-XYZ, CIE-LAB, and YCbCr color spaces and use a thresholding approach to separate the lesion from the background skin. Similarly, Yuan et al. <cit.> use RGB, HSV, and CIE-LAB color spaces as input channels to a fully convolutional-deconvolutional neural network in order to segment the images.Hair removal is a very common preprocessing step used to facilitate skin lesion segmentation. The presence of dark hairs on the skin often significantly degrades the performance of most automatic segmentation algorithms. For example, one very popular software, Dullrazor <cit.>, identifies the dark hair locations using morphological operations. Afterwards, the hair pixels are replaced with nearby skin pixels and the resulting image is smoothed in order to remove the hair artifacts. Many recent techniques uses similar morphology or diffusion methods to remove hair <cit.>.Once the image has been preprocessed, many different methods have been proposed for segmentation. Since most skin lesions are darker than the surrounding skin, thresholding is a very commonly used technique. However, further image enhancement algorithms need to be performed in order to make thresholding effective. For example, Fan et al. <cit.> propose first enhancing the skin lesion image using color and brightness saliency maps before applying a modified Otsu threshold method. Similarly, Garnavi et al. <cit.> use a thresholding based approach on 25 color channels from a variety of different color spaces in order to segment the images. Often with histogram thresholding, a connected components analysis needs to be performed after thresholding in order to remove medical gauzes, pen marks, or other artifacts that have been wrongly classified by thresholding.Clustering-based methods are commonly used as well. For instance, Schmid-Saugeon et al. <cit.> propose using a modified Fuzzy C-Means clustering algorithm on the two largest principle components of the image in the LUV color space. The number of clusters for the Fuzzy C-Mean algorithm is determined by examining the histogram computed using the two principal components of the image. Melli et al. <cit.> propose using mean shift clustering to get a set of redundant classes which are then merged into skin and lesion regions.More recently, convolutional neural networks and other deep artificial neural networks are proposed. Commonly, the image is first preprocessed to remove artifacts. Afterwards, a neural network is used to identify the lesion. For instance, in a work proposed by Jafari et al. <cit.>, the image is first preprocessed using a guided filter before a CNN uses both local and global information to output a label for each pixel.§ IMAGE DATA The 2017 IEEE International Symposium on Biomedical Imaging (ISBI) organized a skin lesion analysis challenge for melanoma detection, ISIC 2017 (International Skin Imaging Collaboration). The challenge released a public set of 2000 skin lesion images that have been segmented and classified by dermatologists. There is also a validation and a test set that is not publicly available for the competition. Lesion segmentation is one of three parts of the competition. For evaluation, the Jaccard Index is used to rank the algorithms. This year, the vast majority of the participants, including the top submissions, used a deep neural networks to segment the images. For example, Yuan et al. [10] augmented the 2000 images using rotation, flipping, rotating, and scaling and trained a convolutional-deconvolutional neural network with 500 epochs. Clustering was also used in some submissions, although the performance of clustering algorithms were generally worse compared with that of the deep artificial networks. A variety of different evaluation metrics have been used in the literature. This includes the XOR measure, and Specificity, Sensitivity, Precision, Recall, and error probability <cit.>. The ISBI 2017 challenge used the Jaccard Index in order to evaluate methods. In order to compare the algorithms and methods proposed in this paper, the average Jaccard Index is used to evaluate the proposed method's performance.The ISIC 2017 training set was used to both train and validate our approaches. This publicly available dataset contains a total of 2000 images including 374 melanoma, 254 seborrheic keratosis and 1372 benign nevi skin lesion images in JPG format. The images are in many different dimensions and resolutions. Each skin lesion has also been segmented by a dermatologist; the 2000 binary masks corresponding to each skin lesion image is provided in PNG format.Figure <ref> shows a collection of sample images from the data set. As can be seen from the images, there are many unwanted artifacts including reflections, hair, black borders, pen marks, dermatoscope gel bubbles, medical gauzes, and other instrument markings that make segmentation difficult. Furthermore, there is a high variability in skin lesion shape, size, color and location.§ METHODOLOGY The main objective of this paper is to develop different approaches that can process skin lesion images, and produce a binary mask with 1 corresponding to pixels that are part of the skin lesion, and 0 corresponding to pixels that are not part of the skin lesion. The output format should be identical to the binary masks segmented by the dermatologists in order to evaluate the performance of the segmentation algorithms.In order to compare the performance of algorithms with that of other submissions in the ISBI challenge, the Jaccard Index, which was the challenge evaluation metric, is used. Since all of the approaches in this paper are performed without a GPU, some approaches, in particular, the ones involving a U-Net, take around several hours (in our case 8 hours) to train for each validation fold. In order for the validation to complete within a reasonable amount of time, 5 random cross validation folds were performed with a 90% training (1800 images) and 10% validation (200 images) split. In this methodology section, we first discuss the performance of U-Nets followed by clustering. §.§ U-Net ApproachThe U-Net architecture used in image segmentation uses a Python library for U-Nets by Akeret et al. <cit.> with the open source library TensorFlow. The architecture is based on the network proposed by Ronneberger et al. <cit.> for biomedical image segmentation (Figure <ref>). As discussed in the paper by Ronneberger et al., this network architecture can be trained with very few images and is relatively fast to train and test.Because the goal of this paper is to develop an algorithm that works without the need of a GPU, the number of training iterations that can be done on the U-Net is severely limited. Unlike the top ISBI submissions that used Neural Networks, which augment the data to tens of thousands of images and set the number of epochs to a few hundred <cit.>, the proposed network was trained with a lot less training iterations.The 1800 images are first augmented to 7200 images by a combination of flips and rotations. Afterwards, just over one epoch (10,000 training iterations) is applied to train the network. This is roughly 1% of the number of training iterations used by the top submissions in the ISBI challenge.§.§.§ Architecture and TrainingSimilar to convolutional neural networks, which repeatedly apply convolutions, activation functions and downsampling, the U-Net architecture extends this by including symmetric expansive layers which contain upsampling operators. The proposed architecture contains 4 contracting steps and 4 expansion steps. Each contraction step contains two unpadded 3× 3 convolutions each followed by a ReLU activation function. Finally, a 2× 2 max pooling operation is used for down sampling. In each down sampling step, the number of feature channels doubles.In each expansion step, two unpadded 3× 3 convolutions each followed by a ReLU activation function is used. However, instead of the 2× 2 max pooling, an up-convolution layer is added which halves the number of feature channels. The features from the corresponding contraction layer are also concatenated in the expansion layer.At the end, a final layer of a 1× 1 convolution is used to map the feature vector to a desired binary decision. A pixel-wise soft-max is applied in order to determine the degree of which a certain pixel belongs to the skin lesion.The loss function used is the cross entropy between the prediction and the ground truth. The Adam optimizer with a constant learning rate of 0.0002 is used to train the network. A dropout rate of 0.5 is used.§.§.§ Pre- and Post-ProcessingBefore the data is sent to the U-Net, a few pre-processing steps are performed. Firstly, in order to increase the amount of contrast in the image, the image is converted to HSI color space, and histogram equalization is applied to the intensity (I) channel. After this, the image is converted back to RGB format. The reason for this equalization is that the contrast between the skin lesions (usually darker) is more noticeable compared to the surrounding skin (usually brighter). On top of this, as a 4th channel, the original intensity (I) channel with values normalized to between 0 and 1 is added. Finally, as a 5th channel, a 2-dimensional Gaussian centered at the middle of the image is generated. The full-width half maximum of the 2D Guassian was set to be 125 pixels, which was roughly where most of the skin lesions lie within the center from a visual inspection of the training data.To the authors' best knowledge, this is the first time that histogram equalization and an extra Gaussian channel is used as an input to a U-Net. The image needs to be rescaled and zero-padded so that the U-Net can perform segmentation. Each image is rescaled such that the largest dimension is 250 pixels. The smaller dimension is filled with white padding so that the image becomes 250× 250. Finally, in order to account for cropping due to the unpadded convolutions, the image is white padded with a border of 46 pixels so that the final size is 342× 342 pixels.After training, some post-processing steps are applied to the testing set in order to improve the result. Firstly, a morphology operation (using the “scipy” binary_fill_holes function) is applied to the image in order to fill the holes in the segmented image. After this, in order to find the best threshold to set for the U-Net output, a gradient descent is applied to the threshold using in order to determine the best threshold.§.§.§ U-Net Algorithm In order to test the effect of the pre-processing and post-processing, two different approaches are compared. One with the pre-/post-processing and one without pre-/post-processing. The exact same U-Net architecture, which is described in an earlier section, is used for both cases.The pseudo-code in Figure <ref> shows the algorithm without pre-/post-processing. The pseudo-code in Figure <ref> shows the algorithm with pre/post-processing. The results obtained from the two approaches are discussed in a later section. §.§ Clustering AlgorithmThis approach leverages an unsupervised clustering technique to analyze the image and segment the lesion of interest. It uses Fuzzy C-means clustering in order to split the image up into a number of distinct regions of interest. It then leverages k-means clustering to group the clusters based on color features. Ultimately it selects the group of clusters with the darkest color features and identifies it to be the mole.§.§.§ Pre-ProcessingWhen conducting clustering to the images, there are two major sources of disturbance within the image. Firstly, the presence of hair on-top of the skin causes discrepancies in the algorithms ability to cluster, therefore strands of hair were removed in a pre-processing step which utilizes a morphological top-hat and thresholding technique. This technique is outlined in pseudo code in Figure <ref>.Secondly, a subset of the images were captured using imaging devices with a circular field of view. Resultant images captured using this type of device were circular in nature with black pixels filling the outer boundaries of the circle. These pixels were removed from the area of analysis and the algorithm for doing so can be seen in Figure <ref>. §.§.§ Clustering Algorithm SummaryThe algorithm outlined in the above sections can be found summarized in the pseudo-code in Figure <ref> and includes the pre-processing steps of hair removal and outer-circle removal. This algorithm will be referenced as Algorithm 2 for future sections. Experimentally, when C = 5, at least one of the clusters is the mole itself. For this reason, C = 5 was used for the algorithm.§ RESULTS AND DISCUSSIONS As discussed in the Methodology section, the segmented images are assessed using the Jaccard Index through 5 random folds validation with 90%-10% train-test split. For the clustering method, since it is completely unsupervised, the average Jaccard Index for the entire data set is also calculated.The Jaccard index and dice values for 5 different folds are presented in Table <ref>. Figure <ref> shows sample segmentations.Using U-Nets, for each random fold, the network is trained using the 1800 training images. After training, the average Jaccard Index score is obtained for the remaining 200 test images. This is done for both the U-Net algorithm without pre-/post-processing (Algorithm 1A) and the more complex algorithm with pre-/post-processing (Algorithm 1B).Examining the Table <ref>, it is clear that the preprocessing steps (performing histogram equalization, etc.) significantly improved the segmentation results. The average Jaccard Index for Algorithm 1A was 53% with a standard deviation of 2% while the average Jaccard Index for Algorithm 1B was 62% with a standard deviation of 4.7%.Comparing the U-Net results with state of the art neural networks while it is lower than the top performers at the 2017 ISBI competition, the results were similar to many other submissions using deep networks. For example, the top ISIC 2017 challenge submission, Yuan et al. <cit.> who used fully convolutional-deconvolutional networks obtained an average Jaccard Index of 0.765. Similarly, Berseth <cit.> who also used a U-Net obtained the second place in the 2017 challenge with a Jaccard Index of 0.762. There were also many challenge submissions that received a Jaccard Index between 0.444 and 0.665, some of which also used deep neural networks.The biggest advantage using the proposed algorithm is the number of training iterations required. For example, Yuan et al. trained with 500 epochs on the 2,000 images. Similarly, Berseth trained 200 epochs using 20,000 images (after data augmentation). This means that the two networks above required 1,000,000 and 4,000,000 training iterations, respectively.In comparison, the networks presented in this paper was trained with only 10,000 training iterations (just over 5 epochs of 1800 images). This means that the network only required 1% of the number of training iterations compared with the other deep network methods. A low number of training iterations is used because the algorithms presented in this paper is designed to work when only low computational power is available (computers without GPUs).Figure <ref> shows a comparison of the skin lesion segmentation algorithms for a few select skin lesion images. The first column in the figure shows the input RGB image. The second column contains the Ground Truth (binary mask manually segmented by a dermatologist). Finally, the third and fourth columns show the output of the U-Net trained with algorithms 1A and 1B respectively. As is evident from the figure, while both algorithm 1A and 1B are able to determine the location of the lesion, algorithm 1B is better at determining the border of the skin lesion since algorithm 1A produced a smaller image.The last three rows show some pathological cases where one or both of the algorithms are unable to properly segment the image. This includes the skin lesion being very small, the presence of another object such as medical gauze or sleeve, and the skin lesion being extremely faint. Looking at the pathological cases, it is clear that the algorithm that performs histogram equalization performs better in most circumstances.Given that the clustering based approach was unsupervised and did not contain any “training” components, it was not necessary to conduct cross-validation over multiple folds. The output was always going to be the same for every image. However, for the purposes of comparison against the U-nets based approach, the values were calculated for each of the folds used in the earlier results. The Jaccard Index calculated can be found summarized in the Table <ref>, last column. The Jaccard Index for the entire data set was calculated to be 0.443. This value is quite low in comparison to the values obtained using U-nets. However, it is important to remember that this algorithm is much simpler and quicker to execute and does not require expensive training.Figure <ref> contains a visualization from the clustering algorithm alongside the output from U-Nets for various image types. Clearly the clustering results are good for images in which no external disturbance exists and color variation is minimized. However, the algorithm performs poorly when artifacts are present. For example, the algorithm incorrectly identifies the black border, the medical gauze, and other dark objects as the skin lesion.When comparing this with other submissions in the ISIC challenge, this approach would be on par with the 21st place submission. Amongst the competition entries, only one submission followed a clustering based approach submitted by Alvarez et al. <cit.>. Their solution produced a significantly better result with a Jaccard Index of approximately 0.679. The main reason for this is that their approach uses a much more complex technique to classify the clusters involving an ensemble method which leveraged random forests and support vector machines. When comparing the results from both approaches it is clear by virtue of the Jaccard Index that the U-Net based approach (with preprocessing) is superior.§ CONCLUSION AND FUTURE WORK In conclusion, this paper presented an effective approach to automatically segmenting skin lesions particularly when computers with GPUs may not be available. We described two algorithms, one leveraging U-Nets and another using unsupervised clustering, and put forward effective pre-processing techniques for both approaches to improve the segmentation performance. It is noted that the solutions outlined in this paper are on par with the mid-tier submissions in the ISIC competition while utilizing significantly less training resources than many submissions.After comparing the clustering and U-Net algorithms, it was clear that U-Nets produce a much better segmentation result, especially using the histogram equalization based pre-processing step. The network only required 1% of the number of training iterations compared with the other deep network methods. This could be particularly beneficial when only low computational power is available (computers without GPUs)In the future, it would be interesting to find out if the histogram equalization based pre-processing step also significantly improves segmentation performance when the U-Net is trained over hundreds of epochs, instead of just one epoch, with a more powerful GPU similar to the top ISIC 2017 challenge submissions.99 1 R. A. Schwartz, Skin Cancer: Recognition and Management, 2nd Edition, Massachusetts: Blackwell Publishing Inc., 2008.2 World Health Organization, "Skin Cancers," Ultraviolet radiation and the INTERSUN Programme, 2017. [Online].3 R. Amelard, "High-Level Intuitive Features (HLIFs) for Melanoma Detection," University of Waterloo, Waterloo, 2013. 4 D. Gutman, C. C. F. Noel, E. Celebi, B. Helba, M. Marchetti, N. Mishra and A. Halpern, "Skin Lesion Analysis toward Melanoma Detection: A Challenge at the International Symposium on Biomedical Imaging (ISBI)," 2016. 5 Melanoma Research Foundation, "The ABCDEs of Melanoma," 2017. [Online]. Available: https://www.melanoma.org/understand-melanoma/diagnosing-melanoma/detection-screening/abcdes-melanoma. [Accessed 2017 February 2017]. 6 H. Fan, F. Xie, Y. Li, Z. Jiang and J. Liu, "Automatic Segmentation of Dermoscopy Images using Saliency Combined with Otsu Threshold," Computers in Biology and Medicine, vol. in press, 2017.7 M. E. Celebi, G. Schaefer, H. Iyatomi and W. V. Stoecker, "Lesion Border Detection in Dermoscopy Images," Computerized Medical Imaging and Graphics, vol. 33, no. 2, pp. 148-153, 2009.8 L. Yu, H. Chen, J. Qin and P.-A. Heng, "Automated Melanoma Recognition in Dermoscopy Images via Very Deep Residual Networks," IEEE Transactions on Medical Imaging, vol. 36, no. 4, pp. 994-1004, April 2017.9 R. Garnavi, M. Aldeen, M. E. Celebi, A. Bhuiyan, C. Dolianitis and G. Varigos, "Automatic Segmentation of Dermoscopy Images Using Histogram Thresholding on Optimal Color Channels," International Journal of Medicine and Medical Sciences, vol. 1, no. 2, pp. 126-34, 2010.10 Y. Yuan, M. Chao and Y.-C. Lo, "Automatic Skin Lesion segmentation with fully convolutional-deconvolutional networks," New York, 2017. 11 T. Lee, V. Ng, R. Gallagher, A. Coldman and D. McLean, "Dullrazor: A software approach to hair removal from images," Computers in Biology and Medicine, vol. 27, no. 6, pp. 533-543, 1997.12 P. Schmid-Saugeon, J. Guillod and J.-P. Thiran, "Towards a computer-aided diagnosis system for pigmented skin lesions," Computerized Medical Imaging and Graphics, vol. 27, no. 1, pp. 65-78, 2003.13 R. Melli, G. Costantino and R. Cucchiara, "Comparison of color clustering algorithms for segmentation of dermatological images," Medical Imaging, pp. 61443S-61443S, 15 march 2006.14 M. H. Jafari, N. Karimi, E. N. Esfahani, Samavi, Shadrokh, S. M. R. Soroushmehr, K. R. Ward and K. Najarian, "Skin Lesion Segmentaion in Clinical Images Using Deep Learning," in IEEE International Conference on Pattern Recognition (ICPR), Cancun, Mexico, 2016.15 J. Akeret, C. Chang, A. Lucchi and A. Refregier, "Radio frequency interference mitigation using deep convolutional neural networks," Astronomy and Computing, vol. 18, pp. 35-39, 2017.16 O. Ronneberger, P. Fischer and T. Brox, "U-Net: Convolutional Networks for Biomedical Image Segmentation," in International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer International Publishing, 2015.17 M. Berseth, "ISIC 2017 - Skin Lesion Analysis Towards Melanoma Detection," 2017. 18 D. Alvarez and M. Iglesias, "k-Means Clustering and Ensemble of Regressions: An Algorithm for the ISIC 2017 Skin Lesion Segmentation Challenge," Oviedo, Spain, 2017. 19 Sahba, Farhang, Hamid R. Tizhoosh, and Magdy MMA Salama. "Application of opposition-based reinforcement learning in image segmentation." Computational Intelligence in Image and Signal Processing, 2007. CIISP 2007. IEEE Symposium on. IEEE, 2007. 20 Shokri, Maryam, and Hamid R. Tizhoosh. "Using reinforcement learning for image thresholding." Electrical and Computer Engineering, 2003. IEEE CCECE 2003. Canadian Conference on. Vol. 2. IEEE, 2003. 21 Othman, Ahmed, and Hamid Tizhoosh. "Segmentation of breast ultrasound images using neural networks." Engineering Applications of Neural Networks (2011): 260-269. 22 Cheng, Kuo-Sheng, Jzau-Sheng Lin, and Chi-Wu Mao. "The application of competitive Hopfield neural network to medical image segmentation." IEEE transactions on medical imaging 15.4 (1996): 560-567. 23 Ozkan, Mehmet, Benoit M. Dawant, and Robert J. Maciunas. "Neural-network-based segmentation of multi-modal medical images: a comparative and prospective study." IEEE transactions on Medical Imaging 12.3 (1993): 534-544. | http://arxiv.org/abs/1710.01248v1 | {
"authors": [
"Bill S. Lin",
"Kevin Michael",
"Shivam Kalra",
"H. R. Tizhoosh"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170927224629",
"title": "Skin Lesion Segmentation: U-Nets versus Clustering"
} |
Random Overlapping Communities: Approximating Motif Densities of Large Samantha PettiGeorgia Tech, [email protected]. Supported in part by an NSF graduate fellowship.Santosh Vempala Georgia Tech, [email protected]. Both authors were supported in part by NSF awards CCF-1563838 and CCF-1717349. December 30, 2023 ========================================================================================================================================================================================================================================A wide variety of complex networks(social, biological, information etc.) exhibit local clustering with substantial variation in the clustering coefficient (the probability of neighbors being connected). Existing models of large capture power law degree distributions (Barabási-Albert) and small-world properties (Watts-Strogatz), but only limited clustering behavior. We introduce a generalization of the classical Erdős-Rényi model of random which provably achieves a wide range of desired clustering coefficient, triangle-to-edge and four-cycle-to-edge ratios for any given size and edge density. Rather than choosing edges independently at random, in the Random Overlapping Communities model, a is generated by choosing a set of random, relatively dense sub(“communities”). We discuss the explanatory power of the model and some of its consequences.§ INTRODUCTIONRandomness has been an effective metaphor to model and understand the structure of complex networks. In 1959,Erdős and Rényi <cit.> defined the simple random model G_n, p, where every pair of n vertices is independently connected with probability p. Their seminal work transformed the field of combinatorics and laid the foundation of network science.Mathematicians have extensively studied properties of generated from this model and used it to prove the existence of with certain properties. (See <cit.> for a survey.) The comparison of real-world to G_n,p is a popular tool for highlighting their nonrandom features <cit.>. Moreover, the model has inspired more sophisticated random models, as predicted byErdős and Rényi in the following remark from their pre-internet/pre-social article:This may be interesting not only from a purely mathematical point of view ... if one aims at describing such a real situation, one should replace the hypothesis of equiprobability of all connections by some more realistic hypothesis. It seems plausible that by considering the random growth of more complicated structuresone could obtain fairly reasonable models of more complex real growth processes.The two most influential random models designed to mimic properties of real-world are the Watts-Strogatz small world model <cit.> and the Barabási-Albert preferential attachment model <cit.>. Briefly, the first is a process that randomly rewires connections of a regular ring lattice . The resulting have small diameter and high clustering coefficient (the probability that two neighbors of a randomly selected vertex are adjacent). The second is a growth model that repeatedly adds a new vertex to an existing and connects to existing vertices with probability proportional to their degree. This model exhibits and maintains a power law in the distribution of vertex degrees, another commonly observed phenomenon.These and other existing random models do not capture the following fundamental aspects of local structure: (1) Existing models cannot be tuned to produce with arbitrary density, triangle-to-edge ratio, and four-cycle-to-edge ratio. (2) The clustering coefficients of produced by existing models lie in very limited ranges determined by the density. In reality, the clustering coefficients of a variety of complex (social, biological, information etc.) vary substantially and are not simply a function of the density <cit.>. We introduce the Random Overlapping Communities (ROC) model, a simple generalization of the Erdős-Rényi model, which produces with a wide range of clustering coefficients as well as triangle-to-edge and four-cycle-to-edge ratios. The model generates that are the union of many relatively dense random communities. A community is an instance of G_s,q on a set of s randomly chosen vertices. A ROC is the union of many randomly selected communities that overlap, soeach vertex is a member of multiple communities.The size s and density q of the communities determine clustering coefficient and triangle and four-cycle ratios.Capturing motif densities.A widely-used technique for inferring the structure and function of a is to observe overrepresented motifs, i.e., small patterns (sub) that appear frequently. Recent work describes the overrepresented motifs of a variety of including transcription regulation , protein-protein interaction , the rat visual cortex, ecological food webs, and the internet (WWW),<cit.>. The type of overrepresented motifs has been shown to be correlated with the function <cit.>. A model that produces with high motif counts is necessary for approximating whose function depends on the abundance of a particular motif. Here we focus on the two most basic motifs— triangles and four-cycles.A natural approach to constructing a with high motif density is to repeatedly add the motif on a randomly chosen subset of vertices. However, this process yields low motif to edge ratios for sparse . For example, a on n vertices with average degree less than √(n) built by randomly adding triangles will have a triangle-to-edge-ratio at most 2/3. (See <Ref>.) In <cit.> Newman considers a similar approach which produces with varied degree sequences and triangle to edge ratio strictly less than 1/3. However, it is not hard to construct with arbitrarily high triangle ratio (growing with the size of the ). In the dense setting, a constant-size stochastic block model can be used to approximate with high motif densities, as guaranteed by Szemerédi's regularity lemma (see <cit.>).In a stochastic block model M, each vertex is assigned to one of k classes, and an edge is added between each pair of vertices independently with probability M_i,j where i and j are the classes of the vertices. However, the situation is drastically different for nondense .To construct a sparse with maximum degree at most n^1/3with non-vanishing four-cycle density, the rank of M must grow with the size of the . Let M be a symmetric n × n matrix with entries in [0,1] such that each row sum is at most d. Let G be a graph on n vertices obtained by adding each edge (i,j) independently with probability M_ij. Then the expected number of k-cycles in G at most d^4 rank(M).For example, the d-dimensional hypercube on n=2^d vertices has a log(n)/4 four-cycle-to-edge ratio; a stochastic block model M that produces a of the same size, degree, and ratio must have rank at least O(n/log^2 n).In contrast to the above approaches, the ROC model produces with arbitrary triangle and four-cycle ratios independent of the density or size of the . In <Ref> we show that for almost all triangle and four-cycle ratios arising from some , there exists parameters for the ROC model to produce with these ratios, simultaneously. Moreover, the vanishing set of triangle and four-cycle ratio pairs not achievable exactly can be approximated to within a small error.Clustering coefficient. The clustering coefficient at a vertex v is the probability two randomly selected neighbors are adjacent:C(v)= |{{a,b} : a, b ∈ N(v), a∼ b}|/deg(v)(deg(v)-1)/2.Equivalently the clustering coefficient is twice the ratio of the number of triangles containing v to the degree of v squared. The ROC model is well suited to produce random that reflect the high average clustering coefficients of real world . Figure <ref> illustrates the markedly high clustering coefficients of real-world as compared with Erdős-Rényi (E-R) of the same density.In <Ref>, we prove the average clustering coefficient of a ROC is approximately sq^2/d, meaning that tuning the parameters s and q with d fixed yields wide range of clustering coefficients for a fixed density. Furthermore, <Ref> describes the inverse relationship between degree and clustering coefficient in ROC , a phenomena observed in protein-protein interaction , the internet, and various social <cit.>. Structure of the paper. In <Ref> we introduce the ROC model, and then in <Ref> we show the model's ability to produce with specified size, density, triangle and four-cycle ratios and clustering coefficients. In <Ref> we introduce a variation of the ROC model which produces with various degree distributions and tunable clustering coefficient. We end with a discussion of the model's mathematical interest and explanatory value in real-world settings in Section <ref>. § THE RANDOM OVERLAPPING COMMUNITIES MODELA complex is modeled as the union of relatively dense, random communities. More precisely, to construct a on n vertices with expected degree d, we pick dn/(qs(s-1)) random , each of density q on a random subset of s of the n vertices.0.95 ROC(n,d,s,q).Output: a on n vertices with expected degree d. Repeat dn/(qs(s-1)) times: * Pick a random subset S of vertices (from {1,2,…,n}) by selecting each vertex with probability s/n.* Add the random G_|S|,q on S, i.e., for each pair in S, add the edge between them independently with probability q; if the edge already exists, do nothing.This generalizes the standard E-R model, which is the special case when s=n and a single community is picked. For G ∼ ROC(n,d,s,q) the expected degree of each vertex is d.If d > sq logn then with high probability G will be connected. Moreover if d/p> lognd/s(s-1)p, then with high probability the communities of G will be connected even though there may be isolated vertices. See Section <ref> of the appendix for a further exploration of the connectivity properties of the ROC model.§ APPROXIMATION BY ROC In this section we analyze small cycle counts and local clustering coefficient of ROC . For proofs of the theorems refer to Section <ref> of the appendix.We state our results as they hold asymptotically with respect to n.§.§ Triangle and four-cycle count in ROC .Define R_k as the ratio between the number of k cycles and the edges in a :R_k(G)=C_k(G)/|E(G)|,where C_k(G) denotes the number of k cycles in G. For G ∼ ROC(n,d,s,q), we instead defineR_k(G)=2C_k(G)/nd,the ratio of the expected number of k cycles to the expected number of edges. Let G ∼ ROC(n,d,s,q)and s = ω(1). Thenlim_n →∞R_3(G)= sq^2/3 d=o(√(n))and lim_n →∞R_4(G)= s^2q^3/4 d=o(n^1/3). By varying s and q, we can construct a ROC that achieves any ratio of triangles to edges or any ratio of four-cycles to edges. By setting s=√(log(n))/4 and q=1, we obtain a family of with the hypercube four-cycle-to-edge ratio log(n)/4, something not possible with any existing random model. Moreover, it is possible to achieve a given ratio by larger, sparser communities or by smaller, denser communities. For example communities of size 50 with internal density 1 produce the same triangle ratio as communities of size 5000 with internal density 1/10. Figure <ref> illustrates the range of s and q that achieve various triangle and four-cycle ratios. Note that it is possible to achieve R_3=3 and R_4 ∈{ 100,50,25} but not R_3=3 and R_4 ∈{3, 10}.Next, we show that for almost all achievable pairs of triangle and four-cycle ratios, there exists a ROC construction that matches both ratios asymptotically.The ROC model approximates most pairs of triangle and four-cycle ratios. * If there exists a H with R_3(H)=r_3 and R_4(H)=r_4, then 3r_3(3r_3-1) ≤ 4r_4. *For any r_3 and r_4 such that 9r_3^2≤ 4r_4, and d= o(n^1/3), the randomG ∼ ROC(n,d, 16r_4^2/27r_3^3, 9r_3^2/4r_4) haslim_n →∞R_3(G)= r_3and lim_n →∞R_4(G)= r_4.For every with triangle and four-cycle ratios in the narrow range 3r_3(3r_3-1) ≤ 4r_4 ≤ 9r_3^2, there exists a ROC construction that matches r_3 and can approximate r_4 by 9r_3^2, i.e., up to an additive error 3r_3/4 (or multiplicative error of at most 1/(3r_3 -1) which goes to zero as r_3 increases). §.§ Clustering coefficient.<Ref> gives an approximation of the expected clustering coefficient when the degree and average number of communities per vertex grow with n. The exact statement is given in <Ref> of Section <ref>, and bounds in a more general setting are given by <Ref>.Let C(v) denote the clustering coefficient of a vertex v with degree at least 2 in a drawn from ROC(n,d,s,q) with d=o(√(n)), d<(s-1) q e^sq, d= ω( sq lognd/s), s^2q=ω(1), and sq =o(d). Then C(v)=(1+o(1) ) s q^2/d. Unlike in E-R in which local clustering coefficient is independent of degree, higher degree vertices in ROC have lower clustering coefficient. High degree vertices tend to be in more communities, and thus the probability two randomly selected neighbors are in the same community is lower.Figure <ref> illustrates the relationship between degree and clustering coefficient, the degree distribution, and the clustering coefficient for two ROC with different parameters and the E-R random of the same density. Let C(v) denote the clustering coefficient of a vertex v in a drawn from ROC(n,d,s,q) with d=o(√(n)), s=ω(1) and deg(v) ≥ 2sq. Then C(v)deg(v)=r= sq^2/r(1 +o_r(1) ) The dependence between degree and clustering coefficient is the result of the variation in the numbers of communities a vertex is part of. To eliminate this variation and obtain a clustering coefficient distribution that is not highly dependent on degree, we can modify the ROC construction as follows. Instead of selecting s vertices uniformly at random to make up a community in each step, pre assign each vertex to precisely d/sq communities of size s. In this setting the expected clustering coefficient can easily be computed:C(v)=two randomly selected nhbs are from the same community q =sq^2/d.Note also, that this variant of the ROC model will produce with fewer isolated vertices.§ DIVERSE DEGREE DISTRIBUTIONS AND THE DROC MODELIn this section we introduce an extension of our model which produces that match a target degree distribution in expectation. The extension is inspired by the Chung-Lu configuration model: given a degree sequence d_1, … d_n, an edge is added between each pair of vertices v_i and v_j with probability d_i d_j/∑_i=1^n d_i, yielding a where the expected degree of vertex v_i is d_i <cit.>. In theDROC model,a modified Chung-Lu random is placed instead of an E-R random in each iteration. Instead of normalizing the probability an edge is selected in a community by the sum of the degrees in the community, the normalization constant is the expected sum of the degrees in the community. We use D to denote a target degree sequence t(v_1), … t(v_n), and d to denote the mean.0.95 DROC(n,D,s,q).Output: a on n vertices where vertex v_i has expected degree t(v_i). Repeat n/((s-1)q) times: * Pick a random subset S of vertices (from {1,2,…,n}) by selecting each vertex with probability s/n.* Add a modified C-L random on S, i.e., for each pair in S, add the edge between them independently with probability qt(v_i) t(v_j)/sd; if the edge already exists, do nothing. Given a degree distribution D with mean d and max_i t(v_i)^2 ≤sd/q,DROC(n,D,s,q) yields a where vertex v_i has expected degree t(v_i). We requiremax_i t(v_i)^2 ≤sd/q to ensure that the probability each edge is chosen is at most 1. Instead of requiring a sequence of n target degrees as input to the DROC model, we can define the model with a distribution 𝒟 of target degrees. In this altered version, Step 0 of the algorithm is to select a target degree for each vertex according to 𝒟. Taking the distribution D_d with t(v)=d for all v in the DROC model does not yield ROC(n,d,s,q). The model DROC(n, D_d, s,q) is equivalent to ROC(n,d,s, qd/s). The following corollary shows that it is possible to achieve a power law degree distribution with the DROC model for power law parameter γ>2. We use ζ(γ)=∑_n=1^∞ n^-γ to denote the Riemann zeta function. Let D∼𝒟_γ be the power law degree distribution defined as follows:t(v_i)=k= k^-γ/ζ(γ),for all 1 ≤ i ≤ n. If γ>2 and s/q =ω(1)ζ(γ)/ζ(γ-1) n^1/γ-1,then with high probability D satisfies the conditions of <Ref>, and therefore can be used to produce a DROC . §.§ Clustering Coefficient.We show that by varying s and q we can control the clustering coefficient of a DROC graph.Let C(v) denote the clustering coefficient of a vertex vin drawn from DROC(n,D,s,q) with max t(v_i)^2 ≤sd/q, s= ω(1), s/n=o(q), and t=t(v). Then C(v) = ∑_u ∈ V t(u)^2^2/d^3n^2s (1- e^-t)^2q^2 +c_t q^3,where c_t ∈ [0,6.2) is a constant depending on t.Equation <ref> in the proof of the theorem gives a precise statement of the expected clustering coefficient conditioned on community membership.§ DISCUSSION AND OPEN QUESTIONSModeling real-world . The ROC model captures the degree distribution and clustering coefficient of graphs simultaneously. Previous work <cit.>, <cit.>, and <cit.> provides models that produce power law with high clustering coefficients. Their results are limited in that the resulting are restricted to a limited range of power-law parameters, and are either deterministic or only analyzable empirically. In contrast, the DROC model is a fully random model designed for a variety of degree distributions (including power law with parameter γ >2) and canprovably produce with a wide range of clustering coefficient.Our model therefore may be a useful tool for approximating large .It is often not possible to test algorithms on with billions of vertices (such as the brain, social , and the internet). Instead, one could use the DROC model to generate a smaller with same clustering coefficient and degree distribution as the large , and then optimize the algorithm in this testable setting.Further study of such a small approximation could provide insight into the structure of the large of interest.Modeling a as the union of relatively dense communities has explanatory value for many real-world settings, in particular for social and biological networks. Social networks can naturally be thought of as the union of communities where each community represents a shared interest or experience(i.e. school, work, or a particular hobby); the conceptualization of social networks as overlapping communities has been studiedin<cit.>, <cit.>. Protein-protein interaction networks can also be modeled by overlapping communities, each representing a group of proteins that interact with each other in order to perform a specific cellular process. Analyses of such networks showproteins are involved in multiple cellular processes, and therefore overlapping communities define the structure of the underlying <cit.>, <cit.>, <cit.>.ROC vs mixed membership stochastic block models. Mixed membership stochastic block models have traditionally been applied in settings with overlapping communities<cit.>, <cit.>, <cit.>. The ROC model differs in two key ways. First, unlike low-rank mixed membership stochastic block models, the ROC model can produce sparse with high triangle and four-cycle ratios. As discussed in the introduction, the over-representation of particular motifs in a is thought to be fundamental for its function, and therefore modeling this aspect of local structure is important. Second, in a stochastic block model the size and density of each community and the density between communities are all specified by the model. As a result, the size of the stochastic block model must grow with the number of communities, but the ROC model maintains a succinct description. This observation suggests the ROC model may be better suited for in which there are many communities that are similar in structure, whereas the stochastic block model is better suited for with a small number of communities with fundamentally different structures. Below we discuss extensions of the ROC model that maintain a succinct description and produce more diverse community structures.Open questions. * Consider the following extension. Instead of adding communities of size s and density q, we define a probability distribution on a set of pairs (s_i, q_i), and in each iteration choose a pair of parameters (s_i, q_i) from the distribution and build the community G_s_i,q_i on s_i randomly selected vertices. Does this modification provide a better approximation for real-world graphs? * A further generalization involves adding particular subfrom a specified set according to some distribution instead of E-R graphs in each step (e.g., perfect matchings or Hamiltonian paths). Does doing so allow for greater flexibility in tuning the number of various types of motifs present (not just triangles and four-cycles)? * A fundamental question in the study of graphs is how to identify relatively dense clusters. For example, clustering protein-protein interaction networks is a useful technique for identifying possible cellular functions of proteins whose functions were otherwise unknown <cit.>. An algorithm designed specifically to identify the communities in a drawn from the ROC model has potential to become a state-of-the-art algorithm for clustering real-world networks with overlapping community structure. * The asymptotic thresholds for properties of E-R have been studied extensively, see <cit.> for a survey. Such questions are yet to be explored on ROC , e.g., does every nontrivial monotone property have a sharp threshold? * How do graph algorithms behave on ROC graphs? For instance, what is the covertime of a random walk on a ROC graph? plain § LIMITATIONS OF PREVIOUS APPROACHES Let G be a on n vertices obtained by repeatedly adding triangles on sets of three randomly chosen vertices. If the average degree is less than √(n), the expected ratio of triangles to edges is at most 2/3. Let t be the number of triangles added and d the average degree, so d = 6t/n. To ensure that d < √(n), t< n^3/2/6. The total number of triangles in the is t + (d/n)^3 n3= t +d^3/6= t+36t^3/n^3. It follows that the expected ratio of triangles to edges is at mostt +36(t/n)^3/3t≤2/3.(of <Ref>) Let σ_1 …σ_rank(M) denote the eigenvalues of M.# k-cycles =∑_i_1 ≠i_2 …≠i_k M_i_1i_2M_i_2 i_3… M_i_k i_1≤ Tr(M^k)= ∑_i=1^rank(M)σ_i^k≤ rank(M) d^k.§ CONNECTIVITY OF THE ROC MODELWe describe the thresholds for connectivity for ROC(n,d,s,q) networks. A vertex is isolated if it is has no adjacent edges. A community is isolated if it does not intersect any other communities. Here we use the abbreviation a.a.s. for asympotically almost surely. An event A_n happens a.a.s. ifA_n→ 1 as n→∞.For (s-1)q (ln n+c)≤ d≤ (s-1)qe^sq(1-), a from ROC(n,d,s,q) a.a.s. has at most e^-c/1- isolated vertices.We begin by computing the probability a vertex is isolated,vis isolated = ∑_i=0^nd/s^2qvis in i communities (1-q)^si= ∑_i=1^nd/s^2qnd/s(s-1)q isn^i 1-s/n^nd/s(s-1)q -i e^-sqi≤ e^-d/(s-1)q∑_i=0^nd/s^2qde^-sq+s/n(s-1)q^ i =e^-d/(s-1)q∑_i=1^nd/s^2qde^-sq(s-1)q^ i = e^-d/(s-1)q11-.Let X be a random variable that represents the number of isolated vertices of a drawn from ROC(n,d,s,q). We computeX>0≤X =ne^-d/(s-1)q11-= e^-c1-. A fromROC(n,d,s,q) with s=o(√(n)) has no isolated communities a.a.s. if d/q> lognd/s^2q. We construct a “community " and apply the classic result that G(n,p) will a.a.s. have no isolated vertices when p> (1+)logn/n for any >0<cit.>. In the “community " each vertex is a community and there is an edge between two communities if they share at least one vertex; a ROC has no isolated communities if and only if the corresponding “community " is connected. The probability two communities don't share a vertex is (1-s/n)^s. Since communities are selected independently, the “community " is an instance of G(nd/s(s-1)q, 1-(1-s/n)^s). By the classic result, approximating the parameters by nd/s^2q, 1-e^s^2/n, this is connected when1-e^-s^2/n> lognd/s^2q/nd/s^2q.Since s= o(√(n)) is small, the left side of the inequality is approximately s^2/n, yielding the equivalent statementd/q>lognd/s^2q.Note that the threshold for isolated vertices is higher, meaning that if a ROC a.a.s has no isolated vertices, then it a.a.s has no isolated communities. These two properties together imply the is connected.§ SECTION <REF> PROOFS (of <Ref>.) Let G ∼ ROC(n,d,s,q) and u,v ∈ V(G). First note that without information about whether u and v are in community together u ∼ v =d/n=o(1) because each edge is equally likely. However, u ∼ v u, v are in a common community =q+o(1). We show that both the triangle count and the four-cycle count are dominated by cycles contained entirely in one community. We compute C_3(G) by counting the total number of triangles in G. Let T_1 be the number triangles with all three edges originating in one community, T_2 be the number of triangles with two edges originating in the same community and the third edge originating in a different community, and T_3 be the number of triangles with edges originating in three different communities. We compute T_1 = (# com.)triangles in a com.=nd/s(s-1)qs^3q^3/6=ndsq^2/6 T_2 =(# com.)#two paths u ∼ v, v ∼ w in a com.u ∼ w in other com.=nd/s(s-1)qss-12 q^2 d/n=d^2sq/2 T_3 =(# triples u,v,w ∈ V(G))u,v,w form a triangle= n3(d/n)^3= d^3/6. ThereforeR_3(G)=2T_1+T_2+T_3/nd=sq^2/3(1+o(1)). Similarly, we compute C_4(G) by summing over different categories of four-cycles based on the shared community membership of the vertices. For simplicity suppose the a,b,c,d are the vertices of the four-cycle and let C_1, … C_4 denote different communities. If {a,b,c,d}∈ C_1, the cycle is type 1. If {a,b,c}∈ C_1 and {a,c,d}∈ C_2, the the cycle is type 2. If {a,b, d}∈ C_1, {b,c}∈ C_2, {c,d}∈ C_3, then the cycle is type 3. If {a,b }∈ C_1, {b,c}∈ C_2, {c,d}∈ C_3, {d,a}∈ C_4, then the cycle is type 4. Let F_i be the number of cycles of type i.We computeF_1 =(# com.)# four-cycles in a com.=nd/s(s-1)q3s^4q^4/24=nds^2q^3/8 F_2 =# vertex pairs u,v in two of the same coms. (# common nhbs of u,v in a com.)^2=n2sn^4nd/s(s-1)q 2( (s-2)q^2)^2= s^2q^2 d^2/4 F_3 = (# com.)#two paths u ∼ v, v ∼ w in a com. |V(G)| x ∼ w and x ∼ u=nd/s(s-1)q ss-12 q^2( d/n)^2=sd^3/2 F_4 =(# quadruples u,v,w,x ∈ V(G))ways u,v,w,x form a four-cycle= n4 3 (d/n)^4= d^4/8. ThereforeR_3(G)=2F_1+F_2+F_3+F_4/nd=s^2q^3/3(1+o(1)). (of <Ref>.) (1) For each edge in H, let t_e be the number of triangles containing e, so ∑_e ∈ E(H) t_e = 3 C_3(H)= 3 r_3 |E(H)|. If triangles abc and abd are present, then so is the four-cycle acbd. This four-cycle may also be counted via triangles cad and cdb. Therefore C_4(H) ≥1/2∑_ e∈ E(H)t_e2. This expression is minimized when all t_e are equal. We therefore obtainr_4|E(H)|=C_4(H)≥|E(H)|/23 r_32= 3r_3(3r_3-1) |E(H)|/4.It follows that 3r_3(3r_3-1)/4r_4≤1.(2) Since the hypothesis guarantees q ≤ 1, applying <Ref> to G ∼ ROC(n,d, 16r_4^2/27r_3^3, 9r_3^2/4r_4) implies the desired statements.<Ref> gives bounds expected clustering coefficient up to factors of (1 + o(1)).The clustering coefficient at a vertex is only well-defined if the vertex has degree at least two. Given the assumption in<Ref> that d= ω(sq lognd/s), d< (s-1)q e^sq, and s=ω(1), <Ref> implies that the fraction of vertices of degree strictly less than two is o(1). Therefore we ignore the contribution of these terms throughout the computations for <Ref> and supporting <Ref>. In addition we divide by deg(v)^2 rather than by deg(v)(deg(v)-1) in the computation of the clustering coefficient since this modification only affects the computations up to a factor of (1 + o(1)).If d= ω( sqlognd/s), s=ω(1), s=o(n), and d<(s-1)q e^sq, then a from ROC(n,d,s,q) a.a.s. has no vertices of degree less than 2.<Ref> implies there are no isolated vertices a.a.s. We begin by computing the probability a vertex has degree one. deg(v)=1=∑_i=1^nd/s^2qvis in i communities q (1-q)^si-1=∑_i=1^nd/s^2qnd/s(s-1)q isn^i 1-s/n^nd/s(s-1)q -i q (1-q)^si-1≤∑_i=1^nd/s^2qnds(s-1)q^ i sn^i e^-d/sq+si/n q e^-qsi+q=q e^-d/sq∑_i=1^nd/s^2qde^-sq(s-1)q^ i =Ode^-sq-d/sq/sLet X be a random variable that represents the number of degree one vertices of a drawn from ROC(n,d,s,q). When d= ω( sqlognd/s), we obtainX>0≤X = Onde^-sq-d/sq/s=o(1). Let C(v) denote the clustering coefficient of a vertex v of degree at least 2 in a drawn from ROC(n,d,s,q) with d=o( √(n)) and d= ω(sq lognd/s). ThenC(v)=(1 + o(1) ) ( ∑_i=1^nd/s^2qnd/s^2q i( s/n)^i (1- s/n)^nd/s^2q-is(s-1)q^3k /( sqk + 2-2q)^2). For ease of notation, we ignore factors of (1+ o(1)) throughout as described in <Ref>. First we compute the expected clustering coefficient of a vertex from an ROC(n,d,s,q) given v is contained in precisely k communities. Let X_1, … X_k be random variables representing the degree of v in each of the communities, X_i ∼ Bin(s,q). We haveC(v)| v in k communities=∑_i=1^kX_i(X_i-1) q/( ∑_i=1^k X_i)^2= qk X_1(X_1-1) /( sq(k-1) + X_1)^2= qk X_1^2 /( sq(k-1) + X_1)^2- qk X_1 /( sq(k-1) + X_1)^2.Write X_1= ∑_i=1^s y_i where y_i ∼ Bernoulli(q). Using linearity of expectation and the independence of the y_i's we haveX_1 /( sq(k-1) + X_1)^2 = s y_1 /( sq(k-1) + (s-1)q+y_1)^2 = sq /( sq(k-1) + (s-1)q+1)^2,and X_1^2 /( sq(k-1) + X_1)^2 = (∑_i=1^s y_i)^2 /( sq(k-1) +∑_i=1^s y_i)^2= s y_1^2 /( sq(k-1) +q(s-1) +y_1)^2+s(s-1)(y_1y_2)^2 /( sq(k-1) + (s-2)q+y_1+y_2)^2=sq /( sq(k-1) +q(s-1) +1)^2+s(s-1)q^2 /( sq(k-1) + (s-2)q+2)^2.Substituting in these values into <Ref>, we obtainC(v)|v ∈ kcommunities =qk(s(s-1)q^2 /( sq(k-1) + (s-2)q+2)^2)=s(s-1)q^3k /( sqk + 2-2q)^2. Let M be the number of communities a vertex is in, so M∼ Bin(nd/s^2q,s/n). It follows C(v) =∑_i=1^nd/s^2qv in k communities C(v)|v in k communities =∑_i=1^nd/s^2qnd/s^2q i( s/n)^i (1- s/n)^nd/s^2q-is(s-1)q^3k /( sqk + 2-2q)^2.The proof of <Ref>, relies on the follow two lemmas regarding expectation of binomial random variables.Let X ∼ Bin(n,p). Then* 1/X+1 X ≥ 1 = 1- ( 1-p)^n+1-(n+1)p(1-p)^n/p(n+1) and* 1/X+1= 1- ( 1-p)^n+1/p(n+1).Observe 1/X+1 X≥ 1 = ∑_i=1^n nip^i (1-p)^n-i/i+1= 1/p(n+1)∑_i=1^nn+1i+1 p^i+1 (1-p)^n-i= 1- ( 1-p)^n+1-(n+1)p(1-p)^n/p(n+1).Similarly 1/X+1 = ∑_i=0^n nip^i (1-p)^n-i/i+1 = 1/p(n+1)∑_i=0^nn+1i+1 p^i+1 (1-p)^n-i = 1- ( 1-p)^n+1/p(n+1). Let X ∼ Bin(n,p). Then1/X X ≥ 1≤1/p(n+1)(1 +3/p(n+2)).Note that when X ≥ 1,1/X≤1/X+1+3/(X+1)(X+2).By Lemma <ref>, 1/X+1 X ≥ 1≤1/p(n+1).We compute 1/(X+1)(X+2) X≥ 1 = ∑_i=1^n ni p^i (1-p)^n-i/(i+1)(i+2)= 1/p^2 (n+2)(n+1)∑_i=1^n n +2i+2 p^i+2(1-p)^n-i≤1/p^2 (n+2)(n+1).Taking expectation of <Ref> gives1/X X≥ 1≤1/p(n+1)(1 +3/p(n+2)). (of <Ref>.) For ease of notation, we ignore factors of (1 + o(1)), as described in <Ref>. It follows from <Ref> in the proof of <Ref> thatq/k+1≤C(v)|v ∈ kcommunities ≤q/k,where the left inequality holds when q(s-1)≥ 5.We now compute upper and lower bounds on C(v), assuming v is in some community. Let M be the random variable indicating the number of communities containing v, M ∼ Bin(nd/s(s-1) q, s/n). It followsC(v)=∑_k=1^nd/s^2qM=k C(v)|M=k q1/M+1 M ≥ 1≤C(v)≤q1/M M ≥ 1.Applying Lemmas <ref> and <ref> to the lower and upper bounds respectively, we obtain q(1-(1-s/n)^nd/s(s-1)q+1- nd/s(s-1)q+1(1-s/n)^nd/s(s-1)q)/d/(s-1)q+s/n≤C(v)≤q/d/(s-1)q +s/n( 1 + 3/d/(s-1)q +2s/n)which for s=o(n) simplifies to(s-1)q^2/d( 1-nd/s (s-1) qe^-d/((s-1)q)) ≤C(v) ≤(s-1)q^2/d( 1+ (s-1)q/d) .Under the assumptions s^2q =ω(1) and sq = o(d), we obtain our desired resultC(v)=(1+o(1) )( sq^2/d).The following lemma will be used in the proof of <Ref>.The X be a nonnegative integer drawn from the discrete distribution with density proportional to f(x)=x^r-xe^-ax. Let z= f(x). Then|x-z| ≥ 2t√(z)≤ e^-t+1. First we observe that f is logconcave:d^2/dx^2ln f(x) = d/dx(-a + r/x-1 - ln x) = -r/x^2-1/xwhich is nonpositive for all x ≥ 0. We will next bound the standard deviation of this density, so that we can use an exponential tail bound for logconcave densities. To this end, we estimate max f. Setting its derivative to zero, we see that at the maximum, we have a+1= r/x-ln x.The maximizer z is very close to r/(a+1)+lnr/(a+1)+ln(r/(a+1)),and the maximum value z satisfies z^r-ze^-az=z^r e^-r+z.Now we consider the point z+δ where f(z+δ)=f(z)/e, i.e., (z+δ)^r-z-δe^-az-aδ/z^r-ze^-az≤ e^-1.The LHS is (1+δ/z)^r-zz^-δ(1+δ/z)^-δe^-aδ ≤ e^δ(r/z-1-a-ln z)e^-δ^2/z≤ e^-δ^2/zwhere in the second step we used the optimality condition (<ref>).Thus for δ = √(z), f(x + δ) ≤ f(x)/e. By logconcavity[which says that for any x,y and any λ∈ [0,1], we have f(λ x +(1-λ)y)≥ f(x)^λ f(y)^1-λ] we have f(x+ δ)= f 1-1/t x + 1/t ( x +tδ) ≥ f(x)^1-1/t f(x+tδ)^1/tfor any t ≥ 1.It follows f(x+tδ) ≤ f(x)/e^t for all t (since we can apply the same argument for z-δ). Taking x=z in <Ref> and usingthe observation ∑_x ∈^+ f(x) ≥ f(z), it follows thatx= z+ t √(z)≤e^-t andx= z- t √(z)≤ e^-tand so|x-z| ≥ t√(z)≤ 2 e^-t≤ e^-t+1. (of <Ref>).Let M denote the number of communities a vertex v is selected to participate in. We can writeC(v)|deg(v)=r = ∑_k=r/s^r C(v)|deg(v)=r, M=kM=k|deg(v=r=∑_k=r/s^r C(v)|deg(v)=r, M=kdeg(v)=r|M=kM=k/deg(v)=r.First we compute the expected clustering coefficient of a degree r vertex given that it is k communities: C(v)|deg(v)=randM=k=∑_i ≠ j, i,j ∈ N(v) q(i, jpart of same community)/deg(v)(deg(v)-1) =q/k.Next we note that M is a drawn from a binomial distribution, and the degree of v is drawn from a sum of k binomials, each being Bin(s,q). Therefore,M=kdeg(v)=r|M=k = nd/s(s-1)q k(s/n)^k(1-s/n)^nd/s(s-1)q-kskr q^r(1-q)^sk-r.Using this we obtainC(v)|deg(v)=r = ∑_k=r/s^r q/kM=kdeg(v)=r|M=k/∑_k=r/s^rM=kdeg(v)=r|M=k= q ∑_k=r/s^r 1/k·(d/(s-1)qk)^ke^-d/(s-1)q+sk/n(skq/r)^re^-qsk+qr/∑_k=r/s^r(d/(s-1)qk)^ke^-d/(s-1)q+sk/n(skq/r)^re^-qsk+qr= q ∑_k=r/s^r 1/k·(d/(s-1)q)^k k^r-ke^-qsk/∑_k=r/s^r (d/(s-1)q)^k k^r-ke^-qsk. Writing a=qs-ln(d/(s-1)q), this is q ∑_k=r/s^r1/k· k^r-ke^-ak/∑_k=r/s^r k^r-ke^-ak.Therefore <Ref> is the same as q1/x when x is a nonnegative integer drawn from the discrete distribution with density proportional to f(x)=x^r-xe^-ax. We let z be as in <Ref> of <Ref>, so z ≈r/sq. We use <Ref> to bound | 1/x-1/z| ≤∑_t=1^∞(1/z-1/z+ t √(z))e^-t +∑_t=1^√(z)-1(1/z- t √(z)- 1/z)e^-t=∑_t=1^∞t√(z)e^-t/z(z+t√(z))+∑_t=1^√(z)-1t√(z)e^-t/z(z-t√(z))≤1/z∑_t=1^∞te^-t/√(z)+1+√(z)/z∑_t=1^√(z)/33te^-t/2z +∑_t=√(z)/3^√(z)-1 te^-t=O(1)/z√(z) + O(1)/ z√(z) + O√(z)/3 e^-√(z)/3 = O(1)/z √(z).Using this and approximating z by r/sq, the expectation of x with respect to the density proportional to f can be estimated:q 1/x=q/z(1+O(1/√(z)))=sq^2/r(1+O(√(sq/r)))=(1+o_r(1))sq^2/ras claimed. § SECTION <REF> PROOFS (of <Ref>.) Let v be a vertex with target degree t= t(v), and let k denote the number communities containing v.First we claim deg(v)∼ Bin ((s-1)k, t q/s).Let s be an arbitrary vertex of a community S containing v. s ∼ vinS = ∑_u ∈ Vs=uv ∼ uinS = ∑_u ∈ V1/n t(u) t q/ds= tq/s. A vertex in k communities has the potential to be adjacent to (s-1)k other vertices, and each adjacency occurs with probability t q /s. Next, let N_u be the event that a randomly selected neighbor of vertex v is vertex u. We compute N_u = ∑_r u ∼ vdeg(v)=rdeg(v)=r/r=∑_r u ∼ vdeg(v)=ru ∼ v/r=u ∼ v1/deg(v) u ∼ v= sn^2n/(s-1)q t(u)t q/sd(1-e^-tqk/tk q)= t(u)1-e^-tqk/qkdn. To see <Ref>, note that by the first claim 1/deg(v) u ∼ v= 1/X+1 where X ∼ Bin((s-1)k -1, tq /s). Applying <Ref> and assuming s=ω(1), we obtain 1/deg(v) u ∼ v=1-(1-tq /s)^(s-1)k /((s-1)k )tq /s= 1-e^-tqk/tk q. Now we compute the expected clustering coefficient conditioned on the number of communities the vertex is part of under the assumption that s/n=o(q). ObserveC(v) v in k communities =∑_u,w N_u N_w u ∼ wu ∼ vandw ∼ v =∑_u,wt(u)t(w)1-e^-tqk^2/(qkdn)^21/k + sn^2n/(s-1)qt(u)t(w)q/sd= 1-e^-tqk^2 ∑_u ∈ V t(u)^2^2/qd^3k^3n^2s. Next compute the expected clustering coefficient without conditioning on the number of communities. To do so we need to compute the expected value of the function f(k)=(1- e^-kqt)^2/k^3. We first use Taylor's theorem to give bounds on f(k). For all k, there exists some z∈ [1/q, k] such thatf(k)= f1q + f'1qk-1/q +f”(z)/2k-1/q^2.Note that for z ∈ [1/q, k] f”(z) = 12 (1 - e^-k q t)^2/k^5 - 12 e^-k q t(1 - e^-k q t) q t/k^4 + 2 e^-2 k q t q^2 t^2/k^3 - 2 e^-k q t (1 - e^-k q t) q^2 t^2/k^3≤12 (1 - e^-k q t)^2/k^5 + 2 e^-2 k q t q^2 t^2/k^3≤ q^5 12+ 2t^2e^-2t,andf”(z) ≥ 0.It follows that f1q + f'1qk-1/q≤ f(k)≤f1q + f'1qk-1/q + q^5 6+ t^2e^-2tk-1/q^2. Let M ∼ Bin(n/(sq), s/n) be the random variable for the number of communities a vertex v is part of. (Since s= ω(1) replacing the number of communities by n/(sq) changes the result by a factor of .) We use <Ref> to give bounds on the expectation of f(M),f(M) ≤ f1q + f'1qM-1/q + q^5 12+ 2t^2e^-2tM-1/q^2=(1- e^-t)^2q^3 + 1/q1-s/n q^56+ t^2e^-2t≤ (1- e^-t)^2q^3 + q^46+ t^2e^-2t andf(M)≥ f1q + f'1qM-1/q =(1- e^-t)^2q^3. Therefore f(M)= (1- e^-t)^2q^3 + c_t q^4 for some constant c_t ∈ [0, 6.2). Finally, we compute C(v) = ∑_k M=k1-e^-tqk^2 ∑_u ∈ V t(u)^2^2/qd^3k^3n^2s= ∑_u ∈ V t(u)^2^2/qd^3n^2sf(M)= ∑_u ∈ V t(u)^2^2/d^3n^2s (1- e^-t)^2q^2 +c_tq^3. (of <Ref>.) Let d= mean(D). We computed = ∑_k=1^∞k^-γ+1/ζ(γ) = ζ(γ-1)/ζ(γ). Next we claim that with high probability the maximum target degree of a vertex is at most t_0=n^2/(γ-1). Let X be the random variable for the number of indices i with t(v_i)>k_0. max_i t(v_i) > t_0≤X = nt(v_1) > t_0≤ n ∑_i=t_0+1^∞i^-γ/ζ(γ)≤ n∫_i=t_0^∞i^-γ/ζ(γ)= 1ζ(γ)(γ-1) n t_0^1-γ=o(1). It follows that max_i t(v_i)^2 ≤ n^1/γ-1, and so max_i t(v_i)^2 ≤sd/q. | http://arxiv.org/abs/1709.09477v1 | {
"authors": [
"Samantha Petti",
"Santosh Vempala"
],
"categories": [
"cs.DM",
"cs.SI",
"math.CO"
],
"primary_category": "cs.DM",
"published": "20170927125006",
"title": "Random Overlapping Communities: Approximating Motif Densities of Large Graphs"
} |
1,2]E. [email protected] 1]J. Greaves 3]H. J. Fraser 2]D. L. Clements 2]L.-N. Alconcel [1]School of Physics and Astronomy, Cardiff University, Cardiff CF24 3AA, UK. [2]Imperial College London, Blackett Lab., Prince Consort Rd, London SW7 2AZ, UK. [3]School of Physical Sciences, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK.Ground-based detection of a cloud of methanol from Enceladus: When is a biomarker not a biomarker? [================================================================================================== Saturn's moon Enceladus has vents emerging from a sub-surface ocean, offering unique probes into the liquid environment.These vents drain into the larger neutral torus in orbit around Saturn.We present a methanol (CH_3OH) detection observed with IRAM 30-m from 2008 along the line-of-sight through Saturn's E-ring.Additionally, we also present supporting observations from the Herschel public archive of water (ortho-H_2O; 1669.9 GHz) from 2012 at a similar elongation and line-of-sight.The CH_3OH 5(1,1)-4(1,1) transition was detected at 5.9σ confidence.The line has 0.43 km/s width and is offset by +8.1 km s^-1 in the moon's reference frame. Radiative transfer models allow for gas cloud dimensions from 1750 km up to the telescope beam diameter ∼73000 km.Taking into account the CH_3OH lifetime against solar photodissociation and the redshifted line velocity, there are two possible explanations for the CH_3OH emission:methanol is primarily a secondary product of chemical interactions within the neutral torus that (1) spreads outward throughout the E-ring or (2) originates from a compact, confined gas cloud lagging Enceladus by several km s^-1. We find either scenario to be consistent with significant redshifted H_2O emission (4σ) measured from the Herschel public archive.The measured CH_3OH:H_2O abundance (>0.5%) significantly exceeds the observed abundance in the direct vicinity of the vents (∼0.01%), suggesting CH_3OH is likely chemically processed within the gas cloud with methane (CH_4) as its parent species.§ INTRODUCTION The discovery of liquid water below the icy surfaces of several moons orbiting Jupiter and Saturn is an exciting prospect for complex chemical and even biological activity taking place within them.However, accessing these subsurface oceans is problematic. The subsequent discovery <cit.> of water vapour plumes venting from Saturn's moon Enceladus (R_Enc∼ 250 km) through geyser-like structures near the moon's south pole <cit.> , can help probe the interior processes of this particular object.Most work to date on the molecular content of material ejected from Enceladus has been conducted through in situ observations by the Cassini spacecraft and by its Ion and Neutral Mass Spectrometer (INMS; ).Molecules like water, carbon dioxide, methane, methanol, ammonia and formaldehyde have been detected in the plumes; a further component of molecular mass 28 (CO or N_2) has also been seen. These observations are conducted in flybys at different altitudes above the surface and on different trajectories, so the properties of the entirety of the plume and any chemical processing within the plume (e.g. by Solar ultraviolet light) must be inferred.On larger scales, a neutral OH torus orbiting Saturn was found by <cit.>, which is fed by these active H_2O plumes<cit.>.The neutral torus is assumed to be centred on Enceladus' orbit (3.95 Saturn radii or R_S; R_S=60268 km), extending from 2.7 to 5.2 R_S <cit.>.H_2O in the neutral torus has been further explored by the Herschel Space Observatory <cit.>, while low signal-to-noise has limited Cassini from performing in situ measurements.The Cassini mission is nearing its end and conducted its final flyby of Enceladus at the end of 2015. With little or no prospect of a new mission to Saturn before 2030, further studies and monitoring of the Enceladus plume must be done remotely from Earth.Fortunately, submillimetre spectroscopy is well suited to such studies since many of the organic molecular species of interest have transitions at these wavelengths.A disadvantage is that single-dish telescopes, while suited to temporal monitoring, trace larger regions than the size of the plumes.In light of further developments in the plume composition from Cassini and neutral gas environment surrounding Enceladus by Herschel, we present the first results from a programme of submillimetre spectroscopic observations of a gas cloud near Enceladus using ground-based observatories from early 2008.This paper is organised as follows: Section <ref> details the ground-based observations targeting methanol (CH_3OH) and Section <ref> presents the methanol spectrum.Section <ref> details the radiative transfer and dynamical models used to constrain the methanol abundance and the likely region from which the methanol originates.Lastly, we summarise our results and discuss their implications for using methanol as a biomarker in solar system objects and exoplanet environments in Section <ref>. § OBSERVATIONSThe observations were made at the Instituto de Radioastronomie Milimetrica (IRAM) 30 m telescopeat Pico Veleta, Spain at 01-08 hours UT on 10 Jan 2008 using the HERAcamera and the VESPA spectrometer. The CH_3OH 5(1,1)-4(1,1) line at 239.7463 GHzwas observedwith 80 MHz passbands with a telescope beam size of ∼10.5” FWHM.The observations were made by frequency-switching over a narrow 3.45 MHz interval to maintain flat baselines using 80 kHz channels in VESPA. Figure <ref> shows the location of Enceladus w.r.t. Saturn at the time of observations. The angular size of Enceladus was 0.08” with Saturn 8.6 AU from theEarth. The ring opening angle was -6.9^∘.We note the only possible contaminant within the beam was the moon Mimas towards the end of the observing period.The telescope could track Enceladus but the acquisition software was only able to track the velocity of Saturn.Saturn was offset by ≥ 30” during the observations to prevent spectral contamination of the Enceladus data. Velocity shifts to place the data in the Enceladus rest-frame were then made in reduction software for each observation.In total 48 observations were made, with the drift between observations being <0.1 km/s on average (worst case of 0.4 km/s). The applied shift varied from -9.2 km/s at the start to -12.6 km/s at maximum elongation followed by reversal back to -10.6 km/s at the end of the track. Hence, any artefacts associated with particular spectrometer channels would be smeared over a few km s^-1 in the co-added data, and so not be able to create a false-positive narrow line.Details of the data reduction (i.e. despiking, baseline-fitting and baseline-subtraction) are given in the Appendix (see Figure <ref>).The spectra are automatically calibrated to a T_A^∗ antenna temperature scale.However, we must correct the antenna temperature scale for telescope inefficiency.The efficiencies were not measured during our run, but the HERA User Manual (V2.0; Nov 2009) stated the beam efficiency is 0.52 at 230 GHz and the forward efficiency is 0.90. We calculate a main-beam temperature T_MB = F_eff/ B_eff T_A^∗, i.e. T_A^∗ is divided by 0.58 to give T_MB.In addition to the CH_3OH spectrum, we also include publicly available observations taken by Herschel (originally from project ID OT2_elellouc_3, PI Lellouch) of the ortho-H_2O 2_12–1_01 line at 1669.9 GHz, taken on 27 June 2012 at ∼22:00 UT over a period of ∼0.33 h using the HIFI instrument.The observations were taken in the fast dual beam switch (DBS) raster mapping mode, where we extracted the spectrum using only the observation centred directly on Enceladus.The telescope beam FWHM is ∼13” at ∼1660 GHz.During this observation, Enceladus was at similar elongation from Saturn as the observations from IRAM, where the rings were edge-on.§ RESULTSThe co-added CH_3OH spectrum is shown in Figure 2a with 0.1 km s^-1 velocity channels and in a frame co-moving with Enceladus.The peak antenna temperature T_A^∗ is 0.038±0.011 K (3.6σ) and integrated intensity ∫T_A^∗ dv is 0.018±0.003 K km/s (5.9σ).This is the only significant feature within the passband, where all other signals summed across 0.8 km/s intervals (totalling 125 intervals across the 80 MHz passband) are ≤3σ (i.e. ≤0.009 K km s^-1).See the Appendix for more detail regarding the robustness of the CH_3OH detection (i.e. Figure <ref>). Accounting for the beam and forwards efficiency, the peak main-beam temperature T_MB is 0.066 K and integrated intensity ∫T_MB dv is 0.031 K km/s.The line FWHM is 0.43 km/s as measured by fitting a Gaussian to the methanol detection (Figure 2), and the line is centred at +8.1 km/srelative to the moon. This indicates that we are not looking at material in the direct vicinity of the plume origin, but material that has been redshifted by some means.The ortho-H_2O 2_12–1_01 line is shown in Figure 2b with 0.3 km s^-1 velocity channels in a co-moving Enceladus frame.The full line profile must be fitted with a two-component Gaussian due to a significant redshifted H_2O line-wing.This redshifted emission spans ∼14 velocity channels (i.e. up to ∼ 7 km s^-1) with a total signal ∼0.421 ± 0.105 K km s^-1 (i.e. 4σ).Even though the venting process is variable <cit.>, the Herschel H_2O observation is redshifted by a similar amount as the CH_3OH detection four years earlier. § MODELSEven though methanol has been found within the plume components <cit.>, the region in which IRAM can detect methanol must be on a larger scale.The diameter of the telescope beam subtends ∼73,000 km (∼1.2R_S), which is much larger than Enceladus' diameter of only ∼500 km.Even if the methanol cloud is optically thick and warm (e.g. plume temperatures are ∼180 K; ), the minimum size we are sensitive to is ∼1750 km, or 7 R_Enc.On the other hand, diffuse gas in the larger neutral torus could fill the beam (where the torus has a radial extent of R_Enc±1.25R_S and a vertical extent of ±0.4R_S;).For generality, we consider a cloud of methanol with length l, and use the non-local thermodynamic equilibrium (non-LTE) radiative transfer code RADEX to find solutions for the radiation temperature (T_R).Details of RADEX and the full parameter space of the models that fit the CH_3OH observation can be found in Section <ref>.For clouds with size scales smaller than the telescope beam (i.e. l<73,000 km), we must account for beam dilution which causes the main-beam temperature to be lower than the expected radiation temperature by T_MB= η_dil T_R, where η_dil= l^2/l^2 + l^2_beam and l_beam is the beam diameter.For the radiative transfer models, we set the gas kinetic temperature T_gas = 100 K (e.g. ) and the line FWHM at 0.43 km s^-1 as observed (Section <ref>).While plume temperatures reach ∼180 K <cit.>, we expect plumes to cool to Saturn's thermal field (100 K) at the scales traced by our observations (seefor modelling of the gas temperature within the neutral torus).From the gas kinetic temperature, we can calculate the non-thermal (turbulent) velocity dispersion of methanol by σ_NT^2 = σ_CH_3OH^2 - σ_T ^2, where σ_NT is the non-thermal line width, σ_CH_3OH is the observed line width of CH_3OH (calculated from σ_CH_3OH= v_FWHM / √(8ln(2))) and σ_T is the thermal component of the line width (assuming a 100 K cloud).Removing the thermal component leaves a turbulent velocity dispersion of ∼0.1 km s^-1, suggesting the methanol gas is dispersing at ≤0.1 km s^-1.Solutions for T_MB were sought for different CH_3OH:H_2O abundance ratios (X), where we relate the H_2O density, n(H_2O), to the CH_3OH column density, N(CH_3OH) by n(H_2O) = N(CH_3OH)/l X.The CH_3OH:H_2O abundance is constrained as X≤ 5%, which is the expected C:H_2O budget (estimated from the CO_2:H_2O abundance within Enceladus' plumes; ).If methane CH_4 is the parent species of CH_3OH, then the preferred CH_3OH:H_2O abundance is X≤1%.In RADEX, the collision partner is H_2, so we set density to be 10× higher to account for the higher mass of the actual collision partner H_2O.Additionally, we find solutions for the cloud length l that reproduce T_MB(CH_3OH) within the allowed CH_3OH:H_2O abundance.Furthermore, a maximum H_2O column density was applied.The maximum assumes that all of the H_2O molecules vented (estimated to be 10^28 molecules s^-1; ) will survive during the photodissociation timescale ∼2.5 months (e.g. ) within the specific region of size l.Figure <ref> depicts the models that match the expected CH_3OH radiation temperature (i.e. the main-beam temperature corrected for beam dilution) within a conservative 50% uncertainty for each cloud length l. We investigate the CH_3OH:H_2O abundance by comparing the methanol and water column densities, N(CH_3OH) and N(H_2O) for each cloud length.We find allowed CH_3OH cloud dimensions l to be 1750–72500 km.At smaller cloud lengths (l<1750 km), methanol becomes increasingly beam diluted and high densities and column densities are needed to produce the methanol detection.However, CH_3OH also becomes optically thick and higher gas temperatures are needed to adequately reproduce the beam-diluted detection.We find the modelled CH_3OH-to-H_2O abundance is relatively independent of cloud length, ranging from 0.5–5%.This abundance is ∼50× higher than what is found in the direct vicinity of the plumes <cit.>, implying chemical processing has occurred.The cloud size scenarios are summarised in the Appendix (Table <ref>), including the range of H_2O densities from the RADEX models that fit the CH_3OH observation. For smaller cloud lengths (e.g. l=1750 km), we have H_2O densities ranging from n_RADEX(H_2O)=1.5×10^9–1.1×10^10 cm^-3.This density range is reasonably in agreement with past Cassini measurements of the E3 and E5 flybys, which found the peak H_2O density to reach ∼10^8–10^9 cm^-3 near the plumes (Teolis et al. 2010; Smith et al. 2010).At larger cloud lengths (e.g. l=72500 km), we find the H_2O density range to be n_RADEX(H_2O) = 4.5×10^4–1.6×10^5 cm^-3, slightly above <cit.>, which found neutral densities in the torus ∼10^4 cm^-3.We can estimate how far the CH_3OH is able to travel outward from Enceladus from the photodissociation timescales under solar irradiation.If the gas is dispersing at ≤0.1 km s^-1 and the photodissociation timescale is t_photo = 1.7×10^6 s (accounting for Saturn's distance at 9.3 AU; ), then we expect a methanol cloud extent up to ≤173,000 km (i.e. ∼ 3 R_S).This means the methanol is easily able to spread into the neutral torus and fill the beam of the telescope.We note that H_2O and CH_4 have even longer photodissociation timescales than CH_3OH and are not a limiting factor.Therefore, there are two possible explanations for the narrow CH_3OH line observation:methanol is made both by Enceladus and through other chemical pathways within the neutral torus that (1) spreads outward or (2) remains in a more compact, confined gas cloud that trails Enceladus' orbit by several km s^-1.The velocities at which methanol is seen are consistent with a torus model, in particular with molecules spreading outwards from Enceladus' orbit <cit.>.At the time of observations, Saturn's rings were tilted such that a beam pointed at Enceladus' position passed preferentially through the far-side of the neutral torus and even broader E-ring (3-9 R_S).Molecules further out than Enceladus will orbit with slower speeds.If we are observing the far-side of the torus or E-ring, molecular line observations will be redshifted w.r.t. Enceladus as it approaches.This is seen in the observations in Figure <ref>.The magnitude of the line-of-sight velocity shift w.r.t. Enceladus can be fitted assuming the molecules are in Keplerian rotation in the ring plane, which was tilted w.r.t. Earth at 7^∘ in 2008.The detected methanol must be at an orbit of ∼8 R_S to be in the telescope beam, assuming most of the spectral emission was contributed around the time of maximum elongation.However, methanol ejected directly from Enceladus through plumes may only be able to spread by ∼ 3R_S accounting for a methanol photodissociation time at the distance of Saturn (i.e. out to a radius ∼7 R_S accounting for the distance between Saturn and Enceladus; ).Therefore, the detected methanol would have to be a secondary product of chemical processing within the larger neutral torus in this scenario, as discussed below.The second possibility is that a compact, confined gas cloud trails Enceladus' orbit by several km s^-1.An example of a trailing gas cloud can be seen in a model by <cit.>, which was reconstructed from Cassini flyby measurements of the plasma.In Figure <ref>, we show a diagram of Enceladus with the plasma velocities and density contours (based on Figure 2 in ).The plume emerges from the southern pole of Enceladus (to -Z) and streams along the X-axis, which is in the line-of-sight for our observations.Electrons south of Enceladus' plumes form a denser cloud at length l≲1750-2000 km (7–8 R_Enc) and trail Enceladus by ≲6 km s^-1.We have observed CH_3OH redshifted by a similar +8.1 km s^-1 and over a similar-sized region.However, further modelling and observations are needed to explain the mechanism in which gas would be confined by Saturn's magnetosphere.Additionally, there is independent evidence of a significant (4σ; see Section <ref>) redshifted molecular component from archival ortho-H_2O spectrum taken when Enceladus was at a similar elongation from Saturn.Since the active plumes are thought to feed the neutral torus in orbit around Saturn, it is expected that H_2O will fill the Herschel telescope beam and be present at both small and large scales surrounding Enceladus.In particular, this redshifted H_2O component can result from gas spreading <cit.> or the presence of a confined cloud of gas lagging the orbital velocity of both the moon and the co-rotating neutral torus, where the latter process depends on the exact velocity of outgassing material at the time.If either gas spreading or a confined cloud are the causes of the redshifted molecular emission, we do not necessarily expect the H_2O and CH_3OH observations to be redshifted by the same amount because H_2O may be tracing different radii than the CH_3OH detection and/or temporal variations observed from the plumes <cit.>.Careful monitoring of Enceladus needs to be done in the future to better understand how the plume processes change over time.Considering that water, hydrogen, oxygen and hydroxyl are abundant in the torus, the rate of methanol production may depend on the availability of methane, which is a ∼1.6% CH_4:H_2O abundance <cit.>, or other hydrocarbons.Using public astronomical networks (KIDA[http://kida.obs.u-bordeaux1.fr], UMIST; ), we suggest possible routes to methanol through gas-phase associate detachment:CH_3 + OH^- → CH_3OH + e^- While other routes for the creation of methanol are possible, these involve more complex molecules that are less abundant and may not be present in the plume environments.<cit.> find that dissociative electron attachment can produce negative water-group ions (e.g. OH^-, O^-, H^-) in Enceladus' plumes from neutral H_2O, where O^- and H^- are produced by polar photodissociation of H_2O by photons in ∼36-100 eV.While H^- has a relatively short photodissociation timescale (∼6 seconds at Saturn; see ), <cit.> find that H^- quickly reactions with H_2O to produce OH^- + H_2 in the comet 1P/Halley environment, where both OH^- and O^- are abundant within the coma (≤1000 km from the nucleus).This cometary environment may be similar to Saturn, where Saturn is a soft X-ray source (scattering X-rays from the Sun; ) that could form these water-group ions.While the methyl radical is not directly modelled from the Cassini INMS data <cit.>, it is a key molecule in the production of more complex hydrocarbons (e.g. Titan; ) and is likely present within Enceladus' plumes or created once material is ejected <cit.>.One possible pathway is through associative attachment from ethylene (CH_2) and water-group ions, where traces of ethylene have noted in the plume environment <cit.>:CH_2 + H^- → CH_3 + e^- Similarly, reactions directly with methane and other molecules abundant in the plume environment can produce the methyl radical.For example, ion-neutral reactions:CH_4 + O^- → CH_3 + OH^- CH_4 + H_2O^+ → CH_3 + H_3O^+,where H_2O^+ and H_3O^+ have been found directly (; altitudes ∼200 km).Neutral-neutral reactions with methane are also possible with neutral water-group atoms and molecules present in the E-ring environment:CH_4 + OH → H_2O + CH_3 CH_4 + O → OH + CH_3 In all of these proposed scenarios, the plasma density appears to be an integral part of the gas-phase chemical production of CH_3OH and enhancement in CH_3OH abundance once the plume material is ejected from Enceladus.This further supports the offsets in velocity we see in the CH_3OH spectrum and the plasma within 2000 km from the south pole of Enceladus <cit.>, where this material is travelling more slowly than the moon.Plasma densities are found to peak at ∼10^2 cm^-3, indicating an electron abundance ≥10^-7 w.r.t. modelled H_2O (see Table <ref>), which is similar to electron abundances found in the ISM (w.r.t. H_2; ).As stated above, further observations are needed to constrain possible variability in the plume and surrounding environment which may lead to changes in density, chemical composition and velocity of the ejected material over time.We note that <cit.> finds a potential CH_3OH ice surface feature, though this can also be interpreted as hydrogen peroxide H_2O_2 <cit.>.However, past work has also shown that CH_3OH tends to be destroyed in ice pathways, making it less likely that CH_3OH is produced from ice on Enceladus' surface <cit.>.§ DISCUSSION AND CONCLUSIONS We report on the first ground-based detection of a molecule (CH_3OH) in Enceladus' plumes.Radiative transfer models suggest the origin of methanol can be from a cloud with length ranging from 1750 km up to a size comparable to the neutral torus in orbit around Saturn.There are two possible explanations for producing the redshifted CH_3OH and H_2O, taking into account the solar photodissociation timescale of CH_3OH and the observed redshifted CH_3OH emission.First, methanol may be a secondary product of chemical processing within the neutral torus and spread further out into the E-ring to a radius ∼ 8R_S (e.g. ) at velocities smaller than Enceladus' rest-frame velocity (i.e. due to Keplerian rotation).Alternatively, methanol may original from a smaller, beam-diluted region nearer to Enceladus (i.e. l<2000 km).To further distinguish between these two mechanisms, further modelling and observations are needed. What is the origin of the methanol cloud surrounding Enceladus?Since methanol is expected to be present in an organic environment <cit.>, an exciting prospect is that the observed methanol is being produced by living organisms within Enceladus' subsurface ocean.In Earth's oceans, <cit.> finds that methanol is produced at a rate as high as 0.3% w.r.t. the total cellular carbon.If we assume that methanol production within subsurface oceans on Enceladus is similar to Earth and all of the carbon measured in Enceladus' vents is expelled by microbes in the subsurface ocean (i.e. C:H_2O∼5%; ), then we would expect a CH_3OH:H_2O abundance within Enceladus' subsurface oceans to be ∼0.015% in an organic environment.This result is similar to the CH_3OH abundance measured in the direct vicinity of the vents by Cassini at ∼0.01% <cit.>, indicating that it is possible for the CH_3OH found in the vents to be a biomarker in an extreme case with the carbon predominantly of biological origin.In contrast at larger scales, our CH_3OH observation suggests the gas cloud trailing Enceladus has a CH_3OH:H_2O abundance that is ∼50× higher than in the vents.Therefore, it is likely methanol is being produced once the material is ejected from the subsurface ocean, making it improbable as abiomarker signature in this particular case.In the future, caution should be taken when reporting on the presence of supposed biomarkers, in both solar system and exoplanet environments.The most robust method for investigating the complex chemistry, particularly in subsurface ocean environments, is obtaining observations close to the vents.§ ACKNOWLEDGEMENTSED acknowledges funding from Cardiff University.This work uses observations from project number 220-07 with the IRAM 30m Telescope. 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H. & Magee, B. 2010, EPSC, 305§ METHODS AND DETECTION OVERVIEW We present an overview of the methods for reducing the CH_3OH spectrum and determining the robustness of the CH_3OH detection.Noticeable spikes towards the ends of the band were blanked before the baseline was fit and subtracted from the spectrum (Figure <ref>).Both the positive and negative response of the CH_3OH line can be seen at a velocity separation of 8.6 km s^-1 due to frequency-switching (see Section <ref> for more details).The robustness of the detection was investigated by averaging the CH_3OH positive and negative response and investigating how likely these features are caused by noise (Figure <ref>).< g r a p h i c s > Top Left:Full spectrum from the IRAM 30-m telescope centred on 239.7463 GHz prior to fitting and subtracting the baseline.The spectrum was produced by shifting individual observations in velocity to Enceladus' rest frame and then averaged. Noticeable spikes (in both positive and negative due to frequency-switching) can be seen at velocity channels towards the ends of the band.Top Right:Baseline-fitting was done with a 14th order polynomial (the highest available in SPLAT) over the regions within the green boxes; the stepped parts show where channels are blanked in the spike removal.Bottom Left:CH_3OH spectrum after despiking and baseline-subtracting.Bottom Right:We investigate the significance of both the positive and negative CH_3OH response of the full spectrum.On the y-axis, we define the average of the CH_3OH positive and negative response [∑T_A^∗(line) - ∑T_A^∗(inverse)]/2 (i.e. [the sum of antenna temperature T_A^∗ over 7 contiguous velocity channels minus the same sum over velocity channels that are 8.63 km s^-1 lower] / 2).The red point denotes the peak of the CH_3OH line at +8 km s^-1.Black points denote the same calculation for other velocity bins, excluding channels where the calculation would involve part of the bandwidth with spikes, or (for clarity) the line or its negative response.Hence, these black points show the statistics for a potentially false line due to noise.The dotted blue lines show the mean ±3σ bounds derived from all of the plotted points.We find the red point denoting the average CH_3OH line and negative response to be the only point that is significant (at the 5σ level).§ FULL RADEX PARAMETER SPACE RADEX is a one-dimensional non-LTE radiative transfer code that assumes isothermal and homogenous medium without large-scale velocity fields using the escape probability method (i.e. the probability a photon will break out of the surrounding medium).The program is iterative, finding a solution for the level populations and calculating the radiation temperature T_R using the following method: * Input parameters are: molecular data file from LAMDA (, which include term energies, statistical weights, Einstein coefficients and rate coefficients for collisional de-excitation), frequency range of the transition, kinetic temperature of the region, number of collision partners (typically H2 as the only collision partner, which has been scaled so that H_2O is the collisional partner for our models), H_2O density, temperature of the background radiation field, column density of the molecule being modelled and FWHM line width.* An initial estimate of the level populations is made by assuming optically thin emission and statistical equilibrium considering the background radiation field (typically 2.73 K blackbody representing the cosmic microwave background or CMB).* The optical depths are then calculated for the molecular line.* The program iteratively continues to calculate new level populations with new optical depth values until both converge on a consistent solution.* The program outputs are: background-subtracted molecular line intensities, excitation temperature and optical depth. The full parameter space of the RADEX models that fit the CH_3OH observation can be found in Table <ref>. | http://arxiv.org/abs/1709.09638v1 | {
"authors": [
"E. Drabek-Maunder",
"J. Greaves",
"H. J. Fraser",
"D. L. Clements",
"L. -N. Alconcel"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170927172112",
"title": "Ground-based detection of a cloud of methanol from Enceladus: When is a biomarker not a biomarker?"
} |
Introduction.In a recent paper <cit.> the NNPDF Collaboration presented NNPDF3.1, a new set of parton distribution functions (PDFs) <cit.>which features several improvements with respect to the previous global analysis, NNPDF3.0 <cit.>, both in terms of methodology and data.On the methodological side, the charm PDF was parametrised on the same footing as the other light quark and gluon PDFs.On the data side, a number of new measurements, especially from Large Hadron Collider (LHC) experiments, were included.Among these, some are directly sensitive to the gluon PDF at medium-to-large values of Bjorken x, which allowed us to determine it with a muchimproved precision.This is a remarkable feature of NNPDF3.1, as a detailedknowledge of the gluon PDF at large x is increasingly crucial in order togenerate precise predictions of both the signal and the backgrounds in searchesfor new massive particles at the LHC.In this contribution we present two new analyses that were not included in NNPDF3.1. First, we study the stability of the NNPDF3.1 fit upon the inclusion of eitherof the five bins in which the inclusive 7 TeV ATLAS 2011 jetdata <cit.> are provided.We demonstrate that the particular choice made in the default NNDPF3.1 fit, i.e. the central rapidity bin, does not affect the ensuing gluon PDF.Second, we provide a quantitative comparison among the constraints provided by the three different datasets included in NNDPF3.1 that aresensitive to the gluon PDF at large x: inclusive jet cross section,Z-boson transverse momentum distribution, and top rapidity distribution data.We make explicit the impact of each of these observables by adding thecorresponding datasets, one at a time, to a baseline dataset that does notinclude any of them.This is different from what was presented in the NNPDF3.1 paper, in which onedataset at a time was removed from the global fit.Stability of NNPDF3.1 upon the choice of the jet bin.The NNPDF3.1 analysis included for the first time the single-inclusivejet cross sections measured in the 2011 run at 7 TeV with R=0.6 by ATLAS and at 2.76 TeV with R=0.7 by CMS. These were added on top of four measurements already included in NNPDF3.0,namely: CDF Run II kT, CMS 2011, 2010 ATLAS 7 TeV and ATLAS 2.76 TeV,including correlations to the 7 TeV data (see <cit.> for the experimental references). Although next-to-next-to-leading order (NNLO) corrections to the inclusive jetproduction cross section are now known <cit.> (in theleading-colour approximation), the exact results are not yet available for all jet datasets included in NNPDF3.1.Therefore, they were included in the NNPDF3.1 NNLO PDF fit using NNLO PDFevolution but next-to-leading order (NLO) matrix elements. A fully correlated theoretical systematic uncertainty, accounting for the missing higher order corrections in the matrix element, was added to the covariance matrix.We also note that the sign and the size of the NNLO corrections stronglydepend on the central scale used in the predictions. If the jet transverse momentum p_T is taken as the central scale, theNNLO/NLO K-factors vary between -5% and +10% in the range measured at theLHC, 100 GeV≲ p_T≲ 2 TeV <cit.>.While no cuts were applied to all jet datasets included in NNPDF3.1, for the 2011 ATLAS 7 TeV dataset a good agreement between data and theory was obtained when fitting only the central rapidity bin, |y_ jet| < 0.5.Concurrently, it was found that achieving a good description of the ATLAS 2011 7 TeV dataset would be impossible, if all five rapidity bins wereincluded simultaneously and if all cross-correlations amongrapidity bins were taken into account accordingly.It is therefore important to demonstrate that the gluon PDF is stable upon thechoice of any of the other rapidity bins.In order to investigate on this, we have performed five additional fits,with the same theoretical settings of the default NNPDF3.1 NNLO PDF fit,in which we have included in turn the second, third, fourth, fifth and sixth jet rapidity bin (0.5<|y_ jet| < 1.0,1.0<|y_ jet| < 1.5, 1.5<|y_ jet| < 2.0,2.0<|y_ jet| < 2.5 and 2.5<|y_ jet| < 3.0 respectively)instead of the central bin.In Table <ref>, we report the value of the χ^2 per data point, χ^2/N_ dat, for the individual 2011 ATLAS 7 TeV data set, before andafter each of the five variants of the default fit, and for the total data setafter the fits.In Fig. <ref>, we show the distance <cit.>between the central value and the uncertainty of the gluon PDF at Q=100 GeVin the default NNLO NNPDF3.1 fit and in each of the five variants of the fitincluding higher rapidity bins. All sets are made of N_ rep=100 replicas.Distances of d≃ 1 correspond to statistically equivalent fits,while for sets of 100 replicas d≃ 10 corresponds to a differenceof one sigma in unity of the corresponding variance. In Fig. <ref> we compare the NNLO gluon PDF obtained from theNNPDF3.1 default fit and the variants in which the second and third rapiditybins of the 2011 ATLAS 7 TeV jet data are fitted instead of the central bin.Very similar plots are found for fits with higher rapidity bins, therefore they are not shown. As is apparent from Table <ref> andFigs. <ref>-<ref>, the description of each separate bin is equally good, the central valuesof the gluon PDF are well within its uncertainty for each fit and PDFs are statistically equivalent. Therefore, we conclude that the gluon PDF in the NNPDF3.1 set isindependent of the choice of the ATLAS 2011 jet data bin used in the fit. Impact of various datasets on the gluon PDF at large x.On top of the inclusive jet data, the two leadingobservables sensitive to the medium-to-large x gluon PDF, that were included in NNDPF3.1, are the total rate and differentialdistributions of top-pair production and the transverse momentum distributionsof the Z boson. All of them have been measured by the ATLAS and CMS experiments at the LHCwith high precision recently.The corresponding NNLO QCD corrections have been computedfor top-pair production cross sections at the level of total rates in <cit.> and of differential cross sectionsin <cit.>; for the transverse momentum of the Z bosonin <cit.>. The impact on the gluon PDF of the newly available data sets for each ofthese observables has been studied in detail <cit.>.They have been shown to provide complementary and compatible constraints ontothe gluon PDF at medium and large x, and that this constraint is competitivewith that provided by inclusive jet data.Specifically, the new data sets directly sensitive to the gluon PDF at largex that were included in NNPDF3.1 are the following: the Z boson(p_T^Z,y_Z) and (p_T^Z,M_ll) double differential distributions at 8 TeVfrom ATLAS and the Z boson (p_T^Z,y_Z) double differential distribution at 8 TeV from CMS; the top-pair production normalised y_tdifferential distribution at 8 TeV from ATLAS, the top-pair productionnormalised y_tt̅ differential distribution at 8 TeV from CMS and the total inclusive cross sections for top-pair production at 7, 8 and13 TeV from ATLAS and CMS (for all references to the experimental papers,see <cit.>).In NNPDF3.1, the three classes of measurements (inclusive jets, Zp_T and top data) have all been included at the same time in the global fit. This was compared to variants in which either of the three measurements wasremoved at a time from the global dataset.This shows only indirectly the impact of each of these datasets.A much more direct comparison is the one in which each dataset is addedindividually to a baseline dataset made up of all the data in NNPDF3.1 exceptthe piece sensitive to the gluon at large x.Therefore, we run four additional fits, with the same theoretical settings as in the NNLO NNPDF3.1 default fit. In the baseline fit, all jet, Z p_T and top data are removed from theNNPDF3.1 NNLO fit.In the other three fits, we add each of these measurements individually on top of the baseline dataset.The distance in the central value and uncertainty between each of these three fits and the baseline is shown in Fig. <ref>.The corresponding gluon PDFs are compared in Fig. <ref>.At the level of the central value, we observe fromFigs. <ref>-<ref> that the three sets ofobservables favour a slightly softer gluon PDF then the baseline above x∼ 0.2.Such an effect, up to half a σ, is more pronounced forjet and top data.At the level of ucnertainties, the transverse momentum distribution of theZ boson decreases the gluon PDF uncertainty by almost a factor of two for10^-2≲ x≲ 10^-1,while keeping the central value well within the baseline error band. Top-pair and jet production data have a bigger impact,as the relative uncertainty is reduced by almost 100% for allx≳10^-2. All data sets consistently pull the gluon central value in the same direction; most notewordly, top data provide a constraint on it competitive with inclusivejet data. In summary, the combined effect of Z p_T, top and jet data is toconsistently constrain the gluon PDF at medium-to-large values of x with unprecedented, few percent, precision. The unprecedented level of precision in the knowledge of the large-x gluon is a remarkable achievement of the LHC experimental program, which can only further improve thanks todata at higher luminosity and centre-of-mass energy in the future.We are grateful to our colleagues in the NNPDF Collaboration, especially toS. Carrazza, S. Forte, J. Rojo and L. Rottoli. E.R.N. is supported by the the STFC grant ST/M003787/1. M.U. is supported by a Royal Society Dorothy Hodgkin Research Fellowship and partially by the STFC grant ST/L000385/1.Plots in Figs. <ref> and <ref> were drawn using thecode of Ref. <cit.>. 99 Ball:2017nwa R. D. Ball et al. 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Mitov, JHEP 1605 (2016) 034.Czakon:2016dgf M. Czakon, D. Heymes and A. Mitov,JHEP 1704 (2017) 071.Boughezal:2015ded R. Boughezal et al., Phys. Rev. Lett.116 (2016) 152001.Ridder:2016nkl A. Gehrmann-De Ridder et al., JHEP 1607 (2016) 133.Gehrmann-DeRidder:2016jns A. Gehrmann-De Ridder et al., JHEP 1611 (2016) 094.Beneke:2012wb M. Beneke, P. Falgari, S. Klein, J. Piclum, C. Schwinn, M. Ubiali and F. Yan,JHEP 1207 (2012) 194.Czakon:2013tha M. Czakon, M. L. Mangano, A. Mitov and J. Rojo, JHEP 1307 (2013) 167.Czakon:2016olj M. Czakon, N. P. Hartland, A. Mitov, E. R. Nocera and J. Rojo, JHEP 1704 (2017) 044.Boughezal:2017nla R. Boughezal, A. Guffanti, F. Petriello and M. Ubiali, JHEP 1707 (2017) 130.zahari_kassabov_2019_2571601 Z. Kassabov,https://doi.org/10.5281/zenodo.2571601https://doi.org/10.5281/zenodo.2571601. | http://arxiv.org/abs/1709.09690v2 | {
"authors": [
"Emanuele R. Nocera",
"Maria Ubiali"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170927183712",
"title": "Constraining the gluon PDF at large x with LHC data"
} |
Point Spread Function Estimation in X-ray Imaging with Partially Collapsed Gibbs Sampling Submitted to the editors 25 September 2017. Kevin T. JoyceSignal Processing and Applied Mathematics, Nevada National Security Site, P.O. Box 98521, M/S NLV078, Las Vegas, NV, 89193-8521, USA ([email protected], [email protected]).,Johnathan M. BardsleyDepartment of Mathematical Sciences, University of Montana, Missoula, MT, 59812, USA ([email protected]).Aaron Luttman[2]December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================ The point spread function (PSF) of a translation invariant imaging system is its impulse response, which cannot always be measured directly. This is the case in high energy X-ray radiography, and it must be estimated from images of calibration objects indirectly related to the impulse response. When the PSF is assumed to have radial symmetry, it can be estimated from an image of an opaque straight edge. We use a non-parametric Bayesian approach, where the prior probability density for the PSF is modeled as a Gaussian Markov random field and radial symmetry is incorporated in a novel way. Markov Chain Monte Carlo posterior estimation is carried out by adapting a recently developed improvement to the Gibbs sampling algorithm, referred to as partially collapsed Gibbs sampling. Moreover, the algorithm we present is proven to satisfy invariance with respect to the target density. Finally, we demonstrate the efficacy of these methods on radiographic data obtained from a high-energy X-ray diagnostic system at the U.S. Department of Energy's Nevada National Security Site.Inverse Problems; Computational Imaging; Uncertainty Quantification; Bayesian Inference; Markov Chain Monte Carlo Methods65C05, 65C40, 68U10 § INTRODUCTIONImage enhancement and reconstruction is often framed within the modelb = Ax + εwhere A is a model operator that maps a quantity of interest, x, to measured data, b, which is subject to additive measurement noise given by ε. A ubiquitous example is image deconvolution; where x is an ideal un-blurred image; b is the blurred data which has been corrupted by additive measurement error ε; and A is a convolution operator whose point response is referred to as the point spread function (PSF).In situations where the PSF is unknown, the same model may be used to solve the dual problem: estimate the PSF with a known image derived from some kind of calibration. The estimation of the PSF has its own intrinsic importance beyond its use in deconvolution, since an accurate estimate of the PSF with meaningful quantification of uncertainty serves as a useful diagnostic of the imaging system.For instance, a drastic increase in the width of the PSF might indicate a malfunction in the system.To be more specific, the inverse problem is to estimate the PSF of A, say p, from a known calibration image x which we think of as operating on p.Expressing the convolution in (<ref>) in terms of p, b = x * p + ε, where `*' denotes the convolution operation and Axx*p. A direct estimate of p would be available if x were to represent an impulse response or point source, but in many cases, this is not feasible. This is acutely the case in high-energy X-ray radiography, where due to physical limitations, an impulse response cannot be obtained from calibration imagery. Instead, we use a vertical aperture to produce an opaque profile of an edge to estimate p from the resulting integral equation. Several established methods use exactly this type of PSF estimation, but with parametric forms of p derived from modeling the physics of the system <cit.>.These methods are often non-linear (exacerbating difficulties in the estimation and quantification of uncertainty), and parametric forms that can be solved are often not adequate to capture an accurate representation of blur that is the result of many components that act in aggregate.This work takes a Bayesian approach to estimation by modeling p as a stochastic quantity where our a priori uncertainty is modeled with a Gauss-Markov random field. We incorporate measured data using a posteriori analysis, where we've modeled the measurement error with an additive likelihood model as in (<ref>). Additionally, the prior modeling for p and the parameters defining it and the measurement error are done in a hierarchical fashion, as in <cit.>, so that a Gibbs sampling scheme is readily applicable. It has been shown that this hierarchy in certain circumstances can result in highly correlated Markov chains when discretization levels limit toward the continuum <cit.>. We present methods that alleviate this correlation and show that this provides effectively uncorrelated samples at an equivalent computational effort.Moreover, our model for the PSF provides a new method for encapsulating radial symmetry in a Gauss-Markov random field, by developing a one-dimensional precision operator that acts on the radial profile of the PSF. We also provide an analysis of the algorithm's convergence and computational efficiency on real and synthetic data that indicates significant efficiencies over standard Gibbs sampling as well as improvements to other newly developed methods. In <Ref>, we introduce a novel mathematical model for an isotropic point spread function reconstruction.This results in a linear integral equation, for which the PSF can be estimated non-parametrically by discretizing the integral operator. In this section, we describe the hierarchical model for estimating p, and using Bayes' Theorem, give an explicit formulation for the posterior density of the quantities of interest. <Ref> outlines three MCMC approaches to analyzing the posterior.We first outline the standard Gibbs sampling approach, then present a recently studied approach called marginal then conditional (MTC) sampling <cit.>, and show it's relationship to Gibbs sampling. Then, we present the partial collapsed Gibbs sampler as it applies to hierarchical models, and show how it is related to the Gibbs and MTC samplers.Finally, <Ref> compares each algorithm numerically on synthetic data and on actual measured PSF data a from high-energy X-ray imaging system at the U.S. Department of Energy's Nevada National Security Site..§ MODELING IMAGE BLURWhen image blur is translation invariant, it can be modeled as convolution with a PSF that represents the impulse response of the system <cit.>. A direct estimate is available by taking a calibration image representing the impulse response <cit.>; for example, in astronomical imaging, it is often estimated by imaging single stars which approximate point sources <cit.>. In our applications, imaging a point source is not feasible, so instead, we model the system response from an image of an aperture that retains the extent information of the action of blurring. The measurement is inherently indirect, and requires the solution of an inverse problem. If the action of the blur is isotropic, then the PSF will be radially symmetric, and it can be estimated as a one-dimensional function of distance. More specifically, let s = (s_1,s_2)∈^2 denote a position in space indexing the intensity of the response of the blurring operator (denote similarly s' for the domain of the PSF).If k( s') is the value of the PSF, then it is given by a function in one variable through the composistion k( s') = p(s'_^2 )where s'_^2 denotes the Euclidean distance in ^2. In this case, we use a beveled vertical aperture which produces a uniformly opaque vertical edge at a known fixed location in the imaging plane. See <Ref>.To model this mathematically, let E denote the half plane, and note the characteristic function, χ_E(s_1',s_2'), depends only on the horizontal coordinate of s_1', so the convolution in (<ref>) can be written asb( s) = ∬ p(s'_^2)χ_(0,∞)(s_1-s_1')d s' + ε,where b represents the noisy and blurred measured edge, and p is the radial profile of the PSF. Since (<ref>) does not depend on s_2 (the edge has vertical translation symmetry), the output of the integral operator in (<ref>) is only a function of s_1. Thus, we may represent b(s_1,s_2) = b(s_1), and for the sake of clarity, we denote b(s_1) = b(s). Further, denote r= s'_^2. In this way, the inverse problem has been reduced to estimating functions on subsets of– that is, estimate the radial profile of the PSF, p(r), from a horizontal cross-section of an image of a blurred edge, b(s). Using the change of variables s_1' = rcos v, s_2'= rsin v in (<ref>) and integrating out the v variable results in the integral operator on the radial profileb(s) = ∫_0^∞ p(r) g(s,r) r dr,whereg(s,r) = {[0 s< - r; 2(π - cos^-1(s/r))|s| ≤ r; 2πs> r. ].This situation is illustrated in <Ref>.Observe that g(s,r) is continuous, but not differentiable (it has a discontinuity in its directional derivatives across the line r=s).Again, since the operator is compact, its discretization results in a matrix with singular values that cluster near zero <cit.>, as evidenced by <Ref>. Hence, discrete estimation in the presence of measurement error will be unstable <cit.>. For such ill-posed problems, prior knowledge about the solution must be incorporated to make the problem well-posed. By representing the PSF as a radial profile p(r), the space and geometry for the domain of the model operator must reflect this representation, and prior notions of smoothness of the PSF must be expressed appropriately in this space. That is, since p depends on the distance r= s'_^2, integration-based regularization operators (the viewpoint in <cit.>), and precision operators in Gaussian based probabilistic frameworks (the viewpoint in <cit.>) will involve a change of variables. Both of these methods typically result in solving the penalized least square problemp_λ,δ=_p{λ𝒢p - b_L^2^2+δ F(p)},where F is the corresponding regularizing norm. A discrete version of (<ref>) is derived in a probabalistic framework in the next section.For our application, imposing Laplacian based smoothness on the PSF is an appropriate prior assumption. Hence, if one denotes the 2D Laplacian by Δ, then r = √(s_1^2 + s_2^2) impliesΔ(p ∘ r) = r^-1·( d/dr(r ·dp/dr) ).Note that this is the radial component of the Laplacian in two-dimensional polar coordinates. Denote the differential operator Rp d/dr(r ·dp/dr).Defining F in terms of the L^2 inner product, induces a similar change of variables; i.e.F_α(p) α⟨ p∘ r, Δ^n (p∘ r)⟩_L^2= 2πα∫_0^∞ p(r)·(R^np(r))· r^1-ndr.So, Laplacian regularization of order n smoothness on the PSF induces a regularization operator on its radial representation of the form r^1-n R^n p. A more rigorous development of these notions is carried out in <cit.>.For boundary conditions, we assume regularity of the PSF at the origin so that.d/dr|_0^+ p(r) = 0.We also assume that the PSF decays away from the origin such that for any klim_r→∞ r^kp(r) = 0, which, when discretized, we assume the imaging field of view is such that the radial profile is sufficiently small in magnitude to assume a zero right boundary condition on the domain of the solution.In the probabilistic framework, the solution to (<ref>) is equivalent to a maximum a posteriori (MAP) estimate when the PSF is assumed to be a Gaussian, and taking n=2 guarantees that the corresponding prior covariance operator is trace class <cit.>. Since data and estimates are inherently discrete quantities, we proceed by discretizing (<ref>) and (<ref>).§.§ Numerical discretization The data are intensity values of image pixels from a fixed horizontal cross-section of the edge, sampled at M=2N+1 points s_i∈ [-1,1] with s_i = i/N and -N≤ i ≤ N such that s_0. Denote the grid spacing as h1/N and the vector of data as b ∈^2N+1 with entries b_ib(s_i). Since g, the integral kernel in (<ref>), is supported on {(r,s): r≥ -s,r≥ 0}, the bounds of integration depend on s_i.Hence, a midpoint quadrature rule for b(s_i) places r_j on the midpoints, i.e., r_j = h(j - 1/2) for 1≤ j ≤ N, _ij g(s_i,r_j), and p_j = p(r_j) gives∫_s_i^∞ p(r)g(s_i,r)dr≈ h∑_j=1^N _ij p_jNote that due to the symmetry of the integration kernel g, imposing the same sampling resolution on p as b results inbeing a (2N+1) × N matrix.The differential operator R in (<ref>) is discretized using centered differencing <cit.>.Explicitly, for r_j±1/2 = r_j ± h/2, the matrix stencil for R is 1/h^2[[ -(r_j-3/2 + r_j-1/2)r_j-1/20;r_j-1/2 -(r_j-1/2 + r_j+1/2)r_j+1/2;0r_j+1/2 -(r_j+1/2 + r_j+3/2);]] [[ p_j-1; p_j; p_j+1; ]]. The left boundary condition given in (<ref>) and radial symmetry implies that the discretization of R has a reflective left boundary condition, hence[ R p]_1 = 2 r_1/2 p_1.We assume that the imaging field of view is sufficiently large so that the right boundary condition in (<ref>) is satisfied to numerical precision; i.e., [ R p]_M = r_N+1/2 p_N.Finally, the discrete precision matrix for the prior is given by Lr^-1⊙ R^2 (since n=2), the coordinate-wise multiplication of reciprocals of the grid points r_j composed with R^2. §.§ Bayesian inference with hierarchical modeling Our approach is to form a probabilistic model to estimate the unknown discrete representation of the PSF as well as the parameters involved in defining each of the distributions. That is, in addition to modeling uncertainty in p with a random field, it has become common to develop hierarchical models to let the data inform the level of regularization <cit.>.This analyisis will be conducted on the the discrete model = p+,wheremodels the discrete measurement error.Assuming that the discrete measurement error is independent Gaussian noise ∼( 0,λ^-1 I), the likelihood is a probability density satisfyingπ(| p,λ)∝λ^M/2exp(-λ/2‖ p-‖^2).When the quantity of interest is assumed to have a discrete Gauss-Markov random field prior with precision δ L, then p∼(,(δ)^-1), in which case the prior density satisfiesπ( p|δ)∝δ^N/2exp(-δ/2 p^T p),where the inverse covariance δ is the previously derived radially symmetric squared-Laplacian scaled by δ. Applying Bayes' theorem, the probability density function for p|,λ,δ is given π( p|,λ,δ)∝π(| p,λ)π( p|δ) = λ^M/2δ^N/2exp(-λ/2‖ p-‖^2-δ/2 p^T p).The maximum a posteriori (MAP) estimator is the maximizer of (<ref>), which is also the minimizer -ln π( p|,λ,δ). By expanding the inner products and centering the quadratic form in terms of p, the density in (<ref>) can be shown to be the Gaussianp|λ,δ,∼((λ^T+δ)^-1λ^T,(λ^T+δ)^-1). Following the Bayesian paradigm, the unknown parameters λ and δ are also modeled as random quantities. A common hyper-prior model for these parameters is a Gamma distribution because of the mutual conjugacy it shares with the Gaussian, making their conditional distributions with respect to the data easy to simulate <cit.>. Moreover, the flexibility of the Gamma distribution allows for a relatively unobtrusive hyper-prior probability density when little a priori information about λ and δ is available. Thus,π(λ)∝λ^α_λ-1exp(-β_λλ), π(δ)∝δ^α_δ-1exp(-β_δδ).We choose the hyper-prior parameters to be α_λ=α_δ=1 and β_λ=β_δ=10^-6 so that the prior distributions of λ and δ cover a broad range of values that have been estimated for similar problems <cit.>. Hence, the full posterior probability density function for ( p,λ,δ|) isπ( p,λ,δ|)∝π(| p,λ)π( p|δ)π(λ)π(δ)=λ^M/2+α_λ-1δ^N/2+α_δ-1exp(-λ/2‖ p-‖^2-δ/2 p^T p-β_λλ-β_δδ). § MCMC ALGORITHMS FOR POSTERIOR INFERENCEThe primary goal of this work is to draw statistical inference on the joint variable p,λ,δ| b by characterizing the joint-posterior density in (<ref>). Due to the hierarchical modeling of λ and δ, the density in (<ref>) does not have a common distributional form, so explicit characterization is not readily available. Monte Carlo methods that utilize the invariance of a Markov process, so called Markov Chain Monte Carlo (MCMC), have become standard because of their computational efficiency and broad applicability. Gibbs sampling, in particular, has found great utility <cit.> due to its direct application to hierarchical modeling with conjugate random variables and ease of implementation with relatively little tuning. Moreover, investigating the Gibbs sampler and its convergence properties have been explored in <cit.>. This work builds upon this literature, by providing an algorithm that is shown to empirically improve the convergence of Gibbs sampling. The application of partial collapse to a general Gibbs sampling scheme was studied in <cit.>, and they show that partial collapse must be done with care, since the resulting Markov chain may no longer be invariant. The loss of invariance is case dependent, and this work provides a proof of the invariance of the Markov chain and directly addresses the potential pitfalls alluded to in <cit.>. §.§ Gibbs samplingIn a Gibb's sampling framework, the full conditionals of each component of the posterior density are used to compute samples of the posterior density. We have already characterized π( p| λ, δ) in (<ref>) in establishing the MAP estimator for fixed λ and δ. The full conditionals for λ and δ are computed by removing proportional terms from (<ref>), and observing that the remaining conditional densities satisfyπ(λ| p,δ,)∝λ^M/2+α_λ-1exp([-1/2‖ p-‖^2-β_λ]λ), π(δ| p,λ,)∝δ^N/2+α_δ-1exp([-1/2 p^T p-β_δ]δ).These are scalings and shifts of the corresponding hyper-priors, and are thus Γ-distributed.With each of the full conditional distributions characterized in (<ref>), (<ref>), and (<ref>), the requisite simulations required to establish the hierarchical Gibbs sampler are known, and steps for the Gibbs sampler are given in <Ref>. This algorithm has been used successfully for other inverse problems applications in computational imaging, and its efficacy has been demonstrated in <cit.>.algorithmHierarchical Gibbs Sampler for PSF reconstructionGiven λ_k,δ_k and p^k, simulate 1. λ_k+1∼Γ(M/2+α_λ,1/2‖ p^k-‖^2+β_λ);2. δ_k+1∼Γ(N/2+α_δ,1/2( p^k)^T p^k+β_δ);3. p^k+1∼((λ_k+1^T+δ_k+1)^-1λ_k+1^T,(λ_k+1^T+δ_k+1)^-1). In steps 1 and 2, several well-established algorithms exist for simulating draws from Γ distributions, and are readily available in most statistical software packages <cit.>.The simulation in step 3 can be achieved by solving the system p^k+1 = (λ_k ^T+ δ_k)^-1 (λ_k^T+ ),∼(,λ_k ^T+ δ_k). In a generic Gibbs sampling framework, any permutation of the steps is still a proper algorithm in the sense that they all produce chains that are invariant with respect to the joint random variable.However, in the hierarchical framework, there is a natural ordering that separates the hierarchical variables and the quantity of interest, which we show in the next section. Moreover, when partial collapse is applied, a permutation actually leads to an algorithm where invariance is lost. §.§ Blocking and MTC Sampling Observe that the conditional densities in steps 1 and 2 of <Ref> are conditionally independent.That is, the normalizing constant in (<ref>) is an integral that does not depend on δ and vice versa for λ in (<ref>); hence, π(λ,δ| p, b) = π(λ| p, b) π(δ| p, b).This has the effect that the hierarchical Gibbs sampler in <Ref> naturally blocks δ and λ. Explicitly, if we denote θ = (λ,δ), then <Ref> is equivalent to <Ref>:algorithmTwo-stage Gibbs Sampler for PSF reconstructionGiven θ^k = (λ_k,δ_k) and p^k, simulate 1. θ^k+1∼π(λ,δ| p^k, b) by (<ref>)2. p^k+1∼π( p|λ_k+1,δ_k+1, b) by (<ref>) Note that any non-trivial permutation with step 3 of <Ref> makes it impossible to block λ and δ, and the methods are no longer equivalent.When Gibbs sampling happens in two stages, the two separate components θ^k and p^k are themselves Markov chains whose stationary distribution is given by the corresponding marginalized density <cit.>. Moreover, the transition kernel associated with {θ^k} is ∫_^nπ( p| λ,δ, b) π(λ',δ'| p, b) d p.This makes the analysis divide naturally into considering the quantity of interest, p, and the hierarchical parameters, λ and δ.In <cit.>, the Gibbs sampling approach was analyzed theoretically on a class of hierarchical models that converge to an infinite dimensional limit. It was shown that the Markov chain degrades as the discretization of the forward operator converges to the continuous limit.In particular, when λ is fixed, the {δ_k} component of the chain makes smaller and smaller expected moves as the discretization converges, effectively slowing the exploration of the δ component of the posterior and creating highly autocorrelated samples.A complementary relationship between λ and p was shown, where the analogous sampler that fixes δ produces a Markov chain in {λ_k} that centers immediately on the true noise precision that are less and less correlated.We observed similar results empirically for PSF reconstruction using Gibbs sampling; that is, as the discretization limits in (<ref>) increase, the δ_k component of the Markov chain moves more slowly and is more highly correlated, whereas the {λ_k} component of converges rapidly and looks like nearly independent samples centered on the true noise precision. See <Ref>.The dependence between the hierarchical components and the quantity of interest, specifically δ_k and p^k, is what drives the slowing of the {δ_k} chain.A straight-forward approach to addressing this dependence, would be to marginalize the dependence of the blocked variable θ in p.That is, sample λ,δ| b rather than λ,δ|θ^k, b. It turns out that the resulting sampling algorithm still provides a Markov chain that converges to the desired posterior.A sampling scheme of this form has been studied by others <cit.>, and its application to a hierarchically modeled linear inverse problem is the subject of the recent work in <cit.>. An algorithm utilizing it for PSF reconstruction is given below.algorithmMTC Sampler for PSF reconstructionGiven θ^k = (λ_k,δ_k) and p^k, simulate 1. θ^k+1∼π(λ,δ| b)2. p^k+1∼π( p|λ_k+1,δ_k+1, b)The density π(λ,δ| b) is given by computing the marginal density of π(λ,δ, p| b) by integrating (<ref>) with respect to p. To see this explicitly, first write the full posterior (<ref>) as the Gaussian density defined in (<ref>) times terms that only involve λ and δ. Then, integrating with respect to p yields π(λ,δ|)∝ λ^M/2π(λ)δ^N/2π(δ)exp(-1/2a(δ,λ)-1/2 b(δ,λ)),wherea(λ,δ) = ln((λ^T+δ)), b(λ,δ) = λ(^T-^T(λ^T+δ)^-1^T).The marginalization comes at a cost, however, as the density π(λ,δ| b) is no longer one where an efficient algorithm is available, and rejection-based methods such Metropolis-Hastings must be used to simulate samples. Hence, the convergence in the chain is now driven by the efficiency of sampling the blocked variable θ^k. In <cit.>, they explore methods to accelerate this process in a deconvolution application, where because the forward operator is a convolution, Fourier based factorizations can be utilized which are not available in the application of PSF reconstruction.The key difference between this method and the partially collapsed sampler, that we present next, is that MTC removes the dependence between λ_k and p^k which negates the predicted efficiency in sampling the λ component when the dependence remains <cit.>, which results in a loss of efficiency in sampling λ. Partial collapse integrates only the δ component with respect to p, leaving the dependence between λ_k and p^k, whereas MTC keeps the variables blocked.Our method requires Metropolis-Hastings only in one dimension, {δ_k}, which requires less tuning and in the proposal. §.§ Partially collapsing the Gibbs sampler for PSF reconstruction Our sampling approach is similar to the MTC algorithm outlined in <Ref>, only that we retain the dependence of λ_k and p^k, and as predicted by the theoretical work in <cit.>, this dramatically improves autocorrelation of the {λ_k} component of the Markov process. Moreover, because Metropolis-Hastings is now only on {δ_k}, gaining efficiency in that component is more easily attained by tuning a one-dimensional random walk proposal.The algorithm falls into a category of samplers that are investigated in <cit.>, where they show using spectral methods that partial collapse can improve chain convergence. Additionally, they show that care must be taken when modifying steps in the Gibbs sampler, since changes could result in a Markov chain whose stationary distribution is no longer the target posterior. In particular, they show that a partially collapsed Gibbs sampler may no longer have the same stationary distribution as the original Gibbs sampler. We will explicitly show that partial collapse maintains the posterior as the stationary distribution. algorithmPartially Collapsed Gibbs for PSF reconstructionGiven λ_k,δ_k and p^k, simulate * λ^k+1∼π(λ| p^k, b)* δ^k+1∼π(δ|λ^k+1, b)* p^k ∼π( p| λ^k+1,δ^k+1, b) First note that the coupling between components in <Ref> is more complicated than <Ref> and <Ref>, and that the algorithm is stationary with respect to π( p,λ,δ| b) is not obvious.<Ref> is stationary with respect to π( p,λ,δ| b). Denote π_(·) = π(·| b) for a conditional density depending on the data b.The transition kernel associated with this algorithm isK(λ,δ, p; λ',δ', p') = π_( p'|λ', δ') π_(δ'|λ') π_(λ'|δ, p).Using (<ref>) to substitute π_(λ|δ, p) = π_(λ| p), the action of transition on the density π_(λ, δ,p) is∫_^N∫_∫_K(λ,δ, p; λ',δ', p') π_(λ,δ, p) dλ dδ d p= π_( p'|λ', δ')π_(δ'|λ') ∫_^N∫_π_(λ'|δ, p) ∫_π_(λ,δ, p) dλ dδ d p= π_( p'|λ', δ')π_(δ'|λ') ∫_^N∫_π_(λ',δ, p)/π_(δ, p)π_(δ, p) dδ d p = π_( p'|λ', δ')π_(δ'|λ') π_(λ') = π_( p',λ',δ').The critical thing to note in this simple computation is that any permutation of the steps of <Ref> will result in a sampler that is no longer invariant with respect to π( p,λ,δ| b). This is in contrast to the Gibbs sampler, in which the steps can be permuted in any order without changing the stationary distribution of the Markov chain.Both <Ref> and <Ref> have been stated in terms of sampling exactly the densities π(λ,δ| b) and π(δ|λ, b), where we have mentioned that these are not readily available, and we employ Metropolis-Hastings to sample them. The density π(δ|λ,b) is obtained by removing the λ-proportional terms from (<ref>).A Metropolis-Hastings method for computing both densities requires repeated evaluations of (<ref>), and the computational cost is dominated by finding the determinant in (<ref>) and computing the matrix solve in (<ref>).For the scales relevant to PSF reconstruction, these can be accomplished by a Cholesky factorization. That is, we utilize the computation of an upper-triangular Cholesky factor R_λ,δ which satisfies R_λ,δ^T R_λ,δ = λ^T + δ,and can be used to compute the quantities in (<ref>) and (<ref>) bya(δ,λ)= ^T(-R_λ,δ^-1 R_λ,δ^-T^T λ) b(δ,λ)= 2∑_i=1^N ln|r_i,i(λ,δ)|.The matrix solves in (<ref>) are accomplished via backwards- and forwards-substitution (O(n^2)), and r_i,i(λ,δ) are diagonal elements of R_λ,δ. Hence, the computational cost of evaluating π(δ| b,λ) is dominated by the Cholesky algorithm (O(n^3)), and in order to emphasize this dependence, we denotec( R_λ,δ)a(δ,λ) + b(δ,λ). A random walk in log-normal space is used for the proposal of the Metropolis-Hastings step in both the MTC and PC Gibbs algorithms. A full statement of each algorithm using this notation is in <Ref> and <Ref>.algorithmMetropolis-Hastings MTC Sampler for PSF ReconstructionGiven λ_k,δ_k, p^k, and a random walk covariance C, simulate * Set λ = λ_k, δ = δ_k and compute R_λ,δ.For j = 1… n_mh(i) Simulate w ∼(0, I_2×2) and set [ λ'; δ' ] = exp( C^1/2 w + [ lnλ; lnδ ]). (ii) Compute R_λ',δ'. (iii)Simulate u∼ U(0,1). If log u < min{ 0, c( R_λ',δ') - c( R_λ,δ) }, then set λ = λ ',δ = δ' and R_λ,δ =R_λ',δ'.Set λ_k+1 = λ and δ_k+1=δ. * Simulate p^k+1∼(λ_k+1 R^-T_λ_k+1,δ_k+1 R_λ_k+1,δ_k+1^-1^T,( R^T_λ_k+1,δ_k+1 R_λ_k+1,δ_k+1)^-1). Observe that in <Ref>, the Cholesky computation used for the last accepted proposal can be re-used in order to simulate p^k+1. In fact, because θ^k does not depend on p^k+1, the two components can be computed in serial; i.e. θ^k can be computed (and potentially thinned) and then samples of p^k+1 can be computed, as suggested in <cit.>. This is in contrast to <Ref>, where an additional Cholesky factor must be computed in order to sample λ_k. We show in <Ref> that the added efficiency is worth this additional computation.algorithmMetropolis-Hastings PC Gibbs Sampler for PSF ReconstructionGiven λ_k,δ_k, p^k, and σ^2, simulate * Simulate λ_k+1∼Γ(M/2+α_λ,1/2‖ p^k-‖^2+β_λ).* Set δ = δ_k and compute R_λ_k+1,δ.For j = 1… n_mh(i) Simulate w ∼(0,1) and set δ' = exp(σ w + ln(δ)). (ii) Compute R_λ_k+1,δ'. (iii)Simulate u∼ U(0,1). If log u < min{ 0, c( R_λ_k+1,δ') - c( R_λ_k+1,δ) }, then set δ = δ' and R_λ_k+1,δ =R_λ_k+1,δ'.Set δ_k+1=δ. * Simulate p^k+1∼(λ_k+1 R^-T_λ_k+1,δ_k+1 R_λ_k+1,δ_k+1^-1^T,( R^T_λ_k+1,δ_k+1 R_λ_k+1,δ_k+1)^-1).The theoretical justification for the use of Metropolis-Hastings as a sub-step in samplers can be found in<cit.>. Note that each proposal step requires a computationally expensive Cholesky solve. The authors in <cit.> suggest that for Metropolis-within-Gibbs, additional proposals (n_ MN>1) may not be worth the computational cost, while others have suggested more sub-steps <cit.> to improve convergence. The situation likely depends on the problem, and due to the lack of objective criteria, we investigate empirical evidence that suggests that in the case of PSF reconstruction, more than one step can improve convergence.§ RESULTSIn this section, each of the three methods described are used to analyze synthetically generated and real data from a diagnostic radiographic imaging systems in operation at the Nevada National Security Site. We first establish the metrics by which we compare them in order to fairly compare the algorithms. In particular, we briefly describe a statistical method for determining whether the chain has reached stationarity and passed the so-called burn-in stage and how efficiently the chains explore the invariant density by estimating the stationary autocorrelation. Our measure of efficiency also takes into account computational effort, and we show that PC Gibbs for hierarchical sampling performs significantly better than standard Gibbs sampling and at least as well as the recently developed MTC sampler. §.§ Statistical measures of convergence. The stationarity of the partially collapsed Gibbs sampler guarantees that Monte Carlo realizations of the Markov process converge in distribution to realizations from π( p, λ,δ|), but this asymptotic result does not address the practical fact that only a finite number of simulations can be computed.Two aspects of convergence are addressed in this section. First, the initial samples must converge, or burn-in, to the desired stationary distribution of the Markov chain, and second, since the process produces identically distributed but dependent samples, how effectively uncorrelated the samples are determines how well they characterize the stationary distribution. In this section, we give a brief overview of two statistical estimators that address these two aspects. Both estimators inform how long to run the MCMC algorithm to effectively analyze the chain as a robust sample from π( p,λ,δ|).The first issue is concerned with how close the Markov chain is to the target invariant density. In practice, the Markov chain is initialized with simulations that are not from the target density, π( p, λ,δ|), and <cit.> provides statistically motivated approach that uses a statistical test to evaluate the test hypothesis that the joint mean value of an early section of the Markov chain is equal to that of a latter portion. Formally, for a given univariate component of a stochastic process, {X^1,…, X^N}, let N_m denote the mth percentile of N, μ_m to be the mean of {X^1,…, X^N_m} and μ_m' the mean of {X^N_m'+1,…, X^N}. Following <cit.>, we choose the 10th and 50th percentiles to establish the estimators for μ_10 and μ_50', whichareX̅_10=1/N_10∑_k=1^N_10X^k, andX̅_50'=1/N-N_50'∑_k=N_50+1^N X^k.For the test H_0:μ_10 = μ_50', <cit.> shows the corresponding convergence diagnostic test statistic satisfiesR_ GewekeX̅_10-X̅_50'/√(Ŝ_10(0)/N_10+Ŝ_50'(0)/N_50)d⟶(0,1), as N→∞,where Ŝ_10(0) and Ŝ_50'(0) denote consistent spectral density estimates for the variances of {X^1,…, X^N_10} and {X^N_51,…, X^N}, respectively.These can be estimated via a periodogram estimator, and in our results, we use a Danielle window of width 2π/(0.3p^1/2) as recommended by <cit.>. The test provides a method for evaluating a portion of the realizations of the Markov chain that are suitable for a rigorous analysis. For the results in this paper, each algorithm was run for a fixed number of iterations, and the last half of the simulations were tested.The process is then assumed to be in stationarity if the test provides no statistical evidence for a difference in the quantile means.The second estimator we establish measures how efficiently the stationary Markov chain characterizes the posterior density. That is, after identifying the burn-in portion of the chain, successive simulations may be highly correlated and result in an excessively slow exploration of the target density.Improving this aspect of convergence is the primary motivation for partial collapse. Following <cit.>, we use the notion of integrated autocorrelation time to quantify how much the Monte Carlo samples have explored the target density relative to a hypothetical independent sample. Summarizing that work, suppose that {X_1,X_2,…} is an identically distributed correlated stochastic process with individual variance σ^2, then the Monte Carlo error for the estimator X̅_N=1/N∑_k=1^NX^i can be divided into a contribution from inherent variance in X_j, and covariance between X_i and X_j for j≠i; i.e.Var(X̅_N)=σ^2/N(1+2∑_k=1^N-1(1-k/N) Cov(X^1,X^1+k)/σ^2).The autocorrelation function at lag k of the process is ρ(k) Cov(X^1, X^|k|)/σ^2, and so for large N, the Monte Carlo error can be approximated withVar(X̅_N) ≈σ^2/N∑_k=-∞^∞ρ(k) σ^2/Nτ_ int.The approximation is based on the assumption that the autocorrelation lag of the process dies off fast enough so that k/N does not contribute to (<ref>).Since σ^2/N would be the variance of the Monte Carlo estimator had {X_1,…,X_N} been uncorrelated, we think conceptually of the parameter τ_int as the equivalent number of Markov chain simulations required to obtain an effectively independent sample from the target density (in terms of Monte Carlo error of sample mean estimation). This analogy motivates what is sometimes called the essential sample size of the chain N_ ESS N/τ_ int.To estimate these parameters, <cit.> gives the following unbiased estimator for the normalized autocorrelation function,τ̂_ int=∑_k=-N̅^N̅ρ̂(k),where N̅< N-1 is some window length, and ρ̂(k) is the empirical normalized covariance estimator over that interval. That is,ρ̂(k) Ĉ(k)/Ĉ(0), whereĈ(k)=1/N-k∑_i=1^N-k (X_i-X̅_̅N̅)(X_i+k-X̅_̅k̅).The choice we use suggested by <cit.> for the window size is the smallest integer such that N̅≥ 3 τ̂_ int. Finally, our estimate the essential sample size, denoted ESS, is given by substituting the estimator τ̂_ int for τ_ int in (<ref>).The ESS estimate can be used in a couple of ways.A standard approach, when samples are relatively cheap to compute, is to do as follows: (1) compute a very long MCMC chain (we choose 10^4 below); (2) remove the first half of the chain as burn-in and verify using Geweke's test that the second half of the chain is in equilibrium; and (3) estimate the number of effectively independent samples in the second half of the chain using K̂_ ESS.In cases in which each sample is expensive to compute, however, there is incentive to make the chain as short as possible.In such instances, both chain convergence and autocorrelation can be monitored online, so that a minimal number of samples are discarded in the burn-in stage, and also so that K̂_ ESS is not larger than it needs to be in order to perform the desired uncertainty analysis.The ESS is not the complete answer to the efficiency of the algorithm. Highly uncorrelated chains for δ can be achieved in the MTC and PC Gibbs algorithm by increasing the number of inner Metropolis-Hastings steps n_ MH, however, the addition of each step increases the number of expensive matrix factorizations by a factor of the chain length. To take the extra computational effort into account, we use the number of Cholesky factorizations as a metric for computational effort since this computation dominates the computational time per MCMC iteration. That is, we use the number of Cholesky solves divided by the ESS, which we interpret as the computational effort to obtain an equivalent uncorrelated sample. If we assume that a chain thinned according to estimated integrated autocorrelation is equivalent to an uncorrelated sample, this measure says how computationally costly it is to obtain each sample. §.§ Synthetic examples of PSF reconstruction We first establish the efficacy of our approach on a simulated example where the true profile is explicitly known, and the data is artificially corrupted with simulated noise. To simulate synthetic data, we reconstruct the radial profile of a two-dimensional Gaussian kernelx(r) = (2πσ^2)^-1 e^-r^2/2σ^2,where σ = 1/15 is chosen so that the effective width of the kernel is about 20% of the image width when scaled to [-1,1]. Observe that in the case of a two-dimensional Gaussian, the action of the forward operator in (<ref>) is the scaled error functionb(s) = 1/√(2π)σ∫_-∞^s e^-s'^2/2σ^2 ds',which can be numerically calculated with very high precision. Gaussian measurement error with noise strength that is 2% of the strength of the signal is synthetically generated and added to b(s).All three MCMC algorithms were run with a chain length of N=10^4, and the last 5×10^3 simulations were tested with the Geweke statistic for stationarity. For each algorithm, the resulting p-values were all greater than 0.9, hence we use the last 5× 10^3 simulations as burned-in MCMC samples from the posterior density.To estimate the PSF and the hierarchical parameters, we use the sample mean of the burned-in samples. The true PSF falls well within the distribution of MCMC samples, and the mean MCMC estimator for the PSF matches the truth quite well; see the left panel of <Ref>. Note that the most uncertain region of the reconstruction are the initial discretization points corresponding to the height of the PSF.Since the simulated data has a known solution, if we interpret the problem variationally with the Tikhonov regularization parameter δ/λ, we can characterize a “best” regularization by minimizing the L^2 norm of the residual with respect to the Tikhonov regularization parameter. In right panel of <Ref>, a plot of the log L^2 norm of the residuals versus the Tikhonov regularization parameter are given with the MCMC estimate δ̂/λ̂ indicated by an asterisk.Note that the MCMC estimator nearly falls on the minimum of the curve. In the left panel of <Ref>, note that increases in the Metropolis-Hastings sub-steps of MTC generally decreases the efficiency of the sampler. This is because of the discussion in <Ref>, where it was shown that the sampling of θ^k does not depend on p^k and only on the previous θ^k-1. Since the efficiency of sampling θ^k depends only on the Metropolis-Hastings algorithm with no influence from p^k, the difference is merely how the number Cholesky factorizations are accounted for per iteration of the algorithm. Said another way, increasing n_ MH of MTC is equivalent to simulating a chain of θ^k of length M · n_ MH, with p^k simulated in a chain of length M. In the case of the PC Gibbs sampler on the other hand (right panel of <Ref>), extra Metropolis-Hastings steps appear to increase the efficiency of the sampler. This is because only δ has been marginalized, and even though λ_k and δ_k can still be blocked into θ^k, it depends on p^k through λ_k. This means that the PC Gibbs transition kernel improves with better samples of δ_k that increased Metropolis-Hastings steps provide. The efficiency statistics for each algorithm are given in <Ref>. In order to give a common basis for comparison, the proposal parameters for each Metropolis-Hastings random walk in MTC and PC Gibbs were derived from the burned-in posterior samples of the hierarchical Gibbs sampler. Specifically, two times the empirical covariance of the burned-in θ^k from a realization of the hierarchical Gibbs sampler was used as the proposal covariance for MTC. Two times the variance of δ_k from the same realization was used as the proposal variance for PC Gibbs.These choices result in acceptance rates near 0.3 for MTC and 0.45 for PC Gibbs.First observe that each sampler generally agrees in terms of the MCMC means generated. As predicted, the λ_k sub-chains are all sampled very efficiently in Gibbs and PC Gibbs, while its efficiency is driven by Metropolis-Hastings in MTC. Moreover, the efficiency of the PC Gibbs algorithm with n_ MH=4 is estimated to be slightly more efficient than MTC. We remark that due to the variability in the proposal tuning, small differences in efficiency are likely not definitive, but this result serves as evidence that the efficiencies of PC Gibbs and MTC are roughly equivalent. The advantage of PC Gibbs is that its Metropolis-Hastings proposal need only be tuned in one dimension. §.§ PSF reconstruction with measured radiographic data Next we reconstruct the point spread function of a high energy X-ray imaging system at the U.S. Department of Energy's Nevada National Security Site.The real edge data is shown in <Ref> (upper left) along with a horizontal cross-section across the edge (upper right). The mean MCMC reconstruction is shown in <Ref> (lower left), along with the 10%, 25%, 50%, 70%, and 90% quantiles of the chain x^k. We estimated the PSF at grid points using the chain-wise mean after burn-in, p̂= 2/M∑_k=M/2+1^Mp^k. Since the true PSF is unknown, we evaluate the accuracy of the estimation by its discrepancy; i.e. we compared forward mapping of the estimate p̂ with the given data b.This is shown in both linear and logarithmic scales in <Ref> (lower right). In both cases the discrepancy is quite low, except at very low intensities where the data is dominated by the noise, which can be seen in the logarithmic scale. Observe that the chain efficiency statistics in the third through fifth columns of <Ref> are similar to those derived on the synthetic example.Similar to the synthetic data, the PC Gibbs algorithm with n_mh=4 and MTC perform with roughly equivalent overall efficiency, however, with λ_k being the least efficient component of MTC.§ CONCLUSIONSPSF reconstruction provides an excellent medium scale inverse problem to test state-of-the-art MCMC algorithms for posterior estimation. This work shows how modifying the hierarchical Gibbs sampler first presented in <cit.> can result in the MTC algorithm which is equivalent to the one derived in <cit.> and the hierarchical PC Gibbs algorithm. Both methods have their advantages: MTC having θ decoupled from p makes it possible to sample p^k at the rate of the integrated autocorrelation time and makes analysis of the algorithm easier; where PC Gibbs is a straight-forward modification of the Hierarchical Gibbs algorithm with an easily tuned one-dimensional Metropolis-Hastings step, which can easily be tuned to be very efficient. We have provided statistical evidence that both algorithms are essentially equivalent for PSF reconstruction in terms of an estimator that measures computational effort per uncorrelated sample.This work contributes two novel aspects to the relevant literature. First is in the application of PSF reconstruction to X-ray imaging, which to our knowledge, has not appeared elsewhere in the inverse problems literature. This involved a novel approach to incorporating radial symmetry in prior modeling the PSF with a Gauss-Markov random field. The results illustrate the effectiveness of a sample-based approach on real data for uncertainty quantification. The second contribution of this work is in new advances of addressing the autocorrelated δ component of a hierarchical Gibbs sampler in <cit.>.The work builds upon <cit.> by collapsing only the δ_k component of Gibbs and retaining the efficiency in sampling λ_k gained by its dependence on p^k. We showed that this fits into the framework of partial collapse presented in <cit.>, and heeding their warnings of creating an improper sampler, we prove that our algorithm is still invariant with respect to the desired posterior density. Finally, MCMC methods were verified using both a synthetic test case and real data. The PC Gibbs sampler is readily adapted to other linear Bayesian inverse problems modelled hierarchically and has potential applications to more general prior modeling (i.e., non-conjugate priors). In general, this work provides evidence that in models with a Gaussian noise likelihood, it is advantageous to employ an MCMC transition that retains the dependence between the parameter defining the likelihood and the data, rather than completely decoupling them.§ ACKNOWLEDGMENTSThe authors would like to thank Peter Golubstov for helpful comments and suggestions on the work and manuscript. This manuscript has been authored by National Security Technologies, LLC, under Contract No. DE-AC52-06NA25946 with the U.S. Department of Energy, National Nuclear Security Administration, [NNSA Subprogram Office funding source]. 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, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The U.S. 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>). The views expressed in the article do not necessarily represent the views of the U.S. Department of Energy or the United States Government. DOE/NV/25946–3373siamplain | http://arxiv.org/abs/1709.09105v1 | {
"authors": [
"Kevin T. Joyce",
"Johnathan M. Bardsley",
"Aaron Luttman"
],
"categories": [
"math.NA",
"65C05, 65C40, 68U10"
],
"primary_category": "math.NA",
"published": "20170926160837",
"title": "Point Spread Function Estimation in X-ray Imaging with Partially Collapsed Gibbs Sampling"
} |
[ Romuald Méango The first version is of 22 April 2012. The present version is of December 30, 2023. This research was supported by SSHRC Grants 410-2010-242, 435-2013-0292 and 435-2018-1273, NSERC Grant 356491-2013, and Leibniz Association Grant SAW-2012-ifo-3. The research was conducted in part, while Marc Henry was visiting the University of Tokyo and Isma^̂22el Mourifié was visiting Penn State and the University of Chicago. The authors thank their respective hosts for their hospitality and support. They also thank Désiré Kédagni, Lixiong Li, Karim N'Chare, Idrissa Ouili and particularly Thomas Russell and Sara Hossain for excellent research assistance. Helpful discussions with Laurent Davezies, James Heckman, Hidehiko Ichimura, Koen Jochmans, Essie Maasoumi, Chuck Manski, Ulrich M^̂22uller, Aureo de Paula, Azeem Shaikh and very helpful and detailed comments from five anonymous referees, from numerous seminar audiencesand the 2018 Canadian senate open caucus on women and girls in STEM are also gratefully acknowledged. Correspondence address: Department of Economics, Max Gluskin House, University of Toronto, 150 St. George St., Toronto, Ontario M5S 3G7, Canada =================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================empty empty Turn-taking is essential to the structure of human teamwork. Humans are typically aware of team members' intention to keep or relinquish their turn before a turn switch, where the responsibility of working on a shared task is shifted. Future co-robots are also expected to provide such competence. To that end, this paper proposes the Cognitive Turn-taking Model (CTTM), which leverages cognitive models (i.e., Spiking Neural Network) to achieve early turn-taking prediction. The CTTM framework can process multimodal human communication cues (both implicit and explicit) and predict human turn-taking intentions in an early stage. The proposed framework is tested on a simulated surgical procedure, where a robotic scrub nurse predicts the surgeon's turn-taking intention. It was found that the proposed CTTM framework outperforms the state-of-the-art turn-taking prediction algorithms by a large margin. It also outperforms humans when presented with partial observations of communication cues (i.e., less than 40% of full actions). This early prediction capability enables robots to initiate turn-taking actions at an early stage, which facilitates collaboration and increases overall efficiency. § INTRODUCTION Turn-taking provides humans the fundamental structure to organize and coordinate their collaborative actions in conversations <cit.>, collaborative task-solving <cit.>, and shared haptic control <cit.>. During collaboration, each agent has the capability to comprehend the on-going task progress and their peer's multi-modal communication cues, in order to predict whether, when, and how to size the next turn. Having a fluent and natural turn-taking process can enhance the collaboration efficiency and strengthen the communication grounding among team members <cit.>. Such coordinated turn-taking behaviors stand out clearly in high-risk and high-paced tasks like surgery in the Operating Room (OR). Surgeons and nurses perform fast, fluent, and precise turn-taking actions when exchanging surgical instruments. Therefore, the OR scenario was selected as the test-bed for investigating human robot turn-takings.The same turn-taking behaviors observed during human-human interactions are expected in human-robot interactions. When robots are designed to collaborate with humans, they are expected to understand the human's turn-taking intentions, in order to determine when is a good time to engage in an interaction. When designing robotic nurses to collaborate with surgeons, they are expected to understand both the surgeon's implicit communication cues (e.g., change of body posture like leaning forward) and explicit cues (e.g., uttering the word “scissors”) <cit.>. Figure <ref> shows our human-robot system including the capability to understand both implicit and explicit communication cues for turn-taking.A typical process where humans delegate the turn to a robot is illustrated in Figure <ref>. As the human is approaching the end of her turn, she starts exhibiting implicit cues (i.e., physiological and physical cues), followed by explicit cues (i.e., utterance). The collection of these multimodal signals indicates her intention to relinquish the turn. Simultaneously, the robot needs to capture these subtle communication cues and predict the end of her turn. Early and accurate predictions of such turn-giving intentions allow robots to begin preparatory movements to facilitate the turn transition process.There has been research on developing algorithms to automatically recognize human's turn-taking intentions, in both conversations <cit.> and physical collaborations <cit.>. However, current turn-taking recognition algorithms build on mathematically derived models such as Support Vector Machines <cit.>, Decision Trees <cit.>, and Conditional Random Field <cit.>. Although a certain level of recognition accuracy is achieved, it still has not reached the level of human competence <cit.>. Moreover, a model derived computationally and mathematically cannot be explained and interpreted well by humans due to different reasoning processes. Therefore, there is a need of a cognitive turn-taking reasoning model that reaches the level of human competence and can be easily interpreted. To bridge this gap, the Cognitive Turn-Taking Model (CTTM) is proposed to simulate and predict human's turn-taking intentions. The CTTM is built upon the brain-inspired Spiking Neural Network (SNN). The SNN framework has plasticity structure <cit.> to model turn-taking processes and predicts human's turn-taking intentions based on multimodal observations. Compared to traditional neural networks, SNN has the advantage of modeling conduction delays of variable lengths. Therefore, it is suitable for time sensitive sequence modelling tasks such as gesture recognition <cit.>, speech recognition <cit.>, and seizure detection <cit.>. However, there are still challenges in using SNN for multimodal turn-taking prediction such as how to map the raw signals into neurons, how to deal with multimodal fusion problems, and how to create descriptive features for early prediction. Those challenges are tackled by the proposed CTTM framework. To summarize, this paper makes the following three-fold contributions: 1) presents a formulation to rigorously define the human-robot collaborative task and the associated turn-taking events; 2) proposes the cognitive framework CTTM for early turn-taking prediction; 3) designs a multimodal interaction system between surgeons and robotic nurses.The paper is organized as the following: Section <ref> presents the related work, followed by definitions of the collaborative task and turn-events in Section <ref>. The proposed CTTM framework is introduced in Section <ref>, and the experimental setup and results are presented in Section <ref>. Finally, Section <ref> summarizes the paper with conclusions and future work. § RELATED WORK This section presents the related work about turn-taking analysis, in the context of conversational turn-taking and physical turn-taking. In conversational turn-taking analysis, linguistic cues such as pause duration <cit.> and pitch levels <cit.> have been found to play a key role in turn-taking regulation. The study of de Ruiter el al. <cit.> revealed that syntax and semantics cues are also important in projecting the end of a speaker's turn. Non-verbal behaviors such as gaze and posture shiftshave been investigated and found to be relevant to turn regulations <cit.>. Physical turn-taking refers to the process where a hybrid human-robot team take turns on a physical task. Turn-taking has been studied in robotics and HRI to regulate shared resources among team members. Those resources include time (i.e., only one person can work on the shared task at a time) and space (i.e., only one person can access the working space at a time). In the context of human robot interaction, the types and usage frequencies of various implicit communication cues have been studied in a robot-assisted assembly task <cit.>. The timing in multimodal turn-taking (i.e., speech, gaze, and gesture) was investigated through a collaborative Towers of Hanoi challenge accomplished by a hybrid human robot team <cit.>. Nonverbal cues for timing coordination in physical turn-taking were studied in manufacturing <cit.>. Such research focuses on turn-taking process modelling and the robot control, while neglecting predictions of the turn-taking timing and intention.§ COLLABORATIVE TASK AND TURN-EVENTS DEFINITION This section introduces the definitions of collaborative task and associated turn-events to formulate the early turn-taking prediction problem. Consider a human agent H working with a robot agent R on a collaborative task 𝒲. H and R conduct and alternate through a sequence of subtasks w_k^A, where subscript k indicates the subtask index and superscript A indicates the agent responsible for this subtask (i.e., A ∈{H,R}). The subtask is defined as w_k^A≜(g_k,z_k^b,z_k^f), where g_k∈ G is the action label, z_k^b is the beginning time, and z_k^f is the finishing time of the subtask. G is the set containing all the action labels, such as delivering, retrieving, and exchanging tools between agents. The subtask w_k^A is treated as the “atomic” component in the definition since a turn only happens during the transition of two subtasks. As the human H conducts subtask w_k^H and time goes on from z_k^b to z_k^f, she gets closer in finishing the subtask and expresses an increased level of intention to give out the turn. Since this paper focuses on enabling robotic assistants to take turns from humans, only the turn transitions from H to R are considered. Each turn transition from w_k^H to w_k+1^R defines a turn-event E_k, in which the human is showing an intention to give out the turn (denoted as ℰ^give). On the other hand, for the majority of subtask w_k^H, the human is focusing on the current action and shows no intention to relinquish the turn. This period implicitly defines a turn-event E_k, in which human intends to keep the turn (denoted as ℰ^keep). Each turn-event E_k ∈{ℰ^give,ℰ^keep} spans a time window [t_k^s,t_k^e], which are the starting and ending time for this turn-event. The collaborative task 𝒲, subtask w_k^A, and turn-events are illustrated in Figure <ref>. As the human is conducting subtask w_k^H, she is monitored through M sensor channels which include physiological, physical, and neurological signals. The captured signal at time t is denoted as s⃗(t) ∈ℝ ^ 1 × M. Given a turn-event E_k which spans time [t_k^s,t_k^e], the captured signals within this time window are stacked together to form a matrix representation X_k ≜ [s⃗ (t_k^s:t_k^e) ] ∈ℝ^L_k × M, where L_k is the discrete event length (i.e., L_k=t_k^e-t_k^s). For each stacked segment X_k, a label y_k∈{0,1} is assigned to indicate either turn-giving (i.e., y_k=1 when E_k ∈ℰ^give) or turn-keeping (i.e., y_k=0 when E_k ∈ℰ^keep). The turn-taking prediction algorithm ϕ(·) then returns an estimate ŷ_k = ϕ(X_k) ∈{0,1}. Moreover, the turn-taking intention should be predicted in an early stage given only partial observations (i.e., with only implicit cues before explicit cues even start). This way, the human's intent to relinquish the turn can be recognized early and the robot can start moving early to facilitate the transition. Therefore, given partial observation X_k^ τ∈ℝ^(τ L_k) × M for only the beginning τ fraction (0<τ≤1), an early decision is made according to ŷ_k^τ = ϕ(X_k^τ) ∈{0,1}. The resultant dataset D^τ={(X_k^τ,y_k,ŷ_k^τ)| X_k^τ∈ℝ^(τ L_k) × M,y_k ∈{0,1}, ŷ_k^ τ = ϕ(X_k^τ ) ∈{0,1}} is then used for the following turn-taking analysis. § COGNITIVE TURN-TAKING MODELS (CTTM) This section presents the CTTM framework: Section <ref> discusses the network structure, Section <ref> presents the neuron mapping methods, and Section <ref> discusses the various aspects of SNN training.§.§ SNN Structure Conventional neural networks enforce the same conducting delay between consecutive layers, thus all the neurons of the same layer can only fire at the same time. This rigid structure is insufficient when modeling multimodal temporal sequences [20]. SNN, however, can model the variability of axonal conduction delays between neurons. Firings will take different amounts of time to traverse around the network based on the different conduction delays. This way, the asynchronous effect of multimodal signals can be modeled. The adopted SNN structure and parameters are similar to the work of Izhikevich <cit.>. The network has 250 neurons, with 200 excitatory neurons (i.e., can be stimulated) and 50 inhibitory neurons (i.e., cannot be stimulated). Each excitatory and inhibitory neuron has 25 post synapses, connecting to 25 other neurons following a uniform distribution. Each synapse has a conduction delay in the range of [1, 20] ms, following a uniform distribution. The weights of the synaptic connections are initialized to be +6 for all post synapses after excitatory neurons, and -5 for all post synapses after inhibitory neurons. Those weights represent how strong the synaptic connection is between two neurons, and are updated based on the Spike Timing-Dependent Plasticity (STDP) rule <cit.>. The maximum weight for each synaptic connection is set to be 10.Each neuron model is depicted by a set of formulas, as given in (<ref>). This neuron model, together with the constant parameters are the same with the simple spiking model proposed in <cit.>:dv/dt = 0.04v^2 + 5v + 140 - u + Idu/dt = a(bv-u) with the auxiliary after-spike resetting:ifv ≥+30mV,then v ← cu ← u+d The variable v represents membrane potential of the neuron, and u represents a membrane recovery variable which provides negative feedback to v. Variable I is the input DC current to this neuron, which was set to 20 mA when this neuron is stimulated by the input multimodal data. As illustrated in Figure <ref>, for recovery variable u, a represents the time scale, b represents the sensitivity, and d represents after-spike reset increment. For membrane potential variable v, c represents the after-spike reset value. The excitatory neurons are configured to exhibit regular spiking (RS) firing patterns with parameters (a,b,c,d)=(0.02,0.2,-65,8), and the inhibitory neurons are configured to exhibit low-threshold spiking (LTS) firing patterns with parameters (a,b,c,d)=(0.02,0.25,-65,2). Those parameters were chosen since they performed the best out of all excitatory-inhibitory neuron combinations proposed by <cit.> in our case. §.§ Neuron Mapping To model human turn-taking behaviors, it is necessary to map the input multimodal data to the SNN neurons. Previous research has attempted to map the discrete input data into the neurons on a one-to-one basis. For example, one image pixel corresponds to one neuron <cit.> or one level of orientation maps to 5 randomly chosen neurons <cit.>. However, mapping multimodal continuous-valued signals into SNN neurons requires a different approach since the network dimension grows exponentially to encode all possible combinations of continuous inputs. A small example of 5 discrete levels and 10 multimodal channels would lead to 10^8 neurons, which is intractable. This problem is solved by resorting to automatic channel quantization and decision-level fusion methods.First, each of the M channels was quantized into V discrete levels. The 1-percentile threshold (r_1) and the 99-percentile threshold (r_99) are used as fences to remove outliers. The V bins were divided within the 1%∼ 99% range to encode the continuous sensor signals. Given a sensor reading value s, it was quantized to level q(q=0,1,...,V-1), following q=int(s-r_1/r_99-r_1*V). The quantization process is applied to each of the M channels of X_k ∈ℝ^L_k × M. The result is X̃_k ∈ℚ_V^L_k × M, where ℚ_V represents the quantized space with V levels. Now the partial observation is X̃_k^τ∈ℚ_V^(τ L_k) × M. For each quantized level q, 5 excitatory neurons in the SNN were randomly allocated following a uniform distribution to represent it. When level q is active, all its 5 corresponding neurons are stimulated one by one at 1 ms intervals. The DC current of 20 mA provides the stimulation to variable I in (<ref>). The value of V was set to be 40, to reach a total of 40*5=200 excitatory neurons so that each excitatory neuron is mapped to a discrete level. To encode the multimodal inputs, one SNN is constructed for each of the M channels. The outcomes from each SNN are combined in the end using a decision-level fusion. This approach follows the human brain mechanism (i.e., vision and hearing are independently processed and are fused in later cognitive decision-making stage). §.§ SNN Training Once the mappings between input data X̃_k^τ and SNN neurons are established, the network needs to be trained. The SNN training includes three stages, where the first stage trains the SNN network weights by feeding unsupervised training data repeatedly into the network. Each training observation activates corresponding neurons in sequence, and the network weights are updated following the STDP rule. The second stage consists of soliciting network responses for different turn-taking classes, and the third stage focuses on constructing features from SNN responses.§.§.§ SNN Synapse Weight Training This training phase aims to tune the synaptic weights for SNN so that it can properly encode the input spatio-temporal signals. For that purpose, the STDP training is used. Under STDP, the synaptic weights are updated based on timing differences of the neural firings <cit.>. The synaptic weights between those neurons which always fire together are strengthened. More specifically, the weight of synaptic connection from pre- to postsynaptic neuron is increased if the post-neuron fires after the presynaptic spike and is decreased otherwise. Parameters for STDP training are set based on <cit.>. During this training stage, each quantized training data X̃_k ∈ℚ_V^L_k × M is mapped to its corresponding neurons in the SNN, and the synaptic weights are updated in each 1ms interval based on the STDP rules. The time allocated to simulating each X̃_k is 250ms. Since the input data length L_k is less than 40, it requires less than 200ms to stimulate the network. Then the network continues to run for 50ms without being provided any input, to allow the spike trains to propagate the network. Patterns X̃_̃k̃ for both turn-taking classes (y_k ∈{0,1}) are presented to the SNN network during this training phase, following a random repeated order. The network was simulated for 250s, which includes a total of 1000 training inputs X̃_k, each of which takes 250ms to simulate. After the simulation, the synaptic weights in the network converge into a steady state. §.§.§ Signature Firing Patterns TrainingThis training phase aims to generate the stereotypical network responses for different types of inputs X̃_k. As mentioned earlier, one SNN network was constructed for each channel of information (i.e., one column of X̃_k ∈ℚ_V^L_k × M, denoted as X̃_ki for column i). Therefore, there are M SNN constructed in total, forming a SNN group and is denoted as 𝒮={S_i}, i=1,...,M. Given input X̃_k, its response to this SNN group is denoted as 𝒢_k=𝒮(X̃_k). 𝒢_k consists of M individual responses (G_ki) for each of the SNN networks, i.e., 𝒢_k ≜{G_ki}, i=1,...,M where responses G_ki≜ S_i(X̃_ki). G_ki denotes the firing maps (which neurons fired at what time) when input X̃_ki is presented to the model S_i. As mentioned earlier, given an input X̃_ki, the network was simulated for T milliseconds and each millisecond forms an atomic time unit. There are in total N neurons in the network which can be potentially fired. Therefore, G_ki is formed as a N by T Boolean matrix (i.e., G_ki∈𝔹^N× T), where a value of 1 at cell (n,t) indicates that neuron n(1≤ n ≤ N) fired at time t(1 ≤ t ≤ T), i.e.,:G_ki(n,t) =1neuron n fired at time t0neuron n did not fire at time t The firing map 𝒢_k = {G_ki} forms a compact and rich representation of the original signal X_k, and is used to predict the turn-taking type (ŷ_k ∈{0,1}). When given partial observation X_k^τ, its discretized version X̃_k^τ is fed into the SNN group 𝒮, generating a partial response 𝒢_k^τ = 𝒮(X̃_k^τ). This response is used to predict its turn-taking type ỹ_k^τ.§.§.§ Descriptive Feature ConstructionThis training phase focuses on creating descriptive features for the SNN network response G_ki. Although G_ki can be directly used for classification purposes, it would cost unnecessary computational time and memory usage, as G_ki is a large sparse matrix with many zeros. To solve that problem, we proposed the Normalized Histogram of Neuron Firings (NHNF) descriptor to compactly represent G_ki. Specifically, the total number of neurons (i.e. N) are evenly divided into B bins, where bin b(b=0,…,B-1) covers neurons indexes in the range [bN/B,(b+1)N/B]. During a simulation of T ms, the total number of firings for neurons within bin b is counted and then divided by the simulation duration T to generate the descriptor (h_ki)_b for sample X_k and feature i:(h_ki)_b = 1/T∑_t=1^T∑_n=bN/B^(b+1)N/BG_ki(n,t) for i=1,...,M;k=1,...,K;b=0,...,B-1. Dividing the histogram by the simulation duration T can make this descriptor duration-invariant and thus can work robustly for variable signal lengths. An illustration of the NHNF descriptor for a sample firing output is given in Figure <ref>. In the figure, the red dotted lines indicate the boundary of bins, and the green bars represent the histogram values. Since there are M channels of information in total, M sets of histograms (h_ki)_b are generated for a given sample X_k. The histograms (h_ki )_b for each bin (b) and each channel (i) are concatenated together to form the final feature descriptor for input X_k, denoted as H_k. Then H_k is used to predict the turn-event type ŷ_k. When only partial responses 𝒢_k^τ are available, the NHNF descriptors were extracted from it (denoted as H_k^τ) and used to predict the turn-event type ŷ_k^τ. § EXPERIMENTS The proposed CTTM framework was tested in a robotic scrub nurse scenario, where the surgeon's turn-taking intentions must be predicted ahead of time. This section discusses the relevant aspects in the experiment setup, including surgical task setup (Section <ref>), human sensing and signal processing (Section <ref>), the performance (Section <ref>), and finally the network visualization (Section <ref>). §.§ Surgical Task Setup A simulation platform for surgical operations was used to capture turn-taking cues of surgeons, as shown in Figure <ref>. The platform consists of a patient simulator and a set of surgical instruments to conduct a mock abdominal incision and closure task <cit.>. In this collaborative task, the surgeon and nurse collaborate by exchanging surgical instruments. The surgeon performs operations while the nurse searches, prepares, and delivers the requested surgical instrument. Participants were recruited to perform a mock surgical task. Twelve participants were recruited from a large academic institution, with the age range of 20 to 31 years (M=25.7, SD=2.93). After inform consent was given (IRB protocol 1305013664), participants completed a training session. Surgical instruments were introduced by repeated recitation of their names, and a training video of step-by-step instructions of the mock abdominal incision and closure task (10 mins) was given. After the video, the participant attempted a “warm-up” trial. Each participant repeated the surgical task 5 times to reach performance proficiency. Although a surgeon population was not used, the training sessions and repeated trials led to high face-fidelity data. This dataset is realistic enough to validate the early turn-taking prediction capability. Each execution of the surgical task included in average 14 surgical instrument requests. The surgical request actions were annotated as turn-giving events (ℰ^give), and the surgical operation actions were annotated as turn-keeping events (ℰ^keep). Two annotators were presented with the recorded videos of the surgical tasks. The annotators independently determined the starting and ending time for each turn-event as well as its type. The main annotator labeled the entire dataset while the second annotator labeled a random 10% of selected segments. An inter-rater reliability of Cohen's κ=0.95 <cit.> was found, indicating high agreement between two sets of annotations. Overall, 846 turn-giving events (y_k=1) and 1305 turn-keeping events (y_k=0) are generated for the following turn-taking analysis. §.§ Multimodal Human Sensing and Signal Processing The participant in the surgeon role had her communication cues collected during the surgical operation for turn-taking analysis. Three sensors were used to capture the multimodal communication cues: the Myo armband, Epoc headset, and Kinect sensor. Each sensor captured multiple channels of information from the human, as illustrated in Figure <ref>. There were in total M (M=50) channels of raw signals, which weresynchronized at a frequency of 20 Hz. Preprocessing techniques were used to smooth and normalize the raw multimodal signals. Each of the M channels of information was smoothed with Exponentially Weighted Moving Average approach with an empirical weight of 0.2 <cit.>. Then, each of the M channels was normalized to zero mean and unit variance.The feature construction and selection algorithm proposed by <cit.> was adopted. Each channel of the normalized signals was first convolved with a filter bank containing 6 filters, i.e., identity transformation, Sobel operator, Canny edge detector, Laplacian of Gaussian detector, and two Gabor filters. Then the correlation between each encoding with the turn-event labels was calculated through χ^2 test of independence. The m features of the largest test statistic values were retained as the final feature set to form the initial representation X_k. In this experiment, the value of m was set to 10 empirically. The selected top 10 features are shown in TABLE <ref>.§.§ CTTM Performance To evaluate the performance of CTTM in predicting surgeons' turn-taking intentions, computational experiments were conducted. The experiment setup followed the leave-one-subject-out (loso) cross validation. In each fold, the data from 11 subjects was used for training and data from the last subject was separated for testing. Such evaluation scheme can evaluate the algorithm's generalization capability on unseen subjects. For accuracy measurement between prediction resultŷ_k ∈{0,1} and ground truth y_k∈{0,1}, the F_1 score for turn-giving class was calculated (i.e., harmonic mean of precision and recall). The CTTM can recognize the type of the turn-event given only partial observation X_k^τ∈ℝ^(τ L_k) × M. An early decision was then made according to ŷ_k^τ = ϕ(X_k^τ) ∈{0,1}. To evaluate the algorithm's early prediction performance, the F_1(τ) value for τ = 0.1, 0.2, ..., 1.0 was calculated and presented. The NHNF descriptor H_k^τ was extracted from the beginning τ fraction of input (i.e., X_k^τ), with 50 bins for the histogram (i.e., B=50). Then the descriptor H_k^τ was normalized so that each dimension had zero mean and unit variance. The normalized feature was then fed into a SVM classifier, which gave the prediction of turn-taking event type ŷ_k^τ. The hyper-parameters for the SVM was set based on a 5-fold within-group grid search process. Four benchmark algorithms were implemented to compare and evaluate the proposed framework, as explained below.The first benchmark, Dynamic Time Warping (DTW), is a traditional time-series modelling algorithm. It has been successfully applied to speech recognition <cit.> and early gesture recognition <cit.> but never to turn-taking tasks. The multi-dimensional DTW proposed in <cit.> was used with 1-norm distance measurement for two multi-dimensional signals. The k-nearest-neighbor classification scheme was applied (with 20 templates for each class). The DTW distance between the two feature vectors was calculated (i.e., X_u and X_k) and the label of the closer sample was used as the predicted label. The second benchmark (Ishii) is a turn-taking prediction algorithm applied to human conservation <cit.>. Even though this framework targets a different turn-taking application, it can still be adopted into this case with minor modifications, as described below. In Ishii's framework, each signal channel from X_k was normalized into the range of [μ-σ, μ +σ] and then three descriptive features were extracted (i.e., average movement, average amplitude, and average frequency of movement) for each channel of signal. The SVM algorithm was then used for classification, with hyper-parameters selected based on 5-fold grid search. The third benchmark (SNN-PNG) is a SNN-type algorithm <cit.> and is the closet algorithm to the proposed approach. The major difference is that the SNN-PNG framework is a template-based technique and uses PNG as features. In <cit.>, PNG extracted from G_ki were used as features, and the Jaccard similarity and Longest Common Sequence (LCS) algorithm are used to measure the distance between the unknown pattern and the training templates. A nearest-neighbor approach was used for classification purposes, with 20 templates for each class. The fourth benchmark (Human) reflects human performance. We used a “button-press” paradigm <cit.>, where recorded videos of the surgical operation were played back to participants and paused at random times. At every pause, the participant was asked what he thinks that the surgeon wants to do (keep or relinquish the turn). The participants in this experiment used a cross-participant setting for data annotation (no self-annotation). The performances of the proposed CTTM framework with the four benchmarks are shown in Figure <ref>. The CTTM framework greatly outperforms all the computational benchmark methods. Additionally, the CTTM performance surpasses human performance when little action is given (i.e., when τ<0.4). After providing more observations, the human performance is better than CTTM, with an average F1 score margin of 0.05. This prediction behavior can allow inference of human's turn-taking intentions in an early stage, since only a few anchor neurons are required to fire to generate a sequence of signals that traverse through the network, forming a stereotypical response <cit.>. Similar observations have been reported in hand digit recognition tasks <cit.> and gesture recognition <cit.> with SNN. §.§ Visualization of Fired Neurons Visualizing the SNN responses allows better understanding of the patterns learned by the model. Figure <ref> shows 6 neurons firing maps (i.e., G_ki) for each class of input. The SNN model for the first feature was selected here for visualization. The responses to turn-keeping inputs (ℰ^keep) are on the top two rows, and the responses to the turn-giving inputs (ℰ^give) are at the bottom two rows. Visual inspection reveals that the SNN responds differently to ℰ^keep and ℰ^give inputs. The ℰ^give inputs in general can fire more neurons in the trained SNN network compared to ℰ^keep inputs. This could mean that humans exhibit a coherent pattern when relinquishing their turn. The neurons in the CTTM framework fire in the presence of such pattern. On the other hand, the intention to keep the current turn (i.e., focusing on operation) can be diverse (since the operations can be very diverse) and cannot trigger enough firing. Additionally, responses in ℰ^give generally have a column-wise structure (either one column or two columns). This structure is generated when a group of neurons fire together in a time-locked pattern, forming a PNG as a signature of early turn-taking intent.§ CONCLUSIONS In human robot interaction, turn-taking capability is critical to enable robots to interact with humans seamlessly, naturally, and efficiently. However, current turn-taking algorithms cannot help accomplish early prediction. To bridge this gap, this paper proposes the CTTM, which leverages cognitive models for early turn-taking prediction. More specifically, this model is capable of reasoning human turn-taking intentions, based on the neurons firing patterns in the SNN. The CTTM framework can reason about the multimodal human communication cues (both implicit and explicit) and predict a person's intention of keeping or relinquishing the turn in an early stage. Such prediction can then be used to control robot actions. The proposed CTTM framework was implemented in a surgical context, where a robotic scrub nurse predicted the surgeon's turn-taking intentions to determine when to deliver surgical instruments. The algorithm's turn-taking prediction performance is evaluated based on a dataset, acquired through a simulated surgical procedure. The proposed CTTM framework outperformed computational state-of-the-art algorithms and can surpass human performance when only partial observation is available (i.e., less than 40% of full action). Specifically, the proposed framework achieves a F_1 score of 0.68 when only 10% of full action is presented and a F_1 score of 0.87 at 50% presentation. Such early prediction capability is partially due to the implemented cognitive models (i.e., SNN) for early prediction. Such behavior would enable co-robots to work in a hybrid environment side by side with humans. Future work includes 1) including more contextual information to improve early prediction capability (e.g., current task progress); 2) validating the framework in real surgeries; 3) transfer the CTTM framework to other scenarios, such as robot-assisted manufacturing.ieeetr | http://arxiv.org/abs/1709.09276v1 | {
"authors": [
"Tian Zhou",
"Juan P. Wachs"
],
"categories": [
"cs.RO"
],
"primary_category": "cs.RO",
"published": "20170926223150",
"title": "Early Turn-taking Prediction with Spiking Neural Networks for Human Robot Collaboration"
} |
What you lose when you snooze: how duty cycling impacts on the contact process in opportunistic networks Elisabetta Biondi, Chiara Boldrini,Andrea Passarella, Marco Conti[All authors are with IIT-CNR, Via G. Moruzzi 1, Pisa, Italy]==================================================================================================================================== In opportunistic networks, putting devices in energy saving mode is crucial to preserve their battery, and hence to increase the lifetime of the network and foster user participation. A popular strategy for energy saving is duty cycling. However, when in energy saving mode, users cannot communicate with each other. The side effects of duty cycling are twofold. On the one hand, duty cycling may reduce the number of usable contacts for delivering messages, increasing intercontact times and delays. On the other hand, duty cycling may break long contacts into smaller contacts, thus also reducing the capacity of the opportunistic network. Despite the potential serious effects, the role played by duty cycling in opportunistic networks has been often neglected in the literature. In order to fill this gap, in this paper we propose a general model for deriving the pairwise contact andintercontact times measured when a duty cycling policy is superimposed on the original encounter process determined only by node mobility. The model we propose is general, i.e., not bound to a specific distribution of contact and intercontact times, and very accurate, as we show exploiting two traces of real human mobility for validation. Using this model, we derive several interesting results about the properties of measured contact and intercontact times with duty cycling: their distribution, how their coefficient of variation changes depending on the duty cycle value, how the duty cycling affects the capacity and delay of an opportunistic network. The applicability of these results is broad, ranging from performance models for opportunistic networks that factor in the duty cycling effect, to the optimisation of the duty cycle to meet a certain target performance. § INTRODUCTION The widespread availability of smart, handheld devices like smartphones and tablets has stimulated the discussion and research about the possibility of new concepts for supporting communications between users. Particularly appealing, towards this direction, is the opportunistic networking paradigm, in which messages arrive to their final destination through consecutive pairwise exchanges between users that are in radio range of each other <cit.>. As such, unlike MANETs, opportunistic networks do not assume a continuous end-to-end path between source and destination, and paths are built dynamically and incrementally by intermediate nodes when new contacts (i.e., new forwarding opportunities) arise. While originally studied as a standalone solution, opportunistic networks are now being exploited in synergy with the cellular infrastructure in mobile data offloading scenarios <cit.>, as an enabling technology for the Internet of Things <cit.>, and they are being enhanced to also exploit a cloud infrastructure when available <cit.>. User mobility, and especially user encounters, is the key enabler of opportunistic communications. Unfortunately, ad hoc communications tend to be very energy hungry <cit.> and no user will be willing to participate in an opportunistic network if they risk to see their battery drained in a few hours. However, there are very few contributions that study how power saving mechanisms impact on the contacts that can be exploited to relay messages. These power saving mechanisms range from completely turning off devices periodically or, more commonly, to tuning the frequency at which the network interface is used (e.g., reducing neighbour discovery activities). We generically refer to all these strategies as duty cycling. With duty cycling, messages can be exchanged only when two nodes are in one-hop radio range and they are both in the active state of the duty cycle. So, power saving may reduce forwarding opportunities, because contacts are missed when at least one of the devices is in a low-energy state. Since some contacts may be missed, the measured intercontact times, defined as the time interval between two consecutive detected encounters between the same pair of nodes, is, in general, larger than the original intercontacts (i.e., those defined exclusively by the nodes mobility process) and this may clearly affect the delay experienced by messages. The measured contacts (i.e., the length of a contact while the two nodes are in radio range and active) may also be affected, if one of the two nodes becomes inactive during a contact.Owing to the extent at which, in principle, measured contacts and intercontact times may affect the performance of opportunistic networks, we argue that it is essential to better understand how they are characterised and how they depend on the duty cycling policy in use. Unfortunately, the effects of duty cycling on the measured pairwise contact process have been largely ignored in the literature.The goal of this work is to characterise the distribution of the measured contact and intercontact times, starting from a given distribution of original contact and intercontact times, and a duty cycling scheme. To this aim, the contribution of this paper is threefold. First, in Section <ref>, assuming that contact duration is negligible, we derive a mathematical model of the measured intercontact times between nodes.For general distributions of the original intercontact times we derive mathematical expressions that can be solved numerically to obtain the first two moments of the measured intercontact times.We can thus approximate any distribution of the measured intercontact times using hyper- or hypo- exponential approximations <cit.>. Under the two most popular intercontact time distributions considered in the literature, exponential and Pareto, the closed forms of the first two moments admit analytic solutions, making the model even more flexible.As a second contribution, in Section <ref> we extend the above model to include the effects of non-negligible contact duration, again under any distribution of contact and intercontact times, thus making the model as general as possible. We extensively validate this model using as input the distribution of contact and intercontact times obtained from traces of real user mobility. Finally, in Section <ref> we show that the results obtained assuming a deterministic duty cycling (as described in Section <ref>), actually provide a very good characterisation also when stochastic duty cycling is used.Focusing on a tagged node pair, the key findings presented in this paper are the following: * The measured contact time C̃ cannot be longer than the duration of the active state of the duty cycle, hence the data transfer capacity of the opportunistic network is generally reduced, even significantly. However, if the duty cycling policy is such that nodes refrain from entering the low-power state when they detect a contact, the measured contact duration (hence the capacity) may be only minimally affected by the duty cycle.* When contact duration C is negligible, if the original intercontact times S are exponential with rate λ, the measured intercontact times S̃ are exponential with rate λΔ, where Δ is the percentage of time nodes keep the wireless interface active (duty cycling parameter). Instead, if the original intercontact times S are Pareto with exponent α, the measured intercontact times S̃ do not feature a well-known distribution but they decay as a Pareto random variable with the same exponent α. This implies that all the properties (e.g., the delay convergence <cit.>) that depend on the shape of the tail of the Pareto distribution of intercontact times are not affected by duty cycling. * The duty cycle can affect measured intercontact times in such a way that low-variability (i.e., with coefficient of variation smaller than one) original intercontact times can turn into highly variable measured intercontact times (and vice versa), thus potentially altering the convergence of the expected delay. * A stochastic duty cycling can be approximated with a deterministic duty cycling for which the length of the active and inactive intervals corresponds to the average length of the same intervals in the stochastic duty cycling. This means that our results about C̃ and S̃ hold for a very large class of duty cycling policies. To the best of our knowledge, as discussed in Section <ref>, this work represents the first comprehensive analysis of how the measured contact process is altered by power saving techniques.§ PRELIMINARIES In this section we introduce the duty cycling process that we take as reference and we describe how the contacts between users can be modelled. §.§ The duty cycling processWe use duty cycling in a general sense, meaning any power saving mechanism that hinders the possibility of a continuous scan of the devices in the neighbourhood. We assume that nodes alternate between the ON and OFF states. In the ON state, nodes are able to detect contacts with other devices. In the OFF state (which may correspond to a low-power state or simply to a state in which devices are switched off) contacts with other devices are missed. Using this generalisation, we are able to abstract from the specific wireless technology used for pairwise communications.Duty cycling policies can be deterministic or stochastic, depending on how the length of their ON and OFF states is chosen (fixed, in the former case, varying according to some known probability distribution in the latter). In the literature also non-stationary duty cycling policies can be found, in which the length of ON and OFF states depends on some properties of the network (or a node's neighborhood) at time t. All these approaches are discussed in Section <ref>. We base our model on the deterministic duty cycling case (which requires a coarse synchronisation between devices), then we later prove in Section <ref> how this model captures the average behaviour of the stochastic case (which does not require synchronisation) as well. Finally, we also discuss how the model captures some notable cases of non-stationary duty cycling policies.In the following, we assume that the duty cycle process and the real contact process are independent and, considering a tagged node pair, we denote with τ the length of the time interval in which both nodes are ON, and with T the period of the duty cycle. Thus, T-τ corresponds to the duration of the OFF interval and Δ=τ/T is the duty cycle parameter.In general, the ON interval can start anywhere within T but here, without loss of generality, we assume it starts at the beginning of interval T. In addition, considering a generic detected contact, we count duty cycle periods from the first one where the contact is detected.Hence, ON intervals will be of type [i T, τ + i T), with i≥ 0 and OFF intervals of type [τ + iT, (i+1)T), with i ≥ 0 (Figure <ref>). In the following, ON and OFF intervals will be denoted with ℐ^ON and ℐ^OFF, respectively. Hence, the set of all ON (OFF) intervals is given by ⋃_n=0^∞ℐ^ON_n (⋃_n=0^∞ℐ^OFF_n).Focusing on a tagged node pair, we can represent how the duty cycle function evolves with time asd(t) = {[ 1 iftT ∈ [0, τ); 0 otherwise ].. When d(t)=1, both nodes are ON, thus their contacts, if any, are detected.Reference values for τ and T depend also on practical aspects. A result derived in <cit.> shows that, for frequencies of switching between ON and OFF states beyond 1/100 s^-1, energy consumption on smartphones drastically increases, making the discovery process not energy efficient. Thus, for the purpose of this paper, we will consider values of T around 100s and will experiment with different values of τ when evaluating the proposed models. This is also the reason why in the paper we do not consider duty cycling schemes switching the wireless interfaces at a much finer granularity, in the order of milliseconds or less.§.§ The contact processSimilarly to the related literature <cit.>, we assume that, from the mobility standpoint, node pairs are independent. From the modelling standpoint, the contact process of each node pair (u,w) can be approximated as an alternating renewal process <cit.>. In this case, the node pair alternates between the CONTACT state in which the two nodes are in radio range, and a state in which they are not (Figure <ref>). The time interval between the beginning and the end of the i-th contact is denoted as C_i^(u,w). The time interval between the end of a contact and the beginning of the next one corresponds to the intercontact time and it is denoted as S_i^(u,w). Hence, the alternating renewal process corresponds to the independent sequence of random variables { C_i^(u,w), S_i^(u,w)}, with i ≥ 1, which is an approximation of the real contact process as C_i^(u,w) and S_i^(u,w) can be dependent for a fixed i but must be independent for different i. Note that assuming independence of consecutive contact and intercontact times is also customary in the literature. Since in the following we focus on a tagged node pair,for the sake of clarity we hereafter drop superscript (u,w) from our notation. Please note, however, that the contact process we consider is heterogeneous, i.e., the distribution of C_i and S_i can be different for different pairs of nodes. Exploiting this notation, we have that the time X_i at which the i-th contact begins is ∑_j=1^i S_j + ∑_j=0^i-1 C_j, ∀ i ≥ 1, while the time Y_i at which it ends is given by ∑_j=1^i S_j + ∑_j=0^i C_j, ∀ i ≥ 1.§ MEASURED INTERCONTACT TIMES WHEN CONTACT DURATION IS NEGLIGIBLEWe now discuss how the measured contact process depends on the contact process described in Section <ref>. We start analysing the case where the contact durations are negligible with respect to the duty cycling period. In real traces, this is often a reasonable approximation, given that a significant part of observed distributions is concentrated on contact durations of a few seconds. Under this assumption, the alternating renewal process of Figure <ref> becomes a simple renewal process <cit.> where S_i are the renewal intervals. We later relax this assumption in Section <ref>.Recall that the effect of duty cycling on contacts is that some contacts between nodes may be lost. So, we first study in Section <ref> what are the characteristics of the process of measured contacts and how it can be modelled. The main outcome of this section is that the measured intercontact time can be obtained as the sum of N (with N stochastic) real intercontact times, assuming that the PDF of S has certain properties that can simplify our derivations. Building upon these results, in Section <ref> we compute the PMF of N and then in Section <ref> we finally derive the measured intercontact times S̃_i. §.§ Problem settingLet us denote with S̃_j the time between the (j-1)-th and the j-th detected contact (corresponding to the j-th measured intercontact time) and assume that at time X̃_j-1 a contact has been detected, as shown in Figure <ref>. For convenience of notation, in Figure <ref> and in the following, the sequence number of the duty cycling interval in which the i-th contact takes place is denoted with n_i, while the sequence number of the ON interval in which the j-th detected contact takes places is denoted with ñ_j. Then, the measured intercontact time S̃_j is the time from X̃_j-1 until the next detected contact at X̃_j, which can be obtained by adding up the intercontact times S_i between X̃_j-1 and the next detected contact. Denoting with N_j the random variable measuring the number of intercontact times between the (j-1)-th and the j-th detected contact (e.g. in Figure <ref>, N_j = 4), we obtain S̃_j = ∑_i=1^N_j S_i, hence (S̃_j = x) = ∑_k=1^∞(∑_i=1^k S_i = x)(N_j=k). In general, S̃_j are not i.i.d., because N_j are in general not i.i.d. However, to keep the analysis tractable, we derive a condition under which N_j can be assumed i.i.d., and thus S̃_j also become i.i.d. This makes also the detected intercontact time process a renewal one. Intuitively (see <cit.> for a formal proof[Content in <cit.> not included in this paper is provided as supplemental material.]) N_j are i.i.d. when the probability density of S does not vary much inside an ON or OFF interval, and thus the exact time within an interval when a contact happens can be approximated as uniformly distributed in that interval, regardless of when the previous contacts took place. When this happens, in fact, since intercontact times S_i are assumed i.i.d., the process forgets about its past and regenerates itself.When f_S_i varies slowly[In the context of this paper, a function f(x) varies slowly in a given interval if, for any x_1,x_2 belonging to that interval, f_S(x_1)/f_S(x_2)∼ 1. We are not implying that the function is slowly varying in the sense of <cit.>.] in intervals of length τ, {N_j }_j ≥ 1 can be modelled as i.i.d. (hence, N_j ∼ N), and the displacement Z^ON of a detected contact within an ON interval is approximately distributed as Unif(0,τ). Lemma <ref> above applies to detected contacts. For modelling purposes, it is also convenient to approximate the displacement of missed contacts as uniformly distributed in the OFF interval in which they are missed. For this reason, we introduce the following lemma, which can be proved using similar arguments as those used in the proof of Lemma <ref>. When f_S_i varies slowly in intervals of length T-τ, it holds that the displacement of a missed contact within its OFF interval is distributed as Z^OFF∼ Unif(0,T-τ). When Lemma <ref> holds true, the measured contact process is, at least approximately, a renewal process. Thus, we can express S̃ as a random sum of i.i.d. random variables, , i.e. S̃ = ∑ S_i. Random sums have some useful properties that we will exploit in Section <ref> in order to derive the first two moments of S̃. Please note that this formula is general, i.e., holds for any type of continuous intercontact time distribution and for any type of duty cycling policy. To complete the analysis of S̃, in Section <ref> we compute the distribution of N, and in Section <ref> we find the moments of S̃.Before continuing, note that, as described in more detail in <cit.>, we can derive simple conditions under which Lemma <ref> and Lemma <ref> hold, for two popular cases of intercontact times distributions, i.e. exponential and Pareto. Specifically, sufficient conditions are λ T ≪1 where λ is the rate of exponential intercontact times, and T ≪ b where b is the scale of Pareto intercontact times. Using these two conditions λ T ≪ 1 and T ≪ b, in Section <ref>, we will see how close theoretical predictions for measured intercontact times are to simulation results depending on whether these conditions are satisfied or not.§.§ Deriving the distribution of N In this section, we derive the probability distribution of N, defined as the number of contacts needed, after a detected contact, in order to detect the next one. Since we are assuming the detected contact process to be renewal, we can focus on the portion of this process between two detected contacts. Specifically, we can focus, without loss of generality, on what happens between the first and second detected contact. Then, the rationale behind the derivation of N is pretty intuitive. In fact, N=1 corresponds to the case where the first intercontact time after a detection ends in an ON interval. For case N=2, the first intercontact ends in an OFF interval, while the following one ends in a following ON interval.All other cases follow using the same line of reasoning. Please recall that, in the following, ON and OFF intervals will be denoted with ℐ^ON and ℐ^OFF, respectively. The derivation of the PMF of N in Theorem <ref> below quantifies the probability P{N=k}. The line of reasoning for deriving this result is as follows. Let us define random variable E_k, which is equal to one when the k-th contact is in an ON interval, equal to zero otherwise. It is easy to see that the following holds true: ℙ(N=k) =ℙ(E_1 =0, ..., E_k-1=0, E_k = 1). Similarly to the argument developed in Section <ref>, this is because the probability that an intercontact falls in an ON or OFF interval depends on the point in time when the previous contact finishes. In <cit.> we considered these dependencies and derived closed form solutions for exponential intercontact times. Hereafter we provide a solution for general intercontact time distributions under the conditions of Lemma <ref> and Lemma <ref>. Intuitively, when S is slowly varying in intervals of length τ and T-τ, the probability of E_k=0 depends, for any k, only on the length of intercontact times, starting from a point in time that is uniformly distributed in an ON or OFF interval, while all dependencies on previous events can be neglected. The joint probability in Equation <ref> becomes the product of the marginal probabilities, and the marginal probabilities admit simple expressions. So we obtain Theorem <ref>.When the PDF of intercontact times S_i is slowly varying in any interval of length max{τ, T-τ},the probability mass function of N can be approximated as follows: {[ℙ{N=1} = g; ℙ{N=k} = (1-g)(1-p)^k-2p,k≥ 2 ]. whereg=∑_n_1=0^∞ℙ(Z^ON + S ∈ℐ^ON_n_1), p=∑_n_2=1^∞ℙ(Z^OFF + S ∈ℐ^ON_n_2), and Z^ON∼ Unif(0, τ), Z^OFF∼ Unif(0, T-τ) as shown in Lemma <ref> and Lemma <ref>. Let us provide an intuitive explanation for this result (the detailed proof can be found in <cit.>). Focusing on the first event (E_1), it is easy to see that ℙ{E_1=1}=ℙ{N=1}. We synthetically denote this quantity as g. The value of g can be computed after noticing that it corresponds to the probability of detecting a contact knowing that the previous contact had been detected (i.e., that it happened in an ON interval). Please note that, thanks to the slowly varying assumption, we can assume that the previous contact took place Z^ON seconds after the beginning on the ON interval in which it was detected.For ℙ{N=k}, the first k-1 contacts are missed. The first one starts during an ON interval, and therefore the probability that it is missed is 1-g. The next k-2 start during an OFF interval. Conceptually similar to g, p denotes the probability of detecting a contact when the previous one starts during an OFF (instead of an ON) period. Thus, the probability of the k-2 events is (1-p)^k-2. Finally, we detect the k-th contact, which happens with probability p. Starting from the distribution of N in Theorem <ref>, we can easily derive N's first two moments and its coefficient of variation, which will be later used to compute the moments of S̃. The first two moments and the coefficient of variation of the approximate PMF of N are 𝔼[N] = 1-g+p/p, 𝔼[N^2] = -g (p+2)+p^2+p+2/p^2, cv_N^2 = (1-g) (g-p+1)/(-g+p+1)^2. In Section <ref> we will see how to derive g and p for two popular distributions of intercontact times (exponential and Pareto). This basically comes down to deriving quantities Z^ON + S and Z^OFF + S, which are the sum of a uniform random variable and S, and then to compute, either symbolically or numerically, the infinite sums. However, the expressions for g and p can be further simplified, as shown in Corollary <ref> below (whose proof can be found in <cit.>), again relying on the slowly varying property of f_S.When f_S is a constant in intervals of length max{τ, T-τ}, it holds that g and p are both approximately equal to τ/T, hence the distribution of N becomes approximately Geometric with parameter τ/T and its moments and coefficient of variation become 𝔼[N] = T/τ, 𝔼[N^2] = T (2 T-τ )/τ^2, cv_N^2 = T-τ/T. Note that there is a subtle difference between Corollary <ref> and Theorem <ref>. The former needs that f_S is constantin intervals of length max{τ,T-τ}, while the latter only requires that f_S is slowly varying in these intervals. Therefore, in principle results from Theorem <ref> are less approximate than those from Corollary <ref>, although in practice the two conditions on f_S are often equivalent.In the related literature <cit.>, the probability to skip a contact is often used to control and optimise the duty cycle, by making sure that the number of missed contact remains below a given threshold. This probability can be also computed with our model, and it simply corresponds to ℙ(N≥2) = 1- g.§.§.§ Validation of ℙ{N=k}In this section we validate our approximation for the PMF of N against simulation results. We set T to 100s, which, as discussed in Section <ref>, was found to be a good trade-off between device discoverability and energy consumption in <cit.>. Then, we explore the parameter space assuming that intercontact times are either exponential or Pareto distributed. Monte-Carlo simulations are performed drawing intercontact time samples from these distributions and filtering them according to the reference duty cycling process (considering two cases, τ=20 and τ=80, corresponding to 20% and 80% duty cycling). The number of contacts filtered out between two detected contacts gives us one sample of N. We collect 100,000 samples for N. When all samples for N are collected, we use them to compute the empirical PMF, which is compared against the analytical predictions in Theorem <ref> and Corollary <ref>.We first consider the case of exponential intercontact times and we explore four values for the rate λ: 10, 0.1, 0.01, and 0.001 s^-1. When real intercontact times are exponential, g and p in Theorem <ref> can be derived in closed-form: g= e^-λτ+λτ +e^-λτ(e^λτ-1)^2/e^λT-1-1/λτp= -e^λτ-e^λ(T-τ )+e^λT+1/λ(e^λT-1) (T-τ ) . Because of the sufficient condition λ T ≪ 1, we expect the model to provide very good approximations when λ = 0.001s^-1 (λ T = 0.1 ≪ 1), reasonable approximationswhen λ = 0.01s^-1 (λ T = 1), and discrepancieswhen λ≥ 0.1s^-1 (because λ T >1). This is exactly what we observe in Figures <ref>-<ref>, except that the model is pretty close to simulation results also when λ=0.1s^-1 and it predicts even better for λ=10 s^-1. To understand why, we need to observe closely what happens when λ T ≫ 1. Specifically, in this case many contacts happen within the same interval (either ON or OFF). This means that there are long sequences of contacts detected one after the other one, but, as soon as the first contact falls in an OFF interval, there will also be many missed contacts. This results in a distribution of N that is almost exclusively concentrated on N=1, with a small peak in the tail (e.g., for λ=10s^-1 the tail of N accumulates at 800s, which is exactly the average number of missed contacts when the OFF interval has length 80s). Theorem <ref> is able to predict the accumulation at N=1 (see Figure <ref> for λ=10s^-1 and 0.1s^-1). The inaccuracy in predicting the peak in the tail is not very critical, as anyway the total mass of probability under these peaks is typically very limited (we will show in Section <ref> that it has no practical effect in predicting S̃). As far as the Geometric approximation in Corollary <ref> is concerned, we observe that it works very well whenλ T≪ 1, but it degrades quickly when this assumption does not hold, especially for smaller values of N (Figures <ref>-<ref>). We have already anticipated the reason for this behaviour. In fact, Theorem <ref> exploits the uniformity of contact displacement in the intervals in which they take place but then uses the actual functional form of f_S. For this reason, when the slowly varying assumption does not hold, predictions are generally more accurate with Theorem <ref> than with Corollary <ref>. Finally, as shown in <cit.>, where we fully develop the case of exponential intercontact times, the better performance of the Geometric approximation for the tail is an artefact of the fact that, in case of exponential intercontact times, the tail decays as (T-τ/T)^k-1. However, this is paid with a very significant inaccuracy in predicting the body of the distribution (and specifically N=1), which is way more important given the accumulation of probability mass around that value. For the Pareto case, the sufficient condition for f_S to be slowly varying in any interval of length T is T/𝔼[S](α-1)≪ 1 or equivalently T/b≪ 1. Again, we take T=100s and two different values for τ (20s and 80s). Since the Pareto distribution is characterised by two parameters,here we have an additional degree of freedom. So, for the purpose of validation, we have selected α = 1.01 for the Pareto, which is right after the threshold α=1 for the convergence of the expectation <cit.>. With α fixed, we have selected b values in order to extensively explore the parameter space. Thus, we have chosen b ∈{1, 10, 100, 1000}. Simulations are performed as discussed for the exponential case. Also in the Pareto case g and p in Theorem <ref> can be derived in closed form: g = T^-α(T b^α(ζ(α -1,b+T-τ/T)+ζ(α -1,b+T+τ/T))-2 T^α +1(b/T)^αζ(α -1,b+T/T)+T^α((α -1) τ +(b+τ ) (b/b+τ)^α-b))/(α -1) τp =-T (b/T)^α(-ζ(α -1,b+T+τ/T)+ζ(α -1,b+T/T)+ζ(α -1,b+T-(T-τ)+τ/T)-ζ(α -1,b+T-(T-τ)/T))/(α -1) (T-τ), where ζ(·, ·) denotes the Hurwitz zeta function.Given the condition T/b≪ 1, we expect discrepancies between the model and simulation results for the first two values of b, a reasonable approximation for b=100, and an accurate prediction for b=1000.From Figure <ref> we observe that predictions are actually very good also in those cases in which they were expected to be less accurate. This happens for the same reason as for the exponential case. Also similarly to the exponential case, the predictions of the Geometric approximation in Corollary <ref> become quickly worse as b shrinks. Summarising, for both exponential and Pareto intercontact times, the predictions of Theorem <ref> are generally very accurate, in particular for the body of the PMF of N. As for the tail of the distribution, it is accurately captured only when the slowly varying conditions are satisfied. In Section <ref> we will see whether this inaccurate prediction affects the results for the measured intercontact times S̃. Finally, the Geometric approximation, despite being very convenient for its simplicity, is safe to use only when the slowly varying conditions are satisfied. When conditions are not satisfied, the Geometric approximation is not able to capture the behaviour of the body of the distribution and, more in general, the variability of N. §.§ Measured intercontact timesExploiting the results in the previous section, here we discuss how to compute the first and second moment of the measured intercontact time S̃ for a generic node pair. We know that the relation between S and S̃ is S̃ = ∑_i=1^N S_i. Thus, S̃ is a random sum of random variables, and we can exploit well-known properties to compute its first and second moment. With the results in Proposition <ref> below we can exploit commonly used representations of positive random variables as hyper/hypo-exponential random variables. The rationale of this approach is to use the moments of S̃ to fit a hyper- or a hypo-exponential distribution (depending on whether cv_S̃^2 is greater or smaller than 1, respectively) and then use this representation of S̃ to derive important metrics (e.g., the delay and number of hops) characterising the performance of networking protocols in the opportunistic network. The first and second moment of S̃ are given by the following: 𝔼[S̃]= 𝔼[N] E[S]𝔼[S̃^2]= 𝔼[N^2]𝔼[S]^2 + 𝔼[N]𝔼[S^2]-𝔼[N]𝔼[S]^2 cv_S̃^2= cv_S^2/𝔼[N] + cv_N^2.In the following we will use these formulas to get some general insights on the possible behaviour of S̃ depending on the distribution of S and the duty cycle Δ. By “behaviour" we mean whether the distribution of S̃ exhibits a hypo-exponential, exponential or hyper-exponential nature. When real intermeeting times are exponential and f_S is slowly varying according to Lemma <ref> and Lemma <ref>, we found that measured intercontact times feature a coefficient of variation that is approximately equal to 1, from which it follows that S̃ can be approximated with an exponential distribution. In Lemma <ref> below we summarise this result and we also provide an expression for the rate of S̃ in this case. When intercontact times S_i are exponential with rate λ and condition λ T ≪ 1 holds, the measured intercontact times are again exponential but with rate λΔ.When λ T ≪ 1 we can apply Corollary <ref>. Thus, recalling that 𝔼[S] = 1/λ and cv_S^2 =1, and substituting them into the equations in Proposition <ref>, we obtain 𝔼[S̃] = 1/λΔ and cv_S̃^2 = 1, which corresponds to an exponentially distributed random variable with rate λΔ. Please note we had already obtained this result in <cit.> using a more complex model and derivation. It has been proved in the literature <cit.> that the expected delay of routing protocols may not converge in opportunistic networks with Pareto intercontact times depending on the value of their shape parameter α, which determines the behaviour of the tail. Lemma <ref> below tells us that if there are convergence issues with S_i, the same problems will show up also with the measured intercontact times, since they decay as a Pareto with the same exponent.When intercontact times are Pareto with exponent α, the CCDF of S̃ decays as a Pareto random variable with exponent α.Since S̃ = ∑_i=1^N S_i, we can write the following: ℙ(S̃ > x) = ∑_k=1^∞ℙ{N=k}ℙ(S_1 + ⋯ + S_k > x). It has been shown <cit.> that, when S_i are i.i.d. Pareto random variables, ℙ(S_1 + ⋯ + S_k > x) goes as L(x) k (b/x)^α when x →∞, where L(x) is a slowly varying (in the sense of <cit.>) function of x. Using this relationship, we can rewrite the above equality as: ℙ(S̃ > x)∼L(x) (b/x)^α∑_k=1^∞ k ℙ{N=k}∼L(x) (b/x)^α𝔼[N] ∼ (b/x)^α Now, in Lemma <ref> (whose proof can be found in <cit.>) we want to answer a more general question: can duty cycling transform a hypo-exponential intercontact time into a hyper-exponential measured intercontact time, and vice versa? The answer is yes, and whether this happens or not depends on the values of cv_S^2, g and p. This result has a serious implication for opportunistic networks: if exponential intercontact times can turn into hyper-exponential measured intercontact times, delay convergence issues may show up also in apparently safe exponential mobility scenarios if a duty cycling policy is deployed. When f_S is slowly varying in intervals of length max{τ, T-τ}, the measured intercontact times behave according to Table <ref>, where ξ(g,p) = 1 - 2g/p + 2/1-g+p and ω(g) = 3/2(g-1)+1/2√(9-10g+g^2). Despite their apparent complexity, the conditions in Table <ref> have an intuitive explanation. Take for example the important case where hypo-exponential intercontact times can turn into hyper-exponential measured intercontact times, corresponding to the bottom right entry of Table <ref>. As worked out in <cit.>, condition p < ω(g) corresponds to the case cv_N^2 > 1, which, according to Equation <ref> is enough for the coefficient of variation of S̃ to be greater than 1. In practice, what happens with cv_N^2 > 1 is that both very small and very large values of N are possible, hence S̃ will be a combination of the sums of a small number of intercontact times S and the sums of a large number of intercontact times S, thus introducing a lot of variability in the distribution of S̃. Vice versa, when condition p < ω(g) does not hold, cv_S^2 starts to weigh in in Equation <ref> and can tilt the balance in favour of hyper-exponentiality depending on how it relates to ξ(g,p) (cfr second condition in the bottom right entry of Table <ref>).Finally, it is interesting to note from Table <ref>, that when g=p (which we know happens under the conditions of Corollary <ref>) S̃ inherits the behaviour of S: if S is hyper-exponential then S̃ is hyper-exponential, and vice versa. In Section <ref> we use this result to assess the behaviour of S̃. Hence, with Corollary <ref> we would never observe a change in the behaviour of S̃.§.§.§ Validation of S̃In order to validate the model, we compare simulation results against theoretical predictions for S̃. Specifically, we generate the measured intercontact times via Monte Carlo simulation, similarly to what we have done for the validation of N. The theoretical predictions are obtained as discussed in the previous section.We start our validation with the exponential case and in Figure <ref> we plot the distribution of S̃ for the same parameter values (λ∈{ 0.001s^-1, 0.01s^-1, 0.1s^-1, 10s^-1}) used in the validation of the PMF of N. We plot in orange the distribution predicted using Theorem <ref> and in green the distribution of an Exponential random variable with rate λΔ (corresponding to the prediction of Lemma <ref>, which in turn relies on Corollary <ref>). Since the PMF of N was predicted with very good accuracy by our model in Theorem <ref>, we also expect that the distribution of S̃ matches its theoretical prediction. This is indeed the case, both for λ values such that λ T ≪ 1 but also in the opposite case, when λ T ≥ 1. This is due to the fact that when λ is large (with respect to τ) ℙ(N = 1) becomes large, and this effect is captured by Theorem <ref> (as we have discussed in Section <ref>). Vice versa, when λ is large the predictions based on the Geometric approximation in Corollary <ref> fail to capture the high probability of event N=1. In Figure <ref> we also plot the tail of the CCDF of S̃. We observe that both models accurately predict the tail when λ T ≪ 1 holds. When the condition does not hold, we need to apply Lemma <ref>. Specifically, computing p, g and ω(g), Lemma <ref> predicts that S̃ is hyper-exponential, while Lemma <ref> would predict a squared coefficient of variation equal to 1. Indeed, the squared coefficient of variation measured in simulation is 142.We complete the validation of the results for S̃ by considering Pareto intercontact times (with α=1.01, as in Section <ref>). When the slowly varying condition holds (b=100, b=1000), we observe a perfect match between simulation results and predictions (with both proposed models, as can be seen in Figure <ref>). When the hypotheses of the model are not met (small b), the model is nevertheless able to well approximate simulation results, though with a slight mismatch in the initial part of the CDF, where the proposed model underestimates the presence of small values of S̃. This is due to the long tail of N under the model in Theorem <ref>, which we had observed in Figure <ref>. Owing to this fact, the model overestimate the presence of large values of N, and this has an impact on the predictions for S̃. However, this effect is quite limited. The above considerations also apply to the tail of S̃ in Figure <ref>. Here we have also plotted a curve for (b/x)^α, in order confirm that S̃decays as predicted by Lemma <ref>.§ DETECTED CONTACT PROCESS WHEN CONTACT DURATION IS NON-NEGLIGIBLE While this is typically a reasonable approximation, contact durations might in general not be negligible with respect to duty cycle periods. Therefore, in Section <ref> we revise the slowly varying conditions for the case of non-negligible contact duration (this boils down to extending the slowly varying assumption to the distribution of the contact duration, C_i). Then in Section <ref> we compute the distribution of the measured contact duration C̃. This measured contact duration cannot, by definition, exceed the length of an ON interval, and its distribution depends on how the real contact duration intersects with the ON interval. Building upon these results, in Section <ref> we derive the measured intercontact times S̃_̃ĩ. The main differences with respect to when contact duration is negligible is that the portions of contact duration not overlapping with ON intervals have to be included into the measured intercontact time. For both C̃ and S̃we also provide a model for the case when nodes refrain from entering the OFF state upon contact detection, as in <cit.>. In this case, the effect of the duty cycle on the measured contact duration is generally limited, since only the initial part of the original contact is missed. As for the measured intercontact times, we do not observe anymore the contribution of the portion of contact duration not overlapping with ON intervals. Finally, in Section <ref> we validate this extended model, showing that its predictions are once again very accurate. §.§ Preliminaries In order to make the model tractable, similarly to what we have done in Section <ref>, we need to model the measured contact process as a renewal process, this time of type alternate-renewal. The measured contact process {C̃_i, S̃_i} is alternate renewal if {C̃_i, S̃_i} are independent sequences of i.i.d. random variables. Based on the discussion for the negligible contact case, it is clear that also in this case the measured contact process is not in general renewal.However, using an argument similar to the one used in Lemma <ref> it is possible to prove (Lemma <ref>) that also in this case we can assume, approximately, the independence for {C̃_i, S̃_i}, provided that the PDF of S_i is slowly varying in intervals of length max{T-τ, τ}.From this, it follows that the displacement of the beginning of a real contact in the ON/OFF interval in which it takes place is uniformly distributed in that interval. We omit the proof of this result as it would be a repetition of the same concepts discussed in Lemma <ref>. When f_S is a slowly varying function in any interval of length max{T-τ, τ}, the measured contact process can be approximated as an alternating renewal process {C̃_i, S̃_i} and the displacement of the beginning of a contact within its ON (or OFF) interval can be approximated as uniformly distributed in that interval. For our derivations, it may also be convenient to assume that not only the beginning of a real contact but also its end is approximately uniformly distributed in the interval in which it takes place. Thus, if needed, we will leverage the additional assumption that the PDF of C_i is slowly varying in intervals of length max{T-τ, τ} (note, though, that this assumption is not needed for the independence of {C̃_i, S̃_i}). When both f_S and f_C are slowly varying functions in any interval of length max{T-τ, τ}, the displacement of the beginning (end) of a contact within its ON (OFF) interval can be approximated as uniformly distributed in that interval.Exploiting the fact that the measured contact process is approximately renewal under the conditions in Lemma <ref>, we can again focus on what happens between two detected contacts. Differently from the previous case with negligible contact duration, the detected real contact corresponding to the measured contact can now start also in an OFF interval. For this reason, operating a shift of index as we have done in the previous section, the starting point of our analysis will be the interval [0, T] in which we know a detected contact has taken place, where its OFF interval is [0, T-τ] and its ON interval [T-τ, T]. All following OFF (ON) intervals will be of type [(i-1)T, iT-τ] ([iT-τ, iT]) with i > 1. Based on Lemma <ref>, we also know that the displacement of the beginning of a contact in an ON interval is distributed as Z^ON∼ Unif(0, τ) while that of a contact in an OFF interval as Z^OFF∼ Unif(0, T-τ). Similarly, under Corollary <ref>, the end of a contact in ON and OFF intervals will be displaced as Z^ON and Z^OFF, respectively. Given the distribution of displacements, we can easily quantify (Lemma <ref> and Corollary <ref> below) the probability of a contact beginning (ending) in an ON interval, and that of a contact beginning (ending) in an OFF interval, which we denote as p_s^ON, p_e^ON, p_s^OFF, p_e^OFF, respectively.When f_S is a slowly varying function in intervals of length max{T-τ, τ}, the probability p_s^ON of a contact beginning in an ON interval and the probabilityp_s^OFF of a contact beginning in an OFF interval are equal to τ/T and T-τ/T, respectively. When Lemma <ref> hold, the starting point of contacts are uniformly distributed in the duty cycling interval in which they take place. Thus, the probability of having an event in an ON or OFF interval depends on the length of that interval with respect to the overall duty cycle interval.If also f_C is slowly varying in intervals of length max{T-τ, τ}, then the probability p_e^ON of a contact ending in an ON interval and the probabilityp_e^OFF of a contact ending in an OFF interval are equal to τ/T and T-τ/T, respectively.It follows from Corollary <ref> that the end point of contacts are uniformly distributed in the duty cycling interval in which they take place. Hence the same argument used in the previous proof can be exploited. §.§ Measured contact timesWe start our derivation from the measured contact times, which are defined as the time intervals during which the two nodes are in contact and both in the ON state. Therefore, real contact times, by definition, overlap with at least one ON interval. We denote with H the number of ON intervals spanned by the detected contact. The distribution of H is very important for the rest of the analysis. The probability that H takes a specific value h depends on τ, T, and on the distribution of the contact time, as intuition suggests and as we show in Lemma <ref> below. The PMF of H, defined as the number of ON intervals spanned by a detected contact, is given by the following: ℙ{H = 1} = p_s^ONℙ(Z^ON + C < T ) + p^OFFℙ(Z^OFF + C^hit < 2T -τ),ℙ{H = h} = p_s^ONℙ((h-1)T < Z^ON + C < h T ) + + p_s^OFFℙ((h-1)T-τ < Z^OFF + C^hit < hT -τ), ∀ h >1, where C^hit can be computed as follows:ℙ(C^hit = c) = f_C(c) ∫_T-τ-c^T-τ1/1-F_C(T-τ-z)T-τ/Td z. The detailed proof can be found in <cit.>. Its rationale is to find the probability associated with the different combinations under which a given event can occur. For example, case H=1 occurs either when the contact starts in an ON interval and ends in the same ON interval or when it starts in an OFF interval, lasts until the next ONinterval and ends in that same ON interval. The probabilities of these two combinations of events correspond to the first and second terms in the summation for case H=1 in Equation <ref>. Exploiting the PMF of H, we can now derive the measured contact time in Theorem <ref> below. Figure <ref> shows some possible instances of the problem when H=1. Note that each detected contact spanning H ON intervals introduces H samples of measured contact duration. In the following lemma, we investigate how these samples are characterised and how frequently they occur. The measured contact time can be approximated with the following expression: =0.5pt2.2 C̃ = {[ C^shortℙ(H=1)/∑_h=1^∞ h ℙ(H=h)τ/Tℙ(Y^ON≤τ)/ℙ(Y^ON≤ T); C^res ℙ(H=1)/∑_h=1^∞ h ℙ(H=h)τ(T-τ)/Tℙ(T-τ < Y^OFF≤ T)/ℙ(T-τ < Y^OFF≤ 2T - τ);Z^ONℙ(H=1) τ/T(1- ℙ(Y^ON≤τ)/ℙ(Y^ON≤ T))/∑_h=1^∞ h ℙ(H=h) + ℙ(H≥ 2) (2τ/T)/∑_h=1^∞ h ℙ(H=h); τ ℙ(H=1) (T-τ)/T(1-ℙ(T-τ < Y^OFF≤ T)/ℙ(T-τ < Y^OFF≤ 2T - τ))/∑_h=1^∞ h ℙ(H=h) + ℙ(H≥ 2) (2 (T-τ)/T)+ ∑_h=3^∞(h-2) ℙ(H=h)/∑_h=1^∞ h ℙ(H=h) ].,where the PMF of H is given in Lemma <ref>, Y^ON = Z^ON + C, Y^OFF = Z^OFF + C, and the distribution of C^short and C^res can be computed as follows: ℙ(C^short = c) ={[ 1/τ f_C(c) ∫_c^τ1/F_C(u)d u 0 < c < τ; 0 otherwise ].,ℙ(C^res≤ c) ={[ F_Y^OFF(c + T - τ) - F_Y^OFF(T - τ) /F_Y^OFF(T) - F_Y^OFF(T - τ)0 < c < τ;0otherwise ].. The complete proof is available at <cit.>. In the following we will provide an intuitive explanation of its derivations. We start by noting that there are basically four ways, illustrated in Figure <ref>, in which a contact can intersect with an ON interval. The contact can be fully contained in the ON interval (case a), partially overlapping (cases c,d), or completely overlapping with the ON interval (case b).In the latter case the measured contact is deterministically equal to τ. In the other cases, the measured contacts are equal to C_short, Z^ON, and C_res respectively. Z^ON is uniformly distributed in the ON interval, while quantities C_short and C_res are derived in <cit.>. Each of the four components weights differently in the distribution of the measured contact times. Due to lack of space, the exact derivation of these weights is left to <cit.>. With the above lemma we are able to fully characterise the measured contact duration in a general duty cycling scenario. When the network is sparse, i.e., the density of users is so low that the probability that a user has more than one neighbour at a given time is low as well, we can derive a useful additional result. Specifically, in some practical applications like the one in <cit.>, users are required to refrain from entering the low-power mode of the duty cycle if a new contact is detected in the current ON interval. This is done to maximise the amount of information that can be transferred between user pairs. In the general case, it can be complex to model this scenario, since a newly detected contact between node i and j effectively alters the joint duty cycle of i and j with all other nodes (because the duration of the joint ON intervals depends on this newly detected contact). However, when the network is sparse, we can ignore the effect that a detected contact between i and j has on the joint duty cycles with all other nodes. Thus, we can derive the following lemma (the proof can be found in <cit.>). We assume, for simplicity, that after the contact ends, the two nodes synchronise with the joint duty cycling as originally planned. When nodes remain active after a contact is detected, the measured contact time can be approximated with the following expression: C̃ = {[Cτ/T; C^res*T-τ/T ]., where the distribution of C^res* can be computed as follows: ℙ(C^res*≤ c) ={[ F_Y^OFF(c + T - τ) - F_Y^OFF(T - τ) /1 - F_Y^OFF(T - τ) 0 < c < τ; 0 otherwise ].. and Y^OFF=Z^OFF + C.In the next section, we study the measured intercontact times in order to complete the characterisation of the measured contact process. §.§ The measured intercontact timeMeasured intercontact times S̃_i are defined as the time interval between two consecutive measured contacts (which we have characterised in the previous section). Thus, intuitively, measured intercontact times are a composition of portions of real contact times (those that do not intersect any ON interval) and intercontact times. Actually, there is another component, which only shows up when the real contact spans more than one ON interval, which we will discuss later on. In order to make the derivation of S̃ more tractable, in the following we exploit the slowly varying approximation for C_i (Corollary <ref>). Hence, both the start time and the end time of a real contact can be assumed uniformly distributed in the ON/OFF interval in which they take place.We start with the simplest case, focusing on a situation in which the last detected contact overlaps with only one ON interval(i.e., H=1), as shown in Figure <ref>. The end of the previous measured contact can be the actual end point of its corresponding detected original contact (if the latter ends in an ON interval, as in Figure <ref>(b)) or otherwise the end point of the ON interval (Figure <ref>(a)). In the latter case, the portion (corresponding to the blue area in Figure <ref>(a)) of the detected contact between the end of the ON interval and the time at which the real contact ends contributes to the measured intercontact time. This quantity corresponds to the displacement of the endpoint of the real contact in the OFF interval, which is distributed as Z^OFF if we assume C_i to be slowly varying (Corollary <ref>). A similar line of reasoning holds for the other extreme, resulting in an additional time interval (labelled (2) in the figure), again distributed as Z^OFF.Therefore, recalling that p_s^OFF=p_e^OFF=T-τ/T when both S_i and C_i are slowly varying in OFF intervals, these residuals can be modelled as R below: R = {[Z^OFF with prob. T-τ/T;0 with prob. τ/T ]. Let us now focus on what happens between the two detected contacts. It is easy to see (Figure <ref>) that, regardless of when detected contact times start and end, the central part of the measured intercontact time contains a certain number N of intercontact times S_i (this is analogous to the negligible contacts case). In addition, this central part can contain missed contacts, i.e., contacts that do not overlap with any ON interval. The probability density of these missed contacts (which we call C_i^miss) can be written as ℙ(C_i^miss=c) = ℙ(C_i=c | Z^OFF + C_i < T-τ), hence it can be obtained similarly to C^short in Theorem <ref>. Putting together our observations, S̃_i is equal to R + ∑_i=1^N S_i + ∑_i=2^N C_i^miss + R when H=1.When H ≥ 2, contacts are long and span more than one ON interval. However, these long contacts cannot be used in their entirety for communication, unless under the assumptions of Lemma <ref>. Specifically, their portion overlapping with OFF intervals cannot be used. Hence, each long contact is split into smaller measured contacts separated by what we call pseudo-intercontact times, i.e. measured intercontacts of length T-τ, as shown in Figure <ref>. What happens after the end of the long detected contact is exactly the same as what we discussed for case H=1: there will be a sequence of intercontact times S_i and missed contacts C_i^miss, whose number depends on how many contacts are missed before the next one is detected. This corresponds to formula R + ∑_i=1^N S_i + ∑_i=2^N C_i^miss + R, which we have derived in the previous section. Based on the above discussion, we know that measured intercontact times when contact duration is non-negligible can be either equal to T-τ (which is the contribution of pseudo-intercontact times) or to R + ∑_i=1^N S_i + ∑_i=2^N C_i^miss + R, thus the following theorem holds. The weights of the two components of the mixture distribution are derived in <cit.> according to the line of reasoning discussed above. The measured intercontact time can be approximated as follows:S̃ = {[ R + ∑_i=1^N S_i +∑_i=2^N C_i^miss+ R∑_h=1^∞ P(H=h)/∑_h=1^∞ h P(H=h);T - τ∑_h=2^∞ (h-1) P(H=h)/∑_h=1^∞ h P(H=h);]. where R is defined in Equation <ref> and the PDF of C^miss is given by the following: ℙ(C^miss = c) ={[ 1/T-τ f_C(c) ∫_c^T-τ1/F_C(u)d u 0 < c < T - τ; 0 otherwise ].. At this point, in order to obtain S̃ the only missing piece is the distribution of N when contact duration is not negligible. Thus we derive it in the lemma below. The proof follows the same line of reasoning as the proof of Lemma <ref>, and it can be found in <cit.>. When contact duration is non-negligible, the probability mass function of N can be approximated by the following:{[ ℙ{N=1} = ĝ, k = 1; ℙ{N=k} = (1-ĝ)(1-p̂)^k-2p̂,k≥ 2 ]. where ĝ = 1 - ∑_n_1=2^∞ℙ( Z + S ∈ℐ^OFF_n_1) ℙ( Z^OFF + C < T-τ) andp̂=1- ∑_n_2=2^∞ℙ( Z^OFF + S ∈ℐ^OFF_n_2) ℙ( Z^OFF + C < T-τ). Going back to Theorem <ref>, it is easy to see that, in most practical applications, S̃ is still dominated by the component ∑_i=1^N S_i. Thus, all the properties discussed in Section <ref> (e.g., the conversion from hypo-exponential behaviour to a hyper-exponential one) will not change significantly. Please note, however, that the distribution of N in the two cases is different, so the result obtained in the negligible contact duration case does not apply as is to the non-negligible contact duration case.Finally, Theorem <ref> can be easily modified to take into account the effect of nodes not entering the OFF state after a contact has been detected, which we have discussed in Section <ref>. In this case, pseudo-intercontact times will not be present in S̃, since nodes do not enter the OFF state when a contact is ongoing. Thus, we obtain S̃ = R + ∑_i=1^N S_i +∑_i=2^N C_i^miss+ R.§.§ ValidationWe now validate the results we have obtained for the measured contact (Theorem <ref>) and measured intercontact time (Theorem <ref>). Specifically, as these are the most common distributions found in real traces, we consider real human mobility traces and extract pairs for which exponential or Pareto distributions fit intercontact or contact times. We use standard Maximum Likelihood estimation and goodness-of-fit techniques to this end.Specifically, the method discussed in <cit.> for estimating both scale and shape for the Pareto distribution, and the Cramér-von Mises test with significance level ϕ = 0.01. For statistical reliability, only pairs with more than 9 samples are considered, as in <cit.>). Several traces of real contacts between nodes are publicly available[E.g., at <http://crawdad.cs.dartmouth.edu/>] and have been often used in the related literature (Table <ref> in <cit.> summarises the most popular ones). However, it is usually neglected the fact that all of them implement a form of duty cycling in the neighbour discovery process, owing to the technology-dependent scanning period (typically in the order of 100s). Hence, what they track is actually the measured contacts and intercontact times, rather than the real ones. However, there are a few datasets that use quite a small duty cycling (in the order of a few seconds) and hence can realistically approximate the real contact and intercontact times in practice (i.e., assuming that both contact and intercontact times last for longer than a few seconds). These traces are PMTR <cit.> and RollerNet <cit.>, and they will be the focus of our analysis[In <cit.> we also provide a discussion on other traces (Infocom and Reality Mining), which are very popular in the literature but that were not suitable for our validation due to their long duty cycle.]. The PMTR trace has been obtained from the readings across 19 days (in November 2008) of 44 Pocket Mobile Trace Recorders (PMTRs), custom devices built for contact detection and distributed to faculty members, PhD students, and technical staff at the University of Milan. Contacts are sampled every 1 seconds. The RollerNet experiment was carried out to analyse the mobility of rollerbladers in Paris. The dataset was collected on August 20, 2006, and it is composed of two sessions of 80 minutes, interspersed with a break of 20 minutes. The 62 Bluetooth sensors (iMotes) were distributed to organisers' friends, members of rollerblading associations and members of staff. Here contacts are sampled every 15 seconds.By applying the fitting technique described at the beginning of the section to each node pair in the PMTR and RollerNet traces, we obtained the results in Tables <ref>-<ref> in <cit.>. In the remaining of the section, we focus only on those pairs for which a given hypothesis (either exponential or Pareto) is not rejected, and we apply our theoretical framework to representative user pairs (i.e. we configure the model using the MLE parameters of the selected pair). In this analysis, we use the same duty cycle configuration that we used in Section <ref>, i.e., τ=20s, T=100s. Assuming that the contact and intercontact times are either exponential (Sec. <ref>) or Pareto (Sec. <ref>), we draw contact and intercontact times samples for either distribution, then we filter them according to the reference duty cycling process (hence, we simulate the effects of the duty cycling on the original contact process that we have extracted from the trace). Please note that the fitting results are used to configure the distribution from which contact and intercontact times are sampled for relevant pairs in the datasets.§.§.§ The exponential caseWe first consider the measured contact duration C̃, which we can approximate as discussed in Theorem <ref>. The probabilities of observing each component of the mixture predicted in Theorem <ref> only depend on τ, T (that are constant in our case), and on the distribution of real contact times. Since we have fixed τ and T, the interplay between the different C̃ components is regulated only by C_i, which we are assuming exponential with rate μ (a thorough discussion on this dependence is provided in <cit.>). Thus, in the following, for different μ values, we compare the predictions of Theorem <ref> for the measured contact time against simulation results. Similarly to Section <ref>, we draw contact and intercontact times (30,000 samples each) from an exponential distribution and we filter them using our reference duty cycling process with τ=20s, T=100s. We set the rate λ of intercontact times equal to the mean rate in the corresponding trace (PMTR or RollerNet). Then, we set the rate of the exponential distribution of contact times equal to significant points of the μ distribution (corresponding to the minimum and maximum values, first and third quartiles, median and mean). Parameter μ varies a lot across pairs in the PMTR trace, spanning several orders of magnitude (from 10^-5 s^-1 to 10^-1 s^-1). For the RollerNet trace, the body of the distribution of μ is quite compact, and only the minimum and maximum values are more distant. Thus, in this case we omit the plot for the median and the first and third quartiles. Please note that we only consider those pairs for which the exponential hypothesis was not rejected by the Cramér-von Mises test. The results for the measured contact times are shown in Figure <ref> for the PMTR trace and in Figure <ref> for the RollerNet trace. In both cases, predictions are generally accurate. The largest discrepancies appear for values of μ around 10^-2. The figures also plot the distribution of the original contact time without duty cycling. As we have set τ=20 and T=80, the maximum contact duration we observe with duty cycling is 20s, while contact durations without duty cycling can be also much longer. Moreover, the model predicts a discontinuity in the density for contact around the value of τ. This justifies the differences in the corresponding curves in the figures. It is also interesting to note that when μ is large, the discontinuity in the CDF of the measured contact disappears, because contacts are so short that they are generally fully contained in ON intervals. Because of this, we expect that the original contact time distribution, C, is not significantly modified by the duty cycling. This is indeed the case, as we can observe[For large μ the main component of C̃ is C^res, which, in the exponential case, converges to C for large μ.] in Figure <ref>.We now focus on the measured intercontact times S̃_̃ĩ. Measured intercontact times have two components, one stochastic component which is the sum of several random variables (see Theorem <ref>) and one constant component T-τ (which we have called pseudo-intercontact time). The probability of selecting either component only depends on μ, τ, and T (because it is a function of H). Fixing, as usual, τ=20s and T=100s, we plot in Figure <ref> the probability of observing pseudo-intercontact times in S̃_̃ĩ. Pseudo-intercontact times start appearing for μ values smaller than 0.06. In this range, long contacts are split into many shorter measured contacts, and the portions of real contacts not overlapping with an ON interval become pseudo-intercontact times. In order to validate Theorem <ref>, we start with the PMTR trace and we plot the CDF of the measured intercontact times keeping μ fixed (specifically, equal to the mean value) and varying λ. In all cases, theoretical predictions and simulations results are overlapping (Figure <ref>). We omit the plots for the RollerNet case, since the same considerations apply and theoretical predictions remain virtually indistinguishable from simulation results. We expected this good result, since λ values are rather small (and, in all case but one, smaller than 1/T), and therefore the slowly varying assumption of Theorem <ref> is verified. In these plots we also observe the contribution of pseudo-intercontact times to the CDF, corresponding to its initial bump (pseudo-intercontact times are of length T-τ=80s, hence they affect the very beginning of the distribution).This was also expected, as, with τ=20 and T=100, the region of μ where the probability of pseudo intercontact times is not negligible is μ<0.06s^-1, and the mean value of μ in PMTR is 0.02s^-1.Note also that with μ∼0.02s^-1, contact duration is not negligible, hence the predictions (green curve in Figure <ref>) that we would obtain using the results for the negligible contact case (i.e., using Lemma <ref>) are not at all accurate. §.§.§ The Pareto case We now perform a similar analysis for those pairs for which the Pareto hypothesis for contact and intercontact times was not rejected by the Cramér-von Mises test (more than 97% of pairs overall, see <cit.>). Here we only focus on measured contact and measured intercontact times. Further results, such as the analysis of H, can be found in <cit.>. In Figure <ref> we plot the CDF of measured contact times obtained from simulations against the theoretical predictions of Theorem <ref>. Simulations are performed as described for the exponential case, except that here we sample from Pareto distributions. We start with the PMTR dataset. In the first set of plots in Figure <ref> we fix b to the average value b=245.40 observed in the dataset and we vary α. As expected, predictions are very accurate, owing to the fact that b > T and thus the slowly varying assumption holds true. In Figure <ref> we vary b fixing α to its average value (α =1.898, which smaller than the threshold α=2 discussed in <cit.> for having accurate predictions of C̃). As expected, the only case when predictions are not very accurate are for α and b small, when the slowly varying assumption does not hold.Similar considerations hold for the RollerNet dataset, for which we omit the plot.Finally, we study the behaviour of measured intercontact times when contact and intercontact times are Pareto. In Figure <ref> we plot the measured intercontact times (simulations vs theoretical predictions) fixing the Pareto parameters of the contact times to their average values and varying the parameters b and α of intercontact times. We observe that in all cases the predictions are very accurate. Note how, in the PMTR+Pareto case, contacts tend to be long and pseudo-intercontact times are observed often, as the big jump at T-τ = 80s shows.§ FROM DETERMINISTIC TO STOCHASTIC DUTY CYCLING POLICIESIn this section we show that the fixed duty cycling model studied so far in the paper provides a good approximation also for non deterministic duty cycling. Although the method can be generalised, for the sake of example, we assume that individual stochastic duty cycles alternate between OFF and ON intervals whose lengths are both exponentially distributed with rate α and β, respectively.The first step in the analysis is the derivation of the joint duty cycling (which is in the ON state when both nodes are ON, in the OFF state otherwise). T_ON and T_OFF denote the length of ON and OFF phases in the joint duty cycle. To this aim, we model the states of our system using a Continuous Time Markov Chain (CTMC). We obtain (the detailed derivation can be found in <cit.>) that T_ON is exponentially distributed with rate 2β, and that T_OFF has the following first and second moments: 𝔼[T_OFF] = 2α + β/2α^2, 𝔼[T_OFF^2] = 10 α ^3+11 α ^2 β +6 αβ ^2+β ^3/2 α ^4 (α +β ). We assume that both α and βare strictly greater than zero[When equal to zero we have the two extreme cases of nodes either always ON or always OFF. In the first case, there is no need to study the effect of duty cycling, in the second case nodes are never able to detect each other.]. For β approaching zero, the squared coefficient of variation of T_OFF approaches 4. For α approaching zero, it approaches 1. Hence, we can conclude that the duration of the OFF interval of the joint duty cycle ranges from a hyper-exponential behaviour to an exponential behaviour, depending on the values of α and β.The fixed joint duty cycling analysed in Sec. <ref> can be considered as an approximation of the stochastic joint duty cycle studied in this section, setting the fixed duty cycling parameters τ and T-τ equal to the average ON and OFF durations of the joint duty cycle. Figure <ref> validates this statement by comparing simulation results obtained with exponential duty cycling against the prediction obtained assuming the duty cycling is deterministic. Specifically, we consider τ=20s and T=100s for the fixed joint duty cycling (as in Section <ref>), thus we obtain for stochastic individual duty cycling β=0.025 and α=0.02 (with cv^2=1.96 for the joint OFF intervals). We assume that real intercontact times are exponentially distributed with rates λ∈{0.001, 0.01, 0.1, 10 } and that contact duration is negligible. We sample measured intercontact times using Monte Carlo simulations. We test both stochastic and fixed duty cycling, and we compare the results obtained against the theoretical predictions of Theorem <ref>, showing (Figure <ref>) that the model with fixed duty cycling well approximate also cases where duty cycling is stochastic, which is popular in the literature. § RELATED WORK Ad hoc communications in opportunistic networks traditionally use either the WiFi or Bluetooth interfaces, which can consume a significant fraction of the smartphone's battery depending on the their current state (idle, scanning, or connected) <cit.>. For this reason, duty cycling techniques have been introduced in order to save energy by putting devices into a low-power state whenever possible. With Bluetooth, this low-power state corresponds to the discoverable state, which is entered by a device after a scanning phase (energy hungry) is concluded. Hence, duty cycling with Bluetooth implies striking the right balance between keeping as much as possible the devices in the discoverable state and not missing too many contacts. The situation is different with ad hoc WiFi,which has no particularly energy-efficient state[For 802.11 cards used in infrastructure mode energy consumption in the idle state has been drastically reduced by the introduction of the Power Saving Mode (PSM) <cit.>. Unfortunately, PSM for ad hoc is typically not implemented in smartphones' 802.11 interfaces.]. In this case, the best power saving strategy is simply to switch off the network interface entirely. Recently, two innovative WiFi-based ad hoc communications modes, namely, WiFi Direct and WLAN-Opp <cit.>, have been proposed to address the problems of ad hoc communications in off-the-shelf smartphones. Unfortunately, their energy consumption is still quite high, in particular as far as neighbour discovery is concerned <cit.>. As discussed in <cit.>, reducing the scanning frequency remains the only viable power saving option for both WiFi Direct and WLAN-Opp. Abstracting the specific communication technology used and building upon the idea that neighbour discovery is an energy-expensive operation in general, the vast majority of papers dealing with power saving issues in opportunistic networks have focused on the contact probing phase. As seen above, reducing and optimising the probing frequency is equivalent to implementing a duty cycling policy in which nodes switch between low-power and high-power states, corresponding to OFF intervals (in which contacts are not detected) and ON intervals (in which contacts are detected), respectively. Contact probing schemes can be classified into fixed, when the ON/OFF duration of the duty cycle is established at the beginning and never changed <cit.>, or adaptive, when the frequency of probing is increased or decreased according to some policy <cit.>. Both fixed and adaptive strategies can be context-oblivious <cit.>, if they do not exploit information on user past behaviour or position, or context-aware <cit.> otherwise. Differently from the above contributions, in this work we do not aim at deriving an optimised power saving strategy for DTN. Instead, our goal is to understand, given a duty cycling strategy, how the measured contact process between nodes is changed by this strategy. Below, we briefly contrast the most relevant related literature against our contribution.The model we introduce in this paper is more general than the one discussed in <cit.> as it is not bound to the RWP model but it can be applied to any distribution for intercontact times. If the intercontact times distributions are Pareto or exponential – also in heterogeneous cases where the parameters change across pairs of nodes – our model can be solved with closed form expressions (note thatPareto and exponential are the two most popular assumptions for contact and intercontact times in the related literature). Otherwise, for any other intercontact time distribution,numerical solutions can be found. In addition, <cit.> only consider a fixed probing every T seconds as their duty cycling strategy. Instead, we generalise the duty cycling process, by considering it composed of two phases (the ON and OFF phases) and also covering the non-fixed duration case.<cit.> evaluate only how link duration (or contact duration, in our terminology) is affected by the contact probing interval. Instead, we investigate the effect of duty cycling (which, as already discussed, can be easily translated into a contact probing problem) both on measured contact duration (i.e., link duration) and measured intercontact time, acknowledging that both components have a huge impact on opportunistic communications (on network capacity and message delay, respectively). Also, as already discussed, despite its simplicity, our duty cycling function with ON/OFF states allows for more flexibility than the simple scanning every T seconds performed in <cit.>. The effective link duration (equivalent to our measured contact duration) is also derived in <cit.>, assuming that nodes wake up every T seconds and remain active for a configurable random amount of time. In this work, the authors implicitly discard the correlations between consecutive contacts (for which we have provided a thorough discussion in <ref>), do not provide closed-form results for particularly relevant case, and do not investigate nor validate in detail the model (because the focus of the work is more on the energy-goodput trade-offs than on the effects of duty cycling on the contact process). In addition, measured intercontact times are not studied in <cit.>, despite their importance. Another set of works that share similarities with our proposal are <cit.>. Analogously to <cit.>, their focus is more on striking the right balance between energy consumption and forwarding performance rather than on the complete characterisation of the measured contact process. <cit.> assume the same kind of duty cycling process with ON/OFF periods that we study in this paper, while for the contact process they assume Pareto contact duration and exponential intercontact times. Under these assumptions, <cit.> derive that the exponential intercontact times are altered by duty cycling in such a way that their rate is scaled by a factor that they call contact probability. This result is analogous to our result in Lemma <ref>. However, this result only holds for contact duration negligible with respect to T, and the model in <cit.> is not able to address what happens when contact duration is instead not negligible. Our model is able to provide a complete characterisation also for this case. In addition, <cit.> do not provide an expression for the distribution of the measured contact and intercontact times under generic distributions ofcontact and intercontact times. Instead, we address this case and provide a general technique, based on the slowly varying approximation, for handling distributions that are not memoryless.Differently from the works discussed so far, in which the wake up schedules of nodes are fixed, <cit.> and <cit.> propose techniques to adaptively schedule the wake-up and sleep states of nodes. To this aim, and differently from our work,<cit.> do not characterise the contact process using a probability distribution but instead rely directly on the history of past encounters. Therefore, our model is more general, as it can represent in a mathematical form the contact process, and the effect on it of duty cycling. <cit.> do not provide a complete characterisation of the impact of duty cycling on the measured contact process either, but focus on probabilistically predicting the next contact. This probability is then used to design their adaptive wake-up schedule. <cit.> assume that intercontact times are exponential, and also neglect contact duration. Instead, we additionally consider the Pareto case for intercontact times and we include the effect of contact duration in the model. Based on the above review, we can conclude that our contribution represents the first comprehensive analysis of how the measured contact process is altered by power saving techniques, both in terms of the effects on the measured contact duration and on the measured intercontact times. This work is an extension of our previous work in <cit.>, where we had focused on the negligible contact case with exponential real intercontact times only, and we had studied how their distribution was affected by the duty cycling policy. In <cit.> we have used a complex model, which was not suitable to be solved with distributions different from the exponential. Specifically, closed-form solutions could not be found when intercontact times were not exponential, and also numerical solution took a lot of time to be obtained. The main outcome of the model in <cit.> is what we have here summarised in Lemma <ref>.§ CONCLUSIONSPower saving mechanisms reduce the forwarding opportunities and the capacity of an opportunistic networks, but this effect has not been yet quantified in a general setting in the related literature. To fill this gap, in this work we have investigated the effects of deterministic duty cycling on contact and intercontact times in opportunistic networks. Specifically, we have proposed two models for characterising the measured contact process (i.e., the contact process after duty cycling has been factored in) between pairs of nodes. These models have been extensively validated, and have been shown to provide very good approximations even when the assumptions under which they have been derived do not hold exactly. The first model can be used when the contact duration for the pair of nodes is negligible with respect to the length of the ON and OFF intervals of the duty cycle. With this model we can derive the first two moments of the measured intercontact times for any distribution oforiginal intercontact times. Exploiting this model, we have discovered that if the original intercontact times are exponential, then the measured intercontact times are also exponential but with a different rate. If original intercontact times are Pareto, measured intercontact times remain Pareto with the same exponent in the tail but, overall, they do not feature a well-known distribution. More in general, we have shown that the measured intercontact times can flip their “behaviour” depending on the duty cycle value and on the distribution of the original intercontact times. Specifically, hyper-exponential measured intercontact times can appear even when the original intercontact times are hypo-exponential, and vice versa.The second model, which is more complex but also more realistic, should be used when contact duration is not negligible. With this second model, we are able to derive the distribution of the measured contact duration, considering both the case in which nodes keep their scheduled duty cycle upon a new encounter and the case in which they do not. In the first case, a measured contact cannot last longer than an ON interval. Since contact duration determines the amount of data that can be transferred, the capacity of the opportunistic network can be significantly affected by duty cycling. Vice versa, in the second case, only a small portion of the contact is missed, hence the capacity can be preserved. We have also derived the measured intercontact times, highlighting the fact that they have two components: one conceptually very similar to the measured intercontact time with negligible contact duration, one very different. We called the second component pseudo-intercontact time, as it is due to long contacts that are split into many shorter contact and intercontact times by the duty cycle.Finally, we have generalised our results, showing that a deterministic duty cycle can be assumed to be a good approximation of a stochastic duty cycle with the same average duration for the ON/OFF intervals. Building upon this finding, the two models presented in the paper can be used to derive the measured contact and intercontact times under any distribution for the contact process and under general duty cycling strategies (deterministic/stochastic, synchronous/asynchronous). abbrv | http://arxiv.org/abs/1709.09551v2 | {
"authors": [
"Elisabetta Biondi",
"Chiara Boldrini",
"Andrea Passarella",
"Marco Conti"
],
"categories": [
"cs.NI"
],
"primary_category": "cs.NI",
"published": "20170927142358",
"title": "What you lose when you snooze: how duty cycling impacts on the contact process in opportunistic networks"
} |
1 Electronic mail address: [email protected] [Present address: ]Chemical Physics Department, Weizmann Institute of Science,76100 Rehovot, Israel. Departamento de Física Aplicada, Universidad de Alicante, Campus de San Vicente del Raspeig, E-03690 Alicante, Spain.Departamento de Física Aplicada, Universidad de Alicante, Campus de San Vicente del Raspeig, E-03690 Alicante, Spain. Department of Physics, Science Campus, University of South Africa, Private Bag X6, Florida Park 1710, South Africa Departamento de Física Aplicada, Universidad de Alicante, Campus de San Vicente del Raspeig, E-03690 Alicante, Spain. CIC NanoGune, E-20018, Donostia, San Sebastian, Spain Ikerbasque, Basque Foundation for Science, 48013 Bilbao, SpainDepartamento de Física Aplicada, Universidad de Alicante, Campus de San Vicente del Raspeig, E-03690 Alicante, Spain. Departamento de Física Aplicada, Universidad de Alicante, Campus de San Vicente del Raspeig, E-03690 Alicante, Spain. Electronic mail address: [email protected] Departamento de Física Aplicada, Universidad de Alicante, Campus de San Vicente del Raspeig, E-03690 Alicante, Spain.We study in detail, by experimental measurements, atomistic simulations and DFT transport calculations, the process of formation and the resulting electronic properties of atomic-sized contacts made of Au, Ag and Cu. Our novel approaches to the data analysis of both experimental results and simulations, lead to a precise relationship between geometry and electronic transmission – we reestablish the significant influence of the number of first neighbors on the electronic properties of atomic-sized contacts. Our results allow us also to interpret subtle differences between the metals during the process of contact formation as well as the characteristics of the resulting contacts. 73.63.-b, 62.25.+g, 68.65.-k, 68.35.Np The role of first neighbors geometry in the electronic and mechanical properties of atomic contacts M. J. Caturla December 30, 2023 ===================================================================================================§ INTRODUCTION Single atoms and molecules have been widely hailed as potential electronic devices over the last twenty years <cit.>. To make such devices a reality, metallic contact formation and the electrical characteristics of few-atom contacts, need to be understood in depth at the atomic level. The electrical conduction in single-atom contacts has been broadly studied both from an experimental and theoretical point of view <cit.>, and single-atom contacts have been proposed as elementary circuit components, such as quantized resistors, capacitors <cit.> or switches. <cit.> The conductance of few-atom contacts is given by the sum of contributions from quantized transport modes propagating at the contact junction andthe number and transmission probabilities of those modes are determined by the size and chemical valence of the central part of the constriction <cit.>. For example, both a single-atom contact and a monoatomic chain of Au exhibit a resistance of around a quantum of conductance G_0=2e^2/h, which is in this case the signature of electronic transport through a single, fully open, quantum channel <cit.>.However, variations in the geometrical configuration of the leads<cit.>, i.e., the number of neighboring atoms in the constriction, give rise to fluctuations of up to 20 percent in the conductance of a single atomic contact.Not only the electrical properties of single atom contacts are strongly influenced by their coordination to the leads, but also their mechanical properties. When two electrodes in the tunneling regime eventually come into contact, it is known, for certain materials and geometries, that the process of contact formation happens as a sudden jump.Nonetheless, jump to contact is not a generalized phenomenon and the process of formation may be smooth. <cit.> The probability of occurrence of jump to contact and the details of this process have already been suggested to strongly depend not only on the bulk mechanical properties of the material, such as its cohesive energy and Young's modulus<cit.>,but also, for certain materials, e.g., Au or Cu, on the specific geometry of the contacting leads <cit.>.In this article, we focus on the influence of the first-neighbor configurations on the process of formation of single-atom contacts made of Au, Ag and Cu, as well as their associated conductance values.To this end, we combine atomistic simulationsand quantum transport calculations <cit.> with a detailed analysis of experimental results.We improve the statistical analysis carried out by Untiedtet al. <cit.> for Au, and compare our results with those obtained from the atomistic simulations we perform to determine the most likely first-neighbor structures at first contact, and corresponding conductance values we calculate from Density Functional Theory (DFT) methods. <cit.> From such a comparison between simulation and experimental results, we can relate the distribution of contact conductances to specific geometries. In agreement with the results published in Refs. <cit.> we find the most likely geometries to lie within four classes: monomers, dimers, Double Contacts (D.C) and Triple Contacts (T.C). Furthermore, we identify more specific structures within these classes and more interestingly, find the dispersion in conductance values for each of these classes to be a consequence of the variations in the number of first neighbors.Our analysis provides a precise assignment of the conductance values reported for these configurations, and remarkably, yields a broader distribution of conductance values for the monomer than in previous works on Au, ultimately explaining previous disagreements between experiments and theory. The reason for this can be traced to a higher dependence of the monomer's conductance on the number of first neighbors. We complete our study by carrying out a similar analysis for Ag and Cu.§ METHODS§.§ Experimental methods Our atomic contacts are fabricated by performing several cycles of indentation and separation of two electrodes made of the same high purity (99.999%) metal,Au, Ag or Cu, under cryogenic vacuum at 4.2K.The electrical conductance of the junctions (obtained as the measured current divided by the applied voltage of 100 mV) is recorded while the two electrodes are carefully brought into contact in a scanning tunneling microscope (STM) setup, as described in previous works. <cit.>The traces of conductance as a function of electrode distance (Fig.<ref>(a)) contain valuable information about the process of contact rupture and formation. When electrodes are close enough but not yet in contact, electrons may tunnel between them. In the tunneling regime,the conductance increases exponentially as the separation between leads decreases. The conductance increases smoothly until a sudden jump occurs, from the tunneling regime up to a clear plateau at around 1 G_0, indicatingthe formation of a monoatomic contact <cit.>. Examples of rupture and formation traces are displayed in Fig.1(a). Every realization of an atomic-size contact produces a slightly different conductance trace, which is suggestive of a variation in structural configurations. Therefore, a statistical analysis of the data is key to extracting information about the most probable configurations.An approach that is widely used in the literature <cit.> is the construction of a conductance histogram (such as the one in Fig. 1b for the case of rupture traces of Au), to determine the conductance values associated with the most probable configurations of the single-atom contact.A more specific method for the study of contact formation was introduced by Untiedt et al. <cit.>As sketched in Fig. 1a, for each formation trace, the highest jump in conductance between two consecutive points is monitored. Two conductance values are then recorded, G_a, from which the jump occurs and G_b, the final value immediately after the jump. A density plot of the pairs (G_a,G_b) (main panel in Fig. 2) displays the values of greatest probability from and to which the conductance jump occurs. As mentioned above, prior to contact formation, the tunneling conductance depends exponentially on the distance between electrodes as G≃ Ke^-√(2mϕ)/hd, whereK is a proportionality constant which depends on the cross-sectional area and density of states at the Fermi level of the electrodes, m corresponds to the electron mass and ϕ is the work function of the material. Since G_a is the conductance in the tunneling regime immediately before jump to contact, its logarithm log(G_a) [log denotes here the common logarithm (base 10)] is proportional to the distance between the electrodes from which the jump occurs.When the G_a axis is plotted on a logarithmic scale, the density plot corresponding to formation of Au contacts, reveals shapes of the maxima that can be more easily interpreted than those previously reported in Ref. <cit.> §.§ Data analysis The projections of the density plot data on both log(G_a) and G_b axes (Fig. 2) can be fitted to a sum of gaussian peaks. This suggests that the density plot is formed by a number of maxima which are normally distributed in both variables. Therefore, we fit the data to the sum of three bivariate normal distributions, sketched as ellipses in Fig. 2 and labeled D1, D2 and D3, with different relative probabilities p. Each of these distributions is described by the expression: f(x,μ,Σ)=1/√(|Σ|)(2π)^2e^-1/2(x-μ)'Σ^-1(x-μ)where x=(log(G_a),G_b), μ=(μ_a, μ_b) andΣ=( [ σ_a^2 ρσ_aσ_b; ρσ_aσ_b σ_b^2 ]). μ_i and σ_i represent the 2D equivalents of the unidimensional mean and standard deviation, respectively, and ρ is the correlation parameter between variables logG_a and G_b. § EXPERIMENTAL RESULTSThe experiments and analysis described in the previous section were repeated during the fabrication of over 2000 contacts made of Au, Ag and Cu. The output fitting parameters for the three materials are summarized in Table I.The characteristic parameters of the distributions can be graphically represented by an ellipse (for example, as the overlays in Fig. 2). The center of the ellipse (μ_a,μ_b) represents the (logG_a,G_b) position of the mean of the distribution, and the axes of the ellipse represent the standard deviations (σ_a,σ _b) in the respective conductance axis. The tilt of the ellipse is proportional to the correlation (ρ) between the two variables. The identification of three maxima is in good agreement withRef. <cit.> for Au. This new analysis provides an opportunity to revise those results and carry out a more precise quantitative analysis of the data. In analogy with Ref. <cit.>, we find an isolated distribution with a low probability of occurrence, well above G_0 (labeled D3 in Fig. 2), while, at around 1 G_0, we find the sum of two distributions. Here we disentangle those two distributions and provide an estimate of their relative probabilities (p). Distribution D1 contains more than 50 percent of the data, while D2 contains around 30 percent. In this instance, the results for all three materials are similar.Moreover, on comparing the three materials, we discover a striking result: there is an important difference between the jump distance of Au versus Ag and Cu, represented by their mean values of log(G_a/G_0) denoted for simplicity as μ_a. This is the focus of a separate study <cit.>, in which we show that the origin of this phenomenon can be traced to the different strengths of relativistic effects in these materials.Besides the information given by the mean of each distribution, the standard deviation also provides a measure of the dispersion in each. Regarding the dispersion in the G_a axis (σ_a), which may appear to be much larger in the case of Au, if scaled to the mean value, it is actually similar to the dispersion for Ag and Cu. This indicates, in all cases, that the variation in jump distance is a percentage of the average distance, which, in turn, supports the interpretation provided in Ref. <cit.>, that the dispersion inconductance originates from the large number of possible geometrical configurations. While the dispersion in conductances before jump (σ_a) remain similar in all three distributions for each material, remarkably, the dispersions in G_b (σ_b) exhibit significant differences. Distribution D1, in contact conductance G_b, is rather broad, while distribution D2, the second-most probable, exhibits a rather narrower dispersion in this parameter, as is evident from the widths of the ellipses in the G_b axis σ_b. This point will be discussed further in light of atomistic simulations, but it already suggests that the conductance in contact of one of the distributions is considerably less sensitive to geometrical variations than the other. Finally, we note that the correlation between G_a and G_b, ρ (visible from the tilt of the ellipses) is very similar not only for the three distributions, but also for all three materials, indicating a slight tendency for contacts associated with shorter jump distances to exhibit higher conductances. Besides the notable discrepancies in μ_a, a comparison of the metals yields also a number of subtle differences that are connected to the longer jump distance in the case of Au. Firstly, the means μ_a of D1 and D2 for Au, occur at about the same distance, while D3's mean has a slightly different value. However, for Ag and Cu, distributions D1 and D3 are centered at similar values of log G_a, while the contacts corresponding to D2 are established from a greater jump distance. Regarding the value of μ_Gb, note the lower conductance value for D1 in the case of Au with respect to the other two distributions, as well as with respect to the corresponding values for Ag or Cu.Although differences between D1 and D2 are small and perhaps within error margins, this behavior is expected for the more "stretched out" structures formed in Au <cit.>. § MOLECULAR DYNAMIC SIMULATIONS AND AB-INITIO CALCULATIONS§.§ MethodologyWe have not found any analysis of experimental measurements of electronic transport in the literature, which can provide information about the geometry at or the instant just before contact is established. Therefore, in order to have an appreciation of the importance of the configuration of the atoms in the immediate vicinity of few-atom contacts, we simulate the experiments by means of classical molecular dynamics (CMD) and first-principles quantum transport calculations. An alternative approach is used in Refs. <cit.>, in which a potential energy surface is calculated as an adiabatic trajectory by DFT. Metal junctions composed of small opposing fragments of Au, Ag or Cu are elongated/separated in small steps with a geometry optimization at each step Molecular dynamics simulations are based on solving Newton's second law for all the atoms, as they evolve from their initial positions. In such simulations, the potential used to model interactions between the atoms is semi-empirical. <cit.> The initial structure in the present work is independent of the metal and consists of 4736 atoms, oriented along the [100] crystallographic direction, as shown in panel a) of Fig. <ref>. The result of solving Newton's second law for this system is that we can obtain the classical trajectories of all the atoms in the structure, as it is ruptured and brought back into contact over many cycles. Extracting from these trajectories, then, the structure at first contact, as well as the one immediately before it, will, via DFT transport calculations <cit.>, yield the conductance at the moment that contact is re-established. As mentioned above, all the simulations involving Au, Ag and Cu are based on the same initial seed structure. The simulations are run in a way that reproduces cyclic loading of the nanowire in analogy with a typical STM or mechanically controllable break junction (MCBJ) experiment. This is also an approach that was followed in our previous works. <cit.> The interactions between the metal atoms are modeled by the semi-empirical, embedded atom method (EAM) potential. <cit.> All the simulations have been realized by means of the Large-scale Atomic/Molecular Massively Parallel Simulator LAMMPS. <cit.> The potential parameters used for Au, Ag and Cu in this work, are taken from Ref. <cit.> The potential itself is derived in Ref. <cit.>Additionally, in order to mimic the conditions of the experiment as closely as possible, we simulate at the boiling point of liquid Helium, 4.2 K. The Nose-Hoover thermostat <cit.> serves to maintain the temperature constant during the cycles of retraction and approach of the nanoelectrodes in the simulations. Thermostatting is performed every 1000 simulation time steps, the time interval that is recommended by the developers of LAMMPS.<cit.>The atoms that are located in the first three crystallographic planes from the top of the initial seed structure, as well as the corresponding three planes at the bottom, are pinned to their equilibrium bulk lattice positions so as to constrain their relative positions. The remaining atoms respond dynamically to the bulk motion of these “frozen" planes. Following, the entire structure is stretched lengthwise (vertically) by moving the frozen layers in opposite directions at a constant speed of ∼1 m/s. The arrows in Fig. <ref>, panels a) and b), illustrate the directions of the applied forces on both ends (top/bottom) during contact rupture and formation. A speed of ∼1 m/s may be many orders of magnitude greater than that employed in the experiments, but we argue that there is enough time for the structures to reach equilibrium, and not merely meta-stable states, because this speed is at least three orders of magnitude lower than that of sound in the bulk metals. <cit.> The low temperature used in our simulations also ensures that processes that would otherwise be important at microsecond time-scales, such as surface diffusion, remain negligible. In fact, at 4.2 K, surface diffusion is inhibited by activation energies that are 3-4 orders of magnitude higher than the thermal energy of the atoms. <cit.>To perform cyclic loading in CMD, the simulation domain is divided longitudinally into slices of equal height, corresponding to the interlayer spacing within the bulk crystal. In a face-centered cubic crystal, this spacing is half the lattice parameter along the [100] crystallographic axis. The slice containing the least number of atoms then corresponds to the minimum cross section of the nanocontact. Hence, the structure is stretched until the minimum-atom slice and either of the slices adjacent to it no longer contain any atoms as shown in Fig. <ref> c). At this point, the motion is reversed and the two ruptured tips are brought back together at the same speed with which the structure was first broken. When the minimum-atom layer contains more than 15 atoms, the motion is once more reversed and the nanocontact is stretched until it breaks. This process is repeated at least 20 times. To clarify our terminology, we denote by one “cycle" a single rupturing and re-forming of the contact. It is crucial in our simulations to know at which time step during approach, first contact occurs. We detect this moment by monitoring the value of the minimum cross section, which happens when there are more than 0 atoms in the contact cross section. This means that contact has been (re-)established. Incidentally, the semi-empirical potentials describing the interactions between the atoms in the simulations, lead to first-contact distances ranging up to half-way between first and second neighbors in a bulk FCC lattice: ∼3.5 Åin the cases of Au and Ag, and ∼3.0 Åin the case of Cu. In past works, this has also been used as the criterion to identify the moment of first contact.<cit.> Figure <ref> c) and d) show the structure prior to and immediately after first contact, respectively. Finally, to calculate the conductance of structures extracted from molecular dynamics simulation trajectories, we have used the electronic transport code ANT.G, <cit.> which depends on DFT parameters calculated by GAUSSIAN09. <cit.> The structures obtained from CMD contain more than 4000 atoms. Therefore, in order to compute the conductance of these structures within a reasonable time via DFT calculations, it has been necessary to trim the region of interest down to around 500 atoms, keeping only those atoms that lie within a box smaller than the original simulation domain, and centered on the region of first contact, or minimum cross section. However, obtaining accurate conductance values required, in addition, that we had to assign a larger basis set of 11 valence electrons to 40 atoms in the contact region. The rest of the atoms were assigned a basis set of one valence electron. §.§ Molecular Dynamics Results For the analysis of the CMD results obtained after 20 cycles of contact rupture and formation, we have used a simple algorithm that counts the number of atoms in layers spaced vertically along the simulation domain. By keeping in mind that the three layers on opposite ends of the structures remain “frozen" internally during the simulations, i.e., that the lattice parameter of these layers stays fixed at the bulk value, we discretizethe entire structure into a number of layers half the bulk lattice parameter in thickness. As lattice parameters, we used 4.08 Åfor Au and Ag, and 3.61 Åfor Cu. Consequently, during an approach (contact formation) phase, for example, we count, at every step, the number of atoms in each layer.Figure <ref> a) shows how the layers are distributed along the length of the nanocontact. The plot in Fig. <ref> c) was constructed by counting the number of atoms in each layer. Thus, in principle, a zoom-in of the atoms in the minimum cross section in a), located somewhere between layers 24 and 29, should lead us to conclude that the contact type is “4-1-1-4".Panel b) is such a zoom-in of panel a) and shows clearly what the contact type is. It therefore confirms, via visual inspection, the result inferred from panel c). The trace in Fig. <ref> d) has been constructed by plotting the minimum of the parabola in c) against simulation time step.The resemblance to an experimental conductance trace is, at the very least, suggestive. Furthermore, we would like to point out that panel c) contains more information than is used for the purposes of the present article. In fact, such a plot can also give us an idea about the evolution of the sharpness of the contact. For example, blunt electrodes will give rise to broader parabolas than sharper tips. This tool could open the way to a novel analysis of the evolution of the contact in CMD, one that renders direct visualization unnecessary. In addition, a better counting algorithm could take advantage of it. All the results in Fig. <ref> have been extracted from cycle 5 of the simulation involving Au, in which contact occurs at time step 85000. The methodology followed to count atoms in the cross section is not unique. Other algorithms, such as the one developed by Bratkovsky et al. <cit.> do not count an integer number of atoms and neighbors in the contact minimum cross section. In this work, we have modified the Bratkovsky algorithm to suit our purposes and count an integer number of atoms in the layers. We are well aware of the limitations of our method, therefore,to obtain complementary information, we calculate the conductance of the CMD structures via DFT and if, in the worst of cases, it differs very much from the expected value, we recheck the structure by visual inspection, and where necessary reassign an appropriate contact type.Thus, we have employed the approach summarized in Fig. <ref>, to study the 3 metals and the 20 cycles of contact rupture-formation they undergo during the simulations. By following the criterion that is outlined in the next paragraph, we have been able to identify different types of contacts as well as their first neighbors, as detailed in Fig.<ref>.Our criterion for identifying the contacts as single, double or triple involves counting the number of atoms in the minimum cross section between the leads, at the very moment when the corresponding layers become populated during the simulation. All three contact types can occur in a monomeric or dimeric configuration, as illustrated in Fig. <ref>. The “low" and “high" coordination designations, irrespective of whether the contacts are monomeric or dimeric, depend on the number of first neighbors found by our algorithm, on either side of the minimum-atom layer. We have established the limit of first neighbors based on an exposed (001) FCC surface layer, which, as is known, is puckered by four-fold hollows, such that an adsorbed atom will have 4 first neighbors immediately beneath it. <cit.> Then, “low" coordination means equal to or less than 4 first neighbors, in both electrodes. As soon as the limit of 4 first neighbors is exceeded at one of the electrodes, that side is designated as “high" coordination. Figure <ref> summarizes the typical contacts encountered in our simulations. For some of the contacts that form, there is an indeterminate number of possible configurations, and therefore, to simplify the statistical analysis, we use an X to represent combinations with more than 4 first-neighbor atoms. Likewise, we use a Y in combinations where the number of first-neighbor atoms are in a similar range as or larger than X (See Fig. <ref>).Hence, we have simulated contact evolution over continuous loading cycles, and studied the electronic transport during contact formation by means of DFT calculations. After 20 cycles, some of the contact types are reproduced several times, while other contact types appear only once. Table <ref> records, for every cycle, the contact type and number of first-neighbor atoms according to the nomenclature outlined in Fig. <ref>. In the same table, we have corrected the type of contact through visual inspection. Raw data about the type of contact, i.e., in the absence of visual inspection, is collected in table <ref>, in the appendix.Finally, the double and triple asterisks in table <ref> refer to those curious cases in which 2 or 3atoms close to forming a contact, contribute to the conductance across the junction, but directly via tunneling. §.§ DFT Calculations based on CMD simulations All the MD frames that have been analyzed from the point of view of the geometry in table <ref>, have also been analyzed via DFT conductance calculations. The results are shown in table <ref> and are, in addition, included in Fig. <ref>. The structures obtained from CMD simulations, which are limited in their ability to predict realistic structures, require interpretation via electronic transport calculations (if meaningful comparisons with the experimental results are to be made). Following this, upon comparing the calculated conductance and experimental density plots, we can extract information about the type of contact that is formed as well as the configuration of the first-neighbor atoms around it. The electronic transport across all the structures has been calculated by means of ANT.G <cit.>, which interfaces with GAUSSIAN09. <cit.>We have grouped the various contacts by type, and their mean conductance values and standard deviations are plotted in Fig. <ref> as dots and vertical bars, respectively. § DISCUSSIONIn this work, our aim is to find the origin of the subtle differences between materials, and identify the properties of types of contacts defined by their specific geometry. Elsewhere, we prove that relativistic effects are responsible for the large discrepancy between the jump-to-contact distances of Au and Ag <cit.>, represented by the respective means of their G_a values.To approach this problem, we use CMD as a tool to visualize the moment of first contact and identify the number and arrangement of the first neighbors. We cannot rely on CMD in the case of tunneling because the potentials only account indirectly for the effects of electrons, and hence, relativity, and, then, only very crudely. Furthermore, it is not possible, experimentally, to know the structure and geometry of the electrodes in the tunneling regime. In CMD, the structure before contact is, at times, preserved in contact, as illustrated in Figs. <ref> c) and d). At other times, significant rearrangements occur and the before-contact structures are no longer preserved. For this reason, we confine our analysis to the first neighbors in the contact regime.The conductance values obtained via DFT from the CMD structures are summarized in table <ref>. The comparison of these results with the experimental distribution of values (Figure <ref>) allows us to interpret our results in terms of the simulated geometry of the contacts. Double and triple contacts are simplified in Fig. <ref>, i.e., we don't distinguish between high or low, or monomer or dimer. Thus, the blue dot and triangle represent mean values, and their error bars, the standard deviations obtained through grouping.In spite of the reduced statistics (we have performed 20 loading cycles in CMD, on each metal), we observe how the distribution of conductance for the calculated geometries, classified as monomer, dimer and higher order contacts, mostly coincide with the three distributions obtained from the experimental data.We can therefore confirm the assignment by Untiedt et al. <cit.>, of distributions D1 as monomer, D2 as dimer, and D3 as higher coordination.Our new simulations allow us to further classify the contactsinto high- and low-coordination.This classification does not provide much additional interpretation of the experimental results due to the reduced statistics, but it does highlight the determining role of coordination on the conductance of atomic contacts.Our results for Au and Cu display a higher dispersion in conductance for monomers (D1) than dimers (D2) (Table <ref>, Fig. <ref>). The values listed in Table II exemplify how variations on the number of neighbors, for a dimer, have little repercussion on the value of the conductance. For a monomer, on the other hand, the number of neighbors result in large changes in conductance.We also find that distribution D3 likely arises from a combination of double- and triple-contact structures. This leads to a wider distribution in conductance values, as can be seen also from the experimental data. Among these structures, we have identified, through conductance calculations, the triple contact, whose conductance values are in the 2-3 G_0 range. In any event, there may be other structures that have not yet been identified, but that could be discovered by means of the new analysis methods introduced in this work. Another important difference in the conductance values obtained from the simulations, is the small dispersion in G_b of the monomeric and dimeric Cu structures as compared to Au. As alluded to earlier, in Cu, the dispersion in calculated G_b of the monomer is twice that of the dimer, which is in agreement with the broader D1 profile relative to D2 in the experimental projections. In Au, the (experimental) D2 profile exhibits a very narrow distribution, similar to the narrow dispersion in values obtained for the low-coordinated single dimers from the simulations. This may suggest that these are actually the predominant structures occurring experimentally. Finally, in the case of Au, we found a particularly good match between experimental and calculated means and standard deviations, particularly for the dimer, while, for the monomer, the calculated means are slightly over estimated.Since the simulations do not accurately capture the jump to contact, contact distances are probably shorter, leading to higher expected conductances. § SUMMARY By introducing a new statistical approach that permits identifying properties of atomic-sized contacts with greater precision, it has been possible to study, in detail, the process of formation of Au, Ag and Cu nanocontacts. This analysis allow us to identify with higher precision the distribution of values of conductance associated to different geometries, but also to extract information on the distance of contact formation for those geometries. Furthermore, we have used molecular dynamics to simulate the formation of atomic-sized contacts in STM/MCBJ experiments. These simulated contacts were, in turn, analyzed by means of a novel methodology that permits classifying their type and finding the number of first-neighbor atoms in their immediate vicinity. DFT transport calculations on the simulated structures provided a means of comparing theoretical results with the experimental data. We have demonstrated that the type of contact and the geometry of its first neighbors (shape, distance between first-neighbor atoms, and between them and the atomic contact itself) play decisive roles in electronic transport across the simulated contacts. Through a combination of the above three methods, we have found that the electronic transport across the atomic-sized contacts depends crucially on the number of first-neighbor atoms. § ACKNOWLEDGMENTS This work has been funded from the Spanish MEC through grants FIS2013-47328 and MAT2016-78625. C.S. gratefully acknowledges financial support from SEPE Servicio Público de Empleo Estatal. W.D. acknowledges funding from the National Research Foundation of South Africa through the Innovation Doctoral scholarship programme, Grant UID 102574. W.D. also thanks J. Fernandez-Rossier and J.J. Palacios for fruitful discussions. § APPENDIX The methodology described in section <ref> and illustrated in Fig. <ref> has been applied to the three materials during 20 MD rupture-formation cycles. Table <ref> summarizes the obtained results. It records, for Au, Ag and Cu (in blue, red and green, respectively), the time step (in kilosteps, or, more precisely, picoseconds) when contact is established as well as the type of first contact that is formed during every cycle. Data marked with asterisks indicate that the algorithm has detected a contact when it has not really occurred. Through visual inspection we have selected the correct CMD timeframe in which contact actually occurred and also identified the type of contact. | http://arxiv.org/abs/1709.09557v1 | {
"authors": [
"C. Sabater",
"W. Dednam",
"M. R. Calvo",
"M. A. Fern/'andez",
"C. Untiedt",
"M. J. Caturla"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170927144426",
"title": "The role of first neighbors geometry in the electronic and mechanical properties of atomic contacts"
} |
Timing and Charge measurement of single gap Resistive Plate Chamber Detectors for INO-ICAL ExperimentAnkit Gaur[Corresponding author: [email protected]] , Ashok Kumar, Md. Naimuddin Department of Physics and Astrophysics, University of Delhi,Delhi 110007, India.December 30, 2023 ================================================================================================================================================================================== The recently approved India-based Neutrino Observatory will use the world's largest magnet to study atmospheric muon neutrinos. The 50 kiloton Iron Calorimeter consists of iron alternating with single-gap resistive plate chambers. A uniform magnetic field of ∼1.5 T is produced in the iron using toroidal-shaped copper coils. Muon neutrinos interact with the iron target to produce charged muons, which are detected by the resistive plate chambers, and tracked using orthogonal pick up strips. Timing information for each layer is used to discriminate between upward and downward traveling muons. The design of the readout electronics for the detector depends critically on an accurate model of the charge induced by the muons, and the dependence on bias voltages. In this paper, we present timing and charge response measurements using prototype detectors under different operating conditions. We also report the effect of varying gas mixture, particularly SF_6, on the timing response. Keywords: ICAL, Timing, Gaseous Detector, RPC § INTRODUCTION In the Standard Model of particle physics, neutrinos are leptons with zero-charge, so they do not participate in either electromagnetic or strong interactions.As leptons, they have spin one-half, and come in three flavors (electron, muon, or tau) named after the charged-lepton flavor with which they partake in weak interactions.Once assumed to have zero-mass, the observation of neutrino flavor-oscillation revealed that at least some flavors of neutrino must have a tiny, but non-zero mass. A concerted worldwide effort to study neutrino oscillations and their masses <cit.>-<cit.> has made tremendous progress unraveling the properties of these most elusive of particles.However, there remain many unresolved questions about the nature of neutrinos.The Indian physics community is considering the construction of a world-class underground neutrino facility, the India-based Neutrino Observatory (INO) <cit.>, in order to answer some of the most important questions about the nature of neutrinos including their mass hierarchy, the extent of CP violation in their interactions, and whether or not they are Majorana particles (i.e. their own anti-particles).The INO facility will host multiple experiments including the Iron Calorimeter (ICAL) detector, which is optimized for the study of atmospheric neutrinos.The ICAL detector, will consist of three 17 kt modules each measuring 16 m×16 m×14.5 m and containing 151 iron plates, 56 mm each, interleaved with Resistive Plate Chambers (RPCs).The iron plates provide sufficient mass to cause passing atmospheric neutrinos to interact,while the RPC detectors provide tracking andfast timing measurements.The excellent timing and spatial resolution <cit.>-<cit.> of RPC detectors allows for highly accurate time of flight measurement <cit.>-<cit.>. The ICAL detector will distinguish between upward and downward traveling muon neutrinos in order to enhance the sensitivity of its physics program. As neutrinos travel at close to the speed of light, a timing resolution of the order of a few tens of nanoseconds is needed to infer the direction of travel from the arrival time at opposite ends of the ICAL detector. In addition to the timing performance, an accurate model for the charge content of pulses in the ICAL detector is needed to design the readout electronics. In this paper, we report on both the timing resolution and charge measurements on prototypes of the ICAL RPC detectors under a variety of operating conditions. § THE ICAL RPC DETECTORRPC detectors are characterized by a high efficiency and a fast response time, achieved at relatively low construction cost.Each RPC consists of two resistive plates, an anode and a cathode,typically fabricated from either bakelite or glass, separated by a gap filled with a gaseous mixture.During operation, a high voltage applied across the electrodes induces an electric field within the gaseous volume.Charged particles traversing the gap ionize the gas, producing electron-ion pairs along their trajectory.The electron-ion pairs accelerate, in opposite directions, under the influence of the electric field, reaching sufficient kinetic energy to further ionize the gas.At a sufficiently high voltage this process produces an avalanche which significantly amplifies the charge that eventually reaches the resistive plates.The readout electronics converts the charge that reaches each plate into electrical signals which encode the time and charge distribution of each pulse.More details on the operation of RPC detectors can be found in <cit.>, while the detailed geometry and construction procedures are in <cit.>-<cit.>. The massive 50 kt ICAL sampling Calorimeter requires 28,000 RPC detectors, the surface of which is 2 m× 2 m. These RPCs are constructed with 3 mm float glass and read out by 2.8 cm wide strip, made up of copper. For efficient and reliable operation of the RPCs under a variety of environmental conditions, a dedicated R&D effort is crucial for optimizing the operating parameters. In this study, we examine the effect of operational parameters on the timing resolution and charge spectra.The performance study of RPCs have also been carried out under different gaseous mixtures, details of which are provided in section <ref>.§ EXPERIMENTAL SETUPThe performance of a prototype ICAL RPC detector was characterized using cosmic ray muons detected with a plastic scintillator hodoscope. The RPC detector is placed between two large scintillators, each consisting of a polyvinyl toluene (PVT) polymer and instrumented with a Hamammastu H3178-51 Photomultiplier tube (PMT). An additional thin "finger" scintillator, the size of a single readout strip in the RPC detector allows for more precise location of incoming cosmic ray muons. The analog output pulses from the three scintillators are fed into a CAEN V814 Leading Edge Discriminator with minimum width and pulse heights optimized to reject instrumental noise while maintaining high-efficiency for cosmic ray muons.In the case of the large scintillators, the pulse width is set at 40 ns and the pulse height threshold is set at 25 mV, while for finger scintillator the settings are 40 ns and 50 mV. The discriminator output corresponding to each scintillator is fed into a CAEN V976 logic unit, where a three-way coincidence provides the trigger signal which is used as a START command for a CAEN V775 time-to-digital converter (TDC).In order to amplify short raw output pulses from the RPC, they arefed into a preamplifier, followed by an amplifier circuit characterized by a bandwidth of 0.1-1 GHz and a gain of ∼60. The output pulses from the amplifier are then fed into a CAEN V814 Leading Edge Discriminator, whose output after providing the appropriate delay with respect to the trigger signalis utilized as a common STOP command for theCAEN V775 TDC. The time interval between the START and STOP signal is converted into voltage level using the built-in TAC (time to analog converter) feature of the multichannel TDC module.The output of the TAC sections are multiplexed and subsequently converted by two fast ADC (analog to digital converter) modules. A TDCproduces output which is the absolute time difference between the START and STOP and fluctuation in their value provides the estimate of time resolution. The block diagram for the measurement of timing resolution is shown inFig. <ref>. We show inFig. <ref>, the distribution of the time difference between the START and the STOP signal. The width of this distribution provides an estimate of timing resolution. The development of ionization charge and its amplification within the RPC detector depend, among others, on the composition of the gaseous mixture. In the avalanche mode, typical gases Tetrafluoroethane (TFE), isobutane and sulphur hexaflouride are used, yielding raw signals (without preamplifier) of an amplitude of 2-5 mV under appropriate electric fields. The study of charge development due to ionization, resulting in avalanche, and their nature with respect to applied voltage gives a panoramic view of avalanche to streamer transition. A charge to digital converter (QDC) has been used for the study of charge produced under various operating conditions. The schematic of the setup for charge measurement is shown inFig. <ref>. The output pulses (analog) from each scintillator is fed into aCAEN V814 Leading Edge Discriminator for analog to digital conversion. The discriminated outputs of the scintillators are sent to CAEN V976 logic unit to obtain GATE pulse for the CAEN V965A QDC, while the analog output of the RPC after providing the appropriate delay with respect to the trigger signal used as an input for the GATE.Fig. <ref> and Fig. <ref>show an example of charge distribution for with and without SF_6gas mixture and at a particular bias voltage of 10.2 kV.§ PERFORMANCE STUDY WITH VARYING GAS MIXTURESThe RPC detector performance is strongly linked to the gas mixture it uses. It has been shown that the addition of sulphur hexafluoride (SF_6) to the mixture in the avalanche mode of operation, results in the reduction of charge produced inside the detector following the passage of charge particle along with the suppression of streamers <cit.>. Although the exact mechanism is not yet well understood, the electron affinity ofSF_6 could explain the charge reduction by the absorption of part of the produced electrons. The suppression of streamers results in streamer free operation across a large voltage range ( i.e., in the plateau region) and more stable operation at higher voltages. As the total charge produced in the gas during each event is also reduced, the presence of SF_6 also slows the aging of detector electrodes, a crucial consideration for long term stable operation of an RPC. However, this stable operation does come at the cost of a reduced efficiency at low voltages, as the same charge suppression effect leading to increased stability also suppresses small charge production at lower voltages. To determine the optimal gaseous mixture for the ICAL RPC detector, the performance of prototype RPCs was evaluated for a range of gas mixtures.Single gap RPCs were operated in avalanche mode, using five different gas mixtures, primarily varying the SF_6 fraction: * First Mixture : R134a (95.0%), C_4H_10 (5.0%), SF_6 (0.0%).* Second Mixture: R134a (95.0%), C_4H_10 (4.7%), SF_6 (0.3%).* Third Mixture: R134a (95.0%), C_4H_10 (4.5%), SF_6 (0.5%).* Fourth Mixture : R134a (95.0%), C_4H_10 (4.3%), SF_6 (0.7%).* Fifth Mixture : R134a (95.0%), C_4H_10 (4%), SF_6 (1%).The performance of the RPCs with respect to leakage current, count rate, and efficiency was measured, as a function of the applied voltage, for each gas mixture. Detectors were build from both Asahi and Saint-Gobain glass, and the effect of the type of glass used was also measured.The results of these performance measurements are shown in Figs. <ref>-<ref>.During the course of the experiment the relative humidity varied between 35-40% and the temperature varied between20^∘C and22^∘C. Across all five gas mixtures, the leakage current varies between 20 and 80 nA for RPC built from Saint-Gobain glass. The gas mixture with maximum SF_6 concentration (1%) has the lowest leakage current and the lowest count rate. The error on the current measurement is 2% of mean value ± 9 nA. The count rate is maximal, approximately 5 Hz/cm^2, for a gas mixture with no SF_6.The efficiency also decreases with the increase of the SF_6 concentration. In case of 0% SF_6, the efficiency curve turns on at8.8 kV while at higher concentration it turns on at higher values. However, at the operating region near 10.6 kV, all gas mixtures are more than 90% efficient. The RPC built with Asahi glass shows similar characteristics with a slight increase in leakage current and count rate at the cost of delayed turn-on for the efficiency.These results are qualitatively as expected, as the charge suppression from the addition of SF_6 is expected to reduce leakage current and spurious counts with the slight reduction in efficiency at low voltages. The dataset used for estimating efficiency and count rate is quite large, with statistical uncertainty of the order of a percent, and are within the markers size of the related figures. § TIMING RESOLUTION AND CHARGE DISTRIBUTIONIn order to differentiate upward from downward-traveling neutrinos, the ICAL detector will precisely measure the arrival time at each RPC in order to distinguish which end was hit first. When sufficiently well instrumented, the timing resolution of an RPC detector is dominated by fluctuations in the time required for the avalanche to reach the readout electronics.Generally a wider gaseous region increases the amplitude and results in a higher efficiency, more charge and less time walk.However, a wider gas gap also increases the time needed by the avalanche to reach the resistive plates, and the associated jitter resulting in a degradation of the timing resolution.Previous studies have shown that the timing resolution of RPCs decreases as the gas gap increases in both the avalanche and streamer mode operation <cit.>. The timing resolution as a function of applied voltage, of prototype RPCs, built with Asahi and Saint-Gobain glass, using a variety of gases, was determined by fitting a Gaussian function to the distribution of STOP-START times, as explained in section 3 and subtracting the estimated contribution from the scintillation counters used in the hodoscope.The smallest timing resolution is obtained under gas mixture having SF_6 of 0.3%. For other gas mixturesthere is degradation of time resolution which might be due to the production of less charge along with the deterioration of effective electric field and fluctuations. We measure a timing resolution of 1.6 ns for Saint-Gobain and 1.7 ns for Asahi at a 10.6 kV bias voltage for the gas mixture with SF_6 fraction of 0.3%. Fig. <ref> shows the corrected timing resolution for both Saint Gobain and Asahi prototypes RPCs. Note, however, that the timing resolution at SF_6 concentration of 0.5% is very close to that of 0.3% SF_6.The effect of varying the discriminator threshold value for the output detector pulse on the timing pulse was also determined. The threshold was varied between 30 and 70 mV and for third gas mixture i.e., R134a (95.0%), C_4H_10 (4.5%), SF_6 (0.5%). The timing resolution improves with increase in the threshold value. However, at higher bias voltages, the timing resolution does not vary significantly between the 50 and 70 mV thresholds. Fig. <ref> shows the timing resolution of prototype RPCs as a function of applied voltage at various discriminator thresholds under third gas mixture. The timing resolution varies widely at lower bias voltages while at higher voltages the variation is much smaller.A discriminator threshold of 70 mV gives best timing resolution, but at the cost of a degraded efficiency. The charge content and distribution of output pulses from the RPC were collected using the QDC as described above. The triple coincidence pulse (40 ns) of the three scintillators was used as a gate signal for the QDC and the amplified analog pulse from RPC (including delay) was used as an input. The charge output as a function of applied voltage for different prototypes is shown in Fig. <ref>. Fig. <ref> shows the root mean square values of the charge for the Saint Gobain and Asahi RPC as a function of applied voltage. § CONCLUSIONS The INO facilities ICAL experiment received final approval from the Indian government in early 2015.The preparations for construction of the tunnel and cavern are currently underway. The collaboration isset to begin construction of28,000 RPC detectors of 2 m × 2 m in size. It is important, at this point, to optimize all the operating parameters of these RPC detectors. The studies reported in this paper are a first attempt to perform a comprehensive characterization of timing and charge measurements for the 3 mm Asahi and Saint-Gobain glass RPC detectors, which are the final candidates for the first modules of ICAL detector . We found these RPCs to have more than 90% efficiency under all gas mixtures that we studied. The count rate and leakage current are found to be within reasonable limits. The gas mixtures with 0.5% and 0.3% SF_6 concentration provides comparable time resolutions, which are best amongst all the gas mixturesstudied.We measured the timing resolution at 1.6 ns for Saint-Gobain RPC and 1.7 ns for Asahi RPCs at 0.3% SF_6 concentration and at operating bias voltage of 10.6 kV. The discriminator threshold studies shows that threshold of 30 and 50 mV gives comparable timing resolution and better efficiency compared to a 70 mV threshold at which slight degradation of efficiency is observed . The charge spectra shows peak charge collection between 0.5 and 2.5 pC for all the gas mixtures studied. The charge output is nearly constant with bias voltage in the absence of SF_6, but it increases with bias voltage with the addition of SF_6 concentration in the gas mixtures. The SF_6 concentration of 0.5% results in a mean charge of approximately 0.6 pC and 2.5 pC at 9.8 kV and 11 kV bias voltage respectively. The behavior of charge measurement at 0% SF_6 concentration needs to be further investigated and better understood. In conclusion, SF_6 concentration of 0.3% provides the best timing resolution while maintaining maximum efficiency, but it suffers from relatively larger count rate and current compared to SF_6 concentration of 0.5%. Moreover, timing resolution and efficiency for SF_6 concentration of 0.5% is comparable to that of 0.3% SF_6 mixture. Since 0.5% SF_6 concentration gives lower count rate, therefore this gas composition is recommended for the INO RPC detectors.§ ACKNOWLEDGMENTS We would like to thank the Department of Science and Technology (DST), India for generous financial support. We would also like to thank the University of Delhi for providing R&D grants which have been extremely helpful in the completion of these studies. Our sincere thanks are also due to Prof. Michael Mulhearn of University of California at Davis for proof reading our article.99kam S. Abe, et al. [KamLAND Collaboration], Precision Measurement of Neutrino Oscillation Parameters with KamLAND, Phys. Rev. Lett. 100 (2008) 221803.supe R. Wendell, et al. [Super-Kamiokande Collaboration], Atmospheric neutrino oscillation analysis with subleading effects in Super-Kamiokande I, II, and III Phys. Rev. D 81 (2010) 092004 .dayF. An, et al. [DAYA-BAY Collaboration], Observation of electron-antineutrino disappearance at Daya Bay, Phys. Rev. Lett. 108 (2012) 171803, arXiv:1203.1669. ren J. Ahn, et al. [ RENO collaboration], Observation of Reactor Electron Antineutrino Disappearance in the RENO Experiment, Phys. Rev. Lett. 108 (2012) 191802, arXiv:1204.0626.ada P. Adamson, et al. [MINOS Collaboration], Measurement of Neutrino and Antineutrino Oscillations Using Beam and Atmospheric Data in MINOS, Phys. Lett 110 (2013) 251801. abe K. Abe, et al. [T2K collaboration], Precise Measurement of the Neutrino Mixing Parameter θ_23 from Muon Neutrino Disappearance in an Off-axis Beam, Phys. Rev. Lett. 112 (2014) 181801. cap F.Capozzi, et al., Status of three-neutrino oscillation parameters, circa 2013, Phys. Rev. D 89 093018. ino S. Atthar, et al. [INO Collaboration], Technical design report of INO 2006, INO Project Report, INO/2006/01, June 2006, http://www.ino.tifr.res.in/ino/OpenReports/INOReport.pdf.san R. Santonico, R. Cardarelli, Development of Resistive Plate Counters, Nucl. Instr. and Meth. A 187 (1981) 377. car R. Cardarelli, R. Santonico, Progress in Resistive Plate Counters, Nucl. Instr. and Meth. A 263 (1988) 20. aba A. Abashian, et al., The K(L) / mu detector subsystem for the BELLE experiment at the KEK B factory, Nucl. Instr. and Meth. A 449 (2000) 112. atl M. Aaboud, et al.[ATLAS Collaboration], Muon Spectrometer Technical Design Report, CERN-LHCC/97-22, 1997.cms M. Della Negra, et al. [CMS Collaboration], The Muon Project Technical Design Report, CERN/LHCC/ 97-32, 1997. blanco A. Blanco, et al., Development of large area and of position-sensitive timing RPCs, Nuclear Instruments and Methods in Physics Research A 478 (2002) 170–175blanco2 A. Blanco, et al., Single-gap timing RPCs with bidimensional position-sensitive readout for very accurate TOF systems, Nuclear Instruments and Methods in Physics Research A 508 (2003) 70–74daljit D. Kaur, et al., Characterization of 3 mm Glass Electrodes and Development of RPC Detectors for INO-ICAL Experiment, Nucl. Instr. and Meth. A 774 (2015) 74. glass Md Naimuddin, et al., Characterisation of glass electrodes and RPC detectors for INO-ICAL experiment, 2014 JINST 9 C10039, doi:10.1088/1748-0221/9/10/C10039. bakelite A. kumar, et al., Study of RPC bakelite electrodes and detector performance for INO-ICAL, 2014 JINST 9 C10042, doi:10.1088/1748-0221/C10042.abe2 K.Abe, et al., Performance of glass RPC operated in streamer mode with SF-6 gas mixture, Nucl. Instr. and Meth. A 455 (2000) 397.cerr E. Cerron Zeballos, et al., A comparison of the wide gap and narrow gap resistive plate chamber, Nucl. Instrum. Methods Phys. Res., Sect. A 373, 35 (1996). | http://arxiv.org/abs/1709.08946v1 | {
"authors": [
"Ankit Gaur",
"Ashok Kumar",
"Md. Naimuddin"
],
"categories": [
"physics.ins-det",
"hep-ex"
],
"primary_category": "physics.ins-det",
"published": "20170926112837",
"title": "Timing and Charge measurement of single gap Resistive Plate Chamber Detectors for INO-ICAL Experiment"
} |
[ C.H. Jeffrey Pang December 30, 2023 =====================With the ever growing amounts of textual data from a large variety of languages, domains and genres, it has become standard to evaluate NLP algorithms on multiple datasets in order to ensure consistent performance across heterogeneous setups. However, such multiple comparisons posetransition - should this be pose or poses? significant challenges to traditional statistical analysis methods in NLP and can lead to erroneous conclusions.In this paper we propose a Replicability Analysis framework for a statistically sound analysis of multiple comparisons between algorithms for NLP tasks. We discuss the theoretical advantages of this framework over the current, statistically unjustified, practice in the NLP literature, and demonstrate its empirical value across four applications: multi-domain dependency parsing, multilingual POS tagging,cross-domain sentiment classification and word similarity prediction. [Our code is at: https://github.com/rtmdrr/replicability-analysis-NLP .]§ INTRODUCTION The field of Natural Language Processing (NLP) is going through the data revolution. With the persistent increase of the heterogeneous web, for the first time in human history, written language from multiple languages, domains, and genres is now abundant. Naturally, the expectations from NLP algorithms also grow and evaluating a new algorithm on as many languages, domains, and genres as possible is becoming a de-facto standard. For example, the phrase structure parsers of Charniak:00 and Collins:03 were mostly evaluated on the Wall Street Journal Penn Treebank <cit.>, consisting of written, edited English text of economic news.In contrast, modern dependency parsers are expected to excel on the 19 languages of the CoNLL 2006-2007 shared tasks on multilingual dependency parsing <cit.>, and additionalchallenges, such as the shared task on parsing multiple English Web domains <cit.>, are continuouslyproposed.Despite the growing number of evaluation tasks, the analysis toolbox employed by NLP researchers has remained quite stable. Indeed, in most experimental NLP papers, several algorithms are compared on a number of datasets where the performance of each algorithm is reported together with per-dataset statistical significance figures.However, with the growing number of evaluation datasets, it becomes more challenging to draw comprehensive conclusions from such comparisons. This is because although the probability of drawing an erroneous conclusion from a single comparison is small, with multiple comparisons the probability of making one or more false claims may be very high.The goal of this paper is to provide the NLP community with a statistical analysis framework, which we term Replicability Analysis, that will allow us to draw statistically sound conclusions in evaluation setups that involve multiple comparisons. The classical goal of replicability analysis is to examine the consistency of findings across studies in order to address the basic dogma of science, that a finding is more convincingly true if it is replicated in at least one more study <cit.>. We adapt this goal to NLP, where we wish to ascertain the superiority of one algorithm over another across multiple datasets, which may come from different languages, domains and genres. Finding that one algorithm outperforms another across domains gives a sense of consistency to the results and a positive evidence that the better performance is not specific to a selected setup.["Replicability" is sometimes referred to as "reproducibility". In recent NLP work the term reproducibility was used when trying to get identical results on the same data <cit.>. In this paper, we adopt the meaning of "replicability" and its distinction from "reproducibility" from Peng:11 and Leek:15 and refer to replicability analysis as the effort to show that a finding is consistent over different datasets from different domains or languages, and is not idiosyncratic to a specific scenario.]In this work we address two questions: (1) Counting: For how many datasets does a given algorithm outperform another? and (2) Identification: What are these datasets?When comparing two algorithms on multiple datasets, NLP papers often answer informally the questions we address in this work. In some cases this is done without any statistical analysis, by simply declaring better performance of a given algorithm for datasets where its performance measure is better than that of another algorithm, and counting these datasets. In other cases answers are based on the p-values from statistical tests performed for each dataset: declaring better performance for datasets with p-value below the significance level (e.g. 0.05) and counting these datasets. While it is clear that the first approach is not statistically valid, it seems that our community is not aware of the fact that the second approach, which may seem statistically sound, is not valid as well. This may lead to erroneous conclusions, which result in adopting new (probably complicated) algorithms, while they are not better than previous (probably more simple) ones. In this work, we demonstrate this problem and show that it becomes more severe as the number of evaluation sets grows, which seems to be the current trend in NLP.We adopt a known general statistical methodology for addressing the counting (question (1)) and identification (question (2)) problems, by choosing the tests and procedures which are valid for situations encountered in NLP problems, and giving specific recommendations for such situations.Particularly, we first demonstrate (Sec. <ref>)that the current prominent approach in the NLP literature: identifying the datasets for which the difference between the performance of the algorithms reaches a predefined significance level according to some statistical significance test, does not guarantee to bound the probability to make at least one erroneous claim. Hence this approach is error-prone when the number of participating datasets is large. We thus propose an alternative approach (Sec. <ref>). For question (1), we adopt the approach of benjamini2009selective to replicability analysis of multiple studies, based on the partial conjunction framework of benjamini2008screening. This analysis comes with a guarantee that the probability of overestimating the true number of datasets with effect is upper bounded by a predefined constant. For question (2), we motivate a multiple testing procedure which guarantees that the probability of making at least one erroneous claim on the superiority of one algorithm over another is upper bounded by a predefined constant. In Section <ref> we review the statistical background for the proposed framework, which includes hypothesis testing based on multiple datasets and the partial conjunction framework introduced in <cit.>. In Section <ref> we then present two estimators, recommended for two different setups in NLP, for the number of domains on which one algorithm is superior to another, and a method for identifying these domains.In Sections <ref> and <ref> we demonstrate how to apply the proposed frameworks to two synthetic data toy examples and four NLP applications: multi-domain dependency parsing, multilingual POS tagging, cross-domain sentiment classification and word similarity prediction with word embedding models.Our results demonstrate that the current practice in NLP for addressing our questions is error-prone, and illustrate the differences between it and the proposed statistically sound approach.We hope that this work will encourage our community to increase the number of standard evaluation setups per task when appropriate (e.g. including additional languages and domains), possibly paving the way to hundreds of comparisons per study. This is due to two main reasons. First, replicability analysis is a statistically sound framework that allows a researcher to safely draw valid conclusions with well defined statistical guarantees. Moreover, this framework provides a means of summarizing a large number of experiments with a handful of easily interpretable numbers (see, e.g., Table. <ref>). This allows researchers to report results over a large number of comparisons in a concise manner, delving into details of particular comparisons when necessary. Central to the entire research in Natural Language Processing (NLP) is the practice of comparing the performance of two algorithms and arguing which one is better on a specific task. It is almost standard to compare the results after applying the algorithms on multiple datasets that are typically distinct by domain or language. For example, in the task of dependency parsing it is common to report a parsing method performance on all datasets from the CoNLL 2006-2007 shared tasks on multilingual dependency parsing <cit.> that consists of more than fifteen datasets in different languages.Lack of adequate understanding of how to connect the results from different datasets in order to determine and quantify the superiority of one algorithm over the other brought us to write this paper that introduces a proper statistical analysis to answer these questions. A notable work in the field of NLP is the work of <cit.> that presents a statistical test for determining which algorithm is better based on the difference in results between the algorithms applied on multiple datasets. However this method could not state in how many datasets one algorithm was superior to the other or point out in which datasets this effect was shown.This quantification of superiority of one algorithm over the other is of great importance to the field of NLP, mostly because one of the main directions of research these days is domain adaptation, meaning developing algorithms that train on one language with abundant train examples but are expected to perform well on datasets from varied languages or domains. Adding a quantifier of superiority allows researchers to better understand how the algorithm performs on different domains or languages. The statistical analysis we introduce to the NLP community in this paper is called replicability analysis. The classical goal of replicability analysis is to examine the consistency of findings across studies in order to address the basic dogma of science, that a finding is more convincingly a true finding if it is replicated in at least one more study, see <cit.> for discussion and further research.We adapt this goal to NLP, where we wish to ascertain the superiority of one algorithm over another across multiple datasets, which come from different domains or languages. Identifying that one algorithm outperforms another in multiple domains gives a sense of consistency of the results and an evidence that the better performance is not specific to a selected domain or language.benjamini2009selective suggested using the partial conjunction approach to replicability analysis. We espouse this approach for the analysis of performance across datasets in NLP. In Section <ref> we review the statistical background for the proposed analysis, which includes hypothesis testing based on multiple datasets and the partial conjunction framework introduced in <cit.>. In Section <ref> we present the estimator for the number of domains on which one algorithm was superior to the other, the domains are represented by the different datasets, and a method for naming these domains. The naive method of simply counting the number of datasets where the p-value was below the desired significance level α is not statistically valid as will be demonstrated in Section <ref>.Reporting on the suggested estimator, which is a single number, could also replace the commonly applied method for publishing results on each dataset in a huge table. This is an equivalent form of reporting results which is more practical and concise, hence we hope to motivate researchers to check the performance of their algorithm on a much larger collection of datasets that consists of hundreds or even thousands of different domains. Finally, in Sections <ref> and <ref> we demonstrate how to perform this analysis on one toy example and three NLP applications. Additionally, we elaborate on the conclusions that can be reported for each task. For completeness we publish a python and R implementations of all methods described in this paper for encouraging NLP researchers to apply them and report their findings accordingly.§ PREVIOUS WORKOur work recognizes the current trend in the NLP community where, for many tasks and applications, the number of evaluation datasets constantly increases. We believe this trend is inherent to language processing technology due to the multiplicity of languages and of linguistic genres and domains. In order to extend the reach of NLP algorithms, they have to be designed so that they can deal with many languages and with the various domains of each. Having a sound statistical framework that can deal with multiple comparisons is hence crucial for the field. This section is hence divided to two. We start by discussing representative examples for multiple comparisons in NLP,focusing on evaluation across multiple languages and multiple domains. We then discuss existing analysis frameworks for multiple comparisons, both in the NLP and in the machine learning literatures, pointing to the need for establishing new standards for our community. Multiple Comparisons in NLPMultiple comparisons of algorithms over datasets from different languages, domains and genres have become a de-facto standard in many areas of NLP.Here we survey a number of representative examples. A full list of NLP tasks is beyond the scope of this paper.A common multilingual example is, naturally, machine translation, where it is customary to compare algorithms across a large number of source-target language pairs. This is done, for example, with the Europarl corpus consisting of 21 European languages<cit.> and with the datasets of the WMT workshop series with its multiple domains (e.g. news and biomedical in 2017), each consisting of several language pairs (7 and 14 respectively in 2017).[http://www.statmt.org/wmt17/] Multiple dataset comparisons are also abundant in domain adaptation work. Representative tasks include named entity recognition <cit.>, POS tagging <cit.>, dependency parsing <cit.>, word sense disambiguation <cit.> and sentiment classification <cit.>. More recently, with the emergence of crowdsourcing that makes data collection cheap and fast <cit.>, an ever growing number of datasets is being created. This is particularly noticeable in lexical semantics tasks that have become central in NLP research due to the prominence of neural networks. For example, it is customary to compare word embedding models <cit.> on multiple datasets where word pairs are scored according to the degree to which different semantic relations, such as similarity and association, hold between the members of the pair <cit.>. In someworks (e.g. <cit.>) these embedding models are compared across a large number of simple tasks.Another recent popular example is work on compositional distributional semantics and sentence embedding. For example, Wieting:15 compare six sentence representation models on no less than 24 tasks.As discussed in Section <ref>, the outcomes of such comparisons are often summarized in a table that presents numerical performance values, usually accompanied with statistical significance figures and sometimes also with cross-comparison statistics such as average performance figures. Here, we analyze the conclusions that can be drawn from this information and suggest that with the growing number of comparisons, a more intricate analysis is required.The results from each dataset are usually displayed in a comparison table and the statistically significant differences are marked. However, the tests usually applied for statistical evaluation in NLP are compatible only for a comparison made on a single dataset. I.e., it is not statistically valid to count the number of significant results and report them, since the true amount of significant outcomes can be much smaller in reality.Existing Analysis Frameworks Machine learning work on multiple dataset comparisons dates back to dietterich1998approximate who raised the question: "given two learning algorithms and datasets from several domains, which algorithm will produce more accurate classifiers when trained on examples from new domains?".The seminal work that proposed practical means for this problem is that of demvsar2006statistical. Given performance measures for two algorithms on multiple datasets, the authors test whether there is at least one dataset on which the difference between the algorithms is statistically significant. For this goal they propose methods such as a paired t-test, a nonparametric sign-rank test and a wins/losses/ties count, all computedacross the results collected from all participating datasets.In contrast, our goal is to count and identify the datasets for which one algorithm significantly outperforms the other, which provides more intricate information, especially when the datasets come from different sources.However, in the presence of a large number of comparisons, the proposed methods are not adequate for answering questions such as on which datasets the differences are significant or even how many such datasets exist.Intuitively, this is because when a large number of comparisons is performed even rare events are likely to happen.In NLP, several studies addressed the problem of measuring the statistical significance of results on a single dataset (e.g. <cit.>). sogaard2013estimating is, to the best of our knowledge, the only work that addressed the statistical properties of evaluation with multiple datasets. For this aim he modified the statistical tests proposed in demvsar2006statistical to use a Gumbel distribution assumptionon the test statistics, which he considered to suit NLP better than the original Gaussian assumption. However, while this procedure aims to estimate the effect size across datasets,it answers neither the counting nor the identification question of Section <ref>.1. Not clear to me what is the "reduced error". 2. Is sogaard2013estimating the only work that addresses this question ? If so, we should say that, instead of saying he was the first.changed here - reduced error to effect size. Is this better? In the next section we provide the preliminary knowledge from the field of statistics that forms the basis for the proposed framework and then proceed with its description. § PRELIMINARIESWe start by formulating a general hypothesis testing framework for a comparison between two algorithms.This is the common type of hypothesis testing framework applied in NLP, its detailed formulation will help us develop our ideas.§.§ Hypothesis TestingWe wish to compare between two algorithms, A and B. Let X be a collection of datasets X = {X^1, X^2, …, X^N}, where for all i ∈{1, … ,N}, X^i = {x_i,1,…, x_i,n_i} . X ={[X^1= (x_1,1,…, x_1,n_1),;X^2= (x_2,1,…, x_2,n_2),; X^m (x_m1,…, x_m,n_m);X^n=(x_n,1,…, x_n,n_n) ]}.Each dataset X^i can be of a different language or a different domain. We denote by x_i,k the granular unit on which results are being measured, which in most NLP tasks is a word or a sequence of words. The difference in performance between the two algorithms is measured using one or more of the evaluation measures in the set ℳ = {ℳ_1,…,ℳ_m}.[To keep the discussion concise, throughout this paper we assume that only one evaluation measure is used. Our framework can be easily extended to deal with multiple measures.]Let us denote ℳ_j(ALG,X^i) as the value of the measure ℳ_j when algorithm ALG is applied on the dataset X^i. Without loss of generality, we assume that higher values of the measure are better. We define the difference in performance between two algorithms, A and B, according to the measure ℳ_j on the dataset X^i as:δ_j(X^i) = ℳ_j(A,X^i) - ℳ_j(B,X^i). Finally, using this notation we formulate the following statistical hypothesis testing problem:H_0i(j):δ_j(X^i) ≤ 0 H_1i(j):δ_j(X^i) > 0.The null hypothesis, stating that there is no difference between the performance of algorithm A and algorithm B, or that B performs better, is tested versus the alternative statement that A is superior.If the statistical test results in rejecting the null hypothesis, one concludes that A outperforms B in this setup.Otherwise, there is not enough evidence in the data to make this conclusion. Rejection of the null hypothesis when it is true is termed type 1 error, and non-rejection of the null hypothesis when the alternative is true is termed type 2 error. The classical approach to hypothesis testing is to find a test that guarantees that the probability of making a type 1 error is upper bounded by a predefined constant α, the test significance level, while achieving as low probability of type 2 error as possible, a.k.a achieving as high power as possible. Erroneous rejection of the null hypothesis when it is true is known as performing a type 1 error. The significance level of the test, denoted by α, is the upper bound for the probability of making this type of error.A statistical test is called valid if it controls a certain type 1 error criterion, i.e., it guarantees to bound the error criterion, such as the significance level, by a predefined constant. However, one can achieve validity by never rejecting any null hypothesis, hence the quality of a statistical test is also being measured by its power: the probability that it would reject a false-null hypothesis. In general, we wish to design tests that are both valid and powerful.We next turn to the case where the difference between two algorithms is tested across multiple datasets. §.§ The Multiplicity ProblemEquation <ref> defines a multiple hypothesis testing problem when considering the formulation for all N datasets.If N is large, testing each hypothesis separately at the nominal significance level may result in a high number of erroneously rejected null hypotheses.This is the first place in the text where "nominal significance level" is used. This term has not been explained before. In our context, when the performance of algorithm A is compared to that of algorithm B across multiple datasets, and for each dataset algorithm A is declared as superior based on a statistical test at the nominal significance level α, the expected number of erroneous claims may grow as N grows. For example, if a single test is performed with a significance level of α = 0.05, there is only a 5% chance of incorrectly rejecting the null hypothesis. On the other hand, for 100 tests where all null hypotheses are true, the expected number of incorrect rejections is 100· 0.05 = 5. Denoting the total number of type 1 errors as V, we can see below that if the test statistics are independent then the probability of making at least one incorrect rejection is 0.994:ℙ(V>0) = 1- ℙ(V=0) =1-∏_i=1^100ℙ(no type 1 error in i) =1-(1-0.05)^100This demonstrates that the naive method of counting the datasets for which significance was reached at the nominal level is error-prone.Again, I do not understand the usage of the word "may" in this context. I cannot map this sentence to a meaning. Similar examples can be constructed for situations where some of the null hypotheses are false.The multiple testing literature proposes various procedures for bounding the probability of making at least one type 1 error, as well as other, less restrictive error criteria (see a survey at <cit.>). In this paper, we address the questions of counting and identifying the datasets for which algorithm A outperforms B, with certain statistical guarantees regarding erroneous claims. While identifying the datasets gives more information when compared to just declaring their number, we consider these two questions separately.As our experiments show, according to the statistical analysis we propose the estimated number of datasets with effect (question 1) may be higher than the number of identified datasets (question 2).We next present the fundamentals of the partial conjunction framework which is at the heart of our proposed methods. §.§ Partial Conjunction HypothesesWe start by reformulating the set of hypothesis testing problems of Equation <ref> as a unified hypothesis testing problem. This problem aims to identify whether algorithm A is superior to B across all datasets. The notation for the null hypothesis in this problem is H_0^N/N since we test if N out of N alternative hypotheses are true: H_0^N/N: ⋃_i=1^NH_0i is true vs.H_1^N/N: ⋂_i=1^NH_1i is true.Requiring the rejection of the disjunction of all null hypotheses is often too restrictive for it involves observing a significant effect on all datasets, i ∈{1,…,N}.Instead, one can require a rejection of the global null hypothesis stating that all individual null hypotheses are true, i.e., evidence that at least one alternative hypothesis is true. This hypothesis testing problem is formulated as follows:H_0^1/N: ⋂_i=1^NH_0i is true vs.H_1^1/N: ⋃_i=1^NH_1i is true.Obviously, rejecting the global null may not provide enough information: it only indicates that algorithm A outperforms B on at least one dataset. Hence, this claim does not give any evidence for the consistency of the results across multiple datasets.A natural compromise between the above two formulations is to test the partial conjunction null, which states that the number of false null hypotheses is lower than u, where 1≤ u ≤ N is a pre-specified integer constant. The partial conjunction test contrasts this statement with the alternative statement that at least u out of the N null hypotheses are false.Consider N ≥ 2 null hypotheses: H_01,H_02,…,H_0N, and let p_1,…,p_N be their associated p-values. Let k be the true unknown number of false null hypotheses, then our question "Are at least u out of N null hypotheses false?" can be formulated as follows:H_0^u/N:k<uvs.H_1^u/N:k≥ u .In our context, k is the number of datasets where algorithm A is truly better, and the partial conjunction test examines whether algorithm A outperforms algorithm B in at least u of N cases.From here to the beginning of 4.1 quite a lot is changed, to avoid repititions. Please review the new text.benjamini2008screening developed a general method for testing the above hypothesis for a given u. They also showed how to extend their method in order to answer our counting question. We next describe their framework and advocate a different, yet related method for dataset identification.§ REPLICABILITY ANALYSIS FOR NLP Referred to as the cornerstone of science <cit.>, replicability analysis is of predominant importance in many scientific fields including psychology <cit.>, genomics <cit.>, economics <cit.> and medicine <cit.>, among others.Findings are usually considered as replicated if they are obtained in two or more studies that differ from each other in some aspects (e.g. language, domain or genre in NLP). The replicability analysis framework we employ <cit.> is based on partial conjunction testing. Particularly, these authors have shown that a lower bound on the number of false null hypotheses with a confidence level of 1-α can be obtained by finding the largest u for which we can reject the partial conjunction null hypothesis H_0^u/N along with H_0^1/N,…, H_0^(u-1)/N at a significance level α.This is since rejecting H_0^u/N means that we see evidence that in at least u out of N datasets algorithm A is superior to B. This lower bound on k is taken as our answer to the Counting question of Section <ref>.In line with the hypothesis testing framework of Section <ref>, the partial conjunction null, H_0^u/N, is rejected at level α if p^u/N≤α, where p^u/N is the partial conjunction p-value. Based on the known methods for testing the global null hypothesis (see, e.g., <cit.>), benjamini2008screening proposed methods for combining the p-values p_1,…,p_N of H_01,H_02,…,H_0N in order to obtain p^u/N.Below, we describe two such methods and their properties. §.§ The Partial Conjunction p-valueThe methods we focus at were developed inbenjamini2008screening, and are based on Fisher's and Bonferroni's methods for testing the global null hypothesis. For brevity, we name them Bonferroni and Fisher. We choose them because they are valid in different setups that are frequently encountered in NLP (Section <ref>): Bonferroni for dependent datasets and both Fisher and Bonferroni for independent datasets.[For simplicity we refer to dependent/independent datasets as those for which the test statistics are dependent/independent. We assume the test statistics are independent if the corresponding datasets do not have mutual samples, and one dataset is not a transformation of the other.]Bonferroni's method does not make any assumptions about the dependencies between the participating datasets and it is hence applicable in NLP tasks, since in NLP it is most often hard to determine the type of dependence between the datasets. Fisher's method, while assuming independence across the participating datasets, is often more powerful than Bonferroni's methodor other methods which make the same independence assumption (see <cit.> for other methods and a comparison between them). Our recommendation is hence to use the Bonferroni's method when the datasets are dependent and to use the more powerful Fisher's method when the datasets are independent. Let p_(i) be the i-th smallest p-value amongp_1,…,p_N. The partial conjunction p-values are: p^u/N_ Bonferroni = (N-u+1)p_(u)p^u/N_ Fisher = ℙ(χ^2_2(N-u+1)≥ -2∑_i=u^Nln p_(i))where χ^2_2(N-u+1) denotes a chi-squared random variable with 2(N-u+1) degrees of freedom. To understand the reasoning behind these methods, let us consider first the above p-values for testing the global null, i.e., for the case of u=1.Rejecting the global null hypothesis requires evidence that at least one null hypothesis is false. Intuitively, we would like to see one or more small p-values. Both of the methods above agree with this intuition. Bonferroni's method rejects the global null if p_(1)≤α/N, i.e. if the minimum p-value is small enough, where the threshold guarantees that the significance level of the test is α for any dependency among the p-values p_1,…,p_N. Fisher's method rejects the global null for large values of -2∑_i=1^Nln p_(i), or equivalently for small values of ∏_i=1^N p_i.That is, while both these methods are intuitive, they are different. Fisher's method requires a small enough product of p-values as evidence that at least one null hypothesis is false. Bonferroni's method, on the other hand, requires as evidence at least one small enough p-value.Now let us consider testing the partial conjunction hypothesis for u>1. If the alternative is true, i.e., at least u null hypotheses are false, then one will find one or more false null hypotheses in any subset of n-u+1 hypotheses.Thus in order to see evidence that the alternative is true, it is intuitive to require rejection of the global null (the intersection hypothesis) for all subsets of n-u+1 hypotheses. Both Bonferroni's and Fisher's methods agree with this intuition. The p-value in Equation <ref> is below α if for every subset of n-u+1 hypotheses the Bonferroni's global null p-value is below α hence the global null is rejected at the significance level of α for every subset of n-u+1 hypotheses.Similarly, Fisher's method rejects H_0^u/n if for all subsets of n-u+1 null hypotheses, the Fisher's global null p-value is below α.For the case u=N, i.e., when the alternative states that all null hypotheses are false, both methods require that the maximal p-value is small enough for rejection of H_0^N/N. This is also intuitive because we expect that all the p-values will be small when all the null hypotheses are false. For other cases, where 1<u<N, the reasoning is more complicated and is beyond the scope of this paper.The partial conjunction test for a specific u answers the question "Does algorithm A perform better than B on at least u datasets?" The next step is the estimation of the number of datasets for which algorithm A performs better than B.§.§ Dataset Counting (Question 1)Recall that the number of datasets where algorithm A outperforms algorithm B (denoted with k in Definition <ref>) is the true number of false null hypotheses in our problem. benjamini2008screening proposed to estimate k to be the largest u for which H_0^u/N, along with H_0^1/N,…,H_0^(u-1)/N is rejected. Specifically, the estimator k̂ is defined as follows:k̂ = max{u:p^u/N_*≤α},where p^u/N_* = max{p^(u-1)/N_* ,p^u/N}, p^1/N=p^1/N_*and α is the desired upper bound on the probability to overestimate the true k.We do not give any explanation as to why they use this non-standard formulation.the only explanation we have written before is that using p* instead of p will guarantee that ℙ(k̂>k) ≤α, I don't know if we want to write this down. It is guaranteed that ℙ(k̂>k) ≤α as long as the p-value combination method used for constructing p^u/N is valid for the given dependency across the test statistics.[This result is a special case of Theorem 4 in <cit.>.]When k̂ is based on p^u/N_Bonferroni it is denoted with k̂_Bonferroni, while when it is based on p^u/N_Fisher it is denoted withk̂_Fisher.A crucial practical consideration when choosing between k̂_ Bonferroni and k̂_ Fisher is the assumed dependency between the datasets. As discussed in Section <ref>, p^u/N_Fisher is recommended when the participating datasets are assumed to be independent, while when this assumption cannot be made only p^u/N_Bonferroni is appropriate. As the k̂ estimators are based onthe respective p^u/Ns, the same considerations hold when choosing between them. With the k̂ estimators, one can answer the counting question of Section <ref>, reporting that algorithm A is better than algorithm B in at least k̂ out of N datasets with a confidence level of 1-α.Regarding the identification question, a natural approach would be to declare the k̂ datasets with the smallest p-values as those for which the effect holds. However, with k̂_Fisher this approach does not guarantee control over type 1 errors. In contrast, for k̂_Bonferroni the above approach comes with such guarantees, as described in the next section. §.§ Dataset Identification (Question 2) As demonstrated in Section <ref>, identifying the datasets with p-value below the nominal significance level and declaring them as those where algorithm A is better than B may lead to a very high number of erroneous claims. A variety of methods exist for addressing this problem.A classical and very simple method for addressing this problem is named the Bonferroni's procedure, which compensates for the increased probability of making at least one type 1 error by testing each individual hypothesis at a significance level of α' = α /N, where α is the predefined bound on this probability and N is the number of hypotheses tested.[Bonferroni's correction is based on similar considerations asp^u/N_ Bonferroni for u=1 (Eq. 2). The partial conjunction framework (Sec. <ref>) extends this idea for other values of u.]While Bonferroni's procedure is valid for any dependency among the p-values, the probability of detecting a true effect using this procedure is often very low, because of its strict p-value threshold.Many other procedures controlling the above or other error criteria and having less strict p-value thresholds have been proposed.Below we advocate one of these methods: the Holm procedure <cit.>. This is a simple p-value based procedure that is concordant with the partial conjunction analysis when p^u/N_Bonferroni is used in that analysis. Importantly for NLP applications,Holm controls the probability of making at least one type 1 error for any type of dependency between the participating datasets (see a demonstration in Section <ref>).Let α be the desired upper bound on the probability that at least one false rejection occurs, let p_(1)≤ p_(2)≤…≤ p_(N) be the ordered p-values and let the associated hypotheses be H_(1)… H_(N). The Holm procedure for identifying the datasets with a significant effect is given in below. ) Holm() Let k be the minimal index such that p_(k)>α/N+1-k.Reject the null hypotheses H_(1)… H_(k-1) and do not reject H_(k)… H_(N). If no such k exists, then reject all null hypotheses.The output of the Holm procedure is a rejection list of null hypotheses,the corresponding datasets are those we return in response to the identification question of Section <ref>. Note that the Holm procedure rejects a subset of hypotheses with p-value below α. Each p-value is compared to a threshold which is smaller or equal to α and depends on the number of evaluation datasets N. The dependence of the thresholds on N can be intuitively explained as follows. The probability of making one or more erroneous claims may increase with N, as demonstrated in Section <ref>. Therefore, in order to bound this probability by a pre-specified level α, the thresholds for p-values should depend on N.It can be shown that the Holm procedure at level α always rejects the k̂_Bonferroni hypotheses with the smallest p-values, where k̂_Bonferroni is the lower bound for k with a confidence level of 1-α. Therefore, k̂_Bonferroni corresponding to a confidence level of 1-α is always smaller or equal to the number of datasets for which the difference between the compared algorithms is significant at level α. This is not surprising in view of the fact that without making any assumptions on the dependencies among the datasets, k̂_Bonferroni guarantees that the probability of making a too optimistic claim (k̂ > k) is bounded by α, while when simply counting the number of datasets with p-value below α, the probability of making a too optimistic claim may be close to 1, as demonstrated in Section <ref>. Framework Summary Following Section <ref> we suggest to answer the counting question of Section <ref> by reporting either k̂_Fisher (when all datasets can be assumed to be independent) or k̂_Bonferroni (when such an independence assumption cannot be made). Based on Section <ref> we suggest to answer the identification question of Section <ref> by reporting the rejection list returned by the Holm procedure. Our proposed framework is based on certain assumptions regarding the experiments conducted in NLP setups. The most prominent of these assumptions states that for dependent datasets the type of dependency cannot be determined. Indeed, to the best of our knowledge, the nature of the dependency between dependent test sets in NLP work has not been analyzed before. In Section <ref> we revisit our assumptions and point on alternative methods for answering our questions. These methods may be appropriate under other assumptions that may become relevant in future. We next demonstrate the value of the proposed replicability analysis through toy examples with synthetic data (Section <ref>) as well as analysis of state-of-the-art algorithms for four major NLP applications (Section <ref>). Our point of reference is the standard, yet statistically unjustified, counting method that sets its estimator, k̂_count, to the number of datasets for which the difference between the compared algorithms is significant with p-value≤α (i.e. k̂_count=#{i:p_i ≤α}).[We use α in two different contexts: the significance level of an individual test and the bound on the probability to overestimate k. This is the standard notation in the statistical literature.]The k̂ estimator value with the rejection list from the Holm procedure define the replicability analysis output we suggest to report every time a dataset multiplicity is tested in NLP. In the next two sections, we show on a variety of scenarios how to report replicability. For each experiment we calculate k̂ and report the output of the Holm procedure. Before that, we shortly refer to another case of multiplicity that is common in NLP, the case of reporting results from multiple datasets using multiple evaluation metrics. We elaborate on the changes that need to be applied on the replicability analysis we presented here, so that one will be able to report the k̂ estimator for this case as well.§.§ Replicability Analysis with Multiple Measures Another type of multiplicity that can be found in NLP is the usage of multiple measures. For example, in dependency parsing it is common to report both unlabeled and labeled attachment scores (UAS and the LAS, respectively), and even exact matches between the induced and the gold dependency trees. Other examples are machine translation and text summarization where it is standard to report results with BLEU, ROUGE and other measures. In this section we hence define replicability analysis when multiple measures are considered for each dataset using the notations presented in <cit.>.Consider n ≥ 2 null hypotheses at each metric j ∈{1,…,m}: H_01(j),H_02(j),…,H_0n(j), and let p_1(j),…,p_n(j) be their associated p-values. Let k(j) be the true unknown number of false null hypotheses for metric j, then the question "For a given metric j, are at least u out of n null hypotheses false?" can be formulated as follows:H_0^u/n(j):k(j)<uvs.H_1^u/n(j):k(j)≥ u . Since the typical number of measures commonly reported for NLP tasks is quite small compared to the number of features in the application considered in <cit.>, we propose a simpler method for addressing multiple measures. Specifically, we integrate the Bonferroni correction <cit.> into the k̂ estimator.Hence for every metric j ∈{1,…,m} we define the k̂(j) estimator as:k̂(j) = max{u:p^u/n_*(j)≤α/m}. This method of adjusting for multiplicity guarantees that ℙ(∃ j: k̂(j)>k(j)) ≤α. This property follows from the fact that ℙ(k̂(j)>k(j)) ≤α for each metric j, and from the union bound inequality.In the next sections we demonstrate the value of the proposed replicability analysis. We start with a toy example (Section <ref>) and continue with the analysis of state-of-the-art methods for three major NLP applications (Section <ref>). Our point of reference is the standard counting method that sets its estimator k̂_count to the number of datasets for which the difference between the compared algorithms is significant with p-value< α. § TOY EXAMPLESFor the examples of this section we synthesize p-values to emulate a test with N =100 hypotheses (domains), and set α to 0.05. changed here - added footnote We start with a simulation of a scenario where algorithm A is equivalent to B for each domain, and the datasets representing these domains are independent. We sample the 100 p-values from a standard uniform distribution, which is the p-value distribution under the null hypothesis, repeating the simulation1000 times.Since all the null hypotheses are true then k, the number of false null hypotheses, is 0.Figure <ref> presents the histogram of k̂ values from all 1000 iterations according to k̂_Bonferroni, k̂_Fisher and k̂_count.The figure clearly demonstrates that k̂_count provides an overestimation of k while k̂_Bonferroni and k̂_Fisher do much better. Indeed, the histogram yields the following probability estimates: P̂(k̂_count>k)=0.963, P̂(k̂_Bonferroni>k)=0.001 and P̂(k̂_Fisher>k)=0.021 (only the latter two are lower than 0.05). This simulation strongly supports the theoretical results of Section <ref>.Notice that for k̂_count α is used twice - once at the level of the individual dataset (i.e. only datasets with p-value lower than α are counted) and once as a confidence level for the counting (i.e. we would like the probability that k̂_count > k to be lower than α . Should we say something about this ? I am quite perplexed by this issue. To consider a scenario where a dependency between the participating datasets does exist, we consider a second toy example. In this example we generate N=100 p-values corresponding to 34 independent normal test statistics, and two other groups of 33 positively correlated normal test statistics with ρ=0.2 and ρ=0.5, respectively.We again assume that all null hypotheses are true and thus all the p-values are distributed uniformly, repeating the simulation 1000 times. To generate positively dependent p-values we followed the process described in Section 6.1 of benjamini2006adaptive.There are a couple of details that I do not understand here due to lack of statistical knowledge. For example, how can the p-values be evenly distributed but still have positive correlations. Or, which type of correlation test does the ρ refer to and so on. I wonder if we want to get into this. We estimate the probability that k̂>k=0 for the three k̂ estimators based on the 1000 repetitions and get the values of: P̂(k̂_count>k)=0.943, P̂(k̂_Bonferroni>k)=0.046 and P̂(k̂_Fisher>k)=0.234. This simulation demonstrates the importance of using Bonferroni's method rather than Fisher's method when the datasets are dependent, even if some of the datasets are independent.§ NLP APPLICATIONS In this section we demonstrate the potential impact of replicability analysis on the way experimental results are analyzed in NLP setups. We explore four NLP applications: (a) two where the datasets are independent: multi-domain dependency parsing and multilingual POS tagging; and (b) two where dependency between the datasets does exist: cross-domain sentiment classification and word similarity prediction with word embedding models.§.§ Data Dependency Parsing We consider a multi-domain setup, analyzing the results reported in choi2015depends. The authors compared ten state-of-the-art parsers from which we pick three: (a) Mate <cit.>[<code.google.com/p/mate-tools>.] that performed best on the majority of datasets; (b) Redshift <cit.>[<github.com/syllog1sm/Redshift>.] which demonstrated comparable, still somewhat lower, performance compared to Mate; and (c) SpaCy <cit.>[<honnibal.github.io/spaCy>.] that was substantially outperformed by Mate.All parsers were trained and tested on the English portion of the OntoNotes 5 corpus <cit.>, a large multi-genre corpus consisting of the following 7 genres: broadcasting conversations (BC), broadcasting news (BN), news magazine (MZ), newswire (NW), pivot text (PT), telephone conversations (TC) and web text (WB). Train and test set size (in sentences) range from 6672 to 34492 and from 280 to 2327, respectively (see Table 1 of <cit.>). We copy the test set UAS results of choi2015depends andcompute p-values using the data downloaded from <http://amandastent.com/dependable/>.For the multilingual setup we experiment with the TurboParser<cit.> on all 19 languages of the CoNLL 2006 and 2007 shared tasks on multilingual dependency parsing <cit.>. We compare the performance of the first order parser trained with the MIRA algorithm to the same parser when trained with the perceptron algorithm, both implemented within the TurboParser. POS Tagging We consider a multilingual setup, analyzing the results reported in <cit.>. The authors compare their Mimick model with the model of Ling:15, denoted with char→tag. Evaluation is performed on 23 of the 44 languages shared by the Polyglot word embedding dataset <cit.> and the universal dependencies (UD) dataset <cit.>. Pinter:17 choose their languages so that they reflect a variety of typological, and particularly morphological, properties. The training/test split is the standard UD split. We copy the word level accuracy figures of <cit.> for the low resource training set setup, the focus setup of that paper. The authors kindly sent us their p-values. Sentiment ClassificationIn this task, an algorithm is trained on reviews from one domain and should classify the sentiment of reviews from another domain to the positive and negative classes. For replicability analysis we explore the results of ziser2016neural for the cross-domain sentiment classification task of blitzer2007biographies. The data in this task consists of Amazon product reviews from 4 domains: books (B), DVDs (D), electronic items (E), and kitchen appliances (K), for the total of 12 domain pairs, each domain having a 2000 review test set.[<http://www.cs.jhu.edu/ mdredze/datasets/sentiment/index2.htm>] ziser2016neural compared the accuracy of their AE-SCL-SR model to MSDA <cit.>, a well known domain adaptation method, and kindly sent us the required p-values. Word Similarity We compare two state-of-the-art word embedding collections: (a) word2vec CBOW <cit.> vectors, generated by the model titled the best "predict" model in Baroni:14;[<http://clic.cimec.unitn.it/composes/semantic-vectors.html>. Parameters: 5-word context window, 10 negative samples, subsampling, 400 dimensions.] and (b) Glove <cit.> vectors generated by a model trained on a 42B token common web crawl.[http://nlp.stanford.edu/projects/glove/. 300 dimensions.] We employed the demo of faruqui-2014:SystemDemo to perform Spearman correlation evaluation of these vector collections on 12 English word pair datasets: WS-353 <cit.>, WS-353-SIM <cit.>, WS-353-REL <cit.>, MC-30 <cit.>, RG-65 <cit.>, Rare-Word <cit.>, MEN <cit.>, MTurk-287 <cit.>, MTurk-771 <cit.>, YP-130 <cit.>, SimLex-999 <cit.>, and Verb-143 <cit.>.§.§ Statistical Significance Tests We first calculate the p-values for each task and dataset according to the principals of p-values computation for NLP discussed in yeh2000more, berg2012empirical and sogaard2014s.For dependency parsing, we employ the a-parametric paired bootstrap test <cit.> that does not assume any distribution on the test statistics. We choose this test because the distribution of the values for the measures commonly applied in this task is unknown. We implemented the test as in <cit.> with a bootstrap size of 500 and with 10^5 repetitions.For multilingual POS tagging we employ the Wilcoxon signed-rank test <cit.>. The test is employed for the sentence level accuracy scores of the two compared models. This test is a non-parametric test for difference in measures, which tests the null hyphotesis that the difference has a symmetric distribution around zero. It is appropriate for tasks with paired continuous measures for each observation, which is the case when comparing sentence level accuracies. For sentiment classification we employ the McNemar test for paired nominal data <cit.>. This test is appropriate for binary classification tasks and since we compare the results of the algorithms when applied on the same datasets, we employ its paired version. Finally, for word similarity with its Spearman correlation evaluation, we choose the Steiger test <cit.> for comparing elements in a correlation matrix. We consider the case of α = 0.05 for all four applications. For the dependent datasets experiments (sentiment classification and word similarity prediction) with their generally lower p-values (see below), we also consider the case where α=0.01.§.§ ResultsLanguge P-value Tag acc Mim acc ta 0.000108 0.8506 0.8216 lv 0.062263 0.8384 0.823 vi 0.035912 0.8467 0.8421 hu 1.12E-08 0.8597 0.8896 tr 0.146036 0.8478 0.8491 bg 0.195676 0.9242 0.9216 sv 0.093904 0.9241 0.9187 ru 0.008147 0.8955 0.9039 da 0.101559 0.8937 0.8809 fa 0.444998 0.9351 0.9346 he 0.102531 0.9164 0.9137 en 0.0208 0.8552 0.853 hi 0.028848 0.8811 0.8781 it 0.481173 0.9237 0.9235 es 0.117623 0.9106 0.9035 cs 2.91E-05 0.8999 0.8961Table <ref> summarizes the replicability analysis results while Table <ref> – <ref> present task specific performance measuresand p-values. Independent Datasets Dependency parsing (Tab. <ref>) and multilingual POS tagging (Tab. <ref>) are our example tasks for this setup, where k̂_Fisher is our recommended valid estimator for the number of cases where one algorithm outperforms another.For dependency parsing, we compare two scenarios: (a) where in most domains the differences between the compared algorithms are quite large and the p-values are small (Mate vs. SpaCy); and (b) where in most domains the differences between the compared algorithms are smaller and the p-values are higher (Mate vs. Redshift).Our multilingual POS tagging scenario (Mimick vs. Char→Tag) is more similar to scenario (b) in terms of the differences between the participating algorithms.Table <ref> demonstrates the k̂ estimators for the various tasks and scenarios.For dependency parsing, as expected, in scenario (a) where all the p-values are small, all estimators, even the error-prone k̂_count, provide the same information. In case (b) of dependency parsing, however, k̂_Fisher estimates the number of domains where Mate outperforms Redshift to be 5, while k̂_count estimates this number to be only 2. This is a substantial difference given that the total number of domains is 7. The k̂_Bonferroni estimator, that is valid under arbitrary dependencies and does not exploit the independence assumption as k̂_Fisher does, is even more conservative than k̂_count and its estimation is only 1. Perhaps not surprisingly, the multilingual POS tagging results are similar to case (b) of dependency parsing. Here, again, k̂_count is too conservative, estimating the number of languages with effect to be 11 (out of 23) while k̂_Fisher estimates this number to be 16 (an increase of 5/23 in the estimated number of languages with effect). k̂_Bonferroni is again more conservative, estimating the number of languages with effect to be only 6, which is not very surprising given that it does not exploit the independence between the datasets. These two examples of case (b) demonstrate that when the differences between the algorithms are quite small, k̂_Fisher may be more sensitive than the current practice in NLP for discovering the number of datasets with effect.To complete the analysis, we would like to name the datasets with effect.As discussed in Section <ref>, while this can be straightforwardly done by naming the datasets with the k̂ smallest p-values, in general, this approach does not control the probability of identifying at least one dataset erroneously. We thus employ the Holm procedure for the identification task, noticing thatthe number of datasets it identifies should be equal tothe value of the k̂_Bonferroni estimator (Section <ref>).Indeed, for dependency parsing in case (a), the Holm procedure identifies all seven domains as cases where Mate outperforms SpaCy, while in case (b) it identifies only the MZ domain as a case where Mate outperforms Redshift. For multilingual POS tagging the Holm procedure identifies Tamil, Hungarian, Basque, Indonesian, Chinese and Czech as languages where Mimick outperforms Char→Tag. Expectedly, this analysis demonstrates that when the performance gap between two algorithms becomes narrower, inquiring for more information (i.e. identifying the domains with effect rather than just estimating their number), may come at the cost of weaker results.[For completeness, we also performed the analysis for the independent dataset setups with α = 0.01. The results are (k̂_count, k̂_Bonferroni, k̂_Fisher): Mate vs. Spacy: (7,7,7); Matevs. Redshift (1,0,2); Mimick vs. Char→Tag: (7,5,13). The patterns are very similar to those discussed in the text.]Dependent Datasets Incross-domain sentiment classification (Table <ref>) and word similarity prediction (Table <ref>), the involved datasets manifest mutual dependence. Particularly, each sentiment setup shares its test dataset with 2 other setups, while in word similarity WS-353 is the union of WS-353-REL and WS-353-SIM. As discussed in Section <ref>, k̂_Bonferroni is the appropriate estimator of the number of cases one algorithm outperforms another.I am not sure if the sharing of training domains is also relevant here. I am afraid the reader will be confused. The results in Table <ref> manifest the phenomenon demonstrated by the second toy example in Section <ref>, which shows that when the datasets are dependent, k̂_Fisher as well as the error-prone k̂_count may be too optimistic regarding the number of datasets with effect. This stands in contrast to k̂_Bonferroni that controls the probability to overestimate the number of such datasets. Indeed, k̂_Bonferroni is much more conservative, yielding values of 6 (α = 0.05) and 2 (α = 0.01) for sentiment, and of 6 (α = 0.05) and 4 (α = 0.01) for word similarity. The differences from the conclusions that might have been drawn by k̂_count are again quite substantial. The difference between k̂_Bonferroniand k̂_count in sentiment classification is 4, which accounts to 1/3 of the 12 test setups. Even for word similarity, the difference between the two methods, which account to 2 for both α values, represents 1/6 of the 12 test setups.The domains identified by the Holm procedure are marked in the tables. Results OverviewOur goal in this section is to demonstrate that the approach of simply looking at the number of datasets for which the difference between the performance of the algorithms reaches a predefined significance level,gives different results from our suggested statistically sound analysis. This approach is denoted here with k̂_count and shown to be statistically not valid in Sections <ref> and <ref>. We observe that this happens especially in evaluation setups where the differences between the algorithms are small for most datasets. In some cases, when the datasets are independent, our analysis has the power to declare a larger number of datasets with effect than the number of individual significant test values (k̂_count). In other cases, when the datasets are interdependent, k̂_count is much too optimistic.Our proposed analysis changes the observations that might have been made based on thepapers where the results analysed here were originally reported. For example, for the Mate-Redshift comparison (independent evaluation sets), we show that there isevidence that the number of datasets with effect is much higher than one would assume based on counting the significant sets (5 vs. 2 out of 7 evaluation sets), giving a stronger claim regarding the superiority of Mate. In multingual POS tagging (again, an independent evaluation sets setup) our analysis shows evidence for 16 sets with effect compared to only 11 of the erroneous count method - a difference in 5 out of 23 evaluation sets (21.7%). Finally, in the cross-domain sentiment classification and the word similarity judgment tasks (dependent evaluation sets), the unjustified counting method may be too optimistic (e.g. 10 vs. 6 out of 12 evaluation sets, for α = 0.05 in the sentiment task), in favor of the new algorithms.Dependency ParsingUAS and p-values for multi-domain dependency parsing are presented in Table <ref>.Based on these results and under the assumption that, taken from different domains, the datasets are independent, we compute the k̂_Fisher estimator.For the Mate vs. SpaCy comparison, both k̂_Fisher and k̂_count agree the differences are significant for all 7 domains. This is not surprising given the low p-values and the selected α. Moreover, the Holm procedure also identifies all datasets as cases where Mate performs significantly better.As expected, this analysis indicates that when the p-values are very small compared to the required confidence level, the traditional count analysis is sufficient.1. Isn't there any value in providing the k̂_Bonferonni number and discussing its relevance here ? Even if its statistical assumption does not hold, won't it help to provide its estimate and explain why not use it ?The comparison between Mate and Redshift allows us to explore the value of replicability analysis in a scenario where the differences between the two compared parsers are not as substantial. Using the same significance level, in this case we get that k̂_count = 2 while k̂_Fisher = 5. This result demonstrates that replicability analysis can sometimes reveal statistical trends that are not clear when applying methods, such as counting, that do not thoroughly consider the statistical properties of the experimental setup.This is a good example. Yet, as a reader I would ask myself why I should trust k̂_Fisher and not k̂_count. A quick reminder of the relevant properties of k̂_Fisher is in place here to complete the discussion.To complete the analysis, we would ideally like to identify those datasets for which Mate provides a significant improvement over Redshift. While when applying the counting method this is straightforward, as discussed in Section <ref> with replicability analysis the situation is more complicated. In our case, the Holm algorithm identifies only one dataset: MZ, which is in line with k̂_Bonferroni that is also equal to 1. Somewhat expectedly, this analysis demonstrates that inquiring for more information (i.e. identifying the datasets rather than just counting them), may come with the cost of weaker results with respect to some criteria.There are a few points that require clarification here: 1. from the previous text I could not infer why the holm should agree with Bonferroni. 2. You did not mention explicitly before that k̂_Bonferroni = 1. Is this true ?Results for multilingual dependency parsing are presented inTable <ref>. Following the same dataset independence assumption of the multi-domain parsing setup, we focus on the k̂_Fisher estimator.Opposite to the multi-domain case, in this example not only does k̂_Fisher and k̂_count almost agree with each other, providing the results of 15 and 16 respectively, but the Holm procedure is able to identify 15 languages in which the TurboParser trained with MIRA outperforms its perceptron trained variantdo not write significance level here, it is not the correct term with the required significance.[All languages except from Dutch, German, Japanese and Turkish.]1. Can we say why this happens here ? I think it is important to go beyond results presentation. 2. Isn't there any value in providing the k̂_Bonferonni number and discussing its relevance here ? 3. Here k̂_count > k̂_Fisher while in the multi-domain setup we saw the opposite trend. Is there anything to say about this ? Cross-domain Sentiment Classification For this task, with its lower p-values (Table <ref>), we set the confidence level to be 1-α=0.99. Following the same considerations as in dependency parsing, we focus on the k̂_Fisher estimator. With this level of confidence we get that k̂_Fisher = 2 and the Holm procedure identifies the DVD target domain as one for which AE-SCL-SR outperforms MSDA.We are missing a discussion here - the text is now very thin. What can we learn from this ? Is there a fundamental difference between this setup and the parsing setup ? Why don't we report the k̂_Bonferonni ? What about k̂_count ? Is it due to an independence assumption ? If so, is it possible that we are not going to use k̂_Bonferonni at all ? as I said above, it is possible that we need to report k̂_Bonferonni and discuss it in any case.maybe say that this is the form of reporting this analysis in the general case, and emphasize that it is much more condensed than the regular form of reporting results, hence one can employ their experiments on a much larger number of domains.Word Similarity Results are presented in Table <ref>. As opposed to the previous tasks, the datasets here are not independent. For example, WS-353 is the union of WS-353-REL and WS-353-SIM. Hence in this task the k̂_Fisher is not a valid estimator of k. We hence reserve to k̂_Bonferroni which allows dependencies between the datasets. I am not sure we are using the notion of "independence" properly. It seems that as long as the datasets do not contain overlapping examples you assume independence. If this is the case then in practice k̂_Bonferroni will be rarely used. Also, in this particular analysis, we may be able to compute k̂_Fisher for all other datasets except from WS and WS-*.In this analysis, k̂_count = 10 while k̂_Bonferroni=9, demonstrating again that the k̂_count might overestimate the true k.I would give k̂_Fisher for completeness and we can see how inaccurate it is. The Holm procedure is in line with k̂_Bonferroni, identifying the 9 datasets in which the Predict model outperforms the Count model: WS-353, WS-353-SIM, WS-353-REL, RG-65, Rare-Word, MEN, MTurk-771, YP-130, and SIMLEX-999.As before, I do not feel that I read a well-rounded story. What exactly do I learn from this ? We have to make our point sharper - possibly discussing again the statistical properties and explain what gain we get here.§ DISCUSSION AND FUTURE DIRECTIONSWe proposed a statistically sound replicability analysis framework for cases where algorithms are compared across multiple datasets. Our main contributions are: (a) analyzing thelimitations of the current practice in NLP work: counting the datasets for which the difference between the algorithms reaches a predefined significance level; and(b) proposing a new framework that addresses both the estimation of the number of datasets with effect and the identification of such datasets.The framework we propose addresses two different situations encountered in NLP: independent and dependent datasets. For dependent datasets, we assumed that the type of dependency cannot be determined. One could use more powerful methods if certain assumptions on the dependency between the test statistics could be made. For example, one could use the partial conjunction p-value based on Simes test for the global null hypothesis <cit.>, which was proposed by <cit.> for the case where the test statistics satisfy certain positive dependency properties(see Theorem 1 in <cit.>).Using this partial conjunction p-value rather than the one based on Bonferroni, one may obtain higher values of k̂ with the same statistical guarantee. Similarly, for the identification question, if certain positive dependency properties hold, Holm's procedure could be replaced by Hochberg's or Hommel's procedures <cit.> which are more powerful. An alternative, more powerful multiple testing procedure for identification of datasets with effect, is the method in benjamini1995controlling, that controls the false discovery rate (FDR), a less strict error criterion than the one considered here. This method is more appropriate in cases where one may tolerate some errors as long as the proportion of errors among all the claims made is small, as expected to happen when the number of datasets grows.We note that the increase in the number of evaluation datasets may have positive and negative aspects. As noted in Section <ref>, we believe that multiple comparisons are integral to NLP research when aiming to develop algorithms that perform well across languages and domains. On the other hand, experimenting with multipleevaluation sets that reflect very similar linguistic phenomena may only complicate the comparison between alternative algorithms. In fact, our analysis is useful mostly where the datasets are heterogeneous, coming from different languages or domains. When they are just technically different but could potentially be just combined into a one big dataset, then we believe the question of demvsar2006statistical, whether at least one dataset shows evidence for effect,is more appropriate. § ACKNOWLEDGEMENT The research of M. Bogomolov was supported by the Israel Science Foundation grant No. 1112/14. We thank Yuval Pinter for his great help with the multilingual experiments and for his useful feedback. We also thank Ruth Heller, Marten van Schijndel, Oren Tsur, Or Zuk and the ie@technion NLP group members for their useful comments.acl2012 | http://arxiv.org/abs/1709.09500v1 | {
"authors": [
"Rotem Dror",
"Gili Baumer",
"Marina Bogomolov",
"Roi Reichart"
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"published": "20170927133141",
"title": "Replicability Analysis for Natural Language Processing: Testing Significance with Multiple Datasets"
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Carrier and strain tunable intrinsic magnetism in two-dimensional MAX_3 transition metal chalcogenides Jeil Jung December 30, 2023 =========================================================================================================In this paper we introduce a new structure to Generative Adversarial Networks by adding an inverse transformation unit behind the generator. We present two theorems to claim the convergence of the model, and two conjectures to nonideal situations when the transformation is not bijection. A general survey on models with different transformations was done on the MNIST dataset and the Fashion-MNIST dataset, which shows the transformation does not necessarily need to be bijection. Also, with certain transformations that blurs an image, our model successfully learned to sharpen the images and recover blurred images, which was additionally verified by our measurement of sharpness.§ INTRODUCTIONIn recent two years generative adversarial networks (GAN) have been increasingly concerned <cit.>. GAN introduce two perceptrons that behave against each other: the generator learns the probability distribution of training data, while the discriminator learns to tell the difference. The conciseness of GAN makes it possible to amend the structure in order to improve its performance, or make it able to achieve our additionally desired effects. While many works focused on the first point (see related works), this paper focuses on the second aspect. We add an inverse transformation unit behind the generator, and make it possible to generate data with the "inverse" effect of the input transformation function. Our architecture is quite useful when we want to generate samples with some additional effects which is hard to implement but the inverse is easy to achieve. This need is natural and common in certain situations. For instance, we want to generate clear images, but we only know the way to implement its inverse – how to blur them.In this paper, we make the following contributions:∙ We presented a new architecture for generative adversarial networks by adding an inverse transformation unit behind the generator.∙ We made rigorous theoretical analysis on our structure: we found the optimal discriminator for a fixed generator when the transformation is a continuous bijection. We also claimed the convergence of the algorithm in such situation.∙ We made two conjectures for cases when the transformation is not bijection.∙ We applied our method to MNIST dataset <cit.> and the Fashion-MNIST dataset <cit.> with different transformation functions. A general survey on various transformation functions was done; and with some special transformation functions, the model showed its ability to sharpen the images and recover blurred images. § RELATED WORKS In recent two years a lot of works on generative adversarial networks (GAN) have appeared. They have researched various aspects of GAN, from theory to applications, and made great improvement to the original method. Many works put their emphasis on improving the performance of GAN, by introducing new loss functions <cit.>, integrating it with other deep learning architectures <cit.>, or making amendments to the original GAN with strong theoretical analysis <cit.>. A number of works also apply GAN to practical issues and solved problems in those domains <cit.>. The purpose of our paper is to survey a new architecture of GAN which makes it possible to learn the samples with certain desired effect.Graph generation has been a popular topic for years, and people have tried different methods to generate graphs with their desired effects. Convolutional Neural Networks (CNN) and GAN are two popular methods in this domain. In <cit.>, researchers successfully train a model to transfer an image's texture style to another image using CNN. In <cit.>, an integration of CNN and GAN is made, and the model turns out to have a better performance in generating images than the original GAN. In <cit.>, three methods including CNN, GAN and Variational Auto Encoders (VAE) are used to learn the typographical style and generate images of letters with new styles.In order to learn the graphical samples with certain desired effect, we add an inverse transformation unit T to the generator, based on the intuition that the generator will learn some additional effect, such as T^-1 if T is invertible, to offset the effect of T. This intuition also appears in <cit.> and <cit.>. In <cit.>, the mapping f from the data distribution to the latent distribution is learned. The function f needs to be invertible and stable, and its inverse f^-1 maps samples from the latent distribution to the data distribution. With f, an unsupervised learning algorithm with exact log-likelihood computation, sampling, inference of latent variables, and an interpretable latent space, is developed to model natural images. In <cit.>, a pair of transformation functions, F and G, are introduced to be the bridges between the source domain X and the target domain Y. Both F and G are unknown and learned to satisfy that F(X) is indistinguishable from Y and F(G(X))≈ X. This pair (cycle), F and G, demonstrates great ability to transfer and enhance the photo style. In our paper, the inverse transformation T is not required to be invertible though theoretical analysis only apply for invertible T's. Also, when we learn the inverse effect of T, T is given explicitly. § GAN WITH INVERSE TRANSFORMATION UNITGAN <cit.> is an excellent architecture for training generative models. It includes two networks, each "fighting with" the other, and both of them are improved during the process. Specifically, the generator G captures the distribution of training data while the discriminator D distinguishes between samples from G and the training data. In our approach, we add an Inverse Transformation Unit T, or a "filter" after G generates a sample distribution. Figure <ref> demonstrates our model compared to the original GAN <cit.> architecture. V(D,G) in equation (1) is our value function; we maximize it over D and minimize it over G: min_Gmax_D V(D,G)=𝔼_x∼ p_data(x)log D(x)+𝔼_z∼ p_z(z)log(1-D(T(G(z)))).Here is an intuitive explanation for the name of T, the inverse transformation unit. If we train on G=T∘ G, this is exactly the original GAN, and G will learn the probability distribution of training data. In this sense, the generator G is creating samples that contain information of "inverse of T", if it exists. For example, the generator G learns how to generate dogs, while the discriminator D learns to judge if it's a true image of dog. Suppose T makes the image blurred. Since T∘ G learns the distribution of true dogs, G will generate samples that are clear enough to eliminate the blurring effect.However, things are complicated when T^-1 doesn't exist. It might be the case that G learns information of T̂ where T̂∘ T is almost identity mapping; however, G may also fail to learn it. In the rest part of the paper, both theocratical analysis and experiments are made to investigate such situations. § THEORETICAL RESULTS In this section, we show that when T is bijection with invertible Jacobian matrix, then the generator G does create samples similar to T^-1 of data. The optimal discriminator D_G^* is given explicitly, and the convergence is analyzed. However, when T is not bijection, the optimal discriminator either does not exist or cannot be written explicitly. Two conjectures are posted about the optimal discriminator when G fixed, with respect to two situations when T is not surjection/injection.Theorem 1. Suppose the transformation function T is a bijection from ℝ^n to ℝ^n. If T has an invertible Jacobian matrix J, then for G fixed, the optimal discriminator D is given byD_G^*(x)=p_data(x)/p_data(x)+p_g(T^-1(x))|J^-1(x)|, a.e.Proof.For G fixed, the discriminator D is trained to maximize[ V(D,G) =∫_ℝ^np_data(x)log D(x)d x+∫_ℝ^np_z(z)log(1-D(T(G(z))))d z; ;=∫_ℝ^n(p_data(x)log D(x)+p_g(x)log(1-D(T(x))))d x. ]The variational method is used to solve the problem. For any H:ℝ^n→ℝ^n with compact support, function h is defined by h(t)=V(D+tH,G), t∈ℝ. Then the optimal discriminator can be found by solving the equation h'(0)=0 with constraint h”(0)<0.The function h(t) can be expanded ash(t)=∫_ℝ^n(p_data(x)log(D(x)+tH(x))+p_g(x)log(1-D(T(x))-tH(T(x))))d x.Thus, h'(t) can be calculated ash'(t)=∫_ℝ^n(p_data(x)·H(x)/D(x)+tH(x)+p_g(x)·-H(T(x))/1-D(T(x))-tH(T(x)))d x.Let h'(0)=0, we have0=h'(0)=∫_ℝ^n(p_data(x)·H(x)/D(x)+p_g(x)·-H(T(x))/1-D(T(x)))d x.Through substitutions y=T(x) and x=T^-1(y), we have∫_ℝ^np_g(x)·-H(T(x))/1-D(T(x))d x =∫_ℝ^np_g(T^-1(y))·-H(y)/1-D(y)·|J^-1(y)|d y.As a result,0=h'(0)=∫_ℝ^n(p_data(x)·H(x)/D(x)-p_g(T^-1(x))·H(x)/1-D(x)·|J^-1(x)|)d x.Since H is arbitrary, it follows thatp_data(x)/D(x)-p_g(T^-1(x))/1-D(x)·|J^-1(x)|=0, a.e.,which leads to the result that the optimal discriminator D is given byD_G^*(x)=p_data(x)/p_data(x)+p_g(T^-1(x))|J^-1(x)|, a.e.Additionally, h”(0)<0 is trivial. Theorem 2. Let C(G)=max_DV(D,G)=V(D_G^*,G). The global minimum of C(G) is achieved if and only if p_data=|J^-1|p_g∘ T^-1; at that point, C(G) achieves the value -log 4.Proof.Let G=T∘ G be the transformed generator, and we see that p_g and p_g̃ has the following relationship according to Theorem 1:p_g̃=|J^-1|p_g∘ T^-1.Then, the rest of the proof is exactly the same as the proof of Theorem 1 (not Theorem 1 in this paper) by Goodfellow, et al. <cit.> Furthermore, if we change the view using equation 11, the convergence is guaranteed under the same conditions of Proposition 2 by Goodfellow, et al<cit.>.When T is not bijection, things are much more complicated. Usually, we can't find the optimal discriminator D_G^* for G fixed. Following is some analysis as well as two conjectures. When T is not surjection, let the range of T is ℝ^n∖ A. Then, the right side of equation 7 will be the integration on ℝ^n∖ A, which leads to the result that equation 8 is changed to[0=h'(0)= ∫_ℝ^n∖ A(p_data(x)·H(x)/D(x)-p_g(T^-1(x))·H(x)/1-D(x)·|J^-1(x)|)d x; +∫_Ap_data(x)·H(x)/D(x)d x. ]However, we usually can't make p_data(x)/D(x)=0 for x∈ A.Conjecture 1.When T is not surjection, there is no explicit optimal discriminator for G fixed. However, the following condition should be satisfied for a good discriminator (i.e. if a discriminator doesn't satisfy the following condition, it's always better to change it into this condition):D_G^*(x)=p_data(x)/p_data(x)+p_g(T^-1(x))|J^-1(x)|,x∈ℝ^n∖ A, a.e. When T is not injection, define the set T^-1(x)={y∈ℝ^n:T(y)=x}. Then, in equation 7 the expression p_g(T^-1(y)) doesn't exist anymore. Instead, it is p_g(t_y), where t_y∈ T^-1(y) is a point in the set but we don't know what it is. This makes the system hard to analysis. Another point of view is that suppose T(x_1)=T(x_2)=y_0, then the following two discriminators perform exactly the same:D_G^1(y_0)=p_data(y_0)/p_data(y_0)+p_g(x_1)|J^-1(y_0)|;D_G^2(y_0)=p_data(y_0)/p_data(y_0)+p_g(x_2)|J^-1(y_0)|. Conjecture 2. When T is not injection, there is no explicit optimal discriminator for G fixed. However, the following discriminator is good enough (i.e. there might be better solutions in different situations, but it's the best one that can be written explicitly):D_G^*(x)=p_data(x)/p_data(x)+p_g(T^-1(x))|J^-1(x)|, a.e.where p_g(T^-1(x)) refers to the average value of p_g(y) for y∈ T^-1(x).The existence of the inverse transformation unit T makes generative adversarial networks possible to generate a wider range of samples with our additionally desired effects. The bijection case is proved to work well, but other cases still need deeper theoretical analysis so that we could figure out when our architecture is effective.§ EXPERIMENTSIn this section we made several experiments to show in which conditions our architecture is able to work and some possible effects our architecture is able to produce. First, in order to know if our architecture works (i.e., being successfully trained) for transformation functions with different properties, we made a general survey on T's with different properties. Then, we showed that our architecture is able to learn the opposite effect of "blurring" with certain transformations. Specifically, it is able to generate sharpened images and generate recovered images that were initially blurred. Additionally, we introduced a measurement of sharpness χ_s and verified the effect of our architecture using χ_s. Our architecture is realized by adding an inverse transformation unit based on DCGAN <cit.>, and all experiments are done on the MNIST dataset <cit.> and the Fashion-MNIST dataset <cit.>. §.§ A General Survey on T's with Different PropertiesThe intuition is that if T is bijection, the training is likely to be successful. This intuition is also supported by Theorem 2 in that the global minimum is achieved with p_data given explicitly. But what if T is not bijection? This question leads to a survey on the effect through various T's with a set of properties shown in Table <ref>, whereÎ_n× n(i,j)=1 if i+j=n and otherwise 0, and σ(x)=1/(1+exp(-x)). Since the images are gray and each pixel can be compressed into interval [-1,1], all T's are mappings from [-1,1] to [-1,1]. The plots of these functions are demonstrated in Fig <ref>. Among the nine transformation functions selected in Table <ref>, four demonstrate good effects (i.e., the corresponding models are able to generate the numbers with desire effect). With T_1, a bijection that maps an image to its mirror image, the model indeed generates the mirror image of the original numbers. With T_22 and T_23, two injections that compress the interval [0,1], the models are able to generate numbers with stronger contrast: more white and black but less gray pixels. With T_32, a surjection that is not one-to-one, the model can also generate images of numbers. Fig <ref> shows the images these four models generate and images transformed by the transformation functions. As we can see, easier transformation functions are more likely to take effect, while complicated ones will have more problems during the training process, leading to bad local optimum or misconvergence. Specifically, T_21 fails because it cannot reach negative values, and for T_4, T_51, T_52, there is a large range of y∈[0,1] that is reached by at least two x's through the transformation, which confuses the model. §.§ Sharpening and Recovery of Blurred images One easy way to blur an image is weighted averaging each pixel and its neighbor pixels. This can be achieved by using the convolutional kernel. Consider an image as an n× n matrix X. To deal with the edges well, we first design a method of extension. The first step is extending a row on the top and the bottom respectively, with values of the rows next to the original edges. This yields an (n+2)× n matrix. The second step is extending a column on the left and the right, using the values of the columns next to them. This yields an (n+2)× (n+2) matrix. After the extension, we perform a convolution using a 3× 3 convolution kernel K, which yields an n× n matrix, representing the blurred image. The whole process is demonstrated in Fig <ref>. If we take this function as T, the generator is expected to learn the inverse effect of this blurring. In other words, the generator may learn to sharpen the images so that after T, the generated images are similar to the images from the dataset. Fig <ref> and Fig <ref> show several samples generated by the model with convolution kernel K_sharpen=( [ 0.01 0.08 0.01; 0.08 0.64 0.08; 0.01 0.08 0.01;]). As expected, there are much fewer gray pixels in the images which make the image smooth. Although the effect of our model is not as obvious as that of standard techniques in image processing, the results show the ability of our architecture to deal with such tasks. Furthermore, if the images are already blurred by the previous method with some convolution kernel K_blur, we can recover the blurred images with our model with convolution kernel K_rec. Essentially, K_blur and K_rec do not need to be exactly the same. Fig <ref> and Fig <ref> demonstrate the blurred and recovered images under several different pairs of (K_blur, K_rec). The results show that images indeed can be recovered even when K_rec≠ K_blur, which implies that our model can be used in practical situations when K_blur is unknown but can be roughly estimated. The selected convolution kernels areK_blur,K_rec^1=1/9( [ 1 1 1; 1 1 1; 1 1 1; ]), K_rec^2=( [0.1 0.120.1; 0.12 0.13 0.12;0.1 0.120.1;]), K_rec^3=( [ 0.08 0.12 0.08; 0.12 0.19 0.12; 0.08 0.12 0.08;]).§.§ Measurement of SharpnessIn order to examine the effect of the sharpening and recovery, a measurement of sharpness χ_s is introduced in this section. χ_s is a function that maps an image (essentially a matrix with all elements ∈[-1,1]) into the interval [0,1]. For an image P, the larger χ_s(P) is, the sharper the image is, according to the meaning of this function. Now we give the definition of χ_s.Let P be a matrix that represents an image, where P_ij∈[-1,1]∀ i,j. Let Δ P be a matrix with exactly the same size as P, where (Δ P)_ij is the average value of the absolute differences of P_ij and its neighbours. That is,(Δ P)_ij=1/4( |P_ij-P_i-1,j|+|P_ij-P_i+1,j| +|P_ij-P_i,j-1|+|P_ij-P_i,j+1| )and similar for P_ij's on the edge or corner of the image. In one sentence, Δ is the absolute average difference operator. Then, χ_s is defined as the average value of the second-order absolute average difference of P:χ_s(P)=Δ(Δ P).Using this measurement, we examined the sharpness of images from six different groups with respect to the two datasets. For each dataset, the six groups includes the original MNIST (or Fashion–MNIST) images, sharpened images, blurred images and recovered images with various convolution kernels. For each group, 108 samples were extracted randomly, and the distributions of the values of sharpness are demonstrated in Figure <ref>. The results almost conform with our theory. For the MNIST dataset, the sharpened images have a higher value of χ_s; although the blurred images have a much lower χ_s, the recovered ones with all three K_rec's tend to have almost the same χ_s as the original images. For the Fashion–MNIST dataset, despite that recovered images with convolution kernel K_rec^3, which is the farthest away from K_blur, have higher χ_s, the other five groups of images still yield good results similar to those of the MNIST dataset.§ CONCLUSIONIn this paper, we presented a new architecture of Generative Adversarial Networks by adding an inverse transformation unit behind the generator. We made rigorous theoretical analysis to our model: we explicitly solved the optimal discriminator given the generator G, and proved the convergence of the algorithm under certain conditions. We also made several experiments. The first experiment was a general survey on models with various transformation functions. The survey illustrates that when the T is not bijection, the model may still work. In the second experiment, we demonstrated that our architecture is able to generate sharpened images, and able to recover blurred images without using the same convolution kernel. In the third experiment, we defined a measurement of sharpness χ_s and compared this value with respect to different groups of images; the results also imply that our model works well for generating images with sharpening or recovering them at the same time. In the future, we plan to apply our model to a wider range of transformation functions in computer vision, such as various filters, and survey the inverse effects of them.GAN Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, Yoshua Bengio: Generative Adversarial Nets. In: Advances in Neural Information Processing Systems, pp. 2672-2680. (2014)MNIST LeCun, Y., Bottou, L., Bengio, Y., and Haffner, P.: Gradient-based learning applied to document recognition. In: Proceedings of the IEEE, vol. 86(11), pp. 2278-2324. (1998)fashion_MNIST Han Xiao, Kashif Rasul, Roland Vollgraf. Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms.arXiv preprint arXiv:1708.07747 (2017)DCGAN Alec Radford, Luke Metz, Soumith Chintala: Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks. arXiv preprint arXiv:1511.06434 (2015)bGAN Masatosi Uehara, Issei Sato, Masahiro Suzuki, Kotaro Nakayama, Yutaka Matsuo: B-GAN: Unified Framework of Generative Adversarial Networks. In: <https://openreview.net/pdf?id=S1JG13oee> (2016)fGAN Sebastian Nowozin, Botond Cseke, Ryota Tomioka: f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization. arXiv preprint arXiv:1606.00709 (2016)InfoGAN Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, Pieter Abbeel: InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets. In: Advances in Neural Information Processing Systems, pp. 2172-2180 (2016)GRAN Daniel Jiwoong Im, Chris Dongjoo Kim, Hui Jiang, Roland Memisevic: Generating images with recurrent adversarial networks. arXiv preprint arXiv:1602.05110 (2016)LAPGAN Emily Denton, Soumith Chintala, Arthur Szlam, Rob Fergus: Deep Generative Image Models using a Laplacian Pyramid of Adversarial Networks. In: Advances in Neural Information Processing Systems, pp. 1486-1494 (2015) EnergyGAN Junbo Zhao, Michael Mathieu and Yann LeCun: Energy-Based Generative Adversarial Networks. arXiv preprint arXiv:1609.03126 (2016)WGAN Martin Arjovsky, Soumith Chintala, and Leon Bottou: Wasserstein GAN. arXiv preprint arXiv:1701.07875 (2017)LSGAN Guo-Jun Qi: Loss-Sensitive Generative Adversarial Networks on Lipschitz Densities. arXiv preprint arXiv:1701.06264 (2017) seqGAN Lantao Yu, Weinan Zhang, Jun Wang, Yong Yu: SeqGAN: Sequence Generative Adversarial Nets with Policy Gradient. In: Thirty-First AAAI Conference on Artificial Intelligence (2017)cross-domain Yaniv Taigman, Adam Polyak, Lior Wolf: Unsupervised Cross-Domain Image Generation. arXiv preprint arXiv:1611.02200 (2016) transfer_CNN Leon A. Gatys, Alexander S. Ecker, Matthias Bethge: Image Style Transfer Using Convolutional Neural Networks. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2414-2423. (2016)letter_CNN Shumeet Baluja: Learning Typographic Style. arXiv preprint arXiv:1603.04000 (2016)letter_GAN TJ TORRES: <http://multithreaded.stitchfix.com/blog/2016/02/02/a-fontastic-voyage/> (2016)letter_VAE Paul Upchurch, Noah Snavely, Kavita Bala: From A to Z: Supervised Transfer of Style and Content Using Deep Neural Network Generators. arXiv preprint arXiv:1603.02003 (2016)realNVP Laurent Dinh, Jascha Sohl-Dickstein, Samy Bengio: Density Estimation using Real NVP. arXiv preprint arXiv: 1605.08803 (2016)cycleGAN Jun-Yan Zhu, Taesung Park, Phillip Isola, Alexei A. Efros: Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks. arXiv preprint arXiv:1703.10593 (2017) § APPENDIX | http://arxiv.org/abs/1709.09354v1 | {
"authors": [
"Zhifeng Kong",
"Shuo Ding"
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"published": "20170927063830",
"title": "Generative Adversarial Networks with Inverse Transformation Unit"
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§ INTRODUCTIONThe next generation of Imaging Atmospheric Cherenkov Telescopes, the Cherenkov Telescope Array, will probe the γ-ray sky above 20 with an unmatched sensitivity and angular resolution using more than 100 telescopes distributed on the South (Chile) and on the North (La Palma) sites. To reach this goal, three different types of telescopes will be used : the 23 diameter Large-Size Telescopes (LST), the 12 diameter Medium-Size Telescopes (MST) and the 4 diameter Small-Size Telescopes (SST). They are respectively adapted to detect photons of 20200 (field of fiew of ∼4), 0.110 (field of fiew of ∼7) and 1300 (field of fiew of ∼9).With its improved sensitivity and higher field of view compared to current experiments, namely MAGIC, H.E.S.S. and VERITAS, CTA will improve our current knowledge on active galactic nuclei (AGN) and especially on blazars. Nowadays, population studies are quite limited by the low/biased sample of blazars detected at very high energy (E ≥ 100). The high quality spectral resolution of CTA will help to look for hadronic signatures in AGN spectra <cit.>. Furthermore, it will also help to better constrain the intergalactic medium opacity to γ-rays, and especially the flux density of the diffuse extragalactic background light (EBL). The first VHE detection of γ-ray bursts (GRBs) might shed light on the radiative processes and the particle content in those accelerators and might provide a new class of distant objects to constrain the EBL density. CTA is expected to operate for thirty years and unlike current experiments it will be an open observatory, meaning that high level data, such as event lists and instrument response functions (IRFs), will be made available to the scientific community. [http://docs.gammapy.org/http://docs.gammapy.org/en/latest/] is a community-developed, open-source Python package for γ-ray astronomy <cit.>. It is built on widely-used scientific packages (Astropy, Numpy, Scipy) and provides tools to simulate and analyse IACT data. It was thought to produce high level data, such as spectra, maps, light curves and even catalogues, taking as inputs lower level data such as event lists and instrument response functions. Classical methods to analyse data in very high energy astronomy, such as background substraction or full-forward folding spectral reconstruction are implemented and were highly tested with real data, especially in the H.E.S.S. experiment. Cube-style analysis, in which a spectral and a spatial model are jointly adjusted, is also implemented <cit.>. In this contribution, we show how can easily beused to tackle some extragalactic science cases of CTA. We first present in Section <ref> how point-like source simulations are made within . Section <ref> describes how it can contribute to the ongoing efforts to estimate the number of expected blazars that could be detected with CTA. Section <ref> is a technical discussion on how Gammapy can be used to constrain the EBL scale factor. Section <ref> briefly presents an application about GRB detectability. The Python scripts used to produce all results and figures presented here are available at https://github.com/gammapy/icrc2017-gammapy-cta-egalhttps://github.com/gammapy/icrc2017-gammapy-cta-egal.§ SIMULATIONS In the following, we will focus on point-like source simulations. To reach this goal, simulations are generated with a set of instrument response functions, which are produced by the simulation group in the consortium <cit.>, including : * effective area as function of true energy,* background rate as function of reconstructed energy,* energy migration matrix from true to reconstructed energyAn energy dependent-angular cut is applied to maximise the significance in each interval of reconstructed energy. To compute an expected excess for a given spectral models and exposure we multiply the desired input spectra with the absorption of one of the available EBL model <cit.> in . We sum the randomised expected excess and background counts to get the data in the ON region of the source. Finally, we randomised the background counts multiplied by the normalisation between the ON and the OFF regions, taken as 0.2 in the following, to get the data in the OFF region. The average time to run 100 simulations is less than 900 on a personal computer[This estimation has been realised with the Timit Python package and a 2.9 Intel Core i7 processor.]. In the following sections, the level of significance is estimated with the formula 17 of Li & Ma <cit.>. § APPLICATION EXAMPLE : EXTRAPOLATION OF BLAZARS FROM FERMI/LAT CATALOGUESThe Fermi/LAT collaboration recently released the 3FHL (The Third Catalog of Hard Fermi-LAT Sources) catalogue <cit.>. It contains 1558 sources detected above 10 in 84 months and analysed with the improved PASS8 analysis. The blazars dominate the sample of sources (78), followed by the galactic sources (9) and the unassociated ones (13). One of the main tasks of the extragalactic group of the CTA consortium, in the preparatory phase, is to estimate the number of detectable AGN by CTA, using as a starting point the 3FHL catalogue, and to measure its impact on population studies. A realistic analysis is available in <cit.>.To illustrate capabilities, we studied the impact of adding an hypothetical exponential cut-off to the 3FHL Fermi/LAT spectra on the detection level. We selected a sample of 466 blazars, firmly identified or associated BL Lacs or flat spectrum radio quasars (FSRQs), having a redshift estimation in the catalogue and whose meridian transit occurs with zenith distance less than 30 at least in one CTA site (north or south). We applied an exponential cut-off at energy 1 / (1+z) to the Fermi/LAT spectrum. To assess the detection level of a source, the following procedure was used : * the spectrum is absorbed by the EBL model Dominguez et al. <cit.>* the IRF labelled as “5h”, production 3b, are used according to the declination of the source (δ≥ 0 is north site)* the simulation is done for 20 of observation time* the counts are integrated from 50 to 100* the final significance is averaged on 20 simulationsThe results of the simulations are shown on Figure <ref>. The fraction of sources detected above 5σ is ∼53 (151/290) for BL Lacs and ∼10 (15/156) for FSRQs. Adding the cut-off decreases theses numbers to 46 and 8, showing the impact of the choice of the intrinsic spectral shape on detectability by CTA. § APPLICATION EXAMPLE: THE EBL SCALE FACTOROne of the main cosmological topics with the Cherenkov Telescope Array is to improve the constraints obtained with blazars on the diffuse extragalactic background light. A general and complete discussion can be found in <cit.>. In this section we will focus on the technical aspects to constrain an EBL scale factor for a given absorption model with and the Python library [http://cxc.cfa.harvard.edu/contrib/sherpa/http://cxc.cfa.harvard.edu/contrib/sherpa/].has been designed to provide flexibility in data-analysis to the end users. Simulations can be fed to widely-used tools that have been developed for X-ray astronomy, such as the library, to produce higher data level. gives access to the full-forward folding method to handle spectral reconstruction and provides a convenient way to do arithmetic with spectral models. Furthermore, each model can be adjusted on its own dataset and each parameter can be linked between different models, which makes the fit of the EBL scale factor straightforward. To obtain strong constraints on the EBL scale factor it is necessary to adjust multiple sources for different redshift ranges <cit.>. The choice of the modelling of the source will have an impact onour ability to reconstruct the EBL scale factor.As an illustration, we studied the impact of an exponential cut-off, fixed at energy 1 / (1+z),on the determination of the EBL scale factor, called α, for different redshifts. To proceed, we used the same sample of blazars described in <cit.> and adjusted for each source the intrinsic spectra ϕ_int and α resulting in an observed spectrum :ϕ(E)=ϕ_int(E) × e^(-ατ_m(E,z)) where τ_m is the γ-ray opacity of the intergalactic medium given by the model m. For each source, the following procedure was used to study the EBL scale reconstruction without (with) a cut-off : * the intrinsic spectrum is given by the 3FHL catalogue, e.g. a power law or a log-parabola (and multiplied by a cut-off)* the spectrum is absorbed by the EBL model <cit.>* the IRF labelled as “5h”, production 3b, is chosen according to the declination of the source (δ≥ 0 is north site)* the simulation is done for 100 of observation time* the spectrum is fitted with the full-forward folding method in a fixed energy range from 50 to 5, with the intrinsic spectra model (and multiplied by a cut-off)* the confidence interval bounds are computed for each parameter We show on the technical Figure <ref>, the reconstructed scale α as a function of the redshift of the sources for the two different spectral models. For low redshift, z ≤ 0.1, the EBL absorption begins in the energy domain where the exponential cut-off is already present (see Figure <ref>) so it makes it harder to constrain the EBL scale factor (the error bars increase). This could be in part compensated by a higher number of sources at low redshift. For the intermediate redshift range, typically from z ∼ 0.1 to z ∼ 0.2, the two effects are competitive and we are able to distinguish the two processes involved in the photon attenuation.At higher redshifts, the EBL extinction intervenes well before the exponential cut-off and the latter becomes less pertinent and thus under-constrained (for those cases, the maximisation likelihood procedure fails). That shows that , in connection with the library, is particularly well suited for these kinds of studies involving custom complex spectral models.§ APPLICATION EXAMPLE: OBSERVATIONAL WINDOW FOR Γ-RAY BURSTSγ-ray bursts are transient and explosive phenomena originating from cosmological distances. For the brief episode of emission their total energy output reaches 10^51-10^53 ergs, placing them among the most powerful presently observed objects. A burst is characterised by a prompt phase dominated by X-ray and γ-ray photons (typically between 10 keV and a few MeV), lasting for a few milli-seconds to a few hundreds of seconds. It is followed by a second emission phase, called the afterglow, during which the electromagnetic emission is shifted to lower energies (X-rays, visible, radio) and rapidly decaying with time. The origin of the bursts is not fully understood. Nevertheless, the combination of the short variability timescale and the huge energy release suggest that GRBs are the results of cataclysmic events in the Universe, most likely associated with the births of stellar size black holes or rapidly spinning, highly magnetized, neutron stars. Up to now, the Fermi/LAT detected more than 100 γ-ray bursts above 75, whereas IACTs never detected a significant γ-ray emission from a GRB. With its high effective area and its big field of view, CTA might detect a few of these events and help to better constrain their intrinsic properties. To get a more detailed picture about high energy and very high energy GRBs the reader may refer to <cit.> and <cit.>, respectively.Here, we used to reproduce the result from <cit.> showing the detection significance of the bright γ-ray burst GRB 080916C as a function of the observational window. The spectral template is described as a power law of spectral index of 2 with an integrated flux in the 0.110 range at a time t_p of Φ (t_p=6.5) = 500e-5□. We further assume that the GRB is located at a distance[The measured redshift of GRB 080916C is z=4.35 ± 0.15 <cit.>.] corresponding to z=3. The flux decay over time is parameterized by a decay index δ=1.7. In order to study the significance of the source as a function of the observational window, we defined 20 logarithmic time intervals representing the beginning and the end of the observations, from 20 to 2 days. Starting from t_0 = 20 after the peak flux, the significance was computed for each physical logarithmic time interval in the following way : * the average time interval t_int is computed by taking into account the flux decay law* the flux normalisation is computed for t_int* the spectrum is absorbed with the EBL model from <cit.>* the North IRF, labelled as “0.5h” from the production 2 are used* the counts are integrated from 30 to 1Results of the simulations are shown on Figure <ref>. The variation of the significance results from two effects, the flux decay of the source and a constant level of background as a function of time. § CONCLUSION We have shown how capabilities can be used tohandle extragalactic science cases with the Cherenkov Telescopes Array. As mentioned in the introduction, we note that the Python scripts used to produce all results and figures presented in this contribution are available at https://github.com/gammapy/icrc2017-gammapy-cta-egalhttps://github.com/gammapy/icrc2017-gammapy-cta-egal.This means anyone can reproduce and check the results, and use the scripts as examples to start their own studies. This work was conducted in the context of the CTA Consortium. Julien Lefaucheur would like to thank Régis Terrier (APC/CNRS) for useful discussions about data analysis with the Python library .JHEP | http://arxiv.org/abs/1709.10169v2 | {
"authors": [
"Julien Lefaucheur",
"Catherine Boisson",
"Zeljka Bosnjak",
"Matteo Cerruti",
"Christoph Deil",
"Jean-Philippe Lenain",
"Santiago Pita",
"Andreas Zech"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170926142148",
"title": "Gammapy: high level data analysis for extragalactic science cases with the Cherenkov Telescope Array"
} |
#1#1.#1#1. | http://arxiv.org/abs/1709.09222v3 | {
"authors": [
"Anish Ghoshal",
"Anupam Mazumdar",
"Nobuchika Okada",
"Desmond Villalba"
],
"categories": [
"hep-th",
"hep-ph"
],
"primary_category": "hep-th",
"published": "20170926190108",
"title": "On the Stability of Infinite Derivative Abelian Higgs"
} |
a1]K. Szałowskicor1 [email protected]]P. Kowalewska[a1]Department of Solid State Physics, Faculty of Physics and Applied Informatics,University of Łódź, ulica Pomorska 149/153, 90-236 Łódź, Poland [cor1]Corresponding author The paper presents a computational study of the ground-state properties of a quantum nanomagnet possessing the shape of a finite two-legged ladder composed of 12 spins S=1/2. The system is described with isotropic quantum Heisenberg model with nearest-neighbour interleg and intraleg interactions supplemented with diagonal interleg coupling between next nearest neighbours. All the couplings can take arbitrary values. The description of the ground state is based on the exact numerical diagonalization of the Hamiltonian. The ground-state phase diagram is constructed and analysed as a function of the interactions and the external magnetic field. The ground-state energy and spin-spin correlations are extensively discussed. The cases of ferro- and antiferromagnetic couplings are compared and contrasted. < g r a p h i c s > magnetic cluster nanomagnet quantum spins Heisenberg model magnetic phase diagram ground state Ground-state magnetic properties of spin ladder-shaped quantum nanomagnet: Exact diagonalization study [ December 30, 2023 ======================================================================================================= § INTRODUCTION Low-dimensional systems attract increasing attention of solid state physicists. The most intensive studies focus on various nanostructures and within this class a considerable attention is paid to magnetic nanosystems <cit.>. The principal motivation for studies of the smallest nanostructured magnetic systems is the possibility of arranging them within bottom-up approach from single atoms of surfaces <cit.>. This approach allows the design and engineering of artificial nanomagnets with high precision. Moreover, their properties can also be carefully characterized at the nanoscale <cit.>. What is crucial, the geometry <cit.> and underlying magnetic interactions in such systems <cit.> can be tuned to achieve the desired characteristics. It has been demonstrated that the nanomagnets arranged of single atoms can serve as memory devices, what proves the high potential for applications <cit.>. Moreover, the nanomagnetic systems are also hoped to be useful for quantum computations <cit.>, to mention, for example, spin clusters representing the qubits <cit.>. This route is particularly promising when based on molecular nanomagnets <cit.>. The mentioned facts serve as a strong motivation for theoretical studies of a variety of nanomagnetic systems.One of the interesting classes of such systems is nanomagnets possessing the shape of spin ladder with finite length. This structure was the subject of experimental interest in Ref. <cit.> and built a prototypical memory device. It should be mentioned that major attention in the literature is paid to infinite spin ladders with various number of legs, constituting one-dimensional systems <cit.>. In that context the notion of Haldane gap and the dependence of excitations on spin magnitude and the number of legs in the ladder should be mentioned <cit.>. However, highly interesting properties can be shown also by the finite systems themselves. Although the magnetic ordering is excluded in such structures, yet they can exhibit interesting magnetic phases and cross-overs between them. Among the studies of such zero-dimensional structures, the works based on exact methods should be especially mentioned <cit.>. It is worth emphasizing that rigorous and exact numerical solutions are, so far, available only for a very limited class of models (especially when the quantum version is considered) <cit.>. In order to explore the magnetic properties of the finite structures, it is first vital to examine their ground states, taking into account various possible interactions between the spins as well as the external magnetic field. This is the aim of the present paper, in which we investigate a nanomagnet being a two-legged finite spin ladder with 12 spins S=1/2. For this purpose we select an approach based on exact diagonalization, which provides an approximation-free picture of the physics of the studied system. The further parts of the paper contain a detailed description of the system in question, the theoretical approach and the review of the obtained results. § THEORETICAL MODEL The system of interest in the present study is a nanomagnet having the shape of a finite ladder with two (equivalent) legs. The schematic view of the system is presented in Fig. <ref>. It consists of N=12 quantum spins S=1/2, coupled with isotropic, Heisenberg-like interactions. Therefore, it is described with the following quantum Hamiltonian: ℋ = -J_1∑_⟨ i,j⟩^𝐒_𝐢·𝐒_𝐣-J_2∑_⟨ i,j⟩^𝐒_𝐢·𝐒_𝐣-J_3∑_⟨⟨ i,j⟩⟩^𝐒_𝐢·𝐒_𝐣 -H∑_i^S^z_i.The operator 𝐒_𝐢=(S^x_i,S^y_i,S^z_i) denotes a quantum spin S=1/2, located at site labelled with i (i=1,…,12), with S_i^α=σ^α/2, where σ^α is the appropriate Pauli matrix and α=x,y,z is the direction in spin space. Moreover, the product 𝐒_𝐢·𝐒_𝐣=S^x_iS^x_j+S^y_iS^y_j+S^z_iS^z_j. The exchange integral between nearest-neighbour spins in the same leg of the ladder amounts to J_1, while the interactions between the ladder legs are denoted by J_2 for nearest neighbours (rung interactions) and J_3 for next-nearest neighbours (crossing interactions); see the scheme Fig. <ref>. All the exchange integrals J_1,J_2,J_3 are allowed to take arbitrary values, both positive (ferromagnetic) and negative (antiferromagnetic). The external magnetic field acting in z direction in spin space is introduced by H.In the present study, the interest is focused on the ground-state properties of the system, at zero temperature. In order to perform the description, the full Hamiltonian (Eq. <ref>) is constructed in a form of a matrix (of the size 2^N× 2^N=4096× 4096) and diagonalized numerically <cit.>. This procedure yields the eigenvalues E_k and eigenvectors |ψ_k> (which can be degenerate). Among the eigenenergies, the ground-state energy E_0 is selected, with eigenvectors |ψ_0^p>, where p=1,…,d and d is the degeneracy. At the zero temperature, each of the degenerate ground states is equally probable. Therefore, the ground-state average of the arbitrary quantum operator A can be evaluated on the basis of the following formula: <A>=1/d∑_p=1^d<ψ_0^p|A|ψ_0^p>. The observable of special interest is here the total z component of spin of the nanomagnet, with the average value of S_T=⟨∑_i=1^NS^z_i⟩. In that context, another quantum number can be defined, namely the total spin quantum number, defined by S̃_T(S̃_T+1)=⟨∑_i=1^N𝐒^2_i⟩. Further important quantities are spin-spin correlation functions c_ij^αβ=<S^α_iS^β_j>, where α,β=x,y,z.The presented theoretical formalism serves as a basis for numerical calculations of the crucial ground-state properties of the studied nanomagnet, which will be discussed in the following section of the paper.§ NUMERICAL RESULTS AND DISCUSSIONAll the calculations presented in this section rely on the exact numerical diagonalization of the system Hamiltonian (Eq. (<ref>)) performed with the Mathematica software <cit.>. The discussion will be subdivided into subsections related to the key characteristics of the ground state.§.§ Ground-state phase diagram Let us commence the analysis from the investigation of the ground-state phase diagram for the system in the external magnetic field. The phases correspond here to various values of the z component of the total spin, denoted by S_T. The phase diagram presenting the stability areas for phases with different values of S_T as a function of J_2/|J_1| and the magnetic field H/|J_1| is shown in Fig. <ref>, for different values of J_3/|J_1|. The cases of antiferromagnetic J_1<0 and ferromagnetic J_1>0 are shown separately, as the absolute value |J_1| is taken as the convenient energy to normalize other quantities. Let us analyse first the diagram for J_1<0 and J_3/|J_1|=0.0 [Fig. <ref>(a)]. At J_2=0.0 we deal with a pair of non-interacting finite spin chains, so the total spins take only even values. Introducing a finite, non-zero interaction J_2 restores the presence of the phases with all possible spins. What is interesting, for strongly antiferromagnetic J_2 the phase boundaries linearize and have identical slopes. This can be explained due to the fact that in the limit of dominant J_2 coupling the critical fields do not depend on J_1 (but only on J_2 itself). On the other hand, for strongly ferromagnetic J_2 the critical fields H/|J_1| cease to depend on J_2. Let us mention that the diagram bears some resemblance to Fig. 2 in Ref. <cit.>, where the two-legged finite ladder was studied with the aim of characterizing an infinite system. If the antiferromagnetic crossing inter-leg interaction J_3 is switched on [as shown in Fig. <ref>(b)], the ladder legs are no longer non-interacting for J_2=0.0. As a consequence, for the full range of exchange integrals J_2 we pass through all the states with spins 0,…, 6 when the magnetic field increases. However, close to some critical values of J_2 (slightly decreasing with the considered spin) the phases with odd spins are suppressed. The limiting behaviour of the diagram for strongly ferromagnetic and strongly antiferromagnetic coupling J_2 remains similar to the case of J_3=0.0. Contrary to the case of antiferromagnetic J_3, for ferromagnetic value J_3/|J_1|=0.2 [Fig. <ref>(c)], the range with only even values of spin expands from points J_2=0.0 to some finite intervals observable for positive J_2 (with their widths slightly rising when the involved spins increase). The diagram is completely different for ferromagnetic interaction J_1>0. In the absence of crossing inter-leg interaction J_3=0.0, as shown in Fig. <ref>(d), the spin is equal to 6 for H>0 and ferromagnetic J_2 (a saturated ferromagnetic state). For J_2<0 the states with all the possible spins are present, but the saturation is reached at considerably weaker field than in the case of antiferromagnetic J_1. If the antiferromagnetic crossing coupling J_3<0 is added [Fig. <ref>(e)], the critical magnetic fields separating the phases with various spins tend to increase. The field ranges corresponding to stability of a given spin are also wider. Moreover, the value of J_2, above which H>0 switches from spin 0 directly to spin 6, is increased (and corresponds to J_2>0). The narrow range where H>0 causes the switching from spin 0 to spin 6 with exclusion of some intermediate values is also noticeable. On the contrary, if J_3>0, as shown in Fig. <ref>(f), the critical magnetic fields are reduced and the field ranges for stability of a given spin are narrow. The critical value of J_2 above which switching directly between spin 0 and spin 6 takes place when H increases is shifted towards antiferromagnetic (negative) values. Like in the case of J_3>0, also here a narrow interval in which some of the total spin values are missing when the field increases can be observed.Let us comment on the quantum state corresponding to all the ranges considered in the described phase diagram (Fig. <ref>). In all the cases when S_T>0 in H>0, the quantum number of z component of total spin S_T is equal to the quantum number of the total spin S̃_T. Moreover, the state is non-degenerate, as the lowest energy is achieved by a single state (other projections S_T<S̃_T are energetically unfavourable in the presence of the field). The situation is quite different for S_T=0 state. Two cases can be separated here. The state with S_T=0 seen in the case of H=0 and J_1>0 (changing to S_T=6 for arbitrarily weak field H>0 - see for example the range of J_2>0 at Fig. <ref>(d)) corresponds to S̃_T=6 and bears 13-fold degeneracy. This is due to the fact that for H=0 all the 2S̃_T+1 possible values of total z spin component (-S̃_T,S̃_T+1,…,S̃_T) lead to the same energy. Moreover, since we deal with canonical ensemble at ground-state, all such states are equally probable, so that all the possible values of total spin z component average to S_T=0. On the contrary, the state with S_T=0 stable for the finite interval of magnetic fields up to the finite first critical magnetic field corresponds to the non-degenerate state with S_T=0 and S̃_T=0.§.§ Ground-state energy The minimization of the energy decides on the form of the ground state taken by the system. However, it can be interesting to analyse the ground-state energy itself as a function of model parameters. The total normalized ground-state energy is plotted as a function of normalized rung interaction J_2/|J_1 and normalized magnetic field H/|J_1| in Fig. <ref> (for antiferromagnetic J_1<0 and in the absence of crossing coupling J_3). In the plot, the colour scale is used to indicate the ground-state energy; moreover, contour lines of constant energy are shown. In addition to these contours, gray lines correspond to the boundaries between phases with various S_T (see Fig. <ref>(a)). It can be noticed that the presence of the magnetic field in principle always decreases the ground-state energy. One exception is the phase with S_T=0, for which the total energy is H-independent. The discontinuities are seen in the slope of the constant energy lines at the phase boundaries. The energy can be observed to decrease much faster with absolute value of crossing coupling |J_2| when J_2<0. The cross-sections of Fig. <ref> at constant magnetic fields H/|J_1|=0.0 and H/|J_1|=0.9 can be followed in Fig. <ref>. The data presented in the plot correspond to both signs on intraleg coupling, both antiferro- and ferromagnetic. Moreover, two kinds of reference energies are plotted. The first one is the optimized 'classical' ground-state energy, expressed byE_classical=-1/4(10|J_1|+6|J_2|+10|J_3|)-1/2NH. This is the minimum energy which can be achieved via the classical states, for which ⟨𝐒_i·𝐒_j⟩ =1/4. Such a correlation value is assumed for every pair of interacting spins to optimize the hypothetical ground-state energy (note that such construction would also minimize the energy of possibly frustrated bonds).Another reference energy is the energy of a set of uncoupled Heisenberg dimers in their ground-state (singlet). Such state energy for a dimer with ℋ_dimer=-J𝐒_𝐢·𝐒_𝐣for J<0 amounts to E_singlet=-3J/4, with correlation ⟨𝐒_i·𝐒_j⟩ = -3/4; the correlation is isotropic in spin space. If J_2 is the dominant coupling, we can decompose the system into 6 dimers (each rung of the ladder is dimerized), having the total energy E_dimer=6E_singlet=-9|J_2|/2.In Fig. <ref> it can be seen that for ferromagnetic J_1>0 and J_2>0 the ground-state energy is equal to the classical energy E_classical. This is the trace of the classical ferromagnetic state. On the other side, for J_2<0 and |J_2|≫ |J_1| the ground-state energy tends to E_dimer, indicating the tendency to forming a rung-dimerized ground state. In all the cases the energy for J_1<0 is smaller than for J_1>0. Moreover, the ground-state energy is less sensitive to external magnetic field for antiferromagnetic J_1 than for ferromagnetic J_1 coupling. If H>0 and J_2<0 for ferromagnetic J_1>0, the ground-state energy is some range of J_2 is higher that the optimum classical energy E_classical. §.§ Spin-spin correlations The magnetic ground state of the quantum nanomagnet can be characterized in a more detailed way by calculating the spin-spin correlations. Since the considered nanostructure lacks translational symmetry, the correlations are site-dependent. Therefore, values of correlation functions for all the inequivalent spin pairs can be examined. An example of such selection of the correlation functions in the absence of the magnetic field, for the antiferromagnetic J_1<0, is shown in Fig. <ref>. The figure presents the inequivalent correlations between nearest neighbours in the same leg of the ladder, between interleg nearest-neighbours as well as crossing interleg correlations. The numerical indices label the pairs of sites at which the spins are located (for explanation see Fig. <ref>). It should be mentioned that for H=0 all the correlations are completely isotropic in spin space (c_ij^xx=c_ij^yy=c_ij^zz). In the limit of strongly antiferromagnetic J_2, all the correlations between nearest-neighbouring spins in different legs (rung correlations) tend to -1/4, as it is expected for quantum dimers (see also the discussion of the ground-state energy). Other correlations (between nearest neighbours in the same leg and crossing correlations) tend to vanish under the same conditions (the intraleg correlations are negative, while the interleg crossing correlations are positive). When J_2 is reduced in magnitude, the interleg correlations become weaker (and achieve 0 at J_2=0), whereas the nearest-neighbour interleg correlations are peaked. In all the cases the correlations for inequivalent spin pairs differ (which is a finite size effect in the nanoscopic system), however, the effect becomes particularly strongfor weak J_2. A trace of dimerization behaviour within each leg can be noticed for weak J_2, as every second correlation is significantly smaller. On the ferromagnetic side of J_2, the rung correlations tend to 1/12 (and the sum of correlations c_ij^xx=c_ij^yy=c_ij^zz=⟨𝐒_i·𝐒_j⟩ amounts to 1/4 as for the classical ferromagnetic state). Other correlations are negative and tend slowly to some limiting values.The behaviour of correlations is completely different for the case of ferromagnetic J_1>0, as shown in Fig. <ref>(b). There, for J_2<0, the rung correlations again tend to -1/4 when J_2 becomes dominant, and other correlations tend to vanish. For ferromagnetic J_2, all the correlations take the common value of 1/12, which is further independent on J_2. At crossing the value of J_2=0 the interleg correlations behave discontinuously, while the intraleg nearest-neighbours correlation varies in continuous manner. For vanishing J_2 from the antiferromagnetic side, all the interleg correlations take the common value. In the case of J_1>0 the differences between the correlations for inequivalent pairs of spins are much less pronounced than in the case of J_1<0.Let us remind that in the absence of the magnetic field we deal with the states characterized by S_T=0 and either S̃_T=6 (which corresponds to J_2>0 in Fig. <ref>(b)) or S̃_T=0(for the rest of Fig. <ref>).The introduction of the external magnetic field exerts an important effect on the behaviour of the correlations. The most important influence is inducing the spin-space anisotropy. For the case of J_1<0 the effect of the magnetic field can be traced in Fig. <ref> (this corresponds to the cross-section of the phase diagram presented in Fig. <ref>(a)). The part Fig. <ref>(a) shows the inequivalent correlations of the type c_ij^zz as a function of normalized J_2/|J_1|. For the most negative values of J_2 we deal with the state S_T=0 (see Fig. <ref>(a)) and the behaviour of the correlations resembles the case from Fig. <ref>(a), close to dimerization with respect to ladder rungs. For weakly negative J_2 we deal with a cross-over to the state S_T=1, and further increase in J_2 causes the state S_T=2 to be achieved. In the latter state, in can be seen that the intraleg nearest-neighbour correlations in the middle of the ladder are strongly suppressed, while similar correlations closer to the edges take more pronounced (negative) values. The situation is somehow similar for the rung correlations and crossing correlations. In general, states with higher spins characterize themselves with more significant differences between correlations for inequivalent spin pairs of the same type.The spin-space anisotropy in correlations can be followed in Fig. <ref>(b), where the correlations of all three types are shown for spin pairs close to the edge. In the state S_T=0 the correlations are isotropic, while for S_T>0 a difference between c_ij^zz and c_ij^xx=c_ij^yy emerges. It is interesting that for S_T=1 the absolute values of zz correlations are smaller than the xx and yy correlations. On the contrary, for S_T=2, the (positive) zz rung correlations dominate over xx and yy components, while the remaining correlations are negative and zz components are weaker than the other in that case.The effect of the magnetic field on the correlations for the case of J_1>0 can be traced in Fig. <ref>. As it is seen in the corresponding phase diagram (Fig. <ref>(d)), increase in J_2 causes crossing of the borders between all the states starting from S_T=0 up to S_T=6. Once more, for strong rung interactions, the correlations resemble the case presented in Fig. <ref>(b). The rung correlations (negative) dominate and tend gradually to switch to positive (ferromagnetic) character when J_2 increases. The remaining correlations show slightly different behaviour. For S_T≥ 4 all the correlations become positive. What is interesting, for S_T≥ 5 all the correlations take the common values (and for S_T=6 this value is 1/4, corresponding to the classical saturated ferromagnetic state). For S_T=1,…,4 the differences between correlations for various inequivalent locations of spin pairs of the same type are more pronounced. Fig. <ref>(b) allows the analysis of the spins pace anisotropies developing under the influence of the magnetic field. For S_T≥ 5 all the zz components take the common positive values, whereas xx and yy components vanish. For the intermediate cases of S_T=1,2,3 the zz components of all correlations are weaker in magnitude than zz components.The dependence of correlations for spin pairs close to the edge on the normalized magnetic field is presented for two selections of interaction parameters in Fig. <ref>, in both cases for J_1<0. In general, it can be seen that the correlations change in step-wise manner, so each state with given S_T corresponds to one plateau. In Fig. <ref>(a) a cross-section of the phase diagram Fig. <ref>(a) reveals that all the intermediate states between S_T=0 and S_t=6 are achieved when H rises. The intraleg nearest-neighbour correlations and rung correlations increase up to saturation, while the crossing correlations first tend to be reduced in their magnitude and then increase. In the case of nearest-neighbour intraleg correlations and rung correlations the zz component can be positive when xx and yy component take negative values, proving the pronounced anisotropy in spin space.In Fig. <ref>(b) the selection of the interaction parameters corresponds to such cross-section of the phase diagram (Fig. <ref>(c)) that only the states with even S_T are crossed when H increases. This is reflected in the number of correlations plateaux. The qualitative behaviour of the correlations resembles the previous case. However, this time the only positive correlations at weak field are the rung correlations. Moreover, at weak field only the nearest neighbour intraleg correlations take the significant values.§ FINAL REMARKS The calculations presented above constitute an extensive analysis of the ground-state behaviour of the selected magnetic nanocluster, of the shape of two-legged finite ladder consisting of 12 spins S=1/2. The interactions in the system corresponded to isotropic quantum Heisenberg model. The obtained results originate from exact numerical diagonalization of such model and provide a view free from approximation artefacts. An influence of the external magnetic field and of the rung coupling on the total spin was illustrated in a phase diagram, calculated for both positive and negative intraleg coupling and for various crossing interactions. The ground-state energy was analysed as a function of the rung coupling and magnetic field. Moreover, the behaviour of the spin-spin correlations for all inequivalent spin pairs of the same type was characterized as a function of the model parameters. In particular, the emergence of spin-space anisotropies under the action of magnetic field was analysed. The comparison of the cases of ferromagnetic and antiferromagnetic intraleg nearest-neighbour couplings was performed, yielding crucial differences between both classes of nanomagnets.The obtained results show an interesting behaviour and a variety of quantum phases exhibited by the studied nanomagnet, demonstrating the large degree of tunability. The possible extensions of the present work involve mainly the consideration of the thermal properties. 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"title": "Ground-state magnetic properties of spin ladder-shaped quantum nanomagnet: Exact diagonalization study"
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[email protected] Physics Division, Faculty of Pure and Applied Sciences, University of Tsukuba, Tennodai Tsukuba 305-8571, Japan We discuss how one calculates the coherent path integrals for locally interacting systems, where some inconsistencies with exact results have been reported previously. It is shown that the operator ordering subtlety that is hidden in the local interaction term modifies the Hubbard-Stratonovich transformation in the continuous time formulation, and it helps reproduce known results by the operator method.We also demonstrate that many-body effects in the strong interaction limit can be well characterized by the free-particle theory that is subject to annealed random potentials and dynamical gauge (or phase) fields.The present treatment expands the conventional paradigm of the one-particle description, and it provides a simple, viable picture for strongly correlated materials of either bosonic or fermionic systems.03.65.Db, 03.70.+k, 05.30.-d, 71.27.+aExact path-integral evaluation of locally interacting systems: The subtlety of operator ordering Nobuhiko Taniguchi December 30, 2023 ================================================================================================ § INTRODUCTIONThe path integral formulation <cit.> has been widely used in many areas of physics and has now become an indispensable tool in formulating, investigating and understanding quantum physics.Its variant, the coherent-state path integral <cit.>, is particularly useful and versatile for analyzing quantum many-body theories where the Hamiltonian is expressed in normal-ordered products of creation and annihilation operators <cit.>. It helps ushandle either bosons or fermions, perform a perturbational expansion, treat nonperturbative contributions like topological effects, and grasp relevant physics intuitively.The downside of the path integral approach is that its direct evaluation tends to demand more effort than that of the operator method. Even for noninteracting quadratic Hamiltonians, great care is needed to tackle the operator ordering subtlety or a seemingly divergent determinant.The situation gets exacerbated for interacting systems, even for the simplest possible interacting system, namely the one-site Bose-Hubbard model.When one uses the time-continuous coherent-state path integral to evaluate, say, [e^-β U c^† c^† c c/2] with a single bosonic field, one may well be deceived into reaching the wrong answer ∑_n=0^∞ e^-β U n^2/2, instead of the correct one ∑_n=0^∞ e^-β U n(n-1)/2 <cit.>. The form of discrepancy strongly suggests that the approach may be plagued by some operator ordering subtlety that the quartic term may have.The same problem prevails in many-particle systems with local interaction, either of bosons or of fermions.In order to remedy this embarrassing situation, <cit.> surmised that an additional correction is present in the representation of the normal-ordered interaction in a way to reproduce the exact result.<cit.> subsequently proposed a possible, consistent redefinition of the coherent-state path integral formulation that successfully reproduces the correct results of the one-site Bose-Hubbard model.The scheme is non-standard, though. Starting with a normal-ordered Hamiltonian, they expressed normal-ordered operators in terms of the coordinate-momentum representation (or the Weyl symbol).The procedure is inconvenient and nontrivial when one tries to treatmany-particle bosonic systems whose degree of freedom is large, not to mention systems that involve many fermions.Because the many-body Hamiltonian is normal-ordered, complying with the standard coherent-state formulation has a clear benefit.It is worth understanding what goes wrong in its conventional treatment and finding a simple, reliable way of reaching correct results. *PurposeIn this paper, we reexamine the coherent-state path integral for locally interacting many-body systems where constituent particles are either bosons or fermions.Like the one-site Bose-Hubbard model, the coherent-path integral seemingly fails to reproduce the exact results of the partition function and Green function, if one uses the conventional definition.We scrutinize the evaluation process and identify the cause in the operator ordering subtlety hidden in the interaction term.We find that circumventing that subtlety makes us modify the Hubbard-Stratonovich (HS) transformation.Accordingly, one can readily reproduce the known exact results in the standard definition of the coherent-state path integral.Our discussion focuses on a simple type of local interaction defined in Eq. (<ref>), but the same argument can straightforwardly apply to a more general form of the interaction among mutually commuting operators, while treating the interaction between mutually non-commuting operators is nontrivial (see Appendix <ref>). Locally interacting models can be viewed as the strong interaction limit of correlated materials where the interaction is much greater than the band width so that each site is effectively isolated. Charge-blocking, many-body Mott physics dominates, and the one-particle picture gets inappropriate.Propagating degrees of freedom responsible for such dynamical gap is elusive <cit.>.In the process of evaluating the path integral, we will encounter a certain free-particle theory that is subject to dynamical phase fields and random potentials. This supplement to the free-particle theory is of great interest because it tells how the free-particle theory can accommodate non-perturbative many-body correlation.One can describe strongly correlated materials by using emergent gauge field <cit.>.Because of charge blocking, the phase degree fluctuates dynamically far beyond Gaussian, and so does the gauge field, which is the time derivative of the phase field.In this respect, one may regard the present calculation as a concrete example of how to analyze dynamical fluctuations of the emergent gauge field in the strong interaction limit. § LOCALLY INTERACTING SYSTEMS§.§ Model We consider a multi-level (or multi-site) system of bosonic or fermionic particles (ψ_α,ψ_α^†), which interact locally.The Hamiltonian is given byĤ = ∑_αϵ_αn̂_α+ U/2N̂(N̂-1),where the label α refer to levels and/or spins, and N̂ = ∑_αn̂_α = ∑_αψ_α^†ψ_α is the total number operator.In spite of the interaction being present, one can exactly calculate thermodynamics and various Green functions by help of the operator method (see Appendix <ref>).Yet, with the coherent-state path integral, one must be cautious to reach those results. §.§ Subtlety disclosed We start by revealing a subtlety hidden in the standard manipulation of the coherent-path integral. Taking the grand partition function Ξ_U(μ)= [ e^-β (Ĥ-μN̂)], we can establish the exact identity between Ξ_U(μ) and the noninteracting counterpart Ξ_0(μ): Ξ_U(μ) =∫^∞_-∞ d[φ̃]e^- βφ̃^2/2U Ξ_0(μ + U2 - iφ̃),which is derived in Eq. (<ref>).Here φ̃ denotes a time-independent Gaussian variable with variance U/β and the measure d[φ̃]=√(β/2π U)dφ̃ includes the normalization. Relation (<ref>) holds for either bosons or fermions. One can see Eq. (<ref>) come from the operator identity [see Eq. (<ref>)],e^-βU/2N̂^2 = ∫^∞_-∞ d[φ̃]e^-βφ̃^2/2U - iβφ̃N̂. The formula can be viewed as an operative version of the HS transformation.An important observation is that when we take the coherent-path integral representation of Eq. (<ref>), it contradicts the standard form of the HS transformation. Indeed, the decomposition concerning e^-βĤ becomes (see Appendix <ref> for the derivation) ∫𝒟[ψ,ψ̅]e^- 𝒮/ħ = ∫𝒟[ϕ̃] 𝒟[ψ,ψ̅]e^- (𝒮_e+𝒮_ϕ)/ħ,where the Euclidean actions 𝒮 and 𝒮_e,ϕ are defined by𝒮 = ∑_α,β∫^βħ_0 dτ ψ̅_α[ ( ħ∂_τ+ ϵ_α) δ_αβ + U/2ψ̅_βψ_β] ψ_α, 𝒮_e= ∫^βħ_0 dτ∑_αψ̅_α( ħ∂_τ + ϵ_α - U/2 + iϕ̃) ψ_α, 𝒮_ϕ =∫^βħ_0 dτ ϕ̃^2/2U.The above formula differs from the standard HS formula by the presence of -U/2 in 𝒮_e.It is caused by circumventing the operator ordering subtlety hidden in the standard derivation of the HS transformation (see Appendix <ref>), and tells us to modify the HS transformation, when we comply with the standard definition of the coherent state path integral.With the modified representation of the interaction, we can readily evaluate the path integral expression of Ξ_U(μ) by following each step of Appendix <ref> reversely.In addition to the operator ordering subtlety, the HS transformation has been known to be plagued by the ambiguity in selecting relevant channels <cit.>.When truncating relevant fluctuations, it causes a serious problem that might give a different physical result.In the present treatment, however, we don't have such a problem, because we carry out the complete integration of the auxiliary fields without any approximation, thanks to the gauge transformation.Moreover, perturbative treatment often brings divergent contributions due to interaction, and it therefore needs an additional renormalization procedure.This is not the case here, because the knowledge of Ξ_0(μ) is sufficient to calculate Ξ_U(μ) exactly. §.§ Green functions We now turn our attention to evaluating various one-particle Green functions of locally interacting systems.Below, we use the closed-time path integral formalism <cit.> to formulate real-time correlation functions.We show how we can exactly evaluate those path integral representations by using a gauge transformation technique <cit.>.Such approach was undertaken in <cit.> to investigate the tunneling density of states at Coulomb-blockade peaks of fermionic locally interacting systems, but its exposition is too succinct to clarify the subtlety of the coherent-state path integrals. We demonstrate how the modified HS transformation [Eq. (<ref>) below], which extends Eqs. (<ref>) to include real-time paths, enables us to evaluate them.Later in Sec. III, we will show that they are identical to what are calculated by the operator method.Moreover, we find that Green functions for a locally interacting system can be connected and determined by the knowledge of noninteracting systems, like the grand partition function [see Eq. (<ref>) or (<ref>) below].We define four types of real-time Green functions, [G_α (t,0) G^<_α(t,0); G^>_α(t,0)G̃_α(t,0) ] = 1/iħ[ ⟨ T ψ_α(t) ψ_α^†⟩±⟨ψ_α^†ψ_α (t) ⟩; ⟨ψ_α (t) ψ_α^†⟩ ⟨T̃ψ_α (t) ψ_α^†⟩ ],where ± refers to bosonic or fermionic systems, and ⟨⋯⟩ is the thermal average specified by chemical potential μ and the inverse temperature β.The operator T is the time-ordering operator, while T̃ is the anti-time-ordering one.We can compactly write them by the contouring-ordering operator T_c along the Keldysh path ∫_K asG_α (t_1,t_2) = 1/iħ⟨ T_cψ_α (t_1)ψ_α^† (t_2) ⟩,= 1/iħΞ_U∫𝒟[ψ,ψ̅] ψ_α(1) ψ̅_α(2) e^i/ħ S,where the path is composed of three segments (see Fig. <ref>): the forward-going (denoting -) from the initial time t_i to the final time t_f, the backward-going (denoting +) from t_f to t_i, and the thermal one from t_i to t_i- iβħ.Since the interaction UN̂(N̂-1)/2 is normal-ordered, the action S that appears in the coherent-state path integral becomesS = ∫_K∑_α,βψ̅_α[( iħ∂_t - ϵ_α) δ_αβ - U/2ψ̅_βψ_β] ψ_α. The next step is crucial: we decompose the interaction term via the modified HS transformation along the Keldysh path.The transformation ise^-i/ħ∫_KU/2 N^2(t) = ∫𝒟[ϕ]e^i/ħ (S_ϕ + S_e).whereS_ϕ = ∫_Kϕ^2(t)/2U,S_e = ∫_K∑_αψ̅_α[iħ∂_t - ϵ_α - ϕ(t)+ U/2 ] ψ_α.The term U/2 is mandatory in S_e, as in Eq. (<ref>).After we have managed the operator ordering subtlety, we may follow the observation in <cit.> to employ the local gauge transformation to absorb most of the effect of ϕ(t). To make this work, however, we have to carefully specify the boundary condition: the periodicity of ϕ(t) must be imposed on each of the three segments of the Keldysh path, to ensure new field operators (Ψ_α below) to remain canonical.We then decompose ϕ-fields on each segment into the static zero-modes φ=(φ_∓,iφ̃), and the dynamical phase fields θ(t) = θ_∓(t) satisfying the periodic boundary condition: ψ_α(t) = e^iθ (t)Ψ_α(t); ϕ(t) = φ - ħθ̇(t).One can safely gauge away the dynamical field on the thermal segment. Now the action becomes S_e = ∫_K∑_αΨ̅_α[iħ∂_t - ϵ_α - φ +U/2 ] Ψ_α,S_ϕ = iβħ/2Uφ̃^2 + Δ t/2U( φ_-^2 - φ_+^2) +∫_Kħ^2θ̇^2/2U,with t_f-t_i = Δ t. The dynamical phase fields may be regarded as compact U(1) gauge fields that commonly emerge in strongly correlated matter <cit.>.One may examine nonperturbative correlation effect by studying nontrivial field configurations that carry finite winding numbers.In the present approach, the dynamical fields θ_∓ describe the fluctuating part on top of nontrivial field configurations, while φ = (φ_±, iφ̃ affects the thermodynamics and its dynamics nonperturbatively.The result of the Ψ-integral can be written by the Green functions multiplied by the grand partition function of the noninteracting particles.We still need to complete the φ- and θ-integrals, but in isolated systems here, those integrals are found to be decoupled <cit.>. Symbolically, one can write the result asG_α (1,2) = 1/Ξ_U⟨Ξ^φ G_α^φ (1,2) ⟩_φ⟨ e^iθ(1) e^-iθ(2)⟩_θ,where ⟨⋯⟩_φ refers to the Gaussian average over the three static Gaussian variables (φ_∓, φ̃), while ⟨⋯⟩_θ, to the path integration over dynamical θ. The explicit forms of Ξ^φ and G_α^φ are nothing but the noninteracting ones, Ξ_0 and G_0,α, whereΞ_0^φ = Ξ_0 ({ϵ_α^φ, μ^φ}) = ∏_α[ 1 ∓ e^-β (ϵ_α^φ - μ^φ)]^∓ 1,G_α^φ(t,0) = G_0,α(t; {ϵ_α^φ,μ^φ}), with incorporating φ-dependence by shifting ϵ_α and μ byϵ_α^φ = ϵ_α - U/2 + φ_c; μ^φ = μ - iφ̃ + φ_c - i Δ t/βħφ_q.Here the convention φ_c = (φ_-+φ_+)/2 and φ_q = φ_- - φ_+ is used. When we recover the Keldysh structure, the part ⟨ e^iθ(1) e^-iθ(2)⟩_θ in Eq. (<ref>) means the contour-ordered vertex correlator.We can calculate it as the action regarding θ is free-particlewith mass U/ħ^2.Though local fluctuation ⟨θ^2 (t) ⟩ diverges, it is finite and equal to ⟨ T_ce^-iθ(t) e^iθ(0)⟩_θ= [ e^-iU/2ħ|t|e^iU/2ħt; e^-iU/2ħte^iU/2ħ|t| ].Combining all the above, we can evaluate exactly all one-particle Green functions for locally interacting systems.Let us briefly illustrate how it operates in practice.The lesser component of Eq. (<ref>) givesG_α^< (t,0) = 1/Ξ_U⟨Ξ_0^φ G^φ,<_α(t,0) ⟩_φ e^iU/2ħ t,and the noninteracting lesser Green function isG^φ,<_α (t,0)= ±e^-i/ħϵ_α^φ t/iħ n_α^φ.The occupation n_α^φ=⟨n̂_α⟩ has to be determined by the partition function Ξ^φ via the standard relation, n_α^φ = - 1/β∂/∂ϵ_αlnΞ_0^φ.It means that G_α^<(t,0) of locally interacting systems is expressed in a form of the annealed average over three random (static) Gaussian variables φ=(φ_∓, φ̃):G_α^< (t,0)= ∓1/iħΞ_U ⟨ e^-i/ħ (ϵ_α^φ- U/2)t ∂Ξ_0^φ/β∂ϵ_α⟩_φ.We can likewise find the greater Green function, G_α^>(t,0)= 1/iħΞ_U⟨ e^-i/ħ (ϵ_α^φ+ U/2)t [ Ξ_0^φ∓∂Ξ_0^φ/β∂ϵ_α] ⟩_φ.From these results of G_α^< and G_α^>, we can construct all the other one-particle Green functions. § EQUIVALENCE TO THE OPERATOR METHOD We now check that the results Eqs. (<ref>) and (<ref>) actually reproduce the Green functions evaluated by the operator method in Appendix <ref>.To see it, we expand Ξ_0^φ in terms of the canonical partition function Z_N of non-shifting levels ϵ_α,Ξ_0^φ = ∑_N=0^∞ Z_Ne^N β (μ + U/2- iφ̃ - iΔ t/βħφ_q). We find that the integration over φ_q simply enforces φ_c/U to non-negative integers N in the limit of Δ t →∞.∫ d[φ_q] e^i Δ t/ħ Uφ_cφ_q e^-iΔ t/ħφ_q N= δ(φ_c -U N). Accordingly, we may say that φ_c/U plays a role of winding numbers of the emergent compact gauge field configuration; a naive saddle-point (or Hartree-Fock) approximation regarding φ misses such nonperturbative contribution. We need to take account of all the contribution of N on principle (see Ref. <cit.> for its implication on the tunneling density of states). By completing the remaining Gaussian average over φ̃, we organize the result as G_α^< (t,0)= ± 1/iħ∑_N=0^∞ e^-i/ħ[ϵ_α+U(N-1)]t n_α|N,G_α^< (ε)= ∓ 2iπ∑_N=0^∞ n_α|N δ( ε - ϵ_α -U(N-1) ).Here we have introduced the quantity n_α|N,the “fractionalparentage” of the occupation number onto the fixed N.It is defined by n_α|N = -1/βΞ_U∂ Z_N/∂ϵ_α e^β Nμ - βU/2N(N-1),and satisfies ⟨n̂_α⟩ = ∑_N=0^∞ n_α|N.Similarly, we find the greater Green function to beG_α^>(t,0) = 1/iħ∑_N=0^∞ e^-i/ħ (ϵ_α + UN)t p_α|N,G_α^>(ε) = -2iπ∑_N=0^∞ p_α|N δ(ε - ϵ_α - UN), by using p_α|N, the fractional parentage of the hole occupation onto a fixed N, defined by p_α|N = 1/Ξ_U[ Z_N∓1/β∂ Z_N/∂ϵ_α] e^β Nμ - βU/2N(N-1).The spectral function ρ_α(ε) is straightforwardly calculated asρ_α (ε) = ∑_N=0^∞[ p_α|N δ(ε - ϵ_α - U N)∓ n_α|N δ(ε - ϵ_α - U(N-1))].In these forms, one can confirm the equivalence with the ones by the operator method in Appendix <ref>.§ DISCUSSIONWe have shown that we can treat a locally interacting system correctly using the standard definition of the coherent-path integral. The results are connected with their noninteracting counterpart. [See Eqs. (<ref>) for the partition function, and (<ref>)-(<ref>) for Green functions.]The relation (<ref>) shows that the thermodynamics of a locally interacting system is exactly equivalent to the annealed average of the noninteracting Hamiltonian with random imaginary potential φ̃.Such simple correspondence, however, cannot be held for Green functions (<ref>)-(<ref>) — they are still written by a free-particle model under the influence of static random fields, as is seen in Eq. (<ref>), but we can assign no single random Hamiltonian for its dynamics, because three independent random variables are needed: φ_∓ along the two real-time paths and φ̃ on the thermal path.We stress that this supplement to the free-particle theory can fully capture various many-body characteristics like atomic correlations, non-rigid bands, asymmetry of particle and hole excitations.While a spectral function in the conventional one-particle/quasiparticle picture has only a single peak, the function ρ_α(ε) of Eq. (<ref>) has multiple peaks with different weights at ε = ϵ_α + UN. At those energies, the retarded self-energy diverges and the retarded Green function vanishes, which signals the demise of the quasiparticle picture <cit.>.To treat non-perturbative many-body effect, it is important to take account of two aspects: discreteness of the particle number and large phase fluctuations beyond quadratic order. They are closely related. We can implement discreteness of N by compactifying the conjugate phase Θ modulo 2π (satisfying [N̂, Θ̂] = i). Non-positive nature of N makes Θ non-Hermite <cit.>.Since the phase Θ(t) couples linearly with Ṅ(t), we may take the HS field ϕ(t) as ϕ(t) = ħΘ̇(t). It means that we need to treat fluctuations of ϕ(t) consistently by respecting such nontrivial nature of Θ.A common practice after introducing the HS field ϕ(t) is to complete the quadratic integration over the field (ψ_α, ψ̅_α), and then to take the saddle-point approximation regarding ϕ.Assuming a uniform solution ϕ(t) = φ_sp, one can determine the self-consistent saddle-point solution φ_sp by the average number ⟨N̂⟩ = φ_sp/U in that approximation.This contrasts with the exact locking of φ_c/U to non-negative integers inEq. (<ref>).A physical picture given by the saddle-point approximation is fundamentally wrong, having no dynamical gap generation and retaining the non-interacting Fermi-Dirac form of the occupation ⟨ n_α⟩.We find the gauge transformation technique is effective to incorporating many-body effects.Without any additional ansatz of the slave-particle, one can describe many-body charge-blocking physics.In hindsight, it is because the local occupation number is conserved that one can solve locally interacting systems exactly.When we couple a locally interacting system linearly with external environments (reservoirs), the local occupation is no longer conserved, and the integrals over φ and θ are coupled unlike Eq. (<ref>). It seems unlikely that we can complete the remaining path integrals exactly.Nevertheless, the present analysis of path integrals provides a useful and systematic means to describe the local strong correlation that perturbation theory cannot treat.In a quantum dot coupled to the leads, two types of strongly correlated phenomena are known to emerge: the Coulomb blockade (or charge-blocking due to correlation) and the Kondo physics <cit.>.When we surmise a decoupling approximation in evaluating the φ and θ integrals as in Eq. (<ref>), repeating the same calculation leads us to the spectral function that is similar to Eq. (<ref>). The only difference is that the delta functions in Eq. (<ref>) now acquire finite width due to the coupling with the reservoirs.It corresponds to the spectral function of the Coulomb blockade regime <cit.>.It was further suggested that if one implements a self-consistent decoupling scheme, one may well understand the Kondo physics <cit.>.It is interesting to see how such decoupling approximation can be improved by taking account of the compact and non-Hermitian nature of phase fluctuations. Our work in this direction is underway.§ SUMMARY To summarize, we have demonstrated how one can evaluate the coherent-state path integrals for locally interacting systems, following its standard definition and bewaring of the operator ordering subtlety.The results agree with the ones by the operator method.In the process of calculating, we find that locally interacting systems is equivalent to certain free-particle models embellished with dynamical phase as well as static random variables. Since we can view locally interacting models the strong interaction limit of a wide-range of strongly correlated materials, it is hoped, we use such free theories as an alternative yet viable simple description for strongly correlated materials.The author gratefully acknowledges financial support from Grant-in-Aid for Scientific Research (C) No. 26400382 from MEXT, Japan. § CALCULATION VIA THE OPERATOR METHOD §.§ Grand partition function Since the effect of the interaction is to increase the energy by U N(N-1)/2 for fixed-N states, we can express the grand partition function of the Hamiltonian (<ref>) asΞ_U (μ) = [ e^-β (Ĥ - μN̂)]= ∑_N=0^∞ Z_Ne^βμ N -βU/2 N(N-1),where Z_N is the canonical partition function of the noninteracting system, defined by Ξ_0(μ) = ∑_N=0^∞ Z_Ne^βμ N = ∏_α[ 1 ∓ e^-β(ϵ_α-μ)]^∓ 1. The sign ∓ 1 refers to bosonic or fermionic systems.One can write the explicit form of Z_N via the inverse transformation of the above asZ_N = ∫^2π_0dθ/2πe^-iNθ Ξ_0 (μ=iθ). §.§ Green functions We can solve exactly various one-particle Green functions for the locally interacting Hamiltonian (<ref>).The system is not needed to be in thermal equilibrium; a generic stationary state will suffice.A quick way to proceed is to examine the equation of motion for a fieldoperator ψ_α:iħ∂ψ_α(t)/∂ t= ( ϵ_α + U N̂) ψ_α(t),which is true for either bosonic or fermionic systems.We can immediately solve its time-evolution asψ_α(t) = e^-i/ħ (ϵ_α + U N̂)tψ_α = ψ_αe^-i/ħ [ϵ_α + U (N̂-1)]t.With this property, we can calculate various Green functions. For instance, the lesser and greater Green functions are found to beG^<_α(t,0)= ±1/iħ⟨n̂_αe^-i/ħ [ϵ_α + U(N̂-1)] t⟩,G^>_α(t,0)= 1/iħ⟨ e^-i/ħ (ϵ_α + U N̂)t( 1 ±n̂_α) ⟩, where the average ⟨⋯⟩ refers to some stationary state average.In the energy space, they becomeG_α^<(ε) = ∓ 2 i π⟨n̂_α δ(ε - ϵ_α - U (N̂-1) ⟩,G_α^>(ε) = -2iπ⟨ (1±n̂_α)δ(ε - ϵ_α - U N̂)⟩. We can construct all other Green functions using the results of G_α^<,>.The spectral function ρ_α(ε) = -G_α^R(ε)/π is found to beρ_α (ε)= ⟨ (1 ±n̂_α)δ(ε - ϵ_α - U N̂)∓n̂_α δ(ε - ϵ_α - U(N̂-1)) ⟩. For fermionic systems, the results take particularly simple forms resembling the free-particle, by the property n̂_α^2 = n̂_α. Indeed, the spectral function becomesρ_α(ε) = ⟨δ(ε - ϵ_α - U N̂'_α) ⟩,with introducing N̂'_α = N̂ -n̂_α.All Green functions likewise have free-fermion forms only with replacing ϵ_α↦ϵ_α + U N̂'_α.When we further assume that the system is in thermal equilibrium with μ and β, the Kubo-Martin-Siggia relation makes the average occupation number be characterized bythe Fermi-Dirac distribution as⟨n̂_α⟩= ⟨1/e^β (ϵ_α + U N̂'_α - μ) + 1 ⟩, though local interaction makes it considerably deviate from the Fermi-Dirac function regarding ϵ_α-μ.§ DERIVATIONS OF EQS. (<REF>)–(<REF>)In this appendix, we present the step-by-step derivations of Eqs. (<ref>)–(<ref>) in the main text.We start with the Gaussian integral formulae^-βU/2 N^2 = ∫ d[φ̃]e^-βφ̃^2/2U - iβφ̃ N,where d[φ̃] include the normalization factor and N is just a number. By using the above and Eqs. (<ref>)–(<ref>), we immediately prove Eq. (<ref>) as ∫^∞_-∞ d[φ̃]e^-βφ̃^2/2U Ξ_0(μ + U2 - iφ̃)= ∫^∞_-∞ d[φ̃]e^-βφ̃^2/2U ∑_N=0^∞ Z_N e^β (μ +U/2-iφ̃) N ,= ∑_N=0^∞ Z_N e^β(μ+U/2) N -β U N^2/2 = Ξ_U(μ). We can extend the Gaussian formula (<ref>) to the operator identity by inserting the complete basis of the occupation number representation |{n_α}⟩ with the total numberN=∑_α n_α:e^-βU/2N̂^2 = ∑_{ n_α} |{ n_α}⟩ e^-βU/2 N^2⟨{ n_α}| ,= ∑_{ n_α} |{ n_α}⟩∫ d[φ̃]e^-βφ̃^2/2U - iβφ̃ N⟨{ n_α}| ,= ∫ d[φ̃]e^-βφ̃^2/2U - iβφ̃N̂.This proves the operator identity (<ref>) in the text. With this identity, we can rewrite the operator e^-βĤ ase^-βĤ =∫ d[φ̃]e^-βφ̃^2/2U -β∑_α (ϵ_α +iφ̃ -U/2) n̂_α. We now represent both sides of Eq. (<ref>) to establish the modification of the Hubbard-Stratonovich transformation in the coherent-state path integral. Since the Hamiltonian Ĥ is normal-ordered, the left-hand side of Eq. (<ref>) is simply represented as(LHS) = ∫𝒟[ψ,ψ̅]e^-𝒮/ħ, 𝒮 = ∑_α,β∫^βħ_0 dτ ψ̅_α[ ( ħ∂_τ+ ϵ_α) δ_αβ + U/2ψ̅_βψ_β] ψ_α.Now we can express the right-hand side of Eq. (<ref>) as(RHS) = ∫ d[φ̃] ∫𝒟[ψ,ψ̅] e^- β/2Uφ̃^2 - 𝒮_e/ħ , = ∫𝒟[θ] ∫ d[φ̃] ∫𝒟[ψ,ψ̅]e^- β/2Uφ̃^2 -𝒮_1/ħ -𝒮_θ/ħ.Here the Euclidean action Lagrangian 𝒮_1 and 𝒮_θ are defined as𝒮_1 = ∫^βħ_0 dτ∑_αψ̅_α( ħ∂_τ + ϵ_α - U/2 + iφ̃) ψ_α, 𝒮_θ = ∫^βħ_0dτ ħ^2/2U (∂_τθ)^2,and, on Eq. (<ref>), we have inserted the path integral over bosonic field θ that satisfies the periodic boundary condition, ∫𝒟[θ]e^- 𝒮_θ/ħ = 1.Next, we introduce a new (dynamical) field ϕ̃(τ) = φ̃ - ħ∂_τθ(τ) to combine φ̃ and θ, and redefine field ψ_α to absorb the phase factor. This is the reverse manipulation of the gauge transformation in <cit.>, with the corresponding Jacobian 𝒟[θ]d[φ̃] = 𝒟[ϕ̃].It enables us to express the right-hand side of Eq. (<ref>) as(RHS) = ∫𝒟[ϕ̃] 𝒟[ψ,ψ̅]e^-𝒮_e/ħ - 𝒮_ϕ/ħ,where 𝒮_e and 𝒮_ϕ are defined in Eqs. (<ref>c,d); this proves Eqs. (<ref>a–d) in the text.§ SUBTLETY OF THE HUBBARD-STRATONOVICH DECOUPLING IN THE CONTINUOUS TIME FORMULATIONWe explicitly point out where matters the subtlety of the Hubbard-Stratonovich transformation in the continuous time formulation. Below we write for the one-site bosonic system but the same argument applies equally to multi-level extension as well as fermionic systems.We examine how one can evaluate the matrix element ⟨ z | e^-i t/ħU/2n̂^2 | w⟩ regarding the bosonic coherent state |z⟩ = e^z̅ b - b^† z |0⟩, with or without the Hubbard-Stratonovich transformation.Direct evaluation of the matrix element leads to⟨ z | e^-i t/ħU/2n̂^2 | w⟩ = e^-1/2(z̅ z + w̅ w)∑_n=0^∞(z̅ w)^n/n! e^-i t/ħU/2 n^2. We now decompose the interaction term using the operator identity.e^-i t/ħU/2n̂^2 = ∫^∞_-∞ d[φ] e^i t/ħ (φ^2/2U - φn̂) = ⟨ e^-it/ħφn̂⟩_φ,where d[φ] includes the normalization factor and ⟨⋯⟩_φ indicates the Gaussian average over φ. One can check the correctness of this decomposition byputting it on the left-hand side of Eq. (<ref>) and using the Wick theorem with ⟨φ^2⟩_φ = ħ U/(-it): ⟨ z| ⟨ e^-it/ħφn̂⟩_φ|w⟩ = e^-1/2 (z̅z + w̅ w)∑_n=0^∞(z̅ w)^n/n!exp[ ⟨12(-itħφ n )^2⟩_φ].So far so good. Now the subtlety appears when we try to formulate it using the path integral.When we expand the expression for infinitesimal time δ t up to the linear order, we see it behave as⟨ z | e^-i δ t/ħU/2n̂^2 | w⟩ ≈ e^-1/2(z̅ z + w̅ w)∑_n=0^∞(z̅ w)^n/n![ 1 - i δ t/ħU/2 n^2].Yet, this correct behavior cannot be reproduced when we truncate Eq. (<ref>) up to the linear order of δ t.The corresponding contribution comes from the quadratic order term proportional to (δ t)^2⟨φ^2⟩.In other words, if we naively formulated the continuous-time path integral just by expanding it regarding the linear δ t and exponentiating it, we would get a wrong result. The missing U/2 term exactly results from this slack manipulation; the use of the modified Hubbard-Stratonovich transformation resolves the issue by avoiding such manipulation carefully.§ EXTENSIONS OF THE OPERATIVE HUBBARD-STRATONOVICH DECOUPLINGOur discussion relies on the operator version of the Hubbard-Stratonovich transformation and its path integral representation.We can generalize the argument to more general forms interaction composed by a set of mutually commuting operators, such as {n̂_α}.It is because we can find the simultaneously diagonalized basis |{n_α}⟩ and the operator identity can be formulated straightforwardly [see Eq. (<ref>)].Therefore, the following operator identity is established:e^-β1/2∑_α,β U_αβn̂_αn̂_β = ∫ d[φ̃⃗̃]e^-β/2∑_αβφ̃_α(U^-1)_αβφ̃_β - iβ∑_αφ̃_αn̂_α.The path-integral representation of the Hubbard-Stratonovich transformation should be modified accordingly to be consistent with this operator identity.The situation gets tricky when one treats a term involving mutually non-commuting operators. A common example is the spin exchange term Ŝ⃗̂^2, which one sometimes tries to decompose in a spin-rotational way using the Hubbard-Stratonovich transformation. The decomposition relies on theintegral identitye^β J S⃗^2 = ∫ d[m⃗] e^-βm⃗^2/4J -β m⃗·S⃗,where m⃗ refers to a three-component vector that obeys the Gaussian distribution respectively and d[m⃗] includes the normalization.We emphasize that though the above identity is correct for any vector S⃗, one cannot promoted it to an operator identity with the spin operator Ŝ⃗̂, because of its non-commutative nature.One can easily check this fact by taking the trace of both sides of Eq. (<ref>) for spin one-half operator — the left-hand side yields 2e^3/4β J, whereas the right-hand side, 2e^β J/4 ( 1 + β J/2).Therefore applying such types of the HS decoupling involving non-commutative operators should be cautioned in the path integral formulation. apsrev4-1 24 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[Feynman and Hibbs(1965)]FeynmanBook65 author author R. P. Feynman and author A. R. Hibbs, @nooptitle Quantum Mechanics and Path Integrals (publisher McGraw-Hill, address New York, year 1965)NoStop [Kleinert(2009)]KleinertBook09 author author H. Kleinert, @nooptitle Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets,edition 5th ed. (publisher World Scientific,address Singapore, year 2009)NoStop [Klauder and Skagerstam(1985)]KlauderBook85 author author J. R. Klauder and author B.-S. Skagerstam, @nooptitle Coherent states, Applications in physics and Mathematical Physics (publisher World Scentific Pub. Co., address Singapore, year 1985)NoStop [Zhang et al.(1990)Zhang, Feng, and Gilmore]Zhang90 author author W.-M. Zhang, author D. H. Feng, and author R. Gilmore, 10.1103/RevModPhys.62.867 journal journal Rev. Mod. Phys. volume 62, pages 867 (year 1990)NoStop [Altland and Simons(2010)]AltlandBook10 author author A. Altland and author B. Simons, 10.1017/CBO9780511789984 title Condensed Matter Field Theory, edition 2nd ed.(publisher Cambridge University Press, address Cambridge, year 2010)NoStop [Wilson and Galitski(2011)]Wilson11 author author J. H. Wilson and author V. Galitski, 10.1103/PhysRevLett.106.110401 journal journal Phys. Rev. Lett. volume 106, pages 110401 (year 2011)NoStop [Kordas et al.(2014)Kordas, Mistakidis, and Karanikas]Kordas14 author author G. Kordas, author S. I. Mistakidis,and author A. I. Karanikas, 10.1103/PhysRevA.90.032104 journal journal Phys. Rev. A volume 90, pages 032104 (year 2014)NoStop [Phillips(2010)]Phillips10 author author P. Phillips, 10.1103/RevModPhys.82.1719 journal journal Rev. Mod. Phys. volume 82, pages 1719 (year 2010)NoStop [Lee et al.(2006)Lee, Nagaosa, and Wen]Lee06 author author P. A. Lee, author N. Nagaosa,andauthor X.-G. Wen, 10.1103/RevModPhys.78.17 journal journal Rev. Mod. Phys. volume 78, pages 17 (year 2006)NoStop [Kleinert(2011)]Kleinert11 author author H. Kleinert, @noopjournal journal Electronic Journal of Theoretical Physics volume 8, pages 57 (year 2011), http://arxiv.org/abs/arXiv:1104.5161 arXiv:1104.5161 NoStop [Kleinert(2016)]KleinertBook16 author author H. Kleinert, @nooptitle Particles and Quantum Fields (publisher World Scentific, address Singapore, year 2016)NoStop [Kadanoff and Baym(1962)]KadanoffBook62 author author L. P. Kadanoff and author G. Baym, @nooptitle Quantum Statistical Mechanics (publisher Addison-Wesley, address New York, year 1962)NoStop [Keldysh(1965)]Keldysh65 author author L. V. Keldysh, @noopjournal journal Sov. Phys. JETP volume 20, pages 1018 (year 1965)NoStop [Rammer(2007)]RammerBook07 author author J. Rammer, 10.1017/CBO9780511618956 title Quantum Field Theory of Non-equilibrium States (publisher Cambridge University Press, address Cambridge,year 2007)NoStop [Kamenev(2011)]KamenevBook11 author author A. Kamenev, 10.1017/CBO9781139003667 title Field Theory of Non-Equilibrium Systems (publisher Cambridge University Press, address Cambridge, year 2011)NoStop [Kamenev and Gefen(1996)]Kamenev96 author author A. Kamenev and author Y. Gefen, 10.1103/PhysRevB.54.5428 journal journal Phys. Rev. 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Rep. volume 256, pages 367 (year 1995)NoStop [Aleiner et al.(2002)Aleiner, Brouwer, and Glazman]Aleiner02 author author I. L. Aleiner, author P. W. Brouwer,and author L. I. Glazman, @noopjournal journal Phys. Rep. volume 358, pages 309 (year 2002), http://arxiv.org/abs/cond-mat/0103008 cond-mat/0103008 NoStop [Ingold and Nazarov(1992)]Ingold92 author author G.-L. Ingold and author Y. Nazarov, title Charge tunneling rates in ultrasmall junctions, in 10.1007/978-1-4757-2166-9_2 booktitle Single Charge Tunneling, series NATO ASI Series, Vol. volume 294, editor edited by editor H. Grabert andeditor M. Devoret (publisher Springer US, address New York, year 1992) Chap. chapter 2, pp. pages 21–107NoStop [Schoeller and Schön(1994)]Schoeller94 author author H. Schoeller and author G. Schön, 10.1103/PhysRevB.50.18436 journal journal Phys. Rev. B volume 50, pages 18436 (year 1994)NoStop | http://arxiv.org/abs/1709.09303v1 | {
"authors": [
"Nobuhiko Taniguchi"
],
"categories": [
"quant-ph",
"cond-mat.str-el"
],
"primary_category": "quant-ph",
"published": "20170927015620",
"title": "Exact path-integral evaluation of locally interacting systems: The subtlety of operator ordering"
} |
Institute of Molecular Physics, Polish Academy of Sciences M. Smoluchowskiego 17, 60-179 Poznań, Poland The aim of presented first principles study of La_0.5Bi_0.5NiO_3 is to investigate electronic structure of orthorhombic phase Pbnm. The calculations show that metallicity and magnetism of the system are strongly related with hybridization between Ni 3d and O 2p. To improve the quality of the electronic structure description of the system, especially the treatment of correlation for the Ni 3d, we employ GGA, LDA, and GGA+U, LDA+U. The LSDA results give good agreement with experiment. Thus, the screening effects originating from the hybridized 3d and O 2p electrons are sufficiently strong that they reduce the electronic correlations in the La_0.5Bi_0.5NiO_3, making it a weakly correlated metal. 71.15.Mb, 71.20.-b, 71.20.Nr, 77.84.BwOrthorhombic phase of La_0.5Bi_0.5NiO_3 studied by first principles J. Kaczkowski[corresponding author e-mail:[email protected]], M. Pugaczowa-Michalska, A. Jezierski December 30, 2023 ================================================================================================================§ INTRODUCTIONThe bismuth perovskite oxides have attracted attention due to the different interesting properties like e.g. multiferroicity (BiFeO_3 <cit.>), charge ordering (BiNiO_3 <cit.>), or as a lead-free ferro- and piezoelectric materials (BiAlO_3 <cit.>) to name the few. Variation of valence states can lead to notable phenomena such as superconductivity, magnetoresistance, negative thermal expansion in transition metal oxides. Much efforts has been focused on BiNiO_3 proved to undergo a phase transition accompanied by the charge redistribution between Bi and Ni ions <cit.>. In ambient conditions, BiNiO_3 in triclinic structure (P-1) displays the antiferromagnetic insulating ground state (G-type AFM, T_N=300 K) <cit.>, as well as exhibits an unusual charge distribution of Bi^3+_0.5Bi^5+_0.5Ni^2+O_3 <cit.>. This charge disproportionation (CD) can be suppressed by applying an external pressure of several GPa or substituting Bi sites partially with La, which turns the system into the orthorhombic (Pbnm) metallic phase <cit.>. According to the analyses of X-ray absorption spectra for the orthorhombic phase of Bi_1-xLa_xNiO_3 <cit.>,the valence of Ni ions is neither +2 nor +3 but in between of them, reflecting the nature of charge fluctuation between the A-site (i.e. Bi) and the B-site (i.e. Ni) ions. The end members of the solid solution (i.e. BiNiO_3 and LaNiO_3) have differing ground state electronic properties. LaNiO_3 is a metal and the only member of the perovskite nickelates family (RNiO_3, where R is a rare earth) lacking any magnetic order in its bulk form.Our previous studies <cit.> revealed the complex nature of the insulating band gap in BiNiO_3 in the P-1 structure that arises not only from the correlation effect of Ni 3d orbitals, but also from CD at Bi sites. The correlation effect of Ni 3d was described by DFT+U method. This method is generally regarded to be the most computationally feasible means to reproduce the correct insulating ground states in correlated systems. The theoretical description of reduction in CD at Bi sites by La substitution at Bi (i.e. Bi^5+ site) of BiNiO_3supports the decrease in the band gap value. However, that it is not sufficient to close the insulator band gap in the system in P-1 structure. Here we present the results of orthorhombic phase ofLa_0.5Bi_0.5NiO_3 studied by first principles. The orthorhombic Pbnm structure is the ground state of La_0.5Bi_0.5NiO_3 <cit.>. This issue will be discussed along with the connection of our results with the experiments <cit.>. § METHOD OF CALCULATIONSThe calculations are performed using the projector augmented wave method (PAW) <cit.>, which is implemented in the Vienna ab initio Simulation Package (VASP) <cit.>. The Perdew-Burke-Ernzerhof (PBE) <cit.> generalized gradient and Perdew-Zunger (PZ) <cit.> local density approximations (LDA) with and without on-site Coulomb interactions (DFT+U) were used for the exchange-correlation potentials. The DFT+U method is generally employed to reproduce the correct ground states in correlated systems (like an insulator or semiconductor). The typical application of the DFT+U method introduces a correction to the DFT energy by introducing a single numerical parameter U. The value of the on-site Coulomb repulsion parameter was chosen as U = 7 eV for Ni 3d (as for BiNiO_3). In the frame of the DFT+U we have used the rotationally invariant approach of <cit.>. § RESULTSWe performed the total energy calculations of Bi_0.5La_0.5NiO_3 for various possible occupations of Bi and La sites in two crystallographic phases: triclinic P-1 and orthorombic Pbnm. Our calculations show that the total energy of Bi_0.5La_0.5NiO_3 in the orthorhombic Pbnm structure is lower that the total energy of the system in the triclinic P-1 phase for both DFT and DFT+U approaches. The differences between the total energies of these structures are at least 41.8 meV/f.u. and 36 eV/f.u.from DFT and DFT+U approaches, respectively. In order to study the site preference of La ion in BiNiO_3 in the low-energy Pbnm structure we substitute some of Bi atoms in the supercell of the parent system by La. In Bi_0.5La_0.5NiO_3 the effect of the substitutional disorder was studied by simulating explicitly all inequivalent Bi/La arrangements within the 40-atom cell of orthorombic phase. Fig.<ref> shows a sketch of Bi sublattice in our study of Bi_0.5La_0.5NiO_3. Six inequivalent Bi/La arrangements are possible for eight positions in the Fig.<ref>. According to the sketch, La ions may occupy the following Bi positions: (1,3,6,8), (2,4,6,8), (4,5,6,8), (1,2,3,4), (3,5,6,8) or (2,5,6,8). Moreover, four types of magnetic ordering at Ni cations sites are considered for the studied orthorombic phase: an ferromagnetic (FM) one and three antiferromagnetic (A-, C-, and G-AFM) ones. The detailed description of these spin arrangements of Ni sublattices are the same as in <cit.>. For all the Bi_0.5La_0.5NiO_3 confgurations investigated, the atomic structure was fully relaxed until residual forces become smaller than 0.01 eV/Å. As a general rule, we note that the system prefers Bi/La arrangements that respect the "rocksalt order", i.e. (2,4,6,8). Furthermore, all of the employed exchange-correlation approaches do not change this feature. Others Bi/La arrangement of Bi_0.5La_0.5NiO_3 correspond to high energy (>22 meV/f.u. for DFT and >19 meV/f.u. for DFT+U) and therefore, we focused on the Bi/La arrangement of Bi_0.5La_0.5NiO_3 with the "rocksalt order". Experimental investigations of Bi_1-xLa_xNiO_3 have shown that the sample with x=0.5 is metallic down to 5 K <cit.>. Temperature dependence of the inverse magnetic susceptibility for the same sample with x=0.5 is more consistent with S=1/2 (Ni^3+) <cit.>. We found that LDA functional reproduces the experimental metallic ground-state (GS) with 0 μ_B on Ni in Bi_0.5La_0.5NiO_3. For calculations with PBE functional we were enabled to achieve metalic GS of the system for low energy FM ordering, but the magnetic moments on Ni are in range of 0.378 ÷ 0.582 μ_B. The consequence of including correlation on 3d shell of Ni atom for both LDA+U and PBE+U calculations is that the GS of the system in Pbnm is the insulating with the band gap at least of 0.403 eV. The G-AFM order is preferable in DFT+U due to the value of total energy of the Bi_0.5La_0.5NiO_3. We note, that this configuration, however, has not been experimentally reported for Bi_1-xLa_xNiO_3 with x ≥ 0.2 in <cit.>. The charge disproportionation (CD) of Bi ions into Bi^3+ and Bi^5+ in the triclinic phase of BiNiO_3 has been discussed by the electron localization function in <cit.>. Our previous studies on BiNiO_3 revealed that the CD of Bi is suppresed by La ions within P-1 structure. However, total energy calculations presented here confirmed experimental results of Ishiwata et al. <cit.> that random substitution of La for Bi causes transition to higher symmetry, i.e. Pbnm. The experimental results were interpreted as a transfer of the oxygen holes from the Bi—O sublattice to the Ni—O sublattice to be expressed as Bi^3++Bi^5++2Ni^2+ (triclinic) → 2Bi^3++2Ni^2+L (orthorhombic) <cit.>. Thus, we focus on results of electronic structure calculations, especially on orbital occupancy of electrons. The main motive of the crystal Pbnm structure are NiO_6 octahedra which are connected via all vertices. As a result the bands are broader and the electron states more itinerant.Fig.<ref> presents the density of states (DOS) of Bi_0.5La_0.5NiO_3 in Pbnm phase calculated using LDA approach. The largest part of the valence band within the energy range -7 eV to 0 eV is dominated by O-2p and Ni-3d states with some small contributions from Bi and La electronic states. From the partial DOS of O-2p and Ni-3d (Fig.<ref>(b)), one can see that the Fermi level comes through the O-2p states having significant admixture of Ni-3d, especially e_g-like orbitals (Fig.<ref>). In the studied system the localized Ni-3d states (mainly t_2g-like) centered at 1.2 eV below the Fermi level are visible as a strong peaks in atom-resolved DOS for spin-↑ and spin-↓ directions. A set of delocalized Ni 3d states (mostly e_g-like) crosses the Fermi level. The Ni t_2g orbitals are occupied for both spin-↑ and spin-↓ DOS and found in energy range -6.5 to 0.0 eV. The Ni e_g states are partially filled. A sharp peak is visible close to the Fermi level. From inspection of projected DOS obtained by LDA we may infer that in the Bi_0.5La_0.5NiO_3 (with Pbnm) the valence of Ni behaves approximately as (2+δ)+, rather than 3+.The computed structural parameters of Bi_0.5La_0.5NiO_3 in Pbnm structure are given in Table 1. § CONCLUSIONSOur study has shown that the presence of 50% La in BiNiO_3, as well as random substitution of La by Bi leads to higher symmetry of the system. The calculated total energy of Bi_0.5La_0.5NiO_3 in the orthorombic Pbnm structure is lower than one in the triclinic P-1. This result is in agreement with experimental evidence. The metalic ground state is obtained for LDA and PBE calculations. The Coulomb repulsion in Ni-3d orbital for both LDA+U and PBE+U calculations (U=7 eV) provides the insulating ground state in Bi_0.5La_0.5NiO_3 with G-AFM magnetic ordering. Based on our calculations, we show that the electronic screening effect from the delocalized Ni-3d and O-2p states should mitigate the electronic correlations of Ni atoms, making Bi_0.5La_0.5NiO_3 a weakly correlated metal.References 99 CatalanG. Catalan, J.F. Scott,Adv. Mater. 21 2463 (2009). http://dx.doi.org/10.1002/adma.200802849 Chybczynska K. Chybczyńska E. Markiewicz, M. Błaszyk, B. Hilczer, B. Andrzejewski, J. Alloys Compd. 671 493 (2016). https://doi.org/10.1016/j.jallcom.2016.02.104 Azuma1 M. Azuma, W-T. Chen, H. Seki, M. Czapski, O. Smirnova, K. Oka, M. Mizumaki, T. Watanuki, N. Ishimatsu, N. Kawamuta, S. Ishiwata, M.G. Tucker, Y. Shimakawa, P. Attfield, Nature Commun. 2 347 (2011). doi: 10.1038/ncomms1361 Azuma2 M. Azuma, S. Carlsson, J. Rodgers, M.G. Tucker, M. Tsujimoto, S. Ishiwata, S. Isoda, Y. Shimakawa, M. Takano, J.P. Attfield, J. Am. Chem. Soc. 129 14433 (2007).DOI: 10.1021/ja074880u Kaczkowski J. Kaczkowski, Mater. Chem. Phys. 177 405 (2016). https://doi.org/10.1103/PhysRevB.80.233104 Mizumaki M. Mizumaki, N. Ishimatsu, N. Kawamura, M. Azuma, Y. Shimakawa, M. Takano, T. Uozumi, Phys. Rev. B 80 233104 (2009). http://dx.doi.org/10.1103/PhysRevB.80.233104 McLeod J.A. McLeod, Z.V. Pchelkina, L.D. Finkelstein, E.Z. Kurmaev, R.G. Wilks, A. Moewes, I.V. Solovyev, A.A. Belik, E. Takayama-Muromachi, Phys. Rev. B 81 144103 (2010). http://dx.doi.org/10.1103/PhysRevB.81.144103 Carl S.J.E. Carlsson, M. Azuma, Y. Shimakawa, M. Takano, A. Hewat, J.P. Attfield, J. Solid State Chem. 181 611 (2008). http://dx.doi.org/10.106/j.jssc.2007.12.037 Ishiwata1 S. Ishiwata, M. Azuma, M. Takano, E. Nishibori, M. Takata, M. Sakata, K. Kato, J. Mater. Chem. 12 3733 (2002). DOI: 10.1039/B206022A Ishiwata2 S. Ishiwata, M. Azuma, M. Hanawa, Y. Moritomo, Y. Ohishi, K. Kato, M Tanaka, E. Nishibori, M. Sakata, I. Terasami, M. Takano, Phys. Rev. B 72 045104 (2005). http://dx.doi.org/10.1103/PhysRevB.72.045104 WadatiH. Wadati, M. Takizawa, T.T. Tran, K. Tanaka, T. Mizokawa, A. Fujimori, A. Chikamatsu, H. Kumigashira, M. Oshima, S. Ishiwata, M. Azuma, M. Takana, Phys. Rev. B 72 155103 (2005). https://doi.org/10.1103/PhysRevB.72.155103 MPM M. Pugaczowa-Michalska, J. Kaczkowski, Comput. Mater.Sci. 126 407 (2017). https://doi.org/10.1016/j.commatsci.2016.10.014 Ishiwata3S. Ishiwata, M. Azuma, M. Takano, E. Nishibori, M. Takata, M. Sakata, Physica B 329-333 813 (2003). http://dx.doi.org/10.1016/S0921-4526(02)02585-1 Blochl P.E. Blöchl, Phys. Rev. B 50 17853 (1994). http://dx.doi.org/10.1103/PhysRevB.50.17953 Kresse1 G. Kresse, D. Joubert, Phys. Rev. B 59 1758 (1999). DOI:10.1103/PhysRevB.59.1758 Kresse2 G. Kresse, J. Furthmüller, Phys. Rev. B 54 11169 (1996). DOI:10.1103/PhysRevB.54.11169Perdew1 J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 3865 (1996). DOI: http://dx.doi.org/10.1103/PhysRevLett.77.3865 Perdew2 J.P. Perdew, A. Zunger, Phys. Rev. B 23 5048 (1981). DOI: http://dx.doi.org/10.1103/PhysRevB.23.5048 Dudarev S. L. Dudarev, G.A. Botton, S.Y. Savrasov, C.J. Humphreys, A.P. Sutton, Phys. Rev. B 57 1505 (1998). DOI: http://dx.doi.org/10.1103/PhysRevB.57.1505 | http://arxiv.org/abs/1709.09469v1 | {
"authors": [
"J. Kaczkowski",
"M. Pugaczowa-Michalska",
"A. Jezierski"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170927121809",
"title": "Orthorhombic phase of La$_{0.5}$Bi$_{0.5}$NiO$_{3}$ studied by first principles"
} |
Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research,36/P, Gopanpally Village, Serilingampally Mandal, RR District, Hyderabad, 500019, India Understanding the effect of glassy dynamics on the stability ofbio-macromolecules and investigating the underlying relaxation processes governingdegradation processes of these macromolecules are of immense importancein the context of bio-preservation. In this work we have studied the stabilityof a model polymer chain in a supercooled glass-forming liquid at different amountof supercooling in order to understand how dynamics of supercooled liquids influence the collapse behavior of the polymer. Our systematic computer simulation studies find that apart from long time relaxation processes (α relaxation), short time dynamics of the supercooled liquid, known as β relaxationplays an important role in controlling the stability of the modelpolymer. This is in agreement with some recent experimental findings.These observations are in stark contrast with the common belief that only long time relaxation processes are the sole player. We find convincing evidence that suggest that one might need to review thethe vitrification hypothesis which postulates that α relaxationscontrol the dynamics of biomolecules and thus α-relaxation time should be considered for choosing appropriate bio-preservatives.We hope that our results will lead to understand the primary factorsin protein stabilization in the context of bio-preservation.Role of α and β relaxations in Collapsing Dynamics of a Polymer Chain in Supercooled Glass-forming Liquid Smarajit Karmakar December 30, 2023 =========================================================================================================§ INTRODUCTIONMany organisms can survive in dehydrated state for long period of times byaccumulating large amount of sugars (sometime 20-50 % of the dry weight) <cit.>. These carbohydrates (mainly trehaloseand sucrose) stabilize proteins and membranes in dry state <cit.>.There are many hypotheses for this protein stabilization and they mainlyfocus on the vitrification of the stabilizing sugar matrix along with thebiomolecules and water replacement from the neighbourhood of the biomoleculesby the sugar <cit.>. In the waterreplacement hypothesis, it is believed that water molecules are replacedby sugar which provides appropriate hydrogen bonds to polar residues ofmacromolecules thereby stabilizing them thermodynamically. A slightlyrefined hypothesis is water entrapment hypothesis, in which it is arguedthat interfacial waters provide stabilization of local conformations ofbiomolecules. Some regions on the surface of the biomolecule are morehydrophilic than others which leads to preferential binding of watermolecules at biomolecule-sugar interface.On the other hand, vitrification hypothesis is purely based on kinematics.It is assumed that the carbohydrates form glasses at high concentrationsor in dry state and thereby slow down the degradation process of biomolecules.This hypothesis mainly focuses on how glassy materials relaxes at longer timescale. A well-known example of such a phenomena is the preservation of insects in amber for millions of years, suggesting that vitrification is one ofthe best choices for nature for bio-preservation. A recent hypothesis whichis a variant of the vitrification hypothesis, suggests that the shortertime-scale β relaxation rather than slower and longer time αrelaxation of the glass-forming liquid is actually responsible for thedegradation of the biomolecule in sugar glasses<cit.>.All of these hypotheses suggest rather different approaches to designappropriate sugar glass model to optimally increase the stability ofbiomolecules for preservation. A clear understanding towards this directionwarrants consideration of all the relevant relaxation processes in glassforming liquids,which is summarized belowRelaxation of density fluctuations in supercooled liquids is hierarchicaland happens in multiple steps as the putative glass transition temperatureis approached. After a fast initial decay the correlation functions approachesa plateau and then at subsequent long-time it decay to zero. The relaxationthat happens in the plateau like regime is called β relaxation andthe longer time decay from the plateau to zero is called α relaxation <cit.>. It is well-known that the α-relaxation is very heterogeneous and cooperative in nature with a associated growth ofa dynamic heterogeneity length scale <cit.>. On theother hand β-relaxation is believed to be more local process withoutany significant growth of correlation length <cit.>, but recent studies havesuggested that shorter time relaxation processes are probably alsocooperative in nature with length scale that grow very similarly as thelong time dynamic heterogeneity length scale <cit.>. This indicates that if cooperative motions are required for certain relaxationprocess to happen in a molecules embedded in supercooled liquid, then both shortand long time relaxation processes will probably play equally important role. Forexample, if α-relaxation plays a key role in degradation of proteinmolecules in glassy matrix due to its cooperative nature to induce mobilityin these biomolecules which are much larger compare to the solvent molecules,then shorter time β-relaxation process will also be able to induce suchmobility especially at lower temperatures where the time scale related toβ-relaxation does not become super exponentially larger compare toα-relaxation time. Indeed in a recent experiment, it is suggestedthat β-relaxation plays very important role in the preservation ofprotein in sugar glasses <cit.>.Although the physical and chemical processesthat degrade a macromolecule is known, the microscopic mechanismsof how glassymatrix helps to slow down these physical and chemical degradation process of a biomolecules is not clearly understood. A clear understanding of these microscopic mechanisms will reduce the trial-and-error aspect of lengthy andtedious long-term stability studies in many fields such as food, pharmaceuticals. The goal of this work is to understand how the dynamics of biomacromolecule might coupleto the dynamics of supercooled liquids and how rates of different processesare modified by the embedding liquid as it is supercooled with decreasingtemperature. In this regard, we have quantified the collapse dynamics ofa model polymer chain<cit.> in a well-known glass formingliquids<cit.> using extensive molecular dynamics simulations. The use of ahomopolymer as a system of choice avoids the inherent molecular heterogeneityof diverse amino-acids in a protein, where isolating individual contributionsto a glassy-matrix induced change in stability is a difficult task and alsoprovides an incentive for exploring the action of glassy matrices onhydrophobic interaction, one of the central driving forces for protein folding. The model helps us to clearly understand how crowding due to thedense packing of embedding glassy liquid molecules particularly influencethe dynamics of model biomolecule and whether the dynamics of the biomoleculecan be slaved to the dynamics of the supercooled liquids. We find that indeedthe dynamics of a polymer chain can be slaved to the dynamics of the supercooled liquid even when the polymer interacts very weakly with theliquid molecules. The rest of the paper is organized as follows. First we will discuss aboutthe models studied and the details of the simulations and then introducecorrelation functions that we have calculated to characterize the relevantrelaxation processes in glass forming liquids. Finallywe show our results and discuss the implications of these results in thecontext of bio-preservation.§ MODELS AND METHODSWe have studied the well known Kob-Anderson 80:20 binary glass former Lennard-Jonesmixture <cit.> as the solvent. The interaction potential in this model isgiven by,V_AB(r) = 4ϵ_AB[(σ_AB/r)^12 - (σ_AB/r)^6]where ϵ_AA = 0.997, ϵ_AB = 1.4955, ϵ_BB = 0.4985,σ_AA = 0.34, σ_AB = 0.272, σ_BB = 0.2992. The units ofϵ is kJ/mole and unit ofσ is in nm (all are transformed to realunit in terms of Argon).This binary mixture works as a solvent for a 32 bead model polymer. Thepolymer model closely resembles that of Berne and coworkers <cit.>.The constituent polymer beads are connected to the covalently bonded neighborby a harmonic potential, with an equilibrium bond length of0.153 nm (thesame as CH2-CH2 bond length). Theangle between adjacent covalent bonds isrepresented by a harmonic potential, with an equilibrium angle of 111^o(the same as CH2-CH2-CH2 bond angle). The polymeris uncharged and the beads interact among themselves and with their environmentvia Lennard-Jones potentials.The bead diameter is fixed at σ_b=0.4nmand the bead-bead interaction is fixed at ϵ_b=11 kJ/mol. Non-bondedinteractions between a bead and its first and second nearest neighbors wereexcluded, and no dihedral interaction terms were included. Thehydrophobic character of the chain can be tuned by varying the intermediateinteraction between polymer beads and particles constituting glassy matricesusing geometric combination rules. Specifically, the polymer-glass interactionpotential is given by √(ϵ_p * ϵ_AA) and √(ϵ_p *ϵ_BB), where we have independently varied the value of ϵ_p=0.1,1.0 and 3.0 kJ/mol in separate simulations for tuning polymer-liquidinteractions.In essence, ϵ_p denotes the polymer contribution towardsthe inter polymer-glass interaction. On the other hand, the inter-polymer-glassinteraction-range has been calculated by √(σ_b * σ_AA) and√(σ_b * σ_BB) . A cutoff of 1.2 nm was used to treat thenonbonding interactions and periodic boundary condition condition was implemented inall dimensions.The temperature range studied for this model is 50-120K. Number of particles wehave chosen for the binary mixture is 9600 in a cubic box of dimension 6.8 nm.The same average density was maintained throughout the simulations. All the molecular dynamics simulations have been performed using GROMACS5.1.4 software.We have solvated the energy-minimized extended configuration of the polymerchain into the energy-minimized binary mixture for each case. The systemswere first energy minimized by steepest descent algorithm and then equilibratedfor 100 ps at 260Kin NVT ensemble and then in NPT ensemble for 200 ps.The systems were then annealed to desired temperatures at a cooling rate of0.5 K/ps and then subjected to a NPT equilibration for 20-1500 ns depending on the temperatures. Note that equilibration runs for each temperatures are atleast 100τ_α or more longer.τ_α is the α relaxation time(defined later) of the solvent glass forming liquid. Finally the systemswere subjected to production run in NPT ensemble. The reference pressurefor NPT simulations in last part of equilibration and production run wasthe average pressure obtained from the equilibrated binary mixture withoutthe polymer. The integration time step used is dt = 0.002 ps.Berendsen and V-rescale thermostat has been used respectively duringequilibration and production runs to maintain the average temperature.On the other hand Berendsen and Parrinello-Rahman barostat to keeppressure fixed during equilibration and production runs respectively.§ RESULTS AND DISCUSSIONAll our simulations start with an extendedconfiguration of the polymer(Radius of gyration R_G = 1.2 nm) as shown in Fig.<ref> in awell equilibrated supercooled liquid state of the solvent mixture at differentstudied temperatures in the range T∈ [50K,120K] and we explore thetransition of the polymer from extended to collapsed conformation during thecourse of the simulation. Time profile of Radius of gyration of the polymeris calculated (as shown in top right panel of Fig.<ref> forT = 50K ) to quantify the collapse-dynamics of the polymer and thecollapse time, τ_c (defined later) is estimated by identifying thetime of sharp transition from extended to collapsed conformation atdifferent supercooling temperature. This collapsing timescale isthen compared with the intrinsicrelaxation time scale (τ_α) of the glassy liquid. Inleft top panel of Fig.<ref>, we have shown one such instanceof the collapsed configuration of the polymer. The solvent binarysupercooled liquid molecules are also shown by reducing their actualsize for clarity. The choice of a large intra-bead interaction parameter (ϵ_b =11 kJ/mol) renders a strong propensity for the polymer-collapse and henceallows us to observe the collapse behavior of the polymer for entire rangeof temperature of interest T∈ [50K,120K] within simulation time scale. So in gas phase the polymer collapses very quickly with a very low temperaturedependence as shown in bottom left panel of Fig.<ref> (green squaresymbols). The reason for choosing such polymer parameters is to explore whetherdynamics of supercooled liquid can slave the dynamics of the polymer evenwhen the polymer interacts weakly with the liquid compare to its owninteraction strength. We also have shown a comparison of the collapsingtimescale in the same panel (red circle) when the polymer is immersed insupercooled liquids with particular interaction (discussed in details later).The changes in the collapsing timescale compare to the gas phase timescaleis really dramatic. This clearly proves why glassy matrices are chosen forbio-preservation. We have used three different ϵ_p value to control the interactionsbetween glass molecules and polymer beads. The ϵ_p values forpolymer-liquid interactions are 0.1, 1.0 and 3.0 kJ/mol. Before goingin to discussing our main observations, we will briefly discuss howcharacterization of the supercooled liquid is done. Relaxation timeis measured from the decay of a modified version of the two pointdensity-density correlation function Q(t), also known as overlap correlation function <cit.>. It is defined asQ(t) = ∑_i=1^N w(|r⃗_i(0)-r⃗_i(t)|)where r⃗_i(t) is the position of particle i at time t, N is thetotal number of particles. The window function w(x) = 1 if x ≤ a and0 otherwise, where a is a cut-off distance at which the the root meansquare displacement (MSD) of the particles as a function of time exhibits aplateau before increasing linearly with time at long time. The precise choiceof a is qualitatively unimportant. This window function is chosen to remove any de-correlation that might happen due to vibrational motions of the solventparticles inside the cages formed by their neighbours. In this study we havetaken a^2 = 0.006nm. The relaxation time τ_α is defined as ⟨ Q(t = τ_α)⟩ = 1/e, where ⟨…⟩refers to ensemble average. The collapse time (τ_c) of the polymer chain is obtained from the timedependence of the radius of gyration (R_G) of the polymer chain. In toppanels of Fig.<ref>, we have shown the R_G as a function of timeclosed to the collapsing transition forT = 50K as an illustrative collapseprofile. In all our analysis, we have considered the chain to be in thecollapsed state when R_G became 0.50nm.In bottom right panel of Fig.<ref>, we have shown the temperaturedependence of the collapsed time for three different cutoff radius of gyrationfor collapsed state of the polymer as 0.45, 0.50 and 0.60 nm. As evident, a different choice of the cut off radius gyration todefine the collapsed state does not change the results qualitatively. Next we compare α-relaxation time, τ_α of the supercooledliquid and the collapse time, τ_c for a situation where the solventsupercooled molecules interacts somewhat strongly with the polymer chainmolecules. Specifically the polymer-solvent intermediate interactionis tuned by using ϵ_p =3.0kJ/mol, keeping polymer bead-beadinteraction fixed at ϵ_b=11 kJ/mol. In Fig.<ref>we have plotted collapse time of the polymeralong with the αrelaxation time of the liquid for different temperatures for this particularchoice of the parameter. It is clear that, at least in the studied temperatureregime, τ_α controls the degradation rate, supporting the“Vitrification Hypothesis”. One may infer that better stability needslarger value of τ_α of the preservative. We have fitted bothτ_α and τ_c by Vogel-Fulcher-Tamman (VFT) formula <cit.>, defined as τ = τ_0 exp(A/(T-T_0)), where τ_0, Aand T_0 are free parameters. T_0 is known as VFT divergence temperatureand is very closed to the Kauzmann Temperature <cit.>. The divergencetemperatures for both τ_α and τ_c are found to be closeto 38K suggesting a strong coupling between the dynamics of supercooledliquid and the collapsing dynamics of the polymer chain. Note that thepolymer chain collapse very rapidly in gas phase, whereas its dynamics nowis slaved to the dynamics of the solvent glassy liquids.Next we look at the other extreme in which we choose an interactionparameter such that the polymer chain interacts very weakly with thesolvent liquid molecules. We choose the value of ϵ_pcontributing to solvent-polymer interaction tobe 0.1kJ/mol, so in this limit polymer dynamics will be mainlyaffected (if at all) by the crowding effect of the solvent glassymolecules. In Fig.<ref>, we show the temperaturedependence of the two timescales and surprisingly, they cross eachother at some intermediate temperature, T ∼ 55K in this case.The corresponding VFT fits also suggest that the extrapolated divergencetemperatures are very different from each other. This is now in contrastwith our previous observation, where both the timescales are more orless proportional to each other. This new result suggests that onlyα-relaxation time is not the main controlling parameter,especially at lower temperatures, where a much faster relaxation process,probably β-relaxation process seems to play a role in the dynamicsof the polymer chain. This is in complete agreement with recentexperimental observations <cit.>, where it is suggested that proteinpreservation in sugar glasses is directly linked to high frequency β-relaxation process as protein stability seems to increasealmost linearly with τ_β when τ_β is increased byadding anti-plasticizing additives. These additives are found to increasethe β-relaxation time even though it decreases α-relaxationtime <cit.>. In a bid to further understand whether it is the glassy dynamics that isslowing down the collapsing dynamics of the polymer chain, we performedquenching studies in which we decrease the temperature of the supercooled liquid rapidly from its initial equilibrium temperature. It is well knownthat if we quench a glass forming liquid to low temperature then it showsaging and initially it relaxes almost at the same timescale as that ofthe initial temperature from which it is quenched. The relaxation timethen gradually increases with increasing waiting time. Now, if the dynamicsof the polymer chain is slaved to the dynamics of the supercooled liquid,then if we quench the whole system, the polymer should still be able tocollapse in a timescale which is almost same as the initial temperaturefrom which it is cooled. In Fig.<ref>, we show that collapse time (referredhere as τ_c^q) seemsto depend on the initial temperature (T = 120K) from which it isquenched, irrespective of the final temperatures (green diamonds,T = 70, 60, 55, 50K respectively). In all these quench studies,the equilibrium collapse time is many orders of magnitude largerthan the time obtained if the system is quenched to thesetemperatures from high temperature. This observation seems tocorroborate with an old experimental finding <cit.>,where it was noted that survival probability of frozen and thawedyeast is orders of magnitude more if it is cooled very slowly.We then increase the polymer-solvent interactions a bit more byincreasing the value to an intermediate value ϵ_p to1kJ/mol to see whether these two timescales still cross each otherat an accessible temperature range. In Fig.<ref>,we indeed see the crossing of these two timescales but now at atemperature lower than that observed in the previous case when thepolymer-solvent interaction is 0.1kJ/mol. With this particularparameter, the crossover temperature moves to 50K. Thus we canexpect that at some intermediate parameter of ϵ_p =1-3kj/mol guiding the polymer-solvent interaction, thecollapsing dynamics of the polymer chain will be completelycontrolled by α-relaxation and below that short time β relaxation will also be important. At this point we can not ruleout the other possibility of cross over of these two timescales atall parameter range, as our conclusions are based on theextrapolation done using VFT formula.In conclusion, we have shown that dynamics of supercooled glassforming liquids play a major role in controlling the collapsingdynamics of a polymer chain at various temperature. At certainpolymer-solvent interaction strength, the polymer can be completelyslaved to the long time α-relaxation of the glassy liquid, on the other hand at low polymer-solvent interaction strength,at which the polymer is passive to the liquid and only packingof the solvent molecules around the polymer molecule is relevant,both short time β and long time α relaxations playintricate role at different temperature regimes. We also haveshown that coupling between the solvent dynamics and polymerbecomes weak if one does quenches from high temperatures dueto aging in the glassy liquids. This suggests that flash freezingmight not be a good method if one wants to preserve a biomoleculesin glassy matrix. Thus “Vitrification Hypothesis” although mightbe valid for some biomacromolecules, need serious revision to includethe effect of shorter time scale processes like β-relaxationin order to better understand bio-preservation in glassy sugarmatrix. In a recent work <cit.>, it is shown thatreaction kinetics of polymer collapsing dynamics depends onviscosity of supercooled liquids with a fractional power. Thisagain supports our findings reported in this work very strongly.Finally, in our model studies all complicated interactionslike hydrogen bonding and complex structural aspects of thebiomolecules are not incorporated, thus it will be important todo further studies to understand how these different parametersinfluence the results reported here.We would like to thank Walter Kob for suggesting us to do the agingstudy and Sashi Thutupalli for bringing Ref.<cit.> toour notice. We would also like to thank Srikanth Sastry, JuergenHorbach and Surajit Sengupta for many useful discussions. 100 CarpenterCroweCrowe98 J.F. Carpenter, J.H. Crowe and L.M. Crowe, The role of vitrificationin anhydrobiosis. Annu. Rev. Physiol., 60, 73–103 (1998).review1 M.A. Mensink, H.W. Frijlink, K.V. Maarschalk, W.L.J. Hinrichs, How sugars protect proteins in the solid state and during drying(review): Mechanisms of stabilization in relation to stress conditions, European Journal of Pharmaceutics and Biopharmaceutics 114, 288–295, (2017).science1 A. Ansari, C.M. Jones, E.R. Henry, J. Hofrichter, W.A. Eaton, The role of solvent viscosity in the dynamics of protein conformational changes, Science256, 1796-1798 (1992).KDCRIJPharm99 V.L. Kett, M.L.H. Duncan, Q.M. Craig, P.G. Royall, The relevance of theamorphous state to pharmaceutical dosage forms: glassy drugs and freezedried systems. International Journal of Pharmaceutics, 179, 179–207, (1999).CiceroneDouglasSoftMatter2012 M.T. Cicerone and J.F. Douglas, β-relaxation governs protein stabilityin sugar-glass matrices. Soft Matter, 8, 2983, (2012).CiceroneDouglasBioPhyJ2004 M.T. Cicerone and C.L. Soles, Fast dynamics and stabilization of proteins:Binary glasses of trehalose and glycerol, Biophysical Journal, 86,3836–3845, (2004).11BB L. Berthier and G. Biroli, Theoretical perspective on the glass transitionand amorphous materials, Rev. Mod. Phys. 83,587–645, (2011)arcmp S. Karmakar, C. Dasgupta, and S. Sastry, Growing length scales and their relation to timescales in glass-forming liquids,Annu. Rev. Condens. Matter Phys. 5, 255 (2014).KDSROPP16 S. Karmakar, C. Dasgupta and S. Sastry, Length scales in glass-forming liquidsand related systems: a review, Rep. Prog. Phys., 79, 2016.SK2016 S. Karmakar, An Overview on Short and Long Time Relaxations in Glass-formingSupercooled Liquids, Journal of Physics: Conf. Series 759, 012008 (2016).KDS S. Karmakar, C. Dasgupta, and S. Sastry, Growing length andtime scales in glass-forming liquids, Proc. Nat. Acad. Sci (USA) 106, 3675 (2009).JG G. P. Johari and M. Goldstein,J. Chem. Phys.53 2372 (1970).betaPRL S. Karmakar, C. Dasgupta and S. Sastry, Short-time beta relaxation inglass-forming liquids is cooperative in nature,Phys. Rev. Lett. 116, 085701 (2016).footnote The β relaxation discussed here is defined accordingto the mode coupling theory [W. Götze, Complex Dynamics of Glass-FormingLiquids: A Mode-Coupling Theory (Oxford University Press, 2009)]. It is believed to be very different from the well studiedJohari-Goldstein process Ref.<cit.>.BZZJACS2009 B.J. Berne R. Zangi, R. Zhou, Urea’s action on hydrophobic interactions,J. Am. Chem. Soc., 131, 1535–1541, (2009).KA W. Kob and H.C. Andersen, Testing mode-coupling theory for asupercooled binary Lennard-Jones mixture I: The van Hove correlation function Phys. Rev. E 51, 4626 (1995).vftH. Vogel, Z. Phys. 22 645 (1921), G.S. Fulcher,J. Amer. Ceram. Soc. 8 339 (1925), D. Tammann J. Soc. Glass Technol.9 166 (1925).kauz W. Kauzmann Chem. Rev. 48 219 (1948).mazurSchmidt P. Mazur and J.J. Schmidt, Interactions of cooling velocity,temperature, and warming velocity on the survival of frozen and thawedyeast, Cryobiology 5 1-17 (1968).KwonPRL2017 S. Kwon, H.W. Cho, J. Kim, and B.J. Sung, Fractional Viscosity Dependence of Reaction Kinetics inGlass-Forming Liquids, Phys. Rev. Lett. 119, 087801 (2017). | http://arxiv.org/abs/1709.09475v1 | {
"authors": [
"Mrinmoy Mukherjee",
"Jagannath Mondal",
"Smarajit Karmakar"
],
"categories": [
"cond-mat.soft",
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.soft",
"published": "20170927124200",
"title": "Role of $α$ and $β$ relaxations in Collapsing Dynamics of a Polymer Chain in Supercooled Glass-forming Liquid"
} |
The Co-Evolution of Test Maintenance and Code Maintenance through the lens of Fine-Grained Semantic Changes Stanislav Levin The Blavatnik School of Computer Science Tel Aviv UniversityTel-Aviv, [email protected] Amiram Yehudai The Blavatnik School of Computer Science Tel Aviv UniversityTel-Aviv, [email protected] 30, 2023 ================================================================================================================================================================================================================================================ It is often recommended that identifiers for ontology terms should be semantics-free or meaningless. In practice, ontology developers tend to use numeric identifiers, starting at 1 and working upwards. Here we describe a number of significant flaws to this scheme, and the alternatives to them which we have implemented in our library, identitas.Software is available from <https://github.com/phillord/identitas>. During the years that ontologies have moved to becoming a standard part of the biomedical chain, a set of standard practices have build up which are used to enable their good management, including the addition of standardised metadata about each ontology term, including labels, definitions, editorial status and so forth.One key piece of metadata is the identifier. For most ontological technologies this is in the form of an IRI (Internationalized Resource Identifer), or something that is convertable into one. Much has been written about the nature of identifier and how they should be chosen. The percieved wisdom is that identifiers should be semantics-free or meaningless. The key aim here is to enable persistence of access to a term <cit.>; an identifier which is based on some semantics associated with the term may need to be changed when that aspect changes, even if the change does not reflect a change in the ontological semantics.As an example, OBO Foundry principles <cit.> provide guidelines for identifiers; these include both management principles (“The ID-space / prefix must be registered with the OBO library in advance.”), syntactic constraints (“The URI should be constructed from a base URI, a prefix that is unique within the Foundry (e.g. GO, CHEBI, CL) and a local identifier (e.g. 0000001).”), in addition to a strong commitment to semantics-free IDs (“The local identifier should not consist of labels or mnemonics meaningful to humans.”). No specific advice is given on the form of the local identifier; however, in practice OBO identifiers use numeric IDs, 8 numerals long, approximately increasing monotonically.While semantics-free identifiers have their advantages there are distinct disadvantages as well, especially for humans. They are poorly mnemonic, hard to differentiate from each other and relatively difficult to read. For this reason, for example, many bioinformatics databases provide both semantic-free accession numbers (which are essentially the same thing as an identifer in ontology terminology), and an identifier (which is rather like a compressed, syntactically predicatable label). It is also interesting to note that, with software development, programmers emphasise the use importance of semantically-meaningful identifiers, and use other techniques to manage change.In this paper, we ask whether it is possible to overcome these and some related issues with monotonic, numeric identifiers while remaining semantics-free. We describe our solutions, along with the identitas library which implements these. Racing: One unusual aspect of ontological identifiers is that they are usually monotonically increasing. This causes a significant race condition if two developers are building a single ontology in parallel. If both attempt to add a new term, they both must coin a new identifier, which must be unique. This is impossible to achieve without some degree of co-ordination. One typical strategy is for developers have to pre-coordinate to build the ontology by using pre-allocation schema. For example, one developer allocated with the IDs from 1 to 1000, another allocated with 1000 to 2000 and so on. This approach is effective, however it requires developers to manage the ID space accurately, and also reduces the overall ID space since preallocated IDs cannot be used elsewhere. Another approach is to just-in-time co-ordinate; for example, the URIGen <cit.> server enables this approach in Proteǵe.́ Projects such as EFO (Experimental Factor Ontology) and SWO (Software Ontology) use this to manage their namespace. A final approach is to use temporary IDs, and then allocate final IDs at a single, co-ordinated point in the development process; URIGen also does this to enable off-line working.We propose a much simpler approach which is to simply use random IDs not just as temporary identifiers. While randomness does not a priori completely remove the potential race condition, given a large enough identifier space, the chances of collision can be reduced to provide world (or universe) uniqueness. This approach is commonly used with random UUIDs (Universal Unique Identifiers) being perhapsthe most common example.Pronouncing: The use of randomness raises a secondary issue. These identifiers are likely to be relatively long, exacerbatting the problems of memorability and pronounceability. One solution to this problem is to just not show the identifiers to humans. With tools like Proteǵethis is possible, of course, because it has a view which may be different from the underlying model. With text file-formats, including OBO format, the various OWL serialisations or the Tawny-OWL <cit.> programmatic representation, this is rather harder (although the latter does provide an mechanism for achieving this). It is also difficult to do this for programmers developing tools like Proteǵe,́ who are themselves using general tools such as IDEs, debuggers and version control systems.We have considered using a dictionary-based approach, to replace numeric identifiers with English words. However, this approach raises the probability of selecting a word which is inappropriate or unfortunate – consider the Sonic Hedgehog gene mutations which causes holoprosencephaly in humans. Instead, we are investigating a solution in the form of the proquint <cit.>. This is a library build to encode numbers as a set of strings of alternating consonants and vowels. Each consonant provide four bits of information, each vowel only two bits, as shown in Figure <ref>. Thus, sixteen bits can be represented using five letters (3 consonants, 2 vowels).For example a numeric identifier 10 associated with some term in a given ontology would be translated to |babab-babap|, 11 would be translated to |babab-babar| by using proquint function which is quite readable, spellable and pronounceable string. In practice, if used to represent random numbers, the proquints would rarely be so close in alphabetic space. Note that proquints map directly to a single number, so can be freely converted in either direction, and that they are alphabetically ordered. Mappings between integer values are shown in Figure <ref>.In a simple extension, to the original algorithm, we have also provided conversions from the Java short and long data types which provides either a larger identifier space, or less typing; conversions are shown in Figure <ref>.We note that the short range at 2^16 numbers is large enough for most ontologies current in operation. However, it is far too small when combined with randomness as due to the birthday problem is very likely to result in collisions even for small ontologies <cit.>. The long range, meanwhile at 2^64 numbers is likely to cope for all ontological applications where the identifiers are allocated as a result of human action; it has half the bit-length of a UUID (which has a 2^128 range).Checking: We note that monotonic numeric ideas suffer from a final problem. As well as being unmnenomic, if a numeric ID is misunderstood, it is very likely that the incorrect ID is stil actually a valid one; for instance, OBI:0001440 (“all pairs design”) and OBI:0001404 (“genetic characteristics information”) are IDs which differ in one one number.A solution to this problem is well-understood with the use of a checksum. For the identitas library, we use the Damm algorithm <cit.>. This algorithm is design to operate on numbers, but it will work on proquints also, as they can be converted to numbers. Examples of valid or invalid numbers are shown in Figure <ref>. Of course, the Damm algorithm incorporates a checksum so reduces the total space of valid identifiers, in this case by an order of magnitude, which will have implications if combined with randomness. Under these circumstances, the larger numeric spaces (int or long) are likely to be necessary.In this paper we present a critique of current ontology semantics-free identifiers; monotonically increasing numbers have a number of significant usability flaws which make them unsuitable as a default option, and we present a series of alternatives. We have provide an implementation of these alternatives which can be freely combined. We are now starting to integrate these into ontology development environments such as Tawny-OWL <cit.>, and will later provide an implementation for Proteǵe.́ This form of identifier space could significantly improve the management of ontologies with very little cost.plain | http://arxiv.org/abs/1709.09021v2 | {
"authors": [
"Nizal Alshammry",
"Phillip Lord"
],
"categories": [
"cs.DL"
],
"primary_category": "cs.DL",
"published": "20170926135647",
"title": "Identitas: A Better Way To Be Meaningless"
} |
http://arxiv.org/abs/1709.09468v3 | {
"authors": [
"Matias Bejas",
"Hiroyuki Yamase",
"Andres Greco"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170927121658",
"title": "Dual structure in the charge excitation spectrum of electron-doped cuprates"
} |
|
x[1] >p#1 | http://arxiv.org/abs/1709.09274v1 | {
"authors": [
"Devesh K Jha",
"Nurali Virani",
"Jan Reimann",
"Abhishek Srivastav",
"Asok Ray"
],
"categories": [
"stat.ML"
],
"primary_category": "stat.ML",
"published": "20170926221126",
"title": "Symbolic Analysis-based Reduced Order Markov Modeling of Time Series Data"
} |
Large cone angles on a punctured sphere.] Geodesic intersections and isoxial Fuchsian groups. G. McShane] Greg McShane UFR de Mathématiques Institut Fourier 100 rue des maths BP 74, 38402 St Martin d'Hères cedex, France [email protected][2010] Primary 57M27, Secondary 37E30, 57M55The set of axes of hyperbolic elements in a Fuchsian group depends on thecommensurability class of the group. In fact, it has been conjectured that it determines the commensurability classand this has been verified in for groups of the second kind by G. Mess andfor arithemetic groups by by D. Long and A. Reid. Here we show that the conjecture holds for almost all Fuchsian groups and explain why our method fails for arithemetic groups.[ [ December 30, 2023 =====================§ INTRODUCTIONLetbe a closed orientable hyperbolic surface.The free homotopy classes of closed geodesics on conjugacy classes of hyperbolic elements in Γ.Ifγ∈Γis a hyperbolic element,then associated toγis an axis⊂. The projection of todetermines a closed geodesic whose length is . We shall denote the set of axes of all the hyperbolic elements in Γby .It's easy to check that ifg ∈ then we have the relation (gΓ g^-1) = g .§.§ Isoaxial groups Following Reid <cit.>we say that a pair of Fuchsian groupsΓ_1 and Γ_2are isoaxial iff (Γ_1) = (Γ_2). One obtains a a trivial example of an isoaxial pairby taking Γ_1any Fuchsian group and Γ_2 < Γ_1any finiteindex subgroup. Thisexamplecan be extendedto a more general setting as follows. Recall that a pair of subgroups Γ_1 and Γ_2 are commensurable if Γ_1 ∩Γ_2 is finite index in bothΓ_1 and Γ_2. Thus if Γ_1 and Γ_2 are commensurable then they are isoaxial because:(Γ_1) =( Γ_1 ∩Γ_2) = (Γ_2),It is natural to ask whether the converse is true: If Γ_1 and Γ_2 are isoaxial then are theycommensurable? In what follows we shall say simply that the group Γ_1 is determined (up to commensurability) by its axes.We shall show that this conjecture holdsfor almost all Fuchsian groups: For almost every pointρ in Teichmueller space of a hyperbolic surface the corresponding Fuchsian representationthe fundamental group Γ is determined by its axes.§.§ SpectraWe define the length spectrum ofto be the collection of lengths ℓ_α ofclosed geodesics α⊂ counted with multiplicity. In fact, sinceis compact, the multiplicity of any value in the spectrum is finite and moreover the set of lengths is discrete. Let α,β be primitive closed geodesicswhich meet at a point z∈, we denote byα∠_zβ,the angle measured in the counter-clockwise direction fromα to β.Let α,β be primitive closed geodesicswhich meet at a point z∈, we denote byα∠_zβ,the angle measured in the counter-clockwise direction fromα to β.Following Mondal <cit.>,<cit.> wedefine anangle spectrum to be the collection ofall suchangles (counted with multiplicity).Thelength spectrum has proved usefulin studying many problems concerningthe geometry of hyperbolic surfaces. The angle spectrum is very different from the length spectrum: theset of angles isobviously not discrete and, as we shall see, the there are surfaces for whichevery value has infinite multiplicity. However,when considering the question ofwhether groups are isoaxial, the angle spectrum has a distinct advantage for itis easy to see that: * There are isoaxial groups whichdo not have the same set oflengths, that is, the same anglespectrum without multiplicities.* If two groups are isoaxial thenthey have the sameset of angles, that is, the same anglespectrum without multiplicities. Using properties of angleswe will deduce Theorem <ref> from the the following lemma inspired by a result of G. Mess (see paragraph <ref> ).Define the group of automorphisms ofto bethe group of hyperbolic isometries which preserve . If has avalue in itsangle spectrum with finite multiplicity thenΓ is finite indexin the group of automorphisms of .It remains to prove that there are such points of , we show in fact that they are generic:For almost every pointρ∈ there is a value in the angle spectrumwhich has multiplicity exactly one.Our method applies provided there is some value in the angle spectrum that has finite multiplicity.Unfortunately,for arithemetic surfaces, the multiplicity of every value is infinity (Lemma <ref>). §.§ Sketch of proofThe method of proof of Theorem <ref> follows the proof of the first part of Theorem 1.1 in<cit.>: Thissays thatthe set of surfacesin Teichmeuller space where every value in thesimple length spectrum has multiciplity exactly one is dense and its complement is measure zero( for the natural measure on Teichmueller space.) §.§.§ Two properties of (simple) length functionsRecall that the simple length spectrum is defined to be the collection of lengths of simple closedgeodesics counted with multiplicity. There are two main ingredients usedin <cit.>: * The analyticity of the geodesic lengthℓ_α as a function over Teichmeuller space;* The fact that if α,β are a pair of distinct simple closedgeodesics then the difference ℓ_α - ℓ_βdefines a non constant(analytic)function on theTeichmeuller space .It is clear that the set of of surfaces where every value in thesimple length spectrum has multiciplity exactly one is the complement ofZ := ∪ _(α, β){ℓ_α -ℓ_β= 0},where the union is over all pairs α, β of distinct closed simple geodesics. Each of the sets on the left is nowhere dense andits intersection with any open setis measure zero. SinceZ is countable union of such sets, its complement is dense and meets everyopen set in a set of full measure.We note in passing that the second of these properties is not true without the hypothesis "simple". Indeed, there are pairs ofdistinct closed unorientedgeodesics α≠β such that ℓ_α = ℓ_β identically on(see <cit.> for an account of their construction).§.§.§ Analogues for angles We will deduce Theorem <ref>using the same approach but instead of geodesiclength functionsweuseangle functions. The most delicate point is to show that if α_1, α_2 are a pair of simpleclosedgeodesics that meet in a single point z andβ_1, β_2 are a pair ofclosedgeodesics that meet in a point z' then the differenceα_1∠_z α_2 -β_1∠_z'β_2 defines a non constant function on Teichmueller space.We do this by establishing the analogue of the following property of geodesic length functions: A closed geodesic α⊂ is simple if and only if the the image of thegeodesic length function ℓ_α is ]0,∞[. Our main technical result (Theorem <ref>) is an analogue of this property. Weconsiderpairs of simple closed geodesicsα_1, α_2which meet in a point z –this configuration will be the analogue of a simple closed geodesic.Now, for any such pair we finda subset X ⊂ such that, for any other pair of closed geodesics β_1, β_2 which meet in z' ≠ z: * the image of Xunderβ_1∠_z'β_2 is a proper subinterval of ]0,π[ * whilst its image under α_1∠_z α_2is the wholeof ]0,π[.§.§ Further remarksSince one objective of this work is to compare systematically the properties of geodesic length and angle functions we include an expositionof geodesic length functions and give an account ofthe characterisation of simple geodesics mentioned above our Proposition <ref>.Mondal <cit.> has obtained a rigidity resultby using a richer collection ofdata than we use here. He defines a length angle spectrumand proves that this determines a surface up to isometry. However, the set of axes does not determine the lengths of closed geodesics and so commensurability is the best one can hope for in the context we consider here.In paragraph <ref> we answer a question of Mondal in <cit.> concerning multiplicities by observing that arithemetic surfaces arevery special: the multiplicity of any anglein the angle spectrum is infinite.§ AUTOMORPHISMS AND COMMENSURATORSTo study this question we define, following Reid, two auxilliary groups. The first is thegroup of automorphisms of : := {γ∈, γ() =}.The second isthe commensurator of Γdefined as:Comm(Γ) := {γ∈: γΓγ^-1 is directly commensurable with Γ}.We leave it to the reader to check that and (Γ) areindeed groupsand that they contain Γ asa subgroup. In fact any element γ∈(Γ) is anautomorphism of .To see this, if γ∈(Γ), then Γ and γΓγ^-1 are commensurablesoare isoaxial. Now by (<ref>)one has= (γΓ x^-1 ) = γso γ∈.In summary one has a chain of inclusions of subgroups: Γ < (Γ) << . We shall be concerned with two cases: * Γ is finite index in .*is dense in so that Γ is necessarily aninfinite index subgroup.The first case arises for the class of Fuchsian groups of the second kind studied by G. Mess and the second for arithemetic groups. §.§ Fuchsian groups of the second kindG. Mess in an IHES preprint studied a variety of questions relatingtonotably proving the following resultIf Γ_1 and Γ_2are isoaxial Fuchsian groups of the second kind thenthey are commensurable.The proof of this resultis a consequence of the factthat, under the hypotheses,is a discrete, convex cocompact Fuchsian group. It is easy to deduce from this that Γ is finite index in .To show thatis discrete it suffices to find a discrete subset of ,containing at least two points, on which it acts. Recall thattheconvex hull of the limit setof Γ,is a convex subset ⊂. If Γ is a Fuchsian groups of the second kind then its limit setis a proper subset of ∂ andis a proper subset ofwhose frontier ∂ consists of countably manycomplete geodesics which we call sides.By definition is -invariantand so is too since, in fact,it is the minimal convex set containing . Now choose a minimal length perpendicular λbetween edgesof ;such a minimising perpendicular exists because the double of / Γ is a compact surface without boundary,every perpendicular between edges of gives rise to a closed geodesic on the doubleand the length spectrum of the double is discrete.Let L be the-orbit of λand observe thatL ∩∂ is a discrete setwhich contains at least two points.§.§ Arithemetic groups In the case of Fuchsian groups of the first kindLong and Reid <cit.> proved the conjecture for arithemetic groups. If a Fuchsian groupis arithmetic thenits commensurator is exactly thegroup of automorphisms of the group. Also notice that ifΓ_1 and Γ_2are isoaxial Fuchsian groups, then for any γ∈Γ_2 (Γ_1) =(γΓ_1γ^-1), and therefore γ∈. Hence Γ_2 < .So bythe above discussionΓ_2 < (Γ_1),andif Γ_2 is also arithmetic, then Γ_1 and Γ_2 are commensurable.Thus they obtain as a corollary: Any pair of isoaxial arithmetic Fuchsian groups is commensurable. §.§.§ Multiplicities for arithemetic groups Let Γ be an arithemetic Fuchsian group since its commensuratoris dense in set of geodesic intersctions is “locally homogenous" in thefollowing sense: Let θ = α∠_z β be an intersection of closed geodesics then for any open subset U ⊂ there isa pair of closed geodesicsα_u, β_u such that: α_u ∠_z_uβ_u, z_u ∈ U. Choosehyperbolic elements a,b ∈Γsuch that the axis of a (resp. b) is a lift ofα (resp β)toand so that the axes meet in alift ẑ∈ of z. Since (Γ) is dense in , there is some element g ∈(Γ)so that g(ẑ) ∈Û for some lift of U to . By the commensurability of the groupsΓ and gΓ g^-1there is a positive integer m such that (gag^-1)^m,( gbg^-1)^m ∈Γso that the axes of these elements project to closed geodesics α_u, β_u onmeeting in a point z_u as required. An immediate corollary is:The multiplicity of any angle θ in the spectrum of an arithemetic surface / Γ is infinite. § FUNCTIONS ON TEICHMEULLER SPACE Recall thatthe Teichmeullerof a surface , , is the set of marked complex structures and that, by Rieman's Uniformization Theorem, this is identified with a component of the character variety of -representations of . Thus we think of a point ρ∈ as an equivalence class of -representationsof . We remarkthat:=/⟨ -I_2 ⟩ so thatalthoughthe trace ρ(a) is notwell definedfor a∈, the square of the trace^2 ρ(a) isandso is| ρ(a)|. In fact, there is a natural topologysuch that for each a ∈,ρ↦^2 ρ(a) is a real analytic function. §.§ Geodesic length If a∈ is non trivialthen there is a unique oriented closed simple geodesic α in the conjugacy class [a]determined by a. The length of α, measured in the Riemannian metric on = /ρ()), can be computed fromρ(a) using the well-known formula| ρ(a)|=2 cosh( ℓ_α/2).There is a natural function, ℓ :×{ homotopy classes of loops}→ ]0,+∞[ which takes the pair ρ,[a] to the lengthℓ_α of the geodesic in the homotopy class [a].It is an abuse, though common in the literature, to refer merely to the length of the geodesic α (rather than, more properly, the length of the geodesic in the appropriate homotopy class).We define the length spectrum ofto be the collection of lengths ℓ_α ofclosed geodesics α⊂ counted with multiplicity. In fact, sinceis compact, the multiplicity of any value in the spectrum is finite and moreover the set of lengths is discrete.§.§.§ AnalyticityA careful study of properties of length functions was made in <cit.>whereone of the key ingredientsis the analyticity of this class offunctions:For each closed geodesic α, the function → ]0,+∞[, ρ↦ℓ_α is a non constant, real analytic function.See <cit.> for a proof of this. Note that, to prove that such a functionis non constant, it is natural to consider two casesaccording to whether the geodesic α is simple or not: * if α is simple then by including it as a curve in a pants decomposition one can view ℓ_α as one of the Fenchel-Nielsen coordinates so it is obviously non constantand, moreover, takes on any value in]0,+∞[* if α is not simple then it suffices to find a closed simple geodesic β such that α and β meet and use the inequality (see Buser <cit.>) sinh(ℓ_α/2)sinh(ℓ_β/2) ≥ 1to see that if ℓ_β→ 0 then ℓ_α→∞ and so is non constant. §.§.§ Characterization of simple geodesicsThere is always a simple closed geodesic shorterthan any given closed geodesic. More precisely, if β⊂ is a closed geodesic whichis not simple thenby doing surgery at the double pointsone can construct a simple closed geodesic β' ⊂with ℓ_β' < ℓ_β.For ϵ > 0 define theϵ-thin part of theTeichmeuller spaceto be the set : = {ℓ_β < ϵ, ∀β closed simple}⊂. By definition, on the complement of the thin part ℓ_β≥ϵ for all simple closed geodesics and since, by the preceding remark,there is always a simple closed geodesic shorterthan any given closed geodesic,ℓ_β≥ϵfor allclosed geodesics. Letbe a finite volumehyperbolic surface. Then a closed geodesic α⊂ is simple if and only if the infimum overofthe geodesic length function ℓ_α is zero. In one direction, if α is simple thenℓ_αis one of the Fenchel-Nielsen coordinates for some pants decomposition ofso there is some (non convergent) sequence ρ_n ∈ such that ℓ_α→ 0.Now suppose that α is not simple and we seeka lower bound for its length.There are two cases depending on whetherthere exists a closed simple geodesic βdisjoint from α or not. If there is no such geodesic then α meetsevery simple closed geodesic β⊂ and it is cusomary to call such a curve a filling curve. Choose ϵ > 0 and consider the decomposition of the Teichmeuller space into the ϵ-thin part and its complement. On the thick part ℓ_α≥ϵwhilst on the thin part,by the inequality (<ref>), it is bounded below byarcsinh (1/ sinh(ϵ/2) ).If there is an essentialsimple closed geodesicdisjoint from α thenwe cut along this curve to obtain a possibly disconnected surface with geodesic boundary. We repeat this process to construct a compactsurface C(α) such thatα is afilling curve inC(α). By construction C(α) embeds isometricallyas a subsurface ofand since α is not simple C(α) is not an annulus. On the other hand, by taking the Nielsen extension of C(α) thencapping off with a punctured disc we obtain a conformalembedding C(α) ↪ C(α)^*where C(α)^* is a punctured surface with a natural Poincaré metric. By the Ahlfors-Pick-Schwarz Lemmathere is acontraction between themetrics induced onC(α) from the metric onand from the Poincaré metric onC(α)^*. A consequence of this is thatthegeodesic in the homotopy classdetermined by α on C(α)is longer thanthe one in C(α)^*. So, to bound ℓ_α it suffices to boundthe length of every filling curve on a punctured surface. There are two cases.* If C(α) has an essential simple closed curve then we have already treated this case above.* If C(α) has no essential simple closed curvesthen it is a3 punctured sphere anthe bound is trivial since the Teichmueller spaceconsists of a point. § FENCHEL-NIELSEN TWIST DEFORMATIONWhilst make no claim as to the originalityof the material in this sectionit is included to set upnotation give an exposition of two results which we use in Section <ref>.§.§ The Fenchel-Nielsen twistWe choose a simple closed curve α⊂. Following <cit.>, cut along this curve,and take the completion of the resulting surfacewith respect to the path metricto obtain apossibly disconnectedsurface with geodesic boundary '. Obviously, one can recover the original surface from ' by identifying pairs of points ofone from each of the boundary components. More generally, if t ∈ then a(left) Fenchel-Nielsen twist along α allows one to construct a new surface _t,homeomorphic toby identifyingthe twoboundary componentswith a left twist of distance t, i.e. the pair of points whichare identified to obtainare nowseparated by distance t along the image of α in_t.Thus this construction gives rise to amap,which we will call the time t twist along α,: →_t, discontinuous for t≠ 0 and mapping ∖α isometrically onto_t ∖α.Note thatis not unique but this will not be important for our analysis,what is important, and easy to see from the construction, isthat the geometryof _t ∖α does not vary with t as we will exploit this to obtain our main result. §.§ The lift of the twist to Let Γ beFuchsian group such that:=/ Γ is a closed surface, α⊂a non separating simple closed geodesic andx ∉α a basepoint for .Now let A ⊂denote the set of all lifts of αand x̂∈a lift of x. Then the complement of A consists of an infinite collection ofpairwise congruent, convex sets. Moreover,if P denotes the connected component of the complement of A containing x̂,then P can be identified with the universal cover of the surface ∖α and the subgroup Γ^P < Γ that preserves P is isomorphic to the fundamental group of this subsurface. Since the geometry of _t does not change with t∈ the geometry of P does not change either. This observation is the key to establishing uniform boundsinthe proof of Theorem <ref>. Each of the other connected components of ∖ Pcan be viewedas a translate of g_i(P) forsomeelementg_i ofΓ and soistiled by copies of P. Let us consider how this tiling evolves under the time ttwist along α. There is a unique lift : → which fixes x̂ and hence P. We can calculate the image of a translate ofPunder the lift of by a recursive procedure. Suppose that for someg_1,… g_n ∈Γ; * ∪ g_i(P̅) is connected,* we have determined the images ofg_1(P),… g_n(P).Let g_n+1(P) be a translate of P such that g_n+1(P̅) ∩g_n(P̅) = α̂. andwe considertwo cases: * If g_n(P) = Pthen the image of g_n+1(P) isϕ^t( g_n+1(P)) where ϕ^t is a hyperbolic translation of length t with axis α̂.* Ifg_n(P) ≠ P and itsimage underis h(P) then the image of g_n+1(P) ish∘ϕ^t∘g_n^-1 (g_n+1(P))) where ϕ^t is a hyperbolic translation of length t with axis g_n^-1(α̂) ⊂ A. This procedure allows us to prove the following: Let ^P ⊂∂ denote the limit set of Γ^P. Thenadmits acanonical extension : ⊔∂→⊔∂which is continuous on ∂.Further:* For any w ∈Γ^P one has (w)= w;* For any w ∈∂ one has lim_t →±∞(w) ∈^P and further this is an endpoint of an edge of . It is standard from the theory of negatively curvedgroupsthat thelift admits a unique extension to ⊔∂, continuous on the boundary , since / Γ is compact and sothe restriction ofthe lift tothe set of lifts of a base point x ∈,Γ.{x̂} is Lipschitz.Since the extension is continuous,to prove (1) it suffices to note thatthe lift of the Fenchel-Nielsen deformationfixes the endpoints of the edges ofand these are dense in ^P.For (2) letw ∈∂ and suppose thatit is not a point of .Then there is an edgeα̂ ofsuch that w is a point of the intervaldetermined by the endpoints of this geodesic. It is easy to check using our recursive descriptionof the action ofon that w converges to the appropriate endpoint of α̂.We note that (2) can also be proved as follows.For t = n ℓ_α, n ∈ the Fenchel-Nielsen twist coincides with a Dehn twist. If β is a loop, disjoint from α then (up to homotopy)it is fixedby the Dehn twist. If β is a loop which crosses α then under iterated Dehn twists ^n it limitstoa curve onthat spirals to α. That is, lifting toand considering the extension of the lift of the Dehn twist^n : ⊔→⊔, an endpoint of ^n(β) converges toan endpoint of some lift of α. It is not difficult to pass to general t using the fact that theextends to ahomeomorphism on ⊔. §.§ Separated geodesics We say that a pair ofgeodesicsγ̂_1,γ̂_2 ⊂ are separated by a a geodesic γ̂ with end points γ̂^±∈ if the idealpoints ofγ̂_1,γ̂_2 are in different connected components of ∖{γ̂^±}. Note that γ̂_1,γ̂_2 are necessarily disjoint.If γ_1, γ_2 ⊂ are a pair of simple closed geodesics, such that α,γ_1,γ_2 are disjointandwe choose an arc β betweenγ_1 and γ_2 that meets α transverselyin a single point then this configuration lifts to as γ̂_1,γ̂_2 separated bya lift α̂ of α. It is easy to convince oneself that,as we deform by the Dehn twist ^n,the length of β goes to infinity. Essentially, our next lemma says that this is true for any pair of geodesics γ_1, γ_2 inadmitting an arc that meets α in an essential way. Let γ̂_1,γ̂_2 ⊂ be a pair of geodesics which are separated by some lift of α then the distance between(γ̂_1) and (γ̂_2) tends to infinityas t→±∞. Let α̂ be a lift of α whichseparates γ̂_1,γ̂_2 ⊂. Let P_1 and P_2 be the pair ofcomplementary regions which have α̂ as a common edge and we label these so that γ̂_i is on the same side ofα̂ as P_i for i=1,2. We choose the lift of the base point to be in P_1 and lift the Fenchel-Nielsen deformation.First consider theorbit (y) of an ideal endpoint yofγ̂_2 as t →∞. Since x ∈ P_1, the region P_2gets translatedand so, for any sideβ of P_2, the sequence (β) converges tothe endpoint α̂^+. Now there is a pair of edges β_1, β_2 such that the endpoints of γ̂_2 are contained in the closed intervalcontaining the endpoints of β_1, β_2. Since each of the β_i converge to α̂^+ under the deformation it iseasy to see that (γ̂_2) must converge to α̂^+ too. Now consider the orbit of an endpoint yofγ̂_1under the deformation. It suffices to show that, under this deformation,y does not convergeto α̂^+. There are two cases according to whether or not y belongs to the limit set ^P_1 of thesubgroup of Γwhich stabilises P_1.* Ify ∈^P_1 then it is invariant under theFenchel-Nielsen deformation.* Ify ∉^P_1 then it limits to apoint in y_∞∈^P_1 which is an endpoint of one of the edges of P_1.By hypothesisγ̂_1 does not meet α̂ and so y_∞ is not α̂^+.§ GEODESIC ANGLE FUNCTIONSWe present two methods for computing(functions of) the anglebetweenα_1,α_2 at z. The first method, just like the formula (<ref>) for geodesic length, is a closedformulainterms of traces(equation (<ref>) whilst the secondis interms of end points oflifts of α_1,α_2 to the Poincaré disk(equation (<ref>)). This second formula will prove useful for obtaining estimates for the variation of angles along a Fenchel-Nielsen deformation. In either case, we start as befor byidentifyingwiththe quotient / Γ whereΓ = ρ(), ρ∈. We choose z as a basepoint forand associate elements a_1,a_2 ∈π_1(,z) such that α_i is the unique oriented closed geodesic in the conjugacy class [a_i] in the obvious way. §.§ Traces and analyticityAs explained in the introduction we shall need an analogue of Fact <ref>so we give a brief account of the analyticity of the angle functions: Ifρ∈ is a point in Teichmuller space then → ]0, 2π[, ρ↦,is areal analytic function. With the notation abovewe have the following expression for the angle:sin^2(α_1 ∠_z α_2) = 4( 2 -tr [ρ(a_1),ρ(a_2) ]) /( tr^2 ρ(a_1)-4)( tr^2 ρ(a_2)-4).This equation is actually implicit in<cit.>but it is not claimed to be new there and seems to have been well known.The left hand side of(<ref>)is clearly an analytic function on and it follows from elementary real analysisthe the angle varies real analytically too. Note that, though we will not need this,(<ref>) shows thatthe square of the sineis in fact a rational function of traces (see Mondal <cit.> for applications of this). §.§.§ Cross ratio formula It will useful to to have another formula for the anglein terms of a cross ratio .This formula is well-known, see for example,The Geometry of Discrete Groups,by A.F. Beardon butwe since we will use it extensively to obtain boundswe give a short exposition. If θ is the angle between two hyperbolic geodesics α̂,β̂⊂ then tan^2 (θ/2) can be expressed as a cross ratio. One can prove this directly by takingα̂ tohave endpointsα^± = ± 1 and β̂ endpointsβ^± = ± e^iθin the Poincaré disc model. Then( α^+-β^+/α^+-β^-)( α^--β^ -/α^--β^+ )=( 1 - /1+ ) (-1+ / -1 - ) =( 1 - /1+ )^2 = tan^2 (θ/2).§ ANGLES DEFINED BY CLOSED GEODESICS §.§ Variation of anglesIn this paragraph we give an improved version of the following well known fact: Letα,β⊂ be a pair of closed simple geodesics that meet in a point z∈. If α is simple thenfor any θ∈ ] 0,π[there exists ρ∈ such thatα∠_xβ = θ.Under the hypothesis,there is a convex subsurface ' ⊂ homeomorphic to a holed torus which contains α∪β. The fact follows by presenting ' as the quotient ofby a Schottky group.Using the preceding discussion of the Fenchel-Nielsen deformation we can relax the hypothesis on β even whilst taking the restriction of the angle functionto a one dimensional submanifold of . The proof will should alsoserve to familiarise the reader with the notation and provide intuition as to why this case is different to that of an intersectionof a generic pair of closed geodesics treated in Theorem <ref> Letα,β⊂ be a pair of closed geodesics that meet in a point z∈. If α is simple thenfor any θ∈ ] 0,π[and anyρ_0 ∈ there exists ρ_t ∈ obtained from ρ_0by a time t Fenchel-Nielsen twist along α such that α∠_xβ = θ. Moreover,lim_ t →±∞α∠_xβ∈{ 0, π}.With the notation of subsection <ref>, thereis a convex region P inboundedby lifts of α as before. Let α̂ be an edge of ,and choose a corresponding lift β̂ which intersects α̂. There is an element of the covering group g ∈Γ such that α̂ = ∩ g().We lift the Fenchel-Nielsen deformationand consider, as before, its extension: ⊔∂→⊔∂ .Now, arguing as in Lemma <ref>, we see that; * the endpoints of α̂ are fixed by ,* the endpoint ofβ̂ on the same side of α̂ as P converges to a point z ≠α^+as t → -∞,* the other endpoint of β̂ converges toα^+as t → -∞It follows that,after possibly changing the orientation of β,that the angle between α̂ and β̂, and hence α∠_x β, tends to 0. Likewise, as t → +∞ the angle between α̂ and β̂, and hence α∠_x β, tends to π.Thus, by continuity, the range of the angle function is ]0,π[.Let β_1,β_2 a pair of closed geodesics andy ∈β_1 ∩β_2. Then for any simple closedgeodesic α, different from bothβ_1 and β_2, the angle function β_1∠_yβ_2 is bounded away fromπ along the Fenchel-Nielsen orbit ofρ∈.If α andβ_1∪β_2 are disjoint then β_1∠_yβ_2 is constant along the-orbit so the result is trivial. Suppose now that α andβ_1∪β_2 are not disjoint and choose x as a basepoint of . Then, with the notation of paragraph <ref>, thereis a convex region P inboundedby lifts of α.We nowconsiderthree casesaccording to the number ofedges ofthat β̂_̂1̂∪β̂_̂2̂meets. We first deal with the simplest case. Suppose thatβ̂_̂1̂∪β̂_̂2̂meets in four distinct edges denotedC_1,C_2,C_3,C_4 ⊂, and, after possibly relabelling these, β̂_̂1̂ meets C_1,C_2 whilst β̂_̂2̂ meets C_3,C_4 as in Figure <ref>. Now we deform ρ_0 bya Fenchel-Nielsen twist along α to obtain a 1-parameter family of ρ_t ∈, t∈. As we have seen above, undersuch a deformationthe length ofα does not change nor does the geometry ofin particular the positions of theC_i remain unchanged. From our discussion of theand its extension to ⊔ it is clear that, ∀ t ∈, (̂β̂_̂1̂)̂ meets C_1,C_2 whilst (̂β̂_̂2̂)̂ meets C_3,C_4. Thus, if the diameters of the circles were small,the angle at ẑcannot not vary much from its value at ρ_0 since the radii of the circles are small. More generally,we can bound the size of the angle using the cross ratio formula. Labeling the endpoints as in Figure <ref> one has:tan^2 (θ/2) = | β_1^+-β_2^+/β_1^+-β_2^-|| β_1^--β_2^ -/β_1^--β_2^+ |Note first that each of the four points lies on the unit circleand sothat itsdiameter, that is 2, is a trivial upper bound for each of the four distances appearing on the left hand side of this equation. Now under the deformation each of the endpoints(β_i^±) stays in one of four disjointeuclidean discs defined by one of the C_j. In particular,there existsδ_4 > 0 such that for all t ∈δ_4 ≤ |(β_1^±) -(β_2^±) |≤2δ_4 ≤|(β_1^±) -(β_2^∓) |≤ 2and this is sufficient to obtain boundsonthe cross ratio:1/2 δ_4 ≤tan (θ/2) ≤2/δ_4If β̂_̂1̂∪β̂_̂2̂meets in just twoedges, C_1,C_2⊂ say. Althoughwe no longer have a uniformlower bound for |(β_1^±) -(β_2^±) |in this case, there stillexists δ_2 > 0 such thatfor all t ∈,δ_2 ≤ |(β_1^±) -(β_2^∓) |.Thus, for all t ∈,0 ≤tan (θ/2) ≤2/δ_2.Finally, if β̂_̂1̂∪β̂_̂2̂meets∂ P in exactlythreeedges then it is easy to see that, using the same reasoning as for the two edge case, there is δ_3 such thatδ_3 ≤ |(β_1^±) -(β_2^∓) |. Let α_1, α_2pairs of simple closed geodesicswhich meet in a singlepoint zand β_1, β_2 primitive closed geodesics which meet in z'. If the differenceα_1 ∠_z α_2 - β_1 ∠_z'β_2is constant then the angles are equal and, after possibly relabelling the geodesics, α_i = β_i and z = z'. Note that we cannot suppose that z,z' are distinct because of the following phenomenon. Jorgenson studied intersections of closed geodesics proving in particular that if z ∈ was the intersectionof a pair of distinct closed geodesics then it is the intersection of infinitely many pairs of distinct closed geodesics. Such intersections are stablein that, if (α_i)_i is a family of geodesics obtained from Jorgenson's procedurethat meet in a point z on some hyperbolic surfacethen, for any ρ∈,there is z_ρ∈/ ρ()such that the family(α_i)_imeet inz_ρ. We first consider the case where four geodesics are distinct then, under the Fenchel Nielsen twistalongα_1,the image of α_1 ∠_z α_2is ]0,π[ whilst, by Theorem <ref>, the imageβ_1 ∠_z'β_2isa strict subinterval.It is easy to see α_1 ∠_z α_2 - β_1 ∠_z'β_2 cannot be a constant. Now suppose,α_1 = β_1. if α_2= β_2 then,since α_1 and α_2 meet in a single point,we must have z = z'and the angles must be the same. On the other hand, if α_2 ≠β_2then z,z' may or may not be distinct* If z = z' then, byLemma <ref>, both α_1 ∠_z α_2 and β_1 ∠_z'β_2 tend to 0 or π as the Fenchel-Nielsen parameter t →±∞. Therefore, if the difference is constant it must be 0 or π and so, up to switching orientation,β_1 = β_2. * If z = z' then, by Lemma <ref> and Theorem <ref> β_1 ∠_z'β_2 is a proper subintervalof the range ofα_1 ∠_z α_2so the difference cannot be constant.of Lemma <ref> Suppose that has a valuein its angle spectrum, θ say,with finite multiplicity. Let x_1,x_2 … x_n ∈ be a complete list of points such that there are pair of slosed geodesics meetingat x_i at angle θ. Then the set of preimagesof thex_i under the covering map → is a discrete set which is invariant under . Thusis discrete andhas Γ as a finiteindex subgroup.99abbookAlan F. Beardon The Geometry of Discrete Groups, Springer BuserP. Buser, Geometry and spectra of compact Rieman surfaces,Birkhauserkerk S. Kerckhoff,The Nielsen Realization problem, Ann. of Math. (2) 117:2 (1983),235–265. MR 85e:32029 Zbl 0528.57008 ARD. Long, A. Reid, On Fuchsian groups with the same set of axes, Bull. L.M.S, 30 (1998), 533–538.MessG. Mess, IHES preprint circa 1991. mcpGreg McShane, Hugo Parlier, Multiplicities of simple closed geodesics and hypersurfaces in Teichmüller space, Geometry and Topologypjm Greg McShane. Length series in teichmuller space. Pacific Journal of Math, 231(2):461–479, 2007.mondal1 S. Mondal,Rigidity of length-angle spectrum for closed hyperbolic surfaces <https://arxiv.org/abs/1701.08829>mondal2S. Mondal, An arithmetic property of the set of angles between closed geodesics on hyperbolic surfaces of finite type, <https://arxiv.org/abs/1703.02478>reidAlan W. Reid, Annales de la faculté des sciences de Toulouse Mathḿatiques (2014), Volume: 23, Issue: 5, page 1103-1118 | http://arxiv.org/abs/1709.08958v1 | {
"authors": [
"Greg McShane"
],
"categories": [
"math.GT",
"57M27, 37E30, 57M55"
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"primary_category": "math.GT",
"published": "20170926120358",
"title": "Geodesic intersections and isoxial Fuchsian groups"
} |
Sławomir Cynk: Institute of Mathematics, Jagiellonian University Łojasiewicza 6, 30-348 Kraków, POLANDInstitute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, [email protected] van Straten: Fachbereich 08, AG Algebraische Geometrie, Johannes Gutenberg-Universität, D-55099 Mainz, GERMANY [email protected] We compute numerical approximations of the period integrals for eleven rigid double octic Calabi–Yau threefolds and compare them with the periods of corresponding weight four cusp forms and find, as to be expected, commensurabilities. These give information on character of the correspondences of these varieties with the associated Kuga-Sato modular threefolds.Periods of double octic Calabi–Yau manifolds D. van Straten December 30, 2023 ============================================§ INTRODUCTIONLet X be a rigid Calabi–Yau threefold and ω∈ H^3,0(X) a regular 3–form on X. For a 3-cycle γ∈ H_3(X,) on Xwe can form the period integral∫_γω∈ .The set of these period integrals form a latticeΛ:={∫_γω: γ∈ H_3(X,)}⊂and hence determine an elliptic curve /Λ=H^3,0(X)^*/H_3(X,)=:J^2(X)which is just an example of the intermediate Jacobian of Griffiths <cit.>. It is now known that any rigid Calabi–Yau threefold defined overis modular <cit.>,<cit.>, in the sensethat one has an equality of L-functions: L(H^3(X),s)=L(f,s) .Here f ∈ S_4(Γ_0(N)) is a weight four cusp form for some level N. It is known more generally that a cusp-form f ∈ S_k+2(Γ_0(N)) can be interpreted as a k-form on the associated Kuga-Sato variety, which is (a desingularisation of) the k-fold fibre product of the universal elliptic curve over the modular curve X_0(N), <cit.>.So in our case one expects theequality of L-functions to come from a correspondence between the rigid Calabi-Yau and the Kuga-Sato variety Y, which resolves the fibre product E ×_C E of the universal elliptic curve over the modular curve C:=X_0(N). Now the periods∫_0^i∞ f(τ)τ^k dτof the modular form determine the periods of the Kuga-Sato variety Y and the correspondence between X andY wouldimply that theperiod-lattice of X is commensurable to the lattice derived from themodular form. In this note we shall compute numerical approximations of period integrals for certain rigid double octic Calabi–Yau threefolds, i.e. Calabi–Yau threefolds constructed as a resolution of a double cover of projective space ℙ^3, branched along a surface of degree eight. More specifically, we will look at the eleven arrangements of eight planes defined by linear forms with rational coefficients,described in the PhD thesis of C. Meyer <cit.>. Thesearrangements define eleven non–isomorphic rigid Calabi–Yau threefolds.He also determined the weight four cusp forms f for these eleven rigiddouble octics using the counting of points in _p for small primes p. Each arrangement of real planes defines a partition of the real projective space ℙ^3() into polyhedral cells and using these cells one can construct certain polyhedral 3-cycles on the desingularisation of the double octic. Using the explicit equations for the planes of the arrangement, one can write the period integral as an explicit sum of multiple integrals,which can be integrated numerically. It turned out to be difficult to identify a complete basis of H_3(X,) interms of polyhedral cycles. But any two non–proportional periods of a rigidCalabi–Yau threefolds define a subgroup of finite index of Λ andhence an elliptic curve isogeneous to the intermediate jacobian J^2(X).From such a numerical lattice one can compute the lattice constants g_2 and g_3, and hence a Weierstrass equation and j–invariant of the curve defined by latticespanned by the polyhedral 3–cycles.We expect that a more refined topological analysis of the above situationwill lead to more precise information on the nature of the correspondencesbetween these varieties.§ DOUBLE OCTIC CALABI–YAU THREEFOLDS By a double octic we understand a variety X given as a double coverπ: Xof , ramified over a surface D ⊂ of degree eight. Such a double octic X can be given by an equation in weighted projective space [](4,1,1,1,1) of the formu^2=F(x,y,z,t) ,where the polynomial F defines the ramification divisor D. If the surfaceD is smooth, then X is a smooth Calabi-Yau threefold, but we will bedealing here with the case that D is a union of eight planes, so thepolynomial actors as into a product of linear forms:F=L_1 L_2 L_3 L_4 L_5 L_6 L_7 L_8 .The associated double octic X then is singular along the lines of intersection of the eight planes D_i:={L_i=0}. In case these planes have the property thatno six intersect in a point, no four intersect along a lineone can construct a Calabi-Yau desingularisation of X. To do so, one first constructs a sequence of blow–ups withsmoothcenters T:and a divisor D insuch thatD is non–singular (in particular reduced), D is even as an element of the Picard group Pic(),by blowing–up the singularities of D in the following order:* fivefold points,* triple lines,* fourfold points,* double lines.In the first two cases we replace the branch divisor by its reduced inverse image: the strict transform plus the exceptional divisor. In the last two cases we replace the branch divisor by its strict transform.The double cover π:Xofbranched along D is now a smooth Calabi–Yau manifold, which we will call the double octic Calabi-Yau threefold of the arrangement. We will also need to consider a particular partial resolution X̂ of X, obtained as double cover of a spaceobtained by performing only the blow–ups in fivefold point, triple lines and doublecurves, so leaving out step (3) in the above procedure. Note thatthere are two types of fourfold points. The fourfold points on triple lines get removed in step (2), but the fourfold points that appear at the intersection of four generic planes produce an ordinary double point if we blow up consecutively the six curves of intersection of the strict transforms of these planes.After the blow-up of the first double line the strict transforms of the remaining two planes (not containing this line) intersect along a sum of two intersecting lines. So we have to blow-up four lines and a cross, the latter producing a node on the threefold . The space doubly coveringand ramified over D̂ is a variety X̂ with twice as many nodes. To summarise the situation, one can consider the following diagram: =1.2cm X̃e,tσs,lπ̃X̂e,tρs,lπ̂Xs,rπ e,tSe,tRThe vertical maps are two-fold covers, the map σ: XX̂ is a small resolution of the nodes of X̂ and ρ: X̂ X is apartial resolution of double octic variety X. The composition R S is the map T: we started with, and τ:=ρσ: X X is a resolution of singularities. In fact the resolutions X̃ and X̂ depend on the choice of the order of blow-up of lines, in the above diagram wechoose the same order of lines for both resolutions. We will be concerned with 11 special arrangements that were studied by C. Meyer <cit.>. Their resolution X of the associated double octic lead to 11 different rigid Calabi-Yau varieties. For the convenience of the reader, we list here the arrangement numbers, second Betti-number and the equations from <cit.>. [ [-0.5mm]0mm4mmb_2(X̃) λ;1 70 xyzt(x + y)(y + z)(z + t)(t + x) -1;3 62 xyzt(x + y)(y + z)(y -t)(x - y -z + t)1; 19 54 xyzt(x + y)(y + z)(x - z - t)(x + y + z - t)2; 32 50 xyzt(x + y)(y + z)(x - y - z - t)(x + y - z + t) -1; 69 50 xyzt(x + y)(x - y + z)(x - y - t)(x + y - z - t) -1; 93 46 xyzt(x + y)(x - y + z)(y - z - t)(x + z - t)2;238 44 xyzt(x + y + z - t)(x + y -z + t)(x - y + z + t)(-x + y + z + t)1;239 40 xyzt(x + y + z)(x + y + t)(x + z + t)(y + z + t)1;240 40 xyzt(x + y + z)(x + y - z + t)(x - y + z + t)(x - y - z - t) -2;241 40 xyzt(x + y + z + t)(x + y - z - t)(y - z + t)(x + z - t)1;245 38 xyzt(x + y + z)(y + z + t)(x - y - t)(x - y + z + t) -2;] The meaning of the number λ in the last column will be explained later. § 3–CYCLES ON A DOUBLE OCTIC In the above eleven examples of rigid double octic Calabi–Yau threefolds defined over ℚ are given. In all these examples the eight planes are given by equations with integralcoefficients. In general, an arrangement defined by real planes gives a decomposition of (ℝ) into a finite number of polyhedral cells. By combining these cells one can construct certain polyhedral cycles on the smooth model X. To explain this, let us fix one of these cells C and consider its double covering C, that is, its preimage under the 2–fold covering mapπ: X. Then C is a 3–cycle in X and determines an element in H_3(X,), up to a sign determined by a choice of an orientation. Question: Do the 3–cycles C generate H_3(X,)? However, C will in general not be a 3–cycle on the desingularisation X, as the canonical map τ_*: H_3(X,)H_3(X,)will not be surjective in general. To see this geometrically, we follow thefate of the cycle C under the blow–up maps and see that its getstransformed into a chain on X that we will still denote by C. Its boundary ∂ C, as a chain on X, is a sum of 2–cycles contained in the exceptional loci∂ C=⋃_iΓ_i. Now observe that C is anti–symmetric with respect to thecovering map π, and as a consequence, these cycles Γ_i are anti–symmetric as well. On the other hand the exceptional divisor corresponding to a fivefold point or a triple line is fixed by the involution, while the exceptional locus corresponding to a double line is a blow-up of a conic bundle, so in all the three cases second cohomology group is symmetric.Henceeach cycle Γ_i contained in the exceptional divisor corresponding to a double line, triple line or a fivefoldpoint there is a boundary, i.e. there is a 3–chain C_i such that δ C_i=Γ_i.Hence, if we subtract from the chain C the chains C_i we get a chain with boundary contained in the exceptional divisors corresponding to the fourfold points. So we see that C can be lifted to a cycle C on the partial resolution X̂ and we have shown: Proposition: The mapρ_*: H_3(X̂,)H_3(X,)is surjective.Special role of the fourfold points So we see that we need to analyse the situation of a fourfold point in more detail. We only have to consider fourfold points p that are not on triple lines, so at which four planes intersect in general position.Near p the space ^3is decomposed into 2^4=16 cells, which are in one-to-onecorrespondence with the sign patterns ((L_1), ( L_2),(L_3), (L_4) )where the linear forms define the planes meeting at p. Each cell has an opposite cell, obtained by a reversing all signs.< g r a p h i c s >If we blow–up the point p, the exceptional divisor is a copy of [2], on which we find four lines in general position,corresponding to the four planes through p; these four lines decompose the real projective plane in seven regions, that can be colored into three 'black' and four 'white' regions. < g r a p h i c s >On the double cover we find an exceptional divisor E that is a double cover of this [2] ramified along these four lines, and each of the eight regions R determine a 2-cycle R in E.These regions are in one–to–one correspondence the pairs of opposite cells. If C is a cell corresponding to a region R, then the chain C on the blow–up, then the boundary of this chain is precisely the 2-cycle corresponding to R:∂ C = ± R What we learn from this is that we can cancel this boundary term of a cell by adding to it the boundary term of the opposite cell!Hence, we can define a group of polyhedral cycles PC^3 consisting of elements∑_C n_CC, n_C ∈for which for all fourfold points p one has:∑_ p ∈C n_C =0 We can assume that the four planes we are considering have equations x=0, y=0,z=0,x+y+z=0 in appropriate coordinates. We will analyse what happens if wesmooth out the fourfold point shifting the fourth plane tox+y+z=ϵ, resolve this and then specialise back to ϵ=0.Let us first blow–up the two disjoint lines x=y=0,z=x+y+z-ϵ=0.In one of the affine charts the blow–up of ℙ^3 is given by the equation x(y+1)-z(v-1)-ϵ. The threefold is smooth unless ϵ=0 when it acquires a node at x=0,y=-1,z=0,v=1. Since the surface x=0,z=0 isa Weil divisor on the threefold which is not Cartier(it is a component of the exceptional locus of the blow–up) the node admits a projective small resolution. Since the node does not lie on the branch divisor it gives two nodes on the double cover. Consequently. we get the partial resolution X̂ from the end of the section <ref>.§ DETERMINATION OF H_3(X̂,) Denote by X_t a smoothing of X̂. By <cit.>, the deformations of X correspond to deformations of the arrangements of eight planes that preserve the incidences between the planesin D. The deformations of the arrangement thatpreserves all the incidencesexcept for the fourfold points correspond to smoothings of X̂. By the work of J. Werner <cit.>, the nodal variety X̂is homotopy equivalent to its small resolution X with 3–cells gluedalong the the exceptional lines (which topologically are 2–spheres).The nodal variety X̂ is also homotopy equivalent to its smoothingwith 4–cells glued along the vanishing 3–cycles. As a consequence, one arrives at the following equations relating topological invariants of X_t, X and X̂:b_4(X_t)+2p_4+b_3(X̂)=b_4(X̂)+b_3(X_t) b_2(X_t)=b_2(X̂)b_3(X)+2p_4+b_2(X̂)=b_3(X̂)+b_2(X) b_4(X)=b_4(X̂)and henceb_3(X̂)=b_3(X)+b_2(X_t)-b_2(X)+2p_4= b_4(X)+b_3(X_t)-b_4(X_t)-2p_4where p_4 is the number of (smoothed) fourfold points in D that do not lie on a triple line. For the eleven rigid double octics from <cit.> we get1.3[No.b_3(X̂) b_3(X_t)p_4^0;1341;3583; 196 104; 327 125; 697 125; 938 146;238 11 20 12;239 11 20 10;240 11 20 10;241 11 20 10;245 11 209;]§ AN EXAMPLEIn order to find two independent cycles, we draw projections ofintersections of all arrangement planes onto the (x,y)–plane,and consider the equations of the planes not perpendicular to itas functions in z. The easiest case is the arrangement No 1.which hasa single p_0^4 point. We will go through some details of this example . The equation of this arrangement isxyzt(x+y)(y+z)(z+t)=0and the only p_0^4 point is (1,-1,1,-1). The affine change of variablest⟼ t-x,maps this point to the plane at infinity. The arrangement is then givenin affine coordinates by the equationxyz(1-x)(x+y)(y+z)(-x+z+1)=0while the p_0 ^4 point is the point at infinity (1,-1,1,0).The planes defined the first, second, fourth and fifth factor of the above product are perpendicular to the (x,y)–plane and intersect the plane inlines x=0,y=0,x=1,x+y=0.The planes in the arrangement that are not perpendicular to the (x,y)–planecan be seen as graphs over the (x,y)-plane and are given byz= f_3(x,y):=0 z= f_6(x,y):=-y z= f_7(x,y):=x-1The projections of the lines of intersection of these planes are given byf_3=f_6 :y=0f_3=f_7 :x=1f_6=f_7 :x+y=1 In the (x,y)–plane we have two bounded domainsI: x>0, y>0, x+y<1II: x<1, y<0, x+y>0 < g r a p h i c s > . For points (x,y) in these regions, the functions f_3, f_6, f_7satisfy there the following inequalitiesI: f_7<f_6<f_3II: f_7<f_3<f_6Consequently, the domains lying over triangle I are given byx>0, y>0, x+y<1, z>x-1, z<-y x>0, y>0, x+y<1, z>-y,z<0As the “right” (horizontal) edge of the triangle II is the projection the intersection of planes no. 3 and 7, the only cycle lying over that triangle is given byx<1, y<0, x+y>0, z>x-1, z< 0,the other domain x<1, y<0, x+y>0, z> 0, z<-yis not bounded by arrangement planes, if we want to use it we would have to add the unbounded domain “across the edge” x>1, x+y>0, x+y<1. Instead we can choose a domain over triangle IIx>0, y<0, x+y>0, z>x-1, z<0. § PERIOD INTEGRALSWhen we are given a degree eight polynomial F(x,y,z,t), then the double octic X ⊂[4](4,1,1,1,1) defined by the equationu^2-F(x,y,z,t)=0comes with a preferred section ω∈Γ(X,ω_X) of its sheaf of dualising differentials. In the affine chart t ≠ 0 it can bewritten as ω:= dxdydz/u=dxdydz/√(F). The period integrals of X are thus of the form∫_γω=∫_γdxdydz/√(F)where γ is a three-cycle in X. If in particular F defines a real arrangement of eight planes and we have a bounded cell C in ℝ^3 yielding a 3–cycleC in Calabi-Yau threefold X,the period integral∫_Cωis just equal to three-fold integral2∭_Cdx dy dz/√(F), In the case consideredin previous section (arrangement No. 1), the twoperiods integral are given by∫ _0^1∫_0^1-x∫_x-1^-y1/√(xyz(1-x)(x+y)(y+z)(-x+z+1))dzdydx ∫ _0^1∫_-x^0∫_x-1^01/√(xyz(1-x)(x+y)(y+z)(-x+z+1))dzdydx. To compute such integrals numerically, we used Maple. However, the function F can have zeros of multiplicity 5 at a vertexof a polyhedron of integration and thus the integrand is unbounded.As a result, a direct numerical integration usually does not yield asatisfactory precision in reasonable time. We used the followingsimple trick which allows us to get 12 digits precision withoutmuch effort, which is sufficient for the our purposes. Using an affine coordinate change, we reduce computations to the case of a integration over a cube0 ≤ x,y,z ≤ 1, with the function F vanishing only for xyz=0. Then substituting (x,y,z) ↦ (x^k,y^k,z^k) in the tripleintegral transforms the integral to the integration of a bounded function. Note that depending on the sign of the function F in a given polyhedral cell C, we get either a real or a purely imaginary number. The computation time in thelatter case can be reduced considerably by just using the function -F!It should be noted that if we multiply F by a constant factor λ,the correspondingperiod integral changes by a factor √(λ). In particular, if we change the sign of F the real and imaginary periods are interchanged. In the case of arrangement nr. 1 everything works nicely, but in the othercases the picture of the decomposition of ℙ^3 becomes much morecomplicated and more generic fourfold points to take into account.We wrote a simple Maple code to produce a linear–cylindricdecomposition and form cycles from the polyhedral cells. Then we used several changes of variables moving each of the planes of the arrangement to infinity, which allowed us to compute the integrals for most of the cycles. In all cases ratios of any two real and any two complex integrals were rational numbers (with numerator and denominator ≤ 6). Below is a table that summarises all different period integrals that appeared in our calculations. It should be kept in mind that all period integrals get multiplied by a common factor if we change the polynomial F defining the arrangement. For these calculations we used the equations F as listed in <cit.> scaled by λ from the last column of table at the end of section <ref>.Arr. No.Real integralsImaginary integrals 1 55.9805041334, 111.96100826769.3694986501i3 80.3028893419, 160.60577868 41.4134587444i, 82.8269174889i 124.240376233i,289.89421121i19 72.1085316451, 144.217063291 72.1085316451i, 144.217063291i 216.325594935 216.325594935i32 55.9805041335, 111.961008267 34.6847493250i, 69.3694986501i138.738997300i, 208.1084959i69 55.9805041335, 111.9610083 34.6847493252i, 138.738997300i 223.922016533 277.4779945i93 55.9805041334 17.3423746625i, 69.3694986502i138.738997300i238 55.9805041334, 111.96100826734.6847493250i239 48.5252148713, 145.575644614 35.2275632784i, 105.682689835i240 43.7468074540, 131.240422363 28.8234453872i, 57.6468907743i241 223.922016533 69.3694986503i245 21.8734037270, 87.4936149079 28.8234453872i, 115.293781548i 131.240422362For each arrangement, the computed period integrals generatea lattice in , which in turn defines an elliptic curve. This lattice might be a proper sublattice of H_3(X,), but in any case it defines a elliptic curve that is isogeneous with the intermediate Jacobian J^2(X)of the corresponding Calabi–Yau threefold. In the following table welist lattice generators, j–invariant and coefficients of theclassical Weierstrass equation y^2=4 x^3-g_2 x-g_3of the elliptic curve, which are easily computednumerically viag_2=60 ∑_0 ≠ m ∈Λ1/m^4,g_6= 140 ∑_0 ≠ m ∈Λ1/m^6,This is a standard functionality in MAGMA. Arr. No τ/i j(τ) g_2 g_3 1 1.239172453413236.13720434 142.879810750 224.3785726833 0.515715674539196267.167917 1838.35630102 -15102.27412619 1 1728 189.072720130 032 0.619586226703 26112.0318779 889.658497527 -4934.9816241669 0.619586226703 26112.0318779 889.658497527 -4934.9816241693 0.309793113352 643142260.966 14101.0467615 -322251.215146238 0.619586226704 26112.0318791 889.658497527 -4934.98162416239 0.725964086338 6517.46790207 487.190579154 -1774.06947556240 0.658869688205 14612.0507801701.139041736-3355.01890381241 0.309793113354 643142256.756 14101.0467615 -322251.215146245 1.31773937641 4737.95402281137.802991416248.136467781 § COMPARISON WITH MODULAR PERIODSRecall that for an Hecke-eigenform f ∈ S_k(Γ_0(N)) with q-expansionf=∑_n=1a_n q^nthe L-function is defined by the series:L(f,s)=∑_n=1^∞a_n/n^s .It converges for Re(s) >1+k/2 and the completed L-functionΛ(f,s):=(√(N)/2π)^s Γ(s)L(f,s)satisfies the functional equationΛ(f,s)=w i^k Λ(f,k-s)where w is the sign of f under the Atkin-Lehner involution. We note that Λ(f,s) =(√(N))^s ∫_0^∞f(it) t^s dt/t In our case we k=4 and w=1, so that the functional equation just readsΛ(f,s)=Λ(f,4-s)This means in particularΛ(f,1)=Λ(f,3)from which we get the equalityL(f,3) = (2π)^2/NΓ(1)/Γ(3) L(f,1)=2π^2/NL(f,1)Furthermore, we see from the functional equation that L(f,k)=0 for k=0,-1,-2,-3,-4,….By direct point counting (and correcting for the singularities of course), C. Meyer was able to determine cusp-forms f=∑_n=1^∞ a_n q^n ∈ S_4(Γ_0(N))such thata_p =Tr(Fr_p:H^3(X)H^3(X))In other words, one has equality of L-functionsL(H^3(X),s)=L(f,s)The result is summarised in the following table (we multiplied the equation of the octic arrangement by the factor λ to obtain modular form of minimal level).[; 6/1 q - 2q^2 - 3q^3 + 4q^4 + 6q^5 + 6q^6 - 16q^7 + O(q^8)240, 245; 8/1q - 4q^3 - 2q^5 + 24q^7 - 11q^9 - 44q^11 + O(q^12) 1, 32, 69, 93, 238, 241;12/1 q + 3q^3 - 18q^5 + 8q^7 + 9q^9 + 36q^11 + O(q^12) 239;32/1 q + 22q^5 - 27q^9 + O(q^12)19;32/2 q + 8q^3 - 10q^5 + 16q^7 + 37q^9 - 40q^11 + O(q^12) 3; ] It is a remarkable fact that only five different modular forms appear.In cases where two varieties X, X' have the same modular form, one expectsthere exists a correspondence ϕ between X and X' that explains it.It is gratifying to see that the numerical evaluation of the period integrals lead to the very same grouping of our examples. Here we summarise the calculations of the critical L-values[ fL(f,1)L(f,2); 6/1 0.22162391559067350824671004425 0.50971042336159397988737819140; 8/1 0.35450068373096471876555989149 0.69003116312339752511910542021;12/1 0.61457902590673022954002802969 0.93444013814191444281042898230;32/1 1.82653044425089816105284840591 1.43455365630418076432004680798;32/2 2.03409594950627923591429024672 1.64778916742512594127684239683; ]If we express the real and imaginary periods of the double octics we get, at least at the numerical level, nice proportionalities withπ L(f,2), π^2 L(f,1)for the corresponding modular form. [3|c|Form 6/1; 240 43.7468074540…=20π^2 L(f,1) 28.8234453871…=18π L(f,2); 245 21.8734037270…=10π^2 L(f,1) 28.8234453871…=18π L(f,2);3|c|Form 8/1; 1 55.9805041334…=16π^2 L(f,1) 69.3694986501…=32π L(f,2);32 55.9805041334…=16π^2 L(f,1) 34.6847493250…=16π L(f,2);69 55.9805041334…=16π^2 L(f,1) 34.6847493250…=16π L(f,2);93 55.9805041334…=16π^2 L(f,1)17.3423746625…=8π L(f,2); 23855.9805041334…=16π^2L(f,1) 34.6847493250…=16π L(f,2); 3|c|Form 12/1; 239 48.5252148713…=8 π^2 L(f,1) 35.2275632785…=12π L(f,2); 3|c|Form 32/1;1972.1085316452…=4π^2 L(f,1) 72.1085316452…=16π L(f,2); 3|c|Form 32/2; 380.3028893419…=4π^2 L(f,1)41.4134587443…=8π L(f,2); ]§ OUTLOOKThe above calculations show that it is possible to verify numerically the relation between the periods of a rigid Calabi-Yau and thecorresponding L-values of the attached modular form. However, one would like to push these calculations to a further level.One important problem that was left untouched by our calculations is the complete determination of the group 3-cycles in terms of polyhedral cycles. We identified some polyhedral cycles, but there isno guaranty that these generate the whole third homology groupH_3(X̃,). It follows from Poincaré-duality, that this group is generated by any two cycles with intersection ±1. This leads to the following question to determine the inteeresection number ⟨δ, γ⟩ between two polyhedral cycles δ, γ in purely combinatorial terms of the cell appearing in δ and γ, and their mutual position inside the arrangement. Apart from the eleven double octics with rational coefficients there are two other with coefficients in(√(-3)) and one in (√(5)) (cf. <cit.>). However, in these cases presented method did not yield any reasonable approximation of the period integrals.Acknowledgments: Part of this research was done during thestay of the first named authoras a guestprofessor at the Schwerpunkt Polen of the JohannesGutenberg–Universität in Mainz. He would like to thank the department for the hospitality and excellent working conditions. This research was supported in part by PL–Grid Infrastructure.The first named author was partially supported by NCN grant no. N N201 608040.99 CvSS. Cynk, D. van Straten,Infinitesimal deformations of double covers of smooth algebraic varieties.Math. Nach. 279 (2006), 716–726. CvS2 S. Cynk, D. van Straten, Small Resolutions and Non-Liftable Calabi-Yau threefolds, manuscriptamath. 130 (2009), no. 2, 233 - 249CvS3 S. Cynk, D. van Straten,Calabi-Yau conifold expansions. Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds, 499–515, Fields Inst. Commun., 67, Springer, New York, 2013. Del P. Deligne, Formes modulaires et représentations ℓ-adiques, Séminaire N. Bourbaki, 1968-1969, exp. no. 355, 139–172.Die L. Dieulefait, On the modularity of rigid Calabi-Yau threefolds: epilogue. J. Math. Sci. (N.Y.) 171 (2010), no. 6, 725–727. GY F. Gouvêa, N. Yui, Rigid Calabi–Yau threefolds overare modular, Expositiones Mathematicae 29 (2011), 142–149.Griff P. Griffiths, Periods of integrals on algebraic manifolds. I. Construction and properties of the modular varieties, American Journal of Mathematics, 90 (2). 568-626. Man Y. Manin, Periods of parabolic forms and p-adic Hecke series,Math. USSR Sbornik Vol. 21 (1973), No. 3, 371–393.Meyer C. Meyer, Modular Calabi-Yau threefolds. Fiebilds Institute Monographs, 22. American Mathematical Society, Providence, RI, 2005. Yui N. Yui, Modularity of Calabi-Yau varieties: 2011 and beyond. Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds, 101–139, Fields Inst. Commun., 67, Springer, New York, 2013. Ver H. Verrill, Transportable modular symbols and the intersection pairing., Algorithmic number theory (Sydney, 2002), 219–233, Lecture Notes in Comput. Sci., 2369, Springer, Berlin, 2002.Werner J. Werner, Kleine Auflösungen spezieller dreidimensionaler Varietäten, Bonner Math. Schriften 186 (1987). | http://arxiv.org/abs/1709.09751v1 | {
"authors": [
"Slawomir Cynk",
"Duco van Straten"
],
"categories": [
"math.AG",
"14J32"
],
"primary_category": "math.AG",
"published": "20170927222443",
"title": "Periods of double octic Calabi--Yau manifolds"
} |
Department of Applied Mathematics, Brown University, Providence, RI 02912Department of Chemistry, Brown University, Providence, RI 02912Department of Chemistry, Brown University, Providence, RI 02912Fractional derivatives are nonlocal differential operators of real order that often appear inmodels of anomalous diffusion and a variety of nonlocal phenomena. Recently, a version of the Schrödinger Equation containing a fractional Laplacian has been proposed. In this work, we develop a Fractional Path Integral Monte Carlo algorithm that can be used to study the finite temperature behavior of the time-independent Fractional Schrödinger Equation for a variety of potentials. In so doing, we derive an analytic form for the finite temperature fractional free particle density matrix and demonstrate how it can be sampled to acquire new sets of particle positions. We employ this algorithm to simulate both the free particle and ^4He (Aziz) Hamiltonians. We find that the fractional Laplacian strongly encourages particle delocalization, even in the presence of interactions, suggesting that fractional Hamiltonians may manifest atypical forms of condensation. Our work opens the door to studying fractional Hamiltonians with arbitrarily complex potentials that escape analytical solutions. Fractional Path Integral Monte Carlo Brenda M. Rubenstein December 30, 2023 ====================================§INTRODUCTION Fractional calculus is the study and use of differential operators of real order. Such operators are nonlocal except for the case of nonnegative integer order. Space- and time-fractional differential equations have emerged as the correct way to model anomalous diffusion, <cit.> nonlocal phenomena,<cit.> memory effects,<cit.> and many non-equilibrium <cit.> and turbulent<cit.> systems. Anomalous diffusion is any diffusion in which the mean square displacement is not linear in time, i.e., ⟨ x^2 ⟩∼ t^2/α,where α≠ 2, in contrast to the α = 2 case, which corresponds to normal/Brownian diffusion. If α < 2, the diffusion is superdiffusive, while if α > 2, the diffusion is subdiffusive. One can think of superdiffusion for 1<α<2 as a regime between normal diffusion and ballistic motion (for which α = 1, implying that the average displacement is proportional to time). A simple example of superdiffusion is a plasma whose particles would ordinarily be expected to undergo diffusion because of their fluid nature, but instead exhibit superdiffusive behavior because of the presence of a magnetic field that favors ballistic motion.<cit.> On the other hand, traps (such as in porous media<cit.>) or formations of eddies in a fluid<cit.> may introduce waiting times that slow down the fluid's particles, pushing their motion toward subdiffusion. In general, both tendencies can occur, resulting in a unique process with “competition” between super- and sub-diffusive overall behavior. Anomalous diffusion differs significantly from normal diffusion, in more ways than can be expected from the scaling relation given by Relation (<ref>). Anomalous diffusion is often characterized by heavy tails, infinite variance, and inhomogeneity in space and time. These are characteristics of Continuous Time Random Walks (CTRWs), a broad family of anomalous diffusions which model both waitings times and long jumps. <cit.>The simplest subfamily of CTRWs is isotropic α-stable Levy motion. Just as the standard Laplacian (-Δ) in ℝ^n is the generator of Brownian motion, a random walk in which the jumps can be drawn from a normal distribution, for 0 < α < 2, the fractional Laplacian (Δ)^α/2 := -(-Δ)^α/2 in ℝ^n is the generator of isotropic α-stable Levy motion.<cit.> In this motion, jumps are drawn from an α-stable distribution, which exhibits power-law tails. The isotropic α-stable distribution is defined by a position parameter and a scale parameter, which for the normal α = 2 case, reduces to the mean and variance, respectively. The field of physics where anomalous diffusion is most widespread is perhaps plasma physics, where anomalous diffusion has been observed for decades.<cit.>This has prompted theoretical discussions on how the equations of magnetohydrodynamics can drive anomalous diffusion<cit.> and applications of CTRWs toplasma physics.<cit.> Anomalous diffusion has moreover been observed<cit.> in percolation clusters, which has since been explained theoretically. <cit.> Inspired by this finding, the role of anomalous diffusion in disordered superconductors (and other disordered media) has also been elucidated.<cit.> Recently, a nonlinear fractional Schrödinger equation has been rigorously derived from a mathematical physics perspective as the continuum limit of a system of discrete nonlinear Schrödinger equations with increasingly long-range interactions.<cit.> Laskin<cit.> first developed fractional quantum mechanics, following the early considerations of Kac.<cit.> Starting from the Feynman path-integral formulation of quantum mechanics, Laskin replaced integrals over Brownian paths with integrals over α-stable Levy paths.He showed that this amounts to replacing the standard Laplacian by the (Riesz)[ We distinguish the Riesz fractional Laplacian from other forms because there are several ways to define the fractional Laplacian. Although all of the popular fractional Laplacians agree with each other in ℝ^n , they no longer agree on a bounded domain.<cit.> The use of the Riesz Laplacian Δ^α/2:=-(-Δ)^α/2, which can be characterized in ℝ^n as having Fourier symbol -|p⃗|^α,is required by the derivation of fractional quantum mechanics. ]fractional Laplacian in the Schrödinger Equation, leading toiħ∂ψ(r⃗, t)/∂ t = - ħ D_αΔ^α/2ψ(r⃗, t)+ V(r⃗)ψ(r⃗, t).The equation is called the space-fractional Schrödinger equation. Laskin showed that this equation enjoys the same basic properties that allow the usual Schrödinger equation to serve as a foundation for quantum mechanics; for example, since -Δ^α/2 is self-adjoint, observables are real valued.<cit.>To date, solutions to the fractional Schrödinger Equation have been derived for a variety of analytically tractable Hamiltonians, including the free particle, square well, and particle in a box Hamiltonians.<cit.> Solutions to Hamiltonians containing more realistic potentials have so far been few and far between, limiting the study of potentially rich fractional mathematics and physics.In this paper, we introduce a new fractional Path Integral Monte Carlo method capable of modeling the finite temperature physics of a wide variety of space-fractional Hamiltonians. The method computes finite temperature observables using the Path Integral Monte Carlo (PIMC) algorithm popularly used to characterize boson and Boltzmannon phase diagrams <cit.> to sample the many body fractional partition function. Key to being able to use the PIMC algorithm in this context is being able to express and sample the free finite temperature fractional density matrix. We therefore derive an analytical form for this fractional density matrix and show how it can be both computed and sampled. Interestingly, we find that the probability distribution function that represents the fractional density matrix manifests dramatically different asymptotic behaviors depending upon the fractional exponent present in the Hamiltonian. Whereas for fractional exponent α=2 the distribution reduces to the usual Gaussian case, for α<2, the distribution possesses heavy tails, while for α>2, the distribution becomes negative. Setting aside the α>2 case, which is typically excluded[There is no α-stable process for α > 2.] in discussions of the fractional laplacian, in the remainder of this work, we illustrate how our algorithm may be used to compute finite temperature observables including energies and radii of gyration for two illustrative Hamiltonians containing α≤ 2 fractional Laplacians: 1) the free particle Hamiltonian; and 2) the ^4He Hamiltonian using the Aziz<cit.> potential. We end with a discussion of the general applicability of our method and the intriguing fractional physics it may reveal.§THEORY AND ALGORITHMS§.§Overview of the Path Integral Monte Carlo Method Since our method may be viewed as a generalization of the conventional Path Integral Monte Carlo algorithm, we begin this section with a review of PIMC. As described in significantly more detail by Ceperley,<cit.> the finite temperature equilibrium expectation values of a system observable, Ô, in the position basis may be expressed as⟨Ô⟩ = Z^-1∫ dR⃗ dR⃗^'ρ(R⃗,R⃗^'; β) ⟨R⃗ | Ô | R⃗^'⟩,where Z denotes the system's partition functionZ = ∫ dR⃗ρ(R⃗, R⃗; β)at inverse temperature β = 1/(k_BT). In the above, R⃗={r⃗_1, r⃗_2, ...,r⃗_N} is the position vector of all N particles' coordinates, where r⃗_i represents the position of the ith particle. One of the key quantities present in both Equations (<ref>) and (<ref>) is ρ(R⃗, R⃗^'; β), the finite temperature density matrix. In position space, the density matrix may be expanded into ρ(R⃗, R⃗^'; β)= ⟨R⃗| e^-βĤ | R⃗^'⟩= ∑_iϕ_i^*(R⃗) ϕ_i(R⃗^') e^-β E_i,where Ĥ is the Hamiltonian that describes the system, and E_i and ϕ_i respectively denote its eigenvalues (energies) and eigenfunctions (wave functions). Although obtaining an exact expression for the density matrix at an arbitrary temperature would require diagonalizing the many body Hamiltonian, a task generally beyond our current computational capabilities, the density matrix may be simplified into more tractable expressions via Trotter factorization. In general, the convolution of two density matrices remains a density matrix such that ρ(R⃗_1, R⃗_3; β_1+β_2) = ∫ dR⃗_2ρ(R⃗_1, R⃗_2; β_1) ρ(R⃗_2, R⃗_3; β_2),which implies that the expression for the density matrix at inverse temperature, β, may be expressed as the convolution of M density matrices at inverse temperature τ = β/M ρ(R⃗_0, R⃗_M; β )= ∫ ... ∫ dR⃗_1 dR⃗_2...dR⃗_M-1ρ(R⃗_0, R⃗_1; τ)ρ(R⃗_1, R⃗_2; τ)...ρ(R⃗_M-1, R⃗_M; τ).The M different N-particle position vectors represent the particle positions at M different so-called “imaginary times.” Because R⃗_0 is the same as R⃗_M in the expression for the partition function, according to the quantum-classical isomorphism, the particles may be thought of as polymers constructed of M different beads linked together. If τ is sufficiently small, assuming the system Hamiltonian may be split into kinetic (K̂) and potential pieces (V̂), Ĥ=K̂ + V̂, the Suzuki-Trotter factorization<cit.> may be used to break up the exponential of the Hamiltonian into corresponding kinetic and potential exponentials e^-τĤ = e^-τ (K̂ + V̂)≈ e^-τK̂ e^-τV̂such that the position-space density matrices may also be broken into ρ(R⃗_0, R⃗_2; τ) ≡∫ dR⃗_1⟨R⃗_0| e^-τK̂ | R⃗_1⟩⟨R⃗_1|e^-τV̂ | R⃗_2⟩.Because potential operators are generally diagonal in the position representation, evaluating the position density matrix is trivial ⟨R⃗_1|e^-τV̂ | R⃗_2⟩ = e^-τV̂(R⃗_1)δ(R⃗_2-R⃗_1).The evaluation of the kinetic density matrix is less trivial and requires finding the eigenvalues and eigenfunctions of K̂. Assuming periodic boundary conditions and the α=2, non-fractional form for the kinetic operator K̂ = -ħ^2/2m∇^2 = -λ∇^2,in which Δ denotes the Laplacian, ∇ denotes the differential operator, and λ = -ħ^2/(2m), the eigenvalues of the three-dimensional kinetic operator are λK⃗_n^2 and the eigenfunctions are L^-3N/2e^iK⃗_nR⃗, with K⃗_n = 2 πn⃗/L. The kinetic density matrix may therefore be expressed as ⟨R⃗_0 | e^-τK̂ | R⃗_1⟩ = ∑_n⃗ L^-3N e^-τλ K_n^2 - i K⃗_n (R⃗_0-R⃗_1) =(4 πλτ)^-3N/2 e^-(R⃗_0-R⃗_1)^2/4 λτ.As discussed in more detail in Section <ref>, the finite temperature partition function may then be sampled via Monte Carlo by sampling the Gaussian kinetic density matrix probability distribution function for new particle coordinates and then using the potential density matrices to accept/reject those coordinates according to their potential energy contributions. As a prelude to the subsequent discussion, it should be noted that Equation (<ref>), and the eigenvalues and eigenfunctions used to evaluate it, is only valid for the ∇^2 Laplacian. If the differential operator is of a different power, the Suzuki-Trotter factorization is still valid (see Appendix II), but new expressions for the density matrix are required. These expressions are derived and discussed below.§.§The Fractional Schrödinger Equation As first presented by Laskin,<cit.> the fractional Schrödinger Equation is a generalization of the Schrödinger Equation in which the Hamiltonian is generalized to a fractional form, Ĥ_α, that contains a fractional Laplacian, Δ^α/2. In one dimension, Ĥ_α, may be expressed as Ĥ_α = - D_αħ^αΔ^α/2 + V̂(x).In the above, V̂(x) denotes the potential and D_α=(1/2m)^α/2 denotes the constant that falls in front of the Laplacian. In general, D_α must have SI units of [D_α] =m^2-α kg^1-α/s^2-α.The fractional exponent, α, may in principle assume any positive value, including non-integer, fractional values, as its name implies.<cit.> For reference, the fractional Hamiltonian reduces to the conventional case when α=2. Please also note that in the next few sections we consider the one-dimensional version of the Hamiltonian to simplify our discussion of the derivation of the fractional density matrix; we will return to a discussion of the full three-dimensional form in later sections.Most previous work on the fractional Schrödinger Equation has explored the properties of fractional Hamiltonians with potentials that are analytically soluble, including the delta potential,<cit.>, the fractional hydrogen atom,<cit.> the fractional oscillator,<cit.> and the fractional square well.<cit.> In addition to the fact that these models manifest atypical physics, such as atypical energy spectra, they are also of intrinsic mathematical interest. Considering that that the ordinary Schrödinger equation is already extremely challenging to solve, the inclusion of a (nonlocal) fractional derivative makes equation (<ref>) even less tractable. More recently, it has been suggested that fractional Hamiltonians may naturally arise from strong many body interactions, which prompted a mean field exploration of fractional Bose-Einstein condensation.<cit.> Nevertheless, no robust algorithm for examining the properties of the fractional Schrödinger Equation for arbitrary potentials has yet been proposed.§.§Spectrum of the Fractional LaplacianIn order to generalize the Path Integral Monte Carlo algorithm discussed in Section <ref> to fractional Hamiltonians, expressions for the fractional density matrix must be obtained. As discussed above, this requires finding an analytical form for the kinetic (non-interacting) density matrix that is also amenable to sampling. This can be achieved by finding the eigenvalues and eigenfunctions of the fractional Laplacian. The eigenvalue spectrum of the Riesz fractional Laplacian in one dimension is given byΔ^α/2cos (Cx)= -|C|^αcos(C x)Δ^α/2sin (Cx)= -|C|^αsin(C x)for any C ∈ℝ in the unbounded case.<cit.> In fact, this can be viewed as the definition of the real power of the Laplacian operator via the spectral theorem.<cit.>The standard boundary conditions used in condensed phase simulations are periodic boundary conditions. The introduction of periodic boundary conditions restricts the continuous family of eigenfunctions to a discrete family of eigenfunctions that satisfy the required periodicity.[We warn the reader that this is not the case for the Riesz fractional Laplacian in a bounded domain with Dirichlet (or Neumann) boundary conditions. There, the eigenfunctions of the Riesz fractional Laplacian are not those of the standard Laplacian in the same domain. See Chapter 12 of Hermann<cit.> for a discussion of this issue, which is due to the nonlocality of fractional derivatives. The direct spectral power of the fractional Laplacian in a bounded domain (that preserves the eigenfunctions of the standard operator) results in a different fractional Laplacian, the spectral fractional Laplacian. However, this is not the same fractional Laplacian that is prescribed by the derivation of the fractional Schrödinger Equation of Laskin – except on ℝ^n.] For a periodic box of length L, C=2π k/L, where k=1,2,3,...,the related eigenvalues of the Laplacian are thereforeλ = - |C|^α =- (2 π k/L)^αand the related eigenfunctions are sin(Cx) and cos(Cx). As in the standard derivation for the particle in a box Hamiltonian,<cit.> the sums and differences of these trigonometric functions may be combined and normalized to yield eigenfunctions of the formϕ_k^per,L(x) = √(1/L) e^-iCx.Inserting these expressions into Equation (<ref>), the final Hamiltonian eigenvalues (energies) may be obtained -D_αħ^αΔ^α/2ϕ_k^per,L(x)=D_αħ^α |C|^α√(1/L) e^-i C x=E_k^per,L√(1/L) e^-i C x .Thus,E_k^per, L = D_αħ^α |C|^α = D_α|2 π k ħ/L|^α.Consequently, the energies in the fractional periodic setting are similar in form to the α=2, non-fractional case, except that the frequencies are now raised to a fractional power instead of being squared.§.§Derivation of the Fractional Density Matrix Following Ceperley and Krauth,<cit.> the kinetic density matrix in one dimension may be obtained along the same lines as in Equation (<ref>) by summing over the Hamiltonian's eigenvalues and eigenfunctions ρ^per, L(x,x',β) = ∑_k=-∞^∞ϕ_k^per,L(x) e^-β E_k^per,Lϕ_k^*,per,L(x') = 1/L∑_k=-∞^∞ e^iC(x'-x) e^-β D_αħ^α |C|^αAs usual, these sums may be turned into integrals by substituting the implied Δ k in the summation with Δ k = Δ C L/(2 π) and letting the box size, L, go to infinityρ^per, L(x,x',β) =lim_L→∞1/L∑_C=...,-2π/L,0,2π/L,...( Δ C L/2 π) e^i C(x-x')e^-β D_αħ^α |C|^α = 1/2 π∫_-∞^∞ dC e^i C(x-x')e^-β D_αħ^α |C|^α.If α=2, this integral boils down to the usual Gaussian case in which completing the square yields the final line of Equation (<ref>). For more general values of α, it is impossible to compute this integral in closed form, but it can be represented as a special function defined by a power series.The integral may be identified as the characteristic function of the generalized normal random variable X_GN. This is the random variable given by the probability density functionf_κ(μ, σ; x) = ξ(κ) = κ/2 σΓ(1/κ) e^-| x-μ/σ|^κ.The characteristic function of ξ(κ) for κ>1was first computed explicitly in terms of special functions by Pogany and Nadarajah in 2006.<cit.> It may be expressed asE{ e^itX_GN} =√(π) e^itμ/Γ(1/κ)Ψ^1_1[ (1/κ, 2/κ); (1/2, 1); -(σ t)^2/4],where Ψ^1_1 denotes the Fox-Wright generalized hypergeometric function. This Fox-Wright function is ageneralization of the generalized hypergeometric function, and is a special case of the Fox H-function. It is specified by an arbitrary number, p+q, of parameters, and isdefined as the seriesΨ^p_q[ (α_1, A_1), ..., (α_p, A_p); (β_1, B_1), ..., (β_q, B_q); z ] = ∑_n=0^∞∏_j=1^pΓ(α_j + A_jn) /∏_j=1^qΓ(β_j + B_jn) z^n/n!.Comparing the generalized normal distribution in the expression for the density matrix given by Equation (<ref>) with the definition of the generalized normal distribution given by Equation (<ref>),we have |x-μ| = |C|, σ = 1/β^1/α D_α^1/αħ, and μ = 0. Making these substitutions, we obtain ρ^per, L (x,x',β) = 1/2 π∫_-∞^∞ dC e^i C(x-x')e^-β D_αħ^α |C|^α= 1/2π 2 Γ(1/α)/β^1/α D_α^1/αħα√(π) e^i (x-x') μ/Γ(1/α) ×Ψ^1_1[ (1/α, 2/α); (1/2, 1);- (x-x')^2/4 β^2/α D_α^2/αħ^2] =1 /√(π)β^1/α D_σ^1/αħα ×∑_n=0^∞Γ(1/α + (2/α)n) /Γ(1/2 + n)[ - (x-x')^2/4 β^2/α D_α^2/αħ^2]^n/n!. The generalized expression for the fractional kinetic density matrix is therefore proportional to the Fox-Wrightfunction of argument (x-x')^2.Similar representations were obtained by Laskin<cit.> for the free-particle density matrix on ℝ^n without boundary conditions.Laskin used the more general Fox H-function in his representation, which for the given indices reduces to the Fox-Wright function shown here. In another context, similar results were derived by Mainardi and Pagnini,<cit.> where the integral form of (<ref>) arises as the fundamental solution to the space-fractional diffusion equation. At this point, an efficient way to sample this Fox-Wright density is required to properly sample the kinetic density matrix.§.§ Computing and Sampling the Fractional Density Matrix The Fox-Wright density given by Equation(<ref>) is interpreted as a probability density function (PDF) of the jump distance, |x-x'|, for the random walkers.For α = 2, the Fox-Wright density reduces to a Gaussian, which is typically sampled using the Box-Muller transformation.<cit.>For simplicity and portability, we have used discrete inverse transform sampling to sample the distribution (<ref>). The method is entirely sufficient for the initial computations presented here. It is both fast and the most versatile method;since it does not rely on any special transformations, it can be used to sample any random variable for which the probability density function can be computed reliably. Our method is summarized in Figure <ref> and described in full detail in this section. The parametersβ, D_α, and ħ are not essential to the sampling methodology. We begin by reducing them to unity. If the density given in Equation (<ref>) is denoted by ρ_β, D_α, ħ, thenρ_1,1,1(x-x') =1/2π∫_-∞^∞ e^iC(x-x') e^-|C|^α dC. By rescaling, ρ_β, D_α, ħ may be reduced to ρ_1,1,1(x-x').Letting C = C'(ħ^αβ D_α)^-1/α and dC = dC' (ħ^αβ D_α)^-1/α, so that β D_αħ^α |C|^α = |C'|^α,we getρ_β, D_α, ħ (x-x') = 1/2π∫_-∞^∞ (β D_αħ^α)^-1/αe^-C' (β D_αħ^α)^-1/α(x-x') e^-|C'|^α dC' = 1/2π∫_-∞^∞ (β D_αħ^α)^-1/α e^-C' ( x-x'/(β D_αħ^α )^1/α) e^-|C'|^α dC' =(β D_αħ^α)^-1/αρ_1,1,1( x-x'/(β D_αħ^α)^1/α)Therefore, in what follows, we can assume withoutloss of generality thatβ = D_α = ħ = 1. As mentioned, the PDF ρ_1,1,1 can be sampled using inverse transform sampling, which requires that the related cumulative density function (CDF) be inverted. The general result is that, if a random variable X has CDF F, then X = F^-1[𝒰(0,1)] in distribution,<cit.> where 𝒰(0,1) denotes the uniform random variable on (0,1). In our case, the CDF of the random variable given by ρ_1,1,1 does not admit analytic inversion, so various numerical methods can be used. Options include binary search on a finely-discretized function table or a Newton method. Since PIMC simulations require samples of x-x' each of the millions of times a bead is displaced, the numerical inversion must be carried out as efficiently as possible.Any method for inverting the CDF will require accurate computation of the CDF, and therefore the PDF ρ_1,1,1. There are two direct ways to compute the PDF of ρ_1,1,1: direct quadrature of the integral in the first line of Equation (<ref>) or truncation of the Taylor series in the last line of Equation (<ref>). The truncated Taylor series is a convenient method when the argument of the distribution is small. For larger arguments, the series (which is an expansion at x = 0) requires increasingly higher-degree terms to be accurate, but this leads quickly to numerical blow-up. The combination of high-degree monomials, large x, and gamma functions and factorials with large argumentsresults in overflow and precision issues. One work around isthe following numerical “order of operations” trick:x^n/n! =( x/1) ( x/2) ... ( x/n-1) ( x/n).This prevents the overflow arising from computing the very large numbers x^n and n! separately before dividing by pairing each large factor x with a large factor k, k = 1, 2, ... n. With this trick, one can use the Taylor series to compute the Fox-Wright function ρ_1,1,1 accurately and quickly for arguments up to about 10,using 100 terms in the series. Beyond that point, numerical issues become more difficult to resolve and the computational costtoo high. Thus, for larger arguments, direct quadrature becomes more favorable.Due to the rapid decay of e^-|C|^α, the truncated integral1/2π∫_-L^L e^iC(x-x')e^-|C|^α dC = 1/2π∫_-L^L cos[C(x-x') ]e^-|C|^α dCconverges rapidly in L. At L ∼ 50, the integrand is far below machine precision. Note that in Equation (<ref>), the complex exponential has been replaced by a cosine because the related sine term is odd and therefore does not contribute to the final integral. The truncated integral can then be approximated by any quadrature rule, e.g., thetrapezoid rule.For large x-x', this is appealing as there are no overflow/precisionproblems. The only hazard is that the frequency of the integrandis proportional to x-x', so the quadrature must be performed with care to ensure that the result is converged in the number of quadrature points for large arguments. Quadrature is relatively costly in terms of computational time.Our final algorithm for sampling is as follows. Treating the PDF ρ_1,1,1 as discrete, we compute the CDF on [-100,100] in intervals of 0.01. This is done using the power series for |x| < 2 and quadrature for 2 < |x| < 100. The choice of at which value to transition from using the power series to using quadrature is immaterial as long as both techniques are accurate in the given region. For |x| > 100, where the Fox-Wright density approaches machine precision, we truncate. The resulting CDF is normalized to correct the minute mass lost by truncation and stored in memory. To generate a random sample x, we use binary search on the stored table to invert the equation y = CDF(x), where y is drawn from a uniform distribution on [0,1]. This method incurs high overhead when initializing the discrete Fox-Wright CDF on a fine grid, but avoids computation of the Fox-Wright density ρ_1,1,1 after initialization. Because the Fox-Wright CDF is computed before Monte Carlo moves are made, this approach avoids the computational bottlenecks the computation of the CDF would otherwise present.It is not necessary to fully discretize the density matrix; in principle, one could use a Newton solver with a coarse binary search for the initial guess to invert the CDF. However, we find the computational overhead to be similar to that needed for discretization/a binary search. In the fully discrete approach, after initialization, no expensive computations of the Fox-Wright density are ever performed, while in the Newton method, the Fox-Wright density must be computed several times at new values to generate each sample. If a specific computation proves to be extremely sensitive to the heavy tails of the Fox-Wright distribution, and it is desired to avoid truncation, the more expensive Newton method may be required. We mention that the inversion of the Fox-Wright CDF could be simplified and accelerated if a simple asymptotic expression as a power law were known. Since this function enters into a power-law regime fairly quickly for α≠ 2 (see Figures <ref> and <ref>), this is a reasonable hope. Any asymptotic expansion that admits closed-form, analytic inversion would make inversion of the Fox-Wright CDF trivial for many samples. Although there exists a vast literature on asymptotic expansions for such special functions, we have not found the desired expression that will serve this purpose.Finally, we remark that inverse-transform sampling is not the only method that applies here. The CMS (Chalmers-Mallows-Stuck) transform method <cit.> for sampling α-stable variables does not rely on the explicitcomputation of the PDF given by Equation (<ref>). The CMS method can be thought of the closest approach to the Box-Muller transform for stable procsses. However, as the developers of that method note,<cit.> it is not clear what method is best for very large-scale Monte Carlo simulations. This is an interesting point for future survey. Regardless, as shown below, sample generation using discrete inverse transform sampling is quite fast after the CDF has been initialized, generating 𝒪(10^4) samples per second on a single core. Accurate tables of the CDF can be saved, so initialization is not required for future calculations. Moreover since the discrete-inverse transform method does not rely on special transforms, it can be readily applied to an arbitrary distribution. This is the main advantage of inverse-transform sampling. For example, if it was desired to use tempered distributions<cit.> (see section <ref>) rather than pure α-stable distributions, it would only be required to swap the PDF subroutines. Thus, the implementation of the inverse-transform method is worthwhile for any future work on fractional PIMC.§.§Path Integral Monte Carlo Simulations Based Upon the Fractional Density Matrix §.§.§Performing PIMC Moves As described in greater detail elsewhere,<cit.> in Path Integral Monte Carlo, a random walk is performed through the space of N-particle paths at the M different time slices, {R⃗_1, R⃗_2,...,R⃗_M}. A random walk may be constructed by randomly displacing the positions of the different beads within the particles. While many different move possibilities exist,<cit.> one of the simplest moves is to displace a subset of the beads in the M-bead path of a given particle. Let the portion of a particle i's path to be updated be denoted by r⃗_i,k+1...r⃗_i,k+s-1, where s is the number of beads in the path to be displaced. Let X⃗={R⃗_0,R⃗_1,...,R⃗_M} denote the old set of particle bead coordinates and let X⃗^'={R⃗_0,...R⃗_k,R⃗_k+1^'...R⃗_k+s-1^',R⃗_k+s,...,R⃗_M} denote the updated (new) set of particle bead coordinates. Then, based upon Equation (<ref>), the probability, P, that this displacement will be accepted from sampling the partition function is P= ∏_j=0^s-1ρ_K(r⃗_k+j', r⃗_k+j+1', Δτ)/∏_j=0^s-1ρ_K(r⃗_k+j, r⃗_k+j+1, Δτ)× e^-Δτ∑_j=1^s-1 V(R⃗_k+j') / e^-Δτ∑_j=1^s-1 V(R⃗_k+j)T(X⃗' →X⃗)/ T(X⃗→X⃗^').Here, T(X⃗^'→X⃗) represents the transition probability, which can be selected with great freedom. The most convenient choice for the transition probability is to set it equal to the product of the kinetic density matrices, so that it can be evaluated using the expressions derived above and can cancel out other contributions to P. If this choice is made,T(X⃗→X⃗^') = ∏_j=0^s-1ρ_K(r⃗_k+j', r⃗_k+j+1', Δτ),and Equation (<ref>) reduces to P=e^-Δτ∑_j=1^s-1 (V(R⃗_k+j^')-V(R⃗_k+j)).These equations imply that, in a practical PIMC simulation, coordinates should first be sampled from the product of the kinetic density matrices (the length of that product depends upon the number of beads to be moved) and then the move should be accepted/rejected based upon the ratio of the potential energy in the new configuration to the potential energy in the old configuration.There are many ways to sample the transition probability and construct multiple bead moves.<cit.> One of the most common ways to propose multi-bead moves in the typical, α=2 case is to use the staging algorithm,<cit.> which samples a hierarchy of position moves based upon the fact that a product of Gaussians is a Gaussian. Because our fractional kinetic density matrices are not in general Gaussians and products of Fox-Wright functions do not yield Fox-Wright functions, the staging algorithm cannot be straightforwardly adapted to this formalism. For the purpose of this paper (we discuss alternatives capable of more efficiently sampling multi-bead and permutation moves may in our Conclusions), we therefore use single-bead and center of mass moves, despite the ergodicity problems single-bead moves may cause at low temperatures, particularly for strong interactions. For center of mass moves, all beads in each particle are moved at once, as usual. For single-bead moves, a bead on a particle is randomly selected. New unscaled x', y', and z' coordinates are then independently sampled from their Fox-Wright Distributions as described in Section <ref> and rescaled by σ^α = ħτ^1/α/2^1/α m^1/αto yield coordinates of the correct dimensions. As further detailed in Appendix I, the x, y, and z coordinates may be independently sampled because the many particle density matrices corresponding to the free fractional Hamiltonian may be re-expressed as the product of single particle density matrices in each dimension. The resulting potential energy from this move is lastly determined and used to accept/reject the proposed move. §.§.§Computing Observables The fundamental purpose of performing PIMC simulations in most cases is to obtain measures ofdifferent systems' finite temperature observables, such as their energies, radial distribution functions, and radii of gyration. Because such observables as the potential energy and radial distribution functions are diagonal in the position representation and therefore do not directly depend upon the fractional exponent, they can be measured as in the α=2 case. Appropriate care must simply be taken for α<2 cases because these averages are computed from distributions with heavy tails. Computing quantities based upon derivatives of the partition function, such as the kinetic energy, however, is much less straightforward. As derived in Appendix III, expressions dependent on derivatives of the partition function are proportional to slightly more complex Fox-Wright distributions. The kinetic energy, for example, may be written as ⟨K̂Ê⟩_link = ⟨3N/τα + 2 N/τα[-C_α(R⃗-R⃗'⃗)^2] .. [ ∑_n=0^∞Γ(1/α + (2/α) (n+1))/Γ(1/2+(n+1))[ -C_α(R⃗-R⃗'⃗)^2]^n/n! ]/ .. [ ∑_n=0^∞Γ(1/α + (2/α) n)/Γ(1/2+n)[ -C_α (R⃗-R⃗'⃗)^2]^n/n! ] ⟩_linkwhere C_α=2^2/αm^2/α/4 τ^2/αħ^2 and ⟨⟩_link denotes the average over links between beads. Note that the additional complexity results from the appearance of factors of n+1 as opposed to n in the gamma functions. As a result of this additional complexity, the series is less manageable. It is therefore more convenient to evaluate the thermodynamic estimator of the average kinetic energy based upon the formula⟨ KE ⟩ = m/β∂ln Z/∂ mand to use a three-point forward difference formula in m to estimate the derivative (see Section <ref>).§RESULTS AND DISCUSSION§.§Fractional Density Matrix Distributions In this section, we compute the Fox-Wright density as described above to analyze its functional form and the performance of our sampling technique. In Figure <ref>, we illustrate the overall behavior of the probability density function that represents the uni-dimensional fractional density matrix. The density exhibits “heavy” algebraic tails for fractional orderα<2. Compared to the normal case, the fractional cases exhibit a larger probability of both small and large jumps, and a lower probability for medium jumps. In Figure <ref>, we look more closely at the tail behavior. As α decreases, the related probability density functions exhibit heavier and heavier tails. In Figures <ref> and <ref>, we have also plotted the density ρ_1,1,1 for α = 2.25 as a curiosity. Of course, α-stable distributions are only defined for 0 < α≤ 2; this illustration shows that for α > 2, ρ_1,1,1 takes negative values and is therefore not a density at all. In particular, while α < 2 corresponds to superduffision, α > 2 does not correspond in any way to subdiffusion.This may be surprising at first, but it has the following heuristic explanation. Unlike superdiffusion, one cannot obtain subdiffusion simply by modifying the spatial jump density of Brownian motion. In superdiffusion, the normal jumps are replaced by “faster”, infinite variance jumps. If one attempts to replace the jumps of Brownian motion by “slower jumps,” such a process would necessarily have finite variance and thus, by the central limit theorem, reduce to Brownian motion. Therefore, it is not surprising that for α > 2, ρ_1,1,1 is meaningless as a density. The only way to obtain subdiffusion is to introduce waiting times into the process, which results in a time fractional derivative.<cit.> In Figure <ref>, we plot the same densities on a log-log scale to better understand their tail behavior. We can see from this that the density enters a power-law regimefairly early, for |x-x'| ∼ 10. Finally, in Figure <ref>, we illustrate the accuracy of the discrete inverse transform sampling method described in Section <ref>. This figure compares histograms representative of the probability density functions obtained for α=1.5 and α=2 obtained using our inversion algorithm. Note that our inversion method preserves the heavy tails that are central to these distributions.The numerical costs of our inversion process are presented in Table <ref>. We see how expensive it is to evaluate the density ρ: it takes about 1s to evaluate the density at 25 different points. On the other hand, for the grid used, it takes binary search 1s to generate 6000 samples. Moreover, since binary search is logarithmic in the number of discretization points, similar performance may be expected even for much finer grids. Therefore, if the CDF can be initialized to the desired accuracy, binary search is superior to Newton methods when a very large number of samples is required, i.e., orders of magnitude more than the number of discretization points. A Newton method would require several additional evaluations of ρ for each sample; on a single core, this would generate <10 samples per second, compared to the roughly thousands of samples per second for binary search.§.§Fractional Kinetic Energy Using the forward difference scheme described in Appendix III, we analyze the functional form of the kinetic energy as a function of the interbead distance squared for a range of α's. Interestingly, whereas the kinetic energy decreases linearly with interbead distance for α=2, it first decreases, then increases, and finally plateaus for smaller α, as depicted in Figure <ref>. Similar functional forms are observed for varying β in Figure <ref>. The plateaus observed in the kinetic energy may be understood by considering the power-law behavior of our Fox-Wright distribution for large argument. In general, the density matrix connecting two consecutive imaginary times, ρ_i(τ) may be approximated as (see Appendix III) ρ_i(τ) = (2^1/α m^1/α/√(π)τ^1/αħα)^3 N × [ ∑_n=0^∞Γ(1/α + 2/α n)/Γ(1/2+n)[ - ( R⃗_i-R⃗_i+1)^2 2^2/α m^2/α/4 τ^2/αħ^2]^n/n! ].Setting all parameters except m, α, and (R⃗_i-R⃗_i+1)^2 equal to 1 in the above, we can writeρ_i(τ) ∼ (m^1/α)^3N F_α(m^1/α√( (R⃗_i-R⃗_i+1)^2)),where F_α is the Fox-Wright distribution given by Equation (<ref>). We note that Equation (<ref>) has its argumentsquared in the Taylor series (see Equation (<ref>)), so to write ρ in terms of F_α we must take the square-root of the expression [ - ( R⃗_i-R⃗_i+1)^22^2/α m^2/α/4 τ^2/αħ^2]to use as an argument of F_α.From Figure <ref> above, we know that for large x,F_α (x) ∼ x^-P, for some P that depends on α. Thus,ρ_i(τ) ∼ (m^3N/α) (m^1/α√( (R⃗_i-R⃗_i+1)^2 ))^-P.As a result, dρ_i(τ)/dm∼ (m^3N/α-1) (m^1/α√( (R⃗_i-R⃗_i+1)^2))^-P-P (m^3N/α) (m^1/α√( (R⃗_i-R⃗_i+1)^2))^-P-1√( (R⃗_i-R⃗_i+1)^2)(m^1/α-1)and so m/τρ_i(τ)dρ_i(τ)/dm ∼ m^-1/τ -P m^1/α - 1/τ(m^1/α√( (R⃗_i-R⃗_i+1)^2))^-1 ×√( (R⃗_i-R⃗_i+1)^2)= m^-1/τ -P m^-1/τ (m^-1/α ) which is a constant value.Thus, ⟨ KE ⟩ will initially decay and then approach this value for large (R⃗_i-R⃗_i+1)^2. As may be expected on a conceptual basis and is confirmed in the above plots, the average interbead distance for which the kinetic energy reaches its minimum shifts to larger values with increasing α and β. This is because larger α imply more diffuse paths, while larger β correspond to lower temperatures and therefore larger de Broglie wave lengths. The presence of a minimum in the kinetic energy is a strictly fractional feature which appears for α infinitesimally smaller than two and suggests that fractional particles may exhibit intriguing behavior as they attempt to minimize their potential and kinetic energies – which have competing minima – at once.§.§Fractional Free Particle PIMC Results Before presenting results for ^4He, we first demonstrate our algorithm by simulating the free-particle Hamiltonian. Our interest in the free-particle Hamiltonian stems from the fact that it is amenable to analytic solutions and free-particle PIMC serves as the bedrock for interacting simulations. More specifically, we consider the Hamiltonian Ĥ_free,α = -ħ^α/(2m)^α/2∇^α = -ħ^α D_α∇^αfor α≤ 2 to avoid sampling from negative distributions. In order to make direct comparisons with our helium results presented in the next section, we set our mass equal to that of helium. In both sections, we moreover simulate N=64 particles at a density of ρ=.00323 bohr^-3 at all of the temperatures and α values presented.As a qualitative check on our simulations and underlying theory, we first consider the conformations of the polymers representing our particles. One of the principle motivations for this work was the fact that a fractional Laplacian for α<2 should lead to significantly more diffuse particles than in the typical α=2 case. As illustrated in Figure <ref>, this is because the α<2 fractional density matrix probability distributions possess heavy tails that should permit more frequent sampling of larger interbead distances, |R⃗_i-R⃗_i+1|. Figure <ref> demonstrates that this is indeed the case: as α decreases, the average square of the interbead distance, ⟨ (R⃗_i-R⃗_i+1)^2⟩, for fixed β and number of beads steadily increases. As may be expected, this increase is most dramatic at the low temperatures at which, in the absence of a potential, increased delocalization should be favored. Similar trends may be observed in the particles' average radii of gyration, R_G = ⟨ (R⃗_i - R⃗_cm)^2⟩, as depicted in Figure <ref>. Example paths illustrating these features for varying temperatures and fractional exponents are depicted in Figure <ref>. What can be gleaned from these plots is that measures of both the average interbead distance and the radius of gyration are accompanied by increasingly large error bars as α decreases. This can be expected as the variance of an α-stable law is infinite for 0 < α < 2, with the mean itself becoming infinite for α < 1 (we only consider the 1 < α < 2 case). Although we have not implemented this in our current code, these variances may be made finite by sampling tempered distributions <cit.> at the cost of approximating the distributions' tails.<cit.> Although related to the interbead distance, a more quantitative measure of our fractional paths is the average kinetic energy. In Figure <ref>, we plot the simulated PIMC kinetic energy vs. temperature for a range of fractional exponent values.As may be expected from the increased interbead distances, the average kinetic energy increases with decreasing α. The magnitudes of the kinetic energy may be rationalized by using the fractional version of the equipartition theorem: ⟨ KE ⟩ = 3 N k_B T/α, which can be readily derived from Equation (<ref>) in the high temperature limit. The larger kinetic energies for smaller α suggests that there is a comparative kinetic energy penalty for fractional cases. While not illustrated here, if the average kinetic energy obtained from PIMC is plotted against the the average interbead distance squared, functional forms possessing minima and plateaus similar to those depicted in Figures <ref> and <ref> are obtained.§.§Fractional He-4 PIMC Results Building upon our free particle simulations, we also simulate fractional ^4He to demonstrate our PIMC algorithm in the presence of interactions. ^4He is a particularly illuminating example because, even in the α=2 case, its particle paths become so diffuse at low temperatures that it can form a superfluid.<cit.> In the following, we model ^4He with the Aziz potential.<cit.> While the average interacting interbead distance remains similar to the non-interacting interbead distance, the average interacting radius of gyration seems to decrease significantly when compared to the non-interacting radius of gyration. This implies that the interacting particles are more localized (see Figure <ref>). This is also borne out in the decreased average kinetic energy depicted in Figure <ref>.Irrespective of the increased localization due to the potential, the overarching trend that delocalization is favored with decreasing α persists.Interestingly, for ^4He, we do not observe a pronounced competition between minimizing the kinetic and potential energies. This competition may be more pronounced, leading to more intriguing behavior, for interparticle potentials with deeper minima. §SUMMARY AND OUTLOOKIn this work, we have presented a methodology that generalizes the Path Integral Monte Carlo algorithm to fractional Hamiltonians, thereby enabling the computational exploration of fractional Hamiltonians with potentials that are not readily amenable to analytical treatments. We have derived and shown how to sample fractional free particle density matrices by developing a novel approach to sampling Fox-Wright functions. This overall approach may be generalized beyond fractional path integrals to the sampling of density matrices of non-Gaussian forms. We have furthermore employed our algorithm to explore the impact of a fractional Laplacian on free-particle and ^4He path observables, such as the radius of gyration and the average interbead distance. We have demonstrated that the fractional kinetic energy possesses a pronounced minimum as a function of average interbead distance for fractional exponents less than two (α<2) and that this minimum shifts to larger interbead distances with decreasing fractional exponents. As such, fractional particles become increasingly more diffuse with decreasing fractional exponents as a consequence of sampling a fractional density matrix distribution with heavier tails than the usual Gaussian distribution. Because the onset of phenomena related to condensation heavily depends on the diffusivity of particle paths, our findings suggest that fractional Hamiltonians may manifest intriguing superfluid and Wigner crystallization phase transitions. Such transitions have only previously been studied for the fractional Schrödinger Equation based upon mean field theories.<cit.> Fractional Path Integral Monte Carlo will enable a more complete understanding of such phenomena. Before the effects of fractional operators on condensation phenomena may be explored, however, several complications revealed by our work must be resolved. First and foremost, because the conventional staging algorithm<cit.> cannot readily be applied to non-Gaussian distributions, we sampled our partition functions using highly inefficient single bead moves in this work. While, for the potentials explored, we verified that our simulations were able to converge to their equilibrium distributions, we would not expect this to be the case for more general potentials. It is furthermore easy to imagine that single bead moves would be suboptimal for sampling permutations among particles with quantum statistics. Future work therefore necessitates the development of generalized multi-bead moves. Fourier path sampling may permit the generality we require.<cit.>As Figures <ref> and <ref> illustrate, the density matrix distributions sampled for x-x' in our algorithm are furthermore significantly more complex than the Gaussian distribution sampled conventionally. For α<2, these distributions possess heavy tails, while for α>2, these distributions can be negative. The discrete inverse-transform sampling method we presented can accurately sample the heavy tails, but there remain theoretical questions about the effect of heavy tails in the theory. Namely, whether the infinite variance (for α<2) and infinite mean (for α≤1) of the fractional density matrix requires the use of other notions of central tendency (such as median) for observables. This is a fascinating question for future study. In this direction, it may be fruitful to explore “tempered-fractional quantum mechanics,” which has not been studied to our knowledge. This would conceivably employ tempered Levy distributions instead of pure heavy-tailed distributions, and temperated fractional derivatives instead of standard fractional derivatives.<cit.> Tempered distributions enjoy the heavy tail to a certain argument, but then decay exponentially to avoid issues with infinite variance.It may also be of interest to consider the physics of the time-fractional Schrodinger Equation,<cit.> and even the space-and-time-fractional Schrodinger Equation.<cit.> These equations introduce waiting times in addition to long jumps. However, unlike the space-fractional equation, the properties of the time-fractional Schrodinger equation suggest that it cannot be used as a probabalistic theory of Quantum Mechanics.<cit.> Thus, more work is needed before any implementation of a time-fractional model is justified. In summary, we have proposed a novel fractional Path Integral Monte Carlo algorithm. Our algorithm provides a clear numerical path toward unraveling the fractional physics of interacting systems, much of which could only be approximated analytically in the past. We look forward to employing this algorithm to explore where our fully interacting predictions differ from previous approximate analytical solutions and to the study of such novel phenomena as fractional superfluidity. The authors would like to dedicate this work to the late Jimmie Doll. Jimmie initially convened the authors to brainstorm about this topic, but unfortunately, did not live to see this work to its fruition. The authors would also like to thank George Karniadakis, who suggested to M.G. that he explore this topic and contact Jimmie. Finally, we thank Paul Dupuis, Matthew Harrison, and Mark Meerschaert for valuable discussions. M. G. would like to acknowledge support from the OSD/ARO/MURI grant “Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics and Applications (W911NF-15-1-0562).” Part of this research was conducted using computational resources and services at the Center for Computation and Visualization, Brown University. §APPENDIX I: SEPARABILITY OF THE FRACTIONAL DENSITY MATRIX IN MULTIPLE DIMENSIONS In this work, we have derived and demonstrated how to sample the one-dimensional form of the free fractional density matrix. Here, we reaffirm that, just as in the α = 2 case, a multi-dimensional free density matrix is simply a product of one-dimensional free density matrices. Recalling Equation (<ref>), the free multi-dimensional density matrix may be written as ρ(R⃗, R⃗^'; τ)= ⟨R⃗| e^-τK̂ | R⃗^'⟩= ∑_jϕ_k,j^*(R⃗) ϕ_k,j(R⃗^') e^-τ E_k,j,where ϕ_k,j(R⃗) denote the eigenvectors and E_k,j denote the eigenvalues of the Hamiltonian, which in this case, reduces to the kinetic operator. In Equation (<ref>), k denotes that these are kinetic eigenvectors/eigenvalues, while j represents the eigenvector/eigenvalue index. Assuming particle masses are the same, the many body, multi-dimensional Laplacian in Equation (<ref>) may be expanded intoK̂= - λ∇^α = -λ∑_i^N( ∇_i,x^α + ∇_i,y^α + ∇_i,z^α),where i denotes the particle number. Since the differential operators act on different particles in orthogonal directions, they commute with one another. Moreover, because the Hamiltonian is non-interacting, the many body wave function is separable |R⃗⟩ = |x_1⟩⊗ | y_1⟩⊗ | z_1⟩⊗ ...⊗ |x_N⟩⊗ | y_N⟩⊗ | z_N⟩.This implies thatρ(R⃗, R⃗^'; τ) = ⟨R⃗| e^-τK̂ | R⃗^'⟩= ∏_i^N⟨ x_i | e^τλ∇_i,x^α | x_i' ⟩⟨ y_i | e^τλ∇_i,y^α | y_i' ⟩⟨ z_i | e^τλ∇_i,z^α | z_i' ⟩= ∏_i^Nρ(x_i, x_i'; τ)ρ(y_i, y_i'; τ) ρ(z_i, z_i'; τ),where ρ(x_i, x_i', τ) denotes a uni-dimensional density matrix of the same form as given by Equations (<ref>) and (<ref>). Sampling the many body, multi-dimensional density matrix thus reduces to independently sampling the one-dimensional density matrices for all x_i, y_i, and z_i coordinates.§APPENDIX II: SUZUKI-TROTTER FACTORIZATION USING FRACTIONAL LAPLACIANSIn order to use the Path Integral Monte Carlo algorithm for the fractional case, it is essential to first demonstrate that the fractional density matrices obey the density matrix convolution principle given by Equation (<ref>). Starting with the one-dimensional expression for the kinetic density matrix for simplicity, ρ^per,L(x,x',β) = 1/2π∫_-∞^∞ dC e^iC(x-x')e^-β D_σħ^α |C|^αthis factorization may be verified via substitution ρ^per,L(x,x',β) = ∫_-∞^∞ dx”ρ^per,L(x,x”,β/2) ρ^per,L(x”,x',β/2) = 1/2π∫_-∞^∞ dx”∫_-∞^∞ dC e^iC(x-x”) e^-(β/2) D_σħ^α|C|^α 1/2π∫_-∞^∞ dC' e^iC'(x”-x') e^-(β/2) D_σħ^α |C'|^α= 1/4π^2∫_-∞^∞∫_-∞^∞ dC dC' e^iCx e^-iCx' [∫_-∞^∞ dx” e^ix”(C'-C)] e^-(β/2) D_σħ^α |C|^α e^-(β/2) D_σħ^α |C'|^α= 1/4π^2∫_-∞^∞∫_-∞^∞dC dC' e^iCx e^-iCx'[2 πδ(C-C') ]e^-(β/2) D_σħ^α |C|^αe^-(β/2) D_σħ^α |C'|^α= 1/2π∫_-∞^∞ dC e^iC(x-x') e^-(β/2) D_σħ^α |C|^α e^-β/2 D_σħ^α |C|^α= 1/2π∫_-∞^∞ dC e^iC(x-x') e^-β D_σħ^α |C|^α= ρ^per,L(x,x'),where the substitution e^ix”(C'-C) = 2πδ(C'-C) was made. This demonstrates that the fractional kinetic density matrix at a large imaginary time, β, may be factored into a convolution over short imaginary time density matrices.In order to factor the full short time propagators into kinetic and potential propagators, it must moreover be verified that Suzuki-Trotter factorization<cit.> can be performed on the fractional Laplacian. As discussed by Simon,<cit.> Suzuki-Trotter factorization is valid as long as the kinetic and potential operators are self-adjoint<cit.> and bounded from below, which holds by the definition of the (negative) fractional Laplacian in ℝ^n as the power, in the sense of the spectral theory, of the Laplacian. This results in a self-adjoint operator<cit.> with positive eigenvalues.<cit.> Because the density matrix convolution property and the Trotter product formula hold, we can factor for α<2 cases, as usual.§APPENDIX III: DERIVATION OF THE EXPRESSION FOR THE FRACTIONAL KINETIC AND TOTAL ENERGIES The thermodynamic estimator for the kinetic energy is the mass derivative of the partition function ⟨K̂⟩ = m/β∂ Z/∂ m.Following Ceperley,<cit.> all R⃗_i-R⃗_i+1 links may be viewed as equivalent and we may therefore take the derivative of the single link density matrix, as opposed to the full partition function, and average over all links. Neglecting the potential contribution to the Hamiltonian for now, let ρ_i(τ) ≡ρ(R⃗_i, R⃗_i+1; τ)= (2^1/α m^1/α/√(π)τ^1/αħα)^3 N[ ∑_n=0^∞Γ(1/α + 2/α n)/Γ(1/2+n)[ - ( R⃗_i-R⃗_i+1)^22^2/α m^2/α/4 τ^2/αħ^2]^n/n! ].Here, it should be noted that the R⃗_i and R⃗_i+1 beads are meant to be adjacent and the corresponding vectors will be replaced by R⃗ and R⃗' in what follows. ⟨ KE ⟩_link = ⟨m/τρ_i(τ)∂ρ_i(τ)/∂ m⟩_link= ⟨m/τρ_i(τ)3 N/α m( 2^1/α m^1/α/√(π)τ^1/αħα)^3N[∑_n=0^∞Γ(1/α + 2/α n)/Γ(1/2 + n)[ -(R⃗-R⃗'⃗)^22^2/α m^2/α/4 τ^2/αħ^2]^n/n! ] ⟩_link+ ⟨m/τρ_i(τ)(2^1/α m^1/α/√(π)τ^1/αħα)^3 N[ ∑_n=1^∞Γ(1/α + 2/α n)/Γ(1/2+n)2n/α m[ - (R⃗-R⃗'⃗)^2 2^2/α m^2/α/4 τ^2/αħ^2]^n/n! ] ⟩_link= ⟨m/τ3N/α m⟩_link+ ⟨m/τ2/α m[ ∑_n=1^∞Γ(1/α + 2/α n)/Γ(1/2+n) n [ -(R⃗-R⃗'⃗)^2 2^2/α m^2/α/4 τ^2/αħ^2]^n/n! ]/ [ ∑_n=0^∞Γ(1/α + 2/α n)/Γ(1/2+n)[ - (R⃗-R⃗'⃗)^22^2/α m^2/α/4 τ^2/αħ^2]^n/n! ] ⟩_link= ⟨m/τ3N/α m⟩_link + ⟨m/τ2/α m[ ∑_n=0^∞Γ(1/α + 2/α (n+1))/Γ(1/2+(n))[ - (R⃗-R⃗'⃗)^22^2/α m^2/α/4 τ^2/αħ^2]^n+1/n! ]/ .. [ ∑_n=0^∞Γ(1/α + 2/α n)/Γ(1/2+n)[ - (R⃗-R⃗'⃗)^22^2/α m^2/α/4 τ^2/αħ^2]^n/n! ] ⟩_link= ⟨m/τ3N/α m⟩_link⟨m/τ2/α m[- (R⃗-R⃗'⃗)^22^2/α m^2/α/4 τ^2/αħ^2] [ ∑_n=0^∞Γ(1/α + 2/α (n+1))/Γ(1/2+(n+1))[ - (R⃗-R⃗'⃗)^22^2/α m^2/α/4 τ^2/αħ^2]^n/n! ]/ .. [ ∑_n=0^∞Γ(1/α + 2/α n)/Γ(1/2+n)[ - (R⃗-R⃗'⃗)^22^2/α m^2/α/4 τ^2/αħ^2]^n/n! ] ⟩_link= ⟨3N/τα⟩_link⟨2/ατ[- (R⃗-R⃗'⃗)^2 2^2/α m^2/α/4 τ^2/αħ^2][ ∑_n=0^∞Γ(1/α + 2/α (n+1))/Γ(1/2+(n+1))[ - (R⃗-R⃗'⃗)^22^2/α m^2/α/4 τ^2/αħ^2]^n/n! ]/ .. [ ∑_n=0^∞Γ(1/α + 2/α n)/Γ(1/2+n)[ - (R⃗-R⃗'⃗)^22^2/α m^2/α/4 τ^2/αħ^2]^n/n! ] ⟩_link In the above, the average ⟨⟩_link denotes an average over all links within the polymers. As a check, this simplifies to the usual expression for the kinetic energy upon substitution of α=2⟨ KE ⟩_link = 3 N/2 τ -m ⟨ (R⃗ - R⃗')^2⟩_link/2 τ^2ħ^2. Because Equation (<ref>) contains a factor of n+1 in its gamma functions, this makes its related series more difficult to sum to convergence. We have likewise resorted to using a three-point forward difference formula in which each density matrix is evaluated at multiple values of the mass. When the density matrices become exceedingly large, it is often more numerically tractable to make the replacementm/τρ_i(τ)∂ρ_i(τ)/∂ m = m/τ∂lnρ_i(τ)/∂ m, and to therefore take the derivative of the lnof the partition function instead. | http://arxiv.org/abs/1709.09089v1 | {
"authors": [
"Mamikon Gulian",
"Haobo Yang",
"Brenda M. Rubenstein"
],
"categories": [
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170926152804",
"title": "Fractional Path Integral Monte Carlo"
} |
Transversal switching between generic stabilizer codes Michael Newman^2, December 30, 2023 ====================================================== emptyAn efficient particle Markov chain Monte Carlo methodology is proposed for the rolling-window estimation of state space models. The particles are updated to approximate the long sequence of posterior distributions as we move the estimation window.To overcome the well-known weight degeneracy problem that causes the poor approximation, we introduce a practical double-block sampler with the conditional sequential Monte Carlo update where we choose one lineage from multiple candidates for the set of current state variables. Our proposed sampler is justified in the augmented space through theoretical discussions. In the illustrative examples, it is shown to be successful to accurately estimate the posterior distributions of the model parameters.Keywords: Double-block sampler; Forward and backward sampling; Importance sampling; Particle Gibbs; Particle Markov chain Monte Carlo; Particle simulation smoother; Rolling-window estimation; Sequential Monte Carlo; State space model; Structural change§ INTRODUCTIONState space models have been popular and widely used in the analysis of economic and financial time series. These models are flexible and capture the dynamics of the complex economic structure. However, several structural changes have been noted in long-term economic series. If the precise time of a structural change is known, we could divide the sample period into two periods, before and after the structural change. However, this time point is usually unknown, and the change may occur gradually from one state to another.Although there are various statistical models for the structural change in the literature, the rolling-window estimation is the simple and common way to reflect the recent change in the forecasting without delay where we fix the number of observations to estimate model parameters and update the dataset to improve the forecasting performance.In non-linear or non-Gaussian state space models, the likelihood is often not obtained analytically, and the maximum likelihood estimation is difficult to implement. The Markov Chain Monte Carlo (MCMC) method is a popular and powerful technique used to estimate model parameters and state variables by generating random samples from the posterior distribution given a set of observed data for various complex state space models. However, for rolling estimation, simply applying the MCMC method would be too time-consuming given the need to estimate a long sequence of posterior distributions.To overcome this difficulty, we take an alternative approach based on the sequential Monte Carlo (SMC) sampler discussed in DelMoral2006.This is effective because, in the rolling-window estimation, we can utilize the weighted samples from one posterior distribution to approximate the next posterior distribution instead of reiterating the same MCMC algorithm with the slightly different dataset. The particles consist of realized values of state variables and static parameters, which are updated when including a new observation and excluding the old observation. As we shall show in the illustrative examples of Section 4, a simple rolling-window sampler that is derived in a straightforward manner from the previous literature leads to the severe weight degeneracy problem, suggesting that the updating step should be constructed carefully. To fix this problem, we adopt the idea of block sampling (e.g. Doucet2006, Polson2008), in which state variables at multiple time points are updated simultaneously when learning new information. It is highly efficient in the sense that it substantially increases the effective sample size. Based on this idea, we propose the novel sampling method, called the double-block sampler, where we sample a block of state variables when both including and excluding the information.However, unless the time series model has a relatively simple form, finding an appropriate proposal distribution for these update steps may be difficult. Hence, instead of generating only one candidate from the proposal distribution, we generate multiple candidates and choose one of them using the conditional SMC of the particle MCMC (AndrieuDoucetHolenstein(10)). This nested structure is similar to that of SMC^2 (Chopin2013, Fulop2013) and nested SMC naesseth2015nested, but our proposed algorithm differs in that it is derived from the particle Gibbs instead of the particle MH (Metropolis-Hastings) algorithm.As a special case of our new method, our proposed double-block sampler can be used to implement the ordinary sequential analysis by keeping all past observations. It contrasts with SMC^2 in that it originates from different types of the particle MCMC algorithms.The remainder of the paper is organized as follows. In Section 2, we introduce the simple rolling-window sampler for state space models and point out that such a sampler derived from the conventional filtering algorithm causes the serious weight degeneracy phenomenon. Section 3 introduces a double-block sampler toovercome this difficulty.Section 4 provides illustrative examples and, in Section 5, theoretical justifications of the proposed method are provided. Section 6 concludes the paper. § PARTICLE ROLLING MCMC IN GENERAL STATE SPACE MODELS §.§ Rolling-window estimation in general state space model Consider the state space model which consists of a measurement equation, a state equation with an observation vector y_t, and an unobserved state vector x_t given a static parameter vector θ. For the prior distribution of θ, we let p(θ) denote its prior probability density function.Further define x_s:t≡ (x_s,x_s+1,…,x_t) and y_s:t≡ (y_s,y_s+1,…,y_t). We assume that the distribution of y_t given (y_1:t-1, x_1:t, θ) depends exclusively on x_t and θ and that the distribution of x_t given (x_1:t-1, θ) depends only on x_t-1 and θ. The corresponding probability density functions are noted as follows: p(y_t | x_1:t, y_1:t-1, θ)=p(y_t | x_t,θ) ≡ g_θ(y_t | x_t),t=1,…,n,p(x_t | x_1:t-1,θ)=p(x_t | x_t-1,θ) ≡ f_θ(x_t | x_t-1),t=2,…,n, where p(x_1 |θ) ≡μ_θ(x_1) denotes a known density function of the stationary distribution given θ.We also incorporate the correlation between y_t and x_t+1, which is conditional on x_t since we consider such an example, the realized stochastic volatility (RSV) model,for the financial time series (see e.g. a seminal work by Takahashi2009) in our illustrative example. It is a stochastic volatility model with an additional measurement equation for the realized volatility. Let y_t=(y_1,t,y_2,t)' where y_1,t and y_2,t denote the daily log return and the logarithm of the realized volatility (variance) at time t. Let x_t denote the latent log volatility which is assumed to follow the stationary AR(1) process. The RSV model is defined as follows: y_1,t = exp(x_t / 2)ϵ_t, ϵ_t ∼𝒩(0,1), t = 1,…,T y_2,t =x_t + ξ + u_t, u_t ∼𝒩(0, σ_u^2), t = 1,…,T x_t+1 = μ + ϕ(x_t - μ) +η_t, η_t ∼𝒩(0,σ^2_η), t = 1,…,T,x_1= μ + 1/√(1-ϕ^2)η_0, η_0 ∼𝒩(0,σ^2_η),|ϕ|<1,where([ ϵ_t; u_t; η_t ]) ∼𝒩( [[ 0; 0; 0 ]], [[ 1 0ρσ_η; 0 σ^2_u 0;ρσ_η 0 σ^2_η ]] ),𝒩(μ,Σ) denotes a normal distribution with mean μ and covariance matrix Σ,and θ=(μ,ϕ,σ^2_η,ξ,σ^2_u,ρ)' is the static parameter vector. The correlation ρ between ϵ_t and η_t is introduced to express the leverage effect. The effect is often negative in empirical studies, which implies that the decrease in the today's log return is followed by the increase in the log volatility on the next day (e.g. Omori2007).In this case, we express the dependence of y_t on x_t+1 (or x_t+1 on y_t) as follows. p(y_t | x_1:t+1,y_1:t-1,θ)=p(y_t | x_t,x_t+1,θ)≡ g_θ(y_t | x_t, x_t+1),t=1,…,n,p(x_t+1| x_1:t,y_1:t, θ)=p(x_t+1| x_t, y_t,θ)≡ f_θ(x_t+1| x_t,y_t),t=1,…,n-1.In the rolling-window estimation of time series, the number of observations (or the window size) in the sample period is fixed and is set equal to, e.g., L+1. We estimate the posterior distribution of θ and x_s:t given the observations y_s:t with t=s+L for s=1,2…, and its probability density function is given by π(x_s:t,θ| y_s:t)∝p(θ)μ_θ(x_s)g_θ(y_s| x_s){∏^t_j=s+1 f_θ(x_j | x_j-1,y_j-1)g_θ(y_j | x_j) },or, equivalently,π(x_s:t,θ| y_s:t) ∝p(θ)μ_θ(x_s){∏^t_j=s+1 f_θ(x_j | x_j-1)g_θ(y_j-1| x_j-1,x_j) }g_θ(y_t | x_t). §.§ Simple rolling-window sampler We first describe a simple rolling-window sampler that is derived in a straightforward manner from the previous literature.The estimation procedure consists of two steps, each of which can be described in the framework of the SMC sampler in DelMoral2006 as follows. In Step 1, suppose we have samples from the old target density π(x_s-1:t-1,θ| y_s-1:t-1) with importance weight W_[s-1,t-1] at time t-1where the subscript [s-1,t-1] implies that the weight is based on observations y_s-1:t-1. After we include an observation y_t,our new target density isπ(x_s-1:t,θ| y_s-1:t). Using the proposal kernel K((x_s-1:t-1,θ),(x_s-1:t,θ)), we update the weightW_[s-1,t] =π(x_s-1:t, θ| y_s-1:t)L((x_s-1:t, θ),(x_s-1:t-1, θ))/π(x_s-1:t-1, θ| y_s-1:t-1)K((x_s-1:t-1, θ),(x_s-1:t, θ))× W_[s-1,t-1],where L((x_s-1:t, θ),(x_s-1:t-1, θ)) is the artificial backward Markov kernel with L≡ 1 in this step.The first factor on the right hand side of Equation (<ref>) is the incremental weight to adjust that of the previous step. In Step 2, we have samples from the old target density π(x_s-1:t, θ| y_s-1:t) with the importance weight W_[s-1,t] from Step 1. After we exclude the observation y_s-1, our new target density is π(x_s-1:t, θ| y_s:t). Using the backward kernel L((x_s:t, θ),(x_s-1:t, θ))=π(x_s-1:t, θ| y_s:t)/π(x_s:t, θ| y_s:t)=π(x_s-1| x_s:t, y_s:t, θ), we update the weightW_[s,t] =π(x_s:t, θ| y_s:t)L((x_s:t, θ),(x_s-1:t, θ))/π(x_s-1:t, θ| y_s-1:t)K((x_s-1:t, θ),(x_s-1:t, θ))× W_[s-1,t],where K((x_s-1:t, θ),(x_s-1:t, θ)) is the artificial proposal kernel with K≡ 1, and discard x_s-1.Additionally, we can refresh all the particles with the MCMC when we observe the weight degeneracy, which is also regarded as an importance sampling step in the SMC sampler with the unnormalized weight equal to one. We note that one can also update particles using the particle Gibbs sampler (AndrieuDoucetHolenstein(10)). Details are given below. Step 1.Assume that, at time t-1, we have a collection of particles (x_s-1:t-1^n,θ^n) with the importance weight W_[s-1,t-1]^n, (n=1,…,N) which is a discrete approximation of π(x_s-1:t-1, θ| y_s-1:t-1).We include a new observation y_t in the information set and aim to sample from π(x_s-1:t, θ| y_s-1:t). Given the current sample (x_s-1:t-1, θ) from π(x_s-1:t-1, θ| y_s-1:t-1), we propose a candidate x_t using some proposal density q_t,θ(x_t | x_t-1,y_t). Since the incremental weight is π(x_s-1:t, θ| y_s-1:t)/π(x_s-1:t-1, θ| y_s-1:t-1)q_t,θ(x_t | x_t-1,y_t) = p(x_t, y_t| x_s-1:t-1, y_s-1:t-1,θ)/q_t,θ(x_t | x_t-1,y_t) p(y_t| y_s-1:t-1)∝ f_θ(x_t | x_t-1, y_t-1)g_θ(y_t | x_t)/q_t,θ(x_t | x_t-1,y_t),we generate x^n_t ∼ q_t,θ^n(x^n_t| x^n_t-1,y_t) and compute the importance weightW_[s-1,t]^n ∝ f_θ^n(x_t^n | x_t-1^n, y_t-1)g_θ^n(y_t | x_t^n)/q_t,θ^n(x_t^n | x_t-1^n,y_t)× W_[s-1,t-1]^n.Finally, we compute some degeneracy criteria such as the effective sample size (ESS), ESS_[s-1:t]≡[∑^N_n=1{W^n_[s-1,t]}^2]^-1,and the particles are resampled if ESS <cN (e.g. c = 0.5). Step 2. We exclude the old observation y_s-1 from the information set, and aim to sample from π(x_s:t,θ| y_s:t) where the backward kernel is π(x_s-1:t,θ| y_s:t)/π(x_s:t,θ| y_s:t)=π(x_s-1| x_s:t,y_s:t, θ)=p(x_s-1|x_s, θ) given byp(x_s-1| x_s, θ)= μ_θ(x_s-1) f_θ(x_s | x_s-1 )/μ_θ(x_s).Since the current sample is from π(x_s-1:t, θ| y_s-1:t) whereπ(x_s-1:t, θ| y_s-1:t) = p(x_s-1:t,y_s-1,θ| y_s:t)/p(y_s-1| y_s:t) =π(x_s-1:t, θ| y_s:t)p(y_s-1| x_s-1:t,y_s:t,θ)/p(y_s-1| y_s:t), the (unnormalized) incremental weight isπ(x_s:t, θ| y_s:t)p(x_s-1|x_s, θ)/π(x_s-1:t, θ| y_s-1:t) = p(y_s-1| y_s:t)/p(y_s-1|x_s-1:t,y_s:t,θ)∝ g_θ(y_s-1| x_s-1,x_s)^-1, Thus, we update the importance weightW_[s,t]^n∝ g_θ^n(y_s-1| x_s-1^n,x_s^n)^-1× W_[s-1,t]^n,and discard x_s-1^n.If some degeneracy criteria are fulfilled, resample all the particles by implementing the MCMC algorithm as in Step 1.The above procedure is summarized in Algorithm 1.§.§ Weight degeneracy problem Using the importance weight (<ref>) in Step 2 is obviously problematic because it would take an extremely high value when g_θ is close to 0. This causes the ESS to rapidly drop and triggers the MCMC update steps many times, which makes the estimation time-consuming. Further, in Step 1, one might think it will work without any problem as long as we choose an appropriate proposal distribution q_t,θ. However, as we shall see in illustrative examples in Section <ref>, this step also causes a serious degeneracy problem. In Section <ref>, we overcome this difficulty of the weight degeneracy by proposing a novel sampling method called “a double-block sampler”with the conditional SMC update. § PARTICLE ROLLING MCMC WITH DOUBLE-BLOCK SAMPLER We consider sampling a block of state variables when we add the new observation or remove the old observation. For example, we update values of {x^n_t-K:t-1}_n=1^N in addition to generating {x^n_t}_n=1^N when we learn the information of y_t. We call this process the forward block sampling (Step 1), and the backward block sampling (Step 2) can also be defined in a similar manner. The double-block sampler addresses the weight degeneracy problem by reducing the path dependence between the new particle and the old particle values that are not updated.§.§ Idealised double-block sampler We first consider the `idealised' double-block sampler where we assume an appropriate K+1 dimensional proposal distribution is available for the block sampling. In the framework of the SMC sampler, in Step 1, we have samples from the old target density π(x_s-1:t-1, θ| y_s-1:t-1) and generate x^†_t-K:t fromthe proposal kernel π(x^†_t-K:t|x_s-1:t-K-1,y_s-1:t,θ). The new target density is π(x_s-1:t-K-1, x^†_t-K:t, θ| y_s-1:t) with the backward kernel π(x_t-K:t-1| x_s-1:t-K-1, y_s-1:t-1,θ). In Step 2, we have samples fromthe old target density π(x_s-1:t, θ| y_s-1:t) and generate a candidate x_s:s+K-1^† using the proposal kernel π(x_s:s+K-1^†|x_s+K:t,y_s:t,θ). Our new target density is π( x_s:s+K-1^†, x_s+K:t,θ| y_s:t) with the backward kernel π(x_s-1:s+K-1| x_s+K:t,y_s-1:t,θ). Finally, we discard x_s-1. Details are given below.Step 1.We include a new observation y_t in the information set, and sample fromπ(x_s-1:t, θ| y_s-1:t). Given the current sample (x_s-1:t-K-1, θ) from π(x_s-1:t-K-1, θ| y_s-1:t-1), we generate x_t-K:t^†∼π(x_t-K:t^†| x_s-1:t-K-1,y_s-1:t,θ).Since π(x_s-1:t-K-1, x_t-K:t^†,θ|y_s-1:t) = p(x_s-1:t-K-1, x_t-K:t^†, y_t, θ|y_s-1:t-1)/p(y_t|y_s-1:t-1)= π(x_s-1:t-K-1, θ|y_s-1:t-1)×p(x_t-K:t^†,y_t|x_s-1:t-K-1,y_s-1:t-1,θ)/p(y_t|y_s-1:t-1)= π(x_s-1:t-K-1, θ|y_s-1:t-1)π(x_t-K:t^†|x_s-1:t-K-1,y_s-1:t,θ)×p(y_t|x_s-1:t-K-1,y_s-1:t-1,θ)/p(y_t|y_s-1:t-1)the unnormalized incremental weight isπ(x_s-1:t-K-1, x_t-K:t^†,θ|y_s-1:t)π(x_t-K:t-1|x_s-1:t-K-1,y_s-1:t-1,θ)/π(x_s-1:t-1, θ|y_s-1:t-1)π(x_t-K:t^†|x_s-1:t-K-1,y_s-1:t,θ) =p(y_t|x_s-1:t-K-1,y_s-1:t-1,θ)/p(y_t|y_s-1:t-1). Thus we let x_s-1:t=(x_s-1:t-K-1,x_t-K:t^†) and update the importance weight asW_[s-1,t]^n ∝p(y_t|x_t-K-1^n,y_s-1:t-1,θ^n) × W_[s-1,t-1]^n,noting that p(y_t|x_s-1:t-K-1,y_s-1:t-1,θ)=p(y_t|x_t-K-1,y_s-1:t-1,θ). Step 2.We remove the old observation y_s-1 from the information set, and sample from π(x_s-1:t, θ| y_s:t). Given the current sample from π(x_s-1:t, θ|y_s-1:t), we generate x_s:s+K-1^†∼π(x_s:s+K-1^†|x_s+K:t,y_s:t,θ).The unnormalized incremental weight isπ(x_s:s+K-1^†, x_s+K:t,θ| y_s:t)π(x_s-1:s+K-1| x_s+K:t,y_s-1:t,θ)/π(x_s-1:t, θ|y_s-1:t)π(x_s:s+K-1^†|x_s+K:t,y_s:t,θ) = p(y_s-1|y_s:t)/p(y_s-1|x_s+K:t,y_s:t,θ),since π(x_s:s+K-1^†, x_s+K:t,θ| y_s:t) = π(x_s:s+K-1^†|x_s+K:t,y_s:t,θ)π(x_s+K:t,θ| y_s:t)andπ(x_s-1:t, θ|y_s-1:t) = π(x_s-1:s+K-1|x_s+K:t,y_s-1:t,θ)π(x_s+K:t, θ|y_s-1:t) = π(x_s-1:s+K-1|x_s+K:t,y_s-1:t,θ) ×p(x_s+K:t, y_s-1, θ|y_s:t)/p(y_s-1|y_s:t)= π(x_s-1:s+K-1|x_s+K:t,y_s-1:t,θ)π(x_s+K:t,θ|y_s:t)×p(y_s-1|x_s+K:t,y_s:t,θ)/p(y_s-1|y_s:t). Noting that p(y_s-1| x_s+K:t,y_s:t,θ)=p(y_s-1| x_s+K,y_s:t,θ), we set x_s:t=(x_s:s+K-1^†,x_s+K:t) and update the importance weightW_[s,t]^n ∝p(y_s-1|x_s+K^n,y_s:t,θ^n)^-1× W_[s-1,t]^n,and discard x_s-1^n. The sampling algorithm is summarized inAlgorithm 2.§.§ Practical double-block samplerIn practice, it is often difficult to find an `idealised' proposal distribution for the block sampling. Hence, we adopt the approach of the conditional SMC update for the particle Gibbs sampler(AndrieuDoucetHolenstein(10)), which considers the artificial target density ad generates a cloud of values for one particle path. In Step 1, we have samples fromthe old target density π(x_s-1:t-1, θ| y_s-1:t-1), and generate the indices k_t-K:t-1=(k_t-K,…,k_t-1) and a cloud of particles from the proposal kernelψ_θ defined in (<ref>). The new target density is π(x_s-1:t-K-1, x^†_t-K:t, θ| y_s-1:t) with the backward kernel π̂/π(x_s-1:t-K-1, x^†_t-K:t, θ| y_s-1:t) where π̂ is defined in (<ref>)[The marginal density of π̂ is π(x_s-1:t-K-1, x^†_t-K:t, θ| y_s-1:t) as shown in Proposition <ref> with x^†_t-K:t=(x_t-K^k_t-K^*,…,x_t^k_t^*).]. We set x_s-1:t=(x_s-1:t-K-1, x^†_t-K:t) which is the sample from the new target density with the unnormalized incremental weightp̂(y_t | x_t-K-1^n,y_s-1:t-1,θ^n) in (<ref>). In Step 2, we have samples fromthe old target density π(x_s-1:t, θ| y_s-1:t), and generate the indices k_s-1:s+K-1 and a cloud of particles from the proposal kernelψ̅_θ defined in (<ref>). The new target density is π(x_s:s+K-1^†, x_s+K:t, θ| y_s:t) with the backward kernel π̌/π(x_s:s+K-1^†, x_s+K:t, θ| y_s:t) where π̌ defined in (<ref>)[The marginal density of π̌ is π(x_s:s+K-1^†, x_s+K:t, θ| y_s:t) as shown in Proposition <ref> with x^†_s:s+K-1=(x_s^k_s^*,…,x_s+K-1^k_s+K-1^*).]. We set x_s:t=(x_s:s+K-1^†, x_s+K:t) which is the sample from the new target density with the unnormalized incremental weightp̂(y_s-1| x_s+K^n,y_s:t,θ^n)^-1 in (<ref>).Details are given below.§.§.§ Forward block sampling (Step 1)We first generate a number of candidates x_t-K:t-1^n,m (m=1,…,M) with the current values x_t-K:t-1^n fixed using the conditional SMC. Then, for each x_t-K:t-1^n,m, we generate x^n,m_t. In this `local particle filtering', we resample the particles at t-K+1,…,t. This operation is equivalent to choosing the `parent' x_j^n,m for x_j+1^n,m (j=t-K,…,t-1). Using this terminology, if we choose one particle x_t^n,m, its `ancestors' are uniquely determined from x_j^n,m (j = t-K,…,t-1). We call this descendant and its ancestors the `lineage'. In the conditional SMC step, fixing the current values x_t-K:t-1^n is seen as fixing one lineage by choosing their indices k_j (j=t-K,…,t-1)(where we drop the superscript n for simplicity) which follows the rulea^k_j+1_j = k_j, j = t-K,…,t-1.In addition, the index of their descendant is determined as k_t = 1.After generating x_t-K:t^n,m (m=1,…,M), we choose one lineage to store as the next values of x^n_t-K:t. This is equivalent to sampling a random index k^*_t for the candidate x^n,m_t and identifying the ancestors for which indices are obtained by following the rulea^k^*_j+1_j =k^*_j,j = t-K,…,t-1 .Moreover, we can improve its efficiency by implementing `smoothing' for the generated candidates following the algorithm reported in WhiteleyAndrieuDoucet(10). In this smoothing step, we again choose k^*_j for j=t-K,…,t-1 randomly.This manipulation of breaking the relationship between the parent and the child in the lineage is effective in improving the mixing property, or sampling values of x^n_t-K:t-1 that may be different from the lineages obtained in the previous step.The detailed algorithm is provided below. We fix one lineage in Step 1-1(a) and implement the conditional SMC in Steps 1-1(b) and 1-1(c). The candidates for x^n_t are generated in Step 1-1(d) and we compute the importance weight for the n-th particle in the `global particle filtering' in Step 1-2. The smoothing is implemented in Step 1-3.* We generate x_t-K:t^n ∼π(x_t-K:t^n | x_s-1:t-K-1^n,y_s-1:t,θ^n) using the conditional SMC update: (a) Sample k_j from {1,…,M} with probability 1/M (j = t-K,…,t-1)and set (x^n,k_t-K_t-K,…,x^n,k_t-1_t-1) =x_t-K:t-1^n, (a^k_t-K+1_t-K,…,a^k_t_t-1)=(k_t-K,…,k_t-1),where x_t-K:t-1^n is a current sample with the importance weight W_[s-1,t-1]^n. (b)Set x^n,a^m_t-K-1_t-K-1 = x_t-K-1^n for all m according to the convention, and sample x^n,m_t-K∼ q_t-K,θ^n(·| x_t-K-1^n,y_t-K) for each m∈{1,…,M}∖{k_t-K}. Let j=t-K+1.(c)Sample a^m_j-1∼ℳ(V^1:M_j-1,θ^n) and x^n,m_j ∼ q_j,θ^n(·| x_j-1^n,a^m_j-1,y_j) for each m∈{1,…,M}∖{k_j} where V^1:M_j-1,θ^n≡ (V^1_j-1,θ^n,…,V^M_j-1,θ^n) andV^m_j,θ^n = v_j,θ^n(x^n,a^m_j-1_j-1,x_j^n,m)/∑^M_i=1v_j,θ^n(x^n,a^i_j-1_j-1,x_j^n,i),v_j,θ^n(x^n,a^m_j-1_j-1,x_j^n,m) = f_θ^n(x^n,m_j| x^n,a^m_j-1_j-1,y_j-1)g_θ^n(y_j| x_j^n,m)/q_j,θ^n(x_j^n,m| x_j-1^n,a^m_j-1,y_j),m=1,…,M.(d)If j <t-1, set j← j+1 and go to (c). Otherwise, sample x^n,m_t (m=1,…,M) and k_t^* as follows. (i) Sample x^n,1_t ∼ q_t,θ^n(·| x^n,k_t-1_t-1,y_t).(ii) Sample a^n,m_t-1∼ℳ(V^1:M_t-1,θ^n) and x^n,m_t ∼ q_t,θ^n(·| x_t-1^n,a^m_t-1,y_t) for each m∈{2,…,M}.(iii) Sample k^*_t ∼ℳ(V^1:M_t,θ^n) and obtaink^*_j (j=t-1,…,t-K) using(<ref>).* Let x_s-1:t^n=(x_s-1^n,…,x_t-K-1^n, x_t-K^n,k_t-K^*,…,x_t^n,k_t^*) and compute the importance weight[we use the notation p̂(y_t | x_t-K-1^n,y_s-1:t-1,θ^n) since it is an unbiased estimator of p(y_t | x_t-K-1^n,y_s-1:t-1,θ^n) as we shall show in Proposition <ref>.]W^n_[s-1,t] ∝ p̂(y_t | x_t-K-1^n,y_s-1:t-1,θ^n)× W^n_[s-1,t-1]. p̂(y_t | x_t-K-1^n,y_s-1:t-1,θ^n)=1/M∑^M_m=1v_t,θ^n(x^n,a^m_t-1_t-1,x_t^n,m),where p̂(y_t | x_t-K-1^n,y_s-1:t-1,θ^n) can be seen as the estimate of the intractable incremental weight p(y_t | x_t-K-1^n,y_s-1:t-1,θ^n)in (<ref>) for the idealised double-block sampler.* Implement the particle simulation smoother to sample (k^*_t-K,k^*_t-K+1,…,k^*_t) jointly. Generate k^*_j∼ℳ(V̅_j,θ^1:M), j=t-1,…,t-K, recursively whereV̅^m_j,θ≡V^m_j,θf_θ(x^k^*_j+1_j+1| x^m_j,y_j+1)/∑_i=1^MV^i_j,θf_θ(x^k^*_j+1_j+1| x^i_j,y_j+1),m=1,…,M,and set x_s-1:t^n=(x_s-1^n,…,x_t-K-1^n, x_t-K^n,k_t-K^*,…,x_t^n,k_t^*). Figure <ref> illustrates an example with K=2, M=4 and the current sample (x_s-1:t-1^n,θ^n). * (a) Sample k_t-2 and k_t-1 from {1,2,3,4} with probability 1/4 and suppose k_t-2=k_t-1=1.We set x_t-2^n,1=x_t-2^n, x_t-1^n,1=x_t-1^n (with the red rectangle) and (a_t-2^1,a_t-1^1)=(1,1).(b) Set x^n,a^m_t-3_t-3 = x_t-3^n for all m (with the black rectangle), and sample x^n,m_t-2∼ q_t-2,θ^n(·| x_t-3^n,y_t-2) for each m∈{2,3,4} (with the black circle). (c) Sample a^m_t-2∼ℳ(V^1:4_t-2,θ^n)for m∈{2,3,4} and suppose a^2_t-2=2, a^3_t-2=3, a^4_t-2=3. Generate x^n,m_t-1∼ q_t-1,θ^n(·| x_t-2^n,a^m_t-2,y_t-1) form∈{2,3,4} (with the black circle). (d)(i) Sample x^n,1_t ∼ q_t,θ^n(·| x^n,1_t-1,y_t).(ii) Sample a^n,m_t-1∼ℳ(V^1:4_t-1,θ^n)for m∈{2,3,4} and suppose a^2_t-1=2, a^3_t-1=1, a^4_t-1=4. Generate and x^n,m_t ∼ q_t,θ^n(·| x_t-1^n,a^m_t-1,y_t) for m∈{2,3,4}.(iii) Sample k^*_t ∼ℳ(V^1:4_t,θ^n) and suppose k^*_t=3. Using (<ref>), we obtain k_t-1^*=k_t-2^*=1 and select (x_t^n,3,x_t-1^n,1,x_t-2^n,1) with red lines.* Let x_s-1:t^n=(x_s-1^n,…,x_t-3^n, x_t-2^n,1,x_t-1^n,1,x_t^n,3) and compute the importance weight.* Implement the particle simulation smoother to sample (k^*_t-2,k^*_t-1,k^*_t) jointly. Generate k^*_j∼ℳ(V̅_j,θ^1:4), j=t-1,t-2, recursively (with dotted lines) and suppose k^*_t-1=3 and k^*_t-2=3. We set x_s-1:t^n=(x_s-1^n,…,x_t-3^n, x_t-2^n,3, x_t-1^n,3,x_t^n,3). As the proposal density q_j,θ, we can either use the prior density f_θ or more sophisticated density that incorporates the information of the likelihood g_θ. Even if we use the prior f_θ as the proposal, the above sampling becomes much more efficient than the simple rolling-window sampler as shown in Section <ref>. §.§.§ Backward block sampling (Step 2)Before we describe the backward block sampling which generates a cloud of particles based on (x^n_s+K:t,θ^n), we define the notation for the particle index as noted in the forward block sampling but in the reverse order. A `parent' particle of x^m_j is chosen from x^1:M_j+1 (not from x^1:M_j-1) and consequently a^m_j+1 denotes its parent's index. In this case, the relationship of a^m_j+1 and k_j is given as follows:a^k_j_j+1 = k_j+1,j=s+K-2,…,s-2. For each n, we first generate M particle paths, x_s-1:s+K-1^n,1:M≡(x_s-1:s+K-1^n,1,…,x_s-1:s+K-1^n,M), and sample one path, x_s:t^n,from x_s:s+K-1^n,1:M as noted below.* We generate x_s-1:s+K-1^n ∼π(x_s-1:s+K-1^n | x_s+K:t^n,y_s:t,θ^n).(a) Sample indices k_j from {1,…,M} with probability 1/M (j = s+K-1,s+K-2,…,s-1) and set(x^n,k_s-1_s-1,…,x^n,k_s+K-1_s+K-1) =x_s-1:s+K-1^n, (a^k_s-2_s-1,…,a^k_s+K-2_s+K-1)=(k_s-1,…,k_s+K-1),where x_s-1:s+K-1^n is a current sample with the importance weight W_[s-1,t]^n. (b) Set x_s+K^n,a_s+K^m=x_s+K^n for all m according to the convention, and sample x^n,m_s+K-1∼ q_s+K-1,θ^n(·| x_s+K^n, y_s+K-1) for each m∈{1,…,M}∖{k_s+K-1}. Let j=s+K-2. (c) Sample a^m_j+1∼ℳ(V^1:M_j+1,θ^n) and x^n,m_j∼ q_j,θ^n(·| x_j+1^n,a^m_j+1, y_j) for each m∈{1,…,M}∖{k_j} where V^1:M_j+1,θ^n=(V^1_j+1,θ^n,…, V^M_j+1,θ^n) andV^m_j,θ^n=v_j,θ^n(x_j^n,m,x^n,a^m_j+1_j+1)/∑^M_i=1v_j,θ^n(x_j^n,i,x^n,a^i_j+1_j+1),v_j,θ^n(x_j^n,m,x^n,a^m_j+1_j+1) = p(x^n,m_j| x^n,a^m_j+1_j+1,θ)g_θ^n(y_j| x_j^n,m,x^n,a^m_j+1_j+1)/q_j,θ^n(x_j^n,m| x_j+1^n,a^m_j+1,y_j),m=1,…,M.(d) If j>s-1, set j← j-1 and go to (c)[Note that we need to generate x_s-1^n,m to compute p̂ in (<ref>).]. Otherwise,sample k^*_s∼ℳ(V^1:M_s,θ^n) and obtain k_j^* (j=s+1,…,s+K-1) using (<ref>).* Let x_s:t^n=(x_s^n,k_s^*, …,x_s+K-1^n,k_s+K-1^*,x_s+K^n,…,x_t^n)and compute its importance weightW^n_[s,t]∝ {[ 1/p̂(y_s-1| x^n_s+K,y_s:t,θ^n)W^n_[s-1,t],,; 0,, ].wherep̂(y_s-1| x_s+K^n,y_s:t,θ^n)= 1/M∑^M_m=1v_s-1,θ^n(x_s-1^n,m,x_s^n,a^m_s),and p̂(y_s-1| x_s+K^n,y_s:t,θ^n)^-1 can be seen as the estimate of the intractable incremental weight p(y_s-1| x_s+K^n,y_s:t,θ^n)^-1in (<ref>) for the idealised double-block sampler.* Implement the particle simulation smoother to sample (k^*_s,k^*_s+1,…,k^*_s+K-1) jointly. Generate k^*_j∼ℳ(V̅_j,θ^n^1:M), j=s+1,…,s+K-1, recursively whereV̅_j,θ^n^m= V_j,θ^n^mp(x_j-1^k_j-1^*| x_j^m,θ^n)/∑_i=1^MV_j,θ^n^ip(x_j-1^k_j-1^*| x_j^i,θ^n), m=1,…,M.and set x_s:t^n=(x_s^n,k_s^*, …,x_s+K-1^n,k_s+K-1^*,x_s+K^n,…,x_t^n).Figure <ref> illustrates an example with K=2, M=4 and the current sample (x_s-1:t^n,θ^n).* (a) Sample indices k_s+1, k_s, k_s-1 from {1,2,3,4} with probability 1/4 and supposek_s+1=1, k_s=1, k_s-1=1. We set (x^n,1_s-1, x^n,1_s,x^n,1_s+1) =x_s-1:s+1^n (with the rectangle)and (a^1_s-1,a^1_s,a^1_s+1)=(1,1,1).(b) Set x_s+2^n,a_s+2^m=x_s+2^n for all m (with the thick black rectangle), and sample x^n,m_s+1∼ q_s+1,θ^n(·| x_s+1^n, y_s+1) for m∈{2,3,4} (with the black circle). (c) Sample a^m_s+1∼ℳ(V^1:4_s+1,θ^n) and suppose a^2_s+1=1, a^3_s+1=3, a^4_s+1=3. Generate x^n,m_s∼ q_s,θ^n(·| x_s+1^n,a^m_s+1, y_s) for m∈{2,3,4}. (d) Sample a^m_s∼ℳ(V^1:4_s,θ^n) and suppose a^2_s=2, a^3_s=2, a^4_s=4. Generate x^n,m_s-1∼ q_s-1,θ^n(·| x_s^n,a^m_s, y_s-1) for m∈{2,3,4}. (e) Sample k^*_s∼ℳ(V^1:4_s,θ^n) and suppose k^*_s=2. Using (<ref>), we obtain k_s+1^*=1, and select (x_s^n,2,x_s+1^n,1) with red lines.* Let x_s:t^n=(x_s^n,2, x_s+1^n,1,x_s+2^n,…,x_t^n)and compute its importance weight. * Implement the particle simulation smoother to sample (k^*_s,k^*_s+1) jointly. Generate k^*_s+1∼ℳ(V̅_s+1,θ^n^1:4), and suppose k^*_s+1=2. We set x_s:t^n=(x_s^n,2, x_s+1^n,2,x_s+2^n,…,x_t^n). In Algorithm 3, we assume we can evaluate p(x_j-1| x_j,θ) given in (<ref>).In the simple rolling-window sampler, we reweighted the particles according to the likelihood g_θ(y_s-1|x_s-1,x_s) in Step 2, while the unbiased estimate of the conditional likelihood p̂(y_s-1| x^n_s+K,y_s:t,θ^n) is used in the practical double-block sampler. Algorithm 3 substantially improves the weight degeneracy since we condition on y_s:t and integrate out (x_s-1,…,x_s+K-1).§.§ Sequential MCMC estimation without rolling the window In the above discussion, it is implicitly assumed that the initial particlesapproximating π(x_1:L+1,θ| y_1:L+1) are obtained.To sample from this initial posterior distribution, using MCMC-based methods is straightforward as in the warm-up period for the practical filtering described in Polson2008.Moreover, we could simplyuse MCMC samples from the initial posterior distribution. However, based on our proposed method for the rolling estimation, we can obtain samples of x_1:L+1 and θ sequentially, simply by skipping Step 2.The advantage of using our SMC-based method is that we can obtain the estimate of marginal likelihood p(y_1:L+1) as a by-product (the initializing algorithm and the marginal likelihood estimator are described in detail in the Supplementary Material B.). This initializing algorithm can be used for the ordinary sequential learning of π(x_1:t,θ| y_1:t) (t = 1,…,T). We note that this approach is derived from the particle Gibbs scheme in AndrieuDoucetHolenstein(10), and hence our approach is different from that of SMC^2 which applies the particle MH scheme as noted in Chopin2013 and Fulop2013. § ILLUSTRATIVE EXAMPLES This section demonstrates the efficiencies of our proposed algorithm using two illustrative examples. The simple rolling-window sampler suffers from the serious weight degeneracy problem, while (the idealised and the practical) double-block samplersovercome such difficulties. To evaluate the weight degeneracy in each of Steps 1 and 2, we define two ratios: R_1t = ESS_[s-1:t]/ESS_[s-1:t-1],R_2t = ESS_[s:t]/ESS_[s-1:t].The ratio R_1t measures the relative change of ESS in Step 1 after adding y_t when compared with that of the previous step. If the distribution of particles is close to the posterior distribution from which we aim to sample in the step, R_1twould be close to 1. On the other hand, in the presence of the weight degeneracy problem, it will be close to 0. Similarly, the ratio R_2t measures the relative change of ESS in Step 2 after removing y_s-1 compared with that of the previous step. §.§ Linear Gaussian state space model We first consider the following univariate linear Gaussian state space model:y_t=x_t + ϵ_t, ϵ_t ∼𝒩(0, σ^2), t = 1,…,2000 x_t+1 = μ + 0.25 (x_t - μ) +η_t, η_t ∼𝒩(0,2σ^2), t = 1,…,2000,x_1= μ + η_0/√(1-0.25^2), η_0 ∼𝒩(0,2σ^2),where θ = (μ,σ^2)' is a parameter vector. We adopt weak conjugate priors, μ|σ^2 ∼𝒩(0,10σ^2) and σ^2 ∼ℐ𝒢(5/2,0.05/2) where ℐ𝒢(a,b) denotes an inverse gamma distribution with shape parameter a and scale parameter b. The rolling estimation is conducted with a window [t-999,t],t=1001,…,2000 and N=1000 using the particle rolling MCMC with and without the double-block sampling. We choose K=1,2,3,5 and 10 to investigate the effect of the block size. Since an idealised double-block sampler is feasible in the linear Gaussian state space model,we compare the following three samplers:* Simple rolling-window sampler (as a benchmark).* Idealised double-block sampler.* Practical double-block sampler with M = 100,300 and 500. Table <ref> shows the number of resampling steps for three samplers. For the simple rolling-window sampler, the resampling steps are triggered 1027 times, while they are drastically reduced for the double-block samplers. They decrease as we increase K where the magnitude of the reduction is largest at K=2.For K=2, they are around 0.8% and 7.2% of the simple rolling-window sampler for the idealised and practical double-block samplers respectively.Additionally, the number of resampling steps of the practical double-block sampler decreases to that of the idealised double-block sampler as M increases. Figure <ref> shows histograms of R_1t and R_2tfor the simple rolling-window sampler and the practical double-block sampler with K=2 and M=100. The ratios R_1tand R_2t measure the relative magnitude of the effective sample size in Step 1 and Step 2after adding y_tand removing y_s-1 respectively when compared with that of the previous step at time t. The R_1t values for the practical double-block samplerare larger and less dispersed compared with those for the simple rolling-window sampler, suggesting that the forward block sampling is more efficient.Additionally, the R_2t values for the practical double-block samplerare much larger and much less dispersed than those for the simple rolling-window sampler, which implies that the backward block sampling is much more efficient.Further, the scatter plots of R_1t and R_2t are shown in Figure <ref> for two sampling methods. These results demonstrate that our practical double-block sampler is more efficient at both Steps 1 and 2 of each rolling step. Table <ref> shows the summary statistics of R_1t and R_2t.The average of R_1t for the practical double-block sampler is slightly larger than that for the simple rolling-window sampler, but the standard deviation for the former is less than half of that for the latter. Moreover, the average of R_2t for the double-block sampling is six times larger than that for the simple sampling, while the standard deviation for the former is approximately half of that for the latter. Thus the practical double-block sampler drastically alleviate the weight degeneracy compared with the simple rolling-window sampler. Finally, to assess the accuracy of the practical double-block sampler (with K=2 and M = 100), we compare the estimation results with their corresponding analytical solutions. The particles are `refreshed' in the MCMC update step so that the approximation errors do not accumulate over time. In Figure <ref>, the algorithm seems to correctly capture both means and 95% credible intervals of the target posterior distribution.In Figure <ref>, true log marginal likelihoods and their estimates are shown in with errors. The estimation errors are very small overall, implying that the proposed algorithm estimates the marginal likelihood p(y_t-999:t) accurately for t=1001,…,2000.§.§ Realized stochastic volatility model This subsection considers the RSV model given by (<ref>)-(<ref>) where the idealised double-block sampler is not feasible.For the static parameter θ=(μ,ϕ,σ^2_η,ξ,σ^2_u,ρ)', we assume the prior distributions as in Takahashi2009: ϕ + 1/2∼ℬ(20,1.5),c ∼𝒩(0,10), σ_u^2 ∼ℐ𝒢(5/2,0.05/2), Σ∼ℐ𝒲(5,Σ_0),Σ_0 =( 5 [ [1 -0.3 × 0.1; -0.3 × 0.1 0.01 ]] )^-1.using the transformationσ_ϵ = exp(μ / 2),c= ξ + μ, Σ = [ [ σ^2_ϵ ρσ_ϵσ_η; ρσ_ϵσ_η σ^2_η ]],where ℬ(a,b) and ℐ𝒲(r,S) denote a beta distribution with parameters (a,b), and an inverse Wishart distribution with r degrees-of-freedom and the scale matrix S respectively.For y_1t and y_2t, we use Standard and Poor's (S&P) 500 index data, which are obtained from the Oxford-Man Institute Realized Library[ The data is downloaded at http:realized.oxford-man.ox.ac.ukdatadownload ] created by Heber2009 (see shephard2010realising for details). The initial estimation period is from January 1, 2000 (t=1) to December 31, 2007 (t=1988) with L+1= 1988.The rolling estimation started after this initial sample period and moved the window until December 30, 2008 (T=2248). Thus the first estimation period is before the financial crisis caused by the bankruptcy of Lehman Brothers and the last estimation period includes the crisis.We first implement the simple rolling-window sampler. If the ESS is less than the threshold (0.5× N), the particles are refreshed with the MCMC update 10 times. (seeTakahashi2009 for the details of the MCMC sampling). We set N=1000 and construct the proposal density q_t,θ(x_t | x_t-1,y_s-1:t) based on the normal mixture approximation (see Omori2007), which is expected to improve the weight degeneracy. Table <ref>presents a summary of R_1t and R_2t. As expected, R_2t's are low, so the update with MCMC kernel should be implemented in almost every step. The results for R_1t's also indicate that the ESS will be often less than the threshold to resample all the particles. In fact, due to these problems, the resampling steps are implemented 271 times for 260 data windows.Next, we implement the practical double-block sampler with[We also tried using other values of M but the computation time is the shortest with M= 300.] M=300 and N=1000. Further we always implement 10 MCMC iterations below unless otherwise stated. As a proposal density, we simply use a prior density q_t,θ(x_t | x_t-1,y_s-1:t) = f_θ(x_t | x_t-1) to demonstrate that the practical double-block sampler improves even when using the simple proposal.The summary statistics of R_1t and R_2t are shown in Table <ref> where we useK=5,10 and 15.In contrast to the simple rolling-window algorithm, both means are close to 1 demonstrating that our proposed algorithm succeeded inovercoming the weight degeneracy problem. As K increases, R_1t and R_2t become larger and less dispersed, but the difference becomes smaller for K=10 and K=15. Figure <ref> shows the trace plot of estimated posterior means and 95% credible intervals for θ=(μ,ϕ,σ^2_η,ξ,σ^2_u,ρ)' from December 31, 2007 (t=1988) to December 30, 2016 (t=4248). From the rolling estimation results, we are able to observe the transition of the economic structure and the effect of the financial crisis ( t=2150,…,2213 correspond to September, October and November in 2008) .The posterior distribution of μ seems to be stable before t=4000 (January 7, 2016), but its mean and 95% intervals decrease after t=4000. The average level of log volatility started to decrease sharply toward the end of the sample period. The autoregressive parameter, ϕ, continues to decrease throughout the sample period indicating that the latent log volatility becomes less persistent. The variances, σ_η^2 and σ_u^2, of error terms in the state equation and the measurement equation of the log realized volatility continue to increase, while the bias adjustment term, ξ, and the leverage effect, ρ, become closer to zero during the sample period. The leverage effects in the stock market are weaker after the financial crisis. Figure <ref> shows three cumulative computation times (wall time) for the same period corresponding to K=5,10 and 15. The computation times with K=5 and K = 15 are longer than that with K=10. This finding implies that, when K=5, the effect of the blocking is not sufficient to reduce the path dependence between x_t and x_t-K-1 (similarly,x_s-1 and x_s+K). When K = 15, the Monte Carlo error in the local conditional SMC increased the variance of the importance weights even though there is a certain decrease in the variance due to the increase in K (we shall see more details in Section 5)[Also see Supplementary Material Cfor the comparison of the computation time of the practical double-block sampler with those of the MCMC and the particle MCMC.].Finally, in Figure <ref>, we investigate the effect of the number of iterations in the MCMC steps on the estimation accuracy of the posterior distribution function of θ for the proposed sampling algorithm. The estimation period is from January 1, 2000 to December 31, 2007 (t=1,…,1988). First, the MCMC sampling is conducted to obtain the accurate estimates of the distribution functions (solid gray). Then we apply our practical double-block sampler with K = 10, M=300 and N=1000 for three cases: one, five and ten MCMC updates.Among three cases, the estimates obtained by5 or 10 iterations are close to those obtained by the exact MCMC sampling. Ifonly one iteration is performed in the MCMC update step, the estimation results are found to be inaccurate because the MCMC iterations not only diversify the particles but also correct approximation errors introduced by the particle algorithm, which basically update only a part of the vector x^n_s-1:t. The estimation errors for the distribution function of μ are most serious, probably because the mixing property of MCMC sampling in the RSV model is poor especially with respect to μ as discussed in the numerical studies of Takahashi2009. Thus these results suggest that MCMC iterations should be implemented a sufficient number of times in the MCMC update steps such that the particles can trace the correct posterior distributions.§ THEORETICAL JUSTIFICATIONIn this section, we provide theoretical justifications of our proposed algorithm in Section <ref>. We prove that our posterior density is obtained as a marginal density of the artificial target density. §.§ Forward block sampling The artificial target density and its marginal density. We prove that our posterior density of (x_s-1:t^n,θ^n) given y_s-1:t is obtained as a marginal density of the artificial target density in the forward block sampling. The superscript n will be suppressed for simplicity below. In Step 1-1(a) of Section <ref>, the probability density function of (x^k_t-K_t-K,…,x^k_t-1_t-1)=x_t-K:t-1 and (a^k_t-K+1_t-K,…,a^k_t_t-1) given (x_t-K-1,θ) and y_t-K:t-1 isp(x_t-K:t-1,a^k_t-K+1_t-K,…,a^k_t_t-1| x_t-K-1, y_t-K:t-1,θ)= π(x_t-K:t-1| x_t-K-1, y_t-K:t-1,θ)/M^K.Let a_j^1:M=(a_j^1,…,a_j^M), x_j^1:M=(x_j^1,…,x_j^M) and a^-k_j+1_j≡ a^1:M_j∖ a_j^k_j+1= a^1:M_j∖ k_j for j=t-K,…, t-1 where we note a^k_j+1_j=k_j and k_t=1 in (<ref>).Further, let a_t-K:t-1^1:M={a_t-K^1:M,…, a_t-1^1:M}, and x_j^-k_j={x_j^a_j^1,…,x_j^a_j^M}∖ x_j^k_j. Then, in1(b), 1(c) and 1(d) of Step 1, given x_t-K-1,(x^k_t-K_t-K,…,x^k_t-1_t-1) =x_t-K:t-1 and (a^k_t-K+1_t-K,…,a^k_t_t-1)=(k_t-K,…,k_t-1), the probability density function of all variables is defined as ψ_θ(x_t-K^-k_t-K,…,x_t-1^-k_t-1,x^1:M_t,a_t-K^-k_t-K+1,…,a_t-1^-k_t,k^*_t | x_t-K-1:t-1, a_t-K^k_t-K+1,…,a_t-1^k_t, y_t-K:t)=∏^M_m=1m≠ k_t-Kq_t-K,θ(x_t-K^m | x_t-K-1,y_t-K) ×∏^t-1_j=t-K+1∏^M_m=1m≠ k_j V_j-1,θ^a^m_j-1q_j,θ(x_j^m | x_j-1^a^m_j-1,y_j) ×q_t,θ(x_t^1 | x_t-1^k_t-1,y_t)×∏^M_m=2 V_t-1,θ^a^m_t-1q_t,θ(x_t^m | x_t-1^a^m_t-1,y_t) × V^k^*_t_t,θ. In Step 1-2, wemultiply W_[s-1,t-1] by p̂(y_t | x_t-K-1^n,y_s-1:t-1,θ^n) to adjust the importance weight for W_[s-1,t].Let x_t-K:t^1:M=(x_t-K^1:M,…,x_t^1:M) and a_t-K:t-1^1:M=(a_t-K^1:M,…,a_t-1^1:M).Our artificial target density (before the particle smoother step) is written asπ̂(x_s-1:t-K-1,x^1:M_t-K:t,a^1:M_t-K:t-1,k^*_t,θ| y_s-1:t)≡ π(x_s-1:t-K-1, x_t-K^k_t-K,…,x_t-1^k_t-1,θ| y_s-1:t-1)/M^K ×0.8ψ_θ (x_t-K^-k_t-K,…,x_t-1^-k_t-1,x^1:M_t,a_t-K^-k_t-K+1,…,a_t-1^-k_t,k^*_t| x_t-K-1:t-1, a_t-K^k_t-K+1,…,a_t-1^k_t, y_t-K:t) ×p̂(y_t | x_t-K-1,y_s-1:t-1,θ)/p(y_t | y_s-1:t-1)=π(x_s-1:t-1, θ| y_s-1:t-1)/M^K ×∏^M_m=1m≠ k_t-Kq_t-K,θ(x_t-K^m | x_t-K-1,y_t-K) ×∏^t-1_j=t-K+1∏^M_m=1 m≠ k_j V_j-1,θ^a^m_j-1q_j,θ(x_j^m | x_j-1^a^m_j-1,y_j)×q_t,θ(x_t^1 | x_t-1^k_t-1,y_t) ×∏^M_m=2 V_t-1,θ^a^m_t-1q_t,θ(x_t^m | x_t-1^a^m_t-1,y_t) × V^k^*_t_t,θ ×p̂(y_t | x_t-K-1,y_s-1:t-1,θ)/p(y_t | y_s-1:t-1).Note that p(y_t | y_s-1:t-1) is the normalizing constant of this target density, which will be shown in Proposition <ref>. The proposed forward block sampling is justified by proving that the marginal density of (x_s-1,…,x_t-K-1, x_t-K^k_t-K^*,…,x_t^k_t^*,θ) in the above artificial target density π̂ is π(x_s-1,…,x_t-K-1, x_t-K^k_t-K^*,…,x_t^k_t^*,θ| y_s-1:t). The artificial target density π̂ for the forward block sampling can be written as π̂(x_s-1:t-K-1,x^1:M_t-K:t,a^1:M_t-K:t-1,k^*_t,θ| y_s-1:t)= π(x_s-1:t-K-1,x^k_t-K^*_t-K,…,x^k_t^*_t,θ| y_s-1:t)/M^K+1×∏^M_m=1m≠ k^*_t-Kq_t-K,θ(x^m_t-K| x_t-K-1,y_t-K)×∏^t_j=t-K+1∏^M_m=1 m≠ k^*_jV_j-1,θ^a^m_j-1q_j,θ(x_j^m| x_j-1^a^m_j-1,y_j),andthe marginal density of (x_s-1:t-K-1, x_t-K^k_t-K^*,…,x_t^k_t^*,θ) is π(x_s-1:t-K-1, x_t-K^k_t-K^*,…,x_t^k_t^*,θ| y_s-1:t).Proof. See Supplementary Material A. Proposition <ref> implies that we can obtain a posterior random sample (x_s-1:t,θ) given y_s-1:t (with the importance weight W_[s-1,t]) by sampling from the artificial target distribution π̂. This justifies our proposed forward block sampling scheme. Properties of the incremental weight. We consider the mean and variance of the (unnormalized) incremental weight, p̂ (y_t | y_s-1:t-1,x_t-K-1,θ). Proposition <ref> shows that this weight can be considered an unbiased estimator. If (x_s-1:t-K-1, x_t-K^k_t-K,…,x_t-1^k_t-1,k_t-K:t-1,θ) ∼π(x_s-1:t-K-1, x_t-K^k_t-K,…,x_t-1^k_t-1,θ| y_s-1:t-1)/M^K and(x_t-K^-k_t-K,…,x_t^-k_t-1,x^1:M_t,a_t-K^-k_t-K+1,…,a_t-1^-k_t,k^*_t) ∼ ψ_θwhere ψ_θ is given in (<ref>), thenE[p̂(y_t | x_t-K-1,y_s-1:t-1,θ)|y_s-1:t] =E[p(y_t | x_t-K-1,y_s-1:t-1, θ)|x_t-K-1,y_s-1:t, θ] =p(y_t | y_s-1:t-1). Proof. See Supplementary Material A. This shows that the incremental weight p̂(y_t | x_t-K-1,y_s-1:t-1,θ) is an unbiased estimator ofthe conditional likelihood p(y_t | x_t-K-1,y_s-1:t-1, θ) given (x_t-K-1,θ). It is also an unbiased estimator of the marginal likelihood p(y_t | y_s-1:t-1) unconditionally, which implies that p(y_t | y_s-1:t-1) is a normalizing constant for the artificial target density π̂. Further, from the law of total variance, we obtain the decomposition of the variance as follows. Var[p̂(y_t | x_t-K-1,y_s-1:t-1,θ)| y_s-1:t]=Var[p(y_t | x_t-K-1,y_s-1:t-1,θ)| y_s-1:t] + E[Var[p̂(y_t | x_t-K-1,y_s-1:t-1,θ) | x_s-1:t-K-1,y_s-1:t, θ]]. The variance of the incremental weight consists of two components, including variance of the conditional likelihood and (expected) variance which is introduced using M particles to approximate the conditional likelihood. This decomposition identifies factors that influences the ESS of the particles.Regarding the first component, for any positive integers, K_1,K_2, with K_1<K_2, the following inequality holds:Var[p(y_t | x_t-K_1-1,y_s-1:t-1,θ)] ≥Var[p(y_t | x_t-K_2-1,y_s-1:t-1,θ)],which is a straightforward result from the law of total variance for p(y_t | x_t-K_1-1,y_s-1:t-1,θ) usingE[ p(y_t | x_t-K_1-1,y_s-1:t-1,θ) | x_s-1:t-K_2-1,θ] =p(y_t | x_t-K_2-1,y_s-1:t-1,θ).On the other hand, the second component may become large as K increases, but it is expected to be controlled by changing the number of particles M.§.§ Backward block sampling The artificial target density and its marginal density. This subsection proves that our posterior density of (x_s:t^n,θ^n) given y_s:t is obtained as a marginal density of the artificial target density in the backward block sampling. The superscript n will be suppressed for simplicity below.In Step 2-1(a) of Section <ref>, the probability density function of(x^k_s-1_s-1,…,x^k_s+K-1_s+K-1)=x_s-1:s+K-1 and (a^k_s-2_s-1,…,a^k_s+K-2_s+K-1) given (x_s+K,θ) and y_s-1:t isp(x_s-1:s+K-1,a^k_s-2_s-1,…,a^k_s+K-2_s+K-1| x_s+K, y_s-1:t,θ)= π(x_s-1:s+K-1| x_s+K, y_s-1:t,θ)/M^K+1.In 1(b), 1(c) and 1(d) of Steps 2, given x_s+K,(x^k_s-1_s-1,…,x^k_s+K-1_s+K-1) =x_s-1:s+K-1, (a^k_s-2_s-1,…,a^k_s+K-2_s+K-1)=(k_s-1,…,k_s+K-1) and y_s-1:s+K-1, the probability density function of all variables is defined as ψ̅_θ (x_s-1^-k_s-1,…,x_s+K-1^-k_s+K-1,a_s^-k_s-1,…,a_s+K-1^-k_s+K-2,k^*_s| x_s-1:s+K, a_s-1^k_s-2, …,a_s+K-1^k_s+K-2,y_s-1:s+K-1) =∏^M_m=1 m≠ k_s+K-1 q_s+K-1,θ(x^m_s+K-1| x_s+K,y_s+K-1) ×∏^s+K-2_j=s-1∏^M_m=1m≠ k_j V^a^m_j+1_j+1,θq_j,θ(x^m_j | x^a^m_j+1_j+1,y_j) × V^k^*_s_s,θ.In Step 2-2, wedivide W_[s-1,t] by p̂(y_s-1| x_s+K^n,y_s:t,θ^n) to adjust the importance weight for W_[s,t]. Similarly to the discussion in Section <ref>, we consider an extended space with the artificial target density written as π̌(x_s-1:s+K-1^1:M,x_s+K:t,a_s:s+K-1^1:M,k_s-1,k^*_s,θ| y_s-1:t)≡ π(x_s-1:t, θ| y_s-1:t)/M^K+1× ∏^M_m=1 m≠ k_s+K-1 q_s+K-1,θ(x^m_s+K-1| x_s+K,y_s+K-1) ×∏^s+K-2_j=s-1∏^M_m=1m≠ k_j V^a^m_j+1_j+1,θq_j,θ(x^m_j | x^a^m_j+1_j+1,y_j) ×V^k^*_s_s,θ×p(y_s-1| y_s:t)/p̂(y_s-1| x_s+K,y_s:t,θ),where p(y_s-1| y_s:t)^-1 is the normalizing constant of this target density as shown in Proposition <ref>. Below we state Proposition <ref> for the backward block sampling, which correspond to Proposition <ref> for the forward block sampling. The artificial target density π̌ for the backward block sampling can be rewritten asπ̌(x_s-1:s+K-1^1:M,x_s+K:t,a_s:s+K-1^1:M,k_s-1,k^*_s,θ| y_s-1:t) = π(x_s^k_s^*,…,x_s+K-1^k_s+K-1^*,x_s+K:t,θ| y_s:t)/M^K×∏^M_m=1m≠ k^*_s+K-1 q_s+K-1,θ(x^m_s+K-1| x_s+K,y_s+K-1:t)×∏^s+K-2_j=s∏^M_m=1m≠ k^*_j V^a^m_j+1_j+1,θq_j,θ(x^m_j | x^a^m_j+1_j+1,y_j) ×∏^M_m=1V^a^m_s_s,θq_s-1,θ(x^m_s-1| x^a^m_s_s,y_s-1) × V^k_s-1_s-1,θ,and the marginal density of (x_s^k_s^*,…, x_s+K-1^k_s+K-1^*, x_s+K:t,θ) is π(x_s^k_s^*,…, x_s+K-1^k_s+K-1^*, x_s+K:t, θ| y_s:t). Proof. See Supplementary Material A.Although the probability density (<ref>) in Proposition <ref>has a bit different form from that of (<ref>) in Proposition <ref>, its marginal probability density is found to be the target posterior density π(x_s:t, θ| y_s:t).Properties of the incremental weight. Similar results to Proposition <ref> hold for the backward block sampling, and are summarized in Proposition <ref>.If (x_s-1^k_s-1,…,x_s+K-1^k_s+K-1, x_s+K:t,k_s-1:s+K-1,θ) ∼π(x_s-1^k_s-1,…,x_s+K-1^k_s+K-1, x_s+K:t,θ| y_s-1:t)/M^K+1and(x_s-1^-k_s-1,…,x_s+K-1^-k_s+K-1,a_s^-k_s-1,…,a_s+K-1^-k_s+K-2,k^*_s) ∼ ψ̅_θ,where ψ̅_θ is given in (<ref>), thenE[p̂(y_s-1| x_s+K,y_s:t,θ)^-1] =E[p̂(y_s-1| x_s+K,y_s:t,θ)^-1| x_s+K,y_s:t,θ] =p(y_s-1| y_s:t,θ)^-1. Proof. See Supplementary Material A.§.§ Particle simulation smootherIn WhiteleyAndrieuDoucet(10) and the discussion of Whiteley following AndrieuDoucetHolenstein(10), the additional step is introduced to explore all possible ancestral lineages. This is expected to improve the mixing property of the particle Gibbs (see e.g. Chopin2015, LeeSinghVihola(20)).We also incorporate such a particle simulation smoother into the double block sampling based on the following proposition.The joint conditional density of(k_t-K^*,…,k^*_t)is given byπ̂(k_t-K^*,…,k^*_t|x_s-1:t-K-1,x_t-K:t^1:M,a_t-K:t-1^1:M,y_s-1:t,θ) =π̂(k^*_t|x_s-1:t-K-1,x_t-K:t^1:M,a_t-K:t-1^1:M,y_s-1:t,θ)× ∏_t_0=t-1^t-Kπ̂(k_t_0^*|x_s-1:t-K-1,x_t-K:t_0^1:M,a_t-K:t_0-1^1:M,x_t_0+1^k_t_0+1^*,…,x_t^k_t^*,k^*_t_0+1:t, y_s-1:t,θ),whereπ̂(k_t_0^*|x_s-1:t-K-1,x_t-K:t_0^1:M,a_t-K:t_0-1^1:M,x_t_0+1^k_t_0+1^*,…,x_t^k_t^*,k^*_t_0+1:t, y_s-1:t,θ)= V̅^k^*_t_0_t_0,θ, V̅^m_j,θ≡V^m_j,θf_θ(x^k^*_j+1_j+1| x^m_j,y_j+1)/∑_i=1^MV^i_j,θf_θ(x^k^*_j+1_j+1| x^i_j,y_j+1). Proof. See Supplementary Material A.Suppose we have (x_s-1:t-K-1,x_t-K:t^1:M,a_t-K:t-1^1:M,k^*_t,θ) ∼π̂ where π̂ is defined in (<ref>). In Step 1-1(d), the lineage k^*_t-K:t is automatically determined when k^*_t is chosen. The particle simulation smoother breaks this relationship and again samples k^*_t-K:t jointly by generating k^*_j∼ℳ(V̅_j,θ^1:M), j=t-1,…,t-K, recursively. § CONCLUSIONIn this paper, we propose a novel efficient estimation method to implement the rolling-window particle MCMC simulation using a SMC framework and refreshing steps with MCMC kernel. The weighted particles are updated to learn and discard the information of the new and old observations using the forward and backward block sampling based on the conditional SMC algorithm, which effectively circumvent the weight degeneracy problem. The proposed estimation methodology is also applicable to the ordinary sequential estimation with parameter uncertainty.The illustrative examples show that our proposed sampler outperforms the simple rolling-window sampler. § ACKNOWLEDGEMENTAll computational results in this paper are generated using Ox metrics 7.0 (see Doornik(09)). This work was supported by JSPS KAKENHI Grant Numbers 25245035, 26245028, 17H00985, 15H01943, 19H00588. Supplementary Material § PROOFS §.§ Proof of Proposition 5.1 We first establish the following lemma which describes a property of the local conditional SMC. For any t and t_0 (t-K ≤ t_0 ≤ t) ,π(x_s-1:t-K-1,x_t-K^k_t-K,…,x_t_0^k_t_0,θ| y_s-1:t_0) /M^t_0 - (t-K) +1 ×∏^M_m=1m≠ k_t-Kq_t-K,θ(x^m_t-K| x_t-K-1,y_t-K) ×∏^t_0_j=t-K+1∏^M_m=1 m≠ k_jV_j-1,θ^a^m_j-1q_j,θ(x_j^m| x_j-1^a^m_j-1,y_j) = π(x_s-1:t-K-1, θ| y_s-1:t-K-1)×∏^M_m=1q_t-K,θ(x^m_t-K| x_t-K-1,y_t-K)×∏^t_0_j=t-K+1∏^M_m=1V_j-1,θ^a^m_j-1q_j,θ(x_j^m| x_j-1^a^m_j-1,y_j) × V^k_t_0_t_0,θ×∏^t_0_j=t-Kp̂(y_j | x_t-K-1,y_s-1:j-1,θ)/p(y_j | y_s-1:j-1),where p̂(y_j | x_t-K-1,y_s-1:j-1,θ)= 1/M∑^M_m=1v_j,θ(x_j-1^a^m_j-1,x_j^m),j=t-K,…,t_0,with x^a^m_t-K-1_t-K-1 = x_t-K-1 anda^k_j_j-1 = k_j-1, j = t-K+1,…,t_0.The probability density (<ref>) corresponds to the target density π_t^* of SMC^2 in Chopin2013 which includes the random particle index. For the particle filtering, the forward block sampling considers the density of x_t-K:t-1^1:M conditional on (x_s-1:t-K-1, θ), while SMC^2 considers that of x_1:t^1:M conditional on θ. Further, the former updates the importance weight for (x_s-1:t, θ) and the latter updates that for θsequentially.Proof of Lemma <ref>. Using Bayes' theorem and v_j,θ(x^a^m_j-1_j-1,x_j^m) = f_θ(x^m_j| x^a^m_j-1_j-1,y_j-1)g_θ(y_j| x_j^m)/q_j,θ(x_j^m| x_j-1^a^m_j-1,y_j),j=1,…,M,the numerator of the first term in (<ref>) is π(x_s-1:t-K-1,x_t-K^k_t-K,…,x_t_0^k_t_0,θ| y_s-1:t_0)= π(x_s-1:t-K-1, θ| y_s-1:t-K-1)/p(y_t-K:t_0| y_s-1:t-K-1)∏^t_0_j=t-K f_θ(x^k_j_j | x^k_j-1_j-1,y_j-1)g_θ(y_j | x^k_j_j) = π(x_s-1:t-K-1, θ| y_s-1:t-K-1)/p(y_t-K:t_0| y_s-1:t-K-1)∏^t_0_j=t-K v_j,θ(x_j-1^k_j-1,x^k_j_j) ∏^t_0_j = t-Kq_j,θ(x_j^k_j| x^k_j-1_j-1,y_j).Thus we obtain(<ref>) =π(x_s-1:t-K-1,x_t-K^k_t-K,…,x_t_0^k_t_0,θ| y_s-1:t_0)/M^t_0 - (t-K) +1×∏^t_0_j=t-K∏^M_m=1 m≠ k_jq_j,θ(x_j^m| x_j-1^a^m_j-1,y_j)×∏^t_0_j=t-K+1∏^M_m=1m≠ k_jV_j-1,θ^a^m_j-1= π(x_s-1:t-K-1, θ| y_s-1:t-K-1)/M^t_0 - (t-K) +1p(y_t-K:t_0| y_s-1:t-K-1)×∏^t_0_j=t-K∏^M_m=1q_j,θ(x_j^m| x_j-1^a^m_j-1,y_j) ×∏^t_0_j=t-K+1v_j-1,θ(x_j-2^k_j-2,x^k_j-1_j-1)∏^M_m=1m≠ k_jV_j-1,θ^a^m_j-1× v_t_0,θ(x_t_0-1^k_t_0-1,x^k_t_0_t_0) = π(x_s-1:t-K-1, θ| y_s-1:t-K-1)/∏^t_0_j=t-K p(y_j| y_s-1:j-1)×∏^t_0_j=t-K∏^M_m=1q_j,θ(x_j^m| x_j-1^a^m_j-1,y_j)×∏^t_0_j=t-K+1∏^M_m=1V_j-1,θ^a^m_j-1× V^k_t_0_t_0,θ× ∏^t_0_j=t-Kp̂(y_j | x_t-K-1,y_s-1:j-1,θ)and the result follows where we substitute (<ref>) in the second equality, and used the definition of p̂(y_j | x_t-K-1,y_s-1:j-1,θ) in the third equality. Using Lemma <ref>, we obtain Proposition 5.1 as follows. Proof of Proposition 5.1.By applying Lemma <ref> with t_0 = t-1 to the first three terms of (44),we obtainπ̂(x_s-1:t-K-1,x^1:M_t-K:t,a^1:M_t-K:t-1,k^*_t,θ| y_s-1:t)= π(x_s-1:t-K-1, θ| y_s-1:t-K-1) ×∏^M_m=1q_t-K,θ(x^m_t-K| x_t-K-1,y_t-K) ×∏^t-1_j=t-K+1∏^M_m=1V_j-1,θ^a^m_j-1q_j,θ(x_j^m| x_j-1^a^m_j-1,y_j) × V^k_t-1_t-1,θ×∏^t-1_j=t-Kp̂(y_j | x_t-K-1,y_s-1:j-1,θ)/p(y_j | y_s-1:j-1) ×q_t,θ(x_t^1 | x_t-1^k_t-1,y_t) ×∏^M_m=2 V_t-1,θ^a^m_t-1q_t,θ(x_t^m | x_t-1^a^m_t-1,y_t) × V^k^*_t_t,θ×p̂(y_t | x_t-K-1,y_s-1:t-1,θ)/p(y_t | y_s-1:t-1)= π(x_s-1:t-K-1, θ| y_s-1:t-K-1) ×∏^M_m=1q_t-K,θ(x^m_t-K| x_t-K-1,y_t-K)×∏^t_j=t-K+1∏^M_m=1V_j-1,θ^a^m_j-1q_j,θ(x_j^m,x_j-1^a^m_j-1| y_j) ×∏^t_j=t-Kp̂(y_j | x_t-K-1,y_s-1:j-1,θ)/p(y_j | y_s-1:j-1)× V^k^*_t_t,θ,where we note a^1_t-1=k_t-1 and k_t=1. Apply Lemma <ref> with t_0 = t and k_t_0=k_t^* to the last equation and the result follows.§.§ Proof of Proposition 5.2 We first define the probability density functionψ_θ,0(x_t-K:t^1:M,a_t-K:t-1^1:M,k^*_t | x_t-K-1,y_s-1:t)≡ π(x_t-K:t-1| x_t-K-1,y_s-1:t-1,θ)/M^K ×ψ_θ (x_t-K^-k_t-K,…,x_t-1^-k_t-1,x^1:M_t,a_t-K^-k_t-K+1,…, a_t-1^-k_t,k^*_t| x_t-K-1:t-1, a_t-K^k_t-K+1,…,a_t-1^k_t, y_t-K:t),where (x_t-K^k_t-K,…,x_t-1^k_t-1)=x_t-K:t-1 andπ(x_t-K:t-1| x_t-K-1,y_s-1:t-1,θ)= π(x_s-1:t-1, θ| y_s-1:t-1)/π(x_s-1:t-K-1, θ| y_s-1:t-1).Noting that p̂(y_t | x_t-K-1,y_s-1:t-1,θ)ψ_θ,0(x_t-K:t^1:M,a_t-K:t-1^1:M,k^*_t | x_t-K-1,y_s-1:t) = π̂(x_s-1:t-K-1,x^1:M_t-K:t,a^1:M_t-K:t-1,k^*_t, θ| y_s-1:t)p(y_t | y_s-1:t-1)/π(x_s-1:t-K-1, θ| y_s-1:t-1),where we used the definition of π̂ in (44), E_ψ_θ,0[ p̂(y_t | x_t-K-1, y_s-1:t-1,θ) | x_t-K-1, y_s-1:t, θ]= ∫p̂(y_t | x_t-K-1, y_s-1:t-1,θ)ψ_θ,0(x_t-K:t^1:M,a_t-K:t-1^1:M,k^*_t| x_t-K-1,y_s-1:t) d x_t-K:t^1:Md a_t-K:t-1^1:Md k^*_t = ∫π̂(x_s-1:t-K-1,x^1:M_t-K:t,a^1:M_t-K:t-1,k^*_t,θ| y_s-1:t) d x_t-K:t^1:Md a_t-K:t-1^1:Md k^*_t p(y_t | y_s-1:t-1)/π(x_s-1:t-K-1, θ| y_s-1:t-1)= π(x_s-1:t-K-1, θ| y_s-1:t)p(y_t | y_s-1:t-1)/π(x_s-1:t-K-1, θ| y_s-1:t-1)=p(y_t | x_s-1:t-K-1, y_s-1:t-1, θ)= p(y_t | x_t-K-1, y_s-1:t-1, θ).Also it is easy to see E[p(y_t | x_t-K-1, y_s-1:t-1, θ)| y_s-1:t] =∫ p(y_t | x_t-K-1, y_s-1:t-1, θ)π(x_t-K-1,θ| y_s-1:t-1) dθ dx_t-K-1 = p(y_t | y_s-1:t-1). §.§ Proof of Proposition 5.3 We first establish the following lemma as in the proof of Proposition 5.1. For any t, s_0 ,and s (s-1 ≤ s_0 ≤ s+K-1), π(x_s_0^k_s_0,…,x_s+K-1^k_s+K-1,x_s+K:t,θ| y_s_0:t)/M^(s+K-1)-s_0+1 ×∏^M_m=1 m≠ k_s+K-1 q_s+K-1,θ(x^m_s+K-1| x_s+K,y_s+K-1) ×∏^s+K-2_j=s_0∏^M_m=1 m≠ k_jV^a^m_j+1_j+1,θq_j,θ(x^m_j | x^a^m_j+1_j+1,y_j)=π(x_s+K:t,θ| y_s+K:t)×∏^M_m=1 q_s+K-1,θ(x^m_s+K-1| x_s+K,y_s+K-1)×∏^s+K-2_j=s_0∏^M_m=1V^a^m_j+1_j+1,θq_j,θ(x^m_j | x^a^m_j+1_j+1,y_j) × V^k_s_0_s_0,θ×∏^s+K-1_j=s_0p̂(y_j | x_s+K, y_j+1:t,θ)/p(y_j | y_j+1:t) with x_s+K^a_s+K^m = x_s+K and a^k_j_j+1 = k_j+1 (s_0≤ j ≤ s+K-2), wherep̂(y_j | x_s+K,y_j+1:t,θ)= 1/M∑^M_m=1v_j,θ(x_j^m,x_j+1^a_j+1^m). Proof of Lemma <ref>. Sinceπ(x_s_0^k_s_0,…,x_s+K-1^k_s+K-1,x_s+K:t,θ| y_s_0:t)= π(x_s+K:t,θ| y_s+K:t)/p(y_s_0:s+K-1| y_s+K:t)∏_j=s_0^s+K-1p(x_j^k_j|x_j+1^k_j+1,θ)g_θ(y_j|x_j^k_j,x_j+1^k_j+1) = π(x_s+K:t,θ| y_s+K:t)/p(y_s_0:s+K-1| y_s+K:t)∏_j=s_0^s+K-1v_j,θ(x_j^k_j, x_j+1^k_j+1) ×∏_j=s_0^s+K-1q_j,θ(x_j^k_j|x_j+1^k_j+1,y_j),we obtainπ(x_s_0^k_s_0,…,x_s+K-1^k_s+K-1,x_s+K:t, θ| y_s_0:t)/M^(s+K-1)-s_0+1 ×∏^M_m=1 m≠ k_s+K-1 q_s+K-1,θ(x^m_s+K-1| x_s+K,y_s+K-1) ×∏^s+K-2_j=s_0∏^M_m=1 m≠ k_jV^a^m_j+1_j+1,θq_j,θ(x^m_j | x^a^m_j+1_j+1,y_j) =π(x_s+K:t,θ| y_s+K:t)/M^(s+K-1)-s_0+1p(y_s_0:s+K-1| y_s+K:t)×∏_j=s_0^s+K-1∏_m=1^Mq_j,θ(x_j^m|x_j+1^a_j+1^m,y_j)×∏_j=s_0-1^s+K-2v_j+1,θ(x_j+1^k_j+1, x_j+2^k_j+2) ×∏^s+K-2_j=s_0∏^M_m=1 m≠ k_jV^a^m_j+1_j+1,θ =π(x_s+K:t,θ| y_s+K:t)/p(y_s_0:s+K-1| y_s+K:t)×∏_j=s_0^s+K-1∏_m=1^Mq_j,θ(x_j^m|x_j+1^a_j+1^m,y_j)×∏_j=s_0^s+K-1{1/M∑_i=1^Mv_j,θ(x_j^i,x_j+1^a_j+1^i) }×∏^s+K-2_j=s_0∏^M_m=1V^a^m_j+1_j+1,θ× V^k_s_0_s_0,θ =π(x_s+K:t,θ| y_s+K:t)/∏_j=s_0^s+K-1p(y_j| y_j+1:t)×∏_j=s_0^s+K-1∏_m=1^Mq_j,θ(x_j^m|x_j+1^a_j+1^m,y_j)×∏^s+K-2_j=s_0∏^M_m=1V^a^m_j+1_j+1,θ× V^k_s_0_s_0,θ×∏_j=s_0^s+K-1p̂(y_j|x_s+K,y_j+1:t,θ) Proof of Proposition 5.3. By applying Lemma <ref> with s_0 = s-1 to the first three terms of the target distribution in (49), we haveπ̌(x_s-1:s+K-1^1:M,x_s+K:t,a_s:s+K-1^1:M,k_s-1,k^*_s,θ| y_s-1:t)= π(x_s+K:t,θ| y_s+K:t) ×∏^M_m=1 q_s+K-1,θ(x^m_s+K-1| x_s+K,y_s+K-1:t)×∏^s+K-2_j=s-1∏^M_m=1 V^a^m_j+1_j+1,θq_j,θ(x^m_j | x^a^m_j+1_j+1,y_j) × V^k_s-1_s-1,θ×∏^s+K-1_j=s-1p̂(y_j | x_s+K,y_j+1:t,θ)/p(y_j | y_j+1:t) × V^k^*_s_s×p(y_s-1| y_s:t)/p̂(y_s-1| x_s+K, y_s:t,θ)=π(x_s+K:t,θ| y_s+K:t) ×∏^M_m=1 q_s+K-1,θ(x^m_s+K-1| x_s+K,y_s+K-1:t)×∏^s+K-2_j=s∏^M_m=1V^a^m_j+1_j+1,θ q_j,θ(x^m_j | x^a^m_j+1_j+1,y_j)× V^k^*_s_s×∏^s+K-1_j=sp̂(y_j | x_s+K, y_j+1:t,θ)/p(y_j | y_j+1:t) ×∏^M_m=1V^a^m_s_s,θ q_s-1,θ(x^m_s-1| x^a^m_s_s,y_s-1) × V^k_s-1_s-1,θ =π(x_s^k_s^*,…,x_s+K-1^k_s+K-1^*,x_s+K:t,θ| y_s:t)/M^K×∏^M_m=1m≠ k_s+K-1^* q_s+K-1,θ(x^m_s+K-1| x_s+K,y_s+K-1:t)×∏^s+K-2_j=s∏^M_m=1m≠ k_j^*V^a^m_j+1_j+1,θ q_j,θ(x^m_j | x^a^m_j+1_j+1,y_j) ×∏^M_m=1V^a^m_s_s,θ q_s-1,θ(x^m_s-1| x^a^m_s_s,y_s-1) × V^k_s-1_s-1,θwhere we again applied Lemma <ref> with s_0 = s and k_s-1=k^*_s-1 in the last equality.§.§ Proof of Proposition 5.4Proof of Proposition 5.4. We first define the probability density functionψ̅_θ,0(x_s-1:s+K-1^1:M,a_s:s+K-1^1:M,k_s-1, k^*_s| x_s+K,y_s-1:s+K-1)≡ π(x_s-1:s+K-1| x_s+K,y_s-1:t,θ)/M^K+1× ψ̅_θ (x_s-1^-k_s-1,…,x_s+K-1^-k_s+K-1,a_s^-k_s-1,…,a_s+K-1^-k_s+K-2,k^*_s| x_s-1:s+K,a_s-1^k_s-2,…,a_s+K-1^k_s+K-2, y_s-1:s+K-1),and note thatπ(x_s-1:s+K-1| x_s+K,y_s-1:t,θ) =π(x_s-1:s+K-1, x_s+K:t, θ| y_s-1:t)/π(x_s+K:t,θ| y_s-1:t).Since1/p̂(y_s-1| x_s+K, y_s:t,θ)ψ̅_θ,0(x_s-1:s+K-1^1:M,a_s:s+K-1^1:M,k_s-1, k^*_s| x_s+K,y_s-1:s+K-1) = π̌(x_s-1:s+K-1^1:M,x_s+K:t,a_s:s+K-1^1:M,k_s-1,k^*_s,θ| y_s-1:t)1/p(y_s-1| y_s:t)π(x_s+K:t,θ| y_s-1:t),where we used the definition of π̌ in (49), we obtainE_ψ̅_θ,0[ p̂(y_s-1| x_s+K, y_s:t,θ)^-1| x_s+K, y_s-1:t, θ]= ∫1/p̂(y_s-1| x_s+K, y_s:t,θ)ψ̅_θ,0(x_s-1:s+K-1^1:M,a_s:s+K-1^1:M,k_s-1, k^*_s| x_s+K,y_s-1:s+K-1)d x_s-1:s+K-1^1:Md a_s:s+K-1^1:Mdk_s-1 d k^*_s= ∫π̌(x_s-1:s+K-1^1:M,x_s+K:t,a_s:s+K-1^1:M,k_s-1,k^*_s,θ| y_s-1:t)d x_s-1:s+K-1^1:Md a_s:s+K-1^1:Mdk_s-1 d k^*_s ×1/p(y_s-1| y_s:t)π(x_s+K:t,θ| y_s-1:t)= π(x_s+K:t,θ| y_s:t)/p(y_s-1| y_s:t)π(x_s+K:t,θ| y_s-1:t) =1/p(y_s-1| x_s+K:t,y_s:t,θ) = 1/p(y_s-1| x_s+K, y_s:t,θ)where we use Proposition 5.3 in the third equality.Further, E[p(y_s-1| x_s+K, y_s:t,θ)^-1| y_s-1:t] = ∫π(x_s+K,θ| y_s-1:t)/p(y_s-1| x_s+K,y_s:t,θ) dx_s+Kdθ= ∫π(x_s+K,θ| y_s:t)/p(y_s-1| y_s:t) dx_s+Kdθ =p(y_s-1| y_s:t)^-1. §.§ Proof of Proposition 5.5Proof of Proposition 5.5. Consider the joint marginal density of (45):π̂(x_s-1:t-K-1,x^1:M_t-K:t_0,a^1:M_t-K:t_0-1, x^k^*_t_0+1_t_0+1,…,x^k^*_t_t, k^*_t_0:t,θ| y_s-1:t)= π(x_s-1:t-K-1,x^k^*_t-K_t-K,…,x^k^*_t_t,θ| y_s-1:t)/M^K+1×∏^M_m=1m ≠ k^*_t-K q_t-K,θ(x^m_t-K| x_t-K-1,y_t-K)×∏^t_0_j=t-K+1∏^M_m=1m ≠ k^*_jV^a^m_j_j-1 q_j,θ(x^m_j | x^a^m_j_j-1,y_j),for t_0=t-1,…,t-K+1, and π̂(x_s-1:t-K-1,x^1:M_t-K, x^k^*_t-K+1_t-K+1,…,x^k^*_t_t, k^*_t-K:t,θ| y_s-1:t) =π(x_s-1:t-K-1,x^k^*_t-K_t-K,…,x^k^*_t_t,θ| y_s-1:t)/M^K+1×∏^M_m=1m ≠ k^*_t-K q_t-K,θ(x^m_t-K| x_t-K-1,y_t-K).Then we obtainπ̂(k_t_0^*| x^1:M_s-1:t-K-1,x^1:M_t-K:t_0,a^1:M_t-K:t_0-1,x^k^*_t_0+1_t_0+1,…,x^k^*_t_t, k^*_t_0+1:t, y_s-1:t,θ)∝ π̂(x^1:M_s-1:t-K-1,x^1:M_t-K:t_0,a^1:M_t-K:t_0-1, x^k^*_t_0+1_t_0+1,…,x^k^*_t_t, k^*_t_0:t,θ| y_s-1:t)∝ π(x_s-1:t-K-1,x^k^*_t-K_t-K,…,x^k^*_t_0_t_0,θ| y_s-1:t_0)/M^t_0-(t-K)+1×∏^M_m=1m ≠ k^*_t-K q_t-K,θ(x^m_t-K| x_t-K-1,y_t-K) × ∏^t_0_j=t-K+1∏^M_m=1 m≠ k^*_jV^a^m_j_j-1 q_j,θ(x^m_j | x^a^m_j_j-1,y_j) ×∏^t_j=t_0+1f_θ(x_j^k_j^*| x^k_j-1^*_j-1,y_j-1)g_θ(y_j | x^k_j^*_j)= π(x_s-1:t-K-1,θ| y_s-1:t-K-1)×∏^M_m=1q_t-K,θ(x^m_t-K| x_t-K-1,y_t-K)×∏^t_0_j=t-K+1∏^M_m=1V_j-1,θ^a^m_j-1q_j,θ(x_j^m| x_j-1^a^m_j-1,y_j) × V^k_t_0^*_t_0,θ×∏^t_0_j=t-Kp̂(y_j | x_t-K-1,y_s-1:j-1,θ)/p(y_j | y_s-1:j-1), ×∏^t_j=t_0+1f_θ(x_j^k_j^*| x^k_j-1^*_j-1,y_j-1)g_θ(y_j | x^k_j^*_j)∝ V^k_t_0^*_t_0,θ× f_θ(x_t_0+1^k_t_0+1^*| x^k_t_0^*_t_0,y_t_0),where we use Lemma A.1at the equality. § SEQUENTIAL MCMC ESTIMATION WITHOUT ROLLING THE WINDOWWe first give the initializing algorithm which is obtained by skipping the discarding step (Step 2) in the particle rolling algorithm. Next, we describe how to estimate the marginal likelihood.§.§ Algorithm(1) At time j=1, sample (x^n_1,θ^n) from π(x_1, θ| y_1) for n=1,…,N. * Sample θ^n ∼ p(θ), andx^n,m_1 ∼ q_1,θ^n(·| y_1) for each m ∈{1,…,M}.* Sample k_1 ∼ℳ(V^n,1:M_1,θ^n) whereV^n,m_1,θ^n = v_1,θ^n(x^n,m_1) /∑^M_i=1v_1,θ^n(x^n,i_1) , v_1,θ^n(x^n,m_1) = μ_θ^n(x^n,m_1)g_θ^n(y_1| x_1^n,m)/q_1,θ^n(x_1^n,m| y_1). * Set x_1^n=x_1^n,k_1 and store (x^n_1,θ^n) with its importance weightW_1^n∝ p̂(y_1 |θ^n), p̂(y_1 |θ^n) = ∑^M_m=1v_1,θ^n(x^n,m_1).(2) At time j=2,…,L+1, implement the forward block sampling to generate x_1:j^n and θ^n, and compute its importance weightW^n_j ∝ p̂(y_j| x_j-K-1^n,y_1:j-1,θ^n)× W^n_j-1,p̂(y_j| x_j-K-1^n,y_1:j-1,θ^n)=1/M∑^M_m=1v_j,θ^n(x^n,a^n,m_j-1_j-1,x_j^n,m).For j<K, we set K=j-1, and all particles of x_1:j^n are resampled.Especially when j is small and the dimension of x_1:j is smaller than that of θ, the MCMC update of θ could lead to unstable estimation results. We may need to modify the MCMC kernel or skip the update in such a case.§.§ Estimation of the marginal likelihood As a by-product of the proposed algorithms, we can obtain the estimate of the marginal likelihood defined asp(y_s:t)= ∫ p(y_s:t| x_s:t,θ)p(x_s:t|θ)p(θ)dx_s:td θ,so that it is used to compute Bayes factors for model comparison. Since it is expressed asp(y_s:t)= p(y_t | y_s-1:t-1)/p(y_s-1| y_s:t)p(y_s-1:t-1),we obtain the estimate p̂(y_s:t) recursively byp̂(y_s:t)= p̂(y_t | y_s-1:t-1)/p̂(y_s-1| y_s:t)p̂(y_s-1:t-1),wherep̂(y_t | y_s-1:t-1) =∑^N_n=1 W^n_[s-1,t-1]p̂(y_t | x^n_t-K-1,y_s-1:t-1,θ^n),p̂(y_s-1| y_s:t) =∑^N_n=1 W^n_[s-1,t]p̂(y_s-1| x^n_s+K,y_s:t,θ^n),using (28), (29), (34) and (35). The initial estimate p̂(y_1:L+1), L=t-s is given byp̂(y_1:L+1)= p̂(y_1)∏^L+1_j=2p̂(y_j | y_1:j-1),where we use (<ref>), (<ref>) and (<ref>) to obtainp̂(y_1) =∑^N_n=1p̂(y_1 |θ^n), p̂(y_j | y_1:j-1) = ∑^N_n=1 W_j-1^np̂(y_j | x_j-K-1^n,y_1:j-1, θ^n).§ ADDITIONAL COMPARISON IN THE RSV MODELWe compare the computation time and the ESS of the practical double-block sampler with those of the MCMC and the particle MCMC. For the initial sample period (using y_1:1988), the MCMC sampling is implementedwith 10,000 iteration (2,000 MCMC samples in the burn-in period are discarded).Table <ref> shows the computation times[The total computation time for the MCMC and the particle MCMC to complete the rolling-window estimation is obtained by multiplying the computation time for the initial sample period by 2261. Thus we obtain 1,293 × 2,261= 2,923,473 and 3,189 × 2,261=7,210,329 respectively.]and ESSs[The ESS is computed as the average of the ESSs during the rolling estimations for our double-block sampler, while that for each parameter is computed as the MCMC sample size (10,000) divided by the inefficiency factor (defined as 1 + 2∑^∞_s=1ρ_s, where ρ_s is the MCMC sample autocorrelation at lag s).]for three methods.The recursive estimation using the standard MCMC or the particle MCMC takes 20-50 times longer than our proposed method. If we take account of the ESS, it would take 400-900 times longer. These results show that the computation time for our proposed method is much smaller compared with recursive estimations using the standard MCMC or the particle MCMC. | http://arxiv.org/abs/1709.09280v5 | {
"authors": [
"Naoki Awaya",
"Yasuhiro Omori"
],
"categories": [
"stat.CO",
"stat.ME"
],
"primary_category": "stat.CO",
"published": "20170926225740",
"title": "Particle rolling MCMC with double-block sampling"
} |
^1Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, Shaanxi Province,and Department of Applied Physics of Xi'an Jiaotong University, Xi'an 710049, P.R. China ^2Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics of University of Science and Technology of China, Hefei 230026, P.R. China One of the interesting topics in quantum contextuality is the construction for various non-contextual inequalities. By introducing a new structure called hyper-graph, we present a general method, which seems to be analytic and extensible, to derive the non-contextual inequalities for the qutrit systems. Based on this,several typical families of non-contextual inequalities are discussed. And our approach may also help us to simplify some state-independent proofs for quantum contextuality in one of our recent works. Graphical Non-contextual Inequalities for Qutrit Systems Weidong Tang^1 and Sixia Yu^2 December 30, 2023 ========================================================§ INTRODUCTION It is well known that a violation of a Bell inequality<cit.> can be used for refuting the local realism assumption ofquantum mechanics. More generally, any quantum violation for a non-contextual inequality<cit.> can be used to disprove the non-contextual assumption forquantum mechanics, and can be considered as another version of the proof for quantum contextuality or the Kochen-Specker(KS) theorem<cit.>. One of the proofs for the KS theorem is to find a contradiction of a KS value assignment — which claims that the value assignment to an observable (can only be assigned to one of its eigenvalues) is independent of the context it measured alongside — to a set of chosen rays. And a non-contextual inequality can be considered as a bridge to connect a logical proof of the KS theorem and a corresponding experimental verification<cit.>.Recently, to give a universal construction for the state-independent proof for quantum contextuality, we introduced a (6n+2)-ray model<cit.>.As a kind of special state-dependent proofs for quantum contextuality, this family of models, can induce a type of basic non-contextual inequalities. Based on this, we can analytically derive numerous non-contextual inequalities.In this paper, starting with the (6n+2)-ray model,weintroduce a kind of generalized graph which is called a “ hyper-graph". Then we give several interesting families of non-contextual inequalities from each kind of hyper-graphs. Finally, we give a rough analysis for the possible quantum violations for these non-contextual inequalities and find that our graphical KS inequality approach may help us to improve some state-independent proofs for the Kochen-Specker theorem in our anther work<cit.>.§ DESCRIPTION OF THE (6N+2)-RAY MODEL Conventionally,the notation of a ray(normalized unless emphasized) in the topic of quantum contextuality is more commonly used than its twoalternatives: a complex vector in the Hilbert space and a normalized rank-1 projector on the vector. To be specific, a ray |ψ_i⟩=α_i|0⟩+β_i|1⟩+γ_i|2⟩ (α_i,β_i,γ_i∈) can represent r_i=(α_i,β_i,γ_i) or or P=|ψ_i⟩⟨ψ_i|. Accordingly, the orthogonality and normalization, ⟨ψ_i|ψ_j⟩=δ_ij,can be written as r_i^∗r_j=δ_ij. For any two rays P_ϕ=|ϕ⟩⟨ϕ| and P_ψ=|ψ⟩⟨ψ|, if |⟨ψ|ϕ⟩|≤n/n+2, we can always add 2n complete orthonormal bases to build a (6n+2)-ray model<cit.>.The graphical representation for this model is shown in FIG.<ref>, where p_n and q_n stand for the rays P_ϕ and P_ψ respectively. And one can easily see the orthogonal relations for all these rays from this graph. Clearly, this model can be considered as a generalization of the Clifton's 8-ray model<cit.> as the latter is exactly the case for n=1. Analogous to the Clifton's 8-ray model, if we assign value 1 to two rays p_n and q_n simultaneously by the non-contextual hidden variable theory, it is not difficult for us to get a contradiction that two orthogonal rays p_0 and q_0 should also be assigned to value 1. This is why the (6n+2)-ray model can be considered as a proof for quantum contextuality, although in a state-dependent manner.We can also get a non-contextual inequalityfrom the (6n+2)-ray model. For some systems with complicated algebraic structures, a regular method to get the upper bounds for the non-contextual inequalities is by the computer search<cit.>. Though the (6n+2)-ray model is somewhat complicated,we have already derived the upper bound in Ref.<cit.> by a exact algebraic approach rather than by a computer search. Next, we give a brief introduction to this approach.First, it is clear that the case for the value assignment to a single ray or several independent (unconnected) rays is trivial. Thus the two-ray value assignment inequality fromFIG.<ref>-(a)can be considered as the simplest and nontrivial one. In spite offailure toshow thequantum contextuality by the inequality itself, it is quite useful in constructingmorecomplicatednon-contextual inequalities.We denote by each index of the vertex in FIG.<ref> the related ray. Then value 0 or 1 can be assigned to each of them, andan inequalityfor FIG.<ref>-(a) can be given by⟨_2⟩=p_1+p_2-p_1p_2≤1.The proof is straightforward.Let p̅_i=1-p_i∈{0,1}, i=1,2,then we have ⟨_2⟩=1-p̅_1p̅_2≤1.Note that we have omit the bra and ket notations in the expansion of ⟨_2⟩ as it will not cause any confusion in the classical case. Hereafter we may follow the same convention for simplicity. For FIG.<ref>-(b), the value assignment inequality is⟨_△⟩=∑_i=1^3p_i-∑_i<j∈{1,2,3}p_ip_j≤1.We can get the value assignment upper bound for ⟨_△⟩ with the help of ⟨_2⟩.For ∑_i=1^3p_i≤1, we can see clearly that ⟨_△⟩≤1 from its expansion in Eq.(<ref>), and for ∑_i=1^3p_i≥2, we can also get the same conclusion from ⟨_△⟩=3⟨_2⟩-∑_i=1^3p_i≤3-2=1.Note that here 3⟨_2⟩≡⟨_2⟩_1+⟨_2⟩_2+⟨_2⟩_3, where ⟨_2⟩_i=p_i+p_i+1-p_ip_i+1, (i=1,2,3) and p_4≡ p_1. In what follows, the similar notations will be used unless specified. Before discussing the KS value assignment inequality for the (6n+2)-ray model, we would like to introduce a special observable, which will be considered as a “hyper-edge" operator in the following text and can be defined asC(p_n,q_n)=∑_i∈v_i-∑_i<j,(i,j)∈v_iv_j-p_n-q_n,whereis the index set for all the vertices of the graph in FIG.<ref>-(c) andis an index representation forthe edge set. In other words, (i,j)∈ indicates that (v_i,v_j) is in the edge set of the graph. Then we can get the following value assignment inequality⟨ C(p_n,q_n)⟩≤2n.This can be derived from another form of ⟨ C(p_n,q_n)⟩, namely,⟨ C(p_n,q_n)⟩= 2n⟨_△⟩ -∑_i=1^np_i(α_i^++α_i^-) -∑_i=1^nq_i(β_i^++β_i^-)-p_0q_0≤2n.Let us return to the (6n+2)-ray model in FIG.<ref>-(c).Lemma. — The KS value assignment inequality for the (6n+2)-ray model can be given by⟨_n⟩=∑_i∈v_i-∑_i<j,(i,j)∈v_iv_j≤ 1+2n. Proof.— Here we give a proof which is different from the approach in Ref.<cit.>.First, Eq.(<ref>) holds for n=0 since ⟨_0⟩=⟨_2⟩.Assume that the statement is also true for n-1 (n≥1), namely, ⟨_n-1⟩≤2n-1. Then we we should prove that it holds for n. Notice that ⟨_n⟩ can also be written as⟨_n⟩= ⟨_n-1⟩+α_n^++α_n^-+β_n^++β_n^-+p_n+q_n -p_n-1(α_n^++β_n^+)-q_n-1(α_n^-+β_n^-) -p_n(α_n^++α_n^-) -q_n(β_n^++β_n^-) -α_n^+β_n^+-α_n^-β_n^-,or⟨_n⟩= ⟨_n-1⟩+2⟨_△⟩+4⟨_2⟩ -(α_n^++α_n^-+β_n^++β_n^-+p_n+q_n) -(p_n-1+q_n-1)= ⟨ C(p_n-1,q_n-1)⟩+2⟨_△⟩+4⟨_2⟩ -(α_n^++α_n^-+β_n^++β_n^-+p_n+q_n). (i)For the case of α_n^++α_n^-+β_n^++β_n^-+p_n+q_n≤2,we have ⟨_n⟩≤⟨_n-1⟩+2≤2n+1 by Eq.(<ref>); (ii)and when α_n^++α_n^-+β_n^++β_n^-+p_n+q_n≥3, according to Eq.(<ref>), we can get ⟨_n⟩≤2(n-1)+2+4-3=2n+1.Therefore, ⟨_n⟩≤2n+1 holds for any non-negative integer n.♯Next, some definitions from graph theory should be given before discussing our main results. § AN INTRODUCTION TO THE HYPER-GRAPHS It is known that one of the original constraints for the KS value assignment requires that two mutually orthogonal rays cannot be assigned to value 1 simultaneously. This constraint can be generalized to two ordinary rays by the (2n+6)-ray model. To be specific, if two rays |ϕ⟩ and |ψ⟩ satisfy |⟨ψ|ϕ⟩|≤n/n+2, they can always generate a (2n+6)-ray model by adding 2n auxiliary complete orthonormal bases, such that |ϕ⟩ and |ψ⟩ can not be both assigned to value 1. This motivates us to defined a new graphical structure to enrich theoriginal graphical representation. Later we will see that thisstructure will facilitate us to analytically derive the upper bounds for various non-contextual inequalities. Notice that we have already presented a systematic and programmable approach to construct a state-independentproof of the KS theoremfor the first time in Ref.<cit.> based on the (6n+2)-ray model. Actually, a state-dependent proof which seems much simpler, can also be constructed by the same method via reducing some constraints. Next we give a brief review on this approach.First, we choose several nonspecific (usually nonparallel or nonorthogonal) rays as a fundamental ray set ={|ψ_i⟩}_i∈ I (or ={P_i|P_i=|ψ_i⟩⟨ψ_i|}_i∈ I), where I is an index set and the number of rays inis |I|. Then for a fixed N (N>0,N∈), considering any two rays |ψ_k⟩ and |ψ_l⟩ from , if they satisfy |⟨ψ_k|ψ_l⟩|∈(n-1/n+1,n/n+2] (0<n≤ N,n∈), we can economically build a (6n+2)-ray model by adding 2n extra complete orthonormal bases. Take FIG.<ref> or FIG.<ref>-(c) for example, what we need to do is just to replace p_n and q_n with P_k=|ψ_k⟩⟨ψ_k| and P_l=|ψ_l⟩⟨ψ_l| respectively. And an n-weighted hyper-edge linking the two rayscan be defined as all the rays from the complete orthonormal bases together with all the edges from the original graphical representation for this (6n+2)-ray model, see FIG.<ref>-(b), where each orange line represents a hyper-edge.Repeat this operation to other pairs of rays in , and we can construct a proof for the KS theorem and get the corresponding hyper-graph G. Clearly, the simplest nontrivialhyper-graph is exactly the representation for the (6n+2)-ray model(FIG.<ref>-(a)).We denote by V and Ethe vertex set and the hyper-edge set of a hyper-graph G respectively. Without loss of generality, we denote V as V=={P_i|i=1,2,...,|V|} and we have |V|=|I|. Note that a 0-weighted hyper-edge between two vertices is an edge of a normal graph. And do not confuse two unconnected vertices with a two-vertex hyper-graph whose hyper-edge is 0-weighted. From this point of view, a normal graph is only a special case of a hyper-graph. This is why we use the same notation G to denote them for simplicity.For each hyper-graph G,we can associate with the following KS observable with respect to the non-contextual inequality<cit.>G=∑_i=1^|V|P_i+∑_i=1^|V|-1∑_j∈_i,j>i C(P_i,P_j),where C(P_i,P_j)defined by Eq.(<ref>) can be referred to as a hyper-edge observable, and _i stands for index set for the neighborhood of the vertex P_i, i.e., if j∈_i, then (P_i,P_j)∈ E. For our optimal construction of a proof for theKS theorem (by adding the minimum number of complete orthonormal bases between any two rays in V),C(P_i,P_j) vanishes if |⟨ψ_i|ψ_j⟩|>N/N+2 and involves 2n_ij complete orthonormal bases wheren_ij=⌈2|⟨ψ_i|ψ_j⟩|/1-|⟨ψ_i|ψ_j⟩|⌉.From Eq.(<ref>), we can get ⟨ C(P_i,P_j)⟩≤ 2n_ij. But for a non-optimal construction, the number of the orthonormal bases corresponding to C(P_i,P_j) isusually larger than n_ij and C(P_i,P_j) vanishes if (P_i,P_j)∉ E. In what follows, we only care about the non-contextual inequality from a given hyper-graph rather than the construction of a proof for KS theorem. Hence other problems such as the optimization for n_ij will be ignored. A vertex set U (U⊂ V)is called a maximal unconnected vertex set of a hyper-graph G if, (i) for any two vertices P_i and P_j in U, (P_i,P_j)∉ E; (ii) for any other set U^'⊂ V satisfies (i), the vertex number |U^'|≤|U|. Usually, this set is not unique. Such an example is given in FIG.<ref>.Let us return to the case of normal graphs. A subgraph G^' of a graph G is also a graph whose vertex set satisfies V^'⊂ V and whose edge set E^' consists of all of the edges in E that have both endpoints in V^'. But here V and E only stand for the vertex set and the edge set of the normal graph G, which is different from the notations referred above. This definition can be easily generalized to the hyper-graph case replacing the edgewith hyper-edge. And we are supposed tocall it “sub-hyper-graph", but for convenience we would still refer to it as “subgraph".FIG.<ref> gives us an example for all the five-vertex subgraphs from a six-vertex hyper-graph.If we denote by G^i the subgraph obtained by removing the vertex P_i and all the edges with one of the endpoint P_i in G, then it holds⟨ G⟩=⟨ G^i⟩+⟨ P_i⟩+∑_j∈_i⟨C(P_i,P_j)⟩.And we can get|V|⟨ G⟩ =∑_i=1^|V|⟨ G^i⟩+∑_i=1^|V|⟨ P_i⟩+∑_i=1^|V|∑_j∈_i⟨C(P_i,P_j)⟩ =∑_i=1^|V|⟨ G^i⟩-∑_i=1^|V|⟨ P_i⟩+2⟨ G⟩or a more compact form<cit.>(|V|-2)⟨ G⟩ = ∑_i=1^|V|⟨ G^i⟩-∑_i=1^|V|⟨ P_i⟩.This can be considered as a relation of the subgraph decomposition.§THREE TYPICAL NON-CONTEXTUAL INEQUALITIES Let us consider some typical hyper-graphical structures and the related non-contextual inequalities. Here we mainly discuss three different families of hyper-graphs. Based on the hyper-graphical representation for the (6n+2)-ray model or Lemma, i.e., ⟨_2⟩=⟨_2⟩=⟨_n⟩≤2n+1 (see FIG.<ref>), we can derive the non-contextual inequalities analytically from the hyper-graphs with more vertices in FIG.<ref>. §.§ Complete hyper-graphs Analogous to the definition of a complete graph in graph theory, a complete hyper-graph is a hyper-graph in which every pair ofvertices is connectedby a hyper-edge. Then we can get the following theorem.Theorem 1.— The non-contextual inequality associated witha k-vertex complete hyper-graph (k≥2) can be written as⟨_k⟩=∑_i=1^kp_i+∑_i=1^k-1∑_j=2,j>i^k⟨ C(p_i,p_j)⟩≤2∑_i=1^k(k-1)/2n_i+1,where p_i is the i-th ray (vertex) and n_i stands for the weight for the i-th hyper-edge in FIG.<ref>-(a).Proof.— Clearly the statement is true for k=2.Assume thatEq.(<ref>) holds for any ⟨_k-1⟩ (k≥3).We should prove thatEq.(<ref>) also holds for ⟨_k⟩. From Eq.(<ref>), we have(k-2)⟨_k⟩=∑_i=1^k⟨_k^i⟩-∑_i=1^kp_i .(i)If ∑_i=1^kp_i≤1, we can see thatEq.(<ref>) still holds from Eq.(<ref>) and the expansion for ⟨_k⟩ in Eq.(<ref>).(ii)If ∑_i=1^kp_i≥2, from another form of ⟨_k⟩ referred above, we have(k-2)⟨_k⟩≤ k+(k-2)·2∑_i=1^k(k-1)/2n_i-2,namely,⟨_k⟩≤2∑_i=1^k(k-1)/2n_i+1.Therefore, Eq.(<ref>) holds for any ⟨_k⟩ (k≥2). ♯ §.§ Linear hyper-graphs Likewise, for the linear hyper-graph shown in FIG.<ref>-(b), we have the following theorem.Theorem 2.— For a k-vertex linear hyper-graph, the non-contextual inequality can be given by⟨_k⟩=∑_i=1^kp_i+∑_i=1^k-1⟨ C(p_i,p_i+1)⟩≤2∑_i=1^k-1n_i+⌈k/2⌉,where n_i is the weight for the relevant hyper-edge in FIG.<ref>-(b).Proof.— Clearly, Eq.(<ref>) holds for k=2.Assume that it is also true for ⟨_k-1⟩ (k≥3). Next let us prove that Eq.(<ref>)holds for ⟨_k⟩. (i)From the expansion of ⟨_k⟩ in Eq.(<ref>), it is clear that the inequality holds for the case ∑_i=1^kp_i≤⌈k/2⌉.(ii)If ∑_i=1^kp_i≥⌈k/2⌉+1, we can use another form of⟨_k⟩, which reads ⟨_k⟩=(k-1)⟨_2⟩+p_1+p_k-∑_i=1^kp_i. Therefore, ⟨_k⟩≤2∑_i=1^k-1n_i+k-1+2-∑_i=1^kp_i≤ 2∑_i=1^k-1n_i+k-⌈k/2⌉≤ 2∑_i=1^k-1n_i+⌈k/2⌉. Hence Theorem 2 holds for all possible KS value assignments to the related rays. ♯ §.§ Cyclic hyper-graphs Anothernon-contextual inequality from the cyclic hyper-graph in FIG.<ref>-(c) is shown in below.Theorem 3.— For a k-vertex cyclic hyper-graph, the non-contextual inequality can be given by⟨_k⟩=∑_i=1^kp_i+∑_i=1^k⟨ C(p_i,p_i+1)⟩≤2∑_i=1^kn_i+⌊k/2⌋.where n_i represents the weight for the hyper-edge linking the rays p_i and p_i+1, and p_k+1≡ p_1.Proof.— Similar to the proof in Theorem 2, we can also write ⟨_k⟩ in another form⟨_k⟩=k⟨_2⟩-∑_i=1^kp_i .(i)By the original expression for ⟨_k⟩ in Eq.(<ref>), it is clear that the inequality holds for the case ∑_i=1^kp_i≤⌊k/2⌋.(ii)If ∑_i=1^kp_i≥⌊k/2⌋+1, we can use the above second form of⟨_k⟩. That is⟨_k⟩≤2∑_i=1^kn_i+k-∑_i=1^kp_i≤ 2∑_i=1^kn_i+k-1-⌊k/2⌋≤ 2∑_i=1^kn_i+⌊k/2⌋. Therefore, Theorem 3 holds for any KS value assignment. ♯§ NON-CONTEXTUAL INEQUALITY FOR AN ORDINARY HYPER-GRAPH For any ordinary hyper-graph, we give a theorem which is equivalent to a conclusion in Ref.<cit.> to describe the non-contextual inequality.Theorem 4.— For a k-vertexhyper-graph G_k,we can always find at least one maximal unconnected vertex set U. If we denote by p_i and n_j the i-th vertex and the weight for the j-th hyper-edge, then we have the following non-contextual inequality⟨ G_k⟩=∑_i=1^kp_i+∑_i=1^k-1∑_j∈_i,j>i⟨ C(p_i,p_j)⟩≤2∑_i=1^|E|n_i+|U|,where E is the hyper-edge set.Proof.—Equivalently, we can prove it by verifying another proposition. That is, if such an inequality can be proved to be true for all the possible hyper-graphs with a fixed maximal unconnected vertex set U (|U|<k), then it also holds for a hyper-graph with k vertices.We denote by V the vertex set of the hyper-graph G_|V|, and label the vertices in U by p_1,p2,...,p_|U|.The case for |V|=|U| is trivial.For |V|=|U|+1, at least one hyper-edge with endpoints p_|U|+1 and some vertex in U can be found by the definition of the maximal unconnected vertex set. If E={(p_|U|+1,p_α_1),(p_|U|+1,p_α_2),...,(p_|U|+1,p_α_l)}. Then we have⟨ G_|U|+1⟩= ∑_i=1,i≠α_1^|U|p_i+⟨_2⟩_p_|U|+1p_α_1 +∑_i=2^l⟨ C(p_|U|+1,p_α_i)⟩≤ |U|-1+2n_1+1+2∑_i=2^|E|n_i = 2∑_i=1^|E|n_i+|U|. Assume thatEq.(<ref>) holds for all the hyper-graphs with |V|=|U|+m (m>1). Then for |V|=|U|+m+1, if ∑_i=1^|U|+m+1p_i≤|U|, it is clear that Eq.(<ref>) is true. Therefore, we only need to check the case for ∑_i=1^|U|+m+1p_i≥|U|+1. From the subgraph decomposition relation Eq.(<ref>), we have[(|U|+m+1)-2]⟨ G_|U|+m+1⟩= ∑_α=1^|U|+m+1⟨ G_|U|+m^α⟩-∑_i=1^|U|+m+1p_i ≤ (|U|+m+1)|U|+2(|U|+m+1-2)∑_i=1^|E|n_i-(|U|+1)= (|U|+m-1)(|U|+2∑_i=1^|E|n_i)+(|U|-1).Thus ⟨ G_|U|+m+1⟩≤2∑_i=1^|E|n_i+|U|+|U|-1/|U|+m-1. Since for any value assignment, ⟨ G_|U|+m+1⟩ should be an integer, and |U|-1/|U|+m-1<1, then ⟨ G_|U|+m+1⟩≤2∑_i=1^|E|n_i+|U|. Hence theEq.(<ref>) holds for any hyper-graph with a maximal unconnected vertex set U.As a special case, it also holds for G_k.♯By Theorem 4, one can also derive the Eqs.(<ref>,<ref>,<ref>) by counting the number of the vertices in their maximal unconnected vertex sets.§ NON-CONTEXTUAL INEQUALITIES FOR SOME FRACTAL STRUCTURES If the vertex number of an ordinary hyper-graph is large, then for any KS value assignment, the upper bound for its non-contextual inequality is very difficult to calculate. But in some special cases, analytical formulas for the upper bounds can be recursively derived. We have already given three families of such examples in previous sections. Here we presenttwo more examples, which come from the fractal hyper-graphs.Considering a fractal hyper-graph family, e.g. FIG.<ref>-(a), it is not difficult for us to notice that some former methods to derive the upper boundof a non-contextualinequalitymay not work effectively in this scenario, e.g., the way used in Theorem 2 and Theorem 3. But fortunately another approach by Theorem 4 seems to be a nice choice. As for some fractal hyper-graph structures, it is easy to find out their maximal unconnected vertex sets.For the fractal hyper-graph families {_k}_k∈ℕ and {_k}_k∈ℕFIG.<ref>, if we denote by U_^k (U_^k) the maximal unconnected vertex set for the k-th graph in {_k}_k∈ℕ ({_k}_k∈ℕ), then|U_^k|= 2^k·1-(1/4)^⌊k/2⌋+1/1-1/4 =4/3·(2^k-2^k2-2); |U_^k|= 2^k-1.Therefore, we have⟨_k⟩= ∑_i=1^2^k+1-1p_i+∑_i=1^2^k-1(⟨ C(p_i,p_2i)⟩+⟨ C(p_i,p_2i+1))≤ 2∑_i=1^2^k+1-2n_i+|U_^k|,and⟨_k⟩ = ∑_i=1^3·(2^k-1)p_i+⟨ C(p_1,p_2)⟩+⟨ C(p_1,p_3)⟩+⟨ C(p_2,p_3)⟩ +∑_i=1^3·(2^k-1-1)(⟨ C(p_i,p_2i+2)⟩+⟨ C(p_i,p_2i+3)⟩ +⟨ C(p_2i+2,p_2i+3)⟩)≤ 2∑_i=1^9·(2^k-1-1)+3n_i+|U_^k|. § NON-CONTEXTUAL INEQUALITIES FOR SOME LATTICE HYPER-GRAPHS In the end, let us consider two typical lattice hyper-graph families, see FIG.<ref>.We denote by _m_xm_y and _m_xm_y the square latticehyper-graph and the torus latticehyper-graph of m_x× m_y vertices respectively. We can get the classical upper bounds of their non-contextual inequalities by calculating the numbers of vertices, |U_^m_xm_y| and |U_^m_xm_y|, in their maximal unconnected vertex sets U_^m_xm_y and U_^m_xm_y. It is clear that |U_^m_xm_y| and |U_^m_xm_y| can be written as|U_^m_xm_y|= ⌈m_xm_y/2⌉;|U_^m_xm_y|= ⌊min{m_x,m_y}/2⌋·max{m_x,m_y}.Then, from Theorem 4, the non-contextual inequality for the square latticehyper-graph can be given by⟨_m_xm_y⟩= ∑_i=1^m_x∑_j=1^m_yp_i,j +∑_i=1^m_x-1∑_j=1^m_y⟨ C(p_i,j,p_i+1,j)⟩ +∑_i=1^m_x∑_j=1^m_y-1⟨ C(p_i,j,p_i,j+1)⟩≤ 2(∑_i=1^m_x-1∑_j=1^m_yn_i,j;i+1,j+∑_i=1^m_x∑_j=1^m_y-1n_i,j;i,j+1) +|U_^m_xm_y|,where p_ij is the vertex on the site (i,j) and n_i,j;i+1,j (n_i,j;i,j+1) is theweight of the hyper-edge (p_i,j,p_i+1,j) ((p_i,j,p_i,j+1)).Likewise, for the torus latticehyper-graph,as m_x, m_y≥3, we can get the following non-contextual inequality⟨_m_xm_y⟩= ∑_i=1^m_x∑_j=1^m_yp_i,j +∑_i=1^m_x∑_j=1^m_y(⟨ C(p_i,j,p_i+1,j)⟩ +⟨ C(p_i,j,p_i,j+1)⟩)≤ 2(∑_i=1^m_x∑_j=1^m_yn_i,j;i+1,j+∑_i=1^m_x∑_j=1^m_yn_i,j;i,j+1) +|U_^m_xm_y|, where p_m_x+1,j≡ p_1,j (p_i,m_y+1≡p_i,1) and n_m_x,j;m_x+1,j≡ n_m_x,j;1,j (n_i,m_y;i,m_y+1≡ n_i,m_y;i,1).Other models such like cubic lattice hyper-graphs can also be discussed by using the same method. § QUANTUM VIOLATIONS To see the quantum violation for the non-contextual inequality for a k-vertex ordinary hyper-graph G_k, the key is to calculate the range of the eigenvalues for ∑_i=1^kp_i. As for any hyper-edge observable C(p_i,p_j), from the view of the complete orthonormal bases,the quantum expectation is strictly equal to 2n_ij, where n_ij is the corresponding hyper-edge weight. We denote by λ_min the minimal eigenvalue for ∑_i=1^kp_i. Then We have ⟨ G_k⟩_q^min=2∑_i=1^|E|n_i+λ_min, where the expression for G_k can be found in Eq.(<ref>) and the notation ⟨·⟩_q represents the quantum expectation. If λ_min>|U|,then Eq.(<ref>) provide us a state-independent non-contextual inequality. An equivalent conclusion can also be found in Ref.<cit.>. For other cases,it is at best a state-dependent non-contextual inequality.Besides calculating the eigenvalue for the sum of all the vertices in a hyper-graph, it seems that the relative size(compared with the vertex number of the hyper-graph) for a maximal unconnected vertex set of a hyper-graph may be one of the key factors in testing the quantum violation for the non-contextual inequality. Although sometimes it may not works very well, it can help us to get a preliminary estimation for the possibility of a quantum violation. From this point of view, the non-contextualinequality for the complete hyper-graph family in FIG.<ref>-(a) seems to be the most possible case for a quantum violation. As the maximal unconnected vertex set is just a single vertex, and ⟨∑_i=1^kp_i⟩_q≥λ_min≥1 is easy to be satisfied. But the main shortcoming is that the number of the hyper-edges might be too large. To balance that, we try to choose the hyper-graphs with less hyper-edges, but might still have a state-independent quantum violation. Here wegive an example in FIG.<ref> From Theorem 4, as |U|=2, the non-contextual inequality can be given by⟨_7⟩ =∑_i=1^7p_i+∑_i=1^7(⟨ C(p_i,p_i+1)+⟨ C(p_i,p_i+3)⟩) ≤2∑_i=1^14n_i+2,where p_8,9,10≡ p_1,2,3 and n_i is the weight for the i-th hyper-edge. To see the quantum violation, we choose {p_1,p_2,p_3,p_4} to be the 4 core rays(vertices) of the Yu-Oh model<cit.>, namely, p_1,p_2,p_3,p_4 are orienting to 4 vertices of a regular tetrahedron, and let {p_5,p_6,p_7} be an approximate orthonormal basis (e.g. with an error of δ<0.01), and also make sure that there are no parallel or antiparallel relationships for these rays,then ⟨∑_i=1^7p_i⟩_q=4/3+1+O(δ)≈7/3>|U|. And we can get a state-independent non-contextual inequality Eq.(<ref>), with a reduction of 7 hyper-edges compared with the extreme case of a 7-vertex complete hyper-graph.We can see that the inequality approach based on hyper-graphs sometimes may help us to optimize the method for construction of a state-independent proof for quantum contextuality in Ref.<cit.> from the above example. In other words, by this method we may get a more economical proof for quantum contextuality with less auxiliary complete orthonormal bases (hyper-edges) but still in a state-independent manner.Constraints for state-dependent non-contextual inequalities by other models referred in previous sections are listed in the following table,where λ_max is the maximal eigenvalue for the corresponding ∑_i=1^|V|p_i (or ∑_i=1^m_x∑_j=1^m_yp_i,j) term. § CONCLUSION AND DISCUSSION We have discussed a general method for deriving the non-contextual inequalities based on the hyper-graphs for the qutrit systems.Several interesting families of non-contextual inequalities are given. Our method can be applied to any hyper-graph by a subgraph decomposition relation. This relation might be very useful in looking for further interesting relations from other possible correlated structures. We also give the conditions for quantum violations of different types of non-contextual inequalities. Besides, our graphical methods might be helpful to improve the construction for state-independent proofs for quantum contextuality in our anther recent work<cit.>. Moreover, we notice that the mathematical structures of certain non-contextual inequalitiesand the Hamiltonians for some systems in condensed matter physics are similar.This might motivate us to give a further research on the link between them and try to look for a new method to learn some many-body physical systems.This work is supported by the NNSF of China (Grant No. 11405120) and the Fundamental Research Funds for the Central Universities. 99 BellJ. S. Bell, Physics 1, 195 (1964). KCBS A. A. Klyachko, M. A. Can, S. Binicioǧlu, and A. S. Shumovsky, Phys. Rev. Lett. 101, 020403 (2008). cabello0801A. Cabello, S. Filipp, H. Rauch, and Y. Hasegawa, Phys. Rev. Lett. 100, 130404 (2008). cabello08 A. Cabello, Phys. Rev. Lett. 101, 210401 (2008). yu-ohS. Yu and C.H. Oh, Phys. Rev. Lett. 108, 030402 (2012). cabello1201M. Kleinmann, C. Budroni, J.-Å. Larsson, O. Gühne, and A. Cabello, Phys. Lett. Lett. 109, 250402 (2012). TYO W. Tang, S. Yu, and C. H. Oh, Phys. Rev. Lett. 110, 100403 (2013). NS condi A. Cabello, M. Kleinmann, and C. Budroni, Phys. Lett. Lett. 114, 250402(2015). Bell2J. S. Bell, Rev. Mod. Phys. 38, 447 (1966). KSS. Kochen and E.P. Specker, J. Math. Mech. 17, 59 (1967). mermin1N.D. Mermin, Rev. Mod. Phys. 65, 803 (1993).Lapkiewicz R. Lapkiewicz, P. Li, C. Schaeff, N. K. Langford, S. Ramelow, M. Wieśniak, and A. Zeilinger, Nature (London) 474, 490 (2011). Zu C. Zu, Y.-X.Wang, D.-L. Deng, X.-Y. Chang, K. Liu, P.-Y. Hou, H.-X. Yang, and L.-M. Duan, Phys. Rev. Lett. 109, 150401 (2012). Vincenzo V. D'Ambrosio, I. Herbauts, E. Amselem, E. Nagali, M. Bourennane, F. Sciarrino, and A. Cabello,Phys. Rev. X3, 011012 (2013). XiangZhang X. Zhang, M. Um, J.H. Zhang, S. An, Y. Wang, D.-L. Deng, C. Shen, L.-M. Duan, and K. Kim, Phys. Rev. Lett. 110, 070401 (2013). Huang2 Y.-F. Huang, M. Li, D.-Y. Cao, C. Zhang, Y.-S. Zhang, B.-H. Liu, C.-F. Li, and G.-C. Guo, Phys. Rev. A 87, 052133 (2013).Tang-yuWeidong Tang, Sixia Yu, arXiv:1707.05626.Clifton R. Clifton, Am. J. Phys. 61, 443 (1993). | http://arxiv.org/abs/1709.09706v1 | {
"authors": [
"Weidong Tang",
"Sixia Yu"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170927191613",
"title": "Graphical Non-contextual Inequalities for Qutrit Systems"
} |
These authors contributed equally to this work.These authors contributed equally to this [email protected] Department of Physics, Isfahan University of Technology, Isfahan, 84156-83111, IranEvolutionary algorithm is combined with full-potential ab-initio calculations to investigate conformational space of (MoS_2)_n and (MoSe_2)_n (n=1-10)nanoclusters and to identify the lowest energy structural isomers of these systems. It is argued that within both BLYP and PBE functionals, these nanoclusters favor sandwiched planar configurations, similar to their ideal planar sheets. The second order difference in total energy (Δ_2E) of the lowest energy isomers are computedto estimate the abundance of the clusters at different sizes and todetermine the magic sizes of (MoS_2)_n and (MoSe_2)_n nanoclusters. In order to investigate the electronic properties of nanoclusters,their energy gap is calculated by several methods, including hybrid functionals (B3LYP and PBE0), GW approach, and Δscf method. At the end, the vibrational modes of the lowest lying isomers are calculated by using the force constants method and the IR active modes of the systems are identified. The vibrational spectra are used to calculate the Helmholtz free energy ofthe systems and then to investigate abundance of the nanoclusters at finite temperatures.First-principles study of MoS_2 and MoSe_2 nanoclusters in the framework of evolutionary algorithm and density functional theory Hadi Akbarzadeh December 30, 2023 ========================================================================================================================================§ INTRODUCTION Transition Metal Dichalcogenides (TMDCs) with chemical formula MX_2 are semiconducting compounds made of hexagonal sheets,where inside a sheet one layer of metal atom M (Mo, W, Ga, V, Sn, Te)is sandwiched between two layers of chalcogen atom X (S, Se, Te). The weak Van der Waals interaction between MX_2 sheets provides the opportunityto extract two dimensional (2D) semiconductors with fascinating applications in supercapacitors <cit.>, solar cells <cit.>, lithium batteries <cit.>, hydrogen evolution in fuel cells <cit.>, optoelectronics devices <cit.> and gas sensor <cit.>. Energy gap of these 2D semiconductors lies between 1-2 eV and compared with graphene, exhibit about 25% more resistance to pressure and strain <cit.>.In addition to the 2D MX_2 sheets, other forms of MX_2 nanostructures have also attracted considerable attention <cit.>. Nanoparticles of these materials exhibit great potential applications as catalysts in hydrogen evolution reaction and hydrogen desulfurization <cit.>. For instance, MoS_2 nanoparticles have been used as catalyst in desulfurizationprocesses of raw oil <cit.>. MoSe_2 nanofilms and nanosheets may also catalyze hydrogen evolution reactionand regeneration of I^- species <cit.>. Very recently biosensor applications of MoS_2 nanocompositeswere also reported <cit.>. Moreover, the high chemical stability of these nanostructures make them very proper lubricants in strongly oxidizing environments <cit.>.In this work we investigate structural and electronic properties ofMoS_2 and MoSe_2 nanoclusters. Experimental observations show that these nanoclusters at large sizesprefer triangular flat plates shape <cit.>. However, the available experimental techniques may not be ableto identify the atomic structure of very small clusters, hence accurate computational simulations are valuable complementary techniques for investigation of the atomic structureof very small atomic nanoclusters <cit.>. Recent proposed algorithm for theoretical structure search are found to be very powerful and reliable for identifying the most stable and metastable atomic configuration of crystals and nanostructures <cit.>. We will use evolutionary algorithm and first-principles calculationsto identify the lowest energy atomic configurations of (MoX_2)_n (X = S, Se) nanoclusters for n<10. After finding the lowest energy configurations, the magic numbers, electronic properties, and vibrational spectraof these nanoclusters will be discussed in detail. § COMPUTATIONAL DETAILS Our structure search was performed in the framework of evolutionary algorithmdeveloped by Oganov et al. andfeaturing local optimization, real-space representation and flexiblephysically motivated variation operators <cit.>.This approach starts with a generation of trial structures, which are randomly generated or taken from some seed structures, and utilizes an auxiliary total energy code to minimize the energy of the structures. Then a series of refining and production operators such asheredity and mutation are applied to the lowest energy structures ofthe previous generation to create an improved generation of trial structures. This process is repeated until achieving a stop criteria,which is persistence of a specific number of lowest lying (in energy) structuresin a certain number of generations. In order to avoid algorithm to be trapped in the local valleys ofthe Born-Oppenheimer energy landscape <cit.>, proper operators for detecting and deleting equivalent structures beforestructural optimization was applied. For bigger clusters, more diverse structures are possible, and more cases should be investigated. So, in order to ensure the reliability of the structure search, we increased the value of population size from 10 to 45, and used stricter stop criteria, meaning that the number of generations and lowest energy isomers inthe stop criterion were increased from 15 to 55 and from 8 to 15, respectively. It should be noted that for generating a new population of trial structures, during the structure searches, 50% of the new structures were generated by heredity operator, 10% by random operator, 10% by permutation, and 20% by softmutation operator. Moreover, the structures obtained from Ref.<cit.> have been added to our structure search, as seed structures.The total energy calculations and minimizations of structures were performed in the frameworkof Kohn-Sham density functional theory (DFT) by using the all-electron full-potential methodimplemented in FHI-aims package<cit.>.This code employs numeric atom-centered orbital (NAO) basis functionswhich are very efficient for computation of non-periodic systems. In order to increase the computational performance, the geometrical relaxation ofthe clusters was done in four steps:initial relaxation with a light basis set, ignoring relativistic and spin-polarization effects, secondary relaxation by turning on the relativistic effects at scalar level and a tighter basis set,third step with addition of spin-polarization effect, and final relaxation with a higher number of basis functions (tight+tier2). The relaxation process of all structures was performed down toresidual atomic forces of less than 10^-3eV/Å, while for calculating vibrational frequencies and IR spectra ofthe lowest energy isomers, a maximum atomic force of 10^-4eV/Å, was considered.The structural relaxations were performed within Becke-Lee-Yang-Parr (BLYP)<cit.> functional, which, compared with Perdew-Burke-Ernzerhof (PBE) functional,sounds to be more accurate for studying molecular systems. <cit.>. It is argued that computational error of exchange energywithin BLYP and PBE functionals are roughly equal, while correlation energy of molecular systemswithin BLYP is about 4 kcal/mol more accurate <cit.>.For more accurate study of the electronic properties,the HOMO-LUMO gap of the lowest energy isomers was calculated within four functionals;two generalized gradient approximations (GGA), BLYP and PBE, and two hybrid functionals, namely B3LYP <cit.> and PBE0 <cit.>. Moreover, the many body based G_0W_0 and the total energy based ΔSCFtechniques were applied to increase the accuracy of the calculated energy gaps.§ RESULTS AND DISCUSSION§.§ Stable and Metastable Isomers The obtained lowest energy structural isomers of (MoS_2)_n and(MoSe_2)_n (n=1-10) clusters are shown in figures <ref>and <ref>, respectively, sorted by their minimized energywithin the BLYP functional.In the size of n=1, the equilibrium X-Mo-X angle in MoS_2 and MoSe_2is 113^ o and the equilibrium Mo-X bond length is 2.13Å and 2.27Å, respectively. The larger length of Mo-Se bond is due to larger atomic radius of Se, compared with S. The Mo-S and Mo-Se bond lengths in the ideal MoX_2 sheet are 2.45Å and 2.58Årespectively,which are approximately 12% longer than cluster values. Lower coordination number of atoms in the nanoclusters gives rise to compressed atomic bonds, compared with the corresponding periodic system. In contrast to murugan2005a, we found that linear configuration of these clusters are dynamically unstable. In the second size (n=2), a direct Se-Se bond is seen in Mo_2Se_4, while the situation in Mo_2S_4 is different. It may be attributed to the lower electronegativity of Se (2.43, Allen scale), compared with S (2.58), which gives rise to less ionic charge transfer from Se to Mo and thus more chemical activity of Se ion, compared with S ion. It should be noted that the second isomer of murugan2005a for Mo_2X_4 coincides with our sixth isomer, evidencing reliability of our structure search.From the third size (n=3), some sandwiched configurations startto appear in the lowest energy isomers.In these configurations, similar to an ideal MoX_2 sheet, a plane of Mo atoms is sandwiched between layers of X atoms. The remarkable point is that with increasing the cluster size,their tendency to form two dimensional triangular structures increases. Some examples are 6b, 7b, 8b, and 8c isomers of MoS_2, 6c, 6d, 7a, 8a, 8b, and 8c isomersof MoSe_2, and stable and several metastable isomers ofsizes 9 and 10.For better understanding of the bonding properties of the systems, we follow the conventions of murugan2005ato classify the X atoms of MoX_2 clusters in three different groups:the X atom can bond toI. one Mo atom (terminal atom, X_T),II. two Mo atoms (bridging atom, X_B), and III. three Mo atoms (face capping atom, X_C). Consequently, four kinds of atomic bonds may occur in the systems, Mo-Mo, Mo-X_T, Mo-X_B, and Mo-X_C bonds. The Mo-X_C bond happens at the central part of the clusters, while Mo-X_B and Mo-X_T bonds appear at the cluster edges and corners. Some bonding properties of MoS_2 and MoSe_2 clusters are presented in table <ref>. Generally, increment in the cluster size, increases the number of Mo-X_C bonds. The average Mo-X_C bond length in MoS_2 and MoSe_2 clusters is 2.37Åand 2.5Å, respectively, which is close to the corresponding values in the ideal sheets. It is seen that the shortest bond length in all clusters belongs to Mo-X_T bonds. Lower coordination number of X_T atoms enhances the strength of Mo-X_T bonding, compared with Mo-X_B and Mo-X_C bondings.For more accurate identification of the planar isomers,the Root Mean Square Deviation (RMSD) of Mo atoms from a perfect planar geometry for some lowest energy isomers of the clusters is calculatedand listed in Figs. <ref> and <ref>. It is generally concluded that, unlike the most stable isomersof MoS_2 in previous studies<cit.> whichinvolve polyhedral core of Mo atoms covered with sulfur atoms, sandwiched planar configurations are favored by MoS_2 and MoSe_2 nanoclusters.Experimental observations in larger MoX_2 nanoclusters (n=10-100)confirm favorability of flat triangular shapes in these systems <cit.>. However, it should be noted that our predicted flat configurations are more similar to the 1T structure of TMDC sheets, which is metastable compared withthe stable 2H structure of these 2D compounds. Observation of 1T configuration is attributed to the fixed stoichiometry of the investigated nanoclusters, while stabilizing 2H triangular clusters requires extra chalcogen atomsat the edges <cit.>.While direct structural observation of very small clusters is hardly feasible, our accurate structure search provides reliable evidence for favorability offlat triangular configurations even in very small MoS_2 and MoSe_2 nanoclusters. In contrast to carbon clusters which prefer cage-like or fullerene-likestructures to saturate their surface dangling bonds, MoX_2 nanoclusters tend to form planar structureswith possible extra X atoms at the edges and cornersto saturate dangling bonds <cit.>.Comparing the stable and metastable isomers of MoS_2 and MoSe_2 with those reported for WS_2 nanoclusters <cit.>, clarifies that from structural point of view, MoS_2 isomers are more similar to MoSe_2 isomers, compared with WS_2. This results reflects the importance role of transition metals in determiningthe stable geometry of TMDCs nanoclusters. We also noticed that, WS_2 nanoclusters exhibit less tendency to planner configurations,because even at larger sizes, WS_2 nanoclusters are not quite planner <cit.>.In order to check the effect of exchange-correlation functional on atomic configuration of MoX_2 clusters, some lowest energy isomers of all nanoclusters were relaxedwithin the PBE and B3LYP functionals. The obtained new energy orders within these two functionals are compared with BLYP in table <ref>. In the case of MoS_2, we observe that PBE displaces the first isomer of 6, 7 and 8 clusters,while in other sizes, most stable isomers are the same within PBE and BLYP. A more precise consideration show that the lowest energy isomers of6, 7, and 8 clusters within BLYP are not quite flat and triangular, it seems that PBE is trying to stabilize more triangular structures. From another point of view, more flat isomers with more number ofMo-S_C bonds are better favored within PBE. For example in the size of n=6, PBE favors 6b isomerwhich has very small RMSD value (Figs. <ref>) and more number of Mo-S_C bonds, compared with other isomers. Then the 6c, 6a, and 6d isomers with respectively 3, 2, and 1 Mo-S_C bond,occupy the next places in the obtained energy order within PBE.The same trend is observed in MoSe_2 clusters (table <ref>).Compared with BLYP, PBE only displaces stable isomer of clusters 4 and 5. Some metastable structures are also displaced according tothe above-mentioned trend.For example, in the size of n=9, the first isomer is not changed within PBE, compared with BLYP, but the isomers 9d, 9e, and 9c with seven Mo-Se_C bondsare more stable than 9b, within PBE. The isomer 9b has the lowest number of Mo-Se_C bonds in this group. On the other hand, among the three isomers with the same number of Mo-Se_C bonds, those with a more triangular shape are more stable.Hence the general statement is that PBE, compared with BLYP,favors more flat triangular structures of MoX_2 nanoclusters. It might be related to the better performance of PBE functionalin description of periodic structures <cit.>,because flat structures are closest cluster configurations to the arrangement of atoms in the ideal octahedral coordination of 1T-MoX_2 sheets. We also re-calculated several isomers within the B3LYP hybrid functionaland found that stability order of isomers with this functional is very similar to BLYP. The same feature has been observed for WS_2 nanoclusters <cit.>.It is already observed that nanoclusters of a non-magnetic materialmay exhibit magnetism <cit.>. In order to find the stable magnetic state of the clusters, their total energy were minimized by considering different initial magnetization and then the lowest energy state was reported as the stable magnetic state. It was found that the lowest energy isomer of the first size (n=1)of MoS_2 and MoSe_2 clusters have a total spin moment of 2 μ_B,within both BLYP and PBE. The lowest energy isomer of MoSe_2 clusters at all other investigated sizes are nonmagnetic, while those of MoS_2 clusters at the second and sixth sizes exhibit a total spin moment of 2 μ_B, within both BLYP and PBE. §.§ Relative Stability of Nanoclusters In order to address the stability of the nanoclusters,their binding energy (BE) per MoX_2 unit was calculated, as follows: BE(n)=E_tot(n)-n(E_at( Mo)+2E_at(S))/n where E_tot(n) is the minimized total energy of the most stable isomer ofthe cluster of size n and E_at(X) is the total energy of a free X atom. The binding energies were calculated within four differentfunctionals and presented in Fig. <ref>. According to this diagram, the binding energy (BE) is a monotonic increment with size,that indicating more stability of larger clusters,which is due to the reduction of the relative number of surface dangling bonds. Taking into account the tendency of the lowest lying isomers to haveplanar/semi-planar configurations, it is expected that BE plots converge to the binding energy ofideal MoS_2 and MoSe_2 sheets,which were calculated 13.7 eV and 10.63 eV within BLYP and 15.11 eV and11.91 eV within PBE, respectively. There is a local minimum at n=9 in all functionals, which indicates more relative stability of this size compared with neighboring sizes.In the case of MoSe_2, another local minimum is visible at n = 6, while n = 7,8 occur on a local bulge of the BE plot,indicating lower relative stability of the 7th and 8th clusters.For more accurate description of the relative stability of the systems and identifying the small magic sizes of the nanoclusters,the second-order difference in energy is defined as follows: Δ_2E=E_tot(n+1)+E_tot(n-1)-2E_tot(n) This parameter is conventionally expected to be comparable withthe mass spectrometry measurements on nanoclusters.The reason is that from a physical point of view, a positive value of Δ_2E indicates higher relative stability and consequently more abundance ofthe corresponding cluster, compared with the neighboring sizes.Hence the local peaks of the Δ_2E plot is expectedto happen at the magic sizes of the clusters. The calculated Δ_2E values for MoX_2 clusters withinfour functionals are shown in Fig. <ref>. It is seen that the n=3,9 sizes of MoS_2 nanoclusters exhibit high abundance within all functionals, while the 5th cluster shows good relative stability only within hybrid functionals. On the other hand, the sizes of n=2,8 show very low abundance within all functionals. These findings contradict with previous reports onthe magic sizes (n=2,4,6) of small MoS_2 clusters <cit.>.This is due to the different stable isomers found in this work. The lowest energy isomers in the work of murugan2005a usually coincide with our high energy metastable isomers, found in our comprehensive systematic structure search. In the case of MoSe_2 nanoclusters, all functionals predictmagic sizes of n=3,6,9, while the sizes of n=2,7,8 exhibit very low relative stability. The large value of Δ_2E at the 9th size of both systems along with the observed local minimum at this size in the binding energy plots, confirm that n=9 is an important magic size of MoS_2 and MoSe_2 nanoclusters. An interesting point is that, except for the lowest magic size (n=3), other magic sizes occur on the systems with higher relative number of Mo-Se_C bonds, compared to neighboring sizes (see table <ref>). §.§ Energy Gap The HOMO-LUMO energy gap of the lowest energy isomers of MoX_2 nanoclusters was calculated within the BLYP, PBE, B3LYP, and PBE0 functionals (Fig. <ref>). It is seen that PBE and BLYP give very similar and rather uniform energy gaps, while hybrid B3LYP and PBE0 functionals display significantly enhanced values. Despite the success of LDA/GGA functionals in predicting structural properties,presence of self-interaction error (SIE) gives rise to significantly underestimatedenergy gaps within these functionals <cit.>. On the other hand, hybrids functionals partially resolve the SIE problem through combination of Hartree-Fock exchange with the GGA one, thus enhancing the energy gap compared with GGA <cit.>. One of the most effective methods for correcting the DFT HOMO-LUMO gap is the many-body based GW approximation<cit.>. In this method, a week screened coulomb interaction is switched on betweenthe fictitious Kohn-Sham particles to perturbatively obtain the quasiparticlespectra of the system by using the Green's function technique. We performed non-self-consistent G_0W_0 calculation,where the mean field Green's function, G_0, and the screened Coulomb interaction, W_0, are determined from the Kohn-Sham eigenvalues and eigenvectors. Therefore, the accuracy of the non-self-consistent G_0W_0 calculationare sensitive to the starting point exchange-correlation functional. By proper selection of the starting point, the accuracy of the G_0W_0 results approaches that ofthe full-self-consistent GW <cit.>.It is argued that hybrid functionals are better starting points, compared with LDA, GGA, and Hartree-Fock method for G_0W_0 calculation. <cit.> However, it is observed that the calculated energy gaps of 4d and 5d materials by usingthe G0W0@PBE0 method show very small deviation from experiment <cit.>.Figure <ref> displays the calculated energy gaps within different functionals after application of the many-body G_0W_0 correction. As expected, the G_0W_0 correction significantly enhancesthe value of the band gaps within all functionals. The enhancement is more pronounced in the smaller clusters, which is due to the lower electronic screening in these systems. As mentioned before, in the G_0W_0 method, a screened coulomb interaction is switched on between the Kohn-Sham particles. Therefore, lower screening in the smaller clusters enhances the effect of G_0W_0 correction in the energy gap. Moreover, it is seen that the G_0W_0 correction decreases the difference between the band gap within hybrid and GGA functionals, in such a way that after G_0W_0, PBE0 and B3LYP give very similar energy gaps. Hence, we deduced that B3LYP may also be a proper starting point for G_0W_0 calculations in the MoX_2 clusters. It is observed that MoS_2 nanoclusters have greater energy gap, compared with MoSe_2 nanoclusters, which is due to the stronger M-X bonding inthe MoS_2 nanoclusters (table <ref>). The same trend is observed in the ideal MoS_2 and MoSe_2 sheets. From a qualitative point of view, energy gap is a measure ofthe chemical hardness of the system <cit.>. The larger the band gap, the more energy required to disrupt the electronic structure of the system, and the lower chemical activity of the system.An alternative approach for calculation of energy gap is ΔSCF method. In this method, an electron is added to and subtracted from the neutral system, andthen total energy of the ionized systems are computed to obtain the energy gap as follows: gap=E_tot(cation)+E_tot(anion)-2E_tot(neutral) If the energy of the ionized system is calculated without relaxation, (exactly at the relaxed geometry of the neutral cluster),the method is called vertical ΔSCF, while in the adiabatic version of the method, the ionized systems are fully relaxed. This method gives very accurate energy gap for molecular systems <cit.>. We used adiabatic and vertical ΔSCF technique along with the BLYP functional to calculate energy gap of MoX_2 nanoclusters (Fig. <ref>). It is seen that adiabatic ΔSCF gives energy gaps very close to the G_0W_0@PBE0 results,while vertical ΔSCF predicts slightly larger energy gaps.The obtained energy gap of nanoclusters displays a smooth decrease as a function of the cluster size, which may be explainedby the quantum confinement effect. Quantum confinement in nanostructures enhances their energy gapwith respect to the periodic structures. Since our flat stable isomers are approaching the 1T phase ofthe corresponding periodic systems, which is found to be metallic, the energy gap of the flat stable isomers is expected to convergeto zero at large sizes. §.§ Vibrational Frequency The dynamical properties of the lowest lying isomers were investigatedby identifying the vibrational modes of the systems inthe framework of the force constant method.In this method, a finite displacement (δ) is applied to the atomic positionsof the fully relaxed structures and then the force set on the atoms is accurately computed with an error of less than 10^-4 eV/Å.The obtained Hessian matrix is then diagonalized to obtainthe vibrational modes and frequencies of the clusters. It should be noted that the vibrational properties arecalculated within the BLYP functional. The absence of imaginary vibrational modes in the systems indicates dynamical stability of the lowest lying isomers.All presented meta-stable isomers are also expected to be dynamically stable, because these configurations are of very low symmetry, obtained through accurate structural relaxation. Generally, it is very unlikely that a low symmetry configuration of atoms traps on a saddle point of the potential energy surface after full atomic relaxation. The obtained vibrational modes are in the frequency range of 0-390 and 0-510 cm^-1 for MoS_2 and MoSe_2 nanoclusters, respectively. The observed difference is due to the somewhat stronger/shorteraverage bonding in MoS_2 with respect to MoSe_2.IR intensity of the lowest energy isomers at their vibrational modes are computed by considering the variation of cluster dipole moments along the vibrational modes. The obtained IR intensities, presented in Fig. <ref>, shows that in practice which IR photons are better absorbed andconverted to thermal vibration in the systems. It is seen that the highest IR absorption happens at the high frequency modes, which corresponds to the strongest Mo-X bonds in the clusters. It was argued that among various Mo-X bonds, higher degree of hybridizationhappens between Mo and terminal X_T atom,and hence terminal Mo-X bonds are stronger and shorter (table <ref>). As a result, these bonds are the origin ofthe hard vibrational modes of the clusters. Since these bonds are on the edges of the clusters,their dipole moment is more flexible and hence they have more IR absorption intensity. The obtained IR spectra indicate that the major part of the IR photons are thermally adsorbed by the hard edge bonds of the nanoclusters. (please note the vertical logarithmic scale in Fig. <ref>). The other vibrational modes, distributed below the high frequency mode, are mainly originated form the collective vibration ofMo-X_B and Mo-X_C bonds in the systems. Vibration of these bonds creates less polarization in the systems, compared with the terminal bonds, and hence their IR intensity is considerably lower. With increasing the cluster size, the number of Mo-X_B and Mo-X_C bonds is increased in the systems and hence more IR active modes are appeared in the IR spectra.In order to investigate the influence of thermal vibrations onrelative stability of the clusters at elevated temperatures,the vibrational Helmholtz free energy (F_vib) of the clusters was calculated as follows: F_vib = E_tot - E_at + 1/2∑_iε_i + k_BT∑_i ln(1-e^ε_i/k_BT) where E_tot is the minimized total electronic energy of the lowest energy isomers,E_at is the sum of the free atom energies of the system,i runs over the number of vibrational modes,ε_i is the energy of the ith mode,k_B is the Boltzmann constant, T is the Kelvin temperature. The second order difference in Helmholtz free energy (Δ_2F)was calculated in different temperatures,ranging from zero to 500 K (Fig. <ref>).In the case of MoS_2, we observe that thermal effects enhance Δ_2Fof even clusters (except n=2), which is more pronounced in the 6th cluster.It is likely attributed to the more low-energy vibrational modes of the 6th cluster, compared with the neighboring systems (Fig. <ref>).With increasing temperature, these low-energy vibrational modes give riseto faster entropy increase and consequently faster free energy decrease ofthe system in such a way that above room temperature,n=6 becomes an important magic number of MoS_2 nanoclusters. In the case of MoSe_2 nanoclusters, slight thermal effectsis seen up to 500 K and only cluster of size n=4 showsslightly enhanced relative stability at above room temperatures.In addition to the relative stability of the isomers,thermal effects may also influence the lowest energyatomic configuration of the clusters. In other words, increasing temperature may change the stability order of the lowest energy isomers of a cluster and induce a structural transition from one isomer to another one, similar to structural transitions reported in MoX_2 single layers <cit.>. In order to address this issue, we focus on the n=4,6 sizes of MoS_2 andn=4,5,6 of MoSe_2 nanoclusters, because in these systems the first metastable isomer has a very close energy to the lowest energy isomer (table <ref>)and hence thermal effects are expected to be more visible in these cases. The Helmholtz free energy difference between the stable and the first metastableconfigurations of these systems was calculated at three different temperatures, presented in table. <ref>. The results indicate absence of any structural transition in these systems up to 500 K, because the stable isomer remains lower in energy with respect to the first metastable isomer at temperatures below 500 K. Only in the 4th size of MoS_2, we observe that the second isomer isgetting very close to the first isomer at 500 K and hence one may expect a structural transition above this temperature. However, we did not continue our free energy calculations to the higher temperatures, because our vibrational studies are in the harmonic approximation limit, which might not be valid at very high temperatures.§ CONCLUSIONS In this paper, we used evolutionary algorithm along with full potential density functional calculations to identify the stable and metastablestructures of (MoS_2)_n and (MoSe_2)_n (n=1-10) nanoclusters. The structure search was done within the BLYP functional and then the energy order of the lower energy isomers was verified within PBE. It was argued that the clusters favor sandwiched planar triangular structureseven in small sizes, similar to the ideal sheets of these systems. The results show that the lowest energy isomer is usually the same within BLYP and PBE, while PBE prefers more flat metastable structures with higher numberof face capping bonds. The second-order difference in minimized energy of the systemsindicate that the robust magic sizes of (MoS_2)_n and (MoSe_2)_nnanoclusters are n=3,9 and n=3,6,9, respectively.It was argued that vibrational excitations at finite temperatures enhancerelative stability of the 6th size of MoS_2 nanoclusters to become a magic sizeat above room temperatures, while the magic sizes of MoS_2 nanoclusters remain unchanged up to 500 K. The HOMO-LUMO energy gap of the lowest energy isomerswas obtained by using the hybrid PBE0 and B3LYP functionals,the many body based G_0W_0 technique, and the total energy based ΔSCF method. It was argued that G_0W_0@PBE0 is expected to givereliable energy gaps for the investigated nanoclusters. The energy gap of the systems was found to increase with decreasing the cluster size, because of the quantum confinement effect. Finally, we calculated the vibrational spectra and the IR active modes of the lowest energy isomers. It was revealed that the most IR absorption happens at a frequency ofabout 510 and 370 cm^-1 in MoS_2 and MoSe_2 nanoclusters, respectively. These IR photons are mainly absorbed by the strong terminal Mo-S and Mo-Se bonds located at the corners of the nanoclusters. This work was jointly supported by the Vice Chancellor of Isfahan University of Technology (IUT) in Research Affairs and Centre of Excellence for Applied Nanotechnology. It should also be acknowledged that the first two authors have equal contributions in this paper. | http://arxiv.org/abs/1710.00052v1 | {
"authors": [
"Zohre Hashemi",
"Shohreh Rafiezadeh",
"Roohollah Hafizi",
"S. Javad Hashemifar",
"Hadi Akbarzadeh"
],
"categories": [
"cond-mat.mtrl-sci",
"physics.atm-clus",
"physics.chem-ph"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170927085138",
"title": "First-principles study of MoS$_2$ and MoSe$_2$ nanoclusters in the framework of evolutionary algorithm and density functional theory"
} |
Corresponding authors: [email protected]; [email protected] Key Laboratory of Strongly-Coupled Matter Physics, Chinese Academy of Sciences, and Hefei National Laboratory for Physical Science at Microscale, and Department of Physics, University of Science and Technology of China. Hefei, Anhui, 230026, P.R.China. By efficient nanoscale plasma etching, the nitrogen-vacancy (NV) centers in diamond were brought to the sample surface step by step successfully. At each depth, we used the relative ratios of spin coherence times before and after applying external spins on the surface to present the decoherence, and investigated the relationships between depth and ratios. The values of relative ratios declined and then rised with the decreasing depth, which was attributed to the decoherence influenced by external spins, surface spins, discrete surface spin effects and electric field noise. Moreover, our work revealed a characteristic depth at which the NV center would experience relatively the strongest decoherence caused by external spins in consideration of inevitable surface spins. And the characteristic depth was found depending on the adjacent environments of NV centers and the density of surface spins. Valid PACS appear here Depth dependent decoherence caused by surface and external spins for NV centers in diamond Wenlong Zhang, Jian Zhang, Junfeng Wang, Fupan Feng, Shengran Lin, Liren Lou, Wei Zhu^* and Guanzhong Wang Received September 26, 2017; accepted November 14, 2017 ==============================================================================================================§ INTRODUCTION The negatively charged nitrogen-vacancy (NV) center in diamond has attracked broad interest owing to its prominent properties. One particular property is the long room-temperature spin coherence time, which is essential for NV centers being used in various applications. Recently, the use of NV centers as sensors to detect external spins <cit.> has been demonstrated, and both the coherence time and detected signal strength are found to be critically dependent on the depth of the NV center <cit.>. In view of this, the depth dependent properties of NV centers as well as the preparation methods of NV centers of different depths have been widely investigated.Conventionally, low-energy nitrogen-implantation<cit.> and the epitaxial growth of a high quality nitrogen-doped CVD diamonds followed by electron <cit.> or ion irradiation <cit.> are the methods to make shallow NV centers. Moreover, in order to bring an NV center closer to the diamond surface step by step to investigate the depth dependence of its properties, thermal oxidation<cit.> and plasma etching <cit.> methods have been developed and widely used in the recent years. Compared with thermal oxidation which performs an etching on the entire diamond sample surface, plasma etching method makes it possible to etch a specific area of the sample by previously depositing a mask on the surface <cit.>.In this paper, we performed plasma etching on a bulk diamond to precisely control the depth of NV centers with respect to the sample surface. Then we studied depth dependence of spin coherence times of the NV centers for samples with external nuclear or electronic spin baths around the surface. In particular, by using NV center array and position marks, we could track the very same single NV center at different depths, which enabled us to keep a stable internal adjacent environment of the center and made the depth and external spins the only two variables.§ METHODS We used an electronic grade (100)-oriented, 2 × 2 ×0.5 mm3 sized diamond substrate from Element Six ([13C]=1.1%, [N]<5ppb) for the experiments. By using electron beam lithography, an arrary made of 60 nm diameter apertures, enclosed with 2 μm wide vacant strips (serving also as position marks <cit.>), were patterned on a 300 nm thick polymethyl methacrylate (PMMA) layer previously deposited on the diamond plate surface <cit.>. The NV center array in the diamond was created by ion implantation with the 14N2+ molecule energy of 50 keV and a fluence of 0.65 × 1011 14N2+per cm2 through the apertures and strips on the PMMA layer. The implanted sample was annealed at 1050 ℃ in vacuum at 2 × 10 -5 Pa for 2 h to form long spin coherence time centers <cit.>. Then the sample, after oxidation for 2 h in air at 430 ℃, was cleaned with acidic mixture (sulfuric, nitric, and perchloric acid in a 1 : 1 : 1 ratio) at 200 ℃ for 1.5 hours. The plasma-related processes were performed using an ICP RIE reactor (Oxford PlasmaPro NGP80 machine equipped with ICP source). In order to obtain an efficient etching rate, we performed the plasma etching on the diamond sample in conditions of 200 W ICP power, 30 mTorr chamber pressure, 10 sccm of oxygen, 5 sccm of argon, which was different from that used in the previous works <cit.>. By depositing a mask (lithography-patterned AZ 6112 photoresist) on a part of the sample's surface to protect it from etching, we could get a reference point of the initial depth, with respect to which we could determine the etching rate and depth from surface topography analysis with the atomic force microscope (AFM). Fig.1 demonstrates a representative result for sample of 20 s etching time. Fig.1(a) shows a 6.8 × 6.8 μm2 area of the etched diamond sample surface, which demonstrates a clear boundary near the middle left. A plane fit was performed on the 1.2 × 1.2 μm2 square delineated in Fig.1(a) to eliminate the effect of slant angle caused by placing sample non horizontally.The resulting corrected image is presented in Fig.1(b). The distribution of height along the randomly selected blue line in Fig.1(b) is shown in Fig1.(c), which indicates an etching depth of about 4 nm at the boundary. After removing remarkable spikes and streaks, the distribution along X axis of the Y-axis-average height is shown in Fig.1(d) from which an etching depth of about 3.8 nm can be obtained. Deriving the etching depths for ten etching experiments of different etching times, we found that the etching rate under the conditions mentioned above was 11.8 ± 1 nm/min. Therefore, we determined to perform a 20-second plasma etching on the sample (corresponding to an etching depth of about 4 nm) each time when the NV centers were distant from the surface, and a short etching time (corresponding to the etching depths of about 2 nm or 1 nm) when they were shallow. In this way, we made the centers approaching to the sample surface step by step until the centers disappeared, from which the initial depths of NV centers could be derived. We tracked 227 single NV centers to investigate the distribution of center depth. Respectively, Fig.2(a), (b) and (c) show the fluorescence images of the same representative region of the tracked area of the sample etched for three different etching times (corresponding to the etching depths 0 nm, 20 nm, 44 nm). The strip-shaped bright regions on the left and up sides of the images, implying where NV center clusters exist, correspond to the position marks, and the lightspots in white circles represent the tracked single NV centers. Obviously, after etching 20 nm, the position mark strips became less bright, and only 23 centers among the initially tracked 33 single NV centers remained discernible. And after etching 44 nm, the bright strips became interrupted and much less bright, and only 5 single NV centers remained. The evolution of the remaining number of all 227 tracked single NV centers are shown in Fig.2(d). We found that the number of remaining centers reduced slightly when etching depth was 20 nm or less. However, when the etching depth increased further, in a range from 20 to 40 nm, the number reduced dramatically. Finally, when the etching depth was above 40 nm, the numberreduced slowly again. By subtracting the numbers of adjacent etching depths, the distribution histogram was obtained andshown in Fig.2(e) which corresponded to the SRIM simulation for an implantation nitrogen atom energy of 25 keV. The result supported our estimated etching rate mentioned above.§ RESULTS At each depth, spin coherence times were measured both before and after applying external spins to the diamond surface. The two liquids used were microscope immersion oil and Cu2+ solution, providing external nuclear<cit.> and electronic<cit.> spins respectively (Though Cu2+ provides nuclear spins as well, the electronic spin leads the main effect for decoherence due to the higher gyromagnetic ratio). The sample was placed in custom built confocal microscope system with the applied magnetic field (B = 55 ± 5 G) paralleling to the detected single NV center axes. Twenty five single centers (labeled NV-0125) were randomly selected for the measurements, and initially, the T2 of three centers of them were less than 50 μs while of the rest were between 120 to 250 μs, indicating that most of the selected centers were deep inside the sample with a spin bath environment of 13C impurities <cit.>. Fig.3 exhibits the results of coherence times of the sample with microscope immersion oil applied on the surface. It has been acknowledged that when the surface is exposed to air, the coherence time will decline as NVs approaching to the surface, owing to the influence of surface spin baths existing naturally <cit.> (Since it was difficult to remove the surface spins completely). The coherence times obtained with no external liquid applied to the surface would be called T2air (which served as a reference), and that with oil applied would be called T2oil. Noteworthily, we used the ratio Rn = (T2oil - T2air) / T2air to represent the decoherence caused by external nuclear spins, which reflected the intensity of interaction between NV and external nuclear spins with respect to the inevitable intrinsic spins around the diamond surface. Fig.3(a) shows four representative results of spin echo measurements of NV-12 at different depths. This center disappeared after etching 38 nm, indicating its initial depth to be d38 nm. Before etching, the T2air of NV-12 was 214.1 μs. After applying oil to the surface, the coherence time T2oil changed to 208.2 μs, and Rn was about -0.03, suggesting that the external nuclear spins had little influence on the coherence time of the center located at a depth about 38 nm. Then, after etching 24 nm (d14 nm), the T2air declined to 160.1 μs, and T2oil declined to 117.9 μs, makingRn to be -0.26, indicating the external nuclear spins indeed had huge effect on coherence time when NVs became shallow. Even more, T2air and T2oil of the same center declined to 59.6 and 34.1 μs, respectively, for sample etched for 30 nm, which corresponded to a depth of NV-12 d8 nm. The Rn reduced to -0.43 which meant the external nuclear spins caused strong decoherence. Then, another 30-second-etching was performed to make NV-12 only about 2 nm to the surface, which caused T2airdeclined to 6.84 μs, T2oil to 6.24 μs However, we found that for the sample thus treated, the Rn increasd to -0.09. The T2air and T2oil of NV-12 for various depths to the sample surface are presented in Fig.3(b) top, and they both have similar evolution with decreasing depth. As shown in the figure, both T2air and T2oil decreased slowly at the depth above 20 nm, then decreased rapidly until NV-12 finally disappeared, which was the same as our previous work <cit.>. Besides, the T2oil is less than T2air, revealing the external nuclear spins have enormous influence on coherence time in most cases in addition to intrinsic surface spins. The Rn for the center with various depths are presented in Fig.3(b) bottom, which shows that the Rn curve goes down with the center depth reducing till about 8 nm and then rises . Another representative result of coherence time measurements of single center NV-13 is shown in Fig.3(c). Similar to NV-12, both T2air and T2oil of NV-13 decreased slowly with the decrease of its depth in the range above 15 nm, and decreased rapidly in the last 20 nm etching. However, the values of Rn at the depths very near the surface were positive, 0.03 and 0.12 (meaning T2 increased when external nuclear spins applied), suggesting the decoherence was suppressed by applying oil on the surface. We summarized the depth dependent Rn of all 25 NV centers in Fig.3(d). For depth above 6 nm, we divided the depth range into sequential intervals (4 nm for the first 8 intervals, and 6 nm for the last 3 intervals), and figured out the mean Rn in every interval. The obtained mean Rn declined from -0.01 to -0.42 with the depth decreasing. For depth under 6 nm, the data of all 25 NVs scattered between -0.54 and 1.43, most around 0.2, revealing that the decoherence caused by applied oil was unsteady, even suppressed when NVs were very near the surface. The result will be discussed in detail in the following part. Fig.4 exhibits the results of coherence times with Cu2+ solution (providing external electronic spins) applied on the sample surface. Likewise, the ratio Re =(T2Cu2+ - T2air) / T2air was used to represent the variation of coherence times caused by external electronic spins. Spin echo measurements of NV-02 with four representative etching depths are shown in Fig.4(a). The center NV-02 initially sat at d ~ 32 nm, according to the fact that it disappeared after etching 32 nm. Before etching, the T2air of NV-02 was 219.9 μs. After applying Cu2+ to the sample surface, T2 ( T2Cu2+) changed to 201.8 μs, and Re was about -0.08. After etching 16 nm (d16 nm), the T2air declined to 122.3 μs, and T2Cu2+ declined to 94.2 μs, making Re to be -0.23. T2air and T2Cu2+ declined to 15.6 and 6.36 μs respectively when NV-02 was etched about 8 nm to the surface. Strong decoherence caused by external electronic spins was detected as indicated by the reduced Re of -0.59. However, with etching depth being 30 nm in total, NV-02 was only about 2 nm to the surface, and T2air and T2Cu2+ declined to 5.13 and 2.33 μs, respectively, making Re changed to -0.55. The T2air and T2Cu2+ of NV-02 for various depths to the sample surface are presented in Fig.4(b) top, and the ratios (Re) of them are in Fig.4(b) bottom. Fig.4(c) shows the results of NV-21. Both figures show that the coherence times declined dramatically when NV centers are under 20 nm, for sample surface with or without Cu2+ applied. Furthermore, T2Cu2+ was less than T2air at each depth, leading to the fact that Re was negative, indicating decoherence caused by external electronic spins was always existing. We also found that Re of the two NVs increased when they were brought very near the surface by etching the sample surface. Re of all the 25 single NV centers at different depths were demonstrated in Fig.4(d).Similar to the processing of Rn, the data of Re in the figure were also divided into two parts, with depths above and under 6 nm. Compared with the evolution of Rn, Re behaved similarly at the depths above 6 nm, also declined regularly. However, when the depth was less than 6 nm, the data points of Re mostly distributed around -0.6, with a few located in the range from -0.42 to -0.06. § DISCUSSION The results demonstrated above showed that the interaction between NV centers and external spins varied with NV center depth. We took the evolution of Rn as an instance to elaborate the various reasons leading to the results. For the NV centers in diamond, it is inevitable that surface spin bath exists, which contributes to the decoherence of NV centers along with spins in bulk <cit.>. Correspondingly, the coherence time is as follows : T_2^air=1/[γ_NV^2(B_bulk^2τ_bulk+B_surf^2τ_surf)+1/2T_1] If external spins are applied on the surface of diamond, the decoherence would also be influenced by the external spins, in addition to the surface spins and spins in bulk. Thus, T2 with external spins around the sample surface (T2ext) can be written in the form : T_2^ext=1/[γ_NV^2(B_bulk^2τ_bulk+B_surf^2τ_surf+B_ext^2τ_ext)+1/2T_1] In the above expressions, γ_NV is the gyromagnetic ratio of NV center, and B_bulk^2, B_surf^2, B_ext^2 are the MS magnetic field signal produced by internal (bulk), surface and external spins respectively, and τ_bulk, τ_surf, τ_ext are the internal (bulk), surface and external spin baths' autocorrelation times respectively, which can be regarded as constant parameters at a given temparature. Noteworthily, the contribution of 1/2T_1 to 1/T_2 is far less than that of various spin noises, so it can be neglected. Then, the expression of ratio can be obtained from Eq.(1) and Eq.(2) as : R=T_2^ext-T_2^air/T_2^air=-B_ext^2τ_ext/B_bulk^2τ_bulk+B_surf^2τ_surf+B_ext^2τ_ext It is worth mentioning again that, by using NV center array and position marks, we could track each paricular NV center no matter how the sample surface was etched and the external spins were applied. For each tracked NV center, the internal adjacent environment was unchanged as the center depth is not very shallow. Therefore, the quantity B_bulk^2τ_bulk in Eq.(3) was constant for the tracked NV center, and would be denoted by Cbulk hereafter. Moreover, for each kind of spins, the autocorrelation time is invariable, i.e. the τ_surf and τ_ext in Eq.(3) can be regarded as constants (independent of center depth) as well.Then we focused on the center depth dependent B_surf^2 and B_ext^2. When NVs are distant from the surface, the relationship between B_surf^2 and center depth can be well described based on a model of a 2D layer of surface g = 2 spins in case with the surface exposed to air <cit.>. In particular, for the (100)-oriented diamond, we have : B_surf^2=σ(gμ_0μ_B/4π)^2(3π/8d^4) where d is the NV center depth and σ is the mean surface spin density. We note that besides the variable d and a unknown quantity of σ , the other parameters in Eq.(4) can be combined as a constant, Csurf. For the B_ext^2 , using a model of the sample surface covered with liquid of infinite thickness that provides homogeneous external nuclear spins <cit.>, it can be derived:B_ext^2=B_oil^2=ρ(μ_0γ_n/4π)^2(3π/4d^3) where ρ is the nuclear spin number density, d is the NV center depth, and γ_n is nuclear gyromagnetic ratio. The unknown quantity in Eq.(5) is ρ, the value of which depends on the applied oil. However, the rest parameters besides variable d can also be treated as a constant, Cext. With Eq.(4) and (5) substituted into Eq.(3), a simplified expression of Rn can be obtained : Rn=-ρτ_extC_ext/d^3/C_bulk+στ_surfC_surf/d^4+ρτ_extC_ext/d^3 Eq.(6) showed the relationship between Rn and NV center depth. With the estimated values of Cin, στ_surf and ρτ_ext in a reasonable range, Eq.(6) was found to fit the data well. The results of Rn and Re (averiged in each interval) are demonstrated in Fig.5. It is noticed that the data points of Rn at the depth above 10 nm conform to the simulated curve in Fig.5(a), while the data points under 10 nm show a deviation from the simulated curve. Moreover, it can be learnt from Eq.(6) that the positive Rn is nonexistent, which is incompatible with the experiment results when NVs were brought near to the surface. The discrepancy was attributed to the following effects. Firstly, the surface spin baths can not be simply treated as a 2D uniform layer when NVs are very shallow (d10 nm) owing to the existence of discrete surface spin effect or spin clustering <cit.>,which makes the surface spin density inhomogeneous, and consequently, Eq.(4) is no longer valid. In this case, B_surf^2 becomes sensitive to the surface spin distribution and the denominator value of Eq.(6) fluctuates, leading to the Rn value scattering around the simulated curve. Furthermore, recent works reveal that electric field noise plays a significant role in decoherence of near-surface NV centers <cit.>, so the fact that Rn can have positive values is related to the microscope immersion oil, nonconductor of high dielectric constant (κ = 2.3), which reduces the electric field noise and suppresses the decoherence <cit.>.The expression of the ratio Re (T2Cu2+ - T2air) / T2air is similar to Eq.(6), and the simulated curve presented in Fig.5(b) demonstrates an appropriate fit with the data. In spite of this, the data deviation from the simulated curve is shown at the depth under 5 nm as well, suggesting that the discrete surface spin effect does influnce the values of the ratios. A difference from Rn is that no positive Re is found, which can be attributed to that the plentiful ions in Cu2+ solution near the sample surface would enlarge the electric field <cit.>, instead of decreasing the electric field noise that would suppress the decoherence.Our results demonstrated that Eq.(6) could explain the depth dependent decoherence behaviours caused by external spins with respect to the inevitable intrinsic spins around the diamond surface. Each simulated curve in Fig.5 shows a minimum at the depth about 6 nm. As an example, by taking the derivative of Rn in Eq.(6) with respect to the center depth d, it was found that Rn would take minimum value at the depth d0 : d_0=√(στ_surfC_surf/3C_bulk) Eq.(7) reveals that the characteristic depth (d0), at which the ratio Rn (and Re as well) meets the minimum, only depends on the the adjacent environments of NV centers and the density of diamond surface spins. In external spin detection, the NV center should be brought closing to the sample surface as far as possible to strengthen the detected signal. However, taking surface spins into consideration, the depth at which the external spins cause relatively the most intense decoherence is several nanometers away from the surface. Since the adjacent environment of NV center is changeless, it can be obtained from Eq.(7) that d0 decreases with σ reducing. Therefore, decreasing the density of surface spins by proper surface treatments can lower d0 and hence the external spin detection with less influence of surface spins can be realized for the shallow NV center sensors.§ SUMMARY We investigated the depth dependence of the coherence times of NV centers for diamond plate with or without external nuclear and electronic spins around the surface. By using NV center array and position marks, each particular NV in the diamond plate etched for different depths could be recognized and tracked. As the internal adjacent environments of the tracked NVs were kept unchanged upon etching, our results obtained byT2 tracking was more persuasive than that by measuring NVs initially in different depths. We performed plasma etching to control the depths of NVs with an efficient etching rate. Based on this, we applied microscope immersion oil and Cu2+ solution on the surface to obtain external nuclear and electronic spins. We introduced the coherence time ratios of Rn and Re to present the decoherence caused by external spins, and found the depth dependent ratios behaved in the form of a function having a minimum. The characteristic depth at which the NV centers experienced relatively the strongest decoherence caused by external spins, as indicated by the minimum ratio R = (T2ext - T2air)/ T2air, was found depending on the adjacent environment of NV center and the density of surface spins, which could be useful in the further study and detection of external spins, surface spin noise and so forth with NV centers in diamond.§ ACKNOWLEDGEMENTS We thank X.X.Wang, J.L.Peng, D.F.Zhou and W.Liu from the USTC Center for Micro- and Nanoscale Research and Fabrication for the technical support of Plasma etching and AFM. We also thank G.P.Guo and J.You from the Key lab of Quantum Information for the support of electron beam lithography. This work was supported by the National Basic Research Program of China (2011CB921400, 2013CB921800) and the National Natural Science Foundation of China (Grant Nos. 11374280, 50772110).unsrtnat @ifundefinedendmcitethebibliography30 f subitem (mcitesubitemcount)[Staudacher et al.(2013)Staudacher, Shi, Pezzagna, Meijer, Du, Meriles, Reinhard, and Wrachtrup]staudacher2013nuclear T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard and J. Wrachtrup, Science, 2013, 339, 561–563 [Mamin et al.(2013)Mamin, Kim, Sherwood, Rettner, Ohno, Awschalom, and Rugar]mamin2013nanoscale H. Mamin, M. Kim, M. Sherwood, C. Rettner, K. Ohno, D. Awschalom and D. Rugar, Science, 2013, 339, 557–560 [Müller et al.(2014)Müller, Kong, Cai, Melentijević, Stacey, Markham, Twitchen, Isoya, Pezzagna, Meijer,et al.]muller2014nuclear C. Müller, X. Kong, J.-M. Cai, K. Melentijević, A. Stacey, M. Markham, D. Twitchen, J. Isoya, S. Pezzagna, J. 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Santamore, Journal of Applied Physics, 2016, 120, 383–182 | http://arxiv.org/abs/1709.09070v1 | {
"authors": [
"Wenlong Zhang",
"Jian Zhang",
"Junfeng Wang",
"Fupan Feng",
"Shengran Lin",
"Liren Lou",
"Wei Zhu",
"Guanzhong Wang"
],
"categories": [
"cond-mat.mes-hall",
"physics.app-ph"
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"primary_category": "cond-mat.mes-hall",
"published": "20170926145311",
"title": "Depth dependent decoherence caused by surface and external spins for NV centers in diamond"
} |
[email protected] of Physics, Florida State University, Tallahassee, FL 32306 [email protected] Department of Physics, Florida State University, Tallahassee, FL 32306BackgroundNuclear astrophysics centers on the role of nuclear physics in the cosmos. In particular, nuclear masses at the limits of stability are critical in the development of stellar structure and the origin of the elements.PurposeTo test and validate the predictions of recently refined nuclear mass models against the newly published AME2016 compilation. MethodsThe basic paradigm underlining the recently refined nuclear mass models is based on existing state-of-the-art models that are subsequently refined through the training of an artificial neural network. Bayesian inference is used to determine the parameters of the neural network so that statistical uncertainties are provided for all model predictions.Results We observe a significant improvement in the Bayesian Neural Network (BNN) predictions relative to the corresponding “bare" models when compared to the nearly 50 new masses reported in the AME2016 compilation. Further, AME2016 estimates for the handful of impactful isotopes in the determination of r-process abundances are found to be in fairly good agreement with our theoretical predictions. Indeed, the BNN-improved Duflo-Zuker model predicts a root-mean-square deviation relative to experiment of σ_ rms≃400 keV. ConclusionsGiven the excellent performance of the BNN refinement in confronting the recently published AME2016 compilation, we are confident of its critical role in our quest for mass models of the highest quality. Moreover,as uncertainty quantification is at the core of the BNN approach, theimproved mass models are in a unique position to identify those nucleithat will have the strongest impact in resolving some of the outstandingquestions in nuclear astrophysics.Validating neural-network refinements of nuclear mass models J. Piekarewicz December 30, 2023 ============================================================ § INTRODUCTION As articulated in the most recent US long-range planfor nuclear science <cit.> “nuclearastrophysics addresses the role of nuclear physics in ouruniverse", particularly in the development of structure andon the origin of the chemical elements. In this context,fundamental nuclear properties such as masses, radii, andlifetimes play a critical role. However, knowledge of thesenuclear properties is required at the extreme conditionsof density, temperature, and isospin asymmetry found inmost astrophysical environments. Indeed, exotic nuclei near the drip lines are at the core of several fundamentalquestions driving nuclear structure and astrophysicstoday: what are the limits of nuclear binding?, where do the chemical elements come from?, and what is the nature of matter at extremedensities? <cit.>.Although new experimental facilities have been commissionedwith the aim of measuring nuclear masses, radii, and decaysfar away from stability, at present some of the requiredastrophysical inputs are still derived from often uncontrolled theoreticalextrapolations. And even though modern experimental facilitiesof the highest intensity and longest reach will determine nuclearproperties with unprecedented accuracy throughout the nuclearchart, it has been recognized that many nuclei of astrophysicalrelevance will remain beyond the experimentalreach <cit.>. Thus, reliance on theoretical models that extrapolate into unknownregions of the nuclear chart becomes unavoidable. Unfortunately, these extrapolations are highly uncertain and may ultimately lead to faulty conclusions <cit.>. However, one should not underestimate the vital role that experiments play and will continueto play. Indeed, measurements of even a handful of exotic short-livedisotopes are of critical importance in constraining theoreticalmodels and in so doing better guide the extrapolations.Although no clear-cut remedy exists to cure such unavoidableextrapolations, we have recently offered a path to mitigatethe problem <cit.>primarily in the case of nuclear masses. The basic paradigm behindour two-pronged approach is to start with a robust underlyingtheoretical model that captures as much physics as possiblefollowed by a Bayesian Neural Network (BNN)refinement that aims to account for the missingphysics <cit.>. Several virtues were identified in such a combined approach. First, we observed a significantimprovement in the predictions of those nuclear masses thatwere excluded from the training of the neural network—evenfor some of the most sophisticated mass models available inthe literature <cit.>. Second, mass models of similar quality that differ widely in their predictions far away from stability tend to drastically and systematically reduce their theoretical spread after theimplementation of the BNN refinement. Finally, due totheBayesian nature of the approach, the refined predictions are always accompanied by statistical uncertainties. Thisphilosophy was adopted in our most recentwork <cit.>, which culminated with thepublication of two refined mass models: the mic-mac modelof Duflo and Zuker <cit.> and the microscopicHFB-19 functional of Goriely and collaborators <cit.>. As luck would have it, shortly after the submission of our latestmanuscript <cit.> we became aware of the newlypublished atomic mass evaluation AME2016 <cit.>. This is highly relevant given that the training of the neural networkrelied exclusively on a previous mass compilation(AME2012) <cit.>. Thus, insofar as the nearly 50new masses appearing in the newest compilation, ours arebonafide theoretical predictions. Confronting the newly refinedmass models against the newly published data is the main goalof this brief report.This short manuscript has been organized as follows. First, no further physics justification nor detailed account of the BNN framework are given here, as both were extensively addressed in our most recent publication <cit.>. Second, theresults presented in Sec. <ref> are limited to those nuclei appearing in the AME2016 compilation whose masses were notreported previously or whose values, although determined fromexperimental trends of neighboring nuclides, have a strong impacton r-process nucleosynthesis. As we articulate below, the main outcome from this study is the validation of the novel BNN approach. Indeed, we conclude that the improvement reported inRef. <cit.> extends to the newly determined nuclearmasses—which in the present case represent true model predictions.We end the paper with a brief summary in Sec. <ref>.§ RESULTS In Ref. <cit.> we published refined masstables with the aim of taming the unavoidable extrapolationsinto unexplored regions of the nuclear chart that are criticalfor astrophysical applications. Specifically, we refined thepredictions of both the Duflo-Zuker <cit.> andHFB-19 <cit.> models using the AME2012 compilation in the mass region from ^40Ca to^240U. The latestAME2016 compilationincludes mass values for 46 additional nuclei in the^40Ca-^240U region, and these are listed in Table <ref> alongside predictions from variousmodels. These include the“bare” models (i.e., beforeBNN refinement) HFB-19 <cit.>,Duflo-Zuker <cit.>, FRDM-2012 <cit.>, HFB-27 <cit.>, and WS3 <cit.>.Also shown are the predictions from the BNN-improvedDuflo-Zuker and HFB-19 models <cit.>.The last column displays the total binding energy as reportedin the AME2016 compilation <cit.>; quantitiesdisplayed in parentheses in the last three columns representthe associated errors. Note that we quote differencesbetween the model predictions and the experimental massesusing only the central values. Finally, the last row containsroot-mean-square deviations associated with each of themodels. The corresponding information in graphical form is also displayed in Fig.<ref>, but only for the five baremodels discussed in the text.The trends displayed in Table <ref> and even more clearly illustratedin Fig. <ref> are symptomatic of a well-known problem, namely, thattheoretical mass models of similar quality that agree in regions where massesare experimentally known differ widely in regions where experimental data isnot yet available <cit.>. Given that sensitivity studies suggest thatresolving the r-process abundance pattern requires mass-model uncertaintiesof the order of ≲100 keV <cit.>,the situation depicted in Fig. <ref> is particularly dire. However, despite thelarge scattering in the model predictions, which worsens as one extrapolatesfurther into the neutron drip lines, significant progress has been achieved inthe last few years. Indeed, in the context of density functional theory, theHFB-27 mass model of Goriely, Chamel, and Pearson predicts a rather smallrms deviation of ∼0.5 MeV for all nuclei with neutron and proton numberslarger than 8 <cit.>. Further, in the case of the Weizsäcker-SkyrmeWS3 model of Liu, Wang, Deng, and Wu, the agreement with experiment is evenmore impressive: the rms deviation relative to 2149 known masses is a mere∼0.34 MeV <cit.>. Although not as striking, the success ofboth models extends to their predictions of the 46 new masses listed inTable <ref>: σ_ rms=0.72MeV andσ_ rms=0.51MeV, respectively. This represents asignificant improvement over earlier mass models that typically predict arms deviation of the order of 1 MeV; see Table <ref> and Fig. <ref>.However, our main focus is to assess the improvement in the predictionsof two of these earlier mass models (HFB-19 and DZ) as a result of the BNNrefinement. In agreement with the nearly a factor-of-two improvement reportedin Ref. <cit.>, we observe a comparable gain in the predictionsof the 46 nuclear masses listed in Table <ref>; that is, σ_ rms=(1.093→0.587) MeV andσ_ rms=(1.018→0.479) MeV forHFB-19 and DZ, respectively. Of course, an added benefit of the BNN approachis the supply of theoretical error bars. Indeed, when such error bars are taken intoaccount—as we do in Fig. <ref>—then all of the refined predictionsare consistent with the experimental values at the 2σ level. For reference, also included in Fig. <ref> are the impressive predictions of the WS3 model, albeit without any estimates of the theoretical uncertainties. We close this section by addressing a particular set of nuclear massesthat have been identified as “impactful" in sensitivity studies of theelemental abundances in r-process nucleosynthesis. These include avariety of neutron-rich isotopes in palladium, cadmium, indium, and tin;see Table I of Ref. <cit.>. As of today, none of thesecritical isotopes have been measured experimentally. However, manyof them have been “flagged" (with the symbol “#") in the AME2016compilation to indicate that while not strictly determined experimentally,the provided mass estimates were obtained from experimentaltrends of neighboring nuclides <cit.>. In Table <ref>theoretical predictions are displayed for those isotopes that have been both labeled as impactful and flagged. Predictions are provided for the WS3 <cit.>, FRDM-2012 <cit.>,DZ <cit.>, and BNN-DZ <cit.> mass models.Root-mean-square deviations of the order of 1 MeV are recorded for all models, except for the improved Duflo-Zuker model where the deviation is only 369 keV. This same information is depicted in graphical form inFig. <ref>. The figure nicely encapsulates the spirit of our two-prongapproach, namely, one that starts with a mass model of the highest quality(DZ) that is then refined via a BNN approach. The improvement in thedescription of the experimental data together with a proper assessmentof the theoretical uncertainties are two of the greatest virtues of the BNNapproach. Indeed, the BNN-DZ predictions are consistent with all massesof those impactful nuclei that have been determined from the experimentaltrends. § CONCLUSIONSNuclear masses of neutron-rich nuclei are paramount to a varietyof astrophysical phenomena ranging from the crustal composition of neutron stars to the complexity of r-process nucleosynthesis. Yet, despite enormous advances in experimental methods andtools, many of the masses of relevance to astrophysics lie wellbeyond the present experimental reach, leaving no option but torely on theoretical extrapolations that often display large systematic variations. The current situation is particularly troublesomegiven that sensitivity studies require mass-model uncertaintiesto be reduced to about ≲100 keV in order to resolver-process abundances. There are at least two different approaches currently used to alleviatethis problem. The first one consists of painstakingly difficult measurementsnear the present experimental limits that aim to inform and constrainmass models. The second approach is based on a global refinementof existing mass models through the training of an artificial neural network. This is the approach that we have advocated in this short contribution. Giventhat the training of the neural network relied exclusively on the AME2012compilation, our approach was validated by comparing our theoretical predictions against the new information provided in the most recentAME2016 compilation. The comparison against the newly available AME2016 data was highlysuccessful. For the nearly 50 new mass measurement reported in the^40Ca-^240U region, the rms deviation of the two BNN-improvedmodels explored in this work (Duflo-Zuker and HFB-19) was reduced bynearly a factor of two relative to the unrefined bare models. Further, for themasses of several impactful isotopes for the r-process, the predictionsfrom the improved Duflo-Zuker model were fully consistent with the new AME2016 estimates and in far better agreement than some of the mostsophisticated mass models available in the literature. Finally and as important, all nuclear-mass predictions in the BNN approach incorporate properly estimated statistical uncertainties. When these theoretical error bars are incorporated, then all of our predictions are consistentwith experiment at the 2σ level.Ultimately, improvements in mass models require a strong synergy between theory an experiment. Next-generation rare-isotope facilities will produce newexotic nuclei that will help constrain the physics of weakly-bound nuclei. In turn, improved theoretical models will suggest new measurements on a few critical nuclei that will best inform nuclear models. We are confident thatthe BNN approach advocated here will play a critical role in this endeavor, particularly in identifying those nuclei that have the strongest impact inresolving some outstanding questions in nuclear astrophysics. We are hopeful that in the near future mass-model uncertainties—both statisticaland systematic—will be reduced to less than 100 keV, which representsthe elusive standard required to resolve the r-process abundance pattern. We are thankful to Pablo Giuliani for many stimulating discussions. This material is based upon work supported by the U.S. Department of Energy Office of Science, Office of Nuclear Physics under Award Number DE-FG02-92ER40750. | http://arxiv.org/abs/1709.09502v1 | {
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"J. Piekarewicz"
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"primary_category": "nucl-th",
"published": "20170927133418",
"title": "Validating neural-network refinements of nuclear mass models"
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ESRF, The European Synchrotron, 71 Avenue des Martyrs, 38000 Grenoble, France ESRF, The European Synchrotron, 71 Avenue des Martyrs, 38000 Grenoble, France ESRF, The European Synchrotron, 71 Avenue des Martyrs, 38000 Grenoble, France [Present address: ]Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. ESRF, The European Synchrotron, 71 Avenue des Martyrs, 38000 Grenoble, France ESRF, The European Synchrotron, 71 Avenue des Martyrs, 38000 Grenoble, France Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, PL-02093 Warsaw, Poland Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, PL-02093 Warsaw, Poland Nikolaev Institute of Inorganic Chemistry, Siberian Branch of the Russian Academy of Sciences, Akademician Lavrentiev Prospekt 3, Novosibirsk 90, 630090, Russia Novosibirsk State University, Pirogova Street 2, Novosibirsk 90, 630090 Russia [email protected] ESRF, The European Synchrotron, 71 Avenue des Martyrs, 38000 Grenoble, FranceRecent theoretical predictions of “unprecedented proximity” of the electronic ground state of iridium fluorides to the SU(2) symmetric j_eff=1/2 limit, relevant for superconductivity in iridates, motivated us to investigate their crystal and electronic structure. To this aim, we performed high-resolution x-ray powder diffraction, Ir L_3-edge resonant inelastic x-ray scattering, and quantum chemical calculations on Rb_2[IrF_6] and other iridium fluorides. Our results are consistent with the Mott insulating scenario predicted by Birol and Haule [Phys. Rev. Lett. 114, 096403 (2015)], but we observe a sizable deviation of the j_eff=1/2 state from the SU(2) symmetric limit. Interactions beyond the first coordination shell of iridium are negligible, hence the iridium fluorides do not show any magnetic ordering down to at least 20 K. A larger spin-orbit coupling in iridium fluorides compared to oxides is ascribed to a reduction of the degree of covalency, with consequences on the possibility to realize spin-orbit-induced strongly correlated physics in iridium fluorides.On the possibility to realize spin-orbit-induced correlated physics in iridium fluorides M. Moretti Sala 22 September 2017 ========================================================================================§ INTRODUCTION The strong motivation behind the intense effort devoted to the investigation of iridium oxides (iridates) resides in the correlated nature of their physical properties. The identification of a spin-orbit-induced Mott insulating state in Sr_2IrO_4 <cit.> triggered a number of theoretical and experimental studies aiming at isolating even more exotic phenomena, such as Kitaev-type magnetism <cit.> or Weyl semi-metallicity <cit.>. For the specific case of Sr_2IrO_4, increasing experimental evidences of similarities with the high-temperature superconducting cuprates have been found in the structural, magnetic and electronic properties <cit.>. It was therefore proposed that the low-energy physics of Sr_2IrO_4 could be described by a (pseudo)spin 1/2 particle in a one-band Hubbard model <cit.>, similarly to the cuprate antiferromagnetic (AFM) parent compounds, where the active orbital is branched off from the 5d–t_2g states by virtue of strong spin-orbit coupling and it is usually termed the j_eff=1/2 state <cit.>. Starting from the assumption that the one-orbital Hubbard model is a good approximation of the electronic structure of Sr_2IrO_4 and that high-temperature superconductivity in doped cuprates is described by the one band Hubbard model, unconventional superconductivity was said to be possible in doped iridates <cit.>. Superconductivity was theoretically predicted for both electron- <cit.> and hole-doped Sr_2IrO_4 <cit.>. These results motivated a substantial experimental campaign to look for superconductivity in iridates, with encouraging results. It was shown that the AFM Mott insulating phase in Sr_2IrO_4 is destroyed upon electron doping and replaced by a paramagnetic phase with persistent magnetic excitations, strongly damped and displaying anisotropic softening <cit.>, in a way reminiscent of paramagnons in hole doped cuprates <cit.>. Angle-resolved photoemission spectroscopy measurements showed that electron doped Sr_2IrO_4 displays typical features of superconducting cuprates, such as Fermi arcs <cit.> and a d-wave gap <cit.> in the intermediate and low temperature phases, respectively. Despite exciting theoretical predictions and promising experimental findings, however, superconductivity has not been observed yet in doped Sr_2IrO_4 or any other iridium oxide. The theoretical finding of spin-orbit-induced correlated physics in a novel class of materials exhibiting an “unprecedented proximity” to the ideal SU(2) limit is therefore extremely welcome <cit.>. Indeed, recently Birol and Haule <cit.> proposed the exciting idea that spin-orbit-induced correlated physics can be found in a novel class of materials, namely rhodium and iridium fluorides. A first indication that this may be the case comes from Pedersen et al. <cit.> who showed that the magnetism of ideal model system molecular iridium fluorides is consistent with the j_eff=1/2 scenario and that they can be used as building-blocks to synthesize electronic and magnetic quantum materials <cit.>, such as those proposed by Birol and Haule. Rb_2[IrF_6] is particularly appealing because it is said to host a j_eff=1/2 ground state with “unprecedented proximity” to the SU(2) symmetric limit, with possible implications for superconductivity in iridates <cit.>. The main motivation behind our study is to understand differences and analogies between the physics of iridium fluorides and oxides. To this aim we investigate the crystal and electronic structure of several iridium fluorides (Rb_2[IrF_6], Na_2[IrF_6], K_2[IrF_6], Cs_2[IrF_6], and Ba[IrF_6]) by means of high-resolution x-ray powder diffraction (XRPD), resonant inelastic x-ray scattering (RIXS), and quantum chemical calculations. Our results are consistent with the predictions of a wide gap j_eff=1/2 Mott insulator retaining a paramagnetic state <cit.> down to 20 K. Indeed, we find that the low-energy electronic structure of these systems is mostly dictated by the local coordination of the IrF_6 octahedra, with no evidence of interactions beyond the first coordination shell of iridium. We observe nevertheless a sizable deviation of the j_eff=1/2 state from the SU(2) symmetric limit suggesting that the distortions in the electronic structure due to a non-cubic environment are larger than predicted. Our experimental results are supported by quantum chemical calculations.§ EXPERIMENTAL DETAILS High-resolution XRPD measurements were performed at beamline ID22 of the European Synchrotron Radiation Facility (ESRF, France). The incoming x-rays were monochromated to λ = 0.3999 Å by a Si(111) double-crystal monochromator. The x-rays diffracted by the sample were collimated by 9 Si(111) analyzers and collected by a Cyberstar scintillation detector. Iridium L_3-edge RIXS spectra were measured at the inelastic x-ray scattering beamline ID20 of the ESRF. ID20 is particularly suited for RIXS experiments due to its energy resolution capabilities. This is as good as 15 meV at 11.2165 keV when a Si(844) back-scattering channel-cut is used to monochromate the incident photon beam. The spectrometer is based on a single Si(844) diced-crystal analyzer (R = 1 m) in Rowland scattering geometry and equipped with a two-dimensional Maxipix detector <cit.>. The overall energy resolution was set to 35 meV for this experiment <cit.>.Samples were grown in the Nikolaev Institute of Inorganic Chemistry (Novosibirsk, Russia). Na_2[IrF_6] was prepared following the method described in Ref. Pedersen2016. After dissolving 1.009 g of Na_2[IrF_6] in 10 ml of H_2O, 10 ml of cation-resin H^+ were added to the solution. After 30 min of mixing and filtering, H_2[IrF_6] was obtained. K_2[IrF_6], Rb_2[IrF_6], Cs_2[IrF_6] and Ba[IrF_6] were prepared by filtering the solutions obtained after reaction of stoichiometric quantities of H_2[IrF_6] and KF, RbF, CsF and BaCO_3, respectively. Single crystals of Rb_2[IrF_6] were grown by slow counter diffusion of Na_2[IrF_6] (0.3 M) and RbF (20 M) solutions in 1% agar gel. § COMPUTATIONAL METHOD Quantum chemical calculations were performed using the ORCA software package <cit.>. State-averaged complete active space self-consistent field (CASSCF) and N-electron valence perturbation theory (NEVPT2) <cit.> calculations were used to determine the energy of the excited states, and the spin-orbit-coupling constant <cit.>. The active space included five electrons distributed over the five d-orbitals. The spin-orbit coupling was treated a posteriori using the quasi-degenerate perturbation theory <cit.> and the mean-field approximation of the Breit-Pauli spin-orbit coupling operator <cit.>. Scalar relativistic effects were included using the zero-order regular approximation (ZORA) <cit.>. Polarized triple-ζ basis sets were used for all elements <cit.>. All computations were done using the embedded cluster approach in order to account for the environment <cit.>. Models constructed starting from the XRPD structure consisted of a central IrF_6 octahedron (the quantum cluster (QC)) surrounded by point charges (PC). A boundary region (BR) containing repulsive capped effective core potentials was introduced to avoid electron flow from the central subunit towards the point charges.§ RESULTS AND DISCUSSION Figure <ref>(a) shows the high-resolution XRPD pattern of Rb_2[IrF_6] at T = 295 K. The black solid line corresponds to the Rietveld refinement of the experimental data (red dots). We find that Rb_2[IrF_6] belongs to the space group P3̅m1 with lattice parameters a = b = 5.9777(0) Å and c = 4.7986(5) Å at 295 K. Crystallite size broadening between 4.8 and 5.2 has been estimated using an instrumental peak shape function implemented in Topas 5 <cit.> and convolving sample size term on top <cit.>. As it can be seen in the inset, the crystal structure of Rb_2[IrF_6] consists of isolated IrF_6 octahedra. The Ir-F bond lengths are all the same and equal to 1.975 Å, while the F-Ir-F bond angles are 87^∘ and 93^∘. As a result, the octahedra are slightly compressed along the crystallographic c axis, thus inducing a trigonal distortion of 3.9%, defined as (β-β_0)/β_0, where β_0≈54.74^∘ and β is the angle between the Ir-F bond and the trigonal axis <cit.>. A similar analysis has been carried out for all the compounds. The results of the crystal structure refinement, summarized in Table <ref>, are in agreement with existing literature <cit.>, except for Ba[IrF_6], for which we converged to the R3̅ space group <cit.>. The common feature to all systems is the presence of isolated IrF_6 units with comparable distortions of the octahedral cage. In the case of Rb_2[IrF_6], XRPD measurements were performed at several temperatures in the range between 100 and 400 K. Fig. <ref>(b) shows the temperature dependence of the diffraction peaks associated to the (100) and the (001) reflections. Their continuous variation can be directly associated to a smooth change of the a and c lattice parameters, as reported in Fig. <ref>(c), and suggests that no structural phase transition occurs in the investigated temperature range.After characterizing the samples from a structural point of view, we now turn to the investigation of their electronic structure. Figure <ref>(a) shows a representative Ir L_3-edge RIXS spectrum of Rb_2[IrF_6] single crystal measured at momentum transfer 𝐐 = (1.5, 0, 6) r.l.u. and T = 20 K. The incident photon energy was fixed at 11.2165 keV, i.e. ∼ 3 eV below the main absorption line, where intra-t_2g excitations are enhanced <cit.>. The black dots in Fig. <ref>(a) correspond to the background-subtracted data points, while the red solid line is the fit to the data. We highlight the absence of features below 0.7 eV. At higher energy losses, two features (A and B) are clearly distinguished. They are fitted to two Pearson VII functions <cit.> (blue dashed lines in Fig. <ref>(a)) and their energy positions are 805± 0.4 meV (A) and 915± 1.3 meV (B). Considering the resonance behavior of the two features, we ascribe them to transitions from the j_eff=1/2 to thej_eff=3/2 states, in line with previous RIXS studies of iridium oxides <cit.>. In addition, the two features do not show any detectable momentum or temperature dependence within the experimental uncertainties, as shown in Figs. <ref>(b) and (c), suggesting that the IrF_6 octahedra behave as isolated units. Our findings support the scenario of a strong insulating character of iridium fluorides, with narrow bands and no tendency to develop long-range magnetic order, in line with theoretical predictions <cit.>. Similar measurements were carried out for all samples in powder form. Figure <ref> shows a stack of the corresponding RIXS spectra. Interestingly, the overall shape is very similar and closely resembles the RIXS spectrum of Fig. <ref>(a). However, before discussing the small but meaningful differences between the different compounds, we notice that they all show a large splitting of the j_eff = 3/2 states. This is indicative of a sizable lifting of the 5d–t_2g states degeneracy <cit.> and contrasts with the prediction of an isotropic electronic state close to the SU(2) limit for Rb_2[IrF_6] <cit.>.To get a better insight into the electronic structure of iridium fluorides, we have performed quantum chemical calculations using the embedded cluster approach. The calculated splittings of the j_eff = 3/2 excited states are compared to the experimental values in Table <ref>. Although the agreement with the experiment is not quantitative, the calculations reproduce the trend among the different compounds. As can be seen in Fig. <ref>, the calculated and experimental splittings are found to correlate very nicely (we exclude Na_2[IrF_6] from this analysis because there are two inequivalent iridium sites in this compound). Overall, the agreement between experiments and quantum chemical calculations suggests that there is little or no influence of the alkali (earth) metal on the low-energy electronic structure of iridium fluorides, but rather that it is solely dictated by the local coordination of the IrF_6 octahedra. This is mainly the consequence of the fact that iridium fluorides are composed by disconnected IrF_6 units. We note that while chemical substitution is not effective at modifying the low-energy electronic structure of iridium fluorides, physical pressure may be.In order to compare iridium fluorides and iridates, we have expressed the experimental results presented above in terms of single-ion model parameters, such as the effective trigonal distortion Δ of the cubic crystal field, and the spin-orbit-coupling constant ζ, often used in the literature of iridates <cit.>. By constraining the energies of the j_eff = 3/2 excited states as calculated from the single-ion model to the energy positions of the features A and B, and by taking into account the sign of the octahedral distortion as determined by XRPD, we estimated the values of Δ and ζ for our systems. As reported in Table <ref>, estimates for |Δ| vary between 0.15 and 0.19 eV, while ζ≈ 0.57 eV for all the iridium fluorides studied here. The latter is 10–40% larger than in iridium oxides, where it ranges between 0.38 <cit.> and 0.52 eV <cit.>. Ligand-field parameters, spin-orbit-coupling constant, and interelectronic repulsion terms have also been calculated by fitting the full configuration interaction matrix elements obtained from the CASSCF/NEVPT2 calculation to the matrix elements of a model Hamiltonian containing those interactions <cit.>. In agreement with the experimentally fitted values discussed above, we calculate ζ≈ 0.55 eV for all iridium fluorides, and ζ≈ 0.52 eV for Sr_2IrO_4 using the same theoretical approach. In order to rule out the possibility that the reduction of spin-orbit coupling in iridium oxides compared to fluorides arises from differences in their crystal structure, we calculated the spin-orbit-coupling constant for an IrF_6 octahedron where the F ions have been placed at the positions of the O ions in Sr_2IrO_4. We obtain ζ≈ 0.55 eV, suggesting that the nature of the coordinating ion, rather than the crystal structure, determines the differences in the spin-orbit-coupling constant. The larger reduction of ζ compared to the free ion value (relativistic nephelauxetic effect <cit.>) in iridium fluorides than in oxides reflects the smaller degree of covalency in the chemical bonds of the former. This is of particular interest because covalency in Sr_2IrO_4 is thought to be responsible for strong orbital anisotropies, in view of the increased spatial extent of the 5d–t_2g orbitals reaching the nearest neighbor iridium atoms and beyond <cit.>. Iridium fluorides might therefore be the ideal playground for studying spin-orbit-induced correlated physics because correlation effects might be enhanced by the more localized nature of the electronic states, whereas long-range anisotropies, which contribute to deviate the j_eff = 1/2 from the SU(2) symmetric limit, are strongly suppressed. We note that a reduction of covalency would lower the ratio of the energy scales of the magnetic over the charge fluctuations. The latter effect might become important may the iridium fluorides be electron- or hole-doped.As a final remark, we would like to discuss oxides and fluorides in relation to the similarities between cuprates and iridates. We start by considering La_2CuO_4 and K_2CuF_4. Although they share the same K_2[NiF_4]-type crystal structure and are insulating, their magnetic properties are very distinct: La_2CuO_4 is an AFM insulator <cit.>, while K_2CuF_4 has a ferromagnetic (FM) ground state <cit.>. Indeed, in La_2CuO_4 a strong tetragonal crystal field splits the 3d–e_g states and stabilizes the x^2-y^2 orbital, which gives rise to ferro-orbital ordering and ultimately to AFM coupling on a straight (180^∘) bond. On the contrary, in the undistorted K_2CuF_4 the degeneracy of the 3d–e_g states is essentially preserved and the Cu^2+ ion is Jahn-Teller active. The so-called cooperative Jahn-Teller effect sets in and x^2-z^2/y^2-z^2 alternating orbital ordering is stabilized leading to FM long-range order in the ground state <cit.>. When moving to Sr_2IrO_4, much of the physics of La_2CuO_4 is retained, namely the Mott insulating AFM state with dominant Heisenberg-like interactions <cit.>. Effectively, the only active orbital is the j_eff=1/2, which is branched off from the 5d–t_2g by virtue of the strong spin-orbit coupling. One important consequence of such strong spin-orbit coupling is that, no matter how undistorted the system is, the Jahn-Teller mechanism is not supported in the Ir^4+ compounds <cit.>. We therefore speculate that while the lack of magnetism in the studied iridium fluorides can basically be attributed to the isolation of the IrF_6 units, the ground state of a hypothetical iridium fluoride with an ideal K_2[NiF_4]-type crystal structure would probably never support FM order (unlike copper fluorides). Instead, it might even be closer to the Heisenberg AFM state found in copper oxides than iridium oxides. In this respect, the parallelism between copper oxides/fluorides and iridium oxides/fluorides is broken.§ CONCLUSIONS We investigated the crystal and electronic structure of Na_2[IrF_6], K_2[IrF_6], Rb_2[IrF_6], Cs_2[IrF_6] and Ba[IrF_6] by means of high-resolution XRPD, Ir L_3-edge RIXS, and quantum chemical calculations. Our results support the theoretical predictions that Rb_2[IrF_6] is characterized by a j_eff=1/2 electronic ground state <cit.>. The absence of low-energy features, as well as momentum and temperature dependence in the RIXS spectra of Rb_2IrF_6 single crystal suggests that interactions beyond the first coordination shell of iridium ions are negligible, thus precluding long-range magnetic order down to at least 20 K. However, the splitting of the j_eff = 3/2 excited states is indicative of a deviation from the SU(2) symmetric limit. Consistently, quantum chemical calculations on a single IrF_6 cluster reproduce the experimental trend observed among the various compounds and elucidate that the low-energy electronic structure of iridium fluorides is ascribed to characteristic local distortions of the IrF_6 cage with no significant influence from neighboring ions.We also report an increase of the spin-orbit coupling in iridium fluorides as compared to iridium oxides. This finding is corroborated by quantum chemical calculations and suggests that the larger electronegativity of fluorine compared to oxygen reduces the degree of covalency in the system. This has important consequences: i) the spatial extension of 5d–t_2g orbitals is reduced and correlation effects might indeed be enhanced; ii) long-range anisotropies are mitigated and an isotropic j_eff=1/2 ground state is more likely to stabilize. If synthesized in crystal structures with connected IrF_6 units, such hypothetical iridium fluorides might indeed support features characteristic of spin-orbit-induced strongly correlated physics and might even more closely resemble the low-energy physics found in copper oxides than it is the case of iridium oxides.We acknowledge the European Synchrotron Radiation Facility (ESRF, France) for providing beamtime. T. K. thanks the ESRF for kind hospitality. K. W. acknowledges support by Narodowe Centrum Nauki (NCN, National Science Centre) under Project No. 2012/04/A/ST3/00331 and Project No. 2016/22/E/ST3/00560. The authors are grateful to C. Henriquet and B. Detlefs for technical assistance during the experiment, and to M. Krisch for critical reading of the manuscript. | http://arxiv.org/abs/1709.09027v1 | {
"authors": [
"M. Rossi",
"M. Retegan",
"C. Giacobbe",
"R. Fumagalli",
"A. Efimenko",
"T. Kulka",
"K. Wohlfeld",
"A. I. Gubanov",
"M. Moretti Sala"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170926141045",
"title": "Possibility to realize spin-orbit-induced correlated physics in iridium fluorides"
} |
Azimuthal angle correlations at large rapidities: revisiting density variation mechanism [ December 30, 2023 =========================================================================================== This report documents the program and the outcomes of GI-Dagstuhl Seminar 16394 “Software Performance Engineering in the DevOps World”. § EXECUTIVE SUMMARY[Andre van Hoorn,Pooyan Jamshidi,Philipp Leitner, andIngo Weber] Andre van Hoorn (University of Stuttgart, DE) Pooyan Jamshidi (Imperial College London, GB) Philipp Leitner (University of Zurich, CH) Ingo Weber (Data61, CSIRO, AU)The seminar addressed the problem of performance-aware DevOps. Both, DevOps and performance engineering have been growing trends over the past one to two years, in no small part due to the rise in importance of identifying performance anomalies in the operations (Ops) of cloud and big data systems and feeding these back to the development (Dev). However, so far, the research community has treated software engineering, performance engineering, and cloud computing mostly as individual research areas. We aimed to identify cross-community collaboration, and to set the path for long-lasting collaborations towards performance-aware DevOps.The main goal of the seminar was to bring together young researchers (PhD students in a later stage of their PhD, as well as PostDocs or Junior Professors) in the areas of (i) software engineering, (ii) performance engineering, and (iii) cloud computing and big data to present their current research projects, to exchange experience and expertise, to discuss research challenges, and to develop ideas for future collaborations.§ OVERVIEW OF TALKS Performance Engineering for DevOps Using Survivability Modeling of High Availability Systems (Keynote) [Alberto Avritzer]Alberto Avritzer (Sonatype, USA, [email protected])As our society evolves, more and more aspects of our daily life depend on large-scale infrastructures such as computer infrastructures, rails and road networks, gas networks, water networks, power networks, and telecommunication networks, including the Internet, wired and wireless telephony. Critical infrastructures are everywhere and they are becoming increasingly more interconnected and interdependent. As networks become smarter, they rely more heavily on information and communication technologies, also known as ICT. For this reasons, a failure in the ICT network can cause problems in different critical infrastructures. As another example, a failure in the power network can cause disruptions in a number of different networks. We present a Nexus Sonatype based DevOps approach for continuous performance improvement of high-availability systems that is composed of several important components: i) large repositories representing the customer environment, ii) automated analysis of performance tests, iii) JMeter load testing scripts and associated data.Wepresent high-availability models of critical infrastructures and we connect our DevOps continuous performance improvement process with our approach tothe system availability assessment, e.g., through reduced failure recovery times. DevOps—How a Fortune 500 Company Translates Theory in Reality (Keynote) [Oliver Beck]Oliver Beck (SAP, Germany, [email protected])In this talk, Oliver Beck presented SAP's approach to managing DevOps at scale, and how they address practical challenges of changing culture in a large, well-established organization. Oliver Beck is a Vice President at SAP, where he is responsible for global DevOps initiatives, among others. (Text written by Ingo Weber) Performance Regression Analysis in the DevOps World [Cor-Paul Bezemer]Cor-Paul Bezemer (Queen's University, Canada, [email protected])A performance regression occurs when an application update unintendedly slows down the application. For example, the CPU usage or execution time of a task increases considerably after deploying an update. The goal of performance regression analysis (PRA) is to detect such performance regressions and analyze what causes them.Most approaches for PRA follow a similar process to detect and analyze performance regressions. For every application update, a performance test is executed while the performance of the application is monitored. Then, the performance of that version of the application is compared with the performance of the previous version, and analyzed if there are significant differences.One problem of the process above is that the performance test must be executed several times to reduce variation in the monitored data. Especially in the DevOps world, where the deployment rate of applications is often considerably higher than in the “traditional” software world, the repeated execution of performance tests for each application update poses severe operational challenges.In my presentation, I will give an overview of the approaches that we study to overcome these challenges and make PRA feasible in the DevOps setting. Exploiting with Integrity—Mining User Data to Improve Software Engineering in the Light of Information Ethics [Markus Borg]Markus Borg (RISE SICS AB, Lund, Sweden, [email protected])Software-intensive products developed in market-driven contexts must quickly respond to requirements from users. Also, delivering software that meets the users' expectations of quality is fundamental. Contemporary approaches to tackle the “need for speed” include agile and lean software development <cit.>, and more recently development with a strong focus on data from operations has gained attention through DevOps <cit.>. There is often a considerable gap between development and operations, but bridging the two activities has the potential to support both responsiveness to market expectations and software evolution in general.Previously, we have directed considerable research effort toward closing another gap in software engineering, namely between requirements engineering (RE) and testing <cit.>. Aligning RE and testing, the “two ends” of traditional software development projects, is critical to ensure efficient development of high-quality software—such alignment is important also in iterative development contexts. Among other solution proposals, we have explored supporting RE and testing alignment by a data-driven approach, i.e., identifying connections between resolved issue reports and requirements and test cases <cit.>. By applying machine learning in the footprints of previous issue resolution activities, we provided recommendations of which requirements would be affected during software evolution, and which test cases should be executed to verify the changes.Now we turn our attention away from issue reports and instead target user data from operations. Our new research direction is still related to recommendation systems highlighting requirements and test cases. However, instead of supporting changes during issue resolution, we plan to predict user needs. In particular, we aim at supporting: i) RE by prioritizing features, and ii) regression testing by improving test case selection. Our approach to tackle this challenge is to continuously mine usage and project data, i.e., data external and internal to the projects. We aim at developing a demonstrator using a mobile platform made available by an industry partner — a company already specializing in distributed data collection. The envisioned data collection includes: position data, data transfer, CPU workload, and logged user actions.While there will be technical challenges involved in the research, such as generating actionable recommendations from large amounts of user data, we believe the challenges could be overcome during the project. We should be able to develop a demonstrator that continually feeds data mined from operations directly to a recommendation system for RE and testing activities. On the other hand, another important consideration surfaces: should we do it? Mining rich user data threatens the privacy of the users, which over time could undermine the users' trust. How much privacy are users willing to give up to improve RE and testing? How does it vary depending on how the data is going to be used? To properly address these issues, we collaborate with socio-legal researchers specializing in studies on trust in the digital society <cit.>. By combining technical and socio-legal aspects, we hope to bring valuable contributions to DevOps research.Can We Make Performance Visible to Developers? [Lubomir Bulej]Lubomir Bulej (Charles University Prague, Czech Republic, [email protected])The phrase “Premature optimization is the root of all evil.” has become a common wisdom and a best practice in software development. Randal Hyde even argues <cit.> that the phrase has become an excuse for not caring about performance at all, under the assumption that performance is a mere optimization, and that we can always go back and optimize the problematic code.In his 1974 article for ACM Computing Surveys <cit.>, Donald Knuth actually wrote: “We should forget about small efficiencies, say 97 % of the time: premature optimization is the root of all evil. Yet we should not pass up our opportunities in that critical 3 %.” In the same paper, Knuth also wrote: “In established engineering disciplines, 12 % performance improvement, easily obtained, is never considered marginal, so why would so many people in computer science pronounce it insignificant?” Apparently, good engineering practice is to understand the system performance and not give it up through sloppiness. Yet in computer science, or software development in general, we have been often doing exactly that—because of the Moore's law, spending human time to make programs fast was more expensive than buying faster hardware every other year.However, the scale and complexity of software systems has increased to a point where performance is not a local issue anymore. If we consider the critical 3 % that Knuth mentioned in his paper—in contemporary systems, those 3 % may be a lot of code spread throughout the whole software stack that no single person can completely understand anymore. Performance has become deeply ingrained in the software architecture and design, and fixing a performance problem often requires making design changes at different levels of abstraction.One problem with performance is that is largely invisible to developers, and what is invisible is difficult to manage. In the past, this was also true for software quality, but it has changed when testing became a best practice in modern software development. Testing is not a silver bullet, but it makes certain aspects of quality immediately visible to developers (who dislike failing tests). Moreover, incorporating testing into software development exerts pressure on making code testable, which improves low-level software design. It also makes refactoring safe (rather than feared), which allows developers to evolve software design to match the actual requirements. In the end—besides the testing itself which tells us that a certain feature works as expected—software testing has improved software quality in many other ways by making quality visible.Dealing with software performance is difficult. While performance can be (with certain difficulties) measured reasonably well (a necessary condition to making it visible and manageable), it is also difficult to interpret. Performance measurements cannot be easily interpreted in a binary fashion and the measurements we reason about have to correspond to a workload that is relevant. While we can think of ways to construct performance tests (similar, but not quite like functional unit tests) that can be evaluated automatically, it is the relevance of these tests that matters, which is where the DevOps culture could help. If the Devs can find (through Ops) what is relevant to performance, then they can write relevant performance tests, and thus make performance visible in the software development (and possibly continuous deployment) process. Can we help make performance visible? Dealing with Uncertainty in Developer Targeted Analytics [Jürgen Cito]Jürgen Cito (University of Zurich, Switzerland, [email protected])Runtime information of deployed software has been used by business and operations units to make informed decisions under the term“analytics”. However, decisions made by software engineers in the course of evolving software have, for the most part, been based on personal belief and gut-feeling <cit.>. This could be attributed to software development being, for the longest time, viewed as an activity that is detached from the notion of operating software in a production environment. In recent years, this view has been challenged by the emergence of the DevOps movement, which aims to promote cross-functional capabilities of development and operations activities within teams. This shift in mindset requires analytics tools that specifically target software developers. We investigate how to support developers in their decision-making process by incorporating runtime information in source code (“Developer Targeted Analytics”) <cit.>. In this approach, we also provide live feedback in IDEs by inferring the impact of code changes on software performance. However, no prediction is perfect. Every inference model comes with a level of uncertainty. To make integrated runtime feedback a proper basis for decision-making, we need methods to properly quantify uncertainty and consequently communicate it to software developers. In this seminar, I want to discuss sources of variation in software performance (e.g., cloud instance variability <cit.>) and possible ways to model uncertainty. Further, I want to review different ways to communicate runtime feedback through visualizations. Transforming Operations for the Cloud [Georgiana Copil]Georgiana Copil (TU Vienna, Austria, [email protected])In a culture where development and operations merge, sustained by new computing models such as cloud computing and management automation technologies, the operations processes need to evolve. Although extensive work is performed addressing the perspective of the cloud provider <cit.>, IT service management, and particularly operation management, is disregarded for the cloud customer side.Well established standards like ITIL <cit.>, BSI ISO 20000 <cit.>, and FitSM <cit.> are currently used for IT service management. However, these standards need to be adapted or completely re-designed to accommodate disruptive trends in technology, as emphasized by Forsgren et al. <cit.> and Fuggetta et al. <cit.>. A plethora of configuration management and automation tools (e.g., Chef, Puppet, Ansible, Brooklin, Vagrant, or rSYBL <cit.>) are being created for providing support both in development and operation. In companies using cloud services for developing their products, some organization roles will have to change, or be replaced by automated tools. For instance, procurement engineers might need to work with configuration management engineers in order to keep track with the costs that depend on system's configuration, or might simply be discarded from the organization chart. Although change is needed in processes and service management organizational charts, tools should be adapted as well in order to integrate better within the organization. For instance, controllers could notify the responsible operation engineers or managers <cit.>, with the correct content and frequency, on abnormal behavior, or on detected incidents. VRealizeand AppDynamicsprovide role-based alerting, and so-called “Virtual War Rooms” for enhancing Dev & Ops collaboration. Although these are a promising start, further investigation is necessary for understanding how operation-time events of various types (e.g., changes, faults, or incidents) can be automatically classified, and used for predicting such events, and analyzing root causes, and providing tighter connection with the developers or operators involved in these aspects.The next challenging aspect is the decoupled ownership of operations management, and especially of incident management, due to the decoupled ownership of cloud services, and systems using these services <cit.>. As stated before, systems using cloud services would need an adapted IT Service management processes, and integrate information coming from cloud providers into these processes. However, the reduced control over data <cit.>, absence of standard format and logs <cit.>, multi-tenancy <cit.>, make this integration very challenging. Better operation management possibilities on the cloud customer side, which includes information related to cloud services operation, can reduce the trust concerns and increase the adoption of cloud technologies.Challenges in Architectural Modeling for Performance-aware DevOps [Robert Heinrich]Robert Heinrich (Karlsruhe Institute of Technology, Germany, [email protected])Building software applications by composing cloud services promises many benefits such as flexibility and scalability. However, it leads to major challenges like increased complexity, fragility and changes during operations that cannot be foreseen in development phase. Cloud-based applications change rapidly and thus require increased communication and collaboration between software developers and operators as well as strong integration of building, evolving and adaptation activities.While previous research focused on automated system adaptation, increased complexity, heterogeneity and limited observability, makes evident that we need to allow operators (humans) to engage in the adaptation process. Architectural models such as those of the Palladio approach <cit.> are a foundation for involving humans and conducting analysis, e.g., for performance and privacy. During operations the system often drifts away from its development models. Run-time models are kept in-sync with the underlying system. However, typical run-time models are close to an implementation level of abstraction which impedes understandability for humans.DevOps practices enable software developers and operators to work more closely <cit.>. The software application architecture is a central artifact for developers and operators. New architectural styles such as microservices are proclaimed to satisfy requirements like scalability, deployability and continuous delivery. By merely introducing new architectural styles, however, current problems in the collaboration and communication among stakeholders of the development and operations phases are not solved. Existing architectural models used in the development phase differ from those used in the operation phase in terms of purpose (finding appropriate design vs. reflecting current system configurations), abstraction (component-based vs. close to implementation level) and content (static vs. dynamic). These differences result in limited reuse of development models during operations and limited phase-spanning consideration of the software architecture.We are developing the iObserve approach to address architectural challenges in performance-aware DevOps. iObserve provides a megamodel <cit.> to bridge the divergent levels of abstraction in architectural models used in development and operations. We employ descriptive and prescriptive architectural run-time models for realizing the MAPE loop. Including dynamic content, like in-memory objects and their communications, will help operators to observe and adapt the application when anomalies exceed automated planning routines <cit.>. Efficient Resilience Benchmarking of Microservice Architectures [André van Hoorn]André van Hoorn (University of Stuttgart, Germany, [email protected])The microservice architectural style <cit.> is gaining more and more prevalence in industrial practice when constructing complex, distributed systems. One of its guiding principles is design for failure, which means that a microservice is able to cope with failures of other microservices and its surrounding software/hardware infrastructure. This is achieved by employing architectural patterns such as circuit breaker and bulkhead <cit.>. Resilience benchmarking <cit.> aims to assess failure tolerance mechanisms—for instance, via fault injection <cit.>. Meanwhile, resilience benchmarking is not only conducted in development and staging environments, but also during a system's production use <cit.>—for instance, via Netflix's Simian Army <cit.>. Existing resilience benchmarks for microservice architectures are ad-hoc and based on randomly injected faults.In this talk, I will sketch the vision for efficient resilience benchmarking of microservice architectures. Resilience vulnerabilities shall be detected more efficiently, i.e., faster and with fewer resources, by incorporating architectural knowledge as well as knowledge about the relationship between performance/capacity/stability (anti) patterns and suitable injections. The idea builds on existing works on model-based and measurement-based dependability evaluation of component-based software systems.Benchmarking Quality of Performance Evaluation in the DevOps World [Vojtěch Horký]Vojtěch Horký (Charles University Prague, Czech Republic, [email protected])Performance in the DevOps world today typically revolves around the issue of collecting performance data from running applications and analyzing them (by the dev-and-ops) to detect performance anomalies. But are we able to say what is the ratio of detected and undetected issues, i.e., how reliable are the tools we use?The nature of continuous delivery complicates answering this. The software is updated too often—we are not able to collect enough information before a new version is introduced and we do not care that much about regressions in a week-old version. So how about writing a benchmark emulating the production of a DevOps-driven application? The benchmark would allow us to evaluate quality of tools and approaches we use. * The benchmark provides the baseline truth and we can quantify how many issues went undetected. Ranking the issues (e.g., how critical a problem is) gives even more information about the precision of the tested approach.* We can (at least partially) evaluate the solution without touching the production environment (the agility of DevOps suggests that we try the tools directly but that may not be always possible).* Having a benchmark would allow a comparison of different approaches that is otherwise virtually impossible because of the variances between the systems under test.The benchmark can test the solution from three different angles. As a database of measurements of various software components in different versions it evaluates how well are we able to detect anomalies. Also the benchmark could emulate a volatile environment where new versions of different components are loaded over time—a score indicates whether the measurement framework survives in such environment and how precise results it provides. Model-based approaches can be used here as well. And finally such emulation could also be used for testing the continuous delivery infrastructure and related processes—effectively benchmarking quality assurance processes.The current status is somewhere at the level of “collecting ideas for implementation”. The emulated application shall be component-based with a well-defined architecture; a good start might be the CoCoME <cit.> or PNMR <cit.>. We would then need to create several versions differing at various levels and add some kind of self-measurement infrastructure. Then we can quantify how well the tested solution adapts to the changes and how precise data it produces <cit.>. Collecting data for the database of measurements is the easier step—the difficult one is establishing the base truth: what were the actual regressions. Around the database we would then create a harness where individual solutions could be plugged-in and run on different data sets. Looking forward to see whether this might be interesting for someone else as well. Machine Learning Meets DevOps [Pooyan Jamshidi]Pooyan Jamshidi (Imperial College London, United Kingdom, [email protected])Today's mandate for faster business innovation, faster response to changes in the market, and faster development of new products demand a new paradigm for software development. DevOps is a set of practices that aim to decrease the time between changing a system in Development, and transferring the change to the Operation environment, and exploiting the Operation data back in the Development <cit.>. DevOps practices are typically relying on large amount of data coming from Operation. The amount of data depends on the architectural style <cit.>, the underlying development technologies and deployment infrastructure. For instance, big data distributed systems consist of an extensible execution engine (e.g., MapReduce), pluggable distributed storage engines (e.g., Apache Cassandra), and a range of data sources (e.g., Apache Kafka). Each of these produce a considerable amount of data regularly in each fraction of second.However, in order to make effective decisions in Development, e.g., architectural refactoring in order to make the system architecture sustainable <cit.> during its lifetime, there has to be efficient processing of such large amount of data in place in order to process the operational data. In this situation where data streams are increasingly large-scale, dynamical and heterogeneous, mathematical and algorithmic creativity are required to bring statistical methodology to bear. Statistical machine learning merges statistics with the computational sciences. Statistical machine learning can fill the gap between operation and development with some more efficient analytical techniques. The data efficient techniques can provide more deep knowledge and can uncover the underlying patterns in the operational data in order to detect anomalies in the operation or detect performance anti-patterns <cit.>. This knowledge can be very practical if detected ontime in order to refactor the development artifacts including code, architecture and deployment <cit.>.In this talk, I present our recent work on configuration tuning of big data software, where I primarily applied Bayesian Optimization and Gaussian Processes, a data efficient statistical machine learning method, in order to quickly find optimum configurations <cit.>. I also talk about transfer learning to exploit complimentary and cheap information (e.g., past measurements regarding early version of the system) to enable learning accurate models quickly and with considerably less cost <cit.>. Evaluating the Effectiveness of Different Load Testing Analysis Techniques [Zhen Ming (Jack) Jiang]Zhen Ming (Jack) Jiang (York University, Toronto, Canada, [email protected])Large-scale software systems like Amazon and eBay must be load tested to ensure they can handle hundreds and millions of current requests in the field. Load testing usually lasts for a few hours or even days and generates large volumes of system behavior data (execution logs and counters). This data must be properly analyzed to check whether there are any performance problems in a load test. However, the sheer size of the data prevents effective manual analysis. In addition, unlike functional tests, there is usually no test oracle associated with a load test. To cope with these challenges, there have been many analysis techniques proposed to automatically detect problems in a load test by comparing the behavior of the current test against previous test(s). Unfortunately, none of these techniques compare their performance against each other.In this talk, I describe our work on the empirical evaluation of the effectiveness of different test analysis techniques <cit.>. We have evaluated a total of 23 test analysis techniques using load testing data from three open source systems. Based on our experiments, we have found that all the test analysis techniques can effectively build performance models using data from both buggy or non-buggy tests and flag the performance deviations between them. It is more cost-effective to compare the current test against two recent previous test(s), while using testing data collected under longer sampling intervals (≥ 180 seconds). Among all the test analysis techniques, Control Chart, Descriptive Statistics and Regression Tree yield the best performance. Our evaluation framework and findings can be very useful for load testing practitioners and researchers. To encourage further research on this topic, we have made our testing data publicity available to download. How I Learned to Stop Worrying and Love Capacity Shortages [Cristian Klein]Cristian Klein (Umea University, Sweden, [email protected])My research focuses on engineering cloud applications so as to maintain responsiveness despite infrastructure capacity shortages. This has several benefits. First, the risk of user experience degradation is reduced when the application observes a sudden increase in popularity. Second, it allows the infrastructure to operate at higher utilization, which helps reduce costs. The main idea is to mark certain code of the application as optional and selectively deactivate its execution, as required to maintain a target response time. The problem can be decomposed into two questions: What code to deactivate and when to deactivate such code?To answer the “what” question, we proposed a methodology to retrofit admission control into existing cloud applications <cit.>. The first step is to model the user behavior based on production logs in a manner that is both realistic, but also flexible enough to test various “what-if” scenarios. Next, the model can be used to generate an amplified workload, identify bottlenecks and add circuit breakers around such bottlenecks, as directed by business objectives. Several iterations can be performed to add multiple circuit breakers, until the resilience of the application is judged to be sufficient. We evaluated the methodology against a production cloud application, featuring a large code base. To answer the “when” question, we proposed “brownout” a software engineering methodology to decide when to disable optional content, such as recommendations or comments. We used control-theory to design a controller that measures application response time and adjusts the probability of serving optional content, as required to maintain a given target response time <cit.>. While brownout itself is rather uninstrusive to the developer, the application's struggle with capacity shortage can no longer be observed by measuring utilization or response time, hence brownout-unaware components around the application may take errornous decisions. To tackle this issue, we proposed a brownout-aware load-balancer <cit.> and admission controller <cit.>.Our techniques allow software engineers to deprioritize performance-related non-functional requirements, such as scalability, and reduce time-to-market. This is important in the context of lean thinking, which advocates the creation of minimum viable products to discover customer requirements. Performance Modeling Challenges while Modernizing Existing Software towards Microservices [Holger Knoche]Holger Knoche (Christian-Albrechts-Universität zu Kiel, Germany, [email protected])In order to fully realize the promises of DevOps, namely the fast and continuous delivery of high-quality software, an appropriate software architecture is required. Currently, Microservices are considered the premier software architecture for this purpose <cit.>. As a consequence, many companies consider the introduction of Microservices to their existing software by implementing new features as Microservices or replacing existing functionality by Microservices. Further information on this topic can be found in <cit.>. Due to their highly distributed nature, Microservices introduce several new challenges for both development and operations. Developers now have to cope with potential performance degradation due to remote invocations, the possibility of partial failure, and the loss of ACID transactionality <cit.>.From an operations point of view, deploying, running, and monitoring a large number of service instances pose the greatest challenge. To address the development challenges described above, performance simulations are a promising option. As the mentioned effects are largely determined by the service design, the use of design-time models such as the Palladio Component Model <cit.> presents itself. The operational challenge is addressed by a group of tools which facilitate the deployment and operation of highly distributed applications, the most prominent being Kubernetes <cit.> and Mesos <cit.>. These tools automatically deploy services to a pool of nodes based on given constraints, and dynamically adjust the deployment should it become necessary (e.g., due to node failure). Unfortunately, however, current design-time performance models lack an appropriate abstraction for such dynamic, self-adaptive deployments. What adds further difficulty is the fact that in migration contexts, both “traditional” applications and Microservices interact with each other, and the performance impact of this interaction is of particular importance. Therefore, a formalism capable of modeling and simulating both worlds in an appropriate way is required. In my talk, I am going to further illustrate the performance challenges of migrating towards Microservices using examples from industrial practice. Furthermore, I will give a brief presentation of Mesos as an example of a modern deployment platform, and point out the performance modeling challenges that arise from such platforms.The Importance of Data Science for DevOps and Continuous Delivery [Philipp Leitner]Philipp Leitner (University of Zurich, Switzerland, [email protected])Academic research in software engineering (SE) is at a pivotal point. With the advent of DevOps, the SE research community needs to expand and augment its traditional topics (e.g., requirements elicitation, software architecture, formal specification, or testing), to include new runtime-related challenges. These include, but are not limited to, cloud computing, continuous delivery and deployment (CD), edge computing and the Internet of Things, or the Facebook-style “move fast” development philosophy that emphasizes the importance of quick delivery over strenuously verified correctness.One implication of this new “runtime focus” is that SE can, and will, become more data-driven. Today, however, data science in SE is still the domain of dedicated specialists more than part of day-to-day development. Developers still struggle to make systematic use of the deluge of data available through Application Performance Monitoring tools <cit.>, rollout decisions in continuous deployment and live testing are based on intuition rather than collected empirical evidence <cit.>, and software performance engineering techniques are still not widely found in industrial practice <cit.>. We argue that one reason for this is that performance monitoring and analysis tools are currently not “developer-targeted”. Their results and visualizations are not actionable for developers, and they do not provide robust value without expert knowledge in statistics and data science.In my talk, I will focus on three DevOps-related SE topics and how they relate to data science. Firstly, I will (briefly) introduce our work on Feedback-Driven Development (FDD), a concept that aims to make runtime performance data more useful for software developers <cit.>. Secondly, I will discuss the importance and challenges of data science for Continuous Delivery, Continuous Deployment, and live testing. In this segment, I will focus on the trade-off between release velocity and confidence <cit.>. Finally, I will discuss the challenge of cost-aware operation of applications on top of public IaaS clouds. I will present recent results in this domain <cit.>, and discuss problems and challenges. Industrial-grade DevOps: DevOps in the Digitalized Industrial World [Fei Li]Fei Li (Siemens AG, Austria, [email protected])Industrial systems, such as factory automation, infrastructure management, utility management, require highly reliable and near real-time solutions. The lifespan of such systems is long, and traditionally, during their lifetime the focus of software development and operation is to ensure high reliability and performance, and to continuously comply with various regulations. Therefore, new versions are released on a yearly basis or even less frequently, and whole-sale update has to be carried out for each release. Correspondingly, monolithic architecture is dominant among industrial software.However, the recent digitalization movement in industrial systems presents significant challenges to the traditional way of industrial software delivery. In a digitalized industrial world, flexibility and time to market are the key to ensure business success, while at the same time the requirements on software quality cannot be compromised. The current development methodology, software architecture and tools cannot support such demands in a digitalized industrial world.To this end, the concept of Industrial-grade DevOps has been proposed by the Corporate Technology department of Siemens to address the challenges of software delivery in the digitalized world. The vision of industrial-grade DevOps is to “move from monolithic products with long release cycles and wholesale updates to componentized products with continuous feature release and evolution in run.”In this scope, the following key research topics will be investigated.* Microservice architecture in industrial systems. The key is to ensure software quality while continuously evolving the microservice architecture. In particular, we focus on five software qualities: * Availability—Throughout continuous updates of independent microservices* Roll-backability—To roll back software and physical systems to earlier states in case of update failure* Resilience—In case single or even multiple microservices fail, the system as a whole should not fail* Security—To secure information as well as physical assets* Changeability—To understand the impact of individual changes of independent changes* Knowledge-drive architectural decision making methods. Development and operational knowledge is the cornerstone to make optimal decisions in the fast evolving DevOps environments. * Extracting operational routines, workflows and data structures as patterns and document them like architectural patterns* Building knowledge base that associates development and operational activities with software qualities.* Use operational patterns to address requirements from Ops* React to quality problems by using the result of operational data analysisThe Challenges and Benefits of Synthesizing and Theorizing the DevOps Phenomenon in Software Engineering [Lucy Ellen Lwakatare]Lucy Ellen Lwakatare (University of Oulu, Finland, [email protected])Software-intensive companies constantly try to improve their software development process for better software quality and a faster time to market. The continuous delivery paradigm represents a more recent mainstream software development practice that is predominant in the web domain and has a huge potential in its adoption in other domains <cit.>.DevOps—collaboration between development and operations activities—is necessary for adopting and enabling continuous delivery and deployment. However, the state-of-art on the DevOps phenomenon is driven by industry, and has very limited contributions in research <cit.>.This is evident in the article on a systematic mapping study of continuous deployment that has also shown that the research is lagging behind in systematizing knowledge and in validating many of the claims advocated in continuous deployment and DevOp practices <cit.>.Using a multi-vocal “grey” literature review and case study approaches in software engineering, a more focused study on the DevOps phenomenon is conducted with the aim of synthesizing, systematizing and providing evidence of DevOps practices <cit.>. The findings show that, even though it is possible to abstract the wide and diverse set of practices advocated by DevOps into useful patterns that can be adopted by other companies seeking to implement DevOps, from the research respective it remains challenging to generate theories. Releasing High-performance Software More Rapidly with Lower Costs using Continuous Test Optimization [Dusica Marijan]Dusica Marijan (Simula, Norway, [email protected])Organizations that follow DevOps practice often benefit from improved efficiency, more reliable releases, higher product quality, or better user experience. However, successfully implementing DevOps entails a number of challenges. One such challenge includes implementing an automated and cost-effective continuous testing, as a prime driver for DevOps. We analyze this challenge in the context of testing industrial communication systems. In this context, any supporting technique/tool has to integrate within CI/CD workflows, enable automation, and support collaboration between development and QA. In our work, we are interested in techniques for reaching a high degree of test automation and optimization for a given production environment, to enable faster deployments and more frequent releases of features and bug fixes, ultimately enabling rapid releases of products. We analyze how can historical data, usage profiles, sampling-based techniques, and other techniques within and outside of a production environment be efficiently used to collect performance metrics, recognize patterns and performance bottlenecks, in order to parameterize optimization algorithms for improved and more cost-effective testing. Joining Adaptation and Evolution Control Loops to Manage Performance in a DevOps Setting [Claus Pahl]Claus Pahl (Free University of Bozen-Bolzano, Italy, [email protected])An important problem is how to organise and manage the different performance engineering tasks within a DevOps setting. The core of the proposal here is a process model combining an adaptation and evolution loop of systems change, which is an extension of the ADEPS model <cit.> that joins two feedback loops over time. It consists of two intertwined change spirals, drawn out over time: An evolution helix reflecting long-term changes, usually with clear reference back into a development stage An adaptation helix reflecting short-term changes, usually dynamically in systems in an automated fashion. This change process model is governed by goal continuity as the key principle: in dynamic systems, performance goals need to be validated and maintained continuously. However, goal continuity is challenged by uncertainties arising from stakeholders, platform and information incompleteness and inaccuracy. Goal continuity is enabled through feedback that links Operations back into Development concerns if required. Tools and mechanisms that enact this are(i) controllers of dynamic adaptation in the Operations domain (e.g., through platform adaptation) and (ii) experiments and prototyping in continuous evolution in the Development domain (e.g., as part of refactoring and re-architecting activities)Both are critical for the enablement of performance engineering. In some way, a MAPE-K loop is in place for both feedback loops, one in an autonomous setting, the other in a more human controlled form. What is needed is a uniform model framework to manage both.Our objective should be to share the same mechanisms of monitoring, analysis and decision making consistently for both cycles. What a controller does for autonomic computing needs to be provided by managed experiments, prototypes and recommenders for an evolutionary scenario. A joint formal, methodological and technical basis shall be the aim to support both with the same rigour. We found for instance an abstraction of problem situations in terms of patterns to be beneficial <cit.>. The proposal is therefore to identify a set of common performance management models, which can build on these patterns. The wider context of this model, i.e., what drives the DevOps process, also needs to be understood better. Technical and economic sustainability is the aim of Continuous Development and Operations <cit.>. A software system is sustainable if it is resilient to emerging uncertainty. Cost and performance are intertwined.Cloud computing shall be selected as a specific case that highlights the relevance of performance engineering in a service-oriented architecture that is constrained by the service-level agreements between solution service provider and consumer. Performance Engineering in Fog Computing — An Overview [Stefan Schulte]Stefan Schulte (TU Wien, Austria, [email protected])First coined by Cisco <cit.>, the term fog computing describes the usage of well-known principles from cloud computing at the edge of the network, most importantly the virtualization of resources provided by networked devices. By applying fog computing, it is possible to lease and release networked devices as virtualized assets in an on-demand, utility-like fashion and to enable rapid elasticity through scaling these leased assets up and down, if necessary. Also, fog devices may be orchestrated in order to cooperatively process data <cit.>. Fog computing partially resembles ideas of mobile computing, however it does focuses on Internet of Things (IoT) entities like sensor nodes, smart objects, or cyber-physical systems. Apart from the term fog computing, edge computing is also frequently used to describe this approach.One main motivation for the advent of fog computing is its alignment with the basic structure of the IoT: Within the IoT, a plethora of technologically heterogeneous, connected devices are interacting with each other <cit.>. While IoT devices are highly (geo-)distributed, cloud resources are actually provided in quite a centralized way. This is not surprising, since the success of cloud computing can be attributed (amongst other reasons) to the economies of scale achieved in centralized, very large data centers. In contrast, fog resources are located at or nearby the edge of the network and thus in the vicinity of IoT devices. By exploiting already available resources, it is possible to do computational tasks “on-site”, i.e., close to the data sources and/or sinks, instead of executing these tasks in the cloud. For instance, preprocessing in data stream scenarios or data prefiltering in Big Data scenarios may be done in the fog instead of the cloud. Actually, decreased latency is one major reason for the usage of fog-based computational resources <cit.>, therefore fog computing is both an enabler and beneficiary of performance engineering.Since fog computing is still a very recent research topic, there are (to the best of our knowledge) no dedicated approaches for performance engineering in the fog. In fact, there are not even DevOps approaches for fog-based applications yet. This talk will therefore present open questions in the field, discussing also the differences between fog and cloud computing and why there is a need for specific performance engineering in the fog. In addition, some first ideas towards solving these questions will be presented.For this, we will introduce the notion of glocal living applications (GLAs), which are fog- and cloud-native applications by design. By applying container technologies, GLAs are isolated, portable applications, which can be hosted on any virtualized infrastructure. GLAs are able to move freely within and between data centers, both in the cloud and in the fog, e.g., by acquiring computing resources from a different cloud service or cloud region closer to the customer, or by offloading computational tasks from the fog to the cloud and vice versa, through migrating application code via the mobile code paradigm. This allows GLAs to minimize latency and enables region-specific service customization. Improving the Performance of Database-centric Applications through DevOps [Weiyi Shang]Weiyi Shang (Concordia University, Canada, [email protected]) There is a growing gap between the software development and operation, especially for database-centric applications. Software developers of database-centric applications typically leverage Object-Relational Mapping frameworks such as Hibernate to ease the access of database, and caching frameworks to optimize the performance of database access. However, the use of such framework brings extra challenges, such as performance overhead. Developers may not understand the field performance impact of executing the source code due to the use of the complex Object-Relational Mapping frameworks. On the other hand, developers often configure the caching framework based on their own experiences and gut feelings. A suboptimal configuration can cause even worse performance than without leveraging any caching framework.The introduction of DevOps bridges the gap between software development and operation and brings developers the rich field information, such as actual impact of source code execution in the field and the usage of the system by real users. With the field information, we propose techniques that can improve the performance of database-centric applications. In particular, one technique leverages the field data from end users to optimize the configuration of caching frameworks <cit.>. Instead of configuring based on developers' experiences, by leveraging the field information from software operation, developers can optimize the caching framework configuration by knowing the data-access patterns of database tables. Significant performance improvements are shown by evaluating the above technique with open source software and with our industrial partners in practice.SPE Meets DevOps: Best Friends or Consensual Enemies? [Catia Trubiani]Catia Trubiani (Gran Sasso Science Institute, Italy, [email protected])DevOps is a novel trend that aims at bridging the gap between development and operations, while providing the control of deployment and application runtime in the hands of developers. When applied in the context of software performance engineering, it raises new challenges related to which performance data should be carried back-and-forth between runtime and design-time and which feedback should be provided to developers to support them in the diagnosis of performance results. There is an obvious trade-off in the performance evaluation of early model abstractions where detected problems are cheaper to fix but the amount of information is limited, and late performance monitoring on running artifacts, where the results are more accurate but several constraints have been added on the structural, behavioral, and deployment aspects of a software system. The goal of this talk is to point out the research challenges in this domain thus to understand at which extent the meeting between SPE and DevOps leads to make them “best friends” by exploiting some synergy or “consensual enemies” by discovering some incompatibility. Performance-aware DevOps Through Declarative Performance Engineering [Jürgen Walter]Jürgen Walter (Julius Maximilians Universität Würzburg, Germany, [email protected])Performance is of particular relevance to software system design, operation, and evolution because it has a major impact on key business indicators. Over the past decades, various methods, techniques, and tools for modeling and evaluating performance properties of software systems have been proposed covering the entire software life cycle <cit.>. However, the application of performance engineering approaches to solve a given user concern is still rather challenging and requires expert knowledge and experience. There are no recipes on how to select, configure, and execute suitable methods, tools, and techniques allowing to address the user concerns. The application of performance engineering approaches is challenging even in classical slow and heavy-weight processes with long-term release cycles. DevOps automates the process of software delivery and infrastructure changes. This results in short-term build and release cycles. Consequently, there is a more frequent need for performance evaluation.Reinforced by faster developments cycles, software and system engineering requires an easy, holistic integration and automation of performance engineering techniques. Declarative Performance Engineering (DPE) <cit.> aims to reach this decoupling of the description of the user concerns to be solved (performance questions and goals) from the task of selecting and applying a specific solution approach. The strict separation of “what” versus “how” enables the development of different techniques and algorithms to automatically select and apply a suitable approach for a given scenario. The goal is to hide complexity from the user by allowing users to express their concerns and goals without requiring any knowledge about performance engineering techniques. Realizing the DPE vision my research includes amongst others* performance concern language design* reference architecture for automated deduction of concerns* tool decision support and capability model* automated performance model learningMonitoring DevOps Processes and Experimental Process Improvement [Ingo Weber]Ingo Weber (Data61, CSIRO, Australia, [email protected])For the last 4 years, my team has been working in the DevOps space, and the lack of readily available material on this topic motivated us to write our DevOps book <cit.>. In the first part of my talk I summarized the key points from the book. The second part of my talk focused on two topics of our DevOps research that are relevant for performance engineering, specifically (i) correlation of DevOps process event occurrence with changes in metrics, and (ii) applying DevOps methods to process improvement and measuring outcomes.*Correlating metrics with DevOps process events The Process-Oriented Dependability (POD) framework has been developed to deal with failures of application operations, since they are one of the main causes of system-wide outages in cloud environments. This particularly applies to DevOps operations, such as backup, redeployment, upgrade, customized scaling, and migration that are exposed to frequent interference from other concurrent operations, configuration changes, and resources failure. However, previous practices failed to provide a reliable assurance of correct execution for these kinds of operations. In this work, we devised an approach to address this problem that adopts a regression-based analysis technique to find the correlation between an operation's activity logs and the operation activity's effect on cloud resources. The correlation model is then used to derive assertion specifications, which can be used for runtime verification of running operations and their impact on resources. The research has been published <cit.>.*Experimental process improvement Software systems that support Business Process Management are in widespread use. They play an important role in facilitating process automation and process improvement. Yet, there is hardly any insight into whether the implementation of a supposedly improved process model leads to an actual improvement in the process. The research problem this work addresses is: how can we determine if a new variant of a process model is an improvement over a previous variant, with respect to relevant measures? To this end, we suggest to build on recent software engineering concepts from the DevOps movement, as well as adopt learnings from performance engineering for measurements and analysis. On this basis, we want to develop novel techniques that provide the infrastructure for assessing in how far a specific business process change leads to an improvement. A vision paper on this topic has been published <cit.>.Performance of Continuous Delivery Pipelines [Johannes Wettinger]Johannes Wettinger (University of Stuttgart, Germany, [email protected])During the last 3–4 years, my research activities focused on diverse challenges concerning DevOps and continuous delivery. I was diving into these topics as part of my works and efforts as a PhD candidate at the Institute of Architecture of Application Systems (IAAS) at the University of Stuttgart. The motivation for these research topics is obvious: especially users, customers, and other stakeholders in the fields of Cloud services, Web applications, mobile apps, and the Internet of things expect quick responses to changing demands and occurring issues. Consequently, shortening the time to make new releases available becomes a critical competitive advantage. In addition, tight feedback loops involving users and customers based on continuous delivery ensure to build the “right” software, which eventually improves customer satisfaction, shortens time to market, and reduces costs.Independent of the chosen approach to establish continuous delivery by tackling cultural and organizational issues, a high degree of technical automation is required. This is typically achieved by implementing an automated continuous delivery pipeline (also known as deployment pipeline), covering all required steps such as retrieving code from a repository, building packaged binaries, running tests, and deployment to production. Such an automated and integrated delivery pipeline improves software quality, e.g., by avoiding the deployment of changes that did not pass all tests. Moreover, the high degree of automation typically leads to significant cost reduction because the automated delivery process replaces most of the manual, time-consuming, and error-prone steps. Establishing a continuous delivery pipeline means implementing an individually tailored automation system, which considers the entire delivery process. Furthermore, a separate pipeline has to be established for each independently deployable unit, e.g., a monolith or microservice. As a result, a potentially large and growing number of individual pipelines have to be established. Therefore, my research focuses on dynamically and systematically establishing such continuous delivery pipelines.The key building blocks of a continuous delivery pipeline are diverse application environments such as a development environment, test environment, and production environment. A specific goal is to provide discovery, consolidation, utilization, and orchestration approaches to choose and combine the most appropriate solutions and implementations (cloud services, infrastructure resources, deployment scripts, etc.) to establish a particular environment, which depends on individual requirements: a development environment typically aims to be lightweight with minimum overhead, e.g., hosting the entire application stack on a single virtual machine, whereas a production environment has to be highly available and elastic, e.g., comprising a cluster of virtual servers.Considering performance engineering in the DevOps world, diverse performance aspects of applications are investigated, for example, in the form of an automated performance testing stage as part of a continuous delivery pipeline. However, beside the actually delivered application, its associated continuous delivery pipeline comprising diverse stages and application environments represents a software system on its own. That system is not static but must evolve to deliver new iterations of an evolving application. If, for instance, new kinds of components or technologies are added to the application stack, at least some of the involved application environments need to be updated. Consequently, performance engineering in the DevOps world is not limited to evolving applications, but also needs to be considered when maintaining continuous delivery pipelines as separately evolving software systems. Beside reliability, a key performance aspect of such pipelines is their speed, i.e., the time it takes from committing a change to the code repository until it is fully tested, packaged, and ready to be put into production. This is the foundation for providing fast and immediate feedback loops to continuously improve the delivered application.Technically, different approaches can be applied to improve a pipeline's performance. For example, the execution of tests can be split, so that an initial test stage only runs tests that are fast, while following test stages run tests that take longer; this accelerates feedback loops and saves resources by failing as fast as possible. Moreover, binary artifacts such as container images should only be built once and then stored in artifact repositories such as a Docker container registry to be reused in following stages of the pipeline. Another optimization approach could be to consolidate application environments, e.g., to run the build and unit test stages in the same environment. While this helps to lower resource consumption, the degree of isolation is reduced. As a result, this approach cannot be applied if isolation is key (e.g., performance testing vs. production). Another performance aspect is the provisioning time of underlying application environments, e.g., the time it takes to make the unit test environment available to run the unit tests based on it. This could potentially be optimized by reusing environments instead of completely provisioning them for each pipeline run. Performance metrics must be defined for each pipeline, e.g., to measure and record speed and resource consumption of pipelines. This also helps to identify performance regressions of pipelines. Further approaches to optimize the performance of continuous delivery pipelines may be identified and discussed as part of these research efforts. An initial goal is to come up with a set of patterns and best practices that can be applied to improve the performance of continuous delivery pipelines. Towards Application-aware Cloud Provisioning for Enterprise Applications [Felix Willnecker]Felix Willnecker (fortiss, Germany, [email protected])Enterprise applications are typically implemented as distributed systems composed of several independent components or services <cit.>. Modern development paradigms allow to develop such applications using elastic infrastructure as deployment targets. These enterprise applications are designed to scale-out if load peaks occur <cit.>. Load peaks are balanced by spawning new virtual servers or containers in an elastic infrastructure [3]. Current strategies are reactive and thus spawn new instances late and often keep this instances for hours after the load peak occurred <cit.>. In such cases, the infrastructure is over-provisioned for most of the time. Over-provisioned deployments can cause unnecessary costs, as infrastructure providers such as Amazon Web Services (AWS) offer their services based on uptime, load and/or traffic. Better deployment strategies can decrease the runtime costs of enterprise applications by re-using already provisioned server instances or selecting server instances that fit the current demand.Elastic cloud infrastructures automatically scale deployments based on the load and eliminate the need for sophisticated deployment topologies <cit.>. The infrastructures leverage the effect of virtualization and over-commit their hardware. Thus, they create more virtual servers as available hardware would allow. However, the benefits of this over-commitment are solely taken by the infrastructure providers, especially when the virtual servers are poorly utilized. Moreover, if one virtualized server's resources exceed, another instance spawns which increase the costs. Resources of other virtual servers of the same enterprise application are usually not considered to contain load peaks and reduce total runtime costs.We propose an application-aware cloud provisioning based on a model based deployment topology optimizer <cit.>. We introduced a deployment topology optimizer, which selects an optimized topology for a fixed workload. The optimizer takes available and unused resources as well as costs and performance into account. In combination with workload prediction and performance models, this optimizer can reorganize distributed enterprise applications based on expected load and already provisioned infrastructure. Thus, reducing the number of instances, which reduces costs and shares the savings of virtual servers in elastic infrastructures with the operator of the enterprise applications.§ WORKING GROUPS Performance Engineering Challenges for Microservices [ Robert Heinrich,Andre van Hoorn,Holger Knoche,Fei Li,Ellen Lwakatare,Claus Pahl,Stefan Schulte,Johannes Wettinger ] Robert Heinrich (Karlsruhe Institute of Technology, Germany)Andé van Hoorn (University of Stuttgart, Germany)Holger Knoche (Christian-Albrechts-Universität zu Kiel, Germany)Fei Li (Siemens Corporate Technology, Austria)Lucy Ellen Lwakatare (University of Oulu, Finland)Claus Pahl (Free University of Bozen-Bolzano, Italy)Stefan Schulte (TU Vienna, Austria)Johannes Wettinger (University of Stuttgart, Germany) §.§.§ Discussed Problems Microservices are an emerging architectural style, complementing approaches like DevOps and continuous delivery in terms of software architecture.To some degree microservices resemble concepts and technologies from other established styles—particularly, service-oriented architectures (SOA).The goal of this breakout group was to discuss challenges and possible approaches in performance engineering for microservices. We started by identifying differences between microservices and SOAs to see what architectural characteristics stop us from just reusing existing performance engineering concepts. Identified differences includei) the technology and protocol stack (e.g., heavy-weight middleware vs. light-weight REST-based communication),ii) the purpose (e.g., integration of systems vs. system decomposition), and iii) deployment models and scalability. Testing, monitoring, and modeling were identified and discussed as primary research areas in performance engineering for microservices. In practice, additional challenges are imposed when existing legacy applications are migrated into a microservice architecture. However, this topic was excluded from the discussion. §.§.§ Possible Approaches The core results from the discussion about performance testing, monitoring, and modeling for microservices were: * Testing.Early-stage testingfor microservices follows common software engineering practices.More comprehensive testing (e.g., integration/system tests) is compensated or replaced by fine-grained monitoring of production environments.Performance testing gets simplified in the first place, i.e., the performance of each microservice (typically deployed as a single container) can be measured and monitored in isolation. The continuous delivery practice (e.g., multiple deployments per day) does not allow time-consuming performance testing of the entire application before each single deployment.Once a microservice is put into operation and used, the monitored performance data (e.g., using established application performance monitoring (APM) approaches) can in turn be utilized to devise and improve performance regression testing. * Monitoring.State-of-the-art APM toolssupport the collection of various measures of the entire system stack.In microservice architectures, basically the same techniques for data collection can be used as in traditional architectures.A technical instrumentation challenge is imposed by the microservice characteristic of polyglot technology stacks, particularly involving the use ofemerging programming paradigms and languages (e.g., Scala). In microservice architectures, additional measures are of interest in order to monitor specific architectural patterns at runtime. Examples include the state of resilience mechanisms, such as circuit breakers.In microservice architectures, it becomes difficult to determine a normal behavior used for anomaly detection. The reason is that due to the frequent changes (updates of microservices, scaling actions, virtualization) no “steady-state” exists. Existing techniques may raise many false alarms. * Modeling. Component-based performance modeling and prediction can be seen as the starting point from which techniques for microservices can be reused. Application-level modeling used to be the focus in traditional component-based performance modeling. Good results could have been obtained by abstracting from the middleware. An example here is work in the Palladio context.Another perspective is the platform/infrastructure level. Microservice platforms turn out to be more complex and the behaviour of the platform is an influencing factor in the overall setting. Thus, the need emerges to explicitly represent the platform with its different layers (e.g., physical machines, VM, Docker, cluster) on which the application is placed. Creating models by hand is not an option anymore due to, firstly, the inherent complexity of the platform and, secondly, due to the frequent changes in the environment. To learn the model in an environment that is bound to change becomes a challenge. Machine learning and model extraction techniques need to be studied to find a solution for the determination and updating of complex, changing models. Work in the context of models@runtime can provide input here. These models can feed into a closed adaptation loop where for instance auto-scaling techniques are applied. Performance models need to guide monitoring data analysis and decision making (MAPE loop). Controllers must cope with the elasticity rules. New processing policies must be specified.§.§.§ Conclusions Despite the obvious importance of a sufficient level of performance, there is still a lack of performance engineering approaches explicitly taking into account the particularities of microservices. So far, performance engineering for microservices has not gained attraction in the relevant communities and the identified challenges impose a number of interesting research questions and directions. Meanwhile, the results of the breakout group have been summarized in a workshop contribution <cit.>. Performance Test Prioritization [ Georgiana Copil, Philipp Leitner, Ingo Weber, Felix Willnecker ] Georgiana Copil (TU Vienna, Austria)Philipp Leitner (University of Zurich, Switzerland) Ingo Weber (Data61, CSIRO, Australia)Felix Willnecker (fortiss, Germany)§.§.§ Discussed ProblemsPerformance testing is currently not integrated into the Continuous Delivery pipeline. This is mainly on account of the multitude of performance tests that need to be executed on a high number of system states, resulting in a vast amount of testing configurations. Several approaches, such as BlazeMeter[<https://www.blazemeter.com>], LoadImpact[< https://loadimpact.com>], or RadView[<http://www.radview.com>] offer PaaS solutions for performance testing under production-like configurations, with possibilities to integrate them into normal continuous delivery pipelines (e.g., BlazeMeter Jenkins plugin). However, the cost and time needed for executing performance tests can be prohibitive, leading to these tests being absent from most continuous delivery pipelines.Figure <ref> shows the defining dimensions influencing runtime performance, which characterize the System Under Test: (i) the code that is being tested, (ii) the workload, (iii) the test data, and (iv) the configuration. The latter includes all software system and hardware related configurations, defining the environment in which the system under test will be running. §.§.§ Possible Approach In the breakout group, we have identified test case prioritization as a promising approach to deal with these issues: prioritizing which of the performance tests need to be run in order to determine most important regressions, and excluding tests that determine the same performance regression. Executing the performance tests covering the most important performance regressions can lead to improved testing time and reduced cost, or can support software developers with limited time or limited budget in discovering possible performance regressions. The core idea of this work is to determine which of a number of microbenchmarks are indicative of true runtimeperformance of an application. After obtaining that knowledge, we can use it for prioritization of performance microbenchmarks to make the most of a given time budget for performance testing, as well as for utilizing the performance measurements obtained as a side effect of regular unit testing for performance regression testing. The means by which we plan to obtain this information is machine learning or statistical techniques such as correlation, as shown in Figure <ref>. Specifically, we plan to use these approaches to determine which differences of performance in microbenchmarks from one version of the software to the next are indicative of actual performance differences between these versions.§.§.§ Conclusions We plan to use machine learning and statistical correlation analysis to identify particularly “promising” performance tests to quickly execute as part of a Continuous Delivery pipeline for fast performance feedback. We hope that this work will lead not only to useful methods for microbenchmark evaluation, but also to immediately useful tools that can be integrated into CI/CD pipelines (e.g., Jenkins plugins). Finally, we expect to improve our understanding of how to write “expressive” microbenchmarks in the first place.Uncertainty in a Performance-Aware DevOps Context [ Markus Borg, Jürgen Cito, Pooyan Jamshidi, Zhen Ming (Jack) Jiang, and Catia Trubiani] Markus Borg (RISE SICS AB, Lund, Sweden)Jürgen Cito (University of Zurich, Switzerland)Pooyan Jamshidi (Imperial College London, United Kingdom)Zhen Ming (Jack) Jiang (York University, Toronto, Canada)Catia Trubiani (Gran Sasso Science Institute, Italy) Uncertainty is a very relevant challenge to performance-aware DevOps. Performance measurements and predictions are essential in DevOps solutions to the extent that many different tools rely on the performance measurements and predictions <cit.>. Various techniques to evaluate the performance properties of software systems in early stages of development exist, e.g., based on architectural models enriched by performance-relevant information. However, in early stages, various parameters of the systems are uncertain, e.g., regarding implementation details and environment in which the software systems are deployed. This imposes a challenge on early performance prediction. In the software performance domain, uncertainties concern, for example, the usage profile (including workload intensity, navigational profiles, and input data), resource demand characteristics of software services, and properties of the deployment and execution environment (including hardware) on which the software will be deployed. In the break out group, we discussed the uncertainty challenges in the context of DevOps and we based our general definition of uncertainty as <cit.>: “any deviation from the unachievable ideal of completely deterministic knowledge of the relevant system”. Such deviations can lead to an overall “lack of confidence” in the obtained predictions based on the monitoring data that they might be “incomplete, blurred, inaccurate, unreliable, inconclusive, or potentially false”. To make informed decisions, DevOps teams need to be aware of uncertainties in the whole DevOps life-cycle to be able to interpret data, models, and results accordingly. We have identified sources of uncertainty in a typical performance-aware DevOps scenario. §.§.§ Discussed ProblemsGoal of this break out group was to identify the sources of uncertainty due to the application of DevOps in the performance evaluation of software systems. In fact, there is an obvious trade-off in the performance evaluation of early model abstractions where the amount of information is limited, and late performance monitoring on running artifacts where the results are more accurate but some constraints have been added. We were discussing our experiences on case studies showing different sources of uncertainty that span on multiple characteristics, such as the structural, behavioural, and deployment aspects of a software system and it emerged that performance results are heavily affected by these aspects. §.§.§ Possible ApproachesTo make informed decisions, DevOps teams need to be aware of uncertainties in the whole DevOps life-cycle to be able to interpret data, models, and results accordingly. To provide support in this direction, possible approaches have to: (i) identify sources of uncertainty in a typical performance-aware DevOps scenario; (ii) elaborate how these uncertainties manifest in input data, design models and operational results; (iii) make suggestions as how to interpret knowledge (i.e., input data, design models and operational results) given these sources of uncertainty. In this way it is possible to figure out the performance trend of the system exposed to such uncertainties. §.§.§ ConclusionsOur conclusions were that it is relevant to bring the sources of uncertainty up-front in the performance-aware DevOps process to support developers in the interpretation of performance evaluation results. Indeed, more research is needed in this direction; we plan to investigate this topic further to investigate how the explicit specification of uncertainties benefit the performance analysis process. How Can We Facilitate Feedback from Operations to Development? [ Markus Borg, Jürgen Cito, Fei Li, Lucy Ellen Lwakatare, Johannes Wettinger ] Markus Borg (RISE SICS AB, Lund, Sweden)Jürgen Cito (University of Zurich, Switzerland)Fei Li (Siemens Corporate Technology, Austria)Lucy Ellen Lwakatare (University of Oulu, Finland)Johannes Wettinger (University of Stuttgart, Germany)§.§.§ Discussed ProblemsFeedback from operations is important to drive informed decisions especially in modern software development approaches of fast release cycles. When software is operated in production, it produces a plethora of data that ranges from log messages emitted by the developer from within the code to performance metrics observed by monitoring tools. All this data gathered at runtime serves as valuable feedback to various stakeholders to improve the software itself and the process overall.However, there are some challenges organizations face when attempting to facilitate proper feedback channels between operations and the rest of the software development life-cycle. Some of these challenges include: organizational size, nature of business, presentation of feedback and other technical challenges. We discussed these challenges in more detail and looked at possible solutions.§.§.§ Possible ApproachesFigure <ref> illustrates a generic feedback process that attempts to abstract the process of retrieving feedback from operations across different organizational boundaries. We derived the process from informally discussing industrial use cases to ensure that the process covers concerns and needs of different company structures.Taking this generalized feedback process as a basis for discussion, we formulate the following considerations: * Role of Deployment for Feedback.The feedback process is initially kicked-off with deployment of software to make it available to end-users. Deployment can range from an automated, continuous delivery/deployment process to releasing a software unit that requires more complex (often manual) processes to roll out. In the former case, it stays within a company’s own organizational boundaries (public/private cloud or data center). Product development together with DevOps/operations engineers from the same organization are responsible for the operability. In the latter case, it is delivered through consultants/solution architects as on-premise software.The complexity to enable a proper feedback process thus depends on the complexity of the deployment pipeline. * Feedback Governance. There needs to be control over which kind of operations feedback is available to which kind of stakeholder. This kind of governance should explicitly provide high-level rules on how data is handled either in organizations or as a cross-organizational concern. These rules are then implemented and enforced by the DevOps/operations engineers by filtering and controlling runtime data. The consequence of this part of the process is that privacy is being enforced and product development only has access to data that exhibits no threat of violations or non-compliance. * Closing the Feedback Loop: Decision-making in Development. Eventually, once operations data passes through governance it becomes valuable feedback to stakeholders in product development. Figure <ref> illustrates two broad examples of stakeholders benefiting from runtime feedback. Product/project managers can now use runtime feedback to better plan their features and optimize their project plan. Software developers and DevOps engineers have a full picture of how users experience their software (e.g., performance metrics, usage counters) and can tweak program and infrastructure code to improve the overall experience. Here, the feedback loop starts again with deploying changes to software that were informed by better decisions through runtime feedback. §.§.§ Conclusions Runtime aspects of software observed and collected as metrics and events can provide valueable feedback in the development cycle of software. We discussed different considerations of establishing a feedback process that depends on the complexity of the existing deployment structures and requires its own feedback governance to be established (especially in larger organizations). We plan to further investigate industrial case studies to identify the challenges when establishing such a feedback process.Performance Engineering for Blockchain-based Applications [ Philipp Leitner, Stefan Schulte, Ingo Weber ] Philipp Leitner (University of Zurich, Switzerland)Stefan Schulte (TU Vienna, Austria)Ingo Weber (Data61, CSIRO, Australia) §.§.§ Discussed ProblemsBlockchain is often named as a way to realize trust between anonymous parties without the need of a trusted third party. Instead, trust is established as an emergent property of the blockchain technology, i.e., a distributed ledger which stores transactions in a permanent and indisputable way <cit.>. Blockchains are not maintained by a single organization, but hosted and enacted in a peer-to-peer manner. Blockchains can be used in arbitrary applications in order to provide provenance of data, e.g., about business transactions. For instance, blockchains have been applied in order to execute business processes as smart contracts <cit.>, thus documenting the process execution in a blockchain and using smart contract features to ensure that the participants in the collaborative process do not deviate from the agreed-upon process model.Within this breakout group, our goal was to identify performance engineering issues for blockchain-based applications and to think about respective solution approaches. Indeed, there has only been limited discussion on how the usage of blockchain technologies influences application design <cit.>. A particular aspect that has not been discussed in detail are timing issues. New blocks in blockchains are usually issued with a particular time distance between two blocks, i.e., the so-called interblock time. For instance, in the Bitcoin blockchain <cit.>, the median interblock time is set to 10 minutes; in the Ethereum blockchain <cit.>, it is set to about 13 seconds. Furthermore, there is a lot of variance: individual interblock times for Bitcoin can easily exceed an entire hour. Applications which want to apply the blockchain need to cope with this and accept it as given. In the following subsection, we discuss how these timing issues as well multi-step transactions lead to (performance) uncertainty in blockchain-based applications. Afterwards, we propose how to overcome these issues.§.§.§ Possible Approaches * Performance Uncertainty. A core performance challenge in blockchain-based applications is caused by the peer-to-peer nature of the network, as well as the cryptographic properties of the different blockchain protocols. Concretely, we have discussed two sources of performance variability and therefore uncertainty in our breakout group: * Uncertainty with regards to the interblock time. While protocols such as Ethereum aim to normalize this time to a well-defined value in the median, substantial outliers exist. For instance, for Bitcoin, in a 10-minute interval the next block is found with a probability of 63%, but roughly 5% of the interblock times are above 30 minutes[Source: <https://en.bitcoin.it/wiki/Confirmation>]. Ethereum suffers from similar variance. As can be seen, interblock times are uncertain, and may vary to a very large extent <cit.>. * Uncertainty with regards to the confirmation of a block. A transaction being part of a single block does not guarantee that the transaction will remain part of the chain. The transaction may still “fall off” if the blockchain decides that the block the transaction is part of becomes deprecated, because of a fork of the chain. However, the more follow-up blocks (also called confirmation blocks) have been added to the blockchain after the transaction has been added, the more unlikely this becomes. In consequence, the probability of a transaction actually being and staying part of a blockchain increases with time. In order to overcome the issues arising because of performance uncertainty, it is first necessary to empirically examine the blockchain's behavior. This requires observing the blockchain networks and to statistically model interblock times as well as the likelihood and frequency of forks. Once this has been done, the likelihood of forks can be modeled using the means of Markov chains, or, assuming that the probability remains constant, as a simple conditional probability. A first short paper in this direction has been published after the seminar <cit.>. * Uncertainty in Multi-Step Transactions. Uncertainty may also arise from applications which include multi-step transactions. If we assume transactions to not only be simple one-shot interactions with the blockchain, but rather multi-step (business) processes <cit.>, uncertainties become even more problematic. For instance, in a business process, process state transitions are triggered by events. Each event can be stored in a blockchain. However, due to the uncertainties mentioned above, a process participant may be required to wait until an event has n confirmation blocks on the blockchain before she considers an event to “have happened for sure”. This leads to considerable delays in state transition times if every transition needs to be delayed until enough time has passed and confirmation blocks have been accumulated to reach sufficient certainty.Alternatively, a process participant may decide that she “accepts” a state transition after a shorter period of time in order to speed up the process, for instance because she can observe the effects of the change in the physical world, or because the specific state transition is not crucial to her. However, this leads to interesting challenges, both from a business process modeling and execution perspective: * From a modeling perspective, it raises the question how a process participant would model what level of certainty she requires for each transition. * From an execution perspective, the blockchain-based engine needs to be aware of the possibility that it is “wrong” about the current state of the process for any given instance. Note that this may have domino effects, as participants may have already taken follow-up actions based on previous state information that may not be valid anymore after an update. A blockchain-based process modeling language and engine should have means to specify and execute suitable compensations for such cases, which bring the process back into a consistent state. §.§.§ ConclusionsTo the best of our knowledge, despite the rapidly increasing application of blockchain technology by the research community and industry, performance engineering considerations around the use of blockchains in application design and development have not gained much attention yet.As concrete follow-up steps of the discussions within the breakout group,we will carry out the needed empirical studies and take the results from these studies into account in order to devise uncertainty-aware, blockchain-based applications and suitable performance engineering approaches. Implications of DevOps and Self-Adaptivity [ Georgiana Copil, Pooyan Jamshidi, Cristian Klein, Claus Pahl ] Georgiana Copil (TU Vienna, Austria)Pooyan Jamshidi (Imperial College London, United Kingdom)Cristian Klein (Umea University, Sweden)Claus Pahl (Free University of Bozen-Bolzano, Italy)§.§.§ Discussed Problems Self-adaptivity is a software engineering concept aiming to reduce runtime uncertainty at design time by designing the application to adapt to runtime changes. For example, the uncertainty in the number of users accessing a particular application can be reduced by designing the application with auto-scaling capabilities. At runtime, the application will adapt to the actual number of users, by acquiring and releasing computing capacity, usually presented as virtual machine instances, or enabling/disabling optional features of the system.We discussed two possible relationships between self-adaptivity and DevOps: (a) adding DevOps to a self-adaptive application (DevOps4SA) and (b) adding self-adaptivity to a DevOps pipeline (SADevOps). For both topics, we discussed motivation, challenges and opportunities.§.§.§ DevOps4SA Using DevOps for delivering a self-adaptive application is motivated by bringing the benefits of evolving both the application and its controller in a reliable manner. However, this would also bring more challenges. First, there would be more sources of uncertainty, bugs and regressions, which could come from the application, the controller or the interaction between the two. Second, controllers and applications are often tightly coupled. Hence evolving both of them would be challenging. Finally, due to the frequent nature of deployments in the DevOps culture, the controller would have shorter learning periods between consecutive changes to the application. Indeed, new code added to the application could invalidate the model learned by the controller. The model learning can be done in an incremental fashion.We also discussed opportunities brought by DevOps4SA. First, the DevOps pipeline could be accelerated by reducing testing time, since uncertainties are compensated for at runtime. Second, the controller could reuse the learned model from previously deployed version of the application, hence not needing to relearn a model of the application from scratch.§.§.§ SA4DevOps One of the main ideas of DevOps is to use measurements from operations as feedback for development. Making the DevOps pipeline self-adaptive would allow taking automated low-level decisions, freeing up humans to deal with higher-level decision. The system would be composed of two loops. The human-in-the-loop would focus on delivering value and deal with long-term predictions, aiming to evolve requirements, design and implementation of the application, as promoted by the BizDevOps concept. The controller-in-the-loop would focus on dull and error-prone processes and deal with short-term predictions, aiming to evolve the configuration. The Dev and Ops would essentially acts as supervisors for the self-adaptive controller.The discussion then naturally steered towards what kind of DevOps activities could be taken over by self-adaptive loops:* Remap for Performance: A controller could identify hotspots during operation — for example, two components that frequently feature high CPU utilization simultaneously — and add anti-co-location constraints to be taken into account during the next deployment.* Remap for Resilience: Similarly to the example above, a controller could identify hotspots that appeared after fault injection and add anti-co-location constraints.* Fault Injection: Based on the log of source code changes, a controller could inject faults targeted at application components that have recently changed, since these are more likely to be plagued by bugs.* Monitoring: Similarly to the example above, a controller could increase the monitoring frequency of recently changed application components in order to get more information at a certain time period.* Anomaly Detection: Taking as input source code changes and test results, a controller could change alarm thresholds and whether to alert developer or operators. A recently changed component is more likely to feature a bug within developer's responsibility, whereas a stable component is more likely to feature erroneous behaviour due to changes in operations.* Tracing: Tracing large-scale distributed systems can be very expensive, therefore, tracing is usually turned off or set to a low level of detail. Taking as input source code changes and test results, a controller could enable tracing or increase the level of detail for application components that recently changed. Most of the above topics consider code change in the system as enviromental uncertainty that requires an appropriate reaction from the down-the-line DevOps toolchain. §.§.§ Conclusions We found many promising research directions arising from the interaction between DevOps and self-adaptivity. DevOps can enable co-evolution of controllers and the controlled system. On the other hand, self-adaptivity can provide opportunities to automate manual and labor intensive tasks in the DevOps pipeline.A Systematic Process for Performance Antipattern Detection and Resolution in DevOps based on Operational Data and Load Testing [ Alberto Avritzer, André van Hoorn, Catia Trubiani, Holger Knoche ] Alberto Avritzer (Sonatype, USA)André van Hoorn (University of Stuttgart, Germany)Catia Trubiani (Gran Sasso Science Institute, Italy)Holger Knoche (Christian-Albrechts-Universität zu Kiel, Germany)§.§.§ Discussed ProblemsWe discussed the problem of providing the performance assessment of complex software systems since such systems are subject to many variabilities, such as workload fluctuation and resource availability. The main issue is that such variabilities may introduce flaws that affect the system quality and generate negative consequences, such as delays and failures. Hence, it is necessary to put in place some methodologies that allow to recognize performance flaws and generate software refactorings able to overcome such flaws.§.§.§ Possible ApproachesWe are investigating an approach based on load testing and profiling data, to identify performance flaws and generate software refactorings to developers, thus to support them in the interpretation of performance measurements. These activities are supported by performance antipatterns that are well known to document common development mistakes leading to performance flaws as well as their solutions. To this end, we are investigating a real-world case study provided by an innovative company in the open-source domain.§.§.§ ConclusionsWe found that the analysis of load testing and profiling data is not trivial, and the domain expert knowledge is fundamental to progress on such activity. The automation of software refactorings is feasible up to a certain extent, there are some peculiarities that cannot be handled by tools since they are not machine-processable due to their nature.Models@DevOps [ Robert Heinrich, Jürgen Walter, Felix Willnecker ] Robert Heinrich (Karlsruhe Institute of Technology, Germany)Jürgen Walter (Julius Maximilians Universität Würzburg, Germany)Felix Willnecker (fortiss, Germany)We see a culture clash of DevOps and traditional performance modelling (cf. Table <ref>). Performance modeling in traditional Software Performance Engineering is characterized by manual efforts for creating and parameterizing performance models during design time. Usually there is no continuous performance analysis and design optimization. In contrast, DevOps practices are characterized by short, fast automated release cycles, agile processes, short feedback-loops and continuous executions of elaborated, automated, and staged test chains. In dynamic systems, performance goals need to be validated and maintained continuously.§.§.§ Discussed ProblemsDue to differences the question comes up: “Do we still need performance models in an agile DevOps world?” Models are not yet common industrial practice. However, there are open DevOps challenges that can be addressed using performance models. Further, we see DevOps as an enabler for performance models. The need is also driven by new application scenarios, e.g., Internet of Tings (IOT) causes a huge number of different devices for which one cannot built a complete test bed. *DevOps Challenges and Model-based SolutionThe results of our discussion are presented in Table <ref> §.§.§ Possible Approaches *Pipeline Integration of Performance ModelsWe discussed where models can be used in the DevOps Pipeline (cf. Figure <ref>). Fully automated * Regression analysis* Runtime adaptation / self adaptation* Recovery models* Resilience analysis* Forecast-based decisions Human in the loop * Design space exploration / Decision making* Performance models as simplified view on APM data* Parallel evaluation of multiple design alternativesPerformance models and prediction can be integrated to automate certain decisions but also to assist developers, designers, architects or operators. Performance models can be used to explore a design space by evaluating a number of design decisions by alternating an extracted model and simulate the effects. Such simulations can be executed in parallel and thus evaluate multiple system variations at the same time. Performance models are also suitable for communication as they represent a simplified view on the system but are at the same time based on detailed measurements. Such models can be used for fully automated regression analysis. Integrated in a continuous delivery pipeline, such evaluations can be conducted where classical performance tests take too long. The resilience and recovery capabilities of a system model can be conducted and evaluate recent changes in regards of fault tolerance of the system. Forecasts and analysis on these models and predictions allow to improve self-adaption of systems and especially with forecasts to speed-up scaling decisions. A resource shortage can be predicted before it occurs and replica systems can thus be spawned and warmed up before the actual shortage occurs.§.§.§ Conclusions *DevOps as Door Opener for Performance ModelsHypothesis: DevOps is an enabler for Performance Modelling. * Especially continuous monitoring is an enabler for automatic performance model generation * Model extraction can be placed in the pipeline * Huge amounts of APM data available to use machine learning to significantly improve models * New use cases and scenarios for performance modelsApplication performance monitoring/management software is nowadays integrated in a vast amount of systems, especially in the DevOps context. This monitoring allows to ensure and enforce software quality with regards to their Service Level Agreements (SLAs) even though fast release cycles change the software quite often. The availability of such monitoring data is an enabler for performance modelling as it allows to generate performance models from these application specific traces. On the other hand, these models can be again applied to the DevOps pipeline to replace and/or enhance performance evaluations that usually take too long to execute in an automated delivery pipeline. *What Has to Change Models did not yet arrive in the DevOps world. Heavy weight processes. In order to address the challenge, we postulate things that have to change …* More automation* Automatically test validity of models* DevOps provides chance to learn application/changes* Accuracy tuning for extraction mechanisms* Different granularity levels (views + solving) * Focus on dynamic behavior instead of steady state* Models that are feasible for automated processing and comprehensible by humans* Partly model updates Performance Testing with a Limited Budget [ Cor-Paul Bezemer, Lubomír Bulej, Vojtěch Horký, Zhen Ming `Jack' Jiang, Dusica Marijan, Weiyi `Ian' Shang ] Cor-Paul Bezemer (Queen's University, Canada) Lubomír Bulej (Charles University, Czech Republic)Vojtěch Horký (Charles University, Czech Republic)Zhen Ming `Jack' Jiang (York University, Canada) Dusica Marijan (Simula, Norway) Weiyi `Ian' Shang (Concordia University, Canada)In comparison to functional unit testing, performance testing is still not a widely adopted software development practice. This is likely because there are considerably many barriers that hinder adoption of performance testing. Measuring and comparing performance on modern platforms is objectively difficult, and great attention must be paid to proper collection and analysis of the results, especially when measuring time quantities with sub-millisecond resolution. A performance unit test is not black and white (which makes it more difficult to interpret), it usually takes much more time (which makes it unsuitable for quick testing), and is more difficult to construct (developers need to know which features need to be tested for performance and submit those to a realistic workload). In contrast, functional unit tests are considerably easier to create and provide clear benefits that are now understood by many. In addition to providing evidence that the software works as expected, they are used to drive the design and enable design changes that are often necessary to suit the everchanging requirements. While the understanding of measurement and analysis techniques has improved over the last few years, some of the issues remain, and they are related to the difficulty the developers have with interpreting performance test results, finding the right features to test, and finding representative workloads. However, we expect this to change as more and more teams adopt the DevOps culture, which brings together developers and operators to ensure continuous delivery of high-quality code. We assume that increased adoption of the DevOps culture will increase the demand for performance testing, and at the same time provide developers with information from “Ops” that will guide performance testing activities. This includes information that might have been previously lacking, such as usage data for features, platforms, and configurations, as well as representative workloads, without which developers were reluctant to invest effort into performance testing. Assuming there is demand for performance tests and that we have access to information from “Ops” to guide performance testing, the discussion in this group focused on our ability to conduct performance testing in a timely manner. §.§.§ Discussed ProblemsPerformance testing takes a lot of time and often requires dedicated hardware resources. Performance unit tests often measure time intervals in the order of milliseconds or below, and if we want to automatically detect performance differences in the order of 10% or smaller, we need to use sound statistical methods. This in turn requires steady-state samples from multiple test runs, which also requires spending considerable time (with respect to the duration of the measured operation) on system warm-up. If we focus on large-scale performance or load tests to detect only critical and easy to detect performance regressions (to avoid deploying broken software), we may not need as many samples and test runs, but such tests take long time to setup and execute anyway (e.g., replaying a 24-hour long request log). In addition to the number and type of performance tests, there are other dimensions that greatly influence the amount of time needed for performance testing. These include different hardware platforms and hardware configurations, multiple software platforms and their configurations, and last, but not least, stream of changes to the software resulting in new versions that need to be tested. Ideally, we want to fully test each version found in a source code repository. However, just multiplying the number of tests, the required number of test runs, the number of hardware platform configurations, and the number of software platform configurations is bound to produce a number of test executions that we cannot hope to fit into a limited time and cost budget. Therefore we need to schedule tests to make the most of the available budget.§.§.§ Possible ApproachesSelecting the right tests under given time or cost constraints is essentially a planning problem, except that the cost and importance of tests is not completely static. Instead, it is influenced by various inputs. We therefore need a test scheduler that can, for each software or configuration change that may affect performance, maximize the chance to find performance regressions under the given time and cost constraints using the following:* The available tests and their duration* Testing requests from developers* Testing guidance from operators* A history of results for a particular test Figure <ref> shows a coarse architecture of such a test scheduling system. The test scheduler itself is only one of several components, and is intended to make decisions based on information coming from multiple sources. The operators determine the resource budgets as well as the general focus of testing, while developers may influence test selection or test priority in response to actual needs. Besides the input from developers and operators, the activity of the scheduler is triggered by incoming code/configuration changes, and the test selection and prioritization takes into account previously collected information about tests. The latter comes from a repository which is mainly updated by other processes based on test results. The processes related to “test execution” and “regression detection” are assumed to be well understood, and are not discussed here in detail—the role of these processes is to provide the system with raw data from test execution and to detect performance regressions. The other processes provide opportunity for optimization,and include prefiltering of changes, test case prioritization, test case selection, and test case execution time. We now discuss these processes in more depth. Change Classification If we were to run the full battery of changes on each commit observed by the test scheduler, the system would most likely fail to test the system for performance regressions in a timely manner. It is therefore important to understand the nature of observed changes and adapt the test selection to the “test worthiness” of a particular change. For example, changes in source code that only contain formatting or comment changes should be ignored, as they are not expected to change the performance of the resulting system.A more complex change analysis may involve mapping of source code to nodes in a feature model, which can be further annotated with information about tests covering a particular feature (e.g., <cit.>). This may allow the scheduler to only schedule tests related to a particular feature or features that depend on it. There is a potentially significant body of related work in the software engineering community, related to change impact analysis and subsequent selection of unit tests (e.g., <cit.>). However, these works are mostly concerned with functional unit testing, while performance tends to be a cross-cutting concern, so it may be more difficult to isolate the code the performance of which is supposed to be influenced by a code change and map it to a particular performance test. Nevertheless, the classification of changes for the purpose of test selection or prioritization is an important aspect of the system that determines the general direction of performance testing, and will benefit from operator (and possibly developer) input. We assume each test to have two kinds of priorities. The first priority is long-term (or static), based on importance of certain features (based on information from Ops) and the history of results of a particular test case (i.e., how often a particular test case identified a performance regression). The second priority is short-term, based on the results of change analysis, on requests from developers for executing a specific test case (to accommodate their gut feelings), and on regression analysis results (so that a change in which a regression was detected can be tested more thoroughly, including surrounding changes in the project’s development history). Equivalence Group Detection Ideally, software features will be covered by multiple targeted performance tests to ensure that changes to the software do not cause performance regressions. However, in many scenarios we envision using performance tests that are not targeted at a specific feature, and instead exercise the system at a more coarse scale. Benchmarks from various benchmark suites can often serve as performance tests, and may be used to track performance and detect performance regressions in a software project. While coarse-grained tests (benchmarks) are undoubtedly useful, their results for many changes may be correlated, suggesting that only a subset of the tests may need to be executed. Tests that detect the same performance anomalies should therefore be organized into equivalence groups, and the scheduler should only select a subset of tests that covers all equivalence groups. To prevent selection bias, the tests within an equivalence groups should be selected in a round-robin fashion. However, a single test may be potentially a member of multiple equivalence groups, which requires a two-level selection process to ensure that different permutations of tests across all equivalence groups are used to avoid test starvation.We also assume that the group membership may change over time, and will therefore require periodic re-evaluation, with the period depending on a particular project. This requires the equivalence group evaluation process to be automated as much as possible. Test Utility Profiling While no test that is part of the code base should be entirely avoided by the scheduler (if a test is considered completely useless, it should be removed), the scheduler should attempt to maximize the likelihood of finding performance regressions by scheduling tests with a history of detecting performance regressions more often. To simplify the scheduler internals, we assume that the information that determines the priority/weight of a particular test case should be available to the scheduler from the test information repository. The long-term weight/priority of a test should reflect its history of detecting performance regressions, and we assume that it can be produced by an independent process that consumes the results of the regression detection process.If the global information about test history does not provide sufficient information for test prioritization, it may be necessary to associate the history of test results with additional information about the change that triggered the testing, such as code locations or feature model nodes. Test Warm-up Profiling An obvious way to reduce testing time is to reduce the duration of test execution. However, the challenge is to determine for how long a test needs to execute. Tests in which the workloads take milliseconds or less to execute (benchmarks and microbenchmarks) typically repeat the workload many times so that the measured duration of the operation can be processed in a robust fashion using statistical methods. Prior to collecting the measurements, a test harness typically performs a warm-up phase to get rid of various transients associated with the start up of a system or an execution platform, such as the Java Virtual Machine. This needs to be repeated multiple times by executing the workloads in newly spawned processes to take into account changes in performance due to layout of code and data in memory, and thus properly sample the observable variance in the measured data.Often, the samples from different test runs contribute more information to the estimate of the observable variance than the samples from a single execution. The warm-up phase of a test run then becomes a significant overhead in test execution.Each test can have a different warm-up period, depending on the activity it exercises and potentially also on the software platform (stack) on top of which it executes. The warm-up period is typically determined manually by analysis of data from a very long run, which makes periodic re-evaluation due to changes in the underlying software stack costly. A performance test scheduling would benefit from an automated process that would perform warm-up profiling of test cases added to the system. Even if the initial warm-up profiling took much longer than the typical duration of a test, the savings in subsequent test executions could be substantial. One approach to such warm-up profiling could be based on finding/identifying repetitive patterns in the collected test metrics over time. The automated process would determine the necessary warm-up time and measurement time so as to properly sample the observable variance within a single run. This is a non-trivial challenge, because many benchmarks (or performance) do not exhibit a classic warm-up behavior, and the collected metrics (most often iteration/response time) may oscillate even in steady state <cit.>.As mentioned earlier, complex long-running performance tests may focus on identifying critical performance regressions in the order of 100% or more, typically by requiring that the metric of interest is within certain bounds. Since the time scale and the range of such bounds is usually significantly larger than what is common for small-scale tests, it may not be necessary to execute complex long-running tests many times. However, due to the nature of such performance tests, they will still take a long time to execute. The question is then whether all the activity performed by the test is relevant for finding the performance regression, or whether a representative subset would be sufficient. Finding this subset and capturing it in a performance test is non-trivial and challenging, especially if it is to be automated.§.§.§ ConclusionsWe assume that the adoption of a DevOps culture will provide both demand and useful information for developers to engage in performance testing. To provide timely feedback based on performance testing to developers, performance tests will have to be scheduled to fit timing and cost constraints, while maximizing the likelihood of discovering a true performance regression. This further relies on the ability to select and prioritize performance tests based on analysis of software changes, utility profile of individual test cases, as well as their membership in equivalence groups of tests producing correlated results. In addition, approaches to determine test-specific warm-up profiles as well as identification of relevant subset of real-world workloads should help in reducing execution times of individual tests, because these are then greatly amplified by the need to test along many dimensions (code changes, configuration changes, hardware platforms, multiple test runs), which leads to testing time explosion. By finding solutions to these problems we should be able to provide a test scheduler with the information necessary to make good scheduling decisions.Alberto Avritzer, Sonatype, USA Oliver Beck, SAP, Germany Cor-Paul Bezemer, Queen's University, Canada Markus Borg, RISE SICS AB, Sweden Lubomír Bulej, Charles University Prague, Czech Republic Jürgen Cito, University of Zurich, Switzerland Georgiana Copil, TU Vienna, Austria Robert Heinrich, Karlsruhe Institute of Technology, Germany Andre van Hoorn, University of Stuttgart, Germany Vojtěch Horký, Charles University Prague, Czech Republic Pooyan Jamshidi, Imperial College London, United Kingdom Jack Jiang, York University, Canada Cristian Klein, Umea University, Sweden Holger Knoche, Christian-Albrechts-Universität zu Kiel, Germany Philipp Leitner, University of Zurich, Switzerland Fei Li, Siemens Corporate Technology, Austria Lucy Ellen Lwakatare, University of Oulu, Finland Dusica Marijan, Simula, Norway Claus Pahl, Free University of Bozen-Bolzano, Italy Stefan Schulte, TU Vienna, Austria Weiyi Shang, Concordia University, Canada Catia Trubiani, Gran Sasso Science Institute, Italy Jürgen Walter, Julius Maximilians Universität Würzburg, Germany Ingo Weber, Data61, CSIRO, Australia Johannes Wettinger, University of Stuttgart, Germany Felix Willnecker, fortiss, Germany 1cm | http://arxiv.org/abs/1709.08951v1 | {
"authors": [
"Andre van Hoorn",
"Pooyan Jamshidi",
"Philipp Leitner",
"Ingo Weber"
],
"categories": [
"cs.PF"
],
"primary_category": "cs.PF",
"published": "20170926114234",
"title": "Report from GI-Dagstuhl Seminar 16394: Software Performance Engineering in the DevOps World"
} |
Computing and Mathematical Sciences, Caltech {acoladan,jalex}@caltech.edu Robust self-testing for linear constraint system games Jalex Stark====================================================== We study linear constraint system (LCS) games over the ring of arithmetic modulo d.We give a new proof that certain LCS games (the Mermin–Peres Magic Square and Magic Pentagram over binary alphabets, together with parallel repetitions of these) have unique winning strategies, where the uniqueness is robust to small perturbations. In order to prove our result, we extend the representation-theoretic framework of Cleve, Liu, and Slofstra <cit.> to apply to linear constraint games over _d for d≥ 2. We package our main argument into machinery which applies to any nonabelian finite group with a “solution group” presentation. We equip the n-qubit Pauli group for n≥ 2 with such a presentation; our machinery produces the Magic Square and Pentagram games from the presentation and provides robust self-testing bounds. The question of whether there exist LCS games self-testing maximally entangled states of local dimension not a power of 2 is left open. A previous version of this paper falsely claimed to show self-testing results for a certain generalization of the Magic Square and Pentagram mod d≠ 2. We show instead that such a result is impossible. gobble arabic§ INTRODUCTION In <cit.>, Mermin and Peres discovered an algebraic coincidence related to the 3×3 “Magic Square” of operators on ^2⊗^2 in Figure <ref>.If we pick any row and take the product of the three operators in that row (note that they commute, so the order does not matter), we get the identity operator. Similarly, we can try this with the columns. Two of the columns give identity while the othergives -1 times identity. Thus, the product of these nine operators depends on whether they are multiplied row by row or column by column. This can be exploited to define a two-player, one-referee game called the Mermin–Peres Magic Square game <cit.> (see Definition <ref> and Figure <ref> for a formal definition). Informally, the Mermin–Peres Magic Square game mod 2 is as follows. The players claim to have a 3× 3 square of numbers in which each row and each of the first two columns sums to 0 2, while the third column sums to 1 2. (The players are usually called “provers”, since they try to prove that they have such a square.) The referee asks the first player to present a row of the supposed square and the second to present a column. They reply respectively with the 3 entries of that row and column in {0,1}. They win if their responses sum to 0 or 1 as appropriate, and they give the same number for the entry where the row and column overlap. This game can be won with probability 1 by provers that share two pairs of maximally entangled qubits of dimension 2, but provers with no entanglement can win with probability at most 8/9. Games which are won in the classical case with probability < 1 but are won in the quantum case with probability 1 are known as pseudotelepathy games. How special is this “algebraic coincidence” and the corresponding game? We can refine this question into a few sub-questions. Are there other configurations of operators with similarly interesting algebraic relations? Do they also give rise to pseudotelepathy games? Arkhipov <cit.> gives a partial answer to this question by introducing the framework of magic games. Starting from any finite graph, one can construct a magic game similar to the Magic Square game. Arkhipov finds that there are exactly two interesting such magic games: the Magic Square (derived from K_3,3, the complete bipartite graph with parts of size 3) and the Magic Pentagram (derived from K_5, the complete graph on 5 vertices).Subsequently, Cleve and Mittal <cit.> introduced linear constraint system games (hereafter referred to as LCS games), which can be thought of as a generalization of Arkhipov's magic games from graphs to hypergraphs. Moreover, they proved that any linear constraint game exhibiting pseudotelepathy requires a maximally entangled state to do so. Their result also suggested that there may be other interesting linear constraint games to find. Indeed, Ji showed <cit.> that there are families of linear constraint games requiring arbitrarily large amounts of entanglement to win. The easiest proof of correctness for a Magic Square game strategy uses the fact the observables measured by the players satisfy the appropriate algebraic relations. Is this a necessary feature of any winning strategy? In order to answer questions like this, Cleve, Liu, and Slofstra <cit.> associate to each LCS game an algebraic invariant called the solution group (see Section <ref> for a precise definition), and they relate the winnability of the game to the representation theory of the group. In particular, they show that any quantum strategy winning the game with probability 1 corresponds to a representation of the solution group—in other words, that the observables in a winning strategy must satisfy the algebraic relations captured by the group. This reduces the problem of finding LCS games with interesting properties to the problem of finding finitely-presented groups with analogous representation-theoretic properties, while maintaining combinatorial control over their presentations. Slofstra used this idea together with techniques from combinatorial group theory to resolve the weak Tsirelson problem <cit.>. By including some techniques from the stability theory of group representations, he improved this result to show that the set of quantum correlations is not closed <cit.>. In words, he constructed an LCS game which can be won with probability arbitrarily close to 1 with finite-dimensional quantum strategies, but cannot be won with probability 1 by any finite (or infinite) dimensional quantum strategy (in the tensor product model). We introduced the magic square operators and then noticed that they satisfy certain algebraic relations. Do these algebraic relations characterize this set of operators? Could we have picked a square of nine different operators, possibly of much larger dimension, satisfying the same relations? This question was resolved by Wu et. al <cit.>. They showed that any operators satisfying the same algebraic relations as those in the Magic Square game are equivalent to those in Figure <ref>, up to local isometry and tensoring with identity. This is sometimes referred to as rigidity of the Magic Square game. Moreover, they showed that the Magic Square game is robustly rigid, or robustly self-testing. Informally, we say that a game is rigid with O((̣))-robustness and perfect completeness if whenever Alice and Bob win the game with probability at least 1-, then there is a local isometry taking their state and measurement operators O((̣))-close to an ideal strategy, possibly tensored with identity. Our contributions Our main result is a robust self-testing theorem which applies to any linear constraint game with sufficiently nice solution group; this is stated as Theorem <ref>. Our proof employs the machinery of <cit.> and <cit.>. We apply the general self-testing result to conclude robust rigidity for the Magic Square game, the Magic Pentagram game, and for a certain repeated product of these two games. We informally state these results now. We emphasize that these results are not new, but it is new that we can achieve all three as simple corollaries of the main self-testing machinery. The general result holds for LCS games mod d, but the only nontrivial application we have is for LCS games mod 2. The Magic Square game is rigid with O()-robustness and perfect completeness. The ideal state is two copies of the maximally entangled state of local dimension 2, and the ideal measurements are onto the eigenbases of the operators in Figure <ref>.This recovers the same asymptotics as in <cit.>. Note that they state their robustness as O(√()); this is because they use the Euclidean distance |ψ⟩- |ideal⟩, while we use the trace-norm distance of density operators ρ - ρ_ideal_1. The Magic Pentagram game (see Figure <ref> for a definition) is rigid with O()-robustness and perfect completeness. The ideal state is three copies of the maximally entangled state of local dimension 2, and the ideal measurements are onto the eigenbases of the operators in Figure <ref>.This recovers the same asymptotics as <cit.>, up to translation between distance measures. Applying our general self-testing theorem to the LCS game product [This is defined precisely in Definition <ref>. This is similar to but not the same as playing multiple copies of the game in parallel.]of many copies of the Magic Square game yields a self-test for n maximally entangled pairs of qubits and associated n-qubit Pauli measurements. For any n≥ 2, there is a linear constraint system game with O(n^2) variables, O(n^2) equations, and _2-valued answers which is rigid with O(n^10)-robustness and perfect completeness. The ideal state is n copies of the maximally entangled state of local dimension 2. The ideal measurements are onto the eigenbases of certain Pauli operators of weight at most 5. The polynomial scaling in n is similar to previous works that self-test n pairs of maximally entangled qubits via copies of the magic square game <cit.>, but we obtain our bound by a simple application of our general self-testing theorem. §.§ Proof OverviewWe step away from games and back towards algebra to discuss Question <ref>. Suppose we wanted a 3× 3 square of operators, call them e_1 through e_9, with the same relations as those in the Magic Square. Concretely, those relations are as follows: 0.2! < g r a p h i c s > * The linear constraints of each row and column: e_2e_5e_8 = -I, e_1e_2e_3 = e_4e_5e_6 = e_7e_8e_9 = e_1e_4e_7 = e_3e_6e_9 = I. * Commutation between operators in the same row or column: e_1e_2 = e_2e_1, e_1e_3 = e_3e_1, e_2e_3=e_3e_2, …, e_3e_6 = e_6e_3, e_3e_9=e_9e_3, e_6e_9=e_9e_6. * Associated unitaries have 2 eigenspaces: e_i^2= I for all i. These are just multiplicative equations. We can define an abstract group whose generators are the e_i and whose relations are those above. This is, in a sense, the most general object satisfying the Magic Square relations. More precisely, any square of operators satisfying these relations is a representation of this group.It's not hard to compute that this group is isomorphic to the group of two-qubit Pauli matrices, a friendly object. (This is proven as Proposition <ref>.)This group is the solution group of the magic square game. We study the representation theory of the solution group of the magic square game, and we apply <cit.> to deduce the exact version of our self-testing Theorem <ref> (i.e. the =0 case). One might view our proof via solution groups as an “algebrization” of the proof in <cit.>. In order to get the robustness bounds, we must work significantly harder. Tracing through the proof of the main result of <cit.>, a finite number of equalities between various operators are applied. Knowing how many equalities are needed, one can get quantitative robustness bounds by replacing these with approximate equalities and then applying finitely many triangle inequalities. In order to carry out this counting argument, we introduce a measure of complexity for linear constraint games and then upper bound the robustness parameter as a function of this complexity.This complexity measure depends on the use of van Kampen diagrams, a graphical proof system for equations in finitely-presented groups. Van Kampen diagrams are introduced in <ref>. Several of our main proofs reduce to reasoning visually about the existence of such diagrams. Manipulating the chains of approximate equalities requires us to develop familiarity with a notion of state-dependent distance; this is done in <ref>.§.§ Organization In Section <ref>, we establish basic tools that we'll use without comment in the main body of the paper. In Section <ref>, we give the definition and basic properties of linear constraint games over _d. Those familiar with linear constraint games over _2 will not find surprises here. In Section <ref>, we establish our measure of LCS game complexity and prove our general robust self-testing result, Theorem <ref>. We warm up first by proving the =0 case of the theorem in <ref>. We then introduce two new ingredients to obtain a robust version. In <ref>, we give a proof by Vidick <cit.> of a so-called stability theorem for representations of finite groups (Lemma <ref>). Such a result first appeared in <cit.>. In <ref>, we show how to extract quantitative bounds on lengths of proofs from van Kampen diagrams, and in <ref>, we complete the proof of the general case. In Section <ref>, we specialize our robust self-testing theorem to the case of the Magic Square and Magic Pentagram games, establishing Theorems <ref> and <ref>. We go on to exhibit a way to compose LCS games in parallel while controlling the growth of the complexity, proving Theorem <ref>.§ ACKNOWLEDGEMENTS An early version of Theorem <ref> used a more complicated linear constraint game. We thank William Slofstra for pointing out that the same analysis goes through for the Magic Square.The arxiv version 1 of this paper falsely claimed that in a certain 3× 3 square of operators, every pair of operators sharing a row or column commute. We thank Richard Cleve, Nadish De Silva and Joel Wallman for pointing out that one pair of them did not.We thank Richard Cleve and Joel Wallman for sharing with the authors a proof that the magic square game mod d for d≠ 2 is not a pseudotelepathy game. More details about this impossibility are provided in section <ref>.We thankWilliam Ballinger, William Hoza, Jenish Mehta, Chinmay Nirkhe, William Slofstra, Thomas Vidick, Matthew Weidner, and Felix Weilacher for helpful discussions. We thank Martino Lupini for pointing us to reference <cit.> and Scott Aaronson for pointing us to reference <cit.>.We thank Arjun Bose, Chinmay Nirkhe,andThomas Vidick for helpful comments on preliminary drafts of the paper.We thank Thomas Vidick for various forms of guidance throughout the project. A.C. was supported by AFOSR YIP award number FA9550-16-1-0495. J.S. was supported by NSF CAREER Grant CCF-1553477 and the Mellon Mays Undergraduate Fellowship.Part of this work was completed while J.S. was visiting UT Austin.§ PRELIMINARIES We assume a basic familiarity with quantum information, see e.g. <cit.>. We introduce all necessary notions from the fields of nonlocal games and self-testing, but we don't reproduce all of the proofs.§.§ NotationWe write [n] to refer to the finite set 1,…, n with n elements. We write [A,B] for ABAB, the group commutator of A and B. We use the Dirac delta notation δ_x,y := 1, if x=y 0, otherwise.𝐇 will refer to a hypergraph, whileH will refer to a Hilbert space. L( H) is the space of linear operators on the Hilbert space H. ρ will always refer to a state on a Hilbert space, whileand τ are reserved for group representations. _d:= e^2π i /d will always refer to the same dþ root of unity.When we have multiple Hilbert spaces, we label them with subscripts, e.g. as H_A,H_B. In that case, we may also put subscripts on operators and states to indicate which Hilbert spaces they are associated with.When the Hilbert space is clear from context, I refers to the identity operator on that space.I_d will always refer to the identity operator on ^d.:=1/√(d)∑_i^d|ii⟩ refers to the maximally entangled state on ^d⊗^d. We use the shorthand _ρ(X) =Xρ.We use the following notion of state-dependent distance, which we'll recall, and prove properties of, in <ref>.ρXY = √(_ρ(X-Y)^†(X-Y)).X_p denotes the p-norm of X, i.e. X_1 = √(XX^†) and X_2 = √( XX^†). §.§ Nonlocal games For our purposes, a nonlocal game G is a tuple (A,B,X,Y,V,π), where A,B,X,Y are finite sets of answers and questions for Alice and Bob, π:X× Y→ [0,1] is a probability distribution over questions, and V:A× B × X × Y→01 is the win condition. If G is a nonlocal game, then a strategy for G is a probability distribution p:A× B× X× Y→ [0,1]. The value or winning probability of a strategy is given by (G;p) := ∑_a,b,x,yπ(x,y)p(a,bx,y)V(a,b,x,y). If the value is equal to 1, we say that the strategy is perfect. If the probability distribution is separable, i.e. p(a,bx,y) = ∑_i p_i(ax)q_i(by) for some probability distributions p_i, q_i, then we say that the strategy is local. We think of a local strategy as being implemented by using only the resource of public shared randomness. Alternatively, the local strategies are the strategies which are implementable by spacelike-separated parties in a hidden variable theory of physics. We say that a strategy p:A× B× X× Y→ [0,1] is quantum of local dimension d if there exist projective measurements A_x^a_a_x, B_y^b_b_y on ^d and a state ρ∈ L(^d⊗^d) such that p(a,bx,y) = _ρ(A_x^a⊗ B_y^b) (By projective measurement we mean that for all x,y,a,b we have (A_x^a)^2 = A_x^a = (A_x^a), (B_y^b)^2 = B^b_y= (B^b_y), and for all x,y, we have ∑_a A_x^a = I = ∑_bB_y^b.) We say that a strategy is quantum if it is quantum of local dimension d for some d.We denote by _*(G) the optimal quantum value of G, i.e. the supremum over all quantum strategies of the winning probability. If the value of a strategy is _*(G), we say that the strategy is ideal. For quantum strategies, we use the term strategy to refer interchangeably to the probability distribution or to the state and measurement operators producing it. We say that a non-local game G self-tests a quantum strategy S = (A_x^a_a_x, B_y^b_b_y, |Ψ⟩) if any quantum strategy S' that achieves the optimal quantum winning probability w_* is equivalent up to local isometry to S.By local isometry we mean a channel Φ:L( H_A⊗ H_B) → L( H_A'⊗ H_B') which factors as Φ(ρ) = (V_A⊗ V_B) ρ(V_A⊗ V_B)^†, where V_A:H_A→ H_A', V_B:H_B→ H_B' are isometries. We say that a non-local game G is (,δ())-rigid if it self-tests a strategy S = (A_x^a_a_x, B_y^b_b_y, |Ψ⟩), and, moreover, for any quantum strategy S̃ = (Ã_x^a_a_x, B̃_y^b_b_y, ρ) that achieves a winning probability of w_*(G) -, there exists a local isometry Φ such that Φ(Ã_x^a⊗B̃_y^b ρÃ_x^a⊗B̃_y^b) - (A_x^a⊗ B_y^b |Ψ⟩⟨Ψ| A_x^a⊗ B_y^b )⊗ρ_extra_2≤δ()where ρ_extra is some auxiliary state, and δ() is a function that goes to zero with . §.§ Groups We work with several groups via their presentations. For the basic definitions of group, quotient group, etc. see any abstract algebra text, e.g. <cit.>. Let S be a set of letters. We denote by F(S) the free group on S. As a set, F(S) consists of all finite words made from s,s s∈ S such that no ss or ss appears as a substring for any s. The group law is given by concatenation and cancellation. Let S be finite and R a finite subset of F(S). Then G = S:R is the finitely presented group generated by S with relations from R. Explicitly, G =F(S)/⟨R|$⟩, where/is used to denote the quotient of groups, and⟨R|$⟩ denotes the subgroup generated by R. We say that an equation w = w' is witnessed by R if w'w^-1 (or some cyclic permutation thereof) is a member of R. We emphasize that in this work, we sometimes distinguish between two presentations of the same group. If G = ⟨S:R|,⟩ G' = ⟨S':R'|$⟩ are two finitely presented groups, we reserve equality for the caseS = S'andR = R', and in this case we'll sayG = G'. We'll say thatG ≅G'if there is a group isomorphism between them. Let G = ⟨S:R|$⟩ be a finitely presented group and: G→ F(S)be an injective function. We say thatis a canonical form forGif the induced map: G→ F(S)/⟨R|$⟩ is an isomorphism. In other words, we require that (g)(h) = (gh) as elements of G, but not as strings. Now and throughout the paper, for a groupG, we'll denote by1its identity, and we'll let[a,b] := ababdenote the commutator ofaandb. The group presentations of interest in this paper will take a special form extending the “groups presented over_2” from <cit.>. Let d∈ and let _d = J:J^d be the finite cyclic group of order d. A group presented over _d is a group G = S':R', where S' contains a distinguished element J and R' contains relations [s,J] and s^d for all s∈ S. For convenience, we introduce notation that suppresses the standard generator J and the standard relations. G= S:R__d = S∪ J: R ∪ s^d,J^d,[s,J] s∈ SIn the group representations of interest, we'll haveJ ↦e^2πi/d—we should always just think ofJas adþroot of unity. We'll think of relations of the formJ[a,b]as “twisted commutation” relations, since they enforce the equationabab = e^2πi/d. The Pauli group on one d-dimensional qudit can be presented as a group over _d. P_d^⊗ 1 = x,z: J[x,z]__d§.§ Group pictures Suppose we have a finitely presented groupG = ⟨S:R|$⟩ and a word w∈ F(S) such that w = 1 in G.Then by definition, there is a way to prove that w = 1 using the relations from R. How complicated can such a proof get? Group pictures give us a way to deal with these proofs graphically, rather than by writing long strings of equations. In particular, we will use group pictures to get quantitative bounds on the length of such proofs. (For a more mathematically rigorous treatment of group pictures, see <cit.>. These are dual to what are usually known as van Kampen diagrams.) Let G = S:R__d be a group presented over _d. A G-picture is a labeled drawing of a planar directed graph in the disk. Some vertices may lie on the boundary. The vertices that do not lie on the boundary are referred to as interior vertices. A G-picture is valid if the following conditions hold: * Each interior vertex is labeled with a power of J. (We omit the identity label.) * Each edge is labeled with a generator from S. * At each interior vertex v, the clockwise product of the edge labels (an edge labeled s should be interpreted as s if it is outgoing and as s if it is ingoing) is equal to the vertex label, as witnessed by R. (Since the values of the labels are in the center of the group, it doesn't matter where you choose to start the word.) Note that the validity of a G-picture depends on the presentation of G. Pictures cannot be associated directly with abstract groups.If we collapse the boundary of the disk to a point (“the point at infinity”), then the picture becomes an embedding of a planar graph on the sphere (see Figure <ref>). The following is a kind of “Stoke's theorem” for group pictures, which tells us that the relation encoded at the point at infinity is always valid. Suppose P is a G-picture. The boundary word w is the product of the edge labels of the edges incident on the boundary of P, in clockwise order. Suppose P is a valid G-picture with boundary word w. Let J^a be the product of the labels of the vertices in P. Then w=J^a is a valid relation in G. Moreover, we say that the relation w=J^a is witnessed by the G-picture P.The proof is elementary and relies on the fact that the subgroup ⟨J|$⟩ is abelian and central, so that cyclic permutations of relations are valid relations. By counting what goes on at each step in the induction of a proof of the above lemma, one can extract a quantitative version. This is stated and proved in <ref>. Recall the group 1 from Example <ref>. It's easy to see that (xz)^d = 1 in this group. In Figure <ref>, we give two proofs of this fact, for the case d =3. The examples are chosen to illustrate that shorter proofs are more natural than longer proofs in the group picture framework. §.§ Representation theory of finite groupsWe'll study groups through their representations. We collect here some basic facts about the representation theory of finite groups. For exposition and proofs, see e.g. <cit.>. Throughout,Gwill be a finite group. It should be noted that some of these facts are not true of infinite groups. A d-dimensional representation of G is a homomorphism from G to the group of invertible linear operators on ^d. A representation is irreducible if it cannot be decomposed as a direct sum of two representations, each of positive dimension. A representation is trivial if its image is I, where I is the identity matrix. The character of a representationis the function defined by g↦((g)). Two representations ρ_1 and ρ_2 are equivalent if there is a unitary U such that for all g, Uρ_1(g) U^† = ρ_2(g).Notice that a1-dimensional representation and its character are the same function, and that1-dimensional representations are always irreducible. We sometimes write “irrep” for “irreducible representation.” The next fact allows us to check equivalence of representations algebraically.ρ_1 is equivalent to ρ_2 iff they have the same character. The following is immediate: Let = ⊕_i _i be a direct sum decomposition ofinto irreducibles. Let ∘ denote composition of maps, and let χ = ∘, χ_i = ∘_i be the characters corresponding to the representations . Then χ = ∑_i χ_i. Furthermore, define χ̃= 1/χ and χ̃_i = 1/_iχ_i as the normalized characters of ,_i. Then the normalized character ofis a convex combination of the normalized characters of _i. χ̃= ∑_i _i/χ̃_i. There is a simple criterion to check whether a representation of a finite group is irreducible:is an irreducible representation of G iff G = ∑_g∈ G(g) (g). The commutator subgroup[G,G] of G is the subgroup generated by all elements of the form [a,b] := abab for a,b∈ G. The indexG:H of a subgroup H ≤ G is the number of H-cosets in G. Equivalently for finite groups, the index is the quotient of the orders G:H=G/H. G has a number G:[G,G] of inequivalent 1-dimensional irreducible representations, each of which restricts to the trivial representation on [G,G]. For a finite group G, the size of the group is equal to the sum of the squares of the dimensions of the irreducible representations. In other words, for R any set of inequivalent irreps, G = ∑_∈ R ()^2iff R is maximal.By “maximal”, we mean that any irreducible representation is equivalent to one fromR. This fact can be used to check whether one has a complete classification of the irreducibles ofG. This is a special case of the following forx = 1. [Second orthogonality relation for character tables] Let x∈ G. Letvary over a maximal set of inequivalent irreps of G, and let n_ be the dimension of . Then 1/ G∑_ n_((x)) = δ_x,1. [Schur's lemma] Let τ: G→ U(^d) be an irrep and X ∈ L(^d) be a linear operator. Suppose that Xτ(g) = τ(g)X for all g∈ G. Then X = λ I is a scalar multiple of identity.§ LINEAR CONSTRAINT SYSTEM GAMES OVER _D We recall several definitions from previous works of Cleve, Liu, Mittal, and Slofstra <cit.>. Following a suggestion from <cit.>, we define the machinery over_dinstead of_2. A hypergraph𝐇 = (V,E,H) consists of a finite vertex setV, a finite edge setE and an incidence matrixH: V× E →. We think ofVas a set of-linear equations,Eas a set of variables, andH(v,e)as the coefficient of variableein equationv. Following Arkhipov <cit.>, some of our hypergraphs of interest will be graphs. Unlike previous works, we introduce signed coefficients (outgoing edges have a positive sign in the incidence matrix, while ingoing edges have a negative sign). This is because previous works considered equations over_2, where1 = -1. Given hypergraph 𝐇, vertex labelling l: V →, and some modulus d∈, we can associate a nonlocal game which we'll call the linear constraint game(𝐇, l, _d). Informally, a verifier sends one equation x to Alice and one variable y to Bob, demanding an assignment a:E→_d to all variables from Alice and an assignment b∈_d to variable y from Bob. The verifier checks that Alice's assignment satisfies equation xd, and that Alice and Bob gave the same assignment to variable y. Formally, we have the following question and answer sets: X = V, Y = E, A = _d^E, B = _d. The win condition selects those tuples (a,b,x,y) satisfying: a(y)= b (Consistency)∑_e∈ E H(x,e)a(e)≡ l(x)d. (Constraint satisfaction)We introduce the two primary LCS games of interest in this paper. The magic square LCS (mod d) has vertex set v_1,…, v_6, edge set e_1,…, e_9, vertex labeling l(v_5) = 1, l(v_i) = 0 for i≠ 5. See Figure <ref> for the full description of the hypergraph and the associated set of linear equations. The magic pentagram LCS (mod 2) has vertex set v_1,…, v_5, edge set e_1,…, e_10, vertex labeling l(v_5) = 1, l(v_i) = 0 for i≠ 5. See Figure <ref> for the full description of the hypergraph and the associated set of linear equations. The following is the main tool we use to understand linear constraint system games. For an LCS game (𝐇, l, _d) with 𝐇 = (V,E,H), the solution group(𝐇, l, _d) has one generator for each edge of 𝐇 (i.e. for each variable of the linear system), one relation for each vertex of 𝐇 (i.e. for each equation of the linear system), and relations enforcing that the variables in each equation commute. Formally, define the sets of relations R_c, the local commutativity relations, and R_eq, the constraint satisfaction relations as R_c := [e,e'] H(v,e) ≠ 0 ≠ H(v,e') for somev∈ V R_eq :=J^-l(v)∏_e∈ Ee^H(v,e)v∈ V.Then define the solution group as(𝐇, l, _d) := E:R_c∪ R_eq__d.(Notice that the order of the products defining R_eq is irrelevant, since each pair of variables appearing in the same R_eq relation also have a commutation relation in R_c.)When the LCS game is clear from context, we'll just writeto denote its solution group.Our aim is to prove that for some specific linear constraint system games, strategies that win with high probability are very close to some ideal form. We start by observing that for any LCS game, any strategy already has a slightly special form. Suppose that p(a,bv,e) =_ρÃ_v^a⊗B̃_e^b is a quantum strategy for an LCS game over _d with hypergraph 𝐇 = (H,V,E). Then there are unitaries A_e^(v)e∈ E, v∈ V and B_e e∈ E such that for all v,e, (A_e^(v))^d = I = B_e^d; for any fixed v, the A_e^(v) pairwise commute; moreover, the provers win with probability 1 iff for all v,e, _ρ A_e^(v)⊗ B_e= 1, andfor all v, _ρ∏_e(A_e^(v))^H(v,e)⊗ I_B = _d^l(v). We refer to the operatorsA_e^(v), B_etogether with the stateρas a strategy presented via observables. Typically the word “observable” is reserved for Hermitian operators. Nonetheless, we call our operators observables because they capture properties of the projective measurements from which they're built in a useful way. Operationally, we think of Bob as measuring the observableB_eand reporting the outcome when asked about variableeand of Alice measuring the observablesA_e^(v)and reporting the outcome for eachewhen asked about equationv. The fact that Alice's observables pairwise commute at each equation means that Alice can measure them simultaneously without ambiguity.A version of this lemma is proved in the course of the proof of Theorem 1 of <cit.>. We give essentially the same proof, just over_d. Define the observables as B_e:= ∑_j_d^-jB̃^i_e A_e^(v):= ∑_i _d^i∑_a: a(e) =iÃ_v^a. It's clear that each of these operators is a unitary whose eigenvalues are dþ roots of unity. To see that A_e^(v) commutes with A_e'^(v), notice that they are different linear combinations of the same set of projectors. Now we compute, for any v,e, _ρ A_e^(v)⊗ B_e = ∑_i,j_d^i-j_ρ(∑_a: a(e) =iÃ_v^a)⊗B̃^j_e= ∑_k _d^k[a(e) - b ≡ k|questionsx=v, y=e]. Notice that the last line is a convex combination of the dþ roots of unity. Hence, it equals 1 if and only if [a(e) ≡ b |questionsx=v, y=e] = 1. A similar computation reveals: _d^-l(v)_ρ∏_e(A_e^(v))^H(v,e)⊗ I =∑_k_d^k-l(v)_ρ∑_a ∑_e H(v,e)a(e)≡ kÃ_v^a⊗ I =∑_k _d^k-l(v)[∑_eH(v,e)a(e)≡ k|questionx=v]Again, the last line is a convex combination of the dþ roots of unity. Hence it equals 1 if and only if [∑_eH(v,e)a(e)≡ l(v) |questionx=v] = 1. Note that we can always recover the original strategy in terms of projective measurements by looking at the eigenspaces of the observables. Therefore, we restrict our attention to strategies presented by observables without loss of generality. Next, we state a simple sufficient condition for the existence of a perfect quantum strategy for an LCS game. An operator solution for the game (𝐇, l, _d) is a unitary representationof the group (𝐇, l, _d) such that (J) = _dI. A conjugate operator solution is a unitary representation sending J↦_̅d̅I.Notice that ifis an operator solution, then for any choice of basis the complex conjugate: g↦(̅g̅)̅is a conjugate operator solution. The existence of an operator solution is sufficient to construct a perfect quantum strategy. [Operator solution for magic square] See the square of group generators in Figure <ref>. Let _2 be the solution group of the Magic Square. Consider the map _2 → U(^d⊗^d) generated by sending each generator in this square to the operator in the corresponding location of Figure <ref>. This map is an operator solution. [Operator solution for magic pentagram] See the pentagram of group generators in Figure <ref>. Let _3 be the solution group of the Magic Pentagram. Consider the map _3 → U(^d⊗^d⊗^d) generated by sending each generator in this pentagram to the operator in the corresponding location of Figure <ref>. This map is an operator solution. Let : → U(^D) be an operator solution. Define a strategy by setting |ψ⟩= |E̊P̊R̊_D⟩, A_e^(v) = (e) for all e,v, and B_e = (̅e̅)̅ for all e. Provers using this strategy win with probability 1. By a well-known property of the maximally entangled state, we have ⟨ψ | (e) ⊗(̅e̅)̅ | ψ|=⟩⟨ψ | (e)(̅e̅)̅^T ⊗ I| ψ|=⟩ 1, where ^T denotes the transpose. Therefore, the consistency criterion (<ref>) is satisfied. Sinceis an operator solution, we have ∏_e(A_e^(v))^H(v,e)= (∏_e (e)^H(v,e)) = (J^l(v)) =_d^l(v)I, so the constraint satisfaction criterion (<ref>) is satisfied. We'll see both exact and approximate converses to this proposition in Section <ref>. § GENERAL SELF-TESTING In this section, we introduce our main robust self-testing theorem for linear constraint system games with solution groups of a certain form. In <ref>, to ease understanding, we start by stating and proving an exact version of the theorem. In <ref> through <ref>, we introduce the necessary tools to prove an approximate version of the self-testing theorem. <ref> introduces the state-dependent distance and some of its properties. <ref> proves a stability lemma for representations of finite groups, which allows us to deduce that the action of a strategy winning with high probability is close to the action of a representation of the solution group.<ref> presents a quantitative version of the van Kampen Lemma from Section <ref>, which is key in bounding the robustness of the main theorem. <ref> shows that if a joint state is approximately stabilized by the action of the Pauli group on two tensor factors, then it is close to the maximally entangled state on the two tensor factors. In <ref> we combine these tools to prove our robust self-testing theorem. §.§ Exact self-testing Throughout, let(𝐇, l, _d), 𝐇 = (V,E,H)be an LCS game with solution group. Supposeis finite and all of its irreducible representations with J↦_d I are equivalent to a fixed irrep :→ U(^d). Suppose A_e^(v), B_e, ρ∈ L( H_A⊗ H_B) is a perfect strategy presented via observables for the game.Then there are local isometries V_A, V_B such that *for all e,v, V_AA_e^(v)V_A^† = (e)⊗ I ⊕Â^(v)_e, where Â^(v)_e V_Aρ V_A^† = 0, and *for all eV_BB_eV_B^† =(̅e̅)̅⊗ I ⊕B̂_e, where B̂_e V_Bρ V_B^† = 0. Awkwardly, we must pick a basis to take the complex conjugate in. Fortunately, we only care about our operators up to isometry. So to make sense of the theorem statement, we pick the basis for complex conjugation first, and then the isometryV_Bdepends on this choice.We break the proof into two lemmas. Supposeis finite and all of its irreducible representations with J↦_dI are equivalent to a fixed irrep :→ U(^d). Then every operator solution is equivalent to ⊗ I and every conjugate operator solution is equivalent to ⊗ I, where the complex conjugate can be taken in any basis. Suppose A_e^(v), B_e, ρ∈ L( H_A⊗ H_B) is a perfect strategy presented via observables for the game. Then, there are orthogonal projections P_A, P_B such that * (P_A⊗ P_B)ρ(P_A⊗ P_B) = ρ; *for each e, P_AA_e^(v)P_A = P_AA_e^(v')P_A, provided that H(v,e) ≠ 0 ≠ H(v',e) (we now write P_AA_eP_A without ambiguity); *the map _A:→ P_A generated by e↦ P_AA_eP_A (and j ↦_dI) is an operator solution; *the map _B:→ P_B generated by e↦ P_BB_eP_B (and j ↦_̅d̅I) is a conjugate operator solution. Take the maps _A and _B from Lemma <ref>; note that their ranges are the subspaces determined by P_A, P_B. From Lemma <ref> we get partial isometries W_A, W_B such that W_A_A(e)W_A^† = (e)⊗ I and W_B_B(e)W_B^† = (̅e̅)̅⊗ I. To complete the proof, let V_A and V_B be any isometric extensions of W_A and W_B, and set Â_e^(v) = V_A(I-P_A)A_e^(v)(I-P_A)V_A^†, B̂_e = V_B(I-P_B)B_e(I-P_B)V_B^†. Checking that these operators satisfy the equations in the theorem is a simple computation. Let τ be an operator solution, i.e. a representation ofwith τ(J) = _d I. Let τ = ⊕_i=1^kτ_i be a decomposition of τ into k irreducibles. As in Lemma <ref>, let χ̃: g↦1/ττ(g) be the normalized character of τ and χ̃_i be the same for τ_i. One can check that χ̃_i(g)≤ 1 for all g∈. Furthermore, χ̃(g) is a convex combination of the χ_i(g). Therefore, χ̃_i(J) = _d for each i. Then also τ_i(J) = _dI for each i, since this the only d-dimensional unitary with trace d_d. We conclude thatτ is equivalent to ⊕_i=1^k = ⊗ I_k. Now suppose that τ' is a conjugate operator solution. Then taking the complex conjugate in any basis, τ̅'̅ is an operator solution. By the above, τ̅'̅ is equivalent to ⊗ I. Therefore, τ' is equivalent to ⊗ I. This is essentially the same proof as given in <cit.> (their treatment is a bit more complicated since they wish to cover the infinite-dimensional case). Let A be the set of finite products of unitaries from A_e^(v), and similarly let B be the set of finite products of unitaries from B_e. Let ρ_A = _B ρ and ρ_B = _Aρ. Define Ĥ_̂ = ρ_A, and Ĥ_̂B̂ = ρ_B, and let P_A and P_B be the projectors onto these spaces. Notice that (P_A⊗ P_B) ρ (P_A⊗ P_B) = ρ. From the consistency criterion (<ref>), we have 1 = _ρ A_e^(v)⊗ B_e, soA_e^(v)|ϕ⟩= B_e^†|ϕ⟩ for |ϕ⟩∈ρ. Let A∈ A be arbitrary. Then, the above implies that there is B∈ B be such that (A⊗ I) ρ (A^†⊗ I) = (I⊗ B^†)ρ(I⊗ B). We compute Aρ_AA^† = _B (A⊗ I)ρ(A^†⊗ I) = _B (I⊗ B^†)ρ(I⊗ B) = _B ρ =ρ_A, from which we conclude that A fixes Ĥ_̂Â. This implies that (PA_1P)(PA_2P) = PA_1A_2P for A_1,A_2∈ A. Next, we compute 1 = _ρ A_e^(v)(A_e^(v'))^†⊗ I = _ρ_A A_e^(v)(A_e^(v'))^†, from which we conclude that P_AA_e^(v)P_A=P_AA_e^(v')P_A. We now write P_AA_eP_A without ambiguity. Finally, we compute 1 = _ρ w_d^-l(v)∏_e:H(v,e)≠ 0 A_e⊗ I = _ρ_A w_p^-l(v)∏_e:H(v,e)≠ 0 A_e⊗ I, from which we conclude that the map e↦ P_AA_eP_A is an operator solution. The same argument shows that e↦ P_BB_eP_B is a conjugate operator solution. (The conjugation comes from equation (<ref>).) Here we constructed representations directly, projecting onto the support of a known state. In the approximate case, this work will be subsumed by an application of the stability lemma <ref>.§.§ State-dependent distanceWe now begin to collect the necessary tools to generalize the previous subsection to the approximate case. To start, we need a convenient calculus for manipulating our notion of state-dependent distance. Recall the definition ofρ··as ρXY = √(_ρ(X-Y)^†(X-Y)) We use the same notation as the Kullback-–Leibler divergence despite the fact that ourρ··is symmetric in its arguments. We do this because we will write complicated expressions in the place ofXandY; the notation becomes harder to parse if the symbolis replaced by a comma. Notice that ifρ_ABis the maximally entangled pure state, thenρ_ABX⊗I_BY⊗I_Bis exactly the usual2-norm distanceX-Y_2. Much like the fidelity of quantum states, the squared distanceρ··^2is often more natural than the distance. We collect computationally useful properties ofρ··in the following lemma. Let H =H_A⊗ H_B be a Hilbert space. Let U,U_i be unitary operators on H. Let Z,Z_i be arbitrary operators on H. Similarly, let A_i, B_i be unitary operators on H_A,H_B respectively. Let X_i, Y_i be arbitrary operators on H_A,H_B, respectively. Let ρ be a state on H_A⊗ H_B. Let V:H→ H' be an isometry and U' a unitary operator on H'. Then * ρUI^2 =2 - 2_ρ U. More generally, ρ ZI = 1 + _ρ Z Z - 2_ρ Z. * ρUZI = ρZU^†. In particular, ρUI=ρU^†I. * ρZ_1Z_3≤ρZ_1Z_2 + ρZ_2Z_3. * ρZU_2U_3≤ρZI + ρU_2U_3. If U_2 commutes with U_3 (in particular if U_3 = I), then also ρU_1U_2U_3≤ρU_2I + ρU_1U_3. * ρ∏_iA_i⊗ I_B∏_iI_A ⊗ B_i≤∑_i ρA_i⊗ I_B I_A⊗ B_i. *If ρI_A⊗ WBI≤ν and ρA⊗ BI≤η, then* ρi U_iI≤iρU_i I. * ρA⊗ I_BI_AB = ρ_AAI_A, where ρ_A = _B ρ. * ρZ_1Z_2 = Vρ V^†VZ_1V^†VZ_2V^†. * If P is a projection such that Pρ = P, then ρXPI = ρXI = ρ XP. * ρ UV^† U' V = Vρ VVUVU'. We'll use (<ref>) and (<ref>) to convert proofs of group relations into proofs of approximate relations between operators which try to represent the group.The reader interested in following theρ··computations in the rest of the paper may find it useful to find their own proofs of the preceeding facts. For completeness, we provide detailed arguments in the sequel. *We complete the square. ρXI^2 = _ρ(X-I)^†(X-I) = _ρ(2I - X - X^†) = 2 - (_ρ X + _̅ρ̅ ̅X̅) = 2 - 2_ρ X.*In the second equality, we use that the map ρ↦ X^†ρ X is trace-preserving. ρXYI^2 =(XY - I)ρ (XY - I)^†=(Y - X^†)ρ (Y - X^†)^†= ρYX^†^2.*First, suppose ρ = ψ is pure. Then ρZ_1Z_3 = Z_1|ψ⟩- Z_3|ψ⟩ and the triangle inequality for the Hilbert space norm applies. Next, notice that ρZ_1Z_3^2 is linear in ρ. Let ρ = ∑_i _i i be a convex combination of pairwise orthogonal pure states. Then we apply linearity and Cauchy-Schwarz: ρZ_1Z_3^2 = ∑_i _iiZ_1Z_3^2 ≤∑_i _i [ iZ_1Z_2^2+ iZ_2Z_3^2+2 iZ_1Z_2 iZ_2Z_3] = ρZ_1Z_2^2+ρZ_2Z_3^2 +2∑_i √(_ii|(Z_1-Z_2)^†(Z_1-Z_2)|i)√(_ii|(Z_2-Z_3)^†(Z_2-Z_3)|i)≤ρZ_1Z_2^2+ρZ_2Z_3^2 +2√(∑_i _i i|(Z_1-Z_2)^†(Z_1-Z_2)|i∑_j_j j|(Z_2-Z_3)^†(Z_2-Z_3)|j)= ρZ_1Z_2^2+ρZ_2Z_3^2 +2ρZ_1Z_2ρZ_2Z_3= (ρZ_1Z_2 + ρZ_2Z_3)^2.*Applying (<ref>) and (<ref>), ρXYZ = ρXYZ^†I= ρXZY^†≤ρXI + ρI ZY^†= ρXI + ρY Z. If Y commutes with Z, then we have ρXYZ = ρXYZ^†I= ρXZ^† YI= ρYZX^†≤ρYI + ρX Z.*We apply (<ref>) and then apply (<ref>) once for each i. ρ∏_iA_i⊗ I_B∏_iI_A⊗ B_i = ρ∏_iA_i⊗ B_i^†I≤∑_i ρA_i⊗ B_i^†I= ∑_i ρA_i⊗ I_BI_A ⊗ B_i.*This follows from (<ref>) and (<ref>) by writing I_A ⊗ BW = (A)(I_A)(A^†)⊗ (B)(WB)(B^†). *By linearity and (<ref>), we have ρiU_iI^2 = 2 - 2_ρ[i U_i] = i[ 2 - 2_ρ U_i] = iρU_iI^2. Then Jensen's inequality completes the proof. *We use that the trace of the partial trace is the trace. ρA⊗ I_BI^2 = 2 - 2_ρ A⊗ I_B = 2 - 2_ρ_AA = ρ_AAI^2.*We apply cyclicity of trace and unitarity, i.e. V V = I. Vρ VVZ_1VVZ_2V^2 =V(Z_1-Z_2) V V(Z_1-Z_2)V Vρ V=V(Z_1-Z_2) (Z_1-Z_2)ρ V=(Z_1-Z_2) (Z_1-Z_2)ρ.*Again, we apply cyclicity of trace. ρXPI^2 = _ρ (XP- I)(XP-I) =(PX- I)(XP-I)ρ=(PX- I)(X-I)ρ= ρ(PX- I)(X-I) = _ρ(X- I)(X-I) = ρ XI^2. This gives the first equality; a similar manipulation gives the second. *By unitary of U, we can apply (<ref>) to get ρ UV U' V = ρU V U' V I. Next we apply (<ref>) to obtain ρ UV U' V = Vρ VVU V U' V VVV. Now we notice that VV is a projection with (VV) Vρ V = Vρ V, so we apply both parts of (<ref>): Vρ VVU V U' V VVV =Vρ VVU V U'I. Finally, by unitary of U', we can apply (<ref>) to get Vρ VVU V U'I =Vρ VVU V(U'). Taking adjoints and chaining equalities recovers the desired equation. We now use some of the properties of the state-dependent distance to give an approximate version of Lemma <ref> from Section <ref>. Suppose that A_e^(v), B_e, ρ is a strategy presented via observables. Let p_con be the probability that Alice and Bob pass the consistency check, p_sat be the probability that Alice and Bob pass the constraint satisfaction check, and p_win be the probability that they pass both checks. Then we have the immediate bounds p_sat + p_con - 1 ≤ p_win≤minp_sat, p_con, together with the following bounds on p_sat and p_con in terms of the strategy: η = v,e1/4ρA_e^(v)⊗ B_eI^2, η ≤ 1-p_con≤ d^2η, μ = v1/4ρ∏_e(A_e^(v))^H(v,e)⊗ I_d^l(v)I^2, μ ≤ 1-p_sat≤ d^2μ. As in the proof of the exact case, let B̃^i_e and Ã_v^a be projectors onto the eigenspaces of the observables, as in the following spectral decomposition: B_e:= ∑_j_d^-jB̃^i_e A_e^(v):= ∑_i _d^i∑_a: a(e) =iÃ_v^a. Now, we compute v,e_ρ A_e^(v)⊗ B_e = v,e∑_i,j_d^i-j_ρ(∑_a: a(e) =iÃ_v^a)⊗B̃^j_e= v,e∑_k _d^k[a(e) - b ≡ k|questions x=v, y=e]. = p_con + ∑_k∈_d 0_d^k[a(e) - b ≡ k]. Taking real parts and applying the inequalities of complex numbers <ref>, <ref>, we recover equation (<ref>): 1 - 2 (1 - p_con) ≤ v,e_ρ A_e^(v)⊗ B_e ≤ 1 - 2d^-2(1-p) 4 (1-p_con) ≥ v,eρA_e^(v)⊗ B_eI^2 ≥ 4 d^-2(1-p_con). (To get from the first line to the second, we applied Lemma <ref>(<ref>).) With a similar computation, we get: v_d^-l(v)ψ |∏_e(A_e^(v))^H(v,e)⊗ I | ψ =v∑_k_d^kψ |∑_a ∑_e H(v,e)a(e)≡ kÃ_v^a⊗ I |ψ =v∑_k _d^k-l(v)[∑_eH(v,e)a(e)≡ k|question x=v] = p_sat + ∑_k∈_d 0_d^k[∑_eH(v,e)a(e)≡ k+l(v)]. Again, (<ref>) follows from the above via Lemmas <ref> and <ref>.§.§ The stability lemmaWe'll use a general stability theorem for approximate representations of finite groups, which will let us take the following approach to robustness. From a quantum strategy winning with high probability, we extract an “approximate representation” of the solution group, i.e. a map from the group to unitaries which is approximately a homomorphism. The stability theorem lets us conclude that this function is close to an exact representation in the way that the unitaries act on the joint state of the provers, up to a local isometry. Once we have a representation, we'll be able to start applying reasoning analagous to that of <ref>. We were first made aware of results of this type by <cit.>. The result of interest was restated more conveniently in <cit.>. In what follows,U(H)will denote the group of unitary operators on the Hilbert spaceH. Let G be a finite group and f: G→ U(^n) be such that f(x)f(y) - f(xy)_2 ≤√(n) for all x,y∈ G. Then there exists m ≤ (1+^2)n, an isometry V: ^n→ C^m, and a unitary representation : G → U(^m), such that f(x) - V^†(x)V_2 ∈ O(√(n)) for every x∈ G. Applying this theorem directly requires a guarantee on the Hilbert-Schmidt distance between operators. However, experiments with nonlocal games will only give us guarantees on the state-dependent distanceD_ρbetween operators, whereρis the state used by the provers. The following variant addresses this concern. The statement and proof are due to Vidick. Let G be a finite group, f: G → U( H_A) be such that f(x) = f(x)^†, ρ_AB a state on H_A⊗ H_Band x,y∈ Gρf(x)f(yx)^† f(y)⊗ I_BI_AB≤η. Then there is some Hilbert space H_Â, an isometry V:H_A→ H_Â, and a representation τ: G→ U( H_Â) such that x∈ Gρf(x)⊗ I_B V^†τ(x)V⊗ I_B ≤η, or equivalentlyx∈ G(V⊗ I_B)ρ (V⊗ I_B)Vf(x)V⊗ I_Bτ(x)⊗ I_B ≤η.Notice the lack of a dimension bound onÂ. From the proof one can check that the dimension ofÂis at mostG^2times the dimension ofA. We won't use any dimension bound explicitly, and proving a tight dimension bound takes considerable effort.We give a self-contained proof of Lemma <ref>. Letvary over irreducible representations of G. For each , let n_ be the dimension of . We define a generalized Fourier transform of f, which acts on irreps of G, by f̂() = x∈ G f(x) ⊗(̅x̅)̅∈ L(C^d⊗^n_). Let H_A_1A_2 = ⊕_^n__A_1⊗^n__A_2. (Notice that the dimension of H_A_1A_2 is G by Fact <ref>.) For each , define a state |E̊P̊R̊_⟩ = 1/√(n_)∑_i^n_|ii⟩ in the -summand of H_A_1A_2. (Notice that the |E̊P̊R̊_⟩ form an orthonormal family.) Let H_A_3 = |⟩ be a Hilbert space of dimension equal to the number of inequivalent irreps of G. Finally, we define the Hilbert space H_Â, isometry V:H_A → H_Â, and representation τ: G→ U( H_Â) from the statement of the lemma. H_ =H_A⊗ H_A_1⊗ H_A_2⊗ H_A_3, V= ∑_ n_(f̂()_AA_1⊗ I_A_2)(I_A⊗|⟩_A_1A_2⊗|⟩_A_3), τ(x)= I_AA_1⊗∑_ ((x)_A_2⊗_A_3). It's clear that τ is a unitary representation. We check that V is an isometry: V^† V = ∑_ n_^2(I_A ⊗⟨|)f̂ ()^†f̂()(I_A⊗|⟩) = ∑_ n__A_1f̂ ()^†f̂() = E̱_x f (x)^†f(x) = I_A. Now we compute the pullback of τ along V: V^†τ(x) V = ∑_ n_^2(I_A ⊗⟨|) (f̂ ()^†f̂() ⊗(x))(I_A⊗|⟩) = ∑_ n_^2 E̱_y,z∈ Gf(y)^† f(z)⟨|(y)^T(̅z̅)̅⊗(x) |⟩= ∑_ n_E̱_y,z∈ G((x)^T(y)^T(̅z̅)̅)f(y)^† f(z) = E̱_y∈ G∑_ n_E̱_z∈ G((yxz))f(y)^† f(z) = E̱_y∈ G f(y)^† f(yx), where the last equality follows from Fact <ref>. Then it follows from properties of ρ·· that x∈ Gρf(x)⊗ I_B V^†τ(x) V⊗ I_B = x,y∈ Gρf(x)f(yx)^† f(y)⊗ I_BI≤η. The equivalence of the two forms of the conclusion follows from Lemma <ref>(<ref>).Notice that we can also use the lemma with the isometry acting on the state instead of the representation, since x∈ Gρf(x)⊗ I_BV^†τ(x) V⊗ I_B =x∈ Gρf(x)V^†τ(x)^† V⊗ I_B I=x∈ GVρ V^†Vf(x)V^†τ(x)^† VV^†⊗ I_BI=x∈ GVρ V^†Vf(x)V^†τ(x)^†⊗ I_BI=x∈ GVρ V^†Vf(x)V^†⊗ I_Bτ(x)⊗ I_B. Here the last two equalities are applications of Lemma <ref>(<ref>,<ref>). §.§ Quantitative van Kampen lemma In order to apply the stability lemma of the previous subsection, we need an error bound averaged over the whole solution group. From playing an LCS game, we learn an error bound averaged over the generators and relations. In order to go from the latter to the former, we need a bound on how much work is required to build up the individual group elements from its generators and relations. In particular, we'll use the following quantitative version of the van Kampen lemma introduced in <ref>. Suppose G = S:R__d and P is a G-picture witnessing the equation w = J^a. Then the equation w = J^a is true, and can be proven by starting with the equation 1= 1 and applying the following steps in some order: *at most twice for each appearance of generator s in P, conjugate both sides of the equation by s, and *exactly once for each appearance of the relation J^-ar∈ R in P, right-multiply the left-hand side of the equation by r and multiply the right-hand side by J^a. It suffices to prove this only for group pictures whose edges and vertices form a connected graph. For graphs with more than one connected component, we can split the picture into subpictures, apply the lemma, and then glue them back in the obvious way.The proof proceeds via a simple algorithm—we prove the validity of the relation witnessed by the group picture by starting from a subpicture (which witnesses a different relation), and inductively growing it to the whole picture. This can be thought of as a graphical way to prove the validity of the equation witnessed by the group picture, with each step in the algorithm corresponding to a rearrangement of the starting relation. The algorithm then terminates when the subpicture has grown to the full picture, and the starting relation has been transformed into the relation witnessed by the picture. We will then keep track of the steps in the algorithm to verify that Proposition <ref> is true. We describe the algorithm precisely in <ref>. We expect, however, that most readers will be satisfied by examining the example application of the algorithm in Figure <ref>.In order to define the algorithm, we set up some terminology: The bubble is the boundary of the expanding subpicture. A bubble-intersection is the intersection between the bubble and an edge of the picture. The pointer is a (vertex, edge) pair. In our diagrams, we'll draw it as a dot at the bubble-intersection at the edge near the vertex. To advance the pointer is to move the pointer from its current location to the next bubble-intersection clockwise around the bubble. We'll work informally with smooth curves. This approach can be rigorized with notions from differential topology—see e.g. <cit.> for an introduction to the subject. See <cit.> for a more careful topological treatment of group pictures.Alternatively, one can use graph embeddings where all the vertices lie at integer coordinates and all curves are piecewise linear, and then argue constructively from there. | http://arxiv.org/abs/1709.09267v2 | {
"authors": [
"Andrea Coladangelo",
"Jalex Stark"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170926215241",
"title": "Robust self-testing for linear constraint system games"
} |
apsrev | http://arxiv.org/abs/1709.09034v2 | {
"authors": [
"Ewa Kocuper",
"Jerzy Matyjasek",
"Kasia Zwierzchowska"
],
"categories": [
"gr-qc"
],
"primary_category": "gr-qc",
"published": "20170926141951",
"title": "Stress-energy tensor of quantized massive fields in static wormhole spacetimes"
} |
Department of Physics & Astronomy, University of Pennsylvania, Philadelphia, PA 19104 USA [email protected] of Physics & Astronomy, University of Pennsylvania, Philadelphia, PA 19104 USA [email protected] We probe the effects of particle shape on the global and local behavior of a two-dimensional granular pillar, acting as a proxy for a disordered solid, under uniaxial compression. This geometry allows for direct measurement of global material response, as well as tracking of all individual particle trajectories. In general, drawing connections between local structure and local dynamics can be challenging in amorphous materials due to lower precision of atomic positions, so this study aims to elucidate such connections. We vary local interactions by using three different particle shapes: discrete circular grains (monomers), pairs of grains bonded together (dimers), and groups of three bonded in a triangle (trimers). We find that dimers substantially strengthen the pillar and the degree of this effect is determined by orientational order in the initial condition. In addition, while the three particle shapes form void regions at distinct rates, we find that anisotropies in the local amorphous structure remain robust through the definition of a metric that quantifies packing anisotropy. Finally, we highlight connections between local deformation rates and local structure. Anisotropic Particles Strengthen Granular Pillars under Compression Douglas J. Durian December 30, 2023 ===================================================================§ INTRODUCTIONWhen a disordered solid is subject to a mechanical load, various characteristics of its local structure and composition directly impact the observed response and performance. For example, composite metallic glasses with interspersed dendrites that arrest shear bands and cracks can counteract the standard trade-off between material strength and fracture toughness in brittle materials <cit.>. Other materials can fail in a ductile fashion, in which material failure is marked by local plastic flow and/or growth and coalescence of voids within the bulk <cit.>.In general, characteristics of the local interactions between constituent elements are critical in determining the response of a disordered system. These descriptors can include bond strength, dissipation, and elasticity. These considerations may require, for instance, additional terms in the development of a constitutive model for the disordered solid, in order to best predict creep and the onset of failure. For example, the Gurson-Tvergaard-Needleman (GTN) model currently serves as a basis for constitutive modeling of ductile failure that can incorporate either void coalescence or plastic flow <cit.>. We would like to focus on one aspect that does not inherently alter the interaction between material components, but can still substantially influence behavior: particle shape. If the shape of constituent particles (or grains) is changed, that alone may not necessarily alter the inherent physics of how particles interact with one another. The underlying mechanisms of their interactions will remain, but one must consider effects that the shapes have on contact distance, surface curvature, and rotational frustration. Indeed, the effects of grain shapes can be observed in a wide variety of systems, spanning several decades of particle sizes. These phenomena include the toughening of disordered nanoparticle assemblies with elongated particles <cit.> and colloidal packings of polygons whose shape frustrates crystalline order <cit.>. On even larger length scales, in which thermal fluctuations are negligible, effects of particle shape become crucially important. Many recent studies have considered the implications of grain shape in granular flows, such as dense driven systems in which nematic ordering can spontaneously occur <cit.>, as well as gaseous states in which random collisions impart both translational and rotational motion <cit.>. To better understand the stability of packings of arbitrarily shaped particles, there has also been interest in characterizing (near-)jamming characteristics, such as contact numbers and vibrational modes, of elongated noncircular particles <cit.>. Recently, grain shapes have been explored as a way to generate free-standing architectural structures <cit.>. Examples include highly elongated and U-shaped particles with the capability to form geometrically constrained contacts <cit.> and custom particle fabrication that is facilitated by evolutionary searches for the strongest shapes under a specified load <cit.>. While an overall strength can be prescribed, stress relaxation events, or avalanches, occur as a granular system is slowly driven <cit.>. The distribution of sizes of these drops, defined either in terms of a global pressure or energy, often falls on a power law with commonly observed exponents <cit.>. Coarse-grained and depinning models have been proposed to associate stress fluctuations with local plastic rearrangements <cit.>. Particle shape is thought to contribute to the micromechanics of localized slip events <cit.>, but to our knowledge has yet to be explicitly studied within this framework.When a granular material is slowly driven, it can behave like a slowly deforming solid and provide a bridge to better understanding much of the microscopic behavior within disordered solids. Granular materials are, by definition, assemblies of discrete macroscopic particles, so their constituents can be directly imaged in certain geometries, allowing for a full characterization of microstructure that is not possible in other materials. Other properties of disordered solids, such as bond strength between component particles, can be represented using fluid capillarity <cit.> or inter-particle bonding with a cured polymer <cit.>. In this study, we focus on altering particle shapes, with varying amounts of circularity, that comprise a dry granular packing. This article is organized as follows. In Section <ref>, we describe the experimental apparatus, the granular system that is used as a model disordered solid, the particle shapes we study, and the techniques used to collect data on the global and local responses of the material under uniaxial compression. In Section <ref>, we summarize the primary findings of our study. Specifically, in Section <ref> we describe the effect of particle shapes on the overall material strength, and in Section <ref> we discuss stress relaxation events, or avalanches, that occur during compression. Then, in Section <ref>, we describe how we characterize structural anisotropies within the packings through the adaptation of a previously defined metric for non-circular grains. In Section <ref> we show how we quantify local plastic strains within the system and test their relationship with avalanches. In Section <ref>, we draw connections between local structure and local dynamics in terms of how anistropies in local plastic strain are correlated with structural anisotropies. Finally, we discuss the broader implications of these findings and motivate further study in Section <ref>.§ MATERIALS & METHODSThe granular system consists of bidisperse acetal (delrin) rods with diameters 0.25 in (6.4 mm) and 0.1875 in (4.8 mm) and uniform height 0.75 in (19 mm), standing upright on an acrylic substrate. The large and small rods are mixed with a number ratio of 1:1. In order to alter the grain shape, we bond individual rods together to form a composite shape that is overall noncircular, but retains surfaces with a constant radius of curvature. Particle shapes that are comprised of bonded, sometimes overlapping, combinations of circles or spheres is a common technique to explore generic grain shapes, especially in simulation <cit.>, so the technique used here is another iteration of this general approach. While simulations have been used to study the response of circular particles in the apparatus described in this article <cit.>, we choose to focus on experiments to establish shape-dependent behaviors before determining how to best incorporate material properties and shape-dependent formulations of contact forces. From here, arbitrary particle number, shape, and size can provide fruitful ventures for simulation study.The specific shapes we study in this article are monomers (individual plastic rods), dimers (pairs of bonded rods), and trimers (groups of three bonded rods in a triangular shape), as shown in Fig. <ref>. Different particle shapes are constructed by gluing rods together using a cyanoacrylate adhesive. The entire fabrication procedure for a dimer is shown in Fig. <ref>. A small amount of adhesive is placed near the top of a rod standing upright on a horizontal table. Then, a second rod, also standing upright, is brought into contact with the first. To ensure both rods are straight, they are confined to stand within the jaws of a vernier caliper set to the rod diameter. The adhesive spreads down the pair of rods through capillarity, while also curing to form a strong bond between the rods. The amount of adhesive used is not precise, but it must be substantial enough so that the cured bond is strong, yet limited so the adhesive does not spread all way down the rods, bonding them to the table. When fully cured, the pair of rods now form a dimer. To make a trimer, this same adhesive procedure is repeated with a third rod brought in to form a triangle. This type of trimer is preferred, as a linear chain of more than two tall macroscopic rods is generally difficult to achieve by hand with sufficient accuracy and consistency. Furthermore, this allows us to isolate dimers as our case study in elongated particles, while the trimers are more axially symmetric, but with characteristic bumps. After allowing the adhesive to cure overnight, the dimers and trimers require substantial effort to break apart by hand.Without precisely measuring shear and/or flexural strength, we observe that internal stresses within each experiment never cause breakage.Our experimental apparatus is shown in Fig. <ref>(a), with its various components labeled. This is the same apparatus used in Refs. <cit.> to study granular pillar deformation. The entire apparatus lies on a horizontal table-top, so gravity does not directly drive or hinder the motion of grains. The grains, all of one chosen shape, are arranged into a tall, narrow pillar with an aspect ratio of approximately 2:1 using a rigid frame. An initial pillar configuration is shown in Fig. <ref>(a)-(b). Since the particle shapes have distinct area fractions when packed randomly, we choose to keep the width of the pillar consistent (W_0 = 4.875 in = 12.4 cm), while the initial pillar height (H_0 ∼ 9.75 in = 24.8 cm) can vary slightly from one trial to another, much less from one shape to another. Differences in H_0 between trials are especially apparent in pillars comprised of dimers and trimers, since the particle geometries frustrate random close-packing as opposed to the circularly symmetric monomers. The initial area fraction of monomers is ϕ = 0.823 ± 0.004, dimers, ϕ = 0.809 ± 0.007, and trimers, ϕ = 0.805 ± 0.005. The uncertainties in ϕ are determined from the range covered over all trials.The pillar is unaxially compressed from the top by a slowly moving bar (v_c = 0.033 in/s = 85 μm/s), while a static bar remains in contact with the pillar bottom. As the pillar is compressed and laterally spreads out, its interior structure constantly evolves due to interspersed local plastic flow and the creation and collapse of voids. These aspects are commonly present in materials undergoing ductile failure <cit.>, so our apparatus can serve as a model system for this type of material failure. An important distinction between this apparatus and other uniaxially driven granular systems <cit.> is that we do not restrict expansion of the system with any sort of hard boundary or soft membrane, nor is the compression direction along or against the direction of gravity. We performed 5 trials for each type of pillar composition, with the specifics of the microscopic initial structure varying from run to run, but initial dimensions remaining constant as described above. Keeping the system dimensions consistent across shapes also requires altering the total number of particles. N = 1000 for monomers, N = 500 for dimers, and N = 334 for trimers. We chose to keep the pillar size constant, rather than the discrete particle count, in order to draw fair comparisons of material strength and behavior. In fact, large pillars comprised of 1000 dimers or 1000 trimers would present practical challenges for the present apparatus. Similar studies that can control for pillar size, particle count, and particle mass, via simulation or custom particle fabrication, would make for interesting studies.While the compressing bar is in motion, we acquire 4.2 megapixel (2048x2048) images of the pillar deformation using a JAI/Pulnix TM-4200CL camera with a frame rate of 8 fps. For each image, we simultaneously record the forces exerted on the moving and static bars using Omega Engineering LCEB-5 force sensors. After acquiring images, we locate all circular particles using a circular edge-finding algorithm <cit.>. Fig. <ref>(c) demonstrates the sharp intensity contrast between the painted caps of the particles and the background illumination. The displacement of the compressing bar between successive frames is about 10^-3R, where R is the large monomer radius, so linking position coordinates together into particle tracks is a straightforward process. To suppress noise in particle positions, we apply a Gaussian filter with a time window equal to the time over which the compression bar moves 2/15R. This becomes the effective time interval between filtered frames. We also use this Gaussian smoothing to differentiate positions, yielding approximations of instantaneous velocities. When analyzing pillars with dimers or trimers, we group rods together by measuring interparticle distances over time. Since every dimer and trimer consists of equal sized rods, we can deduce some of the combinations just from the initial packing. Within portions of the pillar that significantly deform over the full run (over which the bar moves about halfway down the initial height), we usually find there is only one possible combination to link dimers or trimers together. For regions that do not significantly deform, especially large clusters of like-sized particles at the bottom of the pillar, we group particles such that interparticle distance fluctuations are minimized. Ultimately, we can successfully group every dimer and trimer together, particularly those that exhibit motion beyond our noise level in calculating positions. The centroid positions of the dimers and trimers are directly calculated from the positions of their constituent rods, smoothed, and differentiated as described above.After particle tracking and the identification of monomers, dimers, and trimers, we can measure various aspects of local structure and motion. These shall be discussed in further detail in Sections <ref> and <ref>.§ EXPERIMENTAL RESULTS §.§ Material StrengthAs an analog to standard tests of material strength, we measure the stress-strain response of the pillar as it is compressed. Following the procedure set in Ref. <cit.>, we quantify the compressive stress, σ, as F/W where F is the driving force exerted by the moving bar on the pillar and W is the current width of the pillar in contact with the moving bar, making σ a measurement of true stress. A rod is considered to be in contact with the moving bar if its vertical position is within 0.25R of the rod at the top of the pillar, where R is the large rod radius. The pillar width W is calculated as the end-to-end horizontal distance of these contacting rods. The forces on the static bar are negligible for monomer runs, so to be consistent across all trials we choose to focus on just the force actively driving the pillar. As expected, both F and W tend to continuously increase over the course of a pillar compression. While W tends to grow steadily over the course of a compression, it can exhibit a jump discontinuity if new particles(s) come into or out of contact at either end of the pillar top, while new contacting particles within the interior of the pillar top, the primary mechanism of width increase, do not result in W discontinuities or fluctuations. Indeed, jumps in W only occur about 5 times in a single run, so it primarily behaves as a smooth function without inducing substantial fluctuations in stress, σ. In every plot showing stress, we quantify σ in derived units of mgμ/D, where m and D are the mass and diameter, respectively, of a large rod, g is the acceleration due to gravity, and μ is the grain-substrate coefficient of friction, measured to be 0.23 ± 0.01 <cit.>. Effectively, these units represent the stress required to move an individual large monomer at constant speed. The vertical strain γ is given by Δ H/H_0, where H_0 is the initial height of the pillar and Δ H is the difference in height between the initial pillar and the deformed system, H_0 - H. In Fig. <ref>, we show the stress-strain behavior for a single compression trial of a pillar comprised of dimers and highlight three regimes of pillar deformation: (1) an elastic-like initial compression, which occurs over a very short strain range (γ≲ 0.01), too short to confirm a linear response, (2) a yield transition around γ∼ 0.01 when stress reaches a maximum value, and (3) long-term (γ≳ 0.01) deformation and failure that is marked by a fairly constant material strength, with irregular stress fluctuations. In Section <ref> we will consider the distribution of stress drops, but for now we are motivated by their relative size and irregular frequency to consider trial averages as a way of better gauging the material strength of pillars comprised of our three particle shapes. We average 5 trials together to generate stress-strain curves, shown in Fig. <ref>, significantly reducing the prevalence of stress fluctuations during long-term deformation. Note that Fig. <ref> is presented with a horizontal log scale, emphasizing low-strain behavior. Before averaging, the point of zero strain, γ = 0, in each trial is set to minimize initial strain readings that result from the motion of individual particles within the top layer of particles. We see in Fig. <ref>(a) that dimers exhibit more strength than monomers, in terms of a compressive modulus that can be estimated from the quasi-elastic regime, a larger yield stress, as well as the stress required to continually deform the pillar at large strains. Pillars comprised of trimers retain an average long-term strength that is comparable to that of monomers.Dimers are clearly the strongest shape tested in this study, so we would like to further investigate why this is the case. The specific question we would like to answer is: can we prepare a pillar using dimers in a way that either strengthens the pillar to a further degree or diminishes the apparent strengthening effect? To do so, we note the unidirectional driving of the system, in junction with the elongation of the dimers, to prepare two types of highly ordered packings of dimers. In addition to the disordered dimers previously measured, we prepare a set of packings in which dimers are preferentially ordered horizontally, along the compressing and static bars, as well as a set of packings with dimers preferentially ordered vertically, along the compression direction. These pillars are meticulously created layer-by-layer, building upward in the horizontal case and to the right in the vertical case, in an effort to minimize the presence of orientational defects. The pillar dimensions are kept consistent as before, which necessitates the presence of some defects. Due to the high degree of ordering, the initial packing fractions for ordered dimers is higher in both cases, with ϕ = 0.813 ± 0.002 for horizontal dimers and ϕ = 0.814 ± 0.003 for vertical dimers. We also quantify the degree of orientational order present in the initial pillars, and during compression, as shown in Fig. <ref>.We note a marked distinction in the material response for these three types of dimer packings, illustrated in Fig. <ref>(b). Specifically, vertical dimers are substantially weaker than the randomly packed dimers. From there, we see that horizontal dimers reach an even higher compressive strength at γ∼ 0.02.The presence of noise in the low-strain behavior of the ordered pillars should be noted. This noise can be attributed to the presence of orientational defects, specifically those near the top of the pillar, in individual trials. These compound the difficulty of differentiating low strain behavior to define a compressive modulus. Nevertheless, the pillar strength is substantially impacted not only by the grain shape, but also the procedure by which the packing is generated.Looking at raw snapshots of these packing types under compression, illustrated in Fig. <ref>, we also observe distinct local behaviors as the dimers are compressed. These differences are also apparent in full movies in the Supplementary Material <cit.>. The movies in the Supplementary Material include overlays with D^2_min, a metric that quantifies plastic deformation and local rearrangement around each discrete particle <cit.>. We overlay with D^2_min, which is assigned to individual particles, rather than J_2, a measurement of local instantaneous strain rate (discussed at length in Section <ref>), which is defined for regions of three particles. J_2 overlays would thus obscure dimer positions and orientations. The horizontal dimers buckle outward, breaking into separate columns with little slip between particles, as shown in Fig. <ref>(b). Also, the shape of the pillar expands with rough edges, the furthest outward extents lying about a quarter of the way down the pillar. Meanwhile, the vertical dimers deform much more gradually, shown in Fig. <ref>(c) with a smooth symmetric plume right at the very top of the pillar. When dimers are packed randomly, as in Fig. <ref>(a), contributions from both types of deformation are present. The amount of structural rupture occurring within the interior of the pillar is quantified in Section <ref>.We can now state that the material strength gained from dimer packings comes directly from dimers that preferentially lie ordered to each other, specifically interlocking along the horizontal direction as to resist outward expansion of the pillar. We can even see in the right side of Fig. <ref>(b), in junction with Fig. <ref>, that as the random and vertical dimer pillars are continually deformed, dimers rearrange so that those in contact with the bar are mostly horizontal, while the strength of the pillar continually increases. In fact, they are trending toward the strength exhibited by pillars with horizontal dimers to begin with. This result provides further motivation to investigate the relationship between local structural and deformation features, which shall be discussed in Sections <ref>, <ref>, and <ref>. §.§ Avalanches & Stress RelaxationAs previously mentioned in Section <ref>, the stress-strain curve for each compression trial exhibits fluctuations about an average strength during the regime of large strain. The same trend is seen in other amorphous systems, the mechanism of which can be owed to the build up of local stresses, followed by a relaxation that is associated with slip rearrangements <cit.>. In this section, we consider the sizes of stress relaxation events, or avalanches, and their frequency as a function of particle shape. Later, in Section <ref>, we will consider potential origins of the stress fluctuations at a more local scale.To be clear, when we refer to avalanches and their sizes, we are exclusively referring to continuous drops in stress, as illustrated in Fig. <ref>. Avalanches are generally preceded by a build-up of stress within the system, which the avalanche at least partially relaxes away. It is worth noting that the representative data in Fig. <ref> includes stress accumulations and avalanches that are roughly symmetric with respect to strain. This aspect of symmetry is likely due to the hardness of rods preventing elastic energy from being stored locally, along with the lack of a confining boundary permitting the pillar to constantly dilate globally. In each individual run, we locate intervals over which the stress is decreasing, truncated by peaks, valleys, and/or plateaus in σ. The dimensionless magnitude of the stress difference of the entire interval is defined to be S = Δσ/(mgμ/D). In Fig. <ref>, we show the distributions of S for the three particle shapes, measured over all trials. The lower bound of the plotted range in S is chosen to neglect a region where the distributions are increasing, which coincides with avalanches that are below the noise level in our stress measurements. As we expect from other studies of avalanche distributions within amorphous systems, we see that the distributions could be described by a power law. In fact, the exponent for distributions over the range S>1 is approximately -3/2, which has been observed in other amorphous systems <cit.> and predicted by a coarse-grained model <cit.>. We should note that, while we are estimating -3/2 as the exponent, we cannot confidently calculate this exponent given the narrow range of S. This is due to both the noise level in measuring stress, as well as substrate friction, as the maximum observed value for all avalanche sizes is determined by the force required to move 𝒪(10) particles. Furthermore, while the applicability of power laws in other amorphous systems motivates the conjecture of a -3/2 power law, we find that the complementary cumulative distribution function of avalanche sizes can be fit over its full range with a compressed exponential function, exp(-(S/S_0)^β). While the avalanche distribution for all shapes have approximately the same rate of decay, Fig. <ref> shows unique features of the distributions. Monomers exhibit smaller avalanches, while the distributions for dimers and trimers are similar. This is also reflected in compressed exponential fits for the complementary cumulative distribution function. Monomers have β = 1.5 ± 0.2 and S_0 = 0.89 ± 0.05, dimers have β = 1.4 ± 0.2 and S_0 = 2.4 ± 0.2, and trimers have β = 1.4 ± 0.2 and S_0 = 1.7 ± 0.1. In Ref. <cit.>, increased particle friction is observed to result in larger upper thresholds in avalanche size. Since the bumpy concave shapes of dimers and trimers effectively increase particle friction, we observe a similar trend. We do not show the avalanche distributions for the highly ordered dimer packings, as they are virtually identical to the avalanche distribution of randomly packed dimers.Particle shape thus directly influences the global material response, both in terms of averages and fluctuations of stress. In Sections <ref>, <ref>, and <ref>, we further explore the effects of particle shape on both local structure and dynamics. §.§ Local StructureWhile the granular pillars are initially set with consistent dimensions, there are bound to be heterogeneities in the packing efficiency, much less additional structural heterogeneities that are introduced as the pillar is compressed. Furthermore, voids that form or collapse over time are crucial componments of ductile failure. To quantify these aspects of local structure, we use the dimensionless quantity Q_k, previously defined in Ref. <cit.>, to highlight anisotropies in the Voronoi tessellation of the packing.In the simple case of monomers, we perform a radical Voronoi tessellation of the particle positions using the software package voro++ <cit.>, followed by a Delaunay triangulation. Then, we define a vector field 𝐂 that points from the rod center to the centroid of its own Voronoi cell. Finally, we define Q_k for a triangle k from the divergence of this vector field, Q_k = ∇·𝐂_k A_k/⟨ A ⟩, where A_k is the area of triangle k and ⟨ A ⟩ is the average area of all triangles. Scaling the divergence by area sets ⟨ Q_k ⟩ = 0, with some residual contribution from the finite boundaries of the experimental data. To minimize these boundary effects, we ignore all triangles that lie on the boundary of the pillar. Q_k is highly correlated with relative free area fraction, where Q_k < 0 corresponds to under-packed regions, while Q_k > 0 corresponds to over-packed regions. Furthermore, the distribution of Q_k values measured for either experimental hard disks or simulated soft disks is nearly Gaussian and centered at Q_k = ⟨ Q_k ⟩ = 0, in sharp contrast to distributions of local free volume. The deviation from Gaussianity in the tail of Q_k indicates a surplus of underpacked particles, with both standard deviation and skewness of Q_k exhibiting kinks at the jamming point ϕ_c <cit.>. Calculating Q_k with the centroid positions of dimers and trimers requires a small amount of adaptation in the method, as performing the Voronoi tessellation of non-spherical particles can often result in non-convex Voronoi cells <cit.>. Fortunately, given that the dimers and trimers both have circular curvature, we can rely heavily on the initial Voronoi analysis. Starting with the Voronoi tessellation for rods generated from voro++, we can simply delete edges that cut across bonded particles. This leaves a larger effective cell that now surrounds the entire dimer pair or trimer group. A new triangular tessellation is then computed, using knowledge of particles that share Voronoi edges. Fig. <ref> illustrates the two approaches that can be used for computing Q_k for a region of dimers. While the triangulation of dimers and trimers is no longer dual with its Voronoi diagram, this remains an effective way to determine a packing tessellation with no gaps or overlaps. Moreover, Fig. <ref> indicates that Q_k measured in this “molecular” sense retains a Gaussian-like profile on a linear scale. When characterizing local structure in the dimer and trimer packings, we have actually found both pictures can be enlightening: one where Q_k is calculated based on individual rod positions (“Atoms") and one where we instead use the centroid of the composite shape (“Molecules").The “atomistic" Q_k, illustrated on the left side of Fig. <ref>, highlights absolute areas of vacancies and has been shown to correlate well with local free area <cit.>. Fig. <ref>(a) shows the probability density functions for Q_k measured in regions that have been driven at least one large monomer radius from its initial position. From monomers to dimers to trimers, we see that using larger shapes results in distinctly larger voids during deformation. At the same time, bonded rods also allow for additional regions that are overpacked, especially when a triangle corresponds to a discrete trimer particle. These effects are plainly visible by eye in the raw experimental data and are quantified using this method of Q_k measurement.However, relative structural anisotropies are less apparent when accounting for both the shape and orientation of the discrete particles. While the dimers and trimers create large voids, are they consistent with the fact that dimers and trimers are themselves larger? Another question lies in whether similarities in the random preparation protocol for all shapes can be captured in a structural quantity. These questions can be addressed by measuring the “molecular" Q_k, illustrated on the right side of Fig. <ref>, with distributions shown in Fig. <ref>(b). Remarkably, despite the randomness of dimer and trimer packings resulting in more physical void space, we see that the Q_k distributions are strikingly similar. All the distributions are nearly Gaussian in the vicinity of Q_k = ⟨ Q_k ⟩ = 0 and retain similar widths despite the manifestation of distinct global dilation rates. The collapse of these distributions suggests that Q_k, as a metric for local packing anisotropy, may serve well beyond characterizing local free area in packings of circles. Rather, Q_k seems to demonstrate promise to characterize local packing structure with arbitrary particle shape, and that random close packings of symmetric and asymmetric particles can exhibit similar local structural fluctuations.To quantify the collapse of Q_k distributions that results from moving from the “atomistic” picture to the “molecular” picture, we compute the skewness and kurtosis of Q_k distributions shown in Table <ref>. Indeed, similar values are reported for monomers and the “molecular” dimers and trimers. It is also worth noting the physical interpretations of skewness and kurtosis in the context of Q_k. Skewness provides a measurement of the asymmetry of a distribution, while kurtosis quantifies the presence of tails, either fat or broad relative to a Gaussian distribution. While Q_k appears near Gaussian in the linear plots shown in Fig. <ref>, there are necessary deviations in its skewness and kurtosis. For one, there is a finite limit to how closely hard particles, such as the ones used in this study, can pack together, while void space in underpacked regions is only restricted by the boundaries of the system, which in this case are open. This allows a wider accessible range in negative Q_k values, resulting in a negative skewness. Figs. <ref> and <ref>(b) illustrate this asymmetry, since the empirical data in the left tail for monomers lies slightly above the ideal Gaussian curve, while the right tail more closely follows the ideal curve. In turn, the wider range of negative Q_k values requires its tail to decay slower than the Gaussian curve, which is apparent throughout Fig. <ref>. Hence, the kurtosis of Q_k will be higher than that of a Gaussian. As expected, these aspects of Q_k are reflected in Table <ref> for the collapsed “molecular” distributions. For a particular non-circular shape, Q_k can also indicate distinct structural characteristics. Fig. <ref>(c)-(d) shows Q_k distributions for the different dimer packings, using the “atomistic” and “molecular” views of Q_k for deformed regions. As previously suggested in Section <ref>, horizontally ordered dimers strengthen the pillar, while also giving way to additional local rupture. While ordered dimers are initially packed with similar global area fractions, Fig. <ref>(c)-(d) indicates the formation of additional void space when dimers are initially packed horizontally. Vertically packed dimers form voids at a more gradual rate, while randomly packed dimers lie at a rate between the two ordering procedures.While the Q_k distributions for different particle shapes collapse very well in Fig <ref>(b), it is worth nothing that some deviation is seen for highly under-packed regions, where Q_k ≲ -0.3. This kink is even exacerbated in the case of horizontally ordered dimers, shown in Fig. <ref>(d). To seek a dynamical explanation for this feature, we now shift our attention to local deformation. §.§ Local DynamicsIn addition to local structure, we can also quantify local plastic strain within the pillar, another important feature of ductile failure. In this study, we choose to quantify local deformation by the deviatoric strain rate, J_2, which describes how the shape of a small region deforms. The procedure of calculating J_2 is as follows.Over a triangle that is derived from particle positions and Delaunay triangulation, one of the same triangles used in calculating Q_k, we calculate J_2 using the constant strain triangle formalism <cit.>. We must first note that for all results related to J_2 discussed, unless specified, we are focusing on the “molecular" form of triangulation as defined in Section <ref>. As such, we are treating each point in the triangle as discrete particles, capable of moving independent of each other. For the three particles that make up the triangle, we note the velocity of each particle, each having horizontal component v_x and vertical component v_y. Subtracting off the average velocity of the three particles, which is prescribed to the center of mass of the triangle, we determine the local strain tensor e,([ v_x(x,y)-v_x,CM; v_y(x,y)-v_y,CM ]) = ([ e_11 e_12; e_21 e_22 ]) ([ x; y ]), where x and y are Cartesian coordinates relative to the center of the triangle. One way to conceptualize this formalism is to place pins at the particle centroid locations, with some sort of continuous triangular mesh in the middle. We can deform the mesh by moving the pins relative to each other, causing it to stretch, deform, rotate, or some combination thereof. For this study, we choose not to incorporate particle rotations, which are certainly present, into the formulation of this strain tensor, in part because they substantially complicate the local strain tensor. Also, Fig. <ref> indicates that particle motion within dimer and trimer pillars is primarily attributed to translational motion, so a simple strain based on translations alone is likely sufficient to characterize local deformations in this study. Given that acetal rods are slippery compared to the acrylic substrate, grain-grain friction is likely too small to induce rotational velocities that are comparable to translational velocities.Further studies could explicitly incorporate particle rotations as a component of a more complex local strain, especially in systems of highly frictional grains.From the empirical local strain tensor e, we deduce the symmetric strain tensor ε, ε_ij = e_ij+e_ji/2. This local linear strain tensor has a number of invariant quantities that characterize the local relative motion of the grains; for instance, the trace defines the dilation rate. We choose to focus on the deviatoric strain rate, J_2, as a measurement of the amount of local plastic deformation,J_2 = 1/2√((ε_11-ε_22)^2+4ε_12^2). With J_2 defined, we should note that there exist other metrics that can fill the role of quantifying local plastic deformation in a similar fashion, e.g., D^2_min <cit.>. For this study, we choose to focus on J_2 for a few reasons. First, as we shall soon discuss, we would like to make direct comparisons with stress, which is a single measurement made at each time point. Thus, we would like to select a kinematic quantity that can also be prescribed to a single time. By definition, D^2_min requires the choice of a substantial time interval over which to measure plastic displacements. J_2 is calculated from velocities obtained through differentiation over a small time interval as described in Section <ref>, so it can naturally coincide with the same time point of a stress measurement. Second, D^2_min requires the choice of an interaction cutoff length, while J_2, derived from Delaunay triangulation, requires no such cutoff. Third, while calculated over the area surrounding a single grain, D^2_min is assigned to each individual grain. J_2 is rather assigned to a region connected to three grains, so it is a slightly coarse-grained measurement, in line with the approach of established avalanche models <cit.>.Note that J_2 is a strain rate, so it has dimensions of inverse time. J_2 is thus scaled relative to the inverse time required to compress the pillar by one large monomer radius, v_c/R. This is done for all grain shapes, which have distinct sizes but are all undergoing the same global strain rate. For the sake of comparisons with the global measurement of stress, we take the ensemble average ⟨ J_2 ⟩ as a way to quantify the total amount of plastic deformation throughout the system. We exclude stationary triangles, those that have moved less than a large monomer radius, from the ensemble average ⟨ J_2 ⟩. To further confirm the utility of J_2 in quantifying plastic strain, we consider its relationship with stress fluctuations discussed previously in Sections <ref> and <ref>. In Fig. <ref>(a), we see that peaks and troughs of σ and ⟨ J_2 ⟩ over the course of a single dimers trial generally correlate with each other. We explore the relationship between σ and ⟨ J_2 ⟩ further in Fig. <ref>(b)-(d), by plotting the two quantities from all trials directly against each other. We see that in the case of monomers, in Fig. <ref>(b), there is a general positive correlation between the two quantities. This is indicative of particle rearrangements within these pillars as significantly contributing to the presence of avalanches. That is, as the pillar is driven, stress builds up for some period of time. These periods of large σ tend to be associated with large ⟨ J_2 ⟩, suggesting that built up levels of stress are subsequently relaxed away by particle rearrangements within the pillar. Fig. <ref>(c) shows that the correlation between σ and J_2 for dimers is less pronounced. In Fig. <ref>(d), we see that trimers do not exhibit much of a correlation between σ and ⟨ J_2 ⟩. The Pearson correlation coefficient ρ of the three sets of data shown in Fig. <ref>(b-d) are as follows: monomers, ρ = 0.66, dimers, ρ = 0.33, and trimers, ρ = 0.087. Since the dimers and trimers are incrementally more massive than the monomers, the shifts in correlation may be due to the fact that stick-slip motion between individual particles and the substrate becomes more prevalent due to body friction. Still, the fact that we see correlations for monomers, and even dimers to a degree, is indicative that avalanches, derived from either global stress or local strains, can be applied to systems of either symmetric or elongated particles.§.§ Structure-Dynamics ConnectionsFinally, we discuss connections that can be made between our previous results to quantify both local structure and local dynamics. As Q_k quantifies local under- and over-packing relative to the surrounding neighborhood of a localized region, we can also measure the deviatoric strain rate J_2 in a way to highlight regions that are deforming relative to its surroundings <cit.>. In this way, we emphasize rearrangements that are highly localized as well as rigid areas that are adjacent to shear bands. J_2,rel, a relative deviatoric strain rate, for a given triangle is defined by the difference of its J_2 and the average of its neighbors, J_2,rel = J_2 - ⟨ J_2,neighbors⟩. Neighboring triangles are defined to be those which share at least one vertex, i.e., particle, with a given triangle. In Fig. <ref>, we show the raw bivariate histograms for values of Q_k and J_2,rel, highlighting the amount of spread in J_2,rel at each value of Q_k. In general, one should expect under-packed regions are more likely to undergo strain than over-packed regions, since a void region with open space can collapse more easily. Meanwhile, over-packed regions are more constrained by its neighbors, so those can be expected to be less likely to strain. However, one must note that individual structural metrics can be poor predictors of particle rearrangements <cit.>. Indeed, it is difficult to observe any trend in Fig. <ref>, although a slight negative correlation between J_2,rel and Q_k may be visible. To specify an overall trend of J_2,rel versus Q_k, we bin the data in Fig. <ref> by intervals in Q_k and take averages of corresponding values of J_2,rel to generate Fig. <ref>(b). Here, the trend of J_2,rel with Q_k is much more apparent. The vertical error bars represent standard deviation of the mean J_2,rel within each Q_k bin. Given that bins near Q_k = 0 contain 𝒪(10^5) data points, these error bars are vastly suppressed compared to the actual spread in raw data.Indeed, we observe a negative correlation between these two quantities, which is approximately linear in the region around Q_k = 0. This is in line with the expected trend of how likely under- and over-packed region are expected to deform. That is, voids are readily collapsing, while constrained regions are persisting. The linear dependence is more pronounced among monomers, likely due to the absence of geometrical constraints such as elongation and bumpy surfaces. However, we note two important deviations from this general behavior that occur at the opposite extremes in Q_k. At highly positive Q_k, we see a dramatic upturn in J_2,rel for monomers and trimers, indicative of the onset of Reynolds dilatancy <cit.>. The acetal rods are quite hard, so there is a finite limit to how closely they can be packed until they must deform locally. However, dimers do not demonstrate such a dramatic increase, capturing the ability of many dimer pairs to interlock and actually form rigid structures, as demonstrated in Fig. <ref>(b). The second notable deviation we see is at highly negative Q_k values, which indicate large voids in the packing, we see a dip in J_2,rel toward zero for dimers, indicating that the void regions that form in dimers can actually persist for some time, unlike large voids in monomers and trimers that readily collapse. The dip in J_2,rel for discrete dimers in present in Fig. <ref>(b), but is especially apparent when measured using the “atomistic” approach shown in Fig. <ref>(a). This deviation in the dynamical behavior of dimer “molecules” in Fig. <ref>(b) coincides with a kink in the dimer Q_k distribution shown in Fig. <ref>(b) over the same highly negative Q_k region, starting near Q_k ∼ -0.3. To interpret these results, we note that highly packed regions and large voids are actually quite inter-related. When a large void opens up in the packing, for any particle shape, it is always surrounded by a ring of tightly packed grains. As such, the persistence of a void region requires similar persistence of neighboring over-packed regions. § DISCUSSIONIn this article, we presented an experimental study into how a granular pillar, acting as a model disordered solid, deforms under uniaxial compression with varying particle shapes created from bonded groups of circular rods. We see that dimers constitute the strongest pillars, the additional strength originating from dimers that align and interlock perpendicular to the compression direction. The capability of horizontally oriented dimers to bear substantial loads can be seen in pillars in which the initial configuration contains dimers that are preferentially ordered horizontally. While dimers are clearly capable of interlocking as a form of inter-particle friction, it would be interesting to investigate elongated convex shapes, such as ellipses, to determine whether pillar strengthening, in addition to other results presented, can be reproduced regardless of convexity. For example, simulated systems with a similar geometry have seen increased material strength that results from increased contact area yielding additional sliding contacts <cit.>, so it would be informative to performexperimental tests. Furthermore, when convex shapes are used, perhaps rotational frustration will play a larger role in determining the response of the pillar.As is the case in other driven amorphous systems, the stress response of our granular pillars exhibits stress relaxations, or avalanches, over time. Furthermore, we see that particle shape does not affect the exponent of the power law distribution that tends to be representative of avalanche sizes within a wide range of amorphous systems. We do see that the more frictional shapes, dimers and trimers, allow for larger avalanches. While the local mechanisms for avalanches are seemingly unaffected, particle shapes do affect the local threshold stress that precedes local deformation, possibly by way of increased interparticle friction for our concave shapes.We also characterize local structure within the pillar using the previously defined structure metric Q_k <cit.> to highlight local packing anisotropies, which manifest as both voids and compacted regions. When Q_k is computed based on the positions of composite dimers or trimers, rather than component circular rods, we see that the Q_k metric retains its Gaussian-like characteristic. This indicates promise in the utility of Q_k as a randomly distributed measure of local free area in disordered packings of arbitrary particle shape and size. As previously stated, convex shapes would also prove to be a valuable test for Q_k, since voronoi tessellation would require curved facets and a more complex computation of local free area <cit.>.Finally, we measure local strain rates within the pillar and draw correlations with local structural anisotropies. For all shapes we note a general average trend that under-packed regions tend to be more likely to rearrange or undergo strain. Meanwhile, an over-packed region is less likely, on average, to deform, up to the limit where maximally packed grains must undergo dilatancy <cit.>. While this trend is generally true for all shapes, there is a clear deviation for highly under-packed dimers, which are not as likely to deform. This feature captures the observation that dimers can readily form voids that remain even for large strains. We expect that the aggregation of local rigidity can also play a key role in the global strengthening of the pillar. Using some of the techniques described in this article, in junction with the machine learning approach introduced in Ref. <cit.>, we are interested in pursuing studies that further connect local structural defects with particle-scale rearrangements of asymmetric particles. A probabilistic description of the likelihood of a region within a material to fail, in junction with structure functions that account for grain shape, may then illuminate strategies to prevent vulnerable local structures from forming in a wide range of disordered solids. While this study focuses mostly on particle trajectories and local structural features, further studies that include force measurements between grains would elucidate local stress capacities and add another consideration that influences whether a region is likely to deform locally. Furthermore, measuring forces between grains would allow for direct experimental comparisons to established models of ductile failure, with uniquely direct knowledge of the material microstructure. As mentioned in Section <ref>, ductile failure occurs due to local plastic flow and/or the growth and coalescence of voids <cit.>, both of which are present in the deformation of our pillars. Ductile failure can be modeled using the constitutive GTN model <cit.> and modifications thereof. However, these models are governed by a balance between locally applied stresses and yield stresses, both of which are currently inaccessible in the present study. Still, these prospective studies show promise for a more thorough understanding of material failure in disordered solids, and can illuminate methods for mitigating or avoiding catastrophic failure events. § ACKNOWLEDGMENTSWe thank J. M. Rieser for technical assistance. This work was supported by NSF grants MRSEC/DMR-1120901 and MRSEC/DMR-1720530. | http://arxiv.org/abs/1709.09511v2 | {
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Exploring Cascading Outages and Weather via Processing Historic Data Ian Dobson Nichelle'Le K. Carrington[-1mm] Kai Zhou Zhaoyu Wang ECpE Department, Iowa State University [-1mm] Ames Iowa USA [email protected] Benjamin A. CarrerasBACV Solutions Oak Ridge TN USA[-1mm] [email protected] José M. Reynolds-BarredoDepartamento de FísicaUniversidad Carlos III Madrid, [email protected] 30, 2023 ===================================================================================================================================================================================================================================================================================================================================================================================[c]preprint; to appear at Hawaii International Conference on System Sciences, January2018, Big Island,Hawaii[L] Preprint 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. [C] fancyWe describe some bulk statistics of historical initial line outages and the implications for forming contingency lists and understanding whichinitial outages are likely to lead to further cascading. We use historical outage datato estimate the effect of weather on cascadingvia cause codes and via NOAA storm data. Bad weather significantly increases outage rates and interacts with cascading effects, and should be accounted forin cascading models and simulations.We suggest howweather effects can be incorporated into the OPA cascading simulation and validated. There are very goodprospects for improving data processingand models for the bulk statistics ofhistorical outage data so that cascading can be better understood and quantified. IntroductionCascading failure can be defined as a sequence of dependent outages that successively weakens or degrades the power transmission system<cit.>. Although the power transmission system is carefully designed and operated to be robust to multiple outages, cascading outages that are large enough to cause load shedding and blackouts do occur. The large cascading blackouts that are of the greatest concern are infrequent, but likely enough to have substantial risk <cit.>.=-1 Cascading is the general way that transmission blackouts become widespread and there are many mechanisms that contribute to initial outages or the subsequent propagation of outages. There are a correspondingly large variety of models, approximations, simulations, andprocedures to assess and mitigate cascading outages <cit.>. One way to evaluate and improve these efforts is validation with observed historical data <cit.>. There is now much more systematic and automated collection of outage data by utilities, but the challenges of extracting and processing useful information from the data remain.In this paper, we report on some bulk statistical processing of 14years of transmission line outage data from a large North American utility to describe initial line outages and to start to explore the effect of weather on cascading. Our data-driven analysis of the effect of weather on the bulk statistics of cascading and aspects of our bulk statistical analysis of initial line outages are novel. Incorporating some of these effects in the OPA (Oakridge-Pserc-Alaska) cascading blackout simulation is also considered (see summary of OPA in section <ref>).Instead of working directly with data as in this paper, one can make simulation models that use or are tuned to typical parameter values. Several authors have taken this approach to propose models of weather effects in cascading simulations<cit.>. While historical data processing has many advantages, including no modeling assumptions and a very favorable grounding in reality, it should be noted that the grid evolves over 14 years, and that statistical analysis of historical cascades necessarily describes cascading risk averaged over the time period of observation.Historical outage data and its processingThe transmission line outage data consistsof 42 561 automatic and planned line outages recorded by a North American utility over a period of 14 years starting in January 1999 <cit.>. The data includes the outage start time (to the nearest minute), names of the buses at both ends of the line, and the dispatcher cause code. The automatic line outages are identified.All this data is standard and routinely collected by utilities.For example, this data is reported by North American utilities inNERC's Transmission Availability Data System (TADS) <cit.> and is also collectedin other countries. Having formed the network model fromthe automatic and planned line outages <cit.>, the analysis of cascading focuses ononly the automatic outages. There are 10 942 automatic outages in the data. The network model has 614 lines, 361 buses and is a connected network. The structure of cascading is that each cascade starts with initial outages in the first generationfollowed by further outages grouped into subsequent generationsuntil the cascade stops <cit.>. The first step in processing the line outages is to group the line outagesinto individual cascades, and then within each cascadeto group the outages that occur in close succession into generations. The grouping of the outages into cascades and generations within each cascade is done based on the outage start timesaccording to the methods of<cit.>. We summarize the procedurehere and refer to<cit.> for the details. The grouping is done by looking at the gaps in start time between successive outages.If successive outages have a gap of one hour or more, then the outage after the gap starts a new cascade. (That is, suppose o_1, o_2, ... are the outage start times in their order of occurrence. A gap of more than one hour is defined as a time interval between successive outage start times o_i, o_i+1 such that o_i+1-o_i≥ 1 hour. The time before the firstoutage start time o_1 and the time after the last outage start time are also considered to be gaps of more than hour. Let g_1, g_2, ... be all the gaps of more than one hour in their order of occurrence. Then cascade number k is defined to be all the outages that have start times between the gaps g_k and g_k+1. An alternative and equivalent definition is that a cascade is a maximal series of outages with successive outage start times for which the time difference between successive outage start timesin the series is less than one hour. A cascade may consist of one outage or many outages.)Within each cascade, if successive outages have a gap of more than one minute, then the outage after the gapstarts a new generation of the cascade. (That is, suppose o_k,1, o_k,2, ... are the outage start times in their order of occurrence for all the outages in cascade k . A gap of more than one minute is defined as a time interval between successive outage start times o_k,i, o_k,i+1 such that o_k,i+1-o_k,i> 1 minute. The time before the firstoutage start time o_k,1 in the cascade k and the time after the last outage start timeincascade k are also considered to be gaps of more than minute. Let g_k,1, g_k,2, ... be all the gaps of more than one minute in cascade k in their order of occurrence. Then generation number ℓ of cascade number k is defined to be all the outages in cascade k that have start times between the gaps g_k,ℓ and g_k,ℓ+1.) Since the outage times are only knownto the nearest minute, theorder of outages within a generation often cannot be determined.This simple method of defining cascades and generations of outages appears to effectiveand has gap thresholds consistentwith power system time scales sinceoperator actions are usually completed within one hour andfast transients and protection actions such asauto-reclosingare completed within one minute. <cit.> examines the robustness of cascade propagation with respect to varying these gap thresholds.This data processing applied to the 10 942 automatic outages yields 6687 cascades. Most of the cascades are short: 84% of the cascades have only the first generation of outages and do not spread beyond these initial outages. It is important for a fair statistical analysis to include the short cascades (even if they are for other purposes not thought of as cascades); the short cascades usually represent a successful case of resilience in which no load is shed. That is, excluding the short cascades would misleadingly bias the results towards themore damaging cascades that do not stop quickly. The grouping of outages in each cascade into generations allows the initial outages in the first generation to be distinguished from the subsequently cascading outages in the following generations. This is of interest because the mechanisms and mitigations of the initial line outages differ significantly from the interactions between line outages that are involved in the subsequent cascade. Most of the initial line outages are single outages, but there are also multiple initial outages. In other words, there are single, double, triple, etc. contingencies. The probability distribution of the number of initial outages is shown by the black dots in Figure <ref>. The distribution of the initial outages is one way of looking at the severity of initial events: 12% of the initial events have more than one outage (the probability of one initial outage is 88%), 1.5% of the initial events have more than 3 outages, and 0.2% havemore than 5 outages.Cascading increases the probabilities of multiple line outages. The distribution of the total number of outagesafter cascading is shown by the red squares in Figure <ref>. 26% of the total number of outages have more than one outage, 6.6% of the total number of outages have more than 3 outages, 2.7% have more than 5outages, and 0.7% have 10 or more outages.The effect of cascading is progressively larger for the cascades with more outages. For example, while cascading approximately doubles the probability of more than one outage, cascading increases the probability of more than 5 outages by an order of magnitude. The generations of outages in the cascades are analogous to human generations; parents in one generation give rise to children in the next generation. The average propagation per parent λ is the total number of child outages in all the generations divided by the total number of parent outages in all the generations. λ calculated from our data is 0.28. That is, each outage in a generation will, on average, be followed by 0.28 outages in the next generation.λ quantifies in an overall way how much cascading increases the number of line outages starting from the initial line outages. § STATISTICS OF INITIAL OUTAGES We examine the basic statistics of the automatic initial line outages (those outages in the first generation of cascading). The annual outage frequencies μ_1, μ_2, μ_3, …, μ_614 for the 614 lines range from zero outagesto 23 outages per year, with a mean annual frequency μ= 0.92 outages per year. The large variation in initial line outage frequency μ_i in this data has several implications. It is clear that cascading simulations that aim to quantify cascading risk should sample from realistic initial line outage frequencies. One way to accomplish this is to simulate a real power system and use the observed historical frequency of line outages.Another way to accomplish this is to understand and model the factors or characteristics that largely determine the frequency of initial line outages so that they can be represented in artificial power system models. For example, it might be expected that outage frequency has some dependence on line length and other characteristics. (Our data suggests a mild correlation of 0.3 between outage frequency and line length for lines between 1 and 50 miles long.)Another implication is that it may be difficult to classify the probability of higher order initial outages (when assumed roughly independent) by the number of outages using the order of magnitude of the probabilities <cit.> because the single outages vary so much in their probability. The data in Figure <ref> shows a substantial probability of multiple initial outages. The empirical probability of two initial outages is 0.084 with standard deviation 0.0034. To determine whether this can arise from independent single line outages, we suppose that each of the 614 lines has initial outages according to a Poisson process of rate 368/614=0.60 outages per year and that the Poisson processes for different lines are independent. Then the outages of any of the lines is a Poisson process of rate 368 outages per year, which matches the rate extracted from data in the next paragraph. Multiple initial line outages in the data require at least 2 line outages to occur at times that are either in the same or adjacent minutes when the times are quantized to minutes. Given the first outage time, this requires the second outage to occur within a 3 minute interval. (For example, if the quantization works by quantizing the time t in minutes to ⌊ t ⌋, the greatest integer number of minutes less than t, then t_1 and t_2 are in the same or adjacent quantized minutes if and only if ⌊ t_1 ⌋-1<t_2≤⌊ t_1 ⌋+2.) The probability that the second outage occurs within a given 3 minute interval is 1-exp[-(3× 368)/525600]=0.002. Therefore the probability of multiple initial outages and in particular the probability of 2 initial outages are both bounded above by 0.002.Since the empirical probability of 2 initial outages is 0.084, an order of magnitude greater than 0.002, the double initial outages are dependent and cannot be regarded as mainly arising from independent single line outages. It also follows that multiple initial line outages are dependent. A similar claim of dependence for all outages (not just the initial outages)is established in previous work <cit.>.The previous paragraph assumes that initial outages are a Poisson process of rate 368 outages per year.The assumption that initial outages are a Poisson process is supported by examining the distribution of the logarithm of time differences between successive outage times in Figure <ref>. For an exact Poisson process, the time differences follow an exponential distribution with the same rate as the Poisson process, so that the logarithm of the survival function of the time differences is a linear function with the slope of the line determining the rate. The time differences of both the initial outages and all the outages are approximately linear except for the smaller time differences of order one hour or less. (The increased, superlinear number of smaller time differences in all the outages may be attributed to cascading. There are no time differences between initial outages more than one minute and less than one hour because of the data processing that defines the start of new cascades. The initial and all outages also show a fraction of outages that have a time difference less than one minute or zero.) The dotted line in Figure <ref> approximates the slope of the time differences except for the smaller time differences of order one hour or less and has a slope corresponding to368 outages per year.The Poisson process model for initial outages is similar to the Poisson process model of blackout start times analyzed in <cit.>, except that here we do not consider any adjustment to the Poisson process to account for a slowly varying rate.The multiple initial outages have significant spatial dependence. Consider the classification ofinitial double outages in Table <ref>.The double outages are either two adjacent lines (lines with exactly one bus in common), two parallel lines (two buses in common), two separated lines (no common buses), or are repeat outages of the same line. More than half (55%) of the double outages are adjacent lines and only 16% of the double outages are two lines that are separated in the network. In contrast, randomly sampling double line outages by choosing each of the double lines randomly proportional to their outage frequency yields only 2%that are two adjacent linesand 97%that are two lines that are separated in the network. Combining the adjacent, parallel, and repeated outages shows that 84% of the initial double line outages form connected subgraphs.Outages can be caused by line, bus or breaker faults. Line faults are isolated by the breakers at each end of the line so that they usually cause single line outages, while the bus or breaker faults can cause multiple outages because of the substation protection system design.Although we do not know any specifics of the substation designs, one likely cause of the high proportion of adjacent double line initial outages is substation breaker schemes that disconnect two lines for certain faults. 2% of the initial outages are triple outages and 75% of these initial triple outages are connected subgraphs. 4% of the initial outages have 3 or more outages and 68% of these initial multiple outages are connected subgraphs. Spatially close components that are assumed to always outage together for cascading failure analysis are called protection control groups <cit.> or functional groups <cit.>. While there clearly would be some overlap, we do not yet know how exactly the connected subgraphs that form the majority of these initial outages are related to the functional groups that can be applied to approximate the protection system actions. The historical data samples from functional groups, but also samples from rarer or more unusual conditions.It can also be helpful for risk analysis to find out whether different types of multiple initial outages can cause subsequent outages. Our data shows that separated initial line outages are more likely to trigger subsequent cascading outages: 25% of separated initial line outages have subsequent outages, while only 16% of connected initial line outages have subsequent outages. The Mann-Whitney test shows this difference is significant at the 0.01 level (p-value is 0.00355). For momentary initial outages (duration less than one minute) versusnon-momentary initial outages, our data shows little difference in the proportion of subsequent cascading outages: 18% of momentary outages have subsequent outages, and 15% of non-momentary outages have subsequent outages. The Mann-Whitney test shows that this difference is not significant at the 0.01 level (p-value is 0.0197).This suggests that momentary and non-momentary initial outages be treated equally in assessing the risk of further cascading. The distinctionin the processed line outage data between initial outages and subsequently cascading outages allows us to find out and compare which lines are most involved in these two different processes. The top 10 lines involved in initial outages overlap, but do not coincide with the top 10 lines involved in subsequent cascading; there are 6 lines in common but there are 4 lines in each list that differ. Similarly, the top 20 lines involved in initial outages have 10 lines in common with the top 20 lines involved in subsequent cascading and 10 lines in each list that differ. Similar results were obtained by processing line outage data from a neighboring utility <cit.>. It should be noted that statistically prominent outage problems in an initial portion of the historical data may have been already mitigated.§ EFFECT OF WEATHER AND OTHER INFLUENCES VIA CAUSE CODESThe dispatcher outage cause codes allow classification of the cascades of outages into two groups: weather related and non-weather related. (For definiteness when the field and dispatcher causes differ, we do not consider the field cause codes in this analysis.) A cascade of outages is definedas weather related when at least one of the outages in the cascade has one ofthe cause codes “Weather", “Lightning", “Galloping Conductors", “Ice", “Wind",or “Tree blown".(#1,#2)height #1pt depth #2pt width 0pt How the annual cascade rate, average propagation of cascading outages, and cascade size distribution depend on weather are shown inTable <ref> and Figure <ref>. According to Table <ref>,only 21%(101/478) of the cascades are weather related.This implies that only 21% of the initial outages occurred in a weather-related cascade. And Figure <ref> shows that the distribution of initial outages is similar for weather and non-weather related cascades. However,in Table <ref>weather related outages have greatly increased propagation from 0.13 (non-weather related) to 0.55(weather related) and in Figure <ref> thereis a corresponding large difference in the distribution of the total outages in a cascade after the cascading. That is, a minority of cascades are weather related, but they propagate far more to form larger cascades.With the method of processing outages into cascades that we use <cit.>, propagation can arise both from outages causing further outages through interactions in the network (encompassing electrical physics, control systems, and human actions) and through independent outages occurring during the cascade that are similar in mechanism to the initial outages. Note that the processing method defining the subsequentcascading studiously avoids determining the causes or explicit dependences of further outages and simply accounts for outages that occur within one hour of previous outages <cit.>. Indeed <cit.> states that “One caution is that it is unknown to what extent exogenous forcing from weather is augmented by additional dependent cascading effects." Reference <cit.> analyzes all the outages together, determines the average rate of independent outages,and proceeds to quantify the contribution of statistically independent outages towards the λ measure of average propagation, concluding that 4-6% of outages are independent and classified as cascading outages. This seems an acceptable error in the contention that the subsequently cascading outages are dependent outages. The contention essentially relies on the independent outages occurring at a slow enough rate relative to the typical cascade duration.However, the same contention for the subset of weather related outages need not have an acceptable error becausethere is a much higher rate of independent outages during bad weather. The methods ofSection <ref> are not conclusive in this regard, but the results of Section <ref> areconsistent with this conclusion. More importantly, traditional risk analysis does supporta much higher rate of independent outages during bad weather <cit.>. This raises a question of the validity of the method of cascading processing applied to weather related outages when the cascades and propagation areinterpreted as dependent outages occurring through network interactions. However, if the concern is simply the number of subsequent outages during a one hour period without regard to cause, then the method could retain some validity for weather related outages. This is the case, for example, when the concern is the total number of outages, regardless of cause, that the operators have to deal with within a one hour period.The month and time of day can also be used to classify cascades into the summer peaking months ofJune, July,August, September, and the remainder of the year, or those cascades that start during the peak hours between 3 pm and 8 pm and cascades starting outside these peak hours. Table <ref> and Figures <ref> and <ref> show the effects of the summer months and thepeak hours. (The equivalent annual rate shown in Table <ref> is the rate if the condition such as summer months applied all year.) Outages in the summer months of June, July, August, September havea modestly increased propagation from 0.25 (not summer) to 0.31 (summer).Outages in the peak hours between 3 pm and 8 pm have increased propagation from 0.25 (not peak hours) to 0.36 (peak hours).Note that the cascades also depend on the initial outages.Indeed, the data in the summer months shows a 39% higher cascade rate and a 41% higher rate of initial outages.Overall,there is a moderate increase in cascade propagation during peak hours and only a small increase in propagation, but an increased rate of initialoutages in the summer months. However weather effects are larger than either of these factors. Effect of weather via NOAA storm data =-1 One problem with analyzing the effect of weather with outage cause codes is thatcause codes cannot describe the weather when there is no outage. Therefore the line outage rate during bad weather, a key quantity, cannot be estimated from cause code analysis.Also, the outage cause codes are manually entered, rely on subjective judgment aboutclassifying causes,and in any case include a sizable proportion (22% of the dispatcher outage cause codes) of causes “Unknown".One way to address these problems with a different bad weather criterion is to coordinate in time and space the outage data with storm weather records.The National Oceanic and Atmospheric Administration (NOAA) Storm Events Database is a collection of the occurrence of storm events and other significant weather phenomena recorded by NOAA's National Weather Service from 1950 to present <cit.>.The NOAA historical storm data records for 1999 to 2013 were obtained for analyzing the storm weather effects influencing our outage data. The NOAA storm data includes the event type, event start and end time, and the location within the state by county or zone. The storm event types that we choose to define as a stormare “Blizzard", “Freezing Fog", “Hail", “Heavy Rain", “Heavy Snow",“High Wind", “Ice Storm", “Lightning", “Sleet", “Strong Wind",“Thunderstorm Wind", “Tornado", “Winter Storm", and “Winter Weather".To associate the line outages with the storm data, we map the buses onto the county they are located in, and describe each zone by the main counties it intersects. A line is defined to be in a county if either its sending or receiving end bus is in that county. A line is defined to be in a zone if that zone includes a county that the line is in.This associates each line with a set of counties. In some cases this set of counties only contains one county.A line outage is then classified as a storm outage if it occurs during a storm event in one of the counties in the set of counties associated with that line. It is straightforward to count the number of storm outages of line number k over the period of observation. Also, the total storm durations for a county is the sum of the durations of the storms in that county during the period of observation, and the total storm time for line k over the period of observation is computed as the average over the counties of thetotal storm durations for thecounties that the line is in. Then the line k storm outage rate R^ storm_ line k= (number of storm outages of line k)/(total storm time for line k). Finally, the average storm outage rate R^ storm= (number of lines)^-1∑_k R^ storm_ line k. The non-storm line outage rate andtheaverage non-storm line outage rateR^ nostorm are computed similarly.This data processing approximates the average non-storm outage rate as R^ nostorm=1.1 outages per yearand the average storm outage rate asR^ storm=8.1 outages per year. This significant increase in the outage rate during storm weather has important implications for processing historical data and simulating cascading. First of all, models and simulations of cascading should distinguish and separately consider storm weather and non-storm weather periods. This conclusion based on cascading historical data is not surprising given the attention to this distinction in conventional power system reliability <cit.> and in <cit.>. Secondly, it is also clear from conventional power system reliability that the initial outage rateis higher during bad weather <cit.>. The high outage rate during storms could be primarilydue to increases in the initial outage rate alone or to increases in both the initial outage rate and the propagation. The distinction matters to mitigation of cascading because the initial outage mechanisms differ from the mechanisms for the propagation of outages through the network. Thirdly, as already discussed in Section <ref>, the significant increase of the average storm line outage rate will also impact the processing of cascading outages into generations. It seems that better high-level models that not only distinguish weather and non-weather conditions but also capture and distinguish the rates of independent and dependent outages are needed. The effect of the storm weather on the distributions of initial and total number of cascaded outagesis shown in Figure <ref>. In comparing Figure <ref> to Figure <ref>, it should be noted that bad weather or its severity is differently defined by the weather cause codes and the storms in NOAA data.Cause and effect in cascading analysis While an attribution of cause for outages is attractive since it often gives possibilities for mitigation, it should be recognized that the whole notion of detailed cause (and especially single cause)for cascading outages can be murky and ill-defined. Causes for initial outages are less problematic, and oftena single cause or multiple causes can be defined. On the other hand, outages propagating via power system interactionsafterthe initial outages generally depend on the initial system condition, the initial outages, andthe preceding outages <cit.>.To suggest an overall methodological context, we can consider two approaches to cascading: A “bottom-up" approachspecifies a particular cause and effect mechanism of cascading, makes adetailed model of that mechanism and then tries to get data for that mechanism. A “top-down" approach examines available data, at first without regard to detailed cause or mechanism, and then tries to divide the data into classes of causes or mechanisms. This paper is top-down and weather is one simple example of a class of mechanisms. Thebottom-up and top-down approaches are complementary. The bottom-up approach enables understanding and often insight into mitigation of that mechanism, but there are dozens of different mechanisms of cascading, and many of themore unusual and complicated events that often occur in blackoutsare hard to approach in this way, and very often there is no data availableto find the model parameters. The top-down approach already has the data, and can give a useful overall statistical description and quantification of cascading,but gives much less detailed insight and as a consequence work towards mitigation is much less direct. However, the operators will have to deal with all the outages in real timeregardless of whether there are detailed cause-effectrelationships establishedor not. We hope that bottom-up and top-down approaches will gradually converge towards each other as the field progresses to better realize the full range of possibilities of data-driven reliability analysis.Modeling weather effects in OPA The OPA model <cit.> is a simulation that calculates the long-term risk of cascading blackouts of a power transmission system under the slow, complex dynamics of an increasing power demand and the engineering responses to blackouts. The individual cascades are modeled by probabilistic line overloads and outages in a DC load flow model with linear programming generation redispatch.We previously validated OPA on a 1553 bus model ofthe Western North American interconnection against observed data with some success <cit.>.Here we start to explore parameterizing the weather effects in OPA on the 1553 bus model by modifying OPA andshowing the fit with observed data. For details of OPA, parameters,and the 1553 bus model we refer to<cit.>. Inspired by the historical data in the previous sections, weintroducenew OPA parameters p_w, p_5 and some spatial correlation between multiple initial outages. p_w is the probability that in a given day the weather is the cause of outages. p_5 isthe probability that an outage produced by weather will happen in a given iteration of the cascadeon the days that weather is a factor. The spatial correlation of the multiple initial outages is introduced by first using OPA parameter p_0 to determine some initial line outages. Once initial line outages have been calculated, we go through the adjacent lines and probabilistically determine their failure. Then this process is repeated once.The result is that the initial random line outages are sometimes augmented with one or more adjacent lines.To reexamine the previous fit with the observed data in <cit.> with the new parameters, we take p_w=0.25, which is close to the 21% proportion of weather cascadesestimated in section <ref>. We are not yet able to estimate p_5 from data, but p_5 = 0.0002 gives a goodmatch of the cascades with the observed distribution of the number of outages in weather and non-weather cascades as shown in Figure <ref>. (For p_5 <0.0002 the weather driven cascades tend to be too short, and for p_5 >0.0002 the slope of the weather PDF from the OPA results tends to decrease relative to the slope of the PDF of the data.)With these parameters, in Figure <ref>there is a good match between OPA and the historical data for the distribution of load shed, and inFigure <ref> there is a match for the propagation λ in each generation of cascading that improves upon the match in <cit.>.These results suggest that weather effects can be included in OPA and validated against the observed data. Future work should find a way to estimate p_5 from the historical data instead of calibrating p_5 with the distribution of the number of outages in weather cascades.Conclusion In this paper we start to explore processing historical outage data to characterizeinitial outages and subsequent cascading propagation and determine the effects of weather on cascading. Although only one 14 year data set from a large North American utility is analyzed and our specific conclusions are of course limited to that data set,similar data is routinely collected by many utilities worldwide, so that it is straightforward, given access to the data, to apply the data processing methods broadly.A simple processing method based on outage timing allows us to distinguish the initial outages from the subsequent cascading. Most of these initial outages are single outages that do not have following outages. However, the data also shows significant numbers of multiple initial outages and initial outages that cascade further. The initial outages have considerable variation in outage frequency, are dependent, and the multiple outages tend to occur in adjacent lines.However the separated initial multiple outages have more of a tendency to cascade further. Momentary outages appear to cascade further at a similar rate as longer outages. As might be expected from the differing mechanisms involved in initial and propagating outages, the lines most involved in initial outages have some overlap with, but do not coincide with those most involved with subsequent cascading. The bulk statistics of historical initial outages can inform the contingency lists for risk-based or deterministic cascading studies.The effects of weather on historical cascading outage data are studied by means of the weather-related dispatcher cause codes in outage dataand NOAA storm data.A minority of cascades are weather-related, but using the processing methods of the paper, show a significantly increased propagation from the initial outages and a significantly greater outage rate. This suggests that, following traditional power systems risk analysis, cascading models and analysis will need to somehow define and acknowledge bad weather and good weather regimes. An increased outage rate during bad weather is confirmed by traditional power system risk analysis, but its interaction with cascading propagation remains unclear.In particular, the increased outage rate does not allow the increased propagation to be mostly attributed to propagation of cascading via network effectsbecause of limitations of the processing method. New bulk cascading models and data-processing methods are needed for bad weather conditions. Peak hours and peak months of operation show less impact on cascading propagation than bad weather, but there is a higher rate of cascades during these peak conditions.Historical outage data is very valuable for validating and calibrating simulations of cascading outages.The OPA model of long-term cascading risk is one of the few simulations with some validation with bulk statistics of historical data <cit.>. We have started to represent weather effects in OPA and extend the validation to this case.Our bulk statistical data processing methods for historical outage data and NOAA data are initial approaches that are subject to future improvements. However, our results already show the value of this processing for understanding and quantifying key factors in initial outages and subsequent cascading, and the prospects for improved methods and further insights are very good.Acknowledgements:We gratefully thank Bonneville Power Administration andthe National Oceanic and Atmospheric Administrationformaking publicly available the outage and weather datathat made this paper possible. The analysis and any conclusions are strictly those of the authors and not of BPA or NOAA. We gratefully acknowledge support in part from NSF grant 1609080. 99TaskForcePESGM08 IEEE PES CAMS Task Force on Cascading Failure, Initial review ofmethods for cascading failure analysis in electric power transmission systems, IEEE PES General Meeting, Pittsburgh PA USA, July 2008.DobsonCH07 I. Dobson, B.A. Carreras, V.E. Lynch,D.E. Newman, Complex systems analysis of series of blackouts: cascading failure, critical points, and self-organization, Chaos, vol. 17, no. 2, 026103 June 2007.HinesEP09 P. Hines, J. Apt, S. Talukdar, Large blackouts in North America: historical trends and policy implications,Energy Policy, vol. 37, 2009, pp. 5249-5259.NewmanREL11 D.E. Newman, B.A. Carreras, V.E. Lynch, I. Dobson,Exploring complex systems aspects of blackout risk and mitigation,IEEE Trans. Reliability, vol. 60, no. 1, March 2011, pp. 134-143. CarrerasPS16 B.A. Carreras, D.E. Newman,I. Dobson, North American blackout time series statistics and implications for blackout risk, IEEE Trans. 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Slone, Top risks to transmission outages, IEEE PES General Meeting, National Harbor, MDUSA, July 2014.DobsonPS16 I. Dobson, B.A. Carreras, D.E. Newman, J.M. Reynolds-Barredo, Obtaining statistics of cascading line outages spreading in an electric transmission network from standard utility data, IEEE Trans. Power Systems, vol. 31, no. 6, November 2016, pp. 4831-4841.DobsonEPES17 I. Dobson, D.E. Newman, Cascading blackout overall structure and some implications for sampling and mitigation, International Journal of Electrical Power Energy Systems, vol. 86, pp. 29-32, 2017.RenCAS08 H. Ren, I. Dobson, Using transmission line outage data to estimate cascading failure propagation in an electric power system, IEEE Trans. Circuits and Systems Part II, 2008, vol. 55, no. 9, pp. 927-931. DobsonPS12 I. Dobson, Estimating the propagation and extent of cascading line outages from utility data with a branching process, IEEE Trans. Power Systems, vol. 27, no. 4, November 2012, pp. 2146-2155. 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Dobson, B.A. Carreras, V. Lynch, D.E. Newman, An initial model for complex dynamics in electric power system blackouts, 34th Hawaii International Conference on System Sciences (HICSS), Maui, Hawaii, January 2001.CarrerasCH04 B.A. Carreras, V.E. Lynch, I. Dobson, D.E. Newman, Complex dynamics of blackouts in power transmission systems, Chaos, vol. 14, no. 3, Sept. 2004, pp. 643-652.RenPS08 H. Ren, I. Dobson, B.A. Carreras, Long-term effect of the n-1 criterion on cascading line outages in an evolving power transmission grid,IEEE Trans. Power Systems, vol. 23, no. 3, Aug. 2008, pp. 1217-1225. | http://arxiv.org/abs/1709.09079v1 | {
"authors": [
"Ian Dobson",
"NichelleLe K. Carrington",
"Kai Zhou",
"Zhaoyu Wang",
"Benjamin A. Carreras",
"Jose M. Reynolds-Barredo"
],
"categories": [
"physics.soc-ph",
"cs.SY",
"physics.data-an"
],
"primary_category": "physics.soc-ph",
"published": "20170926150838",
"title": "Exploring cascading outages and weather via processing historic data"
} |
D. Paul]Debdutta [email protected] Tata Institute of Fundamental Research, IndiaModelling the luminosity function of long Gamma Ray Bursts using and [ December 30, 2023 ======================================================================== AstroSat E_p T_90 χ_red^2 I have used a sample of long Gamma Ray Bursts (GRBs) common to both and to re-derive the parameters of the Yonetoku correlation. This allowed me to self-consistently estimate pseudo redshifts of all the bursts with unknown redshifts. This is the first time such a large sample of GRBs from these two instruments are used, both individually and in conjunction, to model the long GRB luminosity function. The GRB formation rate is modelled as the product of the cosmic star formation rate and a GRB formation efficiency for a given stellar mass. An exponential cut-off powerlaw luminosity function fits the data reasonably well, with ν = 0.6 and L_b = 5.4 ×10^52 erg.s^-1, and does not require a cosmological evolution. In the case of a broken powerlaw, it is required to incorporate a sharp evolution of the break given by L_b∼0.3×10^52(1+z)^2.90 erg.s^-1, and the GRB formation efficiency (degenerate up to a beaming factor of GRBs) decreases with redshift as ∝(1+z)^-0.80. However it is not possible to distinguish between the two models. The derived models are then used as templates to predict the distribution of GRBs detectable by CZTI on board , as a function of redshift and luminosity. This demonstrates that via a quick localization and redshift measurement of even a few CZTI GRBs,will help in improving the statistics of GRBs both typical and peculiar.gamma ray burst: general – astronomical databases: miscellaneous – methods: statistical – cosmology: miscellaneous.§ INTRODUCTION For any detector of gamma ray bursts (GRBs), an interesting estimable quantity is the number of GRBs observed, as a function of measurable parameters. This depends on instrumental parameters like duration-of-operation, T and field-of-view ΔΩ, as well as the observed GRB production-rate and the distribution over intrinsic properties of GRBs. Following the formalism outlined in <cit.>, let us assume that the rate of GRBs beamed towards an observer on earth from an infinitesimal co-moving volume dV, is given by .R(z)dV/1+z, where z denotes the redshift, and the factor (1+z)^-1 takes care of the cosmological time dilation.Most generally, the number of GRBs detected by the instrument in the luminosity (L) range L_1 to L_2 and redshift range z_1 to z_2 is given by, N(L_1,L_2;z_1,z_2)=T ΔΩ4π _max[L_1,L_c]^L_2Φ_z(L)dL _z_1^z_2.R(z)1+zdV, where L_c denotes a lower-cutoff in the intrinsic luminosity of GRBs (see Section <ref>). The function Φ_z(L) is formally called the `luminosity function' (henceforth LF), having the units of ( erg.s^-1 )^-1, the subscript refering to an implicit dependence on the redshift. In view of the fact that GRBs are end-products of massive stars in galaxies, the GRB formation rate .R(z) can be written as.R(z)=f_BC .ρ_⋆, where .ρ_⋆ gives the cosmic star-formation rate (CSFR) in M_⊙Gpc^-3yr^-1, C gives the efficiency of GRB production given a certain stellar mass, in units of M_⊙^-1, and f_B encodes the beaming effect of the relativistic jets producing the burst. All of these terms may be functions of the redshift.The dependence of the detected number distribution of a certain class of astrophysical object on its luminosity function, is quite general. It has been extensively studied in the context of galaxies, galaxy clusters (see <cit.> and references therein), white dwarfs (see <cit.> for a recent review), quasars (see <cit.> and references therein), high mass Xray binaries (see <cit.> and references therein) etc. The methodologies depend on the observational window available for the study of the particular objects of interest (e.g. while <cit.>, <cit.>, <cit.> etc. use the infrared bands to calculate the absolute magnitude of galaxies, <cit.> use the optical B-band, and <cit.>, <cit.>, etc. use the UV bands), but the central theme is similar for all of the objects – to measure the intrinsic properties of the sources and statistically study their cosmological evolution. Moreover, the LF of the various objects are related to each other, making this a difficult quantity to measure. For example, the cosmic star-formation rate (CSFR) derived from the galaxy LF, and the GRB LF, are related via Equations <ref> and <ref>. This will be discussed in more details below.The measurement of the redshift (hence distance) of a GRB is essential for measuring its intrinsic luminosity. In the era of the Burst and Transient Source Experiment (BATSE) on board the Compton Gamma Ray Observatory (CGRO), which detected around 2700 GRBs in a span of 9 years (approximately one GRB per day, see https://heasarc.gsfc.nasa.gov/docs/cgro/batse/https://heasarc.gsfc.nasa.gov/docs/cgro/batse/), the measurement of redshift of a detected GRB depended on coincident detection by other instruments with greater localization capabilities. In 1997, the Italian-Dutch satellite BeppoSAX provided the redshift of a burst for the first time via afterglow observations, that of GRB970508 (see <cit.>, <cit.>, <cit.> and references therein). However, the number of GRBs with redshifts measured by BeppoSAX remained only a handful over the years <cit.>. The situation changed entirely with the advent of the Burst Alert Telescope (BAT) on board <cit.>, launched in 2004. In addition to detecting roughly 1 GRB every 3 days, it has fast on board algorithms to localize the burst and follow it up swiftly with other on-board instruments, the X-Ray Telescope (XRT) and UltraViolet/Optical Telescope (UVOT), as well as other ground-based missions. This provides redshifts via emission lines, absorption lines and photometry of the host-galaxies and/or afterglow, for roughly 1/3^ rd of the GRBs, making it possible to study the intrinsic properties of ∼300 GRBs till date (https://swift.gsfc.nasa.gov/archive/grb_table/https://swift.gsfc.nasa.gov/archive/grb_table/).<cit.> used the measured redshift and spectral parameters of 12 BeppoSax GRBs from <cit.> and an additional 11 GRBs detected by BATSE to derive the `Yonetoku correlation' between the 1-sec peak luminosity and the spectral energy break in the source frame. Using this correlation, they estimated the `pseudo redshift' of 689 BATSE long GRBs with unknown redshifts. Subsequently, they discussed the GRB formation rate and found that a constant LF does not fit the data.<cit.> studied the logN-logP diagram of BATSE GRBs and the peak-energy distribution of bright BATSE and HETE-2 GRBs, as well as carried out extensive simulations for GRBs and applied them to early data to predict that long GRBs show significant cosmological evolution. <cit.> and <cit.> investigated the peak-flux distribution of BATSE GRBs in different scenarios regarding the CSFR, the evolution of the GRB LF, and the metallicity of the GRB formation environments. They then compared the predicted peak-flux distribution of GRBs primarily with z>2 with available data to conclude that the two satellites observe the same distribution of GRBs, the GRB LF shows significant cosmological evolution, and that the GRB formation is limited to low metallicity environments.Since then, GRBs with measured redshifts have been studied extensively to model the long GRB LF. To do this, <cit.> directly inverted the observed luminosity-redshift relationship. <cit.> carried out a phenomenological study of the observational biases on doing this, concluding that a broken powerlaw model of the long GRB LF is preferred, with pre and post break luminosity of 2.5×10^52erg.s^-1 given by 1.72 and 1.98 respectively. They also point to the requirement of cosmological evolution of the LF at high metallicity environments. <cit.> used a flux-complete sample of 58 GRBs, with a redshift completeness of 90%, to conclude that the broken powerlaw model is degenerate with the exponential cut-off powerlaw model. They also conclude that the GRB LF evolves with redshift, claiming that this conclusion is independent of the used model. <cit.> however used a sample of 112 GRBs to disfavour strong cosmological evolution of the formation rates of GRBs at z<4, and concluded that the best-fit trend of the evolution strongly over-predicts the CSFR at z>4 when compared to UV-selected galaxies, alluding to unclear effects in addition to metallicity and the GRB formation environment. <cit.> used two new observation-time relations and accounted for the complex triggering algorithm of -BAT to reduce the degeneracy between the CSFR and the GRB LF. They satisfactorily fit a non-evolving LF with a powerlaw broken at 0.80±0.40×10^52erg.s^-1 by pre and post indices of 0.95±0.09 and 2.59±0.93 respectively, while not entirely ruling out the possibility of an evolution in the break luminosity. <cit.> used a sample of 200 redshift measured GRBs to carry out a non-parametric determination of the quantities related to the CSFR and the GRB LF. They claimed that the LF evolves strongly with z, satisfactorily fit to a broken powerlaw model with pre and post break indices 1.5 and 3.2 respectively. They also estimated a GRB formation rate an order of magnitude higher than that expected from CSFR at redshifts z<1, but matching with the CSFR at higher redshifts, contrary to all previous studies. On the other hand, <cit.> carried out an extensive study of the observational biases on the flux-truncation, trigger probability, redshift measurement, etc. with 258 GRBs, concluding that it is not possible to argue for a robust cosmological evolution of the long GRB LF. The major limitations in the study of the GRB LF with GRBs is the narrow energy band of BAT, which does not allow an accurate determination of the spectral parameters of the GRBs, since most of the bursts have spectral cutoffs at energies greater than the BAT high-energy threshold of 150 keV. The conclusions of several of these studies are moreover in contradiction to each other. Regardless, several authors have discussed the implications of these results in the context of the structure of the GRB jets, for both BATSE (<cit.>, <cit.>, <cit.>) and GRBs <cit.>. The redshift distribution of bursts emerging from the study of the LFs, assuming different metallicity environments of GRBs, has been discussed by <cit.>.The two major limitations of studies that use GRBs with measured redshifts to constrain the GRB LF are: (1) the number of such available sources is rather small to tightly constrain the LF or statistically answer questions related to its evolution with redshift, leading to a variety of conclusions; (2) the measurement of redshifts always suffers from observational biases. To overcome these limitations, <cit.> used 220 BATSE GRBs with redshifts inferred from an empirical luminosity-variability relation <cit.>. This was extended by <cit.> who carried out a joint fit of these GRBs along with the observed peak-flux distribution of more than 3300 Ulysses/BATSE GRBs. The conclusions always favoured a cosmological evolution of the GRB LF, although the data was not able to distinguish between single powerlaw and double-powerlaw models. <cit.> proposed a multivariate log-normal distribution which he fitted for 2130 BATSE GRBs. <cit.> on the other hand used an empirical lag-luminosity correlation to constrain the GRB LF and the CSFR independently from the study of 900 GRBs, favouring a single powerlaw fit to the GRB LF. Incidentally, similar methods have also been applied for galaxies to study the galaxy LF (see <cit.> and references therein).<cit.> uses the Yonetoku correlation to estimate the pseudo redshifts of 498 GRBs. This avoids the observational bias of the redshift measurements, and the flux truncation is corrected for during the modelling. First they test the correlation parameters by comparing the redshift distribution of 172 GRB whose redshifts are known. They find that the best-fit parameters do not predict the redshift distribution of this sub-sample well. So they choose the values for which the distributions of known and pseudo redshifts of these GRBs are statistically similar. Since the bandpass is too narrow to determine the spectral parameters of GRBs, they use the <cit.> catalog in which the Band function <cit.> parameters are estimated with a Bayesian technique. They conclude that the GRB LF is inconsistent with a simple powerlaw, demanding a fit with a broken powerlaw with pre and post break indices given by 0.8 and 2.0 respectively. In addition, the break itself evolves cosmologically as L_b=1.2×10^51erg.s^-1(1+z)^2, and the GRB formation rate evolves as ∝(1+z)^-1 in contradiction to all previous studies. They do not look into the accuracy of pseudo redshifts of the GRBs individually, and the analysis is entirely based only on a statistical comparison of the redshift distributions.In the present work, I carry out a detailed study of the estimation of pseudo redshifts, using long GRBs that have firm redshift measures from , as well as spectral parameter measurements from . The reason such a sample is useful is because it combines the wide spectral coverage of (which however does not provide redshift) with the redshift measurements from follow-ups. This reduces the errors on the correlated quantities compared to the Butler catalog, which allows me to test the correlation itself, and also carefully examine the accuracy of the pseudo redshifts estimated from the correlation. I then use it to estimate the pseudo redshiftof all and GRBs, and place constraints on the long LF from a combined study of all these 2067 GRBs. Previously, <cit.> has used a combined sample of 127 long GRBs with spectra from and Konus-Wind, and redshift from , to independently model the CSFR and the GRB LF. They used the GRBs irrespective of whether the spectral peak is actually seen in the instrumental waveband. In the present work, I choose only those bursts in which the spectral peak is accurately modelled, to re-derive the parameters of the Yonetoku correlation, which is then used to include a much larger number of sources.This paper is organized as follows. In Section <ref>, the Yonetoku correlation is re-derived. In Section <ref>, I describe the use of the correlation to generate pseudo redshifts of all remaining and GRBs. The GRB LF is modelled in Section <ref>, and in Section <ref>, I present concluding remarks. Throughout this paper, a standard ΛCDM cosmology with H_0 = 72km.s^-1.Mpc^-1 , Ω_m = 0.27 and Ω_Λ = 0.73 has been assumed. All the scripts used and important databases generated in the work are publicly available at <https://github.com/DebduttaPaul/luminosity_function_of_LGRBs_using_Swift_and_Fermi>. § THE YONETOKU CORRELATIONIt is the correlation seen between the peak luminosity L_p and the spectral energy break E_p <cit.> in the source frame.The peak luminosity is defined as L_p=P. 4π d_L(z)^2× k(z;spectrum),where P denotes the peak flux modelled by the Band function during the burst duration, given in erg.cm^-2s^-1, and k(z)=∫_1keV^10^4keVE.S(E)dE∫_(1+z)E_min^(1+z)E_maxE.S(E)dEfor GRBs. In case of bursts, where the peak flux is given in ph.cm^-2s^-1,k(z)=∫_1keV^10^4keVE.S(E)dE∫_(1+z)E_min^(1+z)E_maxS(E)dE . To accurately derive the Yonetoku correlation, I first select the sub-sample of all and bursts that have accurate estimations of the Band function <cit.> parameters during the prompt emission, by , as well as accurate redshift measurement by follow-up. Previous works have relied on modeling the spectral parameters by , which suffers from the limited wavelength range of BAT. I use the accurate spectral parameters from instead, reducing the inaccuracy of the estimates of luminosity. Moreover, due to the same reason, I also notice that the k correction is very close to unity for these bursts, unless the redshift is not too large (even for z=10, the factor is less than 1.5). This is illustrated in left of Fig <ref>. In comparison, the k-correction of is much larger for larger redshifts.§.§ Selecting the common GRBs The updated list of GRBs are selected from the catalog[https://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.htmlhttps://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html] till GRB170501467, which includes 2070 GRBs. Firstly, I choose only those bursts from the catalog that have spectroscopically measured parameters for the GRB Band function, which includes 1729 such cases. Then only those with small errors on the spectral parameters are chosen. For this, it is noted that the primary parameter that drive the error estimates in the luminosity is the . Choosing only those with errors less than 100% in E_p, 1566 bursts are retrieved.The updated list of GRBs are selected from the catalog[https://swift.gsfc.nasa.gov/archive/grb_table/https://swift.gsfc.nasa.gov/archive/grb_table/] till GRB 170428A. The total number is 1021, out of which those with firm redshift measure are 312.Since the nomenclature of and GRBs are different, I select the following criteria for selecting the common GRBs. The difference between the trigger times are selected to be less than 10 minutes, and they are restricted to within 10^∘×10^∘ in RA and Dec for the two instruments. These numbers are empirically chosen, such that the common number of GRBs converge within a reasonable range of these cutoffs. This ensures I do not mistake two GRBs which are well separated in time and space to be the same GRB. Consequently I get 68 common GRBs. Applying thecriterion for identifying short versus long bursts <cit.> separately for the two missions, I note that 65 are long according to both and , two are short in both, while only one is short only in , GRB090927422 (nomenclature). Its - is 0.512±0.231 sec while that of is 2.2 sec. -s are calculated at higher energies and hence known to be systematically smaller in a handful of GRBs. Fig <ref> illustrates this effect. Hence, I choose this as a long burst. Moreover, this also gives me confidence to make the distinction between long and short GRBs based on the -criterion whenever it is available, i.e. for the other common GRBs (without redshift estimates from ). For the ones that are detected only by , I resort to applying the criterion based on the -.§.§ Testing the correlation When I plot L_p versus E_p(1+z) (the factor of (1+z) takes care of the transformation into the co-moving frame) for all the 68 GRBs, I notice that the only burst with systematically smaller L_p than the rest, is a short burst. Moreover, the sample of short GRBs with accurate spectral and redshift measures consists of only two cases. Hence, I do not attempt to study the correlation for short bursts separately. Moreover, I do not find any burst with luminosity lower than 10^49erg.s^-1, nor with > 10^3sec, and hence I do not attempt to segregate the possible separate classes of low-luminosity long GRBs (see e.g. <cit.>), or ultra-long GRBs (e.g. <cit.>).I retrieve the Yonetoku correlation from the 66 long bursts to a high degree of confidence (a null-hypothesis of the Spearman correlation co-efficient of 0.623 being false, ruled out with p=2.368×10^-8), as shown in Fig <ref>. The errors on L_p consist of errors in the flux as well as a conservative estimate of 70% systematic error added to all bursts, to take care of the inaccuracy in the spectral parameters. These parameters are non-linear and hence the errors cannot be calculated directly. The systematic error is chosen conservatively, since the changes in the spectral parameters always affect the estimates in L_p within a factor of 1.5 even for the highest redshift bursts (see Fig <ref> for reference). Also, if linear errors are propagated, the mean errors are again of the same order.For the Yonetoku correlation defined as L_p10^52erg.s^-1=A.[E_p MeV(1+z)]^η,I get the best-fit parameters of A=4.780±0.123 and η=1.229±0.037. The corresponding redshift distributions for the same GRBs, both statistically and individually, are shown in Fig <ref>. It is noticed that although the method does not reproduce the redshifts on an individual basis, it is statistically reliable. The pseudo and observed redshifts has a median ratio of 1.002 ± 0.721, i.e. the number is consistent with unity. This is not an effect of normalization, as all the normalization factors are defined explicitly via Equation <ref>. The reason of it being statistically reliable is that, the method produces the pseudo redshifts of a larger sample by assuming gross parameters from a smaller sample which is however unbiased. The systematic discrepancies for individual bursts can be ascribed to the scatter around the Yonetoku correlation, as discussed below.<cit.> uses the set of parameters that reduce the discrepancy between the distributions of the observed and pseudo redshifts. This method tries to reconcile the problem by changing the parameters, while circumventing the actual problem, that the Yonetoku correlation is intrinsic scattered. This is best illustrated by the left panel of Fig <ref>. Moreover to verify their method, I run it on the current dataset, to find no global minimum of the discrepancy between the distributions. Hence, instead of modifying the parameters, I investigate the possible reasons for the scatter.To investigate the presence of systematics in the discrepancy between the observed and the pseudo redshifts, I look for possible correlations of the ratio of the predicted luminosity from the Yonetoku correlation with the physical parameters E_p(1+z) and the measured redshift. No correlation is found with the former, which confirms that the scatter in the Yonetoku correlation is intrinsic. However, I find a strong anti-correlation between the ratio and the measured redshift, as shown in Fig <ref>, with a null hypothesis of the Spearman correlation co-efficient of -0.533 being false, ruled out with p=4.056×10^-6. The following qualitative hypothesis is proposed to explain this trend. The luminosities predicted by the best-fit parameters of the observed correlation are the better physical estimates of the luminosity, physically correlating with the spectral peak. The scatter in the observed correlation between the quantities L_p and E_p (in the source frame) is due to the inadequacy of the definition of the luminosity, which needs to be corrected for physical factors like the beaming of the burst and the burst environment. This explanation, however, is qualitative and requires an in-depth analysis via modeling the possible physical effects, not attempted in the current work. § THE ESTIMATED LUMINOSITIESI next calculate the luminosities of all the detected bursts. This includes the 66 GRBs already used in Section <ref>, and the rest with spectral estimates from but without redshift estimates from (irrespective of they are detected by ). For the latter cases, pseudo redshifts are predicted via the Yonetoku correlation, using flux and k-corrections. However the - criterion is applied to those with -detections to distinguish between the short and long classes. For the GRBs with only detections along with measured redshifts, I directly calculate the luminosity from the flux and redshifts from the same catalog, and the k-corrections derived from the Band function parameters fixed at the average values of the distribution, given by <E_p> =181.3 keV, <α> =-0.566, <β> =-2.823. It is to be noted that the k-correction is not sensitive to these parameters, as long as they are within a reasonable range (see e.g. <cit.> for the study of BATSE bursts). For those bursts detected only by and further lacking redshift measurements, I estimate the pseudo redshifts via the k-corrections and the Yonetoku correlation. Since E_p features explicitly in the correlation, they are randomly sampled from the distribution of the bursts. The justification for such an approach is again that the being a wide-band detector, samples out all possible values of E_p. In Fig <ref> is shown the L-z distribution of all these cases. The instrumental sensitivities are given by Equation <ref> with P=8.0×10^-8erg.cm^-2.s^-1 for and P=0.2ph.cm^-2.s^-1 for (for a 100 keV photon, this is equivalent to 3.2×10^-8erg.cm^-2.s^-1). These numbers are chosen empirically from the respective catalogs, and describe the lower cutoff well. This places confidence on the used method and the estimated luminosities, and I proceed to use them for modeling the luminosity function (in Section <ref>). The slopes of the two correlations are 1.584±0.002 forand 1.834±0.002 for . A few bursts (eight) fall below the sensitivity line, which may be ascribed to the fact that the spectral parameters are sampled randomly from the distribution, whereas the flux is measured by ; also, the k-correction increases sharply with z for . These bursts are removed from the sample for subsequent analysis.On an average, the pseudo redshifts have ∼ 20 % errors and the luminosities calculated from them have ∼ 40 % errors, after propagating errors in all the estimation steps. Theoretically, the redshifts and hence luminosities of the bursts have much larger uncertainties, because their E_ps are not known, but this fact is ignored, to use these bursts in the statistical sense, laying no claim to the accuracy of the individual pseudo redshifts.I also note that the distribution of pseudo redshifts and corresponding luminosities are relatively insensitive to the exact value of the parameters used for the Yonetoku correlation, as long as they are not significantly different from the best-fit estimates. The advantage of using this method lies in the fact that it evades the complex observational biases that plague and limit the study of redshift measured bursts. Also, it allows the model to take care of the instrumental thresholds while modeling the luminosity function via Equation <ref>, to which I turn next. § MODELING THE LONG GRB LUMINOSITY FUNCTIONFor the purpose of modeling the luminosity function, the GRBs that have pseudo redshift greater than 10 are not considered (27). The final number of GRBs used are showed in Table <ref>. Also, the modeling is carried out separately for and , since the cut-off luminosities which feature in the model, via Equation <ref>, are different for the two instruments, as discussed in Section <ref>. For each instrument, I bin the data into three equipopulous redshift bins: 0<z<1.538, 1.538≤ z<2.657, 2.657≤ z<10.0 for , and 0<z<1.809, 1.809≤ z<3.455, 3.455≤ z<10.0 for . It is to be noted that the errors on N(L) are proportionally large, due to the large percentage errors on the derived luminosities, which are propagated across the bins.In the most recent work on GRB LF, <cit.> discusses various kinds of models. In particular, they test models in which the GRB formation rate is tied to a single population of progenitors via the cosmic star formation rate, another similar but distinct model where low and high luminosity GRBs are separated into two distinct classes, and a third kind where no assumption of the GRB formation rates are made. They conclude that a clear distinction between the three kinds of models cannot be asserted however. In the present work, I do not attempt to classify low and high luminosity GRBs for the reason that there is no clear evidence from the study in Section <ref>. Moreover, I assume that the GRB formation rate is proportional to the star-formation rate, because after all it is massive stars formed in the galaxies that later end their lives in GRBs. There may be an additional dependence on the redshift: most generally represented via Equation <ref>. I take the cosmic star-formation rate .ρ_⋆(z) from <cit.> (see references therein for the values at different redshifts), and model additional dependencies of the normalization, that is the GRB formation rate per unit cosmic star formation rate (or the GRB formation efficiency), asf_BC(z)∝(1+z)^ϵ. It is to noted that the detailed processes involved in the formation of GRBs do not affect this treatment, which is similar to that followed by <cit.>. Within this framework, I attempt to fit two models: the exponential cut-off powerlaw (ECPL) model, described by Φ_z(L)=Φ_0(L/L_b)^-νexp[- (L/L_b) ], and the broken powerlaw (BPL) model, given as Φ_z(L)=Φ_0(L/L_b)^-ν_1, L≤ L_b (L/L_b)^-ν_2, L>L_b. Moreover, most generally the `break-luminosity' L_b is allowed to vary with redshift, as L_b=L_b,0(1+z)^δ, with the quantity L_b,0 describing the normalization at zero redshift, and δ describing the evolution with redshift. The quantity Φ_0 normalizes the probability density function Φ(L), and is an implicit function of the redshift z via the dependence on L_b. The models are then described by Equations <ref>, <ref>, <ref>, <ref>, <ref> and <ref>, along with .ρ_⋆(z) extracted numerically from <cit.>.I look for the best-fit parameters of each model for and GRBs separately, because they have different L_cut(z) as shown in Fig <ref> (refer to Table <ref> for the classes).For the case of the ECPL, it is noticed that any non-zero values of δ or χ (or both) decreases the quality of fit, for both and . This allows me to decrease the parameter-space into a 2-dimensional space of ν and L_b,0 (which is equal to L_b for δ = 0.) In the case of the BPL however, the data strongly requires the inclusion of a positive-definite δ and anegative-definite χ. It is to be noted that the ECPL has one parameter less than the BPL, but allows the break to vary naturally, explaining why the data requires the additional dependencies on the parameters δ and χ for the BPL model.I search for the solutions by computing d_z^2=∑_L[N_ model(L,z)-N_ observed(L,z)]^2 for each redshift bin, then evaluating the discrepancy d^2=∑_zd_z^2, and finally looking for the model parameters that reduces d^2. I optimize the search by first choosing a large grid of parameters with sufficiently small bins, and then gradually converge on the best-fit parameters by decreasing the search-space and bin-size at each run.In the case of the ECPL, both the and runs converge to similar values of parameters, and are consistent within the deduced errors. The fits are generally poorer for the latter case, and also because detects a larger number of GRBs at higher redshifts due to its higher sensitivity compared to . This, however, is not directly taken into account in the modelling, being a limitation of the present work. This is because the exact mathematical form of the detection probabilities at various fluxes is not known. Hence, I tabulate the parameters from only the fits, in Table <ref>. The data is generally over-fitted, with thethefor the two instruments being 0.116 for and 0.539 for . This is because of the large number of bursts with similar luminosities, all with similarly large uncertainties.In the case of the BPL, there is no oscillation of any of the five parameters, justifying that the solutions are global. However it is found that and have different best-fits, significant differences being only in the related parameters ν_2, L_b,0 and δ. The solutions require extreme evolution of the break luminosity (δ=3.95), and raises suspicion of being an artefact of unaccounted systematics. To understand this, I model the detection probabilities of the two instruments by a simple flux powerlaw model, and plugging in the retrieved parameters of the two instruments, find that the difference can be explained by the variation of the detection probabilities with redshift and luminosity. On further investigation, I find that the solutions are in fact degenerate with the solutions. The d^2 contours in the L_b,0-δ space have similar global shapes, and also behave similar locally around the solutions. Thus I conclude that the best-fit solutions obtained for are driven by complications of its detection probability, and hence choose the best-fits as the accepted solutions, thus breaking the degeneracy. These are tabulated in Table <ref>. The corresponding fits for the two instruments are shown in Fig <ref>.The larger proportional errors for make thecomparable for the two instruments however, 0.362 for and 0.364 for . This demonstrates that the use of bursts helps in solving the degeneracy of the parameter space of the model.Since the constant in the RHS of Equation <ref> is not known a priori, it is calculated via the solutions of the models. It is known that for , T∼8.5 yr and for , T∼12 yr. I assume ΔΩ/4π∼1/3 for and 1/10 for , to get ratios of the observed and modelled numbers, which are converted to get f_BC(0)= 12.329×10^-8M_⊙^-1, Fermi,12.842×10^-8M_⊙^-1, Swift.for the ECPL model, and f_BC(0)= 7.498×10^-8M_⊙^-1, Fermi,8.200×10^-8M_⊙^-1, Swift.for the BPL model.These numbers are consistent with each other, and in rough agreement with those quoted by <cit.>.The ECPL shows agreement with the most recent work of <cit.>. The BPL model shows a sharp change at its break, which itself evolves quite strongly with redshift as L_b∼0.3×10^52(1+z)^2.90 erg.s^-1, in general agreement with <cit.>. The GRB formation rate for a given star-formation rate decreases with increasing redshift as f_BC ∝ (1+z)^-0.80 (the normalization is given by Equation <ref>), in agreement to the reports of <cit.>. Whereas the ECPL automatically takes into account the variation of the break, this needs to be incorporated viastrong evolutions with redshift in the BPL model. However, it is not possible to distinguish between the two models based on the fits. One of the reasons is that the data is generally over-fitted due to the large uncertainties, and another possible reason being that the discrepancies between data and model could be a result of the complex nature of detection probabilities of the instruments, which I have not attempted to model directly.It is to be noted that the present work is empirical; it does not attempt to provide an understanding of the models used, nor of the derived values of the parameters. A thorough understanding of the observed GRB number distribution requires one to justify the models via the phenomenology of long GRBs, taking into consideration the beaming of GRB jets and the GRB formation environment. This the scope of future work.§.§ Predictions for CZTI The CZT Imager or CZTI <cit.>, on the Indian multi-wavelength observatory<cit.> is capable of detecting transients at wide off-axis angles, localizing them to a few degrees, and carrying out spectroscopic and polarization studies of GRBs, as demonstrated in <cit.>. A preliminary analysis done with the weakest GRB detected by CZTI suggests that it is at least as sensitive as , which detects roughly 3 times the number of GRBs per year compared to . Similar to , the CZTI is also a wide-field detector. Moreover, it covers a wide energy range, being the most sensitive between 50 and 200 keV. Thus, it is reasonable to assume that its GRB detection rate is at least comparable to that of . Assuming this, I make predictions for CZTI over the redshift bins that were chosen for . The best-fit ECPL model predicts that CZTI should detect 150 GRBs per year. The best-fit BPL model predicts detection-rate of around 140 GRBs per year, with the equipopulous redshift bins almost equipopulous for CZTI as well. In ∼1.3 years of operation, ∼ 120 GRBs has been detected by CZTI by triggered searches alone,[See a comprehensive list at http://astrosat.iucaa.in/czti/?q=grbhttp://astrosat.iucaa.in/czti/?q=grb.] however the exact number is subjective. An automated algorithm to detect GRBs is being thoroughly tested and implemented, the details of which will be reported elsewhere. In the view of this, the predictions point out the fact ∼ 20-30 GRBs are yet undiscovered from the CZTI data. This is encouraging for the efforts on automatic detection, as well as that of the quick localization and follow-up, which will also be reported elsewhere. § CONCLUSIONSPreviously, BATSE and GRBs have been used to constrain the GRB luminosity function. Only a few BATSE GRBs had redshift measurements, so indirect methods were used to study the luminosity function of these GRBs. On the other hand, about 30% of the GRBs have redshift measurements. However, the measurement of the spectral parameters are also crucial to the measurement of the luminosity, via the k-correction factor. Being limited in the energy coverage, estimates of the spectral parameters have large uncertainties. Moreover, the number of GRBs with redshift measures are not as large as the entire BATSE sample. is a GRB detector with large sky coverage, a detection rate roughly 3 times more than , and wide energy coverage, thus measuring the broad-band spectrum of a large fraction (∼75%) of the detected GRBs to sufficient accuracy. However, its poor localization capabilities makes it impossible to make -like follow up observations, and hence the measurement of redshifts.In this work, I show that one of the methods proposed to solve the absence of redshift measures for BATSE GRBs can be used self-consistently to estimate the luminosities of and GRBs without redshift measurements. This method works on the premise that the `Yonetoku correlation' is applicable to all GRBs. For this, I have first used the most updated common sample of 66 long GRBs detected by these two instruments, to re-derive the parameters of this correlation. By a careful study of the discrepancies, I find a significant trend between the ratio of the observed and predicted luminosities with the measured redshift. The exact reason for this trend is not clear, but it highlights the fact that the weakness of the correlation is intrinsic, being driven by physical effects and not measurement uncertainties. I conclude that although the large scatter in the Yonetoku correlation rules out the possibility of using GRBs as distance-indicators, the statistical distribution of observed redshifts is reproduced well, and there is no need to modify the extraction of the correlation parameters as has been suggested previously <cit.>.Next, the method is shown to self-consistently predict `pseudo redshifts' of all long GRBs without redshift measurements. This allows calculation of the luminosities of a total of 2067 GRBs from these instruments, including the subsample (of 66 bursts) that has direct measurements of both redshift and spectra. I then use this large sample to model the GRB luminosity function, and place constraints on two models. The GRB formation rate is assumed to be a product of the cosmic star formation rate and a GRB formation efficiency for a given stellar mass. Whereas an exponential cut-off powerlaw model does not require a cosmological evolution, a broken powerlaw model requires strong cosmological evolution of both the break as well as the GRB formation efficiency (degenerate upto the beaming factor of GRBs). This is the first time GRBs have been used independent of measured redshifts from to study the long GRB luminosity function. Moreover, this is the first time such a large sample of GRBs have been used. The use of the large sample of GRBs helps in placing sufficient confidence on the derived parameters of the broken powerlaw model, when GRBs alone suffer from degeneracies and observational biases. Comparison with recent studies shows reasonable agreement for both the models, however it is not possible to distinguish between them.<cit.> has proposed on increasing the sample of GRBs by taking individual pulses of the same bursts as physically separate entities. In the future, perhaps a conglomeration of their method with the one here can be implemented to increase the sample size even further, to further test the parameters of the models and also carry out an in-depth analysis of the detection probabilities of the two instruments, which is presently quite a daunting task. This work also does not attempt to provide a physical understanding of the empirical models or the parameter values derived, which should be addressed in future works.Finally, I have used the derived models as templates to make predictions about the detection rate of GRBs by CZTI on board . The predictions are encouraging for the ongoing efforts of this collaboration. The quick localization of the few bursts that are predicted to be detected only by CZTI can increase the GRB database even further, and reveal interesting answers about the GRB phenomenon in both the local and the distant universe.§ ACKNOWLEDGEMENTS I sincerely thank my Ph.D. advisor A. R. 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"authors": [
"Debdutta Paul"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170926173105",
"title": "Modelling the luminosity function of long Gamma Ray Bursts using SWIFT and FERMI"
} |
Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, 12489 Berlin, GermanyWe perform Monte Carlo simulations of the CP^N-1 model on the square lattice for N=10, 21, and 41. Our focus is on the severe slowing down related to instantons. To fight this problem we employ open boundary conditions as proposed by Lüscher and Schaefer for lattice QCD. Furthermore we test the efficiency of parallel tempering of a line defect. Our results for open boundary conditions are consistent with the expectation that topological freezing is avoided, while autocorrelation times are still large. The results obtained with parallel tempering are encouraging.Fighting topological freezing in the two-dimensional CP^N-1 model Talk given at the 35th International Symposium on Lattice Field Theory, 18 - 24 June 2017, Granada, Spain. Martin Hasenbusch ============================================================================================================================================================================== § INTRODUCTION The CP^N-1 model shares fundamental properties such as asymptotic freedom and confinement with QCD. Therefore it serves as a toy model of QCD. It has been shown <cit.>that the model has a non-trivial vacuumstructure with stable instanton solutions. It turned out that these topological objects pose a particular problem in the simulation of the lattice CP^N-1 model, similar to lattice QCD.On the torus, in the continuum limit, the configuration space is decomposed into sectors that are characterized by their topological charge. At finite lattice spacing, the free energy barriers between such sectors increase as the lattice spacing decreases. For Markov chain Monte Carlo algorithms that walkin a quasi continuous fashion through configuration space this means that they become essentially non-ergodic and slowing down becomes dramatic. Numerical results are compatible with an increase of autocorrelation times that is exponential in the inverse lattice spacing. In the case of the CP^N-1 model this is numericallyverified, for example, in refs. <cit.>. Modelling the autocorrelation times with a more conventional power law Ansatz, large powers are needed to fit the data. From a practical point of view, the consequence is that it becomes virtually impossible to access lattice spacings below a certain threshold. The numerical studies show that in the case of the CP^N-1 model the problem becomes worse with increasing N. Since it is much less expensive to simulate the two-dimensional model than lattice QCD, it is a good test bed for new ideas and algorithms that could overcome the severe slowing down of the topological modes. For example simulated tempering <cit.> has been studied in ref. <cit.> with moderate success. More recently, “trivializing maps in the Hybrid Monte Carlo algorithm” <cit.> or the “Metadynamics” method <cit.> have been tested.A very principle solution of the problem had been suggested in ref. <cit.>. By abandoning periodic boundary conditions in one of the directions in favour of open ones, barriers between the topological sectors are abolished. The proposal has been further tested <cit.> and adopted in large scale simulations of lattice QCD with dynamical fermions <cit.>. Here we shall probe in detail how open boundary conditionseffect the slowing downin the case of the CP^N-1 model. Since the CP^N-1 model is much cheaper to simulate than lattice QCD, a larger range of lattice spacings can be studied and autocorrelation functions can be computed more accurately.Furthermore, we shall explore parallel tempering <cit.> as a solution to our problem. Parallel tempering is a well established approach in statistical physics to overcome effective non-ergodicity due to a ragged free energy landscape. The idea of parallel temperingand similar methods is to enlarge the configuration space such that the hills can be easily by-passed. A prototype problem is the study of spin-glasses, where parallel tempering is mandatory. For recent work see for example ref. <cit.>. Typically a global parameter such as the temperature or an external field is used as parameter of the tempering. Here instead, we shall discuss a line defect. Finally we like to mention that for the CP^N-1 model dual formulations can be found. These can be simulated by using the worm algorithm <cit.>. In these dual formulations there are no topological sectors and hence severe slowing down does not occur in the simulation. § THE MODELWe consider a square latticewith sites x=(x_0,x_1), wherex_i ∈{0,1,2,...,L_i-1}. The lattice spacing is set to a=1. This means that we trade a decreasing lattice spacing for an increasing correlation length. The action is S = - β N ∑_x,μ(z̅_x+μ̂ z_x λ_x,μ + z_x+μ̂z̅_x λ̅_x,μ -2),where z_x is a complex N-component vector with z_x z̅_x = 1 and λ_x,μ is a complex number with λ_x,μλ̅_x,μ = 1. The gauge fields live on the links, which are denoted by x, μ, where μ∈{0,1} gives the direction and μ̂ is a unit vector in μ-direction.In 1-direction we always consider periodic boundary conditions. In0-direction either open or periodic boundary conditions are considered. We implement open boundary conditions in a crude way, simply settingβ=0 for the links that connect x_0=L_0-1 and x_0=0. §.§ The observablesWe measure the energy, the magnetic susceptibility, the second momentand the exponential correlation length.For the definition of thesequantities see for example ref. <cit.> or section II A ofref. <cit.>.Our main focus is on the topology of the field. Motivated by eq. (33) of ref. <cit.>we define the plaquette angleθ_plaq,x =θ_x,μ +θ_x + μ̂,ν-θ_x + ν̂,μ - θ_x,ν - 2 n π ,μν ,where θ_x,μ= {z̅_x z_x+μ̂} and the integer n is chosen such that -π < θ_plaq,x≤π. We define the topological charge densityq_x=1/2 πθ_plaq,x.The topological charge on the lattice with periodic boundary conditions is defined byQ = ∑_x q_x = 1/2 π∑_xθ_plaq,x .The topological susceptibility is then given byχ_t = 1/V⟨ Q^2 ⟩= 1/L_0 L_1⟨∑_x y q_x q_y⟩ = 1/L_0 L_1⟨∑_x_0 x_1 q_x_0,x_1∑_y_0 y_1 q_y_0 y_1⟩ = 1/L_0⟨∑_x_0q̃_x_0∑_y_0q̃_y_0⟩ ,where we define q̃_x_0 = 1/√(L_1)∑_x_1 q_x_0,x_1.Note that the definition of the topological charge given in ref. <cit.> and (<ref>) are not equivalent at finite lattice spacing. We checked numerically that the difference between the two definitions decreases quickly with increasingβ. Also cooling of the configurations strongly reduces the difference.On the lattice with periodic boundary conditions, Q can take only integer values. Naively, θ_plaq,x adds up to zero, since each link angleappears with both signs. A nontrivial result is due to the fact thatθ_plaq,x is thrown back to the interval [-π, π).In the case of open boundary conditions, the definitions of susceptibilitieshave to be adapted. In order to avoid large finite size effects, the sites with a distance less than l_0from the open boundaryare not takeninto account, when computing the observables.Motivated by the rightmostpart of eq. (<ref>) we arrive atχ_t,open = 1/L_0-2 l_0∑_x_0 = l_0^L_0-l_0-1⟨q̃_x_0^2 ⟩+2 ∑_w=1^l_max1/L_0-2 l_0 -w∑_x_0 = l_0^L_0-l_0-w-1⟨q̃_x_0q̃_x_0+w⟩.§ BASIC ALGORITHMSAs basic algorithmwe use a hybrid of the Metropolis, the heat bath and the microcanonical overrelaxation algorithm.To a large extend, we follow section III of ref. <cit.>. Let us first discuss the updates of the site variables and then the updates of the gauge fields. In an elementary step of the algorithmwe update the variable at a single site x, while keeping the gauge fields and the variables at all othersites fixed. The part of the action that depends on this site variable can be written as asS̃(z_x) = - z_x F̅_x, F_x = 2 N β ∑_μ [λ̅_x,μ z_x+μ̂ + λ_x-μ̂,μ z_x-μ̂].Note that the problem at this point is identical to the update of an O(2 N) invariant vector model with site variables of unit length. Instead of F̅_x we would have to deal with the sum of the variables on the nearest neighbour sites. The microcanonical update keeps S̃(z_x) fixed, while the new value of z_x has maximal distance from the old one. It is given by eq. (43a) of ref. <cit.>:z_x' = 2 z_x F̅_x/|F_x|^2 F_x- z_x.In addition to these updates, we have to perform updates that change the value of the action. To this end we implemented a heat bath algorithm that is applied to the subset of three of the 2 N components of z_x, where we count boththe real and the imaginary parts. The heat bath updateis identical to the one used in the simulation of the O(3)-Heisenberg model on the lattice or for the update of SU(2) subgroups in the simulation of pure SU(N) lattice gauge models <cit.>. We run through all N complex components of z_x taking the real and the imaginary part of the component as first two components for the heat bath. The third component is randomly chosen among the real or imaginary parts of the remaining N-1 components of z_x. Note that the CPU-time required by the microcanonical overrelaxation update is about one order of magnitude less than that for the heat bath update.For fixed variables z the gauge fields can be updated independently of each other. The action readsS̃_g(λ_x,μ) = - λ_x,μf̅_x,μ ,f_x,μ =2 N βz_x+μ̂z̅_x.Here we perform a four hit Metropolis update, where the stepsize was chosen such that the acceptance rate is roughly 50%, and a microcanonical update, see eq. (43b)of <cit.>. §.§ Autocorrelation timesThe performance of a Markov chain Monte Carlo algorithm is characterized by the autocorrelation time. There are different definitions of the autocorrelation time. These are based on the autocorrelation function. The autocorrelation function of an estimator A is given byρ_A(t) = ⟨ A_i A_i+t⟩ - ⟨A ⟩^2/⟨ A^2 ⟩ - ⟨A ⟩^2 .The modulus of the autocorrelation function is bounded from above by an exponentially decaying function. In practice one often finds thatthe autocorrelation function at large t is given byρ_A(t) ≃c_A exp(-t/τ_exp,A).The integrated autocorrelation time of the estimator A is given byτ_int,A = 0.5 + ∑_t=1^∞ρ_A(t).The summation in eq. (<ref>) has to be truncated at some finitet_max. Since ρ_A(t) is falling off exponentially at large distances, the relative statistical becomes large at large distances. Therefore it is mandatory to truncate the summation at some point that is typically much smaller than the total length of the simulation.In the literature one can find various recommendations how this upper bound should be chosen.Fore example, Wolff <cit.> proposes to balance the statistical error with the systematic one that is due to the truncation of the sum.§ SIMULATIONS WITH OPEN BOUNDARY CONDITIONSIn order to keep the fraction of discarded sites small,it seems useful to chose L_0 ≫ L_1. On the other hand, since thetime needed for topological objects to diffuse to the centre of thelattice or back to the boundary increases with increasing L_0, toolarge values of L_0 are not advisable.After performing preliminary simulations we decided to take L_0=4 L_1 throughout.Furthermore wetake l_0 ≈ 10 ξ_2nd and l_max=l_0. For N=10, using standard simulations and periodic boundary conditions, ξ_2nd≈ 23 can be reached<cit.>. Hence it is hard to demonstrate a clear advantage for open boundary conditions. Instead for N=21 it is virtually impossibleto go beyond ξ_2nd≈ 6 by using periodic boundary conditions and standard simulations. Therefore in the following wefocus onour simulations for N=21. We find that forL_1 ⪆ 16 ξ_2nd finite L_1 effects can be ignored at the level of our statisticalaccuracy.We performed simulations for a large number of β-values,ranging from β=0.625 up to 0.95. For each value of β, weperformed 2× 10^6 update cycles. For the larger valuesof β, we discarded 50000 update cycles at thebeginning of the simulation. One update cycle consists of one sweep over all sites of the lattice using the heat bath algorithm,the 4 hit Metropolis updateof the gauge fields, and finally n_ov sweeps using the overrelaxationalgorithm for both the site variables and the gauge fields.We perform a measurement of the observables for each cycle. Autocorrelation timesare quoted in units of these update cycles. The number of overrelaxationupdates n_ov is chosen to be proportional to the correlation length. It increases from n_ov=3 for β=0.625up to n_ov=28 for β=0.95. The second moment correlation length increases from ξ_2nd=2.2968(5) at β=0.625 up to18.2419(43) at β=0.95. Let us discuss the autocorrelation times of the topologicalsusceptibility (<ref>,<ref>). For periodic boundary conditions, τ_int increases very rapid,compatible with exponential in the correlation length. Instead, for open boundary conditions, we first see an increase that is similar to that for periodic boundary conditions. The difference here can be attributed to the different definitions of the topologicalsusceptibility (<ref>,<ref>). Then, for ξ_2nd⪆ 5 the autocorrelation time levels off for open boundary conditions.The behaviour in the case of open boundary conditions can be explained along the lines of ref. <cit.>. For ξ_2nd⪅ 5 changes of the topological charge are dominantly due to the creation and destruction of instantons in the bulk. Then for ξ_2nd⪆ 5 the diffusion from and to the boundaries completely dominates. This diffusion is not effected by the severe slowing down. Our numerical results for N=41 confirm theconclusions drawn here for N=21. § PARALLEL TEMPERING IN A LINE DEFECT In a parallel tempering simulation one introduces a sequence of N_t systemsthat differ in one parameter of the action. For each system there is a configuration {z,λ}_t. The tempering parameter might have a physical meaning. In statistical physicssimulations this parameter is mostly the temperature. However it could also be a parameter that is introduced only for the sake of the simulation, as it is the case here. At one end of the sequence there is the system that wewant to study. In our case this is a lattice with L_0=L_1, periodicboundary conditions in both directions, and the coupling constant is the same for all links. For the system at the other end, it should be easy to sample the whole configuration space. Motivated by the success of the simulations with open boundary conditions, we use a system with a line defect to this end. Such a line defect issketched in fig. <ref>. For l_d=L_1 we recover open boundary conditions. In addition to updates of the individual systems there are exchanges ofconfigurations between the systems.A swap of configurations {z,λ}_t_1'={z,λ}_t_2,{z,λ}_t_2'={z,λ}_t_1 between t_1 and t_2 is accepted with the probabilityA_swap=[1,exp(-S_t_2({z,λ}_t_1) -S_t_1({z,λ}_t_2)+S_t_1({z,λ}_t_1)+S_t_2({z,λ}_t_2))].In our simulations we run from t_1=0 up to t_1=N_t-2 in steps of one, proposing to swap the configurations at t_1 and t_2=t_1+1. The number of replica N_t is chosen such that the acceptance rate for the swap of configurations is larger than 30 % for all t_1. Note that S_t_2 and S_t_1 differ only on the defect line. Typically the swap of configurations is alternating with standard updates of the individual configurations. Typically, when the tempering parameter is homogeneous in space, a sweep over the whole lattice is performed. In contrast, here we temper in a defect that takes only a small fraction of the lattice. Therefore it is advisable to update only some part of the lattice that is centred around the defect.To this end, we introduce a sequence of rectangles of decreasing size, each associated with a level of our update scheme. For details seeFig. <ref>. In one update cycle, at a given level,we sweep over the rectangle, updating all N_t configurations: Once using theheat bath algorithm for the site variables and the 4 hit Metropolis update for the gauge fields. Then follow n_ov,i overrelaxation sweeps of the site variables and the gauge fields. For small i, n_ov,i is the same as for our simulations with open boundary conditions. For larger i, smaller values are taken. The update cycle at agiven level is completed by a swap of configurations (<ref>). We chose n_i such that for each levelof the update scheme, roughly thesame amount of CPU time is spent. The larger l_d, the more topological objects can be generated or destroyed. On the other hand, for increasing l_d,N_t has to be enlarged to keepthe acceptance rate above 30 %.Our numerical study shows that l_d ≈ξ is the optimal choice. We perform a translation of the configuration for t=0 after each swap.This way changes in the topology are injected at any location on the lattice and diffusion is not needed.We performed simulations for N=10, 21, and 41 and various values of β.Let us discuss the simulation for N=21 and β=0.95 in more detail. We used l_d=16 and N_t=32.The complete update cycle over all levels is characterized by n_1=24, n_2=n_3=n_4=3, and n_5=n_6=2. The number of overrelaxation updates per cycle is 28, 14, 7, 7, 3, and 3 at levels 1, 2, 3, 4, 5, and 6, respectively. We find that the acceptance rate is about 81.4 % for the pair t=0 and 1. It drops to39.4 % for the pair t=26 and 27. Then it increases again to 47.3 % for the pair t=30 and 31.The simulation, consisting of 50370 complete update cycles over all levels, took 25 days on a 4 core PC running with 8 threads.This is about the same CPU time that is used for the corresponding run with open boundary conditions. The error bar of the topological susceptibility is smaller by a factor of 2.3 compared with the simulation with open boundary conditions. § PHYSICS RESULTS AND COMPARISON WITH THE LARGE N-EXPANSION Following ref. <cit.>ξ_2nd/ξ_exp = √(2/3) + O(N^-2/3).Our results obtained for N=10, 21 and 41, which are plotted inFig. <ref> a are still quite far from this asymptotic value. Therefore we abstain from estimating the coefficient of theO(N^-2/3) corrections.The product χ_t ξ^2 shouldhave a finite continuum limit. For the exponential correlation length the 1/N-expansion gives <cit.>χ_t ξ_exp^2 = 3/4 π N + O(N^-5/3) . For the second moment correlation length a faster convergence with increasing N is obtained <cit.>χ_t ξ_2nd^2 = 1/2 π N(1 - 0.38088.../N) +O(N^-3).In Fig. <ref> b we plot ξ_2nd^2 χ_t as a function of ξ_2nd.Looking at the figure, the numerical data seem to converge nicely to the scaling limit.Corrections to scaling seem to be smaller for larger values of N. Taking simply the largest values of β for each N we get ξ_2nd^2 χ_t = 0.01737(8), 0.00767(5), and 0.00391(2) for N=10, 21, and 41, respectively. This can be compared with results quoted in the literature. For N=10 one finds for example ξ_2nd^2 χ_t = 0.01719(10)(3) and 0.0175(3) in refs. <cit.>, respectively. For N=21 one finds ξ_2nd^2 χ_t =0.0080(2) and 0.0076(3)in refs. <cit.>, respectively. For N=41 we find in ref. <cit.> the results ξ_2nd^2 χ_t = 0.0044(4) and 0.0036(4) for β=0.57 and 0.6, respectively. Our estimates are essentially consistent with those presented in the literature. In particular for large values of N, we improved the accuracy of the estimates. To see the effect of leading corrections, it is useful to multiply ξ_2nd^2 χ_t by 2 π N. Using our numbers, we get 1.091(5), 1.012(7), and 1.007(5) for N=10, 21, and 41, respectively. As already discussed in ref. <cit.> it is a bit puzzling that the numbers suggest a 1/N correction with the opposite sign as that of eq. (<ref>).§ SUMMARY AND CONCLUSIONS We have shown that the severe slowing down in the simulation of thelattice CP^N-1 model can be avoided by using open boundary conditions in one of the directions. We studied parallel tempering in a line defectas an alternative. Our numerical results areencouraging. Focussingon the statistical error of the topological susceptibility, the simulation with open boundary conditions is outperformedby a factor of about 4. The crucial question is, of course,whether parallel tempering in a defect structure is helpful in simulations of lattice QCD.A more detail account of this study is given in <cit.>.§ ACKNOWLEDGMENTSI thank Stefan Schaefer for discussions. This work was supportedby the Deutsche Forschungsgemeinschaft (DFG) under the grant No HA 3150/4-1. 28DAdda:1978vbw A. D'Adda, M. Lüscher, P. Di Vecchia, Nucl. Phys. B146, 63 (1978)Witten79 E. Witten, Nuovo Cim. A51, 325 (1979)Campostrini M. Campostrini, P. Rossi, E. Vicari, Phys. Rev. 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M. (Interface Foundation, Fairfax Station, 1991), p. 156Hukushima:1996xx Hukushima, Koji and Nemoto, Koji, J.Phys.Soc.Jpn. 65, 1604 (1996), Earl:2005xx Earl, David J. and Deem, Michael W., Phys.Chem.Chem.Phys. 7, 3910 (2005)Janus2013 M. Baity-Jesi et al., Phys.Rev.B 88, 224416 (2013), Prokofev:2001ddj N. Prokof'ev, B. Svistunov, Phys. Rev. Lett. 87, 160601 (2001)Ulli U. Wolff, Nucl. Phys. B832, 520 (2010), MyCPN M. Hasenbusch, Phys. Rev. D96, 054504 (2017), Berg:1981er B. Berg, M. Lüscher, Nucl. Phys. B190, 412 (1981)CM82 N. Cabibbo, E. Marinari, Phys. Lett. 119B, 387 (1982)Cr80 M. Creutz, Phys. Rev. D21, 2308 (1980)Wolff:2003sm U. Wolff (ALPHA), Comput. Phys. Commun. 156, 143 (2004), [Erratum: Comput. Phys. Commun.176,383(2007)], CaRo91 M. Campostrini, P. Rossi, Phys. Lett. B272, 305 (1991)ML78 M. Lüscher, Phys. Lett. 78B, 465 (1978) | http://arxiv.org/abs/1709.09460v1 | {
"authors": [
"Martin Hasenbusch"
],
"categories": [
"hep-lat"
],
"primary_category": "hep-lat",
"published": "20170927114934",
"title": "Fighting topological freezing in the two-dimensional CP$^{N-1}$ model"
} |
< g r a p h i c s >Control over the spontaneous emission of light through tailored optical environments remains a fundamental paradigm in nanophotonics. The use of highly-confined plasmons in materials such as graphene provides a promising platform to enhance transition rates in the IR-THz by many orders of magnitude. However, such enhancements involve near-field plasmon modes or other kinds of near-field coupling like quenching, and it is challenging to use these highly confined modes to harness light in the far-field due to the difficulty of plasmonic outcoupling. Here, we propose that through the use of radiative cascade chains in multi-level emitters, IR plasmons can be used to enhance far field spectra in the visible and UV range, even at energies greater than 10 eV. Combining Purcell-enhancement engineering, graphene plasmonics, and radiative cascade can result in a new type of UV emitter whose properties can be tuned by electrically doping graphene. Varying the distance between the emitter and the graphene surface can change the strength of the far-field emission lines by two orders of magnitude. We also find that the dependence of the far-field emission on the Fermi energy is potentially extremely sharp at the onset of interband transitions, allowing the Fermi energy to effectively serve as a “switch” for turning on and off certain plasmonic and far-field emissions.§ INTRODUCTION One of the most fundamental results of quantum electrodynamics is that the spontaneous emission rate of an excited electron is not a fixed quantity; rather, it is highly dependent on the optical modes of the surroundings <cit.>. This result, known by most as the Purcell effect <cit.>, is the basis for the active field of spontaneous emission engineering, which has become paradigmatic in nanophotonics and plasmonics. One system that has emerged as a promising platform for studying strong plasmon induced light-matter interactions is graphene, which features low-loss, extremely sub-wavelength infrared surface plasmons with a dynamically tunable dispersion relation <cit.>. The combination of these features makes graphene prime for a wide array of applications such as tunable perfect absorbers, x-ray sources with tunable output frequency, tunable phase shifters with 2π phase control, tunable Casimir forces for mechanical sensing <cit.>, tunable light sources via the plasmonic Cerenkov effect <cit.>, and electrical control over atomic selection rules by taking advantage of access to conventionally forbidden transitions <cit.>. Additionally, because graphene plasmons can be confined to volumes 10^8 times smaller than that of a diffraction-limited photon <cit.>, an infrared emitter in the vicinity of graphene can experience extreme enhancement of spontaneous decay through both allowed and forbidden channels via the Purcell effect <cit.>.The Purcell effect and its consequences are almost universally studied in the framework of two-level systems. Surprisingly, the indirect effect of Purcell engineering on radiative cascade dynamics has rarely been exploited. Realistic emitters of course have many levels, and it is therefore possible to influence the spontaneous emission spectrum of an atom far beyond what a simple two-level analysis reveals, as we extensively make use of in this work. In particular, we find that by interfacing the (electrically and chemically) tunable IR Purcell spectrum of 2D plasmonic materials like graphene with multilevel emitters, it may be possible to design an electrically tunable UV frequency emitter, even when no plasmons exist at those frequencies. Remarkably, we find that it is possible to use the Purcell effect at IR frequencies to enhance far-field emission of 100 nm wavelength light by nearly two orders of magnitude. § METHODS In our work, we consider the spectrum produced by emitters near tunable plasmonic environments such as graphene. For calculational concreteness, we take a hydrogenic emitter, whose spectrum shares many features with more generic atoms. The hydrogen atom has a set of electronic states indexed by quantum numbers |n,l,m_l,m_s⟩, where n is the principal quantum number, l is the orbital quantum number, m_l gives the orbital angular momentum, and m_s is the spin of the electron. The effect of spin-flip transitions is negligible for our purposes, and thus we will not keep track of the electron spin quantum number m_s. Additionally, we will not consider fine structure splitting, so the energies of states are indexed solely by the principle number n. Electrons have the ability to transition between states through emission and absorption of light. Strictly speaking, the electromagnetic fluctuations of the system are mixtures of far-field photons and near-field plasmons, but in practice, every excitation can be well-defined as either a photon or a plasmon to a good approximation. Here, excitations at IR frequencies will be considered purely plasmonic in nature, while excitations in the visible and UV will be treated as far-field photons.We consider a multi-state system with a pump from the ground state |g⟩ of Hydrogen to some excited state |e⟩. Both optical and electrical pumping could be considered. Once excited, the electron can then radiatively cascade back down to the ground state, emitting photons of one or more frequencies in the process. The dynamics of the system are governed by the rate equation d𝐍/dt = A𝐍,where 𝐍 is a vector of length n containing the occupation numbers of the n electronic states, and A is a rate matrix with entries A_ij = (1-δ_ij)Γ_ij - δ_ij∑_k=1^n Γ_kj, where Γ_ij is the rate of transition between states i and j. The rate Γ_ge describes the pumping of photons from the ground state to the starting excited state. The rates of all upward transitions with the exception of the pump Γ_ge are assumed to be zero. After sufficient time, a steady state equilibrium is established between the pump and the cascading photons, i.e. d𝐍_s/dt = A𝐍_s = 0, where 𝐍_s contains the steady state populations of the emitter levels.The total rate of production of a particular frequency photon is obtained by summing over all channels of the same frequency of emission to account for potential degeneracies. That is,dp_ω/dt = ∑_ω_ij = ωΓ_ijN_i,where ω_ij = ω_j - ω_i. Then, the observed power output of a frequency ω is given asP(ω) = ħωdp_ω/dt.The observed differential spectrum dP(ω)/dω will be subject to broadening effects, such as doppler broadening and inhomogeneous broadening which we do not consider here. This model is easily extended by adding non-radiative loss channels directly into the rate equations. Also note that we assume electrons in the 2s state return to the ground state without the emission of a photon since this effect is negligible at first order <cit.>.The spontaneous emission rate from state |i⟩ to |j⟩ near the surface of graphene is given as Γ_ij = Γ_0F_p(ω), where Γ_0 is the rate of transition in the vaccum, and F_p(ω) is the Purcell factor. The Purcell factor for p-polarized modes is given as<cit.>F_p(ω) = 1 + f 3c^3/2ω^3∫ q^2 exp(-2qz_0)Im[1/1 - qσ(q,ω)/2iωϵ_0] dq,where ω is the plasmon frequency, q is the plasmon in-plane wavevector, z_0 is the distance between the graphene and the emitter, σ(q,ω) is the conductivity of graphene, and f = 1 (1/2) for dipoles perpendicular (parallel) to the graphene plane. In Figure 1, we see such an emitter near the surface of graphene that is able to radiate into plasmonic surface modes as well as into the far-field. We also see the energy levels of hydrogen with the possible decay pathways from the 4d state shown as given by the dipole selection rules. By calculating the rates, we compute the power spectrum, which we claim can undergo a drastic shift as the emitter is brought into sufficient proximity of the graphene surface.In our study, we consider two models of conductivity: the Drude model, σ_D(ω) = i(e^2 E_F/πħ^2)/(ω + iτ^-1), and the local interband conductivity σ(ω) = σ_D(ω) + σ_I(ω), where the contribution from interband effects is σ_I(ω) = e^2/4ħ(θ(ħω - 2E_F) - i/πln|2E_F + ħω/2E_F - ħω|)<cit.>. In the above, E_F is the Fermi energy of the graphene substrate, which is directly related to the electron carrier density, and τ is the empirical relaxation time corresponding to losses that can generally be a function of frequency and vary with the Fermi energy <cit.>. In this work we neglect the dependence of τ on E_F but it can be accounted for using results of density functional theory analysis.<cit.> While the local model is more precise and has been demonstrated to well-describe flourescence quenching experiments in graphene <cit.>, we also consider the Drude model to connect to other 2D metals and also other Drude metals featuring high local density of states. We comment on the neglect of nonlocality later in the text.properly§ RESULTS AND DISCUSSION Using the rate equation formalism of the previous section, we now explore how proximity to graphene can enhance atomic spectra through radiative cascade. We consider a hydrogen atom pumped from the 1s state to a 4d state. In order to understand the dynamics of enhancement at work, we consider the Purcell factors which enhance each transition frequency at various distances from emitter to the graphene surface. In Figure 2(a) we show the Purcell factor given in Eq. <ref> for a dipole at frequecy ω at a distance z_0 from a graphene surface doped to E_F = 1.0 eV using the Drude model of conductivity. Losses are taken into an account with Drude relaxation time of τ = 10^-13 s. At low frequency and low z_0, loss induced quenching causes the Purcell factor to exhibit divergent behavior. At mid frequency range 0.05 - 0.8 eV, the Purcell enhancement comes primarily from plasmonic emission, and can easily reach 10^6 at experiementally realizable z_0 such as 5 nm. At sufficiently high frequencies (≥ 0.8 eV), proper plasmonic modes cease to exist, and the relevant electromagnetic fluctuations instead can couple to particle-hole excitation. In this region, the supported modes are simply the far-field free space modes. Panel (2c) shows the Purcell factor using the full local RPA model of graphene conductivity which accounts for interband transitions. The main difference between the Drude and full local RPA model can be seen in the 10^5 - 10^6 Hz (0.66 - 6.6 eV) frequency range. The RPA model exhibits a sharp dip in the Purcell factor at a critical frequency characterized by the condition 2E_F = ħω, corresponding to the divergence of the imaginary part of the conductivity.Panels (b) and (d) show the rate of photon production for a hydrogenic dipole emitter approaching the surface of graphene, calculated for the conductivity models described in (a) and (c) respectively. When the emitter is placed within nanometers of the graphene sheet, infrared plasmon emission is enhanced by the Purcell effect. In fact, the direct plasmonic enhancement of IR transitions is much greater than that of vis-UV transitions, leading to dominance of IR decay pathways in the presence of graphene. This alteration of the decay pathways can induce substantial modification of the far-field spectrum. As an example, at distances closer than 20 nm the 3p → 1s (103 nm) UV transition dominates the 2p → 1s transition (121 nm) which is normally prominent in free space. Note that this dominance comes not from direct enhancement of the local density of states at the 3p→ 1s transition frequency, but rather the enhancement of the IR transition 4d→ 3p at 1875 nm associated with plasmon emission. The enhanced IR transition populates 3p with electrons which then prefer to decay into the 1s state. We note that this principle is easily extended to other emitters with much higher frequency transitions (such as helium with a 30+ eV transition and perhaps even EUV transitions). It would also be of interest to extend this idea to higher energy core-shell transitions in heavier atoms.In Figure 3, we see the calculated spectral power output of the emitter at four different distances z. At a distance z=100 nm shown in panel (d), the emitter is too far from the surface of graphene to couple to plasmonic modes, so the output spectrum is effectively that of an emitter in free space. As the emitter nears the surface, IR plasmons are excited at 1875 nm, re-routing the power output of the emitter into the 103 nm UV channel. The strengthening of the 103 nm line can be seen at distance z=20 nm shown in panel (c). At distances below 10 nm, more than 90% of the power output is directed into the 103 nm channel. Both panels (a) and (b) show the vast majority of spectral output in the 103 nm line. Indirect coupling of IR and UV transition rates is a highly efficient method of UV enhancement, as an order of magnitude or more power is channeled into the far-field UV emission than the excitation of the supporting IR plasmon. We now exploit features of the full local conductivity model to work toward a spectrum that can be both drastically modified and delicately tuned by doping graphene. As we recall from Fig. 2(c), the Purcell factor calculated in the full local conductivity model exhibits a sharp dip near 2E_F = ħω as a result of the corresponding logarithmic singularity in σ_I(2E_F/ħ). By tuning the Fermi energy such that 2E_F/ħ corresponds to a characteristic transition frequency ω_0 of the emitter, the rates of other transition frequencies can be greatly enhanced relative to that of frequency ω_0. What results is a dramatic relative slowing of the ω_0 transition, which can cause reduced intensity not only in the emission line ω_0, but also in emission lines corresponding to transitions enabled by radiative cascade after the ω_0 transition. In Figure 4, we see the photon emission rates at four distances z_0 as a function of varying Fermi energy E_F. The two lowest energy transition in this system are at 1875 nm and 656 nm, corresponding to critical E_F values of 0.34 eV and 0.95 eV respectively. Crossing these boundaries causes critical changes in the branching ratios of the system. For example, at a distance of z_0=20 nm shown in panel (d), a Fermi energy of 0.34 eV suppresses the 4d → 3p (1875 nm) transition, whereas a Fermi energy of 0.32 eV allows the 1875 nm transition to dominate. Since electrons in the 3p state are far more likely to transition into 1s than 2s, crossing this Fermi energy boundary acts not only as a “switch” for the 1875 line, but also one which amplifies the 103 nm UV line, and suppresses the visible line at 486 nm. As the emitter nears the surface at distances of z_0 = 1 nm or 5 nm, as shown in panels (a) and (b), the system dynamics change. Namely, with increasing proximity to the surface, the 1875 nm line remains dominant for most values of E_F while the 103 nm line decreases in intensity. However, after crossing the critical threshold corresponding to suppression of the 656 nm line, the 103 nm line is once again enabled to match the intensity of the 1875 nm channel. As another example, consider that by changing the distance from z=5 nm to z=10 nm, one can change which channel the 656 nm line follows. At z_0=5 nm (b), the 656 nm line matches the rate of the 1875 nm line for E_F below the 656 nm suppression threshold. In contrast, at z_0=10 nm (c), the 656 nm line instead follows the 103 nm line. We see that crossing critical doping boundaries allows significant modification of the spectral structure, while tuning between critical points allows smooth and controlled deformation.Using the full nonlocal RPA conductivity model, one can estimate the effects of nonlocality on our calculations.<cit.> The effects of nonlocality are most significant at low Fermi energies. In these regimes, the Purcell factors near ω = 2E_F/ħ can become larger than in the local approximation by around 2 orders of magnitude. The result is a less drastic but stil critical behavior at the expected points which should be observable. At E_F > 0.5 eV, the nonlocal corrections are comparatively much smaller. Additionally, note that nonlocal considerations should not significantly impact the spectrum as a function of distance z variation so long as no important transitions lie in the conductivity divergent frequency range. In other words, strong indirect enhancement of far-field transitions should still be achievable, even in the presence of nonlocal effects.From the analysis here, it is clear that the effect of the Fermi energy of graphene serves not only as a knob to alter plasmonic coupling, but also one which can modify coupling into far-field modes where there are no plasmons - all due to the radiative cascade effect which effectively correlates the emission of IR and UV frequencies. By varying both the Fermi energy and the distance to substrate z, a wide variety of spectral regimes can be accessed. Changing either of these paramaters has the ability to change the fundamental behavior of the emitter. As another example, consider that by changing the distance from z=5 nm to z=10 nm, one can change which channel the 656 nm line follows. At z_0=5 nm (b), the 656 nm line matches the rate of the 1875 nm line for E_F below the 656 nm suppression threshold. In contrast, at z_0=10 nm (c), the 656 nm line instead follows the 103 nm line.Excitation of the emitter to higher energy level states can greatly increase the number of decay pathways available to an electron, as we demonstrate in Figure 5. As an example, consider an emitter excited to the state 5p. At distances z_0 = 20 nm from the surface (c), the emission rate of 95 nm photons into the far-field can increase by a factor of 50 or more across a wide range of carrier densities. In particular, the critical threshold corresponding to the 4050 nm plasmon has the effect of exchanging the spectral dominance of the 95 nm and 103 nm lines. In this case, bringing the atom much closer than 10 nm from the surface changes decay dynamics in a way which actually suppresses this effect, which can be seen in panel (a). At a distance as close as 5 nm, plasmons of lower frequency than those which enable the fast UV transition become excited, competing with the desired high frequency transitions. Perhaps counterintuitively, the strongest enhancement of UV far-field emission occurs at moderate, rather than extremely close or extremely far distances of emitter to surface. At extremely close distances to the surface, plasmons of higher frequency become excited, working against the ability of very low frequency IR plasmons to focus electrons through a very specific decay channel. This emphasizes that strong high frequency enhancement requires not only the enhancement of IR plasmons which enable UV transitions, but also the relative dominance of this enabling IR transition to other competing mechanisms of decay.§ CONCLUSIONS AND OUTLOOK In summary, we have demonstrated that by modulating the carrier density and proximity of an emitter to graphene, the far-field emission spectrum can be fundamentally altered by greatly enhancing the rates of coupling to the electromagnetic fluctuations of the graphene sheet. We note that while we have emphasized the enhancement of UV emission, our results make it evident that the entire spectrum can be drastically altered. This includes both the partial or nearly complete suppression of ordinarily present spectral lines, as well as the amplification of lines that are slow under free-space conditions. Since UV transitions generating far field photons may be indirectly enhanced through cascade, it should be possible to observe evidence of enhancement in an experimental setting, even without the ability to outcouple plasmons.Polaritons other than surface plasmon-polaritons on graphene also provide a broad range of scenarios in which similar effects can be observed, such as phonon and exciton-polaritons. For example, surface phonon-polaritons on hexagonal boron nitride can provide similar Purcell enhancement-type effects. In fact, the Restrahlen band of hBN could allow for even more selective enhancement of first or higher order decay mechanisms, enabling yet another mechanism of control over which transitions are enhanced <cit.>.An alternate possibility for controlling the far-field spectrum is by not only engineering the Purcell enhancement of allowed dipolar transitions, but also utilizing forbidden transitions <cit.>. As shown in <cit.>, it should be possible to use the highly confined plasmons in graphene to make highly forbidden emitter transitions occur at rates competitive with those of electric dipole transitions. Then by radiative cascade, it should be possible to extract a far-field UV signal from these forbidden transitions. Our analysis presented here thus serves as a crucial starting point for designing experiments to detect transitions whose observations have proved elusive since the discovery of spontaneous emission.§ FIGURESResearch supported as part of the Army Research Office through the Institute for Soldier Nanotechnologies under contract no. W911NF-13-D-0001 (photon management for developing nuclear-TPV and fuel-TPV mm-scale-systems). Also supported as part of the S3TEC, an Energy Frontier Research Center funded by the US Department of Energy under grant no. DE-SC0001299 (for fundamental photon transport related to solar TPVs and solar-TEs). shown in this document. | http://arxiv.org/abs/1709.09112v1 | {
"authors": [
"Jamison Sloan",
"Nicholas Rivera",
"Marin Soljačić",
"Ido Kaminer"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170926162109",
"title": "Tunable UV-Emitters through Graphene Plasmons"
} |
familyname_firstname 13889 DESY-PROC-2017-XX Patras 2017First Axion Dark Matter Search with Toroidal Geometry Byeong Rok KoCenter for Axion and Precision Physics Research (CAPP), Institute for Basic Science (IBS), Daejeon 34141, Republic of Korea December 30, 2023 ======================================================================================================================================================We report the first axion dark matter search with toroidal geometry. Exclusion limits of the axion-photon coupling g_aγγ over the axion mass range from 24.7 to 29.1 μeV at the 95% confidence level are set through this pioneering search. Prospects for axion dark matter searches with larger scale toroidal geometry are also given. § INTRODUCTION In the last Patras workshop at Jeju Island in Republic of Korea, we, IBS/CAPP, introduced axion haloscopes with toroidal geometry we will pursue <cit.>. At the end of our presentation, we promised that we will show up at this Patras workshop with “CAPPuccino submarine”. The CAPPuccino submarine is a copper (cappuccino color) toroidal cavity system whose lateral shape is similar to a submarine as shown in Fig. <ref>.r0.45< g r a p h i c s >Lateral (left) and top (right) views of the CAPPuccino submarine. Note that it is a cut-away view to show details of the system. We are now referring to the axion dark matter searches with toroidal geometry at our center as ACTION for “Axion haloscopes at CAPP with ToroIdal resONators” and the ACTION in this proceedings is the “simplified ACTION”. In this proceedings, we mainly show the first axion haloscope search from the simplified ACTION experiment and also discuss the prospects for larger scale ACTION experiments <cit.>.§ SIMPLIFIED ACTION The simplified ACTION experiment constitutes a tunable copper toroidal cavity, toroidal coils which provide a static magnetic field, and a typical heterodyne receiver chain. The experiment was conducted at room temperature. A torus is defined by x=(R+rcosθ)cosϕ, y=(R+rcosθ)sinϕ, and z=rsinθ, where ϕ and θ are angles that make a full circle of radius R and r, respectively. As shown in Fig. <ref>, R is the distance from the center of the torus to the center of the tube and r is the radius of the tube. Our cavity tube's R and r are 4 and 2 cm, respectively, and the cavity thickness is 1 cm. r0.45< g r a p h i c s >Magnetic field as a function of radial position |R+r| at θ=0. Dashed (blue) lines are obtained from the finite element method and correspond to the toroidal cavity system, and solid lines (red) correspond to the cavity tube. Dots with error bars are measurement values. The results at positive R+r are along a coil, while those at negative R+r are between two neighboring coils.The frequency tuning system constitutes a copper tuning hoop whose R and r are 4 and 0.2 cm, respectively, and three brass posts for linking between the hoop and a piezo linear actuator that controls the movement of our frequency tuning system. The quasi-TM_010 (QTM_1) modes of the cavity are tuned by moving up and down our frequency tuning system along the axis parallel to the brass posts. Two magnetic loop couplings were employed, one for weakly coupled magnetic loop coupling and the other for critically coupled magnetic loop coupling, i.e. β≃1 to maximize the axion signal power in axion haloscope searches <cit.>. r0.45< g r a p h i c s >Form factors of the QTM_1 mode of the toroidal cavity as a function of the QTM_1 frequency.A static magnetic field was provided by a 1.6 mm diameter copper wire ramped up to 20 A with three winding turns, as shown in Fig. <ref>. Figure <ref> shows good agreement between measurement with a Hall probe and a simulation <cit.> of the magnetic field induced by the coils. The B_ avg from the magnetic field map provided by the simulation turns out to be 32 G.With the magnetic field map and the electric field map of the QTM_1 mode in the toroidal cavity, we numerically evaluated the form factor of the QTM_1 mode as a function of the QTM_1 frequency, as shown in Fig. <ref>, where the highest frequency appears when the frequency tuning system is located at the center of the cavity tube, such as in Fig. <ref>. As shown in Fig. <ref>, we found no significant drop-off in the form factors of the QTM_1 modes, which is attributed to the absence of the cavity endcaps in toroidal geometry.Our receiver chain consists of a single data acquisition channel that is analogous to that adopted in ADMX <cit.> except for the cryogenic parts. Power from the cavity goes through a directional coupler, an isolator, an amplifier, a band-pass filter, and a mixer, and is then measured by a spectrum analyzer at the end. Cavity associates, ν (resonant frequency), and Q_L (quality factor with β≃1) are measured with a network analyzer by toggling microwave switches. The gain and noise temperature of the chain were measured to be about 35 dB and 400 K, respectively, taking into account all the attenuation in the chain, for the frequency range from 6 to 7 GHz.The signal-to-noise ratio (SNR) in the simplified ACTION experiment isSNR=P_a,g_aγγ∼6.5×10^-8 GeV^-1/P_n√(N),where P_a,g_aγγ∼6.5×10^-8 GeV^-1 is the signal power when g_aγγ∼6.5×10^-8 GeV^-1, which is approximately the limit achieved by the ALPS collaboration <cit.>. P_n is the noise power equating to k_B T_n b_a, and N is the number of power spectra, where k_B is the Boltzmann constant, T_n is the system noise temperature which is a sum of the noise temperature from the cavity (T_n, cavity) and the receiver chain (T_n, chain), and b_a is the signal bandwidth. We iterated data taking as long as β≃ 1, or equivalently, a critical coupling was made, which resulted in about 3,500 measurements. In every measurement, we collected 3,100 power spectra and averaged them to reach at least an SNR in Eq. (<ref>) of about 8, which resulted in an SNR of 10 or higher after overlapping the power spectra at the end. r0.45< g r a p h i c s >Excluded parameter space at the 95% C.L. by this experiment together with previous results from ALPS <cit.> and CAST <cit.>. No limits are set from 6.77 to 6.80 GHz due to with a quasi-TE mode in that frequency region and the TE mode is also confirmed by a simulation <cit.>.Our overall analysis basically follows the pioneer study described in Ref. <cit.>. With an intermediate frequency of 38 MHz, we take power spectra over a bandwidth of 3 MHz, which allows 10 power spectra to overlap in most of the cavity resonant frequencies with a discrete frequency step of 300 kHz. Power spectra are divided by the noise power estimated from the measured and calibrated system noise temperatures. The five-parameter fit also developed in Ref. <cit.> is then employed to eliminate the residual structure of the power spectrum. The background-subtracted power spectra are combined in order to further reduce the power fluctuation. We found no significant excess from the combined power spectrum and thus set exclusion limits of g_aγγ for 24.7<m_a<29.1 μeV. No frequency bins in the combined power spectrum exceeded a threshold of 5.5σ_P_n, where σ_P_n is the rms of the noise power P_n. We found σ_P_n was underestimated due to the five-parameter fit as reported in Ref. <cit.> and thus corrected for it accordingly before applying the threshold of 5.5σ_P_n. Our SNR in each frequency bin in the combined power spectrum was also combined with weighting according to the Lorentzian lineshape, depending on the Q_L at each resonant frequency of the cavity. With the tail of the assumed Maxwellian axion signal shape, the best SNR is achieved by taking about 80% of the signal and associate noise power; however, doing so inevitably degrades SNR in Eq. (<ref>) by about 20%. Because the axion mass is unknown, we are also unable to locate the axion signal in the right frequency bin, or equivalently, the axion signal can be split into two adjacent frequency bins. On average, the signal power reduction due to the frequency binning is about 20%. The five-parameter fit also degrades the signal power by about 20%, as reported in Refs. <cit.>. Taking into account the signal power reductions described above, our SNR for g_aγγ∼6.5×10^-8 GeV^-1 is greater or equal to 10, as mentioned earlier. The 95% upper limits of the power excess in the combined power spectrum are calculated in units of σ_P_n; then, the 95% exclusion limits of g_aγγ are extracted using g_aγγ∼6.5×10^-8 GeV^-1 and the associated SNRs we achieved in this work. Figure <ref> shows the excluded parameter space at a 95% confidence level (C.L.) by the simplified ACTION experiment. r0.5< g r a p h i c s >Expected exclusion limits by the large (solid lines with a single-cavity and dashed lines with a 4-cavity) and small (dotted lines with a single-cavity) ACTION experiments. Present exclusion limits <cit.> are also shown. § PROSPECTS FOR AXION DARK MATTER SEARCHES WITH LARGER SCALE TOROIDAL GEOMETRY The prospects for axion dark matter searches with two larger-scale toroidal geometries that could be sensitive to the KSVZ <cit.> and DFSZ <cit.> models are now discussed. A similar discussion can be found elsewhere <cit.>. One is called the “large ACTION”, and the other is the “small ACTION”, where the cavity volume of the former is about 9,870 L and that of the latter is about 80 L. The B_ avg targets for the large and small ACTION experiments are 5 and 12 T, respectively, where the peak fields of the former and latter would be about 9 and 17 T. Hence, the large and small toroidal magnets can be realized by employing NbTi and Nb_3Sn superconducting wires, respectively. The details of the expected experimental parameters for the ACTION experiments can be found in <cit.> and Fig. <ref> shows the exclusion limits expected from the large and small ACTION experiments. § SUMMARY In summary, we, IBS/CAPP, have reported an axion haloscope search employing toroidal geometry using the simplified ACTION experiment. The simplified ACTION experiment excludes the axion-photon coupling g_aγγ down to about 5×10^-8 GeV^-1 over the axion mass range from 24.7 to 29.1 μeV at the 95% C.L. This is the first axion haloscope search utilizing toroidal geometry since the advent of the axion haloscope search by Sikivie <cit.>. We have also discussed the prospects for axion dark matter searches with larger scale toroidal geometry that could be sensitive to cosmologically relevant couplings over the axion mass range from 0.79 to 15.05 μeV with several configurations of tuning hoops, search modes, and multiple-cavity system. § ACKNOWLEDGMENTS This work was supported by IBS-R017-D1-2017-a00.99 BRKo_PATRAS2016 B. R. Ko, “Contributed to the 12th Patras workshop on Axions, WIMPs and WISPs, Jeju Island, South Korea, June 20 to 26, 2016”; <arXiv:1609.03752>.SIMPLE_ACTION J. Choi, H. Themann, M. J. Lee, B. R. Ko, and Y. K. Semertzidis, Phys. Rev. D 96, 061102(R) (2017).sikivie P. Sikivie, Phys. Rev. Lett. 51, 1415 (1983).CST <http://www.cst.com>.ADMX_NIM H. Peng et al., Nucl. Instrum. Methods Phys. Res., Sect. A 444, 569 (2000).ALPS2010 K. Ehret et al., Phys. Lett. B 689, 149 (2010).haloscope4 C. Hagmann et al., Phys. Rev. Lett. 80, 2043 (1998); S. J. Asztalos et al., Phys. Rev. D 64, 092003 (2001).HF B. M. Brubaker et al., Phys. Rev. Lett. 118, 061302 (2017).CAST2017 V. Anastassopoulos et al. (CAST Collaboration), Nature Physics 13, 584-590 (2017).haloscope2 S. DePanfilis et al., Phys. Rev. Lett. 59, 839 (1987); W. U. Wuensch et al., Phys. Rev. D 40, 3153 (1989).haloscope3 C. Hagmann, P. Sikivie, N. S. Sullivan, and D. B. Tanner, Phys. Rev. D 42, 1297 (1990).haloscope5 S. J. Asztalos et al., Astrophys. J. Lett. 571, L27 (2002); Phys. Rev. D 69, 011101(R) (2004); Phys. Rev. Lett. 104, 041301 (2010).KSVZ1 J. E. Kim, Phys. Rev. Lett. 43, 103 (1979). KSVZ2 M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 166, 493 (1980).DFSZ1 A. R. Zhitnitskii, Sov. J. Nucl. Phys. 31, 260 (1980). DFSZ2 M. Dine, W. Fischler, and M. Srednicki, Phys. Lett. B 140, 199 (1981).DIPOLE Oliver K. Baker et al., Phys. Rev. D 85, 035018 (2012). | http://arxiv.org/abs/1709.09437v1 | {
"authors": [
"B. R. Ko"
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"categories": [
"hep-ex"
],
"primary_category": "hep-ex",
"published": "20170927103806",
"title": "First Axion Dark Matter Search with Toroidal Geometry"
} |
[email protected]@damtp.cam.ac.uk We study the cosmological effects of adding terms of higher-order in the usual energy-momentum tensor to the matter Lagrangian of general relativity. This is in contrast to most studies of higher-order gravity which focus on generalising the Einstein-Hilbert curvature contribution to the Lagrangian. The resulting cosmological theories give rise to field equations of similar form to several particular theories with different fundamental bases, including bulk viscous cosmology, loop quantum gravity, k-essence, and brane-world cosmologies. We find a range of exact solutions for isotropic universes, discuss their behaviours with reference to the early- and late-time evolution, accelerated expansion, and the occurrence or avoidance of singularities. We briefly discuss extensions to anisotropic cosmologies and delineate the situations where the higher-order matter terms will dominate over anisotropies on approach to cosmological singularities. DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd., Cambridge CB3 0WA, U.K. Cosmological Models in Energy-Momentum-Squared Gravity John D. Barrow December 30, 2023 ======================================================§ INTRODUCTION The twin challenges of naturally explaining two periods of accelerated expansion during the history of the universe engage the attentions of many contemporary cosmologists. The first period may have had a beginning and necessarily came to an end when the universe was young and hot: it is called a period of `inflation' and it leaves observable traces in the cosmic microwave background radiation that are believed to have been detected. The second period of acceleration began only a few billion years ago and is observed in the Hubble flow traced by type IA supernovae; it is not known if it will ever come to an end or is changing in any way. There are separate non-unique mathematical descriptions of each of these periods of acceleration but there is no single explanation of both of them, nor any insight into whether or not they are related, or even random, occurrences. For these reasons, there is continuing interest in all the different ways in which expanding universes can undergo periods of accelerated expansion. In the case of late-time acceleration the simplest description of an effectively anti-gravitating stress, known as `dark energy', is provided by introducing a cosmological constant (Λ) into general relativity with a value arbitrarily chosen to match observations.The best-fit theory of this sort is dubbed ΛCDM and in its simplest form is defined by six constants (which determine Λ) that can be fixed by observation. One of those parameters is Λ and its required value is difficult to explain: it requires a theory that contributes an effective vacuum stress of magnitude Λ∼ (t_pl/t_0)^2 ∼ 10^-120 at a time of observation t_0∼ 10^17s, where t_pl∼ 10^-43s is the Planck time <cit.>. Other descriptions that lead to slowly evolving scalar fields in place of a constant Λ have also been explored, together with a range of modified gravity theories that contribute anti-gravitating stresses. There are many such modifications and extensions of Einstein's general relativity and they can be tuned to provide acceleration at early or late times. So far, almost all of these modifications to general relativity have focussed on generalising the gravitational Lagrangian away from the linear function of the spacetime curvature, R, responsible for the Einstein tensor in Einstein's equations. A much-studied family of theories of this sort are those deriving from a Lagrangian of the form F(R), where F is some analytic function. By contrast, in this paper we will explore some of the consequences of generalising the form of the matter Lagrangian in a nonlinear way, to some analytic function of T_μνT^μν, where T_μν is the energy-momentum tensor of the matter stresses. This is more radical than simply introducing new forms of fluid stress, like bulk viscosity or scalar fields, into the Einstein equations in order to drive acceleration in Friedmann-Lemaître-Robertson-Walker (FLRW) universes.In <ref> we discuss and motivate higher order contributions to gravity from matter terms. In <ref> we derive the equations of motion for a generic F(R,T_μνT^μν) modification of the action with bare cosmological constant, before specialising to the case F(R,T_μνT^μν)=R+η (T_μνT^μν)^n. We then investigate several features of the isotropic cosmology in this theory in <ref> and, finally, move to the anisotropic Bianchi type I setting in <ref>.§ BACKGROUND§.§ Field equations Einstein's theory of general relativity (GR) with cosmological constant Λ can be derived from the variation of the action, S=1/2κ∫√(-g)(R-2Λ )d^4x+∫√(-g) L_md^4x,where κ =8π G and L_m is the matter Lagrangian, which we will take to describe a perfect fluid; R≡ R_a^a, where R_b^a is the Ricci tensor, and g is the determinant of the metric itself. Here, and in all that follows, we use units in which c=1.An isotropic and homogeneous universe may be described by the FLRW metric: ds^2=-dt^2+a^2(t)( dr^2/1-kr^2+r^2( dθ ^2+sin ^2θ dϕ ^2) ) ,where k, the curvature parameter, takes the values {-1, 0, +1} corresponding to open, flat and closed 3-spaces, respectively; t is the comoving proper time and a(t) is the expansion scale factor.There are many proposals to modify or extend the ΛCDM cosmological picture. These fall broadly into two categories, depending on which side of the Einstein field equations is modified. We can modify the right-hand side of the Einstein equations by adding new forms of matter that will drive expansion either at early times, as in the theory of inflation, or at late times, such as in quintessence or k-essence scenarios <cit.>. Alternatively, we can modify the left-hand side of the Einstein equations in order to modify the effect of gravity itself. There are several ways to do this, including F(R) theories <cit.> in which the Ricci scalar in (<ref>) is replaced by some function f(R), so-called F(T) theories in which we modify the teleparallel equivalent of general relativity <cit.>, or scalar-tensor theories in which a scalar field is coupled to the Ricci scalar. §.§ Higher-order matter contributions The type of generalisation of general relativity we will explore in this paper looks to include higher-order contributions to the right-hand side of the Einstein equations, where the material stresses appear. This results in field equations that include new terms that enter at high densities and pressures, which may be anti-gravitational in their effects. Typically, they affect the cosmological model at high densities and may alter the conclusions regarding the appearance of spacetime singularities in the finite cosmological past. Conversely, we might expect their effects at late times and low cosmological densities to be very small. Even within general relativity, there is scope to include high-order matter contributions, as the Einstein equations have almost no content unless some prescription or constraint is given on the forms of matter stress. Thus, in the general-relativistic Friedmann models, we can introduce non-linear stresses defined by relations between pressure, p, and density, ρ, of the form ρ +p=γρ ^n, <cit.>, or f(ρ ), <cit.>, where γ≥ 0 and n are constants, or include a bulk viscous stress into the equation of state of the standard form p=(γ -1)ρ -3Hς (ρ ),where H is the Hubble expansion rate and ς≥ 0 is the bulk viscosity coefficient <cit.>. The so-called Chaplygin and generalised Chaplygin gases are just special cases of these bulk viscous models, and choices of n or ς∝ρ ^m introduce higher-order matter corrections. Similarly, the choice of self-interaction potential V(ϕ ) for a scalar field can also introduce higher-order matter effects into cosmology. Analogously, in scalar-tensor theories like Brans-Dicke (BD) which are defined by a constant BD coupling constant, ω, generalisations are possible to the cases where ω becomes a function of the BD scalar field. In all these extensions of the standard relativistic perfect fluid cosmology there will be several critical observational tests which will constrain them. In particular, in higher-order matter theories the inevitable deviations that can occur from the standard thermal history in the early radiation era will change the predicted abundances of helium-4 and deuterium and alter the detailed structure of the microwave background power spectrum. Also, as we studied for Brans-Dicke theory <cit.>, changes in the cold dark matter dominated era evolution can shift the time when matter and radiation densities are equal. This is the epoch when matter perturbations begin to grow and sensitively determines the peak of the matter power spectrum. At a later nonlinear stage of the evolution, higher-order gravity theories will effect the formation of galactic halos. This has been investigated for bulk viscous cosmologies by Li and Barrow <cit.>. These observational constraints will form the subject of a further paper and will not be discussed here.If we depart from general relativity, then various simple quantum gravitational corrections are possible, and have been explored. The most well known are the loop quantum gravity (LQG) <cit.> and brane-world<cit.> scenarios that contribute new quadratic terms to the Friedmann equation for isotropic cosmologies by replacing ρ by ρ (1± O(ρ ^2)) in the Friedmann equation, where the - contribution is from LQG and the + is from brane-world scenarios. The impact on anisotropic cosmological models is more complicated and not straightforward to calculate <cit.>. In particular, we find that simple forms of anisotropic stress are no longer equivalent to a p=ρ fluid as we are used to finding in general relativity. Our study will be of a type of higher-order matter corrections which modify the Friedmann equations in ways that include both of the aforementioned types of phenomenological modification to the form of the Friedmann equations, although the underlying physical theory does not incorporate the LQC or brane-world models or reduce to them in a limiting case.Standard F(R) theories of gravity <cit.> can be generalised to include a dependence of the form S=1/2κ∫√(-g)F(R,L_m)d^4x. This is in some sense an extremal extension of the Einstein-Hilbert action, as discussed in <cit.>. If the coupling between matter and gravity is non-minimal, then there will be an extra force exerted on matter, resulting in non-geodesic motion and a violation of the equivalence principle. This type of modification has been investigated in several contexts, particularly when the additional dependence on the matter Lagrangian arises from F taking the form F(R,𝒯) where 𝒯 is the trace of the energy-momentum tensor <cit.>.A theory, closely related to F(R,𝒯) gravity, that allows the gravitational Lagrangian to depend on a more complicated scalar formed from the energy-momentum tensor is provided by F(R,𝐓^2), where 𝐓^2≡ T_μνT^μν is the scalar formed from the square of the energy-momentum tensor. This was first discussed in <cit.> , and the special case with F(R,𝐓^2)=R+η𝐓^2,where η is a constant, was also discussed in <cit.>, where the authors investigated the possibility of a bounce at early times when η <0 (although in that paper they used the opposite sign convention to us for η), and also found an exact solution for charged black holes in the extended theory. In <cit.> a similar form, with additional cross terms between the Ricci and Energy-momentum tensors, was discussed as arising from quantum fluctuations of the metric tensor. Recently the authors of <cit.> investigated the late time acceleration of universes described by this model in the dust-only case, and used observations of the Hubble parameter to constrain the parameters of the theory.We would expect the theory derived from (<ref>) to provide different physics to the F(R,𝒯) case. Indeed, one example of this is the case of a perfect fluid with equation of state p=-1/3ρ. The additional terms in F(R,𝒯) will vanish as 𝒯=0, but in the F(R,𝐓^2) theory the extra terms in 𝐓^2 will not vanish and we will find new cosmological behaviour. In <ref>, we will investigate the cosmological solutions in a more general setting, where the 𝐓^2 term may be raised to an arbitrary power.§ FIELD EQUATIONS FOR F(R,T_ΜΝT^ΜΝ) GRAVITY WITH COSMOLOGICAL CONSTANT In <cit.> the Friedmann equations were derived in the case where F is given by (<ref>), for a flat FLRW cosmology. A `bare' cosmological constant was also included on the left-hand side of the field equations (rather than as an effective energy-momentum tensor for the vacuum). In <cit.>, the field equations were derived without a cosmological constant and specialised to two particular models. We first derive the equations of motion with a cosmological constant for general F, before specialising to theories where the additional term takes the form (𝐓^2)^n, and determining the FLRW equations with general curvature. In GR, the cosmological constant can be considered to be, equivalently, either a `bare' constant on the left-hand side of the Einstein equations, or part of the matter Lagrangian. As discussed in <cit.>, the two are no longer equivalent in this theory, due to the non-minimal nature of the curvature-matter couplings. A similar inequivalence also occurs in other models that introduce non-linear matter terms, including loop quantum cosmology. We will assume that the cosmological constant arises in its bare form as part of the gravitational action. This gives the modified action S=1/2κ∫√(-g)(F(R,T^μνT_μν)-2Λ )d^4x+∫ L_m√(-g) d^4x, where L_m is taken to be the same as the matter component contributed by T_μν. Since the gravitational Lagrangian now depends on 𝐓^2, we note that the new terms in the variation of the action will arise from the variation of this square, via δ (T_μνT^μν). To calculate this, we define T_μν by T_μν=-2/√(-g)δ(√(-g)L_m)/δ g^μν . We enforce the condition that L_m depends only on the metric components, and not on their derivatives, to find T_μν=g_μνL_m-2∂ L_m/∂ g^μν. Varying with respect to the inverse metric, we define θ _μν≡δ (T_αβT^αβ)/δ g^μν=-2L_m(T_μν-1/2g_μνT)-TT_μν+2T_μ^αT_να-4T^αβ∂ ^2L_m/∂ g^μν∂ g^αβ,where T is the trace of the energy-momentum tensor. Varying the action in this way, we find δ S=1/2κ∫{F_Rδ R+F_T^2δ (T_μνT^μν)-1/2g_μνFδ g^μν+Λ +1/√(-g)δ (√(-g)L_m)}d^4x,where subscripts denote differentiation with respect to R and 𝐓^2, respectively.From this variation we obtain the field equations: F_RR_μν-1/2Fg_μν+Λ g_μν+(g_μν∇_α∇^α-∇_μ∇_ν)F_R= κ(T_μν-1/κF_𝐓^2θ_μν). These reduce, as expected, to the field equations for F(R) gravity in the special case where F(R,𝐓^2)=F(R) <cit.> and to the Einstein equations with a cosmological constant when F(R,𝐓^2)=R.We will assume that the matter component can be described by a perfect fluid, T_μν=(ρ +p)u_μu_ν+pg_μν,where ρ is the energy density and p the pressure; hence T_μνT^μν=ρ^2+3p^2. Furthermore, we take the Lagrangian L_m=p. This means that the final term in the definition of θ _μν vanishes and allows us to calculate the form of θ _μν independently of the function F . Substituting (<ref>) into (<ref>), we find θ_μν=-(ρ^2+4pρ+3p^2)u_μ u_ν. We now proceed to specify a particular form for F(R,𝐓^2) which includes and generalises the models used in <cit.> and for energy-momentum-squared gravity in <cit.> (EMSG). This form is F(R,T_μνT^μν)=R+η (T_μνT^μν)^n, where n need not be an integer. This corresponds to EMSG in the case n=1 , and to Models A and B of <cit.> when n=1/2 and n=1/4, respectively; it reduces the field equations to R_μν-1/2Rg_μν+Λ g_μν=κ(T_μν+η/κ(T_αβT^αβ)^n-1[1/2 (T_αβT^αβ)g_μν-nθ_μν]), which we rewrite as G_μν+Λ g_μν=κ T^eff_μν, where G_μν is the Einstein tensor, to show the relationship to general relativity. Continuing with the perfect fluid form of the energy-momentum tensor, this expands to give: G_μν+Λ g_μν=κ ((ρ+p)u_μu_ν+pg_μν)+η (ρ ^2+3p^2)^n-1[1/2(ρ ^2+3p^2)g_μν+n(ρ +p)(ρ +3p)u_μu_ν]. § ISOTROPIC COSMOLOGYIf we assume a FLRW universe with curvature parameter k, we find the generalised Friedmann equation, ( ȧ/a) ^2+k/a^2=Λ/3 +κρ/3+η/3(ρ ^2+3p^2)^n-1[(n-1/2)(ρ ^2+3p^2)+4nρ p] , and acceleration equation ä/a=-κρ +3p/6+Λ/3-η/3(ρ ^2+3p^2)^n-1[ n+1/2(ρ ^2+3p^2)+2nρ p ] . If the matter field obeys a barotropic equation of state, p=wρ with w constant, then the non-GR terms are all of the form ρ ^2n multiplied by a constant. Thus, the generalised Friedmann equation becomes ( ȧ/a) ^2+k/a^2=Λ/3 +κρ/3+ηρ ^2n/3A(n,w),where A is a constant depending on the choice of n and w, given by A(n,w)≡ (1+3w^2)^n-1[(n-1/2)(1+3w^2)+4nw],and the acceleration equation becomes ä/a=-κ1+3w/6ρ +Λ/3-ηρ ^2n/3B(n,w),where B a constant given by B(n,w)≡ (1+3w^2)^n-1[n+1/2(1+3w^2)+2nw]. Finally, we determine the generalised continuity equation, by differentiating the generalised Friedmann equation, ρ̇=-3ȧ/aρ(1+w)[κ+ηρ^2n-1n(1+3w)(1+3w^2)^n-1/κ+2ηρ^2n-1nA(n,w)] ,where we have written it in a form that makes clear the generalisation of the GR case.We can see immediately that there is an interesting difference between the FLRW equations in GR and in EMSG. When η =0 there are solutions with finite a,ȧ,and ρ but infinite values of p and ä. These are called sudden singularities <cit.> and can be constructed explicitly. In EMSG, where η≠ 0, the appearance of the pressure, p, explicitly in the Friedmann equation changes the structure of the equations and the same type of sudden singularity is no longer possible at this order in derivatives of a. §.§ Integrating the continuity equation We now attempt to determine the cosmological behaviour of some cases where the modified continuity equation can be integrated exactly. We find four simply integrable cases: two of these are for fixed w independent of the value of n, the other two occur for specific values of w dependent on the choice of n, although we note that some of these integrable cases may coincide, depending on our choice of the exponent, n.The first case that can be integrated is for the equation of state corresponding to dark energy, w=-1, where the entire right-hand side of(<ref>) vanishes, and so ρ≡ρ _0, a constant. In this case we expect to find a solution to the modified Friedmann equation that is the same as the solution in GR except with altered constants, which results in a de Sitter solution where H≡ȧ/a= constant, and the universe expands exponentially.Next, we consider the case w=-1/3, which corresponds to an effective perfect fluid representing a negative curvature, so the numerator in the modified continuity equation becomes simply κ , and we can integrate (<ref>) since ρ̇(1/ρ+2η n A(n,-1/3)/κρ^2n-2)= -2ȧ/a,d/dt(ln(ρ)-η n (4/3)^n/(2n-1)κρ^2n-1)= d/dt(ln(a^-2)),ρexp(-η n (4/3)^n/(2n-1)κρ^2n-1)= Ca^-2,with C>0 a constant of integration.We can also integrate the continuity equation when the correction factor in(<ref>) is equal to 1, which occurs when (1+3w)(1+3w^2)^n-1=2A(n,w).The continuity equation then reduces to the standard GR form for these special values, w=w_∗, and so we have ρ =Ca^-3(1+w_∗). The final possibility that we consider is when n(1+3w)(1+3w^2)^n-1=A(n,w),in which case we can write (<ref>) as d/dt(ln(κρ+nηρ^2n(1+3w_*)(1+3w_*^2)^n-1))=d/dt(ln(a^-3(1+w_*))),which integrates to κρ+nηρ^2n(1+3w_*)(1+3w_*^2)^n-1=Ca^-3(1+w_*). We note that, depending on the choice of exponent n, some of the second pair of solutions may exist for multiple choices of w, or may coincide with each other, or with the w=-1, w=-1/3 cases. Also, for some choices of n, there may be no solutions at all.Finally, note that only one of these solutions allows easy integration of the modified Friedmann equation (<ref>). This is the case when w=-1 and so ρ =ρ _0. In this case the Friedmann-like equation becomes ( ȧ/a) ^2+k/a^2=α (Λ ,n),where α is a constant given by α (Λ ,n)≡Λ/3+κρ _0/3-ηρ _0^2n/64^n. The solution to the modified Friedmann equation is then given by a(t)=1/2√(α)(C√(α)+k/C√(α) )cosh (√(α)t)± (C√(α)-k/C√(α))sinh (√(α)t)where C is a new constant of integration. Equivalently, we can write this solution in terms of exponentials as a(t)=1/2√(α)( C√(α)e^√(α)t+ k/C√(α)e^-√(α)t)as well as its time reversal, t→ -t. Assuming α >0, we can see that this reduces to the expected de Sitter solution from general relativity in the case k=0, as we would expect. If α <0 then, writing instead α→ -α, there is a real solution only for negative curvature, where we must choose k=-C^2α, giving the anti-de Sitter solution a(t)=Ccos (√(α)t).It is important to note that because of the form of α, unlike in the unmodified case, we do not necessarily require a negative cosmological constant to find this solution. We would expect this anti-de Sitter analogue to appear whenever η >0, for suitable choices of ρ _0 and n.This solution is very similar to the case of w=-1 in GR, where we can rewrite the cosmological constant as a perfect fluid with this equation of state. This is possible in GR because the continuity equations for non-interacting multi-component fluids decouple, allowing us to treat them independently. Unfortunately, because of the additional non-linear terms arising in these F(R,𝐓^2) models (except in the special case n=1/2), we cannot decouple different fluids in this way and then subsequently superpose them in our Friedmann-like equations. This means that we cannot replace the curvature or cosmological constant terms with perfect fluids with w=-1/3 and -1 as in classical GR. However, for some choices of n and η , the correction terms can themselves provide an additional late-time or early inflationary repulsive force, removing the need for an explicit cosmological constant. §.§ Energy-momentum-squared gravity: the case n=1 If we fix our choice of n, then we can say more about the behaviour of the specific solutions that arise. In what follows we consider primarily the case n=1 which was originally discussed in <cit.>, under the name `energy-momentum squared gravity'. After specialising to n=1, we can say more about the solutions to the continuity equation found in the previous section, and investigate the modified Friedmann equations. The form of the Friedmann equations, after setting n=1 in (<ref>),(<ref>) and (<ref>), is: ( ȧ/a) ^2+k/a^2= Λ/3 +κρ/3+ηρ ^2/6(3w^2+8w+1)ä/a= Λ/3-κ1+3w/6ρ -ηρ ^2/3(3w^2+2w+1)ρ̇=-3ȧ/aρ (1+w)κ +ηρ (1+3w)/κ +ηρ (3w^2+8w+1) The new terms in the Friedmann equations are quadratic in the energy density, which we would expect to dominate in the very early universe as ρ→∞. Additionally, if we choose η <0, then the modified Friedmann equations in this model are similar to the effective Friedmann equations arising in loop quantum cosmology, <cit.>, where ( ȧ/a) =κ/3ρ( 1-ρ/ρ _crit) ,which may warrant further investigation. An analogous higher-order effect occurs in brane world cosmologies, where there is an effective equation of state with <cit.> p^eff=1/2λ(ρ ^2+2pρ );λ >0 constant. We briefly summarise the values of w for which the results of the previous section allow us to integrate the Friedmann equation and find the values of w that satisfy (<ref>) and (<ref>). If we set n=1 then(<ref>) reduces to 3w^2+5w=0,which has solutions w=-5/3 and w=0. The w=0 solution describes `dust' matter. The case w=-5/3 corresponds to some form of phantom energy, which will result in a Big Rip singularity, <cit.>, at finite future time.Alternatively, solving (<ref>) for n=1 gives 3w^2+2w-1=0,which has solutions w=-1 and w=1/3. The first of these has already been found for all n as the first case above, whilst the second gives a solution corresponding to blackbody radiation. Hence, we have exact solutions to the continuity equation for the cases w={-5/3,-1,- 1/3,0,1/3} which include the physically important cases of dust and radiation.The equation of state p=0 corresponds to pressureless dust or non-relativistic cold dark matter, and as shown above, we recover the same dependence of the energy density on the scale factor as in the GR case, ρ =Ca^-3. If we combine this with the modified acceleration and Friedmann equations for w=0 we find aä+2ȧ^2+k=Λ/2a^2+κ/4Ca^-1. If we consider only flat space (k=0) then we find a(t)=(4Λ )^-1/3((C^2+D+1)cosh( √(3Λ/2)t) +(C^2+D-1)sinh( √(3Λ/2)t) -2C)^1/3, where D is a constant of integration, and we have eliminated a further constant by a covariant translation of the time coordinate. We can then findρ explicitly, using (<ref>). We can see, however, that this form of the solution does not capture the case Λ =0. In this case, instead we find the solution a(t)=( 3/8C) ^1/3(C^2t^2-16D)^1/3, whichgives the GR dust behaviour of a∼ t^2/3 at large t.In the case of w=-1/3, we can write ρexp( -4η/3κρ) =Ca^-2.After differentiation and multiplication by a^2, we can write ȧ/a=ρ̇/ρ(1-4η/3κρ )and so in the case k=0 we can write the Friedmann equation in terms of ρ without any exponentials, as ( ρ̇/ρ) ^2(1-4η/3κρ )^21/4C^2=Λ/3+κ/3ρ -2η/9ρ ^2.Finally, in the case of w=1/3, which corresponds to radiation,<cit.> gave a solution in the case of flat space, a(t)∝√( cosh (α t)) where α≡√(4Λ/3). We see that in this case we can write the continuity equation as κρ +2ηρ ^2=Ca^-4,and that in the Friedmann and acceleration equations, the density terms are of equal magnitude but opposite sign. We can then sum the two to find our equation for a(t) ( ȧ/a) ^2+ä/a+k/a^2= 2Λ/3, which we solve by use of the substitution y=a^2 to find a^2(t)=1/4Λ((1+9k^2-12Λ D)cosh (α t)+(1-9k^2+12Λ D)sinh (α t)+6k)for all Λ, k non-zero, with α≡√(4Λ/ 3), as above. In the Λ =0 subcase, we find the solutions a^2(t)={[ Dt-kt^2k≠ 0;Dt k=0; ] . .§.§ de Sitter-like solutions de Sitter solutions arise in EMSG theory. They have constant density and Hubble parameter, which includes the case w=-1. In ΛCDM we expect this to arise in two situations. The first is when we have ρ≡ 0, that is an empty universe whose expansion is controlled solely by Λ , and the second is the similar dark-energy equation of state w=-1 for which the perfect fluid behaves as a cosmological constant. In EMSG we find that there is an extra family of de Sitter solutions. We describe them first for general n, then specialise to EMSG.Since we are searching for solutions with H≡ H_0 and ρ≡ρ _0, from (<ref>) we must have k=0, and the Friedmann equation then reduces to an algebraic one for H^2 in terms of ρ _0. Similarly, since Ḣ=0, (<ref>) reduces to another relation for H^2. Equating the two to remove H^2 and simplifying, we find that ρ _0 must satisfy ρ _0(1+w)(κ +nη (1+3w^2)^n-1(1+3w)ρ _0^2n-1)=0.There are the two standard solutions, w=-1 and ρ _0=0, but the additional factor gives us another family of solutions, with ρ _0^2n-1=-κ/nη (1+3w^2)^n-1(1+3w).In the case of EMSG, when we choose n=1, this condition reduces to ρ _0=-κ/η (1+3w)which gives us a constant density, exponentially expanding solution for every equation of state, w, excluding w=-1/3, for an appropriate sign of η. The existence of this extra de Sitter solution is reminiscent of its appearance in GR cosmologies with bulk viscosity <cit.>This unusual situation suggests that we investigate the stability of these n=1 solutions. We consider a linear perturbation about the constant density solution by writing ρ = ρ _0(1+δ ), H=H_0(1+ϵ ),The perturbed continuity equation is then given by ρ _0δ̇=-3(1+w)H_0(1+ϵ )ρ _0(1+δ ) κ +ηρ _0(1+δ )(1+3w)/κ +ηρ _0(1+δ )(3w^2+8w+1), If we use the expression for ρ _0 given in (<ref>), we can reduce this to δ̇=-3(1+w)(1+3w)H_0(1+ϵ )(1+δ )δ1/ (3w^2+5w)(1+3w^2+8w+1/3w^2+5wδ ). From the perturbation of the modified Friedmann equation we find that ϵ∼δ which means that after expanding to first order in δ, we have δ̇=-3H_0δ(1+w)(1+3w)/(3w+5)w,so small perturbations evolve as δ∝exp( -3H_0(1+w)(1+3w)/(3w+5)w) The exponent in (<ref>) is plotted in Figure <ref>, where we can see that these de Sitter-like solutions are indeed stable for a wide range of w values. This gives us an exponentially expanding universe for (almost) any equation of state as long as we set the density to the correct constant value. In particular, these solutions will be stable for w<-5 /3, -1<w<-1/3 and w>0, and unstable for -5/3<w<-1 and -1/3<w<0. It is also the case that, depending on the sign of the parameter η, some of these solutions will be unphysical, as they require negative energy density. For η <0, there will be no physical solutions for w<-1/3, whilst for η >0 there will be no solutions for w>-1/3. §.§ Early times: the bounce and high-density limits Examining the modified Friedmann equation (<ref>) in the case k≥ 0, we can see that as the left-hand side of the equation is a sum of positive terms, we must have Λ +κρ +ηρ ^2A(1,w)≥ 0, which can be split into two cases, for η A(1,w)<0 and η A(1,w)>0, respectively. The first case occurs for η <0and {w<α _- or w>α _+},η >0and {α _-<w<α _+}, where α _±=--4±√(13)/3are the roots of A(1,w)≡ 3w^2+8w+1=0. In this case we have a maximum possible density given by ρ _max=κ/2A(1,w)η( -1+√(1-4ηΛ A(1,w)/κ ^2)) ,indicating that a bounce occurs in this case, avoiding an initial singularity. In the second case, where η A(1,w)>0, there is no bounce and no maximum energy density.We now consider the solutions when k=0 in the high-density limit, where we assume the correction terms dominate over the ρ and Λ terms. We consider the case of general n, and find an analytic solution. The Friedmann and acceleration equations reduce to ( ȧ/a) ^2= η/3ρ ^2nA(n,w),ä/a=-η/3ρ ^2nB(n,w). From these, we can eliminate ρ to find ( ȧ/a) ^2+A(n,w)/B(n,w)ä/a =0. which has the solution a(t)=D[(A+B)t-C]^A/A+B where C and D are new constants of integration. We can then solve for the density: ρ (t)=( 3A/η) ^1/2n((A+B)t-C))^-1 /n.This solution is real (and thus not unphysical) only if A(n,w)/η is positive. In the case of EMSG, this condition reduces to the requirement that η and 3w^2+8w+1 must have the same sign. So, the two regions where this solution exists are, η >0and {w<α _- or w>α _+},η <0and {α _-<w<α _+}, where α _±=--4±√(13)/3are the roots of 3w^2+8w+1. These are complementary to the conditions for the bounce to occur, as previously discussed. This is as we would expect, with the high-density approximation failing at a maximum density, as in the case of a bounce.§ ANISOTROPIC COSMOLOGY There are several ways of introducing anisotropy into our cosmological models. We will consider the simplest generalisation of FLRW, in which we have a flat, spatially homogeneous universe, with anisotropic scale factors. This is the Bianchi type I universe, with metric given by <cit.> ds^2=-dt^2+a^2(t)dx^2+b^2(t)dy^2+c^2(t)dz^2,where a(t), b(t) and c(t) are the expansion scale factors in the x,y and z directions, respectively.Assuming that the energy-momentum tensor takes the form of a perfect fluid with principal pressures, p_1, p_2 and p_3, so L_m=1/3 (p_1+p_2+p_3), we can derive the field equations for Bianchi I universes in our higher-order matter theories: ȧḃ/ab+ḃċ/bc+ċȧ/ca =κρ +η/6(ρ ^2+∑_i=1^3p_i^2)^n-1[ (6n-3)ρ ^2+8nρ∑_i=1^3p_i+2n(∑_i=1^3p_i)^2-3∑_i=1^3p_i^2 ],ḃċ/bc+b̈/b+c̈/c=-κ p_1+η/6(ρ ^2+∑_i=1^3p_i^2)^n-1[2n(ρ +p_1-p_2-p_3)(2p_1-p_2-p_3)-3∑_i=1^3p_i^2],ċȧ/ca+c̈/c+ä/a=-κ p_2+η/6(ρ ^2+∑_i=1^3p_i^2)^n-1[2n(ρ +p_2-p_3-p_1)(2p_2-p_3-p_1)-3∑_i=1^3p_i^2],ȧḃ/ab+ä/a+b̈/b=-κ p_3+η/6(ρ ^2+∑_i=1^3p_i^2)^n-1[2n(ρ +p_3-p_1-p_2)(2p_3-p_1-p_2)-3∑_i=1^3p_i^2]. In the case of an isotropic pressure fluid (p_1=p_2=p_3=p): ȧḃ/ab+ḃċ/bc+ċȧ/ca =κρ +η/2(ρ ^2+3p^2)^n-1((2n-1)ρ ^2+8nρ p+(6n-3)p^2),ḃċ/bc+b̈/b+c̈/c=-κ p- η/2(ρ ^2+3p^2)^n-13p^2,ċȧ/ca+c̈/c+ä/a=-κ p- η/2(ρ ^2+3p^2)^n-13p^2,ȧḃ/ab+ä/a+b̈/b=-κ p- η/2(ρ ^2+3p^2)^n-13p^2. The first of these is the generalised Friedmann equation.Qualitatively, we expect that the higher-order density and pressure terms will dominate at early times to modify or remove (depending on the sign of η) the initial singularity when n>1/2, but will have negligible effects at late times, when the dynamics will approach the flat isotropic FLRW model. At early times, we know that in GR the singularity will be anisotropic and dominated by shear anisotropy whenever -ρ /3<p<ρ. In order to determine the dominant effects as t→ 0 we will simplify to the case of isotropic perfect fluid pressures (p_1=p_2=p_3=wρ). Now, we determine the dependence of the highest-order matter terms on the scale factors, a,b and c from the generalisation of the conservation equation (<ref>) with an anisotropic metric (<ref>). For the case with general n, this is ρ̇=- ( ȧ/a+ḃ/b+ċ/c ) ρ (1+w)[ κ +ηρ ^2n-1n(1+3w)/κ +2ηρ ^2n-1A(n,w)] ,and so the behaviour of the density is just ρ∝ (abc)^-Γ,where Γ (n,w)=(1+w)[ κ +ηρ ^2n-1n(1+3w)/κ +2ηρ ^2n-1A(n,w)] .The higher-order density terms will dominate the evolution at early times when n>1/2 and we see that, in these cases, Γ is independent of ρ and η as ρ→∞ ,since in this limit, Γ (n,w)→n(1+3w)(1+w)/2A(n,w). In the cosmology obtained by setting n=1 in (<ref>)-(<ref>) we will have domination by the nonlinear matter terms, which will drive the expansion towards isotropy as t→ 0 if ρ ^2 diverges faster than (abc)^-2 as abc→ 0. Thus, the condition for an isotropic initial singularity in n=1 theories is that Γ (1,w)>2, or (1+3w)(1+w)/2A(1,w)>2 When this condition holds as t→ 0, the dynamics will approach the flat FLRW metric with a(t)∝ b(t)∝ c(t)∝ t^2/Γ (1,w). When Γ (1,w)<2, the dynamics will approach the vacuum Kasner metric with (a,b,c)= (t^q_1,t^q_2,t^q_3),∑_i=1^3q_i = ∑_i=1^3q_i^2=1. This condition simplifies to four cases: w>0 anisotropic singularityα _+<w<0 isotropic singularityα _-<w<α _+ anisotropic singularityw<α _- isotropic singularity Here, the constants α _+ and α _- take the values determined earlier in (<ref>)).In general, for arbitrary n, the higher-order correction terms on the right-hand side of the field equations (<ref>) are proportional to ηρ ^2n when p=wρ , and so the condition for an isotropic singularity as t→ 0 becomes Γ (n,w)>2n,and the dynamics approach a(t)∝ b(t)∝ c(t)∝ t^2/Γ (n,w). The case for general n and w is problematic to simplify succinctly due to the exponential dependence on n. However, we can consider specific physically relevant equations of state individually.For dark energy (w=-1) and curvature (w=-1/3) `fluids', we find that Γ (n)=0, for all n, and so the condition for an isotropic initial singularity will depend only on whether n itself is positive or negative.For w=0, dust, we find Γ (n,0)=n/2n-1,for n≠1/2. which leads to isotropy only when 1/2<n< 3/4.For radiation, w=1/3, an isotropic singularity will occur if n/(4/3)^n-1(2n-1/2)>2n, whilst for w=1 we find that the condition for isotropy is n/4^n-1(2n-1/2)>2n. In both of the latter cases, we require n≠1/4.A similar effect will occur in more general anisotropic universes, like those of Bianchi type VII_h or IX, which are the most general containing open and closed FLRW models, respectively. In type IX, the higher-order matter terms will prevent the occurrence of chaotic behaviour with w<1 fluids on approach to an initial or final singularity in a 𝐓^2n theory when n>1. Thus we see that in these theories the general cosmological behaviour on approach to an initial and (in type IX universes) final singularity is expected to be isotropic in the wide range of cases we have determined, when Γ (n,w)>2n. This simplifying effect of adding higher-order effects can also be found in the study of other modifications to GR, for example those produced by the addition of quadraticR_abR^abterms to the gravitational Lagrangian, <cit.>. These also render isotropic singularities stable for normal matter (unlike in GR). If T_ab is not a perfect fluid but has anisotropic terms (for example, because of a magnetic field or free streaming gravitons <cit.> ) they will add higher-order anisotropic stresses.§ CONCLUSIONS We have considered a class of theories which generalise general relativity by adding higher-order terms of the form (T^μνT_μν)^n to the matter Lagrangian, in contrast to theories which add higher-order curvature terms to the Einstein-Hilbert Lagrangians, as in f(R) gravity theories. The family of theories which lead phenomenologically to higher-order matter contributions to the classical gravitation field equations of the sort studied here includes loop quantum gravity, and bulk viscous fluids, k-essence, or brane-world cosmologies in GR. This generalisation of the matter stresses is expected to create changes in the evolution of simple cosmological models at times when the density or pressure is high but to recover the predictions of general relativistic cosmology at late times in ever-expanding universes where the density is small. However, we find that there is a richer structure of behaviour if we generalise GR by adding arbitrary powers of the scalar square of the energy-momentum tensor to the action. In particular, we find a range of exact solutions for isotropic universes, discuss their behaviours with reference to the early- and late-time evolution, accelerated expansion, and the occurrence or avoidance of singularities. Finally, we discuss extensions to the simplest anisotropic cosmologies and delineated the situations where the higher-order matter terms will dominate over the anisotropic stresses on approach to cosmological singularities. This leads to a situation where the general cosmological solutions of the field equations for our higher-order matter theories are seen to contain isotropically expanding universes, in complete contrast to the situation in general relativistic cosmologies. In future work we will discuss the observational consequences of higher-order stresses for astrophysics. AcknowledgementsThe authors are supported by the Science and Technology Facilities Council (STFC) of the UK and would like to thank T. Harko and O. Akarsu for helpful comments.The authors would also like to thank Sebastian Bahamonde, Mihai Marciu and Prabir Rudra for their helpful comments. 99 bshaw J. D. Barrow and D. J. Shaw, Phys. Rev. Lett. 106, 101302 (2011).dyndarkeng E. J. Copeland, M. Sami, and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753, (2006).frtheories A. D. Felice and S. Tsujikawa, Living Rev. Relativity13, 3 (2010).ftbarrow A. Paliathanasis, J. D. Barrow, and P. G. L. Leach, Phys. Rev. D 94, 023525 (2016).jbgen J.D. 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"authors": [
"Charles V. R. Board",
"John D. Barrow"
],
"categories": [
"gr-qc",
"astro-ph.CO",
"hep-th"
],
"primary_category": "gr-qc",
"published": "20170927133250",
"title": "Cosmological Models in Energy-Momentum-Squared Gravity"
} |
PlanetCam absolute photometry Mendikoa et al. Departamento de Física Aplicada, Escuela de Ingeniería, Universidad del País Vasco, Plaza Torres Quevedo 1, E-48013 Bilbao, [email protected] of Theory of Signal & Communications, Universidad Carlos III de Madrid, Avenida de la Universidad 30, Leganés, E-28911 Madrid, [email protected] provide measurements of the absolute reflectivity of Jupiter and Saturn along their central meridians in filters covering a wide range of visible and near-infrared wavelengths (from 0.38 to 1.7 μm) that are not often presented in the literature. We also give measurements of the geometric albedo of both planets and discuss the limb-darkening behavior and temporal variability of their reflectivity values for a period of four years (2012-2016). This work is based on observations with the PlanetCam-UPV/EHU instrument at the 1.23 m and 2.2 m telescopes in Calar Alto Observatory (Spain). The instrument simultaneously observes in two channels: visible (VIS; 0.38-1.0 μm) and short-wave infrared (SWIR; 1.0–1.7 μm). We obtained high-resolution observations via the lucky-imaging method. We show that our calibration is consistent with previous independent determinations of reflectivity values of these planets and, for future reference, provide new data extended in the wavelength range and in the time. Our results have an uncertainty in absolute calibration of 10–20%. We show that under the hypothesis of constant geometric albedo, we are able to detect absolute reflectivity changes related to planetary temporal evolution of about 5-10%. Temporal and spatial variations of the absolute reflectivity of Jupiter and Saturn from 0.38 to 1.7 μm with PlanetCam-UPV/EHU Tables A1-A4 and B1-B8 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ I. Mendikoa 1, A. Sánchez-Lavega 1, S. Pérez-Hoyos 1, R. Hueso 1, J.F. Rojas 1, J. López-Santiago 2Received May 4, 2017; accepted July 22, 2017 =================================================================================================================================================================================================================================================================================================================== § INTRODUCTIONKnowledge of the absolute reflectivity of planetary images is essential for precise modeling of the light scattered by planetary atmospheres. This is particularly important for Jupiter and Saturn, given the spatial and temporal variability of their atmospheres. The combination of observations at different wavelengths from the visible to the near-infrared allows for probing of a wide range of altitudes in the atmospheres of these planets but requires an absolute calibration of the images. The idea of using images of the giant planets acquired in different methane absorption bands to gain insight into the horizontal and vertical structure of their atmospheres is not new <cit.>, but acquiring enough multispectral images at high spatial resolutions is generally difficult. When using multispectral imaging, the loss of spectral resolution can be compensated by the gain in spatial resolution, particularly if filters are properly defined to match the absorption bands and adjacent continuums and other wavelengths of interest, such as the so-called chromophore absorptions at short wavelengths defining the red colors of Jupiter and pale tones of Saturn <cit.>.Although there are many works dealing with calibrated observations in the visible part of the spectrum for Jupiter and Saturn, including Hubble Space Telescope (HST) calibrated observations <cit.> or spacecraft imaging <cit.> and ground-based multispectral photometry <cit.>, the near-infrared side of the spectrum (SWIR; short wavelength infrared) has been analyzed less often <cit.>. This part of the spectrum is of particular interest to prepare for James Webb Space Telescope (JWST) image observations with the NIRCAM instrument of Jupiter to Neptune since JWST will achieve a spatial resolution better than Hubble Space Telescope (HST) from 0.7 to 2.1 microns <cit.>. In this work, we profusely refer to a number of previous studies that served as an external reference for the absolute reflectivity values we obtain in this work. No seasonal variability has been reported in the atmosphere of Jupiter except for planetary-scale changes at particular belts and zones not linked to seasonal changes, <cit.>; however, Saturn is known to display hemispherical seasonal changes andchanging viewing or illumination geometry owing to the tilt of its rotation axis <cit.>. For this reason, Saturn photometric values usually only serve as a snapshot for a particular season. For visual wavelengths (380-1000 nm, covered by our VIS channel), <cit.> provided the giant planets geometric albedo through medium-resolution spectroscopy, while <cit.> covered a wide visible wavelength range providing Jupiter absolute reflectivity values. A similar work for the near-infrared part of the spectrum was presented by <cit.>.More recently, and at the same time as the PlanetCam observations, visible absolute reflectivity data were available from HST for both Saturn (July 2015) and Jupiter from the Outer Planet Atmospheres Legacy program (OPAL[https://archive.stsci.edu/prepds/opal/]; February 2016). Because data in the SWIR channel are much scarcer, we use the calibrated data presented in <cit.> for Jupiter and disk-integrated spectra at medium resolution from <cit.> as a reference for both Jupiter and Saturn. This makes clear that an updated reference for the reflectivity of Jupiter and Saturn in an ample wavelength range is of interest for future works dealing with calibrated images from these planets and other giant planets outside our solar system.Moreover, the availability of Jupiter and Saturn calibrated images in a consistent set of filters and through a number of observation campaigns can also provide support to space missions in the study of their atmospheres, as occurred in other ground-based observations of Jupiter and Saturn at the time of Voyager 1 and 2 flybys <cit.>. In the case of Jupiter, the NASA Juno mission, in orbit around Jupiter since June 2016 <cit.>, is currently receiving ground-based support. Ground-based observations provide information at low phase angles complementary to space missions observing the planet at geometries with higher phase angles, as Jupiter during the Cassini flyby in 2000 <cit.> and the near-to-end nominal mission at Saturn <cit.>.Here we present a multiwavelength (0.38 – 1.7 μm) study of the absolute reflectivity of Jupiter and Saturn using high-spatial-resolution, ground-based images obtained with the instrument PlanetCam-UPV/EHU <cit.>. We present a set of observations from 2012 to 2016, which has allowed the creation of a wide database of Jupiter and Saturn calibrated images in reflectivity. This database will hopefully be enlarged with further observations to be retrieved in the future. Radiative transfer modeling from the calibrated series (i.e., sets of images with all the filters) presented here is not included and will be reported elsewhere.The organization of the paper is as follows. The observations performed so far are presented in section 2. In Sect. 3, we focus on a set of PlanetCam photometric images of Jupiter and Saturn for all the filters available in both VIS and SWIR channels and describe the calibration procedure. We present PlanetCam calibration results in section 4, followed by a review of the different sources of uncertainty. Finally, in section 5 we compare these results with available reference values and discuss the temporal evolution of some atmospheric features over the four year period of these observation campaigns. § PLANETCAM OBSERVATIONS §.§ PlanetCam instrument PlanetCam-UPV/EHU <cit.> is an astronomical camera designed for high-resolution imaging of solar system targets <cit.>. The instrument uses the lucky-imaging technique. This consists of the acquisition in a short time of several hundredths to thousands of short exposures (10’s of milliseconds) selected by their quality and stacked into a single image <cit.> for broadband filters, and longer exposures (i.e., a few seconds each and also stacked for building high signal-to-noise ratio images) for narrowband imaging. The final stacked images are processed to obtain high-resolution observations that largely remove the blur from atmospheric effects <cit.>. The selection of the best 1-10% of frames without adding any additional image processing results in high-quality images that retain the photometry of the object observed.The instrument contains several filters selected for color, broadband continuum, methane absorption bands, and their adjacent narrow continuum wavelengths chosen because of their interest for planetary studies. The camera works in the wavelength range from 0.38 to 1.7 μm supporting scientific research in atmosphere dynamics and vertical cloud structure of solar system planets. The design comprises two channels that work simultaneously by means of a dichroic beam splitter <cit.>. First channel (visible; VIS) covers wavelengths below 1 μm and the second channel (short-wave infrared; SWIR) works above this value and up to 1.7 μm. PlanetCam images have been used so far for several works on Jupiter <cit.>, Saturn <cit.>, Venus <cit.>, and Neptune <cit.>. §.§ Observation campaigns We performed several Jupiter and/or Saturn observation campaigns since the commissioning of PlanetCam in 2012 at Calar Alto Observatory in Spain via both the 1.23 m and 2.2 m telescopes. Table <ref> provides details on these campaigns, including the telescope and PlanetCam configuration <cit.>. The PlanetCam1 (PC1) configuration means that only the visible channel (VIS) was available at the time, while PlanetCam2 (PC2) means that both the visible and infrared (SWIR) channels were available. There are also some differences in the optics between both configurations, but they are of no interest here since both the planet and standard stars are always observed with the same configuration. Table <ref> also shows the standard stars used for absolute reflectivity calibration including their spectral type.A timeline representation of all these observation campaigns is shown in Figure <ref>, indicating Jupiter and Saturn opposition dates as well as HST relevant observations of Saturn (July 2015) and Jupiter (Feb 2016) used in this paper as reference values forabsolute reflectivity. The number of calibrated filter sequences is indicated below each campaign date.As standard stars, we used a selection of stars from the Isaac Newton Group of telescopes spectrophotometric database[http://catserver.ing.iac.es/landscape/] for the VIS channel and from the IRTF spectral library[http://irtfweb.ifa.hawaii.edu/IRrefdata/sp_catalogs.php] for the SWIR channel <cit.>. These databases provide medium- to low-resolution spectra of the stars that can be convolved with our filters plus system response, thus getting accurate calibration factors as we explain later. Whenever this was not possible for all wavelengths, we completed the values using the VizieR photometry tool on the SIMBAD database[http://simbad.u-strasbg.fr/simbad/] <cit.>. The impact of such very low-resolution data for the standard stars that are discussed in section <ref>. The standard stars used for the absolute reflectivity calibration of Jupiter and Saturn were selected according to different criteria. First, the availability of a star with high-resolution spectrum was considered. Second, the position of the star in the sky relative to the planet that is calibrated was also considered, trying to get the standard star as close as possible to the planet with similar airmass at the corresponding observation time.§ ABSOLUTE REFLECTIVITY MEASUREMENTS §.§ Photometric calibration In planetary sciences, it is common to use the absolute reflectivity I/F of a given surface or atmosphere <cit.>. Omitting here the wavelength dependency, this quantity is defined as the ratio of surface brightness (I) to that of a flat Lambertian surface (F), where π F is the solar flux (Wm^-2nm^-1) at the given planetary distance. A planet image can be converted from digital counts into absolute reflectivity I/F with the intensity spectrum of a standard star as reference, from the expression below <cit.>: ( I/F) _i = (C_p)_i/Σ C_star∫_λπ F_star· T(λ)Φ(λ)dλ/∫_λπ F_⊙· T(λ)Φ(λ)dλ D^2 (π/θ^2) e^k/2.5(X_p-X_star). This expression provides the absolute reflectivity I/F at the pixel i in the planet image. Here (C_p)_i represents the count rate at the planet image pixel i, Σ C_star the total number of count rates from the standard star, π F_star the standard star flux outside atmosphere of the Earth (Wm^-2nm^-1) as retrieved from the database, π F_⊙ the solar flux at Earth (Wm^-2nm^-1) given by <cit.>, D (in AU) is the distance from the Sun to the target at the observation time, T(λ) the filter plus optics transmittance curve, Φ(λ) the detector sensitivity curve, and θ the image scale (arcsec/pixel) <cit.>. The last term is added to consider the atmospheric effects, where X_p and X_star are the air masses for the planet and star, respectively, and k the extinction coefficient of atmosphere of the Earth (magnitudes/air mass) at the time the images were taken. §.§ Image acquisition and processing The calibration process begins with the capture of the sequence of frames of the science object (planet or calibration star) that are used to build the final image. This requires suitable exposure times for each filter to get a good signal-to-noise ratio that maintains detector linearity, while being fast enough to perform lucky-imaging at least for the wider filters. Images are processed with a pipeline written in IDL (Interactive Data Language) and specifically designed for PlanetCam. Thesoftware, called PLAYLIST (PLAnetarY Lucky Images STacker), analyzes full directories of data locating for each science file the calibration files containing dark currents and flat fields (i.e., those matching exposure times and camera gain settings) to be used in the image reduction substracting the dark current and dividing by the flat field of each individual frame. Hot and cold pixels are removed then with an adaptive median filter technique. The quality of the resulting frame is analyzed by first finding the scientific object in the frame using a center of brightness algorithm that identifies the center of the planet (or star) and its size by defining a region of interest (ROI) of constant size through the sequence. Next, the quality of each frames is provided by a numeric metric based on the Sobel differential filter <cit.>.Sharp images have higher values of their spatial derivatives while unfocused or smooth images provide lower values. By summing up the absolute values of the Sobel filtered ROI containing the scientific target a good estimation of the frame quality is obtained. The software allows Lucy Richardson deconvolution <cit.> on a frame-by-frame basis, which is used with bright images with low noise levels. Images are co-registered with a multiscale image correlation algorithm that matches images with a precision of one pixel. For each image sequence PLAYLIST generates several FITS files with different trade-offs between image quality in terms of spatial resolution and dynamic range. For instance, the best 1% of all individual frames typically contain fewer atmospheric blurring but also low dynamic range, but the co-registered stack of all frames contains the maximum dynamic range but the highest level of atmospheric blurring. The software automatically produces selections with different percentages of the best frames (1%, 2%, 5%, 10%, 30%, and all frames). Photometric images are saved in FITS files containing double precision real numbers representing the total digital counts per second for each pixel. The SWIR images are typically best represented by the images based on the best 1%-5% frames, while visible images in short wavelengths are generally better represented by the versions containing the best 10% and very dark acquisitions in the strongest methane absorption filters generally require the versions based on the stack of all frames.Figures <ref> to <ref> show photometric images of Jupiter and Saturn for different PlanetCam filters in both VIS and SWIR channels processed with PLAYLIST. These images preserve photometric information but can be high-pass filtered to show faint structures and study dynamics <cit.>. All the PlanetCam filters are listed by their short name and we provide below effective wavelength in nanometers for narrow filters <cit.>. The VIS channel includes filters U, Vio, B, V, R, M1 (619), C1 (635), M2 (727), C2 (750), M3 (890), and C3 (935), while SWIR channel includes YC (1090), YM (1160), V1 (1190), V2 (1220), JC (1275), JM (1375), V3 (1435), HC (1570), HM (1650), J, and H. Figure <ref> shows photometric images of Jupiter taken with PlanetCam2 configuration at the 2.2 m telescope at Calar Alto observatory (March 2016) for absolute reflectivity calibration with VIS channel, while Figure <ref> shows those corresponding to SWIR channel. Figure <ref> shows photometric images of Saturn taken with PlanetCam2 at the 2.2 m telescope at Calar Alto observatory (May 2016) for absolute reflectivity calibration with VIS channel. Figure <ref> shows Saturn images in the SWIR channel where image saturation at the rings in some filters does not affect the atmosphere photometric analysis.§ RESULTS Following equation <ref> we photometrically calibrated Jupiter and Saturn images, computing I/F values for each pixel on the images. We show absolute reflectivity I/F scans along central or sub-observer meridian (i.e., I/F versus latitude) for all available filters at a resolution of 1^∘ in latitude for each observation campaign, using different calibration standard stars detailed in table <ref>. Even though values at a fixed resolution of 1^∘ are given, our photometric images without further processing have effective resolutions of 0.5–1” which, in the case of Saturn, means that only 5–10^∘ resolution is achieved. However, this also depends on the filter, with resolutions better that 0.5” for wide filters used for lucky-imaging and overall better behavior in the SWIR channel <cit.>.Because of the zonal banding in the giant planets, this method is well suited to compare spatial and temporal reflectivity changes in different spectral regions. We obtain a mean I/F scan for each filter and observation campaign. We computed inter-annual mean values from the yearly means from planetographic latitudes -80^∘ to +80^∘. Figures <ref> to <ref> show the mean absolute reflectivity versus latitude at central meridian for each filter, including the standard deviation from the analysis of different image series in each campaign. The error sources are mainly due to image navigation and the uncertainties in the calibration, as we discuss in section <ref>. We omitted the mean I/F scans from some campaigns when the latitudinally averaged deviation from the overall mean I/F scan was higher than the standard deviation at each latitude. After this filtering, we recomputed the overall mean I/F scans and standard deviations, which resulted in the latitudinal scans shown in Figures <ref> to <ref>. We provide a tabulated version of these data in supplementary tables A1 to A4, as they can be used as a reference for future works.We used the empirical Minnaert law for diffuse reflection <cit.> to determine the I/F variation with the cosines of the incidence (μ_0) and emission (μ) angles <cit.>, ( I/F) (μ , μ_0) = ( I/F)_0 μ_0^k μ^k-1, where (I/F)_0 represents the absolute reflectivity in absence of darkening effects at nadir viewing and k is the limb-darkening coefficient. In this way images can be corrected from limb-darkening effects and photometric results from the central meridian can be extended to other longitudes. The mean values of Minnaert coefficients (from observations from 2012 to 2016) for each latitude and PlanetCam filter are presented in the form of tables in supplementary tables B1 to B8. §.§ Absolute reflectivity values Jupiter mean absolute reflectivity I/F values along its central meridian (not using images with the Great Red Spot close to it) from observational campaigns in the visible channel between December 2012 and May 2016 (Figure <ref>) are determined with different levels of uncertainty at different wavelengths. We calculated the latitudinal I/F deviation from the mean for each filter from the ensemble of valid observations campaigns. For some filters only three campaigns provided valid data, while for others our sample grows to five campaigns. In the visible channel, the overall uncertainties considering the mean of standard deviations from all latitudes, for the filters represented in Fig. <ref>, are as follows: UV (three campaigns; 16%), Vio (three; 18%), B (five; 23%), V (four; 16%), M1 (three; 20%), C1 (three; 20%), M2 (three; 11%), C2 (three; 20%), M3 (five; 18%), and C3 (three; 18%). These uncertainty values include, among different causes discussed in section <ref>, possible physical changes due to temporal evolution, as discussed in section <ref>.In a similar way we show in Figure <ref> the overall mean absolute reflectivity and associated uncertainty for SWIR filters corresponding to observations run between March 2015 and July 2016. Filters JM and V3 are not included in Fig. 7 because they are affected by telluric water absorption, while the HM filter is affected by the cutoff of the sensitivity of the PlanetCam SWIR detector (Mendikoa et al., 2016). The number of campaigns considered and average uncertainties for these filters are as follows: YC (four; 24%), YM (four; 19%), V1 (four; 12%), V2 (three; 22%), JC (four; 18%), and HC (four; 11%).We show the overall mean absolute reflectivity at the central meridian of Saturn and its associated uncertainty in Figure <ref> for the filters in the VIS channel. The rings of Saturn cover part of the disk of the planet preventing us from observing the southern hemisphere. Latitudinal values shown here are limited to the northern hemisphere where the influence of Saturn rings is negligible. Our results correspond to observation campaigns run from April 2013 to May 2016. The number of campaigns considered and average uncertainties for these filters are as follows: UV (five; 12%), Vio (four; 13%), B (four; 10%), V (four; 12%), M1 (three; 11%), C1 (four; 16%), M2 (four; 9%), C2 (three; 12%), M3 (three; 23%), C3 (two; 24%).Similar results for Saturn in the SWIR channel filters available are shown in Figure <ref> and they correspond to observations acquired between May 2015 and July 2016. The number of campaigns considered and average uncertainty for these filters are as follows: YC (four; 7%), YM (four; 12%), V1 (four; 17%), V2 (three; 11%), JC (three; 3%), and HC (four; 6%). §.§ Uncertainty sourcesThere are several sources of uncertainty and variability in the retrieved I/F mean values at each latitude: the intrinsic uncertainty in the calibration procedure itself, uncertainty in the determination of the latitudes of each pixel due to small inaccuracies in the planet navigation, variations of the reflectivity at different longitudes on particular bands and, finally, possible temporal changes in the reflectivity of some planet bands observed on different epochs.The uncertainty on I/F produced by the image calibration procedure can be estimated from the errors in the different parameters that appear in Equation 1. In previous sections we estimated the calibration values dispersion up to around 20%. Comparing count rates for a standard star with the same telescope and optics at different observation nights but similar airmass, we can find variations of 1% to 10% and, rarely we can find variations up to 20%. While most of the error can be attributed to the photometrical conditions along the observing run for the planet and calibration star, other sources include detector noise anduncertainties in the knowledge of the spectrum of the standard star used in each case; detector noise is particularly important for some filters in deep methane absorption bands in the SWIR channel resulting in very low signals, such as the HM filter.Fluctuations in flat-fielding and dark current subtraction can affect count rates for the planet and standard stars. However, these variations are well below those due to atmospheric conditions of the Earth (i.e., transparency and seeing). However, such instrumental sources of uncertainties rarely account for more than 2% of the signal <cit.>. Extinction coefficients at Calar Alto Observatory are regularly monitored <cit.> and they imply a relative error of around 5%. In order to minimize errors from atmospheric extinction, the standard stars were usually observed at elevations with differences below 0.3 air masses to the elevation of the planet.Spectral data available for the standard stars also contain a certain degree of inaccuracy that contributes to the total calibration error. In particular, for some standard stars their calibrated flux was only available at low spectral resolution for some wavelength ranges. This could imply significant calibration errors for narrow filters for which spectral bands are not properly resolved in the spectrum of the standard star. As an example, for star HD19445 different spectrum sources were compared (i.e., Isaac Newton Group, SIMBAD, and X-Shooter[http://xsl.u-strasbg.fr/]) including some synthetic models based on Kurucz <cit.> and BT-Nextgen <cit.>. For visible filters, the total star flux showed a variation of around 15% depending on the stellar parameters assumed. For this reason, it would be advisable to promote a calibration campaign with simultaneous medium- to high-resolution spectra taken with another instrument for some selected stars.Uncertainties introduced from inaccuracies in the planet navigation can be estimated after examining results with different navigations of the same planetary image. Our tests show that the uncertainty of a band width can be around 10% and the latitudinal uncertainty in the position of the maximum brightness of a particular band can be around 1^∘. These are typical values for both VIS and SWIR unprocessed photometric images but of course processed images acquired under very good seeing conditions (< 0.7 arcsec) largely minimize these errors.The different I/F scans always correspond to the central meridian, but, because of the rotation of the planet, every scan corresponds to a different planetary longitude, and given that planetary bands have fluctuations in their albedo edges and that they may include discrete features (spots), the band width and reflectivity may vary. As in the disk navigation issue, we estimated uncertainty in band widths due to these variations to be ∼ 10% and positions in around 1^∘, by analyzing several I/F versus latitude scans at different longitudes in the same navigated image.Finally, variations in I/F scans can also be due to intrinsic changes of the planet reflectivity over time at particular bands in Jupiter <cit.> or over large regions in Saturn as seasonal variations proceed <cit.>. Characterizing real global or local color variations in Jupiter and Saturn clouds is a key scientific area and relevant results from PlanetCam are discussed in section <ref> after we appropriately constrained artificial sources of variations and uncertainties.In summary, these uncertainty sources contribute to an overall systematic uncertainty that has been estimated to be around 10-20% in absolute reflectivity and much lower (possibly below 2%) in relative photometry. Not perfectly photometric conditions are the main cause responsible for this level of uncertainty, while the contribution of other sources, while having a minor impact, cannot be fully discarded.§ DISCUSSION§.§ Latitudinal reflectivity dependence The mean Jupiter reflectivity central meridian scans (Figures <ref> and <ref>) have been compared to available reference values from the literature <cit.>. Ground-based calibrated observations presented by <cit.> and HST calibrated images obtained in February 2016 as part of the Outer Planets Atmospheres Legacy (OPAL) program <cit.> constitute excellent sources of comparison for our Jupiter data in the visible. For the SWIR spectral range there are very few reference values of the reflectivity of Jupiter available in the literature. We compared our I/F scans to values obtained by <cit.> that used filters comparable to those used in PlanetCam SWIR. When there was no perfect match between filters wavelength, we used the closest match. The differences in reflectivity between our data and the reference values may be caused by real changes in the cloud structure or in the optical properties of haze particles but part of these differences could be due to the calibration uncertainty. In general, differences above a 10% uncertainty can be assigned to real changes in the hazes and upper Jovian cloud as in particular when compared to the reflectivity scans obtained in July 1994 by <cit.> and our PlanetCam data obtained about 20 years later.Regarding Saturn, HST observations obtained in July 2015 are used as reference. Figure <ref> shows a comparison of the central meridian scans corresponding to the nearest filters to PlanetCam from mean scans from April 2013 to May 2016. Significant differences can be seen in M2, where the effective wavelength and FWHM are slightly different in PlanetCam (727.3nm, 5nm) with respect to that of HST (727nm, 7.3nm). This can cause reflectivity differences in such narrowbands a M2. By contrast, the M3 band is wider and therefore more insensitive to small filters differences between PlanetCam (890.8nm, 5nm) and HST (889nm, 8.9nm). §.§ Albedo§.§.§ Jupiter The full disk geometric albedo for the visual spectral range is shown in Figure <ref> and is compared to reference values from July 1995 <cit.> and February 2016 <cit.>. The agreement is good for most cases, although small deviations are found at the wavelengths of the filters; i.e., U, M2, and C3.Jupiter albedo in the SWIR spectral range is shown in Figure <ref>. In this case, only the central part of the disk has been considered (reflectivity from latitudes -20^∘ to +20^∘) to compare our results with those from <cit.>. Again a reasonable agreement is found between both sets of values.§.§.§ Saturn Saturn overall mean albedo from PlanetCam campaigns is shown in Figure <ref> and is compared to ground-based data obtained in July 1995 <cit.> and recent HST July 2015 observations <cit.>. Because of the viewing angle geometry only the ring-free northern hemisphere can be studied (see Figure 5 and 6) in both PlanetCam and HST albedo values. Our results are comparable to those retrieved from HST photometry.There are, however, some differences when compared to full disk geometric albedo values from <cit.>, which were obtained at Saturn's equinox showing intrinsic brightness differences produced by seasonal changes in the hazes and clouds <cit.> and by the different viewing or illumination geometry.Similarly to Jupiter, the SWIR albedo measurements of Saturn were only calculated for the central part of the disk (only latitudes from -10^∘ to +10^∘ due to the ring presence) and were compared to data from <cit.> (Figure <ref>), where a larger central disk was considered because of the different ring tilt angle. This time the comparison is closer than in the VIS case, given the similar seasonal situation of the planet at both epochs since the subsolar latitude of Saturn was around -17^∘ at the time of <cit.> observations in 1977 and between +17^∘ and +25^∘ for PlanetCam observations (2012-2016), while it was around 0^∘ by the time of Karkoschka observations in 1995. §.§ Temporal evolutionThe temporal evolution of the properties of the cloud cover of Jupiter and Saturn can be analyzed from the absolute reflectivity I/F scans comparing results obtained in different observing campaigns. A key issue in this analysis is to estimate the uncertainty values from the different sources as described in section <ref>, so that the observed variations can be clearly attributed to physical atmospheric variations over time.In order to minimize the systematic uncertainty in absolute reflectivity calibration, we scaled the different I/F scans by a ratio given by the corresponding planet albedo to the overall mean albedo. Thus, we assumed that full disk geometric albedo is constant over the four years of PlanetCam observations and we scaled I/F curves according to this assumption. As long as this assumption is valid and the other uncertainty sources related to image navigation (particularly remarkable at high latitudes) and planet bands width variations along the whole longitudes are known, the remaining variations between the different observing campaigns can be interpreted as caused by temporal changes in the cloud cover of these planets.§.§.§ Jupiter Figure <ref> shows the Jupiter central meridian reflectivity evolution of the belts and zones from December 2012 to May 2016. Some changes are remarkable for example in the northern hemisphere. In the violet (Vio) filter we observe at 20^∘ planetographic latitude that the reflectivity increases from a peak I/F of 0.41 in December 2012 to 0.64 in April 2014, decreasing down to 0.56 in March 2016. We can also compare these I/F peak values to the averaged reference value of 0.55, so a variation of up to 40% can be observed in this spectral band.We estimate that the uncertainties due to the error sources above discussed can be on the order of 10%.Similar variations can be observed in the blue (B) filter, where I/F peak at latitude +20^∘ varies from 0.69 in Dec 2012 (0.98 times the average of this peak in B filter) to 0.77 in April 2014 (1.1 is the global average) and back to 0.68 in March 2016 (0.97 is the global average).A dark belt at planetographic latitude +25^∘ present in December 2012 vanishes in the following years (Figure <ref>). This corresponds to the evolution of an eruption in the NTB in 2012 close to solar conjunction that could not be observed properly. A similar phenomena was observed in 2007 <cit.> and 2017 <cit.>.The reflectivity in the methane band filters show the opposite behavior at these latitudes. The I/F scans in M3 show a peak at latitude +20^∘ continuously decreasing from 0.098 in Dec 2012 (1.14 times the average) to 0.086 in April 2014 (1.00 times the average) and 0.075 in March 2016 (0.87 the average), i.e., a total reflectivity decrease of 27% from the average. Smaller variations can be seen in the 2^∘ wide band at latitude around 45º with a variation of 10% with respect to the four-year average at that latitude. The nearby continuum filter C3 does not show any of these reflectivity values decrease, suggesting altitude variations in the top of a haze layer. On the other hand, variations over the time at this latitude are also seen in the continuum HC filter changing from 0.46 in March 2015 (1.09 times the average) to 0.40 in March 2016 (0.95 the average). Ongoing analyses of these data and their interpretation in terms of radiative transfer models will be presented elsewhere.§.§.§ Saturn In Figure <ref> we show the spectral reflectivity scans for the ring-free northern hemisphere under the viewing angle geometry of the period 2013-2016. We have to make similar considerations as described in the case of Jupiter, regarding the different possible sources of the yearly averaged variations. The most remarkable differences we detect between these campaigns occur between the brightest part near the equator and the polar zones. In addition, smaller variations in some specific belts and zones can be observed.Globally, the northern hemisphere of Saturn showed variations of its temporal reflectivity along the observing period (4 years), in particular at visual wavelengths, which is in agreement with previous observations <cit.>. Temporal variability at different latitudes is detected in particular in the blue filter; this is probably related to changes in the upper aerosol haze layers (stratospheric and tropospheric) sensitive to short wavelength absorption <cit.>. The equatorial zone (0^∘ to +20^∘) and the polar area (+60^∘ to +90^∘) are the regions in which most of the reflectivity temporal variability is seen. Changes in the blue and red filters can reach up to 30% relative to the mean values. Curiously, the variability is much less pronounced in the SWIR channel, while changes around 5% can also be detected for example at the filter V2. A study of the origin of this variability will be performed and presented elsewhere.§ SUMMARY AND CONCLUSIONS The main conclusions from our study are summarized in the following points:* We presented spatially resolved absolute reflectivity measurements of Jupiter and Saturn in a wide wavelength range at specific filters of planetary interest for the study of the cloud structures of both planets from the ultraviolet (380 nm) to short near-infrared wavelengths (1.7 μm). * We performed photometry with a lucky-imaging camera, PlanetCam-UPV/EHU, specifically adapted for the study of solar system objects at high spatial resolution. Our photometrically calibrated images, without any further processing, reaches spatial resolution of up to 0.5”, with typical values of around 1” for narrow filters. * We determined the absolute reflectivity I/F with a wide range of standard stars. Although the precision of these measurements depends on wavelength, we typically found that a precision of 10-20% is achieved in the determination of absolute I/F values. * We presented Minnaert coefficients, which are useful to characterize the reflectivity of giant planet atmospheres under different viewing angles. * We validated the photometric calibration through a comparison with published data available in the literature. Photometrically calibrated images or albedo values of giant planets are scarce in the scientific literature and the capability of determining the absolute reflectivity of Jupiter and Saturn in different years using the lucky-imaging method can contribute relevant data to this field. * The I/F north-south scans along the central meridian of Jupiter and Saturn for each spectral filter show temporal changes in the upper aerosol haze layers along the period studied (2012-2016). * The high spatial resolution provided by the lucky-imaging method allows us to follow the spectrally dependent reflectivity changes in belts and zones, but also in specific features like the Great Red Spot, oval BA, and other cyclones, anticyclones, and storms in Jupiter and Saturn. * PlanetCam-UPV/EHU is providing a long baseline of absolutely calibrated observations of the giant planets with at least two to three observation campaigns each year. * We are able to detect relative changes in reflectivity of around 5-10% when assuming the invariability of the geometric albedo of the planet at a given wavelength.Some of the results we have presented will be further analyzed in future publications in terms of a radiative transfer model to retrieve information related to the vertical structure of the cloud systems of Jupiter and Saturn. Our database of planetary calibrated images, such as those presented here, is expected to be enlarged with additional observations in the future. We expect that PlanetCam observations will be able to support space mission such as Akatsuki at Venus <cit.>, Mars Reconnaissance Orbiter, Mars Express and Exomars at Mars <cit.>, Juno at Jupiter <cit.>, and to help the study of the legacy of the Cassini mission. Observations in the SWIR channel (1-1.7 microns) might be usefulwhen comparing with future JWST observations starting in 2019. The long-term observations will serve to cover the study of those planets where no space missions are expected in the near future as for Saturn, after the Cassini grand finale in September 2017, along with Uranus and Neptune.This work was supported by the research project AYA2015-65041-P (MINECO/FEDER, UE), Grupos Gobierno Vasco IT-765-13, UPV/EHU UFI11/55 and ”Infraestructura” grants from G.Vasco and UPV/EHU. I. Mendikoa has been supported by Aula Espazio Gela funded by Diputación Foral de Bizkaia. J.L.-S. acknowledges the Office of Naval Research Global (award no. N62909-15-1-2011) for support. This research has made use of the VizieR catalog access tool, CDS, Strasbourg, France. The original description of the VizieR service was published in A&AS 143, 23. aa | http://arxiv.org/abs/1709.09664v1 | {
"authors": [
"I. Mendikoa",
"A. Sánchez-Lavega",
"S. Pérez-Hoyos",
"R. Hueso",
"J. F. Rojas",
"J. López-Santiago"
],
"categories": [
"astro-ph.EP",
"astro-ph.IM"
],
"primary_category": "astro-ph.EP",
"published": "20170927145317",
"title": "Temporal and spatial variations of the absolute reflectivity of Jupiter and Saturn from 0.38 to 1.7 $μ$m with PlanetCam-UPV/EHU"
} |
[ [ December 30, 2023 =====================Let N and p be primes such that p divides the numerator of N-1/12. In this paper, we study the rank g_p of the completion of the Hecke algebra acting on cuspidal modular forms of weight 2 and level Γ_0(N) at the p-maximal Eisenstein ideal. We give in particular an explicit criterion to know if g_p ≥ 3, thus answering partially a question of Mazur.In order to study g_p, we develop the theory of higher Eisenstein elements, and compute the first few such elements in four different Hecke modules. This has applications such as generalizations of the Eichler mass formula in characteristic p.§ INTRODUCTION AND RESULTSLet p≥ 2 and N be two prime numbers such that p divides the numerator of N-1/12, whose p-adic valuation is denoted by t≥ 1. This is the situation of an Eisenstein prime extensively studied in <cit.>. Let ν = (N-1,12). We fix in all the article a surjective group homomorphism log : (𝐙/N𝐙)^×→𝐙/p^t𝐙. Let 𝕋̃ (resp. 𝕋) be the 𝐙_p-Hecke algebra acting on the space of modular forms (resp. cuspidal modular forms) of weight 2 and level Γ_0(N). Let Ĩ (resp. I) be the ideal of 𝕋̃ (resp. 𝕋) generated by the Hecke operators T_n - ∑_d d, where the sum is over the divisors of n prime to N.Let 𝔓̃ = Ĩ+(p) and 𝔓 = I+(p); these are maximal ideals. The kernel of the natural map 𝕋̃→𝕋 is 𝐙_p· T_0 for some T_0 ∈𝕋̃. A particular choice of T_0 will be made later using modular forms.Let 𝐓̃ (resp. 𝐓) be the 𝔓̃-adic (resp. 𝔓-adic) completion of 𝕋̃ (resp. 𝕋). Let g_p ≥ 1 be the rank of 𝐓 as a 𝐙_p-module. Barry Mazur asked what can be said about g_p, and more generally about the Newton polygon of 𝐓 <cit.>. This is one of the main motivation of this paper, and we provide a partial answer to Mazur's question. Loïc Merel was the first to give explicit information about g_p. For simplicity, in the rest of the introduction, we assume that p ≥ 5.<cit.> We have g_p>1 if and only if ∑_k=1^N-1/2 k ·log(k) ≡ 0(modulo p).We prove the following deceptively simple generalization.We have g_p>2 if and only if ∑_k=1^N-1/2 k ·log(k) ≡∑_k=1^N-1/2 k ·log(k)^2 ≡ 0(modulo p). The obvious generalization does not hold. More precisely, there seems to be no link between the vanishing of ∑_k=1^N-1/2 k ·log(k)^3 and the fact that g_p>3. For instance, if p=5 and N=3671, we have g_p=5 but ∑_k=1^N-1/2 k ·log(k)^3 ≢0(modulo p), and if p=7 and N=4229, we have g_p=3 and ∑_k=1^N-1/2 k ·log(k)^3 ≡ 0(modulo p).Frank Calegari and Matthew Emerton have identified 𝐓̃ with a universal deformation ring for the residual representation ρ = [ χ_p 0; 0 1 ], where χ_p is the reduction modulo p of the pth cyclotomic character.They deduce a characterization of g_p in terms of the existence of certain Galois deformations of ρ. Using class field theory, they were able to prove the following result.If g_p ≥ 2 then the p-Sylow subgroup of the class group of 𝐐(N^1/p) is not cyclic.The converse of Theorem <ref> happens to be false in general if p > 5. Recently, Preston Wake and Carl Wang–Erickson <cit.> have built on the work of Calegari and Emerton which tackled the determination of g_p through the theory of deformations of Galois representations. It would be interesting to compare their results to ours. In particular, they give another proof of Theorem <ref> and they proposed our Theorem <ref> as a conjecture.Our work is of a different nature. If we compare to the standard conjectures on special values of L-functions, we work on the “analytic side” of the problem, while Calegari–Emerton and Preston–Wake study the “algebraic side”.Let us say a few words about the proof of Merel's theorem. The essential point is the computation of the Eisenstein element of H_1(X_0(N), , 𝐐)_+ (the fixed part by complex conjugation of the singular homology relative to the cusps of the modular curve X_0(N) of level Γ_0(N)). It is an element annihilated by Ĩ. The main idea of this paper is to determine, in well-chosen Hecke modules, the so called higher Eisenstein elements, which have the property to be annihilated by a power of Ĩ. Results such as Theorem <ref> and Theorem <ref> are by-products of our study of higher Eisenstein elements. Unfortunately we could only determine a few of them so that we do not have a general formula for g_p.We first describe the higher Eisenstein elements in the space of modular form. If f is a modular form, let a_0(f) be its constant coefficient at the cusp ∞. For simplicity, we assume that p ≥ 5. There are modular forms f_0, f_1, ..., f_g_p in M_2(Γ_0(N), 𝐙/p𝐙), such that the following property hold. For any prime number ℓ not dividing N and any integer i such that 0 ≤ i ≤ g_p, we have:(T_ℓ-ℓ-1)(f_i) = ℓ-1/2·log(ℓ) · f_i-1 (modulo 𝐙· f_0 + ... + 𝐙· f_i-2).By convention, we let f_-1=0. This determines f_0 up to an element of (𝐙/p𝐙)^×. We normalize f_0 so that its q-expansion at the cusp ∞ isN-1/24+∑_n≥ 1(∑_d | n (d,N)=1 d)· q^n(modulo p).This is, of course, the unique (normalized) Eisenstein series of weight 2 and level Γ_0(N). Note that the constant coefficient of f_0 is 0 modulo p. The image of f_i in M_2(Γ_0(N), 𝐙/p𝐙)/( 𝐙· f_0 + ... + 𝐙· f_i-2) is uniquely determined.We can show that the constant coefficients of f_1, ..., f_g_p-1 are 0 modulo p, and that the constant coefficient of f_g_p is non-zero modulo p. In particular, the following assertions are equivalent. * We have a_0(f_1) = 0.* We have g_p ≥ 2.* We have ∑_k=1^N-1/2 k ·log(k) ≡ 0(modulo p). If g_p ≥ 2, then f_2 exists and the following assertions are equivalent. * We have a_0(f_2) = 0.* We have g_p ≥ 3. * We have ∑_k=1^N-1/2 k ·log(k) ≡∑_k=1^N-1/2 k ·log(k)^2 ≡ 0(modulo p). We prove the following finer result.* We havea_0(f_1) ≡1/6·∑_k=1^N-1/2 k ·log(k)(modulo p). * Assume that g_p ≥ 2. We havea_0(f_2) = 1/12·∑_k=1^N-1/2 k ·log(k)^2(modulo p).Contrary to point (i), point (ii) does not seem to follow from manipulation on the q-expansion of modular forms.There is a notion of higher Eisenstein elements modulo p^r for any integer r such that 1 ≤ r ≤ t. Theorem <ref> generalizes modulo p^r, but we shall work modulo p in this introduction because this is most relevant with respect to the study of g_p.The quantity ∑_k=1^N-1/2 k ·log(k) can thus be interpreted as the constant coefficient of an higher Eisenstein series, and also as a derivative of an L-function. Similarly, we can think of ∑_k=1^N-1/2 k ·log(k)^2 as the second derivative of an L-function. This point of view has been made precise in <cit.>, where these quantities are related to derived Stickelberger elements and to the class group of the cyclotomic field 𝐐(ζ_p, ζ_N), proving a kind of class number number formula. Here, ζ_p and ζ_N are respectively pth and Nth primitive roots of unity.These results on class groups will in fact be essential for us to construct the second higher Eisenstein element in the space of odd modular symbols.We construct higher Eisenstein elements in three other Hecke modules. Before we give our results, we briefly define what we mean by higher Eisenstein elements in general. Let M be a 𝕋̃-module such that M ⊗_𝕋̃𝐓̃ is free of rank one over 𝐓̃. Let M^0 ⊂ M be the submodule annihilated by T_0 ∈𝕋̃. There is a sequence of elements e_0, e_1, ..., e_g_p in M/p· M, called the higher Eisenstein elements, satisfying the following properties. * We have e_0 ≠ 0.* For all prime number ℓ not dividing N and all integer i such that 0 ≤ i ≤ g_p, we have:(T_ℓ-ℓ-1)(e_i) = ℓ-1/2·log(ℓ)· e_i-1 (modulo 𝐙· e_0 + ... + 𝐙· e_i-2)(with the convention e_-1=0). In the case p=ℓ=2 (which is excluded in this introduction but will be considered in the paper), the term ℓ-1/2·log(ℓ) is replaced by log(x) where 2=x^2 modulo N. The following additional properties hold.a) The element e_0 is unique, up to a scalar in (𝐙/p𝐙)^×.There is an element ẽ_0 ∈ M congruent to e_0 modulo p, and which is annihilated by Ĩ. Furthermore, ẽ_0 is unique up to multiplication by an element of 1+p𝐙_p.b) We have e_0, ..., e_g_p-1 ∈ M^0/p· M^0 and e_g_p∉M^0/p· M^0. c) If we fix e_0 in (M^0/p· M^0)[I], then the image of e_i in (M^0/p· M^0)/( 𝐙· e_0 + ... + 𝐙· e_i-2) is uniquely determined.Again, there is an analogous theory modulo p^r, for any integer r such that 1 ≤ r ≤ t, which we avoid in this introduction. Many of our definitions and theorems are valid modulo appropriate powers of p. The reader should consider all statements modulo p as simplifications.Most of our paper is devoted to the study of higher Eisenstein elements in three different modules, using completely different methods. §.§ The supersingular moduleConsider the free 𝐙_p-module M:=𝐙_p[S] on the set S of isomorphism classes of supersingular elliptic curves over 𝐅_N. If E ∈ S, let j(E) ∈𝐅_N^2 be the j-invariant of E and [E] ∈𝐙_p[S] be the element corresponding to E. The element ẽ_0 (unique up to 𝐙_p^×) is well-known:ẽ_0 = ∑_E∈ S1/w_E· [E] ∈ Mwhere w_E ∈{1,2,3} is half the number of automorphism of E. We determine completely the element e_1 (which is unique modulo (𝐙/p𝐙)· e_0). Let H(X) = ∑_i=0^N-1/2N-1/2i^2· X^i ∈𝐅_N[X] be the classical Hasse polynomial andP(X) = _T(H'(T), 256· (1-T+T^2)^3-T^2· (1-T)^2· X) ∈𝐅_N[X]where _X means the resultant relatively to the variable X. Since p>2, we can extend log to a morphism 𝐅_N^2^×→𝐙/p𝐙, still denoted by log. We have, modulo (𝐙/p𝐙)· e_0:e_1 = 1/12·∑_E ∈ Slog(P(j(E))) · [E].There is a bilinear pairing∙ : M × M →𝐙_psuch that [E] ∙ [E']=0 if E≠ E' and [E] ∙ [E] = w_E. This induces a perfect pairing:∙ : M/p· M × M/p· M →𝐙/p𝐙 . It is not hard to show that e_i ∙ e_j only depends on i+j and that e_i ∙ e_j ≡ 0(modulo p) g_p≥ i+j+1.Thus, the determination of g_p is equivalent to the determination of the e_i ∙ e_j modulo p. We now state two results about this pairing.* We havee_1 ∙ e_0 = 1/12·∑_λ∈ Llog(H'(λ)). * We havee_2 ∙ e_0 = e_1 ∙ e_1 =∑_λ∈ L1/24·log(H'(λ))^2 - 1/18·log(λ)^2where L ⊂𝐅_N^2^× is the set of roots of H (these are simple roots). This gives us criteria to determine wether g_p ≥ 2 and g_p ≥ 3 respectively. One can in fact compute directly in an elementary way the discriminant of H (modulo N), which gives us in particular the following formula:e_1 ∙ e_0 = 1/3·∑_k=1^N-1/2 k ·log(k).This gives an other proof of Theorem <ref>.The quantity e_i ∙ e_0 is the degree of e_i. For i=0, Eichler mass formula (in characteristic 0) tells us thate_0 ∙ e_0 ≡N-1/12≡ 0(modulo p). We thus feel justified to call our formulas for e_0 ∙ e_i “higher Eichler's formulas”. We can only state them for i ∈{1,2} (Theorem <ref>).§.§ Odd modular symbols Consider M^- = H_1(Y_0(N), 𝐙_p)^-, the largest torsion-free quotient of H_1(Y_0(N), 𝐙_p) on which the complex conjugation acts by multiplication by -1, where Y_0(N) is the open modular curve. Let M_+ = H_1(X_0(N), , 𝐙_p)_+, the fixed subspace by the complex conjugation of the homology (relative to the cusps) of the classical modular curve X_0(N). There is a perfect 𝕋̃-equivariant bilinear pairing, called the intersection pairing, M_+ × M^- →𝐙_p. We denote this pairing by ∙. Let ξ_Γ_0(N) : 𝐙_p[Γ_0(N) \_2(𝐙)] → H_1(X_0(N), , 𝐙_p)be the usual Manin surjection, given byξ_Γ_0(N)( Γ_0(N) · g ) = {g(0), g(∞)}where, if α,β∈𝐏^1(𝐐), we denote by {α, β} the cohomology class of the image of the geodesic path between α and β in the modular curve X_0(N) (via the complex upper-half plane parametrization). There is a natural identification Γ_0(N) \_2(𝐙) 𝐏^1(𝐙/N𝐙) given byΓ_0(N)·[ a b; c d ]↦ [c:d].We denote by m_i^-, i=1, ..., g_p the higher Eisenstein elements in M^-/p· M^-. The element m̃_0^- is easy to describe as a generator of the kernel of the map H_1(Y_0(N), 𝐙_p)^- → H_1(X_0(N), 𝐙_p)^-. Since X_0(N) has two cusps, this kernel is isomorphic to 𝐙_p. We normalize m̃_0^- so that{0, ∞}∙m̃_0^- = -1. The element m_1^- was essentially determined by Mazur <cit.>. For any x ∈ (𝐙/N𝐙)^×, we have in 𝐙/p𝐙:( (1+c)·ξ_Γ_0(N)([x:1])) ∙ m_1^- = log(x),where c is the complex conjugation. The analogous statement modulo p^t shows that I· H_1(X_0(N), 𝐙_p)_+ = {(1+c)·∑_x ∈ (𝐙/N𝐙)^×λ_x ·ξ_Γ_0(N)([x:1])such that ∑_x ∈ (𝐙/N𝐙)^×λ_x ·log(x) ≡ 0(modulo p^t)} .This shows that I· H_1(X_0(N), 𝐙_p)_+ is generated by the elements(1+c)·ξ_Γ_0(N)([x· y:1]-[x:1]-[y:1])for x,y ∈ (𝐙/N𝐙)^×. Our main result about the higher Eisenstein elements in M^- is the determination of m_2^-, which exists if and only if g_p≥ 2. We describe m_2^- with coefficients not in 𝐙/p𝐙, but in a certain K-group, which is cyclic of order p (see below). If A is a ring, let K_2(A) be the second K-group of A defined by Quillen. Let Λ = 𝐙_p[𝐙/p𝐙] and J be the augmentation ideal of Λ. Let K be the unique subfield of 𝐐(ζ_N) of degree p over 𝐐 where ζ_N is a primitive Nth root of unity. The group (K/𝐐) is canonically isomorphic to (𝐙/N𝐙)^×⊗𝐙/p𝐙 via the Nth cyclotomic character. Since we have fixed a choice of log, we can identify canonically (K/𝐐) with 𝐙/p𝐙. We let 𝒪_K be the ring of integer of K and 𝒦 = K_2(𝒪_K[1/Np])/p· K_2(𝒪_K[1/Np]). Note that 𝒦 is equipped with a canonical action of Λ. We prove the following result about the structure of 𝒦.* The group 𝒦/J·𝒦 is cyclic of order p.* The group J·𝒦/J^2·𝒦 is cyclic of order dividing p, with equality if and only if ∑_k=1^N-1/2 k ·log(k) ≡ 0(modulo p).If x and y are in 𝐙[ζ_N, 1/Np], we denote by {x,y}∈ K_2(𝐙[ζ_N, 1/Np]) the Steinberg symbol associated to x and y. We also let (x,y) be the image of {x,y} via the norm map K_2(𝐙[ζ_N, 1/Np]) →𝒦.There is a unique group isomorphism ι : 𝒦/J·𝒦≃𝐙/p𝐙 such that for all u, v ∈ (𝐙/N𝐙)^×, we haveι(1-ζ_N^u, 1-ζ_N^v) ≡log(u/v)(modulo p). Thus, we have a K-theoretic version of m_1^-, given by ι^-1( ((1+c)·∑_x ∈ (𝐙/N𝐙)^×λ_x ·ξ_Γ_0(N)([x:1]) )∙ m_1^- ) = ∑_x ∈ (𝐙/N𝐙)^×λ_x ·(1-ζ_N^x, 1-ζ_N) = (∏_x ∈ (𝐙/N𝐙)^× (1-ζ_N^x)^λ_x, 1-ζ_N).Let Δ = [1]-[0] ∈Λ; it is a generator of J. The multiplication by Δ induces a surjective morphism 𝒦/J·𝒦→ J·𝒦/J^2·𝒦 since Δ is a generator of J. By Theorem <ref>, it is an isomorphism if and only if g_p ≥ 2. In this case, let δ : J·𝒦/J^2·𝒦𝒦/J·𝒦 be the inverse isomorphism, and δ' = δ∘ι^-1 : 𝐙/p𝐙 J·𝒦/J^2·𝒦.We found our formula for m_2^- under the influence of the work of Alexander Goncharov and Romyar Sharifi (<cit.> et <cit.>). Our formula is conditional on a conjecture inspired by conjectures of Sharifi. Letξ_Γ_1(N) : 𝐙_p[Γ_1(N) \_2(𝐙)] → H_1(X_1(N), , 𝐙_p)be the Manin surjective map, given byξ_Γ_1(N)(Γ_1(N)· g ) = {g(0), g(∞)} .LetM_Γ_1(N)^0 = {Γ_1(N)·[ a b; c d ]∈Γ_1(N) \_2(𝐙), (c· d, N)=1}and let C_Γ_1(N)^0 be the set of cusps of X_1(N) above the cusp Γ_0(N) · 0 of X_0(N). The Manin surjective map induces a surjectionξ_Γ_1(N)^0 : M_Γ_1(N)^0 → H_1(X_1(N), C_Γ_1(N)^0, 𝐙_p). Let ϖ' : M_Γ_1(N)^0 → K_2(𝐙[ζ_N, 1/Np]) ⊗_𝐙𝐙_p be the 𝐙_p-linear map defined byϖ' (Γ_1(N)·[ a b; c d ]) = {1-ζ_N^c, 1-ζ_N^d}⊗ 1(it does not depend on the choice of [ a b; c d ]).One easily shows that ϖ' factors through ξ_Γ_1(N)^0, thus inducing a group homomorphismϖ : H_1(X_1(N), C_Γ_1(N)^0, 𝐙_p) →K_2(𝐙[ζ_N, 1/Np])⊗_𝐙𝐙_p.The natural analogue of Sharifi's conjecture (proved by Fukaya Takako and Kazuya Kato in <cit.>) would be the following conjecture.The map ϖ is annihilated by the Hecke operators T_n - ∑_dn/d·⟨ d ⟩ for any integer n ≥ 1 (where the sum is over the divisors of n prime to N and ⟨ d ⟩ is the dth diamond operator). Conjecture <ref> seems to have been considered by Sharifi himself. We refer to sections <ref> and <ref> for a more detailed discussion about Conjecture <ref>. Assume that Conjecture <ref> holds.Assume g_p ≥ 2,∑_k=1^N-1/2 k ·log(k) ≡ 0(modulop). Let x, y ∈ (𝐙/N𝐙)^×. Then, we have the following equality in J·𝒦/J^2·𝒦:δ'(( 2· (1+c)·ξ_Γ_0(N)([x· y:1]-[x:1] - [y:1]) )∙ m_2^- ) = (1-ζ_N^x, 1-ζ_N)-(1-ζ_N^x, 1-ζ_N^y) - (1-ζ_N^y, 1-ζ_N) . §.§ Even modular symbolsWe denote by m_i^+, i=1, ..., g_p the higher Eisenstein elements in M_+/p· M_+. Merel determined the element m̃_0^+ (unique up to 𝐙_p^×) in terms of Manin symbols. We recall his result below, using a slightly different formula. We will need to use the Bernoulli polynomial functions. Recall that _1 : 𝐑→𝐑 is the function defined by _1(x) = x-⌊ x ⌋-1/2 if x∉𝐙 and _1(x) = 0 if x ∈𝐙.Consider the boundary map ∂ : H_1(X_0(N), , 𝐙_p) →𝐙_p[]^0 given by ∂({α, β}) = (Γ_0(N) ·β) - (Γ_0(N) ·α).Let F_0,p : 𝐏^1(𝐙/N𝐙) →𝐙_p be such that if x = [c:d] ∈𝐏^1(𝐙/N𝐙), we have:6· F_0,p(x) =∑_(s_1,s_2) ∈ (𝐙/2N𝐙)^2(d-c)s_1+(d+c)s_2 ≡ 0(moduloN) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N).This is independent of the choice of c and d such that x = [c:d].We have m̃_0^+ =∑_x ∈𝐏^1(𝐙/N𝐙) F_0,p(x) ·ξ_Γ_0(N)(x) ∈ M_+ . Furthermore, one has ∂m̃_0^+=N-1/12·( (Γ_0(N)· 0) - (Γ_0(N) ·∞)), m̃_0^+ ∙m̃_0^- = N-1/12.We warn the reader that m̃_0^+ is 1/12·ℰ where ℰ is computed in <cit.>.Our main result about M_+ is a formula for m_1^+. Let F_1,p : 𝐏^1(𝐙/N𝐙) →𝐙/p𝐙 be such that12 · F_1,p([c:d])=∑_(s_1,s_2) ∈ (𝐙/2N𝐙)^2(d-c)s_1+(d+c)s_2 ≡ 0(moduloN) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log(s_2/d-c) - ∑_(s_1,s_2) ∈ (𝐙/2N𝐙)^2(d-c)s_1+(d+c)s_2 ≢0(moduloN) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log((d-c)s_1+(d+c)s_2))if [c:d]≠ [1:1] and F_1,p([1:1]) = 0. We have the following equality in M_+/pM_+:m_1^+ = ∑_x ∈𝐏^1(𝐙/N𝐙) F_1,p(x) ·ξ_Γ_0(N)(x).Furthermore, one has ∂ m_1^+ =( 1/3∑_k=1^N-1/2 k ·log(k) ) · ( (Γ_0(N)· 0) - (Γ_0(N) ·∞)),m_1^+ ∙ m_0^- = 1/3∑_k=1^N-1/2 k ·log(k).The last assertion, combined with the first equality of point (i) of Theorem <ref> below gives another proof of Theorem <ref>.§.§ ComparisonWe get additional results by comparing our results in the three above settings. First, combining our results in the spaces of even and odd modular symbols, a computation gives us the following formulas. * We have m_0^+ ∙ m_1^- = m_1^+∙ m_0^- = 1/3·∑_k=1^N-1/2 k ·log(k).* We have m_1^+ ∙ m_1^- = m_2^+ ∙ m_0^- = m_0^+ ∙ m_2^- = 1/6·∑_k=1^N-1/2 k ·log(k)^2.This proves Theorem <ref>. Note that all equalities but the last of each line follows formally from algebraic properties of pairing of Hecke modules.In particular, the boundary of m_2^+ is (1/6·∑_k=1^N-1/2 k ·log(k)^2 ) · ((Γ_0(N) · 0) - (Γ_0(N)·∞)). However, we do not know an expression for m_2^+ in terms of Manin symbols similar to Theorems <ref> and <ref>. It is unclear to us whether such an expression is to be expected as a quadratic expression of logarithms or as a formula in some algebraic number theoretic group considered in section <ref>.We also get the following result relating the three Hecke modules. Let i and j be integers such that 0 ≤ i,j≤ g_p and i+j ≤ g_p. We have:m_i^+ ∙ m_j^- = e_i ∙ e_j.Furthermore, this quantity depends only on i+j. It is 0 if i+j < g_p and it is non-zero if i+j = g_p. The combination of Theorem <ref>, Theorem <ref> and Theorem <ref> allows us to deduce the following identity. We were not able to find an elementary proof of it. Assume that ∑_k=1^N-1/2 k ·log(k) ≡ 0(modulo p). We then have∑_λ∈ L1/4·log(H'(λ))^2 - 1/3·log(λ)^2 ≡∑_k=1^N-1/2 k ·log(k)^2(modulo p) .This is most advanced instance (known to us) of what we call an higher Eichler mass formula. We do not know how to connect the algebraic objects of section <ref> with supersingular j (or λ) invariants.Obviously, there is a theory of higher Eisenstein elements for all levels and all weights. We finish this introduction by giving some reasons as to why we focus on the prime level and weight 2 cases. * Mazur's original question about the rank g_p was in this setting.* It is known that in this case, the Hecke algebra has nice properties. For example it is a Gorenstein ring at the Eisenstein maximal ideals, and the Eisenstein ideal I is locally principal, I/I^2 is cyclic.* We know, thanks to Mazur, some multiplicity one results for the homology of X_0(N).* There is an explicit and simple description of m_1^- coming from the Shimura covering X_1(N) → X_0(N).Mazur deduced from this a description of I/I^2 in terms of modular symbols.* In this case, the supersingular module presents itself. Our methods in turn has applications to supersingular elliptic curves.* The Galois cohomology group arising in the determination of m_2^- is well-understood thanks to the results of <cit.>.It is not clear to us in which settings we can obtain similar results. Mazur's question makes sense in any weight and level (although there are possibly several maximal Eisenstein ideals). However, we do not know multiplicity one in general, and Eisenstein ideals are not locally principal in general neither. See the list of alternative settings already proposed by Mazur in <cit.>.§ THE FORMALISM OF HIGHER EISENSTEIN ELEMENTS§.§ Algebraic settingIn this section, we develop the theory of higher Eisenstein elements in a tentative axiomatic setting. Let 𝕋̃ be a 𝐙_p-algebra which is free of finite rank as a 𝐙_p-module. Let Ĩ be an ideal of 𝕋̃ and _𝕋̃(Ĩ) be the annihilator of Ĩ in 𝕋̃. Let 𝕋 = 𝕋̃/_𝕋̃(Ĩ) and I ⊂𝕋 be the image of Ĩ in 𝕋. Let 𝐓̃ be the Ĩ-adic completion of 𝕋̃, and 𝐓 be the I-adic completion of 𝕋. Let t ≥ 1 be an integer. We assume the following hypotheses. *We have 𝕋̃/Ĩ≃𝐙_p. In particular, the ideal 𝔓̃ := Ĩ+(p) of 𝕋̃ is the unique maximal ideal containing Ĩ.*The 𝐙_p-module _𝕋̃(Ĩ) is free of rank one.*The 𝐙_p-module 𝕋 is free.*The group 𝕋/I is cyclic of order p^t.* The 𝕋/I-module I/I^2 is free of rank one. We fix a group isomorphism e : I/I^2 𝐙/p^t𝐙. If η∈Ĩ, we denote by e(η) the image by e of the class of η in I/I^2.*The ring 𝐓 is Gorenstein, the 𝐓-module _𝐙_p(𝐓, 𝐙_p) is free of rank one. Since 𝕋̃ is a Noetherian ring, if M is a finitely generated 𝕋̃-module then the Ĩ-adic completion of M is canonically isomorphic to M ⊗_𝕋̃𝐓̃ <cit.>.Let M be a 𝕋̃-module which is free of finite rank over 𝐙_p. Assume that M ⊗_𝕋̃𝐓̃ is free of rank one over 𝐓̃. Let M^0 ⊂ M be the submodule annihilated by _𝕋̃(Ĩ). Let r be an integer such that 1 ≤ r ≤ t.There exists a maximal positive integer n(r,p) and a sequence of elements e_0, e_1, ..., e_n(r,p) in M/p^r· M, called theof M, such that the following properties hold. * We have e_0 ∉p · (M/p^r· M)* For all η∈Ĩ, we have η(e_i) ≡ e(η)· e_i-1 (modulo 𝐙· e_0+ ... + 𝐙· e_i-2)(with the convention e_-1=0). We have the following properties.a) The element e_0 is unique up to (𝐙/p^r𝐙)^×. There exists ẽ_0 ∈ M[Ĩ] whose class in M/p^r· M is e_0. Furthermore, the element ẽ_0 is unique up to 𝐙_p^×, and ẽ_0 ∉M^0.b) We have e_0, ..., e_n(r,p)-1 ∈ M^0/p^r · M^0 and e_n(r,p)∉M^0/p^r· M^0 . c) If we fix e_0 in (M^0/p^r · M^0)[I], then for every integer i such that 1 ≤ i ≤ n(r,p) the image of e_i in (M/p^r· M)/(𝐙· e_0 + ... + 𝐙· e_i-2) is uniquely determined.d) The integer n(r,p) is the largest integer n ≥ 1 such that the group Ĩ^n· (𝕋̃/p^r·𝕋̃)/Ĩ^n+1· (𝕋̃/p^r·𝕋̃) is cyclic of order p^r. Furthermore, we have n(r,p) ≤_𝐙_p(𝐓) with equality if r=1. In particular, the integer n(r,p) only depends on 𝕋̃ and r, and not on the choice of M.By hypothesis (<ref>), there is a unique maximal ideal 𝔓̃ containing Ĩ in 𝕋̃. We denote by M_𝔓̃ the 𝔓̃-adic(or equivalently Ĩ-adic) completion of M.* The algebra 𝐓̃ is flat as a 𝕋̃-module.* The canonical map M → M_𝔓̃ is surjective.* Let M be a 𝕋̃-module which is free as a 𝐙_p-module. Then the natural morphism of 𝐓̃-modulesHom_𝐙_p(M_𝔓̃, 𝐙_p) →Hom_𝐙_p(M, 𝐙_p)_𝔓̃is an isomorphism.The ring 𝕋̃ has finitely many maximal ideals, is a semi-local ring since it is a finitely generated 𝐙_p-module. The ring 𝕋̃ is p-adically complete and semi-local, so is the direct sum of its completions at its maximal ideals <cit.>. Thus, any module over 𝕋̃ is the direct sum of its completions at the maximal ideals of 𝕋̃. Lemma <ref> follows immediately. Let M^*:=_𝐙_p(M, 𝐙_p), equipped with its natural structure of 𝕋̃-module. The 𝐓̃-module (M^*)_𝔓̃ is free of rank one by Lemma <ref> (iii) and hypothesis (<ref>). Since M is free over 𝐙_p, we have:M/p^r· M = (M^*/p^r· M^*, 𝐙/p^r𝐙).Thus, we have: (M/p^r· M)[Ĩ^n] = (M^*/(p^r+ Ĩ^n)· M^*, 𝐙/p^r𝐙) .We get a canonical group isomorphism(M/p^r· M)[Ĩ^n]/(M/p^r· M)[Ĩ^n-1] (Ĩ^n-1· (M^*/p^r· M^*)/Ĩ^n· (M^*/p^r· M^*), 𝐙/p^r𝐙) .Since (M^*)_𝔓̃ is free of rank one over 𝐓̃ and that 𝔓̃ is the unique maximal ideal of 𝕋̃ containing Ĩ, we have group isomorphisms(M^*/p^r· M^*)/Ĩ^n· (M^*/p^r· M^*)≃𝐓̃/(p^r+Ĩ^n)·𝐓̃≃𝕋̃/(p^r+Ĩ^n)·𝕋̃ .Let J be the image of Ĩ in 𝕋̃/p^r·𝕋̃.By (<ref>) and (<ref>), we get a (non-canonical) group isomorphism(M/p^r· M)[Ĩ^n]/(M/p^r· M)[Ĩ^n-1] (J^n-1/J^n, 𝐙/p^r𝐙).In particular, for n=1, we have group isomorphisms:(M/p^r· M)[Ĩ] ≃(𝕋̃/(p^r+Ĩ)·𝕋̃, 𝐙/p^r𝐙) ≃𝐙/p^r𝐙 ,where the last isomorphism follows from Hypothesis (<ref>).This shows the existence of e_0 ∈ (M/p^r· M)[Ĩ] such that e_0 ∉p · (M/p^r· M). Furthermore e_0 is unique up to (𝐙/p^r𝐙)^×. Similarly, we have isomorphisms of 𝐙_p-modules:M[Ĩ] ≃_𝐙_p(𝕋̃/Ĩ, 𝐙_p) ≃𝐙_p.This shows the existence of ẽ_0 ∈ M[Ĩ] which reduces to e_0 modulo p^r, and its unicity up to a scalar.We have M[Ĩ] = _𝕋̃(Ĩ) · M. The inclusion _𝕋̃(Ĩ) · M ⊂ M[Ĩ] is obvious. We first claim that M/_𝕋̃(Ĩ) · M is torsion-free. For any ideal 𝔔 of 𝕋̃ different from 𝔓̃, we have_𝕋̃(Ĩ) · M_𝔔 = 0.By the proof of Lemma <ref>, M is the direct sum of its completions at the maximal ideals of 𝕋̃. Thus, we have a canonical isomorphism of 𝕋̃-modules:_𝕋̃(Ĩ) · M _𝕋̃(Ĩ) · M_𝔓̃ .By (<ref>) and (<ref>), we have an isomorphism of 𝐙_p-modulesM/_𝕋̃(Ĩ)· M ≃( M_𝔓̃/_𝕋̃(Ĩ)· M_𝔓̃) ⊕_𝔔≠𝔓̃ M_𝔔 ,where the sum is over the maximal ideals 𝔔 of 𝕋̃ different from 𝔓̃. Since M_𝔓̃ is free of rank one over 𝐓̃, we have an isomorphism of 𝐓̃-modules M_𝔓̃/_𝕋̃(Ĩ)· M_𝔓̃≃𝐓̃/_𝕋̃(Ĩ)·𝐓̃ .As in (<ref>), we have an isomorphism of 𝐙_p-modules 𝕋̃/_𝕋̃(Ĩ)·𝕋̃≃𝐓̃/_𝕋̃(Ĩ)·𝐓̃⊕_𝔔≠𝔓̃𝕋̃_𝔔 .By hypothesis (<ref>), the former 𝐙_p-module is torsion-free. By (<ref>),(<ref>) and (<ref>), the 𝐙_p-module M/_𝕋̃(Ĩ)· M is torsion-free. By (<ref>) the 𝐙_p-module M[Ĩ] is free of rank one.Since M[Ĩ] is a direct summand of the 𝐙_p-module M, the 𝐙_p-module M[Ĩ]/_𝕋̃(Ĩ)· M is a free 𝐙_p-module of rank zero or one. If the rank of M[Ĩ]/_𝕋̃(Ĩ)· M is one, then _𝕋̃(Ĩ)· M = 0. By (<ref>), we get _𝕋̃(Ĩ) ·𝐓̃=0. As above, this implies _𝕋̃(Ĩ)=0, so 𝕋̃=𝕋. This contradicts hypothesis (<ref>). This concludes the proof of Lemma <ref>.We have M^0 = Ĩ· M. The inclusion Ĩ· M ⊂ M^0 is obvious. Since M_𝔓̃ is free of rank one over 𝐓̃, we have isomorphisms of 𝐙_p-modules:M/Ĩ· M ≃𝐓̃/Ĩ≃𝐙_p.We thus have a surjective group homomorphism𝐙_p ≃ M/Ĩ· M → M/M^0 .The 𝐙_p-module M/M^0 is torsion-free by definition of M^0 and the fact that M is a free 𝐙_p-module. Thus, we have either M^0 =Ĩ· M or M^0 =M. If M^0=M, then we have _𝕋̃(Ĩ) · M = 0, which is impossible by Lemma <ref>. Denote by φ : M → M_𝔓̃ the canonical map. The element φ(ẽ_0) is a generator of M_𝔓̃[Ĩ]. In particular, the element e_0 is sent to a generator of (M_𝔓̃/p^r· M_𝔓̃)[Ĩ]. This follows from Lemma <ref> sinceM[Ĩ] = _𝕋̃(Ĩ)· M ≃_𝕋̃(Ĩ)· M_𝔓̃ = M_𝔓̃[Ĩ],where the middle isomorphism follows from (<ref>).The canonical map Ĩ→ I induces a group isomorphism Ĩ/Ĩ^2I/I^2.We have, by definition and by Hypothesis (<ref>), a surjective group homomorphism Ĩ/Ĩ^2 → I/I^2 ≃𝐙/p^t𝐙. It remains to show that the kernel of the map Ĩ/Ĩ^2 → I/I^2 is zero. This kernel is the image of Ĩ∩_𝕋̃(Ĩ) inĨ/Ĩ^2. Thus, it suffices to prove that Ĩ∩_𝕋̃(Ĩ)=0. By hypothesis (<ref>), there exists T_0 ∈𝕋̃ such that _𝕋̃(Ĩ) = 𝐙_p· T_0. Let x ∈Ĩ∩_𝕋̃(Ĩ). For the sake of a contradiction, assume that x ≠ 0. Then there exists n ∈𝐙_p \{0} such that x = n · T_0. Since the image of x in 𝕋̃/Ĩ≃𝐙_p is zero, we have T_0 ∈Ĩ, so _𝕋̃(Ĩ) ⊂Ĩ. This contradicts hypothesis (<ref>). By Lemma <ref>, the ideal Ĩ·𝐓̃ is principal. In particular, for every n ≥ 1 the group Ĩ^n-1· (𝐓̃/p^r·𝐓̃)/Ĩ^n· (𝐓̃/p^r·𝐓̃) is cyclic of order ≤ p^r. Thus, J^n-1/J^n is cyclic of order ≤ p^r. We let n(r,p) be the largest integer n≥ 1 such that the group J^n/J^n+1 has order p^r. By (<ref>), there exists e_0, e_1, ..., e_n(r,p) satisfying properties (ii), and n(r,p) is the largest such integer.To prove property (a), it suffices to prove that ẽ_0 ∉M^0. Assume for the sake of a contradiction that ẽ_0 ∈ M^0. By Lemma <ref>, we have ẽ_0 ∈Ĩ· M. Using Lemma <ref>, we get _𝕋̃(Ĩ)· M ⊂Ĩ· M. Since M_𝔓̃ is free of rank one over 𝐓̃, we have _𝕋̃(Ĩ)·𝐓̃⊂Ĩ·𝐓̃. As in (<ref>), we get _𝕋̃(Ĩ) ·𝕋̃⊂Ĩ·𝕋̃, which contradicts Hypothesis (<ref>). Property (c) follows from (<ref>).Lemma <ref> shows that e_0, ..., e_n(r,p)-1 ∈ M^0/p^r· M^0. Assume e_n(r,p)∉M^0/p^r· M^0. By Lemma <ref>, we have e_n(r,p)∈Ĩ· (M/p^r· M). Let f_i be the image of e_i in M_𝔓̃/p^r· M_𝔓̃. There exists f_n(r,p)+1∈ M_𝔓̃/p^r· M_𝔓̃ satisfying Property (ii). By Lemma <ref>, the group (𝐓̃/p^r·𝐓̃)[Ĩ^n(r,p)+2]/(𝐓̃/p^r·𝐓̃)[Ĩ^n(r,p)+1] is cyclic of order p^r. Thus, J^n(r,p)+1/J^n(r,p)+2 is cyclic of order p^r, which contradicts the definition of n(r,p).We finally show Property (d). The map r ↦ n(r,p) is obviously a decreasing function of r. Thus, it suffices to prove n(1,p) = _𝐙_p(𝐓). Let η be a generator of Ĩ·𝐓̃ and R(X) ∈𝐙_p[X] be the characteristic polynomial of η acting on the free 𝐙_p-module 𝐓̃. We then have a ring isomorphism𝐓̃≃𝐙_p[X]/(R(X)).We have _𝐙_p(𝐓̃) = _𝐙_p(𝐓)+1 by hypothesis (<ref>), so deg(R) =_𝐙_p(𝐓)+1. By Hypothesis <ref>, we have R(0)=0. Since 𝐓̃ is local, we have R(X) ≡ X^_𝐙_p(𝐓)+1 (modulo p ). Thus, we have a ring isomorphism:𝐓̃/p·𝐓̃≃ (𝐙/p𝐙)[X]/(X^_𝐙_p(𝐓)+1)such that η (modulo p) is sent to X. Thus, J^n/J^n+1 is isomorphic to (X^n)/(X^n+1), which is non-zero if and only if n ≤_𝐙_p(𝐓), and also if and only if n ≤ n(1,p) by definition. Thus we have n(1,p) = _𝐙_p(𝐓). This concludes the proof of Theorem <ref>.Another characterization of n(r,p) that will be used later. The integer n(r,p) is the largest integer k ≥ 1 such that p^t is in I^k+p^r· I. We apply Theorem <ref> with M=𝕋̃, which obviously satisfies the hypotheses of the theorem. Let T_0 be a generator of the 𝐙_p-module _𝕋̃(Ĩ). By hypothesis (<ref>), we can assume that T_0 - p^t ∈Ĩ. We can choose ẽ_0 = T_0 by Lemma <ref>. Since r ≤ t, the image of T_0-p^t in M/p^r· M is e_0. By Lemma <ref>, we have M^0 = Ĩ.The integer n(r,p) is the largest integer k≥ 1 such that e_0 ∈Ĩ^k · (M/p^r· M). By Theorem <ref>, if η∈Ĩ\Ĩ^2, there exists c ∈ (𝐙/p^r𝐙)^× such that η^n(r,p)· e_n(r,p) = c· e_0. Thus, we have e_0 ∈Ĩ^n(r,p)· (M/p^r· M). Assume for the sake of a contradiction that e_0 ∈Ĩ^n(r,p)+1· (M/p^r· M), then we have η^n(r,p)· e_n(r,p)∈Ĩ^n(r,p)+1· (M/p^r· M). The groupĨ^n(r,p)·(𝕋̃/p^r·𝕋̃)/Ĩ^n(r,p)+1·(𝕋̃/p^r·𝕋̃)is cyclic of order p^r, so we have e_n(r,p)∈Ĩ· (M/p^r· M). By Lemma <ref>, we have e_n(r,p)∈ M^0/p^r· M^0. This contradicts Theorem <ref> b). By lemma <ref>, the integer n(r,p) is the largest integer k≥ 1 such that T_0-p^t ∈Ĩ^k + p^r·𝕋̃. Since T_0-p^t ∈Ĩ and Ĩ∩ (p^r) = p^r·Ĩ, it is also the largest integer k≥ 1 such that T_0-p^t ∈Ĩ^k + p^r·Ĩ. This concludes the proof of Proposition <ref> since (𝕋̃→𝕋) = 𝐙_p· T_0 and that the unique α∈𝐙_p such that α· T_0 - p^t ∈Ĩ is α=1.§.§ The Newton polygon of 𝐓̃As in (<ref>), there is a (non-canonical) ring isomorphism𝐓̃≃𝐙_p[X]/(R(X))for some monic polynomial R = ∑_i=0^_𝐙_p(𝐓̃)+1 a_i · X^i ∈𝐙_p[X]. Recall that the Newton polygon of R is the lower convex hull of the points {(i, v_p(a_i)), i ∈{0, ..., _𝐙_p(𝐓̃)+1}} (where v_p is the usual p-adic valuation and we omit the points with a_i=0). The Newton polygon of R only depends on 𝐓̃. The Newton polygon of 𝐓̃ is by definition the Newton polygon of R, and we denote it by (𝐓̃). We now recall a finer invariant of 𝐓̃ than (𝐓̃), introduced in by Wake–Wang-Erickson in <cit.>. By convention, we let v_p(0) = ∞. Define a sequence z_0, ..., z__𝐙_p(𝐓̃)+1 inductively by z_0 = v_p(a_0) and z_i = min(z_i-1, v_p(a_i)) (by convention, min(z, ∞)=z for any z ∈𝐙∪{∞}). One easily sees that (𝐓̃) is the lower convex hull of the points {(i, z_i),i∈{0, ...,_𝐙_p(𝐓̃)+1}} (we omit the points with z_i = ∞). Let i ∈{0, ..., _𝐙_p(𝐓̃)+1}. Then z_i is the supremum of the set of integers r≥ 1 such that there exists a surjective ring homomorphismf: 𝐙_p[X]/(R(X)) ↠ (𝐙/p^r𝐙)[x]/(x^i+1)such that f(X) = x (this supremum can be ∞)<cit.>. We choose R so that X corresponds to a generator of Ĩ·𝐓̃ via (<ref>). By Hypothesis (<ref>) we have a_0 = 0. By Hypothesis (<ref>) and Lemma <ref>, we have v_p(a_1) = t. Thus, we have z_0 = ∞ and z_i ≤ t for all i ≥ 1.By Theorem <ref> d), one easily sees that for all integer r such that 1 ≤ r ≤ t, the integer n(r,p) is the largest integer i ≥ 1 such that there exists a surjective ring homomorphismf': 𝐙_p[X]/(R(X)) ↠ (𝐙/p^r𝐙)[x]/(x^i+1)such that f'(X) = x. Thus, for all r ∈{1, ..., t} and i ∈{1, ..., _𝐙_p(𝐓̃)+1}, we have:n(r,p) = max{j ∈{1, ..., _𝐙_p(𝐓̃)+1}, z_j ≥ r }andz_i = max{r' ∈{1, ..., t}, n(r',p) ≥ i } .Hence, knowing the sequence (n(r,p))_1 ≤ r ≤ t amounts to knowing the sequence (z_i)_0 ≤ i ≤_𝐙_p(𝐓̃)+1. Thus, the sequence (n(r,p))_1 ≤ r ≤ t is a finer invariant of 𝐓̃ than (𝐓̃) (<cit.> for an example where the Newton polygon does not determine the (z_i)_0 ≤ i ≤_𝐙_p(𝐓̃)+1). §.§ Pairing between higher Eisenstein elementsThe proof of the following is easy and left to the reader.Let M and M' be two 𝕋̃-modules satisfying the assumptions of Theorem <ref>. Let 1 ≤ r ≤ t be an integer. Let e_0, ..., e_n(r,p) (resp. e_0', ..., e_n(r,p)') be the elements of M/p^r· M (resp. M'/p^r· M') constructed in Theorem <ref>. Let ∙ : M × M' →𝐙/p^r𝐙be a perfect 𝕋̃-equivariant bilinear pairing. Let i ∈{0, ..., n(r,p)}. Then an element m (resp. m') of Ĩ^i· (M/p^r· M) (resp. Ĩ^i· (M'/p^r· M')) is in Ĩ^i+1· (M/p^r· M) (resp. Ĩ^i+1· (M'/p^r· M')) if and only if m ∙ e_i' ≡ 0(modulo p^r) (resp. e_i ∙ m' ≡ 0(modulo p^r)).Keep the notation of Proposition <ref> and consider the elements e_0, ..., e_n(r,p), e_0', ..., e_n(r,p)' modulo p^r. The producte_i ∙ e_j'only depends on i+j, is zero modulo p^r if i+j < n(r,p) and non-zero modulo p^r if i+j=n(r,p). (Note that if i+j>n(r,p), the product is not well-defined since e_i is only defined modulo e_0, ..., e_i-1 and the same for e_j'). In particular, we have n(r,p) ≥ 2 if and only if e_1 ∙ e_0' ≡ 0(modulo p ), and _𝐙_p(𝕋) ≥ 3 if and only if e_1 ∙ e_0' ≡ e_1 ∙ e_1' ≡ 0(modulo p ). §.§ The special case of weight 2 and prime level In this section, as well as in the rest of the article, let 𝕋̃ be the the Hecke algebra with 𝐙_p coefficients acting faithfully on the space ofmodular forms of weight 2 and level Γ_0(N). The ideal Ĩ is the Eisenstein ideal, generated by the Hecke operators T_ℓ-ℓ-1 if ℓ≠ N and by U_N-1. We easily see that 𝕋 is the Hecke algebra acting faithfully on the space of cuspidal modular forms of weight 2 and level Γ_0(N). We now check the hypotheses of section <ref>.Hytothesis (<ref>) is obvious. Hypothesis (<ref>) follows from the fact that there is a unique Eisenstein series of weight 2 and level Γ_0(N) since N is prime. Hypothesis (<ref>) is obvious. Hypothesis (<ref>) follows from <cit.>. Hypothesis (<ref>) follows from <cit.>. Hypothesis (<ref>) follows from <cit.>. Thus, we can apply Theorem <ref>.As in <cit.>, we let T_0 ∈𝕋̃ be such that _𝕋̃(Ĩ) = 𝐙_p· T_0 and T_0-N-1/ν∈Ĩ (where ν = (N-1, 12)).By <cit.>, there is a group isomorphism e : I/I^2𝐙/p^t𝐙 such that for all prime ℓ not dividing N, we have:e(T_ℓ-ℓ-1) = ℓ-1/2·log(ℓ)where, if p=ℓ=2, this equality meanse(T_2-3) = log(x)where x^2 ≡ 2 (modulo N). In order to study g_p, only higher Eisenstein elements modulo p (and not modulo p^r for r>1) are important. In practice, when we construct the e_i's, we need only to check condition (ii) of Theorem <ref> for Hecke operators T_ℓ-ℓ-1 for primes ℓ outside a finite set. We keep the notation of Theorem <ref>. Let S be a finite set of rational primes containing N. Let i be an integer such that 1 ≤ i ≤ n(r,p) and m ∈ M/p^r· M be such that for all ℓ∉S prime, we have:(T_ℓ-ℓ-1)(m) = ℓ-1/2·log(ℓ)· e_i-1 .Then we have m ≡ e_i(modulo the subgroup generated by e_0, ..., e_i-1).The element m-e_i∈ M/p^r is annihilated by T_ℓ-ℓ-1 for all prime ℓ∉S. It thus suffices to prove the following result.Let W ⊂ M/p^r· M be set of elements annihilated by T_ℓ-ℓ-1 for ℓ prime not in S. Then W = (𝐙/p^r𝐙) · e_0. We prove it by induction on r. Let T be the 𝐙_p-subalgebra of 𝕋̃ generated by the operators T_ℓ for ℓ∉S. Assume first that r=1. Let w ∈ W. By the Deligne–Serre lifting lemma <cit.>, there is a finite extension 𝒪⊂𝐐_p of 𝐙_p (𝒪 is a discrete valuation ring) and an element w̃∈ M ⊗_𝐙𝒪 which is proper for the action of T and such that the eigenvalue of w̃ for T_ℓ modulo π (an uniformizer of 𝒪) is the eigenvalue of w for T_ℓ if ℓ∉S.The strong multiplicity one theorem shows that T ⊗_𝐙_p𝐐_p = 𝕋⊗_𝐙_p𝐐_p.Thus, there is a ring homomorphismφ : 𝕋→𝐐_psuch that for all t ∈ T, φ(t) is the eigenvalue of t on w̃. There is an associated semi-simple Galois representation ρ_φ : (𝐐/𝐐) →GL_2(𝐐_p). The associated semi-simple residual representation must be the direct sum of the trivial character and the Teichmüller character. This shows that, in fact, for all prime ℓ not dividing N, T_ℓ-ℓ-1 annihilates w. We conclude that w and e_0 are proportional modulo p, which concludes the case r=1.Now, let r ≥ 1 be any integer. By the case r=1, there exists λ∈𝐙 such thatw - λ· e_0 ∈ p· M/p^r · M.The the elementx-λ· e_0/p∈ M/p^r-1· Mis annihilated by T_ℓ-ℓ-1 for all ℓ∉S. By induction this concludes the proof of Lemma <ref>. § THE SUPERSINGULAR MODULEWe keep the notation of Chapters <ref> and <ref>. In particular, p ≥ 2 is a prime such that p^t divides exactly the numerator of N-1/12. Let r be an integer such that 1 ≤ r ≤ t. §.§ Preliminary results and notation Let S be the set of isomorphism classes of supersingular elliptic curves over 𝐅_N, and M := 𝐙_p[S] be the free 𝐙_p-module with basis the elements of S. If E ∈ S, we denote by [E] the corresponding element in M and we let w_E = ((E))/2∈{1,2,3}. The ring 𝕋̃ acts on M in the following way. If n≥ 1, we haveT_n([E]) = ∑_C [E/C]where C goes through the cyclic subgroup schemes of order n in E. If n and N are coprimes, these subgroup schemes correspond to the subgroups of E(𝐅_N) isomorphic to 𝐙/n𝐙. We also have U_N([E]) = [E^(N)]where U_N is the Nth Hecke operator and E^(N) is the Frobenius transform of E.<cit.> The 𝐓̃-module M ⊗_𝕋̃𝐓̃ is free of rank one. Thus, we can apply Theorem <ref> to M. One easily sees that M^0 is the augmentation subgroup of M. There is a 𝕋̃-equivariant bilinear pairing∙ : M × M →𝐙_pgiven by [E] ∙ [E'] = 0 if [E] ≠ [E'] and [E] ∙ [E] = w_E.We letẽ_0 = ∑_E ∈ S1/w_E· [E] ∈𝐐_p[S].The following result is well-known.* We have ẽ_0 ∈ M[Ĩ].* We have:ẽ_0 ∙ẽ_0 = N-1/12 . We first check that ẽ_0 ∈ M. This is obvious if p ≥ 5, so we need to check it when p ∈{2,3}. If p=2 (resp. if p=3), there is some E in S with w_E=2(resp. w_E=3) if and only if 4 divides N-3 (resp. 3 divides N-2). Since p divides the numerator of N-1/12, we have ẽ_0 ∈ M. The fact that ẽ_0 is annihilated by Ĩ is an easy and well-known computation. The last assertion is equivalent to Eichler mass formula:∑_E ∈ S1/w_E = N-1/12 . We let e_0 be the image of ẽ_0 in M/p^r· M. We denote by e_0, e_1, ..., e_n(r,p) the higher Eisenstein elements in M/p^r· M. In this chapter, we give an explicit formula for e_1. One idea would be to consider the element whose coefficient in [E] is log(S'(j(E))), where S(X) ∈𝐅_N[X] is the monic polynomial whose roots are the j-invariants of elements in S. It turns out that this element is not e_1. For reasons we do not fully understand, this idea leads us to e_1 after adding an auxiliary Γ(2)-structure. In other words, we replace the j-invariants by the Legendre λ-invariants. The analogue of the polynomial S in this context is the well-known Hasse polynomial. In the next section, we mainly study the relation between the Hasse polynomial and the Hecke operators.§.§ The Hasse polynomialRecall the definition of the Hasse polynomial:H(X) = ∑_i=0^N-1/2N-1/2i^2· X^i ∈𝐅_N[X].LetP(Y) = _X(H'(X), 256·(1-X+X^2)^3-X^2· (1-X)^2· Y) ∈𝐅_N[Y]. §.§.§ The discriminant of the Hasse polynomialThe computation of e_1 ∙ e_0 is related to the discriminant (H) of the Hasse polynomial. Let m = N-1/2. We have, in 𝐅_N:(H) = (-1)^m(m-1)/2/m!·∏_k=1^m k^4k .In particular, we have (H) ≠ 0, so the roots of H are simples. The following Picard–Fuchs differential equation is well-known (the proof of <cit.>).Let A=4X(1-X) and B=4(1-2X). Then for all n ≥ 0, we have the following differential equation:A· H^(n+2)+(n+1)· B· H^(n+1)-(2n+1)^2· H^(n)=0where H^(n) is the nth derivative of H.For all n ≥ 0, let r_n = (H^(n), H^(n+1)), whereis the resultant. Note that r_0 = (-1)^m(m-1)/2Disc(H).We can rewrite the differential equation (<ref>) asA· H^(n+2)+(n+1)· B· H^(n+1)=(2n+1)^2· H^(n)and then take the resultant of both sides with H^(n+1).Thus, for all 0 ≤ n ≤ m-2, we have: r_n = (-4)^m-n-1· (2n+1)^2n+2· H^(n+1)(0)· H^(n+1)(1) · r_n+1 .Note that r_m-1 = (H^(m-1), m!) = m!. We have to compute a_n:=H^(n)(0) and b_n:=H^(n)(1). We immediately see that a_n = n!·mn^2. We can again use the differential equation (<ref>) to compute b_n. For all 0 ≤ n ≤ m-1, we get: b_n = -4(n+1)/(2n+1)^2· b_n+1 .Furthermore, we have b_m = m!. We thus have, for all 0 ≤ n ≤ m:b_n = m!·∏_i=n^m-1-4·(i+1)/(2i+1)^2 .Thus, we have:∏_i=1^m-1H^(i)(0)· H^(i)(1) = m!^m-1·∏_i=1^m-1i!·mi^2·(-4· (i+1)/(2i+1)^2)^i.This gives us, after simplification:(H,H') = r_0 = m!^m ·∏_i=1^m-1 (2i+1)^2· (i+1)^i · i! ·mi^2.A direct computation finally shows that:(H,H') = m!^-1·∏_k=1^m k^4k .This concludes the proof of Theorem <ref>. §.§.§ Relation between the Hasse polynomial and the modular polynomials We denote by L the set of roots of H in 𝐅_N. By Theorem <ref>, the roots of H are simples, so (L) = N-1/2. Let L' be the set of isomorphism classes of triples (E, P_1, P_2) where E is a supersingular elliptic curve over 𝐅_N and (P_1, P_2) is a basis of the 2-torsion E[2](𝐅_N). Each such triple is isomorphic to a unique triple of the form (E_λ : y^2=x(x-1)(x-λ), (0,0),(1,0)) where λ∈𝐅_N \{0,1}. We call λ the lambda-invariant of (E, P_1, P_2). The relation between the j-invariant and the lambda-invariant is:j = 256· (1-λ+λ^2)^3/λ^2·(1-λ)^2 .We identify L' with the set of supersingular lambda-invariants. It is well-known that λ∈ L' if and only if H(λ)=0 <cit.>. Thus, one can (and do) identify L' and L. Since the supersingular Legendre elliptic curves in characteristic N are defined over 𝐅_N^2, we have L ⊂𝐅_N^2. If λ and λ' are in 𝐅_N \{0,1} (not necessarily in L) and ℓ≥ 3 is a prime number different from N, we say that λ and λ' are ℓ-isogenous, and we write λ∼_ℓλ', if there is a rational isogeny ϕ : E_λ→ E_λ' of degree ℓ preserving the Γ(2)-structure, ϕ(0,0) = (0,0) and ϕ(1,0)=(1,0).It is well-known (for instance <cit.>) that there exists a unique polynomial φ_ℓ∈𝐙[X,Y] which is monic in X and Y, and such that we have λ∼_ℓλ' if and only if φ_ℓ(λ, λ')=0. The polynomial φ_ℓ(X,Y) is the “Legendre version” of the classical ℓth modular polynomial. The following result is the key to compute e_1. Let ℓ be a prime number not dividing 2· N and let λ∈ L. We have, in 𝐅_N^2:∏_λ' ∼_ℓλ H'(λ') = ℓ^ℓ-1· H'(λ)^ℓ+1 .Equation (<ref>) makes sense even if λ∈𝐅_N \({0,1}∪ L), but it does not hold in general. The supersingularity of λ is used crucially in Lemma <ref> below. The idea is to reinterpret H(λ) and H'(λ) as modular forms of level 1, and more precisely as Eisenstein series. If k ≥ 2 is an integer, letE_k = 1+(-1)^k-1· 4k/B_k·∑_n≥ 1σ_k-1(n)q^nbe the unique Eisenstein series of weight k and level _2(𝐙) with constant coefficient 1.Recall that a modular form of level _2(𝐙) with coefficients in an𝐅_N-algebra A can be considered, following Deligne and Katz <cit.>, as a rule on pairs (E, ω) were E is an elliptic curve over A and ω is a differential form generating H^0(E, Ω^1) over A. Recall that if λ∈𝐅_N \{0,1}, we have denoted by E_λ the Legendre elliptic curve y^2=x(x-1)(x-λ). We let ω_λ = dx/y. The following fact, essentially due du Katz, is crucial to interpret H'(λ).Let m = N-1/2. For all λ∈𝐅_N \{0,1}, we have the following equality in 𝐅_N:E_N+1(E_λ, ω_λ) = 3/m+1·(m· H(λ) - (λ-1) · H'(λ)) + (λ+1)· H(λ).Recall the following (unpublished) theorem of Katz <cit.>. Let (E,ω) be a pair of the form (y^2=4x^3-g_2x-g_3, dx/y). Then, 1/12· E_N+1(E, ω) is the coefficient of x^N-2 in the polynomial (4x^3-g_2x-g_3)^N-1/2.Let λ∈𝐅_N \{0,1}, and f(x) = 4x(x-1)(x-λ). The pair (y^2=f(x), dx/y) is isomorphic to the pair (y^2=g(x), dx/y) with g(x) = f(x+c)=4x^3-g_2x-g_2 (for some g_2 and g_3), where c = λ+1/3. We haveE_N+1(E_λ, ω_λ) = E_N+1(y^2=f(x), 2·dx/y) = 2^-(N+1)· E_N+1(y^2=g(x), dx/y) ,so 2^N+1/12· E_N+1(E_λ, ω_λ) is the coefficient of x^N-2 in g(x)^N-1/2=f(x+c)^N-1/2. We have (f^N-1/2)^(N-2)(c) = (f^N-1/2)^(N-2)(0)+c·(f^N-1/2)^(N-1)(0)where (f^N-1/2)^(k) is the kth derivative of f^N-1/2, since (f^N-1/2)^(k)=0 if k≥ N. A direct computation finally shows that2^N+1/12· E_N+1(E_λ, ω_λ) = 2^N-1/m+1·(m· H(λ)-(λ-1)· H'(λ)) + λ+1/3· 2^N-1· H(λ). Let ℓ be a prime number not dividing 2· N. If φ: E_λ→ E_λ' is an isogeny of degree ℓ preserving the Γ(2)-structure thenφ^*(ω_λ') = c_φ·ω_λforsome c_φ∈𝐅_N^×.Note that there are only two ℓ-isogenies E_λ→ E_λ'. These are φ and -φ. Consequently, c_φ^2 only depends on λ and λ' (satisfying φ_ℓ(λ, λ')=0). In fact one can show that c_φ^2=:c(λ, λ')is a rational function of λ and λ'. More precisely, we have <cit.>:c_φ^2 = -ℓ·λ(1-λ)/λ'(1-λ')·∂_Y φ_ℓ(λ', λ) /∂_X φ_ℓ(λ', λ).Note that this equality is true in 𝐐(λ, λ') and is deduced in 𝐅_N(λ, λ') by reducing modulo N our objects (elliptic curves, isogenies...).We have, over any field of characteristic ≠ 2, without assuming that E_λ is supersingular:∏_φ c_φ^2 = ℓ^2where the product is over a choice of non isomorphic degree ℓ+1 isogenies with origin E_λ and preserving the Γ(2)-structure. Equivalently, we have the following equality in 𝐐(λ):_X(∂_Y φ_ℓ(X, λ), φ_ℓ(X,λ))/_X(∂_X φ_ℓ(X, λ), φ_ℓ(X,λ)) = ℓ^-(ℓ-1) . Let F be a separably closed field of characteristic different from 2. For all λ∈ F \{0,1}, we have in F:∏_λ' ∼_ℓλλ' = λ^ℓ+1and∏_λ' ∼_ℓλ (1-λ') = (1-λ)^ℓ+1 .This is well-known, for instance <cit.>. The equivalence between the two equalities comes from the relation∏_φ c_φ^2 = ℓ^ℓ+1·_X(∂_Y φ_ℓ(X, λ), φ_ℓ(X,λ))/_X(∂_X φ_ℓ(X, λ), φ_ℓ(X,λ))which is a direct consequence of (<ref>) and Lemma <ref>. Let F_ℓ(λ) = ∏_φ c_φ^2. This is a rational fraction in λ, which has no poles or zeros outside 0 and 1. Thus, we have F_ℓ(λ) = C_ℓ·λ^n_ℓ·(λ-1)^m_ℓ where n_ℓ, m_ℓ ∈𝐙 and C_ℓ∈𝐅_N^×.The relationsφ_ℓ(X,Y) = X^ℓ+1Y^ℓ+1·φ_ℓ(X^-1,Y^-1)and∏_λ's.t. φ_ℓ(λ', λ)=0λ' = λ^ℓ+1show that F_ℓ(λ) = F_ℓ(λ^-1). This last relation shows that n_ℓ = -(n_ℓ+m_ℓ). One can easily show that φ_ℓ(1-X, 1-Y) = φ_ℓ(X,Y). This is deduced for example from the fact that λ(-1/z) = 1-λ(z) if z is in the complex upper-half plane. The relation φ_ℓ(1-X, 1-Y) = φ_ℓ(X,Y) shows that F_ℓ(1-λ) = F_ℓ(λ), so n_ℓ=m_ℓ. Since n_ℓ=-(n_ℓ+m_ℓ), we have n_ℓ = m_ℓ = 0, so F_ℓ is a constant. In order to show that C_ℓ = ℓ^2, it suffices to prove it over 𝐂 using the q-expansions, where q = e^iπ z. More precisely, let λ = λ(q) be fixed, where q = e^i π z. The ℓ+1 roots of φ_ℓ(λ, X) are the λ_i = λ(e^iπ·(z+2i)/ℓ) where i ∈{0,1,..., ℓ-1} and λ_ℓ = λ(q^ℓ). We thus haveC_ℓ = c(λ, λ_ℓ)^2·∏_i=0^ℓ-1c(λ, λ_i)^2where c was defined in (<ref>).This last factor is a constant function of z. Note that c(λ, λ_ℓ) = ℓ/c(λ_ℓ, λ). By <cit.>, for all i ∈{0,1,..., ℓ-1} we have:c(λ, λ_i) = θ_3(e^iπ·(z+2i)/ℓ)^2/θ_3(e^iπ z)^2 ,where θ_3(q) = ∑_n∈𝐙 q^n^2. Thus, c(λ, λ_i) goes to 1 when z goes to i∞. Similarly, c(λ_ℓ, λ) goes to 1 when z goes to i∞. This shows C_ℓ=ℓ^2. In order to conclude the proof of Theorem <ref>, we need one last result, which is only true in characteristic N and which uses the supersingularity is used in an essential way. <cit.> Let λ∈ L and φ: E_λ→ E_λ' be an isogeny of degree ℓ. We have:E_N+1(E_λ', ω_λ') = ℓ· E_N+1(E_λ, φ^*(ω_λ')).Using Lemma <ref> and (<ref>), we get∏_λ' ∼_ℓλ H'(λ') = ℓ^ℓ+1·∏_φ c_φ^-(N+1)· H(λ)^ℓ+1 .Lemma <ref> shows that ∏_φ c_φ^N+1 = (ℓ^2)^N+1/2 = ℓ^2, which concludes the proof of Theorem <ref>. We letφ_2(X,Y) := Y^2· (1-X)^2 + 16· X· Y - 16· X ∈𝐙[X,Y] .The following result motivates the definition of φ_2. Let z ∈𝐂 and λ = λ(q) (where q = e^i π z). We have in 𝐂[Y]:(Y-λ(e^iπ z/2)) ·(Y-λ(e^iπ (z+2)/2) ) = 1/(1-λ)^2·φ_2(λ, Y). We have:(Y-λ(e^i π z/2))(Y-λ(e^i π (z+2)/2)) = Y^2 - (λ(q^1/2)+λ(-q^1/2))· Y+λ(q^1/2)λ(-q^1/2)where q^1/2 := e^i π z/2. For clarity, let λ_1 = λ(q^1/2) et λ_2=λ(-q^1/2). We have the following q-expansion identity:λ(q) = 16q∏_n=1^∞( 1+q^2n/1+q^2n-1)^8. We follow <cit.> by letting Q_0=∏_n=1^∞ (1-q^2n), Q_1=∏_n=1^∞ (1+q^2n), Q_2=∏_n=1^∞ (1+q^2n-1) and Q_3=∏_n=1^∞ (1-q^2n-1).We have:λ(q) = 16q·(Q_1/Q_2)^8andλ(q^1/2)·λ(-q^1/2) = -16^2q∏_n=1^∞(1+q^n)^16/(1-q^2n-1)^8 = -16^2q·((Q_1· Q_2)^2/Q_3)^8.We have<cit.>:Q_1· Q_2· Q_3=1and Q_2^8=Q_3^8+16q· Q_1^8. By (<ref>) and (<ref>), we have:λ(q^1/2)·λ(-q^1/2) = -16^2q(Q_1· Q_2)^3· 8 = -16^2q·(Q_1/Q_2)^3· 8· Q_2^6· 8 .By (<ref>) and (<ref>), we have:Q_2^3· 8·(Q_1/Q_2)^8·(1-16q·(Q_1/Q_2)^8)=1which gives Q_2^6· 8 = 16^2q^2/λ^2·(1-λ)^2 .By (<ref>) and (<ref>), we have:λ(q^1/2)·λ(-q^1/2) = -16·λ(q)/(1-λ(q))^2that is λ_1λ_2 = -16λ/(1-λ)^2. By <cit.>, we have λ(-q)=λ(q)/1-λ(q), which gives(1-λ_1)· (1-λ_2)=1,that isλ_1+λ_2=λ_1·λ_2 = -16·λ/(1-λ)^2 .This concludes the proof of Proposition <ref>. The following result is the analogue of Theorem <ref> for ℓ=2.Let λ∈ L and let λ_1 and λ_2 be the roots of the polynomial φ_2(X, λ). We have:λ^N-1· H'(λ_1)· H'(λ_2) = λ^2· (λ-1)/4· H'(λ)^2. Keep the notation of Theorem <ref>, except that λ∈𝐅_N\{0,1} is now arbitrary. We have:λ^N-1· H(λ_1)· H(λ_2) = H(λ)^2It suffices to prove the following identity, in 𝐅_N[X]:_X(H(X), φ_2(X,Y))=H(Y)^2.We have_X(H(X), φ_2(X,Y))=∏_λ' ∈ Lφ_2(λ', Y) =∏_λ' ∈ Lλ'^2· (λ_1'-Y)·(λ_2'-Y),where λ_1' et λ_2' are the roots of φ_2(λ', Y) = 0 (if φ_2(λ', Y) has double roots then we let λ_1' = λ_2'). Since ∏_λ' ∈ Lλ' = 1, we get _X(H(X), φ_2(X,Y)) = ∏_λ' ∈ L (λ_1'-Y)·(λ_2'-Y).If λ' ∈ L then λ_1' and λ_2' are also in L, since by Proposition <ref> the elliptic curves E_λ_1' and E_λ_2' are 2-isogenous to E_λ. If λ' ∈𝐅_N \{0,1,-1} (resp. λ' = -1) then the polynomial φ_2(λ', Y) has two distinct roots (resp. a double root Y=2). Conversely, if λ_1 ∈𝐅_N \{0, 1, 2} (resp. λ_1=2) then the polynomial φ_2(X, λ_1) has two distinct roots (resp. a double root X=-1). Thus, we have∏_λ' ∈ L (λ_1'-Y)·(λ_2'-Y) = ∏_λ' ∈ L (λ'-Y)^2 = H(Y)^2. We are going to differentiate (<ref>) two times. Let K = 𝐅_N(λ) be the function field of 𝐏^1(λ). We have a derivation d/dλ which sends λ to 1. Let F = K(λ_1) where λ_1 has minimal polynomial X^2+-2λ^2+16λ-16/λ^2X+1 over K.The 𝐅_N-derivation d/dλ of K extends in an unique way to a 𝐅_N-derivation ∂/∂λ of F. More precisely, we have in F:∂λ_1/∂λ = -∂/∂ Yφ_2(λ_1, λ)/∂/∂ Xφ_2(λ_1, λ) .This follows from the fact that φ_2 is irreducible of degree 2 as a polynomial in X over 𝐅_N(Y), and that N is prime to 2.Let λ_2 be the other root of X^2+-2λ^2+16λ-16/λ^2X+1 in F. We have:∂λ_1/∂λ·∂λ_2/∂λ = 4/λ· (λ-1)It is just a formal computation.By differentiating (<ref>) two times (λ is considered as a formal variable), we get:4·λ^N-1/λ· (λ-1)· H'(λ_1)· H'(λ_2) = H'(λ)^2+ G(λ, λ_1, λ_2)where G(λ, λ_1, λ_2) is a sum of two terms of the form H(λ), H(λ_1) or H(λ_2) times a polynomial in λ, λ_1, λ_2, ∂λ_1/∂λ et ∂λ_2/∂λ. If λ∈ L then we have H(λ)=H(λ_1)=H(λ_2)=0. By (<ref>), we get4·λ^N-1/λ· (λ-1)· H'(λ_1)· H'(λ_2) = H'(λ)^2.This concludes the proof of Theorem <ref>.§.§ The supersingular module of Legendre elliptic curvesKeep the notation of sections <ref> and <ref>. We denote by v the p-adic valuation of N^2-1 (thus, v=t if p≥ 5, v=t+1 if p=3 and v=t+3 if p=2).In view of the properties satisfied by the Hasse polynomial, the higher Eisenstein element e_1 is more easily determined after adding an auxiliary Γ(2)-structure. Let 𝕋̃' be the full 𝐙_p-Hecke algebra acting faithfully the space of modular forms of weight 2 and level Γ_0(N) ∩Γ(2). If n ≥ 1 is an integer, we denote by T_n' the nth Hecke operator in 𝕋̃'. The ring 𝕋̃ acts on the 𝐙_p-module M' := 𝐙_p[L]. If ℓ is a prime not dividing 2· N and λ∈ L, we have in M':T_ℓ'([λ]) = ∑_λ' ∈ L φ_ℓ(λ, λ')=0 [λ'] .Let ẽ_0' = ∑_λ∈ L [λ] ∈ M'.Let Λ : 𝐅_N^2^×→𝐙/p^v𝐙 be a surjective group homomorphism.Let e_0' be the image of ẽ_0' in M/p^v· M and lete_1' = ∑_λ∈ LΛ(H'(λ))· [λ] ∈ M'/p^v· M'.For all prime ℓ not dividing 2· N, we have in M'/p^v· M':(T_ℓ'-ℓ-1)(e_0')=0 and(T_ℓ'-ℓ-1)(e_1') = (ℓ-1)·Λ(ℓ) · e_0'.The first equality is obvious. The second equality is a direct consequence of Theorem <ref>. We let π : M' → M be the forgetful map. It is a 𝐙_p-equivariant group homomorphism defined by π([λ]) = [E_λ] ∈ S (recall that E_λ is the Legendre elliptic curve y^2 = x(x-1)(x-λ)). For all prime ℓ not dividing 2· N and all x ∈ M', we have in M:π(T_ℓ'(x)) = T_ℓ(π(x)).We have, in M:π(ẽ_0') = 6 ·ẽ_0.Theorem <ref>, (<ref>) and (<ref>) allow us to compute e_1. However, some complications arise when p ∈{2,3}, so we treat the cases p ≥ 5, p=3 and p=2 separately.§.§ The case p≥ 5In this section, we assume p≥ 5. Keep the notation of sections <ref>, <ref> and <ref>. Assume that p ≥ 5. We extend log to a surjective group morphism log : 𝐅_N^2^×→𝐙/p^r𝐙. * We have, in M/p^r· M modulo the subgroup generated by e_0:12· e_1 = ∑_E ∈ Slog(P(j(E))) · [E]where j(E) is the j-invariant of E (the fact that P(j(E)) ≠ 0 is included in the statement). * We have, in 𝐙/p^r𝐙:e_1∙ e_0= 1/12·∑_λ∈ Llog(H'(λ))= 1/3·∑_k=1^N-1/2 k ·log(k). * Assume e_1 ∙ e_0 = 0. We have, in 𝐙/p^r𝐙:72 · e_1 ∙ e_1 = 3 ·( ∑_λ∈ Llog(H'(λ))^2 ) -4·( ∑_λ∈ Llog(λ)^2 ). We prove (i). We choose Λ : 𝐅_N^2^×→𝐙/p^v𝐙 so that for all x ∈𝐅_N^2^×, we have Λ(x) ≡log(x)(modulo p^r). We abuse notation and still denote by π : M'/p^r· M' → M/p^r· M the forgetful map. Let e_1” be the image of e_1' in M'/p^r· M'. Let ℓ be a prime not dividing 2· N. By Theorem <ref>, (<ref>) and (<ref>) we have in M/p^r· M:(T_ℓ-ℓ-1)(π(e_1”)) = 6· (ℓ-1)·log(ℓ)· e_0.By Proposition <ref>, we have π(e_1”) = 12· e_1(modulo the subgroup generated by e_0 ).Let E ∈ S. The coefficient of π(e_1”) in [E] is by definition∑_λ∈ Lj(E_λ) = j(E)log(H'(λ)).By (<ref>), we have∑_λ∈ Lj(E_λ) = j(E)log(H'(λ)) = log(P(j(E))).This concludes the proof of Theorem <ref>.We prove (ii). We have:12· e_1 ∙ e_0= ∑_E ∈ Slog(P(j(E))=∑_λ∈ Llog(H'(λ))= log((H))= 4·∑_k=1^N-1/2 k ·log(k).The first equality follows from (i). The last equality follows from Theorem <ref>, using the fact that log((N-1/2)!) = 0 (since p>2 and (N-1/2)!^4 = 1 in 𝐅_N).We prove (iii). We identify an element of L with its λ-invariant. For E∈ S, let F_E ⊂ L be the fiber above E, the set of λ∈ L such that π([λ]) = [E]. There exists λ∈ L such thatF_E= {λ, 1/λ, 1-λ, λ-1/λ, λ/λ-1, 1/1-λ}(we do not count multiplicity). Let c_E ∈{1,2,3,6} be the cardinality of F_E. We have w_E = 6/c_E. We have, by definition of P, for any E ∈ S and λ∈ F_E:P(j(E))^w_E =H'(λ) · H'(1/λ) · H'(1-λ) · H'( λ-1/λ) · H'(λ/λ-1) · H'(1/1-λ).We have H(X) = (-1)^m· H(1-X) and that H(1/X) = X^-m· H(X) where m = N-1/2. Indeed, H is monic and the roots of H are permuted by the transformation λ↦ 1-λ and λ↦1/λ. By differentiating these two relationswith respect to X and using (<ref>), we get:P(j(E))^w_E =H'(λ)^6 ·λ^4 · (1-λ)^4 .Thus, we have, in 𝐙/p^r𝐙:12^2· e_1 ∙ e_1= ∑_E ∈ S w_E ·log(P(j(E)))^2 =1/6·(∑_λ∈ L6 ·log(H'(λ)) + 4 ·log(λ) + 4 ·log(1-λ) )^2= 6 ·( ∑_λ∈ Llog(H'(λ))^2 ) -8·( ∑_λ∈ Llog(λ)^2 )This concludes the proof of Theorem <ref> * Theorem <ref> (ii) will be proved independently using modular symbols in Section <ref>. * Theorem <ref> (ii) is the first instance of what we call a higher Eichler mass formula, by analogy with the classical Eichler mass formula (<ref>).* In Corollary <ref>, we will show (using modular symbols) that if e_1 ∙ e_0 = 0, then e_1 ∙ e_1 = 12·∑_k=1^N-1/2 k·log(k)^2. This is the second instance of a higher Eichler mass formula. We have note been able to prove this identity directly. See Conjecture <ref> (iii) for a generalization when e_1 ∙ e_0 ≠ 0.§.§ The case p=3In this section, we assume p=3. Keep the notation of sections <ref>, <ref> and <ref>. Extend and lift log to a surjective group homomorphism log: 𝐅_N^2^×→𝐙/3^r+1𝐙. We have, in (M/3^r+1· M)/(𝐙· (3· e_0) ):12 · e_1 ≡ 2·log(2) ·ẽ_0 +∑_E ∈ Slog(P(j(E))) · [E].It obviously suffices to prove Theorem <ref> when r=t, which we assume until the end of the proof. Note that v=t+1, so we can let Λ=log. We abuse notation and still denote by π : M'/3^t+1· M' → M/3^t+1· M the forgetful map. Let ℓ be a prime not dividing 2· N. By Theorem <ref>, (<ref>) and (<ref>) we have in M/3^t+1· M:(T_ℓ-ℓ-1)(π(e_1')) = 6· (ℓ-1)·log(ℓ)· e_0.By Proposition <ref>, we have π(e_1') = 12· e_1(modulo the subgroup generated by the image of ẽ_0in M/3^t+1· M ).Let E ∈ S. The coefficient of π(e_1') in [E] is by definition∑_λ∈ Lj(E_λ) = j(E)log(H'(λ)).By (<ref>), we have∑_λ∈ Lj(E_λ) = j(E)log(H'(λ)) = log(P(j(E))).Thus, there exists C_3 ∈𝐙/3^t+1𝐙, uniquely defined modulo 3, such that we have in M/3^t+1· M:12· e_1 = C_3 ·ẽ_0 +∑_E ∈ Slog(P(j(E))) · [E]. By pairing(<ref>) with ẽ_0, we get in 𝐙/3^t+1𝐙:12· e_1 ∙ e_0 =C_3 ·ẽ_0 ∙ẽ_0 + log(Disc(H)). By Corollary <ref>, Theorem <ref>, Eichler mass formula (<ref>), Theorem <ref> and Lemma <ref> (which are independent of the results of this section), we have in 𝐙/3^t+1𝐙:- 3 ·∑_k=1^N-1 k^2·log(k) = C_3·N-1/12-3 ·∑_k=1^N-1 k^2·log(k) -N-1/6·log(2).This concludes the proof of Theorem <ref>.One could use Theorem <ref> to compute e_1 ∙ e_1 if t≥ 2 and r ≤ t-1.§.§ The case p=2In this section, we assume p=2. Keep the notation of sections <ref>, <ref> and <ref>. Let Λ̃ : 𝐅_N^2^×→𝐙/2^t+3𝐙 be a surjective group homomorphism such that for all x ∈𝐅_N^×, we have Λ̃(x) ≡ 2·log(x)(modulo 2^r+1). Let ϵ_2 ∈{1,-1} be defined byΛ̃((N-1/2)!) ≡ 2^t+1·ϵ_2(modulo 2^t+3). Let Λ be the reduction of Λ̃ modulo 2^r+3. We have, in M/2^r+3· M modulo the subgroup generated by 8· e_0:24 · e_1 =2 · C_2 ·ẽ_0 + ∑_E ∈ SΛ(P(j(E))) · [E].where C_2 ≡2^t+2/N-1·ϵ_2(modulo 4).It obviously suffices to prove Theorem <ref> when r=t (so v=r+3), which we assume until the end of the proof.We abuse notation and still denote by π : M'/2^t+3· M' → M/2^t+3· M the forgetful map. Let ℓ be a prime not dividing 2· N. By Theorem <ref>, (<ref>) and (<ref>) we have in M/2^t+3· M:(T_ℓ-ℓ-1)(π(e_1')) = 12· (ℓ-1)·log(ℓ)· e_0 .By Proposition <ref>, there exists C_2' ∈𝐙/2^t+3𝐙 (uniquely defined modulo 8) such that we have, in M/2^t+3· M modulo the subgroup generated by the image of ẽ_0 in M/2^t+3· M:24· e_1 = C_2' ·ẽ_0 + ∑_E ∈ SΛ̃(P(j(E))) · [E].By pairing (<ref>) with ẽ_0, we get in 𝐙/2^r+3𝐙:24· e_1 ∙ e_0 = C_2' ·ẽ_0 ∙ẽ_0+Λ̃(Disc(H))(modulo 2^t+3). By Corollary <ref>, Theorem <ref>, Eichler mass formula (<ref>) and Theorem <ref> (which are independent of the results of this section), we have in 𝐙/2^t+3𝐙: -2^t+2 +8·(∑_k=1^N-1/2 k ·log(k) ) ≡ C_2'·N-1/12+8·(∑_k=1^N-1/2 k ·log(k) ) - Λ̃((N-1/2)!).Thus, we have in 𝐙/2^t+3𝐙:2^t·(4+ C_2'/3·N-1/2^t+2 - 2·ϵ_2) = 0.Thus, we have C_2' ≡ 6·2^t+2/N-1· (ϵ_2-2) ≡ 2 · C_2(modulo 8).This concludes the proof of Theorem <ref>. The following result is an elementary consequence of Theorem <ref>, for which we have not found an elementary proof.There exists λ∈ L such that H'(λ) is not a square of 𝐅_N^2^×. If E∈ S, let F_E be defined as in (<ref>). By (<ref>), for all λ∈ F_E we have: P(j(E))^w_E =H'(λ)^6 ·λ^4 · (1-λ)^4 .Thus exists E ∈ S such that P(j(E)) is not a fourth power in 𝐅_N^2^× if and only if there is λ∈ L such that H'(λ) is not a square of 𝐅_N^2^×. Such a E exists by Theorem <ref> since we have C_2 ∈ (𝐙/4𝐙)^×. The following conjecture was checked numerically for N<2000 (without assuming N ≡ 1(modulo 8) anymore). We do not know the significance of this empirical fact.Assume N ≡ 1(modulo 4). For all λ∈ L, H'(λ) is not a square of 𝐅_N^2^×. Conjecture <ref> does not hold if N ≡ 3(modulo 4), since in this case there exists λ∈𝐅_N ∩ L, so H'(λ) ∈𝐅_N^× is a square of 𝐅_N^2^× <cit.>.§.§ Conjectural identities satisfied by the supersingular lambda invariantsIn this section, we collect various (often conjectural) arithmetic properties satisfied by the elements of L. These identities are motivated by the theory of the Eisenstein ideal. Assume p ≥ 5. Extend log to a surjective group homomorphism log : 𝐅_N^2^×→𝐙/p^r𝐙.* We have ∑_λ∈ Llog(λ)^2 = -32 ·log(2) ·( ∑_k=1^N-1/2k·log(k) ).* There exists λ∈ L such that log(λ)≠ 0.* We have ∑_λ∈ Llog(H'(λ))^2 = 4·(∑_k=1^N-1/2 k ·log(k)^2) - 3 · 16·log(2) ·(∑_k=1^N-1/2 k ·log(k)). * Assume ∑_k=1^N-1/2 k ·log(k) = 0. For all λ∈ L, we have∑_λ' ∈ L \{λ}log(λ'-λ) ·log(H'(λ')) = log(H'(λ))^2.* Assume ∑_k=1^N-1/2 k ·log(k) = 0.For all λ∈ L, we have∑_λ' ∈ L \{λ}log(λ'-λ) ·log(λ') = log(λ)^2. * These conjectures have been numerically checked (using SAGE) for N < 1000.* Using (derivatives of) the relations H(1-X) = (-1)^m· H(X) and H(1/X) = X^-m· H(X) (where m = N-1/2), we easily prove that∑_λ∈ Llog(H'(λ))·log(λ) = ∑_λ∈ Llog(H'(λ))·log(1-λ) = -∑_λ∈ Llog(λ)^2 = -2 ·∑_λ∈ Llog(λ)·log(1-λ)which allows us to reformulate Conjecture <ref> (i).* The term log(2) in Conjecture <ref> (i) comes from a criterion of Ribet concerning the existence of congruences between cuspidal newforms of weight 2 and level Γ_0(2N), and Eisenstein series <cit.>.* It seems that Akshay Venkatesh found a proof of Conjecture <ref> (i) using the Hecke operator U_2. * We will show in Proposition <ref> that (ii) holds if log(2) ≢0(modulo p).* If ∑_k=1^N-1/2 k ·log(k)=0, Conjecture <ref> (iii) is proved in Corollary <ref>, using modular symbols. * Conjecture <ref> (iv) and (v) are suggested by the theory of the refined ℒ-invariant of de Shalit <cit.>, Oesterlé, Mazur–Tate <cit.> and Mazur–Tate–Teitelbaum <cit.>, and its generalization to level Γ(2) ∩Γ_0(N) studied in <cit.>. We end this section with (conjectural) relations analogous to the ones of Conjecture <ref> for p=3 and p=2.Assume p=3. Extend and lift log to a surjective group homomorphism log : 𝐅_N^2^×→𝐙/p^r+1𝐙.* There exists λ∈ L such that log(λ) ≢0(modulo 3) (λ is not a cube of 𝐅_N^2^×).* We have∑_λ∈ Llog(λ)^2 = 4 ·log(2) ·( ∑_k=1^N-1/2k·log(k) )(modulo 9) .Furthermore, both sides of the equality are congruent to 0 modulo 3.* For al λ∈ L, we have:∑_λ' ∈ L \{λ}log(λ'-λ) ·log(H'(λ')) = log(H'(λ))^2(modulo 3)* For al λ∈ L, we have:∑_λ' ∈ L \{λ}log(λ'-λ) ·log(λ') = log(λ)^2(modulo 3)* Conjecture <ref> (i) should be true even if N ≢1(modulo 9) (which is assumed since p=3).* Conjecture <ref> (ii), (iii) and (iv), although similar to <ref>, is not modulo 3^r+1 in general. The following result, in which N is an arbitrary odd prime, is kind of opposite to Conjecture <ref> (i).For all λ∈ L, λ(1-λ)/2 is a cube of 𝐅_N^2^×. Since j = 256· (λ^2-λ+1)^3/λ^2· (1-λ)^2, it suffices to show that supersingular j-invariants are cubes of 𝐅_N^2^×. This is well-known <cit.>. We conclude this section by stating a result which goes in the opposite direction of Conjecture <ref> (ii). <cit.>Every λ∈ L is a fourth power modulo N (we do not assume anything on N). If N^2 ≡ 1 (modulo8 ), then every λ∈ L is a eighth power modulo N.§.§ Eisenstein ideals of level Γ_0(N) ∩Γ(2)Assume p≥ 3. Keep the notation of sections <ref>, <ref> and <ref>. Extend log to a surjective group homomorphism log : 𝐅_N^2^×→𝐙/p^r𝐙.In view of the role played by the Hasse polynomial, we reformulate the problem of Eisenstein elements in the context of the congruence subgroup Γ_0(N) ∩Γ(2). The modular curve X(Γ_0(N) ∩Γ(2)) has 6 cusps. Thus, the space of Eisenstein series of weight 2 and level Γ_0(N) ∩Γ(2) has dimension 5. It admits a basis of eigenforms for the Hecke operators T_ℓ (ℓ prime ≠ 2,N), U_2 and U_N. These Eisenstein series are characterized by the pair (a_N, a_2) of their Fourier coefficients (at the cusp ∞) at N and 2, which belongs to {(1,1), (1, 2), (1, 0), (N, 1), (N, 0)}. Since p^t divides N-1, these coefficients are in {(1, 1), (1, 2), (1, 0)} modulo p^t. We define three Eisenstein ideals.For α∈{0,1,2}, let I_α be the ideal of the Hecke algebra acting on M_2(Γ(2) ∩Γ_0(N)) generated by the T_ℓ-ℓ-1 (ℓ prime number different from 2 and N), U_2-α and U_N-1. Recall that these Hecke operators generate a 𝐙_p-algebra 𝕋̃' acting on M' = 𝐙_p[L]. We have described the action of T_ℓ (given by the modular polynomial φ_ℓ) and U_N (given by [λ] ↦ [λ^N]). We are going do describe U_2 (this might be well-known, be we could not find a reference). For all λ∈ L, we have in M':U_2([λ]) = [λ_1]+ [λ_2]where λ_1 et λ_2 are the roots of the polynomial φ_2(λ, Y). This follows from Proposition <ref>. Define the following elements in M'/p^r· M', if p ≥ 5:* e_0^0 = ∑_λ∈ Llog(λ)· [λ]* e_0^1 = ∑_λ∈ L (log(1-λ) - 2 ·log(λ) - 4·log(2)) · [λ]* e_0^2 = ∑_λ∈ L [λ]If p=3, we define in the same way e_0^0 and e_0^2. To define e_0^1, note that by Proposition <ref>, for all λ∈ L there exists a_λ∈𝐅_N^2^× such that 1-λ/λ^2· 2^4 = a_λ^3. We then lete_0^1 = ∑_λ∈ L a_λ· [λ] ∈ M'/3^r· M'(this does not depend on the choice of a_λ since r ≤ t < v = v_3(N-1) = t+1). For all α∈{0,1,2}, e_0^α is annihilated by the ideal I_α. The property for T_ℓ-ℓ-1 for a prime ℓ≠ 2, N follows from Lemma <ref>. The fact that these elements are killed by U_N-1 is obvious. The property for U_2 is a formal computation using Proposition <ref>.The elements e_0^1 and e_0^0 are non-zero modulo p. * The fact that e_0^0 ≠ 0 is equivalent to Conjecture <ref> (ii) if p≥ 5 and to Conjecture <ref> (i) if p=3. * By <cit.>, the 𝐙/p^r𝐙-module (M'/p^r· M')[I_α] is free of rank one. Thus,if Conjecture <ref> is true, we have (M'/p^r· M')[I_α] = 𝐙/p^r𝐙· e_0^α. Assume that 2 is not a pth power modulo N, log(2) ≢0(modulo p). Then Conjecture <ref> holds. Assume that for all λ∈ L, log(λ) ≡ 0(modulo p). We use the same notation as in Proposition <ref>. We havelog(λ_1·λ_2) ≡0 ≡log(-16·λ/(1-λ)^2) ≡log(-16)(modulo p) .This is a contradiction since p ≥ 3 and log(2) ≠ 0. Thus, we have e_0^0 is non-zero modulo p.Assume that for all λ∈ L,log(1-λ)-2·log(λ) - 4 ·log(2) ≡ 0(modulo p ).We use the same notation as in Proposition <ref>. We then have 0≡log((1-λ_1)(1-λ_2)) ≡ 2·log(λ_1) + 2 ·log(λ_2) + 8 ·log(2) ≡ 2·log(-16 ·λ/(1-λ)^2) + 8 ·log(2)(modulo p).Thus, we have log(λ) - 2 ·log(1-λ) + 8 ·log(2) ≡ 0(modulo p), so -3 ·log(λ) ≡ 0(modulo p). If p>3, this is a contradiction by the previous case. If p=3, we just work modulo 9 instead of working modulo 3. We now want to determine (M'/p^r· M')[I_α^2]. The only result we got in this direction is the following.Lete_1^2:=e_0^0+1/2· e_0^1+1/2·∑_λ∈ Llog(H'(λ))· [λ].* We have, for all prime number ℓ not dividing 2· N:(T_ℓ-ℓ-1)(e_1^2) = ℓ-1/2·log(ℓ)· e_0^2. * We have:(U_2-2)(e_1^2) = log(2)· e_0^0 * We have(U_N-1)(e_1^2)=0. In particular, we have e_1^2 ∈ (M'/p^r· M')[I_α^2]. Point (i) follows from Proposition <ref> and Theorem <ref>. Point (ii) follows from Proposition <ref>. Point (iii) is obvious. Although we will not need it, results of Yoo show that (M'/p^r· M')[I_2^2] = 𝐙/p^r𝐙· e_0^2 ⊕𝐙/p^r𝐙· e_1^2. We have note been able to conjecture any explicit formula for the higher Eisenstein elements corresponding to I_0 and I_1. We have no conjecture for the analogous element e_2^2 annihilated by I_2^3 (when it exists, when n(r,p) ≥ 2). The degree of e_2^2 is e_1 ∙ e_1, which is1/24·( ∑_λ∈ Llog(H'(λ))^2 ) - 1/18·( ∑_λ∈ Llog(λ)^2 ) if p≥ 5 by Theorem <ref> (iii).One idea would be to express e_2^2 in terms of log(H'(λ))^2, log(λ)^2, log(1-λ)^2 or more general quadratic formula in the logarithms of differences of supersingular invariants. More precisely, define the following elements of M'/p^r· M': *α_1 = ∑_λ∈ L(∑_λ' ∈ L λ' ≠λlog(λ'-λ)^2 )· [λ] *α_2 = ∑_λ∈ L(∑_λ', λ”∈ L λ' ≠λ, λ'≠λ”, λ≠λ”log(λ'-λ)·log(λ”-λ) )· [λ] *α_3 = ∑_λ∈ L(∑_λ', λ”∈ L λ' ≠λ, λ”≠λ', λ≠λ”log(λ'-λ)·log(λ”-λ') )· [λ] *α_4 = ∑_λ∈ Llog(λ)^2· [λ] *α_5 = ∑_λ∈ Llog(1-λ)^2· [λ] *α_6 = ∑_λ∈ Llog(λ)·log(1-λ)· [λ] Then we have checked numerically for N=181 and p=5 that e_2^2 modulo p is not a linear combination of the elements α_i. § ODD MODULAR SYMBOLS: EXTENSION OF THE THEORY OF SHARIFI We keep the notation of chapters <ref> and <ref>. In all this chapter, unless explicitly stated, we assume p ≥ 5. Let r be an integer such that 1 ≤ r ≤ t. §.§ Odd and even modular symbolsIn this section, we do not assume p≥ 5. Thus, we only assume that p ≥ 2 is a prime such that p^r divides the numerator of N-1/12. Let M^- = H_1(Y_0(N), 𝐙_p)^- (resp. H^- = H_1(X_0(N), 𝐙_p)^-) be the largest torsion-free quotient of H_1(Y_0(N), 𝐙_p) (resp. H_1(X_0(N), 𝐙_p)) anti-invariant by the complex conjugation. Let M_+=H_1(X_0(N), , 𝐙_p)_+ (resp. H_+ = H_1(X_0(N), 𝐙_p)_+) be the subspace of H_1(X_0(N), , 𝐙_p) (resp. H_1(X_0(N), 𝐙_p)) fixed by the complex conjugation. With the notation of Theorem <ref>, we easily see that H_+ = (M_+)^0.The intersection product induces perfect 𝕋̃-equivariant pairings:∙ :M_+ × M^- →𝐙_pand∙ :H_+ × H^- →𝐙_p. In order to apply the theory of higher Eisenstein elements as in chapter <ref>, we need to check the hypotheses of Theorem <ref> as a variant of Mazur's well-known result <cit.>. The 𝐓̃-modules M_+ ⊗_𝕋̃𝐓̃ and M^-⊗_𝕋̃𝐓̃ are free of rank 1. Since M_+ and M^- are dual 𝕋̃-modules, it suffices to prove that M_+ ⊗_𝕋̃𝐓̃ is free of rank one over 𝐓̃ by Lemma <ref> (iii).By <cit.>, H_+ ⊗_𝕋𝐓 is free of rank one over 𝐓. We now prove that this implies that M_+ ⊗_𝕋̃𝐓̃ is free of rank one over 𝐓̃. Consider the exact sequence of 𝐙_p-modules0 → H_+ → M_+ 𝐙_p → 0,where the map π sends {0,∞} to 1. It is a 𝕋̃-equivariant exact sequence if we identify 𝐙_p with 𝕋̃/Ĩ with its obvious 𝕋̃-module structure.By Lemma <ref> (i), 𝐓̃ is flat over 𝕋̃. Thus, (<ref>) gives an exact sequence of 𝐓̃-modules0 → H_+⊗_𝕋𝐓→ M_+⊗_𝕋̃𝐓̃𝐓̃/Ĩ·𝐓̃→ 0. We claim that the map e :𝐓̃→ M_+ ⊗_𝕋̃𝐓̃ given by T ↦{0,∞}⊗ T is an isomorphism of 𝐓̃-modules. This follows (<ref>), the fact that π'({0, ∞}⊗ T ) is the image of T in 𝐓̃/Ĩ·𝐓̃ and the fact that the restriction of e to Ĩ·𝐓̃ gives an isomorphism of 𝐓-modules Ĩ·𝐓̃→ H_+ ⊗_𝕋̃𝐓̃. The latter fact comes from <cit.> and the fact that the map Ĩ→ I is an isomorphism of 𝕋̃-modules. Thus, Theorem<ref> holds for M^- and M_+. Recall (Section <ref>) that we have denoted by m_0^-, m_1^-, ..., m_n(r,p)^- the higher Eisenstein elements in M^-/p^r· M^-. Recall also that we have fixed an element m̃_0^- ∈ M^-[Ĩ] reducing to m_0^- modulo p^r, such that {0, ∞}∙m̃_0^- = -1. The kernel of the map M^- → H^- is 𝐙_p·m̃_0^-. If i ∈{1, ..., n(r,p)}, we denote by m_i^- the image of m_i^- in H^-. Since m_1^- is uniquely defined modulo the subgroup generated by m_0^-, the element m_1^- is uniquely defined and is annihilated by the Eisenstein ideal I. If i>1, the element m_i^- is uniquely defined modulo the subgroup generated by m_1^-, ..., m_i-1^-.By intersection duality, the elements m_i^- can be considered as elements of _𝐙(H_+, 𝐙/p^r𝐙). If k ∈{1, ..., n(r,p)} then m_k^- modulo p^r can also be considered as a group homomorphismI^k-1· (H_+/p^r · H_+)/I^k · (H_+/p^r · H_+) →𝐙/p^r𝐙 . Indeed, we have an exact sequence of abelian groups:0 →((H_+/p^r · H_+)/I^k · (H_+/p^r · H_+)/ , 𝐙/p^r𝐙)→((H_+/p^r · H_+)/I^k+1· (H_+/p^r · H_+)/, 𝐙/p^r𝐙)→(I^k· (H_+/p^r · H_+)/I^k+1· (H_+/p^r · H_+), 𝐙/p^r𝐙) → 0. Equivalently, for all k ∈{0,1, ..., n(r,p)} the element m_k^- is uniquely determined by its pairing with Ĩ^k· (M_+/p^r· M^+). The element m_1^- was essentially determined by Mazur <cit.> (although it was not formulated in this way). Recall the Manin surjective mapξ_Γ_0(N) : 𝐙[𝐏^1(𝐙/N𝐙)] → H_1(X_0(N), , 𝐙_p)whose definition is recalled in Section <ref>. The 𝐙_p-module H_1(X_0(N), 𝐙_p) is generated by the element ξ_Γ_0(N)([x:1]) for x ∈ (𝐙/N𝐙)^× <cit.>. The group H_1(X_0(N), 𝐙_p) is acyclic for the complex conjugation <cit.>.In particular, we have H_+ = (1+c)· H_1(X_0(N), 𝐙_p), where c is the complex conjugation. Thus, the element m_1^- is uniquely determined by its pairing with (1+c)·ξ_Γ_0(N)([x:1]) for x ∈ (𝐙/N𝐙)^×. This was essentially computed by Mazur<cit.> (although Mazur does not take into account the complex conjugation). For all x ∈ (𝐙/N𝐙)^×, we have in 𝐙/p^r𝐙:((1+c)·ξ_Γ_0(N)([x:1]) ) ∙ m_1^-= log(x).We easily check that there is a unique group homomorphism f : H_+ →𝐙/p^r𝐙 such that for all x ∈( 𝐙/N𝐙)^×, we have in 𝐙/p^r𝐙: f((1+c)·ξ_Γ_0(N)([x:1]) ) = log(x).We easily see (<cit.>) that the map f annihilates I · H_+.Let ℓ be a prime not dividing 2· N. A simple computation shows that we have, in H_+:(T_ℓ-ℓ-1)({0, ∞}) = -(1+c)·∑_i=1^ℓ-1/2{0, i/ℓ} .Thus, we have in 𝐙/p^r𝐙:f( (T_ℓ-ℓ-1)({0, ∞}) ) = ℓ-1/2·log(ℓ) ·({0, ∞}∙ m_0^-) = ((T_ℓ-ℓ-1)({0, ∞})) ∙ m_1^- .The elements (T_ℓ-ℓ-1)({0, ∞}) generate H_+/I· H_+ when ℓ goes through the primes not dividing 2· N <cit.>. Thus, for all x ∈ H_+ we have in 𝐙/p^r𝐙:f(x) = x ∙ m_1^-.This concludes the proof of Theorem <ref>. Let f : H_+ → A where A is an abelian group. By intersection duality, f corresponds to an element f̂∈ H_1(X_0(N), A)^-. Merel gave a formula for f̂in terms of Manin symbols (Lemma <ref> for the case where 3 is invertible in A). This allows us to compute m_1^- in terms of Manin symbols. If p≠ 3 the formula is given in Lemma <ref> (i).The aim of this chapter is to determine the element m_2^- modulo p^r when n(r,p) ≥ 2 and p≥ 5. We have seen that m_2^- can be considered as a group isomorphismI· (H_+/p^r· H_+) / I^2· (H_+/p^r· H_+) →𝐙/p^r𝐙 .We will construct such a map, using the modular curve X_1(N) and its Eisenstein ideals.§.§ Refined Hida theoryIn this section, we will use the following notation.* σ = [0 -1;10 ] and τ = [0 -1;1 -1 ] ∈_2(𝐙). * If Γ is a subgroup of Γ_0(N) containing Γ_1(N), let X_Γ be the compact modular curve associated to Γ.* C_Γ^0 (resp. C_Γ^∞) is the set of cusps of X_Γ above the cusp Γ_0(N) · 0 (resp. Γ_0(N) ·∞) of X_0(N).* C_Γ = C_Γ^0 ∪ C_Γ^∞ * H̃_Γ' = H_1(X_Γ, C_Γ, 𝐙_p), H̃_Γ = H_1(X_Γ, C_Γ^0, 𝐙_p) and H_Γ = H_1(X_Γ, 𝐙_p).* ∂ : H̃_Γ'→𝐙_p[C_Γ]^0 is the boundary map, sending the geodesic path {α, β} to [β]-[α] where α, β ∈𝐏^1(𝐐). * (H̃_Γ)_+ is the subgroup of elements of H̃_Γ fixed by the complex conjugation. A similar notation applies to H_Γ. * D_Γ⊂ (𝐙/N𝐙)^× is the subgroup generated by the classes of the lower right corners of the elements of Γ and by the class of -1.* Λ_Γ = 𝐙_p[(𝐙/N𝐙)^×/D_Γ]. * If Γ_1 and Γ_2 are subgroups of _2(𝐙) such that Γ_1(N) ⊂Γ_1 ⊂Γ_2 ⊂Γ_0(N), we let J_1→ 2 = (Λ_Γ_1→Λ_Γ_2). It is a principal ideal of Λ_Γ_1, generated by [x]-1 where x is a generator of ((𝐙/N𝐙)^×/D_Γ_1→ (𝐙/N𝐙)^×/D_Γ_2). * 𝕋̃_Γ' (resp. 𝕋̃_Γ, resp. 𝕋_Γ) is the 𝐙_p-Hecke algebra acting faithfully on H̃_Γ' (resp. H̃_Γ, resp. H_Γ) generated by the Hecke operators T_n for n ≥ 1 and the diamond operators. The dth diamond operator is denoted by ⟨ d ⟩. By convention, it corresponds on modular form to the action of a matrix whose lower right corner is congruent to d modulo N. We will need some “refined Hida control” results, describing the kernel of the various maps in homology induced by the degeneracy maps between the various modular curves. Manin proved <cit.> that we have a surjectionξ_Γ : 𝐙_p[Γ\_2(𝐙)] → H_1(X_Γ, C_Γ, 𝐙_p)such that ξ_Γ(Γ· g) is the class of the geodesic path {g(0), g(∞)}, where X_Γ is the compact modular curve associated to Γ. Furthermore, he proved that the kernel of ξ_Γ is spanned by the sum of the (right) σ-invariants and τ-invariants.Recall that Γ_1(N) ⊂Γ⊂Γ_0(N). Consider the bijectionκ : Γ\_2(𝐙) ((𝐙/N𝐙)^2\{(0,0)}) / D_Γgiven by κ(Γ· g)= [c,d] where g = [ a b; c d ] and [c,d] is the class of (c,d) modulo D_Γ. By abuse of notation, we identify Γ\_2(𝐙) and ((𝐙/N𝐙)^2\{(0,0)}) / D_Γ.The map (𝐙/N𝐙)^×/ D_Γ→ C_Γ^0 (resp. (𝐙/N𝐙)^×/D_Γ→ C_Γ^∞) given by u ↦⟨ u ⟩· (Γ· 0) (resp. u ↦⟨ u ⟩· (Γ·∞)) (where ⟨·⟩ denotes the diamond operator) is a bijection. If u ∈ (𝐙/N𝐙)^×/D_Γ, we denote by [u]_Γ^0 (resp. [u]_Γ^∞) the image of u in C_Γ^0 (resp. C_Γ^∞). In other words, we have [u]_Γ^0 = Γ·c/d for some coprime integers c and d not divisible by N, and such that the image of d in (𝐙/N𝐙)^×/D_Γ is u. Similarly, [u]_Γ^∞ = Γ·a/N· b for some coprime integers a and b not divisible by N, and such that the image of a in (𝐙/N𝐙)^×/D_Γ is u^-1. Let [ a b; c d ]∈_2(𝐙). We describe ∂(ξ_Γ([c,d])) in the various cases that can happen.* If c ≡ 0(modulo N) then a ≡ d^-1 (modulo N). Thus, we have ∂(ξ_Γ([c,d])) = [d]_Γ^∞ - [d]_Γ^0. * If d ≡ 0(modulo N) then we have b ≡ -c^-1 (modulo N). Thus, we have ∂(ξ_Γ([c,d])) = [c]_Γ^0 - [c]_Γ^∞. * Ifc· d ≢0(modulo N) then we have ∂(ξ_Γ([c,d])) = [c]_Γ^0 - [d]_Γ^0. In particular, the set of [c,d] such that ∂(ξ_Γ([c,d])) ∈𝐙[C_Γ^0] coincides with the set of [c,d] such that c· d ≢0(modulo N ). Let M_Γ^0 be the sub-𝐙_p-module of 𝐙_p[((𝐙/N𝐙)^2\{(0,0)}) / D_Γ] generated by the symbols [c,d] with c· d ≢0(modulo N). The following statement is well-known, but we could not find a reference. The map ξ_Γ induces a surjective homomorphismξ_Γ^0 : M_Γ^0 →H̃_Γwhose kernel is R_Γ^0 = (M_Γ^0)^τ + (M_Γ^0)^σ + ∑_d ∈ (𝐙/N𝐙)^×𝐙_p · [-d,d] where (M_Γ^0)^τ (resp. (M_Γ^0)^σ) is the subgroup of elements of M_Γ^0 fixed by the right action of τ (resp. σ). Let ξ_Γ' = ξ_Γ∘κ^-1 :𝐙_p[((𝐙/N𝐙)^2\{(0,0)}) / D_Γ] →H̃_Γ' and ξ_Γ^0 be the restriction of ξ_Γ' to M_Γ^0. The computation of ∂ shows that ξ_Γ^0 takes values in H̃_Γ.Let y ∈H̃_Γ. Since ξ_Γ' is surjective, there is some element x = ∑_[c,d] ∈((𝐙/N𝐙)^2\{(0,0)}) / D_Γλ_[c,d]· [c,d] ∈𝐙_p[((𝐙/N𝐙)^2\{(0,0)}) / D_Γ] such that ξ_Γ'(x)=y. Since ∂ξ_Γ'(x) ∈𝐙_p[C_Γ^0], we have λ_[d,0] = λ_[0,d] for all d ∈((𝐙/N𝐙)^2\{(0,0)}) / D_Γ. Since ξ_Γ'([0,d]+[d,0]) = 0, the element y is in the image of ξ_Γ^0. Thus, we have proved that ξ_Γ^0 is surjective.Let x = ∑_[c,d] ∈((𝐙/N𝐙)^2\{(0,0)}) / D_Γλ_[c,d]· [c,d] - μ_[c,d]· [c,d] ∈(ξ_Γ^0) = (ξ_Γ') ∩ M_Γ^0 with λ_[c,d]=λ_[c,d]·τ and μ_[c,d]=μ_[c,d]·σ for all [c,d] ∈((𝐙/N𝐙)^2\{(0,0)}) / D_Γ.We also have λ_[d,0] = μ_[d,0] and λ_[0,d] = μ_[0,d] for all d ∈ (𝐙/N𝐙)^×/D_Γ. Note that for all d ∈ (𝐙/N𝐙)^×/D_Γ, we have:[d,-d]=([d,0]+[0,d]+[d,-d])-([d,0]+[0,d]) ∈(ξ_Γ^0).Hence, x - ∑_d ∈ (𝐙/N𝐙)^×/D_Γλ_[d,0]· [d,-d] ∈ (M_Γ^0)^σ + (M_Γ^0)^τ and x has the form sought after. The map H̃_Γ_1→H̃_Γ_2 is surjective. The map M_Γ_1^0 → M_Γ_2^0 is surjective. We conclude using Proposition <ref>. The ring Λ_Γ_i acts naturally on R_Γ_i^0, M_Γ_i and H̃_Γ_i (for i=1,2).* The kernel of the homomorphism H̃_Γ_1→H̃_Γ_2 is J_1 → 2·H̃_Γ_1. * The kernel of the homomorphism H_Γ_1→ H_Γ_2 is J_1 → 2· H_Γ_1. We prove point (i). Consider the following commutative diagram, where the rows are exact:0 [r]R_Γ_1^0 [r] [d]M_Γ_1^0 [r] [d] H̃_Γ_1[r] [d]00 [r] R_Γ_2^0 [r]M_Γ_2^0[r]H̃_Γ_2[r] 0 It is clear that the kernel of the middle vertical arrow is J_1 → 2· M_Γ_1^0. The cokernel of the left vertical map is zero by Proposition <ref> (using p>3). The snake lemma concludes the proof of point (i).We now prove point (ii). Using point (i), it suffices to show that H_Γ_1∩(J_1→ 2·H̃_Γ_1) = J_1→ 2· H_Γ_1. Consider the following commutative diagram, where the rows are exact:0 [r] H_Γ_1[r] [d] H̃_Γ_1[r] [d] 𝐙_p[C_Γ_1^0] ^0[r] [d]00 [r]H_Γ_1[r] H̃_Γ_1[r]𝐙_p[C_Γ_1^0]^0 [r] 0Here, the vertical maps are induced by the action of [d]-1 where d is a fixed generator of ((𝐙/N𝐙)^×/D_Γ_1→ (𝐙/N𝐙)^×/D_Γ_2), and 𝐙_p[C_Γ_1^0] ^0 is the augmentation subgroup of 𝐙_p[C_Γ_1^0]. Note that J_1 → 2 is principal, generated by [d]-1. Thus, to prove (ii) it suffices to show (using the snake Lemma) that the map H̃_Γ_1[J_1 → 2] →𝐙_p[C_Γ_1^0] ^0[J_1 → 2] is surjective. It suffices to show that the boundary map M_Γ_1^0[J_1 → 2] →𝐙_p[C_Γ_1^0] ^0[J_1 → 2] is surjective. Since we can identify C_Γ_1^0 with (𝐙/N𝐙)^×/D_Γ_1, the action of (𝐙/N𝐙)^×/D_Γ_1 on C_Γ_1^0 is free. Thus, any element of 𝐙_p[C_Γ_1^0] ^0[J_1 → 2] is of the form ∑_x ∈ C_Γ_1^0λ_x · (∑_k=0^m-1 [d^k-1])· [x] where m is the order of d. We have ∑_x ∈ C_Γ_1^0λ_x = 0. Thus, 𝐙_p[C_Γ_1^0] ^0[J_1 → 2] is spanned over 𝐙_p by the elements (∑_k=0^m-1 [d^k-1])· ([u]-[v]) for u, v ∈ C_Γ_1^0. If we identify u and v with elements of (𝐙/N𝐙)^×/D_Γ_1 and lift them to elements of (𝐙/N𝐙)^×, (∑_k=0^m-1 [d^k-1])· ([u]-[v]) is the boundary of the Manin symbol (∑_k=0^m-1 [d^k-1])· [u,v], which is annihilated by J_1 → 2. This concludes the proof of point (ii).We have the analogous (certainly well-known) statement for the Hecke algebras. The kernel of the restriction map 𝕋̃_Γ_1→𝕋̃_Γ_2 (resp. 𝕋_Γ_1→𝕋_Γ_2) is J_1 → 2·𝕋̃_Γ_1 (resp. J_1 → 2·𝕋_Γ_1). We prove the first assertion. There is an isomorphism of 𝕋̃_Γ_i-modules_𝐙_p(𝕋̃_Γ_i, 𝐙_p) ≃ M_2(Γ_i, 𝐙_p),where M_2(Γ_i, 𝐙_p) = M_2(Γ_i, 𝐙) ⊗_𝐙𝐙_p and M_2(Γ_i, 𝐙) is the set of modular form for Γ_i whose q-expansion at the cusp ∞ has coefficients in 𝐙. We haveM_2(Γ_2, 𝐙_p)= M_2(Γ_1, 𝐙_p)[J_1 → 2],where the action of Λ_Γ_1 on M_2(Γ_1, 𝐙_p) is via the diamond operators.This proves the first assertion using (<ref>). The proof of the second assertion is identical, replacing M_2(Γ_i, 𝐙_p) by the space of cusp formsS_2(Γ_i, 𝐙_p). §.§ Eisenstein ideals of X_1^(p^r)(N)We keep the notation of section <ref> and add the following ones. * P_r=(𝐙/N𝐙)^×/( (𝐙/N𝐙)^×)^p^r. * P_r' =( (𝐙/N𝐙)^×)^p^r. * Λ_r = 𝐙_p[P_r]. * J_r ⊂Λ_r is the augmentation ideal. * J_r' = (𝐙_p[(𝐙/N𝐙)^×] →Λ_r ). * Γ_1^(p^r)(N) ⊂Γ_0(N) is the subgroup of Γ_0(N) corresponding to the matrices whose diagonal entries are in P_r' modulo N.* If Γ = Γ_1^(p^r)(N), we let X_1^(p^r)(N)=X_Γ, H̃^(p^r) = H̃_Γ, (H̃^(p^r))_+ = (H̃_Γ)_+, H^(p^r) = H_Γ, (H^(p^r))_+ = (H_Γ)_+, 𝕋'^(p^r) = 𝕋̃_Γ', 𝕋̃^(p^r) = 𝕋̃_Γ, 𝕋^(p^r) = 𝕋_Γ, C_0^(p^r) = C_Γ^0 and C_∞^(p^r) = C_Γ^∞. * If Γ = Γ_0(N), we recall that H̃ = H̃_Γ, H = H_Γ, 𝕋̃ = 𝕋̃_Γ and 𝕋= 𝕋_Γ. * We define two Eisenstein ideals in 𝕋'^(p^r), 𝕋̃^(p^r) and 𝕋^(p^r). The set C_∞^(p^r) is annihilated by the Eisenstein ideal Ĩ_∞' of 𝕋'^(p^r) generated by the operators T_n - ∑_d | n, (d,N)=1⟨ d ⟩·n/d. Similarly, C_0^(p^r) is annihilated by the Eisenstein ideal Ĩ_0' of 𝕋'^(p^r) generated by the operators T_n - ∑_d | n,(d,N)=1⟨ d ⟩· d. * We denote by Ĩ_∞ and Ĩ_0 (resp. I_∞ and I_0) the respective images of Ĩ_∞' and Ĩ_0' in 𝕋̃^(p^r) (resp. 𝕋^(p^r)). The main goal of this section is to give an explicit description of H̃^(p^r)/Ĩ_∞·H̃^(p^r). The Hecke algebra 𝕋'^(p^r) (resp. 𝕋̃^(p^r), 𝕋^(p^r)) acts faithfully on the space of modular forms of weight 2 and level Γ_1^(p^r)(N) (resp. which vanish at the cusps in C_0^(p^r), resp. which are cuspidal). Letζ^(p^r) = ∑_x ∈ (𝐙/N𝐙)^×_2(x/N)· [x^-1] ∈Λ_randδ^(p^r) =∑_x ∈ P_r[x] ∈Λ_r. The following lemma will be useful in our proofs. It is an immediate consequence of Nakayama's lemma, since Λ_r is a local ring. Let f : M_1 → M_2 be a morphism of finitely generated Λ_r-modules. Let f :M_1 → M_2/J_r· M_2 be the map obtained from f. Then f is surjective if and only if f is surjective. The following result is analogous to Mazur's computation of 𝕋/I <cit.>. In fact, our proof uses Mazur's results and techniques. Assume that p ≥ 5. * The map Λ_r→𝕋̃^(p^r) given by [d] ↦⟨ d ⟩ gives an isomorphism of Λ_r-modulesΛ_r/(ζ^(p^r)) 𝕋̃^(p^r)/Ĩ_∞ .* The map Λ_r→𝕋^(p^r) given by [d] ↦⟨ d ⟩ gives an isomorphism of Λ_r-modulesΛ_r/(ζ^(p^r), p^t-r·δ^(p^r)) 𝕋^(p^r)/I_∞ .* The groups 𝕋̃^(p^r)/Ĩ_∞ and 𝕋^(p^r)/I_∞ are finite.The assertion (iii) follows from (i), (ii) and the well-known property of Stickelberger elements (non vanishing of L(χ,2) for any even Dirichlet character χ).We first prove the following result.Let I be an ideal of Λ_r and E_∞ :=∑_n ≥ 1(∑_d | n (d,N)=1[d] ·n/d)· q^n ∈Λ_r[[q]].* The series N-1/24· p^r·δ^(p^r) + E_∞ is the q-expansion at the cusp ∞ of a modular form of weight 2 and level Γ_1^(p^r)(N) over Λ_r.* The following assertions are equivalent.* There exists α∈Λ_r/I such that the image F_α of α + E_∞ in (Λ_r/I)[[q]] satisfies: * F_α is the q-expansion at the cusp ∞ of a modular form of weight 2, level Γ_1^(p^r)(N) over Λ_r/I which vanishes at the cusps in C_0^(p^r).* For all d ∈ P, we have ⟨ d ⟩ F_α = [d] · F_α. * We have ζ^(p^r)∈ I. * The following assertions are equivalent.* There exists α∈Λ_r/I such that the image F_α of α + E_∞ in (Λ_r/I)[[q]] satisfies: * F_α is the q-expansion at the cusp ∞ of a cuspidal modular form of weight 2, level Γ_1^(p^r)(N) over Λ_r/I.* For all d ∈ P, we have ⟨ d ⟩ F_α = [d] · F_α. * We have ζ^(p^r)∈ I and p^t-r·δ^(p^r)∈ I.Fix an embedding of 𝐐_p ↪𝐂. For any non-trivial character ϵ : P_r →𝐂^×, the q-expansion∑_n ≥ 1(∑_d | n, (d,N)=1ϵ(d) ·n/d)· q^n ∈𝐂[[q]]is the q-expansion at the cusp ∞ of an Eisenstein series of weight 2 and level Γ_1^(p^r)(N), which we denote by E_ϵ,1 (for instance <cit.>). Furthermore, we have already seen that N-1/24 + ∑_n ≥ 1(∑_d | n, (d,N)=1n/d)· q^n is the q-expansion at the cusp ∞ of an Eisenstein series of level Γ_0(N), denoted by E_2. Using our fixed embedding 𝐐_p ↪𝐂, we get a natural injective ring homomorphism ι : Λ_r→∏_ϵ∈P̂_r𝐂 where P̂_r is the set of characters of P_r.Thus, we have shown that F_∞:=N-1/24 · p^r·δ^(p^r) + E_∞∈Λ_r[[q]] is the q-expansion at the cusp ∞ of a modular form of weight 2 and level Γ_1^(p^r)(N) (still denoted by F_∞) over ∏_ϵ∈P̂_r𝐂. By the q-expansion principle <cit.>, such a modular form is over Λ_r. This proves point (i) of Lemma <ref>. We now prove point (ii). Since for all d ∈ P_r we have ⟨ d ⟩ E_ϵ,1 = ϵ(d) · E_ϵ, 1, the q-expansion principle shows that⟨ d ⟩F_∞ = [d]· F_∞ .Let α∈Λ_r such that point (a) of (ii) holds and let F_∞ be the image of F_∞ modulo I. The element F:= F_α - F_∞ = α - N-1/24 · p^r·δ^(p^r)∈Λ_r/I ⊂ (Λ_r/I)[[q]] is the q-expansion at the cusp ∞ of a modular form of weight 2 and level Γ_1^(p^r)(N) over Λ_r/I. Furthermore, for all d∈ P_r, we have by the assumptions on F_α and (<ref>): ⟨ d ⟩ F = ⟨ d ⟩ F_α - ⟨ d ⟩F_∞ = [d]· F_α - [d] ·F_∞ = [d]· F.Let I be an ideal of Λ_r and F ∈Λ_r/I. Assume that F is the q-expansion of a modular form of weight 2 and level Γ_1^(p^r)(N) over Λ_r/I such that for all d ∈ P_r, we have ⟨ d ⟩ F = [d] · F. Then we have F=0. We prove by induction on n≥ 0 that F ∈ J_r^n · (Λ_r/I). This is true if n=0. Assume that this is true for some n ≥ 0. By the q-expansion principle, F is the q-expansion at the cusp ∞ of a modular form over the 𝐙[1/N]-module J_r^n · (Λ_r/I) (<cit.> for the notion of a modular form over an abelian group). Let F be the image F by the map J_r^n · (Λ_r/I) ↠ J_r^n · (Λ_r/I)/J_r^n+1· (Λ_r/I). The diamond operators act trivially on F. Thus, F is the q-expansion at the cusp ∞ of a modular form of weight 2 and level Γ_0(N) with coefficients in the module J_r^n · (Λ_r/I)/J_r^n+1· (Λ_r/I). Note that J_r^n · (Λ_r/I)/J_r^n+1· (Λ_r/I) is a quotient of J_r^n/J_r^n+1≃𝐙/p^r𝐙. Since p ≥ 5 and (N,p)=1, <cit.> shows that F=0, we have F ∈ J_r^n+1· (Λ_r/I). This concludes the induction step. Since ⋂_n ≥ 0 J_r^n · (Λ_r/I) = 0, we have F=0. This concludes the proof of Lemma <ref>. By Lemma <ref>, we have F=0, F_α= F_∞. Let w_N be the Atkin–Lehner involution.By <cit.>, the q-expansion at the cusp ∞ of w_N(E_ϵ, 1) is1/N·(∑_x ∈ (𝐙/N𝐙)^×ϵ(x) · e^2 i π x /N) ·(-N/4·( ∑_x ∈ (𝐙/N𝐙)^×ϵ(x)^-1·_2(x/N) )+∑_n ≥ 1∑_d | n (d,N)=1ϵ(n/d)^-1·n/d· q^n ) ∈Λ_r[[q]].Furthermore, we have w_N(E_2) = -E_2.Let μ∈𝐙_p be the primitive Nth root of unity corresponding to e^2 i π/N under our fixed embedding 𝐐_p ↪𝐂. Let Λ_r' = (𝐙_p[μ])[P_r] and 𝒢' = ∑_x ∈ (𝐙/N𝐙)^×μ^x · [x] ∈Λ_r'. The element 𝒢' is invertible in Λ_r' since its degree is -1, which is prime to p, and Λ_r' is a local ring whose maximal ideal is J'+(ϖ) where J' is the augmentation ideal of Λ_r' and ϖ is a uniformizer of 𝐙_p[μ].By reformulating Weisinger's formula, the q-expansion principle shows that the q-expansion at the cusp ∞ of F_0:=w_N(F_∞) is -1/4·𝒢' ·ζ^(p^r) + 𝒢' ·∑_n≥ 1(∑_d | n,gcd(d,N)=1[n/d]^-1·n/d)· q^n. Since the constant coefficients of the modular form associated to F_α = F_∞ at the cusps of C_0^(p^r) are zero, this proves that 𝒢'·ζ^(p^r)∈ I. Since 𝒢' is a unit of Λ_r, we have ζ^(p^r)∈ I.Conversely, if ζ^(p^r)∈ I then the image F_∞ of F_∞ in (Λ_r/I)[[q]] is the q-expansion at the cusp ∞ of a modular form of weight 2 and level Γ_1^(p^r)(N) over Λ_r/I whose constant coefficient at the cusp 0 is zero. Since the diamond operators act transitively on C_0^(p^r), equation (<ref>) shows that the constant terms of the modular form associated to F_∞ at any cusp of C_0^(p^r) is zero. This proves point (ii).The proof of point (iii) is similar to the proof of point (ii). This concludes the proof of Lemma <ref>. We now prove that Theorem <ref> follows from Lemma <ref>. The ring homomorphism Λ_r→𝕋̃^(p^r)/Ĩ_∞ is surjective since we have T_n - ∑_d | n,gcd(d,n)=1n/d·⟨ d ⟩∈Ĩ_∞ for all n ≥ 1. Let K be its kernel. There is some α∈Λ_r such that the image of α + E_∞ in Λ_r/K is the q-expansion at the cusp ∞ of a modular form of weight 2 and level Γ_1^(p^r)(N) over Λ_r/K satisfying the two properties of Lemma <ref> (ii) (a), and K is the smallest such ideal of Λ_r. By Lemma <ref> (i), we have K = (ζ^(p^r)). Point (ii) of Theorem <ref> follows in a similar way. In the following result, we extend the winding homomorphism of Mazur <cit.>. The Λ_r-module (H̃^(p^r))_+/Ĩ_∞·(H̃^(p^r))_+ is isomorphic to Λ_r/(ζ^(p^r)). We first define a homomorphism of Λ_r-modules ẽ: Ĩ_∞→(H̃^(p^r))_+ as follows.We have a map Ĩ_∞' →(H̃^(p^r))_+ given by η↦η·{0,∞}.This induces the desired map Ĩ_∞→(H̃^(p^r))_+.Indeed, if η∈Ĩ_∞' maps to 0 in Ĩ_∞ then η∈Ĩ_∞' ∩Ĩ_0', so in particular η annihilates all the Eisenstein series of M_2(Γ_1^(p^r)(N), 𝐂). There is an Eisenstein series E ∈ M_2(Γ_1^(p^r)(N), 𝐂) such that the divisor of the meromorphic differential form E(z)dz is (0)-(∞). This Eisenstein series induces (via integration) a morphism H_1(Y_1^(p^r)(N), 𝐙) →𝐂. By intersection duality, we get an element ℰ∈ H_1(X_1^(p^r)(N), C_0^(p^r)∪ C_∞^(p^r), 𝐂). Since E is annihilated by Ĩ_∞' ∩Ĩ_0', so is ℰ. Since ℰ - {0,∞}∈ H_1(X_1^(p^r)(N), 𝐂) and η acts trivially on H_1(X_1^(p^r)(N), 𝐂), we see that η·{0,∞}=0. Let e: I → H_+ be the winding homomorphism of Mazur, denoted by e_+ in <cit.>. We denote by a bold letter the various 𝕋̃^(p^r) or 𝕋-modules involved completed at Ĩ_∞ or at I. Thus, for example, (H̃^(p^r))_+ (resp. H_+) is the Ĩ_∞ (resp. I)-adic completion of (H̃^(p^r))_+ (resp. H_+). Let ẽ: Ĩ_∞→(H̃_1)_+ (resp. e : I→H_+) be the map obtained after completion at the ideal Ĩ_∞ (resp. I).The map (H̃^(p^r))_+ → H_+ defines by passing to completion a group isomorphism (H̃^(p^r))_+/J_r·(H̃^(p^r))_+ H_+. By point (ii) of Lemma <ref>, we have (H̃^(p^r))_+ = (H̃^(p^r))_+/(⋂_n ≥ 0Ĩ_∞^n) ·(H̃^(p^r))_+ and H_+ = H_+/(⋂_n≥ 0 I^n) · H_+. By Proposition <ref>, it suffices to show that the image of ⋂_n≥ 0Ĩ_∞^n in 𝕋 is ⋂_n ≥ 0 I^n. Since Λ_r is a local ring, by Theorem <ref> (i) there is a unique maximal ideal of 𝕋̃^(p^r) containing Ĩ_∞, which we denote by 𝔪̃.By Theorem <ref> (iii), the group 𝕋̃^(p^r)/Ĩ_∞ is finite, so there exists an integer n_0 ≥ 1 such that 𝔪̃^n_0⊂Ĩ_∞. Thus, we have ⋂_n ≥ 0Ĩ_∞^n = ⋂_n ≥ 0𝔪̃^n. Similarly, there is a unique maximal ideal 𝔪 of 𝕋 containing I, and we have ⋂_n ≥ 0 I^n = ⋂_n ≥ 0𝔪^n. Any maximal ideal 𝔪̃' of 𝕋̃^(p^r) contains the image of J_r by the map Λ_r →𝕋̃^(p^r). Indeed, we have an injective ring homomorphism Λ_r/ 𝔪' ↪𝕋̃^(p^r)/𝔪̃' where 𝔪' is the pre image of 𝔪̃' in Λ_r. Since 𝕋̃^(p^r)/𝔪̃' is a finite field, Λ_r ∩𝔪̃' is a maximal ideal of Λ_r and therefore contains J_r. If 𝔪' is a maximal ideal of 𝕋, there exists a unique maximal ideal 𝔪̃' of 𝕋̃^(p^r) projecting to 𝔪'. The existence is obvious and the unicity follows from the fact that if 𝔪̃_1 and 𝔪̃_2 are two such ideals, then 𝔪̃_1 = 𝔪̃_1 + J_r = 𝔪̃_2 + J_r = 𝔪̃_2 by the previous remark and Proposition <ref>. The rings 𝕋̃^(p^r) and 𝕋 are semi-local and p-adically complete, so we have 𝕋̃^(p^r) = ⊕_𝔪̃' ∈SpecMax(𝕋̃^(p^r)) (𝕋̃^(p^r))_𝔪̃' and 𝕋 = ⊕_𝔪' ∈SpecMax(𝕋) 𝕋_𝔪', where the subscript means the completion. By the previous discussion, the map (𝕋̃^(p^r))_𝔪̃'→𝕋_𝔪' is surjective if 𝔪̃' projects to 𝔪'. We have ⋂_n ≥ 0𝔪̃^n =⊕_𝔪̃' ∈SpecMax(𝕋̃^(p^r)) 𝔪̃' ≠𝔪̃ (𝕋̃^(p^r))_𝔪̃' and⋂_n ≥ 0𝔪^n =⊕_𝔪' ∈SpecMax(𝕋) 𝔪' ≠𝔪𝕋_𝔪̃'. Thus, the map ⋂_n ≥ 0𝔪̃^n →⋂_n ≥ 0𝔪^n is surjective. Thus, we get a commutative diagram:Ĩ_∞[r]^ẽ[d] (H̃^(p^r))_+[d] I[r]^e H_+ Since e is surjective (it is even an isomorphism by <cit.>), Lemmas <ref> and <ref> show that ẽ is surjective. By the Eichler–Shimura isomorphism (over 𝐂), the 𝐙_p-rank of these two modules must be equal to the 𝐙_p-rank of 𝐓̃_1. Thus, ẽ is an isomorphism. By passing to the quotient map, ẽ gives rise to an isomorphism of Λ_r-modules Ĩ_∞ / Ĩ_∞^2≃(H̃^(p^r))_+/Ĩ_∞. The Λ_r-module (H̃^(p^r))_+/Ĩ_∞, and so Ĩ_∞ / Ĩ_∞^2, is cyclic since it is cyclic modulo J_r by Proposition <ref>. By Nakayama's Lemma the Ĩ_∞ is principal. Since a generator of Ĩ_∞ is not a zero-divisor, we get:𝕋̃^(p^r)/Ĩ_∞≃Ĩ_∞/Ĩ_∞^2.This concludes the proof of Theorem <ref> by Theorem <ref>. The following result will be useful later.We have Ĩ_0 ·(H̃^(p^t))_+ = (H^(p^t))_+. The inclusion Ĩ_0 ·(H̃^(p^t))_+ ⊂(H^(p^t))_+ is obvious. We have (H̃^(p^t))_+/(H^(p^t))_+ = 𝐙_p[C_0^(p^t)]^0. The Λ_t-module 𝐙_p[C_0^(p^t)]^0 is isomorphic to J_t. Let M =(H̃^(p^t))_+/Ĩ_0 ·(H̃^(p^t))_+; this is a Λ_t-module. By Proposition <ref>, we have M/J_t· M = H_+/I· H_+. In particular, M is a cyclic Λ_t-module, which we write Λ_t/K where K ⊂Λ_t is an ideal.We have K ⊂δ^(p^t)·Λ_t. Indeed, we have a surjection of Λ_t-modules Λ_r/K ↠ J_t ≃Λ_t/δ^(p^t)·Λ_t. This induces a surjection Λ_t/K ↠Λ_t/((δ^(p^t))+J) ·Λ_t≃𝐙/p^t𝐙. Since Λ_t is a local ring with maximal ideal J_t + (p), the image of 1 ∈Λ_t in Λ_t/K is mapped to the image of a unit of Λ_t in Λ_t/δ^(p^t)·Λ_t. Thus we have K ⊂δ^(p^t)·Λ_t. We now prove the reverse inclusion. Since M/J_t· M ≃𝐙/p^t𝐙, there is an element β = δ^(p^t)·α in K such that α∈ 1 + J_t + (p). In particular, α is a unit and K = δ^(p^t)·Λ_r.Thus, we have M ≃ J_t and the surjective map (H̃^(p^t))_+/Ĩ_̃0̃·(H̃^(p^t))_+ →(H̃^(p^t))_+/(H^(p^t))_+ is an isomorphism of free 𝐙_p-modules.The natural map H̃→ H gives an isomorphism(H^(p^t))_+/(I_0 + J_t)·(H^(p^t))_+I· H_+/I^2· H_+ .This follows from Propositions <ref> and <ref>.Note that Lemma <ref> gives another proof of the fact that the image of (H^(p^t))_+ in H_+ is I· H_+, which was first proved by Mazur <cit.>. §.§ Algebraic number theoretic criterion for n(r,p) ≥ 2 We keep the notation of section <ref>, and add the following ones.* Fix an algebraic closure 𝐐 of 𝐐.* If n is a positive integer, let μ_n be the group of nth roots of unity in 𝐐.* Let ζ_p^r, ζ_N ∈𝐐 be primitive p^rth and Nth roots of unity respectively.* Let χ_p : (𝐐/𝐐) →𝐙_p^× be the pth cyclotomic character and ω_p : (𝐐(ζ_p)/𝐐)(𝐙/p𝐙)^× be the Teichmüller character.* If M is a 𝐙_p[(𝐐(ζ_p^∞)/𝐐)]-module, and i ∈𝐙,we denote by M_(i) the maximal quotient of M where (𝐐(ζ_p^∞)/𝐐) acts by χ_p^i. LetM^(i) = {m ∈ M, ∀ g ∈(𝐐(ζ_p^∞)/𝐐), g(m) = χ_p^i(g)· m} .Note that if the action of (𝐐(ζ_p^∞)/𝐐) on M factors through (𝐐(ζ_p^r)/𝐐) and if i ≠ 0, then M^(i) is necessarily a 𝐙/p^r𝐙-module.* We normalize class field theory by sending geometric Frobenius substitutions to uniformizers.* For simplicity, we denote the augmentation ideal J_r of Λ_r by J. Let K_r be the unique degree p^r-extension of 𝐐 contained in 𝐐(ζ_N), and L_r = K_r(ζ_p^r). Let 𝒪_r = 𝒪_K_r[1/Np] where 𝒪_K_r is the ring of integers of K_r and let 𝒜_r=𝒪_r[ζ_p^r]. Let S_r (resp. T_r) be the set of infinite places of K_r (resp. L_r) and finite places of K_r (resp. L_r) dividing Np. Let K_S be the maximal extension of K_r unramified outside S_r. Let Γ_r = (K_S/K_r) and Γ_r' = (K_S/L_r). The group Γ_r' is a normal subgroup of Γ_r, and we have canonical group isomorphismsΓ_r/Γ_r' ≃(L_r/K_r) ≃(𝐐(ζ_p^r)/𝐐) ≃ (𝐙/p^r𝐙)^×(the last one coming from the p^rth cyclotomic character). The Nth cyclotomic character gives a group isomorphism (K_r/𝐐) ≃ (𝐙/N𝐙)^×/((𝐙/N𝐙)^×)^p^r. Thus, we have a canonical ring isomorphismΛ_r 𝐙_p[(K_r/𝐐)].We let 𝒦_r = K_2(𝒪_r)/p^r · K_2(𝒪_r); it is equipped with a structure of Λ_r-module since (K_r/𝐐) acts on K_2(𝒪_r). The aim of this section is to understand (partially) the filtration of 𝒦_r given by the powers of J. Recall that 1/p∈𝒪_r. The Chern character induces an isomorphism of Λ_r-modulesK_2(𝒪_r)/p^r · K_2(𝒪_r)≃ H^2_ét(𝒪_r, μ_p^r^⊗ 2) <cit.>. We have a canonical isomorphismH^2_ét(𝒪_r, μ_p^r^⊗ 2) ≃ H^2(Γ_r, μ_p^r^⊗ 2) .Since the real places and the places dividing p of K_r are in S_r, the p-cohomological dimension of Γ_r is ≤ 2 <cit.>. Thus, by <cit.>, the corestriction gives an isomorphismH^2(Γ_r', μ_p^r^⊗ 2)_Γ_r/Γ_r' H^2(Γ_r, μ_p^r^⊗ 2)where H^2(Γ_r', μ_p^r^⊗ 2)_Γ_r/Γ_r' is the group of coinvariants of H^2(Γ_r', μ_p^r^⊗ 2) for the action of Γ_r/Γ_r'.* The group 𝒦_r/J·𝒦_r is cyclic of order p^r.* The group following assertions are equivalent. * The group J·𝒦_r/J^2·𝒦_r is cyclic of order p^r.* We have n(r,p) ≥ 2, ∑_k=1^N-1/2 k ·log(k) ≡ 0(modulo p^r).* The group J·𝒦_r/J^2·𝒦_r is cyclic in any case, since 𝒦_r/J·𝒦_r is cyclic. * We give another criterion equivalent to (a) and (b) in Lemma <ref> in terms of certain norm residue symbols.* We give in Proposition <ref> an explicit isomorphism between 𝒦_r/J·𝒦_r and (𝐙/N𝐙)^×[p^r].* If r=1, one can show that Theorem <ref> is a consequence of <cit.>.Kummer theory gives us an exact sequence of (L_r/𝐐)-modules:0 →(𝒜_r) ⊗_𝐙𝐙/p^r𝐙→ H^2(Γ_r', μ_p^r) → H^2(Γ_r', 𝒜_r^×)[p^r]→ 0.By <cit.>, we have an isomorphism of (L_r/𝐐)-modules:H^2(Γ_r', 𝒜_r^×)[p^r] ≃(⊕_𝔮∈ T_r𝐙/p^r𝐙→𝐙/p^r𝐙).where (L_r/𝐐) acts on the right-hand side by permuting the elements of T_r via the natural left action of(L_r/𝐐) on T_r. Thus, we get the following (L_r/𝐐)-equivariant exact sequence(by tensoring by μ_p^r):0 →(𝒜_r) ⊗μ_p^r→ H^2(Γ_r', μ_p^r) ⊗μ_p^r→(⊕_𝔮∈ T_rμ_p^r→μ_p^r) → 0.Over L_r, we have μ_p^r≃𝐙/p^r𝐙. Thus H^2(Γ_r', μ_p^r) ⊗μ_p^r= H^2(Γ_r', μ_p^r^⊗ 2). Taking (L_r/K_r)-coinvariants in (<ref>), we get (using (<ref>)) an exact sequence of (L_r/𝐐(ζ_p^r))-modules:H_1((L_r/K_r), M) →((𝒜_r) ⊗_𝐙𝐙/p^r𝐙)_(-1)→𝒦_r → M_(0)→ 0whereM:=(⊕_𝔮∈ T_rμ_p^r→μ_p^r). We have H_1((L_r/K_r), M) =0 and M_(0)≃𝐙/p^r𝐙. Recall that (L_r/K_r) ≃(𝐐(ζ_p^r)/𝐐) ≃(𝐙/p^r𝐙)^× is a cyclic group. Thus we have H_1((L_r/K_r), M) ≃ M^(L_r/K_r)/(M) where (M) = {∑_g ∈(L_r/K_r) g · m, m∈ M}. Recall that the action of (L_r/K_r) on M permutes the primes in T_r and acts on μ_p^r via the pth cyclotomic character. Fix a prime 𝔫∈ T_r dividing N. Any other such prime equals g(𝔫) for some g ∈(L_r/K_r). We haveM^(L_r/K_r) ={⊕_g(𝔫) ∈ T_rg ∈(L_r/K_r)ζ^χ_p(g), ζ∈μ_p^r}where ⊕_g(𝔫) ∈ T_rg ∈(L_r/K_r)ζ^χ_p(g) is the element of M whose component at g(𝔫) is ζ^χ_p(g). Furthermore, this is by definition an element of (M). This shows that M^(L_r/K_r)=(M), and thereforeH_1((L_r/K_r), M) =0. The computation ofM_(0) =H_0((L_r/K_r), M) is similar. By Lemma <ref> and(<ref>), we have a canonical short exact sequence of Λ_r-modules:0 →((𝒜_r) ⊗_𝐙𝐙/p^r𝐙)_(-1)→𝒦_r →𝐙/p^r𝐙→ 0.The group 𝒦_r/J·𝒦_r is cyclic of order p^r. Furthermore, the image of the injective map ((𝒜_r) ⊗_𝐙𝐙/p^r𝐙)_(-1)→𝒦_r is J ·𝒦_r. As before, by <cit.>, the corestriction gives a group isomorphism𝒦_r/J·𝒦_rH^2_ét(𝐙[1/Np], μ_p^r^⊗ 2).The latter group is (as before by Tate) isomorphic to K_2(𝐙[1/Np])/p^r· K_2(𝐙[1/Np]). Since 𝐙[1/Np] is euclidean, by <cit.> and the well-known description of K_2(𝐐), we get that K_2(𝐙[1/Np])/p^r· K_2(𝐙[1/Np]) is cyclic of order p^r.The second assertion follows from the short exact sequence (<ref>) since J acts trivially on the right-hand side term 𝐙/p^r𝐙. Lemma <ref> proves point (i) of Theorem <ref>. Fix a prime 𝔫 above N in 𝐐(ζ_p^r). There is a canonical (L_r/𝐐)-equivariant group isomorphism (𝒜_r) 𝒞_r/C where 𝒞_r is the class group of L_r and C is the subgroup of 𝒞_r generated by the classes of the primes in T_r (for instance<cit.>). Thus, Lemma <ref> gives a canonical group isomorphismJ·𝒦_r/J^2 ·𝒦_r ( 𝒞_r/(C+J·𝒞_r + p^r ·𝒞_r))_(-1) .We have denoted (abusively) by J the augmentation ideal of 𝐙[(L_r/𝐐(ζ_p^r))], which acts on 𝒞_r, although J is properly speaking the augmentation ideal of Λ_r ≃𝐙_p[(L_r/𝐐(ζ_p^r))] (the point of this abuse is that 𝒞_r is not a p-group a priori, so we have to work with 𝐙-coefficients).Let _N𝒞_r be the kernel of the norm from 𝒞_r to the class group of 𝐐(ζ_p^r). The group _N𝒞_r/J ·𝒞_r is well understood in this case, thanks to genus theory. We now explain the result, following <cit.>.Let (L_r/𝐐(ζ_p^r))^((𝐐(ζ_p^r)/𝐐)) be the product of copies of (L_r/𝐐(ζ_p^r)) indexed by the elements (𝐐(ζ_p^r)/𝐐) (which we identify with (L_r/K_r) as usual). It is equipped with an action of (𝐐(ζ_p^r)/𝐐) given as follows. If τ_0 ∈(𝐐(ζ_p^r)/𝐐) and (g_τ)_τ∈(𝐐(ζ_p^r)/𝐐)∈(L_r/𝐐(ζ_p^r))^((𝐐(ζ_p^r)/𝐐)), then we letτ_0 · (g_τ)_τ∈(𝐐(ζ_p^r)/𝐐) = (g_τ·τ_0^-1)_τ∈(𝐐(ζ_p^r)/𝐐) .Let f : 𝐐(ζ_p^r)^×→(L_r/K)^((𝐐(ζ_p^r)/𝐐)) be given byf(x) = ( x, L_r/𝐐(ζ_p^r)τ(𝔫))_τ∈(𝐐(ζ_p^r)/𝐐)where · , L_r/𝐐(ζ_p^r)τ(𝔫) is the norm residue symbol of the extension L_r/𝐐(ζ_p^r) at the prime τ(𝔫). The group homomorphism f is (𝐐(ζ_p^r)/𝐐)-equivariant. This follows from the equality, for all x ∈𝐐(ζ_p^r)^× and τ∈(𝐐(ζ_p^r)/𝐐), in (L_r/𝐐(ζ_p^r)):x, L_r/𝐐(ζ_p^r)τ(𝔫) = τ^-1(x), L_r/𝐐(ζ_p^r)𝔫 . We let U = f(𝐙[ζ_p^r]^×). By local class field theory, the kernel of the restriction of f to U is the set of elements of U which are everywhere locally norms of element of L_r. Since L_r/𝐐(ζ_p^r) is cyclic, the Hasse norm theorem shows that this kernel is 𝐙[ζ_p^r]^×∩_L_r/𝐐(ζ_p^r)(𝐙[ζ_p^r]^×). Thus, we have a canonical isomorphism U 𝐙[ζ_p^r]^×/( 𝐙[ζ_p^r]^×∩_L_r/𝐐(ζ_p^r)(𝐙[ζ_p^r]^×) ). Let 𝔞 be an ideal class in _N𝒞_r and I be a fractional ideal of L_r whose class is in 𝔞. There is some α∈𝐐(ζ_p^r)^× such that _L_r/𝐐(ζ_p^r)(I) = (α). One easily checks that the map 𝔞↦ f(α) modulo U is well-defined, is independent of the choice of I and α. We have thus constructed a canonical (𝐐(ζ_p^r)/𝐐)-equivariant group homomorphism f̂ : _N𝒞_r →(L_r/𝐐(ζ_p^r))^((𝐐(ζ_p^r)/𝐐))/U.The following result follows from <cit.> applied with E = 𝐐(ζ_p^r) and F=L_r.* The kernel of f̂ is J ·𝒞_r.* The image of f̂ is (L_r/𝐐(ζ_p^r))^((𝐐(ζ_p^r)/𝐐)),0/U, where(L_r/𝐐(ζ_p^r))^((𝐐(ζ_p^r)/𝐐)),0 = {(g_τ)_τ∈(𝐐(ζ_p^r)/𝐐)∈(L_r/𝐐(ζ_p^r))^((𝐐(ζ_p^r)/𝐐)), ∏_τ∈(𝐐(ζ_p^r)/𝐐) g_τ = 1 } .We get a group isomorphism f̂' :( (_N𝒞_r/J·𝒞_r) ⊗_𝐙𝐙_p )_(-1)((L_r/𝐐(ζ_p^r))^((𝐐(ζ_p^r)/𝐐)),0/U )_(-1) .There is a group isomorphism ( (L_r/𝐐(ζ_p^r))^((𝐐(ζ_p^r)/𝐐)),0)_(-1)(L_r/𝐐(ζ_p^r))given by (g_τ)_τ∈(𝐐(ζ_p^r)/𝐐)↦∏_τ∈(𝐐(ζ_p^r)/𝐐) g_τ^χ_p^-1(τ) .Note that (L_r/𝐐(ζ_p^r)) ≃𝐙/p^r𝐙 is a p-group, so g_τ^χ_p^-1(τ) makes sense. By <cit.>, we have (𝐙[ζ_p^r]^×⊗_𝐙𝐙_p)_(-1)=0 (we are using the fact that χ_p^-1≢χ_p(modulo p^r) since p>3). Thus, the image of U in ( (L_r/𝐐(ζ_p^r))^((𝐐(ζ_p^r)/𝐐)),0)_(-1) is trivial. Consequently, we have( (L_r/𝐐(ζ_p^r))^((𝐐(ζ_p^r)/𝐐)),0/U )_(-1)≃𝐙/p^r𝐙 . Since p does not divide the numerator of B_2 = 1/6, the Herbrand–Ribet theorem shows that ((𝐙[ζ_p]) ⊗_𝐙𝐙_p)_(-1) is trivial. By Nakayama's lemma, it follows that ((𝐙[ζ_p^r])⊗_𝐙𝐙_p)_(-1)=0Thus, we have (𝒞_r ⊗_𝐙𝐙_p)_(-1) =(_N𝒞_r ⊗_𝐙𝐙_p)_(-1) and ( (_N𝒞_r/J·𝒞_r) ⊗_𝐙𝐙_p )_(-1) = ( (𝒞_r/J·𝒞_r) ⊗_𝐙𝐙_p)_(-1) .Since the primes above p in L_r are fixed by (L_r/K_r), we have (𝒞_r/(C+J·𝒞_r + p^r·𝒞_r ))_(-1) = (𝒞_r/(C'+J·𝒞_r + p^r·𝒞_r ))_(-1) where C' ⊂𝒞_r is generated by the classes of primes above N in 𝒞_r. The multiplication by a generator of J gives a surjective group homomorphism 𝒦_r/J·𝒦_r ↠ J·𝒦_r/J^2·𝒦_r. Thus, J·𝒦_r/J^2·𝒦_r is a cyclic group of order dividing p^r. By (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), the group J·𝒦_r/J^2·𝒦_r is cyclic of order p^r if and only if f̂'((C' ⊗_𝐙𝐙_p)_(-1))=0. Let ℐ be the group of fractional ideals of 𝐐(ζ_p^r). By (<ref>) and (<ref>), we have (ℐ⊗_𝐙𝐙_p)_(-1) = (𝐐(ζ_p^r)^×⊗_𝐙𝐙_p)_(-1) .Recall that we have fixed a prime ideal 𝔫 above N in ℐ. By (<ref>), the image of 𝔫 in (ℐ⊗_𝐙𝐙_p)_(-1) is some x ∈ (𝐐(ζ_p^r)^×⊗_𝐙𝐙_p)_(-1). We let ũ be any lift of x in 𝐐(ζ_p^r)^×⊗_𝐙𝐙_p and u be the image of ũ in 𝐐(ζ_p^r)^×⊗_𝐙𝐙/p^r𝐙. By the discussion above, J·𝒦_r/J^2·𝒦_r is a cyclic group of order p^r if and only if we have, in (L_r/𝐐(ζ_p^r)):∏_τ∈(𝐐(ζ_p^r)/𝐐)u, L_r/𝐐(ζ_p^r)τ(𝔫)^χ_p^-1(τ) = 1.(this does not depend on the choice of ũ). By (<ref>), this is equivalent to∏_τ∈(𝐐(ζ_p^r)/𝐐)τ^-1(u)^χ_p^-1(τ), L_r/𝐐(ζ_p^r)𝔫 = 1.Note that we have written the 𝐙_p-module structure of 𝐐(ζ_p^r)^×⊗_𝐙𝐙/p^r𝐙 multiplicatively. Letu_χ_p^-1 = ∏_τ∈(𝐐(ζ_p^r)/𝐐)τ(u)^χ_p(τ)∈ (𝐐(ζ_p^r)^×⊗_𝐙𝐙/p^r𝐙)^(-1) .(again, this does not depend on the choice of ũ).To conclude the proof of Theorem <ref>, it suffices to prove the following result. The following assertions are equivalent. * We have u_χ_p^-1, L_r/𝐐(ζ_p^r)𝔫 = 1.* We have ∑_k=1^N-1/2 k ·log(k) ≡ 0(modulo p^r).We first relate u_χ_p^-1 to a Gauss sum. Let ω_N : (𝐙/N𝐙)^×→𝐙_N^× be the Teichmüller character, characterized by ω_N(a) ≡ a(modulo N) for all a ∈ (𝐙/N𝐙)^×. Let𝒢_r= ∑_a ∈ (𝐙/N𝐙)^×ω_N(a)^-N-1/p^r·ζ_N^a ∈𝐙[ζ_p^r, ζ_N],where we view ω_N^N-1/p^r as taking values in the p^rth root of unity of 𝐙[ζ_p^r] using the choice of 𝔫. Galois theory shows that 𝒢_r^p^r∈𝐐(ζ_p^r). Thus, we have 𝒢_r ∈ L_r. Let𝒢_r,χ_p^-1 = ∏_τ∈(𝐐(ζ_p^r)/𝐐)τ(𝒢_r)^χ_p(τ)∈ (L_r^×⊗_𝐙𝐙/p^r𝐙)^(-1) .The inclusion 𝐐(ζ_p^r) ↪ L_r gives a group isomorphism(𝐐(ζ_p^r)^×⊗_𝐙𝐙/p^r𝐙)^(-1)((L_r^×⊗_𝐙𝐙/p^r𝐙)^(-1))^(L_r/𝐐(ζ_p^r))where the (L_r/𝐐(ζ_p^r)) in the exponent means the invariants by (L_r/𝐐(ζ_p^r)). The long exact sequence of cohomology attached to the short exact sequence of (L_r/𝐐(ζ_p^r))-modules 1 →μ_p^r→ L_r^× (L_r^×)^p^r→ 1 gives us, using Hilbert 90:1 →μ_p^r→𝐐(ζ_p^r)^×𝐐(ζ_p^r)^×∩ (L_r^×)^p^r→ H^1((L_r/𝐐(ζ_p^r)), μ_p^r) → 1and1 → H^1((L_r/𝐐(ζ_p^r)), (L_r^×)^p^r) → H^2((L_r/𝐐(ζ_p^r)), μ_p^r). The long exact sequence of cohomology attached to the short exact sequence of (L_r/𝐐(ζ_p^r))-modules 1 →(L_r^×)^p^r→ L_r^×→ L_r^×⊗_𝐙𝐙/p^r𝐙→ 1 gives us, using Hilbert 90:1 →𝐐(ζ_p^r)^×∩ (L_r^×)^p^r→𝐐(ζ_p^r)^×→ (L_r^×⊗_𝐙𝐙/p^r𝐙)^(L_r/𝐐(ζ_p^r))→ H^1((L_r/𝐐(ζ_p^r)), (L_r^×)^p^r) → 1.Since p>3, we have χ_p^-1≢χ_p(modulo p), so we get:H^1((L_r/𝐐(ζ_p^r)), μ_p^r)^(-1)=H^2((L_r/𝐐(ζ_p^r)), μ_p^r)^(-1)=1.Combining (<ref>) and (<ref>), we get(𝐐(ζ_p^r)^×∩ (L_r^×)^p^r/(𝐐(ζ_p^r)^×)^p^r)^(-1)=1.Combining (<ref>) and (<ref>), we getH^1((L_r/𝐐(ζ_p^r)), (L_r^×)^p^r)^(-1)=1.Combining (<ref>) and (<ref>), we get((L_r^×⊗_𝐙𝐙/p^r𝐙)^(-1))^(L_r/𝐐(ζ_p^r)) =(𝐐(ζ_p^r)^×/𝐐(ζ_p^r)^×∩ (L_r^×)^p^r)^(-1) .We have an exact sequence1 →𝐐(ζ_p^r)^×∩ (L_r^×)^p^r/(𝐐(ζ_p^r)^×)^p^r→𝐐(ζ_p^r)^×/(𝐐(ζ_p^r)^×)^p^r→𝐐(ζ_p^r)^×/𝐐(ζ_p^r)^×∩(L_r^×)^p^r→ 1. We haveH^1((𝐐(ζ_p^r)/𝐐),(𝐐(ζ_p^r)^×∩ (L_r^×)^p^r/(𝐐(ζ_p^r)^×)^p^r) ⊗_𝐙/p^r𝐙μ_p^r) = 1.Since (𝐐(ζ_p^r)/𝐐) is cyclic, the Herbrand quotient of (𝐐(ζ_p^r)^×∩ (L_r^×)^p^r/(𝐐(ζ_p^r)^×)^p^r) ⊗_𝐙/p^r𝐙μ_p^r is well-defined, and is trivial since this last group is finite by Kummer theory. By (<ref>), we have H^0((𝐐(ζ_p^r)/𝐐),(𝐐(ζ_p^r)^×∩ (L_r^×)^p^r/(𝐐(ζ_p^r)^×)^p^r) ⊗_𝐙/p^r𝐙μ_p^r) = 1. Using the previous remark on the Herbrand quotient, this proves thatH^1((𝐐(ζ_p^r)/𝐐),(𝐐(ζ_p^r)^×∩ (L_r^×)^p^r/(𝐐(ζ_p^r)^×)^p^r) ⊗_𝐙/p^r𝐙μ_p^r) = 1.Combining Lemma <ref>, (<ref>) and (<ref>), we get a group isomorphism:(𝐐(ζ_p^r)^×/(𝐐(ζ_p^r)^×)^p^r)^(-1)(𝐐(ζ_p^r)^×/𝐐(ζ_p^r)^×∩(L_r^×)^p^r)^(-1) .To conclude the proof of Lemma <ref>, it suffices to combine (<ref>), (<ref>) and (<ref>).We have 𝒢_r,χ_p^-1∈((L_r^×⊗_𝐙𝐙/p^r𝐙)^(-1))^(L_r/𝐐(ζ_p^r)). By Lemma <ref>, there is a unique g_r,χ_p^-1∈ (𝐐(ζ_p^r)^×⊗_𝐙𝐙/p^r𝐙)^(-1) such that 𝒢_r,χ_p^-1 is the image of g_r,χ_p^-1 in (L_r^×⊗_𝐙𝐙/p^r𝐙)^(-1).There exists α∈ (𝐙/p^r𝐙)^× such that g_r,χ_p^-1=u_χ_p^-1^α. Let ℐ_𝐐(ζ_p^r) (resp. 𝒞_𝐐(ζ_p^r)) be the group of fractional ideals (resp. the class group) of 𝐐(ζ_p^r). Similarly, let ℐ_L_r be the group of fractional ideals of L_r. We have an exact sequence 1 →𝐙[ζ_p^r]^×→𝐐(ζ_p^r)^×→ℐ_𝐐(ζ_p^r)→𝒞_𝐐(ζ_p^r)→ 1 .The snake lemma gives an exact sequence 1 →𝒞_𝐐(ζ_p^r)[p^r] → (𝐐(ζ_p^r)^×/𝐙[ζ_p^r]^×) ⊗_𝐙𝐙/p^r𝐙→ℐ_𝐐(ζ_p^r)⊗_𝐙𝐙/p^r𝐙 . We have (𝒞_𝐐(ζ_p^r)[p^r])^(-1)=1. It suffices to show that (𝒞_𝐐(ζ_p^r)[p^r])^(-1)[p]=1, that (𝒞_𝐐(ζ_p^r)[p])^(-1)=1. If M is a (𝐐(ζ_p)/𝐐)-module and i ∈𝐙, let M^[i] (resp. M_[i]) be the subgroup (resp. the maximal quotient) of M on which (𝐐(ζ_p)/𝐐) acts by ω_p^i. It suffices to show that (𝒞_𝐐(ζ_p^r)[p])^[-1]=1. It suffices to show that (𝒞_𝐐(ζ_p^r))^[-1]=1, or equivalently that (𝒞_𝐐(ζ_p^r))_[-1]=1. By Nakayama's lemma, it suffices to show that (𝒞_𝐐(ζ_p))_[-1]=1, which follows from the Herbrand–Ribet theorem since p does not divide B_2 = 1/6.By (<ref>), we get an embedding((𝐐(ζ_p^r)^×/𝐙[ζ_p^r]^×)⊗_𝐙𝐙/p^r𝐙)^(-1)↪(ℐ_𝐐(ζ_p^r)⊗_𝐙𝐙/p^r𝐙)^(-1) .The map(𝐐(ζ_p^r)^×⊗_𝐙𝐙/p^r𝐙)^(-1)→((𝐐(ζ_p^r)^×/𝐙[ζ_p^r]^×)⊗_𝐙𝐙/p^r𝐙)^(-1)is an isomorphism. We have an exact sequence1 →𝐙[ζ_p^r]^×/𝐙[ζ_p^r]^×∩ (𝐐(ζ_p^r)^×)^p^r→𝐐(ζ_p^r)^×⊗_𝐙𝐙/p^r𝐙→ (𝐐(ζ_p^r)^×/𝐙[ζ_p^r]^×)⊗_𝐙𝐙/p^r𝐙→ 1.To prove Lemma <ref>, it suffices to prove thatH^0((𝐐(ζ_p^r)/𝐐), (𝐙[ζ_p^r]^×/𝐙[ζ_p^r]^×∩ (𝐐(ζ_p^r)^×)^p^r) ⊗_𝐙/p^r𝐙μ_p^r)=1andH^1((𝐐(ζ_p^r)/𝐐), (𝐙[ζ_p^r]^×/𝐙[ζ_p^r]^×∩ (𝐐(ζ_p^r)^×)^p^r) ⊗_𝐙/p^r𝐙μ_p^r)=1.Since (𝐐(ζ_p^r)/𝐐) is cyclic and (𝐙[ζ_p^r]^×/𝐙[ζ_p^r]^×∩ (𝐐(ζ_p^r)^×)^p^r) ⊗_𝐙/p^r𝐙μ_p^r is a finite group, a Herbrand quotient argument like in Lemma <ref> shows that it suffices to prove (<ref>). We have an exact sequence1 →𝐙[ζ_p^r]^×∩ (𝐐(ζ_p^r)^×)^p^r/(𝐙[ζ_p^r]^×)^p^r→𝐙[ζ_p^r]^×/( 𝐙[ζ_p^r]^×)^p^r→𝐙[ζ_p^r]^×/ 𝐙[ζ_p^r]^×∩ (𝐐(ζ_p^r)^×)^p^r→ 1.To prove (<ref>), it thus suffices to prove that(𝐙[ζ_p^r]^×⊗_𝐙𝐙/p^r𝐙)^(-1)=1and H^1((𝐐(ζ_p^r)/𝐐), (𝐙[ζ_p^r]^×∩ (𝐐(ζ_p^r)^×)^p^r/(𝐙[ζ_p^r]^×)^p^r) ⊗_𝐙/p^r𝐙μ_p^r)=1.The equality (<ref>) follows from <cit.>. As before, a Herbrand quotient argument shows that to prove (<ref>), it suffices to proveH^0((𝐐(ζ_p^r)/𝐐), (𝐙[ζ_p^r]^×∩ (𝐐(ζ_p^r)^×)^p^r/(𝐙[ζ_p^r]^×)^p^r) ⊗_𝐙/p^r𝐙μ_p^r)= 1,which follows from (<ref>). By (<ref>) and Lemma <ref>, we get an embedding(𝐐(ζ_p^r)^×⊗_𝐙𝐙/p^r𝐙)^(-1)↪(ℐ_𝐐(ζ_p^r)⊗_𝐙𝐙/p^r𝐙)^(-1) . By (<ref>), in order to prove Lemma <ref> it suffices to prove that there existsα∈ (𝐙/p^r𝐙)^×such that u_χ_p^-1 and g_r,χ_p^-1^α have the same image in (ℐ_𝐐(ζ_p^r)⊗_𝐙𝐙/p^r𝐙)^(-1). Let ℐ_N ⊂ℐ_𝐐(ζ_p^r) be the subgroup generated the ideals above N in 𝐐(ζ_p^r). This is a direct summand of the free 𝐙-module ℐ_𝐐(ζ_p^r). Thus, we have an embedding (ℐ_N ⊗_𝐙𝐙/p^r𝐙)^(-1)↪(ℐ_𝐐(ζ_p^r)⊗_𝐙𝐙/p^r𝐙)^(-1). Furthermore, the group (ℐ_N ⊗_𝐙𝐙/p^r𝐙)^(-1) is cyclic of order p^r. By definition, the image of u_χ_p^-1 in(ℐ_𝐐(ζ_p^r)⊗_𝐙𝐙/p^r𝐙)^(-1) is a generator of (ℐ_N ⊗_𝐙𝐙/p^r𝐙)^(-1). To conclude the proof of Proposition <ref>, it suffices to prove that the image of g_r,χ_p^-1 in (ℐ_𝐐(ζ_p^r)⊗_𝐙𝐙/p^r𝐙)^(-1) is a generator of (ℐ_N ⊗_𝐙𝐙/p^r𝐙)^(-1). We first prove that this image lies in (ℐ_N ⊗_𝐙𝐙/p^r𝐙)^(-1). Let 𝔑 be the unique prime ideal of L_r above the fixed prime ideal 𝔫 of 𝐐(ζ_p^r) dividing N. If τ∈(𝐐(ζ_p^r)/𝐐), let _r(τ) be the unique integer in {1, ..., p^r-1} such that τ(ζ_p^r) = ζ_p^r^_r(τ). We have the following prime ideal decomposition in L_r (for instance <cit.>):(𝒢_r) = ∏_τ∈(𝐐(ζ_p^r)/𝐐)τ(𝔑)^_r(τ^-1) .This allows us to compute the image of 𝒢_r,χ_p^-1 in ℐ_L_r⊗_𝐙𝐙/p^r𝐙, which is(∏_τ∈(𝐐(ζ_p^r)/𝐐)τ(𝔑)^_r(τ^-1))^∑_k=1^p^r-1 k^2 = 1since ∑_k=1^p^r-1 k^2≡ 0(modulo p^r). The kernel of the map ℐ_𝐐(ζ_p^r)⊗_𝐙𝐙/p^r𝐙→ℐ_L_r⊗_𝐙𝐙/p^r𝐙 is ℐ_N⊗_𝐙𝐙/p^r𝐙. By (<ref>), the image of g_r,χ_p^-1 in(ℐ_𝐐(ζ_p^r)⊗_𝐙𝐙/p^r𝐙)^(-1) is in (ℐ_N ⊗_𝐙𝐙/p^r𝐙)^(-1). Thus, there exists α∈𝐙/p^r𝐙 such that g_r,χ_p^-1=u_χ_p^-1^α. In order to prove that α∈ (𝐙/p^r𝐙)^×, it suffies to prove the following result.The element g_r,χ_p^-1 is not a pth power in (𝐐(ζ_p^r)^×⊗_𝐙𝐙/p^r𝐙)^(-1). For the sake of a contradiction, assume that g_r,χ_p^-1 is a pth power in (𝐐(ζ_p^r)^×⊗_𝐙𝐙/p^r𝐙)^(-1). In particular, the image of 𝒢_r, χ_p^-1 in L_r^×⊗_𝐙𝐙/p𝐙 is trivial. Thus, we have∏_τ∈(𝐐(ζ_p)/𝐐)τ(∏_τ' ∈(𝐐(ζ_p^r)/𝐐(ζ_p))τ'(𝒢_r))^ω_p(τ) = 1in(L_r^×⊗_𝐙𝐙/p𝐙)^(-1) .Note that ∏_τ' ∈(𝐐(ζ_p^r)/𝐐(ζ_p))τ'(𝒢_r) = _L_r/K_r(ζ_p)(𝒢_r).* The map (K_r(ζ_p)^×⊗_𝐙𝐙/p𝐙)^(-1)→ (L_r^×⊗_𝐙𝐙/p𝐙)^(-1) is injective.* The map (L_1^×⊗_𝐙𝐙/p𝐙)^(-1)→ (K_r(ζ_p)^×⊗_𝐙𝐙/p𝐙)^(-1) is injective.This follows by considering the cohomology of the exact sequences 1 →μ_p → L_r^×→(L_r^×)^p → 1 and 1 →μ_p → K_r(ζ_p)^×→ (K_r(ζ_p)^×)^p → 1 as in the proof of Lemma <ref>.By (<ref>), we get that∏_τ∈(𝐐(ζ_p)/𝐐)τ( _L_r/K_r(ζ_p)(𝒢_r) )^ω_p(τ) is a p th power in (K_r(ζ_p)^×⊗_𝐙𝐙_p)^(ω_p^-1) . Here, if M is a 𝐙_p[(𝐐(ζ_p)/𝐐)], module, we let M^(ω_p^-1) be the submodule of M on which (𝐐(ζ_p)/𝐐) acts by ω_p^-1.We have, in (K_r(ζ_p)^×⊗_𝐙𝐙_p)^(ω_p^-1):∏_τ∈(𝐐(ζ_p)/𝐐)τ(_L_r/K_r(ζ_p)(𝒢_r))^ω_p(τ) = ∏_τ∈(𝐐(ζ_p)/𝐐)τ(𝒢_1)^ω_p(τ) .Using (<ref>), one checks that both sides have the same image in (ℐ_K_r(ζ_p)⊗_𝐙𝐙_p)^(ω_p^-1). One then uses the fact that the map (K_r(ζ_p)^×⊗_𝐙𝐙_p)^(ω_p^-1)→ (ℐ_K_r(ζ_p)⊗_𝐙𝐙_p)^(ω_p^-1) is injective by <cit.> since ω_p^-1≠ω_p and ω_p^-1 is odd.By Lemma <ref> (ii), (<ref>) and Lemma <ref>, we get that ∏_τ∈(𝐐(ζ_p)/𝐐)τ(𝒢_1)^ω_p(τ) is a pth power in L_1^×⊗_𝐙𝐙_p. This contradicts <cit.>.This concludes the proof of Lemma <ref>.By Lemma <ref>, in order to conclude the proof of Lemma <ref>, it suffices to prove the following result.The following assertions are equivalent. * We have g_r,χ_p^-1, L_r/𝐐(ζ_p^r)𝔫 = 1.* We have ∑_k=1^N-1/2 k ·log(k) ≡ 0(modulo p^r).Let ℒ : 𝐐_N(ζ_N)^×⊗_𝐙𝐙/p^r𝐙→𝐙/p^r𝐙 be the group homomorphism defined byℒ(a ⊗ b) = log(a/(1-ζ_N)^v_N(a))· bwhere a ∈𝐐_N(ζ_N)^×, v_N(a) is the N-adic valuation of a (normalized by v_N(1-ζ_N)=1), a/(1-ζ_N)^v_N(a) is the reduction of a/(1-ζ_N)^v_N(a) modulo (1-ζ_N) and b ∈𝐙/p^r𝐙. We haveN = ∏_i=1^N-1(ζ_N^i-1),so ℒ(N ⊗1) = ℒ(N/(ζ_N-1)^N-1⊗1) ≡log((N-1)!) ≡ 0(modulo p^r ).Let 𝔑' be the prime above 𝔫 in 𝐐(ζ_p^r, ζ_N). We let ℒ' :𝐐(ζ_N, ζ_p^r)^×⊗_𝐙𝐙/p^r𝐙→𝐙/p^r𝐙 be the composition of ℒ with the group homomorphism 𝐐(ζ_N, ζ_p^r) ⊗_𝐙𝐙/p^r𝐙→𝐐_N(ζ_N)^×⊗_𝐙𝐙/p^r𝐙 induced by the 𝔑'-adic completion. Let ℒ” :𝐐(ζ_p^r)^×⊗_𝐙𝐙/p^r𝐙→𝐙/p^r𝐙 be the composition of ℒ' with the group homomorphism 𝐐(ζ_p^r)^×⊗_𝐙𝐙/p^r𝐙→𝐐(ζ_N, ζ_p^r)^×⊗_𝐙𝐙/p^r𝐙.One easily sees that the 𝔑-adic completion of L_r is the extension of 𝐐_N obtained by adjoining a p^rth root of N. The classical properties of the norm residue symbol (recalled for instance in <cit.>) show that g_r,χ_p^-1, L_r/𝐐(ζ_p^r)𝔫 = 1 if and only if ℒ”(g_r,χ_p^-1 - N ⊗ v )=0 where v ∈𝐙/p^r𝐙 is the 𝔫-adic valuation of g_r,χ_p^-1. By (<ref>), this is equivalent to ℒ”(g_r, χ_p^-1)=0. By definition of g_r, χ_p^-1, this is equivalent to ℒ'(𝒢_r, χ_p^-1)=0 where 𝒢_r, χ_p^-1 is viewed abusively as an element of 𝐐(ζ_N, ζ_p^r)^×⊗_𝐙𝐙/p^r𝐙.It is well-known (for instance<cit.>) that we have, for all τ∈(𝐐(ζ_p^r)/𝐐):τ(𝒢_r)/(ζ_N-1)^_r(τ)·N-1/p^r≡ -1/(_r(τ)·N-1/p^r)! (modulo 𝔑').where we recall that _r(τ) is the unique integer in {0, 1, ..., p^r-1} such that τ(ζ_p^r)=ζ_p^r^_r(τ). Thus, we have in 𝐙/p^r𝐙:ℒ'(𝒢_r, χ_p^-1) = ∑_a=1 (a, p)=1^p^r-1 a·log(-1/(a·N-1/p^r)!).In order to conclude the proof of Lemma <ref>, it suffices to prove the following identity. We have:∑_a=1 (a, p)=1^p^r-1 a·log((a·N-1/p^r)!)≡ -2· (p-1)/3·∑_k=1^N-1/2 k ·log(k)(modulo p^r) .We prove this by induction on r. Let s be an integer such that 1 ≤ s ≤ r. We follow the computations of the forthcoming work of Karl Schaefer and Eric Stubley, but we do them modulo p^s and not modulo p as they do. We have, in 𝐙/p^s𝐙:∑_k=1^N-1 k^2·log(k)= ∑_a=1^p^s-1∑_b=0^N-1/p^s-1 (a+b· p^s)^2·log(a+b· p^s)= ∑_a=1^p^s-1∑_b=0^N-1/p^s-1 a^2·log(a+b· p^s) =∑_a=1^p^s-1a^2·∑_b=0^N-1/p^s-1log(a/p^s+b)where in the last equality we used the fact that ∑_a=1^p^s-1 a^2 ≡ 0(modulo p^s).Let Γ_N: 𝐙_N →𝐙_N^× be the Morita N-adic Gamma function. This is the unique continuous function 𝐙_N →𝐙_N^× satisfying Γ_N(n) = (-1)^n·∏_1 ≤ i ≤ n-1,pgcd(n,N )=1 i if n>1 is an integer.We will use the following properties of Γ_N:* Γ_N(0)=1.* If x, y ∈𝐙_N are such that x ≡ y(modulo N), then Γ_N(x) ≡Γ_N(y)(modulo N).* If x ∈𝐙_N is such that for all integer k in {0, ..., N-1/p^s-1} we have x + k ≢0(modulo N), then we have∏_k=0^N-1/p^s-1Γ_N(x+k) = (-1)^N-1/p^s·Γ_N(x+N-1/p^s)/Γ_N(x) . Thus, we have:∑_k=1^N-1 k^2·log(k)= ∑_a=1^p^s-1a^2·log(Γ_N(a/p^s+N-1/p^s)/Γ_N(a/p^s))=∑_a=1^p^s-1a^2·log(Γ_N(a-1/p^s)/Γ_N(a/p^s)) = ∑_a=1^p^s-1((a+1)^2-a^2)·log(Γ_N(a/p^s)) =∑_a=1^p^s-1((a+1)^2-a^2)·log(Γ_N(-N-1/p^s· a)) = ∑_a=1^p^s-1((p^s-a+1)^2-(p^s-a)^2)·log(Γ_N(1+a·N-1/p^s)) = ∑_a=1^p^s-1((p^s-a+1)^2-(p^s-a)^2)·log((a·N-1/p^s)!) .Similarly, we have in 𝐙/p^s𝐙:∑_k=1^N-1k·log(k)=∑_a=1^p^s-1∑_b=0^N-1/p^s-1 (a+b· p^s)·log(a+b· p^s)= ∑_a=1^p^s-1 a·∑_b=0^N-1/p^s-1log(a/p^s+b)=∑_a=1^p^s-1 a·log(Γ_N(a/p^s+N-1/p^s)/Γ_N(a/p^s))=∑_a=1^p^s-1 a·log(Γ_N(a-1/p^s)/Γ_N(a/p^s))= ∑_a=1^p^s-1 (a+1-a)·log(Γ_N(a/p^s)) Since ∑_k=1^N-1 k ·log(k) = 0, we get ∑_a=1^p^s-1log(Γ_N(a/p^s)) ≡0(modulo p^s). Thus, we have:∑_k=1^N-1 k^2·log(k) ≡ -2 ·∑_a=1^p^s-1 a ·log((a·N-1/p^s)!).By Lemma <ref> (which is independent of everything else in this paper), we have:∑_a=1^p^s-1 a ·log((a·N-1/p^s)!) ≡2/3·∑_k=1^N-1/2 k ·log(k) (modulo p^s).This proves Lemma <ref> for r=1. Assume that Lemma <ref> is true for all 1 ≤ s < r. We have:∑_a=1^p^r-1 a ·log((a·N-1/p^r)!) ≡∑_s=0^r-1∑_a=1 (a,p)=1^p^r-s-1 p^s· a ·log(N-1/p^r· p^s· a) (modulo p^r).By the induction hypothesis and (<ref>), we have:∑_a=1 (a,p)=1^p^r-1 a ·log((a·N-1/p^r)!) ≡2/3· (1-∑_s=1^r-1 p^s· (1-p))Since ∑_s=1^r-1 p^s ≡ -p/p-1 (modulo p^r), this concludes the induction, and thus the proof of Lemma <ref>.This concludes the proof of Lemma <ref>.This concludes the proof of Theorem <ref>.By point Theorem <ref> (i), we have a group isomorphism 𝒦_r/J·𝒦_r ≃𝐙/p^r𝐙. Such an isomorphism follows canonically from the choice of log we have made throughout the article. If x,y ∈𝐙[ζ_N, 1/Np]^×, we let (x, y)_r be the image of the Steinberg symbol {x,y}∈ K_2(𝐙[ζ_N, 1/Np]) in 𝒦_r via the norm map K_2(𝐙[ζ_N, 1/Np]) → K_2(𝒪_r).There is a unique group isomorphismι_r :𝒦_r/J·𝒦_r 𝐙/p^r𝐙such that for all u and v in (𝐙/N𝐙)^×, we haveι_r ( (1-ζ_N^u, 1-ζ_N^v)_r) ≡log(u/v) ∈𝐙/p^r𝐙 .By Matsumoto's Theorem <cit.>, we have a map K_2(𝐐(ζ_N)) →𝐙/p^r𝐙 given by the Hilbert symbol{x,y}↦log( y^v(x)/x^v(y)).Here, v is the N-adic valuation and y^v(x)/x^v(y) is the reduction modulo the prime above N in 𝐙[ζ_N] of y^v(x)/x^v(y). Composing with the (injective) map K_2(𝐙[ζ_N, 1/Np]) → K_2(𝐐(ζ_N)), we get a map φ : K_2(𝐙[ζ_N, 1/Np]) →𝐙/p^r𝐙 such that φ({1-ζ_N^u, 1-ζ_N^v}) ≡log(u/v) modulo p^r. The map φ is (𝐐(ζ_N)/𝐐)-equivariant, where the action of (𝐐(ζ_N)/𝐐) on 𝐙/p^r𝐙 is trivial. By <cit.>, this induces the map ι_r of the statement (which is unique because the elements log(u/v) generate 𝐙/p^r𝐙). Let Δ_r = [x]-[1] ∈Λ_r where x ∈ P_r is such that log(x) ≡ 1 (modulo p^r). The element Δ_r is a generator of J. The multiplication by Δ_r gives a natural surjective homomorphism δ_r' : 𝒦_r/J·𝒦_r → J·𝒦_r/J^2·𝒦_r, which is an isomorphism if and only if n(r,p) ≥ 2. In this case, we define δ_r : 𝐙/p^r𝐙 J·𝒦_r/J^2·𝒦_r by δ_r = δ_r' ∘ι_r^-1.§.§ Refined Sharifi theoryWe follow the notation of sections <ref>, <ref> and <ref>.In this section, we explain, inspired by Sharifi <cit.>, the link between H_1(X_1^(p^r)(N), C_0^(p^r), 𝐙_p) and the K-group 𝒦_r studied in section <ref>. As in section <ref>, if u, v ∈ (𝐙/N𝐙)^×, we have the Manin symbol ξ_Γ_1(N)([u,v]) ∈H̃_Γ_1(N). Recall that M_Γ_1(N)^0 = 𝐙_p[((𝐙/N𝐙)^×)^2/±]. Following Goncharov <cit.>, consider the mapϖ' : M_Γ_1(N)^0 → K_2(𝐙[ζ_N, 1/Np]) ⊗_𝐙𝐙_pgiven by ϖ'([u,v] ) ={1-ζ_N^u, 1-ζ_N^v}⊗ 1.The map ϖ' is even with respect to the complex conjugation c. It factors through ξ_Γ_1(N). Hence we get a mapϖ : ( H̃_Γ_1(N))_+ → K_2(𝐙[ζ_N, 1/Np]) ⊗_𝐙𝐙_psuch thatϖ((1+c)·ξ_Γ_1(N)([u,v])) = {1-ζ_N^u, 1-ζ_N^v} .The proof is the same as the proof of <cit.> (also independently found by Sharifi). We give the proof here for the convenience of the reader. By Proposition <ref>, it suffices to prove the following equalities for all u, v ∈ (𝐙/N𝐙)^×: * ϖ'([-u,v]) = ϖ'([u,v]).* ϖ'([u,v]) + ϖ'([-v,u]) = 0.* Ifu+v ≠ 0, we have ϖ'([u,v]) + ϖ'([-(u+v),u]) + ϖ'([v,-(u+v)]) = 0.* ϖ'([u,-u])=0. These equalities follow from those general identities.* The Steinberg relations. If x ∈𝐙[ζ_N, 1/Np]^× and 1-x ∈𝐙[ζ_N, 1/Np]^×, then we have {x, 1-x}=0.* Antisymmetry. For all x, y ∈𝐙[ζ_N, 1/Np]^×, we have {x,y}=-{y,x}. * For all root of unity ζ of order prime to p and all x ∈𝐙[ζ_N, 1/Np]^×, we have {ζ,x}=0. We prove (i). We have ϖ'([-u,v]) = {1-ζ_N^-u, 1-ζ_N^v} = {ζ_N^u-1, 1-ζ_N^v} - {ζ_N^u, 1-ζ_N ^v} = {ζ_N^u-1, 1-ζ_N^v} = {1-ζ_N^u, 1-ζ_N^v} = ϖ'([u,v]). We prove (ii). We have ϖ([-v,u]) = {1-ζ_N^-v, 1-ζ_N^u} = {1-ζ_N^v, 1-ζ_N^u}= -{1-ζ_N^u, 1-ζ_N^v} = -ϖ([u,v]). We prove (iii). Note that we haveζ_N^v· (1-ζ_N^u)/1-ζ_N^u+v+ 1-ζ_N^v/1-ζ_N^u+v=1.This shows that {ζ_N^v· (1-ζ_N^u)/1-ζ_N^u+v, 1-ζ_N^v/1-ζ_N^u+v} = 0. Using the facts above and the bilinearity of {·, ·}, this proves (iii). We have: {1-ζ_N^-u, 1-ζ_N^u} = {1-ζ_N^u, 1-ζ_N^u}= 0, which proves (iv). The following conjecture is inspired by the work of Sharifi. We refer to <cit.> for details about Sharifi's conjectures. However, our situation is not considered by Sharifi and Fukaya–Kato, who consider modular curves of level divisible by p <cit.>. The map ϖ is annihilated by the Hecke operator T_n-∑_d | n, (d,N)=1n/d·⟨ d ⟩ for all n ≥ 1 such that (n,p)=1. We end this section by stating a conjecture about our K-groups generalizing Theorem <ref>. The norm map gives a group homomorphism K_2(𝐙[ζ_N, 1/Np]) ⊗_𝐙𝐙_p→𝒦_r sending {x,y}⊗ 1 to (x,y)_r. ByProposition <ref>, the map ϖ induces a Λ_r-module homomorphismϖ^(p^r) : (H̃^(p^r))_+ →𝒦_rsuch thatϖ^(p^r)((1+c)·ξ_Γ_1^(p^r)(N)([u,v])) = (1-ζ_N^u, 1-ζ_N^v)_rfor all u,v ∈ (𝐙/N𝐙)^×. By Proposition <ref>, the map ϖ^(p^r) is surjective modulo J, so it is surjective. The map ϖ^(p^r) induces an isomorphism of Λ_r-modules(H̃^(p^r))_+/(Ĩ_∞+(p^r)) ·(H̃^(p^r))_+𝒦_r.The Λ_r-module 𝒦_r is cyclic generated by (1-ζ_N^x, 1-ζ_N^y)_r for all x, y ∈ (𝐙/N𝐙)^× such that xy^-1 is not a pth power. Using Stickelberger theory, one can show that the annihilator of 𝒦_r in Λ_r contains (p^r)+(ζ^(p^r)). Theorem <ref> gives us the following structure theorem for the Λ_r-module 𝒦_r. Assume Conjecture <ref>. Then Conjecture <ref> is true if and only if the annihilator of 𝒦_r in Λ_r is (ζ^(p^r))+(p^r).If r=1, this is true if and only if the 𝐅_p-rank of 𝒦_1 is the largest integer i ∈{1, 2, ..., p-1} such that for all 1 ≤ j < i, we have∑_k∈ (𝐙/N𝐙)^×_2(k/N) ·log(k)^j ≡ 0(modulo p)(this is a particular case for χ = ω_p^-1 of <cit.>). Using the computer software PARI/GP, Nicolas Mascot checked the truth of the last condition (when r=1) of Proposition <ref> for p=5 and N ≤ 12791, thus proving Conjecture <ref> (assuming Conjecture <ref>) for theses values of N and p. He did the computation with 𝒦_1 replaced by the ω_p^-1-part 𝒞_(ω_p^-1) of the class group of L_r modulo p, since one can show that 𝒞_(ω_p^-1) and 𝒦_1 are isomorphic Λ_1-modules. §.§ Evidence in favor of Conjecture <ref>We follow the notation of section <ref>.We give some evidence in favor of Conjecture <ref>. The following result was proved in level Γ_1(p) and weight 2 by Busioc <cit.> (and independently by Sharifi) and her proof can be adapted directly to our case. We reproduce it here for the convenience of the reader.The map ϖ is annihilated by the Hecke operators T_2-2-⟨ 2 ⟩ and T_3-3-⟨ 3 ⟩. The proof relies on explicit formulas for Hecke operators acting on Manin symbols <cit.>. For each integer n ≥ 1 prime to N, let 𝒳_n = {[ a b; c d ]∈ M_2(𝐙),a>b≥ 0 ,d>c ≥ 0, ad-bc=n} .This is a finite set <cit.>. By <cit.>, for all Γ_0(N)·γ∈Γ_0(N) \_2(𝐙) we have in H_1(X_1(N), , 𝐙):T_n ( ξ_Γ_1(N)(Γ_0(N)·γ) ) = ∑_α∈𝒳_nξ_Γ_1(N)(Γ_0(N)·γ·α). We first prove that ϖ is annihilated by the Hecke operator T_2-2-⟨ 2 ⟩. We have𝒳_2 = {[ 1 0; 0 2 ], [ 2 0; 0 1 ], [ 1 0; 1 2 ], [ 2 1; 0 1 ]} .Let u,v ∈ (𝐙/N𝐙)^×. By (<ref>), we have:T_2(ξ_Γ_1(N)([u,v]) ) = ξ_Γ_1(N)([u,2v])+ξ_Γ_1(N)([2u,v])+ξ_Γ_1(N)([u+v,2v])+ξ_Γ_1(N)([2u,u+v]).Assume first that u + v ≠ 0. By (<ref>), we have in 𝒦_r:ϖ( (1+c)· T_2(ξ_Γ_1(N)([u,v])) )={1-ζ_N^u, 1-ζ_N^2v} + {1-ζ_N^2u, 1-ζ_N^v} + {1-ζ_N^u+v, 1-ζ_N^2v} + {1-ζ_N^2u, 1-ζ_N^u+v} .The following identity was discovered by McCallum and Sharifi <cit.>:(1-ζ_N^u+v)· (1-ζ_N^u)/1-ζ_N^2u+ζ_N^u· (1-ζ_N^2v)·(1-ζ_N^u)/(1-ζ_N^2u)· (1-ζ_N^v) = 1.Using (<ref>) and the properties of {·, ·} stated in the proof of Lemma <ref>, wehave in K_2(𝐙[ζ_N, 1/Np]):0 ={(1-ζ_N^u+v)· (1-ζ_N^u)/1-ζ_N^2u,ζ_N^u· (1-ζ_N^2v)· (1-ζ_N^u)/(1-ζ_N^2u)· (1-ζ_N^v)}={1-ζ_N^u+v, 1-ζ_N^2v}+{1-ζ_N^u+v, 1-ζ_N^u}-(1-ζ_N^u+v, 1-ζ_N^2u}-{1-ζ_N^u+v, 1-ζ_N^v}+{1-ζ_N^u, 1-ζ_N^2v}-{1-ζ_N^u, 1-ζ_N^2u}-{1-ζ_N^u, 1-ζ_N^v}-{1-ζ_N^2u, 1-ζ_N^2v}-{1-ζ_N^2u, 1-ζ_N^u}+{1-ζ_N^2u, 1-ζ_N^v}= {1-ζ_N^u, 1-ζ_N^2v} + {1-ζ_N^2u, 1-ζ_N^v} + {1-ζ_N^u+v, 1-ζ_N^2v} + {1-ζ_N^2u, 1-ζ_N^u+v} +{1-ζ_N^u+v, 1-ζ_N^u} - {1-ζ_N^u+v, 1-ζ_N^v} - {1-ζ_N^u, 1-ζ_N^v}-{1-ζ_N^2u, 1-ζ_N^2v} .Furthermore, we have seen that the Manin relations hold, :{1-ζ_N^u, 1-ζ_N^v}+ {1-ζ_N^u+v, 1-ζ_N^u} - {1-ζ_N^u+v, 1-ζ_N^v} = 0.Using <ref>, we get:ϖ( (1+c)· T_2(ξ_Γ_1(N)([u,v])) ) = 2·{1-ζ_N^u, 1-ζ_N^v} + {1-ζ_N^2u, 1-ζ_N^2v} .Since ⟨ 2 ⟩·ξ_Γ_1(N)([u,v]) = ξ_Γ_1(N)([2u,2v]), we have:ϖ( (1+c)· (T_2-2-⟨ 2 ⟩)(ξ_Γ_1(N)([u,v])) ) = 0. If u+v=0, then we have ξ_Γ_1(N)([u,v]) = 0 by the Manin relations. Thus, we also have in 𝒦_r:ϖ_r( (1+c)· (T_2-2-⟨ 2 ⟩)(ξ_Γ_1(N)([u,v])) ) = 0. The proof that ϖ' is annihilated by the Hecke operator T_3-3-⟨ 3 ⟩ is the same as for T_2-2-⟨ 2 ⟩, using the identity (discovered by Sharifi in an unpublished work)ζ_N^v-u· (1-ζ_N^3u)· (1-ζ_N^v)/(1-ζ_N^u)· (1-ζ_N^3v)+(1-ζ_N^v-u)· (1-ζ_N^v)· (1-ζ_N^u+v)/1-ζ_N^3v=1.We were not able to prove that T_5-5-⟨ 5 ⟩ annihilates ϖ_r.Another evidence in favor of Conjecture <ref> is that the analogous conjecture when p divides the level has been proved by Fukaya and Kato <cit.>. Their methods are p-adic, so it is not clear how to generalize them in our setting. We end this section by recalling briefly a construction due to Goncharov, used by Fukaya and Kato. If u ∈ (𝐙/N𝐙)^×, one can defined a Siegel unit g_0, u/N∈𝒪(Y_1(N)_𝐙[ζ_N, 1/N])^×⊗_𝐙𝐙[1/N], where 𝒪(Y_1(N)_𝐙[ζ_N, 1/N])^× is the ring of global sections of the open modular curve Y_1(N) over 𝐙[ζ_N, 1/N]. We refer to <cit.> for its definition. The specialization of g_0, u/N at the cusp Γ_1(N)·∞ is 1-ζ_N^u. We define a map g : 𝐙[((𝐙/N𝐙)^×)^2/± 1] → K_2(𝒪(Y_1(N)_𝐙[ζ_N, 1/N])^×) ⊗_𝐙𝐙[1/N]by g([u,v]) = {g_0, u/N, g_0, v/N} where {·, ·} is the Steinberg symbol. Goncharov proved that an analogous map at level Γ(N) factors through the Manin relations <cit.>. His proof in fact shows that g factors through ξ_Γ_1(N). Thus, we get a mapΓ : H_1(X_1(N), C_Γ_1(N)^0, 𝐙) →K_2(𝒪(Y_1(N)_𝐙[ζ_N, 1/N])^×) ⊗_𝐙𝐙[1/N] .We expect that Γ commutes with the action of the Hecke operators T_n and ⟨ n ⟩ for n prime to N. This was proved by Fukaya and Kato when p divides the level <cit.>. If this is true, then Conjecture <ref> is true by specializing at the cusp Γ_1(N)·∞. This is the idea behind the proof of <cit.>. However, we were not able to prove that Γ is Hecke-equivariant. §.§ Construction of the element m_2^- under Conjecture <ref> In this section, we assume Conjecture <ref> but not yet that n(r,p) ≥ 2. We follow the notation of sections <ref>, <ref>, <ref> and <ref>.We first construct an explicit group homomorphism:ψ : I· H_+/I^2· H_+ → J_t·𝒦_t/J_t^2 ·𝒦_t . Recall the Eisenstein ideals Ĩ_0 and Ĩ_∞ defined in section <ref>, annihilating the cusps Γ_1^(p^t)(N) · 0 and Γ_1^(p^t)(N)·∞ respectively. Since we assume that Conjecture <ref> holds, ϖ^(p^t) induces a surjective morphism of Λ_t-modules φ: (H̃^(p^t))_+ →𝒦_t annihilating Ĩ_∞·(H̃^(p^t))_+. Let φ' be the restriction of φ to Ĩ_0 ·(H̃^(p^t))_+. We claim that the image of φ' is J_t ·𝒦_t. Since φ is surjective, it suffices to note that we have T_ℓ- ℓ·⟨ℓ⟩-1 ∈Ĩ_0 and T_ℓ - ⟨ℓ⟩ - ℓ∈Ĩ_∞, so(⟨ℓ⟩-1)· (ℓ-1) ∈Ĩ_0 + Ĩ_∞for all prime number ℓ different from N. Similarly, we see that φ' induces a surjective group homomorphismφ” : Ĩ_0 ·(H̃^(p^t))_+/(Ĩ_0^2 +J·Ĩ_0)·(H̃^(p^t))_+ → J_t ·𝒦_t/J_t^2·𝒦_t. By Proposition <ref> and Corollary <ref>, we have Ĩ_0 ·(H̃^(p^t))_+/(Ĩ_0^2 +J·Ĩ_0)·(H̃^(p^t))_+ =(H^(p^t))_+/(I_0 + J)·(H^(p^t))_+ ≃ I· H_+/I^2· H_+.Thus, we have a canonical group isomorphismψ' : I· H_+/I^2· H_+ Ĩ_0 ·(H̃^(p^t))_+/(Ĩ_0^2 +J·Ĩ_0)·(H̃^(p^t))_+.We then let ψ = φ”∘ψ' : I· H_+/I^2· H_+ → J_t·𝒦_t/J_t^2 ·𝒦_t. As in section <ref>, we abuse notation and denote J_r by J. If n(r,p) ≥ 2, the map ψ induces a surjective group homomorphism:ψ_r: I· (H_+/p^r· H_+)/I^2· (H_+/p^r· H_+) → J·𝒦_r/J^2·𝒦_r.The norm map yields a canonical surjective group homomorphism J_t ·𝒦_t/J_t^2 ·𝒦_t → J ·𝒦_r/J^2 ·𝒦_r. Thus, ψ gives a surjective group homomorphismψ_r : I· H_+ / I^2· H_+ → J ·𝒦_r/J^2 ·𝒦_r.To construct ψ_r, it suffices to show that ψ_r vanishes on the image of (p^r· H_+) ∩ (I· H_+) in I· H_+/I^2· H_+. It suffices to prove that(I^2· H_+ + (p^r· H_+) ∩ (I· H_+) ) /( I^2· H_+ + p^r· I·H_+) = 0 Since I ·(I^2· H_+ + (p^r· H_+) ∩ (I· H_+) )⊂( I^2· H_+ + p^r· I·H_+), it suffices to show (<ref>) after completion at I. Using the winding isomorphism <cit.>, it suffices to show that((p^r· I)∩ I^2 + I^3 )/( p^r· I^2 + I^3 ) ⊗_𝕋𝐓= 0.We have ( (p^r· I) ∩ I^2 ) ⊗_𝕋𝐓 = ( p^t· I + p^r· I^2 )⊗_𝕋𝐓. We have p^t· I + p^r· I^2 ⊂(p^r· I) ∩ I^2 since p^t ∈ I and r ≤ t. Let η be a generator of I·𝐓, and x ∈ (p^r· I)∩ I^2. We can write x = p^r ·η· u = η^2· v for some u,v ∈𝐓. Since η is not a zero divisor in 𝐓, we have p^r· u = η· v ∈ I. Let m∈𝐙 such that u-m ∈ I. We have p^r· m ∈ I ∩𝐙_p = p^t·𝐙_p. Thus, we have x ∈ p^t· I + p^r· I^2.Since n(r,p) ≥ 2, we have by Proposition <ref>, p^t· I ⊂ I^3+p^r· I^2. This ends the proof of the proposition. If n(r,p) ≥ 2, the group J·𝒦_r/J^2·𝒦_r is cyclic of order p^r by Theorem <ref>. As explained in Section <ref>, the map ψ_r gives the construction of the higher Eisenstein element m_2^-. §.§ Explicit computation of m_2^- In this section, we assume that Conjecture <ref> holds and that n(r,p) ≥ 2. We keep the notation the previous sections of Chapter <ref>. Recall that by intersection duality, we can consider m_2^- as a group homomorphismI · (H_+/p^r· H_+) / I^2· (H_+/p^r· H_+) →J·𝒦_r/J^2·𝒦_r.By Proposition <ref>, an element of I · H_+ is the image of an element of (H^(p^r))_+, which can be written as∑_[u,v] ∈( (𝐙/N𝐙)^×)^2/P_r'λ_[u,v]· (1+c)·ξ_Γ_1^(p^r)(N)([u,v])for some λ_[u,v]∈𝐙_p, with the boundary condition ∑_[u,v]λ_[u,v]·([v]-[u])=0 in Λ_r.Equivalently, we haveI· H_+ = {∑_x ∈ (𝐙/N𝐙)^×λ_x· (1+c)·ξ_Γ_0(N)(x), ∑_xλ_x ·log(x) ≡ 0(modulo p^t)} . An element ∑_x ∈(𝐙/N𝐙)^×λ_x · [x] ∈𝐙_p[(𝐙/N𝐙)^×] satisfies∑_x ∈(𝐙/N𝐙)^×λ_x ·log(x) ≡ 0(modulo p^t) if and only if it is in the subgroup generated by the square of the augmentation ideal of 𝐙_p[(𝐙/N𝐙)^×] and by [1]. Thus, any such element is a linear combination of elements of the form [x· y]-[x]-[y]. Therefore, m_2^- is determined by the values((1+c)·ξ_Γ_0(N)([x· y:1]-[x:1]-[y:1]))∙ m_2^-forall x,y ∈ (𝐙/N𝐙)^×.Assume that Conjecture <ref> holds. Assume that n(r,p) ≥ 2, that∑_k=1^N-1/2 k ·log(k) ≡ 0(modulop^r). Let x, y ∈ (𝐙/N𝐙)^×. Then we have the following equality in J·𝒦_r/J^2·𝒦_r:δ_r(( 2· (1+c)·ξ_Γ_0(N)([x· y:1]-[x:1] - [y:1]) ) ∙ m_2^-) = (1-ζ_N^x, 1-ζ_N)_r -(1-ζ_N^x, 1-ζ_N^y^-1)_r -(1-ζ_N^y^-1, 1-ζ_N)_rwhere (·, ·)_r : 𝐙[ζ_N, 1/Np]^××𝐙[ζ_N, 1/Np]^×→𝒦_r and δ_r : 𝐙/p^r𝐙J·𝒦_r/J^2·𝒦_r were defined in Section <ref>. Let π : (H̃^(p^r))_+ → H_+ be the canonical map. The group homomorphismφ : (H̃^(p^r))_+ →𝒦_tgives a surjective group homomorphismφ_r : (H̃^(p^r))_+ →𝒦_r/J·𝒦_r.Similarly,the group homomorphismφ' : Ĩ_0 ·(H̃^(p^r))_+ →𝒦_tgives a surjective group homomorphismφ_r' : Ĩ_0 ·(H̃^(p^r))_+ →J·𝒦_r/J^2·𝒦_r. For all x, y ∈ (𝐙/N𝐙)^×, we have inH_+:ξ_Γ_0(N)([x· y:1]-[x:1]-[y:1]) = π(ξ_Γ_1^(p^r)(N)([x,y^-1]-[x,1]+[y^-1,1])).Furthermore ξ_Γ_1^(p^r)(N)([x,y^-1]-[x,1]+[y^-1,1]) ∈(H^(p^r))_+ since∂( ξ_Γ_1^(p^r)(N)([x,y^-1]-[x,1]+[y^-1,1]))= [x]_Γ_1^(p^r)(N)^0-[y^-1]_Γ_1^(p^r)(N)^0- ([x]_Γ_1^(p^r)(N)^0-[1]_Γ_1^(p^r)(N)^0 )+ [y^-1]_Γ_1^(p^r)(N)^0-[1]_Γ_1^(p^r)(N)^0 = 0(the computation of ∂ in Section <ref>). Thus, to prove Theorem <ref> it suffices to prove that for all h ∈Ĩ_0 ·(H̃^(p^r))_+, we have in J·𝒦_r/J^2·𝒦_r:δ_r( 2 ·π(h) ∙ m_2^-) = -φ_r'(h). Let ℓ be a prime not dividing N. Note thatT_ℓ - ℓ·⟨ℓ⟩ -1 = T_ℓ-ℓ-⟨ℓ⟩ - (ℓ-1)· (⟨ℓ⟩-1) ∈ I_∞ -(ℓ-1)· (⟨ℓ⟩-1) .Since Conjecture <ref> is assumed to be true, for all u ∈(H̃^(p^r))_+ we have in J·𝒦_r/J^2·𝒦_r:φ_r'((T_ℓ-ℓ⟨ℓ⟩-1)(u)) = -(ℓ-1)· ([ℓ]-1) ·φ_r(u).By construction, for all [x,y] ∈ M_Γ_1^(p^r)(N)^0, we have in 𝐙/p^r𝐙: ι_r( φ_r((1+c)·ξ_Γ_1^(p^r)(N)([x,y])) )= log(x/y) = ( (1+c)·ξ_Γ_0(N)([x:y]))∙ m_1^- .Combining (<ref>) and (<ref>), we have in J·𝒦_r/J^2·𝒦_r:φ_r'((T_ℓ-ℓ⟨ℓ⟩-1)(u))= -δ_r(2·ℓ-1/2·log(ℓ)· (π(u) ∙ m_1^-)) =-δ_r(2· (T_ℓ-ℓ-1)(π(u)) ∙ m_2^- ) = -δ_r(2·π((T_ℓ-ℓ⟨ℓ⟩-1)(u)) ∙ m_2^- ). By Proposition <ref>, the elements (T_ℓ-ℓ⟨ℓ⟩-1)(u) span (H^(p^r))_+ when ℓ and u varies. This concludes the proof of (<ref>). § EVEN MODULAR SYMBOLSIn this chapter, unless explicitly stated, we allow p=2 and p=3. We keep the notation of chapters <ref>, <ref> and<ref>. We assume as usual that p divides the numerator of N-1/12. We let v be the p-adic valuation of N-1. We extend log to a group homomorphism (𝐙/N𝐙)^×→𝐙/p^v𝐙 (still abusively denoted by log).Recall that in Theorem <ref>, we defined an element m̃_0^+ ∈ H_1(X_0(N), , 𝐐)_+ (independent of the choice of p). We have, in fact, m̃_0^+ ∈ H_1(X_0(N), , 𝐙_p)_+ even if when p ∈{2,3} the formula defining F_0,p is not p-integral. We fix this choice of m̃_0^+ for the rest of the paper. Let r be an integer such that 1 ≤ r ≤ t. We denote by m_0^+, ..., m_n(r,p)^+ the higher Eisenstein elements in M_+/p^r· M_+, where m_0^+ is the image of m̃_0^+ in M_+/p^r· M_+.In this chapter, we give an explicit formula for m_1^+ modulo p^t if p ≥ 3, and a formula for m_1^+ modulo p^t-1 if t≥ 2 and p=2. In particular, we prove Theorem <ref>. The formula when p=3 (resp. p=2) is given in Theorem <ref> (resp. Theorem <ref>). §.§ Some results about the homology of X_0(N) and X_1(N) In this section, we gather some useful results about the homology of X_0(N) and X_1(N). Let n ≥ 1 be an integer. * The inclusion M_+ ↪ H_1(X_0(N), , 𝐙_p) gives a 𝕋̃-equivariant group isomorphismM_+/p^n· M_+H_1(X_0(N), , 𝐙/p^n𝐙)_+. * The surjection H_1(Y_0(N), 𝐙_p) ↠ M^- gives a 𝕋̃-equivariant group isomorphismH_1(Y_0(N), 𝐙/p^n𝐙)^-M^-/p^n· M^-. Point (ii) follows from point (i) by intersection duality. Thus, we only need to prove (i). The multiplication by p^n gives an exact sequence of 𝕋̃-modules0 → H_1(X_0(N), , 𝐙_p)H_1(X_0(N), , 𝐙_p) → H_1(X_0(N), , 𝐙/p^n𝐙) → 0 .There is an action of 𝐙/2𝐙 on each of the groups involved in (<ref>), given by the complex conjugation.Furthermore, (<ref>) is 𝐙/2𝐙-equivariant. The long exact sequence of cohomology associated to 𝐙/2𝐙 yields an exact sequence:0 → M_+M_+ → H_1(X_0(N), , 𝐙/p^r𝐙)_+ → H^1(𝐙/2𝐙, H_1(X_0(N), , 𝐙_p))[p^n].We have a 𝐙/2𝐙-equivariant exact sequence0 → H_1(X_0(N), 𝐙_p) → H_1(X_0(N), , 𝐙_p) →𝐙_p → 0,where the action of 𝐙/2𝐙 on 𝐙_p is trivial. The long exact sequence in cohomology yields an exact sequenceH^1(𝐙/2𝐙, H_1(X_0(N), 𝐙_p)) → H^1(𝐙/2𝐙, H_1(X_0(N), , 𝐙_p)) → H^1(𝐙/2𝐙,𝐙_p).We have H^1(𝐙/2𝐙, 𝐙_p)=0 and H^1(𝐙/2𝐙, H_1(X_0(N), 𝐙_p))=0 <cit.>. Thus, we have H^1(𝐙/2𝐙, H_1(X_0(N), , 𝐙_p)) = 0, which concludes the proof of Proposition <ref> by (<ref>).Thus, we will abuse notation and write H_1(X_0(N), , 𝐙/p^n𝐙)_+ (resp. H_1(Y_0(N),𝐙/p^n𝐙)^-) for M_+/p^n· M_+ (resp. M^-/p^n· M^-). Let π : X_1(N) → X_0(N) be the standard degeneracy map. This produces by pull-back and push-forward two maps π^* : H_1(X_0(N), , 𝐙) → H_1(X_1(N), , 𝐙)and π_* : H_1(X_1(N), , 𝐙) → H_1(X_0(N), , 𝐙).We will freely abuse notation and still denote the pull-back maps for different coefficient rings than 𝐙 by π^* and π_*. * For any integer n ≥ 1, the kernel of π^* : H_1(X_0(N), , 𝐙/p^n𝐙) → H_1(X_1(N), , 𝐙/p^n𝐙) is cyclic of order p^min(n,t), annihilated by the Eisenstein ideal Ĩ and by 1+c where c is the complex conjugation. If p≠ 3, a generator of this kernel is p^max(n-t,0)·ℰ_p^-, whereℰ_p^- := 1/3·∑_x ∈ℛ x ≁[1:1]log(x-1/x+1) ·ξ_Γ_0(N)(x) ∈ H_1(X_0(N), 𝐙/p^t𝐙).Here, ℛ is the set of equivalences classes in 𝐏^1(𝐙/N𝐙) for the equivalence relation [c:d] ∼ [-d:c].* If p ≥ 3, then for any integer n ≥ 1, the pull-back mapM_+/p^n· M_+ → H_1(X_1(N), , 𝐙/p^n𝐙)is injective.* For any integer n ≥ 1, the kernel of the pull-back mapM_+/2^n· M_+ → H_1(X_1(N), , 𝐙/2^n𝐙)is spanned by the reduction of 2^n-1·m̃_0^+ modulo 2^r.We prove (i). Let U = (𝐙/N𝐙)^×/μ_12, where μ_12 is the 12-torsion subgroup of (𝐙/N𝐙)^×. We have a commutative diagram whose rows are exact and whose vertical maps are surjective:Γ_1(N) [r][d]^γ↦{z_1, γ(z_1)} Γ_0(N) [r]^[ a b; c d ]↦ d[d]^γ↦{z_0, γ(z_0)}(𝐙/N𝐙)^×[d] [r] 0 H_1(Y_1(N), 𝐙) [r] H_1(Y_0(N), 𝐙) [r]^φU[r] 0where z_1 ∈ Y_1(N) (resp. z_0 ∈ Y_0(N)) is any fixed point. The complex conjugation acts on Γ_0(N) via [ a b; c d ]↦[ -10;01 ]^-1[ a b; c d ][ -10;01 ] = [a -b; -cd ]. Thus, the map φ is annihilated by 1+c. The map φ is also annihilated by Ĩ <cit.>. By intersection duality, we get an exact sequence0 →(U, 𝐙/p^n𝐙) → H_1(X_0(N), , 𝐙/p^n𝐙)H_1(X_1(N), , 𝐙/p^n𝐙).The intersection duality is Hecke equivariant and changes the sign for the complex conjugation. This proves the first assertion of (i). The map φ : H_1(Y_0(N), 𝐙) → U corresponds by intersection duality to an element ℰ^- of H_1(X_0(N), , U), which is a generator of the kernel of π^* : H_1(X_0(N), , U) → H_1(X_1(N), , U). The following general result, essentially due to Merel <cit.> and Rebolledo <cit.>, allows us to compute ℰ^- in terms of Manin symbols.Keep the notation of Section <ref>. Let A be an abelian group in which 3 is invertible.Let f : H_1(X_Γ, C_Γ, 𝐙) → A be a group homomorphism and f̂∈ H_1(Y_Γ, A) be the element corresponding to f by intersection duality. Let ℛ_Γ be the set of equivalence classes in Γ\_2(𝐙) for the equivalence relation Γ· g ∼Γ· g ·σ. The image of f̂ in H_1(X_Γ, A) is∑_Γ· g ∈ℛ_Γ f(ξ_Γ(Γ· g))·ξ_Γ(Γ· g) +1/3·∑_g·Γ∈Γ\_2(𝐙)(2· f(ξ_Γ(Γ· g·τ))+f(ξ_Γ(Γ· g·τ^2))) ·ξ_Γ(Γ· g) .We follow the notation of <cit.>. Note that Merel assumes that [ -10;0 -1 ]∈Γ, but he uses the coset Γ\_2(𝐙). Since we have Γ\_2(𝐙) = Γ\_2(𝐙), this assumption of Γ is not important. Let ℌ be the upper-half plane and π : ℌ∪𝐏^1(𝐐) → X_Γ be the canonical surjection. Let ρ = e^2π i/3 and δ be the geodesic path between i and ρ. Let R = π(_2(𝐙)·ρ) and I = π(_2(𝐙)· i). These sets are disjoints. If Γ· g ∈Γ\_2(𝐙), let ξ_Γ'(Γ· g) be the class of π(g·δ) in H_1(Y_Γ, R∪ I, 𝐙). Let f' : H_1(X_Γ - (R ∪ I), , 𝐙) → A be the composition f with the canonical map H_1(X_Γ - (R ∪ I), , 𝐙) → H_1(X_Γ, , 𝐙). By intersection duality, f' corresponds to an element f̂' ∈ H_1(Y_Γ, R∪ I, A). By <cit.>, we havef̂' = ∑_g·Γ∈Γ\_2(𝐙) f(ξ_Γ(Γ· g)) ·ξ_Γ'(Γ· g).We have f̂' ∈ H_1(Y_Γ, A) since f' factors through H_1(X_Γ, , 𝐙). Consider the image f̂” of f̂' in H_1(X_Γ, R∪ I, A). We have f̂”∈ H_1(X_Γ, A). Lemma <ref> follows from the following result. This is a slight generalization of <cit.> (Rebolledo's result assumes that 6 is invertible in A).Letx =∑_Γ· g ∈Γ\_2(𝐙)λ_Γ· g·ξ_Γ'(Γ· g) ∈ H_1(X_Γ, R∪ I, A).Assume that x ∈ H_1(X_Γ, A). We have, in H_1(X_Γ, , A):x = ∑_Γ· g ∈ℛ_Γλ_Γ· g·ξ_Γ(Γ· g)+ 1/3·∑_Γ· g ∈Γ\_2(𝐙) (2·λ_Γ· g·τ + λ_Γ· g·τ^2)·ξ_Γ(Γ· g).For simplicity, we write λ_g for λ_Γ· g. We have in H_1(X_Γ, ∪ R ∪ I, A):x =∑_Γ· g ∈Γ\_2(𝐙)λ_g·{g(i), g(ρ)} =∑_Γ· g ∈Γ\_2(𝐙)λ_g·{g(i), g(∞)}- ∑_Γ· g ∈Γ\_2(𝐙)λ_g·{g(ρ), g(∞)} = ∑_Γ· g ∈ℛ_Γ( λ_g·{g(i), g(∞)} + λ_g·σ·{g·σ(i), g·σ(∞)}) -1/3·∑_Γ· g ∈Γ\_2(𝐙)( λ_g ·{g(ρ), g(∞)} +λ_g·τ·{g·τ(ρ), g·τ(∞) }+λ_g·τ^2·{g·τ^2(ρ), g·τ^2(∞) }) = ∑_Γ· g ∈ℛ_Γ( λ_g·{g(i), g(∞)} + λ_g·σ·{g·σ(i), g·σ(∞)}) -1/3·∑_Γ· g ∈Γ\_2(𝐙)( λ_g ·{g(ρ), g(∞)} +λ_g·τ·{g·τ(ρ), g·τ(∞) }+λ_g·τ^2·{g·τ^2(ρ), g·τ^2(∞) }).Note that σ(i) = i, σ(∞) = 0, τ(ρ)=ρ and τ(∞)=0. Since the boundary of x is zero, we have λ_g ·σ = -λ_g and λ_g = -λ_g=-λ_g·τ -λ_g·τ^2 <cit.>. Thus, we have in H_1(X_Γ, ∪ R ∪ I, A):x =∑_Γ· g ∈ℛ_Γλ_g·( {g(i), g(∞)} -{g(i), g(0)}) -1/3·∑_Γ· g ∈Γ\_2(𝐙)( -(λ_g·τ +λ_g·τ^2) ·{g(ρ), g(∞)} +λ_g·τ·{g(ρ), g(0) }+λ_g·τ^2·{g(ρ), g·τ(0) })=∑_Γ· g ∈ℛ_Γλ_g·{g(0), g(∞)}-1/3·∑_Γ· g ∈Γ\_2(𝐙)λ_g·τ·{g(∞), g(0) }+λ_g·τ^2·{g(∞), g·τ(0) } .We have:{g(∞), g·τ(0)} = {g(∞), g·τ(∞)}+ {g·τ(∞), g·τ(0)} = -{g(0), g(∞)}- {g·τ(0),g·τ(∞) } .Thus, we have:x =∑_Γ· g ∈ℛ_Γλ_g·{g(0), g(∞)}+1/3·∑_Γ· g ∈Γ\_2(𝐙)λ_g·τ·{g(0), g(∞)} +1/3·∑_Γ· g ∈Γ\_2(𝐙)λ_g·τ^2·{g(0), g(∞)} +1/3·∑_Γ· g ∈Γ\_2(𝐙)λ_g·τ·{g(0), g(∞)} .This concludes the proof of Lemma <ref>.This concludes the proof of Lemma <ref>.We identify H_1(X_0(N), , U) with H_1(X_0(N), , 𝐙) ⊗_𝐙 U. The homomorphism φ : H_1(Y_0(N), 𝐙) → U induces a map φ' : H_1(X_0(N), 𝐙) → U. It is given byφ'(ξ_Γ_0(N)(Γ_0(N) ·[ a b; c d ])) = c·d^-1 ,where (c,N)=(d,N)=1 and if x ∈𝐙 is prime to N, then x is the image of x in U. By Lemma <ref>, we have in H_1(X_0(N), , 𝐙) ⊗_𝐙 U:ℰ_-= ∑_x ∈ℛ x ≁∞ξ_Γ_0(N)(x) ⊗x+ 1/3·∑_x ∈𝐏^1(𝐙/N𝐙)x ≠ 0, ∞ξ_Γ_0(N)(x) ⊗-1/x· (x+1) = 1/3·∑_x ∈ℛ x ≁∞ξ_Γ_0(N)(x) ⊗x-1/x+1 .This concludes the proof of point (i).Point (ii) is an immediate consequence of point (i). We now prove point (iii). We have φ((1+c)· H_1(Y_0(N), 𝐙)) = U^2, where U^2 is the subgroup of squares in U. We have an exact sequence:H_1(Y_1(N), 𝐙) → H_1(Y_0(N), 𝐙)/(1+c)· H_1(Y_0(N), 𝐙) → U/U^2 → 0.Let H_1(Y_0(N), 𝐙)^- (resp. H_1(Y_1(N), 𝐙)^-) be the largest torsion-free quotient of H_1(Y_0(N), 𝐙) (resp. H_1(Y_1(N), 𝐙)) annihilated by 1+c.The kernel of the map H_1(Y_0(N), 𝐙) → H_1(Y_0(N), 𝐙)^- is (1+c)· H_1(Y_0(N), 𝐙) where c is the complex conjugation. The kernel of the map H_1(X_0(N), 𝐙) → H_1(X_0(N), 𝐙)^- is (1+c)· H_1(Y_0(N), 𝐙) by <cit.>. We have an exact sequence 0 → H_1(X_0(N), 𝐙) → H_1(X_0(N), , 𝐙) →𝐙→ 0. Since the cusps are fixedby the complex conjugation, this gives on the plus subspaces an exact sequence:0 → H_1(X_0(N), 𝐙)_+ → H_1(X_0(N), , 𝐙)_+→𝐙→ 0. By intersection duality, we get a commutative diagram whose rows are exact:0 [r]𝐙[r][d]H^1(Y_0(N), 𝐙)[r][d]H^1(X_0(N),𝐙) [r][d] 0 0 [r]𝐙[r] H^1(Y_0(N), 𝐙)^- [r] H^1(X_0(N),𝐙)^- [r] 0 .By the snake lemma, the kernel of the map H_1(Y_0(N), 𝐙) → H_1(Y_0(N), 𝐙)^- is (1+c) · H_1(Y_0(N), 𝐙). By Lemma <ref>, we have an exact sequence:H_1(Y_1(N), 𝐙)^- → H_1(Y_0(N), 𝐙)^- → U/U^2 → 0.By intersection duality, we have an exact sequence0 →(U/U^2, 𝐙/2^n𝐙) → M_+/2^n· M_+ → H_1(X_1(N), , 𝐙_2)_+⊗_𝐙_2𝐙/2^n𝐙 .Note that H_1(X_1(N), , 𝐙_2)_+⊗_𝐙_2𝐙/2^n𝐙 is a subgroup of H_1(X_1(N), , 𝐙_2)⊗_𝐙_2𝐙/2^n𝐙, which we identify with H_1(X_1(N), , 𝐙/2^n𝐙). The map φ : H_1(Y_0(N), 𝐙) → U is annihilated by the Eisenstein ideal. The map H_1(Y_0(N), 𝐙)^- → U/U^2 is also annihilated by the Eisenstein ideal. Thus, the image of (U/U^2, 𝐙/2^n𝐙) in M_+/2^n· M_+ has order 2 and is Eisenstein, so is generated by 2^n-1·m̃_0^+ modulo 2^n. Proposition <ref> (iii) is another way to express that the Shimura subgroup and the cuspidal subgroup of J_0(N) intersect at a point of order 2 <cit.>. §.§ The method used to compute m_1^+ In this section, we give the idea that lead us to the proof of Theorems <ref> and <ref>. This section is not necessary for the proof, and is mainly for the convenience of the reader. We begin by describing the first higher Eisenstein element in the space of modular forms. Following <cit.>, let 𝒩 be the 𝐙-module of weight 2 modular forms f of level Γ_0(N) and weight 2 for which a_n(f)∈𝐙 if n≥ 1 and a_0(f) ∈𝐐, where ∑_n ≥ 0 a_n(f) · q^n is the q-expansion of f at the cusp ∞. We let ℳ = 𝒩⊗_𝐙𝐙_p. By <cit.> and <cit.>, the 𝐓̃-module ℳ⊗_𝕋̃𝐓̃ is free of rank one. By <cit.>, the pairing ℳ×𝕋̃→𝐐_p given by (f,T) ↦ a_1(T(f)) takes values in 𝐙_p and induces a canonical 𝕋̃-equivariant isomorphism ℳ_𝐙_p(𝕋̃, 𝐙_p).Let T_0 ∈𝕋̃ be such that a_1(T_0(f)) = 24/ν· a_0(f)for all f ∈𝒩. We have (𝕋̃→𝕋) = 𝐙_p · T_0. By <cit.>, we have T_0 - N-1/ν∈Ĩ .This is coherent with the choice of section <ref>.Let f̃_0 ∈ℳ be the modular form whose q-expansion at the cusp ∞ isN-1/24+∑_n≥ 1(∑_d| n (d,N)=1 d)· q^n.Let f_0, ..., f_n(t,p) be the higher Eisenstein elements of ℳ/p^t·ℳ.We normalize f_0 to be the image of f̃_0 in ℳ/p^t·ℳ. The image of f_1 in (ℳ/p^t·ℳ)/𝐙· f_0 is uniquely determined. There is an element f_1' in ℳ/p^t·ℳ such that for all prime number ℓ not dividing N we havea_ℓ(f_1) = ℓ-1/2·log(ℓ)(with the usual interpretation if ℓ=p=2, chapter <ref>). The images of f_1' and f_1 in ( ℳ/p^t·ℳ)/𝐙· f_0 coincide. Let ϕ : 𝕋̃→𝐙_p be the 𝐙_p-linear homomorphism corresponding to f̃_0, given byϕ(T_n) = ∑_d | n (d,N)=1^n dand ϕ(T_0)=N-1/ν. If T ∈𝕋̃, we have by definition T-ϕ(T) ∈Ĩ. The winding homomorphism of Mazur yields a map α : Ĩ/Ĩ^2 →𝐙/p^t𝐙 sending T_ℓ-ϕ(T_ℓ) to ℓ-1/2·log(ℓ) for all prime ℓ not dividing N.The map ψ : 𝕋̃→𝐙/p^t𝐙 given by ψ(T) = α(T-ϕ(T)) is a group morphism, so defines a modular form f_1' with a_n(f_1') = ψ(T_n) for all integers n ≥ 0. In particular, we have a_ℓ(f_1') = ℓ-1/2·log(ℓ). It is straightforward to check that (T_ℓ-ℓ-1)(f_1') = ℓ-1/2·log(ℓ) · f_0 for all prime ℓ not dividing N. This concludes the proof of Proposition <ref>. Theorem <ref> (which will be proved in chapter <ref>) shows that we havea_0(f_1') = a_0(f_1) =1/6·∑_k=1^N-1/2 k ·log(k) .The key fact is that f_1' is related to an Eisenstein series of level Γ_1(N) and weight 2. If ϵ : (𝐙/N𝐙)^×→𝐂^× is an even character, there is an Eisenstein series E_ϵ, 1∈ M_2(Γ_1(N), ϵ) whose q-expansion at the cusp ∞ is:2·∑_n≥ 1 ( ∑_d| nϵ(d)·n/d)q^n(for instance <cit.>). Similarly, there is an Eisenstein series E_1, ϵ∈ M_2(Γ_1(N), ϵ) whose q-expansion at the cusp ∞ is:L(-1, ϵ) + 2·∑_n≥ 1 ( ∑_d| nϵ(d)· d )q^n .Let ϵ : (𝐙/N𝐙)^×→𝐂^× be the even character defined byϵ(x) = e^2iπ·log(x)/p^t.We now defineE = E_1, ϵ-τ(ϵ)· E_ϵ^-1,1/4where τ(ϵ)= ∑_a=1^N-1ϵ(a) · e^2· i ·π· a/N. The key observation is that the q-expansion of E is f̃_0 + π· f_1' + O(π^2) where π = 1-ζ_p^t is a uniformizer of 𝐙_p^unr[ζ_p^t]. Thus, there should be a way to compute m_1^+ using Eisenstein elements in H_1(X_1(N), , 𝐂)_+. Formulas for all the Eisenstein elements of level Γ(N) have recently been given in <cit.>, where Γ(N) ⊂_2(𝐙) is the subgroup of matrices congruent to the identity modulo N. §.§ Eisenstein elements of level in H_1(X_1(N), , 𝐂)Recall that we have defined _1: 𝐑→𝐑 by _1(x) = x-⌊ x ⌋-1/2if x ∉𝐙 and _1(x)=0 else, where ⌊ x ⌋ is the integer part of x. Following <cit.>, let D_2: 𝐑→𝐑 be given by D_2(x) = 2·( _1(x) - _1(x + 1/2)). It is a periodic function with period 1 such that D_2(0) = D_2(1/2)=0, D_2(x) = -1 if x ∈ ]0, 1/2[ and D_2(x) = 1 if x ∈ ]1/2, 1[. For the convenience of the reader, we state the main result of <cit.>. As in section <ref>, letξ_Γ(N) : 𝐙[Γ(N) \_2(𝐙)] → H_1(X(N), , 𝐙)be the Manin surjective map. Since N is odd, we have canonical identifications_2(𝐙/N𝐙)/±1 ≃Γ(N) \_2(𝐙) ≃±Γ(2N) \Γ(2). Let F : (𝐙/N𝐙)^2 →𝐂 be defined byF(x,y) = ∑_(s_1,s_2) ∈ (𝐙/2N𝐙)^2 e^2i π· ( s_1(x'+y') +s_2(x'-y'))/2N·_1(s_1/2N) _1(s_2/2N)where (x', y') ∈ (𝐙/2N𝐙)^2 is a lift of (x,y) such that x'+y' ≡ 1(modulo 2). We have F(x,y)=F(-x,-y).If P ∈ (𝐙/N𝐙)^2, let ℰ̃_P = ∑_γ∈_2(𝐙/N𝐙)/±1 F(γ^-1P) ·ξ_Γ(N)(γ) ∈ H_1(X(N), , 𝐂).We have ℰ̃_P = ℰ̃_-P. Let ℰ_P be the image of ℰ̃_P to H_1(X_1(N), , 𝐂).The following result is an easy consequence of the work of Debargha and Merel (<cit.>). Let ℓ be a prime not dividing N. For all x ∈ (𝐙/N𝐙)^×, we have:T_ℓ(ℰ_(x,0)) = ℓ·ℰ_(x,0) + ℰ_(ℓ^-1x, 0)andT_ℓ(ℰ_(0,x)) = ℓ·ℰ_(0,ℓ x) + ℰ_(0,x)where T_ℓ is the ℓth Hecke operator. Recall the notation of section <ref> for the cusps of X_1(N). For all prime ℓ not dividing N, the Hecke operator T_ℓ - ℓ-⟨ℓ⟩ (resp. T_ℓ - ℓ⟨ℓ⟩ - 1) annihilates C_Γ_1(N)^∞ (resp. C_Γ_1(N)^0), where ⟨ℓ⟩ is the ℓth diamond operator. By definition we have ⟨ℓ⟩· [x]_Γ_1(N)^∞ = [ℓ· x]_Γ_1(N)^∞ and ⟨ℓ⟩· [x]_Γ_1(N)^0 = [ℓ· x]_Γ_1(N)^0.Let E_N be the 𝐂-vector space generated by the elements ℰ_(x,0) and ℰ_(0,x) for all x ∈ (𝐙/N𝐙)^×/± 1. Let ∂_N : H_1(X_1(N), , 𝐂) →𝐂[C_Γ_1(N)]^0 be the boundary map. By <cit.>, the restriction of ∂_N to E_N gives a Hecke-equivariant isomorphism E_N 𝐂[C_Γ_1(N)]^0. By <cit.>, for all x ∈ (𝐙/N𝐙)^×/± 1 we have∂_N(ℰ_(x,0)) = 2N·(∑_μ∈ (𝐙/N𝐙)^× (F(μ· (1,0))+1/4)· [(μ· x)^-1]_Γ_1(N)^∞) - N/2·(∑_c ∈ C_Γ_1(N) [c] )and∂_N(ℰ_(0,x)) = 2N·(∑_μ∈ (𝐙/N𝐙)^× (F(μ· (1,0))+1/4)· [μ· x]_Γ_1(N)^0) - N/2·(∑_c ∈ C_Γ_1(N) [c] ).Thus, for all prime ℓ not dividing N we have∂_N( ⟨ℓ⟩ℰ_(x,0)) =⟨ℓ⟩∂_N(ℰ_(x,0)) = ∂_N(ℰ_(ℓ^-1x,0)).By injectivity of ∂_N, we have ⟨ℓ⟩ℰ_(x,0) = ℰ_(ℓ^-1x,0). Similarly, we have (T_ℓ- ℓ - ⟨ℓ⟩)(ℰ_(x,0)) = 0 since it is true after applying ∂_N. This proves the first relation of Theorem <ref>. The second relation is proved in a similar way. The counterparts of the modular forms E_ϵ,1 and E_1, ϵ of section <ref> are the following modular symbolsℰ_∞ = ∑_x ∈ (𝐙/N𝐙)^×/± 1[x] ·ℰ_(x,0)∈ H_1(X_1(N), , 𝐂[(𝐙/N𝐙)^×/± 1])andℰ_0 = ∑_x ∈ (𝐙/N𝐙)^×/± 1[x]^-1·ℰ_(0,x)∈ H_1(X_1(N), , 𝐂[(𝐙/N𝐙)^×/± 1]). As an immediate application of Theorem <ref>, for all prime ℓ not dividing N we have(T_ℓ-ℓ-[ℓ])(ℰ_∞)=0and(T_ℓ-ℓ· [ℓ]-1)(ℰ_0)=0. Letℰ' =π^*(m̃_0^+) ∈ H_1(X_1(N), , 𝐙_p).Recall that there is a canonical bijection Γ_1(N) \_2(𝐙)(𝐙/N𝐙)^2\{(0,0)}/± 1 (Section <ref>), and that we denote by [c,d] the class of (c,d) in (𝐙/N𝐙)^2\{(0,0)}/± 1 for any (c,d) ∈ (𝐙/N𝐙)^2\{(0,0)}. We sometimes abusively view [c,d] as an element of Γ_1(N) \_2(𝐙). We have 6·ℰ'= ∑_[c,d] ∈ (𝐙/N𝐙)^2\{(0,0)}/± 1( ∑_(s_1, s_2) ∈ (𝐙/2N𝐙)^2s_1(d-c)+s_2(d+c) ≡ 0(modulo N) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ) ·ξ_Γ_1(N)([c,d]) = - ∑_[c,d] ∈ (𝐙/N𝐙)^2\{(0,0)}/± 1( ∑_(s_1, s_2) ∈ (𝐙/2N𝐙)^2s_1(d-c)+s_2(d+c) ≢0(modulo N) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N))·ξ_Γ_1(N)([c,d]).The second equality follows from the first equality and from the identity∑_(s_1, s_2) ∈ (𝐙/2N𝐙)^2(-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) = (∑_s ∈ (𝐙/2N𝐙)^2 (-1)^s ·_1(s/2N))^2 = 0.Let (t_1,t_2) ∈ (𝐙/N𝐙)^2. Then ∑_(s_1,s_2) ∈(𝐙/N𝐙)^2(s_1,s_2) ≡ (t_1, t_2)(modulo N) (-1)^s_1+s_2_1(s_1/2N) _1(s_2/2N) = 1/4· D_2(t_1/N) D_2(t_2/N).Thus, we have∑_[c,d] ∈ (𝐙/N𝐙)^2\{(0,0)}/± 1( ∑_(s_1, s_2) ∈ (𝐙/2N𝐙)^2s_1(d-c)+s_2(d+c) ≡ 0(modulo N) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ) ·ξ_Γ_1(N)([c,d]) = ∑_[c,d] ∈ (𝐙/N𝐙)^2\{(0,0)}/± 1(∑_(t_1,t_2)∈ (𝐙/N𝐙)^2(t_1:t_2)=(c+d:c-d) ∈𝐏^1(𝐙/N𝐙)1/4· D_2(t_1/N)D_2(t_2/N) ) ·ξ_Γ_1(N)([c,d]).Note that for all [u:v] ∈𝐏^1(𝐙/N𝐙) = Γ_0(N) \_2(𝐙), we haveπ^*(ξ_Γ_0(N)([u:v])) = ∑_[c,d] ∈ (𝐙/N𝐙)^2\{(0,0)}/± 1[c:d] = [u:v]ξ_Γ_1(N)([c,d]).The first equality of the lemma then follows from the above computation Theorem <ref>. Define two maps G_∞, G_0 : Γ_1(N) \_2(𝐙) →𝐙[1/2N][(𝐙/N𝐙)^×/±1] as follows. Let γ=[ a b; c d ]∈_2(𝐙). We defineG_∞(Γ_1(N) ·γ) = ∑_(s_1, s_2) ∈ (𝐙/2N𝐙)^2(d-c)s_1+(d+c)s_2 ≢0(modulo N) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) · [(d-c)s_1+(d+c)s_2]^-1andG_0(Γ_1(N) ·γ) = ∑_(s_1, s_2) ∈ (𝐙/2N𝐙)^2(d-c)s_1+(d+c)s_2 ≡ 0(modulo N) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) · [(b-a)s_1+(b+a)s_2].Letℰ_∞' = ∑_Γ_1(N) ·γ∈Γ_1(N) \_2(𝐙) G_∞(Γ_1(N) ·γ) ·ξ_Γ_1(N)(Γ_1(N) ·γ) ∈ H_1(X_1(N), , 𝐙[1/2N][(𝐙/N𝐙)^×/± 1])andℰ_0' = ∑_Γ_1(N) ·γ∈Γ_1(N) \_2(𝐙) G_0(Γ_1(N) ·γ) ·ξ_Γ_1(N)(Γ_1(N) ·γ)∈ H_1(X_1(N), , 𝐙[1/2N][(𝐙/N𝐙)^×/± 1]).For all prime ℓ not dividing N, we have(T_ℓ-ℓ-[ℓ])(ℰ_∞')=0and(T_ℓ-ℓ· [ℓ]-1)(ℰ_0')=0.Consider the following two elements of 𝐂[(𝐙/N𝐙)^×/± 1]:𝒢 = ∑_x ∈ (𝐙/N𝐙)^×e^2iπ x/N· [x]andδ = ∑_x ∈ (𝐙/N𝐙)^× [x]. An easy computation shows that we have, in H_1(X_1(N), , 𝐂[(𝐙/N𝐙)^×/± 1]):2/N·ℰ_∞ =∑_[c,d] ∈ (𝐙/N𝐙)^2 \{(0,0)} / ±1ξ_Γ_1(N)([c,d]) ·∑_x ∈ (𝐙/N𝐙)^× (s_1,s_2) ∈ (𝐙/2N𝐙)^2 (-1)^s_1+s_2·_1(s_1/2N)_1(s_2/2N)[2x]· e^2iπ x((d-c)s_1+(d+c)s_2)/N= 6 ·δ·ℰ'+[2]·𝒢·ℰ_∞'. The last equality follows from Lemma <ref>. By (<ref>), the operator T_ℓ-ℓ-[ℓ] annihilates ℰ_∞. Furthermore, T_ℓ-ℓ-1 annihilates ℰ' and [ℓ]-1 annihilates δ, so T_ℓ-ℓ-[ℓ] annihilates δ·ℰ'. Thus, T_ℓ-ℓ - [ℓ] annihilates 𝒢·ℰ_∞'.The element 𝒢 is not a zero divisor of 𝐂[(𝐙/N𝐙)^×/± 1]. It suffices to prove that if α : (𝐙/N𝐙)^×→𝐂^× is any character such that α(-1)=1, then we have ∑_x ∈ (𝐙/N𝐙)^× e^2i π x/N·α(x) ≠ 0. This is a well-known fact on Gauss sums. By Lemma <ref>, the operator T_ℓ-ℓ - [ℓ] annihilates ℰ_∞', as wanted. The proof that T_ℓ-ℓ· [ℓ]-1 annihilates ℰ_0' is similar. Let 𝒰 = ℰ_0'+ℰ_∞'/2∈ H_1(X_1(N), , 𝐙[1/2N][(𝐙/N𝐙)^×/± 1]). This is the modular symbol counterpart of (<ref>). Let J ⊂𝐙[(𝐙/N𝐙)^×/± 1] be the augmentation ideal. By Lemma <ref>, we have 𝒰∈ J· H_1(X_1(N), , 𝐙[1/2N][(𝐙/N𝐙)^×/± 1]). Lemma <ref> shows that we have(T_ℓ-ℓ-1)(𝒰)=([ℓ]-1)·ℰ_∞' + ℓ·ℰ_0'/2This is the fundamental equality which allow us to compute m_1^+. We will study the cases p≥ 5, p=3 and p=2 separately. §.§ The case p ≥ 5The following theorem is a generalization of Theorem <ref> modulo p^t. Assume that p≥ 5. Let F_1,p : 𝐏^1(𝐙/N𝐙) →𝐙/p^t𝐙 be defined as follows. Let [c:d] ∈𝐏^1(𝐙/N𝐙). If [c:d] ≠ [1:1], let12 · F_1,p([c:d])=∑_(s_1,s_2) ∈ (𝐙/2N𝐙)^2(d-c)s_1+(d+c)s_2 ≡ 0(moduloN) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log(s_2/d-c)-∑_(s_1,s_2) ∈ (𝐙/2N𝐙)^2(d-c)s_1+(d+c)s_2 ≢0(moduloN) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log((d-c)s_1+(d+c)s_2)).This is independent of the choice of c and d. Let F_1,p([1:1]) = 0. We have: m_1^+ ≡∑_x ∈𝐏^1(𝐙/N𝐙) F_1,p(x) ·ξ_Γ_0(N)(x)in (M_+/p^t· M_+)/𝐙· m_0^+Let β : J/J^2 →𝐙/p^t𝐙 be given by [x]-1 ↦log(x) for x ∈ (𝐙/N𝐙)^×. This induces a map β_* : J· H_1(X_1(N), , 𝐙[1/2N][(𝐙/N𝐙)^×/± 1]) → H_1(X_1(N), , 𝐙/p^t𝐙). By (<ref>) and Lemma <ref>, we have:(T_ℓ-ℓ-1)(β_*(𝒰)) ≡6 ·ℓ-1/2·log(ℓ) ·ℰ'(modulo p^v) .Let F_1,p' : 𝐏^1(𝐙/N𝐙) →𝐙/p^t𝐙 be defined by12 · F_1,p'(x)= ∑_(s_1,s_2) ∈ (𝐙/2N𝐙)^2(d-c)s_1+(d+c)s_2 ≡ 0(moduloN) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log((b-a)s_1+(b+a)s_2)- ∑_(s_1,s_2) ∈ (𝐙/2N𝐙)^2(d-c)s_1+(d+c)s_2 ≢0(moduloN) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log((d-c)s_1+(d+c)s_2))where as above a and b are such that [ a b; c d ]∈_2(𝐙/N𝐙). This expression does not depend on the choice of c and d by Lemma <ref>. This also does not depend on the choice of a and b. For all [c:d] ∈𝐏^1(𝐙/N𝐙), we have F_1,p'([-c:d]) = F_1,p'([c:d]). Thus, we have ∑_x ∈𝐏^1(𝐙/N𝐙) F_1,p'(x)·ξ_Γ_0(N)(x) ∈ H_1(X_0(N), , 𝐙/p^t𝐙)_+. The element 1/6·β_*(𝒰) is the image of ∑_x ∈𝐏^1(𝐙/N𝐙) F_1,p'(x)·ξ_Γ_0(N)(x) via the pull-back map H_1(X_0(N), , 𝐙/p^t𝐙)_+ → H_1(X_1(N), , 𝐙/p^t𝐙)_+. By (<ref>) and Proposition <ref> (ii), for any prime ℓ not dividing N we have in H_1(X_0(N), , 𝐙/p^t𝐙)_+:(T_ℓ-ℓ-1)(∑_x ∈𝐏^1(𝐙/N𝐙) F_1,p'(x)·ξ_Γ_0(N)(x) ) = ℓ-1/2·log(ℓ) · m_0^+. If [ a b; c d ]∈_2(𝐙) and (s_1, s_2) ∈ (𝐙/2N𝐙)^2 are such that (d-c)s_1 + (d+c)s_2 ≡ 0(modulo N) and d ≢c(modulo N), then we have (b-a)s_1+(b+a)s_2 ≡2/d-c· s_2 (modulo N). By Lemma <ref>, we have:12·∑_x ∈𝐏^1(𝐙/N𝐙) F_1,p(x)·ξ_Γ_0(N)(x) = 12·∑_x ∈𝐏^1(𝐙/N𝐙) F_1,p'(x)·ξ_Γ_0(N)(x) - 6·log(2)· m_0^+.By (<ref>), we have:(T_ℓ-ℓ-1)(∑_x ∈𝐏^1(𝐙/N𝐙) F_1,p(x)·ξ_Γ_0(N)(x) ) = ℓ-1/2·log(ℓ) · m_0^+.This concludes the proof of Theorem <ref>.§.§ A few identities in (𝐙/N𝐙)^× and in((𝐙/N𝐙)^×)^⊗ 2In this section, we establish a few identities which will be useful to determine m_1^+ when p ∈{2,3} and also in chapter <ref>. We do not impose any restriction on p. If i is a non-negative integer, we letℱ_i = ∑_k=1^N-1/2log(k)^i ∈𝐙/p^v𝐙 .We have ℱ_0 = ℱ_1=0 if p>2 and 2·ℱ_0 = 4·ℱ_1=0 if p=2. We have ℱ_2 = 0 if p>3, 3·ℱ_2 = 0 if p=3 and 4·ℱ_2 = 0 if p=2. If 0 ≤ i ≤ 2, we easily check that ℱ_i = ∑_k=N+1/2^N-1log(k)^i (we use the fact that N-1/2 is even if p=2). We have, in 𝐙/p^v𝐙:4·∑_k=1^N-1/2 k ·log(k) = -3·∑_k=1^N-1 k^2 ·log(k) - log(2)·N-1/6 - ℱ_1.Let f : 𝐑→𝐑 be defined by f(x) = (x-⌊ x ⌋)^2 and let g : 𝐑→𝐑 given by g(x) = f(2x) - 4f(x). If x ∈ [0, 1/2[, we have g(x) = 0 and if x ∈ [1/2, 1[, we have g(x) = -4x+1. We thus have, in 𝐙/p^v𝐙:∑_k ∈ (𝐙/N𝐙)^× g(k/N) ·log(k)= -4 ·∑_k=N+1/2^N-1 k ·log(k) + log((-1)^N-1/2·(N-1/2)! ) = -4 ·∑_k=1^N-1/2(N-k) ·log(-k) + log((-1)^N-1/2·(N-1/2)! )=4 ·∑_k=1^N-1/2 k ·log(k) + ℱ_1.On the other hand, we have in 𝐙/p^v𝐙:∑_k ∈ (𝐙/N𝐙)^× g(k/N) ·log(k) = ∑_k ∈ (𝐙/N𝐙)^×( f(2k/N)- 4· f(k/N)) ·log(k) = -3·∑_k ∈ (𝐙/N𝐙)^× f(k/N) ·log(k) - log(2)·∑_k ∈ (𝐙/N𝐙)^× f(k/N) = -3 ·∑_k=1^N-1k^2·log(k) - log(2) ·N-1/6 .We have the following variant of lemma <ref>.We have, in 𝐙/p^v𝐙:4·∑_k=1^N-1/2 k ·log(k)^2 =-3 ·∑_k=1^N-1 k^2 ·log(k)^2 + log(2)^2 ·N-1/6 - 2 ·log(2) ·∑_k=1^N-1 k^2·log(k) + 3·ℱ_2 .Let f,g : 𝐑→𝐑 be as in the proof of Lemma <ref>. We have:∑_k ∈ (𝐙/N𝐙)^× g(k/N) ·log(k)^2= -4·∑_k=N+1/2^N-1 k ·log(k)^2 + ℱ_2 = -4 ·∑_k=1^N-1/2 (N-k) ·log(-k)^2 + ℱ_2 = 4·∑_k=1^N-1/2 k ·log(k)^2- 3·ℱ_2 .On the other hand, we have in 𝐙/p^v𝐙: ∑_k ∈ (𝐙/N𝐙)^× g(k/N) ·log(k)^2 = ∑_k ∈ (𝐙/N𝐙)^×( f(2k/N)- 4· f(k/N)) ·log(k)^2=∑_k ∈ (𝐙/N𝐙)^× (-4)· f(k/N) ·log(k)^2 + f(2k/N)·(log(2k)-log(2))^2= -3 ∑_k ∈ (𝐙/N𝐙)^× f(k/N) ·log(k)^2 +log(2)^2·∑_k ∈ (𝐙/N𝐙)^× f(2k/N)- 2 ·log(2) ·∑_k ∈ (𝐙/N𝐙)^× f(2k/N) ·log(2k)= -3 ·∑_k=1^N-1 k^2 ·log(k)^2 + log(2)^2 ·N-1/6 - 2 ·log(2) ·∑_k=1^N-1 k^2·log(k). We have, in 𝐙/p^v𝐙:∑_t_1,t_2=1 t_1≠ t_2^N-1/2log(t_1-t_2) = -2 ·∑_k=1^N-1/2k ·log(k)and∑_t_1, t_2=1^N-1/2log(t_1+t_2) = 2·∑_k=1^N-1/2k ·log(k) -ℱ_1.We first compute S_1 := ∑_t_1,t_2=1 t_1≠ t_2^N-1/2log(t_1-t_2) ∈𝐙/p^v𝐙. When t_1 and t_2 vary in {1, ..., N-1/2}, the quantity t_1-t_2 varies in X:={-N-1/2+1, ..., N-1/2-1}. If k≠ 0 ∈ X, then the number of such t_1 and t_2 such that k = t_1-t_2 is min(N-1/2-k, N-1/2)-max(1-k,1)+1. If 1 ≤ k ≤N-1/2-1, this number is N-1/2-k. If -N-1/2+1 ≤ k ≤ -1, this number is N-1/2+k. Thus, we haveS_1=∑_k=-N-1/2^-1(k+N-1/2)·log(k) + ∑_k=1^N-1/2(-k+N-1/2)·log(k) =2·∑_k=1^N-1/2(-k+N-1/2)·log(k)= -2 ·∑_k=1^N-1/2k ·log(k).The proof of the second equality is similar and is left to the reader.In a similar way, we have the following result (whose proof is left to the reader).We have, in 𝐙/p^v𝐙:∑_t_1,t_2=1 t_1≠ t_2^N-1/2log(t_1-t_2)^2 = -2 ·∑_k=1^N-1/2k ·log(k)^2and∑_t_1, t_2=1^N-1/2log(t_1+t_2)^2 = 2·∑_k=1^N-1/2k ·log(k)^2 -ℱ_2 . We have, in 𝐙/p^v𝐙:∑_(t_1, t_2) ∈ (𝐙/N𝐙)^2 t_1 ≠ t_2 D_2(t_1/N)· D_2(t_2/N)·log(t_1-t_2)= -∑_(t_1, t_2) ∈ (𝐙/N𝐙)^2 t_1 ≠ -t_2 D_2(t_1/N)· D_2(t_2/N)·log(t_1+t_2) = -8 ·∑_k=1^N-1/2 k·log(k) + 2·ℱ_1= 6·∑_k=1^N-1 k^2·log(k) + log(2) ·N-1/3 .The first equality of Lemma <ref> follows from the change of variable t_2 ↦ -t_2, using D_2(-x) = -D_2(x) for all x ∈𝐑. We have, in 𝐙/p^v𝐙:∑_(t_1, t_2) ∈ (𝐙/N𝐙)^2 t_1 ≠ t_2 D_2(t_1/N)D_2(t_2/N)log(t_1-t_2)= 2 ∑_t_1,t_2=1 t_1≠ t_2^N-1/2log(t_1-t_2)- 2∑_t_1,t_2=1^N-1/2log(t_1+t_2) = -8∑_k=1^N-1/2k ·log(k) +2·ℱ_1 .where in the last equality, we have used Lemma <ref>. This shows the second equality of Lemma <ref>. The third equality follows from Lemma <ref> and 4·ℱ_1=0. Similarly, using Lemmas <ref> and <ref> we get the following result (which will be used in chapter 6).We have, in 𝐙/p^v𝐙:∑_(t_1, t_2) ∈ (𝐙/N𝐙)^2 t_1 ≠ t_2 D_2(t_1/N)· D_2(t_2/N)·log(t_1-t_2)^2= -∑_(t_1, t_2) ∈ (𝐙/N𝐙)^2 t_1 ≠ -t_2 D_2(t_1/N)· D_2(t_2/N)·log(t_1+t_2)^2 = -8 ·∑_k=1^N-1/2 k·log(k)^2 + 2·ℱ_2= 6 ·∑_k=1^N-1k^2 ·log(k)^2 - log(2)^2·N-1/3+ 4 ·log(2) ·∑_k=1^N-1 k^2·log(k) -4·ℱ_2.The following identity will be useful in chapter <ref>.For any a ∈ (𝐙/N𝐙)^×, we have in 𝐙/p^v𝐙:∑_k ∈ (𝐙/N𝐙)^× k ≠ alog(k-a)·log(k)=- log(a)^2 + log(-1)·log(a)+ ℱ_2.We make the change of variable k = a · s. We get, in 𝐙/p^v𝐙:∑_k ∈ (𝐙/N𝐙)^× k ≠ alog(k-a)·log(k) =∑_s ∈ (𝐙/N𝐙)^× s≠ 1 log(s-1)·log(s) -log(a)^2+log(a) ·∑_s ∈ (𝐙/N𝐙)^× s≠ 1log(s)+ log(a) ·∑_s ∈ (𝐙/N𝐙)^× s ≠ 1log(s-1) =∑_s ∈ (𝐙/N𝐙)^× s≠ 1 log(s-1)·log(s) -log(a)^2+2 ·log(a) ·∑_s ∈ (𝐙/N𝐙)^×log(s)- log(-1)·log(a)= -log(a)^2 + log(-1)·log(a) + ∑_s ∈ (𝐙/N𝐙)^× s≠ 1 log(s-1)·log(s) .Since 2·∑_s ∈ (𝐙/N𝐙)^×log(s) =log((N-1)!^2) = 0, we have in 𝐙/p^v𝐙:∑_k ∈ (𝐙/N𝐙)^× k ≠ alog(k-a)·log(k) = -log(a)^2 + log(-1)·log(a) + ∑_s ∈ (𝐙/N𝐙)^× s≠ 1 log(s-1)·log(s) .We have, in 𝐙/p^v𝐙:∑_s ∈ (𝐙/N𝐙)^× s≠ 1 log(s-1)·log(s) = ℱ_2.We treat the cases p=2 and p>2 separately. Assume first that p>2. In this case, we have ℱ_2=0. On the other hand, the left-hand side is easily seen to be zero, using the change of variable s ↦1/s. During the rest of the proof, we assume that p=2 (so N ≡ 1(modulo 8)). We define three equivalence relations ∼_1, ∼_2 and ∼_3 in (𝐙/N𝐙)^×\{1, -1}, characterized by x ∼_1 -x, x∼_2 1/x, x∼_3 1/x and x ∼_3 -x for all x ∈ (𝐙/N𝐙)^×. For i ∈{1, 2, 3}, let R_i ⊂ (𝐙/N𝐙)^× be a set of representative for ∼_i. We can and do choose R_1, R_2 and R_3 so that R_3 ⊂ R_1 ∩ R_2. We denote by R_i the complement of R_i in (𝐙/N𝐙)^×\{1, -1}. Let ζ_4 ∈ R_2 be the unique element of order 4. If x ∈ R_3 and x ≠ζ_4, there is a unique element [x] of R_2 such that [x] ≠ x and [x] ∼_3 x. We have [x] = -x or [x] = -1/x. We get a partitionR_2 = {ζ_4}_x ∈ R_3 x ≠ζ_4{x, [x]} . We have, in 𝐙/p^v𝐙:∑_s ∈ (𝐙/N𝐙)^× s≠ 1 log(s-1)·log(s)=∑_s ∈ (𝐙/N𝐙)^× s≠ 1, -1 log(s-1)·log(s)=∑_s ∈ R_2log(s-1)·log(s)+∑_s ∈R_2log(s-1)·log(s) = ∑_s ∈ R_2log(s-1)·log(s) - log(1/s-1)·log(s).In the first equality, we have used the fact that log(-1)·log(-2)=0 since log(2) ≡ 0(modulo 2) by the quadratic reciprocity law (recall that N ≡ 1(modulo 8)). Thus, we have:∑_s ∈ (𝐙/N𝐙)^× s≠ 1 log(s-1)·log(s) = ∑_s ∈ R_2log(s)^2 - log(-1)·log(s).We have:∑_s ∈ R_2log(s) ≡log(ζ_4) + ∑_s ∈ R_3s ≠ζ_4log(s)+log([s])(modulo 2).We have log(s) + log([s]) ≡ 0(modulo 2), and log(ζ_4) ≡ 0(modulo 2). Thus, we have:∑_s ∈ R_2log(-1)·log(s) = 0.By (<ref>) and (<ref>), have:∑_s ∈ (𝐙/N𝐙)^× s≠ 1 log(s-1)·log(s) = ∑_s ∈ R_2log(s)^2.We have:ℱ_2= ∑_s ∈ R_1log(s)^2 = log(ζ_4)^2 + 2·∑_s ∈ R_3s≠ζ_4log(s)^2=log(ζ_4)^2 +∑_s ∈ R_3s≠ζ_4log(s)^2 + log([s])^2= ∑_s ∈ R_2log(s)^2.Combining the latter equality with (<ref>), this concludes the proof of Lemma <ref>.Lemma <ref> follows from (<ref>) and Lemma <ref>. The following identity will be useful to compute m_1^+ when p=2 (Theorem <ref>). Assume that p=2, so that N ≡ 1(modulo 8).For all x ∈ (𝐙/N𝐙)^×\{1, -1}, we have in 𝐙/2𝐙:∑_s_1,s_2=1(1-x)s_1+(1+x)s_2≡ 0(modulo N)^N-1/2 1=log(x+1/x-1)= ∑_s_1,s_2=1(1-x)s_1+(1+x)s_2≡ 0(modulo N)^N-1/2log(2/1-x· s_2)+ ∑_s_1,s_2=1(1-x)s_1+(1+x)s_2≢0(modulo N)^N-1/2log( (1-x)s_1+(1+x)s_2) .The integer∑_s_1,s_2=1(1-x)s_1+(1+x)s_2≡ 0(modulo N)^N-1/2 1is the number of s_2 ∈{1, 2, ..., N-1/2} such that the representative of x+1/x-1· s_2 ∈𝐙/N𝐙 in {1, 2, ...., N-1} is in {1, 2, ..., N-1/2}. By Gauss's Lemma <cit.>, this number is congruent to N-1/2-log(x+1/x-1) modulo 2. Since N-1/2 is even, we get in 𝐙/2𝐙: ∑_s_1,s_2=1(1-x)s_1+(1+x)s_2≡ 0(modulo N)^N-1/2 1 =log(x+1/x-1). To conclude the proof of Lemma <ref>, it suffices to prove the following equality in 𝐙/2𝐙:∑_s_1,s_2=1(1-x)s_1+(1+x)s_2≡ 0(modulo N)^N-1/2log(2/1-x· s_2) + ∑_s_1,s_2=1(1-x)s_1+(1+x)s_2≢0(modulo N)^N-1/2log( (1-x)s_1+(1+x)s_2)= log(x+1/x-1) .We denote by S the left hand side of (<ref>). Since N ≡ 1(modulo 8), the class of 2 in (𝐙/N𝐙)^× is a square log(2) ≡ 0(modulo 2), and we have log(-1)≡ 0(modulo 4). We have, in 𝐙/2𝐙 (using the first equality):S= log(x+1/x-1) ·log(x-1) + ∑_s_1,s_2=1(1-x)s_1+(1+x)s_2≡ 0(modulo N)^N-1/2log(s_2)+ ∑_s_1,s_2=1(1-x)s_1+(1+x)s_2≢0(modulo N)^N-1/2log( (1-x)s_1+(1+x)s_2)=log(x+1/x-1) ·log(x-1) + ∑_s_1,s_2=1^N-1/2log(s_2) + ∑_s_1,s_2=1(1-x)s_1+(1+x)s_2≢0(modulo N)^N-1/2log( (1-x)·s_1/s_2+(1+x)) = log(x+1/x-1) ·log(x-1) + ∑_s_1,s_2=1(1-x)s_1+(1+x)s_2≢0(modulo N)^N-1/2log( (1-x)·s_1/s_2+(1+x))=∑_s_1,s_2=1(1-x)s_1+(1+x)s_2≢0(modulo N)^N-1/2log( s_1/s_2-x+1/x-1) .In the last equality, we have used:∑_s_1,s_2=1(1-x)s_1+(1+x)s_2≢0(modulo N)^N-1/21 ≡∑_s_1,s_2=1(1-x)s_1+(1+x)s_2≡ 0(modulo N)^N-1/21 ≡log(x+1/x-1) (modulo 2),which follows from:∑_s_1,s_2=1^N-1/2 1 = (N-1/2)^2 ≡ 0(modulo 2).Let f : (𝐙/N𝐙)^×→𝐙 be such that if y ∈ (𝐙/N𝐙)^×, f(y) is the number of elements (s_1,s_2) ∈{1, 2, ..., N-1/2}^2 such that s_1/s_2≡ y(modulo N).We have shown that we have, in 𝐙/2𝐙:S = ∑_y ∈ (𝐙/N𝐙)^× y ≠x+1/x-1log(y-x+1/x-1)· f(y).As above, by Gauss's lemma and the fact that N ≡ 1(modulo 4), we have f(y) ≡log(y)(modulo 2). By (<ref>), we have in 𝐙/2𝐙:S = ∑_y ∈ (𝐙/N𝐙)^× y ≠x+1/x-1log(y-x+1/x-1)·log(y). By Lemma <ref> and the fact that N ≡ 1(modulo 8), we have in 𝐙/2𝐙:S = -log(x+1/x-1)^2+N-1/12 = log(x+1/x-1).This concludes the proof of Lemma <ref>.§.§ The case p=3 In this section, we focus on the case p=3. We determine the image of m_1^+ in (M_+/3^t· M_+)/𝐙· m_0^+. It suffices to determine the image of 6 · m_1^+ in ( M_+/3^t+1· M_+)/𝐙· 3· m_0^+. The formula given is a minor variation of Theorem <ref>. Recall that we lift log to a group homomorphism (𝐙/N𝐙)^×→𝐙/3^t+1𝐙 (still denoted by log). Assume that p=3. We have, in ( M_+/3^t+1· M_+)/𝐙· 3· m_0^+:6 · m_1^+≡log(2) ·m̃_0^+ + ∑_x ∈𝐏^1(𝐙/N𝐙) F_1,3(x) ·ξ_Γ_0(N)(x) ,where F_1,3 : 𝐏^1(𝐙/N𝐙) →𝐙/3^t+1𝐙 is defined byF_1,3([c:d])= 1/2∑_(s_1,s_2) ∈ (𝐙/2N𝐙)^2(d-c)s_1+(d+c)s_2 ≡ 0(moduloN) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log(s_2/d-c) - 1/2∑_(s_1,s_2) ∈ (𝐙/2N𝐙)^2(d-c)s_1+(d+c)s_2 ≢0(moduloN) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log((d-c)s_1+(d+c)s_2))if [c:d] ≠ [1:1] and F_1,3([1:1])=0. Let β : J/J^2 →𝐙/3^t+1𝐙 be given by [x]-1 ↦log(x) for x ∈ (𝐙/N𝐙)^×. This induces a map β_* : J· H_1(X_1(N), , 𝐙[1/2N][(𝐙/N𝐙)^×/± 1]) → H_1(X_1(N), , 𝐙/3^t+1𝐙). Let F_1,3': 𝐏^1(𝐙/N𝐙) →𝐙/3^t+1𝐙 be defined by:F_1,3'([c:d])= 1/2∑_(s_1,s_2) ∈ (𝐙/2N𝐙)^2(d-c)s_1+(d+c)s_2 ≡ 0(moduloN) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log((b-a)s_1+(b+a)s_2) - 1/2∑_(s_1,s_2) ∈ (𝐙/2N𝐙)^2(d-c)s_1+(d+c)s_2 ≢0(moduloN) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log((d-c)s_1+(d+c)s_2))where [ a b; c d ]∈_2(𝐙). By Lemma <ref>, this does not depend on the choice of[ a b; c d ]. We also have, for all [c:d] ∈𝐏^1(𝐙/N𝐙):F_1,3'([c:d])= 1/8∑_(s_1,s_2) ∈ (𝐙/N𝐙)^2(d-c)s_1+(d+c)s_2 ≡ 0(moduloN) D_2(s_1/N)D_2(s_2/N)·log((b-a)s_1+(b+a)s_2) - 1/8∑_(s_1,s_2) ∈ (𝐙/N𝐙)^2(d-c)s_1+(d+c)s_2 ≢0(moduloN) D_2(s_1/N)D_2(s_2/N) ·log((d-c)s_1+(d+c)s_2)). For all [c:d] ∈𝐏^1(𝐙/N𝐙), we have F_1,3'([-c:d]) = F_1,3'([c:d]). Thus, we have∑_x ∈𝐏^1(𝐙/N𝐙)F_1,3'(x)·ξ_Γ_0(N)(x) ∈ H_1(X_0(N), , 𝐙/3^t+1𝐙)_+.By construction, the pull-back of∑_x∈𝐏^1(𝐙/N𝐙)F_1,3'(x)·ξ_Γ_0(N)(x) in H_1(X_1(N), , 𝐙/3^t+1𝐙)_+ is β_*(𝒰). By (<ref>), Lemma <ref> and Proposition <ref> (ii), for all prime ℓ not dividing N we have in M_+/3^t+1· M_+:(T_ℓ-ℓ-1)(∑_x ∈𝐏^1(𝐙/N𝐙) F_1,3'(x) ·ξ_Γ_0(N)(x)) = 6·ℓ-1/2·log(ℓ)· m_0^+.Thus, there exists K_3 ∈𝐙/3^t+1𝐙 such that we have:6 · m_1^+≡ K_3 · m_0^+ + ∑_x ∈𝐏^1(𝐙/N𝐙) F_1,3'(x) ·ξ_Γ_0(N)(x)in (M_+/3^t+1· M_+)/𝐙· 3· m_0^+.Note that K_3 is only uniquely defined modulo 3. We have, in 𝐙/3^t+1𝐙:6 · m_1^+ ∙ m_0^-= K_3 · (m̃_0^+ ∙m̃_0^-)-F_1,3'([0:1])+F_1,3'([1:0]).Recall that m̃_0^+ ∙m̃_0^- = N-1/12. We also have in 𝐙/3^t𝐙:m_1^+ ∙ m_0^-= m_0^+ ∙ m_1^- =-1/4·(N-1/6·log(2) + ∑_k=1^N-1k^2·log(k) ).The first equality follows from Corollary <ref> and the second equality follows from Theorem <ref> (these results essentially come from <cit.> and are thus independents of the results of this chapter).We have, using Lemma <ref>: F_1,3'([0:1])= -1/8∑_(s_1,s_2) ∈ (𝐙/N𝐙)^2s_1 + s_2 ≢0(modulo N) D_2(s_1/N)D_2(s_2/N)·log(s_2+s_1)=3/4∑_k=1^N-1 k^2·log(k) + log(2)/2·N-1/12Note also that we have F_1,3'([1:0]) = -F_1,3'([0:1]). We thus get, in 𝐙/3^t+1𝐙:-1/2·( N-1/2·log(2) + 3∑_k=1^N-1 k^2·log(k)) =(K_3-log(2))·N-1/12-3/2∑_k=1^N-1k^2·log(k).We thus have K_3 ≡log(2)(modulo 3). If [ a b; c d ]∈Γ(2) and (s_1, s_2) ∈ (𝐙/2N𝐙)^2 are such that (d-c)s_1 + (d+c)s_2 ≡ 0(modulo N) and d ≢c(modulo N), then we have (b-a)s_1+(b+a)s_2 ≡2/d-c· s_2 (modulo N). By Lemma <ref>, we have in M_+/3^t+1· M_+: ∑_x ∈𝐏^1(𝐙/N𝐙) F_1,3(x)·ξ_Γ_0(N)(x) = ∑_x ∈𝐏^1(𝐙/N𝐙) F_1,3'(x)·ξ_Γ_0(N)(x)- 3·log(2) · m_0^+.This concludes the proof of Theorem <ref>. §.§ The case p=2We now study the case p=2 using a similar method, although our result is only partial. Recall that ℛ is the set of equivalence classes in 𝐏^1(𝐙/N𝐙) for the equivalence relation [c:d] ∼ [-d:c].We first give a formula for m̃_0^+ in terms of Manin symbols. Assume that p=2. Let F_0,2 : 𝐏^1(𝐙/N𝐙) →𝐙_p be given byF_0,2([c:d]) = -N-1/12+1/3·∑_s_1,s_2=1(d-c)s_1+(d+c)s_2 ≡ 0(modulo N)^N-1/2 1. If [c:d] ∈𝐏^1(𝐙/N𝐙) \{[1:1], [-1:1]}, we have F_0,2([-d:c]) = -F_0,2([c:d]). Thus, for all x ∈𝐏^1(𝐙/N𝐙) the element F_0,2(x) ·ξ_Γ_0(N)(x) of M_+ only depends on the class of x in R. We have in M_+:m̃_0^+ = ∑_x ∈ℛ F_0,2(x) ·ξ_Γ_0(N)(x).Recall <cit.> that we have in H_1(X_0(N), , 𝐐_p)_+:12 ·m̃_0^+ = ℰwhereℰ = ∑_x ∈𝐏^1(𝐙/N𝐙)F_0,2'(x) ·ξ_Γ_0(N)(x)andF_0,2'([c:d]) =1/2·∑_(s_1,s_2) ∈ (𝐙/N𝐙)^2(d-c)s_1+(d+c)s_2 ≡ 0(modulo N) D_2(s_1/N)· D_2(s_2/N).Assume that c≠± d. We have, in 𝐙:F_0,2'([c:d])= ∑_s_1,s_2=1 (d-c)s_1+(d+c)s_2 ≡ 0(modulo N)^N-1/2 1 -∑_s_1=1^N-1/2∑_s_2=N+1/2 (d-c)s_1+(d+c)s_2 ≡ 0(modulo N)^N-1 1= 2 ·∑_s_1,s_2=1 (d-c)s_1+(d+c)s_2 ≡ 0(modulo N)^N-1/2 1 - ∑_s_1=1^N-1/2∑_s_2=1(d-c)s_1+(d+c)s_2 ≡ 0(modulo N)^N-1 1 = -N-1/2 + 2 ·∑_s_1,s_2=1 (d-c)s_1+(d+c)s_2 ≡ 0(modulo N)^N-1/2 1.Thus, for all [c:d] ≠ [1:± 1] ∈𝐏^1(𝐙/N𝐙) we haveF_0,2'([c:d]) = 6· F_0,2([c:d]).Thus, we haveℰ = 6·∑_x ∈𝐏^1(𝐙/N𝐙)F_0,2(x) ·ξ_Γ_0(N)(x).By (<ref>), we get:m̃_0^+ = 1/2·∑_x ∈𝐏^1(𝐙/N𝐙)F_0,2(x) ·ξ_Γ_0(N)(x).For all [c:d] ∈𝐏^1(𝐙/N𝐙), we have F_0,2'([-d:c])= 1/2·∑_(s_1,s_2) ∈ (𝐙/N𝐙)^2(c+d)s_1+(c-d)s_2 ≡ 0(modulo N) D_2(s_1/N)· D_2(s_2/N) =1/2·∑_(s_1,s_2) ∈ (𝐙/N𝐙)^2(c+d)s_1+(d-c)s_2 ≡ 0(modulo N) D_2(s_1/N)· D_2(-s_2/N)=- 1/2·∑_(s_1,s_2) ∈ (𝐙/N𝐙)^2(c+d)s_1+(d-c)s_2 ≡ 0(modulo N) D_2(s_1/N)· D_2(s_2/N) =-F_0,2'([c,d]).By (<ref>) and (<ref>), for all [c:d] ≠ [1:±1] ∈𝐏^1(𝐙/N𝐙), we have F_0,2([-d:c]) = -F_0,2([c:d]). Since ξ_Γ_0(N)([-1:1]) = ξ_Γ_0(N)([1:1])=0, we have in M_+:m̃_0^+ = ∑_x ∈ℛ F_0,2(x) ·ξ_Γ_0(N)(x).By combining Theorem <ref>, Lemma <ref> and Proposition <ref> (iii), we get a new proof of Proposition <ref> (i) when p=2 and r=1.Assume that p=2 and that t ≥ 2, N ≡ 1(modulo 16). We have:6 · m_1^+ ≡ m_0^+ + ∑_x ∈ℛ F_1,2(x) ·ξ_Γ_0(N)(x)in (M_+/2^t· M_+)/𝐙· 2· m_0^+where F_1,2 : 𝐏^1(𝐙/N𝐙) →𝐙/2^t+1𝐙 is defined byF_1,2([c:d])= ∑_s_1, s_2=1(d-c)s_1+(d+c)s_2 ≡ 0(modulo N)^N-1/2log(2/d-c· s_2) - ∑_s_1, s_2=1(d-c)s_1+(d+c)s_2 ≠ 0(modulo N)^N-1/2log((d-c)s_1+(d+c)s_2)if [c:d]≠ [1:1] and F_1,2([1:1])=0 (this does not depend on the choice of c and d). Let γ=[ a b; c d ]∈_2(𝐙) with c ≠± d. We haveG_∞(Γ_1(N) ·γ)= 1/4∑_(s_1, s_2) ∈ (𝐙/N𝐙)^2(d-c)s_1+(d+c)s_2 ≢0(modulo N) D_2(s_1/N)D_2(s_2/N) · [(d-c)s_1+(d+c)s_2]^-1 =1/2∑_s_1,s_2=1(d-c)s_1+(d+c)s_2 ≢0(modulo N)^N-1/2[(d-c)s_1+(d+c)s_2]^-1 - 1/2∑_s_1=N+1/2^N-1∑_s_2=1(d-c)s_1+(d+c)s_2 ≢0(modulo N)^N-1/2[(d-c)s_1+(d+c)s_2]^-1=∑_s_1,s_2=1(d-c)s_1+(d+c)s_2 ≢0(modulo N)^N-1/2[(d-c)s_1+(d+c)s_2]^-1 - 1/2∑_s_1=1^N-1∑_s_2=1(d-c)s_1+(d+c)s_2 ≢0(modulo N)^N-1/2[(d-c)s_1+(d+c)s_2]^-1 =-N-1/2·δ + 1/2·∑_s_2=1^N-1/2 [(d+c)s_2]^-1+ ∑_s_1,s_2=1(d-c)s_1+(d+c)s_2 ≢0(modulo N)^N-1/2[(d-c)s_1+(d+c)s_2]^-1 where δ = ∑_x ∈ (𝐙/N𝐙)^×/±1[x]. We thus getG_∞(Γ_1(N) ·γ)=(1/2-N-1/2)·δ + ∑_s_1,s_2=1(d-c)s_1+(d+c)s_2 ≢0(modulo N)^N-1/2[(d-c)s_1+(d+c)s_2]^-1 . Similarly, noting (d-c)s_1+(d+c)s_2 ≡ 0(modulo N) implies (b-a)s_1+(b+a)s_2 ≡2· s_2/d-c (modulo N), we get:G_0(Γ_1(N) ·γ) = -1/2·δ +∑_s_1, s_2=1(d-c)s_1+(d+c)s_2 ≡ 0(modulo N)^N-1/2[2/d-c· s_2]. Note that for all [c,d] ∈ (𝐙/N𝐙)^2 \{(0,0)} / ±1, we have G_∞([-d,c]) = -G_∞([c,d]) and G_0([-d,c]) = -G_0([c,d]). Recall also that ξ_Γ_1(N)([-d,c]) = - ξ_Γ_1(N)([c,d]) and that ξ_Γ_1(N)([1,1])=0. Let R_1 be the set of equivalence classes in (𝐙/N𝐙)^2 \{(0,0)} / ±1 under the equivalence relation [c,d] ∼ [-d,c]. We thus have:𝒰 = ∑_[c,d] ∈ R_1[c,d]≁[1,1]ξ_Γ_1(N)([c,d]) · ( -N-1/2·δ + ∑_s_1, s_2=1(d-c)s_1+(d+c)s_2 ≠ 0(modulo N)^N-1/2[(d-c)s_1+(d+c)s_2]^-1 + ∑_s_1, s_2=1(d-c)s_1+(d+c)s_2 ≡ 0(modulo N)^N-1/2[2/d-c· s_2] ) . By Lemma <ref> and the definition of 𝒰, we have 𝒰∈ J· H_1(X_1(N), , 𝐙[(𝐙/N𝐙)^×/± 1]), where as above J is the augmentation ideal of 𝐙[(𝐙/N𝐙)^×/± 1]. Let β : J/J^2 →𝐙/2^t+1𝐙 be the group homomorphism given by [x]-[1]↦log(x). Letβ_* : J· H_1(X_1(N), , 𝐙[(𝐙/N𝐙)^×/± 1]) → H_1(X_1(N), , 𝐙/2^t+1𝐙)be the map induced by β. We have: β_*(𝒰)= ∑_[c,d] ∈ R_1[c,d]≁[1,1] F_1,2'([c,d]) ·ξ_Γ_1(N)([c,d]),whereF_1,2': (𝐙/N𝐙)^2 \{(0,0)}/± 1 →𝐙/2^t+1𝐙 is defined by F_1,2'([c,d])= ∑_s_1, s_2=1(d-c)s_1+(d+c)s_2 ≡ 0(modulo N)^N-1/2log(2/d-c· s_2) - ∑_s_1, s_2=1(d-c)s_1+(d+c)s_2 ≢0(modulo N)^N-1/2log((d-c)s_1+(d+c)s_2).Formula (<ref>) makes sense because F_1,2'([-d,c])=-F_1,2'([c,d]) ,so F_1,2'([c,d]) ·ξ_Γ_1(N)([c,d]) does not depend on the choice of [c,d] in its equivalence class in R_1. By (<ref>), for all prime ℓ not dividing 2· N we have in H_1(X_1(N), , 𝐙/2^t+1𝐙):(T_ℓ-ℓ-1)(β_*(𝒰)) = 6 ·ℓ-1/2·log(ℓ) ·ℰ'.Note that for all λ∈ (𝐙/N𝐙)^× and [c,d] ∈ (𝐙/N𝐙)^2 \{(0,0)}/± 1, we have F_1,2'([λ· c, λ· d]) = F_1,2'([c,d]).Thus F_1,2' induces the map F_1,2 : 𝐏^1(𝐙/N𝐙) →𝐙/2^t+1𝐙 defined in the statement of Theorem <ref>. Let𝒱 =∑_x ∈ℛ F_1,2(x) ·ξ_Γ_0(N)(x) ∈ H_1(X_0(N), , 𝐙/2^t+1𝐙).One easily sees that for all[c,d] ∈ (𝐙/N𝐙)^2 \{(0,0)}/± 1, we have F_1,2'([-c,d]) = F_1,2'([c,d]).Thus, we have 𝒱∈H_1(X_0(N), , 𝐙/2^t+1𝐙)_+.Let ζ_4 be a primitive fourth root in (𝐙/N𝐙)^×. For all λ,x ∈ (𝐙/N𝐙)^×, we have (using the Manin relations) in H_1(X_1(N), , 𝐙):ξ_Γ_1(N)([x, x·ζ_4]) = -ξ_Γ_1(N)([x·ζ_4, -x])andξ_Γ_1(N)([x, -x·ζ_4]) = -ξ_Γ_1(N)([x·ζ_4, x]).We also have, in H_1(X_0(N), , 𝐙):ξ_Γ_0(N)([ζ_4]) = ξ_Γ_0(N)([-ζ_4]) = 0.By (<ref>) and (<ref>), the element α := ∑_x ∈ (𝐙/N𝐙)^×/⟨ζ_4 ⟩ξ_Γ_1(N)([x, x·ζ_4])+ ξ_Γ_1(N)([x, -x ·ζ_4])is well-defined in H_1(X_1(N), , 𝐙/2𝐙), where ⟨ζ_4 ⟩ is the subgroup generated by ζ_4. Thus, the element 2^t·α is well-defined in H_1(X_1(N), , 𝐙/2^t+1𝐙). We have2^t·α∈ (1+c)· H_1(X_1(N), , 𝐙/2^t+1𝐙),where c is the complex conjugation. By (<ref>), we have F_1,2'([1, ζ_4])=-F_1,2'([ζ_4,-1]). Since [ζ_4,-1] = [ζ_4· 1, ζ_4 ·ζ_4], by (<ref>) we also have F_1,2'([ζ_4,-1]) = F_1,2'([1, ζ_4]). Thus, we have 2 · F_1,2'([1, ζ_4]) = 0, F_1,2'([1, ζ_4]) ≡ 0(modulo 2^t ). Thus, there exists K_2 ∈𝐙/2𝐙 such that for all x ∈ (𝐙/N𝐙)^×, we have F_1,2'([x, x·ζ_4]) = K_2 · 2^t.By (<ref>), for all x ∈ (𝐙/N𝐙)^×, we have F_1,2'([x, -x·ζ_4]) = K_2 · 2^t. For any integer r ≥ 1, let π_r^* : H_1(X_0(N), , 𝐙/2^r𝐙) → H_1(X_1(N), , 𝐙/2^r𝐙) be the pull-back map. By construction and by (<ref>), we haveπ_t+1^*(𝒱) = β_*(𝒰)+K_2 · 2^t·α .We have, in H_1(X_1(N), 𝐙/2^t+1𝐙):2^t·α = π_t+1^*(∑_x ∈ℛ x ≁1log(x+1/x-1) ·ξ_Γ_0(N)(x)).Recall the notation of Lemma <ref>, applied to Γ = Γ_1(N). We identify as before Γ_1(N) \_2(𝐙) with (𝐙/N𝐙)^2\{(0,0)}/±1. We let ℛ_1 ⊂ℛ_Γ_1(N) be the subset of equivalence classes of elements [c,d] with c · d ≢0(modulo N). We have, in H_1(X_1(N), , 𝐙/2^t+1𝐙):π_t+1^*(∑_x ∈ℛ x ≁1log(x+1/x-1) ·ξ_Γ_0(N)(x)) = 2^t·α +∑_[c,d] ∈ℛ_1log(d-c/d+c) ·ξ_Γ_1(N)([c,d]).To conclude the proof of Lemma <ref>, it suffices to prove the following result.We have, in H_1(X_1(N), 𝐙/2^t+1𝐙):∑_[c,d] ∈ℛ_1log(d-c/d+c) ·ξ_Γ_1(N)([c,d]) = 0.Consider the morphism h : 𝐙[Γ_1(N) \_2(𝐙)]→𝐙/2^t+1𝐙 given by [c,d] ↦log(c/d) if c · d ≢0(modulo N) and [c,d]↦ 0 else. This factors through ξ_Γ_1(N), and we let g : H_1(X_1(N),, 𝐙) →𝐙/2^t+1𝐙 be the induced map. Let f : H_1(X_1(N),𝐙) →𝐙/2^t+1𝐙 be the restriction of g to H_1(X_1(N),𝐙), and f̂∈ H_1(X_1(N), 𝐙/2^t+1𝐙) be the element corresponding to f by intersection duality. By Lemma <ref>, we have:f̂ = ∑_[c,d] ∈ℛ_1( log(c/d)+ 2/3·log( d/-(c+d)) + 1/3·log( -(c+d)/c) - 2/3·log( c/d-c)-1/3·log( c-d/d) ) ·ξ_Γ_1(N)([c,d]) =1/3·∑_[c,d] ∈ℛ_1log(d-c/d+c) ·ξ_Γ_1(N)([c,d]). To prove that f̂=0, it suffices to prove that f = 0. Let∑_[c,d] ∈ (𝐙/N𝐙)^2 \{(0,0)}/± 1c· d ≢0(modulo N)λ_[c,d]·ξ_Γ_1(N)([c,d]) ∈ H_1(X_1(N), 𝐙).We have (Section <ref>), in 𝐙[(𝐙/N𝐙)^×/± 1]:∑_[c,d] ∈ (𝐙/N𝐙)^2 \{(0,0)}/± 1c· d ≢0(modulo N)λ_[c,d]· ([c]-[d]) = 0.Thus, we havef( ∑_[c,d] ∈ (𝐙/N𝐙)^2 \{(0,0)}/± 1c· d ≢0(modulo N)λ_[c,d]·ξ_Γ_1(N)([c,d])) =∑_[c,d] ∈ (𝐙/N𝐙)^2 \{(0,0)}/± 1c· d ≢0(modulo N)λ_[c,d]·( log(c)-log(d) )= 0.This concludes the proof of Lemma <ref>.This concludes the proof of Lemma <ref>.By (<ref>), Lemma <ref> and (<ref>), for all prime ℓ not dividing 2· N, we have in H_1(X_1(N), , 𝐙/2^t+1𝐙):π_t+1^* ( (T_ℓ-ℓ-1)(𝒱 - K_2·∑_x ∈ℛ x ≁1log(x+1/x-1) ·ξ_Γ_0(N)(x) )- 6·ℓ-1/2·log(ℓ)· m_0^+) = 0. Thus, we have in H_1(X_1(N), , 𝐙/2^t+1𝐙):π_t+1^*( (T_ℓ-ℓ-1)(𝒱 - 6· m_1^+ - K_2·∑_x ∈ℛ x ≁1, 0log(x+1/x-1) ·ξ_Γ_0(N)(x)) ) =0.By (<ref>) and Proposition <ref> (i), for all prime ℓ not dividing 2· N, we have in H_1(X_0(N), , 𝐙/2^t+1𝐙):(1+c)· (T_ℓ-ℓ-1)(𝒱 - 6· m_1^+) = 0,where c is the complex conjugation. By (<ref>), we have (1-c) · (T_ℓ-ℓ-1)(𝒱 - 6· m_1^+) = 0. Thus, for all prime ℓ not dividing 2· N, we have in H_1(X_0(N), , 𝐙/2^t𝐙):(T_ℓ-ℓ-1)(𝒱 - 6· m_1^+) = 0.Thus, there exists a constant C_2, uniquely defined modulo 2, such that we have:6 · m_1^+ ≡ C_2· m_0^+ + 𝒱 in (M_+/2^t· M_+)/𝐙· 2· m_0^+.To conclude the proof of Theorem <ref>, it suffices to prove that m_0^+ ≡𝒱 modulo 2. This follows from Theorem <ref> and Lemma <ref>.§ INTERPLAY BETWEEN HIGHER EISENSTEIN ELEMENTS AND CONNECTION WITH GALOIS DEFORMATIONSIn this chapter, we relate the various Hecke modules (and their higher Eisenstein elements) that were considered in chapters <ref>, <ref> and <ref>. We assume as usual that p divides the numerator of N-1/12 (we allow p=2 and p=3). We keep the notation of chapters <ref>, <ref>, <ref> and <ref>. In particular M=𝐙_p[S] is the supersingular module, M_+ = H_1(X_0(N), , 𝐙_p)_+ and M^- = H_1(Y_0(N), 𝐙_p)^-. Recall also that we let ν = (N-1,12). We fix an integer r such that 1 ≤ r ≤ t. We denote by f_0, f_1, ..., f_n(r,p) the higher Eisenstein elements if ℳ/p^r·ℳ (where ℳ was defined in section <ref>). The q-expansion of an element f ∈ℳ at the cusp ∞ is denoted as usual by ∑_n ≥ 0 a_n(f) · q^n. §.§ Higher Eisenstein elements in the module ℳ of modular forms * We have, for all m∈ M:T_0(m)= 12/ν· (m∙ẽ_0)·ẽ_0. * We have, for all m_+∈ M_+:T_0(m_+)=12/ν· (m_+∙m̃_0^-)·m̃_0^+. * We have, for all m^-∈ M^-:T_0(m^-)=12/ν· (m̃_0^+∙ m^-)·m̃_0^-. We haveĨ· T_0=0since the Hecke operators of Ĩ· T_0 annihilate all the modular forms of weight 2 and level Γ_0(N).We first prove (i). By (<ref>), for all m ∈ M the element T_0(m) is annihilated by Ĩ, so is proportional to ẽ_0. Thus, for all m ∈ M there exists C_m ∈𝐙_p such thatT_0(m) = C_m ·ẽ_0.We thus have, in 𝐙_p:C_m · (ẽ_0 ∙ẽ_0) = T_0(m) ∙ẽ_0 = m ∙ T_0(ẽ_0). By (<ref>), we have T_0(ẽ_0) = N-1/ν·ẽ_0. Recall that by Eichler mass formula, we have ẽ_0 ∙ẽ_0 = N-1/12.By (<ref>), we get:C_m = 12/ν· (m ∙ẽ_0). We now prove (ii). As above, for all m_+ ∈ M_+ there exists C_m_+∈𝐙_p such that T_0(m_+) = C_m_+·m̃_0^+.We thus have, in 𝐙_p:C_m_+· (m̃_0^+ ∙m̃_0^-) = T_0(m_+) ∙m̃_0^- = m_+ ∙ T_0(m̃_0^-).By (<ref>), we have T_0(m̃_0^-) = N-1/ν·m̃_0^-. Recall that we have m̃_0^+ ∙m̃_0^- = N-1/12.By (<ref>), we get:C_m_+ = 12/ν· (m_+∙m̃_0^-).The proof of (iii) is similar. Let M_1 and M_2 be 𝕋̃-modules equipped with a 𝕋̃-equivariant bilinear pairing ∙: M_1 × M_2 →𝐙_p. By (<ref>), for any m_1 ∈ M_1 and m_2 ∈ M_2 we have an element of ℳ whose q-expansion at the cusp ∞ isν/24·(m_1∙ T_0(m_2)) + ∑_n ≥ 1(m_1∙ T_n(m_2)) · q^n.We apply this remark to the cases (M_1,M_2)=(M,M) and (M_1,M_2)=(M_+,M^-). We choose the pairing ∙ already described in the previous chapters for these couples. Let x ∈ M (resp. x_+ ∈ M_+, resp. x^- ∈ M^-) be such that ẽ_0 ∙ x = 1 (resp. x_+ ∙m̃_0^-=1, resp. m̃_0^+ ∙ x^- = 1). For all integers i such that 0 ≤ i ≤ n(r,p), letE_i = e_i∙ e_0/2 +∑_n≥ 1 (e_i ∙ T_n(x)) · q^n , E_i^+ = m_i^+ ∙ m_0^-/2 + ∑_n≥ 1 (m_i^+∙ T_n(x^-) )· q^nandE_i^- =m_0^+ ∙ m_i^-/2+ ∑_n≥ 1 (T_n(x_+) ∙ m_i^-)· q^nin (1/2𝐙⊕ q·𝐙[[q]]) ⊗_𝐙𝐙/p^r𝐙. Note that the image of E_i in((1/2𝐙⊕ q·𝐙[[q]]) ⊗_𝐙𝐙/p^r𝐙)/(𝐙· E_1+ ... + 𝐙· E_i-1)is uniquely determined, and similarly for E_i^+ and E_i^-.For all integers i such that 0 ≤ i ≤ n(r,p), we have:∑_n ≥ 0 a_n(f_i) · q^n ≡ E_i ≡ E_i^+ ≡ E_i^-in ((1/2𝐙⊕ q·𝐙[[q]]) ⊗_𝐙𝐙/p^r𝐙)/(𝐙· E_1+ ... + 𝐙· E_i-1).By (<ref>), the image of the group homomorphism ℳ→𝐐_p given by f ↦ a_0(f) is contained in ν/24·𝐙. Thus, the q-expansion at the cusp ∞ gives an injective group homomorphism ι : ℳ/p^r·ℳ↪(1/2𝐙⊕ q·𝐙[[q]]) ⊗_𝐙𝐙/p^r𝐙.The fact that E_i, E_i^+ and E_i^- lie in ι( ℳ/p^r·ℳ) follows from Proposition <ref> and (<ref>). We abuse notation and consider E_i, E_i^+ and E_i^- as elements of ℳ/p^r·ℳ.One easily sees that we have E_0 = E_0^+ = E_0^- = f_0. Furthermore, for all prime ℓ not dividing 2· N and all 1 ≤ i ≤ n(r,p), we have(T_ℓ-ℓ-1)(E_i) ≡ℓ-1/2·log(ℓ) · E_i-1 in( ℳ/p^r·ℳ)/( 𝐙· E_0 + ... + 𝐙· E_i-1).Proposition <ref> follows immediately by induction on i. By comparing the constant coefficients of the various modular forms of Proposition <ref>, we get the following comparison result, which is Theorem <ref> if r=1. For all 0 ≤ i,j≤ n(r,p) such that i+j≤ n(r,p), we have in 𝐙/p^r𝐙:e_i ∙ e_j = m_i^+ ∙ m_j^-.By Corollary <ref>, the common quantity of Corollary <ref> depends only on i+j, is 0 if i+j<n(r,p) and is non zero if i+j=n(r,p). §.§ Computation of m_i^+ ∙ m_j^- when i+j=1 By Corollary <ref>, we have in 𝐙/p^r𝐙:e_0 ∙ e_1 = e_1 ∙ e_0 = m_1^+ ∙ m_0^- = m_0^+ ∙ m_1^-. Theorem <ref> shows that if p≥ 5 then e_1∙ e_0 = 1/3·∑_k=1^N-1/2 k ·log(k). In fact, one can compute m_0^+ ∙ m_1^- directly even for p ≤ 3, using Merel's work. We have m_0^+ ∙ m_1^- = -1/12·log(ϵ·ζ·∏_k=1^N-1/2 k^-4· k) ∈𝐙/p^r𝐙where ϵ=1 if N ∉1+8𝐙, ϵ=-1 if N ∈ 1+8𝐙, ζ=1 if N ∉1+3𝐙 and ζ = 2^N-1/3 if N ∈ 1+3𝐙.If p=2, ϵ·ζ·∏_k=1^N-1/2 k^-4· k≡ x^4(modulo N) is a 4th power of (𝐙/N𝐙)^×, and the meaning of the right-hand side is -1/3·log(x), which is-1/3· (2^t-1-∑_k=1^N-1/2 k·log(k)).If p=3, ϵ·ζ·∏_k=1^N-1/2 k^-4· k≡ x^3(modulo N) is a 3rd power of (𝐙/N𝐙)^×, and the meaning of the right-hand side is -1/4·log(x), which is-1/4∑_k=1^N-1 k^2·log(k).This follows from <cit.>. The last assertion in the case p=3 follows from Lemma <ref>.Assume p=2. We have n(r,p) ≥ 2 if and only if ∑_k=1^N-1/2 k·log(k) ≡ 2^t-1 (modulo 2^r).In the case r=1, g_2=n(1,p) was determined completely in terms of the class group of 𝐐(√(-N)) in <cit.>: we have g_2=2^m-1-1 where m is the 2-adic valuation of the order of thisclass group.Assume p=3. We have n(r,p) ≥ 2 if and only if ∑_k=1^N-1 k^2·log(k) ≡ 0(modulo 3^r).§.§ Computation of m_i^+ ∙ m_j^- when i+j ≤ 3 and p ≥ 5In all this section, we assume p≥ 5. Theorem <ref> is a consequence of the following result. Assume that we have 1 ≤ r ≤ t and n(r,p) ≥ 2 (∑_k=1^N-1/2 k ·log(k) ≡ 0(modulo p^r)). We have:m_1^+ ∙ m_1^- ≡1/6∑_k=1^N-1/2 k ·log(k)^2(modulo p^r).By Theorems <ref> and <ref>, we have in 𝐙/p^r𝐙:24·m_1^+ ∙ m_1^-= ∑_x ∈ (𝐙/N𝐙)^× x≠ 1 log(x) ·(∑_(s_1,s_2)∈ (𝐙/2N𝐙)^2 (1-x)s_1 + (1+x)s_2 ≡ 0(modulo N) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log(s_2/1-x) )- log(x) ·( ∑_(s_1,s_2)∈ (𝐙/2N𝐙)^2 (1-x)s_1 + (1+x)s_2 ≢0(modulo N) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log((1-x)s_1 + (1+x)s_2) )= 1/4∑_x ∈ (𝐙/N𝐙)^× x≠ 1 log(x) ·(∑_(s_1,s_2)∈ (𝐙/N𝐙)^2 (1-x)s_1 + (1+x)s_2 = 0 D_2(s_1/N)D_2(s_2/N)·log(s_2/1-x) ) -log(x) ·(∑_(s_1,s_2)∈ (𝐙/N𝐙)^2 (1-x)s_1 + (1+x)s_2 ≠ 0 D_2(s_1/N)D_2(s_2/N) ·log((1-x)s_1 + (1+x)s_2) )= 1/4∑_(s_1,s_2) ∈ (𝐙/N𝐙)^2s_1 ≠± s_2 D_2(s_1/N)D_2(s_2/N)· (log(s_1+s_2/s_1-s_2) ·log(s_2-s_1/2) - ∑_x ≠ 0, s_1+s_2/s_1-s_2log(x)·log((x-1)s_1+(x+1)s_2) ) .Since n(r,p) ≥ 2, we have ∑_k=1^N-1/2 k ·log(k) ≡ 0(modulo p^r). By Lemma <ref>, we have:∑_(s_1,s_2) ∈ (𝐙/N𝐙)^2s_1 ≠± s_2 D_2(s_1/N)D_2(s_2/N)·log(s_1+s_2/s_1-s_2)≡ 0(modulo p^r). Thus, we have:96·m_1^+ ∙ m_1^-= ∑_(s_1,s_2) ∈ (𝐙/N𝐙)^2s_1 ≠± s_2 D_2(s_1/N)D_2(s_2/N)· (log(s_1+s_2/s_1-s_2) ·log(s_2-s_1) - ∑_x ≠ 0, s_1+s_2/s_1-s_2log(x)·log((1-x)s_1+(1+x)s_2) ) . Lemma <ref> shows that ∑_x ≠ 0, s_1+s_2/s_1-s_2log(x)·log((1-x)s_1+(1+x)s_2) = -log(s_2-s_1)·log(s_1+s_2/s_1-s_2)- log(s_1+s_2/s_1-s_2) ^2.We conclude that96 · m_1^+∙ m_1^-= ∑_(s_1,s_2) ∈ (𝐙/N𝐙)^2s_1 ≠± s_2 D_2(s_1/N)D_2(s_2/N) ·( log(s_1+s_2/s_1-s_2) log(s_2-s_1) + log(s_2-s_1)log(s_1+s_2/s_1-s_2)+ log(s_1+s_2/s_1-s_2) ^2)=∑_(s_1,s_2) ∈ (𝐙/N𝐙)^2s_1 ≠± s_2 D_2(s_1/N)D_2(s_2/N) ·( log(s_1+s_2)^2 - log(s_1-s_2)^2 )=16 ·∑_k=1^N-1/2 k·log(k)^2where the last equality follows from Lemma <ref>. Combining Corollary <ref>, Theorem <ref> (iii) and Theorem <ref>, we get the following identity.Assume n(r,p) ≥ 2, ∑_k=1^N-1/2 k ·log(k) ≡ 0(modulo p^r). We have, in 𝐙/p^r𝐙:∑_λ∈ L 3·log(H'(λ))^2 - 4·log(λ)^2 = 12·∑_k=1^N-1/2 k ·log(k)^2 .We now compute the pairings m_0^+ ∙ m_2^- and m_1^+ ∙ m_2^- using the formula for m_2^- given in Theorem <ref>. Recall that we defined function F_0,p, F_1,p : 𝐏^1(𝐙/N𝐙) →𝐙/p^r𝐙 in Theorems <ref> and <ref> (F_0,p really takes values in 𝐙_p, but we abuse notation and consider it modulo p^r). Recall the group isomorphism δ_r : 𝐙/p^r𝐙→ J·𝒦_r/J^2·𝒦_r defined in Section <ref>. Assume that Conjecture <ref> holds.Assume that n(r,p) ≥ 2, that ∑_k=1^N-1/2 k ·log(k) ≡ 0(modulo p^r). Let g ∈ (𝐙/N𝐙)^× be a generator such that log(g) ≡ 1(modulo p^r). We have:δ_r(m_0^+ ∙ m_2^-)= 1/4∑_i=1^N-1(∑_j=i^N-1 F_0,p(g^j) ) ·(1-ζ_N^g^i-1, 1-ζ_N/1-ζ_N^g^-1)_r.We have, in H_1(X_0(N), , 𝐙/p^r𝐙):m_0^+ = ∑_i=1^N-1 F_0,p(g^i)·ξ_Γ_0(N)([g^i:1]).The hypothesis n(r,p) ≥ 2 can be rewritten as∑_i=1^N-1 i· F_0,p(g^i) ≡ 0(modulo p^r).Thus, we have in H_1(X_0(N), , 𝐙/p^r𝐙):m_0^+ = ∑_i=1^N-1 F_0,p(g^i) ·(ξ_Γ_0(N)([g^i:1]) - i·ξ_Γ_0(N)([g:1]) ).We have ξ_Γ_0(N)([g^i:1]) - i·ξ_Γ_0(N)([g:1])=( ξ_Γ_0(N)([g^i:1])-ξ_Γ_0(N)([g^i-1:1])- ξ_Γ_0(N)([g:1]))+( ξ_Γ_0(N)([g^i-1:1]) -(i-1)·ξ_Γ_0(N)([g:1]) ).By induction on i, we get (using ξ_Γ_0(N)(1)=0):ξ_Γ_0(N)([g^i:1]) - i·ξ_Γ_0(N)([g:1])= ∑_j=1^i( ξ_Γ_0(N)(g^j)-ξ_Γ_0(N)(g^j-1)- ξ_Γ_0(N)([g:1])). Thus, we have in H_1(X_0(N), , 𝐙/p^r𝐙)_+:2· m_0^+ = ∑_i=1^N-1(∑_j=i^N-1 F_0,p(g^j) ) · (1+c)·(ξ_Γ_0(N)([g^i:1]) - ξ_Γ_0(N)([g^i-1:1]) - ξ_Γ_0(N)([g:1]) ),where c is the complex conjugation. By Theorem <ref>, we have:(1+c)· (ξ_Γ_0(N)([g^i:1]) - ξ_Γ_0(N)([g^i-1:1]) - ξ_Γ_0(N)([g:1]) ) ∙ m_2^- = 1/2·((1-ζ_N^g^i-1, 1-ζ_N)_r-(1-ζ_N^g^i-1, 1-ζ_N^g^-1)_r - (1-ζ_N^g^-1, 1-ζ_N)_r )By the bilinearity of (·, ·)_r, we have:(1-ζ_N^g^i-1, 1-ζ_N)_r-(1-ζ_N^g^i-1, 1-ζ_N^g^-1)_r = (1-ζ_N^g^i-1, 1-ζ_N/1-ζ_N^g^-1)_r.Using (<ref>), this concludes the proof of Theorem <ref>. Since m_1^+∙ m_1^- = m_0^+ ∙ m_2^-, Theorems <ref> and <ref> give us the following identity. Assume that Conjecture <ref> holds.Assume that n(r,p) ≥ 2, that ∑_k=1^N-1/2 k ·log(k) ≡ 0(modulo p^r). Let g ∈ (𝐙/N𝐙)^× be a generator such that log(g) ≡ 1(modulo p^r). We have the following equality in J·𝒦_r/J^2·𝒦_r:δ_r( ∑_k=1^N-1/2 k ·log(k)^2) = 3/2·∑_i=1^N-1(∑_j=i^N-1 F_0,p(g^j) )·(1-ζ_N^g^i-1, 1-ζ_N/1-ζ_N^g^-1)_r . We have not been able to prove directly this identity, without using the theory of higher Eisenstein elements. Similar computations give us the following result.Assume that Conjecture <ref> holds.Assume that n(r,p) ≥ 3, that ∑_k=1^N-1/2 k ·log(k) ≡∑_k=1^N-1/2 k ·log(k)^2 ≡ 0(modulo p^r). Let g ∈ (𝐙/N𝐙)^× be a generator such that log(g) ≡ 1(modulo p^r). We have:δ_r(m_1^+ ∙ m_2^-)= 1/4∑_i=1^N-1(∑_j=i^N-1 F_1,p(g^j) )·( 1-ζ_N^g^i-1, 1-ζ_N/1-ζ_N^g^-1)_r.As a corollary, we get our most advanced numerical criterion for bounding n(r,p) from below. Assume that Conjecture <ref> holds.Assume that n(r,p) ≥ 3, that ∑_k=1^N-1/2 k ·log(k) ≡∑_k=1^N-1/2 k ·log(k)^2 ≡ 0(modulo p^r). Let g ∈ (𝐙/N𝐙)^× be a generator such that log(g) ≡ 1(modulo p^r). We have n(r,p) ≥ 4 if and only if we have, in J·𝒦_r/J^2·𝒦_r:∑_i=1^N-1(∑_j=i^N-1 F_1,p(g^j) )·( 1-ζ_N^g^i-1, 1-ζ_N/1-ζ_N^g^-1)_r= 0.§.§ Computation of m_1^+ ∙ m_1^- when p=3In this section, we assume p=3. We give a formula for m_1^+ ∙ m_1^- modulo 3^t. We are only able to simplify the formula for this intersection product modulo 3^t-1. Note that we do not have an explicit formula for m_2^-, but nevertheless it is possible to compute m_0^+ ∙ m_2^-=m_1^+ ∙ m_1^-.For a and b in (𝐙/N𝐙)^×, we let [a,b] be the reduction modulo N of the interval [a,b], where a and b are representatives of a and b in {-N,..., -1} and {1,2,...,N} respectively.Let μ_3' the set of cubic primitive roots of unity in (𝐙/N𝐙)^×. Recall that σ = [0 -1;10 ] and τ = [0 -1;1 -1 ] ∈_2(𝐙). There is a right action of _2(𝐙) on 𝐏^1(𝐙/N𝐙) given by x ·[ a b; c d ] = ax+c/bx+d if x ∈𝐏^1(𝐙/N𝐙). The set(𝐙/N𝐙)^×\{1, -1}⊂𝐏^1(𝐙/N𝐙) is stable under σ.Let log_3 : 𝐙[ (𝐙/N𝐙)^×\{1, -1}]→𝐙/3^t+1𝐙 be given bylog_3(∑_x ∈(𝐙/N𝐙)^×\{1, -1}λ_x · [x]) = ∑_x_3 ∈μ_3'∑_y ∈ [-x_3,x_3] (λ_yσ - λ_y)·log(x_3). The following lemma summarizes the properties log_3 we will need.* If a ∈ (𝐙/N𝐙)^×\{1, -1} is fixed by τ, then log_3([a]) ≡log(a)(modulo 3^t+1).* If a ∈(𝐙/N𝐙)^×\{1, -1} is fixed by σ, then log_3(a) ≡ 0(modulo 3^t+1).Point (i) (resp. point (ii)) follows from <cit.> (resp. <cit.>). By Lemma <ref>, the group homomorphism 𝐙[(𝐙/N𝐙)^×\{1, -1}] →𝐙/3^t+1𝐙 given by [x] ↦log(x) - log_3(x) annihilates the Manin relations, and thus induces a group homomorphismφ_3 : H_1(X_0(N), 𝐙/3^t+1𝐙) →𝐙/3^t+1𝐙 . Assume p=3. Assume n(r,p) ≥ 2, ∑_k=1^N-1 k^2·log(k) ≡ 0(modulo 3^r). Then we have in 𝐙/3^r+1𝐙:12 · m_1^+ ∙ m_1^- = log(2) ·φ_3(m_0^+) + φ_3(∑_x ∈ (𝐙/N𝐙)^×\{1, -1} F_1,3(x) ·ξ_Γ_0(N)([x:1]))where F_1,3 : 𝐏^1(𝐙/N𝐙) →𝐙/3^t+1𝐙 is defined in Theorem <ref>. By Theorem <ref>, we have H_1(X_0(N), , 𝐙/3^t+1𝐙)_+:6· m_1^+ = log(2) · m_0^+ + ∑_x ∈ (𝐙/N𝐙)^×\{1, -1} F_1,3(x) ·ξ_Γ_0(N)([x:1]).Let G_0 : (𝐙/N𝐙)^×\{1, -1}→𝐙/3^t𝐙 be such that m_0^+ = ∑_x ∈ (𝐙/N𝐙)^×\{1, -1} G_0(x) ·ξ_Γ_0(N)([x:1]).Let G_1 : (𝐙/N𝐙)^×\{1, -1}→𝐙/3^t𝐙 be such that m_1^+ = ∑_x ∈ (𝐙/N𝐙)^×\{1, -1} G_1(x) ·ξ_Γ_0(N)([x:1]). By Theorem <ref> and <cit.>, we have the following equality in (𝐙/3^t+1𝐙)[(𝐙/N𝐙)^×\{1, -1} ]:6·∑_x ∈ (𝐙/N𝐙)^×\{1, -1} G_1(x) · [x] = log(2) ·∑_x ∈ (𝐙/N𝐙)^×\{1, -1} G_0(x) · [x] +∑_x ∈ (𝐙/N𝐙)^×\{1, -1} F_1,3(x) · [x] + a_σ+a_τwhere a_σ (resp. a_τ) is an element of (𝐙/3^t+1𝐙)[(𝐙/N𝐙)^×\{1, -1} ] fixed by σ (resp. by τ). By definition, we have in 𝐙/3^r𝐙:m_1^+ ∙ m_1^- = 1/2·∑_x ∈ (𝐙/N𝐙)^×\{1, -1} G_1(x) ·log(x).By Lemma <ref> (i), we get in 𝐙/3^r+1𝐙:12 · m_1^+ ∙ m_1^- = log(2) ·φ_3(m_0^+) + φ_3(∑_x ∈ (𝐙/N𝐙)^×\{1, -1} F_1,3(x) ·ξ_Γ_0(N)([x:1])) which concludes the proof of Theorem <ref>.Assume t ≥ 2, N ≡ 1(modulo 27). Assume 1 ≤ r ≤ t-1. If n(r,p) ≥ 2, if ∑_k=1^N-1 k^2 ·log(k) ≡ 0(modulo3^r), then we have in 𝐙/3^r𝐙:m_1^+ ∙ m_1^- = -1/4·∑_k=1^N-1k^2·log(k)^2.Since r ≤ t-1, the map log_3 is zero modulo 3^r+1. Thus, for all x ∈ (𝐙/N𝐙)^×\{1, -1} we have φ_3(ξ_Γ_0(N)(x)) ≡log(x)(modulo 3^r+1). In particular, we have φ_3(m_0^+) ≡ 2 · m_0^+ ∙ m_1^-(modulo 3^r+1).We get, by Theorem <ref>:φ_3(m_0^+) ≡ - 1/2·∑_k=1^N-1 k^2·log(k)(modulo 3^r+1). Thus, we have in 𝐙/3^r+1𝐙:6 · m_1^+ ∙ m_1^- = -1/2·log(2) ·∑_k=1^N-1 k^2·log(k) +∑_x ∈ (𝐙/N𝐙)^×\{1, -1} F_1,3(x) ·log(x). Recall that F_1,3 : 𝐏^1(𝐙/N𝐙) →𝐙/3^t+1𝐙 is defined byF_1,3([c:d])= 1/2∑_(s_1,s_2) ∈ (𝐙/2N𝐙)^2(d-c)s_1+(d+c)s_2 ≡ 0(moduloN) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log(s_2/d-c) -1/2∑_(s_1,s_2) ∈ (𝐙/2N𝐙)^2(d-c)s_1+(d+c)s_2 ≢0(moduloN) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log((d-c)s_1+(d+c)s_2))if [c:d] ≠ [1:1] and F_1,3([1:1])=0. We have, in 𝐙/3^t+1𝐙: 2· ∑_x ∈ (𝐙/N𝐙)^×\{1, -1} F_1,3(x) ·log(x)= ∑_x ∈ (𝐙/N𝐙)^× x≠ 1 log(x) ·(∑_(s_1,s_2)∈ (𝐙/2N𝐙)^2 (1-x)s_1 + (1+x)s_2 ≡ 0(modulo N) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log(s_2/1-x) ) - log(x) ·( ∑_(s_1,s_2)∈ (𝐙/2N𝐙)^2 (1-x)s_1 + (1+x)s_2 ≢0(modulo N) (-1)^s_1+s_2_1(s_1/2N)_1(s_2/2N) ·log((1-x)s_1 + (1+x)s_2) )= 1/4∑_x ∈ (𝐙/N𝐙)^× x≠ 1 log(x) ·(∑_(s_1,s_2)∈ (𝐙/N𝐙)^2 (1-x)s_1 + (1+x)s_2 ≡ 0(modulo N) D_2(s_1/N)D_2(s_2/N)·log(s_2/1-x) )-log(x) ·(∑_(s_1,s_2)∈ (𝐙/N𝐙)^2 (1-x)s_1 + (1+x)s_2 ≢0(modulo N) D_2(s_1/N)D_2(s_2/N) ·log((1-x)s_1 + (1+x)s_2) ) = 1/4∑_(s_1,s_2) ∈ (𝐙/N𝐙)^2s_1 ≠± s_2 D_2(s_1/N)D_2(s_2/N)· (log(s_1+s_2/s_1-s_2) ·log(s_2-s_1/2) - ∑_x ≠s_1+s_2/s_1-s_2log(x)log((1-x)s_1+(1+x)s_2) ) . Since r+1 ≤ t, Lemma <ref> shows that we have, in 𝐙/3^r+1𝐙:∑_x ≠s_1+s_2/s_1-s_2log(x)log((1-x)s_1+(1+x)s_2) = - log(s_2-s_1)log(s_1+s_2/s_1-s_2)- log(s_1+s_2/s_1-s_2) ^2.Thus we have, in 𝐙/3^r+1𝐙: 2∑_x ∈ (𝐙/N𝐙)^×\{1, -1} F_1,3(x) ·log(x) =1/4∑_(s_1,s_2) ∈ (𝐙/N𝐙)^2s_1 ≠± s_2 D_2(s_1/N)D_2(s_2/N)·( log(s_1+s_2)^2-log(s_1-s_2)^2 - log(2) ·log(s_1+s_2/s_1-s_2) )Using Lemmas <ref> and <ref> and the assumption r+1 ≤ t, we get in 𝐙/3^r+1𝐙:∑_x ∈ (𝐙/N𝐙)^×\{1, -1} F_1,3(x) ·log(x) = -3/2·∑_k=1^N-1k^2·log(k)^2 +1/2·log(2) ·∑_k=1^N-1 k^2·log(k).By (<ref>), we have in 𝐙/3^r+1𝐙:6 · m_1^+ ∙ m_1^- = -3/2·∑_k=1^N-1k^2 ·log(k)^2.This concludes the proof of Theorem <ref>.Assume that t ≥ 2, that N ≡ 1(modulo 27). Assume that 1 ≤ r ≤ t-1. The following assertions are equivalent: * n(r,p) ≥ 3* ∑_k=1^N-1 k^2 ·log(k) ≡∑_k=1^N-1 k^2 ·log(k)^2 ≡ 0(modulo 3^r ). Corollary <ref> does not hold in general if N ≢1(modulo 27 ). For instance, if N = 1279 and r=1 then g_3=2 and ∑_k=1^N-1 k^2 ·log(k) ≡∑_k=1^N-1 k^2 ·log(k)^2 ≡ 0(modulo 3 ). IfN=1747 and r=1 then g_3=3 and ∑_k=1^N-1 k^2 ·log(k)^2 ≢0(modulo 3 ).§.§ Computation of m_1^+ ∙ m_1^- when p =2In this section, we assume that p=2. We give a formula for m_1^+ ∙ m_1^- modulo 2^t-1. Note that we do not have an explicit formula for m_2^- modulo 2^t-1, but nevertheless it is possible to compute m_0^+ ∙ m_2^- = m_1^+ ∙ m_1^- modulo 2^t-1. Assume that t≥ 2, that N ≡ 1(modulo 16 ). Let r be an integer such that 1 ≤ r ≤ t-1 and n(r,p) ≥ 2 (∑_k=1^N-1/2 k ·log(k) ≡ 0(modulo 2^r) by Corollary <ref>). We have, in 𝐙/2^r+1𝐙:6· m_1^+ ∙ m_1^- =∑_k=1^N-1/2 k ·log(k) +∑_k=1^N-1/2 k ·log(k)^2.Recall that we have normalized m̃_0^-, and thus m_1^-, so that for all x ∈𝐏^1(𝐙/N𝐙) \{0, ∞} we have, in 𝐙/2^t𝐙:(1+c) ·ξ_Γ_0(N)(x) ∙ m_1^- = log(x).For all x ∈ (𝐙/N𝐙)^×, we have in 𝐙/2^t+1𝐙:F_1,2(x) = F_1,2(-x).By Theorem <ref>, Theorem <ref> and (<ref>), we have in 𝐙/2^r+2𝐙:12· m_1^+ ∙ m_1^- = -2/3·(2^t-1 - ∑_k=1^N-1/2 k ·log(k) ) + ∑_x ∈ R' F_1,2(x) ·log(x),where R' is any set of representative in (𝐙/N𝐙)^×\{1, -1} for the equivalence relation x ∼ -1/x.Let F̃_1,2 : (𝐙/N𝐙)^×→𝐙/2^t+2𝐙 be defined byF̃_1,2(x)= ∑_s_1, s_2=1(1-x)s_1+(1+x)s_2 ≡ 0(modulo N)^N-1/2log(2/1-x· s_2) - ∑_s_1, s_2=1(1-x)s_1+(1+x)s_2 ≠ 0(modulo N)^N-1/2log((1-x)s_1+(1+x)s_2).By definition, for all x∈ (𝐙/N𝐙)^× we have F_1,2(x) ≡F̃_1,2(x)(modulo 2^t+1).For all x∈ (𝐙/N𝐙)^×, we have in 𝐙/2^t+2𝐙:F̃_1,2(-1/x) = -F̃_1,2(x) + N-1/2+N-1/2·log(x-1/x+1).Let x∈ (𝐙/N𝐙)^×. We have, in 𝐙/2^t+1𝐙:F̃_1,2(x)+F̃_1,2(-1/x)= ∑_s_2=1^N-1/2∑_s_1=1(1-x)s_1+(1+x)s_2 ≡ 0(modulo N)^N-1/2log(2/1-x· s_2 )+∑_s_2=1^N-1/2∑_s_1=N+1/2 (1-x)s_1+(1+x)s_2 ≡ 0(modulo N)^N-1log(2· x/1-x· s_2 )- ∑_s_2=1^N-1/2∑_s_1=1 (1-x)s_1+(1+x)s_2 ≢0(modulo N)^N-1/2log((1-x)s_1+(1+x)s_2) - ∑_s_2=1^N-1/2∑_s_1=N+1/2 (1-x)s_1+(1+x)s_2 ≢0(modulo N)^N-1log(1/x·((1-x)s_1+(1+x)s_2 ))= ∑_s_2=1^N-1/2∑_s_1=1(1-x)s_1+(1+x)s_2 ≡ 0(modulo N)^N-1log(2/1-x· s_2 ) + log(x) ·𝒩(x) - ∑_s_2=1^N-1/2∑_s_1=1 (1-x)s_1+(1+x)s_2 ≢0(modulo N)^N-1log((1-x)s_1+(1+x)s_2) + log(x) ·( (N-1/2)^2-𝒩(x)) ,where 𝒩(x) is the number of s ∈{1, 2, ..., N-1/2} such that the representative of s ·x-1/x+1 in {1, ..., N-1} is in {N+1/2, ..., N-1}. Thus, we have in 𝐙/2^t+2𝐙:F̃_1,2(x)+F̃_1,2(-1/x)= ∑_s_2=1^N-1/2∑_s_1=1(1-x)s_1+(1+x)s_2 ≡ 0(modulo N)^N-1log(2/1-x· s_2 )- ∑_s_2=1^N-1/2∑_s_1=1 (1-x)s_1+(1+x)s_2 ≢0(modulo N)^N-1log((1-x)s_1+(1+x)s_2) = N-1/2·log(2/1-x) + log((N-1/2)! )- N-1/2· (N-2)·log(1+x) -∑_s_2=1^N-1/2∑_s_1=1 (1-x)s_1+(1+x)s_2 ≢0(modulo N)^N-1log(s_2 - x-1/x+1· s_1).Since N ≡ 1(modulo 8), we have log(2) ≡ 0(modulo 2) by the quadratic reciprocity law. Thus we have in 𝐙/2^t+2𝐙:F̃_1,2(x)+F̃_1,2(-1/x)= N-1/2·log(x+1/x-1) + log((N-1/2)! ) - ∑_s_2=1^N-1/2(log((N-1)!) - log(s_2) ) =N-1/2·log(x+1/x-1) + 2·log((N-1/2)! )= N-1/2+ N-1/2·log(x+1/x-1) .This concludes the proof of Lemma <ref>.Let ζ_4 be an element of order 4 in (𝐙/N𝐙)^×. Since t≥ 2, we have 2^t ·log( ζ_4 )= 0 in 𝐙/2^t+2𝐙. Since F_1,2(ζ_4)≡ 0(modulo 2^t), we have in 𝐙/2^t+2𝐙:F̃_1,2(ζ_4)·log(ζ_4) = 0Similarly, we have in 𝐙/2^t+2𝐙:F̃_1,2(-1)·log(-1) = 0By Lemma <ref>, (<ref>) and (<ref>), we have in 𝐙/2^t+2𝐙:∑_x ∈ (𝐙/N𝐙)^×F̃_1,2(x) ·log(x)= ∑_x ∈ R'F̃_1,2(x) ·log(x) + F̃_1,2(-1/x) ·log(-1/x) = 2·∑_x ∈ R'F̃_1,2(x) ·log(x) +∑_x ∈ R'N-1/2·log(x) + N-1/2·log(x) ·log(x-1/x+1)+ log(-1)·∑_x ∈ R'F̃_1,2(x).By (<ref>), we have ∑_x ∈ R' F_1,2(x) ≡ 0(modulo 2). By Lemma <ref> and the fact that log(-1)·log(-2) ≡ 0(modulo 4), we have 2·∑_x ∈ R'log(x) ·log(x-1/x+1) ≡∑_x ∈ (𝐙/N𝐙)^×\{1, -1}log(x) ·log(x-1/x+1) ≡ 0(modulo 4).Thus, we have in 𝐙/2^t+2𝐙:∑_x ∈ (𝐙/N𝐙)^×F̃_1,2(x) ·log(x) = 2·∑_x ∈ R'F̃_1,2(x) ·log(x) . By (<ref>) and (<ref>), we have in 𝐙/2^t+2𝐙:24· m_1^+ ∙ m_1^- = -4/3·(2^t-1 - ∑_k=1^N-1/2 k ·log(k) ) + ∑_x ∈ (𝐙/N𝐙)^×F̃_1,2(x) ·log(x).We have, in 𝐙/2^t+2𝐙:∑_x ∈ (𝐙/N𝐙)^×F̃_1,2(x) ·log(x)=∑_x ∈ (𝐙/N𝐙)^×∑_s_1,s_2=1 (1-x)s_1+(1+x)s_2 ≡ 0(modulo N)^N-1/2log( 2/1-x· s_2)·log(x) -∑_x ∈ (𝐙/N𝐙)^×∑_s_1,s_2=1(1-x)s_1+(1+x)s_2 ≢0(modulo N)^N-1/2log((1-x)s_1+(1+x)s_2)·log(x) = ∑_s_1,s_2=1s_1 ≠ s_2^N-1/2log(s_2-s_1)·log(s_1+s_2/s_1-s_2) -∑_s_1,s_2=1^N-1/2∑_x ∈ (𝐙/N𝐙)^× x≢s_1+s_2/s_1-s_2 (modulo N)log((1-x)s_1+(1+x)s_2)·log(x) =∑_s_1,s_2=1s_1 ≠ s_2^N-1/2log(s_2-s_1)·log(s_1+s_2/s_1-s_2) - ∑_s_1,s_2=1s_1 ≠ s_2^N-1/2log(s_2-s_1)·(log((N-1)!)-log(s_1+s_2/s_1-s_2))+∑_s_1,s_2=1s_1 ≠ s_2^N-1/2∑_x ∈ (𝐙/N𝐙)^× x≢s_1+s_2/s_1-s_2 (modulo N)log(x - s_1+s_2/s_1-s_2)·log(x) =2·∑_s_1,s_2=1s_1 ≠ s_2^N-1/2log(s_2-s_1)·log(s_1+s_2/s_1-s_2)+ log(-1)·∑_s_1,s_2=1s_1 ≠ s_2^N-1/2log(s_2-s_1) + ∑_s_1,s_2=1s_1 ≠ s_2^N-1/2∑_x ∈ (𝐙/N𝐙)^× x≢s_1+s_2/s_1-s_2 (modulo N)log(x - s_1+s_2/s_1-s_2)·log(x). By Lemmas <ref> and<ref>, we have in 𝐙/2^t+2𝐙:∑_x ∈ (𝐙/N𝐙)^×F̃_1,2(x) ·log(x)=2·∑_s_1,s_2=1s_1 ≠ s_2^N-1/2log(s_1-s_2)·log(s_1+s_2/s_1-s_2) + ∑_s_1,s_2=1s_1 ≠ s_1^N-1/2( log(-1)·log(s_1+s_2/s_1-s_2) - log(s_1+s_2/s_1-s_2)^2+ℱ_2 ).Since N ≡ 1(modulo 8), 4·ℱ_2 = 4 ·ℱ_1 = 0 and log(-1) ≡ 0(modulo 4), we have in 𝐙/2^t+2𝐙:∑_x ∈ (𝐙/N𝐙)^×F̃_1,2(x) ·log(x) =2·∑_s_1,s_2=1s_1 ≠ s_2^N-1/2log(s_1-s_2)·log(s_1+s_2/s_1-s_2) - ∑_s_1,s_2=1s_1 ≠ s_1^N-1/2log(s_1+s_2/s_1-s_2)^2 .We have in 𝐙/2^t+2𝐙:∑_s_1,s_2=1s_1 ≠ s_2^N-1/2log(s_1+s_2)^2=∑_s_1,s_2=1^N-1/2log(s_1+s_2)^2 - ∑_s=1^N-1/2log(2· s)^2= ∑_s_1,s_2=1^N-1/2log(s_1+s_2)^2 - ℱ_2 .By Lemma <ref> and the fact that 2·ℱ_2 = N-1/2, we have in 𝐙/2^t+2𝐙:∑_s_1,s_2=1s_1 ≠ s_2^N-1/2log(s_1+s_2)^2 =N-1/2 +2·∑_k=1^N-1/2 k·log(k)^2and∑_s_1,s_2=1s_1 ≠ s_2^N-1/2log(s_1-s_2)^2 =-2·∑_k=1^N-1/2 k·log(k)^2 .By (<ref>), (<ref>) and (<ref>), we have in 𝐙/2^t+2𝐙:∑_x ∈ (𝐙/N𝐙)^×F̃_1,2(x) ·log(x) = N-1/2+4·∑_k=1^N-1/2 k·log(k)^2 + 4·∑_s_1,s_2=1s_1 ≠ s_1^N-1/2log(s_1+s_2)·log(s_1-s_2) .We have, in 𝐙/2^t+2𝐙:2·∑_s_1,s_2=1s_1 ≠ s_1^N-1/2log(s_1+s_2)·log(s_1-s_2) = ∑_s_1=1^N-1/2∑_s_2=1s_2 ≠ s_1, N-s_1^N-1log(s_1+s_2)·log(s_1-s_2)= ∑_s_1=1^N-1/2∑_s_2 ∈ (𝐙/N𝐙)^× s_2 ≠± s_1log(s_1+s_2)·log(s_1-s_2)= ∑_s_1=1^N-1/2∑_s_2 ∈ (𝐙/N𝐙)^× s_2 ≠ 2· s_1, s_1log(s_2)·log(2· s_1-s_2)=∑_s_1=1^N-1/2∑_s_2 ∈ (𝐙/N𝐙)^× s_2 ≠ 2· s_1log(s_2)·log(s_2-2· s_1) +log(-1)·∑_s_1=1^N-1/2log(2· s_1) - ∑_s_1=1^N-1/2log(s_1)^2 = -ℱ_2+ ∑_s_1=1^N-1/2∑_s_2 ∈ (𝐙/N𝐙)^× s_2 ≠ 2· s_1log(s_2)·log(s_2-2· s_1) .In the last equality, we have used the fact that log(2) ≡ 0(modulo 2) by the quadratic reciprocity law, since N ≡ 1(modulo 8). 2·∑_s_1,s_2=1s_1 ≠ s_1^N-1/2log(s_1+s_2)·log(s_1-s_2) = -ℱ_2+ ∑_s_1=1^N-1/2( log(-1)·log(2· s_1) -log(2· s_1)^2 + ℱ_2) =-ℱ_2 -∑_s_1=1^N-1/2log(2· s_1)^2= -2·ℱ_2 .By (<ref>), we have in 𝐙/2^t+2𝐙:∑_x ∈ (𝐙/N𝐙)^× F_1,2(x) ·log(x) = N-1/2+4·∑_k=1^N-1/2 k ·log(k)^2.By (<ref>), we have in 𝐙/2^r+3𝐙:24· m_1^+∙ m_1^-= 4/3·∑_k=1^N-1/2k ·log(k) + 4·∑_k=1^N-1/2 k ·log(k)^2= 4·∑_k=1^N-1/2k ·log(k) + 4·∑_k=1^N-1/2 k ·log(k)^2.In the last equality, we have used the fact that ∑_k=1^N-1/2k ·log(k) ≡ 0(modulo 2^r). This concludes the proof of Theorem <ref>.Assume that t ≥ 2, that N ≡ 1(modulo 16). Let r be an integer such that 1 ≤ r ≤ t-1. Assume that n(r,p) ≥ 2, (∑_k=1^N-1/2 k ·log(k) ≡ 0(modulo 2^r) by Corollary <ref>). The following assertions are equivalent: * We have n(r,p) ≥ 3.* We have∑_k=1^N-1/2 k ·log(k) +∑_k=1^N-1/2 k ·log(k)^2 ≡ 0(modulo 2^r+1) By Remark (<ref>), the integer n(1,p) is odd when p=2. In particular, if n(1,p) ≥ 2 then n(1,p) ≥ 3. If N ≡ 1(modulo 16), this means by Corollary <ref> that ∑_k=1^N-1/2 k ·log(k) ≡ 0(modulo 2) implies ∑_k=1^N-1/2 k ·log(k) +∑_k=1^N-1/2 k ·log(k)^2 ≡ 0(modulo 4). We have not found an elementary proof of this fact.§.§ The q-expansion of the second higher Eisenstein element f_2 via Galois deformationsRecall that we keep the notation of chapter <ref>, and in particular those of section <ref>. In this section, we assume p ≥ 5, r=1 and n(r,p)≥ 2. Theorem <ref> below aims at giving a rather explicit description of the higher Eisenstein element f_2.Our proof relies on the results of <cit.> and <cit.>, some of which we now recall. If i ∈𝐙, let K_(i) be the number field defined in <cit.>. In particular, K_(i) is an abelian Galois extension of 𝐐(ζ_p) of exponent p, unramified outside N and p over 𝐐 and such that (𝐐(ζ_p)/𝐐) acts by multiplication by ω_p^i on (K_(i)/𝐐(ζ_p)). We have K_(1) = 𝐐(N^1/p, ζ_p) and K_(0) = L_1. If F is a number field, we let 𝒞_F = (𝒪_F) ⊗_𝐙𝐙/p𝐙, where (𝒪_F) is the class group of F. The following result follows easily from the proof of <cit.>.Let i ∈{-1,0,1}, so that in particular we have (𝒞_𝐐(ζ_p))_(i) = 0. Then (K_(i)/𝐐(ζ_p)) is cyclic of order p, and global class field theory gives a canonical (𝐐(ζ_p)/𝐐)-equivariant group isomorphismβ_(i) : (K_(i)/𝐐(ζ_p))U^⊗ 1-i⊗μ_p^⊗ i ,where U := (𝐙/N𝐙)^×/((𝐙/N𝐙)^×)^p is equipped with the trivial action of (𝐐(ζ_p)/𝐐) (the group law of U is denoted multiplicatively). As in the proof of <cit.>, global class field theory gives a canonical (𝐐(ζ_p)/𝐐)-equivariant group isomorphismα_(i) : ( ( 𝐙[ζ_p]/(N) )^×⊗_𝐙𝐙/p𝐙)_(i)(K_(i)/𝐐(ζ_p)).Let S be the set of prime ideals above N in 𝐙[ζ_p]. The Chinese remainder theorem shows that there is a canonical group isomorphismf : ( 𝐙[ζ_p]/(N) )^×∏_𝔫∈ S( 𝐙[ζ_p]/𝔫)^× .We let (𝐐(ζ_p)/𝐐) acts on ∏_𝔫∈ S( 𝐙[ζ_p]/𝔫)^×, via the formulag · (x_𝔫)_𝔫∈ S =(g(x_g^-1(𝔫)))_𝔫∈ S ,where g ∈(𝐐(ζ_p)/𝐐). The isomorphism f is then (𝐐(ζ_p)/𝐐)-equivariant.To conclude the proof of Proposition <ref>, it suffices to construct a canonical (𝐐(ζ_p)/𝐐)-equivariant group isomorphism γ_(i) :( ∏_𝔫∈ S( 𝐙[ζ_p]/𝔫)^×⊗_𝐙𝐙/p𝐙)_(i) U^⊗ 1-i⊗μ_p^⊗ i . We then let β_(i) := γ_(i)∘α_(i)^-1. There is a canonical group isomorphism( 𝐙[ζ_p]/𝔫)^×⊗_𝐙𝐙/p𝐙( 𝐙[ζ_p]/𝔫)^×[p]given by x ⊗ 1 ↦ x^N-1/p .We thus get a canonical group isomorphism( ∏_𝔫∈ S( 𝐙[ζ_p]/𝔫)^×⊗_𝐙𝐙/p𝐙)_(i)( ∏_𝔫∈ S( 𝐙[ζ_p]/𝔫)^× [p])_(i) .Note that there is a canonical group isomorphism U^⊗ 1-i⊗μ_p^⊗ i V ⊗(μ_p ⊗V̂)^⊗ i where V = (𝐙/N𝐙)^×[p] and V̂ = (V, 𝐙/p𝐙).Thus, it suffices to construct a canonical group isomorphismγ_(i)' : ( ∏_𝔫∈ S( 𝐙[ζ_p]/𝔫)^× [p])_(i) V ⊗ (μ_p ⊗V̂)^⊗ i .Let ζ_p ∈μ_p. If 𝔫∈ S, let ζ_𝔫∈ V be the reduction of ζ_p modulo 𝔫 and ζ̂_𝔫∈V̂ be given by ζ̂_𝔫(ζ_𝔫)=1. The element ζ_p ⊗ζ̂_𝔫∈μ_p ⊗V̂ only depends on 𝔫, and not on the choice of ζ_p. We letγ_(i)'((x_𝔫)_𝔫∈ S) = ∑_𝔫∈ S x_𝔫⊗ (ζ_p ⊗ζ̂_𝔫)^⊗ i .This is independent of the choice of ζ_p, so this is canonical. This concludes the proof of Proposition <ref>. Let ℓ be a rational prime not dividing N such that ℓ≡ 1(modulo p). Let λ be any prime above (ℓ) in 𝐙[ζ_p]. We define β_ℓ = β_(1)(_λ) ·β_(-1)(_λ) ∈μ_p ⊗( U^⊗ 2⊗μ_p^⊗ -1)= U^⊗ 2 . This does not depend on the choice of λ dividing (ℓ).Let K = 𝐐(N^1/p). Genus theory shows that 𝒞_K is non-zero <cit.>. By <cit.>, if g_p ≥ 2 then the rank of the 𝐅_p-vector space 𝒞_K is ≥ 2. The group 𝒞_K_(1) has an action of (K_(1)/𝐐(ζ_p)) (a cyclic group of order p) and of (K_(1)/K) = (𝐐(ζ_p)/𝐐) =(𝐙/p𝐙)^× (via the Teichmüller character).Let J be the augmentation ideal of (𝐙/p𝐙)[(K_(1)/𝐐(ζ_p))]. If χ : (K_(1)/K) → (𝐙/p𝐙)^× is a character, lete_χ = 1/p-1·∑_g ∈(K_(1)/K)χ^-1(g)· [g] ∈ (𝐙/p𝐙)[(K_(1)/K)]be the idempotent associated to χ. Let χ_0 be the trivial character and ω_p be the Teichmüller character (considered as a character (K_(1)/K) via the canonical restriction isomorphism (K_(1)/K) (𝐐(ζ_p)/𝐐)). If V is a (𝐙/p𝐙)[(K_(1)/K)]-module, we let V^(χ) = e_χ· V ⊂ V and V_(χ) = V/(⊕_χ' ≠χ e_χ'· V ). The map V^(χ)→ V_(χ) is a group isomorphism.Since [K_(1):K] = p-1 is prime to p, the natural map 𝒞_K→𝒞_K_(1) is injective, and its image is (𝒞_K_(1))^(χ_0).There exists a generator Δ of J such that for all character χ as above, we have <cit.>: Δ· e_χ = e_χ·ω_p·Δ .Thus, the multiplication by Δ induces a natural surjective group homomorphism(𝒞_K_(1)/Δ·𝒞_K_(1))^(ω_p^-1)→ (Δ·𝒞_K_(1)/Δ^2 ·𝒞_K_(1))^(χ_0) which gives a surjective group homomorphism:δ : (𝒞_K_(1)/Δ·𝒞_K_(1))_(ω_p^-1)→ (Δ·𝒞_K_(1)/Δ^2 ·𝒞_K_(1))_(χ_0) .The group (𝒞_K_(1)/Δ·𝒞_K_(1))_(ω_p^-1) is cyclic of order p.It was proven in <cit.> that the above map is an isomorphism if and only if g_p ≥ 2 (which is assumed in this section). Thus, we have a map𝒞_K = (𝒞_K_(1))_(χ_0)→ (𝒞_K_(1)/Δ^2 ·𝒞_K_(1))_(χ_0)and an exact sequence0 → (Δ·𝒞_K_(1)/Δ^2 ·𝒞_K_(1))_(χ_0)→ (𝒞_K_(1)/Δ^2 ·𝒞_K_(1))_(χ_0)→ (𝒞_K_(1)/Δ·𝒞_K_(1))_(χ_0)→ 0with(Δ·𝒞_K_(1)/Δ^2 ·𝒞_K_(1))_(χ_0)≃ (𝒞_K_(1)/Δ·𝒞_K_(1))_(χ_0)≃𝐙/p𝐙 . If ℓ is a prime such that ℓ≢1(modulo p), let 𝔭_ℓ be the unique prime of 𝒪_K above ℓ with residual degree 1 and 𝔭_ℓ be the image of the class of 𝔭_ℓ in (𝒞_K_(1)/Δ^2 ·𝒞_K_(1))_(χ_0). There is a unique group isomorphism α : (𝒞_K_(1)/Δ·𝒞_K_(1))_(χ_0) U such that for any prime ℓ≢1(modulo p), we have α(𝔭_ℓ) = ℓ^1/2 ,where ℓ is the image of ℓ in U. We letπ :(𝒞_K_(1)/Δ^2 ·𝒞_K_(1))_(χ_0)→ Ube the morphism induced by α. Assume that g_p ≥ 2, so that f_2 ∈ℳ/p·ℳ exists (and is uniquely determined modulo the subgroup generated by f_0 and f_1).Let 𝒫 be the set of prime numbers different from N. Let A_0 = (ℓ+1)_ℓ∈𝒫 and A_1=(ℓ-1/2·log(ℓ))_ℓ∈𝒫 in (𝐙/p𝐙)^(𝒫).There exists C ∈ (𝐙/p𝐙)^× and a group homomorphism π' :(𝒞_K_(1)/Δ^2 ·𝒞_K_(1))_(χ_0)→ U^⊗ 2 whose image in ( (𝒞_K_(1)/Δ^2 ·𝒞_K_(1))_(χ_0), U^⊗ 2)/ π⊗ U is non-zero and uniquely determined, such that we have, in (U^⊗ 2)^(𝒫) modulo the subgroup generated by A_0 ⊗ U^⊗ 2 and A_1 ⊗ U^⊗ 2: (a_ℓ(f_2) ⊗γ^⊗ 2)_ℓ∈𝒫 = ((ℓ^⊗ 2)^1/4·ϵ_ℓ)_ℓ∈𝒫where γ∈ U is such that log(γ) ≡ 1(modulo p) and ϵ_ℓ∈ U^⊗ 2 is defined as follows.* If ℓ≡ 1(modulo p), then we let ϵ_ℓ = β_ℓ^C.* If ℓ≢1(modulo p ), then we let ϵ_ℓ = ( π'(𝔭_ℓ)· (ℓ^⊗ 2)^1/8)^ℓ-1 . By <cit.>, if g_p ≥ 2 then there exists a Galois representation:ρ : (𝐐/𝐐) →GL_2((𝐙/p𝐙)[x]/x^3)which satisfies certain deformation conditions defined in <cit.>. We denote by F the number field cut out by the kernel of ρ. By <cit.>, one can assume that ρ has the following form:ρ =[ ω_p· (1 + a · x + a'· x^2)x ·β_1 + x^2· b;x·β_-1 + x^2· c1 +d· x + d'· x^2 ] .where a, a', b, c, d, d' and β_-1 are maps (𝐐/𝐐) →𝐙/p𝐙.We can choose ρ such that for all ℓ≠ N, if _ℓ is any Frobenius element at ℓ, we have:a(_ℓ) = log(ℓ)/2 .The cocycle β_-1 : (𝐐/𝐐) →𝐙/p𝐙 satisfies, for all g, g' ∈(𝐐/𝐐):β_-1(g· g') = β_-1(g)+ ω_p(g)^-1·β_-1(g').Furthermore, the restriction of β_-1 to (𝐐/𝐐(ζ_p)) is uniquely determined since we have fixed the choices of β_1, a and the conjugacy class of ρ.Since det(ρ) = ω_p, we have:a+d=0anda'+d' =ω_p^-1·β_1·β_-1 - a· d. Furthermore, as in <cit.> the subgroup ρ((F/𝐐)) of GL_2((𝐙/p𝐙)[x]/x^3) is generated by{[ α 0; 0 1 ], α∈ (𝐙/p𝐙)^×}∪(_2((𝐙/p𝐙)[x]/x^3) →_2(𝐙/p𝐙)).Since ρ is a group homomorphism, we have for all g, g' ∈Gal(𝐐/𝐐):a'(gg') = a'(g)+a'(g')+a(g)· a(g')+ω_p(gg')^-1·β_1(g)·β_1(g').Since the restriction of β_1 to (𝐐/K) is trivial, the restriction of a”:=a'-a^2/2 to (𝐐/K) is a group homomorphism. The commutator subgroup of ρ((F/K)) is generated by the matrices [ 1 x^2; 0 1 ], [ 1 0; x 1 ] and [ 1 0; x^2 1 ], and the abelianization of ρ((F/K)) is isomorphic to (𝐙/p𝐙)^2 × (𝐙/p𝐙)^×. The subgroup ρ((F/K)) of ρ((F/𝐐)) consists of those matrices in ρ((F/𝐐)) whose upper-right coefficient is 0 modulo x^2. Let A = [ a_1 0; a_3 a_4 ], B=[ b_1 b_2; b_3 b_4 ], C=[ c_1 0; c_3 c_4 ] and D = [ d_1 d_2; d_3 d_4 ]be in M_2(𝐙/p𝐙) and let α, β in 𝐙/p𝐙^×. Let Δ_α = [ α 0; 0 1 ] and Δ_β = [ β 0; 0 1 ]. A formal computation shows that we have, in GL_2((𝐙/p𝐙)[x]/x^3):(Δ_α + x· A + x^2· B)·(Δ_β + x· C + x^2· D) ·(Δ_α + x· A + x^2· B)^-1·(Δ_β + x· C + x^2· D)^-1= [ 1 0; 0 1 ] + x·[00; β-1/αβ· a_3+ 1-α/αβ· c_30 ]+x^2·[ 0 (1-β)· b_2 + (α-1)· d_3; X 0 ]where X = 1/αβ·(1-β/α· a_1a_3 + 1/β· a_3c_1 - 1/α· a_1c_3 + a_4c_3+α-1/β· c_1c_3-a_3c_4+ (β-1)· b_3 + (1-α)· d_3 ) .Thus, the commutator of ρ((F/K)) is {[ 1 u · x^2; v· x + w· x^2 1 ], u,v,w ∈𝐙/p𝐙} .Thus,the commutator of ρ((F/K)) has order p^3 and is generated by [ 1 x^2; 0 1 ], [ 1 0; x 1 ] and [ 1 0; x^2 1 ]. The second claim follows immediately.By Lemma <ref>, the kernel of the couple (a,a”) cuts out an abelian extension H of K such that (H/K) ≃ (𝐙/p𝐙)^2. By <cit.>, H is unramified everywhere over K. Furthermore, as a subquotient of the image of ρ, (H/K) is generated by the image of the matrices [ 1+x^2 0; 0 1-x^2 ] and [1+x0;0 (1+x)^-1 ]. Let ℋ be the maximal abelian extension of K_(1) unramified everywhere such that (ℋ/K_1) is a group of exponent p, and let 𝒢 = (ℋ/K_(1)). Global class field theory gives us a canonical (K_1/𝐐)-equivariant group isomorphism 𝒞_K_(1)≃𝒢. We claim that the compositum K_(0)K_(1) is unramified everywhere over K_(1). One only needs to check that K_(0)K_(1) is unramified at the primes above N in K_(1). This follows from the fact that ρ(I_N) is cyclic of order p, where I_N ⊂(𝐐/𝐐) is the inertia at N, and that K_(1) totally ramified of ramification index p at the primes above N in 𝐐(ζ_p). Thus, we have K_(0)K_(1)⊂ℋ. We have the following inclusions of subgroups of 𝒢:(ℋ/K_(0)K_(1)) = e_χ_0(Δ·𝒢) ·∏_χ≠χ_0 e_χ(𝒢)and(ℋ/H(ζ_p)) = e_χ_0(Δ^2 ·𝒢) ·∏_χ≠χ_0 e_χ(𝒢).Here, χ runs trough the set of characters (K_(1)/K) → (𝐙/p𝐙)^×. The first equality comes from genus theory: K_(0)K_(1) is the largest extension of K_(1) which is unramified everywhere,abelian over 𝐐(ζ_p) and such that (K_(1)/K) ≃(𝐐(ζ_p)/𝐐) acts by χ_0. Since (K_(1)/K) acts trivially on (H(ζ_p)/K_(1)), we have∏_χ≠χ_0 e_χ(𝒢) ⊂(ℋ/H(ζ_p)).Since (ℋ/H(ζ_p)) is a subgroup of index p of (ℋ/K_(0)K_(1)), it suffices to prove the following inclusion:e_χ_0(Δ^2 ·𝒢) ·∏_χ≠χ_0 e_χ(𝒢)⊂(ℋ/H(ζ_p)).Since Δ· e_ω_p^-1 = e_χ_0·Δ, it suffices to show the following lemma. The compositum K_(1)K_(-1) contained in ℋ. We have(ℋ/K_(1)K_(-1)) = e_ω_p^-1(Δ·𝒢) ·∏_χ≠ω_p^-1 e_χ(𝒢)andΔ·(ℋ/K_(1)K_(-1)) ⊂(ℋ/H(ζ_p)).The compositum K_(1)K_(-1) is unramified everywhere over K_(1) <cit.>, so is contained in ℋ The first equality comes from genus theory: K_(-1) is the largest extension of K_(1) which is unramified everywhere,abelian over 𝐐(ζ_p) and such that (𝐐(ζ_p)/𝐐) acts by ω_p^-1. Let M = K_(1)· K_(-1)· H; this is an abelian extension of K_(1) contained in F. The group ρ((F/M)) is the set of matrices of the form Id + x^2·[ 0 a; b 0 ] where a and b are in 𝐙/p𝐙. Thus, (F/M) is a normal subgroup of (F/𝐐(ζ_p)), which means that M is Galois over 𝐐(ζ_p). There is an action of (K_(1)/𝐐) on (M/𝐐(ζ_p)). It suffices to show that we haveΔ·(M/K_(1)K_(-1)) ⊂(M/H(ζ_p)) .In fact, we shall prove that we have Δ·(M/K_(1)K_(-1))= 0. The group ρ( (M/K_(1)K_(-1))) is generated by the images of [ 1+x^2 0; 0 1-x^2 ] and [1+x0;0 (1+x)^-1 ]. This follows indeed from the description of ρ((H/K)) given above.Let s be the image of [ 1 x; 0 1 ] in ρ((M/𝐐(ζ_p))). The image of s in ρ((K_(1)/𝐐(ζ_p)) ) generator of ρ((K_(1)/𝐐(ζ_p)) ). We have to show that the commutators[ 1 x; 0 1 ]·[ 1+x^2 0; 0 1-x^2 ]·[ 1 x; 0 1 ]^-1·[ 1+x^2 0; 0 1-x^2 ]^-1and[ 1 x; 0 1 ]·[1+x0;0 (1+x)^-1 ]·[ 1 x; 0 1 ]^-1·[1+x0;0 (1+x)^-1 ]^-1are in ρ((F/M)). If A, B, C and D are in M_2(𝐙/p𝐙), the commutator of 1+x· A + x^2· B and 1+x· C+x^2· D is 1+x^2· (AC-CA) (in GL_2((𝐙/p𝐙)[x]/x^3)). Thus, these two commutators are respectively 0 and [ 1 -2x^2; 0 1 ]. These matrices are in ρ((F/M)), which concludes the proof of Lemma <ref>.By Lemma <ref>, Global class field theory gives a canonical group isomorphism ϖ : (𝒞_K_(1)/Δ^2 ·𝒞_K_(1))_(χ_0)(H/K).The group homomorphism log∘π : (𝒞_K_(1)/Δ^2 ·𝒞_K_(1))_(χ_0)→𝐙/p𝐙 is the composition of the restriction of a to (H/K) with ϖ. We letπ' : (𝒞_K_(1)/Δ^2 ·𝒞_K_(1))_(χ_0)→ U^⊗ 2be the group homomorphism such that log^⊗ 2∘π' is the composition of the restriction of a” to (H/K) with ϖ, where log^⊗ 2 : U^⊗ 2𝐙/p𝐙 is defined by log^⊗ 2(a⊗ b) = log(a)·log(b). The group homomorphisms π' and π⊗γ are not proportionals (recall that γ is a generator of U such that log(γ)=1). For each prime ℓ≠ N, let a_ℓ∈𝐙/p𝐙 be defined by the following equality in (𝐙/p𝐙)[x]/x^3:(ρ(_ℓ)) = ℓ+1 + x·ℓ-1/2·log(ℓ) + x^2· a_ℓ ,where _ℓ is any Frobenius substitution at ℓ in (F/𝐐).We have, in (𝐙/p𝐙)^(𝒫) modulo the subgroup generated by A_0:=(a_ℓ(f_0))_ℓ∈𝒫 and A_1:=(a_ℓ(f_1))_ℓ∈𝒫:(a_ℓ(f_2))_ℓ∈𝒫 = (a_ℓ)_ℓ∈𝒫 .By <cit.>, the Galois representation ρ corresponds to a ring homomorphism φ : 𝐓̃→ (𝐙/p𝐙)[x]/x^3 where we recall that 𝐓̃ is the completion of the full Hecke algebra of weight 2 and level Γ_0(N) at the p-maximal Eisenstein ideal. Furthermore, we have by construction φ(T_ℓ) = (ρ(_ℓ)). We have normalized ρ such that (ρ(_ℓ)) ≡ℓ+1 + x·ℓ-1/2·log(ℓ)(modulo x^2). Thus, we have φ(T_ℓ-ℓ-1) ≡ x ·ℓ-1/2·log(ℓ)(modulo x^2 ) .The morphism φ corresponds a to a modular form F in M_2(Γ_0(N), (𝐙/p𝐙)[x]/x^3). For each n≥ 1, let a_n(F) be the nth Fourier coefficient of F at the cusp ∞. We writea_n(F) = a_n^0(F) + x· a_n^1(F)+x^2· a_n^2(F)in (𝐙/p𝐙)[x]/x^3 where a_n^i(F) ∈𝐙/p𝐙 for i ∈{0,1,2}. The q-expansion ∑_n ≥ 0 a_n^0(F) · q^n in 𝐙/p𝐙[[q]] is the q-expansion of the (normalized) Eisenstein series f_0 of weight 2 and level Γ_0(N). Thus, x ·∑_n ≥ 0 (a_n^1(F)+x· a_n^2(F))· q^n is the q-expansion of a modular form (namely F-f_0). By the q-expansion principle <cit.>, ∑_n ≥ 0 (a_n^1(F)+x· a_n^2(F))· q^n is the q-expansion at ∞ of a modular form in M_2(Γ_0(N), (𝐙/p𝐙)[x]/x^2). Thus, by specializing at x=0, we get that ∑_n ≥ 0 a_n^1(F) · q^n is the q-expansion at ∞ of a modular form in M_2(Γ_0(N), 𝐙/p𝐙), which we call F_1. Again, by the q-expansion principle, ∑_n ≥ 2 a_n^2(F)·q^n is the q-expansion at ∞ of a modular form in M_2(Γ_0(N), 𝐙/p𝐙), which we call F_2.We have, for every prime ℓ≠ N:(T_ℓ-ℓ-1)(F) = (T_ℓ-ℓ-1)(f_0)+x· (T_ℓ-ℓ-1)(F_1)+x^2· (T_ℓ-ℓ-1)(F_2) = x· (T_ℓ-ℓ-1)(F_1)+x^2· (T_ℓ-ℓ-1)(F_2) . Thus, we have:(T_ℓ-ℓ-1)(F) ≡ x · (T_ℓ-ℓ-1)(F_1)(modulo x^2) .On the other hand, we have (T_ℓ-ℓ-1)(F) = φ(T_ℓ-ℓ-1)· F. By (<ref>), we get:(T_ℓ-ℓ-1)(F_1) = ℓ-1/2·log(ℓ) · f_0 . Thus, F_1 is the first higher Eisenstein element f_1 (modulo the subgroup generated by f_0). Similarly, F_2 is the second higher Eisenstein element f_2 (modulo the subgroup generated by f_0 and f_1), which concludes the proof of Lemma <ref>. By Lemma <ref> and (<ref>), we have in (𝐙/p𝐙)^(𝒫) modulo the subgroup generated by A_0 and A_1:(a_ℓ(f_2))_ℓ∈𝒫 = ( ℓ· a'(_ℓ)+d'(_ℓ) )_ℓ∈𝒫 = ((ℓ-1)· a'(_ℓ) + β_1(_ℓ) ·β_-1(_ℓ)/ℓ + a(_ℓ)^2)_ℓ∈𝒫 = ((ℓ-1)· (a”(_ℓ) + log(ℓ)^2/8) + β_1(_ℓ) ·β_-1(_ℓ)/ℓ + log(ℓ)^2/4)_ℓ∈𝒫where in the second equality we used (<ref>) and (<ref>) and in the last equality we used (<ref>) and the definition of a”. If ℓ≢1(modulo p), then we may choose _ℓ in (F/K). With this choice, we have β_1(_ℓ) = 0 and a”(_ℓ) = log^⊗ 2(π'(𝔭_ℓ)). If we change π' by a multiple of π, we only change the right-hand side of (<ref>) by a multiple of A_1. Thus, the choice of π' is only well-defined up to a multiple of π. If ℓ≡ 1(modulo p), we have _ℓ∈(F/𝐐(ζ_p)). The restriction of β_1 to (F/𝐐(ζ_p)) factors through (K_(1)/𝐐(ζ_p)), thus giving a group isomorphism β_1' : (K_(1)/𝐐(ζ_p)) 𝐙/p𝐙. Fix a group isomorphism log_ζ : μ_p 𝐙/p𝐙. Then β_1' and log_ζ∘β_(1) are proportionals. Similarly, β_-1 gives a group isomorphism β_-1' : (K_(-1)/𝐐(ζ_p)) 𝐙/p𝐙 which is proportional to ( log^⊗ 2⊗log_ζ^⊗ -1) ∘β_(-1), where log_ζ^⊗ -1 : μ_p^⊗ -1𝐙/p𝐙 is induced by log_ζ. Thus, there exists C ∈ (𝐙/p𝐙)^× (independant of ℓ) such that β_1(_ℓ) ·β_-1(_ℓ) =C ·log^⊗ 2(β_ℓ).This concludes the proof of Theorem <ref>.The constant C is an invariant of the conjugacy class of the modular Galois representation ρ : (𝐐/𝐐) →_2(𝐓/(p·𝐓+I^2)) coming from J_0(N)[I^2+(p)](𝐐). The forthcoming thesis of Jun Wang (a Phd student of Sharifi) seems to imply that, at least conditionnally on a conjecture of Sharifi (which seems related to our Conjecture <ref>), we have C=1.Let 𝒫' be the set of prime numbers ℓ satisfying the following conditions. * We have ℓ≠ N,p* We have ℓ≡ 1(modulo p).* The prime N is a pth power modulo ℓ.The element (a_ℓ(f_2))_ℓ∈𝒫' of (𝐙/p𝐙)^(𝒫') is well-defined modulo the element X=(ℓ+1)_ℓ∈𝒫'. We have the following equality modulo (𝐙/p𝐙) · X in (𝐙/p𝐙)^(𝒫'):(a_ℓ(f_2))_ℓ∈𝒫' = (log(ℓ)^2/4)_ℓ∈𝒫' .This follows from Theorem <ref> and the fact that if ℓ≡ 1(modulo p), the following assertions are equivalent: * We have β_(1)(_ℓ) = 1. * The prime ℓ splits completely in 𝐐(ζ_p, N^1/p) * We have ℓ≡ 1(modulo p) and N is a pth power modulo ℓ. § TABLES AND SUMMARY OF OUR RESULTSThe following table for g_p was extracted from the data of <cit.> (Naskrecki extended his computations to N <13000, and kindly sent the result to us). We give the 5-uples (N, p, t, g_p, m), where N, p, t and g_p were defined in the article, and m is the number of conjugacy class of newforms which are congruent to the Eisenstein series modulo p. The range is N < 13000, and we only display the data where p≥ 5 and g_p ≥ 3. Let r be an integer such that 1 ≤ r ≤ t = v_p(N-1/12). The following tables summarize our results about the integer n(r,p) using the three 𝕋̃-modules M, M^- and M_+ studied in this paper (the equalities below take place in 𝐙/p^r𝐙 unless explicitly stated otherwise). plain | http://arxiv.org/abs/1709.09114v2 | {
"authors": [
"Emmanuel Lecouturier"
],
"categories": [
"math.NT"
],
"primary_category": "math.NT",
"published": "20170926162450",
"title": "Higher Eisenstein elements, higher Eichler formulas and rank of Hecke algebras"
} |
red [cor1]Corresponding author: [email protected][label1]Institute of Experimental Physics, Slovak Academy of Sciences, Watsonova 47, 040 01 Košice, Slovakia [label2]Department of Theoretical Physics and Astrophysics, Faculty of Science, P. J. Šafárik University, Park Angelinum 9, 040 01 Košice, Slovakia [label3]Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, SK-845 11, Bratislava, Slovakia An alternative model for a description of magnetization processes in coupled 2D spin-electron systems has been introduced and rigorously examined using the generalized decoration-iteration transformation and the corner transfer matrix renormalization group method. The model consists of localized Ising spins placed on nodal lattice sites and mobile electrons delocalized over the pairs of decorating sites. It takes into account a hopping term for mobile electrons, the Ising coupling between mobile electrons and localized spins as well as the Zeeman term acting on both types of particles. The ground-state and finite-temperature phase diagrams were established and comprehensively analyzed. It was found that the ground-state phase diagrams are very rich depending on the electron hopping and applied magnetic field. The diversity of magnetization curves can be related to intermediate magnetization plateaus, which may be continuously tuned through the density of mobile electrons. In addition, the existence of several types of reentrant phase transitions driven either by temperature or magnetic field was proven. strongly correlated systems Ising spins mobile electrons reentrant phase transitions metamagnetic transitions § INTRODUCTION Correlated spin-electron systems belong to the intensively studied materials in the condensed matter physics due to the variety of unconventional structural, electronic, magnetic, and transport properties <cit.> implying wide application potential. An exhaustive understanding of their origins thus opens a new way in technological applications, where suitable and unconventional properties could be used simultaneously.To achieve this goal, various types of models <cit.> in combination with less or more sophisticated methods <cit.> have been used. In spite of enormous effort, some of physical phenomena still lack full understanding since they have not been reliably explained so far. In the present paper we will consider two-dimensional (2D) coupled spin-electron systems consisting of localized Ising spins and mobile electrons using a relatively simple analytical method based on the Fisher mapping idea <cit.>. In this concept an arbitrary statistical-mechanical system, which merely interacts either with two or three outer Ising spins, may be replaced with effective interactions between the outer Ising spins through the generalized decoration-iteration or star-triangle mapping transformations <cit.>. This procedure was successfully applied to simulate magnetic properties of various two-component spin-electron systems in one <cit.> or two dimensions <cit.> with a good qualitative coincidence of magnetic behavior in real materials. For example, a 2D coupled spin-electron model could be used as a simplified theoretical model of selected rare-earth compounds, manganites or intermetallics with a quasi-2D character in which the existence of metamagnetic or reentrant transitions was observed <cit.>. These unconventional magnetic phenomena put aforementioned two-component spin-electron systems into the center of research interest, because one may easily control their magnetic states and thus manage various processes driven through a relatively simple change of external parameters, as for instance, temperature and/or magnetic field.In the present work we concentrate our attention to a coupled spin-electron model on the square lattice, which contains the localized Ising spins situated at the nodal lattice sites while mobile electrons are delocalized over the pairs of decorating sites placed at each of its bonds. Our previous study of the identical model with the absence of external magnetic field <cit.> has shown that this relatively simple model is able to describe a rich spectrum of unconventional physical properties, like the existence of various magnetic phases or presence of interesting reentrant phase transition in both the ferromagnetic and the antiferomagnetic limit. The rich spectrum of the zero-field model has motivated us to inspect the behavior of the same model under the influence of external magnetic field with the main goal to examine the existence of magnetization plateaus accompanied by the presence of metamagnetic transitions. Besides, we also concentrate on the existence of reentrant phase transitions in order to observe influence of external magnetic field and temperature on phase-transition stability. The organization of the paper is as follows. In Section <ref> we will introduce the investigated model and the method used for its solution. The ground-state and finite-temperature phase diagrams will be discussed in detail in Section <ref> along with the thermal dependencies of magnetization. In this section, we focus on possible emergence of the reentrant phase transitions as well as the intermediate magnetization plateaus in magnetization curves. Finally, the most significant results will be summarized together with future outlooks in the Section <ref>. § MODEL AND METHODLet us define an interacting spin-electron system on doubly decorated 2D lattices. The investigated model contains one localized Ising spin at each nodal lattice site and a set of mobile electrons delocalized over the pairs of decorating sites (dimers) placed at each bond. The total Hamiltonian of investigated spin-electron modelcan be written as a sum over allbond Hamiltonians Ĥ=∑_k=1^Nq/2Ĥ_k, where the symbol q denotes the coordination number of the underlying 2D lattice and N denotes the total number of its nodal lattice sites. Each bond Hamiltonian Ĥ_k contains the kinetic energy of mobile electrons on the k-th bond, the exchange interaction between the mobile electrons and their nearest-neighbor Ising spins, and the Zeeman terms describing the influence of external magnetic field on the magnetic moment of mobile electrons and localized spins (see Fig. <ref>) Ĥ_k= -t(ĉ^†_k_1,↑ĉ_k_2,↑^ +ĉ^†_k_1,↓ĉ_k_2,↓^ + ĉ^†_k_2,↑ĉ_k_1,↑^ +ĉ^†_k_2,↓ĉ_k_1,↓^ ) -Jσ̂^z_k_1(n̂_k_1,↑-n̂_k_1,↓)- Jσ̂^z_k_2(n̂_k_2,↑-n̂_k_2,↓) -h_e(n̂_k_1,↑-n̂_k_1,↓)- h_e(n̂_k_2,↑-n̂_k_2,↓) -h_i/q(σ̂^z_k_1+σ̂^z_k_2) . The symbols ĉ^†_k_α,γ and ĉ_k_α,γ^ (α=1,2; γ=↑,↓) in Eq. (<ref>) represent the creation and annihilation fermionic operators for the mobile electron. The respective number operators are denoted by n̂_k_α,γ=ĉ^†_k_α,γĉ_k_α,γ^ and n̂_k_α=n̂_k_α,↑+n̂_k_α,↓. σ̂^z_k_α denotes the z-component of the Pauli operator with the eigenvalues σ=±1. The first term in Eq. (<ref>) corresponds to the kinetic energy of mobile electrons delocalized over a couple of decoratingsites k_1 and k_2 from the k-th dimer modulated by the hopping amplitude t. The second and the third terms represent the Ising interaction between the mobile electrons and their nearest-neighbor Ising spins described by the parameter J. Finally, the last three terms in Eq. (<ref>) correspond to the Zeeman energy of themagnetic moments relevant tothe localized spins (h_i) and the delocalized electrons (h_e). The dissimilarity between the magnetic fields is implemented for the purpose of analytic calculations only, however, both the magnetic fields are considered equal to one another, h_i=h_e=h at the final stage of our analysis.For illustration, a schematic representation of the k-th bond of the aforementioned model is displayed in Fig. <ref> for a special case of the doubly decorated square lattice. It should be noted, however, that all derivations presented in Section <ref> are general and hold for an arbitrary 2D lattice. To study the ground-state as well as thermodynamic properties of the coupled spin-electron system defined by the Hamiltonian (<ref>), it is necessary to consider the grand-canonical partition function Ξ Ξ=∑_{σ}exp[-β∑_k=1^Nq/2(Ĥ_k- μn̂_k) ], where β=1/(k_BT), k_B is the Boltzmann's constant, T is the absolute temperature, n̂_k=n̂_k_1+n̂_k_2 is the number operator of mobile electrons delocalized over the k-th decorating dimer and μ is the chemical potential. The summation in Eq. (<ref>) runs over all possible spin configurations {σ} of the nodal Ising spins and the symbol Tr stands for the trace over the degrees of freedom of the mobile electrons only. Assuming the mutual commutativity of two bond Hamiltonians, i.e. [Ĥ_i,Ĥ_j]=0, one can partially factorize the grand-canonical partition function into the product of bond partition functions Ξ_kΞ =∑_{σ}∏_k=1^Nq/2_kexp(-βĤ_k) exp(βμn̂_k)=∑_{σ}∏_k=1^Nq/2Ξ_k . Here, the symbol Tr_k stands for the trace over the degrees of freedom of the mobile electrons from the k-th decorating dimer. This simplification allows us to calculate the partition function Ξ exactly out of the eigenvalues of the bond Hamiltonian (<ref>). Validity of the commutative relation between the bond Hamiltonians Ĥ_k and the number operator of mobile electrons per bond (n̂_k) implies that the matrix form of the bond Hamiltonian Ĥ_k can be divided into several disjoint blocks H_k(n_k) corresponding to the respective orthogonal Hilbert subspaces, which are characterized by different number of the mobile electrons (n_k) per bond. Thus, the eigenvalues of the bond Hamiltonians Ĥ_k can be calculated straightforwardly [n_k=0: E_k_1=-h_iL/q ,;n_k=1:E_k_2= -JL/2+√(J^2P^2+4t^2)/2-h_iL/q-h_e ,; E_k_3=-JL/2-√(J^2P^2+4t^2)/2-h_iL/q-h_e ,; E_k_4=+JL/2+√(J^2P^2+4t^2)/2-h_iL/q+h_e ,; E_k_5=+JL/2-√(J^2P^2+4t^2)/2-h_iL/q+h_e ,;n_k=2: E_k_6=-JL-h_iL/q-2h_e ,; E_k_7=+JL-h_iL/q+2h_e ,; E_k_8=E_k_9=-h_iL/q ,; E_k_10=+√(J^2P^2+4t^2)-h_iL/q ,; E_k_11=-√(J^2P^2+4t^2)-h_iL/q ,;n_k=3: E_k_12= -JL/2+√(J^2P^2+4t^2)/2-h_iL/q-h_e ,;E_k_13=-JL/2-√(J^2P^2+4t^2)/2-h_iL/q-h_e ,;E_k_14=+JL/2+√(J^2P^2+4t^2)/2-h_iL/q+h_e ,;E_k_15=+JL/2-√(J^2P^2+4t^2)/2-h_iL/q+h_e ,;n_k=4:E_k_16=-h_iL/q . ] For simplification, we have defined here two parameters L=σ_k_1+σ_k_2 and P=σ_k_1-σ_k_2.After tracing out the degrees of freedom of mobile electrons, the bond grand-canonical partition function Ξ_k depends only on the spin states of two localized Ising spins, whereas its explicit form can be replaced with the generalized decoration-iteration transformation <cit.> Ξ_k =∑_i=1^16exp(-β E_k_i)exp[βμ n_k(E_k_i)] =exp(β h_iL/q){1+4(z+z^3)cosh[β(JL/2+h_e)]×cosh[β/2√(J^2P^2+4t^2)]+ 2z^2{ 1+cosh[β (JL+h_e)]. .+cosh[β√(J^2P^2+4t^2)]} + z^4}= Aexp(β Rσ_k_1σ_k_2)exp(β h_efL/q) . Here, z=exp(βμ) is used to denote the fugacity of the mobile electrons. The physical meaning of this decoration-iteration transformation (<ref>) lies in replacement of a more complicated system (<ref>) by its simpler counterpart (<ref>) with new effective interactions. The evaluation of the mapping parameters A, R, and h_ef are given by "self-consistent" condition of the decoration-iteration transformation (<ref>), which must hold for all four combinations of the two Ising spins σ_k_1 and σ_k_2 requiring A=(V_1V_2V_3^2)^1/4,β R=1/4ln(V_1V_2/V_3^2), β h_ef=q/4ln(V_1/V_2), whereV_1 =exp(2β h_i/q){1+z^4+4(z+z^3)cosh[β(J+h_e)]×. .cosh(β t)+ 2z^2[ 1+cosh[2β(J+h_e)] +cosh(2β t)]}, V_2 =exp(-2β h_i/q){1+z^4+4(z+z^3)cosh[β(J-h_e)]×. .cosh(β t)+2z^2[ 1+cosh[2β(J-h_e)] +cosh(2β t)]}, V_3 =1+z^4+4(z+z^3)cosh(β√(J^2+t^2))cosh(β h_e) +2z^2[ 1+cosh(2β√(J^2+t^2))+cosh(2β h_e)] . Substitutingthe transformation (<ref>) - (<ref>) into the expression (<ref>), one obtains a simple mapping relation between the grand-canonical partition function Ξ of the interacting spin-electron system on the doubly decorated 2D lattices and, respectively, the canonical partition function Z_IM of a simple Ising model on the corresponding undecorated lattice with an effective nearest-neighbor interaction R and effective field h_ef Ξ(β,J,t,h)=A^Nq/2Z_IM(β,R,h_ef) . Obviously, the mapping parameter A cannot cause non-analytic behavior of the grand-canonical partition function Ξ. Hence, the investigated spin-electron system becomes critical if and only if the corresponding Ising model becomes critical as well.To study the model behavior in context of various electron concentrations, it is necessary to determine the equation of state relating its mean value ⟨ n_k⟩ with respect to the model parameters. The mean electron concentration ⟨ n_k⟩ can be straightforwardly derived from the grand potential Ω=-k_BTlnΞ ρ ≡⟨ n_k⟩=-(∂Ω/∂μ)_T=z/Nq/2∂/∂ zlnΞ=z∂/∂ zln A+zε∂/∂ zβ R +z/q⟨σ_k_1+σ_k_2⟩∂/∂ zβ h_ef=z/4(V'_1/V_1+V'_2/V_2+2V'_3/V_3)+z/4ε(V'_1/V_1+V'_2/V_2-2V'_3/V_3) +z/2⟨σ_k_1⟩(V'_1/V_1-V'_2/V_2), where ε=⟨σ_k_1σ_k_2⟩ denotes the nearest-neighbor pair correlation function and V'_1 =∂ V_1/∂ z=4exp(2β h_i/q){(1+3z^2)cosh(β (J+h_e))×. .cosh(β t)+z[1+cosh(2β (J+h_e))+cosh(2β t)]. .+z^3} , V'_2 =∂ V_2/∂ z=4exp(-2β h_i/q){(1+3z^2)cosh(β (J-h_e))×. .cosh(β t)+z[1+cosh(2β (J-h_e))+cosh(2β t)]. .+z^3} , V'_3 =∂ V_3/∂ z=4(1+3z^2)cosh(β√(J^2+t^2))cosh(β h_e) +4z[1+cosh(2β√(J^2+t^2))+cosh(2β h_e)]+4z^3 . As mentioned above, the main goal of this paper is to analyze the magnetization processes of the coupled 2D spin-electron model, and for this purpose, we separately derive expressions for the uniform sublattice magnetizations of localized spins m_i and the mobile electrons m_e per elementary unit cell m_i=-(∂Ω/∂ h_i)_z,m_e=-(∂Ω/∂ h_e)_z. The final formulas for the uniform sublattice magnetizations relate with the partial derivatives of all mapping parameters with respect to the relevant local fields h_i and h_e m_i =q/2[∂ln A/∂β h_i+ε∂β R/∂β h_i +2m_IM/q∂β h_ef/∂β h_i]=m_IMm_e =q/2[∂ln A/∂β h_e+ε∂β R/∂β h_e +2m_IM/q∂β h_ef/∂β h_e]=q/8[(1+ε) (W_1/V_1+W_2/V_2)+2(1-ε)W_3/V_3. +.2m_IM(W_1/V_1-W_2/V_2)], where the coefficients W_1, W_2 and W_3 are defined as follows: W_1 =∂ V_1/∂β h_e=4 exp(2β h_i/q)[ (z+z^3)cosh(β t)×. .sinh(β (J+h_e))+z^2sinh(2β (J+h_e))] ,W_2 =∂ V_2/∂β h_e=-4 exp(-2β h_i/q)[ (z+z^3)cosh(β t)×. .sinh(β (J-h_e))+z^2sinh(2β (J-h_e))] ,W_3 =∂ V_3/∂β h_e=4 [ (z+z^3)cosh(β√(J^2+t^2))sinh(β (h_e)). +.z^2sinh(2β (h_e))]. The total uniform magnetization of the coupled spin-electron model normalized with respect to its saturation value is then given by m_tot=m_i+m_e/1+ρ. It should be mentioned that the uniform magnetization is a convenient order parameter of the ferromagnetic ordering, however, it is inapplicable to the antiferromagnetic type of ordering. For this reason, we define new order parameters known as the staggeredsublattice magnetizations of localized spins m^s_i and the mobile electrons m^s_em^s_i =1/2⟨σ_k_1-σ_k_2⟩=m^s_IM, m^s_e =⟨1/Ξ_k[∂Ξ_k/∂β Jσ_k_1-∂Ξ_k/∂β Jσ_k_2]⟩=4Jm^s_i/V_3√(J^2+t^2)[(z+z^3)sinh(β√(J^2+t^2))cosh(β h_e). +.z^2sinh(2β√(J^2+t^2))], and, in analogy to the former case, the total staggered magnetization normalized to its saturation value is defined m^s_tot=m_i^s+m_e^s/1+ρ.All the derived analytical expressions depend on the uniform (<ref>) and the staggered (<ref>) magnetizations of the effective Ising model as well as on the nearest-neighbor correlation function ε, cf. Eqs. (<ref>) and (<ref>). To evaluate those quantities accurately, we have adapted the Corner Transfer Matrix Renormalization Group (CTMRG) method <cit.> to all the subsequent calculations. The CTMRG is a numerical algorithm applicable to 2D classical lattice spin models, which is build on ideas of the Density Matrix Renormalization Group method <cit.>. It enables to calculate all the thermodynamic functions efficiently and accurately. The main advantage of CTMRG lies in higher numerical accuracy of the thermodynamic functions (if compared with the Monte Carlo simulations <cit.>) when analyzing phase transitions and their critical behavior in various 2D spin systems. § RESULTS AND DISCUSSIONThe following section introduces the most interesting results obtained from the study of the magnetization processes in the coupled spin-electron model on doubly decorated square lattice with the coordination number q=4and the ferromagnetic nearest-neighbor interaction J>0 between the localized spins and mobile electrons. Without the loss of generality, the magnitude of this interaction and the Boltzmann constant are set to unity (i.e., J=1, k_B=1). The number of the free parameters can be further reduced by considering h_i=h_e=h>0. Finally, the electron density per decorating dimer may be restricted up to the half filling at most, i.e. 0≤ρ≤ 2 since the particle-hole symmetry applies for ρ>2. §.§ Ground stateWe start our discussion with the ground-state phase diagrams established in the μ-h plane for a wide range of the hopping parameter t. The detailed description of all possible phases forming the ground-state phase diagrams is listed in Tab. <ref> along with the associated ground-state energies E. After excluding the two trivial phases of the bond subsystem with ρ=0 (zero electron occupancy) and ρ=4 (full-filling), we identify three types of ground-state phase diagramswith typical examples presented in Fig. <ref>.The first type, which is represented in Fig. <ref> for t=0.15, occurs whenever the hopping term is below the critical value t_c( II_1- II_3)=√(h(1+1/q)[2J+h(1+1/q)]). Under this condition, the fully polarized (F) spin-electron state is present. Surprising existence of the F state within the half-filled case, for which the quantum antiferromagnetic (AF) state at h=0 with a perfect Néel order of the nodal spins was previously detected <cit.>, can be explained as follows: rather weak correlations between the mobile electrons induced by the hopping term t become insignificant in comparison with the exchange interaction J between the spin and electron subsystems. Hence, it follows that an arbitrary magnetic field h≠0 aligns all the spins into field direction. The minimal effect of the hopping term is likewise reflected in stabilizing the phase II_1 with two mobile electrons per bond, which is dominant in the phase diagram, contrary to the phases I and III with odd number of the mobile electrons per bond existing in narrow regions only.The competitionamong the hopping term t, the exchange coupling J, and the magnetic field h becomes more intricate above the critical value of the hopping term t>t_c( II_1- II_3). Then, the quantum AF state II_3 (with the Néel spin order), which is observable for ρ=2 and h=0, persists whenever the magnetic fieldis smaller than q(√(J^2+t^2)-J)/(1+q). At this value, the effect of the hopping term t is completely suppressed by the magnetic field, and the system undergoes a discontinuous phase transition to the F state II_1 (see the case t=1 in Fig. <ref>). It is evident from Tab.<ref> that the occurrence probabilities of microstates emerging within the phase II_3 strongly depend on the parameters t and J, but they do not depend on the magnetic field h. In addition, Fig. <ref> demonstrates that the most probable spin orientation of the mobile electrons follows the spin orientation of the localized Ising spins, i.e. the occurrence probability of the microstate |↑,↓⟩ is always the highest within the phase II_3. Another interesting observation is that stronger correlations between the mobile electrons stabilize the phases I and III with odd number of the mobile electrons. To conclude, the second type of the ground-state phase diagram includes six different ground-state phases and can be found for moderate values of the hopping term t_c( II_1- II_3)<t<t_c( II_2- II_3)=q[J^2-(h/q)^2]/2h. Last but not least, the competition between the model parameters may generate an extra phase II_2 at the half-filled band case whenever the hopping term exceeds the critical value t_c( II_2- II_3). The third type of the phase diagram thus totally involves seven ground states, as represented by the special case t=4 in Fig. <ref>. The novel phase occurs in between the quantum AF phase II_3 and the classical F phase II_1. It can be regarded as an intermediate phase with the mixed character of F-AF order. Namely, the external magnetic field primarily forces the localized Ising spins to align into a magnetic field, but the electronic subsystem still displays a quantum AF order owing to a strong electron correlation mediated by the hopping term t. The phase boundaries among all the three phases II_1, II_2, and II_3 with two mobile electrons per bond are given by the following conditions: : h=t-J,: h=q(√(J^2+t^2)-J)/(1+q) ,: h=q(√(J^2+t^2)-t). Moreover, it can beobserved from Fig. <ref> thatthe stability regions of the phases I and III with odd number of mobile electrons per bond are in the third type of the ground-state phase diagram repeatedly wider in comparison with two aforementioned cases, which leads to the conclusion that the quantum-mechanical hopping energetically favors the configurations with odd number of mobile electrons per bond. For the sake of completeness, analytical expressions for the other ground-state phase boundaries associated with discontinuous (first-order) phase transitions are derived by comparing energies from Tab. <ref> resulting in : μ=-J-h-t, : μ=-J-h+t, : μ=J+h-t, : μ=J+h+2h/q+t-2√(J^2+t^2), : μ=J+h-t, : μ=-J-h+t, : μ=-J-h-2h/q-t+2√(J^2+t^2), : μ=J+h+t. To analyze possible metamagnetic transitions caused by the external magnetic field h and the electron density ρ, the total uniform magnetization m_tot of the spin-electron model on the doubly decorated square lattice is plotted in Fig. <ref> at low temperature T=0.021. In general, one detects a close coincidence between the low-temperature magnetization curves shown in Fig. <ref> and the ground-state phase diagrams depicted in Fig. <ref>. As a matter of fact, the total magnetization m_tot normalized with respect to its saturation value always reaches unity (excluding the zero-field case h=0) for t=0.15, which corroborates F order within both subsystems for all 0<ρ≤ 2. The total magnetization saturates upon strengthening the magnetic field at low electron densities 0≤ρ≤ 1 assuming t=1. At the same time, it exhibits an intermediate plateau before being saturated at the higher electron densities 1<ρ≤ 2. The stability of this intermediate plateau (in the whole range of the electron concentration 1< ρ≤ 2) coincides with the phase II_3, for which the critical field h_c is given by Eq. (<ref>). The observed magnetization plateau indeed turns into a zero magnetization plateau in the half-filling case ρ=2. It agrees with the AF nature of the phase II_3, whereas the critical field h_c shows only a small shift towards lower magnetic fields upon decreasing of the electron concentration ρ. It is noteworthy that the height of the intermediate magnetization plateau can be continuously tuned according to the formula m_tot=2(2-ρ)/(1+ρ) upon varying of the electron density within the range 1< ρ≤ 2. The striking dependence of the height of intermediate plateau on the electron density can be attributed to a competition between the local F order supported by a single hopping electron per bond and the local AF order supported by a hopping of two mobile electrons per bond. Owing to this fact, the height of intermediate magnetization plateau interpolates betweenm_tot=1 and m_tot=0 when the electron density changes from a quarter filling to a half filling.The ground-state phase diagram shown in Fig. <ref> suggests that the magnetization curves at t=4 should include two intermediate magnetization plateaus for the electron densities ρ>1 in concordance with the 3D magnetization plot displayed in Fig. <ref>. The first intermediate plateau emergent at lower magnetic fields at m_tot=2(2-ρ)/(1+ρ) has the same origin, as described above for the case with a moderately strong hopping term,while the second intermediate plateau originates from the mutual competition between the magnetic field and the hopping term. In the latter case, the external magnetic field is strong enough to polarize the localized Ising spins, although it does not suffice to break the AF correlation of two mobile electrons supported by the quantum-mechanical hopping process. In fact, the sublattice magnetization of the spin subsystem m_i=1 is saturated within this intermediate magnetization plateau unlike the sublattice magnetization of the electron subsystem depending on the electron concentration according to the formula m_e=(2-ρ). Owing to this fact, the other magnetization plateau appears at the following value of the total magnetization m_tot=(3-ρ)/(1+ρ). The critical magnetic fields, at which the investigated spin-electron system undergoes steep changes of the magnetization connected to the appearance of the phase II_2, coincide with the critical fields given by Eqs. (<ref>) and (<ref>).For completeness, we also investigate the staggered magnetization as an order parameter for the AF type of ordering. It turns out that the staggered magnetization is non-zero for t>t_c( II_1- II_3) and ρ→ 2 in accordance with the stability region of the phase II_3. It was found, that the non-zero value of staggered magnetization m^s_tot is accompanied withinthe phase II_3 by the non-zero value of the uniform magnetization m_tot in response to a presence of magnetic field, and thus a new type of the AF ordering with F features (AF^*) is formed. In contrast to the uniform magnetization m_tot, the staggered magnetization m^s_tot does not exhibit any stepwise dependence on magnetic field. Instead, it exhibits a plateau, whose height strongly depends on the electron concentration ρ. Here we notice that the non-zero staggered magnetization m^s_tot≠ 0 is observed at lower electron fillings in comparison with the zero-field limit h=0. This is an indication for existence of reentrant phase transitions. In addition, the external magnetic field can also cause the reentrant phase transitions while fixing the electron concentration ρ. Hence, one of the following two sequences of the reentrant transitions is generated: F_m_tot=a–AF^*–F_m_tot=1 or F_m_tot=a–AF^*–F_m_tot=b–F_m_tot=1 for a<b<1 depending on the strength of the hopping term t. Except the aforementioned reentrances, the investigated spin-electron model also exhibits additional field-induced phase transitions for the electron concentration close to a quarter and half filling: [_m_tot≠ 1_m_tot=1t_c( II_1- II_3) < t < t_c( II_2- II_3),; ρ→ 1,; _m_tot=a_m_tot=b_m_tot=1 t>t_c( II_2- II_3),; ρ→ 1,; _m_tot=1t_c( II_1- II_3) < t < t_c( II_2- II_3),;ρ→2,;_m_tot≠ 1_m_tot=1 t>t_c( II_2- II_3),;ρ→2,; _m_tot=1t_c( II_1- II_3) < t < t_c( II_2- II_3),;ρ=2,;_m_tot≠ 1_m_tot=1 t>t_c( II_2- II_3),;ρ=2. ]§.§ Thermodynamics Let us analyze magnetic behavior of the investigated system at finite temperature using the CTMRG method <cit.>, which is designed to compute the canonical partition function Z_IM=∑exp[-β H_IM(R,h_ef)], cf. Eq. (<ref>), within sufficiently high numerical accuracy. We primarily focus on the cooperative phenomena originating from the competition between the kinetic term, exchange coupling, magnetic field and temperature. Provided that the hopping term t<t_c( II_1- II_3), only the F ground states are present in the finite-temperature phase diagramregardless of the electron filling. For t>t_c( II_1- II_3), the phase diagram in the T-ρ plane at zero magnetic field involves paramagnetic (P), AF, and F phases, as comprehensively studied in our previous work <cit.>.The effect of the external magnetic field is expected to be most pronounced within the P phase, where randomly oriented spins are forced to align in the magnetic-field direction. Moreover, it is reasonable to assume that the AF phase shrinks within the finite-temperature phase diagram in response to strengthening of the external magnetic field. In accordance with our expectations, the critical temperature of the AF phase with m^s_tot≠ 0 reduces upon the strengthening of the magnetic field h for most of theelectron concentrations. As already mentioned above, the non-zero magnetic fields generate the AF spin arrangement at slightly lower electron concentrations with respect to the zero-field case. Therefore, the phase diagram in Fig. <ref> at t=1 shows interesting thermally-induced reentrant phase transitions at low magnetic fields (h=0 and h=0.01), where three consecutive phase transitions separate the sequence of the phases AF^*–F–AF^*–F for the electron concentration ρ≈1.84. However, the reentrance completely vanishes at greater values of the hopping term as exemplified on theparticular case t=4. Besides the usual thermal reentrant phase transitions, we also plotted field-induced reentrant phase transitions at fixed non-zero temperature in Fig. <ref>.Let us recall that the AF^* phase is used to denote such a parameter space, which is typical for both the non-zerouniform m_tot≠ 0 as well asthe staggered m_tot^s≠ 0 magnetizations. The existence of the AF^* phase is a consequence of the two opposite competing effects: (1) the AF order, originating from a quantum hopping process of the mobile electrons, and (2) the F order caused by theexternal magnetic field. To get a deeper insight, we have analyzed thermal behavior of both sublattice magnetizations of the spin and electron subsystems along with the total magnetization. The results are presented in Fig. <ref> and Fig. <ref>. Evidently, the existence of the AF^* phase is dominantly conditionedby the electron filling ρ and, of course, by the competition of all present interactions. Both the electron filling and the value of the electron hopping t strongly determine the number of AF and non-magnetic bonds in system (clearly visible from the evolution of probability in T=0, Fig. <ref>), which is reflected in the value ofm_e^s(T→ 0). Since the m_i^s(T→ 0) is independent on t, the m^s_tot(T→ 0) strongly depends on electron hopping processes. As our analyses showed, the existence of non-zero m_totaffected by the external magnetic field h is indirectly conditioned by the electron subsystem, because the lower (higher) number of AF bonds produces the smaller (stronger) damped forces to reorient themagnetic moments into the field direction. Consequently, the existence and character of the AF^* phase is strongly determined by the features of electron subsystem. Another interesting observation is that the AF^* phase maintains its mixed ferro-antiferromagnetic character up to higher temperatures at higher magnetic fields. However, the most interesting thermal behavior of the magnetization can be detected when thermal reentrant phase transitions take place. Typical thermal variations of the magnetization with successive reentrant phase transitions are presented in Fig. <ref> for the electron density ρ=1.84 and the hopping term t=1.It is obvious from Fig. <ref> that there exist two regions with the non-zero staggered magnetization m_tot^s ≠ 0 and the non-zero uniform magnetization m_tot≠ 0 for the sufficiently small magnetic field h=0.006. As one can see, the increasing temperature basically reduces the total uniform magnetization m_tot as well as the total staggered magnetization m_tot^s. The latter AF order parameter m_tot^s becomes zero at moderate temperatures, while the former F order parameter m_tot retains non-zero albeit relatively small value due to the non-zero external magnetic field. The AF order re-appears at higher temperatures as evidenced by a sudden uprise of the total staggered magnetization m_tot^s, which finally disappears at third (highest) critical temperature. The thermal reentrance naturally vanishes at higher magnetic fields (e.g. h=0.2), whereas the total uniform and staggered magnetizations m_tot and m_tot^s then become almost thermally independent at low enough temperatures. Our thermal analysis can imply a great potential of the studied spin-electron system for technological applications, because different magnetic states are controllable by various external parameters, such as temperature, magnetic field and/or electron density. To summarize, one may tune the investigated spin-electron model across several types of the magnetic phase transitions with respect to the electron concentration:[ ,; ,; ,; . ]§ CONCLUSIONS In the present paper we have examined the coupled spin-electron model on the doubly decorated square lattice in presence of the external magnetic field by combining the analytic decoration-iteration mapping transformation with the numerical CTMRG method. Our analysis was primarily concentrated on the magnetization processes elucidating the intermediate magnetization plateaus, metamagnetic transitions, and the reentrant phase transitions. Both the ground-state and the finite-temperature phase diagrams were studied in detail with respect to the electron filling. It has been found that a spin arrangement emerges within individual ground states and strongly depends on the mutual interplay among the hopping term, the exchange interaction, the external magnetic field, and the electron concentration. The non-zero values of external magnetic field result in the richer spectrum of magnetic ground-state phase diagrams. Three types of the ground-state phase diagrams were identified, which depend on the electron hopping term. The first type of the phase diagram solely exhibits the F type of ordering in both the spin and electronic subsystems within the entire parameter space. The remaining two types of the ground-state phase diagrams contain magnetic states with the AF ordering in the both subsystems; an even more strikingly, a combined F ordering of the localized spins accompanied with the AF ordering of the mobile electrons. These novel ground states are responsible for the appearance of the intermediate plateaus in low-temperature magnetization curves including metamagnetic transitions in between them. In addition, it has been shown that the intermediate magnetization plateaus emerge above the quarter filling (ρ>1) only, and the height of magnetization plateaus is continuously tunable by the electron doping as evidenced by the derived exact formulae.The most remarkable finding refers to theAF^* phase detected close to the half-filling case ρ→ 2, which simultaneously carries non-zero uniform and staggered total magnetizations m_tot≠ 0 and m_tot^s ≠ 0, respectively. The existence of such a phase is the direct consequence of the present magnetic field because its existence has not been determined in the zero-field counterpart yet. Moreover, it turns out that the AF^* phase can re-appear at higher temperatures on account of reentrant phase transitions driven either by temperature or magnetic field. The most surprising finding is that a relatively simple spin-electron model can describe the existence of phase with the F as well as AF features along with other significant magnetic phenomena of cooperative nature, whichhave been experimentally observed in several real magnetic materials. In particular, doped manganites exhibit quasi-2D character <cit.> and the magnetic behavior basically depending on the electron doping, whereas the AF and F orders may indeed coexist together in some manganites <cit.>. It is also generally known that the manganites also exhibit other unconventional phenomena <cit.>, which may originate from a competition between the localized and mobile magnetic particles. In this regard, our simple model reproduces severalmagnetic features such as multistep magnetization curves, metamagnetic transitions, and reentrant phase transitions, which all arise from the mutual competition of the kinetic term, the exchange coupling, the magnetic field, and the electron density. Our theoretical achievements presented in this work thus have obvious potential to contribute significantlyexplaining the unconventional cooperative phenomena of the correlated spin-electron systems. This work was supported by the Slovak Research and Development Agency (APVV) under Grants No. APVV-0097-12, APVV-0808-12 and APVV-16-0186. The financial support provided by the VEGA under Grants No. 1/0043/16and 2/0130/15 is also gratefully acknowledged. 00Kanamori J. Kanamori, Prog. Theor. Phys. 30 (1963) 275. Takada K. Takada, H. Sakurai, E. Takayama-Murinachi, F. Izumi, R. A. Dilanian, and T. Sasaki, Nature (London) 422 (2003) 53. Honecker A. Honecker, J. Schulenburg, and J. Richter, J. 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Mihalik, Z. Jagličić, R. Vilarinho, J. Agostinho Moreira, E. Queiros, P. B. Tavares, A. Almeida, and M. Zentková, Physica B 506 (2017) 163. Nashino T. Nishinoand K. Okunishi, J. Phys. Soc. Jpn. 65 (1996) 891-894; J. Phys. Soc. Jpn. 66 (1997) 3040.White S. R. White, Phys. Rev. Lett. 69 (1992) 2863; Phys. Rev. B 48 (1993) 10345. Binder K. Binder, ”Monte Carlo Simulation in Statistical Physics: An Introduction”, Springer Series in Solid-State Sciences 80 (Springer Berlin, 2002).Ganguly P. Ganguly, J. Solid State Chem. 53 (1984) 193. Moreo A. Moreo, S. Yunoki, and E. Dagotto, Science 283 (1999) 2034. Salamon M. B. Salamon, Rev. Mod. Phys. 73 (2001) 583. Reis M. S. Reis, A. M. Gomes, J. P. Araujo, P. B. Tavares, J. S. Amaral, I. S. Oliveira, and V. S. Amaral,Mater. Sci. Forum 455-456 (2004) 148. | http://arxiv.org/abs/1709.09341v1 | {
"authors": [
"Hana Čenčariková",
"Jozef Strečka",
"Andrej Gendiar"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170927053551",
"title": "Magnetization processes and existence of reentrant phase transitions in coupled spin-electron model on doubly decorated planar lattices"
} |
Dykstra splitting and approximate proximal point]Dykstra splitting and an approximate proximal point algorithm for minimizing the sum of convex functions [2010]90C25, 65K05, 68Q25, 47J25We show that Dykstra's splitting for projecting onto the intersection of convex sets can be extended to minimize the sum of convex functions and a regularizing quadratic. We give conditions for which convergence to the primal minimizer holds so that more than one convex function can be minimized at a time, the convex functions are not necessarily sampled in a cyclic manner, and the SHQP strategy for problems involving the intersection of more than one convex set can be applied. When the sum does not involve the regularizing quadratic, we discuss an approximate proximal point method combined with Dykstra's splitting to minimize this sum.We acknowledge grant R-146-000-214-112 from the Faculty of Science, National University of Singapore. We gratefully acknowledge discussions with Ting-Kei Pong on Dykstra's splitting which led to this paper. Department of Mathematics National University of Singapore Block S17 08-11 10 Lower Kent Ridge Road Singapore [email protected][ C.H. Jeffrey Pang December 30, 2023 ===================== § INTRODUCTION Throughout this paper, let X be a finite dimensional Hilbert space. Consider the problem of minimizing the sum of convex functions ∑_i=1^rh_i(·),where h_i:X→ℝ∪{∞} are closed proper convex functions. The aim of this paper is to combine Dykstra's splitting and an approximate proximal point algorithm in order to minimize (<ref>). §.§ Dykstra's algorithm For closed convex sets C_i, where i∈{1,…,r}, Dykstra's algorithm <cit.> solves the problem min_x1/2x-x_0^2+∑_i=1^rδ_C_i(x),where δ_C_i(·) is the indicator function of the set C_i. Note that (<ref>) is also equivalent to the problem of projecting the point x_0 onto ∩_i=1^rC_i. The projection onto the intersection ∩_i=1^rC_i may be difficult, but each step of Dykstra's algorithm requires only the projection onto one set C_i at a time. Its convergence to a primal minimizer without constraint qualifications was established in <cit.>. Separately, Dykstra's algorithm was rediscovered in <cit.>, who noticed that it is block coordinate minimization on the dual problem, and proved the convergence to a primal minimizer, but under a constraint qualification. This dual perspective was also noticed by <cit.>, who built on <cit.> and used duality to prove the convergence to a primal minimizer without constraint qualifications. Dykstra's algorithm can be made into a parallel algorithm by using the product space approach largely attributed to <cit.>. But this parallelization is slower than the original Dykstra's algorithm because the dual variables are not updated in a Gauss Seidel manner. (In other words, the dual variables are not updated with the most recent values of the other dual variables.) It was also noticed in <cit.> (among other things) that the projections onto the sets C_i need not be performed in a cyclic manner to achieve convergence. In <cit.>, we studied a SHQP (supporting halfspace and quadratic programming) heuristic for improving the convergence of Dykstra's algorithm by noticing that the projection operations onto the sets C_i generate halfspaces containing C_i, and the intersection of these halfspaces can be a better approximate of ∩_i=1^rC_i than each C_i alone. We now refer to the natural extension of Dykstra's algorithm for minimizingmin_x1/2x-x_0^2+∑_i=1^rh_i(x),where h_i(·) are generalized to be closed convex functions, as Dykstra's splitting. Instead of projections, one now uses proximal mappings. (See (<ref>) for an example.) Dykstra's splitting was studied in <cit.> and <cit.> for the case of r≥2, and they proved the convergence (to the primal minimizer) under constraint qualifications. It was also proved in <cit.> that Dykstra's splitting converges for the case of r=2 without constraint qualifications. Dykstra's algorithm is related to the method of alternating projections for finding a point in the intersection more than one closed set. For more information on the various topics in Dykstra's algorithm mentioned so far, we refer to <cit.>.§.§ Block coordinate minimization For the problem of minimizing f(x)+g(x), where f(·) is smooth and g(·) is block separable, one strategy is to minimize one block of the variables at a time, keeping the others fixed. This strategy is called block coordinate minimization, or alternating minimization. Nonasymptotic convergence rates of O(1/k) to the optimal value were obtained for when the smooth function is not known to be strongly convex in <cit.>. We refer to these papers for more on the history of block coordinate minimization. The smooth portion of the dual problem in Dykstra's algorithm is a specific quadratic function, so block coordinate minimization for this problem coincides with a block coordinate proximal gradient approach in <cit.>. Convergence properties of minimizing over more than one block at a time were discussed. There is too much recent research on block coordinate minimization and block coordinate proximal gradient, so we refer the reader to the two recent references <cit.> and their references within.§.§ Proximal point algorithm The proximal point algorithm attributed to <cit.> is a method for finding minimizers of min_xf(x) by creating a sequence {x_j}_j such that [ x_j+1≈_f(x_j):=min_xf(x)+1/2x-x_j^2. ]It was noticed in <cit.> that one can use the proximal point algorithm to solve (<ref>) by approximately solving a sequence of problems of the form (<ref>) using Dykstra's algorithm. The rules there for moving to a new proximal center x_j involves finding a primal feasible point that satisfies the optimality conditions approximately. But such a feasible point might not be found in a finite number of iterations when some of the functions h_i(·) are indicator functions, so a separate rule for moving the proximal center is needed.§.§ Other methods for minimizing the sum of functions When the constraint sets are either too big and have to be split up as the intersection of more than 1 set, or when these constraint sets are only revealed as the algorithm is run, it is beneficial to write these problems in the form (<ref>) where two or more of the h_i(·) are indicator functions. In such a case, as remarked in <cit.>, the accelerated methods of <cit.> and further developed by <cit.> do not immediately apply (to the primal problem). We now recall other methods and observations on minimizing (<ref>) when more than one of the functions h_i(·) are indicator functions and the algorithm can operate on a few of the functions h_i(·) at a time. As we have seen earlier, Dykstra's algorithm is one such example. In the case where all the functions h_i(·) in (<ref>) are indicator functions, then this problem coincides with the problem of finding a point in the intersection of convex sets, which is a problem of much interest on its own. (See for example <cit.>.) We refer to this as the convex feasibility problem. The convex feasibility problem can be solved by the method of alternating projections and the Douglas-Rachford method. A discussion of the effectiveness of methods for the convex feasibility problem is <cit.>.Beyond the convex feasibility problem, various extensions of the subgradient method in <cit.> can solve problems of the form (<ref>). Another recent development is in superiorization (See for example <cit.>), where an algorithm for the convex feasibility problem is perturbed to try to reduce the value of the objective function. The result is an algorithm that seeks feasibility at a rate comparable to algorithms for the feasibility problem, while achieving a superior objective value to what an algorithm for the feasibility problem alone would achieve. A comparison of projected subgradient methods and superiorization is given in <cit.>. A typical assumption on the constraint sets is that they have a Lipschitzian error bound, which is also equivalent to the stability of the intersection under perturbations. See for example <cit.>. Lastly, another method for minimizing (<ref>) is the ADMM <cit.>. The ADMM is an effective method, but we feel that Dykstra's splitting still has its own value. For example, as we shall see later, the different agents can minimize in any order, and convergence doesn't even require the existence of a dual minimizer. In problems where the different agents are assumed not to be able to freely communicate between each other or if communications between two agents are one dimensional, methods derived from subgradient algorithms can still be a method of choice <cit.>, even though many algorithms are preferred over the subgradient algorithm in large scale problems with less restrictive communcation requirements <cit.>.We refer to the survey <cit.> for other proximal techniques for minimizing (<ref>). §.§ Contributions of this paper Firstly, in Section <ref>, we extend Dykstra's splitting for minimizing (<ref>) so that (A)the proof of convergence does not require constraint qualifications,(B)the r in (<ref>) is any number greater than or equal to 2, and (C)h_i(·) can be any closed convex function instead of the indicator function.As mentioned earlier, <cit.> and <cit.> have features (A) and (B),<cit.> has (B) and (C), and <cit.> has (A) and (C). We are not aware of Dykstra's splitting being proved to have features (A), (B) and (C). In addition, our analysis incorporates these features that are now rather standard in block coordinate minimization algorithms. (D)the convex functions h_i(·) are not necessarily sampled in a cyclic manner like in <cit.>,(E)more than one convex function h_i(·) can be minimized at one time in the Dykstra's splitting, and(F)the SHQP strategy in <cit.> is applied.The proof is largely adapted from <cit.>. This paper also updates the discussion of the SHQP strategy in <cit.> by pointing out that if the convex functions δ_C_i^*(·) are not necessarily sampled in a cyclic manner, then we just need one set of the form C̃^n,w in Algorithm <ref> instead of multiple sets of this type as was done in <cit.>. Secondly, in Section <ref>, we show that one can minimize problems of the form (<ref>) where the feasible region is a compact set by combining Dykstra's splitting on problems of the kind (<ref>) and an approximate proximal point algorithm where the proximal center is moved once the KKT conditions are approximately satisfied. The compactness of the feasible region allows us to remove the constraint qualifications on the constraint sets for our results.In Section <ref>, we show that if a dual minimizer exists and some processing is performed so that the dual multipliers related to the indicator functions are uniformly bounded throughout all iterations, an O(1/n) convergence of the dual problem (which leads to an O(1/√(n)) convergence to the primal minimizer) can be attained.§.§ Notation We use “∂” to refer to either the subdifferential of a convex function, or the boundary of a set, which should be clear from context. The conjugate δ_C^*(·) of the indicator function has the form δ_C^*(y)=sup_x∈ C⟨ y,x⟩, and is also known as the support function.§ DYKSTRA SPLITTING FOR THE SUM OF CONVEX FUNCTIONS Consider the primal problem [ (P)α=x∈ Xmin1/2x-x_0^2+i=1r_1∑f_i(x)+i=r_1+1r_2∑g_i(x)+i=r_2+1r∑δ_C_i(x), ]where X is a finite dimensional Hilbert space, and(A1)f_i:X→ℝ are convex functions such that f_i(·)=X for all i∈{1,…,r_1}.(A2)g_i:X→ℝ are lower semicontinuous convex functions for all i∈{r_1+1,…,r_2}.(A3)C_i are closed convex subsets of X for all i∈{r_2+1,…,r}.In this section, we generalize the proof in <cit.> to show that Dykstra's splitting algorithm can be used to minimize problems of the form (<ref>).We note that the functions δ_C_i(·) and f_i(·) can be written as g_i(·). But as we will see later, we will treat the functions of the three types differently in Algorithm <ref>.For convenience of future discussions, let h:X→ℝ and h_i:X→ℝ be the convex functions defined by h(·)=∑_i=1^rh_i(·)h_i(·)= f_i(·)i∈{1,…,r_1}g_i(·)i∈{r_1+1,…,r_2} δ_C_i(·)i∈{r_2+1,…,r},so that the objective function in (<ref>) can be written simply as 1/2x-x_0^2+h(x). §.§ Algorithm description and commentary The (Fenchel) dual of problem (<ref>) is (D)β=max_z∈ X^rF(z),where F:X^r→ℝ is defined by [ F(z)=-1/2‖(i=1r∑z_i)-x_0‖ ^2-i=1r∑h_i^*(z_i)+1/2x_0^2. ]By weak duality, we have β≤α. (Actually β=α is true; We will see that later.) If C̃ is any closed convex set such that C̅⊂C̃, where the set C̅ is defined by C̅:=[∩_i=r_2+1^rC_i]∩[∩_i=r_1+1^r_2 g_i(·)],then problem (<ref>) has the same (primal) minimizer as[ (P_C̃)α=x∈ Xmin1/2x-x_0^2+i=1r∑h_i(x)+δ_C̃(x). ]The dual of (P_C̃) is (D_C̃)β=max_z∈ X^r+1F_C̃(z),where F_C̃:X^r+1→ℝ is defined by [ F_C̃(z)=-1/2‖(i=1r+1∑z_i)-x_0‖ ^2-i=1r∑h_i^*(z_i)-δ_C̃^*(z_r+1)+1/2x_0^2. ]As detailed in <cit.>, this observation leads us to construct a set C̃^n,w that changes in each iteration of our extended Dykstra's algorithm in Algorithm <ref> below. We list some observations of Algorithm <ref>. The choice of S_n,w in line 6 of Algorithm <ref> allows for more than one block of z to be minimized in (<ref>). If w̅=r+1, the sets S_n,w are chosen to be {w}, and r_1=r_2=0, then Algorithm <ref> reduces to the extended Dykstra's algorithm that was discussed in <cit.>. (Choice of H^n,w) An easy choice for H^n,w in line 11 of Algorithm <ref> is to choose a halfspace with outward normal z_r+1^n,w that supports the set C̃^n,w. Another example of H^n,w is the intersection of the halfspace mentioned earlier with a small number of halfspaces containing C̅ defined in (<ref>) that will allow H^n,w to approximate C̅ well.We have the following identities to simplify notation: v^n,w:=[ j=1r+1∑z_j^n,w ] x^n,w:=[ x_0-v^n,w. ]For all i∈ S_n,w, we have(a)-x^n,w+∂ h_i^*(z_i^n,w)∋0,(b)-z_i^n,w+∂ h_i(x^n,w)∋0, and(c)h_i(x^n,w)+h_i^*(z_i^n,w)=⟨ x^n,w,z_i^n,w⟩. By taking the optimality conditions in (<ref>) with respect to z_i for i∈ S_n,w, we deduce (a). The equivalences of (a), (b) and (c) is standard. Dykstra's algorithm is traditionally written in terms of solving for the primal variable x. For completeness, we show the equivalence between (<ref>) and the primal minimization problem. (On solving (<ref>)) If a minimizer z^n,w for (<ref>) exists, then the x^n,w in (<ref>) satisfies x^n,w=[ x∈ Xmini∈ S_n,w∑h_i(x)+1/2‖ x-(x_0-i∉ S_n,w∑z_i^n,w)‖ ^2. ]Conversely, if x^n,w solves (<ref>) with the dual variables {z̃_i^n,w}_i∈ S_n,w satisfying[ z̃_i^n,w∈∂ h_i(x^n,w)x^n,w-x_0+i∉ S_n,w∑z_i^n,w+i∈ S_n,w∑z̃_i^n,w=0, ]then {z̃_i^n,w}_i∈ S_n,w solves (<ref>). For the first part, note that [ ∂(h+1/2·-(x_0-i∉ S_n,w∑z_i^n,w)^2)(x^n,w) ]⊃ [ i∈ S_n,w∑∂ h_i(x^n,w)+[x^n,w-(x_0-i∉ S_n,w∑z_i^n,w)] ]∋ [ i∈ S_n,w∑z_i^n,w+x^n,w-x_0+i∉ S_n,w∑z_i^n,w(<ref>)=0. ]For the second part, note that the first part of (<ref>) implies that x^n,w∈∂ h_i^*(z̃_i^n,w), while the second part of (<ref>) implies that 0 lies in the subdifferential of the objective function in (<ref>). (Information needed to calculate (<ref>)) We note that in (<ref>), one only needs to have knowledge of the variables v^n,w-1 and z_i^n,w-1 for i∈ S_n,w. Thus Dykstra's splitting may be suitable for problems where the communication costs is high compared to the costs of solving the proximal problems. (On line 18 of Algorithm <ref>) If M=∞ in Algorithm <ref>, then z^n+1,0 and H^n+1,0 can be set to be z^n,w̅ and H^n,w̅ respectively. We had to add this line to Algorithm <ref> because the boundedness condition (<ref>) is necessary for our O(1/n) convergence result in Section <ref>. This detail can be skipped for the discussions in this section and Section <ref>. We need the following fact before we discuss how to find z^n+1,0 and H^n+1,0 satisfying (<ref>).(Aggregating halfspaces) Consider two halfspaces, say H_1 and H_2, which have (outward) normals z_1 and z_2. Assume that {z_1,z_2} are linearly independent. Construct a third halfspace H_3 with normal z_1+z_2 such that H_3⊃ H_1∩ H_2 and ∂ H_3∩[H_1∩ H_2]≠∅. Let x be any point on ∂ H_1∩∂ H_2. We see that x∈∂ H_3. We have δ_H_1^*(z_1)+δ_H_2^*(z_2)=⟨ z_1,x⟩+⟨ z_2,x⟩=⟨ z_1+z_2,x⟩=δ_H_3^*(z_1+z_2).If {z_1,z_2} is linearly dependent instead, then H_3:=H_1∩ H_2 is a halfspace, and δ_H_3^*(z_1+z_2)≤δ_H_1^*(z_1)+δ_H_2^*(z_2). Moreover, the inequality is strict if, for example, z_1≠0, z_2≠0 and H_1⊊ H_2. This fact can be generalized for more than two halfspaces.We state some notation necessary for further discussions. For any i∈{1,…,r+1} and n∈{1,2,…}, let p(n,i) bep(n,i)=max{m:m≤w̅,i∈ S_n,m}.In other words, p(n,i) is the index m such that i∈ S_n,m but i∉ S_n,k for all k∈{m+1,…,w̅}. It follows from lines 9 and 13 of Algorithm <ref> that z_i^n,p(n,i)=z_i^n,p(n,i)+1=⋯=z_i^n,w̅.We now show one way to find z^n+1,0 and H^n+1,0 satisfying (<ref>).(On satisfying (<ref>)) Set z_i^n+1,0=α_iz_i^n,w̅ for all i∈{r_2+1,…,r}, where α_i is a number in [0,1], so that (<ref>) is satisfied. Then z_r+1^n+1,0(<ref>),(<ref>)=∑_i=r_2+1^r+1z_i^n,w̅-∑_i=r_2+1^rz_i^n+1,0=z_r+1^n,w̅+∑_i=r_2+1^r(1-α_i)z_i^n,w̅.For i∈{r_2+1,…,r}, recall that by the construction of z_i^n,p(n,i) in (<ref>) and (<ref>), the condition (<ref>) implies that H̃^n,i⊃ C_i, where the halfspace H̃^n,i is defined by H̃^n,i:={x:⟨ x-x^n,p(n,i),z_i^n,p(n,i)⟩≤0}(<ref>)= {x:⟨ x-x^n,p(n,i),z_i^n,w̅⟩≤0}.and x^n,w is as defined in (<ref>). We can check that δ_C_i^*(α z_i^n,w̅)=δ_H̃^n,i^*(α z_i^n,w̅)α≥0i∈{r_2+1,…,r}.Let I_n⊂{r_2+1,…,r} be the set of indices i such that z_i^n+1,0≠ z_i^n,w̅. Let H^n+1,0 be the halfspace with outward normal z_r+1^n+1,0 such that[ H^n+1,0⊃ H^n,w̅∩⋂_i∈ I_nH̃^n,i ] [ ∂ H^n+1,0∩[H^n,w̅∩⋂_i∈ I_nH̃^n,i]≠∅. ]Then (<ref>) is satisfied. Furthermore, (<ref>) is actually an equality if the normals {z_i^n,w̅}_i∈ I_n∪{r+1} are linearly independent. The conclusion can be deduced from Fact <ref>.One can check that the construction in Proposition <ref> also leads to the conditions in (<ref>). In particular, (<ref>) can be inferred from (<ref>) via ∑_i=1^r+1z_i^n+1,0=z_r+1^n+1,0+∑_i=1^r_2z_i^n+1,0+∑_i=r_2+1^rz_i^n+1,0(<ref>)≤ z_r+1^n,w̅+∑_i=1^r_2z_i^n,w̅+∑_i=r_2+1^r(α_i+(1-α_i))z_i^n,w̅ =∑_i=1^r+1z_i^n,w̅. The other items in (<ref>) are clear. §.§ Convergence of Algorithm <ref> We now prove the convergence of Algorithm <ref>. We first list assumptions that will ensure convergence to the primal minimizer. We make a few assumptions on Algorithm <ref>: (a)The objective value α in (<ref>) is a finite number.(b)The sets S_n,w⊂{1,…,r+1} are chosen such that for all n, ∪_w=1^w̅S_n,w={1,…,r+1}.(c)There are constants A and B such that for all n, ∑_i=1^r+1z_i^n,w≤√(n)A+B.(d)Minimizers of (<ref>) can be obtained in each step. We give a brief commentary on Assumption <ref>. Assumption <ref>(a) together with the strong convexity of the primal problem says that (<ref>) is feasible and a unique primal minimizer exists. As we will see later, the structure of the functions f_i(·) for i∈{1,…,r_1} implies that z_i^n,w is uniformly bounded for all i∈{1,…,r_1}. In Proposition <ref>, we shall introduce a condition on the choice of S_n,w that will ensure that Assumption <ref>(c) is satisfied.We follow the proof in <cit.> to show that lim_n→∞x^n,w̅ exists and is the minimizer of (P). For any x∈ X and z∈ X^r+1, the analogue of <cit.> is 1/2x_0-x^2+∑_i=1^rh_i(x)+δ_C̃(x)-F_C̃(z_1,…,z_r,z_r+1)(<ref>)= 1/2x_0-x^2+∑_i=1^r[h_i(x)+h_i^*(z_i)]-⟨ x_0,∑_i=1^r+1z_i⟩ +1/2‖∑_i=1^r+1z_i‖ ^2+δ_C̃(x)+δ_C̃^*(z_r+1)≥ 1/2x_0-x^2+∑_i=1^r+1⟨ x,z_i⟩-⟨ x_0,∑_i=1^r+1z_i⟩ +1/2‖∑_i=1^r+1z_i‖ ^2 =1/2‖ x_0-x-∑_i=1^r+1z_i‖ ^2≥0.The theorem below generalizes <cit.> for the setting (<ref>).Suppose Assumption <ref> holds. For the sequence {z^n,w}_1≤ n<∞ 0≤ w≤w̅⊂ X^r+1 generated by Algorithm <ref> and the sequences {v^n,w}_1≤ n<∞ 0≤ w≤w̅⊂ X and {x^n,w}_1≤ n<∞ 0≤ w≤w̅⊂ X deduced from (<ref>), we have: (i)The sum ∑_n=1^∞∑_w=1^w̅v^n,w-v^n,w-1^2 is finite and {F_H^n,w̅(z^n,w̅)}_n=1^∞ is nondecreasing.(ii)There is a constant C such that v^n,w^2≤ C for all n∈ℕ and w∈{1,…,w̅}. (iii)There exists a subsequence {v^n_k,w̅}_k=1^∞ of {v^n,w̅}_n=1^∞ which converges to some v^*∈ X and that lim_k→∞⟨ v^n_k,w̅-v^n_k,p(n_k,i),z_i^n_k,w̅⟩=0i∈{1,…,r+1}.(iv)For the v^* in (iii), x_0-v^* is the minimizer of the primal problem (P) and lim_k→∞F_H^n_k,w̅(z^n_k,w̅)=1/2v^*^2+∑_i=1^rh_i(x_0-v^*). The properties (i) to (iv) in turn imply that lim_n→∞x^n,w̅ exists, and x_0-v^* is the primal minimizer of (<ref>). We first show that (i) to (iv) implies the final assertion. For all n∈ℕ we have, from weak duality, [ F_H^n,w̅(z^n,w̅)≤β≤α≤1/2x_0-(x_0-v^*)^2+i=1r∑h_i(x_0-v^*), ]hence β=α=1/2x_0-(x_0-v^*)^2+h(x_0-v^*), and that x_0-v^*=min_xh(x)+1/2x-x_0^2. Since the values {F_H^n,w̅(z^n,w̅)}_n=1^∞ are nondecreasing in n, we have [ n→∞limF_H^n,w̅(z^n,w̅)=1/2x_0-(x_0-v^*)^2+i=1r∑h_i(x_0-v^*), ]and (substituting x=x_0-v^* in (<ref>)) [ 1/2x_0-(x_0-v^*)^2+h(x_0-v^*)-F_H^n,w̅(z^n,w̅) ](<ref>),(<ref>)≥ [ 1/2x_0-(x_0-v^*)-v^n,w̅^2 ](<ref>)= [ 1/2x^n,w̅-(x_0-v^*)^2. ]Hence lim_n→∞x^n,w̅ is the minimizer in (P). It remains to prove assertions (i) to (iv).Proof of (i): We note that if r+1∈ S_n,w, then F_H^n,w-1(z^n,w-1)C̃^n,w⊂ H^n,w-1≤ [ F_C̃^n,w(z^n,w-1) ](<ref>),(<ref>)≤ [ F_C̃^n,w(z^n,w)-1/2v^n,w-v^n,w-1^2 ]= [ F_H^n,w(z^n,w)-1/2v^n,w-v^n,w-1^2. ]The first inequality comes from the fact that since C̃^n,w⊂ H^n,w-1 (from line 8 of Algorithm <ref>), then δ_C̃^n,w^*(·)≤δ_H^n,w-1^*(·). The second inequality comes from the fact that {z_i^n,w}_i∈ S_n,w is a minimizer of the mapping [ {z_i}_i∈ S_n,w↦i∈ S_n,w∑h_i^*(z_i)+1/2‖(i∈ S_n,w∑z_i)-(x_0-i∉ S_n,w∑z_i^n,w)‖ ^2, ]with the 1/2v^n,w-v^n,w-1^2 arising from the quadratic term.When r+1∉ S_n,w, then we can make use of the fact that z_r+1^n,w=z_r+1^n,w-1 and C̃^n,w=H^n,w-1=H^n,w to see that the inequality (<ref>) carries through as well. Recall that through (<ref>), F_H^n+1,0(z^n+1,0)≥ F_H^n,w̅(z^n,w̅). Combining (<ref>) over all m∈{1,…,n} and w∈{1,…,w̅}, we have [ F_H^1,0(z^1,0)+m=1n∑w=1w̅∑v^m,w-v^m,w-1^2≤ F_H^n,w̅(z^n,w̅). ]Next, F_H^n,w̅(z^n,w̅)≤α by weak duality. The proof of the claim is complete.Proof of (ii): Substituting x in (<ref>) to be the primal minimizer x^* and z to be z^n,w, we have[ 1/2x_0-x^*^2+i=1r∑h_i(x^*)-F_H^1,0(z^1,0) ]≥ [ 1/2x_0-x^*^2+i=1r∑h_i(x^*)-F_H^n,w(z^n,w) ](<ref>)≥ [ 1/2‖ x_0-x^*-i=1r+1∑z_i^n,w‖ ^2(<ref>)=1/2x_0-x^*-v^n,w^2. ]The conclusion is immediate.Proof of (iii):We first make use of the technique in <cit.> (which is in turn largely attributed to <cit.>) to show that [ n→∞lim inf[(w=1w̅∑v^n,w-v^n,w-1)√(n)]=0. ]Seeking a contradiction, suppose instead that there is an ϵ>0 and n̅>0 such that if n>n̅, then (∑_w=1^w̅v^n,w-v^n,w-1)√(n)>ϵ. By the Cauchy Schwarz inequality, we have [ ϵ^2/n<(w=1w̅∑v^n,w-v^n,w-1)^2≤w̅w=1w̅∑v^n,w-v^n,w-1^2. ] This contradicts the earlier claim in (i) that ∑_n=1^∞∑_w=1^w̅v^n,w-v^n,w-1^2 is finite. Next, we recall Assumption <ref>(c) that there are constants A and B such that ∑_i=1^r+1z_i^n,w̅≤ A√(n)+B for all n. Through (<ref>), we find a sequence {n_k}_k=1^∞ such that lim_k→∞[(∑_w=1^w̅v^n_k,w-v^n_k,w-1)√(n_k)]=0. Thus [ k→∞lim[(w=1w̅∑v^n_k,w-v^n_k,w-1)z_i^n_k,w̅]=0i∈{1,…,r+1}. ] Moreover, |⟨ v^n_k,w̅-v^n_k,p(n_k,i),z_i^n_k,w̅⟩|≤ [ v^n_k,w̅-v^n_k,p(n_k,i)z_i^n_k,w̅ ]≤ [ (w=1w̅∑v^n_k,w-v^n_k,w-1)z_i^n_k,w̅. ]By (ii), there exists a further subsequence of {v^n_k,w̅}_k=1^∞ which converges to some v^*∈ X. Combining (<ref>) and (<ref>) gives (iii).Proof of (iv):From earlier results, we obtain [ -i=1r∑h_i(x_0-v^*)-δ_H^n_k,w̅(x_0-v^*) ](<ref>)≤ [ 1/2x_0-(x_0-v^*)^2-F_H^n_k,w̅(z^n_k,w̅) ]= [ 1/2x_0-(x_0-v^*)^2-F_H^n_k,p(n_k,r+1)(z^n_k,w̅) ](<ref>),(<ref>)= [ 1/2v^*^2+i=1r∑h_i^*(z_i^n_k,p(n_k,i))+δ_H^n_k,p(n_k,r+1)^*(z_r+1^n_k,w̅) ] [ -⟨ x_0,v^n_k,w̅⟩+1/2v^n_k,w̅^2 ],i∈ S_n,p(n,i)= [ 1/2v^*^2+i=1r+1∑⟨ x_0-v^n_k,p(n_k,i),z_i^n_k,p(n_k,i)⟩ ] [ -i=1r∑h_i(x_0-v^n_k,p(n_k,i))-⟨ x_0,v^n_k,w̅⟩+1/2v^n_k,w̅^2 ](<ref>)= [ 1/2v^*^2-i=1r+1∑⟨ v^n_k,p(n_k,i)-v^n_k,w̅,z_i^n_k,w̅⟩ ] [ -i=1r∑h_i(x_0-v^n_k,p(n_k,i))-i=1r+1∑⟨ v^n_k,w̅,z_i^n_k,w̅⟩+1/2v^n_k,w̅^2 ](<ref>)= [ 1/2v^*^2-1/2v^n_k,w̅^2-i=1r+1∑⟨ v^n_k,p(n_k,i)-v^n_k,w̅,z_i^n_k,w̅⟩ ] [ -i=1r∑h_i(x_0-v^n_k,p(n_k,i)). ]Since lim_k→∞v^n_k,w̅=v^*, we have lim_k→∞1/2v^*^2-1/2v^n_k,w̅^2=0. The term ∑_i=1^r+1⟨ v^n_k,p(n_k,i)-v^n_k,w̅,z_i^n_k,w̅⟩ converges to 0 by (iii). Next, recall from (<ref>) that x_0-v^n_k,p(n_k,i)∈ C_i. Recall from the end of the proof of (iii) that x_0-v^*=lim_k→∞x_0-v^n_k,p(n_k,i), so x_0-v^*∈ C_i. Hence x_0-v^*∈∩_i=r_2+1^rC_i. Since H^n_k,w̅ was designed so that ∩_i=r_2+1^rC_i⊂ H^n_k,w̅, we have x_0-v^*∈ H^n_k,w̅, so δ_H^n_k,w̅(x_0-v^*)=0. Lastly, by the lower semicontinuity of h_i(·), we have -lim_k→∞∑_i=1^rh_i(x_0-v^n_k,p(n_k,i))≤-∑_i=1^rh_i(x_0-v^*).Therefore (<ref>) becomes an equation in the limit, which leads to lim_k→∞F_H^n_k,w̅(z^n_k,w̅)=1/2v^*^2+∑_i=1^rh_i(x_0-v^*). We now show some reasonable conditions that guarantee Assumption <ref>(c).(Satisfying Assumption <ref>(c)) Assumption <ref>(c) is satisfied when all of the following conditions on S_n,w hold: * There are only finitely many S_n,w for which S_n,w∩{r_1+1,…,r+1} contains more than one element.* There are constants M_1>0 and M_2>0 such that the size of the set {(m,w):m≤ n, w∈{1,…,w̅}, |S_m,w|>1}is bounded by M_1√(n)+M_2 for all n. We only need to prove this result for when only condition (2) holds and S_n∩{r_1+1,…,r+1} always contains at most one element. We have ∑_i=1^r+1z_i^n,w̅ ≤ ∑_i=1^r+1z_i^n,0+∑_i=1^r+1∑_w=1^w̅z_i^n,w-z_i^n,w-1(<ref>)≤ ∑_i=1^r+1z_i^n-1,w̅+∑_i=1^r+1∑_w=1^w̅z_i^n,w-z_i^n,w-1.Hence ∑_i=1^r+1z_i^n,w̅(<ref>)≤∑_i=1^r+1z_i^1,0+∑_m=1^n∑_i=1^r+1∑_w=1^w̅z_i^n,w-z_i^n,w-1.So it suffices to show that there are numbers A' and B' such that ∑_m=1^n∑_i=1^r+1∑_w=1^w̅z_i^m,w-z_i^m,w-1≤ A'√(n)+B'.The sum of the left hand side of (<ref>) can be written as ∑_(m,w)∈S̅_n,1∑_i=1^r+1z_i^m,w-z_i^m,w-1+∑_(m,w)∈S̅_n,2∑_i=1^r+1z_i^m,w-z_i^m,w-1,where S̅_n,1={(m,w):|S_m,w|=1, m≤ n, w∈{1,…,w̅}}, S̅_n,2={(m,w):|S_m,w|>1, m≤ n, w∈{1,…,w̅}}.First, there is a constant M_3 such that ∑_(m,w)∈S̅_n,1∑_i=1^r+1z_i^m,w-z_i^m,w-1|S_n,w|=1(<ref>),(<ref>)= ∑_(m,w)∈S̅_n,1v^m,w-v^m,w-1(<ref>)≤ ∑_w=1^w̅∑_m=1^nv^m,w-v^m,w-1≤ √(w̅n)√(∑_w=1^w̅∑_m=1^nv^m,w-v^m,w-1^2)≤ √(n)M_3.Next, we estimate the second sum in (<ref>). For each (m,w)∈S̅_n,2, by condition (1), there is a unique i_m,w∈ S_m,w∩{r_1+1,…,r+1}. We have z_i_m,w^m,w-z_i_m,w^m,w-1(<ref>),(<ref>)=v_i_m,w^m,w-v_i_m,w^m,w-1-∑_j∈ S_m,w\{i_m,w}(z_j^m,w-z_j^m,w-1).For each j∈ S_m,w\{i_m,w}, we have z_j^m,w∈∂ f_j(x^m,w)(<ref>)=∂ f_j(x_0-v^m,w).Together with the fact that v^m,w is bounded from Theorem <ref>(ii) and the fact that f_j(·) are Lipschitz on bounded domains, we deduce that z_j^m,w and z_j^m,w-1 are bounded for all j∈{1,…,r_1} by standard convex analysis. Since S_m,w\{i_m,w}⊂{1,…,r_1}, every term on the right hand side of (<ref>) is bounded, so there is a constant M_4>0 such that z_i^m-z_i^m-1≤ M_4. Therefore condition (2) implies ∑_(m,w)∈S̅_n,2∑_i=1^r+1z_i^m,w-z_i^m,w-1≤ M_4(M_1√(n)+M_2).Combining (<ref>) and (<ref>) into (<ref>) gives the conclusion we need. § O(1/N) CONVERGENCE WHEN A DUAL MINIMIZER EXISTS In this section, we show that for the problem (<ref>), if Algorithm <ref> is applied with some finite M and a minimizer for the dual problem exists, then the rate of convergence of the dual objective function is O(1/n), which leads to the O(1/√(n)) rate of convergence to the primal minimizer.We recall a lemma on the convergence rates of sequences.(Sequence convergence rate) Let α>0. Suppose the sequence of nonnegative numbers {a_k}_k=0^∞ is such that a_k≥ a_k+1+α a_k+1^2k∈{1,2,…}. * <cit.> If furthermore, [ a_1≤1.5/αa_2≤1.5/2α ], then [ a_k≤1.5/α kk∈{1,2,…}. ]* <cit.> For any k≥2, [ a_k≤max{(1/2)^(k-1)/2a_0,4/α(k-1)} . ]In addition, for any ϵ>0, if [ [ k≥max{2/ln(2)[ln(a_0)+ln(1/ϵ)],4/αϵ} +1, ] ]then a_n≤ϵ. Instead of condition (A2) after (<ref>), we assume a stronger condition on g(·): (A2')g_i:X→ℝ are convex functions such that g_i(·) are open sets for all i∈{r_1+1,…,r_2}.In other words, the functions g_i(·) are such that if lim_j→∞x_j lies in ∂g_i(·), then lim_j→∞g_i(x_j)=∞.We have the following theorem.(O(1/n) convergence of dual function) Suppose conditions (1) and (2) in Proposition <ref> and Assumption <ref> are satisfied and Algorithm <ref> is run with finite M. If a dual minimizer to (<ref>) exists, then the convergence rate of the dual objective value is O(1/n). This in turn implies that the convergence rate of {x^n,w̅-x^*}_n is O(1/√(n)). Let V_n=-F_H^n,w̅(z^n,w̅). Recall that {V_n} is nonincreasing by Theorem <ref>(i). We want to show that V_n-(-β)≤ O(1/n). First, from line 8 of Algorithm <ref>, we have H^n,w⊃C̃^n,w+1, so [ 1/2v^n,w-x_0^2+i=1r∑h_i^*(z_i^n,w)+δ_H^n,w^*(z_r+1^n,w)-1/2x_0^2 ]H^n,w⊃C̃^n,w+1≥ [ 1/2v^n,w-x_0^2+i=1r∑h_i^*(z_i^n,w)+δ_C̃^n,w+1^*(z_r+1^n,w)-1/2x_0^2 ](<ref>),(<ref>)≥ [ 1/2v^n,w+1-x_0^2+i=1r∑h_i^*(z_i^n,w+1)+δ_C̃^n,w+1^*(z_r+1^n,w)-1/2x_0^2 ] [ +1/2v^n,w+1-v^n,w^2 ]= [ 1/2v^n,w+1-x_0^2+i=1r∑h_i^*(z_i^n,w+1)+δ_H^n,w+1^*(z_r+1^n,w)-1/2x_0^2 ] [ +1/2v^n,w+1-v^n,w^2. ]In view of the above and the definitions of F_H(·) in (<ref>) and V_n, we have [ V_n≥ V_n+1+1/2w=1w̅∑v^n,w-v^n,w-1^2. ]We then look at the subgradients generated in each iteration. Recall how z^n,w were defined in (<ref>). We have, for each i∈{1,… r+1}, v^n,w̅-v^n,p(n,i)^s_i:= (<ref>)=v^n,w̅-x_0+x^n,p(n,i)∈v^n,w̅-x_0+∂ h_i^*(z_i^n,p(n,i))(<ref>)=v^n,w̅-x_0+∂ h_i^*(z_i^n,w̅).Let the vector s∈ X^r+1 be defined so that the ith component s_i∈ X is as in (<ref>). Then [ s_i≤w=p(n,i)+1w̅∑v^n,w-v^n,w-1≤w=1w̅∑v^n,w-v^n,w-1. ]Let z^*∈ X^r+1 be a minimizer of -F_H^n,w(·) with z_r+1^*=0. (Such a minimizer can be constructed by appending z_r+1^*=0 to a minimizer of (<ref>), which was assumed to exist.) Making use of the elementary fact that s∈∂(-F_H^n,w̅)(z^n,w̅), we have V_n-V^*=[ -F_H^n,w̅(z^n,w)-(-F_H^n,w̅(z^*)) ]≤ [ -⟨ s,z^n,w̅-z^*⟩ ]≤ [ i=1r+1∑s_iz_i^n,w̅-z_i^* ](<ref>)≤ [ w=1w̅∑v^n,w-v^n,w-1i=1r+1∑z_i^n,w̅-z_i^*. ]Claim: There is a constant M_4 such that z_i^n,w≤ M_4 for all n≥0, w∈{1,…,w̅} and i∈{1,…,r+1}. Step 1: The claim is true for all n≥0, i∈{1,…,r_2} and w∈{0,…,w̅}.The limit lim_n→∞x^n,w̅ must lie in the interior of the domains of f_i(·) and g_i(·) for all i∈{1,…,r_2} (by Assumption (A2')). It is well known that the subgradients of a convex function is bounded in the interior of its domain, so there is a constant M_1 such that z_i^n,w≤ M_1 for all i∈{1,…,r_2} and w∈{0,…,w̅}. Step 2: The claim is true for all n≥0, i∈{1,…,r+1} and w=0. Since we assumed that Algorithm <ref> was run with a finite M, by (<ref>), z_i^n,0≤ M for all i∈{r_2+1,…,r} and n≥0. Next, we show that M_1 can be made larger if necessary so that z_r+1^n,0≤ M_1 for all n≥0. Seeking a contradiction, suppose that there is a subsequence {n_k} such that lim_k→∞z_r+1^n_k,0=∞. Then this would mean that lim_k→∞∑_i=1^r+1z_i^n_k,0=∞. Recalling (<ref>) for the special case where x=x^*(the primal minimizer), we have [ 1/2 x_0-x^*-i=1r+1∑z_i^n,0^2 ] ≤ [ 1/2x_0-x^*^2+i=1r∑h_i(x^*)-F_H^n,0(z^n,0) ]≤ [ 1/2x_0-x^*^2+i=1r∑h_i(x^*)-F_H^1,0(z^1,0), ]which is a contradiction.Step 3: The claim is true for all n≥0, i∈{r_2+1,…,r+1} and w∈{1,…,w̅}.Next, we recall from Theorem <ref>(i) that ∑_n=1^∞∑_w=1^w̅v^n,w-v^n,w-1^2 is finite. This implies that there is a M_2>0 such that v^n,w-v^n,w-1≤ M_2 for all n≥0 and w∈{1,…,w̅}. Next, for each n≥0 and i∈{r_2+1,…,r}, we want to show that there is a constant M_3 such that z_i^n,w≤ M_3 for all n≥0 and w∈{1,…,w̅}. Since the z_i^n,w were chosen by condition (1) in Proposition <ref>, then if n is large enough, if S_n,w∩{r_2+1,…,r+1}≠∅, then there is a i_n,w∈ S_n,w such that S_n,w\{i_n,w}⊂{1,…,r_2}. We have z_i_n,w^n,w(<ref>)=z_i_n,w^n,w-1+v^n,w-v^n,w-1-∑_j∈ S_n,w\{i_n,w}[z_j^n,w-z_j^n,w-1].Then we have z_i_n,w^n,w (<ref>)≤ z_i_n,w^n,w-1+v^n,w-v^n,w-1+∑_j∈ S_n,w\{i_n,w}[z_j^n,w+z_j^n,w-1]S_n,w\{i_n,w}⊂{1,…,r_2}≤ z_i_n,w^n,w-1+M_2+2r_2M_1.This would easily imply that z_i^n,w≤ M_4 for some M_4>0 for all n≥0, i∈{1,…,r+1}, and w∈{1,…,w̅} as needed, ending the proof of the claim.Now, [ w=1w̅∑v^n,w-v^n,w-1≤√(2w̅)√(1/2w=1w̅∑v^n,w-v^n,w-1^2)(<ref>)≤√(2w̅)√(V_n-V_n+1). ]Then combining the above, we have V_n-V^* (<ref>),(<ref>)≤ [∑_i=1^r+1z_i^n,w̅+∑_i=1^r+1z_i^*]√(2w̅)√(V_n-V_n+1)≤ [(r+1)M_4+∑_i=1^r+1z_i^*]√(2w̅)√(V_n-V_n+1).Letting M_5 be (r+1)M_4+∑_i=1^r+1z_i^* and rearranging (<ref>), we have [ V_n-V^*≥ V_n+1-V^*+1/2w̅M_5^2(V_n+1-V^*)^2. ]Applying Lemma <ref> gives the first statement of our conclusion. The second statement comes from substituting x=x^* in (<ref>) and noticing that x_0-x^*-∑_i=1^r+1z_i(<ref>)=x^n,w̅-x^*.(Nonexistence of dual minimizers) An example of a problem where dual minimizers do not exist is in <cit.>. Lemma 2 in <cit.> shows that a fast convergence rate to the primal minimizer implies the existence of dual minimizers. § APPROXIMATE PROXIMAL POINT ALGORITHM Consider the problem of minimizing [ i=1r∑h_i(x). ]If one of the functions h_i(·) can be split as h_i(·)=h̃_i(·)+c_i/2·-x_0^2 for some convex h̃_i(·) and c_i>0, then (<ref>) can be minimized using Dykstra's splitting algorithm of Section <ref>. In this section, we propose an approximate proximal point method for minimizing (<ref>) without splitting h_i(·). We first present Algorithm <ref> and prove that all its cluster points are minimizers of the parent problem. Then, in Subsection <ref>, we show that the Dykstra splitting investigated in Section <ref> can find an approximate primal minimizer required in Algorithm <ref>.§.§ An approximate proximal point algorithm Consider the problem of minimizing h:X→ℝ, where [ h(·)=δ_D(·)+i=1r_2∑h_i(·), ]and each h_i:X→ℝ is a closed convex function whose domain is an open set, and D is a compact convex set in X. This setting is less general than that of (<ref>), since it does not allow for all lower semicontinuous convex functions, and we only allow for one compact set D instead of r-r_2 sets. Algorithm <ref> shows an approximate proximal point algorithm, where one solves a regularized version of (<ref>) and shifts the proximal center x_k when an approximate KKT condition is satisfied. If r_2=1, D=X and h_1(·) were allowed to be any lower semicontinuous convex function, then Algorithm <ref> would resemble the classical proximal point algorithm. Define the operator T:X→ X by [ T(x):=_h(x):=x'minh(x')+1/2x'-x^2. ] This operator has some favorable properties in monotone operator theory. We prove our first result. (Approximate of T(·)) Consider the problem (<ref>). Let T(·) be as defined in (<ref>). Suppose D⊂ X is compact and convex. For all ϵ>0, there is a δ>0 such that for all x,x^+∈ X and z,e∈ X^r+1 such that d(x,D)≤δ and [ z_i ] ∈ ∂ h_i(x^++e_i)i∈{1,…,r_2}, [ z_0 ] ∈N_D̃(x^++e_0), [ (x^+-x)+i=0r_2∑z_i ] ≤ δ, [ e_i ] ≤ δi∈{0,…,r_2},where D̃⊃ D is a closed convex set, then x^+-T(x)≤ϵ.Seeking a contradiction, suppose otherwise. Then there exists a ϵ>0 such that for all positive integers k, there are x_k,x_k^+∈ X, z^(k),e^(k)∈ X^r+1 and a closed convex set D^(k)⊃ D such that d(x_k,D)≤1/k,z_i^(k) ∈ ∂ h_i(x_k^++e_i^(k)),z_0^(k) ∈N_D^(k)(x_k^++e_0^(k)), (x_k^+-x_k)+∑_i=0^r_2z_i^(k)_d_k ≤1/k, e_i^(k) ≤1/ki∈{0,…,r_2},but x_k^+-T(x_k)≥ϵ.Letting k↗∞, we can assume (by taking subsequences if necessary) that lim_k→∞x_k=x̅lim_k→∞x_k^+=x̅^+lim_k→∞e^(k)=0.There are two cases we need to consider.Case 1: x̅^+ lies in the interior of h_i for all i∈{1,…,r_2}.Making use of the fact that convex functions are locally Lipschitz in the interior of their domains, we obtain the boundedness of {z^(k)}. We can assume (by taking subsequences if necessary) that lim_k→∞z^(k)=z̅. Taking the limits of (<ref>) as k→∞ would give us z̅_i∈∂ h_i(x̅^+), z̅_0∈ N_D(x̅^+) and x̅^+-x̅+∑_i=0^r_2z̅_i=0, which would in turn imply that x̅^+=T(x̅). It is well known that T(·) is nonexpansive and hence continuous, so 0<ϵ(<ref>)≤lim_k→∞x_k^+-T(x_k)=x̅^+-T(x̅)=0,a contradiction. Case 2: x̅^+ lies on the boundary of h_i for some i∈{1,…,r_2}.We cannot use the method in Case 1 as some components of {z^(k)} might be unbounded. We now consider the perturbed functions h_i,k(·) defined by h_i,k(x):=h_i(x+e_i^(k)).Let h̃_k:X→ℝ be defined by h̃_k(·)=δ_D^(k)(·)+∑_i=1^r_2h_i,k(·). Then the conditions (<ref>) imply that [ x_k^+=_h̃_k(x_k+d_k)=xmin h̃_k(x)+1/2x-(x_k+d_k)^2, ]where d_k is marked in (<ref>). Suppose i̅ is such that x̅^+ lies on the boundary of h_i̅. Then we have that lim_k→∞h_i̅(x_k^+)=∞. Since D is bounded, inf_x∈ Dh_i(x) is a finite number for all i, which implies that [ k→∞limh̃_k(x_k^+)+1/2x_k^+-(x_k+d_k)^2=∞. ]Next, let x'_k=_h(x_k) and x'=_h(x̅). By the continuity properties of T(·)=_h(·) and lim_k→∞x_k=x̅ in (<ref>), we must have lim_k→∞x'_k=x'. It is clear that x' lies in (h_i). Since we assumed that (h_i) is open, x'∈(h_i) for all i. We then have [ k→∞limh̃_k(x')+1/2x'-x_k^2=h(x')+1/2x'-x̅^2<∞. ] But on the other hand, since lim_k→∞d_k=0, we have[ k→∞limh̃_k(x')+1/2x'-(x_k+d_k)^2(<ref>)≥k→∞limh̃_k(x_k^+)+1/2x_k^+-(x_k+d_k)^2(<ref>)=∞. ]Formulas (<ref>) and (<ref>) are contradictory, so x̅^+ must lie in the interior of all h_i for all i, which reduces to case 1.Thus we are done.To simplify notation in the next two results, we define the set A to be A:=min_xh(x). We have another lemma.For all ϵ>0, there exists δ>0 such that for all w such that d(w,D)≤δ, we have T(w)-w≤δd(w,A)≤ϵ.Seeking a contradiction, suppose otherwise. In other words, there is a ϵ̅>0 such that for all k>0, there is a w_k such that d(w_k,A)>ϵ̅ but T(w_k)-w_k≤1/k. By taking subsequences if necessary, let w̅=lim_k→∞w_k, which exists by the compactness of D. Taking limits as k↗∞ gives us w̅=T(w̅), which will in turn imply that d(w̅,A)=0, a contradiction.(Cluster points of Algorithm <ref>) All cluster points of {x_j} in Algorithm <ref> are minimizers of h(·). For any ϵ_1>0, we make use of Lemma <ref> and obtain δ_1>0 such that T(w)-w≤δ_1d(w,A)≤ϵ_1.By Lemma <ref>, there exists some K large enough so that x_k+1-T(x_k)≤ϵk≥ K.Let ϵ>0 be small enough so that 4ϵ^2+δ_1^2/4ϵ>(D). Then since lim_k→∞d(x_k,D)=0 and A⊂ D, we can increase K if necessary so that [ d(x_k,A)≤(D)<4ϵ^2+δ_1^2/4ϵk≥ K. ]Let x̅_k=P_A(x_k) so that d(x_k,A)=x_k-x̅_k. It is well known from the theory of monotone operators that T(·) is firmly nonexpansive (see for example <cit.>), so we have T(x_k)-x̅_k^2+x_k-T(x_k)^2≤x_k-x̅_k^2.Suppose k≥ K. We split our analysis into two cases.Case 1: d(x_k,A)>ϵ_1. Then (<ref>) implies T(x_k)-x_k>δ_1. We have d(x_k+1,A)x̅_k∈ A≤ x_k+1-x̅_k≤ x_k+1-T(x_k)+T(x_k)-x̅_k(<ref>),(<ref>)≤ ϵ+√(x_k-x̅_k^2-x_k-T(x_k)^2) <ϵ+√(d(x_k,A)^2-δ_1^2)(<ref>)≤d(x_k,A)-ϵ.Case 2: d(x_k,A)≤ϵ_1. We have d(x_k+1,A)x̅_k∈ A≤x_k+1-x̅_k ≤ x_k+1-T(x_k)+T(x_k)-x̅_k(<ref>),(<ref>)≤ ϵ+x_k-x̅_k=ϵ+d(x_k,A)≤ϵ+ϵ_1.The analysis in these two cases implies d(x_k+1,A)≤ϵ_1+ϵ for all k large enough. Since ϵ_1 and ϵ can be made arbitrarily small, any cluster point x' of {x_k}_k=1^∞ must thus satisfy d(x',A)=0, or x'∈ A. §.§ Satisfying (<ref>) using Dykstra splittingConsider the problem of minimizing h:X→ℝ, where[ h(x)=i=1r_2∑h_i(x)+i=r_2+1r∑δ_C_i(x). ]and each h_i:X→ℝ is a closed convex function whose domain is an open set, and each C_i is a closed convex set such that ∩_i=r_2+1^rC_i is compact. This formulation is slightly more general than that of (<ref>). Theorem <ref> below shows that Algorithm <ref> can find approximate minimizers to (<ref>) that will satisfy the conditions for moving to a new proximal center in Algorithm <ref>.Consider the problem (<ref>). For any δ>0, there is some n>0 such that when Algorithm <ref> is applied to solve [ h(x)=i=1r_2∑h_i(x)+i=r_2+1r∑δ_C_i(x)+1/2x-x_0^2, ]we have a set D^(n), and points x^(n)∈ X and z̃∈ X such that [ x^n,w̅-x^n,w ] ≤ δw∈{1,…,w̅}, [ z_i^n,w̅ ] ∈ ∂ h_i(x^n,p(n,i))i∈{1,…,r}, [ i=1r_2∑z_i^n,w̅+z̃+x^n,w̅-x_0 ] ≤ δ, [ ∩_i=r_2+1^rC_i ] ⊂D^(n), [ z̃ ] ∈N_D^(n)(x^(n)), [ x^(n)-x^n,w̅ ] ≤ δ.Define z_0^n,w̅ to be [ z_0^n,w̅=i=r_2+1r+1∑z_i^n,w̅. ]Theorem <ref> says that for any δ>0, we can find n>0 such that the first 3 conditions in (<ref>) hold if z̃ were chosen to be z_0^n,w̅. We separate into two cases, and discuss how the set D^(n) (which will actually be either the whole space X or a halfspace) and the point x^(n) are constructed. Case 1: lim inf_n→∞z_0^n,w̅=0.By taking subsequences {n_k}_k, we can assume that lim_k→∞z_0^n_k,w̅=0. Then the set D^(n) can be chosen to be X, and x^(n) can be chosen to be x^*, the minimizer of (<ref>). The vector z̃ can be chosen to be zero, and the inequalities in (<ref>) can be easily seen to be satisfied.Case 2: lim inf_n→∞z_0^n,w̅>0.Recall that for i∈{r_2+1,…,r}, the dual vector z_i^n,w̅ was constructed so that z_i^n,w̅∈ N_C_i(x^n,p(n,i)), and that z_r+1^n,w̅∈ N_H^n,w̅(x^n,p(n,r+1)). For i∈{r_2+1,…,r+1}, define the halfspace H̃^n,i to be the halfspace with x^n,p(n,i) on its boundary and outward normal vector z_i^n,w̅ like in (<ref>). Note that H̃^n,i⊃ C_i for i∈{r_2+1,…,r}, and H̃^n,r+1⊃ H^n,w̅. Let D^(n) be the halfspace with outward normal z_0^n,w̅ such that D^(n)⊃⋂_i=r_2+1^r+1H̃^n,i, and ∂ D^(n)∩⋂_i=r_2+1^r+1H̃^n,i≠∅. Through Fact <ref>, this choice of D^(n) would give us [ δ_D^(n)^*(z_0^n,w̅)≤i=r_2+1r∑δ_C_i^*(z_i^n,w̅)+δ_H^n,w̅^*(z_r+1^n,w̅). ]We now show how to satisfy (<ref>) and (<ref>) with z̃=z_0^n,w̅. Let x^* be the optimal primal solution of (<ref>). We now want to show that we can choose a further subsequence if necessary so that lim_k→∞d(∂ D^(n_k),x^*)=0. From the definition of the support function and the fact that x^*∈⋂_i=1^rC_i⊂ D^(n), we have[ δ_D^(n)^*(z_0^n,w̅)-⟨ x^*,z_0^n,w̅⟩=δ_D^(n)-x^*^*(z_0^n,w̅)=z_0^n,w̅d(∂ D^(n),x^*). ]We now mimic (<ref>) to obtain [ 1/2x_0-x^*^2+i=1r_2∑h_i(x^*)+i=r_2+1r∑δ_C_i(x^*)-F_H^n,w̅(z^n,w̅,,z_r+1^n,w̅) ](<ref>)= [ 1/2x_0-x^*^2+i=1r_2∑[h_i(x^*)+h_i^*(z_i^n,w̅)]-⟨ x_0,i=1r∑z_i^n,w̅⟩ ] [ +1/2i=1r∑z_i^n,w̅^2+i=r_2+1r∑δ_C_i^*(z_i^n,w̅)+δ_H^n,w̅^*(z_r+1^n,w̅) ](<ref>)≥ [ 1/2x_0-x^*^2+[i=1r∑⟨ x^*,z_i^n,w̅⟩-i=r_2+1r∑⟨ x^*,z_i^n,w̅⟩] ] [ -⟨ x_0,i=1r∑z_i^n,w̅⟩+1/2i=1r∑z_i^n,w̅^2+δ_D^(n)^*(z_0^n,w̅) ](<ref>),(<ref>)= [ 1/2i=1r∑z_i^n,w̅+x^*-x_0^2+z_0^n,w̅d(∂ D^(n),x^*)≥0. ]Theorem <ref>(i)(iv) shows that the formula in the first line of (<ref>) has a limit of zero, so by the squeeze theorem, the last term in (<ref>) also has limit zero as n→∞. This means that lim_k→∞d(∂ D^(n_k),x^*)=0. Thus x^(n) can be chosen as the projection of x^* onto ∂ D^(n). We then have lim_k→∞x^(n)-x^*=0, and so the inequalities in (<ref>) can be easily seen to be satisfied.(Achieving (<ref>)) A final detail is to find an implementable way for checking that d(x^(n),∩_i=r_2+1^rC_i) is indeed small. Note that x^n,p(n,i)∈ C_i for all i∈{r_2+1,…,r}, so we have the estimate d(x^n,w̅,C_i)≤x^n,w̅-x^n,p(n,i). One can easily make use of the compactness of ∩_i=r_2+1^rC_i to prove that for all δ_1>0, there is a δ_2>0 such that max_r_2+1≤ i≤ rd(x,C_i)≤δ_2d(x,∩_i=r_2+1^rC_i)≤δ_1.Putting this fact together with Theorem <ref> allows us to find a point satisfying (<ref>) in Algorithm <ref>.(Treating multiple sets in Theorem <ref>) We remark that if we had analyzed Theorem <ref> for the case where there are more than one indicator functions, then analyzing (<ref>) would mean having to introduce constraint qualifications (for example, a Lipschitzian error bound assumption) needed to deal with issues related to the stability of sets under perturbations.amsalpha | http://arxiv.org/abs/1709.09499v1 | {
"authors": [
"C. H. Jeffrey Pang"
],
"categories": [
"math.OC"
],
"primary_category": "math.OC",
"published": "20170927133135",
"title": "Dykstra splitting and an approximate proximal point algorithm for minimizing the sum of convex functions"
} |
Column densities and variability in the X-ray spectrum * Sean J. Taylor and Dean Eckles December 30, 2023 ==================================§ INTRODUCTIONActive galactic nuclei (AGN) are the most persistent luminous objects in the universe. Observed in all wavelengths from Radio to X-rays, they are powered by accretion of matter on to a super massive black hole.Among the plethora of phenomenon they exhibit, 50% of type 1 AGN feature ionized outflows. The launching mechanism of these winds remains in debate, and suggestions vary from thermal evaporation <cit.> to line driving <cit.> and magnetic hydrodynamics <cit.>. These AGN winds are observed in a multitude of absorption lines of different ions, in both UV and X-rays <cit.>. These lines areubiquitously blueshifted with respect to the rest-frame of the host galaxy, with velocities often consistent between the X-rays and the UV <cit.>, suggesting they are part of the same kinematic structure.If these outflows are indeed associated with the AGN, an important question is whether the energy or mass they deposit is important for galacticevolution by means of energy feedback. The kinetic power of these outflows scales with v^3, the outflow velocity, which is typically a few 100 km s^-1 <cit.>. These low velocities limit the efficiency of these outflows as a feedback mechanism. However, some outflows feature velocities of a few 1000 km s^-1, NGC 7469 for example exhibits a fast component at a blueshift of 2000 km s^-1.AGN winds have been the focus of studies relating change in absorption troughs in AGN spectra to the distance and density of the associated outflows. Examples in both X-rays and UV analysis can be found in <cit.>. <cit.> for example constrain the distance of the outflow in NGC 5548 to be at least a few pc from the AGN source, with distances up to more than 100 pc. These large distances lead to anambiguity of whether the AGN is responsible for driving these outflows directly.This is the second paper as part of a multi-wavelength observation campaign on NGC 7469. <cit.> derived outflow parameters using global fitmodels of photo-ionized plasmas.We continue the examination of the XMM-Newton RGS spectrum focusing on measurement of the column densities.In addition to the seven observations observed on a logarithmic timescale during the 2015 campaign, we analyze archival data from 2004. With these data we compare changes on timescales of years, months and days,with the intent of seeking variability in absorption troughs,and through this to constrain the distance of the outflow from the AGN. This, along with a measurement of the kinetic power of the outflow will determine the role the outflow plays in coupling the AGN to its host galaxy.§ DATAXMM-Newton observed NGC 7469 as part of the multi-wavelength campaign 7 times during 2015 for a total duration of 640 ks.The observation log is shown in Table <ref>, including previous observations published in <cit.>. We use the RGS (1 and 2) data from all observations to constrain variability in absorption troughs. The RGS spectra are reduced using `rgsproc' within the software package SAS 15[<http://xmm-tools.cosmos.esa.int>] and combined using the standard RGS command, `rgscombine'. The reduction is detailed in <cit.>. The spectral fitting in the present paper is done on grouped spectra, re-binned to 20 mÅ (grouping two default SAS bins). The full 2015 RGS spectrum (black) and best-fit model (red) are shown in Fig. <ref>, and the model is described in Sec. <ref>.The EPIC-pn lightcurve of NGC 7469 is presented in Fig. <ref>. An interesting featureis the rapid change of photon flux on an hourly basis, while the average seems to remain constant over years. The mean EPIC-pn count rate (count s^-1) for the 2004 observations is 24.7, with a standard deviation of σ=1.9, and for 2015 the mean is 23.2 with σ=3.5. § SPECTRAL MODELING§.§ MethodWe first model the 2015 combined spectrumsince column densities between observations in the campaign are consistent within 90% uncertainties (See Sec. <ref>). This agreement between the different observations, within the larger uncertainties of individual observations, is a clear indication in favor of using the combined spectrum, at least initially. All uncertainties we quote in this paper are 90% confidence intervals.Following the ion-by-ion fitting approach by <cit.>,we fit the continuum I_0 along with the ionic column densities, N_i, which are this paper's main goal. The transmission equation is given by:I(λ)/I_0=1-(1-e^-∑_i N_iσ_i(λ))Cwhere I(λ) is the observed continuum intensity, I_0 is the unabsorbed continuum intensity, σ_i is the absorption cross section depending on photon energy.The coveringfraction is C, with C=0 indicating no absorption and C=1 indicating the source is entirely covered by the outflow. Some results in the UV suggest the covering fraction is ion dependent or even velocity dependent<cit.>, but the much smaller X-ray source is not expected to be partially covered. The X-ray continuum of NGC 7469 in the RGS band can be modeled by a single powerlaw. A complete X-ray continuum model based on the EPIC spectra will be presented by Middei et al. (in preparation). The powerlaw is given by:I_0(E)=A(E/1keV)^-Γwith the norm A and the slope Γ as free parameters.On top of the absorbed continuum I(λ) we observe emission lines.These lines were modeled by <cit.>, and include both photo-driven and collisionally excited lines. They are fixed in our model and are assumed not to be absorbed by the outflow.The absorption cross section is given byσ_i(λ)= σ_i^edge(λ)+σ_i^line(λ)== σ_i^edge(λ)+(π e^2/m_e c)∑_j<k f_jkϕ (λ-λ_jk)Here σ_i^edge describes the ionization edge of ion i, ϕ(λ) is the Voigt line profile and the sum is over all the strong ion line transitions j→ k, e is the electron charge, m_e the electron mass, and f_jk are the oscillator strengths.All transitions are assumed to be from the ground level. We use the oscillator strengths and ionization edges calculated using the HULLACatomic code <cit.> as used in <cit.>.The parameters determining the profile shape and position ϕ are ion temperature, turbulent velocity, and outflow velocity.The temperature and turbulent velocity broadenings seen in the UV <cit.> are below theRGS resolution of Δλ≈70mÅ. Thus, in order to constrain simultaneously the covering factor, the turbulent velocity, and the ion column density one needs 3 measurable lines of a given ion (See Eq. <ref>). N^+6 is the best ion providing 3 lines unambiguously visible in the spectrum. These are observed at wavelengths of 25.18Å, 21.25Å, and 20.15Å.Nonetheless, the best fit favors a covering factor of 1.0 with the 90% confidence interval ranging down to 0.8 when all ions are taken into account. The uncertainty in the continuum adds another level of uncertainty here, so we make no claims regarding covering factor and hold it frozen to 1.0. Since constraining the line profile parameters is not the goal of this paper, we fix the ion temperature at 0.1 keV.We then fit only the O^+7 Lyα doublet line atthe observed wavelength of 19.2Å with the outflow and turbulent velocities thawed and set initially to the values of <cit.> in order to determinethem.Fig. <ref> shows the contribution of each velocity component to the absorption profile. The fit favors a 3 velocity model over 2 in accordance with these two papers, decreasing reduced χ^2 (χ^2/d.o.f.) by 0.5 from the 2-component to the 3-component model. The best-fit three components have velocities of -620, -960, and -2050 km s^-1 and turbulent velocities of 80, 40, 50 km s^-1 respectively.Three components are also favored by <cit.> and <cit.>. Though the fit converges we are not able to obtain meaningful uncertainties on these parameters.We leave them frozen for the rest of the fit, freeing them for one final iteration after the ion column densities are constrained.The fitted model parameters are thus the powerlaw normalization, the powerlaw slope, and the column density per ion. In addition, the three outflow velocities and three turbulent velocities are constrained once at the beginning according to O^+7, and one more time at the end. The strength of this model[The code for the model can be found in <https://github.com/uperetz/AstroTools>, including a full graphical suite for fitting models to fits files. The README details the contents of the directory.] lies in the independence of the ionic free parameters.§.§ Column densitiesThe full 2015 spectrum (black) and best-fit model (red) is seen in Fig. <ref>, with a best-fit reduced χ^2 of 1.4.For the 2004 spectra we obtain areduced χ^2 of 1.28. There are 1450 spectral bins and 64 free parameters.We also re-measure column densities from the 2004 spectra previously done by <cit.>. This is done in order to maintain consistency in the comparison with the 2015 spectra using the same code and atomic data.<cit.> finds 2 velocities, but we retain the 3 velocity model for a consistent comparison with 2015. There is no increase of reduced χ^2 compared to the two velocity model, suggesting the kinematics remain similarover a timescale of years.In Table <ref> the continuum parameters of both epochs, 2004 and 2015, are presented.Finally the summed (across velocity components) column densities of the two epochs are given in Table <ref>. These are compared graphically in Fig. <ref>, as well as with the <cit.> measured column densities for reference. While the different velocities may be associated with different physical components, the current measurement is not sensitive to ionic column density changes in individual components due to the limited spectral resolution. This is manifested in an inherit degeneracy of column densities between the velocity components, and the sum allows us to increase the sensitivity to change. A clear match can be seen, with 30/34 ion column densities within 90% confidence. Only N^+6, O^+4, Fe^+17, and S^+12 are discrepant between observations, but with 90% uncertainties 3-4 measurements are expected to be discrepant. Moreover, other similar-ionization ions do not vary, indicating no absorption variability between the two epochs. §.§ Absorption Measure DistributionWe characterize the ionization distribution of the absorber plasma using the Absorption Measure Distribution <cit.>, defined as:AMD≜d N_H/dlogξwhere N_H is the column density and ξ=L/(n_e R^2) is the ionization parameter. Here n_e is the electron number density and R is the distance of the absorber from the source. We can reconstruct the AMD using the measured ionic column densities:N_i=∫ A_Z f_i(ξ) d N_H/dlogξdlogξwhere A_Z is the solar abundance of the element <cit.> and f_i(ξ) is the fractional abundance of the ion as a function of ξ. We use a multiple thin shell model produced by XSTAR version 2.38[<http://heasarc.gsfc.nasa.gov/docs/software/lheasoft/xstar/xstar.html>, along with AMD analysis code in <https://github.com/uperetz/AstroTools>, see README.] to determine the ionic fractions as a function of logξ. The thin shell model assumes each logξ is exposed to the unabsorbed continuum directly. This is justified by observing that the broad band continuum is not significantly attenuated by the absorption as seen by the relatively shallow edges (See Fig. <ref>). Our model grid is calculated from logξ=-3.9 to logξ=3.9 with Δlogξ=0.1.We use a Spectral Energy Distribution (SED) from 1 to 1000 Ry extrapolated from our multi-wavelength observations and corrected for galactic absorption (M. Mehdipour et al., in preparation). An estimate of the AMD can be obtained assuming that each ion contributes its entire column at the ξ_max where the ion's relative ionic abundance peaks.The total equivalent N_H for each logξ_max is then estimated by each ion:N_H = N_i/A_Z f_i(ξ_max)This is a lower limit on column densities since in general f(ξ)≤ f(ξ_max). The estimate is plotted in Fig. <ref>, and shows a slight increase in column with logξ consistent with <cit.>. Different ions from different elements in the same logξ bin should agree, and discrepancies reflect deviations from solar abundances.In order to compute the AMD we want to solve the discretized set of equations (<ref>)N=A_Zf ( AMD ⊗Δlogξ)where A_Zf is the matrix of ionic fractions given by XSTAR multiplied by A_Z, Nis the vector of measured ionic column densities, and AMD ⊗Δlogξ is the vector of H column densities we want to find multiplied by the vector of AMD bins. Note the AMD vector is re-binned manually and may be uneven, enlarging the size of the bin until significant constraints are obtained for each bin.The predicted columns are N^p=A_Zf ( AMD ⊗Δlogξ). We use C-statistics <cit.> to fitthe AMD as we expect zero-value bins and there are less than 30 d.o.f.We minimize the C_stat in order to find a best fit for the AMD:C_stat=2∑_k^d.o.f.(N^p-NlnN^p)_kThe uncertainties of the measured ionic column densities are propagated stochastically.We use 1000 Monte-Carlo runs on the vector N, where each column density isrolled from a triangular probability distribution ranging through the 90% confidence interval peaking at the best fit.The resulting AMD is plotted in Fig. <ref>, and resembles the AMD of <cit.>. This is also well in agreement with the usual bi-modal shape commonly observed in AGNs <cit.>. The consistency of the AMD structure along with the individual ionic column measurements increases our confidence that the absorber is unchanged between the 2004 and 2015 observations. § VARIABILITY AND ELECTRON DENSITYFollowing the previous work of <cit.>, and <cit.> we constrain a lower-limit on distance of the source to the outflow using the fact that no variability is measured in ionic column densities. From this we can estimate upper limits on n_e. In Appendix <ref> a rigorous derivation of the equations used in this section is provided for reference. §.§ Days time scale variabilityThe NGC 7469 lightcurve, created using the high statistics of the EPIC-pn, shows NGC 7469 has a variable continuum. In Fig. <ref> the 9EPIC-pn lightcurves are presented, two from 2004 and the rest from 2015, with the count rate varying by up to a factor of 2 within a day. This rapid variability (compare with the year time scales, Sec. <ref>) suggests the possibility of constraining the minimum response time to a change in ionizing flux of NGC 7469, and giving a lower limit on n_e and thus an upper limit on the distance of the outflow from the AGN. This would only be possible if ionic column densities would be observed to change within the timescales of the continuum variability. In our case no variability can be detected on scales of days and longward, and thusonly lower limits on distance and upper limits on n_e may be obtained.Since we can constrain the column densities at best to 50%, evaluated by comparing the uncertainties to the best fit values, weaker variability is not ruled out. Conversely, the lack of detected variability in ∼30 individual ions, as well as a lack of a systematic trend in the discrepancies between best-fit values, implies that if any change exists, it is small and may not be attributed to a change of the ionizing flux.UV observations are more sensitive to variation in absorption troughs, and a detailed UV analysis of the epochs of NGC 7469 will be presented in a separate paper (Arav et al. in preparation).In order to check the stability of the absorption due to the ionized outflow we apply the best-fit model on the combined spectra as a starting point for the fit of each individual spectrum. Though the lower S/N of a single observation hampers tight constraints, the results are consistent within the 90% uncertainty intervals across observations (Fig. <ref>), even better. The only exceptions are Ne^+8 and Fe^+20deviating for one observation, but not the same one. Beyond constancy among observations, when considering the best-fit values it is evident that there is no clear trend - the ordering of column densities of different ions of similar ionization parameter between observations is not uniform. This indicates there is no observable change, in fact, of the ionic column densities during the last half year of 2015. §.§ Intra-day time scale variability: comparing high and low statesThe stochastic nature of the ionizing flux may lead to an hypothesis that any outflow which is not dense and close to the source would not respond quickly enough to the changes, at least not measurably.By summing spectra of predominantly high and low states of the AGN separately, more subtle changes can be measuredby improving the S/N of small absorption troughs which change in a consistent manner, on a daily basis.We divide the states according to the EPIC-pn lightcurve, around the mean count rate for the 2015 observations (which is nearly identical to the median one) of 23.2 counts s^-1.Retaining all photons in favor of statistics and in order to secure similar RGS S/N in the high and low states, we cut the eventsat 0.4σ above the mean EPIC-pn count rate (σ=3.5 counts s^-1 is thestandard deviation of the light curve).These spectra are presented in Fig. <ref>, showing very similar troughs. As was the case for the individual epoch analysis, we begin a fit from the best-fit model of the combined spectra. Results are presented in Fig. <ref>. Here 3 ions only deviate, which is expected within the 90% statistics.Once again the NGC 7469 outflow proves to be remarkably stable such that when observing only times of high flux and comparing to times of low flux, no change is observed in column densities and thus the outflow ionization distribution. Here, variability is constrained at best to 25% (by comparing the uncertainties to the best fit values), and again, weaker variations may be present. §.§ Year time scale variabilityWhile on timescales of days and less we see that the continuum of NGC 7469 is variable in EPIC-pn lightcurves (Fig. <ref>), ionic column densities remain unchanged over timescales of days and months, observed during 2015. In addition, the column densities are comparable to those of 2004, despite the 25% difference in flux (See Table <ref>).Thus, we make the assumption that column densities remain unchanged for the entire T=10 years. This assumption allows us to constrain the distance R of the outflow from the ionizing source assuming τ>T, where τ is the ionization equilibrium time <cit.>. A full derivation of the dependence of R on T is detailed in Appendix <ref>. While the power of this derivation cannot be fully utilized for NGC 7469 as we detect no variability, a useful result for this case is:R^2>(α_i L/ξ_max+ℒ_i)TGiven an ion i the recombination rate coefficient is α_i.The photoionization cross-section and rate are, respectively, σ_i and ℒ_i/R^2, whereℒ_i=1/4π∫_1 Ry^1000 Ryσ_i(E)L(E)/Ed E L(E) is the luminosity density in erg s^-1 keV^-1 and L is the ionizing AGN luminosity: L=∫_1Ry^1000RyL(E)d E L(E) and L are estimated from the SED, which yields L=1.4× 10^44 erg s^-1. Recombination and photoionization coefficients are taken from the CHIANTI software package[http://www.chiantidatabase.org/] <cit.>. Once we obtain a lower limit on the distance, we may use the definition of ξ to extract an upper limit on n_e:n_e < L/ξ R^2_minwhere R_min is the minimal value obtained from eq. <ref>, and the same ξ and L values are used.Distances and electron densities measured from several ions are given in Table <ref>. The outflow is constrained to be at least 12 pc away from the source for Fe^+22 and31 pc for N^+6. This constraint is not strong enough to dis-associate it from the AGN completely, or associate it with the starburst region seen in NGC 7469 <cit.>, which is approximately 1 kpc from the source.§ ENERGY DEPOSITUsing mass conservation in a continuous conical outflow with opening angle Ω, d m=Ω R^2 n_e v μ m_p d t,we define the kinetic power of the outflow asĖ_K=1/2d m/d t v^2=Ω/2 n_e R^2 v^3 μ m_p=Ω/2L/ξv^3 μ m_pWhere μ=1.4 is the mean molecular weight and m_p is the proton mass.We assuming here a bi-conical flow of Ω=2π. We use the maximal velocity component of -2000 km s^-1and the lowest ionization observed at that velocity, logξ=1 (excluding ions with column density consistent with 0). These values are seen for example in C^+5 and O^+6. This yields a maximal possible value ofĖ_K=8.2×10^44 erg s^-1≈0.6L_Eddwhere L_Edd=1.4×10^45 erg s^-1. Using a high logξ=2.5 but leaving the velocity of-2000 km s^-1 (observed for example in Fe^+22) will reduce this value by two orders of magnitude: Ė_K=6.2×10^42 erg s^-1≈0.004L_Edd Substituting in the lowest velocity of -600 km s^-1 will reduce Ė_K by another 1.5 orders of magnitude, and an opening angle less than 2π would reduce it even further.The fact that a range of ξ values is ubiquitously observed in AGN outflows indicates the wind cannot have a conical n_e∝ R^-2 density profile. Multiple ionization winds have been discussed in the models of <cit.>. Eq. <ref> results in an increase of power with decreasing ionization.Other definitions of kinetic luminosity, such as that of <cit.>, assume a thin shell of thickness Δ R<R rather than a continuous outflow, dividing the mass by the traversal timescale, R/v. In that case the kinetic luminosity would be lower by Δ R/R.One may also assume Δ R=R, such that Ė_K∝ n_eR^2=N_HR. In this case we can use the measured lower limits on distance (Table <ref>) and the measured equivalent H column densities (Eq. <ref>). Lower bounds for Ė_K from N^+6, O^+7, Fe^+22 respectively are 1.1×10^43, 4.2×10^41, and 2.6×10^42erg s^-1.Note for each ion we use the fastest velocity where measured column density is inconsistent with 0, namely –600 km s^-1 for N^+6, and –2000 km s^-1 for O^+7 and Fe^+22. The lowest estimate is even lower than that of Equation <ref>.Assuming the highest estimate of the kinetic power (Eq. <ref>) is the true energy carried by the outflow would imply significant feedback. However, the fact is that a starburst regionis observed at 1 kpc <cit.> and does not seem to be affected by theoutflow. This would lead to the conclusion that the outflow is spatially de-coupled from the starburst region.If the outflow power is much lower as in Eq. <ref>, thiswould naturally explain why the starburst region is unaffected.§ CONCLUSIONSThe X-rays absorption spectra of NGC 7469 is remarkably stable on all of the measured time scales. In observations spread over years, months and days column densities associated with the ionized absorber are not observed to change.On the other hand, the intrinsic variability of the source is large, changing by up to a factor of two in the course of a single day.In addition, the average soft X-ray powerlaw slope changes between 2004 and 2015 from 2.1 to 2.3, again, with no observed absorption variability.The kinematic components of the outflow are also constant between the 2004 and 2015 observations, and between the X-ray and the UV bands. Constancy of the outflow can also be observed in the reconstructed AMD, featuring one high ionization component and one low ionization component with the same column densities in both 2004 and 2015. Admittedly, the broad and relatively flat AMD makes ionization changes much harder to detect than in a single-ξ component. To that end, we would expect to notice changes only in the highest and lowest ionization states. Nonetheless, the UV spectra of this campaign (Arav et al. in preparation) confirms for the most part the lack of absorption variability, except for minor changes that are detected in a few velocity bins in the UV, but are much below the current X-ray sensitivity.The flux variations on different timescales with no effect whatsoever on the outflow imply a distant outflow, several pc away from the AGN at least. Beyond the large distance, the velocities, luminosity, and observed ionization parameters suggest the outflow may carry as much as 2/3 of the Eddington AGN power, which is significant in terms of feedback. However, this is dependent on ξ (Eq. <ref>) as expected for non-conical outflows, and is 2 orders of magnitude lower for high ξ values, making these estimates ambiguous and inconclusive as estimators of feedback without a physical model associated with Ė_K .We found no evidence the AGN is responsible for driving the outflow, since the distance scales are beyond the torus <cit.> and comparable to the region of narrow (∼500 km s^-1) line emission. The obtained constraints on distance and power of the outflow need to be examined in other AGNs in order to understand if these outflows are unimportant to the galactic scale, and what is their connection to the AGN itself. This work was supported by NASA grant NNX16AC07G through the XMM-Newton Guest Observing Program, and through grants for HST program number 14054 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555.The research at the Technion is supported by the I-CORE program of the Planning and Budgeting Committee (grant number 1937/12).EB received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 655324.SRON is supported financially by NWO, the Netherlands Organization for Scientific Research. NA is grateful for a visiting-professor fellowship at the Technion, granted by the Lady Davis Trust.SB and MC acknowledge financial support from the Italian Space Agency under grant ASI-INAF I/037/12/0. BDM acknowledges support from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 665778 via the Polish National Science Center grant Polonez UMO-2016/21/P/ST9/04025. LDG acknoweledges support from the Swiss National Science Foundation. GP acknowledges support by the Bundesministerium für Wirtschaft undTechnologie/Deutsches Zentrum für Luft- und Raumfahrt(BMWI/DLR, FKZ 50 OR 1408 and FKZ 50 OR 1604) and the Max Planck Society. 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C., et al. 2005, , 634, 193[Stern et al.(2014)Stern, Behar, Laor, Baskin, & Holczer]Stern14 Stern, J., Behar, E., Laor, A., Baskin, A., & Holczer, T. 2014, , 445, 3011[Suganuma et al.(2006)Suganuma, Yoshii, Kobayashi, Minezaki, Enya, Tomita, Aoki, Koshida, & Peterson]Suganuma06 Suganuma, M., Yoshii, Y., Kobayashi, Y., et al. 2006, , 639, 46§ EQUILIBRIUM TIMEFollowing the works of <cit.> we define the two inverse timescales for ionization and recombination respectively:𝒥_i =∫_ν_0^∞σ_i(ν)J(ν)/hνdν= =1/4π R^2∫_ν_0^∞σ_i(ν)L(ν)/hνdν≜ℒ_i/R^2 ℛ_i =α_i n_eThe ionization/recombination/equilibrium time τ used in this paper is the decay time of the exponential solution of the system of equations for the ionic populations n_i:ṅ_i=-(𝒥_i+ℛ_i)n_i+𝒥_i-1 n_i-1+ℛ_i+1 n_i+1for charge states 0<i<Q and the boundary defined by n_Q+1=n_-1=0 or: ṅ_0= -𝒥_0n_0+ℛ_1 n_1 ṅ_Q= -ℛ_Qn_Q+𝒥_Q-1n_Q-1These equations assume all charge states are exposed to the same radiation field J(ν). In the general case where the radiation field J(ν) is non-uniform this approximation breaks down. §.§ Assumptions and caveatsIn general, <ref> must be solved for atime varying set of 𝒥_i,ℛ_i, making the full solution much more difficult, and is formally given in <cit.>. This is less practical when we want to use our measurements to constrain unobserved quantities, such as n_e. In this case, we often want to consider a system in equilibrium, with a given inital set 𝒥_i^0,ℛ_i^0, where we abruptly change the external conditions using a new set of 𝒥_i^final - making the assumption that the continuum changed as a step function, and we are observing much after the step (See Sec. <ref>), or conversely that the system is in equilibrium and this abrupt change has yet to be observed. Though often not the case, this is a good assumption when observing the outflow much before and much after such a change in seed flux, such that the continuum observed is steady for times greater than the τ. Consider now a short scale oscillating variation in seed flux,t_short≪τ AGNs in general (indeed, NGC 7469 is a good example) may change drastically on timescales of days, with no observable change in column densities. In this case we may assume that the effective continuum on the plasma is in fact a steady one, given by the time averaged flux.𝒥_i = ∫_0^T≥ t_short𝒥_i d t/T≥ t_short=1/T∫_0^T d t ∫_ν_0^∞σ_i(ν)J(ν)/hνdν== ∫_ν_0^∞σ_i(ν)J(ν)/hνdνSo we make 3 assumptions when analyzing this photoionized plasma: * If no column densities are observed to change while flux varies on short (hours,days) time scales, a steady time averaged continuum may be assumed.* If column densities are changed between two observations, and the flux is shown to be steady, we will assume T_final>T_start+τ where T_final is the final observation and T_start is the time where the continuum started to change, after the first observation. In this case we assume a step function change for the continuum.* Finally, if column densities remain unchanged between two observations but flux is shown to have changed and remain steady, we will assume τ>δ T, the time between observations. §.§ SolutionFrom the form of the equations, or from solving the simple 2-level system one may quickly come to the conclusion a general solution should be of the form (assuming constant 𝒥_i,ℛ_i, as per Sec. <ref>):n_i=A_ie^-t/τ+B_iFirst-order differential equations have only one free coefficient depending on the initial conditions.τ must be independent of charge, and this can easily be shown by substituting different τ_i,τ_j for consecutive charge states into the equation for ṅ_i, <ref>, assuming A_i and B_i are constants.Some properties of this solution are evident immediately. Assuming steady state before t=0 and at t→∞ leads to the conclusion:B_i =n_fiA_i =n_ii-n_fiwhere n_fi are the equilibrium densities at ∞ and n_ii are the initial equilibrium densities. An important consequence is that B_i are not integration coefficients. These are the final equilibrium solutions, explicitly given by 𝒥_i,ℛ_i, as seen in the Section <ref>. Substitute in our form <ref> to <ref>:-τ^-1A_ie^-t/τ= -(𝒥_i+ℛ_i)(A_ie^-t/τ+B_i)+ +𝒥_i-1(A_i-1e^-t/τ+B_i-1)++ℛ_i+1(A_i+1e^t/τ+B_i+1)→ → A_i+1e^t/τ +B_i+1==1/R_i+1((𝒥_i+ℛ_i-τ^-1)A_i-𝒥_i-1A_i-1)e^-t/τ++1/R_i+1((𝒥_i+ℛ_i)B_i-𝒥_i-1B_i-1)Grouping the coefficient for the exponent and constant results in the formulas for the coefficients:A_i+1 =1/R_i+1((𝒥_i+ℛ_i-τ^-1)A_i-𝒥_i-1A_i-1)B_i+1 =1/R_i+1((𝒥_i+ℛ_i)B_i-𝒥_i-1B_i-1)It is easy to prove that <ref> results in B_i/B_i-1=𝒥_i-1/ℛ_i which we know must be true, as B_i are an equilibrium solution (see Sec. <ref>). What will be interesting to us is the relation of A_i to τ.§.§ Equilibrium timeA closed form solution for A_i is more difficult, but we are only interested in τ, which may be obtained from <ref> using any observed ionization triad:τ= ((𝒥_i+ℛ_i)-ℛ_i+1A_i+1+𝒥_i-1A_i-1/A_i)^-1== ( (𝒥_i+ℛ_i)- . - . ℛ_i+1(n_ii+1-n_fi+1)+𝒥_i-1(n_ii-1-n_fi-1)/n_ii-n_fi)^-1Measuring 3 ions of an element and seed flux of 2 different observation epochs will allow us to constrain n_e. In terms of what we measure:n_ii-1-n_fi-1/n_ii-n_fi=δ n_i-1/δ n_i=lδ N_i-1/δ N_iwhere N_i are the column densities of the specific ions and l is the ratio of widths over which the two ions extend. We will assume l=1 as ξ is inversely proportional to R and <cit.> shows most adjacent ion stages tend to extend over similar ξ ranges, and indeed may exist in the same part of the plasma, though this does not have to be the case. An interesting thing to note is that the equilibrium constants n_i,f are also dependent on the ℛ and 𝒥, and obviously each is a different set of constants as both n_e and J(ν) have changed, but only those of n_fi are the same as the explicit 𝒥 and ℛ appearing in <ref>. Finally, substituting the expressions for 𝒥_i,ℛ_i obtain the relationship we need: τ= ((α_i-δ N_i+1/δ N_iα_i+1)n_e+ . +. (ℒ_i-δ N_i-1/δ N_iℒ_i-1)R^-2)^-1We note this equation is the same as eq. 10 in <cit.> when (ℒ_i-δ N_i-1/δ N_iℒ_i-1)R^-2== -J(t>0)/J(t=0)(α_i-δ N_i+1/δ N_iα_i+1)n_eand δ N = N, tying a step change in ionization flux to recombination.Note that the ionization parameter is an observable that is found independently:ξ=L/n_e R^2= ∫_1 Ry^1000 Ry L(ν)dν/n_e R^2 While at first glance this may seem like it would be embedded somehow in <ref>, note that ξ is a purely equilibrium characteristic of the plasma, while τ is of course the time scale characterizing the system out of equilibrium. This gives us physical justification to say <ref> and <ref> are independent equations, and may be solved simultaneously for n_e and R^2:n_e= ((α_i-δ N_i+1/δ N_iα_i+1)+ . +. (ℒ_i-δ N_i-1/δ N_iℒ_i-1)ξ/L)^-1τ^-1 R^2= ((α_i-δ N_i+1/δ N_iα_i+1)L/ξ+ . +. (ℒ_i-δ N_i-1/δ N_iℒ_i-1))τAn interesting consequence is that the coefficients of τ must be positive. If this is not the case, then these solutions are wrong and our assumptions need to be put to test. Note that for a two level system this must be true as the ratio of column change is always negative. §.§ ApplicationsWhile most parameters insofar are either measurable independently (ℒ_i,L,δ N_i,ξ) or known (α_i) we in general only have a limit on τ as we do not observe the plasma continuously. To practically apply this result to observational data we need inequalities, not equalities. Assume we know τ is lower than some constant T, a time between two observations. This happens often when we see an AGN in a steady low/high state at one time, and a high/low in another, with different columns. We can then use:n_e> ((α_i-δ N_i+1/δ N_iα_i+1)+. +. (ℒ_i-δ N_i-1/δ N_iℒ_i-1)ξ/L)^-1T^-1 R^2< ((α_i-δ N_i+1/δ N_iα_i+1)L/ξ+ . +. (ℒ_i-δ N_i-1/δ N_iℒ_i-1))T to constrain a maximal R, and minimal electron density. If on the other hand no variability is measured we are struck with a problem. While we would know τ>T, so constraints would be reversed, we do not know the final column densities. One way to handle this is to make the assumption δ N_i+1/δ N_i is, as a two level system, always negative, allowing an estimate:R^2> ((α_i-δ N_i+1/δ N_iα_i+1)L/ξ+(ℒ_i-δ N_i-1/δ N_iℒ_i-1))T>> (α_i L/ξ+ℒ_i)T and consequently following from eq. <ref> we have: n_e < L/ξ R^2_min where R_min is obtained from the lower limit given by eq. <ref>. This is the approximation used in Sec. <ref>. §.§ EquilibriumWe add this section for completeness' sake only. This problem can trivially be solved for the case ṅ_i=0, where by induction if n_i-1/n_i=ℛ_i/𝒥_i-1 and0=-(𝒥_i+ℛ_i)n_i+𝒥_i-1n_i-1+ℛ_i+1n_i+1Substituting in the induction assumption we have the well known result:0= -(𝒥_i+ℛ_i)n_i+ℛ_in_i+ℛ_i+1n_i+1→ → n_i/n_i+1=ℛ_i+1/𝒥_iThis is easy to show for the first pair using <ref>=0. This recursive solution is quickly generalized for the relationship between n_i and n_j, where i<j and i>j respectively:n_i= ℛ_i+1/𝒥_in_i+1=ℛ_i+1/𝒥_iℛ_i+2/𝒥_i+1n_i+2=...= = n_j∏_k=i^j-1ℛ_k+1/𝒥_kn_i= 𝒥_i-1/ℛ_in_i-1=𝒥_i-1/ℛ_i𝒥_i-2/ℛ_i-1n_i-2=...= = n_j∏_k=j^i-1𝒥_k/ℛ_k+1Finally we note that our system when summed is telescopic, that is:∑_i=0^T ṅ_i= 0→N= ∑ n_i=n_i(1+(∑_j<i+∑_j>i)n_j/n_i)where we have defined N as the constant number of particles. Thus we obtain a complete closed form solution, starting from equation <ref> and solving for n_i:n_i=N/(1+∑_j<i∏_k=j^i-1ℛ_k+1/𝒥_k+ ∑_j>i∏_k=i^j-1𝒥_k/ℛ_k+1) | http://arxiv.org/abs/1709.09633v1 | {
"authors": [
"U. Peretz",
"E. Behar",
"G. A. Kriss",
"J. Kaastra",
"N. Arav",
"S. Bianchi",
"G. Branduardi-Raymont",
"M. Cappi",
"E. Costantini",
"B. De Marco",
"L. Di Gesu",
"J. Ebrero",
"S. Kaspi",
"M. Mehdipour",
"R. Middei",
"S. Paltani",
"P. O. Petrucci",
"G. Ponti",
"F. Ursini"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170927171758",
"title": "Multi-wavelength campaign on NGC7469 II. Column densities and variability in the X-ray spectrum"
} |
Facility for Rare Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany Department of Physics, Arizona State University, Tempe, Arizona 85287, USA Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany School of Science, Huzhou University, Huzhou 313000, China We report accurate quantum Monte Carlo calculations of nuclei up to A=16 based on local chiral two- and three-nucleon interactions up to next-to-next-to-leading order. We examine the theoretical uncertainties associated with the chiral expansion and the cutoff in the theory, as well as the associated operator choices in the three-nucleon interactions. While in light nuclei the cutoff variation and systematic uncertainties are rather small, in [16]O these can be significant for large coordinate-space cutoffs. Overall, we show that chiral interactions constructed to reproduce properties of very light systems and nucleon-nucleon scattering give an excellent description of binding energies, charge radii, and form factors for all these nuclei, including open-shell systems in A=6 and 12.Properties of Nuclei up to A=16 using Local Chiral Interactions X. B. Wang December 30, 2023 ===============================================================Introduction.—Predicting the emergence of nuclear propertiesand structure from first principles is a formidable task. An important open question is whether it is possible to describe nuclei and their global properties, e.g., binding energies and radii, from microscopic nuclear Hamiltonians constructed to reproduce only few-body observables, while simultaneously predicting properties of matter, including the equation of state and the properties of neutron stars. Despite advanced efforts, definitive answers are not yet available <cit.>.Several properties of nuclei up to [12]C have been successfully described using the phenomenologicalArgonne v_18 (AV18) nucleon-nucleon (NN) potential combined with Illinois models for the three-body interactions <cit.>. Unfortunately these phenomenological modelshave at least two main limitations. They do not provide a systematic way to improve the interactions or to estimate theoretical uncertainties. In addition, they provide a too soft equation of state of neutron matter <cit.>, with the consequence that the predicted structure of neutron stars is not compatible with recent observations of two solar-mass stars <cit.>.The Argonne-Illinois models have been constructed to be nearly local: The dominant parts of the interaction do not depend on the momenta of the two interacting nucleons but only on their relative distance, spin, and isospin. This construction was motivated by the ease of employing such potentials in continuum quantum Monte Carlo (QMC) methods, such as the Green's function Monte Carlo (GFMC) and auxiliary field diffusion Monte Carlo (AFDMC) methods. The advantage of QMC methods is that they can be used to solve accurately and nonperturbatively the many-body problem without requiring the use of softer Hamiltonians. The GFMC and AFDMC methods can besuccessfully used only for nearly local Hamiltonians because of the sign problem <cit.>.In the last two decades, chiral effective field theory (EFT) has paved the way to the development of nuclear interactions and currents in a systematic way <cit.>. Chiral EFT expands the nuclear interaction in the ratio of a small scale (e.g., the pion mass or a typical momentum scale in the nucleus) to a hard scale (the chiral breakdown scale). Such an expansion provides several advantages over the traditional approach, including the ability to improve the interaction order by order, means to estimate theoretical uncertainties, and the fact that many-body forces and currents are predicted consistently. The long-range pion-exchange contributions are determined by pion-nucleon couplings, while the short-range contributions (given by so-called low-energy constants) are fit to reproduce experimental data. Usually, chiral EFT interactions are formulated in momentum space, but recently Gezerlis et al. demonstrated a way to produce equivalent local formulations of chiral NNinteractions up to next-to-next-to-leading-order(N^2LO) <cit.>. Consistent three-body forces were constructed in Refs. <cit.>, as well as chiral interactions with explicit Delta degrees of freedom <cit.>.To solve for the ground state of nuclei, we use the AFDMC method with local chiral interactions that have been determined from fits to NN scattering, the alpha particle binding energy, and n–α scattering <cit.>. This method has previously been used to determine the properties of homogeneous and inhomogeneous neutronmatter <cit.>,and nuclear matter and finite nuclei using simplified potentials <cit.>.In this Letter we present several new important achievements: (i) the first application of the AFDMC method to calculate properties of nuclei using chiral Hamiltonians at N^2LO, including three-body forces, (ii) a systematic investigation of the chiral expansion, including truncation error estimates, in selected nuclei from A=3 to A=16, and (iii) an investigation of the cutoff dependence and the use of different three-body operators for A≥6.Hamiltonian and AFDMC method.—The Hamiltonian is of the formH=-ħ^2/2m∑_i ∇_i^2+∑_i<jv_ij+∑_i<j<kV_ijk ,where the two-body interaction v_ij also includes Coulomb and other electromagnetic effects.The two-body potentials v_ij and three-body potential V_ijk are as in Refs. <cit.>. The general form of the variational state is the following:|Ψ⟩=[F_C+F_2+F_3]|Φ⟩_J,T ,where F_C accounts for all the spin- and isospin-independent correlations,and F_2 and F_3 are linear in spin- and isospin-pair two- and three-body correlations as described in Ref. <cit.>.The term |Φ⟩ is taken to be a shell-model-like state with total angular momentum J and total isospin T. Its wave function consists of a sum of Slater determinants D constructed using single-particle orbitals:⟨ RS |Φ⟩_J,T = ∑_n c_n(∑ D{ϕ_α(r_i,s_i)})_J,T ,where r_i are the spatial coordinates of the nucleons and s_irepresent their spins. Each single particle orbital ϕ_α consists of a radial function φ(r) coupled to the spin and isospin states. The determinants are coupled with Clebsch-Gordan coefficients to total J and T, and the c_n are variationalparameters multiplying different components having the same quantum numbers. The radial functions φ(r) are obtained by solving for the eigenfunctions of a Wood-Saxon well, and all parameters are chosen by minimizing the variational energy as described in Ref. <cit.>. In order to improve |Φ⟩, we include single particle orbitals up to the sd shell.A complete description of the AFDMC method using two-body interactions is given in Refs. <cit.>.Here we describe how three-body interactions are included. The main limitation of the AFDMC method is thatthe standard Hubbard-Stratonovich transformation used to propagate the wave function in imaginary time can only be applied to potentials that are quadratic in spin and isospin operators. The three-body coordinate-space dependence is straightforward to include, as are several important terms in thethree-body interaction that depend on the spin and isospin of two nucleons at a time.Terms depending on the spin and isospin of all three nucleons are included in an effective way in the propagation, and then fully accounted for in the final results. In practice, we determine a Hamiltonian H' thatmimics the full Hamiltonian, as discussed in the following, and then we calculate as a perturbation the difference⟨ H'-H⟩. This procedure goes beyond the standard normal ordering that averages the dependence of the third nucleon's position, spin, and isospin.The chiral three-body interactions at N^2LO contain terms that can be organized asV=V_a^2π,P+V_c^2π,P+V^2π,S+V_D+V_E.The first, second, and third terms correspond to the two-pion exchange diagrams in P and S waves [Eqs. (A.1b), (A.1c) and (A.1a), respectively, of Ref. <cit.>]. The subscripts a and c refer to the fact that these contributions can be written in terms of an anticommutator or commutator, respectively. We can rewrite V_a,c^2π,P by separating it into long-, intermediate-, and short-range parts:V_a,c^2π,P=V_a,c^XX+V_a,c^Xδ+V_a,c^δδ ,where X and δ refer to the X_ij(r) and δ_R_3N(r) functions defined in Ref. <cit.>. V_D contains an intermediate-range one-pion-exchange-contact interaction [Eq. (24b) of Ref. <cit.>], while V_E contains a short-range term.In this work, we consider two alternative forms for V_E: namely, V_Eτ and V_E1[Eqs. (26a) and (26b), respectively, of Ref. <cit.>]. They differ in the operator structure, according to the Fierz-rearrangementfreedom in the selection of local contact operators in the three-body sector up to N^2LO <cit.>. Eτ refers to the choice of the two-body operator τ_i·τ_j,while E1 to the choice of the identity operator 1.The terms V_a^2π,P, V^2π,S, V_D, and V_E arepurely quadratic in spin and isospin operators, and can be included exactly in the AFDMC propagator. The term V_c^2π,P contains instead explicit cubic spin and isospin operators.These terms cannot be fully included in the AFDMC propagation; however, their expectation value can be calculated. We determine the Hamiltonian H' that can be fully propagated asH'=H-V_c^2π,P+α_1 V_a^XX+α_2 V_D+α_3 V_E.The three constants α_i are adjusted in order to have⟨ V_c^XX⟩ ≈⟨α_1 V_a^XX⟩ ,⟨ V_c^Xδ⟩ ≈⟨α_2 V_D⟩ ,⟨ V_c^δδ⟩ ≈⟨α_3 V_E⟩ ,where the identifications are suggested by the similar ranges and functional forms. The average ⟨ ⋯⟩ indicates an averageover the propagated wave function. Once the ground state Ψ of H' is calculated with the AFDMC method, the expectation value of the Hamiltonian H is given by⟨ H⟩ ≈⟨Ψ|H'|Ψ⟩-⟨Ψ|H'-H|Ψ⟩ ,where the last quantity in the previous expression isevaluated perturbatively. Adjusting the constants α_i in such a way thatthe correction is small suggests that the correction is perturbative. The same estimate is used in GFMC calculations to determine the small contributions from nonlocal terms that arepresent in the AV18 potential, and in that case thedifference v_8'-v_18 is calculated as a perturbation <cit.>.In order to test the technique described above, we first determined the optimal parameters α_i for a given system, then changed their values by up to 10%, and verified that the final result of ⟨ H⟩ is nearly independent of such a variation. For example,for ^16O such a variation changes ⟨ H'-H⟩ from ≈ 1 to ≈ 15MeV, but the final estimate of the ground-state energy iswithin 2MeV. In addition, we benchmarked the energies of A=3 and A=4 nuclei using the AFDMC method, by comparing with the GFMC results of Refs. <cit.>, where the three-body interactions are included fully in the propagation and found very good agreement within a few percent. Note that in many other approaches the three-body force is replaced by an effective two-body interaction (this is achieved by normal ordering) neglecting the residual three-body term <cit.>. However, this approximation has only been benchmarked for softer interactions <cit.>.The AFDMC method used here is limited by a sign problem <cit.>. The sign problem is initially suppressed by evolving the wave function in imaginary time using the constrained-path approximation, where the configurations are constrained to have positive real overlap with the trial function, as described in Ref. <cit.>. After an initialequilibration of the configurations using the constrained-path approximation, the constraint is removed, and then the evolution in imaginary-time is performed until the sign-problem dominates and the variance of the results becomes severely large. The final (statistical) error strongly depends on the quality of the trial wave function. We have made several tests to check the results and the dependence on the initial trial wave function, and have concluded that the systematicuncertainties due to releasing the constraint give results correct to ∼5% for [16]O. Initial attempts to improve the wave functionfor [16]O show a lowering of the energy by about 4-5MeV, but since the computational cost is much higher and statistical errors aresimilar to this difference, we leave more detailed studies to future work.Results.—We consider chiral Hamiltonians at leading-order (LO),next-to-leading order (NLO), and N^2LO. In this way, following Ref. <cit.>, we can assign theoretical uncertainties to observables coming from the truncation of the chiral expansion. Uncertainties for an observable X are estimated asΔ X^N^2LO=max(Q^4×|X^LO|, Q^2×|X^NLO-X^LO|, Q×|X^N^2LO-X^NLO|), where we take Q=m_π/Λ_b with Λ_b=600MeV (see Ref. <cit.> for a detailed discussion on uncertaintyestimates with local chiral interactions).In Table <ref> we report the AFDMC results for the ground-state energies and charge radii for nuclei with A≥ 6 at N^2LO. In particular, we used the two different cutoffs R_0=1.0 and R_0=1.2fm (approximately corresponding to cutoffs in momentum spaceof 500 and 400MeV <cit.>,note, however also Ref. <cit.>),and two of the three available V_E interactions constructed in Ref. <cit.>. We find that, starting from local chiral Hamiltonians fit to NN scattering data <cit.>and three-body interactions fit to light nuclei <cit.>, energies andradii for nuclei up to A=16 are qualitatively well reproduced.In particular, we find that the two cutoffs employed here, R_0=1.0 and R_0=1.2fm, reproduce experimental binding energies and charge radii up to A=6 within a few percent. An exception is for the charge radius of[6]Li that is slightly underestimated for both cutoffs. Sizably different is the case of the softer interaction (R_0=1.2fm)for larger systems, which can significantly overbind [16]O, resulting in a very compact system. In this case the theoretical uncertainties on the energy are large, dominated by the severeoverbinding at LO (≈ -1110MeV).We also find that the two different forms (Eτ, E1) for the three-body interaction give similar results (agreeing within the EFT uncertainty) for nuclei up to A=16. This suggests that the theoretical uncertainties coming from the truncation of the chiral expansion are sufficient to account for the violation of the Fierz rearrangement <cit.>.In Fig. <ref> we present the ground-state energies per nucleon of selected nuclei with 3≤ A≤16, calculated at LO, NLO, and N^2LO (Eτ) with the cutoff R_0=1.0fm. The error bars are estimated by including the statistical uncertainties given by the AFDMC calculations as well as the error given by the truncation of the chiral expansion. The ground-state energies per nucleon are in agreement with experimental data up to A=6, while for [12]C and [16]O the energies are somewhat underpredicted. The uncertainties are reasonably small, dominated by the truncation error.In Fig. <ref> we compare the charge radii calculated at LO, NLO, and N^2LO (Eτ) with the R_0=1.0fm cutoff to experimental data.These results show that a qualitative description of binding energies and charge radii is possible starting from Hamiltonians constructed using only few-body data. We note, however, that the radius of [6]Li is slightly smaller than the experimental measurement. It is interesting to note that the charge radius of [6]Li calculated with the GFMC method employing the AV18 and Illinois VII (IL7) three-body interactions is also underestimated <cit.>.We show in Fig. <ref> the charge form factors of [12]C and [16]O compared to experimental data. The [12]C form factor is also compared to previous GFMC calculations with the AV18+IL7 potentials. Our form factor calculations have beenperformed using one-body charge operators only. Two-body operators are expected to give small contributions only at momenta larger than≈ 500MeV <cit.>, as they basically include relativistic corrections. It is interesting to compare the curves given by the two different cutoffs. In the figure, the result obtained using R_0=1.0fm at N^2LO (Eτ) (solid blue line) includes the uncertainty from the truncation of the chiral expansion (shaded blue area). The agreement with experimental data is very good.For R_0=1.2fm at N^2LO (Eτ) (dotted red line), the radius is too small and the first diffraction minimum occurs at a significantly higher momentum than experimentally observed, consistent with the overbinding obtained for this interaction.Finally, in Fig. <ref> we present the Coulomb sum rules for [12]C and [16]O. The AFDMC result for [12]C is compatible both with the available experimental data as extracted in Ref. <cit.> and with the GFMC result for AV18+IL7 <cit.>.The differences between the AFDMC and GFMC results at high momentum are due to two-body currents,fully implemented to date only in the GFMC calculations.For [16]O, the result for the harder interaction with R_0=1.0fm is very close to that of [12]C, and is compatible with the findings of Ref. <cit.> for the AV18+UIX potential. The softer interaction with R_0=1.2fm produces instead a significantly different result, as for the charge form factor.Summary.—We have performed QMC calculations of selected nuclei up to A=16 usinglocal chiral interactions at LO, NLO, and N^2LO for different cutoffs and three-body interactions. We conclude that these Hamiltonians, constructed only from NN data and properties of few-body nuclei, can give a good description of ground-state properties of nuclei up to A=16,including binding energies, charge radii, form factors, and Coulomb sum rules.This is true in particular for the harder interaction considered here, corresponding tocoordinate-space cutoff R_0=1.0fm.For the larger cutoff R_0=1.2fm, we find in [16]O a strongdependence of the energy uncertainty coming from the truncation of the chiral expansion, and a large overbinding and compactness. The latter two could be a consequence of the large c_Dcoupling in the Eτ parametrization of the three-body force <cit.>,resulting in a sizable attractive contribution not present in the hard interaction (c_D=0). More detailed analysis to further investigate this behavior will be performed in future works.We thank I. Tews, A. Lovato, A. Roggero, C. Petrie, and K. Hebeler for many valuable discussions. The work of D.L. was was supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under the FRIB Theory Alliance Grant Contract No. DE-SC0013617 titled “FRIB Theory Center - A path for the science at FRIB,” and by the NUCLEI SciDAC program. The work of J.C. and S.G. was supported by the NUCLEI SciDAC program, by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract No. DE-AC52-06NA25396, and by the LDRD program at LANL. The work of J.E.L. and A.S. was supported by the ERC Grant No. 307986 STRONGINT and the BMBF under Contract No. 05P15RDFN1. K.E.S. was supported by the National Science Foundation Grant No. PHY-1404405. X.B.W. thanks the hospitality and financial support of LANL and the National Natural Science Foundation of China under Grant No. U1732138, No. 11505056, and No. 11605054. 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"authors": [
"D. Lonardoni",
"J. Carlson",
"S. Gandolfi",
"J. E. Lynn",
"K. E. Schmidt",
"A. Schwenk",
"X. B. Wang"
],
"categories": [
"nucl-th"
],
"primary_category": "nucl-th",
"published": "20170926172720",
"title": "Properties of nuclei up to $A=16$ using local chiral interactions"
} |
addressref=aff1,aff2,[email protected]]Y.F.Yunfang Cai 0000-0002-4956-4320addressref=aff1,[email protected]]Z.Zhi Xu 0000-0001-7084-3511addressref=aff1,[email protected]]Z.G.Zhenggang Li 0000-0002-4635-984Xaddressref=aff1,[email protected]]Y.Y.Yongyuan Xiang 0000-0002-5261-6523addressref=aff1,aff2,[email protected]]Y.C.Yuchao Chen 0000-0001-8279-7014addressref=aff1,[email protected]]Y.Yu Fu 0000-0002-1570-1198addressref=aff1,corref,[email protected]]K.F.Kaifan Ji 0000-0001-8950-3875[id=aff1]Yunnan Observatories, Chinese Academy of Sciences, Kunming 650011, China [id=aff2]University of Chinese Academy of Sciences, Beijing 100049, China Y.F. CaiPrecise Reduction of Solar SpectraWe present a precise and complete procedure for processing spectral data observed by the 1-meter New Vacuum Solar Telescope (NVST). The procedure is suitable for both the sit-and-stare and raster-scan spectra. In this work, the geometric distortions of the spectra are firstly corrected for subsequent processes. Then, considering the temporal changes and the remnants of spectral lines in the flat-field, the original flat-field matrix is split into four independent components for ensuring a high precision flat-fielding correction, consisting of the continuum gradient matrix, slit non-uniform matrix, CCD dust matrix, and interference fringe matrix. Subsequently, the spectral line drifts and intensity fluctuations of the science data are further corrected. After precise reduction with this procedure, the measuring accuracies of the Doppler velocities for different spectral lines and of the oscillation curves of the chromosphere and photosphere are measured. The results show that the highest measuring accuracy of the Doppler velocity is within 100 m s-1, which indicates that the characteristics of the photosphere and chromosphere can be studied co-spatially and co-temporally with the reduced spectra of NVST.§ INTRODUCTION The solar grating spectrometer is one of the most classical equipments used to obtain solar spectral information. At present, many famous solar telescopes are equipped with grating spectrometers, such as the GREGOR Infrared Spectrograph (GRIS) <cit.>, and the Fast Imaging Solar Spectrograph (FISS) of the New Solar Telescope (NST) (). The New Vacuum Solar Telescope (NVST) of China () was built and put into observation in 2010, with two spectrometers which are also grating spectrometers.As is well known, the spectral observations taken by the grating spectrometer are always subject to the optical system of the telescope and the cameras used. To acquire physical parameters of the solar atmosphere accurately, the raw spectrum data requires post-processing. In the earlier years, the Swedish Vacuum Solar Telescope (SVST) team proposed some approaches to solve the basic problems encountered by the grating spectrograms <cit.>. After that, based on these approaches, different improvements were introduced according to the different problems existed in different spectrometers. For example, the Vacuum Tower Telescope (VTT) team separates the slit non-uniformity from the flat-field pattern () to eliminate the influence of the spectrograph drift and temporal changing of the flat-field. The NST team acquires several groups of flat data () in which the position of the spectral line on the detector is changed by setting the different grating angles. They use those groups of flat data to demodulate the final flat-field matrix (). The NVST team proposed a set of reduction procedure (), which is suitable for processing the science data whose acquisition time is very close to the flat data. For the spectrums acquired in long time series of several hours, the time-varying factors should be considered. In addition, the remnants of spectral lines always exist in the obtained flat-field and bring new errors for processing science data. In this paper, we propose a new procedure to solve these problems and to achieve higher processing accuracy.The basic purpose of spectrum reduction is to achieve high accuracy measurements of physical parameters of the solar atmosphere. The Doppler velocity is one of the most important parameters, which is greatly different at different solar atmosphere heights. For example, the average Doppler velocity in the chromosphere is above 1 km s-1 <cit.>, while in the photosphere it is about 0.1 – 0.4 km s-1. The NVST multi-band spectrometer has a dispersion power of about 130,000 in the Hα band, and the spectral sampling rate is about 0.024 Å pixel-1, so each pixel corresponds to 1.1 km s-1 Doppler velocity. To study the characteristics both of the chromosphere and photosphere simultaneously, the measuring accuracy of the Doppler velocity is required to reach the level same as that of photosphere, namely up to the one-tenth pixel level.The NVST and its spectrometers are introduced in Section <ref>, and the observations are described in Section <ref>. The detailed processing methods are exhibited in Section <ref>. The reduced results and the measuring accuracy of the Doppler velocity are given in Section <ref>, and the conclusion is presented in Section <ref>.§ INSTRUMENTSThe NVST () is a vacuum solar telescope with a 985 mm clear aperture. Its scientific goal is to obtain solar information by high resolution imaging () and spectral observations in the wavelength ranges from 0.3 to 2.5 μm. There are three main groups of instruments, the 151-component adaptive optics (AO) system <cit.> and polarization analyzer (PA), the high resolution imaging system and two vertical grating spectrometers: the Multi-Band Spectrometer (MBS) and the High Dispersion Spectrometer (HDS).The MBS uses a blazed grating working in the visible bands with 6 m focal length, and the HDS uses an echelle grating working in the near-infrared bands with 9 m focal length. They share the same slit but cannot work at same time, because their structures are perpendicular to each other. To switch between the two spectrometers, the collimators should be changed, and the slit direction should be rotated. In addition, the spectrometers are equipped with a slit-jaw imaging system and a scanning mechanism to do raster-scan observations. The MBS can cover any part of the visible wavebands simultaneously, but only three bands, the Hα 6562.8 Å band, Ca II 8542 Å band and Fe I 5324 Å band, are in daily observation. The basic parameters of MBS are listed in Table <ref>.§ OBSERVATIONS The data used in this paper were observed by the MBS in the Hα band with the aid of the AO system on 2 March 2016. The width of incident slit is 60 μm, corresponding to 0.3”. The camera used is the PCO4000. Its original sensor size is 4008 × 2672 pixels^2, and the pixel size is 9 μm. In the observation, a binning factor of 2 × 4 is used. Thus the size of the observation data is 2004 × 668 pixels^2. The Hα 6562.8 Å line is observed in the 1st order. Therefore, the spectral sampling rate is about 0.024 Å pixel-1 and the spatial sampling rate is 0.164” pixel-1.Three kinds of observation data are acquired: dark data, flat data, and science data. The X and Y axes of the data denote the dispersion and space directions, respectively. The dark data are acquired when the dome is closed. The integration time was 60 ms, the same as that of the flat data and science data. One hundred frames of dark data are acquired continuously and only the average frame D_ mean is saved. The flat data are taken when the telescope is in fast and random movement. The field of view (FOV) is kept in a quiet region near the disk center. The average result of more than five hundred frames, F_ mean, is used for the follow-up purpose of flat-fielding and calibration. The science data was a set of sit-and-stare data acquired with the AO system, focused on the umbra of a sunspot from 10:27:38 UT to 10:33:14 UT. In this paper, this set of data is used as an example to demonstrate the complete reduction process, which can be used in both the sit-and-stare and raster-scan spectrum modes.§ DATA PROCESSINGIn this section, we analyse the causes of non-uniformities, and describe the detailed steps of data processing, including distortion correction, flat-field correction, spectral line position alignment, intensity normalization, and wavelength calibration. Before processing, all the science data and the F_ mean are subtracted with the D_ mean.§.§ Distortion Correction The spectral distortion is mainly caused by the slight inaccuracy of the spectrometer system. For example, the CCD edges and the dispersion direction or spatial direction are not strictly parallel or vertical, which leads to inclination of the spectral lines. The deviation between the incident grating angle and the center of the grating causes curvature of the spectral lines. In order to acquire an accurate flat-field, these distortions have to be corrected. During the distortion correction process, the inclination is corrected by the horizontal black stripe that is caused by defects of the slit edge or dust attached to the slit. The curvature is corrected by the terrestrial lines. The horizontal stripe and the terrestrial lines are all from the F_ mean.§.§.§ Correction of the X InclinationThe deepest horizontal black stripe is chosen as the baseline for the horizontal inclination correction. The positions of the stripe in each column are calculated with a sub-pixel centroid algorithm. The median of these positions is used as the reference value, and the relative positions are obtained as shown in Figure <ref>a. The figure shows that the curve of relative positions exhibits a linear trend. Then a linear fitting (red line in Figure <ref>a) is used to get the inclination angle, which is 0.167^∘. The corrected matrix F_ meanX is obtained by rotating F_ mean with the inclination angle. The incomplete rows at the top or the bottom of the image are discarded.§.§.§ De-stretching of the Y CurvatureAfter the X-direction inclination corrected, the curvature of spectral lines along the Y-direction still exist. In order to accurately correct the curvature, the four terrestrial lines (6547.7 Å, 6548.6 Å, 6552.6 Å, 6572.1 Å) located on both sides of the Hα line are selected to calculate the curvature offsets. By using the sub-pixel centroid algorithm, the positions of the four terrestrial lines in each row are determined. The relative position curves in Figure <ref>b show that the curvature of the four terrestrial lines are very similar and arenonlinear. So a second-order polynomial approximation is used to fit the median of the four relative positions (red line in Figure <ref>b), and then the curvature displacement of each row is obtained. Each row of F_ meanX is shifted along the X-direction with the opposite direction of its displacement. Similarly, the incomplete columns are also cut away. The F_ meanX after curvature correcting is named as F_ meanXY. §.§ Flat-fielding§.§.§ Problems with the Flat-fielding The purpose of flat-fielding is to correct the non-uniformities of the system response, including the non-uniform sensitivity of the pixels across the chip of the CCD, and the non-uniform illumination. Comparing with the flat data of the imaging observation, the spectral flat data always contains not only non-uniformities, but also spectral lines. In order to obtain the correct flat-field, the spectral lines have to be removed from the F_ meanXY. The traditional methods divide each row of F_ meanXY by the mean profile which is the average of all rows. In this paper, this matrix is defined as the primary flat-field. However, when the primary flat-field is used to process our spectrum, the following problems still exist that will reduce the precision of the physical parameters.The first problem is that the primary flat-field cannot be used to correct the horizontal continuum gradient of science data, because the gradient in the primary flat-field is eliminated when each row of F_ meanXY is divided by the mean profile. However, this gradient still exists in the science data. The F_ meanXY in Figure <ref>a shows the obvious intensity gradient of data. Figure <ref>b shows the primary flat-field, in which the obvious intensity gradient is deducted.The second problem is that remnants of spectral lines exist in the primary flat-field, which caused by the following reasons. First, the solar structure in spectral lines is not completely smoothed, especially in the broad and deep Hα spectral line, which can be clearly seen in Figure <ref>b. Second, the solar differential rotation within the FOV results in the inclination of the solar spectral lines. As mentioned above, the four terrestrial lines are used to obtain the curvature displacements of the spectral line. But we find that even though all the modified terrestrial lines are almost parallel to the Y axes, the modified photospheric lines still have a slight inclination. This phenomenon may be caused by solar differential rotation within the FOV. The magnitude of this inclination is related to the position of the slit on the solar surface during the observation. The third reason is that although the curvature displacements of spectral lines are accurately calculated, there still exist a small amount of unavoidable deviation, and partial spectral lines are not deducted entirely. These remnants in the obtained flat-field are not considered in traditional approaches, but they will introduce new errors when the flat-field is used for processing the science data.The last problem is that there are two temporal changes in the primary flat-field, horizontal stripes and interference fringes. The drift of horizontal stripes is caused by the spectrograph drift during the observations. This phenomenon is similar with the spectrograph of VTT (). Besides the drift of horizontal stripes, the interference fringes are the other time-varying factor in the NVST spectral data, especially in the infrared wavelengths. Moreover, the shapes of interference fringes are irregular, and the intensity and phase change with the incidence angle. If the flat data are taken frequently, or the acquisition time of flat data is very close to that of science data, this phenomenon can be neglected. In many cases, a long time and continuous observation is required to study the evolution of solar structure information, especially for raster-scan observations. The flat data could be taken only in the beginning and the end of the observations, in which case the flat-field conditions are changed. In order to avoid the influences of the time-varying factors, they should be extracted from the full flat-field and further corrected before being used to process the science data.§.§.§ Extraction and Correction of the Non-uniform MatricesBased on the above three problems, if the primary flat-field is used to process the science data, not only some non-uniformities still remain in the result, but also a lot of extra false signals are introduced. So we need to make some improvements to the traditional approaches. The best method to solve those problems is to separate the full flat pattern into several independent matrices and then deduct them from the data one by one. According to the characteristic of non-uniformities in the NVST spectra, the ideal flat-field is defined with the following equation, F =M_1·M_2·M_3·M_4 .Where F denotes the full flat-field matrix, M_1 is the continuum gradient matrix, M_2 is the slit non-uniformity matrix, M_3 is the CCD dust matrix, and M_4 is the interference fringe matrix. In order to keep the continuum gradient when removing spectral lines from the flat-field, F_ meanXY is divided by a new profile, which is normalized by fitting a curve to the continuum of the mean profile. The result, defined as F_ extend, contains M_1, M_2, M_3, M_4 and spectral lines remnants. To removed the time-varying factors from the flat-field, M_2 and M_4 should be extracted from F_ extend. Since M_2 has a relatively stable shape and just drifts along the Y axes (horizontal drift can be ignored), it can be directly extracted from F_ extend. M_4 is very difficult to extract directly from F_ extend, because its shape is irregular, and its intensity and phase change with the incidence angle. To solve this problem, the method used in this paper is to extract M_1, M_2 and M_3 respectively, and leave the remnants of spectral lines and the M_4 behind. To avoid the errors caused by the curve fitting with the continuum in F_ extend, F_ meanXY is used as the extracted matrix instead of F_ extend. The detailed steps are the following:The continuum gradient is mainly caused by the non-uniform illumination and stray light of optical path. considering the physical parameters, like spectral line symmetry, are affected by the continuum gradient, the following procedure is used to correct M_1. At first, the F_ meanXY is normalized. Then, the regions of normalized F_ meanXY, where only contain continuum, are used to fit the continuum gradient matrix M_1 with a fifth-order surface polynomial. The fitting result is shown in Figure <ref>a. A new matrix is named as F_1, which is the result of dividing F_ meanXY by M_1.The slit non-uniformities appear as horizontal black lines and bright lines in data, which are caused by imperfection of the slit as mentioned above. The average of F_1 along the X-direction can be used as the slit pattern. However, considering that CCD dust and interference fringes may bring some systematic errors, the median of all the columns of F_1 is used as the slit profile. Then, this profile is replicated along the X-direction to get the slit non-uniformity matrix M_2 which is shown in Figure <ref>b. The matrix F_2 is the result of F_1 divided by M_2.The CCD dust matrix is caused by the dust particles lying on the CCD surface. A morphological method is used in extracting M_3. For a better extraction result, the intensity distribution of the extracted matrix needs to be relatively uniform except M_3, while the matrix F_2 still contains spectral lines, CCD dust and interference fringes. To remove the spectral lines, each row of F_2 is divided by the mean profile, and the result is defined as F_3. After this process, some remnants of spectral lines, as well as the interference fringes and CCD dust, remain in F_3. So we have to smooth these remnants and fringes by following two steps: The first step is that a large scale (40 pixel) median-filter processing is adopted for each column of F_3 to obtain a new matrix, and F_3 is divided by this matrix to give the matrix S. The second step is to carry out the same processing for each row of S, and the smoothed matrix F_3s is obtained, which just contains the CCD dust and a small amount of interference fringes. To extract the CCD dust matrix from F_3s, a proper threshold (the mean minus 5 times the standard deviation) is taken for F_3s to get a binary matrix, in which the values less than the threshold are set to 1 and the others are 0. Then, the binary matrix is subject to morphological dilation with a 5 × 5 pixels^2 structure to restore the shape of CCD dust. The CCD dust gray matrix is the restored binary matrix multiplied by F_2. In this gray matrix, the values of CCD dust intensities are restored. The CCD dust matrix M_3 is obtained by keeping the values of CCD dust in the gray matrix and setting others values to 1.0, and the result is shown in Figure <ref>c. After F_3 is divided by M_3, the result is M_4s, in which the interference fringes and remnants of spectral lines are left. This is displayed in Figure <ref>d.Based on the methods explained above, we get four independent flat matrices (M_1, M_2, M_3 and M_4s). These matrices are extracted from F_ meanXY, with the geometric distortion in both horizontal and vertical directions corrected. Thus, before flat-fielding, the science data should be processed the same geometric distortion as F_ mean to align with the four flat-field matrixes. After the geometric correction, M_1 and M_3, which are relatively stable, can be directly divided from the science data. However, M_2 is changed in the science data as described in Section <ref>. In order to correct the slit non-uniformity completely, the deepest horizontal strips which exist in both F_ mean and the science data are chosen to calculate their positions by the sub-pixel centroid algorithm. The relative displacements of M_2 are obtained by using the position of F_ mean as a reference value. Then, M_2 is modified by vertical shifting with the opposite relative displacements, respectively. After that, the slit non-uniformity in the science data can be eliminated accurately by using the shifted M_2. In principle, further steps should be made to process the interference fringes. However, because of the particularity of interference fringes as described in Section <ref>, it is very difficult to extract the pure M_4 from the flat-field, even from the science data. So we temporarily do not deal with the interference fringes. Their impacts will be analyzed in Section <ref>. §.§ Additional Correction to the Science Data After flat-fielding, there still exist two additional temporal changes in the science data: the position drift of spectral lines and relative intensity fluctuation. The position drift of spectral lines is caused by the spectrometer drift, which also brings the above slit non-uniformity matrix drift. The correction of this drift is to align the wavelength of each data frame, which helps to study the evolution of physical parameters at a fixed spatial position (like solar oscillation, etc.), and prepares for the future processing of raster-scan spectra (for example, ensuring the monochromaticity of the raster images composed by the science data). The correction of intensity fluctuation is important for long-period observations. It is useful for comparing the physical parameters between frames and for increasing the spatial resolution of the raster images.For most sit-and-stare spectral data, the correction of spectral line drift and relative intensity fluctuation could be ignored, and the drift correction could be applied in deriving the spectral parameters associated with the positions of spectral lines. But to build the universal procedure for sit-and-stare and raster-scan spectrum and to obtain more standard three-dimensional spectral arrays (X, Y, wavelength), both corrections are applied to science data.§.§.§ Spectral Line Position Aligning To correct the drift of spectral lines, the positions of the deepest terrestrial lines in F_ mean and the science data are calculated by the sub-pixel centroid algorithm.The position of F_ mean is used as the reference value to obtain the drift offsets, which are shown in Figure <ref>a. This figure shows obvious fluctuation of the position, and the ranges are about -0.2 – 0.5 pixels corresponding to the Doppler velocities of -220 – 550 m s-1. The science data are shifted horizontally with the inverse drift offsets, and this ensures consistency of spectral lines.§.§.§ Intensity NormalizationIn science data, the average intensities in the same limited region (the rectangular box in Figure <ref>a) in each frame are calculated. Then, the median of all the average intensities is used as the reference value, and the normalization factors are obtained by the average intensity of each limited region divided by the reference value. The relative intensity distribution of the limited region is shown in Figure <ref>b. Because the data used only last about five minutes, the fluctuations of the intensity distribution in Figure <ref>b are not obvious. After the correction of intensity fluctuations, the science data all are normalized by the normalization factors.§.§ Wavelength Calibration To acquire precise wavelengths of the spectral lines and the spectral sampling rate of the spectrum, wavelength calibration is applied after the above processing. Because the position of spectral lines in flat data and science data are aligned, wavelength calibration can be obtained by the reduced F_ mean. The reference spectral lines for wavelength calibration need to be as stable as possible, and the terrestrial lines are usually a good choice. The positions of four terrestrial lines (the same lines used for geometric correction)in the NVST spectrum are firstly calculated. Meanwhile, the wavelengths of the four terrestrial lines are obtained from the standard solar spectrum atlas which is observed by the Fourier Transform Spectrometer (FTS) at the McMath-Pierce Solar Telescope. The wavelength calibration is applied by fitting the positions found in NVST spectrum and wavelengths obtained from FTS with the least square method. The spectral sampling rate of the NVST spectrum is also acquired, which is 0.0244 ± 0.0001 Å pixel-1. Figure <ref> shows a comparison result between the NVST reduced F_ mean profile (red line) and the FTS spectrum (black line). § RESULTS AND DISCUSSIONThis section shows the result of fine processed science data. The measuring accuracies of the Doppler replacements are use to evaluate whether the reduced NVST spectrum can reach the scientific requirement. Here the Doppler replacements are the same as the above relative positions of spectral lines, calculated by the sub-pixel centroid algorithm. Figure <ref>a shows the reduced 100th frame science data. It shows that the continuum gradient, the slit non-uniformity and the CCD dust are well deducted. The relative standard deviations of the rectangular regions, mentioned in Section <ref>, in continuum area of raw data and reduced data are 2.1% and 1.4%, respectively. This means the noise is well suppressed after fine processing. Figure <ref>b shows the Doppler displacement curves of four terrestrial lines. As can be seen from the figure the geometric curvature of spectral lines is well corrected. The Doppler displacement curves of three other photospheric lines, shown in Figure <ref>c, shows a strong correlation. The average shift range is about -1.24 – 1.21 pixels corresponding to -1.36 – 1.33 km s-1 Doppler velocities. That reflects the motion characteristics of the solar photosphere at this moment. Figure <ref>d shows the Doppler displacement of the Hα line center, and the range is -5.41 – 14.73 km s-1. In the reduced data, the non-uniformities of continuum gradient, slit and CCD dust are all corrected except the interference fringes. Here, we will analyze the influence of the interference fringes on the accuracy of our measurements. The F_ mean is the averaged result of the multi-frame flat data, and each of the flat data was acquired during the random movement of the telescope, so the fine solar structures in the FOV of the slit are smoothed. The root mean square (RMS) of the Doppler displacements in the reduced F_ mean mainly reflects the impact of the interference fringes. The RMS of the seven spectral lines are listed in Table <ref>, and the range is 0.021 – 0.038 pixels, corresponding to Doppler velocities of 23.1 – 41.8 m s-1. Of course, due to the time-varying property of the interference fringes, it is hard to calculate the interference fringe influence exactly at each position of data. In some positions, the impact may reach up to 50 m s-1, or even greater. The final measuring accuracy of reduced science data is our greater concern. In order to avoid the influence of the solar structures on the measuring accuracy calculation, the measuring accuracy is inferred by a set of second-order finite differences, which are calculated from three neighboring pixels. The depth of the spectral line has a great influence on the measuring accuracy of the Doppler velocity. Therefore, the measuring accuracy and the depth of the terrestrial lines and the photospheric lines are calculated respectively, and the result are listed in Table <ref>. It shows that there is a strong inverse correlation between the depth and the measuring accuracy in both terrestrial lines and photospheric lines. The correlation coefficients are -0.98 and -0.97, respectively. Namely the deeper the spectral lines, the higher the accuracy of Doppler velocity measurement. In Table <ref>, it is also shown that when the depth of photospheric lines is within 0.314 – 0.615, the measuring accuracy can reach up to 71.5 – 157.3 m s-1. In the Hα spectral line center, the measuring accuracy is 0.121 pixels, corresponding to 133 m s-1.In addition, the Doppler velocity of the Hα line center in the sunspot umbra region of all the reduced science data is also obtained, and the result is shown in Figure <ref>a. This figure shows the three-minute oscillation of the solar chromosphere, and the range of the fluctuation is -5.5 – 3.0 km s-1. Figure <ref>b shows the Doppler velocity evolution curve of a quiet region in the photosphere. It exhibits an oscillation with a cycle of five minutes, and the range of this five-minute oscillation is -0.5 – 0.1 km s-1. The above results are consistent with the basic facts of solar physics. The measuring accuracies of the two oscillation curves are 0.186 pixels and 0.069 pixels, corresponding to 204.6 and 75.9 m s-1, respectively.In conclusion, the results shows by this precise reduction that the highest Doppler measuring accuracy of photospheric lines that can be achieved is within 100 m s-1, and the Doppler measuring accuracy of the Hα spectral line is about 100 – 200 m s-1. The actual measuring accuracies are always affected by systematic errors and random noise. In this paper, the corrected non-uniformities are systematic errors. The interference fringes and random noise needs to be further suppressed. § CONCLUSION In this paper, the precise reduction methods and procedure presented can effectively correct particular non-uniformities of the sit-and-stare and raster-scan spectra of NVST. The separation method is used to do an accurate flat-fielding, which can not only reduce most of the influences of the non-uniformities, but also avoid introducing additional errors. The additional corrections are applied to the science data to obtain a standard three-dimensional reduced spectral arrays. After the fine processing, the measuring accuracies of Doppler velocities with different spectral lines, and of the curves of the chromospheric three-minute oscillation and photospheric five-minute oscillation, demonstrate the validity of this procedure. The Doppler velocity measuring accuracy of the Hα line center can reach up to 100 – 200 m s-1. The accuracies of photospheric lines depend on their line depths, and the highest Doppler measuring accuracy is within 100 m s-1. This shows that by this fine processed spectrum, the characteristics of the chromosphere and the photosphere can be studied with reduced NVST data co-spatially and co-temporally. Some work remains to do, like removing interference fringes and image de-noising, and this will be studied in future work.We are appreciated for all the help from the colleagues in the NVST team. We are also thankful to the unknown referee for their useful comments. A lot of thanks to Mr. Song Feng, Ms. Yanxiao Liu and Dr. Huanwen Peng for their kind assistance and helpful comments on this manuscript. The FTS atlas used in this paper were produced by the NSO/NOAO. This work is supported by the National Natural Science Foundation of China (NSFC) under grant numbers 11773072, 11573012 and 11473064. Disclosure of Potential Conflicts of Interest The authors declare that they have no conflicts of interest. [Chae2004]Chae2004 Chae, J.: 2004,221, 15. [Chae 2013]Chae2013 Chae, J., Park, H.-M., Ahn, K. : 2013,288, 1.[Collados 2012]Collados2012 Collados, M., López, R., Páez, E. : 2012, Astron Nachri. 338, 872.[Hong 2014]Hong2014 Hong, J., Ding, M.D., Li, Y. : 2014,792, 13.[Johannesson1992]Johannesson1992 Johannesson, A.: 1992, Technical report.[Kiselman1994]Kiselman1994 Kiselman, D.: 1994,104.[Liu 2014]Liu2014 Liu, Z., Xu, J., Gu, B.-G. : 2014, Res. Astro. 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"authors": [
"Yunfang Cai",
"Zhi Xu",
"Zhenggang Li",
"Yongyuan Xiang",
"Yuchao Chen",
"Yu Fu",
"Kaifan Ji"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170926145146",
"title": "Precise Reduction of Solar Spectra Observed by the 1-meter New Vacuum Solar Telescope"
} |
calc, positioning | http://arxiv.org/abs/1709.09098v2 | {
"authors": [
"Daniel Becker",
"Chris Ripken",
"Frank Saueressig"
],
"categories": [
"hep-th",
"gr-qc"
],
"primary_category": "hep-th",
"published": "20170926155221",
"title": "On avoiding Ostrogradski instabilities within Asymptotic Safety"
} |
Azimuthal angle correlations at large rapidities: revisiting density variation mechanism [ December 30, 2023 ===========================================================================================* Glass and Time, IMFUFA, Department of Science and Environment, Roskilde University, Postbox 260, DK-4000 Roskilde, Denmark * Institute of Physics, University of Silesia, ul. Uniwersytecka 4, 40-007 Katowice, Poland* Institut Laue-Langevin, 71 avenue des Martyrs, CS 20156, 38042 Grenoble cedex 9, FranceThe glass transition plays a central role in nature as well as in industry, ranging from biological systems such as proteins and DNA to polymers and metals <cit.> . Yet the fundamental understanding of the glass transition which is a prerequisite for optimized application of glass formers is still lacking <cit.>.Glass formers show motional processes over an extremely broad range of timescales, covering more than ten orders of magnitude, meaning that a full understanding of the glass transition needs to comprise this tremendous range in timescales <cit.>.Here we report on first-time simultaneous neutron and dielectric spectroscopy investigations of three glass-forming liquids, probing in a single experiment the full range of dynamics. For two van der Waals liquids we locate in the pressure-temperature phase diagram lines of identical dynamics of the molecules on both second and picosecond timescales. This confirms predictions of the isomorph theory <cit.> and effectively reduces the phase diagram from two to one dimension. The implication is that dynamics on widely different timescales are governed by the same underlying mechanisms. Glasses are formed when the molecular motions of a liquid become so slow that it effectively becomes a solid. When the glass transition of a liquid is approached, the dynamics of the molecules spreads out like a folding fan covering more than ten orders of magnitude. There are at least three overall contributions to the dynamics of glass-forming liquids: 1) vibrations, 2) fast relaxations on picosecond timescales, and 3) the structural alpha relaxation, which has a strongly temperature-dependent timescale. The glass transition occurs when the alpha relaxation is on a timescale of hundreds of seconds and therefore completely separated from the two fast contributions to dynamics. Nonetheless, both theoretical <cit.> and experimental results <cit.> have suggested that fast and slow dynamics are intimately connected.A complete understanding of the glass transition therefore necessitates a full understanding of all these dynamic processes.During the past 15 years, pressure has increasingly been introduced to study dynamics of glass-forming liquids in order to disentangle thermal and density contributions to the dynamics. The most striking finding from high-pressure studies is that the alpha relaxation, both its timescale and spectral shape is, for a large number of different liquids, uniquely given by the parameter Γ=ρ^γ/T, where γ is a material specific constant <cit.>. This scaling behaviour can be explained by the isomorph theory <cit.>, which, moreover, predicts the value of γ (Gundermann11). The fundamental claim of isomorph theory is the existence of isomorphs. Isomorphs are curves in the phase diagram along which all dynamic processes and structure are invariant.Put in other words, the phase diagram is predicted to be one dimensional with respect to structure and dynamics on all timescales, with the governing single variable being Γ. Isomorph theory has been very successful in describing Lennard-Jones type computer simulated liquids, e.g. refs. (Bacher14,Pedersen16). Experimental studies of isomorph theory predictions require high-precision high-pressure measurements andare still limited <cit.>. Consequently, it remains open whether isomorph theory holds for real molecular liquids.The only systems that obey isomorph theory exactly are those with repulsive power law interaction potentials which do not, of course, describe real systems. Hence, isomorph theory is approximate in its nature and expected to work for systems without directional bonding and competing interactions <cit.>.With this in mind, we have studied the dynamics on three well-studied glass formers representing non-associated liquids and liquids with directional bonding, two van der Waals bonded liquids (vdW-liquids): PPE (5-polyphenyl ether) and cumene (isopropyl benzene), and a hydrogen bonding (H-bonding) liquid: DPG (dipropylene glycol).In this work, we use a new high-pressure cell for simultaneous measurements of the fast dynamics by neutron spectroscopy and the alpha relaxation by dielectric spectroscopy to demonstrate that for the studied vdW-liquids, the three mentioned distinct dynamic components are invariant along the same lines in the phase diagram. This is the first direct experimental evidence for the existence of isomorphs, and it proves that the phase diagram of simple vdW-liquids is one dimensional with respect to dynamics. Unlike the scaling behaviour of the alpha relaxation dynamics, which is often found to hold surprisingly well in H-bonding systems <cit.>, we only find invariance of the fast relaxational and vibrational dynamics on picosecond timescales in the vdW-liquids. For the single investigated H-bonding system, DPG, we make a different observation, as expected based on isomorph theory.Dynamics from picosecond to kilosecond cannot be measured with one single technique; several complementary techniques are required. A glass-forming liquid is in metastable equilibrium and the dynamics is very sensitive to even small differences in pressure and temperature. The high viscosity of the liquid close to the glass transition temperature, T_g, makes the transmission of isotropic pressure non-trivial, as pressure gradients are easily generated. In order to ensure that the different dynamics are measured under identical conditions, we have therefore developed a cell for doing simultaneous dielectric spectroscopy (DS) and neutron spectroscopy (NS) under high pressure (Fig. 1). The experiments were carried out on spectrometers at the Institut Laue-Langevin (ILL) on the time-of-flight (TOF) instruments IN5 and IN6. The different NS instruments access different timescales with IN5 giving information on the ∼10 ps scale, while a backscattering (BS) instrument like IN16 accesses ∼1 ns dynamics. DS provides fast (minutes) and high accuracy measurements of the dynamics from microsecond to 100 s.The dynamics measured with the different techniques are illustrated in Fig. 1a and b for PPE. The center panels of (a) and (b) are sketches of the incoherent intermediate scattering function, I(Q,t), while the top and bottom panel show raw data. At T_g (Fig. 1a), no broadening is observed on nanosecond timescales (IN16) corresponding to a plateau in I(Q,t), on picosecond timescales from IN5 we observe contributions from fast relaxational processes and vibrations, whereas the alpha relaxation is seen in DS at much longer timescales, a difference of more than 10 orders of magnitude. As the temperature is increased, the processes merge (Fig. 1b), and relaxation dominates the signal in all three spectrometers.Picosecond dynamics measured on IN5 and IN6 are presented in Fig. 2. We observe the same trend for all spectra for all values of wave vector Q (Extended Data Fig. 2), and have summed over Q to improve statistics. All spectra are shown on the same S(ω̃)-axis. Motivated by isomorph theory, the energy scale is shown in reduced units, effectively ω̃ = ωρ^-1/3T^-1/2 <cit.>. The effect of scaling is small, though visible, in the studied range of ρ and T. The data is shown on an absolute energy scale in the Extended Data Fig. 1.For all three samples in row (a) Fig. 2, which shows dynamics in the liquid, we observe the extreme scenarios sketched in Fig. 1. At low pressure, relaxational contributions are dominating (Fig. 1b). At the glass transition, we find only fast relaxational and vibrational contributions (Fig. 1a).The fast relaxational contributions decrease in the glassy state (Fig. 2b), leaving the excess vibrational density of states, which shows up as the so-called Boson peak <cit.>, as the dominant contribution (black full lines in Fig. 2b). For all three samples, we observe different dependencies on temperature and pressure for the three contributions to the dynamics, such that their relative contributions vary along both isobars and isotherms.Sokolov et al. <cit.> studied seven different liquids with Raman spectroscopy, both H-bonding and vdW-liquids. In contrast to what we see, they observe a correlation between pressure-induced variations in the fast relaxation and the Boson peak. Our observations are in agreement with those shown by neutron spectroscopy for ortho-terphenyl in (Patkowski03), namely that the T and P dependencies are different for the three contributions to the dynamics. All glass formers have isochrones which are lines in the (T,P)-phase diagram with constant alpha relaxation time, τ_α. If isomorphs exist in a liquid, these coincide with the isochrones, since all dynamic processes on all timescales and of all dynamic variables are invariant along an isomorph. Thus, experimentally we can identify candidates for isomorphs by the isochrones. We use DS effectively as a 'clock', which identifies the alpha relaxation time from the dipole-dipole correlation function, while simultaneously measuring the picosecond dynamics with incoherent neutron scattering probing the particle self-correlation.Row (c) in Fig. 2 shows the picosecond dynamics measured along the glass transition isochrone T_g(P) defined as when τ_α=100 s found from DS. We observe superposition of the spectra at the picosecond timescale, thus invariance of the dynamics, for the two vdW-liquids (PPE and cumene). This is in agreement with the prediction of the isomorph theory and it is particularly striking because at T_g(P), fast relaxational and vibrational motion are completely separated in timescale from the alpha relaxation as illustrated in Fig. 1a. In contrast, for the H-bonding liquid (DPG) we find a clear shift towards higher energy and an intensity decrease of the Boson peak along the T_g(P) isochrone. The lack of superposition in the H-bonding system demonstrates that the superposition seen in the vdW-liquids is non-trivial. Thus, the superposition observed in the vdW-liquids is a genuine signature of the isomorphs in these liquids.To compare more state points, including shorter alpha-relaxation time isochrones (τ_α<1 μs), isotherms and isobars, we plot the corresponding inelastic intensities at a fixed reduced energy (ω̃=0.06) as a function of temperature and pressure (Fig.3a and b). Along the investigated isochrones the inelastic intensity at ω̃=0.06 or t∼ 1 ps is found to be invariant for the vdW-liquids. Again the H-bonding liquid behaves differently and its picosecond dynamics varies along the isochrones. This confirms the isomorph prediction of spectral superposition of fast relaxation, vibrations/Boson peak and alpha relaxation for the vdW-liquids: both when there is timescale separation and when the processes are merged.The power of isomorph theory is that the phase diagram becomes one dimensional; the dynamics only depend on which isomorph a state point is on and not explicitly on temperature and pressure (or density). Since isomorphs are isochrones the other dynamic components should become a unique function of the alpha relaxation time. In Fig. 3(c) we plot the intensity of the picosecond dynamics along isotherms and isobars as a function of the dielectric alpha relaxation time at fixed reduced energies (ω̃=0.02, 0.04, 0.06, 0.08, 0.1). The data from each energy collapses in this plot, illustrating that, as predicted, all the dynamics is governed by one parameter.Previously suggested connections between fast and slow dynamics, e.g. via the temperature dependence of properties <cit.>, often suggest a causality, where the fast dynamics controls the slow dynamics.In contrast, the connection between slow and fast dynamics shown in this work does not tell us if one controls the other or if they are simply controlled by the same underlying mechanism.Isomorph theory is approximate in its nature, and the isomorphs of real physical systems are approximate. The fact that the isomorph prediction works for dynamics which is separated in timescale by more than ten orders of magnitude, tells us that whatever governs this dynamics is controlled by properties of the liquid that obey the isomorph scale invariance. There is a consensus that the alpha relaxation is cooperative, fast relaxations are normally understood as cage rattling, whereas it has been heavily debated whether or not the Boson peak is localized <cit.>. It is clear from the spectra in Fig. 2 that fast relaxation and Boson peak are different in nature because their relative intensities vary along isotherms and isobars, and that these two types of fast dynamics are distinctively different from the structural alpha relaxation. Yet all of these dynamic features are controlled by the single parameter Γ=ρ^γ/T. Our finding implies that a universal theory for the glass-transition needs to be consistent with the one-dimensional phase diagram coming out of isomorph theory.10 url<#>1urlprefixURLKhodadadi15 authorKhodadadi, S. & authorSokolov, A. 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M. titleCritical point measurements for fourteen compounds by a static method and a flow method. journalJournal of Chemical & Engineering Data volume41, pages1252–1254 (year1996). <http://dx.doi.org/10.1021/je960052x>. http://dx.doi.org/10.1021/je960052x.* This work was supported by the Danish Council for Independent Research (Sapere Aude: Starting Grant).We gratefully acknowledge J. Dyre for fruitful discussions and technical support from the workshop at IMFUFA and the SANE group at the ILL. Competing Interests The authors declare that they have no competing financial interests. Correspondence Correspondence and requests for materials should be addressed to K. Niss. (email: [email protected]). 0.0cm 0.2cm 16cm21cm 1.0cm24pt § METHODS SECTION FOR "EVIDENCE OF A ONE-DIMENSIONAL THERMODYNAMIC PHASE DIAGRAM FOR SIMPLE GLASS-FORMERS" §.§ H.W. Hansen^1, A. Sanz^1, K. Adrjanowicz^2, B. Frick^3 & K. Niss^1∗§.§.§ Materials Isopropyl benzene (cumene) and dipropylene glycol (DPG) were purchased from Sigma Aldrich, and 5-polyphenyl ether (PPE) was purchased from Santolubes. All three samples were used as acquired.The glass transition for the three samples is found from dielectric spectroscopy and defined as when the maximum of the loss peak corresponds to τ_α=100 s, where τ_α=1/2πν”_max. At atmospheric pressure, the glass transition temperature T_g is 245 K for PPE, 126 K for cumene, and 195 K for DPG.The traditional way of quantifying how much the alpha relaxation time, or the viscosity, as a function of temperature deviates from Arrhenius behavior is given by the fragility <cit.>, defined as:m = dlog_10τ_α/d(T_g/T)|_T_g.For the two van der Waals liquids, the fragility at ambient pressure is m≈80 for PPE and m≈70 for cumene. The hydrogen-bonding liquid DPG has fragility m≈60. §.§.§ Methods All experiments were carried out at the Institut Laue-Langevin (ILL) on the time-of-flight instruments IN5 and IN6. In neutron spectroscopy, the different instrumental energy resolutions give access to different dynamical timescales; the coarser the energy resolution the faster the time window accessible: Δ E_res≈ 0.1 meV on IN5 and IN6 corresponds to ∼10 ps. DS provides fast (minutes) and high accuracy measurements of the dynamics from microsecond to 100 s. Details on the high-pressure cell for doing simultaneous dielectric and neutron spectroscopy can be found in the forthcoming publication.In Fig. 2, we presented data on PPE and cumene from IN5 and on DPG from IN6 (hence the difference in statistics). All data was measured with a wavelength of 5 Å and an energy resolution ∼0.1 meV, corresponding to a timescale of approximately 10 picoseconds. All spectra have been corrected in the conventional way by normalizing to monitor and vanadium, subtracting background, and correcting for self-shielding, self-absorption and detector efficiency using LAMP, a data treatment program developed at the ILL. The data has then been grouped for constant wavevector Q in steps of 0.1 Å^-1 in the range 1.2-1.9 Å^-1 for the IN5 data (PPE and cumene) and in the range 1.2-1.7 Å^-1 for the IN6 data (DPG) and is presented in Fig. 2 as a sum over Q. Data are shown as measured in Extended Data Fig. ED1 on an absolute energy scale. No scaling has been done on the y-axis in S(ω) of any of the spectra, i.e. all spectra are plotted on the same scale. Comparing Fig. S1 to Fig. 2, the effect of plotting data in reduced units is visible at higher energy transfer. The same picture is observed for all spectra for each value of Q, and we have therefore summed over Q in the data shown in Fig. 2 to improve statistics. An example of spectra along the glass transition at different Q for PPE and DPG is shown in Extended Data Fig. ED2.§.§.§ Reduced units According to isomorph theory, the relevant scale to look at is in reduced units ^11. The reduced energy units used in Fig. 2 and Extended Data Fig. ED2 are given byω̃ = ω t_0 = ωρ^-1/3√(m/(k_BT)) where ω is the energy transfer, setting ħ=1. k_B is Boltzmann's constant and T is temperature. Here, ρ is the number density and m is the average particle mass, the latter assumed constant. We set m=k_B=1. Effectively, this becomes ω̃ = ωρ^-1/3T^-1/2,where ρ is now the volumetric mass density.Just like reduced energy units, wave vector or momentum transfer Q should also be presented in reduced units: Q̃ = Q ρ^-1/3 But as the density changes are in the percent range in this study, scaling of Q will be around 1% and will be within the uncertainty of the data and is therefore neglected. §.§.§ Calculating density Equations of state (EOS) have been used to calculate the temperature and pressure dependence of the density. For DPG, the EOS is taken from <cit.>. For PPE, a fit to the Tait equation from PVT data from <cit.> has been used to obtain the density:ρ(T,P) = (V_0exp(α_0T){1-Cln[1+P/b_0exp(-b_1T)]})^-1,where ρ is in g/cm^3 and equal to 1/V_sp, the specific volume, P is pressure in MPa and T is temperature in ^∘C. The fitting parameters are V_0=0.82, α_0=6.5·10^-4, C=9.4·10^-2, b_0=286 and b_1=4.4·10^-3. For cumene, density has been reported in <cit.> in the temperature range 150-320 K at atmospheric pressure, a linear dependence was found in this range encapsulated in the equation,ρ/ρ_r=1-a(T-T_r),where a is a constant given for cumene as a=0.000954 K^-1, and ρ_r is the density at a reference temperature of T_r=273.2 K, ρ_r=0.879. This linearity is assumed to hold tothe glass transition temperature at ambient pressure, 126 K. The pressure dependence of density was measured in <cit.> up to 4 GPa at room temperature. To obtain an EOS, the compressibility and expansivity in the whole temperature range is found using the approach from <cit.> for toluene. The pressure and temperature dependence of the α(P) is calculated at temperatures higher than 240 K using the formula in <cit.> for toluene rescaled to cumene by using its critical temperature and density, T_c=631.1 K and P_c=321 kPa <cit.>.For all three samples, the density changes are in the percent range in the temperature-pressure range of this study. The PVT data and EOS are only used for scaling on the energy axis of the data, where the reduced energy units contain the cubic root of the density. Hence, the scaling is in practice mainly with temperature, and the use of EOS therefore does not alter with the overall conclusion.§.§.§ Higher temperature isochrones High temperature isochrones with alpha relaxation time τ_α≪100 s shown in Fig. 3 were done for PPE and DPG, but not for cumene; because cumene crystallizes easily in the region above T_g and below the melting point. In Extended Data Fig. ED3 and ED4, examples of higher temperature isochrones, where relaxation is dominating in the picosecond dynamics, are given for PPE. In Extended Data Fig. ED3, the imaginary part of the capacitance used to find isochrones are shown, both at the glass transition and for faster relaxation times. Extended Data Fig. ED4 shows the corresponding NS spectra to the two high-temperature isochrones shown in Extended Data Fig. ED3. We observe total invariance of the spectra along isochrones. Again, no scaling has been done in the y-direction, and in this interval, the effect of plotting in reduced units on the x-axis is negligible. | http://arxiv.org/abs/1709.08953v1 | {
"authors": [
"Henriette Wase Hansen",
"Alejandro Sanz",
"Karolina Adrjanowicz",
"Bernhard Frick",
"Kristine Niss"
],
"categories": [
"cond-mat.soft",
"cond-mat.mtrl-sci",
"physics.chem-ph"
],
"primary_category": "cond-mat.soft",
"published": "20170926115343",
"title": "Evidence of a one-dimensional thermodynamic phase diagram for simple glass-formers"
} |
plain ThmTheorem[section] Cor[Thm]Corollary Conj[Thm]Conjecture Pro[Thm]Problem MainMain Theorem Lem[Thm]Lemma Claim[Thm]Claim Prop[Thm]Proposition ExamExample ToDoTo Dodefinition Def[Thm]Definition Exer[Thm]Exercise Rem[Thm]RemarkremarkDennis Tseng, Harvard University, Cambridge, MA 02138 [email protected] show the Kontsevich space of rational curves of degree at most roughly 2-√(2)/2n on a general hypersurface X⊂P^n of degree n-1is equidimensional of expected dimension and has two components: one consisting generically of smooth, embedded rational curves and the other consisting of multiple covers of a line. This proves more cases of a conjecture of Coskun, Harris, and Starr and shows the Gromov-Witten invariants in these cases are enumerative. A note on rational curves on general Fano hypersurfaces Dennis Tseng December 30, 2023 =======================================================§ INTRODUCTIONOur investigation is motivated by the following conjecture by Coskun, Harris, and Starr.[<cit.>]Let X⊂P^n be a general hypersurface of degree d≤ n and dimension at least 3. Then, the open locus R_e(X) in the Hilbert scheme of X parameterizing smooth rational curves of degree e is irreducible of dimension e(n+1-d)+n-4. Furthermore, if d≤ n-1, then the evaluation map _0,1(X,e)→ X is flat. There has been progress on Conjecture <ref> by using induction on e via bend and break <cit.>. Most recently, Riedl and Yang showed that Conjecture <ref> holds when d≤ n-2 <cit.>. For work on rational curves on arbitrary smooth hypersurfaces, see <cit.>. In this note, we will look at rational curves of low degree on general hypersurfaces of degree n-1 in over an algebraically field of characteristic zero. Specifically, we will showIf X⊂P^n is a general hypersurface of degree n-1 for n≥ 4, ande<n-1+√(n^2-n-15)/2,then the evaluation map _0,1(X,e)→ X is flat and the Kontsevich space _0,0(X,e) is a local complete intersection stack of pure dimension 2e+n-4 and has two components. One of the components consists of e to 1 covers of a line and the other component consists generically of smooth rational curves.To see the e≥n/4 suffices, use AM-GM to show n^2+2e(e+1)+8≥(1+4e)n is implied by n^2+2e(e+1)+8≥(1+4e)^2/2+n^2/2, then expand and concludeAnother method that has been successful in controlling the dimensions of R_e(X) has been to consider the incidence correspondence between hypersurfaces and rational curves in projective space. In the setting of Conjecture <ref>, R_e(X) is smooth and of the expected dimension for e≤ d+2 by work of Gruson, Lazarsfeld and Peskine <cit.>. Furukawa made this connection explicit and gave a weaker bound that also works in positive characteristic <cit.>. An advantage in considering the Kontsevich space _0,0(X,e) instead of R_e(X) is the connection with Gromov-Witten invariants. The space _0,0(X,e) is given by the zero section of a vector bundle on _0,0(P^n,e) and the virtual fundamental class of _0,0(X,e) is the Euler class <cit.>. In the cases covered by Theorem <ref>, the fundamental class of _0,0(X,e) agrees with the virtual fundamental class as the dimensions agree. Furthermore, it is possible to see that each component of _0,0(X,e) is generically smooth, as we will identify smooth points parameterizing highly reducible curves in every component in our analysis. In particular by Kleiman-Bertini, given subvarieties V_1,…, V_r of P^n, then the number of degree e rational curves in X meeting each of V_1,…,V_r is ∫_[_0,r(X,e)]^ vir_1^*[V_1']∩⋯∩_r^*[V_r']provided the integrand in (<ref>) is expected dimension zero and V_i' is a general PGL_r+1 translate of V_i. Finally, we remark that hypersurfaces of degree n-1 in P^n are examples of Fano varieties of pseudo-index 2.Given a Fano variety M, the pseudo-index of M is defined asmin{-K_M· C| C is a rational curve of M}.If f: C→ M is a rational curve on M with -K_M· C=2 with f_*[C]=β, then the expected dimension of _0,0(M,rβ) is the same as the expected dimension of r-fold covers of stable maps in _0,0(M,β). From this point of view, it is clear why the two components given in Theorem <ref> are necessary. See <cit.> for another example of a Fano variety of pseudoindex 2 and a description of the components of its space of rational curves. §.§ MethodsIn order to apply bend and break in families as in <cit.>, we will need to apply a couple of results from the author <cit.> to control the locus of hypersurfaces with more lines through a point than expected and the locus of hypersurfaces with positive dimensional singular loci. Also, the space of lines through a point is expected to be finite, and so in particular is not irreducible. This will present a technical obstacle in showing irreducibility of the main component in Theorem <ref>. To deal with this, we will show that the space of conics through a general point is irreducible, and then use an argument that is similar in spirit to the irreducibility argument in <cit.> but will require us to specialize further. We work in characteristic zero, but it seems likely that the techniques extend to positive characteristic.§.§ OutlineThe argument will have two parts. In Section <ref>, we will show the fibers of the evaluation map _0,1(X,e)→ X have the expected dimension in Theorem <ref>. We will look at the irreducible components of the general fiber in Theorem <ref> in Section <ref>. Theorem <ref> will follow from Theorems <ref> and <ref>. In Section <ref>, we are mostly interested in dimension, so it suffices to work with the coarse moduli space of _0,1(X,e), but we will need look at smoothness in Section <ref>, so we will need to work with _0,1(X,e) as a stack. In general, the fact that the Kontsevich space _0,1(X,e) and, more generally, the Behrend-Manin stacks <cit.> are Deligne-Mumford stacks in characteristic zero will allow us to avoid technical difficulties with stacks by passing to an étale cover. § ACKNOWLEDGEMENTSThe author would like to thank Jason Starr for help in identifying the irreducible components in Theorem <ref> and Eric Riedl and David Yang for helpful conversations. The author would also like to thank the referee for detailed suggestions on how to improve the exposition and for the additional references to the literature.§ CONVENTIONS Throughout the paper, we will work over an algebraically closed field of characteristic zero. We will let * X⊂P^n be a hypersurface of degree 2≤ d≤ n-1* N=d+nn-1 so P^N parameterizes hypersurfaces of degree d in P^n* X→P^N be the universal hypersurface X={(p,X)|p∈ X}⊂P^n×P^N* _0,r(X,e) is the Kontsevich space parameterizing stable maps C→ X of degree e, where C is a genus 0 curve with r marked points* _0,r(X/P^N,e) is the relative Kontsevich space parameterizing stable maps mapping into the fibers of X→P^N.We are interested in when d=n-1. § FLATNESS The goal of the first half of the note is to proveIf e<n-1+√(n^2-n-15)/2, then for a general hypersurface X of degree d=n-1 in P^n, the evaluation map_0,1(X,e)→ Xis flat and _0,1(X,e) is a local complete intersection stack. §.§ DefinitionsGiven a projective scheme X, let _0,r(X,e) denote the Kontsevich space parameterizing stable maps C→ X of degree e, where C is a genus 0 curve with r marked points.Given a point p∈ X, there is an evaluation map : _0,1(X,e)→ X, and we will refer to ^-1(p) loosely as the rational curves through p. We will denote the boundary of _0,1(X,e), consisting of reducible curves, as ∂_0,1(X,e).Given a projective morphism X→ S, let _0,a(X/S,e)→ S be the relative Kontsevich space parameterizing stable maps mapping into the fibers of X→ S.In our case, X will be a hypersurface and X→ S will be the universal hypersurface. For the rest of this section, we let X⊂P^n be a hypersurface of degree d for d≤ n-1.We recall the notion of e-level <cit.>, with some slight modifications. Given a point p∈ X⊂P^n of a hypersurface of degree d, p is called e-level if one of the following holds: * p is a smooth point and the space of degree e rational curves through p has dimension at most e(n-d+1)-2* p is a singular point and the space of degree e rational curves through p has dimension at most e(n-d+1)-1.By the space of rational cuves through p, we mean the fiber of the evaluation map _0,1(X,e)→ X over p∈ X. When p fails to be e-level, the space of rational curves through p has larger dimension than expected. The notion of e-levelness extends the notion of flatness of the evaluation map to singular points in a manner that allows us bound the locus of points that are not e-level. A hypersurface X is e-level if, * the singular locus is finite, and* every point of X is k-level for all k≤ e.Instead of requiring only finitely many singular points, what is actually being used in the dimension counts of <cit.> <cit.> is that there is no rational curve C⊂ X of degree less than e, for which the space of rational curves of degree k<e through every point p∈ C has dimension exceeding k(n-d+1)-2. For example, the original definition of e-level <cit.> replaced the condition of finitely many singular points with the condition that there is no rational curve of degree at most e in the singular locus. §.§ Basic lemmasWe collect here some crucial facts needed to run the argument.For any map ϕ: T→_0,r(X,e) from an irreducible scheme T, the pullback ϕ^-1(∂_0,r(X,e))⊂ T is empty or has codimension at most 1. This follows from the fact that ∂_0,0(P^n,e) is a divisor in _0,0(P^n,e) <cit.>.Explicitly, consider an etale cover π: Y→_0,0(P^n,e) by a scheme and note that ∂_0,0(P^n,e) pulls back to a divisor on the scheme T×__0,0(P^n,e)Y in the fiber diagrams below. T×__0,0(P^n,e)Y [d] [r]π^-1(_0,r(X,e)) [r, hook] [d] Y [d, "π"]T [r, "ϕ"] _0,r(X,e) [r, hook] _0,0(P^n,e) For any map ϕ: T→M_0,r(X,e) from an irreducible scheme T, the pullback ϕ^-1(∂M_0,r(X,e))⊂ T has codimension at most 1. This is because the boundary of M_0,r(P^n,e) is a divisor <cit.>. We will also need a version of bend and break. <cit.>If T is a closed locus in _0,0(P^n,e) of dimension at least 2n-1, then T contains maps with reducible domains. §.§ Codimension of the locus of hypersurfaces that are not 1-level As described in <cit.>, the idea of the argument is to borrow rational curves from nearby hypersurfaces to apply bend and break. To run the argument, we need to know the locus of hypersurfaces that are not 1-level has high codimension. For the rest of this section, let P^N be the space of all hypersurfaces of degree d in P^n. We are primarily interested in the case where d=n-1.[<cit.>] Let U⊂P^N be the open locus of smooth hypersurfaces. For 4≤ d=n-1, a largest component of the closed locus Z⊂ U of hypersurfaces that are not 1-level consists of hypersurfaces containing a 2-plane. We will not need this but the largest component is unique when n-1=d>4. We also need to consider hypersurfaces with a larger dimensional family of lines through a singular point than expected. Let U⊂P^N be the open locus of hypersurfaces with at most finitely many singular points. Let Z⊂P^N be the locus of hypersurfaces X for which there exists a singular point p containing an (n-d+1)-dimensional family of lines in X. For d≤ n-1, the codimension of Z in P^N is at least n+12.This is proven in <cit.>. Even though they assume d≤ n-2, the analysis for the case of a singular point goes through. The main obstruction to extending <cit.> to the case where d=n-1 (case of smooth points) is covered by Theorem <ref>.To prove a hypersurface is e-level, we need to rule out a positive dimensional singular locus. [<cit.>] For d≥ 7, the unique largest component of the closed locus Z⊂P^N of hypersurfaces with positive dimensional singular locus consists of hypersurfaces singular along a line. Suppose 7≤ d=n-1. Let Z⊂P^N be the locus of hypersurfaces that are not 1-level. Then, the codimension of Z is n+12-3(n-2). We need to compare the contributions of hypersurfaces that * have a larger dimensional family of lines than expected through a smooth point,* have a larger dimensional family of lines than expected through a singular point,* or have a positive-dimensional singular locus.The first case happens in codimension n+12-3(n-2) by Theorem <ref>. The dimension of hypersurfaces singular along a line is dn-2n+3=n^2-3n+3 <cit.> and bounds the dimension of hypersurfaces with positive-dimensional singular locus by Theorem <ref>.The third case is covered by Proposition <ref>. Comparing these bounds yields the result.§.§ Bend and break in familiesLet P^N be the space of degree d hypersurfaces in P^n, and S_e⊂P^N denote the closure of the hypersurfaces that are not e-level. We have a chain of inclusionsS_1⊂ S_2⊂ S_3⊂⋯⊂P^N.We will use the argument in <cit.> to bound the codimension of each inclusion S_e⊂ S_e+1.Suppose A⊂ B× C with projections π_1: A→ B and π_2:A→ C. If C is a projective scheme and H⊂ C is a general linear section. Then, (π_1(π_2^-1(H)))=(π_1(A)) if π_1: A→ B has positive-dimensional fibers, respectively (π_1(π_2^-1(H)))=(π_1(A))-1 if π_1: A→ B is generically finite onto its image. If A→ B has positive dimensional fibers, choose H so that it cuts down the dimension of a general fiber of π_1 by 1. new plan: we're going to apply this obvious lemma to _0,0(X/P^N,e)⊂P^N×_0,0(P^n,e) For each e, let T_e be the defined as the closure of the hypersurfaces that either * have a point that is not k-level for k≤ e or* are singular along a rational curve of degree k for k≤ e-1.By definition, we have the chainT_1⊂ S_1⊂T_2⊂ S_2⊂ T_3⊂ S_3⊂⋯⊂P^N.The codimension of S_e-1⊂ S_e is at most 2n-(n-d+1)e.expected fibers: 2e-2 moduli needed: 2n-1 dimension needed to apply bend and break: (2n-1)-(2e-1)=2n-2eThe argument is contained in <cit.>, but we will give the proof again for clarity[The bound has changed by 1 because <cit.> claims to prove e-layeredness of all the points in a general hypersurface. We have corresponded with the authors of <cit.>, and it is not clear how their argument implies e-layeredness of all points, but it shows Theorem <ref>, which is enough for their main theorem.]. Let a=e(n-d+1)-2 be the expected dimension of a fiber : _0,1(X/P^N,e)→X. _0,1(X/P^N,e) [r] [bend left=20, rr] [d][bend left=20, dd] _0,1(P^n,e) [r] _0,0,(P^n,e)X [d]P^NSuppose A⊂_0,1(X/P^N,e) is an irreducible component of {[C] ∈_0,1(X/P^N,e)| ([C]) is a singular point, dimension of fiber ofat [C]≥ a+2}.If A consists of covers of lines, then A→P^N maps into S_1. If A contains a map that is not a cover of a line, then the image of A→_0,0(P^n,e) has dimension at least 3n-3 from PGL_n+1-invariance, as we can interpolate a curve from A through 3 general points by taking the PGL-translates of a single curve. (It is easy to do a bit better, for example by considering the dimension of the space of conics, but it is only important to us that the dimension of the image of A→_0,0(P^n,e) is at least 2n-1.)Now, we apply Lemma <ref> to the image of A under the forgetful map A⊂_0,1(X/P^N,e)→_0,0(X/P^N,e)⊂P^N×_0,0(P^n,e).to cut A by hyperplane sections in P^N to produce A'⊂A such that A'→_0,0(P^n,e) has image dimension 2n-1 and (A'→_0,0(X/P^N,e))→_0,0(P^n,e) is generically finite onto its image, so (A'→_0,0(X/P^N,e)) is also dimension 2n-1. By Lemma <ref>, A' contains maps from reducible curves. The image of A'→_0,0(X/P^N,e)→P^N has dimension at most 2n-1-(a+2).If (A'→X)→P^N is generically finite onto its image, then A'→_0,0(P^n,e) is generically finite onto its image. Therefore, A' has dimension 2n-1 and the image of A→P^N has dimension at most 2n-1-(a+2). If (A'→X)→P^N has positive dimensional fibers, then A' has dimension at most 2n, but now the fibers of A'→P^N have dimension at least (a+2)+1, so again the image of A'→P^N has dimension at most 2n-1-(a+2).Finally, we assume, for the sake of contradiction, that the codimension of S_e-1 in the image of A→P^N is at least 2n-(n-d+1)e+1=2n-(a+2)+1. Then, by construction, A'→P^N misses the locus S_e-1 completely. Applying <cit.> shows the locus of reducible curves in A' has dimension at most e(n-d+1)-2=a in each fiber of the evaluation map A'→X. This implies the locus of reducible curves in A' has codimension at least 2, contradicting Proposition <ref>. Similarly, to finish we need to consider the case where a smooth point is not level. We let A be an irreducible component of the closure of{[C] ∈_0,1(X/P^N,e)| ([C]) is a smooth point, dimension of fiber ofat [C]≥ a+1}.If A consists of covers of lines, then A→P^N maps into S_1. Otherwise, as before, A→_0,0(P^n,e) has dimension at least 3n-3. Let A by hyperplane sections in P^N to produce A'⊂A such that A'→_0,0(P^n,e) has image dimension 2n-1 and the image of A'→P^N is at most 2n-1-(a+1). As before, Lemma <ref> shows A' contains reducible curves. If we assume, for the sake of contradiction, that the codimension of S_e-1 in the image of A→P^N is at least 2n-(n-d+1)e+1=2n-(a+1)+1, then A'→P^N misses the locus S_e-1 completely. Applying <cit.> shows the locus of reducible curves in A' has dimension at most e(n-d+1)-2=a in each fiber of the evaluation map A'→X over a singular point and has dimension at most a-1 in each fiber over a smooth point. Since the general point in the image of A'→X is smooth, again we have the locus of reducible curves in A' has codimension at least 2, contradicting Proposition <ref>._0,1(X/P^N,e) [r] [bend left=20, rr, "π"] [d][bend left=20, dd, "ρ"] _0,1(P^n,e) [r] _0,0,(P^n,e)X [d]P^N (1) look at a component in _0,1(X/P^N,e) lying above X where the fibers exceed the expected dimension (let the dimension of the generic fiber in this component be a) (2) let the image of that component in P^N be B. There are two cases: (i) B consists of only singular points (ii) B consists generically of smooth points (3) cut B by generic hyperplanes to get B' so that B' is contained in the (e-1) level hypersurfaces (4) want to show that the stuff lying above B hit something of dimension at least 2n-1 in M(P^N,e) (5) codimension of general fiber of _0,1(X/P^N,e)→_0,0,(P^n,e) in _0,1(X/P^N,e) is at least 3n-3 (if a general element does not map to a line) (b) (6) codimension of general fiber in P^N is at least3n-3-a-(n-1)=2n-1-a(b-a-(n-1)) (7) dimension of B' should be at least 2n-1-a, so the two intersect in finitely points in general(b-a-(n-1)) §.§ Conclusion of argument (of Theorem <ref>) To prove Theorem <ref>, it suffices to show that the fibers of the evaluation map are of the expected dimension <cit.>. First suppose d≥ 7. We apply Theorem <ref> to show S_e is not all of P^N. By Corollary <ref>, we have S_1 has codimension at least n+12-3(n-2) in P^N. Theorem <ref> implies then that S_e has codimension at least n+12-3(n-2)-∑_e'=2^e(2n-2e') =n+12-3(n-2)-2n(e-1)+e(e+1)-2=1/2(n^2-n-4ne+2e^2+2e+8).Solving for the values of e for which n^2-n-4ne+2e^2+2e+8>0 yields Theorem <ref> in the case d≥ 7. If d<7, then e≤ 2. We can assume d∈{5,6} <cit.>. As mentioned in Remark <ref>, we chose to restrict to hypersurfaces with finitely many singular points to simplify the argument for d≥ 7. When e=2, we can replace S_1 with the locus of hypersurfaces for which all the points are 1-level and the singular locus does not contain a line. Hypersurfaces singular along a line have codimension n^2-3n+3 <cit.>, which is at least the codimension of hypersurfaces containing a 2-plane for n≥ 3. Then, we can run the same argument in Proposition <ref> to see flatness in Theorem <ref> as <cit.> still applies.§ IRREDUCIBLE COMPONENTS §.§ Conics through a pointWe will work with the Hilbert scheme of conics instead of the Kontsevich space to focus on degree 2 maps that are not covers of a line. We let* _2t+1(P^n) denote the Hilbert scheme of conics in P^n* _2t+1(X) denote the Hilbert scheme of conics in X* _2t+1(X/P^N) denote the relative Hilbert scheme of conics in the fibers of X→P^N* C→_2t+1(X/P^N) be the universal curve.Note that _2t+1(P^n) is smooth, as a P^5 bundle over G(2,n), _2t+1(X/P^N) is smooth as _2t+1(X/P^N)→_2t+1(P^n) is a P^N-5-bundle, and C is smooth as C→P^n is a G(1,n-1)×P^4×P^N-5 bundle. The goal of this section is to prove The general fiber of C→X is smooth and connected.The proof of <cit.> regarding lines on a general hypersurface is very similar, and so is good preparation for the proof of Proposition <ref>. The proof of Proposition <ref> will reduce to Propositions <ref> and <ref> below. Since C→X is a surjective map between smooth varieties, the general fiber is smooth by generic smoothness. To see connectedness, let C'→X be the Stein factorization of C→X. Let D⊂C' be the singular locus of C'→X. The inverse image of D in C is contained in the singular locus of the map C→X. By Proposition <ref>, D has codimension at least 2 when pulled back to C hence codimension at least 2 in C'. By the Purity theorem <cit.>, C'→X is étale. To finish, it suffices to see that X is étale simply-connected. This follows from the fact that X→P^n is a P^N-1 bundle and the homotopy exact sequence for étale fundamental groups <cit.>. For a>0, let W_a=H^0(P^1,Ø_P^1(a)) be the degree a forms in 2 variables. The case a=1 of Lemma <ref> below is given in <cit.>.Consider the multiplication map m: W_a× W_b→ W_a+b. For a linear hyperplane V⊂ W_a+b, let D(V):={f∈ W_b| m(W_a×{f})⊂ V}⊂ W_b.Then, (D(V))=min{a,b,ℓ}, where ℓ is the smallest positive integer such that V∈P(W_a+b)^* lies on the ℓ-secant variety S_ℓ-1(C),. Here C⊂P(W_a+b)^* is the rational normal curve given as the image P^1→P(W_a+b)^* where p∈P^1 maps to {A∈ W_a+b| A(p)=0}. Let the elements of W_b, and W_a+b be written as b_0s^b+b_1s^b-1t+⋯+b_b t^b, and c_0s^a+b+c_1s^a+b-1t+⋯+c_a+bt^a+b respectively. So f= b_0s^b+b_1s^b-1t+⋯+b_b t^b is in D(V) if and only if b_0s^b+it^j+b_1s^b+i-1t^j+1+⋯+b_b s^it^b+j∈ V for all i,j≥ 0 with i+j=a. If V is written as defined by d_0c_0+⋯+d_a+bc_a+b=0, then f∈ D(V) if and only if[ c_0 c_1 ⋯ c_b; c_1 c_2 ⋯ c_b+1; ⋮ ⋮ ⋱ ⋮; c_a c_a+1 ⋯ c_a+b ][ b_0; b_1; ⋮; b_b ] =[ 0; 0; ⋮; 0 ]Therefore, (D(V)) is the rank of the matrix in (<ref>) with i,j entry c_i+j for 0≤ i≤ a, 0≤ j≤ b. By <cit.>, the locus in P(W_a+b)^* where the matrix is rank ℓ is given by the ℓ-secant variety S_ℓ-1(C) of the rational normal curve C⊂P(W_a+b)^* given by the rank 1 matrices. Therefore, C is given as the image of P^1→P(W_a+b)^* mapping [s:t] to [s^a+b: s^a+b-1t: ⋯ : t^a+b]. So [s:t] maps to the hyperplane in P(W_a+b) defined by c_0s^a+b+c_1s^a+b-1t+⋯+c_a+bt^a+b, which is precisely {A∈ W_a+b| A(p)=0} for p=[s:t].Suppose n≥ 4 and d<n. Then, the closed locus Z of forms (G,F_3,…,F_n) for which the map W_3× W_1^n-2→ W_2d-1 given by (A,L_3,…,L_n)↦ AG+L_3F_3+⋯+L_nF_n is not surjective has codimension at least 2. Consider the incidence correspondence Φ={(V,G,F_3,…,F_n)| GW_3+F_3W_1+⋯+F_nW_1⊂ V}⊂P(W_2d-1)^*× W_3× W_1^n-2→ W_2d-1.It suffices to show (Φ)≤(W_3× W_1^n-2). To do so, we will consider the projection π: Φ→P(W_2d-1)^*, stratify P(W_2d-1)^* and bound the loci in Φ lying over the strata separately using Lemma <ref>. As in Lemma <ref>, let E⊂P(W_2d-1)^* be the rational curve parameterizing hyperplanes in P(W_2d-1) of the form {A∈ W_a+b| A(p)=0} and let E=S_1(E)⊂ S_2(E)⊂ S_3(E)⊂PW_2d-1^* its first, second and third secant varieties. Now, we have four cases: * The locus S_1(E) is 1-dimensional. For V∈ S_1(E), π^-1(V) is codimension n-1 in W_3× W_1^n-2 by Lemma <ref>. Therefore, (W_3× W_1^n-2)-(π^-1(S_1(E)))=n-2. * The locus S_2(E) is 3-dimensional. For V∈ S_2(E)\ S_1(E), π^-1(V) is codimension 2(n-1) in W_3× W_1^n-2 by Lemma <ref>. Therefore, (W_3× W_1^n-2)-(π^-1(S_2(E)\ S_1(E))=2n-5. * The locus S_3(E) is at most 5-dimensional. For V∈ S_3(E)\ S_2(E), π^-1(V) is codimension 2(n-1)+1 in W_3× W_1^n-2 by Lemma <ref>. Therefore, (W_3× W_1^n-2)-(π^-1(S_3(E)\ S_2(E))=2n-6.If d=3 and n=4, then * For V∈P(W_2d-1)^*\ S_3(E),π^-1(V) is codimension 2(n-1)+2 in W_3× W_1^n-2 by Lemma <ref>. Therefore, (W_3× W_1^n-2)-(π^-1(P(W_2d-1)^*\ S_3(E))=2n-2d+1. Since n≥ 4 and d<n, each case yields a bound that is at least 2, which is what we wanted. By Lemma <ref> in the case a=1 and b=2d-2, the locus of F_i'∈ W_2d-2 for which the map W_1→ W_2d-1 given by multiplication by F_i' is contained in V has codimension 2. Let S_1⊂ S_2⊂ S_3⊂PW_2d-1^* be the first, second and third secant varieties to the rational normal curve described in Lemma <ref> in the case a=3 and b=2d-4. Since we have restricted ourselves to the open locus U of forms with no common zero, we do not have to consider the case where V is in S_1. We have to consider S_2\ S_1, S_3\ S_2,S_3\PW_2d-1^*. From Lemma <ref> in the case a=3 and b=2d-4, we see the hyperplanes V in each of those sets impose 2+2(n-2),3+2(n-2),4+2(n-2) conditions on (G',F_3',…,F_n') respectively for the image of W_3× W_1^n-2→ W_2d-1 to be contained in V. If d≥ 4, then S_3⊂PW_2d-1^* is a proper subvariety and combining these cases yields the locus Z⊂ U has codimension at least 4+2(n-2) -(2d-1)=2n-2d+1. If d=3 and n=4, then S_3=PW_2d-1^*, so the codimension is at least 3+2(n-2) -(2d-1)=2n-2d=2. Let _2t+1(X/P^N)⊂_2t+1(X/P^N) denote the open locus of smooth conics. The singular locus of C→X has codimension at least 2 for n≥ 4.Since all smooth conics are projectively equivalent, we can fix the smooth conic C⊂P^n with ideal (Q(X_0,X_1,X_2),X_3,⋯,X_n) and parameterization P^1→P^n given by (s,t)→ (s^2,st,t^2,0,…,0). It suffices to show that, in the P^N-5 hypersurfaces X that contain C, the locus of hypersurfaces X with h^1(N_C/X(-p))≠ 0 has codimension at least 2 for p∈ C. Let F(X_0,…,X_n) cut out a hypersurface X containing C. Then, F can be written asF= Q(X_0,X_1,X_2)G(X_0,X_1,X_2)+X_3F_3(X_0,…,X_n)+⋯+X_nF_n(X_0,…,X_n),where G is degree d-2 and each F_i is degree d-1. The polynomials G and F_i can be chosen independently.For p∈ C,∂_i F(p)= G(p)∂_iQ(p)if 0≤ i≤ 2F_i(p)if3≤ i≤ n,so X is smooth along C if and only if G,F_3,…,F_n do not have a common zero on C, in particular the locus of hypersurfaces singular at a point of C has codimension at least n-2 in P^N-5. Therefore, it suffices to restrict to the open locus in P^N-5 of hypersurfaces smooth along C. Consider the short exact sequence0 [r] N_C/X[r] N_C/P^n [r] [equal,d] N_X/P^n|_C [r] [equal,d]0Ø_C(2H)⊕Ø_C(H)^⊕(n-2) [r] Ø_C(dH)To determine the map Ø_C(2H)⊕Ø_C(H)^⊕ (n-2)→Ø_C(dH), we consider the dual map. We have the conormal bundles N_C/P^n^∨=(Q,X_3,…,X_n)/(Q,X_3,…,X_n)^2 and N_X/P^n^∨|_C=(F)/(F)(Q,X_3,…,X_n) as quotients of ideal sheaves. The map N_X/P^n^∨|_C→ N_C/P^n^∨ is induced by inclusion (F)⊂ (Q,X_3,…,X_n). Dualizing, the map Ø_C(2H)⊕Ø_C(H)^⊕ (n-2)→Ø_C(dH) is given by multiplication by the vector (G,F_3,…,F_n). The long exact sequence in cohomology implies H^1(N_C/X(-p))=0 if and only if H^0(Ø_C(2H-p)⊕Ø_C(H)^⊕ (n-2))→ H^0(Ø_C(dH)) is surjective. By pulling back via the parameterization P^1→P^n, we are done by Lemma <ref>.For n≥ 4, the singular locus of C→X has codimension at least 2 in C. The space _2t+1(P^n) can be stratified into three loci: smooth conics, unions of two distinct lines, and doubled lines. These strata are of codimensions 0, 1 and 2, respectively. Since the Hilbert function is constant on _2t+1(P^n), the map C→_2t+1(P^n) is a Zariski-local projective bundle. This means the pullbacks of these strata in C are also of codimensions 0, 1, and 2. Therefore, it suffices to show Proposition <ref> when restricted to the smooth conics and to the unions of two distinct lines. By Proposition <ref>, we know Proposition <ref> is true when we restrict C to the locus where C→_2t+1(X/P^N) is smooth. Now, consider the locally closed subset Y⊂C consisting of pointed curves (C,p), where C is a union of two distinct lines L_1,L_2, p∈ L_1\ L_2. The locus Y is irreducible, as we can parameterize Y by a smooth, irreducible variety Z→Y by specifying (C,p) by first choosing p∈P^n, the two plane P containing C, a line L_1 containing p and contained in P, a second line L_2 contained in P, and finally a hypersurface X containing L_1∪ L_2. Since the complement of the union of Y and the smooth locus of C→_2t+1(X/P^N) has codimension 2, it suffices to show the singular locus of C→X has codimension 1 in Y. A parameter count involving the normal bundle of a line in a hypersurface similar to <cit.> shows the singular locus of the map from the universal line _0,1(X/P^N,1)→X is singular in codimension at least min{n-2,n-d}=n-d. A more complicated version of this parameter count was given in detail in the proof of Proposition <ref> above. This means the map Z→X is smooth at a general point. Since the image Y of Z→C has codimension 1 and Z→Y has finite reduced fibers, this means C→X is smooth at a general point of Y. By <cit.>, it suffices to see X is étale simply-connected and the singular locus of C→X has codimension at least 2. The first condition follows from the fact that X→P^n is rational as a Zariski-locally trivial P^N-1 bundle and the birational invariance of étale fundamental groups.The second condition follows Proposition <ref> below. §.§ LayerednessLet X⊂P^n be a hypersurface of degree d, where d≤ n-1.A point p∈ X is called e-layered if it is 1-level and for every 1<k≤ e, every irreducible component parameterizing degree k rational curves through p contains reducibles. A hypersurface X is e-layered if it is e-level and a general point is e-layered. We have corresponded with the authors of <cit.> and it is not clear how their argument as written shows e-layeredness at all points as claimed, but they brought us to the attention that e-levelness at all points and e-layeredness at a general point suffices to prove their main theorem. §.§ Existence of e-layered hypersurfacesLet P^N be the space of degree d hypersurfaces in P^n, and T_e⊂P^N denote the closure of the hypersurfaces that are not e-layered. We have a chain of inclusionsT_1⊂ T_2⊂ T_3⊂⋯⊂P^N.The codimension of T_e-1⊂ T_e is at most 2n-(n-d+1)e.If f:X→ Y is a map between Noetherian schemes of finite presentation and Z⊂ X is a closed subset, then the set A⊂ X of all x∈ X such that a geometric component of the fiber f^-1(f(x)) containing x is disjoint from Z is constructible. Since this is a statement about the underlying topological spaces, we can assume X and Y are reduced. By restricting to each component of X, we can assume X is integral. By Noetherian induction on X, it suffices to prove Proposition <ref> after restriction to some open subset of Y, so we can assume Y= Spec(A) is affine and integral. Let η∈ Y be the generic point and X_η be the fiber over the generic point. Following <cit.>, we will find a finitely presented surjection Y'→ Y such that each component of the generic fiber of X×_YY' is irreducible as follows. Since X_η has finitely many components, there is a finite separable extension L of K(A) such that each component of X_η× _ Spec(K(A)) Spec(L) is geometrically irreducible <cit.>. Since L is separable over K(A), it is generated by some element α, which we can assume to be in A after multiplying by an element of A. Then, if the field extension L over K(A) is defined by the monic polynomial p, then let A'=A[T]/p(T) and Y'= Spec(A').If we prove Proposition <ref> for the morphism X×_YY'→ Y', then we prove Proposition <ref> via Chevellay's Theorem <cit.> applied to X×_YY'→ X. Therefore, we can assume in addition that X_η has geometrically irreducible components.Let X_1,η,…,X_r,η denote the irreducible components of X_η and X_1,…,X_r the closures of X_1,η,…,X_r,η in X respectively. Since X\ (X_1,η∪⋯∪ X_r,η)→ Y misses η∈ Y, Chevellay's Theorem again implies we can restrict Y to a standard affine that avoids the image. Therefore, we can assumeX is the set-theoretically the union of X_1,…,X_r.By generic flatness <cit.> and the fact that flat morphisms are open <cit.>, we can replace Y by an open subset so that each X_i→ Y is flat and surjective. By generic flatness again, we can replace Y with a standard open to assume X_i∩ X_j→ Y is flat for each pair 1≤ i,j≤ r. By equidimensional of the fibers of a flat morphism <cit.>, we know f^-1(p)∩ (X_i∩ X_j)⊂ f^-1(p)∩ X_i is nowhere dense for i≠ j. By replacing Y with an open subset, we can assume that each geometric fiber of X_i→ Y is irreducible <cit.>. Finally, if we let S⊂{1,…,r} be the subset of indices i such that X_i does not intersect Z, then A is the union ⋃_i∈ SX_i.From Theorem <ref>, we know the codimension of T_e-1⊂ T_e-1∪ S_e is at most 2n-(n-d+1)e. Using the same setup as the proof of Theorem <ref>, let B be{[C]∈_0,1(X/P^N,e)| a component of ^-1(([C])) containing [C] has no reducibles}.Note that B is constructible by Proposition <ref>, so in particular every component of B contains an open dense subset contained in B. Consider the map B→X→P^N, and consider the closed locus in (B→X) where the fiber dimension is n-1. LetB'⊂B be the inverse image of that closed locus in B. Hence, B' are the stable maps in B lying above the hypersurfaces [X]∈P^N that have a component covered by B→X under the evaluation map. We want to control the image B'→P^N. Let A⊂B' be an irreducible component. If A contains a map that is not a cover of a line, then the image of A→_0,0(P^n,e) has dimension at least 3n-3. Applying Lemma <ref> as before allows us to cut A by hyperplane sections in P^N to obtain A' such that the image of A'→_0,0(P^n,e) has dimension 2n-1. The image of the generic fiber of A'→_0,0(P^n,e) in P^N is finite. Lemma <ref> shows A' contains stable maps from reducible curves. This means (A'→_0,0(X/P^N,e)) is also dimension 2n-1. Since the fiber dimension of (A'→X)→P^N is n-1, the fiber dimension of (A'→_0,0(X/P^N,e))→P^N is at least (n-1)+(a-1), where a=(n-d+1)e-2 as in the proof of Theorem <ref>. The image of A'→P^N has dimension at most 2n-1-(n+a-2).Let a=e(n+1-d)-2. We claim the image of A'→P^N has dimension at most 2n-1-(a+1). The general fiber of A'→_0,0(P^n,e) has dimension at most 1. If the general fiber dimension is 0, then the dimension of A' is 2n-1 and, since the general fiber of A'→P^N has dimension at least a+1, the image of A'→P^N has dimension at most 2n-1-(a+1). Then, by assumption the image A'→P^N misses the locus S_e-1, which contradicts <cit.> and Proposition <ref>. If the general fiber dimension of A'→_0,0(P^n,e) is 1, then we claim the general fiber dimension of A'→P^N is at least a+2. Otherwise, the image of A'→P^N has dimension 2n-(a+1). Let X⊂P^n be a general hyperplane in the image of A'→P^N. From <cit.>, a general element of A” is irreducible. Since a general element of A” is not a cover of a line, we know from the argument in <cit.> or <cit.> bounding rational curves that are multiple covers, the general curve in A” is injective. But then the (a+1)-dimensional family of curves in A' inside of X all cover the same rational curve, which is a contradiction.Assume, for the sake of contradiction, that the codimension of T_e-1⊂ T_e is at least 2n-(n-d+1)e+1. Then, since n≥ 3 the image A'→P^N misses the locus S_e-1, which contradicts <cit.> and Proposition <ref>. Here, the point is <cit.> shows the locus parameterizing reducible curves intersects every fiber of A'→X in codimension at least 1, the general fiber of A'→X parameterizes no reducible curves, and singular points occur in the image of A'→X in codimension at least n-1 by the definition of (e-1)-levelness. This means the locus in A' parameterizing reducible curves is codimension at least 2, which contradicts Proposition <ref>.If e<n-1+√(n^2-n-15)/2, then a general hypersurface X of degree d=n-1 in P^n is e-layered. If d≥ 7, then this follows from the same dimension computation in Section <ref> applied in the proof of Theorem <ref> using Theorems <ref> and <ref>. When d≤ 6, it suffices to consider the case e≤ 2, in which case Proposition <ref> suffices. §.§ Behrend-Manin stacksLet X⊂P^n be a smooth hypersurface of degree d. In order to keep track of the combinatorics of the components of reducible rational curves, we will use Behrend-Manin stacks. We refer the reader to <cit.> for the precise definitions of a stable A-graph τ and the associated Behrend-Manin stack (X,τ). Also see <cit.> for a shorter account that suffices for our purposes. Roughly, a stable A-graph keeps track of the combinatorics of the irreducible components of a stable map C→ X, including the dual graph of how they intersect, the marked points on each component, the degree of the map restricted to each component, and the genus of each component. Since we are dealing with rational curves, all the stable A-graphs we consider will have genus zero, meaning the genus of each vertex is zero and the underlying graph is a tree. Associated to a stable A-graph τ, there is a set of vertices (τ), a set of edges (τ) connecting them, and a set of tails (τ), which can be thought of half edges attached to vertices. There is also a map β: (τ)→Z_≥ 0, assigning a degree to each vertex. We also let β(τ)=∑_v∈(τ)β(v) and the expected dimension(X,τ):=(n+1-d)β(τ)+#(τ)-#(τ)+(X)-3<cit.>. Finally, there is a set of flags (τ), where we have two flags corresponding to each edge in (τ), corresponding to the two endpoints, and one flag for each tail. In particular, #(τ)=2#(τ)+#(τ). The Behrend-Manin stack (X,τ) parameterizes stable maps C→ X, where the curve C consists of prestable curves C_v <cit.>, one for each vertex of τ, that glue together and map to X according to the data in τ. The open locus of (X,τ) of strict maps is quicker to define. See <cit.> for more details.A stable map C→ X in (X,τ) is a strict map if C_v≅P^1 for each v∈(τ). The locus of strict maps is an open substack M(X,τ)⊂(X,τ).A point in (X,τ)⊂(X,τ) can be specified by the data ((C_v)_v∈(τ),(h_v:C_v→ X)_v∈(τ),(q_f)_f∈(τ)) such that q_f∈ C_v, where v is the vertex to which q_f is attached. Each map h_v: C_v→ X is specified to have degree β(v). Since we want to think of (τ) as parameterizing marked points, for each f∈(τ), we have an evaluation map _f: (X,τ)→ X<cit.>. Similarly, we have _f for all f∈(τ), as the remaining flags correspond to the points of intersection between different prestable curves C_v that piece together to give the domain of a stable map C→ X, and we can ask for the image of such an intersection point.§.§ A criterion for smoothness Let τ_r(e) be the stable A-graph that has one vertex v, no edges, r tails such that β(τ)=β(v)=e. By definition, (X,τ_r(e)) is the Kontsevich space _0,r(X,e). Given any A-stable graph with β(τ)=e and #(τ)=r, there is a morphism (X,τ)→_0,r(X,e) canonical up to relabeling the tails.If we repeatedly specialize a rational curve C→ X so that it breaks up into more and more components, then we eventually end up with a tree of lines. Since we will care about rational curves through a general point p∈ X given by a tree of lines, we make the following definition. By abuse of notation, it is different than the one given in <cit.>.Let a stable A-graph τ be called a basic A-graph if β(v)∈{0,1} for all v∈(τ) and #(τ)=1.Let a basic A-graph be called nondegenerate if β(v)=1 for all v∈(τ).The argument in <cit.> applied in our case givesLet τ be a basic A-graph and X⊂P^n be a smooth e-level hypersurface of degree d=n-1 with an irreducible Fano scheme of lines. Then, the morphism (X,τ)→_0,1(X,e) maps a general point of (X,τ) to a point in the smooth locus of : _0,1(X,e)→ X. Applying the argument in <cit.>, reduces the question of checking whether a point (h: C→ X)∈(X,τ) is a smooth point of _0,1(X,e)→ X to checking whether H^1(C,h^*T_X(-p))=0, where p∈ C corresponds to the unique tail in (τ). The tangent space to a fiber of _0,1(X,1)→ X at a pair (ℓ,p), where p∈ℓ⊂ X and ℓ is a line is H^0(N_ℓ/X(-p)), and this is of the expected dimension if and only if H^1(N_ℓ/X(-p))=0. By generic smoothness, this holds for a general pair (ℓ,p). From the short exact sequence, 0→ Tℓ→ TX|_ℓ→ N_ℓ/X→ 0, H^1(TX|_ℓ(-p))=0 for a general point (ℓ,p)∈_0,1(X,1). Applying <cit.> allows us to conclude. Instead of arguing via the smoothness of the nonseparated Artin stack of prestable curves as in <cit.> in the beginning of the proof of Proposition <ref>, an equivalent way is to add c marked points to C→ X so the prestable curve C is actually stable. Then, it suffices to show smoothness of _0,1+c(X,e)→M_0,1+c× X at C→ X as this implies the map _0,1+c(X,e)→ X is smooth at C→ X. Note that e-levelness guarantees flatness of the evaluation map <cit.>, so smoothness at a point is equivalent to being smooth in its fiber. The condition on the Fano scheme is automatically satisfied in our case since the Fano scheme of lines is smooth and connected for a general hypersurface <cit.> if the degree d of X⊂P^n is at most 2n-4 and X is not a quadric surface. §.§ Rational curves through a pointTheorem <ref> will follow from Theorem <ref> for the statement on dimension, and Corollary <ref> and Theorem <ref> for the statement on irreducible components.Let e≥ 2 and d=n-1. If there exist e-layered hypersurfaces, then for a general hypersurface X, the fiber F_p of the evaluation map_0,1(X,e)→ Xover a general point p∈ X has only one component that contains curves C→ X that are not multiple covers of a line.A diagram of the proof of Theorem <ref> in the case e=3 is depicted in Figure <ref>.The case e=2 is Proposition <ref>, so suppose e>2. We will use strong induction on e. By e-levelness, each component of F_p has the same dimension. If C→ X is a rational curve in X through p, we can use e-layeredness to specialize C→ X to C_0→ X, so that C_0→ X lies in M(X,τ), where τ is a nondegenerate basic A-graph. By Proposition <ref>, we can assume C_0→ X is a smooth point of F_p. Each component of the fiber of (X,τ)→ X over p lies in a unique component of F_p. What we need to show is that as we vary over all nondegenerate basic A-graphs τ we only get one component of F_p that contains curves that are not covers of a line. To do this, we will reduce ourselves to looking at “combs” of lines, where the backbone gets collapsed. One can get this by specializing a tree of lines to a “broom”, where all the lines pass through p. For clarity, we will instead first reduce to the case of chains of lines and then specialize the chain of lines to a comb. As before, let C_0→ X be in M(X,τ), where τ is a nondegenerate basic A-graph. Let (C_v)_v∈(τ) be the components of C_0. Note that each C_v≅P^1. Let v_0 be the vertex to which the unique tail of τ is attached. By abuse of notation, we call the marked point in C_v_0 that maps to p under C→ X also as p∈ C_v_0. Let q_a_1,…,q_a_r∈ C_v_0 correspond to the edges attached to v_0 in τ, or the points of attachment of the other components of C to C_v_0. [gray, thick] (0,0) – (8,0); [black] (1,0) circle (2pt) node[anchor=south]p; [black] (2,0) circle (2pt) node[anchor=south]q_a_1; [black] (4,0) circle (2pt) node[anchor=south]q_a_2; [black] (5,0) circle (2pt) node[anchor=south]q_a_3; [black] (6,0) circle (2pt) node[anchor=south]q_a_4; [black] (8.5,0) circle (1pt) node; [black] (9,0) circle (1pt) node; [black] (9.5,0) circle (1pt) node; [black] (4,-.5) circle (0pt) nodeC_v_0;[gray, thick] (2,0) – (2+.5,0+3/2); [black] (2+.5+1/6,3/2+1/2) circle (1pt) node; [black] (2+.5+2/6,3/2+2/2) circle (1pt) node; [black] (2+.5+3/6,3/2+3/2) circle (1pt) node;[gray, thick] (4,0) – (4+.5,0+3/2); [black] (4+.5+1/6,3/2+1/2) circle (1pt) node; [black] (4+.5+2/6,3/2+2/2) circle (1pt) node; [black] (4+.5+3/6,3/2+3/2) circle (1pt) node;[gray, thick] (5,0) – (5+.5,0+3/2); [black] (5+.5+1/6,3/2+1/2) circle (1pt) node; [black] (5+.5+2/6,3/2+2/2) circle (1pt) node; [black] (5+.5+3/6,3/2+3/2) circle (1pt) node;[gray, thick] (6,0) – (6+.5,0+3/2); [black] (6+.5+1/6,3/2+1/2) circle (1pt) node; [black] (6+.5+2/6,3/2+2/2) circle (1pt) node; [black] (6+.5+3/6,3/2+3/2) circle (1pt) node; Now, we specialize the points q_a_1,…,q_a_r one by one to a fixed general point q_a∈ C_v_0. Let the resulting curve be C_0'→ X, given by gluing together the prestable curves (C_v')_v∈(τ). From the argument in <cit.>, if we want to understand what happens to C_v_0 in the limit as we specialize, it suffices to understand what happens to the map (C_v_0,q_a_1,…,q_a_r)→ X from the pointed curve (C_v_0,q_a_1,…,q_a_r) as we specialize the points q_a_1,…,q_a_r. Then, C_v_0 gets replaced with the prestable curve C_v_0' that is C_v_0 with a chain of rational curves attached at q_a. Proposition <ref> tells us that C_0'→ X is a smooth point of F_p. By induction, the space of degree e-1 curves through q_a contains only one component with curves that do not cover a line.This means C_0'→ X is in the same component of F_p as the curve we get when we take C_v_0≅P^1 and attach a general chain of lines to q_a in the same component of degree e-1 curves through q_a. This chain of lines may be a cover of a line. In this way, we have reduced to the case where τ is a chain. Now, let v_0,…, v_e-1 be (τ), where each v_i is connected to v_i+1 for 0≤ i≤ e-2. Let the unique tail of τ be attached to v_0 and h: C→ X given by (h_i: C_v_i→ X) be a point of (X,τ)∩ F_p, general in its component. As before, each h_i: C_v_i≅P^1→ X is an embedding of a line. Now, we want to specialize the points of attachment. [gray, thick] (0,0) – (3,1); [gray, thick] (2,1) – (5,0); [black] (1,1/3) circle (2pt) node[anchor=north]p ; [black] (2.5,2.5/3) circle (2pt) node[anchor=west] ; [black] (1,1) circle (0pt) nodeC_v_0; [black] (3.5,1) circle (0pt) nodeC_v_1; [black] (4.5,.5/3) circle (2pt) node;[gray, thick] (4,0) – (7,1); [gray, thick] (6,1) – (9,0); [black] (2.5+4,2.5/3) circle (2pt) node[anchor=west] ; [black] (1+4,1) circle (0pt) nodeC_v_2; [black] (3.5+4,1) circle (0pt) nodeC_v_3; [black] (4.5+4,.5/3) circle (2pt) node;[gray, thick] (8,0) – (11,1); [gray, thick] (10,1) – (13,0); [black] (2.5+8,2.5/3) circle (2pt) node[anchor=west] ; [black] (1+8,1) circle (0pt) nodeC_v_4; [black] (3.5+8,1) circle (0pt) nodeC_v_5; [black] (4.5+8,.5/3) circle (2pt) node;[gray, thick] (12,0) – (13,1/3); [black] (14,1/2) circle (1pt) node; [black] (14.5,1/2) circle (1pt) node; [black] (15,1/2) circle (1pt) node; Let p_0=p and p_i be the point of C_v_i that is attached to C_v_i-1 under h:C→ X for 1≤ i≤ e-1. For 0≤ i≤ e-2, let q_i∈ C_v_i be the point that is attached to C_v_i+1. Now, we take a 1-dimensional family that specializes q_0 to p_0. Then, we specialize q_1 to p_1. Let C'→ X given by (h_i': C_v_i'→ X) be the result after specializing q_i to p_i for all 0≤ i≤ n-2. Then, each C_v_i' for 0≤ i≤ e-2 becomes a union of two rational curves, where the one containing q_i and p_i is collapsed under the map to X.[gray] (0,-1) .. controls (2,0) .. (4,-1); [gray] (3,-1) .. controls (5,0) .. (7,-1); [gray] (6,-1) .. controls (8,0) .. (10,-1); [gray] (9,-1) .. controls (11,0) .. (13,-1);[gray, thick] (2.5,0) – (2.5,-3); [gray, thick] (5.5,0) – (5.5,-3); [gray, thick] (8.5,0) – (8.5,-3); [gray, thick] (11.5,0) – (11.5,-3);[black] (.5,-.75) circle (2pt) node[anchor=south]p;[black] (3.5,-.75) circle (2pt) node[anchor=south]q_0=p_1; [black] (6.5,-.75) circle (2pt) node[anchor=south]q_1=p_2;; [black] (9.5,-.75) circle (2pt) node[anchor=south]q_2=p_3;;[black] (1.5,-1) circle (0pt) node[anchor=north]C_v_0'; [black] (4.5,-1) circle (0pt) node[anchor=north]C_v_1'; [black] (7.5,-1) circle (0pt) node[anchor=north]C_v_2'; [black] (10.5,-1) circle (0pt) node[anchor=north]C_v_3';[black] (13.5,-1/2) circle (1pt) node; [black] (14,-1/2) circle (1pt) node; [black] (14.5,-1/2) circle (1pt) node; Proposition <ref> tells us that C'→ X is a smooth point of F_p. However, we note that C'→ X is also a specialization of a strict map D→ X lying in ℳ(X,τ_ comb)∩ F_p, where (τ_ comb)={v_ center,v_0,…,v_e-1} with β(v_ center)=0 and β(v_i)=1, and the edges of τ_ comb connect v_ center to each of v_0,…,v_e-1. The unique tail of τ_ comb is attached to v_ center. Proposition <ref> says that a general choice of a strict map D given by (g_v: D_v→ X)_v∈(τ_ comb) is a smooth point of F_p. In fact, since the choice of the maps D_v_i→ X is discrete, every choice of a strict map D→ X is smooth.[gray] (0,0) –(12,0);[black] (6,0) circle (0pt) node[anchor=south]D_v_ center; [black] (1,0) circle (2pt) node[anchor=north]p;[gray] (2,1) –(2,-3); [black] (2,-1) circle (0pt) node[anchor=east]D_v_0;[gray] (4,1) –(4,-3); [black] (4,-1) circle (0pt) node[anchor=east]D_v_1;[gray] (8,1) –(8,-3); [black] (8,-1) circle (0pt) node[anchor=east]D_v_2;[gray] (10,1) –(10,-3); [black] (10,-1) circle (0pt) node[anchor=east]D_v_3; [black] (12.5,0) circle (1pt) node; [black] (13,0) circle (1pt) node; [black] (13.5,0) circle (1pt) node; The backbone D_v_ center gets collapsed to p under the map to X and each D_v_i gets mapped to a line through p. To finish, we need to show that all strict maps D→ X, where not all of the D_v_i get mapped to the same line, are in the same component of F_p. Let S⊂{v_0,…,v_e-1} be a strict subset, with # S≥ 2 and such that the maps D_v→ X for v∈ S do not all embed as the same line. Then, we can specialize D→ X to D'→ X, where D_v_ center' is now a rational curve with two components, one that is attached to D_v' for all v∈ S and the other that contains p and is attached to D_v' for v∉ S. The maps D_v_i≅ D_v_i'→ X are unchanged. Then, using the induction hypothesis, this map D'→ X is in the same component of F_p as the map we would get by modifying the lines D_v'→ X for v∈ S, so long as D_v'→ X for v∈ S do not all embed as the same line. The end result is that D→ V is in the same component as the map we would get by modifying D_v→ X for v∈ S, so long as we maintain D_v→ X for v∈ S do not all embed as the same line. In this way, we see that all strict maps D→ X, where not all of the D_v_i get mapped to the same line, are in the same component of F_p.plain | http://arxiv.org/abs/1709.09740v2 | {
"authors": [
"Dennis Tseng"
],
"categories": [
"math.AG",
"14H10, 14J45, 14J70"
],
"primary_category": "math.AG",
"published": "20170927213024",
"title": "A note on rational curves on general Fano hypersurfaces"
} |
tablep positioning relbare[1][0.32 * ] [baseline=([yshift=-.5ex]current bounding box.center)] 0-#1#10;relmatrix[3][0.32 * ] [ [#1](1-0.5,1-.4) – (#2 +.5, #3 +.4);] plain theoremTheorem[section] corollary[theorem]Corollary lemma[theorem]Lemma question[theorem]Question proposition[theorem]Proposition conjecture[theorem]Conjecture maintheoremTheorem definition definition[theorem]Definition example[theorem]Example notation[theorem]Notation *example*Exampleremark remark[theorem]Remark *remark*Remarkequationsection section#2#1#3 Syzygies of the determinant and permanent]Syzygies of the apolar ideals of thedeterminant and permanent J. Alper]Jarod Alper Department of Mathematics University of Washington Box 354350 Seattle, WA 98195-4350, USA [email protected] R. Rowlands]Rowan Rowlands Department of Mathematics University of Washington Box 354350 Seattle, WA 98195-4350, USA [email protected] [2010]Primary 13D02; Secondary 13P20, 68Q17, 14L30 We investigate the space of syzygies of the apolar ideals _n^⊥ and _n^⊥ of the determinant _n and permanent _n polynomials.Shafiei had proved that these ideals are generated by quadrics and provided a minimal generating set.Extending on her work, in characteristic distinct from two, we prove that the space of relations of _n^⊥ is generated by linear relations and we describe a minimal generating set.The linear relations of _n^⊥ do not generate all relations, but we provide a minimal generating set of linear and quadratic relations.For both _n^⊥ and _n^⊥, we give formulas for the Betti numbers β_1,j, β_2,j and β_3,4 for all j as well as conjectural descriptions of other Betti numbers.Finally, we provide representation-theoretic descriptions of certain spaces of linear syzygies. [ [ December 30, 2023 =====================§ INTRODUCTION This paper began as an investigation into the difference in complexity between the determinant and permanent polynomials by exploring homological properties of their apolar ideals.To set up our notation, letbe a field and n be a positive integer.Let 𝐱 =(x_i,j)_1≤ i,j≤ n denote an n × n matrix of indeterminates.The determinant and permanent polynomials are defined as_n = (𝐱) =∑_σ∈_n(σ) x_1, σ(1) x_2, σ(2)… x_n, σ(n),and_n = (𝐱) = ∑_σ∈_n x_1, σ(1) x_2, σ(2)… x_n, σ(n).respectively, where _n is the symmetric group on n letters.If we denote the -vector space of n × n matrices as M_n(), then the polynomials _n and _n are elements in the vector space ^n M_n()^∨ of homogenous polynomials on M_n() of degree n.Understanding the difference between the determinant and permanent polynomials is of central interest in theoretical computer science and, in particular, algebraic complexity theory.See <ref> for more details on the connection to complexity theory. §.§ ApolarityIn this paper, we will investigate homological properties of the apolar ideals of _n and _n.We begin by recalling the definition of the apolar ideal. Let W be a vector space overof dimension n and f(x_1, …, x_n) ∈^d W^∨ be a homogenous polynomial on W of degree d, where we have chosen a basis x_1, …, x_n of W^∨.Let R = ^* W^∨ and S = ^* W; we will identify R with the polynomial ring[x_1, …, x_n], and S with thesubring [x_1^-1, …, x_n^-1] of the fraction field (R) of R.Using this identification, multiplication induces an S-module structure ⋆ S × R → R as follows: for f ∈ R and g ∈ S ⊂(R), then g ⋆ f = gf if gf ∈ R and 0 otherwise. Note that g acts like “differentiation without coefficients”: that is, if h ∈ R is a polynomial independent of x_i, thenx_i^-1⋆ h x_i^n = h x_i^n-1 = 1/n∂/∂ x_i h x_i^nfor n > 0 as long as n is invertible in , and x_i^-1⋆ h = 0 = ∂/∂ x_i h. The apolar ideal of f is the idealf^⊥ { g|g ⋆ f = 0 }⊆ S. The quotient ^* W / f^⊥ is a graded Artinian Gorenstein algebra with socle in degree d.A theorem of Macaulay <cit.> states that the assignment f ↦ f^⊥ gives a one-to-one bijection between homogeneous polynomials f ∈^d W^∨ up to scaling and homogenous ideals I ⊂^* W such that the quotient ^d W / I is a graded Artinian Gorenstein algebra with socle in degree d. In particular, the homogeneous ideal f^⊥ determines f uniquely up to scaling.Thus one can attempt to distinguish the determinant and permanent polynomials via studying their apolar ideals.Specifically, we ask: What are the minimal graded free resolutions of S/_n^⊥ and S/_n^⊥? This question was the starting point for our investigations.It is a theorem of Shafiei <cit.> that for every n ≥ 2, both the apolar ideals _n^⊥ and _n^⊥ are minimally generated by n + 1 2^2 quadrics.These quadrics can be explicitly described — see <ref> for a summary of Shafiei's work.The main result of this paper determines the relations between these generators (i.e., the first syzygies): *If () ≠ 2, all relations of _n^⊥ are minimally generated by 4 n+13n+23 linear relations.*In arbitrary characteristic, all relations of _n^⊥ are minimally generated by 4 n+13n+23 linear relations and 2 n2n4 quadratic relations. Observe that in characteristic 2, the determinant and permanent polynomials are equal, and thus the relations of _n^⊥ are described by Part <ref>.We find it interesting that in ()=2, the syzygies of the apolar ideal of _n seem to resemble those of the _n in arbitrary characteristic.As in Shafiei's result, we provide an explicit minimal generating sets of relations for both _n^⊥ and _n^⊥; see <ref>.Moreover, we compute the dimension of the space of linear second syzygies of both _n and _n (<ref>) and conjecture on the dimension of linear higher syzygies (<ref>).In <ref>, we provide Macaulay2 computations of the full or partial Betti tables for small n.Finally, the free resolution of the apolar ideal _n^⊥ is naturally a free resolution of representations of the symmetry group G__n of the determinant _n.In <ref>, we provide a complete representation-theoretic description of the spaces of linear generators, relations and second syzygies (<ref>).An outline of the paper is provided in <ref>. §.§ Motivation Understanding the difference between the “complexity” of the determinant and permanent polynomials is of fundamental significance to algebraic complexity theory.There are several notions of complexity of a homogeneous polynomial such as determinantal complexity, Waring rank, and product rank. Valiant conjectured in <cit.> and <cit.> that the determinantal complexity of the n × n permanent _n is not bounded above by any polynomial in n and, moreover, showed that this conjecture implies a separation between the algebraic complexity classes _ e and, which are algebraic analogues of P and NP.Since the apolar ideal uniquely determines a homogenous polynomial up to scaling, it is natural to ask the following vaguely formulated question:Can algebraic or homological properties of the ideal f^⊥ be used to give lower or upper bounds for any complexity measure? This question was our original motivation and we decided to focus on the minimal free graded resolutions of f^⊥.For instance, in <cit.>, it was shown that if f^⊥ is generated in degree D, then the Waring rank of f is bounded below by 1/D_k S/f^⊥.One might hope that there is a stronger lower bound involvinghigher syzygies. Our search for new lower bounds of the Waring rank, determinantal complexity, and other complexity measures in terms of the minimal graded free resolutions of f^⊥ has so far been elusive. Therefore, we unfortunately have no positive answers to <ref>.Nevertheless, we find the problem of determining the syzygies of the apolar ideals of _n and _n intrinsically interesting and our work has generating appealing connections to combinatorics and representation theory. §.§ AcknowledgementsDuring the preparation of this paper, the first author was partially supported by the Australian ResearchCouncil grant DE140101519. The second author was partially supported by David Smyth's Australian Research Council grant DE140100259.Both authors would like to thank Daniel Erman for providing helpful suggestions.§ BACKGROUND§.§ NotationAs in the introduction, 𝐱 =(x_i,j)_1≤ i,j≤ n will denote an n × n matrix of indeterminates.We set R = [x_i,j], a polynomial ring in n^2 variables.The polynomials _n and _n are elements of this ring.Let M_n() be the -vector space of n × n matrices with basis { X_i,j} dual to {x_i,j} for 1≤ i,j≤ n.Define S = ^* M_n() = [X_i,j]. The apolar ideals _n^⊥ and _n^⊥ are ideals in this ring.The operation ⋆ S × R → R defined in <ref> gives R the structure of an S-module.Since no variable appears in _nwith degree greater than one, we have that X_i,j⋆_n = ∂/∂ x_i,j_n so that we may identify _n^⊥ with the ideal of polynomials g ∈ S such thatg ( ∂/∂ x_1,1, ∂/∂ x_1,2, …, ∂/∂ x_n,n)( _n) = 0. The same applies for _n. The first observation to make is that the action of ∂/∂ x_i,j on _n is the same as taking the i,jth minor, up to sign; that is,X_i,j⋆_n 𝐱 = ∂/∂ x_i,j_n 𝐱 = (-1)^i+j_n-1𝐱(i; j)where 𝐱(i; j) denotes the submatrix of 𝐱 obtained by deleting the ith row and jth column. Similarly,X_i,j⋆_n 𝐱 = ∂/∂ x_i,j_n 𝐱 = _n-1𝐱(i; j)Therefore, since (S / _n^⊥)_d and (S / _n^⊥)_d are isomorphic to the spaces of dth derivatives of _n and _n respectively, and since there are nd^2 minors or permanent-minors of order d for an n × n matrix and they are linearly independent, we obtain the following fact: The dimensions of (S / _n^⊥)_d and (S / _n^⊥)_d are each nd^2.Now, consider the minimal graded free resolutions…→ F_2F_1F_0 → S / _n^⊥→ 0 and…→ F'_2F'_1F'_0 → S / _n^⊥→ 0 where F_i and F_i' are free graded S-modules of the form ⊕_j S(-j)^β_i,j. Elements of the kernel of d_i+1 or d_i+1' are called ith syzygies.The numbers β_i,j are called the graded Betti numbers.Clearly F_0 = F_0' = S. The summands of F_1 (resp. F'_1) correspond to a minimal set of generators of _n^⊥ (resp. _n^⊥).§.§ Generators Shafiei determined sets of minimal generators for _n^⊥ and _n^⊥:<cit.>The apolar ideal _n^⊥ is minimally generated by the following polynomials:X_i,j^2, for i,j = 1, …, n; X_i,j X_i,k, for i,j,k = 1, …, n, j ≠ k; X_i,j X_k,j, for i,j,k = 1, …, n, i ≠ k; and X_i,j X_k,l + X_i,l X_k,j, for i,j,k,l = 1, …, n and i ≠ k, j ≠ l.The apolar ideal _n^⊥ is minimally generated by the following polynomials:X_i,j^2, for i,j = 1, …, n; X_i,j X_i,k, for i,j,k = 1, …, n, j ≠ k; X_i,j X_k,j, for i,j,k = 1, …, n, i ≠ k; and X_i,j X_k,l - X_i,l X_k,j, for i,j,k,l = 1, …, n and i ≠ k, j ≠ l.In particular, both ideals are generated byβ_1,2= n+12^2 quadrics, and all other graded Betti numbers β_1,j for j ≠ 2 are zero.§.§ RelationsThe main goal of this paper is to describe F_2 and F_2', the relations between these generators. Elements in F_1 = ⊕_j S(-j)^β_2,j and F_1'may be thought of as formal S-linear combinations of the generators of _n^⊥ or _n^⊥, and we will write them as such: e.g.X_1,2 (X_2,1^2) + X_1,1 (X_2,1 X_2,2) - X_2,1 (X_1,1 X_2,2 + X_1,2 X_2,1)If the S-coefficients all have degree 1, we call the relation linear, and if the S-coefficients have degree 2, it is quadratic. §.§ Gradings There are three gradings of S = [X_i,j] that will be important in this paper: * standard grading: This is the grading by ℤ where each X_i,j has degree 1. * multigrading:This is the grading by ℤ^n ×ℤ^n, where X_i,j has degree e_i + f_j, i,j ∈{1, …, n}, where e_i and f_j are the standard basis elements of each copy of ℤ^n. * monomial grading: This is the grading by ℤ^n × n where X_i,j has degree e_i,j.We will sometimes write monomial degrees in the form of a matrix, in the obvious way, by writing the coefficient of e_i,j in position (i,j). Each of these gradings is strictly finer than the last: any element that is homogeneous with respect to monomial degree is homogeneous with respect to multidegree, and similarly for multidegree and standard degree. For example, the polynomial X_1,1^2 X_1,2 X_2,2 in [X_1,1, …, X_2,2] has standard degree 4, multidegree 3e_1 + e_2 + 2f_1 + 2f_2 = ( (3,1), (2,2) ), and monomial degree 2e_1,1 + e_1,2 + e_2,2 or equivalently [ 2 1; 1 ] in matrix notation.We can extend these gradings to F_1 and F_1': the degree of an element f · (g), where f ∈ S and g is a generator of _n^⊥ or _n^⊥, is the sum of the degrees of f and g. Note that “linear” elements of F_1 and F_1' actually have standard degree 2+1 = 3 in this sense, and “quadratic” elements have standard degree 4, since all generators g of _n^⊥ and _n^⊥ have standard degree 2.Our proof of <Ref> is divided into two cases according to the following definition:If a multidegree is a tuple consisting only of 0's and 1's, we will call it singular; otherwise, it is plural.§.§ Symmetries The symmetries of the determinant and permanent will play an important role in this paper.The determinant is invariant (up to scaling) under multiplying the n × n matrixof indeterminates on the left and right by any two n × n matrices and under transposing.It is a theorem of Frobenius <cit.> that these are all the symmetries.That is,if we consider M_n() = VW where V and W are n-dimensional vectors spaces over , then (M_n()) acts on the vector space R_n = ^n M_n()^∨ and the stabilizer of _n viewed as an element in the projective space (R_n) isG__n = ((V) ×(W))/^* ⋊/2,where ^* ⊂(V) ×(W) is the subgroup consisting of pairs (α I_n, α^-1 I_n) for α∈^* (and where I_n denotes the identity matrix).An element (A,B) in the first factor of G__n acts on M_n() via M ↦ AMB^⊤ and the non-identity element in the second factor acts via transposition M ↦ M^⊤.Similarly, the permanent is invariant (up to scaling) under transposing, permuting the rows and columns, and multiplying the rows and columns by non-zero scalars — these are all of the symmetries <cit.>.That is, if we let T_V ⊂(V) (resp. T_W ⊂(W)) be the subgroup of diagonal matrices and N(T_V) (resp., N(T_W)) be its normalizer, then the stabilizer of _n ∈(R_n) isG__n = (N(T_V) × N(T_W))/^* ⋊/2,where elements act in a similar fashion to the determinant.The symmetries of transposing and permuting rows and columns will be particularly important to us; we think of the latter of these as a group action of _n ×_n.Observe that Shafiei's lists of generators for _n^⊥ and _n^⊥ (<ref>) are setwise invariant under transposing as well as under permuting rows and columns, and also that every generator is homogeneous with respect to the standard grading and multigrading. These properties are also inherited by the relations.The symmetries allow us to write the multidegrees and monomial degrees of syzygies more concisely. Since the space of syzygies is symmetric under permuting rows and columns, the order of the entries in the two n-tuples comprising a multidegree is somewhat irrelevant. We can therefore think of the multidegree of a syzygy as a pair of partitions of the integer m, where m is the syzygy's standard degree.For instance, the multidegree of X_1,1^2 X_1,2 X_2,2 corresponds to the pair of partitions (3+1,2+2).§.§ Outline of the proof of <ref> In <ref>, we show that all relations of _n^⊥ of singular multidegree are generated by linear relations (<ref>).To establish this, we identify relations of a fixed singular multidegree with certain -labelings of the Cayley graph of the symmetric group _m and then study the combinatorics of the Cayley graph. In <ref>, we show that all relations _n^⊥ of plural multidegree are generated by linear relations.This allows us finish the proof of <ref><ref>.In <ref>, we perform the necessary adjustments to <ref> to characterize the module of relations of _n^⊥ and thus establishing <ref><ref>.Unlike for the determinant, both linear and quadratic relations are needed to generate all relations of the permanent. § SINGULAR MULTIDEGREE CASEIn this section, we begin our investigation of the space of relations of the minimal quadratic generators of _n^⊥ as listed in <ref> by focusing on relations of singular multidegree.The main result is <ref> which asserts that all relations of singular multidegree are generated by linear relations.This theorem is established as follows.First, we identify monomials in S = [X_i,j] of standard degree m and of fixed multidegree with permutations in _m; see <ref>.Using the Cayley graph Γ(_m) of the symmetric group _m, we identify the space of relations of standard degree m and of fixed multidegree with the space of certain -labelings called zero-magic labelings (see <Ref>) on Γ(_m) (<ref>).By a result of Doob (<ref>), the space of zero-magic labelings is spanned by cycle labelings (see <ref>).Finally, by studying the combinatorics of the Cayley graph, we show that any cycle labeling of Γ(_m) is the sum of certain commutator labelings corresponding to linear relations (<ref>).§.§ Singular multidegree and permutations Recall from <Ref> that a multidegree is singular if it is a tuple of only 0s and 1s. Firstly, observe that the generators (X_i,j^2), (X_i,j X_i,k) and (X_i,j X_k,j) each contribute 2 to the multidegree in some row or column, so elements with singular multidegree can only involve the generator (X_i,j X_k,l + X_i,l X_k,j). Secondly, if a monomial in S of standard degree m has singular multidegree e_i_1 + ⋯ + e_i_m + f_j_1 + ⋯ + f_j_m, then the monomial necessarily has the form X_i_1, j_σ(1)⋯ X_i_m, j_σ(m) for some permutation σ∈_m.In other words, if we fix a singular multidegree with standard degree m, there is a one-to-one correspondence between monomials in S with this multidegree (up to scaling) and elements of the symmetric group _m.§.§ Cayley graphsTo progress further with this train of thought, we must first define a certain graph. The Cayley graph of a group A together with a set of generators G is the directed graph whose vertex set is A, with a directed edge from a to a g for each a ∈ A and each generator g ∈ G.Any Cayley graph is connected. Indeed, for vertices a_1, a_2 ∈ A, the element a_1^-1 a_2 must be expressible as g_1 … g_r for some g_1, …, g_r ∈ G, since G is a generating set. Therefore g_1, …, g_r describe a path between a_1 and a_1 (a_1^-1 a_2) = a_2.The symmetric group _m is generated by the set of transpositions, that is, the permutations of the form (ij) for distinct i,j ∈{1, …, m}. Therefore we may construct the Cayley graph of _m with this generating set. In this case, since every transposition is its own inverse, each directed edge between vertices a and a' given by a transposition τ has a matching edge from a' to a given by τ^-1 = τ, so we will instead consider the simpler undirected Cayley graph of _m made by merging each of these pairs of edges into a single, undirected edge. Call this graph Γ(_m). The graph Γ(_m) is bipartite. The vertices with odd and even sign as permutations form a bipartition, since every edge necessarily connects a vertex with odd sign to one with even sign. Let us now link this back to the apolar ideal. For a fixed singular multidegree μ of standard degree m, we already saw that the set of monomials (up to scaling) with multidegree μ is in bijection with _m. Consider an element of the form f · (X_i,j X_k,l + X_i,l X_k,j) ∈ F_1 with multidegree μ where f ∈ S is a monomial. Let σ_1, σ_2 ∈_m be the permutations corresponding to the monomials f X_i,j X_k,l∈ S and f X_i,l X_k,j∈ S under the above bijection. Moreover, the monomial degrees of f X_i,j X_k,l and f X_i,l X_k,j differ by a transposition, namely (jl).Thus we associate f · (X_i,j X_k,l + X_i,l X_k,j) to the edge (jl) betweenσ_1 andσ_2 in the Cayley graph Γ(_m).To specify an element of F_1 of multidegree μ, we must specify only the S-coefficients of generators of the form (X_i,j X_k,l + X_i,l X_k,j) or, equivalently, the -coefficients of terms of the form f · (X_i,j X_k,l + X_i,l X_k,j) where f ∈ S is a monomial.This in turn precisely corresponds to a -labeling of the edges of Γ(_m).The condition that the element is a relation, i.e., an element of (F_1F_0), is that for each monomial in S, the sum of the coefficients of the terms in F_1 involving it in the relation is zero.In the Cayley graph interpretation, this equivalently means that at every vertex v, the sum of the labels of the edges that meet v is zero.This last condition on a graph is important enough to warrant its own definition: A -edge-labeling of a graph is called zero-magic if for every vertex, the sum of the labels of the edges meeting this vertex is zero. We give the set of edge labelings of Γ(𝐒_m) a -vector space structure in the obvious way, by making addition and scalar multiplication act edgewise. In this way, the set of zero-magic labelings forms a subspace.To summarize the above discussion, we have:For a singular multidegree μ of standard degree m, the space of relations of multidegree μ is isomorphic to the vector space of zero-magic labelings of Γ(_m). Given a bipartite graph G and a cycle C ⊆ G with a distinguished edge e, we define the cycle labeling Λ(C, e) to be the -labeling where every edge outside C is labeled 0, and the edges along C are given the alternating labels 1 and -1, starting with the label 1 for e. Since G is bipartite, all cycles have even length, so this definition makes sense.Also, we allow a cycle to travel along the same edge multiple times; if this occurs, the labels of such an edge are added together.Every cycle labeling is a zero-magic labeling. However, Doob established a far stronger result:<cit.>Let G be a connected bipartite graph. The vector space of zero-magic labelings of G is spanned by the cycle labelings.Moreover, if T is a spanning tree of G, adding any single edge from E(G) ∖ E(T) to T must introduce a cycle; if we pick one cycle C_e ⊆ T ∪{e} for each edge e ∈ E(G) ∖ E(T), then the cycle labelings Λ(C_e, e) form a basis for the vector space of zero-magic labelings. In particular, the dimension of the vector space of zero-magic labelings is | E(G) ∖ E(T) |, which is often called the circuit rank of G.§.§ Commutator cycles<Ref> allows us to restrict our attention to only those relations of singular multidegree that correspond to cycle labelings in Γ(_m). These have a simple description in terms of permutations: a cycle in Γ(_m) may be specified by a starting vertex and a sequence of transpositions in _m that compose to give the identity permutation. Since Cayley graphs are clearly vertex-transitive, any sequence of transpositions that compose to the identity may define a cycle at any starting vertex, so we will often ignore the datum of the starting point.We now define a specific class of cycles in Γ(_m). Let i,j,k ∈{1, …, m} be distinct. We have the following composition of transpositions:(ij) (ik) (ij) (jk) = (1).This sequence of transpositions defines a length-4 cycle in Γ(_m), given a starting vertex. A cycle of this form is called a commutator cycle, and a cycle labeling built on this cycle is called a commutator labeling. Note that a length-4 cycle in Γ(_m) has a choice of four starting points and two directions, so(ik) (ij) (jk) (ij) = (1)is also a commutator cycle. (The other six choices of starting point and direction can be made from these two sequences by interchanging i, j and k.) More abstractly, commutator cycles can be writtena b a [aba] or a b [bab] bfor transpositions a and b that are distinct but not disjoint, where [aba] means the single transposition that is b conjugated by a.The prototypical commutator cycle is the cycle specified by the sequence(12) (13) (12) (23)starting at the vertex (1). In the correspondence between zero-magic labelings and relations, this corresponds to the linear relationρ_S = X_3,3 (X_1,1 X_2,2 + X_1,2 X_2,1) - X_1,2 (X_2,1 X_3,3 + X_2,3 X_3,1) + X_2,3 (X_1,1 X_3,2 + X_1,2 X_3,1) - X_1,1 (X_2,2 X_3,3 + X_2,3 X_3,2)when m = 3, and a monomial times this when m > 3. Note that for a fixed n and singular multidegree, any commutator cycle of standard degree m can be obtained from this cycle by renaming the indices, that is, by permuting rows and columns. All cycle labelings on cycles of length 4 in Γ(_m) are sums of commutator labelings, or zero.A cycle of length 4 corresponds to a sequence of 4 transpositions a b c d that composes to the identity. This means that cd = (ab)^-1.There are three possibilities for a and b: * a = b. In this case we must have c = d, so the cycle labeling gives each of the edges a = b and c = d the label 1 - 1 = 0, so this is the zero labeling. * a and b are distinct but not disjoint. Then a = (ij), b = (ik), for some i,j,k. Thus ab = (ikj), so cd = (ijk); this means one of the following: * c = (ij) and d = (jk). Then abcd = (ij) (ik) (ij) (jk), a commutator cycle of the first type.* c = (jk) and d = (ik). Then abcd = (ij) (ik) (jk) (ik), which is a commutator cycle of the second type.* c = (ik) and d = (ij). Then abcd = (ij) (ik) (ik) (ij), so the cycle goes along the same two edges forwards then backwards. Thus the labeling cancels to zero on both edges, so it is the zero labeling.* a and b are disjoint. Say a = (ij) and b = (kl). Then either c = (kl) and d = (ij), in which case the cycle goes along the same edges forwards and backwards and thus cancels to zero; or c = (ij) and d = (kl), so abcd = (ij) (kl) (ij) (kl). Consider the following subgraph of Γ(_m):(1) @-[rrr]^(ij)@-[ddd]_(kl)@-[rd]_(jk)(ij)@-[ddd]^(kl)@-[ld]^(ik) (jk) @-[r]^(ij)@-[d]^(jl)(ikj) @-[d]^(i l) (jlk) @-[r]^(ij)(ilkj) (kl) @-[rrr]_(ij)@-[ru]^(jk)(ij)(kl) @-[ul]_(ik) The outer square is abcd, but each of the five inner squares is a commutator cycle. By choosing signs appropriately, we may add together the labelings given by these five inner commutator cycles so that the labels on the internal edges cancel, leaving only the cycle labeling of the outer square. Hence the cycle labeling for (ij) (kl) (ij) (kl) is a sum of commutator labelings. (We assumed that abcd started at the vertex (1) in this diagram, but vertex-transitivity means that we equally could have started at any vertex.) §.§ Commutation rulesA consequence of this proposition is that we obtain commutation rules for transpositions. Suppose we have some loop containing adjacent edges a and b. If a = b, the labels on this edge sum to zero, so we can effectively cancel these two transpositions with each other. If a and b are distinct but not disjoint, then we may commute the edges following the rule ab = [aba] a = b [bab] in the cycle, by adding the commutator labelings given by the cycles aba[aba] or ab[bab]b, with the right choice of sign:[vertex/.style=circle, inner sep=1.8pt, fill, auto, on grid](left dots) …;[vertex] (sigma) [right=of left dots] ;[vertex] (a sigma) [above right=of sigma] ;[vertex] (ab sigma) [below right=of a sigma] ;[vertex] (aba sigma) [below right=of sigma] ;(right dots) [right=of ab sigma] …;(left dots) to (sigma);(sigma) to node a (a sigma) to node b (ab sigma);(ab sigma) to (right dots);[dashed] (sigma) to node [swap] aba (aba sigma) to node [swap] a (ab sigma);[below=of a sigma] ↓; [vertex/.style=circle, inner sep=1.8pt, fill, auto, on grid](left dots) …;[vertex] (sigma) [right=of left dots] ;[vertex] (a sigma) [above right=of sigma] ;[vertex] (ab sigma) [below right=of a sigma] ;[vertex] (b sigma) [below right=of sigma] ;(right dots) [right=of ab sigma] …;(left dots) to (sigma);(sigma) to node a (a sigma) to node b (ab sigma);(ab sigma) to (right dots);[dashed] (sigma) to node [swap] b (b sigma) to node [swap] bab (ab sigma);[below=of a sigma] ↓; And if a and b are disjoint, then we can commute ab = ba by adding on the labeling of the length-4 cycle abab, which we just saw could be built from commutator cycles:[vertex/.style=circle, inner sep=1.8pt, fill, auto, on grid](left dots) …;[vertex] (sigma) [right=of left dots] ;[vertex] (a sigma) [above right=of sigma] ;[vertex] (ab sigma) [below right=of a sigma] ;[vertex] (b sigma) [below right=of sigma] ;(right dots) [right=of ab sigma] …;(left dots) to (sigma);(sigma) to node a (a sigma) to node b (ab sigma);(ab sigma) to (right dots);[dashed] (sigma) to node [swap] b (b sigma) to node [swap] a (ab sigma);[below=of a sigma] ↓; We finally reach the following proposition:Any cycle labeling in Γ(_m) is the sum of commutator labelings, or zero.We need to show that any sequence of transpositions in _m that composes to the identity can be reduced to the identity using the commutation rules introduced above. We proceed by induction on m. Observe that when m = 2, Γ(_2) only has one edge, between (1) and (12), so all cycles simply travel along this edge backwards and forwards, and all cycle labelings cancel to 0.Now, let m > 2. Suppose we have some sequenceτ_1 τ_2 …τ_s = (1)of transpositions in _m. We describe how to reduce this to a sequence whose transpositions are all contained in the subgroup isomorphic to _m-1 of permutations that fix m. Every transposition either fixes m or moves it. If all transpositions in our sequence fix m, we are done, so assume that some transpositions move m, and consider the left-most one of these: suppose it is τ_r = (im). We want to commute this towards the right. If τ_r and τ_r+1 are disjoint, use the rule(im) (jk) = (jk) (im).If τ_r and τ_r+1 are distinct but not disjoint and τ_r+1 fixes m, use the rule(im) (ij) = (ij) (jm).If τ_r and τ_r+1 are distinct but not disjoint and τ_r+1 moves m, use the rule(im) (jm) = (ij) (im).If τ_r and τ_r+1 are equal, they cancel:(im) (im) = (1).After applying any of these rules, the left-most m-moving transposition is strictly closer to the right-hand end of the sequence. Therefore if we repeat this process, we may continue for no more than s-1 steps, until either there are no more m-moving transpositions, or there is a single one and it is at the very right-hand end of the string. But this latter case is impossible: a sequence of the formτ_1 …τ_s'-1τ_s' = (1)where τ_1, …, τ_s'-1 fix m, but τ_s' moves m, must as a whole move m, so it cannot equal (1), and we have a contradiction. Therefore our algorithm must produce a sequence of transpositions where none involve m; that is, a string contained in _m-1. But by the induction hypothesis, every loop in _m-1 can be reduced to the identity by these commutator rules. All relations of singular multidegree are generated by linear relations and specifically by the orbit of ρ_S (introduced in (<ref>)), under the symmetry action of permuting rows and columns. By <ref>, the vector space of relations of a fixed singular multidegree is isomorphic to the space of zero-magic labelings of Γ(_m). By <ref>, the latter space is spanned by the cycle labelings, and by <ref>, every cycle labeling is a sum of commutator labelings. But every commutator labeling is an S-multiple of an element in the orbit of ρ_S under _n ×_n. The vector space of linear relations of a fixed singular multidegree has dimension 4.By <ref>, the vector space of linear relations of a fixed singular multidegree is isomorphic to the space of zero-magic labelings of Γ(_3). By <ref>, the dimension of this space is the circuit rank of Γ(_3). But Γ(_3) is isomorphic to the complete bipartite graph K_3,3: it is a bipartite graph with | _3 | = 6 vertices, and each vertex has degree 3, since there are 32 = 3 transpositions in _3. The circuit rank of K_3,3 is 4. (The graph Γ(_3) is shown in <ref>, with a spanning tree highlighted.) § PLURAL MULTIDEGREE CASEIn this section, we prove <ref><ref> asserting that all relations of the apolar ideal _n^⊥ are generated by linear relations as long as ≠ 2.Moreover, in <ref>, a minimal generating set of linear relations is given.In this section, we find it useful to describe relations using a `dots-and-boxes' notation as detailed in <ref>.After determining a basis of the linear relations in <ref>, we turn our attention to relations of standard degree ≥ 2 and plural multidegree, since relations of singular multidegree were fully investigated in the previous section.We first show that quadratic relations of plural multidegree are generated by linear relations (<ref>) and then establish that all relations of plural multidegree are generated by linear relations (<ref>). §.§ Dots-and-boxes notationIt will be helpful to develop a shorthand notation to express the relations among the generators of _n^⊥ listed in<ref>. Recall the free resolution of S/_n^⊥ from (<ref>) and that elements of F_1 are formal S-linear sums of the generators of _n^⊥. We can display this information pictorially in a matrix by showing the generator as a rectangle whose corners are at the locations of the variables that comprise it, and denoting monomials in S with dots in the positions corresponding to the variables (or numbers instead of dots, to indicate multiplicity greater than 1). We will only display the minimal submatrix in which all the variables appear. For example:X_2,1 (X_1,2 X_2,2) = 22122221X_1,2 X_3,3 (X_1,1 X_2,2 + X_1,2 X_2,1) = 3311221233X_2,2 X_2,3 (X_1,1^2) - X_1,1^2 (X_2,2 X_2,3) = 2311112223 - 232223[2]11. §.§ Linear relations The space of linear relations is generated by the orbit of the following six relations under the symmetry of permuting rows and columns and transposing:ρ_1= X_1,2 (X_1,1^2) - X_1,1 (X_1,1 X_1,2)ρ_2= X_1,3 (X_1,1 X_1,2) - X_1,2 (X_1,1 X_1,3)ρ_3= X_2,1 (X_1,1 X_1,2) - X_1,2 (X_1,1 X_2,1)ρ_4= X_1,1 (X_1,1 X_2,2 + X_1,2 X_2,1) - X_2,1 (X_1,1 X_1,2) - X_2,2 (X_1,1^2)ρ_5= X_1,3 (X_1,1 X_2,2 + X_1,2 X_2,1) - X_2,2 (X_1,1 X_1,3) - X_2,1 (X_1,2 X_1,3)ρ_S= [t] X_3,3 (X_1,1 X_2,2 + X_1,2 X_2,1) - X_1,2 (X_2,1 X_3,3 + X_2,3 X_3,1) + X_2,3 (X_1,1 X_3,2 + X_1,2 X_3,1) - X_1,1 (X_2,2 X_3,3 + X_2,3 X_3,2).In dots-and-boxes notation, these relations areρ_1= 12111112 - 12111211 ρ_2= 13111213 - 13111312 ρ_3= 22111221 - 22112112 ρ_4= 22112211 - 22111221 - 22111122 ρ_5= 23112213 - 23111322 - 23121321 ρ_S= 33112233 - 33213312 + 33113223 - 33223311.To list the linear relations, we will split F_1 into multidegree-homogeneous components. Recall from <ref> that we may specify a degree m multidegree up to symmetry by giving a pair of partitions of m. Linear relations have m = 3, and there are 3 partitions of 3 (namely 3, 2+1 and 1+1+1); therefore, there are 3^2 = 9 multidegrees to consider:(3,3) (3,2+1) (3,1+1+1) (2+1,3) (2+1,2+1) (2+1,1+1+1) (1+1+1,3) (1+1+1,2+1) (1+1+1,1+1+1)The multidegree (1+1+1,1+1+1) is singular — we have seen that that the space of relations of this multidegree are generated by the orbit of ρ_S (<ref>) and its dimension is 4 (<ref>).Furthermore, three of the multidegrees above can be obtained from the others by transposing. This leaves five cases left to consider, up to symmetry:(3,3) (3,2+1) (3,1+1+1)[(2+1,3)][c]-(2+1,2+1) (2+1,1+1+1)[(1+1+1,3)][c]- [(1+1+1,2+1)][c]- [(1+1+1,1+1+1)][c]- We will identify a basis for the vector space of linear relations by considering these multidegrees case by case. For concreteness, we will write the variables as, say, X_1,2 instead of X_i,j, and generalize by symmetry. * (3,3). The only term with this multidegree (up to symmetry) is X_1,1 (X_1,1^2), which is the very compact picture 11111111 in the dots-and-boxes notation. Since there is only one possible term, and its image in F_0 is the non-zero monomial X_1,1^3, a relation of this multidegree must be the zero relation. * (3,2+1). There are two possible terms with this multidegree (up to symmetry): X_1,2 (X_1,1^2) = 12111112 and X_1,1 (X_1,1 X_1,2) = 12111211. Both of these get mapped to X_1,1^2 X_1,2 in F_0. Therefore all relations of this multidegree are a scalar multiple of the relationρ_1 = X_1,2 (X_1,1^2) - X_1,1 (X_1,1 X_1,2) = 12111112 - 12111211.It will be useful later to know the dimensions of these spaces of relations, so note that this relation spans a vector space with dimension 1. * (3,1+1+1). There are three terms with this multidegree:X_1,3 (X_1,1 X_1,2) = 13111213, X_1,2 (X_1,1 X_1,3) = 13111312,andX_1,1 (X_1,2 X_1,3) = 13121311.These all map to X_1,1 X_1,2 X_1,3∈ F_0. The relations among them are spanned byρ_2 = X_1,3 (X_1,1 X_1,2) - X_1,2 (X_1,1 X_1,3) = 13111213 - 13111312, X_1,2 (X_1,1 X_1,3) - X_1,1 (X_1,2 X_1,3) = 13111312 - 13121311,andX_1,1 (X_1,2 X_1,3) - X_1,3 (X_1,1 X_1,2) = 13121311 - 13111213.Note that the second and third relations are permutations of the first, ρ_2.These relations are linearly dependent since they sum to 0, but no single one of them generates the entire space by itself, so this space of relations has dimension 2. * (2+1,2+1). There are four terms with this multidegree:X_2,2 (X_1,1^2) = 22111122, X_2,1 (X_1,1 X_1,2) = 22111221, X_1,2 (X_1,1 X_2,1) = 22112112, X_1,1 (X_1,1 X_2,2 + X_1,2 X_2,1) = 22112211.Note that the first maps to X_1,1^2 X_2,2 in F_0, the second and third map to X_1,1 X_1,2 X_2,1, and the fourth maps to the sum of these two polynomials. Thus for a linear combination of these to be a relation, if the -coefficient of X_1,1 (X_1,1 X_2,2 + X_1,2 X_2,1) is c, the coefficient of X_2,2 (X_1,1^2) must be -c, and the coefficients of X_2,1 (X_1,1 X_1,2) and X_1,2 (X_1,1 X_2,1) must sum to -c. Therefore the relations of this multidegree are spanned byρ_3 = X_2,1 (X_1,1 X_1,2) - X_1,2 (X_1,1 X_2,1) = 22111221 - 22112112 andρ_4 = X_1,1 (X_1,1 X_2,2 + X_1,2 X_2,1) - X_2,1 (X_1,1 X_1,2) - X_2,2 (X_1,1^2) = 22112211 - 22111221 - 22111122.The space of relations of this multidegree has dimension 2. * (2+1,1+1+1). There are six terms with this multidegree:X_2,3 (X_1,1 X_1,2) = 23111223↦ X_1,1 X_1,2 X_2,3 = A, X_2,2 (X_1,1 X_1,3) = 23111322↦ X_1,1 X_1,3 X_2,2 = B, X_2,1 (X_1,2 X_1,3) = 23121321↦ X_1,2 X_1,3 X_2,1 = C, X_1,3 (X_1,1 X_2,2 + X_1,2 X_2,1) = 23112213↦ B + C, X_1,2 (X_1,1 X_2,3 + X_1,3 X_2,1) = 23112312↦ A + C, X_1,1 (X_1,2 X_2,3 + X_1,3 X_2,2) = 23122311↦ A + B.If the -coefficient of the ith term is c_i in a relation, then by comparing terms in F_0, we have that c_1+c_5+c_6=0, c_2+c_4+c_6=0, and c_3+c_4+c_5=0.We see that the relations are spanned byρ_5 = X_1,3 (X_1,1 X_2,2 + X_1,2 X_2,1) - X_2,2 (X_1,1 X_1,3) - X_2,1 (X_1,2 X_1,3) = 23112213 - 23111322 - 23121321, X_1,2 (X_1,1 X_2,3 + X_1,3 X_2,1) - X_2,3 (X_1,1 X_1,2) - X_2,1 (X_1,2 X_1,3) = 23112312 - 23111223 - 23121321, X_1,1 (X_1,2 X_2,3 + X_1,3 X_2,2) - X_2,3 (X_1,1 X_1,2) - X_2,2 (X_1,1 X_1,3) = 23122311 - 23111223 - 23111322.The second and third of these are permutations of ρ_5. The space of relations of this multidegree has dimension 3. This covers all multidegrees possible for a linear relation up to symmetry, so the set of permutations and transposes of ρ_1, …, ρ_5, ρ_S generates all linear relations.The dimension of the vector space of linear relations isβ_2,3 = 4 n+13n+23 = 1/9 n^2 ( n+1 )^2 ( n-1 ) ( n+2 ). The space of all linear relations is the direct sum of the spaces of linear relations of fixed multidegree. We calculated the dimension of these spaces for each multidegree in <ref> and the proof of <ref>, so it only remains to add these together, with multiplicity. These dimensions are reprinted in <ref> for reference.Given an n × n matrix, we therefore need to know how many ways there are of picking a multidegree corresponding to each partition pair (p, p'), for p,p' ∈{3,2+1,1+1+1}. Note that we may allocate the rows and columns independently, so we just need to know how many n-tuples of non-negative integers have non-zero entries corresponding to each partition p. There are n1 = n ways to get the partition 3, one for each tuple (0, …, 0, 3, 0, …, 0); there are n1n-11 = n (n-1) ways to get 2+1, first picking a position for the 2 and then a position for the 1; and there are n3 ways to get 1+1+1.Therefore the dimension of the space of all linear relations is0 n^2 + 1 n^2 (n-1) + 2 n n3 + 1 n^2 (n-1) + 2 n^2 (n-1)^2 + 3 n (n-1) n3+ 2 n n3 + 3 n (n-1) n3 + 4 n3^2= 1/9 n^2 (n+1)^2 (n-1) (n+2).§.§ Relations of higher degree Let L ⊂ F_1 be the submodule generated by the linear relations; by <ref>, we know that L is generated by the permutations and transposes of ρ_1, …, ρ_5, ρ_S. We say that elements h_1 and h_2 are “equivalent modulo L”, or that “h_1 ≡ h_2 modulo L”, if h_1-h_2 ∈ L.We call the three classes of generators given by X_i,j^2, X_i,j X_i,k and X_i,j X_k,j the monomial generators, to distinguish them from X_i,j X_k,l + X_i,l X_k,j.The first step is to show that it suffices to consider relations of plural multidegree whose terms only involve the monomial generators. Consider an element of the form f (g)∈ F_1, where f ∈ S and g is a generator from <ref>.If f(g) ∈ F_1 is multi-homogeneous of plural multidegree, then it is equivalent modulo L to an S-linear combination of monomial generators.This statement is trivial if g is a monomial generator, so assume that g=X_i,j X_k,l + X_i,l X_k,j.Since the multidegree of f(g) is plural, at least one of the rows or columns in the dots-and-boxes notation has either two dots or a dot and a corner of the rectangle. In other words, up to symmetry, the element f(g) ∈ F_1 must be divisible by one of22112211, 23112213, 331122[2]33or3411223334.It therefore suffices to show that each of these elements is equivalent to an S-linear combination of monomial generators modulo L.The first two of these are straightforward: ρ_4 tells us that 22112211≡22111221 + 22111122 modulo L, and ρ_5 says that 23112213≡23111322 + 23121321 modulo L.For the third case, observe that331122[2]33 - 3333331122 - 3333331221 = ( 3322331133 - 3333331122 - 3332331123) + ( 3321331233 - 3333331221 - 3331331223)- ( 3311322333 - 3332331123 - 3331331223) - ( 3322331133 + 3321331233 - 3311322333 - 331122[2]33)where each summand on the right hand side is in L: the first three are each permutations of ρ_4 times some X_i,j, and the fourth summand is ρ_S times X_3,3. Therefore 331122[2]33≡3333331122 + 3333331221 modulo L.In the fourth case, we have3411223334 - 3433341122 - 3433341221 = ( 3422331134 - 3433341122 - 3432341123) + ( 3421331234 - 3433341221 - 3431341223) - ( 3411322334 - 3432341123 - 3431341223) - ( 3422331134 + 3421331234 - 3411322334 - 3411223334)where each summand on the right hand side is in L — the first three are permutations of ρ_5 times some X_i,j, and the fourth is ρ_S times X_3,4 — so 3411223334≡3433341122 + 3433341221 modulo L.The first step in considering higher dimension relations is to examine the quadratic ones.If () ≠ 2, then all quadratic relations of plural multidegree are generated by the linear relations.Once again we work case by case; however, by <ref>, it suffices to consider relations only involving the monomial generators, which allows us to use the finer monomial grading instead of multigrading.To establish this result, it suffices to show that all elements in F_1 of the form f(g) with the same monomial degree, where f is a monomial and g is a monomial quadratic generator, are equivalent modulo L.We will classify the monomial degrees by the multidegrees they are a refinement of.We will use the matrix description to describe monomial degrees as introduced in <ref>.* (4,4). There is only one monomial degree possible with this multidegree (up to symmetry), namely [ 4 ] (displayed in matrix form), and the only possible term with this degree is X_1,1^2 (X_1,1^2), i.e.,111111[2]11 in the dots-and-boxes notation. * (4,3+1). The only monomial degree of this multidegree is [ 3 1 ]. The two possible terms with this degree are 1211111112 and 121112[2]11, which are equivalent by ρ_1. * (4,2+2). The only monomial degree is [ 2 2 ]. We have121111[2]12 ≡1211121112(by ρ_1)≡121212[2]11(by ρ_1)and these are all the terms with this monomial degree. * (4,2+1+1). The only monomial degree is [ 2 1 1 ]. We have1311111213 ≡1311121113(by ρ_1)≡1311131112(by ρ_2)≡131213[2]11(by ρ_2)and these are all the terms with this monomial degree. * (4,1+1+1+1). The only monomial degree is [ 1 1 1 1 ]. The terms with this monomial degree are the permutations of 1411121314. Any two of these permutations that have a 11 in the same position are equivalent by ρ_2, and any of the permutations that don't share a 11 are each mutually equivalent to another of the permutations by the same rule: e.g.1411121314≡1412131114≡1413141112* (3+1,3+1). The two monomial degrees for this multidegree are * [ 3; 1 ]: in this case, the only monomial term is 2211111122.* [ 2 1; 1 ]: we have2211121121 ≡2211111221≡2211211112(by ρ_1) * (3+1,2+2). The only monomial degree up to symmetry here is [ 2 1; 1 ]. We have2211111222 ≡2211121122(by ρ_1)≡221222[2]11(by ρ_3)* (3+1,2+1+1). The monomial degrees up to symmetry are: * [ 2 1; 1 ]: here we have2311111223 ≡2311121123(by ρ_1) * [ 1 1 1; 1 ]: we have2311211213 ≡2311122113(by ρ_3)≡2311132112≡2312132111(by ρ_2) * (3+1,1+1+1+1). Up to symmetry the only monomial degree is [ 1 1 1; 1 ]. But2411121324 ≡2411131224≡2412131124(by ρ_2)* (2+2,2+2). The monomial degrees are * [ 2; 2 ]: by adding relations from L, specifically permutations of ρ_4 and ρ_3, we can obtain( 2211221122 - 2212221121 - 222222[2]11) - ( 2211221122 - 2211122122 - 221111[2]22) + ( 2212221121 - 2211122122) = 221111[2]22 - 222222[2]11and thus 221111[2]22≡222222[2]11 modulo L. These are the only terms possible with this monomial degree.* [ 1 1; 1 1 ]: we have2221221112 ≡2212221121≡2211122122≡2211211222(by ρ_3) * (2+2,2+1+1). The possible monomial degrees are: * [ 2; 1 1 ]: we can use ρ_5, ρ_4 and ρ_3 to write:( 2311221123 - 2321231112 - 232223[2]11) - ( 2311221123 - 2311112223 - 2311211223) + ( 2321231112 - 2311211223) = 2311112223 - 232223[2]11so 2311112223≡232223[2]11 modulo L, and these are the only terms with this monomial degree.* [ 1 1; 1 1 ]: we have2311122123 ≡2311211223≡2321231112(by ρ_3) * (2+2,1+1+1+1). The only monomial degree up to symmetry is [ 1 1; 1 1 ]. We can form the following sum using ρ_5:( 2411241223 - 2411122324 - 2411132224) - ( 2411241223 - 2423241112 - 2422241113) + ( 2413241122 - 2421241213 - 2422241113) - ( 2413241122 - 2412132124 - 2411132224) + ( 2412241123 -2412132124 - 2411122324) - ( 2412241123 - 2421241213 - 2423241112) = 2 ( 2423241112 - 2411122324)and since≠ 2, we must have 2423241112≡2411122324 modulo L. (This is the only place in the argument where the hypothesis that ≠ 2 is used.) * (2+1+1,2+1+1). The possible monomial degrees here are * [ 2; 1; 1 ]: in this case, the only possible term is 3311112233.* [ 1 1; 1; 1 ]: we have3311122133 ≡3311211233(by ρ_3) * [ 1 1; 1; 1 ]: using ρ_5 and ρ_3 we can write( 3311221331 - 3311311322 - 3321311213) - ( 3311221331 - 3311132231 - 3312132131) + ( 3311311322 - 3311132231) = 3312132131 - 3321311213so 3312132131≡3321311213 modulo L.* (2+1+1,1+1+1+1). Up to symmetry, the only possible monomial degree is [ 1 1; 1; 1 ], and 3411122334 is the only term with this degree.This covers all cases, up to symmetry.We now generalize this to all plural relations. If () ≠ 2, then all relations of plural multidegree are generated by the linear relations.By using <ref>, it suffices to consider relations involving monomial generators.As in the proof of <ref>, it suffices to show that any two elements h_1=f_1(g_1) and h_2=f_2(g_2) of the same monomial degree, where each f_i is a monomial and each g_i is a monomial quadratic generator, are equivalent modulo L. Since h_1 and h_2 have the same monomial degree, they involve the same variables, with multiplicity.By <ref> and <ref>, we may assume that the standard degree m of h_1 and h_2 is greater than 4. Note that exactly two variables appear in each generator, so f_1 and f_2 each involve m-2 variables. Therefore, by the pigeonhole principle, at least m-4 of the variables must appear in both f_1 and f_2. Thus we can writeh_1 - h_2 = p · (h'_1 - h'_2)where p is a degree m-4 monomial in S, and h'_1 and h'_2 are elements in F_1 which have plural multidegree and standard degree 4, and only involve monomial generators. But <ref> says that h'_1 and h'_2 are equivalent modulo L, and therefore h_1 and h_2 are equivalent modulo L.This finally establishes our main result: <Ref> covers the singular case, and <ref> covers the plural case.The dimension of the space of linear relations is computed in <ref>.If () ≠ 2, then all Betti numbers β_2,j for j ≠ 3 are zero.§ ANALOGOUS RESULTS FOR THE PERMANENTMost of the results from <ref> also apply to the apolar ideal of _n with some slight modifications. Recall from <ref> that the generators of _n^⊥ include polynomials of the form X_i,j X_k,l - X_i,l X_k,j (instead of X_i,j X_k,l + X_i,l X_k,j) as well as the monomial generators X_i,j^2, X_i,j X_i,k and X_i,j X_k,j.The main result of this section is <ref> characterizing the module of relations between these generators. §.§ Keeping track of parityIn <ref>, we associated singular multidegree terms with generator X_i,j X_k,l + X_i,l X_k,j to edges of the Cayley graph Γ(_m), but with the generator X_i,j X_k,l - X_i,l X_k,j instead, we must take care to distinguish between X_i,j X_k,l - X_i,l X_k,j and its negative, X_i,l X_k,j - X_i,j X_k,l. To ensure consistency, given a term f · (X_i,j X_k,l - X_i,l X_k,j) where f is a monomial, we choose its sign so that the monomials f X_i,j X_k,l and f X_i,l X_k,j get the same sign as the parity of the corresponding permutations of _m.With this convention, the coefficient of a monomial corresponding to an odd permutation is the negative of the sum of the weights of edges meeting the corresponding vertex of the Cayley graph. But this still gives rise to a relation if and only if these weights sum to zero at every vertex, so relations still correspond to zero-magic graphs.Thus <ref> still holds and the rest of <ref> applies intact. In particular, the relation corresponding to the prototypical commutator cycle (12) (13) (12) (23) isρ_S' = X_3,3 (X_1,1 X_2,2 - X_1,2 X_2,1) - X_1,2 (X_2,3 X_3,1 - X_2,1 X_3,3) + X_2,3 (X_1,2 X_3,1 - X_1,1 X_3,2) - X_1,1 (X_2,2 X_3,3 - X_2,3 X_3,2). This issue of parity applies to the dots-and-rectangles notation as well. To remove ambiguity, when discussing the permanent, the generator X_i,j X_k,l - X_i,l X_k,j will be shown as a rectangle with a dotted line connecting the corners from the monomial with positive sign. Thus(X_1,1 X_2,2 - X_1,2 X_2,1) = 221122(X_1,2 X_2,1 - X_1,1 X_2,2) = 221122 = - 221122§.§ Linear relationsIn <ref>, we must modify the six linear relations from <ref> to the linear relations:ρ_1'= X_1,2 (X_1,1^2) - X_1,1 (X_1,1 X_1,2)ρ_2'= X_1,3 (X_1,1 X_1,2) - X_1,2 (X_1,1 X_1,3)ρ_3'= X_2,1 (X_1,1 X_1,2) - X_1,2 (X_1,1 X_2,1)ρ_4'= X_1,1 (X_1,1 X_2,2 - X_1,2 X_2,1) + X_2,1 (X_1,1 X_1,2) - X_2,2 (X_1,1^2)ρ_5'= X_1,3 (X_1,1 X_2,2 - X_1,2 X_2,1) - X_2,2 (X_1,1 X_1,3) + X_2,1 (X_1,2 X_1,3)ρ_S'= [t] X_3,3 (X_1,1 X_2,2 - X_1,2 X_2,1) - X_1,2 (X_2,3 X_3,1 - X_2,1 X_3,3) + X_2,3 (X_1,2 X_3,1 - X_1,1 X_3,2) - X_1,1 (X_2,2 X_3,3 - X_2,3 X_3,2).<ref> hold with only minor modifications.However, <ref> is not true for the permanent. Recall also that <ref> breaks down if ()=2, and the place where the proof breaks is showing that 2423241112≡2411122324 modulo L in the case of multidegree (2+2,1+1+1+1). It turns out that the for the apolar ideal _n^⊥ in arbitrary characteristic(as well as _n^⊥ if ()=2), there are quadratic relations not generated by linear relations. For _n^⊥ in arbitrary characteristic, the quadratic relationρ_Q' = X_2,3 X_2,4 (X_1,1 X_1,2) - X_1,1 X_1,2 (X_2,3 X_2,4) = 2423241112 - 2411122324is not generated by the linear relations (<ref>).Let L' be the submodule generated by the orbit of ρ_1', …, ρ_5', ρ_S' (analogous to L, defined earlier).The multidegree of the relation ρ_Q' is (2+2,1+1+1+1). Therefore, if this relation can be written as a homogeneous sum of the linear relations, the only linear relation that can be involved is ρ_5': the relations ρ_1' and ρ_2' have multidegree 3 in one row, ρ_3' and ρ_4' have multidegree 2 in both a row and a column, and ρ_S' has positive multidegree in three rows and three columns.Under the action of _2 ×_4 on the 2 × 4 submatrix where this relation lives, the possible terms with this multidegree form two distinct orbits:2411122324and2412231124are representative elements of each.The orbit of the linear relation ρ_5' can generate only two relations (up to scaling) that contain the term 2412231124, made by taking each of the dots as constant: they are2412231124 - 2411122324 + 2411132224= X_2,4( 24122311 - 24111223 + 24111322)and2412231124 - 2423241112 + 2422241113= X_1,1( 24122324 - 24232412 + 24222413).Suppose some -linear combination of permutations of these relations gives ρ_Q'. Since 2412231124 does not appear in ρ_Q', if one of these has scalar coefficient c, the other must have coefficient -c.Therefore, if ρ_Q' is generated by the linear relations, it must be a -linear combination of the 𝐒_2 ×𝐒_4 orbit of( 2412231124 - 2423241112 + 2422241113) - ( 2412231124 - 2411122324 + 2411132224) = 2411122324 - 2423241112 - 2411132224 + 2422241113.This relation tells us that 2411122324 - 2423241112≡2411132224 - 2422241113 modulo L', thus permutations of this relation simply say that swapping columns of 2411122324 - 2423241112 is allowed modulo L'. There is no way of reducing ρ_Q' to 0 by swapping columns, so since these are all the relations we have at our disposal, ρ_Q' is not in L'.For the determinant, relations of this multidegree are linear combinations of( 2412231124 - 2423241112 - 2422241113) - ( 2412231124 - 2411122324 - 2411132224) = 2411122324 - 2423241112 + 2411132224 - 2422241113,which means that swapping columns of 2411122324 - 2423241112 while also reversing sign is allowed modulo L. It is possible to swap columns of this relation an odd number of times and get back what we started with — for example, we swap columns 2 and 3, then 1 and 2, then 1 and 3, which overall swaps columns 1 and 2, and has no effect — so for the determinant, we see that 2411122324 - 2423241112 is equivalent to its negative modulo L, thus it must be equivalent to 0 except in characteristic 2. This is how the decomposition in <ref> was constructed, and why it was necessary to assume that () ≠ 2.For the permanent, this exception is the only major modification we need to make to <ref>.In other words, the argument of <ref> establishes that in any characteristic, all quadratic relations are generated by linear relations ρ'_1, ⋯, ρ'_5, ρ'_S together with the quadratic relation ρ'_Q under permuting rows and columns and transposing.§.§ Relations of higher degree The proof of <ref> needs no changes as long as the statement is amended to include the quadratic relations obtained from ρ'_Q from permuting and transposing.This establishes the following theorem:All relations in the apolar ideal of the permanent (or the determinant if () = 2) are generated by the orbit of the linear relations ρ'_1, ⋯, ρ'_5, ρ'_S from (<ref>) and the orbit of the quadratic relation ρ_Q' from (<ref>) under permuting rows and columns and transposing.The dimension of the vector space of linear relations isβ_2,3 = 4 n+13n+23 = 1/9 n^2 ( n+1 )^2 ( n-1 ) ( n+2 ).The dimension of the vector space of quadratic relations modulo the subspace generated by the linear relations isβ_2,4 = 2 n2n4 = 1/24 n^2 (n-1)^2 (n-2) (n-3).All other Betti numbers β_2,j for j ≠ 3,4 are zero.As already pointed out, <ref> holds for _n with only minor modifications; this gives the dimension of the vector space of linear relations.The additional quadratic relations are generated by the orbit of ρ_Q' under _n ×_n and transposing. We saw in <ref> that permutations of the columns of this relation are equivalent modulo L'. Also, swapping the rows is the same as multiplying by -1 and permuting the columns. Therefore we need only compute the number of multidegrees with the partition form (2+2,1+1+1+1). This number is n2n4: we must choose 2 of the n rows to give multidegree 2, and 4 of the n columns to give multidegree 1. After including a factor of 2 to account for transposing, we have the dimension2 n2n4.The statement about the Betti numbers for j ≠ 3,4 is <ref>.The theorem follows from combining <ref> and <ref>.§ HIGHER SYZYGIESIn this section, we compute β_3,4, the dimension of the space of linear second syzygies, for the apolar ideals _n^⊥ and _n^⊥ (<ref>) and provide a conjectural description of β_r,r+1, the dimension of the space of linear higher syzygies (<ref>) of _n^⊥.We also record Macaulay2 computations of the graded Betti tables for small n in <ref>. §.§ Linear second syzygiesUsing <ref>, we can compute the Betti number β_3,4, the number of linear second syzygies, of S / _n^⊥ and S / _n^⊥.The dimension of the space of linear second syzygies of S / _n^⊥ isβ_3,4 = 1/192 (n-1) n^2 (n+1)^2 (n+2) (5n^2 + 5n - 18) = 6 n+14n+34 + 9 n+24^2.The dimension of the space of linear second syzygies of S / _n^⊥ isβ_3,4 = 1/192 (n - 1) n^2 (5 n^5 + 25 n^4 + 35 n^3 - 85 n^2 + 8n - 84) = 6 n+14n+34 + 9 n+24^2 + 2 n2n4. Recall that if M is a finitely generated graded S-module and ⋯→ F_1 → F_0 → M → 0 is a minimal graded free resolution with F_i = ⊕_j S(-j)^β_i,j, then we have the following formulas for the Hilbert function H_M of M (c.f. <cit.>)H_M(d) = ∑_i (-1)^i H_F_i(d) = ∑_i,j (-1)^i β_i,j H_S(d-j)in terms of the Hilbert functions H_F_i and H_S.Since S is a polynomial ring in n^2 variables and using <ref>, we know that H_S(d) = n^2 + d - 1dand H_S/_n^⊥(d) = H_S/_n^⊥(d) = nd^2.For the determinant, (<ref>) implies thatH_S / _n^⊥(4) = β_0,0 H_S(4) - β_1,2 H_S(2) + β_2,3 H_S(1) - β_3,4 H_S(0)since all other β_i,j are either 0, or have j > 4 so H_S(4-j) = 0.We know the value of each β_i,j here (in particular, β_0,0 = 1, and β_1,2 and β_2,3 are computed in <ref> respectively). Solving this equation for β_3,4 gives the formula above.For the permanent, β_2,4 is non-zero, so (<ref>) implies thatH_S / _n^⊥(4) = β_0,0 H_S(4) - β_1,2 H_S(2) + β_2,3 H_S(1) + β_2,4 H_S(0) - β_3,4 H_S(0)and solving this gives the formula above for β_3,4.The information we know about the Betti numbers of S / _n^⊥ and S / _n^⊥ is summarized in <ref>. The formulae for β_r,r+1 for the determinant seem to follow a pattern. We conjecture that the pattern continues:The Betti number β_r,r+1 for S / _n^⊥ is given byβ_r,r+1 = r ∑_i=1^r N_r,in+ir+1n+r-i+1r+1where N_r,i are the Narayana numbers (sequence http://oeis.org/A001263A001263 at <cit.>):N_r,i= 1/rriri-1.We know this conjecture is true for r ≤ 3 by <ref>.The first few Narayana numbers are shown in <ref>. See <ref> for a refinement of this conjecture.§.§ Betti tables S / _n^⊥ and S / _n^⊥ for low n Using the Macaulay2 software on our laptops[Thanks to the help of Scott Morrison, we also tried these computations on a more powerful computer, but unfortunately this didn't allow for the computation of any additional Betti numbers.] we computed the graded Betti tables of S / _n^⊥ and S / _n^⊥. Some of these tables are incomplete due to computer limitations; where this is the case, every column shown is complete, but there may be more columns that aren't shown.The computations presented in tab:Betti-2tab:Betti-7 support <ref>. Complete Betti tables for S / _n^⊥ and S / _n^⊥ where n = 4[t] S / _4^⊥ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 1 100 800 3075 6496 7700 4800 1225 2 2500 16800 51275 93600 113256 93600 51275 16800 2500 31225 4800 7700 6496 3075 800 100 4 1 [t] S / _4^⊥ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 1 100 800 3087 6688 8400 4320 794 2 12 192 3200 16320 50844 93600 113256 93600 50844 16320 3200 192 12 3794 4320 8400 6688 3087 800 100 4 1 § REPRESENTATION-THEORETIC DESCRIPTION OF SYZYGY MODULES In this section, we will assume that () = 0.Denote V ≅ W ≅^n.Recall from <ref> that the symmetry group of _n isG__n = ((V) ×(W))/^* ⋊/2 where the non-identity element of /2 corresponds to the transpose element of G__n (i.e., M_n() → M_n(), A ↦ A^⊤). The minimal free resolution (<ref>) of S/_n^⊥ is naturally a resolution of G_-representations.The main results of this section are <ref>, which provide a representation-theoretic characterization of the space of generators, the space of relations, and the space of linear second syzygies.Moreover, we provide a conjectural representation-theoretic description of the space of linear higher syzygies of the determinant. §.§ Representation theory A representation of G__n is the data of a (V) ×(W)-representation Q such that the diagonal ^* ⊂(V) ×(W) (consisting of pairs (α I_n, α^-1 I_n) for α∈^*) acts trivially together with an involution ι Q → Q (corresponding to transpose element of G__n)such that (g_1,g_2) ·ι(q) = ι( (g_2, g_1) · q) for (g_1,g_2) ∈(V) ×(W) and q ∈ Q. By standard representation theory, any irreducible representation of (V) ×(W) is V_λ W_η where λ: λ_1 ≥⋯≥λ_n and ηη_1 ≥⋯≥η_n are the highest weights.We denote by |λ| = λ_1 + ⋯ + λ_n the degree of a weight λ. Thus an irreducible representation G__n either has the form V_λ W_λ for a highest weight λ or (V_λ W_η) ⊕ (V_η W_λ) for a pair of distinct highest weights λ, η with |λ| = |η|.§.§ The space of generatorsBy <ref>, we know that _n^⊥ is generated by X_i,i^2, X_i,jX_i,k, X_i,jX_k,j, and X_i,j X_k,l + X_i,l X_k,j.This decomposes the degree 2 component F_1 as a direct sum of one-dimension representations of T_V × T_W, where T_V denote the maximal torus of (V).By examining the weights, we see that X_1,1^2 is the highest weight vector with weight ((2), (2)).Note that V_(2) W_(2)≅^2 V ^2 W is an irreducible G__n-representation where the transpose automorphism acts on this representation by interchanging ^2 V and ^2 W. But since (^2 V ^2 W) = n+12^2 is equal to β_1,2 (i.e., the number of minimal quadratic generators of _n^⊥), we conclude that: There is an isomorphism of G__n-representationsF_1 ≅ (^2 V ^2 W)SIn particular, the degree 2 component of F_1 is isomorphic to the irreducible G__n-representation ^2 V ^2 W.§.§ The space of relationsThe linear relations have standard degree 3. The highest weight possible is ((3), (3)); however, no non-zero relations have this weight, since there is only one element of F_1 with this multidegree, namely X_1,1 (X_1,1^2). The next highest weight possible is ((3), (2,1)), and this weight space is inhabited by ρ_1 = X_1,2 (X_1,1^2) - X_1,1 (X_1,1 X_1,2). Note that ((2,1), (3)) is inhabited by the transpose of this.We see that as (V) ×(W)-representations, the degree 3 component of F_2 contains (V_(3) W_(2,1)) ⊕ (V_(2,1) W_(3)).Since V_(3) = n+23 and V_(2,1) = 2 n+13, we compute that(V_(3) W_(2,1)) ⊕ (V_(2,1) W_(3)) = 4 n+23n+13which is equal to β_2,3.We conclude that:There is an isomorphism of G__n-representationsF_2 ≅((V_(3) W_(2,1)) ⊕ (V_(2,1) W_(3)))SIn particular, the degree 3 component of F_2 is isomorphic to the irreducible G__n-representation (V_(3) W_(2,1)) ⊕ (V_(2,1) W_(3)).§.§ The space of linear 2nd syzygiesThe linear second syzygies have standard degree 4.It's not hard to check that there are no second syzygies of weight ((4), (4)), ((4), (3,1)) and ((4), (2,2)).The second syzygy X_1,3( 13111112 - 13111211) - X_1,2( 13111113 - 13111311) + X_1,1( 13111213 - 13111312)has weight ((4), (2,1,1)) and its transpose has weight ((2,1,1), (4)).This generates an irreducible G__n-representation ( (V_(2,1,1) W_(4)) ⊕ (V_(4) W_(2,1,1)) ) contained in thespace of linear second syzygies (i.e., the degree 4 component of F_3). The next highest weight appearing in the space of linear second syzygies but not in this subrepresentation is ((3,1), (3,1)), which is inhabited byX_2,1( 22111112 - 22111211) - X_1,2( 22111121 - 22112111) + X_1,1( 22111221 - 22112112).We can therefore conclude:The degree 4 component of F_3 is isomorphic as G__n-representations to ( (V_(2,1,1) W_(4)) ⊕ (V_(4) W_(2,1,1)) ) ⊕(V_(3,1) W_(3,1)),the direct sum of two irreducible G__n-representations. §.§ Conjectural description Observing the pattern of <ref>, we can conjecture:The degree r+1 component of F_r is isomorphic as a G__n-representation to⊕_i=1^r V_(r-i+2, 1^i-1)⊗ V_(i+1, 1^r-i),where 1^a means a copies of 1. This is the direct sum of ⌊r+1/2⌋ irreducible G__n-representations.The dimension of (<ref>) agrees with the dimension in <ref>. Indeed, the dimension of the irreducible representation corresponding to a Young diagram λ can be given by the hook length formula:V_λ = ∏_(i,j) ∈λi - j + n/(λ_i,j)where the product is over cells (i,j) in the Young diagram. For a Young diagram shaped like a Γ with c cells, of which d are in the top row, all cells are either the top left cell, or a cell in the rightward arm, or a cell in the downward column, so this formula simplifies to: V_(d, 1^c-d)= ( n/c) ·( (n+1)/(d-1)…(n+d-1)/1) ·( (n-1)/(c-d)…(n-c+d)/1)= (n+d-1)! (n-1)!/c (n-1)! (n-c+d-1)! (d-1)! (c-d)! = c-1d-1n+d-1c Summing this expression over the direct sum in <ref> gives(F_r)_r+1= ∑_i=1^r rr-i+1n+r-i+1r+1·rin+ir+1 = r ∑_i=1^r 1/rriri-1n+ir+1n+r-i+1r+1which is the expression in <ref>. Val79b[Doo74]doob M. Doob. Generalizations of magic graphs. J. Combinatorial Theory Ser. B, 17:205–217, 1974.[Eis05]eisenbud-syzygies D. Eisenbud. The geometry of syzygies, volume 229 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2005.[Fro97]frobenius F. G. Frobenius. Über die Darstellung der endlichen Gruppen durch lineare Substitutionen. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, pages 994–1015, 1897. Available at <http://www.e-rara.ch/doi/10.3931/e-rara-18879>.[Mac16]macaulay F. S. Macaulay. The algebraic theory of modular systems. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1916. Reprinted in 1994.[MM62]marcus-may M. Marcus and F. C. May. The permanent function. Canad. J. Math., 14:177–189, 1962.[RS11]ranestad-schreyer K. Ranestad and F. Schreyer. On the rank of a symmetric form. J. Algebra, 346:340–342, 2011.[Sha15]shafiei S. M. Shafiei. Apolarity for determinants and permanents of generic matrices. J. Commut. Algebra, 7(1):89–123, 2015.[Slo10]OEIS N. J. A. Sloane. The on-line encyclopedia of integer sequences, published electronically at <https://oeis.org/A001263>, 2010.[Val79a]valiant1 L. G. Valiant. Completeness classes in algebra. In Conference Record of the Eleventh Annual ACM Symposium on Theory of Computing (Atlanta, Ga., 1979), pages 249–261. ACM, New York, 1979.[Val79b]valiant2 L. G. Valiant. The complexity of computing the permanent. Theoret. Comput. Sci., 8(2):189–201, 1979. | http://arxiv.org/abs/1709.09286v1 | {
"authors": [
"Jarod Alper",
"Rowan Rowlands"
],
"categories": [
"math.AC",
"math.AG"
],
"primary_category": "math.AC",
"published": "20170926235431",
"title": "Syzygies of the apolar ideals of the determinant and permanent"
} |
SF2A 2017HR 7001LESIA, UMR 8109, Observatoire de Paris Meudon, Place J.Janssen, Meudon, France Department of Physics and Astronomy, Notre Dame University - Louaize, PO Box 72, Zouk Mikael, Lebanon GEPI,UMR 8111, Observatoire de Paris Meudon, Place J.Janssen, Meudon, France Department of Astronomy and Space Sciences, Faculty of Science, Ankara University, 06100, Ankara, TurkeyThe flat bottomed lines of Vega T. Kılıcoğlu December 30, 2023 ===============================Using one high dispersion high quality spectrum of Vega (HR7001, A0V) obtained with the échelle spectrograph SOPHIE at Observatoire de Haute Provence, we have measured the centroids of 149 flat-bottomed lines. The model atmosphere and spectrum synthesis modeling of the spectrum of Vega allows us to provide identifications for all these lines. Most of these lines are due toC I, O I, Mg I, Al I, Ca I, Sc II,Ti II, Cr I, Cr II, Mn I, Fe I, Fe II, Sr II, Ba II, the large majority being due to neutral species, in particular Fe I.stars: individual, stars: Vega, HR 7001§ INTRODUCTION Vega (HR 7001), the standard A0V spectral type, is one of the 47 northern slowly rotating early-A stars studied by <cit.>. The low projected rotational velocity of HR 7001, about 24is due to the very low inclination angle (i ≃ 0) while the equatorial velocity v_e≃ 245 is very large <cit.>. Hence Vega is a fast rotator seen nearly pole-on whose limb almost coincides with the equator of the star. At the equator, the centrifugal forece reduces the effective surface gravity which alters the ionization balance and strengthens the local I_λ profile of certain species. For these species, the distribution of the Doppler shift is bimodal, ie. arises from the two equatorial regions near the limb. We have measured all the centroids of all 149 flat-bottomed lines we could find in the high resolution SOPHIE spectrum of Vega.We have synthesized all lines expected to be present in the SOPHIE spectrum of HR 7001 in the range 3900 up to 6800 Å using model atmospheres and spectrum synthesis and an appropriate chemical composition for Vega as derived by <cit.>. The synthetic spectrum has been adjusted adjusted to the SOPHIE spectrum of HR 7001 in order to identify the flat-bottomed lines of HR 7001 § OBSERVATIONS AND REDUCTION A search of the SOPHIE archive reveals that HR 7001 has been observed 78 times at the Observatoire de Haute Provence using SOPHIE from 03 August 2006 to 06 August 2012. We have used one high resolution (R = 75000) 30 seconds exposure secured with a S/N ratio of about 824 at 5000 Å to search for the flat-bottomed lines. § MODEL ATMOSPHERES AND SPECTRUM SYNTHESISThe effective temperature and surface gravity of HR 7001 were first evaluated using Napiwotzky et al's (1993) UVBYBETA calibration of Stromgren's photometry. The found effective temperatureis 9550 ± 200 K and the surface gravityis3.98 ± 0.25 dex. This temperature is in very good agreement with the fundamental temperature derived by <cit.> from the integrated flux and the angular diameter and with the mean temperature and surface gravity derived by <cit.>.A plane parallel model atmosphere assuming radiative equilibrium, hydrostatic equilibrium and local thermodynamical equilibrium was then computed using the ATLAS9 code <cit.>, specifically the linux version using the new ODFs maintained by F. Castelli on her website[http://www.oact.inaf.it/castelli/]. The linelist was built starting from Kurucz's (1992) gfhyperall.dat file[http://kurucz.harvard.edu/linelists/] which includes hyperfine splitting levels. This first linelist was then upgraded using the NIST Atomic Spectra Database[http://physics.nist.gov/cgi-bin/AtData/linesform] and the VALD database operated at Uppsala University <cit.>[http://vald.astro.uu.se/ vald/php/vald.php]. A grid of synthetic spectra was then computed with a modified version of SYNSPEC49 <cit.> to model the lines. The synthetic spectrum was then convolved with a gaussian instrumental profile and a parabolic rotation profile using the routine ROTIN3 provided along with SYNSPEC49. We adopted a projected apparent rotational velocity v_esin i =24.5 km.s^-1 and a radial velocity v_rad = -13.80 km.s^-1 from <cit.>.§ DETERMINATION OF THE MICROTURBULENT VELOCITY In order to derive the microturbulent velocity of HR 7001, we have derived the iron abundance [Fe/H] by using 36 unblended Fe II lines for a set of microturbulent velocities ranging from 0.0 to 2.5 . Figure<ref> shows the standard deviation of the derived [Fe/H] as a function of the microturbulent velocity. The adopted microturbulent velocity is the value which minimizes the standard deviation ie. for that value, all Fe II lines yield the same iron abundance, which is [Fe/H] = -0.60 ± 0.07 dex. Hence iron is found to be underabundant in HR 7001 in agreement with previous abundance determinations<cit.>.We therefore adopt a microturbulent velocity ξ_t = 1.70 ± 0.04constant with depthfor HR 7001. § THE LIST OF FLAT-BOTTOMED LINES IN HR 7001 An example of flat-bottomed line in the spectrum of HR 7001 is the Ba II line at 4554.04 Å shown in Fig. <ref>.Note that the lines of Cr II at 4554.99 Å and of Fe II at 4555.99 Å have normal profiles. All the flat-bottomed lines are collected together with their identifications in Tab. <ref>.These lines are weak lines due to C I, O I, Mg I, Al I, Ca I, Sc II, Ti II, Cr I, Cr II, Mn I, Fe I, Fe II, Sr II, Ba II, the large majority being due to neutral species, in particular Fe I. Most of the lines we find to be flat-bottomed are also listed in the investigation of weak lines conducted by <cit.> in their high signal-to-noise high resolution spectrum of Vega. § CONCLUSIONSA systematic search for flat-bottomed lines in the high resolution high quality SOPHIE spectrum of HR 7001 yields 149 lines in the range 3900 Å up to 6800 Å which complete the previous list published by <cit.>. Most of these lines are due toC I, O I, Mg I, Al I, Ca I, Sc II,Ti II, Cr I, Cr II, Mn I, Fe I, Fe II, Sr II, Ba II, the large majority being due to Fe I. lllrll Identifications for flat-bottomed lines in Vegaλ_obs (Å)λ_lab (Å) Species log gf E_low Comments 3903.08 3902.945 Fe I -0.47 12560.933 3916.453916.45V II -1.060 11514.760 3918.643918.642Fe I-0.720 24338.7663920.31 3920.258 Fe I -1.75 978.074 3922.943922.912Fe I -1.65 415.9333927.983927.920 Fe I -1.59888.1323930.313930.296 Fe I -1.590 704.007 3930.304 Fe II -4.030 13673.1863932.063932.023 Ti II -1.780 9118.2603935.98 3935.962Fe II -1.860 44915.046 3938.323938.289 Fe II -3.890 13474.411 3938.400Mg I -0.760 35051.263 3944.03 3944.006 Al I -0.620 0.0003945.203945.210 Fe II -4.250 13673.1863956.713956.677Fe I-0.430 21715.731 3961.543961.520Al I-0.320112.061 4002.36 4002.483Cr II-2.060 42897.990?4005.29 4005.242 Fe I -0.610 12560.933 4021.88 4021.866Fe I -0.66022249.4284034.494034.469Mn I-0.810 0.0004034.502 Mn I -0.8100.0004035.67 4035.595Fe I-1.100 34039.315blend4035.627V II-0.96014461.750 4035.694 Mn I -0.190 17281.999 4035.713 Mn I -0.19017281.999 4035.715 Mn I -0.19017281.999 4043.994043.897Fe I -0.83026140.178blend 4043.977Fe I -1.130 26140.178 4044.012Fe II-2.410 44929.549 4057.56 4057.461 Fe II -1.550 58668.256blend 4057.505 Mg I-1.200 35031.263 4068.014067.978 Fe I-0.430 25899.9864070.894070.840 Cr II-0.750 52321.010 4072.384072.502 Fe I-1.440 27666.3454118.57 4118.545 Fe I 0.280 28819.9524122.69 4122.668 Fe II -3.380 20830.5534132.11 4132.058Fe I-0.650 12968.554 4134.67 4134.677Fe I-0.490 22838.320 4143.37 4143.415 Fe I-0.200 24574.652 4143.86 4143.868Fe I-0.450 12560.9334161.584161.535 Ti II-2.360 8744.2504167.344167.271Mg I-1.600 35051.263blend4167.299Fe II -0.56090300.6264175.694175.036Fe I-0.670 22946.815 4176.634176.566Fe I-0.620 27166.817 4181.74 4181.755 Fe I-0.18022838.320 4187.064187.039 Fe I-0.55019757.031 4187.834187.795 Fe I-0.550 19562.437 4191.494191.430Fe I-0.670 19912.494 4198.29 4198.247 Fe I-0.460 27166.817 blend4198.304Fe I-0.720 19350.8914199.11 4199.095 Fe I 0.250 24574.6554202.02 4202.029Fe I -0.710 11976.2384210.404210.343Fe I -0.87020019.633 4210.383Fe I-1.24024772.0164215.604215.519 Sr II -0.140 0.000 4219.40 4219.360Fe I0.12028819.9524222.304222.213 Fe I-0.97019757.031blend 4222.381 Zr II-0.900 9742.800 4226.75 4226.728 Ca I 0.2400.0004227.45 4227.427 Fe I 0.23026874.5464236.00 4235.936 Fe I-0.340 19562.4374238.80 4238.810 Fe I-0.280 27394.689 blend 4238.819 Fe II-2.720 54902.315 4250.15 4250.119 Fe I-0.410 19912.494 4250.80 4250.787 Fe I-0.710 12560.9334273.30 4273.326 Fe II-3.260 21812.0554274.80 4274.797 Cr I-0.230 0.0004275.60 4275.606 Cr II-1.710 31117.390 4278.20 4278.159 Fe II-3.820 21711.9174282.45 4282.403 Fe I-0.810 17550.180blend 4282.490Mn II-1.680 44521.5214284.20 4284.188Cr II -1.86031082.940 4287.904287.872 Ti II -2.0208710.440 4312.90 4312.864Ti II -1.1609518.0604325.0 4324.999 Sc II -0.440 4802.870 4367.64367.659 Ti II -1.270 20891.660 blend 4367.578 Fe I -1.27024118.816 4369.504369.411 Fe II-3.670 22469.8524371.44371.367 C I -2.330 61981.822 4386.94386.844 Ti II-1.260 20951.600 4394.054394.051 Ti II-1.590 9850.9004395.084395.033 Ti II-0.660 8744.250 4400.40 4400.379Sc II-0.510 4883.5704411.10 4411.074Ti II-1.060 24561.0314417.804417.719Ti II-1.4309395.710 4418.40 4418.330Ti II-2.460 9975.9204450.60 4450.482 Ti II-1.4508744.250 4451.60 4451.551 Fe II -1.84049506.9954454.904455.027 Fe I-1.090 31307.244 4464.50 4464.450 Ti II -2.080 9363.620 4466.65 4466.551 Fe I-0.590 22856.320 4473.00 4472.929 Fe II-3.430 22939.357 4476.104476.019Fe I-0.57022946.815blend4476.076Fe I-0.29029732.7354488.40 4488.331 Ti II -0.82025192.791 4494.65 4494.563Fe I-1.140 17726.9284528.704528.614 Fe I-0.820 17550.180 4529.604529.569 Fe II-3.190 44929.5494541.604541.524 Fe II-3.05023031.2994554.204554.033Ba II0.00015 hfs iso4582.85 4582.835 Fe II-3.100 22939.357 4590.04589.958Ti II -1.7909975.920blend4589.967O I -2.390 86631.1534592.10 4592.049 Cr II-1.22032854.949 4616.604616.629 Cr II-1.29032844.7604620.504620.521 Fe II-3.28022810.3564666.804666.758Fe II-3.330 22810.3564703.004702.991Mg I-0.67035051.263blend4702.991Zr II -0.800 20080.3014731.504731.453Fe II -3.360 23317.632 4736.82 4736.773 Fe I -0.740 25899.986very weak4775.90 4775.897 C I-2.67018145.2854780.004779.985Ti II-1.37016518.860 4812.40 4812.468 Fe I -5.400 22249.428 4836.20 4836.229Cr II -2.250 31117.3904890.70 4890.755Fe I -0.430 23192.4974891.504891.492 Fe I-0.140 22996.673 blend 4891.485 Cr II-3.040 31350.901 4919.054918.994 Fe I -0.370 23110.937blend4918.954 Fe I -0.340 33507.1204920.504920.502 Fe I 0.060 22845.868 4932.00 4932.049 C I-1.880 61981.822blend 4932.080 Fe II -1.73083196.488 4934.10 4934.076 Ba II -0.150 0.000 4993.40 4993.565 N I -2.86095475.313 ? 5004.20 5004.195 Fe II0.500 82853.6605052.20 5052.167 C I-1.650 61981.822 very flat bottomed5129.20 5129.152Ti II -1.390 15257.4305133.70 5133.688Fe I0.140 33695.394 5154.00 5154.070Ti II -1.920 12628.731 5171.60 5171.596 Fe I-1.79011976.238 blend, very extended flat core 5171.640 Fe II-4.37022637.2055185.90 5185.913Ti II-1.35015265.6195188.70 5188.680Ti II -1.210 12758.110 5192.405192.442Fe II-2.0205192.4425206.005206.037 Cr I0.020 7593.150 5208.40 5208.425 Cr I0.1607593.150 5217.00 5216.863 Fe II 0.61084527.779 blend5216.854 Fe II 0.39084710.6865226.60 5226.343 Ti II-1.30012628.731 5226.84 5226.862 Fe I-0.550 24506.9145227.305227.481Fe II0.800 84296.829 blend5227.323 Fe II-0.03084344.832 5237.405237.329 Cr II -1.15032854.311 5255.00 5254.929Fe II -3.230 26055.422 5266.60 5266.555 Fe I-0.490 24180.861 5269.60 5269.537Fe I -1.3206928.2685275.01 5274.964Cr II-1.2932836.6805291.655291.666Fe II0.58 84527.779 5313.555313.563 Cr II -1.650 32854.9495324.205324.179 Fe I -0.240 25899.9865325.505325.503 Fe II -2.800 25981.6705329.705329.673 O I -2.20086627.777blend5329.681 O I-1.61086627.777 5329.690 O I-1.41086627.777 5330.805330.726O I-2.570 86631.453blend5330.735 O I-1.710 86631.453 5330.741O I -1.120 86631.4535336.80 5336.771Ti II -1.70012758.1105337.60 5337.732Fe II - 3.890 26055.4225337.772 Cr II-2.030 32854.949 5362.92 5362.869 Fe II-2.740 25805.329blend5362.957Fe II -0.080 84685.798 5367.505367.466Fe I0.350 35611.622 5370.00 5369.961Fe I0.350 35257.323 blend 5370.164Cr II0.320 86782.0115371.50 5371.489Fe I-1.6507728.059 blend 5371.437Fe I-1.24035767.561 5380.35 5380.337C I-1.84097770.1805383.30 5383.369Fe I0.50034782.4205404.15 5404.117 Fe I0.540 34782.420blend5404.151 Fe I0.52035767.5615405.80 5405.663 Fe II-0.440 48708.863 blend 5405.775Fe I-1.840 7985.784 5410.90 5410.910Fe I 0.28036079.371 5415.20 5415.199Fe I0.500 35379.205 5425.20 5425.257Fe II-3.36025805.3295447.00 5446.916 Fe I -1.930 7985.7845455.50 5455.454Fe I 0.300 34843.934blend5455.609Fe I -2.0908154.713 5526.805526.770 Sc II 0.13014261.320 5528.405528.405 Mg I-0.620 35051.2635534.805534.847Fe II-2.94026170.181 blend5534.890Fe II -0.69085048.6205572.755572.842Fe II-0.310 27394.6895586.805586.842Cr II 0.91088001.361 blend 5586.756Fe I-0.210 27166.817 5588.70 5588.619Cr II-5.55031117.390 5615.705615.644 Fe I -0.140 26874.547 5669.0 5668.943C I-2.43068856.328blend5669.038Sc II-1.120 12101.4996147.706147.741 Fe II-2.720 31364.440 6149.306149.258 Fe II-2.720 31968.450 6162.0 6162.173Ca I 0.10015315.943very weak 6238.356238.392 Fe II -2.63031364.440 6247.456247.557 Fe II -2.330 31307.9496417.006416.919Fe II-2.740 31387.949 The authors acknowledge use of the SOPHIE archive (<http://atlas.obs-hp.fr/sophie/>) at Observatoire de Haute Provence. They have used the NIST Atomic Spectra Database and the VALD database operated at Uppsala University (Kupka et al., 2000) to upgrade atomic data. [Castelli & Kurucz (1994)]Castelli94 Castelli, F., Kurucz, R.L. 1994, A&A, 281, 817 [Code et al. (1976)]Code1976 Code, A., D., Davis, J., Bless, R.C., Hanbury Brown, R. 1976, ApJ, 203, 417 [Gulliver et al. (1994))]Gulliver94 Gulliver, A.F., Hill, G., Adelman, S.J. 1994, ApJ, 429L, 81G [Hill et al. (2010)]Hill2010 Hill, G., Gulliver, A.F., Adelman, S.J. 2010, ApJ, 712, 250 [Hubeny & Lanz (1992)]Hubeny92 Hubeny, I., Lanz, T. 1992, A&A, 262, 501 [Hubeny & Lanz (1995)]Hubeny95 Hubeny, I., Lanz, T. 1995, ApJ, 439, 875 [Kupka et al. (2000)]kupka2000Kupka F., Ryabchikova T.A., Piskunov N.E., Stempels H.C., Weiss W.W., 2000, Baltic Astronomy, vol. 9, 590-594 [Kurucz (1992)]Kurucz92 Kurucz, R.L. 1992, Rev. Mexicana. Astron. Astrofis., 23, 45 [Napiwotzki et al. (1993)]Napiwotzki93 Napiwotzki, R., Schoenberner, D., Wenske, V. 1993, A&A, 268, 653 [Royer et al. (2014)]Royer14 Royer, F., Gebran, M., Monier, R., Adelman, S., Smalley, B., Pintado, O., Reiners, A., Hill, G., Gulliver, G. 2014 , A&A, 562A, 84R [Takeda et al. (2008)]Takeda08 Takeda, Y., Kawanomoto, S., Ohishi, N. 2008, ApJ, 678, 446 | http://arxiv.org/abs/1709.09509v1 | {
"authors": [
"R. Monier",
"M. Gebran",
"F. Royer",
"T. Kılıcoğlu"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170927133805",
"title": "The flat bottomed lines of Vega"
} |
000–000 0000 Coherence recovery mechanisms in quantum Hall edge states. Anna S. Goremykina, Eugene V. SukhorukovDecember 30, 2023 ==========================================================We explore the physics of relativistic radiation mediated shocks (RRMSs) in the regime where photon advection dominates over photon generation. For this purpose, a novel iterative method for deriving a self-consistent steady-state structure of RRMS is developed, based on a Monte-Carlo code that solves the transfer of photons subject to Compton scattering and pair production/annihilation. Systematic study is performed by imposing various upstream conditions which are characterized by the following three parameters: the photon-to-baryon inertia ratio ξ_u *, the photon-to-baryon number ratioñ, and the shock Lorentz factor γ_u. We find that the properties of RRMSs vary considerably with these parameters. In particular, while a smooth decline in the velocity, accompanied by a gradual temperature increase is seen for ξ_u*≫ 1, an efficient bulk Comptonization, that leads to a heating precursor, is found for ξ_u*≲ 1. As a consequence, although particle acceleration is highly inefficient in these shocks, a broad non-thermal spectrum is produced in the latter case. The generation of high energy photons through bulk Comptonization leads, in certain cases, to a copious production of pairs that provide the dominant opacity for Compton scattering. We also find that for certain upstream conditionsa weak subshock appears within the flow. For a choice of parameters suitable to gamma-ray bursts, the radiation spectrum within the shock is found to be compatible with that of the prompt emission, suggesting that subphotospheric shocks may give rise to the observed non-thermal features despite the absence of accelerated particles.gamma-ray burst: general — shock waves — plasmas — radiation mechanisms: non-thermal — radiative transfer — scattering§ INTRODUCTION Shocks are ubiquitousin high-energy astrophysics.They are believed to be the sources of non-thermal photons,cosmic-rays and neutrinos observed in extreme astrophysical objects.Two distinct types of astrophysical shocks have been identified: "collisionless" shocks, in which dissipation is mediated by collective plasma process,and "radiation mediated shocks" (RMS), in which dissipation is governed by Compton scattering and under certainconditions also by pair production.A collisionless shock usually forms in a dilute, optically thin plasma,where binary collisions and radiation drag are negligible,its characteristic width is of the order of theplasma skin depth, and it is capable of acceleratingparticles to non-thermal energies in cases where the magnetization of the upstream flow is not too high. A RMS, on the other hand, forms when a fast shock propagates in an optically thick plasma, its width is of the order of theThomson scattering length, and it cannot accelerate particles to non-thermal energies by virtue of its large width, thatexceeds any kinetic scale by many orders of magnitudes (see further discussions below regarding this point).It is worth noting that while the microphysics of collisionless shocks is poorly understood, and any progress in our understanding of these systems relies heavily on sophisticated plasma (PIC) simulations, the microphysics of RMS is fully understood, which considerably alleviates the problem.RMS play a key role ina variety of astrophysical systems, including shock breakout in supernovae (SNe) andlow-luminosity GRBs <cit.>, choked GRB jets<cit.>,sub-photospheric shocks in GRBs<cit.>,and accretion flows into black holes. The environments in which the shocks propagate and their velocity vary significantly between the various systems.For instance, shocks that are generated by various types of stellar explosions propagate in an unmagnetized, photon poor medium, and their velocity prior to breakout ranges from sub-relativistic to ultra-relativistic, depending on the type of the progenitor andthe explosion energy <cit.>.Sub-photospheric internal shocks in GRBs, on the other hand,propagate in a photon rich plasma, conceivablywith anon-negligible magnetization, at mildly relativistic speeds.Consequently, the structure and observational properties of RMS are expected to vary between the different types of sources.Early work on RMS <cit.>was restricted the Newtonian regime, where the diffusion approximation holds. Unfortunately, the limited range of shock velocitiesthat can be analyzed by employing the diffusion approximation renders its applicability to most high-energy transients of little relevance. In the last decade there has been a growing interest in extending the analysis to the relativistic regime, in an attempt to identify observational diagnostics of these shocks, and in particular the early signal expected from shock breakout in various cosmic explosions, and the contribution of sub-photospheric shocks to prompt GRB emission.An elaborated account of the astrophysical motivation is given in Section <ref>. There are vast differences between relativistic and non-relativistic RMS, as described in <cit.> and <cit.>.In brief, in non-relativistic RMS the shockthickness is much larger than the photon meanfree path and theenergy gain in a single collision is small.As a consequence, the diffusion approximation holds, which considerably simplifies the analysis.In contrast, in relativistic and mildly-relativistic RMS the shock thickness is a few Thomson depths, the change in photon momentumin a single collision is large, the optical depth is highly anisotropic, owing to relativistic effects, and pair production is important, even dominant in RRMS with cold upstream.Thus, the analysis of RRMSis far more challenging and requires different methods. Monte-Carlo simulations is an optimal method for computations of steady and slowly evolving shocks. The advantage of this method is its flexibility, that allows a systematic investigation of a large region of the parameter space relevant to a variety of systems.In this paper we present results of Monte-Carlo simulations of infinite planar RMS propagating in an unmagnetized plasma,in the regime where advection of photons by the upstream flow dominates over photon production inside and just downstreamof the shock transition layer (photon rich shocks).The regime of photon starved shocks will be presented in a follow up paper.In Section <ref> we give an overview of the astrophysical motivation.In Section <ref> we derive some basicproperties of a RRMS and compute analyticallyits structure. In Section <ref> we describe the code and the method of solutions.The results are presented in Section <ref>, and the applications toGRBs in Section <ref>. We conclude in Section <ref>. § ASTROPHYSICAL MOTIVATIONThis section briefly summarizes the astrophysical motivation for considering RMS. We focus on shock breakout in stellar explosion, including chocked GRB jets, and photospheric GRB emission. §.§ Shocks generated by stellar explosions The first light that signals the death of a massive star is emitted upon emergence of the shockwave generated by the explosion at the surface of the star. Prior to its breakout the shock propagates in the dense stellar envelope, and ismediated by the radiation trapped inside it. The observational signature of thebreakout event depends on the shockvelocity and the environmental conditions, thus, detection of the breakout signal and the subsequent emission can provide a wealthof information on the progenitor (e.g., mass, radius, mass loss prior to explosion, etc.) and on the explosion mechanism.Recent observational progress has already led to the discovery of a few shock breakout candidates, notably SN 2008D, and next generation transient surveys promise to detect many more.In order to extract this information the structure of the RMS must be computed.A shock that traverses the stellar envelope during a SN explosion accelerates as it propagates down the declining density gradient near the stellar edge, and while the bulk of the material is always non-relativistic in SNe, the accelerating shock can bring, in some cases, a small fraction of the mass to mildly and even ultra-relativistic velocities. For a typical spherical explosion with an energy of ∼ 10^51 erg this happens in compact stars with R_* ≲ R_⊙, while for larger energies, and/or strongly collimated explosions, relativistic shocks are generated also in more extended progenitors <cit.>. <cit.>obtained solutions of infinite planar RMS in the ultra-relativistic limit, under conditions anticipated in SN shock breakouts.These solutions are valid within the star where the shock width is much smaller than the scale over which the density vary significantly.<cit.>employed those solutions to show that a relativistic shock breakout from a stellar surface gives rise to a flash ofgamma-rays with very distinctive properties.Their analysis is applicable to progenitors in which the breakout is sudden, as in caseswhere transition to a collisionless shock occurs near the stellar surface,but not to the gradual breakouts anticipated in situations whereinthe progenitor is surrounded by a stellar wind thick enough to sustain the RMS after it emerges from the surface of the star. In the latter case, the gradual evolution of the shock duringthe breakout phase can significantly alter the breakout signal.This case is of special interest since Wolf-Rayet stars, which are thought to be the progenitors of long GRBs, and are also compactenough to have relativistic shock breakout in extremely energetic SNe (such as SN 2002ap),are known to drive strong stellar winds.Shock breakout from a stellar wind has been studied in the non-relativistic regime <cit.>.Analytic solutions forthe structure of ultra-relativistic RMS with gradual photon leakage have also been found recently <cit.>. We plan to carry out a comprehensive analysis of such shocks is the near future.§.§ Implications for high-energy neutrino production in GRBs It has been proposed that the interaction of photons with protons accelerated to high energy in shocks during jet propagation through the stellar envelop may produce, for both GRB jets and slower jets that may be present in a larger fraction of core collapse SNe, bursts of ∼1 TeV neutrinos<cit.>.However, in early models the fact that internal and collimation shocks that are produced below the photosphere are mediated by radiation has been overlooked. As explained below, in such shocks particle acceleration is extremely inefficient.This has dramatic implications for neutrinoproduction in GRBs <cit.>.This problem can be avoided inultra-long GRBs <cit.>and in low-luminosity GRBs <cit.>, if indeed produced by choked GRB jets, as in the unified picture proposed by <cit.>.In the latter scenario, the progenitor star is ensheathed by an extendedenvelope that prevents jet breakout.If the jet is chocked well above the photosphere, then internal shocks produced inside the jetare expected to be collisionless. The photon density at the shock formation site may still be high enough to contribute the photo-pionopacity required for production of a detectable neutrino flux. Substantial magnetization of the flow may alter the above picture, because in this caseformation of a strong collisionless subshock within the RMS occurs <cit.>. Whether particle acceleration is possible in mildly relativistic internal shocks with at relativelyhigh magnetization is unclear at present <cit.>, but if it does then the problem of neutrino production in GRBs should be reconsidered.§.§ Sub-photospheric shocks in GRBs The composition and dissipation mechanisms of GRB jets are yet unresolved issues. The conventional wisdom has been that those jets are powered by magneticextraction of the rotational energy of a neutron star or an accreting black hole, and that the energy thereby extracted is transportedoutward in the form of Poynting flux, which on large enough scales is converted to kinetic energy flux. An important question concerning the prompt emission mechanism is whether the conversion of magnetic-to-kinetic energy occurs above or well belowthe photosphere <cit.>.Dissipation wellbelow the photosphere is naturallyexpected in case of a quasi-striped magnetic field configuration <cit.>. Rapid dissipation of an ordered magnetic field may ensue in a dense focusing nozzle via the growth of internal kink modes, as demonstrated recently by state-of-the-art numericalsimulations<cit.>. In such circumstances, the GRB outflow is expected to beweakly magnetized when approaching the photosphere. If this is indeed the case, then further dissipation, that produces the observed prompt emission, most likely involves internal and recollimation shocks in the weakly magnetized flow.Substantial dissipation is anticipated just below thephotosphere for typical parameters <cit.>. Various (circumstantial) indications of photospheric emission support this view. These include: (i) detection of a prominent thermal component in several bursts, e.g., GRB090902B <cit.>, and claimed evidence for thermal emission in many others <cit.>; (ii) a hard spectrum below the peak that cannot be accounted for by optically thin synchrotron emission; (iii) evidence (though controversial) for clustering of the peak energy around 1 MeV, that is most naturally explained by photospheric emission;In addition, an attractive feature of sub-photospheric dissipation models is that they can lead to the high radiative efficiency inferred from observations.A large body of work on photospheric emission does not address the nature of the dissipation mechanism and the issue of entropy generation.Earlier work <cit.> attempted to compute the evolution of the photon density below the photosphere, assuming dissipation by some unspecified mechanism.They generally find significant broadening of the seed spectrum if dissipation commences in sufficiently opaque regions and proceeds through the photosphere.<cit.> have reached a similar conclusion, demonstratingthat multiple RMS can naturally generate a Band-like spectrum.More recent work <cit.> combines hydrodynamics (HD) and Monte-Carlo codes to compute the emitted spectrum. In this technique, the output of the HD simulations is used as input for the Monte-Carlo radiative transfer calculations. These calculations illustrate that a Plank distribution, injected at a large optical depth, evolves into aBand-like spectrum owing to bulk Comptonscattering on layers with sharp velocity shears, mainlyassociated with reconfinement shocks. However, one must be cautious in applying those results, since the emitted spectrum is sensitive to the width of the boundary shear layers <cit.>, which is unresolved in those simulations. Furthermore, the radiative feedback on the shear layer is ignored.Ultimately, the structure of those radiation mediatedreconfinementshocks needs to be resolved to check the validity of the results.The Monte-Carlo simulations described in Section <ref>enable detailed calculations of the spectrum produced in a sub-photospheric GRB shock prior to its breakout. The implications for prompt GRB emission are discussed in Section <ref>.§ GENERAL CONSIDERATIONS AND SUMMARY OF PREVIOUS WORKConsider an infinite planar shock propagating in an unmagnetized plasma at a velocity β_u (henceforth measured in units of the speed of light c). In the frame of the shock, the jump conditions read:n_uγ_uβ_u=n_dγ_dβ_d,w_uγ_u^2β_u^2+p_u=w_dγ_d^2β_d^2+p_d,w_uγ_u^2β_u=w_dγ_d^2β_d,where n denotes the baryon density, w the specific enthalpy, β the fluid velocity with respect to the shock frame, and γ=(1-β^2)^-1/2 the Lorentz factor. Thesubscripts u and d refer to the upstream and downstream values of the fluidparameters, respectively.Radiation dominance is established in the downstream plasma when the shock velocity satisfiesβ_u > 4×10^-5 n_u 15,where n_u 15 = n_u / 10^15 cm^-3. At velocities well in excess of this value the shock becomesradiation mediated <cit.>.This readily implies that under conditions anticipated in essentially all compact astrophysical systems,relativistic and mildly relativistic shocks that form in opaque regions are mediated by radiation.It is insightful to compare different scales that govern microphysical interactions.The width of the RMS transition layer is typically on the order of the photon diffusion length, l^'_s∼ (n_u σ_Tβ_u)^-1≃10^9(β_u n_u 15)^-1 cm, here measured in the shock frame. As shown below, it can be substantially smaller when pair production is important, but by no more than three orders of magnitudes. The skin depth isδ∼ c/ω_p≃ 1n_15^-1/2 cm, where ω_p is the plasma frequency,and the gyroradius of relativistic protons of energy ϵ_p is, r_L∼3(ϵ_p/m_pc^2) (B/10^6G)^-1 cm. Evidently, under typical conditions kinetic processes are expected to play no role in RMS, with the exception ofsubshocks that form within the shock transition layer in certain cases (see below). In particular, theextremely small ratio, r_L/l_s^'∼ 10^-8, implies that particle acceleration is unlikely in RMS.Kinetic processes become important, of course, during the transition phase from RMS to collisionless shocks.The properties of the downstream flow of RMS depend on the parameters of the upstream flow, and in particularits velocity, magnetization, specific entropy and optical depth.The following discussion elucidates theeffect of each of these parameters.§.§ Photon sourcesThe main sources of photons inside and just downstream of the shock transition layer are photon advection by the upstream flow, and photon generation by Bremsstrahlung and double Compton emission.Each process dominates in a different regime. In what follows "photon rich" shocks refer to RMS in which photon generation is negligible and "photon starved" shocks to RMS in which photon advection is negligible. The advected radiation field is henceforth characterized by two parameters: the photon-to-baryon density ratio far upstream, ñ=n_γ u/n_u, and the fraction ξ_γ of the total energy which is carried by radiation in the unshocked flow.Under the conditions anticipated in RRMS the thermal energy of the upstream plasma isnegligible, so that to a goodapproximation one has ξ_γ=γ_u e_γ u/(γ_u-1)m_pc^2n_u+γ_u e_γ u, where e_γ=3p_γ denotes the energy density of the radiation field.The specific photon generation rate by thermal Bremsstrahlung emission can be expressed asṅ_ff = 2^3/2α_fσ_Tcn_i^2/√(3π) Θ^1/2{(1+2x_+)Λ_ep+ [x_+^2+(1+x_+)^2]Λ_ee+x_+(1+x_+)Λ_+-},here n_i is the ion density, x_+=n_+/n_i is the pair multiplicity, specifically the number of positrons per ion,Θ=kT/m_ec^2 denotes the plasma temperature in units of theelectron mass, and α_f is the fine structure constant.The total number of electrons is dictated by charge neutrality, n_e=(1+x_+)n_i. The terms labeled ep and +- account for the contributions of e^± p and e^+ e^+ encounters, respectively, and the term ee for the contribution of e^- e^- and e^+ e^+ encounters.The quantities Λ_ep, Λ_ee and Λ_+- are functions of the temperature Θ, and are given explicitly in <cit.>. The rate of double Compton (DC) emission is ṅ_DC=16/πα_fσ_Tcn_en_γΘ^2Λ_DC, with Λ_DCgiven in <cit.>. As the ratio of the two rates is ṅ_DC/ṅ_ff∼ (n_γ/n_e)Θ^5/2, it is readily seen that DC emission is only important in regions where the photon density largely exceeds the total lepton densityn_γ≳ n_e Θ^-5/2.Such conditions prevail in the near downstream of sufficiently photon rich shocks <cit.>. In photon starved shocks, where n_γ≃ n_e and Θ≃ 0.2, DC emission is negligible.As shown in Section <ref>, this is also true for photon rich shocks with small enough ñ, in which the density of pairs produced by nonthermal photonsbecomes substantial, x_+ ≫ 1. We now give a crude estimate of the advected photon density above which the shock is expected to be photon rich and below which it is photon starved <cit.>.We note first that photons which are produced downstream candiffuse back and interact with the upstreamflow provided they were generated roughly within one diffusion length, L_d=(σ_T n_lβ_d)^-1, from the shock transition layer, where n_l=(1+2x_+)n_d is the total lepton density in the immediate post shock region.The density of these photons isδ n_γ=ṅ_γ L_d/cβ_d.Since for marginally rich shocks photon generation is dominated by Bremsstrahlung, we have to a good approximationδ n_γ/n_γ d≃2^3/2α_f n_d/√(3π)Θ_d^1/2β_d^2n_γ d× {Λ_ep+[x_+^2+(1+x_+)^2]Λ_ee/(1+2x_+)+x_+(1+x_+)Λ_+-/(1+2x_+)},where Equation (<ref>) has been used with n_i=n_d.Assuming that the shock is photon rich, we employ Equations (<ref>)and(<ref>) below to getδ n_γ/n_γ d≃3×10^-3/√(ñγ_uβ_u)× {Λ_ep+[x_+^2+(1+x_+)^2]Λ_ee/(1+2x_+)+x_+(1+x_+)Λ_+-/(1+2x_+)}.Typically, the terms Λ_12 lie in the range10< Λ_12 < 20, and since ñ>10^3 for photon rich shocks(see Equation (<ref>) below) it isevident that photon generation is important only when pair loading is substantial (x_+ ≫ 1). Adopting for illustrationΛ_ee+Λ_+-/2=30 we estimate that photon generation will dominate over photon advection whenx_+>10√(ñ γ_uβ_u). §.§ Photon rich regime As shown in Section <ref>, in photon rich shocks the density of pairs produced by nonthermal photons is typically much smaller thanthe density of the radiation, and while under certain conditions the pairs can dominate the opacity inside the shock and affectits structure, they contribute very little to the totalenergy budget downstream.The specific enthalpy is then well approximated by w=n m_pc^2+4p_γ, where p_γ denotes the radiation pressure. Adopting the latter equation of state, the jump conditions, Equations (<ref>)-(<ref>), can be solved to yield the parameters of the downstream flow.Solutions for the downstream velocity, β_d, and the specific radiation energy, e_γ d=3p_γ d,are exhibited in Fig. <ref> in the limit ξ_γ≪ 1. The radiation energy in Fig. <ref> is normalized to the value obtained in the ultra-relativistic limit (i.e., for β_d=1/3):e_γ d=2 n_uγ_u^2β_u^2m_pc^2.As seen, this asymptotic value is a good approximation also at mild Lorentz factors, and is adopted for illustrationin the following discussion. The downstream region of a RRMS is inherently non-uniform, because the thermalization lengthover which the plasma reaches full thermodynamic equilibrium is larger than the width of the shock transition layer.However, for typical astrophysical parameters, the thermalization length exceeds theshock width by several orders of magnitudes <cit.>, so that for any practical purpose photon generation in the downstream plasma can be ignored. This readily implies that to a good approximation the photon number is conserved across theshock transition layer, whereby n_γ dγ_dβ_d=n_γ uγ_uβ_u. Combining withEquation (<ref>), one finds ñ=n_γ u/n_u=n_γ d/n_d. The temperature can be computed using Equations (<ref>) and (<ref>), yielding Θ_d =e_γ d/3n_γ d m_ec^2=2m_p/3 m_e(γ_uβ_u)(γ_dβ_d)/ñ ≃ 430 γ_uβ_u/ñ, where β_d=1/3 was adopted to obtain the numerical factor in the rightmost term.Thus, Θ_d ≪ 1 as long as ñ≫ 430γ_uβ_u.Next, we estimate the minimum value of ñ required in order that counter-streaming photons will be able to decelerate the upstream flow. Let η denotes the fraction of downstream photons thatpropagate towards the upstream. The average energy each counter-streaming photon can extract in a single collisionis at most γ_um_ec^2. Thus, the number of downstream photons required to decelerate the upstream flow satisfies γ_d n_γ d>η^-1 (m_p/m_e)γ_u n_u (assuming ξ_γ≪ 1).By employing Equation (<ref>) we find that the shock can be mediated by the advectedphotons provided the photon-to-baryon number ratio far upstream satisfies ñ>ñ_crt≡m_p/m_eβ_d/ηβ_u≃m_p/m_e, adopting β_d/η =1, which is roughly the value obtained from the Monte-Carlo simulations. At the critical number density the average photon energy, 3 k T_d ≃ 2η m_ec^2 γ_uβ_u,is in excess of the electron mass.Under this condition a vigorous pair production is expected to ensue insideand just downstream of the shock, that will significantly enhance photon generation, therebyregulating the downstream temperature. §.§.§ Analytic shock profileLetn_γ→ u denotes the density of photons moving in upstream direction (i.e., from downstream to the upstream), as measured in the shock frame, and n, n_e, n_± the proper densities of baryons, electrons and e^± pairs, respectively.We suppose that the flow moves along the z-axis, chosen such thatits positive direction is towards the upstream. Thechange in the photon density is governed by the equation dn_γ→ u/dz=-σ_KNγ (n_e+n_±) n_γ→ u. For sufficiently photon-rich shocks the scattering of bulk photons is inthe Thomson regime. In terms of the optical depth, dτ_* = σ_T (n_e+n_±)γ dz, and the energy density ofthe counter streaming photons, u_γ→ u=<ϵ_γ>n_γ→ u, one then has du_γ→ u/dτ_*=-u_γ→ u. The total inverse Compton power emitted by a single electron (positron) inside the shock is approximatelyP_Comp= κ_γ cσ_T (γβ)^2 u_γ→ u, where the pre-factor κ_γ ranges from 4/3 for isotropic radiation to 4 for completely beamed radiation. For simplicity, we shall assume that it is constant throughout the shock.Neglecting the internal energy relative to baryon rest mass energy, the energy flux of the plasma can be expressed asT_b^0z=m_pc^2n γ^2 =J γ, in terms of the conserved mass flux, J=m_pc^2 n γ. Using Equation (<ref>) we obtaind T_b^0x/dz = γ(n_e+n_±) c^-1 P_comp= κ_γσ_T γ(n_e+n_±)(γ^2-1) u_γ→ u,orJ d γ/dτ_*=κ_γ (γ^2-1) u_γ→ u. The boundary condition is γ(τ_*→∞)=γ_u. Denoting α=κ_γ u_γ→ u(τ_*=0)/J, andη(τ_*)=ln(γ_u+1/γ_u-1)+2α e^-τ_* the solution of Equations (<ref>) and (<ref>) reads: γ(τ_*)=e^η+1/e^η-1. From the jump conditions we have α=2κ_γϵγ_u, whereϵ=u_γ→ u/u_γ d is roughly thefraction of downstream photons that propagate backwards. The black solid line inFig. <ref> shows the shock profile obtained for κ_γϵ=0.2. The red line is the result of the full simulation.§.§ Photon starved regimeAt ñ<ñ_crt photon advection is negligible, and the prime source of photons inside and downstream of theshock transition layer is free-free emission.For sufficiently fast shocks, β_u>0.6, a pair production equilibriumis established downstream, keeping the temperature at Θ_d≲ m_ec^2/3 <cit.>.In relativistic shocks, γ_u ≫ 1, the downstream photon density is obtained fromEquation (<ref>) and the relation e_γ d≃ m_ec^2 n_γ d':n_γ d≃ 2m_p/m_en_u γ_u^2. This, in turn, implies that the transition from photon rich to photon starved shocks should in fact occur at ñ≃ñ_crtγ_u.Because the average energy of downstream photons is roughly m_ec^2, scattering of counter streaming photons off electrons (positrons) in the shock transition layer is in the Klein-Nishina regime. As a consequence, the temperatureat any position z inside the shock is expected to be comparable to the local bulk energy of the leptons,specifically Θ(z)≃γ(z). Indeed, this result has been verified by detailed simulations <cit.>.When this relation is adopted it is possible to compute the shock structure analytically in the limitγ_u ≫ 1 <cit.>. The analysis indicates that the shock width increases with Lorentz factor according to l_s^'≃10^-2γ_u^2 (σ_T n_u)^-1,where the numerical coefficient is somewhat arbitrary <cit.>.This scaling stems from Klein-Nishina effects, and is different than the scaling obtained for photon rich shocks.The photon spectrum exhibits (in the shock frame) a peak at hν_peak≃ m_ec^2, with a broad (a rough power law) extension up to energies > γ_u m_ec^2 <cit.>.§.§ Effect of magnetic fields Substantial magnetization of the upstream flow can significantly alter the shock profile andemission. A prominent feature of such shocks is the formation of a relatively strong subshock<cit.>. The results exhibited in Section <ref>indicate that subshocks form also in unmagnetized shocks undercertain conditions (see Figs. <ref>, <ref> and <ref>), but those are generallyweak and have little effect on the overall shock structure and emission, with the exception of thebreakout transition, where photon leakagebecomes important.In magnetized shocks formation of strong subshocks is anticipated even in regions of large optical depth, which can considerably alter the energy distribution of particles in the shock if particle acceleration at the collisionless subshock ensues.The net amount of energy that can be transferred to the nonthermal particles depends primarily on the relative strength of the subshock, and needs to be quantified. The formation of a strong subshock in RRMS may have profound implications for emission of sub-photosphericGRB shocks, as well as for neutrino production in chocked jets, as described above. §.§ Finite shocks and breakoutThe structures computed in <cit.> and in Section <ref> are applicable to RRMS propagating in a medium of infinite optical depth.In cases where the shock propagates in a medium of gradually decreasing optical depth, e.g., stellar wind, it will eventually reacha point at which the radiation trapped inside it starts escaping to infinity.The leakage of radiation leads to a steepening of the shock,at least in photon starved RRMS. Nonetheless, the shock remains radiation mediated also at radii at which the optical thicknessof the medium ahead of the shock is much smaller than unity,owing to self-generation of its opacity through accelerated pair creation.Breakoutoccurswhen the Thomson thickness of the unshocked medium becomes smaller than (m_e/m_p)γ_u, provided γ_u>1 ,or else in the Newtonian regime <cit.>. How this affects the emitted spectrum is yet to be explored. A similar process may take place also during the breakout of a photon rich shock, since photon escape from the mediumahead of the shock (i.e., the upstream plasma) is expected to lead to the gradual decline of ñ over time.If pair creation and photon generation occur over a time shorter than the breakout time, thena transition from photon rich to photon starved shock is expected prior to breakout, at least in cases where the shock remains relativistic in the frame of the unshocked medium.Otherwise the shock will evolve in some complex manner. In any case,the spectrum emitted during the breakout phase may be altered.§ MONTE-CARLO SIMULATIONS OF RRMS §.§ Description of the Monte-Carlo codeThe Monte-Carlo code used by us enables computations of mildly relativistic and fully relativistic radiation mediated shocks in a planar geometry, for arbitrary upstream conditions.It incorporates an energy-momentum solver routine that allows adjustments of the shock profile in each iterative step.A photon source is placed sufficiently far upstream, and is tuned to account for the assumed photon density advected by the upstream flow. In each run, an initial shock profile is imposed (usually some parametrized analytic function that satisfies the shock jump conditions) during some initial stage at which photons that were injected upstream and crossed the shock are accumulated downstream. Once the photon density downstream reaches a level that ensures stability of the system, the energy-momentum solver is switched on, and the shock profile is allowed to change iteratively, until a steady state is reached whereby energy and momentum conservation of the entire system of particles (i.e., photons, baryons and electron-positron pairs) is satisfied at every grid point.Choosing the initial shock profile such that it satisfies the jump conditions in the frame of the simulation box guarantees that the final shock solution is stationary in this frame (otherwise it propagates across the box accordingly). Since the jump conditions depend only on the parameters of the upstream flow, they can be determined a-priori for any given set of upstream conditions.The present version of the code includes the following radiation processes: Compton scattering, pair productionand annihilation, and energy-momentumexchange with the bulk plasma. Its applicability is therefore restricted to photon rich shocks. We are currentlyin the process of incorporating also internal photon sources, specifically relativistic Bremsstrahlung and double Compton scattering, that would allow simulations of shocks for any upstream conditions, and in particular photon starved shocks.Magnetic fields can also be included upon a simple modification of the energy-momentum solver, and is planned for a future work.In developing the prescription for the iteration in our code, wemimic the method used in the context ofrelativistic cosmic ray modified shocks <cit.>. The difference is that, while we track photons, they track cosmic rays using Monte-Carlo technique andevaluate the feedback on the bulk shock profile.§.§ Numerical setupIn our calculations, the input parameters are the following quantities at thefar upstream region: (i) thephoton-to-baryon inertia ratio,ξ_u *≡ e_γ u / (n_u m_p c^2),(ii) the photon-to-baryon number ratio,ñ≡ n_γ u/n_u, and (iii) the bulk Lorentz factor of the upstream flow with respect to the shock frame, γ_u.Once these parameters are determined, all the physical quantities at the far upstream regionare specified under the assumption that the radiation and the plasma (protons and electrons) have identical temperature, T_u.We further assume thatthe photons and the plasma constituents in the upstream region have Wien and Maxwell distributions, respectively.For given upstream conditions, we derive the corresponding steady shock solution using the iterative scheme described in the previous section.[ Note that the shock structure can be determined without specifying the absolute value of the baryon density (or, equivalently, photon number density) whenexpressed as a function of optical depth.The obtained solution is scale-free in which the number densities of photon and pairs are only describedin terms of the ratio to that of the baryons. The determination of the baryon number density gives the absolute values of these quantities as well as the physical spatial scale. This is valid as long as effects such as free-free absorption that break the scalability are ignored.] In each iterative step, we solve the radiation transfer using Monte-Carlo method under a given plasma profile, and evaluate theenergy-momentum exchange between the photon and plasma.We continue the iteration until the deviation of the total energy-momentum flux at every grid point from that of the steady state value becomes small.In the calculations presented in this paper, the errors in the conservation of momentum and energy fluxesafter the iteration are mostly within a few % throughout the entire structure (< 15 % at most). The total energy and momentum fluxes at each grid point are evaluated as F_m=γ^2 (ρ_pl c^2 + e_pl + p_pl) β^2 + p_pl + F_m,γ,andF_e=γ^2 (ρ_pl c^2 + e_pl + p_pl) β + F_e,γ,respectively. Here ρ_ pl = n m_p + (n + n_±) m_e, e_pl = 3/2 n k T + 3/2 f(T) (n + n_±) k T, p_pl = (2n + n_±) k T are therest mass density, internal energy density and pressure of the plasma, respectively, where n_± is the number density of the created electron-positron pairs andf(T) =tanh[( lnΘ + 0.3) / 1.93] + 3/2 is an analytical function of temperature defined in <cit.>, obtained from a fit to the exactequation of state of pairs at an arbitrary temperature (f=1for Θ≪ 1 and f≈ 2 for Θ≫ 1). It is assumed that the protons and pairs have identical local temperature at every grid point. The last terms in the above equations, F_m, γ and F_e, γ, denote the momentum and energy fluxes of radiation that are directly computed by summing up the contributions of individual photon packets tracked in the Monte-Carlo simulation. The steady state values of momentum, F_m,u, and energy fluxes, F_e,u, are evaluated by substituting the enthalpy w_u = n_u(m_p + m_e) c^2 + [(7/2 + 3/2 f(T_u))n + 4 n_γ u] k T_u and pressure p_u = (2n + n_γ) k T_u in the left side terms of Equations (<ref>) and (<ref>).At first,the above iteration is performed under the assumption that a subshockis absent in the system. If it converges to steady flow, we simply employ the solution and consider that theshock dissipationis solelydue to photon plasma interaction.On the other hand, when we find that the flow does not reachthe steady stateunder the assumption (error in energy-momentum flux islarger than ∼ 20 %),we introduce a subshock in the system.The subshock is treated as adiscontinuity in the plasma profilewhich satisfies the Rankin-Hugoniot conditionsunder the assumption that bulk plasma is isolated from the radiation.This setup is justified due to the fact that,since the plasma scale is much shorterthan that of the photon mean free path,photons cannot feel the continuous change in the transition layer of shock formed via plasma interactions.Once we introduce the subshock in the system, we also vary the immediate upstream velocity in each iterative stepsand continue the computation until it approaches to steady solution.Regarding the microphysical processes, Compton scattering is evaluated using the full Klein-Nishina cross section. It is noted that,in each scattering, bulk motion as well as thermal motion of the pairs are properly taken into account, under the assumption that the pairs have a Maxwellian distribution at the local temperature. The rate of pair production is calculated based on the local photon distribution using the cross section given in <cit.>.The pair annihilation rate is computed as a function of the local number densityand temperature. Here we use the same analytical function employed in <cit.> which is based on the formula given by <cit.>. As for the spectra of photons generated via the pair annihilation process, we employ an analytical formula derived in <cit.> which is given as a function of temperature. The details of the processes incorporated in our code are summarized in the appendix.In this study we systematically explore the properties of RRMS, with a particular focus on the role of the three parameters defined above. We performed 15 model calculations that cover a wide range of parameters (10^-2≤ξ_u *≤ 10, 10^3 ≤ñ≤ 10^5, and 2 ≤γ_u ≤ 10).Table <ref> summarizes the imposed values for each calculation. The total number of injected photon packets varies among the models, but is typically in therange N_ pack∼ 10^8 - 10^9, which is sufficiently large to avoid significant statistical errors. § RESULTSIn this section, we demonstrate how the differentparametersaffectthe shock properties.Hereafter we refer to the models with γ_u = 2 and ñ = 10^5 as fiducial cases (g2e1n5, g2e0n5, g2e-1n5 and g2e-2n5), sincesuchconditions are likely to prevail in sub-photospheric GRB shocks.Note that the cases ξ_u *≥ 1 andξ_u * < 1correspond to shocks formed below and above the saturation radius, respectively,in the context of the fireball model. §.§ Dependence onξ_u*As forthe fiducial models (γ_u = 2 and ñ = 10^5), wecompute the cases for ξ_u*=10, 1, 0.1, and 10^-2. The obtained shock structures aredisplayedin Fig. <ref>.The horizontal axis in all plots shows the angle averaged, pair loaded optical depth for Thomson scattering, as measured in the shock frame:τ_* = ∫γ (n + n_±) σ_T dz,wheredz denotes the distance element along the flow direction. It is measured from the subshock, when present, where τ_* = 0, and from the location where the bulk velocity has first reached the far downstream value,β≃β_d, when the subshock is absent. As a function of τ_*, the vertical axisshows the4-velocity, γβ, temperature, T,and the pair-to-baryon density ratio, n_±/n. Together with the plasma temperature, we also display the quantity T_γ, eff = I_0^'/3I_0^' for reference, where I_0 and I_0 are, respectively, the 0th moments of the intensity I_ν and the photon flux density I_ν/hν.Henceforth,quantities with and without the superscript prime are measured in the comoving frameand shock frame, respectively.The nth moments of the intensity and photon flux density are defined as follows:I_n = 2 π∫∫ I_ν cos^nθ dν dΩ, I_n = 2 π∫∫I_ν/h ν cos^nθ dν dΩ, (n=0,1,2),where θ is the angle between the flow velocity and thephoton direction.Note that T_γ, eff can be regarded as the actual temperature of the radiation when the distribution is Wien, I_ν∝ν^3exp[hν/(kT)], or Planck.Henceforth, we refer to T_γ,eff as the effective radiation temperature. The angle integrated spectral energy distribution (SED),∫ν I_ν dΩ,computed in the shock frame at a given location,is exhibited in Fig.<ref> for each model at different locations. In Fig. <ref> we showa comparison of the 4-velocity profiles of the different fiducial models, togetherwith the comoving 1st and 2nd moments of the radiation intensity normalized by the 0th moment, I_1^'/I_0^' andI_2^'/I_0^'.When the radiation field is isotropic in the comoving frame, we have I_2^'/I_0^' =1/3 and I_1^'=0, while completely beamed radiation leads toI_2^'/I_0^' =1 and I_1^'/I_0^'=-1 or 1.Although there is some difference between the models,the deceleration of the shock occurs overan optical depth τ* of a few in all cases. This stems from the fact that in the relativistic case the plasma crossing time of the shock is nearly equal to the light crossing time.The relativistic motion of the plasma is also the sole reason why inside the shock transition layerthe radiation appears highly anisotropicin the rest frame of the fluid, as seen in Fig. <ref>.As mentioned in Section <ref>,this is in marked difference to non-relativistic shocks in which the diffusion length is much longer, and the radiation is nearly isotropic. During the deceleration, Compton scattering alsoheats up the plasma to higher temperatures.Regarding the radiation spectra,while a Wien distribution at the local temperature is establishedat the far upstream and downstreamregions,non-thermal distributions originating from bulk Comptonization are produced near the shock transition layer.Apart from these general trends, the details of shock dissipation vary considerably with ξ_u*. In model g2e1n5, the inertia of the flow at far upstream is largely dominated by the radiation(ξ_u*=10). In this case,strong anisotropy cannot develop within the shock,since a small departure fromisotropy issufficient to give significant impact on the bulk flow of the plasma. As a result, the velocity profile is relatively smooth, reflecting gradual decelerationcompared with the cases of lower ξ_u*. As shown in <cit.>,in the extreme limit of ξ_u*=∞, the radiation fieldmust satisfy the force-free condition I_1^'=0.Here (model g2e1n5) the plasma has a finite contribution to the inertia(∼ 10 % of the total), therefore a small but finite anisotropy is present. In this model, the temperature of the plasma coincides withthe effective temperature of the radiation, T_γ, eff, at any position.This is due to the fact thatthe photon distribution is close to Wien, so thatCompton equilibrium (see Section <ref> for details) isestablished throughout the shock. The spectra of photons do not largely depart from the Wien distribution becausebulk Comptonization,which mediates the shock, need not be significant. The resulting temperature shows gradual increase from k T ∼ 30 keV to ∼ 60 keV across the deceleration zone (see upper left panel of Fig. <ref>).Although the change in the temperature is only afactor of ∼ 2 across the shock, significant increaseis found in the pair number density.This is because the pair production rate by photons in a Wien distribution is a sensitive function oftemperature in thisrange (k T ∼ 30-60keV),since only thehigh energy population around the exponential cutoff exceeds the threshold energy for pair creation.As a result,the pair loading, n_±/n, increasesby three orders of magnitude across the transition layer.At the far upstream and downstream regions, pair production and annihilation are in balance andthe number density of pairscan be well approximated by that of Wien equilibrium atnon-relativistic temperatures (see Section <ref>). From Equation (<ref>) (n_±/n ∼ñΘ^-3/2 exp(-Θ^-1)), we obtain n_±∼ 0.35 (n_±∼ 6.2 × 10^2) for a far upstream (downstream) temperature of kT ∼ 30 keV (∼ 60 keV).Indeed, this is in good agreement with our simulation. As the value of ξ_u* decreases, the velocity gradient,d γβ / dτ_*,steepens.This is mainly due to the fact that for a given density ratio ñ, the average energy of photons decreases withdecreasing ξ_u*, since the upstream temperature satisfies T_u∝ξ_u *.As a result, Klein-Nishina effects are diminished, andthe average mean free path of photonsis reduced, ultimately approaching the Thomson limit.A steeper velocity gradient is also required for lowervalues of ξ_u * in orderto increase the efficiency at which the bulk kinetic energy is extracted by bulk Comptonization.As shown in Fig. <ref>,when the photon-to-baryon inertia ratio is reduced to the valueξ_u * = 10^-2 (model g2e-2n5),a smooth velocity profile is no longersufficient to achieve energy-momentum conservation to the required accuracy at every grid point, andour calculations imply the formation of a subshock in the system. It is noted, however, that the subshock is quite weak, in the sense that it carries only a small fraction (a few percents at most)of the entire shock energy, and so do not play an important rolein the dissipation process. Therefore, its impact on the radiation properties is also negligible. Therefore, in what follows we mainly focus onthe global properties of the shock, that are not affected by the subshock.The detailsof the subshock structure will be given later on, in Section <ref>. The bulk Comptonization in the deceleration zone becomes significantas ξ_u * decreases, and results in the emergence of a non-thermalspectrum.As shown inFig. <ref>, the spectral slope is harder for smaller values of ξ_u *.Concomitant with the hardening of the spectrum, the departure from isotropy (as seen in the comoving frame) that develops inside the shock becomes more prominent (bottom panel of Fig. <ref>). The maximum energy attainable through bulk Comptonization is limited bythekinetic energy of the electrons/positrons to about(γ_u - 1) m_e c^2 ∼ 500 keV. When the pair content is small, this corresponds roughly to thecutoff energy of the non-thermal photons at high energies(e.g., models g2e0n5 and g2e-1n5).On the other hand, when gamma ray production via pair annihilation is important, the conversion of rest mass energy leads to a moderate increase inthe cutoff energy, roughly toγ_u m_e c^2 ∼ 1 MeV (e.g., model g2e-2n5).Also note that, although the temperature is non-relativistic,thermal motions slightly shift the energy to higher valuesandproduce a broadening of thespectrum at the highest energies (Fig. <ref>). Since Compton equilibrium is achievedthroughout the shock(except for the immediate post subshock region), as explained in Section <ref>, and higher energy photonscan exchange theirenergy more efficiently via scattering, the presence of non-thermal photons will result in an abrupt heating of the plasma up to a temperature well in excess of T_γ, eff.Therefore, while no departure is found for model g2e1n5 (T∼ T_γ, eff),the deviation of the plasma temperaturefromT_γ, eff becomes more substantialasξ_u *decreases (see Fig. <ref>). This implies the presence of a prominent plasma heating precursor at the onset ofthe shock transition layer for relatively low values of ξ_u *. The pair density profile also changes significantly with ξ_u*.While there is a significant amount of pairs in model g2e1n5,they are negligible in models g2e0n5 and g2e-1n5 (n_±/n ≪ 10^-10).This is a direct consequence of thelower peak energy (approximately 3 k T_γ, eff), that gives rise to an exponential suppression of the number of photons abovethe pair creation threshold.On the other hand, while T_γ, eff is still low, the production of a prominent non-thermal component leads to enhanced pair production in model g2e-2n5.The pairs only appear in the vicinity of the transition layer, since the pair production opacity contributed by the bulk Comptonized photons peaks there.It should be noted that the existence of pairs can changethe spatial width ofthe shock considerably once their density exceeds the baryon density (n_±/n ≳ 1)and begins to govern the scattering opacity inside the shock.For example, the physical length scale per optical depth dz/dτ_*at far upstream is longer than that of the fardown stream by roughly 3 orders of magnitude.Therefore, one should bear in mind that, while the shock width in terms of dτ_* is similar among the models,it could largely differ when measured in terms of the physical length scale dz, even when thesame far upstream density n_u is invoked. §.§.§ Emergence of a weak subshockAs mentioned earlier, emergence of aweak subshock seems necessary in model g2e-2n5.Although its contribution to the overall dissipation is quitesmall, its existence is required to achieve steady flowsolutions (seeSection <ref> for details). As described in Section <ref>,we treat the subshock as a discontinuityin the flow parametersthat satisfy the Rankin-Hugoniot condition for a plasma isolated from the radiation. A notable feature of the subshock is a sharpspike followed by a dip in the velocity and temperature profiles.The drop in the velocity to a value smaller than thefar downstream velocity is an inevitable consequence of the plasma sound speed, c_s ≈ [5 P_pl/3ρ_pl]^1/2,being small(c_s/c ∼ 0.09 for kT∼ 500 keV and n±/n ∼ 10). The rise of the temperature just behind the subshock, up to k T_d,sub∼ 500 keV, is caused by theself-generated heat of the plasma within the subshock.Since the photons cannot interact with particles over the plasma scale, the post shock temperature is well above that obtained in Compton equilibrium. Consequently, followingshock heating, the pairs exposed tothe intense radiation field rapidly cool via Compton scatteringuntil the temperature reaches the equilibrium value (roughly equals to that ahead of the subshock). As a result, a structure that resembles an isothermal shock is formed(see a magnified view in Fig. <ref>). Within the cooling layer (τ_*≲ 0.001), the bulk plasma rapidlyaccelerates,predominantly by its pressure gradient force. Above the cooling layer, the acceleration continues more gradually, mainly due tothe radiation force, up to the distancewhereit reaches the far downstream velocity (at τ_*∼ 0.6) A crude evaluation of the thickness of the cooling layer, dτ_*,cool,can be derived as follows:The number of scatterings per unit time for asingle electron/positronis given by ∼ n_γc σ_Tin the comoving frame.Hence, given the energy loss per scattering, ∼ 4 <hν> k T_d,sub / m_e c^2,and the downstream thermal energy per electron/positron, 3 k T_d,sub,the cooling time is derived ast_cool∼ 3/4 (<hν>/ m_e c^2)^-1(n_γc σ_T)^-1,where <hν> and k T_d,sub denote theaverage photon energy and the temperature at the immediate downstream of the subshock. In terms of the effective temperature,the average photon energy at the subshock can be expressed as <hν> ∼ 3 k T_γ, eff. While the photon number density is approximately n_γ∼ñn over most of the RRMS layer, it is given byn_γ∼ñn β_u,sub/β_d,sub at the immediate downstream of the subshock, owing to thesudden compression of the plasma there, where β_u,sub (β_d,sub) is the velocity at the immediate upstream (downstream) of the subshock.Taking into account the above factors, the cooling layer thickness can be expressed asdτ_*,cool ∼ β_d,sub c (n+n_±) σ_T t_cool∼ 1.5 × 10^-3(kT_γ,eff/2 keV)^-1(ñ/10^5)^-1(n+n_±/n/10) ×(β_u,sub/β_d,sub/8) (β_d,sub/0.03) . Pair creation and annihilation wereignored in the above derivation, as they are negligible over the cooling layergiven its small thickness relative to the entire RRMS transition layer (see bottom right panel of Fig. <ref>). We can confirm fromFig. <ref> (as well as from Figs. <ref>, <ref> and <ref>for the other models with subshocks)that this rough estimation is in agreementwith our numerical results within a factor of a few. Note that there are several factors that were ignored in our crude estimation of the cooling layer thickness, and which can lead to some differences between the analytic and numerical results. For example, we have neglected the effect of adiabatic cooling as well as the effect of broad radiation spectrum.Moreover, in evaluating the cooling rate, we have used an expression which is only valid in the non-relativistic limit, kT_u,sub,kT_d.sub≪ m_e c^2, while the temperature is typically mildly relativistic. In view of these simplifications, we find the mild disagreementbetween the numerical result and the analytic result derived above reasonable. It is worth noting that,while this weak subshock strongly affects the properties of the plasma in its vicinity,it has almost noinfluence on the radiation.This is simply because the thermal energy generated by the subshock,3 (n + n_±) k T_d,sub,is negligible compared withthat contained in the radiation,3 n_γ k T_γ, eff.Therefore, the weak subshock does not affectthe overall energetics of the system nor the radiation properties.This is also true for all the other cases in which subshocks were found, and for which the photon-to-baryon number ratio is sufficiently above the critical valueñ_crt given in Equation (<ref>)(see Section <ref>).While our numerical simulations predicttheir existence,we could not identify the physical origin of the “weak” subshocksthat we found in the regime ñ≳ n_crt(models g2e-2n5, g4e-2n5, g10e-1n5 and g10e-2n5),unlike the case of a photon starved shock, ñ < n_crt (models g2e-1n3 and g2e-2n3), where formation of a“strong” subshock is dictated by inefficient energy extraction thoroughCompton scattering, as will be discuss in greater detailin Section <ref> below. Though non trivial, this presumably indicates thatno steady, continuous flowsolutions exist in a certain regime of the parameter space. As seen in Fig. <ref>, the flow velocity gradient tendsto steepen as the value of ξ_u * is reduced. Our result suggests that there is a threshold value of ξ_u * below which the continuous steepening of thevelocity profile ultimately turns into a weak subshock at theedge of the RRMS transition layer. It is worth mentioning that <cit.> also found a weak subshock in their simulations of photon starved RRMS (in which photon generation is included). It should be stressed, however, that these weak subshocks are merely small disturbancesin the global shock structure, and their physics is not important in evaluating the overall dynamics of the bulk flowas well as the radiation properties. §.§ Dependence on ñTo explore the dependence of the shock propertieson thephoton-to-baryon number ratio, we performed several calculations that invoke smaller values of ñ (10^4 and 10^3) than that used in the fiducial models, but the same values of γ_u and ξ_u*. In the models with ñ=10^4, three cases with different values of photon-to-baryon inertia ratio(ξ_u * = 1, 0.1 and 0.01) are considered (g2e0n4, g2e-1n4 and g2e-2n4).Their overall structures and spectra are summarized inFigs. <ref> and<ref>.As seen, the general trends are quite similar to those ofthe fiducial models; the decrease in ξ_u * results in a steepening of their velocity gradient dγβ/dτ_*and in the enhancement of the non-thermal spectrum. Apart from the similarities, there are also interesting differences from thefiducial models (ñ = 10^5). For a fixed value of ξ_u *, lower ñ results in ahigher temperature (T ∝ñ^-1), since the same amount of energy is shared by a smaller number of particles (photons, protons and pairs). Hence, the overall temperature and average photon energy are roughly 10 times higher in these models. Thisshifts the average mean free path of photons to larger values owning to the increase in the population of photons that are scattered in the Klein-Nishina regime. As a result, thedeceleration lengths are found to be longer than those inthe corresponding fiducial models (g2e0n5, g2e-1n5, g2e-2n5). The higher temperature and photon energy are probably the reason for the absence of a weak subshock in model g2e-2n4,in difference from model g2e-2n5(that has the sameξ_u * value).We speculate that the smoother velocity profile in model g2e-2n4, that resultsfrom the larger penetration depth of the photons, enables the existence of steady solutions with no subshock.However, it is expected that a weak subshock will form also in these models forsufficiently low values ofξ_u * (< 0.01).The larger temperature or, equivalently,average photon energy, also leads to enhanced pair production rate. In particular, while the pair content is negligible for ξ_u * =1 andξ_u * =0.1 in the fiducial models (g2e0n5 and g2e-1n5), the models with ñ=10^4 and the same ξ_u * values (g2e0n4 and g2e-1n4) give rise to a significant amount of pairs.Likewise, the pair density in model g2e-2n4 is higher by an order of magnitudethan thatin the fiducial model g2e-2n5. Comparing the structures, the profiles in model g2e0n4 are similar to those in model g2e1n5, rather than in model g2e0n5 that has the sameξ_u * value.Accordingly, as inmodel g2e1n5, the radiation and pairs are well approximated to be in Wien equilibrium at far upstream and downstream, while the in the transition layer they depart from the equilibrium due to a slight deviation fromthe Wien distribution. On the other hand, the shape of the spectrum inmodels g2e-1n4 and g2e-2n4 is similar to that of the counterpart fiducial models with same ξ_u *(g2e-1n5 and g2e-2n5),but its average energy is shifted toward higher energies, by a factor of ∼ 10, while the cutoff energy remains unchanged ∼γ_u m_e c^2 ∼ 1 MeV. The higher photon energies in models g2e-1n4 and g2e-2n4 give rise to a higher pair production rate than in the fiducial models. Therefore, in all of these models, we find a non-negligible pair content.The properties of the shocks drastically change in the models with ñ=10^3.In the present study, two cases with the values ξ_u * = 0.1 and 0.01 are computed (g2e-1n3 and g2e-2n3).Their overall structures and spectra are exhibited inFigs. <ref> and<ref>, respectively.The notable difference from the models with higher ñ is theformation of a “strong” subshock. Unlikethe “weak” subshocks found in some of the other models(see Section <ref> for details),the physical origin of the strong subshocks is understood, and will be described in detailin Section <ref>.As Equation (<ref>) predicts, the temperature downstream of the subshock in the models with ñ=10^3 approaches the pair equilibrium value, kT ∼ 200 keV, as seen in Fig. <ref>.The pair density increases rapidly inside the shock and approaches the Wien equilibrium value, n_±≈ n_γ K_2(Θ^-1)/ Θ^2, just downstream of the subshock. At this temperature the pair density becomes comparable to the photon density, n_±∼ n_γ.Since in the absence of an internal photon source the number of quanta is conserved, we have ñ=(n_γ+n_±)/n in the downstream region,which effectively reduces the number of photons that can extract energy, and strengthens the subshock further.One should keep in mind that while the subshock is relatively strong, it dissipates only about 30% of the entire shock energy (in model g2e-2n3).Thus, a moderate increment in the photon density downstream (by no more than a factor of a few) will considerably weaken, or completely eliminate, the subshock.We anticipatethis to happen once internal photon sources (in particular free-free emission by the hot pairs) are included. Shock solutions that correspond to the fiducial models with fixed ξ_u*=0.1 are compared, for clarity, in Fig. <ref>.The distinct properties of the marginally starved shock (g2e-1n3) stand out. The discontinuity in the profiles of the moments I_1^'/I_0^' andI_2^'/I_0^' in the marginally starved shockarises from the sudden change in the velocityof fluid elements (and, hence, in the frame in which these moments are computed) across the subshock.§.§.§ Transition to the photon starved regime Next, let us examine the transition from the photon rich to the photon starved regime in some greater detail.In section <ref> it has been argued thatonce the photon-to-baryon number ratio far upstream becomes smaller than the criticalvalue ñ_crt,the advected photons cannot support the shock anymore, and the shock becomes photon starved. In the absenceof photon production processes oneexpects that the strength of the subshock will dramatically increase as ñ approaches ñ_crt≃ 10^3. This isthe situation in models g2e-1n3 and g2e-2n3.Fig. <ref> exhibits results obtained for these models, verifying that the subshock is indeed substantially stronger than in the runs with ñ>ñ_crt. As also seen, the downstream temperature reaches 200 keV (except for the spike produced by the subshock),in accord with Equation (<ref>), leading to a vigorous pair creation in the shock transition layer. The pair-photon plasma downstream quickly reaches equilibrium, with roughly equal densities, n_±/n_γ≃1.From Equation (<ref>) it is anticipated that under these conditions photon generation (not included in our simulations) will start dominating over photonadvection, so that in reality the shock will be supported by photons produced inside and just behind the shock, and the subshock will disappear or remain insignificant.For higher upstream Lorentz factors, γ_u ≫ 1, we expect that photon generation will dominate at somewhat higher ñ values, roughly by a factor of γ_u/2,since even though the shock can be supported by the advected photons the temperature downstream exceeds the pair production threshold, at which e^± pairequilibrium is established. The results of <cit.> confirm this. We are currently in the process of modifying the code to include free-free anddouble Compton emissions.Results of simulations of photon starved shocks will be presented in a future publication.§.§ Dependence on γ_uTo investigate the dependence of the shock propertieson the Lorentz factor, we have calculated two sets of models with higher γ_u (4 and 10), but with the values of ñ and ξ_u* being identical to those in the fiducial models. In both cases, three calculations that invoke different ξ_u* values (1, 0.1 and 0.01) were performed, and are described next.The structures and spectra obtained in the models with γ_u = 4 and γ_u = 10 are displayed in Figs. <ref> - <ref>.Like in the fiducial models, also here the velocity profile steepens as ξ_u* is reduced.The trends of the temperature profile are also similar to those in the fiducial models, albeit with a higher downstream temperature, since it is roughly proportional to the 4-velocity far upstream when ξ_u *≲ 1(see Equation (<ref>)).At low values of ξ_u* a weak subshock appears (see Figs. <ref> and<ref>for a magnified view), as in the fiducial models. The larger γ_u the larger the value of ξ_u* at which the subshock forms (ξ_u*≤0.01 for γ_u=4 and ξ_u*≤ 0.1 for γ_u=10).The reason for this is unclear at present.It might be related to the fact that the condition for starvation is proportional to γ_u(see Equation (<ref>) ).The main effects caused by increasing the shock Lorentz factor can be observed in the resulting spectra and pair populations, andcan be summarized as follows: (i) The heating precursor broadensand the peak temperature increases as γ_u increases, and likewise the width of the shock transition layer. (ii)The pair content rises sharply as γ_u increases, as is evident from a comparison of Figs. <ref> and <ref>. This is a direct consequence of the fact that the number of bulk Comptonized photons that surpass the pair production threshold and, hence, the pair production rate, are sensitivefunctions of γ_u. The large pair enrichment gives rise to a pronounced signature of the 511 keV pair annihilation line in thespectrum(the small spectral bumps seen in Figs. <ref> and <ref>). (iii) For fixed values of ñ and ξ_u* the high energy cutoff of the spectrum isroughly proportional to γ_u, as naively expected. To summarize the dependence of the shock structure on the bulk Lorentz factor, we compare, in Fig. <ref>, the profiles of γβ,I_1^'/I_0^' andI_2^'/I_0^' in the three models (g2e-1n5, g24-1n5 and g10e-1n5) thathave different values for γ_u but same valuesofξ_u* (=0.1) and ñ (=10^5).As seen, the shock width slowly increases with increasing γ_u, in rough agreement withthe analytic solution derived in Section <ref>.The level of anisotropy of the photon distribution and its extentalso become larger as γ_u is increased. In the highest Lorentz factor case, the radiation intensity achieves nearly complete beaming (I_2^'/I_0^'=1, I_1^'/I_0^'=-1).This reflects the rise in the population of high energy photons that penetrate against the flow to larger distances upstream. § APPLICATIONSSo far, we have focused on the fundamental properties of RRMSs. Here let us consider the applicationsto GRBs. Since RRMSs are expected to form in sub-photospheric regions, they should havesubstantial imprints on the resulting emissions <cit.>.As shown in the previous section, when the energy density of the radiation at far upstreamis much larger than that of therest mass energy of the plasma, viz., ξ_u *≫ 1, thermal spectra with roughly the same peak energy and flux are found at any location in the shock(see top left panel of Fig. <ref>). This implies that observed spectra produced by a sub-photospheric shock (even if strong)should benearly thermal when the photosphere is located far below the saturation radius. On the other hand, significant broadening is expected when the rest mass energy is comparable or larger than that of the radiation at far upstream (ξ_u *≲ 1). This corresponds to shocks that form around or above the saturation radius. To gain some insight into how the radiation will be seen by an observer during the breakout of a RRMS under such conditions,we plot, in Fig. <ref>,spectrathat were averaged over a certainphysical interval Δ z, for models g2e-1n5 and g2e-2n5. In addition to the angle integrated spectra that are computed by summing up the contribution of photonsin all directions (4 π steradians), we also display cases where only the photons in a half hemisphere (2 π steradians) propagating along (θ < π /2) andagainst (θ > π / 2) the flow are summed up. The former (latter) represents the spectra emitted during the breakout of the reverse (forward) shock that is advancing relativistically in the central engine frame.Hereafter we (loosely) refer to the cases that correspond to photons propagating along and against the flow asreverse and forward shock, respectively.We emphasize that oblique shocks, that are likely to form near the photosphere,are also referred to here as reverse shocks, as their radiation escapes to infinity along the flow[Note that upon appropriate Lorentz transformation oblique shocks can be transformed into perpendicular shocks.].We further point out that the spectra exhibited in Fig. <ref> are computed in the rest frame of the shock. In GRBs this frame moves at a high Lorentz factor with respectthe observer, and so the observed emission is strongly beamed. Thus, photons moving along the flow may significantlycontribute to the observed spectrum also in forward shocks, depending on viewingangle.Our definition of "forward" and "reverse" in regards to the integrated spectrumis merely for explication.As expected, there is a prominent hard component extending above the peakin the case of emission from a reverse shock. It is produced by bulk Comptonization around the RRMS transition layer. The spectrum emitted by a forward shock, on the other hand, lacks sucha component (although it is broader than an exponential cutoff), since the high energy photons produced bybulk Comptonization move preferentially along the bulk flow.In both cases, the portion of the spectrum below thepeak is softer (broader) than a thermal spectrum. This isdue to the moderately bulk Comptonized component in which energy gain by scattering is notso significant,as well as due to the superposition of thermal-like spectra emitted from the upstream and downstream regions.A substantial hardening is also seen at the lowest energies (below the thermal peak of the radiation upstream), sincenone of the above mentioned broadening effects can play a role. In comparison withobservations, the spectral slopes below and above the peak energyin reverse shocks fall well within the range of detected values.For example, the reverse shock in model g2e-2n5 has low energy andhigh energy photon indices (dlogI_ν/dlogν - 1) in the range-1.5 ≲α≲-1 and -3 ≲β≲-2.5 for the cases shown inFig. <ref> (middle right panel), which are indeed ingood agreement with the observations <cit.>.Similar values are found also for the low energy spectral index α in forward shocks. On the other hand, unlike in reverse shocks, in forward shocks the spectrum above the peak shows a sudden drop off, and is incompatible with a power-law fit. While this is in conflict with Band-like spectra, it is consistent with models that prefer an exponential-cutoff to fit observations, although it could be that those models are misled by an artifact of poor photon statistics at high energies <cit.>. It should be noted, however, that in general the shape of thespectra emitted from a certain fluid shellvary with the width of the shell, or in other words, with spatial interval Δ z over which they are averaged.As we extend the length of this interval,the contribution from the far downstream and/or upstream regions increases and, therefore,the average spectrum asymptotes to a thermal spectrum.[More accurately, it asymptotes to the superposition oftwo thermal componentsthat have far upstream and downstream temperatures. The relative strength is determined by the ratio of spatially integrated intensity in each region.] The reason is thatthe bulk Comptonized component is confined to the vicinity of the shock transition layer by virtueofefficient downscattering of high energy photons by the downstream plasma. This means thatin case of a single shock that formed at a distance below the photosphere which is muchlarger than the width of the shock transition layer, while the signal aroundthe time when the shock reaches photosphere canbe highly non-thermal, the time integrated spectrum, that is, the spectrum integrated over the entire duration of burst, would appear quasi-thermal. The broadening at sufficiently low energies might still prevail even then, since its energy exchange rate with electrons is slower than that of the high energy photons. Hence, the low energy broadening has a largerchance to be observed in the overall spectrum. A more detailed analysis of the properties of photospheric GRB emission requires some knowledge abouthow shocks are distributed within the outflow, which in turn determines the relative importance of each emission region. This is set by the nature of the central engine as well as by the environment into which theoutflow is propagating.Moreover, our calculations are restricted to infinite, steady shocks in planar geometry. While our analysis can describe the shocks at regions well beneath the photosphere, it cannot adequately address the breakout phase during which the photons diffuse out from the system. One must take into accountthe drastic change in the shock structure during breakout<cit.> for a more accurate analysis of the released emission. To that end, dynamical calculations must be performed which is beyond the scope of the present study. Nevertheless, we emphasize that our steady-state simulations confirm that abroad, non-thermal spectrum is an inherent feature of RRMSswhichshould also be present at the breakout phase. Although more sophisticated computations are necessary for a firm conclusion, we suggest that sub-photospheric shocks may provide a possible explanation for the non-thermal shape in the observed prompt emission spectraof GRBs.§ SUMMARY AND CONCLUSIONSWe performed Monte-Carlo simulations of relativistic radiation mediated shocks for a broad range of upstream conditions.Since photon generation is not included in the current version of the code our results are applicable only to photon-rich shocks, for which the shock is supported by scattering ofback streaming photons that were advected by the upstream flow.To gain insight into the physical processes that shape the structure and spectrum of the shock,the results of the simulations are compared with analytic results whenever possible.Our simulations confirm that the transition from photon rich to photon starved regime occurs when the photon-to-baryon number ratio far upstream satisfies ñ≃ (m_p/m_e)γ_u, as expected from an analytic comparison of the advection rate and the photon generation rate by the downstream plasma. At this critical value the downstream temperature approachesthe saturation value, roughly 200 keV,at which it is regulated by vigorous pair creation <cit.>. At sufficiently higher values of ñ the downstream temperature is much lower, pair loading is significantly reduced, and the shock is supported by the advected photons.We find that the deceleration of the bulk plasma occurs over a scale ofa few pair loaded Thomson depths, with only a weak dependence on upstream conditions; the actual physical scale may be much smaller in cases where vigorous pair production ensues.The shock width increases, but only slightly,when the relative contribution of high energy photons, that are scattered in the KN regime,becomes larger. This is in difference to photon starved shocks in which the shock width is essentially governed by KN effects <cit.>. We also find that in the photon rich shocks we studied,the temperature of the plasma is determined almostsolely by the Compton equilibrium throughout the shock, owing tothe high photon-to-baryon number ratio (ñ≫ 1),with the exception of the immediate downstream temperature of the subshock whenever it is present. Apart from these common features, our simulations indicatethat the properties of the shock and its emission has a notable dependence on the upstream parameters, ξ_u *, ñ, and γ_u.Below we summarize our main findings:*When the energy density of the radiation far upstream largely exceeds the rest mass energy density (ξ_u *≫ 1), the net increase in the radiation energyacross the shock is small. The dominance of the radiation renders the Lorentz factor profile smooth and broad;any attempt of steepening is readily smeared out by the large radiation drag acting upon the plasma. Since the plasma cannot affectmuch the radiation, the photon distribution is well described by aWien distribution with a temperature that is equal to that of the local plasma temperature throughout the shock.In our fiducial model with ñ=10^5 the large value of ξ_u * renders the temperature high enough to induce significant pair production. The resulting population of pairs in this case can bewell approximated by the Wien equilibrium.* The situation changes drastically when ξ_u *≲ 1. In this regime the radiation inside the shock becomes highly anisotropic, and a significant fraction of the upstream bulk energyis converted, viabulk Comptonization of counter streaming photons, to high-energy photons. The consequent photon spectra exhibit a broad, non-thermal componentthat extends up to an energy of ∼γ_u m_e c^2.The spectrum inside the shock becomes harder for lower values ofξ_u *,leading to enhanced pair creation by virtue of the increased number of photons with energies in excess of the pair production threshold. The large pair enrichment in models with high γ_u and low ξ_u* gives rise to a signature of the 511 keV annihilation line in the spectrum. * It is also found that for sufficiently lowvalues of ξ_u *a weak subshock appears,although we were not able to identify its physical origin.Its effect on theoverall structure and emission of the shock is negligible,since it only dissipates a small amount of the total bulk kinetic energy.Thus, in practice the presence of the subshock is unimportant for the overall analysis of RRMS, and it is merely of academic interest. The above discussion excludes the cases with ñ≃ m_p/m_e (models g2e-1n3 and g2e-2n3), thatdelineate a transition between photon rich and photon starved RRMS, and for which a strong subshock was found.As explained above, the presence of a strong subshock in those models is an artifact that stems from the omission ofphoton production process in our code.Once included, this subshockshould become weak <cit.>.* In order that advected photons will be able to extract the entire upstream bulk energy, the number of advected photonsper baryonshould exceed m_p/m_e (i.e., ñ>m_p/m_e).As stated above,this value marks the transition between photon rich and photon starved shocks. In the photon rich regime the value of ñ merely determines the downstream temperature, that scales as T∝ñ^-1.As a consequence, the pair production rate depends sensitively on the value of ñ.* The dependence on the bulk Lorentz factor γ_u is relatively monotoniccompared to the other two parameters.As the Lorentz factor increases, themaximum photon energy attainable throughbulk Comptonization increases as ∼γ_u m_e c^2.Thus, the resulting photon spectra extends to higher energies.Theemergence of high energy photons thatcan diffuse backto larger distances in the upstream region also leads to an increase in the shock width,in the peaktemperature in the heating precursor, and in the density of pairs inside the shock.We also considered the application to GRBs, and haveshown that spectracompatible with the observations can be produced within RRMSs.In particular, we demonstrate thatthe significantspectral broadening occurring in RRMS with ξ_u * < 1 can reproducethe typical Band-like spectrum.This result suggests that RRMS may be responsible, at least in part,for the non-thermal features found in the prompt emission spectra.However, our analysis is limited to infinite, planar shocks, and cannot accountfor the change in shock structure and emission caused by photon escape duringthe breakout phase, that may alter the observed spectrum. We intend to carry out detailed analysis ofbreakout emission in a future work. § ACKNOWLEDGMENTSWe thank A. Beloborodov, I. Vurm, C. Lundman, D. Ellison and E. Nakar for fruitful discussions. This work is supported by the Grant-in-Aid for Young Scientists (B:16K21630) from The Ministry of Education, Culture, Sports, Science and Technology (MEXT). Numerical computations and data analysis were carried out on XC30 and PC cluster at Center for Computational Astrophysics, National Astronomical Observatory of Japan and at the Yukawa Institute Computer Facility.This work is supported in part by the Mitsubishi Foundation, a RIKEN pioneering project `Interdisciplinary Theoretical Science (iTHES)' and 'Interdisciplinary Theoretical & Mathematical Science Program (iTHEMS)'. AL acknowledges support by a grant from the Israel Science Foundation no. 1277/13 99[Abdo et al.(2009)]Abd09 Abdo A. 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P., 1967, Physics of shock waves and high-temperature hydrodynamic phenomena§ MONTE-CARLO RADIATION TRANSFER CALCULATIONThe Monte-Carlocode used in this study handles transfer of photons in a medium at which Compton scattering,pair production, and pair annihilation takes place.We iteratively perform many calculation runs, in order to find the steady-state shock profile. In each iterative step, the photon transfer is solved under a givenprofile of number density, n and n_±, temperature, T, and velocity, β of the plasma as follows:We track the propagation of packets which are ensemble of photons that have identical 4-momentum, P_γ^μ=(hν/c, (hν/c)n), where n denotes the unit 3-vector along the propagation direction. The photon packets are injected at the inner and outer boundaries which are located at the far upstream and downstream regions, respectively. In addition, pair annihilation processes adds photons into the calculation domain. After the injection,the evolution of the injected packetsare computed until theyreach theboundaries of the calculation domain or become absorbed via pair production process. During the propagation, they are subject to multiple scatterings by the pair plasma. Between each scattering event, the packet travels in the direction along the 3-vector n. After the scatterings, their4-momenta are updated based on the differential cross section of Compton scatterings, and the propagation directionis changed to newly determined n. Once we finish the calculation, quantities such as total energy-momentum flux and distribution function of photons are evaluated by sampling all of the simulated packets at each grid point, For each photon packet,distances prior to the scattering and absorption events are determined by drawing the corresponding optical depths, δτ_ sc and δτ_±. The probability for the selected optical depth to be in the range of [δτ, δτ + dτ] is given by exp(- δτ)dτ. From the given optical depths, path lengths to the scattering or absorption events in the laboratoryframe (shock rest frame)are determined from theintegration of opacity along the ray of photons, which can be expressed asδτ = ∫^l_0D^-1α^' dl,where D = [γ (1 - β cosθ)] is the Doppler factor.Here α^' denotes the opacity for the corresponding process in the comoving frame of the plasma which can be evaluated from the local physical conditions (see below for detail). Below, we summarize how the injection of photon packets as well asscattering and absorption processes are treated in our code.Hereafter, we label quantities that are measured in the comoving frame of the bulk plasma with the superscript prime symbol.§.§ Boundary Conditions At the boundaries located far upstream and downstream, we assume the photons are isotropic in the comoving frame andhavea Wien distribution characterized by thelocal plasma temperature.Therefore,the photon flux densityat the boundaryin the laboratory frame is a function of thephoton number density and temperature, and can be written asdN_γ/dt dν dΩ dS =D^2dN_γ/dt^' dν^' dΩ^' dS^' ,where dN_γ/dt^' dν^' dΩ^' dS^' = n_γ/8 π(h/k T)^3 ν^'2 exp( - h ν^'/k T) . Thus, for a given range of solid anglesdΩ and frequencies dν, dN_γ/dt dν dΩ dS (n_γ, pack)^-1 cosθ dΩ dν gives the injection rate ofthe packet number per unit area of the boundary surface, where n_γ,pack is the number of photons contained in a single packet. §.§ Pair annihilationThe pair annihilation rate per unit volumeis evaluated as a function of the pair number density and temperature:( dN_±/dt dV)_ ann= - (n + n_±/2) (n_±/2) c σ_T r_± (Θ),whereσ_T is the Thomson cross section. Here r_± is an analytical function introduced by <cit.> based on the formula shown in <cit.>, which is given byr_± = 3/4[1 + 2 Θ^2/ ln(2 η_E Θ + 1.3)]^-1,where η_E = e^-γ_E≈ 0.56146 and γ_E ≈ 0.5772 is the Euler's constant.It is noted thatthe above quantityis Lorentz invariant (i.e., dN/dt dV = dN/dt^' dV^').As for the energy spectrum of thephotons produced via pair annihilation, we use an fitting formula given in <cit.>, which approximates the exact emissivity in a wide range of temperatures. By normalizing the given function to be consistent with the Equation (<ref>), it can be written as( dN_γ/dt dν dΩ dV)_ ann = D( dN_γ/dt^' dν^' dΩ^' dV^')_ annwhere(dN_γ/dt^' dν^' dΩ^' dV^')_ ann = {[Q_1 Θ^0.5 x_ν^'3/2 exp(- x_ν^' + x_ν^'-1/Θ) C(x_ν^'Θ)/K_2(1/Θ)^2 for x_ν^'Θ≤ 20 ,; Q_2 x_ν^' (ln4 η_E x_ν^'Θ - 1)exp( -x_ν^'/Θ) C(x_ν^'Θ)/K_2(1/Θ)^2 for x_ν^'Θ > 20 .; ] .Herex_ν^' = h ν^' / (m_e c^2),andK_2 denote the2nd order modified Bessel function of the second kind. In evaluating the Bessel function, we used the approximate formula K_2(1/Θ)^2 = 4 Θ^4exp(-2/Θ) × [1 + 2.0049 Θ^-1 + 1.4774 Θ^-2+ π (2Θ)^-3 ],which is also given in <cit.>.The function C is an analytical function given byC(y) = {[ 1 + 6.8515487 y + 1.4351694 y^2 + 0.001779014 y^3/1 + 4.63115589 y + 1.5253007 y^2 + 0.04522338 y^3for y ≤ 20 ,;1 + 2.712 y^-1 - 55.6 y^-2 + 1039.8 y^-3 - 7800 y^-4 for y > 20 .; ].The normalization factors Q_1 and Q_2 are determined from the condition∫∫dn_γ/dt dν dΩ dV dν dΩ = -(N_±/dt dV)_ ann. In our code, for a given range of solid anglesdΩ and frequencies dν, ∫∫( dN_γ/dt dν dΩ dV)_ ann (n_γ, pack)^-1dΩ dν gives the injection rate ofthe packet number per unit volume in the calculation domain. §.§ Compton scatterings In evaluating the opacity of photons to Compton scattering,we fully take into account the thermal motion of the plasma and Klein-Nishina effects. As a function of the photon frequency,local density of pairs and temperature, it is calculated asα_ sc^'(ν^') = ∫∫∫ F_ sc( P_e, T, ν^')d P_e^3,whereF_ sc( P_e, T, ν^') = (1 - β_ecosθ_e γ) (n + n_±) f_B( P_e, T) σ_ sc(ν^”) ,andf_B(P_e, T) = 1/4π (m_e c)^3 ΘK_2(1/Θ) exp(- E_e/kT) ,isthe Maxwell-Jüttner distribution function andσ_ sc(ν)=3/4σ_T [ 1 + x_ν/x_ν^3{2 x_ν (1+x_ν)/1+2 x_ν -ln(1+2x_ν) } +1/2x_ν ln(1+2 x_ν) - 1 + 3x_ν/(1+2x_ν)^2] ,is the total cross section for Compton scattering. Here ν^” denotes the frequency in the rest frame of pairs. The quantitiesP_e, β and θ_e γ are, respectively, the spatial components of the 4-momentum, the 3-velocity measured in units of the light speed, andthe angle between the photon and pair directions measured in the comoving frame, andP_e = | P_e| and E_e = ( (m_e c^2)^2 + (P_e c)^2 )^0.5.By plugging in the evaluated opacity in Equation (<ref>), wedetermine the distance for the photon to propagate before scattering. Once the position of the scattering event is determined, we choose the4-momentum of a thermal pair that will interact with the photon. The probability for the pair within a range d P_e^3to be drawn is given by F_ sc d P_e^3 / α_ sc. Then we transform the 4-momentum of photons to the rest frame of thechosen electron/positron anddetermine the 4-momentumafter the scattering based onthe probability given by thedifferential cross section of Compton scattering:dσ_ sc/dΩ=3/16πν_1^2/ν^2( ν/ν_1 + ν_1/ν -sin^2 θ_ sc) ,where ν_1 = ν [1 + x_ν(1-cosθ_ sc)]^-1 is the frequency after the scattering and θ_ sc is the angle between the propagation directions of the incident and scattered photon. Finally we transform back the 4-momentum ofthe scattered photontothe laboratory frame and repeat the above cycleuntil the packet is either absorbed or reaches the surface of the computation boundaries.§.§ Pair productionFor a given 4-momentum of incident photon in the comoving frame,P_γ^' μ=(hν^'/c, hν^'/cn^'), the opacity for the pair productionis calculated asα_γγ^'(ν^',n^') = ∫∫∫(1-cosθ_γγ)f_γ^'(P̃_γ^') σ_γγ(ν^', ν̃^', θ_γγ) d̃P̃_γ^3,where P̃_γ = h ν̃/c ñ and θ_γγ denote the spacial component of the target photon 4-momentum andthe angle between the propagation directions of the incident and target photons, respectively. Here f_γ^'(P̃_γ^') is the distribution function of photons in the comoving frame.The cross section for the interaction, taken from <cit.>, is given byσ_γγ(ν, ν̃, θ_γγ) =3/16σ_T (1- β_ cm^2) × [ (3 - β_ cm^4) ln( 1 + β_ cm/1 - β_ cm)- 2 β_ cm (2- β_ cm^2) ] for ν≥ν_ thr,and σ_γγ = 0 forν < ν_ thr, wherehν_ thr = 2 m_e c^2 [h ν (1 -cosθ_γγ) ]^-1 is the threshold energy for pair production and β_ cm= √(1 - 2 m_e^2 c^4/ [(1- cosθ_γγ)h^2νν̃])is the velocity of the pairs in the center of momentum frame. As in the case of Compton scattering, the distance for the photon to propagatebefore being absorbed via the pair production process is computed by substituting the above opacity in Equation (<ref>).In our code, the local photon distribution functionf_γ is determined by recording photon packet in each grid point during a single run of the simulation. Since we cannot a-priori know the distribution of the current run before calculation,in each run, we use thephoton distribution obtained in the previous step in evaluating the opacity.To ensure the self-consistently of our calculation, we continue the iteration until the difference between thephoton distribution evaluated in the current and previous run becomes sufficiently small. § CALCULATION OF THE PAIR DENSITY PROFILE In our code, independent from the energy-momentum conservation, we must determine thepair density profile whichsatisfy the steady state condition: γβ c d(γ n_±β)/dz = ( dN_±/dt dV)_ ann +( dN_±/dt dV)_ pr,where dz denotes the distance element and( dN_±/dt dV )_ pr is the pair production rate. While the annihilation rate is calculated based on the local quantities as shown in Equation (<ref>), the production rate is evaluated by summing up the number of packets thatare absorbed during the propagation in each grid points. During the iteration, pair density profile is modified from the previous iteration step in order to minimize the deviation from the above condition. We continue the iteration until the error becomes sufficiently small. §.§ Wien equilibrium When the thermal temperature exceeds k T ∼ 30 KeV, copious pairs are produced and can reach the equilibrium state where pair production and annihilation is balanced (Wien equilibrium). In this case, the number density of pairs can be derived asn_± = n ( √(1 +(n_γ/n)^2( K_2(Θ^-1)/Θ^2)^2 ) - 1 ) . In the limit of n_γ≫ n, it asymptotes ton_±≈ n_γK_2(Θ^-1)/Θ^2 .If the temperature is non-relativistic,Θ≪ 1, the above equationcan be further approximated asn_±≈√(π / 2) Θ^-3/2 exp (- Θ^-1) .§ CALCULATION OF THE TEMPERATURE PROFILE In principle, the plasma temperature profile can be self-consistently obtained bydetermining the plasma profile for which energy-momentum conservations is satisfied. It is, however, numerically difficult to constrain the temperature profile accuratelyfrom the condition F_m =const, F_e=const using our iterative method. This stems from the fact thatsince the contribution of the thermal energy is always much smaller than that of the rest mass energy (ρ_pl≫ e_pl), small numerical errors can lead to large errors in the temperature. On the other hand, the radiation field responds non-linearly to changes in temperature, thereby renderingthe calculations unstable and preventingconvergence to the steady state solution. In order to overcome this numerical difficulty, we impose an additional constraint that can be derived from the energy-momentum conservation equations. The equation solves the evolution ofinternal energy density in comoving framewhich is given byγβ c de_pl/dz=( dE^'/dt^'dV^')_ sc + ( dE^'/dt^'dV^')_ ann + ( dE^'/dt^'dV^')_ pr +(e_pl + P_pl)γβ c dn/dz,where( dE^'/dt^'dV^' )_ sc,( dE^'/dt^'dV^' )_ ann, and ( dE^'/dt^'dV^' )_ pr denotethe net heating/cooling rate of the plasma by Compton scattering, pair annihilation, and production, respectively. The last term corresponds to the contribution of adiabatic heating (cooling) due to compression (expansion). While the adiabatic cooling term is evaluated from the local density gradient, ( dE^'/dt^'dV^' )_ sc,( dE^'/dt^'dV^' )_ ann, and ( dE^'/dt^'dV^')_ pr, are evaluated by summing up the contributions of scattering, annihilation and production of photon packets ineach grid point. By using the above equation, we could successfully obtain an accurate temperature profile. In our calculations,the temperature is almost solely determined by the condition ofCompton equilibrium (net heating and cooling by Compton scattering is balanced, that is.,( dE^'/dt^'dV^')_ sc∼ 0. This is owning to the fact that the heat capacity of the plasma is extremelysmall compared to that of the radiation field, since the photon-to-lepton density ratiois large n_γ/(n+n_±)≫ 1. As a result, any small deviation from the equilibrium willinevitably washed out by the numerous photons within a length scale much smaller thanthe Thomson mean free path τ≪ 1. Notable deviation from the Compton equilibrium temperature is only seen at the immediate downstream region of subshockwhere instantaneous viscous heating occurs.Note that the downstream of the subshocks in models g2e-1n3 andg2e-2n3 has a large extent (τ_*≳ 1) where equilibrium is not established, owing to the vigorous pair enrichment that renders the heat capacity of the plasma comparable to that of the radiation (n_γ∼ n_±). | http://arxiv.org/abs/1709.08955v2 | {
"authors": [
"Hirotaka Ito",
"Amir Levinson",
"Boris E. Stern",
"Shigehiro Nagataki"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170926120153",
"title": "Monte-Carlo simulations of relativistic radiation mediated shocks: I. photon rich regime"
} |
[email protected] Department of Physics, School of Sciences, HNBG Central University, Srinagar, Uttarakhand 246174, INDIA. [email protected] Department of Physics, Himachal Pradesh University, Shimla 171005, INDIA. [email protected] Department of Physics, Himachal Pradesh University, Shimla 171005, INDIA.We investigate the texture structures of lepton mass matrices with four (five) non-zero elements in the charged lepton mass matrix and three (four) vanishing cofactors in the neutrino mass matrix. Using weak basis transformations, all possible textures for three and four vanishing cofactors in M_ν are grouped into 7 classes and predictions for the unknown parameters such as the Dirac CP violating phase and the effective Majorana mass for the phenomenologically allowed textures have been obtained. We, also, illustrate how such texture structures can be realized using discrete Abelian flavor symmetries.Neutrino mass matrices with three or four vanishing cofactors and non diagonal charged lepton sector Radha Raman Gautam December 30, 2023 ====================================================================================================§ INTRODUCTIONSolar, atmospheric, and reactor neutrino experiments in addition to the more recent neutrino production from acceleration-based beams have provided some novel results over the last two decades or so and, invariably, strengthened the flavor Standard Model (SM). However, some critical ingredients like leptonic CP violation, neutrino mass hierarchy and neutrino masses are still missing. Furthermore, the nature of neutrinos (Majorana/Dirac) and absolute neutrino masses are still open issues. While the developments over the past two decades have brought out a coherent picture of neutrino mixing, the neutrino mass hierarchy, which is strongly correlated with the neutrino masses and the CP phase δ is still unknown. Specifically, the sign of |Δ m^2_31|=|m^2_3-m^2_1| is still unconstrained and is the focal issue for several ongoing and forthcoming experiments. In addition, recent neutrino oscillation data hint towards a non-maximal atmospheric mixing angle (θ_23) which implies two possibilities: θ_23 < π/4 or θ_23 > π/4 <cit.> which when combined with the θ_13-δ <cit.> and the ±Δ m^2_31-δ degeneracy <cit.> leads to an overall eight fold degeneracy <cit.>. In the SM, all fermion masses are Dirac masses which are generated via the Higgs mechanism. In order to have massive Dirac neutrinos, one has to necessarily enlarge the SM particle content by introducing right-handed neutrinos ν_R. The ν_L and ν_R form a Dirac spinor Ψ_ν=ν_L+ν_R where ν_R are the additional spin states for the neutrinos. However, the gauge singlets ν_R can have a Majorana mass term ν_R^T C^-1 M_Rν_R where M_R, in general, is not diagonal in the flavor basis where M_D is diagonal. Diagonalizing the full mass term leads to Majorana neutrinos and new mass eigenstates. Of course, one can attempt to forbid M_R by postulating an additional symmetry such as lepton number conservation. The mass matrix for Majorana neutrinos is, in general, complex symmetric containing nine physical parameters which include the three mass eigenvalues (m_1, m_2, m_3), the three mixing angles (θ_13,θ_12, θ_23) and the three CP-violating phases (α, β, δ). The two mass-squared differences (Δ m^2_12, Δ m^2_23) and the three neutrino mixing angles (θ_12, θ_23, θ_13) have been measured in solar, atmospheric and reactor neutrino experiments. While the Dirac-type CP-violating phase δ will be probed in the forthcoming neutrino oscillation experiments the neutrinoless double beta decay searches will provide additional constraints on neutrino mass scale. The last unknown mixing angle (θ_13) has been measured with a fairly good precision by a number of recent <cit.> experiments. It is, however, clear that the neutrino mass matrix which encodes the neutrino properties has several unknown neutrino parameters which will remain undetermined even in the near future. Thus, the phenomenological approaches aimed at reducing the number of independent parameters are bound to play a crucial role in further development. There are several classes of predictive models in the literature such as texture zeros <cit.>, vanishing cofactors <cit.>, hybrid textures <cit.> and equality between elements <cit.> which explain the presently available neutrino oscillation data. Neutrino mass matrices with texture zeros and vanishing cofactors are particularly interesting due to their connections to flavor symmetries. Neutrino mass models with texture zeros and vanishing cofactors have been widely studied in literature <cit.> for this reason. The lepton mass matrices with texture zeros and vanishing cofactors in both the charged lepton mass matrix M_l and the neutrino mass matrix M_ν have been systematically studied in Refs. <cit.>. Lepton mass matrices where the charged lepton mass matrix M_l has four (five) non-zero elements while Majorana neutrino mass matrix M_ν have three (two) non-zero matrix elements have been studied recently in Ref. <cit.> and an inverted neutrino mass ordering and a non-maximal Dirac-type CP-violating phase δ are predicted for all viable textures. The recent confirmation of a non-zero and not so small reactor mixing angle θ_13 has emerged as an important discriminator of neutrino mass models and many models based on discrete symmetries have been discarded as these models require breaking of these symmetries to accommodate the current neutrino data.In the present work, we consider new textures of lepton mass matrices and systematically, investigate their predictions for the unknown parameters. We show that these new textures can be realized on the basis of discrete Abelian Z_n symmetries. Specifically, we investigate textures of lepton mass matrices with four (five) non-zero elements in M_l and three (four) vanishing cofactors in M_ν. The textures considered in the present work are as predictive as the textures with two texture zeros/two vanishing cofactors in the flavor basis. Moreover, vanishing cofactors in M_ν can be seen as zero entries in M_R and M_D within the framework of type-I seesaw mechanism <cit.>:M_ν=- M_D M_R^-1 M_D^T.where M_D is the Dirac neutrino mass matrix and M_R is the right-handed Majorana neutrino mass matrix.The texture structures related by weak basis transformations lead to the same predictions for neutrino parameters, hence, one cannot distinguish mass matrix structures related by weak basis transformations. The charged lepton mass matrix having 4 non-zero matrix elements with non-zero determinant (as none of the charged lepton masses is zero), can have the following form: M_l=( [ × 0 0; 0 × ×; 0 0 × ]).Other possible structures can be obtained by considering all possible reorderings of rows and columns of M_l. The Hermitian products H_l=M_l M_l^† corresponding to charged lepton mass matrices are given by H_l1=( [ m_e^2 0 0; 0 × ×; 0 × × ]), H_l2=( [ × 0 ×; 0 m_μ^2 0; × 0 × ]), H_l3=( [ × × 0; × × 0; 0 0 m_τ^2 ])where the diagonalization of H_l gives the value of V_l as V_l^† H_l V_l=diag(m_e^2,m_μ^2,m_τ^2).The neutrino mass matrices with three vanishing cofactors have following 20 distinct possible structures which have been classified into six classes:Class-IM_ν 1=( [ × Δ Δ; Δ × Δ; Δ Δ × ]), Class-IIM_ν 2=( [ Δ Δ ×; Δ × Δ; × Δ × ]), M_ν 3=( [ × Δ Δ; Δ Δ ×; Δ × × ]), M_ν 4=( [ Δ × Δ; × × Δ; Δ Δ × ]),M_ν 5=( [ × Δ Δ; Δ × ×; Δ × Δ ]), M_ν 6=( [ × × Δ; × Δ Δ; Δ Δ × ]), M_ν 7=( [ × Δ ×; Δ × Δ; × Δ Δ ]), Class-IIIM_ν 8=( [ × × Δ; × Δ ×; Δ × Δ ]), M_ν 9=( [ × Δ ×; Δ Δ ×; × × Δ ]), M_ν 10=( [ Δ × ×; × × Δ; × Δ Δ ]),M_ν 11=( [ Δ Δ ×; Δ × ×; × × Δ ]), M_ν 12=( [ Δ × ×; × Δ Δ; × Δ × ]), M_ν 13=( [ Δ × Δ; × Δ ×; Δ × × ]), Class-IVM_ν 14=( [ Δ Δ Δ; Δ × ×; Δ × × ]), M_ν 15=( [ × × Δ; × × Δ; Δ Δ Δ ]), M_ν 16=( [ × Δ ×; Δ Δ Δ; × Δ × ]), Class-VM_ν 17=( [ Δ × ×; × Δ ×; × × Δ ]), Class-VIM_ν 18=( [ × × ×; × Δ Δ; × Δ Δ ]), M_ν 19=( [ Δ × Δ; × × ×; Δ × Δ ]), M_ν 20=( [ Δ Δ ×; Δ Δ ×; × × × ])where Δ at (i,j) position represents vanishing cofactor corresponding to element (i,j) while × denotes a non-zero arbitrary entry. Neutrino mass matrices in each class are related by S_3 permutation symmetry as M_ν→ S^T M_ν S, where S denotes the permutation matrices corresponding to S_3 group:S_1 = ( [ 1 0 0; 0 1 0; 0 0 1 ]), S_123=( [ 0 0 1; 1 0 0; 0 1 0 ]), S_132=( [ 0 1 0; 0 0 1; 1 0 0 ]),S_12 = ( [ 0 1 0; 1 0 0; 0 0 1 ]), S_13=( [ 0 0 1; 0 1 0; 1 0 0 ]), S_23=( [ 1 0 0; 0 0 1; 0 1 0 ]).§ ANALYSIS The mass term for charged leptons and Majorana neutrinos can be written as-ℒ_mass=l_L M_l l_R- 1/2ν^T_L C^-1M_νν_L+H.c.where C is the charge conjugation matrix. The charged lepton and the Majorana neutrino mass matrix can be diagonalized asV_l^† M_lM_l^† V_l = (M_l^D)^2, V_ν^T M_ν V_ν = M_ν^Dwhere M^D_l= diag(m_e,m_μ,m_τ), and M^D_ν= diag(m_1,m_2,m_3). V_l and V_ν are unitary matrices connecting mass eigenstates to the flavor eigenstates. The lepton mixing matrix also known as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix <cit.> is given byU=V_l^† V_νwhich can be parametrized in terms of three mixing angles and three CP-violating phases in the standard parametrization as <cit.>U=( [ c_12 c_13 c_13 s_12 e^-i δ s_13; -c_23 s_12-e^i δ c_12 s_13 s_23c_12 c_23-e^i δ s_12 s_13 s_23 c_13 s_23;s_12 s_23-e^i δ c_12 c_23 s_13 -e^i δ c_23 s_12 s_13-c_12 s_23 c_13 c_23 ]) ( [ 1 0 0; 0 e^i α 0; 0 0 e^i (β+δ) ])where c_ij=cosθ_ij and s_ij=sinθ_ij. α and β are the two Majorana CP-violating phases and δ is the Dirac-type CP-violating phase.The CP violation in neutrino oscillation experiments can be reflected in terms of Jarlskog rephasing invariant quantity J_CP <cit.> withJ_CP=Im{ U_11 U_22 U^*_12 U^*_21}=sinθ_12sinθ_23sinθ_13cosθ_12cosθ_23cos^2θ_13sinδ .The effective Majorana neutrino mass |m_ee|, which determines the rate of neutrinoless double beta decay is given by|m_ee|=|m_1 U^2_e1+m_2 U^2_e2+m_3 U^2_e3|.There are a large number of experiments such as CUORICINO <cit.>, CUORE <cit.>, MAJORANA <cit.>, SuperNEMO <cit.>, EXO <cit.> which aim to achieve a sensitivity up to 0.01 eV for |m_ee|.Recent cosmological observations provide more stringent constraints on absolute neutrino mass scale. Planck satellite data <cit.> combined with WMAP, cosmic microwave background and baryon acoustic oscillation experiments limit the sum of neutrino masses ∑_i=1^3 m_i≤ 0.23 eV at 95% confidence level (CL).§ NUMERICAL ANALYSISIn this section we present detailed numerical analysis along with the main predictions for viable textures. The charged lepton mixing matrices corresponding to structures H_l1, H_l2, H_l3 are given by V_l1 = ( [100;0 cosθe^i ϕ_lsinθ;0 - e^-i ϕ_lsinθ cosθ ]),V_l2 = ( [ cosθ0e^i ϕ_lsinθ;010; - e^-i ϕ_lsinθ0 cosθ ]),V_l3 = ( [ cosθe^i ϕ_lsinθ0; - e^-i ϕ_lsinθ cosθ0;001 ]) respectively, with cosθ=√(m_y-m/m_y-m_x). The parameters m_x, m_y are defined as m_x=m_μ^2, m_y=m_τ^2 for the structure H_l1, m_x=m_e^2, m_y=m_τ^2 for structure H_l2 and m_x=m_e^2, m_y=m_μ^2 for structure H_l3. Also, the parameter m is constrained as m_x<m<m_y. The charged lepton mass eigenvalues are m_e=0.510998928 MeV, m_μ=105.6583715 MeV and m_τ=1776.86 MeV <cit.>.Neutrino mass matrices of Class-I and Class-II lead to one or more zeros in the lepton mixing matrix U and, hence, both classes are phenomenologically excluded. Neutrino mass matrices of Class-VI lead to two degenerate neutrino mass eigenvalues, which is inconsistent with the current experimental data and hence this class is, also, phenomenologically ruled out. Therefore, we focus on the other three non trivial classes i.e., Class-III, IV and V.The 10 possible structures for M_ν from Classes-III, IV, V along with 3 structures for H_l, lead to a total of 10 × 3=30 possible combinations of charged lepton and neutrino mass matrices. But all possible combinations for H_l and M_ν are not independent of each other as the transformations M_ν→ S M_ν S^T and H_l→ S H_l S^† relate some of the textures with each other. Table <ref> contains all possible independent texture structure of M_ν and H_l and their viabilities for Normal mass Ordering (NO) and Inverted mass Ordering (IO). §.§ Class-III Texture III-(H): First, we analyze the texture structure III-(H) in which M_ν has vanishing cofactors corresponding to (1,3), (2,2) and (3,3) elements and H_l has texture zeros at (1,3) and (2,3) positions. The texture structures of neutrino and charged lepton mass matrices have the following form:M_ν 8=( [ × × Δ; × Δ ×; Δ × Δ ]) and H_l 3=( [ × × 0; × × 0; 0 0 m_τ^2 ]).All the non-zero elements of M_ν are, in general, complex. The neutrino mass matrix can be made real as M_ν=P_ν M_ν^r P_ν^T, with the phase matrix P_ν = diag(e^i ψ_1,e^i ψ_2,e^i ψ_3). The matrix M_ν^r is diagonalized by orthogonal matrix O_ν asM_ν^r=O_ν M_ν^D O_ν^T and the neutrino mixing matrix is V_ν=P_ν O_ν. The PMNS mixing matrix is given byU=V^†_l V_ν=V^†_l P_ν O_ν.We use invariants Tr[M_ν^r], Tr[M_ν^r^2] and Det[M_ν^r] to redefine mass matrix elements in terms of mass eigenvalues. The eigenvalues of the neutrino mass matrix for NO are m_1, -m_2 and m_3. The eigenvalues m_2 and m_3 can be calculated from the mass-squared differences Δ m_21^2 and Δ m_31^2 using the relationm_2=√(Δ m_21^2+m_1^2) and m_3=√(Δ m_31^2+m_1^2). The orthogonal mixing matrix O_ν for NO is given byO_ν 8| NO=( [-√(m_2 m_3)√((m_2-m_1)(m_1+m_3))/√((m_1+m_2)(m_3-m_1) a) -√(m_1 m_3)√((m_2-m_1) (m_3-m_2))/√((m_1+m_2)(m_2+m_3) a) √(m_1 m_2)√((m_1+ m_3) (m_3-m_2))/√((m_3-m_1)(m_2+m_3) a);√(m_1(m_3-m_2))/√((m_1+m_2) (m_3-m_1)) -√(m_2 (m_1+m_3))/√((m_1+m_2)(m_2+m_3)) √((m_1-m_2) m_3)/√((m_1-m_3) (m_2+m_3));m_1 √(m_1 (m_3-m_2))/√((m_1+m_2)(m_3-m_1) a)m_2 √(m_2 (m_1+m_3))/√((m_1+m_2)(m_2+m_3) a) m_3 √((m_2-m_1) m_3)/√((m_3-m_1) (m_2+m_3) a) ]) where a=m_1 (m_2-m_3)+m_2 m_3. For IO, with neutrino mass eigenvalues (-m_1, m_2, m_3), the orthogonal matrix O_ν is given by O_ν 8|IO=( [-√((m_2-m_1) (m_1-m_3))√(m_2 m_3)/√(b (m_1+m_2)(m_1+m_3)) √(m_1 m_3)√((m_2-m_1) (m_2+m_3))/√(b(m_1+m_2) (m_2-m_3)) -√(m_1 m_2)√((m_1-m_3)(m_2+m_3))/√(b (m_2-m_3)(m_1+m_3));-√(m_1 (m_2+m_3))/√((m_1+m_2)(m_1+m_3))√(m_2(m_1-m_3))/√((m_1+m_2)(m_2-m_3))√((m_2-m_1) m_3)/√((m_2-m_3) (m_1+m_3)); m_1 √(m_1 (m_2+m_3))/√(b(m_1+m_2) (m_1+m_3))m_2√(m_2 (m_1-m_3))/√(b(m_1+m_2) (m_2-m_3)) m_3 √((m_2-m_1) m_3)/√(b(m_2-m_3) (m_1+m_3)) ]) where b=m_1 (m_2+m_3)-m_2 m_3. The mass eigenvalues can be calculated by using the relationsm_2=√(Δ m_23^2+m_3^2) and m_1=√(Δ m_13^2+m_3^2) .The charged lepton mixing matrix is given byV_l3=( [ cosθ_le^i ϕ_lsinθ_l0; - e^-i ϕ_lsinθ_l cosθ_l0;001 ])where cosθ_l=√(m_μ^2-m/m_μ^2-m_e^2) and m_e^2<m<m_μ^2.The lepton mixing matrix for texture structure III-(H) can be written asU = V_l 3^†V_ν 8 = V_l 3^† P_ν O_ν 8|NO(IO)= ( [ cosθ_l -e^i ϕ_lsinθ_l0; e^-i ϕ_lsinθ_l cosθ_l0;001 ])( [ e^i ψ_1 0 0; 0 e^i ψ_2 0; 0 0 e^i ψ_3 ]) O_ν 8|NO(IO)= P^'( [ cosθ_l -e^i ηsinθ_l0; sinθ_le^i ηcosθ_l0;001 ]) O_ν 8|NO(IO)where η =ψ_2-ψ_1+ϕ_l and the phase matrix P^'= diag(e^i ψ_1,e^i (ψ_1-ϕ_l),e^i ψ_3).Texture III-(I):For texture structure III-(I), M_ν has vanishing cofactors corresponding to (1,2), (2,2), and (3,3) elements while H_l has zeros at (1,2) and (2,3) elements. The neutrino and the charged lepton mass matrices for texture III-(I) have the following form:M_ν 9=( [ × Δ ×; Δ Δ ×; × × Δ ]) and H_l2=( [ × 0 ×; 0 m_μ^2 0; × 0 × ]).The neutrino mass matrix M_ν 9 is related to M_ν 8 by permutation symmetry, M_ν 9=S_23 M_ν 8 S_23^T. Therefore, the neutrino mixing matrix for texture M_ν 9 can be written in terms of V_ν 8 as V_ν 9=S_23 V_ν 8. Therefore, the PMNS mixing matrix for texture III-(I) is given by U = V^†_l2 V_ν 9 = V^†_l2 S_23 V_ν 8 = V^†_l2 S_23 P_ν O_ν 8 = ( [ cosθ_l0 -e^i ϕ_lsinθ_l;010; e^-i ϕ_lsinθ_l0 cosθ_l ])( [ 1 0 0; 0 0 1; 0 1 0 ])( [ e^i ψ_1 0 0; 0 e^i ψ_2 0; 0 0 e^i ψ_3 ]) O_ν 8|NO(IO) =P^'( [ cosθ_l -e^i ηsinθ_l0;001; sinθ_le^i ηcosθ_l0 ]) O_ν 8|NO(IO).For NO and IO, the orthogonal matrix O_ν 8 is given in Eqs. (22) and (23), respectively.The three lepton mixing angles in terms of the lepton mixing matrix elements are given by sinθ_13=|U_13|, sinθ_23= |U_23|/√(1-|U_13|^2) and sinθ_12=|U_12|/√(1-|U_13|^2). The CP violating phase δ can be calculated using Eq. (15):sinδ = Im(U_11 U_22 U_12^† U_21^†)/sinθ_12sinθ_23sinθ_13cosθ_12cosθ_23cos^2θ_13where the elements of U are given in Eq. (26) for texture III-(H) and Eq. (28) for texture III-(I).Similarly, the three mixing angles can be calculated by substituting the elements of U from Eqs.(26) and (28) into Eq.(29)for textures III-(H) and III-(I), respectively. The effective Majorana mass |m_ee| can be calculated using Eq. (16)There exist following relations between neutrino oscillation parameters of texture structures III-(H) and III-(I):θ_12^H=θ_12^I, θ_13^H=θ_13^I, θ_23^H=π/2-θ_23^I. For the numerical analysis, we have generated random numbers of the order of ∼ 10^7 for parameters Δ m^2_21 and Δ m^2_31 (Δ m^2_32) for NO (IO) within their experimentally allowed 3 σ ranges. m_1 (m_3) have been generated randomly between 0 - 0.33 eV. We vary the parameter η randomly within the range (0-2π) and parameter m has been varied randomly with in the range (m_e^2 - m_μ^2) for texture III-(H) and (m_e^2 - m_τ^2) for texture III-(I). We use the experimental constraints on neutrino oscillation parameters as given in Table <ref>. In this analysis, the upper bound on sum of neutrino masses is set to be ∑ m_i≤ 1 eV. It turns out that textures III-(H) and III-(I) are, phenomenologically, viable only for normal mass ordering. Figs. <ref> and <ref> depict the predictions for textures III-(H) and III-(I), respectively.It can be seen from Fig. <ref> that sinδ lies in the range (-1 - 1) and the Jarlskog CP invariant parameter J_CP varies in the range (-0.036 - 0.036) for NO in III-(H) whereas for texture III-(I) the ranges for sinδ and J_CP as shown in Fig. <ref> are (-1 - 1) and ±(0.004 - 0.036), respectively. The correlation plots in (θ_23-sinδ) plane in Figs. <ref> and <ref> show that δ is more favored to lie near δ∼±π/2. These results are consistent with the recent observations in the long baseline neutrino oscillation experiments like T2K and NOvA <cit.> which show a preference for the CP violating phase δ to lie around δ∼ -π/2. The range for smallest neutrino mass m_1 is (0.0087 - 0.015) eV for texture III-(H) and (0.0084 - 0.014) eV for texture III-(I). The sum of neutrino masses ∑ m_i lies in the ranges (0.071 - 0.086) eV and (0.07 - 0.085) eV for textures III-(H) and III-(I), respectively. The charged lepton correction θ_l for Class-III turns out to be very large and lies in the range (62^∘ - 69^∘) for both allowed textures. Parameter |m_ee| is constrained to lie in the range (0.0025 - 0.0071) eV for texture III-(H) and (0.0024 - 0.007) eV for III-(I). All textures of Class-III with inverted mass ordering are ruled out at 3σ CL. §.§ Class-IVThe textures IV-(A), IV-(B) and IV-(C) are, phenomenologically, non-viable for the current 3σ ranges of neutrino oscillation parameters due to the presence of one zero element in the lepton mixing matrix U. We discuss the viable textures of Class-IV and their phenomenology below.Texture IV-(D) Here M_ν has vanishing cofactors corresponding to elements (1,1), (1,2), and (1,3) which is equivalent to scaling neutrino mass matrix <cit.> and charged lepton mass matrix H_l has zeros at (1,3) and (2,3) positions. The mass matrices M_ν and H_l are given byM_ν 14 = ( [ Δ Δ Δ; Δ × ×; Δ × × ]) ≡( [ A e^i ϕ_a B e^i ϕ_bB e^i ϕ_b /c; B e^i ϕ_b D e^i ϕ_d D e^i ϕ_d/c;B e^i ϕ_b /c D e^i ϕ_d/c D e^i ϕ_d/c^2 ]),H_l3 = ( [ × × 0; × × 0; 0 0 m_τ^2 ]).The neutrino mass matrix which, in general, is complex symmetric for this class can not be diagonalized directly due to the presence of non-removable phase. Instead, we diagonalize the Hermitian product M_ν M_ν^† which gives neutrino mixing matrix asV_ν^† M_ν 14 M^†_ν 14 V_ν = diag(m_1^2,m_2^2,m_3^2). Therefore, we have following Hermitian matrixM_ν 14 M^†_ν 14=( [ab e^i ϕb e^i ϕ/c; b e^-i ϕdd/c; b e^-i ϕ/cd/cd/c^2 ])= P^†_ν( [ a b b/c; b d d/c; b/c d/c d/c^2 ]) P_νwhere a= A^2+B^2(1+1/c^2),b e^i ϕ =A B e^-i (ϕ_a + ϕ_b)+B D e^i (ϕ_b + ϕ_d)(1+1/c^2),d=B^2+D^2(1+1/c^2)and P_ν= diag(e^i ϕ,1,1) is the phase matrix. The mixing matrix for M_ν 14 is given byV_ν 14 = P_ν O_ν 14where O_ν 14 is an orthogonal unitary matrix which diagonalizes the real symmetric mass matrix given in Eq.(35).For NO, the neutrino mass eigenvalues are m_1=0, m_2=√(Δ m^2_21), m_3=√(Δ m^2_31) and the orthogonal matrix O_ν 14 is given byO_ν 14|NO=( [0 a c^2-(c^2+1) d-√(x)/√(4 b^2 c^4+4b^2 c^2+(-a c^2+d c^2+d+√(x))^2) ac^2-(c^2+1) d+√(x)/√(4 b^2 c^4+4 b^2c^2+(a c^2-(c^2+1) d+√(x))^2);-1/√(c^2+1)2 b c^2/√(4 b^2 c^4+4 b^2c^2+(-a c^2+d c^2+d+√(x))^2)2 bc^2/√(4 b^2 c^4+4 b^2 c^2+(a c^2-(c^2+1)d+√(x))^2); c/√(c^2+1)2 b c/√(4 b^2 c^4+4 b^2c^2+(-a c^2+d c^2+d+√(x))^2)2 bc/√(4 b^2 c^4+4 b^2 c^2+(a c^2-(c^2+1)d+√(x))^2) ])where x = (c^2 (d-a)+d)^2+4 b^2 (c^4+c^2),c = √(d)/√(-a-d+m_2^2+m_3^2),and b = √(d)√((m_3^2-a)(a-m_2^2))/√(-a+m_2^2+m_3^2).The free parameters a,d are constrained to lie in the range m_2^2<a<m_3^2 and d<m_3^2. For IO, the mass eigenvalues are m_1=√(Δ m^2_23-Δ m^2_21), m_2=√(Δ m^2_23) and m_3=0 and the corresponding orthogonal matrix O_ν 14 is given byO_ν 14|IO=( [ a c^2-(c^2+1) d-√(x)/√(4 b^2 c^4+4 b^2c^2+(-a c^2+d c^2+d+√(x))^2) ac^2-(c^2+1) d+√(x)/√(4 b^2 c^4+4 b^2c^2+(a c^2-(c^2+1) d+√(x))^2)0;2 b c^2/√(4 b^2 c^4+4 b^2 c^2+(-a c^2+dc^2+d+√(x))^2) 2 b c^2/√(4 b^2 c^4+4b^2 c^2+(a c^2-(c^2+1) d+√(x))^2)-1/√(c^2+1);2 b c/√(4 b^2 c^4+4 b^2 c^2+(-a c^2+dc^2+d+√(x))^2) 2 b c/√(4 b^2 c^4+4 b^2c^2+(a c^2-(c^2+1) d+√(x))^2) c/√(c^2+1) ])wherex = (c^2 (d-a)+d)^2+4 b^2 (c^4+c^2),c = √(d)/√(-a-d+m_1^2+m_2^2),b = √(d)√((m_2^2-a)(a-m_1^2))/√(-a+m_1^2+m_2^2)with m_1^2<a<m_2^2 and d<m_2^2.The PMNS mixing matrix for texture IV-(D) is given byU = V_l3^† V_ν 14= V_l 3^† P_ν O_ν 14|NO(IO)= ( [ cosθ_l -e^i ϕ_lsinθ_l0; e^-i ϕ_lsinθ_l cosθ_l0;001 ])( [ e^i ϕ 0 0; 0 1 0; 0 0 1 ]) O_ν 14|NO(IO)= ( [ e^i ϕ 0 0; 0 e^i (ϕ-ϕ_l) 0; 0 0 1 ]) ( [ cosθ_l -e^i (ϕ_l-ϕ)sinθ_l0; sinθ_le^i (ϕ_l-ϕ)cosθ_l0;001 ]) O_ν 14|NO(IO)= P^'( [ cosθ_l -e^i ηsinθ_l0; sinθ_le^i ηcosθ_l0;001 ]) O_ν 14|NO(IO)where η =ϕ_l-ϕ and cosθ_l=√(m^2_μ-m/m^2_μ-m^2_e) for m^2_e<m<m^2_μ. The phase matrix is given by P^'=diag(e^i ψ,e^i (ϕ-ϕ_l),1). For IO of texture IV-(D), neutrino oscillation parameters are related assinθ_13=sinθ_l/√(c^2+1), tanθ_23=cosθ_l/c . Texture IV-(E)Texture IV-(E) has three vanishing cofactors in M_ν corresponding to elements (1,1), (1,2), (1,3) andtwo zeros in H_l at (1,2) and (2,3) positions. The mass matrices of neutrinos and charged leptons are given byM_ν 14=( [ Δ Δ Δ; Δ × ×; Δ × × ]) and H_l2=( [ × 0 ×; 0 m_μ^2 0; × 0 × ]). The charged lepton mixing matrix for H_l2 is given byV_l2=( [cosθ_l 0 e^i ϕ_lsinθ_l; 0 1 0; -e^-i ϕ_lsinθ_l 0cosθ_l ])where cosθ_l=√(m_τ^2-m/m_τ^2-m_e^2) for m_e^2<m<m_τ^2. The PMNS mixing matrix for this texture is given byU = V_l2^† V_ν 14,=V_l2^† P_ν O_ν 14|NO(IO), =P^'( [ cosθ_l0 -e^i ηsinθ_l;010; sinθ_l0e^i ηcosθ_l ]) O_ν 14|NO(IO)where η =ϕ_l-ϕ and P^'= diag(e^i ψ,e^i (ϕ-ϕ_l),1). The orthogonal matrix O_ν 14|NO(IO) is given in Eqs.(38) and (40) for normal and inverted mass orderings. Neutrino oscillation parameters for IO of texture IV-(E) are related as follows:sinθ_13=sinθ_lc/√(c^2+1), tanθ_23=θ_l/c. Texture IV-(F)In this texture structure, the neutrino mass matrix has three vanishing cofactors corresponding to (1,2), (2,2), and (2,3) positions and the charged lepton mass matrix has zero elements at (1,2) and (1,3) positions. M_ν and H_l have the following structure:M_ν 16=( [ × Δ ×; Δ Δ Δ; × Δ × ]) and H_l1=( [ m_e^2 0 0; 0 × ×; 0 × × ]). The neutrino mass matrix M_ν 16 is related to M_ν 14 by S_3 permutation symmetry as M_ν 16=S_12 M_ν 14 S_12^T, where S_12 is an element of the permutation group S_3. Therefore, the mixing matrix for structure M_ν 16 is given byV_ν 16=S_12 V_ν 14.The PMNS mixing matrix for texture IV-(F) is given byU = V_l1^† V_ν 16=V_l1^† S_12 V_ν 14=P^'( [010; cosθ_l0 -e^i ηsinθ_l; sinθ_l0e^i ηcosθ_l ]) O_ν 14|NO(IO)where Eqs.(38) and (40) give O_ν 14 for NO and IO, respectively. For IO of texture IV-(F), neutrino oscillation parameters are related assinθ_13=1/√(c^2+1), tanθ_23= tanθ_l.The numerical results of Class-IV are presented in Figs. <ref>, <ref> and <ref> for both normal and inverted mass orderings. It can be seen from these figures that sinδ spans the range (-1 - 1) except for NO of texture IV-(F) for which sinδ is bounded by (-0.6 - 0.6). The Jarlskog CP invariant parameter J_CP varies in the range (-0.02 - 0.02) for NO in texture IV-(F) whereas for other viable textures the range is (-0.036 - 0.036). There is a strong correlation between the charged lepton mixing angle θ_l and (θ_12, θ_23) for both mass orderings for all allowed textures of this class. Similar correlations are, also, present for |m_ee| with θ_12 and θ_23 as shown in Figs. <ref>, <ref> and <ref>. For textures IV-(D), IV-(E) and IV-(F), the effective Majorana mass |m_ee| is highly constrained to lie in the ranges (0.001 - 0.0045)eV for NO and (0.014 - 0.05)eV for IO. For viable textures of Class-IV, charged lepton correction θ_l is very large for normal mass ordering. For inverted mass ordering θ_l is small and the results are in agreement with Ref.<cit.>, where charged lepton corrections were taken to be of the order of the Cabibbo angle. The sum of neutrino masses ∑ m_i varies in the range (0.057 - 0.061) eV for NO and (0.097 - 0.102) eV for IO. Textures IV-(A), IV-(B) and IV-(C) of Class-IV are phenomenologically incompatible with the 3σ neutrino oscillation data. §.§ Class-V In this class, there are three possible texture structures viz. M_ν 17 H_l1, M_ν 17 H_l2 and M_ν 17 H_l3. H_l and M_ν for this class have the following form: H_l1 = ( [ m_e^2 0 0; 0 × ×; 0 × × ]), H_l2=( [ × 0 ×; 0 m_μ^2 0; × 0 × ]), H_l3=( [ × × 0; × × 0; 0 0 m_τ^2 ]),M_ν 17 = ( [ Δ × ×; × Δ ×; × × Δ ]) ≡ P_ν( [ a√(a b) -√(a d);√(a b) b√(b d); -√(a d)√(b d) d; ]) P_ν^T = P_ν M_ν 17^r P_ν^T where P_ν is a diagonal phase matrix. The real symmetric matrix M_ν 17^r can be diagonalized by the orthogonal matrix O_ν 17:M_ν 17^r=O_ν 17 M_ν 17^D O_ν 17^T. O_ν 17 is given byO_ν 17=( [√(a d)(m_1-2 b)/(a+b-m_1) m_1√(x+1) √(a d)(m_2-2b)/(a+b-m_2) m_2√(y+1) √(a d)(m_3-2b)/(a+b-m_3) m_3√(z+1);√(b d)(2 a-m_1)/(a+b-m_1) m_1√(x+1)√(b d)(2 a-m_2)/(a+b-m_2)m_2 √(y+1) √(b d)(2 a-m_3)/(a+b-m_3) m_3 √(z+1); 1/√(x+1) 1/√(y+1) 1/√(z+1) ])where x=b d (2 a-m_1)^2/m_1^2(a+b-m_1)^2+a d(m_1-2 b)^2/m_1^2(a+b-m_1)^2,y=b d (2 a-m_2)^2/m_2^2(a+b-m_2)^2+a d^2(m_2-2 b)^2/m_2^2(a+b-m_2)^2,z=b d (2 a-m_3)^2/m_3^2(a+b-m_3)^2+a d(m_3-2 b)^2/m_3^2(a+b-m_3)^2and the parameters a, b are related to neutrino masses m_1, m_2 and m_3 as a = -d^2+d (m_1+m_2+m_3)+√(d (d(-d+m_1+m_2+m_3)^2+m_1 m_2m_3))/2 d, b =-d^2-d (m_1+m_2+m_3)+√(d (d(-d+m_1+m_2+m_3)^2+m_1 m_2m_3))/2 d. Three vanishing diagonal cofactors of M_ν relate mass eigenvalues as m_1 m_2+m_2m_3+m_1m_3=0. All texture structures of Class-V are inconsistent with the present neutrino oscillation data at 3σ level.Texture structures III-(A) to III-(G) and V-(A) to V-(C) cannot simultaneously satisfy the experimental constraints on the mass squared differences and mixing angles and thus are inconsistent with neutrino oscillation data. In Table <ref> we have summarized the parameter space for the three mixing angles associated with each disallowed texture. One can see that the three mixing angles cannot simultaneously have values lying with in their experimental 3σ ranges for the disallowed textures. §.§ Neutrino mass matrices with four vanishing cofactorsAnother possibility for lepton mass matrices includes four vanishing cofactors in the neutrino mass matrices with five non-zero elements in the charged lepton mass matrices. These textures have total eight degrees of freedom and such textures should have the same predictability as the two texture zero neutrino mass matrices in flavor basis. There are a total of 15 possible structures for M_ν having four vanishing cofactors. In a 3×3 complex symmetric matrix, vanishing of any set of four cofactors leads to either the vanishing of the fifth or all six cofactors, simultaneously. A neutrino mass matrix where all six cofactors vanish leads to two degenerate neutrino masses which is incompatible with the experimental data. The remaining possible structures of M_ν which have five vanishing cofactors and non-degenerate mass eigenvalues are given below:Class-VII M_ν 21=( [ Δ Δ Δ; Δ Δ Δ; Δ Δ × ]), M_ν 22=( [ Δ Δ Δ; Δ × Δ; Δ Δ Δ ]), M_ν 23=( [ × Δ Δ; Δ Δ Δ; Δ Δ Δ ]).The charged lepton mass matrices with five non-zero matrix elements are of the following form:M_l=( [ × 0 0; 0 × ×; × 0 × ]) with all possible reorderings of rows and columns of M_l. The Hermitian products H_l=M_l M_l^†, whose diagonalization gives charged lepton mixing matrix V_l, are given below:H_l1=( [ × 0 0; 0 × ×; 0 × × ]), H_l2=( [ × 0 ×; 0 × 0; × 0 × ]), H_l3=( [ × × 0; × × 0; 0 0 × ]),H_l4=( [ × × 0; × × ×; 0 × × ]), H_l5=( [ × 0 ×; 0 × ×; × × × ]), H_l6=( [ × × ×; × × 0; × 0 × ]),H_l7=( [ × × Δ; × × ×; Δ × × ]), H_l8=( [ × Δ ×; Δ × ×; × × × ]), H_l9=( [ × × ×; × × Δ; × Δ × ]) .where Δ at ij position represents the vanishing cofactors corresponding to ij elements.If M_ν is one of Eq.(57) and H_l is one of Eq.(59), the lepton mixing matrix U has one of its elements equal to zero which is inconsistent with current experimental data and, hence, the charged lepton mass matrices listed in Eq.(59) are phenomenologically ruled out for four vanishing cofactors in M_ν. Therefore, we focus on other forms of H_l given in Eqs.(60) and (61). There are 18 possible combinations of M_ν and H_l but not all are independent as interchanging rows and columns of M_ν is equivalent to reordering the rows and columns of M_l. Table <ref> gives independent combinations of M_ν and H_l along with their viabilities. The neutrino mass matrices for this class cannot be diagonalized directly due to the presence of a non-removable phase and, hence, we diagonalize the Hermitian product M_ν M_ν^†, which gives neutrino mixing matrix V_ν. The neutrino mass matrix M_ν can be written as M_ν 21 = ( [ Δ Δ Δ; Δ Δ Δ; Δ Δ × ]) ≡( [ A e^i ϕ_1 B e^i ϕ_3 0; B e^i ϕ_3 D e^i ϕ_2 0; 0 0 0 ])and the corresponding Hermitian matrix is given byM_ν 21 M^†_ν 21=( [ab e^i ϕ0; b e^-i ϕd0;000 ])= P^†_ν( [ a b 0; b d 0; 0 0 0 ]) P_νwhere a=A^2+B^2,b e^i ϕ =A B e^-i(ϕ_3-ϕ_1)+B D e^i (ϕ_3-ϕ_2),d=B^2+D^2and P_ν= diag(e^i ϕ,1,1). For NO, the neutrino masses are m_1=0, m_2=√(Δ m^2_21), m_3=√(Δ m^2_31) and for IO m_1=√(Δ m^2_23-Δ m^2_21), m_2=√(Δ m^2_23), m_3=0. The orthogonal matrix O_ν21 for NO and IO is given byO_ν 21|NO= ( [ 0-√(d-m_2^2)/√(m_3^2-m_2^2) √(m_3^2-d)/√(m_3^2-m_2^2); 0 √(m_3^2-d)/√(m_3^2-m_2^2) √(d-m_2^2)/√(m_3^2-m_2^2); 1 0 0; ]) and O_ν 21|IO= ([ -√(d-m_1^4)/√(m_2^4-m_1^4) √(m_2^4-d)/√(m_2^4 -m_1^4)0; √(m_2^4-d)/√(m_2^4 -m_1^4)√(d-m_1^4)/√(m_2^4-m_1^4)0;001;])where m_2^2<d<m_3^2 for NO and m_1^2<d<m_2^2 for IO.The mixing matrix for M_ν 21 is given byV_ν 21 = P_ν O_ν 21.For texture M_ν22, which is related to M_ν21 by permutation symmetry: M_ν22→ S_23M_ν21S_23^T, neutrino mixing matrix is given byV_ν 22 = S_23 P_ν O_ν 21.For M_ν23, neutrino mixing matrix is given byV_ν 23 = S_123 P_ν O_ν 21.The charged lepton mixing matrix for structures given in Eqs.(60) and (61) can be parametrized asV_l=P_l O_l, with O_l=( [c_12c_13c_13s_12s_13; -c_23s_12- c_12s_13s_23c_12c_23- s_12s_13s_23c_13s_23;s_12s_23- c_12c_23s_13 - c_23s_12s_13-c_12s_23c_13c_23 ])where s_ij=sinχ_ij, c_ij=cosχ_ij and P_l= diag(e^i ϕ^',1,e^i ϕ^”) is unitary phase matrix.For charged lepton mass matrix H_l4, a zero at the (1,3) position impliesm_e^2 O_l 11 O_l31+ m_μ^2 O_l12 O_l32+ m_τ^2 O_l13 O_l33=0which givessinχ_13=(m_e^2-m_μ^2) sinχ_12cosχ_12tanχ_23)/m_e^2 cos^2 χ_12+m_2^2 sin^2χ_12-m_τ^2.For charged lepton mass matrix H_l7, vanishing cofactor at (1,3) position impliesH_l|_21 H_l|_32-H_l|_22 H_l|_31=0which givessinχ_13=m_τ^2(m_e^2-m_μ^2) sin 2χ_12tanχ_23/m_τ^2 (m_e^2-m_μ^2) cos 2χ_12+m_e^2 (2 m_μ^2-m_τ^2)-m_μ^2 m_τ^2.The PMNS mixing matrix for this class is given byU = V_l^† V_ν,= O_l^T P_l^† P_ν O_ν |NO(IO),= O_l^T P O_ν |NO(IO),where P= diag(e^i (ϕ-ϕ^'),1,e^-iϕ^”). The orthogonal matrices O_ν and O_l are given in Eqs.(65) and (69). In our numerical analysis, the parameters χ_12, χ_23, d, ϕ, ϕ^' and ϕ^” have been generated randomly and 3σ experimental constraints on oscillation parameters have been used. We found that all textures of this class are viable except VII-(D) which is unable to fit neutrino oscillation data for both mass orderings. For texture VII-(D), θ_12 becomes too large in case of NO and θ_13 becomes very small for IO. The predictions of all viable textures VII-(A), VII-(B), VII-(C), VII-(E) and VII-(F) for oscillation parameters are very similar for NO and IO. The Jarlskog CP invariant parameter J_cp for this class varies in the range (-0.04 - 0.04) and sinδ spans the range (-1 - 1) for both mass orderings. The ranges for |m_ee| are (0.0032 - 0.0042)eV and (0.01-0.05)eV for NO and IO, respectively. The parameter ∑ m_i lies within the ranges (0.057 - 0.0605)eV and (0.097 - 0.102)eV, for NO and IO, respectively.§ SYMMETRY REALIZATIONIt has been shown in Ref. <cit.> that vanishing cofactors and texture zeros in the neutrino mass matrix can be realized with extended scalar sector by means of discrete Abelian symmetries. We present a type-I seesaw realization of three vanishing cofactors using discrete Abelian flavor symmetries in the non-flavor basis.Abelian group Z_6 can be used for symmetry realization of mass matrix textures of Class-III. To obtain texture structure III-(H), one of the simplest possibilities is to have the following structures for M_D, M_R and M_l:M_D=( [ × 0 0; 0 × 0; 0 0 × ]), M_R=( [ × × 0; × 0 ×; 0 × 0 ]), and M_l=( [ × × 0; 0 × 0; 0 0 × ]).Here, M_D is diagonal and texture zeros in M_R propagate as vanishing cofactors in the effective neutrino mass matrix M_ν. In addition to the SM left-handed SU(2)_L lepton doublets D_l L= ( [ ν_lL;l_L ]),(l=e,μ,τ) and the right-handed charged lepton SU(2)_L singlets l_R, we introduce three right handed neutrinos ν_l R. In the scalar sector, we need two SU(2)_L Higgs doublets ϕ's and two scalar singlets χ's.We consider the following transformation properties of various fields under Z_6 for texture III-(H):D_eL → ω^5 D_eL, e_R→ω^2 e_R, ν_eR→ων_eR D_μ L → D_μ L, μ_R→ωμ_R, ν_μ R→ω^5 ν_μ R D_τ L → ω^3D_τ L, τ_R→ω^3τ_R, ν_τ R→ω^2ν_τ Rwith ω=e^2 π i/6 as generator of Z_6 group. The bilinears D_l L l_R, D_l Lν_l R and ν_l Rν_l R relevant for M_l, M_D and M_R, respectively, transform asD_l L l_R∼( [ ω 1 ω^2; ω^2 ω ω^3; ω^5 ω^4 1 ]), D_l Lν_l R∼( [ 1 ω^4 ω; ω ω^5 ω^2; ω^4 ω^2 ω^5 ]) and ν_l Rν_l R∼( [ ω^2 1 ω^3; 1 ω^4 ω; ω^3 ω ω^4 ]).For M_R, (1,2) element is invariant under Z_6 and hence the corresponding mass term is directly present in the Lagrangian without any scalar field. However, (1,1) and (2,3) matrix elements require the presence of two scalar singlets χ_1 and χ_2 which transform under Z_6 as χ_1→ω^4χ_1 and χ_2→ω^5χ_2, respectively. The other entries of M_R remain zero in the absence of any further scalar singlets. To achieve non diagonal charged lepton mass matrix M_l, two scalar Higgs doublets are needed which transform under Z_6 as ϕ_1→ϕ_1, ϕ_2→ω^5ϕ_2. A diagonal M_D is obtained since scalar Higgs doublets ϕ̃_̃j̃(≡σ_2ϕ_j^∗) transform under Z_6 as ϕ̃_̃1̃→ϕ̃_̃1̃ and ϕ̃_̃2̃→ωϕ̃_̃2̃. Thus, the Z_6 invariant Yukawa Lagrangian for texture III-(H) is given byℒ_Y= -Y_ee^l D_eLϕ_2 e_R - Y_e μ^l D_eLϕ_1 μ_R - Y_μμ^l D_μ Lϕ_2 μ_R- Y_ττ^l D_τ Lϕ_1 τ_R - Y_ee^DD_eLϕ̃_̃1̃ν_eR - Y_μμ^DD_μ Lϕ̃_̃2̃ν_μ R -Y_ττ^DD_τ Lϕ̃_̃2̃ν_τ R + Y_ee^R/2ν_eR^T C^-1ν_eRχ_1 + M_e μ^R/2 (ν_e R^T C^-1ν_μ R +ν_μ R^T C^-1ν_e R)+ Y_μτ^R/2 (ν_μ R^T C^-1ν_τ R + ν_τ R^T C^-1ν_μ R) χ_2 +H. c.For texture III-(I), the structures for M_D, M_R, M_l are given byM_D=( [ × 0 0; 0 × 0; 0 0 × ]), M_R=( [ × 0 ×; 0 0 ×; × × 0 ]), M_l=( [ × 0 ×; 0 × 0; 0 0 × ])and the leptonic fields are required to transform as D_eL → ω^5D_eL, e_R→ω^2 e_R, ν_eR→ων_eR D_μ L → ω^3 D_μ L, μ_R→ω^3 μ_R, ν_μ R→ω^2 ν_μ R D_τ L → D_τ L, τ_R→ωτ_R, ν_τ R→ω^5ν_τ Runder Z_6. The bilinears D_l L l_R, D_l Lν_l R and ν_l Rν_l R corresponding toM_l, M_D and M_R transform asD_l L l_R∼( [ ω ω^2 1; ω^5 1 ω^4; ω^2 ω^3 ω ]), D_l Lν_l R∼( [ 1 ω ω^4; ω^4 ω^5 ω^2; ω ω^2 ω^5 ]) and ν_l Rν_l R∼( [ ω^2 ω^3 1; ω^3 ω^4 ω; 1 ω ω^4; ]) .The two scalar fields required for non-zero elements in M_R, transform as χ_1→ω^4 χ_1, χ_2→ω^5χ_2 under Z_6. The desired form of M_l requires the Higgs fields to transform as ϕ_1→ϕ_1 and ϕ_2→ω^5 ϕ_2. The scalar Higgs doublets acquire non-zero vacuum expectation value (VEV) at the electroweak scale, while scalar singlets acquire VEV at seesaw scale.Similarly, the symmetry realization for Class-IV can also be achieved using Z_6 group. The transformation properties of leptonic and scalar fields under Z_6 group are given in Table <ref>.For four vanishing cofactors in M_ν and five non-zero elements in M_l, the symmetry realization of textures can be achieved by cyclic group Z_9. Table <ref> depicts the transformation properties of leptonic and scalar fields under Z_9 group for Class-VII. § CONCLUSION We have investigated some new texture structures for lepton mass matrices. We have studied texture structures with three (four) vanishing cofactors in the neutrino mass matrix M_ν with four (five) non-zero elements in the charged lepton mass matrix M_l. There are 3 possible structures for H_l and 20 possible structures of M_ν grouped into Classes-I, II, III, IV, V and VI, for three vanishing cofactors in M_ν and four non-zero elements in M_l. It is found that among six classes only Class-III and IV are phenomenologically viable. We also found that there are 5 viable textures having four vanishing cofactors in M_ν with five non-zero elements in M_l. By using the recent global neutrino oscillation data and data from cosmological experiments, a systematic phenomenological analysis has been done for each viable texture. 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"authors": [
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"Desh Raj",
"Radha Raman Gautam"
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"published": "20170926151532",
"title": "Neutrino mass matrices with three or four vanishing cofactors and non diagonal charged lepton sector"
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Flexibility in fixed conformal classes]Flexibility of geometrical and dynamical data in fixed conformal classesDepartment of Mathematics and Statistics, Queen's University, Kingston, ON [email protected] Department of Mathematics, Pennsylvania State University, State College, PA [email protected] a smooth closed surface M of fixed genus ⩾ 2 with a hyperbolic metric σ of total area A. In this article, we study the behavior of geometric and dynamical characteristics (e.g., diameter, Laplace spectrum, Gaussian curvature and entropies) of nonpositively curved smooth metrics with total area A conformally equivalent to σ. For such metrics, we show that the diameter is bounded above and the Laplace spectrum is bounded below away from zero by constants which depend on σ. On the other hand, we prove that the metric entropy of the geodesic flow with respect to the Liouville measure is flexible. Consequently, we also provide the first known example showing that the bottom of the L^2-spectrum of the Laplacian cannot be bounded from above by a function of the metric entropy. We also provide examples showing that our conditions are essential for the established bounds. [ Alena Erchenko==================§ INTRODUCTIONLet M be a closed surface equipped with a Riemannian metric σ. An old and classical problem in geometry is to wonder how dynamical or geometrical invariants can change when one deforms the metric σ in a certain class of metric. Among the invariants of interests are the Laplace spectrum, both on the surface, and the L^2-spectrum on the universal cover, the systole, the entropies (metric, harmonic, or topological) of the geodesic flow, or the Kaimanovich entropy and linear drift of the Brownian motion. Since anything can be changed by scaling, a necessary condition to make the problem non-trivial is to fix, for instance, the volume. There has been a lot of work done around that problem, in particular when the deformation is taken among all the negatively (or nonpositively) curved metrics. Another often studied class is the conformal class of a fixed metric. There seem, however, to have been relatively little work for metrics in the intersection of these two classes. Hence, in the present article, we aim to investigate how the geometric and dynamical invariants behave when we consider all negatively, or non positively, curved metrics in a fixed conformal class, with fixed area. In <cit.>, Katok proved that for any hyperbolic metric σ on M with total area A and any negatively curved metric g=e^2uσ, with total area A, we have h_μ(g)⩽ h_μ(σ)∫_M e^ud v_σ/A andh_top(σ)(∫_M e^ud v_σ/A)^-1⩽ h_top(g),where h_μ(g) and h_top(g) are, respectively, the metric entropy with respect to the Liouville measure, and the topological entropy of the geodesic flow. In particular, Katok obtains that, for any negatively curved metric g with total volume A, we haveh_μ(g)⩽(2π|χ(M)|/A)^1/2⩽ h_top(g).Furthermore, he shows that either equality above holds if and only if g is a hyberbolic metric. Moreover, in <cit.> the second author and Katok proved that equation (<ref>) gives the only restriction on the possible pair of entropies in the class of negatively curved metrics with total area A. They do not, however, control the conformal classes of the metrics that they build. Another famous set of inequalities (due to Guivarch <cit.> and Ledrappier <cit.>), that are satisfied for any metrics on a compact manifold, is the following4λ_1(g) ≤ h(g)≤ l(g) h_top(g)≤ h_top(g)^2,where λ_1(g) is the bottom of the L^2-spectrum of the Laplacian on the universal cover M, and h(g) and l(g) are respectively the Kaimanovich entropy and the linear drift of the associated Brownian motion (see, for instance <cit.> for the definitions). Moreover, in dimension 2, when g is assumed to be negatively curved, we have a strong rigidity result: any equality in the chain of inequalities above imply that g is hyperbolic <cit.>. When considering a fixed conformal class, Ledrappier <cit.> proved that, for any hyperbolic metric σ on M with total area A and any negatively curved metric g=e^2uσ, we havel(g)⩽ l(σ)∫_M e^ud v_σ/A,h(g)=h(σ)andh_harm(g)⩾ h_harm(σ)(∫_M e^ud v_σ/A)^-1,where h_harm(g) is the entropy of the harmonic measure, and is equal to h(g)/l(g) <cit.>. Moreover, h_top(g) ⩾ h_harm(g) (by the variational principle) and equality holds if and only if g is a hyperbolic metric <cit.>. Notice that, in spite of all these related results, the relationship, if any, between λ_1 and the entropy of the Liouville measure, h_μ, has remained very mysterious so far. In what appears to be one of the first steps in this direction, we will show (Theorem <ref>) that h_μ cannot be an upper bound for λ_1.Our aim in this study is to try to uncover the further restrictions, if any, for all of the above invariants that may appear when a conformal class is fixed, in the spirit of A. Katok flexibility program (see <cit.>). Notice that bounds on all of these invariants can be obtained if one assumes a lower bound on the curvature. However, the curvature is unbounded below in a conformal class, hence none of these results are available to us.For any positive constant A, we let [g]_A^⩽ (resp. [g]_A^<) be the family of nonpositively (resp. negatively) curved metrics conformally equivalent to g and with total area A. In the remainder of this text, a hyperbolic metric will always refer to a metric of constant negative curvature equal to -1. Our first result, which can be seen as a slight generalization of Schwarz Lemma (see, for instance, <cit.>), is the following[Theorem <ref>]Let (M,σ) be a hyperbolic surface of genus ⩾ 2 and total area A. Then, there exists a positive constant C = C(σ) such that for any smooth function u M → such that g= e^2uσ∈ [σ]_A^⩽ we have u(x)⩽ C for any x∈ M. This result admits many corollaries. First, we obtain an upper bound on the diameter of M for any g∈ [σ]_A^⩽ depending only on σ (independently of that result, we also obtain an upper bound on the diameter under the weaker assumption that g has no conjugate point, see Theorem <ref>). Second, thanks to the Min-Max principle (see Subsection <ref>), we also get a lower bound on the Laplace spectrum: [Corollary <ref>] Let (M,σ) be a hyperbolic surface of genus ≥2 and total area A.There exists a constant C>0 such that the following holds: For any g= e^2uσ∈ [σ]_A^⩽ and any k, we haveλ_k(g)⩾ Cλ_k(σ). In particular, the function g ↦λ_1(g) is uniformly bounded away from 0 on the space [σ]_A^⩽.Similarly, on the universal cover, we have λ_1( g) ⩾ C λ_1(σ). Let us stress again that, as opposed to, for instance, Li and Yau's lower bound for the Laplace spectrum (see <cit.>), our result does not require a lower bound on the Gaussian curvature (see Section <ref> for an example of metric in a fixed conformal class with arbitrarily negative Gaussian curvature). We also study in more details a certain family g_ of metrics build in <cit.>. This family is such that the metric entropy of g_ comes arbitrary close to zero (forgoing to 0) and the topological entropy is bounded above by a fixed constant. By <cit.>, the linear drift of these examples is also bounded away from zero by a fixed constant.We prove that the family of hyperbolic metrics σ_ in the conformal class of g_ must stay in a compact part of the Teichmüller space (see Theorem <ref> and its proof). As a result of this study and our Corollary above, we obtain the first known example of a negatively curved metric g such that h_μ(g)^2 < 4λ_1(g): [Corollary <ref>] Let >0 and f→^+ be any real continuous function such that f(0)=0, then there exists a negatively curved metric g such thatf(h_μ(g)) < λ_1(g), andh_μ(g)< .Furthermore, we can choose g∈ [σ]^<_A, where σ is a hyperbolic metric inside a fixed compact set of the Teichmüller space. In terms of the flexibility program, Theorem <ref> shows that one can stay in the conformal classes of hyperbolic metrics in a fixed compact of the Teichmüller space (or any small neighborhood of at least one particular hyperbolic metric), and still obtain all the possible values for the metric entropy. We believe however that the stronger result below should be true.Let (M,σ) be a hyperbolic surface of genus ⩾ 2 and total area A. Then, inf_g∈[σ]_A^<(h_μ(g))=0. Another goal of our study was to determine whether the topological entropy is as flexible as the metric entropy in a conformal class. It turns out that the flexibility of the topological entropy is linked to the flexibility of two other invariants: the linear drift l(g) and the systole (g), i.e., the length of the shortest geodesic.On one side, the topological entropy is, in negative curvature, always bounded above by a constant depending only on the area and the (inverse of the) systole (see Theorem <ref>). So in particular, if h_top(g) is unbounded, then (g) must go to zero.Conversely, Besson, Courtois and Gallot <cit.>, proved that if a family of negatively curved metrics with bounded diameter (this is always satisfied in our case due to Theorem <ref>) is such that if (g) goes to zero, then h_top(g) must be unbounded (this result also follows from the arguments of <cit.>). Finally, the variational principle and equation (<ref>) implies that, if l(g) goes to zero in a conformal class, then h_top(g) must be unbounded (and hence, (g) must go to zero).In this article, we obtained some partial results regarding the systole. We did not manage to show that the systole stays bounded away from zero in a class [σ]_A^⩽. However, we did prove that the “obvious” way of building a family of metric with systole going to zero cannot be done in a conformal class without positive curvature. More precisely, writing l_g(γ) for the g-length of a curve γ, we showLet (M,σ) be a hyperbolic surface of genus ⩾ 2 and total area A. For any N>0, and any closed σ-geodesic γ, such that l_σ(γ) ⩽ M, there exists a positive constant =(N) such that, for any g∈ [σ]_A^⩽, we have l_g(γ) ⩾. This result is presumably far from optimal, since our lower bound goes to zero as the length of the original σ-geodesic goes to infinity, and it seems at the very least counter-intuitive that one could shrink a very long curve without shrinking short ones. Although we did not pursue it, one could adapt our arguments to show that for any topologically non trivial closed curve γ of controlled σ-length (i.e., bounded above), and controlled σ-geodesic curvature (both above and below), one can obtain a positive lower bound for the g-length of γ, with g in [σ]_A^⩽. But again, the bound obtain via our technique will go to zero as the σ-length or σ-geodesic curvature goes to infinity. Nonetheless, we feel confident that the following should be true. Let (M,σ) be a hyperbolic surface of genus ⩾ 2 and total area A. Then, there exists a constant C = C(σ)>1 such that, for any g∈[σ]_A^⩽, (g)>C^-1, h_top(g) <C, and C^-1< l(g)< C.Finally, in Section <ref> we build examples, some folkloric, others new, showing that the conditions of conformality and negative curvature are essential for all our established and conjectural bounds.We will use the following notations. The pair (M, g) denotes a smooth closed Riemannian surface with metric g, and v_g is the associated Riemannian measure on M. The universal cover of M is denoted by M, g is the lifted metric, and d_ M(·,·) is the associated distance function. We write Δ_g = div_g(_g ) for the Laplacian of g and 0 = λ_0(g)< λ_1(g)⩽λ_2(g)⩽→∞ for its spectrum. Throughout the text, since we will often have to switch between objects defined for different metrics, we write g-length, g-ball, g-geodesic, g-area, etc., to refer to the length, ball, geodesic, or area defined by the metric g. Finally, we will also sometimes abuse terminology and refer to a shortest closed geodesic as “a systole”. § RESTRICTIONS FOR METRICS WITH FIXED TOTAL AREA IN A FIXED CONFORMAL CLASSIn this section, we will prove Theorems <ref> and <ref>, as well as other bounds that we can obtain from our conditions. The proofs in subsections <ref> and <ref> follow essentially one overarching easy idea that we sketch now.Suppose σ is a fixed hyperbolic metric and g=e^2uσ is a metric in its conformal class. Then, the Gaussian curvature K_g of g satisfy to the followingΔ_σ u + 1 + K_g e^2u = 0.In particular, g has nonpositive curvature if and only if Δ_σ u ⩾ -1.Now, for a function of two variables u, equation (<ref>) still allows a lot of flexibility. However, if u was one-dimensional, then equation (<ref>) becomes very stringent. So, in the arguments, we average u over some closed curves in local charts to obtain a one-dimensional function that satisfies equation (<ref>) and then leverage the condition that the g-area stays constant to get our bounds. §.§ Upper bound on the conformal factorLet (M,σ) be a hyperbolic surface of genus ⩾ 2 and total area A. Consider a smooth function u M → such that g= e^2uσ∈ [σ]_A^⩽. Let u be the lift of u to M and x_max∈ M be a point, where a global maximum of u is achieved. Then,∫_S_σ(x_max,R) u dl_σ⩾2πsinh R (max_Mu - 2log(coshR/2)),where S_σ(x_max,R) is the σ-sphere of radius R centered at x_max.Passing to the universal cover, let (r,θ) be hyperbolic polar coordinates on the σ-ball B_σ(x_max,ρ) ⊂ M. Recall that the Laplacian in polar coordinates is given byΔ_σ = ∂^2/∂ r^2 + cosh r/sinh r∂/∂ r + 1/sinh^2 r∂^2/∂θ^2.Writing u for the lift of u to the universal cover M, we can use Green's theorem to get∫_B_σ(x_max,ρ)Δ_σ u dv_σ = ∫_0^2π∂ u/∂ r(ρ,θ)sinhρdθ.The fact that g has nonpositive curvature is equivalent to Δ_σ u ⩾ -1. Therefore, we have∫_B_σ(x_max,ρ)Δ_σ u dv_σ⩾ -(B_σ(x_max,ρ)) = -2π(coshρ - 1).As a result, by (<ref>) and (<ref>) we obtain∫_0^2π∂ u/∂ r(ρ,θ)dθ⩾ -2πcoshρ - 1/sinhρ.Integrating the above inequality with respect to ρ from 0 to R gives the proposition:∫_0^R ∫_0^2π∂ u/∂ r(ρ,θ)dθ dρ = ∫_0^2πu(R,θ)dθ -2πmax_Mu ⩾- 4πlog(coshR/2).Let (M,σ) be a hyperbolic surface of genus ⩾ 2 and total area A. Then, there exists a positive constant C = C(σ) such that for any smooth function u M → such that g= e^2uσ∈ [σ]_A^⩽ we have u(x)⩽ C for any x∈ M.This result can be seen as a slight generalization of Schwarz lemma (see, for instance, <cit.>), where we do not assume that the curvature is bounded above by a negative constant.Again passing to the universal cover, let (r,θ) be hyperbolic polar coordinates on the σ-ball B_σ(x_max,(σ)) ⊂ M, where (σ) is the injectivity radius of (M,σ), and x_max∈ M is a point where the global maximum of the lift u of u is achieved.By Proposition <ref> and Jensen's inequality, for every 0<R⩽(σ) we havelog(∫_0^2πe^2u(R,θ) dθ)⩾log (2π) + 1/2π∫_0^2π2u(R,θ) dθ⩾log(2π)+2max_M u - 4log(coshR/2).Therefore, we obtain∫_0^2πe^2u(R,θ) dθ⩾ 2πexp(2max_M u)/cosh^4R/2. The g-area of B(x_max, (σ)) is bounded above by A, since the projection to M restricts to a bijection on B(x_max, (σ)). So, using the above inequalities, we obtainA ⩾∫_0^(σ)∫_0^2πe^2u(r,θ)sinh r dθ dr ⩾ 2πexp(2max_M u)∫_0^(σ)sinh r/cosh^4r/2 dr= 2πexp(2max_M u)∫_0^(σ)2sinhr/2coshr/2/(cosh^2r/2)^2 dr = 4πexp(2max_M u)(1-1/cosh^2(σ)/2). Therefore, max_M u ⩽1/2logA/4πtanh^2(σ)/2. Let (M,σ) be a hyperbolic surface of genus ⩾ 2 and total area A. Then, for any integer N⩾ 1 there exists a positive constant K_1 = K_1(N, (M)) such that, for any open set O⊂ M with (O)⩽ N(M) and any smooth function u M → such that g= e^2uσ∈ [σ]_A^⩽, we have∬_O udv_σ⩾ -K_1,where u is the lift of u to the universal cover.Given our assumptions on O, there exists a σ-ball B(x_max,R) centered at a point x_max, where R = (N+1)(M) the global maximum of u is achieved, such that O⊂ B(x_max,R). Then,∬_O udv_σ= ∬_B(x_max,R) udv_σ - ∬_B(x_max,R)∖ O udv_σ.Also, we notice that max_M u⩾ 0 as g has the same area as σ. In the polar coordinates (r,θ) for the metric σ in the ball B(x_max, R), by Proposition <ref> we have∬_B(x_max, R) udv_σ = ∫_0^R∫_0^2π usinh rdθ dr⩾ -4π∫_0^Rsinh rlog(coshr/2) dr = = -2π-4π((cosh R+1)log(coshR/2)-1/2cosh R). From the inequality above and Theorem <ref>, it is easy to deduce the Corollary.Thanks to Theorem <ref>, we can bound the Laplace spectrum of nonpositively curved metrics in a given conformal class.Recall that 0=λ_0(g)< λ_1(g) ≤λ_2(g) ≤… denotes the Laplace spectrum of the metric g on M, and λ_1( g) is the bottom of the L^2-spectrum of the lifted Laplacian on the universal cover M. The Min-Max principle state that, for any k ∈, λ_k(g) = inf_V_ksup{R_g(f)| f≠ 0, f∈ V_k},where V_k runs through all the k+1-dimensional subspaces of the Sobolev space H^1(M) and R_g(f) is the Rayleigh quotient of f, i.e.,R_g(f) = ∫_M|∇_g f|_g^2 dv_g/∫_Mf^2 dv_g.Note that, when g = e^2uσ, the Rayleigh quotient becomesR_g(f) = ∫_M|∇_σ f|_σ^2 dv_σ/∫_Me^2uf^2 dv_σ. Hence, using the Min-Max principle, we immediately obtain the following corollary from Theorem <ref>.Let (M,σ) be a hyperbolic surface of genus ≥2 and total area A.There exists a constant C = C(σ)>0 such that, for any Riemannian metric g∈ [σ]^⩽_A, and all k ∈ℕ,λ_k(g)⩾ Cλ_k(σ). In particular, for any nonpositively curved metric in a fixed conformal class, the function λ_1(g) _g(M) is uniformly bounded away from 0.Similarly, on the universal cover, we have λ_1( g) ⩾ C λ_1(σ). §.§ Lower bounds on the length of some curves In this section, we prove lower bounds for the integral of u on a number of special curves. This results can be translated using Jensen's inequality to lower bounds on the g-length of these curves. In particular, one gets Theorem <ref> from Proposition <ref> in that way.Let (M,σ) be a hyperbolic surface of genus ⩾ 2 and total area A. Let x∈ M and 0<R_1≤ R_2 arbitrary. There exists a positive constant K_2= K_2(σ, R_1, R_2) such that for any smooth function u M → such that g= e^2uσ∈ [σ]_A^⩽ and any r∈ [R_1,R_2], we have ∫_S_σ(x,r) u dl_σ⩾ - K_2,where S_σ(x,r) is the σ-sphere of radius r centered at x and u is the lift of u to the universal cover. Let (r,θ) be hyperbolic polar coordinates on the σ-ball B_σ(x,R_2) ⊂ M.Let 0≤⩽ρ be arbitrary. Let A_σ(,ρ) = { (r,θ) |⩽ r ⩽ρ}. There exists an integer N(ρ)⩾ 1 depending on ρ and the hyperbolic metric σ such that the annulus A_σ(,ρ) is contained in at most N(ρ) fundamental domains for M in M. Call this minimal union of fundamental domains F. Since Δ u ⩾ -1 and ∫_F Δ u dv_σ = N(ρ) ∫_M Δ u dv_σ= 0,we get that ∫_A_σ(,ρ)Δ u dv_σ = - ∫_F∖ A_σ(,ρ)Δ u dv_σ⩽∫_F∖ A_σ(,ρ) dv_σ⩽AN(ρ). Now, ∫_A_σ(,ρ)Δ u dv_σ = ∫_0^2π∫_^ρΔ u sinh r dr dθ= ∫_0^2πsinhρ∂ u (ρ, θ)/∂ r - sinh∂ u (, θ)/∂ r dθ. Hence, taking =0 and changing the order of the integration and differentiation, we obtain∂/∂ r. ∫_0^2π u(r, θ) dθ |_r=ρ⩽AN(ρ)/sinh(ρ). Since we also have that ∫_A_σ(,ρ)Δ u dv_σ≥ - (A_σ(,ρ)), we deduce ∂/∂ r. ∫_0^2π u(r, θ) dθ |_r=ρ⩾ - (A_σ(0,ρ))/sinh(ρ). From the control of derivatives given by the two equations above and Corollary <ref>, it is easy to deduce the Proposition. Let (M,σ) be a hyperbolic surface of genus ⩾ 2 and total area A. Let 0<R_1⩽ R_2 be arbitrary. There exists a positive constant K_3= K_3(σ, R_1,R_2) such that for any smooth function u M → such that g= e^2uσ∈ [σ]_A^⩽, any x∈ M and any r∈ [R_1,R_2], we have ∫_L_r u dl_σ⩾ - K_3,where L_r⊂ S_σ(x,r) is a piece (a union of pieces) of arcs of the σ-sphere S_σ(x,r) of radius r centered at x and u is the lift of u to the universal cover.The result follows from Proposition <ref> and Theorem <ref>.∫_L_r u dl_σ = ∫_S_σ(x,r) u dl_σ - ∫_S_σ(x,r)∖ L_r u dl_σ⩾ -K_2-Cl_σ(S_σ(x,r)∖ L_r) Let (M,σ) be a hyperbolic surface of genus ⩾ 2 and total area A. For any closed σ-geodesic γ, there exists a positive constant K_4 = K_4(l_σ(γ)) such that, for any smooth function u M → such that g= e^2uσ∈ [σ]_A^⩽, we have ∫_γ u dl_σ≥ - K_4. Our proof gives a counter-intuitive dependency of K_4=K_4(l_σ(γ)) in the geodesic γ. Indeed, due to the use of Corollary <ref>, K_4(l_σ(γ)) goes to infinity as l_σ(γ) goes to infinity. One could presumably obtain a uniformly bounded constant by being a bit more careful, but we choose not to pursue that direction as it does not lead to any significant improvement regarding Conjecture <ref>. Note also that the proof shows that for any curve α that is obtained as an equidistant curve from a σ-geodesic γ, we have ∫_α u dl_σ≥ - K_4. Let M_c be a cylindrical cover of M with fundamental group generated by the homotopy class of γ. Consider hyperbolic normal polar coordinates (r,θ) on M_c, where γ is described by the equation r=0 and θ∈[0;l_σ(γ)]. Let u_c be the lift of u to M_c. Let 0≤≤ρ be arbitrary and A_σ(-ρ, -) = { (r,θ) | -ρ≤ r ≤ -}.Green's theorem implies that∬_A_σ(-ρ, -)Δ_σu_c dv_σ = ∫_0^l_σ(γ)∂ u_c/∂ r(-,θ)coshdθ - ∫_0^l_σ(γ)∂ u_c/∂ r(-ρ,θ)coshρdθ = cosh∂/∂ r. ∫_0^l_σ(γ) u_c(r,θ) dθ |_r=- - coshρ∂/∂ r.∫_0^l_σ(γ) u_c(r,θ) dθ |_r=-ρ. Recall that g being nonpositively curved implies that Δ_σu⩾ -1, so Δ_σu_c⩾ -1. Therefore, ∬_A_σ(-ρ, -)Δ_σu_c dv_σ⩾ -(A_σ(-ρ, -)) = -(sinhρ - sinh)l_σ(γ) Now, by Corollary <ref> applied to A_σ(-(M),0) and the equality ∬_A_σ(-(M), 0)u_c dv_σ = ∫_0^(M)∫_0^l_σ(γ)u cosh r dθ dr,there exists a positive constant K_5 = K_5((M), l_σ(γ),σ) and R_1∈[0;(M)] such that ∫_0^l_σ(γ)u_c(-R_1,θ)dθ⩾ -K_5/(M).The constant K_5 goes to infinity as the length of γ goes to infinity due to the use of Corollary <ref>. If R_1=0, then the Proposition is proven.So we suppose that R_1≠ 0. Then, there exists a constant R_2⩾ R_1 and smaller than some constant R = R((M), K_5, σ) such that ∫_0^l_σ(γ)u_c(-R_2,θ)dθ⩾ -K_5/(M), ∂/∂ r.∫_0^l_σ(γ)u_c(r,θ)dθ |_r=-R_2⩾ -K_5. Indeed, either R_1 itself works, or∂/∂ r.∫_0^l_σ(γ)u_c(r,θ)dθ |_r=-R_1< -K_5 ≤ 0.So, at least locally around -R_1, the function ∫_0^l_σ(γ)u_c(r,θ)dθ is strictly decreasing.Then, for all r ≥ R_1, sufficiently close to R_1, we have∫_0^l_σ(γ)u_c(-r,θ)dθ> ∫_0^l_σ(γ)u_c(-R_1,θ)dθ⩾-K_5/(M).However, since u is uniformly bounded above (by Theorem <ref>), for all r we have, ∫_0^l_σ(γ)u_c(-r,θ)dθ⩽ Cl_σ(γ).Hence we cannot have∂/∂ r.∫_0^l_σ(γ)u_c(r,θ)dθ |_r< -K_5,for all r≥ R_1. Thus R_2 exists as claimed.Now take ρ = R_2 in equation (<ref>) and recall that R_2⩽ R. Assume that for any D>0 there exists u such that ∫_γ u dl_σ< -D. Then, there exists ∈[0;R_2] such that ∂/∂ r.∫_0^l_σ(γ) u_c(r,θ) dθ |_r=-⩽ -D-K_5/(M)/R. The constant D can be chosen arbitrary large. As a result, we obtain a contradiction as the left-hand side of the equality (<ref>) is bounded below and the right-hand side of it can be arbitrary small. Therefore, the Proposition follows. Using Jensen's inequality we can restate all the results of this section, replacing the integral of u over the curves by the g-length of the curves. In particular, Theorem <ref> is a direct corollary of Proposition <ref>. §.§ Upper bound on diameter for metrics without conjugate points As mentioned in the introduction, a direct consequence of the upper bound on the conformal factor given by Theorem <ref> is that the diameter stays bounded in a class [σ]_A^⩽. But we can further relax our assumptions and still get a bound on the diameter.Let (M,σ) be a compact hyperbolic surface of genus ⩾2 and total area A. Then, there exists a constant D= D(σ)>0 such that for any smooth metric g without conjugate points and total area A that is conformally equivalent to σ, we havediam(M,g)⩽ D. The proof of the Theorem above relies on the following lemma, which says that one cannot increase infinitely the length inside a fixed disc without creating conjugate points. We state the lemma in a more general context as the one needed for Theorem <ref>, as we will also be using that lemma in the proof of Theorem <ref>.Let (M,h) be a nonpositively curved surface of genus ⩾ 2 and total area A. Let >0 be arbitrary. Suppose that for some x∈ M, the metric h is invariant under rotations in the h-ball B_h(x,2) ⊂ M. Then, there exists a constant C>0, depending only onand the metric h such that, for any Riemannian metric g= e^2uh with no conjugate points and total area A, we have, for all y,z∈ B_h(x,),d_g(y,z) ≤ C.As we will see in the proof, the constant C actually depends only onand an upper bound for the function √( h) in B_h(x,2)Let (r, θ) be polar coordinates in the ball B_h(x,2). The g-area of an annulus T() = {(r,θ)| ⩽ r⩽ 2, 0⩽θ⩽ 2π} is smaller than A and is equal to∬_T()e^2udv_σ = ∫_^2∫_0^2πe^2u(c_θ(r))f(r) dθ dr,where f(r) = √( g), g is the determinant of the metric tensor and c_θ(r) is a h-geodesic radius corresponding to the angle θ. The function √( g) depends only on r in B_h(x,2) because h is assumed to be invariant under rotations in that ball.Therefore, there exists r̅∈[; 2] such that ∫_0^2πe^2u(c_θ(r̅))f(r̅) dθ⩽A/.It follows that∫_0^2πe^2u(c_θ(r̅))dθ⩽A/ f(r̅).Moreover, if we denote C_r̅ = {(r, θ) | r=r̅,0⩽θ< 2π}, then the g-length of C_r̅ is controlled:l_g(C_r̅) = ∫_0^2πe^u(c_θ(r̅))f(r̅) dθ⩽ f(r̅)√(2π)(∫_0^2πe^2u(c_θ(r̅))dθ)^1/2⩽√(max_r∈[;2] f(r))(2π A/)^1/2=:W.As a result, any two points p,q ∈ C_r̅ can be connected by two arcs of C_r̅ of g-length less than or equal to W. Fix any two points p,q ∈ C_r̅. Let z ∈ B_h(x,). We want to show that either d_g(z,p) ≤ W, or d_g(z,q) ≤ W. Once this is established, the lemma will follow with C= 3W.So, suppose this is not the case, i.e., suppose that the g-length of any path connecting p to q and passing through z is strictly greater than W.Let B_h(x,) be a lift of B_h(x,) in the universal cover M. Let p,q,z ∈ B_h(x,) the associated lifts of p,q and z. By what we proved so far and our assumption, the space of curves connecting p to q of g-length less than or equal to W has two connected components. Therefore, there exist two different geodesics connecting p and q. This contradicts the fact that g has no conjugate points. Hence, z is at g-distance less than W from either p or q.The proof of Theorem <ref> follows easily from Lemma <ref> Choose a minimal cover of M by balls of radius (σ), the injectivity radius of σ. Let N(σ) be the number of balls in the cover. Let z_1,z_2∈ M. The σ-geodesic between z_1 and z_2 is covered by at most N(σ) balls. Now, since σ is invariant under rotations in every balls of the cover, we can apply Lemma <ref> with h=σ. It gives us the existence of a constant C((σ)) such that d_g(z_1,z_2) ≤ N(σ)C((σ)).§ FLEXIBILITY FOR NEGATIVELY CURVED METRICS WITH FIXED TOTAL AREA IN A FIXED CONFORMAL CLASSIn this section, we show that the metric entropy is flexible inside a conformal class and prove Theorem <ref>. Consider a surface M of genus ⩾ 2 and total area A. There exists a compact set 𝒦 in the Teichmüller space on M on which the following holds. For any positive constant >0 there exists a negatively curved Riemannian metric g on M, such that:* the total area of M with respect to g is A; * the metric g is conformally equivalent to a hyperbolic metric in 𝒦; * the metric entropy, h_μ(g), of g, i.e., the entropy of the geodesic flow of g with respect to its Liouville measure, satisfy 0< h_μ(g) ⩽.As in <cit.>, one might want to chose g in the theorem above in such a way that its topological entropy h_top(g) is as close to (2π|χ(M)|/A)^1/2 as one wishes. However, if we try to do this following <cit.> here, then the compact set 𝒦 that we obtain will, a priori, have to become infinitely big. In particular, we do not know whether small values of h_μ forces a gap for h_top or not.Thanks to Theorem <ref>, we immediately obtain theConsider a surface M of genus ⩾ 2 and A>0. There exists a hyperbolic metric σ on M of total area A such that in any neighborhood 𝒰 of σ in the Teichmuller space for any >0 there exists a hyperbolic metric σ'∈𝒰 of total area A with inf_g∈[σ']_A^<h_μ(g)<. Notice that one can apply the normalized Ricci flow to a metricg∈[σ']_A^< such that h_μ(g)<, and, using the continuity of the metric entropy (see <cit.>), get that all the values in [, (2π|χ(M)|/A)^1/2] are realized as the metric entropy of some metric in [σ']_A^<.For σ as in Corollary <ref>, we actually expect that inf_g∈[σ]_A^<h_μ(g)=0. Indeed, for any n, one can pick σ_n a hyperbolic metric and g_n = e^2u_nσ_n such that σ_n converges in the C^∞ norm to σ, and h_μ(g_n)<1/n, then, by continuity, one would expect that the metric entropy of h_n = e^2u_nσ converges to 0. There are however two big problems to try to make that argument works. First, h_n has no reason a priori to be negatively curved, as the curvature of g_n has to converge to zero. Second, even if we knew that h_n was negatively curved, we also know, from the proof of Theorem <ref>, that the functions u_n can be C^0, but not C^1 close as n goes to infinity. Hence one cannot use the known results about continuity of the metric entropy.In spite of this, we do believe that the metric entropy should be totally flexible in a conformal class. We even believe (Conjecture <ref>) that inf_g∈[σ]_A^<h_μ(g)=0 for any hyperbolic metric σ on M with total area A. Let >0 and f→^+ be any real continuous function such that f(0)=0, then there exists a negatively curved metric g such thatf(h_μ(g)) < λ_1(g),andh_μ(g) < .Furthermore, we can choose g∈ [σ]^<_A, where σ is a hyperbolic metric inside a fixed compact set of the Teichmüller space.This follows directly from the combination of Corollary <ref> and Theorem <ref>. In Section 3.1 of <cit.>, the second author and Katok builds a family of negatively curved metrics satisfying the conditions 1) and 3). We will show here that this family of example also satisfy the condition 2). To prove that, we show that the hyperbolic metrics which are in the same conformal class have diameter uniformly bounded above, and hence, stay in a compact part of the Teichmüller space. The proof of the uniform bound on the diameter follows the same line as the proof of Theorem <ref>.We start by a quick description of the construction given in <cit.> of the relevant metrics. Let σ be a fixed hyperbolic metric on M of total area A. Let d>0 be sufficiently small.Let 𝒯 be a triangulation by σ-geodesic triangles of M such that each edge is a geodesic segment of σ-length between d/3 and d. Such a triangulation exists: We can take any geodesic triangulation and refine it in such a way that it satisfies the condition. To each triangle T in 𝒯, consider its comparison triangle T_∗ in Euclidean space, i.e., the Euclidean triangle T_∗ has sides of Euclidean length equal to the σ-length of the corresponding side in T. From all the comparison triangles T_∗, we construct a polyhedral surface (homeomorphic to M) with conical singularities by gluing them according to the incidence of the original triangles in . In particular, we thus obtain a singular metric g_0 on M with zero curvature everywhere except from the conical singularities at the vertices ofwhere the angle is larger than 2π. For sufficiently small d, the ratio of the area between a triangle T ∈ and its comparison triangle T_∗ becomes arbitrarily close to 1 (see <cit.>). Hence, we can choose the triangulationin such a way that the total area of M for g_0 is comprised between, say, (1-1/1000)A and (1+1/1000)A.For all α>0 sufficiently smaller than d, we can find a family of smooth Riemannian metrics {g_}, ∈(0,α], obtained by smoothing of g_0, such that (see <cit.> for the details):* each metric g_ has nonpositive curvature; * the metric g_ coincide with g_0 outside of some -neighborhoods (for g_0) of the conical singularities of g_0; * the total area of M for g_ is between (1-1/100)A and smaller than (1+1/100)A. Let p be a conical point of g_0. Forfixed, consider the polar coordinates (r,θ) centered at p for the metric g_. As usual, θ is the angular coordinate and r is the g_-distance to p. The smoothing procedure in Lemma 3.3 of <cit.> is such that a sufficiently small g_-ball centered at the conical point of g_0 is invariant under rotations with respect to the center. Therefore, all such g_0-balls are also g_-balls for any(but of varying radius). Moreover, the construction gives uniform upper and lower bounds on the g_-radii of such balls in terms of α and(see <cit.> for details). Furthermore, since the metrics g_ and g_0 coincide outside of neighborhoods of the conical points, the g_0-balls which do not intersect α-neighborhoods of singular points of g_0 are also g_-balls of the same radii.Let {B_g_0(x_i,r_i)}_i=1,…, N be an open cover of M by g_0-balls such that, for every i, B_g_0(x_i,2r_i) is either entirely contained in a flat part of g_0, or x_i is a conical point of g_0 and B_g_0(x_i,2r_i) does not contain any other conical points. We further assume that {B_g_0(x_i,r_i)}_i=1,…, N is a minimal such cover, i.e., it has the smallest number of balls among all covers satisfying to our conditions. So N is a number depending only on g_0, or equivalently, depending only on the topology of M, the starting hyperbolic metric σ and the choice of d. Let R=max_i r_i. Similarly, R depends only on g_0.We now choose α smaller if necessary, so that all the balls B_g_0(x_i,2r_i) that are entirely contained in a flat part of g_0 are disjoint from every α-neighborhood of the conical points of g_0.Let C = sup{√( g_)|∈(0,α] }. Given the smoothing procedure used, C is finite and depends only on g_0, , α and R (this follows from formula (3.5) in <cit.>). Since, by construction, for ∈(0,α], all of the g_ are rotationally invariant in each ball B_g_0(x_i,r_i), we can apply Lemma <ref> to all of them. Moreover, according to Remark <ref>, the constant C given by the lemma depends only on g_0, , C, α and R. Thus, we deduce as in the proof of Theorem <ref> that the diameter of any metric without conjugate points conformally equivalent to one of g_, ∈(0,α], and of total area A is bounded above by NC. In particular, for any , the hyperbolic metric σ_ of total area A, that is conformally equivalent to g_, has diameter bounded by N C. Hence, for any , the hyperbolic metrics σ_ must stay in a compact subset of the Teichmüller space. It is shown in Section 3.3 of <cit.> that the metric entropy of g_ tends to 0 astends to 0.Now, by construction, the total area of each g_ is bounded between (1-1/100)A and (1+1/100)A. Therefore, the metrics g̅_ = g_A/_g_(M) are smooth nonpositively curved metrics, of total area A, and their metric entropy still converges to 0 with .Finally, to obtain metrics that are negatively curved, instead of just nonpositively, we can smoothly approximate the metrics g̅_ by metrics of negative curvature with total area A that are conformally equivalent to g̅_ by essentially taking some negative curvature from the neighborhoods of the conical points and distribute it among regions where we have zero curvature (see Proposition 5.1 in <cit.> for more details). Another way to avoid nonpositive curvature is to replace initial triangles of zero curvature by triangles of constant negative curvature close to 0 and repeat the construction.§ FLEXIBILITY FOR METRICS WITH FIXED TOTAL AREA UNDER WEAKER CONDITIONSIn this section, we construct a number of examples, some relatively well-known, some potentially new, showing that none of the bounds obtained in Section <ref> still hold when we relax our hypothesis.§.§ Examples of metrics with arbitrary short systole Let M be a surface of genus ⩾ 2 equipped with a hyperbolic metric σ of total area A. Then, for every >0 there exists a Riemannian metric g on M of total area A that is conformally equivalent to σ such that (g) ≤ and (M,g) ≤ 2(M,σ). Notice that the metric we construct for this lemma would fall into the scope of Proposition <ref> if it was nonpositively curved. Hence these particular examples must have some positive curvature.Let >0. Let γ be a systole for σ on M, i.e., such that l_σ(γ)=(σ). We assume that <(σ). For any δ, to be specified later, we consider a smooth bump function ϕℝ→ℝ such that ϕ(0)=1 and ϕ(t) = 0 for every t∉ (-δ, δ). We define a smooth function u M→ byu(x) = -log((σ)/)ϕ(d_σ(x,γ))+C(1-ϕ(d_σ(x,γ)))where C is some real number that will be determined later and d_σ(x, γ) is the σ-distance from a point x to the curve γ. In particular, u is equal to -log((σ)/) on γ and equal to C outside the δ-neighborhood (for σ) of γ.Set g:=e^2uσ. For an appropriate choice of δ and C, we will show that the metric g satisfy the lemma. First, notice that the g-length of γ is equal to . This implies that (g)⩽. As g is in the conformal class of σ, all we have left to show is that the g-volume of M is A, and that the g-diameter of M is at most twice the σ-diameter.Call V(γ,δ) the δ-neighborhood of γ for the hyperbolic metric σ. For sufficiently small δ we have _σ(V(γ,δ))≤ A/2. Then, the g-volume of M, which is equal to ∫_Me^2udv_σ, satisfy ∫_Me^2udv_σ ⩾ e^2C_σ(M ∖ V(γ,δ) )⩾e^2CA/2>A ifC> log 2/2 ∫_Me^2udv_σ ⩽ e^2C_σ(M ∖ V(γ,δ)) + _σ(V(γ,δ))⩽ e^2CA+A/2<A ifC<-log 2/2. Therefore, using the above inequalities and their continuous dependency on C, we can deduce that there exists C(δ) such that _g(M) = A. Moreover, C(δ) converges to zero when δ does.Given the construction, it is also clear that, as δ goes to zero, the diameter of the metric g tends to the diameter of σ. Hence, we can choose δ small enough so that our last claim is satisfied. If we do not require the condition that g has to be conformally equivalent to σ in Lemma <ref>, then by Section 2.1 in <cit.> we have the followingSuppose M is a surface of genus ⩾ 2 and A>0. Then, there exists D>0 such that, for every >0, there exists a negatively curved Riemannian metric g on M of total area A, diameter less than D, and the g-length of its systole is smaller than . Here, the hyperbolic metrics in the conformal class of the examples of Lemma <ref> must escape every compact of the Teichmüller space. Indeed, they are build in such a way (see <cit.>) that Proposition <ref> can apply to each of them, i.e, their systole is along a geodesic of the hyperbolic metric in their conformal class. Hence, if the hyperbolic metrics in their conformal class were to stay in a compact of the Teichmüller space, then Proposition <ref> would give a lower bound on the systole, a contradiction with Lemma <ref>.§.§ Examples of metrics with arbitrary large diameter Let M be a surface of genus ⩾ 2, equipped with a hyperbolic metric σ, of total area A. Then, for every D>0, there exists a Riemannian metric g on M of total area A that is conformally equivalent to σ and (M,g)>D. The gist of the construction is quite easy: Consider a small neighborhood around some point and change the metric conformally in such a way that this neighborhood is stretched, but without changing the volume too much. As a result one gets a surface with a long and very thin nose. The only difficulty is to actually produces a function that has exactly the right type of behavior.As before, we fix a point p on M and consider standard hyperbolic polar coordinates (r, θ). Let us consider a family of smooth bump functions ϕ(r; a)ℝ→ℝ, where a>0, such that ϕ(r; a)=1 when r∈[-a/2; a/2] and ϕ(r; a)=0 when r⩾ a. Fix a positive constant >0 such that the -neighborhood (for σ) of p is inside the coordinate chart. Then, for any positive δ<-1/log-log 2 (i.e., δ is such that (/2)^-δ<e) we define a smooth function ρ(x;δ, C) on M by, for every x∈ M inside the coordinate chart ρ(x;δ, C) = ϕ(r; )(/2)^-δ/2(Aδ/16π)^1/2(ϕ(r;e^-1/δ)(e^-1/δ/2)^-(1-δ/2)+(1-ϕ(r;e^-1/δ))r^-(1-δ/2)) +(1-ϕ(r; ))C,and ρ(x;δ, C)=C for every x∈ M outside the coordinate chart. As in the proof of Lemma <ref>, C is a positive constant that will be chosen so that the total area of M for the metric g_δ = ρ^2(x;δ, C)σ is equal to A.Let B(p,) denote the -neighborhood of p for the hyperbolic metric σ. Notice that for sufficiently smallwe have _σ(B(p,))≤ A/2. Then, the g-volume of M, which is equal to ∫_Mρ^2dv_σ, satisfy ∫_Mρ^2(x;δ, C)dv_σ ⩾ C^2_σ(M ∖ B(p,) )⩾C^2A/2>A ifC^2> 2∫_Mρ^2(x;δ, C)dv_σ ⩽ C^2_σ(M ∖ B(p,)) + _g_δ(B(p,δ))⩽ C^2 A+A/2<A ifC^2<1/2. Therefore, using the above inequalities and their continuous dependency on C, we deduce that there exists C(δ,) such that _g_δ(M) = A. Moreover, C(δ,) converges to 1 whentends to zero. The inequality _g_δ(B(p,δ))⩽A/2, forsmall enough, is obtained by direct computations. Indeed, we haveArea_g_δ(B(p,/2)) = ∫_0^2π∫_0^/2ρ^2(r;δ)sinh(r) dr dθ⩽∫_0^2π∫_0^/2(/2)^-δ(Aδ/8π)r^-(1-δ) dr dθ = = (/2)^-δ(Aδ/4)1/δ(/2)^δ⩽A/4,for sufficiently small . And, if C^2<1 and , δ are sufficiently small, the area of the annulus {(r, θ)| r∈[/2; ], θ∈ [0;2π)} satisfies∫_0^2π∫_/2^ρ^2(r,δ)sinh(r)drdθ⩽ 4π∫_/2^((/2)^-δ/2(Aδ/16π)^1/2r^-(1-δ/2)+C)^2rdr⩽ 4π(A(2^δ-1)/16π+C^2^2/2+2C/2^-δ/2(1+δ/2)(Aδ/16π)^1/2)⩽A/4. So all we have left to do is to prove that the diameter of g_δ is greater than D. First, notice that the radial curves (for σ) in the annulus B_σ(/2)∖ B_σ(e^1/δ) are minimal geodesics. Indeed, since the conformal coefficient in B_σ(/2)∖ B_σ(e^1/δ) depends only on r and not on θ, the shortest curves between the boundaries are those such that the angle component stays constant, i.e., radial curves.Now, let l be such a radial curve in B_σ(/2)∖ B_σ(e^1/δ). Then its g_δ-length satisfieslength_g_δ(l) = ∫_e^-1/δ^/2ρ(r;δ)dr⩾∫_e^-1/δ^/2r^-(1-δ/2)(/2)^-δ/2(Aδ/16π)^1/2dr ⩾(/2)^-δ/2(Aδ/16π)^1/22/δ((/2)^δ/2-e^-1/2) ⩾1/δ^1/2(A/4π)^1/2(1-(/2)^-δ/2e^-1/2) → +∞ as δ→ 0. Therefore, for every positive constant D there exists a positive number δ such that the diameter of M with respect to a Riemannian metric g_δ (defined above) of total area A and which is conformally equivalent to σ is larger than D.Notice that one can prove the above result more theoretically but less explicitely. The idea, used in <cit.>, goes as follows: A Riemannian surface (Σ,g) is, locally, almost Euclidean. Hence, it is, locally, almost conformal to the sphere with its standard metric. Therefore, starting with any metric h conformal to the standard sphere, it is possible to construct a conformal deformation of (Σ,g) around a point to make it arbitrarily close to h in that neighborhood. Applying that scheme when h is a long and thin sphere gives an example as in Lemma <ref>. One can also use this method to build conformal Cheeger Dumbell. Again, we will instead give an explicit example in the proof of Lemma <ref>.We recall that if we do not require the condition that g has to be conformally equivalent to σ in Lemma <ref> then there exists a Riemannian metric of constant negative curvature of fixed total area such that its diameter is larger than any a priori given positive number. §.§ Examples of metrics with arbitrary small first eigenvalue of Laplacian Our next example shows that if one drops the nonpositive curvature assumption, then Corollary <ref> fails, i.e., one can get arbitrarily small λ_1 in a fixed conformal class.Let M be a surface of genus ⩾ 2, equipped with a hyperbolic metric σ, of total area A. Then, for every >0 there exists a Riemannian metric g of total area A that is conformally equivalent to σ and λ_1(g)<. As mentioned in Remark <ref> above, such examples can also be obtained using the techniques of Colbois and El Soufi in <cit.>. Let p, q be two points in M at σ-distance strictly greater than . We define a metric g_δ that has total area A and is conformally equivalent to σ in the following way. Let ρ(· ;δ) be the function that is constant outside of -neighborhoods of p and q and defined by equation (<ref>) inside of the neighborhoods. We define g_δ := ρ^2(·;δ)σ.An argument similar to the proof of Lemma <ref> shows that one can pick the value of the function ρ outside of the -neighborhoods in such a way that g_δ has total area A. Recall from Lemma <ref> that a σ-ball centered at p or q of radius smaller than or equal to /2 is a g_δ-ball a priori of some other radius. Let (R, t) be polar coordinates for g_δ in the neighborhood of p. Denote by R_1 and R_2 the g_δ-radii of σ-balls of radii e^-1/δ and /2, respectively. Notice that equation (<ref>) givesR_2-R_1 = 1/δ^1/2(A/4π)^1/2(1-(/2)^-δ/2e^-1/2) → +∞ as δ→ 0 Let f_1 be a function on M such that it is equal to 1 in the R_1-neighborhood of p (for g_δ) and 0 outside of the R_2-neighborhood of it. In the annulus {(R,t)| R∈[R_1; R_2], t∈[0;2π)} we define f_1(R,t) = -R/R_2-R_1+R_2/R_2-R_1. Let f_2 be the function obtained as f_1, but considering q instead of p. Notice that f_1, f_2∈ H^1(M) as they are continuous and piecewise continuously differentiable. Furthermore, f_1 and f_2 have disjoint supports by construction.The fact that the first eigenvalue of Laplacian for g_δ tends to 0 as δ tends to 0 will follow from the Min-Max principle (see equation (<ref>)). Let V_2 be the 2-dimensional subspace of the Sobolev space H^1(M) spanned by the functions f_1 and f_2. Any F∈ V_2 can be written as F = af_1+bf_2, where a,b∈ℝ. Then,∫_M F^2 dv_g_δ = a^2∫_M f^2_1dv_g_δ+2ab∫_M f_1f_2dv_g_δ+b^2∫_M f^2_2dv_g_δ = a^2∫_M f^2_1dv_g_δ+b^2∫_M f^2_2dv_g_δ,as f_1,f_2 have disjoint supports. Similarly,∫_M |∇_g_δ F|_g_δ^2 dv_g_δ = a^2∫_M |∇_g_δ f_1|_g_δ^2dv_g_δ+b^2∫_M |∇_g_δ f_2|_g_δ^2dv_g_δ.Therefore, R_g_δ(F) = ∫_M|∇_g_δ F|_g_δ^2 dv_g_δ/∫_Mf^2 dv_g_δ = a^2∫_M |∇_g_δ f_1|_g_δ^2dv_g_δ+b^2∫_M |∇_g_δ f_2|_g_δ^2dv_g_δ/a^2∫_M f^2_1dv_g_δ+b^2∫_M f^2_2dv_g_δ⩽ R_g_δ(f_1)+R_g_δ(f_2). The Min-Max principle and the estimate above on R_g_δ(F) imply that λ_1(g_δ)⩽ R_g_δ(f_1)+R_g_δ(f_2). As a result, if we show that R_g_δ(f_1) and R_g_δ(f_2) tend to 0 as δ tends to 0, then λ_1(g_δ) tends to 0 as δ tends to 0 and Lemma <ref> is proven.We only estimate R_g_δ(f_1), since the computations for f_2 are the same. Let S(R_1;R_2) = {(R,t)|r∈[R_1;R_2], t∈[0;2π)}. By definition of f_1 and equation (<ref>), for sufficiently smallwe have∫_M |∇_g_δ f_1|_g_δ^2dv_g_δ = 1/(R_2-R_1)^2Area_g_δ(S(R_1;R_2)) ⩽A/4(R_2-R_1)^2. Notice, since R_2-R_1 tends to +∞ as δ tends to 0, that ∫_M |∇_g_δ f_1|_g_δ^2dv_g_δ tends to 0 as δ tends to 0.So all we have left to do is show that ∫_M f_1^2 dv_g_δ is bounded below by a strictly positive constant.For R_2>R>R_1, the coordinates R and r, which are respectively the g_δ- and σ-distances from p, are related by the following equation. R = R_1 + ∫_e^-1/δ^r (/2)^-δ/2(Aδ/16π)^1/2r^-(1-δ/2)dr = R_1 + (/2)^-δ/2(A/4πδ)^1/2(r^δ/2-e^-1/2). So direct computation shows that, for R_2>R>R_1,f_1(R,t) = R_2-R/R_2-R_1 = (/2)^δ/2 - r^δ/2/(/2)^δ/2-e^-1/2and there exists a positive constant F=F(A) such that ∫_M f_1^2 dv_g_δ>F(A) for sufficiently small δ. Therefore, R_g(f_1) tends to 0 as δ tends to 0, and the same is true for f_2.It is easy to construct hyperbolic metrics with arbitrarily small λ_1, just by leaving compacts in the Teichmüller space. However, these examples have also unbounded diameter. Here, we show that one can build negatively curved surface with bounded diameter and arbitrarily small λ_1. Notice that by Corollary <ref>, the hyperbolic metrics in the conformal class of these examples must also leave every compacts of the Teichmüller space.Suppose M is a surface of genus 2 and A>0. There exists D>0 such that, for every >0, there exists a negatively curved Riemannian metric g of total area A, the diameter less than D, and λ_1(g)<.Let σ be a metric on M of constant negative curvature and total area A. Let γ be a closed σ-geodesic such that l_σ(γ) =(σ). We assume that we chose σ such that it is symmetric with respect to γ and has a rotationally invariant cylindrical neighborhood C of sufficiently large size around γ. In <cit.>, it was shown that it is possible to modify σ on this cylindrical neighborhood and obtain a family of negatively curved metrics of total area A such that the infimum of the systole in this family of metrics is 0 and the supremum of the diameter in it is bounded above by a constant D, which depends only on A and on the initial choice of σ. We recall that construction and point out what modifications are needed in order to get Lemma <ref>. Without loss of generality, we may assume that σ has curvature -1 (so A= 2π|χ(M)|). Let a:=(σ)/2π. The metric tensor in the normal coordinates (r, θ) for σ with respect to the σ-systole has the form ( 1 00 a^2cosh^2(r)), where θ is the coordinate along the σ-systole, which runs from 0 to 2π, and r is the distance to the σ-systole. The metric g that we build matches with σ outside of the cylindrical neighborhood C of γ and is also symmetric and rotationally invariant in it. In general, if g is a smooth metric that admits a rotationally invariant cylindrical neighborhood of a closed geodesic, then in the normal coordinates (r,θ) with respect to that geodesic the metric tensor for g has the form ( 1 00 f^2(r)). Moreover, g is negatively curved if and only if f”(r)>0, that is, f(r) is convex.It is easy to construct (see Figure <ref>) a function f(r) that is convex, even, matches smoothly acosh(r) outside of a fixed neighborhood of 0, and such that f(r) is arbitrarily small on another neighborhood of 0 of fixed length. The metric g obtained from f is then negatively curved, coincides with σ outside of C, and contains an almost flat cylinder of fixed area and arbitrarily small radius. We can further normalize g so that the total volume is A, without affecting any of the essential features of g: (M,g) is uniformly bounded by some constant D, negatively curved and contains an almost flat cylinder of fixed area and arbitrarily small radius (see <cit.> for more details).Hence the classical proof that the Cheeger dumbbell has arbitrarily small λ_1 applies to g and yields the lemma (see, for instance <cit.>). § TOPOLOGICAL ENTROPY VS SYSTOLE As we mentioned in the introduction, putting together Theorem <ref> below and a result of Besson, Courtois and Gallot (<cit.>) implies that, for a family of negatively curved metrics of fixed total area with diameter uniformly bounded above, then the topological entropy goes to infinity if and only if the systole goes to zero.We were not able to find a reference for Theorem <ref> in the literature, therefore, we provide a proof here, based on suggestions by Keith Burns. Suppose M is a smooth closed Riemannian manifold of dimension n with negative sectional curvature and total volume V. There exists an upper bound on the topological entropy of the geodesic flow that depends only on the injectivity radius of the metric, n and V. Let ρ be the injectivity radius of M and set = ρ/4. Then, the length of the shortest closed geodesic is 2ρ = 8. The volume of a ball of radiusin M is bounded from below by the volume of a ball of radiusin ℝ^n, i.e, ν(,n) = (√(π))^n/Γ(n/2 + 1). Let N = ⌊V/ν(,n)⌋.There can be at most N pairwise disjointballs in M. Since the centers of a maximal collection of disjointballs in M are a 2-spanning set, it is possible to choose N points, p_1,…,p_N such that the balls B(p_i,2) cover M. For each p ∈ M choose j(p) ∈{1,…,N} such that p ∈ B(p_j(p),2). A geodesicγ [0,T] → M can be coded by the sequence j(γ(0)), j(γ()), j(γ(2)),…, j(γ(n_(T)), j(γ(T)),where n_(T) is the largest integer such that n_(T) < T. Suppose that γ_1 [0,T_1] → M and γ_2 [0,T_2] → M are two closed geodesics with the same coding sequences. For each n the geodesic segments γ_1|_[n,(n+1)] and γ_2|_[n,(n+1)] lie in a common ball of radius 3 < ρ. The same is true of the segments γ_1|_[n_(T_1), T_1] and γ_2|_[n_(T_2), T_2]. It follows that the Hausdorff distance between the two geodesics is at most ρ. Consequently, the two closed geodesics must be the same (up to reparametrization). Hence, the number of closed geodesics of length ⩽ T will be at most the number of possible codings, which is bounded above by N^(T/ + 1). Now,lim_T→ +∞1/Tlog(N^(T/ + 1)) = 1/log N.Since g is negatively curved, the topological entropy is given by the exponential growth rate of the number of closed geodesics of length ⩽ T <cit.>, <cit.>. Hence (log N)/ is an upper bound on the topological entropy and depends only on V, n and =ρ/4. 1 [BKRH]BKRHJ. Bochi, A. Katok, and F. Rodriguez Hertz, Flexibility of Lyapunov exponents. Preprint [BGM71]BGM M. Berger, P. Gauduchon,and E. Mazet, Le spectre d'une variété riemannienne. Lecture Notes in Mathematics, Vol. 194 Springer-Verlag, Berlin-New York (1971) pp. vii+251.[BCG03]BCG_margulis G. Besson, G. Courtois, and S. Gallot, Un lemme de Margulis sans courbure et ses applications, prépublications de l'Institut Fourier (2003). https://www-fourier.ujf-grenoble.fr/?q=fr/content/595-un-lemme-de-margulis-sans-courbure-et-ses-applicationshttps://www-fourier.ujf-grenoble.fr/sites/default/files/REF_595.pdf [B72]B72 R. Bowen, Periodic orbits for hyperbolic flows, Amer. Jour. Math., 94 (1972), pp. 1–30. [CE03]CE03 B. Colbois, A. El Soufi, Extremal Eigenvalues of the Laplacian in a Conformal Class of Metrics: The “Conformal Spectrum”,Ann. Global Anal. Geom., 24 (2003), pp. 337–349. [Er17]Er17 A. Erchenko, Flexibility of exponents for expanding maps on a circle, preprint. https://arxiv.org/abs/1704.00832arxiv:1704.00832 [EK]EK A. Erchenko, A. Katok, Flexibility of entropies on surfaces of negative curvature, preprint. [Gu80]Gu80 Y. Guivarc’h, Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire, Astérisque, 74 (1980), pp. 47–98. [Ka82]K A. Katok, Entropy and closed geodesics, Ergod. Th. & Dynam. Sys., 2 (1982), pp. 339–367. [KKPW90]KKPW A. Katok, G. Knieper, M. Pollicott, and H. Weiss, Differentiability of entropy for Anosov and geodesic flows. Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 2, pp. 285–293.[Le87]L F. Ledrappier, Propriété de Poisson et courbure négative, C. R. Acad. Sci. Paris, 305, Série I, (1987), pp. 191–194. [Le90]Led90 F. Ledrappier, Harmonic measures and Bowen-Margulis measures, Israel J. Math. 71 (1990), pp. 275–287 [Le10]Led10 F. Ledrappier, Linear drift and entropy for regular covers, Geom. Func. Anal. 20 (2010), pp. 710–725. [M69]M69 G.A. Margulis, Applications of ergodic theory to the investigation of manifolds of negative curvature, Funct. Anal. Appl., 3 (1969), pp. 335–336.(Translated from Russian.) [SY10]SY R. Schoen, S.-T. Yau, Lectures on Differential Geometry, International Press (2010). [Y78]Y S.-T. Yau, A general Schwarz Lemma for Kahler Manifolds, Amer. Jour. Math., 100 (1978), pp. 197–203. | http://arxiv.org/abs/1709.09234v1 | {
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1 Yunnan Observatories, Chinese Academy of Sciences, 396 Yangfangwang, Guandu District, Kunming, 650216, P. R. China2 University of Chinese Academy of Sciences, Beijing 100049, P. R. China3 Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, 396 Yangfangwang, Guandu District, Kunming, 650216, P. R. China4Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing, 100012, P. R. China^Corresponding author: H. T. Liu, e-mail: [email protected] present our observations of the optical intra-day variability (IDV) in γ-ray BL Lac object Mrk 501. The observations were run with the 1.02 m and 2.4 m optical telescopes at Yunnan Observatories from 2005 April to 2012 May. The light curve at the R band on 2010 May 15 passes both variability tests (the F test and the ANOVA test). A flare within the light curve on 2010 May 15 has a magnitude change Δ m = 0.03 ± 0.005_stat± 0.007_sys mag, a darkening timescale of τ_d= 26.7 minutes, and an amplitude of IDV Amp=2.9%±0.7%. A decline described by 11 consecutive flux measurements within the flare can be fitted linearly with a Pearson's correlation coefficient r = 0.945 at the confidence level of > 99.99%. Under the assumptions that the IDV is tightly connected to the mass of the black hole, and that the flare duration, being two times τ_d,is representative of the minimum characteristic timescale, we can derive upper bounds to the mass of the black hole. In the case of the Kerrblack hole, the timescale of Δ t_min^ob= 0.89 hours gives M_∙ 10^9.20 M_⊙, which is consistent with measurements reported in the literature. This agreement indicates that the hypothesis about M_∙ and Δ t_min^ob is consistent with the measurements/data. § INTRODUCTION Blazars are the most violently variable subclass of active galactic nuclei (AGNs), which include BL Lacertae objects (BL Lacs) and flat spectrumradio quasars (FSRQs). They are characterized by rapid and strong variability at all wavelengths of the whole electromagnetic spectrum, strongand variable polarization from radio to optical bands, non-thermal emission dominating at all wavelengths. Blazars usually exhibit the core-dominated radio shape. These extreme properties are generally believed to be generated from a relativistic jet with a viewing angle lessthan 10^∘ <cit.>. The broadband spectral energy distributions (SEDs) of blazars usually show a double-peak profile.The low-energy component spans the IR-optical-UV bands and the high-energy component extends to GeV/TeV gamma-ray bands <cit.>. The first peak is generally interpreted as the synchrotron radiation of relativistic electrons in the jet. The secondpeak is generally believed to be generated by the inverse-Compton scattering of the same electron distribution responsible for the synchrotron radiation <cit.>. Photometric technique is a powerful tool to investigate the nature of blazars. The variability including amplitude, duty cycles, timescales andso on, could help us to study the structure, dynamics, radiation mechanism and even phase of blazars <cit.>.Previous observations show that blazars exhibit variability on diverse timescales, which can be broadly divided into three classes: intra-dayvariability (IDV) or micro-variability, short-term variability (STV), and long-term variability (LTV). For IDVs, the timescales range from a fewminutes to several hours, and the variability of flux changes by a few tenths of magnitude <cit.>. STVs and LTVs have timescalesof days to months and months to years, respectively <cit.>. Over the last three decades, a great mount ofwork for variability of blazars has been reported at different timescales in different bands <cit.>.The IDVs of the beamed emission of jets can be used to constrain the central black hole masses of blazars <cit.>. Mrk 501 is a typical nearby BL Lac object (at redshift z=0.034), one of the nearest Jy BL Lac objects<cit.> and one of thebrightest extragalactic sources in the X-ray/TeV sky <cit.>. It has attracted much attention. Since the first detection by Whippleobservatory <cit.>, Mrk 501 was the second extragalactic object identified as a very high energy (VHE) γ-ray emitter <cit.>. In the second year since it was discovered, Mrk 501 went into a surprisingly high activity and strong variabilitystate. Fluxes over 10 times brighter than 10 Crab were reported in several groups <cit.>. After the outburst state,the mean VHE γ-rays dropped by an order of magnitude during 1998–1999 <cit.>. In 2005, an interesting phenomenon,the VHE γ-ray flux varies on a timescale of minutes, was detected <cit.>. The recent report about a multi-wavelengthstudy in 2009 shows fast variability with ∼ 15 minute doubling times in the VHE gamma rays <cit.>. The variability of Mrk 501on the whole electromagnetic wavelengths has been extensively studied <cit.>.The IDVs of Mrk 501 have been reported over the entire electromagnetic spectrum <cit.>. Most of previous workswere on timescales of tens to one hundred minutes for the optical IDVs. A rapid X-ray flare from Mrk 501 detected with the Rossi X-Ray Timing Explorer in 1998 shows the 60% increase in 200 seconds (∼ 3 minutes) <cit.>. <cit.> reported a R band flare on a timescaleof 105 minutes. <cit.> found a R band flare around 106 minutes. Compared to the X-ray and VHE flares on timescales of minutes,the optical IDVs are on timescales of hours. Shorter optical IDVs may be expected. In this paper, we present observations and investigate the variability in B, I, R, and V bands from 2005 to 2012. To find out the fastestoptical IDV, most exposure times are less than 3 minutes. The structure of this paper is as follows: in section 2 we present the observations anddata reduction; the results of variability are exhibited in section 3; section 4 is for discussion.§ OBSERVATIONS AND DATA REDUCTIONThe optical monitoring program of Mrk 501 was carried out with two telescopes: the 1.02 m and 2.4 m optical telescopes at Yunnan Observatories, China. The 1.02 m telescope is located at Kunming, China. From 2006 to 2009, the telescope equipped with a RAC CCD (1024 pixels × 1024 pixels) at f/13.3 Cassegrain focus, and the entire CCD chip covered ∼6.5 × 6.5 arcmin^2. So, the projected angle on the sky ofeach pixel corresponded to 0.38 arcsec in both dimensions. The readout noise and gain were 3.9 electrons and 4.0 electrons/ADU, respectively.After 2009, the telescope was equipped with an Andor AW436 CCD (2048 pixels × 2048 pixels) camera at the f/13.3 Cassergrain focus.The field of view of the CCD is ∼7.3 × 7.3 arcmin^2, and the pixel scale is 0.21 arcsec in both dimensions. The readout noise andgain were 6.33 electrons and 2.0 electrons/ADU, respectively <cit.>. The 2.4 m telescope is located at Lijiang, China. The telescope was equipped with an Princeton Instruments VersArray1300B CCD (1300 pixels × 1340 pixels) camera at f/8 Cassegrain focus,and the entire CCD chip covered ∼4.40 × 4.48 arcmin^2. The readout noise and gain were 6.5 electrons and 1.1 electrons/ADU, respectively. For both of the two telescopes, we selected standard Johnson broadband filters to carry out the observations in the B, V, R,and I bands. During our monitoring program from 2005 to 2012, 1532 CCD frames were obtained in 50 nights. For most nights, the exposure times are40–400 seconds for 1.02 m telescope and 10–50 seconds for 2.4 m telescope. The complete observation log is listed in Table 1. For eachimage, the standard stars and object were always in the same field. The standard stars considered do not change, so the brightness of theobject was obtained from the standard stars <cit.>. Because star 1 is the brightest of all comparison stars andthe magnitudes have been measured in all bands in other works <cit.>, it was selected to calculate the object magnitude. However,there are some uncertainties which may cause the standard stars to change, so we chose another standard star to characterize the change,and we used the standard deviation of the two comparison stars [σ(star1-star6)] as the error of photometry. Star 6, closest to the objectand with all four bands measured at the same time with star 1, is used as the second standard star. The standard deviation of the differential instrumental magnitude of star1-star6 is given in Table 1. The IDV of the target object was investigated using two statistical methods: the Ftest <cit.> and the one-way analysis of variance <cit.>. The value of F test inour work is calculated as: F = Var(BL-star1)/Var(star1-star6), where BL, star1, and star6 are the magnitudes of BL Lac object, star1, and star6, respectively. Var(BL-star1) and Var(star1-star6) are the variancesof differential instrumental magnitudes for 'BL-star1' and 'star1-star6', respectively. The critical value of the F test can be compared with theF value for which the significance level was set as 0.01 in F-statistics. ANOVA is a powerful tool to detect IDVs, and does not rely on error measurements. We divided data points into groups. Each group contains three data points, and if the last group has less than three data points,we would merger them with the previous group. The ANOVA critical value can be obtained from F^α_ν1,ν2 in the F-statistics,where ν1 (ν1 = k-1, and k is the number of groups) is the degree of freedom between groups and ν2 (ν2 = N - k, and Nis the number of measurements) is the degree of freedom within these groups, and α is the significance level <cit.>. When calculating the values of F and ANOVA, we only used the light curves with observational data points ≥ 9 in a night, and only the lightcurves satisfied with both criteria were considered to be variable.Table 1 shows the results of F test and ANOVA for all the observable nights satisfying the criteria. All of the photometric data were reduced using the standard procedure in the Image Reduction and Analysis Facility (IRAF) software. Foreach night, we combined all the bias frames and then obtained a master bias. All of the object image frames and flat-field image frameswere subtracted by the master bias. Then we generated the master flat-field for each band by taking the median of all flat-field image framesfor each band. After the flat-field correction, aperture photometry was performed by the APPHOT task. Because Mrk 501 is an extended sourceand the standard stars are point like sources, we used two different criteria to determine the aperture size for the object and comparison stars.For the standard stars, we found that the best signal-to-noise ratio was obtained with the aperture radius of 1.2 full width at half maximum(FWHM), i.e. a dynamic aperture, which is generally applied to the point like sources consisting of AGNs and their host galaxies, and is largeenough to cover the hosts. For extended sources, such as nearby AGNs, the change of seeing may affect the dynamic aperture and then the contribution of the hosts. If the host is considered to be constant for Mrk 501, the photometric results could reflect a combined contributionof the aperture effect and the intrinsic variations of AGN. We chose 16 different aperture radii among 1.2–8.0 FWHMs, and the variability ofMrk 501 is similar to each other when the apertures are greater than or equal to than 3 FWHMs (see the upper left panel in Figure 1). The lightcurves with apertures of 6.0–8.0 FWHMs have almost the same outline, i.e., the light curve profile is stable as the aperture is enough large.It seems that there is an arc dip on 2010 May 17 lasting for about 90 minutes. However, the aperture is dynamic with the FWHM of standardstar, and the light curve profile may be biased by the FWHM variability. The lower left panel in Figure 1 shows the FWHM variability. This FWHM variability is very similar to those of the light curves in 1.2–2.5 FWHMs, especially the arc dip. The smaller FWHM results in the smaller aperture,and the fewer host contribution. This will result in the darkening of Mrk 501. A flare in the I band on 2009 May 11 has a large magnitude changeof Δ I = 0^m.28 and a symmetrical profile, when the dynamical aperture is used. However, the corresponding FWHMs nearly have the same variability as the light curve of Mrk 501 (see Figure 1). Thus, the dynamical aperture significantly influences the variability results of the photometry due to the FWHM variability, and the fixed aperture is used to perform photometry. The fixed aperture can avoid this host influenceon the measured magnitude variability of Mrk 501. In order to contain the most contribution of AGN and avoid the effect of background, we used3.5 arcsec aperture radius, which is more than 1.5 × typical FWHM for most of images. Though the fixed aperture can avoid the host influence of the dynamic aperture on the variability of Mrk 501, the seeing effect influencesthe FWHM, and then the fraction of the host light contained by the fixed aperture. Because the profile of the host galaxy is not flat <cit.>,the different point spread functions influence the contribution of the host in a fixed aperture. However, this influence is much smaller than thatof the dynamic aperture mentioned above. For this host, the larger seeing results in the larger point spread function that leads to the smaller faction of the host light contained by the fixed aperture. The increasing FWHM will generate the darkening of measured magnitude. The observational data and the fit results confirm the deduced FWHM–magnitude relation (see Figure 2). From 2010 May 15 to 2010 May 18, four night observations are used to test this relation. Also, there is a similar relation for these four night data. Thus, the FWHM–magnitude relation is corrected for each night data. The host contribution is subtracted according to the method of <cit.> for the data on 2010 May 18, and the FWHM–magnitude relation appears more significant. The host-corrected magnitudes depend on point spread function or seeing (FWHM). This FWHM–magnitude relation is corrected to get the final result for each night data. Figure 3 shows the light curves in the I band on 2009 May 11 for a fixed aperture without and with correcting the FWHM–magnitude relation. Comparison of the light curves in Figure 3 shows that the seeing effect is improved. The IDV seems to exist in the corrected light curve in Figure 3. The F test or the ANOVA test show six possible IDVs, and only the IDV on 2010 May 15 is confirmed by the two tests (see Table 1). Two interesting light curves with the possible IDVs are presented in Figure 4.§ RESULTSTable 1 shows six nights for which the the light curve gave positive in the F or ANOVA tests, from which only one night, 2010 May 15, gave positive in both tests. As an example, Figure 4 shows two nights, 2010 May 15 and 17.These light curves have a peak-to-peak flux change of Δ R ∼ 0.03 mag, while the observational error estimated by the standard deviation from the magnitude differences between standard star 1 and star 6 is σ_R =0.005. Therefore, the peak-to-peak flux change Δ R ∼ 6σ_R. We followed the method in <cit.> to determine the systematic error of the fluxes. First, we used a median filter to smooth the light curve, and then subtracted it from the original light curve. Second, we calculated the standard deviation of the residual light curve, and adopted this deviation as an estimate of the systematic uncertainty. The value estimated with this method is 0.007 mag. If this systematic error is combined quadratically with the statistical error, we find a total uncertainty in the flux variation of σ_T=0.009, which leads to a peak-to-peak flux change of Δ R ∼ 3.3 σ_T. The detection success rate of the optical IDVs seems to be very low for Mrk 501. This may be due to the intrinsic weak IDVs of Mrk 501 and/or the relatively brighter host galaxy. For the large amplitude variations of BL Lacertae objects, an effective variation on a short timescale (from a few hundred seconds to several hours) requires that the amplitude of optical variability must be more than 5σ, where σ is the maximum total observational rms error <cit.>. The rms error used in <cit.> is the same as the standard deviation of the two comparison stars we used in this paper. The ratio of Δ m / σ>5 (where σ relates to the statistical uncertainty) was used as a necessary criterion rather than a sufficient and necessary condition for optical variability. Though, the light curves on 2010 May 15 and 17 match the criterion Δ m > 5σ, they do not necessarily have IDVs because of the systematic uncertainties mentioned above. The long-term light curves are presented in Figure 5 for our observational data.The larger amplitudes of variability are found in the long-termlight curves. The continuous observations in the light curves appear to be a cluster, which may show a larger amplitude of variability. A variability amplitude of Δ B ∼ 0.8 mag is around MJD=54900 for the B band. We find a variability amplitude of Δ I ∼ 0.7 mag forthe I band around MJD=55250. The long-term light curve in the R band is similar to that in the I band. However, the R band FWHMs are larger than 5.3 arcseconds on 2006 April 3, which may be due a worse observational condition. The I band FWHMs are smaller than thoseof the R band on 2006 April 3, and the I band observations were completed before the R band observations.This may result in a worse photometric magnitude in the R band, darker than 18 mag (see Figure 5). The other data points of the R band are not particularly surprising. There is a variability amplitude of Δ R ∼ 0.7 mag around MJD=55250.For the V band around MJD=54200, a variability is found with an amplitude of Δ V∼ 0.7 mag. Also, a variability is found with an amplitude of Δ V∼ 0.7 mag around MJD=56000.These variations with large amplitudes of Δ m ∼0.7–0.8 mag have durations within ∼ 100 days. Also, the similar timescale variations are found in Seyfert galaxies, e.g., NGC 5548 <cit.>. These timescales within ∼ 100 days may be from the lighthouse effect ofa jet, where the jet precession will result in the forward beaming of the emission <cit.>.§ DISCUSSION Except for the jet origin of the optical variations, there is an alternative way to explain the optical variability and the IDVs or micro-variabilityin BL Lac objects, i.e. the optical IDVs are likely from accretion disks <cit.>. The accretion disk instabilities are able to explainsome of the phenomena seen in the optical–X-ray bands, but cannot explain the radio IDVs <cit.>. The latest research on BL Lac object PKS 0735+178 shows that the blazar variability on timescales from years down to hours–i.e., both the long-term large amplitude variability and the micro-variability–is generated by the underlying single stochastic process (at the radio and optical bands), or a linear superposition ofsuch processes (in the gamma-ray regime), within a highly non-uniform portion of the flow extending from the jet base up to thepc-scale distances <cit.>. <cit.> searched for short timescale variability, and identified an interesting event in the J band with a durationof ∼ 25 minutes for BL Lac object PKS 0537-441. In both the low and high states, the emission appears to be dominated by the jet, and no evidence of a thermal component is apparent for PKS 0537-441. The spectral energy distributions of PKS 0537-441 are interpreted in terms of the synchrotron and inverse Compton mechanisms within a jet, where the plasma radiates via internal shocks and the dissipation depends on the distance of the emitting region from the central engine <cit.>. The optical emission of Mrk 501 is neither the thermal component from accretion disk nor the nonthermal component from the jet <cit.>. The optical emission is dominated by the host galaxy, and the UVemission is from the jet for Mrk 501 <cit.>. Thus, it is not possible that the optical IDVs are from accretion disk for Mrk 501. The IDVs discussed here may be directly related to shock processes in a jet. The shock-in-jet model, the most frequently cited model usedto explain the IDVs, is based on a relativistic shock propagating down a jet and interacting with a highly non-uniform portion in the jet flow <cit.>. The featureless optical continuum is the typical characteristic of BL Lac objects,and quasars show many strong broad emission lines. The broad-line region (BLR) seems to not be in BL Lac objects <cit.>.Broad emission lines were observed only in a few BL Lac objects <cit.>. Thus, the broad emission lines seem to be weakso that the broad lines were rarely observed in BL Lac objects. The accretion rates are very low for BL Lac objects <cit.>.The absence of broad emission lines in most of BL Lac objects may be due to the very weak emission of accretion disk. The BLR clouds areright there though observations have not detected the broad lines in most of BL Lac objects. Thus, the optical variability in BL Lac objectsis not dominated by the emission of accretion disk. On the contrary, the relativistic jets are likely dominating the optical variability. Thougha minority of BL Lac objects have the broad emission lines, they have Full width at half maximum v_FWHM∼ 1300–5500 kms^-1 <cit.>. Their widths are very similar to those of Seyfert galaxies. This similarity indicates a fundamental difference in accretionrate between these two kinds of AGNs. This accretion rate difference implies that BL Lac objects do not have the same origin as Seyfert galaxieshave an accretion disk origin of the optical variability. Mrk 501 has a featureless continuum, and then its optical IDVs do not have the accretiondisk origin. The variability timescales were defined in different ways. The doubling timescale is usually used to estimate the variability timescale of thelarge amplitude change in gamma rays <cit.>. A fast variability in the VHE gamma rays shows doubling times ∼ 15 minutesfor Mrk 501 <cit.>. But the fastest variability observed on Mrk 501 at VHE is ∼ 2 minutes <cit.>. These VHE gamma rays arefrom the relativistic jets. These variability timescales of ∼ 2–15 minutes could give upper limits to the diameter sizes of gamma-ray emitting regions, D_γ c Δ t_min^obδ /(1+z), where c is the speed of light, Δ t_min^obis the minimum variability timescale observed, and δ is the corresponding Doppler factor. In general, the lower energy bands willshow the smaller variability amplitudes for Mrk 501 <cit.>. The early common definition of the variability timescale is τ =F /|Δ F / Δ t|, and the more conservative approach is τ =|Δ t / Δln F|, where F is the flux and Δ F is the variability amplitudeon the timescale Δ t <cit.>. These definitions were used to the variability with a large amplitude change, e.g.,0.1 mag. Another choice is the interval between the adjacent local minima at the adjacent valleys in the light curve, respectively (i.e., the flare duration),or two times the interval between the adjacent local minimum and maximum at the adjacent valley and peak, respectively, if the flare is not complete. As a shock passes through the emitting region (knot or blob) in the jet, this passage will generate a flare with the duration comparableto the passing timescale of the shock. In the case, the variability timescale corresponds to the flare duration, i.e., the interval between the adjacent local minima at the adjacent valleys in the light curve. Another possibility is that as the blob-like emitting region becomes optical thin, it generatesa flare with a duration about equal to the light crossing time of the emitting region. These definitions should locate by hand the correspondingdata points in the short-term light curve. The minimum timescale was determined by hand for the short-term light curve <cit.>.In an analogous fashion, we searched possible flares in the light curves on 2010 May 15 and 17. There is a darkening in the light curve on2010 May 15 (see Figure 6). This darkening consists of 11 data points, has Δ R=0.030± 0.005 mag and lasts for 26.7 minutes. The corresponding rate of the change is 0.067 mag/hr. The 11 data points can be fitted linearly with a Pearson's correlation coefficient r = 0.945at the confidence level of more than 99.99 per cent. Thus, the random fluctuation origin of this darkening can be eliminated at a highconfidence level. There seems no significant adjacent rising phase before this darkening (see Figure 6). We do not get the totalduration of the flare from the sum of the rising and darkening timescales. The two times of the darkening timescale may be assumed asa representative value for the duration, 0.89 hours. According to the necessary condition of optical variability Δ m > 5σ usedin <cit.>, an IDV might be indicated for this flare, because the uncertainty σ does not take into account thesystematic uncertainty. The variability amplitude can be calculated by Amp = √((A_max - A_min)^2 - 2σ^2)<cit.>, where A_max and A_min are the maximum and minimum magnitudes of the light curve being considered, respectively, and σ is the measurement error. There are Amp=4.3%± 0.7% for the light curve on 2010 May 15, and Amp=2.9%± 0.7% for the flare mentioned above. The auto-correlation function method could search for the characteristic timescale of the largeamplitude variability in the long-term light curve <cit.>. The F test and the ANOVA test indicate the IDV in the lightcurve on 2010 May 15, but cannot give the IDV details, e.g, timescale. The structure function (SF), introduced by <cit.>, has been employed to quantify the characteristic timescale for a light curve with confirmed variability (e.g., meeting some tests) <cit.>. The first-order SF is defined as SF^(1)(Δ t)=1/N∑^N_i=1[m(t_i)-m(t_i+Δ t)]^2, where m(t) is the magnitude at time t, and Δ t is the time separation. The characteristic timescale in a light curve is indicatedby a local maximum of the SF. The first local maximum was used, i.e., the one with the shortest time, in the case of multiple local maximain the SF <cit.>. Figure 7 displays the first-order SF for the light curve on 2010 May 15. However, the SF cannot give "a real breakor peak" (a large break or peak), i.e., a robust variability characteristic timescale. The SF is not able to determine the variability timescaleof the micro-amplitude flare shown in Figure 6. The SF increases with the timescale, which implies that there is more power of variability at the longer timescales (which is a typical characteristic in blazars and AGNs in general). The IDV is an intrinsic phenomenon and tightly constrains the sizes of the emitting regions in blazars <cit.>. The timescales of variationsin the optical band might have an underlying connection with the black hole masses of the central engines in blazars. The minimum timescalesof variability were used to determine the masses of the central black holes in AGNs <cit.>. <cit.> and <cit.> determined the black hole masses M_∙ for non-blazar-like AGNs or some AGNs with a weaker blazar emission componentin fluxes relative to an accretion disk emission component, based on the assumption that an accretion disk is surrounding a supermassive blackhole, and the optical flux variations are from the accretion disk. <cit.> proposed a sophisticated model to constrain M_∙ using therapid variations for blazars. The model is suitable to constrain M_∙ in blazars using the minimum timescale Δ t_min^ob of variations of the beamed emission from the relativistic jets. <cit.> gave M_∙ 5.09× 10^4 δΔ t_min^ob/1+zM_⊙( j∼ 1),M_∙ 1.70 × 10^4 δΔ t_min^ob/1+zM_⊙( j=0), where Δ t_min^ob is in units of seconds, and j=J/J_max is the dimensionless spin parameter of a black hole withthe maximum possible angular momentum J_max=G M_∙^2/cand G being the gravitational constant. For the light curveof Mrk 501 on 2010 May 15, the optical IDV has a darkening timescale of τ_d= 26.7 minutes with a micro-amplitude of Δ m=0.03 mag. We made the assumption of considering the duration of 0.89 hours, two times τ_d, as a representative value for the variability timescale to be used in formulae (3a) and (3b). Other studies used other prescriptions to estimate a variability timescale (like flux-doubling times, or characteristic times in SF), which cannot be used with the optical data reported in this paper. The optical–γ-ray emission is mostly the Doppler boosted emission of jets for γ-ray blazars <cit.>. A value of δ∼ 10 was adopted for GeV gamma-ray blazars <cit.>. As in <cit.>, δ = 10 is taken to estimate M_∙ with formulas (3) for Mrk 501. In the case of Δ t_min^ob= 0.89 hours, we have M_∙ 10^9.20 M_⊙ for the Kerr black hole [formula (3a)] and M_∙ 10^8.72 M_⊙ for the Schwarzchild black hole [formula (3b)]. The flare duration of 0.89 hours gives M_∙ 10^8.72–10^9.20 M_⊙. <cit.> and <cit.> measured the stellar velocity dispersion and estimated M_∙= 10^9.21± 0.13 M_⊙ and M_∙=10^8.93± 0.21 M_⊙, respectively, by the black hole mass–stellar velocity dispersion relation. <cit.> reported a mass M_∙=10^8.84 M_⊙. The mass upper limit of M_∙10^9.20 M_⊙ is basically consistent with the mass estimates M_∙= 10^8.93 ± 0.21 M_⊙, 10^9.21± 0.13 M_⊙, and 10^8.84 M_⊙. <cit.> showed the large uncertainties in the determination of the black hole mass of Mrk 501, ranging from ∼ 10^7.8 M_⊙ to 10^8.7 M_⊙, which are also consistent with the mass upper limits. We made the assumption that IDV is tightly connected to the black hole, in order to set constraints to theblack hole mass using the method proposed in <cit.>. The derived upper bounds indicate that these two hypotheses are consistent with the measurements/data. The presence of systematic uncertainties caused by poor weather conditions, telescope tracking inaccuracies, etc, will maskthe optical IDV finding for Mrk 501. Based on the method used in <cit.>, we estimated the systematic uncertainty from the standard deviation of the residuals, which were obtained by subtracting the median-smoothed light curve from the original light curve. 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M. 2008, ApJS, 174, 111 lcccccccc 90pc Observation log and results of IDV observations of Mrk 501 Date FiltersNσ(star1-star6)F F(99)ANOVA ANOVA(99)Telescope (1) (2) (3) (4) (5) (6)(7) (8) (9)2005 Apr 05 B 4 0.074 1.02 I 5 0.015 1.02 R 4 0.01 1.02 V 5 0.015 1.02 2005 Apr 06 B 1 1.02 I 9 0.013 0.23 6.03 3.37 10.92 1.02 R 10 0.018 0.19 5.35 0.41 9.55 1.02 V 9 0.033 0.02 6.03 0.29 10.92 1.02 2006 Apr 03 I 4 0.054 1.02 R 4 0.024 1.02 2007 Mar 27 I 39 0.012 0.48 2.16 3.45 2.96 1.02 V 28 0.076 0.03 2.51 1.13 3.63 1.02 2007 Mar 28 I 5 0.008 1.02 V 3 0.01 1.02 2007 Mar 29 I 8 0.005 1.02 V 4 0.023 1.02 2007 Mar 30 I 8 0.006 1.02 V 5 0.027 1.02 2007 Apr 15 I 13 0.015 0.46 4.16 2.81 6.99 1.02 2007 Apr 22 B 2 0.013 1.02 I 10 0.006 0.63 5.35 0.31 9.55 1.02 V 10 0.04 0.22 5.35 1.54 9.55 1.02 2007 Apr 23 I 4 0.005 1.02 2007 Apr 24 I 15 0.007 2.42 3.7 0.54 5.99 1.02 V 14 0.052 0.21 3.91 6.22 6.55 1.02 2007 May 08 I 10 0.012 1.21 5.35 0.42 9.55 1.02 2007 May 09 B 1 1.02 I 9 0.009 0.58 6.03 1.68 10.92 1.02 V 5 0.032 1.02 2008 May 06 I 4 0.037 1.02 2008 May 07 I 87 0.009 1.05 1.69 1.81 2.08 1.02 2008 May 08 I 82 0.008 1.38 1.69 3.08 2.11 1.02 2009 Mar 21 B 2 0.001 2.4 R 3 0.002 2.4 V 3 0.003 2.4 2009 Mar 22 B 3 0.004 2.4 R 5 0.005 2.4 V 5 0.003 2.4 2009 Mar 23 B 5 0.007 2.4 R 4 0.002 2.4 V 4 0.001 2.4 2009 Mar 26 B 5 0.013 2.4 R 5 0.006 2.4 V 5 0.003 2.4 2009 Apr 14 I 61 0.011 0.6 1.84 2.3 2.39 1.02 2009 Apr 16 I 20 0.007 2.41 3.03 0.78 4.7 1.02 R 20 0.023 0.15 3.03 0.83 4.7 1.02 2009 May 11 I 30 0.009 1.32 2.42 2.43 3.46 1.02 2009 May 16 I 7 0.02 1.02 2009 May 17 I 75 0.013 0.6 1.75 9.39 2.18 1.02 2010 Feb 21 I 5 0.004 1.02 R 5 0.009 1.02 V 5 0.012 1.02 2010 Feb 22 I 4 0.003 1.02 R 1 1.02 V 5 0.011 1.02 2010 Feb 26 B 10 0.008 0.95 5.35 1.31 9.55 2.4 I 8 0.002 2.4 R 7 0.001 2.4 V 10 0.006 0.18 5.35 18.37 9.55 2.4 2010 Feb 28 B 15 0.143 0.01 3.7 0.42 5.99 2.4 I 15 0.018 1.84 3.7 1.16 5.99 2.4 R 15 0.015 0.17 3.7 1.75 5.99 2.4 V 15 0.025 0.22 3.7 1.9 5.99 2.4 2010 Mar 11 B 1 2.4 I 1 2.4 R 1 2.4 V 1 2.4 2010 May 04 R 19 0.007 0.92 3.13 0.95 4.86 1.02 V 18 0.01 0.37 3.24 0.77 5.06 1.02 2010 May 15 R 88 0.005 5.14 1.69 4.37 2.08 1.02 2010 May 16 R 88 0.007 1.03 1.69 1.67 2.08 1.02 2010 May 17 R 80 0.005 5.31 1.75 1.75 2.17 1.02 2010 May 18 R 70 0.005 1.12 1.79 2.26 2.27 1.02 2011 May 07 R 60 0.007 1.44 1.93 1.18 2.39 1.02 2011 May 09 R 63 0.007 2.11 1.84 2.15 2.37 1.02 2011 May 10 R 38 0.012 0.47 2.21 0.69 3.02 1.02 2011 Aug 21 B 5 0.046 1.02 I 5 0.013 1.02 R 5 0.019 1.02 V 5 0.019 1.02 2011 Aug 22 B 5 0.054 1.02 I 5 0.065 1.02 R 4 0.06 1.02 V 5 0.075 1.02 2012 Feb 27 I 5 0.009 1.02 R 5 0.012 1.02 V 5 0.021 1.02 2012 Feb 28 I 5 0.011 1.02 R 5 0.023 1.02 V 4 0.011 1.02 2012 Apr 02 I 4 0.004 1.02 R 5 0.026 1.02 V 4 0.039 1.02 2012 Apr 29 B 4 0.056 1.02 I 4 0.009 1.02 R 4 0.009 1.02 V 4 0.019 1.02 2012 May 01 I 5 0.007 1.02 R 5 0.005 1.02 V 5 0.028 1.02 2012 May 03 B 4 0.03 1.02 I 4 0.022 1.02 R 4 0.011 1.02 V 4 0.021 1.02 2012 May 11 I 5 0.011 1.02 R 5 0.014 1.02 V 5 0.025 1.02 2012 May 13 V 1 1.02 2012 May 16 R 31 0.02 0.26 2.39 1.11 3.4 1.02 2012 May 17 R 26 0.015 0.58 2.6 2.13 3.84 1.02 Column 1: observation dates; Column 2: observation filters; Column 3: observation numbers; Column 4: observational errors estimated by standard star 1 and star 6; Column 5: the F values in the F test for the observation data; Column 6: F(99) is the critical F value at a 99% confidence level; Column 7: the ANOVA values in the ANOVA test for the observation data; Column 8: ANOVA(99) is the critical ANOVA value at a 99% confidence level; Column 9: 1.02 and 2.4 represent 1.02 m and 2.4 m telescopes, respectively.lccccccccccc 120pc The observational data for Mrk 501 3cB 3cI 3cR 3cV1-3 4-6 7-9 10-12MJD - 50000Mag FWHMMJD - 50000Mag FWHMMJD - 50000Mag FWHMMJD - 50000 Mag FWHM3465.856933 14.459 3.39 3465.847570 12.579 2.65 3465.865243 12.942 3.36 3465.853114 13.382 3.24 3465.872859 14.431 3.39 3465.862686 12.576 2.66 3465.881111 12.938 3.27 3465.868264 13.361 3.15 3465.887859 14.450 4.45 3465.878669 12.573 2.48 3465.895648 12.943 3.26 3465.883577 13.368 3.39 3465.902095 14.457 3.30 3465.893345 12.586 2.77 3465.909653 12.942 3.20 3465.898079 13.368 3.57 3466.806713 14.602 2.49 3465.907292 12.587 2.37 3466.800787 13.146 1.85 3465.912072 13.364 3.36 4205.897373 14.254 3.29 3466.798473 12.459 1.69 3466.813588 13.137 2.55 3466.802917 13.579 2.31 4205.900324 14.254 3.18 3466.811401 12.460 2.29 3466.822234 13.132 2.28 3466.815973 13.592 2.37 4229.711204 14.478 1.71 3466.819977 12.467 2.22 3466.831366 13.146 2.29 3466.824537 13.583 2.58 ... ... ... ... ... ... ... ... ... ... ... ... This table is available in its entirety in a machine-readable form in the online journal. A portion is shown here for guidance regarding its form and content. FWHM is for the standard stars, and is in units of arc-seconds. Mag denotes magnitude. | http://arxiv.org/abs/1709.09308v1 | {
"authors": [
"Hai-Cheng Feng",
"H. T. Liu",
"X. L. Fan",
"Yinghe Zhao",
"J. M. Bai",
"Fang Wang",
"D. R. Xiong",
"S. K. Li"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170927022603",
"title": "Search for intra-day optical variability in Mrk 501"
} |
DSS of Jets to Vortices]Direct Statistical Simulation of Jets and Vortices in 2D Flows [email protected] Department of Applied Mathematics, University of Leeds. Leeds, LS2 9JT, [email protected] Department of Physics, Box 1843, Brown University, Providence, RI 02912-1843 USA In this paper we perform Direct Statistical Simulations of a model of two-dimensional flow that exhibits a transition from jets to vortices.The model employs two-scale Kolmogorov forcing, with energy injected directly into the zonal mean of the flow. We compare these results with those fromDirect Numerical Simulations. For square domains the solution takes the form of jets, but as the aspect ratio is increased a transition to isolated coherent vortices is found. We find that a truncation at second order in the equal-time but nonlocal cumulants that employs zonal averaging (zonal CE2) is capable of capturing the form of the jets for a range of Reynolds numbers as well as the transition to the vortex state, but, unsurprisingly, is unable to reproduce the correlations found for the fully nonlinear (non-zonally symmetric) vortex state. This result continues the program of promising advances in statistical theories of turbulence championed by Kraichnan. 47.27.De, 47.32.cd, 47.27.eb, 94.05.Jq [ J. B. Marston December 30, 2023 =====================§ DIRECT STATISTICAL SIMULATION: THE LEGACY OF KRAICHNAN Robert Kraichnan's vision of a statistical mechanics of turbulence notably emphasized essential differences between flows in two and three spatial dimensions <cit.>.Two dimensional flows are striking for the frequent emergence of coherent structures.The structures are of two basic types:Vortices and jets<cit.>.The Juno mission to Jupiter has recently returned beautiful images of both types of structures, with jets dominating at low latitudes, and a proliferation of vortices near the poles <cit.>.In this paper we investigate a simple model of two dimensional fluid flow that exhibits a transition between jets and vortices.We employ both Direct Numerical Simulation (DNS) and Direct Statistical Simulation (DSS).DSS is a rapidly developing set of tools that attempt to describe, directly, the statistics of turbulent flows, bypassing the traditional way of accumulation of those statistics (for example mean flows and two-point correlation functions) by DNS.These statistical methods lead to a deeper understanding of fluid flows that should guide researchers to regimes not accessible through DNS.The pioneering work of Kraichnan largely focused on flows with isotropic and homogeneous statistics <cit.>.The statistical description of forward and/or inverse cascades of energy between different scales, the topic explored in his seminal 1967 paper <cit.>, is particularly clear in this context.Equally important, translational and rotational symmetries reduce the technical complexity of statistical theories.Most fluid flows in nature, however, are both anisotropic and heterogeneous. In DSS this is seen as a feature, rather than a defect, as anisotropy and inhomogeneity can lessen nonlinearity of the flows and make the statistics accessible to perturbative computation.We show that a particularly simple version of DSS, one in which the equations of motion for the spatially-averaged statistics are closed at the level of second-order moments or cumulants <cit.>, is already able to reproduce many features of the model two-dimensional flow.The Kolmogorov forcing we employ is purely deterministic and no parameters are tuned at the level of the second-order cumulant expansion (CE2), permitting a fair and unbiased comparison between DNS and DSS.We introduce the model in Section <ref>.Results from DNS and DSS are presented in Section <ref>.Comparison between the two approaches is made in Section <ref> and some conclusions and possible directions for further exploration are discussed in Section <ref>. § SET-UP OF THE MODEL: FORMULATION, EQUATIONS, AND FORCING. The models we study are of incompressible fluid moving on a two-torus — i.e atwo dimensional Cartesian domain (0 ≤ x < L_x, 0 ≤ y < L_y = 2 π) with periodic boundary conditions in both directions.The fluid motion is damped by viscosity ν and driven by a time-independent forcing (described below).Owing to the two-dimensionality of the system the dynamics is completely described bythe time-evolution of the vorticity ζ≡ẑ· (∇⃗×v⃗), which(in dimensional units) is given byζ̇ + J(ψ,ζ) = ν∇^2 ζ + g(y),where g(y) is the forcing term and J(A,B) is the Jacobian operator given by J(A,B)= A_x B_y - A_y B_x. We note that, in contrast to some earlier models, here we do not consider the effects of rotation via a β-effect.The forcing g(y) is a generalisation of the Kolmogorov forcing (see e.g. lk2014) to two meridional wavenumbers; that is we setg(y)= A_1 cos(y)+ 4.0*A_4 cos (4y).We set A_1=-1 and A_4=-2, which leads to non-trivial dynamics in the fluid system. We note here that this choice of deterministic forcing injects energy directly into the zonal flow, and should be contrasted with previous studies that impose stochastic forcing in the small zonal scales. Once the forcing and the length scale in y (say) is fixed then the dynamics (and indeed statistics) of the flow is determined by the choice of the viscosity ν (which controls the Reynolds number of the flow, which may be calculated a posteriori) and the aspect ratio (determined by L_x).The aim of this paper is to determine how successful Direct Statistical Simulation truncated at second order (sometimes termed CE2) is at describing the transitions that occur as the parameters are varied. In particular we shall investigate how well DSS reproduces the strength of the mean shear flows and the transition from solutions dominated by jets to those dominated by coherent vortex pairs. § RESULTSIn this section we describe results from DNS (in subsection <ref>) and those obtained from DSS using CE2 (in subsection <ref>) before comparing them in Section IV. §.§Direct Numerical Simulation Direct Numerical simulation is performed using a pseudo-spectral code optimised for use on parallel architectures with typical resolutions of 512^2. In all cases the resolution is increased to this level to obtain until convergent results. Initially we integrate the equations forthree different values of the viscosity in a square domain with L_x=L_y=2π. The time series for the resulting spatially-averaged enstrophy ζ^2 and kinetic energy ψ_y^2+ψ_x^2 whereA≡1/L_x L_y∬ Adxdy, are shown in Figure <ref> for three values of the viscosity. This figure clearly shows that, as expected, as the viscosity is decreased (with the forcing fixed) both the enstrophy (top panel) and kinetic energy (bottom panel) of the solutions increases. Figure <ref> (multimedia view) shows snapshots from movies of the evolution of the vorticity for the cases with L_x=2 π (which are included in the supplementary material).The flow is reasonably laminar although the solution has already undergone a bifurcation from a steady state. After some initial transients the solution becomes time periodic, with a strong band/jet of positive vorticity in the domain and weaker negative vorticity (in the form of a vortex) at the edges. This corresponds to a rightward jet in the upper half of the domain and a reverse jet in the lower half (see later). Both the jet and the vortex remain fixed in space though pulse in time. Though time-dependence is present, these vortex regions possess a well defined zonalmean, which can be calculated by averaging over a suitably long time. The average Reynolds number for this flow is given by Re ≈ 730.As the viscosity is decreased from this solution the dynamics becomes more irregular and time-dependent, as shown in thetwo movies. Decrease of the viscosity leads to stronger patches of vorticity and faster flows. For ν = 0.022, Re ≈ 1370, whilst for ν=0.02, Re ≈ 1650.For both of these solutions the non-zonally symmetric part i.e. the part of the solution with k_x0 and the vortex travel in space, rather than remaining fixed as for the earlier case. The strength of the vortex patches and jets increases with decreasing ν (as does the corresponding mean flows and vorticities — see later) as the inertial terms play an increasingly important role.When the aspect ratio is increased so that L_x=4 π, the nature of the solution changes. The driving which is independent of x no longer puts substantial power into the k_x=0 modes, but instead drives a fully nonlinear quasi-steady vortex pair solution as shown in Figure <ref> (multimedia view) and the corresponding movie. This state is reminiscent of the localised states analysed extensively in Ref. lk2014. For this state the average enstrophy and vorticity are significantly lower than for the jet states (as shown in the time series in Figure <ref>). Moreover, as we shall see, this state has little energy in the zonally-averaged vorticity and flow and so can not be characterised as a jet state. This remarkable transition appears to be the opposite of a zonostrophic instability (see Ref. sy2012); there a small-scale forcing with zero zonal mean drives flow that interacts with rotation to put significant amount of energy into a zonally averaged jet. Here the forcing is designed to drive strong zonal flows, but nonlinear interactions prefer to put energy into vortex states with weak zonal flows, and may therefore be termed a “vortostrophic instability.”We note that the aspect ratio controls a similar transition between jets and vortices in other two-dimensional models <cit.>. The non-trivial nonlinear dynamics provides an interesting testing ground for the types of statistical theories favoured by Kraichnan and so, in the next section, we compare the results obtained here via Direct Numerical Simulation, with those obtained by Direct Statistical Simulation truncated at second order (CE2).§.§ Direct Statistical Simulation: The Cumulant Equations In this section we perform DSS for the system for the same range of parameters as above.The approach we take is based upon truncating the hierarchy of equations of motion for the equal-time cumulants at low order.It is related to stochastic structural stability theory (S3T) <cit.> and other approaches <cit.> that do not assume spatial homogeneity or isotropy in the statistics.Here we define the cumulants in terms of zonal averages over the x-direction (see Refs. marston65conover,tobias2011astrophysical,tobias2013direct,marston2014direct,ait2016cumulant) as opposed to ensemble averages <cit.>.Thusc_ζ(y) = ⟨ζ⟩,where ⟨⟩ indicates a zonal average, is the first cumulant and c_ζζ(y, y^',ξ) = ⟨ζ^'(x, y)ζ^'(x+ξ, y^') ⟩,is the second cumulant (or two-point correlation function). We note that owing to the translational symmetry of the system (including the forcing) the first cumulant is a function only of y and the second cumulant is a function of three rather than four dimensions <cit.>. There are similar definitions for the first and second cumulants involving the streamfunction (i.e. c_ψ and c_ψζ), but these can be related straightforwardly to the cumulants for the vorticity.The cumulant hierarchy can be derived in a number of ways <cit.>. Truncated at second order (CE2) this takes the form of evolution equations for c_ζζ(y,y^') and c_ζ(y) (see Ref. tobias2013direct):∂/∂ tc_ζ(y)= [ -(∂/∂ y+ ∂/∂ y^')∂/∂ξ c_ψζ(y,y^',ξ)]|_y^'=y, ξ = 0+ g(y)+ν∂^2 /∂ y^2 c_ζ(y), together with∂/∂ tc_ζζ = ∂/∂ y c_ψ(y) ∂/∂ξ c_ζζ(y,y^',ξ)- ∂/∂ y (c_ζ(y)) ∂/∂ξ c_ψζ(y,y^',ξ) - ∂/∂ y^' c_ψ(y^') ∂/∂ξ c_ζζ(y,y^',ξ)- ∂/∂ y^' (c_ζ(y^')) ∂/∂ξ c_ζψ(y,y^',ξ) + ν( 2∂^2/∂ξ^2+∂^2/∂ y^2+∂^2/∂y^'^2) c_ζζ.In the limit of no forcing or dissipation, equations (<ref>–<ref>) conserve energy, enstrophy and the Kelvin impulse; thus the CE2 is a conservative approximation <cit.>.The equations are integrated forward in time numerically using a pseudospectral code with typical resolutions of (n_y,n_y^',n_ξ) = (64,64,16) until the statistics have settled down to a statistically steady state and means and two-point correlation functions are averaged in time. § COMPARISON OF DNS AND DSSThe solutions from the cumulant equations are compared with the corresponding statistics obtained from DNS by averaging in x and time. In all cases the DNS solutions are averaged over the final third of the evolution and the statistics are well converged. The DSS solutions were averaged over the final 10% of the evolution, though the averages rapidly converge in all cases.Figure <ref> shows the comparison between DNS and CE2 for a domain of length 2 π and decreasing viscosity. As the viscosity is decreased the mean vorticity amplitude ⟨ζ⟩ and the mean zonal flow ⟨ u ⟩ increase in amplitude as expected. It is clear that the comparison of the mean flows between CE2 and DNS is excellent. CE2 has a tendency to emphasise turning points in the vorticity that are washed out by eddy + eddy → eddy interactions in the DNS <cit.>. However the agreement in the amplitude and form of the solution is very good. One might expect CE2 to improve as the ratio of energy in the zonal mean flow to that in the fluctuations increases. This expectation is largely met, though in all cases agreement is good.What is remarkable is that CE2 is capable of capturing the transition to vortices as shown in Figure <ref>. When the zonal flow is switched off in DNS by changing the aspect ratio, CE2 predicts the same behaviour. Although CE2 does not get the form of the weak zonal flow completely correct, it predicts the amplitude very well. This is unexpected since zonally averaged CE2 is expected to work poorly in a case where the zonal means are small and subdominant to the fluctuations. A more stringent test of the accuracy of statistical representation involves a comparison of not only the zonal means (first cumulants) but also the two-point correlation functions (second cumulants). These are given in Figures <ref> and <ref>. These show the two-point correlation function c_ζζ(y,3 π/4,ξ), i.e.the correlation in spacewith the point three-eighths of the way up on the left hand side of each plot.In all the cases for the jet solutions this is dominated by a k_x=1, k_y=2 solution, though this is modulated in y. DNS and CE2 can be seen to be in good agreement here as they should be for flows with such strong mean. Figure <ref> shows less good agreement between DNS and CE2, with CE2 failing to match both the amplitude and spatial form of the second cumulant (underestimating the k_y=0 component) for the case where the flow is dominated by strong vortices — the DNS velocity correlation function has a near k=0 symmetry in the meridional direction and a near reflection symmetry in the zonal direction as the solution is dominated by a k_x= 1, k_y=0 vortex mode. This is clearly picked up by the correlation function, which is also dominated by these wavenumbers. The failure of CE2 to match the two-point correlation function for the vortex state is not surprising as the form of this correlation function is presumably determined by eddy + eddy → eddy interactions that are discarded from the quasilinear CE2 description.The importance of cubic terms for the form of the solution is determined by the amplitude of their projection onto the second cumulant. This can be seen by comparing the second cumulants from CE2 and DNS. What is clear is that this projection is small for the cases with a significant zonal mean and large for the case of the vortex We discuss possible strategies for improving the agreement between DNS and DSS for this case next in the Discussion.§ DISCUSSIONWe have shown that an expansion in equal-time and zonally averaged cumulants, truncated at second order, is able to describe both the jet- and vortex-dominated phases of a two-dimensional flow driven by deterministic Kolmogorov forcing.Zonal mean flows in both phases are accurately reproduced, and two-point correlations of the vorticity are also captured in the jet phase, but not in the vortex-dominated phase.It is remarkable that DSS with such a simple closure can capture much of the behavior exhibited by DNS.We expect that more sophisticated closures will be able to describe the vortex phase accurately.Higher-order closures such as CE3^* and CE2.5 (see Ref. marston2014direct) include eddy + eddy → eddy interactions and can improve qualitative agreement in two-point correlations found by DSS in comparison to DNS. The replacement of zonal averages with ensemble averages has been demonstrated to describe non-zonal structures <cit.>, likely including the vortices seen here at aspect ratio L_y / L_x = 2.Finally the generalized quasi-linear approximation (GQL)<cit.> and its associated generalized cumulant expansion (GCE2), by allowing long-wavelength non-zonal structures to interact fully nonlinearly, should also be able to describe the vortex-dominated phase.Each of these variants is more computationally demanding than simple zonal-average CE2, but we plan to test these other forms of DSS for the Kolmogorov forced model.All of these different forms of DSS respect the realizability inequalities studied by Kraichnan <cit.>.They generalize the program of understanding the statistics of turbulence, greatly advanced by Kraichnan, to encompass anisotropy and heterogeneity.Statistical theories of turbulence thus continue to extend their reach, permitting a deeper understanding of fluid flows that may someday allow us to access regimes not currently reachable by DNS.We wish to acknowledge the help of Mark Dixon of The University of Leeds HPC facility team. 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"authors": [
"Steven Tobias",
"Brad Marston"
],
"categories": [
"physics.flu-dyn"
],
"primary_category": "physics.flu-dyn",
"published": "20170927120114",
"title": "Direct Statistical Simulation of Jets and Vortices in 2D Flows"
} |
I. Large-scale kinematic structure and CO excitation propertiesDepartment of Physics and Astronomy, Uppsala University, Box 516, 751 20 Uppsala, [email protected] Department of Earth and Space Sciences, Chalmers University of Technology, SE-43992 Onsala, Sweden Department of Astrophysics, University of Vienna, Türkenschanzstr. 17, 1180 Vienna, Austria South African Astronomical Observatory, P.O. Box 9, 7935 Observatory, South Africa Astronomy Department, University of Cape Town, University of Cape Town, 7701, Rondebosch, South Africa National Institute for Theoretical Physics, Private Bag X1, Matieland, 7602, South Africa Institut d'Astronomie et d'Astrophysique, Université Libre de Bruxelles, Campus Plaine C.P. 226, Boulevard du Triomphe, B-1050 Bruxelles, Belgium European Southern Observatory, Karl-Schwarzschild-Straße 2, 85748 Garching, GermanyThe S-type asymptotic giant branch (AGB) star π^1 Gru has a known companion at a separation of 27 (≈400 AU). Previous observations of the circumstellar envelope (CSE) show strong deviations from spherical symmetry. The envelope structure, including an equatorial torus and a fast bipolar outflow, is rarely seen in the AGB phase and is particularly unexpected in such a wide binary system. Therefore a second, closer companion has been suggested, but the evidence is not conclusive. The aim is to make a 3D model of the CSE and to constrain the density and temperature distribution using new spatially resolved observations of the CO rotational lines.We have observed the J=3-2 line emission from ^12CO and ^13CO using the compact arrays of the Atacama Large Millimeter/submillimeter Array (ALMA). The new ALMA data, together with previously published ^12CO J=2-1 data from the Submillimeter Array (SMA), and the ^12CO J=5-4 and J=9-8 lines observed with Herschel/Heterodyne Instrument for the Far-Infrared (HIFI), is modeled with the 3D non-LTE radiative transfer code SHAPEMOL. The data analysis clearly confirms the torus-bipolar structure. The 3D model of the CSE that satisfactorily reproduces the data consists of three kinematic components: a radially expanding torus with velocity slowly increasing from 8 to 13 km s^-1 along the equator plane; a radially expanding component at the center with aconstant velocity of 14 km s^-1; and a fast, bipolar outflow with velocity proportionally increasing from 14 km s^-1 at the base up to 100 km s^-1 at thetip, following a linear radial dependence. The results are used to estimate an average mass-loss rate during the creation of the torus of 7.7×10^-7 M_⊙ yr^-1. The total mass and linear momentum of the fast outflow are estimated at 7.3×10^-4 M_⊙ and 9.6×10^37 g cm s^-1, respectively. The momentum of the outflow is in excess (by a factor of about 20) of what could be generated by radiation pressure alone, in agreement with recent findings for more evolved sources. The best-fit model also suggests a ^12CO/^13CO abundance ratio of 50. Possible shaping scenarios for the gas envelope are discussed. The extended molecular envelope of the asymptotic giant branch star π^1 Gruis as seen by ALMA L. Doan 1 S. Ramstedt1 W. H. T. Vlemmings2 S. Höfner1E. De Beck2 F. Kerschbaum3 M. Lindqvist2 M. Maercker2 S. Mohamed4,5,6 C. Paladini7 M. Wittkowski8 ===================================================================================================================================================================================================== § INTRODUCTIONAsymptotic giant branch (AGB) stars are believed to evolve from low- to intermediate-mass (0.8–8 M_⊙) main sequence stars. The evolution of AGB starsis governed by the massive wind from the stellar surface <cit.>, which creates an expanding circumstellar envelope (CSE) of molecular gas and dust already on the AGB. The physical processes behind the AGB-star wind are comparatively well understood and current radiation-hydrodynamical models reproduce observed properties well <cit.>. However,several important aspects still need to be investigated further, such as the wind evolution over time and the formation of complex large-scale structures, to establish the formation scenario for planetary nebulae (PNe).Planetary nebulae are large, tenuous emission nebulae glowing at visible wavelengths through recombination and forbidden lines from ionized (atomic) gas. Expansion velocities of PNe are on average about a factor of two larger <cit.> than typical AGB wind velocities, but more extreme velocities in excess of ∼100 km s^-1 are also found in post-AGB stars and PNe <cit.>. <cit.> studied the CO emission from 30 protoplanetary nebulae (P-PNe) and found that almost all of the sample sources have both a slowly expanding envelope (probably the remnant AGB wind) and fast (often bipolar) outflows. The momenta of the envelopes are consistent with a wind driven by radiation pressure on dust grains (as on the AGB), but an alternative mechanism is necessary to accelerate fast outflows. Imaging surveys of PNe <cit.> show that less than 5% of PNe are round, as would be expected if they were the direct result of an isotropic wind, while a majority of the AGB envelopes seem to be spherically symmetric <cit.> on large scales. This further supports the idea that strong dynamical evolution takes place before AGB stars become PNe. While different shaping agents (binary interaction, large planets, and global magnetic fields) have been suggested <cit.>, much work remains before the theoretical models can be confirmed observationally. In this context, molecular line observations of transitional objects are extremely valuable since they trace the remnant material of the AGB wind and give kinematic information <cit.>. The unsurpassed potential of the Atacama Large Millimeter/submillimeter Array (ALMA) to study these transitional objects has already been demonstrated with Early Science capabilities <cit.>. To add to previous studies, we observed four binary stars on the AGB (R Aqr, Mira, W Aql, and π^1 Gru) with ALMA to investigate the dependence of the circumstellar shaping and morphology on the AGB on the binary separation and wind properties. The observations of Mira (o Cet) already revealed how the massive AGB wind has been sculpted by the fast, thinner wind from the companion <cit.>, and in this paper, we present the initial results on the largest separation source of our sample, π^1 Gru. π^1 Gru is an evolved, S-type AGB star <cit.> at a distance of about 150 pc <cit.> and log(L/L_⊙)=3.86 <cit.>. A fast bipolar molecular outflow was discovered by <cit.> in ^12CO J=1-0 and 2-1 emission. <cit.> also observed the ^13CO J=1-0 line emission and found a ^12C/^13C abundance ratio in the range 25-50. π^1 Gru has a known G0V companion <cit.> at 27 separation, but<cit.> already mentioned that a closer unknown companion would be required to explain the observed morphology. The ^12CO J=2-1 emission was mapped by <cit.> and later by <cit.> using the SMA (with a synthesized beam of 22×42). <cit.> built on the model by <cit.> and suggested that the star is surrounded by a thick, low-velocity (11 km s^-1) expanding torus with a faster bipolar outflow that is oriented perpendicular to the torus; this torus is referred to as a flared disk, but we use torus throughout this paper, since the structure shows no sign of rotating. Herschel/PACS observations show a large arc, possibly a spiral arm, reaching out at ∼40to the east of the star <cit.> and possibly shaped by the known 27 companion. <cit.> have also analyzed VLTI/MIDI andAMBER data, together with Hipparcos and Tycho observations to search for a closer companion. Although the second companion is not directly detected in the VLTI observations, they have found support for a closer companion (at 10-30 AU separation) from the combined analysis of the available data. We observedπ^1 Gru in ^12CO and ^13CO J=3-2 with ALMA. In this paper we construct a 3D kinematic and radiative transfer model based on the ALMA Atacama Compact Array (ACA) and Total Power (TP) observations, together with the previously published ^12CO J=2-1 SMA observations from <cit.> via the publicly available radiative transfer code SHAPEMOL <cit.>. We present the new ALMA observations and the previously published observations (SMA and Herschel/HIFI) in Section <ref> and the ALMA and SMA observational results in Section <ref>. The radiative transfer modeling is described in Section <ref> and the model results are given in Section <ref>. We give the Discussion and Summary in Sect. 6 and 7.§ OBSERVATIONS §.§ New CO radio line observations with ALMA The^12CO and ^13CO J=3-2 emission was observed with the ALMA-ACA in 2013. The observations consist of a four-point mosaic. They were performed using four spectral windows with a width of 2 GHz each, centered on 331, 333, 343, and 345 GHz. The u-v coverage of an interferometer is always incomplete. Insufficient u-v coverage can cause artificial features when imaging. Observations with the ALMA-TP array was performed in cycle 2 (in 2015) to recover the most extended emission from the source and produce high fidelity images. This array included three 12m antennas, and the observations were performed in single-dish, on-the-fly mapping mode.We used the Common Astronomy Software Application (CASA) for calibration and imaging <cit.>. Firstly, the interferometric data was calibrated and preliminary imaged to combine with the TP images later on. Quasars J0006-0623 and J2235-4835 were used as bandpass and complex gain calibrators, respectively. Uranus was used for flux calibration. Because of the low signal-to-noise ratio of the ^13CO J=3-2 emission relative to ^12CO J=3-2, the ^13CO J=3-2 visibility data was imaged using natural weighting to improve the sensitivity. The spectral resolution was about 0.5 km s^-1, but has been binned to 2 km s^-1 for ^12CO J=3-2 and 3 km s^-1 for ^13CO J=3-2 to improve the signal-to-noise ratio in the images cubes. The resulting line profiles with a recovered flux in the ALMA-ACA observation are discussed in Section <ref>. Secondly, the TP observation calibration were carried out with quasars J2230-4416 for focusing, J2230-4416 for pointing, and both of these and π^1 Gru for atmospheric calibration. The brightness of the TP observation was first given in main-beam brightness temperature scale (T_mb) and then converted to Jy beam^-1. The conversion factors of the data from K to Jy beam^-1 are 43.2 at 345 GHz and 45.0 at 330 GHz. The beam size of the TP observation is 19 at 345 GHz and the rms noise level of the images is 0.6 Jy beam^-1 for ^12CO J=3-2 and 0.58 Jy beam^-1 for ^13CO J=3-2. The overall uncertainty of the TP calibration is about 5%. The data was finally combined using CASA packages. Owing to the difference in spatial pixel sizes and spectral ranges between the TP map and the ALMA-ACA map, the TP map was first regridded to the coordinate system of the ALMA-ACA map. Then the ALMA-ACA data was re-imaged using the CLEAN package in an iterative procedure with a decreasing threshold parameter. In this procedure the ALMA-TP map was used as a model for initial cleaning. Finally, the ALMA-TP and ALMA-ACA images were combined via the FEATHER package. A summary of the observations and the final image cubes is given in Table <ref>. §.§ Previously published CO radio line observationsSMA Observations: The ^12CO J=2-1 observation was performed in 2004 using the SMA with a 2 GHz bandwidth correlator and a 812.5 kHz channel separation over 256 channels. The source was observed with 28 baselines and the longest baseline was about 83kλ. The calibrators were observed along with the target. The data has previously been published by <cit.>. In this study, the upper sideband data has been recalibrated and reimaged using CASA. The bandpass calibration was carried out using Uranus. The nearby quasar 2258-279 was set as the complex gain calibrator and absolute flux calibrator (instead of Uranus as in <cit.>). We applied the Briggs weighting method for imaging and an active mode for cleaning the dirty maps in which the emitting region was carefully selected to avoid the strong noise speaks. This resulted in some differences in the line profile compared to that found in <cit.> (see Section <ref>). The observations and the final image cube are summarized in Table <ref>.Herschel/HIFI observations: The ^12CO J=5-4, 9-8, and 14-13 observations were part of the Herschel SUCCESS program <cit.> and observed with the onboard instrument HIFI <cit.>. The signal-to-noise ratio of the ^12CO J=14-13 line was about 2 to 3 and because of this high uncertainty we decided to omit this line from our analysis. The telescope beam sizes of ^12CO J=5-4 and 9-8 observations are 361 and 201, respectively. The technical setup, data reduction, and line profiles were published in <cit.>. The line profiles were plotted in main beam temperature scale with a noise rms of 15 mK at a channel resolution of 3 km s^-1. In this paper, we adopted the line profiles without any recalibration. § OBSERVATIONAL RESULTS AND DISCUSSION §.§ The continuum fluxA single continuum source was detected from the line-free channels of the ALMA-ACA and SMA observations. Neither the known companion, nora closer companion, such as that proposed by Mayer et al. (2014), would be resolved. The continuum flux densities are 32.7±6 mJy at 230 GHz and 82.2±4.7 mJy at 343 GHz. Assumingoptically thick blackbody emission from a stellar photosphere with a temperature of 3000 K <cit.> and a stellar radius of 2.2×10^13 cm <cit.>, the flux densities would be 35 mJy and 78 mJy, respectively. The measured continuum flux is consistent with the thermal emission of the stellar photosphere. Since the stellar temperature, radius, and flux measurement are highly uncertain, it is not enough to determine whether there is a contribution from dust continuum emission. §.§ Line profiles Fig. <ref> (left and middle) shows the line profiles of the combined (ACA+TP) ^12CO J=3-2 and ^13CO J=3-2 data generated by integrating over a circular region with the width of the APEX beam (18) centered on the stellar position. The ^12CO J=3-2 line profile from APEX <cit.> is also plotted to evaluate the recovered flux in the combined ALMA data. A comparison shows that the ALMA-ACA observations recovered a fraction of less than 50% of the flux observed in the TP observation. The combined ALMA maps contain ∽90% of the flux observed by APEX in 2005, which is within the calibration uncertainties. The lower sideband containing the ^13CO line was affected by the mirrored ^12CO line from the upper sideband. This resulted in an artificial feature in the blueshifted wing of the ^13CO J=3-2 line profile. The line is therefore only plotted from -35 km s^-1 in Fig. <ref> (middle). The ^12CO J=2-1 emission observed with the SMA <cit.>, shown in Fig. <ref> (right), is convolved with the SMA primary beam (FWHM of 55).All emission lines (except the ^13CO J=3-2) show a double-horned profile with steep sides at intermediate velocity and extended wings. The double-horned core of the spectral line shows a slowly expanding component, while the high-velocity wings, reaching out to ±60 km s^-1 relative to the systemic velocity, are indicative of an outflow that is much faster than a typical AGB wind. There is no sign of an asymmetry in the ^12CO J=3-2 data observed with the mosaic field ∽40. <cit.> suggested that an over-resolved inhomogeneous structure caused the stronger peak at the blueshifted velocities in the ^12CO J=2-1 line profile from their analysis of the data. In contrast, the peak is seen on the redshifted side of both the line profile observed by <cit.> and the line profile from our recalibration of the data. This discrepancy may be due to the different calibration and data reduction strategy, as already mentioned in Section <ref>. §.§ Images^12CO J=3-2: The channel maps of the ^12CO J=3-2 emission from the combined data, shown in Fig. <ref>, were constructed by integrating over 2 km s^-1 close to the systemic velocity (from -30 km s^-1 to 8 km s^-1) and over 4 km s^-1 for the high velocities (from -62 km s^-1 to -46 km s^-1 and from 20 km s^-1 to 36 km s^-1) to increase the signal to noise of the high-velocity channels in which the emission is weaker. The emission at the systemic velocity, V_S=-12 km s^-1, shows what looks like an elongated torus along the east-west (EW) direction. The emission has a maximum close to the stellar position. When moving away from the systemic velocity, the size of the emitting region decreases. Also, the emission gradually moves from the south to the north from blue- to redshifted velocities. This spatial shift of the emission distribution at low relative velocities can be interpreted that the radially expanding torus is inclined relative to the line of sight. The high-velocity channel maps are shown in the first and last rows of Fig. <ref>. In agreement with the ^12CO J=2-1 emission <cit.>, the north-south (NS) orientation of the higher velocity emission is opposite to that at lower velocities. Furthermore, the ^12CO J=3-2 emission seen in the channels from -58 to -46 km s^-1 and 20 to 32 km s^-1 shows an extended region with two separated parts. We propose that this bimodal distribution can be possibly interpreted as emission coming from the lobe walls of a bipolar outflow, while the -62 km s^-1 and 36 km s^-1 channels may show the lobe tips or lobe edges at the highest line-of-sight velocity of the bipolar outflow.In Fig. <ref>, as well as in the position-velocity (PV) diagram in Fig. <ref>, the emission is divided into two components around -46 km s^-1 and +20 km s^-1. At blueshifted velocities, the southern component is just slightly north of the stellar position while the second, northern component is found approximately 5to the north. Assuming that the velocity of the high-velocity outflow increases radially and the system is inclined relative to the line of sight, the gas moving along the edges of lobes have different line-of-sight velocities at the same distance from the equator. For example, the gas moving along the edge of the northern lobe facing the observer, has a higher line-of-sight velocity than the gas moving along the edge away from the observer at the same distance from the equator. This can explain the bimodal distribution seen at, for example -50 km/s. At this velocity channel, the southern emission component would come from the closer lobe edge and the northern emission component would come from the more distant lobe edge, where the same line-of-sight velocity is reached further away from the equator of the system. The exact distribution of the emission as a function of velocity is an intricate function of the gas distribution, i.e., inclination and curvature of the high-velocity outflow and the clumpiness of the gas, and kinematics.^13CO J=3-2: The J=3-2 emission from the less abundant ^13CO isotopologue was imaged by integrating over 3 km s^-1 (Fig. <ref>). The emission is very weak compared to the ^12COemission and the gas is mostly concentrated and excited in the inner parts of the envelope. Atthe systemic velocity, the emission has two peaks on either side of the stellar position along the EW direction. Even if the data was convolved with an identical beam as the ^12CO J=3-2, the image would still have this distribution. Any emission is below the noise level at velocities beyond -25 km s^-1 and 2 km s^-1.^12CO J=2-1: The channel maps from the rereduced ^12CO J=2-1 line emission are given in Fig. <ref>. As previously described <cit.>, the emission at the systemic velocity, V_S=-12 km s^-1, shows a torus structure that is flared and elongated along the EW direction, similar to the ^12CO J=3-2 distribution. The slightly larger spatial size at each channel velocity compared to the J=3-2 emission is caused by the lower minimum kinetic temperature required for the J=2-1 excitation. Moreover, the emission has a two-peak (on either side of the stellar position) distribution that differs from the new ^12CO J=3-2 data. The gap between the two peaks has been attributed to a central cavity <cit.>, but that is not required to explain the new ^12CO J=3-2 data. The orientation of the synthesized beams (see Fig. <ref> and <ref>) can contribute to the different distribution of the ^12CO J=2-1 and ^12CO J=3-2, but not fully explain the difference. This is discussed further in the following sections.§.§ Position-velocity diagramsThe PV diagrams of the ^12CO J=3-2 emission shown in Fig. <ref> were made along a NS cut (one pixel wide at a PA of 0^∘) through the stellar position. A different correlation between the velocity vector field and the position vector is seen for the equatorial torus and the fast bipolar outflow, separately. The NS PV ^12CO J=3-2 diagram is similar to that of ^12CO J=2-1 <cit.>, but less extended in spatial offset than the ^12CO J=2-1 emission (as already seen in the channel maps). The emission at low relative velocity(plotted in the central plot of Fig. <ref>) is consistent with an expanding, inclined equatorial torus, extended to the north at redshifted velocity, and to the south at blueshifted velocity. The emission at higher velocities originates from the fast outflow component (plotted in the far left and right plot of Fig. <ref>) and shows the opposite pattern: a redshifted velocity to the south and blueshifted velocity to the north. In this part of the figure, the position is directly proportional to the velocity, i.e., emission with higher velocity comes from a position further away from the center. The offset emission regions seen at blue- and redshifted velocities (with a Dec-offset beyond about ±5) correspond to the bimodal distributions also seen in the channel maps in Fig. <ref>. We do not detect significant emission beyond -70 km s^-1 in the blueshifted part and +40 km s^-1 in the redshifted part. This corresponds to a maximum projected gas velocity of about 60 km s^-1 in the CSE.§ CIRCUMSTELLAR MODEL §.§ Geometry and velocity field As mentioned above, <cit.> built on the flared-disk model to reproduce the low-velocity component seen in the ^12CO J=2-1 SMA maps, but they never attempted to model the fast component. In this study, we reconstructed the gas envelope using both a low-velocity torus and a fast bipolar outflow to study thefull 3D morphology and kinematics of the system. The combined ALMA data are used as observational constraints, together with the previously published SMA data. The structure of the modeled gas envelope is schematically illustrated in Fig. <ref>. The modeled envelope was constructed as a system with the following three components: (1) A radially expanding torus in the shape of a flared disk with an opening angle of 2φ_0, an inner radius of d/2, and an outer radius of R_1. The radius R_1 is just the maximum radius used in the model computation. It does not necessarily represent a density-cutoff radius, i.e., the physical outer boundary of the torus; nebular layers beyond R_1do not contribute significantly to the observed emission for the density and temperature laws adopted in our model. The outer radius was constrained by the observed angular size at the central channel. The radial gas velocity inside the torus depends on the distance from the center and latitude above or below the equator, v_1=[ v_1a+v_1br/R_1]f_φ,where the constants v_1a and v_1b were chosen to produce line profiles with the same width as the line cores of the observational data and r is the radial distance from the center (r=√(x^2+y^2+z^2)). The f_φ factor increases linearly with latitude. It has the value of f_0^∘ =1 at the equator and the values of f_±φ_0 at the top and bottom edges were determined by fitting the data.(2) A central, radially expanding component originating at the center and placed inside the torus. The velocity is constant, v_2. Its shape is cylindrical with diameter d and height h. (3) A faster bipolar outflow perpendicular to the torus extends from the central component. The lobes have a radially expanding velocity field linearly increasing with the distance from the equator, v_3= v_3a+v_3bz/R_3,wherev_3a andv_3bwere chosen so that the velocity increases from the central component velocity to the highest velocity inferred from the data and z is the vertical distance from the equator. The radius R_3 is just the maximum radius used in the model computation. It is not the physical outer boundary of the outflow. The R_3 value was poorly constrained by the observations because of the low signal-to-noiseratio at very high velocities and the dependence on the inclination of the system. §.§ Density and temperature distributionSome simplifying assumptions are necessary to limit the number of free parameters of the 3D model. The density distribution was chosen assuming the following hypothetical, but realistic, scenario for the shaping of the current CSE: The torus (1) is presumably formed in the earlier AGB phase. At some point, the bipolar outflow was triggered and the faster moving material has plowed through the polar regions giving rise to the central component (2), in which the current dynamics are the result of the interaction between the fast outflow and the slower AGB wind. Further out, the bipolar outflow (3) is the faster moving material that has escaped the AGB envelope. The gas density distribution of a spherically expanding AGB envelope created by a constant mass-loss rate, Ṁ, and expansion velocity,v_e, decreases outward according to n(r)= Ṁ/4πr^2 v_e m,where m is the particle mass. Since the gas velocity inside the torus depends on the radius according to Eq. (<ref>), the H_2 number density of the torus was set to be proportional to r^-3, n_1(r,φ)=n_1a[ r/10^15cm]^-31/f_φ,where n_1a is a scaling factor, r is the radial distance from the center, and the f_φ factor is due to the linear dependence of the torus velocity on the latitude (Eq. <ref>). The constant velocity of the central component (2) motivates an r^-2 dependence of the H_2 number density,n_2(r)=n_2a[ r/10^15cm]^-2.The scaling factors, n_1a and n_2a, were initially set using Eq. (<ref>) with the mass-loss rate chosen as an average of previous results (1×10^-6 M_⊙ yr^-1) and a constant expansion velocity (11 km s^-1), but then varied to fit the data. For bipolar planetary nebulae, a velocity distribution that follows the Hubble law expansion is often found <cit.>. Assuming that the velocity increases radially in the outflow component (3), the density was initially set as an inverse cubic function of the radius, n_3(r)=n_3a[ r/10^15cm]^-3+α,and then both the scaling factor n_a3 and the exponent α were varied until the data could be reproduced at high velocities (see Sect. <ref>). The ^12CO abundance (relative to H_2) was assumed to be constant for all three components and a value of 6.5× 10^-4 <cit.> was adopted. The ^13CO abundance was changed to fit the ^13CO J=3-2 data once all the other parameters had been obtained from fitting the ^12CO lines. A description of the gas kinetic temperature as a function of radius in a spherical envelope was presented by <cit.>. <cit.> and <cit.> successfully applied this temperature distribution when they modeled the torus. A similar dependence on radius,T=T_0[ r/10^15cm] ^-0.7+βK, was adopted for the whole envelope. The scaling factor, T_0, was varied from 100 K to 500 K, while the exponent was slightly varied around -0.7 through the free parameter β. §.§ Radiative transfer modeling and imagingIn the radiative transfer model, the parameters introduced in Eqs. (<ref>)-(<ref>) and(<ref>)-(<ref>), together with the inclination angle of the torus relative to the line of sight, and the PA of the equator were varied until all the available spatially resolved data could be reproduced. The radiative transfer calculation was performed via SHAPE+SHAPEMOL <cit.>, which is a 3D modeling tool for complex gaseous structures. To solve the radiative transfer equations for ^12CO and ^13CO, the code uses tabulated absorption and emission coefficients that are appropriate for different geometries and kinematic, and calculates the non-LTE level populations via the large velocity gradient (LVG) approximation. We believe the LVG approximation is valid for π^1 Gru because the expansion velocity is larger than the local line width (the thermal contribution is less than 1 km s^-1) and owing to the large velocity gradient across the CSE. The output simultaneously depends on the gas density, kinetic temperature, velocity distribution, CO isotopologue abundance, and logarithmic velocity gradient ((dV/dr)(r/V)). A microturbulent velocity of 2 km s^-1 <cit.> was adopted. The calculation is conducted for the 17 lowest rotational transitions of the ground-vibrational state of both^12CO and ^13CO, including collisions with H_2 <cit.>. The envelope was constructed of 128^3 grid cells and rendered through a velocity interval of ±100 km s^-1 around the systemic velocity. The velocity band is divided into 150 channels. The comparison between the model and observations requires that the effects of the interferometer, for example, the missing u-v coverage, noise of theatmosphere, and electric systems, are included before imaging.Using the SIMOBSERVE task in CASA, we simulated the visibilities of the observation using the resulting brightness distribution from the radiative transfer calculation. The SMA and ALMA-ACA simulated data has the on-source duration and the antenna configurations of the original observations. The data was then imaged in the same way as the real observational data. The ALMA-TP simulation was performed inside SHAPE+SHAPEMOL by convolving the 3D model with the ALMA primary beam. The ALMA-ACA simulated images were finally combined with the ALMA-TP simulated images using the same procedure as for the real data.§ MODELING AND INTERPRETATION§.§ Finding the best-fit model All free parameters were first adjusted to fit the ^12CO J=3-2 line emission. With the many adjustable parameters of the 3D model, the goodness of fit has to be evaluated by taking several different aspects into account. The velocity distribution and inclination angle of the torus were set to reproduce the observed line shapes and spatial distribution seen in the channel maps. The sizes of the different components were constrained by fitting the spatial extent seen in the channel maps (by eye) assuming a distance of 150 pc <cit.>. The density and temperature distribution were primarily chosen to fit of the strength of the emission in the^12CO J=3-2 line. Those best values were applied to successfully reproduce the ^12CO J=2-1. Then the high-J transition lines ^12CO J=5-4 and J=9-8 could be fitted by refining the temperature function since the line ratios are sensitive to the kinetic temperature. The ^13CO J=3-2 data was matched by only adjusting the ^13CO/H_2 fractional abundance.Some additional test models with different velocity fields, morphologies, and temperature distributions (see Appendix A) were considered to find the best-fit model. A chi-square measure was used to evaluate the goodness of the fit from the line profiles, χ ^2=∑_i=1^N(I_i^o-I_i^m)^2/Nσ_i^2, where I_i^o and I_i^m are the beam corrected flux of the observations and the model at channel i, respectively, N is the number of channels, and σ is the measurement uncertainty of the observed flux, assumed to be 20% on average.§.§ Best-fit model and comparison to previous resultsThe parameters of the best-fit model are given in Table <ref>. The outer radius (R_1= 3×10^16 cm), and opening angle (φ_0 = 25^∘) of the torus agree with the model by <cit.>. The introduced central component is smaller (angular width ≈ 1.7) than the synthesized beam of the ALMA-ACA observation and does not affect the fit significantly. At a given radial distance, the gas velocity at the top and bottom edges of the torus is 2.5 times (f_φ_0=2.5) higher than the gas velocity along the equator. An inverse cubic density law (i.e., α=0) for the fast bipolar outflow resulted in images very similar to those observed, meaning that the best fit is achieved when the density distribution of the torus and the outflow have the same dependence on radius. Only the scaling factors of the density functions (Eq. <ref> and Eq. <ref>) differ slightly. The best-fit model has a PA of 5^∘ and the inclination of the torus relative to the line of sight is 40^∘, which agrees with what was found in previous studies <cit.>. If the inclination is decreased (or increased) by more than 10^∘ relative to the best-fit value, the model line profiles have a parabolic shape(or a U-shaped profile with a deep center) in contrast to observed data. If the PA is changed by more than 10^∘, the relative intensity of the line peaks is not reproduced. The model shows that the temperature distribution suggested by <cit.> for the torus cannot be applied to reproduce the high-J transition lines as well. The temperature in our model is lower (T_0=190 K instead of 300 K) and decreases outward more rapidly (β = -0.15)than thatfound by <cit.>.The model gives a value of 50 for the ^12CO/^13CO abundance ratio, which is in agreement with upper limit for π^1 Gru derived by <cit.>, but is twice the median value of 25 found for S-type AGB stars <cit.>.The model line profiles are plotted (Fig. <ref>) together with the observed line profiles and overall they match the data very well. The SMA ^12CO J=2-1 observation only recovered less than 50% of the total flux as mentioned by <cit.>. Therefore, in the very right panel of Fig. <ref>, the line intensity has been scaled by a factor of 2 to be able to compare the line shapes. This is only an approximation because the missing flux mainly comes from the extended parts of the envelope with higher velocity, whereas the flux of line core comes from the inner parts and would be better recovered. The peak intensity of the predicted ^12CO J=9-8 line is about 20% less than that from the observation. Our model cannot reproduce the different peak strengths seen in the^12CO J=9-8 and the ^13CO J=3-2 line. This can be an indication that the LVG approximation does not perfectly apply to the torus where the velocity gradient is less steep than in the outflow.Thespatial distribution of the ^12CO J=3-2 andJ=2-1 is shown in the model channel maps in Fig. <ref> and Fig. <ref>. At the systemic velocity, the model has successfully reproduced the two-peaked distribution seen in the ^12CO J=2-1 data without a cavity at the center and the central peak distribution seen in the ^12CO J=3-2 data. This means that the model without the central cavity can reproduce the data features. In general, the model images at each channel are very similar to the observations for low to intermediate velocities. The weak bimodal emission distribution at very high-velocity channels (Fig. <ref>) was not reproduced by our model. As suggested above, the emission could come from the gas at the edges of the two bipolar lobes, but it would depend on the detailed structure and temperature distribution of the outflow, and an exact fit was not attempted.Figure <ref> shows the line-of-sight optical depth (along the system equator at the systemic velocity) as a function of distance from the center for all three modeled lines. Since the emission is optically thin, the peak position of the optical depth indicates where the gas is maximally excited, and agrees with the positions of the emission peaks of the channel map at the systemic velocity (Fig. <ref>). The maximum optical depth of the ^12CO J=3-2 line occurs inside 5and the corresponding two emission peaks are unresolved by the beam. §.§ Outflow linear momentum The terminal gas velocity along the torus equator is about 13 km s^-1. If this value is representative of the expansion velocity of the CSE before the formation of the outflow, this corresponds to an average mass-loss rate of about 7.7×10^-7 M_⊙ yr^-1. This agrees with the value estimated by <cit.>, and is almost half of that estimated in <cit.> when fitting the single-dish ^12CO J=2-1 emission. From the sizes of the structures and the expansion velocity distributions, the kinematic timescale of the torus and outflow is about 730 yrs and 160 yrs, respectively. The bipolar outflow of the best-fit model suggested has a density distribution that is comparable to the torus at the same radial distance. The estimated total mass of the bipolar outflow is M_outflow=7.3×10^-4 M_⊙ when applying an average particle mass of 3×10^-24 g and integrating the density distribution across the structure. The highest deprojected gas velocity is about 100 km s^-1 corresponding to a Doppler shift velocity of about 60 km s^-1 in the lines. This results in a linear momentum of the outflow, which is the total outflow mass multiplied by the outflow velocity integrated over the velocity distribution, ofP_outflow=9.6×10^37 g cm s^-1. If radiation pressure alone is responsible for lifting the outflow, this process would take about 3300 yrs (calculated from P_outflow/(L/c)). This means that the radiation pressure alone would not be sufficient to drive the fast outflow with a kinematic timescale of 160 yrs, which is in agreement with the findings for more evolved sources <cit.>.§ DISCUSSION §.§ Excitation propertiesAll line profiles in this work show a slowly expanding component at the line core and a high-velocity component at the wings. The less extended wings of the ^12CO J=5-4 and J=9-8 line profiles indicate that the CO molecules are mostly excited to the high-J states in the central regions where the temperature and density are high enough. The minimum kinetic temperature (∼250 K) of the ^12CO J=9-8 transition, needed for significant collisional excitation, is only reached in regions close to central star. The transition is mainly radiatively excited.The differences in spatial distribution of the emission from the CO lines are also due to the different excitation requirements. The higher excitation temperature of the J=3-2 transition than the J=2-1 makes the emission more compact at every channel and particularly at the systemic velocity. Our model can reproduce the different apparent spatial distributions of the two ^12CO lines, and the ^13CO line, reasonably well without including a central cavity <cit.>. If the cavity actually exists, it must be smaller than the synthesized beam in the case of the ^12CO J=3-2 emission. The apparent shape is dependent on the observational setup, line excitation, and actual distribution of the gas, and shows that it is very important to perform detailed radiative transfer modeling before drawing conclusions about the physical distribution of the gas in these types of objects.§.§ Envelope shaping mechanismsThe formation of the torus+outflow structure seen in π^1 Gru, poses a challenge to the understanding of stellar evolution theory, as do the typical bipolar structures in PNe. The gravitational perturbation of the known companion of π^1 Gru is not strong enough to concentrate material onto the orbital plane and form the torus <cit.>. Even if the orbit is extremely eccentric and the envelope could be compressed at the periastron passage, the timescale for the creation of the torus is too short for it to still be significantly affected. If the orbit is assumed circular, the period would be 6200 yrs to be compared to the kinematic timescale of the torus on the order of 730 yrs. Stellar wind models for AGB stars (assuming a dust-driven wind) typically show wind velocities less than 30 km s^-1 <cit.>. For π^1 Gru, the fast (≤100 km s^-1) bipolar outflow has a linear momentum that is higher than the maximum value available from the radiation pressure of the star, which suggests that a different mechanism is required to drive the outflow.The torus could be created if the star itself is rotating already on the AGB. <cit.> have shown that the effect of a slow rotation (the order of 2 km s^-1), combined with the strong temperature and density dependence of the dust formation process in AGB stars, can lead to an enhanced equatorial mass loss and produce an elliptical envelope. By applying the wind compressed disk model to AGB stars,<cit.> showed that the coriolis effect, effective if the star is rotating fast enough, can produce a disk-like structure. A close companion or a binary merger can spin-up AGB stars to the rotation rate required for the wind-compressed disk to be formed. However, there is no evidence for rotation observed in π^1 Gru, but a velocity below 2 km s^-1 cannot be excluded. The gravitational effect of a close companion star, or a giant planet, can play an important role in the formation of a torus <cit.>.Whether single stars can form bipolar morphologies was initially investigated in interacting wind models <cit.>. In these models, a fast isotropic wind is launched inside a slowly expanding torus (without investigating the formation of the torus itself) with a density contrast between the pole and the equator. A hot bubble is created in the post-shock gas and the bubble expands at constant pressure. The expansion velocity of the bubble depends on the density distribution of the torus and varies inversely from the pole to the equator <cit.>. The very high gas velocity at the poles eventually launches the bipolar outflow. Wind-interaction models including detail hydrodynamics and microphysics <cit.> have confirmed the dependence of the shaping on the density distribution of the previously ejected gas envelope. A density contrast (between the torus equator and the pole) from 2 to 5 results in a fast bipolar outflow. A higher value results in a highly collimated outflow. Within the modeled torus, the best-fit model of π^1 Gru has a density contrast of 2.5 between the equator and the edge at an angle of 25^∘. If a linear dependence with angle is assumed, the contrast between the equator and the pole is 6.4, which is a reasonable agreement between the results from the interacting wind models and the observed morphology.The kinematic timescales for the torus and fast outflow in our model are similar to the typical values found for PPNe and PNe <cit.>, showing the same circumstellar structures. The momentum excess (compared to what is available from radiation pressure alone) implies the need for an additional wind driver. Wind formation and the collimation of bipolar outflows have also been studied using magneto-hydrodynamics (MHD) models where the wind is driven by magnetic pressure <cit.>. These models are successful in creating bipolar structures and highly collimated jets when the magnetic field is strong enough and the rotational velocity is sufficient. However, recent investigations combining the results of MHD and stellar evolution models, have shown that the rotational velocities retained at the end of the AGB are not sufficient to form bipolar PNe <cit.>. <cit.> used MHD models to show that a modest magnetic field <cit.> is sufficient to form a dense equatorial disk around an AGB star with a slow, massive wind. In combination with the interacting wind models described above, this gives another possible explanation for the observed morphology. Again, the origin of the magnetic field or the launching of the fast wind are not explained, but would require an extra source of angular momentum, e.g., a binary companion. The suggested second companion remains to drive the formation of the observed morphology. If close enough, it could also accrete the wind through an accretion disk and possibly drive the fast outflow. The low velocity of the torus sets favorable conditions for wind Roche-lobe overflow <cit.>, where a slowly expanding wind first fills the Roche lobe of the primary star (the AGB star) and then flows through the inner Lagrangian point onto the accretion disk of the companion. Models including the wRLOF scenario can produce substantial accretion rates also in larger separation binaries <cit.>, where traditional Bondi-Hoyle accretion models fail to reproduce the observations. The exact requirements to drive an outflow with the momentum observed in π^1 Gru will be the subject of a future publication. The CSE of π^1 Gru is not the only case where a torus plus bipolar structure has been observed around an AGB star. The CO line observations of V Hya also shows a similar structure <cit.>. Using the CO J=2-1 and 3-2 line observation,<cit.> have distinguished three components in the CSE of V Hya based on its kinematic properties, which are similar to the results in this study. These authors also suggested a similar explanation for generating an intermediate-velocity component as the central component in our model. Another similar example is the post-AGB star, CRL 618, with a very fast, collimated outflow rising from a low-velocity, dense core <cit.>. Owing to the complex geometry showing features commonly found in stars of the next evolutionary phase, π^1 Gru is an extremely interesting case to study in order to find the missing link between the spherical outflows of AGB stars and the bipolar outflows observed at later stages. § SUMMARY We have presented the analysis of new ALMA-ACA data of the ^12CO J=3-2 and ^13CO J=3-2 line emission, together with previously observed ^12CO J=2-1 data from the S-type AGB star π^1 Gru. The high-sensitivity ALMA observations recovered the extended emission, and for the first time, resolved the high-velocity component. The analysed data, including low-J transitions (from ALMA and SMA observations) high-J transition (from Herschel/HIFI observations) provided sufficient constraints for a 3D radiative transfer model. The best-fit model reconstructing the gas envelope has satisfactorily reproduced the line profiles, channel maps, and suggested a reasonable value for the abundance of ^12CO/^13CO. The gas envelope is modeled as a system of three separate components: a radially expanding torus with the velocity linearly increasing with latitude and radial distance, a central,radially expanding component that may have resulted from the dynamical interaction between the fast outflow and the torus, and a fast bipolar flow perpendicular to the equator with a radially expanding velocity field. The outflow momentum excess found in our model rules out a scenario in which radiation pressure alone can lift the high-velocity outflow. The density contrast between the equator and polar regions suggested from various formation mechanisms can successfully reproduce the data. This supports that the gravitational effect of a close companion is involved the torus formation, while the wind interaction mechanism and/or a bipolar magnetic field could be included when considering the outflow formation. The authors would like to thank the staff at the Nordic ALMA ARC node for their indispensable help and support. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2012.1.00524.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ.We are grateful to T. Danilovichfor helping us with the Herschel/HIFI data.S. Mohamed is grateful to the South African National Research Foundation (NRF) for a research grant. W. H. T. Vlemmings acknowledges support from ERC consolidator grant 614264. C. Paladini is supported by the Belgian Fund for Scientific Re- search F.R.S.- FNRS.aa§ FINDING THE BEST-FIT MODEL Some alternative models that were tested to find the best-fit model are presented here. The models with different velocity fields, morphologies, and temperature distributions were constrained by both the channel maps and line profiles.A model using a constant expansion velocity for the whole torus could not reproduce the line profile nor the channel maps. Indeed the line profile of the ^12CO J=3-2 line from such a model has an intensity at the systemic velocity that is much smaller than that of the peaks (at about ±6 km s^-1), while the center and peak intensities are comparable in the observed line profile. The model channel maps are rather sensitive to the assumed velocity field. If v_1 is reduced by 10%, the ^12CO J=2-1 emission in the velocity channel beyond ±12 km s^-1 is reduced below the noise level. The best-fit velocity distribution is also dependent on the chosen inclination angle of the torus, which was constrained by the flattened shape of the torus seen at the systemic velocity in the channel maps (Fig. <ref>). Finally, the PA of the torus was set to reproduce the relative intensity of the two peaks compared to each other in the observed line profiles (see Section <ref>).A collimated velocity field, which is typically found in P-PNe and PNe in which the outflow gas velocity is perpendicular to the equatorial plane, was first attempted to model the fast bipolar component.This gave line profiles similar to the observed profiles, however, it also resulted in a very large intensity ratio between the central velocity channel and the nearby channels, which is not seen in the observed images.An alternative model, without the central component (2), was also tested. In that model, the torus reaches regions close to the central star and is shaped like a flared disk. The fast bipolar outflow rises on top of the torus. This model successfully reproduced the spatial distribution of the ^12CO J=2-1 and the ^12CO J=3-2 emission, in particular the two-peaked distribution of the ^12CO J=2-1 emission, as well as the single central peak of the ^12CO J=3-2 emission at the systemic velocity (without a central cavity). However, in order to reproduce the data, the innermost outflow velocity had to be lower than the velocity of the torus at the same position, which would be unphysical. The temperature distribution, as suggested by <cit.>, results in a good agreement with the observations in the low-J transition lines, the ^12CO J=2-1 and J=3-2, and ^13CO J=3-2 line. However, it overestimates the intensity of the ^12CO J=5-4 and J=9-8 lines that are excited in the central regions. The two lines are sensitive to temperature and can be simultaneously fitted when the temperature decreases outward more rapidly than that found in<cit.> (see Section <ref>). | http://arxiv.org/abs/1709.09435v1 | {
"authors": [
"L. Doan",
"S. Ramstedt",
"W. H. T. Vlemmings",
"S. Höfner",
"E. De Beck",
"F. Kerschbaum",
"M. Lindqvist",
"M. Maercker",
"S. Mohamed",
"C. Paladini",
"M. Wittkowski"
],
"categories": [
"astro-ph.SR",
"astro-ph.GA"
],
"primary_category": "astro-ph.SR",
"published": "20170927103215",
"title": "The extended molecular envelope of the asymptotic giant branch star $π^{1}$ Gruis as seen by ALMA I. Large-scale kinematic structure and CO excitation properties"
} |
Partial differential systems with nonlocal nonlinearities: Generation and solutions Margaret Beck Anastasia Doikou Simon J.A. Malham Ioannis Stylianidis 30th January 2018 ===================================================================================Abstract Molecular fingerprints, i.e. feature vectors describing atomistic neighborhood configurations, is an important abstraction and a key ingredient for data-driven modeling of potential energy surface and interatomic force. In this paper, we present the Density-Encoded Canonically Aligned Fingerprint (DECAF) fingerprint algorithm, which is robust and efficient, for fitting per-atom scalar and vector quantities. The fingerprint is essentially a continuous density field formed through the superimposition of smoothing kernels centered on the atoms. Rotational invariance of the fingerprint is achieved by aligning, for each fingerprint instance, the neighboring atoms onto a local canonical coordinate frame computed from a kernel minisum optimization procedure. We show that this approach is superior over PCA-based methods especially when the atomistic neighborhood is sparse and/or contains symmetry. We propose that the `distance' between the density fields be measured using a volume integral of their pointwise difference. This can be efficiently computed using optimal quadrature rules, which only require discrete sampling at a small number of grid points. We also experiment on the choice of weight functions for constructing the density fields, and characterize their performance for fitting interatomic potentials. The applicability of the fingerprint is demonstrated through a set of benchmark problems.§ INTRODUCTION Molecular Dynamics (MD) simulations have been widely used for studying atomistic systems, e.g. proteins and catalysts, due to their ability to precisely capture transient events and to predict macroscopic properties from microscopic details <cit.>. In its most prevalent implementation, the trajectory of an atomistic system is integrated in time according to the Newton's law of motion using forces calculated as the negative gradient of a Hamiltonian, whose functional form and parameters are collectively referred to as a force field <cit.>. Traditionally, the pairwise and many-body terms that comprise a force field are derived empirically by fitting to quantum mechanical calculations and experimental data.Three properties directly relate to the applicability of a force field: accuracy, transferrability, and complexity <cit.>. Over the years, a large number of force fields have been developed, each carrying a particular emphasis over these three properties. However, the combinatorial complexity of atomistic systems can easily outpace force field development efforts, the difficulty of which explodes following the curse of dimensionality <cit.>. A deceptively simple system that can demonstrate the situation is water, a triatomic molecule with a well-characterized molecular structure.In fact, all common water models, such as SPC-E, TIP3P, and TIP4P, have only succeeded in reproducing a small number of structural and dynamical properties of water due to the difficulty in modeling strong intermolecular many-body effects such as hydrogen bonding and polarization <cit.>.In lieu of a force field, quantum mechanical (QM) calculations can be employed straightforwardly to drive molecular dynamics simulations. The method achieves significantly better accuracy and transferrability by solving for the electronic structure of the system. However, the computational complexity of QM methods is at least cubic in the number of electrons, and consequently the time and length scales accessible by QM-driven molecular dynamics are severely constrained.Assuming that there is smoothness in the potential energy surface of the atomistic system, one possible strategy to accelerate QM-driven molecular dynamics is to use QM calculations on only a subset of the time steps, and to interpolate for similar atomic configurations <cit.>. A schematic overview of the process is given in Figure <ref>, which is enabled by the recent development of high-dimensional nonlinear statistical learning and regression techniques such as Gaussian process regression <cit.> and artificial neural networks <cit.>.This paper focuses on a particular aspect of the machine-learning-driven molecular computation pipeline, i.e. fingerprint algorithms, whose importance arises naturally from the aforementioned regression protocol. A fingerprint is an encoding of an atomistic configuration that can facilitate regression tasks such as similarity comparison across structures consisting of variable numbers of atoms and elements. As has been pointed out previously <cit.>, a good fingerprint should possess the following properties: * It can be encoded as a fixed-length vector so as to facilitate regression (particularly for artificial neural networks).* It is complete, i.e. different atomistic neighborhood configurations lead to different fingerprints and vice versa, and the `distance' between the fingerprints should be proportional to the intrinsic difference between the atomistic neighborhood configurations.* It is continuous with regard to atomistic coordinates, and the change in fingerprint should be approximately proportional to the structural variation as characterized by, for example, some internal coordinates.* It is invariant under permutation, rotation, and translation.* It is computationally feasible and straightforward to implement. Before we proceed to the details of our work, we will first briefly review several fingerprints that are closely related to our work, i.e. the Smooth Overlap of Atomic Positions (SOAP) kernel <cit.>, the Coulomb matrix <cit.>, and the Graph Approximated Energy (GRAPE) kernel <cit.>. Smooth Overlap of Atomic Positions (SOAP):The SOAP kernel is built on the idea of representing atomistic neighborhoods as smoothed density fields using Gaussian kernels each centered at a neighbor atom. Similarity is measured as the inner product between density fields, while rotational invariance is achieved by integrating over all possible 3D rotations, which can be performed analytically using the power spectrum of the density field. In fact, our fingerprint algorithm is inspired by this idea of treating atoms as smoothed density fields. However, we take a different approach to endorse the fingerprint with rotational invariance, and use the Euclidean distance instead of inner product as a distance metric. Coulomb Matrix:The practice of using graphs to represent atomistic neighbor configurations was first implied by the Coulomb matrix, and later further formulated in the GRAPE kernel, where the diffusion distance was proposed as a similarity measure between different local chemical environments <cit.>. The idea is to construct an undirected, unlabeled graph G = ( V , E ) with atoms serving as the vertices and pairwise interactions weighting the edges. For example, the Coulomb matrix can be treated as a physically-inspired Laplacian matrix <cit.>𝐌 = 𝐃 - 𝐀 𝐃_IJ =0.5 Z_I^2.4 ifI=J0ifI ≠ J𝐀_IJ =0ifI=J-Z_I Z_J/‖𝐑_I - 𝐑_I ‖ ifI ≠ Jwhere the degree matrix 𝐃 encodes a polynomial fit of atomic energies to the nuclear charge, while the adjacency matrix 𝐀 corresponds to the Coulombic interactions between all pairs of atoms. Due to the use of only relative positions between atoms in the adjacency matrix, the Coulomb matrix is automatically invariant under translation and rotation. However, the matrix itself is not invariant under permutation, as swapping the order of two atoms will result in an exchange of the corresponding columns and the rows. To address this, the sorted list of eigenvalues of the Coulomb matrix can be used instead as a feature vector, while an ℓ_p norm can be used as a distance metric. In practice, due to the fact that the number of neighbor atoms may change, the shorter eigenvalue list is padded with zeros in a distance computation. Graph Approximated Energy (GRAPE):The GRAPE kernel evaluates the simultaneous random walks on the direct product of the two graphs representing two atomistic neighborhood configurations. Permutational invariance is achieved by choosing a uniform starting and stopping distribution across nodes of both graphs. However, the cost of distance computation between two graphs scales as 𝒪(N^2) with a one-time per-graph diagonalization cost of 𝒪(N^3).In the sections below, we present our new fingerprint algorithm, namely the Density-Encoded Canonically Aligned Fingerprint (DECAF). The paper is organized as follows: in Section <ref>, we introduce a robust algorithm that can determine canonical coordinate frames for obtaining symmetry-invariant projections; in Section <ref>, we present numerical recipes to use smoothed atomistic density fields as a fingerprint for molecular configuration; in Section <ref>, we demonstrate the capability of the fingerprint via examples involving the regression of atomistic potential energy surfaces; in Section <ref>, we discuss the connection between our algorithm and previously proposed ones; we conclude with a discussion in Section <ref>.§ LOCALIZED CANONICAL COORDINATE FRAME FOR ROTATIONALLY INVARIANT DESCRIPTION OF ATOMISTIC NEIGHBORHOOD §.§ Kernel Minisum Approach To improve model generalization while minimizing data redundancy, a fingerprint algorithm should be able to recognize atomistic structures that differ only by a rigid-body transformation or a permutation of atoms of the same element, and to extract feature vectors invariant under these transformations. As summarized in Table <ref>, a variety of strategies have been successfully employed by common fingerprint algorithms to achieve rotational invariance.However, these approaches do not provide a means for the acquisition of vector-valued quantities in a rotational invariant form. One approach is to only acquire and interpolate the potential energy, a scalar quantity, and then take the derivative of the regression model. This approach, however, triggers the need for methods to decompose the total energy among the atoms, which is a property of the entire system rather than individual atoms <cit.>.Another approach proposed by Li et al. <cit.> is to project vector quantities onto a potentially overcomplete set of non-orthogonal basis vectors obtained from a weighted sum of the atomic coordinate vectors:𝐕_k = ∑_i 𝐱_i exp[-( ‖𝐱_i ‖/ R_c)^p_k].However, the approach may suffer from robustness issues. For example, all of the 𝐕_k generated with different p_k will point in the same direction if the radial distance of the atoms are all equal. Further, the configuration with 4 atoms at (r cosε, r sinε),(0,r),(-r,0),(0,-r) leads to𝐕_k= c·[ (r cosε, r sinε) + (0,r) + (-r,0) + (0,-r) ]= c· r·( 1 - cosε, sinε).Thus, if ε gets close to zero, V_k will always point toward either (0,1) or (0,-1), even if the vector quantity of interest may point in other directions.Here, we present a robust kernel PCA-inspired algorithm for the explicit determination of a canonical coordinate frame, within which the projection of the atomistic neighborhood is invariant under rigid-body rotation. Furthermore, the canonical coordinate frame can be directly used to capture vector-valued quantities in a rotational-invariant form. Given N atoms with position 𝐱_1, …, 𝐱_N ∈ℝ^d, we first formulate the L_p PCA algorithm as an optimization problem where we seek a unit vector 𝐰^* that maximizes the sum of the projections:𝐰^*= argmax_‖𝐰‖ = 1 ∑_i=1^N| 𝐰^𝖳𝐱_i |^p= argmax_‖𝐰‖ = 1 ∑_i=1^N| r_i |^p | 𝐰^𝖳𝐞_i |^p,where r_i = ‖𝐱_i ‖ is the distance from the origin to atom i, 𝐞_i = 𝐱_i / r_i is the unit vector pointing toward atom i, respectively. The optimization process can only uniquely determine the orientation of a projection vector up to a line, because | 𝐰^𝖳𝐞| ≡| -𝐰^𝖳𝐞|. As a consequence, further heuristics are needed to identify a specific direction for the PCA vectors.To overcome this difficulty, we generalize the | r_i |^p term into a weight function g(r) of radial distance and the | 𝐰^𝖳𝐞_i |^p term into a bivariate kernel function κ(𝐰,𝐞) between two vectors. We then attempt to seek a unit vector 𝐰^* that minimizes the kernel summation:𝐰^* = argmin_‖𝐰‖ = 1 ∑_i=1^N g(r_i) κ( 𝐰, 𝐞_i ). In particular, we have found a square angle (SA) kernel and an exponentiated cosine (EC) kernel that perform well in practice:κ_SA(𝐰,𝐞)≐1/2arccos^2( 𝐰^𝖳𝐞 ),κ_EC(𝐰,𝐞)≐exp(-𝐰^𝖳𝐞).As shown in Figure <ref>, both kernels are minimal when 𝐰 and 𝐞 are parallel, and monotonically reach maximum when 𝐰 and 𝐞 are antiparallel. Intuitively, optimizing the minisum objective function generated by the SA kernel will yield a vector that, loosely speaking, bisects the sector occupied by the atoms. The EC kernel exhibits very similar behavior but leads to a smoother objective function. As shown in Figure <ref>, this allows for the determination of a projection vector without ambiguity, even if the atom configuration contains perfect symmetry.A major advantage of the kernel minisum approach versus L^p norm-based PCA, lies in its 1) robustness in the presence of structural symmetry; and 2) continuity of the resulting principal axes with respect to angular movement of the input data. As shown in Figure <ref>, kernel minisum is particularly suitable for atomistic systems where strong symmetries are common and the continuity against angular movement is desired. The minisum framework can also be used with other customized kernels to suit for the characteristics of specific application scenarios. §.§ Solving the Kernel Minisum Optimization Problems The optimization problem can be solved very efficiently using a gradient descent algorithm as detailed below. Square Angle: The objective function of the minisum problem using the square angle (SA) kernel isA_SA(𝐰) ≐1/2∑_i=1^N g(r_i) arccos^2( 𝐰^𝖳𝐞_i ).The gradient of A_SA with respect to 𝐰 is∇_𝐰 A_SA = ∑_i=1^N -g(r_i) arccos( 𝐰^𝖳𝐞_i ) /√( 1 - (𝐰^𝖳𝐞_i)^2 ) 𝐞_i.Note that arccos( 𝐰^𝖳𝐞_i ) /√( 1 - (𝐰^𝖳𝐞_i)^2 ) is singular when 𝐰∥𝐞_i. This can be treated numerically by replacing the removable singularities at 𝐰^𝖳𝐞_𝐢 = 1 with the left-limit lim_𝐰^𝖳𝐞_i → 1^-arccos( 𝐰^𝖳𝐞_i ) /√( 1 - (𝐰^𝖳𝐞_i)^2 ) = 1, while truncating the gradient at a finite threshold near the poles at 𝐰^𝖳𝐞_𝐢 = -1.A local minimum can be iteratively searched for with gradient descent while renormalizing 𝐰 after each iteration. Moreover, due to the locally quadratic nature of the objective function, we have found that the Barzilai-Borwein algorithm <cit.> can significantly accelerate the convergence at a minimal cost. The algorithm is presented in Alg. <ref>. Exponentiated Cosine:The objective function of the minisum problem using the exponentiated cosine (EC) kernel is:A_EC(𝐰) ≐∑_i=1^N g(r_i) exp(-𝐰^𝖳𝐞_i).The gradient of A_EC with respect to 𝐰 is∇_𝐰 A_EC = ∑_i=1^N -g(r_i) exp(-𝐰^𝖳𝐞_i) 𝐞_i.The gradient contains no singularity. However, it is not always locally quadratic or convex. This can cause the Barzilai-Borwein algorithm to generate negative step sizes and consequently divert the search towards a maximum. Luckily, this can be easily overcome by always using the absolute value of the step size generated by the Barzilai-Borwein algorithm. Such enforcement prevents the minimization algorithm from going uphill. The complete algorithm is given in Alg. <ref>.As shown in Table <ref>, both Alg. <ref> and Alg. <ref> converge quickly and consistently across a wide range of representative point configurations commonly found in molecular systems. However, the gradient descent method can only find local optima. Thus, multiple trials should be performed using different initial guesses to ensure that a global minimum can be located. §.§ Complete Set of Orthogonal Projection Vectors as A Canonical Coordinate Frame In 3D, a complete set of orthogonal bases can be found greedily using the protocol as described in Alg. <ref>. Specifically, we use the globally optimal solution of the minisum optimization problem as the first basis 𝐛_α, and the constrained optimal solution in a plane orthogonal to 𝐛_α as the second basis 𝐛_β. Special care must be taken for determining the third basis 𝐛_γ, as the only degree of freedom now is its sign due to the orthogonality constraint. The straightforward approach of choosing the direction that gives the smaller objective function value may fail, for example, when the system contains improper rotational symmetry. In that case, 𝐛_α and 𝐛_β are interchangeable and both perpendicular to the rotation-reflection axis. As a result, the two candidates of 𝐛_γ will both align with the rotation-reflection axis and are thus indistinguishable by kernel minisum. However, the projection of the atoms into the two seemingly equivalent coordinate frames are not identical, but rather mirror images of each other. Fortunately, this can be addressed by choosing the direction of the half-space, as created by the plane 𝐛_α-𝐛_α, that yields the smaller kernel objective function between the bisector of 𝐛_α and 𝐛_β versus the points lies in that half-space. This rule can also handle general situations with/without symmetry. It is difficult to prove global uniqueness of the kernel minisum solution given the non-convex nature of the exponentiated cosine and square angle kernels. In fact, it seems that the only kernel that allows analytical proof of solution uniqueness is κ_COM(𝐰,𝐞) ≐-𝐰^𝖳𝐞, whose solution simply corresponds to the weighted center of mass of the neighbor atoms. Unfortunately, this simple kernel is not robust against reflectional and rotational symmetry. Luckily, the rare cases where two global optimal solutions do coexist can be safely captured by the repeated searching procedure starting from different seeds. Thus, a fingerprint can be extracted using each of the resulting coordidate frame. This may mildly increase the size of the training set, which could even be advantagenous when training data is scarce.§ DENSITY-ENCODED CANONICALLY ALIGNED FINGERPRINT§.§ Density Field and Approximation of Volume Integral The local density field ρ_𝐬(𝐫) around a point 𝐬 is formulated as a superimposition of smoothing kernel functions each centered at a neighbor atom i=1,2,…,N with relative displacement 𝐱_i with regard to 𝐬 and within a cutoff distance R_c:ρ_𝐬(𝐫) = ∑_i,‖𝐱_i - 𝐬‖ < R_c^N (𝐱_i-𝐬) (𝐱_i - 𝐫)This density field, as has been pointed out previously <cit.>, may be used as a fingerprint of the local atomistic environment. Here, we assume that the smoothing kernel (𝐫) takes the form of a stationary Gaussian σ^-1 exp[ -1/2‖𝐫‖ ^2 / σ^2 ]. We also assume that the density scaling function (𝐫), which ensures the continuity of the density field when atoms enter or exit the cutoff distance, is a bell-shaped function with compact support. Further discussion on both (𝐫) and (𝐫) can be found in Section <ref> and Section <ref>, respectively.To achieve rotational invariance, we project the atom coordinates into the canonical coordinate frame 𝐑≐ [𝐛_α, 𝐛_β, 𝐛_γ] as determined by the kernel minisum algorithm, when generating the density field:ρ_𝐬(𝐫) = ∑_i,‖𝐱_i - 𝐬‖ < R_c^N (𝐑^𝖳𝐱_i-𝐬) (𝐑^𝖳𝐱_i - 𝐫).Depending on the specific application, 𝐬 may not necessarily overlap with any of the 𝐱_i. Scalar properties can be acquired directly from the target atom, while vector-valued properties, such as force, can be acquired and interpolated in the local orthogonal coordinates as ỹ = 𝐑^𝖳𝐲. We define the distance between two density fields ρ_i and ρ_j as a weighted volume integral of their pointwise difference:(𝐫)≐ρ_𝐬_1(𝐫) - ρ_𝐬_2(𝐫), d(ρ_𝐬_1,ρ_𝐬_2)≐( ∫_ℝ^3 w(𝐫) |(𝐫)|^2 dV(𝐫) )^1/2.The weight function w(𝐫) provides additional flexibility for emphasizing particular regions of the atomistic neighborhood. It could play an important role for fitting properties with steep gradients, e.g. the repulsive part of the Lennard-Jones potential.We now introduce an optimal quadrature rule to approximate the integral in Eq. <ref> in a computationally tractable manner. A quadrature rule is a numerical recipe in the form ∫ f(x) = ∑_i=0^N w_i f(x_i), which numerically approximates a definite integral using only discrete evaluations of the integrand. To determine the quadrature nodes and weights, we decompose the volume integral in Eq. <ref> into a surface integral over spherical shells and a 1D integral along the radial direction:∫_ℝ^3 w(𝐫) |(𝐫)|^2 dV(𝐫)= ∫_r=0^∞( ∫_φ=0^2π∫_θ=0^π w(r,θ,φ) |(r,θ,φ)|^2 sinθ dθ dφ ) r^2 drThe surface integral can be optimally approximated using the Lebedev quadrature rule <cit.>:(r) ≐∫_φ=0^2π∫_θ=0^π w(r,θ,φ) |(r, θ,φ)|^2 sinθ dθ dφ≈ 4π∑_m=1^N_a(r) w(r ·𝐪_m) β_m |(r ·𝐪_m)|^2,where β_m, 𝐪_m ≐ (x_m, y_m, z_m), and N_a are the weights, positional unit vectors, and number of the Lebedev nodes, respectively. The radial integral fits well into the generalized Laguerre-Gauss quadrature formula with weight function r^2 e^-r <cit.>:∫_0^∞(r) r^2 dr ≈∑_n=1^N_rα_n e^r_n (r_n),where α_n, r_n, and N_r are the weights, coordinates, and number of the Laguerre nodes, respectively. Combining Eq. <ref>–<ref>, a composite quadrature rule can be generated consisting of several spherical layers of nodes. As shown in Figure <ref>, the radial coordinates of the quadrature nodes are determined by the Laguerre quadrature nodes, while the angular positions are determined by the Lebedev quadrature nodes, respectively. This composite quadrature formula translates the 3D volume integral into a summation over discrete grid points:∫_ℝ^3 w(𝐫) |(𝐫)|^2 dV(𝐫)≈ 4 π∑_n=1^N_r∑_m=1^N_a(n)α_n β_m w( r_n ·𝐪_m ) e^r_n |(r_n ·𝐪_m)|^2.Using the right hand side of Eq. <ref> to replace the integral in Eq. <ref>, and use the multi-index notation k = (n,m); 1 ≤ n ≤ N_r, 1 ≤ m ≤ N_a(n) to enumerate over the quadrature nodes located at 𝐫_k = r_n ·𝐪_m with weights w_k = 4 π α_n β_m w(r_n·𝐪_m) e^r_n, we obtain the final discretized formula for computing the distance between the fingerprints:d(ρ_𝐬_1,ρ_𝐬_2)≈[ ∑_k=1^N w_k |ρ_𝐬_1(𝐫_k) - ρ_𝐬_2(𝐫_k)|^2 ]^1/2.For quick reference, we tabulated in Appendix the values for r_n and α_n in the Laguerre quadrature of up to 6 points, and the values for 𝐪_m, β_m in the Lebedev quadrature of up to 50 points.In addition, the quadrature nodes could be radially scaled such that the outer most nodes lie at a radius R^* within some cutoff distance R_c. This allows us to fit a Laguerre quadrature of any order within an arbitrary cutoff distance. The scaled quadrature rule is given by:τ = R^* / max_n(r_n), d^*(ρ_𝐬_1,ρ_𝐬_2)≈[ ∑_k=1^Nτ^3 w_k |ρ_𝐬_1(τ 𝐫_k) - ρ_𝐬_2(τ 𝐫_k)|^2 ]^1/2.Since the scaling is simply constant among all nodes, it can be safely omitted in many regression tasks where only the relative distance between the fingerprints are of significance. §.§ Radial Weight FunctionsIn this section, we examine two radial weight functions that can be used to fine-tune the density field fingerprint: the density scaling function (𝐫) as appears in Eq. <ref> and the weight of integral w(r) as appears in Eq. <ref>.Driven by the interest of reducing computational cost, we would like to use a cutoff distance to select atoms involved in constructing the density field. However, it is important to ensure that atoms will enter and exit the neighborhood smoothly. This naturally requests that the contribution of an atom to be zero outside of the cutoff, and to increase continuously and smoothly when the atom approaching entrance. Correspondingly,(𝐫) should: 1) become unity at the origin; 2) smoothly approach zero at the cutoff distance; and 3) be twice-differentiable to ensure the differentiability of regression models based on the fingerprint. Candidates satisfying the above conditions include, for example, a tent-like kernel(𝐫) = (1 - ‖𝐫‖ / R_c)^t, t>2and a bell-shaped polynomial kernel with compact support(𝐫) =-b (1 - ‖ r ‖ / R_c)^a + a (1 - ‖ r ‖ / R_c)^b /a-b, a>b>2as detailed in Appendix.The approximation of the radial integral using a Laguerre quadrature requires that the integrand, i.e. the pointwise difference between the atomistic density fields, decays sufficiently fast beyond the outermost quadrature nodes in order to achieve acceptable convergence. In addition, the steeply repulsive yet flat attractive interatomic short-range interactions prompt that the sensitivity of fingerprint be adjusted correspondingly in order to avoid numerical difficulties in training a regression model. The weight of the integral, w(r), provides a convenient means for achieving the purpose. Different from (𝐫), w(r) should instead satisfy the following conditions: 1) is normalized such that ∫ w(r) dV(𝐫) = 1; 2) decays sufficiently fast, but not necessarily to 0, beyond the outer most quadrature node; and 3) be sufficiently smooth beyond the outermost quadrature node. Candidates for w(r) includes the tent-like kernel and the bell-shaped kernel for (𝐫), albeit with a different normalization factor. The Laplacian kernelw(r) = exp(-| r | / l) / (8π l^3), l>0with a properly sized length scale l also appears to be a candidate due to its similarity with e^-r part of the weight function of the Laguerre quadrature. Note that the constant kernelw(r) = 3 / (4π R_c^3)may also be a choice as long as the density field already decays fast enough due to the density scaling function (𝐫).In Figure <ref>, we demonstrate the effect of the density scaling function and the weight of integral on the distance matrices between fingerprint obtained from a pair of atoms. The comparison between panel A and B shows that a bell-shaped integration weight allows the distance between fingerprints to change more rapidly when the atoms are closer but more slowly when the atoms are farther apart. The visible discontinuity in the second row clearly demonstrates the importance of the damping function when only atoms within a finite cutoff distance are used to compute the fingerprint.We further examine the impact of the weight functions on the performance of Gaussian process regression (GPR) using the fingerprint of the interatomic force of a minimal system containing two nitrogen atoms. Despite the simplicity of the system, this case is of fundamental importance because of its ubiquity, and because the fast-growing repulsive regime of the Lennard-Jones potential could cause difficulty as a small change in the system configuration can trigger large changes in the regression target function. In Figure <ref>, we compare the performance among the combination of four weights of theintegral and two density scaling functions. The initial training set consists of two samples collected at r_N-N = 1.0 and 6.0. The regression is then refined using a greedy strategy that consecutively learns the point with the largest posterior variance until the largest uncertainty, defined as twice the posterior standard deviation, is less than 0.1 eV/Å. The active learning scheme is able to delivery a GPR model, for each and every combination of the weight functions, that closely fits the target function. However, the numbers of refinement iterations and the achieved accuracy do vary. Therefore, it is important to evaluate and choose the weight functions in the context of specific application scenarios. §.§ Quadrature Resolution and Density KernelDespite the formal convergence of the composite quadrature in DECAF, a cost of 𝒪(NM) distance calculations and kernel evaluations are needed to sample a density field generated by N atoms using M quadrature nodes. A less prominent cost is associated with the L^2 distance calculation, which comes at a cost of 𝒪(M) floating point operations. Thus, in practice it is often desirable to use as few nodes as possible to capture only information of the density field within a certain band limit<cit.>. Accordingly, the integral cutoff R_c, the number of quadrature nodes, and the width of the density kernel need to be tuned to obtain an optimal balance between resolution and computational speed.When designing the composite quadrature rule, we chose the Laguerre quadrature for the radial direction because its nodes are denser near the origin but sparser farther away. This is consistent with our physical intuition that the near field generally has a stronger influence than the far field in an atomistic neighborhood. For example, the Van de Waals potential grows rapidly when atoms are in direct contact, but flattens out of the first coordinate shell. Accordingly, it may be possible for us to use sparser outer-layer grids to reduce the total number of quadrature nodes, while still keeping enough nodes in the inner layers to maintain the sensitivity of the quadrature toward close neighbors. Cooperatively, we can also use non-stationary Gaussian density kernels whose width dependent on the distance from the atom to the origin. In this way, the sparser nodes should still sufficiently sample the smoother far field. Wider kernels at remote atoms also reduce the total difference between the far fields of two fingerprints in a statistical sense. Thus, the contribution of the far field in the integral can be effectively tuned even though the weights on the quadrature nodes remain the same.In Figure <ref>, we demonstrate how a variable-resolution quadrature can be combined with a widening smoothing density kernel to simultaneously reduce the computational complexity and preserve the quality of the fingerprint. In column A, a dense grid is used to sample density fields generated by a wide smoothing length. By examining the distance matrices of fingerprints sampled during bond stretching and angular stretching movements, we note that the radial similarity decreases monotonically while the angular similarity changes nearly constantly. In column B, the number of quadrature nodes is kept the same, but the smoothing length is reduced as an attempt to increase fingerprint sensitivity. Better response in the near field of the radial direction is obtained, but the linearity in the far field in the angular direction is compromised. In column C, the fingerprint performs even worse due to the combination of a sparser quadrature grid and a small smoothing length. In column D, the performance recovered because we let the smoothing length parameter σ of the Gaussian density kernels (𝐫) depend on the distance from the origin to each atom, and simultaneously adjust the quadrature node density according to this pattern.§ DEMONSTRATION §.§ Method Regression tasks throughout this work are performed using Gaussian process regression (GPR), a nonlinear kernel method that treats training data points as discrete observations from an instantiation of a Gaussian process. Predictions are made using the posterior mean and variance of the joint distribution between the test data and the training data. One particular interesting property about Gaussian process is that the posterior variance may be interpreted as a measure of prediction uncertainty, which can be exploited to design active learning algorithms for sampling expensive functions. The actual computation used our own software implementation which was made publicly available on Zenodo <cit.>. We use the square exponential covariance kernel to compute the covariance, i.e. correlation, between the samples:k_SE(x,x') = σ^2 exp[ -1/2d(x, x')^2 / l^2 ]where x and x' are DECAF fingerprints, and d(x, x') the distance between norm as computed by Eq. <ref> or Eq. <ref>. The kernel is stationary, meaning that the covariance depends only on the relative distance between two samples but not their absolute position. The training process searches for the hyperparameters, i.e. the output variance σ and the length scale l, that maximizes the likelihood of the training data. A detailed tutorial on GPR can be found in Ref. <cit.>. An illustration on the complete workflow of using the density field fingerprint to perform regression tasks is given in Figure <ref>. §.§ Potential Energy Surface First, we attempt to fit the potential energy surface of a protonated water dimer system, in a head-to-head configuration, as a function of the oxygen-oxygen distance r_O-O and the dihedral angle φ between the two planes each formed by a water molecule. As shown in Figure <ref>A, the system contains an improperly rotational symmetry, which we wish to capture with the kernel minisum algorithm. A GPR model was seeded with 8 training points corresponding to the combinations of r_O-O=2.2, 2.4, 2.6, 2.8 and φ=0,π/2. Subsequently, an active learning protocol was used to greedily absorb points with the highest uncertainty into the training set. Despite that we restricted all training data to be within the subdomain φ <= π/2, as shown by Figure <ref>B and <ref>C, we are able to accurately reproduce the target function over the entire parameter space after a few active learning steps.The DECAF fingerprint used here is constructed with 3 spherical layers within a cutoff distance R_c of 6.0 Å, each consisting of 14, 26, and 38 Lebedev quadrature nodes, respectively. The weight of integral was chosen as w(r) = W^6,4( 1 - r / R_c ), where W^6,4 is the bell-shaped polynomial as defined in Appendix Eq. <ref>. The density scaling function (𝐫) = (1 - ‖𝐫‖ / R_c)^3, where 𝐫 is the vector from the atom to the fingerprint center, is the tent-like kernel as defined in Eq. <ref> with t = 3. The density kernel that sits on the oxygen atoms assumes the form of a non-stationary Gaussian as discussed in Section <ref>: ^O(𝐫, 𝐫') = σ_O(𝐫)^-1 exp[ -1/2‖𝐫' ‖ ^2 / σ_O(𝐫)^2 ], σ_O(𝐫) = 1.5 + 0.25 ‖𝐫‖ with 𝐫 being the vector from the atom to the fingerprint center and 𝐫' being the vector from the atom to the quadrature node. The density kernel for the hydrogen atoms has a different weight and width to ensure discriminability: ^H(𝐫, 𝐫') = 0.75 σ_H(𝐫)^-1 exp[ -1/2‖𝐫' ‖ ^2 / σ_H(𝐫)^2 ], σ_H(𝐫) = 0.9 + 0.15 ‖𝐫‖. §.§ Geometry Optimization and Vibrational Analysis Next, we demonstrate the usability of fingerprint for fitting vector-valued quantities by performing geometry optimization and vibrational analysis on a single water molecule. The process involves the simultaneous regression of: 1) energy, a molecular scalar quantity; 2) force, a per-atom vector quantity; and 3) dipole, a molecular vector quantity. Correspondingly, we performed GPR of energy and dipole using fingerprints extracted from the center of mass of the molecule, and GPR of force using fingerprints extracted from each atom. Each component of the vector properties is modeled independently as a scalar Gaussian process. The training set consists of 45 configurations uniformly covering the range r_O-H = 0.93, 0.95, 0.97, 0.99, 1.05 Å and θ_H-O-H = 101, 105.5, 111. As shown in Table <ref>, the GPR model can successful drive calculations of the infrared spectrum of the molecule from randomly perturbed initial structures in arbitrary orientation. The fingerprint configuration is the same as in the previous section. §.§ Molecular Dyanmics Trajectory As shown in Figure <ref>, here we attempt to fit for the forces felt by the atoms in a benzene molecule along the MD Trajectories as obtained from a sibling database of QM7 <cit.>. The density kernel for the carbon atoms assumes the same functional form with that of the oxygen atoms, but uses a different smoothing length function σ_C(𝐫) = 1.2 + 0.2 ‖𝐫‖. The rest of the parameters are inherited from the previous examples. The training configurations were chosen adaptively in an iterative process using the sum of the GPR posterior variance and the prediction error as the acquisition function. § CONNECTION TO OTHER FINGERPRINT ALGORITHMSIn Figure <ref>, we compare the ability to distinguish atomistic configurations of our fingerprint as well as SOAP and the Coulomb matrix. Our work is inspired by the SOAP descriptor <cit.>, which proposes the use of smoothed densities to represent atomistic neighborhoods. However, instead of converting the density field into the frequency domain using spherical harmonics, we perform density field sampling and comparison directly in the real space. This is enabled thanks to the available of canonical coordinate frame as computed through the kernel minisum optimization. We have mainly used the L_2 norm to compute the distance between atomistic neighborhoods. However, our fingerprint exhibits very similar behavior to SOAP when used together with an inner product formulad(ρ_𝐬_1,ρ_𝐬_2)≈∑_k=1^N w_k ρ_𝐬_1(𝐫_k) ρ_𝐬_2(𝐫_k) /√(∑_k=1^N w_k ρ_𝐬_1(𝐫_k) ρ_𝐬_1(𝐫_k))√(∑_k=1^N w_k ρ_𝐬_2(𝐫_k) ρ_𝐬_2(𝐫_k))as demonstrated in Figure <ref>A. Thus, our fingerprint could be used in conjunction with a wide variety of covariance functions based on either the Euclidean distance or the inner product similarity.At first sight, DECAF is very different from the Coulomb matrix fingerprint and GRAPE, which are both graph-based algorithms <cit.>. However, instead of trying to capture the overall density field, if we measure the contribution from each individual atom on the quadrature nodes at 𝐳_1,𝐳_2,…,𝐳_M as a row vector, and stacked up the results to yield the matrix𝐄_ij^N× M =k(𝐱_i, 𝐳_j).Then 𝐄 can be regarded as an incidence matrix <cit.> between atoms and the quadrature nodes. This is similar to the graph-based abstraction as seen in the Coulomb matrix and the GRAPE kernel. However, in both cases the vertices in the graph represent atoms while the edges represent pairwise interatomic interactions. Here, the density-based incidence matrix adopts the opposite pattern and constructs a graph with the quadrature nodes being vertices and atoms being edges. The adjacency matrix in this case is computed as the inner product 𝐄^𝖳𝐄:𝐀_ij^M× M = (𝐄^𝖳𝐄)_ij = ∑_k=1^Nk(𝐱_k, 𝐳_i) k(𝐱_k, 𝐳_j).The weight on the edges, as represented by the elements of the adjacency matrix A, can be interpreted as the total flux as contributed by all paths each bridged by an atom k. We have numerically found that the smallest N eigenvalues (except for the 0 eigenvalue) of the symmetric normalized Laplacian𝐋 = 𝐈 - 𝐃^-1/2𝐀𝐃^-1/2, where 𝐃_ii = δ_ij∑_j 𝐀_ijis invariant under rotation up to a certain noise level, even if the quadrature nodes do not rotate with the atoms. Nonetheless, this detour appears to represent a pure theoretical interest rather than any practical value.§ CONCLUSIONIn this paper, we presented the Density-Encoded Canonically Aligned Fingerprint (DECAF) by exploring the idea of using smoothed density fields to represent and compare atomistic neighborhoods. One of the key enabling technique in DECAF is a kernel minisum algorithm, which allows the unambiguous identification of a canonically aligned coordinate frame that can be used for rotationally invariant projection of the density field as well as any associated vector quantities. We have performed detailed analysis to study the behavior of the fingerprint by changing various parameter, such as resolution, smoothing length, and the choice of weight functions. We demonstrate that the fingerprint algorithm can be used to implement highly accurate regressions of both scalar and vector properties of atomistic systems including energy, force and dipole moment, and could be a useful building block for constructing data-driven next generation force fields to accelerate molecular mechanics calculations with an accuracy comparable to those driven by quantum mechanical theories and calculators.§ ACKNOWLEDGMENTThis work was supported by the Department of Energy (DOE) Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4). This work was also supported by the Army Research Laboratory under Cooperative Agreement Number W911NF-12-2-0023. bibstyle§ APPENDIX§.§ Polynomial Smoothing Functions with Compact Support As candidates for the weight of integral and density scaling functions (Section <ref>), a class of compact polynomials that satisfy the criteria <cit.>: * is compactly supported,* is strictly positive within some cutoff distance r_c,* decreases monotonically,* is at least twice continuously differentiablewith minimal number of non-zero terms are:W^a,b(s)= -b s^a + a s^b/σ, a > b > 2, where s = 1 - r / h is the normalized complementary coordinate within the span h of the kernel, andσ = 8 π h^3 ( a/b^3+6b^2+11b+6-b/a^3+6a^2+11a+6)is an optional normalization factor to ensure that the integral of the kernel in a 3D ball of radius h is unity. The parameters a and b are free parameters that can be used to adjust the smoothness and width of the kernel, and can take any real numbers satisfying the condition a>b>2. Note that the kernel W^4,3 is equivalent to the Lucy kernel commonly used in Smoothed Particle Hydrodynamics simulations <cit.>. The kernel can be evaluated very efficiently using only multiplication and addition when both a and b are integers. §.§ Table of Quadrature Nodes and Weights In Table <ref>, we list the nodes and weights of the Laguerre quadrature rules up to N_r = 6, using notations from Eq. <ref>. In Table <ref>, we list the nodes and weights of the Lebedev quadrature rules up to N_r = 6, using notations from Eq. <ref>. The Laguerre and Lebedev quadrature nodes can be combined using Eq. <ref>-<ref> into composite grids for sampling the atomistic density field. | http://arxiv.org/abs/1709.09235v3 | {
"authors": [
"Yu-Hang Tang",
"Dongkun Zhang",
"George Em Karniadakis"
],
"categories": [
"cs.CE",
"physics.chem-ph",
"physics.comp-ph"
],
"primary_category": "cs.CE",
"published": "20170926194932",
"title": "An Atomistic Fingerprint Algorithm for Learning Ab Initio Molecular Force Fields"
} |
FoodNet: Recognizing Foods Using Ensemble of Deep Networks Paritosh Pandey^*, Akella Deepthi^*, Bappaditya Mandal and N. B. Puhan P. Pandey, A. Deepthi and N. B. Puhan are with the School of Electrical Science, Indian Institute of Technology (IIT), Bhubaneswar, Odisha 751013, India. E-mail: {pp20, da10, nbpuhan}@iitbbs.ac.in B. Mandal is with the Kingston University, London, Surrey KT1 2EE, United Kingdom. Email: [email protected] ^* Represents equal contribution from the authors. =============================================================================================================================================================================================================================================================================================================================================================================================================================================================== In this work we propose a methodology for an automatic food classification system which recognizes the contents of the meal from the images of the food. We developed a multi-layered deep convolutional neural network (CNN) architecture that takes advantages of the features from other deep networks and improves the efficiency. Numerous classical handcrafted features and approaches are explored, among which CNNs are chosen as the best performing features. Networks are trained and fine-tuned using preprocessed images and the filter outputs are fused to achieve higher accuracy. Experimental results on the largest real-world food recognition database ETH Food-101 and newly contributed Indian food image database demonstrate the effectiveness of the proposed methodology as compared to many other benchmark deep learned CNN frameworks. Deep CNN, Food Recognition, Ensemble of Networks, Indian Food Database. § INTRODUCTION AND CURRENT APPROACHESThere has been a clear cut increase in the health consciousness of the global urban community in the previous few decades. Given the rising number of cases of health problems attributed to obesity and diabetes reported every year, people (including elderly, blind or semi-blind or dementia patients) are forced to record, recognize and estimate calories in their meals. Also, in the emerging social networking photo sharing, food constitutes a major portion of these images. Consequently, there is a rise in the market potential for such fitness apps products which cater to the demand of logging and tracking the amount of calories consumed, such as <cit.>. Food items generally tend to show intra-class variation depending upon the method of preparation, which in turn is highly dependent on the local flavors as well as the ingredients used. This causes large variations in terms of shape, size, texture, and color. Food items also do not exhibit any distinctive spatial layout. Variable lighting conditions and the point of view also lead to intra-class variations, thus making the classification problem even more difficult <cit.>. Hence food recognition is a challenging task, one that needs addressing.In the existing literature, numerous methodologies assume that the texture, color and shape of food items are well defined <cit.>. This may not be true because of the local variations in the method of food preparation, as well as the ingredients used. Feature descriptors like histogram of gradient, color correlogram, bag of scale-invariant feature transform, local binary pattern, spatial pyramidal pooling, speeded up robust features (SURF), etc, have been applied with some success on small datasets <cit.>. Hoashi et al. in <cit.>, and Joutou et al. in <cit.> propose multiple kernel learning methods to combine various feature descriptors. The features extracted have generally been used to train an SVM <cit.>, with a combination of these features being used to boost the accuracy.A rough estimation of the region in which targeted food item is present would help to raise the accuracy for cases with non-uniform background, presence of other objects and multiple food items <cit.>. Two such approaches use standard segmentation and object detection methods <cit.> or asking the user to input a bounding box providing this information <cit.>. Kawano et al. <cit.> proposed a semi-automated approach for bounding box formation around the image and developed a real-time recognition system. It is tedious, unmanageable and does not cater to the need of full automation. Automatic recognition of dishes would not only help users effortlessly organize their extensive photo collections but would also help online photo repositories make their content more accessible. Lukas et al. in <cit.> have used a random forest to find discriminative region in an image and have shown to under perform convolutional neural network (CNN) feature based method <cit.>.In order to improve the accuracy, Bettadapura et al. in <cit.> used geotagging to identify the restaurant and search for matching food item in its menu. Matsuda et al. in <cit.> employed co-occurrence statistics to classify multiple food items in an image by eliminating improbable combinations. There has been certain progress in using ingredient level features <cit.> to identify the food item. A variant of this method is the usage of pairwise statistics of local features <cit.>. In the recent years CNN based classification has shown promise producing excellent results even on large and diverse databases with non-uniform background. Notably, deep CNN based transferred learning using fine-tuned networks is used in <cit.> and cascaded CNN networks are used in <cit.>. In this work, we extend the CNN based approaches towards combining multiple networks and extract robust food discriminative features that would be resilient against large variations in food shape, size, color and texture. We have prepared a new Indian food image database for this purpose, the largest to our knowledge and experimented on two large databases, which demonstrates the effectiveness of the proposed framework. We will make all the developed models and Indian food database available online to public <cit.>. Section II describes our proposed methodology and Section III provides the experimental results before drawing conclusions in Section IV. § PROPOSED METHODOur proposed framework is based on recent emerging very large deep CNNs. We have selected CNNs because their ability to learn operations on visual data is extremely good and they have been employed to obtain higher and higher accuracies on challenges involving large scale image data <cit.>. We have performed extensive experiments using different handcrafted features (such as bag of words, SURF, etc) and CNN feature descriptors. Experimental results show that CNNs outperform all the other methods by a huge margin, similar to those reported in <cit.> as shown in Table <ref>. It can be seen that CNN based methods (SELC & CNN) features performs much better as compared to others. §.§ Proposed Ensemble Network ArchitectureWe choose AlexNet architecture by Krizhevsky et al. <cit.> as our baseline because it offers the best solution in terms of significantly lesser computational time as compared to any other state-of-the-art CNN classifier. GoogLeNet architecture by Szegedy et al. <cit.> uses the sparsity of the data to create dense representations that give information about the image with finer details. It develops a network that would be deep enough, as it increases accuracy and yet have significantly less parameters to train. This network is an approximation of the sparse structure of a convolution network by dense components. The building blocks called Inception modules, is basically a concatenation of filter banks with a mask size of 1×1, 3×3 and 5×5. If the network is too deep, the inception modules lead to an unprecedented rise in the cost of computation. Therefore, 1×1 convolutions are used to embed the data output from the previous layers.ResNet architecture by He et al. <cit.> addresses the problem of degradation of learning in networks that are very deep. In essence a ResNet is learning on residual functions of the input rather than unreferenced functions. The idea is to reformulate the learning problem into one that is easier for the network to learn. Here the original problem of learning a function H(x) gets transformed into learning non-linearly by various layers fitting the functional form H(x) = Γ(x) + x, which is easier to learn, where the layers have already learned Γ(x) and the original input is x. These CNN networks are revolutionary in the sense that they were at the top of the leader board of ImageNet classification at one or other time <cit.>, with ResNet being the network with maximum accuracy at the time of writing this paper. The main idea behind employing these networks is to compare the increment in accuracies with the depth of the network and the number of parameters involved in training. Our idea is to create an ensemble of these classifiers using another CNN on the lines of a Siamese network <cit.> and other deep network combinations <cit.>. In a Siamese network <cit.>, two or more identical subnetworks are contained within a larger network. These subnetworks have the same configuration and weights. It has been used to find comparisons or relationships between the two input objects or patches. In our architecture, we use this idea to develop a three layered structure to combine the feature outputs of three different subsections (or subnetworks) as shown in Fig. <ref>. We hypothesize that these subnetworks with proper fine-tuning would individually contribute to extract better discriminative features from the food images. However, the parameters along with the subnetwork architectures are different and the task is not that of comparison (as in case of Siamese network <cit.>) but pursue classification of food images. Our proposition is that the features once added with appropriate weights would give better classification accuracies.Let I(w,h,c) represents a pre-processed input image of size w× h pixels to each of the three fine-tuned networks and c is the number of channels of the image. Color images are used in our case. We denote C(m,n,q) as the convolutional layer, where m and n are the sides length of the receptive field and q is the number of filter banks. Pooling layer is denoted by P(s,r), where r is the side length of the pooling receptive field and s is the number of strides used in our CNN model. In our ensemble net we did not use pooling. But in our fine-tuned networks pooling is employed with variable parameters. GoogLeNet for example uses overlapping pooling in the inception module. All convolution layers are followed by ReLU layers (see the text in Sec <ref>) considered as an in-built activation. L represents the local response normalization layer. Fully connected layer is denoted by F(e), where e is the number of neurons. Hence, the AlexNet CNN model after fine-tuning is represented as: Φ_A ≡ I(227,227,3) ⟶ C(11,4,96) ⟶ L ⟶ P(2,3) ⟶ C(5,1,256) ⟶ L ⟶ P(2,3) ⟶ C(3,1,384) ⟶ C(3,1,384) ⟶ C(3,1,256) ⟶ P(2,3) ⟶ F(4096) ⟶ F(4096)⟶ F(e). AlexNet is trained in a parallel fashion, referred as a depth of 2. Details of the architecture can be found in <cit.>. For GoogLeNet we need to define the inception module as: D(c1, cr3, c3, cr5, c5, crM), where c1, c3 and c5 represent number of filter of size 1×1, 3×3 and 5×5, respectively. cr3 and cr5 represent number of 1×1 filters used in the reduction layer prior to 3×3 and 5×5 filters, and crM represents the number of 1×1 filters used as reduction after the built in max pool layer. Hence GoogLeNet is fine-tuned as: Φ_G ≡ I(224,224,3) ⟶ C(7,2,64) ⟶ P(2,3) ⟶ L ⟶ C(1,1,64)⟶ C(3,1,192) ⟶ L ⟶ P(2,3) ⟶ D(64,96,128,16,32,32) ⟶ D(128,128,192,32,96,64) ⟶ P(2,3) ⟶ D(192,96,208,16,48,64) ⟶ D(160,112,224,24,64,64) ⟶ D(128,128,256,24,64,64) ⟶ D(112,144,288,32,64,64) ⟶ D(256,160,320,32,128,128) ⟶ P(2,3) ⟶ D(256,160,320,32, 128,128)⟶ D(384,192,384,48,128,128) ⟶ P^*(1,7) ⟶ F(e), P^* refers to average pooling rather than max pooling used everywhere else. For fine-tuned ResNet, each repetitive residual unit is presented inside as R and it is defined as: Φ_R ≡ I(224,224,3) ⟶ C(7,2,64) ⟶ P(2,3) ⟶ 3 × R(C(1,1,64) ⟶ C(3,1,64) ⟶ C(1,1,256)) ⟶ R(C(1,2,128) ⟶ C(3,2,128) ⟶ C(1,2,512)) ⟶ 3 × R(C(1,1,128) ⟶ C(3,1,128) ⟶ C(1,1,512)) ⟶ R(C(1,2,256) ⟶ C(3,2,256) ⟶ C(1,2,1024)) ⟶ 5× R(C(1,1,256) ⟶ C(3,1,256) ⟶ C(1,1,1024)) ⟶ R(C(1,2,512) ⟶ C(3,2,512) ⟶ C(1,2,2048)) ⟶ 2× R(C(1,1,512) ⟶ C(3,1,512) ⟶ C(1,1,2048)) ⟶ P^*(1,7) ⟶ F(e). Batch norm is used after every convolution layer in ResNet. The summations at the end of each residual unit are followed by a ReLU unit. For all cases, the length of F(e) depends on the number of categories to classify. In our case, e is the number of classes. Let F_i denote the features from each of the fine-tuned deep CNNs given by (<ref>)-(<ref>), where i∈{A, G, R}. Let the concatenated features are represented by Ω(O,c), where O is the output features from all networks, given by:O=concatenate(w_i F_i)|∀ i,where w_i is the weight given to features from each of the networks with the constraint, such that Σ_i w_i = 1. We define the developed ensemble net as the following:Φ_E ≡ Ω(e*η,c) ⟶ ReLU ⟶ F(e) ⟶ SoftMax,where η is the number of fine-tuned networks. The SoftMax function or the normalized exponential function is defined as:S(F)_j=exp^F_j/∑_k=1^e exp^F_k, for j=1, 2, …, e,where exp is the exponential. The final class prediction D∈{1, 2, …, e} is obtained by finding the maximum of the values of S(F)_j, given by:D=max_j(S(F)_j), for j=1, 2, …, e.§.§ Network Details The ensemble net we designed consists of three layers as shown in Fig. <ref>. Preprocessed food images are used to fine-tune all the three CNN networks: AlexNet, GoogLeNet and ResNet. Then the first new layer one concatenates the features obtained from the previously networks, passing it out with a rectified linear unit (ReLU) non-linear activation. The outputs are then passed to a fully connected (fc) layer that convolves the outputs to the desired length of the number of classes present. This is followed by a softmax layer which computes the scores obtained by each class for the input image.The pre-trained models are used to extract features and train a linear kernel support vector machine (SVM). The feature outputs of the fully connected layers and max-pool layers of AlexNet and GoogLeNet are chosen as features for training and testing the classifiers. For feature extraction, the images are resized and normalized as per the requirement of the networks. For AlexNet we used the last fully connected layer to extract features (fc7) and for GoogLeNet we used last max pool layer (cls3_pool). On the ETH Food 101 database, the top-1 accuracy obtained remained in the range of 39.6% for AlexNet to 44.06% for GoogLeNet, with a feature size varying from a minimum of 1000 features per image to 4096 features per image. Feature length of the features extracted out of the last layer is 1000. The feature length out of the penultimate layer of AlexNet gave a feature length of 4096 features, while the ones out of GoogLeNet had a feature length of 1024. All the three networks are fine-tuned using the ETH Food-101 database. The last layer of filters is removed from the network and replaced with an equivalent filter giving an output of the size 1×1×101, i.e., a single value for 101 channels. These numbers are interpreted as scores for each of the food class in the dataset. Consequently, we see a decrease in the feature size from 1×1000 for each image to 1×101 for each image. AlexNet is trained for a total of 16 epochs.We choose the MatConvNet <cit.> implementation of GoogLeNet with maximum depth and maximum number of blocks. The implementation consists of 100 layers and 152 blocks, with 9 Inception modules (very deep!). To train GoogLeNet, the deepest softmax layer is chosen to calculate objective while the other two are removed. The training ran for a total of 20 epochs. ResNet's smallest MatConvNet model with 50 layers and 175 blocks is used. The capacity to use any deeper model is limited by the capacity of our hardware. The batch size is reduced to 32 images for the same reason. ResNet is trained with the data for 20 epochs. The accuracy obtained increased with the depth of the network. The ensemble net is trained with normalized features/outputs of the above three networks. Parametrically weights are decided for each network feature by running the experiments multiple times. A total of 30 epochs are performed. A similar approach is followed while fine-tuning the network for Indian dataset. As the number of images is not very high, jitters are introduced in the network to make sure the network remains robust to changes. Same depth and parameters are used for the networks. The output feature has a length of 1×1×50 implying a score for each of the 50 classes.§ EXPERIMENTAL SETUP AND RESULTSThe experiments are performed on a high end server with 128GB of RAM equipped with a NVDIA Quadro K4200 with 4GB of memory and 1344 CUDA cores. We performed the experiments on MATLAB 14a using the MatConvNet library offered by vlFeat <cit.>. Caffe's pre-trained network models imported in MatConvNet are used. We perform experiments on two databases: ETH Food-101 Database and and our own newly contributed Indian Food Database. §.§ Results on ETH Food-101 DatabaseETH Food-101 <cit.> is the largest real-world food recognition database consisting of 1000 images per food class picked randomly from foodspotting.com, comprising of 101 different classes of food. So there are 101,000 food images in total, sample images can be seen in <cit.>. The top 101 most popular and consistently named dishes are chosen and randomly sampled 750 training images per class are extracted. Additionally, 250 test images are collected for each class, and are manually cleaned. Purposefully, the training images are not cleaned, and thus contain some amount of noise. This comes mostly in the form of intense colors and sometimes wrong labels to increase the robustness of the data. All images are rescaled to have a maximum side length of 512 pixels. In all our experiments we follow the same training and testing protocols as that in <cit.>.All the real-world RGB food images are converted to HSV format and histogram equalization are applied on only the intensity channel. The result is then converted back to RGB format. This is done to ensure that the color characteristics of the image does not change because of the operation and alleviate any bias that could have been present in the data due to intensity/illumination variations. Table <ref> shows the Top-1, Top-5 and Top-10 accuracies using numerous current state-of-the-art methodologies on this database. We tried to feed outputs from the three networks into the SVM classifier but the performance was not good. We have noted only the highest performers, many more results can be found in <cit.>. It is evident that with fine-tuning the network performance has increased to a large extent. Fig. <ref> (a) shows accuracies with the ranks plot up to top 10, where the rank r:r ∈{1, 2, …, 10} shows corresponding accuracy of retrieving at least 1 correct image among the top r retrieved images. This kind of graphs show the overall performance of the system at different number of retrieved images. From Table <ref> and Fig. <ref> (a), it is evident that our proposed ensemble net has outperformed consistently all the current state-of-the-art methodologies on this largest real-world food database.§.§ Results on Indian Food DatabaseOne of the contributions of this paper is the setting up of an Indian food database, the first of its kind. It consists of 50 food classes having 100 images each. Some sample images are shown in Fig. <ref>. The classes are selected keeping in mind the varied nature of Indian cuisine. They differ in terms of color, texture, shape and size as the Indian food lacks any kind of generalized layout. We have ensured a healthy mix of dishes from all parts of the country giving this database a true representative nature. Because of the varied nature of the classes present in the database, it offers the best option to test a protocol and classifier for its robustness and accuracy. We collected images from online sources like foodspotting.com, Google search, as well as our own captured images using hand-held mobile devices. Extreme care was taken to remove any kind of watermarking from the images. Images with textual patterns are cropped, most of the noisy images discarded and a clean dataset is prepared. We also ensured that all the images are of a minimum size. No upper bound on image size has been set. Similar to the ETH Food-101 database protocol, we have randomly selected 80 food images per class for 50 food classes in the training and remaining in the test dataset.Fig. <ref> (b) shows accuracies with the ranks plot up to top 10 and Table <ref> shows the Top-1, Top-5 and Top-10 accuracies using some of the current state-of-the-art methodologies on this database. Both these depict that our proposed ensemble of the networks (Ensemble Net) is better at recognizing food images as compared to that of the individual networks. ResNet under performs as compared to GoogLeNet and AlexNet probably because of the lack of sufficient training images to train the network parameters. For overall summary: as is evident from these figures (Fig. <ref> (a) and (b)) and tables (Tables <ref> and <ref>) that there is no single second best method that outperforms all others methods in both the databases, however, our proposed approach (Ensemble Net) outperforms all other methods consistently for all different ranks in both the databases. § CONCLUSIONSFood recognition is a very crucial step for calorie estimation in food images. We have proposed a multi-layered ensemble of networks that take advantages of three deep CNN fine-tined subnetworks. We have shown that these subnetworks with proper fine-tuning would individually contribute to extract better discriminative features from the food images. However, in these subnetworks the parameters are different, the subnetwork architectures and tasks are different. Our proposed ensemble architecture outputs robust discriminative features as compared to the individual networks. We have contributed a new Indian Food Database, that would be made available to public for further evaluation and enrichment. We have conducted experiments on the largest real-world food images ETH Food-101 Database and Indian Food Database. The experimental results show that our proposed ensemble net approach outperforms consistently all other current state-of-the-art methodologies for all the ranks in both the databases. IEEEtran | http://arxiv.org/abs/1709.09429v1 | {
"authors": [
"Paritosh Pandey",
"Akella Deepthi",
"Bappaditya Mandal",
"N. B. Puhan"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170927101331",
"title": "FoodNet: Recognizing Foods Using Ensemble of Deep Networks"
} |
An Efficiently Searchable Encrypted Data Structure for Range QueriesFlorian Kerschbaum University of WaterlooWaterloo, Ontario, CanadaEmail: [email protected] Anselme Tueno SAP Karlsruhe, Germany Email: [email protected] ===============================================================================================================================================================================================At CCS 2015 Naveed et al. presented first attacks on efficiently searchable encryption, such as deterministic and order-preserving encryption. These plaintext guessing attacks have been further improved in subsequent work, e.g. by Grubbs et al. in 2016. Such cryptanalysis is crucially important to sharpen our understanding of the implications of security models. In this paper we present an efficiently searchable, encrypted data structure that is provably secure against these and even more powerful chosen plaintext attacks. Our data structure supports logarithmic-time search with linear space complexity. The indices of our data structure can be used to search by standard comparisons and hence allow easy retrofitting to existing database management systems.We implemented our scheme and show that its search time overhead is only 10 milliseconds compared to non-secure search. § INTRODUCTION At CCS 2015 Naveed et al. <cit.> presented attacks on order-preserving encryption. Later Grubbs et al. <cit.> improved the precision of these attacks. Further attacks on searchable encryption have been presented <cit.>. Such cryptanalysis is crucially important to sharpen our understanding of the implications of security models, since many of the attacked encryption schemes are proven secure in their specific security models. In this paper we formalize security against these attacks and show a connection to chosen plaintext attacks. We also demonstrate that there exists an encrypted data structure that supports efficient range queries by regular comparisons and that provably prevents these attacks.Comparison using regular comparison operators (e.g. greater-than) as enabled by our scheme and order-preserving encryption has many practical benefits. These encryption schemes can be retrofitted to any existing database management system making them extra-ordinarily fast, flexible and easy-to-deploy. We preserve this property as our implementation demonstrates, but some minor modifications to the search procedure are necessary.Efficiency – logarithmic time and linear space complexity – is also an important property of search over encrypted data. In Table <ref> we provide a comparison of our scheme to the most secure and efficient order-preserving schemes <cit.>, order-revealing encryption <cit.> and range-searchable encryption <cit.> schemes. No searchable encryption scheme – including ours – offers perfect security and efficiency for all functions (equality and range search, insertions, deletions, etc.). It is a research challenge to balance the trade-off between the two objectives, even for a restricted set of functions. We aim at provable security against the recently publicized plaintext guessing attacks while still enabling efficient range search.In this respect, we achieve a novel and preferable trade-off between security and efficiency.In the construction of our scheme we borrow the ideas of previous order-preserving encryption schemes: modular order-preserving encryption by Boldyreva et al. <cit.>, ideal secure order-preserving encoding by Popa et al. <cit.> and frequency-hiding order-preserving encryption by Kerschbaum <cit.>. We assign a distinct ciphertext for each – even repeated – plaintext as Kerschbaum does, but his scheme statically leaks the partial insertion order. So, we compress the randomized ciphertexts to the minimal ciphertext space using Popa et al.'s interactive protocol. Then we rotate around a modulus as Boldyreva et al., but on the ciphertexts and not on the plaintexts.As a result we achieve structural independence between the ciphertexts and plaintexts which is a prerequisite for security against chosen plaintext attacks and plaintext guessing attacks – particularly, if the adversary has perfect background knowledge on the distribution of plaintexts. We formalize this insight as a novel security model (-security) for efficiently searchable, encrypted data structures and we prove our scheme secure in this model. Our security model encompasses a number of recently publicized attacks where attackers broke into cloud system and stole the stored data. Such an attack will reveal no additional information when data is encrypted with our scheme. This will also thwart the attacks by Naveed et al. <cit.> and Grubbs et al. <cit.> mentioned at the beginning of the introduction. The implementation of our scheme shows only 10 milliseconds overhead compared to non-secure search on a database with a million entries. In summary our contribution are as follows: * We formulate a security notion that provably prevents chosen plaintext attacks and plaintext-guessing attacks as those by Naveed et al. and Grubbs et al. Our model provides provable security against attackers with (one-time) snapshot access to the encrypted data as in the most common attacks on cloud computing. * We present an efficiently searchable, encrypted data structure that supports range queries and fulfills this security notion. Our search scheme is retrofittable into existing database management systems and we provide a prototypical implementation. * We evaluate the performance of our scheme in a prototypical implementation. Our scheme shows only roughly 10 milliseconds overhead compared to non-secure search.The remainder of the paper is structured as follows. In the next section we define what we mean by an efficiently searchable, encrypted data structure. In Section <ref> we present and motivate our new security model preventing plaintext guessing attacks. Then, we present our efficiently searchable, encrypted data structure secure in this model in Section <ref>. We evaluate the performance of the implementation of our scheme in <ref>. Finally, we review related work in Section <ref> and summarize our conclusions in Section <ref>.§ EFFICIENTLY SEARCHABLE ENCRYPTED DATA STRUCTURESFirst, we define what we mean by a efficiently searchable encrypted data structure (). We start by defining what we mean by a data structure. We use the fundamental representation of a data structure in random-access memory, i.e. an array. Each cell of the array consists of a structured element. We do not impose any restriction on the structure of the element, but usually this element contains two parts: the data to be searched over and further structural information, such as indices of further entries. Note that structural information may be implicit, i.e. the index where an element is stored itself is structural information albeit not explicitly stored. This implicit structural information may also not be encrypted, but only randomized. An example of explicit structural information are the indices of the cells of the two children in a binary search tree which would be stored in a cell's structure in addition to the data of a tree node. Explicit structural information can be encrypted. We write [j] for the j-th element and if it is clear from the context, we assume it consists only of a ciphertext of the data element (with j being the implicit structural information). A data structureconsists of an array of elements [j] (0 ≤ j < n).For an encrypted data structure there are a number of options on the type of encryption. First, we can choose symmetric or public-key encryption. We can instantiate our encrypted data structure with either one. Letbe a probabilistic symmetric encryption scheme consisting of three – possibly probabilistic – polynomial-time algorithms = (), (, m), (, c). Letbe a probabilistic public-key encryption scheme consisting of three – possibly probabilistic – polynomial-time algorithms = (), (, m), (, c). Let () be a deterministic algorithm that derives the public key from the private key in a public-key encryption scheme. For symmetric key encryption letbe the identity function. Let ∈{, } and we usewhen we leave the choice of encryption scheme open. is defined in Section <ref>. Second, we can either encrypt the data structure as a whole or parts of the data structure – ideally each cell. Our requirement of efficient search rules out the first option. Since in this case each search operation would require decrypting the data structure which is at least linear in the ciphertext size, sublinear search is impossible. Hence, we require each cell to be encrypted as a separate ciphertext.[In case several cells of a simple data structure are encrypted as a whole, we call this combination a cell of another data structure.]Third, for data security it may only be necessary to encrypt the data elements of a cell and not the structural information. In fact, our own proposedis an instance of such a case where the structural information is implicit from the array structure and unencrypted. Hence, we only require the data element of each cell to be encrypted. An encrypted data structure _ consists of an array of elements [j] where at least the data part has been encrypted with .We can now define the operations on a searchable encrypted data structure _. We writewhen the choice encryption scheme is clear from the context. Furthermore we denote sometimes denote the version h (after h insertions) of a data structure as ^h. Our definition is for range-searchable encrypted data structures, but this implies a definition for keyword searchable data structure as well (where the range parameters are equal: a=b). Furthermore, we do not define how operations on ourare to be implemented. These operations can be implemented as algorithms running on a single machine or protocols distributed over a client and server (hiding the secret key from the server). Both choices are covered by our definition. A searchable encrypted data structure _ offers the following operations. * (): Generates a – either secret or private – keyfrom the encryption schemeaccording to the security parameter . * ^h+1(, ^h, m):Encrypts the plaintext m using .((), m) and inserts it into the data structure ^h resulting in data structure ^h+1.[Note that in case of public key encryption our definition does not imply that the entire operation can be completed using only the public key.] * m := (, [j]): Computes the plaintext m for the data part of encrypted cell [j] using key . * { j_0, …, j_ℓ-1} := (, , a, b):Computes the set of indices { j_0, …, j_ℓ-1} for the range [a, b] on the encrypted data structureusing key .For the correctness of encryption we expect in a sequence of operations (, ^0, m_0), …, (, ^n-1, m_n-1) resulting in data structure ^n that ∀ i∃ j m_i = (, ^n[j]). For the correctness of search we expect that for any { j_0, …, j_ℓ-1} := (, , a, b), it holds that ∀ j∈{ j_0, …, j_ℓ-1}⟹(, [j]) ∈ [ a, b ] and∀ j∈{ j | (, [j]) ∈ [ a, b ] }⟹ j ∈{ j_0, …, j_ℓ-1}.We can now finally define an efficiently searchable encrypted data structure. An efficiently searchable encrypted data structureis a searchable encrypted data structure where the running time τ ofis poly-logarithmic in n (plus the size of the returned set of matching ciphertext indices) and the space σ ofis linear in n:τ() ≤polylog(n) + ℓ σ() = n It is now clear that efficient search prevents encrypting the entire data structure and thereby achieving semantic () security. Next, we give our definition of security that implies that each cell's data is encrypted with a semantically secure encryption scheme. Our security definition also prevents all plaintext guessing attacks of the type of Naveed et al. and Grubbs et al. Furthermore, we show that even when the data structure consists of only one semantically secure ciphertext in each cell, this does not guarantee security against these plaintext guessing attacks.§ SECURITY OF Before we define the security of anwe will review recent attacks on cloud infrastructures and searchable encryption scheme to motivate our security model. Particular we review in depth plaintext guessing attacks that only need a (multi-)set of ciphertexts as input (and do not perform active attacks during encryption or search operations). We try to generalize these attacks and show that even if all elements in anare semantically secure encrypted, this does not imply that these attacks are infeasible. §.§ Motivation Our model is motivated by recent attacks on cloud infrastructures and order-preserving or deterministic encryption. Not only the theoretic demonstrations, but also real world incidents show the risks of deterministic – not even order-preserving – encryption. In at least one case passwords were encrypted using a deterministic algorithm and many subsequently broken <cit.>. The cryptanalysis was performed on stolen ciphertexts only (using additional plaintext hints). Many other hacking incidents have been recently publicized, e.g. <cit.>, that resulted in leakage of sensitive information – not necessarily ciphertexts.All these attacks share a common “anatomy”. The hackers are capable to break in, access and copy sensitive information. They used the opportunity of access to gain as much data as possible in a short time, i.e. the adversary obtains a static snapshot. Note that this does not rule out the revelation of more sophisticated, longitudinal attacks in the future, but underpins the pressure to secure our current systems.In this respect our model achieves the following: An attacker gaining access to all ciphertexts stored in an encrypted database does not gain additional information to his background knowledge. We assume even perfect background knowledge, i.e. the adversary has chosen all plaintexts. This may sound contradictory at first – why would someone break into a database which data he knows. However, if we are able to show security against such strong adversaries, security holds even if the adversary has less, e.g. imperfect, background knowledge. §.§ Transformation toMost commonly plaintext guessing attacks are performed on multi-sets of deterministic or order-preserving ciphertexts which do not impose an order as ourdo. However, there exists a natural connection between these encryption schemes and . We present transformations that turn deterministic or order-preserving encryption schemes into anas in Definition <ref> with equivalent leakage. Any attack successful on these encryption schemes will be successful on the corresponding .Our transformation for deterministic encryption is loosely based on the data structure in <cit.>. Letbe the multi-set of plaintexts andbe the set of distinct plaintexts. We denote the size of a multi-set 𝕄 as |𝕄| and the number of occurrences of element m in multi-set 𝕄 as #_𝕄m. Let m̃_i be i-th distinct plaintext and hence #_m̃_i be the number of elements m̃_i in . We also denote these elements as m_i, h for h = 0, …, #_m̃_i - 1. Let p_i,h be the index of m_i, h inand p_i,h = -1, if h ≥#_m̃_i. Since deterministic encryption can be stored in a relational table, we use the row identifiers id_i,h of each ciphertext as the document identifiers and the data m̃_i as the keywords. Let _ be a keyed, pseudo-random function that maps the domain of keywords onto the size n of the .[For ease of exposition we assume no collisions.] Then [_(m̃_i)] ⦜.(, m̃_i), .(, id_i,0), p_i, 1For each data m_i, h where 0 < h < #_m̃_i we store[p_i,h] ⦜.(, m̃_i), .(, id_i,h), p_i, h+1One reveals _(m̃_i), accesses the corresponding bucket (cell) in the data structure and then traverses the list for efficient (keyword) search.For deterministic order-preserving encryption we can use a similar transformation as above, but use the order order(m̃_i) of the plaintext as the element index.Instead of hashing the keyword into a bucket, one can use binary search for efficient search on this .[order(m̃_i)] ⦜.(, m̃_i), .(, id_i,0), p_i, 1In frequency-hiding order-preserving encryption, we no longer have a list of identical ciphertexts, but each ciphertext is unique. Then we can use the randomized order rand-order(m_i,h) where elements are sorted, but ties are broken based on the outcome of a coin flip as defined in <cit.> as the element index. However, we no longer need to store the row-identifier, since each ciphertext is unique and can be found in the relational table.[rand-order(m_i,h)] .(, m̃_i) Theseare susceptible to the same plaintext guessing attacks as those by Naveed et al. <cit.> and Grubbs et al. <cit.> on the respective encryption schemes. We next review these attacks on these encryption schemes. §.§ Plaintext Guessing AttacksNaveed et al. <cit.> present a series of attacks on deterministic and order-preserving encryption. They attack deterministic order-preserving encryption by Boldyreva et al. <cit.>. Grubbs et al. improved the precision of the attacks and also extended them to other order-preserving and order-revealing encryption. Their new attacks are not fundamentally different, but improve the matching algorithm between the assumed and measured frequency.However, Grubbs et al. present thefirst attack on frequency-hiding order-preserving encryption (FH-OPE) – the “bucketing” attack. Letbe the multi-set (a multi-set may potentially include duplicate values) of ciphertexts andbe the multi-set of plaintexts in the background knowledge of the adversary. We assume that the sizes of the multi-sets are equal: n = || = ||.§.§.§ Frequency AnalysisThe frequency analysis attack first computes the histograms Hist() and Hist() of the two multi-sets. Then it sorts the two histograms in descending order: c⃗ := Sort(Hist()) and m⃗ := Sort(Hist()). The cryptanalysis for c_i is m_i, i.e. the two vectors are aligned.Naveed et al. implement the frequency analysis as the l_P-optimization attack. Lacharite and Paterson show that frequency analysis is expected to be the optimal cryptanalysis <cit.>, but also that l_P-optimization is expected to be close to this optimimum.In the l_P-optimization attack the two histograms are not simply sorted and aligned, but a global minimization is run to find an alignment. Let 𝕏 be the set of n × n permutation matrices. The attack then finds X ∈𝕏, such that the distance || c⃗ - X m⃗ ||_P is minimized under the l_P distance. For many distances l_P the computation of X can be efficiently (polynomial in n) performed using an optimization algorithm, such as linear programming. The cryptanalysis for c_i is X[m]_i, i.e. the two vectors are aligned after permutation. The attack works not only for order-preserving encryption, but also for deterministic encryption. The attack is very successful in experimentally recovering hospital data – even for such deterministic encryption. Naveed et al. report an accuracy of 100% for 100% and 95% of the hospitals for the binary attributes of “mortality” (whether a patient has died) and “sex”, respectively, under deterministic encryption. §.§.§ Sorting Attack Let 𝔻 be the domain of all plaintexts in multi-set . Let N = |𝔻| be the size of the domain 𝔻. The sorting attack assumes thatis dense, i.e. contains a ciphertext c for each m ∈𝔻. The adversary computes the unique elements Unique() and sorts them c⃗ := Sort(Unique()) and the domain d⃗ := Sort(𝔻). The cryptanalysis for c_i is d_i, i.e. the order of the ciphertext and the plaintext are matched.The attack is 100% accurate, if the ciphertext multi-set is dense. This is a strong assumption, but already Naveed et al. present a refinement that works also for low-density data. This cumulative attack combines the l_P-optimization and sorting attack. The adversary first computes the histograms c⃗_1 := Hist(), m⃗_1 := Hist() and the cumulative density functions c⃗_2 := CDF(), m⃗_2 := CDF() of the cipher- and plaintexts. The attack then finds the permutation X ∈𝕏, such that the sum of the distances between the histograms and cumulative density functions || c⃗_1 - X m⃗_1 ||_P + || c⃗_2 - X m⃗_2 ||_P is minimized. Again, this can be done using efficient optimization algorithms. The cryptanalysis for c_i is X[m]_i.The attack is very accurate against deterministic order-preserving encryption as demonstrated by Naveed et al. They report an accuracy of 99% for 83% of the large hospitals for the attributes of “age”. The age column is certainly not low-density, but also not dense (as the success rate shows).Grubbs et al. further improve the algorithms in this attack by using bipartite matching. They report an accuracy for their improved attacks of up to 99% on first names and up to 97% for last names which have much more entropy than age.§.§.§ Bucketing Attack on FH-OPEThe extension of the sorting attack – the bucketing attack – on FH-OPE proceeds as follows. The adversary sorts the multi-sets c⃗ := Sort() and m⃗ := Sort(), i.e. it is not necessary to only use unique values. The cryptanalysis for c_i is m_i. Note that in FH-OPE every element c_i ∈ is unique, but after the attack aligned to the cumulative density function ofas in the cumulative attack.Grubbs et al. recover 30% of first names and 6% of last names in their data set. However, it can be even more accurate depending on the precision of the background knowledge . It can be very dangerous to make assumptions about the adversary's background knowledge, since they are hard, if not impossible, to verify and uphold. Hence, in our -security model forwe assume perfect background knowledge of the adversary, i.e. the multi-setis the exact same multi-set as the plaintexts of the ciphertexts. In fact, the multi-setis chosen by the adversary in a chosen plaintext attack. The bucketing attack then succeeds with 100% accuracy.The focus of this paper is preventing these plaintext guessing attacks on efficiently searchable encryption. However, attacks using stronger adversaries, e.g. active modifications or insertions, have been presented in the scientific literature <cit.>. §.§ Security DefinitionWe give our security definition as an adaptation of semantic security to data structures. We show that our adaptation implies that each data value is semantically secure encrypted. However, we also show that even if all cells consist of only one semantically secure ciphertext, our adaptation is not necessarily fulfilled.First, recall the definition of semantic security.A public-key encryption schemehas indistinguishable encryptions under a chosen-plaintext attack, or is -secure, if for all PPT adversariesthere is a negligible functionsuch that , :=| Exp_,^() = 1 - 12| ≤ [center] Exp_,^()⦜,.()⦜m_0, m_1,(, ) b c .(, m_b) b' (, , c, ) b = b' We note that -security only considers a single ciphertext whereas a data structure consists of multiple ciphertexts and hence some structural information. Exactly this structural information can be used in plaintext guessing attacks and we need to adapt semantic security to all ciphertexts. We call our adaptation indistinguishability under chosen-plaintext attacks for data structures or -security for short. Loosely speaking, our security model ensures that an adversary who has chosen all plaintexts encrypted in a data structure cannot guess the plaintext of any ciphertext better than a random guess. Recall that we denote the size of multi-set 𝕄 as |𝕄| and the number of occurrences of element m in multi-set 𝕄 as #_𝕄m. An efficiently searchable encrypted data structureis indistinguishable under a chosen-plaintext attack, or is -secure, if for all PPT adversariesthere is a negligible functionsuch that , :=| Exp_,^() = ⦜1, p - p| ≤ [center] Exp_,^() ⦜,.() ⦜_0, _1,(, ) |_0| ≠|_1|b := m ∈_b .(, m, ) ⦜j', m' (, , , ) ⦜.(, [j']) = m', #__0 ∪_1m'|_0 ∪_1| There are two differences between -security and -security. First, the adversary chooses two multi-sets of plaintexts as input to the challenge instead of two single plaintexts. This enables the adversary to create different situations to distinguish. Assume the adversary returns two disjoint multi-sets as _0 and _1, e.g. _0 = {0, 0} and _1 = {1,1}. Then it can attempt to distinguish which of the two plaintext multi-sets have been encrypted by guessing any plaintext in the data structure. Assume the adversary returns the same multi-set as _0 and _1, but with distinct plaintexts in the (identical) multi-set, e.g. _0 = _1 = {0,1}.This is admissible in the definition of -security, since the only requirement is that the two multi-sets are of the same size. The adversary can then attempt to distinguish at which position in the data structure each plaintext has been encrypted.In order to enable the adversary to win the game when the position in the data structure is not indistinguishable, we made a second change to -security: The adversary's guess is the plaintext of a single ciphertext at any position in the data structure. Hence, the adversary does not necessarily have to distinguish between the two plaintext multi-sets, it is sufficient, if it guesses correctly within the choice of sets (which may be equal). However, even if the position in the ciphertext is indistinguishable, in order to win the adversary only has to guess correctly with a probability non-negligibly better than the frequency of the plaintext in the union of the multi-sets. Hence, if the two multi-sets are not equal and the adversary can guess the chosen multi-set, it can win the game.We next explain the implications of -security and first prove that -security implies -security. Our proof assumes the use of public-key encryption, but the proof for symmetric encryption is analogous using an encryption oracle. We prove this by turning an adversarythat has advantage ϵ in experiment Exp_,^ into an adversarythat has advantage ϵ in experiment Exp_,_^.If _ is -secure, then each ciphertext of the data element in [j] (0 ≤ j < n) must be from a -secure encryption scheme.Letbe an adversary that has advantage ϵ in experiment Exp_,^. We construct an adversaryfor experiment Exp_,_^ as follows.[center](, )⦜m_0, m_1, ' (, )_0 := { m_0 }_1 := { m_1 }:= ' { _0, _1 }⦜_0, _1,(, , , )' { _0, _1 } :=b' (, , [0], ')⦜0, _b'[0] The adversary 's view is indistinguishable from experiment Exp_,^ If adversaryguesses correctly, then 's output is also correct. Hence, if 's advantage is ϵ, then 's advantage is ϵ. However, we also prove that even if each cell inconsists of a single ciphertext from a -secure, public-key encryption scheme , then this does not imply -security. We prove by giving a data structure that consists of a single ciphertext from , but that is not -secure. Again, the proof for symmetric encryption is analogous.If each ciphertext in an efficiently searchable, encrypted data structure _ [j] (0 ≤ j < n) is from a -secure encryption scheme , then _ is not necessarily -secure.Given a multi-set of plaintextsand a -secure, public-key encryption scheme , we construct a data structure as follows. Let rand-order(m_i) be the randomized order of each plaintext m_i ∈. Recall that in a randomized order of a multi-set, elements are sorted, but ties are broken based on the outcome of a coin flip.[rand-order(m_i)] .(, m_i) This data structure has equivalent leakage to frequency-hiding order-preserving encryption (FH-OPE) by Kerschbaum <cit.>. It is easy to see that each cell of the data structure consists of only one semantically secure ciphertext. However, we construct an adversary that succeeds with probability 1 for p = 12 in our experiment Exp_,_^. [center](, ):= { 0, 1 }⦜,(, , , )⦜0, 0 The adversary always wins the game, since in the given encryption scheme plaintext 0 will always be encrypted at position 0. Grubbs et al. showed in <cit.> the practicality of the attack by constructing a plaintext guessing attack – the bucketing attack described in Section <ref> – on FH-OPE. In their experiments it succeeds with probability 30% where the base line guessing probability is only 4%. §.§.§ Relation to Other Security Definitions In searchable encryption a security definition of indistinguishability under chosen-keyword attack (-security) has been defined in <cit.> and used in many subsequent works. Loosely speaking, this security definition states that the data structure is -secure, if it is indistinguishable from a simulator given (a set of) leakage function(s) . However, this can be misleading, since the leakage function does not necessarily clearly state the impact on plaintext guessing attacks. We first state the following corollary:If a public-key encryption schemeis -secure, then there exists a simulator _(,), such that for all PPT adversariesand all PPT distinguishers, ,:=| (c, ) = 1 : ⦜c, RealExp_,^() - .. (c, ) = 1 : ⦜c, SimExp__,^()|≤ [center] RealExp_,^()⦜,.()⦜m_0, m_1,(, )bc .(, m_b)c, SimExp__,^()⦜,.()c _(,)c,It follows that there exists a simulator for an encrypted data structure whose cells consists only of semantically secure ciphertexts which requires a leakage function of only the length n of the data structure and the public key . However, as we have shown in Theorem <ref> such a data structure may not be -secure and susceptible to plaintext guessing attacks.An efficiently searchable, encrypted data structure _ may be indistinguishably simulated with a leakage function = {, n } and be susceptible to plaintext guessing attacks.Consider the data structure from the proof of Theorem <ref>. It is indistinguishable from n ciphertexts produced using public keyand successful plaintext guessing attacks have been shown by Grubbs et al. in <cit.>. Hence, leaking the number of plaintexts may be sufficient for a successful plaintext guessing attack in a simulation-based security proof. Our -security model prevents this by introducing a structural independence constraint. While Curtmola et al. have been careful not to make this mistake in <cit.> and theiris -secure, subsequent work was not as careful. Boelter et al.'s data structure <cit.> has a (correct) simulation-based proof and is not -secure and susceptible to plaintext guessing attacks.[This is easy to see, since they do not encrypt the structural information in their data structure, i.e. the pointers to leaf nodes in the tree, and hence the ciphertexts can be ordered.]§.§.§ Impact on Plaintext Guessing Attacks We can now revisit the plaintext guessing attacks on deterministic and order-preserving encryption. First, our security model fully captures the attack setup. The adversary is given full ciphertext information and can chose the plaintexts such that it has perfect background knowledge[Recall that the adversary is allowed to submit the same plaintext multi-sets in the -security experiment], i.e. the adversary in our model has at least the same information as was used in those attacks. Second, our security definition implies that if the adversary is then able to infer even one plaintext better than with negligible probability over guessing our scheme is broken. Hence security in themodel implies security against all (passive, ciphertext-only) plaintext guessing attacks.§ AN -SECUREFOR RANGE QUERIESWe next present our efficiently searchable, encrypted data structure for range queries that is -secure. We emphasis that using the result from the data structure we can perform range queries in any commodity database management system without modifications. Hence, our data structure is as easy to integrate as order-preserving encryption, yet it is secure against chosen-plaintext attacks. We begin by describing the system architecture and give the intuition of our construction. We then present our encryption algorithm and interactive search protocol. §.§ System Architecture We depict an overview of our architecture in Figure <ref>. In our setup we assume a client holding the secret key .() and a server that holds the data structure . The server may hold several data structures managed independently for each database column, but needs to take care of correlation attacks as in <cit.>. A database table then contains the rows linking the entries by their index in the data structure.After encrypting the plaintexts, the client and the server can interactively perform a search query, e.g. a range query, on the server's data structure which results in two indices j, j'.Then these two indices j, j' can be used in subsequent range queries on the database management system. We assume that the server is semi-honest, i.e. only performs passive attacks. This model is commonly assumed in the scientific literature on database security. §.§ IntuitionOur data structure combines the ideas of three previous order-preserving encryption schemes: First, the scheme by Popa et al. <cit.> provides the basis for managing the order of ciphertexts in a stateful, interactive manner. Of course, this scheme is not secure against the attacks by Naveed et al., since it is deterministic and ordered. Second, we add the frequency-hiding aspect of the scheme by Kerschbaum <cit.>. The scheme itself cannot be used as the basis of an -secure data structure, since it partially leaks the insertion order. Therefore the frequency-hiding idea needs to be fit into Popa et al.'s scheme. We do this by encrypting the plaintext using a probabilistic algorithm (similar to the stOPE scheme in <cit.>) and also inserting a ciphertext for each plaintext using Kerschbaum's random tree traversal. This combined construction would still not besecure. Third, we apply Boldyreva et al.'s modular order-preserving encryption idea <cit.>. This idea rotates the plaintexts around a modulus statically hiding the order. However, modular order-preserving encryption has been developed for deterministic order-preserving encryption. In our probabilistic encryption – as introduced by Kerschbaum – we need to apply the modulus on the ciphertexts. This can be done by updating the modulus after encryption.In summary, intuitively our encryption scheme works as follows: We maintain a list of ciphertexts for each plaintext (including duplicates) sorted by the plaintexts on the server. However, the list is rotated around a random offset (chosen uniformly from the range between 1 and the number of ciphertexts). We then encrypt and search using binary search. However, due to the rotation which can divide a set of identical plaintexts adjacent in the list into a lower and upper part, the search and encryption algorithms become significantly more complex which is apparent in their detailed description below. §.§ Encryption AlgorithmLetbe a standard, probabilistic encryption scheme supporting the following three – possibly probabilistic – polynomial-time algorithms: ,and . We use symmetric encryption, e.g. AES in CBC or GCM mode, for speed, but assume an encryption oracle in the definition of semantic security. Let 𝔻 be the domain of plaintexts and N = |𝔻| its size. We now describe the algorithms and protocols of our efficiently searchable, encrypted data structure: * (): Execute .(). * ^h+1(, m, ^h): We denote ^h asfor brevity, if it is clear from the context. First the client and server identify the index j_m where m is to be inserted (before). Then the client sends the ciphertext of m to the server which inserts it at position j_m. Finally the server rotates the data structure by a random offset. * The client sets l := 0 and u := n - 1. * If n = 0 then go to step <ref>. * The client requests [0] and sets r := (, [0]). * Set j := ⌊ l + u - l/2⌋. The client requests [j] and executes m' := (, [j]). If m' - rN > m - rN, then the client sets l := j + 1. If m' - rN < m - rN, then the client sets u := j. If m' = mN, then the client flips a random coin and sets either l := j + 1 or u := j depending on the outcome of the coin flip. The client repeats this step until l = u. *The client sends c .(, m) to the server. * The server sets[n] :=[n-1][n-1] :=[n-2]⋯ [l+1] :=[l][l] := cThe server sets n := n+1. * The server chooses a random number s ℤ_n-1. The server sets the new encrypted data structure to ^h+1[j] := [j+sn] for 0 ≤ j < n as a result of the encryption operation. This data structure ^h+1 will be used as input to the next encryption operation. * ⦜j, j' := (, , a, b): Wlog. we assume that a ≤ b in the further exposition. In case a > b the query is rewritten as to match all x, such that 0 ≤ x < ba ≤ x < N.Let j_min(v) be the minimal index of plaintext v and j_max(v) be the maximal index of plaintext v.j_min(v) := min(j | (, [j]) = v) j_max(v) := max(j | (, [j]) = v) If j_min(v) = 0 and j_max(v) = n - 1 and there are two distinct plaintexts in the data structure, then we redefine as j_min(v) := j + 1 | (, [j]) < v (, [j+1]) = v) j_max(v) := j - 1 | (, [j]) > v (, [j-1]) = v) If a and b do not span the modulus, i.e. j_min(a) < j_max(b), then a query for x ∈ [a, b] is rewritten to j_min(a) ≤ x ≤ j_max(b). Else, it is rewritten to 0 ≤ x < j_max(b)j_min(a) ≤ x < n'.Both j_min(a) and j_max(b) are found using a separately run, interactive binary search. We next present this protocol. * The client sets l := 0 and u := n - 1. * The client requests [0], [n-1] and sets r := (, [0]). If (, [0]) = (, [n-1]) and searching for j_min(a), it sets r := r + 1. * Set j := ⌊ l + u - l/2⌋. The client requests [j] and executes m := (, [j]). If m - rN < a - rN (or m - rN ≤ b - rN, respectively) then the client sets l := j + 1. Else the client sets u := j. The client repeats this step until l = u. * The client returns j_min(a) := l (or j_max(b) := u, respectively). * m := (, [j]): Set m := .(, [j]). §.§ Security Our efficiently searchable, encrypted data structure _ is -secure. Since all cells of the data structure consists only of ciphertexts from a -secure encryption scheme, we can replace the encrypted data structure by a simulator.Let the simulator _^E_(, n) output n ciphertexts c E_(0). The adversarycannot distinguish the following experiment Exp_,_,_^ from experiment Exp_,_^ except with negligible probability.[center] E_(m) c .(, m) c Exp_,_,_^()⦜,.()⦜_0, _1,^E_()|_0| ≠|_1| _^E_(, |_0|)⦜j', m' ^E_(, , )⦜.([j']) = m', #__0 ∪_1m'|_0 ∪_1|The adversaryin Exp_,_,_^ clearly has no information which plaintext multi-set has been encrypted or about the plaintexts' positions in the data structure. Since in our _ each plaintext has equal probability of being at any index within the data structure, the adversary can at best guess the index j' for any m' ∈. However, the probability of a successful guess is bounded by #__0 ∪_1m'|_0 ∪_1|.§.§ Implementation In order to allow efficient online encryptions, we employ a technique we call decoupled encryptions in our implementation that however temporarily violates -security. A decoupled encryption has the positive effect that an encryption operation returns control almost instantly to the client. First, we store the index j explicitly in a database table along with the ciphertexts. Second, we choose a large domain D for the index, e.g. 256 bit. When we insert a new plaintext m into the data structure, we search for the element [j'] before and the element [j”] after the new element as described before. Then we insert m as [⌊j”-j'2⌋ + j'] = .(k, m). This operation is constant time, however after multiple encryptions the adversary may distinguish the data structures for two distinct sets of plaintexts.To restore security, we operate a background process in the database management system. This background process scans the entire data structure and makes the indices of all neighbouring data cells equidistant. For example, let there be n ciphertexts in the data structure and let |D| be the size of the domain of the index. Then the background process assigns the indices ⌊|D|n+1⌋, 2 ·⌊|D|n+1⌋, 3 ·⌊|D|n+1⌋, … to the data cells. The background process also rotates the data structure around a new random number r. After the background process completes the data structure is -secure.The background process can run incrementally and independently of queries. This allows it to be scheduled adaptively to the load of the database system. Hence, decoupled encryptions allow efficient search and online encryption operations while reaching -security eventually.§ PERFORMANCE EVALUATIONWe prototypically implemented and in a number of experiments evaluated the performance our -secure . In this section we report the results of our experiments measuring the run-time of range searching over encrypted data. §.§ Implementation We used Java for our implementation and evaluation, since many multi-tier applications are implemented in Java. Although a native cryptographic library, such as Intel's AES-NI, promises further performance improvements, programming languages such as C or C++ are more commonly used for systems software (such as database management systems) rather than for database applications (which only issue database queries). However, in our setup encryption and decryption is performed in the database application. We used Oracle's Java 1.8 and all experiments were run on the Java SE 64-Bit Server virtual machine. The database backend was the MySQL replacement MariaDB in version 10.1. When using a database, such as MariaDB, that was not specifically developed for operation on encrypted data, one needs to configure it to prevent the attacks on configuration described by Grubbs et al. <cit.>. All experiments were run on a single machine with a 4-core Intel i7 CPU at 2.9 GHz and 16 GB of RAM on Windows 10 Enterprise. §.§ Experimental Setup We measure the run-time of a typical, simply structured (i.e. a single search term and no conjunctions or disjunctions) database query on a single ordered database column, e.g. a range query or a top-k query. We use synthetic data and queries. However, we adapt our choice of parameters to the data from the DBLP data set. In the spirit of Grubbs et al. <cit.> we considered author names. At the time of our experiments there were about 1.500.000 million distinct author names in DBLP, the most frequent of which appears roughly 80 times.We implement the client interface as it would be used in an application using a database. The application supplies the parameters, e.g. the start a and end b of a range or the k in top-k, and receives the results in plaintext. Thus, our measured run-time includes thealgorithm, the standard query by the database management system and the decryption of the result. We emphasize that in more complex queries, e.g. including multiple search terms combined by conjunction and disjunctions, the relative time for executing the query on the database management system would be proportionally higher. Hence, our experiments put an upper bound on the worst case of the proportional overhead.Our target quantity in our measurements is the absolute run-time in milliseconds. For range queries we measure the dependence of the run-time on different parameters.* Size of the database:We vary the database size from 100.000 to 1.000.000 plaintexts in steps of 100.000, i.e. data items before encryption. * Size of the queried range: We vary the range size and consequently the result set size in the query from 10 to 100 in steps of 10.For top-k queries we measure the dependence of the run-time of the following parameter.* k:We vary the limit k from 10 to 100 in steps of 10.We compare the run-time on encrypted data to the run-time on plaintext data. Note that queries on plaintext only need to execute the query on the database management system, i.e. the time for thealgorithm and decryption of results is 0.We use synthetically generated data and queries. We uniformly choose distinct plaintexts and we uniformly choose a begin of the range query and then compute the end using the fixed size parameter of the experiment.We repeat each experiment 30 times discarding the first 10 experiments in order to allow to adjust the Java JIT compiler. We report the mean and 95% confidence interval for each parameter setting. §.§ Results Database size: Figure <ref> shows the running time over the database size. We use a query range size of 10. The database size increases from 100.000 to 1.000.000 plaintexts in steps of 100.000. The running time is measured in milliseconds. The error bars show the 95% confidence interval. Since our search algorithms run in sub-linear time only a very slight increase (20%) in running time is measurable compared to the increase in database size (900%). The overhead of our encryption is roughly 9 milliseconds. Query Range Size: Figure <ref> shows the running time over the query range size. We use a database size of 1.000.000 plaintexts. The query range size and hence the expected result set size increases from 10 to 100 in steps of 10. The running time is measured in milliseconds. The error bars show the 95% confidence interval. The running time increase is slight and approximately linear in the query range size and there is a constant baseline. We attribute the constant cost to our binary search algorithm which as shown in Figure <ref> behaves almost constant for these database sizes. We attribute this increase to the cost of decryption which is dominated by the cryptographic operations.Top-k queries: Figure <ref> shows the running time for top-k queries over k. We use a database size of 1.000.000 plaintexts. The value of k increases from 10 to 100 in steps of 10. The running time is measured in milliseconds. The error bars show the 95% confidence interval. The constant baseline is lower, since top-k queries can be executed without a search algorithm, when the minimum ciphertext (the rotation value) is stored as part of the key. The linear increase due to decryption of results is now clearly visible. §.§ Discussion We observe an almost constant overhead for the search algorithm of about 9 milliseconds. Then, decryption and filtering is linear in the result size. However, for reasonable result set sizes – up to 100 ciphertexts in our experiments – it stays below 2 milliseconds. Note that decryption is unavoidable in searching over encrypted data and often excluded in other scientific work. The database query time is not measurably affected by our scheme.Hence, we conclude that our -secure data structure adds an overhead of roughly 10 milliseconds (per encrypted column in the query) for reasonable result sizes. This is a very good performance compared to other schemes for range queries over encrypted data <cit.>.§ RELATED WORKOur work is related to other order-preserving encryption schemes, searchable encryption schemes – particularly for range queries –, order-revealing encryption, leakage-abuse attacks and other encryption schemes that in principle can be used to perform range queries. §.§ Order-Preserving Encryption Order-preserving encryption was introduced by Agrawal et al. in <cit.>. The idea is based on running queries using unmodified database management systems using deterministic encryption by Hacigümüs et al. <cit.>. However, Agrawal et al. extended it to range queries. Their original proposal uses an informal security model. Later, Boldyreva et al. <cit.> formalized the security and presented a new construction. They define indistinguishability under ordered chosen plaintext attack (). Note that -security leaks the order of plaintexts by the ciphertexts and is hence strictly less secure than -security. They also show that with constant local storage (the key only) -security requires exponentially sized ciphertexts and therefore settle for a weaker notion. Again, Boldyreva et al. further improve this definition in <cit.>. In this paper, they also introduce modular order-preserving encryption. Mavroforakis et al. show how to improve the security of modular order-preserving encryption against query observation attacks in <cit.>. They introduce fake queries to hide the modulus, but this only works for uniformly distributed plaintexts as shown by Durak et al. <cit.>. An improved security model and construction requiring only constant local storage was introduced by Teranishi et al. in <cit.>. Their idea is to occasionally introduce larger gaps into the ciphertexts. However, this also does not yet achievesecurity and has not been tested against the attacks by Naveed et al. Hwang et al. present a performance improvement for this encryption scheme in <cit.>. They show how to use a more efficient random sampling.The firstsecure order-preserving encryption was presented by Popa et al. in <cit.>. It also forms the firstfor order-preserving encryption, since it imposes a data structure beyond a single ciphertext. They introduce the concept of storing the state (symmetrically encrypted) on the server and make ciphertexts (necessarily) mutable, i.e. adapting to insertions. Schröpfer and Kerschbaum improve the performance of this model in <cit.>. Kerschbaum introduces an even stronger security model – indistinguishability under frequency-analyzing ordered chosen plaintext attack – in <cit.>. We build upon his idea and incorporate the concept of frequency-hiding into our data structure. Roche et al. combine FH-OPE by Kerschbaum with on-demand sorting, i.e. when searches are performed <cit.>. While their encryption is strongly secure before any queries, it deteriorates after queries even on the stored data structure and hence is less secure than . There are many more order-preserving encryption schemes that have been proposed in the literature <cit.> which we do not discuss here, since they lack a formal security analysis. §.§ Searchable Encryption Searchable encryption allows the comparison of a token (corresponding to a plaintext) to a ciphertext. The ciphertext (without any token) is -secure. The token can match plaintexts for equality or the plaintext to a range. Only the secret key holder can create tokens.The concept of searchable encryption has been introduced by Song et al. in <cit.>. It supports equality searches and additions, but requires linear time for searching, since each ciphertext needs to be compared. In order to speed up search an encrypted inverted index can be built. This inverted index is an , since it imposes a data structure. The first encrypted inverted index for equality search was presented by Curtmola et al. in <cit.>. It is an efficiently searchable, encrypted data structure. It supports (expected) constant time search, but all plaintexts (the inverted index) need to be encrypted at once and additions are not supported. Dynamic searchable encryption <cit.> made the data structure mutable in order to support additions. Since then a number of dynamic searchable encryption schemes with indexes have been proposed <cit.>. A recent survey provides a good overview <cit.>.Tackling range queries with searchable encryption is more complex. The first proposal by Boneh and Waters in <cit.> had ciphertext size linear in the size of the domain of the plaintext. The first poly-logarithmic sized ciphertexts scheme was proposed by Shi et al. in <cit.>. However, their security model is somewhat weaker than standard searchable encryption. The construction is based on inner-product predicate encryption which has been made fully secure by Katz et al. in <cit.>. All schemes follow the construction by Song et al. (without inverted indices) and require linear search time. The first attempt to build range-searchable encryption into an index (an ) has been made by Lu in <cit.>. However, the inverted index tree reveals the pointers and is hence no more secure than order-preserving encryption. Demertzis et al. <cit.> map a range query to keyword queries by providing tradeoffs between storing replicated values in each of its ranges and enumerating all values within range query. The search can then be easily performed using the data structure of Curtmola et al. <cit.>. While the scheme is range searchable, its queries are very revealing and it has high storage cost (at least O(n log n)). Boelter et al. <cit.> use garbled circuits to implement the search within a node of the index. They do not encrypt the pointers in the index and are hence susceptible to the attacks by Naveed et al. and are notsecure. The scheme by Hahn and Kerschbaum <cit.> creates an index using the access pattern of the range queries. No other information is leaked, however, this provides amortized poly-logarithmic search time. The scheme is only -secure as long as no queries have been performed (and the index has been partially built). Their scheme is based on inner-product predicate encryption which is too slow for practical use. §.§ Order-Revealing Encryption Order-revealing encryption <cit.> is an alternative to order-preserving encryption. Instead of preserving the order there is a public function that reveals the order of two plaintexts using the ciphertexts only. At first, it may seem paradoxical to combine the disadvantages of order-preserving and searchable encryption: order revelation and modified comparison function. However, order-revealing encryption has also advantages. It allows ansecure encryption with constant-size ciphertexts, constant size client storage and without mutation circumventing impossibility results in <cit.> and <cit.>. However, the first construction was not only impractical due to its disadvantages, but also due to its performance. A different construction with slightly more leakage, but significantly better performance was presented by Chenette et al. in <cit.>. This construction was further improved by Lewi and Wu in <cit.>. They allow comparison only between a token and an -secure ciphertext as in searchable encryption, i.e. the scheme has no leakage when no token is revealed. Their search procedure requires a linear search over all ciphertext and no indexing is possible. Hence, compared to our scheme which has logarithmic search time, order-revealing encryption currently remains impractical. §.§ Leakage-Abuse Attacks We discussed many leakage-abuse attacks on search over encrypted data. There are static attacks on order-preserving encryption <cit.> and attacks using dynamic information that also work on searchable encryption <cit.>.Kellaris et al. <cit.> have presented generic inference attacks on encrypted data using range queries. Their attacks work in a setup where the adversary has compromised the database server and can observe all queries, i.e. they work for dynamic leakage during the execution of queries and are not ciphertext-only attacks. They do not assume a specific cryptographic protection mechanism, but work only on its dynamic leakage profile, such as the access pattern or the result size, i.e. they also apply to ORAM-protected databases. The prerequisite assumption for Kellaris et al.'s attack to work is that the distribution of queries and the distribution of plaintexts differ. Specifically, they assume that each possible query will be executed, but not each possible plaintext is in the database. We note that Kellaris et al. performed all their attacks on synthetic data and queries whereas static ciphertext-only attacks on real data have been publicized <cit.>.There are also some more specific inference attacks. Islam et al. <cit.> and Cash et al. <cit.> have performed inference attacks by observing the queries on encrypted data. Islam et al. assume that the distribution of query keywords is approximately known and then can recover the query keywords using frequency analysis. Cash et al. improve the accuracy of this attack even under slightly weaker assumptions about the knowledge of query distribution, but then also use the information to recover plaintexts from the access pattern. Lacharite et al. <cit.> improve the accuracy of plaintext guessing by incorporating information from observed queries.Next to the ones already discussed plaintext guessing attacks Pouliot and Wright show that adding deterministic encryption to Bloom filters – not surprisingly – does not prevent cryptanalysis <cit.>. Zhang et al. assume that the adversary can actively insert plaintexts and can then recover query plaintexts from the access pattern <cit.>. Grubbs et al. <cit.> also attacked an implementation of multi-user searchable encryption which allows inferences between users leading to a complete breakdown of the security guarantee of encrypted web applications. §.§ Other Encryption Schemes Several attempts were made to build indexes for range queries using deterministic encryption or distance-revealing encryption <cit.>. However, since they do not follow a formal security model and are based on primitives that are easily attackable we do not consider them here.Oblivious RAM <cit.> allows to hide the accesses to disk or memory and hence the access pattern of searchable or order-preserving encryption. However, as Naveed showed in <cit.> the combination is not straightforward. Recently, a new ORAM technique – TWORAM – has been presented by Garg et al. in <cit.> that overcomes these limitations. Kellaris et al. showed in <cit.> that inference attacks even against ORAM-protected range queries exist.In theory search can be implemented without leakage using homomorphic encryption <cit.>. However, since in our model the server returns an arbitrarily sized subset of the data and in homomorphic encryption the worst case determines the cost, the server would always return the entire encrypted data. In terms of performance this can, of course, always be beaten by symmetric encryption and search on the client.§ CONCLUSIONSWe present the -security model – an extension of semantic security – that provably prevents plaintext guessing attacks as those by Naveed et al. <cit.> and Grubbs et al. <cit.>We show how this model implies that each ciphertext of an efficiently searchable, encrypted data structure must be semantically secure. However, we also show that even if all ciphertexts in a data structure are semantically secure, this does not imply -security.Then we present an efficiently searchable (logarithmic time, linear space), encrypted data structure secure in this model. We show that this scheme is practical in our evaluation, since it only has a 10 milliseconds overhead on a range query over a million database entries. This shows that one can built efficient, encrypted databases that withstand break-ins and data theft as we have seen in many recent attacks on cloud infrastructures. §.§ Future Work: Full dynamicity For ease of exposition we excluded deletion from the operations of our efficiently searchable, encrypted data structures . However, given our instantiation for range queries over encrypted data, it should be easy to see that deletion does not pose any major obstacle compared to insertion. Of course, for a fully functional database implementation we also implement deletion. IEEEtranS | http://arxiv.org/abs/1709.09314v1 | {
"authors": [
"Florian Kerschbaum",
"Anselme Tueno"
],
"categories": [
"cs.CR"
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"primary_category": "cs.CR",
"published": "20170927025852",
"title": "An Efficiently Searchable Encrypted Data Structure for Range Queries"
} |
Input:Output:thmTheorem corCorollary lemLemma propProposition conjConjecture defnDefinition examExampleproofProofremRemark[#1]#2 footnote[#2] | http://arxiv.org/abs/1709.09676v4 | {
"authors": [
"Mine Alsan",
"Ranjitha Prasad",
"Vincent Y. F. Tan"
],
"categories": [
"cs.IT",
"cs.LG",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170927180141",
"title": "Lower Bounds on the Bayes Risk of the Bayesian BTL Model with Applications to Comparison Graphs"
} |
Dipartimento di Fisica "E.R. Caianiello", Università di Salerno, I-84084 Fisciano (SA), Italy INFN - Sezione di Napoli, Gruppo collegato di Salerno, I-84084 Fisciano (SA), Italy Van Swinderen Institute, University of Groningen, 9747 AG, Groningen, The Netherlands Dipartimento di Fisica "E.R. Caianiello", Università di Salerno, I-84084 Fisciano (SA), Italy INFN - Sezione di Napoli, Gruppo collegato di Salerno, I-84084 Fisciano (SA), Italy Van Swinderen Institute, University of Groningen, 9747 AG, Groningen, The Netherlands Kapteyn Astronomical Institute, University of Groningen, 9700 AV, Groningen, The Netherlands04.50.Kd, 04.60.-m, 04.80.Cc, 03.65.Ta In this paper we will study for the first time how the wave-packet of a self-gravitating meso-scopic system spreads in theories beyond Einstein's general relativity. In particular, we will consider a ghost-free infinite derivative gravity, which resolves the 1/r singularity in the potential - such that the gradient of the potential vanishes within the scale of non-locality. We will show that a quantum wave-packet spreads faster for a ghost-free and singularity-free gravity as compared to the Newtonian case, therefore providing us a unique scenario for testing classical and quantum properties of short-distance gravity in a laboratory in the near future.Quantum spreading of a self-gravitating wave-packet in singularity free gravityAnupam Mazumdar December 30, 2023 ================================================================================ § INTRODUCTION On large distances and late times the gravitational interaction is well described by the theory of general relativity (GR) that, indeed, has been very successful since Einstein's initial work, being tested to a very high precision in the infrared (IR) <cit.>. The most recent success of GR comes from the observation of gravitational waves from merging of binary blackholes which gave a further confirmation of its predictions <cit.>. Despite these great achievements, our knowledge of the gravitational interaction in the ultraviolet (UV) is still very limited: suffice to say that the inverse-square law of the Newtonian potential has been tested only up to 5.6×10^-5 m in torsion-balance experiments so far <cit.>. This means that any modification from the Newtonian 1/r-fall is expected to happen in the large range of values going from the lower bound 0.004 eV to the Planck scale M_p∼10^19 GeV. This is the place where nature should manifest a different behaviour compared to GR and where either quantum or classical modification from GR should appear in order to solve problems that still remain unsolved as, for example, blackhole and cosmological singularities that make Einstein's theory incomplete in the UV. There have been many theoretical attempts that try to modify GR in the UV regime but none of them have been sufficiently satisfactory so far. In fact, only the experiment will be able to tell us whether the gravitational interaction is really quantum or not, and, in both cases, whether the classical properties are also modified.Recently, a new scenario has been proposed in which by studying the quantum spread of the solitonic wave-packet for a self-gravitating meso-scopic system one can test and constrain modified theories of gravity in the near future <cit.>. In this framework, the so called infinite derivative gravity (IDG) <cit.> was considered as example of alternative theory: it belongs to the class of non-local ghost-and singularity-free theories of gravity. It was shown that in the non-relativistic and in the weak-field regimes the dynamics of the matter-sector is governed by a Schrödinger equation with a non-linear self-interaction term <cit.>: [i∂/∂ tψ(x⃗,t)=[-1/2m∇^2.-Gm^2∫ d^3x'; [ .× Erf(M_s/2|x⃗'-x⃗|)/|x⃗'-x⃗||ψ(x⃗',t)|^2]ψ(x⃗,t), ] ] where G=1/M_p^2 and M_s represents the scale of new physics at which non-locality-effects should become relevant, i.e. 0.004 eV≤ M_s≤ 10^19 GeV. Such an integro-differential equation can have two completely different physical interpretations, correspondingly the wave-function ψ(x⃗,t) can assume two different meanings[See Refs. <cit.> for more details and a review on these two different physical approaches in the case of Newtonian gravity, where the main equation is the Schrödinger-Newton equation.]: * It can appear when gravity is quantized and directly coupled to the stress-energy tensor operator. In this case, it is derived as a Hartree equation in a mean-field approximation; ψ(x⃗,t) has the meaning of wave-function associated to an N-particle state, with large number of particles (N→∞), i.e. a condensate <cit.>. * Moreover, Eq. (<ref>) can be seen as a fundamental equation describing the dynamics of a self-gravitating one-particle system, when considering a semi-classical approach where gravity is coupled to the expectation value of the stress-energy tensor; in this case ψ(x⃗,t) represents a one-particle wave-function. In such semi-classical framework gravity is treated as a classical interaction, while matter is quantized.Note that, in case 1. the non-linearity emerges when considering the limit of large number of particles, i.e. in the mean-field regime; while in 2. one has non-linearity even for a one-particle state, bringing to a modified Schrödinger equation.In this respect, a semi-classical approach to IDG would seem more speculative as, not only we would modify GR, but also quantum mechanics. While considering the case of quantized gravity, and studying the dynamics of a self-gravitating condensate, could be particularly more interesting as the main motivations of IDG concern problems emerging when one tries to quantize gravity. However, in this manuscript we wish to make a general treatment by considering both cases of semi-classical and quantized gravitational interaction, and discuss the experimental feasibility of the model in both cases.It is also worth emphasizing that the non-linear potential term in Eq. (<ref>) can be split in two parts as follow <cit.>: [V[ψ](x⃗)≃-G m^2 M_s/√(π)∫_|x⃗'|<2/M_sd^3 x' |ψ(x⃗',t)|^2;-Gm^2 ∫_|x⃗'|≥ 2/M_s d^3 x' |ψ(x⃗',t)|^2/|x⃗'-x⃗|. ] From the last decomposition one can notice that the first term contains all information about the non-local nature of the gravitational interaction, while the second one has the same form of the usual Newtonian self-potential that also appears in the well-known Schrödinger-Newton equation, see <cit.>.In Ref. <cit.> it was shown that Eq. (<ref>) admits stationary solitonic-like solutions for the ground-state and it was found that in the case of IDG the energy E and the spread σ of the solitonic wave-packet turn out to be larger compared to the respective ones in Newtonian gravity, i.e. E_ IDG≥ E_ N and σ_ IDG> σ_ N; these are effects induced by the non-local nature of the gravitational interaction <cit.>. The expression of the ground-state energy is given by E_ IDG=3/41/mσ^2-√(2/π)Gm^2M_s/√(2+M_s^2σ^2), that in the limitM_sσ>2 recovers the energy of Newton's theory E_ N=3/41/mσ^2-√(2/π)Gm^2/σ <cit.>. The above Eq. (<ref>) shows the action of two kinds of forces that are completely different in nature: a quantum-mechanical kinetic contribution that tends to spread the wave-packet and the gravitational potential which takes into account the attractiveness of gravity coming from the non-linear term of Eq. (<ref>). In a stationary scenario the two contributions balance each other and the soliton-like solution above can be found.In this paper, unlike Ref. <cit.>, we are more interested in studying non-stationary solutions of Eq. (<ref>). In particular we want to understand how the spreading of the wave-packet is affected by the presence of a non-local gravitational self-interaction. The analysis that we will present will apply to both cases of semi-classical and quantized gravity as the main equation is mathematically the same.As pointed out in Refs. <cit.>[In Refs. <cit.> the authors mainly focused on the semi-classical approach, where gravity is treated classically. However, the following treatment will also apply to the case of quantized gravity as one has to consider the same integro-differential equation (<ref>). Let us keep in mind that in the semi-classical approach the quantum wave-packet represents a one-particle wave-function; while, when gravity is quantized, it is associated to the dynamics of a many-particle system, i.e. a condensate.], where numerical studies of the Schrödinger-Newton equation were made, there should exist a threshold mass μ, such that the collapse of the wave-function induced by gravity will take place for any m>μ. In Ref. <cit.> it was noticed that the collapsing behavior appears only if the initial state of the quantum system has negative energy, such that the attractive contribution of self-gravity dominates. From this last observation, we understand that a possible way to find an analytical estimation for the threshold mass μ is to equate kinetic and gravitational contributions in Eq. (<ref>); thus we obtain μ_ IDG=(3/4√(π/2)√(2+M_s^2σ^2)/Gσ^2M_s)^1/3. Note that in the limit when M_s σ > 2, Eq. (<ref>) gives the threshold mass similar to the case of Newtonian theory: μ_ N=(3/4√(π/2)1/Gσ)^1/3. Eqs. (<ref>) and (<ref>) clearly show that non-locality implies a larger value of the threshold mass, i.e. μ_ IDG>μ_ N for any values of M_s and σ. If we choose the current lower bound on the scale of non-locality, M_s=0.004 eV <cit.>, and σ=500 nm, [It is worthwhile to note that the special choice we have made for the value of the spread, σ=500 nm, corresponds to the actual slit separation d in a Talbot-Lau interferometry setup <cit.>. The slit separation is related to both length L of the device and de Broglie wave-length λ =h/mv, where h is the Planck's constant, m the mass of the particle and v is its velocity, through the relation L=d^2/λ. The interference with larger masses requires smaller wavelengths, which in turn means shorter slit-separations. Moreover, the most massive quantum systems which have been seen showing interference are organic molecules with a mass of the order of 10^-22-10^-21 kg <cit.>. ] the values of the threshold masses are μ_ IDG≃ 3.5× 10^-17 kg and μ_ N≃6.7× 10^-18 kg.From an analytical estimation one expects that for masses, m>μ_ IDG, the self-gravitating quantum wave-packet collapses, while for masses, m<μ_ IDG, one would expect no collapse of the wave-packet, but only a slow-down of the spreading compared to that of the free-particle case. We now wish to study the quantum spreading of a self-gravitating wave-packet and understand how it is affected by singularity-free gravity, without taking into account the collapsing phase. It means we will work in a regime in which we can assume that the non-linear contribution in Eq. (<ref>) is smaller compared to the kinetic term: √(2/π)Gm^2M_s/√(2+M_s^2σ^2)<3/41/mσ^2. In this regime non-linearity-effects are sufficiently small and it allows us to find non-stationary solutions by applying the Fourier analysis to Eq. (<ref>). From Eq. (<ref>), we can also define the dimensionless parameterξ:=Gm^3σ^2M_s/√(2+M_s^2σ^2), that quantifies the degree of non-linearity due to self-gravity. It depends on the initial data through the mass m and σ, that can represent the initial spread of the self-gravitating quantum wave-packet. When ξ <1, we can assume that the non-linear effects are sufficiently small. In the Newtonian limit, M_s σ_0 > 2, we will obtain ξ∼ Gm^3σ.A comparison between modified theories of gravity, IDG in our case, and Newton's gravity, will provide us a new and unique framework to test short-distance gravity beyond Einstein's GR. This paper is organized as follows: first of all we will briefly introduce ghost-free and singularity-free IDG; then we will study the spreading solutions of Eq. (<ref>) with the aim of comparing free, Newton and IDG cases; finally there will be a summary and a discussion on current and near future experimental scenarios in both cases of semi-classical and quantized gravity.§ INFINITE DERIVATIVE GHOST-FREE AND SINGULARITY-FREE GRAVITY There have been many attempts to modify GR by introducing higher order derivative contribution in the action, especially a conformal gravity containing quadratic terms in the curvature like ℛ^2, ℛ_μνℛ^μν, ℛ_μνρσℛ^μνρσ. Such quadratic theory of gravity turns out to be conformal as well as renormalizable, but it suffers from the presence of a massive spin-2 ghost field that makes the theory classically unstable and non-unitary at the quantum level <cit.>. Recently, it has been noticed that by considering an infinite number of derivatives in the quadratic curvature gravitational action one can prevent the presence of ghost <cit.>. At the same time, such a ghost-free action also improves the behaviour of the gravitational interaction in the UV regime showing a non-singular potential and a vanishing gravitational force: Φ→const and F_g → 0 as r→ 0, where F_g represents the mutual force between two particles separated by the distance r <cit.>. The most general torsion-free, parity-invariant and quadratic covariant action that contains an infinite number of derivatives has been constructed around constant curvature backgrounds, and reads <cit.>: [S=1/16π G∫ d^4x√(-g)[ℛ+α(ℛℱ_1(_s)ℛ..; [ ..+ℛ_μνℱ_2(_s)ℛ^μν+ℛ_μνρσℱ_3(_s)ℛ^μνρσ)], ] ] where α is a dimensionful coupling, _s≡/M_s^2 and ≡ g^μν∇_μ∇_ν, where μ, ν=0,1,2,3, and the mostly positive metric signature, (-,+,+,+), is chosen. The information about the presence of infinite derivatives is contained in the three gravitational form factors ℱ_i(_s) which have to be analytic functions of , ℱ_i(_s)=[n=0]∞∑f_i,n(_s)^n, thus we can smoothly recover GR when we take the limit → 0. These form factors can be further constrained by requiring general covariance, that no additional dynamical degrees of freedom propagate other than the massless graviton and the ghost-free condition, that around Minkowski background is given by 2 F_1(_s)+F_2(_s) + 2 F_3(_s)=0 <cit.>. Note that around a constant curvature spacetime we can set F_3=0 <cit.>, without loss of generality. As we have already mentioned above, the parameter M_s represents the scale of non-locality where gravity, described by this class of ghost- and singularity-free theories, shows a non-local nature <cit.>. The current constraints on M_s comes from torsion-balance experiments which have seen no departure from the Newtonian 1/r-fall up to adistance of 5.6× 10^-5 meters, that implies M_s ≥ 0.004 eV <cit.>.Furthermore, it is worth noting that IDG-theory can also resolve cosmological singularity, see <cit.>, and the non-local nature of gravity can possibly even play a crucial role in the resolution of blackhole singularity as pointed out in Ref. <cit.>; while at the quantum level it is believed that the action in Eq. (<ref>) describes a gravitational theory that is UV-finite beyond 1-loop <cit.>.The ghost-free condition of IDG demands a special choice for the gravitational form factors, see <cit.>:[The exponential choice e^-/M_s^2 is made in order to have a UV-suppression in the propagator in momentum space. Indeed, the dressed physical propagator turns out to be suppressed either for time-like and space-like momentum exchange <cit.>. Note that if we had assumed to work with the mostly negative signature, we would have had to choose e^/M_s^2. In both cases one obtains a well-defined gravitational potential that recovers the correct Newtonian limit in the IR. Both choices are also compatible with the change of sign of the kinetic term in the graviton Lagrangian depending on the signature convention, h_μν h^μν and -h_μν h^μν, respectively, where h_μν is the graviton field defined as metric perturbation around flat spacetime, g_μν=η_μν+h_μν. Moreover, integrations in momentum space with such exponentials can be performed by following various prescriptions as, for example, Wick rotation to Euclidean space, or for alternative prescriptions see also <cit.> and <cit.>.] α F_1(_s)=-α/2 F_2(_s)=a(_s)-1/,a(_s)=e^-/M_s^2. Generally, a(_s) should be exponential of an entire function <cit.>, in order to avoid additional dynamical degrees of freedom other than the massless spin-2 graviton, therefore no propagating ghost-like states. In fact, any generalised form of exponential of an entire function yields a similar behavior in the UV and IR regimes, namely a similar non-singular modified gravitational potential that for large distances recovers the Newtonian 1/r-fall <cit.>.By linearizing the action in Eq. (<ref>) and going to momentum space one can easily show that the choice in Eq. (<ref>) does not introduce any extra degrees of freedom in the gravity sector. Indeed, as shown in Ref. <cit.>, the gauge-independent part of the propagator corresponding to the linearized action around Minkowski spacetime is given by Π(-k^2)=1/a(-k^2)(𝒫^2/k^2-𝒫^0/2k^2), where 𝒫^2 and 𝒫^0 are the well known spin projector operators that project any symmetric two-rank tensor along the spin-2 and spin-0 components, respectively; Π_ GR=𝒫^2/k^2-𝒫_s^0/2k^2 is the GR propagator. For the special choice a(-k^2)=e^k^2/M_s^2, there are no additional poles in the complex plane and thus only the massless graviton propagates.We are interested in the non-relativistic, weak-field and static spacetime approximations, such that we can compute the gravitational potential from which, in turn, one can write down the Hamiltonian interaction coupling gravity and matter sectors in both cases of semi-classical and quantized gravitational interaction, as it has been done in Ref. <cit.>. As shown in Ref. <cit.>, in the semi-classical approach gravity is coupled to the expectation value of the quantum energy-stress tensor, and the field equation for the potential, with the choice Eq. (<ref>), reads[Since terms with derivatives of order higher than four are usually neglected at small curvature, one could be brought to think that the Taylor expansion of the exponential e^-∇^2/M_s^2 can be truncated. However, it is not the case here: in fact, in Eq. (<ref>) we are considered the most general quadratic-curvature action, and the infinite-order in derivatives comes from the form factors F_i() and not from higher order curvature-invariants. Moreover, even if we wanted to truncate the series, we would suffer from the ghost problem again.]: [ e^-∇^2/M_s^2∇^2Φ=4π G ⟨ψ| τ̂_00| ψ⟩; = 4π Gm | ψ(x⃗,t)|^2, ] whose solution is given by Φ[ψ](x⃗)=-Gm∫ d^3x' Erf(M_s/2|x⃗'-x⃗|)/|x⃗'-x⃗||ψ(x⃗',t)|^2, i.e. one has a classical gravitational potential generated by the probability density |ψ(x⃗,t)|^2 that plays the role of a semi-classical source. In a Hamiltonian formulation the potential inEq. (<ref>) contributes to the self-interaction of matter as described by the modified Schrödinger equation in Eq. (<ref>), see <cit.>.In the case of quantized gravity instead, the graviton field is directly coupled to the quantum energy-stress tensor so that the analog of Eq. (<ref>) reads e^-∇^2/M_s^2∇^2Φ̂=4π G τ̂_00, whose solution is given by Φ̂(x⃗)=-Gm∫ d^3x' Erf(M_s/2|x⃗'-x⃗|)/|x⃗'-x⃗|ψ̂^†(x⃗')ψ̂(x⃗'). Note that we have used τ̂_00≡ρ̂=mψ^†ψ. By calculating also in this case the Hamiltonian interaction, it becomes clear that the quantum gravitational potential in Eq. (<ref>) does not introduce any non-linearity in a N-particle Schrödinger equation, but the non-linear integro-differential equation (<ref>) would emerge when considering mean-field regime for the many-body system (N→∞) <cit.>.In the following section we will study the spreading solutions of Eq. (<ref>) and, by comparing to the case of Newtonian gravity, we will be able to see which is the effect of non-locality on a self-gravitating wave-packet in IDG-theory. The analysis will hold for both cases of semi-classical and quantized gravity as the main dynamical equation is mathematically the same, i.e. Eq. (<ref>).§ SPREADING SOLUTIONS FOR A SELF-GRAVITATING WAVE-PACKETWe now wish to study non-stationary solutions of the non-linear integro-differential equation in Eq. (<ref>) by working in a regime where non-linearity-effects can be considered sufficiently small such that there will be no gravity-induced collapse. Such a regime is the one described by the inequality in Eq. (<ref>) which gives the range of masses (see Eqs. (<ref>) and (<ref>)) for which the self-gravitating wave-packet will not collapse, as the attractive contribution due to gravity is not dominating, but it can allow only the spread of the wave-packet. In this scenario, we are allowed to study Eq. (<ref>) in the Fourier space.Let us suppose we start with an initial Gaussian wave-packet: ψ(x⃗,0)=1/π^3/4σ_0^3/2e^-|x⃗|^2/2σ_0^2,∫ d^3xψ(x⃗,0)^2=1, where σ_0 is the initial spread. A formal expression of the wave-packet at a generic time t>0 can be found in terms of its Fourier transform: [ψ(x⃗,t)= ∫d^3k dω/(2 π)^4ϕ (k⃗,ω) e^i(k⃗·x⃗-ω t); = ∫d^3k/(2 π)^3ϕ(k⃗) e^i(k⃗·x⃗-ω (k⃗) t), ] where we have used ϕ (k⃗,ω)=2πϕ (k⃗) δ (ω -ω (k⃗)), and ϕ (k⃗) can be obtained by calculating the anti-Fourier transform at the initial time t=0: [ ϕ (k)= ∫ d^3x ψ(x⃗,0) e^-ik⃗·x⃗;= 2 √(2)π^3/4σ_0^3/2 e^-1/2k^2σ_0^2, ] where k ≡ |k⃗|. By using Eq. (<ref>), and the expression of the IDG potential in the momentum space,Erf(M_s/2|x⃗'-x⃗|)/|x⃗'-x⃗|= ∫d^3k/(2π)^34π e^-k^2/M_s^2/k^2e^ik⃗·(x⃗'-x⃗), and by acting with ∫ d^3xdt e^-i(k⃗·x⃗-ω t) on both sides of the modified Schrödinger equation in Eq. (<ref>), we obtain the dispersion-frequency ω_ IDG as a function of k: ω_ IDG (k)= k^2/2m - 32Gm^2π^5/2σ_0^3 D_ IDG(k), where[𝒟_ IDG(k)=; [ ∫d^3k'd^3k”/(2π)^3(2π)^3e^-|k⃗”-k⃗'|^2(σ_0^2+1/M_s^2)e^[(k⃗”-k⃗')·k⃗-k⃗”·k⃗'] σ_0^2/|k⃗”-k⃗'|^2.] ] In order to solve the integral in Eq. (<ref>) we can make the following change of integration variables: X⃗:=k⃗”-k⃗', Y⃗:=k⃗”+k⃗', thus k⃗”·k⃗'=(Y^2-X^2)/4 and the integral turns out to be decoupled in two other integrals that can be easily calculated by using polar coordinates: [The special function Erfi(x) is the imaginary error-function and is defined as Erfi(x):=2/√(π)∫_0^x dt e^t^2≡ -iErf(ix).][ 𝒟_ IDG(k)= 1/8∫d^3X/(2π)^3e^-(3/4σ_0^2+1/M_s^2)|X⃗|^2e^X⃗·k⃗σ_0^2/|X⃗|^2∫d^3Y/(2π)^3e^-|Y⃗|^2 σ_0^2/4;= 1/32π^5/2σ_0^51/kErfi(k M_s σ_0^2/√(4+3M_s^2σ_0^2)), ]From Eq. (<ref>) we obtain an expression for the dispersion relation in Eq. (<ref>) in the case of IDG self-interaction: ω_ IDG (k)= k^2/2m - Gm^2/σ_0^21/kErfi(k M_s σ_0^2/√(4+3M_s^2σ_0^2)), note that in the regime M_s σ_0>2, gives the corresponding Newtonian limit: ω_ N (k)= k^2/2m - Gm^2/σ_0^21/kErfi(kσ_0/√(3)). In Fig. <ref> it is shown the behavior of the dispersion relation ω (k) in the free, Newtonian and IDG cases, and one can immediately notice that as the parameter M_s increases the frequency ω_ IDG tends to ω_ N.The dispersion relation in Eq. (<ref>) is crucial in order to determine the time evolution of the wave-packet, indeed from Eq. (<ref>) the solution ψ_ IDG (x⃗,t) is expressed in terms of the frequency ω_ IDG: ψ_ IDG(x⃗,t)= 2√(2)π^3/4σ_0^3/2∫d^3k/(2 π)^3 e^-1/2k^2σ_0^2e^i(k⃗·x⃗-ω_ IDG(k) t). The integral in Eq. (<ref>) cannot be solved analytically, but we will be able to find numerical solutions. First of all, note that in the free-particle case there exists a well know analytical solution that describes a quantum-mechanical spreading of the wave-packet, and the corresponding probability density reads: |ψ_free(x⃗,t)|^2= 1/π^3/2σ_0^3e^-|x⃗|^2/σ_0^2(1+(t/mσ_0^2)^2)/(1+(t/mσ_0^2)^2)^3/2, from which one can see that there exist a time-scale for the spreading, i.e. a time after which the particle turns out to be de-localized, and it is given by: τ_free=m σ_0^2.Note that the time-scale in Eq. (<ref>) can be also obtained by imposing the equality τ_free=1/2ω_free(1/σ_0). We argue that in the same way we can also obtain an analytical estimation for the spreading time-scale of a self-gravitating system, thus by using Eq. (<ref>) and imposing τ_ IDG=1/2ω_ IDG(1/σ_0) we obtain[Although an exact analytical derivation is lacking for such a time-scale, our argument turns out to be consistent with the numerical analyses, some of which are presented in the end of this section.] τ_IDG∼mσ_0^2/1-2Gm^3σ_0Erfi(M_s σ_0/√(4+3M_s^2σ_0^2)), that in the case of Newtonian self-interaction reduces to τ_N∼mσ_0^2/1-2Gm^3σ_0Erfi(1/√(3)). In the opposite regime M_s σ_0<2, when non-locality-effects become dominant, the time-scale in Eq. (<ref>) assume the following form: τ_IDG^(M_s σ_ 0<2)∼mσ_0^2/1-2Gm^3M_s σ_0^2/√(π), that is the same time-scale that we would have if there was a constant gravitational potential. Since we are considering non-linear effects sufficiently small, in order to be consistent with the inequality in Eq. (<ref>) we need to require 2Gm^3σ_0Erfi(M_sσ_0/√(4+3M_s^2σ_0^2)) <1, that can be seen as a quantifier of non-linearity, as the one in Eqs. (<ref>)-(<ref>), and from which we can determine again a threshold value for the mass: χ_ IDG=[2Gσ_0Erfi(M_sσ_0/√(4+3M_s^2σ_0^2))]^-1/3, that in the regime M_sσ_0>2 reduces to the one of Newton's gravity: χ_ N=[2Gσ_0Erfi(1/√(3))]^-1/3. If we choose the values M_s=0.004 eV and σ_0=500 nm for the scale of non-locality and the initial spread, respectively, we obtain χ_ IDG≃ 3.1× 10^-17 kg and χ_ N≃ 6.1× 10^-18 kg. These values are of the same order of the threshold masses μ_ IDG and μ_ N that we have found above by equating kinetic and gravitational energy contributions. Thus, Eq. (<ref>) is consistent with the inequality in Eq. (<ref>).It is very clear by comparing Eqs. (<ref>), (<ref>) and (<ref>) thatτ_free< τ_IDG≤τ_N, namely the gravitational self-interaction causes a slow-down of the spreading of the wave-packet compared to the free-particle case. Moreover, since at short distances IDG interaction is weaker than the Newtonian one, a self-gravitating wake-packet in the IDG case will spread more quickly compared to Newton's theory.We have solved numerically the integral in Eq. (<ref>) for both IDG and Newton's theory, and in Fig. <ref> we have plotted the radial probability density ρ(x⃗,t)=4π|x⃗|^2|ψ(x⃗,t)|^2 in the free, Newtonian and IDG cases at two fixed values of time as a function of the radial coordinate |x⃗|. We can immediately notice that the results in the two plots are in agreement with the analytical estimation made in Eq. (<ref>).Such gravitational inhibitions of the spreading, not only would offer a way to explore classical and/or quantum properties of the gravitational interaction, but would also provide a new framework to test short-distance gravity beyond GR, by investigating the real nature of the gravitational potential. § DISCUSSIONIn the previous section we have found quantum spreading solutions for a self-gravitating wave-packet both in the case of IDG and Newtonian self-interactions. The results we have obtained, especially the analytical estimation in Eq. (<ref>) and the numerical solutions in Fig. <ref>, provide us a unique window of opportunity to test modified theories of gravity in a laboratory, in particular IDG-as an example of singularity-free theory of gravity. Although the previous analysis holds for both cases of semi-classical and quantized gravity, in order to discuss the experimental testability of the model we need to distinguish between the two cases. As we have already mentioned above: * In the case of quantized gravitational interaction, Eq. (<ref>) describes the dynamics of a condensate, where the mutual gravitational interaction of all components would give an effective self-potential-contribution as result of the mean-field approximation; * In the semi-classical approach Eq. (<ref>) can describe the dynamics of a self-gravitating one particle system, as for instance elementary particles, or molecules whose center-of-mass' dynamics would be taken into account. The first case might be more interesting to explore: let us remind that the main motivations of IDG concern problems arising when trying to quantize the gravitational interaction, for example unitarity and renormalizability as already mentioned in Section II. Moreover, a semi-classical approach to IDG, would also imply a modification of quantum mechanics such that the fundamental equation governing the dynamics of a single-particle state would be non-linear. In this respect, case 1. seems less speculative than case 2..However, in this section we wish to study the current and future experimental testability of our predictions about both classical and quantum aspects of gravity.As far as case 1. is concern, there is a very promising experiment that is aimed to test quantum mechanics of weakly coupled Bose-Einstein condensates (BEC) in a freely falling system, where the spread of the quantum wave-packet would be tested in microgravity, see Ref. <cit.>. The used BEC was made of about 10^4 atoms, but technology is progressing and in the near future it will be possible to consider BEC with 10^6, or even more, atoms, allowing us to compare our predictions with the experimental data in such a way to constrain the scale of non-locality M_s, that so far has been only bounded in torsion-balance experiments <cit.>.Regarding case 2., molecule-interferometry <cit.> seems to be one of the most favorable scenario to verify predictions of the semi-classical approach. Although it is not the configuration analyzed in this paper, it is also worth mentioning that another promising scenario aimed to test semi-classical gravity is given by optomechanics as explained in Refs. <cit.>, where one considers many-body systems with a well-localized wave-function for the center of mass.All these kinds of experiments are very sensitive and, unfortunately, there are several sources of noise that need to be taken into account, as for example decoherence effects, see Ref. <cit.> for a review, and they represent a big challenge to overcome.The choices of the initial spread σ_0 and the mass m are very crucial in order to determine the time-scale for the spreading. First of all, to have appreciable gravity-induced effects we need values of the mass that are not much smaller than the threshold mass. Moreover, if we take σ_0=500 nm and m∼χ_ N≃ 6.0× 10^-18 kg, the time-scales are of the order of τ_IDG,τ_N∼𝒪(10^5)-𝒪(10^6) sec and τ_free∼ O(10^4) sec. In an interferometric experimental setup, for example, these values would not be suitable to test short-distance gravity, because the time-scales are too large compared to the coherence-times achieved by a modern matter-wave interferometer (1 -3 seconds) <cit.>. This means that we need smaller initial spreads so that the values of the time-scales will also decrease. For instance, if we choose σ_0=1 nm, m=4.8× 10^-17 kg and M_s=300 eV, the time-scales turn out to be of the order of τ_free∼𝒪(10^-1) sec, τ_IDG∼𝒪(1) sec and τ_N∼𝒪(10) sec, which are more suitable values of time to test and compare modified theories of gravity.In the table I we have shown some values of the three time-scales for fixed values of initial spreads and masses. It is very clear that by decreasing the initial spread, the time-scales also decrease. Those numerical values apply to both cases 1. and 2.; of course the setup can be different depending on the kind of experiments, and so the preparation of the initial state will also differ. In tests with BEC <cit.> the experimental configuration is given by an asymmetric Mach-Zehnder interferometer, while molecular-interferometry can be performed, for instance, with a Talbot-Lau interferometer <cit.>. In both cases the spread σ_ 0 of the initial wave-packet is related in some way to the size of the slit separation d in the interferometric setup. For example, as we have already mentioned in the footnote <ref>, in a Talbot-Lau interferometry setup the initial spread σ_0 is of the same order of the slit separation d, that in turn is expressed in terms of the length L of the interferometric device and of the the wave-length λ of the particle: L=d^2/λ. The smallest slit separation that has been achieved so far is d=500 nm <cit.>, which also implies σ_0∼ 500 nm. However, we have seen that in order to build a very suitable experimental scenario in which we can test modified theories of gravity we need at least a value σ_0=1 nm for the initial spread, which means that technology should decrease at least of two orders of magnitude the slit-separation d. Moreover, we also need masses of the order of 𝒪(10^-18-10^-17) kg, and this is one of the biggest challenge to overcome: in fact, the most massive systems for which interference patterns have been observed are organic molecules with a mass of 10^-22-10^-21 kg, see Ref. <cit.>. In view of this last observation, it is worthwhile to mention that in Ref. <cit.> the authors present a new exciting proposal in which one might be able to perform quantum-interference with superconducting spheres with masses of the order of 10^-14 kg, and in such a regime both gravitational and quantum effects should be not negligible.In this manuscript we have extended the window of opportunity to test classical and quantum properties of modified gravity provided in Ref. <cit.>, where stationary properties of a quantum wave-packet were taken into account. Indeed, here we have made additional predictions that might allow us to test and constrain gravitational theories by studying non-stationary properties, in particular the quantum spreading of the wave-packet associated to a self-gravitating meso-scopic system in alternative theories beyond Einstein's GR. 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"authors": [
"Luca Buoninfante",
"Gaetano Lambiase",
"Anupam Mazumdar"
],
"categories": [
"gr-qc",
"quant-ph"
],
"primary_category": "gr-qc",
"published": "20170926212012",
"title": "Quantum spreading of a self-gravitating wave-packet in singularity free gravity"
} |
Ohm's law is one of the most central transport rules stating that the total resistance of sequential single resistances is additive. While this rule is most commonly applied to electronic circuits, it also applies to other transport phenomena such as the flow of colloids or nanoparticles through channels containing multiple obstacles, as long as these obstacles are sufficiently far apart. Here we explore the breakdown of Ohm's law for fluids of repulsive colloids driven over two energetic barriers in a microchannel, using real-space microscopy experiments, particle-resolved simulations, and dynamical density functional theory. If the barrier separation is comparable to the particle correlation length, the resistance is highly non-additive, such that the resistance added by the second barrier can be significantly higher or lower than that of the first. Surprisingly, in some cases the second barrier can even add a negative resistance,such that two identical barriers are easier to cross than a single one. We explain this counterintuitive observation in terms of the structuring of particles trapped between the barriers.§ INTRODUCTION One of the basic characteristics of any transport situation is the resistance, commonly known from electric circuits, which is in general defined as the ratio of the transport flux and the driving force, typically in the linear-response regime of small drives.For both electric circuits and classical transport, Ohm's law states that when resistors are put in series, their resistances simply add up. However, this macroscopic law is expected to break down on the microscopic scale, in particularwhen the distance between the two obstacles approaches the correlation length of the transported particles. Knowing and controlling flow resistance is of particular importance when tuning the transport of solutes through channels. This type of transport is the basic situation in microfluidics <cit.>, where the transported objects are typically micron-sized colloidal solutes, such thatthermal fluctuations play a significant role <cit.>.Similar transport scenarios include the collective migration of bacteria through channels <cit.>, the transport of nanoparticles through porous media <cit.>, and the transport of ions through membranes via nanopores <cit.>. On the macroscopic scale, the flow of e.g. cars or pedastrians <cit.> or animals <cit.> through crowded environments can lead to similar physics. Obstacles in such channels naturally inhibit the overall steady-state rate at which the particlesare able to traverse the channel, providing an effective resistance to the flow. In channels with multiple obstacles, we expect Ohmic (i.e. additive) behavior of the corresponding resistances when the separation between the obstacles is large, and a breakdown of Ohm's law for smaller distances. The crossover between these regimes is determined by the correlation length in the system, i.e. the length scale associated with local structure in the fluid of transported particles. Detailed knowledge of these non-additive effects is of vital importance for the design of efficient microfluidic devices, as well as for our broader understanding of constricted flow phenomena. Here, we explore the additivity of resistances in mesoscopic colloidal suspensions driven through a microchannel <cit.>.First, as a proof of concept, we perform an experiment on repulsive colloidal particles confined to microchannels containing two step-like barriers on the substrate, and measure the current through the channels as a function of the strength of the gravitational driving force. Our results show that step-like barriers in a microchannel can indeed be interpreted as resistors. We then further explore this concept using Brownian dynamics simulations and dynamical density functional theory and map out the interplay between the two barriers by varying their height and separation. We find strong deviations from additivity for the resistance of two barriers when the separation between the two obstacles is comparable to the correlation length of the system, which is on the order of several interparticle spacings. Amazingly, if the barrier separation is comparable to the interaction range, the resistance contributed by the second barrier can even be negative such that climbing two hills is faster than one. We explain this counterintuitive effect of negative resistancevia the long-ranged particle interactions and the ordering of the particles trapped between the two barriers. When these particles are disordered, they exhibit spontaneous fluctuations which modulate their interactions with particles crossing the barriers, significantly enhancing barrier crossing rates <cit.>. This surprising phenomenon provides a route for tuning and enhancing particle flow over an obstacle by the inclusion of additional barriers, reminiscent of the use of geometric obstacles to assist e.g. the flow of panicked crowds <cit.>.§ RESULTS§.§ Experiment We measure the particle current in the channel as a function of the gravitational driving force, controlled by the tilt angle of the setup, for channels with zero, one, and two barriers. In the absence of barriers, the current shows the expected linear dependence on the driving force, shown by the red line in Fig. <ref>d. For a single barrier (blue line in Fig. <ref>d), we observe a crossover from a zero-flow regime at small driving forces (where the driving force is too weak to push particles across the barrier) to an approximately linear regime for large driving forces <cit.>. Hence, the barrier provides a resistance to the flow, which reduces the particle current. Adding a second barrier to the channel clearly results in a further decrease of the current, as one would expect (orange line in Fig. <ref>d). In order to examine the possibility of non-additive resistance, we also plot in Fig. <ref>d the total resistance of the channel, defined as the current divided by the driving force. Here, we only consider tilt angles where the channels do not get fully blocked. For an empty channel, we find a well-defined constant resistance, consistent with the linear behavior of the current as a function of the tilt angle. The single barrier increases this background resistance. For sufficiently large driving forces, this increase is essentially constant, indicating that we can indeed interpret it as a simple additional resistor added to the channel. For low driving forces, the effective resistance added by the barrier is significantly higher, which we attribute to intermittent blocking of the channel: in this regime, the driving force is not always capable of pushing the particles across the barrier, and thermal fluctuations likely play a role in enabling the flow. Finally, adding a second barrier adds another contribution to the total resistance. Interestingly, this additional resistance is not simply equal to the resistance of the first barrier, even though we are in the regime where the particles flow through the channel without blockage. In particular, the total flux in the two-barrier system at the highest tilt angle is on the same order as the flux in the single-barrier case when its resistance has reached its plateau value (see Fig. <ref>d). Hence, we conclude that the two resistors interact non-additively in this case, indicating a breakdown of Ohm's law. As the barriers in this experiment are separated only by a distance of approximately 2.5 times the typical interparticle distance, which is shorter than the correlation length in the fluid, this breakdown could be the result of microscopic structuring of the fluid between the two barriers. Indeed, as shown by the snapshots in Fig. <ref>c, we consistently find two layers of particles in between the barriers. To explore this concept of non-additivity further, we now turn to a numerical treatment of the problem, where we can more readily explore a wide range of conditions. §.§ Theory and SimulationWe make use of overdamped Brownian dynamics simulations and dynamical density functional theory (DDFT) calculations. We consider a two-dimensional system with periodic boundary conditions along the channel (x-direction), containing N particles interacting via a dipolar repulsionβ V_int(r) = Γ(a/r)^3,where β = 1 / k_B T with k_B Boltzmann's constant and T the temperature, Γ is the dimensionless interaction strength, and a=ρ_0^-1/2 sets the length scale of a typical interparticle spacing of a given mean number density ρ_0. The particles additionally experience a constant driving force F 𝐱̂ pushing the particles along the channel. The confining channel and barriers are modeled as an external potential V_ext(x,y) = V_channel(y) + V_barrier(x). The first term here is a steep repulsive wall potential confining the particles in one direction. V_barrier represents one or two parabola-shaped potential barriers with width a and height V_0 = 10 k_BT, see Fig. <ref>b inset and Methods. We choose the channel width L_y = 4.65 a, and the channel length L_x such that the total number density ρ_0 = N / (L_x L_y) = 1/a^2 for a given particle number N.In our DDFT calculations <cit.>, we choose the Ramakrishnan–Yussouff functional <cit.> to model interacting particles in a fluid state (Γ = 5). In addition to DDFT, we perform Brownian Dynamics simulations of particles experiencing the same potentials and external driving force. As a reference we provide an analytical solution for non-interacting particles (Γ=0). See Methods for details.Using both DDFT and simulations, we explore the relation between the total steady-state particle current J along the channel, the driving force F on the particles, and the distance Δ x between the two barriers. The ratio of the driving force and current characterizes the total resistance of the system, = F/J. In a channel without barriers, the particles trivially adopt the average drift current J_0 = F ρ_0 L_y ξ^-1, where ξ is the friction coefficient of the background solvent, leading to an inherent background resistance = ξ / (L_yρ_0). In a single-barrier system, the resistance R_1 added by the barrier can be extracted from the total resistance _ =+ R_1 by measuring the single-barrier current J_:R_1 = _ -= F(1/J_ - 1/J_0).Similarly, in a double-barrier system (with current J_), the total resistance is _ =+ R_1 + R_2, and the effective resistance of the second barrier R_2 can be written asR_2 = F(1/J_ - 1/J_).In the case of additivity, the resistance R_2 of the second barrier will be equal to R_1 (the resistance of the first barrier), while deviations from this rule will indicate non-additivity.In Fig. <ref>, we plot R_2/R_1 for a range of barrier separations Δ x at different driving forces F, as obtained from analytical theory (see Methods) (a), DDFT calculations (b), and computer simulations (c). For non-interacting particles R_2 is lowest when the two barriers are touching (Δ x = a) and converges exponentially to R_1 for larger distances. In contrast, for interacting particles and for all investigated F, the resistance of the second barrier is highest at Δ x = a. At this separation the resistance added by the second barrier can be many times higher than R_1, signaling strong non-additivity. More interestingly, for slightly larger separations (Δ x ≃ 1.5 a), R_2 becomes smaller than R_1, and even negative for sufficiently weak driving forces. In this regime, the addition of the second barrier reduces the overall resistance in the channel. At larger Δ x, R_2 shows decaying oscillations,converging towards the additive case (R_2 = R_1), as expected at sufficiently large distances.We can understand this observation by considering the interactions between the particles. Since these are dipolar in nature, they are sufficiently long-ranged to span across the barrier. Hence, a particle on top of the barrier experiences forces from particles between the two barriers, which depend on the density and structuring of those particles. In Fig. <ref> we plot the density profile of the particlesρ_x(x), projected onto the long axis of the channel, for various barrier separations Δ x, as well as for a single barrier. In the single-barrier case, we always observe a high density peak in front of the barrier, and a slightly lower peak just after the barrier (see Fig. <ref>a). In the two-barrier cases, the additional peaks in between the two barriers vary in height based on Δ x. For very small separations (Fig. <ref>b), where the resistance of the second barrier is high (R_2 > R_1), we find a single sharp density peak between the barriers, which is significantly higher than the peak observed after a single barrier. Here, particles between the barriers are arranged in a single line with little room for fluctuations, and hence provide a strong and relatively constant force on particles crossing the first barrier, pushing them back. In the regime where R_2 < R_1 (Fig. <ref>c), we instead see two much lower peaks, indicating a structure with two layers and significantly larger fluctuations. These larger fluctuations not only provide space for particles entering via the first barrier, but also modulate the force exerted on particles crossing the barriers, resulting in a fluctuating effective barrier height. For weak driving forces, barrier crossings are rare events, whose rate depends exponentially on the barrier height. Fluctuations in barrier height are known to lead to significantly higher crossing rates <cit.> and hence higher currents. Finally, for larger separations, where R_2 > R_1 again, we observe two higher peaks, indicating a more structured pair of layers between the barriers.We confirm this intuitive picture by plotting in Fig. <ref> the relative height of the first peak after the first barrier δρ^peak = ρ_^peak / ρ_^peak, where ρ_^peak is the height of the first peak after a single barrier, and ρ_^peak is the height of the first peak after the first of two barriers. When plotted as a function of Δ x, the peak height (blue in Fig. <ref>) indeed strongly correlates with the particle current (red) in both the DDFT framework and the simulations. In our particle-resolved simulations, the additional fluctuations of the particles in between the two barriers are clearly visible. Moreover, examining simulation trajectories demonstrates that for most barrier separations, whenever a particle crosses the first barrier, the sudden increase in density between the barriers typically leads to the rapid expulsion of a particle over the second barrier. This observation confirms that the first of the two barriers can indeed be considered as the main bottleneck for the overall flow process. However, for Δ x ≲ 1.3, the bottleneck is instead the crossing of the second barrier. Here, particles form a single narrow layer between the two barriers, which inhibits the possibility of collectively pushing a particle across the second barrier. This may explain the reduced correlation between δρ_peak and R_2 / R_1 for small Δ x in Fig. <ref>. § DISCUSSION We have explored the effect of sequential potential energy barriers on the flow of colloidal particles driven through microchannels. As our experiment shows, two barriers close together can result in drastically higher resistance than twice the resistance of a single barrier. Moreover, via a detailed investigation of this non-additivity using both simulations and dynamical density functional theory, we discover that depending on the barrier spacing, the second barrier can add an effective resistance that is higher than the resistance of a single barrier, lower, or even negative. In the negative regime, the presence of the second barrier helps particles cross the first barrier, contrary to what intuition would suggest. We show that this enhanced barrier-crossing rate can be attributed to the structuring of the layer of particles in between the two barriers: weaker structuring (evidenced by lower peaks in the density profile) increase the current. A vital component for this phenomenon is the requirement that particles on top of the barriers can still interact with the particles aggregated just before and after that barrier, necessitating sufficiently long-ranged interactions. Indeed, preliminary simulations show a clear reduction of the observed non-additivity when the barrier is wider in comparison to the interaction range. Note, however, that the interaction range in our setup is controlled directly via the applied external field, rather then by the inherent properties of the colloidal particles. Such interactions can be induced in a wide range of colloids or nanoparticles, as long as they are susceptible to polarization by an external (electric or magnetic) field. As a second requirement, the density should be high enough to enable significant ordering of particles. In the confined region between the barriers, the ordering will depend sensitively on the ratio of the barrier spacing Δ x and the preferred spacing between neighboring layers of particles, as long as Δ x is small compared to the correlation length in the system. Similar confinement effects have shown to result in oscillatory behavior in forces between plates or spheres immersed in a background of smaller particles <cit.>. Interestingly, the effect of negative resistance is reminiscent of the interplay between reflecting barriers in quantum-mechanical systems, where interference is known to lead to enhanced transmission for certain barrier spacings, as used in e.g. Fabry-Perot interferometers <cit.>. The sensitivity of the resistance to the barrier separation and microscopic particle interactions provide a method to tailor and control the flow of particles through complex environments <cit.>. Indeed, for colloids driven across disordered energy landscapes <cit.>, long-range interactions have been shown to dramatically affect clogging behavior <cit.>. For more periodic energy landscapes, clever choices of the particle interactions and external fields can lead to complex individual or collective dynamics <cit.>, including e.g. an effective negative particle mobility <cit.>. The mitigation of a flow-resisting barrier by placing another barrier near it might have important implications in the design of microfluidic devices, where clogging can be a major issue <cit.>. Moreover, this strategy may also be effective in aiding the flow of particles through geometric constrictions <cit.>, where particles have to pass through a bottleneck rather than over a potential energy barrier. In this scenario additional geometric obstacles – typically placed before the bottleneck – have already been shown to enhance flow <cit.>, as applied in e.g. the design of emergency exits <cit.>. Hence, it seems likely that potential energy barriers, e.g. induced by external fields, could accomplish the same thing. The specificity of this approach to relatively long-ranged interactions suggests an opportunity for separating different particle species, or enhanced flow control via external fields modifying the interactions or particle motion.Further applications include the directed transport of strongly charged dust particles in a plasma <cit.> and congestion in granulates <cit.>, as well as jammed flow situations of colloids <cit.>.An interesting question for future research is whether the effective total resistance could be further tuned by using a combination of three, four, or an infinite number of obstacles <cit.> (forming e.g. a ratchet <cit.>), barriers of different heights or shapes <cit.>, time-dependent barriers <cit.>, or active particles <cit.>.We gratefully acknowledge funding from the German Research Foundation (DFG) within project LO 418/19-1. We thank Arjun Yodh, Laura Filion, and Marco Heinen for helpful discussions.§ METHODSExperimental setup: Our experiments are based on repulsive microscopic particles, gravitationally driven through microchannels. We use superparamagnetic colloidal particles (Dynal M-450, diameter σ =4.50(5) μm, ρ_m = 1500 kg m^3) which are restricted to two-dimensional in-plane motion due to gravity. The cell consists of two rectangular reservoirs of side length 1 mm which are connected by multiple channels. The dimensions of each channel is 2 mm in length, 30 μm in width and 8 μm in height. In the channels, U–shaped step-like barrier structures are implemented along the channel, each of them with width 3 μm and height 500 nm near the channel walls and 250 nm in the middle of the channel.The applied magnetic field 𝐁_ext induces a dipole-dipole repulsion among the colloidal particles, andthe strength of the dipole-dipole interaction can be tuned by changing the magnitude of the magnetic field. The repulsive in-plane interaction potential V(r) is <cit.> V(r) = μ_0 (χ_eff𝐁_ext)^2 /4π r^3,forr ≥σ, ∞,forr < σ, where μ_0 is the vacuum permeabilityand χ_eff = 7.88(8) · 10^-11Am^2T^-1 is the effective magnetic susceptibilityof the particles. Note that for sufficiently high field strengths, the particles never touch, such that the hard-core component of the interaction potential can be neglected.By tilting the whole experimental setup, gravity acts as an external driving force, with a strength controlled by the tilt angle and the buoyancy-corrected effective mass of the particles (m^* = 2.385(80) · 10^-14 kg). Using video microscopy, we measure the total particle flux through channels with zero, one, or two barriers as a function of the strength of the driving force. Channel model: The external potential V_ext(x, y) is composed of a confining channel contribution, (y), and the barrier potential, (x).The steep repulsive potential forming the channel walls is given by(y) = V_c [ 1 - 1/2( y + L_y/2/√(2)w) + 1/2( y - L_y/2/√(2)w) ],with channel width L_y and maximum channel potential height V_c = 1000 k_B T. The parameter w = 0.25 a sets the softness of the walls. We choose L_y = 4.65 a. The channel length L_x = 25.79 a is fixed by the imposed number of particles N=120.A single barrier potential is given by(x) =V_0 [ 1 - (x - x_1/a / 2)^2 ], for|x - x_1| < a / 20,otherwise,where x_1 is the position of the barrier. The double barrier potential is simply the superposition of two non-overlapping single barrier potentials at x_1 and x_2, where |x_1 - x_2| = Δ x ≥ a.Dynamical density functional theory: Within the DDFT framework <cit.>, the number density field ρ( r,t) of the colloidal particles is calculated by solving the differential equation ∂ρ (𝐫,t)/∂ t = D ∇(ρ(𝐫,t) ∇δℱ[ρ(𝐫,t)]/δρ(𝐫,t)), where D=k_BT/ξ is the single particle diffusion constant, ξ the friction coefficient and ℱ[ρ] = ℱ_id[ρ] + ℱ_ext[ρ] + ℱ_exc[ρ] is the total Helmholtz free energy functional. This functional incorporatesthe ideal gas contributionℱ_id[ρ] = k_BT ∫d𝐫 ρ(𝐫) ( log(Λ^2 ρ(𝐫)) - 1 )and the external potential termℱ_ext[ρ] = ∫d𝐫 ρ(𝐫) ( V_ext(𝐫) - x F),where Λ is the thermal de Broglie wavelength.As an approximation for the excess free energy functional ℱ_exc[ρ] we chose the Ramakrishnan–Yussouff functional <cit.> ℱ_exc[ρ] = ℱ_exc^ref (ρ_0) -k_BT/2∫d𝐫∫d𝐫' Δρ(𝐫)Δρ(𝐫') c^(2)_0(|𝐫 - 𝐫'|;ρ_0,Γ). Here, ℱ_exc^ref (ρ_0) is the excess free energy of an isotropic and homogeneous reference fluid at density ρ_0, Δρ(𝐫) = ρ(𝐫) - ρ_0 describes the density difference to the reference density, and c^(2)_0(r;ρ_0,Γ) is a pair (two-point) direct correlation function<cit.> that has been calculated via liquid integral theory with Rogers-Young closure <cit.>. The DDFT is solved numerically by using finite volume difference methods <cit.>. In each run, we first compute the equilibrium configuration of the system at a given barrier configuration in absence of a driving force (F = 0). Then, we switch on the driving force and let the system evolve towards its steady state.Analytical prediction for non-interacting particles:For non-interacting particles the excess free energy vanishes, i.e. ℱ_exc≡ 0, and the DDFT in the steady state can be reduced to a single variable x. The general solution for periodic boundary conditions and tilted potential V(x) = (x) - Fx is <cit.>J = Dρ_0 L_xL_y (1 - )/I_+I_-- (1-)∫_0^L_xx̣ e^-β V(x)∫_0^xx̣'̣ e^-β V(x'),with I_± = ∫_0^L_xx̣ e^±β V(x).For single and double barrier potentials we can find an analytic expression for J and therefore express the ratio of resistances asR_2/R_1 = 1 - K (e^-F(Δ x - a) + e^-F(L_x-Δ x-a)).Here, the value K=P/Q does not dependent on Δ x and is determined by the expressionsP= β^2 F^2 (A_1(1-) - A_2) - (1-)^2,Q= β^2 F^2 (A_1(1-) + A_3 - A_4)- β F a (1-) - (1-)(1-),withγ = a/4√(π/β V_0), ζ_± = √(β V_0)(Fa/4β V_0± 1),= (ζ_+) - (ζ_-),= (ζ_+) - (ζ_-),A_1= γ(e^-ζ^2_+ + e^ζ^2_-),A_2= γ^2,A_3= γ^2((ζ_+) - (ζ_-)),A_4= a^2(1-)/8β V_0(ζ_+^2 Φ(ζ_+^2) - ζ_-^2 Φ(ζ_-^2)),where (x) is the imaginary error function, and Φ(x) = x is the generalized hypergeometric function. Brownian dynamics simulations:In addition to DDFT, we perform overdamped Brownian Dynamics simulations of particles. Here, we numerically solve the equations of motion for the particles, given by𝐫̇_i = -∇_i V_tot/ξ + F 𝐱̂/ξ+ √(2 D)𝐑(t),where 𝐫_i is the position of particle i, and V_tot is the total potential energy of the system, including particle-particle, particle-wall, and particle-barrier interactions. Finally, 𝐑(t) is a delta-correlated random variable with zero mean and unit variance. The dipolar interactions were truncated and shifted at a distance of 5 a.Most simulations were performed using N=120 particles, in a channel with periodic boundary conditions along the x-axis. We have confirmed that our results are qualitatively the same for larger systems of N=600 particles. We gratefully acknowledge funding from the German Research Foundation (DFG) within project LO 418/19-1. We thank Arjun Yodh, Laura Filion, and Marco Heinen for helpful discussions.@ifundefinedendmcitethebibliography58 f subitem(mcitesubitemcount) [Squires and Quake(2005)Squires, and Quake]micro Squires, T. M.; Quake, S. R. 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E 2009, 79, 051404 [Guyer et al.(2009)Guyer, Wheeler, and Warren]FiPy Guyer, J. E.; Wheeler, D.; Warren, J. A. FiPy: partial differential equations with python. Comput. Sci. Eng. 2009, 11 [Risken(1996)]Risken1996 Risken, H. The Fokker-Planck Equation: Methods of Solution and Applications, Vol. 18 of Springer Series in Synergetics, 3 ed.; Springer-Verlag: Berlin, 1996; p 289 | http://arxiv.org/abs/1709.09711v2 | {
"authors": [
"Urs Zimmermann",
"Hartmut Löwen",
"Christian Kreuter",
"Artur Erbe",
"Paul Leiderer",
"Frank Smallenburg"
],
"categories": [
"cond-mat.soft"
],
"primary_category": "cond-mat.soft",
"published": "20170927193736",
"title": "Climbing two hills is faster than one: collective barrier-crossing by colloids driven through a microchannel"
} |
Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen, DenmarkDepartment of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, SwedenWe show that spin-polarized local density of states (LDOS) measurements can uniquely determine the chiral nature of topologically protected edge states surrounding a ferromagnetic island embedded in a conventional superconductor with spin-orbit coupling. The spin-polarized LDOS show a strong spin-polarization directly tied to the normal direction of the edge, with opposite polarizations on opposite sides of the island, and with a distinct oscillatory pattern in energy. Probing chiral edge states in topological superconductors through spin-polarized local density of state measurements Annica M. Black-Schaffer====================================================================================================================The past few years have seen a rapid development of the field of topological superconductivity <cit.>. Motivated both by exploration of a new frontier in physics and the prospect of using Majorana bound states for topological quantum computation <cit.>, substantial theoretical and experimental progress have occurred <cit.>. While many topological superconductors are predicted to host Majorana bound states, they only arise under specific geometric conditions, such as when a topological superconductor forms a one-dimensional (1D) wire or in the superconducting vortex core of a 2D system <cit.>. Still, for other higher-dimensional geometries topologically protected edge statesnaturally also appear, although these states will form dispersing edge states. This is of great practical utility when scanning for candidate topological superconductors as it requires much less stringent experimental conditions. For example, islands of arbitrary shape and size can be studied, rather than highly specialized wire geometries.The most direct and natural way to study real space structures, such as topological superconductor islands, is through scanning tunneling microscopy (STM) measurements. Such an experiment was recently carried out on Pb/Co/Si(111) with clear indications of topologically protected edge states <cit.>. Here Pb provides superconductivity and Rashba spin-orbit interaction, while Co islands provide the necessary magnetism to seemingly generate a topological superconducting state inside the islands. While the evidence are convincing, it is important to remember that even if the edge states are established as due to a non-trivial topology, different topological phases can result in widely differing edge states. The most prominent distinction is that between a single chiral edge state and counter-propagating, or helical, edge states. This distinction is not only crucial for establishing the basic physical properties, but more importantly, only the chiral topological superconductor can host non-degenerate Majorana bound states, providing the most direct route for constructing Majorana bound state devices.Straightforward local density of states (LDOS) measurements cannot easily distinguish between chiral and helical edge states. However, spin-polarization around isolated magnetic impurities have recently been used as a tool to characterize different properties of spin-orbit coupled and topological superconductors <cit.>. It has also recently been shown that a chiral topological superconductor with Rashba spin-orbit interaction exhibits persistent spin-polarized currents along its edges <cit.>. These persistent currents can be understood as a consequence of the chirality, while the spin-polarization is due to the spins in the currents coupling to the Rashba spin-orbit interaction. Together, these results suggests that spin-polarization measurements might be very attractive for determining the nature of the edge states in candidate topological superconductors.In this work we perform extensive numerical calculations of the LDOS and spin-polarized LDOS for ferromagnetic islands embedded in conventional s-wave superconductors with Rashba spin-orbit coupling within the topologically non-trivial phase. We find that the edge states are clearly visible in the LDOS forming a characteristic x-shaped structure in real space when crossing the energy gap. This gives rise to a one-ring structure around the island at zero bias, while at higher energies the edge states naturally form a two-ring structure. This clearly establish the existence of edge states, but it is notably not possible to determine the number of edge states from this data.However, our spin-polarized LDOS results clearly shows that only one spin species cross the Fermi level at any given edge, manifestly proving the chiral nature of the edge states. In particular, we show that this branch has an in-plane spin-polarization directed along the normal to the edge. At higher ingap energies we find that the spin-polarization is transferred to the opposite edge of the island, which provides a very powerful experimentally accessible signature for chiral edge states. For completeness we also investigate the influence of a p-wave superconducting order parameter component instead of the Rashba spin-orbit interaction, as such are often used interchangeably in many theoretical models. However, we find no qualitative difference compared to the chiral state generated by finite spin-orbit interaction, and thus conclude that edge state properties does not change between using a p-wave component or a Rashba spin-orbit interaction. In particular, this is in contrast to arguments presented in Ref. <cit.>, where it was argued that a p-wave order parameter was needed to explain their experimental results. All the relevant signatures are here reproduced in both models and no extension beyond the conventional spin-orbit picture is therefore needed to explain the data. We consider a general 2D superconductor with spin-orbit coupling on a square lattice with additional finite sized ferromagnet islands, see Fig. <ref> and described by the Hamiltonian <cit.>ℋ = ℋ_kin + ℋ_Δ_s + ℋ_so + ℋ_Δ_p + ℋ_V_z.The two first termsare given byℋ_kin = -t∑_⟨𝐢,𝐣⟩,σc_𝐢σ^†c_𝐣σ - μ∑_𝐢,σc_𝐢σ^†c_𝐢σ,ℋ_Δ_s = ∑_𝐢(Δ_s c_𝐢↑^†c_𝐢↓^† +H.c.),and describe a conventional s-wave superconductor, where c_𝐢σ^† (c_𝐢σ) is a creation (annihilation) operator for a σ-spin on site 𝐢, t the nearest-neighbor hopping amplitude, μ the chemical potential, and Δ_s a conventional spin-singlet s-wave superconducting order parameter. The next two terms are ℋ_so = α∑_𝐢𝐛(e^iθ_𝐛c_𝐢+𝐛↓^†c_𝐢↑ +H.c.),ℋ_Δ_p = Δ_p∑_𝐢𝐛(e^iθ_𝐛c_𝐢+𝐛↑^†c_𝐢↓^† +H.c.),which adds a Rashba spin-orbit interaction with strength α and a chiral spin-triplet p-wave superconducting order parameter Δ_p, respectively. Here, 𝐛 runs over the vectors that point along the nearest-neighbor bonds and θ_𝐛 is its polar angle. In a superconductor with spin-orbit interaction such an explicit p-wave component is present if the Cooper pairs are formed in the basis where the original kinetic energy plus the spin-orbit interaction is diagonal <cit.>.Finally, we get the effect of a ferromagnetic island by adding the Zeeman exchange termℋ_V_z = -∑_𝐢,σ,σ'V_z(𝐢)(σ_z)_σσ'c_𝐢σ^†c_𝐢σ'.Unlike all other terms, which are homogeneous throughout the system, the Zeeman term V_z(𝐢) is given a spatially varying value V_z(𝐢) that smoothly transitions from zero outside of the island, to a finite value inside the island. We model the system described by Eq. (<ref>) on a large (201 × 201) square lattice with parameters (in units of t): μ = -3.9, Δ_s = 0.08, and one of Δ_p or α set to 0.28 while the other is set to 0. For the ferromagnetic islands we assume the profile V_z(𝐢) = 0.24(1/2 - atan((r-R)/W)/π), where r is the distance from the center of the system, R is the radius of the island, and W = 5 sets the scale over which the Zeeman term decays to zero at the island boundary. For the finite spin-orbit interaction α model, this choice of parameters put the ferromagnetic island well within a chiral topological phase with a single chiral edge state, since the condition for the topological phase reads (-4t+μ)^2 + |Δ_s|^2 < V_z^2 <cit.>. Note that without finite magnetism this model is always topologically trivial.In the alternative case of a finite p-wave pairing Δ_p, a time-reversal invariant topological phase with helical edge states occurs for very large Δ_p when no magnetism is present <cit.>, but we choose Δ_p such that the surrounding superconductor is decisively within the trivial phase. We have checked that the exact value of all parameters are not of importance and our results have general qualitative validity. This is also true when varying the dimensional parameters R and W. The particular choice of parameters have been chosen to provide quantitatively relevant results for the already experimentally realized system of Co ferromagnetic islands on Pb <cit.>.To solve Eq. (<ref>) we use a Chebyshev polynomial expansion method <cit.> to expand the non-principal part of the Green's function G_σσ'(𝐢,𝐢, E) using 4000 Chebyshev coefficients. The LDOS is then calculated as ρ(𝐢, E) = -∑_σ1/πG_σσ(𝐢,𝐢, E). Note that the imaginary part is here not taken as conventionally done when using the retarded Green's function, since the non-principal part has already been isolated in the Chebyshev expansion <cit.>. Similarly, the spin-polarized LDOS along the spin-polarization axis 𝐧̂ is calculated using ρ_𝐧̂(𝐢, E) = -1/π∑_σσ'(⟨𝐧̂|)_σG_σσ'(𝐢,𝐢, E)(|𝐧̂⟩)_σ', where |𝐧̂⟩ = cos(θ/2)|↑⟩ + sin(θ/2)e^iφ|↓⟩. All calculations were implemented using the TBTK library for discrete second-quantized models <cit.>.Turning to the results, we plot in Fig. <ref> the LDOS for three different island sizes, for the case of non-zero p-wave superconductivity (left) or Rashba spin-orbit interaction (right). As seen, the LDOS exhibits a clear x-shaped feature crossing through the energy gap around the edges of the ferromagnetic island, a feature that is remarkably similar for both models.Notably, these x-shaped states are very localized in space, which provides, even by itself, strong evidence for these states being topologically protected edge states.We note that this is also very similar to both the experimental and numerical results reported in Ref. [arXiv:1607.06353].While we know that the model with finite spin-orbit interaction have a chiral edge state, an LDOS measurement by itself can not give any information regarding whether these states are actually chiral or helical. In fact, from the total LDOS plots in Fig. <ref> it is tempting to conclude that it is two branches crossing the Fermi level, which would mean that the edge states are either helical or that there are two chiral modes, both clearly incorrect for at least the system with finite spin-orbit interaction. In Fig. <ref> we go further and present the the spin-polarized LDOS with the spin-polarization chosen along the positive x-axis. These plots immediately reveal that the x-up spin-polarization branch cross the Fermi surface only on one side of the sample (here left side), while this spin is completely absent on the opposite edge. A reversal of the spin-polarization axis similarly shows that the x-down branch crosses the Fermi level on the opposite edge. This result directly establishes that there is only one branch along the edge, which necessarily implies a single, spin-polarized, chiral edge state. In both Figs. <ref>-<ref> we see that the two models have the very same qualitative behavior for the whole range of grain sizes. We can therefore conclude that the two models are entirely interchangeable as far as the edge state features are concerned. Notably both models produce single chiral edge states around the ferromagnetic island. The only visible difference between the two models is that the model with Rashba spin-orbit interaction has a somewhat higher density of intragap states that pollutes the low energy spectrum inside the island. In the following we continue with a more detailed study of the Rashba spin-orbit interaction model, but note that we have confirmed all results in both models. We choose the spin-orbit interaction model because it has the more polluted low-energy spectrum, which means that low-energy edge state features clearly visible in this model are only clearer in the p-wave model. It is also the Rashba spin-orbit interaction that is the primary source of the non-trivial topology in actual materials, since the p-wave superconductivity is usually induced by the Rashba spin-orbit interaction.To further understand the properties of the chiral edge states in terms of LDOS measurements, we plot in Fig. <ref> both the LDOS and spin-polarized LDOS over the whole surface surrounding a R = 30 ferromagnetic island for three different energies. At E = 0 (bottom panels) the x-up spin-polarization is clearly localized on the left edge, while the total LDOS is symmetrically distributed around the whole edge. At a higher energy (middle panels) the spin-polarized LDOS is however transfered and instead becomes mainly localized on the opposite, right edge. This can be understod as a direct consequence of the avoided crossing for the up-spin branch at the right edge of the island (see center energy-resolved panel). This avoided crossing causes a locally flat dispersion and thus the spin-polarization of this gapped edge state overwhelms all other contributions starting at the avoided crossing energy. The result is a net x-up polarization on the right side of the island at this energy and thus an overall transfer of spin-polarization between the edges of the island as function of increasing energy. We here strongly emphasize that the chiral edge state at zero energy on the right side, with its x-down spin polarization, only exists because the x-up branch have this avoided crossing and is thus fully gapped. Thus this avoided crossing should not be seen as the remnant of any helical state, but it is an intrinsic component of any chiral edge state. A similar increased x-up spin concentration at the right edge also occur at the corresponding negative energy, although it is not displayed in the figure. We point out that the transfer in concentration of the spin-polarized LDOS from one edge to the other as a function of the bias voltage is particularly interesting from an experimental point of view. Namely, the presence of topologically protected chiral edge states can be detected as a characteristic oscillation of the spin-polarized LDOS from one edge to the other as the bias voltage is swept through the superconducting gap. This is particularly useful if the edge states are much more symmetrically distributed than in Fig. <ref>, for example if V_z is strong enough to almost entirely tilt the spins inside the island perpendicular to the surface. The difference between the edges for the spin-polarized signal is then notably smaller. The oscillating nature of the spin-polarized LDOS can be utilized as an additional signature because it provides a method for verifying whether the contrast is large enough to distinguish the values at the two different edges. Notably this can be done without any physical modification of either system or probe, only a change in bias voltage is needed. The oscillating spin-polarized LDOS can also be utilized to detect the chiral nature of the edge state when studying a single straight edge, rather than the edge of a circular island.At even higher energies (Fig. <ref>, top) a two-ring structure develops symmetrically around the whole edge with the spin-polarized LDOS approaching the same symmetric appearance as the total LDOS. Similarly, a two-ring structure also develops at the corresponding negative energies. Such a two-ring structure has also recently been reported experimentally <cit.>. We also note that, while the low-energy spectrum looks notably polluted inside the ferromagnetic island for the line cuts shown in Figs. <ref>-<ref>, the full 2D plots in Figs. <ref> make it clear that the edge features at the edge are in fact clearly dominating. Moreover, the notable ripple pattern seen in Fig. <ref> reveals that the intragap energy features are due to the tails of the edge states stretching into the island and are thus diminishing with increasing island size.We have also performed calculations for different spin-polarization directions. As long as the spin-polarization is in-plane, the spin-polarized LDOS shows similar results as in Fig. <ref>. The only difference is that the position of the maximum intensity rotates together with the spin-polarization axis, such that it always occur when the edge is perpendicular to the spin-polarization axis. However, if the spin-polarization axis is taken to be perpendicular to the surface, then the spin-polarized LDOS is practically identical to the LDOS. This is a consequence of the spin-polarization of the edge states having its origin in the Rashba spin-orbit interaction, which has only an in-plane spin dependence. In summary, we have shown that spin-polarized LDOS measurements can be a very powerful tool for detecting topological superconductors with chiral edge states. For any in-plane spin-polarization axis, the spin-polarized low-energy LDOS is located on only one side of the island in the topologically non-trivial phase. By simply sweeping the bias voltage through the gap the spin-polarization is transferred from one island side to the opposite.We thank J. Cayao, D. Roditchev, and P. Simon for useful discussions. 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"authors": [
"Kristofer Björnson",
"Annica M. Black-Schaffer"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170926143943",
"title": "Probing chiral edge states in topological superconductors through spin-polarized local density of state measurements"
} |
^1Planetarium Osnabrück, Klaus-Strick-Weg 10, D-49082 Osnabrück, Germany ^2Universität Wien, Institut für Astrophysik, Türkenschanzstraße 17, 1180 Wien, Austria tel: +43 1 4277 53800, e-mail: [email protected] (corresponding author)^3Parc Astronòmic Montsec, Comarcal de la Noguera, Pg. Angel Guimerà 28-30, 25600 Balaguer, Lleida, Spain^4Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, C.Martí i Franqués 1, 08028 Barcelona, Spain^5Département de physique, Cégep de Sherbrooke, Sherbrooke, Québec,J1E 4K1, Canada ^6Formerly with US National Park Service, Natural Sounds & Night Skies Division, 1201 Oakridge Dr, Suite 100, Fort Collins, CO 80525, USA^7Leibniz-Institute of Freshwater Ecology and Inland Fisheries, 12587 Berlin, Germany^8Eötvös Loránd University, Savaria Department of Physics, Károlyi Gáspár tér 4, 9700 Szombathely, Hungary^9National Institute for Public Health and the Environment, 3720 Bilthoven, The Netherlands^10DDQ Apps, Webservices, Project Management, Maastricht, The Netherlands^11LightPollutionMonitoring.Net, Urb. Veïnat Verneda 101 (Bustia 49), 17244 Cassà de la Selva, Girona, Spain^12Kuffner-Sternwarte,Johann-Staud-Straße 10, A-1160 Wien, Austria^13Deutsches GeoForschungsZentrum Potsdam, Telegrafenberg, 14473 Potsdam, GermanyMeasuring the brightness of the night sky has become an increasingly important topic in recent years, as artificial lights and their scattering by the Earth’s atmosphere continue spreading around the globe. Several instruments and techniques have been developed for this task. We give an overview of these, and discuss their strengths and limitations. The different quantities that can and should be derived when measuring the night sky brightness are discussed, as well as the procedures that have been and still need to be defined in this context. We conclude that in many situations, calibrated consumer digital cameras with fisheye lenses provide the best relation between ease-of-use and wealth of obtainable information on the night sky. While they do not obtain full spectral information, they are able to sample the complete sky in a period of minutes, with colour information in three bands. This is important, as given the current global changes in lamp spectra, changes in sky radiance observed only with single band devices may lead to incorrect conclusions regarding long term changes in sky brightness. The acquisition of all-sky information is desirable, as zenith-only information does not provide an adequate characterization of a site.Nevertheless, zenith-only single-band one-channel devices such as the “Sky Quality Meter” continue to be a viable option for long-term studies of night sky brightness and for studies conducted from a moving platform. Accurate interpretation of such data requires some understanding of the colour composition of the sky light. We recommend supplementing long-term time series derived with such devices with periodic all-sky sampling by a calibrated camera system and calibrated luxmeters or luminance meters.Atmospheric effectsSite testingLight pollutionTechniques: photometricTechniques: spectroscopic§ INTRODUCTIONThe last decade has seen a rapid increase in research into artificial light in the nighttime environment (Table 1). This increased attention can be attributed to several factors, including: recognition of the impacts of artificial light on ecology and health <cit.>, increasing amounts of artificial light in the environment <cit.>, improved quality of imagery from space <cit.>, the current global change in lighting technology <cit.>, and increasing quality, ease, instruments, and methods for measuring light at starlight intensities <cit.>. While the increase in possible ways to measure artificial light is a positive development, it likely presents a barrier to newcomers to the field of light measurement. Furthermore, both the various terms and units used in standard and astronomical photometry and the typical amounts of light experienced in the environment are not familiar to most scientists. This presents a challenge for interdisciplinary researchers, such as biologists who wish to measure the light exposure during a field experiment (see the discussion in <cit.> and examples of such studies <cit.>). This paper aims to introduce measurements of night sky brightness at visible wavelengths to readers with no background in the field. While the focus is the night sky, we expect this overview will be useful for readers interested in characterization of field sites for biological studies[Such readers should however keep in mind that properly characterizing sites for biological studies also involves measuring glare, and potentially ultraviolet and infrared radiation as well.]. We first provide a background on natural and artificial night sky brightness (Section 2). We then discuss different techniques for measuring sky brightness, starting with the visibility of stars and traditional naked-eye observations (Section 3), followed by measurements with single channel instruments (Section 4), imaging instruments (Section 5), and spectrometers (Section 6).The classic technique of astronomical photometry is discussed in the Appendix, as it is unlikely to be used by researchers outside of the field of astronomy. Unless otherwise stated, the paper assumes that direct sources of light are not present in the field of view of the instrument.It is important to note that in many cases we convert values into SI units for the sake of facilitating understanding, but in most cases this conversion is only approximate. For example, an inexpensive luxmeter that does not perfectly match the CIE spectral sensitivity <cit.> could give reasonably accurate readings for a source with a daylight-like spectrum, but have large errors for sources with a different spectral power distribution (such as sodium or fluorescent lamps). § SKY BRIGHTNESS Near human settlements, the brightness of the night sky is made up of light from both natural and artificial sources. Celestial light can travel directly to an observer (allowing one to see stars, galaxies, etc.), or it can scatter in the atmosphere and contribute to the diffuse glow of the sky. Some celestial sources appear as points or small objects (e.g. stars, planets, the moon), others are extended and diffuse (aurorae, airglow, galactic nebulae, galaxies). Even after setting, the sun can contribute diffuse light through scattering interactions with the atmosphere (twilight and noctilucent clouds) or from dust in the solar system (zodiacal light and Gegenschein). Details about many of these natural sources are discussed by Leinert et al. <cit.> and Noll et al. <cit.>, and typical values for night sky brightness are presented in Table 2.Artificial light emitted from the Earth’s surface can also scatter off molecules or aerosols in the atmosphere and return to Earth as “skyglow” <cit.>. On clear nights, this brightened sky results in a loss of star visibility <cit.>, especially near the horizon where the glow is brightest. It also reduces the degree of linear polarization of the clear moonlit sky <cit.>. Clouds have a strong influence on both natural and artificial sky brightness. In urban areas, clouds can increase skyglow by more than an order of magnitude, whereas in natural areas devoid of artificial light clouds make the sky modestly darker <cit.>.There are two parameters that can be used to quantify light in the environment. The first is “irradiance”, which can be thought of as the total amount of electromagnetic radiation falling on a surface. A well-lit office likely has a high level of irradiance, and relatively uniform values across over the floor and surfaces in the room. The second is “radiance”, which can be thought of as how bright a (usually small) area in your field of view appears. The unpleasant sensation felt when looking directly towards a glaring lamp is due to the large radiance. Spectrometers measure radiance (or irradiance) at many different wavelengths. The radiance per unit wavelength is called the spectral radiance. Other devices have a more restricted range. In this case, the spectral sensitivity of the device must be reported along with the radiance value. Photometric instruments are specially designed so that their spectral response matches a simplified theoretical response curve for the human visual system <cit.>. When the spectral response is perfectly matched to human vision, then measurements from these instruments are called “luminance” (or popular "brightness") and “illuminance”, instead of “radiance” and “irradiance”. For measurements of the sky brightness, it is important to further clarify whether sky radiance is being measured in the areas between the visible stars (“sky background brightness”), or whether starlight is included (“sky brightness”). Irradiance always includes all sources of light.Traditionally, sky brightness is measured by astronomers in the astronomical magnitude system mag/arcsec^ 2 (magnitude per square arcsecond; e.g. <cit.>). The idea behind this system is that if an area on the sky contained only exactly one magnitude X star in each square arcsecond, the sky brightness would be X mag/arcsec^ 2. The magnitude system was introduced by the ancient Greek astronomer Hipparchos, who assigned a magnitude of 1 for the brightest stars visible to the naked eye, and magnitude 6 for the faintest stars visible to the naked eye (in a time before widespread light pollution). For this reason, larger values in mag/arcsec^ 2 indicate darker skies. Once accurate measurement techniques became available, this system was later quantified and extended to stars and other objects in the sky much fainter than 6th magnitude <cit.>.Astronomers measure radiances in different wavelength ranges. Similar to the way the photopic system standardized measurement of “human visible” light, the “UBV system” or “Johnson system” <cit.> of ultraviolet (U), blue (B), and green (visual = V) filters allows astronomers to make and report consistent observations in other color bands. The green V spectral band is not greatly different from the visual photometric spectral band, so astronomical brightness values in mag_V can be approximately transformed to photometric values <cit.>: Luminance[cd/m^2]≈10.8 × 10^4 × 10^-0.4*mag_V The darkest places on earth have a sky background brightness of about 22 mag_V/arcsec^ 2, while in bright cities it is often 16–17 mag_V/arcsec^ 2<cit.>. Typical sky brightness for different locations and meteorological situations are shown in Table 2. We make use of the approximation that the sky brightness is uniform across the entire sky, so the luminance and illuminance are related by a factor of π. In reality, this is a conservative estimate <cit.>, as sky brightness tends to be brightest near the horizon <cit.>, especially on clear nights in predominantly artificially lit areas. The sky background brightness has also been measured from outside of the atmosphere, where the Hubble Space Telescope found 22.1–23.3 mag_V/arcsec^ 2, depending on the galactic latitude (Table 3). § CONSTRAINING THE NIGHT SKY BRIGHTNESS BY VISUAL OBSERVATIONS A traditional method used to qualify the night sky quality is evaluating the “limiting magnitude”, the magnitude of the faintest star visible to the naked eye (e.g. <cit.>). The technique makes use of the contrast threshold of the human visual system: with a bright sky background, only bright stars can be seen, while against a dark background fainter stars are distinguishable. Crumey <cit.> recently derived a new formula to describe the relation between limiting magnitude and sky brightness: m_lim = 0.426 μ - 2.365 - 2.5 log F with m_lim limiting visual star magnitude (in mag), μ sky background brightness in the V band in mag_V/arcsec^ 2, and a field factor F, typically between 1.4 and 2.4, which accounts for elements such as the observer’s experience and visual acuity. Several methods can be used to classify limiting magnitude, for example observing a sequence of stars of decreasing magnitude around the celestial North Pole, or the number of visible stars in some special star fields, which is proportional to the limiting magnitude (e.g. <cit.>). In the Globe at Night <cit.> and “How many stars can we still see?” <cit.> projects, observers compare their view of prominent constellations to a set of star charts with integer limiting magnitudes. This lowers the possible precision of the observations to ±1.2 magnitudes <cit.>, but makes broad participation including non-experts possible.The “Loss of the Night” app for Android and iOS devices<cit.> was developed to allow observers with no astronomical knowledge to make observations with higher precision. The app uses the smartphone’s inertial sensors to display a live-updating star map. It directs observers to look for individual stars in the sky, and report whether they are visible (including with averted vision), not visible, or if they cannot be observed for some other reason (e.g. a tree in the way). As the observer reports results for many stars of differing magnitudes, precision as good as 0.05 magnitudes is possible, and the precision of individual observations can be estimated based on the self-consistency of the data. The star catalog in the Loss of the Night app only extends to about magnitude 5.2, so its useful use is restricted to urban and suburban areas. The data from both Globe at Night and the Loss of the Night app can be displayed and evaluated at the www.myskyatnight.com website. § ONE DIMENSIONAL INSTRUMENTS The following subsections give an overview of devices that measure the sky brightness using a single channel, and typically observing only at zenith. One dimensional instruments measure the sky brightness as a sum of both the sky background brightness and the stars within the viewing field. In order to facilitate data exchange, a standard data format for acquiring data from one dimensional devices was adopted in 2012 <cit.>. §.§ The Sky Quality Meter The Sky Quality Meter (SQM) was originally developed by Unihedron mainly as a tool for amateur astronomers to measure the night sky brightness at their observation sites <cit.>. In recent years, it has been used in a large number of studies of sky brightness (Table 1). The SQM detector consists of a solid state light-to-frequency detector (TAOS TSL237S). The spectral response encompasses the photopic eye response, but is more sensitive for shorter wavelenghts, i.e. more blue sensitive than a truly photopic response. Luminances are reported in the unitmag_SQM/arcsec^ 2, but can be converted into mcd/m^ 2 using Eq. (<ref>) and the approximate relation:mag_V ≈ mag_SQM. Note, however, that differences between the three photometric systems (SQM, Johnson V and visual photometric) can occur for different colors of the night sky <cit.>. The SQM spectral response actually depends on four components: the sensor, an infrared filter, the lens, and any weatherproof screen, and is discussed in detail in <cit.>.Luminance values down to the order of 100 μcd/m^ 2 (∼22.5 mag_SQM/arcsec^2) can be measured. The field of view of the SQM is 20^∘ (full width at half maximum, FWHM) in the case of more commonly used lensed version (SQM-L). Normally measurements are taken near zenith. In areas with little artificial skyglow, nearby lamps (e.g. at 10^∘ above the horizontal) can affect the measurement <cit.>. This can be due to the residual sensitivity of the SQM at large angles, or scattered light from the lamp itself dominating over the natural sky brightness. The SQMs are calibrated against a reference light meter. This reference light meter is a calibrated lux meter that meets CIE (International Commission on Illumination) regulations with an accuracy of 5%. The applied light source was an integrating sphere using a compact fluorescent bulb until February 2011, and afterwards a green Light Emitting Diode (LED) light source with its peak at 520 nm (Tekatch, priv. comm.). The LED calibrated instruments (since serial number 5944) measure about 0.15 to 0.2 mag/arcsec^ 2 systematically brighter than the older ones. The SQMs are calibrated to the same constant surface brightness of 8.71 mag/arcsec^ 2. The dark current is determined in complete darkness. The temperature dependence of the photodiode was determined for a sample of sensors, and is corrected for by the electronics before readout <cit.>.The manufacturer reports an accuracy of ±10%, corresponding to ±0.1 mag_SQM/arcsec^ 2. During the KIck-Off IntercomparisonS Campaign (KIOS 2011), nine SQM’s operated in the Dutch Night Sky Brightness Monitoring network were intercompared at the Cabauw Experimental Site for Atmospheric Research in The Netherlands. The KIOS 2011 campaign showed an initial scatter between the individual instruments of ±14%, ranging from −16% to +20%. With intercalibration methods it was possible to reduce this to 0.5%, and −7% to +9%, respectively <cit.>.The long-term behaviour of SQMs is still an issue to be examined in detail. Pioneering work in this context has been done by So <cit.>, who measured a long-term change in the SQM measurements mainly related to degradation of housing window. Also den Outer et al. <cit.> discuss long-term changes in the response of this kind of devices.Different types of SQM devices are available. The most important difference between the devices is those with lens (SQM-L) and without. We strongly recommend the use of the L version because it reduces the field of view to around 20 degrees (FWHM), leading to more consistent readings when nearby sources of light are present. The SQM is available as a handheld or connected device, via USB, Ethernet or RS232. These connected devices require a computer in the field for data acquisition and storage. A data logging device (SQM-LU-DL) can store data without a computer. All devices for continuous outdoor measurements require weatherproof housing (available off-the shelf) to operate under all weather conditions. Since it is a rather inexpensive device, ready to use and with the possibility to do automatic measurements, the SQM is widely used for monitoring night sky brightness. In long-term observations, the sampling rate should be ideally at least once per minute <cit.>, but a 15 minute interval is sometimes necessary for battery-operated SQM-LU-DL. In the case of scanning observations or relatively rapid changes in the night sky brightness, it is critical that the sampling period is longer than the time needed by the device to record an observation. User-written acquisition software enables the user to extend this range. The minimum sampling rate depends on the amount of light.In the case of making observations with a handheld SQM-L, it is recommended to skip the first 3–4 measurements, and then to average the result of four observations, rotating the SQM (and observer’s body) 90^∘ after each observation to a different compass direction <cit.>. If the SQM-L is affected by stray light, this may minimize or reveal the effect. It also reduces errors due to pointing inaccuracy <cit.>. If the four observations are not self-consistent (maximum range about 0.2 mag_SQM/arcsec^ 2), then it is probably not a good location, and the data should not be recorded. It is not recommended to use SQM devices to report measurements on clear moonlit nights because of the possibility of stray light on the detector<cit.>. Handheld SQM-L observations can be archived with Globe at Night or via the Loss of the Night app, and can be viewed at www.myskyatnight.com.SQM-L meters have also successfully been used to sample multiple points in the celestial hemisphere <cit.>. This permits quantitative monitoring toward multiple azimuths, and thus tracking of the relative contributions from multiple light domes. Another method is the use on a car's roof together with a GPS detector, to collect positions and sky brightness simultaneously over large areas with a software called "Roadrunner" <cit.>. Recently, the European funded project STARS4ALL has developed a new detector that uses the same TSL237 photodiode detector as the SQM. This instrument is called TESS and it has an extended bandpass compared to SQM devices. TESS also uses a dichroic filter, allowing for better coverage of the red band of the visible spectra with good response <cit.>.TESS was designed with the goal of creating a large European network with inexpensive but well tested photometers. In fact, the device is calibrated by the manufacturer, and it contains a complete system to transmit the data to STARS4ALL servers, where it will be accessible for researchers and general public. The device also includes an infrared detector, in order to obtain information about the presence or absence of clouds in the field of view of the detector.§.§ Dark Sky Meter The Dark Sky Meter is an iPhone app which uses the back camera of iPhone 4S (and later models) to collect light and determine a sky brightness value. The app was developed by DDQ, a software company from the Netherlands, and is available via the Apple Store. Older and other models (e.g. iPads and iPods) are not supported because the camera chip does not detect enough light in a single exposure. The iPhone camera is not designed for long exposures, but the most recent versions of the iPhone’s CMOS (Complementary metal–oxide–semiconductor) sensor (Sony IMX145) are sensitive enough to collect light at 21 mag/arsec^ 2, with a practical limit around 21.8 mag/arcsec^ 2. Apple restricts developers to adjust exposure times, so readings are taken as a series of shots.In addition to recording sky radiance, the app records device inclination (to know if the device is pointed correctly towards the zenith), moon phase, cloudiness (input by user), and GPS location. For scientific use, broadband spectral radiances are transmitted separately as three RGB values. The units of measurement are mag_SQM/arcsec^ 2. Estimated naked eye limiting magnitude is also reported. The sensitivity range is between 12 and 22 mag/arcsec^ 2, with the darkest value depending on the iPhone model, as the software restricts readings above 22 mag/arcsec^ 2. The advertised measurement accuracy is ±0.2 mag/arcsec^ 2. The typical absolute calibration differences are 20% for iPhone4S devices and 30% with iPhone5 devices. The field of view is around 20^∘ based on binned 240x240 LRGB images.The data obtained can be automatically submitted to the public Globe at Night database, and are available for further works and analysis. Data under clear night conditions and with the phone pointed towards zenith are available at www.myskyatnight.com, and the complete dataset can be viewed at www.darkskymeter.com.§.§ Solar-cell-based Lightmeter The IYA-lightmeter (International Year of Astronomy 2009) uses quasi-continuous measurements of the photoelectric current of a solar cell (see Fig. 3) to generate proxies for illuminance and irradiance (total radiation) at a site. It is designed for long-term monitoring and uses SI-units to support communication with the public, technical lighting and the Earth- and life sciences. Because artificial night-brightening is strongest with "bad" weather and near the horizon, the instrument is all-sky and all-weather capable. Long term stability of the device is assured by the industrial solar cell, developed for outdoor use. Remaining weathering effects of the cell or the readout electronics are taken into account by repeated on-site, on-the-fly calibrations to the Sun, the Moon and the twilight and an atmospheric model <cit.>. These are made from the monitoring time-series and require no extra measurements or visits to the site.To expand the dynamic range, at bright light levels the lightmeter response is non-linear. Below 0.1 lx, the lightmeter has a linear response, and can be calibrated to another instrument by simultaneously measuring a weak reference source, e.g. the half Moon, and applying a constant factor to the readout. The natural light of moonless, astronomical nights is dominated by the airglow and modulated by the atmosphere. The variation exceeds a factor of 2 and needs separation from artificial brightening. The high-frequency lightmeter time-series (temporal resolution: up to several samples per second) allow the use of cloud-indicators developed for astronomical- and pyranometer-time series, and, when the Sun or the Moon is present, the derivation of atmospheric extinction. The daylight capability allows to separate contributions due to atmospheric trends by comparison with existing IMO-standardised, global, long-term, total-radiation series of climate and meteorological research.The Lightmeter has a sensitivity at a readout rate of up to 10 Hz from about 10 μlx (10^-5 lx) to above 200 000 lx with 1% resolution (Mark 2.3 version). Thus it covers the full range of human perception of light with one sensor. The spectral response is mainly due to the amorphous Si solar cell, with contributions by the protective glass-layer. The field of view is the upper hemisphere (2π sr), with an essentially ideal Lambertian response to contributions away from zenith, giving a FWHM of 120^∘. It needs a data logger with USB support (netbook, Raspberry-Pi). The data format was standardized for the German Astronomical Virtual Observatory (GAVO) “lightweather” database <cit.>.§.§ Luxmeters and Luminance meters Illuminance and luminance can be measured with commercially available instruments, called luxmeters or luminance meters. Higher quality instruments are adapted to the spectral sensitivity curve of V(λ)<cit.> with an accuracy of less than 6% and a total error of 10% according to the German industry norm DIN 5032 class B. For an illuminance meter, the angular sensitivity characteristic must be adapted to the cosine distribution of incident light to at least 3%. For field measurements the homogenous illuminance from the sky is difficult to realize due to obstruction by buildings, trees or disturbances through lamps. The main disadvantage of most such meters is the low sensitivity and low accuracy at low light levels. For a luxmeter, a limiting sensitivity of 0.1 to 0.01 lux is often specified, for luminance meters a lower limit of 0.01 to 0.001 cd/m^ 2. Given that the night sky brightness can reach values close to 1 mlux, this implies that most luxmeters and luminance meters are not sensitive enough for the tasks discussed in this paper. §.§ Digilum “Digilum” is a specially designed luminance meter with a large measurement range from day to night time. It was developed by Henk Spoelstra, and manufactured by Instrument Systems GmbH –- Optronik Division (Germany). The spectral sensitivity is exactly photopic (adjusted to the V_λ sensitivity), and measures luminance between 0.1 mcd/m^ 2 to 20 kcd/m^ 2. The measurement accuracy for night sky brightness values below 1 mcd/m^ 2 ranges from 5% up to about 20% at 0.25 mcd/m^ 2 for temperatures roughly above 5 ^∘C.It is calibrated annually in the laboratory, and the dark current is corrected through a temperature correction. The main advantage of the instrument is the strict spectral sensitivity to the V(λ) curve. It can be read out up to once per second, and the field of view is about 5^∘. § TWO DIMENSIONAL (IMAGING) INSTRUMENTSThe following subsections give an overview of devices and techniques to map and measure the sky brightness by analyzing wide-angle images (preferably images of the whole upper hemisphere). Some observations are based on CCD (charge coupled device) cameras, others on CMOS sensors of commercially available DSLR (digital single-lens reflex) cameras. Depending on the method of reduction, observations based on two dimensional instruments can report either sky luminance or sky background luminance (see Sect. <ref>). In the case of sky background luminance, they necessarily report darker values (larger mag/arcsec^ 2) than one dimensional instruments. §.§ The All-Sky Transmission Monitor (ASTMON) The All-Sky Transmission Monitor (ASTMON) is based on a f=4.5mm fisheye lens and an integrated astronomical CCD camera. It measures the luminance of the complete night sky in several wavelength bands including the standard astronomical unit mag_V/arcsec^2.The system is designed to perform a continuous monitoring of the surface brightness of the night sky background in a fully robotic mode. In addition to the sky background brightness, ASTMON can provide atmospheric extinction and cloud coverage estimates for the entire sky surface at the same time <cit.>.The spectral range of the instrument is directly related to the spectral sensitivity of the detector and the use of the filter wheel, which may contain one, three, or five filters depending on the version. The most common setup is with Johnson B, V, and R astronomical filters, but other filters with a 1.25 inch diameter may be added.Because the detector is an astronomical CCD camera, different exposure times can be used to cover a wide range of sky luminances. When setting the exposure, it is recommended to keep star counts below 40,000 to avoid loss of linearity. The calibration procedure is based on astronomical photometry and the accuracy of the measurement of sky brightness is related to this procedure. Typical accuracies are around 0.15 magnitudes for U and I filters and around 0.02 for B, V and R Johnson filters, which corresponds to about 15% and 2%, respectively.The instrument is available in three different versions: ASTMON Full, ASTMON Lite (see Fig. 4) and ASTMON Micro. The Full version is designed to be installed outdoors as a fully robotic and continuous monitoring station. This version is protected with a complete enclosure and with a solar shutter to prevent sunlight from damaging the system.The Lite version is a portable, with a tripod and weatherproof enclosure. The system can be used outdoors for a few weeks without problems, but it is not as safe as the full permanent version. Finally the last version is Micro, which is the smallest and operates without a filter wheel, so only a Johnson V filter is permanently installed. The system is completely controlled by a computer through a specific software designed by the manufacturer of the instrument. In case of the portable devices, it is necessary to correctly adjust the device's orientation.Every observation image has to provide a good signal-to-noise ratio and enough bright stars to find an astrometric and photometric solution. An algorithm evaluates the image to produce a catalogue of the stars in the field of view, and this catalogue is cross-matched with an astrometrically and photometrically calibrated catalogue. The processing of data includes a typical astronomical image preprocessing (dark, bias, and flat fields). This last step is especially critical because of the field distortion in an all-sky device, so ASTMON is provided with a master flat, and it is possible to update the calibration with a uniformly illuminated sphere like DomeLight <cit.>.The processing of the data to obtain night sky brightness measurements is based on classical astronomical photometry, with the determination of zero points and extinction coefficient (see Appendix). In the case of bad photometric conditions without identified stars, ASTMON uses a default calibration. In case of good photometric conditions, the parameters are determined and are updated as a new default. With all the calibrations and parameters established, the system can determine the luminance of the sky background in mag/arcsec^ 2<cit.> (see Fig. 5). The value of the sky brightness is corrected for field distortion, because in most all-sky devices, each pixel covers a different area of the sky.Typical exposure time values in unpolluted areas are 300 seconds for the U filter and 40 seconds for the others (B, V and R). Exposures must be short enough to prevent smearing due to the rotation of the Earth, but long enough to get a good signal-to-noise ratio of the night sky background. The calibration processes are automatically run after each sequence of observations (one image with every selected filter). The typical time for a complete sequence of 5 filters is around 8 minutes, allowing the user to track nightly variations in sky brightness with 70 independent observations during a standard night. The software provided by the manufacturer is not open source, so some of the parameters are not controllable by the user. For this reason, there are alternative options for the processing of ASTMON images that could be applied for other all sky devices such as DSLR cameras (discussed below). One alternative option is the PyASB software <cit.>. This open source python code is still under development, but can already generate processed all sky maps of sky background brightness, and estimates of cloud coverage and atmospheric extinction according to photometric analysis of the images. Among other places, this software has been used during the Intercomparison Campaigns of Loss of the Night Network <cit.> and as well as in works related to all sky modal analysis decomposition<cit.>.ASTMON has been installed in several National Parks and at astronomical observatories (e.g. Calar Alto Observatory, Canary Islands Observatories, Universidad Complutense de Madrid) <cit.>. §.§ Digital cameras equipped with wide angle and fisheye lenses Similar to ASTMON, a commercial camera with a fisheye lens can provide quantitative information on the luminance of the whole upper hemisphere with a single exposure. Compared to CCD cameras, these systems are more easily accessible, and are highly transportable (<cit.>, <cit.>). The method has been proved to be an effective tool to characterize potential dark sky parks (e.g. <cit.>). Cameras cannot currently be used for measurements 'off-the-shelf', but only after calibration. The precision of the measurements depends on the calibration procedure and on the camera itself. The CMOS sensors and the analogue-to-digital conversion of modern cameras with 14 bit digitization provide linear measuring possibility in a 1-16384 dynamic range with a single exposure. That is usually enough for night sky monitoring, and higher dynamic range can be obtained with HDR imaging <cit.>. For precise measurements, dark frames (images with the same exposure but covered with the lens cap) have to be taken, which can be used to compensate for the dark signal.Calibration is the difficult part of photometry with a camera (<cit.>). There are at least four different data processing steps involved: lens vignetting (flat fielding in CCD photometry), geometric distortion especially with wide field lenses, colour sensitivity of the camera, and the sensitivity function of the camera. Vignetting corrections are essential, as the transmission of a fisheye lens can drop to below 50% at the edges compared to the image centre. The most accurate way to correct for vignetting is to use an integrating sphere in a laboratory as a “flat” source. With this method, a precision of ∼1-2% can be obtained.The colour sensitivity curve of the green (G) filter in cameras has a very large overlap with the astronomical Johnson V photometric band, with some degree of similarity to both the photopic and scotopic curves of human vision (see Fig. 2). Therefore, the most straightforward method to derive luminance from camera images is to extract the brightness values only from the G colour band in the raw data. More accurate luminance values can be derived using a linear combination of the RGB values with suitable coefficients (much smaller coefficients for the R and B compared to the G band).In order to estimate the possible error of the colour transformation, Kolláth <cit.> selected a set of spectra representing common light sources and used average camera sensitivity curves for the R, G, B filters and simulated different scenarios in fitting the coefficients. This showed that the relative error due to colour transformation is not larger than 3%, and even smaller than 2% for the Johnson V filter.There are two different methods to calibrate the DSLR system: (a) laboratory measurements with a luminance meter and standard light sources; (b) stellar photometry to get extinction corrected sky background in astronomical V magnitude (see, e.g., <cit.>). The calibration should be performed at different luminance and exposure levels in order to map the whole dynamic range of the transmission curve. In both cases, it is difficult to reach a precision better than 5%.Summing the error sources of the calibration steps together, we can conclude that the precision of camera photometry can reach the 10% range with thorough measurements and data processing. When the precision is not critical but the directional distribution of sky brightness is important (e.g. complementary data to other instruments), the images can be scaled to the other measurements as an alternative way of calibration <cit.>. For most cameras, the longest exposure time allowed without a remote switch is 30 seconds. In bright locations, a shorter exposure time will be necessary. Test exposures are needed to ensure that the sky exposure is far from both the zero level and saturation. According to our experience, up to 30 second exposures can be used in suburban locations; but under skies without light pollution, at least 2–3 minute long exposures are recommended at ISO 800–1600. The data obtained by cameras are stored in an image format. Compressed formats like jpg and gif cannot be used, as they do not store the original values observed for each pixel. Rather, data must be stored in a raw and/or astronomical FITS format. It is common to generate false colour images of the measurements, as this provides the simplest visualization of the data. See Fig. <ref> for an example of an RGB image a) and a luminance false colour plot b).In addition to the night sky brightness, the correlated colour temperature of the night sky can also be derived by analysing raw DSLR images. For this purpose and many more tasks, a dedicated software, “Sky Quality Camera”, has been developed by Andrej Mohar (priv. comm.). §.§ All-sky mosaics This measurement approach pioneered by the US National Parks Service (NPS) provides all-sky coverage similar to that obtained with a fisheye lens, but by mosaicking multiple wide-field images taken in succession <cit.>.The NPS image scale (93.5 arcsec / pixel) is sufficient to perform accurate stellar photometry required for calibration to known standards, and produces panoramic or fisheye 39.3 mega-pixel images. Detailed images of sky luminance, expressed as astronomical V magnitudes per square arcsecond, can be transformed into photopic measures (e.g. cd/m^ 2) or horizontal or vertical illuminance (expressed as lux) at the ground level.The NPS hardware is a 1024 x 1024 px CCD detector (front-illuminated Kodak 1001E), a standard 50 mm f/2 camera lens, an astronomical photometric Bessel V filter with IR blocker, and a consumer-grade computer-controlled robotic telescope mount (Fig. <ref>). A total of 45 images covers the entire upper hemisphere and the lower hemisphere down to -6^∘, with a full acquisition requiring about 20 minutes. Data collection is orchestrated with a small portable computer, commercial software, and custom scripts.The processing of the images follows the standard astronomical photometry methods, with preprocessing with master flats, bias and dark frames. Determination of zero point and extinction coefficient is done by aperture photometry with standard stars. The typical maximum error in luminance measurement across the entire frame is ±10 μcd/m^ 2. Wide angle rectilinear lenses produce images (see Fig. <ref>) with a varying pixel scale from center to edge. The pixel scale used to determine the number of square arc seconds per pixel for the entire frame is that which occurs on the image where correction by the master flat is unity. The high resolution across the celestial hemisphere obtained by a robotic system is an advantage in quantifying small distant sources of sky glow. It is also beneficial for measuring the luminance of small bright sources accurately, and for detecting change over time,and for quantifying atmospheric extinction. The system is less suited for intensive surveillance monitoring, but well-suited for long-range monitoring from mountain tops and high-value sites.A similar system has been employed by Falchi <cit.> (see also <cit.> and <cit.>). While their detector has a relatively low resolution, the large pixel size (24 μm) allows for relatively short integration times per frame to achieve adequate signal to noise in the sky background with an f/2.8 optical system, even in dark environments (18 seconds maximum exposure).By fitting a strong neutral density filter in front of the lens, the same camera may be used to measure luminance or illuminance of nearby bright sources (such as unshielded outdoor lamps).§ SPECTRA OF THE NIGHT SKY This section discusses the study of the spectral power distribution (SPD) of night sky brightness. The SPD is produced by a combination of natural and artificial sources, and is dependent on nearby environmental characteristics such as atmospheric aerosol content or effects of the reflecting surfaces (ground, trees, buildings). Since artificial light sources often have considerably different SPDs, evaluation of the SPD during nighttime can provide information regarding the kind of sources that are responsible for the generation of night sky brightness. This is particularly beneficial during the current rapid change in lighting technology <cit.>.One approach to evaluate the SPD is the classical usage of telescopes and astronomical spectrometers to decompose the light and identify different contributions on the night sky brightness. This strategy was used, e.g., at the ESO Observatories in Chile <cit.>, in La Palma <cit.>, and at the Vienna Observatory <cit.>. Another option are dedicated devices to evaluate the SPD of the night sky. The Spectrometer for Aerosol Night Detection (SAND) will be presented as an example.SAND was initially developed for work in urban areas, but it was adapted with better sensitivity to operate in dark places <cit.>. The latest version of SAND is shown in Fig. <ref>. SAND-4 is a long slit spectrometer which uses a CCD imaging camera as light sensor. SAND-4 has a spectral resolution of 2 nm, and the spectral range is from 400 to 720 nm. The instrument is fully automated so that it can operate on its own with minimal human intervention.With spectrometers, it is relatively easy to separate total sky radiance into its major contributing sources. Figure <ref> shows a typical spectrum from an area minimally affected by artificial skyglow and without moonlight. This spectrum was taken with SAND in January 2014 at El Leoncito Observatory (Argentina) during a site characterization campaign in the framework of the Cherenkov Telescope Array project <cit.>. An example of a spectrum for a highly light polluted site is shown in Fig. <ref>. In that case, artificial sky brightness is clearly dominant, and the natural contribution can be effectively neglected (at least when the moon is down). The typical integration time for urban sites is on the order of a few minutes, while the integration time rises to about two hours for sites without artificial skyglow. § CONCLUSIONS AND OUTLOOK Quantifying light levels in a given nocturnal environment is important for many different disciplines in science, ranging from ecology to astronomy. One major obstacle for progress in this interdisciplinary field has been the large number of measurement systems and units in use. Conversion between measured quantities is roughly possible in some cases (e.g. mag/arcsec^ 2 to cd/m^ 2), while other times a conversion is only possible under particular assumptions (e.g. luminance cd/m^ 2 to illuminance lux, see <cit.>). The proper choice of the measurement method also depends on the information one wants to get. Is it sufficient to know the sky brightness only at the zenith, shall the illuminance of the whole sky on a horizontal surface be measured, or is it necessary to know the vertical illuminance of light sources near the horizon?To characterize a specific location at a specific time, the ideal observation would report spectrally resolved radiance at relatively fine angular resolution in all possible directions. Unfortunately, this cannot be achieved with currently available instruments at nocturnal luminance levels typical in non-urban environments. Trade-offs must be made according to the experimental task that is being undertaken. Examples of such tasks are monitoring the long-term trend in a location <cit.>, and characterizing an experimental field site <cit.>.During the past two decades, several new experimental techniques for measuring the night sky brightness have been developed. We have presented a number of commonly used instruments which are suitable for this purpose. These instruments can be sorted into several classes: “one-dimensional” versus “imaging” instruments, and broadband versus spectrally resolving instruments. Each of the instruments has differing strengths and weaknesses, which may make them more or less appropriate for a given experimental task. An overview of each of the instruments discussed is given in Tab. 4.One dimensional instruments are the most strongly suited for monitoring temporal (especially long-term) changes in sky brightness. As many are relatively low-cost, it means that they can be used in many different locations, including regular use by citizen scientists (often amateur astronomers). Such instruments have several major weaknesses, however. First, they only measure the sky brightness but not the atmospheric extinction, which influences the astronomical sky quality considerably. Second, in most installations, they only provide information about the sky brightness in a given pointing direction (usually zenith), while real skies are characterized by strong angular gradients (Fig. <ref>). Third, they have broadband spectral response, and this does not correspond to standards such as photometrical V or astronomical Johnson V <cit.>. Finally, under very clear skies, the distinction between brightening of the sky by natural airglow and brightening by a moderate amount of scattered artificial light is almost impossible on the basis of measurements with such devices.A decisive strength of imaging instruments is that they can provide information about the radiation from all directions. The data is intuitive to understand, and if directed properly the instruments can be used to provide information about what a nocturnally active animal actually perceives. Moreover, they provide broadband spectral information in three (RGB) channels. This is helpful to monitor changes in artificial lighting techniques (especially during the transition to solid state lighting <cit.>), and spectral regions beyond the V band could be important to understand their influence on animals or plants affected by artificial light at night. Nocturnal insects, for example, are strongly influenced by ultraviolet and blue light <cit.>. In this respect, DSLR measurements in the B channel can deliver valuable information on the biological impact of light sources.All-sky information can be easily obtained in a single image with 180^∘ fisheye lenses. However, such lenses have strong vignetting, transmitting only about 50 percent of the light at the horizon where most artificial light sources are. To measure these influences accurately, calibration and orientation of fisheye cameras is very critical. Longer focal lengths have fewer problems with vignetting, but distortions must be well understood in order to stitch pictures together to a full sky frame. One of the barriers to making use of cameras is the lack of free software for calibration and display of the data. There has been substantial progress in this area, for example the development of software like PyASB<cit.>, dclum<cit.>, and the National Park Service display system. Nevertheless, there is a clear need for universally accepted calibration algorithms for deriving night sky luminances from camera images, like the ones used in luminance measurements (e.g. LMK mobile air <cit.>). For most instruments, the long term stability is not well known. Digital cameras seem to be very stable, while it is known that the plastic window of the SQM housing loses transmission over time, at least under tropical climates (So 2014). Calibration could either be done under photometric conditions in a laboratory, or using stars as a stable light source whose brightness generally is known to 1% accuracy. In some cases, the decision of which instrument to use will not be either/or. For example, long-term measurements with an SQM would be greatly complemented by periodic validation with an imaging system and/or a spectrometer. It is important to periodically re-calibrate long term systems under laboratory conditions, or as part of intercomparison campaigns<cit.>.To ease interpretation of current results in the long-term future (when the currently available devices are no longer commercially available), it would be useful to establish a database of device and lens properties (e.g. geometrical correction), so that people in the future could recover the radiometric values for comparison to contemporary devices. This database would also be useful at the current time to ease calibration of cameras. It would be ideal to store some sort of “minimally processed” data rather than raw camera images. Based on our experience with each of the techniques our discussions during the development of this review, we conclude that in many situations calibrated digital cameras with fisheye lenses usually provide the best compromise between cost, ease-of-use, and amount of information obtained. The development of standard software for calibration and display of such data should be a high priority of the field of light at night researchers.§ APPENDIX. CLASSICAL ASTRONOMICAL PHOTOMETRY The classical approach to evaluate the night sky background brightness is based on astronomical star photometry in defined photometric systems (eg. UBV Johnson system). Normally this technique is used for stellar brightness measurements with photometers and CCD detectors through filters on telescopes. Some new devices use the same strategy to determine the sky brightness, taking standard astronomical stars as reference candles to estimate the background brightness. Here we will summarize the basic steps of this procedure.First of all, in order to use this method it is necessary to observe during nighttime (after the astronomical twilight when the Sun is below 18^∘ under the horizon). To obtain good photometric results, it is also necessary to observe when the Moon is below 10° under the horizon <cit.>.The best quality results are obtained during nights of photometric quality. These nights follow the conditions of stability that results in a constant ratio between atmospheric extinction (κ) and airmass (χ). The airmass can be summarized as the total amount of air crossed by the light from a star as it propagates through the atmosphere, and it is directly determined by the altitude above the horizon of the selected star as follows (Bouguer method): χ = secz (1 - 0.0012tan^2z) where z is the zenith angle (90^∘ - altitude) of the selected star in the moment of the observation.The observation strategy starts with the observation of standard astronomical stars placed at different airmasses, to cover the whole range of altitude above horizon for which we expect to determine the sky brightness. For best results, it is necessary to use standard stars of different astronomical magnitude. For each field of observation we will determine the airmass for the standard stars in each measurement, and the digital number counts detected with our astronomical device. These counts could be determined simply using the classical double circle technique <cit.> to measure the number of counts of the star without sky contribution, and divide this number of counts by the exposure time to get real counts per second values. Otherwise, every other method of stellar photometry can be used, and a large range of image processing and specialized software is available for different operation systems (Sextractor, Aperture Photometry, DAOPHOT, Iris, AstroImageJ, Astroart, Midas, IRAF).The standard stars have the value of their brightness (magnitude) recorded in photometric catalogues. Combining the catalogue magnitude data (mag) with the airmass (χ) and the number of counts (I) per second that we detect, it is possible to determine the instrumental zero points (ZP) and the extinction coefficient (κ) using the classical equation <cit.>: 2.5log _10 I= ZP - κ·χ The determination of ZP and κ is established using the set of all of the standard astronomical stars observed during the night. If the atmosphere is stable and there are no changes in the instruments, these two parameters remain constant during the observation period.Finally, using the background counts per second, we can evaluate the sky brightness of any image obtained during the night, taking into account the image scale. The best option is to determine the number of counts inside a defined rectangle that does not contain visible stars. The number of counts has to be divided by the angular area of the rectangle and by exposure time to get “counts per second and per arcsec^2” (I_sky) <cit.>.The sky brightness (SB) is finally determined in magnitudes per arcsec^2 for any selected position of the sky using the determined parameters with the equation: SB = ZP -2.5log _10 I _sky This method has been intensively used by Wim Schmidt to measure the sky background brightness in extended regions of the Netherlands <cit.>.§ ACKNOWLEDGEMENTS This article is based upon work from COST Action ES1204 LoNNe (Loss of the Night Network), supported by COST (European Cooperation in Science and Technology). Initial drafts were written during COST sponsored short term scientific missions of Zoltán Kolláth and Salvador J. Ribas to Austria and Germany. We gratefully acknowledge the help of Franz Binder (Vienna) with the formatting of this paper. 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"authors": [
"Andreas Hänel",
"Thomas Posch",
"Salvador J. Ribas",
"Martin Aubé",
"Dan Duriscoe",
"Andreas Jechow",
"Zoltán Kollath",
"Dorien E. Lolkema",
"Chadwick Moore",
"Norbert Schmidt",
"Henk Spoelstra",
"Günther Wuchterl",
"Christopher C. M. Kyba"
],
"categories": [
"astro-ph.IM"
],
"primary_category": "astro-ph.IM",
"published": "20170927144643",
"title": "Measuring night sky brightness: methods and challenges"
} |
0000-0001-8061-216X]Matteo Luisi Department of Physics and Astronomy, West Virginia University, Morgantown WV 26506, USA Center for Gravitational Waves and Cosmology, West Virginia University, Chestnut Ridge Research Building, Morgantown WV 26505, USA0000-0001-8800-1793]L. D. Anderson Department of Physics and Astronomy, West Virginia University, Morgantown WV 26506, USA Center for Gravitational Waves and Cosmology, West Virginia University, Chestnut Ridge Research Building, Morgantown WV 26505, USA Adjunct Astronomer at the Green Bank Observatory, P.O. Box 2, Green Bank WV 24944, USA0000-0002-2465-7803]Dana S. Balser National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville VA 22903-2475, USA0000-0003-0640-7787]Trey V. Wenger National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville VA 22903-2475, USA Astronomy Department, University of Virginia, P.O. Box 400325, Charlottesville, VA 22904-4325, USA Institute for Astrophysical Research, Department of Astronomy, Boston University, 725 Commonwealth Ave., Boston MA 02215, USA We analyze the diffuse ionized gas (DIG) in the first Galactic quadrant from ℓ=18 to 40 using radio recombination line (RRL) data from the Green Bank Telescope. These data allow us to distinguish DIG emission fromregion emission and thus study the diffuse gas essentially unaffected by confusion from discrete sources. We find that the DIG has two dominant velocity components, one centered around 100 associated with the luminousregion W43, and the other centered around 45 not associated with any largeregion. Our analysis suggests that the two velocity components near W43 may be caused by non-circular streaming motions originating near the end of the Galactic bar. At lower Galactic longitudes, the two velocities may instead arise from gas at two distinct distances from the Sun, with the most likely distances being ∼6 kpc for the 100 component and ∼12 kpc for the 45 component. We show that the intensity of diffuse Spitzer GLIMPSE 8.0emission caused by excitation of polyaromatic hydrocarbons (PAHs) is correlated with both the locations of discreteregions and the intensity of the RRL emission from the DIG. This implies that the soft ultra-violet photons responsible for creating the infrared emission have a similar origin as the harder ultra-violet photons required for the RRL emission. The 8.0emission increases with RRL intensity but flattens out for directions with the most intense RRL emission, suggesting that PAHs are partially destroyed by the energetic radiation field at these locations.§ INTRODUCTIONFirst proposed by <cit.>, the warm interstellar medium (WIM) is a widespread component of the interstellar medium (ISM) with density ∼0.1 and temperatures from 6000 to 10000 K <cit.>. At the upper end of this temperature range, the WIM is nearly fully ionized, with a hydrogen ionization ratio n(H^+)/n(H^0) ≥ 13 <cit.>. Thus, the WIM is also known as the “Diffuse Ionized Gas" (DIG). Despite its low density, ∼80–90% of the total free-free emission in our Galaxy is thought to come from the DIG.Though the exact mechanisms are still unknown, it is believed that the DIG maintains its ionization from O-type stars, whose UV radiation leaks out of theregions surrounding them and into the ISM <cit.>. <cit.> confirmed that a large number of ionizing photons are leaking fromregions. <cit.> derived an ionizing radiation leaking fraction of ∼25% for the bubbleregion RCW 120 using Hα data at 656 nm. They also showed that the photodissociation region (PDR) surrounding theregion has distinct “holes" through which photons can escape into the ISM. This suggests that PDRs are generally not homogeneous. Recently, we showed that the non-uniform PDR surrounding the compactregion NGC 7538 allows radiation to escape preferentially along a single direction <cit.>. We calculated a leaking fraction f_R = 15 ± 5% of the radio continuum emission. This leaking emission appears spatially confined within an additional, more distant PDR boundary around NGC 7538 and thus seems to only affect the local ambient medium. Results suggest, however, that giantregions such as W43 may have a much larger effect in maintaining the ionization of the DIG and despite their small numbers may be the dominant source of ionizing radiation in the ISM <cit.>.Together withregions and PDRs, the DIG is a major source of radio recombination line (RRL) emission. Consequently, RRL observations have been used to map its spatial and velocity distribution. Compared to studies of optical emission lines, specifically Hα <cit.>, RRL observations have the advantage of essentially being free from extinction due to interstellar dust. Their disadvantage is reduced sensitivity, restricting RRL detections to gas with higher emission measure than that traced by Hα. The fully-sampled 1.4 GHz RRL survey of <cit.> mapped the plane of the Galaxy at a spatial resolution of 144.They were, however, unable to distinguish the contributions from discreteregions and the DIG for most sight lines. The observing method of the fully-sampled SIGGMA RRL survey <cit.> partially filters out the emission from the DIG. Finally, <cit.> observed the Galactic plane in RRLs near 327 MHz from -28 < ℓ < 89. Despite the low resolution of ∼2, they obtain an upper limit of 12,000 K for the electron temperature of the gas and suggest that the emission originates from low-density ionized gas formingregion envelopes.With the emergence of high-sensitivity RRL surveys, the DIG has been serendipitously detected in observations of discreteregions <cit.>. In the Green Bank TelescopeRegion Discovery Survey <cit.> we identified multiple RRL velocity components in ∼30% of all observed targets. This fraction is too large to be caused by multiple discreteregions along the line of sight <cit.>. We thus infer that the RRL emission at these locations is usually composed of emission from a discrete source and emission from the DIG <cit.>.Here, we use data from past observations <cit.> and previously unpublished data for directions either known to be devoid of discreteregions, or in directions where theregion emission can be distinguished from that of the DIG (see 2 for details on how we distinguish between these two components). This gives us an irregularly-spaced grid of pointings, for which we can extract the intensity and velocity of only the DIG. The advantage of our strategy is that the beam size is relatively small (82) compared with typical spacings between discreteregions so the emission at each pointing is not contaminated withregion emission.The disadvantage of course is that the -space is not fully sampled. By distinguishing the emission from discreteregions and the DIG, our data allows us to essentially filter outregion emission entirely and map only the diffuse component. This gives us an advantage over previous RRL surveys <cit.> as these are at least partially contaminated byregion emission. With this analysis we are able to investigate the relationship between discreteregions and the diffuse gas, and test our hypothesis that largeregions are dominant in maintaining the ionization of the DIG <cit.>.§ DIG RRL EMISSIONOur RRL emission data were taken with the Auto-Correlation Spectrometer (ACS) on the National Radio Astronomy Observatory Green Bank Telescope (GBT).We observed a total of 254 directions between ℓ=18 and 40 and |b| < 1 which yielded 379 sets of line parameters for the DIG. Our data come from two previously published sources, <cit.> and <cit.>, and one previously unpublished source (see below). <cit.> contains directions coincident withregions, as defined by 8 μm Spitzer GLIMPSE emission, for which the diffuse gas velocity can be distinguished from theregion velocity (these data include 98 pointings with 116 sets of diffuse line parameters). The process of distinguishing the diffuse gas velocities fromregion velocities is described in <cit.>. We use previous GBT observations, analyze the derived electron temperature for each velocity component, and search for the molecular emission or carbon recombination lines associated with one RRL component. Sight lines that that do not pass within the 8 μm-definedregion PDR are always considered “diffuse." <cit.> also includes such directions devoid of discreteregions which allows us to directly sample the DIG without confusion (135 pointings; 237 sets of line parameters). We also incorporate observations taken near the giantregion W43 (21 pointings; 26 sets of diffuse line parameters) which we have not published previously. Here, we use our HRDS data to distinguish the diffuse gas velocity from theregion velocity. If the observed direction is spatially coincident with a knownregion, we assume that the velocity component closest to theregion velocity is due to theregion itself. We summarize these data in Table <ref>, which lists the source, the Galactic longitude and latitude, the hydrogen line intensity, the FWHM line width, the local standard of rest (LSR) velocity, and the rms noise in the spectrum, including all corresponding 1σ uncertainties of the Gaussian fits. For directions with multiple velocity components detected along the line of sight, the source names are given additional letters, “a," “b," or “c," in order of decreasing peak line intensity. Velocity components that are due to discreteregions are marked with an asterisk in the table and are not used for our data analysis. For each observed direction, we simultaneously measured 7 Hnα RRL transitions in the 9 GHz band, H87α to H93α, using our standard techniques <cit.>, and averaged all spectra together to increase the signal-to-noise ratio using TMBIDL[V7.1, see https://github.com/tvwenger/tmbidl.git.] <cit.>. We assume that the brightest line emission from the DIG is due to hydrogen and fit a Gaussian model to each line profile. We use the line intensities, full width at half maximum (FWHM) values, and LSR velocities derived from the Gaussian fits for all further analysis.lcccccccccc 0pt RRL Emission Near W43Sourceℓ b T_L σ T_L Δ V σΔ V V_ LSR σ V_ LSR rms Notea(degree) (degree) (mK) (mK) () () () () (mK)G030.400+0.180 30.400 +0.180 8.2 0.1 58.7 1.5 88.1 0.5 0.7 G030.570-0.230 30.570 -0.230 38.8 0.3 19.3 0.2 88.10.1 0.4 * G030.570+0.090a 30.570 +0.090 27.6 0.3 23.2 0.3 42.4 0.1 1.7 G030.570+0.090b 30.570 +0.090 22.0 0.3 24.9 0.4 104.9 0.2 1.7 G030.650-0.150a 30.650 -0.150 53.8 0.2 22.0 0.1 96.0 0.1 0.5 G030.650-0.150b 30.650 -0.150 24.1 0.2 18.4 0.2 119.3 0.1 0.5 G030.740-0.060 30.740 -0.060 617.6 1.5 24.2 0.1 90.70.1 2.1 * G030.740+0.010 30.740 +0.010 612.5 0.7 29.5 0.1 91.10.1 2.3 * G030.740+0.100a 30.740 +0.100 34.8 0.3 21.5 0.2 120.9 0.1 1.6 G030.740+0.100b 30.740 +0.100 24.3 0.3 19.8 0.388.0 0.1 1.6 * G030.740+0.100c 30.740 +0.100 9.9 0.3 20.5 0.7 39.9 0.3 1.6 G030.740+0.180a 30.740 +0.180 12.8 0.2 46.7 1.1 96.6 0.4 1.5 * G030.740+0.180b 30.740 +0.180 12.5 0.3 21.6 0.7 39.5 0.3 1.5 G030.740+0.260a 30.740 +0.260 50.3 0.4 20.7 0.2 100.5 0.1 1.3 * G030.740+0.260b 30.740 +0.260 13.1 0.4 19.7 0.7 37.0 0.3 1.3 G030.740+0.280a 30.740 +0.280 64.2 0.4 19.7 0.1 100.8 0.1 1.2 * G030.740+0.280b 30.740 +0.280 9.9 0.4 20.3 0.8 37.2 0.4 1.2 G030.740+0.300a 30.740 +0.300 34.0 0.3 16.9 0.2 102.4 0.1 1.3 * G030.740+0.300b 30.740 +0.300 11.3 0.2 23.5 1.1 79.0 0.5 1.3 G030.740+0.300c 30.740 +0.300 7.6 0.2 22.3 0.8 36.7 0.3 1.3 G030.740+0.350a 30.740 +0.350 15.8 0.3 14.7 0.5 78.2 0.2 1.2 G030.740+0.350b 30.740 +0.350 15.7 0.3 21.6 0.6 102.3 0.2 1.2 G030.740+0.350c 30.740 +0.350 3.6 0.2 35.4 3.7 39.5 1.2 1.2 G030.740+0.430 30.740 +0.430 8.8 0.3 28.0 1.0 98.8 0.4 1.7 G030.740+0.510a 30.740 +0.510 8.2 0.2 34.8 0.8 92.1 0.3 1.2 G030.740+0.510b 30.740 +0.510 5.7 0.3 10.7 0.6 18.9 0.3 1.2 G030.780-0.020 30.780 -0.020 2179.2 2.6 31.4 0.1 91.70.1 4.8 * G030.780+0.010 30.780 +0.010 227.1 0.4 31.4 0.1 92.60.1 2.2 * G030.820-0.060 30.820 -0.060 280.0 0.4 29.0 0.1 106.30.1 1.8 * G030.820+0.180a 30.820 +0.180 14.0 0.2 18.5 0.3 36.4 0.1 0.8 G030.820+0.180b 30.820 +0.180 11.3 0.1 38.7 0.5 99.6 0.2 0.8 * G030.900-0.060a 30.900 -0.060 33.1 1.7 18.3 0.4 107.3 0.4 1.2 G030.900-0.060b 30.900 -0.060 14.5 0.8 23.5 1.9 90.0 1.3 1.2 G030.900-0.060c 30.900 -0.060 4.6 0.2 31.2 1.8 45.7 0.7 1.2 G030.900+0.340a 30.900 +0.340 7.3 0.1 37.3 0.7 103.4 0.3 0.8 G030.900+0.340b 30.900 +0.340 2.8 0.1 28.6 1.7 38.4 0.7 0.8 G031.070-0.150a 31.070 -0.150 13.8 0.2 28.6 0.4 99.0 0.2 0.9 G031.070-0.150b 31.070 -0.150 4.3 0.2 25.8 1.4 26.2 0.6 0.9 a RRL components associated with discreteregions are marked with an asterisk (*, see text). § DISCUSSION §.§ The Galactic Location of the DIGOver the longitude range considered here, the DIG emission is concentrated near two velocities, 45and 100(Figure <ref>). This suggests that within our observed Galactic longitude range the DIG itself is located at two distinct distances, assuming that the diffuse gas in each velocity range can be assigned a single distance. We summarize the DIG emission properties in Table <ref>.Just as for the discrete sources, however, this diffuse gas also suffers from the kinematic distance ambiguity (KDA). Unfortunately, we cannot use theemission/absorption () method <cit.> for the DIG, both because it is faint and also because of the difficulty in finding a suitable “off” position.Only massive stars can produce ionizing photons energetic enough to create and maintain the DIG <cit.>.We can therefore potentially determine the kinematic distance for the diffuse gas by associating it with massive star formation tracers that have their KDA resolved: massiveregions, molecular gas, and cold . Below, we attempt to find the distance to the two observed velocity components of the DIG by resolving their KDA. In 3.1.1 and 3.1.2 we assume that each velocity component can be assigned a single distance from the Sun. In 3.1.3 we explore the possibility of the two observed velocity components being due to interacting gas clouds at the same distance from the Sun.§.§.§ The 45 km s^-1 Gas Component The KDA leads to two possible distance ranges for each velocity range.The 45gas could be at either 1.7 - 3.7 kpc or 10.6 - 12.7 kpc, if we assume = (30, 0), and use the <cit.> rotation curve (see Table <ref>).lcc0pt DIG ParametersVelocity range45100Number of RRL components (total) 128 211Number of RRL components (on)a 33 63Number of RRL components (off) 95 148Mean velocity () 45.3 100.0Median velocity () 43.0 99.4Std. Dev. velocity () 9.2 10.1Mean T_ A (mK) 12.5 16.8Median T_ A (mK) 9.8 13.6Std. Dev. T_ A (mK) 9.6 11.2Near distance (kpc) 1.7-3.7 4.4-7.2Far distance (kpc) 10.6-12.7 7.2-10.0Assumed distance (kpc) ∼12 ∼6Total integrated flux (Jy) 172.8 246.0Total integrated flux (Jy)b 118.3 220.4 a “on" and “off" correspond to directions coincident withregions (on), and directions devoid of discreteregions (off). b From <cit.>. Assuming that the DIG is maintained by massive stars, we can use the ionization rate ofregions as a tracer to determine the distance to the DIG. In the range ℓ = 18 to 40, there are 205regions with velocities between 25 and 65 , and 127 of these have kinematic distance ambiguity resolutions <cit.>. The total radio flux density of the 94 regions at the far kinematic distance is 10.84 Jy, whereas the total flux density of the 33 regions at the near kinematic distance is only 1.09 Jy. We estimate the ionization rate for each region using our HRDS data <cit.> byN_ ly≈ 4.76 × 10^48( S_ν/Jy) ( T_ e/K) ^-0.45( ν/GHz) ^0.1( d/kpc) ^2, where N_ ly is the ionization rate, the number of emitted Lyman Continuum ionizing photons per second, S_ν is the radio flux density of theregion, T_ e is the electron temperature, ν = 1.4 GHz is the observed frequency <cit.>, and d is the distance to the region. We assume a constant T_ e = 10^4 K and sum the contribution for each individual region to find the total N_ ly forregions at the far and near kinematic distance. This estimate yields N_ ly = 1.0650 s ^-1 for the far distance and only N_ ly = 5.8047 s ^-1 for the near distance. This suggests that most of the DIG near 45is also at the far kinematic distance.There is also over twice as much total CO gas at the far kinematic distance for clouds in the velocity range 25 to 65 compared to the near distance.The average near GRS cloud CO luminosity from <cit.> in units of 10^4 pc^-2 is 0.23 with a standard deviation of 0.31, while it is 1.4 with a standard deviation of 1.7 for the far GRS clouds.The total CO luminosity for the near clouds is 3.45 pc^-2, while it is 8.05 pc^-2 for the far GRS clouds. This again supports the 45DIG being at the far kinematic distance, if it is indeed associated with the molecular gas traced by CO emission that will continue to form massive stars.Finally, we investigate the location of the coldgas using themethod. Only coldforeground to a radio continuum source will causeabsorption, assuming thatself-absorption is negligible. Thespectrum toward an extragalactic radio continuum source can show absorption for allalong the line of sight, while for Galactic sources thespectrum can only show absorption up to the source velocity. Comparing thespectra toward nearby extragalactic and Galacticregion pairs can therefore tell us about thedistribution. If coldgas is foreground to theregion, we expect to see absorption in both spectra. Coldbeyond theregion, however, will only show absorption in the spectrum toward the extragalactic source.Here we use the Very Large Array Galactic Plane Survey <cit.> spectral line data to compare thespectrum for threeregions (G24.47+0.49, G24.81+0.10, and W43) with velocities near 100 . All three have nearby (within ∼40) extragalactic radio continuum sources. Figure <ref> shows the difference between on- and off-target directions for theregions and extragalactic radio sources, where the on- and off-positions are separated by 6. Theregions are located either foreground or background to the 45 gas, depending on their KDARs. As a result,gas at velocities showing extragalactic absorption which is not present in theregion spectra should be background to theregion. This analysis implies that most of thebelow 50 near W43 is at the far kinematic distance. The firstspectrum pair (G24.47+0.49) extracted near ℓ = 24shows partial absorption near 45 that is inconsistent with the absorption features seen in the second pair near ℓ = 24(G24.81+0.10). Therefore, we can not assign a single distance to thenear the ℓ = 24region. These results are somewhat ambiguous, however, since the separation between the line of sight towards theregions and the extragalactic continuum sources are probing differentvolumes. Since both the totalregion ionization rate and the fraction of molecular gas are greater at the far distance, we favor the conclusion that most of the 45diffuse gas is at its far kinematic distance of ∼12 kpc as well. This is a simplified assumption and does not take into account the existence of additional gas at the other distance.§.§.§ The 100 km s^-1 Gas ComponentThe possible distance range for the 100gas is 4.4 - 10.0 kpc for = (30, 0).Because the molecular gas and massive star formation for the 80-100 locus is associated with W43 <cit.> at a distance of 5.49^+0.39_-0.34 kpc <cit.>, we assume throughout the remainder of this paper that the 100DIG is at a distance of ∼6 kpc. Recently, <cit.> observed the DIG along 18 lines of sight between ℓ = 30and 32using the [] 158 μm and [] 205 μm fine structure lines. They find a strong line component near ∼115and argue that this component is due to DIG emission associated with the inner edge of the Scutum spiral arm tangency at a distance of ∼7 kpc. Even if our assumption that the gas is at the distance of ∼6 kpc is poor, our conclusions below are largely unaffected. §.§.§ Interacting Gas Clouds?Our detection of the DIG in two separate velocity ranges suggests that each velocity range is primarily located at either its near or its far kinematic distance. If the two velocity components are indeed interacting, we would expect to observe an interaction signature between them. Such an interaction signature has been suggested by <cit.> for the ^13CO(2-1) emission near the W43 region, as well as for dense gas tracers like N_2H^+. In the Milky Way, however, this picture is further complicated by the vicinity of the 45 component to the Galactic bar and the Scutum arm. Using an extragalactic counterpart to the W43 region, <cit.> argue that gas buildup near the bar/spiral arm interface, where W43 is located, is likely due to crossings between different orbit families. They posit that the observed velocities in the bar/spiral arm interface of NGC 3627 are primarily due to interacting gas clouds.If we assume that the observed velocities toward the ℓ∼ 30 region are due to interacting gas clouds at a single distance, we can use the method described by <cit.> to estimate the expected gas velocities observed along the line of sight and compare these with our observations. The simplest approximation assumes that the observed diffuse gas towards W43 is located at the tip of the Galactic bar, and that the two observed velocity signatures are due to the unperturbed, purely circular gas motion around the Galactic center and gas streaming motions along the bar, respectively. Using the <cit.> rotation curve, we find a circular gas velocity, V_C ∼ 230 for the observed diffuse gas towards W43. This corresponds to a velocity component along the line of sight of 91 , almost identical to the observed velocity of 89.8 for W43 itself. The perturbed velocity component due to streaming motions can be described by determining the bar perturbation to the gravitational potential <cit.>. Since we only consider emission from the end of the bar, the radial streaming velocity component must go to zero, and the resultant azimuthal velocity component, v_ϕ^B, isv_ϕ^B ∼( 1 - 1 - q_ϕ^2/4 q_ϕ^2) V_C , where q_ϕ is the axial ratio of the bar potential. We use 1 - q_ϕ≃1/3(1 - q) from <cit.>, where q = 0.3 - 0.4 is the axial ratio of the density distribution for the Milky Way bar <cit.>. We adopt q = 0.35 ± 0.05 and find that v_ϕ^B = 0.84 ± 0.02 V_C. Observed along the line of sight, this corresponds to a velocity of 57 ± 4 which is near our observed 45 velocity component.Although the above method describes the observed velocity components near ℓ∼ 30 fairly well, the assumption that the gas is located at the end of the Galactic bar breaks down when considering gas emission from the ℓ∼ 24 region further within the bar where we observed a similar velocity distribution. To describe the kinematics of the gas at this location, we must include radial streaming motions along the bar<cit.> which can be estimated byv_r^B ∼2/3( 1 - 1 - q_ϕ^2/4 q_ϕ^2) V_C . We repeat the analysis above for the ℓ∼ 24 region, and find an unperturbed velocity component along the line of sight of 96 , and a perturbed velocity component of 7 ± 2 . In theory, shocks and turbulence could increase the latter to match our observed 45 emission. While we can not quantify the amount of turbulence in the DIG directly, we can compare the observed hydrogen recombination line widths at the diffuse directions with the line widths of directions coincident with discreteregions. Assuming the same electron temperature, differences in line widths should trace the relative strength of turbulence between these directions. We find, however, no statistically significant difference of line widths between directions coincident withregions (FWHM = 24.5 ± 6.4 ) and our diffuse directions (FWHM = 23.7 ± 8.8 ). We show the corresponding FWHM line width distributions in Figure <ref>. This suggests that turbulence does not play a significant role in altering the observed velocity of the gas. As a result, the large difference of the derived 7 velocity component to our observed 45 emission makes it doubtful whether interacting gas clouds at a single distance near ℓ∼ 24 could result in the observed velocity distribution.Although the simple model discussed above suggests that interacting gas clouds can not account for our observed data, a more thorough numerical analysis would be required to confirm this result. <cit.> developed a hydrodynamical simulation of a Milky Way-like galaxy which includes star formation and stellar feedback through photoionization, radiative pressure and supernovae. They find that the leading edges of bars are favorable locations for converging gas flows and shocks. A similar model, focusing on bar kinematics in particular, may provide more insight towards the interaction processes near the bar-spiral arm interface.§.§ Intensity and Distribution of the DIGOur database of RRL parameters from the HRDS also allows us to investigate the spatial distribution of the DIG in the plane of the sky. Using our irregularly gridded data points, we examine the diffuse gas separately for the two velocities, 45and 100 . We create maps of the DIG in these two velocity ranges by interpolating the irregularly-spaced grid of 233 points to create pixels 6 square.We do this by first performing a Delauney triangulation (using the IDL program “qhull”) and then create anmap of the RRL intensity from the DIG using inverse distance weighting (using the IDL program “griddata”). This method has the advantage that the maximum and minimum values in the interpolated surface can only occur at sample points. We assume that the top and bottom edges of the map (b = ± 1) have zero intensity to ensure that the emission is constrained in latitude. We show these images in Figure <ref> for the two velocity ranges. We also show in Figure <ref> the 1.4 GHzParkes All-sky survey RRL map <cit.> averaged over the velocity ranges of the 45and 100 components for comparison. The green circles in Figure <ref> show the locations of discreteregions cataloged by <cit.> that are within the velocity range of interest, while the gray crosses show locations where the DIG was detected within the velocity range.Using the same data set of RRL parameters, we explore the velocity distribution of the DIG in more detail. We create a longitude-velocity diagram of the DIG by interpolating between our grid points (Figure <ref>, top panel), and assume that the velocity edges of the diagram (at 0 and 130 ) have zero intensity so that the emission is constrained in velocity space. This assumption appears valid, since we did not detect any RRL components outside of this velocity range. In fact, our smallest and largest detected velocities at 18 and 124 , respectively, are well within this range. For comparison, we also show a longitude-velocity diagram of ^12CO used to trace molecular clouds <cit.>.§.§.§ The 45 km s^-1 Gas ComponentThe pixel-by-pixel correlation of RRL intensity at 45 between our maps and the <cit.> data is poor (see Figure <ref>, top panel). Our emission towards W43 near ℓ∼ 30 and the ℓ∼ 24 region is disproportionately large in the 45 map, whereas we do not see strong emission near the map edge at ℓ∼ 19. This may be due to interpolation errors between our sparse RRL pointings in this velocity and longitude range. Our low number of pointings may also be the cause of some of the more extended RRL emission between W43 and the ℓ∼ 24 complex that is less pronounced in the <cit.> data. This makes it challenging to distinguish between interpolation errors and actual diffuse gas below the Parkes 1.4 GHz RRL survey's sensitivity threshold for the undersampled regions in our maps. Additionally, the beam size of ∼ 14 in the <cit.> maps is too large to avoidregions at locations where their number density is high. Thus, most of their emission towards W43 and the ℓ∼ 24 region must be caused by discreteregions rather than the DIG. Overall, the total integrated intensity of our maps is 46% larger in the 45 component compared with the <cit.> data. This perhaps indicates that we are more sensitive to the diffuse gas.§.§.§ The 100 km s^-1 Gas ComponentWhile the interpolated 45 map shows poor agreement with the <cit.> data, our 100 map is strongly correlated with the 1.4 GHz RRL emission data (Figure <ref>, bottom panel). By-eye comparison of the two maps (Figure <ref>) indicates that we are more sensitive to the diffuse gas component, especially at lower Galactic longitudes. The total integrated intensity of our data is 10% larger in the 100 component compared to <cit.> (see Table <ref>). Figure <ref> (top panel) shows that much of the 100 emission from the DIG may be associated with the Scutum spiral arm. The higher velocities of the DIG compared to the Scutum arm may indicate that we are observing strong streaming motions in this direction <cit.>. Alternatively, the DIG may be located near the inner edge of the Scutum tangency where it is falling into the arm's gravitational potential, as suggested by <cit.>.The directions of strong emission in the two velocity ranges are slightly correlated, such that locations of strong emission from the DIG near 45mostly have strong emission near 100 as well. The correlation is more significant towards the W43 region, whereas it is weak near ℓ∼ 24 as shown in Figure <ref>. This may suggest that the two velocity ranges towards W43 represent flows of interacting ionized gas (see 3.1.3), whereas the two velocity ranges towards the ℓ∼ 24 region could be caused by DIG emission at two distinct distances. §.§ H I and the Diffuse GasIt is uncertain whether a substantial amount of coldgas can coexist with the diffuse ionized gas in regions with strong RRL emission from the DIG. If the radiation field in such regions is strong enough to ionize a large fraction of the gas, we may be able to observe a depletion inat locations and velocities of strong DIG emission <cit.>.This relationship has been probed by <cit.> who find that thedistribution does not correlate (or anti-correlate) at all with the diffuse Hα emission tracing the ionized gas in the face-on galaxy NGC 157. The angular resolution of theirmap, however, does not match the better resolution of their Hα map. As a result, they would not be able to resolvedepletion cavities much smaller than 1 kpc. A previous study by <cit.> analyzed the same correlation for Hα-emittingclouds in the Milky Way. They find that the neutral and ionized components in these clouds are likely spatially separated. Since they only observed a relatively small region of the sky away from the Galactic plane, their available sample size is limited.We use the VGPSdata to test whether regions with strongemission from the DIG show a deficiency in . The VGPS data cubes have a spatial resolution of 1× 1 and a spectral resolution of 1.56 . For the W43 region near ℓ∼ 31 we find an apparent depletion cavity at 92 (Figure <ref>, top left panel) which is consistent with our strong DIG emission shown in Figure <ref>. We find a similar depletion cavity for the ℓ∼ 23 region at 60 (see Figure <ref>, top right panel) which is, however, offset by ∼1.5 from the strong DIG emission seen in the 45component near ℓ∼ 24.5 (Figure <ref>). Our large number of RRL pointings near the ℓ∼ 24 region makes it unlikely that this offset is an artifact from our interpolation algorithm. We did not find any other strongdepletion cavities in the velocity ranges of significant emission from the DIG for the sky zones in Figure <ref>.Comparison of our RRL data with ^13CO maps casts doubt on whether strong ionized gas emission is usually spatially associated with a deficiency in . Using ^13CO GRS data, we can determine for both our regions whether these deficiencies are caused byself-absorption or due to an actual lack ofgas.self-absorption, first described in detail by <cit.>, is usually correlated with CO emission features <cit.>. We find substantial ^13CO emission at the two directions and velocities (Figure <ref>, bottom panels). In fact, the integrated ^13CO emission found at ℓ∼ 31 and ℓ∼ 23 is among the strongest within the range of the GRS. This suggests that the lack inemission is caused byself-absorption and is not due to an actual deficiency ingas. §.§ Correlation with 8.0 μm IntensityEmission from polycyclic aromatic hydrocarbons (PAHs) within the 8.0band is usually caused by softer ultra-violet (UV) radiation than that responsible for RRL emission fromregions. For example, <cit.> show, using radiative transfer models, that most of PAH heating is provided by B stars, compared to RRL emission typically caused by O stars. While the 8.0emission is often associated with strong PDRs surrounding discreteregions, there exists significant PAH emission that originates from the diffuse gas without nearbyregions. Below, we analyze this “diffuse" PAH emission and its relation to the DIG. In the bottom panel of Figure <ref> we show a map of the point-source subtracted 8.0Spitzer GLIMPSE emission <cit.>. Since there is also strong 8.0emission from discreteregions, we blank out these regions based on their corresponding positions and sizes from the WISE catalog, Version 1.4 <cit.>. In the upper panel of Figure <ref> we show histograms of the location of single-velocity and multiple-velocityregions. We observe a correlation between the location of discreteregions and the intensity of the 8.0emission (see Figure <ref>). Both of these diffuse emission components should be caused by UV photons leaking from the discreteregions. The PAHs responsible for the 8.0emission are destroyed in the hard UV radiation within anregion <cit.>, but can survive where the radiation field is softer, i.e. in the diffuse ISM. The correlation therefore suggests that either a significant amount of the (soft) UV photons responsible for the 8emission is leaking from the discreteregions or that the harder UV radiation produced by the O stars softens as it escapes into the ISM. Such a radiation softening has been suggested by <cit.> and was recently observed indirectly for the compactregion NGC 7538 <cit.>.We also observe a correlation between the hard UV radiation field within the DIG and the softer UV radiation field responsible for PAH emission. Our method probes the radiation field strengths by using the observed RRL emission from the DIG and the 8.0emission intensities as a proxy for the diffuse hard UV and soft UV radiation field strengths, respectively. A correlation between the intensities of these two emission components indirectly tests for a correlation between the radiation fields. We determine the diffuse 8.0background by integrating the 8.0flux in a circular 41 aperture centered at each of the 135 off-target directions. By only using the off-target directions, we ensure that we are only sampling the diffuse 8.0background, and not the emission associated withregion PDRs. We then compute the fluxes using our Kang software[http://www.bu.edu/iar/files/script-files/research/kang/]. Kang is an astronomical visualization and analysis package written in IDL.Its relevant functionality here is that it can compute aperture photometry measurements using arbitrary aperture shapes.We compare the diffuse 8.0emission with the integrated intensity from all hydrogen RRLs at all velocities detected at an off-target direction. We plot in Figure <ref> the correlation between the diffuse 8.0emission and the integrated RRL intensity. Figure <ref> indicates that the hard UV radiation is correlated with the softer UV radiation, as one would expect, although there is quite a large scatter.Furthermore, the correlation appears to change above 35 mK. This may be due to the destruction of the PAH molecules that are largely responsible for the diffuse 8.0emission. A similar effect has been observed by <cit.> in the Galacticregion NGC 3603. Direct PAH destruction typically requires photons with energies >20 eV which exist in sufficient numbers only in the most energetic radiation fields (e.g., withinregions). The binding energy of H atoms to PAHs, however, is only ∼4.8 eV <cit.>. Thus, even less energetic radiation fields can contribute to PAH dissociation. The relatively strong hydrogen ionizing radiation field (≥ 13.6 eV) within regions with substantial RRL emission from the DIG must therefore partly be responsible for PAH dissociation. Considering that this effect limits the abundance of PAHs in these regions, it comes as no surprise that we observe a saturation in PAH emission. Clearly, further study of this correlation is required. Most helpful would be additional pointings toward cleaner sight lines where the source of the UV photons can be more easily be determined. A more direct measurement of the radiation field strength is difficult. Recently, <cit.> described a technique to estimate the UV field intensity using the ratio between two PAH spectral components at 7.6and 7.8 . While this technique has not yet been applied to the diffuse ISM, it may prove useful in constraining properties of the radiation field outsideregions.§ SUMMARYHere, we analyze the DIG using hydrogen RRL emission line spectra in the range ℓ=18 to 40 which are either devoid ofregion emission or have multiple velocity components. Our data set is comprised of 353 RRL emission line components from the DIG. These allow us to determine the intensity and distribution of the diffuse gas. We find that the DIG is spatially concentrated in two areas near ℓ=31 and ℓ=24, with two dominant velocity components (45 and 100) in each of the areas. We investigate the KDA for the two velocity ranges and conclude that the 100 component has a Galactocentric distance of ∼6 kpc, corresponding to the location of W43. This suggests that much of the 100 gas is associated with W43. The origin of the 45component is less clear. The intensity of the emission in the two velocity ranges is slightly correlated near ℓ=31, which may imply that both velocity components originate at a single distance. In this case, the 45component may arise from complex streaming motions near the end of the Galactic bar. For the ℓ=24 region, however, it is unlikely that the observed velocity components are due to this effect. As an alternative, we suggest that the 45emission may have its origin at a Galactocentric distance of ∼12 kpc, or a combination of both. Unfortunately, our current data are insufficient to clearly distinguish between these cases, a problem which may be investigated in future work. Future work may also explore in more detail the connection between the DIG observed in RRL emission, the more diffuse component observed in Hα, and the different environments these data are tracing.Since regions with strong RRL emission from the DIG may show a deficiency in , we examine data from the VGPS fordepletion cavities. We find such a bubble inemission for the W43 region at ℓ∼ 31 at 92 . We also find a second bubble at ℓ∼ 23 and 60 which is, however, offset by ∼1.5 from the direction of strong DIG emission at ℓ∼ 24.5. There is strong ^13CO emission associated with these locations, suggesting that the deficiency inemission is rather caused byself-absorption than an actual lack ofgas.The intensity of the RRL emission from the DIG is also correlated with the intensity of diffuse Spitzer GLIMPSE 8.0emission, implying that the soft UV photons responsible for creating the infrared emission have a similar origin as the harder UV photons required for the RRL emission. The diffuse 8.0emission appears to saturate at locations with the strongest RRL emission suggesting that the PAHs responsible for the 8.0emission are destroyed by the radiation field in these regions.We thank Robert A. Benjamin and Brian L. Babler for providing us with the point-source subtracted 8.0 μm data. Support for TVW was provided by the NSF through the Grote Reber Fellowship Program administered by Associated Universities, Inc./National Radio Astronomy Observatory. Green Bank Telescope. TMBIDL <cit.>, Kang.aasjournal | http://arxiv.org/abs/1709.09232v1 | {
"authors": [
"Matteo Luisi",
"L. D. Anderson",
"Dana S. Balser",
"Trey V. Wenger",
"T. M. Bania"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170926193201",
"title": "Diffuse Ionized Gas in the Milky Way Disk"
} |
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