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[ "Canonical Quasilocal Energy and Small Spheres", "Canonical Quasilocal Energy and Small Spheres" ]	[ "J D Brown \nDepartment of Physics\nNorth Carolina State University\n27695-8202RaleighNCUSA\n\nDepartment of Mathematics\nNorth Carolina State University\n27695-8205RaleighNCUSA\n", "† S R Lau \nDepartment of Physics\nNorth Carolina State University\n27695-8202RaleighNCUSA\n\nDepartment of Physics & Astronomy\nUniversity of North Carolina\nCB# 3255 Phillips Hall, Chapel Hill27599-3255NCUSA\n\nInstitut für Theoretische Physik\nTechnische Universität Wien\nWiedner Hauptstraße 8-10A-1040WienÖsterreich\n", "J W York \nDepartment of Physics\nNorth Carolina State University\n27695-8202RaleighNCUSA\n\nDepartment of Physics & Astronomy\nUniversity of North Carolina\nCB# 3255 Phillips Hall, Chapel Hill27599-3255NCUSA\n", "\nCurrent address: Applied Mathematics Group\nDepartment of Mathematics\nUniversity of North Carolina\nCB# 3250 Phillips Hall, Chapel Hill27599-3250NCUSA\n" ]	[ "Department of Physics\nNorth Carolina State University\n27695-8202RaleighNCUSA", "Department of Mathematics\nNorth Carolina State University\n27695-8205RaleighNCUSA", "Department of Physics\nNorth Carolina State University\n27695-8202RaleighNCUSA", "Department of Physics & Astronomy\nUniversity of North Carolina\nCB# 3255 Phillips Hall, Chapel Hill27599-3255NCUSA", "Institut für Theoretische Physik\nTechnische Universität Wien\nWiedner Hauptstraße 8-10A-1040WienÖsterreich", "Department of Physics\nNorth Carolina State University\n27695-8202RaleighNCUSA", "Department of Physics & Astronomy\nUniversity of North Carolina\nCB# 3255 Phillips Hall, Chapel Hill27599-3255NCUSA", "Current address: Applied Mathematics Group\nDepartment of Mathematics\nUniversity of North Carolina\nCB# 3250 Phillips Hall, Chapel Hill27599-3250NCUSA" ]	[]	Consider the definition E of quasilocal energy stemming from the Hamilton-Jacobi method as applied to the canonical form of the gravitational action. We examine E in the standard "small-sphere limit," first considered by Horowitz and Schmidt in their examination of Hawking's quasilocal mass. By the term small sphere we mean a cut S(r), level in an affine radius r, of the lightcone N p belonging to a generic spacetime point p. As a power series in r, we compute the energy E of the gravitational and matter fields on a spacelike hypersurface Σ spanning S(r). Much of our analysis concerns conceptual and technical issues associated with assigning the zero-point of the energy. For the smallsphere limit, we argue that the correct zero-point is obtained via a "lightcone reference," which stems from a certain isometric embedding of S(r) into a genuine lightcone of Minkowski spacetime. Choosing this zero-point, we find the following results: (i) in the presence of matter E = 4 3 πr 3 [T µν u µ u ν ]\| p + O(r 4 ) and (ii) in vacuo E = 1 90 r 5 [T µνλκ u µ u ν u λ u κ ]\| p + O(r 6 ). Here, u µ is a unit, future-pointing, timelike vector in the tangent space at p (which defines the choice of affine radius); T µν is the matter stress-energy-momentum tensor; T µνλκ is the Bel-Robinson gravitational super stress-energy-momentum tensor; and \| p denotes "restriction to p." Hawking's quasilocal mass expression agrees with the results (i) and (ii) up to and including the first non-trivial order in the affine radius. The non-vacuum result (i) has the expected form based on the results of Newtonian potential theory.	10.1103/physrevd.59.064028	[ "https://arxiv.org/pdf/gr-qc/9810003v1.pdf" ]	13,901,289	gr-qc/9810003	32e7e895370cc70588bbf991a2ca8840371c0461	Canonical Quasilocal Energy and Small Spheres arXiv:gr-qc/9810003v1 1 Oct 1998 J D Brown Department of Physics North Carolina State University 27695-8202RaleighNCUSA Department of Mathematics North Carolina State University 27695-8205RaleighNCUSA † S R Lau Department of Physics North Carolina State University 27695-8202RaleighNCUSA Department of Physics & Astronomy University of North Carolina CB# 3255 Phillips Hall, Chapel Hill27599-3255NCUSA Institut für Theoretische Physik Technische Universität Wien Wiedner Hauptstraße 8-10A-1040WienÖsterreich J W York Department of Physics North Carolina State University 27695-8202RaleighNCUSA Department of Physics & Astronomy University of North Carolina CB# 3255 Phillips Hall, Chapel Hill27599-3255NCUSA Current address: Applied Mathematics Group Department of Mathematics University of North Carolina CB# 3250 Phillips Hall, Chapel Hill27599-3250NCUSA Canonical Quasilocal Energy and Small Spheres arXiv:gr-qc/9810003v1 1 Oct 1998 Consider the definition E of quasilocal energy stemming from the Hamilton-Jacobi method as applied to the canonical form of the gravitational action. We examine E in the standard "small-sphere limit," first considered by Horowitz and Schmidt in their examination of Hawking's quasilocal mass. By the term small sphere we mean a cut S(r), level in an affine radius r, of the lightcone N p belonging to a generic spacetime point p. As a power series in r, we compute the energy E of the gravitational and matter fields on a spacelike hypersurface Σ spanning S(r). Much of our analysis concerns conceptual and technical issues associated with assigning the zero-point of the energy. For the smallsphere limit, we argue that the correct zero-point is obtained via a "lightcone reference," which stems from a certain isometric embedding of S(r) into a genuine lightcone of Minkowski spacetime. Choosing this zero-point, we find the following results: (i) in the presence of matter E = 4 3 πr 3 [T µν u µ u ν ]\| p + O(r 4 ) and (ii) in vacuo E = 1 90 r 5 [T µνλκ u µ u ν u λ u κ ]\| p + O(r 6 ). Here, u µ is a unit, future-pointing, timelike vector in the tangent space at p (which defines the choice of affine radius); T µν is the matter stress-energy-momentum tensor; T µνλκ is the Bel-Robinson gravitational super stress-energy-momentum tensor; and \| p denotes "restriction to p." Hawking's quasilocal mass expression agrees with the results (i) and (ii) up to and including the first non-trivial order in the affine radius. The non-vacuum result (i) has the expected form based on the results of Newtonian potential theory. INTRODUCTION Consider Einstein's non-covariant first-order action [1], the 4-integral of a "bulk" Lagrangian which is quadratic in the Christoffel symbols and thus often called the "ΓΓ action." Starting with the Einstein action, one applies standard techniques associated with Noether's theorem in order to derive, among other things, an energy definition in general relativity: namely, the 2-integral of an Einstein superpotential over some generic 2-surface S in spacetime M. 1 The Einstein energy is well known to be ambiguously defined because it depends on the choice of background coordinates. Nevertheless, one may use the Einstein construction to define sensible notions of total gravitational energy. Indeed, consider the scenario of asymptotic flatness, say, towards future null infinity J + . In this case, S tends to a round, infinite-radius, 2-sphere cut of J + , and the (now suitably unique) choice of asymptotically Cartesian coordinates ensures that the Einstein energy agrees with the accepted Trautman-Bondi-Sachs (tbs) notion of total energy. [2] However, were we to offer the Einstein definition as the energy contained within some quasilocal (that is, finite) 2-surface S, we would still be confronted with the task of choosing a physically meaningful set of background coordinates. The only natural choice would be coordinates which are partially adapted to the embedding of S ⊂ M. However, such a choice wrecks the agreement between the Einstein and tbs energies as S tends towards J + . In fact, choosing such S-adapted coordinates, one finds that the Einstein energy blows up in the said limit. Similar statements can be made regarding other approaches for defining energy, which trade coordinate (that is, holonomic-frame) ambiguity for ambiguity of a different stripe, e. g. tetrad (or rigid) frame, spin frame, or auxiliary vector (or spinor) fields. The traditional party line regarding these issues is the following: there is no over-arching rule, applicable for all quasilocal 2-surfaces, for selecting a (suitably unique) background frame; whence gravitational quasilocal energy is not well-defined. 2 To what extent does the stubborn presence of frame ambiguity in the quasilocal context point to a gap in our understanding of gravitational energy? To address this question and sharpen our thoughts on these issues, let us consider a covariant version of the Einstein construction. Employing a straightforward field-theoretic generalization of the Hamilton-Jacobi (hj) method [5], one may derive from a covariant action functional an expression for canonical quasilocal energy (qle) in general relativity. [6] We call this definition of qle canonical, because, owing to its intimate connection with hj theory, this qle is also the on-shell value of the gravitational Hamiltonian for the choice of unit lapse function and vanishing shift vector at the system boundary. [6] We also note that the canonical qle is the thermodynamic internal energy in a thermodynamical description of a (relativistic) self-gravitating system. 1 See the excellent review article by Goldberg in Ref. [2] for the details of this analysis. 2 A serious contender for such an over-arching rule has been given by Dougan and Mason, who use certain "(anti-)holomorphic" spinor fields in order to define a four-dimensional space of "quasitranslations" pointing on and off an essentially generic 2-surface. [3] Szabados has shown that the Dougan-Mason proposal provides a tidy characterization of pp-wave geometries. [4] [7] The analysis that leads to the canonical qle runs along a somewhat different line than the one followed in a Noether-type analysis, but it also leads to a concept of energy which is not unique. Indeed, as is always the case with energy, the canonical qle is defined only up to the choice of a zero-point. The zero-point ambiguity may be traced to a freedom present in the action principle. Namely, one may always add to the action any functional of the fixed boundary data without affecting the variational principle. As with the situation above, if the goal is to obtain agreement with the accepted notions of gravitational energy at spacelike or null infinity, then there is a suitably unique choice of energy zero-point [6,8], whereas at the quasilocal level there seems to be no preferred choice. 3 While at first sight this seems no better or worse than the situation encountered above, notice that now the ambiguity in the energy has a physical interpretation, and, moreover, is a field-theoretic generalization of the standard ambiguity present in the hj definition of energy in ordinary mechanics. We may now restate the emphasized portion of the party line above as follows: there is no over-arching rule, applicable for all quasilocal 2-surfaces, for selecting a (suitably unique) zero-point. Taking this statement at face value, we claim that it is the physicist's job to select an appropriate energy zero-point, guided by the principle that the selection should be appropriate for the physics of the problem at hand. We would like to point out that this is a common enough state of affairs in general relativity, a many-faceted theory known for its wealth of possible boundary conditions. Indeed, by way of analogy consider the search for solutions of the Einstein field equations. In practice, relativists certainly do not attempt to find the general solution, rather they attempt to find solutions given some additional physical input (boundary conditions, symmetries, etc.). In practice, the same such additional input is needed to associate a meaningful qle with a particular quasilocal 2-surface. Bearing these points in mind, we recall the form of the canonical qle: E = (8π) −1 S dS (k − k\| ref ) , (0.1) where we adopt geometrical units (in which both Newton's constant and the speed of light are set to unity), S is a closed 2-surface, dS is the proper area element on S, and k is the mean curvature of S as embedded in a spanning 3-surface Σ. It is important to realize that this E, while obtained as a proper surface integral over S, is the energy of the gravitational and matter fields which live on Σ, that is to say, E is a functional of the initial data of Σ. This concept of energy is rooted in the 3+1 view of spacetime geometry, and for a fixed S it is slightly sensitive to the choice of spanning Σ. [More precisely, E depends on the equivalence class of spanning 3-surfaces determined by a unit timelike vector on (and pointing orthogonal to) S.] This sensitivity is quite analogous to the observer dependence of energy in special relativity, and a priori we expect its presence. [9] In general, the term k\| ref = k\| ref (σ ab ) represents an arbitrary (local) function of the intrinsic metric σ ab of S and corresponds to the freedom to assign the qle zero-point. Notice that this freedom corresponds to a proper surface integral of what is effectively a free function of two variables. This freedom, stemming from the field-theoretic character of gravity, is rather more subtle than the freedom in simple mechanics of simply adding a constant to the energy. For our analysis here we only consider two energy zero-points, one determined by lightcone reference and the other by Euclidean reference. We make these concepts precise below. In this paper we examine the canonical qle in the standard "small-sphere limit," first considered by Horowitz and Schmidt in their classic examination of Hawking's quasilocal mass. 4 By the term small sphere we mean a cut S(r), level in an affine radius r, of the lightcone N p belonging to a generic spacetime point p. As a power series in r, we compute the energy E of the gravitational and matter fields on a spacelike hypersurface Σ spanning S(r). Much of our analysis concerns conceptual and technical issues associated with assigning the zero-point of the energy, and, therefore, particularly elucidates the points raised in the first two paragraphs above. For the small-sphere limit, we argue that the correct zero-point is obtained via the aforementioned lightcone reference, which stems from a certain isometric embedding of S(r) into a genuine lightcone of Minkowski spacetime and amounts to fixing k\| ref = − 1 2 R 1 + 1 6 (R) −1 ∆ log R + 1 + 1 6 (R) −1 ∆ log R −1 (0.2) in the above qle definition. Here ∆ is the Laplacian operator and R is twice the Gaussian curvature of S(r); therefore, as advertised, this choice for k\| ref depends solely on the intrinsic geometry of S(r). Notice that, due to the presence of the square root in this expression, one expects this choice for k\| ref to be valid only for (topologically spherical) 2-surfaces possessing everywhere positive Gaussian curvature (as is the case both for the small spheres we study here). Choosing the proper surface integral of this choice for (8π) −1 k\| ref as the energy zeropoint, we find the following results for small spheres: in the presence of matter E = 4 3 πr 3 [T µν u µ u ν ]\| p + O(r 4 ) (0.3) and in vacuo E = 1 90 r 5 [T µνλκ u µ u ν u λ u κ ]\| p + O(r 6 ) (0.4) Here, u µ is a unit, future-pointing, timelike vector in the tangent space at p (which defines the choice of affine radius); T µν is the matter stress-energy-momentum tensor; T µνλκ is the Bel-Robinson gravitational super stress-energy-momentum tensor; 5 and \| p denotes "restriction to p." It is interesting to note that, when integrated, the Bel-Robinson "energy" in (0.4) has been proven to be very useful already in the sense of a mathematical "energy" in the study of existence of solutions of hyperbolic equations in Einsteinian gravity. [13][14][15][16] It is noteworthy that it shows up also in the physical limit given in (0.4). Although the full physical significance of the physical limit is not known to us, we think that these mathematical and physical properties go beyond mere coincidences. For both the vacuum and non-vacuum cases, Hawking's quasilocal mass expression agrees with the canonical qle results (0.3) and (0.4) up to and including the first non-trivial order in the affine radius. We find this result rather striking in light of the fact that the Hawking mass has no apparent connection with the gravitational action or Hamiltonian. We show that the non-vacuum result (0.3) has the expected form based on the results of Newtonian potential theory. Compare the small-sphere limit considered here with the large-sphere limit when S(r) tends to a cut of J + , in which case we know that the choice of Euclidean reference yields agreement between E and the tbs energy. [8] In both limiting cases, the 2-surface S(r) of interest is a cut, level in an affine radius r, of an outgoing congruence of null geodesics. There is, however, a crucial distinction to be made. In the small-sphere case, S(r) arises as the cut of a genuine lightcone, while in the large-sphere case this is generally not true. We find it remarkable that this distinction can be mirrored in the choice of zero-points. To grasp this point, consider first the Euclidean reference for either limit. This reference involves an isometric embedding of S(r) into a flat Euclidean 3-space E 3 . Now, in both limiting scenarios S(r) is, in general, slightly distorted from perfect roundness. Therefore, with E 3 viewed in turn as an inertial slice of Minkowski space M, the enveloped S(r) ⊂ E 3 ⊂ M cannot be the cut of a genuine lightcone of M. (Technically, in this case the outward null congruence associated with S(r) is not shear-free, but the lightcones of M are shear-free.) Therefore, in the general large-sphere scenario, neither the S(r) embedding into the physical spacetime M nor its embedding into the reference spacetime M corresponds to a lightcone embedding [that is, in neither case is S(r) the cut of a lightcone]. Of course, for the smallsphere limit we may employ either the Euclidean reference or the lightcone reference. Of these two choices, the lightcone employs flat spacetime to put the reference space on an equal footing with the small sphere construction in the physical spacetime. To define the lightcone reference, we first isometrically embed S(r) into the lightcone N q of a point q ∈ M, and then select a certain 3-surface Σ ⊂ M spanning S(r) [so that S(r) = Σ N q ]. The details of this construction, eventually leading to the expression (0.2), are found in Subsection 2.A. Now, having defined the lightcone reference (tailored to the small sphere limit) and found the resulting closed-form expression (0.2), we may now invert the question, asking whether or not a lightcone-reference k\| ref defines a qle (0.1) possessing a correct large-sphere limit towards J + . We discuss this issue in Appendix A. The organization of this paper is as follows. In Section 1 we lay the foundations for our T µνλκ := C µρλσ C ν ρ κ σ + * C µρλσ * C ν ρ κ σ = 4 + C µρλσ − C ν ρ κ σ , (0.5) where C µρλσ is the Weyl tensor, * C µρλσ := 1 2 ǫ µραβ C αβ λσ is the left-dual of the Weyl tensor, and ± C µρλσ := 1 2 (C µρλσ ∓ 1 2 iǫ µραβ C αβ λσ ) is the self-dual (+) [anti-self-dual (−)] part of the Weyl tensor. Further curvature conventions are discussed in the appendix. examination of the small-sphere limit. We describe the geometry of the limit in Subsection 1.A, fix some general conventions in Subsection 1.B, and make some general observations concerning the embedding of a 2-surface in Minkowski spacetime in Subsection 1.C. No choice of energy zero-point is made in Section 1, although the results of Subsection 1.C are used in the subsequent sections to construct zero-points. In Section 2 we study the small-sphere limit, subject to the choice of lightcone reference, and derive the results (0.3) and (0.4). In Section 3 we discuss the relationships between the main results (0.3), (0.4) and results from Newtonian potential theory. Appendix A contains discussions of the small-sphere limit subject to the choice of Euclidean reference, and the large-sphere limit towards J + subject to the choice of lightcone reference. Throughout our analysis, we use the Newman-Penrose (np) formalism [17,18], with which we assume the reader is familiar. In Appendix B we collect various conventions and results associated with the np formalism which are used in the main parts of the paper. I. PRELIMINARIES A. Geometry of the limit construction Choose a generic spacetime point p ∈ M, as well as a unit, future-pointing, timelike vector u µ which lies in T p (M), the tangent space at p. We may think of u µ as the instantaneous fourvelocity of an Eulerian observer at p. Let N p ⊂ M represent the future lightcone generated by the null geodesics emanating from p. Label the generators of N p by coordinates (θ, φ), or equivalently by (ζ,ζ), where ζ := e iφ cot(θ/2) is the stereographic coordinate. Further, choose the affine parameter r along the generators of the lightcone which at the point p satisfies the following conditions: r = 0 and the null tangent l µ := (∂/∂r) µ to the affinely parameterized generators obeys l µ u µ = −1. By the term small sphere we shall mean a 2surface S(r) ⊂ N p level in the coordinate r. Provided that we restrict our attention to small enough values of r, we need not be troubled by conjugate points and each 2-surface S(r) will be only slightly distorted from perfect roundness. On our lightcone N p we construct a null tetrad {l µ , n µ , m µ ,m µ } as follows. We take n µ as the inward null normal to each S(r) (normalized so that l µ n µ = −1), and assume that the complex space leg m µ points everywhere tangent to each S(r). Further, via enforcement of the conditionm λ l µ ∇ µ m λ = 0, we remove the freedom to perform r-dependent rotations of the complex dyad. Together with the geodetic property of l µ , this implies that the spin coefficient ε = 0 [cf. Eq. (B1a)]. We also consider a standard pseudo-orthonormal tetrad {u µ =: e ⊥ µ , e x µ , e y µ , e z µ } at the point p, in terms of which we have the following expansions at p: l µ = u µ + sin θ cos φ e x µ + sin θ sin φ e y µ + cos θ e z µ (1.1a) n µ = 1 2 [u µ − (sin θ cos φ e x µ + sin θ sin φ e y µ + cos θ e z µ )] . (1.1b) Capital Latin letters, e. g. A, B, C, · · ·, denote pseudo-orthonormal tetrad indices and run over the values (⊥, x, y, z). The expansions l µ = l A e A µ and n µ = n A e A µ in Eq. (1.1) show that the components l A and n A are essentially the first four spherical harmonics. Standard orthogonality properties of the spherical harmonics then yield the following identities: d Ω l A l B = 4 3 u A u B + 1 3 g AB (1.2a) d Ω l A n B = 1 3 u A u B − 1 6 g AB , (1.2b) d Ω l A l B l C l D = 1 5 16u A u B u C u D + 12u (A u B g CD) + g (AB g CD) , (1.2c) which prove quite useful for reducing many of the integral expressions encountered below. Here we make use of the convenient notation d Ω := (4π) −1 dΩ, where dΩ is the area element of a unit-radius round sphere and the integrations in Eq. (1.2) are over the unit-radius round sphere. B. Physical and reference energy surface densities Let us now define the physical energy surface density (8π) −1 k, whose proper surface integral is the total unreferenced quasilocal energy. Our construction requires that we fix a 3-dimensional hypersurface Σ spanning S(r), or, more precisely, an equivalence class of spanning 3-surfaces determined by the choice of a unit, future-pointing, timelike vector u µ on (and pointing orthogonal to) S(r). We choose u µ := 1 2 l µ + n µ ,(1.3) which at p agrees with the four-velocity introduced in the last paragraph. In terms of the convergences ρ and −µ of the null normals defined in the appendix Eqs. (B1i,k), the mean curvature of S(r) as embedded in Σ is given by k = 2µ + ρ . (1.4) We shall write E\| phy := (8π) −1 S(r) dSk (1.5) for the unreferenced qle associated with the physical slice Σ in spacetime M. Likewise, we introduce the reference energy surface density (8π) −1 k\| ref , and with it define the reference contribution to the qle, E\| ref := (8π) −1 S(r) dSk\| ref . (1.6) As mentioned, E\| ref is a functional solely of the intrinsic geometry of S(r), although as yet we have made no definite choice for this functional. In Section 2 we choose the specific functional stemming from the lightcone reference, while in Section 3 we choose the one stemming from the Euclidean reference. The difference E = E\| phy − E\| ref (1.7) is the total referenced qle (0.1). C. The geometry of 2-surfaces in Minkowski spacetime This subsection collects a few basic results concerning the reference embedding S(r) ⊂ M. We shall use these results later when constructing a particular reference energy surface density. For notational convenience here and in what follows, we often use a sans serif notation for objects associated with Minkowski spacetime M, and we may write k in place of k\| ref . We restrict our attention to Minkowski-spacetime references. That is to say, the reference energy surface density (8π) −1 k is determined via an auxiliary isometric embedding of the 2-surface S(r) into a 3-dimensional hypersurface Σ which is itself contained in Minkowski spacetime M. Physically, this means that we assign the zero value of the energy to that portion of the slice Σ contained within S(r). At the end of the day, our expressions for k depend only on the S(r) 2-metric σ ab (with a, b, c, · · · as S(r) indices), and, therefore, there is technically no need to consider the reference spacetime. Nevertheless, in order to motivate and derive our choices, we begin with the chain of inclusions S(r) ⊂ Σ ⊂ M and an overall spacetime point of view associated with it. Let us collect a few definitions. Take u µ as the unit future-pointing normal of Σ as embedded in M. Let k ab be the extrinsic curvature tensor associated with the spacelike, outward-pointing, unit normal of S(r) as embedded in Σ. Denote by K ij the extrinsic curvature tensor (with i, j, k, · · · as Σ indices) of Σ associated with u µ . Projection into S(r) of all of the free indices of K ij defines an extrinsic curvature tensor l ab on S(r) associated with u µ . Tangential-normal projection with respect to S(r) of K ij defines a covector A b on S(r). If S(r) is to arise as a 2-surface in M, then along with its intrinsic metric σ ab the triple {k ab , l ab , A b } of extrinsic data must obey certain constraints. These are integrability criteria relating the intrinsic and extrinsic data of S(r) to the (vanishing) components of the M Riemann tensor. Among the constraints for S(r) ⊂ M are the following: 6 (k) 2 − k ab k ab − (l) 2 + l ab l ab − R = 0 (1.8a) k :b − k a b:a + A b l − A a l ab = 0 . (1.8b) Here R is the scalar curvature of S(r), and the colon denotes covariant differentiation intrinsic to S(r). We shall not have need to consider the other embedding constraints. II. SMALL-SPHERE LIMIT Throughout this section, the term k\| ref stands for the explicit expression (0.2) given in the introduction, and the functional (1.6) is fixed accordingly. A. Lightcone reference To construct the lightcone reference (see the figure at the end of this paper), consider the notations of Subsection 1.C and assume that S(r) is a cut of a genuine lightcone N q belonging to a point q ∈ M. In this case, the geodesic congruence N q is sheer-free, which means that the complex shear ς := k mm + l mm of N q vanishes. Here k mm := k ab m a m b and l mm := l ab m a m b are complex components with respect to the S(r) dyad {m a ,m a }, respectively capturing the trace-free pieces of k ab and l ab . The vanishing of ς thus implies that the trace-free piece of k ab equals minus the trace-free piece of l ab ; whence Eq. (1.8a) becomes (k) 2 − (l) 2 − 2R = 0 . (2.1) We have no need to consider Eq. (1.8b) in this subsection. 7 We define the convergence ̺ := 1 2 (k − l) of the outward null normal to S(r) ⊂ N q , and with it rewrite Eq. (2.1) as (k) 2 − (k − 2̺) 2 − 2R = 0 . (2.2) Now, we are going view ̺ = ̺(ζ,ζ; r) as some specified function. Specification of ̺ (indirectly) fixes a slice Σ (or, more precisely, an equivalence class of slices) spanning S(r), which in turn defines k. With the last equation and some algebra, we find k = 1 2 (̺) −1 R + ̺ (2.3) as the expression for k in terms of the function ̺ which is not yet specified. Now, plugging both the radial expansion (B21) for R and some radial expansion for ̺ into Eq. (2.3), we obtain a radial expansion for k, concerning which we make the following crucial observations. First (relevant for the non-vacuum case), a reference convergence of the form ̺ = −r −1 +O(r) determines an expansion for k up to and including O(r 2 ) (which is actually higher than the order needed to get the non-vacuum limit). Second (relevant for the vacuum case), a reference convergence of the form ̺ = −r −1 + O(r 2 ) determines an expansion for k up to and including O(r 3 ). Since we wish our final expression for k to serve as a proper reference term, we demand that ̺ = ̺(ζ,ζ; r) be built purely from the 2-metric σ ab of S(r) (so that k is). This restriction alone hardly fixes the choice of ̺ (and k). However, for both scenarios of interest the geometry of the embedding of S(r) into the physical spacetime M suggests a natural choice. Let us write down our choice for ̺, verify that it leads to Eq. (0.2), and finally discuss why it is physically meaningful. We pick ̺ = − 1 2 R 1 + 1 3 (R) −1ð ð log R ,(2.k = − 1 2 R 1 + 1 3 (R) −1ð ð log R + 1 + 1 3 (R) −1ð ð log R −1 ,(2.5) as given in Eq. (0.2) for 8π times the reference energy surface density. Now, let us argue that Eq. (2.5) defines a physically sensible reference surface density for the small-sphere limit. Using Eqs. (B16) and (B14) for the ð operator and the radial expansion (B21) for R, we find (whether in vacuo or not) that (2.4) satisfies ̺ = −r −1 − 1 4 r(R 0 + 1 3ð 0 ð 0 R 0 ) + O(r 2 ) ,(2.6) where ð 0 is the unit-radius round-sphere "eth" operator. Hence, we have at least ̺ = −r −1 + O(r) for the reference convergence. To start with, consider the non-vacuum case and glance at the expansion (B4a) for the convergence ρ associated with the physical embedding S(r) ⊂ N p . We see that through O(r 0 ) our choice for ̺ agrees with the physical ρ. Now turn to the vacuum case and notice that the expansion (B5a) for the convergence ρ associated with the physical embedding obeys ρ = −r −1 +O(r 3 ). Combining Eq. (2.6) with the vacuum identity 8 ð 0ð0 R 0 = −3R 0 ,(2.k\| ref =: k = −2r −1 − 1 2 rR 0 − 1 2 r 2 R 1 − 1 2 r 3 R 2 + O(r 4 ) (2.9) for the vacuum radial expansion of Eq. (2.5). B. Non-vacuum limit Turning our attention to the non-vacuum scenario, we begin by using the appendix Eqs. (B4a,f) for the convergences −µ and ρ of the null normals to determine the following expansion for 8π times the physical energy surface density: [cf. Eq. (1.4)] k = −2r −1 + r Ψ 0 2 +Ψ 0 2 + 2Λ 0 + 2 3 Φ 0 11 + O(r 2 ) ,(2.10) Here, Ψ 0 2 , Φ 0 11 , and Λ 0 are r → 0 limits of standard curvature terms from the np formalism. Respectively, they are proportional to a certain null-tetrad component of the Weyl tensor C µνλκ at p, a certain null-tetrad component of the Ricci tensor R µν at p, and the Ricci scalar R at p [cf. Eqs. (B3c,f,k)]. Because we have adapted our null tetrad to the lightcone N p , the components Ψ 0 2 and Φ 0 11 are in fact angle-dependent, i. e. they are functions of (ζ,ζ); however, as the limit of a scalar function, Λ 0 is not angle dependent. Next, we substitute both Eq. (2.10) and the appendix Eq. (B18) for the surface element dS of S(r) into the basic expression (1.5) for E\| phy , and expand out the result in order to obtain E\| phy = −r + r 3 d Ω 1 2 (Ψ 0 2 +Ψ 0 2 ) + Λ 0 + 1 3 Φ 0 00 + 1 3 Φ 0 11 + O(r 4 ) . (2.11) Like before, Φ 0 00 , an angle-dependent term, is proportional to a certain null-tetrad component (B3g) of the Ricci tensor at p. We can perform the integrations remaining in Eq. (2.11). The spherical average of the real part of Ψ 0 2 vanishes identically. Indeed, the r → 0 limit of Eq. (B3c) shows that Ψ 0 2 +Ψ 0 2 = (l A n B l C n D C ABCD )\| p , and by symmetry the average of l A n B l C n D must yield terms either proportional to u A u B u C u D or containing g AB . Therefore, that the average in question vanishes follows from the index symmetries and trace-free character of the Weyl tensor. For the next integration we can straightaway write d Ω Λ 0 = 1 24 R\| p . To evaluate the unit-sphere averages of the other terms in the integrand, we must use the identities (1.2a,b) k\| ref = −2r −1 + 1 3 r 2ð 0 Ψ 0 1 + 2ð 0Ψ 0 1 − ð 0 Φ 0 10 −ð 0Φ 0 10 − Φ 0 00 + O(r 2 ) . (2.13) Now, substitute Eqs. (2.13) and (B18) into Eq. (1.6), do some algebra, and perform a few simple integrations 9 in order to reach 9 Here and in what follows, "simple integrations" refer to the following version (and its complex E\| ref = −r + 1 6 r 3 d Ω Φ 0 00 + O(r 4 ) . (2.14) Like before, we use Eq. (1.2a) to compute the remaining integral, thereby finding the desired result: E\| ref = −r + 1 36 r 3 [4R µν u µ u ν + R]\| p + O(r 4 ) ,(2.15) which should be compared with Eq. (2.12). We may now easily obtain the Introduction's result (0.3) for the non-vacuum limit. Indeed, combination of Eqs. (2.12) and (2.15) yields the following result for the total qle (1.7): E = 1 6 r 3 [G µν u µ u ν ]\| p + O(r 4 ) . (2.16) Notably, this result for E is valid whether or not the field equations hold, i. e. it is a geometric identity. However, with Einstein's equations G µν = 8πT µν we immediately arrive at the Introduction's result (0.3). C. Vacuum limit The derivation of the vacuum limit proceeds along the same lines as those just considered for the non-vacuum limit. With the appendix Eqs. (B5a,f) for the vacuum-case convergences −µ and ρ, we compute the radial expansion for the physical energy surface density, [cf. Eq. (1.4)] k = −2r −1 + 1 3 r ð 0Ψ 0 1 +ð 0 Ψ 0 1 + 1 6 r 2 ð 0Ψ 1 1 +ð 0 Ψ 1 1 +r 3 1 180 Ψ 0 0Ψ 0 0 − 1 20 ð 0 Ψ 0 0 Ψ 0 1 + 4Ψ 2 1 − 1 20ð 0 Ψ 0 0Ψ 0 1 + 4Ψ 2 1 − 1 2 R 2 + O(r 4 ) . (2.17) Next, we substitute this expansion along with the expansion (B19) for the vacuum-case surface element into Eq. (1.5), do some algebra, and perform a few simple integrations. These steps lead to an explicit computation using our definition (0.5) of the Bel-Robinson tensor. Evidently then, we may use Eq. (1.2c) to find E\| phy = −r + r 5 d Ω 7 360 Ψ 0 0Ψ 0 0 + 1 4 1 45 Ψ 0 0Ψ 0 0 − R 2 + O(r 6 ) .E\| phy = −r + 7 450 r 5 [T µνρσ u µ u ν u ρ u σ ]\| p + O(r 6 ) (2.19) as the desired limit expression for E\| phy in vacuo. Turning now to the calculation of E\| ref , we first put together Eqs. (2.9) and (B23), in order to get the following explicit expansion: k\| ref = −2r −1 + 2 3 r ð 0Ψ 0 1 +ð 0 Ψ 0 1 + 5 12 r 2 ð 0Ψ 1 1 +ð 0 Ψ 1 1 + 1 2 r 3 1 45 Ψ 0 0Ψ 0 0 − R 2 − 1 45 Ψ 0 0Ψ 0 0 + O(r 4 ) . (2.20) We substitute this expansion as well as the expansion (B19) into Eq. (1.6), and again do some algebra and a few simple integrations, thereby obtaining E\| ref = −r + r 5 d Ω 1 180 Ψ 0 0Ψ 0 0 + 1 4 1 45 Ψ 0 0Ψ 0 0 − R 2 + O(r 6 ) . (2.21) Finally, calculations identical to those just performed for E\| phy establish that E\| ref = −r + 1 225 r 5 [T µνρσ u µ u ν u ρ u σ ]\| p + O(r 6 ) ,(2. III. NEWTONIAN POTENTIAL THEORY It is interesting to compare our main results with analogous results from Newtonian potential theory. 10 The Newtonian interpretation for the Introduction's result (0.3) is straightforward. Consider a pressureless ball of fluid of radius r and constant (volume) mass density ǫ. The total Newtonian mass for the ball is M = 4 3 πr 3 ǫ. If we identify this radius with the affine radius that appears in the result (0.3), then a further identification between the Newtonian mass M and the quasilocal energy E yields the correspondence ǫ = [T µν u µ u ν ]\| p . This is a solid result, fully in accord with the non-existence of "pure gravitational field energy" at a point, since only matter energy contributes in this limit. The Newtonian analog of the result (0.4) is not clear. Nevertheless, we find the following result rather interesting. Consider again the ball of fluid, but now let us compute the Newtonian gravitational field energy within the ball. Recall that the energy density of the Newtonian gravitational field is given by − ∇Φ· ∇Φ/(8π), where ∇ is the flat-space gradient operator and Φ is the Newtonian potential. Inserting the appropriate expression for Φ into the energy density and integrating over the interior of the ball, we obtain E N = − 1 90 r 5 (4πǫ) 2 (3.1) for the gravitational potential energy inside the ball. Comparison of this result with the small-sphere result (0.4), we see that both expressions depend on the fifth power of radius and both contain a numerical factor of 1 90 . We do not know at this time if the close resemblance between these results has any physical significance. IV. ACKNOWLEDGMENTS For discussions we thank H. Balasin, T. G. Concannon, N.Ó Murchadha, and L. B. Szabados. In particular, SRL thanks L. B. Szabados for comments which led to this investigation and N.Ó Murchadha for discussions concerning the Newtonian interpretation of the vacuum limit. This work was begun while SRL visited the Research Institute for Particle & Nuclear Physics in Budapest, Hungary, and SRL wishes to express his appreciation to this institute both for financial support and a hospitable stay in Budapest. This work has been supported by the National Science Foundation of the USA (NSF grant # PHY-9413207 to the University of North Carolina), and in part by the "Fonds zur Förderung der wissenschaftlichen Forschung" in Austria (FWF Project 10.221-PHY). APPENDIX A: EUCLIDEAN REFERENCE AND LARGE-SPHERE LIMIT One can also carry out our analysis of the small-sphere limit for the choice of Euclidean reference. To define the Euclidean reference, again consider the notations of Subsection I.C, but now assume that Σ is a flat inertial hyperplane E 3 in M. In this case, l ab = 0 = A b and the constraints given in Eqs. (1.8a,b) are simply (k) 2 − k ab k ab − R = 0 (A1a) k :b − k a b:a = 0 .(A1b) To analyze these equations, it proves useful to split k ab into its trace and trace-free pieces. This is readily achieved by working with the components of k ab with respect to the null dyad {m a ,m a }. Indeed, 2k mm := 2m amb k ab and k mm := m a m b k ab respectively capture the trace and trace-free pieces of k ab , and in terms of these quantities the Eqs. (A1a,b) become [k mm ] 2 − k mm kmm − 1 2 R = 0 (A2a) ðk mm −ðk mm = 0 .(A2b) For the case of Euclidean reference, we are unable to obtain a closed-form expression for k (neither in the non-vacuum nor vacuum cases). Nevertheless, it is evident from Eqs. (A1a,b) that the full extrinsic curvature tensor k ab (and hence its trace piece) is determined solely by the intrinsic metric on S(r). One expects that Eqs. (A2a,b) may be solved for k ab , provided S(r) is only slightly distorted from perfect roundness. For small enough values of r, the expansion (B21) for R assures us that this is indeed the case. Moreover, the solution k ab should be unique up to Euclidean translations and rotations. Lacking a closed-form expression for k, we have obtained radial expansions in both the non-vacuum and vacuum scenarios. For both scenarios, we obtain such expansions by first plugging into Eqs. (A2a,b) the Ansätze k mm = −r −1 + rk 1 mm + r 2 k 2 mm + r 3 k 3 mm + O(r 4 ) (A3a) k mm = rk 1 mm + r 2 k 2 mm + r 3 k 3 mm + O(r 4 ) ,(A3b) along with appropriate (vacuum or non-vacuum, as the case may be) expansions for both ð and R. We have performed this calculation, and find that the choice of Euclidean reference also establishes (0.3) in the non-vacuum case. However, with the choice of Euclidean reference, the vacuum small-sphere limit of the qle is not related directly to the Bel-Robinson tensor. Our closing comments concern the large-sphere limit, in which case S(r) tends to a round infinite-radius 2-sphere cut of J + . Thus, we now consider a spacetime M which is asymptotically flat towards J + and a corresponding system of Bondi coordinates (w, r, ζ,ζ). Here w is a retarded time coordinate, and r is an affine radius similar to before. Now S(r) arises as a cut, level in r, of an outgoing null hypersurface N (in general not a genuine lightcone), and twice the Gaussian curvature of S(r) has the following expansion in powers of inverse r: R = 2r −2 + R −3 r −3 + O(r −4 ) ,(A4) where the coefficient R −3 may be expressed in terms of the asymptotic shear σ 0 of N . [8] If we use formula (2.1) (appropriate for a lightcone embedding) along with the "rest-frame" choice l = 0, then we have k = − √ 2R .(A5) Although this expression differs from (0.2), it is also a lightcone reference. from (A5) we find the following radial expansion for the reference mean curvature: k\| ref = −2r −1 − 1 2 r −2 R −3 + O(r −4 ) ,(A6) which agrees through O(r −2 ) with the k\| ref expansion obtained in Ref. [8] via the Euclidean reference. Hence, all of the results found in Ref. [8] are also valid for the this choice of lightcone reference. In particular, the lightcone-referenced qle agrees with the tbs energy in a suitable null limit. Moreover, in the same limit the "smeared" version of the qle [which incorporates a lapse function into the definition (0.1)] agrees with Geroch's supermomentum (when the latter is evaluated in a Bondi conformal frame). See Ref. [8] for further details. To differentiate between the lightcone and Euclidean references in the large-sphere limit, one could examine multipole-moment terms (which arise at higher powers of inverse radius) for stationary spacetimes. We hope to return to this issue elsewhere. APPENDIX B: CONNECTION AND CURVATURE Throughout our discussion, we have taken (−, +, +, +) as the signature of the spacetime metric. We have also adopted the index conventions of Ref. [19] for curvature. With these choices we shall define the standard Newman-Penrose (np) connection and curvature coefficients such that our np equations match those listed by Newman and Tod in the appendix of Ref. [17]. As we use a somewhat non-standard np formalism, let us list some of our basic conventions explicitly. Label our null frame as {l µ , n µ , m µ ,m µ } = {e 1 µ , e 2 µ , e 3 µ , e 4 µ } and name its associated connection coefficients as follows: ε = 1 2 (Γ 121 + Γ 431 ) κ = Γ 131 π = Γ 421 (B1a,b,c) γ = 1 2 (Γ 122 + Γ 432 ) τ = Γ 132 ν = Γ 422 (B1d,e,f) β = 1 2 (Γ 123 + Γ 433 ) σ = Γ 133 µ = Γ 423 (B1g,h,i) α = 1 2 (Γ 124 + Γ 434 ) ρ = Γ 134 λ = Γ 424 , (B1j,k,l) where, for example, Γ 134 = l µmν ∇ ν m µ . For the small-sphere limit the null frame is adapted to the lightcone N p as described in the first section, and, as a result, the following vanish identically for our construction: ρ −ρ = µ −μ = ε = κ = τ −ᾱ − β = π − α −β = 0 .(B2) As we retain the freedom to perform r-independent rotations of the space leg m µ , we shall use those elements of the Geroch-Held-Penrose (ghp) formalism [18] pertaining both to spin-weighted scalars and the "eth" operator ð on S(r). If a quantity Q transforms as Q → exp(2isχ)Q under the rotation m µ → exp(2iχ)m µ (with χ independent of r), then Q is said to be a spin-weighted scalar of spin weight s (in symbols, sw(Q) = s). As the extents of our null normals l µ and n µ have been fixed once and for all, we have no need to consider the concept of boost weight. Next, consider the spacetime Ricci scalar as well as the components of the Weyl and Ricci tensors with respect to the null tetrad. With these define the following standard pieces of the spacetime curvature: Ψ 0 = C 1313 Ψ 1 = C 1213 Ψ 2 = 1 2 (C 1212 + C 4312 ) (B3a,b,c) Ψ 3 = C 1242 Ψ 4 = C 2424 Λ = 1 24 R (B3d,e,f) Φ 00 = 1 2 R 11 Φ 10 = 1 2 R 14 Φ 20 = 1 2 R 44 (B3g,h,i) Φ 01 = 1 2 R 13 Φ 11 = 1 4 (R 12 + R 34 ) Φ 21 = 1 2 R 24 (B3j,k,l) Φ 02 = 1 2 R 33 Φ 12 = 1 2 R 23 Φ 22 = 1 2 R 22 . (B3m,n,o) As mentioned above, with these conventions our np equations are exactly those listed in the appendix of Ref. [17]; however, we shall consider all possible simplification of these equations afforded by Eq. (B2) and the use of ð [cf. Eqs. (B14) and (B16)]. For the small-sphere limit examined in this paper, we consider the pullbacks to the lightcone N p of the curvature components (B3) and assume that each pullback may be expanded as a power series in r along the lightcone. That is, we assume Ψ i = Ψ 0 i + rΨ 1 i + r 2 Ψ 2 i + · · · i = 0, 1, 2, 3, 4 (B4a) Φ ij = Φ 0 ij + rΦ 1 ij + r 2 Φ 2 ij + · · · i, j = 0, 1, 2 (B4b) Λ = Λ 0 + rΛ 1 + r 2 Λ 2 + · · · .(B4c) We shall use these expansions for the curvature components along with the radial np field equations in order to obtain radial expansions along N p for those spin coefficients (B1) used in our analysis. We consider the non-vacuum and vacuum cases separately. Non-vacuum Up to the appropriate order in the affine radius, we have confirmed the non-vacuum asymptotic expansions of the spin coefficients given by Kelly et al [11] and by Dougan. [12] For completeness we recall that this list is ρ = −r −1 + 1 3 rΦ 0 00 + O(r 2 ) (B4a) σ = 1 3 rΨ 0 0 + O(r 2 ) (B4b) α = r −1 α 0 + 1 6 r α 0 Φ 0 00 + 2Φ 0 10 −Ψ 0 1 −ᾱ 0Ψ0 0 + O(r 2 ) (B4c) β = −r −1ᾱ0 − 1 6 r ᾱ 0 Φ 0 00 − 3Ψ 0 1 − α 0 Ψ 0 0 + O(r 2 ) (B4d) λ = 1 6 r 5Φ 0 20 +Ψ 0 0 + O(r 2 ) (B4e) µ = − 1 2 r −1 + 1 2 r Ψ 0 2 +Ψ 0 2 + 2Λ 0 + 2 3 Φ 0 11 − 1 3 Φ 0 00 + O(r 2 ) ,(B4f) where α 0 ≡ 1 8 ζ. For the scenario at hand the expansions (B4c,d) determine the expansions for τ and π up to O(r 2 ). Actually, we need only know that λ = O(r) in order to obtain the O(r) coefficient for µ; however, for completeness we have explicitly given the O(r) coefficient for λ. As they are not needed in this paper, we do not list the expansions for γ and ν. Vacuum For the vacuum scenario, we need some of the required spin-coefficient expansions out to an order higher than given in either of Refs. [11,12]. We obtain the following list: ρ = −r −1 + 1 45 r 3 Ψ 0 0Ψ 0 0 + O(r 4 ) (B5a) σ = 1 3 rΨ 0 0 + 1 4 r 2 Ψ 1 0 + 1 5 r 3 Ψ 2 0 + O(r 4 ) (B5b) α = r −1 α 0 − 1 6 r Ψ 0 1 +ᾱ 0Ψ0 0 − 1 12 r 2 Ψ 1 1 +ᾱ 0Ψ1 0 + 1 360 r 3 7α 0 Ψ 0 0Ψ 0 0 + 8Ψ 0 0 Ψ 0 1 − 18ᾱ 0Ψ2 0 − 3ð 0Ψ 2 0 + O(r 4 ) (B5c) β = −r −1ᾱ0 + 1 6 r 3Ψ 0 1 + α 0 Ψ 0 0 + 1 12 r 2 4Ψ 1 1 + α 0 Ψ 1 0 − 1 360 r 3 7ᾱ 0 Ψ 0 0Ψ 0 0 − 20Ψ 0 0Ψ 0 1 − 18α 0 Ψ 2 0 − 15ð 0 Ψ 2 0 + O(r 4 ) (B5d) λ = − 1 6 rΨ 0 0 + O(r 2 ) (B5e) µ = − 1 2 r −1 + 1 2 r Ψ 0 2 +Ψ 0 2 + 1 3 r 2 Ψ 1 2 +Ψ 1 2 +r 3 1 360 Ψ 0 0Ψ 0 0 − 1 40 ð 0 Ψ 0 0 Ψ 0 1 + 4Ψ 2 1 − 1 40ð 0 Ψ 0 0Ψ 0 1 + 4Ψ 2 1 − 1 4 R 2 + O(r 4 ) ,(B5f) where the coefficient R 2 [cf. Eq. (B21)] found in µ is written out explicitly below in Eq. (B23c). Also, the unit-sphere "eth" operator ð 0 is defined below in Eq. (B15). As before, Eqs. (B5c,d) determine expansions for τ and π up to O(r 4 ), and we do not need the expansions for γ and ν. The particular form of the O(r 3 ) coefficient µ 3 in the expansion (B5f) for µ = − 1 2 r −1 + · · · + r 3 µ 3 + O(r 4 )(B6) plays a crucial role in our calculation of the qle's vacuum limit. Therefore, let us sketch how to obtain the given form of µ 3 , assuming that we have already determined both µ up to and including O(r 2 ) and the remaining expansions (B5) as given. Rather than making a straightforward appeal to the np field equations (as we have done to obtain the other spin coefficients), we shall instead take advantage of the particular geometry of our construction and derive this coefficient via the geometric identity [18] K = −Ψ 2 + Φ 11 + Λ − σλ + µρ . (B7) K is the complex Gauss curvature of S(r), and the Ricci scalar of S(r) is simply R = 2(K +K). Since we work here in vacuo and both ρ and µ are real, Eq. (B7) implies that R = −2(Ψ 2 +Ψ 2 ) − 2(σλ +σλ) + 4µρ .(B8) Now, into this equation insert both Eq. (B6) and the expansions (B5a,b,e), and then isolate the O(r 2 ) piece of the resulting expression, in order to establish that µ 3 = − 1 2 (Ψ 2 2 +Ψ 2 2 ) + 2 45 Ψ 0 0Ψ 0 0 − 1 4 R 2 .(B9) To work this relationship into the desired form, we appeal to the following vacuum np Bianchi identity: DΨ 2 −ðΨ 1 = −λΨ 0 + πΨ 1 + 3ρΨ 2 = 0 ,(B10) where D = l µ ∇ µ and the action ofð on Ψ 1 is defined as the conjugate of ðΨ 1 [with ð as in Eq. (B16) and sw(Ψ 1 ) = −1]. The given form of Eq. (B10) is specific to the geometry of our construction, but it can be deduced from the (more general) appendix Eq. (A.4c) given in Ref. [17]. Now, plug the expansions (B4) and (B5) as well as the radial expansion forð into the Bianchi identity (B10), thereby obtaining a tower of identities (one identity at each power of r). Isolate the particular identity determined at O(r) in the tower, and into this equation make repeated substitutions with (B24a,b). These steps lead to Ψ 2 2 = 1 24 Ψ 0 0Ψ 0 0 + 1 20ð 0 (Ψ 0 0Ψ 0 1 + 4Ψ 2 1 ) ,(B11) which, upon substitution into Eq. (B9), establishes the chosen form of the coefficient µ 3 found in the µ expansion (B5f). Intrinsic geometry of S(r) In this final appendix subsection we consider the intrinsic geometry of S(r) and collect radial expansions along N p for the "eth" operator ð, the surface area element dS, and the intrinsic curvature scalar R [the Gaussian curvature of S(r) being twice R]. Again, we consider the non-vacuum and vacuum cases separately. We obtain the desired intrinsicgeometry expansions as follows. First, writing the complex space leg of the null tetrad as [11] δ ≡ m µ ∂/∂x µ = ξ ∂/∂ζ +η ∂/∂ζ , we determine the r expansions along N p for the tetrad coefficients ξ and η from the np commutator equations, [17,11] Dξ = ρξ + ση (B13a) Dη = ρη +σξ (B13b) δξ − δη = (α −β)ξ − (ᾱ − β)η ,(B13c) and the radial expansions for the spin coefficients [Eqs. (B4a,b,c,d) or Eqs. (B5a,b,c,d) as the case may be]. The given forms of these equations are particular to how our null frame has been adapted to the lightcone N p . Let us first consider the ð operator. With both Eqs. (B4c,d) and the non-vacuum radial expansions for ξ and η, we compute the non-vacuum radial expansion for ð, ð ≡ δ + s(ᾱ − β) = r −1 ð 0 + 1 6 r Ψ 0 0ð 0 − 4sΨ 0 1 + Φ 0 00 ð 0 + 2sΦ 0 10 + O(r 2 ) .(B14) Here, as above, δ is the standard np notation for the directional derivative along m µ , s = sw(ϕ) denotes the spin weight of some spin-weighted scalar ϕ on which ð acts, and in terms of P = 1 + ζζ we define ð 0 ≡ 1 2 P ∂/∂ζ + 2sᾱ 0 . A similar calculation shows that the vacuum ð operator is ð ≡ δ + s(ᾱ − β) = r −1 ð 0 + 1 6 r Ψ 0 0ð 0 − 4sΨ 0 1 + 1 12 r 2 Ψ 1 0ð 0 − 5sΨ 1 1 + 1 360 r 3 7Ψ 0 0Ψ 0 0 ð 0 + 18Ψ 2 0ð 0 − 18s ð 0 Ψ 2 0 − 12sΨ 0 0Ψ 0 1 + O(r 4 ) ,(B16) with δ, s, and ð 0 as before. Next, we substitute the radial expansions we find for ξ and η into area element of S(r), dS = 1 2 dΩP 2 ξξ + ηη ξξ − ηη . Here dΩ is the standard unit-radius round sphere area element. For the non-vacuum scenario dS = dΩr 2 1 − 1 3 r 2 Φ 0 00 + O(r 3 ) ,(B18) These expansions may be checked with the well-known formula D(dS) = −2ρdS for the change in the surface-area element. [10] Let us turn to the Gauss curvature of S(r). An order-by-order examination of the identity (ðð − ðð)ϕ = 1 2 Rϕ (B20) [where sw(ϕ) = 1, but ϕ is otherwise arbitrary, and we assume the expansion ϕ = ϕ 0 + rϕ 1 + r 2 ϕ 2 + · · ·] determines the following expansion for the intrinsic curvature of S(r): R = 2r −2 + R 0 + rR 1 + r 2 R 2 + O(r 3 ) . (B21) In examining Eq. (B20), we must use the appropriate expansion for ð, Eq. (B14) or Eq. (B16) as the case may be. For the non-vacuum scenario, we find the first relevant coefficient to be R 0 = 2 3 Φ 0 00 + ð 0 Φ 0 10 +ð 0Φ 0 10 − 2ð 0Ψ 0 1 − 2ð 0 Ψ 0 1 .(B22) For the vacuum case we need more coefficients. Tedious but straightforward calculation yields the set: R 0 = − 4 3 ð 0Ψ 0 1 +ð 0 Ψ 0 1 (B23a) R 1 = − 5 6 ð 0Ψ 1 1 +ð 0 Ψ 1 1 (B23b) R 2 = 1 45 Ψ 0 0Ψ 0 0 − 3 5 ð 0Ψ 2 1 − 3 5ð 0 Ψ 2 1 − 17 90 ð 0 Ψ 0 0 Ψ 0 1 − 17 90ð 0 Ψ 0 0Ψ 0 1 .(B23c) We remark that, along the way to obtaining these coefficients of R from Eq. (B20), one finds terms involving derivatives of the coefficients of ϕ. However, one knows that such terms must vanish, as the right-hand side of Eq. (B20) involves no derivatives of ϕ. Therefore, one may simply discard these terms. A detailed examination (order by order in r) of the Bianchi identities (non-vacuum or vacuum as the case may be) shows that such terms indeed vanish. Vacuum Bianchi Identities Consider the vacuum Bianchi identities as given in in the appendix Ref. [17]. For the situation at hand, we may simplify these equations by writing them in terms of ð rather than δ, and we do so. Then, considering the equations order by order in r, we find the following at the lowest order in r: Respectively, the top and bottom boxes depict the physical and reference spacetimes. In the top box the shaded surface spanning S(r) is the 3-surface Σ, while in the bottom box the shaded surface spanning S(r) is the 3-surface Σ. Whether viewed as the intersection N p Σ or the intersection N q Σ, the 2-surface S(r) has the same intrinsic 2-metric. Our limit construction gives us Σ, but we must choose Σ; and, moreover, our choice of Σ must be determined solely by the intrinsic 2-metric of S(r). Our choice and its physical motivation are described in Subsection 2.A. However, we note here that, whenever S(r) is at all distorted from perfect roundness (as it generally will be), Σ is not flat Euclidean 3-space E 3 (because the intersection of E 3 with the genuine lightcone N q would be a round sphere). Choosing such a lightcone reference, we assign the zero value of the energy to that (shaded) portion of Σ contained within S(r), and compute the energy of (the shaded portion of) Σ relative to this zero-point. ð 0 Ψ 0 0 = 0ð 0 Ψ 0 0 = 4Ψ 0 1 (B24a,b) ð 0 Ψ 0 1 = − 1 2 Ψ 0 0ð 0 Ψ 0 1 = 3Ψ 0 2 (B24c,d) ð 0 Ψ 0 2 = −Ψ 0 1ð 0 Ψ 0 2 = 2Ψ 0 3 (B24e,f) ð 0 Ψ 0 3 = − 3 2 Ψ 0 2ð 0 Ψ 0 3 = Ψ 0 4 (B24g,h) ð 0 Ψ 0 4 = −2Ψ 0 3ð 0 Ψ 0 4 = 0 . ̺ = −r −1 + O(r 2 ); and, therefore, now through O(r 1 ), our choice for ̺ agrees with the physical ρ. To sum up, we can state that, whether in vacuo or not, our choice (2.4) for ̺ determines an embedding of S(r) into N q ⊂ M which would seem closely related both intrinsically and extrinsically to the physical embedding of S(r) into N p ⊂ M. Let us now put aside the issue of motivation, and simply expand our choice (2.5) for k in powers of r. Since ̺ = −r −1 + O(r) in non-vacuum, from Eq. (2.3) we get k\| ref =: k = −2r −1 − 1 2 rR 0 + O(r 2 ) (2.8) for the non-vacuum radial expansion of Eq. (2.5). Likewise, since ̺ = −r −1 +O(r 2 ) in vacuo, from Eq. (2.3) we now get 8 One proves the identity as follows. First, as we do in the appendix, calculate the vacuum-case O(r 0 ) piece of the S(r) curvature scalar. One finds R 0 = −4(Ψ 0 2 +Ψ 0 2 ) [cf. Eqs.(B23a) and (B24d)]. But, as shown in the final part of the appendix, the radial Bianchi identities [cf. Eq. (B24d,e,f,g)] imply that ð 0ð0 Ψ 0 2 = −3Ψ 0 2 which establishes the result. at the explicit expression (B23c) for the coefficient R 2 [of the O(r 2 ) term in the expansion (B21) for R] shows that the unit-sphere average of the term within the parenthesis above vanishes. To evaluate the final unit-sphere average, we first note that the r → 0 limit of the square of Eq. (B3a) is in fact A l B l C l D T ABCD ]\| p . This may be checked by conjugate) of the divergence theorem. Suppose that sw(f ) = −1 (with sw denoting spin weight). Then with ð 0 representing the "eth" operator on the unit-radius round sphere, we have d Ω ð 0 f = 0. FIG. 1 . 1Geometry of the Lightcone Reference. . The r → 0 of Eq. as the final limiting expression for the unreferenced qle.To obtain an expansion analogous to Eq. (2.12) for E\| ref , we must first explicitly compute the radial expansion for 8π times the reference energy density. Putting together Eqs.(2.8) and (B22), we get(B3g) is Φ 0 00 = 1 2 [l A l B R AB ]\| p , which shows that the average of Φ 0 00 is readily obtained with Eq. (1.2a). By rewriting the r → 0 limit of Eq. (B3k) as Φ 0 11 = [ 1 2 l A n B R AB + 1 8 R]\| p , we likewise obtain the average of Φ 0 11 with Eq. (1.2b). Adding together the individual results for these integrations, we find E\| phy = −r + 1 18 r 3 [5R µν u µ u ν + 2R]\| p + O(r 4 ) (2.12) Other than the choice of a vanishing zero-point. This choice corresponds to the aforementioned choice of S-adapted coordinates for the Einstein definition. Like before, such a choice, leading to an infinite energy, wrecks the agreement between the canonical quasilocal and tbs energies in the large-sphere null limit. This limit is also considered in Ref.[11] by Kelly, Tod, and Woodhouse for Penrose's kinematic twistor and associated quasilocal mass, and in Ref.[12] by Dougan for the Dougan-Mason quasilocal four-momentum.5 Our vacuum-case Bel-Robinson tensor is the following: As our use of power series in r is widespread in this paper, we use parenthesis when a variable, say R, is raised to a power. Hence, (R) 2 = RR and (R) −1 = 1/R, while R 2 always denotes the O(r 2 ) coefficient in the expansion R = · · · + r 2 R 2 + · · ·. The only exception to this rule will be the radius r itself. As there is no possibilty for confusion, we use, for example, r −1 to mean 1/r. For the lightcone reference we construct in this subsection, Eq. (1.8b) and the other embedding constraints not appearing above would be differential equations determining the remaining S(r) extrinsic data from σ ab and our choices for l and k. 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York, Phys. Rev. D47, 1407 (1993). . J W York, Phys. Rev. 332092J. W. York, Phys. Rev. D33, 2092 (1986); . J D Brown, E A Martinez, J W York, Phys. Rev. Lett. 662281J. D. Brown, E. A. Martinez, and J. W. York, Phys. Rev. Lett. 66, 2281 (1991); . J D Brown, J W York, Phys. Rev. 471420J. D. Brown and J. W. York, Phys. Rev. D47, 1420 (1993). . J D Brown, S R Lau, J W York, Phys. Rev. 551977J. D. Brown, S. R. Lau, and J. W. York, Phys. Rev. D55, 1977 (1997). . J D Brown, S R Lau, J W York, Boosted Quasilocal Energy in General Relativity. manuscript in preparationJ. D. Brown, S. R. Lau, and J. W. York, "Boosted Quasilocal Energy in General Rela- tivity" (manuscript in preparation). G T Horowitz, B G Schmidt, Proc. R. Soc. Lond. A. R. Soc. Lond. A381215G. T. Horowitz and B. G. Schmidt, Proc. R. Soc. Lond. A 381, 215 (1982). . R M Kelly, K P Tod, N M J Woodhouse, Class. Quantum Grav. 31151R. M. Kelly, K. P. Tod, and N. M. J. Woodhouse, Class. Quantum Grav. 3, 1151 (1986). . A J Dougan, Class. Quantum Grav. 92461A. J. Dougan, Class. Quantum Grav. 9, 2461 (1992). H Friedrich, Proc. R. Soc. Lond. A. R. Soc. Lond. A375169H. Friedrich, Proc. R. Soc. Lond. A 375, 169 (1981). D Christodoulou, S Klainerman, The Global Nonlinear Stability of Minkowski Space. PrincetonPrinceton University PressD. Christodoulou and S. Klainerman, The Global Nonlinear Stability of Minkowski Space (Princeton University Press, Princeton, 1993). . A Anderson, Y Choquet-Bruhat, J W York, Topological Methods in Nonlinear Analysis. 10353A. Anderson, Y. Choquet-Bruhat, and J. W. York, Topological Methods in Nonlinear Analysis 10, 353 (1997). J W York, Gravitation and Geometry. W. Rindler and A. TrautmanBibliopolis, NaplesJ. W. York, in Gravitation and Geometry, eds. W. Rindler and A. Trautman (Bibliopolis, Naples, 1987). . T Newman, K P Tod, General Relativity and Gravitation. A. Held2Plenum PressT. Newman and K. P. Tod, in General Relativity and Gravitation, vol. 2, edited by A. Held (Plenum Press, New York, 1980). R Penrose, W Rindler, Spinors and Spacetime. CambridgeCambridge University Press1R. Penrose and W. Rindler, Spinors and Spacetime, vol. 1, (Cambridge University Press, Cambridge, 1985). C W Misner, K S Thorne, J A Wheeler, Gravitation. San FranciscoFreemanC. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973).	[]
[ "Non-Decoupling Effects of Higgs Bosons on e + e − → W + L W − L in the Two-Doublet Model", "Non-Decoupling Effects of Higgs Bosons on e + e − → W + L W − L in the Two-Doublet Model" ]	[ "Shinya Kanemura [email protected]:[email protected] \nIntroduction\n\n", "Hide-Aki Tohyama ", "\nDepartment of Physics\nTheory Group\nKEK\n305TsukubaIbarakiJapan\n", "\nOsaka University\n560ToyonakaOsakaJapan\n" ]	[ "Introduction\n", "Department of Physics\nTheory Group\nKEK\n305TsukubaIbarakiJapan", "Osaka University\n560ToyonakaOsakaJapan" ]	[]	The non-decoupling effects of heavy Higgs bosons on the process e + e − → W + L W − L are discussed in the two Higgs-doublet model. The one-loop corrections to the cross section are calculated by using the equivalence theorem and the explicit expressions of its deviation from the standard model are derived. The leading mass contributions to the deviation in expansion by s are related to the T parameter by the low-energy theorem. The next-to leading ones are free from the present data, which should be determined by future experiments. The deviation can amount to ∼ 3 % at √ s = 1 TeV under the constraint from the present data, so that it may give useful information on the Higgs sector in cooperation with data from future linear colliders. 1	10.1103/physrevd.57.2949	[ "https://arxiv.org/pdf/hep-ph/9707454v2.pdf" ]	16,196,739	hep-ph/9707454	16ed92ea07400c0b03e3356cf6ef2cc9c2dd8de6	Non-Decoupling Effects of Higgs Bosons on e + e − → W + L W − L in the Two-Doublet Model September 10, 2018 Shinya Kanemura [email protected]:[email protected] Introduction Hide-Aki Tohyama Department of Physics Theory Group KEK 305TsukubaIbarakiJapan Osaka University 560ToyonakaOsakaJapan Non-Decoupling Effects of Higgs Bosons on e + e − → W + L W − L in the Two-Doublet Model September 10, 2018arXiv:hep-ph/9707454v2 6 Oct 1997 KEK Preprint 97-96 KEK-TH-528 OU-HET 271 The non-decoupling effects of heavy Higgs bosons on the process e + e − → W + L W − L are discussed in the two Higgs-doublet model. The one-loop corrections to the cross section are calculated by using the equivalence theorem and the explicit expressions of its deviation from the standard model are derived. The leading mass contributions to the deviation in expansion by s are related to the T parameter by the low-energy theorem. The next-to leading ones are free from the present data, which should be determined by future experiments. The deviation can amount to ∼ 3 % at √ s = 1 TeV under the constraint from the present data, so that it may give useful information on the Higgs sector in cooperation with data from future linear colliders. 1 Introduction Radiative corrections can be one of the powerful tools for exploration of new physics in cooperation with the experimental precision data. The study of the oblique corrections, which are controlled by three parameters S, T and U [1], has played a crucial role to determine properties of new physics with the data from LEP (and SLC) precision measurement [2]. Standard Model (SM) has not been found to have any substantial deviation between its prediction and the data, while some types of new physics based on the dynamical breaking of electro-weak gauge symmetry have been strongly constrained by the data by virtue of their non-decoupling properties [3]. Generally speaking, radiative corrections can be useful to constrain the new physics in which heavy mass effects do not decouple from a low energy observable. There are models in which non-decoupling effects are expected but also are not completely constrained yet by the present data. To obtain substantial information about such models indirectly, we need to study radiative corrections to other types of observable than oblique ones, which are expected to be measured at LEP 2 [4] or future experiments [5,6]. One of these models may be an extension of SM with two Higgs doublets, in which the non-decoupling effects of heavy Higgs bosons may be expected to appear as well as those of the fermion contributions. The two Higgs-doublet model (THDM) has been studied [7] so far with a lot of motivations such as additional CP phases [8], which are often required for the electro-weak baryogenesis [9], the strong CP problem and Minimal Supersymmetric Standard Model (MSSM). We note that in MSSM, the heavier Higgs bosons (m 2 H 0 , m 2 H ± and m 2 A 0 ) can become heavy only by making the soft-breaking parameter to be large because all the Higgs self-couplings are fixed to be O(m 2 W /v 2 ). As a result, very heavy Higgs bosons, as well as all heavy super-particles, are decoupled from the low energy theory [10,11] and the non-decoupling effects of these particles can be no longer expected in MSSM. We still stress that in the non-suspersymmetric THDM, there remains a parameter region in which the model becomes a non-decoupling theory for heavy Higgs bosons [12]. One of the observables for which we can expect to have precision data next may be the scattering process e + e − → W + W − . This process is not only one of the main target processes at LEP 2 but also expected to be well measured at future e + e − linear colliders. Ahn et al. [13] once showed that the non-decoupling effects of heavy fermions on this process can substantially be enhanced by the factor s/m 2 W in a region m 2 W ≪ √ s < M, where M represents the heavy fermion mass. This enhancement disappears in the high energy limit, M ≪ √ s, because of the unitarity cancellation between s-and t-channel diagrams. The heavy mass effects on this process appear through the corrections to the triple gauge vertices (TGV's) W + W − Z 0 and W + W − γ [14], as well as to the propagators of intermediate Z 0 and γ. Appelquist and Wu have studied the connection between TGV form factors and the chiral Lagrangian operators [15]. The one-loop estimation of TGV's has been made by several authors in SM [13,16], in the model with extra fermions such as might be present in technicolor-type theories [15,17], and in MSSM [18]. The non-decoupling effects of heavy fermions have been one of their main interests. The possibility of the extra fermion generation and technicolorlike models, however, has already been constrained to considerable extent by careful comparison between the oblique parameters and the present precision data [3]. On the other hand, the non-decoupling effects of additional heavy Higgs bosons have not been so studied so far. In spite of the constraint from the present data [19], THDM remains to have large allowed region for the Higgs boson masses and the mixing angles. Thus it is quite interesting to investigate the possibility to constrain the parameter region of THDM through the correction to the observables which are to be measured at future measurements. This is just our motivation to this work. In this paper, we discuss the non-decoupling effects of heavy Higgs boson masses on the scattering of the polarized electron-positron into a longitudinal W boson pair in THDM with a softly broken discrete symmetry. We investigate the cross section in the one-loop level (O(1/(4πv) 2 )) and also estimate its deviation from SM, δ(s) = σ T HDM /σ SM − 1. Note that this quantity is related to the deviation of the TGV form factors κ V (s) and g V 1 (s), which are defined in Ref [14]. In calculation, we make full use of the equivalence theorem [20,21] in the Landau gauge and drop the gauge boson loops because of the suppression by m 2 W /M 2 Higgs . This procedure is quite justified for our purpose to extract the non-decoupling effects of heavy Higgs bosons [12,22]. We obtain the explicit expressions for the leading (quadratic) and the next-to leading (logarithmic) contributions of the heavy Higgs masses to δ(s) in expansion by s/M 2 Higgs . We find that the leading contributions to δ(s) are written in terms of ∆ρ(= α f T ) parameter. This is regarded as a kind of the low energy theorem (Here low energy means √ s ≪ M Higgs .). It is understood from the fact that the physical quadratic contributions of the heavy masses comes from an unique term ∼ β 1 f 2 (trT V µ ) 2 in the chiral Lagrangian [15]. This term disappears in the custodial SU(2) V symmetric limit [23]. The similar phenomenon has been known to appear in the scattering such as W + L W − L → W + L W − L [24] . On the other hand, the next-to leading contributions include new additional parameters other than S, T and U. Thus these contributions can give substantial information about the new physics. In the MSSM like cases, the both contributions turn out to vanish in the heavy Higgs limit, so that the heavy mass effects are decoupled from the low-energy observable consistently. The numerical study shows that the deviation δ(s) can amount to ∼ 2-3 % at √ s = 1 TeV by the non-decoupling effects of the Higgs boson masses even under the constraint from the present data as well as from the perturbative unitarity [25]. We note that the accuracy may be expected to be smaller than ∼ 2 % at √ s ∼ 1 TeV by taking account of the ambiguity due to our approximation as well as the ambiguity of the measurement by the expected statistic and systematic errors at future e + e − linear colliders [26]. Therefore we conclude that such deviation between THDM and SM on the process may be detectable at future experiments and bring useful information on the Higgs sector. In Sec.2, we will summarize the results by the effective Lagrangian briefly. THDM will be defined in Sec.3. Details of calculation will be shown in Sec.4. The concrete expressions for the deviation from SM of the cross section are derived and the non-decoupling properties are discussed in Sec.5. Results will be summarized in the last section. 2 W + W − Z 0 , W + W − γ vertices The contribution of non-decoupling effects to e + e − → W + W − comes from the corrections to the triple gauge vertices (TGV's) [14] as well as the obliquetype ones. All the corrections in other types of diagram are suppressed by the electron masses or gauge boson masses. The effective Lagrangian for TGV's in the C, P and CP conserving case is expressed as L W W V g W W V = ig V 1 (W + µν W −µ V ν − W − µν W +µ V ν ) + iκ V W + µ W − ν V µν +i λ V (4πv) 2 W + µν W −ν ρ V ρµ ,(1) where g W W γ = −e and g W W Z = −e cot θ W . The tree level form factors are given in SM and also in THDM as g V 1 = κ V = 1, λ V = 0.(2) The deviation from these tree-level values is generated at the loop level and they are denoted here as ∆g V 1 = g V 1 − 1 and the same for others. It is known that κ γ and λ γ are related to the magnetic moment and quadrupole moment of W ± -bosons [14]. Appelquist and Wu have derived the relation between the form factors and the coefficients in the chiral Lagrangian [15]. From the context of the power counting method, it is expected that at one loop level there are quadratic mass contributions of inner heavy particles with the mass M in g V 1 and κ V ; namely ∼ O(M 2 /(4πv) 2 ), where v is the vacuum expectation value (VEV). These quadratic contributions actually appear if the new physics does not have the custodial SU(2) V invariance [23]. These occur through the dimension 2 operator [15] L ′ 1 ≡ 1 4 β 1 (4πv) 2 [tr(T V µ )] 2 ,(3) where V µ and T are expressed in terms of the dimensionless unitary unimodular matrix field U(x) as V µ = (D µ U)U † and T = Uτ 3 U † . The dimensionless parameter β 1 , which measures the breaking of SU(2) V , is known to be related to α f T (= ∆ρ) as α f T = 2β 1 , where α f is the fine structure constant. Note that the form factors, g V 1 , κ V and λ V , should be considered as functions of the energy √ s in general. The next-to leading mass contribution in expansion by s to the form factors may become important at high energy region [16]. Since these contributions have not been known yet, this can be expected to bring additional information for new physics in cooperation with the measurement at future linear colliders [5]. The helicity amplitudes for the polarized electron-positron scattering into a W -boson pair are expressed in terms of these form factors [14]. In the case of longitudinally polarized Wboson final states, the helicity amplitudes depend only on the combination of g V 1 (s) + (s/2m 2 W )κ V (s) even if there is no C, P , or CP invariance in the model, where V = γ and Z 0 . We evaluate the correspondence to these form factors later. Two Higgs-Doublet Model Here we define THDM with a softly broken discrete symmetry; Φ 1 → Φ 1 , Φ 2 → −Φ 2 . This model is the most general one in which the natural flavor conservation is realized [27]. Since the effects of CP violation disappear in the process e + e − → W + L W − L [14] as already mentioned in Sec. 2, we consider the CP invariant Higgs sector from the beginning. The Lagrangian of the Higgs sector is then given as L(Φ 1 , Φ 2 ) = µ 2 1 \| Φ 1 \| 2 +µ 2 2 \| Φ 2 \| 2 +2µ 2 3 ReΦ † 1 Φ 2 −η 1 \| Φ 1 \| 4 −η 2 \| Φ 2 \| 4 −η 3 \| Φ 1 \| 2 \| Φ 2 \| 2 −η 4 (ReΦ † 1 Φ 2 ) 2 − η 5 (ImΦ † 1 Φ 2 ) 2 .(4) The Higgs sector then includes the eight parameters in general. We here consider all the parameters to be free. ( Note that MSSM is considered as a special case in this Lagrangian, in which the supersymmetry imposes strong relations between these parameters. In MSSM, all the quartic couplings are constrained into O(g 2 ), where g denotes weak gauge couplings. Thus in the heavy Higgs limit, which is realized by µ 2 3 → ∞, the Higgs mass-effects are decoupled from low-energy observables [10,11]. ) The Higgs doublets, both of which are assigned hypercharge as Y = 1/2, are parametrized as Φ i = w + i 1 √ 2 (v i + h i + iz i ) , (i = 1, 2)(5) where vacuum expectation values v 1 and v 2 satisfy v 2 1 + v 2 2 = v ∼ 246 GeV. The mass eigenstates are obtained by rotating the fields in the following way: h 0 1 h 0 2 = cos α − sin α sin α cos α H 0 h 0 ,(6)w ± 1 w ± 2 = cos β − sin β sin β cos β w ± H ± ,(7) z 0 1 z 0 2 = cos β − sin β sin β cos β z 0 A 0 .(8) By setting tan β = v 2 /v 1 , w ± and z 0 become the Nambu-Goldstone bosons which are to be absorbed into the longitudinally polarized gauge bosons W ± L and Z 0 L , respectively. H ± and A 0 are then massive charged and CP-odd neutral states. On the other hand, h 0 and H 0 are massive CP-even neutral states. The mixing angle α is taken in order h 0 to be lighter than H 0 . The quartic couplings are then represented by using the mass parameters, the mixing angles and VEV as η 1 = 1 2v 2 cos 2 β (m 2 H 0 cos 2 α + m 2 h 0 sin 2 α) − µ 2 3 2v 2 sin β cos 3 β(9)η 2 = 1 2v 2 sin 2 β (m 2 H 0 sin 2 α + m 2 h 0 cos 2 α) − µ 2 3 2v 2 cos β sin 3 β(10)η 3 = sin 2α v 2 sin 2β (m 2 H 0 − m 2 h 0 ) + 2m 2 H ± v 2 − 2µ 2 3 v 2 sin 2β (11) η 4 = − 2m 2 H ± v 2 + 4µ 2 3 v 2 sin 2β (12) η 5 = 2 v 2 (m 2 A 0 − m 2 H ± )(13) In general, THDM does not have the custodial SU(2) V symmetry. But if η 5 is zero, the Higgs sector turns out to be SU(2) V symmetric even after the spontaneous breakdown of SU(2) L ⊗ U(1) Y gauge invariance occurs [28]. Therefore the mass splitting between H ± and A 0 measures SU(2) V -breaking in the Higgs sector [12,28]. Note that there are some cases where we can take the mass splitting to be enough large within the constraint from the present data. For example, if we set α − β = π/2 and m 2 H ± ∼ m 2 H 0 , m 2 A 0 and m 2 h 0 can be chosen freely with keeping α f T ∼ 0. As for the Yukawa couplings, there can be two types of model in THDM (what we call, Model I and II in Ref. [7]) in which natural flavor conservation is realized by imposing discrete symmetries (see eq. (4)). Note that the difference between Model I and II vanishes in such the situation as we will consider later, where the mass of bottom quarks is negligible. 4 e + e − → W + L W − L in THDM In this section, we show the one-loop calculation of the process e − X e + Y → W + L W − L in THDM, where X (Y ) is the helicity of the electron (positron). The calculation is quite simplified by making full use of the equivalence theorem (ET) [20], which says that in the case of √ s ≫ m W the cross section for e + e − → W + L W − L is equivalent to that for e + e − → w + w − up to O(m 2 W /s) 3 . The extension of ET to the loop level has been studied by several authors [21]. They found that the some modification factors, which depend on gauge parameters, should be multiplied. He et al. showed that in SM the Landau gauge is a good choice for such the purpose here because the modification factors no longer depend on the Higgs boson mass and they can be set into unity within the approximation. We note that this situation is not changed even in the case of THDM [12,30]. Thus the radiative correction here is calculated in the Landau gauge. The non-decoupling effects of Higgs boson masses on e + e − → w + w − come from the corrections to the V µ w + w − vertices, (V = γ, Z 0 ) in the s-channel iM µ V ww (s) = iM − V ww (s)(p + − p − ) µ + iM + V ww (s)(p + + p − ) µ ,(14) where p + (p − ) is the momentum of w + (w − ). Since the second term of RHS, which is proportional to (p + + p − ) µ , produces the contribution suppressed by the negligible electron mass squared, we have only to calculate M − V ww (s) for our purpose. It is expressed by iM − V ww (s) = −ig V 1 + G V (s) + Z w + δg V g V ≡ −ig V Γ V (s),(15) where G V (s) are the contributions of one-loop diagrams other than counterterms, Z w denotes the wave function renormalization constant for the external 3 In general, the error is of O(m W / √ s). It turns out to be O(m 2 W /s) in some cases including present process [31]. w ± lines and δg V are the shift of coupling constants defined by g V → g V +δg V . The tree level coupling constants g V are given by g Z = e cot 2θ W , g γ = e,(16) where e and θ W denote the electric charge and the Weinberg angle respectively. The scattering amplitude for the polarized e + e − scattering into w + w − is given in terms of Γ V (s) as iA(e − X e + Y → w + w − ) = ie 2v X γ µ u X Γ γ (s) s + f XY Γ Z (s) s − m 2 Z (p + − p − ) µ ,(17) where f XY is defined according to the e − e + helicities by f LR = cot 2θ W , f RL = − cos 2θ W 2 cos 2 θ W . The total cross section is then expressed in each case as σ(e − X e + Y → w + w − ) = e 4 s 24π Γ γ (s) s + f XY Γ Z (s) s − m 2 Z 2 .(18) The estimation of Γ V (s) is performed in order. Here we concentrate into the contributions from the Higgs sector. At first, the one-loop contributions to G V (s) are calculated as G Z (s) = 1 (4πv) 2 2 cos 2θ W C [A 0 H ± H 0 ] sin 2 (α − β) +C[A 0 H ± h 0 ] cos 2 (α − β) +C[w ± H 0 w ± ] cos 2 (α − β) +C[w ± h 0 w ± ] sin 2 (α − β) +C[H ± H 0 H ± ] sin 2 (α − β) +C[H ± h 0 H ± ] cos 2 (α − β) +C[H ± A 0 H ± ] ,(19) and G γ (s) = 1 (4πv) 2 C [w ± H 0 w ± ] cos 2 (α − β) +C[w ± h 0 w ± ] sin 2 (α − β) +C[H ± H 0 H ± ] sin 2 (α − β) +C[H ± h 0 H ± ] cos 2 (α − β) +C[H ± A 0 H ± ] ,(20) where p 2 + = p 2 − = 0 is taken in the Landau gauge. The functionC[123] is defined byC [123] = (m 2 1 − m 2 2 )(m 2 3 − m 2 2 )(C 11 − C 12 )[123],(21) where C 11 [123] and C 12 [123] can be written in terms of the B 0 and C 0 functions introduced by Passarino and Veltman [29]. We employ these notations here according to the definition in Ref. [3]. Secondly, the wavefunction renormalization Z w is calculated up to O(m 2 W /M 2 Higgs ) as [12,30] Z w = − 1 (4πv) 2 1 2 (2m 2 H ± + m 2 h 0 + m 2 H 0 + m 2 A 0 ) + m 2 H ± m 2 A 0 m 2 A 0 − m 2 H ± ln m 2 H ± m 2 A 0 + cos 2 (α − β) m 2 H ± m 2 h 0 m 2 h 0 − m 2 H ± ln m 2 H ± m 2 h 0 + sin 2 (α − β) m 2 H ± m 2 H 0 m 2 H 0 − m 2 H ± ln m 2 H ± m 2 H 0(22) Thirdly, the renormalization of the coupling constants δg V is expressed as δg Z g Z = 1 2 cos 2 2θ W δm 2 Z m 2 Z − δm 2 W m 2 W − sin 2 2θ W δα f α f ,(23)δg γ g γ = 1 2 δα f α f ,(24) where δα f is the shift of the fine structure constant and δm 2 V are mass renormalizations defined as α f → α f + δα f and m 2 V → m 2 V + δm 2 V (here V = W or Z),δm 2 W m 2 W = −1 (4πv) 2 1 2 (2m 2 H ± + m 2 h 0 + m 2 H 0 + m 2 A 0 ) + m 2 H ± m 2 A 0 m 2 A 0 − m 2 H ± ln m 2 H ± m 2 A 0 + cos 2 (α − β) m 2 H ± m 2 h 0 m 2 h 0 − m 2 H ± ln m 2 H ± m 2 h 0 + sin 2 (α − β) m 2 H ± m 2 H 0 m 2 H 0 − m 2 H ± ln m 2 H ± m 2 H 0 , δm 2 Z m 2 Z = −1 (4πv) 2 1 2 (m 2 h 0 + m 2 H 0 + m 2 A 0 ) + cos 2 (α − β) m 2 A 0 m 2 h 0 m 2 h 0 − m 2 A 0 ln m 2 A 0 m 2 h 0 + sin 2 (α − β) m 2 A 0 m 2 H 0 m 2 H 0 − m 2 A 0 ln m 2 A 0 m 2 H 0 .(25) By inserting all these results into eq. (18), we finish to calculate the total cross sections at one loop level. Non-Decoupling Effects of Higgs Bosons Now we consider the non-decoupling effects of heavy Higgs masses on the ratio R(s) = σ T HDM /σ SM , where the cross section in SM, σ SM (s), is to be calculated in the one-loop level in the same approximation manner as σ T HDM (s). The one-loop corrections to Γ V (see eq. (15)) in SM case can be then immediately calculated as Γ SM γ,Z = 1 + 1 (4πv) 2 C [w ± φ 0 SM w ± ] − 1 2 m 2 φ 0 SM ,(26) where φ 0 SM is the Higgs boson in SM. For the case of e − e + helicity LR (RL), the magnitude of σ SM (s) amounts at one loop level to 355 ∼ 358 (67.6 ∼ 68.3) fb at √ s = 500 GeV and 98.9 ∼ 98.5 (19.7 ∼ 19.6) fb at √ s = 1000 GeV for m φ 0 SM = 140 ∼ 1000 GeV, respectively. These values are consistent with the previous results [13,26] up to the ambiguity due to the use of the equivalence theorem. We did not include the fermion-loop contribution in these values. This is because that it takes the same form between THDM and SM, so that it consequently does not contribute to the deviation at all. It is convenient to parametrize the ratio R(s) as R(s) = σ T HDM σ SM = 1 + δ(s).(27) The deviation-function δ(s) is expressed in terms of the difference of the one loop corrections to Γ V between THDM and SM; δ(s) = 2Re    1 s δΓ T HDM γ − δΓ SM γ + f XY s−m 2 Z δΓ T HDM Z − δΓ SM Z 1 s + f XY s−m 2 Z    ,(28) where one-loop corrections δΓ model V are defined by Γ model V = 1 + δΓ model V . The quark loops do not contribute to δ(s) at all because of the exact cancellation between both models. As we mentioned before, δ(s) is expressed in terms of ∆g V 1 (s) and ∆κ V (s). We have the relation for m 2 W ≪ s as δ(s) = 4γ 2 1 + f XY 1 − ξ XY 2 sin 2 θ W ∆κ T HDM Z (s) − ∆κ SM Z (s) − ∆κ T HDM γ (s) − ∆κ SM γ (s) ,(29) where ξ LR = 1, ξ RL = 0 and γ = √ s/2m W . For extracting non-decoupling effects of heavy Higgs bosons, we expand δ(s) by powers of s as δ(s) = δ (0) + δ (1) s + O(s 2 ).(30) Note that this expansion is valid only in the case with m W ≪ √ s ≪ M Higgs . At one loop level O(1/(4πv) 2 ), all the non-decoupling effects are included in δ (0) and δ (1) . By dimensional counting, we know that δ (0) represents the quadratic Higgs mass effects and δ (1) includes at most logarithmic ones. We show the explicit expressions for δ (0) and δ (1) at one loop level. In calculation, we identify the lighter neutral Higgs boson h 0 in THDM as SM like Higgs boson φ 0 SM . At first, δ (0) is calculated as δ (0) = 1 (4πv) 2 f XY 1 + f XY 2 cos 2θ W + 1 cos 2 2θ W × F (m 2 H ± , m 2 A 0 ) + sin 2 (α − β) F (m 2 H ± , m 2 H 0 ) − F (m 2 A 0 , m 2 H 0 ) + cos 2 (α − β) F (m 2 H ± , m 2 h 0 ) − F (m 2 A 0 , m 2 h 0 ) ,(31) where F (x, y) = x + y 2 − xy x − y ln x y .(32) Comparing eq. (31) to the expression of the ∆ρ (= α f T ) parameter [7], we have a relation . δ (0) = f XY 1 + f XY 2 cos 2θ W + 1 cos 2 2θ W α f T.(33)g 1 (x, y, z) = −1 (x − z) 2 (x − y) 2 x 2 (y − z) x 2 + x(y + z) − 3yz ln x.(37) Note that the function G(x, y, z) vanishes if and only if we set x = y or y = z. We can see from eq. (34) that δ (1) does not vanish except for a few cases. One of the cases where δ (1) vanishes is that with the complete degeneracy between all of the Higgs boson masses. Another one is the case with α − β ∼ π/2 and m 2 H 0 ∼ m 2 H ± ∼ m 2 A 0 . We note that the latter case occurs in MSSM with the large m A 0 limit [7]. Since these cases also imply the custodial SU(2) V invariance in the Higgs sector, we find that the non-decoupling effects both δ (0) and δ (1) vanish simultaneously in these cases and the model then becomes a decoupling theory for Higgs boson masses [10,11]. On the other hand, there are some cases in which δ (1) becomes large to some extent with parameters satisfying the constraint from the present experimental data. For example, if we set m H 0 ∼ m H ± and α − β = π/2, the other masses m h 0 and m A 0 can be chosen freely with keeping α f T ∼ 0. In such cases, we can expect to obtain In the situation such as the momentum √ s comparable to the largest mass of the Higgs bosons, the question whether the non-decoupling effects become relatively large or not may occur. In this case, the expansion above is no longer allowed, so that we have to investigate the non-decoupling effects only through numerical estimation. The √ s dependence of δ(s) is described in Figure 1. The parameters are chosen in order to satisfy the constraint from the present data. We here chose α − β = π/2, m h 0 = 140, m H 0 = 350, and m H ± = 347 GeV for satisfying −3.8 × 10 −3 < α f T T HDM < 2.6 × 10 −4 [3], where T T HDM is the additional contribution to T parameter in THDM. We can see that the behavior changes according to the relative energy scale to M Higgs ∼ m A 0 . The deviation by non-decoupling effects is enhanced by s for √ s < m A 0 but is reduced for very high energy region as m A 0 ≪ √ s. This behavior is consistent with the result for the fermion effects by Ahn et al. [13] that the enhancement disappears in the high energy limit because of the unitarity cancellation between s-and t-channel diagrams. In Figure 2, we show the m A -dependence of δ(s) at √ s = 500 GeV and 1000 GeV in the same choice for other parameters as Fig.1. We can see in Figs. 1 and 2 that the large mass difference (around √ s) between m A 0 and m H 0 ∼ m H ± tends to produce the large deviation. At √ s = 1000 GeV, it amounts to ∼ 3 % for m A 0 = 1200 GeV. The deviation for helicity LR is larger than that for RL in general. Note that all the parameter choice here is also taken account of the constraint from the perturbative unitarity [25]. Conclusion We have discussed the non-decoupling effects of the heavy Higgs bosons on the scattering process e − L e + R (or e − R e + L ) → W + L W − L in THDM. The cross section has been calculated at one-loop level O(1/(4πv) 2 ) by making full use of the equivalence theorem. The effects of heavy Higgs bosons have extracted in the ratio of the cross section between THDM and SM, R(s)(= 1+δ(s)). The leading (quadratic) contributions of the masses to δ(s) become to be written in terms of T -parameter. This phenomenon is regarded as the result by the low energy theorem and can be understood by the chiral Lagrangian approach. On the other hand, the next-to leading (logarithmic) contributions include the additional parameters other than oblique ones. The next-to leading contributions do not vanish in general except for a few cases, so that they may be useful for the indirect exploration of New physics by combining with data from future e + e − colliders. One of the exceptional cases is that with cos 2 (α − β) ∼ 0 and m 2 H 0 ∼ m 2 H ± ∼ m 2 A 0 , which corresponds to MSSM in large m A limit. In these cases, both the leading and the next-to leading contributions simultaneously vanish and the model then becomes a decoupling theory as expected. Otherwise, the non-decoupling effects on δ (1) exist and can become large with keeping the constraint δ (0) ∝ α f T ∼ 0. One example for such cases may be m H ± ∼ m H 0 and α − β ∼ π/2. Then the values of m A 0 and m h 0 can be taken freely with keeping α f T ∼ 0. Actually we have numerically found that in these cases there can be the relatively large deviation from SM, which amounts to 2-3 % (see Fig.2) at √ s = 1 TeV for large m A 0 but within the constraint from the perturbative unitarity. Therefore the non-decoupling effects by heavy Higgs bosons on this process can become large to some extent, so that they may be constrained by the data from future e + e − linear colliders. gauge boson intermediate diagrams. Other types of diagrams (the neutrino exchanged t-channel and box-type diagrams) are always suppressed by powers of the electron mass or gauge boson masses. The oblique corrections in the s-channel diagrams are also neglected because of the suppression factor m 2 W /M 2 Higgs . (Note that this approximation is valid in the situation like m 2 W ≪ s < M 2 Higgs or m 2 W ≪ M 2 Higgs < s.) Thus we have only to calculate corrections to the V µ w + w − vertices for our purpose here. Moreover, it turns out that only the Higgs-Goldstone boson loops contribute to the vertices because the diagrams including a gauge boson loop are relatively suppressed by a factor m 2 W /M 2 Higgs [12, 22]. The V µ w + w − vertices can be decomposed as Figure 1 : 1The √ s dependence of δ(s) for m A 0 = 800 and 1200 GeV. The solid (dashed) lines represent δ(s) for the initial helicity states e − L e + R (e − R e + L ). The other parameters are fixed being taken account of the constraint from T parameter by m h 0 = 140, m H 0 = 350, m H ± = 347 GeV, and α − β = π/2. Figure 2 : 2The m A dependence of δ(s) at √ s = 500 and 1000 GeV. The solid (dashed) lines represent δ(s) for the initial helicity states e − L e + R (e − R e + L ). The other parameters are fixed being taken account of the constraint from T parameter by m h 0 = 140, m H 0 = 350, m H ± = 347 GeV, and α − β = π/2. . some useful information through this process. respectively. Note that δα f is relatively suppressed by the factorm 2 W /M 2 Higgs . On the other hand, δm 2 V are expressed up to O(m 2 W /M 2 Higgs ) as [32] AcknowledgmentsThe authors would like to thank Y. Okada for valuable discussions, K. Hagiwara for useful comments.The second term in the bracket of RHS comes from δg Z in eq.(23). We can see that the leading effects, δ (0) , can be written in terms of T . This phenomenon is due to the low energy theorem and is understood as the concrete realizations of the fact which we discussed in Sec 2. In fact, this leading contribution δ (0) vanishes if mass degeneracy between A 0 and H ± exists. In this case, the Higgs sector becomes custodial SU(2) V symmetric, so that the term (3) is then forbidden. We note all the contributions to eq. (31) come from only δΓ Z . As a result, the leading contribution is found not to be substantial because T has already been fairly constrained by the present data.The next-to leading contributions, δ (1) , may be possible candidates for the probe of the new physics. They include non-decoupling effects like ∼ log M Higgs . These are extracted from eq. (28) aswhere G(x, y, z) is the coefficient of the second term of s-expansion forC function. 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[ "Numerical detection of the Gardner transition in a mean-field glass former", "Numerical detection of the Gardner transition in a mean-field glass former" ]	[ "Patrick Charbonneau \nDepartment of Chemistry\nDuke University\n27708DurhamNorth CarolinaUSA\n\nDepartment of Physics\nDuke University\n27708DurhamNorth CarolinaUSA\n", "Yuliang Jin \nDepartment of Chemistry\nDuke University\n27708DurhamNorth CarolinaUSA\n\nDipartimento di Fisica\nSapienza Universitá di Roma\nINFN\nSezione di Roma I\n\nIPFC -CNR\nPiazzale Aldo Moro 2I-00185RomaItaly\n\nLPT\nUMR 8549\nÉcole Normale Supérieure\nCNRS\n24 Rue Lhomond75005France\n", "Giorgio Parisi \nDipartimento di Fisica\nSapienza Universitá di Roma\nINFN\nSezione di Roma I\n\nIPFC -CNR\nPiazzale Aldo Moro 2I-00185RomaItaly\n", "Beatriz Seoane \nDipartimento di Fisica\nSapienza Universitá di Roma\nINFN\nSezione di Roma I\n\nIPFC -CNR\nPiazzale Aldo Moro 2I-00185RomaItaly\n\nInstituto de Biocomputación y Física de Sistemas Complejos (BIFI)\n50009ZaragozaSpain\n", "Francesco Zamponi \nLPT\nUMR 8549\nÉcole Normale Supérieure\nCNRS\n24 Rue Lhomond75005France\n" ]	[ "Department of Chemistry\nDuke University\n27708DurhamNorth CarolinaUSA", "Department of Physics\nDuke University\n27708DurhamNorth CarolinaUSA", "Department of Chemistry\nDuke University\n27708DurhamNorth CarolinaUSA", "Dipartimento di Fisica\nSapienza Universitá di Roma\nINFN\nSezione di Roma I", "IPFC -CNR\nPiazzale Aldo Moro 2I-00185RomaItaly", "LPT\nUMR 8549\nÉcole Normale Supérieure\nCNRS\n24 Rue Lhomond75005France", "Dipartimento di Fisica\nSapienza Universitá di Roma\nINFN\nSezione di Roma I", "IPFC -CNR\nPiazzale Aldo Moro 2I-00185RomaItaly", "Dipartimento di Fisica\nSapienza Universitá di Roma\nINFN\nSezione di Roma I", "IPFC -CNR\nPiazzale Aldo Moro 2I-00185RomaItaly", "Instituto de Biocomputación y Física de Sistemas Complejos (BIFI)\n50009ZaragozaSpain", "LPT\nUMR 8549\nÉcole Normale Supérieure\nCNRS\n24 Rue Lhomond75005France" ]	[]	Recent theoretical advances predict the existence, deep into the glass phase, of a novel phase transition, the so-called Gardner transition. This transition is associated with the emergence of a complex free energy landscape composed of many marginally stable glassy sub-basins. In this study, we explore several methods to detect numerically the Gardner transition in a simple structural glass former, the infinite-range Mari-Kurchan model. The transition point is robustly located from three independent approaches: (i) the divergence of the characteristic relaxation time, (ii) the divergence of the caging susceptibility, and (iii) the abnormal tail in the probability distribution function of cage order parameters. The results are consistent with previous theoretical predictions. The methods we propose may also be generalized to more realistic numerical models as well as to experimental systems.	10.1103/physreve.92.012316	[ "https://arxiv.org/pdf/1501.07244v3.pdf" ]	9,235,909	1501.07244	2928e281384268677996e0e6c27ddd4ef7c98cbf	Numerical detection of the Gardner transition in a mean-field glass former Patrick Charbonneau Department of Chemistry Duke University 27708DurhamNorth CarolinaUSA Department of Physics Duke University 27708DurhamNorth CarolinaUSA Yuliang Jin Department of Chemistry Duke University 27708DurhamNorth CarolinaUSA Dipartimento di Fisica Sapienza Universitá di Roma INFN Sezione di Roma I IPFC -CNR Piazzale Aldo Moro 2I-00185RomaItaly LPT UMR 8549 École Normale Supérieure CNRS 24 Rue Lhomond75005France Giorgio Parisi Dipartimento di Fisica Sapienza Universitá di Roma INFN Sezione di Roma I IPFC -CNR Piazzale Aldo Moro 2I-00185RomaItaly Beatriz Seoane Dipartimento di Fisica Sapienza Universitá di Roma INFN Sezione di Roma I IPFC -CNR Piazzale Aldo Moro 2I-00185RomaItaly Instituto de Biocomputación y Física de Sistemas Complejos (BIFI) 50009ZaragozaSpain Francesco Zamponi LPT UMR 8549 École Normale Supérieure CNRS 24 Rue Lhomond75005France Numerical detection of the Gardner transition in a mean-field glass former Recent theoretical advances predict the existence, deep into the glass phase, of a novel phase transition, the so-called Gardner transition. This transition is associated with the emergence of a complex free energy landscape composed of many marginally stable glassy sub-basins. In this study, we explore several methods to detect numerically the Gardner transition in a simple structural glass former, the infinite-range Mari-Kurchan model. The transition point is robustly located from three independent approaches: (i) the divergence of the characteristic relaxation time, (ii) the divergence of the caging susceptibility, and (iii) the abnormal tail in the probability distribution function of cage order parameters. The results are consistent with previous theoretical predictions. The methods we propose may also be generalized to more realistic numerical models as well as to experimental systems. I. INTRODUCTION Upon compressions (that are sufficiently rapid to avoid crystallization), a fluid of hard spheres first turns sluggish and then forms a glass [1,2]. This glass can then be further compressed until the system jams [3], which occurs under the application of an infinite confining pressure [4,5]. Glass formation is entropic, i.e., particles vibrate and thus cage each other in place, while jamming is mechanical, i.e., no motion is possible and particles are held steady through direct contacts with each other. Over the last decade, this two-transition scenario has been broadly validated, both numerically and theoretically [5][6][7][8][9][10][11][12][13]. Interestingly, recent advances predict that -at least in the mean-field, infinite-dimensional (d → ∞) limit -there exists a third transition, a socalled Gardner transition, that is intermediate in density and pressure between glass formation and jamming [14][15][16][17]. First discovered in spin glass models [18][19][20][21][22], the Gardner transition corresponds to a single glass state splitting into a complex hierarchy of marginally stable sub-states. The transition is thus akin to the spin-glass transition of the Sherrington-Kirkpatrick (SK) model, wherein a critical temperature separates a paramagnetic phase, in which a single thermodynamic state exists, from a marginal phase, in which a large number of distinct spin-glass states appear [23]. In structural glasses, however, the high-temperature phase corresponds to a given glass basin that has been dynamically selected by a quenching protocol; it is this glass metabasin that then undergoes a spin-glass-like transition. The discovery of a Gardner transition in glasses has already markedly advanced our theoretical understanding of jamming by providing analytical predictions for * [email protected] † [email protected] the associated critical exponents [15,16,24]. It further explains the abundance of soft vibrational modes in that same limit [25,26]. These effects may even underlie the peculiar behavior of the specific heat in quantum glasses [14,15], and various other transport and thermodynamic properties in this regime. In this context, a crucial question is whether the Gardner transition itself, which provably exists in the d → ∞ limit, exists in finite (low) dimensions. A similar line of inquiry has been pursued for decades in the context of spin glasses. Renormalization group results indicate, as for spin glasses, that the transition might disappear or dramatically change of nature in low d [27]. Yet it has also been shown in spin glasses that, whatever the ultimate fate of the phase transition in the thermodynamic limit may be, the d → ∞ scenario provides a very good description of the system over the relevant experimental length and time scales [28]. At present, the only way to assess the relevance of the Gardner transition for the description of experimental glasses is through the use of numerical simulations. It is therefore important to identify the observable consequences of this transition. This study primarily aims to develop procedures and to identify observables in order to reliably detect the Gardner transition. To that effect, we consider a simple structural glass former, the infinite-range Mari-Kurchan model [29,30], which is a mean-field model by construction. The model is quite abstract and in some ways far from realistic models of glasses, but it can be studied analytically in great detail, which is useful for guiding the numerical exploration. It also shares the same phase diagram as hard spheres in the infinite dimensional limit d → ∞. Our aim is to here discern the signatures of the Gardner transition in this well-controlled mean-field model, where we are certain that the transition exists, before later considering more realistic situations where the existence of the transition is not guaranteed. We discuss below several quantities that bear the sig-arXiv:1501.07244v2 [cond-mat.dis-nn] 23 Feb 2015 nature of the Gardner transition. The analogy with spin glasses suggests some of the relevant observables. For instance, at both the spin-glass and the Gardner transitions, the "spin-glass susceptibility" diverges, and the distribution of overlaps (distances) between different replicas becomes non-trivial. In the end, we find that the Gardner transition can be reliably and reproducibly located in the MK model. In the conclusion section, we also discuss other possible measurements to detect and characterize the Gardner transition, which may be more appropriate for numerical simulations of more realistic model glass formers as well as for experiments. II. MODEL AND BASIC PHYSICAL PICTURE We consider a simple glass-former, the infinite-range Mari-Kurchan (MK) model [29,30] -initially proposed by Kraichnan [31], in which N hard spheres (HS) of diameter σ interact through a pair distance shifted by a quenched random vector Λ ij . The total interaction energy is thus U = N i<j u (\|r ij \|) ,(1) where for particles at positions {r i } the shifted distance r ij is defined as r ij = r i −r j +Λ ij , and u(r) is the HS potential, i.e., e −u(r) = θ(r − σ) with r = \|r\|. The random vectors, which are uniformly distributed over the system volume V , induce a quenched disorder that suppresses both crystallization and nucleation between metastable glassy states. The model further enables planting, which is a simple point process for generating equilibrated liquid configurations at all densities [32,33] (see Sec. III A). In all spatial dimensions, the MK model has a mean-field structure by construction, and exhibits a jamming transition of the same universality class as standard HS [30]. The MK model is also fully equivalent to standard HS in the limit d → ∞, where both models can be solved exactly [5,29,[34][35][36][37]. In the rest of this section, we briefly describe the phase diagram in this limit, as obtained from the analytical (mean-field) solution [5,[14][15][16][17] (Fig. 1), as well as some of the finite-dimensional corrections that have thus far been considered [33]. A. Equilibrium states (liquid phase) The liquid phase of the MK model ergodically samples equilibrium configurations following the Gibbs distribution and has a remarkably simple structure. Its pair correlation function is given by where β = 1/T is the inverse temperature, · · · denotes thermal averaging, and · · · denotes averaging over quenched disorder, i.e., over random shifts. The second virial coefficient is g 2 (r) ≡ V N (N − 1) i =j δ(r ij − r) = e −βu(r) = θ(r − σ),(2)B 2 = − 1 2V f (r 12 )dr 1 dr 2 = V d σ d 2 ,(3) where the Mayer function f (r) = e −βu(r) − 1, and V d is the volume of a d-dimensional ball of unit radius. Because no indirect correlations exist, higher-order correlation functions can be factorized in a trivial way, and the corresponding virial coefficients are zero. For example, the three-body correlation function is g 3 (r, r ) ≡ V 2 N (N − 1)(N − 2) i =j =k δ(r ij − r)δ(r ik − r ) = g 2 (r)g 2 (r ),(4) and the third virial coefficient B 3 = − 1 3V f (r 12 )f (r 13 )f (r 23 )dr 1 dr 2 dr 3 = − 1 3V f (r 12 )f (r 13 )× × f (r 13 − r 12 + Λ 12 + Λ 23 − Λ 13 )dr 1 dr 2 dr 3 = 0.(5) Note that f (r 13 − r 12 + Λ 12 + Λ 23 − Λ 13 ) = 0 in the thermodynamic limit, because the random shifts Λ ij are uncorrelated and are typically of the system size, hence \|r 13 − r 12 + Λ 12 + Λ 23 − Λ 13 \| σ. It is straightforward to generalize this argument to show that all higher-order virial coefficients are also zero. Because only the second virial coefficient is non-zero, the reduced pressure p equation of state (EOS) for the liquid becomes p ≡ βP/ρ = 1 + B 2 ρ = 1 + 2 d−1 ϕ,(6) where the combination of inverse temperature β, pressure P , and number density ρ = N/V gives a unitless quantity p whose only dependence is on the liquid volume fraction ϕ = ρV d (σ/2) d . B. Dynamical glass transition Although the structure and thermodynamics of the liquid are trivial, its dynamics is not. In infinite dimension, a dynamical glass transition ϕ d separates two distinct dynamical regimes. For ϕ < ϕ d , the dynamics is diffusive at long times, as expected of any liquid. Upon approaching ϕ d , however, the dynamics grows increasingly sluggish, and above ϕ d , each particle is fully confined within a cage formed by its neighbors. The typical size of that cage is the cage order parameter ∆, which, in that regime, can be extracted from the long-time limit of the mean-squared displacement ∆(t) = 1 N N i=1 \|r i (t) − r i (0)\| 2 .(7) For finite d, the liquid dynamics in the MK model is related to this description, but is markedly richer, with the notable apparition of hopping processes. A detailed study of their effect was recently reported [33]. From the d → ∞ solution, we know that the equilibrium distribution of the order parameter, P eq (∆), has two peaks for φ > ϕ d (see Fig. 2) [5,38]. The first characterizes the distance between two glass configurations within a same metabasin. It is centered around ∆ 1 , which is the typical size of this metabasin. The second characterizes the intra-basin distance. It is centered around ∆ 0 = ∞, because states that belong to different metabasins are completely uncorrelated. Note, however, that the peak at ∆ 1 has an exponentially small weight in N , because there exists an exponentially large number of distinct glass states [39]. Hence, in the thermodynamic limit, P eq (∆) = δ(∆−∆ 0 ) everywhere in the liquid phase. Note also that for the finite-d version of the MK model, the distinction between the two dynamical regimes below and above ϕ d is not sharp, because single-particle (and some cooperative) hopping remains possible in that density regime [33]. However, increasing density far above the (avoided) dynamical glass transition strongly suppresses hopping, which makes the d → ∞ description increasingly accurate. C. Glass State following In d → ∞, each equilibrium configuration at density ϕ 0 > ϕ d is trapped into one of exponentially many glass metabasins. The pressure of an equilibrium configuration at ϕ 0 is given by the liquid EOS, Eq. (6), but if one compresses (or decompresses) such a configuration up (or down) to a density ϕ, the system remains trapped within the metabasin that was initially selected and thus falls out of equilibrium. Because in d → ∞ glass states have an infinite lifetime, one can adiabatically follow the EOS of a glass state, i.e., perform state following (SF) [17,[40][41][42]. For the MK model in finite d, strongly suppressing hopping also gives an arbitrarily long lifetime to glass states, and thus algorithmically enables SF. In absence of these structural relaxations, particles simply vibrate within cages. Upon compression, these cages shrink until the jamming density, ϕ J (ϕ 0 ), is reached. Particles are then mechanically in contact, which makes the system mechanically rigid. Examples of d → ∞ glass EOS obtained by SF are given in Fig. 1 (from Ref. [17]). An exact finite-d solution of the MK model is not yet available (see [33] for a discussion of the challenges in constructing this solution), but the results are expected to be qualitatively similar when hopping is suppressed, and that difference should not affect the key points of this paper. Simply transposing the theoretical results in d = 3 (by setting d = 3 in the rescaled quantities ϕ and p) hence gives a reasonably good agreement with numerical simulations (see Sec. III for details). Note that although the system is formally out of equilibrium when it is confined to a single glass state (metabasin), a "restricted equilibrium" of this state can nonetheless be achieved. Very slow compressions or fast compressions followed by long relaxations, like the ones we perform in this work, achieve just that. Note also that following an equilibrated glassy states gives a distribution P SF (∆) that has a single peak at ∆ 1 ; because the system is dynamically confined to a single glass basin, the peak at ∆ 0 is absent. D. Gardner transition In d → ∞, a compressed state under restricted equilibrium undergoes a Gardner transition at ϕ G (ϕ 0 ), at which point the glass metabasin divides into a hierarchy of sub-basins. In the Gardner phase, a stable solution of the glass free energy is obtained from the full replica symmetry breaking (fullRSB) calculation, as in the Sherrington-Kirkpatrick (SK) spin-glass model [15][16][17][18][19]23]. The equilibrium distribution P eq (∆) then has a peak at ∆ 0 as well as two other peaks centered around ∆ EA and ∆ 1 , connected by a wide continuous band (see Fig. 2). Here, ∆ EA is the typical size of the innermost sub-basins at the lowest hierarchical level, and ∆ 1 is the typical size of the outermost basin. For the same rea-son as above, in the thermodynamic limit only the peak of P eq (∆) at ∆ 0 survives in the liquid phase, while in a restricted equilibrium, P SF (∆) does not show the ∆ 0 peak. Because the cage order parameter changes continuously, the Gardner transition is a continuous critical transition [18,19]. The Gardner phase is also marginally stable, in the sense that a zero mode is always present in the stability matrix of the free energy [23]. Because jamming is located within the Gardner phase, its marginal stability and critical scaling behaviors can consequently be obtained from a fullRSB thermodynamic calculation [15,16]. E. Timescales We conclude this section by summarizing the different timescales that characterize the equilibrium dynamics in the different phases of the MK model, based on the general correspondence between statics and dynamics in spin glasses [43,44] (and hence neglecting hopping). In addition to the general microscopic timescale τ 0 over which dynamics is essentially ballistic, one gets the following. • In the liquid phase below ϕ d , dynamics is characterized by two timescales: a short timescale τ , over which particles explore their cages, and a longer timescale τ α , over which dynamics is diffusive. In d → ∞, τ α is finite in this regime, but diverges at ϕ d . • In the liquid phase above ϕ d , and in the simple 1RSB glass phase as well, the same two timescales exist: τ is the timescale for equilibrating inside a glass basin, while τ α ∼ exp(N ) is the timescale for jumping from one glass state to another. • In the Gardner phase, one can extract three timescales: τ is the timescale for equilibrating inside a single glass sub-basin (∆ < ∆ EA ), τ α ∼ exp(N ) is the timescale for jumping from one glass metabasin to another (∆ ∼ ∆ 0 ), and τ meta is the timescale for exploring the structure of sub-basins within a given glassy metabasin (∆ EA < ∆ < ∆ 1 ). In reality, this last process does not correspond to a simple exponential with a single timescale, but is instead characterized by a complex distribution of free energy barriers and relaxation times. It is expected that τ meta ∼ exp(N α ) with α < 1 (α should be 1/3 in the SK model [23]), hence barriers between sub-basins are much lower than the ones between metabasins and τ meta τ α . Note that if the system is prepared by a quick compression, such that even a restricted equilibrium is not achieved, then the timescales τ meta τ α are both finite, but increase with the time spent waiting t w after preparing the system and before making measurements [43,44]. ) within the 1RSB phase, ϕ0 < ϕ < ϕG (for ϕ0 > ϕ d ), for typical inner-basin ∆1 and intra-basin ∆0 = ∞ distances. (b) In the fullRSB phase, ϕG < ϕ < ϕJ, 1RSB basins split into a fractal hierarchy of sub-basins with typical innermost sub-basin ∆EA and outermost basin ∆1 distances. Schematics of the equilibrium distribution Peq(∆) and restricted equilibrium distribution PSF(∆) (blue area), are given in (c) and (d), for 1RSB and fullRSB phases, respectively. III. NUMERICAL APPROACH In this section, we provide the numerical details used in the simulations of the glass states of the MK model in d = 3. A. Planting An important algorithmic advantage of the MK model is that planting can be used to generate equilibrium liquid configurations at any ϕ 0 [32,33]. This procedure sidesteps the tedious and time-consuming work of first preparing dense equilibrium configurations, as would be needed for typical glass formers, such as HS. The basic idea is to switch the order in which initial particle positions {r i } and random shifts {Λ ij } are determined. As long as the quenched and the annealed averages of the free energy are the same (see Ref. [32] for a more detailed discussion), a planted state automatically satisfies the liquid EOS, Eq. (6). This condition is met along the replica symmetric phase for ϕ 0 < ϕ K , where ϕ K is the Kauzmann point at which the configurational entropy vanishes. Because in the MK model ϕ K = ∞, planting a liquid configuration is thus possible at any density, which dramatically reduces the computational cost of initial equilibration. B. Molecular dynamics (MD) simulations We use event-driven molecular dynamics (MD) [7,33] to simulate MK particles in d = 3. Periodic boundary conditions with the minimum image convention are implemented on the shifted distances \|r i − r j + Λ ij \|. Time t is expressed in units of βmσ 2 , where the particle mass m and diameter σ as well as the inverse temperature β are set to unity. Systems consist of N = 800 particles unless otherwise specified. This system size is large enough to contain a first full shell of neighbors around each particle, and to keep the periodic boundary effects on caging to a minimum [33]. Finite-size effects are studied for one of the observables for the initial liquid density ϕ 0 = 2.5 (see Sec. IV B). To simulate SF, we start from a planted equilibrium configuration (a given {r i } and {Λ ij } defines a sample) at a packing fraction ϕ 0 , and grow the spheres following the Lubachevsky-Stillinger algorithm [6,7] at constant growth rate γ = 0.001, unless otherwise specified, up to a desired ϕ. Once compression is stopped the time evolution of ∆(t) is measured, keeping density and temperature (and, equivalently, energy) constant. This procedure is repeated over N s samples in order to average over thermal and quenched disorders. Errors are computed using the jack-knife method [45]. Depending on the statistical convergence of the different observables, N s is varies from 500 to 75,000, as specified in the discussion of the various measurements. C. Observables The pressure evolution along SF is reasonably well described by a free-volume EOS 1 p = C 1 − ϕ ϕ J ,(8) where C is a fitting parameter. Fitting Eq. (8) to the compression results provides an estimate of ϕ J (see Table II). A sufficiently small γ is chosen, such that no aging is observed (see Fig. 4). Using slower compression rates gives only negligible corrections to the glass EOS (see Fig. 3). Interestingly, upon decompression, the state follows the same EOS up to a threshold density at which it melts into a liquid phase. This phenomenon has been recently predicted by a thermodynamic theory [17,46], and observed numerically in simulated ultrastable glasses [47,48]. To obtain more structural information about the free energy landscape, we also simulate a cloning procedure. The approach consists of taking two exact copies (clones) A and B of the same planted configuration at ϕ 0 , and assigning them different initial velocities, randomly drawn from the Maxwell-Boltzmann distribution. These two copies are then independently compressed up to ϕ, before measuring the mean-squared distance between them ∆ AB (t) = 1 N N i=1 \|r A i (t) − r B i (t)\| 2 ,(9) where r A i (t) and r B i (t) are the positions of particle i at time t in clones A and B, respectively. Although the two clones start from the same initial configuration, they have had different compression histories once ϕ is reached. In the 1RSB phase, the two clones are uncorrelated in the glass basin and ∆ AB = ∆ 1 (with a slight time dependence up to t ∼ τ 0 if the two clones are not sufficiently well equilibrated along the compression). If the end point of the compression falls in the Gardner phase, clones most likely fall into different sub-basins. Their mean-squared distance can then be described by a timedependent probability distribution, P AB (t, ∆), that depends on the way sub-basins are sampled. Calculating these weights is difficult, because the two clones are generally out of equilibrium. What we can say is that in the long-time limit when clones visit all the states in a given glass metabasin with equilibrium weights, i.e., τ meta t τ α , we expect P AB (t, ∆) → P SF (∆), and therefore ∆ AB should tend to the restricted equilibrium value ∆ SF ≡ ∆P SF (∆)d∆ > ∆ EA . IV. DETECTING THE GARDNER TRANSITION In this section, we describe different means of detecting the Gardner transition through SF. We follow two complementary approaches. First, from the long-time dependence of the cage order parameter, we determine the Gardner transition from the divergence of the characteristic relaxation time. Second, from the distribution function P (∆ AB ) at a fixed time, we detect the onset of ergodicity breaking associated with the Gardner transition. A. Time-dependent functions As discussed in Sec. II D, the Gardner transition is characterized by a continuous change in the probability distribution of the cage order parameter ∆. This change should also be associated with a diverging characteristic relaxation time for ∆. We thus define two different measures of the cage order parameter: (i) the standard ∆(t) from Eq. (7), and (ii) the distance between two clones ∆ AB from Eq. (9). Figure 4 shows the time dependence of both quantities for ϕ 0 = 2.5, averaged over N s = 15, 000 to N s = 500 from the lowest and highest ϕ, respectively. Qualitative change in caging and susceptibility Both ∆ and ∆ AB are observed to behave slightly differently below and above ϕ G (ϕ G ≈ 3.00 for ϕ 0 = 2.50, see Fig. 4). For ϕ < ϕ G , ∆(t) first (and up to a microscopic time τ 0 ∼ 0.1) grows quickly because of the ballistic motion of particles [33] and then more slowly in the beta relaxation regime, before eventually reaching the plateau ∆ = ∆ 1 that defines the cage size. This plateau coincides with the (almost time-independent) results for ∆ AB (t), as is qualitatively expected for a system in a 1RSB phase. For ϕ > ϕ G , the situation is a bit more convoluted. As discussed above, beyond the Gardner transition each of the original metabasins is expected to subdivide into a hierarchical distribution of glassy states. Dynamicallyspeaking, the system should thus initially (immediately after stopping the compression) get trapped into a subbasin, and thus ∆(t) should grow until reaching a plateau corresponding to the size of the sub-basin, ∆ EA , for t τ meta . Here, however, the timescale τ meta is not large, because N itself is not very large and because the system is prepared out of equilibrium. After a first shoulder, ∆(t) thus keeps increasing (Fig. 4), and no clear first plateau can be detected. A second plateau should be reached in the limit t τ meta , but this regime is here beyond computational reach. Similarly, ∆ AB (t) should generally correspond to the average distance between sub-basins within a metabasin, but at short times the basins are sampled with non-equilibrium weights, and hence ∆ AB (t) drifts on a (very slow) timescale t ∼ τ meta . Note that because we have argued that even in the very long time limit, ∆ AB ≈ ∆ SF ≡ ∆P SF (∆)d∆ > ∆ EA , for t τ meta we thus expect that ∆(t) < ∆ AB (t). The dynamical behavior of the caging susceptibility is qualitatively similar (Fig. 5). We define a timedependent caging susceptibility as χ(t) = N ∆ 2 (t) − ∆(t) 2 ∆(t) 2 ,(10) and its counterpart of cloned configurations as χ AB (t) = N ∆ 2 AB (t) − ∆ AB (t) 2 ∆ AB (t) 2 .(11) As for the cage order parameters, when ϕ < ϕ G the two susceptibilities become identical in the long time limit, χ(t → ∞) = χ AB (t → ∞). By contrast, in the fullRSB phase (when ϕ > ϕ G ), on a timescale t τ meta , we generally observe that χ(t) < χ AB (t). Note that the magnitude of the susceptibility increases by more than a decade in the density range considered here, which is a clear signature of the Gardner transition. A detailed analysis of this increase is discussed in Sec. IV B 2. Computation of timescales In this subsection we build on the analysis proposed by Ogielski in his classic paper on spin glasses [49], and recently extended to the study of the dynamical transition in the d = 3 Edwards-Anderson model under an external field [50]. The idea consists in obtaining a relaxation timescale τ from the decay of ∆ AB (t) − ∆(t) (Fig. 4). Note, however, that this scheme is only well defined in the 1RSB phase, where ∆ AB (t) ≈ ∆(t → ∞), which suggests the existence of a single intrinsic τ within a metabasin. In order to estimate τ , we fit the results to an empirical scaling form used for spin glasses [49,50] ∆ AB (t) − ∆(t) = c t −a exp (−t/τ ) b , ϕ < ϕ G ,(12) where the parameters a, b and τ depend on both ϕ and ϕ 0 ; τ offers a first estimate of τ . All the fits are very good, as reflected by χ 2 per degree of freedom (d.o.f.) being much less than 1 (e.g., Fig. 4). (Recall that χ 2 = N T i=1 [y i − f (t i ; c, a, τ , b)] 2 /σ 2 i , where N T is the numbers of times t i and y i = ∆ AB (t i ) − ∆(t i ), σ i is the error of y i , and f is the fitting function, Eq. (12)). Because as ϕ approaches ϕ G for a given ϕ 0 , τ is expected to diverge, we fit τ to the power-law form τ ∼ \|ϕ − ϕ τ G \| −γτ .(13) Once again, reasonably good fits are obtained ( Fig. 6; Table I for fit parameters). An alternate estimate of τ can be obtained from the logarithmic scaling of ∆ AB (t) − ∆(t) at long times (see Fig. 7). For ϕ < ϕ G , the fitting form with a density-dependent constant k gives τ [50]. Comparing τ and τ suggests that the divergence of the two timescales is compatible with a same ϕ G and a very similar power-law exponent (see inset of Fig. 7). The insensibility of the estimator of τ to its precise definition adds support to our claim that the observed divergence is due to a true thermodynamic transition. ∆ AB (t) − ∆(t) = k 1 − log(t) log(τ )(14) We note, however, that the results for ∆ AB (t) and ∆(t) at low densities, i.e., ϕ d ≤ ϕ 0 2.2, may be affected by hopping [33]. In these systems the timescale for leaving a metabasin (albeit only through local hopping processes) is comparable to τ even near ϕ G . The estimate of ϕ G in this regime is therefore subject to a larger error, which explains the bigger difference between ϕ τ G and other ϕ G estimates (Table II). For the limit case ϕ 0 = 1.8, we do not even attempt to fit the data because no clear power-law regime can be distinguished. By contrast, for ϕ 0 ≥ 2.5, hopping is negligible on the timescales achieved numerically. B. Static functions In this subsection, we estimate the location of the Gardner transition using an approach based on the change in the probability distribution function (pdf) of cage order parameters. Probability distribution functions We study the pdf of cage order parameters at a fixed timet, and compute ∆(t) and ∆ AB (t) for each sample. Because our estimate of ϕ G is only based on ∆ AB (t) (see below), and because ∆ AB (t) is time independent for ϕ < ϕ G , we could choose any arbitraryt within this regime. Here, we chooset = 0.2V 1/3 ∼ 2, such that τ 0 <t τ meta . Hence the distribution of ∆(t) qualitatively represents P SF (∆) in the 1RSB phase, while it represents the peak around ∆ EA in the Gardner phase (see Fig 2). Below, we abbreviate∆ ≡ ∆(t) and ∆ AB ≡ ∆ AB (t), unless otherwise specified. Figure 8 shows P (∆) for ϕ 0 = 2.5 calculated from N s = 40, 000 -75,000 samples. The shape of P (∆) is Gaussian-like at all ϕ, and the mean value monotonically decreases with increasing ϕ. The shape of P (∆ AB ), however, changes considerably over that same regime. For ϕ < ϕ G , it is Gaussian and analogous to that of P (∆), but near ϕ G it develops an exponential tail akin to a Gumbel distribution. If ϕ is further increased, P (∆ AB ) then becomes broader, which is consistent with the presence of two (unresolved) peaks. From theoretical considerations, we expect one of these peaks to be P (∆) at that same ϕ, while the second peak should be centered at a higher value (see Fig 2). The development of an exponential tail at a critical point has been observed and studied for spin glasses in a field [51,52]. The effect is thought to be due to disorder. Whereas the results for most samples fall within Gaussian fluctuations around a given mean value, a few rare samples have much larger ∆ AB than the mean, giving an exponential tail to the distribution. The smaller the system, the stronger the effect (Fig. 9). These rare fluctuations are hypothesized to originate from the sampleto-sample fluctuations of the critical point [51,53], which then translates into significant sample-to-sample fluctuations of some of the measured observables. We come back to this point in Sec. IV C. The connection between the changing shape of the distribution and criticality suggests that we can determine the critical transition from P (∆ AB ) alone. We propose below two alternative procedures for detecting the Gardner transition using standard moments of the distribution. Caging susceptibility We first consider define a caging susceptibility from the variance of P (∆ AB ) χ = N ∆ 2 AB − ∆ AB 2 ∆ AB 2 ,(15) where the denominator corrects for the fact that ∆ AB changes with ϕ. As in the vicinity of any critical point, the susceptibility is expected to diverge as χ ∝ (ϕ χ G − ϕ) −γ ,(16) where the critical exponent γ is not to be confused with the growth rate of particles used for sample preparation. Because the MK model is mean-field in nature, one expects γ = 1, which we verify in Fig. 10 for different values of ϕ 0 (and thus ϕ G ). We observe a critical scaling for all ϕ 0 except for ϕ 0 = 1.8, where the spacing between ϕ 0 and ϕ G is narrowest, and where hopping is most likely to obfuscate the critical regime (see Fig. 14). For the other ϕ 0 , the Gardner transition is extracted by fitting Eq. (16) with γ = 1 in the ϕ < ϕ G region (Table II and Fig. 14). (2) The definition of χ involves taking the quotient of two quantities that both suffer from strong finite-size corrections. In order to control for this effect we study the behavior of both terms as function of 1/N (see Fig. 11). The denominator, ∆ AB , behaves linearly in 1/N , decreases smoothly with ϕ, and eventually saturates above the Gardner point. The numerator, N ( ∆ 2 AB − ∆ AB 2 ) has, however, a more complex behavior. While it follows a nearly 1/N behavior for ϕ < ϕ G , with a small dependence on ϕ, it grows significantly faster both with in N and ϕ for ϕ > ϕ G . Yet in both cases, at least in the range of sizes studied and below ϕ G , the two quantities behave smoothly in 1/N , which allows us to extract their value at the thermodynamic limit using a second-order polynomial fit. The resulting χ N →∞ obtained using both extrapolations, is included in Fig. 10 Upon approaching ϕG, χ clearly diverges as \|ϕ − ϕ χ G \| −1 , except for ϕ0 = 1.8. Note that the divergence of χ coincides with the maximum of Γ. The N → ∞ extrapolation of the curves in Fig. 11 were used to obtain the solid line. for this system size, the determination of ϕ χ G is fairly well controlled. It is important to point out that one can only measure the critical divergence of χ upon approaching ϕ χ G from the low-density (1RSB) phase. Above the Gardner transition, as a consequence of the appearance of the second peak in P SF (∆) (recall Fig. 2), the susceptibility continues to grow with ϕ, and actually diverges with N in the Gardner (fullRSB) phase. Caging skewness Near the Gardner transition, large sample-to-sample fluctuations give rise to a strong exponential tail in P (∆ AB ). This effect can be quantified by the skewness of the distribution Γ = ∆ AB − ∆ AB 3 ∆ AB − ∆ AB 2 3/2 .(17) (Recall that the skewness is a measure of a distribution's asymmetry and that a Gaussian distribution would have Γ = 0.) Sample-to-sample fluctuations are expected to be maximal at the critical point (see Sec. IV C), which provides an estimate of the Gardner transition, ϕ Γ G (see, e.g., Fig. 10 and Table II). For all ϕ 0 , we can see that ϕ Γ G is very close to the fitted divergence of the susceptibility, ϕ χ G . Finite-size analysis further shows that the rescaled skewness, Γ √ N , collapses the data for different N (Fig. 10). The peak of Γ is thus expected to persist all the way to the thermodynamic limit, consistently with comparable observations in spin glass models [51]. The functional behavior Γ(ϕ, ϕ 0 ) for different ϕ 0 can also be rescaled onto a master curve Γ(ϕ, ϕ 0 ) Γ(ϕ Γ G , ϕ 0 ) = F ϕ − ϕ Γ G δϕ ,(18) where δϕ is the full width at half maximum of Γ(ϕ, ϕ 0 ). We empirically observe that the tails of F (x) are reasonably well fitted by a power-law form F (x) ∼ \|x\| −ω , \|x\| 0(19) with an exponent ω = 0.85 (see Fig. 12). This property suggests that our analysis is robust for any choice of ϕ 0 , but to the best of our knowledge there exists no theoretical justification for this scaling form or its universality. C. Sample-to-sample fluctuations We have assumed above that the abnormal behavior of Γ around the Gardner transition is due to sample-tosample fluctuations. To further motivate this hypothesis, we perform N s = 10, 000 independent clonings and compute Γ for each sample. Sample-to-sample fluctuations are thus only due to differences in quenching history. In Figure 13, the data for the ensemble of samples are compared with those of four individual samples (as in Fig. 10). We note that the density evolution of Γ for the individual samples can have a very different behavior from the ensemble one. In particular, a peak around ϕ G is generally not seen. Sample-to-sample fluctuations have a smaller effect on χ (Fig. 13). While the magnitude and the critical density exhibit some fluctuations, they all display a divergence similar to that of Eq. (16). Because each realization of disorder corresponds to different 1RSB metabasins, our results suggest that the metabasins themselves have slightly different properties. In particular, they exhibit different ϕ G , which is likely the physical origin of the exponential tail of P (∆ AB ) and thus of its anomalous skewness. V. SUMMARY OF RESULTS AND COMPARISON WITH THEORY The Gardner transition at ϕ G was independently identified from: (i) the power-law divergence of the characteristic time τ , (ii) the power-law divergence of the susceptibility χ, and (iii) the maximum of the skewness Γ of the cage order parameter pdf (see Table II). Figure 14 summarizes these results in a phase diagram. The different estimates of ϕ G are generally quantitatively consistent with each other and qualitatively consistent with the mean-field SF results from d → ∞. The agreement with the mean-field calculation improves with density, likely because the Gaussian caging approximation used in the theory becomes a better approximation at higher densities [37]. A couple of reasons underlie the discrepancy between numerical estimates and theory in the vicinity of ϕ d . First, caging becomes imperfect at such low densities, which allows particles to hop between neighboring cages on a timescale comparable with the simulation time [33]. Hopping thus affects the dynamics of the system (∆(t) and ∆ AB (t)) and also softens the sharpness of transitions (at either ϕ d or ϕ G ) in this regime. Second, because ϕ G is expected to converge to ϕ d upon approaching the dynamical glass transition (see Figs. 1 and 14), the critical regime becomes too small to make any fit to a critical power-law scaling. We also compare the density evolution of the cage order parameters with the theoretical predictions in d → ∞. Figure 15 shows that ∆ AB and ∆ coincide when ϕ < ϕ G but separate around ϕ G . This result is consistent with a diverging relaxation timescale, as was found in Sec. IV A. However, one should treat ∆ with caution. As discussed above, because of both the finite-size and out-of-equilibrium nature of the system, the value of ∆ drifts with time as different sub-basins are explored. In practice, we evaluate ∆ by averaging over relatively short times 50 < t < 200, and thus obtain a reasonable estimate of the size of a single sub-basin. These results are similar to the theoretical prediction [17]. We also compare the results for (see Figure 15) ∆ f = 2∆ r − ∆ g − ∆,(20) where ∆ g is the long time limit of the ∆(t) for the planted equilibrium configuration {r 0 i (t)} at ϕ 0 , ∆ g (t) = 1 N N i=1 \|r 0 i (t) − r 0 i (0)\| 2 ,(21) and ∆ r is the long time limit of the relative displacement ∆ r (t) = 1 N N i=1 \|r i (t) − r 0 i (0)\| 2 .(22) Both ∆ r and ∆ g are measured over the same time window as ∆. Here again, the theoretical predictions agree reasonably well with the numerical results (Fig. 15). The calculation in d → ∞ is, however, only done for a 1RSB phase, and this phase becomes unstable beyond the Gardner transition. A fullRSB calculation would be needed to significantly improve the agreement with simulations in that regime. [17]. VI. CONCLUSIONS In this work we have developed a numerical procedure for detecting the Gardner transition in the MK model, for which this transition has been predicted analytically. We have presented three independent approaches for locating the transition, all of which show that the transition exists and is found in a region that is roughly consistent with earlier theoretical predictions. This work also paves the way for studying the Gardner transition in more realistic numerical models of glasses, where the very existence of the Gardner transition is debated [27]. Our approach is also suitable to be reproduced in experiments. SF, for instance, corresponds to a straightforward annealing, and some of the observables should be readily available through standard microscopy or scattering techniques. The Gardner transition thus presents a promising way of testing theoretical results in temperature and density regimes where simple equilibrium thermodynamics is violated. One key hurdle to generalizing our methodology to other systems is the need to equilibrate, without planting (and thus by slow annealing), a glass state well above the (avoided) dynamic glass transition, so that activated processes are strongly suppressed. For numerical simulations this requirement can be particularly computationally onerous, but it may be more easily achievable in experimental systems, where longer timescales can typ-ically be reached. In experiments, the bigger challenge would be to substitute the cloning procedure with a (potentially very) large number of experimental replicates. It is possible that finite-dimensional non-mean-field glass formers display features that are not observable in the MK model. In particular, we expect a diverging length scale to be associated with the Gardner transition in these systems. This length scale is expected to capture static heterogeneity, which represents the spatial inhomogeneity of cage sizes around ϕ G . In principle, this kind of static heterogeneity should be different from both the dynamic heterogeneity around the dynamic glass transition, and the heterogeneity close to jamming, which is related to the soft relaxation modes. These interesting properties will be the object of future studies. FIG. 2 . 2(Color online). (a) Organization of glass states (blue dots FIG. 3 . 3(Color online). Compressions (γ > 0) and decompressions (γ < 0) of an initial equilibrium state at ϕ0 = 2.5. The results are averaged over Ns = 100 samples. On the scale of this figure, results for γ ≤ 0.001 would be indistinguishable, hence only γ = 0.001 is shown. The theoretical curve is from Ref.[17]. FIG. 4 . 4(Color online). SF from ϕ0 = 2.50 gives (top) ∆(t), (inset) ∆AB(t), and (bottom) ∆AB(t) − ∆(t) at different ϕ. Results obtained after a waiting time tw = 10 (dashed lines in top panel) indicate that no aging occurs for ϕ < ϕG. Solid lines are fits to Eq. (12) (bottom panel). FIG. 5 . 5(Color online). Time evolution of the caging susceptibility χ(t) (scatters) and χAB(t) (lines). online). (Main panel) Growth of τ with ϕ τ G −ϕ for ϕ τ G fromTable II. Recall that τ is extracted from fitting ∆(t) − ∆AB(t) to Eq. (12) for different ϕ0 (fits for ϕ0 = 2.5 are displayed inFig. 4). (Inset) Evolution of a and b with ϕ for ϕ0 = 2.5. FIG. 7 . 7(Color online). (Main panel) Logarithmic decay of ∆(t) − ∆AB(t) with time for ϕ0 = 2.5. The results are fitted to Eq. (14) (dashed lines). (Inset) The two estimates of τ , τ and τ , as a function of ϕ τ G (ϕ0) − ϕ. FIG. 8 . 8(Color online). (top) P (∆) and (bottom) P (∆AB) at different ϕ for ϕ0 = 2.5. FIG. 9 . 9(Color online). (top) Finite-size behavior of P (∆AB) at ϕ = 3.0 ≈ ϕG (ϕ0 = 2.5). (bottom) Density evolution of the rescaled skewness Γ for three different N . FIG and compared with the N = 800 results. This extrapolation suggests that although the finite-size effects in ∆ AB are still very strong . 10. (Color online). (top) Inverse caging susceptibility, χ −1 , and (bottom) skewness, Γ, as functions of ϕ for different ϕ0. FIG . 11. (Color online). Finite-size behavior of the two terms of the definition of the susceptibility in Eq. (15): (Top) ∆AB , and (bottom) N ( ∆ 2 AB − ∆AB 2), for ϕ0 = 2.50. The dashed lines are fits of a second-order polynomial in 1/N to the data. The 1/N → 0 extrapolation of these two fits have been used to extract the N → ∞ limit of χ, given inFig. 10by a dark solid line.. FIG . 12. (Color online). Rescaling results from Fig. 10 collapses them onto a master curve (using ϕG from Table II). (Left) Growth of χ upon approaching ϕ χ G , where h(ϕ0) is a ϕ0-dependent constant. The solid line with the power-law exponent −1 is the mean-field scaling. (Right) Using Eq. (18) collapses γ. The tails are fitted to \|x\| −ω with ω = 0.85 (lines). FIG . 13. (Color online). The susceptibility χ and the skewness Γ averaged over the ensemble of clones are compared with those of four individual samples. FIG . 14. (Color online). Inverse reduced pressure 1/p density ϕ phase diagram of the MK model in d = 3. The liquid EOS follows Eq. (6), while a specific state follows a glass EOS from ϕ0 up to jamming at ϕJ. Numerical estimates of the Gardner transition evolve with ϕ0 similarly as the theoretical predictions[17]. FIG. 15. (Color online). Density evolution of ∆, ∆AB, and ∆ f . Numerical results are compared with theoretical predictions(lines) TABLE I . IFit parameters of Eq. (13) for different ϕ0 ϕ0 ϕ τ G γτ χ 2 /d.o.f., 2.2 2.78(3) 2.7(3) 0.78/4, 2.5 3.02(2) 1.2(4) 8.4/2, 3.0 3.529(12) 1.2(2) 1.92/3, 4.0 4.538(12) 0.7(3) 2.6/2. TABLE II . IIEstimates of ϕG and ϕJ for various ϕ0ϕ0 ϕ Γ G ϕ χ G ϕ τ G ϕJ 1.8 2.04(5) 2.081(11) - 2.534(3) 2.2 2.638(6) 2.670(4) 2.78(3) 2.876(4) 2.5 2.995(13) 3.0026(18) 3.02 ACKNOWLEDGMENTS. We wish to thank L. Berthier . 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[ "Peer Review File Scientific multi-agent reinforcement learning for wall-models of turbulent flows REVIEWER COMMENTS Reviewer #1 (Remarks to the Author): Report to the manuscript entitled ``Scientific multi-agent reinforcement learning for wall-models of turbulent flows''", "Peer Review File Scientific multi-agent reinforcement learning for wall-models of turbulent flows REVIEWER COMMENTS Reviewer #1 (Remarks to the Author): Report to the manuscript entitled ``Scientific multi-agent reinforcement learning for wall-models of turbulent flows''", "Peer Review File Scientific multi-agent reinforcement learning for wall-models of turbulent flows REVIEWER COMMENTS Reviewer #1 (Remarks to the Author): Report to the manuscript entitled ``Scientific multi-agent reinforcement learning for wall-models of turbulent flows''", "Peer Review File Scientific multi-agent reinforcement learning for wall-models of turbulent flows REVIEWER COMMENTS Reviewer #1 (Remarks to the Author): Report to the manuscript entitled ``Scientific multi-agent reinforcement learning for wall-models of turbulent flows''" ]	[]	[]	[]	The authors developed a multi-agent reinforcement learning (SciMARL) method for the automated discovery of wall models in large-eddy simulations (LES) of wall-bounded turbulent flows. They treat discretization points as cooperating agents that learn to supply the closure model and propose a control policy by cooperating agents using the recovery of the correct mean wall-shear stress as a reward. The paper shows that the SciMARL method requires limited data by using LES of a turbulent channel flow at moderate Reynolds numbers in the training processes. Two models with different state spaces, including a velocity-based wall model (VWM) and a log-law-based wall model (LLWM), are considered. The paper concludes that the proposed method can be generalized for LES of a turbulent boundary layer and turbulent channel flow at extreme Reynolds numbers. The paper may be considered for publication in Nature communication after a major revision and the following issues must be addressed:	10.1038/s41467-022-28957-7	null	247,520,163	2106.11144	4434b0c5abac5f1395790776dfe44b99e86a77a9	Peer Review File Scientific multi-agent reinforcement learning for wall-models of turbulent flows REVIEWER COMMENTS Reviewer #1 (Remarks to the Author): Report to the manuscript entitled ``Scientific multi-agent reinforcement learning for wall-models of turbulent flows'' Peer Review File Scientific multi-agent reinforcement learning for wall-models of turbulent flows REVIEWER COMMENTS Reviewer #1 (Remarks to the Author): Report to the manuscript entitled ``Scientific multi-agent reinforcement learning for wall-models of turbulent flows'' Open Access This file is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. In the cases where the authors are anonymous, such as is the case for the reports of anonymous peer reviewers, author attribution should be to 'Anonymous Referee' followed by a clear attribution to the source work. The images or other third party material in this file are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit The authors developed a multi-agent reinforcement learning (SciMARL) method for the automated discovery of wall models in large-eddy simulations (LES) of wall-bounded turbulent flows. They treat discretization points as cooperating agents that learn to supply the closure model and propose a control policy by cooperating agents using the recovery of the correct mean wall-shear stress as a reward. The paper shows that the SciMARL method requires limited data by using LES of a turbulent channel flow at moderate Reynolds numbers in the training processes. Two models with different state spaces, including a velocity-based wall model (VWM) and a log-law-based wall model (LLWM), are considered. The paper concludes that the proposed method can be generalized for LES of a turbulent boundary layer and turbulent channel flow at extreme Reynolds numbers. The paper may be considered for publication in Nature communication after a major revision and the following issues must be addressed: 1. Please provide more comments about the advantages of SciMARL models, as well as the dependency of SciMARL models on the training data or the predefined theoretical model. In the situation of the velocity-based wall model (VWM), the training processes are dependent on the LES of a turbulent channel flow at moderate Reynolds numbers, and the model cannot be easily generalized to turbulent channel flow at very high Reynolds numbers. The limitation of the proposed VWM is similar to the situations of supervised learning models, which are limited by the scope of training data. In the situation of the log-law-based wall model (LLWM), the training processes are dependent on the well-known theoretical log-law model. For the general complex flows which are not consistent with the theoretical log-law model, the proposed LLWM could fail. For the flows which are consistent with the theoretical log-law model, the original log-law model is enough. 2. Please provide the computational cost for LES of wall-bounded turbulent flows using the new models, and make a comparison with traditional wall models. 3. Please give some comments about the effects of different parameters of SciMARL models on the accuracy of LES. Reviewer #2 (Remarks to the Author): This paper identifies an important research question in computational fluid mechanics: developing approximate boundary conditions for Large Eddy Simulations without employing experimental data or direct numerical simulation data. The authors have designed two reinforcement learning approaches to solve this problem. One of these methods (LLWM) is performing quite well. This method assumes the logarithmic law for the velocity profile and determines two unknown coefficients kappa and B. The authors have claimed that their approach is a significant advance over the state of the art because rather than assume kappa and B, it has learned them through unsupervised learning. However, the manuscript does not state how the truth values of these turbulence coefficients (kappa and B) are established and provided to the reinforcement learning agents. If these target values are taken from prior experiments or direct numerical simulations (DNS), then I believe this paper is overstating its significance since the authors acknowledge that there is already a model (EQWM) that uses assumed coefficients to make accurate predictions. On the other hand, I would be very interested to learn if the authors have some novel way to establish the truth values for kappa and B (and tau_w for the velocity based model), which does not require prior information (experiment/DNS). Upon close reading of the manuscript, I have not been able to determine how the truth values of kappa and B are determined. I have given the authors the benefit of the doubt and tried to write this review under the assumption that the authors can modify the manuscript to demonstrate that kappa and B are discovered rather than assumed. If this is not the case, the present model, in my opinion, is not any more predictive of the wall stress than the classical EQWM. The principle finding of this paper would then be regarding the correlation of the velocity and wall stress fluctuations. The paper would need to be revised to emphasize that result, which is interesting, though perhaps not suitable for publication in Nature Communications. Overall the paper was well written, but the organization was a bit unintuitive. Perhaps there could be a section between the "introduction" and "results" sections. Or perhaps that is forbidden by the format requirements of this journal. More specific comments regarding this will follow. For future revisions, it would be helpful if the manuscript had line numbers. Comments on the introduction • "such wall models can only provide limited success" It might be helpful to add references that outline some of the regimes where the RANS models are failing (separated flows, predicting transition, etc.) to help motivate the present work. "Limited success" is fairly vague. • "SciMARL does not rely on a priori knowledge" the accuracy of this statement depends on how the authors address my comment regarding truth values kappa, B, and tau_w. Comments on the section "Results" • The "results" section also describes the model. Is this common in this journal? • Eq. 1, the notation for the expectation under policy pi should be explicitly mentioned since presently E_pi is not defined. The summation over n is not defined. n is defined in the methods section, but must be defined when it is used in Eq. 1. It would be helpful to note that gamma will be specified in the methods section since it was unclear that this was a constant parameter when it was introduced in the results section and not specified. • A single policy is made by all the agents. This seems like it is only the case in homogeneous flows. In general problems, there will need to be n agents with n policies. In a channel, it seems like only 1 agent is needed. In a boundary layer, one would need more than 1 agent if kappa and B are thought to vary with the streamwise coordinate. (However, this variation is likely modest.) Are the authors using additional agents in the span just to improve the statistical convergence? Comments on the section "Velocity-based wall model" • It might be helpful to refer the reader to the methodology section since the term "reward" is used without being defined. Especially the bonus to the reward that is designed to promote stability. The reader may not realize that this will be defined later. Comments on the section "State-action map" • I don't see where the authors mention which flow they are plotting. I assume it is a channel. • For figure 2, I think the a simpler explanation will be possible by splitting 2b into 2 panels (2b and 2c) for plots of kappa vs y* and B vs y. This would make the presentation of results consistent with those in figure 2a and would more clearly show if the model is correctly downshifting or upshifting the parameters if they are over-or under-predicted. Perhaps this is inconvenient though if there is some error cancellation of kappa and B that is currently being masked in 2b. Comments on the section "Turbulent channel flow" • The name of this section is confusing since I thought the last section on maps was also about channels. Perhaps it just needs to be more specific, for example, "Testing higher Reynolds number channels." • There is a claim that the LLWM is superior to the classical EQWM because the latter uses an "empirical coefficient tuned for this particular case." This is a bit misleading since the EQWM uses coefficients that are generally not tuned on a case-by-case basis, even if those coefficients were originally tuned using channel data. I would say that present approach also has some assumed truth values for the coefficients (which I am unsure how these are determined). If the EQWM fails because the best choices of its model coefficients differ from the assumed values, I'm not sure if LLWM will outperform unless its assumed truth values are adapted. Such automatic adaptation is not demonstrated in the paper, but somehow seem to be claimed throughout. If manual adaptation is required for the LLWM, please note that manual adaptation is also possible for the EQWM. It would be helpful if the authors show a case where the truth values automatically adapt, like in a pressure gradient boundary layer where kappa and B are different than their default values. Figure 3 • The dotted horizontal line does not go through zero. What is this line? • It is difficult to distinguish much of the data and impossible to discern the magnitude of the error. Please replot with log scale. Or show a separate figure with the VWM data omitted. • I can see that the error for LLWM is growing with Retau. There should be a remark in the text that acknowledges this. This is important since the central claim of this paper is that the LLWM is learning kappa and B and no DNS simulations are required to learn this information. However, it appears at high Retau, that the procedure is not predicting the non-universality (Reynolds number dependence) of kappa and B that has been reported widely in the literature. Why should LLWM perform better at low Retau, if not for an issue with the truth values of kappa and B. • Please compare against the accuracy of the EQWM. Throughout the paper the EQWM is regarded as the state of the art, but it is not compared against in this key plot, so it is impossible for the reader to compare the present approach with the state of the art in this canonical setting. Comments on the section "Potential of SciMARL wall models" • Regarding the limitations of the EQWM, I have a similar comment to the one that I made in the section "Turbulent channel flow." It is unclear to me that their model is less limiting than the EQWM. • "do not require any DNS simulation data." the accuracy of this statement depends on how the authors address my comment regarding truth values kappa, B, and tau_w. • The comparison of LLWM with supervised learning may not be appropriate (depending on how the authors defined their ground truth values as questioned above). If the truth values for kappa and B are from DNS/experiments, then the claimed advantage of RL is not present. • "single model trained from various canonical roughness cases." It is not clear to me how the authors can generalize this model to rough walls without providing some roughness characteristics to the model and also modified truth values. Please provide additional information or equations. Comments on the section "Discussion" • The phrase in the discussion section "present a paradigm shift" is perhaps an overly strong claim. The present approach has not been shown to be more predictive than existing methods and hasn't been tested in complex cases that testing methods have succeeded in. The usefulness of the model has been speculated, but not demonstrated yet. Comments on the section "Methods-Reinforcement learning": • Please explicitly define the term "episode." • Please provide a reference of clipping "far-policy" experiences as this seems ad hoc. "One way …" it is unclear if the authors are proposing this strategy or if it has been used previously. Comments on the section "Methods-Overview of the training setup": • Only one resolution is used in x and z. Demonstrating other resolutions would be useful for establishing robustness. Especially since the boundary layer also has very similar resolution. • Change "uniformly sampled" to "spatially uniformly sampled" • Robustness to numerics is claimed, but this this not tested. • Why is white noise added to the training data set. Is this to create additional realizations? If so, I don't see how these additional realizations would be more informative than a single realization. Please explain. • How did the authors decide on their network topology. Do they have any references or did they conduct a study on the sensitivity to the network architecture? Comments on the sections called "Testing" • Please clarify this sensitivity comment. • Perhaps here the authors should cite Kawai and Larsson for using the third point matching location: "third grid point off the wall in the wall-normal direction" H. J. Bae & P. Koumoutsakos October 8, 2021 We are grateful for the Referees' insightful comments and suggestions for improving the paper. In the revised manuscript we have addressed all the comments of the referee and made modifications to further clarify the value and originality of this contribution. The relevant modifications in response to the comments of the Referee have been made in blue. 1. Please provide more comments about the advantages of SciMARL models, as well as the dependency of SciMARL models on the training data or the predefined theoretical model. In the situation of the velocity-based wall model (VWM), the training processes are dependent on the LES of a turbulent channel flow at moderate Reynolds numbers, and the model cannot be easily generalized to turbulent channel flow at very high Reynolds numbers. The limitation of the proposed VWM is similar to the situations of supervised learning models, which are limited by the scope of training data. In the situation of the log-law-based wall model (LLWM), the training processes are dependent on the wellknown theoretical log-law model. For the general complex flows which are not consistent with the theoretical log-law model, the proposed LLWM could fail. For the flows which are consistent with the theoretical log-law model, the original log-law model is enough. We have followed the suggestion of the Referee, and in the revised manuscript we provide further information on the SciMARL models, their data dependency, and their training. The referee is correct about the advantages and limitations of the VWM. The LLWM is based on the existence of a log law, which can be theoretically shown to exist for any flow that has a scale separation in the inner and outer region (Millikan 1939). However, we wish to clarify that the LLWM does not take as input the log-law parameters and we have added to the text, the following sentence: "We emphasize that this model does not take as input the a priori known values of κ and B from the log-law, but rather derived quantities from the instantaneous flow" by calculating y ∂u * /∂y * and u * − y * ∂ * /∂y * log y * in the current flow field. In addition, any flow with a solid boundary will indeed exhibit a log-law, although with different values of κ and B. We now state that "These flows [flows exhibiting roughness, stratified flows, compressible flows, among many others] usually have different log-law coefficients κ and B that are manually tuned for existing wall models. However, in the present work, these values are adjusted automatically using a SciMARL-based model." We note that this is a key advantage of SciMARL-based models over models such as EQWM, as they can adapt to various flow configurations without manual changing of the coefficients. 1 We also state in the manuscript that "Furthermore, the RL framework can be extended to various flow configurations by adding an additional dimension to the state vector. Since all flow with an inner-outer scale separation exhibit a log law in the overlap region, the current configuration for wall-model development can be extended to flows exhibiting roughness, stratified flows, compressible flows, among many others... This gives the LLWM a distinct advantage over existing models. For example, in cases with varying roughness over the simulation domain, traditional methods will have to assign different model parameters for each patch containing different roughness heights. In contrast, the SciMARL model can smoothly transition between various roughness elements with a single policy trained from various canonical roughness cases when the roughness height is included as a state". As such, additional flow features can be addressed by augmenting the state vector with quantities that best reflect the influence of this feature on the flow field. We hope to address this in a future study, as the present paper represents a proof of concept of the methodology, and such extensions of the model will require extensive simulations. 2. Please provide the computational cost for LES of wall-bounded turbulent flows using the new models, and make a comparison with traditional wall models. We now include a sentence in the manuscript stating that "the evaluation of the LLWM involves evaluating the weights of the trained neural net, which is an order of magnitude faster than the EQWM that solves an ODE at each time step." 3. Please give some comments about the effects of different parameters of SciMARL models on the accuracy of LES. We now include comments regarding the different parameters of SciMARL in the Methods section: "The most notable hyper-parameters used in our description of the MARL set-up are the spatial resolution for the interpolation of the actions onto the grid (determined by ∆ m x /∆ x , and ∆ m z /∆ z ). The default values ∆ m x /∆ x , and ∆ m z /∆ z reduce the number of experiences generated per simulation to O(10 5 ). This value is similar to the number of experiences generated per simulation used for SciMARL of SGS model development. Consistent with previous studies, we found that further reducing the number of agents per simulation reduced the model's adaptability and therefore exhibit slightly lower performance. Because we use conventional reinforcement learning update rules in a multi-agent setting, single parameter updates are imprecise. We found that ReF-ER with hyper-parameters C = 1.5 and D = 0.05 (Eqs. (6) and (7)) stabilizes training. We ran multiple training runs per reward function and whenever we vary the hyperparameters, but we observe consistent training progress regardless of the initial random seed." 2 Response to Referee #2 H. J. Bae & P. Koumoutsakos October 8, 2021 We are grateful for the Referees' critical feedback and detailed suggestions for improving the paper. In the revised manuscript we have addressed all the comments of the referee and made modifications to further clarify the value and originality of this contribution. The relevant modifications in response to the comments of the Referee have been made in red. General comments • The authors have claimed that their approach is a significant advance over the state of the art because rather than assume kappa and B, it has learned them through unsupervised learning. However, the manuscript does not state how the truth values of these turbulence coefficients (kappa and B) are established and provided to the reinforcement learning agents. .....I have given the authors the benefit of the doubt and tried to write this review under the assumption that the authors can modify the manuscript to demonstrate that kappa and B are discovered rather than assumed. We appreciate the comments of the Referee and we apologize that in the previous version of the manuscript it was not clear that the values of κ and B are indeed discovered by the multi-agent reinforcement learning. In the following, we further stress this point as it is a key contribution of the paper. Comments on the introduction • "such wall models can only provide limited success" It might be helpful to add references that outline some of the regimes where the RANS models are failing (separated flows, predicting transition, etc.) to help motivate the present work. "Limited success" is fairly vague. Following the suggestion of the Referee, we now specify that "Such wall models do not function as intended in real-world applications, where various flow states coexist (e.g. separated flows, flow over roughness, predicting transition, etc.)." and we provide additional references. • "SciMARL does not rely on a priori knowledge" the accuracy of this statement depends on how the authors address my comment regarding truth values κ, B, and τ w . The SciMARL-based wall models do not need an explicit input of the log-law coefficients κ and B, which is necessary for the traditional RANS-based models. Instead, it only requires the mean wall-shear stress for a few Reynolds numbers, which can be readily obtained from existing data sets. In canonical flow over flat-plates, the values of κ and B are widely studied and considered constants. However, this is not generalizable for flows configurations with roughness, pressuregradient effects, or compressibility effects. While manually setting the RANSbased wall models can be done, the ultimate purpose of WMLES is to provide predictions where the features of the flow configuration are not known a priori. In the current paper, we show the potential of SciMARL-based wall models to perform as well as RANS-based wall models in canonical zero-pressure-gradient flat-plate boundary layer flow. We also argue that the SciMARL-based wall models "can be extended to various flow configurations [(e.g. roughness, pressure-gradient effects, compressibility effects)] by adding an additional dimension to the state vector." This will allow the SciMARL wall models to adapt naturally to the given flow configuration, unlike RANS-based models that need to be manually tuned. 3. Comments on the section "Results" • The "results" section also describes the model. Is this common in this journal? The journal allows only three sections (Introduction, Results, and Discussion) in the main body of the paper. The separate Methods section, which comes after the References, is supposed to include the methodology used to produce the results. We opted to introduce the models themselves in the Results section in order to explain their training within the main text of the article. • Eq. 1, the notation for the expectation under policy π should be explicitly mentioned since presently E π is not defined. The summation over n is not defined. n is defined in the methods section, but must be defined when it is used in Eq. 1. It would be helpful to note that γ will be specified in the methods section since it was unclear that this was a constant parameter when it was introduced in the results section and not specified. We regret the confusion introduced by this notation. To maintain consistency, with the rest of the manuscript we removed Eq. 1 from the main text. We now write that: "The optimal policy π * (s, a) is found by maximizing the expected utility, which is given by the expected cumulative reward." • A single policy is made by all the agents. This seems like it is only the case in homogeneous flows. In general problems, there will need to be n agents with n policies. In a channel, it seems like only 1 agent is needed. In a boundary layer, one would need more than 1 agent if κ and B are thought to vary with the streamwise coordinate. (However, this variation is likely modest.) Are the authors using additional agents in the span just to improve the statistical convergence? We agree with the Referee that using multiple agents allows for better statistics and in turn a robust, collective formulation of the agents' policy that is also adjusted to their position in the flow field. We note that even in non-homogeneous flows (e.g. flow with varying pressure gradients or roughness height), a single policy can be maintained by all the agents. The state-space will include additional information in this case (such as pressure gradient), which will allow the agents to determine the correct policy based on those additional inputs. This is indeed the primary benefit of SciMARL, which allows automatic adjustment of the log-law coefficients based on flow configuration. We now include an example to highlight this feature: "... the SciMARL model can smoothly transition between various pressure-gradient effects with a single policy trained from various canonical cases when the parameters such as pressure and velocity gradients are included as a state". In the channel case, multiple agents are necessary as the agents themselves act as the wall model. This is equivalent to solving the ODE associated with the EQWM at all wall points in the computational domain. 4. Comments on the section "Velocity-based wall model" • It might be helpful to refer the reader to the methodology section since the term "reward" is used without being defined. Especially the bonus to the reward that is designed to promote stability. The reader may not realize that this will be defined later. We agree and we now refer the reader to the methods section. Comments on the section "State-action map" • I don't see where the authors mention which flow they are plotting. I assume it is a channel. The Referee is correct. We now include a text that mentions that the results are "for the channel flow at friction Reynolds number Re τ = 2000, 4200, 8000". • For figure 2, I think the a simpler explanation will be possible by splitting 2b into 2 panels (2b and 2c) for plots of κ vs y * and B vs y * . This would make the presentation of results consistent with those in figure 2a and would more clearly show if the model is correctly downshifting or upshifting the parameters if they are over-or under-predicted. Perhaps this is inconvenient though if there is some error cancellation of κ and B that is currently being masked in 2b. The main goal of figure 2 was to show the state and action map of the two trained wall models, VWM and LLWM. Since the states for the LLWM are 1/κ and B, we decided that the current form on figure 2b was the most appropriate. 6. Comments on the section "Turbulent channel flow" • The name of this section is confusing since I thought the last section on maps was also about channels. Perhaps it just needs to be more specific, for example, "Testing higher Reynolds number channels." We have changed the subsection title to "Testing: channel flow". We also changed the following subsection to "Testing: Spatially evolving turbulent boundary layer" • There is a claim that the LLWM is superior to the classical EQWM because the latter uses an "empirical coefficient tuned for this particular case." This is a bit misleading since the EQWM uses coefficients that are generally not tuned on a case-by-case basis, even if those coefficients were originally tuned using channel data. I would say that the present approach also has some assumed truth values for the coefficients (which I am unsure how these are determined). If the EQWM fails because the best choices of its model coefficients differ from the assumed values, I'm not sure if LLWM will outperform unless its assumed truth values are adapted. Such automatic adaptation is not demonstrated in the paper, but somehow seem to be claimed throughout. If manual adaptation is required for the LLWM, please note that manual adaptation is also possible for the EQWM. It would be helpful if the authors show a case where the truth values automatically adapt, like in a pressure gradient boundary layer where κ and B are different than their default values. We emphasize that the proposed sciMARL models do not use as input known values of κ or B. In fact 'estimates' of κ and B obtained by calculating y * ∂u * /∂y * and u * − y * ∂ * /∂y * log y * in the current flow field are used as states for the agents. These states are changing over time and they are used to inform the agent on what action to take in order to maximize its reward. On the contrary, EQWM requires the value of κ in the model as well as the damping coefficient, which indirectly relates to the value of B. We mention "empirical coefficient tuned for this particular case" to indicate that the EQWM requires the κ and B as parameters of the model. While the training of RL models requires the true value of τ w at low Reynolds number cases, τ w is a known quantity obtained from the imposed pressure gradient for channel cases. Additional flow features can be addressed by augmenting the state vector with quantities that best reflect the influence of this feature on the flow field. We hope to address this in a future study, as the present paper represents a proof of concept of the methodology, and such extensions of the model will require extensive simulations. 7. Figure 3 • The dotted horizontal line does not go through zero. What is this line? Thank you for catching this mistake. The line was drawn at 1% error by mistake. The dotted line is meant to be zero error and is now located at zero. • It is difficult to distinguish much of the data and impossible to discern the magnitude of the error. Please replot with log scale. Or show a separate figure with the VWM data omitted. Due to the negative error, we cannot plot the error in log scale; however, we now include a zoom-in of the figure from -1 to 5% in figure 3(b). We now also include the EQWM results in the zoomed-in figure for comparison. • I can see that the error for LLWM is growing with Re τ . There should be a remark in the text that acknowledges this. This is important since the central claim of this paper is that the LLWM is learning κ and B and no DNS simulations are required to learn this information. However, it appears at high Re τ , that the procedure is not predicting the non-universality (Reynolds number dependence) of κ and B that has been reported widely in the literature. Why should LLWM perform better at low Re τ , if not for an issue with the truth values of κ and B. We agree with the Referee and we now include a short discussion on this trend: "The error increases with Reynolds number, most likely due to the high variation of the streamwise wall-normal gradient with increasing Reynolds number as well as the departure of (h m ) + from the trained range of values." • Please compare against the accuracy of the EQWM. Throughout the paper the EQWM is regarded as the state of the art, but it is not compared against in this key plot, so it is impossible for the reader to compare the present approach with the state of the art in this canonical setting. We now include the errors of the EQWM for the same Reynolds number and grid resolutions. While the errors for EQWM are smaller than those of LLWM, this is expected since EQWM has information regarding the true log-law embedded in the model itself. We now mention that "Still, the results are comparable to the results obtained from the widely-used equilibrium wall model (EQWM) up to Re τ ≈ 10 5 , which uses an empirical coefficient tuned for this particular flow configuration. This range of Reynolds numbers is sufficient for various external aerodynamic and geophysical flows." 8. Comments on the section "Potential of SciMARL wall models" • Regarding the limitations of the EQWM, I have a similar comment to the one that I made in the section "Turbulent channel flow." It is unclear to me that their model is less limiting than the EQWM. We hope the previous response also addresses this comment and it is satisfactory for the Referee. • "do not require any DNS simulation data." the accuracy of this statement depends on how the authors address my comment regarding truth values κ, B, and τ w . We hope the previous response is satisfactory for the referee. • The comparison of LLWM with supervised learning may not be appropriate (depending on how the authors defined their ground truth values as questioned above). If the truth values for κ and B are from DNS/experiments, then the claimed advantage of RL is not present. We remark that the truth values of κ and B are not used in the RL model. Supervised learning requires (a lot of) instantaneous snapshots that are not readily available publicly. To obtain this data, either a new DNS needs to be run to save the instantaneous flow fields or access to prior databases that store such data is necessary. On the contrary, finding averaged quantities for τ w is much easier and readily available in the literature. Additionally, the storage cost alone is advantageous for RL models. • "single model trained from various canonical roughness cases." It is not clear to me how the authors can generalize this model to rough walls without providing some roughness characteristics to the model and also modified truth values. Please provide additional information or equations. We appreciate this observation of the Referee. Indeed in order to address flows with features such as roughness the training of the RL should include flows that exhibit such features. Additional flow features can be addressed by augmenting the state vector with quantities that best reflect the influence of this feature on the flow field. We hope to address this in a future study, as the present paper represents a proof of concept of the methodology, and such extensions of the model will require extensive simulations. 9. Comments on the section "Discussion" • The phrase in the discussion section "present a paradigm shift" is perhaps an overly strong claim. The present approach has not been shown to be more predictive than existing methods and hasn't been tested in complex cases that testing methods have succeeded in. The usefulness of the model has been speculated, but not demonstrated yet. We respectfully note that the word "paradigm shift" refers to the way the wall model is derived. In classical wall modeling, there is an a priori specification of the form of the wall model and the parameters are obtained by comparing with existing results. The present method allows for on-the-fly adaptation of the wall model using current information. Moreover, while past methodologies first obtain the model and then apply it, here the model is adapted while being applied. We believe that this is indeed a "paradigm shift" and unique to the SciMARL methodology that we have introduced. The Referee is correct in observing that we have not demonstrated that our approach is superior to everything else, but we believe that the methodology is new and can lead to major advances by ourselves and others that would adopt and extend this approach. In the revised manuscript we now specify that "paradigm shift" refers to the methodology and stress the two points mentioned above: "This advance will present a paradigm shift in wall model development for LES in the prediction and control of industrial aerodynamics and environmental flows." 10. Comments on the section "Methods-Reinforcement learning": • Please explicitly define the term "episode." We mention in the text "The flow simulations are distributed across workers who collect, for each agent, experiences organized into episodes, E i = {s (i) n , r (i) n , µ (i) n , σ (i) n , a (i) n } n=0,...,N , where n tracks in-episode RL steps, µ and σ are the statistics of the Gaussian policy used to sample a, and t N is the final time step for each episode." • Please provide a reference of clipping "far-policy" experiences as this seems ad hoc. "One way ..." it is unclear if the authors are proposing this strategy or if it has been used previously. We now state that "We circumvent this issue by..." to make sure to convey that this is how we make sure that the policy does not diverge from the distribution of experiences in the RM. We cite Ref. 52 as the reference for clipping far-policy experiences. More details can be found in the publication. 11. Comments on the section "Methods-Overview of the training setup": • Only one resolution is used in x and z. Demonstrating other resolutions would be useful for establishing robustness. Especially since the boundary layer also has very similar resolution. Figure S1: Mean velocity profile of channel flow at Re τ = 10 5 using LLWM. The resolutions in the streamwise and spanwise directions are ∆x ≈ ∆z ≈ 0.05δ (circle), ∆x ≈ 0.1δ and ∆ z ≈ 0.05δ (cross), and ∆x ≈ 0.05δ and ∆z ≈ 0.1δ (triangle). We have conducted two simulations, using the LLWM, where the resolution in x and z, respectively, are coarsened by a factor of 2. The mean velocity profile for Re τ = 10 5 is shown for the original resolution and the two coarsened cases in figure S1. The difference is almost nonexistent, and any difference should due to the SGS model rather than the wall model. This is because the wall model takes only the information at y = h wm . If the SGS model is capable of providing the correct eddy viscosity, the wall model will produce similar results regardless of resolution. Due to the limitation of the journal format (limit on the total number of figures), we cannot include an additional figure, but we add in the manuscript (Testing: Channel flow) that "While only results using ∆ x ≈ ∆ z ≈ 0.05δ are reported here, using different grid resolutions representative of WMLES also produce similar results." • Change "uniformly sampled" to "spatially uniformly sampled" We use "uniformly sampled" to indicate that the Re τ for the simulation is chosen at random, with equal probability, from the set {2000, 4200, 8000}. We feel that "spatially uniformly sampled" is not accurate for this description. • Robustness to numerics is claimed, but this this not tested. There are two claims to robustness. One is robustness to a posteriori testing as opposed to a priori testing of models. This is a big caveat to supervised learning, where the model may perform well for one time-step predictions, but fails to perform well when the errors accumulate in time. Hence, we claim that "The [SciMARL] wall models are robust with respect to the numerical discretizations, as these errors are taken into consideration in the training process" in the introduction. This is a key finding in reinforcement learning in general. The second is that we "expect" robustness to different numerics as only velocity and velocity gradient information away from the wall is required. We mention in the text "The LLWM is easy to extend to complex geometries and flow simulations utilizing different numerical methods or SGS models, as it only takes as states the instantaneous streamwise (or wall-parallel) velocity, its wall-normal gradient, and the distance from the wall. These quantities do not depend heavily on numerics or SGS models, unlike filtered velocities or eddy viscosity values required in the dynamic model." This is also the reason that wall models such as EQWM are robust to numerics. We feel that this does not overstate our findings with the current model. • Why is white noise added to the training data set. Is this to create additional realizations? If so, I don't see how these additional realizations would be more informative than a single realization. Please explain. The Referee is correct-the additional white noise is to create additional realizations. In particular, this was to avoid the learning process from overtraining on a particular initial condition to the point the actions become deterministic. • How did the authors decide on their network topology. Do they have any references or did they conduct a study on the sensitivity to the network architecture? We now include comments regarding the different parameters of SciMARL models in the Methods section: "The most notable hyper-parameters used in our description of the MARL set-up are the spatial resolution for the interpolation of the actions onto the grid (determined by ∆ m x /∆ x , and ∆ m z /∆ z ). The default values ∆ m x /∆ x , and ∆ m z /∆ z reduce the number of experiences generated per simulation to O(10 5 ). This value is similar to the number of experiences generated per simulation used for SciMARL of SGS model development. Consistent with previous studies, we found that further reducing the number of agents per simulation reduced the model's adaptability and therefore exhibit slightly lower performance. Because we use conventional reinforcement learning update rules in a multi-agent setting, single parameter updates are imprecise. We found that ReF-ER with hyper-parameters C = 1.5 and D = 0.05 (Eqs. (6) and (7)) stabilizes training. We ran multiple training runs per reward function and whenever we vary the hyper-parameters, but we observe consistent training progress regardless of the initial random seed." 12. Comments on the sections called "Testing" • Please clarify this sensitivity comment. We have changed the manuscript to: "For example, if y is located at the midpoint of two computational grid points, a central finite difference can be used to compute the wall-normal derivative ∂u * /∂y * . On the other hand, if y is located on the computational grid point, either a left-or right-finite difference is used. In this case, we chose y values that are midpoints of the two computational grid points. Changing the location of y had minor effects on the results, with the wall-shear stress changing ∼ 5% when the location of y was chosen to be on the computational grid point." We hope the text is more clear now. • Perhaps here the authors should cite Kawai and Larsson for using the third point matching location: "third grid point off the wall in the wall-normal direction" We now cite Kawai and Larsson for the use of the third grid point off the wall. I appreciate that the authors have addressed most of my comments, but I believe that the novelty and utility of using reinforcement learning for this problem is overstated. More specifically, the revised manuscript does not provide a convincing argument why SciMARL should be favored over supervised learning or the classical wall modeling approach. Although the authors have acknowledged this criticism, I believe that this key question remains unresolved. We are grateful for the Referee's critical feedback and detailed suggestions for improving the paper. In the revised manuscript we have addressed all the comments of the Referee and made modifications to further clarify the value and originality of this contribution. We respectfully maintain that the present method provides a major contribution to the development of wall models for turbulent flows. Moreover, by addressing this challenging problem, we believe that we provide evidence for a novel, and in our opinion potent, way of combining numerical methods and learning algorithms to develop closures for under-resolved equations of complex systems. In the following, we list the comments of the Referee in red (using the numbering of the Referee bullets) followed by our answers. The relevant modifications in the manuscript are also noted in red. 2. I maintain that the author's statement "SciMARL does not rely on a priori knowledge" is misleading. Training the model relies on knowledge of the exact wall shear stress. Even in a canonical boundary layer, this requires a high fidelity simulation (DNS) as a-priori knowledge that is used to define the reward for the SciMARL method. The quotation and similar statements throughout the manuscript should be removed. Surrounding discussions should be revised to not claim priority of SciMARL over other methods that also require a-priori knowledge. The Referee is right that RL relies on a-priori knowledge. However we wish to emphasize the very limited amount of information that is necessary for its training. For example sciMARL requires only global state information (such as total drag) and not detailed spatio-temporal information. The limited data is a key feature of SciMARL as a semisupervised learning algorithm in contrast to supervised learning algorithms. We have modified the text to "SciMARL is a semi-supervised learning algorithm that requires information about the flow formulated in terms of a reward rather than detailed spatiotemporal data as in the case of supervised learning. In the case of wall 1 modeling, we emphasize that SciMARL does not rely on a priori knowledge of the loglaw coefficients but rather aims to discover active closure policies according to patterns in the flow physics captured by the filtered equations." 6. I don't accept the author's claims of the advantages of SciMARL over the classical EQWM. By providing the exact shear stress data in the training of SciMARL, the method learns the appropriate values of kappa and B. Similarly, the classical EQWM is provided with empirical values of kappa and B. So both rely on empirical information that is case specific (insofar as kappa and B vary in non-canonical cases). The EQWM and SciMARL will both need to be retrained in flows where the optimal values of kappa and B vary. We respectfully disagree. In fact the comment of the Referee points exactly to the difference between SciMARL and EQWM: SciMARL learns effective representations of the κ and B coefficients while the EQWM is given the values of the κ and B coefficients. In addition, the SciMARL model neither knows a-priori nor learns the coefficient directly. Instead SciMARL learns how to control the wall shear stress based on the instantaneous velocity and velocity gradient measurements from the LES itself. This will be beneficial in cases such as adverse pressure gradient flow, where the resulting adverse pressure gradient due to the geometry is not known a priori. Thus, the model cannot be tuned in advance to capture the correct flow-the SciMARL model would be capable of adapting within the simulation, while the EQWM cannot. Moreover, obtaining the κ and B for non-canonical flows is not a straightforward process, whereas obtaining the mean wall-shear stress for a particular case is an easier task. The consensus on the values of κ and B for the flat plate case was only reached after decades of high Reynolds number experiments on this geometry. For adverse-pressuregradient cases, this will require high-Reynolds number simulations and experiments over various values of adverse-pressure gradients, which may not be feasible in the near future. a. The authors correctly comment that SciMARL is a framework that could be provided with the roughness or pressure gradient to be applied to complex flows. Meanwhile, the law of the wall (or EQWM) has already been extended to depend on roughness parameters (see the review by J. Jimenez "Turbulent flows over rough walls" ARFM). The authors should acknowledge that both SciMARL and the EQWM depend on empirical training data and both can be retrained in complex settings by supplying additional data. We agree with the Referee that SciMARL requires appropriate training in order to generalise to various flows. In fact SciMARK requires physical insight to facilitate its training and does not promise to work without having physical knowledge about turbulent flows. In order to extend SciMARL to other flows physical insight is necessary for the appropriate choice of states and actions that SciMARL entails. In that sense SciMARL provides a new and more flexible set of tools for developing wall models when compared to the EQWM. We iterate that EQWM requires apriori knowledge of the appropriate κ and B while SciMARL does not. However, here we do not argue that EQWM will not be successfully formulated and extended to various flows. Indeed there are several excellent researchers that are working in this directions as discussed in the ARFM article of Jimenez. What we propose is an alternative to the EQWM. b. In the rebuttal, the authors argue that the exact shear stress is not external information since the channel is driven by a constant pressure gradient, so the exact shear stress is available from the pressure gradient. i. Firstly, this is only true in a channel, so this argument does not generalize to other flows, so my criticism stands in general flows. ii. Secondly, I'm not convinced that the channel is actually run with an applied constant pressure gradient. In such a setting, the mean wall stress is determined by the applied pressure gradient, so mean wall stress will be correct regardless of the wall model (by force balance). How then, can providing SciMARL with the exact wall stress be any different than providing it with the mean modeled stress? By this argument, the reward (equation 8) incentivizes a policy that minimizes wall stress fluctuations. What is the physical motivation for such a reward? The Referee is correct. We made an error in the previous rebuttal regarding how the channel flow is driven. The training process was indeed run using a constant mass flow rate, which was obtained from the mean velocity profile of the channel flow. The mean velocity profile for these case is readily available in the literature for low to moderate Reynolds numbers for a wide range of cases (including pressure gradient effects), which is all that is necessary for the training of SciMARL. This provides information for both the mean wall-shear stress and the mass flow rate. The testing process was run with a constant pressure gradient, as we indicated before in our previous response. We hope that the above statements help clarify an issue that appears to be at the core of several questions of the Referee. 8. The authors are comparing their SciMARL approach to a particularly unattractive flavor of supervised learning. They assume that supervised learning will involve storing many instantaneous flow fields to train the model, and thus requires large amounts of data storage. This black-box approach of using the entire instantaneous flow field will probably not be as successful as an alternative approach that assumes the existence of a log law (similar to the authors' SciMARL approach or others in the literature). After making this assumption, the supervised learning approach only depends on the mean velocity profile, which is ubiquitously reported in the literature, and has negligible storage requirements. Using such a profile and assuming a log law, one can fit kappa and B, which is the type of supervised learning that has been traditionally used for this problem (the classical log law). In a more complicated setting, the pressure gradient and wall roughness can also be provided as inputs and a neural network can be used to provide a more complex model for kappa and B. I don't see how the author's method is less costly or more accurate than this physics-informed supervised learning approach. I think it is a false comparison to compare their SciMARL approach to a physics-free black-box approach that uses the whole velocity field and does not make the log law assumption. We report on published supervising learning approaches that use many instantaneous flow fields to develop data for training their models. We do not disagree that what the Referee suggest may be a viable alternative and we will be happy to provide references if they are available. At the same time we wish to iterate that even in that case, supervised learning will require information about the parameters of the law of the wall. On the other hand, the wall model discovered through SciMARL, by including additional states, can adapt to flow conditions without specifying what the log law for that particular flow should be. The EQWM and the suggested supervised learning of the EQWM behavior still need this external information (for example, what type of log-law to follow at a particular location) and need to be re-calibrated based on the flow, which may not be feasible in complex settings. Reviewer #2 (Remarks to the Author):I appreciate that the authors have addressed most of my comments, but I believe that the novelty and utility of using reinforcement learning for this problem is overstated. More specifically, the revised manuscript does not provide a convincing argument why SciMARL should be favored over supervised learning or the classical wall modeling approach. Although the authors have acknowledged this criticism, I believe that this key question remains unresolved. 1. This comment is postponed for address in subsequent remarks. 2. I maintain that the author's statement "SciMARL does not rely on a priori knowledge" is misleading. Training the model relies on knowledge of the exact wall shear stress. Even in a canonical boundary layer, this requires a high fidelity simulation (DNS) as a priori knowledge that is used to define the reward for the SciMARL method. The quotation and similar statements throughout the manuscript should be removed. Surrounding discussions should be revised to not claim priority of SciMARL over other methods that also require a priori knowledge. 3. This comment has been resolved. 4. This comment has been resolved. 5. This comment has been resolved. 6. I don't accept the author's claims of the advantages of SciMARL over the classical EQWM. By providing the exact shear stress data in the training of SciMARL, the method learns the appropriate values of kappa and B. Similarly, the classical EQWM is provided with empirical values of kappa and B. So both rely on empirical information that is case specific (insofar as kappa and B vary in noncanonical cases). The EQWM and SciMARL will both need to be retrained in flows where the optimal values of kappa and B vary. a. The authors correctly comment that SciMARL is a framework that could be provided with the roughness or pressure gradient to be applied to complex flows. Meanwhile, the law of the wall (or EQWM) has already been extended to depend on roughness parameters (see the review by J. Jimenez "Turbulent flows over rough walls" ARFM). The authors should acknowledge that both SciMARL and the EQWM depend on empirical training data and both can be retrained in complex settings by supplying additional data. b. In the rebuttal, the authors argue that the exact shear stress is not external information since the channel is driven by a constant pressure gradient, so the exact shear stress is available from the pressure gradient. i. Firstly, this is only true in a channel, so this argument does not generalize to other flows, so my criticism stands in general flows. ii. Secondly, I'm not convinced that the channel is actually run with an applied constant pressure gradient. In such a setting, the mean wall stress is determined by the applied pressure gradient, so mean wall stress will be correct regardless of the wall model (by force balance). How then, can providing SciMARL with the exact wall stress be any different than providing it with the mean modeled stress? By this argument, the reward (equation 8) incentivizes a policy that minimizes wall stress fluctuations. What is the physical motivation for such a reward? 7. This comment has been resolved. 8. The authors are comparing their SciMARL approach to a particularly unattractive flavor of supervised learning. They assume that supervised learning will involve storing many instantaneous flow fields to train the model, and thus requires large amounts of data storage. This black-box approach of using the entire instantaneous flow field will probably not be as successful as an alternative approach that assumes the existence of a log law (similar to the authors' SciMARL approach or others in the literature). After making this assumption, the supervised learning approach only depends on the mean velocity profile, which is ubiquitously reported in the literature, and has negligible storage requirements. Using such a profile and assuming a log law, one can fit kappa and B, which is the type of supervised learning that has been traditionally used for this problem (the classical log law). In a more complicated setting, the pressure gradient and wall roughness can also be provided as inputs and a neural network can be used to provide a more complex model for kappa and B. I don't see how the author's method is less costly or more accurate than this physics-informed supervised learning approach. I think it is a false comparison to compare their SciMARL approach to a physics-free black-box approach that uses the whole velocity field and does not make the log law assumption. 9. This comment has been resolved. 10. This comment has been resolved. 11. This comment has been resolved. 12. This comment has been resolved.Response to Referee #2H. J. Bae & P. Koumoutsakos December 13, 2021 All of my comments have been addressed. I believe that the revised manuscript clearly represents its novel contribution to this important field of study and that the methods and results are clearly and accurately presented and interpreted. Thank you for the detailed responseThank you for the detailed response. All of my comments have been addressed. I believe that the revised manuscript clearly represents its novel contribution to this important field of study and that the methods and results are clearly and accurately presented and interpreted.	[]
[ "Self-gravitation of massive charge and the Einstein-Maxwell electron radius", "Self-gravitation of massive charge and the Einstein-Maxwell electron radius", "Self-gravitation of massive charge and the Einstein-Maxwell electron radius", "Self-gravitation of massive charge and the Einstein-Maxwell electron radius" ]	[ "H Dekker [email protected] \nInstitute for Theoretical Physics\nPrivate Institute for Advanced Study\nUniversity of Amsterdam Science\nPark 9041098 XHAmsterdamThe Netherlands\n\nRésidence Le Jardin Lijnbaansgracht 209L\n1016 XAAmsterdamThe Netherlands\n", "H Dekker [email protected] \nInstitute for Theoretical Physics\nPrivate Institute for Advanced Study\nUniversity of Amsterdam Science\nPark 9041098 XHAmsterdamThe Netherlands\n\nRésidence Le Jardin Lijnbaansgracht 209L\n1016 XAAmsterdamThe Netherlands\n" ]	[ "Institute for Theoretical Physics\nPrivate Institute for Advanced Study\nUniversity of Amsterdam Science\nPark 9041098 XHAmsterdamThe Netherlands", "Résidence Le Jardin Lijnbaansgracht 209L\n1016 XAAmsterdamThe Netherlands", "Institute for Theoretical Physics\nPrivate Institute for Advanced Study\nUniversity of Amsterdam Science\nPark 9041098 XHAmsterdamThe Netherlands", "Résidence Le Jardin Lijnbaansgracht 209L\n1016 XAAmsterdamThe Netherlands" ]	[]	The existence of stable, charged elementary 'point particles' still is a basically unsolved puzzle in theoretical physics. E.g., in quantum electrodynamics the infinite self-energy of the Dirac point electron is 'swept under the carpet' by renormalizing its mass. The present work takes a fresh look at the problem by including gravity-without resorting to string theory. Using Einstein's equations for the gravitational fields in a general static isotropic metric with the full energy-momentum tensor (for the charged material mass and the electromagnetic fields) as the source term, an exact solution with a well-defined characteristic radius emerges where mass and charge accumulate: rc= rero/2-with re=e 2 /4πǫomc 2 ≈10 −15 m being the 'classical' electron radius and where ro=2mG/c 2 ≈10 −57 m is the Schwarzschild radius belonging to the observable mass m≈10 −30 kg. The novel 'Einstein-Maxwell' gravitational electron radius can also be written as rc=ℓP √ αe, where ℓP= hG/c 3 ≈10 −35 m is the fundamental Planck length and αe=e 2 /4πǫohc≈1/137 the fine-structure constant, which yields r electron c =1.38063×10 −36 m. The implied absence of infinite mass renormalization opens up new perspectives for unifying non-gravitational quantum theory and non-renormalizable general relativity.	10.24297/jap.v14i2.7596	[ "https://arxiv.org/pdf/1408.4796v3.pdf" ]	118,656,129	1408.4796	7a12a1263383ddf103b18e494e64ebdc9ec7df2d	Self-gravitation of massive charge and the Einstein-Maxwell electron radius 20 Aug 2014 August 22, 2014 H Dekker [email protected] Institute for Theoretical Physics Private Institute for Advanced Study University of Amsterdam Science Park 9041098 XHAmsterdamThe Netherlands Résidence Le Jardin Lijnbaansgracht 209L 1016 XAAmsterdamThe Netherlands Self-gravitation of massive charge and the Einstein-Maxwell electron radius 20 Aug 2014 August 22, 2014arXiv:1408.4796v1 [physics.gen-ph].Cv, 04.20.-q, 04.40.Nr, 11.10.Gh The existence of stable, charged elementary 'point particles' still is a basically unsolved puzzle in theoretical physics. E.g., in quantum electrodynamics the infinite self-energy of the Dirac point electron is 'swept under the carpet' by renormalizing its mass. The present work takes a fresh look at the problem by including gravity-without resorting to string theory. Using Einstein's equations for the gravitational fields in a general static isotropic metric with the full energy-momentum tensor (for the charged material mass and the electromagnetic fields) as the source term, an exact solution with a well-defined characteristic radius emerges where mass and charge accumulate: rc= rero/2-with re=e 2 /4πǫomc 2 ≈10 −15 m being the 'classical' electron radius and where ro=2mG/c 2 ≈10 −57 m is the Schwarzschild radius belonging to the observable mass m≈10 −30 kg. The novel 'Einstein-Maxwell' gravitational electron radius can also be written as rc=ℓP √ αe, where ℓP= hG/c 3 ≈10 −35 m is the fundamental Planck length and αe=e 2 /4πǫohc≈1/137 the fine-structure constant, which yields r electron c =1.38063×10 −36 m. The implied absence of infinite mass renormalization opens up new perspectives for unifying non-gravitational quantum theory and non-renormalizable general relativity. Introduction Modern theoretical physics is essentially based on the existence of a finite set of elementary 'point particles'-leptons and quarks-and their electromagnetic, gravitational, and weak or strong interactions (see, e.g., Refs. [1][2][3][4]). Apart from the neutrino's, all fundamental particles carry an electric charge. However, the very concept of a stable 'point charge'-such as the electron-is an old and as yet basically unsolved problem. Namely, why should it be possible to accumulate a finite amount of electric charge in an infinitely small volume? What internal force does the work against the repulsive self-interaction? In fact, without such a force the charged particle should immediately explode. Historically, the problems with a point charge were already recognized in classical physics (see, e.g., Refs. [4][5][6][7][8][9][10][11][12]). For instance, in Poincaré's 'electron model' [6] the electric force on the charged sphere was counteracted by an elastic force of unspecified, non-electromagnetic and non-gravitational nature in order to define a total energy-momentum tensor T αβ satisfying the condition ∂T αβ /∂x β =0 characteristic of a closed system (see, e.g., Ref. [12], Ch. 7.3). And even more than a decade after the advent of general relativity, during a visit to Leiden University in 1925, Einstein asked Lorentz' opinion on a purely electromagnetic model for the electron-i.e., without gravitational forces. Lorentz rejected the idea (Ref. [11], Letter 398). In any case, to quote from Feynman's Lectures (Ref. [9] Vol. I, p. 32-4): "The classical electron radius r e =e 2 /4πǫ o mc 2 ≈10 −15 m no longer has the significance that we believe that the electron really has such a radius". More recently, based on state-of-the-art precision measurements of the electron's gyromagnetic g-factor (and using a simple 'electron model' due to Brodsky and Drell [13]), 1989 Nobel laureate H.G. Dehmelt [14] (see also, e.g., Refs. [15,16]) has pointed out that "Today everybody 'knows' the electron is a Dirac point particle with radius r * =0 and g=2. But is it? The value r * ≈10 −22 m given here constitutes an important new upper limit. [...] Thus, the electron may have size." Nevertheless, in non-gravitational quantum theory the electron can be treated successfully as a structureless point particle-at least, if the problem of its infinite self-energy is being 'swept under the carpet'. Indeed, as is well-known from the very beginning of quantum electrodynamics (see, e.g, Ref. [17]), this 'success' is only possible at a price. Namely, handling the infinite self-energy of a point charge requires an infinite mass renormalization to yield results in terms of the observed mass m (see, e.g., Refs. [2,3,[17][18][19][20][21][22]). Unfortunately, this fundamentally hampers the unification with quantum gravity-as the latter has been found to be non-renormalizable. For instance, to quote from Ref. [2], p. 568: " [...] Even pure quantum gravity is non-renormalizable in four dimensions. In fact, it is commonly believed that the theory remains non-renormalizable, a property which would indicate the breakdown of local quantum theory at Planck's length scale ℓ P = hG/c 3 ≈10 −35 m." It is further worth noticing that modern string theory has been proposed-and is still under development-inter alia to cope with the problem of point-like particles, replacing them by tiny one-dimensional loops of Planck size (see, e.g., Ref. [3,4]). The present work takes a fresh look at the problem by including gravitywithout resorting to string theory. Namely, the enormous amount of electrostatic energy compressed into an infinitely small volume must-according to Einstein's general theory of relativity-give rise to huge local gravitational ef-fects. Therefore, in this article the gravitational field equations for the Ricci tensor R µν are studied for a classical self-gravitating massive charge, i.e., with the full energy-momentum tensor T µν for the material mass and the electromagnetic fields as the source of gravitational energy-in a static isotropic metric (Sec. 2 and 3). The analysis involves an 'electrostatic equilibrium' condition (Sec. 4), and rigorously yields a well-defined massive-charge distribution with (for all charged leptons) a characteristic size r c ≈10 −36 m (Sec. 5 and 6). The gravitational field equations Throughout the present article, it is chosen to keep both the speed of light c and the gravitational constant G explicitly in the formula-rather than using 'geometrized units' where c=G=1. The notation closely follows that of Weinberg's book [1]. The field equations of general relativity may then be written as R µν = − 8π c 4 G µν , (2.1) with G µν = G T µν − 1 2 g µν T λ λ , (2.2) where R µν is the Ricci tensor, T µν =g µλ g νκ T λκ the covariant energy-momentum tensor, g µν the metric tensor, and T λ λ =g µν T µν (with g µν g νλ =δ µ λ ). In mixed components (see, e.g., Sec. 4), Eq. (2.1) becomes R µ ν =−(8π/c 4 )G µ ν with R µ ν = g µλ R λν and G µ ν =G(T µ ν − 1 2 δ µ ν T λ λ ). The covariant tensor g µν defines the Riemannian space-time geometry by means of the proper time dτ , such that dτ 2 = −c −2 g µν dx µ dx ν , (2.3) where, in this paper, time will be labelled by µ=0. For the problem of a massive charge, the energy-momentum tensor consists of two contributions, viz., for the-electrically charged-matter and the electromagnetic field itself. In the standard 'ideal fluid' form (without internal pressure), the energy-momentum tensor for the mass reads T µν m = ρ m dx µ dτ dx ν dτ , (2.4) where ρ m =−c −2 T λ m λ is the proper mass density. In the Dirac 'dust' representation one has ρ m = g −1/2 n m n δ 4 [x − x n (τ )]dτ,(2.5) where g= −g µν is the determinant of the metric tensor, so that the mass M = n m n is given by M = γ 1/2 ρ m d 3 x,(2.6) where γ=−g/g oo is the determinant of the three-dimensional metric tensor γ ij =(g ij −g oi g oj /g oo )-see, e.g., Refs. [12,23]. The system is closed (so that T µν total;ν =0) by including the electromagnetic field energy-momentum, given by T µν em = ǫ o F µ λ F νλ − 1 4 g µν F λκ F λκ , (2.7) where the antisymmetric electromagnetic field tensor F µν is defined by F oi =E i for the electric field and F ij /c =ε ijk B k for the magnetic field, ε ijk representing the usual three-dimensional Levi-Civita symbol and ǫ o being the vacuum permittivity. It is useful to note that, since g µν F µ λ =F νλ and g µν g µν =4, Eq. (2.7) implies that T λ emλ =g µν T µν em =0. The electromagnetic fields satisfy Maxwell's equations ∂ ∂x ν g 1/2 F µν = g 1/2 J µ em /ǫ o ,(2.8) where J µ em =c −1 ρ e dx µ /dτ is the current four-vector-with ρ e being the proper charge density, which in the Dirac 'dust' representation reads ρ e = g −1/2 n e n δ 4 [x − x n (τ )]dτ. (2.9) Since the current satisfies the conservation law J µ em;µ =0, the charge Q= n e n is conserved and given by Q = γ 1/2 ρ e d 3 x. (2.10) Note that both ρ e and ρ m transform like a scalar. The static isotropic case The static isotropic metric may be written in the 'standard' form in spherical coordinates r, θ, ϕ, so that the only nonvanishing components of the metric tensor are g oo = −B(r), g rr = A(r), g θθ = r 2 , g ϕϕ = r 2 sin 2 θ, and g=A(r)B(r)r 4 sin 2 θ. The only nonzero components of the Ricci tensor are (with A ′ =dA/dr, etc.) R oo = − B ′′ 2A + B ′ 4A A ′ A + B ′ B − B ′ rA , (3.2) R rr = B ′′ 2B − B ′ 4B A ′ A + B ′ B − A ′ rA , (3.3) R θθ = −1 − r 2A A ′ A − B ′ B + 1 A ,(3.d dr r 2 √ ABE r = r 2 √ A ρ e ǫ o (3.7) for the only nonzero component E r =F or of the electric field, while for the contributions from Eq. (2.7) one obtains G em oo = ǫ o 2 GAB 2 E 2 r , G em rr = − ǫ o 2 GA 2 BE 2 r , G em θθ = ǫ o 2 GABr 2 E 2 r ,(3.8) while G em ϕϕ =G em θθ sin 2 θ. Note that G λ emλ =g µν G em µν =0, as it should be. The equilibrium condition Einstein's gravitational field equations (2.1) for the static isotropic massivecharge system may thus be written as R oo B = − 4π c 2 Gρ m − 4πǫ o c 4 GABE 2 r , (4.1) R rr A = − 4π c 2 Gρ m + 4πǫ o c 4 GABE 2 r , (4.2) R θθ r 2 = − 4π c 2 Gρ m − 4πǫ o c 4 GABE 2 r . (4.3) Note that R oo /B=−R o o , R rr /A=R r r , and R θθ /r 2 =R θ θ . To find the equilibrium equation, first consider R rr /A+R oo /B. This leads to 1 A A ′ A + B ′ B = 8π c 2 Grρ m . (4.4) Using the R θθ -equation (4.3) to eliminate A ′ , one gets r A B ′ B + 1 A = 1 − 4πǫ o c 4 GABr 2 E 2 r . (4.5) Now differentiating Eq. (4.5) with respect to r and using R rr /A−R oo /B to eliminate B ′′ , one obtains rB ′ 2AB A ′ A + B ′ B = 8πǫ o c 4 GABrE 2 r + 4πǫ o c 4 G d dr ABr 2 E 2 r . (4.6) Once more invoking Eq. (4.4), the result reads B ′ B = 2ABrE 2 r + d(ABr 2 E 2 r )/dr ρ m c 2 r 2 /ǫ o , (4.7) which-putting E=r 2 √ ABE r and noticing the identity d(E 2 /r 2 )/dr=−2E 2 /r 3 + 2(E/r 2 )dE/dr-by invoking the Poisson equation (3.7) can be rewritten as B ′ B = 2A √ BE r ρ e ρ m c 2 , (4.8) which represents the balancing of the repulsive electrostatic self-force and the attractive gravitational self-force. It is the electrostatic counterpart of the usual 'hydrostatic equilibrium' condition for ideal fluids. In fact, by virtue of the Bianchi identities, Eq. (4.8) is a direct consequence of the conservation law T rν total,ν =0. Namely, one gets T rν m;ν =Γ r oo T oo m =Γ r oo ρ m c 2 /B with Γ r oo =B ′ /2A, while by means of the Poisson equation (3.7) one obtains T rν em;ν = √ BE r ρ e . For a structureless massive charge the intrinsic charge-to-mass ratio e n /m n should be an n-independent constant, i.e., e n =km n , which according to Eqs. (2.5) and (2.9) implies ρ e =kρ m for the proper density. Without loss of generality, one may put k=Q/M so that-by virtue of Eq. (2.10) for Q-the as yet undetermined mass M satisfies Eq. (2.6). Hence, the equilibrium equation (4.8) may be rewritten as AE r = µ 2 B ′ B 3/2 ,(4.9) with µ=M c 2 /Q. Now using the Newtonian limit B=1−2mG/rc 2 and the Poisson limit AE r =Q/4πǫ o r 2 for r→∞, one obtains m = Q 2 4πǫ o GM . (4.10) By Eq. (4.9) the problem of the massive charge is reduced to finding the temporal metric function B(r). The temporal metric function Consider Eq. (4.5) for A, and using Eq. (4.9) for E r write it as A = 1 + β + Mβ 2 ,(5.1) where β=rB ′ /B and M=πǫ o Gµ 2 /c 4 . Now again take R rr /A−R oo /B, collect the A ′ terms and once more use Eq. (4.9) for E r . This leads to rB ′′ B + 1 − 1 2 β B ′ B − 1 + 1 2 β A ′ A = 2Mβ B ′ B . (5.2) Substituting A and A ′ from Eq. (5.1), and combining similar terms-some of which add up to zero-Eq. (5.2) is found to factorize such that for M= 1 4 it becomes a trivial zero identity (see Appendix A and, e.g., Ref. [24]) while for M = 1 4 it either leads to the trivial solution B ′ =0 or to a nontrivial temporal metric function B(r) satisfying 1 2 rB ′′ + 1 + 1 2 Mβ 2 B ′ = 0, (5.3) which is akin to the prototype equation dβ/dt=β 3 for the temporal evolution of so-called 'finite-time blow-up' processes in, e.g., chemistry and hydrodynamic turbulence (see, e.g., Ref. [25], p. 353). Noticing that Bβ ′ =rB ′′ +(1−β)B ′ , using Eq. Results The temporal metric function now follows from dB/dζ=B/rζζ ′ . Using Eq. while A(r<r c )=1. For finite r/r c and M→∞, one gets 1/A=1−(r c /r) 2 . Next, the radial electrostatic field is obtained from Eq. (4.9), which yields E r = µ 2r √ B ζ M + ζ + ζ 2 Θ (r − r c ) ,(6.3) where Θ(r) is the unit step function. Fig. 2 shows the exact solution as E r (r)/E c , with E c =µ/2r c √ M. For finite values of r/r c and with M→∞, this function becomes E r /E c =(r c /r) 2 1−(r c /r) 2 , which has its peak value 2 9 √ 3≈0.38 at r/r c = 1 2 √ 6≈1.22. One further gets r 2 √ ABE r = µ 2 r M + ζ + ζ 2 Θ (r − r c ) . (6.4) Finally, the Poisson equation (3.7) leads to r 2 √ Aρ e = µǫ o r c 2 √ M δ(r − r c ) − µǫ o 4ζ Θ(r − r c ) M + ζ + ζ 2 ,(6.5) where δ(r) is the Dirac distribution and which rigorously satisfies Eq. (2.10) for the total charge Q. The singular part of the density follows from the Poisson equation rather than from the gravitational field equations per se. Namely, from Eq. (6.5) one has ρ sing e =(µǫ o /2Mr c ) ζ(r) δ(r−r c ) so that by virtue of ζ(r c )=0 one has T sing oo m =0. Of course, the continuous part of the density ρ cont e =−µǫ o /4r 2 ζ 2 A obeys the Einstein equations for r>r c (for all values of M). E.g., consider Eq. (4.4) and note that its right-hand side amounts to −2M/rζ 2 A, so that it remains to show that r(A ′ /A+B ′ /B)=−2M/ζ 2 -which is easily done since rA ′ /A=ζdA/dζ=−1/ζ−2M/ζ 2 [by Eqs. (5.4) and (6.2)] and rB ′ /B=1/ζ by definition. quantum field theory with non-renormalizable quantum gravity, a formidable task for the future which obviously goes well beyond the scope of the present paper. ϕϕ =R θθ sin 2 θ, and the only nonzero component of the material energymomentum tensor now reads T oo m = ρ m c 2 m ϕϕ =G m θθ sin 2 θ. For the metric (3.1), Eq. (2.8) leads to the Poisson equation (5.3) for B ′′ and defining ζ=1/β, one thus rigorously obtains rζζ ′ = M + ζ + ζ 2 . integration constant r o is set by the Newtonian limit. For r→∞, one has ζ→∞ as well. On the other hand, ζ→0 for r→r c . Namely, expanding Eq. (5.5) in powers of ζ yields ζ 2 /2M≈ln(r/r c ), with r c = r o electron using µ=M c 2 /e in the definition of M in Eq. (5.1) and invoking Eq. (4.10), one obtains M=M/4m=r e /2r o so that r c ≈ r o r e /2. With r e ≈10 −15 m and r o ≈10 −57 m, one obtains r electron c ≈10 −36 m-which may be called the 'Einstein-Maxwell' gravitational electron radius. (r≤r c )=B c . For finite values of r/r c and M→∞, the function B(r)= (B−B c )/(1−B c ) becomes B=(2/π)arccos(r c /r). Fig. 1 shows the exact solution B(r) of Eq. (5.3) for a few values of M. Further, using Eq. (5.5) for ζ, Eq. (6.1) can also be written as B(r)=(r o /r) 2 (M+ζ+ζ 2 ). Similarly, the radial metric function A(r) follows from Eq. Figure 1 : 1The temporal metric function B(r) [for M=2 (bottom line), M=20 (middle), M=200 (top)]. Shown is the solution of the exact Eq. (5.3), as a function of the non-dimensional radial variable r/rc. Figure 2 : 2The radial electrostatic field Er(r) [for M=2 (bottom line), M=20 (middle), M=200 (top)]. Shown is Er/Ec [with Ec=µ/2Mro], using the solution of the exact Eq. (5.3), as a function of the non-dimensional radial variable r/rc. AcknowledgementI am indebted to J.J.J. Kokkedee[27], at the time professor of theoretical physics in Delft, The Netherlands, who taught me the general theory of relativity during an exiting lecture course at Delft University of Technology, in 1974.Final remarksThe massive-charge distribution (6.5) is an exact particle-like solution of the classical Einstein-Maxwell equations (for all values of the parameter M). It emerges from a rigorous balance between electrostatic self-repulsion and gravitational self-attraction. For a 'point charge'-like, e.g., the electron-the electrostatic self-interaction is the well-known source of an infinite self-energy, and it requires the inclusion of gravity to establish equilibrium at the new finite size r c = r e r o /2≈10 −36 m (where r e ≈10 −15 m is the 'classical' electron radius and r o ≈10 −57 m is the electron Schwarzschild radius).The characteristic radius r c is mass independent (i.e., identical for all charged leptons). Namely, one has r c =\|e\| G/4πǫ o c 4 -which is also worth noticing to follow at once from equating the electromagnetic mass m charge c =e 2 /8πǫ o r c c 2 at r=r c to the Schwarzschild gravitational mass m grav c =r c c 2 /2G. Hence, apart from involving the total charge, the novel 'Einstein-Maxwell' radius only depends on the fundamental constants c and G. It is readily written aswhere ℓ P = hG/c 3 ≈10 −35 m is the Planck length (see, e.g., Refs.[2,10]) and α e =e 2 /4πǫ oh c≈1/137 is the fine-structure constant. The resulting numerical value (see, e.g., Ref.[26]) is: r electron c =1.38063×10 −36 m. Since the radius (7.1) is about an order of magnitude smaller than ℓ P , the electron is indeed a point-like charge from the perspective of non-gravitational theory. However, while tiny it is large enough to produce only a relatively small quantum mechanical self-energy δm. For instance, for the free electron (see, e.g., Ref.[19], p. 394) one now qualitatively gets δm/m ≈(3α e /2π)ln(ℓ C /r c )≈0.2 (where ℓ C =h/mc≈10 −13 m is the Compton wavelength)-which removes the infinite mass renormalization from quantum electrodynamics (see also, e.g., Ref.[21,22]) and thus opens up new perspectives for unifying non-gravitational Finally, notice that by Eq. (4.10) for M= 1 4 the observable classical mass amounts to m=\|Q\|/ √ 4πǫ o G, so that in this case the corresponding Schwarzschild radius r o =2mG/c 2 precisely equals the mass-independent 'Einstein-Maxwell' r c =\|Q\| G/4πǫ o c 4 =ℓ P √ α e of Eq. (7.1). However, one now gets m≈10 −9 kg if Q=e-which is obviously far too heavy for the electron and is unlikely to be repaired by quantum effects (see, e.g., Sec. 7). To account for the correct order of magnitude of the electron mass m≈10 −30 kg one needs M≫1, as shown in the main text. S Weinberg, Gravitation and Cosmology. New YorkWileyS. 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Oxford/New YorkClarendon PressA. Pais, Inward Bound (Clarendon Press, Oxford/New York, 1986). The Scientific Correspondence of. A J Kox, H.A. LorentzSpringerINew YorkA.J. Kox, The Scientific Correspondence of H.A. Lorentz (Springer, New York, 2008) Vol. I. C Møller, The Theory of Relativity. Oxford/DelhiClarendon/Oxford University PressC. Møller, The Theory of Relativity (Clarendon/Oxford University Press, Oxford/Delhi, 1952/1977). . S J Brodsky, S D Drell, Phys. Rev. D. 222236S.J. Brodsky and S.D. Drell, Phys. Rev. D 22 (1980) 2236. H G Dehmelt, High Energy Spin Physics. K. HellerNew York319AIP Conf. Proc. No. 187. see also: NobelPrize.org/ Dehmelt-Lecture.pdfH.G. Dehmelt, in: High Energy Spin Physics, Edited by K. Heller (AIP Conf. Proc. No. 187, New York, 1989) p. 319; see also: NobelPrize.org/ Dehmelt-Lecture.pdf, 1989. . B Odom, D Hanneke, D D'urso, G Gabrielse, Phys. Rev. Lett. 9730801B. Odom, D. Hanneke, D. D'Urso, and G. Gabrielse, Phys. Rev. Lett. 97 030801 (2006). . 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[ "Multilinear algebra in the context of diffeology", "Multilinear algebra in the context of diffeology" ]	[ "Ekaterina Pervova " ]	[]	[]	This note is dedicated to the details of multilinear algebra on diffeological vector spaces; most of them are the to-be-expected corollaries of standard constructions and various facts of diffeology collected elsewhere. Most of the attention is paid to the notion of the diffeological dual and the implications of the resulting absence of isomorphism-by-duality. MSC (2010): 53C15, 15A69 (primary), 57R35, 57R45 (secondary).	null	[ "https://arxiv.org/pdf/1504.08186v3.pdf" ]	119,168,210	1504.08186	9e5db62cce461794eba05ed2b5faf2929ce71e87	Multilinear algebra in the context of diffeology 30 Apr 2015 May 1, 2015 Ekaterina Pervova Multilinear algebra in the context of diffeology 30 Apr 2015 May 1, 2015arXiv:1504.08186v1 [math.DG] This note is dedicated to the details of multilinear algebra on diffeological vector spaces; most of them are the to-be-expected corollaries of standard constructions and various facts of diffeology collected elsewhere. Most of the attention is paid to the notion of the diffeological dual and the implications of the resulting absence of isomorphism-by-duality. MSC (2010): 53C15, 15A69 (primary), 57R35, 57R45 (secondary). Introduction The aim of this work is rather modest; it is intended as a (self-contained) supplement to [3], detailing the basics of multilinear algebra on diffeological vector spaces. Most of what is to be found here is rather simple and easily follows from the ingredients already collected in the excellent and comprehensive source [2]; the main idea is to spell out the definitions and their motivations that stem from the intention to extend the notion of the Riemannian metric to the diffeological context. It is necessary to say at this point that the very notion of diffeological tangent bundle has only just appeared in the very recent [1], as the internal tangent bundle with the dvs diffeology; this seems to be the first notion of a diffeological tangent bundle where each fibre is a diffeological vector space (the property that other notions of a tangent bundle of a diffeological space do not generally enjoy; do note that there are other approaches, some of which are summarized in [1], Section 3.4, and also references therein). The appearance of this new notion makes the attempt to define, by analogy with the notion of a Riemannian metric, a kind of diffeological metric a reasonable one; obviously, in order to carry it out, the main issue is to construct a suitable diffeological bundle such that its smooth sections yield, at each point, a scalar product on the internal tangent space at the point (and include all "reasonable" collections of scalar products on internal tangent bundle...) What this latter request means precisely, is the motivation for the present paper. Indeed, again by analogy with Riemannian metrics, it is natural to look for a bundle whose fibres be covariant 2-tensors on the fibres of the internal tangent space. This requires, as a preliminary, the construction of the proper internal cotangent bundle, that is, with fibres the duals of internal tangent spaces; and at this point, as we will see below, the diffeological setting diverges from the usual smooth one (at least a priori), by virtue of the fact that the usual dual of a vector space may not be the same as the diffeologically smooth dual of a diffeological vector space; other similar phenomena follow. This, in brief, is what calls for putting down the details of multilinear algebra on diffeological vector spaces, spelling out what diffeology the dual space or the tensor product must have... None of this claims to be particularly original (it is all easily assembled from the ingredients already known); perhaps the main point is to make, and explain the choices made, in reference to the case of those diffeological vector spaces that can appear as fibres of a diffeological tangent bundle as opposed to the a priori case of an abstract diffeological vector space, and to provide a suitable selection of examples whenever could appear desirable. Finally, we shall note that a description of a tensor bundle construction for diffeological spaces is available in [6], in a rather concise form from the multilinear algebra point of view. 1 We make suitable references/comparisons as we go along, possibly reminding the notions defined therein. The content Our final aim is to obtain constructions where the result be a diffeological vector space with a "reasonable" diffeology. The measure of such reasonableness is a compatibility with the known properties of (fibres of) internal tangent bundles of diffeological spaces, considered with the dvs diffeology [1]. 2 All constructions come from the "elementary" diffeologies described in detail in [2], Chapter 1, and recalled in Section 1, that are applied to the usual constructions of multilinear algebra. The material thus collected is simple but probably (in part) new (at least I was not able to find a reference where it be spelled out). As already mentioned, some of the definitions were given in [6], Section 2.3. Other considerations that influence the content of the paper are, for many if not all constructions of multilinear algebra there is more than one way to carry them out; each of these ways finds its own reflection in elementary constructions of diffeology. From the linear algebra point of view all these different ways to define, let's say, the tensor product are equivalent; but would the same be true also for diffeologies at which we arrive by these different methods? And if not, what does the difference depend on, and most importantly, which is the most reasonable one to prefer?.. 3 Finally, it is mentioned in the preface to Chapter 3 of [2] that many of the possible connections between the notion of a diffeological space and various kinds of vector spaces that naturally appear in (linear algebra) applications have not yet been explored; by providing whenever possible "characteristic" examples (even if some of them are aprioristic rather than practical), we hope to give a small contribution to filling that void. The structure In the first two sections we collect the necessary notions regarding diffeological structures (in the first section) and diffeological vector spaces (in the second one). The other three sections are where the main content lies; therein, we describe the typical constructions of multilinear algebra from the diffeological point of view. In particular, we concentrate on the difference between the diffeological dual and the usual dual, and the consequences thereof; and discuss the (im)possibility to carry over into the diffeological setting some of the classic equivalences of multilinear algebra (such as various ways to define the tensor product, etc). Acknowledgments I came across the notion of diffeology only recently (but I do have to thank xxx.lanl.gov for that), but the time I've spent wondering the ways of mathematical research will soon constitute a half of my total lifetime. This might in part explain why I feel this work owes a lot to a colleague of mine who is not a mathematician and who, at the time of these words being written, has no idea that it exists. The name of my colleague is Prof. Riccardo Zucchi; what this work owes to him is inspiration and motivation, first of all those needed to be brave enough as to invest time and effort into something destined to be imperfect, and to do what, correctly or not, seems right or necessary at one particular moment only, independently of what the next step would be. Background on diffeological spaces This section is devoted to a short background on diffeological spaces, introducing the concepts that we will need in what follows. The concept We start by giving the basic definition of a diffeological space, following it with the definition of the standard diffeology on a smooth manifold; it is this latter diffeology that allows for the natural inclusion of smooth manifolds in the framework of diffeological spaces. Definition 1.1. ([5]) A diffeological space is a pair (X, D X ) where X is a set and D X is a specified collection of maps U → X (called plots) for each open set U in R n and for each n ∈ N, such that for all open subsets U ⊆ R n and V ⊆ R m the following three conditions are satisfied: 1. (The covering condition) Every constant map U → X is a plot; 2. (The smooth compatibility condition) If U → X is a plot and V → U is a smooth map (in the usual sense) then the composition V → U → X is also a plot; 3. (The sheaf condition) If U = ∪ i U i is an open cover and U → X is a set map such that each restriction U i → X is a plot then the entire map U → X is a plot as well. Typically, we will simply write X to denote the pair (X, D X ). Such X's are the objects of the category of diffeological spaces; naturally, we shall define next the arrows of the category, that is, say which maps are considered to be smooth in the diffeological sense. The following definition says just that. Definition 1.2. ( [5]) Let X and Y be two diffeological spaces, and let f : X → Y be a set map. We say that f is smooth if for every plot p : U → X of X the composition f • p is a plot of Y . As is natural, we will call an isomorphism in the category of diffeological spaces a diffeomorphism. The typical notation C ∞ (X, Y ) will be used to denote the set of all smooth maps from X to Y . The standard diffeology on a smooth manifold Every smooth manifold M can be canonically considered a diffeological space with the same underlying set, if we take as plots all maps U → M that are smooth in the usual sense. With this diffeology, the smooth (in the usual sense) maps between manifolds coincide with the maps smooth in the diffeological sense. This yields the following result (see Section 4.3 of [2]). Theorem 1.3. There is a fully faithful functor from the category of smooth manifolds to the category of diffeological spaces. Comparing diffeologies Given a set X, the set of all possibile diffeologies on X is partially ordered by inclusion (with respect to which it forms a complete lattice). More precisely, a diffeology D on X is said to be finer than another diffeology D ′ if D ⊂ D ′ (whereas D ′ is said to be coarser than D). Among all diffeologies, there is the finest one, which turns out to be the natural discrete diffeology and which consists of all locally constant maps U → X; and there is also the coarsest one, which consists of all possible maps U → X, for all U ⊆ R n and for all n ∈ N. It is called the coarse diffeology (or indiscrete diffeology by some authors). Generated diffeology and quotient diffeology One notion that will be crucial for us is the notion of a so-called generated diffeology. Specifically, given a set of maps A = {U i → X} i∈I , the diffeology generated by A is the smallest, with respect to inclusion, diffeology on X that contains A. It consists of all maps f : V → X such that there exists an open cover {V j } of V such that f restricted to each V j factors through some element U i → X in A via a smooth map V j → U i . Note that the standard diffeology on a smooth manifold is generated by any smooth atlas on the manifold, and that for any diffeological space X, its diffeology D X is generated by ∪ n∈N C ∞ (R n , X). Note that one useful property of diffeology as concept is that the category of diffeological spaces is closed under taking quotients. To be more precise, let X be a diffeological space, let ∼ = be an equivalence relation on X, and let π : X → Y := X/ ∼ = be the quotient map. The quotient diffeology ([2]) on Y is the diffeology in which p : U → Y is the diffeology in which p : U → Y is a plot if and only if each point in U has a neighbourhood V ⊂ U and a plotp : V → X such that p\| V = π •p. Sub-diffeology and inductions Let X be a diffeological space, and let Y ⊆ X be its subset. The sub-diffeology on Y is the coarsest diffeology on Y making the inclusion map Y ֒→ X smooth. It consists of all maps U → Y such that U → Y ֒→ X is a plot of X. This definition allows also to introduce the following useful term: for two diffeological spaces X, X ′ a smooth map f : X ′ → X is called an induction if it induces a diffeomorphism X → Im(f ), where Im(f ) has the sub-diffeology of X. Pushforward of diffeology For any diffeological space X, any set X ′ , and any map f : X → X ′ there exists a finest diffeology on X ′ that makes the map f smooth. It is this diffeology that is called the pushforward of the diffeology of X by the map f ; it is denoted by f * (D) where D stands for the diffeology of X. The pushforward diffeology can be characterized as follows. A map f ′ : U → X ′ defined on a domain U is a plot of f * (D) if and only if it satisfies the following condition: for every r ∈ U there exists an open neighbourhood V of r such that either f ′ \| V is a constant map or there exists a plot g : V → X such that f ′ \| V = f • g. Pullbacks of diffeologies Let X be a set, let X ′ be a diffeological space with diffeology D ′ , and let f : X → X ′ be a map. The pullback of the diffeology D ′ by the map f is the coarsest diffeology on X such that f is smooth; this pullback diffeology is usually denoted by f * (D ′ ). Note that p : U → X is a plot for f * (D ′ ) if and only if f • p is a plot for D ′ . Subductions Let X and X ′ be two diffeological spaces, and let f : X → X ′ be some map; this map is said to be a subduction if it satisfies the following conditions: 1) it is surjective, and 2) the diffeology D ′ of X ′ is the pushforward of the diffeology D of X, i.e. D ′ = f * (D). An equivalent description of what it means that f be a subduction is, f must be a smooth surjection such that for every plot g ′ : U → X ′ and for every x ∈ U there exist an open neighbourhood V of x and a plot g : V → X such that g ′ \| V = f • g. Sums of diffeological spaces Let {X i } i∈I be a collection of diffeological spaces, with I being some set of indices. The sum, or the disjoint union, of {X i } i∈I is defined as X = i∈I X i = {(i, x) \| i ∈ I and x ∈ X i }. The sum diffeology on X is the finest diffeology such that the natural injections X i → i∈I X i are smooth for each i ∈ I. The plots of this diffeology are maps U → i∈I X i that are locally plots of one of the components of the sum. The diffeological product Let, again, {X i } i∈I be a collection of diffeological spaces, and let D i , i ∈ I, be their respective diffeologies. The product diffeology D on the product X = i∈I X i is the coarsest diffeology such that for each index i ∈ I the natural projection π i : i∈I X i → X i is smooth. Functional diffeology Let X, Y be two diffeological spaces, and let C ∞ (X, Y ) be the set of smooth maps from X to Y . Let ev be the evaluation map, defined by ev : C ∞ (X, Y ) × X → Y and ev(f, x) = f (x). The words "functional diffeology" stand for any diffeology on C ∞ (X, Y ) such that the evaluation map is smooth; note, for example, that the discrete diffeology is a functional diffeology. However, they are typically used, and we also will do that from now on, to denote the coarsest functional diffeology. Diffeological vector spaces In this section we treat with some detail the notion of a diffeological vector space. The concept and some basic constructions Let V be a vector space over R. The vector space diffeology on V is any diffeology of V such that the addition and the scalar multiplication are smooth, that is, [(u, v) → u + v] ∈ C ∞ (V × V, V ) and [(λ, v) → λv] ∈ C ∞ (R × V, V ), where V × V and R × V are equipped with the product diffeology. A diffeological vector space over R is any vector space V over R equipped with a vector space diffeology. In the diffeological context, we find all the usual constructions of linear algebra, such as spaces of (smooth) linear maps, products, subspaces, and quotients; we now describe these. First of all, given two diffeological vector spaces V and W , we can speak of the space of smooth linear maps between them; this space is denoted by L ∞ (V, W ) and is defined simply as: L ∞ (V, W ) = L(V, W ) ∩ C ∞ (V, W ); this is an R-linear subspace of L(V, W ). Next, a subspace of a diffeological vector space V is any vector subspace of V endowed with the sub-diffeology. Finally, if V is a diffeological vector space and W V is a subspace of it then the quotient V /W is a diffeological vector space with respect to the quotient diffeology. Direct product of diffeological vector spaces Let {V i } i∈I be a family of diffeological vector spaces. Consider the usual direct product V = i∈I V i of this family; then V , equipped with the product diffeology, is a diffeological vector space. Euclidean structure on diffeological vector spaces The notion of a Euclidean diffeological vector space does not differ much from the usual notion of the Euclidean vector space. A diffeological space V is Euclidean if it is endowed with a scalar product that is smooth with respect to the diffeology of V and the standard diffeology of R; that is, if there is a fixed map , : V × V → R that has the usual properties of bilinearity, simmetricity, and definite-positiveness and that is smooth with respect to the diffeological product structure on V × V and the standard diffeology on R. Fine diffeology on vector spaces The fine diffeology on a vector space R is the finest vector space diffeology on it; endowed with such, V is called a fine vector space. Note that any linear map between two fine vector spaces is smooth. The fine diffeology admits a more or less explicit description of the following form: its plots are maps f : U → V such that for all x 0 ∈ U there exist an open neighbourhood U 0 of x 0 , a family of smooth maps λ α : U 0 → R, and a family of vectors v α ∈ U 0 , both indexed by the same finite set of indices A, such that f \| U0 sends each x ∈ U 0 into α∈A λ α (x)v α : f (x) = α∈A λ α (x)v α for x ∈ U 0 . A finite family (λ α , v α ) α∈A , with λ ∈ C ∞ (U 0 , R) and v α ∈ V , defined on some domain U 0 and satisfying the condition just stated, is called a local family for the plot f . Some generating sets for the fine diffeology can also be described explicitly. Let L(R n , V ) be the set of all linear maps from R n into V , and let L * (R n , V ) be the set of all injective linear maps from R n into V . The following two families both generate the fine diffeology of V : F = ∪ n∈N L(R n , V ) and F * = ∪ n∈N L * (R n , V ). Example 2.1. It is easy to see that R n with the standard diffeology is a fine vector space. Note, furthermore, that any fine vector space is isomorphic to the standard R n ([2], p. 71). The following statement should be quite important for our discussion (as we will see in the next section): 3.9) Let V be a fine diffeological vector space. Then for any other diffeological vector space W every linear map from V to W is smooth, i.e., Lemma 2.2. ([2],L ∞ (V, W ) = L(V, W ). Finally, regarding the examples of diffeological vector spaces, we have already mentioned an obvious one, which is any R n with the standard diffeology. We also observe a fact of which we will make use shortly, that any vector space endowed with the coarse diffeology is a diffeological vector space. We will consider various other examples, as we go along. Existing notions of the tensor product There have already been treatments of the multilinear algebra in the diffeological context, for instance in [6], Section 2.3, which we briefly recall. Doing this requires the notion of the weak vector space diffeology, which is a generalization given in [6] of the abovementioned fine vector space diffeology. For reasons of terminology, we recall the following definition given in [6]. Definition 2.3. ([6] , Definition 2.2.1) Let V be a vector space, and let D V be a diffeology on its underlying set. The weak vector space diffeology on V generated by D V is the weakest diffeology generated by the collection of maps of form U ∋ u → n i=1 λ i (u)γ i (u), i.e. finite sums with smooth functional coefficients of plots of D V . In other words, it is the finest vector space diffeology on V containing the given one. Definition 2.4. ([6], Definition 2.3.2) Let V 1 , . . . , V n be diffeological vector spaces, let V 1 ⊗ . . . ⊗ V n be their tensor product as vector spaces, and let φ : V 1 × . . . × V n → V 1 ⊗ . . . ⊗ V n be the universal multilinear map. The tensor product diffeology on V 1 ⊗ . . . ⊗ V n is the weak vector space diffeology generated by the diffeology that is the pushforward by φ of the product diffeology on V 1 × . . . × V n . Modificating a bit the notation in [6], we denote this diffeology by D Vi ; it is then claimed ([6], Theorem 2.3.5) that, if F is another diffeological vector space, V 1 ⊗ . . .⊗ V n is endowed with the diffeology D Vi , and V 1 × . . . × V n is endowed with the product diffeology, the space of all smooth linear maps V 1 ⊗ . . . ⊗ V n → F is diffeomorphic to the space of all smooth multilinear maps V 1 × . . . × V n → F , these two spaces being considered each with the corresponding functional diffeology. Smooth linear and bilinear maps In this section we begin our treatment of the multilinear algebra on diffeological vector spaces; the first step is to consider the various possibilities to define the diffeological dual (this is actually done in the section that follows), which immediately imposes that we consider, more generally, the issue of linear maps and smooth linear maps. As we show, replacing the former with the latter makes a priori a significant difference. Linear maps and smooth linear maps The sometimes significant difference between the two notions mentioned in the title is illustrated by the following example; we present it immediately, so as to motivate the reasoning that follows. Example 3.1. Let us see V such that L ∞ (V, R) is a proper subspace of L(V, R). Set V = R n equipped with the coarse diffeology; we claim that the only smooth linear map V → R is the zero map. Indeed, let f : V → R be a linear map; recall that, by definition, for f to be smooth, the composition f • p must be a plot of R for any plot p of V . What this means is that f • p must be a smooth map U → R for some domain U of some R k ; but by definition of the coarse diffeology, p is allowed to be any set map U → V , so it might not even be continuous. Already by this observation it is intuitively clear that we will find numerous plots p such that f • p is not smooth; but let us be precise. Choose some basis {v 1 , . . . , v n } of V = R n , and a basis {v} of R. 4 With respect to these, f is given by n real numbers, more precisely, by the matrix (a 1 . . . a n ). Let us choose n specific plots, that we call p i for i = 1, . . . , n, by setting p i : R → V and p i (x) = \|x\|v i ; then (f • p i )(x) = a i \|x\|v. The only way for this latter map to be smooth is to have a i = 0; recalling again that if f is smooth then the composition f • p i must be smooth for all i, we conclude that we must have a i = 0 for all i = 1, . . . , n, whence our claim. As we have already indicated, one of the main issues (possibly, the main one) that presents itself when one tries carry the (multi)linear algebra over to the diffeological context is the issue of smoothness of linear maps: the fact (that we have already mentioned a few times) is that, for two arbitrary diffeological spaces V , W , it might a priori occur that L ∞ (V, W ) < L(V, W ). True, this would require some rather surprising vector spaces/diffeologies for this happen; but diffeology was designed for dealing with surprising, or at least unusual from the Differential Geometry point of view, objects, 5 and so some "weird examples" should be welcome. In fact, such surprises are easy to find, as we show by elaborating on the Example 3.1; and this we do in a rather obvious fashion. We stress that what comes in play here is the very essence of what diffeology aims to add to the "usual" setting of Differential Geometry; which is the flexibility of what can be called smooth. In particular, the fact that any given map R k ⊇ U → X can be a plot for some diffeology on a given set X (namely, for the diffeology generated by this map plus, for example, some already existing diffeology on X) easily gives rise to some surprising instances of, say, diffeological vector spaces, as we have already seen in the Example 3.1 and as we discuss in more detail below. Example 3.2. Once again, consider V = R n and some basis {v 1 , . . . , v n } of V ; endow it with the vector space diffeology generated by all smooth maps plus the map p i already mentioned, that is, the map p i : R → V acting by p i (x) = \|x\|v i . Let v be a generator of R ( i.e., any non-zero vector). Using the same reasoning as in Example 3.1, one can show that if f : V → R is linear and, with respect to the bases chosen has matrix (a 1 . . . a n ), then for it to be smooth we must have a i = 0; hence the (usual vector space) dimension L ∞ (V, R) ( i.e., that of the diffeological dual of V ) is at most n − 1. 6 This reasoning can be further extended by choosing some natural number 1 < k < n and a set of k indices 1 i 1 < i 2 < . . . < i k n, and endowing V with the vector space diffeology generated by all smooth maps plus the set {p i1 , . . . , p i k }. Arguing as above, we can easily conclude that dim( L ∞ (V, R)) is at most n − k. The examples just cited 7 show that the diffeological dual can be much different from the usual one. Given the importance of the isomorphism-by-duality in the usual multilinear algebra, the implications of this difference call for some care; which is why below we try to explore them as thoroughly as possible. Bilinear maps and smooth bilinear maps In this section we consider the question analogous to the one considered for linear maps in the previous section: given two diffeological vector spaces, what is the difference between the set of all bilinear maps on one of them with values in the other, and the set of all such bilinear maps that in addition are smooth? Smooth bilinear maps Let V , W be two diffeological spaces. From the linear algebra point of view, a W -valued bilinear map can be interpreted in two ways. One is the straightforward definition of it as a map V × V → W linear in each argument. In the diffeological context we restrict ourselves to the maps that are smooth (with respect to the product diffeology on V × V ), thus facing again the possibility that the set of smooth maps is strictly smaller than that of bilinear maps. Let us first fix some notation. Given V , W two diffeological vector spaces, let B(V, W ) be the set of bilinear maps on V with values in W , and let B ∞ (V, W ) be the set of those bilinear maps that are smooth with respect to the product diffeology on V × V and the given diffeology on W . Example 3.3. The examples seen in the previous section provide readily the instances of V and W such that B ∞ (V, W ) is a proper subspace of B(V, W ). Indeed, let us take V = R n equipped with the coarse diffeology, and let W = R considered with the standard diffeology. It does not take much to extend the reasoning of Example 3.1 to show that for these two spaces B ∞ (V, W ) = 0. Once again, take a basis {v 1 , . . . , v n } of V and a basis {w} of W ; let f ∈ B ∞ (V, W ). Then with respect to the bases chosen f is defined by the matrix (a ij ) n×n where f (v i , v j ) = a ij w. For each i = 1, . . . , n consider the already-seen map p i : R → V given by p i (x) = \|x\|v i ; this map is a plot of V by definition of the coarse diffeology (that includes all set maps from domains of various R k to V ). Now call p ij the product map p ij : R → V × V , i.e. the map given by p ij (x) = (p i (x), p j (x)); it is obviously a plot for the product diffeology on V × V . Putting everything together, we get that (f • p ij )(x) = a ij \|x\|w; recalling that for f to be smooth this composition must be a plot of R, which is equivalent to being smooth, we conclude that a ij = 0. The indices i, j being arbitrary, we conclude that the only way for f to be smooth is for it be the zero map, whence the conclusion. The example just given stresses the importance of making a distinction between bilinear maps and smooth bilinear maps, showing that the two families can be (a priori) quite different, and motivates the next paragraph. The function spaces B ∞ (V, W ) and L ∞ (V, L ∞ (V, W )) As is well-known, in the usual setting each bilinear map can be viewed as a linear map V → L(V, W ). In the diffeological context, since a priori we might have L ∞ (V, W ) < L(V, W ), we need to consider the question of whether any smooth bilinear map can be seen as a smooth map V → L ∞ (V, W ), where the latter is endowed with the functional diffeology. Indeed this follows rather easily from the definition of the product diffeology and that of the functional diffeology, as we now show, starting with the following lemma: Lemma 3.4. Let V , W be two diffeological vector spaces, let f : V × V → W be a bilinear map smooth with respect to the product diffeology on V × V and the given diffeology on W , and let G : V → L ∞ (V, W ) be a linear map that is smooth with respect to the given diffeology on V and the functional diffeology on L ∞ (V, W ). Then: • for every v ∈ V the linear map F (v) : V → W given by F (v)(v ′ ) = f (v, v ′ ) is smooth; • the bilinear map g : V × V → W given by g(v, v ′ ) = G(v)(v ′ ) is smooth. Proof. Let us prove the first statement. Fix a v ∈ V . Recall that F (v) is smooth if and only if for every plot p : U → V the composition F (v) • p is a plot of W ; recall also thatp : U → V × V is a plot for product diffeology if and only if π 1 •p : U → V and π 2 •p : U → V are both plots of V (where π 1 and π 2 are the two natural projections). Fix now a plot p : U → V of V ; definep : U → V × V by settingp(x) = (v, p(x)) for all x ∈ V . Observe thatp is a plot for the product diffeology: indeed, π 1 •p is a constant map in V (and all such maps are plots of any diffeology by its definition), while π 2 •p = p, which is a plot by assumption. It is obvious by construction that F (v) • p = f •p; the latter map is a plot of W since f is smooth by assumption. Since p is arbitrary, this proves that F (v) is smooth. To prove the second statement, it suffices to observe that g writes as the composition g = ev•(G×Id V ); the map Id V being obviously smooth, G being smooth by assumption, their product being smooth by definition of the product diffeology, and, finally, the evaluation map ev being smooth by the definition of the functional diffeology, we get the conclusion. What the above lemma gives us are the following two maps: • the mapF : B ∞ (V, W ) → L(V, L ∞ (V, W )) that assigns to each f ∈ B ∞ (V, W ) the map F of the lemma (i.e., the specified map that to each v ∈ V assigns the smooth linear map F (v) : V → W ). Observe that F now writes as F =F (f ) and that the following relation holds: f = ev • (F × Id V ); • the mapG : L ∞ (V, L ∞ (V, W )) → B ∞ (V, W ) that assigns to each G ∈ L ∞ (V, L ∞ (V, W )) the map g = ev • (G × Id V ) . This latter map now writes as g =G(G). Before going further, we cite the following statement, which we will use immediately afterwards: We are now ready to prove the following lemma: Lemma 3.6. The following statements hold: 1. The mapF takes values in L ∞ (V, L ∞ (V, W )); furthermore, it is smooth with respect to the functional diffeologies of B ∞ (V, W ) and L ∞ (V, L ∞ (V, W )). 2. The mapG is smooth with respect to the functional diffeologies of L ∞ (V, L ∞ (V, W )) and B ∞ (V, W ). The mapsF andG are inverses of each other. Proof. 1. Let us first prove that F : V → L ∞ (V, W ) is smooth. Consider an arbitrary plot p : U → V ; by definition of a smooth map, we need to show that F • p is a plot for the functional diffeology on L ∞ (V, W ). Applying Proposition 3.5 to the composition F • p, we consider the induced map U × V → W that acts by the assignment (u, v ′ ) → (F • p)(u)(v ′ ) = F (p(u))(v ′ ) = f (p(u), v ′ ) = f • (p × Id V )(u, v ′ ). Since p × Id V is obviously a plot for the product diffeology on V × V and f is smooth, f • (p × Id V ) is a plot of W , so it is naturally smooth. The Proposition then allows us to conclude that F • p is a plot for L ∞ (V, W ), which, p being arbitrary, means that F is a smooth map. Let us now show thatF : W )). To do this, we apply again Proposition 3.5: it suffices to consider the map B ∞ (V, W ) → L ∞ (V, L ∞ (V, W )) is smooth; taking p : U → B ∞ (V, W ) a plot of B ∞ (V, W ), we need to show thatF • p is a plot of L ∞ (V, L ∞ (V,U × V → L ∞ (V, W ) acting by (u, v) → (F • p)(u)(v) = F (p(u))(v) = ev•((F •p)×Id V )(u, v). Having already established that F is smooth, we can now conclude thatF is smooth as well. 2. Let us now prove thatG : L ∞ (V, L ∞ (V, W )) → B ∞ (V, W ) is smooth, i.e. , taking an arbitrary plot p : U → L ∞ (V, L ∞ (V, W )), we need to show thatG • p is a plot of B ∞ (V, W ). Applying again Proposition 3.5, we consider the map U × (V × V ) → W defined by (u, (v, v ′ )) → (G • p)(u)(v, v ′ ) = (ev • (p(u) × Id V ))(v, v ′ ) = (ev • (p × Id V ×V ))(u, (v, v ′ )) , which allows us to conclude that the map is smooth, and thereforeG • p is a plot of B ∞ (V, W ); whence the conclusion. 3. This follows immediately from the definitions of the two maps. We now get the desired conclusion, which does mimick what happens in the usual linear algebra case: Proof. The desired diffeomorphism as diffeological spaces is given by the mapsF andG of Lemma 3.6. It remains to note that these two maps are also linear (actually, as vector spaces maps they coincide with the usual constructions), and that all the functional diffeologies involved are vector space diffeologies. The diffeological dual In this section we consider the various ways to define the diffeological dual; this discussion stems from the previous section, which illustrates how the function spaces of linear maps change at the passage to smooth maps. The final notion remains the most natural one, but we discuss also other possibilities and their implications. The dual as the set of smooth linear maps We first discuss the most obvious notion of the dual of a diffeological vector space, obtained by adding "diffeological" (or "smooth") wherever possible. Recall that for an arbitrary vector space V its dual space V * is defined as the (vector) space of all linear maps V → R, that is, V * = L(V, R). Now suppose that V is a diffeological vector space; then defining the dual in the usual manner just stated could a priori take us out of the category of diffeological spaces. 8 The latter would occur in the case L ∞ (V, R) < L(V, R) (with R being considered with the standard diffeology). That this can actually occur is illustrated by Example 3.1. On the other hand, the equality L ∞ (V, R) = L(V, R) does hold for fine diffeological vector spaces, so we should say why we do not restrict the discussion to those. This is due to our interest in tangent spaces of diffeological spaces, not all of which are fine (see [1], Example 4.22). What has just been said thus justifies the following definition: Let V be a diffeological vector space. Then its diffeological dual V * equipped with the functional diffeology is a diffeological vector space. Proof. As is shown in [2], Section 3.3, V * is a vector space; it is furthermore a subspace of both L(V, R) and C ∞ (V, R), the latter being a vector space with respect to the pointwise addition and multiplication by a scalar. By [2], Section 1.58, the functional diffeology on V * coincides with the sub-diffeology of the functional diffeology on C ∞ (V, R), and, as mentioned in [2], pp. 66-67, the latter is a diffeological vector space. By 3.5, [2], we conclude that V * is a diffeological vector space. Unless specified otherwise, the diffeological dual will always be considered with its functional diffeology. Observation 4.3. As follows from the example above, the diffeological dual V * a priori is not diffeomorphic to V , even if the dimension of V is finite. We stress that this occurs because in the case described 9 V * = L ∞ (V, R) is a proper subspace of L(V, R) ∼ = V , which obviously implies that it has a smaller dimension (zero in the case illustrated); so they are not isomorphic even as usual vector spaces. It is natural to wonder at this point: suppose that V is a (say, finite-dimensional) diffeological vector space such that L ∞ (V, R) = L(V, R); does this imply that V is also diffeomorphic to V * = L ∞ (V, R)? The following proposition provides a positive answer to this question. Proof. Let {v 1 , . . . , v n } be a basis of V , and let π i : V → R the projection of V onto the i-th coordinate for all i = 1, . . . , n (i.e., if v = α 1 v 1 + . . . + α n v n then π i (v) = α i ). Note that each π i , being a linear map from V to R, is smooth by assumption. Observe, furthermore, that the following map is smooth: π i : V × V → R given byπ i (v 1 , v 2 ) = π i (v 1 )π i (v 2 ) , the usual product of π i by itself; this follows from the definition of the product diffeology. Indeed, let p : U → V × V , written as p(x) = (p 1 (x), p 2 (x)), be a plot of V × V ; we need to show that π i • p is a plot of R, that is, that it is a usual smooth map U → R. Observe that each of p 1 , p 2 is a smooth map U → V , hence each of π i • p 1 , π i • p 2 is a smooth map U → R. The diffeology of R being the standard one, the usual product map is smooth. This product map beingπ i • p, we get the conclusion. Let us now prove the main statement. Take the dual basis {v 1 , . . . , v n } of V * and define F : V → V * by setting F (v i ) = v i for each i and extending by linearity. The map F is obviously an isomorphism of vector spaces; we need to show that it is also smooth. To do this, take an arbitrary plot p : U → V of V ; we need to show that F • p is a plot for the functional diffeology of V * = L ∞ (V, R). By Proposition 3.5, this is equivalent to the induced map U × V → R, given by (u, v) → F (p(u))(v), being smooth. By the definition of F , we have F (p(u))(v) = n i=1 π i (p(u))π i (v) = n i=1π i (p × Id V )(u, v). Since p × Id V is smooth by definition of the product diffeology (on U × V ) andπ i has already been shown to be smooth, we conclude that F • p is smooth, which proves the proposition. Observe that the definition of the diffeological dual given in [6] coincides with ours; but the discussion in [6] does not deal with the issue of the existence of diffeological spaces such that not every linear realvalued map be smooth: L ∞ (V, R) < L(V, R) (or, more generally, that L ∞ (V, W ) < L(V, W ); see Example 3.1). In the following two sections we turn to considering other a priori possibilities for defining the diffeological dual; in particular, we explore those alternatives that provide hope 10 to preserve as much of the usual isomorphism-by-duality as possible. The dual as the set of linear maps with functional diffeology In this section we treat the following option: • letV * be the usual dual of V , i.e.,V * = L(V, R), endowed with the corresponding functional diffeology. What this option gives us is a dual that is isomorphic as a vector space to the initial V (provided that V has finite dimension). But this is not the only issue in our context; what we should see next is whether it is also diffeomorphic to it. The following example shows that the answer is a priori negative. Example 4.5. What follows stems naturally from an example already seen (Example 3.1). Indeed, consider again V = R n with the coarse diffeology; let {e 1 , . . . , e n } be its canonical basis, and let ·\|· be the canonical scalar product. We have already calledV * = L(R n , R) the space of all linear maps R n → R considered with the functional diffeology. The canonical scalar product defines an isomorphism f : V →V * acting by f (v)(v ′ ) = v\|v ′ ; we now show that f is not smooth. Consider, once again, the map p i : R → V acting by p i (x) = \|x\|e i ; for f to be smooth it is necessary that f • p i be a plot ofV * = L(R n , R). By Proposition 3.5 this is equivalent to the smoothness of the map ϕ : R × V → R given by ϕ(x, v) = (f • p i )(x)(v) = p(x)\|v = \|x\|v i (where v i is the ith component of v). Let us show that ϕ is not smooth. Indeed, for it to be so, for any plot p of the product diffeology on R × V the composition ϕ • p must be a plot of R with the standard diffeology, i.e. it must be smooth in the usual sense, as a map R → R. We fix w ∈ V and define p to be p = Id R × c w , where c w is the constant map that sends everything to w; this is obviously a plot for the product diffeology on R × V . Then (ϕ • p)(x) = \|x\|w i , with w i being a constant; it is obviously not smooth and in particular it is not a plot for R with the standard diffeology. Hence ϕ is not a smooth map; thus f • p i is not a plot ofV * , hence f is not smooth. Now note that the same reasoning holds for any other isomorphism between V andV * : indeed, any such isomorphism is defined by a (non-degenerate) pairing. Let us sketch the details: let g : V →V * be an isomorphism; denoting by {e 1 , . . . , e n } the dual basis, we obtain that g is a linear map given by the matrix (a j i ) in the usual way, g(e i ) = a j i e j . As before, it is sufficient to show that ψ i : R × V → R defined by ψ i (x, v) = (g • p i )(x)(v) is not smooth (for some i). Let us now take n plots R → U × V , denoted by s j and defined by s j = Id R × c j , where c j is the constant map that sends everything to e j ; we obtain that (ψ i • s j )(x) = \|x\|a j i , so it is not smooth as a map R → R, as soon as a j i = 0. Hence the conclusion desired. 11 The example just made does not in and of itself invalidate the notion proposed; 12 later on (when we speak of the tensor product) we provide some comments regarding the implications of this different notion. The dual as the set of linear maps with pushforward diffeology Another way to define the dual is take, once again, the usual space of all linear maps V → R and endow it with the diffeology obtained using an isomorphism of (finite-dimensional) V with its usual dual, as is stated more precisely below: • let V be a finite-dimensional diffeological vector space, and letV * be the usual vector space dual of V endowed with the following diffeology: choose an isomorphismf : V →V * and denote by Df the pushforward 13 of the diffeology of V by the mapf . The definition as posed presents the (somewhat formal) question of being well-posed, i.e., whether the diffeology obtained depends on the choice of the isomorphism. Proof. It is sufficient to show that the composition map g • f −1 : (V * , Df ) → (V * , Dĝ) is smooth with respect to the pushforward diffeology. Let p : U → (V * , Df ) be a plot of (V * , Df ); we need to show that 11 The proof that we have just spelled out is actually much more than what we need for the purposes at hand (it would have been sufficient to start by choosing i and j such that a j i = 0 and speak of those only); it allows us to conclude that the only smooth linear map from V toV * is the zero map. 12 Although one can object to it from the categorical point of view. 13 Even if in the case of an isomorphism it does not make a difference, we mention that instead of a pushforward of the diffeology of V we could speak of its pullback by the inverse isomorphism. (g • f −1 ) • p is also a plot, of (V * , Dĝ). By definition of the pushforward diffeology, p being a plot of (V * , Df ) implies that (up to passing to a smaller negihbourhood) there exists a plot p ′ : U → V of V such that p = p ′ • f . Since f is invertible, we can write now (g • f −1 ) • p = g • (f −1 • p) = g • p ′ ; and the latter map is by definition a plot of the pushforward diffeology onV * by g (that is, it is a plot of (V * , Dĝ)), whence the conclusion. Now, of course, there are obvious limitations to this approach; the biggest one is that not all vector spaces are isomorphic to their duals. Nevertheless later on we will make some comments regarding the possibility of using this notion in the finite-dimensional case. Observation 4.7. One might wonder if there is a non-a priori reason to not limit ourselves to finitedimensional spaces only. Indeed, there is one: it was shown in [1] that even in a rather simple example, such as R n with the so-called wire diffeology, the internal tangent space (at 0) is infinite-dimensional. The dual map In this section we speak of the dual maps of linear maps; recall that, given two vector spaces V , W , and a linear map f : V → W between them, then the dual map is f * : W * → V * acting by f * (g)(v) = g(f (v)). Now suppose that V and W are two diffeological vector spaces, and that f ∈ L ∞ (V, W ). Then, for each of the three possible notions of the dual space, there is the question whether the corresponding f * is a smooth map. It turns out that the answer is positive for V * andV * , but a priori it is negative for V * . Let us prove these two statements. In the first case, we give the full proof for V * only, noting that it essentially uses the fact that it is endowed with the functional diffeology (so can be carried over toV * ). In the second case we give an example. Proposition 4.8. Let V , W be two diffeological vector spaces, and let f : V → W be a smooth linear map. Let f * : W * → V * be the dual map between the diffeological duals, f * (g)(v) = g(f (v)). Then f * is smooth. Proof. Let p be a plot of W * ; we need to show that f * • p is a plot of V * . The diffeology of W * being functional, by Proposition 3.5 p being a plot is equivalent to the smoothness of the map ψ : U × W → R acting by ψ(u, w) = p(u)(w); now, for ψ to be smooth, we must have for any plot (p 1 , p W ) : U ′ → U × W (where p 1 and p W are plots of U ′ and W respectively) that ψ • (p 1 , p W ) is a plot of R, that is, a usual smooth map U ′ → R. For future use, let us write explicitly that (ψ • (p 1 , p W ))(u ′ ) = p(p 1 (u ′ ))(p W (u ′ )). Now, to prove that f * • p is a plot of V * , we need to show that the map ϕ : U × V → R given by ϕ(u, v) = p(u)(f (v)) is smooth, that is, that for any plot (p 1 , p V ) : U ′ → U × V (where p 1 and p V are plots of U ′ and V respectively) the composition ϕ • (p 1 , p V ) is a plot of R. Writing explicitly (ϕ • (p 1 , p V ))(u ′ ) = p(p 1 (u))(f (p V (u ′ ))) and observing that f being smooth and p V being a plot of V , we get that f • p V is a plot of W , so setting p W = f • p V , we deduce immediately the desired conclusion from the analogous expression for ψ (smooth by assumption) and p W . We now briefly describe an example that shows that the dual map f * :Ŵ * →V * may not be smooth. Example 4.9. Let V be R n with the fine diffeology, and let W be R n with the coarse diffeology. Observe that this implies thatV * andŴ * , being pullbacks, also have, respectively, the fine and the coarse diffeology. Let f : V → W be any linear map (it is automatically smooth); then f * is a map from R n with coarse diffeology to the one with fine (that is, standard) diffeology. It suffices to choose, as a plot p ofŴ * , any non-smooth map to R n , to get that f * • p is not a plot ofV * , thus disproving the smoothness of f * . Conclusion To summarize the discussion carried out up to now in the present section, we settle for the first definition of the diffeological dual, i.e., that of the set of all smooth linear maps: V * = L ∞ (V, R). Apart from it being the most natural (and obvious) from the categorical point of view, the other two notions present their own limitations, which preclude a universal choice. To be more precise, we have already said that the main drawback of V * is the (frequent) failure of the isomorphism-by-duality: 14 by Proposition 4.4, its existence (in the finite-dimensional case) is equivalent to the equality L ∞ (V, R) = L(V, R), so can be guaranteed only for fine diffeological vector spaces. This drawback however is not escaped by takingV * , which is indeed isomorphic to V whenever the usual dual is so, but it may not be diffeomorphic to it, as Example 4.5 shows. Finally, this issue is resolved byV * , which however, in addition to not being always defined, lacks the advantages of being considered with its most natural diffeology, the functional one; as is noted above, this in some instances implies that the dual map of a smooth linear map may not be smooth. Therefore our choice; although we do not pretend that other notions cannot be considered. The tensor product In this section we discuss the definition (and the relative properties) of the tensor product, starting with discussing in detail the case of two factors. While there is a definition that we prefer, we briefly mention some alternatives, 15 based on the discussion in the previous section. The tensor product of two diffeological vector spaces We now begin to consider the tensor product, first treating in detail the case of the tensor product of just two diffeological vector spaces; it seems convenient to do so before going into the full generality, as, on one hand, all the main issues are already visible at this point, and, on the other hand, these can be seen in a very concrete fashion. The tensor product: definition Let us first look at the case of the tensor product between two diffeological vector spaces V and W , following a somewhat naive approach. Namely, let us describe step-by-step the direct construction of the tensor product and how it reflects itself in the contruction of the corresponding diffeology. • Consider first the direct product V × W and endow it with the product diffeology (so it, too, becomes a diffeological vector space). • Next, recall that the tensor product V ⊗ W can be defined as the quotient of V × W by the subspace Z generated by all elements of the form ( α 1 v 1 + α 2 v 2 , w) − α 1 (v 1 , w) − α 2 (v 2 , w), (v, β 1 w 1 + β 2 w 2 ) − β 1 (v, w 1 ) − β 2 (v, w 2 ) , where α 1 , α 2 , β 1 , β 2 run over R, v, v 1 , v 2 run over V , and w, w 1 , w 2 run over W . Endow Z with the sub-diffeology of V × W . • Finally, endow V ⊗ W = (V × W )/Z with the quotient diffeology, which we denote by D V ⊗W . The procedure just described certainly yields a diffeology on V ⊗ W , that makes it a diffeological vector space, and possesses the first properties of the tensor product, such as being commutative 16 and having V ⊗ R ∼ = V (as diffeological vector spaces) for any V ; but it is not the only possible and, what is more important, it might not be the same as the one arising from other ways of defining the tensor product. Whether it is, or it is not, is what we shall see next. Comparison with the tensor product diffeology of [6] Let V 1 , V 2 be two diffeological vector spaces; let us make a comparison between the diffeology D Vi of [6] and the diffeology D V1⊗V2 described in the previous paragraph. Proof. This is a direct consequence of the definitions: in fact the diffeology D V1⊗V2 is the quotient diffeology with respect to the kernel of the standard projection φ : V 1 × V 2 → V 1 ⊗ V 2 ; this is the same as the pushforward of the product diffeology by this projection. The latter is actually the diffeology D Vi of [6] (note that by [2], Section 3.5, this pushforward is already a vector space diffeology). We anticipate that this statement holds just the same (and with the same proof) for the tensor product of more than two spaces, the fact that we will state when we arrive to discussing that case. Note that this means that the analogue of the usual universality result (Theorem 2.3.5 of [6]) holds for our (description of the) tensor product, that is: (V ⊗ W ) * = L ∞ (V ⊗ W, R) ∼ = Mult ∞ (V × W, R), where Mult ∞ (V × W, R) is the space of all smooth (for the product diffeology) bilinear maps V × W → R. The tensor product of maps Let us consider two (smooth) linear maps between diffeological vector spaces, f : V → V ′ and g : W → W ′ . As usual, we have the tensor product map f ⊗g : V ⊗W → V ′ ⊗W ′ , defined by (f ⊗ g)( v i ⊗ w i ) = f (v i ) ⊗ g(w i ). We observe that f ⊗ g is a smooth map (with respect to the tensor product diffeologies on V ⊗ W and V ′ ⊗ W ′ ) due to the properties of the product and the quotient diffeologies. The tensor product and the direct sum Let V 1 , V 2 , V 3 be vector spaces; recall that in the usual linear algebra the tensor product is distributive with respect to the direct sum, i.e.: V 1 ⊗ (V 2 ⊕ V 3 ) ∼ = (V 1 ⊗ V 3 ) ⊕ (V 2 ⊗ V 3 ), via a canonical isomorphism, which we denote by T ⊗,⊕ . Now, if V 1 , V 2 , V 3 are diffeological vector spaces, then so are V 1 ⊗ (V 2 ⊕ V 3 ) and (V 1 ⊗ V 3 ) ⊕ (V 2 ⊗ V 3 ). 17 The map T ⊗,⊕ is still an isomorphism of the underlying vector spaces; we shall deal with the question whether it is also a diffeomorphism. Lemma 5.2. Let V 1 , V 2 , V 3 be diffeological vector spaces, and let T ⊗,⊕ : V 1 ⊗ (V 2 ⊕ V 3 ) → (V 1 ⊗ V 3 ) ⊕ (V 2 ⊗ V 3 ) be the standard isomorphism. Then T ⊗,⊕ is smooth. Proof. By the properties of the quotient diffeology, it is sufficient to show that the covering mapT ×,⊕ : V 1 × (V 2 ⊕ V 3 ) → (V 1 × V 3 ) ⊕ (V 2 × V 3 ) is smooth. Let p : U → V 1 × (V 2 ⊕ V 3 ) be a plot; we must show that T ×,⊕ •p is a plot for (V 1 ×V 3 )⊕(V 2 ×V 3 ). Let π 1 : V 1 ×(V 2 ⊕V 3 ) → V 1 and π 2,3 : V 1 ×(V 2 ⊕V 3 ) → (V 2 ⊕V 3 ) be the natural projections; observe that by definition of the sum diffeology, π 2,3 writes (at least locally) as π 2,3 = p 2 ⊕ p 3 , where p 2 is a plot of V 2 and p 3 is a plot of V 3 . Write nowT ×,⊕ • p asT ×,⊕ • p = p ′ ⊕ p ′′ ; observe that p ′ = (π 1 • p, p 2 ), while p ′′ = (π 1 • p, p 3 ). These are plots for the sum diffeology on ( V 1 × V 3 ) ⊕ (V 2 × V 3 ) , hence the conclusion. The tensor product V ⊗ W as a function space As is known, in the usual linear algebra context the tensor product of two finite-dimensional vector spaces V ⊗ W is isomorphic to the spaces L(V * , W ), the space of linear maps V * → W , and L(W * , V ), the space of linear maps W * → V . Recall that, somewhat naively, these isomorphisms are given as: • for f ∈ V * , v ∈ V , and w ∈ W we set (v ⊗ w)(f ) = f (v)w, extending by linearity; • for g ∈ W * , v ∈ V , and w ∈ W we set (v ⊗ w)(g) = g(w)v, extending by linearity. The question that we consider now is whether these isomorphisms continue to exist if all spaces we consider are (finite-dimensional) diffeological vector spaces (in particular, all duals are meant in the diffeological sense), all linear maps are smooth, and all function spaces are endowed with their functional diffeologies. The observations made regarding the sometimes substantial difference between a diffeological vector space V and its diffeological dual V * (in particular, that it might be zero even for V "quite large", for instance, with total space any R n ) suggest that we start by considering again one of our examples. Example 5.3. Once again, consider V = R n for n 2 with the coarse diffeology and W = R with the standard diffeology. Then, as shown in Example 3.1, the diffeological dual of V is trivial: V * = {0}; this obviously implies that L ∞ (V * , W ) = {0}. Recall also that, the diffeology of W being fine, its dual is isomorphic to W , so we have W ∼ = W * ∼ = R; furthermore, as it occurs for all fine diffeological vector spaces (see Section 2), we have L ∞ (W * , V ) = L(W * , V ) ∼ = V . On the other hand, the total space of the diffeological tensor product V ⊗ W is the same as that of the usual tensor product, i.e., it is isomorphic to V . This implies right away that there is not an isomorphism between V ⊗ W and L ∞ (V * , W ), the two spaces being different as sets. On the other hand, L ∞ (W * , V ) and V ⊗ W are isomorphic as usual vector spaces; it is easy to see that they are also diffeomorphic (this follows from the fact that V has the coarse diffeology 18 ). The example just made shows that in general, at least one of these classical isomorphisms might fail to exist (and at a very basic level). We may wish however to see what could be kept of the standard isomorphisms, in the sense that the two maps V ⊗ W → L(V * , W ) and V ⊗ W → L(W * , V ) are still defined; we might wonder if their ranges consist of smooth maps and, if so, whether they are smooth. Proposition 5.4. Let V , W be two finite-dimensional diffeological vector spaces. Then: 1. IfF : V ⊗ W → L(V * , W ) is the map defined, via linearity, by v ⊗ w → [F (v ⊗ w)(f ) = f (v)w] then F takes values in L ∞ (V * , W ). Furthermore, as a map V ⊗ W → L ∞ (V * , W ) between diffeological spaces, it is smooth; 2. IfĜ : V ⊗ W → L(W * , V ) is the map defined, via linearity, by v ⊗ w → [Ĝ(v ⊗ w)(g) = g(w)v] then G takes values in L ∞ (W * , V ). Furthermore, as a map V ⊗ W → L ∞ (W * , V ) between diffeological spaces, it is smooth. Proof. Let us prove 1; we will quite liberally avail ourselves of the commutativity of all the products. We need to show thatF is a smooth map that takes values in L ∞ (V * , W ). To prove the latter, it is enough to show thatF (v ⊗ w) is smooth, for any v ∈ V and w ∈ W . Let us fix v ∈ V and w ∈ W ; we need to show that for any plot p : U → V * the compositionF (v ⊗ w) • p is a plot of W . Writing explicitly (F (v ⊗ w) • p)(u) =F (v ⊗ w)(p(u)) = p(u)(v)w, we recall that any constant map on a domain is a plot for any diffeology, so the map c w : U → W that sends everything in w is a plot of W . Finally, the map (u, v) → p(u)(v) is a smooth map to R, by Proposition 3.5 and because p is a plot of V * = L ∞ (V, R) whose diffeology is functional; recalling that multiplication by scalar is smooth for any diffeological vector space, we get the conclusion. Let us now prove thatF is a smooth map V ⊗ W → L ∞ (V * , W ); by Proposition 3.5 we need to prove that the induced map V * × U → W is smooth. 19 This map acts by sending each (f, u) (where f ∈ V * ) to (F • p)(u)(f ) and so it writes as (f, u) → (ev V * ⊗ Id W )(Id V * × p)(f, u); the diffeology of V * being functional, so that the evaluation map is smooth, we conclude thatF • p is smooth, so the conclusion. The proof of 2 is completely analogous, so we omit it. Observation 5.5. As a final remark to this paragraph, we observe that already Example 5.3 tells us that, in general, there is not an analogue of the classical isomorphism V * ⊗ V ∼ = L ∞ (V, V ): it suffices to consider the same V , that is, R n with the coarse diffeology. Then the product on the left is the trivial space, V * being the trivial space, whereas the space on the right consists of all linear maps V → V (since the coarse diffeology includes any map into V , all of these maps are automatically smooth). Tensor product of duals and the dual of a tensor product Recall, once again, that for usual vector spaces there is a standard isomorphism V * ⊗ W * ∼ = (V ⊗ W ) * ; we are now interested in the question whether the existence of this isomorphism extends to the diffeological context, i.e., whether the corresponding map is (always) smooth. 20 18 Consider the obvious map F : V → L(R, V ) = L ∞ (R, V ) given by F (v)(x) = xv; it is obviously bijective, and it is smooth by Proposition 3.5. Indeed, for any plot p : U → V we need that F • p be a plot, which is equivalent to the map U × R → V given by (u, x) → (F • p)(u)(x) = xp(u) being smooth. But simply due to the fact that it is a map in V , that has the coarse diffeology, it is a plot of it, so the conclusion. 19 Note the change in the order of factors, for formal purposes. 20 This question becomes quite important when one comes to considering scalar products, as is our intention. The standard isomorphism V * ⊗ W * → (V ⊗ W ) * , which in this paragraph we denote by F is defined by setting: F ( i f i ⊗ g i )( j v j ⊗ w j ) = i,j f i (v j )g i (w j ). The first thing that we need to check is whether it does take values in (V ⊗ W ) * , that is, if, fixed some f ⊗ g ∈ V * ⊗ W * , 21 it actually defines a smooth (and not just linear) map V ⊗ W → R. Lemma 5.6. Let V , W be diffeological vector spaces, and let f ∈ V * , g ∈ W * . Then the map F (f ⊗ g) : V ⊗ W → R is smooth. Proof. By Proposition 3.5 we need to check that for any plot p : U → V ⊗ W the composition F (f ⊗ g) • p is a smooth map U → R. Recall that locally (so we assume that U is small enough, so as to avoid complicating the notation) p writes as a composition p = π ⊗ •p, where π ⊗ is the natural projection V × W → V ⊗ W andp : U → V × W is a plot for the product diffeology; furthermore,p writes as p = (p V , p W ), where p V is a plot of V and p W is a plot of W . Putting all of this together, we write (F (f ⊗ g) • p)(u) = f (p V (u))g(p W (u)), that is, F (f ⊗ g) • p is the usual product in R of two maps, f • p V and g • p W . Now, f being smooth by its choice and p V being a plot of V , their composition f • p V is a smooth map in R. The same holds also for g • p W ; the product of two smooth maps being smooth, we get the desired conclusion. By the lemma just proven, F is an injective linear map from the tensor product of the diffeological duals V * , W * into the diffeological dual of the tensor product V ⊗ W . We should check next whether it is smooth. Proposition 5.7. Let V , W be diffeological vector spaces, and let F : V * ⊗ W * → (V ⊗ W ) * be the already defined map between the diffeological duals. Then F is smooth. Proof. For the map F to be smooth, it is required that, for any plot p : U → V * ⊗ W * the composition F • p be a plot of (V ⊗ W ) * . Recall that the latter is equivalent to the following map being smooth: • Φ : U × (V ⊗ W ) → R such that Φ(u, v j ⊗ w j ) = F (p(u))( v j ⊗ w j ). The map Φ being smooth is equivalent to: • for any map (p U , p V ⊗W ) : U ′ → U × (V ⊗ W ) such that p U : U ′ → U is smooth and p V ⊗W : U ′ → V ⊗ W is a plot of V ⊗ W the composition Φ • (p U , p V ⊗W ) is a smooth map U ′ → R. We write explicitly: (Φ • (p U , p V ⊗W ))(u ′ ) = F ((p • p u )(u ′ ))(p V ⊗W (u ′ )). This is the map of which we need to establish the smoothness. To do so, let us write explicitly what it means that p : U → V * ⊗ W * is a plot of the second space. First of all, by definition of the quotient diffeology we have: • for U small enough 22 p lifts to a smooth mapp : U → V * × W * , that is, p = π V * ⊗W * •p, where π V * ⊗W * is the natural projection (smooth by definition); moreover,p writes asp = (p V * , p W * ), where p V * is a plot of V * and p W * is a plot of W * . Recall that p V * being a plot of V * means that the map ϕ V : U × V → R given by ϕ V (u, v) = p V * (u)(v) is smooth; accordingly, p W * being a plot of W * means that the map ϕ W : U × W → R given by ϕ W (u, w) = p W * (u)(w) is smooth. In addition, we should say what it means for p V ⊗W be a plot: • for U small enough, p V ⊗W lifts top V ×W , a plot of V × W , that is, p V ⊗W writes as p V ⊗W = π ⊗ • p V ×W for the appropriate natural projection π ⊗ ; furthermore,p V ×W writes asp V ×W = (p V , p W ), where p V is a plot of V and p W is a plot of W . 21 Extending by linearity is smooth by definition of a diffeological vector space. 22 Which we can always assume Assume now that the domain U is small enough so that all of the above be valid; then we can write, by definition of F , that (Φ • (p U , p V ⊗W ))(u ′ ) = p V * (u ′ )(p V (u ′ )) · p W * (u ′ )(p W (u ′ )) = ev(p V * , p V )(u ′ ) · ev(p W * , p W )(u ′ ), where (p V * , p V ) : U ′ → V * × V and (p W * , p W ) : U ′ → W * × W are the obvious maps. By definition of the product diffeology they are plots for, respectively, V * × V and W * × W ; furthermore, each diffeological dual carrying the functional diffeology, each evaluation map ev is obviously smooth. It follows that Φ • (p U , p V ⊗W ) : U ′ → R writes as the product of two smooth maps U ′ → R; the diffeology of R being the standard one, these maps are smooth in the usual sense, hence so is their product. This implies that Φ is a smooth map, therefore F is smooth, and the Proposition is proven. We are now ready to prove the following statement: Theorem 5.8. Let V , W be two finite-dimensional diffeological vector spaces. Then F : V * ⊗ W * → (V ⊗ W ) * is a diffeomorphism. Proof. It remains to check that F is surjective with smooth inverse, i.e., that for any smooth linear map f : V ⊗ W → R its pre-image F −1 (f ) (which a priori belongs to the tensor product of the usual duals) actually belongs to the tensor product of the diffeological duals. By definition of F , it is sufficient to observe that f being smooth means that for any plot p : U → V ⊗ W the composition f • p : U → R is a (usual) smooth map; furthermore, for U small enough p writes as p = π • (p V , p W ), where π : V × W → V ⊗ W is the natural projection, p V : U → V is a plot of V , and p W : U → W is a plot of W , hence f • p actually writes as (f • p)(u) = f (p V (u) ⊗ p W (u)). Note that F −1 (f ) writes as F −1 (f ) = f i ⊗ g i with f i belonging to the usual dual of V , and g i belonging to the usual dual of W ; we obtain that (F −1 (f ) • (p V , p W ))(u) = (f i • p V )(u)(g i • p W )(u), and we can draw the desired by choosing the appropriate p V , p W . Possible alternatives Since there are more ways to look at the tensor product in the usual (multi)linear algebra, there are a priori more ways to define a diffeology on V ⊗ W (the usual vector space). We now briefly describe these other possibilities. The diffeology stemming from V ⊗W ∼ = L(L(V, R), W ) Let V and W be two finite-dimensional diffeological vector spaces; the existence of classical (multi)linear algebra isomorphisms V ⊗W ∼ = L(L(V, R), W ) ∼ = L(L(W, R), V ) could suggest to define the diffeological tensor product by taking the usual tensor product V ⊗ W and endowing it with the functional diffeology of either L(L(V, R), W ) or L(L(W, R), V ), where L(V, R) and L(W, R) are considered with their respective functional diffeologies. More precisely, the idea is to follow the two steps below: • letV * be the space defined in Section 4.2. Consider L(V * , W ) and endow it with the functional diffeology; • endow V ⊗W with the diffeology D f V ⊗W that is the pullback of the above diffeology by the standard map V ⊗ W → L(V * , W ). The map just referenced is the map analogous toF of the previous section, i.e., acting by the rulê F ( v i ⊗ w i )(f ) = f (v i )w i for f : V → R linear. The construction just described is of course possible; the specific reason why we do not employ it stems from the proof of Lemma 5.6, for which the fact that the elements of the dual are smooth maps is significant. The Lemma being a prerequisite for (for example) Theorem 5.8, we discard the option just described. The exterior product We finally mention the exterior product of diffeological vector spaces. Recall that, given A ∈ Λ p (V ) and B ∈ Λ q (V ), their exterior product is defined as A ∧ B = (p+q)! p!q! Alt(A ⊗ B). The exterior product is smooth as a map Λ p (V ) × Λ q (V ) → Λ p+q (V ) (the former space being considered with the product diffeology), as follows from the definition of the diffeological tensor product and the smoothness of the antisymmetrization operator. Proposition 3. 5 . 5([2], 1.57) Let X, Y be two diffeological spaces, and let U be a domain of some R n . A map p : U → C ∞ (X, Y ) is a plot for the functional diffeology of C ∞ (X, Y ) if and only if the induced map U × X → Y acting by (u, x) → p(u)(x) is smooth. Theorem 3. 7 . 7Let V and W be two diffeological vector spaces, let B ∞ (V, W ) be the space of all smooth bilinear maps V × V → W considered with the functional diffeology, and let L ∞ (V, L ∞ (V, W )) be the space of all smooth linear maps V → L ∞ (V, W ) endowed, it as well, with the functional diffeology. Then the spaces B ∞ (V, W ) and L ∞ (V, L ∞ (V, W )) are diffeomorphic as diffeological vector spaces. Definition 4. 1 . 1Let V be a diffeological vector space. The diffeological dual of V , denoted by V * , is the set L ∞ (V, R) of all smooth linear maps V → R. Proposition 4 . 4 . 44Let V be a finite-dimensional diffeological vector space such that L ∞ (V, R) = L(V, R),i.e., such that every real-valued linear map from V is smooth. Then V is diffeomorphic to L ∞ (V, R) = L(V, R), i.e., to its diffeological dual. Lemma 4 . 6 . 46Let V be a finite-dimensional diffeological vector space, letf : V →V * andĝ : V →V * be two vector space isomorphisms of V with its dual, and let Df and Dĝ be the corresponding pushforward diffeologies. Then Df = Dĝ. Lemma 5 . 1 . 51For any two diffeological spaces V 1 , V 2 the diffeologies D Vi and D V1⊗V2 on V 1 ⊗ V 2 coincide. The description given in[6] applies essentially to the so-called fine diffeological vector space (see Section 2); as is shown in[1], those are not sufficient for the study of tangent spaces. In brief, the total space of the internal tangent bundle is the disjoint union of all the internal tangent spaces; the dvs diffeology is the smallest diffeology such that the corresponding sub-diffeology on each fibre makes it a diffeological vector space.3 We do not really endeavour to give a precise answer to the latter question; rather, we are interested in spelling out explicitly all the different versions, perhaps indicating which one seems preferable in some abstract sense. The true answer (if there exists the true answer) to the question, which one is better, would depend on the context, which the present paper does not provide. Obviously, the respective canonical bases would do the job just fine. This is a story beautifully told in the Preface and Afterword to the excellent book[2].6 One can actually show that f is smooth if and only if a i = 0, and so dim(L ∞ (V, R)) is precisely n − 1; we do not elaborate on this, since we mostly interested in showing that it can be strictly smaller (by any admissible value, as we see below). 7 Artificial as they may appear. There are however ways to do it, of which we speak later on. We will see shortly various other examples of similar kind. 10 Which eventually turns out to be unfounded, from the diffeological viewpoint. Which we actually want to be a diffeomorphism 15 Due to alternative definitions of the diffeological dual.16 Meaning that V ⊗ W and W ⊗ V are isomorphic as diffeological vector spaces, as follows from the commutativity of the product diffeology. By considering, in addition to the tensor product diffeology, the sum diffeology, whenever appropriate. We omit the proof. The spaceV * Another a priori option (once again, in the case of finite-dimensional spaces) is to apply the construction of the previous paragraph, taking instead ofV * the spaceV * defined in Section 4.3. We shall avoid following this path, the motivation being the already-mentioned indications (such as the potential non-smoothness of the dual map) that this construction is too artificial.Scalar productsThe motivation for this work stemming from wishing to have an analogue of Riemannian metric on diffeological bundles with fibres diffeological vector spaces, we wish to pay particular attention to various ways of viewing scalar products on the latter. Recall that in the usual context a scalar product on a vector space V , being a bilinear map (with some extra properties), can be see also as an element of V * ⊗ V * ; in the diffeological context, it is a smooth bilinear map (symmetric and definite positive) and the tensor product is that of the diffeological duals. Thus, a priori there is the question whether similar identification continues to hold. This follows from Theorem 2.3.5 of[6]and Theorem 5.8. Indeed, the former implies that the diffeological space of all smooth bilinear mapsRemark 5.9. We thus get a work-in-progress conclusion that a prospective notion of a diffeological metric can use a bundle with fibre (V ⊗ V ) * , or a bundle with fibre V * ⊗ V * : there would not be any difference (at least, on the level of the total space).The tensor product of n spacesWe now provide quickly the definition of the tensor product of more than two spaces; this construction is easily generalized, and in the most obvious manner, from the case of n = 2. Let V 1 , ..., V n be diffeological vector spaces, let T : V 1 × . . . × V n → V 1 ⊗ . . . ⊗ V n be the universal map onto their tensor product as vector spaces, and let Z V 1 × . . . × V n be the kernel of T . We denote by D ⊗ the following diffeology on V 1 ⊗ . . . ⊗ V n :• endow V 1 × . . . × V n with the product diffeology, and Z with the corresponding subspace diffeology;• let D ⊗ be the quotient diffeology on (V 1 × . .As has already been mentioned, the diffeology D ⊗ can also be described as the pushforward by T of the product diffeology on V 1 × . . . × V n .This construction obviously includes the usual p-covariant and q-contravariant tensors, defined as elements of the tensor product V * ⊗ . . .where V is a diffeological vector space and V * its diffeological dual. As usual, the notation T p q (V ) is used for the above diffeological tensor product.Symmetrization and antisymmetrizationConsidering the space, the usual symmetrization and antisymmetrization operators are defined; and there are their invariant subspaces, the space S p (V ) of symmetric tensors and the space Λ p (V ) of antisymmetric tensors. These spaces are endowed with the subspace diffeology, which, due to the smoothness of the operations of a diffeological vector space, ensures that the two operators are smooth. 23 J D Christensen, -E Wu, arXiv:1411.5425v1Tangent spaces and tangent bundles for diffeological spaces. J.D. Christensen -E. Wu, Tangent spaces and tangent bundles for diffeological spaces, arXiv:1411.5425v1. . P Iglesias-Zemmour, Diffeology, Mathematical Surveys and Monographs. 185AMSP. Iglesias-Zemmour, Diffeology, Mathematical Surveys and Monographs, 185, AMS, Providence, 2013. Construction of metrics on diffeological spaces. E Pervova, in preparationE. Pervova, Construction of metrics on diffeological spaces, in preparation. J M Souriau, Groups différentiels, Differential geometrical methods in mathematical physics (Proc. Conf. Aix-en-Provence/SalamancaSpringer836J.M. Souriau, Groups différentiels, Differential geometrical methods in mathematical physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Mathematics, 836, Springer, (1980), pp. 91-128. Groups différentiels de physique mathématique, South Rhone seminar on geometry. J M Souriau, Astérisque. Numéro Hors SérieJ.M. Souriau, Groups différentiels de physique mathématique, South Rhone seminar on geometry, II (Lyon, 1984), Astérisque 1985, Numéro Hors Série, pp. 341-399. M Vincent, Diffeological differential geometry. University of CopenhagenMaster ThesisM. Vincent, Diffeological differential geometry, Master Thesis, University of Copenhagen, 2008, available at http://www.math.ku.dk/english/research/top/paststudents/ martinvincent.msthesis.pdf	[]
[ "Invalid Methods of Causal Inference in Physics Education Research", "Invalid Methods of Causal Inference in Physics Education Research" ]	[ "M B Weissman \nDepartment of Physics\nUniversity of Illinois at Urbana-Champaign\n1110 West Green Street61801-3080UrbanaIL\n" ]	[ "Department of Physics\nUniversity of Illinois at Urbana-Champaign\n1110 West Green Street61801-3080UrbanaIL" ]	[]	Finding good educational policies requires sound estimates of their potential effects. Methods for making such estimates, i.e. finding causal estimands, have made great progress in the last few decades. Nevertheless, serious errors in causal reasoning have been found previously in papers n a leading physics education journal, Physical Review Physics Education Research. Here we examine three more recent papers from that journal that present explicit methods of causal inference. The methods given include major errors, including in identifying causal mediation, choosing variables to control for, and imputing missing data.	null	[ "https://arxiv.org/pdf/2110.04266v2.pdf" ]	238,531,526	2110.04266	9c8377e1d02d88866a57dddb0bcd7a866ae4f98d	Invalid Methods of Causal Inference in Physics Education Research M B Weissman Department of Physics University of Illinois at Urbana-Champaign 1110 West Green Street61801-3080UrbanaIL Invalid Methods of Causal Inference in Physics Education Research 1/31/22 -1 - 1/31/22 2 Finding good educational policies requires sound estimates of their potential effects. Methods for making such estimates, i.e. finding causal estimands, have made great progress in the last few decades. Nevertheless, serious errors in causal reasoning have been found previously in papers n a leading physics education journal, Physical Review Physics Education Research. Here we examine three more recent papers from that journal that present explicit methods of causal inference. The methods given include major errors, including in identifying causal mediation, choosing variables to control for, and imputing missing data. Introduction Awareness is growing that modern causal inference methods (Imbens and Rubin 2015) are needed to help educators predict the consequences of different choices in teaching methods and in educational policy (Murnane and Willett 2011). Especially in the last four decades, sophisticated methods of causal inference have been developed to estimate such potential outcomes from observational data. (Pearl, Glymour, and Jewell 2016;Hernán and Robins 2020) Field-specific introductions to these methods are now available for education (Murnane and Willett 2011), economics (Varian 2016;2015), , psychology (Foster 2010;Rohrer 2018), epidemiology (Hernán and Robins 2020), biology (Glymour, Zhang, and Spirtes 2019), public health (Glass et al. 2013), sociology (Gangl 2010), political science (Keele 2015), and other fields (Imbens and Rubin 2015), with ref. (Rohrer 2018) providing a particularly accessible primer. One might expect that physics, the most consistently mathematical of the natural sciences, would take the lead in employing valid new methods of quantitative reasoning to address its educational challenges. Nevertheless, in a recent paper (Weissman 2021) I showed that several papers published before 2021 in the leading journal of physics education, Physical Review Physics Education Research (PRPER), made major causal inference errors in particular applications. My previous paper focused on instances in which particular causal patterns were assumed without regard to the wide variety of plausible causal patterns equally consistent with the data. I included a very brief introduction (Weissman 2021) to the diagrammatic methods by which causal relations are often represented, (Pearl, Glymour, and Jewell 2016;Hernán and Robins 2020;Rohrer 2018), which need not be repeated here. Different specific issues are raised by three more recent PRPER papers, published in 2021. (Walsh et al. 2021;Young and Caballero 2021;Verostek, Miller, and Zwickl 2021) These papers include not only examples but explicit methods instructions whose use would lead to seriously biased causal estimands. The issues raised include the rules for which variables should be included or excluded from regression models to obtain unbiased causal estimands, the ways to identify causal mediators, the specification of potential outcomes, and the ways to impute missing data. The errors seriously distort causal implications for at least one major policy decision, the use of standardized tests in admissions decisions. The central conclusion of this paper will be to reinforce my call for improving awareness of causal inference methods throughout the PER field. My arguments here involve no new methods but are just applications of the standard beginning methods of the field. Some of the introductory material included may be too elementary for many readers, but it is intended to make the arguments accessible to PER workers. PRPER has expressed a reluctance to publish any further work that describes some of its papers as "incorrect", indicating that the field may be in need of some outside intervention, to which this paper is meant to contribute. Background on mediation, bias from including inappropriate variables, and treatment of missing data Within the potential outcome or counterfactual approaches to causation, the average causal effect of a change in one variable on another variable is the average change in the second variable that would occur if one were to change the first variable without intervening to change anything else. (Pearl, Glymour, and Jewell 2016;Hernán and Robins 2020) This definition has an obvious correspondence to part of what one needs to know to decide whether to make the change in question, although for a more accurate estimation of the utility of some action one should know the full change in the probability distribution for the outcome rather than just its mean. A causal estimand is a formula estimating that causal effect from various correlations among other variables, including the cause being studied. (Pearl, Glymour, and Jewell 2016;Hernán and Robins 2020) A causal estimand is said to be biased if it systematically mis-estimates the effect. Bias can arise when the estimand either contains terms that do not correspond to results of intervening on the cause or omits relevant terms that do correspond to such changes. (An excellent easily readable introduction to this issue can be found in ref. (Rohrer 2018).) The different coefficients found in different predictive models employing different predictive variables are not themselves "biased"; they are simply the coefficients of different models that would be used with different available predictive information. They become biased only if they are used to estimate something else-the 1/31/22 causal effects. It is meaningless to say which coefficient is "true" and which is "biased" without specifying a causal question and the true causal model. Descriptions of bias in causal estimands often focus on the role of omitted variables (called confounders) that have effects on both the cause in question and the outcome. (Rohrer 2018) In a diagram, the simplest confounder would be one with arrows to both the cause and the outcome. Simple cause-outcome correlations in models omitting the confounders give biased estimates of the actual causal effects because they include correlations between the cause and the outcome induced by the confounders. Controlling for confounders, e.g. by including them in an appropriate regression model, can remove this bias. A causal mediator is a variable which appears as a node on a path from a cause to an outcome. Tar deposits in lungs provide a classic example, because tar appears on an important pathway from smoking to lung cancer. A more topical example would be inflammation as a mediator on one path from SARS-CoV-2 infection to death. (The_RECOVERY_Collaborative_Group 2021) Operationally, a mediator can be identified because the portion of the causal effect that occurs via the mediated path can be shut down if the mediator can somehow be prevented from changing despite changes in the earlier cause. This operational definition of mediation is a reminder of why it can sometimes be important to identify mediators. Sometimes it is easier to change the mediators by some intervention than it is to change the preceding cause. For example, intervening on the SARS-CoV-2! death pathway by suppressing the inflammation mediator with dexamethasone is fairly effective (The_RECOVERY_Collaborative_Group 2021), although not nearly as effective as suppressing Covid itself with vaccines would have been. The effect of this intervention provides strong evidence that inflammation is indeed an important mediator, not just a marker for actual causes, a result that could not be determined simply from previously observed correlations. For a contrasting example, the weaker effectiveness of steroids in reducing mortality from sepsis shows that the inflammation accompanying sepsis is not such an important mediator in that analogous case. (Yao et al. 2019) 1/31/22 When it is suspected that one of two variables mediates the other's effects on the outcome, the time order of the two potential causes can eliminate one of the two possible mediation relations, since the mediator must came after its cause. Mere time order does not, however, establish that a mediation relation exists. When, as often happens, the observed variables are not themselves on the causal path but rather are measurements of underlying latent variables, the time order of the measurements does not even constrain the direction of any possible causal relation between those latent variables. For example, if someone measures that they have a fever and then gets a PCR test that is positive for SARS-CoV-2, it would not be correct to use the time order of the variables temperature measurement and viral measurement to conclude that the fever caused the viral infection. The starting time order of the latent variables temperature and infection is opposite to the time order of their measurements, and in this instance the latent time order gives the true causal direction. The portion of a causal effect that occurs via a measured mediator is often termed an "indirect effect". Any remaining causal effect for which no mediator has been identified is then termed the "direct effect". Despite this terminology, it is important to realize that there is no fundamental difference between these types of causation. On the classical scale, all causation proceeds through time-like event paths. Therefore any classical effect can be converted from "direct" to "indirect" by recording and controlling for a big enough set of mediators. Thus, despite the infelicitous standard terminology, the description of a causal effect as "indirect" should not be interpreted to mean that it is any less real than one described as "direct". In general causal estimates of the magnitude of e.g. smoking!cancer or SARS-CoV-2!death will be severely biased if they are based on models in which one controls for mediators, e.g. by including them in a linear regression model. (Rohrer 2018) The effect of smoking cigarettes on lung cancer rates should not be estimated by excluding the large portion of the effect that occurs via tar deposits, since the tar deposits are themselves usually caused by the smoking. We say SARS-CoV-2 causes death, even though if one controlled for known mediators such as low blood oxygen, lung scarring, blood clots, and systemic inflammation, the "direct effect" would be small. Mediators are not the only variables that should not be controlled for if one wishes to use model coefficients to obtain an unbiased causal estimand, although they may be the most obvious such type of variable. Estimands will also be biased by controlling for variables that are affected both by the suspected cause and by other unmeasured causes that have effects on the outcome. Such variables are called colliders because in a causal diagram arrows from the suspected cause and the unmeasured causes collide on them. Controlling for a collider, e.g. by inclusion in a regression model, thus introduces collider stratification bias to the estimand. (Pearl, Glymour, and Jewell 2016;Greenland 2003;Rohrer 2018) One of the best-known illustrations of collider bias is the low birth weight paradox. (VanderWeele 2014) In a model controlling for birth weight, which is a risk factor for infant mortality, maternal smoking becomes negatively associated with infant mortality. This does not mean that smoking is protective, but rather that the model was biased. Out-of-model causes of low birth weight (e.g. certain diseases or types of drug use) cause low birth weight, as does smoking or some variable associated with smoking. Thus birth weight is a collider between smoking and those out-of-model causes. A non-smoker with a low birth weight infant is more likely to have those other causes than is a mother whose smoking already created a likelihood of low birth weight. It turns out that the other causes are more dangerous than smoking, so among low birth weight infants those with non-smoking mothers are less likely to survive. Omitting birthweight from the model gives a positive association between smoking and infant mortality, which gives a more accurate representation of the causal effect. (VanderWeele 2014) In many real-world observational studies, some of the data are missing. It is often advantageous to impute the probability distributions of the missing data using the other data, both to avoid throwing out information and to avoid biased estimates of the true correlations that can arise when the data are not missing completely at random. An easily readable introduction to such imputation has been published in PRPER. (Nissen, Donatello, and Van Dusen 2019) Incorrect imputation methods can introduce bias even when the data are missing completely at random. To explain the key reason, we can consider a toy example, so simple that one would not be tempted to use imputation in practice because it adds nothing to the information available from the complete-case data set. For simplicity I will describe the logic assuming large-N samples of bivariate normal variables, X and Y with correlation coefficient r and each with distribution N(0,1). Say 25% of the X values are missing completely at random, so that the correlation inferred on the 75% of cases that are complete provides an unbiased estimator of the true correlation. Replacing the missing values with values imputed solely from what is known of X just means using imputed values from N(0,1) that are uncorrelated with Y. Including them will be expected to reduce the correlation coefficient to 0.75r in the sample that includes imputed points, i.e. to give an estimate biased toward zero. If instead the missing X's are each replaced with a value imputed from the complete-case probability distribution conditional on both X and Y then the imputed x is of the form ry+(1-r 2 ) 1/2 e, where e is an N(0,1) random variable uncorrelated with X and Y. The correlation on the resulting sample is then expected to be r, an unbiased estimate of the true value. In a more relevant case there are several predictor Xi's with some correlations, not just one X. For simplicity we still assume that data are missing completely at random so that the correlations found in the complete cases are unbiased estimators of the true correlations, and assume multivariate normality so that the covariance matrix contains all the information. Imputing a missing x1 via linear regression using the probability distribution for X1 conditioned on both Y and the other Xi's will still not bias the coefficient estimates since the imputed distribution shares all covariances with the complete-case distribution, by construction of linear regression. Imputing the missing x1 via a probability distribution conditioned only on the other Xi's would reduce the magnitude of the projection of Y onto the X predictor space, i.e. reduce R 2 of the model fit, just as in the toy one-dimensional case. Thus omitting outcome variables from the conditioning biases estimates to become less predictive even in the simplest, least problematic cases. Choosing Variables to Include in Causal Models A recent methods paper (Walsh et al. 2021) attempts to introduce PRPER readers to some of the issues involved in inferring causation from correlations, focusing on bias that can result from omitting variables. Although much of the verbal introduction is correct, the rules about which variables should be controlled for are stated incorrectly and the examples used undercut the 1/31/22 message that one needs to carefully define causal questions before determining the correct combination of correlations needed to infer a causal effect with minimal bias. The title already conveys that bias arises from omitting variables from a model, but as we have seen bias can equally well arise from including inappropriate variables in a model. (Pearl, Glymour, and Jewell 2016;Hernán and Robins 2020;Greenland 2003) (Here and below "including" means controlling for by inclusion in a regression model, the usage in ref. (Walsh et al. 2021).) The following general rule is stated (Walsh et al. 2021): "A variable included in a model will be biased by an omitted variable if the following two conditions are met: (1) the correlation between the omitted variable and the included variable is nonzero, and (2) the "true" effect of the omitted variable on the dependent variable is nonzero. " This rule is simply untrue, since it would often imply that a causal model should include mediators and colliders. (Rohrer 2018) Following this rule would allow one to show that essentially any causal effect is zero by including enough mediators in the model. The introduction (Walsh et al. 2021) promises to give examples using "explanatory" models as distinct from "predictive" models for "testing causal hypotheses". Nevertheless, the examples (Walsh et al. 2021) consistently contrast different "predictive" models without specifying causal patterns. Coefficients in some models are described as "biased" as compared to coefficients in others. This reflects a misunderstanding of what "bias" means. Predictions can be improved by adding parameters, but the resulting predictive coefficients for some suspected cause may either become more biased or less biased estimates of the causal effects of intervening on that cause, depending on the relations among the different variables in a causal diagram. (Pearl, Glymour, and Jewell 2016;Hernán and Robins 2020;Rohrer 2018) The paper goes on to discuss various cases in which adding a variable to a multiple regression model makes either large or small changes in the coefficients of other variables. (Walsh et al. 2021) Throughout, the assumption is made that the extra variables should always be used when they make large changes, in order to approximate the "true" model (quotes in original). (Walsh et al. 2021) The question investigated is "whether omitting a particular variable will lead to bias…" (Walsh et al. 2021) but the other half of the question, whether including a variable will lead to bias, is simply not mentioned. As we have seen, this approach is fundamentally mistaken because inclusion of colliders or mediators as covariates often systematically biases the relevant model coefficient away from the desired causal coefficient. (Pearl, Glymour, and Jewell 2016;Hernán and Robins 2020;Glymour, Zhang, and Spirtes 2019;Glass et al. 2013;Varian 2016;Foster 2010;Gangl 2010;Keele 2015;Greenland 2003;Rohrer 2018). The first example given (Walsh et al. 2021) follows the same logical pattern as the smoking!tar! cancer example. It concerns improved outcomes for males, compared to females, when labs are switched from real-world to virtual reality (VR), i.e. the difference between VR_benefit for males and VR_benefit for females. (Walsh et al. 2021) The wording of ref. (Walsh et al. 2021) gives the impression that the causal question is the effect of gender on VR_benefit. It is not entirely clear to what interventions knowing the causal effect of gender on VR_benefit would be relevant since it is unlikely that anyone deciding on some genderdetermining action would be much concerned with the effects of their decision on optimal physics lab modes. As a purely predictive relation it could be relevant to deciding which students might benefit from which lab mode, but for purely predictive relations one needs only conditional probabilities, not a causal model. A raw comparison of the distributions of VR_benefit for males and females suffices for predicting the effects on these groups. The paper claims that the model should include another variable, video game experience, which correlates very strongly with gender and strongly with VR_benefit. (Walsh et al. 2021) The model including gaming gives results that are contrasted with any "conclusions about biological or sociological differences in men's and women's ability to learn from VR", (Walsh et al. 2021) although it is unclear why gaming is not considered a "sociological difference". The diagram in Fig. 1 shows a common-sensical causal pattern, in which being male leads, in the particular social context, to video gaming. Being male can lead to VR_benefit by the video game path or by other paths so the causal effect of being male on VR_benefit is just given by the unconditional regression coefficient in this simple linear model. Inclusion of gaming as an independent predictor biases the estimate of the causal effect of maleness on VR_benefit because it eliminates one of the causal paths for that effect. Ref. (Walsh et al. 2021) suggests "running the analysis with video game experience in lieu of gender", which corresponds to erasing not only the arrow from gender to gaming but also the arrow from gender to VR_benefit, in effect treating gender as an irrelevant marker for the gaming. Which coefficients in the common-sense model are relevant depends on what action is being considered. Is the potential action choosing whether to offer VR labs at all, choosing which students to encourage to take them, encouraging students to play video games, or something else? To give a not very serious example, if one wants to know whether students should be encouraged to play video games to prepare for VR labs, then one would want to know to what extent gaming is an actual mediator on the gender!VR_benefit path or just a marker for gender. Because the correlation between maleness and video gaming is said to be very high (Walsh et al. 2021 gaming, and response to virtual-reality labs. Gender is an exogenous variable, i.e. its causes are not shared with other variables because at the time of the study they were almost always a quasirandom conception lottery. The Markov-equivalent second graph is the one implicitly used to obtain the Gender! VR_Benefit coefficient in which gaming and gender are treated as correlated variables neither of which causes the other. One interpretation given by (Walsh et al. 2021) would also drop the arrow from gender to VR_benefit, giving an inequivalent graph that would in principle be distinguishable via observed correlations. Identifying mediators and specifying potential outcomes Nevertheless, it is evident that the choice of which correlated variable, if either, to call a mediator cannot be based just on the correlation itself, since that relation is essentially symmetrical: the least-squares estimate of each normalized variable is simply the correlation coefficient r times the other normalized variable, plus a variable that's orthogonal to the predictor variable and has variance (1-r 2 ). This familiar symmetry is just the simplest case in which different causal graphs have exactly equivalent predictions for correlations. (Pearl, Glymour, and Jewell 2016;Richardson 2003) Following the procedure of (Young and Caballero 2021) would produce evidence of "mediation" in all cases where two causal variables are correlated, regardless of whether any actual causal mediation exists. That procedure confuses the estimation of coefficients for a known causal diagram with the discovery of the proper causal diagram. To give a graphical illustration of the issue, I turn to the SEM on the effects of "Race", Fig. 13 of (Young and Caballero 2021), reproduced here as Fig. 2. (In ref. (Young and Caballero 2021) "Race" stands for an aggregate approximately meaning the same as "under-represented minority".) The correlations were interpreted to show that GRE-P mediates the causal effects of GPA on admission, and thus also the effects of "Race" on admission via UGPA. What the correlations themselves show, however, is that "Race" has very little correlation with GRE-P (not statistically significant in the sample) when UGPA is held constant. No discussion is included as to why the relation UGPA!GRE-P is assumed. (Young and Caballero 2021) The mistaken mathematical conception of mediation leads to at least one strange implication. A literal reading of the original SEM, taking its causal diagram seriously, would say that the way to get rid of the negative effect of Race on Admit would be to eliminate GPA-not just to drop it as an admissions factor, which would still allow a negative Race effect to flow through it to GRE, but to actually eliminate it, perhaps by abolishing grades. That would almost entirely block the path for the negative causal impact of Race on Admit, since almost none of that impact bypasses GPA in that SEM. A more conventional view might be that UGPA and GRE-P are correlated because both are affected by a variety of shared factors, e.g. diligence in studying physics. This alternative is illustrated by the second graph in Fig. 2, which is Markov equivalent to the first graph and therefore exactly equally consistent with the correlations. (Pearl, Glymour, and Jewell 2016;Richardson 2003) This modified SEM does not have any peculiar (presumably unintended) implications about what effect eliminating UGPA would have on GRE-P. Although for the most part UGPA is obtained before GRE-P, that has no implications for the causal order, if any, between the latent variables underlying them. This simple conclusion is not included in the paper. For gender, in contrast, such a substitution would favor females, as can be seen from the coefficients in Fig. 12 of ref. (Young and Caballero 2021). Fig. 2. The SEM graph used to describe the causal effect of "race" on admissions, as given in (Young and Caballero 2021) on top, and using a Markov equivalent and seemingly more plausible graph on the bottom. Bidirectional arrows stand for causation by exogenous unmeasured variables, not two-way causation. The latter graph avoids the false implication that dropping GRE-P would reduce the effects of "racial" differences in GPA on admissions. Another fundamental issue arises in the framing of the causal effects, i.e. potential outcomes, of changing admissions criteria. Unsurprisingly, ref. (Young and Caballero 2021) shows that admission probability tends to increase with higher UGPA and GRE-P. The paper emphasizes that "more applicants could be penalized for having a low physics GRE score despite a high GPA than could benefit from having a high physics GRE score despite a low GPA." (Young and Caballero 2021) The net effect on acceptance of changes in acceptable levels for different criteria, however, depends on the ratio of the hypothetical incremental changes for those two criteria and thus cannot be determined without specifying that ratio. Ref. (Young and Caballero 2021) does that specification via arbitrary definitions. More fundamentally, framing the outcome as the total number of accepted applicants does not consider the actual potential outcomes, in which the total number of accepted applicants who can attend graduate school is not set by admission criteria but by limits on funding, mentorship, and job openings. Any change of admission criteria may be judged on a variety of grounds, but these do not include effects on the total number of new graduate students, which can only be changed by other methods. Changing criteria changes which students are accepted, not how many. The comparison of net acceptance rates for different criteria may, however, make sense if the implicit goal is to increase the number of domestic students by reducing the number of international students, who were omitted from the analysis. (Young and Caballero 2021) Implications for use of GRE scores: collinearity, collider bias, and imputation bias Many physics and astronomy departments have dropped or are considering the possibility of dropping use of GREs for admission, dropping either just GRE-P or both GRE-P and the more widely taken quantitative GRE, GRE-Q. (Chawla 2020;Young 2020) Estimating the causal effects of these actions is therefore worthwhile. Some of the pathways by which these choices might affect various outcomes (e.g. by effects on both graduate and undergraduate curricula and grading) are essentially unmeasurable without trying the experiment. Other pathways, such as the potential performance of students who are not currently admitted, may be roughly estimated via model-based extrapolation from results on how well GREs predict graduation in the currently admitted cohort. (Miller et al. 2019;Weissman 2020b;Miller et al. 2020;Weissman 2020a;Verostek, Miller, and Zwickl 2021) Here I address causal issues in a recent PRPER paper (Verostek, Miller, and Zwickl 2021) that uses this latter approach. Although ref. (Verostek, Miller, and Zwickl 2021) uses somewhat vague language to describe its aims (e.g. asking which metric "offers the most insight"), in context (Miller and Stassun 2014;Chawla 2020;Young 2020) there is no doubt that the treatments under consideration are to drop use of GREs in graduate physics admissions. Unlike for the previous two papers (Walsh et al. 2021;Young and Caballero 2021), each of which was characterized by a small number of core errors in causal reasoning, for this paper I will discuss more conceptually disparate errors, united by sharing a sign of their effect in biasing causal estimates. A causal diagram approximately representing the actual admissions policy choices to be made is shown in Fig. 3. Different criteria may be used or not used in choosing whom to admit. These criteria include the ones that are relatively easy to tabulate and study (UGPA, GRE-Q, and GRE-P) and various other factors that are used by most departments (Potvin, Chari, and Hodapp 2017) but are harder to quantify and tabulate across schools and thus comprise out-of-model-predictors (OOMP). The effect of choosing to use or not use a criterion is conditional on choices for the other criteria, since inclusion of a highly redundant predictor has little effect, and redundancy depends on what else is used. To determine the effects of dropping both relevant GREs (Q and P) on the predictability of graduation, one should compare predictions using those two GREs with predictions excluding them but otherwise the same. That direct comparison for graduation rates is not made in the PRPER paper (Verostek, Miller, and Zwickl 2021) or its predecessors (Miller et al. 2020(Miller et al. , 2019, but we shall see that it may be estimated from the data presented. Fig. 4 shows a causal diagram roughly representing the key points of the model for what causes an individual student to be more or less likely to graduate. Coefficients from this model can then be used implicitly or explicitly to estimate the effects of using different metrics in the policy diagram of fig. 3. Fig. 4 encodes some assumptions, probably not quite exactly true but not especially controversial. One is that whether a student is admitted and if so to what rank of department depends on the explicit metrics available to the admissions committees, (Young and Caballero 2021) not on the traits that they partially measure. Another is that the metrics only partially reflect some unspecified traits of the applicants that make them more or less likely to graduate. Correlations between metrics arise because they measure overlapping traits. I arbitrarily use four such traits in fig. 4, enough to allow linear independence of their different combinations in the four metrics. A causal diagram is included as Fig. 5 in ref. (Verostek, Miller, and Zwickl 2021), but since it includes only a single metric, it does not capture that each causal effect in our Fig. 3 is conditional on which other metrics are also used. That dependence approximately corresponds to the dependence of the predictive coefficients on which other predictors are included in a model for individual students like Fig. 4. Thus the diagram used in (Verostek, Miller, and Zwickl 2021) obscures the key point-that dropping either GRE substantially raises the coefficient and the statistical significance of the other GRE for graduation probability. (Weissman 2020b(Weissman , 2020a The effects of dropping one predictor on the other's coefficient are later given (Verostek, Miller, and Zwickl 2021) but only for a different outcome, graduate GPA (GGPA), for which the effect is small because GRE-Q adds little to its prediction, unlike for graduation. To estimate the causal coefficients of the policy-choice model one needs first to correctly estimate the coefficients and statistical uncertainties of the predictive coefficients of the individual-level model. Ref. (Verostek, Miller, and Zwickl 2021) gives coefficients and confidence intervals for the coefficients of GRE-P and GRE-Q for predicting graduation within the stratum of enrolled students within a model including UGPA and various demographic predictors. Within the main text, it is stated at least 12 times that neither of the separate coefficients meets the conventional cutoff for statistical significance. (Verostek, Miller, and Zwickl 2021) In the Supplement, however, we find that when a somewhat less biased method is used for imputing missing data, both coefficients actually do pass the standard "significance" criterion. (Verostek, Miller, and Zwickl 2021) (Here I do not mean to endorse the common nullhypothesis-significance testing approach of making qualitatively dichotomous interpretations of small differences in coefficient p-values (Amrhein, Greenland, and McShane 2019), but merely go along with it to focus on other issues.) In ref. (Verostek, Miller, and Zwickl 2021) as well as in its predecessors (Miller et al. 2020(Miller et al. , 2019 a substantial amount of missing predictive data was filled in using a previously unspecified (Miller et al. 2019) multiple imputation method. Ref. (Verostek, Miller, and Zwickl 2021) now points out that in its main text as well as in the preceding work the imputation model used omitted outcome variables, claiming that "the imputation approach presented here is theoretically sound". The Supplement (Verostek, Miller, and Zwickl 2021) expresses surprise that the "counterintuitive" inclusion of outcome variables results in less biased estimates, based on concern that "Employing a model of data imputation that uses the outcome variable to predict missing values of the independent variable may seem like a self-fulfilling prophecy, guaranteeing a relationship to exist between them." Nevertheless, the Supplement (Verostek, Miller, and Zwickl 2021) concedes that "research suggests that including all variables, including the outcome variable, in the imputation model in fact tends to produce less biased results." We have seen, however, that one needs only algebraic inspection of the simplest models to reveal that imputation methods that omit outcome variables from the conditioning are not "theoretically sound" because they introduce systematic bias. (This issue may provide a clue as to how proficiency in beginning algebra, as measured by GRE-Q, could predict research performance.) The Supplement of ref. (Verostek, Miller, and Zwickl 2021) now gives imputations including conditioning on GGPA. (Verostek, Miller, and Zwickl 2021) Even that partial correction suffices to make GRE-P and GRE-Q "significant" predictors of graduation in the U.S. cohort. It appears, however, that graduation itself was still not included in the imputation method of the Supplement, so the results would still be biased toward weakening the model's predictive power, especially for that outcome. Estimating from blow-ups of figures in the main text and the Supplement as well as from the decrease in p-values and the partial data given on effect size for GRE-P, just conditioning on GGPA raised the sum of the standardized coefficients of GRE-P and GRE-Q for graduation by ~8%. The standardized coefficient of UGPA went up ~50%. Insufficient data are given to tell how much the coefficients for graduation would go up if that imputation were also conditioned on the more relevant outcome, graduation itself. One may get a rough idea from the effect of GGPA conditioning on the coefficients for predicting GGPA. When GGPA was used in the imputation conditioning R 2 for the model increased from 0.11 to 0.17 and the coefficients of UGPA and GRE-P for GGPA went up ~35%. (Verostek, Miller, and Zwickl 2021) (The coefficient of GRE-Q for GGPA was very small and not significant in either imputation. (Verostek, Miller, and Zwickl 2021)) It would be reasonable to guess that the sum of the GRE coefficients for graduation would also go up another 10% to 30% if graduation were properly included in the imputations for that model. The actual value could be calculated easily by the original authors. Some of the graduation outcome data were also missing because at the end of the data window some students had been enrolled for five years but had not yet graduated. (Verostek, Miller, and Zwickl 2021) Although it was stated that roughly 95% of these were likely to graduate, the model treated their graduation rate as 100%, in effect substituting fiveyear survival for graduation as the outcome. (Verostek, Miller, and Zwickl 2021) A less biased result would be obtained by using imputation based on the 95% estimate. The probable effect of this fairly small unnecessary bias, like that of the larger errors, is to understate the predictive power of GREs. (Weissman 2020b(Weissman , 2020a PhD are negligible except as mediated by the necessary step of getting into at least one program, even though those variables can be important markers of traits that do affect graduation rates. The correlations among the predictor variables arise because they share underlying causes. Since there are no data on students who were not admitted to any programs, some bias from stratifying on the collider "rank_enroll" is unavoidable. Stratification into rank tiers or into the narrower strata of individual programs increases that bias. Once the coefficients relevant to fig. 4 have been established, one must estimate their implications for the coefficients of the policy choices of fig. 3. Since GRE-Q and GRE-P are highly correlated, collinearity effects make the separate point estimates and confidence intervals for their coefficients not directly relevant to the question of whether dropping both predictors would significantly reduce predictive power. Estimating the effects of dropping one or both predictors requires a calculation that uses both the coefficients and the correlations among the predictors. A previous analysis (Weissman 2020b(Weissman , 2020a showed unambiguously that the predictive power for graduation would fall significantly if both GREs were dropped, an effect of more than four times the standard statistical error even using the improperly imputed estimates. We shall see that consideration of the information in the new PRPER paper (Verostek, Miller, and Zwickl 2021) will give the same conclusion but with larger effect sizes. Even correct use of within-sample predictive coefficients would still give somewhat biased estimates of how much predictive power would be lost by dropping each predictor, because the data are necessarily restricted to those students who were admitted and then enrolled. (Weissman 2020b(Weissman , 2020a As we see in Fig. 4, graduate admission is a collider between the model variables (UGPA, GRE-P, GRE-Q, demographics) and OOMP, a summary of other admission criteria used by almost all programs. (Potvin, Chari, and Hodapp 2017) Some collider stratification bias is thus inevitable. (Verostek, Miller, and Zwickl 2021) allows an estimate of the magnitude of such collider effects since it gives the results of the most recent analysis and that of a previous one in which extra collider stratification was introduced by dividing the graduate programs into three tiers of rank. (Miller et al. 2019) (The comparison is very slightly complicated because the new estimates appear to use a probit link function and the old ones use a logit link function, but that should have very minor effect on this comparison. Also the new numbers use only ~80% of the old sample, but we are not given any indication that that change affects the comparison. (Verostek, Miller, and Zwickl 2021)) The new less-stratified graduation predictive effect sizes for UGPA, GRE-P, and GRE-Q were increased from those found in the more-stratified model by factors of 1.7, 1.2, and 1.6, respectively. This substantial change illustrates the importance of collider bias. These factors are far more precise than one might guess from a raw combination of the separate confidence intervals on the more and less stratified model results, because both models use the same set of partially random outcomes. The actual narrow confidence intervals for the factors could be determined by a pairwise bootstrap calculation using the actual data sets. We may apply these correction factors to my previous calculation of the effect sizes for the logit of graduation probability as a function of UGPA and equal-weight GRE sum. (Weissman 2020b(Weissman , 2020a) Based on the highly stratified results, the logit effects on going from the 10 th to 90 th percentile among the U.S. test-taking cohort, holding other predictors constant, were previously found to be 0.60 and 0.72 for UGPA and equal-weight GRE sum, respectively. (Weissman 2020b(Weissman , 2020a After correction for the excess stratification of that model each logit effect is 1.0, coincidentally. I suspect that the remaining stratification bias from the necessary absence of OOMP in the newer less-stratified results is not as big as the difference between them and the highly stratified earlier results, in part because the remaining stratification bias is likely to be partly balanced by some confounding bias. (Greenland 2003;Weissman 2020b;Rohrer 2018) Using the new imputation results, partially repaired by including GGPA conditioning, would raise those estimates to 1.5 and 1.1, respectively, for UGPA and GRE_sum. Fully repairing the imputation method would likely raise both effects substantially more. It would be good to see the actual result of that straightforward calculation, preferably on the full data set used in ref. (Miller et al. 2019). Estimating the causal effects of admitting students who are now excluded because of low GRE scores or because they did not take any GREs requires extrapolation of results from students who are currently admitted to that broader cohort. Extrapolation involves use of some model link function, e.g. either logit or probit, to convert predictors to probability. This extrapolation is uncertain, especially for GRE-Q, for which the range among all the students who take the test and are interested in going on in physics is more than twice as large as in the enrolled group. (Miller et al. 2020) Range restriction is much less severe for GRE-P and UGPA. (Miller et al. 2020) Data on the enrolled group can provide at least some hint as to whether that extrapolation is problematic, e.g. by seeing if outcomes for the bottom quintile of those enrolled fall below expectations from the overall model fit. Since extrapolation to the full U.S. cohort, even via the linear logistic model, predicts bigger effects for GRE-Q than for GRE-P (Miller et al. 2019), it would be particularly important to see a quintile breakdown for GRE-Q. The Supplement to ref. (Verostek, Miller, and Zwickl 2021) provides such graduation results for quintiles of UGPA and GRE-P but omits the potentially more important results for GRE-Q. We can now turn to the question of whether dropping just the GRE-P while retaining GRE-Q would significantly reduce predictive power. According to the analysis in the Supplement (Verostek, Miller, and Zwickl 2021), GRE-P has a significant predictive coefficient in the full model. From results (Miller et al. 2020) for the Akaike Information Criterion (Akaike 1974), dropping both GREs would give a much worse model overall and within the U.S. cohort but dropping just GRE-P only gives a marginally worse U.S. model. The loss of predictive power should be somewhat less than the point estimate of the GRE-P predictive coefficient might lead one to expect. (Weissman 2020b(Weissman , 2020a The reason again is that once one predictor is dropped, the coefficients of the other predictors change. GRE-Q and UGPA, both correlated with GRE-P, would take up some of the slack left by dropping GRE-P. (Weissman 2020b) That's also why if one GRE is dropped, the incremental predictive power obtained from the remaining one is increased, so that dropping the second GRE gives a larger loss than its coefficient in the full model would suggest. (Weissman 2020b(Weissman , 2020a The primary motivation for examining the effects of using GREs on admissions is stated to be concern for under-represented minorities (URM, approximately corresponding to "Race" in ref. (Young and Caballero 2021)) and for females. (Verostek, Miller, and Zwickl 2021) The effect of dropping GREs on URM admissions will depend on the balance of which other factors are used to replace them. Although ref. (Verostek, Miller, and Zwickl 2021) does not specify what combination would be used, it emphasizes the superiority of UGPA as a predictor. We have seen, however, that in the cohort studied in ref. (Young and Caballero 2021) the standardized prediction coefficient for "Race"!UGPA is more negative than that for "Race"!GRE-P. (The coefficient for GRE-Q was not given.) Thus not only would dropping GREs have a greater negative effect on graduation probability than suggested in ref. (Verostek, Miller, and Zwickl 2021), but it could also have a negative effect on URM admission if more weight is placed on UGPA. On the other hand, the GREs (especially GRE-P) , unlike UGPA, do systematically underpredict both female graduation rate and GGPA. (Verostek, Miller, and Zwickl 2021) Female admissions could be increased by dropping or de-emphasizing them. The same goal could also be achieved by gender-norming those scores, without losing the significant information provided by the GREs. It seems likely that something like that norming was already being done in the period for which data were collected, since the overall female GGPA is insignificantly less than the male GGPA. (Verostek, Miller, and Zwickl 2021) Without something like gender normed admissions, the female GGPA would have been expected to be higher than the male GGPA, since it is significantly higher than predicted by a UGPA-GRE model. (Verostek, Miller, and Zwickl 2021) Much of ref. (Verostek, Miller, and Zwickl 2021) is devoted to a "mediation analysis" in which GGPA is treated as a potential mediator on the paths from UGPA, GRE-Q and GRE-P to graduation. The predictive effects of UGPA and GRE-P for graduation are described as almost entirely "mediated" by GGPA because controlling for GGPA removes almost all of the dependence of graduation probability on UGPA and GRE-P in the model. Just as in the mediation cases we examined above, however, that result does not show whether GGPA actually mediates the causal effects on graduation of the traits measured by UGPA and GRE-P or merely serves as another marker for those traits. Only to the extent that GGPA actually mediates the causation or serves as a marker for true mediators, as opposed to being a marker for pre-existing traits, would interventions that raise it (e.g. better classroom teaching) increase graduation rates. (Interestingly, controlling for GGPA scarcely changes the dependence of graduation probability on GRE-Q in the model, suggesting that completing a PhD partly depends on some traits measured by GRE-Q that would not be changed much by changing classroom teaching.) UGPA and GRE-P are described as providing only "indirect prediction", an unusual cross between causal and predictive terminology. The use of causal terms, "mediation" and "indirect", in describing a purely predictive relation, is somewhat misleading. The pre-admission predictors are no more or less predictive of graduation than they would be if post-admission GGPA had not been subsequently recorded. The methodological errors in ref. (Verostek, Miller, and Zwickl 2021) are important in themselves, but in this case, the substantive conclusions are also important, so a summary of the key corrected conclusions may be useful. Unlike UGPA, GREs remain useful in the full cohort, not just the domestic subset. Within the group of all applicants, GREs provide more incremental predictive power for PhD attainment than does UGPA. Within the cohort of U.S. applicants' GREs provide about the same incremental predictive power as UGPA. Within the cohort of domestic students who are interested in physics graduate school and who have the same UGPA, dropping both GREs would lose an odds ratio of at least a factor of three in estimating how the probability for graduation varies between high-scorers and low-scorers. Dropping only GRE-P would lose substantially less predictive power than dropping both GREs. Discussion We have seen several major errors concerning causal inference in recent PRPER papers, including in explicit methods sections. The errors are directly consequential for estimating the effects of policy choices, including a major policy choice under active consideration. These errors reinforce the impression given by different errors in other papers (Weissman 2021) that the PER field needs better knowledge of causal inference techniques. In all the papers discussed here and in my previous PRPER article (Weissman 2021), there was no clear specification of which actual interventions and consequences would be considered. Instead, implicit rather than explicit assumptions were used to justify applying more or less causal language to collections of correlations. Policy recommendations were then intuited or implied, leaning on causal interpretations of some conveniently chosen correlations. Although a similar description might apply to much other social science research, especially from before about 1980, many fields have now moved well beyond that. (Pearl, Glymour, and Jewell 2016;Hernán and Robins 2020;Glymour, Zhang, and Spirtes 2019;Glass et al. 2013;Varian 2016;Foster 2010;Gangl 2010;Keele 2015;Murnane and Willett 2011;Rohrer 2018;Imbens and Rubin 2015) PER should join them. Perhaps the most important lesson is that one should start with an explicit question about reasonably well-defined potential consequences of reasonably well-defined choices of potential actions. (Robins and Weissman 2017) Even when a randomized controlled trial is infeasible, just trying to imagine one can help to clarify what question is being asked. (Hernán 2021) Although explicit assumptions are still needed to pare down the possibilities to something tractable, that starting clarity can go a long way toward guiding the type of causal graphs that are needed. Graphs then help to see what combinations of correlations provide the best estimate of the effects of possible actions. Some procedural changes might help reduce the current problems. One might be to preregister protocols for studies.(Chambers 2019) That allows some refereeing of methods before data are gathered and before particular methods are closely tied to particular conclusions. That helps limit cherry-picking methods and data, reducing p-hacking and reverse p-hacking (Chuard et al. 2019). Even with pre-registration, however, one still needs a community of reviewers and readers who can distinguish between valid and invalid methods. Establishing that may require help from outside PER. ), that may be hard to disentangle. Nevertheless, clarity about what actions are under consideration would help to guide which coefficients in which diagrams are relevant. Although the paper is not proposing anything that would actually affect choices about interventions on gender, the method used for estimating the effects of gender would consistently give incorrect causal coefficients for treatments on which interventions are possible. The causal question addressed is precisely analogous to the more serious one of finding whether inflammation is an actual mediator of SARS-CoV-2! death, which determines whether intervening with dexamethasone is useful. For an educational policy example, inclusion of time spent viewing lectures in a multiple regression model would bias the estimand of the effects of two different video lecture styles on exam scores, because that variable would be a mediator on the path from style to scores. Viewing time would become relevant only if there were another way of intervening on it. Fig. 1 . 1The top figure represents a common-sense view of a causal relation between gender, A recent PRPER paper examined the causes of admissions to selective graduate programs in physics, focusing on the roles of Graduate Record Exams in physics (GRE-P) and undergraduate grade point averages (UGPA), employing a structural equation model (SEM) analysis(Bollen and Pearl 2013) of the causal role of these variables and others on admissions.(Young and Caballero 2021) The causal theme is to see what effects dropping GRE-P would have on admissions, and in particular whether GRE-P is helping some students be accepted who might otherwise be overlooked. I shall discuss here only problems in the translation of statistical claims to causal claims, not those in the conversion of data to statistical claims. These problems include an incorrect rule for identifying causal mediators and unrealistic framing of the potential outcomes for the interventions under consideration.The SEM analysis emphasized testing for mediation.(Young and Caballero 2021) Although the verbal definition of causal mediation is presented correctly, the mathematical method presented for identifying mediators treats one causal variable (e.g. GRE-P) as mediating a path from another cause (e.g. UGPA!admission) whenever the "mediator" can be expressed in terms of the other cause plus a random component, as explicitly shown in their Equation 2.(Young and Caballero 2021) The logistic regression SEM analysis assumed that GRE-P mediates UGPA!admission and concluded that the mediation effect was large because the dependence on UGPA of the logit for the admissions probability changed substantially when GRE-P is held constant.(Young and Caballero 2021) The correlations encoded in the original SEMs have other, less peculiar, policy implications than the SEMs themselves. The original SEM and the revised one are Markov equivalent, so they share the same correlations between Race and the variables UGPA and GRE-P, i.e. -0.48 and-weight on UGPA could plausibly have decreased the admission of under-represented minorities in this cohort, a conclusion that matters for the policy choices under consideration. Fig. 3 . 3This graph illustrates the effects of possible choices of admissions criteria on the net graduation rate for a particular program. Out-of-model predictors (OOMP) are available to admissions committees but not to subsequent modelers. Fig. 4 . 4This graph illustrates how the variables available to the admissions committees and the subset available for the overall model are related to the probability of graduation. 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[ "Asymptotic approach for backward stochastic differential equation with singular terminal condition ", "Asymptotic approach for backward stochastic differential equation with singular terminal condition " ]	[ "Paulwin Graewe \nDeloitte Consulting GmbH\nKurfürstendamm 2310719BerlinGermany\n", "Alexandre Popier \nLaboratoire Manceau de Mathématiques\nLe Mans Université\nAvenue O. Messiaen72085Le Mans cedex 9France\n" ]	[ "Deloitte Consulting GmbH\nKurfürstendamm 2310719BerlinGermany", "Laboratoire Manceau de Mathématiques\nLe Mans Université\nAvenue O. Messiaen72085Le Mans cedex 9France" ]	[ "Mathematics Subject Classification. 34E05, 60G99, 60H10" ]	In this paper, we provide a one-to-one correspondence between the solution Y of a BSDE with singular terminal condition and the solution H of a BSDE with singular generator. This result provides the precise asymptotic behavior of Y close to the final time and enlarges the uniqueness result to a wider class of generators.	10.1016/j.spa.2020.12.004	[ "https://arxiv.org/pdf/1906.05154v2.pdf" ]	186,206,875	1906.05154	f7f3a6e78e04a17e31ac8066e69b8ec55272de78	Asymptotic approach for backward stochastic differential equation with singular terminal condition * 2010 Paulwin Graewe Deloitte Consulting GmbH Kurfürstendamm 2310719BerlinGermany Alexandre Popier Laboratoire Manceau de Mathématiques Le Mans Université Avenue O. Messiaen72085Le Mans cedex 9France Asymptotic approach for backward stochastic differential equation with singular terminal condition * Mathematics Subject Classification. 34E05, 60G99, 60H10 2010Backward stochastic differential equationsingular terminal conditionasymp- totic approachsingular generator In this paper, we provide a one-to-one correspondence between the solution Y of a BSDE with singular terminal condition and the solution H of a BSDE with singular generator. This result provides the precise asymptotic behavior of Y close to the final time and enlarges the uniqueness result to a wider class of generators. Introduction This paper is devoted to the study of the asymptotic behavior of the solution of backward stochastic differential equations (BSDEs) with singular terminal condition. We adopt from [26] and [21] the notion of a weak (super) solution (Y, Z) to a BSDE of the following form − dY t = 1 η t f (Y t )dt + λ t dt − Z t dW t(1) where W is a d-dimensional Brownian motion on a probability space (Ω, F, P) with a filtration F = (F t ) t≥0 . The filtration F is the natural filtration generated by W and is supposed to be complete and right continuous. The function f : R → R is called the driver (or generator) of the BSDE. The particularity here is that we allow the terminal condition ξ to be singular, in the sense that ξ = +∞ a.s. Since the seminal paper by Pardoux and Peng [23] BSDEs have proved to be a powerful tool to solve stochastic optimal control problems (see e.g. the survey article [8] or the book [25]). BSDEs with singular terminal condition provide a purely probabilistic solution of a stochastic control problem with a terminal constraint on the controlled process. The analysis of optimal control problems with state constraints on the terminal value is motivated by models of optimal portfolio liquidation under stochastic price impact. The traditional assumption that all trades can be settled without impact on market dynamics is not always appropriate when investors need to close large positions over short time periods. In recent years models of optimal portfolio liquidation have been widely developed, see, e.g. [1], [2], [10], [11], [15], or [19], among many others. In [4], the following problem is considered: minimizing the cost functional J(X) = E T 0 (µ s \|α s \| p + λ s \|X s \| p ) ds(2) over all progressively measurable processes X that satisfy the dynamics X s = x + s 0 α u du with the terminal constraint that X T = 0 a.s. Here p > 1 and the processes µ and λ are non-negative and progressively measurable. In this framework the state process X denotes the agent's position in the financial market. At each point in time t she can trade in the primary venue at a rate α t which generates costs µ t \|α t \| p incurred by the stochastic price impact parameter η t . The term γ t \|X t \| p can be understood as a measure of risk associated to the open position. J(X) thus represents the overall expected costs for closing an initial position x over the time period [0, T ] using strategy X. In [4], optimal strategies and the value function of this control problem (2) are characterized with the BSDE − dY t = −(p − 1) Y q t µ q−1 t dt + λ t dt − Z t dW t(3) with lim t→T Y t = +∞. Here q > 1 is the Hölder conjugate of p. The generator f is here a polynomial function. Variants of the position targeting problem (2) have been studied in [5], [12], [13] or [29]. Note that these problems are particular cases of the stochastic calculus of variations (see [3]). Let us explain the methodology to obtain a solution for the BSDE (3). The most common approach in the literature is the so-called penalization approach, see, e.g., [26], [27], [4], [12], [21], and the references therein. The idea of the penalization approach is to relax the binding liquidation constraint by penalizing open position in the underlying liquidation problem. In [4], as in [21] for more general driver, the authors use the penalization approach, replacing the singular terminal by a constant n and letting n go to +∞. The convergence is obtained by a comparison principle for solution of BSDEs (see [20] or [24]). In [13], the approach consists in the study of the precise asymptotic behavior at time T of the solution Y of (3). Roughly speaking, the major singular term of Y is then removed to obtain a non-singular problem. The key of this asymptotic approach is to establish sharp a priori estimates of the singular solution at the terminal time. In [13], the authors consider a time-homogeneous Markov setting and obtain the a priori estimates and uniqueness by establishing a general comparison principle for singular viscosity solution to (3). This results are based on timeshifting arguments, applied similar before in [26], which in general do not apply in a nontime-homogeneous setting. However, it is outlined in [13] how the shifting argument may be applied in non-Markov settings to obtain sharp a priori estimates of the singular solution to (3). One major result of [13] is the uniqueness of the solution of (3) (under boundedness assumptions on the coefficients µ and λ). Let us outline in which directions our findings generalize some results from these papers. In [21] the generator may depend also on Z in a non trivial way; here our generator has a special form. However in the previously mentioned papers f is assumed to be a polynomial function or of polynomial growth w.r.t. y, that is f (y) ≤ −y\|y\| q . Here we essentially assume that 1/f is integrable on the neighborhood of +∞. If η and λ are deterministic, the BSDE becomes an ODE and this condition is necessary and sufficient to ensure that the solution can be equal to +∞ at time T , but finite at any time t < T . Under this condition (called (C1)), we prove existence of a minimal solution (Y, Z Y ) of the BSDE (1). The function f (y) = −(y + 1)\| log(y + 1)\| q is an example satisfying (C1) but not covered by the preceding papers. Our second main result concerns the decomposition of this minimal solution. We prove that Y is equal to: Y t = φ E T t 1 η s ds F t + ψ T − t η ⋆ H t , ∀t ∈ [0, T ],(4) where • φ solves the ODE: φ ′ = f • φ with initial condition φ(0) = +∞ and ψ = −φ ′ , • η ⋆ is the deterministic upper bound on the process η. The process H is the minimal non-negative solution of a BSDE with terminal condition 0 and with a singular generator F in the sense of [17] (Theorem 1). As a consequence, we provide a one-to-one correspondence between the BSDE (1) with singular terminal condition and a BSDE with singular generator. We give a self-contained construction of the solution (H, Z H ) (without any reference to Y ) which extends the existence result of [17]. The asymptotic behavior follows from the boundedness of the process ψH/φ on some neighborhood of T : φ(A t ) = φ E T t 1 η s ds F t ≤ Y t ≤ (1 + κ)φ E T t 1 η s ds F t , where the constant κ depends on the coefficients η, λ and f . At this stage it is important to note that there is some asymmetry in (4) since the first term with φ is random, whereas the second with ψ is deterministic. However this method avoids assuming extra assumptions on f . To deal with a symmetric expansion, we suppose that f is concave and we decompose Y as follows: for any t ∈ [0, T ] Y t = φ E T t 1 η s ds F t + ψ E T t 1 η s ds F t H t = φ(A t ) + ψ(A t )H t .(5) Again H solves a BSDE with a singular generator. As in the case (4) (2). Here the proof is only based on the comparison principle for BSDEs. The condition (H) is a stronger Novikov condition. For a general process η, this assumption may be false; some regularity on its Malliavin derivative is required in (H). Thereby, assuming that η is an Itô process, we provide sufficient conditions under which (H) holds. Under this Itô setting with bounded coefficients, we provide another decomposition of Y , where again H is the unique solution of a BSDE with a singular generator, but without the troubling linear part. Up to now, the construction of H is based on the comparison principle for BSDEs and H is the monotone limit of a sequence of solutions of "standard" monotone BSDEs. In the power case f (y) = −y\|y\| q , we follow the arguments of the paper [13] for a PDE and show that the process H can be obtained by Picard iterations in the suitable space H. This construction has two main advantages: first we have a more accurate behavior of H at time T , secondly this construction is more tractable for numerical approximation. In addition to the precise behavior of the solution Y , that is the behavior of the value function of the control problem (2) in the power case, or the uniqueness result for (1), our result establishes a link between Y and H. The main drawback for BSDE with singular terminal condition is the lack of approximation scheme with some rate of convergence. Moreover most of numerical schemes for BSDE are based on backward induction starting at the terminal value. The correspondence between Y and H could be a promising solution for numerical scheme, since the terminal value of H is zero. The singularity of the generator of H is a serious obstacle. But if H is obtained by a fixed point argument in a weighted space, we strongly believe that it could be a way to compute H, and thus Y . This point is left for further research. The paper is decomposed as follows. In the next section we explain our assumptions on the coefficients η, λ and f of the BSDE (1). The reader finds here several examples of functions f , for which the asymptotic behaviour of Y holds. Let us emphasize these assumptions only imply the behaviour of f on an interval [R, +∞) for R sufficiently large. In Section 3, we recall and extend several results concerning the existence of the solution (Y, Z Y ) of the BSDE (1) with singular terminal condition +∞ and provide some a priori estimates on this solution Y and on Z Y . Section 4 is dedicated to the decomposition (4), by proving the existence of the minimal non-negative solution of the BSDE (Equation (21)) with a singular generator and with terminal condition 0 (Theorem 1) and the one-toone correspondence between the minimal solutions Y and H. In Section 5, we study the symmetric decomposition (5) of Y and prove uniqueness of the solutions Y for BSDE (1) and H for the BSDE (42). In the power case we prove that H can be constructed by a Picard approximation scheme. In the last section, we briefly explain the relations between the different expansions of Y . Let us emphasize that all results from Section 4 are ordered from the more general to the less general drivers. In the continuation, unimportant constants will be denoted by C and they could vary from line to line. Assumptions on the generator In the BSDE (1), the generator is of the form: (ω, t, y) → 1 η t (ω) f (y) + λ t (ω). In the rest of the paper, the following conditions hold: (A1) There exist three constants 0 < η ⋆ < η ⋆ and λ ≥ 0 such that a.s. for any t η ⋆ ≤ η t ≤ η ⋆ , 0 ≤ λ t ≤ λ . (A2) The function f is continuous and non increasing, with f (0) = 0 and with continuous derivative. Supposing that f is continuous and non increasing 1 is coherent with the existence and uniqueness results concerning monotone BSDEs (see [24,Chapter 5.3.4]). Note that if f (0) = 0, then f (y) η t + λ t = f (y) − f (0) η t + λ t + f (0) η t = f (y) η t + λ t , provided that λ t ≥ 0. The non-negativity of λ is natural for the control problem (2) and leads to a more accurate expansion of Y . However it is not necessary (see Section 6.1 for a short discussion on this point). Somehow and to summarize, the only stronger condition on our type of generator is the regularity: f ∈ C 1 (R). Now let us consider the ordinary differential equation (ODE): y ′ = −f (y) with the terminal condition y(T ) = +∞. There exists a solution if and only if the function G given by: G(x) := ∞ x 1 −f (t) dt 1 For monotone BSDEs, the classical assumption is: for some µ ∈ R and any (y, is well-defined at least on some interval (κ, +∞), with κ = sup{y ≥ 0, f (y) = 0}, meaning that f ≡ 0 on [0, κ]. Note that the function G is positive, strictly decreasing and convex, such that G(∞) = 0 and the smoothness of f implies that G(κ) = +∞. Then the solution y is given by: y ′ ) ∈ R 2 , (f (y) − f (y ′ ))(y − y ′ ) ≤ µ(y − y ′ ) 2 .y(t) = G −1 (T − t) on [0, T ]. Defining f κ (x) = f (x+ κ), G κ (x) = G(x+ κ) yields that G κ is defined on (0, +∞). Moreover the solution y is given by: y(t) = G −1 (T − t) = (G κ ) −1 (T − t) − κ and solves y ′ = −f κ (y) together with y(T ) = +∞. Hence w.l.o.g. we assume from on now that: (A3) For any x > 0, the function G(x) := ∞ x 1 −f (t) dt is well-defined on (0, ∞). We define the two functions: φ(x) := G −1 (x) > 0, ψ(x) := −φ ′ (x) > 0.(6) The function φ being decreasing and C 2 on (0, ∞) solves φ ′ = f • φ. Under the previous conditions, there exists a minimal non-negative solution (Y, Z Y ) for the BSDE (1) (Proposition 1), and Y verifies the a priori estimate (19). Note that we can extend the result when the generator also depends in a particular way on Z (see Remark 1). For the asymptotic behavior of Y , we also consider the next condition: (C1) There exists a constant δ > 0 and R > 0 such that x → G(x) −δ = ∞ x 1 −f (y) dy −δ is convex on [R, +∞). Let us emphasize that this condition only involves the function f on some interval [R, +∞) and the value of R may be large. From Lemma 5, Condition (C1) is equivalent to the boundedness of x → −x φ ′′ (x) φ ′ (x) on a neighborhood of zero. Example 1 If f (y) = −(y + 1)\| log(y + 1)\| q for some q > 1 and y ≥ 0, all conditions (A1), (A2) and (A3) are verified and if p is the Hölder conjugate of q then ∀x > 0, G(x) = 1 q − 1 log(x + 1) 1−q . Thereby φ(x) = exp ((q − 1)x) 1−p − 1. Direct computations show that − φ ′′ (x) φ ′ (x) x = (p − 1) p x −p + p is not a bounded function near zero and for any δ > 0, G −δ is not convex. Somehow this function f is "not enough non linear". Example 2 Here we study several functions f , ordered by their "non linearity". All of them verify (C1). • If f (y) = −y\|y\| q for some q > 0, then G(x) = 1 qx q and φ(x) = 1 qx 1 q . The assumption (C1) holds for any δ > 0 and − φ ′′ (x) φ ′ (x) x = q + 1 q . • If f (y) = −(exp(ay) − 1) for some a > 0, then G(x) = − 1 a log 1 − e −ax = G −1 (x) = φ(x). And − φ ′′ (x) φ ′ (x) x = axe ax e ax − 1 ∼ x→0 1, is bounded near zero. • If f (y) = − exp(ay 2 ) for some a > 0 and y ≥ 0 (note that f (0) = −1 to simplify the computations). Then G(x) = π a 1 − N (x √ 2a) , φ(x) = 1 √ 2a N −1 1 − x a π , where N (·) is the cumulative distribution function of the normal law. Thereby φ ′ (x) = − exp(aφ(x) 2 ), φ ′′ (x) = 2aφ(x)(φ ′ (x)) 2 , and with √ aφ(x) = z/ √ 2 − φ ′′ (x) φ ′ (x) x = 2axφ(x) exp(aφ(x) 2 ) = z √ 2aG(z/ √ 2a) exp(z 2 /2) = z √ 2π exp(z 2 /2) [1 − N (z)] ∼ z→+∞ 1. using the classical tail estimate of the normal law. Hence xφ ′′ (x)/φ ′ (x) is bounded near zero and (C1) holds (again by Lemma 5). Let us define A t = E T t 1 η s ds F t (7) together with 2 φ t = φ(A t ), ψ t = ψ (A t ) .(8) Let us emphasize that (t → φ t , t ≥ 0) and (t → ψ t , t ≥ 0) are processes. Remark that from the boundedness of η 1 η ⋆ (T − t) ≤ A t = E T t 1 η s ds F t ≤ 1 η ⋆ (T − t).(9) From the monotonicity of φ and ψ, we get φ ⋆ (t) = φ T − t η ⋆ ≤ φ t ≤ φ T − t η ⋆ = φ ⋆ (t).(10) and ψ ⋆ (t) = ψ T − t η ⋆ ≤ ψ t ≤ ψ T − t η ⋆ = ψ ⋆ (t).(11) We introduce our next condition on f : (C2) For some p ≥ 1, some τ < T and for any r ≥ 0 E T τ f (φ t + r) ψ ⋆ (t) dt p < +∞. Since the process φ is bounded from above by φ ⋆ , (C2) holds if T τ −f (φ ⋆ (t) + r) ψ ⋆ (t) dt < +∞ since this integral w.r.t. t is now deterministic. Hence (C2) depends only on the behavior of f on a neighborhood of +∞. In particular if the function −f is submultiplicative: −f (x + y) ≤ C(−f (x))(−f (y)) for some fixed constant C, then − f (φ ⋆ (t) + r) ψ ⋆ (t) ≤ C(−f (r)) −f (φ ⋆ (t)) ψ ⋆ (t) ≤ C(−f (r)). In Remark 3, we show that all functions of Example 2 verifying (C1), also satisfy (C2), even if they are not submultiplicative. In Section 5, we add several conditions on f : (C3) f is concave and of class C 2 on (0, +∞). (C4) If F is the increasing and concave function F : x → G −1 (x −1/δ ) for x > 0, then (−f ) • F is also increasing and concave on a neighborhood of +∞. From (C1), we know that F is increasing. Since f is concave, −f ′ is a non-decreasing function and there exists a rank such that for any x greater than this rank, −f ′ > 0. In other words (−f ) • F is an increasing function, at least on a neighborhood of ∞. Hence the main assumption in (C4) is the concavity of −f • F. We prove that all functions of Example 2 satisfying (C1) also verify (C3) and (C4) (see computations after Lemma 11). Finally our last condition on f is the following. Let us define for some ρ ∈ (0, 1), the non-negative function h(y) = −f (y)G(y) 1−ρ . (C5) There exists ρ ∈ (0, 1) such that the function y → y h(y) remains bounded on a neighborhood of +∞. h ′ (y) = G(y) −ρ −f ′ (y)G(y) − (1 − ρ) = G(y) −ρ G ′′ (y)G(y) + (ρ − 1)(G ′ (y)) 2 (G ′ (y)) 2 , the non-negativity of h ′ is equivalent to the non-negativity of the second derivative of G ρ . In other words h is non decreasing if and only if G ρ is convex. For all functions of Example 2, direct computations show that lim y→+∞ y h(y) = 0, for any ρ ∈ (0, 1). Hence (C5) holds. BSDEs with singular terminal condition In this section the assumptions (A1), (A2) and (A3), except for the last result (Lemma 4) where we suppose besides that f is concave and that (C1) is verified. Let us introduce the following spaces for p ≥ 1. • D p (0, T ) is the space of all adapted càdlàg 3 processes X such that E sup t∈[0,T ] \|X t \| p < +∞. • H p (0, T ) is the subspace of all predictable processes X such that E T 0 \|X t \| 2 dt p 2 < +∞. • S p (0, T ) = D p (0, T ) × H p (0, T ) and S ∞ (0, T ) = p≥1 S p (0, T ). From [21,Theorem 1], if f (y) ≤ −y\|y\| q for some q > 0, we know that the singular BSDE (1) has a minimal solution (Y, Z Y ), in the sense of the next definition: Definition 1 (BSDE with singular terminal condition) The process (Y, Z Y ) is a so- lution of the BSDE (1) with terminal condition +∞ if: • For any ε > 0, (Y, Z Y ) ∈ S ∞ (0, T − ε); • for any 0 ≤ s ≤ t < T , Y s = Y t + t s 1 η u f (Y u ) + λ u du − t s Z Y u dW u ; • Y t ≥ 0 a.s. for any t ∈ [0, T ] ; • a.s. lim t→T Y t = +∞. Minimality means that for any other process ( Y , Z) satisfying the previous four items, a.s. Y t ≥ Y t for any t. Moreover from [13,Theorem 6.3], if f (y) = −y\|y\| q , the solution is unique. To obtain the existence of a solution, we need some a priori estimates on Y (see [4], [13] or [21]). Using the arguments of [13, Proposition 6.1], we obtain that if f (u) ≤ −u\|u\| q , any solution of the BSDE (1) satisfies: Y t ≤ 1 (T − t) q † E T t η s q 1 q + (T − s) q † λ s ds F t(12) where q † is the Hölder conjugate of q + 1. Under our setting and from this estimate we have: Y t ≤ η ⋆ q(T − t) 1 q + (T − t) (q † + 1) λ = φ T − t η ⋆ + (T − t) (q † + 1) λ . The goal of this section is to extend these results to our class of drivers. Let us first begin with a lower bound, similar to [4, Estimate 3.7], but for a more general driver f . Lemma 1 The minimal solution Y satisfies a.s. for any t ∈ [0, T ], Y t ≥ φ t = φ (A t ) .(13) Proof. Indeed the process A satisfies − dA t = 1 η t dt + Z A t dW t(14) for some Z A ∈ H 2 (0, T ). For some L > 0, define A L t = 1 L + A t . Since φ is a smooth function, if U L = φ(A L ), Itô's formula leads to −dU L t = φ ′ (A L t ) 1 η t dt + Z A t dW t − 1 2 φ ′′ (A L t )(Z A t ) 2 dt = 1 η t f (U L t )dt + Θ t dt + Z U L dW t . Note that φ ′′ (x) = f ′ (φ(x))f (φ(x)) ≥ 0, thus Θ t ≤ 0. Since U L T = φ(1/L), from the comparison principle for monotone BSDE (see [24,Proposition 5.34]) and the construction of Y by approximation, we obtain that Y t ≥ U L t . Passing through the limit on L leads to the conclusion. Let us now give an upper bound on Y , similar to [13, Proposition 6.1] but again for a general driver f . Let us consider the function G(x) = ∞ x −1 λ + f (y) η ⋆ dy = η ⋆ ∞ x 1 −C − f (y) dy defined on the interval (Υ = f −1 (−C), +∞) with C = λ η ⋆ . Since f is a function with continuous derivative, G(Υ) = +∞. If we define ϑ = G −1 , this function is well-defined on (0, +∞), with ϑ(0) = +∞ and satisfies: ϑ ′ = λ + f (ϑ) η ⋆ .(15) Note that the function ϑ strongly depends on η ⋆ , λ and f . Lemma 2 Assume that the process (U, Z U ) satisfies the dynamics: for any ε > 0 and 0 ≤ t ≤ T − ε U t + ζ t = U T −ε + T −ε t λ s + 1 η s f (U s ) ds − T −ε t Θ s ds − T −ε t Z U s dW s ,(16) where ζ and Θ are two non-negative processes. Then a.s. for all t ∈ [0, T ), 0 ≤ U t ≤ ϑ(T − t).(17) Proof. We proceed as in the proof of [13, Proposition 6.1], namely we shift the singularity. Take any 0 < ε such that 0 ≤ T − ε < T . The function (ϑ (T − ε − t) , t ∈ [τ, T − ε]) solves the ODE: y(T − θ) = +∞ and y ′ = − λ − f (y) η ⋆ . By the comparison principle again we have that U t ≤ ϑ (T − ε − t) on [0, T − ε]. Since U does not depend on ε, we obtain that a.s. ∀t ∈ [0, T ), U t ≤ ϑ (T − t) . This achieves the proof of the lemma. As a by-product, our proof implies that for any non-negative solution (Y, Z Y ) of the BSDE (1), we have a.s. on [0, T ]: φ ⋆ (t) = φ T − t η ⋆ ≤ φ t ≤ Y t ≤ ϑ t = ϑ(T − t).(18) The first inequality comes from (10). Compared to (12), in the power case f (y) = −y\|y\| q , this estimate is less accurate. However it holds for functions without polynomial growth. For the upper bound, note that if λ = 0, then ϑ = ψ ⋆ . In general we have Lemma 3 For any ε > 0, there exists a deterministic time T ε ∈ [0, T ) such that a.s. for any t ∈ [T ε , T ] Y t ≤ φ T − t (1 + ε)η ⋆ = φ ⋆ ε (t).(19) Proof. We write G(x) = η ⋆ ∞ x 1 −C − f (y) dy = η ⋆ G(x) + η ⋆ C ∞ x 1 (−C − f (y))(−f (y)) dy. Therefore η ⋆ G(x) ≤ G(x) and G(x) ≤ η ⋆ G(x) + η ⋆ C 1 (−C − f (x)) G(x) = η ⋆ G(x) −f (x) −C − f (x) . We deduce that on the interval (f −1 (−(1 + ε)C), +∞), η ⋆ G(x) ≤ G(x) ≤ (1 + ε)η ⋆ G(x), and thereby on the neighborhood of zero (0, G −1 (f −1 (−(1 + ε)C)/η ⋆ )) = (0, Υ),G −1 (x/η ⋆ ) ≤ ϑ(x) ≤ G −1 (x/((1 + ε)η ⋆ )) = φ(x/((1 + ε)η ⋆ )) . Thus provided that T ε = T − Υ ≤ t ≤ T Y t ≤ ϑ(T − t) ≤ φ ⋆ ε (t) . Hence we deduce that there exists a constant τ ∈ [0, T ) such that a.s. for any t ∈ [τ, T ] φ ⋆ (t) ≤ Y t ≤ φ ⋆ 1 (t) with two deterministic functions φ ⋆ and φ ⋆ 1 on a deterministic neighborhood 4 of T . Let us state the following result. Note that if f (y) ≤ −y\|y\| q , there is nothing new here. But since we strength the integrability conditions on η and λ, we can remove this growth condition on f . A typical example is f (y) = −(y + 1)\| log(y + 1)\| q for some q > 1. Proposition 1 Under our setting, the BSDE (1) has a minimal non-negative solution (Y, Z Y ). Proof. The existence of a non-negative solution can be obtained by the same penalization arguments as in [4] or [21]. We use the a priori estimate (19) in order to obtain the convergence of the penalization scheme on any interval [0, T − ε]. Minimality can be proved as in [21,Proposition 4]. Thus we skip the details here. Remark 1 (Generator depending on Z) Assume that the generator has the form: (t, ω, y, z) = f (y) η t (ω) + λ t (ω) + ζ(t, ω, z), where there exists a constant C such that for any (t, ω, z, z ′ ) 0 ≤ ζ(t, ω, 0) ≤ C, \|ζ(t, ω, z) − ζ(t, ω, z ′ )\| ≤ C\|z − z ′ \|. Using the Girsanov theorem, existence of a solution can be derived directly from Proposition 1. Moreover all results in this paper remain valid under some probability measure Q equivalent to P. To finish this section, let us give an estimate of Z Y . Let us also emphasize that this upper bound is valid for any solution of the BSDE (1), since the proof only uses the dynamics on [0, T ) and the a priori estimate (18) on Y , but not the construction by penalization of Y . In the power case (f (y) = −y\|y\| q ), it is known (see [6,26]) that E T 0 (T − s) 2/q (Z Y s ) 2 ds < +∞. Lemma 4 Assume that f is concave and that (C1) holds. Any solution (Y, Z Y ) of (1) satisfies for all p ∈ [1, +∞) E T 0 1 (T − s)(ψ ⋆ (s)) 2 (Z Y s ) 2 ds p < +∞. Let us immediately remark that this estimate is not optimal in the power case since we only have E T 0 (T − s) 2/q+1 (Z Y s ) 2 ds < +∞. Nevertheless it is sufficient for our purpose in Section 5. Proof. The following argument will be used several times through the paper. From the definition of a solution, we have for any ε > 0 E T −ε 0 (Z Y s ) 2 ds p < +∞. Since s → 1 (T −s)(ψ ⋆ (s)) 2 is bounded on [0, T − ε], we have to prove only that E T τ 1 (T − s)(ψ ⋆ (s)) 2 (Z Y s ) 2 ds p < +∞ for some deterministic τ ∈ [0, T ). From (18), Y remains bounded away from zero on [0, T ]. Thus let us apply the function G to Y : (19)). Since G is non-increasing G(Y t ) − G(Y 0 ) = t 0 1 f (Y s ) − 1 η s f (Y s ) − λ s ds + t 0 1 f (Y s ) Z Y s dW s + 1 2 t 0 −f ′ (Y s ) (f (Y s )) 2 (Z Y s ) 2 ds. Hence 0 ≤ 1 2 t 0 −f ′ (Y s ) (f (Y s )) 2 (Z Y s ) 2 ds ≤ G(Y t ) + t 0 1 η s ds − t 0 1 f (Y s ) Z Y s dW s . Now for p ≥ 1, there exists C p such that 0 ≤ t 0 −f ′ (Y s ) (f (Y s )) 2 (Z Y s ) 2 ds p ≤ C p (G(Y t )) p + t p η p ⋆ + sup u∈[0,t] u 0 1 f (Y s ) Z Y s dW s p . Recall that φ ⋆ (t) ≤ Y t ≤ φ ⋆ (t) (Equation0 ≤ G(Y t ) ≤ G(φ ⋆ (t)) and since −f ′ and φ ⋆ are non-decreasing (that is f is concave), for any s ∈ [0, T ] 0 ≤ 1 −f ′ (Y s ) ≤ 1 −f ′ (φ ⋆ (T )) < +∞ Taking the expectation and using BDG's inequality we obtain E t 0 −f ′ (Y s ) (f (Y s )) 2 (Z Y s ) 2 ds p ≤ C p (G(φ ⋆ (t))) p + t p η p ⋆ + C p E sup u∈[0,t] u 0 1 f (Y s ) Z Y s dW s p ≤ C p G(φ ⋆ (t)) p + T p η p ⋆ + C p E t 0 −f ′ (Y s ) (f (Y s )) 2 (Z Y s ) 2 ds p/2 . Therefore E T 0 −f ′ (Y s ) (f (Y s )) 2 (Z Y s ) 2 ds p < +∞. Using the monotonicity of f and f ′ , using (19) we get E T 0 −ψ ′ ⋆ (s) ψ ⋆ (s)(ψ ⋆ (s)) 2 (Z Y s ) 2 ds p = 1 η ⋆ E T 0 −f ′ (φ ⋆ (s)) (f (φ ⋆ (s))) 2 (Z Y s ) 2 ds p < +∞. Recall that under (C1), from Lemma 5, the function s → −ψ ′ ⋆ (s) ψ⋆(s) (T − s) is bounded. This leads to the conclusion. Asymptotic behavior for a general driver f In this section, we assume that the hypotheses (A1) to (A3), (C1) and (C2) hold. Recall that ψ(x) = −φ ′ (x) = −f (φ(x)) ≥ 0 and if φ t = φ(A t ), assume that Y t = φ t + ψ T − t η ⋆ H t = φ t + ψ ⋆ (t)H t .(20) Let us derive formally the dynamics of H. From the proof of Lemma 1 −dY t = φ ′ (A t ) 1 η t dt + Z A t dW t − 1 2 φ ′′ (A t )(Z A t ) 2 dt + 1 η ⋆ ψ ′ T − t η ⋆ H t dt − ψ ⋆ (t)dH t But we also know that −dY t = 1 η t f (Y t )dt + λ t dt − Z Y t dW t . Then −ψ ⋆ (t)dH t = 1 η t f (Y t ) − 1 η t φ ′ (A t ) dt + λ t + 1 2 φ ′′ (A t ) (Z A t ) 2 dt − 1 η ⋆ ψ ′ T − t η ⋆ H t dt − φ ′ (A t ) Z A t + Z Y t dW t . And we deduce −dH t = 1 ψ ⋆ (t)η t [f (φ t + ψ ⋆ (t)H t ) − f (φ t )] dt − 1 η ⋆ ψ ⋆ (t) ψ ′ T − t η ⋆ H t dt + λ t ψ ⋆ (t) dt + 1 2 ψ (A t ) ψ ⋆ (t) φ ′′ (A t ) ψ(A t ) A t (Z A t ) 2 A t dt − 1 ψ ⋆ (t) φ ′ (A t ) Z A t + Z Y t dW t . From Lemma 1, we know that Y t ≥ φ t = φ(A t ), thus H t ≥ 0 a.H t = T t F (s, H s )ds − T t Z H s dW s ,(21) with generator F (t, h) = α t (Z A t ) 2 A t + γ t + β t T − t h + 1 ψ ⋆ (t)η t [f (φ t + ψ ⋆ (t)h) − f (φ t )] 1 h≥0(22) with α t = 1 2 ψ (A t ) ψ ⋆ (t) φ ′′ (A t ) ψ(A t ) A t = 1 2 ψ (A t ) ψ ⋆ (t) − ψ ′ (A t ) ψ(A t ) A t , β t = − T − t η ⋆ ψ ′ ψ T − t η ⋆ , γ t = λ t ψ ⋆ (t) . Let us emphasize that the generator is singular in the sense of [17], since T 0 β t T − t dt = +∞.E   T 0 \|F (s, H s )\|ds + T 0 (Z H s ) 2 ds 1 2   < +∞. The aim of this section is to prove existence of a minimal non-negative solution (H, Z H ) of this BSDE (21), without using the existence of Y , such that the relation (20) Properties of the generator F In order to construct the process H, let us describe the properties of the generator F given by (22). 4.1.1 On the coefficients α, β and γ Lemma 5 The next two assertions are equivalent. Since φ ′′ = (f ′ •φ)φ ′ ≥ 0 and ψ ′ = −φ ′′ ≤ 0, 1. There exists a constant δ > 0 and R > 0 such that x → G(x) −δ is convex on [R, +∞) (condition (C1)). 2. The functions φ and ψ verify the next property: there exists K > 1 and ̺ > 0 such that for all x ∈ (0, ̺] x φ ′′ (x) φ ′ (x) + x ψ ′ (x) ψ(x) ≤ K. The constants are related by: ̺ = 1/R and δ = K − 1. Proof. Remark that ψ ′ (x) = −φ ′′ (x) ⇒ − ψ ′ (x) ψ(x) x = φ ′′ (x) ψ(x) x. Moreover x φ ′′ (x) φ ′ (x) ≤ 0. Hence it is enough to show that there exists K > 0 such that −x φ ′′ (x) φ ′ (x) ≤ K. W.l.o.g. we can assume that K > 1. Now let us define ϕ by ϕ(x) = φ(1/x) for any x > 0. Then φ ′ (x) = − 1 x 2 ϕ ′ (1/x), φ ′′ (x) = 2 x 3 ϕ ′ (1/x) + 1 x 4 ϕ ′′ (1/x). Thus −x φ ′′ (x) φ ′ (x) = 2 + 1 x ϕ ′′ (1/x) ϕ ′ (1/x) . Hence to establish Lemma 5 it is sufficient to prove that there exists K > 1 and ̺ > 0 such that for all t ≥ 1/̺ = R, −t ϕ ′′ (t) ϕ ′ (t) ≥ 2 − K = −(K − 2). Let us rewrite this condition in terms of the so-called Arrow-Pratt coefficient of absolute risk aversion by interpreting ϕ as utility function, α ϕ (t) := − ϕ ′′ (t) ϕ ′ (t) ≥ − (K − 2) t =: α K−2 (t),(23) where the utility function to α K is given (up to positive affine transformations) by u K (t) = t K−1 . By a classical theorem due to Pratt ([28], see also [ for a strictly increasing concave function F. As ϕ = G −1 (1/·), Pratt's condition is equivalent to F(t) := G −1 1 u −1 K (t) = G −1 t − 1 K−1 . defines a strictly increasing concave function. In other words x → G(x) 1−K is strictly increasing and convex. This achieves the proof of the Lemma. Under the condition (C1), using the second assertion of the previous lemma, the process β t is non negative and bounded provided that T −t ≤ η ⋆ ̺ = η ⋆ /R, that is T −η ⋆ /R ≤ t ≤ T . The process λ is bounded and since ψ tends to ∞ when x goes to zero, γ is bounded on [0, T ]. Concerning the process α, using (9), A t ≤ 1/R, if T − η⋆ R ≤ t ≤ T . Thus the process − ψ ′ (A) ψ(A) A is bounded on this interval. Since ψ is non increasing, ψ(A t ) ≤ ψ ⋆ (t). There- fore we deduce that ψ (A) ψ ⋆ (t) is also bounded. Finally under condition (C1) and with our assumption (A1) on η and on λ, α, β and γ are bounded processes at least on some interval [τ, T ] with τ = max T − η ⋆ R , T − η ⋆ R , 0 .(25) For t ∈ [τ, T ] \|α t \| ≤ 1 + K 2 max(1, η K ), \|β t \| ≤ K, \|γ t \| ≤ λ (ψ(T )) −1 . On the rest of the interval [0, τ ] these coefficients are also bounded due to the regularity of f (Conditions (A1) and (A2)). Remark 2 Using the previous lemma, integration leads to: for any y ∈ (0, 1/R) and a ≤ 1, 1 ≤ ψ(ay) ψ(y) ≤ 1 a K . Hence for any δ > 1: 1 ≤ ψ ⋆ δ (t) ψ ⋆ (t) ≤ δ K . If we assume that for some δ > 1 Y t = φ t + ψ T − t δη * H t , we have: H t ≤ H t ≤ δ K H t . Hence, up to some constant, this new development of Y is equivalent to (20). Properties of Z A In the generator F given by (22), we also have to control the process Z A . First note that the martingale t 0 Z A s dW s , t ∈ [0, T ] is a BMO martingale (see [18]) and Z A ∈ H q ((0, T )), q > 1, due to the assumption that η is bounded above and away from zero. Lemma 6 For any ρ ∈ (0, 1) and p > 1, we have E T 0 (Z A s ) 2 (A s ) 1+ρ ds p < +∞.(26) Proof. Let us apply Itô's formula to A 1−ρ on [0, T − ε]: A 1−ρ t = (A T −ε ) 1−ρ + T −ε t (1 − ρ) (A s ) −ρ η s ds − 1 2 T −ε t (1 − ρ)(−ρ) (Z A s ) 2 (A s ) 1+ρ ds + T −ε t (1 − ρ)(A s ) −ρ Z A s dW s . Hence, (1 − ρ)ρ 2 T −ε 0 (Z A s ) 2 (A s ) 1+ρ ds = A 1−ρ 0 − (A T −ε ) 1−ρ − (1 − ρ) T −ε 0 1 η s (A s ) ρ ds (27) − (1 − ρ) T −ε 0 (A s ) −ρ Z A s dW s . Taking the expectation and using (9) and the fact that t → (T − t) −ρ is integrable at time T , we can apply Lebesgue monotone convergence theorem to get E T 0 (Z A s ) 2 (A s ) 1+ρ ds = 2 ρ E (A 0 ) 1−ρ − T 0 1 η s (A s ) ρ ds < +∞. Using (27) for any p > 1 we obtain for some constant C p > 0, 1 C p T −ε 0 (Z A s ) 2 (A s ) 1+ρ ds p ≤ \|(A T −ε ) 1−ρ \| p + \|(A 0 ) 1−ρ \| p + T −ε 0 1 η s (A s ) ρ ds p + T −ε 0 (A s ) −ρ Z A s dW s p ≤ \|(A 0 ) 1−ρ \| p + \|(T /η ⋆ ) 1−ρ \| p + 1 η p ⋆ T 0 1 (A s ) ρ ds p + T −ε 0 (A s ) −ρ Z A s dW s p . From the BDG and Hölder inequalities, taking the expectation leads to 1 C p E T −ε 0 (Z A s ) 2 (A s ) 1+ρ ds p ≤ E \|(A 0 ) 1−ρ \| p + \|(T /η ⋆ ) 1−ρ \| p + 1 η p ⋆ T 0 1 (A s ) ρ ds p + E T −ε 0 ((A s ) −ρ Z A s ) 2 ds p 1/2 . The function x → x 1−ρ is bounded on [0, T /η ⋆ ] by some constant C. Thus 1 C p E T −ε 0 (Z A s ) 2 (A s ) 1+ρ ds p ≤ E \|(A 0 ) 1−ρ \| p + \|(T /η ⋆ ) 1−ρ \| p + 1 η p ⋆ T 0 1 (A s ) ρ ds p + C p/2 E T −ε 0 (Z A s ) 2 (A s ) 1+ρ ds p 1/2 . In other words if γ ε = E T −ε 0 (Z A s ) 2 (As) 1+ρ ds p , then there exists C independent of ε such that 0 ≤ γ ε ≤ C(1 + (γ ε ) 1/2 ), which leads to the existence of some constant C such that γ ε ≤ C. Using the monotone convergence theorem, we obtain the desired estimate. From this estimate on Z A , using (9), we have for any p > 1 E T 0 (Z A s ) 2 A s ds p ≤ T η ⋆ ρp E T 0 (Z A s ) 2 (A s ) 1+ρ ds p < +∞.(28) Construction of the process H Recall that the generator F is given by: F (t, h) = α t (Z A t ) 2 A t + γ t + β t T − t h + 1 ψ ⋆ (t)η t [f (φ t + ψ ⋆ (t)h) − f (φ t )] 1 h≥0 . Let us summarize its properties. • F (t, h) = F (t, 0) ≥ 0 for any h ≤ 0. • F is continuous and monotone w.r.t. h: for any h and h ′ , (h − h ′ )(F (t, h) − F (t, h ′ )) ≤ β t T − t (h − h ′ ) 2 , since f is itself monotone. • For any \|h\| ≤ r, \|F (t, h) − F (t, 0)\| ≤ β t T − t r − f (φ t + ψ ⋆ (t)r) ψ ⋆ (t)η t . • The process F (·, 0) equal to F (t, 0) = α t (Z A t ) 2 A t + γ t belongs to L p ([0, T ] × Ω) for any p > 1 (Inequality (28) and boundedness of the coeffcients α, β and γ due to (C1)). Our aim is to prove that the BSDE (21) has a solution (H, Z H ). However we cannot apply directly the results of [24], since the previous functions t → βt T −t and t → − f (φt+ψ ⋆ (t)r) ψ ⋆ (t)ηt are not necessarily integrable on [0, T ]. The cases of BSDEs with singular generator studied in [16,17] are also not adapted to our problem. In order to solve the problem, we modify the generator F . Let us consider δ > 0 and ε > 0 and define F δ,ε (t, h) = α t (Z A t ) 2 A t + γ t + β t T + ε − t h + 1 ψ ⋆ (t)η t f (φ t + ψ δ (t)h) − f (φ t ) 1 h≥0 (29) with ψ δ (t) = ψ T + δ − t η ⋆ = ψ ⋆ (t − δ). We consider the following BSDE H t = T t F δ,ε (s, H s )ds − T t Z H s dW s(30) on the interval [0, T ]. Lemma 7 Assume that (C1) and (C2) hold. Define µ ε t = t τ β s T + ε − s ds.(h − h ′ )(F δ,ε (t, h) − F δ,ε (t, h ′ )) ≤ β t T + ε − t (h − h ′ ) 2 and the process β is bounded. Moreover if \|h\| ≤ r \|F δ,ε (t, h)\| ≤ \|F (t, 0)\| + β t T + ε − t r − f (φ t + ψ δ (t)r) ψ ⋆ (t)η t = \|F (t, 0)\| + β t T + ε − t r − f (φ t + ψ(δ)r) ψ ⋆ (t)η t = Φ ♯ r (t). From our assumptions, in particular here Condition (C2), the definition of ψ and the properties of η, using Inequality (28), we deduce that E T 0 e µ ε s Φ ♯ r (t)ds p < +∞. Using [24, Proposition 5.24], we deduce that there exists a unique solution (H δ,ε , Z H,δ,ε ) satisfying the desired estimate. Remark 3 (Comments on (C2)) In Example 2, all functions are submultiplicative (and thus (C2) holds), except f (y) = − exp(ay 2 ). Nevertheless for this case −f (φ t + r) ψ ⋆ (t) ≤ C exp(ar 2 ) exp(2arφ t ) = exp(ar 2 ) exp(2arG −1 (A t )). And using (9) T τ −f (φ t + r) ψ ⋆ (t) dt = exp(ar 2 ) T τ exp(2arG −1 (A t ))dt ≤ exp(ar 2 ) T τ exp 2arG −1 T − t η ⋆ dt ≤ η ⋆ exp(ar 2 ) ∞ ζ exp(2arz) exp(−az 2 )dz < +∞. Thereby (C2) holds also in this case. Let us begin with an a priori estimate on H δ,ε . Recall that the function ϑ is defined just before Lemma 2. Lemma 8 For all t ∈ [0, T ), 0 ≤ H δ,ε t ≤ ϑ(T − t) ψ ⋆ (t) .(31) In particular H δ,ε is bounded on any interval [0, T − θ], 0 < θ < T . Proof. For fixed δ and ε, the dynamics of φ t + ψ ⋆ (t)H δ,ε t is given by: −d(φ t + ψ ⋆ (t)H δ,ε t ) = φ ′ (A t ) 1 η t dt − 1 2 φ ′′ (A t )(Z A t ) 2 dt + 1 η ⋆ ψ ′ T − t η ⋆ H δ,ε t dt + ψ ⋆ (t)F δ,ε (t, H δ,ε t )dt + φ ′ (A t )Z A t dW t − ψ ⋆ (t)Z H,δ,ε t dW t . Recall that F δ,ε is given by (29). Therefore we obtain −d(φ t + ψ ⋆ (t)H δ,ε t ) = λ t + 1 η t f (φ t + ψ δ t H δ,ε t ) dt + 1 η ⋆ ψ ′ T − t η ⋆ 1 (T − t) 2 ε T + ε − t H δ,ε t dt + φ ′ (A t )Z A t − ψ ⋆ (t)Z H,δ,ε In other words the process U t = φ t + ψ δ t H δ,ε t satisfies the BSDE: U t + (ψ ⋆ (t) − ψ δ t )H δ,ε t = U T −θ + T −θ t λ s + 1 η s f (U s ) ds − T −θ t Θ s ds − T −θ t Z U s dW s , with a non negative Θ, (ψ ⋆ − ψ δ )H δ,ε ≥ 0. Using Lemma 2, we deduce that ∀t ∈ [τ, T ), U t ≤ ϑ (T − t) . This leads to the conclusion of the Lemma. Again as a by-product, our proof implies that for any solution (H, Z H ) of the BSDE (21) 0 ≤ H t ≤ ϑ(T − t) ψ ⋆ (t) .(32) Now by the comparison principle, for a fixed δ > 0, since β t ≥ 0, (H δ,ε t , ε > 0) is a increasing sequence when ε decreases to zero, and for a fixed ε > 0, (H δ,ε t , δ > 0) is a decreasing sequence when δ decreases to zero. Thereby for any ε 1 < ε 2 and δ 1 < δ 2 ≤ δ for some δ > 0, we have the following inequalities: a.s. 0 ≤ H δ 1 ,ε 1 t ≤ H δ 2 ,ε 1 t ≤ H δ t , and 0 ≤ H δ 1 ,ε 2 t ≤ H δ 1 ,ε 1 t ≤ H δ t , where H δ t = lim ε↓0 H δ,ε t .(33) Note that H δ also satisfies (31). Now for a fixed ε > 0, we define H ε t = lim δ↓0 H δ,ε t ,(34) and H t = lim ε↓0 H ε t .(35) Since H δ,ε T = 0 a.s., we have immediately that a.s. H T = 0 and for all t ∈ [0, T ], H t ≥ 0. Proposition 2 There exists Z H ∈ H p (0, T − θ) for any θ > 0, such that the couple (H, Z H ) solves the BSDE (21) with generator F on the interval [0, T − θ] for any 0 < θ < T . Proof. Let us define c p = p(1 ∧ (p − 1)). Step 1. Given ε 1 < ε 2 and δ 1 < δ 2 , applying Itô's formula to ∆H = H δ 1 ,ε 1 − H δ 2 ,ε 2 on the interval [t, T − θ], θ > 0, leads to: e p µt \|∆H t \| p + c p 2 T −θ t e p µs \|∆H s \| p−2 1 ∆Hs =0 \|∆Z H s \| 2 ds ≤ e p µ T \|∆H T −θ \| p +p T −θ t e p µs \|∆H s \| p−2 1 ∆Hs =0 ∆H s (F δ 1 ,ε 1 (s, H δ 1 ,ε 1 s ) − F δ 2 ,ε 2 (s, H δ 2 ,ε 2 s ))ds −p T −θ t β s θ + 2 p − 1 p e p µs \|∆H s \| p ds − p T −θ t e p µs \|∆H s \| p−2 1 ∆Hs =0 ∆H s ∆Z H s dW s with ∆Z H = Z H,δ 1 ,ε 1 − Z H,δ 2 ,ε 2 and µ t = t 0 β s θ + 2 p − 1 p ds. Remark that from the monotonicity of f : ∆H s (F δ 1 ,ε 1 (s, H δ 1 ,ε 1 s ) − F δ 2 ,ε 2 (s, H δ 2 ,ε 2 s )) = ∆H s β s T + ε 1 − s H δ 1 ,ε 1 s − β s T + ε 2 − s H δ 2 ,ε 2 s + 1 ψ s η s ∆H s f (φ s + ψ δ 1 s H δ 1 ,ε 1 s ) − f (φ s + ψ δ 2 s H δ 2 ,ε 2 s ) ≤ ∆H s β s T + ε 1 − s H δ 1 ,ε 1 s − β s T + ε 2 − s H δ 2 ,ε 2 s + 1 ψ s η s ∆H s f (φ s + ψ δ 1 s H δ 2 ,ε 2 s ) − f (φ s + ψ δ 2 s H δ 2 ,ε 2 s ) ≤ β s T + ε 1 − s (∆H s ) 2 + ∆H s H δ 2 ,ε 2 s β s ε 2 − ε 1 (T + ε 1 − s)(T + ε 2 − s) + 1 ψ s η s ∆H s f (φ s + ψ δ 1 s H δ 2 ,ε 2 s ) − f (φ s + ψ δ 2 s H δ 2 ,ε 2 s ) . We deduce that e p µt \|∆H t \| p + c p 2 T −θ t e p µs \|∆H s \| p−2 1 ∆Hs =0 \|∆Z H s \| 2 ds ≤ e p µ T −θ \|∆H T −θ \| p + p T −θ t e p µs β s T + ε 1 − s − β s θ − 2 p − 1 p \|∆H s \| p ds +p T −θ t e p µs \|∆H s \| p−2 1 ∆Hs =0 ∆H s H δ 2 ,ε 2 s β s ε 2 − ε 1 (T + ε 1 − s)(T + ε 2 − s) ds +p T −θ t e p µs \|∆H s \| p−2 1 ∆Hs =0 ∆H s ψ s η s f (φ s + ψ δ 1 s H δ 2 ,ε 2 s ) − f (φ s + ψ δ 2 s H δ 2 ,ε 2 s ) ds −p T −θ t e p µs \|∆H s \| p−2 1 ∆Hs =0 ∆H s ∆Z H s dW s . By Young's inequality p T −θ t e p µs \|∆H s \| p−2 1 ∆Hs =0 ∆H s H δ 2 ,ε 2 s β s ε 2 − ε 1 (T + ε 1 − s)(T + ε 2 − s) ds ≤ (p − 1) T −θ t e p µs \|∆H s \| p ds + T −θ t e p µs H δ 2 ,ε 2 s β s ε 2 − ε 1 (T + ε 1 − s)(T + ε 2 − s) p ds and p T −θ t e p µs \|∆H s \| p−2 1 ∆Hs =0 ∆H s ψ s η s f (φ s + ψ δ 1 s H δ 2 ,ε 2 s ) − f (φ s + ψ δ 2 s H δ 2 ,ε 2 s ) ds ≤ (p − 1) T −θ t e p µs \|∆H s \| p ds + T −θ t e p µs 1 ψ s η s f (φ s + ψ δ 1 s H δ 2 ,ε 2 s ) − f (φ s + ψ δ 2 s H δ 2 ,ε 2 s ) p ds. Recall that β, 1/η are bounded on [0, T ] whereas 1/ψ is bounded on [0, T − θ]. From Estimate (31), on the interval [0, T − θ], H ε 2 ,δ 2 is also bounded and we have: e p µt \|∆H t \| p + c p 2 T −θ t e p µs \|∆H s \| p−2 1 ∆Hs =0 \|∆Z H s \| 2 ds ≤ e p µ T −θ \|∆H T −θ \| p + C(ε 2 − ε 1 ) p +C T −θ t f (φ s + ψ δ 1 s H δ 2 ,ε 2 s ) − f (φ s + ψ δ 2 s H δ 2 ,ε 2 s ) p ds −p T −θ t e p µs \|∆H s \| p−2 1 ∆Hs =0 ∆H s ∆Z H s dW s .(36) The constant C depends on all bounds of our coefficients and on θ. This constant explodes when θ goes to zero. Step 2. Let us fix ε = ε 1 = ε 2 > 0. Since for δ 2 ≤ δ, H δ 2 ,ε ≤ H δ and H δ satisfies the estimate (31), using the dominated convergence theorem E T −θ 0 e p µs f (φ s + ψ δ 2 s H δ 2 ,ε s ) − f (φ s + ψ δ 1 s H δ 2 ,ε s ) p ds → 0, as δ 1 and δ 2 tend to zero. Therefore using (36) and taking the expectation we deduce that T −θ can be handled as in [7]: for any C > 0 E T −θ 0 e pE [Λ] 1/2 T −θ ≤ C 2 E sup t∈[0,T −θ] e p µt \|∆H t \| p + 1 2C E T −θ 0 e p µs \|∆H s \| p−2 1 ∆Hs =0 \|∆Z H s \| 2 ds . Thereby (H δ,ε , δ > 0) is a Cauchy sequence: E sup t∈[0,T −θ] e p µt \|∆H t \| p → 0 as δ 1 and δ 2 tend to zero. Finally E T −θ 0 e 2 µs \|∆Z H s \| 2 ds p/2 = E T −θ 0 e 2 µs (∆H s ) 2−p (∆H s ) p−2 1 ∆Hs =0 \|∆Z H s \| 2 ds p/2 ≤ E   sup t∈[0,T −θ] e µt \|∆H t \| p(2−p)/2 T −θ 0 e p µs (∆H s ) p−2 1 ∆Hs =0 \|∆Z H s \| 2 ds p/2   ≤ E sup t∈[0,T −θ] e p µt \|∆H t \| p (2−p)/2 E T −θ 0 e p µs (∆H s ) p−2 1 ∆Hs =0 \|∆Z H s \| 2 ds p/2 ≤ 2 − p 2 E sup t∈[0,T −θ] e p µt \|∆H t \| p + p 2 E T −θ 0 e p µs (∆H s ) p−2 1 ∆Hs =0 \|∆Z H s \| 2 ds where we have used Hölder's and Young's inequality with 2−p 2 + p 2 = 1. Hence we obtain that (H δ,ε , Z H,δ,ε ) converges in S p (0, T − θ) to some process (H ε , Z H,ε ). The process H ε is non negative and also satisfies the a priori estimate (31) with H ε T = 0, and we have for any 0 ≤ t ≤ T − θ < T H ε t = H ε T −θ + T −θ t α s (Z A s ) 2 A s + γ s ds + T −θ t β s T + ε − s H ε s ds + T −θ t 1 ψ ⋆ (s)η s [f (φ s + ψ ⋆ (s)H ε s ) − f (φ s )] ds − T −θ t Z H,ε s dW s . Step 3. Let us prove the convergence of (H ε , Z H,ε ) when ε tends to zero. The arguments are almost the same as in the second step. Indeed the formula (36) becomes: e p µt \|∆H t \| p + c p 2 T −θ t e p µs \|∆H s \| p−2 1 ∆Hs =0 \|∆Z H s \| 2 ds ≤ e p µ T −θ \|∆H T −θ \| p + C(ε 2 − ε 1 ) p −p T −θ t e p µs \|∆H s \| p−2 1 ∆Hs =0 ∆H s ∆Z H s dW s . with ∆H = H ε 1 − H ε 2 and ∆Z H = Z H,ε 1 − Z H,ε 2 . The conclusion follows from the same arguments as in step 2. Note that from (31), the arguments to prove (19) and the remark 2, we obtain that a.s. on [0, T ) 0 ≤ H t ≤ C 1 ψ ⋆ (t) φ ⋆ (t) = C φ ⋆ (t) −f (φ ⋆ (t)) ,(37) and 0 ≤ φ t + ψ ⋆ (t)H t ≤ φ ⋆ ε (t) .(38) Since f is a non positive and non increasing function, from (A3), the function y → −1/f (y) is an integrable, non-negative and non increasing function. Thereby we know that (see [14,Section 178]) lim y→+∞ y −f (y) = 0. Thereby using the a priori estimate (37), the process H satisfies a.s. lim t→T H t = 0. Let us emphasize that H solves the BSDE (21), in the sense that for any 0 ≤ t ≤ u < T H t = H u + u t F (s, H s )ds − u t Z H s dW s = u t α s (Z A s ) 2 A s + γ s ds − u t Z H s dW s + u t β s T − s H s ds + u t 1 ψ ⋆ (s)η s [f (φ s + ψ ⋆ (s)H s ) − f (φ s )] ds.(39) It is important to note that if we define H by the relation (20) ψ ⋆ (s)η s \|f (φ s + ψ ⋆ (s)H s ) − f (φ s )\| ds = − u t 1 ψ ⋆ (s)η s [f (φ s + ψ ⋆ (s)H s ) − f (φ s )] ds. Lemma 9 The process β s T − s H s , s ∈ [0, T ) is integrable on [0, T ]. Proof. Indeed using the very definition of β s we have 0 ≤ β s T − s H s = − 1 η ⋆ ψ ′ ψ T − s η ⋆ H s ≤ − 1 η ⋆ ψ ′ ψ T − s η ⋆ φ T −t η ⋆ ψ T −t η ⋆ = 1 η ⋆ φ ′′ φ (φ ′ ) 2 T − s η ⋆ . Now u t 1 η ⋆ φ ′′ φ (φ ′ ) 2 T − s η ⋆ ds = (T −t)/(η ⋆ ) (T −u)/(η ⋆ ) φ ′′ φ (φ ′ ) 2 (x) dx and our result follows if (φ ′′ φ)/(φ ′ ) 2 is integrable at zero. Remark that φ φ ′ ′ = 1 − φ ′′ φ (φ ′ ) 2 . Hence the integrability is equivalent to the existence of the limit at zero of φ/φ ′ , that is the limit at infinity of y → y/(−f (y)), which is zero. This achieves the proof of the lemma. Coming back to (39) and taking the conditional expectation, we get: H t = E u t α s (Z A s ) 2 A s + γ s ds F t + E u t β s T − s H s ds F t + E u t 1 ψ ⋆ (s)η s [f (φ s + ψ ⋆ (s)H s ) − f (φ s )] ds F t . From the previous lemma, we deduce that E T t 1 ψ ⋆ (s)η s \|f (φ s + ψ ⋆ (s)H s ) − f (φ s )\| ds < +∞. In other words taking t = 0 E T 0 \|F (s, H s )\|ds < +∞. Then using again (39), we easily deduce that E sup 0≤u≤T u 0 Z H s dW s < +∞. By Burkholder-Davis-Gundy's inequality, we deduce that Z H is an element of H 1 (0, T ). Let us summarize our results. Theorem 1 Assume that (C1) and (C2) hold. There exists a process (H, Z H ), which the minimal non-negative solution of the BSDE (21), that is: • H is non negative and essentially bounded: for any 0 ≤ t < T , 0 ≤ sup s∈[0,t] H s < +∞ a.s. and E T 0 \|F (s, H s )\|ds < +∞. • The process Z H belongs to H 1 (0, T ) ∩ H p (0, T − θ) for any θ > 0 and p > 1. Proof. The only thing to prove is the minimality. Let ( H, Z) be another solution of the BSDE (21). Let us first show that H is a non-negative process. Applying Itô's formula for the non-positive part of H and the very definition (22) of F leads to: H t − ≤ − T t F (s, H s )1 Hs≤0 ds + T t ( Z s )1 Hs≤0 dW s = − T t α s (Z A s ) 2 A s + γ s 1 Hs≤0 ds + T t ( Z s )1 Hs≤0 dW s . Since α and γ are non-negative, taking the conditional expectation knowing F t yields to the non-negativity of H. Now for ε > 0, the process ∆H = H − H ε satisfies for any θ > 0: ∆H t = ∆H T −θ + T −θ t F (s, H s ) − F ε (s, H ε s ) ds − T −θ t ( Z s − Z ε s )dW s = ∆H T −θ + T −θ t ε β s T − s H s + β s T + ε − s + κ s ∆H s ds − T −θ t ( Z s − Z ε s )dW s where κ s = 1 ψ ⋆ (s)η s f (φ s + ψ ⋆ (s) H s ) − f (φ s + ψ ⋆ (s)H ε s ) 1 ∆H s 1 ∆Hs =0 . This process κ is bounded from above by zero since f is monotone. Thus if Γ t,s = exp s t β u T + ε − u + κ u du , by standard arguments concerning linear BSDE (see [20,Lemma 10] or [24,Proposition 5.31]), we have: ∆H t = E ∆H T −θ Γ t,T −θ + T −θ t ε β s T − s H s Γ t,s ds F t ≥ E ∆H T −θ Γ t,T −θ F t . By Fatou's lemma, letting θ going to zero, we obtain that for any ε > 0, H t ≥ H ε t . Hence the minimality of H is proved. The process (H, Z H ) solves a BSDE with singular driver in the sense of [17]. As mentioned in [17, Proposition 3.1], uniqueness is not an obvious property for such kind of BSDEs. In our case assume that ( H, Z) be another non negative solution of the BSDE (21). Then ∆H t = T t F (s, H s ) − F (s, H s ) ds − T t ( Z s − Z H s )dW s = T t λ s ∆H s ds − T t ∆Z s dW s . Hence we have a linear BSDE with singular generator with λ s = β s T − s + 1 ψ ⋆ (s)η s f (φ s + ψ ⋆ (s) H s ) − f (φ s + ψ ⋆ (s)H s ) 1 ∆H s 1 ∆Hs =0 . Nevertheless we cannot apply the result in [17, Propositions 3.1 and 3.5], since we don't know the sign of the drift λ. Remark 4 If H t = lim δ↓0 H δ t , then H t ≤ H t . The proof of the previous proposition shows that H also satisfies the BSDE (21) (in the sense of the previous theorem). It seems difficult to prove that these two processes are equal. In other words, as remarked above, we don't have any comparison or uniqueness result concerning the BSDE (21). Asymptotics of the minimal solution Let us consider the process Y t = φ t + ψ ⋆ (t)H t on [0, T ). Then from our heuristic study, for any 0 ≤ t ≤ s < T , we have: Y t = Y s + s t f ( Y u ) η u + γ u du − s t Z Y u dW u and a.s. lim t↑T Y t = +∞. Note that this process ( Y , Z Y ) belongs to any S ∞ (0, T − θ) for any θ > 0. If f (y) = −y\|y\| q , by uniqueness proved in [13], Y = Y and thus Inequality (37)). In other words, the non-negative process Y t = 1 qA t 1 q + η ⋆ q(T − t) 1 q +1 H t with 0 ≤ H t ≤ C(T − t) (Y t (A t ) 1 q = Y t E T t 1 η s ds F t 1 q is bounded. In general since (Y, Z) is the minimal non-negative solution of (1) (see Proposition 1), we have a.s. ∀t ∈ [0, T ], φ t ≤ Y t ≤ Y t = φ t + ψ ⋆ (t)H t . But from our heuristic computations, we have Y t = φ t + ψ ⋆ (t) H t . Thus 0 ≤ H t ≤ H t and H satisfies the same BSDE (21), at least on any interval [0, T − ε]. Since H ≤ H, the preceding arguments show that H is also a solution of the same BSDE on the whole interval [0, T ]. Since H is the minimal solution, we have proved that H = H and thus Y = Y . In other words: (1) is given by (20): Theorem 2 The minimal solution (Y, Z Y ) of the BSDEY t = φ t + ψ T − t η ⋆ H t , where (H, Z H ) is the minimal solution of the BSDE (21). Recall that from Inequality (38), on the interval [τ, T ] with τ given by (25), ψ ⋆ (t)H t φ t ≤ φ ⋆ 1 (t) φ ⋆ (t) . Lemma 10 There exists a constant κ depending on η ⋆ and η ⋆ such that for any t ∈ [τ, T ], 0 ≤ ψ ⋆ (t)H t φ t ≤ κ. Proof. Recall that since φ is non increasing, φ(y) ≤ φ(ay). Moreover from Lemma 5 we know that for any y ∈ (0, 1/R) and 0 < a ≤ 1, 1 ≤ ψ(ay) ψ(y) ≤ 1 a K . From the very definition of ψ, this inequality can be written as: −φ ′ (ay) ≤ −a −K φ ′ (y). Integrating this inequality (between y and η) leads to: φ(ay) ≤ a 1−K φ(y) + C for some constant C ≥ 0. Hence 1 ≤ φ(ay) φ(y) ≤ a 1−K + C φ(y) . Since φ(0) = ∞, the conclusions follows from this inequality. Hence we have proved that the minimal solution Y of the BSDE (1) satisfies on the interval [τ, T ]: φ t ≤ Y t ≤ φ t (1 + κ). The concave case In the expansion (20) of Y , there is an asymmetry between ψ t which is random, and the deterministic ψ ⋆ (t). This asymmetry has the advantages to avoid the presence of Z H in the generator of H and of an extra term with the second derivative of ψ. However it leads to the fact that f ′ (φ ⋆ (t)) η t = f ′ (φ t ) = f ′ (φ (A t )) .(40) Thereby we cannot interpret the bracket β t T − t h + 1 ψ ⋆ (t)η t [f (φ t + ψ ⋆ (t)h) − f (φ t )] = 1 ψ ⋆ (t)η t f (φ t + ψ ⋆ (t)h) − f (φ t ) − f ′ (φ ⋆ (t)) η t ψ ⋆ (t)h as the reminder to the first Taylor polynomial of f at φ t . Here we study a possible workaround: define φ t and ψ t symmetrically, i.e. φ t = ψ(A t ) and ψ t = ψ(A t ). Then (40) is satisfied. This, however, leads to an additional linear term in the driver of H. This linear term creates some main difficulties. To overcome them, we add several assumptions on f and on η. Symmetric development for general process η Recall that A satisfies (14), φ verifies: φ ′ = f • φ and ψ = −φ ′ . Setting φ t := φ(A t ) and ψ t := ψ(A t ) yields dφ t = −f (φ t ) 1 η t dt + Z A t dW t + 1 2 φ ′′ (A t )(Z A t ) 2 dt, dψ t = −f ′ (φ t )ψ t 1 η t dt + Z A t dW t + 1 2 ψ ′′ (A t )(Z A t ) 2 dt. Recall that (9) leads to (10): φ ⋆ (t) = φ T − t η ⋆ ≤ φ t ≤ φ T − t 2η ⋆ = φ ⋆ (t). And since φ ′ is non-decreasing, we also have: ψ ⋆ (t) ≤ ψ t ≤ ψ ⋆ (t). For −dY t = 1 η t f (Y t ) dt + λ t dt − Z Y t dW t we make the ansatz Y t = φ t + ψ t H t and hence obtain the heuristic dynamics of H, namely: −dH t = 1 η t ψ t f (φ t + ψ t H t ) − f (φ t ) − f ′ (φ t )ψ t H t dt + λ t ψ t − φ ′′ (A t )A t 2φ ′ (A t ) (Z A t ) 2 A t dt + ψ ′′ (A t )(A t ) 2 2ψ(A t ) Z A t A t 2 H t − A t ψ ′ (A t ) ψ(A t ) Z A t A t Z H t dt − Z H t dW t = 1 η t ψ t f (φ t + ψ t H t ) − f (φ t ) − f ′ (φ t )ψ t H t dt + λ t ψ t + κ 1 t (Z A t ) 2 A t dt + κ 2 t Z A t A t 2 H t + κ 3 t Z A t A t Z H t dt − Z H t dW t(41) where κ 1 t = − φ ′′ (A t )A t 2φ ′ (A t ) κ 2 t = ψ ′′ (A t )(A t ) 2 2ψ(A t ) = −A t ψ ′′ (A t ) ψ ′ (A t ) κ 1 t κ 3 t = − A t ψ ′ (A t ) ψ(A t ) . Note that from (13), H t ≥ 0 a.s. Under (C1), using Lemma 5, we deduce that κ 1 and κ 3 are bounded and non-negative. Compared to the previous section and the BSDE (21), the dynamics (41) of H has a new linear term, namely (t, h, z) → κ 2 t Z A t A t 2 h + κ 3 t Z A t A t z. In the next lemma we prove that under the additional conditions (C3) and (C4) of f , κ 2 is bounded. Proof. Indeed if f is concave, we have ψ ′′ (x) = (−f ′′ • φ)(x)(φ ′ (x)) 2 − (f ′ • φ(x))φ ′′ (x) ≥ 0. Hence xψ ′′ (x)/ψ ′ (x) ≤ 0. The conclusion of the lemma is equivalent to the boundedness from below of −x ψ ′′ (x)/ ψ ′ (x) in the neighborhood of ∞ with ψ(x) = ψ(1/x). From the proof of Lemma 5, we have ψ(x) = (−f )(F • u K ) = F • u K where F and F are increasing and concave. Note that we can assume w.l.o.g. that the constant K is the same. Indeed if (C1) holds for some δ > 0, the same condition holds for any δ ′ ≥ δ. Hence boundedness is equivalent to the existence of K > 1 such that F : x → G −1 (x −1/(K−1) ) (condition (C1)) and (−f ) • F are increasing and concave. Remark 5 Note that under (C3), the boundedness of κ 2 is equivalent to condition (C4). Let us consider again the functions of Example 2. • If f (y) = −y\|y\| q for some q > 0, then we can take F(x) = 1 q 1 q x 1 q+1 , ((−f ) • F)(x) = q + 1 q 1 q x and K = 2 + 1/q. • If f (y) = −(exp(ay) − 1) for some a > 0, then φ(x) = − 1 a log (1 − e −ax ) and ψ( x) = −φ ′ (x) = 1 e ax − 1 . Hence − φ ′′ (x) φ ′ (x) x = axe ax e ax − 1 ∼ x→0 1, − ψ ′′ (x) ψ ′ (x) x = ax(1 + e ax ) e ax − 1 ∼ x→0 2, are bounded near zero. • If f (y) = − exp(ay 2 ) for some a > 0, then φ(x) = 1 √ 2a N −1 1 − x a π , ψ(x) = −φ ′ (x) = exp −aφ(x) 2 . And ψ ′ (x) = −φ ′′ (x) = −2aφ(x)ψ(x) 2 , ψ ′′ (x) = 2a(ψ(x) 3 ) − 4aφ(x)ψ(x)ψ ′ (x). Thus − ψ ′′ (x) ψ ′ (x) x = x ψ(x) φ(x) − 2x ψ ′ (x) ψ(x) . From (C1) and Lemma 5, the second term is bounded. Arguing as at the end of Example 2 yields to: x ψ(x) φ(x) = √ 2a z G z √ 2a e −z 2 /2 −→ z→+∞ 0. In other words all functions considered in Example 2 verify (C3) and (C4). However the preceding lemma is not sufficient to obtain existence of a solution. Indeed if we consider this BSDE (41) with f ≡ 0, we get a linear BSDE. From our best knowledge, existence of a solution is proved only under some exponential moment condition on the coefficients (see [24,Proposition 5.31]). Even if we avoid the final time T , then 1/A is bounded on [0, T −ε] (Inequality (9)), but Z A is only BMO. Hence the stochastic exponential of the martingale M = H t = T t 1 η s ψ s f (φ s + ψ s H s ) − f (φ s ) − f ′ (φ s )ψ s H s 1 Hs≥0 ds + T t λ s ψ s + κ 1 s (Z A s ) 2 A s + κ 2 s Z A s A s 2 H s + κ 3 s Z A s A s Z H s ds − T t Z H s dW s . (42) Proof. Here we do not construct (H, Z H ) from scratch, but we use the existence of a minimal solution (Y, Z Y ) of (1). Indeed our previous computations show that if H = (Y − φ)/ψ, then the process (H, Z H ) verifies: • It satisfies the dynamics given by (41) on any interval [0, T − ε]. • H verifies an a priori estimate similar to (32): 0 ≤ H t ≤ φ ⋆ (t) ψ t ≤ φ ⋆ (t) ψ ⋆ (t) ≤ 2η ⋆ η ⋆ K φ ⋆ (t) ψ ⋆ (t) (we use again Remark 2). • Z H belongs to H p ( τ , T − ε) for any p > 1. Thus we only have to extend the assertions on [0, T ]. Compared to Section 4 and the discussion above Lemma 9, we need to control the additional term: κ 2 t Z A t A t 2 H t + κ 3 t Z A t A t Z H t . We already know that κ 2 and κ 3 are bounded and that Z A satisfies the inequality (26). Let us precise the relation between Z H and Z Y . Since Y = φ + ψH, we have: Z Y t = −f (φ t )Z A thus Z H t = −1 + f ′ (φ t )H t Z A t + 1 ψ t Z Y t and Z H t (A t ) (1−ρ)/2 = −1 + f ′ (φ t )H t Z A t (A t ) (1−ρ)/2 + 1 ψ t (A t ) (1−ρ)/2 Z Y t = −(A t ) ρ − κ 3 t H t (A t ) 1−ρ Z A t (A t ) (1+ρ)/2 + 1 ψ t (A t ) (1−ρ)/2 Z Y t .(43) Using (9), (26) and Lemma 4, for any p > 1, E T 0 (Z A t ) 2 (A t ) 1+ρ dt p + T 0 (Z Y t ) 2 (ψ t ) 2 (A t ) (1−ρ) dt p < +∞. Combining (43) together this estimate, we have E T 0 κ 2 t Z A t A t 2 H t + κ 3 t Z A t A t Z H t < +∞ if we can prove that for some p > 1 E sup t∈[0,T ] H t (A t ) 1−ρ p < +∞. We know that 0 ≤ H t (A t ) 1−ρ ≤ (η ⋆ ) 1−ρ H t (T − t) 1−ρ ≤ (η ⋆ ) 1−ρ 2η ⋆ η ⋆ K φ ⋆ (t) ψ ⋆ (t)(T − t) 1−ρ . The last term is deterministic and if we prove that this term remains bounded on [0, T ], the result follows. Note that (2η ⋆ ) 1−ρ φ ⋆ (t) ψ ⋆ (t)(T − t) 1−ρ = φ ⋆ (x) ψ ⋆ (x)x 1−ρ = φ ⋆ (x) −f (φ ⋆ (x))x 1−ρ = y −f (y)G(y) 1−ρ = y h(y) . with x = (T − t)/(2η ⋆ ) and y = φ ⋆ (x). From Condition (C5), we deduce that (H, Z H ) is a solution of our BSDE on [0, T ]. Let us point out again that in the BSDE (42), the driver has a triple singularity: h, which requires to control the quadratic variation of the previous martingale. • (s, h) → 1 η s ψ s f ′ (φ s )ψ s h ( Hence we add the next condition: (H) There exist a deterministic time τ < T and a positive constant C > 1 2 ∨ κ 2 ∞ ∨ κ 3 2 ∞ such that E exp C T τ Z A s A s 2 ds < +∞. Assuming that (C1) to (C4) and (H) hold and arguing as in the proof of Proposition 2 in Section 4, we construct directly a process (H, Z H ) solving the dynamics (41) H t − ≤ − T t F (s, H s )1 Hs≤0 ds + T t ( Z s )1 Hs≤0 dW s = − T t λ s ψ s + κ 1 s (Z A s ) 2 A s + κ 2 s Z A s A s 2 H s + κ 3 s Z A s A s Z s 1 Hs≤0 ds + T t ( Z s )1 Hs≤0 dW s ≤ T t κ 2 s Z A s A s 2 H s − ds − T t κ 3 s Z A s A s Z s 1 Hs≤0 ds + T t ( Z s )1 Hs≤0 dW s . From the condition (H), the martingale E(Z A ) t = exp t τ κ 3 s Z A s A s dW s + 1 2 t τ (κ 3 s ) 2 Z A s A s 2 ds , t ∈ [ τ , T ],(∆H t ) + ≤ T t 1 η s ψ s f (φ s + ψ s H s ) − f (φ s + ψ s H s ) − f ′ (φ s + ψ s H s )ψ s ∆H s 1 ∆Hs≥0 ds + T t 1 η s f ′ (φ s + ψ s H s ) − f ′ (φ s ) ∆H s 1 ∆Hs≥0 ds + T t κ 2 s Z A s A s 2 ∆H s + κ 3 s Z A s A s ∆Z H s 1 ∆Hs≥0 ds − T t ∆Z H s 1 ∆Hs≥0 dW s ≤ T t 1 η s f ′ (φ s + ψ s H s ) − f ′ (φ s ) 1 Hs≤0 (∆H s ) + ds + T t κ 2 s Z A s A s 2 ∆H s + κ 3 s Z A s A s ∆Z H s 1 ∆Hs≥0 ds − T t ∆Z H s 1 ∆Hs≥0 dW s = T t κ 2 s Z A s A s 2 (∆H s ) + + κ 3 s Z A s A s ∆Z H s 1 ∆Hs≥0 ds − T t ∆Z H s 1 ∆Hs≥0 dW s , About Condition (H) The condition (H) is very strong and seems difficult to be checked in general. However in the Itô setting on the process η, this assumption may hold. Let us suppose that 1 η =: γ is an Itô process dγ t = d γ t dt + σ γ t dW t .(44) Note that with Condition (A1), it is equivalent to assume that η is an Itô process. First of all the next result holds. Lemma 12 For p > 2, if d γ and σ γ belong to L 2p ((0, T ) × Ω), the process Z A /A belongs to H 2p (0, T ), that is E T 0 Z A s A s 2 ds p < +∞.(45) Proof. We consider the processĀ t := A t /(T − t), which satisfies the BSDE −dĀ t = 1/η t −Ā t T − t dt − ZĀ dW t ,Ā T = 1/η T . Since ZĀ t = Z A t /(T − t), to verify (45) it is sufficient to establish ZĀ ∈ H 2p (0, T ). For the later again it is sufficient to establish that the driver toĀ is in L 2p . (Here we used frequently that η is bounded above and away from zero.) To establish (1/η −Ā)(T − ·) ∈ L 2p we first check Kolgomorov's criterion for 1/η: For 0 ≤ t ≤ s ≤ T , by Jensen and BDG inequality, E[\|1/η s − 1/η t \| 2p ] ≤ C\|s − t\| p−1 E s t \|d γ r \| 2p + \|σ γ r \| 2p dr . Hence, by Kolgomorov's criterion, for any α ∈ (0, p−2 2p ) there exits a random variable ξ ∈ L 2p (Ω) such that \|1/η t − 1/η s \| ≤ ξ\|t − s\| α , t, s ∈ [0, T ]. Therefore, using the mean value theorem, E T 0 1/η t −Ā t T − t dt 2p ≤ E T 0 ξ(T − t) α T − t dt 2p ≤ CE[ξ 2p ], which completes the proof. The coefficients in the linear part of the BSDE (41) are in H 2p (0, T ). However it is not sufficient to get (H). Let us remark that: In the next two lemmas we give sufficient conditions on the coefficients of (44) such that (H) holds. Lemma 13 If d γ and σ γ are essentially bounded, then Condition (H) holds. Proof. In this setting, Condition (H) holds if and only if E exp   C T τ Z A s A s 2 ds   < +∞,(47) for some C > C. Since d γ is essentially bounded, then A t ≤ E T t (T − u) d γ du F t = d γ 2 (T − t) 2 . Itô's formula leads to −d A t (T − t) = − A t (T − t) 2 dt + d γ t dt + Z A t (T − t) dW t . Hence we obtain that the martingale M u = u 0 Z A t (T − t) dW t , u ∈ [0, T ] is a BMO mar- tingale: ∀u ∈ [0, T ], \| M T − M u \| ≤ 2 d γ (T − u) ⇒ sup u∈[t,T ] E \| M T − M u \| F u ≤ 2 d γ (T − t).E   exp   C T t Z A s A s 2 ds     < +∞. Precisely C should be smaller than the inverse of the BMO norm of M . Thereby choosing τ sufficiently close to T , we get Condition (47) and the conclusion of the lemma. Let us now assume that the process γ = 1/η solves a SDE: dγ t = d(γ t ) dt + σ(γ t ) dW t(48) The derivative D r γ t satisfies the following linear equation: D r γ t = σ(γ r ) + t r σ(s)D r γ s dW s + t r d(s)D r γ s ds for r ≤ t a.e. and D r γ t = 0 for r > t a.e., where d(s) and σ(s) are two bounded processes, such that if d and σ are of class C 1 , they are given by: d(s) = (∂ x d)(γ s ), σ(s) = (∂ x σ)(γ s ). In this case ξ = T 0 (T − u)d γ u du is in D 1,2 and by the Clark-Ocone formula, we have Z A t = T t (T − u)E D t d γ u F t du = T t (T − u)E d(u)D t γ u F t du. Lemma 14 If γ is a diffusion process solution of a SDE with Lipschitz continuous coefficients and a bounded diffusion coefficient, then Condition (H) holds. Proof. Since ζ u = D t γ u satisfies the linear one-dimensional SDE: Since the function d is supposed to be Lipschitz continuous, d is essentially bounded. Thereby ζ u = σ(γ t ) +Z A t ≤ d T t (T − u)E \|D t γ u \| F t du = d \|σ(γ t )\| T t (T − u)E exp u t d(s)ds F t du ≤ 1 2 d e T d \|σ(γ t )\|(T − t) 2 = C(T − t) 2 \|σ γ t \|. Finally it implies that Z A t T − t = σ γ t (1 + (T − t)ς γ t ) , where ς γ is a bounded process. Therefore by Lemma 13 (for (44)) or Lemma 14 (for (48)), using Proposition 4, we deduce that the BSDE (42) with singular generator has a unique solution (H, Z H ) and, using Theorem 3, that the BSDE (1) with singular terminal condition has also a unique solution (Y, Z Y ). A different asymptotic development under the bounded Itô setting If 1/η is given by (44) with essentially bounded coefficients, we can change our approach. Coming back to (46), we deduce also that A t ≤ γ t (T − t) + d γ 2 (T − t) 2 . Taking the expectation under Q and letting u go to T , we obtain E Q T 0 γ t ψ t f (φ t + ψ t H t ) − f (φ t ) − f ′ (φ t )ψ t H t dt < +∞ and thus E Q sup 0≤t≤T T t Z H s dW Q s dt < +∞. We have proved that (H, Z H ) verifies for any 0 ≤ t ≤ T : H t = T t γ s ψ s f (φ s + ψ s H s ) − f (φ s ) − f ′ (φ s )ψ s H s 1 Hs≥0 ds + T t λ s ψ s + (T − s) κ 1 s + κ 2 s H s + κ 3 s Z H s ds − T t Z H s dW s .(52) Adapting the arguments of the proof of Proposition 4, (H, Z H ) is the unique solution of (52). 5.4 The power case f (y) = −y\|y\| q In this case recall that φ(x) = 1 qx 1/q and ψ(x) = 1 qx 1+1/q and we assume that η is an Itô process, dη t = d η t dt + σ η t dW t ,(53) such that d η ∈ L ∞ ([0, T ] × Ω; R) and σ η ∈ L 2 ([0, T ] × Ω; R d ). Then the process φ t is equal to φ t := η t q(T − t) 1/q = ζ t (q(T − t)) 1/q = ζ t φ(t) and again from condition (A1), ζ is an Itô process with drift d ζ ∈ L ∞ ([0, T ] × Ω; R) and diffusion matrix σ ζ ∈ L 2 ([0, T ] × Ω; R d ). We assume Y t = ζ t φ(t) + ψ(t)H t = (η t ) 1/q φ(t) + ψ(t)H t . and formally obtain the dynamics for H: −dH t = ψ(t) −1 λ t + φ(t)d ζ t dt − Z H t dW t + 1 ψ(t)η t f (ζ t φ(t) + ψ(t)H t ) − f (ζ t φ(t)) − f ′ (ζ t φ(t))ψ(t)H t dt =: F (t, H t ) dt − Z H t dW t ,(55) where F can be rewritten as Let us remark that this generator is again singular and that the second derivative of f is Hence, there exists L > 0 such that F (t, H) = λ t ψ(t) + φ(t) ψ(t) d ζ t + ψ(t)H 2 η tdF dH (t, (T − t) 2 R) ∞ ≤ L ∀t ∈ [T − δ, T ]. The assertion then follows by the mean value theorem. We are now ready to prove that Γ maps B H δ (R) contractiv into itself (for appropriate R and δ ∈ (0, η 1/q ⋆ /R)): For R > 0 specified below choose L > 0 as in Lemma 16. For H, H ′ ∈ B H δ (R) it then holds for all t ∈ [T − δ, T ] \|Γ(H) t − Γ(H ′ ) t \| ≤ E T t \|F (s, H s ) − F (s, H ′ s )\| ds F t ≤(T − t) 3 L H − H ′ H δ . This yields, as long as 0 < δ ≤ 1/(2L), Γ(H) − Γ(H ′ ) H δ ≤ 1 2 H − H ′ H δ . Hence, Γ is an 1/2-contraction on B H δ (R) if δ ≤ 1/(2L). Furthermore, for H ∈ B H δ (R), \|Γ(H) t \| ≤ \|Γ(H) t − Γ(0) t \| + \|Γ(0) t \| ≤ (T − t) 2 R 2 + E T t (q(T − s)) 1+1/q λ s + q(T − s)\|d ζ s \| ds F t ≤ (T − t) 2 R 2 + (T − t) 2 (δ 1/q q 1+1/q λ + q d ζ ∞ ). Thus, choosing R = 2(q 1+1/q λ +q d ζ ∞ ) and δ = min{1, 1/2L, η Proof. Using the property of the map Γ, we deduce that there exists δ > 0 such that there exists a unique process H ∈ H δ such that a.s. for any t ∈ [T − δ, T ]: H t = E T t F (s, H s ) ds F t . By the martingale representation, we obtain Z H and since H ∈ H δ , from Lemma 15, we deduce that the martingale · T −δ Z H dW is a BMO martingale. In particular the random variable H T −δ is bounded. If we consider the BSDE (55) starting at time T − δ from the terminal condition H T −δ , we can apply directly [24,Proposition 5.24] to obtain a unique solution (H, Z H ) on [0, T − δ] such that H is bounded. f ⋆ and terminal condition +∞. And we can adapt the proof of Lemma 2 in order to prove that there exist two functions ϑ ⋆ and ϑ ⋆ = ϑ such that: ϑ ⋆ (T − t) ≤ (Y ⋆ ) t ≤ Y t ≤ ϑ ⋆ (T − t), where ϑ ⋆ is the solution of the ODE: y ′ = λ ⋆ − f (y) η ⋆ with λ ⋆ ≤ f (0) + λ t (ω) ≤ λ and ϑ ⋆ (0) = +∞. Arguing as in the proof of Lemma 3 we get that for any 0 ≤ ε < 1, on some deterministic and non-empty interval [T ε , T ], a.s. φ T − t (1 − ε)η ⋆ ≤ Y t ≤ φ T − t (1 + ε)η ⋆ . the three processes α, β and γ are non-negative. Moreover the functions φ and ψ = −φ ′ are continuous and bounded on [η, +∞) for any η > 0 and ψ never reaches zero on compact subset on (0, ∞). Thereby the coefficients α, β and γ are bounded on any time interval [0, T − θ] for 0 < θ < T . The next result shows that they are also bounded on the whole interval [0, T ]. on the basis of the minimal solution of the BSDE (1), then this process H satisfies all properties described previously. Proposition 2 shows that H can be constructed "from scratch" if (C1) and (C2) hold. Now we prove that this solution (H, Z H ) is a solution of the BSDE (21) (in the sense of Definition 2) and that this solution is minimal. The a priori estimate (37) is crucial here. The processes α and γ are bounded and non-negative on [0, T ]. Hence using Inequality (28+ γ s ds is well defined and is the increasing limit of the same integral on the interval [t, u] for u < T . Since H is non-negative we also have • For lim t→T H t = 0 = H T . • For any ( H, Z) solution of the BSDE (21), a.s. for any t ∈ [0, T ], H t ≥ H t . Lemma 11 11Assume that (C1), (C3) and (C4) hold. Then the term κ 2 (x) := x 2 ψ ′′ (x) ψ(x) is non-negative and bounded on a neighborhood of zero. s is uniformly integrable. But controlling the exponential of the bracket of M is more difficult. If ξ = T 0 1/η s ds is Malliavin differentiable, then we require that its Malliavin derivative has exponential moments. Finally, if we do not control the quantity (Z A ) 2 /A 2 , then from [17, Proposition 3.1], we may have infinitely many solutions. First we show that any solution (H, Z H ) of (41) on [0, T ) is a solution of the BSDE on [0, T ], that is: Proposition 3 Under the hypotheses (C1) to (C5), there exists a non-negative process (H, Z H ) solution (in the sense of Definition 2) of: for any t ∈ [0, T ], is uniformly integrable. Using Girsanov's theorem and the expression of the solution of a linear BSDE (see [24, Proposition 5.31]), we get that a.s. for any t ∈ [ τ , T ], H t − = 0. The arguments used in Section 4 show that the process (H, Z H ) is the minimal non-negative solution of (42), that is if ( H, Z) is another solution of (42), then H t ≥ H t . Now we prove uniqueness of the solution. Since f is concave, and if ∆H = H − H, ∆Z = Z − Z H , then formula for \|ζ u \| reads \|ζ u \| = \|σ(γ t f ′′ (ζ t φ(t) + aψ(t)H)(1 − a) da = F (t, 0) − (q + 1)q ψ(t)H 2 η t 1 0 \|ζ t φ(t) + aψ(t)H\| q−1 sign(ζ t φ(t) + aψ(t)H)(1 − a) da = F (t, 0) − q + 1 η t H 2 (T − t) ζ t + a H T − t (1 − a) da. ⋆ /R} yields Γ(H) H δ ≤ R.Theorem 4 The BSDE (55) has a unique solution (H, Z H ) on [0, T ] such that: H dW is a BMO-martingale. , one singularity comes from the explosion at time T and creates trouble only close to time T . But since H is multiplied now by a random process, there are extra linear terms including the martingale part of the process A. These terms have to be controlled on the whole interval [0, T ], and not only on a neighborhood of T .Nonetheless we prove that under a technical condition, called (H), (H, Z H ) is the unique solution of the BSDE with singular generator and as a by-product, we obtain uniqueness of the solution of the BSDE (1). Let us emphasize that there was only one result about uniqueness, namely [13, Theorem 6.3] for the power case. Uniqueness was proved by showing that any solution (Y, Z Y ) is the value function of the control problem Again this property only depends on the behavior of f near +∞. Note that+∞ y 1 h(t) dt = 1 ρ G(y) ρ , thus 1/h is integrable on [1, +∞). It is known (see [14, Section 178]) that if h is non- decreasing, then we have lim y→+∞ y h(y) = 0. Thus (C5) holds. But since s. In other words H should solve the BSDE: Hence we will adopt their definition ([17, Definition 2.1]) of a solution.Definition 2 (BSDE with singular generator) We say that (H, Z H ) solves the BSDE (21) if the relation (21) holds a.s. for any t ∈ [0, T ] and if holds a.s. on [0, T ]. Then there exists a unique solution (H δ,ε , Z H,δ,ε ) to the BSDE (30) such that a.s. for all t ∈ [0, T ]\|H δ,ε t \| ≤ E T t e µ ε s −µ ε t \|F (s, 0)\|ds F t , and E sup s∈[t,T ] e µ ε s H δ,ε s p + T t e 2µ ε s \|Z H,δ,ε s \| 2 ds p/2 F t ≤ C q E T t e µ ε s \|F (s, 0)\|ds p F t . Finally a.s. for any t ∈ [0, T ], H δ,ε t ≥ 0. Proof. Let us check that all conditions of [24, Proposition 5.24] hold (we keep also the same notations). First we have for all (h, h ′ ) µs \|∆H s \| p−2 1 ∆Hs =0 \|∆Z H s \| 2 dstends to zero when δ 1 and δ 2 go to zero. Moreover remark that ifΛ t = t 0 e p µs \|∆H s \| p−2 1 ∆Hs =0 ∆H s ∆Z H s dW s , then the bracket [Λ] 1/2 on any interval [ τ , T − ε]. To get the existence result similar to Lemma 7, we use [24, Theorem 5.30] and Condition (H) is designed to fulfill the assumptions of this theorem. More important than the existence, we obtain also uniqueness under (H) 5 . Proposition 4 Under the hypotheses (C1) to (C5) and (H), there exists a unique nonnegative process (H, Z H ) solution (in the sense of Definition 2) of the BSDE (42).Proof. Let us show first that any solution ( H, Z) of (42) is non-negative. From the Itô formula for the non-positive part of H we obtain for t ∈ [ τ , T ]: since H is non-negative and f ′ is non-increasing. Arguing as before yields that (∆H) + is equal to zero. Then uniqueness holds on [ τ , T ].Let us extend uniqueness on the whole time interval [0, T ]. If ( H, Z) still denotes another solution, then the two processes solve the same BSDE (42) on [0, τ ] with the same terminal condition H τ = H τ . Since the generator of (42) remains singular on the whole interval (due to the linear term), uniqueness on the rest of the time interval[0, τ ] is not trivial. But if we define Y t = φ t + ψ t H t ,then Y is the first part of the solution of the BSDE (1) on [0, τ ] with the bounded terminal condition φ τ + ψ τ H τ . Since uniqueness holds for the BSDE (1), we deduce that H = H also on [0, τ ]. Theorem 3 Under the hypotheses (C1) to (C5) and (H), the BSDE (1) has a unique solution (Y, Z Y ). This solution is given by: Y t = φ t + ψ t H t a.s. for any t ∈ [0, T ], where H is the unique solution of the BSDE (42). Proof. Let us consider ( Y , Z) solution of the BSDE (1) (in the sense of Definition 1). Then it satisfies (18) and the property of Lemma 4. Therefore if we define H = ( Y − φ)/ψ, then this process H and the related Z H solve the BSDE (42) (in the sense of Definition 2). From the previous proposition, we obtain the desired result. Therefore we can choose τ very close to T such that the BMO norm of M on [ τ , T ] is as small as required. Using the Nirenberg inequality (see[18, Theorem 2.2]), there exists a constant C depending on the BMO norm of M , such that where d and σ are Lipschitz continuous functions defined on R. From [22, Theorems 2.2.1 and 2.2.2], the coordinate γ t belongs to D 1,∞ for any t ∈ [0, T ]. Moreover for any p ≥ 1 \|D r γ t \| p < +∞.sup 0≤r≤T E sup r≤t≤T In the following, Xt denotes a random process, whereas X(t) is a deterministic function. French acronym for right-continuous with left-limit Note that we can consider the solution ϑ of the ODE (15) starting at the point φ ⋆ 1 (τ ). Then defining for t ∈ [0, τ ], φ ⋆ 1 (t) = ϑ(T − t), we can extend the estimate on the whole interval [0, T ]. t dW t . t − f ′ (φ t )ψ t H t Z A t + ψ t Z H t = ψ t Z A t − f ′ (φ t )ψ t H t Z A t + ψ t Z H t Without (H), even the existence of a minimal solution for the BSDE (42) is unclear. If we denotethenIf the quantity ξ = T 0 (T − u)d γ u du is in D 1,2 (see[22]for the notations concerning the Malliavin calculus), then by the Clark-Ocone formula, we haveTherefore since φ is non increasing, we haveDefine.(1), we make the ansatz Y t = φ t + ψ t H t and hence obtain that H t ≥ 0 and:where the coefficients κ 1 , κ 2 , κ 3 are essentially bounded. Compared to (42), the linear term has now bounded coefficients and thus the new BSDE can be solved without any reference to the singular BSDE (1). As in Section 4.2, we may define the generatorThe terminal condition is again equal to zero. From[24,Theorem 5.30], there exists a unique solution (H δ,ε , Z H,δ,ε ) ∈ S p (0, T ), p > 1, to the BSDE:From Lemma 1, we deduce that H is non-negative. The upper bound of Lemma 8 holds since the proof is based on a control on φ t + ψ t H δ,ε t , which satisfies the same dynamics. Hence we can pass to the limit and defineThe sequence Z δ,ε also converges to Z H and clearly (H, Z H ) satisfies the desired dynamics on any interval [0, T ): for any 0 ≤ t ≤ u < Twhere the probability measure Q is equivalent to P with density E( κ 3 s ds) and W Q = W − κ 3 is a Brownian motion under Q. Using Lemma 9, we deduce thatIndeed our upper bound on H is deterministic and thus does not depend on a particular choice of Q equivalent to P. The monotonicity of f leads to. not well-defined at zero if 0 < q < 1. To establish local existence for (55), we don't use monotonicity arguments (as in the preceding sections). But instead, we proceed very similar as in[13]and carry out the Picard iteration in the spaceProof. From our assumptions, the first part of F (t, H t )Note that δ ∈ (0, (η ⋆ ) 1/q /R) ensures that ζ t + aH t /(T − t) > 0 for all t ∈ [T − δ, T ], a ∈ [0, 1]. AndThe lemma is now proved. The preceding lemma allows to define byLemma 16 For every R > 0 there exists a constant L > 0 independent of δ ∈ (0, ηProof. We have for q = 1T − t da.Comparison of the asymptotics and extensionLet us summarize our results.• Under (C1) and (C2), Y and H are related by(20):where H is the minimal solution of the BSDE(21):with a singular generator F given by(22).• Under the additional assumptions (C3) to (C5) and (H), Y can be developed as follows:where H is the unique solution of the BSDE with singular generator (42).• In the Itô setting with bounded coefficients, we get -Uniqueness for H, since (H) holds.-Another possible decomposition of Y :where H is the unique solution of the BSDE (52) and γ solves (50).-In the power case f (y) = −y\|y\| q , we can use (54):where H # solves the BSDE (55) .First let us remark that if η or 1/η is an Itô process, then using (46):From Remark 2 and Condition (A1) and for a bounded process d γ , we deduce that there exists a constants C such thatwith a bounded process κ. In other words, in the Itô setting, all developments coincide. The second point we want to stress is the behavior in the power case f (y) = −y\|y\| q under the Itô setting. From the construction of H # , we know that \|H # t \| ≤ C(T − t) 2 . Using our different asymptotics, the previous development of φ(A) and uniqueness of the (minimal) solution, we obtain that H, H and H verify also this estimate, which is better than (37). However if we use the estimate (12), we havewhere q † is the Hölder conjugate of q + 1. Using thatη = η 1/q is an Itô process with essentially bounded drift d η,q , we haveThus we have the desired result 0 ≤ H t ≤ C(T − t) 2 .Non-negativity of λFrom the comparison principle for monotone BSDE (see[24,Proposition 5.34]), any solution of (1) with a non-negative terminal condition is bounded from below by the solution (Ȳ ,Z) of the BSDE with generator f ⋆ (ω, t, y) = 1 η t (ω) (f (y) − f (0)) − (f (0) + λ t (ω)) − and terminal condition 0.Ȳ is non-positive and if λ is bounded,Ȳ is also bounded. 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[ "ON THE SECOND VARIATION OF THE BIHARMONIC CLIFFORD TORUS IN S 4", "ON THE SECOND VARIATION OF THE BIHARMONIC CLIFFORD TORUS IN S 4" ]	[ "S Montaldo ", "C Oniciuc ", "A Ratto " ]	[]	[]	The flat torus T = S 1 1 2 × S 1 1 2 admits a proper biharmonic isometric immersion into the unit 4-dimensional sphere S 4 given by Φis the minimal Clifford torus and i : S 3 ( 1 √ 2 ) → S 4 is the biharmonic small hypersphere. The first goal of this paper is to compute the biharmonic index and nullity of the proper biharmonic immersion Φ. After, we shall study in the detail the kernel of the generalised Jacobi operator I Φ 2 . We shall prove that it contains a direction which admits a natural variation with vanishing first, second and third derivatives, and such that the fourth derivative is negative. In the second part of the paper we shall analyse the specific contribution of ϕ to the biharmonic index and nullity of Φ. In this context, we shall study a more general compositionΦ =φ • i, whereφ : M m → S n−1 ( 1 √ 2 ), m ≥ 1, n ≥ 3, is a minimal immersion and i : S n−1 ( 1 √ 2 ) → S n is the biharmonic small hypersphere. First, we shall determine a general sufficient condition which ensures that the second variation ofΦ is nonnegatively defined on C φ −1 T S n−1 . Then we complete this type of analysis on our Clifford torus and, as a complementary result, we obtain the p-harmonic index and nullity of ϕ. In the final section we compare our general results with those which can be deduced from the study of the equivariant second variation.	null	[ "https://arxiv.org/pdf/2201.10415v1.pdf" ]	246,275,986	2201.10415	36c3dddc934b1151d4ca56afc0e0079a5e07f4de	ON THE SECOND VARIATION OF THE BIHARMONIC CLIFFORD TORUS IN S 4 25 Jan 2022 S Montaldo C Oniciuc A Ratto ON THE SECOND VARIATION OF THE BIHARMONIC CLIFFORD TORUS IN S 4 25 Jan 2022 The flat torus T = S 1 1 2 × S 1 1 2 admits a proper biharmonic isometric immersion into the unit 4-dimensional sphere S 4 given by Φis the minimal Clifford torus and i : S 3 ( 1 √ 2 ) → S 4 is the biharmonic small hypersphere. The first goal of this paper is to compute the biharmonic index and nullity of the proper biharmonic immersion Φ. After, we shall study in the detail the kernel of the generalised Jacobi operator I Φ 2 . We shall prove that it contains a direction which admits a natural variation with vanishing first, second and third derivatives, and such that the fourth derivative is negative. In the second part of the paper we shall analyse the specific contribution of ϕ to the biharmonic index and nullity of Φ. In this context, we shall study a more general compositionΦ =φ • i, whereφ : M m → S n−1 ( 1 √ 2 ), m ≥ 1, n ≥ 3, is a minimal immersion and i : S n−1 ( 1 √ 2 ) → S n is the biharmonic small hypersphere. First, we shall determine a general sufficient condition which ensures that the second variation ofΦ is nonnegatively defined on C φ −1 T S n−1 . Then we complete this type of analysis on our Clifford torus and, as a complementary result, we obtain the p-harmonic index and nullity of ϕ. In the final section we compare our general results with those which can be deduced from the study of the equivariant second variation. Introduction Harmonic maps are the critical points of the energy functional (1.1) E(φ) = 1 2 M \|dφ\| 2 dv M , where φ : M → N is a smooth map from a compact Riemannian manifold (M m , g) to a Riemannian manifold (N n , h). In particular, φ is harmonic if it is a solution of the Euler-Lagrange system of equations associated to (1.1), i.e., (1.2) − d * dφ = trace ∇dφ = 0 . The left member of (1.2) is a vector field along the map φ or, equivalently, a section of the pull-back bundle φ −1 T N: it is called tension field and denoted τ (φ). In addition, we recall that, if φ is an isometric immersion, then φ is a harmonic map if and only if it defines a minimal submanifold of N (see [7,8] for background). A related topic of growing interest is the study of biharmonic maps. As suggested in [8], [9], these maps, which provide a natural generalisation of harmonic maps, are defined as the critical points of the bienergy functional E 2 (φ) = 1 2 M \|d * dφ\| 2 dv M = 1 2 M \|τ (φ)\| 2 dv M . There have been extensive studies on biharmonic maps. We refer to [5,11,13,15,20,22] for an introduction to this topic. We observe that, obviously, any harmonic map is trivially biharmonic and an absolute minimum for the bienergy. Therefore, we say that a biharmonic map is proper if it is not harmonic and, similarly, a biharmonic isometric immersion is proper if it is not minimal. Our paper is devoted to the study of the second variation of the bienergy functional. In order to introduce this topic, it is convenient to recall some basic facts about the generalised Jacobi operator I φ 2 (V ) and the definition of index and nullity. More specifically, let φ : M → N be a biharmonic map. We shall consider a two-parameter smooth variation {φ t,s } (−ε < t, s < ε, φ 0,0 = φ) and denote by V, W their associated vector fields: V (x) = ∂ ∂t t=0 φ t,0 (x) ∈ T φ(x) N (1.3) W (x) = ∂ ∂s s=0 φ 0,s (x) ∈ T φ(x) N . Note that V and W are sections of φ −1 T N. The Hessian of the bienergy functional E 2 at its critical point φ is defined by H(E 2 ) φ (V, W ) = ∂ 2 ∂t∂s (t,s)=(0,0) E 2 (φ t,s ) . The following theorem was obtained by Jiang and translated by Urakawa [11]: Theorem 1.1. Let φ : M → N be a biharmonic map between two Riemannian manifolds (M m , g) and (N n , h), where M is compact. Then the Hessian of the bienergy functional E 2 at a critical point φ is given by H(E 2 ) φ (V, W ) = I φ 2 (V ), W = M I φ 2 (V ), W dv M , where I φ 2 = : C (φ −1 T N) → C (φ −1 T N) is a semilinear elliptic operator of order 4. When the context is clear, we shall just write I 2 instead of I φ 2 . Next, we want to give an explicit description of the operator I 2 . To this purpose, let ∇ M , ∇ N and ∇ φ be the induced connections on the bundles T M, T N and φ −1 T N respectively. Then the rough Laplacian on sections of φ −1 T N, denoted by ∆, is defined by ∆ = d * d = − m i=1 ∇ φ e i ∇ φ e i − ∇ φ ∇ M e i e i , where {e i } m i=1 is a local orthonormal frame field tangent to M. Let S n (R) denote the Euclidean n-dimensional sphere of radius R. We shall write S n when R = 1. In the present paper, we shall only need the explicit expression of I 2 (V ) in the case that the target manifold is S n . This useful formula, which was first given in [19] and can be deduced from a more general result in [11], is the following: I 2 (V ) = ∆ 2 V + ∆ trace V, dφ· dφ · −\|dφ\| 2 V + 2 dτ (φ), dφ V + \|τ (φ)\| 2 V −2 trace V, dτ (φ)· dφ · −2 trace τ (φ), dV · dφ · − τ (φ), V τ (φ) +trace dφ·, ∆V dφ · +trace dφ·, (trace V, dφ· dφ·) dφ · (1.4) −2\|dφ\| 2 trace dφ·, V dφ · +2 dV, dφ τ (φ) − \|dφ\| 2 ∆V + \|dφ\| 4 V , where · denotes trace with respect to a local orthonormal frame field on M. Next, we recall from the general theory that, since M is compact, the spectrum µ 1 < µ 2 < . . . < µ i < . . . of the generalised Jacobi operator I 2 (V ) associated to the bienergy is discrete and tends to +∞ as i tends to +∞. We denote by V i the eigenspace associated to the eigenvalue µ i . Then we can define the biharmonic index of φ as follows (see, for instance, [24]): (1.5) Index 2 (φ) = µ i <0 dim(V i ) . The biharmonic nullity of φ is defined as the dimension of the kernel of I 2 : (1.6) Null 2 (φ) = dim V ∈ C φ −1 T N : I 2 (V ) = 0 = dim (Ker(I 2 )) . We say that a biharmonic map φ : M → N is stable if Index 2 (φ) = 0. As pointed out in [8], this definition has to be regarded as a notion of second order stability. This notion has a geometric meaning, that is it does not depend on the variation {φ t } but only on the associated vector field V = ∂φ t /∂t t=0 . On the other hand, we know that for a variation {φ t } with associated vector field V which belongs to Ker(I 2 ), it may happen that (1.7) d ℓ dt ℓ t=0 E 2 (φ t ) = 0 , ℓ = 1, . . . , k − 1 ; d k dt k t=0 E 2 (φ t ) < 0 for some even k ≥ 4 . If (1.7) occurs, then the function E 2 (φ t ) has a local maximum at t = 0. In general, the study of the second variation of the bienergy functional is a complicated task and there are not many papers dealing with computations and estimates of the index and nullity of some proper biharmonic submanifolds in the Euclidean unit sphere S n (for instance, see [2,13,14,16]). A natural first step is to investigate the second variation of a proper biharmonic submanifold of S n which lies in the small hypersphere S n−1 ( 1 √ 2 ) as a minimal submanifold. In this order of ideas, one of the simpler examples is the proper biharmonic Clifford torus in S 4 . More precisely, let T be the flat torus with radii equal to 1/2, i.e., (1.8) T = S 1 1 2 × S 1 1 2 . and denote by Φ : T → S 4 the proper biharmonic isometric immersion obtained as the composition of the minimal Clifford torus ϕ : T → S 3 1 √ 2 followed by the proper biharmonic inclusion i : S 3 1 √ 2 → S 4 . This example was partially discussed in [13], where the authors conjectured that its biharmonic index is equal to 1. In this paper we shall complete the study of this example and compute the exact values of both Null 2 (Φ) and Index 2 (Φ). For future use, we point out that the we shall perform all the computations related to Φ = i • ϕ using the following explicit description: Φ : T → S 4 ⊂ R 5 (γ, ϑ) → 1 2 cos γ, 1 2 sin γ, 1 2 cos ϑ, 1 2 sin ϑ, 1 √ 2 , 0 ≤ γ, ϑ ≤ 2π . (1.9) The first result of our paper is: Theorem 1.2. Let Φ : T → S 4 be the proper biharmonic immersion defined in (1.9). Then (i) Index 2 (Φ) = 1 ; (ii) Null 2 (Φ) = 11 . The proof of Theorem 1.2 will be carried out in Section 2. In Section 3 we shall completely determine the structure of Ker(I 2 ) putting in evidence the existence of sections which are not originated from the Killing vector fields. We shall also show that the interesting phenomenon (1.7) does occur for a suitable variation and k = 4. We observe that the Clifford torus ϕ : T → S 3 1 √ 2 is minimal, and so it is a stable critical point for the bienergy. By contrast, the small hypersphere i : S 3 1 √ 2 → S 4 is unstable with biharmonic index equal to 1 (see [13]). Then it seems of interest to study the effects of this composition on the second variation. This type of analysis will be carried out in Section 4, where we shall study the specific contribution of the minimal immersion ϕ to the biharmonic index of the composition Φ = i • ϕ. In this context, we shall study a more general compositionΦ =φ • i, whereφ : M m → S n−1 ( 1 √ 2 ), m ≥ 1, n ≥ 3, is a minimal immersion and i : S n−1 ( 1 √ 2 ) → S n is the biharmonic small hypersphere. First, we shall determine a general sufficient condition which ensures that the second variation ofΦ is nonnegatively defined on C φ −1 T S n−1 . Then we complete this type of analysis on our Clifford torus. As a complementary result, we shall also obtain the p-harmonic index and nullity of the harmonic map ϕ for any p ≥ 2, where a p-harmonic map φ : M → N is a critical point of the p-energy functional (1.10) E (p) (φ) = 1 p M \|dφ\| p dv M . In general, working on the whole C ∞ (M, N), it is very difficult to carry out a complete study of the second variation of a given biharmonic map. Therefore, in order to obtain some partial, but geometrically interesting results, it seems of interest to develop a method of investigation reducing in a suitable way the domain of the bienergy functional. More precisely, when a compact Lie group of isometries G acts on both M and N, we can restrict to the set Σ of all symmetric points with respect to the natural action of G on C ∞ (M, N). In this spirit, in the final section, because the Clifford torus Φ(T) in S 4 is a G-invariant submanifold with G = SO(2) × SO(2), we compare our general results with those which can be deduced from the study of the second variation restricted to the set Σ of all symmetric points of C ∞ (T, S 4 ) with respect to the action of G (see [23]). 4 Proof of Theorem 1.2 The first step is to derive an explicit expression for the operator I 2 : C (Φ −1 T S 4 ) → C (Φ −1 T S 4 ) using formula (1.4). To this purpose, we first introduce suitable vector fields along Φ. More specifically, using Cartesian coordinates y = (y 1 , y 2 , y 3 , y 4 , y 5 ) on R 5 , we define (2.1) V γ = Y γ (Φ) ; V ϑ = Y ϑ (Φ) ; V ν = Y ν (Φ) ; V η = Y η (Φ) , where (2.2) Y γ = 2 (−y 2 , y 1 , 0, 0, 0) = −2y 2 ∂ ∂y 1 + 2y 1 ∂ ∂y 2 ; Y ϑ = 2 (0, 0, −y 4 , y 3 , 0) ; Y ν = √ 2 (y 1 , y 2 , −y 3 , −y 4 , 0) ; Y η = y 1 , y 2 , y 3 , y 4 , − 1 √ 2 . From a geometric viewpoint, we observe that {V γ , V ϑ , V ν , V η } provides an orthonormal basis of T S 4 at each point of the image of Φ. Moreover, we point out that V γ , V ϑ are tangent to the Clifford torus Φ (T), V ν represents the normal direction to the torus in S 3 1 √ 2 , while V η is the normal to S 3 1 √ 2 in S 4 . By way of summary, we conclude that each section V ∈ C (Φ −1 T S 4 ) can be written as (2.3) V = f 1 V γ + f 2 V ϑ + f 3 V ν + f 4 V η , where f j ∈ C ∞ (T), j = 1, . . . , 4. We also point out that X γ and X ϑ , where X γ = 2 ∂ ∂γ , X ϑ = 2 ∂ ∂ϑ , are globally defined vector fields which form an orthonormal basis at each point of T T. Moreover, dΦ(X γ ) = Y γ (Φ) = V γ , dΦ(X ϑ ) = Y ϑ (Φ) = V ϑ . For our purposes, it shall be sufficient to study in detail the case that the functions f j in (2.3) are eigenfunctions of the Laplacian. Our first goal is to compute ∆V γ , ∆V ϑ , ∆V ν , ∆V η . To this purpose, we observe that the choice of vector fields of type (2.2) suggests that the simplest way to compute covariant derivatives along Φ is to use the following well-known formula, where W is a section of the pull-back bundle Φ −1 T S 4 given by the composition of a (local) vector field on S 4 , again denoted by W , with Φ: ∇ Φ X W = ∇ S 4 dΦ(X) W = ∇ R 5 dΦ(X) W + dΦ(X), W Φ . We have: ∇ Φ Xγ V γ = ∇ Φ Xγ Y γ (Φ) = ∇ S 4 Yγ (Φ) Y γ = ∇ R 5 Yγ (Φ) Y γ + Y γ (Φ), Y γ (Φ) Φ = ∇ R 5 −2y 2 ∂/∂y 1 +2y 1 ∂/∂y 2 −2 y 2 ∂ ∂y 1 + 2 y 1 ∂ ∂y 2 + y 1 , y 2 , y 3 , y 4 , y 5 Φ = −4y 1 , −4y 2 , 0, 0, 0 + y 1 , y 2 , y 3 , y 4 , y 5 Φ = −3y 1 , −3y 2 , y 3 , y 4 , y 5 Φ = − √ 2 V ν − V η . 5 Similarly, using the same method of computation we obtain the following identities: (2.4) ∇ Φ Xγ V γ = − √ 2 V ν − V η ; ∇ Φ Xγ V ν = √ 2 V γ ; ∇ Φ X ϑ V γ = 0 ; ∇ Φ X ϑ V ν = − √ 2 V ϑ ; ∇ Φ Xγ V ϑ = 0 ; ∇ Φ Xγ V η = V γ ; ∇ Φ X ϑ V ϑ = √ 2 V ν − V η ; ∇ Φ X ϑ V η = V ϑ . Next, we compute using (2.4) and obtain: (2.5) ∆V γ = 3 V γ ; ∆V ϑ = 3 V ϑ ; ∆V ν = 4 V ν ; ∆V η = 2 V η . By way of example, we detail here the steps to obtain the first equality in (2.5): ∆V γ = − ∇ Φ Xγ ∇ Φ Xγ V γ + ∇ Φ X ϑ ∇ Φ X ϑ V γ = − ∇ Φ Xγ − √ 2 V ν − V η + 0 = √ 2 ∇ Φ Xγ V ν + ∇ Φ Xγ V η = 2 V γ + V γ = 3 V γ . In the sequel, we shall denote by ∆ the Laplace operator on T, i.e., (2.6) ∆ = −4 ∂ 2 ∂γ 2 + ∂ 2 ∂ϑ 2 . Lemma 2.1. Assume that ∆f = λf . Then (i) ∆(f V γ ) = (λ + 3)f V γ + 4 √ 2 f γ V ν + 4f γ V η ; (ii) ∆(f V ϑ ) = (λ + 3)f V ϑ − 4 √ 2 f ϑ V ν + 4f ϑ V η ; (iii) ∆(f V ν ) = −4 √ 2 f γ V γ + 4 √ 2f ϑ V ϑ + (λ + 4)f V ν ; (iv) ∆(f V η ) = −4f γ V γ − 4f ϑ V ϑ + (λ + 2)f V η . Proof of Lemma 2.1. This lemma can be easily proved using (2.4) and (2.5) together with the general formula ∆(f V ) = (∆f ) V − 2 ∇ Φ ∇f V + f ∆V , where ∇f = 2f γ X γ + 2f ϑ X ϑ . 6 Lemma 2.2. Assume that ∆f = λf . Then (i) ∆ 2 (f V γ ) = (λ + 3) 2 f − 48f γγ V γ + [16f γ ϑ ] V ϑ + 4 √ 2(2λ + 7)f γ V ν + [4(2λ + 5)f γ ] V η ; (ii) ∆ 2 (f V ϑ ) = [16f γ ϑ ] V γ + (λ + 3) 2 f − 48f ϑϑ V ϑ − 4 √ 2(2λ + 7)f ϑ V ν + [4(2λ + 5)f ϑ ] V η ; (iii) ∆ 2 (f V ν ) = − 4 √ 2(2λ + 7)f γ V γ + 4 √ 2(2λ + 7)f ϑ V ϑ + (λ 2 + 16λ + 16)f V ν + −16 √ 2f γγ + 16 √ 2f ϑϑ V η ; (iv) ∆ 2 (f V η ) = − [4(2λ + 5)f γ ] V γ − [4(2λ + 5)f ϑ ] V ϑ + −16 √ 2f γγ + 16 √ 2f ϑϑ V ν + (λ 2 + 8λ + 4)f V η . Proof of Lemma 2.2. Since X γ is a Killing field on T, ∆f γ = λ f γ . Similarly, ∆f ϑ = λ f ϑ . Then the lemma can be proved using Lemma 2.1 and again (2.4), (2.5) together with (2.6). Our first key result is: Proposition 2.3. Assume that f ∈ C ∞ (T) is an eigenfunction of ∆ with eigenvalue λ. Then (2.7) (i) I 2 (f V γ ) = [λ(4 + λ)f − 48f γγ ] V γ + 16f γϑ V ϑ + 8 √ 2(2 + λ)f γ V ν + 8λf γ V η ; (ii) I 2 (f V ϑ ) = 16f γϑ V γ + [λ(4 + λ)f − 48f ϑϑ ] V ϑ − 8 √ 2(2 + λ)f ϑ V ν + 8λf ϑ V η ; (iii) I 2 (f V ν ) = − 8 √ 2(2 + λ)f γ V γ + 8 √ 2(2 + λ)f ϑ V ϑ + [λ(12 + λ)f ] V ν + −16 √ 2f γγ + 16 √ 2f ϑϑ V η ; (iv) I 2 (f V η ) = −8λf γ V γ − 8λf ϑ V ϑ + −16 √ 2f γγ + 16 √ 2f ϑϑ V ν + [(λ 2 + 4λ − 16)f ] V η . Proof. The proof of Proposition 2.3 amounts to computing all the 13 terms which appear in the right-hand side of formula (1.4) and adding them up. In this proposition we only have to deal with sections of Φ −1 T S 4 which are of the type V = f V * , where f is an eigenfunction of ∆ on the torus and V * is one amongst the 4 vector fields defined in (2.1). We use Roman numbers to denote the 13 terms in (1.4) and now we show how to compute each of them. Term I. It is of the type ∆ 2 (f V * ) and was computed in Lemma 2.2. Term II. Since Φ is an isometric immersion from a 2-dimensional domain, we have \|dΦ\| 2 = 2. Therefore, ∆ trace (f V * ), dΦ· dΦ · −\|dΦ\| 2 (f V * ) = ∆ (f V * , V γ V γ ) + ∆ (f V * , V ϑ V ϑ ) − 2∆(f V * ) . Since each of the two scalar products V * , V γ , V * , V ϑ is either equal to 0 or to 1, we conclude readily that term II can now be computed using directly Lemma 2.1. Term III. First, we observe that, since Φ = i • ϕ τ (Φ) = di(τ (ϕ)) + trace∇di(dϕ·, dϕ·) = −2V η , where we used the fact that ϕ is minimal and the second fundamental form of the small hypersphere S 3 (1/ √ 2) in S 4 is B(X, Y ) = − X, Y η, with η\| Φ = V η . Then 2 dτ (Φ), dΦ (f V * ) = −4 ∇ Φ Xγ V η , V γ (f V * ) − 4 ∇ Φ X ϑ V η , V ϑ (f V * ) = −8f V * , where the last equality is an immediate consequence of (2.4). Term IV. \|τ (Φ)\| 2 (f V * ) = 4(f V * ). Term V. −2 trace V, dτ (Φ)· dΦ· = 4f V * , ∇ Φ Xγ V η V γ + 4f V * , ∇ Φ X ϑ V η V ϑ = 4f V * , V γ V γ + 4f V * , V ϑ V ϑ . Term VI. −2 trace τ (Φ), d(f V * )· dΦ· = 4 V η , 2f γ V * + f ∇ Φ Xγ V * V γ + 4 V η , 2f ϑ V * + f ∇ Φ X ϑ V * V ϑ . Also this computation can now be ended easily using (2.4). Term VII. − τ (Φ), (f V * ) τ (Φ) = −4f V η , V * V η . Term VIII. trace dΦ·, ∆(f V * ) dΦ· = V γ , ∆(f V * ) V γ + V ϑ , ∆(f V * ) V ϑ . So this term can be calculated using Lemma 2.1. Term IX. Since {V γ , V ϑ } are orthonormal at each point we easily find: trace dΦ·, (trace (f V * ), dΦ· dΦ·) dΦ· = f V * , V γ V γ + f V * , V ϑ V ϑ . Term X. −2\|dΦ\| 2 trace dΦ·, (f V * ) dΦ· = −4f V γ , V * V γ − 4f V ϑ , V * V ϑ . Term XI. 2 d(f V * ), dΦ τ (Φ) = −4 2f γ V * , V γ + f ∇ Φ Xγ V * , V γ + 2f ϑ V * , V ϑ + f ∇ Φ X ϑ V * , V ϑ V η . Therefore the calculation of this term ends easily using (2.4). Term XII. −\|dΦ\| 2 ∆(f V * ) = −2∆(f V * ) . This computation was performed in Lemma 2.1. Term XIII. \|dΦ\| 4 (f V * ) = 4f V * . Now we are in the right position to complete the proof of the proposition. As for (2.7)(i), we follow the lines of computation which we have just described and we obtain the 13 terms 8 I-XIII in the case that V = f V γ . The result is I = (λ + 3) 2 f − 48f γγ V γ + [16f γ ϑ ] V ϑ + 4 √ 2(2λ + 7)f γ V ν + [4(2λ + 5)f γ ] V η II = − (λ + 3)f V γ + 4 √ 2 f γ V ν + 4f γ V η III = −8f V γ IV = 4f V γ V = 4f V γ VI = −4f V γ VII = 0 VIII = (λ + 3)f V γ IX = f V γ X = −4f V γ XI = −8f γ V η XII = −2 (λ + 3)f V γ + 4 √ 2 f γ V ν + 4f γ V η XIII = 4f V γ . Adding up all these 13 terms and simplifying we obtain (2.7)(i). The proof of (2.7)(ii)-(iv) is analogous and so we omit further details. Proof of Theorem 1.2. We recall that the Laplace operator ∆ on T is given in (2.6) and we denote by λ i , i ∈ N, its spectrum. We know that the eigenvalues of ∆ have the form λ i = 4(m 2 + n 2 ), where m, n ≥ 0. Then it is convenient to define S λ i = {f 1 V γ : ∆f 1 = λ i f 1 } ⊕ {f 2 V ϑ : ∆f 2 = λ i f 2 } ⊕ {f 3 V ν : ∆f 3 = λ i f 3 } ⊕ {f 4 V η : ∆f 4 = λ i f 4 } (2.8) As in [13], S λ i ⊥ S λ j if i = j and ⊕ +∞ i=0 S λ i is dense in C (Φ −1 T S 4 ) (note that the scalar product which we use on sections of Φ −1 T S 4 is the standard L 2 -inner product). Moreover, using the explicit description (2.8), it is easy to deduce from Proposition 2.3 that the operator I 2 preserves each of the S λ i . Indeed, this is a consequence of the fact that, if f is an eigenfunction of ∆ with eigenvalue λ, then the same is true for all its partial derivatives with respect to γ, ϑ because ∂/∂γ and ∂/∂ϑ are Killing vector fields on the torus. By way of summary, we can compute the biharmonic index and nullity of I 2 restricted to each of the S λ i 's and then add up the results to complete the proof of Theorem 1.2. First, let us examine the eigenvalue λ 0 = 0. We have S λ 0 = {c 1 V γ : c 1 ∈ R} ⊕ {c 2 V ϑ : c 2 ∈ R} ⊕ {c 3 V ν : c 3 ∈ R} ⊕ {c 4 V η : c 4 ∈ R} and dim S λ 0 = 4. It follows by a direct application of Proposition 2.3 that the restriction of I 2 to S λ 0 gives rise to the eigenvalues µ 1 = −16 with multiplicity 1 (eigenvector V η ), and µ 2 = 0 with multiplicity equal to 3 (eigenvectors {V γ , V ϑ , V ν }). Then, we conclude that the contributions of this subspace to Index 2 (Φ) and Null 2 (Φ) are respectively 1 and 3. Next, let us consider the case that λ > 0. In this case, it is difficult to describe explicitly all the couples (m, n) such λ = 4(m 2 + n 2 ) and so we proceed introducing a further, more suitable, decomposition. More precisely, let us denote by W λ the corresponding eigenspace. In a similar fashion to [2], we decompose (2.9) W λ = W m,0 ⊕ m,n≥1 W m,n ⊕ W 0,n , where it is understood that in (2.9) we have to consider all the possible couples (m, n) ∈ N×N such that λ = 4(m 2 +n 2 ). S m,n = {f 1 V γ : f 1 ∈ W m,n } ⊕ {f 2 V ϑ : f 2 ∈ W m,n } ⊕ {f 3 V ν : f 3 ∈ W m,n } ⊕ {f 4 V η : f 4 ∈ W m,n } . All these subspaces are orthogonal to each other. Moreover, for any positive eigenvalue λ i , we have S λ i = ⊕ 4(m 2 +n 2 )=λ i S m,n . It follows easily from Proposition 2.3 that the operator I 2 preserves each of the subspaces S m,n . Therefore, its spectrum can be computed by determing the eigenvalues of the matrices associated to the restriction of I 2 to each of the S m,n 's. We shall see that dim (S m,n ) is either 8 or 16 and so this approach enable us to provide a rather unified treatment, while the direct study of the S λ i 's would be much trickier. We point out that, in the rest of the proof, there are several long and tedious computations which can be conveniently carried out using a suitable software (we used Mathematica). We separate three cases: Case 1: S m,0 , m ≥ 1. In this case dim (S m,0 ) = 8 and an orthonormal basis {e i } i=1...8 of S m,0 is given by: √ 2 cos(mγ) π V γ , √ 2 sin(mγ) π V γ , √ 2 cos(mγ) π V ϑ , √ 2 sin(mγ) π V ϑ (2.10) √ 2 cos(mγ) π V ν , √ 2 sin(mγ) π V ν , √ 2 cos(mγ) π V η , √ 2 sin(mγ) π V η . Using Proposition 2.3 with λ = 4m 2 and computing we find that the (8 × 8)-matrices which describe the operator I 2 with respect to the basis (2.10), i.e. (I 2 (e i ), e j ), are:             16m 2 (m 2 + 4) 0 0 0 0 16m 2 (m 2 + 4) 0 0 0 0 16 (m 4 + m 2 ) 0 0 0 0 16 (m 4 + m 2 ) 0 8 √ 2m (4m 2 + 2) 0 0 −8 √ 2m (4m 2 + 2) 0 0 0 0 32m 3 0 0 −32m 3 0 0 0 0 −8 √ 2m (4m 2 + 2) 0 −32m 3 8 √ 2m (4m 2 + 2) 0 32m 3 0 0 0 0 0 0 0 0 0 16m 2 (m 2 + 3) 0 16 √ 2m 2 0 0 16m 2 (m 2 + 3) 0 16 √ 2m 2 16 √ 2m 2 0 16 (m 4 + m 2 − 1) 0 0 16 √ 2m 2 0 16 (m 4 + m 2 − 1)             The characteristic polynomial is (2.11) P (x) = x − 16(m 2 + m 4 ) 2 P 3 (x) 2 , where P 3 (x) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 and its coefficients are the following: a 0 = −4096m 2 (m − 1)(m + 1)(m 8 − 3m 6 + m 4 + 4m 2 − 2) a 1 = 256m 2 (3m 6 + 4m 4 + 7m 2 − 9) a 2 = −16(3m 4 + 8m 2 − 1) a 3 = 1 . Because all the roots of P 3 (x) are real, according to the Descartes rule we can conclude that the third order polynomial P 3 (x) possesses three positive roots provided that (2.12) a 0 < 0 , a 1 > 0 , a 2 < 0 , a 3 > 0 . Now, it is easy to check that (2.12) is satisfied provided that m ≥ 2. By contrast, when m = 1 we have a 0 = 0. In this case, it is easy to conclude that P 3 (x) has two positive roots and one root equal to 0, with multiplicity 2 as a root of the characteristic polynomial P (x) in (2.11). In summary, we have proved that the contributions of the subspaces S m,0 , m ≥ 1, to Null 2 (Φ) and Index 2 (Φ) are respectively 2 and 0. Case 2: S 0,n , n ≥ 1. Because of the symmetry of the map Φ, the contribution of the subspaces S 0,n , n ≥ 1, to Null 2 (Φ) and Index 2 (Φ) is precisely as in Case 1 above, i.e., 2 for the nullity and 0 for the index. Case 3: S m,n , m, n ≥ 1. This is the case which requires the biggest computational effort. In this case dim (S m,n ) = 16 and an orthonormal basis {e i } i=1...16 of S m,n is given by: 2 π cos(mγ) cos(nϑ)V γ , 2 π cos(mγ) sin(nϑ)V γ , 2 π sin(mγ) cos(nϑ)V γ , 2 π sin(mγ) sin(nϑ)V γ , 2 π cos(mγ) cos(nϑ)V ϑ , 2 π cos(mγ) sin(nϑ)V ϑ , 2 π sin(mγ) cos(nϑ)V ϑ , 2 π sin(mγ) sin(nϑ)V ϑ , 2 π cos(mγ) cos(nϑ)V ν , 2 π cos(mγ) sin(nϑ)V ν , 2 π sin(mγ) cos(nϑ)V ν , 2 π sin(mγ) sin(nϑ)V ν , 2 π cos(mγ) cos(nϑ)V η , 2 π cos(mγ) sin(nϑ)V η , 2 π sin(mγ) cos(nϑ)V η , 2 π sin(mγ) sin(nϑ)V η . As an application of Proposition 2.3 with λ = 4(m 2 + n 2 ), with the aid of Mathematica we compute the (16 × 16)-matrices (I 2 (e i ), e j ) and find that their characteristic polynomial is P (x) = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + c 4 x 4 4 = Q 4 (x) 4 , where the coefficients of the fourth order polynomial Q 4 (x) are: Next, with the methods used in [16], it is not difficult to show that, if m ≥ 2, n ≥ 1 or m ≥ 1, n ≥ 2, then c 0 > 0 , c 1 < 0 , c 2 > 0 , c 3 < 0 , c 4 > 0 . Because all the roots of Q 4 (x) are real, we conclude that the fourth order polynomial Q 4 (x) admits 4 positive roots in these cases. By contrast, when m = n = 1 we have: c 0 = 0 , c 1 < 0 , c 2 > 0 , c 3 < 0 , c 4 > 0 . It follows easily that, in this case, Q 4 (x) has 3 positive roots and one root equal to 0. This last root corresponds to the zero eigenvalue for I 2 , with multiplicity equal to 4. In summary, we have proved that the contributions of the subspaces S m,n , m, n ≥ 1, to Null 2 (Φ) and Index 2 (Φ) are respectively 4 and 0 and this ends Case 3. Adding up the results of S λ 0 , λ 0 = 0, with those of Cases 1,2,3 we conclude that Null 2 (Φ) = 3 + 2 + 2 + 4 = 11 , Index 2 (Φ) = 1 + 0 + 0 + 0 = 1 and so the proof of Theorem 1.2 is completed. The study of Ker(I 2 ) Let φ : (M, g) → (N, h) be a biharmonic map. First, we recall a basic fact about Ker(I 2 ), that is: if {φ t } is a variation of φ by means of biharmonic maps, then V = d dt t=0 φ t belongs to Ker(I 2 ). In fact, for an arbitrary W ∈ C(φ −1 T N), we can consider a variation {W t } such that W 0 = W and W t ∈ C(φ −1 t T N). Then we define the two parameter variation of φ by φ t,s (x) = exp φt(x) (sW t (x)) . With respect to {φ t,s } we have V = d dt t=0 φ t,0 , W = d ds s=0 φ 0,s . As φ t is biharmonic for all t, ∂ ∂s s=0 E 2 (φ t,s ) = 0 and consequently ∂ 2 ∂t∂s (t,s)=(0,0) E 2 (φ t,s ) = (I 2 (V ), W ) = 0 . We conclude that V belongs to Ker(I 2 ). In particular, if {φ t } is given by composing φ with a one parameter family of isometries of the domain or the codomain, the above properties imply {dφ(X) : X ∈ C(T M), X is Killing} ⊂ Ker(I 2 ) and {Z • φ : Z ∈ C(T N), Z is Killing} ⊂ Ker(I 2 ) . In this section we show that Ker(I 2 ) is the orthogonal sum of a 10-dimensional subspace spanned by Killing vector fields as above and a 1-dimensional subspace spanned by V ν . It is easy to describe the space of Killing vector fields on T: it has dimension 2 and it is spanned by {X γ , X ϑ }. As for the target, we know that the space of Killing vector fields on S n has dimension n(n + 1)/2. In particular, a base for this subspace of C (T S 4 ) can be obtained by restriction 13 of the following 10 vector fields on R 5 : Z 1 = (−y 2 , y 1 , 0, 0, 0) ; Z 2 = (0, 0, −y 4 , y 3 , 0) ; Z 3 = (−y 4 , 0, 0, y 1 , 0) ; Z 4 = (0, −y 3 , y 2 , 0, 0) ; Z 5 = (−y 3 , 0, y 1 , 0, 0) ; Z 6 = (0, −y 4 , 0, y 2 , 0) ; Z 7 = (0, −y 5 , 0, 0, y 2 ) ; Z 8 = (−y 5 , 0, 0, 0, y 1 ) ; Z 9 = (0, 0, 0, −y 5 , y 4 ) ; Z 10 = (0, 0, −y 5 , 0, y 3 ) . Then, we define vector fields V i ∈ C (Φ −1 T S 4 ) as follows: (3.1) V i (P ) = Z i (Φ(P )) , i = 1, . . . 10 . Now, we are in the right position to state the main result of this section. Theorem 3.1. (3.2) Ker(I 2 ) = W (10) ⊕ W (1) , where W (10) is a 10-dimensional subspace spanned by the vector fields V i ∈ C (Φ −1 T S 4 ) defined in (3.1), while dim W (1) = 1 and W (1) is spanned by V ν . Proof. A computation, using the notations in (2.1), shows that: V 1 = 1 2 V γ V 2 = 1 2 V ϑ V 3 = 1 2 sin γ sin ϑ V γ + 1 2 cos γ cos ϑ V ϑ − 1 √ 2 cos γ sin ϑ V ν (3.3) V 4 = − 1 2 cos γ cos ϑ V γ − 1 2 sin γ sin ϑ V ϑ − 1 √ 2 sin γ cos ϑ V ν V 5 = 1 2 sin γ cos ϑ V γ − 1 2 cos γ sin ϑ V ϑ − 1 √ 2 cos γ cos ϑ V ν V 6 = − 1 2 cos γ sin ϑ V γ + 1 2 sin γ cos ϑ V ϑ − 1 √ 2 sin γ sin ϑ V ν V 7 = − 1 √ 2 cos γ V γ − 1 2 sin γ V ν − 1 √ 2 sin γ V η V 8 = 1 √ 2 sin γ V γ − 1 2 cos γ V ν − 1 √ 2 cos γ V η (3.4) V 9 = − 1 √ 2 cos ϑ V ϑ + 1 2 sin ϑ V ν − 1 √ 2 sin ϑ V η V 10 = 1 √ 2 sin ϑ V ϑ + 1 2 cos ϑ V ν − 1 √ 2 cos ϑ V η . From this it is easy to check that, for all 1 ≤ i, j ≤ 10, i = j, we have (3.5) V i , V ν = 0 and V i , V j = 0 and so the V j 's are linearly independent. Therefore, they span a 10-dimensional subspace W (10) of Ker(I 2 ). Now, the statement (3.2) in Theorem 3.1 is an immediate consequence of the fact that Null 2 (Φ) = 11 and V ν ∈ Ker(I 2 ), as shown in the proof of Theorem 1.2. Remark 3.2. (i) The contribution to the nullity of Φ given by the two Killing fields {X γ , X ϑ } on the domain T is included in W (10) . Indeed, the first two equalities in (3.3) show that both dϕ(X γ ) = V γ = 2V 1 and dϕ(X ϑ ) = V ϑ = 2V 2 belong to W (10) . (ii) In the notation of the proof of Theorem 1.2, the vector fields V 3 , V 4 , V 5 , V 6 belong to S 1,1 . Using Proposition 2.3 with λ = 8, together with the explicit equalities (3.3)-(3.4), it is possible to check directly that I 2 (V i ) = 0, i = 3, . . . , 6. Similarly, using λ = 4, the same is true for V 7 , V 8 , which belong to S 1,0 , and V 9 , V 10 ∈ S 0,1 . Next, we shall compute the higher order derivatives for a natural variation {Φ t } such that ∂Φ t /∂t t=0 = V ν , and from this we shall deduce that (1.7) occurs with k = 4. Theorem 3.3. Let Φ t : T → S 4 ⊂ R 5 be defined by (3.6) Φ t = 1 √ 1 + t 2 (Φ + tV ν ) . Then ∂Φ t /∂t t=0 = V ν and we have: d ℓ dt ℓ t=0 E 2 (Φ t ) = 0 , ℓ = 1, 2, 3 . (3.7) d 4 dt 4 t=0 E 2 (Φ t ) = −48π 2 < 0 . Proof. Let us consider the variation Φ t : T → S 4 ⊂ R 5 defined in (3.6). It is immediate to check that ∂Φ t /∂t t=0 = V ν . Then we have to compute explicitly the tension field of Φ t . To this purpose, we observe that τ (Φ t ) = −∆Φ t + dΦ t 2 Φ t , where ∆ was given in (2.6). Now, a routine computation yields: τ (Φ t ) = − √ 2 t + 1 cos γ (t 2 + 1) 3/2 , − √ 2 t + 1 sin γ (t 2 + 1) 3/2 , √ 2 t − 1 cos ϑ (t 2 + 1) 3/2 , √ 2 t − 1 sin ϑ (t 2 + 1) 3/2 , √ 2 (2t 2 + 1) (t 2 + 1) 3/2 . From this we obtain τ (Φ t ) 2 = 4 + 8t 2 (t 2 + 1) 2 . Finally, E 2 (Φ t ) = 1 2 T τ (Φ t ) 2 dV T = 2 + 4t 2 (t 2 + 1) 2 π 2 from which (3.7) follows readily. Further studies: composition and p-harmonic index and nullity of ϕ So far, we computed the biharmonic index and the biharmonic nullity of Φ = i • ϕ. The aim of the first part of this section is to carry out a detailed study of a natural sub-class of variations, i.e., variations of the form Φ s,t = i • ϕ s,t . As a completion of this analysis, in the second part of the section we shall compute the p-harmonic index and nullity of ϕ. Now, more generally, letΦ =φ • i, whereφ : M m → S n−1 ( 1 √ 2 ), m ≥ 1, n ≥ 3, is a minimal immersion and i : S n−1 ( 1 √ 2 ) → S n , is the biharmonic small hypersphere. The mapΦ is proper biharmonic and we shall determine a general sufficient condition which ensures that the second variation ofΦ is nonnegatively defined on C φ −1 T S n−1 ( 1 √ 2 ) . To this purpose, we study 2-parameter variations Φ s,t = i •φ s,t , whereφ s,t : M m → S n−1 ( 1 √ 2 ) is a 2-parameter variation ofφ. Let V, W be vector fields associated toΦ s,t as in (1.3). We denote the subspace of these vector fields as Wφ ⊂ C Φ −1 (T S n ) (note that, geometrically, there is a natural identification between Wφ and C φ −1 T S n−1 ( 1 √ 2 ) ). Now, E 2 Φ s,t = 1 2 M τ Φ s,t 2 dV M and we observe that, in this case, (4.1) τ Φ s,t 2 = di τ (φ s,t ) + trace∇di(dφ s,t ·, dφ s,t ·) 2 = \|τ φ s,t \| 2 + \|dφ s,t \| 4 where the last equality is a consequence of the fact that i is an isometric immersion and the two terms are orthogonal. As a consequence of (4.1), for all V, W ∈ Wφ we have: (IΦ 2 (V ), W ) = (Iφ 2 (V ), W ) + 2(Jφ (4) (V ), W ) = ((Jφ) 2 (V ), W ) + 2 (Jφ (4) (V ), W ) , where Jφ (p) is the Jacobi operator associated to the p-energy and we write Jφ for Jφ (2) . Now, let π : C Φ −1 (T S n ) → Wφ be the orthogonal projection and denote (4.2) IΦ ,π 2 (V ) = π IΦ 2 (V ) = (Jφ) 2 (V ) + 2 Jφ (4) (V ) ∀V ∈ Wφ . We want to study the quadratic form = ∆V + trace 2 − dφ(·), dφ(·) V + V, dφ(·) dφ(·) (4.5) = ∆V + trace 2 − mV + V ⊤ , where, for V ∈ Wφ, V ⊤ and V ⊥ are respectively the tangent and the normal component. The expression of Jφ (p) was computed in [18] for a generic harmonic mapφ. Because, in our context,φ is a minimal immersion, (1.9) of [18] simplifies and gives: (4.6) Jφ (p) (V ) = (p − 2)m p−4 2 d * dV, dφ dφ + m p−2 2 Jφ(V ) . Now, using (4.6) with p = 4 in (4.2) we deduce that (4.7) IΦ ,π 2 (V ) = (Jφ) 2 (V ) + 2m Jφ(V ) + 4 d * dV, dφ dφ ∀V ∈ Wφ . Sinceφ is an immersion, there exists a unique vector field ξ ∈ C(T M) such that dφ(ξ) = V ⊤ . We prove that (4.8) d * dV, dφ dφ , V = M (divξ) 2 dV M . Indeed, let P ∈ M be arbitrarily fixed. We use a geodesic frame field {X i } around P . Then, at P we have: Theorem 4.1 suggests to investigate the spectrum of Jφ, or, more in detail, to compute the index and the nullity of Jφ. The stability of minimal and CMC immersions with respect to the volume functional has been widely studied in the literature (see [1,3] and [6] for more recent developments). On the other hand, any minimal immersion is also a p-harmonic map and so it is natural to study its second variation as a critical point of the p-energy functional (1.10). There are not many papers where index and nullity computations in this context have been carried out. Some interesting results of this type were obtained in [25] where, for instance, the p-harmonic index of the identity map was computed when M is an Einstein manifold. Nevertheless, not much has been done when the map is not the identity and so our first goal is to compute the p-harmonic index and the p-harmonic nullity of the minimal Clifford torus dV, dφ = m i=1 ∇φ X i (V ⊤ + V ⊥ ), dφ(X i ) = m i=1 ∇ M X i ξ, X i + B(X i , ξ), X i + ∇ ⊥ X i V ⊥ , X i − A V ⊥ (X i ), X i = m i=1 ∇ M X i ξ, X i − A V ⊥ (X i ), X i = div(ξ) − m i=1 B(X i , X i ), V ⊥ = div(ξ) − m H, V ⊥ = div(ξ) ,ϕ : T → S 3 ( 1 √ 2 ) ⊂ R 4 (γ, ϑ) → 1 2 cos γ, 1 2 sin γ, 1 2 cos ϑ, 1 2 sin ϑ , 0 ≤ γ, ϑ ≤ 2π . We recall that Index p−harm (ϕ) and Null p−harm (ϕ) are defined precisely as in (1.5) and (1.6) respectively, with I 2 replaced by J ϕ (p) . As in the case of I 2 , when the context is clear we shall simply write J, J (p) instead of J ϕ , J ϕ (p) respectively. Also, when p = 2, we write harm instead of 2−harm . Our first result in this setting is: (i) J (f V γ ) = λf V γ + 4 √ 2f γ V ν (ii) J (f V ϑ ) = λf V ϑ − 4 √ 2f ϑ V ν (iii) J (f V ν ) = −4 √ 2f γ V γ + 4 √ 2f ϑ V ϑ + λ f V ν . The subspaces S 0 and S m,n are defined precisely as in the proof of Theorem 1.2, just omitting V η and working in R 4 . So, now, the subspace S 0 is spanned {V γ , V ϑ , V ν } and dim(S 0 ) = 3. It follows immediately from Proposition 4.4 with λ = 0 that S 0 belongs to Ker(J) and thus its contribution to Null harm (ϕ) is equal to 3. As for S m,0 , m ≥ 1, we construct the 6 × 6-matrices (J(e i ), e j ) and find that their associated characteristic polynomial is P (x) = (−4m 2 + x) 2 (16m 4 + x 2 − 8m 2 (4 + x)) 2 . Then we see that, when m ≥ 2, all the eigenvalues are positive. By contrast, when m = 1, we have one negative eigenvalue µ 1 = 4 − 4 √ 2 with multiplicity 2. The contribution of the subspaces S 0,n is analogous. In summary, the subspaces S m,0 and S 0,n , m, n ≥ 1 do not contribute to Null harm (ϕ) and give a contribution equal to 4 to Index harm (ϕ). Next, one studies the subspaces S m,n , m, n ≥ 1, and finds that there is no contribution to Index harm (ϕ). By contrast, the eigenvalue µ = 0 appears in the study of S 1,1 with multiplicity 4. Adding up all the contributions we end readily the proof of Theorem 4.3. Remark 4.5. In order to complete the analysis which we carried out during the proof of Theorem 4.3, we point out that a basis for the 4-dimensional eigenspace W µ 1 associated to µ 1 is {W 1 , W 2 , W 3 , W 4 }, where W 1 = √ 2 π cos γ V γ + sin γ V ν (4.10) W 2 = √ 2 π − sin γ V γ + cos γ V ν W 3 = √ 2 π − cos ϑ V ϑ + sin ϑ V ν W 4 = √ 2 π sin ϑ V ϑ + cos ϑ V ν . Let W conf ⊂ C ϕ −1 T S 3 1 √ 2 denote the subspace determined by the restrictions to ϕ(T) of the conformal fields on S 3 1 √ 2 . These vector fields can be conveniently described as follows. Let a = (a 1 , a 2 , a 3 , a 4 ) ∈ R 4 . Then the elements of W conf have the following form: V a = a − 2 a, ϕ ϕ . Since ϕ(T) is not contained in any hyperplane of R 4 , it is easy to check that dim(W conf ) = 4. It is well-known that conformal fields have been used to prove the instability of harmonic maps into S n , n ≥ 3. In our example, a computation using (4.5) and standard properties of d and d * shows that, for all a ∈ R 4 , a = 0, (J(V a ), V a ) = 2 T 2[4 a, ϕ 2 − \|a\| 2 ] + V γ , V a 2 + V ϑ , V a 2 dV T = −π 2 \|a\| 2 (V a , V a ) = T \|a\| 2 − 2 a, ϕ 2 dV T = 3 4 π 2 \|a\| 2 and from this (J(V a ), V a ) (V a , V a ) = − 4 3 > µ 1 . We deduce that the Hessian is negatively defined on W conf , but we have W conf ∩ W µ 1 = { 0}. By contrast, we have verified that the Hessian of Φ is positively defined on the subspace determined by the conformal vector fields on S 4 . The proof of Theorem 4.3 has shown that J ϕ admits the negative eigenvalue µ = 4 − 4 √ 2 ∈ (−4, 0). Therefore, in this case, the hypothesis of Theorem 4.1 is not verified. Thus, in this case, the study of the quadratic form (4.3) must be carried out directly. The result of this investigation is: Proposition 4.6. Let H(E 2 ) Φ (V, W ) be the quadratic form defined as in (4.3). Then (I Φ,π 2 (V ), V ) ≥ 0 ∀ V ∈ Wφ . Proof. We know from Theorem 1.2 that Index 2 (Φ) = 1 and I Φ 2 (V η ) =μV η , withμ = −16 < 0. We argue by contradiction and assume that I Φ,π 2 has a negative eigenvalue. Then there exists V * ∈ W ϕ such that I Φ,π 2 (V * ) = µ * V * , with µ * < 0. Because both V * and I Φ 2 (V * ) are orthogonal to V η , it is easy to conclude that I Φ 2 would be negatively defined on the 2-dimensional subspace spanned by V * and V η , a fact which contradicts Theorem 1.2 and completes the proof of the proposition. Remark 4.7. We point out that Proposition 4.6 can also be verified by using the method of Theorem 1.2. Remark 4.8. Obviously, Ker(I Φ 2 ) ∩ W ϕ ⊂ Ker(I Φ,π 2 ) . Using the method of Theorem 1.2, we could verify that actually Ker(I Φ 2 ) ∩ W ϕ = Ker(I Φ,π 2 ). More precisely, we observed first that, in a similar fashion to (2.3), each section V ∈ W ϕ can be written as (4.11) V = f 1 V γ + f 2 V ϑ + f 3 V ν , where f j ∈ C ∞ (T), j = 1, . . . , 3. Then, we defined the subspaces S 0 , S m,n taking into account (4.11). Next, we found that the only contributions to the nullity are 3 from S 0 and 4 from S 1,1 , so that dim Ker(I Φ,π 2 ) = 7. Finally, we observed that V ν and V 1 , . . . , V 6 belong to W ϕ and so they provide a basis for Ker(I Φ,π 2 ). By contrast, V 7 , . . . , V 10 are not in W ϕ . We complete our analysis by means of the p-harmonic extension of Theorem 4.3, a result which shows how index and nullity may depend on p. For simplicity, we shall assume p ≥ 1. (i) J p (f V γ ) = −(p − 2)2 p−4 2 4f γγ V γ + 4f γϑ V ϑ + 2 p−2 2 J(f V γ ) (ii) J p (f V ϑ ) = −(p − 2)2 p−4 2 4f γϑ V γ + 4f ϑϑ V ϑ + 2 p−2 2 J(f V ϑ ) (iii) J p (f V ν ) = 2 p−2 2 J(f V ν ) , where J(V ) is given in Proposition 4.4. The calculations required to prove this proposition just amount to use (4.6) with m = 2 and compute explicitly the d * term. Equivariant Index and Nullity In Theorems 1.2 and 3.1 we obtained a complete description of the second variation of Φ : T → S 4 . From these results it appears that there exist essentially two geometrically significant directions, that is V η , which determines the index, and V ν , which is the only direction of KerI 2 which is not generated by a Killing field either of the ambient or of the domain. The Clifford torus Φ(T) in S 4 is a G-invariant submanifold, with G = SO(2) × SO(2). The main aim of this section is to show that the key directions V η , V ν could also be determined by a direct analysis in the orbit space S 4 /G. It is convenient to consider the following family of maps: Φ η,ν : S 1 (R 1 ) × S 1 (R 2 ) → S 4 ⊂ R 2 × R 2 × R (γ, ϑ) → (sin η sin ν)e iγ , (sin η cos ν)e iϑ , cos η , 0 ≤ γ, ϑ ≤ 2π . (5.1) In a map of the type (5.1), we assume that R 1 , R 2 > 0 are fixed. A map of the type (5.1) is G-equivariant, i.e., Φ η,ν (gx) = gΦ η,ν (x) for all x ∈ T, g ∈ G (here G acts naturally on T, and on the first 4 coordinates of R 4 concerning the target). In fact, the family of all the maps of the type (5.1) form the set Σ of the symmetric points of C ∞ (T, S 4 ) with respect to the action of G (see [23]). In our case, Σ is 2-dimensional and the tangent space to Σ at a symmetric point Φ η,ν is generated by ∂Φ η,ν ∂η , ∂Φ η,ν ∂ν . The orbit space T/G is just a point, while the orbit space Q = S 4 /G is naturally identified with the spherical sector 0 ≤ η ≤ π, 0 ≤ ν ≤ π/2. By way of summary, at the level of orbit spaces, the map Φ η,ν can be identified with the point (η, ν) ∈ Q. To proceed further, we need to compute the bienergy function. To this purpose, we observe that the tension field of a map of type (5.1) can be computed using τ (Φ η,ν ) = −∆Φ η,ν + dΦ η,ν 2 Φ η,ν and, writing M for S 1 (R 1 ) × S 1 (R 2 ), that leads us to E 2 (Φ η,ν ) = 1 2 M τ (Φ η,ν ) 2 dV M = MÊ 2 (η, ν) dV M , where, setting c = 1/(32R 4 1 R 4 2 ), E 2 (η, ν) = c 5R 4 1 − 2R 2 1 R 2 2 + 5R 4 2 + (3R 4 1 + 2R 2 1 R 2 2 + 3R 4 2 ) cos(2η) sin 2 η −2(R 2 1 − R 2 2 ) 2 cos(4ν) sin 4 η + 2(R 4 1 − R 4 2 ) cos(2ν) sin 2 (2η) . We shall callÊ 2 (η, ν) the reduced bienergy function. We point out that, alternatively, the explicit expression of τ (Φ η,ν ) can also be obtained using the reduced energy function and the fact that τ (Φ η,ν ) is tangent to Σ at Φ η,ν (see [17]). Now, we are in the cohomogeneity zero case of the reduction theory introduced in [12]. Then, according to [4,Proposition 2.5], the map Φ η,ν is biharmonic if and only if (η, ν) is a critical point ofÊ 2 , that is (5.2) ∂Ê 2 ∂η (η, ν) = 0 and ∂Ê 2 ∂ν (η, ν) = 0 . Moreover, the map Φ η,ν is an isometric immersion if and only if (5.3) R 1 = sin η sin ν and R 2 = sin η cos ν . As we are looking for proper biharmonic immersions which are not congruent, we can assume that 0 < η, ν < π/2. Then we check that the only possibility to satisfy both (5.2) and (5.3) is η * = π 4 , ν * = π 4 which give R 1 = R 2 = 1/2 and so Φ η * ,ν * = Φ. The counterparts of V η , V ν in the orbit space can be described as follows. Let π : S 4 → Q, sin 2 η dν 2 + dη 2 be the canonical projection. Then V η , V ν are horizontal with respect to π and (5.4) dπ(V η ) = ∂ ∂η , dπ(V ν ) = √ 2 ∂ ∂ν . Now, we can proceed to the study of the equivariant second variation, a notion which was introduced in [16]. To this purpose, we compute the Hessian matrix ofÊ 2 at the critical point (η * , ν * ). Setting R 1 = R 2 = (1/2), we obtain       ∂ 2Ê 2 ∂η 2 ∂ 2Ê 2 ∂η∂ν ∂ 2Ê 2 ∂η∂ν ∂ 2Ê 2 ∂ν 2       (η * ,ν * ) = −16 0 0 0   = 1 Vol(M)   (I 2 (V η ), V η ) 1 √ 2 (I 2 (V ν ), V η ) 1 √ 2 (I 2 (V η ), V ν ) 1 2 (I 2 (V ν ), V ν )     . Therefore, this analysis in the orbit space at the critical point (η * , ν * ) tells us that the equivariant index of Φ is equal to 1 (eigenvalue µ 1 = −16, unit eigenvector ∂/∂η). Moreover, also the equivariant nullity of Φ is equal to 1 (unit eigenvector √ 2 ∂/∂ν). These results, together with (5.4), suggest that in this example the orbit space analysis displays all the significant second variation features of Φ. Remark 5.1. The method of this section can be extended to other examples. For instance, one could apply these arguments to the SO(ℓ + 1) × SO(ℓ + 1)-invariant proper biharmonic immersions Φ ℓ : S ℓ 1 2 × S ℓ 1 2 → S 2ℓ+2 , ℓ ≥ 2. Of course, it would be nice to be able to exclude, when ℓ ≥ 2, that there exist directions different from V η , V ν which are geometrically significant for the study of the second variation of Φ ℓ (index or nullity). Remark 5.2. In the case of Theorem 4.3 the submanifold ϕ(T) is again SO(2) × SO(2)invariant and the orbit space is 1-dimensional. In this example, there are 4 geometrically significant directions which determine the index (see (4.10)), but it is not possible to recover this 4-dimensional subspace by carrying out a simplified analysis in the orbit space. Remark 5.3. We point out that, in our example Φ : T → S 4 , the normal bundle has dimension equal to two and so it is interesting to compare this situation with the recent results proved by Ou concerning the normal stability of certain proper biharmonic hypersurfaces of the Euclidean sphere (see [21]). c 0 0= 65536 m 16 + n 16 + 8(m 14 n 2 + m 2 n 14 ) + 28(m 12 n 4 + m 4 n 12 ) + 56(m 10 n 6 + m 6 n 10 ) +70m 8 n 8 − 3(m 14 + n 14 ) − 21(m 12 n 2 + m 2 n 12 ) − 63(m 10 n 4 + m 4 n 10 ) −105(m 8 n 6 + m 6 n 8 ) + 16(m 10 n 2 + m 2 n 10 ) + 64(m 8 n 4 + m 4 n 8 ) + 96m 6 n 6 +7(m 10 + n 10 ) − 21(m 8 n 2 + m 2 n 8 ) − 98(m 6 n 4 + m 4 n 6 ) − 3(m 8 + n 8 ) +12(m 6 n 2 + m 2 n 6 ) + 94m 4 n 4 − 4(m 6 + n 6 ) − 12(m 4 n 2 + m 2 n 4 ) +2(m 4 + n 4 ) + 12m 2 n 2 c 1 = −4096 4(m 12 + n 12 ) + 24(m 10 n 2 + m 2 n 10 ) + 60(m 8 n 4 + m 4 n 8 ) + 80m 6 n 6 +3(m 10 + n 10 ) + 15(m 8 n 2 + m 2 n 8 ) + 30(m 6 n 4 + m 4 n 6 ) + 15(m 8 + n 8 ) −36(m 6 n 2 + m 2 n 6 ) − 102m 4 n 4 + m 6 + n 6 + 11(m 4 n 2 + m 2 n 4 ) − 15(m 4 + n 4 ) −46m 2 n 2 + 2(m 2 + n 2 ) c 2 = 256 6(m 8 + n 8 ) + 24(m 6 n 2 + m 2 n 6 ) + 36m 4 n 4 + 15(m 6 + n 6 ) +45(m 4 n 2 + m 2 n 4 ) + 14(m 4 + n 4 ) + 44m 2 n 2 − 10(m 2 + n 2 ) c 3 = −16 4(m 4 + n 4 ) + 8m 2 n 2 + 9(m 2 + n 2 ) − 1 c 4 = 1 . 2 )Φ(V, W ) = (IΦ ,π 2 (V ), W ) ∀ V, W ∈ Wφ .Our general result in this context is:Theorem 4.1. Suppose that no eigenvalue of Jφ belongs to the open interval (−2m, 0). Then the quadratic form (4.3) is nonnegatively defined, i.e., IΦ ,π 2 (V ), V ) ≥ 0 ∀ V ∈ Wφ . where B and A are the second fundamental form and the shape operator of M. Thusd * dV, dφ dφ , V = M ( dV, dφ ) 2 dV M = M (divξ) 2 dV Mand so (4.8) is verified. Now, we can end the proof. Let V ∈ Wφ and assume Jφ(V ) = µV . 2mµ)\|V \| 2 + 4(divξ) 2 dV M , and from this the conclusion of Theorem 4.1 follows immediately. Example 4. 2 . 2Theorem 4.1 can be applied whenφ is the totally geodesic embedding ofS m (1/ √ 2) in S n−1 (1/ √ 2), m = 1, 2, n ≥ 3.In fact, in these cases, the first eigenvalues of Jφ are −2m and 0 which do not belong to the open interval (−2m, 0) (see[10], Appendix 1). Theorem 4 . 3 . 43Let ϕ : T → S 3 ( 1 √ 2 ) be the minimal Clifford torus. Then (i) Index harm (ϕ) = 4 ; (ii) Null harm (ϕ) = 7 . Proof. The proof follows the lines of the proof of Theorem 1.2, so we just point out the relevant intermediate steps. First, a computation as in the proof of Proposition 2.3 gives: Proposition 4.4. Let f ∈ C ∞ (T) be an eigenfunction of ∆ with eigenvalue λ. Then Theorem 4. 9 . 9Assume that p ≥ 1 and let ϕ :T → S 3 ( 1 √ 2 ) be the minimal Clifford torus. Then (i) Index p−harm (ϕ) = 4 if 1 ≤ p < 4 ; (ii) Index p−harm (ϕ) = 0 if p ≥ 4 . Proof. The proof follows exactly the lines given for Theorem 4.3 and so we omit the details. To the benefit of the interested reader, we just point out that, in this context, the version of Proposition 4.4 is: 20 Proposition 4.10. Let f ∈ C ∞ (T) be an eigenfunction of ∆ with eigenvalue λ. Then The subspaces of the type W m,0 are 2-dimensional and are spanned by the functions {cos(mγ), sin(mγ)}. Similarly, W 0,n is 2-dimensional and is generated by {cos(nϑ), sin(nϑ)}. 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[ "New Axion Searches at Flavor Factories", "New Axion Searches at Flavor Factories" ]	[ "Xabier Cid Vidal \nInstituto Galego de Física de Altas Enerxías (IGFAE)\nSantiago de CompostelaSpain\n", "Alberto Mariotti \nIIHE/ELEM\nInternational Solvay Institutes\nTheoretische Natuurkunde\nVrije Universiteit Brussel\nPleinlaan 2B-1050BrusselsBelgium\n", "Diego Redigolo \nRaymond and Beverly Sackler School of Physics and Astronomy\nTel-Aviv University\n69978Tel-AvivIsrael\n\nSchool of Natural Sciences\nInstitute for Advanced Study\nEinstein Drive08540PrincetonNJUSA\n\nDepartment of Particle Physics and Astrophysics\nWeizmann Institute of Science\n7610001RehovotIsrael\n", "Filippo Sala \nDESY\nNotkestraße 85D-22607HamburgGermany\n", "Kohsaku Tobioka \nDepartment of Physics\nFlorida State University\n32306TallahasseeFLUSA\n\nTheory Center\nHigh Energy Accelerator Research Organization (KEK)\n305-0801TsukubaJapan\n" ]	[ "Instituto Galego de Física de Altas Enerxías (IGFAE)\nSantiago de CompostelaSpain", "IIHE/ELEM\nInternational Solvay Institutes\nTheoretische Natuurkunde\nVrije Universiteit Brussel\nPleinlaan 2B-1050BrusselsBelgium", "Raymond and Beverly Sackler School of Physics and Astronomy\nTel-Aviv University\n69978Tel-AvivIsrael", "School of Natural Sciences\nInstitute for Advanced Study\nEinstein Drive08540PrincetonNJUSA", "Department of Particle Physics and Astrophysics\nWeizmann Institute of Science\n7610001RehovotIsrael", "DESY\nNotkestraße 85D-22607HamburgGermany", "Department of Physics\nFlorida State University\n32306TallahasseeFLUSA", "Theory Center\nHigh Energy Accelerator Research Organization (KEK)\n305-0801TsukubaJapan" ]	[]	We assess the impact of searches at flavor factories for new neutral resonances that couple to both photons and gluons. These are well motivated by "heavy axion" solutions of the strong CP problem and by frameworks addressing both Dark Matter and the Higgs hierarchy problem. We use LHCb public diphoton data around the Bs mass to derive the current best limit on these resonances for masses between 4.9 and 6.3 GeV. We estimate that a future LHCb dedicated search would test an axion decay constant of O(TeV) for axion masses in the few-to-tens of GeV, being fully complementary to the low mass ATLAS and CMS searches. We also derive the impact of BABAR searches based on Υ decays and the future Belle-II reach.	10.1007/jhep01(2019)113	[ "https://arxiv.org/pdf/1810.09452v2.pdf" ]	119,075,951	1810.09452	4e6b8a066a121971119594681855c6ca4fbac97a	New Axion Searches at Flavor Factories Xabier Cid Vidal Instituto Galego de Física de Altas Enerxías (IGFAE) Santiago de CompostelaSpain Alberto Mariotti IIHE/ELEM International Solvay Institutes Theoretische Natuurkunde Vrije Universiteit Brussel Pleinlaan 2B-1050BrusselsBelgium Diego Redigolo Raymond and Beverly Sackler School of Physics and Astronomy Tel-Aviv University 69978Tel-AvivIsrael School of Natural Sciences Institute for Advanced Study Einstein Drive08540PrincetonNJUSA Department of Particle Physics and Astrophysics Weizmann Institute of Science 7610001RehovotIsrael Filippo Sala DESY Notkestraße 85D-22607HamburgGermany Kohsaku Tobioka Department of Physics Florida State University 32306TallahasseeFLUSA Theory Center High Energy Accelerator Research Organization (KEK) 305-0801TsukubaJapan New Axion Searches at Flavor Factories (Dated: October 24, 2018)PACS numbers: 1480Mz (Axions and other Nambu-Goldstone bosons) We assess the impact of searches at flavor factories for new neutral resonances that couple to both photons and gluons. These are well motivated by "heavy axion" solutions of the strong CP problem and by frameworks addressing both Dark Matter and the Higgs hierarchy problem. We use LHCb public diphoton data around the Bs mass to derive the current best limit on these resonances for masses between 4.9 and 6.3 GeV. We estimate that a future LHCb dedicated search would test an axion decay constant of O(TeV) for axion masses in the few-to-tens of GeV, being fully complementary to the low mass ATLAS and CMS searches. We also derive the impact of BABAR searches based on Υ decays and the future Belle-II reach. I. INTRODUCTION The lack of new physics at the Large Hadron Collider (LHC) and the lack of direct detection signal of dark matter (DM) at present experiments make it necessary to rethink the theoretical questions in the SM from a wider viewpoint and trigger broader experimental searches for new physics (NP). In this paper we make a step in this direction by presenting a NP case for flavor factories at the intensity frontier. These are light resonances below the EW scale which are neutral under the SM gauge group and couple to both gluons and photons. We show that flavor experiments have an unexploited potential to probe these states in a complementary mass range to previously proposed low-mass resonance searches at ATLAS and CMS [1]. Pointing out these gaps in the search program at flavor facilities is now a particularly important question in view of the upcoming LHCb upgrade and the Belle II data taking. The possibility we consider here is that the new physics scale M NP lies beyond the reach of the LHC. If that is the case, NP signals might still arise from pseudo-Nambu-Goldstone bosons (pNGBs) associated to spontaneously broken approximate symmetries. These are often called axion-like particle (ALP) in the literature, they can be sensibly lighter than the NP scale (m a M NP ) and their couplings to the SM are controlled by the inverse of the decay constant 1/f . Generically, one has M NP = g * f with g * being the typical size of the couplings in the NP sector, so that probing weak enough couplings of the pNGB gives an indirect probe of the scale of new physics. The focus of this paper will be on pNGBs with m a between 2 and 20 GeV, a mass window within the reach of flavor experiments. The driving question is whether flavor experiments can be sensitive to couplings of pNGBs small enough to probe new physics beyond the LHC reach. This question has been partially addressed for ALPs which couple to the SM by mixing with the Higgs sector [2,3] but it is surprisingly unexplored for ALPs with only gluon and photon couplings. In the large theory space of all the possible couplings of the ALP to the SM, having a non-zero coupling to gluons is particularly well motivated from the theory perspective. In this paper we will discuss in detail two particularly compelling examples: "heavy" QCD axions [4][5][6][7][8][9][10][11][12][13][14][15][16] and the R-axion [17][18][19] in low energy SUSY-breaking. As we will show, in these two classes of models the gluon coupling is unavoidable, the photon coupling generic, the mass range of interest for this paper can be easily achieved. A TeV decay constant is theoretically favoured by ensuring the quality of the axion potential [20][21][22][23] or by explaining the DM relic abundance via thermal freeze out. Besides these two examples, ALPs with both gluon and photons couplings arise for instance as new pions in Composite Higgs models [24][25][26] or in theories with vector-like confinement [27]. The first observation of this paper is that many existing search strategies for light resonances in the 2−20 GeV range [28][29][30][31][32][33][34] lose sensitivity as soon as the gluon coupling is switched on. The main reason is that the decay width into gluons dominates over the one into photons arXiv:1810.09452v1 [hep-ph] 22 Oct 2018 unless a non-generic hierarchy of couplings is assumed, therefore strongly suppressing the signals expected in the existing strategies. The dominant di-jet final states are much more difficult to distinguish from the SM background than diphotons. 1 As a way to overcome this issue, we show that the large production rate in pp collisions induced by the non-zero gluon coupling can be exploited at LHCb, which already has a low mass diphoton trigger designed to look for the rare decay B s → γγ. To substantiate this point, we use 80 pb −1 of public LHCb diphoton data [37] around the B s mass to derive a limit of O(100) pb on the signal strength of new diphoton resonances. This limit already constitutes the strongest existing probe for ALPs in the mass range between 4.9 and 6.3 GeV and motivates a dedicated LHCb search for diphoton resonances in a broader mass range. We estimate the sensitivity of such a search and show that decay constants at around the TeV scale are within reach of the high-luminosity phase of LHCb. This extends the coverage of low-mass resonance searches down to masses as low as 2 GeV and constitutes a new probe of multi-TeV scale NP which could be difficult to produce directly at the LHC. A similar point was made in Ref. [1] with ATLAS, CMS, and Tevatron diphoton searches, that are however limited by trigger issues to masses roughly above 10 GeV. We finally discuss bounds on light resonances produced from SM meson decays. We estimate the BABAR constraint on Υ(1, 2, 3S) → γa(jj) and assess the future Belle-II sensitivity. This production channel currently constitutes the best probe of ALPs below ∼ 3 GeV. II. RESULTS We consider a spontaneously broken approximate U (1) symmetry in the UV. Integrating out the new physics sector at the scale M NP , we write down the effective interactions between the pNGBs and the SM L eff = 1 2 (∂ µ a) 2 − 1 2 m 2 a a 2 + a f 3 i=1 c i α i 4π F i,µνF µν i ,(1) where i runs over the hypercharge, weak and strong gauge groups,F µν i = µνρσ F i,ρσ /2, α i = g 2 i /4π and α 1 is GUTnormalised (α 1 = 5α y /3). The constants c i are anomaly coefficients which depend on the number of degrees of freedom chiral under the U (1) symmetry and carrying a non-zero charge under the SM gauge group. 2 1 As an example the LEP limit on BR(Z → γa) is 1.7 · 10 −5 from 36.9 pb −1 of data if a is a diphoton resonance [35] and 4.7 · 10 −4 from 5.5 pb −1 of data if a is a dijet resonance [36]. 2 If the SM fermions and the Higgs doublet are uncharged under the U (1) symmetry, the couplings of the pNGB to them arise only from loops of SM gauge bosons and can safely be neglected. [30,37], projections are given for Belle II and future LHC stages. Details are given in Sec. IV. The other bounds are derived from Z width measurements [28,38], heavy ion collisions [39,40], Z → γa(jj) decays at LEP I [29] and diphoton cross section measurements at CDF (relevant only for ma 10 GeV), CMS and ATLAS [1]. For the latter we also give sensitivities up to the HL stage as derived in Ref. [1]. The thin dashed lines indicate theory benchmarks motivated by heavy QCD axion models and by ALPportal Dark Matter described in Sec. III. New coloured and EW states are expected to have masses of order g * f , where g * = 4π/ √ Nmess = 4π/ √ 2 ci. � �� -� �� -� �� -� � � [��] � [��] � �γγ [�� -� ] � � =� � =� � =�� →�(γγ) �� γγ � �� -� �� γγ �� -� γγ �� -� γ γ � � � � � � � Υ → γ � ( �� ) � � �� /� � � � � → � (γ γ ) �� In the NP sector, the strength of the interaction g * generically limits the maximal number of degrees of freedom to be below ≈ (4π) 2 /g 2 * . Therefore, a lower g * allows for large couplings of the ALP to the SM but at the same time it lowers the scale of new physics M NP g * f . For m a M Z , we can write the ALP couplings to photons and gluons below EWSB using the same notation of the QCD axion L eff ⊃ N α 3 4π a f G µνG µν + Eα em 4π a f F µνF µν ,(2) where we have N = c 3 , E = c 2 + 5c 1 /3 , g aγγ = α em πf E ,(3) where g aγγ agrees with the standard formula for the QCD axion after normalizing the decay constant with respect to the QCD coupling f = 2N f PQ . The relevant decay widths of the pNGB are Γ γγ = α 2 em E 2 64π 3 m 3 a f 2 , Γ gg = K gg α 2 s N 2 8π 3 m 3 a f 2 ,(4) where we include NNLO corrections to the gluon width [41] in K gg (see Appendix A for more details). Note that (0.1 mm) −1 Γ tot = Γ gg + Γ γγ m bin γγ over the mass range of our interest. The new resonance decays promptly and has a very narrow width compared to its mass. The LHCb constraint and sensitivities derived in Section IV are displayed on the ALP parameter space in Figure 1, for the benchmark c 1 = c 2 = c 3 = 10. We compute σ(pp → a) with ggHiggs v4 [42][43][44][45] using the mstw2008nnlo pdf set. We compare it with that obtained by the use of different pdf sets and of MadgraphLO v2 6 [46,47] upon implementing the ALP model in FeynRules [48], finding differences from 20% at m a = 20 GeV to a factor of 2 or larger for m a < 5 GeV. As detailed in Appendix A, a more precise determination of the signal would be needed, especially for m a 5 GeV. In Figure 1 we also show i) the 2σ constraint Γ Z − Γ SM Z < 5.8 MeV [28,38]; ii) the LEP limit BR(Z → γa(jj)) < 1 − 5 × 10 −4 [29]; iii) the constraint derived in [1] from the ATLAS [49,50], CMS [51], and CDF [52] inclusive diphoton cross section measurements, corresponding to σ(pp/pp → X a(γγ)) < 10 − 100 pb; iv) the sensitivities derived in [1] from inclusive diphoton cross section measurements at ATLAS and CMS. The HL-LHC reach assumes minimal photon p T cuts of 25 and 22 GeV and minimal photon separation of ∆R = 0.4. These numbers correspond to the 7 TeV measurement in Ref. [49]. Higher p T cuts would increase the minimal value of the invariant mass within the reach of HL-LHC. v) the BABAR constraint BR Υ 2S,3S → γa(jj) < 10 −4 − 10 −6 [30], where we compute BR Υ → γa BR Υ → µμ 8 E 2 α em 4π m Υ 4πf 2 1 − m 2 a m 2 Υ 3 ,(5) where BR Υ 2S,3S → µμ = 1.92%, 2.18%. The above expression corrects a factor of 4 in the result of Ref. [53]. vi) the Belle-II sensitivity in the same channel, that we determine simply by rescaling the expected sensitivities in [30] by a factor of 10. This assumes that the Belle-II reach will be statistics-dominated, and that it will be based on a factor of 100 more Υ(3S) than the BABAR one (i.e. on 1.2 × 10 10 Υ(3S) in total). The current Belle-II run plan for the first years assumes only a factor of 10 for the above ratio [54,55], corresponding to a few weeks of dedicated run at the Υ(3S) threshold. An extra factor of 10 could be obtained in a comparable time with dedicated later runs, because a higher instantaneous luminosity is foreseen [55]. An analogous search could be effectively performed, at Belle-II, also analysing the decays of Υ(1S, 2S). vii) limits from the diphoton final state from heavy ion collisions are extracted from the recent CMS analysis in Ref. [40] and the reinterpretation of the AT-LAS light by light scattering data [39] of Ref. [56]. The lower reach of these measurements is set to m a 5 GeV as a consequence of the minimal cuts on the two photons transverse momenta. ATLAS limits from Z → γa(γγ) [57] are not displayed in Fig 1. They imply BR(Z → γa(γγ)) < 2.2 · 10 −6 and turn out to be comparable to the heavy ions bound for our benchmark in Fig. 1. Similar constraints can be derived from the ATLAS inclusive search in pp → γa(γγ) [57]. The lower invariant mass reach of these ATLAS searches is set by the diphoton isolation requirement of [57], ∆R γγ = 0.15. This corresponds to an ALP mass of 4 GeV as discussed in Ref. [58]. Notice that LEP searches for Z → γa(γγ) [31] are weaker than the ATLAS bound. Future sensitivities from e + e − → γa(γγ) [33,34] do not reach values of f larger than 50 GeV and are not shown. Finally, the proposed search in B → K ( * ) a(γγ)) [33] at Belle-II has some sensitivity in a very limited portion of our mass range and it is not shown to avoid clutter. In Fig. 2 we fix the ALP masses to two representative values m a = 5, 15 GeV and show the impact of the various searches in the plane (N/f, E/f ) which control the ALP's gluon and photon coupling respectively. As one can see from Fig. 2, diphoton searches for a ALP produced in gluon fusion both at ATLAS/CMS (see Ref. [1]) and at LHCb (see Sec. IV) can be sensitive to N/f as small as 10 −4 GeV −1 as long as the coupling to the photons is large enough. Moreover they can cover significant portion of the parameter space where the couplings are of their natural size. Searches taking advantage of uniquely the photon coupling such as the ones in Refs. [32,34,57] become relevant only in the upper left corner of the plane where E/N 50. Such a hierarchy can be realized in clockwork constructions where the photon coupling is enhanced with respect to the gluon one (see for example Ref. [59]). The ATLAS, CMS and LHCb limits and sensitivites shown in Fig. 2 are derived assuming gluon fusion as the ALP production process, so they sharply stop at a given small gluon coupling. If other production processes like vector-boson-fusion are taken into account, the limits and sensitivities would be slightly improved in the upper left corner of Fig. 2. Practically, the Heavy Ion results that we are including will always lead to stronger constraint because of the enhanced photon-fusion production and the loop suppressed background from light-by-light scattering. The bottom right corner where the new resonance mostly couples to gluons is challenging to constrain in this mass range, even though boosted dijet searches at the LHC were recently able to go down to invariant masses of 50 GeV (see Ref. [60]). Of course for N/f (100GeV) −1 one expects color states generating the ALP coupling to be within the reach of the LHC. III. PHYSICS CASES In this section we expand on the two theory lines displayed in Fig. 1. We would like to motivate: 1) the coupling of the axion to gluons and photons, 2) the TeV decay constant, 3) the mass range considered here. a. Heavy axions As a first example, we consider a particular class of axion solutions to the strong CP problem in the SM. First of all, introducing a spontaneously broken Peccei-Quinn symmetry U (1) PQ which is anomalous under QCD [61,62] leads unavoidably to a light axion with non-zero couplings to gluons [63,64]. In this sense, the axion coupling to gluons is deeply connected to its role in solving the strong CP problem. Taking the SM fields to be uncharged under the U (1) PQ , the QCD anomaly is generated by heavy vector-like fermions like in KSVZ type of models [65,66] L PQ ⊃ g * Φψψ + h.c., Φ = f √ 2 e ia/f ,(6) where the fermion charges should satisfy \|q PQ ψ − q PQ ψ \| = q PQ Φ and by writing Eq. (6) we take q PQ Φ = 1. After U (1) PQ gets spontaneously broken by the VEV of Φ, the fermion mass is at M NP = g * f / √ 2. Below the PQ breaking scale we can integrate out the heavy fermions and match to the effective Lagrangian in Eq. (2): N = q PQ Φ ψ C 3 (R ψ ) E = q PQ Φ ψ Q 2 em (R ψ ) . (7) The vector-like fermions are often assumed to carry a non-zero hypercharge in order to allow a non-zero mixing with the SM quarks, to make them decay avoiding cosmological problems. This induces an anomaly of U (1) PQ with respect to the hypercharge, which leads to a non-zero coupling of the axion to photons: E = 0. To fix a benchmark, we add N mess complete SU (5) fundamental representations, that lead to N = N mess /2 and E = 4/3 N mess . This is the scaling assumed in Fig. 1, where we also take N mess = (4π/g * ) 2 to ensure calculability below M NP . In Fig. 2 we go beyond this benchmark and show how E/N can be modified changing the SM representation of the fermions in Eq. (6) (see Ref. [67] for a discussion). Operators breaking U (1) PQ other than the QCD anomaly would in general spoil the axion solution of the strong CP problem [20][21][22][23]. We can parametrize these contributions as new terms in the potential for the scalar Φ: ∆V / PQ = λ ∆ Φ ∆ Λ ∆−4 UV + h.c., λ ∆ = \|λ ∆ \|e iα∆ .(8) In the presence of these new contributions the axion potential below the QCD phase transition is V a −Λ 4 QCD cos N a f + 1 2 ∆ 2 −1 \|λ ∆ \|f ∆ Λ ∆−4 UV cos α ∆ + ∆ a f .(9) Since the new phase α ∆ is in general not aligned with the contribution given by the QCD anomaly, the presence of the UV operator shifts the axion VEV away from the origin, jeopardizing the solution to the strong CP problem. Note that this holds even if the NP sector inducing Eq. (8) preserves CP, because a new phase α ∆ ∼ O(1) is induced by rotating away the phase in the quark mass matrices. Requiring 2 N a/f 10 −10 to satisfy the present bound on the neutron dipole moment [68,69] gives an upper bound on the axion decay constant f Λ UV 10 −10 · N ∆ · Λ QCD Λ UV 4 1/∆ ,(10) where we have assumed \|λ ∆ \| ∼ α ∆ ∼ O(1) and neglected other O(1) factors for simplicity. The upper bound on f depends on the scale of the UV completion Λ U V g * f and on the "quality" of the U (1) PQ , i.e. the dimension ∆ of the lowest dimension operator breaking the symmetry. In the best case scenario, first discussed in Refs [20][21][22][23], the U (1) PQ is only broken by Planck suppressed operators 3 but, more generally, one might argue that all the global symmetries should be an accidental consequence of the gauge and matter content of the theory, exactly like in the SM. In the latter case the Λ UV in Eq. (10) will be below M Pl . Taking Eq. (10) In the usual QCD axion where m a 6 keV · TeV/f (see e.g. [72]), values of the decay constant motivated by the axion quality problem are abundantly excluded by star cooling bounds [73] and K → πa transitions [3,74]. A common solution to this problem is to go to higher values of f and require a U (1) PQ with higher quality. Such a U (1) PQ can be made accidental in extra-dimensions or with more complicated UV completions in 4 dimensions (we refer to Refs [75][76][77][78] for an illustration of the challenges involved in constructing gauge theories with a U (1) PQ with arbitrarily high quality). Alternatively, one can construct QCD axion models where the axion mass is heavier than its QCD value. The idea is to introduce new contributions to the axion potential which are aligned to the QCD one, so that the axion mass gets larger without spoiling the solution to the strong CP problem. A larger m a then relaxes the experimental constraints on f , potentially allowing to satisfy Eq. (10). There are several classes of models of this type which differ from the way the alignment is achieved: mirror axion models with one axion and two mirror QCD's [4][5][6][7][8], models where the QCD running is modified at high energies [9][10][11][12][13][14], and a more recent proposals [15] where the QCD group is embedded in SU (3) N with N axions relaxing each one of the allowed θ-angles. All the solutions of the strong CP problem mentioned above can easily achieve the 2-20 GeV mass range, and 3 Gravity is expected to break global symmetries at the non perturbative level via wormhole solutions swallowing the PQ charge [70]. In this case the Wilson coefficient of the operators in Eq. (8) can be very suppressed for a large enough wormhole action: \|λ ∆ \| ∼ e −S Eucl . The latter has been shown in Ref. [71] to be too small in the Einstein theory of gravity but large enough in theories where the Einstein theory is suitably modified at Planckian distances. result in an axion which generically couples to both gluons and photons with a decay constant at the TeV scale or lower. These are a perfect benchmark for the collider searches discussed here. For illustrative purposes we show in Fig. 1 the value of f corresponding to a U (1) PQ broken by ∆ = 6 operators generated at M GUT = 10 15 GeV. b. ALP-mediated Dark Matter The second example of ALP with coupling and masses of interest for this study comes from demanding it to be the mediator that couples the SM to fermion DM, singlet under the SM gauge group. This possibility has particular interest for colliders because direct detection constraints are totally irrelevant, see e.g. [79]. We write the ALP coupling to DM as in equation (6) and identify the DM as the Dirac fermion (ψ,ψ † ), so that m ψ = g * f / √ 2. The DM annihilation cross section into SM particles, mediated by the ALP, is dominated by final state gluon pairs and reads (σv) gg = 2 π c 3 α s 4π 2 g 2 * f 2 ,(11) where α s is evaluated at the scale µ = 2 m ψ . The cross section for t-channel annihilation into a pair of mediators is p-wave and reads [80] ( σv rel ) aa = v 2 rel 384π g 4 * m 2 ψ ,(12) therefore it is negligible with respect to the annihilation into gluons for the parameter values we are interested in, even for relativistic v rel . Requiring Eq. (11) to match 4.8 × 10 −26 cm 3 /sec, which is the value needed for heavy Dirac DM to reproduce the correct DM abundance via thermal freeze-out [81], we find m ψ 4.6 TeV c 3 10 g * 3 2 ⇒ f 1.9 TeV 3 g * ,(13) where in the second equality we have assumed the scaling c 3 8π 2 /g 2 * . This is the benchmark value we display in Figure 1. It is interesting to note that indirect detection is still far from probing thermal values of the annihilation cross section for DM in this mass range (see e.g. [82][83][84]), thus adding further motivation to test this scenario with colliders. Note that we have neglected the possible Sommerfeld enhancement from exchange of the ALP in the initial state. The precise computation of this effect is still the object of some debate, see e.g. [85] for a recent study with references, so that for simplicity we do not include it here. Its inclusion would result in an O(1) change in the favoured value of f , but would not affect our physics point that pseudoscalar mediated DM motivates ALP searches at flavor factories. c. R-axion in Supersymmetry. We finally notice that the simplified DM model presented above arises naturally in theories of low-scale SUSY breaking. These predict that the lightest supersymmetric particle (LSP) is the Gravitino, whose mass m 3/2 is generically too small to account for the observed DM abundance. Indeed, using the power counting described in [19], one gets m 3/2 = F/( √ 3M Pl ) 11 meV·(g * /3)·(f /4 TeV) 2 . While not reproducing the observed value of DM, Gravitino masses in this ballpark are safe both from collider [86][87][88] and cosmological [89] constraints. In the absence of stable superpartners, the natural DM candidate in these SUSY theories are particles belonging to the messenger or SUSY breaking sectors, see [90] for a first study of this possibility. In this case, as first noted in [91] (see [92] for further model building), the DM phenomenology may be dominated by its interactions with a pseudoscalar that is naturally present in the theory, the R-axion. This arises as the pNGB of the U (1) R symmetry, defined as the only abelian global symmetry which does not commute with the SUSY generators. The spontaneous breaking of the U (1) R is intimately related to SUSYbreaking according to the general results of [17,93]. The R-axion couplings to gluons and photons are unavoidably generated by loops of gauginos, whose Majorana masses are chiral under the U (1) R , and possibly by UV messengers. Couplings to fermions and to the Higgs are less generic and can be suppressed by suitable charge assignment (see [19] for more details). Under these circumstances, the R-axion matches perfectly the Lagrangian in Eq. (1). For f = O(TeV), motivated here not only by DM but also by the naturalness of the Fermi scale, i) its mass is expected to lie in the MeV range [94] or above [19,93], thus motivating searches at flavor factories, ii) superpartners can be taken outside the LHC reach, thus making it potentially the first sign of SUSY at colliders [19]. IV. DIPHOTON SEARCHES AT LHCb LHCb detects photons either as "unconverted", i.e. they reach the electromagnetic calorimeter (ECAL), or as "converted", i.e. they convert to an e + e − pair upon interacting with the detector material before reaching the ECAL. The public LHCb note [37] presents the trigger and cut strategy that will be used to look for B s → γγ, and classifies diphoton events into two unconverted (0CV), one unconverted and one converted (1CV LL and DD, corresponding to conversions occurring in the Vertex Locator region or after it) and two converted (2CV) samples. Searches for B s → γγ benefit from requiring the γγ vertex to be displaced from the pp interaction point, while the resonances we are interested in typically have a lifetime much shorter than the B s one. A displaced γγ vertex is however not imposed on the 0CV sample, because the resolution on the directions of the photons does not allow for a precise enough vertex reconstruction. Therefore this sample can be used to derive a bound on prompt diphoton resonances. Measured diphoton events that pass the cuts are reported in [37] for L = 80 pb −1 of data, for each conversion category, in a diphoton invariant mass interval 4.9 GeV < m γγ < 6.3 GeV and in bins of 14.5 MeV. No known QCD or SM resonance is expected to give a signal within the LHCb reach, explaining why the event distributions in m γγ are very smooth in all categories, so that they constitute an ideal avenue to look for BSM resonances. Therefore, we place an upper limit on the signal cross section of a resonance a decaying to diphotons as N sig (m a ) < 2 N bkg m bin γγ 14.5 MeV ,(14) where N sig = × σ fid × L, σ fid = A × σ(pp → Xa(γγ)) ,(15) with A the geometrical acceptance of the signal in the LHCb detector and the total efficiency of the cuts plus detector effects in a given diphoton category. We use A = 0.15 , 0CV = 0.142 ,(16) where the latter is given in [37] for the SM "signal" B s → γγ, and we determine the former by simulating the signal (see Appendix D for details) and imposing 2 < η < 5 at truth level. Coming to the right-hand side of Eq. (14), N bkg is the number of background events in the 14.5 MeV bin reported in [37], which we take constant as the distribution in m γγ is actually flat well within its statistical uncertainties. 4 m bin γγ is the size of the bin centered on m γγ = m a that we expect to contain most of the signal from the resonance, which we assume to be narrow. In practice we use N bkg = 8000 × L 80 pb −1 , m bin γγ = 4δm γγ ,(17) where δm γγ is the invariant mass resolution for the 0CV category which can be derived from the energy resolution and the granularity of the LHCb ECAL (see Appendix C). Fixing for definiteness m bin γγ /m γγ = 13%, we obtain σ 0CV fid 106 pb · m a 5 GeV · 80 pb −1 L .(18) 4 While this holds for the 1CV and 2CV categories, the distribution in the 0CV category is flat up to mγγ 5.7 GeV, and then drops smoothly. A possible origin of this drop is the use of 2 × 2 ECAL cells to measure the photon energy deposition at the first level of the software trigger (HLT1) [37]. In Appendix B we verified that imposing invariant mass cuts at HLT1 can cause a flat background at HLT1 to develop a dropping shape at higher level, where the invariant mass is defined using 3 × 3 cells. The sensitivities that could be achieved by the current full dataset of 8 fb −1 and by the High Luminosity phase of LHCb with 300 fb −1 of data can be easily obtained from the above equation. 5 We also extend the mass range of the search to 3 < m γγ /GeV < 20, where the lower bound is chosen to make the computation of the signal strength reliable (see also Appendix A) and the upper bound is chosen somehow arbitrarily at 20 GeV, where the reach of the current ATLAS/CMS inclusive diphoton dataset [1] is already stronger than the projections of LHCb. For simplicity we take the signal acceptance and the efficiency to be constant and equal to the ones in Eq. (16). We discuss in Appendix D the motivations for this simplified assumption. Moreover we assume that the background is also constant in the extended mass range and equal to the one in Eq. (17). This simple procedure sets a useful benchmark for the actual search, which is good enough for the purpose of this paper. The resulting reach in the ALP parameter space is shown in Fig. 1. We finally speculate about the limit and reach obtainable if the 1CV photon categories could be used. To set an optimistic reach, we do not take into account the signal loss because of the requirement of vertex displacement in present LHCb search. With this assumption, we repeat the procedure described above, with constant background N 1CV,DD bkg = 1600 and N 1CV,LL bkg = 1300 and constant efficiencies 1CV,DD = 1.35% and 1CV,LL = 1.32% as reported in Ref. [37]. Concerning the mass resolution, we take the one of the 0CV category divided by √ 2 to roughly account for the much better energy resolution of the converted photon. With all these assumptions we combine in quadrature the exclusions from the LL and DD single-converted categories and get σ 1CV fid 283 pb · m a 5 GeV · 80 pb −1 L ,(19) which is almost a factor of 3 weaker than the 0CV bound. In more realistic conditions we expect a sensible loss of signal from the requirement of displacement, although better background discrimination might be also achieved thanks to the converted photon. We do not even study the 2CV photon category because it is plagued by a very small signal efficiency. As a useful input for future more detailed studies, we collect here some considerations about the LHCb reach outside the interval 4.9 GeV < m γγ < 6.3 GeV: As far as the signal is concerned, we do not expect a significant drop in the efficiency going at higher invariant masses. As detailed in Appendix D at higher invariant masses the diphoton final state will be less forward, reducing the geometric acceptance. However, the decreasing boost of the produced particle is more than compensated by the higher efficiency of the photon p T cuts. Practically, the ultimate high mass reach of LHCb is not very relevant for the purposes of discovering new physics, since above 10-20 GeV it is likely to be superseded by the ATLAS/CMS diphoton searches (see [1] for details). The most stringent limitation for scanning masses above ∼12 GeV at LHCb is the current dynamic range of the ECAL. This range, which depends on the electronics and not on the actual configuration of the detector, limits at the moment reconstructing photons with E T above ∼10 GeV (∼6 GeV at the level of the first level of the software trigger HLT1). Therefore, a potential increase in the dynamic range of the ECAL after the LHCb Upgrade would be very benificial to increase LHCb's sensitivity to higher masses. For instance, modifying the electronics to increase the range to 15− 20 GeV would be enough to cover all the mass range for which ATLAS and CMS have a poor sensitivity. As already mentioned, the invariant mass distribution in the 0CV category from the data in Ref. [37] displays a drop for masses larger than approximately m γγ 5.7 GeV. In Appendix B, we argue that such drop is a consequence of the use of 2 × 2 ECAL cells to measure the photon energy deposition at HLT1. If our guess is correct there should be another drop of the background at low invariant masses in a region not showed by the plot of Ref. [37]. Understanding the composition of the diphoton background given in Ref. [37] would require a detailed MC simulation, including detector effects, which is beyond the scope of this paper. In Appendix C we provide a simple kinematical argument which shows that the background from boosted π 0 faking photons is likely to dominate over the one from real photons. A categorization of the data in different η regions would help suppressing this background at small η. This could be used to maximize the reach. A quantitative assessment of this is left for future studies. The precise assessment of the 1CV limit and sensitivities would require a dedicated search for promptly decaying resonances without the requirement of a displaced vertex. In this case one could get an even better reach than the one presented here by combining the 0CV and the 1CV category. We hope that this work could provide enough motivation to explore further the open issues described above and in general the possibility of performing bump hunts on the top of the diphoton background at such low invariant masses. V. CONCLUSIONS The LHC has pushed the energy scale of many motivated SM extensions beyond the TeV range. How to experimentally test NP models at and beyond those scales? A possibility is to look for low energy remnants of such theories, like pseudo-Goldstone bosons (aka ALPs) from an approximate global symmetry. In Section III we showed that ALPs with masses and decay constants of interest for flavor factories arise as a solution to the strong CP problem ("heavy QCD axions") and in frameworks motivated by Dark Matter freeze-out and the Higgs hierarchy problem (e.g. the SUSY R-axion as mediator of DM interactions). These scenarios share the prediction of ALP couplings to gluons and photons, that are currently tested in a particularly poor way for masses below O(10) GeV. In Section IV, we have used 80 pb −1 of public LHCb data to set a bound on diphoton resonances of σ(pp → Xa(γγ)) 100 pb, and we have performed a first study to assess future LHCb sensitivities. This bound is already the strongest existing one on the ALPs discussed above, and shows that LHCb has a very promising potential to test unexplored territory of well-motivated BSM extensions. Technical results that might be useful for future LHCb studies are provided in Appendices C and D. We have also recasted BABAR limits on Υ → γa(jj) on this model, and estimated the associated future capabilities of Belle-II, finding they would be particularly relevant for masses below ≈ 3 GeV. These results are summarised in Figure 1. Our findings provide a strong motivation to pursue the phenomenological and experimental program of testing this class of ALPs at LHCb and Belle-II, thus enriching the physics case of both machines. A. Acknowledgements We thank Sean Benson and Albert Puig Navarro for many useful discussions, in particular about the LHCb note [37], and Marco Bonvini for clarifications about ggHiggs. D.R. thanks Simon Knapen for discussion and clarifications on Ref. [56]. D.R. thanks CERN and the Galileo Galilei Institute for theoretical physics (GGI) for kind hospitality during the completion of this work. F.S. is grateful to the Mainz Institute for Theoretical Physics (MITP), CERN and GGI for kind hospitality at various stages of this work. K.T. thanks MITP for kind hospitality during the completion of this work. Appendix A: More on the Signal We compute the gluon fusion production cross section at N3LO using ggHiggs v4 [42][43][44][45] and at LO using Mad-GraphLO. We compare the two predictions in Figure 3 left, for different choices of the pdf sets, and rescaling the ggHiggs cross section using that c 3 /f = 1/( √ 2 v) with v 246 GeV (anomaly coefficient coming from a top loop). The agreement between these determinations goes from the 20% level at m a = 20 GeV, down to a factor of 2 and worse for m a ≤ 4 GeV. We mention that at such low values the ggHiggs output should be taken with extra care, as it also yields some negative LO and NLO cross sections. This comparison underlines the need for a more precise determination of the production cross section, especially for ALP masses below 5 GeV or so. This task goes however beyond the purpose of this paper. We use the ggHiggs prediction with the mstw2008nnlo pdf set for all the LHC phenomenology in Section II. Coming now to the ALP branching ratios, we use the NNLO QCD correction to the width of a pseudoscalar into gluons from [41]. In the notation of Eq. (4), it Figure 3 right we plot the resulting diphoton branching ratio together with its NLO and LO value and with the one given by Madgraph. NNLO corrections to the diphoton branching ratio reduce its LO value by a factor of 2, over the whole mass range we consider. We use the NNLO expression for all the limits and sensitivities described in Section II. reads K gg = 1 + α (5) s π E A + α (5) s π 2 E A 3 4 E A + β1 β0 , where E A = 97 4 − 7 6 N f , β 0 = 11 4 − N f 6 , β 1 = 51 8 − 19 24 N f . In Appendix B: mγγ distribution of the 0CV category In our analysis, we assume the background yield to be roughly constant with respect to the diphoton invariant mass even outside the mass range reported in Ref. [37]. This is to provide an order-of-magnitude estimate of the background for the LHCb sensitivities to ALPs. The flatness of the data is actually seen in the 1CV and 2CV categories of Fig. 4(b-c) of Ref. [37]. However, in the 0CV category ( Fig. 4(a)), a kink is observed at large invariant masses. In what follows, we argue that this is an artifact due to the trigger level invariant mass cut. In the invariant mass calculation at the trigger level of the 0CV category, two approximations are employed to speed up the calculation: 1) the photon energy is calculated from the energy deposition in 2 × 2 ECAL ∆θ γ1γ2 . We examined these two approximation and concluded that 1) could be the reason for the kink. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • � �� [��] σ �� → �� /� � � [��] � � = �� / � < μ � = μ � < � � � �� • • • • • • • • • • • • • • • • • • • � �� [��] ��(� → γγ) � � =� � =� � � �� It is easy to show that the approximate mass formula is equivalent to the full mass formula with O(0.01) accuracy. This comes from the fact that the diphoton events within the LHCb fiducial volume have a small opening angle ∆θ γ1γ2 = O(0.1), after the E T trigger cuts are imposed. On the other hand one needs to use 3 × 3 cells to capture full energy deposit of a photon, so the information based on 2 × 2 cells underestimates the photon energy, which leads to the lower invariant mass, m trigger γγ < m full γγ . Because the first invariant mass cut is made at the trigger level, 3.5 GeV < m trigger γγ < 6 GeV, bins with a given m trigger γγ migrate to bins with m full γγ > m trigger γγ .This could explain why the reduction of the yield appears above m full γγ ∼ 6 GeV. This argument is confirmed by Fig. 1 bottom of [37], that shows how the trigger level mass distribution of the B s signal shifts to higher values of off-line invariant mass. We further validate the argument modelling the energy smearing of the LHCb ECAL. For simplicity, we focus on the inner ECAL and approximate 2×2 cells as a circle of radius 4 cm. Because the Molière radius of a photon in the LHCb ECAL is 3.5cm, 6 the energy deposit inside the 2×2 cells is expected to be 95% of the total energy deposit on average. In order to model a realistic environment we include a stochastic gaussian smearing from the average value. We choose a standard deviation of 10% 7 such that the shift of the signal at m a = m Bs reproduces Fig. 1 bottom of [37]. This result is shown in Fig. 4 left. Then, we use the same prescription for the background-like events. The result is shown in Fig. 4 right. The invariant mass distribution in terms of m trigger γγ is normalized to be rectangular after the invariant mass cut. When the same dataset is plotted in terms of m full γγ we can see that a kink is induced. Appendix C: Details on the LHCb Calorimeter The Electromagnetic Calorimeter (ECAL) of LHCb has three layers with different granularities and is placed vertically with respect to the beam axis at z Ecal =12.52 m away from the collision point. The ECAL square cells have side lengths of ∆x cell =4.04, 6.06cm, 12.12cm for inner, middle, and outer layer, respectively [95]. The photon reconstruction algorithm uses patterns of 3×3 cells in each layer. Therefore, the inner layer, where most of the energy is expected to be deposited, has the best angular resolution. a. Invariant mass resolution The invariant mass can be written as m 2 γγ = 2E γ1 E γ2 (1 − cos θ γγ ) ,(C1) where E γ1,2 are the energies of the two photons and θ γγ is the angular separation between them. Using the above formula, we can relate the invariant mass smearing to the photon energy smearing and the ECAL granularity δm γγ m γγ 1 2 δm 2 γγ m 2 γγ = 1 2 δE γ1 E γ1 ⊕ δE γ2 E γ2 ⊕ sin θ γγ δθ 1 − cos θ γγ 1 √ 2 δE γ E γ ⊕ δθ θ γγ = 6.4% GeV E γ ⊕ 0.6% ⊕ 0.3% E γ m γγ .(C2) In the second line we assumed for simplicity E γ1 E γ2 E γ and approximated our result at the first order in θ 1. To obtain the second expression in the second line, we used the LHCb ECAL energy resolution δE/E 9% GeV/E ⊕ 0.8% reported in Ref. [96] and the granularity of the inner layer of the ECAL δθ = ∆x cell /z Ecal 0.003. Moreover, we have approximated θ γγ m γγ /E γ to get an expression of the typical energy smearing as a function of the typical photon energy. In computing the invariant mass resolution in the text, we take E γ = 50GeV. We believe this is a realistic benchmark value for this analysis because Separation of photon pairs from π 0 decay as a function of the pion total energy E π 0 . If this photon pair is misidentified as a single (fake) photon, E π 0 is the energy of the fake photon. Cases of inner, middle, and outer layers are plotted in blue, red and magenta respectively. E γ = E T γ cosh η and the LHCb analysis in Ref. [37] imposes E T γ > 3.5 GeV and E T γ1 + E T γ2 > 8 GeV on 2 × 2 cell clusters. b. Background from π 0 faking single photon One of the advantages to study low mass diphoton resonances at LHCb is that low energy fake photons from QCD can be distinguished from real photon candidates. Here we focus on fake photons from π 0 decays whose collimated diphoton decay can mimick a single photon candidate. Photon pairs from π 0 decay have angular separation θ π 0 γγ m π 0 /E γ 2m π 0 /E π 0 . The corresponding separation on a given ECAL layer is then ∆r π 0 γγ z Ecal θ π 0 γγ 2z Ecal m π 0 E π 0 .(C3) If the π 0 is very energetic, the diphoton separation ∆r π 0 γγ is smaller than a single cell size and the object is mostly misidentified as a single photon candidate of energy E π 0 . Viceversa, when a pion is less energetic and the diphoton separation is large, ∆r π 0 γγ > O(2)∆x cell , two photon clusters are separately formed and a pion is resolved. In a regime where 1.8∆x cell > ∆r π 0 γγ 0.5x cell , the shower shape information makes a single energy cluster identified as a π 0 , which is called merged π 0 [97]. The identification efficiency using both resolved and merged π 0 is O(50%) for p T π 0 10 GeV (Fig. 21 left of Ref. [97]). As shown in Fig. 5, the final energy thresholds vary depending on the ECAL layer. For example, in the inner ECAL diphotons with E π 0 < 28 GeV corresponding to a large separation of ∆r π 0 γγ > 3∆x cell can be reconstructed as resolved π 0 s, while the ones with 46 GeV < E π 0 160 GeV could be seen as merged π 0 s. The planned LHCb B s → γγ analysis uses a photon energy threshold of E T γ >3.5 GeV which corresponds to E γ = 13 (260) GeV at η =2 (5). Comparing with the threshold determined above for the pions to be detected as fake photons, one learns that i) the background to the current search contains a non-negligible amount of fake photons; ii) a categorization in η of the data could help in reducing photon fakes. Appendix D: Signal Acceptance and Efficiency In this Appendix we discuss the strategy that we adopted to estimate the acceptance and efficiency of the signal. As mentioned in the main text, we eventually consider a constant value for the product of acceptance times efficiency on the mass range of interest for this paper. As reference value, we have chosen the one at the invariant mass of 5 GeV, corresponding to the B s signal considered in the LHCb note [37]. In order to estimate the acceptance and efficiency of the signal at LHCb, we implement the axion model in FeynRules [48], we generate events with MadgraphLO v2 6 [46,47] and shower them with Pythia 8.1 [98,99], matching up to 1 extra jets [100]. We then perform a simple analysis of the resulting samples using MadAnalysis5 [101]. Note that the signal events which are inside the acceptance of LHCb contain topologies where the axion has acquired a significant longitudinal boost, without the need of extra hard radiation. As a consequence the signal efficiency is essentially not changed by including extra jets (the minimal E T cuts of the LHCb selection can be satisfied with just a small transverse boost). This has to be contrasted with the low invariant mass searches at ATLAS/CMS where the recoil of the resonance against the extra jet increases the signal efficiency of the p T cuts significantly, as it was shown in Ref. [1]. In Table I we report the acceptance and the efficiency that we find in the mass range 5 − 15 GeV by following the selection cuts of refs. [37], that is A : 2 < η(γ) < 5 (D1) :        E T (γ) > 3.5 GeV, E γ1 T + E γ2 T > 8 GeV p T (γ 1 γ 2 ) > 2 GeV (D2) We first observe that the value we find for the product A × , though in the same ballpark than the number reported by the LHCb note (see eqn. (16)), differs by around a factor of 2 on the case of m a = 5 GeV. In order to check wheter the discrepancy could be caused by detector effects, we also processed the same samples using Delphes as fast LHCb detector simulator, but we did not find a substantial improvement in the agreement. However, besides the discrepancy on the benchmark of 5 GeV, our simple analysis provides indications on what could be the expected product of A × for the selection cuts (D2) for different mass values. As one can observe from Table I, increasing the mass of the axion the acceptance generically decreases. This is due to the fact that a heavier resonance will more likely be produced with less boost on the longitudinal axis, and hence the resulting photons will be less into the forward region which is covered by the LHCb detector. On the other hand, for larger values of the axion mass the outgoing photons will be more energetic and will more likely pass the energy and p T cuts, hence resulting in an increase in the signal efficiency. The combination of these two effects result in a product of acceptance times efficiency which actually slightly grows along the mass interval 5 − 15 GeV, but does not changes significantly. This justifies the simplified choice that we have adopted in the main part of the paper. FIG. 2 : 2Constraints on the ALP parameter space for fixed masses ma = 5, 15 GeV in the up, down panel respectively. We fix c1 = c2 so that E in Eq. (3) controls both the Zγ and the γγ coupling. The bounds are shown as shaded regions while the projections as dashed lines. The three grey lines show the "axion window" obtained by integrating out fermions in different representations of the SM gauge group, the central one E = 8N/3 corresponds to the choice of Figure 1. at face value, the most dangerous contribution comes from ∆ = 5 operators, that would require f O(10) GeV even for Λ UV = M Pl . However, if operators of dimension five are forbidden (for example by a discrete Z 2 -symmetry) then ∆ = 6 contributions give f O(10) TeV for Λ UV = M Pl and f O(1) TeV for Λ UV = M GUT , motivating the ranges of decay constant of interest for this paper. Having f around the TeV scale would lead to axion solutions relying on U (1) PQ with the same quality of the baryon number in the SM. and research infrastructure acknowledgements: * X.C.V. is supported by MINECO through the Ramón y Cajal program RYC-2016-20073; * A.M. is supported by the SRP High Energy Physics and the Research Council of the Vrije Universiteit Brussel; A.M is also supported by FWO under the EOS-BE.H project n. 30820817 * D.R. is supported in part by the National Science Foundation under Grant No. NSF PHY17-48958; * F.S is supported in part by a Pier Seed Project funding (Project ID PIF-2017-72); * K.T. is supported by his start-up funding at Florida State University. FIG. 3 :� 3Left: Production cross section of an ALP coupled to GG, as determined with MadGraphLO and with ggHiggs at N3LO, for various choices of the pdf sets, fixing f = 1 TeV and c3 = 1. For ggHiggs we display the band enclosed by µ f = µr = ma/2 and µ f = µr = 2 ma. Right: ALP branching ratio into diphoton at LO, NLO and NNLO, and from MadGraphLO. �� [� �� =�� % ��] γγ [��] ��(�� ) �� →γγ (�� ) FIG. 4: Left: ALP signal event with ma = mB s in diphoton invariant mass m full γγ (yellow) and trigger level diphoton invariant mass m trigger γγ (blue). Fraction of energy, E (2×2) γ = E full γ min[1, P normal (µ = 0.95, σ = 0.1)], is used for the calculation of trigger level invariant mass. Right: SM diphoton event with a cut, 3.5 GeV < m trigger γγ < 6 GeV, in m full γγ (yellow) and in m trigger γγ (blue). 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[ "Compatible Weighted Proper Scoring Rules ", "Compatible Weighted Proper Scoring Rules " ]	[ "Peter G M Forbes \nDepartment of Statistics\nUniversity of Oxford\n1 South Parks RoadOX1 3TGOxfordU.K\n" ]	[ "Department of Statistics\nUniversity of Oxford\n1 South Parks RoadOX1 3TGOxfordU.K" ]	[]	Many proper scoring rules such as the Brier and log scoring rules implicitly reward a probability forecaster relative to a uniform baseline distribution. Recent work has motivated weighted proper scoring rules, which have an additional baseline parameter. To date two families of weighted proper scoring rules have been introduced, the weighted power and pseudospherical scoring families. These families are compatible with the log scoring rule: when the baseline maximizes the log scoring rule over some set of distributions, the baseline also maximizes the weighted power and pseudospherical scoring rules over the same set. We characterize all weighted proper scoring families and prove a general property: every proper scoring rule is compatible with some weighted scoring family, and every weighted scoring family is compatible with some proper scoring rule.	10.1093/biomet/ass046	[ "https://arxiv.org/pdf/1311.1131v1.pdf" ]	88,513,054	1311.1131	b0fa355b082d0f38db702cbc019cdbc1d081537f	Compatible Weighted Proper Scoring Rules * 5 Nov 2013 September 19, 2012 Peter G M Forbes Department of Statistics University of Oxford 1 South Parks RoadOX1 3TGOxfordU.K Compatible Weighted Proper Scoring Rules * 5 Nov 2013 September 19, 2012 Many proper scoring rules such as the Brier and log scoring rules implicitly reward a probability forecaster relative to a uniform baseline distribution. Recent work has motivated weighted proper scoring rules, which have an additional baseline parameter. To date two families of weighted proper scoring rules have been introduced, the weighted power and pseudospherical scoring families. These families are compatible with the log scoring rule: when the baseline maximizes the log scoring rule over some set of distributions, the baseline also maximizes the weighted power and pseudospherical scoring rules over the same set. We characterize all weighted proper scoring families and prove a general property: every proper scoring rule is compatible with some weighted scoring family, and every weighted scoring family is compatible with some proper scoring rule. Introduction Suppose Y is a random variable taking values in {1, . . . , m}. The valid distributions for Y are P = {(p 1 , . . . , p m ) T : 0 ≤ p i ≤ 1, m i=1 p i = 1} ⊂ R m . * This is a pre-copyedited, author-produced PDF of an article accepted for publication in Biometrika following peer review. The definitive publisherauthenticated version (Biometrika 99 (4): [989][990][991][992][993][994]2012) is available online at http://biomet.oxfordjournals.org/cgi/content/abstract/ass046?ijkey=CaRYBhLvVa4XvRY&keytype=ref. A scoring rule s : P × P → R is a function linear in its second argument. The scoring rule is proper if s(p, r) is maximized over p at p = r, and strictly proper if this maximum is unique. Consider a forecaster asked to issue a probabilistic prediction p for Y . She is motivated by a reward of s(p, r) upon observing outcome distribution r. If the forecaster's true belief is p * , her expected score s(p, p * ) is maximized when she predicts p = p * . Hence proper scoring rules encourage honesty. Two scoring rules equivalent if their rewards are linearly related for all p, r ∈ P: s 1 (p, r) = a{s 2 (p, r) + b , r } where · , · is the standard inner product on R m , a > 0 and b ∈ R m . The main characterization theorem for proper scoring rules was stated by McCarthy (1956) and proved by Hendrickson and Buehler (1971). (2) defined on P Λ = {λp : λ > 0, p ∈ P} is convex and satisfies S(p) ≥ s(p, q) for all p, q ∈ P. The scoring rule is strictly proper if and only if S is strictly convex on P. The function S is called the optimal expected score. Grünwald and Dawid (2004) showed that the negative optimal expected score can be interpreted as a generalized entropy. When S is differentiable we have (Hendrickson and Buehler, 1971) s(p, r) = s(λp, r) = ∇ λp S(λp) , r which associates a proper scoring rule with any convex differentiable function S. For the rest of this paper we assume that S is twice differentiable on P Λ , strictly convex on P, and achieves its unique minimum in P + , the interior of P. Equation (3) extends the domain of s to P Λ × P and allows us to differentiate s with respect to its first parameter. Since s(λp, r) = s(p, r) for any λ > 0, we have ∇ p s(p, r) , 1 m = 0 for any p and r ∈ P, where 1 m ∈ R m has all entries equal to one. Consider a sequence of observations y 1 , . . . , y n with empirical distribution r ∈ P. Let p(θ) be some model which takes values in P + and is differentiable over some open convex set Θ. Then any scoring rule defines an optimal score estimator (Gneiting and Raftery, 2007) viã θ(r) = arg max θ∈Θ s{p(θ), r} = arg max θ∈Θ n i=1 s{p(θ), y i }. From (1), all equivalent scoring rules have the same optimal score estimator. The optimal score estimator is well behaved at r ifθ(r) exists and is the unique root of ∇ θ s{p(θ), r} in Θ. When s is the log scoring rule s(p, r) = m i=1 r i log p i , the optimal score estimator becomes the maximum likelihood estimator. A well behaved optimal score estimateθ(r) yields the parameter choice that maximizes the forecaster's expected score under the assumption that the future is similar to the past. Specifically we suppose that our forecaster issues the prediction p(θ) for some θ ∈ Θ. If she believes that the next observation's distribution is r, then p{θ(r)} maximizes her expected score. The optimal score estimator can be generalized so that each y i follows a different probability distribution, as long as these distributions share a common parameter θ ∈ Θ. Thus the optimal score estimator is applicable to regression models that depend on both θ and some additional covariates. For the sake of brevity we consider only the basic optimal score estimator here, though all the results hold in the general case. Results We define the baseline of a strictly proper scoring rule to be the unique q ∈ P + that maximizes the generalized entropy −S(p). For example, the log scoring rule's generalized entropy is the Shannon entropy, which is maximized by the uniform distribution. Proper scoring rules tends to give larger rewards for riskier predictions which vary significantly from the baseline. Given q ∈ P + and a strictly proper scoring rule s(p, r), there is an equivalent rule with baseline q given by s(p, r) − s(q, r). A weighted scoring family s (p, r ·) = {s (p, r q) : q ∈ P + } is a family of strictly proper scoring rules where each member s (p, r q) has baseline q. Two weighted proper scoring rules are equivalent if (1) is satisfied, where now a and b are functions of q. Different members from the same family need not be equivalent. Weighted scoring families allow us to tailor our scoring rule to the problem at hand, as motivated in Jose et al. (2009) and Johnstone and Lin (2011). This tailoring is achieved by modifying the baseline. The baseline is easily interpretable and justifiable in many real world situations. For instance, weighted scoring families are used in Jose et al. (2008) for a optimal portfolio allocation problem, where the baseline corresponds to the market price. Let s(p, r) be a proper scoring rule and s (p, r ·) be a weighted scoring family. We say s (p, r ·) is compatible with s(p, r) if for any q and r ∈ P + , ∇ p s(p, r)\| p=q = a(q)∇ p s (p, r q)\| p=q(4) for some function a(q) > 0. In words, equation (4) says that the tangent of a weighted scoring rule at its baseline q is parallel to the compatible scoring rule's tangent at q. By approximating s (p, r q) with its tangent at p = q and applying (4), we obtain s (p, r q) ≈ s (q, r q) + 1 a(q) ∇ p s(p, r)\| p=q , p − q . The first term corresponds to an equivalence factor b(q) , r . Thus, up to equivalence, every member of the weighted scoring family s (p, r ·) is linearly approximated by the compatible proper scoring rule s(p, r) in the vicinity of its baseline. Theorem 2. Any proper scoring rule is compatible with at least one weighted scoring family. Conversely, every weighted scoring family is compatible with some proper scoring rule, which is unique up to equivalence. Proof. Let s(p, r) be a proper scoring rule. From the definition (4), it is compatible with the weighted scoring family where each member is equivalent to s(p, r): s (p, r q) = s(p, r) − s(q, r). Conversely, consider the weighted scoring family s (p, r ·). From (3) and (4), a proper scoring rule s(p, r) is compatible with this family if and only if its optimal expected score S(p) satisfies ∇ 2 q S(q) = a(q) ∇ 2 p S (p q) p=q(5) for some a(q) > 0 and all q ∈ P + . The right hand side is a positive definite matrix since it is the Hessian of the convex function S (p q). Thus the S satisfying (5) is convex and corresponds to a strictly proper scoring rule. This solution is unique up to equivalence since the solution of a second-order differential equation is unique up to a linear term. Having shown that a compatible proper scoring rule always exists, we now provide an alternative characterization for compatibility which has direct applications to optimal score estimation and decision theory. Lemma 1. A weighted scoring family s (p, r ·) is compatible with the proper scoring rule s(p, r) if and only if ∇ θ s{p(θ), r}\| θ=θ 0 = 0 implies ∇ θ s{p(θ), r p(θ 0 }\| θ=θ 0 = 0 for all differentiable models p(θ), all θ 0 ∈ Θ, and all r ∈ P + . Proof. Choose some model p(θ) and r ∈ P + . Suppose that s (p, r ·) is compatible with s(p, r), so that (4) holds for all q ∈ P + . Then (4) certainly holds when q = p(θ 0 ) for any θ 0 ∈ Θ. Left multiplying both sides of (4) with the matrix ∇ θ p T (θ) θ=θ 0 and using the chain rule, ∇ θ s{p(θ), r}\| θ=θ 0 = a{p(θ 0 )} ∇ θ s{p(θ), r p(θ 0 )}\| θ=θ 0 . Thus if ∇ θ s{p(θ), r}\| θ=θ 0 = 0 then ∇ θ s{p(θ), r p(θ 0 )}\| θ=θ 0 = 0. Conversely, suppose ∇ θ s{p(θ), r}\| θ=θ 0 = 0 implies ∇ θ s{p(θ), r p(θ 0 )}\| θ=θ 0 = 0. When q = r, both sides of (4) are being evaluated at their critical points and hence are zero. We will show (4) holds for q = r by showing that v = ∇ p s (p, r q)\| p=q is parallel to w = ∇ p s(p, r)\| p=q . Using (3) we can rewrite v as v = ∇ 2 p S (p q) p=q r(6) where ∇ 2 p S (p q) is the positive definite Hessian of S (p q). This implies v = 0 since r = 0. Furthermore since v is a gradient of s (p, r q), v , 1 m = 0. The same arguments show that w = 0 and w , 1 m = 0. Suppose v is not parallel to w. Then we can define the non-zero vector b = v − v , w w , w w.(7) By construction b , w = 0. Consider the model p(θ) = q + θb where θ takes values on Θ, an open neighbourhood of zero small enough such that {p(θ) : θ ∈ Θ} ⊂ P + . It follows from v , 1 m = 0 and w , 1 m = 0 that p(θ) is normalized for all θ ∈ Θ. Thus p(θ) is a valid distribution for θ ∈ Θ and, by our choice of w and p(θ), ∇ θ s{p(θ), r}\| θ=0 = ∇ θ p(θ)\| θ=0 , w = b , w = 0. Hence by assumption, ∇ θ s{p(θ), r q}\| θ=0 = 0. By definition of v we have ∇ θ s{p(θ), r q}\| θ=0 = b , v and thus b , v = 0. Substituting this into (7), v , v w , w = v , w 2 and the Cauchy-Schwarz inequality implies that v is parallel to w: w = a(r, q)v. Using (6), we rewrite w = a(r, q)v as ∇ 2 p S (p q) p=q r = a(q, r) ∇ 2 p S(p) p=q r. Since both matrices are positive definite, a(q, r) > 0. Since the left hand side is linear in r, we see a = a(q), which proves (4). Consider a forecaster motivated by a weighted scoring rule with baseline q to issue a prediction p(θ) for Y . She chooses her prediction based on some decision rule p{θ(r)}, where r is the empirical distribution of the previous observations of Y . For instance,θ could be the optimal score estimator for her weighted scoring rule. Her risk function is −s[p{θ(p * )}, p * q], which depends on the unknown true distribution p * of Y . Since p * is unknown it is approximated with the empirical distribution r. Suppose the baseline is determined by the optimal score estimator of the compatible scoring rule, q = p{θ(r)}. Then, assumingθ andθ to be well behaved at r, Lemma 1 implies that the forecaster's risk function is uniquely minimized when she issues the prediction q. The optimal score estimator of the compatible scoring rule dominates any other estimatorθ for this choice of baseline. Examples Define the quasi-Bregman weighted scoring families to be the proper scoring rules with optimal expected scores S (p q) = h m i=1 f (q i )g p i q i − g ′ (1) m i=1 p i f (q i ) q i h ′    g(1) m j=1 f (q j )    ,(8) where g ′ denotes the derivative of g with respect to its parameter, and similarly for h ′ . We require that f is positive, g is twice differentiable and strictly convex, and that h is twice differentiable and strictly increasing. This defines a weighted scoring family for each choice of f , g and h. The expected score S (p q) is strictly convex since g is strictly convex, f is positive and h is increasing. Hence the quasi-Bregman weighted scoring families are strictly proper. The second term of (8) ensures that S (p q) has baseline q, though removing it achieve a simpler, equivalent rule for optimal score estimation. The weighted power and pseudospherical scoring families of Jose et al. (2008), defined by s pow (p, r q) = 1 − m i=1 p β i q 1−β i β − 1 − m i=1 r i p β−1 i q 1−β i β − 1 , s ps (p, r q) = 1 β − 1      m i=1 r i p i /q i m i=1 p β i q 1−β i 1/β − 1      for β > 1, are quasi-Bregman weighted scoring families with f (x) = x and h pow (x) = x − 1 β(β − 1) , g pow (x) = x β , h ps (x) = x 1/β − 1 β(β − 1) , g ps (x) = x β . Johnstone and Lin (2011) proved that ∇ θ s{p(θ), r}\| θ=θ 0 = 0 implies ∇ θ s{p(θ), r p(θ 0 }\| θ=θ 0 = 0 when s (p, r r) is a power or pseudospherical weighted scoring family and s(p, r) is the log scoring rule. From Lemma 1, this is equivalent to showing that the power and pseudospherical weighted scoring families are compatible with the log scoring rule. Corollary 1. The log scoring rule is compatible with any quasi-Bregman weighted scoring family with f (x) = x. This holds for any twice differentiable and strictly convex g, and any twice differentiable and strictly increasing h. Proof. By substituting f (x) = x into (8) and using (3), we obtain s (p, r q) = h ′ m i=1 q i g p i q i m j=1 r j g ′ p j q j .(9) The log scoring rule is s(p, r) = m i=1 r i log p i . Substituting (9) and the log scoring rule into (4) shows that the equality holds with a = h ′ {g(1)} g ′ (1). The functions h and g enter only through their values and first derivatives at 1. We define the Bregman weighted scoring families as the quasi-Bregman weighted scoring families with h(x) = x. By substituting (8) into (3) and using equivalence, the Bregman weighted scoring families take the simple form s (p, r q) = m i=1 f (q i ) g p i q i + g ′ p i q i r i − p i q i .(10) We recover the unweighted Bregman scoring rules of Grünwald and Dawid (2004), i.e., s(p, r) = m i=1 g(p i ) +g ′ (p i )(r i − p i ) ,(11) by using a flat baseline and rescaling g tog (p i ) = f (q i )g(p i /q i ) = f (m −1 )g(mp i ). The unweighted Bregman scoring rules are uniquely specified through the convex functiong alone. Corollary 2. The unweighted Bregman rule specified byg is compatible with all weighted Bregman families with f (x) = x 2g′′ (x). This holds for any twice differentiable and strictly convex g. Proof. We use (4) with s (p, r q) given by (10) with f (x) = x 2g′′ (x) and s(p, r) given by (11). We illustrate the use of this corollary via an example. The unweighted power scoring rule is defined byg(x) = x β /{β(β − 1)} for β > 1. Using the above corollary with f (x) = x β , we see that the unweighted power scoring rule is compatible with all weighted scoring families of the form s (p, r q) = m i=1 q β i g p i q i + g ′ p i q i r i − p i q i(12) for any choice of g. Discussion As an application of compatible proper scoring rules, consider a portfolio allocation problem similar to Jose et al. (2008). There is a market consisting of m assets, and a market maker who sets the prices at q. After one time period asset Y will be worth 1 unit and the other assets will be worthless. The investor purchases a portfolio, spending a proportion of his wealth p i (θ) on each asset and thus receiving p i (θ)/q i (θ) units of each asset. He chooses θ based on the current prices q and the historical outcome distribution r. Suppose the investor's negative risk is given by a weighted scoring rule s{p(θ), r q}. The market maker does not know the form of the investor's scoring rule, but he believes it to come from a weighted scoring family compatible with some known proper scoring rule. The market maker prices the assets using the compatible rule's optimal score estimator, q = p{θ(r)}. Then the market maker's price coincides with the investor's minimal risk portfolio p(θ): when the pricing is done by a compatible proper scoring rule, the investor is best served by buying the same number of units of each asset. Johnstone (2011) interprets this minimal risk portfolio from an economic perspective, for the special case where the compatible rule is the log score. Until now, the only weighted scoring families considered in the literature were the weighted power and pseudospherical scoring rules. Since both are compatible with the log scoring rule, their optimal score estimators are dominated by the maximum likelihood estimator when the baseline is given by the latter. Johnstone and Lin (2011) conjectured the existence of a characterization theorem for all weighted proper scoring families whose optimal score estimators are dominated in this way. They went on to suggest that this theorem might reveal an unrecognized property of the log scoring rule. We have found their conjectured characterization theorem: the optimal score estimator of any weighted proper scoring rule is dominated by the compatible proper scoring rule's optimal score estimator when the baseline is set to the compatible proper scoring rule's optimal score estimate. However, instead of revealing a special property of the log scoring rule, we have shown that every proper scoring rule is compatible with some family of weighted proper scoring rules. Theorem 1 . 1A scoring rule s is proper if and only if the function S(λp) = λs(p, p) AcknowledgmentI thank Steffen Lauritzen, Philip Dawid, Tilmann Gneiting and the referees for their helpful comments. 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Economic interpretation of probabilities estimated by MLE or score. Management Science 57 (2), 308-314. Fitting probability forecasting models by scoring rules and maximum likelihood. D J Johnstone, Y.-X Lin, Journal of Statistical Planning and Inference. 1415Johnstone, D. J. and Y.-X. Lin (2011). Fitting probability forecasting models by scoring rules and maximum likelihood. Journal of Statistical Planning and Inference 141 (5), 1832-1837. Scoring rules, generalized entropy, and utility maximization. V R R Jose, R F Nau, R L Winkler, Operations Research. 565Jose, V. R. R., R. F. Nau, and R. L. Winkler (2008). Scoring rules, gen- eralized entropy, and utility maximization. Operations Research 56 (5), 1146-1157. Sensitivity to distance and baseline distributions in forecast evaluation. V R R Jose, R F Nau, R L Winkler, Management Science. 554Jose, V. R. R., R. F. Nau, and R. L. Winkler (2009). Sensitivity to dis- tance and baseline distributions in forecast evaluation. Management Sci- ence 55 (4), 582-590. Measures of the value of information. J Mccarthy, Proceedings of the National Academy of Sciences. 42McCarthy, J. (1956). Measures of the value of information. Proceedings of the National Academy of Sciences 42, 654-655.	[]
[ "Topological power pumping in quantum circuits", "Topological power pumping in quantum circuits" ]	[ "J Luneau \nLaboratoire de Physique\nUniv Lyon\nENS de Lyon\nCNRS\nF-69342LyonFrance\n", "C Dutreix \nUMR 5798\nUniv. Bordeaux\nCNRS\nLOMA\nF-33405TalenceFrance\n", "Q Ficheux \nDepartment of Physics\nETH Zürich\nCH-8093ZürichSwitzerland\n", "P Delplace \nLaboratoire de Physique\nUniv Lyon\nENS de Lyon\nCNRS\nF-69342LyonFrance\n", "B Douçot \nLaboratoire de Physique Théorique et Hautes Energies\nUMR 7589\nSorbonne Université\nCNRS\n4 place Jussieu75252, Cedex 05ParisFrance\n", "B Huard \nLaboratoire de Physique\nUniv Lyon\nENS de Lyon\nCNRS\nF-69342LyonFrance\n", "D Carpentier \nLaboratoire de Physique\nUniv Lyon\nENS de Lyon\nCNRS\nF-69342LyonFrance\n" ]	[ "Laboratoire de Physique\nUniv Lyon\nENS de Lyon\nCNRS\nF-69342LyonFrance", "UMR 5798\nUniv. Bordeaux\nCNRS\nLOMA\nF-33405TalenceFrance", "Department of Physics\nETH Zürich\nCH-8093ZürichSwitzerland", "Laboratoire de Physique\nUniv Lyon\nENS de Lyon\nCNRS\nF-69342LyonFrance", "Laboratoire de Physique Théorique et Hautes Energies\nUMR 7589\nSorbonne Université\nCNRS\n4 place Jussieu75252, Cedex 05ParisFrance", "Laboratoire de Physique\nUniv Lyon\nENS de Lyon\nCNRS\nF-69342LyonFrance", "Laboratoire de Physique\nUniv Lyon\nENS de Lyon\nCNRS\nF-69342LyonFrance" ]	[]	In this article, we develop a description of topological pumps as slow classical dynamical variables coupled by a quantum system. We discuss the cases of quantum Hall pumps, Thouless pumps, and the more recent Floquet pumps based frequency converters. This last case corresponds to a quantum topological coupling between classical modes described by action-angle variables on which we focus. We propose a realization of such a topological coupler based on a superconducting qutrit suitably driven by three modulated drives. A detailed experimental protocol allowing to measure the quantized topological power transfer between the different modes of a superconducting circuit is discussed.	10.1103/physrevresearch.4.013169	[ "https://arxiv.org/pdf/2109.12897v2.pdf" ]	237,941,061	2109.12897	54e61e6ec5211a54d4b55f876994e1873b1f266e	Topological power pumping in quantum circuits J Luneau Laboratoire de Physique Univ Lyon ENS de Lyon CNRS F-69342LyonFrance C Dutreix UMR 5798 Univ. Bordeaux CNRS LOMA F-33405TalenceFrance Q Ficheux Department of Physics ETH Zürich CH-8093ZürichSwitzerland P Delplace Laboratoire de Physique Univ Lyon ENS de Lyon CNRS F-69342LyonFrance B Douçot Laboratoire de Physique Théorique et Hautes Energies UMR 7589 Sorbonne Université CNRS 4 place Jussieu75252, Cedex 05ParisFrance B Huard Laboratoire de Physique Univ Lyon ENS de Lyon CNRS F-69342LyonFrance D Carpentier Laboratoire de Physique Univ Lyon ENS de Lyon CNRS F-69342LyonFrance Topological power pumping in quantum circuits (Dated: March 4, 2022) In this article, we develop a description of topological pumps as slow classical dynamical variables coupled by a quantum system. We discuss the cases of quantum Hall pumps, Thouless pumps, and the more recent Floquet pumps based frequency converters. This last case corresponds to a quantum topological coupling between classical modes described by action-angle variables on which we focus. We propose a realization of such a topological coupler based on a superconducting qutrit suitably driven by three modulated drives. A detailed experimental protocol allowing to measure the quantized topological power transfer between the different modes of a superconducting circuit is discussed. In this article, we develop a description of topological pumps as slow classical dynamical variables coupled by a quantum system. We discuss the cases of quantum Hall pumps, Thouless pumps, and the more recent Floquet pumps based frequency converters. This last case corresponds to a quantum topological coupling between classical modes described by action-angle variables on which we focus. We propose a realization of such a topological coupler based on a superconducting qutrit suitably driven by three modulated drives. A detailed experimental protocol allowing to measure the quantized topological power transfer between the different modes of a superconducting circuit is discussed. I. INTRODUCTION Topological properties of matter have always been discussed in relation with topological pumping. In an enlightening gedankenexperiment, B. Laughlin related the topological nature of the transverse conductivity of the dimension D = 2 quantum Hall state to a transfer of charge between two edges in a Corbino geometry [1]. A modern interpretation views the Hall sample as effectively wrapped on a cylinder, realizing a D = 1 quantum Hall charge pump as the enclosed magnetic flux is smoothly increased [2,3]. This quantized adiabatic pumping of a time-dependent quantum system was soon generalized beyond the quantum Hall effect to a driven one-dimensional crystal by D. Thouless [4]. In a Thouless pump, exemplified by the Rice-Mele model [5], the time-varying flux of the quantum Hall effect is replaced by a time-periodic phase φ(t) = ωt which accounts generically for the external drive. The corresponding dynamics is periodic both in the linear spatial dimension of the crystal, but also in time. In contrast with previous implementations of geometrical pumps, such topological pumps are characterized by a Chern number and were only recently realized using cold atoms lattices [6][7][8], optical waveguides [9][10][11], magnetically coupled mechanical resonators [12] or stiffness-modulated elastic plates [13]. An extension of this pumping was unveiled recently in quantum systems of effective spatial dimension D = 0, but with two temporal dimensions corresponding to a drive at two different frequencies ω 1 and ω 2 . The initial theoretical proposal arose in Cooper pair pumps realized in Josephson junction driven by two independent superconductors' phase differences [14][15][16][17][18][19] and later on as a frequency converter in a driven two level system [20]. This led to the realization of such a frequency converter using * [email protected] a single nitrogen-vacancy center in diamond [21] or with a superconducting qubit [22], although in both cases measuring the topological transfer was beyond experimental reach. In all existing realizations, the topological pumping is probed indirectly, as a property of the time-dependent quantum system. The dynamics of these quantum systems is assumed to be adiabatic, i.e. restricted to an eigensubspace well-separated in energy from other quantum eigenstates. Pumping manifests itself as an anomalous velocity in real space for the Quantum Hall and Thouless pumps, or an anomalous velocity in the harmonic Floquet space for the frequency converter. This anomalous velocity was initially identified in the case of the quantum Hall effect in a crystal [23][24][25], and originates from a Berry curvature whose average value defines a Chern number characterizing the topological nature of the pump. It was recently measured in the cold atom [6][7][8] and optical waveguides [10] realizations of a Thouless pump. Alternatively to these bulk measurements, the topology of the D = 1 pumps can be inferred from the occurrence of states at the boundary of the chain during a period of the drive. A direct probe of evolution of these edge states allows to characterize the pumping [12]. Indeed, these topological edge states manifest themselves in the scattering description of the quantum pump [26][27][28][29][30]: they lead to resonances of the reflection matrix R during one cycle of the driven quantum chain, whose associated π phase shift lead to the winding of the phase of detR [3]. Such a theoretical approach was recently extended to generalized pumps obtained by wrapping D = 2 topological insulators around a cylinder [31,32]. Although previous descriptions of pumping focused on the driven quantum system, they call for a key generalization that embeds the coupled degrees of freedom of the environment if one wants to model the observable topological transfer itself. In this article, we present such a global framework that allows to characterize the actual pumping and propose a realistic experiment that directly measures this topological pumping. In practice, arXiv:2109.12897v2 [cond-mat.mes-hall] 3 Mar 2022 a classical treatment of the degrees of freedom of the environment is sufficient to observe a topological transfer and we restrict our attention to that limit. While a similar treatment dates back to the Born-Oppenheimer approximation [33][34][35], here we focus on classical modes or action-angle variables which describe the periodic drive of a quantum system. The interaction between these slow classical degrees of freedom and the gapped quantum system, when treated within the adiabatic approximation, effectively reduces on average to a topological coupling leading to a quantized pumping between the classical environments. To illustrate the virtues of this formalism, we propose a realization of a Chern mode coupler enabling to measure the topologically protected power transferred between electromagnetic modes. The proposed setup consists in using a qutrit, realized from the first three levels of a highly anharmonic superconducting circuit, a fluxonium, driven at multiple microwave frequencies. The driven fluxonium realizes a time-dependent version of a Haldane model on a Lieb lattice. The corresponding phase diagram can directly be revealed by measuring the power transfer between the microwave modes. The topological pumping leads to a topological redistribution of energy between the three microwave modes. II. SLOW CLASSICAL MODES COUPLED TO A FAST QUANTUM STATE A. Dynamics of classical degrees of freedom adiabatically coupled to a quantum system We provide a general model of coupled fast and slow degrees of freedom by resorting to a mixed classicalquantum description, see Fig. 1, similar in spirit to that used by M.V. Berry and J. Robbins [35] and by Q. Zhang et al. [36]. The fast degrees of freedom are generically described by a quantum Hamiltonian H. The slow degrees of freedom are associated to an energy separation (much) smaller than that of the fast system, allowing for a classical description by pairs of conjugated variables q α , p α , α = 1, ...N satisfying the Poisson bracket relations {q α , p β } = δ αβ [37]. We consider the common case where the quantum system couples to only one of the variable, q α , of each pair of conjugate variables, which includes in particular the case of a driven quantum system. Hamilton equations of motion The dynamics of each pair of classical variables, prior to the coupling to the quantum system, is assumed to be slow and described by classical dynamics deduced from a classical Hamiltonian H α (q α , p α ) followinġ q (0) α = ∂H α ∂p α andṗ (0) α = − ∂H α ∂q α . (1) Fast Quantum FIG. 1. We consider a general gapped and fast system (in blue) coupled to slow variables of the environment (in red). The energy separation of the fast quantum system is assumed to be much larger than that of the slow degrees of freedom, allowing a description of the latter by classical pairs of conjugate variables. While of different nature, we denote generically these variables by qα, pα. The dynamics of each pair follows from a classical Hamiltonian Hα. The quantum system couples instantaneously to a single variable qα of each degree of freedom of the classical environment. The quantum system is described by a Hamiltonian H({q α }) parametrized by the states of the classical variables q α . We focus on quantum systems that remain gaped during the evolution of the classical modes. This allows to approximate at shorter times the dynamics of the quantum system by an adiabatic evolution, driven by the slow dynamics of the classical modes. More precisely, we consider a quantum system prepared at time t = 0 into one of the eigenstates denoted \|ψ ν (t = 0) of the t = 0 Hamiltonian H({q α (t = 0)}). This amounts to assume an initial correlation between the state {q α (t = 0)} of the classical environment and the quantum system. Through the coupling to q α (t), the quantum system will slowly evolve in time on a timescale dictated by the slow environment. We denote its instantaneous state \|Ψ(t) and assume that it remains approximately within the same eigensubspace of the Hamiltonian, which is the essence of the adiabatic approximation. In return, the coupling of the classical degrees of freedom to the quantum system also perturbs their dynamics and results in an effective coupling between different pairs of slow degrees of freedom which is the focus of this paper. The modified equations of motion for the slow variables areq α =q (0) α (2) p α =ṗ (0) α − Ψ(t)\| ∂H ∂q α \|Ψ(t) .(3) The second term in (3) was discussed as a geometric force when q α is a position in [35]. This decomposition of dynamics into slow and fast degrees of freedom, in the spirit of the Born-Oppenheimer decomposition [33,38,39], reduces to the standard semi-classical analysis for the specific case of the motion of electrons in insulators [40]. Let us stress that the above description ensures the conservation of the total energy of both quantum and classical degrees of freedom. Indeed, the Schrödinger equation in finite dimension corresponds to classical Hamiltonian equations associated to an Hamiltonian function E(Ψ) = Ψ\| H \|Ψ and to a natural Poisson bracket structure on the Hilbert space [41]. Therefore the total phase space also inherits a Poisson bracket structure and the above equations of motion from the Hamiltonian function H({q α }, {p α }, Ψ) = α H α (q α , p α ) + Ψ\| H({q α }) \|Ψ , thus abiding by the conservation of the total energy. To evaluate the matrix elements in (3), we now describe the adiabatic evolution of the quantum state \|Ψ(t) . Due to the slow evolution of the Hamiltonian H({q α (t)}), the state \|Ψ(t) of the quantum system does not identify with the instantaneous eigenstate \|ψ ν (t) defined by H({q α (t)}) \|ψ ν (t) = E ν ({q α (t)}) \|ψ ν (t) . The corresponding correction to the dynamics of the slow classical variables p α in Eq. (3) is now expressed aṡ p α =ṗ (0) α − ∂E ν ∂q α + β =αq (0) β F (ν) qαq β ,(4) see Appendix A for the detailed derivation. The first correction relates to the energy variation of the quantum state and follows from the standard classical -quantum coupling. The last correction is more exotic: it manifests an effective coupling between the different slow variables q α , p α . The strength of this transverse coupling depends on the geometry of the eigenstates \|ψ ν of the quantum system through the components F (ν) qαq β of the two-form Berry curvature which are defined as F (ν) qαq β = i µ =ν ψ ν \| ∂ qα H \|ψ µ ψ µ \| ∂ q β H \|ψ ν (E ν − E µ ) 2 − (α ↔ β). (5) Condition of adiabaticity The above adiabatic approximation in one eigensubspace \|ψ ν holds as long as the transitions to a different eigensubspace \|ψ µ can be neglected. These non-adiabatic transitions can be described as Landau-Zener transitions [42,43], with a probability of transition from one eigenstate to another behaving as exp(−π/(4 max t µν )) where the time dependent parameter µν reads µν = ψ µ \| dH dt \|ψ ν (E µ − E ν ) 2 .(6) In this formula, τ col µν = \|E µ − E ν \|/ ψ µ \| dH dt \|ψ ν corresponds to the characteristic time of the Landau-Zener collision, or equivalently to an energy spread δE = /τ col µν . Transitions occur when this spread is comparable to the gap E µ − E ν , and the parameter µν , which controls the validity of the adiabatic approximation, identifies with this ratio µν = δE/\|E µ − E ν \|. For an alternative derivation of this small parameter µν within the adiabatic expansion [44], see Appendix A. Along a trajectory in phase space, the gap of the quantum system varies. The transition dynamics at each minimum can be described following the above Landau-Zener approach, leading to a series of transitions. It is then possible to determine the characteristic time of validity of the adiabatic approximation τ adiab by constraining the cumulative transition probability to be e.g. of order 0.1. Introducing the mean free time τ mft separating the evolution in phase space between two minima of the gap, we get τ adiab ≈ 0.1 τ mft exp π 4 max t µν(7) where the maximum of the parameter µν is evaluated along the phase space trajectory during τ mft . Note that for the above analysis to be consistent, the collision time τ col µν must be smaller than the mean free time τ mft . Since we focus on aperiodic evolution, we also neglected the effect of relative phases accumulated between the transitions [45]. B. Geometrical power transfer The geometrical coupling in (4) between the different subset of slow variables q α (t), p α (t) is associated with an energy transfer between them. The change of energy of each classical degree of freedom is: dE α dt =q α ∂H α ∂q α +ṗ α ∂H α ∂p α = −q (0) α ∂E ν ∂q α + β =αq (0) αq (0) β F (ν) qαq β .(8) The antisymmetry of the Berry curvature implies the conservation of the total energy. C. Nature of classical degrees of freedom Let us discuss briefly the implications of the previous modification of Hamilton equations for different types of classical slow degrees of freedom coupled to a quantum system. Massive classical particles The initial context of the Born-Oppenheimer approximation, at the origin of the adiabatic approximation, was the description of light particles, the electrons, coupled to heavy particles, the nucleus. In this situation, the slow degrees of freedom described classically are those of the massive particle: its position q α and conjugated momentum p α . The corresponding Hamiltonian is H α = p 2 α /(2M ) + V (q α ), parametrized by the mass M and potential V (q). The equations of motion in this case take the forṁ q α = p α M ,(9)p α = −V (q α ) − ∂ qα E ν + β =α p β M F (ν) qαq β .(10) Equation (10) describes the associated anomalous geometrical force [34,35]. Classical modes While the previous adiabatic formalism was initially designed with classical massive degree of freedom in mind, it also applies to the case of slow action angle φ α , n α variables, which we will call a classical mode. In more details, we consider a variable p α = n α where n α takes only integer values and its canonical phase q α = φ α , a 2π periodic phase. Of particular interest is the situation of a monochromatic mode, corresponding to the Hamiltonian H α = ω α n α = ω α p α whose linearity in n α is the distinctive feature compared with the massive case. In this situation, the modified Hamilton equations of motion readφ α =q α = ω α ,(11)ṅ α =ṗ α = − ∂E ν ∂φ α + β =α ω β F (ν) φαφ β .(12) In the case of classical modes, the Eq. (12) describes the filling rate of mode α. The energy transferred between the different modes corresponds to dE α dt = − ω α ∂E ν ∂φ α + β =α ω α ω β F (ν) φαφ β .(13) III. TOPOLOGICAL PUMPING A. Topological versus geometrical couplings In the above section, we have shown how a geometrical quantity, the Berry curvature, encodes the strength of the effective coupling mediated by a quantum system between classical variables. Of particular interest is the case of two classical variables, e.g. q 1 and q 2 with a compact configuration space denoted [0, 2π] 2 . In this situation, introducing the integer C (ν) 12 Chern number, the quantitŷ [0,2π] 2 dq 1 dq 2 F (ν) q1q2 = 2π C (ν) 12(14) is a topological quantity quantized in units of 2π, i.e. it is insensitive of perturbations of the quantum Hamiltonian H provided the gap between E ν and other states does not close. Noting that (14) is nothing but the averaged Berry curvature over the configuration space, the case of a Hamiltonian H characterized by a topological Chern number corresponds to a quantized averaged coupling in Eq. (4). When the Berry curvature fluctuates in the configuration space around its topological average, a quantized coupling between the classical variables is recovered when averaging over initial position in the configuration space, which is hardly practical. Instead, we can resort to a time average: if the evolution of classical variables is ergodic, an average over the configuration space can be replaced by an average over a sufficiently long time. The above situation of two classical environments topologically coupled through a quantum system correspond to a topological pump. Historically, the relation between a topological Chern invariant and a pumping process originates from the Laughlin's description of a charge transfer between inner and outer edges of a quantum Hall sample in a Corbino geometry [1]. Later on, D. Thouless proposed another topological pumping mechanism by periodically modulating a D = 1 crystal. Much more recently, realizations of a topological pump were proposed as a junction between superconductors [14,15,17] or by driving a two level system at two different frequencies [46]. The topological nature of these processes can be inferred from the dynamics of the quantum degrees of freedom: in all cases the corresponding band structure is gapped, and the eigenmanifold in which the adiabatic dynamics takes place is characterized by a Chern number (14). Alternatively, such topological pumps can be characterized from a scattering point of view by probing the appearance of topological edges during the evolution of the quantum system through phase shifts of the reflection matrix [3,31,32]. In the following, we develop a "Born-Oppeiheimer description" of this topological pumping by showing how they can naturally be described with the formalism of Sec. II of adiabatic topological coupling between classical slow variables of different nature. B. D = 2 Quantum Hall pump In an enlightening argument, B. Laughlin related the quantization of the transverse conductivity of the quantum Hall state to a transfer of charge between edges in a Corbino geometry as the flux threading the disk is increased by one quantum [1]. Later on, Niu, Thouless and Wu introduced the notion of generalized boundary conditions for quantum Hall states [23]. The quantum Hall topological properties are expressed as the Chern number of the ensemble of many-body groundstates over the closed manifold of phase boundary conditions. These boundary conditions parameters were later related to electromotive forces through loops connecting opposite edges of the sample [47,48] effectively generalizing the topology of Laughlin's gedanken experiment to that of a torus and allowing for a dynamical description of the quantum Hall effect over a classical parameter space [49]. Here we consider the classical phases entering the generalized boundary conditions as dynamical variables. This effectively amounts to realize a quantum Hall topological pump between two LC harmonic circuits. Let us consider a quantum Hall sample coupled to two independent electrical circuits in the x and y direction, see Fig. 2(a). The coupling between each circuit and the quantum Hall sample follows from the boundary conditions of Niu et al. [23] on the many-body ground state wave function Ψ(x i , y i ): Ψ(x i + L x , y i ) = e iΦ1 Ψ(x i , y i ),(15a)Ψ(x i , y i + L y ) = e iΦ2 Ψ(x i , y i ). (15b) The two phases Φ 1 , Φ 2 are related to the voltage drop V α in each directions as Φ α (t) = (e/ )´t V α (t )dt . If we model each electric branch associated to this voltage drop as an LC circuit [50], these phases are dynamical classical variables, whose canonically conjugate momenta are the (rescaled) accumulated charge in each circuit Q α = ( /e)´t I α (t )dt , I α being the current in each circuits. The classical Hamiltonian describing each LC circuit is H α = e 2 2 2 C α Q 2 α + 2 2e 2 L α Φ 2 α ,(16) where L α is the inductance and C α the capacitance of the corresponding circuit [50]. Hence the dynamics of an LC circuit identifies to that of a massive particle of position q α = Φ α and momentum p α = Q α , in a harmonic potential. The Hamilton's equations (9 and 10) include a correction to the usual relations between flux and currenṫ Q 1 = e I 1 = − e 2 Φ 1 L 1 + F Φ1Φ2 e V 2(17) via the Berry curvature F Φ1Φ2 of the quantum Hall effect ground states derived by Niu et al. [23]. This Berry curvature being independent of the external fluxes and thus constant over the parameter space, it is related to the quantum Hall Chern number C 12 via F Φ1Φ2 = C 12 /(2π). Thus the corrected classical equation of motion of the LC circuits reads I 1 = − e Φ 1 L 1 + e 2 h C 12 V 2 .(18) The usual case of an ideal ammeter is recovered in the limit L 1 → ∞, C 1 → 0. Note that the energy transferred from one LC circuit to the other, following (8), is dE 1 dt = F Φ1Φ2Φ1Φ2 = C 12 e 2 h V 1 V 2 = δI 1 V 1 .(19) In the limit of an ideal Hall measurement, where the L 1 C 1 circuit corresponds to an ammeter, we get V 1 = 0, and no energy is transferred between the two circuits. C. D = 1 Thouless pump Let us now turn to the canonical example of a topological pump, proposed by D. Thouless [4], which consists in a D = 1 crystal suitably periodically driven in time, such as the Rice-Mele model [5]. The single electron dynamics is thus described by a time-dependent Bloch Hamiltonian H(k, φ(t)) periodic both in momentum over the Brillouin zone, and in 2π periodic phase φ(t) = ωt. This Hamiltonian is assumed to be gapped at all time, and with energy band ν eigenstates \|ψ ν (k, φ) possessing a finite Chern number C (ν) kφ over the 2-torus constituted of the D = 1 Brillouin zone and periodic phase configuration space of phase φ. In such a system, topological pumping is usually described as the appearance of a steady current in the bulk of a closed ring, corresponding to an anomalous geometric velocity for semi-classical states in band ν ẋ = F (ν) kφ ∂ t φ = 2π C (ν) kφ ω [40,51]. We propose an alternative description of such a Thouless pump, by considering an open D = 1 crystal of size L connected on both ends to an LC circuit, analogous to the quantum Hall pump, see Fig. 2(b). The coupling between the charged particles in the crystal and the LC circuit follows from the boundary conditions (15) on the many-body groundstate wavefunction Ψ(x = 0, φ(t)) = e iΦ1 Ψ(x = L, φ(t)). This amounts to couple the crystal to a pair of classical conjugated variables Q 1 , Φ 1 identical to those in the quantum Hall pump, with a classical Hamiltonian (16) describing their dynamics. The periodic driving of the Hamiltonian is now interpreted as the coupling between the charges of the crystal and a dynamical classical variable φ 2 = ωt, conjugated to a variable n 2 . The dynamicsφ 2 = ω,ṅ 2 = 0, correspond to that of a classical mode introduced in Sec. II C 2. In this representation, the topological coupling between the LC circuit and the classical mode leads to modified equation of motion in the LC circuit, manifesting the appearance of a charged current. The modified Hamilton equation (9) readṡ Q 1 = e I 1 = − e 2 Φ 1 L + F Φ1φ2 ω,(20) which leads to a steady charge current I 1 = e2πF Φ1φ2 /T through the driven crystal, corresponding to an average number C (ν) Φ1φ2 = C (ν) kφ of charged transferred across the chain per period T of the drive. This results identifies with the standard steady anomalous velocity in the bulk of the Thouless pump. D. D = 0 Power pump In the previous section, we interpreted a time-periodic quantum Hamiltonian as a coupling between a fast quantum system and a slow classical mode. A natural extension consists in considering a single quantum system coupled to the phases φ α of an arbitrary number N of classical modes, see Fig. 2(c). The quantum dynamics of such a system can be described within Floquet theory [20]. The topological Chern numbers of such a quantum system are defined in Floquet space, and, when non zero, leads to a frequency conversion mechanism. The description of such a quantum mode coupler is natural in terms of an effective classical dynamics of the modes. We assume that the quantum system couples only to the phases of the modes, corresponding to a Hamiltonian H(φ 1 (t), . . . , φ N (t)). In such a case, the equations of motions are given by (11,12) with a power leaving each mode given by Eq. (13). In the particular case of two modes of frequency ω 1 , ω 2 , we recover the result of Martin et al. for the averaged power between two modes given by [20]. In the following section, building on this general description of topological pumping we propose a realization of such a quantum topological coupler between microwaves modes using an artificial 3 level atom, a qutrit. dE 1 /dt = ω 1 ω 2 C (ν) φ1φ2 /(2π) IV. TOPOLOGICAL PUMP IN QUANTUM CIRCUITS : THE TOPOLOGICAL QUTRIT A. Case of a qubit It is possible to apply the concept of geometrical and topological response of gapped states to individual quantum systems. The seminal work of Martin, Refael, and Halperin [46] proposed to use a spin-1/2 under two frequency drives to observe the quantized pumping of energy from one drive to the other. Measuring such a power transfer presents a substantial experimental challenge. Recently, Malz and Smith [22] used the IBM Quantum Experience to observe the inner dynamics of a superconducting qubit state that would correspond to a topological quantum transition. However, their control scheme was mixing the flows of power between the drives -hindering a direct measurement of the power transfer. Similarly, other experiments on NV centers demonstrate a topological transition in the qubit dynamics but could not explore the quantized pumping of power [21]. Essentially, the model proposed in Ref. [46] consists in engineering the Hamiltonian H(φ 1 , φ 2 ) ∝ sin(φ 1 )σ X + sin(φ 2 )σ Y + (M − cos(φ 1 ) − cos(φ 2 )) σ Z ,(21) where σ i are the Pauli operators of the qubit, M is a parameter that drives the topological transition and the phases φ 1 = ω 1 t and φ 2 = ω 2 t are driven at two incommensurate frequencies. We envision three ways to realize this Hamiltonian. • As in Ref. [22], it is possible to drive a single superconducting qubit, i.e. a transmon, with a complex amplitude [ X(t) + iY (t)] e −2iπfqt−2i´t 0 Z(τ )dτ , where f q is the qubit frequency in order to implement any driving term of the sort H(t) = X(t)σ X + Y (t)σ Y + Z(t)σ Z . However, while it is possible to infer what is the transferred power between frequency components at ω 1 and ω 2 from the measured qubit dynamics, this power flow lacks physical embodiment and cannot be measured using any known apparatus. • Alternatively, the σ Z term in the Hamiltonian (21) can be achieved by controlling the frequency of the qubit directly, hence physically separating the source of power from the three terms corresponding to each Pauli operator in Eq. (21). The frequency of flux tunable qubits can be tuned rapidly using a on-chip flux control. However, measuring the power of the drive used for such a flux-tunable bias has never been achieved to our knowledge and requires involved technical development. • Lastly, the frequency of the qubit can be controlled by exploiting the ac-Stark shift created by a drive far detuned from the qubit transition frequency. The ac-Stark shift is a commonly used method to engineer the spectrum of artificial atoms but the relatively low anharmonicity of transmons imposes a finite bandwidth on the control parameter Z(t). All these solutions present serious practical limitations either on the achievable Z-control or -more importantly -on the ability to measure the quantized power transfer. We propose to circumvent this difficulty by extending the size of the Hilbert space. This new pumping schemes uses a qutrit to create gapped states. B. Implementation with a superconducting qutrit Principle of the experiment The experiment we propose consists in driving a superconducting qutrit at several frequencies in order to establish a topologically given power flow between microwave modes at various frequencies. We propose to use a superconducting circuit behaving as a qutrit where every transitions can be addressed individually with a well defined phase. In the following, we denote the transitions \|0 − \|1 as 1, \|1 − \|2 as 2, and \|0 − \|2 as 3 for the sake of simplicity. By modulating the drive amplitude Ω i of each transition i at a frequencyφ i = ω i , it is possible to engineer an effective Hamiltonian in a configuration space defined by the phases φ 1 , φ 2 , φ 3 . By enforcing φ 3 = φ 1 −φ 2 , the dimension is reduced while still enabling the observation of a topological transition in the power transferred between the various driving tones. The difference between any two transition frequencies of the qutrit shall be much greater than any drive amplitude Ω i and modulation frequency ω i in order to enable the direct measurement of the power transfer between any driving modes. Finally, all the above-mentioned timescales should be much smaller than the coherence time of the qutrit transitions. An example of a superconducting circuit satisfying theses requirements is the fluxonium artificial atom. The fluxonium is a highly anharmonic superconducting circuit whose first three energy levels can be used as a qutrit. The circuit is a loop composed of a Josephson junction shunted by a large inductance, see Fig. 3(a). When the loop is threaded by an external magnetic flux corresponding to almost (but not exactly) half a flux quantum, no selection rule prevents the direct driving of all three transitions while the circuit transitions have been shown to display record long coherence times [52][53][54]. The circuit is embedded in a cavity with a single port connected to a transmission line. The cavity is used as an off-resonant readout mode dispersively coupled to the circuit transitions [55]. The cavity also acts as a filter that protects the circuit from direct energy dissipation into the electromagnetic environment of the transmission line, while preserving fast microwave control through to the direct coupling of the input port to the circuit antenna. Incoming microwave modes carry the modulated drives [see Fig. 3(b)] to the qutrit and outgoing modes carry the reflected signal before being measured by a power spectrum analyzer. At the input, each drive at frequency f i is modulated in amplitude at a frequency ω i , resulting in the pairs of sidebands in Fig. 3(b). The geometrical and topological signatures can be observed in the power transfer between these various frequency modes. We model the propagating mode in the transmission lines at frequency f as a classical mode of energy hf n f such that the net photon flux is given by the difference between the outgoing and incoming signals at this frequency (S out [f ] − S in [f ])/hf . Precisely, one first needs to probe the difference ∆S i in power spectral density between two sidebands [see Fig. 3(c)] and convert it in photon fluẋ n i = ∆S i /hf i (see Appendix C for a refined expression). Hamiltonian in the rotating frame The fluxonium Hamiltonian reads [56] H fluxonium = 4E CN 2 + E L 2φ 2 − E J cos(φ − ϕ ext ),(22) whereN is the charge on the capacitor of the circuit,φ is the phase twist across the inductance, ϕ ext is the exter- is addressed by a microwave drive applied on a capacitance, so that each pump induces a term proportional to cos(φ i ) cos(θ i )N in the Hamiltonian, where θ i (t) = 2πf i t + θ 0 i is the phase of each tone and φ i (t) = ω i t is the phase of their amplitude modulation. We denote as \|0 , \|1 , \|2 the first three energy levels of the fluxonium (22). The charge operatorN is off-diagonal in this basis (\|0 , \|1 , \|2 ). Hence the effect of the drives is purely off-diagonal. The frequencies of the three drives f 1 , f 2 and f 3 are constrained to satisfy f 3 = f 1 + f 2 so that each tone drives a single transition of the fluxonium. This constraint can be enforced in the microwave domain by using mixers to generate a tone at f 3 using tones at f 1 and f 2 or by direct numerical synthesis. We move to a rotating frame by applying the unitary diagonal transformation U (t) = diag(1, exp(−2iπf 1 t), exp(−2iπf 3 t)). One can choose the initial phases θ 0 i of the tones so that in the rotating frame and using the rotating wave approximation, the dynamics of the qutrit is governed by the Hamiltoniañ H(φ 1 , φ 2 , φ 3 ) =   0 Ω 1 cos(φ 1 ) −iΩ 3 cos(φ 3 ) Ω 1 cos(φ 1 ) δ 1 Ω 2 cos(φ 2 ) iΩ 3 cos(φ 3 ) Ω 2 cos(φ 2 ) δ 3  (23) which depends on the phases φ i (t) = ω i t of the drive amplitude modulations, where δ 1 = 2πf 01 − 2πf 1 and δ 3 = 2πf 02 − 2πf 3 are the frequency detuning between the fluxonium transition frequencies and the drive fre-quencies (see Appendix B for details). Note that the above constraint on drive frequencies sets δ 2 = δ 3 − δ 1 . Besides, in order to simplify the dynamics, we impose an additional constraint on the modulation frequencies, namely φ 3 = φ 1 − φ 2 . Therefore the effective Hamiltonian can be described by an Hamiltonian evolution controlled by two phases only H(φ 1 , φ 2 ) ≡H(φ 1 , φ 2 , φ 1 − φ 2 ). The rotating wave approximation is valid if the detunings δ i and the drive amplitudes Ω i are much lower than the fluxonium transitions frequencies f ij and the difference between any two transition frequencies, which is one of the requirements detailed above. Chern insulator on the Lieb lattice In this section, we show that the fluxonium qutrit emulates in time the physics in momentum space of a Chern insulator on a Lieb lattice, which is illustrated in Fig. 4(a). This proposal thus provides a missing implementation of a new three level topological model, which in particular supersedes a recent proposal based on molecular enantiomers [57]. In this quantum simulation correspondence, the drive detunings δ 1 and δ 3 correspond to onsite potential energies on the Lieb lattice, while the drive amplitudes Ω 1 and Ω 2 mimic nearest-neighbour tight-binding amplitudes, and Ω 3 simulates a next-nearest-neighbour coupling along one diagonal direction. The relative π phase between the Ω 1 , Ω 2 and Ω 3 terms in Eq. (23) originates from a periodic pattern of staggered magnetic fluxes, as A trivial insulator can be adiabatically connected to an atomic limit, which is necessarily characterized here by positive parity products πν = +1. A band insulator exhibiting negative parity products πν = −1 is therefore topological. Enumerating all the configurations of the parity eigenvalues allowed by the model parameters leads to the topological phase diagram in panel (c) (see Appendix D). (c) Phase diagram of the qutrit model. Each colored region corresponds to a set of band Chern numbers. On the boundaries the gap closes for at least one combination of phases φ1 and φ2. shown in Fig. 4(a). These fluxes break the time-reversal symmetry of the tight-binding model, while preserving the translation invariance of the Bravais lattice. We thus end up with a generalization of the celebrated Haldane's model [58] but on the Lieb instead of the honeycomb lattice. In the sublattice basis (0, 1, 2) [see Fig. 4 δ 3      . The pseudo-momentum k is dimensionless, corresponding to a lattice constant chosen as a length unit. The qutrit Hamiltonian (23) is then recovered through the substitutions k x → 2φ 1 and k y → 2φ 2 . Note that our model also captures the dynamics of other quantum systems such as spin chains [59]. We now aim at determining the ranges of drive parameters Ω i , δ i that lead to topologically nontrivial band structures for the model of Eq. (24). By definition, a topological band structures cannot be smoothly deformed to that of an atomic limit [60]. In contrast, a trivial band structure admits some band representations of the crystal space group on a basis of symmetric localized orbitals [61][62][63][64]. An efficient strategy to detect topological band structures then consists of enumerating all possible band representations of a space group and identifying band structures that do not support such representations. This strategy lies at the heart of the recent paradigm of Topological Quantum Chemistry and led to the predictions of exhaustive catalogues of topological materials [65][66][67][68][69]. We can use this methodology to efficiently determine the phase diagram of model (24). We first determine the band representations of the Lieb lattice in Fig. 4(a) for the atomic limit Ω 1,2,3 = 0. For non-degenerate onsite energies δ 1,3 = 0 and δ 1 = δ 3 , and in the presence of staggered magnetic fluxes, the lattice only has inversion symmetry and belongs to the wallpaper space group p2. The three orbitals occupy the maximal Wyckoff positions q 0 = (1/2, 0), q 1 = (0, 0), and q 2 = (0, 1/2) in the primitive unit cell. Their elementary band representations are determined from the band parities p ν = ±1 -i.e. eigenvalues of the parity operatorat the inversion-invariant momenta in the Brillouin zone depicted in Fig. 4(a) [70]. It leads to the band representations summarised in the top panel in Fig. 4(b). The parity product π ν = p ν (Γ)p ν (X)p ν (Y )p ν (M )(25) of each band ν is always positive in the atomic limit. Therefore, band structures with negative parity products fall outside these band representations and are topological. Away from the atomic limit, i.e. for Ω 1,2,3 = 0, we enumerate all the possible parity configurations of the band structure H(k) at the inversion-invariant momenta (see Appendix D and Fig. 8). Each configuration leads to a colored region in the δ 1 δ 3 -plane in Fig. 4(c). The ivory colored regions correspond to the parity configurations of the atomic limit in Fig. 4(b). Thus, they describe trivial band insulators. In contrast, we find that the red, green, and yellow regions exhibit negative parity products, thus characterizing topological insulators. As an illustration, the bottom panel in Fig. 4(b) specifies the parity configuration of the yellow region, where π ν = −1 for ν = 0 and ν = 2. It shows that the change of parity products between the trivial atomic limit and the topological insulator requires the bands ν = 1 and ν = 0 (ν = 2) to switch parities at point X (Y ) of the Brillouin zone. More generally, a parity switch cannot occur continuously and requires the band gap to close at one of the inversion-invariant momentum (see Appendix D). This provides a very efficient determination of the phase diagram of Fig. 4(c) by considering the gap closing at these inversion-invariant momenta as a function of the Ω i and δ i . Such band crossings mark the borders between topologically distinct regions in the phase diagram represented in Fig. 4(c), where from now on we use the shorthand notation for the Chern number C (ν) = C (ν) 12 . One can further show that this Chern number is non zero when the parity product is negative [71,72]. Since the qutrit Hamiltonian in Eq. (23) is recovered via the substitutions k x,y → 2φ 1,2 , the Chern number for the fluxonium qutrit is four times larger than for the Lieb insulator, hence the values summarized in the table in Fig. 4(c). Topological power transfer between three modes We now discuss how the topological nature of the qutrit pump manifests itself in filling rate of the three modes. The dynamical system consist of three classical modes described by the classical phases φ i conjugated to n i coupled to the qutrit through the Hamiltonian (23) in the rotating frame. The equations of dynamics have the same form in the rotating frameṅ i = − 1 Ψ\| ∂ φiH \|Ψ , i = 1, 2, 3 where the dynamics of the qutrit state \|Ψ(t) in the rotating frame is governed by the Hamiltonian (23), see Appendix E for details. As said above, the frequencies of amplitude modulation satisfies ω 3 = ω 1 − ω 2 such that φ III = φ 1 − φ 2 − φ 3 is a constant of motion, so we can keep φ 1 − φ 2 − φ 3 = 0 at all time. We consider the following canonical transformation n I = n 1 + n 3 , n II = n 2 − n 3 , φ I = φ 1 , φ II = φ 2 . The dynamics of n I and n II is given bẏ n 1 +ṅ 3 = − 1 ∂E ν ∂φ 1 + F (ν) φ1φ2 ω 2 (26a) n 2 −ṅ 3 = − 1 ∂E ν ∂φ 2 − F (ν) φ1φ2 ω 1 (26b) with E ν the energy and F (ν) the Berry curvature of the band ν of the Hamiltonian H(φ 1 , φ 2 ) ≡H(φ 1 , φ 2 , φ 1 − φ 2 ) in which the qutrit is initially prepared (see Appendix E). Then, the topological power transfer between the modes 1, 2, 3 is ω 1 ṅ 1 +ṅ 3 t = C (ν) 2π ω 1 ω 2 = − ω 2 ṅ 2 −ṅ 3 t ,(27) with C (ν) the Chern number of the band ν of the Hamiltonian. From a more experimental point of view, the power transfer to demonstrate is ∆S 1 hf 1 + ∆S 3 hf 3 = C (ν) 2π ω 2 and ∆S 2 hf 2 − ∆S 3 hf 3 = − C (ν) 2π ω 1 . (28) In order to get a sense of how feasible this measurement is, let us set some possible figures for the experiment that fulfill the criterion discussed earlier. The fluxonium frequencies could be set to f 01 = 4 GHz, f 12 = 6 GHz and f 02 = 10 GHz. It is then possible to drive the transitions with Ω 1,2,3 /2π = 100 MHz and a similar range of variation for the detunings. The modulation frequencies could then be ω 1 /2π = 5 MHz, ω 2 /2π 3 MHz. From the simulations below, we see that the topological power transfer can be resolved in about 30 periods 2π/ω i , which is a few µs. This is well below the typical decoherence times of fluxonium qubits, which will thus not limit the dynamics of the system during the measurement. Verifying Eq. (28) thus requires to measure instantaneous powers in the range of hf i ω i . This corresponds to a power of several dozens of aW, which is a level of precision that is now routinely reached experimentally [73][74][75]. C. Numerical analysis of topological pumping Topological power transfer In order to perform numerical simulations of the proposed experiment, we solve the time-dependent Schrödinger equation of the qutrit under the rotating wave approximation (23). We first determine the optimal parameters for the pump: the stability of the adiabaticity evolution requires the largest gap. This is reached at the resonant drive δ 1 = δ 3 = 0, point A in the phase diagram in Fig. 6(a), and with equal driving amplitudes Ω 1,2,3 = Ω, resulting in a gap 0.87 Ω. In these conditions, from the analysis of Fig. 4(c) both bands 0 and 2 have non-zero Chern number C = ±4, whereas the band 1 is topologically trivial with C = 0. Therefore the two lowest energy states do not constitute an effective topological qubit, leading to different dynamics than for a conventional 2 level pump as we will see below. To ensure an ergodic exploration of the classical configuration space of the pump, we choose the ratio between phases frequencies ω 1 /ω 2 = (1 + √ 5)/2. Keeping parameters of the pump to point A in the phase diagram in Fig. 6(a), we initialize the qutrit at t = 0 in its ground state. The evolution of the filling n i of each individual modes is represented in Fig. 5(a), and varies linearly in time. However the filling rateṅ i are not set by the topological nature of the pump, and depend on the precise values of the pump parameters: changing slightly these from point A to point B in the same topological region of Fig. 6(a) leads to different filling rates, as seen in Fig. 5(a). On the other hand, the topological power transfer defined in Eq. (27) is insensitive of the precise values of the pump parameters, as shown in Fig. 5(b). Besides the linear topological evolution in time, this power transfer displays temporal fluctuations which have two different origins as deduced from Eq. (26): a dominant spectral term, corresponding to variation of the energy E 0 of the qutrit, and a geometrical contribution originating from fluctuations of the Berry curvature around its topologically quantized average value (see Appendix F). Thus the order of magnitude of the correlation timescale of these fluctuations correspond to the period of the drive. A reasonable requirement to detect the topological power transfer is to average it over 30 such independent fluctuations, leading to a measurement time of 8 µs, as announced in the previous section. Numerical detection of topological transitions Having established that the average topological power transfer gives access to the Chern number of the band in which the qutrit was initialized, we now address the detection of the topological phase transitions of Fig. 4(c) when the detuning parameters δ 1 , δ 2 are varied. The richness of the adiabatic dynamics of the qutrit pumps require different experimental protocols adjusted to each phase transition. For example, sets A and C of parameters in Fig. 6(a) lead to exactly the same topological power rate for a qutrit initialized in the ground state, as shown in To detect all transitions, we monitor the evolution of the pumps with both initialization in the 0 and 1 states. Figure 6 displays the resulting power rate, determined by a linear fit using Eq. (27), as well as the average populationsp µ = 1 ∆t´∆ t 0 \| ψ µ (t)\|Ψ(t) \| 2 dt of the qutrit state \|Ψ(t) on the three instantaneous eigenstates \|ψ µ (t) . Figures 6(b,e,h) correspond to a pump with qutrit initialized in band 0, while Fig. 6(c,f,i) correspond to an initial preparation in band 1. Fig. 6(a,d), the topological transitions occur between two bands only, whereas along the line FG in Fig. 6(g) the gaps between all three bands close at the transitions. Along any line, the transition is detected by the evolution of the populations. Moreover, a quantized power transfer in a given state appears as a direct test of the adiabatic nature of the evolution, related to the distance to the transitions. In that respect, optimal choice of parameters for the pump correspond to point A in the yellow topological phase of the phase diagram, in which the bands 0 and 2 are non-trivial and spectrally separated by a trivial band 1 and thus generically separated from a trivial phase by two transitions. Far from any transition near point A, the instantaneous energy separation with different eigenstates is large, resulting in an adiabatic evolution: the average population of the qutrit in the initial band remains close to 1 and the power rate is quantized and set by the Chern number of the band. Along the lines DE and HI in In the other topological states of the qutrit, the effects of non-adiabaticity are manifest, resulting from shorter distances to phase transitions and thus small gaps. For example along line DE for a qutrit prepared in band 1, while the Chern number C (1) takes values −4 and +4 for respectively δ 1 /Ω < −1 and δ 1 /Ω > 1, the qutrit does not evolve adiabatically and the dynamics of the classical variables (26) must be corrected, leading to an unquantized power transfer shown in Fig. 6(c). Similarly in Fig. 6(b) the Chern number +4 of band 0 for δ 1 /Ω < 1 manifests itself as a pleateau of power rate for a reduced set of parameters for −1.5 < δ 1 /Ω < 0.5 (between points C and B). Non-adiabatic pumping The importance of the non-adiabatic effects can be anticipated from the estimation of the time of validity of the adiabatic approximation τ adiab introduced in Eq. (7). In this expression, the maximum of the adiabatic parameter µν is taken on all the values of phases φ 1 and φ 2 , and the mean free time τ mft is the order of magnitude of the phases periodicity, we take τ mft = 2π/ω 2 ∼ 0.3 µs. For the point A, B and C of the phase diagram, the Landau-Zener collision time τ col 01 associated to the transition between states 0 and 1 is one order of magnitude below this mean free time, which is consistent with the derivation of the adiabatic time τ adiab discussed in Sec. II A 2. For the point A of maximal stability with a preparation in band 0, we get τ A adiab ∼ 150 ms so the non-adiabatic effects will not be limiting for experiments with these parameters values. This is illustrated in Fig. 7(a), where the evolution for 100 µs of the energy ω 1 (n 1 +n 3 ) and the populations p µ (t) = \| ψ µ (t)\|Ψ(t) \| 2 of the qutrit state \|Ψ(t) on the three instantaneous eigenstates \|ψ µ (t) are displayed. The qutrit stays in the ground state with p 0 (t) > 0.997, and the energy is transferred at the topologically quantized rate. Figure 7(b) corresponds to point B closer to the topological transition line towards the phase where C (0) = 0. The estimated time of adiabaticity for band 0 is τ B adiab ∼ 6 µs. After this typical time, the population on state 1 exceeds 0.1, in agreement with the definition of τ adiab , and the pumping rate deviate from its topologically quantized value. Figure 7(c) corresponds to point C, where we crossed a different topological transition line, where the ground state remains topological but the first excited state switches from trivial to non-trivial with Chern number C (1) = −4. We compute here τ C adiab ∼ 8 µs in agreement with the observed deviations from the adiabatic evolution. At longer times, about 70 µs, the first excited state is mostly populated and the energy pumping is reversed, manifesting the associated change of Chern number. V. DISCUSSION We have proposed an experiment that is able to observe a topologically protected power exchange. The topological properties appear in the measured incoming and outgoing energy flows that drive a quantum system. Our general description of topological quantum pumps includes the classical degrees of freedom that carry these flows. Considering a qutrit instead of a qubit is key for two aspects. First, it solves the tremendous challenge to probe the power flows that drive a two level topological pump. Second, owing to its richer dynamics, which simulates the topological band properties of a 3-band Chern model, it gives access to various protocols of pumping which can be freely chosen by setting the initial state of the qutrit. Besides the fascinating perspective to actually measure this topological power transfer, such a system also opens the paths to the study of the interplay between decoherence and the topological adiabatic evolution of the quantum system. Our framework raises two natural questions. Topological pumping requires an initial correlation between the qutrit and the classical modes. Then how long does the pumping survive in presence of qutrit decoherence? Besides, could we revert the perspective and use the developed framework and measurable pumping rate as a tool to characterize the correlations between the quantum system and the classical modes. Finally, let us stress that while we have proposed an experiment demonstrating the topologically protected transfer of microwave power using a superconducting circuit, our general framework can be applied to any quantum system and its driving environment such as cold atoms, mechanical oscillators or polaritons. Symmetries are essential to classify topological matter and in particular topological pumps [31]. The enforcement of these symmetries to protected topological pumping in these different systems is a stimulating perspective. [2] C. L. Kane, Topological band theory and the Z2 invariant, in Contemporary Concepts of Condensed Matter Science, The first-order correction in the adiabatic limit results iñ a ν (τ ) −ã ν (0) = − i η∆ µ =ν ψ ν (τ )\|∂ τ ψ µ (τ ) E µ (τ ) − E ν (τ )ã µ (τ )e iΓµν (τ ) e 1 η´τ 0 dτ ∆µν (τ ) + i η∆ µ =ν ψ ν (0)\|∂ τ ψ µ (0) E µ (0) − E ν (0)ã µ (0) + O(η 2 ). (A9) If the quantum system is initially prepared in the instantaneous eigenstate \|ψ ν -i.e. a ν (0) = 1 and a µ =ν (0) = 0 -we find the time dependent state \|Ψ(t) = e iΓν (t)+i∆ν (t)   \|ψ ν (t) − i µ =ν ψ µ (t)\| d dt \|ψ ν (t) E ν (t) − E µ (t) \|ψ µ (t)   + i µ =ν e iΓµ(t)+i∆µ(t) ψ µ (t)\| d dt \|ψ ν (t) E ν (t) − E µ (t) t=0 \|ψ µ (t) + O(η 2 ) (A10) where the components on \|ψ µ =ν are first order terms in η. Using the identity ψ µ (t)\| d dt \|ψ ν (t) = ψ µ \| dH dt \|ψ ν /(E ν − E µ ), the correction at first order in η to the equation of motion of the variable p α is − Ψ\| ∂H ∂q α \|Ψ = − ψ ν \| ∂H ∂q α \|ψ ν + i µ =ν ψ ν \| ∂H ∂qα \|ψ µ ψ µ \| dH dt \|ψ ν (E ν − E µ ) 2 − i µ =ν ψ ν \| dH dt \|ψ µ ψ µ \| ∂H ∂qα \|ψ ν (E ν − E µ ) 2 (A11) where the last term of (A10) induces only first order terms rapidly oscillating at the Bohr frequencies (E ν − E µ )/ which has no effect on the dynamics at the time-scale of the slow variables. We note that this terms are due to the choice of initial condition and could be eliminated if we choose a ν (0) = 1 and a µ =ν (0) = −i ψ µ (t)\| d dt \|ψ ν (t) /(E ν (t) − E µ (t)) t=0 . For the implementation with a superconducting qutrit, a qutrit is fully controllable and any quantum state can be prepared [78][79][80]. Using this initial condition, we recover the adiabatic parameter µν introduce in Sec. II A 2 in the population on an excited state \| ψ µ (t)\|Ψ(t) \| 2 = µν (t) 2 . Returning to Eq. (A11), the time dependence of the Hamiltonian H(t) = H({q β (t)}) is due to the coupling to the classical variables q β , whose classical dynamics is not modified by the coupling to the quantum system,q β = q (0) β = ∂H β ∂p β . Thus, using the relation ψ ν \| ∂H ∂qα \|ψ ν = ∂Eν ∂qα and the expression of the components of the Berry curvature two form given in (5), we obtain for the correction ofṗ α − Ψ\| ∂H ∂q α \|Ψ = − ∂E ν ∂q α + β =αq (0) β F qαq β . (A12) where f 01 and f 02 are the transition frequencies of H flux , and Ω i,ab = g i a\|N \|b . We change of reference frame with the unitary transformation U (t) = diag(1, exp(−i2πf 1 t), exp(−i2πf 3 t)). The frequencies of the three pumps satisfy the constraint f 3 = f 1 + f 2 , so the Hamiltonian in the rotating frame is H rot = U † H labo U − i U † dU dt =         0 e −i2πf1t 3 i=1 Ω i,01 cos(φ i ) cos(θ i ) e −i2πf3t 3 i=1 Ω i,02 cos(φ i ) cos(θ i ) c.c. δ 1 e −i2πf2t 3 i=1 Ω i,12 cos(φ i ) cos(θ i ) c.c. c.c. δ 3         (B3) with the detuning δ i given by δ 1 = 2πf 01 − 2πf 1 (B4) δ 2 = 2πf 12 − 2πf 2 (B5) δ 3 = 2πf 02 − 2πf 3 = δ 1 + δ 2 . (B6) The phases of the drives are given by θ i (t) = 2πf i t + θ 0 i . In the rotating wave approximation, we ignore the terms of the Hamiltonian in the rotating frame which oscillate at the frequency of the drives, so we approximate the Hamiltonian by where the other terms oscillates at the frequency f i ± f j . We recover the Hamiltonian (23) by noting the drive amplitudes Ω 1 = 1 2 \|Ω 1,01 \| = 1 2 g 1 \| 0\|N \|1 \|, Ω 2 = 1 2 \|Ω 2,12 \| = 1 2 g 2 \| 1\|N \|2 \| and Ω 3 = 1 2 \|Ω 3,02 \| = 1 2 g 3 \| 0\|N \|2 \|, and by choosing the initial pump phases θ 0 i to set the complex phase of the couplings at the desired value. The rotating wave approximation is valid if the drive amplitudes and detunings are much lower than the frequencies f i ± f j of the oscillating terms, which means that they must be much lower than the difference between any two transition frequencies of the fluxonium as said in the Sec. IV B 1. The inversion eigenvalues labeling the bands at Γ, X, Y , and M are shown in the green boxed insets, as a parity triplet p = (p0, p1, p2) associated with the band energies E0 < E1 < E2. It is only shown for insulating regions that exhibit negative band parity products. In such regions, the band structures do not support any band representation and cannot be adiabatically connected to an atomic limit. The fluxonium qutrit then simulates nontrivial Chern insulators. H rot (t) H(φ 1 , φ 2 , φ 3 ) =   0 1 2 Ω 1,01 cos(φ 1 )e iθ where the dynamics of state \|Ψ labo (t) of the qutrit in the laboratory frame is governed by the Hamiltonian H labo ({φ j , θ j }), introduce before (B2). In the rotating frame, the dynamics of \|Ψ rot (t) = U † (t) \|Ψ labo (t) is governed by H rot = U † H labo U − i U † dU dt which satisfies ∂Hrot ∂φi = U † ∂H labo ∂φi U since the unitary transformation U (t) does not depend on the phase φ i . Thus, the equation of the dynamics of n i has the same form in the rotating framė n i = − 1 Ψ rot (t)\| ∂H rot ∂φ i \|Ψ rot (t) (E2) where in the rotating wave approximation, we consider H rot ({φ j , θ j }, t) H (φ 1 , φ 2 , φ 3 ), with the 3 phases Hamiltonian given by (23). We consider the following canonical change of variable n I = n 1 + n 3 ; φ I = φ 1 (E3a) n II = n 2 − n 3 ; φ II = φ 2 (E3b) n III = −n 3 ; φ III = φ 1 − φ 2 − φ 3(E3c) satisfying {φ A , n B } = 1/ , A, B = I, II, III. Since these new variables are conjugated, the equations of motion arė n A = − 1 ∂E ν ∂φ A + B =Aφ B F (ν) φ A φ B (E4) with E ν the energy and F (ν) the Berry curvature of the band ν of H(φ I , φ II , φ III ) =H(φ I , φ II , φ I − φ II − φ III ). The frequencies of the phases φ 1 , φ 2 , φ 3 are chosen such that ω 3 = ω 1 − ω 2 , thusφ III = 0 and we keep φ III = 0 at all time with the initial condition. Thus, the equations of motion reduce to (26). φ 1 φ 2 − C (ν) 2π ). In green: Topological energy transfer at constant rate ω 1 ω 2 2π C (ν) . The fluctuation of the energy of the qutrit is the predominant source of temporal fluctuation of the energy. Appendix F: Temporal fluctuations During the adiabatic evolution, the time derivative of the energy of a mode can be decomposed in a sum of three terms ω 1 (ṅ 1 +ṅ 3 ) = −ω 1 ∂E ν ∂φ 1 + ω 1 ω 2 (F (ν) φ1φ2 − C (ν) 2π )+ ω 1 ω 2 2π C (ν) . (F1) The first term is the variation of the energy E ν of the band ν of the qutrit, corresponding to an energy exchange between the qutrit and the mode. The second term correspond to the fluctuation of the Berry curvature F (ν) φ1φ2 around its topologically quantized average value C (ν) 2π , with C (ν) the Chern number. This corresponds to the fluctuation of the geometrical transfer of energy between the two modes. The last term is the topological power rate, the only non-zero term in time-average. The two first terms are responsible for the time fluctuation of the energy of the mode. In Fig. 9 is represented the time-integration of each term in the case of resonance δ 1 = δ 3 = 0, point A in Fig. 6(a). We see that the temporal fluctuation of the energy is mainly due to the energy exchange between the qutrit and the mode, and the fluctuation of the Berry curvature is much lower. This is the case for every value of parameters in the region of interest of the phase diagram. FIG . 2. (a) Schematic quantum Hall circuit where two LC branches, described by classical conjugate variables Φ and Q, are connected through a quantum Hall sample. (b) Schematic Thouless pump driven by a phase φ1 = ω1t conjugate to a variable n1, and coupled to an LC branch. Topological pumping gives rise to a current in the LC circuit. (c) Topological mode coupler, or frequency converter, in which two classical modes described by φ1, n1 and φ2, n2 variables are coupled topologically through a quantum system. FIG. 3 . 3nal magnetic flux threading the loop, and E C , E L , E J are respectively the charging energy, inductive energy and Josephson energy of the circuit. The fluxonium (a) Schematic of the experimental setup. A fluxonium circuit is embedded in a host cavity. The transitions between the first three levels of the fluxonium are driven with a detuning δi, an amplitude Ωi, and a modulation frequency ωi (blue, red and green). The power of the outgoing signals are recorded with a power spectrum analyzer that provides the instantaneous photon flux of each frequency mode. (b) Spectrum of the driving tones. Each fluxonium transition is driven with two side-bands used to implement a topologically protected power transfer. (c) For decoherence rates smaller than the modulation frequencies ωi, the power of each sideband can be resolved. The quantized power transferred is expressed as a function of the difference of the spectral power ∆Si in the sidebands of each reflected driving tone according to Eq.(28). FIG. 4 . 4(a) Schematic representations of a Haldane model on the Lieb lattice and its square Brillouin zone with the four inversion-invariant momenta. The sublattices 0, 1, and 2 respectively have on-site potential energies 0, δ1, and δ3. The regular and bold black lines represent the nearest-neighbor tight-binding amplitudes Ω1 and Ω2. The dashed black lines depict the next-nearest-neighbor coupling Ω3. (b) Two allowed configurations of the band parities pν at the inversion-invariant momenta. FIG. 5 . 5Pumping by a qutrit initialized in its ground state. (a) Filling of the different modes ni as a function of time, for three different set of pump parameters corresponding to points A, B and C in Fig. 6(a). The driving amplitudes are chosen all equals Ω1,2,3 = Ω. The filling rate are found to depend on the parameters of the pump. (b) The two different combinations of fillings display the topological power ω1ω2C (0) /2π (grey dashed lines), invariant on the region of stability of the phase diagram, where C (0) = 4 is the Chern number of the ground state for all parameter sets A,B,C. FIG. 6 . 6Topological transitions for bands 0 and 1 in the case of same drive amplitudes Ω1,2,3 = Ω. For each working point of the phase diagram, we fit the time evolution of the energy E1 = ω1(n1 + n3) by a line on a time ∆t = 8µs to construct the reduced power rate 2πĖ1/ ω1ω2 (red dots) when the qutrit is initialized in band 0, panels [(b),(e),(h)], or in band 1, panels [(c),(f),(i)]. The average populations of the qutritpµ = 1 ∆t´∆ t 0 \| ψµ(t)\|Ψ(t) \| 2 dt are displayed to evaluate adiabaticity. [(a)-(c)] Transition line DE, with a transition between bands 1 and 2 at δ1/Ω = −1, and between bands 0 and 1 at δ1/Ω = 1. [(d)-(f)] Transition line HI, with a transition between bands 0 and 1 at δ1/Ω = −1, and between bands 1 and 2 at δ1/Ω = 1. [(g)-(i)] Transition line FG, with a transition between the three bands at δ1/Ω = ± √ 2/2. Fig. 5 ( 5b), but corresponds to a different topological qutrit phase. Indeed, the corresponding qutrit phases differ by the topological nature of the excited bands 1 and 2, while the nature of the ground state is unchanged. Thus detecting this particular transition requires an initialization of the qutrit in the first excited state 1. ACKNOWLEDGMENTS This research was supported by IDEX Lyon project ToRe (Contract No. ANR-16-IDEX-0005). C.D. also acknowledges the supports of Idex Bordeaux (Maesim Risky project of the LAPHIA Programme) and Quantum Matter Bordeaux. We are grateful to Anton Akhmerov, Antoine Essig, Leonid Glazman, Sébastien Jezouin and Christophe Mora for insightful discussions. [ 1 ] 1R. B. Laughlin, Quantized Hall conductivity in two dimensions, Physical Review B 23, 5632 (1981). FIG. 8 . 8Parity product of the bands: The central map represents the minimum value of the lower gap (∆01) or upper gap (∆12) over the Brillouin zone, as a function of the detunings δ1 and δ3. It is obtained from numerical diagonalization of the Bloch Hamiltonian (24). Energy is in units of Ω1 = Ω2 = Ω3 = Ω. The white areas correspond to values of the detuning for which a band gap closes. They are well described by the dashed (dotted) lines obtained analytically from the closing conditions of gap ∆01 (∆12) at the inversion-invariant momenta Γ, X, Y , and M in the Brillouin zone. These degeneracy lines mark the transitions between topologically nonequivalent band insulators, where the parity product πν of certain bands changes signs. FIG. 9 . 9Different terms in the variation of the energy ω1(n1 + n3), in the case of resonance δ1 = δ3 = 0, point A in Fig. 6(a). In blue: Time-integration of the term of variation of the energy of the qutrit −ω1 ∂Eν ∂φ 1 . In orange: Timeintegration of the term of fluctuation of the geometrical coupling ω1ω2(F (ν) Appendix A: First-order adiabatic evolution For the sake of completeness we detail here the derivation of the first-order adiabatic evolution usually derived in different manner in the literature, see for example[4,33,49,76,77]. The quantum Hamiltonian H(t) depends continuously on time through the set of variables {q α (t)}. The time-dependent Schrödinger equation isWe look for a quantum state of the formwhere \|ψ ν (t) are the instantaneous eigenstates of H(t) and satisfyThis leads to the equation of motioṅWe can get rid of the first term in the right-hand side by introducing the phasesand making the substitution a ν (t) = e i∆ν (t)+iΓν (t)ã ν (t). The time integration of Eq. (A4) then results iñ)/( ∆), and ∆ = min µ,t \|E µ (t) − E ν (t)\|/ is assumed to be non zero. Here, we have also introduced a dimensionless "time" τ = η∆t, where η is a small parameter. We can evaluate the time integral in Eq. (A7) by integrating by part on the exponential termη´τ 0 dτ ∆µν (τ ) .(A8)Appendix B: Details on the derivation of the HamiltonianWe detail here the derivation of the Hamiltonian of the qutrit in the rotating frame explained in Sec. IV B 2. The circuit is driven by three drives whose amplitudes are modulated in time according to the drive Hamiltonianwith θ i (t) = 2πf i t + θ 0 i the phase of the the electromagnetic field of frequency f i , φ i (t) = ω i t the phase of the time modulation of the amplitude, and g i the coupling rates. In the basis (\|0 , \|1 , \|2 ) of the three eigenstates of H flux (22) of lowest energy, the diagonal elements ofN are null so the Hamiltonian in the laboratory frame H labo = H flux + H drive has the form:Appendix C: Classical variables coupled to the qutritThe fluxonium is coupled to three drives whose amplitudes are modulated in time. The coupling Hamiltonian in the laboratory frame iswhere θ i (t) = 2πf i t is the phase of the drive and φ i (t) = ω i t is the phase of time-modulation of the amplitude of the drive. The term in parenthesis is proportional to the amplitude of the propagating wave on the lineThus, the transmission line contains six modes at frequencies f ± i = f i ± ω i /2π, for i = 1, 2, 3 (seeFig. 3). As explained in Sec. IV B 1, we model the propagating mode at frequency f ± i as a classical mode of energy hf n ± i , where the phase θ ± i of each mode is conjugated to n ± i , such that the net photon flux is given by the difference between the outgoing and incoming signals at this frequency hf ± iṅ Appendix D: Chern insulator on the Lieb latticeThe three-band insulator on the Lieb lattice satisfies inversion symmetry. We write the inversion operator as P (k)=diag(e ikx ,1,e iky ). It leads to four inversioninvariant momenta in the 2D Brillouin zone: Γ = (0, 0), X = (π, 0), Y = (0, π), and M = (π, π). At these highsymmetry points, the Bloch Hamiltonian commutes with the inversion operator. Thus, the parities p ν -eigenvalues of the inversion operator -are good quantum numbers to label each energy band ν at the inversion-invariant momenta. We sort the energy bands as E 0 < E 1 < E 2 and introduce the triplet π = (π 0 , π 1 , π 2 ), where π ν is the band parity productWe now determine the different configurations of parity products allowed for the three bands emulated by the fluxonium.At momentum Γ, the parity operator reduces to the identity matrix. All energy bands have the same parity, regardless of the Hamiltonian parameters. This leads to the parity triplet p(Γ) = (p 0 (Γ), p 1 (Γ), p 2 (Γ)) = (+, +, +).At momentum X, the parity operator and the Bloch Hamiltonian read P (X) = diag(−1, 1, 1) andThis shows that any change of band parity requires the band gap to close at the inversion-invariant momenta. All the band-parity configurations determined above, as well as their parity products, are summarized in the insets ofFig. 8as a function of δ 1 and δ 3 .Appendix E: Dynamics in the rotating frameThe equations of dynamics of the classical variables n i conjugated to the phases φ i are first written in the laboratory framė t) in the instantaneous eigenstates are displayed for three working point A, B and C in the phase diagram of Fig. 6(a). (a) Point A, case of resonance δ1 = δ3 = 0 where the evolution of the qutrit remains adiabatic during 200 periods so the pumping rate is stable. (b) Point B, limit case beyond which pumping is no longer quantized on Fig. 6(b), where the evolution of the qutrit is no longer adiabatic after approximately 8 µs so the pumping rate is no longer quantized. (c) Point C, other limit case beyond which pumping is no longer quantized on Fig. 6(b. FIG. 7. Non-adiabatic effect at longer time. The time evolution for the qutrit prepared in band 0 of the topological energy combination ω1(n1 + n3) and qutrit populations pµ. where in this phase the Chern number of band 1 is opposite to band 0 so we see pumping in the other direction when band 1 is mainly populated. Downsampling of data has been applied for clarity of presentationFIG. 7. Non-adiabatic effect at longer time. The time evolution for the qutrit prepared in band 0 of the topological energy combination ω1(n1 + n3) and qutrit populations pµ(t) in the instantaneous eigenstates are displayed for three working point A, B and C in the phase diagram of Fig. 6(a). (a) Point A, case of resonance δ1 = δ3 = 0 where the evolution of the qutrit remains adiabatic during 200 periods so the pumping rate is stable. (b) Point B, limit case beyond which pumping is no longer quantized on Fig. 6(b), where the evolution of the qutrit is no longer adiabatic after approximately 8 µs so the pumping rate is no longer quantized. 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[ "Optimal proportional reinsurance and investment for stochastic factor models", "Optimal proportional reinsurance and investment for stochastic factor models" ]	[ "Brachetta M [email protected] ", "C Ceci [email protected] " ]	[]	[]	In this work we investigate the optimal proportional reinsurance-investment strategy of an insurance company which wishes to maximize the expected exponential utility of its terminal wealth in a finite time horizon. Our goal is to extend the classical Cramér-Lundberg model introducing a stochastic factor which affects the intensity of the claims arrival process, described by a Cox process, as well as the insurance and reinsurance premia. Using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman equation we characterize the optimal strategy and provide a verification result for the value function via classical solutions of two backward partial differential equations. Existence and uniqueness of these solutions are discussed. Results under various premium calculation principles are illustrated and a new premium calculation rule is proposed in order to get more realistic strategies and to better fit our stochastic factor model. Finally, numerical simulations are performed to obtain sensitivity analyses.	10.1016/j.insmatheco.2019.03.006	[ "https://arxiv.org/pdf/1806.01223v1.pdf" ]	102,487,176	1806.01223	6dd59f126f5807312a8329aafe8bbf3e7393809b	Optimal proportional reinsurance and investment for stochastic factor models Brachetta M [email protected] C Ceci [email protected] Optimal proportional reinsurance and investment for stochastic factor models Optimal proportional reinsuranceoptimal investmentCox modelstochastic con- trol JEL Classification codes: G220C610G110 MSC Classification codes: 93E2091B3060G5760J75 Declarations of interest: none In this work we investigate the optimal proportional reinsurance-investment strategy of an insurance company which wishes to maximize the expected exponential utility of its terminal wealth in a finite time horizon. Our goal is to extend the classical Cramér-Lundberg model introducing a stochastic factor which affects the intensity of the claims arrival process, described by a Cox process, as well as the insurance and reinsurance premia. Using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman equation we characterize the optimal strategy and provide a verification result for the value function via classical solutions of two backward partial differential equations. Existence and uniqueness of these solutions are discussed. Results under various premium calculation principles are illustrated and a new premium calculation rule is proposed in order to get more realistic strategies and to better fit our stochastic factor model. Finally, numerical simulations are performed to obtain sensitivity analyses. Introduction In this paper we investigate the optimal reinsurance-investment problem of an insurance company which wishes to maximize the expected exponential utility of its terminal wealth in a finite time horizon. In the actuarial literature there is an increasing interest in both optimal reinsurance and optimal investment strategies, because they allow insurance firms to increase financial results and to manage risks. In particular, reinsurance contracts help the reinsured to increase the business capacity, to stabilize operating results, to enter in new markets, and so on. Among the traditional reinsurance arrangements the excess-of-loss and the proportional treaties are of great importance. The former was studied in [Sheng et al., 2014], [Li et al., 2018] and references therein. The latter was intensively studied by many authors under the criterion of maximizing the expected utility of the terminal wealth. Beyond the references contained therein, let us recall some noteworthy papers: in [Liu and Ma, 2009] the authors considered a very general model, also including consumption, focusing on well posedness of the optimization problem and on existence of admissible strategies; in [Liang et al., 2011] a stock price with instantaneous rate of investment return described by an Ornstein-Uhlenbeck process has been considered ; in [Liang and Bayraktar, 2014] the problem has been studied in a partially observable framework by introducing an unobservable Markovmodulated risk process; in [Zhu et al., 2015] the surplus is invested in a defaultable financial market; in [Liang and Yuen, 2016] and [Yuen et al., 2015] multiple dependent classes of insurance business are considered. All these works may be considered as attempts to extend both the insurance risk and the financial market models. In all these articles we can recognize two different approaches to dealing with the surplus process of the insurance company: some authors considered it as a diffusion process approximating the pure-jump term of the Cramér-Lundberg model (see for example [Bai and Guo, 2008, Cao and Wan, 2009, Zhang et al., 2009, Gu et al., 2010, Li et al., 2018 and references therein). This approach is validated by means of the famous Cramér-Lundberg approximation (see [Grandell, 1991]). Other authors (see [Liu and Ma, 2009, Zhu et al., 2015, Liang et al., 2011, Sheng et al., 2014, Yuen et al., 2015 and references therein) took into account the jump term using a compound Poisson risk model with constant intensity, that is the classical Cramér-Lundberg model. On the one hand this is the standard model for nonlife insurance and it is simple enough to perform calculations, on the other it is too simple to be realistic (as noticed by [Hipp, 2004]). As observed by Grandell, J. in [Grandell, 1991], more reasonable risk models should allow the insurance firm to consider the so called size fluctuations as well as the risk fluctuations, which refer respectively to variations of the number of policyholders and to modifications of the underlying risks. This paper aims at extending the classical risk model by modelling the claims arrival process as a doubly stochastic Poisson process with intensity affected by an exogenous stochastic process {Y t } t∈ [0,T ] . This environmental factor lead us to a reasonably realistic description of any risk movement (see [Grandell, 1991], [Schmidli, 2018]). For example, in automobile insurance Y may describe road conditions, weather conditions (foggy days, rainy days, . . . ), traffic volume, and so on. While in [Liang and Bayraktar, 2014] the authors considered a Markov-modulated compound Poisson process with the (unobservable) stochastic factor described by a finite state Markov chain, we consider a stochastic factor model where the exogenous process follows a general diffusion. An additional feature is that the insurance and the reinsurance premia are not evaluated using premium calculation principles, contrary to the majority of the literature; moreover, they turn out to be stochastic processes depending on Y . Furthermore, we highlight that under the most frequently used premium calculation principles (expected value and variance premium principles) some problems arise: firstly, the optimal reinsurance strategy turns out to be deterministic (this is a limiting factor because the main goal of our paper is to consider a stochastic factor model); secondly, the optimal reinsurance strategy does not explicitly depend on the claims intensity. In order to fix these problems, we will introduce a new premium calculation principle, which is called intensity-adjusted variance premium principle. Finally, the financial market is more general than those usually considered in the literature, since it is composed by a risk-free bond and a risky asset with Markovian rate of return and volatility. For instance, in [Bai and Guo, 2008], [Cao and Wan, 2009], [Zhang et al., 2009] and [Liang and Bayraktar, 2014] the authors used a geometric Brownian model, in [Gu et al., 2010] and [Sheng et al., 2014] a CEV model. Nevertheless, some authors considered other general models: in [Irgens and Paulsen, 2004] and [Li et al., 2018] the risky asset follows a jump-diffusion process with constant parameters, in [Liang et al., 2011] the instantaneous rate of investment return follows an Ornstein-Uhlenbeck process, in [Zhu et al., 2015] the authors used the Heston model, in [Xu et al., 2017] the authors introduced a Markov-modulated model for the financial market. However, in these papers the authors considered the classical risk model with constant intensity for the claims arrival process. Using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman equation we characterize the optimal strategy and provide a verification result for the value function via classical solutions of two backward partial differential equations (see Theorem 6.1). Moreover we provide a class of sufficient conditions for existence and uniqueness of classical solutions to the PDEs involved (see Theorems 8.1 and 8.2). Results under various premium calculation principles are discussed, including the intensity-adjusted variance premium principle. Finally, numerical simulations are performed to obtain sensitivity analyses of the optimal strategies. The paper is organized as follows: in Section 2 we formulate the main assumptions and describe the optimization problem; Section 3 contains the derivation of the Hamilton-Jacobi-Bellman equation. In Section 4 we characterize the optimal reinsurance strategy, discussing in Subsections 4.1 and 4.2 how the general results apply to special premium calculation principles (expected value, variance premium and intensity-adjusted variance principles). In Section 5 we provide the optimal investment strategy. Section 6 contains the Verification Theorem. In Section 7 we illustrate some numerical results and sensitivity analyses. In Section 8 existence and uniqueness theorems are discussed for the PDEs involved in the problem. Finally, in Appendix A the reader can find some proofs of secondary results. Problem formulation Assume that (Ω, F, P, {F t }) is a complete probability space endowed with a filtration {F t } t∈[0,T ] , shortly denoted with {F t }, satisfying the usual conditions. We introduce the stochastic factor Y = {Y t } t∈[0,T ] as the solution of the following SDE: dY t = b(t, Y t ) dt + γ(t, Y t ) dW (Y ) t Y 0 ∈ R (2.1) where {W (Y ) t } t∈[0,T ] is a standard Brownian motion on (Ω, F, P, {F t }) . This stochastic factor represents any environmental alteration reflecting on risk fluctuations. For instance, as suggested by Grandell, J. (see [Grandell, 1991], Chapter 2), in automobile insurance Y may describe road conditions, weather conditions (foggy days, rainy days, . . . ), traffic volume, and so on. We suppose that there exists a unique strong solution to (2.1) such that E [∫ T 0 \|b(t, Y t )\| dt + ∫ T 0 γ(t, Y t ) 2 dt ] < ∞ (2.2) sup t∈[0,T ] E[\|Y t \| 2 ] < ∞ (2.3) (for instance, it is true if the coefficients of the SDE (2.1) satisfy the classical Lipschitz and sublinear growth conditions, see [Gihman and Skorohod, 1972]) and denote by L Y its infinitesimal generator: L Y f (t, y) = b(t, y) ∂f ∂y (t, y) + 1 2 γ(t, y) 2 ∂ 2 f ∂y 2 (t, y) f ∈ C 1,2 ((0, T ) × R). Let us introduce a strictly positive measurable function λ(t, y) : [0, T ] × R → (0, +∞) and define the process {λ t . = λ(t, Y t )} t∈[0,T ] for all t ∈ [0, T ]. Under the hypothesis that E [∫ T 0 λ u du ] < ∞ (2.4) we denote by {N t } t∈[0,T ] the claims arrival process, which is a conditional Poisson process having {λ t } t∈[0,T ] as intensity. More precisely, we have that for all 0 ≤ s ≤ t ≤ T and k = 0, 1, . . . P[N t − N s = k \| F Y T ∨ F s ] = (∫ t s λ u du ) k k! e − ∫ t s λu du , where {F Y t } t∈[0,T ] denotes the filtration generated by Y . Then it is easy to show that N t − ∫ t 0 λ s ds is an {F t }-martingale 1 . Now we define the cumulative claims up to time t as follows: C t = Nt ∑ i=1 Z i t ∈ [0, T ],δ (t,∆Ct) (dt, dz) = ∑ n≥1 δ (Tn,Zn) (dt, dz)1 {Tn≤T } , (2.5) where {T n } n=1,... denotes the sequence of jump times of {N t } t∈[0,T ] , then the process {C t } t∈[0,T ] satisfies C t = ∫ t 0 ∫ D 0 zm(ds, dz). (2.6) The following Lemma will be useful in the sequel. Lemma 2.1. The random measure m(dt, dz) given in (2.5) has dual predictable projection ν given by the following: ν(dt, dz) = dF Z (z)λ t dt (2.7) i.e. for every nonnegative, {F t }-predictable and [0, D]-indexed process {H(t, z)} t∈[0,T ] E [∫ T 0 ∫ D 0 H(t, z) m(dt, dz) ] = E [∫ T 0 ∫ D 0 H(t, z) dF Z (z)λ t dt ] . Proof. See Appendix A. Remark 2.1. Let us observe that for any {F t }-predictable and [0, D]-indexed process {H(t, z)} t∈[0,T ] such that E [∫ T 0 ∫ D 0 \|H(t, z)\| dF Z (z)λ t dt ] < ∞ the process M t = ∫ t 0 ∫ D 0 H(s, z) ( m(ds, dz) − dF Z (z)λ s ds ) t ∈ [0, T ] turns out to be an {F t }-martingale. If in addition E [∫ T 0 ∫ D 0 \|H(t, z)\| 2 dF Z (z)λ t dt ] < ∞, then {M t } t∈[0,T ] is a square integrable {F t }-martingale and E[M 2 t ] = E [∫ t 0 ∫ D 0 \|H(t, z)\| 2 dF Z (z)λ t dt ] ∀t ∈ [0, T ]. Moreover, the predictable covariation process of {M t } t∈[0,T ] is given by ⟨M ⟩ t = ∫ T 0 ∫ D 0 \|H(t, z)\| 2 dF Z (z)λ t dt that is {M 2 t − ⟨M ⟩ t } t∈[0,T ] is an {F t }-martingale 2 . 2 For these results and other related topics see e.g. [Bass, 2004]. Remark 2.2. Let {G t } t∈[0,T ] be the filtration defined by G t = F t ∨ F Y T . Then m(dt, dz) defined in (2.5) has {G t }-dual predictable projection ν given in (2.7). In fact, first observe that {λ t } t∈[0,T ] is {F t }-adapted by definition, hence it is {G t }-adapted. Now notice that {λ t } is the {G t }-intensity of {N t } t∈[0,T ] because for any 0 ≤ s ≤ t ≤ T E[N t \| G s ] = N s + E[N t − N s \| G s ] = N s + ∑ k≥1 k (∫ t s λ u du ) k k! e − ∫ t s λu du = N s + ∫ t s λ u du and this implies that In this framework we suppose that the gross risk premium rate is affected by the stochastic factor, i.e. we describe the insurance premium as a stochastic process E[N t − ∫ t 0 λ u du \| G s ] = N s − ∫ s 0 λ u du.{c t . = c(t, Y t )} t∈[0,T ] , where c : [0, T ] × R → (0, +∞) is a nonnegative measurable function such that E [∫ T 0 c(t, Y t ) dt ] < ∞. (2.8) The insurance company can continuously purchase a proportional reinsurance contract, transferring at each time t ∈ [0, T ] a percentage u t of its own risks to the reinsurer, who receives a reinsurance premium q t given by the definition below. in u ∈ [0, 1] and such that 1. q(t, y, 0) = 0 for all (t, y) ∈ [0, T ] × R, because a null protection is not expensive; 2. ∂q(t,y,u) ∂u ≥ 0 for all (t, y, u) ∈ [0, T ] × R × [0, 1], since the premium is increasing with respect to the protection; 3. q(t, y, 1) > c(t, y) for all (t, y) ∈ [0, T ] × R, because the cedant is not allowed to gain a profit without risk. In the rest of the paper ∂q(t,y,0) ∂u and ∂q(t,y,1) ∂u should be intended as right and left derivatives, respectively. Moreover, we assume the following integrability condition: E [∫ T 0 q(t, Y t , u) dt ] < ∞ ∀u ∈ [0, 1]. (2.9) Then the reinsurance premium associated with a reinsurance strategy {u t } t∈[0,T ] (which is the protection level chosen by the insurer) is defined as {q t . = q(t, Y t , u t )} t∈[0,T ] . In addition, we will use the hypothesis that the insurance gross premium and the reinsurance premium will never diverge too much (being approximately influenced by the stochastic factor in the same way), that is there exists a positive constant K such that \|q(t, Y t , u) − c(t, Y t )\| ≤ K P-a.s. ∀t ∈ [0, T ], u ∈ [0, 1]. (2.10) Under these hypotheses the surplus (or reserve) process associated with a given reinsurance strategy {u t } t∈[0,T ] is described by the following SDE: dR u t = [ c(t, Y t ) − q(t, Y t , u t ) ] dt − (1 − u t )dC t = [ c(t, Y t ) − q(t, Y t , u t ) ] dt − ∫ D 0 (1 − u t )z m(dt, dz) R u 0 = R 0 ∈ R + (2.11) Let us observe that by Remark 2.1, since E [∫ T 0 ∫ D 0 u r zλ r dF Z (z) dr ] ≤ E[Z]E [∫ T 0 λ r dr ] < ∞, the process ∫ t 0 ∫ D 0 (1 − u s )z(m(ds, dz) − λ s dF Z (z) ds) turns out to be an {F t }-martingale. Furthermore, we allow the insurer to invest its surplus in a financial market consisting of a risk-free bond {B t } t∈[0,T ] and a risky asset {P t } t∈[0,T ] , whose dynamics on (Ω, F, P, {F t }) are, respectively, dB t = RB t dt B 0 = 1 (2.12) with a fixed R > 0, and T ] and the random measure m(dt, dz) 3 . Let us assume that there exists a unique strong solution to (2.13) such that dP t = P t [ µ(t, P t ) dt + σ(t, P t ) dW (P ) t ] P 0 > 0 (2.13) where {W (P ) t } t∈[0,T ] is a standard Brownian motion independent of {W (Y ) } t∈[0,E [∫ T 0 \|P t µ(t, P t )\| dt + ∫ T 0 P 2 t σ(t, P t ) 2 dt ] < ∞ (2.14) sup t∈[0,T ] E[P 2 t ] < ∞ (2.15) (for example, it is true if the coefficients of the SDE (2.13) satisfy the classical Lipschitz and sub-linear growth conditions, see [Gihman and Skorohod, 1972]). Furthermore, we assume the Novikov condition: E [ e 1 2 ∫ T 0 \| µ(t,P t )−R σ(t,P t ) \| 2 dt ] < ∞,(2.16) which implies the existence of a risk-neutral measure for {P t } t∈[0,T ] and ensures that the financial market does not admit arbitrage. We will denote with w t the total amount invested in the risky asset at time t ∈ [0, T ], so that X t − w t will be the capital invested in the risk-free asset (now X t indicates the total wealth, but it will be defined more accurately below, see equation (2.18)). We also allow the insurer to short-sell and to borrow/lend any infinitesimal amount, so that w t ∈ R. Finally, we only consider self-financing strategies: the insurer company only invests the surplus obtained with the core business, neither subtracting anything from the gains, nor adding something from another business. The insurer's wealth {X α t } t∈[0,T ] associated with a given strategy α t = (u t , w t ) is described by the following SDE: T ] are, respectively, the proportion of reinsured claims and the total amount invested in the risky asset {P t } t∈[0,T ] . dX α t = dR u t + w t dP t P t + ( X α t − w t ) dB t B t = [ c(t, Y t ) − q(t, Y t , u t ) ] dt + w t [ µ(t, P t ) dt + σ(t, P t ) dW (P ) t ] + ( X α t − w t ) R dt − ∫ D 0 (1 − u t )z m(dt, dz) (2.17) with X α 0 = R 0 ∈ R + . Remember that {u t } t∈[0,T ] and {w t } t∈[0, Remark 2.3. It can be verified that the solution of the SDE (2.17) is given by the following: X α t = X α 0 e Rt + ∫ t 0 e R(t−r) [ c(r, Y r ) − q(r, Y r , u r ) ] dr + ∫ t 0 e R(t−r) w r [µ(r, P r ) − R] dr + ∫ t 0 e R(t−r) w r σ(r, P r ) dW (P ) r − ∫ t 0 ∫ D 0 e R(t−r) (1 − u r )z m(dr, dz). (2.18) Now we are ready to formulate the optimization problem of an insurance company which subscribes a proportional reinsurance contract and invests its surplus in a financial market according with a strategy {α t = (u t , w t )} t∈[0,T ] in order to maximize the expected utility of its terminal wealth: sup α∈U E [ U (X α T ) ] where U denotes a suitable class of admissible controls defined below (see Definition 2.2) and U : R → [0, +∞) is the utility function representing the insurer preferences. We focus on CARA (Constant Absolute Risk Aversion) utility functions, whose general expression is given by U (x) = 1 − e −ηx x ∈ R where η > 0 is the risk-aversion parameter. This utility function is highly relevant in economic science and in particular in insurance theory, in fact it is commonly used for reinsurance problems (e.g. see [Bai and Guo, 2008], [Cao and Wan, 2009], [Sheng et al., 2014], and many others). Using the dynamic programming principle we will consider a dynamic problem which consists in finding the optimal strategy α s , for s ∈ [t, T ], for the following optimization problem given the information available at the time t ∈ [0, T ]: sup α∈Ut E [ U (X α t,x (T )) \| F t ] t ∈ [0, T ] where U t denotes the class of admissible controls in the time interval [t, T ] (see Definition 2.2 below). Here {X α t,x (s)} s∈[t,T ] denotes the solution to equation (2.17) with initial condition X α t = x. For the sake of simplicity, we will reduce ourselves studying the function −e −ηx . Another possible choice is to study the corresponding minimizing problem for the function e −ηx , but the first choice is usually preferred in the literature. Definition 2.2. We will denote with U the set of all admissible strategies, which are all the {F t }-predictable processes α t = (u t , w t ), t ∈ [0, T ], with values in [0, 1] × R, such that E [∫ T 0 \|w r \|\|µ(r, P r ) − R\| dr ] < ∞, E [∫ T 0 w 2 r σ(r, P r ) 2 dr ] < ∞. When we want to restrict the controls to the time interval [t, T ], we will use the notation U t . From now on we assume the following assumptions fulfilled. Assumption 2.1. E[e ηZe RT ] < ∞, E[Ze ηZe RT ] < ∞ E[Z 2 e ηZe RT ] < ∞ (2.19) E [ e (E[e ηe RT Z ]−1) ∫ T t λs ds \| F t ] < ∞ ⟨P = 1⟩ ∀t ∈ [0, T ]. (2.20) Proposition 2.1. Under the Assumption 2.1 the control (0, 0) is admissible and such that E[e −ηX (0,0) t,x (T ) \| F t ] < ∞ ⟨P = 1⟩ ∀(t, x) ∈ [0, T ] × R. Proof. See Appendix A. Remark 2.4. Let us observe that Proposition 2.1 implies that ess sup α∈Ut E [ U (X α t,x (T )) \| F t ] > −∞ ⟨P = 1⟩ t ∈ [0, T ] and as a consequence that sup α∈U E [ U (X α T ) ] > −∞. In order to solve this dynamic problem we introduce the value function associated with it v(t, x, y, p) = sup α∈Ut E [ −e −ηX α t,x (T ) \| Y t = y, P t = p ] (2.21) where the function v : V → R is defined in the domain V . = [0, T ] × R 2 × (0, +∞). The following Lemma gives sufficient conditions to extend Proposition 2.1 to all constant strategies. Lemma 2.2. Under the Assumption 2.1, let us suppose σ(t, p) and µ(t, p) are bounded for all (t, p) ∈ [0, T ] × (0, +∞). Then we have that all constant strategies α t = (u, w) with u ∈ [0, 1] and w ∈ R are admissible and such that E[e −ηX α t,x (T ) \| F t ] < ∞ ⟨P = 1⟩ ∀(t, x) ∈ [0, T ] × R. Proof. See Appendix A. Hamilton-Jacobi-Bellman equation Let us consider the Hamilton-Jacobi-Bellman equation that the value function is expected to solve if sufficiently regular ⎧ ⎨ ⎩ sup (u,w)∈[0,1]×R L α v(t, x, y, p) = 0 v(T, x, y, p) = −e −ηx ∀(y, p) ∈ R × (0, +∞) (3.1) where L α denotes the Markov generator of the triple (X α t , Y t , P t ) associated with a constant control α = (u, w). In what follows, we denote by C 1,2 b all bounded functions f (t, x 1 , . . . , x n ), with n ≥ 1, with bounded first order derivatives ∂f ∂t , ∂f ∂x1 , . . . , ∂f ∂xn and bounded second order derivatives w.r.t. the spatial variables ∂ 2 f ∂x 2 1 , . . . , ∂f ∂x 2 n . Lemma 3.1. Let f : V → R be a function in C 1,2 b . Then the Markov generator of the stochastic process (X α t , Y t , P t ) for all constant strategies α = (u, w) ∈ [0, 1] × R is given by the following expression: L α f (t, x, y, p) = ∂f ∂t (t, x, y, p) + ∂f ∂x (t, x, y, p) [ Rx + c(t, y) − q(t, y, u) + w(µ(t, p) − R) ] + 1 2 w 2 σ(t, p) 2 ∂ 2 f ∂x 2 (t, x, y, p) + b(t, y) ∂f ∂y (t, x, y, p) + 1 2 γ(t, y) 2 ∂ 2 f ∂y 2 (t, x, y, p) + pµ(t, p) ∂f ∂p (t, x, y, p) + 1 2 p 2 σ(t, p) 2 ∂ 2 f ∂p 2 (t, x, y, p) + wσ(t, p) 2 p ∂ 2 f ∂x∂p (t, x, y, p) + ∫ D 0 [ f (t, x − (1 − u)z, y, p) − f (t, x, y, p) ] λ(t, y) dF Z (z). (3.2) Proof. See Appendix A. Now let us introduce the following ansatz: v(t, x, y, p) = −e −ηxe R(T −t) φ(t, y, p) where φ does not depend on x and it is a positive function 4 . Then the original HJB problem given in (3.1) reduces to the simpler one given by − ∂φ ∂t (t, y, p) − b(t, y) ∂φ ∂y (t, y, p) − 1 2 γ(t, y) 2 ∂ 2 φ ∂y 2 (t, y, p) + ηe R(T −t) c(t, y)φ(t, y, p) − pµ(t, p) ∂φ ∂p (t, y, p) − 1 2 σ(t, p) 2 p 2 ∂ 2 φ ∂p 2 (t, y, p) + sup u∈[0,1] Ψ u (t, y)φ(t, y, p) + sup w∈R Ψ w (t, y, p) = 0 (3.3) with final condition φ(T, y, p) = 1 for all (y, p) ∈ R × (0, +∞), defining Ψ u (t, y) . = −ηe R(T −t) q(t, y, u) + λ(t, y) ∫ D 0 [ 1 − e η(1−u)ze R(T −t) ] dF Z (z) (3.4) and Ψ w (t, y, p) . = ηe R(T −t) ( (µ(t, p) − R)φ(t, y, p) + pσ(t, p) 2 ∂φ ∂p (t, y, p) ) w − 1 2 σ(t, p) 2 η 2 e 2R(T −t) φ(t, y, p)w 2 . (3.5) It should make it clear that we can split the optimal control research in two distinct problems: the optimization of Ψ u will give us the optimal level of reinsurance (see Section 4), while working with Ψ w we will find the optimal investment policy (see Section 5). Optimal reinsurance strategy In this section we discuss the problem sup u∈[0,1] Ψ u (t, y), (t, y) ∈ [0, T ] × R (4.1) with Ψ u (t, y) given in (3.4). First, let us observe that Ψ u (t, y) is continuous w.r.t. u ∈ [0, 1], for any (t, y) ∈ [0, T ] × R and admits continuous first and the second order derivatives w. r.t. u ∈ [0, 1] ∂Ψ u (t, y) ∂u = −ηe R(T −t) [ ∂q(t, y, u) ∂u − λ(t, y) ∫ D 0 ze η(1−u)ze R(T −t) dF Z (z) ] ∂ 2 Ψ u (t, y) ∂u 2 = −ηe R(T −t) [ ∂ 2 q(t, y, u) ∂u 2 + ηe R(T −t) λ(t, y) ∫ D 0 z 2 e η(1−u)ze R(T −t) dF Z (z) ] . Notice that these derivatives are well defined thanks to (2.19). Now we are ready for the main result of this section. Proposition 4.1. Given Ψ u (t, y) in (3.4), suppose that − ∂ 2 q(t, y, u) ∂u 2 < ηe R(T −t) λ(t, y)E [ Z 2 e η(1−u)Ze R(T −t) ] ∀(t, y, u) ∈ [0, T ] × R × (0, 1). (4.2) Then there exists a unique measurable function u * (t, y) for all (t, y) ∈ [0, T ] × R solution to (4.1). Moreover, it is given by u * (t, y) = ⎧ ⎪ ⎨ ⎪ ⎩ 0 (t, y) ∈ A 0 u(t, y) (t, y) ∈Â 1 (t, y) ∈ A 1 (4.3) where A 0 . = { (t, y) ∈ [0, T ] × R \| λ(t, y)E[Ze ηZe R(T −t) ] ≤ ∂q(t, y, 0) ∂u } A . = { (t, y) ∈ [0, T ] × R \| λ(t, y)E[Ze ηZe R(T −t) ] > ∂q(t, y, 0) ∂u , ∂q(t, y, 1) ∂u > E[Z]λ(t, y) } A 1 . = { (t, y) ∈ [0, T ] × R \| ∂q(t, y, 1) ∂u ≤ E[Z]λ(t, y) } andû(t, y) is the unique solution of the following equation: ∂q(t, y, u) ∂u = λ(t, y) ∫ D 0 ze η(1−u)ze R(T −t) dF Z (z). (4.4) Proof. Since Ψ u (t, y) is continuous in u ∈ [0, 1] and ∂ 2 Ψ u (t,y) ∂u 2 < 0 ∀(t, y, u) ∈ [0, T ] × R × (0, 1) by (4.2), Ψ u (t, y) is strictly concave in u ∈ [0, 1]. As a consequence there exists a unique maximizer u * (t, y) of (4.1), whose measurability follows by classical selection theorems. Observe that A 0 ∪Â ∪ A 1 = [0, T ] × R. In fact, let us define these subsets as follows: A 0 = { (t, y) ∈ [0, T ] × R \| ∂Ψ 0 (t, y) ∂u ≤ 0 } A = { (t, y) ∈ [0, T ] × R \| ∂Ψ 0 (t, y) ∂u > 0, ∂Ψ 1 (t, y) ∂u < 0 } A ! = { (t, y) ∈ [0, T ] × R \| ∂Ψ 1 (t, y) ∂u ≥ 0 } . Now, being ∂Ψ u (t,y) ∂u strictly decreasing in u ∈ (0, 1), for any (t, y) ∈Â ∪ A 1 we have that ∂Ψ 0 (t, y) ∂u > ∂Ψ 1 (t, y) ∂u ≥ 0 ⇒ (t, y) ̸ ∈ A 0 which implies that [0, T ] × R \ A 0 = { (t, y) ∈ [0, T ] × R \| ∂Ψ 0 (t, y) ∂u > 0 } = { (t, y) ∈ [0, T ] × R \| ∂Ψ 0 (t, y) ∂u > 0, ∂Ψ 1 (t, y) ∂u < 0 } ∪ { (t, y) ∈ [0, T ] × R \| ∂Ψ 0 (t, y) ∂u > 0, ∂Ψ 1 (t, y) ∂u ≥ 0 } =Â∪A 1 . Moreover, sinceÂ ∩ A 1 = ∅, then A 0∪Â∪ A 1 = [0, T ] × R. Let us recall that ∂Ψ u (t,y) ∂u is continuous and strictly decreasing in u ∈ [0, 1], for any ∀(t, y) ∈ [0, T ] × R. If (t, y) ∈ A 0 then Ψ u (t, y) is strictly decreasing in u ∈ [0, 1], hence no reinsurance is chosen, i.e. u * (t, y) = 0. If (t, y) ∈Â then there exists a unique u * (t, y) ∈ (0, 1) such that ∂Ψ u (t,y) ∂u = 0, and it is the unique solution to equation (4.4). Finally , if (t, y) ∈ A 1 then Ψ u (t, y) is strictly increasing in u ∈ [0, 1], hence u * (t, y) = 1. Remark 4.1. We also observe for the sake of completeness that if λ(t, y) had been vanished for some (t, y), then ∂Ψ u (t,y) ∂u would have become strictly negative for all u, and in this case u * (t, y) = 0. In fact, the case of λ(t, y) = 0 corresponds to a degenerate situation: the risk premia are paid, but there is no "real" risk to be insured. From the economic point of view, we could say that if the reinsurance is not too much expensive (more precisely, if the price of an infinitesimal protection is below a certain dynamic threshold) and if full reinsurance is not optimal, then the optimal strategy is provided by (4.4), i.e. by equating the marginal cost and the marginal gain; moreover, the following remark points out the relevance of the third case in (4.3). Remark 4.2. In the current literature full reinsurance is always considered sub-optimal, contrary to the result given by formula (4.3). The main reason is that using premium calculation principles many authors force the reinsurance premium to have certain properties, such as the convexity with respect to the protection level. In fact, it can be shown that if the reinsurance premium q(t, y, u) is convex w.r.t. u, full reinsurance is never optimal (see Remark 4.3). Nevertheless, it is reasonable that the insurer firm could regard full reinsurance as convenient for a limited period and in some particular scenarios, because actually the objective is to maximize the expected utility of the wealth at the end of the period. Moreover, from the reinsurer's point of view, there is no reason to prevent the insurer from buying a full protection, providing the cedant is ready to pay a fair price. At the same time, if the reinsurer is not able to sell a full reinsurance, then it is sufficient to choose q(t, y, u) such that A 1 = ∅. Now we provide some sufficient conditions in order to guarantee that condition (4.2) is fulfilled. Lemma 4.1. Suppose that at least one of the following condition holds: 1. ∂q(t,y,0) ∂u = 0 for all (t, y) ∈ [0, T ] × R; 2. ∂ 2 q(t,y,u) ∂u 2 ≥ 0 for all u ∈ (0, 1) and (t, y) ∈ [0, T ] × R; 3. − ∂ 2 q(t,y,u) ∂u 2 < ηλ(t, y)E[Z 2 ] for all u ∈ (0, 1) and (t, y) ∈ [0, T ] × R. Then the inequality (4.2) holds, which implies that the function Ψ u (t, y) is strictly concave in u ∈ (0, 1). Proof. First, let us observe that 1 ⇒ 2 ⇒ 3. In fact, by the fundamental theorem of calculus we have that ∂q(t, y, u) ∂u = ∂q(t, y, 0) ∂u + ∫ u 0 ∂ 2 q(t, y, w) ∂w 2 dw and, being ∂q(t,y,u) ∂u ≥ 0, ∂q(t,y,0) ∂u = 0 implies that the integrand function must be nonnegative, that is 1 ⇒ 2. The implication 2 ⇒ 3 is trivial, being η > 0, λ(t, y) > 0. Now it is sufficient to show that 3 implies (4.2); clearly − ∂ 2 q(t, y, u) ∂u 2 < ηλ(t, y)E[Z 2 ] < ηe R(T −t) λ(t, y)E [ Z 2 e η(1−u)Ze R(T −t) ] and hence ∂ 2 Ψ u (t,y) , 1), the full reinsurance is never optimal. In fact, for any arbitrary couple (t, y) we have that ∂u 2 < 0 for all u ∈ (0, 1), i.e. (4.2) holds, which implies that Ψ u (t, y) is strictly concave in u ∈ (0, 1) Remark 4.3. Under the hypotheses that ∂ 2 q(t,y,u) ∂u 2 ≥ 0 and c(t, y) > E[Z]λ(t, y) for all (t, y, u) ∈ [0, T ] × R × (0q(t, y, 1) = q(t, y, 0) + ∫ 1 0 ∂q(t, y, u) ∂u du. Being q(t, y, 0) = 0 and q(t, y, 1) > c(t, y) > E[Z]λ(t, y) (because the reinsurance is not cheap and using the net-profit condition for the insurance premium), we obtain that ∫ 1 0 ∂q(t, y, u) ∂u du > E[Z]λ(t, y). Since ∂q(t,y,u) ∂u is continuous in u ∈ [0, 1] by hypothesis, from the mean value theorem for integrals we know that there exists u 0 ∈ (0, 1) such that ∂q(t, y, u 0 ) ∂u > E[Z]λ(t, y). Under the hypothesis that ∂ 2 q(t,y,u) ∂u 2 ≥ 0 for all u ∈ (0, 1), ∂q(t,y,u) ∂u is an increasing function of u, and this implies that ∂q(t, y, 1) ∂u ≥ ∂q(t, y, u 0 ) ∂u > E[Z]λ(t, y). From this result we deduce that ∂Ψ 1 (t, y) ∂u = −ηe R(T −t) [ ∂q(t, y, 1) ∂u − E[Z]λ(t, y) ] < 0, (t, y) ∈ [0, T ] × R which implies that A 1 = ∅, i.e. the full reinsurance is never optimal. Let us observe that the preceding Remark requires two special conditions. The first one concerns the concavity of the reinsurance premium and in Subsection 4.1 we will show that it is fulfilled by the most famous premium calculation principles. The second hypothesis is the so called net-profit condition (e.g. see [Grandell, 1991]) and it is usually assumed in insurance risk models to ensure that the expected gross risk premium covers the expected losses. Now we investigate how Proposition 4.1 applies to a special case. ∫ ∞ 0 ze η(1−u)ze R(T −t) ζe −ζz dz = ∂q(t, y, u) ∂u . Taking k = η(1 − u)e R(T −t) − ζ it can be written as λ(t, y) ∫ ∞ 0 ze kz ζ dz = ∂q(t, y, u) ∂u and requiring that ζ η > e RT (4.5) which implies that k < 0, finally equation (4.4) reads as λ(t, y) ζ (η(1 − u)e R(T −t) − ζ) 2 = ∂q(t, y, u) ∂u . (4.6) Summarizing, if Z is an exponential r.v. with parameter ζ > ηe RT , under the condition (4.2) we have that expression (4.3) holds with A 0 . = { (t, y) ∈ [0, T ] × R \| λ(t, y) ζ (ηe R(T −t) − ζ) 2 ≤ ∂q(t, y, 0) ∂u } A . = { (t, y) ∈ [0, T ] × R \| λ(t, y) ζ (ηe R(T −t) − ζ) 2 > ∂q(t, y, 0) ∂u , ∂q(t, y, 1) ∂u > λ(t, y) ζ } A 1 . = { (t, y) ∈ [0, T ] × R \| ∂q(t, y, 1) ∂u ≤ λ(t, y) ζ } and withû(t, y) being the unique solution to equation (4.6). Expected value and variance premium principles Proposition 4.1 clarifies that the optimal reinsurance strategy crucially depends on the reinsurance premium. In this subsection we specialize that result using two of the most famous premium calculation principles: the expected value principle and the variance premium principle. We will show that in both cases we loose the dependence of the optimal reinsurance strategy on the stochastic factor. Moreover, the optimal reinsurance strategy does not explicitly depend on the claims intensity. These will be our motivations for introducing the intensity-adjusted variance premium principle in Subsection 4.2. Lemma 4.2. Under the expected value principle, i.e. if the reinsurance premium admits the following expression q(t, y, u) = (1 + θ r )E[Z]λ(t, y)u ∀(t, y, u) ∈ [0, T ] × R × [0, 1] (4.7) for some constant θ r > 0 (which is called the reinsurance safety loading), there exists a unique maximizer u * (t) for all (t, y) ∈ [0, T ] × R for the problem (4.1). In particular, u * (t) = { 0 t ∈ A 0 u(t) t ∈ [0, T ] \ A 0 (4.8) where A 0 . = { t ∈ [0, T ] \| E[Ze ηZe R(T −t) ] ≤ (1 + θ r )E[Z] } andû(t) is the unique solution to the following equation: (1 + θ r )E[Z] = ∫ D 0 ze η(1−u)ze R(T −t) dF Z (z). (4.9) Proof. From (4.7) we get ∂q(t, y, u) ∂u = (1 + θ r )E[Z]λ(t, y), ∂ 2 q(t, y, u) ∂u 2 = 0 ∀u ∈ (0, 1) which implies that Ψ u (t, y) is strictly concave in u ∈ (0, 1) thanks to Lemma 4.1. Moreover, by the means of Remark 4.3 we know that the full reinsurance is always sub-optimal, in fact the set A 1 in Proposition 4.1 is empty. Now we only have to apply Proposition 4.1. Note that we always have E[Ze ηZe R(T −t) ] > E[Z] for each t ∈ [0, T ], thus A 0 could be an empty set when the reinsurer's safety loading is close to 0. (4.7) the result for exponential claims is even more simplified, in fact we find the following explicit solution: u * (t) = ⎧ ⎨ ⎩ 1 − ζ η ( 1 − 1 √ 1+θr ) e −R(T −t) t ∈ [0, t 0 ∧ T ) 0 t ∈ [t 0 ∧ T, T ] (4.10) where t 0 = T − 1 R log [ ζ η ( 1 − 1 √ 1 + θ r )] . (4.11) The expression for t 0 can be derived from the characterization of the set [0, T ] × R \ A 0 , which in this case reads as follows: ζ − √ ζ (1+θr)E[Z] η < e R(T −t) < ζ + √ ζ (1+θr)E[Z] η where the second inequality is always fulfilled in view of (4.5), hence we get t 0 only from the first inequality. for some constant reinsurance safety loading θ r > 0, the optimization problem (4.1) admits a unique maximizer u * (t) ∈ (0, 1) for all (t, y) ∈ [0, T ] × R , which is the solution to the following equation: 2θ r E[Z 2 ]u = ∫ D 0 ze η(1−u)ze R(T −t) dF Z (z) − E[Z]. (4.13) Proof. Using the expression (4.12) we get that ∂q(t, y, u) ∂u = E[Z]λ(t, y) + 2θ r E[Z 2 ]λ(t, y)u ∀u ∈ (0, 1) and ∂ 2 q(t, y, u) ∂u 2 = 2θ r E[Z 2 ]λ(t, y) > 0 ∀u ∈ (0, 1). By Lemma 4.1 Ψ u (t, y) is strictly concave w.r.t. u and the full reinsurance is never optimal because of Remark 4.3. Moreover, in order to apply Proposition 4.1 we notice that E[Ze ηZe R(T −t) ] > E[Z] ⇒ A 0 = ∅ thus the optimal strategy is unique and it belongs to (0, 1). In order to find such a solution, we turn the attention to the first order condition, which is exactly the equation (4.13). The same result was obtained in [Liang and Bayraktar, 2014], Lemma 3.1. Example 4.3. (Exponentially distributed claims under the variance premium principle) Under the variance premium principle (4.12), suppose that the claims are exponentially distributed with parameter ζ > ηe RT . Then it is easy to show that the optimal strategy is given by u * (t) = 1 − ζ η ( 1 − √ ζ ζ + 4θ r ) e −R(T −t) t ∈ [0, T ]. (4.14) Intensity-adjusted variance premium principle We have shown that both the expected value principle (see Lemma 4.2) and the variance premium principle (see Lemma 4.3) lead us to deterministic optimal reinsurance strategies, which do not depend on the stochastic factor. This is a limiting factor, since the main objective of our paper is to solve the maximization problem under a stochastic factor model. In addition, in both cases the optimal reinsurance strategy does not explicitly depend on the claims intensity. As a consequence, there is a paradox that we clarify with the following example. Let us consider two identical insurers (i.e. with the same risk-aversion, time horizon, and so on) who work in the same insurance business line, for example in automobile insurance, but in two distinct territories with different riskiness. More precisely, let us assume that the two companies insure claims which have the same distribution F Z but occur with different probabilities. Hence it is a reasonable assumption that the claims arrival processes have two different intensities. Now let us suppose that both the insurers use Lemma 4.2 (or Lemma 4.3) in order to solve the maximization problem (4.1). Then they will obtain the same reinsurance strategy, but this is not what we expect. Hence the optimal reinsurance strategy should explicitly depend on the claims intensity. In order to fix these two problems, in this subsection we introduce a new premium calculation principle, which will be referred as the intensity-adjusted variance premium principle. Let us first formalize that there exists a special class of premium calculation principles that lead us to deterministic strategies which do not depend on the claims intensity. , the optimal reinsurance strategy u * t = u * (t, Y t ) given in Proposition 4.1 turns out to be deterministic. Moreover, it does not explicitly depend on the claims intensity. For example, the expected value principle and the variance premium principle admit the factorization (4.15) with, respectively, Q(t, u) = (1 + θ r )E[Z]u and Q(t, u) = E[Z]u + θ r E[Z 2 ]u 2 . Now the basic idea is to find a reinsurance premium {q t } t∈[0,T ] (see Definition 2.1) such that E [∫ t 0 q(s, Y s , u s ) ds ] = E [∫ t 0 u s dC s ] + θ r var [∫ t 0 u s dC s ] ∀t ∈ [0, T ] (4.16) for a given reinsurance safety loading θ r in order to dynamically satisfy the original formulation of the variance premium principle 6 . For this purpose, we give the following result. M u t = ∫ D 0 ∫ t 0 u s z ( m(ds, dz) − dF Z (z)λ s ds ) . Recalling that {C t } t∈[0,T ] is defined in (2.6), the variance of the reinsurer's cumulative losses at the time t ∈ [0, T ] is given by var [∫ t 0 u s dC s ] = E [(∫ t 0 u s dC s ) 2 ] − E [∫ t 0 u s dC s ] 2 = E [ \|M u t \| 2 + (∫ t 0 u s λ s E[Z] ds ) 2 + 2M u t ∫ t 0 u s λ s E[Z] ds ] − E [∫ t 0 u s dC s ] 2 . Denoting with ⟨M u ⟩ t the predictable covariance process of M u t , using Remark 2.1, we find that var [∫ t 0 u s dC s ] = E[⟨M u ⟩ t ] + E[Z] 2 E [(∫ t 0 u s λ s ds ) 2 ] − E[Z] 2 E [∫ t 0 u s λ s ds ] 2 = E[Z 2 ]E [∫ t 0 u 2 s λ s ds ] + E[Z] 2 var [∫ t 0 u s λ s ds ] ∀t ∈ [0, T ]. (4.18) Here we have used that E [ M u t ∫ t 0 u s λ s E[Z] ds ] = 0. In fact we notice that E [ M u t ∫ t 0 u s λ s E[Z] ds ] = E [ E [ M u t ∫ t 0 u s λ s E[Z] ds \| F Y T ]] = E [ E [ M u t \| F Y T ] ∫ t 0 u s λ s E[Z] ds ] and being G 0 = F 0 ∨ F Y T ⊇ F Y T (see Remark 2.2) we have that E [ M u t \| F Y T ] = E [ E [ M u t \| G 0 ] F Y T ] = E [ M u 0 \| F Y T ] = 0 and the proof is complete. Remark 4.5. We highlight that Lemma 4.4 applies to {F Y t } t∈[0,T ] -predictable reinsurance strategies, but this is not restrictive. In fact, from Lemma 4.1 we know that the optimal strategy belongs to the class of {F Y t } t∈[0,T ] -predictable processes. Remark 4.6. In the classical Cramér-Lundberg model, i.e. λ(t, y) = λ, for any deterministic strategy u t = u(t) Under any risk model with stochastic intensity the formula (4.12) neglects the term E[Z] 2 var [∫ t 0 u s λ s ds ] in the equation (4.17). In order to capture the effect of this term, we can find the following estimate: var [∫ t 0 u s λ s ds ] ≤ E [(∫ t 0 u s λ s ds ) 2 ] ≤ E [ T ∫ t 0 u 2 s λ 2 s ds ] . As a consequence, we can choose as premium calculation rule q(t, y, u) = E[Z]λ(t, y)u + θ r E[Z 2 ] [ λ(t, y) + T λ(t, y) 2 ] u 2 (4.19) which will be called intensity-adjusted variance principle in this work; using this formula, we ensure that E [∫ t 0 q(s, Y s , u s ) ds ] ≥ E [∫ t 0 u s dC s ] + θ r var [∫ t 0 u s dC s ] ∀t ∈ [0, T ] for all {F Y t } t∈[0,T ] -predictable reinsurance strategies and for any arbitrary level of reinsurance safety loading θ r > 0. Lemma 4.5. Under the intensity-adjusted variance premium principle (4.19), the optimization problem (4.1) admits a unique maximizer u * (t, y) ∈ (0, 1) for all (t, y) ∈ [0, T ] × R, which is the solution to the following equation: 2θ r E[Z 2 ] [ 1 + T λ(t, y) ] u = ∫ D 0 ze η(1−u)ze R(T −t) dF Z (z) − E[Z]. (4.20) Proof. From the expression (4.19) we get ∂q(t, y, u) ∂u = E[Z]λ(t, y) + 2θ r E[Z 2 ] [ λ(t, y) + T λ(t, y) 2 ] u ∀u ∈ (0, 1) and ∂ 2 q(t, y, u) ∂u 2 = 2θ r E[Z 2 ] [ λ(t, y) + T λ(t, y) 2 ] > 0 ∀u ∈ (0, 1). By Lemma 4.1 Ψ u (t, y) is strictly concave w.r.t. u and full reinsurance is never optimal because of Remark 4.3. Moreover, we notice that A 0 = ∅ as in Lemma 4.3, thus the optimal strategy is unique and it belongs to (0, 1). In order to find such a solution, we turn the attention to the first order condition, which is exactly equation (4.20). Through the numerical simulations in Section 7 we will show that the intensity-adjusted variance premium principle leads to optimal strategies which are consistent with the desired properties obtained under the other premium calculation principles. Moreover, the reinsurance strategies under the intensity-adjusted variance premium principle are not deterministic and explicitly depend on the (stochastic) intensity. Hence the problems described in the beginning of this subsection are fixed. Using the result given in Example 4.3, it is easy to specialize Lemma 4.5 to the case of exponentially distributed claims. Example 4.4. (Exponentially distributed claims under the intensity-adjusted variance premium principle) Under the intensity-adjusted variance premium principle (4.19), suppose that the claims are exponentially distributed with parameter ζ > ηe RT . Then the optimal strategy u * (t, y) ∈ (0, 1) is given by u * (t, y) = 1 − ζ η ( 1 − √ ζ ζ + 4θ r [ 1 + T λ(t, y) ] ) e −R(T −t) (t, y) ∈ [0, T ] × R. (4.21) Remark 4.7. In [Liang and Yuen, 2016] and [Yuen et al., 2015] the authors used, respectively, the variance premium and the expected value principles to obtain optimal reinsurance strategies in a risk model with multiple dependent classes of insurance business. In those papers the optimal strategies explicitly depend on the claims intensities, but it is due to the presence of more than one business line, hence our arguments are not valid there. Nevertheless, in [Yuen et al., 2015] the authors realized that in the diffusion approximation of the classical risk model the variance premium principle lead to optimal strategies which do not depend on the claims intensities. In fact, this was the main motivation of their work. Their observation confirms our perplexities of strategies independent on the claims intensity. Optimal investment policy Lemma 5.1. The problem sup w(t,y,p)∈R Ψ w (t, y, p) where Ψ w (t, y, p) is defined in (3.5), admits a unique solution w * (t, y, p) for all (t, y, p) ∈ [0, T ] × R × (0, +∞) given by w * (t, y, p) = µ(t, p) − R ησ(t, p) 2 e R(T −t) + p ηe R(T −t) ∂φ ∂p (t, y, p) φ(t, y, p) . (5.1) Proof. Since φ(t, y, p) > 0, Ψ w (t, y, p) is strictly concave w.r.t. w and the result follows from the first order condition. We emphasize that the optimal w * is the sum of the classical solution 7 plus an adjustment term due to the dependence of the risky asset price coefficients on the stochastic process {P t }. Remark 5.1. If µ, σ are continuous function and φ ∈ C 1,2 , then w * is a continuous function w.r.t. (t, y, p). Corollary 5.1. Suppose that there exist two functions f (t, y) : [0, T ] × R → (0, +∞) and g(t, p) : [0, T ] × (0, +∞) → R such that φ(t, y, p) = f (t, y)e g(t,p) for all (t, y, p) ∈ [0, T ] × R × (0, +∞), with f (t, y) > 0. Then the optimal investment strategy (5.1) reads as follows: w * (t, p) = µ(t, p) − R ησ(t, p) 2 e R(T −t) + p ηe R(T −t) ∂g ∂p (t, p). (5.2) Verification Theorem Now we conjecture a solution to equation (3.3) of the form φ(t, y, p) = f (t, y)e g(t,p) , with f (t, y) > 0. Using Lemma 5.1, replacing all the derivatives and performing some calculations, the equation (3.3) reads as follows ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ − ∂f ∂t (t, y) − b(t, y) ∂f ∂y (t, y) − 1 2 γ(t, y) 2 ∂ 2 f ∂y 2 (t, y) + [ ηe R(T −t) c(t, y) + max u(t,y)∈[0,1] Ψ u (t, y) ] f (t, y) + f (t, y) [ − ∂g ∂t (t, p) − pR ∂g ∂p (t, p) − 1 2 p 2 σ(t, p) 2 ∂ 2 g ∂p 2 (t, p) + 1 2 ( µ(t, p) − R ) 2 σ(t, p) 2 ] = 0 f (T, y)e g(T,p) = 1 ∀(y, p) ∈ R × (0, +∞) (6.1) It is easy to show that if f, g are two solutions of the following Cauchy problems ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ − ∂f ∂t (t, y) − b(t, y) ∂f ∂y (t, y) − 1 2 γ(t, y) 2 ∂ 2 f ∂y 2 (t, y) + [ ηe R(T −t) c(t, y) + max u(t,y)∈[0,1] Ψ u (t, y) ] f (t, y) = 0 f (T, y) = 1 (6.2) ⎧ ⎪ ⎨ ⎪ ⎩ − ∂g ∂t (t, p) − pR ∂g ∂p (t, p) − 1 2 p 2 σ(t, p) 2 ∂ 2 g ∂p 2 (t, p) + 1 2 ( µ(t, p) − R ) 2 σ(t, p) 2 = 0 g(T, p) = 0 (6.3) then they solve the Cauchy problem (6.1) and v(t, x, y, p) = −e −ηxe R(T −t) f (t, y)e g(t,p) solves the original HJB equation given in (3.1). Before we prove a verification theorem, we must show that our proposed optimal controls are admissible strategies. Lemma 6.1. Suppose that (6.2) and (6.3) admit classical solutions with ∂g ∂p satisfying the following growth condition: ⏐ ⏐ ⏐ ⏐ ∂g ∂p (t, p) ⏐ ⏐ ⏐ ⏐ ≤ C(1 + \|p\| β ) ∀(t, p) ∈ [0, T ] × (0, +∞) (6.4) for some constants β > 0 and C > 0. Moreover, assume that E [∫ T 0 \|µ(t, P t )\|P β+1 t dt + ∫ T 0 σ(t, P t ) 2 P 2β+2 t dt ] < ∞. (6.5) Let be u * (t, y) as given in Proposition 4.1 and w * (t, p) in Lemma 5.1. Let us define the processes u * t . = u * (t, Y t ) and w * t . = w * (t, P t ); then the pair (u * t , w * t ) is an admissible strategy, i.e. (u * t , w * t ) ∈ U. Proof. First let us observe that both u * t , w * t are {F t }-predictable processes since u * (t, u) and w * (t, p) are measurable functions of their arguments and Y is adapted. Moreover, they take values, respectively, in [0, 1] and in R. Furthermore, using the expression (5.2) we have that E [∫ T 0 \|w * t \|\|µ(t, P t ) − R\| dt ] ≤ E [∫ T 0 (µ(t, P t ) − R) 2 ησ(t, P t ) 2 e R(T −t) dt ] + E [∫ T 0 \|µ(t, P t ) − R\|P t ηe R(T −t) ⏐ ⏐ ⏐ ⏐ ∂g ∂p (t, P t ) ⏐ ⏐ ⏐ ⏐ dt ] ≤ E [∫ T 0 (µ(t, P t ) − R) 2 ησ(t, P t ) 2 e R(T −t) dt ] + C E [∫ T 0 \|µ(t, P t )\|(1 + P β t )P t dt ] < ∞ and E [∫ T 0 (w * t σ(t, P t )) 2 dt ] = E [∫ T 0 (µ(t, p) − R) 2 η 2 σ(t, p) 2 e 2R(T −t) dt ] + E [∫ T 0 σ(t, P t ) 2 P 2 t η 2 e 2R(T −t) ( ∂g ∂p (t, P t ) ) 2 dt ] + 2E [∫ T 0 (µ(t, p) − R)P t η 2 e 2R(T −t) ∂g ∂p (t, P t ) dt ] ≤ E [∫ T 0 (µ(t, p) − R) 2 η 2 σ(t, p) 2 e 2R(T −t) dt ] + C E [∫ T 0 σ(t, P t ) 2 P 2 t η 2 e 2R(T −t) ( 1 + P β t ) 2 dt ] + C E [∫ T 0 \|µ(t, p) − R\|P t η 2 e 2R(T −t) (1 + P β t ) dt ] < ∞ where C denotes any positive constant and the expectations are finite because of the Novikov condition (2.16) together with (6.4) and (6.5). Now we are ready for the verification argument. Theorem 6.1 (Verification Theorem). Suppose that (6.2) and (6.3) admit bounded classical solutions, respectively f ∈ C 1,2 ((0, T ) × R)) ∩ C([0, T ] × R)) and g ∈ C 1,2 ((0, T ) × (0, +∞)) ∩ C([0, T ] × (0, +∞)). Let us assume that the conditions (6.4) and (6.5) hold and suppose that ⏐ ⏐ ⏐ ⏐ ∂f ∂y (t, y) ⏐ ⏐ ⏐ ⏐ ≤C(1 + \|y\| β ) ∀(t, y) ∈ [0, T ] × R (6.6) for some constants β > 0 andC > 0. As an alternative, the conditions (6.4), (6.5) and (6.6) may be replaced by the boundedness of ∂g ∂p and ∂f ∂y . Then the function v : V → R defined by the following v(t, x, y, p) = −e −ηxe R(T −t) f (t, y)e g(t,p) (6.7) is the value function of the reinsurance-investment problem and α * (t, Y t , P t ) = (u * (t, Y t ), w * (t, P t )) with u * (t, y) given in Proposition 4.1 and w * (t, p) in (5.2) is an optimal control. Proof. Let f (t, y) : [0, T ]×R → (0, +∞) and g(t, p) : [0, T ]×(0, +∞) → R be functions satisfying the assumptions required by Theorem 6.1 and suppose that they are solutions of the Cauchy problems (6.2) and (6.3). Now consider the function φ(t, y, p) = f (t, y)e g(t,p) . As already observed, it satisfies equation (3.3), i.e. it is a solution of the problem ⎧ ⎨ ⎩ sup (u,w)∈[0,1]×R H α φ(t, y, p) = 0 φ(t, y, p) = 1 ∀(y, p) ∈ R × (0, +∞). (6.8) Now, taking v(t, x, y, p) = −e −ηxe R(T −t) φ(t, y, p), we have that v is a solution of the Cauchy problem (3.1). This implies that, for any (t, x, y, p) ∈ [0, T ] × R × R × (0, +∞) L α v(s, X α t,x (s), Y t,y (s), P t,p (s)) ≤ 0 ∀s ∈ [t, T ] for all α ∈ U t , where {Y t,y (s)} s∈[t, T ] denotes the solution to equation (2.1) with initial condition Y t = y and, similarly, {P t,p (s)} s∈[t,T ] denotes the solution to equation (2.13) with initial condition P t = p. Now, from Itô's formula we have that v(T, X α t,x (T ), Y t,y (T ), P t,p (T )) − v(t, x, y, p) = ∫ T t L α v(s, X α t,x (s), Y t,y (s), P t,p (s)) ds + M T (6.9) where {M r } r∈[t,T ] is the following stochastic process: M r = ∫ r t w s σ(s, P s ) ∂v ∂x (s, X α s , Y s , P s ) dW (P ) s + ∫ r t P s σ(s, P s ) ∂v ∂p (s, X α s , Y s , P s ) dW (P ) s + ∫ r t γ(s, Y s ) ∂v ∂y (s, X α s , Y s , P s ) dW (Y ) s + ∫ D 0 ∫ r t [ v(s, X α s − (1 − u)z, Y s , P s ) − v(s, X α s , Y s , P s ) ]( m(ds, dz) − λ(s, Y s ) dF Z (z) ) . (6.10) Now we prove that {M r } r∈[t,T ] is an {F r }-local martingale. Since the jump term is a real martingale because v is bounded, we only need to show that E [∫ T ∧τn t ( w s σ(s, P s ) ∂v ∂x (s, X α s , Y s , P s ) ) 2 ds ] < ∞ E [∫ T ∧τn t ( P s σ(s, P s ) ∂v ∂p (s, X α s , Y s , P s ) ) 2 ds ] < ∞ E [∫ T ∧τn t ( γ(s, Y s ) ∂v ∂y (s, X α s , Y s , P s ) ) 2 ds ] < ∞ for a suitable non-decreasing sequence of stopping times {τ n } n=1,... such that lim n→+∞ τ n = +∞. Taking into account the expression (6.7), we note that ∂v ∂x (t, x, y, p) = ηe R(T −t) e −ηxe R(T −t) f (t, y)e g(t,p) ∂v ∂y (t, x, y, p) = −e −ηxe R(T −t) e g(t,p) ∂f ∂y (t, y) ∂v ∂p (t, x, y, p) = −e −ηxe R(T −t) f (t, y)e g(t,p) ∂g ∂p (t, p). Let us define a sequence of random times {τ n } n=1,... as follows: τ n . = inf{s ∈ [t, T ] \| X α s < −n ∨ \|Y s \| > n} n = 1, . . . In the sequel of the proof we denote with C n any constant depending on n = 1, . . . . Then we have that E [∫ T ∧τn 0 ( w s σ(s, P s ) ∂v ∂x (s, X α s , Y s , P s ) ) 2 ds ] = E [∫ T ∧τn 0 ( w s σ(s, P s )ηe R(T −s) e −ηX α s e R(T −s) f (s, Y s )e g(s,Ps) ) 2 ds ] ≤ C n E [∫ T 0 ( w s σ(s, P s ) ) 2 ds ] < ∞ ∀n = 1, . . . because w t is admissible and f and g are bounded by hypothesis. Moreover,we have that E [∫ T ∧τn 0 ( γ(s, Y s ) ∂v ∂y (s, X α s , Y s , P s ) ) 2 ds ] = E [∫ T ∧τn 0 ( γ(s, Y s )e −) 2 (1 + \|Y s \| β ) 2 ds ] ≤ C n E [∫ T 0 γ(s, Y s ) 2 ds ] < ∞ ∀n = 1, . . . because g is bounded and using the assumptions (2.2) and (6.6). Finally, we obtain that E [∫ T ∧τn 0 ( P s σ(s, P s ) ∂v ∂p (s, X α s , Y s , P s ) ) 2 ds ] = E [∫ T ∧τn 0 P 2 s σ(s, P s ) 2 ( e −ηX α s e R(T −s) f (s, Y s )e g(s,Ps) ∂g ∂p (s, P s ) ) 2 ds ] ≤ CE [∫ T ∧τn 0 P 2 s σ(s, P s ) 2 ( e −ηX α s e R(T −s) f (s, Y s )e g(s,Ps) ) 2 (1 + \|P s \| β ) 2 ds ] ≤ C n E [∫ T 0 σ(s, P s ) 2 (P 2 s + P 2β+2 s ) ds ] < ∞ ∀n = 1, . . . because f and g are bounded by hypothesis and using conditions (2.14), (6.4) and (6.5). Thus Taking the expected value of both sides of (6.9) with T replaced by T ∧ τ n , we obtain that E[v(T ∧ τ n , X α t,x (T ∧ τ n ), Y t,y (T ∧ τ n ), P t,p (T ∧ τ n )) \| F t ] ≤ v(t, x, y, p) for any α ∈ U t , t ∈ [0, T ∧ τ n ], n ≥ 1. Now notice that E[v(T ∧ τ n , X α t,x (T ∧ τ n ), Y t,y (T ∧ τ n ), P t,p (T ∧ τ n )) 2 ] = E[e −2ηX α t,x (T ∧τn)e R(T ∧τn −t) f (T ∧ τ n , Y T ∧τn ) 2 e 2g(T ∧τn,P T ∧τn ) ] ≤ C e −2ηne R(T ∧τn) ≤ C thus {v(T ∧ τ n , X α t,x (T ∧ τ n ), Y t,y (T ∧ τ n ), P t,p (T ∧ τ n ))} n=1,. .. is a family of uniformly integrable random variables. Hence it converges almost surely. Observing that {τ n } n=1,... is a bounded and non-decreasing sequence, since P[\|X α t \| < +∞] = 1 (see (2.18)) and using (2.3) and (2.15), taking the limit for n → +∞, we conclude that E[v(T, X α t,x (T ), Y t,y (T ), P t,p (T )) \| F t ] = lim n→+∞ E[v(T ∧ τ n , X α t,x (T ∧ τ n ), Y t,y (T ∧ τ n ), P t,p (T ∧ τ n )) \| F t ] ≤ v(t, x, y, p) ∀α ∈ U t , t ∈ [0, T ]. (6.11) To be precise, we have that lim n→+∞ X α t,x (T ∧ τ n ) = X α t,x (T −) = X α t,x (T ) P-a.s. since the jump of {N t } t∈[0,T ] occurs at time T with probability zero. Using the final condition of the HJB equation (3.1), from (6.11) we get E[U (X α t,x (T ))] ≤ v(t, x, y, p) ∀α ∈ U t , t ∈ [0, T ]. Now note that α * (t, y, p) was calculated in order to obtain L α * v(t, x, y, p) = 0; replicating the calculations above, replacing L α with L α * , we find the equality: sup α∈Ut E[U (X α t,x (T )) \| Y t = y, P t = p] = v(t, x, y, p) thus α * (t, Y t , P t ) is an optimal control. After the characterization of the value function, we provide a probabilistic representation by means of the Feynman-Kac formula. In preparation for this result, let us introduce a new probability measure Q ≪ P. Novikov condition (2.16) implies that the process {L t } t∈[0,T ] defined by L t = e − ( 1 2 ∫ t 0 \| µ(s,Ps)−R σ(s,Ps ) \| 2 ds+ ∫ t 0 µ(s,Ps )−R σ(s,Ps ) dW (P ) s ) is an {F t }-martingale and we can introduce the following probability measure Q: dQ dP ⏐ ⏐ ⏐ ⏐ Ft = L t t ∈ [0, T ]. (6.12) By Girsanov theorem we know thatW Ps) ds is a Q-Brownian motion and we can rewrite the risky asset dynamic as (P ) t = W (P ) t + ∫ t 0 µ(s,Ps)−R σ(s,dP t = P t [ R dt + σ(t, P t ) dW (P ) t ] . (6.13) Since the discounted price {P t = P t e −Rt } t∈[0,T ] turns out to be an {F t }-martingale, then Q is a martingale or risk-neutral measure for {P t } 8 . We will denote by E Q the conditional expectation with respect to the martingale measure Q. Proposition 6.1. Suppose that (6.2) and (6.3) admit classical solutions f ∈ C 1,2 ((0, T ) × R)) ∩ C([0, T ] × R)) and g ∈ C 1,2 ((0, T ) × (0, +∞)) ∩ C([0, T ] × (0, +∞)), respectively, both bounded with ∂f ∂y and ∂g ∂p satisfying the growth conditions (6.6) and (6.4). Then f and g admit the following Feynman-Kac representations: f (t, y) = E [ e − ∫ T t ( ηe R(T −s) c(s,Ys)+Ψ u * (s,Ys) ) ds \| Y t = y ] (6.14) g(t, p) = −E Q [∫ T t 1 2 ( µ(s, P s ) − R ) 2 σ(s, P s ) 2 ds \| P t = p ] (6.15) where Ψ u * (t, y) is the function defined by (3.4), replacing u with u * (t, y), and Q is the probability measure introduced in (6.12). Proof. The result is a simple consequence of the Feynman-Kac theorem. In Section 8 we will provide sufficient conditions which ensure that the functions f and g given in (6.14) and (6.15) are, respectively, C 1,2 ((0, T ) × R) and C 1,2 ((0, T ) × (0, +∞)) solutions to the Cauchy problems (6.2) and (6.3). Simulations and numerical results Here we illustrate some numerical results based on the theoretical framework developed in the previous sections. In particular, we perform sensitivity analysis of the optimal reinsuranceinvestment strategy in order to study the effect of the model parameters on the insurer's decision. Reinsurance strategy First, we compare the optimal reinsurance strategy under the expected value principle (see Lemma 4.2) and the intensity-adjusted variance premium principle (see Lemma 4.5). In this subsection the first one will be shortly referred as EVP, while the second one as IAVP. The main difference is that under EVP we loose the dependence on the stochastic factor, while under IAVP we keep this dependence; moreover, IAVP also depends on the second moment of the r.v. Z introduced in Section 2. In what follows we assume that {Z i } i=1,... is a sequence of i.i.d. positive random variables Pareto distributed with shape parameter 1.8182 and scale parameter 0.0545. The stochastic factor is described by the SDE (2.1) with constant parameters b = 0.3, γ = 0.3 and initial condition Y 0 = 1. For the sake of simplicity, we assume that λ(t, y) = λ 0 e 1 2 y , that is {λ t = λ(t, Y t )} t∈[0,T ] solves dλ t = λ t 1 2 dY t λ 0 = 0.1, which guarantees that the intensity is positive. Finally, we consider the model parameters in Table 1, using the notation introduced in Section 2. Parameter Value T 5 Y η 0.5 θ r 0.1 R 5% From Figure 1 we observe that the optimal reinsurance strategy is positively correlated to the risk-aversion parameter; moreover, the strategy under EVP seems to be more sensitive to In Figure 2 we notice that any increase in the reinsurance safety loading leads to a decrease of the reinsured risks. It is a simple consequence of the well-known law of demand: the higher the price, the lower the quantity demanded. It is worth noting that under our assumptions the strategy under IAVP is more sensitive than under EVP. Finally, in Figure 3 we can see that the insurer increases the protection when the time horizon is higher. Again, the strategy under EVP turns out to be more sensitive to any change of the time horizon. It is interesting that over 15 years EVP leads to more conservative strategies. We conclude this subsection investigating the dynamical properties of the reinsurance strategies under EVP and IAVP 9 . Figure 4 shows that the mean behavior of the optimal reinsurance strategy is decreasing over the time interval; nevertheless, under IAVP the strategy crucially depends on the stochastic factor, hence the insurer will react to any movement of the claims intensity, while under EVP she will follow a deterministic strategy. Summarizing the main results of our numerical simulations, we can conclude that any variation of the model parameters has the same effect on the optimal strategy under EVP and IAVP, at least from a qualitative point of view. It is important noting that any quantitative comparison is affected by the parameters initial choice. Nevertheless, we can state that using our model parameters under EVP the strategy is more sensitive with respect to the model parameters, except for the safety loading, but it is dynamically more stable during the time interval [0, T ], because it does not take into account any variation of the claims intensity. Investment strategy Now we illustrate a sensitivity analysis for the investment strategy based on the Corollary 5.1. In our simulations we assumed that the risky asset follows a CEV model, that is dP t = P t [ µ dt + σP β t dW (P ) t ] P 0 = 1 with µ = 0.1, σ = 0.1, β = 0.5, while the risk-free interest rate is R = 5% as in the previous subsection. Let us observe that this model corresponds to (2.13) assuming that µ(t, p) = µ and σ(t, p) = σp β , with constant µ, σ > 0. The numerical computation of the function g(t, p) and its partial derivative ∂g ∂p (t, p) is required by the equation (5.2); for this purpose we used the Feynman-Kac representation given in (6.15) evaluated through the standard Monte Carlo method. In figure 5 we show that the higher is the insurer's risk aversion, the lower is the total amount invested in the risky asset. Figure 6 illustrates that if the volatility increases, then an increasing portion of the insurer's wealth is invested in the risk-free asset. Finally, if the risk-free interest rate grows up, then the insurer will find it more convenient to invest its surplus in the risk-free asset, as shown in figure 7. Similar results can be found in [Sheng et al., 2014]. In particular, figure 6 confirms the result obtained in Figure 3a of that paper; in addition, figures 5 and 7 completes the sensitivity analyses performed there. Existence and uniqueness of classical solutions In this section we are interested in providing sufficient conditions for existence and uniqueness of the solutions to the PDEs involved in the reinsurance-investment problem, see the Cauchy problems (6.2) and (6.3) and as a consequence of a classical solution to HJB equation associated with our problem. First, let us consider (6.3). The following Lemma prepares the main result. Lemma 8.1. Let us define the set D n . = ( 1 n , n) for n = 1, . . . and assume that the functions µ(t, p), σ(t, p) are Lipschitz-continuous in p ∈ D n , uniformly in t ∈ [0, T ]. Moreover, assume that σ(t, p) is bounded from below, i.e. there exists a constant δ σ > 0 such that σ(t, p) ≥ δ σ for all (t, p) ∈ [0, T ] × (0, +∞). Then for each n = 1, . . . the function k : [0, T ] × (0, +∞) → R defined by k(t, p) = ( µ(t, p) − R ) 2 σ(t, p) 2 (8.1) is uniformly Lipschitz-continuous on [0, T ] × D n . Proof. Firstly, using the Lipschitz-continuity of the parabolic function on the bounded domain D n we have that \|k(t, p) − k(t ′ , p ′ )\| = ⏐ ⏐ ⏐ ⏐ ⏐ ( µ(t, p) − R σ(t, p) ) 2 − ( µ(t ′ , p ′ ) − R σ(t ′ , p ′ ) ) 2 ⏐ ⏐ ⏐ ⏐ ⏐ ≤ K n ⏐ ⏐ ⏐ ⏐ µ(t, p) − R σ(t, p) − µ(t ′ , p ′ ) − R σ(t ′ , p ′ ) ⏐ ⏐ ⏐ ⏐ = K n ⏐ ⏐ ⏐ ⏐ σ(t ′ , p ′ )[µ(t, p) − R] − σ(t, p)[µ(t ′ , p ′ ) − R] σ(t, p)σ(t ′ , p ′ ) ⏐ ⏐ ⏐ ⏐ for a positive constant K n > 0 which depends on n. Now, being σ(t, p) bounded from below, settingK n = Kn δ 2 σ we have that \|k(t, p) − k(t ′ , p ′ )\| ≤K n \|σ(t ′ , p ′ )[µ(t, p) − R] − σ(t, p)[µ(t ′ , p ′ ) − R]\| ≤K n R\|σ(t, p) − σ(t ′ , p ′ )\| +K n \|σ(t ′ , p ′ )µ(t, p) − σ(t, p)µ(t ′ , p ′ )\| ≤K n R\|σ(t, p) − σ(t ′ , p ′ )\| +K n \|σ(t ′ , p ′ )µ(t, p) − σ(t ′ , p ′ )µ(t ′ , p ′ )\| +K n µ(t ′ , p ′ )\|σ(t ′ , p ′ ) − σ(t, p)\| and, observing that any Lipschitz-continuous function on a bounded domain is also bounded, the result is a consequence of our hypotheses. Theorem 8.1. Suppose that the following conditions are satisfied: 1. µ(t, p) and σ(t, p) are locally Lipschitz-continuous in p, uniformly in t ∈ [0, T ], i.e. for each n = 1, . . . there exists a positive constant K n such that for instance, it is true if we assume the sub-linear growth for σ(t, p): \|µ(t, p) − µ(t, p ′ )\| + \|σ(t, p) − σ(t, p ′ )\| ≤ K n \|p − p ′ \| ∀p, p ′ ∈ [ 1 n , n ] ,\|σ(t, p)\| ≤ K σ (1 + p) ∀p ∈ (0, +∞), t ∈ [0, T ] together with the other hypotheses of this theorem; 10 3. σ(t, p) is bounded from below, i.e. there exists a constant δ σ > 0 such that σ(t, p) ≥ δ σ for all (t, p) ∈ [0, T ] × (0, +∞); 4. µ(t, p) is bounded, i.e. there exists a constant δ µ > 0 such that \|µ(t, p)\| ≤ δ µ for all (t, p) ∈ [0, T ] × (0, +∞). Then the function g(t, p) given in (6.15) satisfies the Cauchy problem (6.3) and there exists a unique classical solution to (6.3). Moreover, we have that g ∈ C 1,2 ((0, T ) × (0, +∞)). Proof. The proof is a consequence of Theorem 1 and Lemma 2 in [Heath and Schweizer, 2000], toghether with Lemma 8.1. We highlight that in order to use those results, we take D n . = ( 1 n , n) for n = 1, . . . as bounded domains such that (0, +∞) = ⋃ ∞ n=1 D n . Moreover, we observe that the function k defined in (8.1) is bounded, as requested in Lemma 2 of [Heath and Schweizer, 2000], because µ(t, p) is bounded and σ(t, p) is bounded from below. Remark 8.1. In [Sheng et al., 2014] the authors found an explicit solution of the Cauchy problem (6.3) in the particular case of the CEV model, i.e. when µ(t, p) = µ and σ(t, p) = kp β . Now we turn the attention to the second PDE involved in the reinsurance-investment problem, see the Cauchy problem (6.2). Before proving the existence theorem, let us state some preliminary results. Lemma 8.2. Given a compact set K ∈ R let us assume that H(t, y, u) : [0, T ] × R × K is Hölder-continuous in (t, y) ∈ [0, T ] × R uniformly in u ∈ K with exponent 0 < ξ ≤ 1. Then max u∈K H(t, y, u) is Hölder-continuous in (t, y) ∈ [0, T ] × R with exponent 0 < ξ ≤ 1. Proof. Given t, t ′ ∈ [0, T ] and y, y ′ ∈ R, let us define h 1 (u) = H(t, y, u) h 2 (u) = H(t ′ , y ′ , u). Then we have that \|max u∈K h 1 (u) − max u∈K h 2 (u)\| ≤ max u∈K \|h 1 (u) − h 2 (u)\|. (8.2) In fact, observing that \|max u∈K h 1 (u)−max u∈K h 2 (u)\| = { max u∈K h 1 (u) − max u∈K h 2 (u) if max u∈K h 1 (u) ≥ max u∈K h 2 (u) max u∈K h 2 (u) − max u∈K h 1 (u) if max u∈K h 1 (u) < max u∈K h 2 (u) we notice that in the first case and this completes the proof. Corollary 8.1. Let us assume that the following hypotheses hold: • q(t, y, u) is bounded and Hölder-continuous in (t, y) ∈ [0, T ] × R uniformly in u ∈ [0, 1] with exponent 0 < ξ ≤ 1; • λ(t, y) is bounded and Hölder-continuous in (t, y) ∈ [0, T ] × R with exponent 0 < ξ ≤ 1. Then max u(t,y)∈[0,1] Ψ u (t, y) is Hölder-continuous in (t, y) ∈ [0, T ] × R with exponent 0 < ξ ≤ 1. Proof. In view of Lemma 8.2, it is sufficient to show that Ψ u (t, y) is Hölder-continuous in (t, y) ∈ [0, T ] × R uniformly in u ∈ [0, 1] with exponent 0 < ξ ≤ 1. Let us recall equation (3.4): Ψ u (t, y) = −ηe R(T −t) q(t, y, u) + λ(t, y) ∫ D 0 [ 1 − e η(1−u)ze R(T −t) ] dF Z (z) Since e R(T −t) is differentiable and bounded on t ∈ [0, T ], our first hypothesis ensures that the first term ηe R(T −t) q(t, y, u) is Hölder-continuous in (t, y) ∈ [0, T ] × R uniformly in u ∈ [0, 1] with exponent 0 < ξ ≤ 1. For the second term we notice that it is a product of two bounded and Hölder-continuous functions, in fact ⏐ ⏐ ⏐ ⏐ ⏐ ∫ D 0 e η(1−u)ze R(T −t) dF Z (z) − ∫ D 0 e η(1−u)ze R(T −t ′ ) dF Z (z) ⏐ ⏐ ⏐ ⏐ ⏐ ≤ E [ ⏐ ⏐ ⏐e η(1−u)Ze R(T −t) − e η(1−u)Ze R(T −t ′ ) ⏐ ⏐ ⏐ ] . Using Lagrange's theorem, there existst ∈ [0, T ] such that E [ ⏐ ⏐ ⏐e η(1−u)Ze R(T −t) − e η(1−u)Ze R(T −t ′ ) ⏐ ⏐ ⏐ ] ≤ E [ ⏐ ⏐ ⏐Rη(1 − u)Ze R(T −t) e η(1−u)Ze R(T −t) ⏐ ⏐ ⏐ ] \|t − t ′ \| ≤ Rηe RT E [ Ze ηZe RT ] \|t − t ′ \| and the proof is complete. The following theorem is based on the main result of [Heath and Schweizer, 2000]. Theorem 8.2. Suppose that the following conditions are satisfied: 1. b(t, y) and γ(t, y) are locally Lipschitz-continuous in y, uniformly in t ∈ [0, T ], i.e. for each n = 1, . . . there exists a positive constant K n such that \|b(t, y) − b(t, y ′ )\| + \|γ(t, y) − γ(t, y ′ )\| ≤ K n \|y − y ′ \| ∀y, y ′ ∈ [−n, n], t ∈ [0, T ]; 2. for all couple (t, y) ∈ [0, T ] × R the solution {Y t,y (s)} s∈[t,T ] does not explode, i.e. P[ sup s∈[t,T ] Y t,y (s) < ∞] = 1; for instance, it is true when we assume that for some positive constant K 2 \|b(t, y)\| + \|γ(t, y)\| ≤ K 2 (1 + \|y\|) ∀t ∈ [0, T ], y ∈ R together with the previous assumption; 11 3. there exists a constant δ γ > 0 such that γ(t, y) 2 ≥ δ γ ; From the proof of the Proposition 2.1 (see above), we know that for any constant strategy α t = (u, w) with u ∈ [0, 1] and w ∈ R the equation (A.1) holds. Now, using the inequality (2.10), we have that Since h t is bounded, the Novikov condition is satisfied: E[e −ηX α t,x (T ) \| F t ] ≤ ≤ e −ηxe R(T −t) e η K R (e R(T −t) −1) E[e ηE[e 1 2 ∫ T 0 h 2 s ds ] < ∞. This allows us to introduce a new probability measureP using the change of measure given by L t = dP dP ⏐ ⏐ ⏐ ⏐ Ft . Using the Kallianpur-Striebel formula, we obtain that Proof of Lemma 3.1. Looking at (2.17), we apply Itô's formula to the stochastic process f (t, X α t , Y t , P t ): f (t, X α t , Y t , P t ) = f (0, X α 0 , Y 0 , P 0 ) + ∫ t 0 L α f (s, X α s , Y s , P s ) ds + m t where m t = ∫ t 0 w s σ(s, P s ) ∂f ∂x (s, X α s , Y s , P s ) dW (P ) s + ∫ t 0 P s σ(s, P s ) ∂f ∂p (s, X α s , Y s , P s ) dW (P ) s + ∫ t 0 γ(s, Y s ) ∂f ∂y (s, X α s , Y s , P s ) dW (Y ) s + ∫ D 0 ∫ t 0 [ f (s, X α s − (1 − u)z, Y s , P s ) − f (s, X α s , Y s , P s ) ]( m(ds, dz) − λ(s, Y s ) dF Z (z) ) . (A.2) We only need to prove that this is an {F t }-martingale. Let us observe that E [∫ T 0 ( w s σ(s, P s ) ∂f ∂x (s, X α s , Y s , P s ) ) 2 ds ] < ∞ E [∫ T 0 ( P s σ(s, P s ) ∂f ∂p (s, X α s , Y s , P s ) ) 2 ds ] < ∞ E [∫ T 0 ( γ(s, Y s ) ∂f ∂y (s, X α s , Y s , P s ) ) 2 ds ] < ∞ because all the partial derivatives are bounded and using, respectively, the definition of the set U, (2.14) and (2.2). Thus the first three integrals in (A.2) are well defined and, according to the Itô integral theory, they are martingales. Finally, the jump term in (A.2) is a martingale too, being the function f bounded. Then our statement follows by the proof of Lemma 2.1 (see Appendix A) by replacing {F t }predictable and [0, D]-indexed processes with {G t }-predictable and [0, D]-indexed processes. Definition 2.1. (Proportional reinsurance premium) Let us define a function q(t, y, u) : [0, T ] × R × [0, 1] → [0, +∞), continuous w.r.t. the triple (t, y, u), having continuous partial derivatives Example 4.2. (Exponentially distributed claims under the expected value principle) Let us come back to example 4.1. Under the expected value principle Lemma 4. 3 . 3Under the variance premium principle, i.e. if the reinsurance premium admits the following expression q(t, y, u) = E[Z]λ(t, y)u + θ r E[Z 2 ]λ(t, y)u 2 (4.12) Remark 4 . 4 . 44For any reinsurance premium {q t } t∈[0,T ] admitting the following representation q(t, y, u) = λ(t, y)Q(t, u) (4.15) for a suitable 5 function Q : [0, T ] × [0, 1] → [0, +∞) Lemma this case we choose expression (4.12) and the equation (4.16) is satisfied. {M r } r∈[t,T ] is an {F r }-local martingale and {τ n } n=1,... is a localizing sequence for {M r } r∈[t,T ] . Figure 1 : 1The effect of the risk-aversion parameter η on the optimal initial strategy any variation of the risk-aversion. Figure 2 : 2The effect of the reinsurance safety loading θr on the optimal initial strategy Figure 3 : 3The effect of the time horizon T on the optimal initial strategy Figure 4 : 4Dynamical reinsurance strategies. The dashed line represents the optimal (deterministic) strategy under EVP. Figure 5 : 5The effect of the risk-aversion parameter η on the optimal initial strategy Figure 6 : 6The effect of the volatility parameter σ on the optimal initial strategy Figure 7 : 7The effect of the risk-free interest rate R on the optimal initial strategy t ′ , y ′ , u)\| ≤ max u∈K \|H(t, y, u) − H(t ′ , y ′ , u)\| ≤ L(\|t − t ′ \| ξ + \|y − y ′ \| ξ ) R(T −r) (1−u)z m(dr,dz) \| F t ]× × E[e −η ∫ T t e R(T −s) w[µ(s,Ps)−R] ds e −η ∫ T t e R(T −s) wσ(s,Ps) dW (P ) s \| F t ] ≤ C e −ηx e η K R (e R(T −t) −1) E[e ηe RT ∫ T t ∫ D 0 z m(dr,dz) \| F t ] E[e −ηwe RT ∫ T t σ(s,Ps) dW (P ) s \| F t ]where C is a positive constant and the first expectation is finite because of the proof of the Proposition 2.1. Now let us define the stochastic process {h t } t∈[0,T ] as h t = ηwe RT σ(t, P where the sequence of i.i.d. strictly positive F 0 -random variables {Z i } i=1,... represents the amount of the claims. In the sequel we will assume that all the {Z i } i=1,... are distributed like a r.v. Z, independent on {N t } t∈[0,T ] and {Y t } t∈[0,T ] , with distribution function F Z (dz) such that F Z (z) = 1 ∀z ≥ D, with D ∈ R + (eventually D = +∞). Moreover, Z satisfies some suitableintegrability conditions (see (2.19) below). Consider the random measure associated with the marked point process {C t } t∈[0,T ] defined as follows m(dt, dz) = ∑ t∈[0,T ]: ∆Ct̸ =0 Proof. Let us denote with {M u t } t∈[0,T ] the following {F t }-martingale:4.4. For any {F Y t } t∈[0,T ] -predictable reinsurance strategy {u t } t∈[0,T ] we have that for any t ∈ [0, T ] var [∫ t 0 u s dC s ] = E[Z 2 ]E [∫ t 0 u 2 s λ s ds ] + E[Z] 2 var [∫ t 0 u s λ s ds ] . (4.17) Table 1 : 1Simulation parameters t ∈ [0, T ]; 2. for all couple (t, p) ∈ [0, T ] × (0, +∞) the solution {P t,p (s)} s∈[t,T ] does not explode, i.e.P[ sup s∈[t,T ] P t,p (s) < ∞] = 1; Now let us notice thatE[e ηe RT ∫ T t ∫ D 0 z m(dr,dz) \| F t ] = E[e ηe RT ∑ N T i=N t Zi \| F t ] = ∑ n≥Nt E[e ηe RT ∑ n i=N t Zi \| F t ] P[N T = n \| F t ] i=Nt e ηe RT Zi \| F t ] P[N T = n \| F t ] = ∑ n≥Nt E[e ηe RT Z \| F t ] (n−Nt) P[N T = n \| F t ] = ∑ n≥0 E[e ηe RT Z ] n P[N T − N t = n \| F t ] = ∑ n≥0 E[e ηe RT Z ] n EProof of Lemma 2.2. Assume that there exists a positive constant K ′ such that \|µ(t, p)\| + σ(t, p) ≤ K ′ ∀(t, p) ∈ [0, T ] × (0, +∞).= ∑ n≥Nt E [ n ∏ [ (∫ T t λ s ds ) n n! e − ∫ T t λs ds \| F t ] = E [ e (E[e ηe RT Z ]−1) ∫ T t λs ds \| F t ] < ∞ ⟨P = 1⟩ because of the Assumption 2.1. E[e −ηwe RT ∫ T t σ(s,Ps) dW (P ) s \| F t ] = E[e − s ds \| F t ] < ∞and the proof is complete.∫ T t hs dW (P ) s \| F t ] = E[L T e 1 2 ∫ T t h 2 s ds \| F t ] L t ≤ EP[e 1 2 ∫ T t h 2 See e.g.[Brémaud, 1981, II] This is a classical assumption which implies that the financial market is independent on the insurance market. Intuitively, we note that X α t,x (T ) = X α t,0 (T ) + xe R(T −t) and we use the exponential form of the function v. I.e. Q is such that q fulfills the Definition 2.1. 6 See e.g.[Young, 2006]. See e.g.[Merton, 1969]. Let us observe that under Q the dynamics of {Yt} and {Rt} do not change. Under a practical point of view, we simulated the stochastic processes using the classical Euler's approximation method, with dt = T 500 . See[Pascucci, 2011], Theorem 9.11, p. 281. See[Pascucci, 2011], Theorem 9.11, p. 281. AcknowledgementsThe authors would like to thank Prof. Cristina Caroli Costantini for helpul technical suggestions.A. AppendixProof of Lemma 2.1. First, let us start considering all the [0, D]-indexed processes {H(t, z)} t∈[0,T ] of this type:where {H t } t∈[0,T ] is a nonnegative and {F t }-predictable process. Using the independence be-Using[Brémaud, 1981, App. A1, T4 Theorem, p.263] this result can be extended to all nonnegative, {F t }-predictable and [0, D]-indexed process {H(t, z)} t∈[0,T ] and this completes the proof.Proof of Proposition 2.1. For any constant strategy α t = (u, w) with u ∈ [0, 1] and w ∈ R we have thatbecause of the independence between the financial and the insurance markets. In particular, for the null strategy α t = (0, 0), using the inequality (2.10), we have that the intensity function λ(t, y) is bounded and Hölder-continuous in. t, y) ∈ [0, T ] × R with exponent 0 < ξ ≤ 1the intensity function λ(t, y) is bounded and Hölder-continuous in (t, y) ∈ [0, T ] × R with exponent 0 < ξ ≤ 1; Then the function f (t, y) defined in (6.14) satisfies the Cauchy problem (6.2) and there exists a unique classical solution to (6.2). Moreover, we have that f ∈ C. T ) × R). 12Then the function f (t, y) defined in (6.14) satisfies the Cauchy problem (6.2) and there exists a unique classical solution to (6.2). Moreover, we have that f ∈ C 1,2 ((0, T ) × R). Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint. L Bai, J Guo, Insurance: Mathematics and Economics. 42, y) is continuous and bounded from above. References [Bai and GuoProof. 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[ "CONSTRAINING STELLAR MASS BLACK HOLE MERGERS IN AGN DISKS DETECTABLE WITH LIGO", "CONSTRAINING STELLAR MASS BLACK HOLE MERGERS IN AGN DISKS DETECTABLE WITH LIGO" ]	[ "Barry Mckernan \nDept. of Science\nCUNY-BMCC\n199 Chambers St10007New YorkNY\n\nDept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY\n", "K E Saavik Ford \nDept. of Science\nCUNY-BMCC\n199 Chambers St10007New YorkNY\n\nDept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY\n", "J Bellovary \nDept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY\n\nDept. of Physics\nCUNY-QCC\n11364BaysideNew York, NY\n", "N W C Leigh \nDept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY\n", "Z Haiman \nColumbia Astrophysics Laboratory\nColumbia University\n10027New YorkNY\n", "B Kocsis \nInstitute of Physics\nEötvös University\n1117BudapestHungary\n", "W Lyra \nDept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY\n", "M.-M Mac Low \nDept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY\n", "B Metzger \nColumbia Astrophysics Laboratory\nColumbia University\n10027New YorkNY\n", "M O'dowd \nDept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY\n\nDept. of Physics\nCUNY-Lehman\n10468New YorkNY\n", "S Endlich \nStanford Institute for Theoretical Physics\nStanford University\n94306CA\n", "D J Rosen \nDept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY\n" ]	[ "Dept. of Science\nCUNY-BMCC\n199 Chambers St10007New YorkNY", "Dept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY", "Dept. of Science\nCUNY-BMCC\n199 Chambers St10007New YorkNY", "Dept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY", "Dept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY", "Dept. of Physics\nCUNY-QCC\n11364BaysideNew York, NY", "Dept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY", "Columbia Astrophysics Laboratory\nColumbia University\n10027New YorkNY", "Institute of Physics\nEötvös University\n1117BudapestHungary", "Dept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY", "Dept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY", "Columbia Astrophysics Laboratory\nColumbia University\n10027New YorkNY", "Dept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY", "Dept. of Physics\nCUNY-Lehman\n10468New YorkNY", "Stanford Institute for Theoretical Physics\nStanford University\n94306CA", "Dept. of Astrophysics\nAmerican Museum of Natural History\n10028Central Park West, New YorkNY" ]	[]	Black hole mergers detectable with LIGO can occur in active galactic nucleus (AGN) disks. Here we parameterize the merger rates, the mass spectrum and the spin spectrum of black holes (BH) in AGN disks. The predicted merger rate spans ∼ 10 −4 − 10 4 Gpc −1 yr −1 , so upper limits from LIGO (< 212Gpc −1 yr −1 ) already constrain it. The predicted mass spectrum has the form of a broken power-law consisting of a pre-existing BH powerlaw mass spectrum and a harder powerlaw mass spectrum resulting from mergers. The predicted spin spectrum is multi-peaked with the evolution of retrograde spin BH in the gas disk playing a key role. We outline the large uncertainties in each of these LIGO observables for this channel and we discuss ways in which they can be constrained in the future.	10.3847/1538-4357/aadae5	[ "https://arxiv.org/pdf/1702.07818v2.pdf" ]	119,478,929	1702.07818	5941ae51ac29e342088325b8bd974fce397211c9	CONSTRAINING STELLAR MASS BLACK HOLE MERGERS IN AGN DISKS DETECTABLE WITH LIGO 30 Jan 2018 Draft version January 31, 2018 Barry Mckernan Dept. of Science CUNY-BMCC 199 Chambers St10007New YorkNY Dept. of Astrophysics American Museum of Natural History 10028Central Park West, New YorkNY K E Saavik Ford Dept. of Science CUNY-BMCC 199 Chambers St10007New YorkNY Dept. of Astrophysics American Museum of Natural History 10028Central Park West, New YorkNY J Bellovary Dept. of Astrophysics American Museum of Natural History 10028Central Park West, New YorkNY Dept. of Physics CUNY-QCC 11364BaysideNew York, NY N W C Leigh Dept. of Astrophysics American Museum of Natural History 10028Central Park West, New YorkNY Z Haiman Columbia Astrophysics Laboratory Columbia University 10027New YorkNY B Kocsis Institute of Physics Eötvös University 1117BudapestHungary W Lyra Dept. of Astrophysics American Museum of Natural History 10028Central Park West, New YorkNY M.-M Mac Low Dept. of Astrophysics American Museum of Natural History 10028Central Park West, New YorkNY B Metzger Columbia Astrophysics Laboratory Columbia University 10027New YorkNY M O'dowd Dept. of Astrophysics American Museum of Natural History 10028Central Park West, New YorkNY Dept. of Physics CUNY-Lehman 10468New YorkNY S Endlich Stanford Institute for Theoretical Physics Stanford University 94306CA D J Rosen Dept. of Astrophysics American Museum of Natural History 10028Central Park West, New YorkNY CONSTRAINING STELLAR MASS BLACK HOLE MERGERS IN AGN DISKS DETECTABLE WITH LIGO 30 Jan 2018 Draft version January 31, 2018(Received; Revised; Accepted) Submitted to ApJarXiv:1702.07818v2 [astro-ph.HE] Typeset using L A T E X twocolumn style in AASTeX61black holes, LIGO -AGN disks -mergers Black hole mergers detectable with LIGO can occur in active galactic nucleus (AGN) disks. Here we parameterize the merger rates, the mass spectrum and the spin spectrum of black holes (BH) in AGN disks. The predicted merger rate spans ∼ 10 −4 − 10 4 Gpc −1 yr −1 , so upper limits from LIGO (< 212Gpc −1 yr −1 ) already constrain it. The predicted mass spectrum has the form of a broken power-law consisting of a pre-existing BH powerlaw mass spectrum and a harder powerlaw mass spectrum resulting from mergers. The predicted spin spectrum is multi-peaked with the evolution of retrograde spin BH in the gas disk playing a key role. We outline the large uncertainties in each of these LIGO observables for this channel and we discuss ways in which they can be constrained in the future. INTRODUCTION The gravitational wave (GW) events detected by the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) correspond to the merger of stellar mass black holes (BH) considerably more massive than those observed in our own Galaxy. The upper end of the range of BH merger rates derived from LIGO observations of 212 Gpc −3 yr −1 (Abbott et al. 2016b) requires consideration of locations where BH mergers can occur faster than expected from GW emission alone. Among the first few LIGO detections are possible low value spin or misaligned spins, which may be problematic for models of binary evolution (O'Shaugnessey, Gerosa & Wysocki 2017). While BHs with larger than expected masses can occur naturally in the field (Belczynski et al. 2010;deMink & Mandel 2016), they are more likely to form in regions with concentrations of BHs, such as galactic nuclear star clusters (Hopman & Alexander 2006;O'Leary et al. 2009;Antonini & Rasio 2016;Rodriguez et al. 2016). Massive gas disks in active galactic nuclei (AGN) provide natural locations for gas accretion and repeated mergers because the gas disk can drive migration of BH towards migration traps, reduce the inclination of intersecting orbits, enable binary formation, and harden existing binaries. Together, these effects can result in rapid increase in the mass of embedded BHs, potentially to observed values (e.g. McKernan et al. 2012McKernan et al. , 2014Bellovary et al. 2016; Bartos et al. 2017;Stone et al. 2017). In this paper we parameterize the expected merger rate, and the mass and spin distributions from this channel for comparison with the LIGO observations, and we discuss how observations and simulations can constrain these predictions. MODEL OUTLINE Galactic nuclei likely contain some of the densest concentrations of BHs in the Universe (e.g. Morris 1993;Miralda-Escudé & Gould 2000, and references therein), so it is natural to look for BH mergers in galactic nuclei (O'Leary et al. 2009;McKernan et al. 2012;Antonini et al. 2014). While BH binary mergers can occur at modestly enhanced rates (compared to the field) in nuclear star clusters just from dynamical binary hardening (Antonini & Rasio 2016;Rodriguez et al. 2016), or capture from single-single (O'Leary et al. 2009) and binary-single encounters (Samsing et al. 2014), a dense nuclear disk of gas can greatly accelerate the rate of BHB formation and merger (McKernan et al. 2012(McKernan et al. , 2014. The simplest picture of this LIGO channel begins with a spherical distribution of BH, stars and other stellar remnants orbiting in the central pc 3 of a galactic nuclei around a supermassive black hole (SMBH). Next, around the SMBH, we add a massive gas disk, which can be geometrically thin or thick. A fraction f co of the initial number of BH in the nucleus N BH , will have orbits coincident with the disk and approximately half of these orbits should be retrograde compared to the disk gas. Yet another fraction f g of the population N BH intersect the disk on their orbits and are ground down into the plane of the disk within the AGN disk lifetime (τ AGN ). Thus an overall fraction f d = f co + f g of nuclear BH end up embedded in the disk, and quickly have their orbits damped and circularized by gas drag (e.g. McKernan et al. 2012). The net torques from disk gas causes BH to migrate within the disk and encounter each other at low relative velocities (McKernan et al. 2012;Bellovary et al. 2016). BH binaries that form in the disk are expected to merge efficiently due to gas torques (e.g. Haiman et al. 2009;Kocsis et al. 2011;McKernan et al. 2011;Stahler 2010;Baruteau et al. 2011;McKernan et al. 2012). BH mergers may preferentially occur in convergence zones containing migration traps Bellovary et al. (2016) which occur in semirealistic models of AGN disks (Sirko & Goodman 2003;Thompson et al. 2005). Multiple objects trapped in such orbits collide efficiently rather than being ejected (Horn et al. (2012); Secunda, Bellovary et al. (2018) in prep.). In this paper, we examine what constraints can be put on the merger rate and the BH spin and mass distributions for this AGN channel. RATE OF BLACK HOLE BINARY MERGERS IN AGN DISKS We parameterize the rate of BH-BH mergers in AGN disks simply as: R = N GN N BH f AGN f d f b ǫ τ AGN(1) where N GN (Mpc −3 ) is the average number density of galactic nuclei in the Universe, f AGN is the fraction of galactic nuclei that have active AGNs which last for time τ AGN , f d = f co + f g is the fraction of nuclear BH that end up in the disk, f b is the fraction of BH in BH-BH binaries in the disk, and ǫ represents the fractional change in number N BH of BH in the central region (∼ pc 3 ) over a full AGN duty cycle 1 R can be parameterized as: 1 If ǫ ∼ 1 then N BH is approximately conserved between AGN episodes. If ǫ(>) < 1 N BH (grows) shrinks between AGN phases due to the net effect of mergers, infall of new BH, stellar evolution etc.. R = 12Gpc −3 yr −1 N GN 0.006Mpc −3 N BH 2 × 10 4 f AGN 0.1 (2) × f d 0.1 f b 0.1 ǫ 1 τ AGN 10Myr −1 . However, if we want to constrain the constributions of this channel to LIGO observations, it is much more useful to show the allowed range of R and the range of each of the contributing factors from eqn. (1), which we list in Table 1. The N GN lower limit corresponds to galaxies with stellar mass greater than or equal to that of the Milky Way (Baldry et al. 2012) as measured from Schechter function fits to galaxy luminosity functions (e.g. Cole et al. 2001). The N GN upper limit corresponds to dwarf galaxies with stellar mass > 10 9 M ⊙ (Baldry et al. 2012), which includes all locally observed SMBH (≥ 10 5 M ⊙ ) inferred from M −σ studies of galaxies and dwarf galaxies (Reines & Volonteri 2015). Also in Table 1, N BH ∼ 10 3 corresponds to the number of BH allowed ≤ 0.1pc −3 of Sgr A* according to the distribution of the S-star orbits (Antonini et al. 2014), whereas N BH ∼ 10 6 pc −3 seems to be the maximal density allowed by simulations (Antonini et al. 2014). The lower limit to f AGN assumes only quasar disks are efficient BH merger sites and f AGN ∼ 0.3 assumes all LINER galactic nuclei (Ho 2008) consist of advection dominated accretion flows (ADAFs) with high accretion rate (Paczynski & Witta 1980;Narayan & Yi 1995;Lasota et al. 2016), capable of driving BH mergers. The binary fraction of BH f b has been estimated to be as high as f b ∼ 0.2 (Antonini et al. 2014)), but dynamically hot environments such as star clusters, could actually yield very low binary fractions f b ≤ 0.01 over time in the absence of gas (Miller & Davies 2012;Leigh et al. 2016) due to the large number of 'ionizing' interactions, so we choose f b = [0.01, 0.2] in Table 1. Reasonable estimates of τ AGN span 0.1-100Myr (Haehnelt & Rees 1993;King & Nixon 2015;Schawinski et al. 2015). R will be highest if AGN episodes are short-lived but frequently repeated and efficient at BH mergerse. These circumstances ensure that there are multiple opportunities for BH in a galactic nucleus to encounter each other at low relative velocity and merge in a disk. From Table 1, the allowed range from Eqn. (1) is R ∼ 10 −4 -10 4 Gpc −3 yr −1 . The upper bound to the LIGO BH binary merger rate of ∼ 240Gpc −3 yr −1 already rules out upper limits to most parameters in Ta- Thompson et al. 2005). h/R ∼ 0.1-0.7 in super-Eddington ADAFs (Lasota et al. 2016). fg depends on h/R, ρ disk and τAGN . f d = fco + fg. fco comes from h/R, the disk aspect ratio. h/R ∼0.01-0.1 (Sirko & Goodman 2003). h/R ∼ 10 −3 -0.1 ( ble 1 2 and allows actual astrophysical limits to be placed on models of AGN disks by LIGO BH merger detections. Future observational constraints and simulation results will, however, be required to figure out which upper limits are ruled out by LIGO. For example, the upper limit to N GN could be reduced by contrasting activity rates as a function of galactic mass in a complete sample. The inferred N BH can be constrained via population studies of the X-ray emission from binaries around Sgr A* and in M31, as well as via dynamics studies of the number density of BH allowed from the orbital parameters of stars in galactic nuclei. The upper limit on f AGN can be reduced if we can observationally distinguish between high-and low-accretion rate LINERs. Simulations that include a spherical component of individual stars and BH as well as migrating objects in the disk are required to properly constrain f b . Encounters between objects from the spherical dynamical component and the disk dynamical component will occur at relatively high velocity and can therefore ionize sufficiently soft, large radius, binaries. Thus, in order for f b to be moderately large in this channel, we require f g to be large, since otherwise the rate of ionizing encounters can ionize binaries (Leigh et al. 2017). So limits on f g from semi-analytic approaches or simulations (Kennedy et al. 2016) can also help constrain f b . Uncertainties in R are dominated mainly by lack of knowledge of the distribution and number of BH in galactic nuclei, how efficiently gas disks can grind down orbits, and whether geometrically thick disks can efficiently merge BHs. Understanding multiple-object migration and the role of retrograde orbiters is another key area for future work. CONSTRAINING BH MASSES By merging BHs in AGN disks, we expect 'overweight' BH to result (McKernan et al. 2012). To investigate the range of BH masses involved in mergers in this channel, we use a toy model calculation of the evolution of a population of BH embedded and migrating in an AGN disk. We made many simplifying assumptions: there are no BH binaries to begin with (f b = 0), BH remain in the disk after merger, tertiary encounters are neglected, no BHs merge with the SMBH, no new BH are added to the population (f g = 0) and we ignore mass growth due to gas accretion. We began with a uniform distribution of BH drawn from a Kroupa (2002) initial mass function N BH (M ) ∝ M −γ0 , with γ 0 = 2.3 distributed over three mass bins (5,10,15M ⊙ ) and chose normalization N BH (5 M ⊙ ) = 10 3 . A BH on a prograde orbit in an AGN disk with mass M 1 will migrate on a (Type I) timescale (Paardekooper et al. 2010;McKernan et al. 2012) t mig ≈ 38Myr N 3 −1 R b 10 4 r g −1/2 M 1 5M ⊙ −1 × h/R b 0.02 2 Σ 10 5 kgm −2 −1 M SMBH 10 8 M ⊙ 3/2 (3) where N is a numerical factor of order 3. So the toy model population outlined above will evolve over time. If 10 3 BH are uniformly distributed across a disk of radius R d ∼ 10 5 r g , (r g = GM SMBH /c 2 ), BH orbits are separated by ∼ 10 2 r g on average. This separation could be closed in ∼ 0.4 Myr from eqn. (3). Our initial distribution of singleton BH separated by ∼ 10 2 r g on average will therefore evolve from f b = 0 towards f b ∼ 0.5 within ∼ 0.4Myr due to migration. The probability of encounter between BH of masses M 1 , M 2 in time ∆t is P (M 1 \|M 2 ) ∝ N (M 1 )N (M 2 ) t mig (M 1 )t mig (M 2 ) .(4) When a pair of BHs approaches within their binary Hill radius R H = (q/3) 1/3 R b , where q is the binary mass ratio and R b is the radius of the binary center of mass, gas drag can cause them to merge rapidly. Baruteau et al. (2011) showed that binary semi-major axis a b halves due to gas drag in only 200 (1000) orbits about the binary center of mass for a retrograde (prograde) binary compared to gas velocity. Using this result, a BH binary with a b = R H at R b ∼ 10 3 r g has a characteristic timescale for binary hardening of 0.4 kyr (8 kyr) in the retrograde(prograde) case. Only 20-25 such halvings (corresponding to ∼ 0.1-0.2 Myr, naively assuming a constant gas hardening rate) would shrink a b sufficiently that GW emission takes over and the merger happens promptly. The gas hardening rate may be even faster than this estimate since more gas enters the binary's Hill sphere as it shrinks (Baruteau et al. 2011), which may pump binary eccentricity. However, gas torques may decrease in efficiency once the binary has hardened sufficiently that the binary velocity is substantially supersonic compared to most gas within the Hill radius (Sánchez-Salcedo & Chametla 2014). For our toy model, we therefore assume ∼ 0.1Myr is the minimum gas hardening timescale to merger, but we note that the actual gas hardening timescale could take up to an order of magnitude longer. In our toy model, if the typical time for a BH to encounter another BH in the disk is ∼ 0.4Myr, then adding an additional ∼ 0.1 − 1Myr for a gas-hardening timescale, yields a characteristic time to merger of ∼ 0.5 − 1.5Myr in our model. So, we expect that around half the initial population of our toy model will have encountered each other and merged in this time. In calculating the evolution of our toy model, we chose ∆t ∼ 0.1 − 0.3Myr to correspond to a time when ∼ 10% of the initial population of lowest mass BHs (5 M ⊙ ) have encountered each other and merged. All other encounters are normalized to this encounter rate. For simplicity, we assume all binaries formed in ∆t merge within that time, and we neglect the mass-energy loss from the mergers. After ∆t, all BH that merged are removed from their original mass bins, and the newly merged object is added to the appropriate mass bin. Figure 1 demonstrates the simplistic evolution expected as the initial BH distribution (black line) evolves to the red curve in time step ∆t ∼ 0.1 − 0.3Myr, where ∼ 10% of the lowest mass BHs in the initial (black) distribution have merged. The red curve evolves to the Evolution of an initial 5 − 15M⊙ BH mass distribution (black curve) in an AGN disk based on a toy merger model. Black curve corresponds to the initial BH mass distribution. Red and blue curves shows the evolution of the distribution after timesteps corresponding to ∆t ≈ 0.1 − 0.3 Myr and ∆t ′ ∼ 0.2 − 0.6 Myr respectively (see text). A choice of heavier inital mass range will alter upper mass limits. blue curve after an additional ∆t ′ ∼ 0.2 − 0.6Myr, when ∼ 10% of the lowest mass BH on the red curve are expected to merge. The BH mass distribution in our toy model flattens from γ 0 = 2.3 to γ ∼ 2 as low-mass BH are consumed. Now assume that BH from the non-disk spherical population, interact with the disk and their orbits are ground down into the disk, i.e. f g > 0. The addition of some of the (initially) spherical BH population into the disk will support the BH mass distribution in the disk at the low mass end. So an initial power law distribution ∝ M −γ0 of BH mass will evolve towards a broken-power law distribution of the form N BH ∝    N 1 M −γ1 for M < M break N 2 M −γ2 for M > M break ,(5) where γ 2 < γ 1 , N 1 /N 2 ∼ (f g /f co ), where f co is the fraction of BH initially in the disk and on average f g is the fraction of BH ground down into the disk over τ AGN /2 and M break lies near the upper end of the inital mass range (∼ 15 M ⊙ in our toy model). In order to include gas accretion in this toy model, we assumed a gas accretion rate for BH on [retrograde, prograde] orbits ofṀ 1 ∼ [10 −2 , 1]Ṁ Edd , wherė is the Eddington mass accretion rate with m p the proton mass and η the accretion luminosity efficiency. Over an AGN disk lifetime of τ AGN ∼ 10Myr, we can neglect gas accretion onto BH on retrograde orbits. In Table 2 we list parameter ranges for BH masses on the basis of the probabilistic toy model outlined above for three different assumptions: 1) N BH ∝ M −2 (roughly the blue curve in Fig. 1), corresponding to a short lived disk with f co ≫ f g . 2) N BH ∝ M −1 , corresponding either to a long lived disk (τ AGN > 10Myr) or efficient gas hardening with a low rate of orbit grind down (f co ≫ f g ). 3) N BH ∝ M −2 (M −1.5 ) for M < 15M ⊙ (> 15M ⊙ ), corresponding either to efficient orbit grind down (f g ∼ f co ), or efficient stellar formation and evolution in the disk with a new top-heavy IMF. In Table 2 we list the binary mass ratio M b range for each set of assumptions. The lower limit to M b is trivially the lowest possible mass binary drawn from the initial mass distribution, with no growth from gas accretion and the upper limit to M b is simply the highest mass binary in the distribution. Also listed in Table 2 are the range of mass ratios (q) of the binaries in the three different scenarios, with the lower limit given by the range of BH masses allowed in the three different distributions and q = 1 is the trivial upper limit. M Edd = 4πGM 1 m p ηc ≈ 2.2 × 10 −7 M ⊙ yr η 0.1 −1 M 1 10M ⊙(6) If the fraction of BH ground down into the disk f g (t) ≥ f co (t), the fraction of BH coincident with the disk, which will be true for relatively long-lived, thin (h/R ≪ 1) disks, the BH mass spectrum evolves from an initial power-law distribution to a broken power-law as in Eqn. (5) with γ 1 ∼ γ 0 > γ 2 . The uncertainty in mass estimates for this channel is driven mainly by the initial mass distribution of BH in the central region, as well as the ratio of f g (t)/f co (t), which in turn depends on disk density and h/R. RANGE OF BH SPINS As black holes in the AGN disk accrete gas and merge with each other, their initial spin distribution will change with time. Assuming a uniform distribution of spins (a) and angular momenta (L) for BH in galactic nuclei, there will be four distinct populations of BHs in AGN disks as follows: 1. Prograde spin, on prograde orbits, denoted by (a + , L + ). 2. Prograde spin, on retrograde orbits (a + , L − ). 3. Retrograde spin, on prograde orbits (a − , L + ). 4. Retrograde spin, on retrograde orbits (a − , L − ). We expect the fraction f co of BH co-orbital with the AGN disk should have an initial uniform distribution across all four BH populations. The four BH populations will evolve differently due to gas accretion. The (a + , L + ) population rapidly accretes gas, spins up, and aligns spins with the disk gas once the BH has accreted a few % of its own mass (Bogdanovic et al. 2007), i.e. in < τ AGN . An initially uniform spin distribution a + = [0, +0.98] evolves towards a + ∼ 0.98 at an average rate ∼ (τ AGN /40Myr)(ṁ/Ṁ Edd ) whereṁ/Ṁ Edd is the average gas accretion rate as a fraction of the Eddington rate (which takes ≈ 40Myr to double mass). By contrast, the (a + , L − ) population faces a strong headwind, so it accretes very weakly from the gas. An initially uniform distribution of spins in this population will remain uniform over τ AGN . The (a − , L + ) population spins down towards a ∼ 0 after an increase of mass by a factor 3/2 (Bardeen 1970) and will then join the (a + , L + ) population. The (a − , L − ) population spins down more slowly due to the headwind and so an initial uniform distribution of spins remains uniform over τ AGN . BH mergers will further complicate the spin evolution of the four BH populations. The four populations interact due to migration and form binaries if captured within the binary Hill sphere. Binary orbital angular momentum (L b ) is the dominant contributor to the spin of the merged BH binary so equal mass BH mergers yield merger products with \|a\| ∼ 0.7 (Hofmann et al. 2016). Binaries can form with prograde or retrograde orbital angular momentum compared to the disk gas Figure 2. Evolution of an initial uniform BH spin distribution in an AGN disk based on a toy merger model, including gas accretion (see text). Spins are binned per 0.05 of spin parameter (a). Black line corresponds to a uniform BH spin distribution for the initial population. The corresponding initial mass distribution is given by the black curve in Fig. 1. The red solid curve shows the spin distribution after the toy model has evolved for ∆t ≈ 0.1 − 0.3 Myr to include mergers and gas accretion at the Eddington rate. The corresponding mass distribution after this time is given by the red curve in Fig. 1. The red dashed curve is as the solid curve, except we assume super-Eddington accretion at ×5 the Eddington rate. (denoted by L ± b ). If a binary forms with retrograde orbital angular momentum (L − b ), the merger is faster than in the prograde case (Baruteau et al. 2011), and the merger product will have a − = −0.7 (i.e. retrograde spin compared to disk gas). Thus the fastest growing of the four populations of BH in the disk due to mergers will actually be (a − , L ± ). This population evolves towards low spin (a ∼ 0) due to gas accretion, at an average rate ∼ (τ AGN /40Myr)(ṁ/Ṁ Edd ). Among the initial fraction f co of co-orbital BHs, we expect equal numbers of prograde to retrograde orbits. However, since prograde orbits are ground down faster (smaller headwind, greater Bondi radius), we expect (a ± , L + )/(a ± , L − ) ≈ 1 + (f g /f co ). Applying all of this to our toy model above allows us to construct the spin distribution in Fig. 2. An initial uniform spin distribution (black line) evolves towards the solid red curve after ∆t ≈ 0.1 − 0.3Myr. The corresponding mass distribution is the red curve in Fig. 1. The red solid curve in Fig. 2 shows a prominent peak at a = −0.7 due to a ×5 faster merger rate of retrograde binaries and a smaller peak at a = +0.7 due to mergers of prograde binaries. Both peaks are smeared out towards the right by gas accretion during ∆t and will consist of BH masses ≥ 10M ⊙ from the initial mass distribution. Some pile-up is happening at a > 0.95 due to gas accretion onto the already near maximal spinners of the (a + , L + ) population. The red dashed curve shows what happens if we assume gas accretion can occur at super-Eddington rates onto BH in the disk ( ×5 the Eddington rate). In particular the more massive merged population at a ∼ −0.7 gets quickly smeared out and driven towards low spin. Thus, from Fig. 2 if LIGO constrain the spins of most merger precursor BHs to be small, the AGN channel requires super-Eddington accretion onto initially retrograde spin BH to grow this population. Only the (a + , L + ) population will align or anti-align relatively quickly with the AGN disk gas. Assuming the (a + , L + ) population are all aligned or anti-aligned with the disk gas, by drawing randomly from a uniform distribution across (a ± , L ± ), there is a ≈ 1/16 chance that both BH have (anti-)aligned spins and represents our lower limit for the fraction of BH (anti-) aligned with disk gas. If f g (t) ≫ f co (t), then effectively the two populations (a ± , L + ) will dominate so f ±align ≈ 1/4, which is our approximate upper limit for the fraction of BH (anti-) aligned with disk gas. Our estimates of f ±align suggest that a larger population of mergers will be requied to test this channel in population spin studies than estimated by Fishbach et al. (2017); Gerosa & Berti (2017). Anti-aligned binaries in the AGN disk allow LIGO a unique chance to test the spin precession instability (Gerosa et al. 2015). Once a BH binary merges, the resulting merger product can experience a gravitational radiation recoil kick of v kick ∼ 20-400 km s −1 , depending on relative spins and mass ratios (e.g. Merritt et al. 2004;Campanelli et al. 2007). The result of kicks from mergers between aligned and anti-aligned objects is to incline the merger product's orbit relative to the AGN disk by θ = tan −1 (v kick /v orb ) where v orb is the orbital velocity of the binary center of mass. Since v orb ≫ 400km/s in most of the disk, the orbital inclination perturbation is at most a few degrees and the merger product could be ground back down into the disk in time < τ AGN . Mergers of BH with spins out of alignment with the plane of the disk and each other can produce the largest magnitude kicks (up to several thousand kilometers per second) (e.g. Schnittman & Buonnano 2007;Lousto et al. 2012). Such mergers will be rare, but will produce large kicks (∝ q 2 /(1 + q) 4 in the mass ratio q, Campanelli et al. (2007)), escape the disk at angle θ and may not be ground back down within τ AGN . Table 3 summarizes the ranges allowed for spins in this LIGO channel. The typical spin distribution depends on the relative fractions of the four populations of BH in Table 3. Parameter ranges in BH spins. Parameter Lower Upper (1) (2) (3) a + (L + ) 0 0.98 a − (L + ) -0.98 0 a + (L − ) 0.0 0.98 a − (L − ) -0.98 0 amerge -0.7 +0.7 f ±align 0.06 0.25 Note-Parameter ranges allowed for BH spins in this channel (see text). the disk (a ± , L ± ) and their evolution as f g /f co changes, driven in turn by disk aspect ratio (h/R) and the disk gas density and τ AGN . We expect an initial population uniform across (a ± , L ± ), but (a ± , L + ) will grow with the fraction f g (t) of BH ground-down into the disk. Peaks will arise in the spin distribution at a ∼ −0.7, +0.7 due to mergers and gas accretion will drive a − → 0 and a + → 0.98 independent of mergers. Gas accretion at super-Eddington rates plus faster mergers by retrograde binaries may be required to generate a population of overweight, low spin BH in the AGN disk. 6. OBSERVATIONAL CONSTRAINTS: GW Binary black hole mergers in an AGN disk imply unique, testable predictions that would not be expected from other BH merger channels, including: 1. A spin distribution (see §5) that includes aligned/anti-aligned spin binaries and 2. a population of overweight BH or IMBH orbiting SMBHs, generating GWs detectable with the Laser Interferometer Space Antenna (LISA) (McKernan et al. 2014). A circularized IMBH-SMBH binary at a migration trap (a b ∼ 10 2 r g ) around a SMBH with M SMBH < 10 7 M ⊙ will be detectable with LISA at modest signal-to-noise ratio in a year's observation (McKernan et al. 2014). If AGN disks are efficient at gas-driven mergers of BH, we expect that every AGN must contain one or more IMBH-SMBH binaries, implying an approximate rate comparable to that in Portegies Zwart et al. (2006). OBSERVATIONAL CONSTRAINTS:EM The brightest AGN are too bright compared to any short-term EM signal that might result from a BH merger in a gas disk. Low luminosity AGN might permit short timescale EM events from BH mergers to be visible. As IMBHs grow in migration traps, gaps and cavities in the accretion flow can form and oscillations on the dynamical timescale of the accreting IMBH can be detected in optical, UV, and X-ray spectral signatures (e. g. McKernan et al. 2013McKernan et al. , 2014McKernan & Ford 2015). Temporal and energetic asymmetries in the X-ray signatures are best detected using micro-calorimeters, such as the one that will fly on the X-ray Astronomy Recovery Mission succeeding Hitomi. Perturbations of the innermost disk will occur as migrators in the disk plunge into the SMBH and temporarily dominate the local corotating mass, detectable in large UV-optical quasar surveys (Drake et al. 2009) as well as the X-ray band. Large optical surveys of quasar disks can also limit total supernova rates due to migrating/accreting/colliding stars (Graham et al. 2017), in turn placing limits on the disk populations of stars and stellar remnants. Estimates of the rates of transits by bloated stars, best detected in the X-ray band (McKernan & Yaqoob 1998), can put limits on the population on spherical orbits around and passing through AGN disks. As the AGN phase ends, remaining BH will interact dynamically, so the distribution of orbital parameters of the BHs and stars entrained in the disk will relax. Alexander et al. (2007) show that if very massive stars (> 10 2 M ⊙ ) exist in our own Galactic nucleus, they can pump the eccentricity distribution of massive stars to even e ∼ 0.4 within 5 Myrs. However, such stars are short-lived and observed stellar eccentricities reach e ∼ 0.7 (Paumard et al. 2006). On the other hand, a population of overweight BHs caused by merger in an AGN disk can rapidly pump stellar orbital eccentricites post-AGN and inflate the thickness (h/R) of stellar disks in galactic nuclei. Thus, if this BH merger channel is efficient, thin disks of stars will not be observed in post-AGN galactic nuclei. Neutron stars (NS) should also exist in AGN disks, and can migrate. So there should be a correlation between NS-NS and NS-BH mergers in AGN disks and the rate of BH-BH mergers expected from this chan-nel. No correlation has been observed so far between short gamma-ray bursts in the local universe and AGNs (Berger 2014), but so far, only a handful of short gamma-ray bursts have sufficiently accurate positions in the sky to rule out an association with AGN in these cases. The efficiency of this LIGO channel could be further constrained by ongoing studies of the correlation of short gamma-ray bursts with AGN. Future simulations could usefully focus on the expected distribution of NS in mass segregating clusters in galactic nuclei, and ultimately on determining the expected NS merger rate in AGN disks. CONCLUSIONS We parameterize the rate of black hole mergers within AGN disks and the mass and spin distributions that result. The strongest observational constraints can be placed on this channel by: 1. ruling out a population of maximal spin BH via LIGO, 2. ruling out a correlation betwen short gamma-ray bursts and AGN, 3. constraining the rate of obscured supernovae in AGN disks via studies of large samples of AGN, 4. ruling out a population of high accretion rate ADAFs in galactic nuclei and 5. observing very thin disks of stars in nearby Galactic nuclei. Future simulations should focus on 1. the ratio of NS/BH in nuclear star clusters undergoing mass segregation, 2. encounters between prograde and retrograde orbiters in AGN disks and 3. interactions and binary formation between BHs with pro-and retrograde spins and orbits at migration traps in a range of AGN disk models. If AGN are efficient at merging BH, LISA will detect a large population of IMBH in disks around SMBH in the nearby Universe. 9. ACKNOWLEDGEMENTS. Figure 1 . 1Figure 1. Evolution of an initial 5 − 15M⊙ BH mass distribution (black curve) in an AGN disk based on a toy merger model. Black curve corresponds to the initial BH mass distribution. Red and blue curves shows the evolution of the distribution after timesteps corresponding to ∆t ≈ 0.1 − 0.3 Myr and ∆t ′ ∼ 0.2 − 0.6 Myr respectively (see text). A choice of heavier inital mass range will alter upper mass limits. Table 1 . 1Parameter ranges in Eqn. 1.Parameter Lower Upper N a GN (Mpc −3 ) 4 × 10 −3 10 −2 N b BH (pc −3 ) 10 3 10 6 f c AGN 0.01 0.3 f b 0.01 0.2 f d d 0.01 0.7 τAGN (Myr) 1 100 ǫ 0.5 2 R(Gpc −3 yr −1 ) 10 −4 10 3 Note-Range of parameters in Eqn. (1) and range of merger rate (see text). a from Baldry et al. (2012). b from Miralda-Escudé & Gould (2000); Antonini et al. 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[ "TROPICAL DESCENDANT GROMOV-WITTEN INVARIANTS", "TROPICAL DESCENDANT GROMOV-WITTEN INVARIANTS" ]	[ "Hannah Markwig ", "Johannes Rau " ]	[]	[]	We define tropical Psi-classes on M 0,n (R 2 , d) and consider intersection products of Psiclasses and pull-backs of evaluations on this space. We show a certain WDVV equation which is sufficient to prove that tropical numbers of curves satisfying certain Psi-and evaluation conditions are equal to the corresponding classical numbers. We present an algorithm that generalizes Mikhalkin's lattice path algorithm and counts rational plane tropical curves satisfying certain Psi-and evaluation conditions.	10.1007/s00229-009-0256-5	[ "https://arxiv.org/pdf/0809.1102v2.pdf" ]	14,692,769	0809.1102	7adc254d1cf45361f182dd99d14cc30b48234f98	TROPICAL DESCENDANT GROMOV-WITTEN INVARIANTS 29 Nov 2009 Hannah Markwig Johannes Rau TROPICAL DESCENDANT GROMOV-WITTEN INVARIANTS 29 Nov 2009 We define tropical Psi-classes on M 0,n (R 2 , d) and consider intersection products of Psiclasses and pull-backs of evaluations on this space. We show a certain WDVV equation which is sufficient to prove that tropical numbers of curves satisfying certain Psi-and evaluation conditions are equal to the corresponding classical numbers. We present an algorithm that generalizes Mikhalkin's lattice path algorithm and counts rational plane tropical curves satisfying certain Psi-and evaluation conditions. INTRODUCTION Psi-classes ψ i are certain divisor classes on spaces of stable curves or stable maps, M g,n or M g,n (P r , d), which arise as the first Chern class of the line bundle whose fiber over a point (C, x 1 , . . . , x n ) (or (C, x 1 , . . . , x n , f )) is the cotangent space of C at x i . They are for example useful to count curves with tangency conditions. To count curves that satisfy incidence conditions (e.g. pass through given points), one defines evaluation maps on the space of stable maps, ev i : M g,n (P r , d) → P r that send a stable map (C, x 1 , . . . , x n , f ) to the image f (x i ) of the marked point i. Then we can pullback the conditions via the evaluation map. Finally, we can intersect pullbacks of evaluation maps and Psi-classes on M g,n (P r , d). The degrees of such top-dimensional intersection products are called descendant Gromov-Witten invariants and have been studied in Gromov-Witten theory. The aim of this paper is to define tropical analogues of rational descendant Gromov-Witten invariants in the plane and to show that, under certain assumptions on the distribution of the Psi-and evaluation conditions, they coincide with their conventional counterparts. To do so, we use the constructions of moduli spaces of abstract and parameterized rational tropical curves as tropical varieties and the definition of evaluation maps on the latter ones ( [SS04], [GKM07], [Mi07]). Moreover, [Mi07] already defines Psi-classes on the space of abstract tropical curves M 0,n and [KM07] deals with their intersections. In this paper, we define Psi-classes on the space of parameterized tropical curves. Together with the intersection theory of [Mi06] and [AR07], we have all tools to define descendant Gromov-Witten invariants. We show that these invariants are independent of the position and "type" of the evaluation conditions and that they fulfill the string and divisor equation. Then we show that the invariants also fulfill a certain WDVV equation which can be used to determine the numbers recursively. As the classical numbers fulfill the same equations, it remains to compare the initial numbers appearing in the recursion to show that the classical and tropical invariants coincide. These results can only be achieved for invariants such that the Psi-conditions come together with point conditions, and not alone or with line conditions. Note that one should expect such restrictions as we work with a non-compact moduli space that does not parametrize curves with components in the "boundary" of R n . Hence the tropical descendant Gromov-Witten invariants are different from the classical ones in some cases, namely whenever tropical curves are "missing" in the corresponding tropical count. However, we show that this does not happen when Psi-conditions always come together with point conditions. To prove the WDVV equation we show that the weight of a curve in an intersection product can be computed locally as the determinant of a map (which basically collects all evaluation maps) and then proceed similarly to [GM05]. Finally, we present a tropical algorithm similar to Mikhalkin's lattice path count ( [Mi03]) to determine the numbers of rational plane tropical curves passing through points and satisfying Psi-conditions. Our definition of tropical descendant invariants partly agrees with Mark Gross' definition which was found independently in his study of mirror symmetry ( [Gr09]). The paper is organized as follows. In section 2, we recall some facts about tropical moduli spaces and tropical intersection theory that we need. Then we define Psi-classes on the space of parametrized tropical curves and tropical descendant invariants. In section 3 we define what it means for incidence conditions to be general and what consequences arise for our tropical descendant invariants if we choose the conditions to be general. In section 4, we show that every tropical curve in an intersection product of Psi-classes, point and line evaluations and the pullback of a point with a large coordinate in M 0,4 under the forgetful map contains a contracted bounded edge. Thus the tropical curve can be interpreted as a reducible curve by cutting it along this contracted bounded edge. In section 5 we show that the weight of a tropical curve in a zero-dimensional intersection product can be computed using a determinant of a linear matrix. We use this in section 6 to show that the weight of tropical curves with a contracted bounded edge can be (almost) split into two factors corresponding to the irreducible components. In section 7, we show the string equation and the divisor equation for our tropical descendant invariants. In section 8 finally, we collect our results to prove that our tropical descendant invariants satisfy a certain WDVV equation, and we conclude that the tropical invariants are equal to the corresponding classical invariants that satisfy the same recursion. In section 9, we describe an algorithm similar to the lattice path count that determines tropical descendant invariants. We would like to thank A. Gathmann and M. Kerber for useful discussions. DEFINING THE INVARIANTS First of all, let us briefly recall the constructions from [AR07] that we need here: A cycle X is a balanced (weighted, pure-dimensional, rational and polyhedral) complex (resp. fan) in R n . The top-dimensional polyhedra (resp. cones) in X are called facets, the codimension one polyhedra (resp. cones) are called ridges. The integer weights assigned to each facet σ are denoted by ω(σ). Balanced means that the weighted sum of the primitive vectors of the facets σ around a ridge τ ∈ X σ∈X (dim(X)) τ <σ ω(σ)v σ/τ vanishes "modulo τ ", or, precisely, lies in the linear vector space spanned by τ , denoted by V τ . Here, a primitive vector v σ/τ of σ modulo τ is a integer vector in Z n that points from τ towards σ and fulfills the primitive condition: The lattice Zv σ/τ + (V τ ∩ Z n ) must be equal to the lattice V σ ∩ Z n . Slightly differently, in [AR07] the class of v σ/τ modulo V τ is called primitive vector and v σ/τ is just a representative of it. For us, a polyhedron σ is always understood to be closed. The (relative) interior Int(σ) is the topological interior of σ in its affine span (e.g. Int({P }) = {P }). The support of X, denoted by \|X\|, is the union of all facets in X with non-zero weight. A (non-zero) rational function on X is a function ϕ : \|X\| → R that is affine (resp. linear) with rational slope on each polyhedron (resp. cone). The divisor of ϕ, denoted by div(ϕ) = ϕ · X, is the balanced subcomplex (resp. subfan) of X constructed in [AR07,3.3], namely the codimension one skeleton X \ X (dim X) together with the weights ω ϕ (τ ) for each ridge τ ∈ X. These weights are given by the formula ω ϕ (τ ) = σ∈X (dim(X)) τ <σ ω(σ)ϕ σ (v σ/τ ) − ϕ τ σ∈X (dim(X)) τ <σ ω(σ)v σ/τ , where ϕ σ : V σ → R denotes the linear part of the affine function ϕ\| σ . Note that the balancing condition of X around τ ensures that the argument of ϕ τ is an element of V τ . If ϕ is globally affine (resp. linear), all weights are zero, which we denote by ϕ · X = 0. Let the support of ϕ, denoted by \|ϕ\|, be the subcomplex of X containing the points where ϕ is not locally affine. Then we have \|ϕ · X\| ⊆ \|ϕ\|. Furthermore, the intersection product is bilinear (see [AR07,3.6]). As the restriction of a rational function to a subcycle is again a rational function, we can also form multiple intersection products ϕ 1 · . . . · ϕ l · X. In this case we will sometimes omit "·X" to keep formulas shorter. Note that multiple intersection products are commutative (see [AR07,3.7]). By abuse of notation, a cycle is also a class of balanced fans with common refinement and agreeing weights. A rational function ϕ on such a class is just a rational function on a fan X contained in the class. We can generalize our intersection product to such classes of fans [X] by defining ϕ · [X] := [ϕ · X]. In the following, we try to avoid these technical aspects whenever possible. We will also omit the brackets distinguishing between fans and their classes, hoping that no confusion arises. A morphism of cycles X ⊆ R n and Y ⊆ R m is a map f : \|X\| → \|Y \| that comes from a linear map from Z n to Z m and that maps each polyhedron (resp. cone) of X into one of Y . Such a morphism pulls back rational functions ϕ on Y to rational functions f * (ϕ) = ϕ • f on X. Note that the second condition of a morphism, which is not required in [AR07], makes sure that we do not have to refine X further. f * (ϕ) is already affine (resp. linear) on each cone. Furthermore, we can push forward subcycles Z of X to subcycles f * (Z) of Y . This is due to [GKM07, 2.24 and 2.25] in the case of fans and can easily be generalized to complexes. We can omit further refinements here if we assume that f (σ) ∈ Y for all σ ∈ X. The projection formula (see [AR07,4.8]) connects all the above constructions via f * (f * (ϕ) · X) = ϕ · f * (X). Moreover, let us recall the basic facts of rational equivalence introduced in [AR07, section 8]. The degree of a zero-dimensional cycle Z is just the sum of all weights. Hence the push-forward of a zero-dimensional cycle preserves degree. If X is a one-dimensional cycle, and ϕ is a bounded rational function, then deg(ϕ · X) = 0 (see [AR07,8.3]). The pull-back of a bounded rational function is again bounded. Two functions are called rationally equivalent if they differ by the sum of a bounded and a globally linear function. Hence (and by linearity of the pull-back) rational equivalence is preserved when pulled back. An example for functions that are rationally equivalent is given by translations of functions on R n . Proof. Let X be a subdivision of R n on which h is a rational function. For each cone σ ∈ X, let h σ be the linear part of the affine function h\| σ . Take the maximum of the finitely many h σ (v), σ ∈ X and call it c. Now, X subdivides the line segment x + λv, λ ∈ [0, 1] into q line segments of length λ i contained in some polyhedron σ i . This means h(x + v) can be expressed as h(x) + h σ1 (λ 1 v) + . . . + h σq (λ r v), where i λ i = 1. This implies h(x + v) − h(x) ≤ c, which proves that h ′ − h is bounded. In the following, we will apply these constructions and results to the case of Psi-and evaluation classes on the space of rational plane curves. The tropical analogue M 0,n of the space of stable n-marked curves is the space of trees, or (a quotient of) the tropical Grassmanian ( [SS04], [GKM07], [Mi07]). Thus an abstract tropical curve is just a tree with n marked ends and whose bounded edges e are equipped with a length l(e) ∈ R >0 . The fan M 0,n is stratified by cones corresponding to combinatorial types of trees. The facets correspond to 3-valent trees. The tropical analogue M 0,n (R 2 , d) of the space of stable maps has been studied in [GKM07]. An element of M 0,n (R 2 , d) is an abstract tropical curve Γ (i.e. a tree) together with a map h : Γ → R 2 such that the image satisfies the balancing condition and marked ends are contracted to a point. An important feature of this definition is that it also allows to contract bounded edges, as it will happen in section 4 and 6. If we furthermore also label the non-contracted ends, we obtain the space M lab 0,n (R 2 , d). The advantage of this space is that, after choosing the vertex of one marked end as root vertex, we can identify M lab 0,n (R 2 , d) with M n+3d × R 2 , where the second factor describes the position of the root vertex in R 2 (cf. [GKM07]). In particular, in this sense M lab 0,n (R 2 , d) is a tropical variety. For enumerative purposes, its difference to M 0,n (R 2 , d) cumulates in nothing but a factor (d!) 3 by which each invariant in M lab 0,n (R 2 , d) must be divided to get the corresponding one in M 0,n (R 2 , d). Note that, independent of the choice of a root vertex, there exists a forgetful map ft ′ : M lab 0,n (R 2 , d) → M n+3d forgetting just the position of the image of a curve in R 2 . This forgetful map ft ′ : M lab 0,n (R 2 , d) → M n+3d is a morphism of tropical varieties, as after choosing a root vertex and identifying M lab 0,n (R 2 , d) with M n+3d × R 2 , ft ′ is just the projection onto the first factor. Analogues of Psi-classes on tropical M 0,n have been defined recently ( [Mi07]). ψ i with i = 1, . . . , n is the codimension one subcycle that consists of cones corresponding to trees where the marked end i is at a 4or higher-valent vertex. How such Psi-classes intersect is discussed in [KM07]. To do so, Psiclasses ψ i , i = 1, . . . , n are defined as divisors of rational functions f i on M 0,n cf. [KM07, proposition 3.5]. As M 0,n is simplicial, the function f i can be defined by specifying its values on the primitive vectors of the rays contained in M 0,n . These rays are given by curves with only one bounded edge splitting up the marked ends into two sets I · ∪ J = [n]. Let v I\|J be the corresponding primitive vector and assume w.l.o.g. i ∈ I, then f i is defined by f i (v I\|J ) = \|J\| (\|J\| − 1) (n − 1)(n − 2) . Note that we denote by f i a multiple of what is called f i in [KM07], such that we obtain div(f i ) = ψ i . We use these functions to pull back Psi-classes to M lab 0,n (R 2 , d). Definition 2.2 (Psi-classes for parameterized curves). For i = 1, . . . , n we define the i-th Psi-class on M lab 0,n (R 2 , d) to be ψ i := div(ft ′ * (f i )). Remark 2.3. It can be shown that two rational functions on M lab 0,n (R 2 , d) (or M 0,n ) defining the same divisor cycle only differ by the restriction of a globally linear function. Hence, the choice of the functions defining our Psi-classes is not really important for intersection-theoretic purposes. This justifies that throughout our paper we use the specific function ft * (f i ) to describe ψ i and in particular define ψ i · Y := ft ′ * (f i ) · Y, where Y is an arbitrary subcycle of M lab 0,n (R 2 , d). Note also that for our purposes we do not really need that the function describing ψ i is (nearly) unique. The only thing we need to know is contained in the following lemma. Lemma 2.4 (Products of Psi-classes). Let r 1 , . . . , r n be positive integers and let X = n k=1 ψ r k k · M lab 0,n (R 2 , d) be a product of Psi-classes. Then X is the codimensionk r k -subfan of M lab 0,n (R 2 , d) consisting of cones σ corresponding to trees such that for each vertex V we have val(V ) = K(I V ) + 3, where I V denotes the set I V = {k ∈ [n] : end x k is adjacent to V } ⊂ [n] and K(I) is a short notation for K(I) = k∈I r k . The weight of σ equals ω(σ) = V K(I V )! n k=1 r k ! . Proof. Choose a root vertex and identify M lab 0,n (R 2 , d) with M 0,n+3d × R 2 . Then ft ′ is just the projection on the first factor and we can apply [AR07,9.6], i.e. instead of intersecting the pull-backs of the f k on the product, we can just intersect the f k on the first factor and then multiply with R 2 . Thus, X = n k=1 ψ r k k · M 0,n+3d × R 2 , where here ψ k denotes a Psi-class in M 0,n+3d . But in the case of non-parameterized curves, it is proved in [KM07,4.1] that the valence of the vertices and the weights of the facets satisfy the formulas of the statement. Multiplying with R 2 does not disturb this, as the weight of R 2 is one and as the combinatorics of a curve remain unchanged under ft ′ . Remark 2.5. In particular the preceding lemma says that ψ i consists of those curves whose marked end i is adjacent to an at least 4-valent vertex (where bounded edges as well as marked ends and noncontracted ends count towards the valence). Later on, we will also use the forgetful map ft : M lab 0,n (R 2 , d) → M 0,4 , which forgets the map of a given curve C to R 2 and all its ends but the first four marked ends (it also "stabilizes", which means that, after forgetting one marked end, it replaces all two-valent vertices by straight edges while adding up lengths). Lemma 2.6. The forgetful map ft : M lab 0,n (R 2 , d) → M 0,4 is a morphism of cycles. Proof. Let ft n : M 0,n → M 0,n−1 be the forgetful map that just forgets the i-th end of an i-marked non-parameterized curve. It is shown in [GKM07,3.9] that ft n is a morphism for all integers n ≥ 4. As mentioned above, the map ft ′ is a morphism, too. Thus, the statement follows from the formula ft = ft 5 • . . . • ft n • ft ′ . Moreover, we use the evaluation maps ev i : M lab 0,n (R 2 , d) → R 2 assigning to a curve C the position of its i-th marked end. It is shown in [GKM07,4.8] that these maps are also morphisms of cycles. Along these morphisms we will pull back lines and points. Definition 2.7 (Lines). A line G is a one-dimensional cycle in R 2 that is the divisor of a tropical polynomial of degree one. In other words, lines are divisors of translations of the functions max{x, y, 0}, max{x, 0}, max{y, 0} or max{x, y}. max{x, y} max{x, y, 0} max{x, 0} max{y, 0} Lines of type max{x, y, 0} are also called non-degenerated. We would like to pull back lines and points along an evaluation map ev i . However, up to now, pull backs are only defined for functions, not for cycles. Of course, we can choose rational functions cutting out the line resp. point in question and pull them back instead. In the following lemma we will show that, for our purposes, the choice of describing functions plays no role. Notation 2.8. We use the following notation: We have a total number of l + m + n marked ends, which are subdivided into the three sets L · ∪ M · ∪ N = {1, . . . , l + m + n}, such that \|L\| = l, \|M \| = m and \|N \| = n. In the following, the ends i ∈ L are unrestricted, the ends j ∈ M are restricted by lines G j (see 2.7) and the ends k ∈ N have to meet points P k . Furthermore we fix numbers r k , k ∈ N describing how many Psi-classes we require at k ∈ N . Lemma 2.9. Consider the intersection product Z := j∈M ev * j (G j ) k∈N ev * k (P k )ψ r k k · M lab 0,l+m+n (R 2 , d), where ev * j (G j ) stands for ev * j (h) with a function h cutting out G j and ev * k (P k ) stands for ev * k (h 1 ) · ev * k (h 2 ) with function h 1 , h 2 cutting out P k . Then Z is well-defined, i.e. it does not depend on the chosen rational functions. Proof. Let ev := ev i be an evaluation map and G be a line. First we check that the intersection product ev * (G) · M lab 0,l+m+n (R 2 , d) does not depend on the rational function describing G: Choose the vertex of the end i as root vertex and identify M lab 0,l+m+n (R 2 , d) with M 0,l+m+n+3d × R 2 . Then ev is just the projection onto the second factor. By [AR07,9.6] we deduce ev * (G) · (M 0,l+m+n+3d × R 2 ) = M 0,l+m+n+3d × G, which shows independence of the describing function. Now let X = ϕ 1 ·. . .·ϕ r ·M lab 0,l+m+n (R 2 , d) be a cycle given by arbitrary rational functions ϕ 1 , . . . , ϕ r . Then, by commutativity of the intersection product, the cycle ev * (G) · X = ϕ 1 · . . . · ϕ r · ev * (G) · M lab 0,l+m+n (R 2 , d) is also well-defined. The same arguments work if we consider a point P instead of G. But this suffices to conclude inductively that the big intersection product Z is also well-defined. Moreover note that the same argument also shows that our choice of the function f i describing ψ i does not matter in this intersection product. We are now ready to define our tropical descendant Gromov-Witten invariants. Proposition and Definition 2.10. Let d, l, m, n and r k , k ∈ N be positive integers such that l + m + n + 3d − 3 + 2 = m + 2n + k∈N r k . (1) Then we define the tropical descendant Gromov-Witten invariant τ 0 (0) l τ 0 (1) m k∈N τ r k (2) d to be the number τ 0 (0) l τ 0 (1) m k∈N τ r k (2) d := 1 (d!) 3 deg   j∈M ev * j (G j ) k∈N ev * k (P k )ψ r k k · M lab 0,l+m+n (R 2 , d)   . As indicated by the notation, this number only depends on d, l, m, n, r k , k ∈ N , but not on the lines G j and the points P k . Proof. Lemma 2.1 says that we can move around our points and lines arbitrarily, namely by translating the describing functions, without changing the degree. It remains to show that the type of the lines does not matter, for example the type of G 1 . We will show that ev * 1 (G 1 ) · F does not depend on the choice of the line G 1 for a one-dimensional cycle F , where F = j∈M\{1} ev * j (G j ) k∈N ev * k (P k )ψ r k k · M lab 0,l+m+n (R 2 , d) To see this, we have to use lemma 3.7 which requires general conditions and therefore is stated and proven in the next section of this article. It states that ev 1 * (F ) has only standard outer directions −e 1 , −e 2 and e 1 + e 2 . Knowing this, we push forward ev * 1 (G 1 ) · F via ev 1 , which does not change the degree and use the projection formula ( [AR07]). It tells us that ev 1 * (ev * 1 (G 1 )·F ) = G 1 ·ev 1 * (F ). Now, as ev 1 * (F ) has only standard outer directions (at least for general conditions, which we can assume), any line intersects ev 1 * (F ) in the same number of points, not depending on the type. Note that lemma 3.7 does not care about the types of the lines appearing in the product of F . Thus we can apply the above argument inductively and see that the types of all lines G j can be changed arbitrarily without changing the degree of the intersection product. Remark 2.11. The dimension of the space M lab 0,l+m+n (R 2 , d) = M 0,m+n+l+3d × R 2 is l + m + n + 3d − 3 + 2 since a 3-valent tree with m + l + n + 3d ends has l + m + n + 3d − 3 bounded edges. The codimension of the intersection of Psi-classes is k∈N r k . The pullback of a line has codimension 1 and the pullback of a point codimension 2. Hence the requirement (1) is equivalent to a 0-dimensional expected dimension of the intersection. Notation 2.12. We will use the τ -notation in a more general meaning: A product i∈I τ ri (c i ) d (with round brackets) stands for a cycle in M lab 0,\|I\| (R 2 , d), obtained as the intersection product where we replace the i-th factor τ ri (c i ) by ψ ri i ev * (C i ). Here, C i is some point P i if c i = 2, some line G i (of some type) if c i = 1 and the whole space R 2 (which means you can omit this pull-back) if c i = 0; thus c i describes the codimension of C i . If i∈I τ ri (c i ) d is zero-dimensional, we denote, as before, by i∈I τ ri (c i ) d = 1 (d!) 3 deg i∈I τ ri (c i ) d the degree of the product above divided by (d!) 3 . Note that a factor τ 0 (0) can not be dropped in this notation as it stands for a marked end that does not have to meet any condition at all. Remark 2.13. Later on, we will also allow the factor ft * (λ) in this notation, where λ is an element in M 0,4 and ft * (λ) stands for the pull-back of a rational function on M 0,4 describing λ. Two such functions differ by an affine one, and so do the pull-backs. Hence, the intersection product containing ft * (λ) as factor is still well-defined. GENERAL INCIDENCE CONDITIONS The invariants defined in 2.10 are well-defined also for "special" incidence conditions, e.g. if we choose all points P i to coincide. In this case the set of curves fulfilling the conditions is of too big dimension, but our intersection theory ensures that the corresponding intersection product still has the correct dimension and degree. However, many of the following arguments still require a notion of "general incidence conditions" that ensures that our intersection product equals the set-theoretical count of curves fulfilling the incidence conditions (up to weights). Let us start with the case of pulling back a single line in R 2 . Let X be a subcomplex of M lab 0,n (R 2 , d), let f : X → R 2 be a map that is the restriction of a linear map (e.g. morphisms like f = ev i ) and let G be a line in R 2 . Let f −1 (G) be the subcomplex of X containing all polyhedra σ ∩ f −1 (δ) for all σ ∈ X and δ ∈ G (where δ denotes a cone in the polyhedral complex G). Recall that the interior of a polyhedron Int(σ) denotes its topological interior in its affine span. Proof. Let σ be a facet of X and δ a one-dimensional polyhedron of G. Consider the map q • f σ , where f σ : V σ → R 2 is the extension of f \| σ to V σ and q : R 2 → R 2 /V δ is the quotient map. This composition has either rank 1 (in which case ker(q • f σ ) has codimension one in V σ ; hence, for a general translation δ ′ of δ, the polyhedron σ ∩ f −1 (δ ′ ) is either empty or of codimension 1 and intersecting the interior of σ) or has rank 0 (then σ ∩ f −1 (δ ′ ) is empty for a general translation of δ ′ of δ). As there are only finitely many pairs σ, δ, the set of vectors v ∈ R 2 such that these statements are true simultaneously is still open and dense. But note that all facets of f −1 (G ′ ) can be obtained in this way for some pair σ, δ. This shows part (a) and (b). Furthermore, let V ∈ R 2 be the vertex of G (if G is of type max{x, y, 0}). Applying the same argument to V shows that for a general translation V ′ := V + v, the preimage f −1 (V ′ ) has at least codimension 2, which proves part (c). Definition 3.2. Let Z be an intersection product of the form ( i∈I τ ri (c i )) d with incidence conditions C i . Define X := i∈I ψ ri i · M lab 0,\|I\| (R 2 , d). We call the conditions general if the following holds: (a) The subcomplex S of X containing all points C ∈ X fulfilling ev i (C) ∈ C i has dimension dim(S) = dim(Z). (b) The interior of a facet of S is contained in the interior of a facet of X. (c) The interior of a facet σ of S maps to the interior of a facet of C i under ev i . (d) Any intersection C i ∩ C j , i, j ∈ I has expected codimension c i + c j . Remark 3.3. Let S be the subcomplex of X containing all the curves C ∈ X fulfilling ev i (C) ∈ C i . Note that Z is a subcomplex of S. This follows from the facts that the support of an intersection product is contained in the support of the intersecting rational function and that the support of a pull-back is contained in the preimage of the support of the pulled-back function. Note that in general we have S = Z (as sets) if dim(S) = dim(Z) is satisfied, the only thing that can happen in principle is that there are facets of Z which get 0 as a weight in the intersection product, although they are facets of S. For the intersection products we work with, this cannot happen though, since we only have a weight of 0 if the set S is of higher dimension (see section 5). Hence for us the incidence conditions being general implies that \|Z\| equals the set of curves satisfying the incidence conditions, and deg(Z) equals the number of curves satisfying the conditions, counted with weight. Lemma 3.4. The set of general conditions in the space of all conditions (which can be identified with some big R N collecting all the translation vectors) is open and dense. Proof. The set of conditions fulfilling 3.2 (d) is obviously open and dense. The remaining follows from recursively applying 3.1 to X and ev 1 , then X ∩ ev −1 1 (C 1 ) and ev 2 , and so on. More precisely, if c i = 0 we have nothing to do in this step, if C i is a line, we apply 3.1, and if C i is a point, we apply 3.1 twice for two lines intersecting set-theoretically in the single point C i . Remark 3.5. We also consider the following case: Let X be a 1-dimensional subcycle of M lab 0,n (R 2 , d) and consider the forgetful map ft : The following lemma describes the combinatorial type of the curves which satisfy general incidence conditions. M lab 0,n (R 2 , d) → M 0,4 . We call λ ∈ M 0,4 general, if λ / ∈ ft(X (0) )∪ M (0) 0,4 , where X (0) denotes Lemma 3.6. Let Z be an intersection product of the form (τ 0 (0) l τ 0 (1) m k∈N τ r k (2)) d with general conditions. Then (b') For a curve C in the interior of a facet the following holds: All ends k ∈ M ∪ N lie at different vertices and the valence of a vertex is r k + 3 if k ∈ N is adjacent to it and 3 otherwise. Proof. Because of remark 3.3 we know that Z ⊂ S. In addition, condition 3.2 (a) says that Z and S have the same dimension and therefore (b) and (c) also hold for curves in the interior of a facet of Z. Let C be in the interior of a facet of Z. Condition 3.2 (d) implies that ev i (C) = ev j (C) for all i ∈ M ∪ N , j ∈ N , as in this case C i ∩ P j is empty. If i, j ∈ M would lie at the same vertex this would induce either a contracted bounded edge (which contradicts 3.2 (a)) or valence greater than 3 of this vertex (which contradicts 3.2 (b)). Hence all ends in M ∪ N must lie at different vertices. The statement about the valence of the vertices follows from 3.2 (b) and the description of X in 2.4. As a first application of our notion of general conditions we can now prove the lemma which we promised and needed in the independence statement 2.10. Lemma 3.7. Let F be a one-dimensional cycle of the form (τ 0 (0)τ 0 (0) l τ 0 (1) m k∈N τ r k (2)) d with general conditions. Let x denote the marked end corresponding to the first factor τ 0 (0). Then all of the unbounded rays of the push-forward ev x * (F ) have standard directions −e 1 , −e 2 and e 1 + e 2 . Proof. Let σ be a facet of F . For a curve in the interior of σ two possibilities can occur: Either x is adjacent to a higher-valent vertex. Then by 3.6 also an end k ∈ N interpolating the point P k lies at this vertex. Therefore, ev x (σ) = ev k (σ) = {P k }. Secondly, x might be adjacent to a 3-valent vertex. Since x itself is contracted, the two other edges which are adjacent are mapped to lines with opposite direction (because of the balancing condition). That means locally the image looks like a straight line with the marked point h(x) on it. We can deform a curve in σ in a one-dimensional family (thus covering σ) by changing the length of the two adjacent edges and thus making the point h(x) move on the line. This movement is unbounded if and only if one of these two edges is an end. But then ev x (σ) points to the same direction as this end, which is by definition one of the standard directions or 0. CONTRACTED EDGES Let F be a one-dimensional cycle of the form (τ 0 (0) l τ 0 (1) m k∈N τ r k (2)) d with general conditions. Remember that this implies that \|F \| equals the set of curves satisfying the conditions. Notation 4.1. We fix the type of the first four ends in the sense that we assume from now on 1 ∈ L, 2 ∈ M and 3, 4 ∈ N . As before we denote by ft the forgetful map ft : M lab 0,l+m+n (R 2 , d) → M 0,4 , which forgets the embedding and all ends but the first four marked ends. It is the aim of this section to show that for a very large M 0,4 -coordinate λ, the curves in ft −1 (λ) ∩ F (i.e. curves with such a large M 0,4 -coordinate) must contain a contracted bounded edge. We will use the contracted bounded edge in section 6 to split such curves into two components. Definition 4.2. Let C be a curve in M lab 0,l+m+n (R 2 , d). For two different marked ends i 1 , i 2 , we denote by S(i 1 , i 2 ) the smallest connected subgraph of C containing i 1 and i 2 and call it the string of i 1 and i 2 . Such a string S(i 1 , i 2 ) is called movable if i 1 , i 2 ∈ L ∪ E, where E denotes the set of non-contracted ends, and if S(i 1 , i 2 ) does not intersect (the closure) of any k for k ∈ N . Proof. We know dim(F ) = 1, codim(F ) = m + 2n + k∈N r k and dim(M lab 0,l+m+n (R 2 , d)) = l + m + n + 3d − 3 + 2. Plugging in all this in dim(F ) + codim(F ) = dim(M lab 0,l+m+n (R 2 , d)) leads to l + 3d = n + k∈N r k + 2. On the other hand we can compute the number of connected components of Γ \ k∈Nk : Removing k increases the number of connected components by r k + 1 as the valence of the adjacent vertex is r k + 3 by 3.6. So, after removing all n ends, we arrive at 1 + n + k∈N r k connected components. The above equation tells us that there is one more end in L ∪ E then there are connected components and therefore at least two ends i 1 , i 2 ∈ L ∪ E lie in the same component. Hence S(i 1 , i 2 ) is a movable string. By construction all vertices of a movable string are 3-valent. Lemma 4.4. Let σ be a facet of F such that the corresponding interior curves do not contain a contracted bounded edge. Then the image of σ under ft is bounded. Proof. Let C be a curve in the interior of σ. We will deform C in a one-dimensional family inside σ. Since σ is one-dimensional itself, this family covers σ. By lemma 4.3 there exists a movable string S in C. In the following, we show that either σ is bounded (i.e. the deformation of C is bounded) or ft is constant on σ (i.e. the deformation of C does not affect ft). Let V be a vertex in S. We call V degenerated if we can deform C one-dimensionally locally around V , i.e. if (a) either one of the adjacent edges is a marked end i ∈ L, (b) or one of the adjacent edges is a marked end j ∈ M and the linear spans of the corresponding line G j at ev j (C) and of the other two edges adjacent to V coincide (i.e. if the curve C and the line G do not intersect transversally at ev j (C)), (c) or all edges adjacent to V are non-contracted, but their span near V is still only one-dimensional; w.l.o.g. we denote the edge alone on one side of V by v and the two edges on the other side by v 1 , v 2 . (b) (c) (a) v 1 v 1 v 2 v 2 G j j v v 2 v 1 i If such a degenerated vertex exists, the 1-dimensional deformation of the curves inside σ is given by moving this vertex and changing the lengths of the adjacent edges accordingly. We show that this movement is either bounded or, if not, the changed lengths do not influence ft. Consider the cases (a) and (b) and let v 1 , v 2 be the two other edges adjacent to V . At least one of the two edges, say v 1 , is bounded. Then the movement is unbounded only if v 2 is unbounded. But v 2 cannot be contracted, as then v 1 would also be contracted (and bounded). But this means that ft forgets v 2 and therefore also the length of v 1 . Now consider the case (c). The balancing condition says v = v 1 + v 2 (by abuse of notation we denote the direction vectors by the same letters as the edges), which in particular implies that v is not primitive and hence the edge v has to be bounded. Now again, if we require the movement of V to be unbounded, v 1 and v 2 must be unbounded. But they are also non-contracted which means that ft forgets them and the length of v. So we are left with the case that all vertices of S are non-degenerated. We can still describe the deformation of the curves inside σ using the movement of the string: Take one of the ends of the string (which is necessarily non-contracted) and move it slightly in a non-zero direction modulo its linear span. Consider the next vertex V and let v be the adjacent edge not contained in the string. Then two things can happen: If v is non-contracted (case A), our moved end will meet the affine span of v at some point P (as V is non-degenerated). So we change the length of v such that it ends at P (while keeping the position of its second vertex fixed). Then we also move the second edge of the string to P and go on to the next vertex. If v is contracted (case B), our assumptions ensure that it is a marked end j ∈ M and that the corresponding line G j intersects our curve transversally at V . Thus our moved edge will again meet G j at some point and by changing the lengths of the adjacent edges appropriately, the obtained curve will still meet G j . j case B: case A: S S G j In this way we can make our way through the string and finally obtain a deformation of the whole curve. Note that the non-degeneracy of all the vertices ensures that all edges of the string must change their positions modulo their linear span and, hence, that all edges adjacent to, but not contained in the string must change their length. In particular this means we cannot have more non-contracted ends adjacent to our string: Then we would have two different strings providing two independent deformations of the curves inside σ, which is a contradiction as σ is one-dimensional. Let us summarize: Our string S is generated by two unique non-contracted ends i 1 , i 2 , all of its vertices are 3-valent and the adjacent edges not contained in the string are either bounded or marked ends in M , where the corresponding line G j intersects transversally. The deformation only moves the string S; the adjacent edges are shortened or elongated and the other parts of the curve remain fixed. We want to show that, even if the movement is unbounded, the considered M 0,4 -coordinate is bounded. If there are bounded edges adjacent to S to both sides of S as in picture (a) below then the movement of the string is bounded. (This is true because if we move the string to either side, we can only move until the length of one of the adjacent bounded edges shrinks to 0.) So we only have to consider the case when all adjacent bounded edges of S are on the same side of S, say on the right side as in picture (b) below. Label the edges of S (respectively, their direction vectors) by v 1 , . . . , v k and the adjacent bounded edges of the curve by w 1 , . . . , w k−1 as in the picture. As above the movement of the string to the right is bounded. If one of the directions w i+1 is obtained from w i by a left turn (as it is the case for i = 1 in the picture) then the edges w i and w i+1 meet on the left of S. This restricts the movement of the string to the left, too, since the corresponding edge v i+1 then shrinks to length 0. (a) (b) w 3 v 1 w 2 w 3 (c) v 4 v 3 v 2 v 1 w 1 v 1 v 4 w 3 w 2 w 1 (d) v 1 v 2 w 1 (e) v 4 w 1 w 2 v 2 v 3 S S S So we can assume that for all i the direction w i+1 is either the same as w i or obtained from w i by a right turn as in picture (c). The balancing condition then shows that for all i both the directions v i+1 and −w i+1 lie in the angle between v i and −w i (shaded in the picture above). Therefore, all directions v i and −w i lie within the angle between v 1 and −w 1 . In particular, the image of the string S cannot have any self-intersections in R 2 . We can therefore pass to the (local) dual picture (d) where the edges dual to w i correspond to a concave side of the polygon whose other two edges are the ones dual to v 1 and v k . But note that there are no such concave polygons with integer vertices if the two outer edges (dual to v 1 and v k ) are two of the vectors ±(1, 0), ±(0, 1), ±(1, −1) that can occur as dual edges of an end of a plane tropical curve of degree d. Therefore the string can consist at most of the two ends i 1 and i 2 that are connected to the rest of the curve by exactly one bounded edge w 1 . This situation is shown in picture (e). In this case the movement of the string is indeed not bounded to the left. Note that then w 1 is the only internal edge whose length is not bounded. But by our assumptions 1, 3 and 4 cannot lie on S but must lie on the other side of w 1 ; hence its length does not influence ft. This finishes the proof. COMPUTING WEIGHTS In this section we prove that the weight of a curve in a zero-dimensional intersection product can be computed as the (absolute value of a) determinant of the linear map that basically collects all the evaluation morphisms (and forgetful morphism if present). We will use this to express the weight of a reducible curve (in the sense that it contains a contracted bounded edge) in terms of the weights of the two components. The following statement connects intersection products with determinants. Let V = R ⊗ Λ be a real vector space of dimension n with underlying lattice Λ and let h 1 , . . . , h n ∈ Λ ∨ be linear functions on Λ resp. V . By H : V → R n we denote the linear map given by x → (h 1 (x), . . . , h n (x)). Choose lattice bases of Λ and Z n and consider the matrix representation of H with respect to these bases. Obviously, the absolute value of the determinant of this matrix is independent of the choice of bases; hence we denote it by \| det(H)\|. On the other hand, we can consider the rational functions ϕ i = max{h i , 0} on V . To do so, we give V the fan structure consisting of all cones on which each h i is either positive or zero or negative. These rational functions form a zero-dimensional intersection product, which obviously consists of only {0} with a certain weight. Lemma 5.1. The weight of {0} appearing in ϕ n · . . . · ϕ 1 · V is equal to \| det(H)\|. Proof. Let us first assume that h 1 , . . . , h n form a lattice basis of Λ ∨ with dual basish 1 , . . . ,h n . We compute ϕ 1 · V : For each ridge τ of V (with the fan structure described above) there exists a unique j such that τ ⊆ h ⊥ j . Then there are two facets containing τ (where h j is positive resp. negative) and the corresponding (representatives of the) primitive vectors areh j resp. −h j . Therefore the weight of τ in the intersection product ϕ 1 · V can be computed as ω ϕ1 (τ ) = max{h 1 (h j ), 0} + max{h 1 (−h j ), 0} = h 1 (h j ) + 0 = h 1 (h j ). Hence, when omitting cones with weight 0, ϕ 1 · V consists of all cones contained in h ⊥ 1 and all weights are 1. Now we can apply induction on h 2 \| h ⊥ 1 , . . . , h n \| h ⊥ 1 and conclude that ϕ n · . . . · ϕ 1 · V produces {0} with weight 1. On the other hand, the matrix representation of H with respect to the basish 1 , . . . ,h n for Λ and the standard basis for Z n is just the unit matrix. Hence, \| det(H)\| = 1, which proves the statement in the special case. General case: For general h 1 , . . . , h n we can choose a lattice basis l 1 , . . . , l n of Λ ∨ such that h 1 = a 1,1 l 1 , h 2 = a 2,1 l 1 + a 2,2 l 2 , . . . h n = a n,1 l 1 + . . . + a n,n l n , where the a i,j are integers. Then we get \| det(H)\| = \| det((a i,j ))\| = \|a 1,1 · . . . · a n,n \|. On the other hand, let us compute that ϕ n · . . . · ϕ 1 · V produces {0} with weight \|a 1,1 · . . . · a n,n \|. We saw in the special case that max{l 1 , 0} · V is l ⊥ 1 with weight 1. The above equations tell us that max{h 1 , 0} = \|a 1,1 \| · max{l 1 , 0}. Using the linearity of the intersection product, we deduce that ϕ 1 · V is l ⊥ 1 with weight \|a 1,1 \|. Now we apply induction on n again: After restricting all functions to l ⊥ 1 , we can omit all terms a i,1 l 1 in the above equations (in particular we can omit the first equation). Hence we can apply our induction hypothesis and conclude that ϕ n · . . . · ϕ 2 · l ⊥ 1 produces {0} with weight \|a 2,2 · . . . · a n,n \|. Hence, ϕ n · . . . · ϕ 2 · (ϕ 1 · V ) = \|a 1,1 \| · ϕ n · . . . · ϕ 2 · l ⊥ 1 = {0} with weight \|a 1,1 · . . . · a n,n \|. With this tool we can express all weights occurring in a zero-dimensional intersection product in terms of absolute values of determinants. Notation 5.2. Let Z be a zero-dimensional intersection product of the form (τ 0 (0) l τ 0 (1) m k∈N τ r k (2)) d or of the form (ft * (λ) · τ 0 (0) l τ 0 (1) m k∈N τ r k (2)) d with general condition G j and P k (and λ, resp.). The set of curves S that fulfill the incidence conditions set-theoretically is finite by 3.2 (a). For simplicity, let us furthermore assume that all points P k = (p 1 , p 2 ) are described by the rational functions max{x, p 1 } and max{y, p 2 } on R 2 and that all lines G j are vertical, i.e. of type max{x, 0} (i.e. are given by a rational function max{x, c j }). λ can be described by max{x, λ}, where x is the coordinate of the ray in whose interior λ lies (see 3.5). Denote X := k∈N ψ r k k · M lab 0,n (R 2 , d). We then consider the morphisms ev : X → R m × (R 2 ) n , C → ev j (C) x j∈M , ev k (C) k∈N , respectively ft × ev : X → M 0,4 × R m × (R 2 ) n , C → (ft(C), ev j (C) x j∈M , ev k (C) k∈N , where ev j (C) x denotes the first coordinate of the point ev j C ∈ R 2 . Thus, these morphisms evaluate at each end i ∈ M · ∪ N and keep all coordinates if i ∈ N and only the first coordinate if i ∈ M . Let C be a curve in the interior of a facet σ of X (and with ft(C) not being the vertex of M 0,4 ). Then ev (resp. ft × ev) is affine in a neighborhood of C and we define \|det C (ev)\| (resp. \| det C (ft × ev)\|) to be \| det(H)\|, where H is the linear part of ev (resp. ft × ev) at C. Theorem 5.3. The zero-dimensional intersection product Z (as in notation 5.2) can be computed as Z = C∈S \|det C (ev)\| · C, resp. Z = C∈S \|det C (ft × ev)\| · C, i.e. the weight of a curve C ∈ Z is just \|det C (ev)\| (resp. \| det C (ft × ev)\|). Proof. Each C ∈ Z is contained in the interior of a facet σ of X (see 3.2 (a)). In 2.4 the weight of σ in X was computed to be ω(σ) = V K(I V )! n k=1 r k ! . But we know from 3.6 that no two marked ends lie at a common vertex and hence K(I V ) = r k if k is adjacent to V , 0 otherwise. Therefore we can cancel the fraction defining ω(σ) down to 1. As the computation of the weight of C is local, we can replace X by V := R · σ. On the other hand, locally around C, all the pull backs along ev and ft are of the form max{a, c}, where a is an affine function on V and c is a constant. To be more precise, a is exactly one of the coordinate functions of ev (resp. ft × ev). Now, up to translations and subtracting constant terms, we are in the situation of 5.1: The weight of C equals the absolute value of the determinant of linear map H whose coordinate functions are the linear parts of the affine functions a. Thus H is the linear part of ev (resp. ft × ev) on V , and we can conclude that the weight of C in Z is precisely \| det(H)\| = \| det C (ev)\| (resp. = \| det C (ft × ev)\|). Remark 5.4. If we dropped the requirement that Psi-conditions are only allowed at marked ends that are also restricted by a point condition, we could still prove a formula similar to the above one: For a zero-dimensional intersection product of arbitrary Psi-and evaluation classes, the weights can still be computed as the absolute value of an appropriate determinant times the weight of the corresponding facet in X. In particular, this shows that such weights are always positive, as well as the degree of the product. Hence, whenever classical descendant Gromov-Witten invariants are negative (e.g. τ 1 (1)τ 0 (2) alg 1 = −1), we have an example of classical invariants that do not coincide with their tropical counterparts as we define them. SPLITTING CURVES Now we want to think of curves with a contracted bounded edge as reducible curves. We do that basically by cutting the contracted bounded edge. We have to show that the weight of the curve C is (almost) the product of the two weights of the two curves that arise after cutting. We use the description of weight in terms of determinants from section 5. Notation 6.1. As in section 4 we assume 1 ∈ L, 2 ∈ M and 3, 4 ∈ N . Additionally we require from now on L = {1} (i.e. the marked end 1 is the only "free" end) and r 3 ≥ 1 (i.e. the marked end is restricted by at least one Psi-class). Let Z be a zero-dimensional cycle of the form (ft * (λ) · τ 0 (0)τ 0 (1) m k∈N τ r k (2)) d with general condition G j and P k and λ, as in 5.2. Let C ∈ Z. Let Z i := j∈Mi ev * j (G j ) · k∈Ni ev * k (P k ) · ψ r k k · [M lab 0,li+ni+mi+1 (R 2 , d i )] denote the corresponding intersection products. Notation 6.3. We pullback a general point λ ∈ M 0,4 (i.e. not the vertex) via the forgetful map ft : M lab 0,1+m+n (R 2 , d) → M 0,4 . There are 3 types of such general points, corresponding to the 3 types of abstract tropical curves with 4 marked ends. The ends 1 and 2 can be together at a vertex, or the ends 1 and 3, or the ends 1 and 4. We use the following short notation: if ft(C) is in the ray corresponding to the type where 1 and 2 are together at a vertex, we say ft(C) = 12/34 (and analogously in the other cases). (a) dim(Z 1 ) = 0 and dim(Z 2 ) = 2, (b) dim(Z 1 ) = 1 and dim(Z 2 ) = 1, or (c) dim(Z 1 ) = 2 and dim(Z 2 ) = 0. If ft(C) = 13/24 then d 1 , d 2 > 0 and the analogous 3 cases are to distinguish. Proof. If there were two contracted edges, then all evaluations (i.e. 2n + m coordinates) would depend only on 1 + m + n + 3d − 3 − r i = m + 2n − 1 coordinates, so we get \| det C (ev × ft)\| = 0. So we can assume now there is only one contracted bounded edge e. Since e has to count towards the M 0,4 -coordinate to satisfy \| det C (ev × ft)\| = 0, 1, j ⊂ L 1 ∪ M 1 ∪ N 1 and k, l ⊂ L 2 ∪ M 2 ∪ N 2 if ft(C) = 1j/kl. Let us first consider the case where one of the d i 's is zero. This implies that all edges of the corresponding curve C i are contracted. As we cannot have more contracted bounded edges, C i is a star-shaped curve containing only a single vertex V . But 3.6 states that the ends 2, 3, 4 ∈ M ∪ N all lie at different vertices. Thus the cases d 1 = 0, ft(C) = 13/24 and d 2 = 0 cannot occur, whereas in the remaining case d 1 = 0, ft(C) = 12/34 the single vertex V must be 3-valent which is the same as L 1 ∪ M 1 ∪ N 1 = {1, 2}. Let us now assume d 1 , d 2 > 0. It remains to show that dim(Z 1 ) + dim(Z 2 ) = 2 which follows since dim(Z 1 ) + dim(Z 2 ) =3d 1 − k∈N1 r k − m 1 − 2n 1 + 3d 2 − k∈N2 r k − m 2 − 2n 2 = 3d − k∈N r k − m − 2n = 2 where the last equality follows since Z is zero-dimensional and thus 3d− k∈N r k −1 = m+2n+1. Remark 6.5. In the following, we will choose bases in order to write down an explicit matrix representation for the map ev × ft or ev locally on a cone. For a cone σ of M lab 0,n (R 2 , d) corresponding to a combinatorial type (i.e. an abstract graph Γ (without length) together with all direction vectors) we pick a root vertex V of Γ and choose the coordinates of the point h(V ) ∈ R 2 to which this vertex is mapped as two coordinates. The remaining coordinates of σ are given by the lengths of the bounded edges. For the spaces R 2 or R that describe our incidence conditions locally, we choose the standard basis vectors. It follows from remark 3.2 of [GM05] that the absolute value of the determinant does not depend on any of the choices we make. Lemma 6.6. Let C be as in construction 6.2 and stick to the notations from there. If ft(C) = 12/34 and d 1 = 0 we want to show \| det C (ev × ft)\| = (G 2 ·C 2 ) h(e) · \| detC 2 (ev M2∪N2 )\|. For the other three cases from lemma 6.4 we want to show: (a) \| det C (ev × ft)\| = \| det C1 (ev M1∪N1 )\| · \| det C2 (ev M2∪N2 × ev e )\|, (b) \| det C (ev × ft)\| = (C 1 ·C 2 ) h(e) · \| detC 1 (ev M1∪N1 )\| · \| detC 2 (ev M2∪N2 )\|, or (c) \| det C (ev × ft)\| = \| det C1 (ev M1∪N1 × ev e )\| · \| det C2 (ev M2∪N2 )\|. Proof. For all cases, note first that the matrix of \| det C (ev × ft)\| has a column with only zeros except one 1. This is the column corresponding to e. Since e is contracted, it is not needed for any evaluation. But it is needed for the M 0,4 -coordinate, so it has zeros except a 1 in the ft-row. We can delete this row and column without changing the absolute value of the determinant. Call the matrix with the deleted row and column A. Then \| det(A)\| = \| det C (ev × ft)\|. Now let ft(C) = 12/34 and d 1 = 0, it follows L 1 ∪ M 1 ∪ N 1 = {1, 2}. We want to show that the boundary vertex V of e in C 2 is 3-valent, too. Assume it is not, then there has to be a marked end with a Psi-condition adjacent to V . But this marked end is in N and thus required to meet a point. This is a contradiction, since the point is not on the line that 2 is required to meet (cf. 3.2 (d)). So let e 1 and e 2 be the two other edges adjacent to V and assume first that both of them are bounded. Denote their common direction vector (up to sign) by v = (v 1 , v 2 ) and their lengths by l(e 1 ), l(e 2 ). Assume that the root vertex is on the e 1 -side of e. Then the entries of the matrix A corresponding to l(e 1 ) and l(e 2 ) are ↓ evaluation at. . . l(e 1 ) l(e 2 ) 2 (1 row) v 1 0 points reached via e 1 from 2 (1 or 2 rows) 0 0 points reached via e 2 from 2 (1 or 2 rows) v v We see that after subtracting the l(e 2 )-column from the l(e 1 )-column we again get one column with only one non-zero entry v 1 . So for the determinant we get v 1 as a factor, dropping the corresponding row and column (which means removing e and straightening the 2-valent vertex), so we get \| det( A)\| = v 1 · \| detC 2 (ev M2∪N2 )\| = (C 2 · G 2 ) h(e) · \| detC 2 (ev M2∪N2 )\|. Essentially the same argument holds if one of the adjacent edges -say e 2 -is unbounded: in this case there is only an l(e 1 )-column which has zeroes everywhere except in the one 2-row where the entry is v 1 . Next, let dim(Z 1 ) = 0 and dim(Z 2 ) = 2. Denote by a i the dimension of k∈Ni ψ r k k · M lab 0+li+mi+ni+1 (R 2 , d i ), that is, a i = 3d i + l i + m i + n i + 1 − k∈Ni r k − 1. Since dim(Z 1 ) = 0 we have m 1 + 2n 1 = a 1 and since dim(Z 2 ) = 2 we have m 2 + 2n 2 = a 2 − 2. Let the boundary vertex V of e in C 1 be the root vertex for C. Choose the following order of coordinates: start with the root vertex, then bounded edges in C 1 , next bounded edges in C 2 . Start with the marked ends in C 1 and then add the marked ends in C 2 . Then the matrix A is in block form: because the points on C 1 need only the root vertex and the bounded edges of C 1 , they need the first a 1 = m 1 + 2n 1 coordinates, and have 0 after that. So there is a 0 block on the top right, and the top left is just the matrix of ev M1∪N1 at C 1 . So \| det(A)\| = \| det C1 (ev M1∪N1 )\| · \| det(B)\| where B denotes the lower right box. Consider the matrix of ev M2∪N2 × ev e at C 2 , and let the root vertex be the boundary vertex of e in C 2 . Then this matrix has two more rows and columns than B, namely the root vertex columns and the rows corresponding to ev e . But since these two rows start with a 2 × 2 unit matrix block and have zeros after that, we can see that \| det(B)\| = \| det C2 (ev M2∪N2 × ev e )\|. The third case is symmetric. Finally, assume dim(Z 1 ) = 1 and dim(Z 2 ) = 1, i.e. m 1 + 2n 1 = a 1 − 1 and m 2 + 2n 2 = a 2 − 1. First we want to show that the two vertices of e are 3-valent. Assume the vertex in C 1 , V , is not 3-valent, then there must be a marked end i with a Psi-class adjacent to V . But this end is in N then, so it is required to meet a point P i ∈ R 2 . Since dim(Z 1 ) = 1 we can move C 1 locally in a 1-dimensional family such that all incidence conditions are still defined. Let C ′ 1 be an element of this family. Since C ′ 1 has to meet P i as well, we can glue C ′ 1 to C 2 thus producing a curve C in Z. This is a contradiction since the dimension of Z is 0. Since the argument is symmetric it follows that both vertices of e are 3-valent. Denote the two edges adjacent to e in C 1 by e 1 and e 2 and the two edges in C 2 adjacent to e by e 3 and e 4 . Assume first that all of those edges are bounded. Let the boundary vertex V of e in C 1 be the root vertex for C. Then the matrix A reads: lengths in C 1 lengths in C 2 root (a 1 − 4 cols) l(e 1 ) l(e 2 ) l(e 3 ) l(e 4 ) (a 2 − 4 cols) (2n 1 + m 1 ends behind e 1 I 2 * v 0 0 0 0 rows) ends behind e 2 I 2 * 0 −v 0 0 0 (2n 2 + m 2 ends behind e 3 I 2 0 0 0 w 0 * rows) ends behind e 4 I 2 0 0 0 0 −w * where I 2 is the 2 × 2 unit matrix, and * denotes arbitrary entries. Now add v times the root columns to the l(e 2 )-column, subtract the l(e 1 )-column from the l(e 2 )-column and the l(e 4 )-column from the l(e 3 )-column to obtain the following matrix with the same determinant: lengths in C 1 lengths in C 2 root (a 1 − 4 cols) l(e 1 ) l(e 2 ) l(e 3 ) l(e 4 ) (a 2 − 4 cols) (2n 1 + m 1 ends behind E 1 I 2 * v 0 0 0 0 rows) ends behind E 2 I 2 * 0 0 0 0 0 (2n 2 + m 2 ends behind E 3 I 2 0 0 v w 0 * rows) ends behind E 4 I 2 0 0 v w −w * Note that this matrix has a block form with a zero block at the top right. Denote the top left block (of size 2n 1 + m 1 = 2 + a 1 − 4 + 1) by A 1 and the bottom right (of size 2n 2 + m 2 = 3 + a 2 − 4) by A 2 , then \| det(A)\| = \| det A 1 · det A 2 \|. The matrix A 1 is precisely the matrix for the evaluation map ev M1∪N1 ofC 1 (which arises from C 1 after forgetting the marked end corresponding to e) if we choose the other vertex of e 2 as the root vertex. Hence \| det A 1 \| = \| detC 1 (ev M1∪N1 )\|. In the same way the matrix for the evaluation map ev M2∪N2 of C 2 , if we again forget the marked end corresponding to e and now choose the other vertex of e 3 as the root vertex, is the matrix A ′ 2 obtained from A 2 by replacing v and w in the first two columns by the first and second unit vector, respectively. But A 2 is simply obtained from A ′ 2 by right multiplication with the matrix v w 0 0 0 I 2n2+2 which has determinant det(v, w). So we conclude that \| det A 2 \| = \| det(v, w)\| · \| det A ′ 2 \| = (C 1 · C 2 ) h(e) · \|detC 2 (ev M2∪N2 )\|. Remark 6.7. The following "converse" of lemma 6.4 and lemma 6.6 is also true: For each choice of C 2 satisfying all conditions but 2 and each choice of an intersection point ofC 2 with G 2 we can add a contracted bounded edge and the two marked ends 1, 2 on the other side to built exactly one possible C. The curve C then contributes (G 2 ·C 2 ) h(e) \| detC 2 (ev M2∪N2 )\| to the count. By Bézout's theorem ( [RST03]), each choice ofC 2 contributes d 2 · \| detC 2 (ev M2∪N2 )\| = d · \| detC 2 (ev M2∪N2 )\|. For each choice of C 1 satisfying the conditions in L 1 ∪ M 1 ∪ N 1 and each choice of C 2 satisfying the conditions in L 2 ∪ M 2 ∪ N 2 plus in addition the condition h(e) = p we get exactly one possible C by gluing the two curves along e. This curve C contributes to the count with weight \| det C1 (ev M1∪N1 )\| · \| det C2 (ev M2∪N2 × ev e )\| (and the other way round). For each choice ofC 1 andC 2 satisfying the conditions in L 1 ∪ M 1 ∪ N 1 and L 2 ∪ M 2 ∪ N 2 and for each choice of points P ∈C 1 and Q ∈C 2 that map to the same image point in R 2 we can glue P and Q along a contracted bounded edge and thus built exactly one possible C. The curve C contributes to the count with weight (C 1 · C 2 ) h(e) · \| detC 1 (ev M1∪N1 )\| · \| detC 2 (ev M2∪N2 )\|. By Bézout's theorem, each choice ofC 1 andC 2 thus contributes d 1 · d 2 · \| detC 1 (ev M1∪N1 )\| · \| detC 2 (ev M2∪N2 )\|. STRING AND DIVISOR EQUATION In this section we prove two lemmas which deal with the case of an extra end in a top-dimensional intersection product that is restricted either by no condition at all (string equation) or by only a line condition (divisor equation). Lemma 7.1 (String equation). For tropical descendant Gromov-Witten invariants the following equality holds: τ 0 (0) · τ 0 (0) l · τ 0 (1) m · k∈N τ r k (2) d = k∈N r k >0 τ 0 (0) l · τ 0 (1) m · τ r k −1 (2) · k =k ′ ∈N τ r k ′ (2) d Proof. Choose incidence conditions G j , P k such that they are general for all the derived intersection products Z := (τ 0 (0) · τ 0 (0) l · τ 0 (1) m · k∈N τ r k (2)) d , Z k := (τ 0 (0) l · τ 0 (1) m · τ r k −1 (2) · k =k ′ ∈N τ r k ′ (2)) d (note that Z lives in M lab 0,1+l+m+n (R 2 , d), whereas the Z k lives in M lab 0,l+m+n (R 2 , d)). Then 3.2 (a) tells us that the products just consist of the set of curves fulfilling the incidence conditions and having required valences, with the additional data of a weight for each curve. Let C ′ be a curve in Z k . Then we obtain a curve C ∈ Z by attaching the additional end, say x, to the vertex V k at which the end k lies. Let us check that the weight of C ′ in Z k and C in Z coincide. As our conditions are general, C ′ lies in a facet σ ′ of ψ r k −1 k · k =k ′ ∈N ψ r k ′ k ′ · M lab 0,l+m+n (R 2 , d) and C lies in a facet σ of k∈N ψ r k k · M lab 0,1+l+m+n (R 2 , d)). Moreover, the map ft x : M lab 0,1+l+m+n (R 2 , d) → M lab 0,l+m+n (R 2 , d) forgetting the additional end x maps σ Z-isomorphically to σ ′ (the inverse is given by adding 1 to V k as above). The evaluation maps ev k on σ are just obtained as pull-backs ft x * (ev ′ k ), where ev ′ k denotes the corresponding evaluation map on σ ′ . Hence, the weights of C and C ′ coincide. It remains to check that each C ∈ Z is obtained in the above way from C ′ ∈ Z k for unique k ∈ N . Uniqueness is clear, as by 3.6 (b') all ends k ∈ N lie at pairwise different vertices and hence x can not be adjacent to more than one end k ∈ N . On the other hand, to show that it is adjacent to a k ∈ N with r k > 0, it suffices to show that x cannot be adjacent to a 3-valent vertex. If it were, at least one of the other two adjacent edges, say E would be bounded (otherwise the abstract graph were not connected). But then, we could change the length of E (and accordingly the length of the other edge if necessary) without changing the coordinates of the marked ends, which contradicts the fact that the set of curves fulfilling our given conditions is finite by 3.2 (a). Lemma 7.2 (Divisor equation). For tropical descendant Gromov-Witten invariants the following equality holds: τ 0 (1) · τ 0 (0) l · τ 0 (1) m · k∈N τ r k (2) d = d · τ 0 (0) l · τ 0 (1) m · k∈N τ r k (2) d Proof. First we choose general incidence conditions. Because of 2.10 we can assume that the line conditions are all vertical lines, i.e. of type max{x, 0}. Then for all curves C in (τ 0 (1) · τ 0 (0) l · τ 0 (1) m · k∈N τ r k (2)) d we know that their weight is equal to \| det C (ev)\|, where ev denotes the product of all evaluation maps (evaluation of the x-coordinate for all lines, both coordinates for all points) (theorem 5.3). Assume x is the additional marked end with line condition G (but without Psi-condition). x has to be adjacent to a 3-valent vertex (see 3.6). Exactly as in lemma 6.6 we can see that \| det C (ev)\| = (G ·C) h(x) · \| detC (ev x )\| whereC is the curve we get when forgetting x (i.e. removing it from C and straightening the 2-valent vertex) and ev x is the product of all other evaluations. Thus any curve in (τ 0 (1)·τ 0 (0) l ·τ 0 (1) m · k∈N τ r k (2)) d gives us a curve in (τ 0 (0) l ·τ 0 (1) m · k∈N τ r k (2)) d by removing the marked end x. Conversely, given a curveC in (τ 0 (0) l · τ 0 (1) m · k∈N τ r k (2)) d we can pick a point p ∈ (G ·C) and attach a marked end to get a curve C ∈ (τ 0 (1) · τ 0 (0) l · τ 0 (1) m · k∈N τ r k (2)) d . Since (G ·C) = p (G ·C) p = d by tropical Bézout's theorem ( [RST03]), the statement follows. RECURSION Now we sum up the results of the preceding sections to a certain WDVV equation. We also show in this section that this WDVV equation together with the string and the divisor equation are sufficient to show that the tropical invariants coincide with the classical ones. To distinguish our tropical invariants that we denote by τ 0 (0) l τ 0 (1) m k∈N τ r k (2) d from the classical ones, we use the notation τ 0 (0) l τ 0 (1) m k∈N τ r k (2) alg d for the classical invariants. Theorem 8.1. The tropical descendant invariants as defined in 2.10 satisfy the following WDVV equation if r 3 > 0: τ 0 (1) m k∈N τ r k (2) d + D · τ 0 (0)τ 0 (1)τ 0 (1) m1 k∈N1 τ r k (2)τ 0 (e) d1 · τ r3 (2)τ r4 (2)τ 0 (1) m2 k∈N2 τ r k (2)τ 0 (f ) d2 = D · τ 0 (0)τ r3 (2)τ 0 (1) m1 k∈N1 τ r k (2)τ 0 (e) d1 · τ 0 (1)τ r4 (2)τ 0 (1) m2 k∈N2 τ r k (2)τ 0 (f ) d2 where D = (d 1 !) 3 · (d 2 !) 3 d! 3 and the sums range over all e + f = 2, e, f ≥ 0, M 1 · ∪ M 2 = M \ {2}, N 1 · ∪ N 2 = N \ {3, 4} and E 1 · ∪ E 2 = E, E 1 , E 2 = ∅. Here, E denotes the set of non-contracted ends, and E 1 is subset of non-contracted ends such that each of the standard directions −e 1 , −e 2 , e 1 + e 2 appears d 1 times. The equation can be rewritten as τ 0 (1) m k∈N τ r k (2) d + τ 0 (0)τ 0 (1)τ 0 (1) m1 k∈N1 τ r k (2)τ 0 (e) d1 · τ r3 (2)τ r4 (2)τ 0 (1) m2 k∈N2 τ r k (2)τ 0 (f ) d2 = τ 0 (0)τ r3 (2)τ 0 (1) m1 k∈N1 τ r k (2)τ 0 (e) d1 · τ 0 (1)τ r4 (2)τ 0 (1) m2 k∈N2 τ r k (2)τ 0 (f ) d2 where now the sums range over all e + f = 2, M 1 · ∪ M 2 = M \ {2}, N 1 · ∪ N 2 = N \ {3, 4} and d 1 + d 2 = d, d 1 , d 2 > 0. Proof. It follows from rational equivalence that τ 0 (0)τ 0 (1) m k∈N τ r k (2) ft * (λ) d = 1 (d!) 3 deg   j∈M ev * j (G j ) k∈N ev * k (P k )ψ r k k · ft * (λ) · M lab 0,1+m+n (R 2 , d)   does not depend on the choice of λ ∈ M 0,4 . Thus we can pick a very large λ 1 on the ray 12/34 of M 0,4 and a very large λ 2 on the ray 13/24 and set the degree equal for those two values. Denote by Z := deg   j∈M ev * j (G j ) k∈N ev * k (P k )ψ r k k · ft * (λ 1 ) · M lab 0,1+m+n (R 2 , d)   . We show that the left hand side of the above sum equals 1 d! 3 times the degree of Z. Analogously one can show that the right hand side equals 1 d! 3 times the degree of the analogous intersection product with λ 2 , which finishes the proof. By theorem 5.3 we know that Z = C∈S \|det C (ft × ev)\| · C, where S is the set of curves in M lab 0,1+m+n (R 2 , d) satisfying the point and line conditions and mapping to λ 1 under ft. Let F = (τ 0 (0)τ 0 (1) m k∈N τ r k (2)) d , then F is a one-dimensional cycle. Let σ be a cone of F corresponding to curves without a contracted bounded edge. Then lemma 4.4 says that the image of σ under ft is bounded. Since we picked λ 1 to be very large, we therefore know that σ cannot contribute to the degree of Z. Hence all C ∈ S contain a contracted bounded edge. Pick a curve C ∈ S, then we know by 6.4 that we can cut the contracted edge thus producing two curves C 1 and C 2 with an extra marked end e. If the degree of C 1 , d 1 , equals 0 then we know by 6.6 that \|det C (ft × ev)\| = (G 2 ·C 2 ) h(e) · \|detC 2 (ev M2∪N2 )\|, where G 2 denotes the line condition for the marked end 2 andC 2 denotes the curve that we get from C 2 by forgetting the additional marked end e. By 6.7 we know that each choice ofC 2 satisfying all conditions in L 2 ∪ M 2 ∪ N 2 = L ∪ M ∪ N \ {1, 2} contributes d · \|detC 2 (ev M2∪N2 )\| possible curves C (counted with weight). Thus the contribution to Z from curves C such that d 1 = 0 equals d · j∈M\{2} ev * j (G j ) k∈N ev * k (P k )ψ r k k · M lab 0,m−1+n (R 2 , d) which by the divisor equation (7.2) equals j∈M ev * j (G j ) k∈N ev * k (P k )ψ r k k · M lab 0,m+n (R 2 , d). Multiplying by the factor 1 d! 3 , we can see that those curves contribute τ 0 (1) m k∈N τ r k (2) d to 1 d! 3 deg Z. Now assume that d 1 > 0 and denote as in 6.2 Z i := j∈Mi ev * j (G j ) · j∈Ni ev * j (p j ) · j∈Ni ψ rj j · [M lab 0,li+ni+mi+1 (R 2 , d i )]. Then we know by 6.4 that one of the following three cases hold: (a) dim(Z 1 ) = 0 and dim(Z 2 ) = 2 or (b) dim(Z 1 ) = 1 and dim(Z 2 ) = 1 or (c) dim(Z 1 ) = 2 and dim(Z 2 ) = 0. We know by 6.6 that in the first case, \|det C (ev × ft)\| = \|det C1 (ev M1∪N1 )\| · \|det C2 (ev M2∪N2 × ev e )\|, where ev e now denotes the evaluation on both coordinates of the new marked end e. By 6.7 we know that for each choice of C 1 and C 2 satisfying the conditions we get exactly one possible C. But by 5.3 we know that Z 1 = C1 \|det C1 (ev M1∪N1 )\| · C 1 , and analogously ev * e (P ) · Z 2 = C2 \|det C2 (ev L2∪M2∪N2 × ev e )\| · C 2 , Thus we get a contribution of deg( Z 1 ) · deg(ev * e (P ) · Z 2 ) to deg(Z), respectively (d 1 !) 3 · (d 2 !) 3 d! 3 τ 0 (0)τ 0 (1)τ 0 (1) m1 k∈N1 τ r k (2) d1 · τ r3 (2)τ r4 (2)τ 0 (1) m2 k∈N2 τ r k (2)τ 0 (2) d2 to 1 d! 3 deg Z. Analogously, we get a contribution of (d 1 !) 3 · (d 2 !) 3 d! 3 τ 0 (0)τ 0 (1)τ 0 (1) m1 k∈N1 τ r k (2)τ 0 (2) d1 · τ r3 (2)τ r4 (2)τ 0 (1) m2 k∈N2 τ r k (2) d2 in the last case. In the second case, we know by 6.6 that \|det C (ev × ft)\| = (C 1 ·C 2 ) h(e) · \|detC 1 (ev M1∪N1 )\| · \|detC 2 (ev M2∪N2 )\| and by 6.7 we know that each choice ofC 1 andC 2 satisfying the conditions gives us d 1 · d 2 · \|detC 1 (ev M1∪N1 )\| · \|detC 2 (ev M2∪N2 )\|. Since (ft e ) * (Z i ) = C i \|detC i (ev Mi∪Ni )\| ·C i ( where ft e denotes the map which forgets the marked point e) and since d 1 · τ 0 (0)τ 0 (1)τ 0 (1) m1 k∈N1 τ r k (2) d1 = τ 0 (0)τ 0 (1)τ 0 (1) m1 k∈N1 τ r k (2)τ 0 (1) d1 and d 2 · τ r3 (2)τ r4 (2)τ 0 (1) m2 k∈N2 τ r k (2) d2 = τ r3 (2)τ r4 (2)τ 0 (1) m2 k∈N2 τ r k (2)τ 0 (1) d2 by the divisor equation we get a contribution of (d 1 !) 3 · (d 2 !) 3 d! 3 τ 0 (0)τ 0 (1)τ 0 (1) m1 k∈N1 τ r k (2)τ 0 (1) d1 · τ r3 (2)τ r4 (2)τ 0 (1) m2 k∈N2 τ r k (2)τ 0 (1) d2 . Finally, there are d d1 Lemma 8.2. Choose strictly positive integers r, d such that 1 + 3d − 3 + 2 = 2 + r. Then the classical one-marked-point invariant τ r (2) alg d equals τ r (2) alg d = 1 (d!) 3 . Proof. We use two (classical) WDVV equations ( [FP95] or, more detailed but unpublished, [Ko]) with four marked points. If we compute τ 0 (0)τ 0 (1) 2 τ r (2) ft * (λ) alg d for the two special points λ = 12/34 and λ = 13/24 on M 0,4 , then we get τ 0 (1) 2 τ r (2) alg d = τ 0 (0)τ r (2)τ 0 (2)= L L R 2 L L R 2 P, ψ r 0 d d 0 P, ψ r For τ 0 (0)τ 0 (1)τ 0 (2)τ r−1 (2) ft * (λ) alg d we get τ 0 (1)τ r−1 (2)τ 0 (2) alg d = τ 0 (0) 2 τ r−1 (2) alg d−1 as illustrated by P, ψ r−1 = L R 2 L P R 2 0 d − 1 d 1 P degree: P, ψ r−1 Now, applying string and divisor equation where possible and plugging in the left hand side of the first equation in the right hand side of the second equation produces d 3 · τ r (2) alg d = τ r−3 (2) alg d−1 . Together with the initial invariant τ 0 (0) 2 τ 0 (2) alg 0 = 1, this proves the lemma. Lemma 8.3. Choose strictly positive integers r, d such that 1 + 3d − 3 + 2 = 2 + r. Then the tropical one-marked-end invariant τ r (2) trop d equals τ r (2) trop d = 1 (d!) 3 . Proof. Choosing the single marked end e as root vertex, we get M lab 0,1 (R 2 , d) = M 0,1+3d × R 2 and the two projections are ft ′ and ev e . Recall that Psi-classes for parameterized curves are just pull-backs of Psi-classes along ft. Using [AR07, 9.6], we get τ r (2) trop d = 1 (d!) 3 deg (ft ′ (ψ e )) r · ev e (P ) · (M 0,1+3d × R 2 ) = 1 (d!) 3 deg (ψ r e · M 0,1+3d ) × (P · R 2 ) = 1 (d!) 3 , where in the last step we use deg(ψ r e · M 0,1+3d ) = 1 (cf. [KM07, 4.2]). Theorem 8.4. Let d, l, m, n and r k , k ∈ N be positive integers with d > 0 such that l + m + n + 3d − 3 + 2 = m + 2n + k∈N r k . Then the corresponding tropical and classical descendant invariants coincide, i.e. τ 0 (0) l τ 0 (1) m k∈N τ r k (2) trop d = τ 0 (0) l τ 0 (1) m k∈N τ r k (2) alg d . Proof. The tropical string, divisor and WDVV equations proved in the preceding sections are also fulfilled by the corresponding classical invariants. Hence, we can use these equations to reduce our invariants to such ones for which we know or can prove that they coincide. 1. case: r k = 0 for all k ∈ N (i.e. no Psi-classes) After applying string and divisor equation, we can assume that l = 0 = m. Using 5.3, we see that the numbers I trop (d; 0, 0, n; 0) are equal to the numbers N d defined in [GM05, 3.4, 3.9]. It is well-known that these numbers coincide with the classical ones (see [Mi03] and [GM05, 5.6]). 2. case: exists k ∈ N with r k > 0 (i.e. at least one Psi-class) subcase I: n = 1 After applying string and divisor equation, we can assume that l = 0 and m = 0. The two last preceding lemmas show that in this case the classical and tropical invariants coincide. subcase II: n ≥ 2 After applying string and divisor equation, we can assume that l = 0 and m = 1. In particular, if m = 0, we can use the divisor equation to add a line condition, which introduces a factor 1 d and therefore leads to rational numbers. Then we can use both the tropical (see theorem 8.1) and the classical WDVV equation ( [FP95] or [Ko]) and express τ 0 (1) k∈N τ r k (2) d in terms of invariants τ 0 (0) l ′ τ 0 (1) m ′ k∈N ′ τ r ′ k (2) d ′ with n ′ + k∈N ′ r ′ k < n + k∈N r k . Repeating this procedure, we eventually end up with n ′ = 1 (subcase I) or r ′ k = 0 for all k ∈ N ′ , which is the 1. case. LATTICE PATHS In this section, we present a lattice-paths algorithm to determine the numbers k∈N τ r k (2) d = 1 (d!) 3 deg k∈N ev * k (P k )ψ r k k · M lab 0,n (R 2 , d) i.e. numbers of curves with Psi-and point conditions (and no line conditions; all other numbers can be easily computed using string and divisor equation). Note that in this case we need 3d − 1 = n + k∈N r k to get a zero-dimensional cycle. We use the fact that if we choose general point conditions, the intersection product k∈N ev * k (P k )ψ r k k · M lab 0,n (R 2 , d) equals set-theoretically the set of all points corresponding to curves satisfying the Psi-and the point conditions (see 3.3). Each such curve C has to be counted with weight, and it is counted with the weight 1 d! 3 \| det C (ev)\| (see theorem 5.3), where ev denotes the product of all evaluation maps (at both coordinates). Note that no such curve can have a string since this would provide a deformation of the curve described in the proof of 4.4, which contradicts 3.2 (a). We pick a certain configuration of points and count dual Newton subdivisions of curves passing through the points and satisfying the Psi-conditions. The dual Newton subdivisions are in fact dual to the image h(Γ) ⊂ R 2 of the graph in the plane. In particular, the labels of the non-contracted ends are lost. That means we have to count tropical curves without labels on the non-contracted ends, and then multiply with the number of possibilities to set labels. There is a map c : M lab 0,n (R 2 , d) → M 0,n (R 2 , d) which forgets the labels of the non-marked ends. This map is a cover, the number of preimages is the number of ways to set labels. The biggest number of preimages is d! 3 . However, not every point in a facet has this number of preimages: the curve in M 0,0 (R 2 , 2) pictured below has only 4 preimages, not 8, since the two ends in direction (−1, 0) are not distinguishable. R 2 Let C ′ ∈ k∈N ψ r k k ·M lab 0,n (R 2 , d) and let C be the curve after forgetting the labels of the non-contracted ends. Assume that the facet σ in M 0,n (R 2 , d) in which C lies has s preimages under the cover above. Thus C has to be counted with s d! 3 \| det C ′ (ev)\|. Assume that C has t vertices V 1 , . . . , V t such that b ij non-contracted ends of the same direction −e j are adjacent to V i (where j goes from 0 to 2 and e 0 := −e 1 − e 2 ). Then ν C := t i=1 2 j=0 1 bij ! = s d! 3 and we have to count C with ν C \| det C ′ (ev)\|. First, we want to understand this weight locally in terms of vertex multiplicities. We define another weight that we denote by mult(C) and we show that it is equal to ν C \| det C ′ (ev)\|. Definition 9.1. Let C ′ ∈ k∈N ψ r k k · M lab 0,n (R 2 , d) and let C be the curve after forgetting the labels of the non-contracted ends. Define the weight mult(C) as ν C times the product of the multiplicities of those (necessarily 3-valent) vertices without any marked ends on them (see [Mi03], definition 2.16). Example 9.2. Let C be the curve as in the picture below. (For this example, we chose some other degree, not d, to keep the picture nice.) As in remark 6.5 we choose coordinates to write down an explicit matrix for ev. Choose V to be the root vertex. Then the matrix of ev is     1 0 v 1,1 0 0 1 v 1,2 0 1 0 0 v 2,1 0 1 0 v 2,2     . The absolute value of the determinant is equal to \| det(v 1 , v 2 )\|, which is the multiplicity of the 3-valent vertex V . Γ h V p 2 x 1 x 2 p 1 v 2 v 1 R 2 Lemma 9.3. Let C ′ ∈ k∈N ψ r k k · M lab 0,n (R 2 , d) and let C be the curve after forgetting the labels of the non-contracted ends. Then ν C \| det C ′ (ev)\| = mult(C) if C has no string. Proof. We have to show that \| det C ′ (ev)\| equals the product of multiplicities of all 3-valent vertices without marked end. This is an induction on the number of bounded edges. Curves with no bounded edge satisfy \| det C ′ (ev)\| = V mult(V ) = 1 (the product is empty). Curves with 2 bounded edges (as the one in the example 9.2) need to have the two marked ends at the two "outer" vertices, because otherwise there is a string. So as in the example, there is one "interior" 3-valent vertex without a marked end, and \| det C ′ (ev)\| is the multiplicity of the vertex. Now we can assume we have 4 or more bounded edges. (The number of bounded edges is even, 2n − 2.) We choose one such that there are still bounded edges on both sides of it. (If such an edge does not exist, it means we have a "star-shaped" tropical curve with one vertex in the middle and all bounded edges around. If one of those bounded edges was not adjacent to a marked end, C has a string, so we can assume that all bounded edges are adjacent to a marked end. If there is no marked end in the middle, we then have b + 2 = 2b where b is the number of bounded edges, since b + 2 is the number of coordinates of this cone, and 2b is the number of coordinates of the b marked ends. So b = 2 and we are in the situation of the example. If there is a marked end in the middle, we have b + 2 = 2b + 2 so b = 0, so this curve has no bounded edge and counts one. Now we can assume we do not have a star-shaped curve, and there is in fact a bounded edge with bounded edges on both sides.) Then we cut this edge to get two curves C 1 and C 2 . Let I i denote the subset of marked ends on C i and let e i be the number of non-contracted ends of C i . We make the cut edge a new non-contracted end of C i , so C i has in fact e i + 1 non-contracted ends, one of them the special new end. Assume #I 1 ≤ e 1 − 2 − k∈I1 r k , then if we remove all the closures of marked ends (as in lemma 4.3) we get k∈I1 r k + #I 1 + 1 connected components, which is less than or equal to k∈I1 r k + e 1 − 2 − k∈I1 r k + 1 = e 1 − 1. So there must be a connected component which has two non-contracted ends of C (not the new end of C 1 ). Hence C has a string, which contradicts the assumption. We have #I 1 + #I 2 = n = 3d − 1 − r k = e 1 + e 2 − 1 − r k . Therefore #I 1 = e 1 − k∈I1 r k and #I 2 = e 2 − 1 − k∈I2 r k without restriction. As in remark 6.5, we pick coordinates to write down an explicit matrix for ev. C 1 has e 1 + #I 1 − 2 − k∈I1 r k = 2#I 1 − 2 bounded edges. We pick the root vertex to be the boundary vertex of the cut edge in C 1 . We order the basis elements such that the root vertex comes first, then the bounded edges in C 1 , then the cut edge, then the bounded edges in C 2 . We order the basis of R 2n such that the marked ends in C 1 come first and then the marked ends in C 2 . Then the matrix of ev for C is a block matrix. The block on the top left is just the matrix of ev for C 1 -so by induction, the product of multiplicities of 3-valent unmarked vertices of C 1 . The top right block is 0, because no marked end on C 1 needs a bounded edge of C 2 . The bottom right block has the same determinant as the matrix ev for C 2 , when we add a marked end on the cut edge and make its end vertex the root vertex. So the determinant of this block is again by induction the product of multiplicities of 3-valent unmarked vertices in C 2 . This proves the claim. Now we know that 1 (d!) 3 deg k∈N ev * k (P k )ψ r k k · M lab 0,n (R 2 , d) equals the number of all curves C ∈ M 0,n (R 2 , d) satisfying the Psi-and the point conditions, each counted with weight mult(C). We want to simplify this count even further: we do not want to count parametrized tropical curves C = (Γ, x i , h), but we want to count their images in R 2 . Definition 9.4. Let C = (Γ, x i , h) ⊂ k∈N ψ r k k · M lab 0,n (R 2 , d). In the image h(Γ), some edges may lie on top of each other. Mark each edge in the image h(Γ) by a partition reflecting the weights of all edges which map onto this image edge. The image h(Γ) together with those partitions is called the labelled image of C. Example 9.5. The following picture shows a tropical curve and its labelled image. (1, 1) h Γ (1, 1, 1) (3) R 2 Given a labelled image, there can be different possible parametrizations. Ambiguity may for example arise if the labelled image comes from a parametrization that maps vertices on top of each other. We could then also parametrize this labelled image with a graph where the two vertices are replaced by only one. To avoid this ambiguity, we need a slightly more special notion of general conditions, which we call restricted general conditions. This definition is cooked up in such a way that we exactly avoid all ambiguity and make parametrizations unique. Definition 9.7. The subset of R 2n of restricted general conditions is defined to be the subset of the set of general conditions such that only simple curves C ∈ k∈N ψ r k k · M lab 0,n (R 2 , d) pass through the points (i.e. satisfy ev(C) = (P 1 , . . . , P n )). Remark 9.8. It is easy to see that the subset of restricted general conditions is still open and dense. Points which are not restricted general admit a non-simple curve. Being not simple sums up to codimension 1 conditions, hence only the image under ev of certain lower-dimensional subsets of k∈N ψ r k k · M lab 0,n (R 2 , d) is not restricted general. Lemma 9.9. Given a labelled image of a tropical curve through restricted general conditions, there is exactly one abstract tropical curve (Γ, x i ) and one map h parametrizing this labelled image and sending the marked ends to the P i . Proof. Clearly there is a parametrization C = (Γ, x i , h) of the labelled image, we just need to show that it is unique. Since the P i are general, C cannot have a contracted bounded edge. If it had a contracted bounded edge, we could vary the length of this edge without changing the image, in contradiction to 3.2(a). Hence all edges can be seen in the image h(Γ). Due to the conditions for being simple, we can also distinguish the images of vertices and the images of all edges in the labelled image. Because of the labels we know whether edges lie on top of each other. If there are edges lying on top of each other, then we know that they have to share a vertex. If there is a vertex V with two edges of the same direction, it has to be more than 3-valent. If it was 3-valent, then by the balancing condition the 3 edges would be mapped to a line. At least one of the 3 edges is bounded, and so we could change the length of this edge (and accordingly the lengths of the other two edges, if necessary) without changing the image of the curve. That contradicts 3.2(a). Since V is more than 3-valent, there must be a marked end adjacent to it. It is not possible that 2 (or more) of the points P i lie on a line with a direction that can be the direction of an edge. Thus if we have two edges in the labelled image on top of each other, there must be exactly one adjacent vertex which passes through a point P i . Thus we know that the edges have to be connected at that vertex when we built the parametrization. Definition 9.10. Let C be a curve in k∈N ψ r k k · M lab 0,n (R 2 , d) passing through restricted general conditions. Draw a dual Newton subdivision to the image h(Γ) and label the dual edges also with the partitions belonging to the edges of the labelled image h(Γ). This is called a labelled dual Newton subdivision. Mark the polygons dual to vertices which are adjacent to a marked end x i . Those marked polygons in ∆ d together with the partitions belonging to their boundary edges is called the set of dual marked polygons of C. Example 9.11. The following picture shows the labelled dual Newton subdivision to the labelled image from example 9.5. Next to it, we can see the set of dual marked polygons of C. (1, 1, 1) ( (1, 1) (1, 1, 1) (1, 1) Our aim is to count dual marked polygons to curves in k∈N ψ r k k · M lab 0,n (R 2 , d). To do that, we have to choose a special point configuration P as our condition. This configuration is chosen is such a way that the set of dual marked polygons can be described as something like a generalized lattice path that we call a rag rug. We will now first introduce labelled lattice paths and rag rugs, and then show that the count of rag rugs equals the count of labelled images of curves in k∈N ψ r k k · M lab 0,n (R 2 , d) passing through P (with weight mult(C)). Let ∆ d be the triangle with endpoints (0, 0), (d, 0) and (0, d). Fix λ to be a linear map of the form λ : R 2 → R : (x, y) → x − εy, where ε is a small irrational number. Recall that a path γ : [0, n] → R 2 is called a lattice path if γ\| [j−1,j] , j = 1, . . . , n is an affine-linear map and γ(j) ∈ Z 2 for all j = 0 . . . , n. For n = 1, . . . , n, we call γ\| [j−1,j] ([j − 1, j]) a step (the j-th step) of the lattice path γ. A lattice path is called λ-increasing if λ • γ is strictly increasing. Let p := (0, d) and q := (d, 0) be the points in ∆ := ∆ d where λ\| ∆ reaches its minimum (resp. maximum). Let G be a line in R 2 orthogonal to ker(λ). Then G divides the plane into two halfplanes. We will denote the upper one by H + and the lower one by H − . Definition 9.12. A labelled λ-increasing lattice path in ∆ is a λ-increasing lattice path from p to q such that the k-th step is labelled by a partition α k = ((α k ) 1 , . . . , (α k ) r k ) of the integer length of this step, that is (α k ) 1 + . . . + (α k ) r k = #(Z 2 ∩ γ([k − 1, k])) − 1. Remark 9.13. Let δ be a labelled λ-increasing lattice path from p to q whose image is contained in the boundary ∂∆ and whose steps are labelled with partitions consisting of only ones. All those paths will be possible end paths for the recursion defining multiplicity. The following picture shows 3 examples for ∆ 3 . (1) (1) (1) (1, 1) (1, 1, 1) (1) (1) (1) (1) (1, 1) (1, 1) (1) (1) γ 1 γ 2 γ 3 Definition 9.14. We define the positive multiplicity µ + (resp. negative multiplicity µ − ) of a labelled λ-increasing lattice path recursively as follows: (a) For a possible end path δ as in remark 9.13 going clockwise from p to q (resp. counterclockwise) with n steps we define µ ± (δ) := n k=1 1/(\|α k \|!), where \|α k \| denotes the size of the partition of the k-th step (recall it has to be a partition with only ones as entries). (b) For a labelled λ-increasing lattice path γ which is not a possible end path, assume that the k-th and the k + 1-th step form the first left (resp. right) corner of the path γ. (If no such turn exists, we define µ ± (γ) := 0.) Define a finite set of lattice paths as follows: • pick an integer r with 0 ≤ r ≤ min{\|α k \|, \|α k+1 \|}, • pick a set S of r pairs S = [(α k ) i1 , (α k+1 ) j1 ], . . . , [(α k ) ir , (α k+1 ) jr ] such that the multiset {(α k ) i1 , . . . , (α k ) ir } is a subset of the multiset {(α k ) 1 , . . . , (α k ) r k } and the multiset {(α k+1 ) j1 , . . . , (α k+j ) jr } is a subset of the multiset {(α k+1 ) 1 , . . . , (α k+1 ) r k+1 }. For each l = 1, . . . , r, build a triangle T r,S,l with one edge of integer length (α k ) i l and one edge of integer length (α k+1 ) j l (in the direction of the k-th resp. k + 1-th step). Let M r,S be the Minkowski sum of all triangles T r,S,l for l = 1, . . . , r, and edges e s in direction of the k-th step of integer length (α k ) s for all s which are not one of the i l and edges f t in direction of the k + 1-th step of integer length (α k+1 ) t for all t which are not one of the j l . Label each edge E of M r,S with a partition reflecting the integer lengths of edges e s , f t , and edges of triangles T r,S,l that contribute to E. Think of the polygon M r,S as sitting in the corner built by step k and k + 1 of γ, and define a new labelled λ-increasing lattice path γ r,S by going the other way around M r,S . If M r,S does not fit inside the polygon ∆, we define µ ± (γ r,S ) = 0. The positive multiplicity of this new labelled λ-increasing lattice path is known recursively, because it includes a smaller area with the possible end paths. We define µ ± (γ) = r S Area(T r,S,1 ) · . . . · Area(T r,S,r ) · µ ± (γ r,S ), (where Area(T ) is the normalized lattice area, i.e. the area of the simplex with vertices (0, 0), (1, 0) and (0, 1) is defined to be 1). Example 9.15. For the 3 possible end paths from remark 9.13, we have multiplicity µ − (γ 1 ) = 1, µ − (γ 2 ) = 1 4 and µ − (γ 3 ) = 1 12 . Example 9.16. The following picture shows an example of the recursion from definition 9.14 to compute the positive multiplicity of a labelled path γ in ∆ 3 . The first left turn is from step 2 to step 3. We have 3 choices for r: r = 0, r = 1 or r = 2, since both the partition of step 2 as the partition of step 3 contain 2 elements. No matter what we choose for r, there is just one choice for the set S (r pairs consisting of all ones), since both partitions contain only ones. For r = 0 and S = ∅, M 0,∅ is a square of size 2 which does not fit inside ∆ 3 . Therefore the multiplicity of γ 0,∅ = 0. For r = 1 and S = {(1, 1)}, M 1,S is a pentagon. The integer length of each new side is one. The new side of direction (0, 1) is labelled by the partition (1), because it comes from the edge e s in direction of the 2-nd step of integer length 1 which is not one of the i l in the set of pairs S. The new side of M 1,S of direction (1, −1) comes with label (1), because it comes from a side of the triangle T 1,S,1 of integer length 1. The side of direction (1, 0) comes from an edge of the 3-rd step which is not part of S, and gets label (1) as well. The area of T 1,S,1 is one. For r = 2, we have S = {(1, 1), (1, 1)}, and M 2,S is a triangle of size 2 whose new side gets the label (1, 1) because it comes from 2 sides of the two triangles T 2,S,1 and T 2,S,2 . The area of both triangles T 2,S,1 and T 2,S,2 is one. The picture shows how the recursion goes on after the first step. The choices where r = 0 for γ 1,S or where r = 1 for γ 2,S are left out because they yield to a path of multiplicity 0. We end up with one path of multiplicity 1 and one of multiplicity 1 2 , so µ + (γ) = 3 2 . (1, 1) (1, 1) (1, 1) γ (1, 1) (1, 1) (1, 1) γ 0,S γ 1,S γ 2,S (1, 1) Definition 9.17. Let F be a set of n convex polytopes Q 1 , . . . , Q n inside ∆ whose endpoints are lattice points of ∆ and whose boundary edges e are labelled by partitions. It is possible that a polygon Q i is 1-dimensional, i.e. just an edge itself, then it has two partitions as labels, one for each outward pointing normal vector. We call F a rag rug of the form (r 1 , . . . , r n ) if the following conditions are satisfied: (a) the (outside) label α e of an edge e in the boundary of ∆ d is α e = (1, 1, . . . , 1), (b) two polygons Q i and Q j intersect in at most one point, (c) boundary edges whose outward normal vector points into H + (starting at G) (with their corresponding labels) form a labelled λ-increasing lattice path from p to q that we will denote by γ + , (d) boundary edges whose outward normal vector points into H − (with their corresponding labels) form a labelled λ-increasing lattice path from p to q that we will denote by γ − , (e) the order of the polytopes Q 1 , . . . , Q n agrees with the obvious order given by the paths γ + resp. γ − , (f) the sum of the sizes of the partitions of the boundary edges of Q i is equal to r i + 2, e\|e edge of Qi \|α e \| = r i + 2. We define the multiplicity µ(F ) to be µ + (γ + ) · µ − (γ − ). Example 9.18. The following picture shows a rag rug F of the form (2, 2, 0, 0) in ∆ 3 , and the two labelled λ-increasing lattice paths γ + and γ − . For all edges of integer length one, the corresponding partitions are just (1) and we did not mark this in the picture. We have µ + (γ + ) = 3 and µ − (γ − ) = 1 6 , so µ(F ) = 1 2 . γ − γ + HM (1, 1, 1) (3)(3) (1, 1, 1) Definition 9.19. Given d, n and numbers (r 1 , . . . , r n ) we define N rr (d, n, (r 1 , . . . , r n )) to be the number of rag rugs of form (r 1 , . . . , r n ), counted with multiplicity as defined in 9.17. Remark 9.20. Note that this definition generalizes Mikhalkin's lattice path count (see [Mi03]). A λincreasing lattice path γ from p to q is a rag rug of form (0, . . . , 0). We have to attach labels (1) to each edge. The two paths γ + and γ − agree with γ. In the recursion for the lattice path count, we define mult ± (γ) depending on the multiplicity of two other paths γ ′ and γ ′′ . γ ′ is the path that cuts the corner, and γ ′′ is the path that completes the corner to a parallelogram. In our definition, we can choose r = 0 or r = 1. For r = 0, we have S = ∅ as only choice. The polygon M 0,∅ is the parallelogram which is equal to the Minkowski sum of the two steps of the corner. For r = 1, we have S = {(1, 1)} as only choice, and M 1,S is the triangle formed by the two steps of the corner. Since all partitions are just (1), also the end paths have only those partitions, so that there is in fact only one end path, the path δ ± . It has multiplicity 1. Therefore our definition gives the same multiplicity in this case. It is not true that N rr (d, n, (r 1 , . . . , r n )) = 1 (d!) 3 deg k ev * k (P k )ψ r k k · M lab 0,n (R 2 , d) , since we count also reducible curves with the rag rugs. We therefore have to modify the count and throw away the dual subdivisions corresponding to reducible tropical curves. Definition 9.21. Given a rag rug γ and the two corresponding lattice paths γ + and γ − , perform the recursion to compute their multiplicity and keep track of the polygons M r,S that the new paths γ r,S in the recursion enclose with γ ± . This way we end up with a set of labelled Newton subdivisions. We call this the set of possible labelled Newton subdivisions for γ. The recursion allows us to assign a multiplicity to a possible labelled Newton subdivision, so that the multiplicity of γ is equal to the sum of the multiplicities of the possible labelled Newton subdivisions for γ. Definition 9.22. Given a labelled Newton subdivision, draw a dual labelled image and then the unique tropical curve mapping to this image. This is well-defined up its position in R 2 and the lengths of its bounded edges. We say that the Newton subdivision is reducible if the tropical curve mapping to a dual labelled image is reducible (again, this does not depend on the choice of dual labelled image). Otherwise, we say it is irreducible. Remark 9.23. It is possible to express the reducibility condition in terms of the Newton subdivision itself and not in terms of the dual tropical curve. A labelled marked Newton subdivision is reducible if and only if it admits a mixed subdivision where the marked polygons Q i remain unmixed (i.e. come from sums of the form Q i + v 1 + . . . + v t , where the v j are vertices of the subdivision of the j-th summand). For details, see [M08]. Definition 9.24. For a rag rug γ as defined in 9.17 define its irreducible multiplicity mult ′ (γ) to be the multiplicity mult(γ) minus the number of possible reducible Newton subdivision (counted with multiplicity). Given d, n and numbers (r 1 , . . . , r n ) we say N ′ rr (d, n, (r 1 , . . . , r n )) is the number of rag rugs counted with their irreducible multiplicity. Definition 9.25. Let P λ = (P 1 , . . . , P n ) denote n restricted general point conditions on the line G orthogonal to ker(λ) such that the distance between P i and P i+1 is much bigger than the distance between P i−1 and P i . Lemma 9.26. Let C ∈ k ψ r k k · M lab 0,n (R 2 , d) with ev(C) = P λ . Then the set of dual marked polygons of C is a rag rug of the form (r 1 , . . . , r n ). Proof. The polygon Q i dual to P i is convex and has to satisfy e\|e edge of Qi \|α e \| = r i + 2, where α e denotes the partition belonging to e. This is true since the marked end x i ⊂ Γ is adjacent to a vertex of valence r i + 3 and we can see all edges (except the contracted end x i ) in the labelled image (and thus in their labelled dual Newton subdivision, too). (Outside) labels of edges in the boundary of ∆ d have only ones as entries, since the ends of C are all of weight 1. That the boundaries of those polygons form labelled λ-increasing lattice paths follows analogously to [Mi03], 8.27 (or [Ma06], 5.48 for more details). Theorem 9.27. The number N ′ rr (d, n, (r 1 , . . . , r n )) from definition 9.24 equals the intersection product 1 (d!) 3 deg k ev * k (P k )ψ r k k · M lab 0,n (R 2 , d) = k τ r k (2) . Proof. To determine 1 (d!) 3 deg k ev * k (P k )ψ r k k · M lab 0,n (R 2 , d) , we can draw all labelled images of tropical curves that pass through P λ and count them each with their weight mult(C) which is ν C times the product of the multiplicities of non-marked vertices. We show that this count is equivalent to counting irreducible possible labelled Newton subdivision for all rag rugs of the form (r 1 , . . . , r n ). The proof is a generalization of the proof of theorem 2 of [Mi03]. We know that each C leads to a rag rug γ C as in lemma 9.26. Each rag rug yields a set of possible labelled Newton subdivision, with multiplicity. We will show that for each such possible labelled Newton subdivision, there is a dual tropical curve C through P λ of the same weight. At the same time, we show that for each curve C through P λ , the dual labelled Newton subdivision is possible for the rag rug γ C . Let γ be a rag rug. The recursion for γ + yields possible subdivision of ∆ d above the polygons Q i . They correspond to the part of a tropical curve above G. Analogously, possible subdivisions for γ − correspond to the parts of tropical curves below G. From lemma 9.3 it follows that weight of a tropical curve can be computed locally, so the weight of C is equal to the weight of the part above G times the weight of the part below G. The same is true for the multiplicity of the dual Newton subdivision. Therefore it is enough to show that for each subdivision above the Q i , there is a dual part of a tropical curve above G of the same weight, and that each part of a tropical curve above G is dual to a possible subdivision above the Q i . The corresponding statement for subdivisions below the Q i and parts of tropical curves below G follows analogously, and thus the complete statement follows. For each point P i ∈ P λ , draw edges emanating from P i of directions dual to the boundary edges of Q i and with the same partitions as labels. Draw a line G ′ in H + parallel to G, such that the strip between G and G ′ encloses one intersection of the edges we have drawn through the P i . This intersection of edges corresponds to the first left turn of the path γ + , since the distances between the P i are increasing. Let us determine the possibilities how the tropical curve can go on at this point. We should think about both edges as a set of edges of weights given by the partition. Edges can either meet in a 3-valent vertex, or intersect. First, we pick r less than the smaller number of edges in a set to determine how many edges should meet in a 3-valent vertex. Then we pick a set of r pairs of weights to determine which edges should meet in a 3-valent vertex. The other edges intersect. The weight locally in the strip between G and G ′ is equal to the product of areas of triangles dual to the 3-valent vertices because of lemma 9.3. The dual polygon is the Minkowski sum of those triangles and the remaining edges which intersect. Therefore the recursion for the multiplicity of γ + corresponds to the possibilities for a labelled image of a tropical curve in the strip between G and G ′ and keeps track of the weight. The end paths which do not have zero multiplicity are exactly those dual to ends of direction (1, 1) and weight one. The multiplicity of such an end path corresponds to the correction factor ν C with which we have to divide the weight of a tropical curve if more than one non-contracted end is adjacent to the same vertex. Example 9.28. The following picture shows how to count τ 2 (2) 2 τ 0 (2) 2 3 using rag rugs. The left column shows all rag rugs of the form (2, 0, 2, 0) in the triangle ∆ 3 . The middle column shows the possible Newton subdivisions for the rag rugs and their multiplicity. The third column shows sketches of the dual tropical curves. (1, 1) (1, 1) (1, 1) Lemma 3. 1 . 1There exists a open dense subset U ⊂ R 2 such that for v ∈ U and a translation G ′ := G+v of G, it holds:(a) The subcomplex f −1 (G ′ ) is either empty or of pure codimension 1 in X. (b) The interior of a facet of f −1 (G ′ ) is contained in the interior of a facet of X. (c) For an element C in the interior of a facet of f −1 (G ′ ), the image f (C) lies in the interior of a facet of G. the single vertex of M 0,4 . This ensures that all points in ft \| −1 X (λ) lie in the interior of a one-dimensional polyhedron of X. Lemma 4. 3 . 3Let C be a curve in the interior of a facet of F . Then C contains a movable string S. Construction 6. 2 . 2Assume C satisfying \| det C (ev × ft)\| = 0 has a contracted bounded edge e. Cut the bounded edge e, thus producing two marked ends. In this way we get two curves C 1 and C 2 that both have a new marked end in the place of e. Let L i , M i and N i be the subsets of L, M and N of marked ends in C i . Let l i , m i and n i be the sizes of these subsets. Let d i be the degree of C i . Denote by ev Mi∪Ni : j∈Ni ψ rj j · [M lab 0,li+ni+mi+1 (R 2 , d i )] → R mi+2ni the map that evaluates the first coordinate for the points in M i and both coordinates for the ends in N i (as in 5.2). Denote by ev e : j∈Ni ψ rj j · [M lab 0,li+ni+mi+1 (R 2 , d i )] → R 2 the evaluation at e at both coordinates, and by (ev e ) x the evaluation at the first coordinate. Denote bỹ C i the curve C i where we remove the marked end e and straighten the 2-valent vertex which appears. Lemma 6. 4 . 4Let C be as in construction 6.2 and stick to the notations from there. If ft(C) = 12/34 (then {1} = L 1 , so l 1 = 1 and l 2 = 0), then either d 1 = 0 and L 1 ∪ M 1 ∪ N 1 = {1, 2}, or d 1 , d 2 > 0 and there are 3 cases to distinguish (of which the first and last are symmetric): Definition 9.6. A curve C = (Γ, x i , h) is called simple, if it satisfies:(a) the map h is injective on vertices, (b) if h(V ) ∈ h(e) for a vertex V and an edge e then V is adjacent to an edge e ′ which is mapped on top of e, (c) if two edges e and e ′ are mapped on top of each other, then they share a vertex, (d) assume p ∈ R 2 is a point through which more than two edges pass. Divide the edges into equivalence classes depending on the slope of the line to which they are mapped. Then we have at most 2 equivalence classes. Lemma 2.1 (Translations are rationally equivalent). Let h be a rational function on R n , choose v ∈ R n and consider h ′ with h ′ (x) := h(x + v). Then h and h ′ are rationally equivalent. choices of the sets E 1 and E 2 if we fix d 1 and d 2 . First steps in tropical intersection theory. Lars Allermann, Johannes Rau, arxiv:0709.3705Mathematische Zeitschrift. to appear); also atLars Allermann, Johannes Rau, First steps in tropical intersection theory, Mathematische Zeitschrift (to appear); also at arxiv:0709.3705. Notes on stable maps and quantum cohomology. William Fulton, Rahul Pandharipande, Proc. Symp. Pure Math. 62also at arxiv:alg-geom/9608011William Fulton, Rahul Pandharipande, Notes on stable maps and quantum cohomology, Proc. Symp. Pure Math. 62, part 2, 45-96 (1997); also at arxiv:alg-geom/9608011. Mirror symmetry for P 2 and tropical geometry. Mark Gross, arxiv:0903.1378preprintMark Gross, Mirror symmetry for P 2 and tropical geometry, preprint arxiv:0903.1378. Tropical fans and the moduli spaces of tropical curves. Andreas Gathmann, Michael Kerber, Hannah Markwig, arxiv:0708.2268Compos. Math. 1451Andreas Gathmann, Michael Kerber, Hannah Markwig, Tropical fans and the moduli spaces of tropical curves, Com- pos. Math. 145, No. 1, 173-195 (2009); also at arxiv:0708.2268. Kontsevich's formula and the WDVV equations in tropical geometry. Andreas Gathmann, Hannah Markwig, arxiv:math/0509628Adv. Math. 2172Andreas Gathmann, Hannah Markwig, Kontsevich's formula and the WDVV equations in tropical geometry, Adv. Math. 217, No. 2, 537-560 (2008); also at arxiv:math/0509628. Joachim Kock, Notes on Psi classes. Joachim Kock, Notes on Psi classes, available at http://mat.uab.es/ kock/GW/notes/psi-notes.pdf. Intersecting Psi-classes on tropical M 0,n. Michael Kerber, Hannah Markwig, arxiv:0709.3953Int. Math. Res. Not. 2Michael Kerber, Hannah Markwig, Intersecting Psi-classes on tropical M 0,n , Int. Math. Res. Not. 2009, No. 2, 221-240 (2009); also at arxiv:0709.3953. An equivalent condition for the reducibility of tropical curves, preprint. Brian Mann, Ann ArborUniversity of Michiganin preparationBrian Mann, An equivalent condition for the reducibility of tropical curves, preprint, University of Michigan, Ann Arbor, in preparation. The enumeration of plane tropical curves. Hannah Markwig, TU KaiserslauternPhD thesisHannah Markwig, The enumeration of plane tropical curves, PhD thesis, TU Kaiserslautern, 2006; available at http://www.crcg.de/wiki/index.php5?title=Publications Hannah. Enumerative tropical geometry in R 2. Grigory Mikhalkin, arxiv:math/0312530J. Amer. Math. Soc. 18Grigory Mikhalkin, Enumerative tropical geometry in R 2 , J. Amer. Math. Soc. 18, 313-377 (2005); also at arxiv:math/0312530. Tropical geometry and its applications. Grigory Mikhalkin, arxiv:math/0601041Int. Congress Math. IIGrigory Mikhalkin, Tropical geometry and its applications, Int. Congress Math. Vol. II, 827-852 (2006); also at arxiv:math/0601041. Moduli spaces of rational tropical curves. Grigory Mikhalkin, arxiv:0704.0839Proceedings of the 13th Gökova geometry-topology conference. the 13th Gökova geometry-topology conferenceCambridge, MAInternational PressGrigory Mikhalkin, Moduli spaces of rational tropical curves, Proceedings of the 13th Gökova geometry-topology conference, Cambridge, MA, International Press, 39-51 (2007); also at arxiv:0704.0839. Jürgen Richter-Gebert, Bernd Sturmfels, Thorsten Theobald, math/0306366First steps in Tropical geometry, Idempotent Mathematics and Mathematical Physics, Proceedings Vienna. Jürgen Richter-Gebert, Bernd Sturmfels and Thorsten Theobald, First steps in Tropical geometry, Idempotent Mathe- matics and Mathematical Physics, Proceedings Vienna, 2003, math/0306366. . David Speyer, Bernd Sturmfels, arxiv:math/0408099Tropical mathematics. David Speyer, Bernd Sturmfels, Tropical mathematics, preprint arxiv:math/0408099. . Hannah Markwig, Mathematisches Crcg, Georg-August-Universität Institut, Bunsenstr Göttingen, GÖTTINGEN, GERMANYHANNAH MARKWIG, CRCG, MATHEMATISCHES INSTITUT, GEORG-AUGUST-UNIVERSITÄT GÖTTINGEN, BUNSENSTR. 3-5, 37073 GÖTTINGEN, GERMANY	[]
[ "Emergence of Spontaneously Broken Supersymmetry on an Anti-D3-Brane in KKLT dS vacua", "Emergence of Spontaneously Broken Supersymmetry on an Anti-D3-Brane in KKLT dS vacua" ]	[ "Renata Kallosh \nDepartment of Physics\nStanford University\n94305StanfordCAUSA\n", "Timm Wrase \nDepartment of Physics\nStanford University\n94305StanfordCAUSA\n" ]	[ "Department of Physics\nStanford University\n94305StanfordCAUSA", "Department of Physics\nStanford University\n94305StanfordCAUSA" ]	[]	The KKLT construction of de Sitter vacua includes an uplifting term coming from an anti-D3-brane. Here we show how this term can arise via spontaneous breaking of supersymmetry, based on the emergence of a nilpotent chiral supermultiplet on the world-volume of the anti-D3brane. We establish and use the fact that both the DBI as well as the WZ term, with account of orientifolding, acquire a form of the Volkov-Akulov action. For an O3 orientifold involution of R 9,1 we demonstrate the cancellation between the fermionic parts of the DBI and WZ term for the D3-brane action. For the anti-D3-brane we show that the DBI action and the WZ action combine and lead to the emergence of the goldstino multiplet which is responsible for spontaneous breaking of supersymmetry. This provides a string theoretic explanation for the supersymmetric uplifting term in the KKLT effective supergravity model supplemented by a nilpotent chiral multiplet.	10.1007/jhep12(2014)117	[ "https://arxiv.org/pdf/1411.1121v2.pdf" ]	52,971,613	1411.1121	ddfaeda37c1ebd9cf277ec8eb09f6500f103ebb9	Emergence of Spontaneously Broken Supersymmetry on an Anti-D3-Brane in KKLT dS vacua 14 Nov 2014 Renata Kallosh Department of Physics Stanford University 94305StanfordCAUSA Timm Wrase Department of Physics Stanford University 94305StanfordCAUSA Emergence of Spontaneously Broken Supersymmetry on an Anti-D3-Brane in KKLT dS vacua 14 Nov 2014 The KKLT construction of de Sitter vacua includes an uplifting term coming from an anti-D3-brane. Here we show how this term can arise via spontaneous breaking of supersymmetry, based on the emergence of a nilpotent chiral supermultiplet on the world-volume of the anti-D3brane. We establish and use the fact that both the DBI as well as the WZ term, with account of orientifolding, acquire a form of the Volkov-Akulov action. For an O3 orientifold involution of R 9,1 we demonstrate the cancellation between the fermionic parts of the DBI and WZ term for the D3-brane action. For the anti-D3-brane we show that the DBI action and the WZ action combine and lead to the emergence of the goldstino multiplet which is responsible for spontaneous breaking of supersymmetry. This provides a string theoretic explanation for the supersymmetric uplifting term in the KKLT effective supergravity model supplemented by a nilpotent chiral multiplet. Introduction The manifestly supersymmetric effective d = 4 supergravity action describing the KKLT model of the AdS stabilization of the volume modulus in type IIB string theory results from the following Kähler potential and superpotential, [1,2]: W = W 0 + Ae −aρ , K = −3 ln(ρ + ρ) . (1.1) The supersymmetric AdS vacua in KKLT models are defined by the equation D ρ W = 0. The uplifting term was added in the next step in the KKLT construction in the form [1,2] δV = D (ρ +ρ) 3 . ( 1.2) In the string theory model [3] it has been argued that the presence of the anti-D3-branes breaks supersymmetry spontaneously since the anti-D3-branes can decay to a supersymmetric state by annihilating with fluxes. However, it was not clear how to write down an effective N =1 supergravity action: in [1,2] eq. (1.2) was used, which corresponds to a pure bosonic term breaking supersymmetry explicitly. It was explained in [2] that for a D3-brane slowly moving in the background with no anti-branes the net force vanishes due to gravitational and five-form cancellations: the relevant parts of the DBI and the WZ terms cancel. For the anti-D3-brane the force exerted by gravity and the five-form field are of the same sign and add, so we have a factor of 2 for the anti-D3-brane versus 0 for the D3-brane, leading to (1.2). This argument was developed in [2] in the absence of the fermions on the brane. In this paper we will find that when the fermions on the brane are taken into account and supersymmetry is broken spontaneously, the same effect, doubling versus cancellation, of the full Volkov-Akulov goldstino action [4] takes place. This will provide us with a supersymmetric uplifting of the supergravity KKLT models which has an origin in the supersymmetric D-brane physics. Recently, a systematic construction of metastable de Sitter vacua in a broad class of string theory motivated supergravity models was performed in [5]. It has confirmed the standard expectation that supersymmetry is an indispensable tool, which helps to find many dS vacua and simultaneously ensures their local stability. More recently it was pointed out in [6] that in supergravity one could have started with the following supersymmetric model, depending on 2 supermultiplets, ρ and S, where S represents a Volkov-Akulov goldstino multiplet [4] W = W 0 + Ae −aρ − M 2 S , K = −3 ln(ρ + ρ) + SS at S 2 = 0 . (1.3) Here S is the nilpotent 1 chiral supermultiplet [7,8] which provides a manifestly supersymmetric version of the Volkov-Akulov goldstino. After computing the potential, we have to set the scalar part of the superfield S to zero. We find V = V KKLT (ρ,ρ) + M 4 (ρ +ρ) 3 ,(1.4) where V KKLT (ρ,ρ) is the KKLT potential without the uplifting term, at M = 0. This shows that (1.3) corresponds to a manifestly supersymmetric supergravity version of the uplifting term arising from an anti-D3-brane (extending the bosonic expression for the uplifting term from the anti-D3-brane used in [1,2]). For simplicity we consider here the case without warping. This will allow us to study the supersymmetry upon gauge-fixing of κ-symmetry on the world-volume of the brane in a flat type IIB supergravity background, which is a relatively simple case. Generalization to a generic type IIB background will be a next step. From the point of view of d = 4 supergravity, the supersymmetrization of the uplifting due to a chiral nilpotent multiplet is obvious. It is less obvious how all this is related to D-brane physics and to the fact that adding a D3-brane to the system considered in [1,2] will not lead to an uplift, whereas adding an anti-D3-brane, will result in the emergence of a VA multiplet and supersymmetric uplifting. Below we will present a refined relation between our d = 4 supergravity and Dp-brane physics with global supersymmetry and local κ-symmetry [9][10][11][12][13][14][15]. In [6] we referred to the well known argument [12,15] that a Dp-brane action, when gauge-fixed in a certain gauge, always leads to a DBI term which has Volkov-Akulov fermions on its world volume. Not surprisingly, the non-linear VA fermions are superpartners of the Born-Infeld non-linear vectors. In the same gauge the WZ terms vanishes, as was first established in [10]. It appears therefore that the emergence of the VA fermions takes place independently of the charge of the brane: for the Dp-brane and for the anti-Dp-brane we are always getting the VA fermions. However, this is not expected to be true in the context of the KKLT model, where by construction, only the anti-D3-brane can be responsible for the uplifting, a D3-brane will not do the job. There must be a reason why the emergence of the VA fermion on the word-volume of the brane is different for a D3-brane and an anti-D3-brane. And indeed, as we show in this paper, for a large class of models (that include the KKLT scenario) such a reason exists: In order to preserve N = 1 supersymmetry in d = 4 starting from type II N = 2 supersymmetry in d = 10 one has to compactify the theory on a Calabi-Yau manifold, and in addition perform an orientifold projection. However, the standard κsymmetry gauge [10] in which the WZ term for any brane vanishes, is incompatible with an orientifold projection. If, instead, one uses a κ-symmetry gauge fixing that is consistent with the orientifold projection, then the WZ action does not vanish and the emergence/vanishing of the VA fermions on the world-volume indeed depends on the charge of the brane. The remarkable discovery of the fact that the WZ term of the D9-brane with the type I orientifold truncation becomes a Volkov-Akulov goldstino action was made in [16,17]. Therefore, depending on the choice of the charge of the brane, for a given choice of the sign in the orientifolding condition the total action either vanishes or becomes the sum of the two VA actions. This gives a hint on a possible reason for an analogous dependence on the charge of the brane for a D3-/anti-D3-brane in the presence of an O3 orientifold projection. In this paper we perform a generic analysis of the Dp-brane in a flat background and show that the WZ term upon orientifolding becomes exactly the VA action. This gives an analytic explanation of the computational result in [16,17] for the D9-brane case, and also makes this result more general including other cases, like the D3-brane. For our purpose to find the origin of the Volkov-Akulov dynamics with a single goldstino, corresponding to a single nilpotent superfield in our supergravity models (1.3) we find it convenient to study and to compare the cases of a single D3-brane versus a single anti-D3-brane on top of an O3-plane. Hopefully, the phenomenon which we describe here will be preserved in a more realistic string theory setting with many coincident branes, fluxes, curved geometry and with an account of the volume of the compactified manifold. The outline of the paper is as follows: In section 2 we present the classical κ-symmetric D3and anti-D3-brane actions in the flat supergravity background. In section 3 we discuss the issue of a compatibility of orientifolding with κ-symmetry gauge-fixing, following earlier studies in [18]. We also derive in that section the DBI and the WZ actions for D3-/anti-D3-brane with account of orientifolding and show that they both have the same fermion parts, given by the Volkov-Akulov goldstino action. Therefore, depending on the sign in the orientifolding condition, either the D3-or anti-D3-brane action vanishes whereas the other one acquires a VA goldstino action. We also discuss a possible modification of this construction in case that the flat background is replaced by a CY 3 compactification. In Appendix A we describe the generic case of a Dp-or anti-Dp-brane with the corresponding orientifold projection, and show how, in general, one finds that the WZ term upon orientifolding becomes the VA action. Classical actions for D3 and anti-D3 branes A detailed description of classical IIB Dp-branes is given in Appendix A.1 of [15] and we are using the notation of this paper. The κ-symmetric D3-brane action with q = 1 and anti-D3-brane action with q = −1, in a flat background geometry consists of the Dirac-Born-Infeld-Nambu-Goto term S DBI and Wess-Zumino term S (q) WZ with the world-volume coordinates σ µ (µ = 0, . . . , 3): 2 S DBI + S (q) WZ = − d 4 σ − det(G µν + α F µν ) + q Ω 4 . (2.1) Here the longitudinal and transverse coordinates are X m = {X m , φ I } , m = 0, 1, 2, 3 , I = 1, . . . , 6 ,(2.2) where m refers to the 4 worldvolume directions and I refers to the 6 transverse directions and G µν = η m n Π m µ Π n ν + δ IJ Π I µ Π J ν , Π m µ = ∂ µ X m −θΓ m ∂ µ θ , Π I µ = ∂ µ φ I −θΓ I ∂ µ θ . (2.3) The φ I are the scalars on the D3-brane that control its position in the six transverse directions, and the Born-Infeld field strength F µν is given by F µν ≡ F µν − b µν , b µν = α −1θ σ 3 Γ m ∂ µ θ ∂ ν X m − 1 2θ Γ m ∂ ν θ − (µ ↔ ν) . (2.4) Finally, Ω 4 is a particular 4-form [9][10][11]. Here we will describe it using the formalism in the flat supergravity background in [10,13,14]. Namely, we define a closed 5-form I 5 ≡ d Ω 4 = dθT 3 dθ , (2.5) where wedges products are implicit and the 3-form T 3 = σ 1 FΓ + iσ 2Γ 3 3! , (2.6) depends on the matrix-valued 1-form 3 Γ = Γ m Π m = Γ m (dX m +θΓ m dθ) . (2.7) We have also introduced the pull-backs of the flat matrices Γ m to the world-volume: Γ µ ≡ Π µ m Γ m ,Γ µ ≡ G µνΓ ν = Π µ m Γ m , Π µ m = G µν Π n ν η mn , (2.8) where G µν is the inverse of G µν . They satisfy the Clifford algebra relationsΓ µΓν +Γ νΓµ = 2G µν and Π µ m Π m ν = δ µ ν . The brane action (2.1) has a global supersymmetry under which δ Π m = 0 and δ F = 0. Besides the global supersymmetry the action is also invariant under a local κ-symmetry (presented in details in our notation in Appendix A.1 of [15] and in eq. (A.8) in this paper). The κ-symmetry δ κ θ = (1 + qΓ)κ , (2.9) is defined in terms of the hermitian traceless product structure Γ with Tr Γ = 0 , Γ 2 = 1. Note that in the standard κ-symmetry gauge taken in [10,12,15] (1 ± σ 3 )θ = 0 , (2.10) the WZ term (2.5) vanishes since (2.6) involves the off-diagonal σ 1 and σ 2 . The gauge-fixed action of the D3-and anti-D3-brane is the same and is given in eqs. (85)-(88) in [10]. D3-and anti-D3-brane with orientifolding The relation between orientifold truncation and gauge-fixing κ-symmetry for a D3-brane was discussed in detail in [18]. An orientifold action requires that (1 − Γ O )θ = 0. The gauge-fixing condition for κ-symmetry can be given in the form (1−Γ κ )θ = 0. In order for these two conditions to be compatible we need that [Γ O , Γ κ ] = 0 . (3.1) The O3 orientifolding studied in [18] for the D3-brane is defined by Γ O = iσ 2 Γ 0123 . Thus, the general gauge-fixing condition for a Dp-brane (2.10) with Γ κ = ∓σ 3 which leads to a vanishing WZ term is incompatible with the O3 orientifold projection since [Γ O , Γ κ ] = 0 and the gauge-fixing for which the WZ term vanishes cannot be used. To describe the KKLT physics we would like to demonstrate the emergence of the supersymmetric fully non-linear VA fermion action on the anti-D3-brane, and the absence of such a fermion action on the D3-brane under a certain choice of orientifolding. We start with the action (2.1) and impose the supersymmetric truncation constraint (1 + qΓ)θ = 0 , (3.2) together with F µν = 0 , Π I µ = ∂ µ φ I −θΓ I ∂ µ θ = 0 . (3.3) Our κ-symmetry matrix Γ in δ κ θ = (1 + qΓ)κ then simplifies significantly and becomesΓ defined as followsΓ ≡ Γ\| (1+qΓ)θ=Fµν =Π I µ =0 = σ 3 σ 1 1 4! ε µ 1 ...µ 4Γ µ 1 ...µ 4 = σ 3 σ 1 Γ D3 (0) , (3.4) with Γ D3 (0) = Γ 0123 . We have four 1-forms Π m = dX m +θ 1 Γ m dθ 1 +θ 2 Γ m dθ 2 , where m are the 4 world-volume directions and where spinors have been restricted by the condition (3.2). The restricted 1-forms are E m = dX m + 1 2θ Γ m (1 − qΓ)dθ = dX m +λΓ m dλ . (3.5) Here λ = √ 2 θ 1 is a 16-component spinor. When all constraints are taken into account we find that the DBI action takes the form S DBI \| (1+qΓ)θ=Fµν =Π I µ =0 = − d 4 σ − det G µν = − 1 4! E m 0 ∧ ... ∧ E m 3 ε m 0 ...m 3 = − det E . (3.6) The fact that the DBI action reduces in this limit to the VA action has been known for a long time [12] and recently confirmed in [15]. Now we will study the WZ term when the orientifold projection (3.2) together with (3.3) is taken into account. This illustrates the general argument in the Dp-brane case given in Appendix A.3. We start with qĨ 5 ≡ qdΩ 4 = dθqT 3 dθ withT 3 = σ 3 σ 1 (E m Γ m ) 3 3! , (3.7) so that we get qĨ 5 = −E m 1 ∧ E m 2 ∧ E m 3 ∧ dθ qσ 3 σ 1 1 3! Γ m 1 m 2 m 3 dθ . (3.8) Now we use the identity Γ m 1 m 2 m 3 = ε m 1 m 2 m 3 m 0 Γ m 0 Γ D3 (0) ,(3.9) and obtain qĨ 5 = − 1 3! ε m 1 m 2 m 3 m 0 E m 1 ∧ E m 2 ∧ E m 3 ∧ dθΓ m 0 qσ 3 σ 1 Γ D3 (0) dθ . (3.10) Using (3.4) we can rewrite this as follows qĨ 5 = − 1 3! ε m 1 m 2 m 3 m 0 E m 1 ∧ E m 2 ∧ E m 3 ∧ dθΓ m 0 qΓdθ . (3.11) Thus we get using (3.2) and (3.5) qĨ 5 ≡ qdΩ 4 = 4 4! ε m 1 m 2 m 3 m 0 E m 1 ∧ E m 2 ∧ E m 3 ∧ dE m 0 = − 1 4! ε m 1 m 2 m 3 m 0 d(E m 1 ∧ E m 2 ∧ E m 3 ∧ E m 0 ) . (3.12) This can be integrated to qΩ 4 \| (1+qΓ)θ=Fµν =Π I µ =0 = − 1 4! ε m 0 m 1 m 2 m 3 E m 0 ∧ E m 1 ∧ E m 2 ∧ E m 3 = − det E , (3.13) and we learn that our WZ term of the D3-brane or anti-D3-brane under the restrictions (3.2) and (3.3) becomes the VA action. It adds to the DBI term. If we would use, instead, the constraint (1 − qΓ)θ = 0, it would lead to a cancellation between the DBI and the WZ terms since qΩ 4 \| (1−qΓ)θ=Fµν =Π I µ =0 = 1 4! ε m 0 m 1 m 2 m 3 E m 0 ∧ E m 1 ∧ E m 2 ∧ E m 3 = det E . (3.14) In particular, for an anti-D3-brane with q = −1 the constraint which doubles the action is the usual O3 − -plane projection condition (1 + qΓ)θ = (1 −Γ)θ = 0 ⇔ θ 1 = Γ 0123 θ 2 . (3.15) The conditions (3.3) arise, if we place the anti-D3-brane at a fix point locus of the orientifold projection. In this case the world volume vector field A µ and the scalars φ I are projected out, which leads to (3.3) (see appendix A.4 for the vanishing of the fermionic terms). Note, that the fermions on an anti-D3-brane on top of an O3-plane are not projected out, see for example [19], [20]. Since the analysis in [20] was made in the linear approximation, the presence of fermions in absence of bosons was qualified as breaking of all supersymmetries. However, it was stressed in for example the abstract and introduction of [20] that this system is free of tachyons. Meanwhile, as our non-linear analysis shows, we agree on absence of vector and scalars on a single brane, however, we find that the fermions form a goldstino multiplet with spontaneously broken supersymmetry. This fact that D-branes break supersymmetry spontaneously is often overlooked in the string theory literature although it is clearly stated for example on page 140 of [21], where it is also mentioned that the fermions on the brane are the goldstinos. Thus our action of the anti-D3 brane upon orientifolding is (S DBI + S (−1) WZ )\| anti−D3 (1−Γ)θ=F =Π I =0 = −2 d 4 σ − det G µν = −2 E 0 ∧ ... ∧ E 3 = −2 det E , (3.16) where E m = dX m +λΓ m dλ , m = 0, 1, 2, 3 . (3.17) Here λ is a 16-component Majorana-Weyl spinor (related to θ 1 ). This same constraint for a D3-brane at a fixed point locus of the orientifold involution leads to a cancellation of the WZ term and the DBI term. This cancellation is a manifestation of the fact that not only the scalars and the vector on the world volume of the D3-brane are projected out but also the fermions: (S DBI + S (+1) WZ )\| D3 (1−Γ)θ=F =Π I =0 = 0 . (3.18) Compactification A detailed study of the KKLT string theory model, including D-branes in a curved background with ISD fluxes, compactified on a CY 3 manifold will require an additional investigation. Here we will just make some plausible comments on the situation which might be expected on the basis of the results established in this paper. We also like to mention here the relevant earlier work [22]. In section 5 of this paper the authors investigate the possibility that the anti-D3-brane in a KKLT setup breaks supersymmetry spontaneously. They furthermore conjecture that the gaugino is the goldstino. However, since the authors work in the gauge with the vanishing WZ term, they cannot distinguish between the fermionic action of an anti-D3-brane and a D3-brane and the complicated background prevents them from obtaining conclusive results. The comments below go beyond the scope of this work, since we studied explicitly only the case of a single anti-D3-brane on top of an O3-plane in a flat supergravity background. Our expression for the D3-and anti-D3-brane classical action in Sec. 3 corresponds to a dimensional reduction of the D9-and anti-D9-brane classical action on a T 6 when all fields are assumed to be independent of the world-volume coordinates σ 4 , ..., σ 9 and after T-dualizing on all direction of the T 6 , see for example eq. (98) in [10] where the DBI term is given. This means that the spinors on the branes remain 32-component ones in the classical actions and have 16 component upon gauge fixing κsymmetry or upon making a supersymmetric truncation, i.e. imposing an orientifold projection. Before discussing the compactification on a CY 3 manifold we would like to explain here the main feature of the Volkov-Akulov theory. The action in (3.16) can be shown to have a non-linear supersymmetry in a gauge where X m = δ m µ σ µ , see for example Appendix A in [15]. In this gauge E m \| X m =δ m µ σ µ = dσ µ δ m µ +λΓ m dλ , m = 0, 1, 2, 3 . The corresponding non-linear supersymmetry of the action acting on the fermion field λ(σ) is given by the global parameter ζ δ ζ λ(σ) = ζ +λ(σ)Γ µ ζ ∂ µ λ(σ) . (3.20) The first constant term shows that the supersymmetry is spontaneously broken, the second term is quadratic in fermions living on the brane. A beautiful feature of the VA action is that one can present the symmetries of the theory in a much nicer way before gauge-fixing X m = δ m µ σ µ . The manifest supersymmetry of the action (3.16) in a form with E m = dX m +λΓ m dλ is a superspace type transformation in which the fermionic coordinate λ(σ) is shifted by a global spinor ζ and the bosonic coordinates X m (σ) transform to compensate this shift δλ(σ) = ζ , δX m (σ) =ζΓ m λ(σ) . Note that this superspace-type symmetry (3.21) explains that the second term in (3.20) is just a compensating, field dependent, general coordinate transformation with a parameter ξ µ ζ =λ(σ)Γ µ ζ. Note that so far we have a 16-component spinor λ(σ) as well as a 16-component global supersymmetry parameter ζ. This form of the VA action and its symmetries, before we gauge fix X m , are most suitable for the discussion of the compactification on a CY 3 manifold. Let us now present the sixteen component spinor λ(σ) as three four dimensional spinors λ i (σ), i = 1, 2, 3 that transform as 3 under the SU (3) holonomy (similarly to the complex scalars ϕ i (σ) = φ 2i−1 + iφ 2i ) and one spinor λ 0 (σ) that is an SU (3) singlet. The global 16-component supersymmetry parameter ζ is also split into a singlet ζ 0 and a triplet ζ i under SU (3). For a CY 3 manifold the concept of a global spinor has to be replaced by a covariantly constant spinor. Only the singlet ζ 0 is covariantly constant whereas the triplets are not, see for example [23]. Then the above transformations (3.21), with only the four component covariantly constant spinor ζ 0 allowed, become δλ i (σ) = 0 , δλ 0 (σ) = ζ 0 , δX m (σ) =ζ 0 Γ m λ 0 (σ) . (3.22) We now observe that if we do not truncate the triplet spinors on the brane λ i (σ), then the N = 1 VA supersymmetry on the brane is explicitly broken. However, if the compactification on the CY 3 manifold together with the orientifold projection removes the λ i (σ), then we end up with a model with N = 1 VA supersymmetry where the action of the anti-D3-brane is (S DBI + S (−1) WZ )\| anti−D3 ,CY 3 (1−Γ)θ=F =Π I =0 = −2 E 0 ∧ ... ∧ E 3 = −2 det E . (3.23) where E m = dX m +λ 0 Γ m dλ 0 , m = 0, 1, 2, 3 . (3.24) This is the VA action in d=4 corresponding to spontaneously broken N = 1 supersymmetry δλ 0 (σ) = ζ 0 , δX m (σ) =ζ 0 Γ m λ 0 (σ) , (3.25) which is equivalent to a chiral nilpotent superfield. If we would take another step and assume a finite volume for the CY 3 , we would get an action for the anti-D3-brane in Einstein frame which takes into account the volume of the extra dimensions: S anti-D3 = −2 d 4 σe K(ρ,ρ) det E , (3.26) whereas under the same conditions we find S D3 = 0 . (3.27) Discussion In this note we have clarified the relation between the emergence of the nilpotent supermultiplet in d = 4 supergravity and an anti-D3-brane on top of an O3 orientifold plane. The anti-D3-brane has a Volkov-Akulov goldstino multiplet [4] on its word-volume. This construction, developing the one proposed in [6], explains how the manifestly supersymmetric effective action based on the Kähler and superpotential in (1.3) provides the supersymmetric version of the KKLT construction. The de Sitter vacua have a spontaneously broken VA supersymmetry, which in effective supergravity can be described by a chiral nilpotent multiplet [7] corresponding to the emergence of the VA goldstino on the world-volume of the anti-D3 brane. In application to the KKLT model our investigation was performed so far in the simplified model of a single D3-and anti-D3-brane on top of an O3-plane in the flat supergravity background. In such a case it was possible to establish a simple connection to a supergravity effective KKLT model with an additional single nilpotent chiral multiplet corresponding to the Volkov-Akulov goldstino as given in (1.3). However, in a more realistic case of a full string theory one should study models with many coincident branes in a curved supergravity background, including ISD fluxes and further moduli fields like the axio-dilaton and the complex structure moduli. This we postpone to future studies. In cosmological applications the role of the nilpotent multiplet, which has only a fermion and does not have a fundamental scalar, was shown to have various advantages over the better known supergravity models with standard chiral multiplets. In the new models there is no need to stabilize the scalar of the nilpotent multiplet since it is proportional to a bilinear of the fermions and therefore does not affect the cosmological evolution. Another advantage in using the nilpotent multiplet is that it is possible to build simple supergravity models of inflation which have an exit into de Sitter vacua [24]. The issues of cosmology raised our interest to the formal aspects of the D-brane physics and we were able to derive analytically a new result here: the Wess Zumino part of the Dp-brane action with orientifold truncation acquires the form of the Volkov-Akulov action. This includes in particular the D3-brane case. Our derivation of this general result also explains the reason why for a D9-brane it was established computationally in [16,17] that the WZ term becomes the VA action when a consistent supersymmetric orientifolding is applied. It is instructive also to mention here again the recent progress in constructing dS vacua in [5]. In these models the effective supergravity action is manifestly supersymmetric, whereas dS vacua break supersymmetry spontaneously, on solutions, as in early dS models of this type in [25]. In new models in [5] it was possible to achieve the absence of tachyons and local stability of generic dS vacua. The current universe acceleration appears to be well described by a cosmological constant. It is therefore gratifying to find various new parts of the string theory landscape with spontaneously broken supersymmetry and an abundance of dS vacua, such as the ones in [5], in [6,24], and in the advanced version of the KKLT construction presented in this paper. It would be interesting to continue exploring these kind of 'supersymmetric pillars' providing uplifting and local stability of dS vacua in the landscape. 166/1-1) of the German Research Foundation (DFG). A Appendix: Dp-branes and anti-Dp-branes with orientifolding Here we extend the analysis of orientifolding on Dp-superbranes in IIB string theory, which was performed for the D3-brane case in the main part of the paper. The basis for this analysis is Appendix A in [15]. We start with the classical action for a Dp-brane with q = 1 and an anti-Dp-brane with q = −1: S DBI + S (q) WZ = − d p+1 σ − det(G µν + α F µν ) + q Ω p+1 . (A.1) Here G µν is the manifestly supersymmetric induced world-volume metric 4 G µν = η mn Π m µ Π n ν , Π m µ = ∂ µ X m −θΓ m ∂ µ θ , (A.2) and the Born-Infeld field strength F µν is given by F µν ≡ F µν − b µν , b µν = α −1θ σ 3 Γ m ∂ µ θ ∂ ν X m − 1 2θ Γ m ∂ ν θ − (µ ↔ ν) , (A.3) where Ω p+1 is a p + 1-form [9][10][11]. Here we will describe it using the formalism in the flat supergravity background in [10,13,14]. Namely, we define a closed p + 2 form in IIB theory I p+2 ≡ d Ω p+1 = dθT p dθ , (A.4) where in IIB models with odd p the p-form T p is T p = e F l=0 (σ 3 ) l σ 1Γ 2l+1 (2l + 1)! p−form . (A.5) The meaning of this expression is that e F is expanded in powers of the 2-form F and combined with powers of the 1-formΓ. T p is then picking out the p-forms. HereΓ is the following matrix-valued 1-form Γ = Γ m Π m = Γ m (dX m +θΓ m dθ) , (A.6) and the pull-backs of the flat matrices Γ m to the world-volume are: Γ µ ≡ Π µ m Γ m ,Γ µ ≡ G µνΓ ν = Π µ m Γ m , Π µ m = G µν Π n ν η mn . (A.7) The action (A.1) has a global supersymmetry, local κ-symmetry, general coordinate symmetry and a U (1) gauge symmetry: δθ = + (1 + qΓ)κ + ξ µ ∂ µ θ , δX M = −θΓ M +θΓ M (1 + qΓ)κ + ξ µ ∂ µ X M , α δA µ = −θΓ M σ 3 ∂ µ X M + 1 6θ σ 3 Γ M θ Γ M ∂ µ θ + 1 6θ Γ M θ σ 3 Γ M ∂ µ θ +θσ 3 Γ M (1 + qΓ)κ ∂ µ X M − 1 2θ σ 3 Γ M (1 + qΓ)κθΓ M ∂ µ θ − 1 2θ Γ M (1 + qΓ)κθσ 3 Γ M ∂ µ θ + ∂ µ Λ + ξ ν F νµ . (A.8) Note that this implies that δ F = 0 , δ Π m = 0 . (A.9) The local κ-symmetry on fermions is given by δ κ θ = (1 + qΓ)κ , (A.10) where κ 1,2 (σ), is an arbitrary doublet of Majorana-Weyl spinors of the same chirality. Γ satisfies Tr Γ = 0 , Γ 2 = 1 . In the Pauli matrices basis, and acting on positive-chirality spinors θ 1,2 , Γ is given by Γ = 0 β − (−1) n β + 0 , (A.11) where β + and β − are matrices that satisfy β − β + = β + β − = (−1) n , with n = (p − 1)/2. In terms of the pull-backs, the matrices β + and β − are given by β ± ≡ G se ± α 2 FµνΓ µν Γ Dp (0) ≡ G n+1 k=0 (±α ) k 2 k k!Γ µ 1 ν 1 ···µ k ν k F µ 1 ν 1 · · · F µ k ν k Γ Dp (0) , (A.12) and G = \|G\| \|G + α F\| = det δ µ ν + α F µρ G ρν −1/2 . (A.13) Here se is the skew-exponential function, so the expansion has effectively only a finite number of terms. The matrix Γ Dp (0) is defined by Γ Dp (0) = 1 (p + 1)! \|G\| ε µ 1 ...µ p+1Γ µ 1 ...µ p+1 , (Γ Dp (0) ) 2 = (−1) n . (A.14) For p < 9 in expressions above we split the coordinates as follows where m refers to the p + 1 worldvolume directions and I refers to the 9 − p transverse directions and G µν = η m n Π m µ Π n ν + δ IJ Π I µ Π J ν , Π m µ = ∂ µ X m −θΓ m ∂ µ θ , Π I µ = ∂ µ φ I −θΓ I ∂ µ θ . (A.16) Thus, φ I are the scalars on the p < 9 branes. When a consistent dimensional reduction of the D9-brane is performed, the 9 − p scalars are related to 9 − p components of the d=10 vector, namely to A I . A.1 θ 1 = 0, θ 2 = λ, X m = δ m µ σ µ gauge There are 32 global supersymmetries with the parameters 1 , 2 . In the gauge θ 1 = 0, X m = δ m µ σ µ , described in detail in [15] the κ parameters and general coordinate transformation parameters ξ µ (σ) become functions of and fields of the theory, so that this gauge is preserved, namely δθ 1 = 1 + κ 1 + β − κ 2 = 0 (A.17) and δX m = −λΓ m 2 + (−1) nλ Γ m β + 1 + ξ m = 0 . (A.18) The gauge-fixed action has 32 global supersymmetries, 16 can be identified with deformed standard linear transformations of the vector multiplet, see eq. (A.29) in [15], whereas the other 16 when acting on fermions have the form of the Volkov-Akulov non-linear transformations, see eq. (A.30) in [15]. A.2 Supersymmetric truncation F = 0, Π I = 0 , (1 ± qΓ)θ = 0 We define a supersymmetric truncation on the Dp-brane and anti-Dp-brane as follows. First we definẽ Γ ≡ Γ\| F =0, Π I =0 = (σ 3 ) n σ 1ΓDp (0) , (A.19) whereΓ Dp (0) = 1 (p + 1)! ε µ 1 ...µ p+1 Γ µ 1 ...µ p+1 , (Γ Dp (0) ) 2 = (−1) n . (A.20) There are two choices for the constraint on the spinor for actions with κ-symmetry δ κ θ = (1 + qΓ)κ which we consider. In the first case (1 + qΓ)θ = 0 ⇒ S = S DBI + S (q) W Z = 2 S V A , (A.21) we will find that the DBI and WZ term are equal to each other and to the VA action. They therefore add up to produce the VA action. In the second case (1 − qΓ)θ = 0 ⇒ S = S DBI + S (q) W Z = 0 , (A.22) we will find that the action is the difference between the DBI and WZ term, which each are equal to the VA action. Thus they cancel and the action vanishes. These two projections correspond to Op orientifold projection and anti-Op orientifold projections. The usual Op orientifold projection leads to a vanishing action for the Dp-brane and the VA action for the anti-Dp-brane when the brane/antibrane are located at an orientifold fixed point. The reason that the Dp-brane has a vanishing action in this case is that all its degrees of freedom are projected by the orientifold projection. For the anti-Dp-brane the fermionic degrees of freedom survive [19]. Let us consider the first case in detail. We require that the same truncation is valid for the global supersymmetry parameter as is expected for an orientifold involution (1 + qΓ) = 0 , (A.23) and that (1 + qΓ) δ κ θ = 0 ⇒ (1 + qΓ)(1 + qΓ)κ = 2(1 + qΓ)κ = 0 , (A. 24) and therefore (1 + qΓ)κ = 0 . (A. 25) Thus, θ, and κ satisfy the same constraint. These conditions also serve as a gauge-fixing of the κ-symmetry. In the second case we require that (1 − qΓ) = 0 (A.26) and that (1 − qΓ) δ κ θ = 0 ⇒ (1 − qΓ)(1 + qΓ)κ = (1 −Γ 2 )κ = 0 . (A.27) This condition is satisfied without constraining κ, which means that the complete κ-symmetry gaugefixing is not achieved. However in this case the brane action vanishes. A.3 Evaluation of the action with F = 0, Π I = 0 , (1 + qΓ)θ = 0 constraint Imposing the constraint (A.21), we have p + 1 1-forms E m = dX m +θΓ m 1 2 (1 − qΓ)dθ = dX m +λΓ m dλ , (A.28) where λ = √ 2 θ 1 . The DBI action at F = 0, Π I = 0 and (1 + qΓ)θ = 0 becomes S DBI = − d 10 σ − det G µν = − 1 (p + 1)! E m 0 ∧ ... ∧ E m p ε m 0 ...m p . (A.29) Now we look at the WZ action qĨ p+2 ≡ qdΩ p+1 = qdθT p dθ , (A.30) where for p = 2n + 1T p = (σ 3 ) n σ 1 (E m Γ m ) p p! , (A.31) so that we get for our odd p I p+2 ≡ dΩ p+1 = −E m 1 ∧ ... ∧ E m p dθq(σ 3 ) n σ 1 1 p! Γ m 1 ...m p dθ . (A.32) We now use the following identity for odd p A.4 Consistency of the supersymmetric truncation The action of the Dp-brane with p < 9 depends on scalars and vectors via the manifestly supersymmetric combinations F µν and Π I . Here we would like to show that the truncation of the scalars and the vector has to be realized via their supersymmetric combinations, as suggested in eqs. (3.3) and (A.19). We start with the combination Π I µ = ∂ µ φ I −θΓ I ∂ µ θ . The charge conjugation matrix C has the useful property (Γ M ) T C = −CΓ M . Now we use this and find for odd p in our type IIB models (taking into account that q 2 = 1), that θΓ I dθ =θ 1 Γ I dθ 1 +θ 2 Γ I dθ 2 = (θ 2 ) T Γ T p Γ T p−1 . . . Γ T 0 CΓ I Γ 01...p dθ 2 +θ 2 Γ I dθ 2 = (θ 2 ) T CΓ p p−1...10 Γ I Γ 01...p dθ 2 +θ 2 Γ I dθ 2 =θ 2 Γ I Γ p p−1...10 Γ 01...p dθ 2 +θ 2 Γ I dθ 2 = −θ 2 Γ I dθ 2 +θ 2 Γ I dθ 2 = 0 . (A.42) Note, that the argument works also if there is an additional minus sign in the relation (A.41). It is instructive to explain here whyθΓ m dθ, where m = 0, ..., p does not vanish when the same constraint on spinors is applied. The difference lies in the fact that Γ 01...p commutes with Γ I and anti-commutes with Γ m . This removes the minus in the last line of eq. (A.42) so that the contributions from θ 1 and θ 2 instead of canceling as in the Γ I case, actually double. Likewise, we find thatθσ 3 Γ m dθ =θ 1 Γ m dθ 1 − θ 2 Γ m dθ 2 = 0. Together with (A.42) this then implies that β µν = 0 (cf. (A.3)). Our orientifold projection that removes the vector fields therefore leads to a vanishing of the supersymmetric version of the vector field strength F µν = 0. X m = m{X m , φ I } , m = 0, 1, . . . , p , I = 1, . . . , 9 − p , (A.15) Γ m 1 1...m p = ε m 1 ...m p m 0 Γ m 0Γ Dp (0) , (A.33) to replace the p Γ-matrices in eq. (A.32) by their expression in (A.33) and obtainqĨ p+2 ≡ qdΩ p+1 = − (p + 1) (p + 1)! ε m 1 ...m p m 0 E m 1 ∧ ... ∧ E m p dθΓ m 0 q(σ 3 ) n σ 1ΓDp (0) dθ . (A.34)Using (A.19) and (A.28) we can rewrite this as followsqĨ p+2 ≡ qdΩ p+1 = − 1 ...m p m 0 E m 1 ∧ ... ∧ E m p dθΓ m 1 ...m p m 0 E m 1 ∧ ... ∧ E m p ∧ dE m 0 . (A.35)This can be integrated toqΩ p+1 \| (1+qΓ) θ=0 = − 1 (p + 1)! ε m 0 m 1 ...m p E m 0 ∧ E m 1 ∧ ... ∧ E m p = − det E , (A.36)and we learn that our WZ term under restrictions imposed above is proportional to the VA action! Now we apply our findings to the Dp-/anti-Dp-brane action in (A.1). With the supersymmetric truncation/orientifoldingS DBI +S (q) WZ = − d p+1 σ − det(G µν + α F µν ) + q Ω p+1 F =0,Π I =0,(1+qΓ) θ=0(A.37) we find that the action doubles for the choice of truncation in (A.21) (S DBI +S (q) WZ )\| F =0,Π I =0,(1+qΓ) θ=0 = −2 det E . (A.38) Similarly, it vanishes for the opposite choice of truncation: (S DBI +S (q) WZ )\| F =0,Π I =0,(1−qΓ) θ=0 = 0 . (A.39) projection removes the scalars and we show that the fermionic term vanishes as well. We consider an (anti)-Dp-brane extended along 01 . . . p. The orientifolding condition given in (A.19) and (A.21) can be written as (1 + qΓ)θ = 0 θ 1 = −qΓ Dp (0) θ 2 = qΓ 01...p θ 2 . 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[ "New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile", "New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile" ]	[ "Kenneth Edwards [email protected] \nPhysics Department Faculty of Science\nMahidol University\nRama 6 Road10400BangkokThailand\n", "Michael A Allen \nPhysics Department Faculty of Science\nMahidol University\nRama 6 Road10400BangkokThailand\n" ]	[ "Physics Department Faculty of Science\nMahidol University\nRama 6 Road10400BangkokThailand", "Physics Department Faculty of Science\nMahidol University\nRama 6 Road10400BangkokThailand" ]	[]	We consider the tiling of an n-board (a board of size n × 1) with squares of unit width and (1, 1)-fence tiles. A (1, 1)-fence tile is composed of two unit-width square subtiles separated by a gap of unit width. We show that the number of ways to tile an n-board using unit-width squares and (1, 1)-fence tiles is equal to a Fibonacci number squared when n is even and a golden rectangle number (the product of two consecutive Fibonacci numbers) when n is odd. We also show that the number of tilings of boards using n such square and fence tiles is a Jacobsthal number. Using combinatorial techniques we prove identities involving sums of Fibonacci and Jacobsthal numbers in a straightforward way. Some of these identities appear to be new. We also construct and obtain identities for a known Pascal-like triangle (which has alternating ones and zeros along one side) whose (n, k)th entry is the number of tilings using n tiles of which k are fence tiles. There is a simple relation between this triangle and the analogous one for tilings of an n-board. Connections between the triangles and Riordan arrays are also demonstrated. With the help of the triangles, we express the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers as double sums of products of two binomial coefficients.	null	[ "https://arxiv.org/pdf/2009.04649v3.pdf" ]	221,586,083	2009.04649	19cf7110a9d62ba6e440544654c993f45a966dca	New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile 4 Oct 2020 Kenneth Edwards [email protected] Physics Department Faculty of Science Mahidol University Rama 6 Road10400BangkokThailand Michael A Allen Physics Department Faculty of Science Mahidol University Rama 6 Road10400BangkokThailand New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile 4 Oct 20201 Corresponding author. We consider the tiling of an n-board (a board of size n × 1) with squares of unit width and (1, 1)-fence tiles. A (1, 1)-fence tile is composed of two unit-width square subtiles separated by a gap of unit width. We show that the number of ways to tile an n-board using unit-width squares and (1, 1)-fence tiles is equal to a Fibonacci number squared when n is even and a golden rectangle number (the product of two consecutive Fibonacci numbers) when n is odd. We also show that the number of tilings of boards using n such square and fence tiles is a Jacobsthal number. Using combinatorial techniques we prove identities involving sums of Fibonacci and Jacobsthal numbers in a straightforward way. Some of these identities appear to be new. We also construct and obtain identities for a known Pascal-like triangle (which has alternating ones and zeros along one side) whose (n, k)th entry is the number of tilings using n tiles of which k are fence tiles. There is a simple relation between this triangle and the analogous one for tilings of an n-board. Connections between the triangles and Riordan arrays are also demonstrated. With the help of the triangles, we express the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers as double sums of products of two binomial coefficients. Introduction The (n + 1)th Fibonacci number (A000045), defined by F n+1 = δ n,1 + F n + F n−1 , F n<1 = 0, where δ i,j is 1 if i = j and zero otherwise, can be interpreted as the number of ways to tile an n-board (a board of size n × 1 composed of 1 × 1 cells) with 1 × 1 squares (henceforth referred to simply as squares) and 2 × 1 dominoes [4,3]. More generally, the number of ways to tile an n-board with all the r × 1 r-ominoes from r = 1 up to r = k is the k-step (or k-generalized) Fibonacci number F (k) [3]. Edwards [5] showed that it is possible to obtain a combinatorial interpretation of the tribonacci numbers (the 3-step Fibonacci numbers, A000073) as the number of tilings of an n-board using just two types of tiles, namely, squares and ( 1 2 , 1)-fence tiles. A (w, g)-fence tile is composed of two subtiles (called posts) of size w × 1 separated by a gap of size g × 1. n+1 = δ n,1 + F (k) n + F (k) n−1 + · · · + F (k) n−k+1 , with F (k) n<1 = 0 We presented a bijection between the Fibonacci numbers squared (A007598) and the tilings of an n-board with half-squares (i.e., 1 2 × 1 tiles always oriented so that the shorter side is horizontal) and ( 1 2 , 1 2 )-fence tiles [7] and this was used to formulate combinatorial proofs of various identities [7,8]. We also identified a bijection between tiling an n-board with ( 1 2 , g)-fence tiles where g ∈ {0, 1, 2, . . .} and strongly restricted permutations and then used it to obtain results concerning the permutations [6]. Here we show that the number of ways to tile an n-board using square and (1, 1)-fence tiles is a Fibonacci number squared if n is even and a golden rectangle number (the product of two successive Fibonacci numbers, A001654) if n is odd. We also consider the number of ways to tile boards using a total of n of these tiles and refer to this as an n-tiling. We show that enumerating n-tilings yields the Jacobsthal numbers J n≥0 = 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, . . . (A001045) where the nth Jacobsthal number is defined via J n = δ n,1 + J n−1 + 2J n−2 , J n<1 = 0. (1) We use both types of tiling to formulate straightforward combinatorial proofs of identities involving the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers. We also obtain two Pascal-like triangles (one for n-tilings, the other for tilings of an n-board) whose entries are the number of tilings with squares and (1, 1)-fences which use a given number of fences. A number of properties of the triangles are derived including their relation to Riordan arrays. Finally, the triangles are used to express the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers as double sums of a product of two binomial coefficients. Tiling boards with squares and fences When tiling a board with fences it is helpful to first determine the types of metatile since any tiling of the board can be expressed as a tiling using metatiles [5]. A metatile is an arrangement of tiles that exactly covers an integral number of adjacent cells and cannot be split into smaller metatiles [5,6]. When tiling with squares (S) and (1, 1)-fence tiles 1 2 3 4 5 6 7 8 Figure 1: An 8-board tiled with the three possible metatiles: a free square (cell 1), a filled fence (cells 2-4), and a bifence (cells [5][6][7][8]. The symbolic representation of this tiling is SF SF F . (henceforth referred to simply as fences or F ), the simplest metatile is the square. To tile adjacent cells by starting with a single fence we must fill the gap with either a square or the post of another fence. These generate what we will refer to as the filled fence (F S) and bifence (F F ) metatiles, respectively (Fig. 1). The bifence clearly has a length of 4. The filled fence has length 3 and the square inside will be called a captured square. A square which is not captured (and is therefore a metatile) is called a free square. Theorem 1. Let A n be the number of ways to tile an n-board using squares and fences. Then A n = δ n,0 + A n−1 + A n−3 + A n−4 , A n<0 = 0.(2) Proof. We condition on the last metatile [2,6]. If the last metatile is of length l there will be A n−l ways to tile the remaining n − l cells. The result (2) follows from the fact that there are three possible metatiles and these have lengths of 1, 3, and 4. If n = l there is exactly one tiling (which corresponds to that metatile filling the entire board) so we make A 0 = 1. There is no way to tile an n-board if n < l and so A n<0 = 0. A n≥0 = 1, 1, 1, 2, 4, 6,9,15,25,40,64,104,169,273,441,714,1156, . . . is A006498. As we will shortly prove combinatorially, the even (odd) terms of this sequence are the Fibonacci numbers squared A007598 (golden rectangle numbers A001654). Lemma 1. There is a bijection between the fence-square tilings of a 2n-board (a (2n + 1)board) and the square-domino tilings of an ordered pair of n-boards (an (n + 1)-board and an n-board). Proof. Tile an n-board (an (n + 1)-board) with the contents of the odd-numbered cells of the given 2n-board ((2n + 1)-board) fence-square tiling and tile a second n-board (an n-board) with the contents of the even-numbered cells. The posts of any fence (which always lie on two consecutive odd or even cells) get mapped to a domino. The procedure is reversed by splicing the two square-domino tilings. Theorem 2. For n ≥ 0, A 2n = f 2 n , (3a) A 2n+1 = f n f n+1 ,(3b) where f n = F n+1 . Proof. There are f n ways to tile an n-board using squares and dominoes [3]. From Lemma 1, A 2n is the same as the number of ways to tile an ordered pair of n-boards using squares and dominoes which is f 2 n , and A 2n+1 is the same as the number of ways to tile an n-board and (n + 1)-board using squares and dominoes which is f n f n+1 . As it is easily done and the result will be used in future work, in the following theorem we generalize Theorem 2 to the case of tiling an n-board with squares and (1, m − 1)-fences for some fixed m ∈ {2, 3, . . .}. B n = J n+1 .(4) Proof. As in the proof of Theorem 1, we condition on the last metatile. If the last metatile contains m tiles, there are B n−m possible (n − m)-tilings. As the three possible metatiles contain 1, 2, and 2 tiles we have B n = δ n,0 + B n−1 + 2B n−2 , B n<0 = 0.(5) where the δ n,0 is to ensure that B 0 = 1 so that when an n-tiling is just one metatile we count precisely one tiling. Comparing (5) with (1) gives the result. Combinatorial proofs of identities involving the Fibonacci squares and golden rectangle numbers The proofs of Identities 1, 2, and 3 (and of Identities 6, 7, 8, and 9 in the next section) follow the techniques of Benjamin and Quinn [3]. Identity 1. For n ≥ 0, n j=0 f 2 j = f n f n+1 . Proof. How many tilings of a (2n+1)-board are there? Answer 1: A 2n+1 = f n f n+1 . Answer 2: condition on the position of the last metatile which is not a bifence. The tiles after this metatile must be bifences and thus occupy 4m cells (0 ≤ m ≤ ⌊n/2⌋). If the last non-bifence metatile is a square there are A 2n+1−4m−1 = f 2 n−2m tilings. If it is a filled fence there are A 2n+1−4m−3 = f 2 n−1−2m tilings. Summing over all possible values of m for both cases gives the result. Identity 2. For n ≥ 0, 2n−1 j=0 f j f j+1 = f 2 2n − 1. Proof. How many tilings of a 4n-board use at least 1 square? Answer 1: A 4n − 1 = f 2 2n − 1 since the only way to tile a 4n-board without using squares is to use only bifences. Answer 2: condition on the position of the last metatile which is not a bifence. This gives A 4n−4m−1 = f 2n−2m f 2n−2m−1 tilings if the last non-bifence metatile is a square and A 4n−4m−3 = f 2n−2m−1 f 2n−2m−2 if it is a filled fence for 0 ≤ m < n. Summing over m for both cases gives the result. Identity 3. 2 n−2 j=0 f j f j+2 − f n−2 f n−1 = f 2 n − 1, n > 1, (6a) 2 n−2 j=0 f j f j+2 + f 2 n−1 = f n f n+1 − 1, n > 0. (6b) Proof. How many tilings of an m-board use at least 1 fence? Answer 1: A m − 1 since only the tiling using just squares does not use any fences. Answer 2: condition on the location of the last fence. If the last fence is in a metatile of length l starting at cell k + 1 (where 0 ≤ k ≤ m − l) there will be A k possible tilings since the cells after the metatile must all be occupied by squares. The two metatiles containing fences are of length 3 and 4. Summing numbers of tilings over all positions and types of the last fence-containing metatile and equating the two answers gives 2 m−4 k=0 A k + A m−3 = A m − 1. Replacing m in this by 2n and 2n + 1 leads to (6a) and (6b), respectively, after using (3) and f 2 j + f j f j+1 = f j f j+2 . The proofs of Identities 4 and 5 (and of Identities 10, 11, and 12 in the next section) follow the techniques used in [1,8]. As far as we know, all these identities are new. Identity 4. For n ≥ 0, f n f n+1 = 1 + ⌊n/2⌋ + n j=1 jf n−j f n−j+1 . Proof. How many tilings of a (2n+1)-board contain at least two squares? Answer 1: A 2n+1 − 1 2 n − 1 (A 2n+1 − 1 2 (n + 1)) if n is even (odd) since the only possible tilings with less than 2 squares when n is even (odd) is one free square (filled fence) among n/2 ( 1 2 (n − 1)) bifences and there are 1 2 n − 1 ( 1 2 (n + 1)) such tilings. Answer 2: condition on the location of the second square. The metatile containing this must end on an even cell, 2j. Written in terms of symbols (see the caption to Fig. 1), the tiling of the first 2j cells must end in S. This leaves one S that may be placed anywhere among the F symbols which number j − 1. The number of ways to tile the cells to the right of the 2jth cell is A 2n+1−2j . Summing over all possible j gives n j=1 jA 2(n−j)+1 . Equating this to Answer 1 and simplifying gives A 2n+1 − ⌊n/2⌋ − 1 = n j=1 jA 2(n−j)+1 . The identity follows from (3b). Note that if we consider the tilings of a 2n-board that contain at least two squares we obtain Identity 2.1 of [8]. To generalize Identity 4 we first define C (r) n as the number of ways to tile a (2n + 1)-board using 2r + 1 squares (and n − r fences). Lemma 2. For n ≥ r ≥ 0, C (r) n = C (r) n−2 + n + r 2r , C (r) n<0 = 0.(7) Proof. In symbolic form, a tiling can end in either S or F F . If S, the number of ways to place the remaining 2r squares and n − r fences is n+r 2r . If F F , there are C (r) n−2 ways to place the remaining tiles. There are no tilings if n < 0. As will be shown in Theorem 6, C (r) n is the (n, r)th element of the (1/[(1 − x)(1 − x 2 )], x/(1 − x) 2 ) Riordan array (A158909). Identity 5. For p > 0, f n f n+1 = p−1 r=0 C (r) n + n j=p j + p − 1 2p − 1 f n−j f n+1−j . Proof. How many tilings of a (2n + 1)-board have at least 2p squares? Answer 1: the total number of tilings minus the tilings that contain less than 2p squares, i.e., A 2n+1 − p−1 r=0 C (r) n . Answer 2: we condition on the location of the 2pth square. If the metatile containing this lies on the 2jth cell, in the symbolic representation, there are 2p − 1 S and j − p F that precede the 2pth S and hence j+p−1 2p−1 ways to arrange them. There are A 2n+1−2j ways to place the remaining tiles after the 2jth cell. Summing over all possible j and equating the result to Answer 1 gives A 2n+1 − p−1 r=0 C (r) n = n j=p j + p − 1 2p − 1 A 2(n−j)+1 , and the identity follows from (3b). 4 Combinatorial proofs of identities involving the Jacobsthal numbers Identity 6. For n ≥ 0, 2 n r=1 J r = J n+2 − 1. Proof. How many (n + 1)-tilings use at least 1 fence? Answer 1: B n+1 − 1. Answer 2: condition on the last metatile containing a fence. If this last metatile contains the (j + 1)th and (j + 2)th tiles (0 ≤ j ≤ n − 1) then there remain j unspecified tiles. As there are two types of metatile containing a fence there are a total of 2B j tilings for each j. Summing over j we have n−1 j=0 2B j = B n+1 − 1, and the identity follows from Theorem 4. Identity 7. For n ≥ 0, 2n r=1 J r = J 2n+1 − 1. Proof. How many 2n-tilings use at least 1 square? Answer 1: B 2n − 1 since there is only one 2n-tiling without a square (the all-bifence tiling). Answer 2: condition on the last square which must be the 2(n − m)th tile (0 ≤ m ≤ n − 1) since any metatiles after the last square are bifences. If this last square is a free square there are 2n − 2m − 1 remaining unspecified tiles. There are 2n − 2m − 2 if it is inside a filled fence. Summing over all possible m and equating to Answer 1 gives B 2n − 1 = 2n−1 r=0 B r . The identity follows from (3b). Identity 8. For n ≥ 0, 2n−1 r=1 J r = J 2n . Proof. The proof follows that for Identity 7 but we count the number of (2n − 1)-tilings by conditioning on the last square. Identity 9. For m, n ≥ 0, J m+n+1 = J m+1 J n+1 + 2J m J n . Proof. The number of (m + n)-tilings is B m+n . Of these there are B m B n tilings where the mth tile is the last tile in a metatile. If the mth tile is the first tile in a metatile containing two tiles, there are m − 1 unspecified tiles before it and n − 1 unspecified tiles after the metatile. As there are two kinds of two-tile metatiles we have B m+n = B m B n + 2B m−1 B n−1 . The identity then follows from (3b). Identity 10. For n ≥ 0, J n+1 = ⌈ 1 2 (n + 1)⌉ + n−1 j=1 jJ n−j . Proof. How many n-tilings have at least two squares? Answer 1: B n − 1 2 (n + 1) (B n − 1 2 n − 1) when n is odd (even) since the possible tilings with one square when n is odd (even) are one filled fence (free square) placed among 1 2 (n − 1) ( 1 2 (n − 2)) bifences and there are 1 2 (n + 1) (n/2) such tilings, and the only possible tiling with no squares is the all-bifence tiling which only occurs when n is even. Answer 2: condition on the second metatile containing an S. The symbolic representation of the tiling up to and including this must end in an S. If this S is the jth tile, there are j − 1 ways to order the symbols preceding it and thus (j − 1)B n−j n-tilings. Summing over all possible j, equating to Answer 1, and simplifying gives B n − ⌈ 1 2 (n + 1)⌉ = n j=2 (j − 1)B n−j . The identity is obtained on replacing j by j + 1 and using Theorem 4. As before, we can generalize Identity 10. We need the following definition and lemma. Let D (r) n be the number of n-tilings that contain exactly r squares. As the only tilings with no squares are the all-bifence tilings, for n > 0, D (0) n is 1 (0) when n is even (odd). For convenience we make D (0) (0) = 1. Lemma 3. For n ≥ r > 0, D (r) n = D (r) n−2 + n − 1 r − 1 .(8) Proof. The symbolic representation of a tiling must end in either S or F F . If S, we are free to place the remaining n − 1 tiles (of which r − 1 are squares) in any order; this gives n−1 r−1 possibilities. If F F , there are D (r) n−2 ways to place the remaining tiles. As will be shown in Theorem 5, D Identity 11. For p > 0, J n+1 = p−1 r=0 D (r) n + n k=p k − 1 p − 1 J n+1−k . Proof. How many n-tilings have at least p squares? Answer 1: the total number of tilings minus the tilings that contain less than p squares, i.e., Identity 12. For n > 0, J n+1 = n + J n−1 + n k=3 (2k − 5)J n+1−k . Proof. For n > 0, how many n-tilings have at least two fences? Answer 1: B n −1−(n−2+1) since only the all-square tiling and tilings with 1 filled fence among n − 2 squares have less than two fences. Answer 2: condition on the location of the second fence. If it is the kth tile (k = 3, . . . , n − 1) and part of a filled fence or the first tile in a bifence, the first fence is part of a filled fence among k − 3 squares and hence there are 2(k − 2)B n−(k+1) tilings for these cases. If the second fence is the end of bifence and is the kth tile (k = 2, . . . , n), the tiles before the bifence are all squares and hence there are B n−k tilings in this case. Summing over all possible k, changing k to k − 1 in the first sum, and equating to Answer 1 gives B n − n = 2 n k=4 (k − 3)B n−k + n k=2 B n−k = B n−2 + n k=3 (2k − 5)B n−k . The identity then follows from Theorem 4. Identity 13. For n ≥ 0, J n+1 = F n+1 + n j=2 J j−1 F n+1−j . Proof. First note that the number of n-tilings with no bifences is given by S n = δ 0,n + S n−1 + S n−2 and hence S n = F n+1 . How many n-tilings have at least one bifence? Answer 1: B n − S n . Answer 2: condition on the last bifence. When the second fence it contains is the jth tile (j = 2, . . . , n) then the number of tilings is B j−2 S n−j . Summing over all possible j and equating this to Answer 1 gives B n − S n = n j=2 B j−2 S n−j . The identity follows from applying S n = F n+1 and Theorem 4. A Pascal-like triangle giving the number of n-tilings using k fences We define n k as the number of n-tilings which contain exactly k fences. We define 0 0 = 1 so that the result B n = n k=0 n k(9) is valid for n ≥ 0. The first 12 rows of the triangle whose entries are n k are shown in Figure 2. As will be shown later via its connection with a Riordan array, the triangle is sequence A059259. Identity 14. For n ≥ 0, n 0 = 1. Proof. There is only one way to tile without using any fences. Identity 15. For n ≥ 1, n 1 = n − 1. Proof. If only one of the n tiles is a fence, there must be 1 filled fence and n − 2 free squares making a total of n − 1 metatile positions. The filled fence can be placed in any of these. The following two identities describe, respectively, the entries in the first and second diagonal of the triangle. Identity 16. For n ≥ 0, n n = 1, n even; 0, n odd. Proof. An all-fence tiling must be composed of just bifences. This can only occur if the number of tiles is even. Identity 17. For m > 0, 2m − 1 2m − 2 = 2m 2m − 1 = m. Proof. If there are 2m − 1 or 2m fences and 1 square, there must be m − 1 bifences. The remaining metatile is then, respectively, a free square or a filled fence. There are m possible positions for this remaining metatile. The following identity shows that the third diagonal of the triangle is A002620. Identity 18. For m > 0, 2m 2m − 2 = m 2 ; 2m + 1 2m − 1 = m(m + 1). Proof. When 2 out of 2m tiles are squares there must be either m − 1 bifences and 2 free squares (totalling m + 1 metatile positions) or m − 2 bifences and 2 filled fences (giving m metatile positions). There are m+1 2 places to put the squares in the first case and m 2 ways to place the filled fences in the second. The total number of tilings is thus m 2 + m+1 2 = m 2 . When 2 out of 2m + 1 tiles are squares, there must be m − 1 bifences, 1 filled fence, and 1 free square, and thus m + 1 metatile positions. There are therefore 2 m+1 2 = m(m + 1) ways to place the free square and filled fence. The following two identities show that the third and fourth columns of the triangle are A000124 and A003600, respectively. Identity 19. For n ≥ 2, n 2 = n − 2 2 + n − 1. Proof. If there are 2 fences, there are either 2 filled fences or 1 bifence. In the first case there are n − 4 free squares and hence a total of n − 2 metatile positions in which to place the filled fences. There are thus n−2 2 ways to place the filled fences. In the second case there are n − 2 free squares and thus n − 1 metatile positions in which the bifence can be placed. Identity 20. For n ≥ 3, n 3 = n − 3 3 + 2 n − 2 2 . Proof. If there are 3 fences, there are either 3 filled fences or 1 bifence and 1 filled fence. In the first case there are n − 6 free squares and 3 filled fences giving a total n − 3 metatile positions to place the filled fences. In the second case there are n − 4 free squares and thus n − 2 metatile positions to place the filled fence and bifence. Identity 21. For n ≥ k > 0, n k = n k + n − 1 k − 1 .(10) Proof. Interpret n k as the tilings of an (n+ k)-board with k dominoes (D) and n−k squares (S). Proceeding from left to right along the board, replace DD by a bifence, DS by a filled fence, and then leave any of the remaining S as they are. Except for the case of a 'left over' single D at the right end of the board, this generates all possible n-tilings using k fences. If the (n + k)-board ends in an isolated D, ignore it and hence obtain a (n − 1)-tiling with k − 1 fences. In both cases the scheme is reversible. Identity 22. For n > k > 0, n k = n − 1 k + n − 1 k − 1 . (11) Proof. An n-tiling such that n > k must contain a free square or filled fence. Construct a bijection between n-tilings using k fences and (n−1)-tilings using k or k −1 fences as follows. In the n-tiling find the final square. If it is free, remove it to obtain an (n − 1)-tiling with k fences. If the square is part of a filled fence, remove the fence to obtain an (n − 1)-tiling with k − 1 fences. Identity 23. For n ≥ r ≥ 0, n n−r = D (r) n . Proof. The result follows from the definition of D (r) n since n n−r is also the number of n-tilings containing r squares. Identity 24. n k = δ n,0 δ k,0 + n − 1 k + n − 2 k − 1 + n − 2 k − 2 .(12) Proof. We count n k by conditioning on the last metatile on the board. If the metatile contains m tiles of which j are fences, for the remaining tiles the number of (n − m)-tilings is n−m k−j . Summing these for the three types of metatile gives the result. A (p(x), q(x)) Riordan array is a lower triangular matrix whose (n, k)th entry is the coefficient of x n in the series for p(x){q(x)} k [9]. Proof. Let p = 1/(1 − x 2 ), q = x/(1 − x). Then R(n − l, k − j) is the coefficient of x n in the expansion of x l pq k−j . Multiplying the identity q = x + x 2 + x 2 q by pq k−1 and taking the coefficient of x n gives R(n, k) = R(n−1, k −1) + R(n−1, k −1) + R(n−2, k −1) + R(n−2, k) for n > 2, k > 0. Taking R(n < 0, k) = R(n < k, k) = 0 and including terms to arrive at a relation that is also compatible with the values of R(k, n) for 0 ≤ n ≤ 2 and k = 0 gives R(n, k) = δ n,0 δ k,0 + R(n − 1, k − 1) + R(n − 1, k − 1) + R(n − 2, k − 1) + R(n − 2, k),(14) which is then valid for all n and k. Substituting (13) into (12), replacing k by n − k, and noting that δ n,0 δ n−k,0 can be rewritten as δ n,0 δ k,0 , gives (14). From Identity 23, R(n, k) = D (k) n . In other words, a combinatorial interpretation of R(n, k) is the number of n-tilings that use k squares (and n − k (1, 1)-fences). Then from Lemma 3 we have for n ≥ k ≥ 0, R(n, k) = R(n − 2, k) + n − 1 k − 1 .(15) This allows us to prove a conjecture given in the OEIS entry for A059259 concerning A071921 which is the square array a(n, m) given by a(0, m ≥ 0) = 1, a(n, m) = m−1 r=0 n − 1 + 2r n − 1 .(16) Using our notation, the conjecture is as follows. Identity 25. n + 2m 2m = a(n, m + 1). Proof. From Theorem 5, n+2m 2m = R(n + 2m, n). Repeatedly applying (8) gives R(n + 2m, m) = n − 1 + 2m n − 1 + n − 1 + 2(m − 1) n − 1 + · · · + n − 1 + 2 n − 1 + R(n, n). Using the fact that R(n, n) = 1 the result follows from (16). If b, f , and s are, respectively, the numbers of bifences, filled fences, and free squares in an n-tiling using k fences then it is easily seen that n = 2b + 2f + s, (17a) k = 2b + f. (17b) Identity 26. n k =        bmax b=b min n − k + b k − b k − b b , b min ≤ b max , 0, b min > b max ,(18) where b min = max(0, ⌈k − n/2⌉) and b max = ⌊k/2⌋. Proof. For given values of n and k we sum the number of tilings for all possible values of b. The maximum number of bifences b max is obtained from (17b) when f is 0 or 1 depending on whether k is even or odd, respectively. Eliminating f from (17) gives b = 1 2 (2k − n + s). If 2k − n is negative, the minimum possible value of b is zero. Otherwise b min is obtained when s is 0 or 1 when 2k − n is even or odd, respectively. From (25) we have that the total number of metatiles, b + f + s = n − k + b. The number of ways of tiling using b bifences, f filled fences, and s free squares is the multinomial coefficient b+f +s b, f, s which may be re-expressed as a product of binomial coefficients written in terms of b, n, and k. There will be no possible values of b and therefore no tilings if b min > b max . Corollary 1. J n+1 = n k=0 ⌊k/2⌋ b=max(0,⌈k−n/2⌉) n − k + b k − b k − b b . Proof. The result follows from (9), Theorem 4, and Identity 26. The (n, k)th entry, which we will denote here by [ n k ] 1/2 , of the Pascal-like triangle A123521 is the number of ways to tile an n-board using k ( 1 2 , 1 2 )-fences and 2(n − k) half-squares (with the shorter sides always horizontal) [8]. We now show that the [ n k ] 1/2 triangle can be obtained from the n k triangle by removing the odd downward diagonals of the latter which is equivalent to the following identity. Identity 27. For n ≥ k ≥ 0, n k 1/2 = 2n − k k . Proof. The total post length of a ( 1 2 , 1 2 )-fence is 1. The entry [ n k ] 1/2 can also be viewed as counting the number of tilings that use k ( 1 2 , 1 2 )-fences and 2(n − k) half-squares since the total length occupied by the n tiles is k + 2(n − k) 1 2 = n. The entry 2n−k k counts the number of tilings using k (1, 1)-fences and 2(n − k) squares. This latter tiling differs from the former only in that the tiles are twice the length. 6 A Pascal-like triangle giving the number of tilings of an n-board using k fences We define [ n k ] as the number of tilings of an n-board which contain exactly k fences (Fig. 3). We define [ 0 0 ] = 1 so that the result A n = n k=0 n k(19) is valid for n ≥ 0. As a result of the following identity, the upward diagonals of the n k triangle are the rows of the [ column k of the n k triangle downwards by k (and filling the entries above with zeros). Thus we again obtain sequences A000124 and A003600 for the k = 2 and k = 3 columns, respectively (Identities 32 and 33). Identity 28. n k = n − k k . Proof. If a tiling contains n − k tiles of which k are fences, the total length is n. The even rows of the triangle [ n k ] give the triangle [ n k ] 1/2 (defined just before Identity 27). Identity 29. 2n k = n k 1/2 . Proof. The number of tilings of a 2n-board with squares and (1, 1)-fences is the same as the number of tilings of an n-board with tiles of half the length. Identity 30. For n > 0, n 0 = 1. Proof. There is only one way to tile a board without using any fences. Identity 31. For n > 2, n 1 = n − 2. Proof. If there is only one fence, there must be 1 filled fence (and n − 3 free squares). The filled fence can be placed in any of the n − 2 metatile positions. Identity 32. For n > 3, n 2 = n − 4 2 + n − 3. Proof. If there are 2 fences, there are either 2 filled fences or 1 bifence. In the first case there are n − 6 free squares and thus a total of n − 4 metatile positions to place the filled fences. There are thus n−4 2 ways to place the filled fences. In the second case there are n − 4 free squares and thus n − 3 metatile positions in which the bifence can be placed. Identity 33. For n > 5, n 3 = n − 6 3 + 2 n − 5 2 . Proof. If there are 3 fences, there are either 3 filled fences or 1 bifence and a filled fence. In the first case there are n − 9 free squares and 3 filled fences giving a total n − 6 metatile positions for the filled fences. In the second case there are n − 7 free squares and thus n − 5 metatile positions for the filled fence and bifence. Identity 34. For n ≥ r ≥ 0, [ 2n+1 n−r ] = C (r) n . Proof. The result follows from the definition of C (r) n since [ 2n+1 n−r ] is also the number of ways to tile a (2n + 1)-board using 2r + 1 squares. Identity 35. n k = n − 1 k + n − 3 k − 1 + n − 4 k − 2 + δ 0,k δ 0,n .(20) Proof. We count [ n k ] by conditioning on the last metatile on the board. If the metatile is of length l and contains j fences, the number of ways to tile the remaining n − l cells with k − j fences is [ n−l k−j ]. Summing these for the three types of metatile gives the result. To show that C (r) n is a Riordan array we first need a recursion relation that involves only the odd rows of the triangle. Identity 36. 2n + 1 k = 2n − 1 k + 2n − 1 k − 1 + 2n − 3 k − 1 + 2n − 3 k − 2 − 2n − 5 k − 3 + δ 0,k δ 0,n .(21) Proof. Let E(n, k) denote (20). Then E(2n + 1, k) + E(2n, k) − E(2n − 1, k − 1) gives the identity. Proof. Let p = 1/[(1 − x)(1 − x 2 )], q = x/(1 − x) 2 . The recursion relation without incorporating boundary conditions for a Riordan array with this particular q isR(n, k) = R(n − 1, k) +R(n − 1, k − 1) +R(n − 2, k) +R(n − 2, k − 1) −R(n − 3, k) for n > 2, k > 0 [8]. Including terms to obtain a relation valid for all n and k (and taking R(n, k) = 0 if n < 0 or n < k) we arrive at R(n, k) = δ n,0 δ k,0 +R(n−1, k)+R(n−1, k−1)+R(n−2, k)+R(n−2, k−1)−R(n−3, k). (23) Substituting (22) into (21), replacing k by n − k, and noting that δ n,0 δ n−k,0 can be rewritten as δ n,0 δ k,0 , gives (23). From Identity 34,R(n, k) = C (k) n . In other words, a combinatorial interpretation of R(n, k) is the number of tilings of a (2n + 1)-board that use 2k + 1 squares (and 2(n − k) (1, 1)-fences). Then from Lemma 2 we have for n ≥ k ≥ 0, R(n, k) =R(n − 2, k) + n + k 2k . If b, f , and s are, respectively, the numbers of bifences, filled fences, and free squares in a tiling of an n-board using k fences then it is easily seen that n = 4b + 3f + s, (25a) k = 2b + f. (25b) Identity 37. For m ≥ 0, 4m + p 2m + q =      1, p = q = 0; m + 1, p = 1, q = 0 or p = 3, q = 1; (m + 1) 2 , p = 2, q = 0; and for m > 0, 4m + p 2m + q = m, p = −1, q = −1; m(m + 1), p = 0, q = −1. Proof. Substituting n = 4m + p and k = 2m + q into (25) and eliminating m gives Identity 38. p − 2q = f + s,n k =        bmax b=b min n − 2k + b k − b k − b b , b min ≤ b max , 0, b min > b max ,(27) where b min = max(0, ⌈ 1 2 (3k − n)⌉) and b max = ⌊k/2⌋. Proof. For given values of n and k we sum the number of tilings for all possible values of b, the number of bifences. The maximum number of bifences is obtained from (25b) when f is 0 or 1 depending on whether k is even or odd, respectively. Eliminating f from (25) gives b = 1 2 (3k − n + s). If 3k − n is negative, the minimum possible value of b is zero. Otherwise b min is obtained when s is 0 or 1 when 3k − n is even or odd, respectively. From (25) we have that the total number of metatiles, b+ f + s = n−2k + b. The proof is then the same as for Identity 26. Corollary 2. f 2 n = 2n k=0 ⌊k/2⌋ b=max(0,⌈ 1 2 (3k−2n)⌉) 2n − 2k + b k − b k − b b , f n f n+1 = 2n+1 k=0 ⌊k/2⌋ b=max(0,⌈ 1 2 (3k−2n−1)⌉) 2n + 1 − 2k + b k − b k − b b . Proof. The result follows from (19), Theorem 2, and Identity 38. Theorem 3 . 3If A (m) n is the number of ways to tile an n-board using squares and (1, m − 1)fences then for n ≥ 0, , r = 0, . . . , m − 1,where f n = F n+1 .Proof. We identify the following bijection between the tilings of a (mn + r)-board using squares and (1, m − 1)-fences and the square-domino tilings of an ordered m-tuple of r (n + 1)-boards followed by m − r n-boards. For convenience we number the boards in this m-tuple from 0 to m − 1 and the cells in the (mn + r)-board from 0 to mn + r − 1. Tile board j in the m-tuple with the contents (taken in order) of the cells of the given (mn + r)-board fence-square tiling whose cell number modulo m is j. The posts of any (1, m−1)-fence (which will always lie on two consecutive cells with the same cell number modulo m) get mapped to a domino in board j. The procedure is reversed by splicing the square-domino tilings of the m-tuple of boards, hence establishing the bijection. The number of square-domino tilings of the m-tuple of boards is f r n+1 f m−r n and the result follows. Theorem 4 . 4If B n is the number of n-tilings using squares and fences then 2 ), x/(1−x)) Riordan array (A059260). Answer 2: we condition on the location of the pth square. If it is the kth tile, there are k−1 p−1 ways to place the first k tiles and B n−k ways to place the remaining tiles. Summing over all possible k and equating the result to Answer Figure 2 : 2A Pascal-like triangle with entries n k (A059259). Theorem 5 . 5If R(n, k) is the (n, k)th entry of the (1/(1 − x 2 ), x/(1 − x)) Riordan array then n k = R(n, n − k). Figure 3 : 3n k ] triangle. Equivalently, column k of the [ n k ] triangle is obtained by displacing A Pascal-like triangle with entries [ n k ] (A335964). in which the only possible values of f and s are non-negative integers. p = q = 0 we must have f = s = 0 which corresponds to a tiling using m bifences only. When p = 1, q = 0 then we must have f = 0, s = 1 since (26) would imply a non-integral number of bifences if f = 1, s = 0. With the allowed case there are m + 1 metatile positions in which to place the free square. When p = 3, q = 1 we must have f = 1, s = 0 and again there are m + 1 places for the filled fence. For p = 2, q = 0, either s = 2, f = 0 or s = 0, f = 2. In the first case there are m+2 2 ways of placing the 2 free squares. In the second there are m − 1 bifences and hence m+1 2 ways of placing the 2 filled fences. Adding gives the required result. With p = q = −1 the only possibility is s = 0, f = 1. There are then m − 1 bifences and hence m ways to place the filled fence. Finally, if p = 0, q = −1 we must have f = s = 1. With m + 1 metatile positions, there are a total of 2 m+1 2 ways to place the filled fence and free square. Theorem 6 . 6IfR(n, k) is the (n, k)th entry of the (1/[(1 − x)(1 − x 2 )], x/(1 − x) 2) Riordan array then 2n + 1 k =R(n, n − k). (22) Corresponding author. Identity 39. For n ≥ k ≥ 0,where m = min(⌊(n + 1)/2⌋, k).Proof. From Lemma 1, [ 2n+1 k ] is also the number of square-domino tilings of an (n+1)-board and an n-board using k dominoes in total. The number of ways to tile an (n + 1)-board with j dominoes (and n + 1 − 2j squares) is n+1−j j . If the (n + 1)-board has j dominoes then the n-board will have k − j dominoes (and n − 2(k − j) squares). Hence there areways to tile the boards if the (n + 1)-board has j dominoes. Evidently j cannot exceed k or ⌊(n + 1)/2⌋ and so m ≥ j ≥ k − m. We then sum over all possible values of j.where m = min(⌊n/2⌋, k).Proof. The proof is analogous to that of Identity 39.Identity 40 is equivalent to Identity 3.2 in[8]. Summing Identities 39 and 40 over all possible k will, respectively, give alternative ways of expressing f n f n+1 and f 2 n as double sums of products of two binomial coefficients. Unified tiling proofs of a family of Fibonacci identities, Fibonacci Quart. A T Benjamin, J Crouch, J A Sellers, 57A. T. Benjamin, J. Crouch, and J. A. Sellers, Unified tiling proofs of a family of Fibonacci identities, Fibonacci Quart. 57 (2019), 29-31. Linear recurrences through tilings and Markov chains. A T Benjamin, C R H Hanusa, F E Su, Utilitas Math. 64A. T. Benjamin, C. R. H. Hanusa, and F. E. Su, Linear recurrences through tilings and Markov chains, Utilitas Math. 64 (2003), 3-17. Proofs That Really Count: The Art of Combinatorial Proof. A T Benjamin, J J Quinn, Mathematical Association of AmericaWashingtonA. T. Benjamin and J. J. Quinn, Proofs That Really Count: The Art of Combinatorial Proof, Mathematical Association of America, Washington, 2003. A tiling scheme for the Fibonacci numbers. R C Brigham, R M Caron, P Z Chinn, R P Grimaldi, J. Recreational Math. 28R. C. Brigham, R. M. Caron, P. Z. Chinn, and R. P. Grimaldi, A tiling scheme for the Fibonacci numbers, J. Recreational Math. 28 (1996), 10-16. A Pascal-like triangle related to the tribonacci numbers, Fibonacci Quart. K Edwards, 46K. Edwards, A Pascal-like triangle related to the tribonacci numbers, Fibonacci Quart. 46/47 (2008/2009), 18-25. Strongly restricted permutations and tiling with fences. K Edwards, M A Allen, Discrete Appl. Math. 187K. Edwards and M. A. Allen, Strongly restricted permutations and tiling with fences, Discrete Appl. Math. 187 (2015), 82-90. A new combinatorial interpretation of the Fibonacci numbers squared. K Edwards, M A Allen, Fibonacci Quart. 575K. Edwards and M. A. Allen, A new combinatorial interpretation of the Fibonacci num- bers squared, Fibonacci Quart. 57(5) (2019), 48-53. K Edwards, M A Allen, A new combinatorial interpretation of the Fibonacci numbers squared. Part II., Fibonacci Quart. 58K. Edwards and M. A. Allen, A new combinatorial interpretation of the Fibonacci num- bers squared. Part II., Fibonacci Quart. 58 (2020), 169-177. The Riordan group. L W Shapiro, S Getu, W.-J Woan, L C Woodson, Discrete Appl. Math. 34L. W. Shapiro, S. Getu, W.-J. Woan, and L. C. Woodson, The Riordan group, Discrete Appl. Math. 34 (1991), 229-239. Jacobsthal identities, combinatorial proof, combinatorial identities, n-tiling, Pascal-like triangle. Mathematics Subject Classification: Primary 05A19, 11B39 Keywords: Fibonacci identities. Riordan array (Concerned with sequences A000045, A000124, A000930, A001045, A001654, A002620, A003269, A003600, A006498, A007598, A017817, A059259, A059260, A071921, A123521, A158909, and A335964.Mathematics Subject Classification: Primary 05A19, 11B39 Keywords: Fibonacci identities, Jacobsthal identities, combinatorial proof, combinatorial identities, n-tiling, Pascal-like triangle, Riordan array (Concerned with sequences A000045, A000124, A000930, A001045, A001654, A002620, A003269, A003600, A006498, A007598, A017817, A059259, A059260, A071921, A123521, A158909, and A335964.)	[]
[ "A Clustering Based Approach for Realistic and Efficient Data-Driven Crowd Simulation", "A Clustering Based Approach for Realistic and Efficient Data-Driven Crowd Simulation" ]	[ "Mingbi Zhao [email protected] \nParallel and Distributed Computing Center School of Computer Engineering\nNanyang Technological University\n639798Singapore\n" ]	[ "Parallel and Distributed Computing Center School of Computer Engineering\nNanyang Technological University\n639798Singapore" ]	[]	In this paper, we present a data-driven approach to generate realistic steering behaviors for virtual crowds in crowd simulation. We take advantage of both rule-based models and data-driven models by applying the interaction patterns discovered from crowd videos. Unlike existing example-based models in which current states are matched to states extracting from crowd videos directly, our approach adopts a hierarchical mechanism to generate the steering behaviors of agents. First, each agent is classified into one of the interaction patterns that are automatically discovered from crowd video before simulation. Then the most matched action is selected from the associated interaction pattern to generate the steering behaviors of the agent. By doing so, agents can avoid performing a simple state matching as in the traditional example-based approaches, and can perform a wider variety of steering behaviors as well as mimic the cognitive process of pedestrians. Simulation results on scenarios with different crowd densities and main motion directions demonstrate that our approach performs better than two state-of-the-art simulation models, in terms of prediction accuracy. Besides, our approach is efficient enough to run at interactive rates in real time simulation.	null	[ "https://arxiv.org/pdf/1506.04480v1.pdf" ]	224,282	1506.04480	f4c029044afa6cb3b08d5e47701d532b3aed9a40	A Clustering Based Approach for Realistic and Efficient Data-Driven Crowd Simulation Mingbi Zhao [email protected] Parallel and Distributed Computing Center School of Computer Engineering Nanyang Technological University 639798Singapore A Clustering Based Approach for Realistic and Efficient Data-Driven Crowd Simulation In this paper, we present a data-driven approach to generate realistic steering behaviors for virtual crowds in crowd simulation. We take advantage of both rule-based models and data-driven models by applying the interaction patterns discovered from crowd videos. Unlike existing example-based models in which current states are matched to states extracting from crowd videos directly, our approach adopts a hierarchical mechanism to generate the steering behaviors of agents. First, each agent is classified into one of the interaction patterns that are automatically discovered from crowd video before simulation. Then the most matched action is selected from the associated interaction pattern to generate the steering behaviors of the agent. By doing so, agents can avoid performing a simple state matching as in the traditional example-based approaches, and can perform a wider variety of steering behaviors as well as mimic the cognitive process of pedestrians. Simulation results on scenarios with different crowd densities and main motion directions demonstrate that our approach performs better than two state-of-the-art simulation models, in terms of prediction accuracy. Besides, our approach is efficient enough to run at interactive rates in real time simulation. I. INTRODUCTION Crowd simulation has become an active research field that has increasing applications in many areas such as virtual environment [1], [2],object tracking [3]- [5] and abnormal detection [6]. A fundamental issue in crowd simulation is to simulate the movements of individuals (agents) in virtual worlds in a human-like manner so that the simulation looks realistic. Although various models to simulate realistic crowd motions have been proposed over the last few years, most of them are based on simplifying assumptions (e.g. least biomechanical energy spent [7], collision-free velocities closest to the preferred velocities [8] and so on) and a set of rules that employed by all agents in crowd [9]. These models produce similar steering behaviors for all agents and thus fails to reproduce the complexity and variety of steering behaviors in the real pedestrians. Many researchers attempt to replicate complex crowd behaviors by introducing personal attributes such as personality, emotion, group interactions and cultural difference [10]- [13], or by defining more finely tuned situation-specific rules and strategies [1], [14]- [16]. However, identifying the correct rules that influence pedestrian's motions requires a great amount of manual work and can be non-trivial. What is more, the wide variety of steering behaviors are difficult to generate by considering limited number of factors. Recently, some data-driven modeling techniques have been proposed to solve the above problems [17]- [19]. The key idea is to represent the implicit rules in crowd videos by using state-action samples extracted from videos. Here, state refers to the information agents observed and action is the future velocities. Compared to those pre-defined rules, stateaction samples implicitly contains a large number of different situations and reactions learned from the source crowd videos. Besides, they are more adaptable to different scenarios. Velocities are predicted basically by selecting the nearest sample in the original crowd videos. Therefore the definition of a distance measure (or a matching function) is crucial to the simulation results. The matching function should capture the subtle details of agents' states that trigger various actions and at the same time should be calculated efficiently in simulation to meet the real-time constraint. However, there are trade-offs between quality and efficiency. To fully model the states of agents, more features are required but more features lead to larger computation time. Furthermore, rules implicitly defined by state-action samples are lack of strategic meanings and these example-based models work like copy-paste systems without the ability of thinking and reasoning like human cognitive process. Crowds generated by these example-based models often perform random-like behaviors. Our approach is also data-driven. However, we first discover interaction patterns from crowd videos, so that in simulation, agents can first select an interaction pattern and then select an action among the pattern. The benefit of doing this is twofold. First, these interaction patterns can be viewed as strategies or rules learned from crowd videos rather than predefined, which are more realistic and flexible. We apply videoleaned rules to control the behavior of agents instead of simply matching to one of the samples. Second, by selecting an interaction pattern, the searching space of agents to search for an action is reduced and searching actions in the same interaction pattern requires less features. Therefore, the calculation time can be reduced compared to other data-driven models. The contribution of the paper are as follows. • A data-driven crowd modeling approach is proposed to generate realistic steering behaviors of pedestrians. Rather than specifying behavior rules manually, our approach can generate a wide variety of steering behaviors based on pedestrian behaviors from crowd videos. • An unsupervised method is proposed to automatically identify the new interaction patterns to capture the features of steering behaviors of pedestrians. • A hierarchical matching mechanism is proposed to reduce the search space when agents selecting action to extend their movements. This mechanism can improve the simulation efficiency to meet the real-time response. We compare our algorithm with state-of-the-art models both qualitatively and quantitatively in two real world scenarios with various crowd densities and main motion directions. The experimental results demonstrate that the proposed approach gives more accurate predictions of agents' steering behaviors. Furthermore, the proposed approach is efficient enough to satisfy real-time response. The paper is organized as follows: We first report related work in Section 2 together with an overview of the proposed approach in Section 3. Section 4 describes our method to automatically discover interaction patterns, followed by pattern learning and action selection in Section 5. Section 6 describes the velocity prediction procedure for agents in simulation. Section 7 presents the experimental results and Section 8 concludes the paper. II. RELATED WORK Numerous crowd simulation models have been proposed over the years. One of the early works, rule-based models proposed by Reynolds [9], [20], define a set of steering rules that agents will follow based on their local view of their state. Many works followed thereafter focus on introducing more rules in the aspect of group interaction [10], higher level navigational behaviors [1] and real-time simulation in complex and structured environment [21]. The Cellular Automation (CA) based models discretize space into squares, triangular, hexagonal and so on. The values of all cells depend on the values of their neighboring cells and are updated synchronously at each time step [22]- [25]. In the social force models [26]- [29], agents and obstacles impose forces to others and are influenced by the forces of others at the same time. The motion of agents is then derived according to Newton's Law of Motion. In recent works, Van den Berg et al. [8], [30]- [35] proposed models which can calculate the velocities that guarantee collision-free motion in next τ time while closest to the preferred velocities in an efficient and robust way to handle dense scenarios with tens or hundreds or thousands of agents. Most recently, real-world videos are applied to estimate the parameters of pedestrian models to improve the accuracy of motion prediction [2], [4], [36]- [41]. However, a handspecified model is still needed before the calibration process. Other data-driven approaches [17]- [19], [42] directly extract state-action samples from crowd videos to form a sample database. In simulation, agents match their current states to the nearest ones in the database to decide future positions. In order to improve the matching efficiency, a graph-based data structure is formed before simulation in which similar states are interconnected with actions to avoid continuous database searches [19]. III. OVERVIEW Similar to other data-driven models, we focus on simulating low level pedestrians motion, where a pedestrian's velocity is mainly affected by other pedestrians and obstacles, and the preferred velocities of agents (direction and speed) are obtained from higher level path planning models. In this paper, the preferred velocities are estimated from input data as accurate as possible. The proposed model needs to be trained from input data before simulation. At pre-processing stage, interaction patterns are discovered in an unsupervised way from input data and learned by a neural network classifier, which is used to assign interaction patterns for agents in simulations. Actions of different interaction patterns are stored in k-d trees using a vector consisting of 4 features, in order to achieve efficient action selection in run-time simulations. A. Pre-processing Based on the trajectories annotated from crowd videos, interactions between two pedestrians are extracted (we call this the pairwise interactions), which are essential to understand steering behaviors and is described in Section IV-A. We then employ an unsupervised learning approach to discover patterns in the pairwise interactions which can be used as rules to control agents in simulations (Section IV-C). These interaction patterns are learned by training a neural network classifier and one k-d tree is built for each pattern to store feature vectors of actions as index. Actions, each consisting of two sequences of velocities of two interacting pedestrians are represented in their own local coordinate system (Section V). B. Simulation In simulation, pedestrians are represented as circular agents. The start positions, goal positions and preferred speed of agents are pre-defined together with the time step of entering the scenario. At every time step, agents observe the scenario (i.e., other pedestrians and obstacles) to generate pairwise interactions similarly to the pre-processing stage. These pairwise interactions are classified to one pattern using the neural network classifier trained in the pre-processing stage. And one action is selected by searching the k-d tree of that pattern. An action includes future velocities for both interacting agents. If no interaction is detected, agents simply move towards their goals with its preferred speed. IV. PAIRWISE INTERACTION PATTERN DISCOVERY We first describe how to identify pairwise interactions from crowd trajectories. Pairwise interactions contain information about two pedestrians interact with each other. It starts with determining one nearby pedestrian who has profound influence. Then a clustering method is proposed to discover underlying patterns among them. A. Pairwise Interaction Identification The trajectory of a person in a scenario is described as a sequence of observations (s, t), where s is the position vector of a person at time t. A pedestrian i can sense other pedestrians in his sensing range. Among these nearby pedestrians, one pedestrian who has the key influence on i's future motion behaviors is called the core neighbor (CN i ) of pedestrian i. In order to quantify the influence of nearby pedestrians, we define a variable Inf i (j, t) to denote the influencing factor of pedestrian j on pedestrian i at time t, which is inverse proportional to the smallest distance between i and j throughout the entire future temporal overlap Γ of i and j . Inf i (j, t) = 1 min t s t i − s t j t ∈ Γ(1) where s t i and s t j are the positions of i and j at time t respectively. The intuition behind this distance-based influencing factor is that if two pedestrians will be very close to each other in some future time, it is highly possible that they have interactions, e.g., collision avoidance. If more than thr group percent of the steps in Γ, the distance between pedestrians i and j are smaller than thr dist , which indicates i and j walk as a group, the influence factor of j is set to positive infinite, Inf i (j, t) = +∞. Pedestrians from a group tend to keep similar velocities in order to walk together towards their goal, while collision avoidance behaviors between pedestrians are much more complicated and therefore worth more attention. By doing this group identification, pedestrians in the same group would be exclude from core neighbor consideration to give other nearby pedestrians more chances. The influencing factors of all the other pedestrians in the sensing range of pedestrian i are calculated and among them, all agents where Inf i (j, t) > thr Inf are selected as the candidate set C CN i (t). C CN i (t) = {j\|Inf i (j, t) > thr Inf }(2) If there are more than one agent in the candidate set C CN i (t), we select the one with the smallest time to interaction T T I i (j, t) with i as the core neighbor of i at time t. CN i (t) = arg min j T T I i (j, t) j ∈ C CN i (t)(3) where T T I i (j, t) is defined as the number of time step from current time t until i and j has the smallest distance. T T I i (j, t) = arg min t s t i − s t j − t t ∈ Γ(4) Let's take pedestrians interactions in Figure 1 as an example, which illustrates one case in crowd videos. Pedestrian i has sensing radius r and 270 • field of view. Figure 2 draws the distance between i and his nearby pedestrians as a function of time and Figure 3 illustrates the influencing factors of nearby pedestrians as a function of time. At t 1 , there are 7 pedestrians in the sensing range of i (see Figure 1a). The future trajectories of all pedestrians are shown in Figure 1b and the calculated influencing factors of nearby pedestrians are shown with various colors in Figure 1c. The pedestrians with higher influencing factors are shown with darker colors. Among them, pedestrian a is the only one with influencing factor larger than thr Inf at t 1 (Figure 3) and therefore is the core neighbor of i at t 1 . At t 2 , pedestrian b enters the sensing range ( Figure 1d) and also has an influencing factor larger than thr Inf . However, pedestrian a has a smaller time to interaction T T I i (a, t 2 ) = t 3 − t 2 than pedestrian b, T T I i (b, t 2 ) = t 4 − t 2 . Thus pedestrian a is still identified as the core neighbor of agent i at t 2 . For pedestrian i, his core neighbor CN i at every time step t (CN i (t)) is a time series of most influencing nearby pedestrians. The consecutive time steps at which i have the same core neighbor are combined as interaction period. During one of these time periods, pedestrian i interacts with the same core neighbor CN i and thus is called as a pairwise interaction. Pairwise interactions extracting from crowd videos describe the knowledge of how human solve interactions fluidly and naturally. By learning and reproducing them, naturally motions can be generated in simulation. The pairwise interactions between i and CN i are recorded B. Pairwise Interaction Similarity A pairwise interaction between pedestrian i and j is a temporal sequence of observations P I ij = {(rel ij , t)}, where the observations rel ij at time step t is the relative position of the core neighbor j in a coordinate system centered on i. One simple way to calculate the distance between two P Is is to add up all the spatial distances between observations at the same time step. However, considering the small annotation time interval of crowd videos and the tracking noise, comparing the observations at the same time step is a strict constraint. Therefore, we adopt the dynamic time warping method (DTW) with a window size w to allow an observation to find its closest match within certain time window. Let Γ be the longer temporal duration of two pairwise interactions a and b. The distance between these two interactions d(a, b) is calculated as follows:d (a, b) = DT W (a, b),(5)δ t (a, b) = 1 ifd t (a, b) ≤ thr dist 0 otherwise,(6)ρ(a, b) = t∈Γ δ t (a, b),(7)d(a, b) =d (a, b) ρ(a, b) \|Γ\|(8) The distance is scaled by the number of times that the spatial difference between two observations is smaller than the threshold thr dist , similar to [43]. By doing this, two pairwise interactions that have similar observations over a long period of time would have a smaller distance. This helps to get better and meaningful clustering result against tracking noise by imposing aggregated measurement over time. C. Pairwise Interaction Clustering Before clustering, we first delete the outlier pairwise interactions, which are usually caused by tracking noise or personal abnormal behaviors. For example, a pedestrian suddenly decides to turn back and walks in the opposite direction during the interaction. These anomalous behaviors may be of interest in video surveillance area. But they are not the typical behaviors of pedestrians and the patterns discovered by the clustering algorithm will be more accurate if those outliers are removed. We delete those pairwise interactions whose a) average distance to its N nearest neighbors is large; b) length is smaller than thr len . We apply a bottom-up hierarchical clustering algorithm [44] which starts with treating each pairwise interactions as separate clusters and gradually merge pairwise interactions with smallest distances to build larger clusters. In order to measure distance between groups of pairwise interactions, we extend the distance metric described in section IV-B by using the modified Hausdorff distance similar to [45]. For two groups M and N , the directed distance from M to N is h(M, N ) = 1 \|M \| × \|Ψ i \| i∈M j∈Ψi d(i, Ψ i (j))(9) where Ψ i is a set consisting of P Is in group N whose distances to i are in the larger half among all the distances d(i, l), l ∈ N . Hence \|Ψ i \| = \|N \|/2 . Then the symmetric distance between M and N is H(M, N ) = 1 2 (h(M, N ) + h(N, M ))(10) We use the average value of the directed distances between two groups instead of the maximum value used in the original Hausdorff distance in order to average out noise. One of the advantages to use the bottom-up hierarchical clustering is that the number of clusters is not required before clustering. To automatically discover the number of clusters, we define a stopping threshold to stop merging clusters if the symmetric distance between two clusters defined in equation 10 is larger than the threshold. V. LEARNING PAIRWISE INTERACTION PATTERN USING CLASSIFICATION After clustering, patterns of pairwise interactions are detected according to the relative positions of two pedestrians involved. Each pattern can be viewed as one rule describing how two people react with the presence of the other. In order to classify each agent into one interaction pattern to obtain a rule, a feed-forward neural network classifier is trained using any thr len consecutive observations in pairwise interaction sequences. The output of the classifier is the pattern membership of the input observations. The neural network has two layers. The first layer uses the tan-sigmoid transfer function and the second layer uses the linear transfer function. Among all the sample data, 85 percent are used to train the neural network and the remaining 15 percent are used for validation to prevent overfitting. The network is trained using Levenberg-Marquardt optimization [46]. The pairwise interactions are reformatted from relative positions to the velocity sequences of two pedestrians action = (v i (t), v j (t)). The velocities are transformed to their own local coordinate system oriented along their current motion directions respectively. In order to select the optimal action from all actions in the same pattern, actions are associated with an additional vector containing four features: To improve efficiency, we store actions in the same interaction pattern into a k-d tree using feature vectors described above. Therefore, after pre-processing, a neural network classifier and various k-d trees (the same number as the number of patterns) are generated for utilization in simulation. VI. SIMULATION In simulation, for each agent at every time step, the core neighbor is first identified using the method described in section IV-A, where the influencing factor of each nearby pedestrian is calculated by comparing spatial distances throughout the entire future temporal overlap. In simulation, future positions of an agent is estimated using the linear extrapolation of its current velocity and the number of future time steps used in the influencing factor calculation is estimated by the average travel time of all agents using their preferred velocities. If no core neighbor is found, the agent is considered interaction-free and the preferred velocity is selected. Otherwise, interaction is detected. If the agent has previously selected velocities to execute and the core neighbor remains the same, no classification is required and velocity is obtained by executing the current action. Otherwise, new action is selected in two phases. First, the relative positions between the agent and its core neighbor in past thr len time steps are classified into one pairwise interaction pattern using the neural network trained in Section V. And then one action with the nearest feature vector is selected using the k-d tree of that pattern. Two velocities sequences in the action are assigned to both the agent and its core neighbor for execution. The stored velocities in the action are transformed from its own local coordinate system to the global coordinate system when execution. Algorithm 1 Velocity Prediction VII. RESULT We implement our method in Java using MASON multiagent simulation library [47]. We also implement other two state-of-the-art models, RVO2 model [8] and social force model [27] on the same Java framework. A. Datasets We collect two unique datasets consisting of two public scenarios to validate our proposed method. The detailed information are shown in Table I together with the values of parameters used in the experiment. Crowd density is calculated as the average number of neighboring pedestrians who lie in the circle of 1 m radius around a pedestrian. In corridor dataset, pedestrians are walking through a store street, recorded in a shopping center ( Figure 5). There are two main walking directions in this scenario. In crossing dataset (Figure 6), pedestrians are crossing Oxford Circus in central London from four directions at the same time, collected from Internet. All videos are annotated at the rate of 10 frames per second manually to eliminate tracking errors. After mapping the trajectories from image coordinate to real coordinate, each dataset is split into two part according to annotation time. First half of the trajectories in each dataset are used to train the model separately and the rest trajectories are used as the ground truth to test the proposed model's performance. The proposed model is evaluated on the same scenario that trains it. B. Pairwise Interaction Pattern Discovery The discovered pairwise interaction patterns of two datasets are shown in Figure 4. The pairwise interactions in corridor scenario are clustered into four different patterns. Because pedestrians walk in two opposite directions, they avoid opposing pedestrians either from left-hand side or right hand side, which are represented by two clusters in green and magenta. The remaining two clusters shown in yellow and cyan demonstrate group walking behavior in corridor dataset. The pairwise interactions in crossing scenario are clustered into 14 patterns because of more moving directions and higher crowd density. Among them, four major patterns (which have larger number of instances) are avoiding opposite walking pedestrians from two sides and interactions with pedestrians from the perpendicular directions. Group walking behaviors are also discovered. C. Quality Comparisons We compare the proposed model with two state-of-the-art models, social force model [27] and RVO2 model [8] in simulating crowd motion in corridor and crossing scenarios. Our proposed data-driven model is trained using motion data from the same scenario, while social force model and RVO2 model do not need training. We compare the simulated trajectories of three models with the ground truth trajectories using three error metrics introduced in [48]. The average position error quantifies the position prediction accuracy at every time step. The average area error measures the difference in shape between two trajectories, indicating the accuracy of predicting angular velocity, while the average speed error quantifies the accuracy of predicting how fast agents move. The experimental result of corridor scenario is shown in Table II. The proposed model performs better than other stateof-art models in average position error and average area error, while the average speed error is slightly worse than RVO2 model. Therefore, the proposed model can predict where a pedestrian will walk more accurately than RVO2 model and social force model. However, it does not improve the prediction accuracy of how fast a pedestrian will walk. Besides, RVO2 model outperforms the social force model in all three error metrics. Figure 5 shows the trajectories simulated by three models and the ground truth. A pedestrian is walking towards the right side of the scenario and another pedestrian is moving in the opposite direction. The social force model (SF) starts to avoid collision too late and has a sudden position change because of the repulsive force. The RVO2 model predicts a straight line indicating no velocity modification is needed because RVO2 model seeks the collision-free velocities that closest to the preferred velocities. Only our proposed model follows the trajectory from the ground truth. The error metrics of simulating crossing scenario is shown in Table III. Compared to the corridor scenario, all models have larger errors because of a more complicated scenario and a higher crowd density that increase the difficulty of motion prediction. And the performance gap between models are also reduced. The proposed model still performs better than social force model and RVO2 model in position error and area error but worse than RVO2 model in speed error, similar to those in corridor scenario. RVO2 model still performs better than social force model but the difference is small. Figure 6 compares the simulated trajectories of all models with the ground truth in the crossing scenario. Our model successfully detects the collision and produces a trajectory similar to the ground truth, while RVO2 model predict a path with smaller deviation and social force model has a sudden fluctuation and also steers in the wrong direction (Figure 6a). Sometimes however suggests meaningful, our proposed model estimates in the wrong direction (Figure 6b), increasing the prediction errors. D. Timing Comparisons We report the simulation frame rates (f rm/sec) of three models on two scenarios in agents in the corridor scenario at every time step is 4.6 and the crossing scenario contains 50 agents at every time step on average. All simulations are single threaded and run on a single core. In both scenarios, the proposed data-driven model run slower than other two models, but still meets the real-time mark. The frame rates of crossing scenario is smaller than corridor scenario due to higher densities. VIII. CONCLUSION In this paper, we propose a data-driven model to predict velocities by learning interaction patterns from crowd videos. We propose a clustering approach to automatically discover interaction patterns from pedestrian's trajectories and apply them as interaction rules in the real-time simulation. We test our approach with two state-of-the-art simulation models on different types of scenarios. The simulation results demonstrate that our approach achieves better performance. Fig. 2 . 2Distance between pedestrian i and his nearby pedestrians as a function of t. Fig. 3 . 3Influencing factors of pedestrian i's nearby pedestrians as a function of t. Fig. 1 . 1Influencing factor calculation and core neighbor identification. by the relative positions of CN i in a coordinate system centered on i and oriented along i's current facing direction. A pedestrian's facing direction is quantized according to the number of main motion directions of the scenario. For example, in a corridor scenario, pedestrians walk bi-directionally. So the facing directions are quantized into 2 sets. • Self speed: The average speed of a pedestrian in the past thr len time steps. • Intended walking direction: The angle of a vector pointing from a pedestrian's current position to the goal position. • Relative position: The current relative position of a pedestrian's core neighbor in the coordinate system centered on the pedestrian and oriented along the pedestrian's current walking direction. • Average position error (m): the average spatial distances between the simulated positions and the ground truth positions. • Average area error (m 2 ): the area between the simulated path and the ground truth path after the simulation has completed, averaged across all simulated agents. • Average speed error (m/s): the average difference in speed at every time step between the simulated agent and the ground truth agent. Fig. 4 .Fig. 6 . 46Interaction patters clustered from corridor and crossing datasets. (a): all pairwise interactions in corridor dataset with outlier interactions in red; (b): four pairwise interaction patterns in corridor scenario; (c): all pairwise interactions in crossing scenario with outliers in red; (d): fourteen pairwise interaction patterns in crossing dataset. Motion prediction of all models compared to the ground truth in the cross scenario. Best viewed in color. TABLE II. ERRORS IN MOTION PREDICTION OF CORRIDOR SCENARIO Fig. 5. Motion prediction of all models compared to ground truth in the corridor scenario. Best viewed in color.corridor Error metrics Model Position Error Area Error Speed Error Social Force 0.4779 2.282 0.1986 RVO2 0.3557 1.531 0.1287 Proposed 0.3355 1.504 0.1295 Table IV . IVThe average number of TABLE III. ERRORS IN MOTION PREDICTION IN THE CROSS SCENARIO. THE ORIGINAL VIDEO IS ACCELERATED. 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[ "On the Inefficiency of the Uniform Price Auction ", "On the Inefficiency of the Uniform Price Auction " ]	[ "Evangelos Markakis [email protected] \nDepartment of Informatics\nAthens University of Economics and Business\nGreece\n", "Orestis Telelis [email protected] \nDepartment of Informatics\nAthens University of Economics and Business\nGreece\n" ]	[ "Department of Informatics\nAthens University of Economics and Business\nGreece", "Department of Informatics\nAthens University of Economics and Business\nGreece" ]	[]	We present our results on Uniform Price Auctions, one of the standard sealed-bid multiunit auction formats, for selling multiple identical units of a single good to multi-demand bidders. Contrary to the truthful and economically efficient multi-unit Vickrey auction, the Uniform Price Auction encourages strategic bidding and is socially inefficient in general, partly because of a "Demand Reduction" effect; bidders tend to bid for fewer (identical) units, so as to receive them at a lower uniform price. Despite its inefficiency, the uniform pricing rule is widely popular by its appeal to the natural anticipation, that identical items should be identically priced. Application domains of its variants include sales of U.S. Treasury notes to investors, trade exchanges over the internet facilitated by popular online brokers, allocation of radio spectrum licenses etc.In this work we study equilibria of the Uniform Price Auction for bidders with (symmetric) submodular valuation functions, over the number of units that they win. We investigate pure Nash equilibria of the auction in undominated strategies; we produce a characterization of these equilibria that allows us to prove that a fraction 1 − e −1 of the optimum social welfare is always recovered in undominated pure Nash equilibrium -and this bound is essentially tight. Subsequently, we study the auction under the incomplete information setting and prove a bound of 4 − 2 k on the economic inefficiency of (mixed) Bayes Nash	null	[ "https://arxiv.org/pdf/1211.1860v4.pdf" ]	17,503,441	1211.1860	95214c772e77c83f0a739dbc5f0eb3b7c836148c	On the Inefficiency of the Uniform Price Auction * 19 Jun 2013 Evangelos Markakis [email protected] Department of Informatics Athens University of Economics and Business Greece Orestis Telelis [email protected] Department of Informatics Athens University of Economics and Business Greece On the Inefficiency of the Uniform Price Auction * 19 Jun 20131 We present our results on Uniform Price Auctions, one of the standard sealed-bid multiunit auction formats, for selling multiple identical units of a single good to multi-demand bidders. Contrary to the truthful and economically efficient multi-unit Vickrey auction, the Uniform Price Auction encourages strategic bidding and is socially inefficient in general, partly because of a "Demand Reduction" effect; bidders tend to bid for fewer (identical) units, so as to receive them at a lower uniform price. Despite its inefficiency, the uniform pricing rule is widely popular by its appeal to the natural anticipation, that identical items should be identically priced. Application domains of its variants include sales of U.S. Treasury notes to investors, trade exchanges over the internet facilitated by popular online brokers, allocation of radio spectrum licenses etc.In this work we study equilibria of the Uniform Price Auction for bidders with (symmetric) submodular valuation functions, over the number of units that they win. We investigate pure Nash equilibria of the auction in undominated strategies; we produce a characterization of these equilibria that allows us to prove that a fraction 1 − e −1 of the optimum social welfare is always recovered in undominated pure Nash equilibrium -and this bound is essentially tight. Subsequently, we study the auction under the incomplete information setting and prove a bound of 4 − 2 k on the economic inefficiency of (mixed) Bayes Nash Introduction We study the Uniform Price Auction, a standard multi-unit auction format, for allocating multiple units of a single good to multi-demand bidders within a single auction process. Multi-unit auctions are being applied in a variety of diverse trade exchanges, including online sales over the Internet held by various brokers [24], allocation of radio spectrum licenses [21], sales of U.S. Treasury notes to investors [25], and allocation of advertisement slots on Internet sites [8]. The particular feature of the Uniform Price Auction is a single price for every unit allocated to any bidder; this makes it a proper representative of a wider category of uniform pricing auctions, as opposed to discriminatory pricing ones, that sell identical units of a single item at different prices [24,16]). As observed by Milgrom in [21], resurgence of interest in auction design is owed to a large extent to the success of multi-unit and -particularly -uniform price auction formats. Charging a uniform price for identical items, apart from appealing to the intuitive anticipation that identical items should be identically priced, it eases the worries of proxy agents that bid on behalf of their employers; they do not have to explain why they did not achieve a better price than their competitors. The design of mechanisms for auctioning multiple units of a single good to multi-demand bidders dates back to the seminal work of Vickrey [33]. Since then, three sealed-bid standard multi-unit auction formats have been identified in Auction Theory [16] [Chapter 12]: the Multi-Unit Vickrey Auction, the Uniform Price Auction, and the Discriminatory Price Auction. A significant volume of research in economics has been dedicated to identifying the properties of these standard formats [23,9,1,28,3]. All three auctions have the same (sealed) bidding format and allocation rule, and have been studied mostly for bidders with "downward sloping" (symmetric submodular [17]) valuations; these prescribe that the marginal value that a bidder has for each additional unit is non-increasing. Therefore each bidder is asked to issue such a non-increasing sequence of marginal bids for the k available units. The k highest marginal bids win the auction and each winning bid grants its issuing bidder a distinct unit. The Multi-Unit Vickrey Auction charges according to an instantiation of the Clarke payment rule [6] and is a generalization of Vickrey's celebrated single-item Second-Price mechanism. The Discriminatory Price Auction charges the winning bids as payments and it is a generalization of the First-Price Auction. The Uniform Price Auction, proposed by Friedman [12], charges per allocated unit the highest rejected (losing) marginal bid. Among these three formats, the Multi-Unit Vickrey Auction for submodular bidders retains the characteristics of the single-item Second-Price Mechanism, i.e., optimizes the Social Welfare and is truthful (it is a -weakly -dominant strategy for every bidder to report his marginal values truthfully). Neither the Discriminatory nor the Uniform Price auctions are truthful; they encourage strategic bidding. One of the downsides of the Uniform Price Auction is the effect of Demand Reduction, observed in [23,9] and formalized in a general model for multi-unit auctions by Ausubel and Cramton [1]. Bidders may have an incentive to shade their marginal bids for some units, only to win fewer ones in a lower uniform price. This effect leads to diminished revenue and inefficient allocations in equilibrium. In particular, it is known that the socially optimal allocation cannot be generally implemented in an equilibrium in (weakly) undominated strategies. Despite this effect, the variants of the Uniform Price Auction have seen extensive applications, contrary to the Vickrey auction, which has been largely overlooked in practice; implementations of variants of the standard format are offered by several online brokers 1 [24,15] and are also being used for sales of U.S. Treasury notes to investors since 1992 [25]. We also note that despite the Demand Reduction effect, the Uniform Price Auction does retain some interesting characteristics: overbidding any marginal value is a weakly dominated strategy, and so is any misreport of the marginal bid for the first unit. In this work we give a detailed account of the properties of undominated pure Nash equilibria for the Uniform Price Auction, when bidders have submodular valuation functions. Subsequently, we study the economic inefficiency of pure Nash and mixed Bayes-Nash equilibria, incurred by the effect of demand reduction in the former case, and by demand reduction and incomplete information in the latter. Contribution. We study pure Nash and (mixed) Bayes-Nash equilibria of the Uniform Price Auction. We focus first on bidders with submodular valuation functions and in Section 4 we give a detailed description of (pure) undominated strategies in the standard model of Uniform Price Auctions. Although these properties are mentioned or partially derived in previous works on Uniform Price Auctions, our analysis aims at clarifying some ambiguity between assumptions and implications. Additionally, we give a characterization of a subset of pure Nash equilibria in undominated strategies, that facilitates our analysis of economic inefficiency later on. In Section 5, we study the social inefficiency of pure Nash equilibria (PNE) for submodular bidders, in undominated strategies, i.e., the Price of Anarchy (PoA) over the subset of such equilibria. We derive an upper bound of e e−1 , which states that the auction recovers in (undominated) PNE at least a fraction 1 − e −1 of the welfare of the socially optimal allocation. We note here that the auction does have a socially optimal equilibrium (discussed in Section 3, but not in undominated strategies; undominated PNE are known to be socially inefficient in general. As noted earlier, this is largely due to the effect of Demand Reduction [1], whereby a bidder shades his bids for additional units, so as to pay a lower price for the units he wins. Our analysis can thus be viewed as a quantification of this effect. For any number of units k ≥ 9, we provide an almost matching lower bound, equal to 1 − e −1 + 2 k −1 . We also discuss the social inefficiency of the auction for k < 9 units. In Section 6, we consider (mixed) Bayes-Nash equilibria in the incomplete information model of Harsanyi. For mixed Bayes-Nash equilibria that emerge from randomized bidding strategy profiles containing only undominated pure strategies in their support, we prove an upper bound of 4 − 2 k on the Price of Anarchy 2 . Related Work Mult-Unit Auctions. The Uniform Price Auction has received significant attention within the economics community. Noussair [23] and Engelbrecht-Wiggans and Kahn [9] gave characterizations of pure Bayes-Nash equilibria under the model of independent private values of bidders, drawn from continuous distributions. They also made some initial observations on the effect of demand reduction. Ausubel and Cramton formalized demand reduction for a more general model of multiunit auctions in [1], that allows also interdependent private values. Bresky showed in [3] existence of pure Bayes-Nash equilibria in the independent private values model (with continuous valuation distributions) for a large class of multi-unit auctions, including all three standard formats. Simultaneous Auctions. There has been a growing recent interest in the computer science community in analyzing auction schemes that -although not necessarily truthful -have an appealing simplicity and appear to achieve increased economic efficiency in equilibrium, compared to what is achievable with truthful mechanisms [5,2,14,13,18]. Our work presents conceptual and technical resemblance to these works. Christodoulou, Kovács and Schapira initialized in [5] the study of Simultaneous Auctions for bidders with combinatorial demands, where they proposed that, a set of distinct goods is sold by having an independent single-good Second-Price auction for each of them in parallel. For bidders with fractionally subadditive valuation functions (see [10] for a definition), they prove a tight bound of 2 for the Price of Anarchy, even for (mixed) Bayes-Nash equilibria. They show how pure Nash equilibria can be computed efficiently for submodular valuation functions and describe an iterative best response algorithm that converges to a pure Nash equilibrium for fractionally subadditive ones. Bhawalkar and Roughgarden proved in [2] an upper bound of 2 on the Price of Anarchy of pure Nash equilibria, for bidders with subadditive valuation functions. Feldman et al. proved for this same setting an upper bound of 4 on the Price of Anarchy of mixed Bayes-Nash equilibria (thus, improving on a previous bound of O(log m) from [2]). Finally, Fu, Kleinberg and Lavi [13] showed recently a tight upper bound of 2 on the Price of Anarchy of pure Nash equilibria (when they exist), for arbitrary valuation functions. Hassidim et al. [14] considered simultaneous First-Price auctions; first, they showed that pure Nash equilibria are efficient, when they exist, for arbitrary valuation functions. For fractionally subadditive valuation functions they proved upper bounds of 2 and 4 for the pure and mixed Bayes-Nash Price of Anarchy respectively. For subadditive and arbitrary valuation functions their corresponding upper bounds are O(log m) and O(m). Feldman et al. [11] and Syrgkanis [30] improved the upper bound of the mixed Bayes-Nash Price of Anarchy to respectively: 2, for subadditive valuation functions and e e−1 , for fractionally subadditive ones. Sequential and Greedy Auctions. Very recently, Syrgkanis and Tardos studied in [18] sequential First-and Second-Price auctions, motivated by the practical issue that supply may not be readily available at once. They showed that sequential First-Price auctions are efficient in subgame-perfect equilibrium. In [31] they extended their results in the incomplete information setting. Lucier and Borodin [19] analyzed the social inefficiency at (mixed) Bayes-Nash equilibrium, of combinatorial auctions for multiple distinct goods, incorporating Greedy allocation algorithms (and using appropriate adaptations of "first" and "second" pricing rules). They showed that these auctions have Price of Anarchy fairly comparable to the approximation factors of the greedy allocation algorithms, for the underlying welfare optimization problem. Truthful Mechanism Design. From the mechanism design perspective, Vickrey designed in [33] the first truthful mechanism for auctioning multiple units "in one go", so as to maximize the social welfare. The Vickrey multi-unit auction is computationally efficient for a bounded number of units. Since then, computationally efficient truthful approximation mechanisms for multi-demand bidders were given by Mu'alem and Nisan in [22] and by Dobzinski and Nisan in [7]. These works considered several different classes of valuation functions, much more general than symmetric submodular ones. Very recently, Vöcking gave a randomized universally truthful polynomial-time approximation scheme for bidders with general valuations [34] (a universally truthful mechanism is a probability distribution over deterministic truthful mechanisms), thus almost closing the problem. It is worth noting that in these works, the bids are accessed by the allocation algorithms through polynomially bounded many value queries to the bidders, for specific bundles of items (with the exception being the case of k-minded bidders, who have non-zero value for at most k sizes of bundles). Definitions and Preliminaries We consider auctioning k units of a single good to a set of n bidders, denoted by [n]. Every bidder i ∈ [n] has a private valuation function, defined over the quantity of units he receives, i.e., v i : ({0} ∪ [k]) → + , i = 1, . . . , n, where v i (0) = 0 and each v i is non-decreasing. In this work we consider submodular valuation functions: Definition 1 A valuation function f : ({0}∪[k]) → + is called submodular, if for every x < y, f (x) − f (x − 1) ≥ f (y) − f (y − 1). Proposition 1 Given x, y ∈ [k] with x ≤ y, any submodular valuation function f satisfies f (x)/x ≥ f (y)/y.b i = (b i (1), b i (2), . . . , b i (k)), satisfying b i (1) ≥ b i (2) ≥ · · · ≥ b i (k). Here b i (j) is the declared marginal value of i, for obtaining the j-th unit of the good. Given a bidding configuration b = (b 1 , . . . , b n ), the allocation algorithm produces an allocation x(b) = (x 1 (b), x 2 (b), . . . , x n (b)), as follows: each of the k highest issued marginal bids grants a unit to its issuing bidder. After the allocation is completed, every bidder i pays a uniform price p(b) per received unit, which equals the highest rejected (marginal) bid. That is, if under bidding configuration b, bidder i is allocated x i (b) units in total and the uniform price is p(b), i pays a total of x i (b) · p(b). The utility that i derives under b is u i (b) = v i (x i (b)) − x i (b) · p(b) . Given a bidding profile b, we will denote by β j (b) the j-th lowest winning (maginal) bid, for j = 1, . . . , k, i.e., the lowest winning bid is β 1 (b) and the highest one is β k (b). For submodular bidders, the Uniform Price Auction admits an efficient pure Nash equilibrium; let x * = (x * 1 , ..., x * n ) be an optimal allocation 3 of units to the bidders. Consider the profile where any winner i of at least one unit in x * bids b i = (m i (1), ..., m i (x * i ), 0, . .., 0) and any loser bids the zero vector. It is straightforward to verify that this a Nash equilibrium. However, the strategies of losing bidders in this profile are weakly dominated, as we shall see. Pure Nash equilibria in undominated strategies are known to suffer from a demand reduction effect [1]; bidders may have an incentive to understate their marginal increase for the j-th unit onwards, for some j > 1. This implies that equilibria in undominated strategies are generally inefficient. We show that, despite this effect, the Uniform Price Auction does quite well in approximating the optimal Social Welfare. Incomplete Information. In Section 6 we will move to an incomplete information setting, where each bidder faces uncertainty over the other bidders' valuation functions. In particular, we will assume that every bidder i ∈ [n] obtains his type/valuation function from a finite set V i of valuation functions, through a discrete probability distribution π i : V i → [0, 1], independently of the rest of the biddders; for any particular v ∈ V i we write v ∼ π i to signify that it is drawn randomly from distribution π i . The valuation function of every bidder is private. A valuation profile v = (v 1 , . . . , v n ) ∈ V = × i V i is drawn from a publicly known distribution π = × i π i , π : V → [0, 1]. We write accordingly v ∼ π. Every bidder i knows his own valuation function v i -drawn from V i according to π i , but does not know the valuation function v i drawn by any other bidder i = i. Bidder i may only use his knowledge of π to estimate v −i . Given the publicly known distribution π, the (possibly mixed) strategy of every bidder is a function of his own valuation v i , denoted by B i (v i ). B i maps a valuation function v i ∈ V i to a distribution B i (v i ) = B vi i , over all possible bid vectors (strategies) for i. In this case we will write b i ∼ B vi i , for any particular bid vector b i drawn from this distribution. We also use the notation B v−i −i , to refer to the vector of randomized strategies of bidders other than i, under valuation profile v −i for these bidders. A Bayes-Nash equilibrium (BNE) is a strategy profile B = (B 1 , . . . , B n ) such that, for every bidder i and for every valuation v i , B i (v i ) maximizes the utility of i in expectation, over the distribution of the other bidders' valuation functions w −i given v i and over the distribution of i's own and the other bidders' strategies, B (vi,w−i) . That is, for every pure strategy c i of i: E w−i\|vi E b∼B (v i ,w −i ) u i (b) ≥ E w−i\|vi E b−i∼B w −i u i (c i , b −i ) where we use the notation E v and E w−i\|vi to denote the expectation over the distribution π and over π(·\|v i ) respectively, i.e., given v i (instead of using E v∼π and E w−i∼π(·\|vi) , since the analysis is always in the context of π and π i ). Fix a valuation profile v ∈ V and consider a (mixed) bidding configuration B v , under v. The Social Welfare SW (B v ) under B v is defined in expectation over the bidding profiles chosen by the bidders collectively, from their randomized strategies: SW (B v ) = E b∼B v i v i (x i (b)) The expected Social Welfare in Bayes-Nash Equilibrium is E v∼π [SW (B v )]. The socially optimal assignment under valuation profile v ∈ V will be denoted by x v . The expected optimal social welfare is then E v [SW (x v )], by slight abuse of notation. Under these definitions, we will be studying the Bayes-Nash Price of Anarchy, i.e., the worst case ratio E v [SW (x v )]/E v [SW (B v )] over all possible distributions π and Bayes-Nash equilibria B. As in previous works [5,11], we ensure existence of Bayes-Nash equilibria in our auction format by assuming that bidders have bounded and finite strategy spaces, e.g., derived through discretization. Our bounds on the Bayesian inefficiency hold for sufficiently fine discretizations (see also the discussion in Appendix D of [11]). Undominated Equilibria In this work we consider only bidders with submodular valuation functions, so that m i (1) ≥ ... ≥ m i (k), for every bidder i. As already explained in Section 3, the bidding interface of the Uniform Price Auction requires that each bidder submits a sequence of non-increasing marginal bids b i = (b i (1), . . . , b i (k)), with b i (1) ≥ b i (2) ≥ · · · ≥ b i (k) (see e. g., the related chapters in [16] and [21]). By the auction's definition, it follows that, under any strategy profile b, the uniform price p(b) never exceeds any of the winning (marginal) bids. Lemmas 1 and 2 below state two well known facts about the Uniform Price Auction with submodular bidders (see e.g., [16,21]). For the sake of clarity and completeness, we state them and prove them here to clarify some ambiguities and emphasize their dependence on the requirement that bidders issue non-increasing marginal bids. Their proofs are provided in Appendix A. Lemma 1 For bidders with submodular valuation functions, and for any j ∈ [k], it is a weakly dominated strategy to declare a bid b i (j) with b i (j) > m i (j). An assumption that is recently being used in various other auction formats is that bidders do not overbid their value for a set of goods (e.g., [2,4,5,27]). The justification for this is that such strategies may be dominated by other strategies and hence should be avoided. In our context, this would mean that for any number of r units, r j=1 b i (j) ≤ v i (r) = r j=1 m i (j) . We note here that Lemma 1 shows that a weakly undominated strategy in our setting implies a stricter notion of conservative behavior than the usual "no-overbidding" assumption of the literature. To distinguish from the usual no-overbidding assumption, we call a bidder i who does not bid beyond m i (j) for any j ∈ [k] conservative in marginal values. Lemma 2 In an undominated strategy, a bidder with a submodular valuation, never declares a bid b i (1) = v i (1). We now give a characterization of a subset of undominated pure Nash equilibria, which will be utilized for the analysis of their social inefficiency in Section 5. Lemma 3 Let b be a pure Nash equilibrium strategy profile of the Uniform Price Auction in undominated strategies for submodular bidders, with uniform price p(b). There always exists a pure Nash equilibrium b in undominated strategies, satisfying x(b ) = x(b) and: 1. b i (x) = m i (x), for every bidder i and every x ≤ x i (b). 2. p(b ) ≤ p(b) and p(b ) is either 0 or equal to v i (1) for some bidder i. Proof. Since b is an equilibrium in undominated strategies, for every bidder i we have b i (1) = v i (1). Also, b i (x) ≤ m i (x) for every x ≥ 2. For every winning marginal bid b i (x) of any bidder i and for x = 2, . . . , x i (b), we may set b i (x) = m i (x) without altering neither the uniform price nor the units obtained by the winners under b. This is because the losing bids remain unchanged whereas the winning bids only increase, hence the price is the same as before. Finally, the new profile is easily shown to be an equilibrium simply because we started with an equilibrium b. Thus the first condition b i (x) = m i (x) for x ≤ x i (b) can be satisfied for every winner i. For the second condition, consider an equilibrium b satisfying the first condition. Assume that b is such that p(b) = b i (j), for some j ≥ 1. If j = 1, then it must be x i (b) = 0 and then p(b) = b i (1) = v i (1), because b is undominated. Consider the case where j ≥ 2. Then x i (b) ≥ 1. If the value b i (j) = p(b) is unique in b, then either p(b) = 0, or i could lower all his marginal bids for the j-th unit and onwards to 0. This way, he would strictly increase his utility, by lowering the uniform price to the next highest marginal bid, which contradicts that b is a PNE. Thus we may assume that more than one highest losing bids of the same value exist, that determine the price. If none of them are equal to v i (1) for some i, we can zero out the bids of all bidders who have the highest losing bid (from that bid onwards) and again obtain a bidding configuration b with a price p(b ) < p(b). The new vector b is still an undominated PNE. The utilities of all winners have now increased, but none of them may increase his utility more by deviating unilaterally, because we have not altered any of the winning bids. It is the last winning bid that will determine the new uniform price in case of any bidder trying to win an extra unit by deviating. But since there was no incentive to do such a deviation under b the same is true for b too. Hence, we have managed to reduce the price and have some bidders zero out their losing bids. We can repeat this procedure for the configuration b , until we reach a a configuration satisfying the second property. 2 Inefficiency of Undominated Pure Nash Equilibria We develop a welfare guarantee for pure Nash equilibria in undominated strategies, of the Uniform Price Auction, for bidders with submodular valuation functions. Recall that, given a configuration b, we denote the (k highest) winning bids by β j (b), j = 1, . . . , k, so that β 1 (b) ≤ β 2 (b) ≤ · · · ≤ β k (b). We extend this notation to partial configurations b −i for any bidder i ∈ [n]. Let x * = (x * 1 , . . . , x * n ) be a socially optimal allocation and b be a bidding configuration corresponding to an undominated pure Nash equilibrium of the auction. Under b the allocation is x(b) = (x 1 (b), . . . , x n (b)). To simplify notation we use x for x(b) and x i for x i (b). Given any allocation x, define the set W(x) = {i\|x i ≥ 1} to be the subset of winners, i.e. bidders that receive at least one item unit. We also define 3 additional sets, W 0 (x), W 1 (x), W 2 (x), all with reference to x * as follows: W 0 (x) = {i ∈ W(x * )\|x i ≥ x * i }, W 1 (x) = {i ∈ W(x * )\|x i < x * i }, and W 2 (x) = W(x) \ W 0 (x) ∪ W 1 (x) . We note that given any assignment x = x(b) for a pure Nash equilibrium configuration b, W 0 (x) ∪ W 1 (x) ∪ W 2 (x) is a partition of W(x) , given the assumption of undominated strategies about b; every winner i ∈ W(x * ) will still be a winner also under b, because of specifying his v i (1) truthfully (by Lemma 2), thus obtaining at least one unit. Hence none of W 0 (x), W 1 (x), W 2 (x) may contain non-winning bidders of x. First we present a general upper bound on the Price of Anarchy for undominated pure Nash equilibria. Lemma 4 Let b denote any undominated pure Nash equilibrium of a Uniform Price Auction for k units and x * be an assignment that maximizes the social welfare. The Price of Anarchy is: P oA ≤ sup b max i:x * i −xi(b)>0   vi(x * i ) ·   v i x i (b) + x * i −xi(b) j=1 β j (b)   −1   (1) The proof of the lemma is given in Appendix B. We can now present our constant bound on the Price of Anarchy: Theorem 1 The Uniform Price Auction recovers in undominated pure Nash equilibrium a fraction of at least 1 − e −1 of the optimal Social Welfare, for multi-demand bidders with symmetric submodular valuation functions. Proof. Without loss of generality, it suffices to upper bound the social inefficiency of undominated equilibria b satisfying the properties of Lemma 3. Let p(b) be the uniform price payed under equilibrium b, i.e. the value of the highest losing (marginal) bid. In order to estimate a lower bound on the Social Welfare of b, we consider possible deviations of bidders i ∈ W 1 (x(b)). We may assume that W 1 (x(b)) = ∅ for, otherwise, W 0 (x(b)) = W(x * ) and b is socially optimal, i.e. SW (b) = SW (x * ). For every bidder i ∈ W 1 (x(b)), define r i (b) = x * i − x i (b) . For every bidder i ∈ W 1 (x(b)) and for every value j = 1, . . . , r i (b), there exists a deviation that will grant him j more units, additionally to the ones he already wins under b; this is justified by the fact that at equilibrium b, all bidders play marginal bids equal to their actual marginal values, or 0 (as prescribed by the properties given in Lemma 3). Since the optimal assignment results from a simple sorting of the actual marginal values, every winner i ∈ W(x * ) may feasibly deviate under b, so as to obtain any total number of units between x i and x * i . A deviation of i ∈ W 1 (x(b)) for obtaining any number of j = 1, . . . , r i (b) additional units will raise the uniform price to exactly β j (b) and cannot be profitable for i, i.e.: v i (x i (b) + j) − (x i (b) + j) · β j (b) ≤ v i (x i (b)) − x i (b) · p(b) To simplify notation, we use hereafter x i for x i (b), p for p(b), r i for r i (b), and β j for β j (b), (always with respect to an undominated pure Nash equilibrium b). Then we deduce that for every i ∈ W(x * ): β j ≥ 1 j + x i · v i (x i + j) − v i (x i ) + x i · p , for j = 1, . . . , r i(2) We can now proceed to upper bound (1) from Lemma 4, using (2) as follows: v i (x i ) + ri j=1 β j ≥ v i (x i ) + ri j=1 1 j + x i · v i (x i + j) − v i (x i ) (3) = v i (x i ) + ri j=1 j j + x i · v i (x i + j) − v i (x i ) j ≥ v i (x i ) + v i (x * i ) − v i (x i ) x * i − x i · ri j=1 j j + x i (4) = v i (x i ) + v i (x * i ) − v i (x i ) x * i − x i ·   x * i − x i − x i · ri j=1 1 j + x i   = v i (x * i ) − v i (x * i ) − v i (x i ) x * i − x i · x i · ri j=1 1 j + x i (5) ≥   1 − x i x * i · ri j=1 1 j + x i   · v i (x * i ) ≥ 1 − x i x * i · x * i xi 1 y dy · v i (x * i )(6)= 1 + x i x * i · ln x i x * i · v i (x * i ) ≥ (1 − e −1 ) · v i (x * i )(7) (3) occurs by substitution of β j from (2) and after dropping the x i · p ≥ 0 term. (4) follows by submodularity of the valuation functions, particularly that vi(xi+j)−vi(xi) j ≥ vi(x * i )−vi(xi) x * i −xi , for any j = 1, . . . , r i where r i = x * i −x i . For (6) we used vi(x * i )−vi(xi) x * i −xi ≤ vi(x * i ) x * i , given v i (0) = 0; we bounded the sum of harmonic terms with the integral, using n k=m f (k) ≤ n m−1 f (x)dx, for a monotonically decreasing positive function. We obtain the final result by minimizing f (y) = 1 + y ln y over (0, 1) for y = e −1 . The claimed bound for the Price of Anarchy follows by Lemma 4. 2 We will produce an almost matching lower bound for the result of theorem 1, which holds for any number of units k ≥ 9. We pause here to discuss first three simple tight examples for k = 2, 3, 4 units. Examples. We give a detailed example for k = 3 first. We show a simple lower bound of 18 13 . Consider 3 item units and 3 bidders, with valuation functions v 1 (x) = x, v 2 (x) = 2 3 , v 3 (x) = 1 2 . The(b ) = v 1 (x i (b )) − x i (b ) × 2 3 = 3 − 3 × 2 3 = 1 = u 1 (b) . The Price of Anarchy in this example is 18 13 . For k = 2 and k = 4 our examples follow a similar pattern to the one discussed for k = 3. We can take n = k = 2 bidders with valuation functions v 1 (x) = x and v 2 (x) = 1 2 . This instance yields a Price of Anarchy at least 4 3 ; in the social optimum, bidder 1 gets both units, while in the (undominated) equilibrium profile, b 1 = (1, 0) Notice that our lower bound for k = 4 is smaller than for k = 3. None the less, all these examples can be shown to provide tight lower bounds for their corresponding cases k = 2, 3, 4, by usage of simple arguments and explicit treatment of the left-hand side of (6). For k = 2, our argument is that the worst-case instance occurs when there is a single winner in the social optimum; otherwise, if there are 2 winners, they also remain winners in undominated pure Nash equilibrium (due to Lemma 2) and the Price of Anarchy is 1. In the worst-case instance though, the bidder being a winner (of 2 units) in the social optimum remains a winner (of exactly 1 unit) in the undominated pure Nash equilibrium. Taking x i = 1, x * i = 2, r i = 1 in the left-hand side of (6), sufficies to obtain 3 4 v i (x * i ). For the case of k = 3 the reasoning is similar. There have to be strictly less than 3 winners in the social optimum for, otherwise, the allocation coincides with that of an undominated pure Nash equilibrium. Now, if there are 2 winners, one of them obtains only 1 unit (out of k = 3) in both, socially optimal allocation and undominated equilibrium allocation. Thus, the social inefficiency is due to the other winner losing one unit in equilibrium (of the 2 that he obtains in the social optimum). By our experience with k = 2(and by Lemma 4), this example cannot have Price of Anarchy more than 4/3. Thus, we may assume that there exists a single winner in the social optimum. To achieve maximum welfare "damage" in undominated pure Nash equilibrium, this single winner of 3 units loses 2 of them in equilibrium and we apply (6) appropriately, to obtain an upper bound of 18/13. For k = 4 the reasoning uses our experience from both previous cases. We exclude instances with 4, 3 and 2 winners in the social optimum, as they cannot have Price of Anarchy more than 1, 4/3, 18/13 respectively. For maximum welfare damage, we assume that the single winner loses 3 out of 4 units in undominated pure Nash equilibrium. , b 2 = ( 1 2 , 0) (and SW (b) = 3 2 ). For k = 4, we take n = k = 4 bidders with valuation functions v 1 (x) = x, v 2 (x) = 1 2 , v 3 (x) = 2 3 , v 4 (x) = The following more general lower bound is valid for at least k ≥ 9 units. For the remaining values of k = 5, 6, 7, 8, we do not have any tighter upper and lower bounds. Theorem 2 For any k ≥ 9, the Uniform Price Auction recovers in undominated pure Nash equilibrium at most a factor (1 − e −1 + 2 k ) of the optimal social welfare, even for 2 submodular bidders. Proof. Consider k ≥ 9 units and 2 bidders. For q = e −1 · k − 1 (notice that q ≥ 1) define the valuation functions to be: v 1 (x) = x and v 2 (x) = x − q · (H k − H k−x ) x ≤ k − q k − q · (1 + H k − H q ) x > k − q where H m is the m-th harmonic number. Notice that the marginal values of bidder 2 are equal to 0 for x > k − q. It can be verified that v 2 is symmetric submodular in x: v 2 (x) = x − q · H k − H k−x = x j=1 1 − q k − j + 1 = x j=1 r − j + 1 k − j + 1 where r = k − q. Then r−j+1 k−j+1 ≤ r−j+2 k−j+2 = r−(j−1)+1 k−(j−1)+1 , thus v 2 (x) − v 2 (x − 1) ≤ v 2 (x − 1) − v 2 (x − 2), for x ≤ k − q; for x > k − q, v 2 (x) = v 2 (x − 1), thus v 2 is submodular. For the optimal social welfare we grant all units to bidder 1, i.e. x = (k, 0, . . . , 0) and obtain a total welfare SW (x * ) = k. For the equilibrium configuration b we set: b 1 (j) = 1, for j ≤ q 0, for j > q b 2 (j) = r−j+1 k−j+1 , for j ≤ r = k − q 0, for j > r Thus, under b, q units are obtained by bidder 1 and k − q units by bidder 2. We show that b is a pure Nash equilibrium. Notice that bidder 2 is essentially truthful in this profile and may not increase his bids further so as to obtain another unit (given that he plays undominated strategies). On the other hand, the uniform price is 0 in this setting, so bidder 2 does obtain the maximum of his utility for the won units. Bidder 1 also pays the uniform price of 0, so he does not have incentive to drop any of his units. Should bidder 1 try to retain any j ≤ r of the r = k − q units held by bidder 2, the uniform price would become j k−r+j and bidder 1 will hold a total of k − r + j units. The marginal gain from bidder 1 obtaining the extra j units is cancelled out by a total payment equal to j; thus bidder 1 does not have incentive to deviate under b. For the social welfare of b we have: SW (b) = v 1 (q) + v 2 (r) = q + r − q · H k − H q = k − q · H k − H q = k · 1 − q k · (H k − H q ) Then, the Price of Anarchy is at least: k k · 1 − q k · (H k − H q ) = 1 − q k · H k − H q −1 ≥ 1 − e −1 · k − 2 k · k q+1 1 y dy −1 = 1 − e −1 · k − 2 k · ln k e −1 k − 1 + 1 −1 ≥ 1 − e −1 + 2 k −1 where we used H k −H q = k r=q+1 1 r ≥ k+1 q+1 1 y dy ≥ k q+1 1 y dy, for monotonically decreasing positive functions; the final derivation follows by q + 1 ≤ e −1 · k and e −1 k − 1 + 1 ≥ e −1 k 2 Bayes-Nash Inefficiency with Undominated Support In this section we investigate the social inefficiency of (mixed) Bayes-Nash equilibria for bidders with submodular valuation functions. Just as in the case of pure equilibria we focused on undominated strategies, here we will focus on mixed Bayes-Nash equilibria that are supported by pure undominated strategies. We wish to note, however, that the assumption of undominated support is only marginally restrictive for the main (social inefficiency) result described in this section; it guarantees the properties given by Lemmas 1 and 2, that allow us to prove a tighter inefficiency bound. In the end of this section we discuss how our analysis leads to a similar (slightly worse) bound, under the standard assumption of no-overbidding used, e.g., in [5,2,11]. Following [5,2], to ensure the existence of mixed Bayes-Nash equilibria, we make the assumption of a finite bidding space for bidders under sufficiently fine discretization. We claim that such mixed Bayes-Nash equilibria supported by undominated pure strategies exist for the case of bidders with submodular valuation functions. Indeed, consider a Bayesian game where the strategy space of each player is bounded (e.g. of the form [0, U ], where U is a sufficiently large upper bound on the values of all bidders) and finite, through some sufficiently fine discretization. We first claim that in our setting there always exists a Bayes-Nash equilibrium where all strategies used in its support are not weakly dominated. Then we will claim that these strategies are conservative w.r.t. marginal bids. To argue about these statements, we use two well known facts from game theory. The first one is that a Bayes-Nash equilibrium B can be seen as a Nash equilibrium of a complete information game (see e.g. [26][Chapter 9]), where the set of players is the set of pairs (i, v i ) for every i = 1, ..., n and v i ∈ V i , and the strategy space of player (i, v i ) is the same as the strategy space of i in the Bayesian game. For a given mixed strategy profile B in this game, the utility function of player (i, v i ), denoted by u vi i (B), is: u vi i (B) = E v−i\|vi E b∼B (v i ,v −i ) u i (b) = v−i∈V−i π(v −i )E b∼B (v i ,v −i ) u i (b) Note that the utility of a player (i, v i ) does not depend on the action of other pairs that involve i, (i.e., on the other types of player i). Since this complete-information game is a finite game it possesses a mixed Nash equilibrium. It is well known that in any finite game there always exists a mixed equilibrium where no weakly dominated action is contained in the support of each player's strategy (see [26] [Section 4.4]). Putting everything together, we have that there always exists a Bayes-Nash equilibrium where every strategy used in its support is not weakly dominated. It is an easy exercise to verify the validity of Lemma 1 for a sufficiently fine discretization of the strategy space and that Lemma 2 holds, for any such discretization that does not exclude the actual marginal values from the bidders' strategy space. We introduce some auxiliary notation for the analysis that follows. For any valuation profile v ∈ V let x v = (x v 1 , . . . , x v n ) denote the socially optimal assignment. For any particular bidder i ∈ [n] let U i ⊆ V denote the subset of valuation profiles v ∈ V where x v i ≥ 1, i.e., U i = {v ∈ V\|x v i ≥ 1} ; these are the profiles under which i is a "social optimum winner". Accordingly, we let W v denote the subset of "social optimum winners" in valuation profile v ∈ V. Given any (pure) bidding profile b, we use the "operator" β j (b) here as well, to denote the j-th lowest winning bid in b, as in Section 5. The following Lemma will serve the purpose of lower bounding the Social Welfare of a profile b by a sum of the winning bids (much like inequalities (19) and (20) along with Lemma 1 did, in the proof of Lemma 4 in the previous section). Lemma 5 For a valuation profile v, let b denote an arbitrary pure bidding profile in undominated strategies, p(b) be the uniform price under b and let x * be the efficient (socially optimal) assignment of k units to ≤ k winning bidders w.r.t. v. Fix an arbitrary ordering of the winning bidders under x * and define t i = x * i 2 , i = 1, . . . , . Then: 1≤i≤ t i β ti (b −i ) ≤ p(b) + k − 1 k SW (b)(8) To ease our way towards the main result of this section, we defer the technical proof of this Lemma to the end of our exposition, along with a discussion on how to replace the undominated support assumption with the no-overbidding assumption. The following Lemma facilitates the expression of BNE conditions regarding unilateral deviations, and has been proved in a different form and under a different context (for simultaneous single-unit auctions with combinatorial bidders) also in [5,2]. We provide its proof here for completeness. u i (m [j] i , b −i ) ≥ v i (j) − j · β j (b −i ). The proof of this lemma for our setting is given in Appendix C. Now we can show the main result of this section: v = (v i , v −i ) ∈ V. For t v i = x v i 2 , and for any valuation profile w −i ∈ V −i and strategy b −i ∼ B w−i −i , we apply Lemma 6. Then, we take the expectation over the randomized strategies of the other bidders and, subsequently, over all valuation profiles w −i ∈ V −i , to obtain: E w−i\|vi E b−i∼B w −i −i [u i (m [t v i ] i , b −i )] ≥ v i (t v i ) − t v i · E w−i\|vi E b−i∼B w −i −i [β t v i (b −i )] ≥ v i (x v i ) 2 − E w E b∼B w t v i · β t v i (b −i )(9) The second inequality is justified as follows. By independence of the distributions {π i \| i ∈ [n]}, we have that for any i ∈ [n] and any v i ∈ V i : w−i π(w −i \|v i ) = w−i π(w −i ) = w−i π(w −i ) wi π(w i ) = (wi,w−i) π(w −i \|w i )π i (w i ) = 1 = w π(w) Also, by submodularity (Proposition 1) and monotonicity of valuation functions : v i (t v i ) = v i ( x v i 2 ) ≥ 1 2 v i (x v i ). Because under BNE B, bidder i does not have an incentive to deviate, we have: E w−i\|vi E b∼B (v i ,w −i ) [u i (b)] ≥ E w−i\|vi E b−i∼B w −i −i [u i (m [t v i ] i , b −i )] Thus: E w−i\|vi E b∼B (v i ,w −i ) [u i (b)] + E w E b∼B w t v i · β t v i (b −i ) ≥ v i (x v i ) 2 We take expectation of both sides over the distribution of v ∈ V and summing over all bidders yields the final expression: i v∈V π(v) · E w−i\|vi E b∼B (v i ,w −i ) u i (b) + i v∈V π(v) · E w E b∼B w t v i · β t v i (b −i ) ≥ i v∈V π(v) · v i (x v i ) 2 = v∈V π(v) i∈W v v i (x v i ) 2 = 1 2 E v [SW (x v )](10) The last equality holds since it is enough to sum over i ∈ W v , to compute the welfare produced at the optimal assignment with respect to v. We show in Appendix C that the first summand of the left-hand side of (10) satisfies: i v∈U i π(v) · E w−i\|vi E b∼B (v i ,w −i ) u i (b) = E v E b∼B v i u i (b)(11) Similarly, we have for the second summand on the left-hand side of (10): i v∈U i π(v) · E w E b∼B w t v i · β t v i (b −i ) = i v∈U i π(v) w∈V π(w) · E b∼B w t v i · β t v i (b −i ) = v∈V π(v) i∈W v w∈V π(w) · E b∼B w t v i · β t v i (b −i ) = v∈V π(v) w∈V π(w) · E b∼B w i∈W v t v i · β t v i (b −i ) = E v E b∼B v i∈W v t v i · β t v i (b −i )(12) Note that in the first term above, we sum only over v ∈ U i , since for v ∈ U i , t v i = 0. By (27), (12) and (10), we obtain: E v E b∼B v i u i (b) + i∈W v t v i · β t v i (b −i ) ≥ 1 2 E v [SW (x v )](13) To finish the proof, we substitute the second sum inside the expectation by its upper bound as given by Lemma 5 in (8). Notice that p(b) appearing in (8) is absorbed by the payment appearing in the utility u i (b) of at least one bidder. Thus we obtain: E v E b∼B v i v i x i (b) + k − 1 k SW (b) ≥ 1 2 E v [SW (x v )] which essentially concludes the proof, by i v i x i (b) = SW (b). 2 To complete our arguments for the proof of Theorem 3, we prove Lemma 5. Subsequently, we comment on how our arguments can be adjusted for the case of Bayes-Nash equilibria supported by general no-overbidding strategies (i.e., not restricted to undominated ones), to yield an upper bound of 4 for the Price of Anarchy. Proof of Lemma 5. For the winning bidders i = 1, . . . , we have x * i ≥ 1. Define ψ i = j≤i x * j . First, we prove by induction on i that: 1≤j≤i t j β tj (b −j ) ≤ β t1 (b −1 ) + 1≤j≤ψi−1 β j (b).(14) For the basic step of the induction, consider i = 1; then: t 1 β t1 (b −1 ) ≤ β t1 (b −1 ) + (t 1 − 1)β t1 (b) (15) ≤ β t1 (b −1 ) + 2t1−2 j=t1 β j (b) = β t1 (b −1 ) + j≤ψ1−1 β j (b) Assuming (14) holds for i > 1, we show it remains true for i + 1. We have: j≤i+1 t j β j (b −j ) = j≤i t j β j (b −j ) + t i+1 β ti+1 (b −(i+1) ) ≤ β t1 (b −1 ) + j≤ψi−1 β j (b) + t i+1 β ti+1 (b) ≤ β t1 (b −1 ) + j≤ψi+1−1 β j (b)(16) where (16) is justified as follows: for any value of ψ i ≥ 1, we have β ti+1 (b) ≤ β j (b), for all j = ψ i − 1 + x * i+1 2 , . . . , ψ i − 1 + x * i+1 (a total of at least x * i+1 2 = t i+1 inequalities). By ψ i+1 = ψ i + x * i+1 , we obtain (16) and, thus, (14). Setting i = , thus, ψ = k in (14), we have: 1≤i≤ t i β ti (b −i ) ≤ β t1 (b −1 ) + k−1 j=1 β j (b) ≤ β t1 (b −1 ) + SW (b) − β k (b)(17) For every bidder i with x * i ≥ 1, it must be also x i (b) ≥ 1, because the profile b consists of undominated strategies, thus, b i (1) = m i (1) = v i (1). Because β k (b) = max i,j b i (j), by submodularity of valuation functions and by no-overbidding with respect to marginal bids, there exists a bidder i 1 such that b i1 (1) = v i1 (1) = m i1 (1) = β k (b). Then, β k (b) = m i1 (1) is also the largest marginal value contributed to SW (b) and m i (1) ≥ SW (b)/k. Moreover, for every bidder i, it must hold that b i (1) = m i (1) = v i (1) ≥ β x * i (b), i.e., b i (1) is at least the x * i -th bid in β 1 (b), . . . , β k (b); otherwise, at least one bid in b is beyond the marginal value of some bidder, which also contradicts b consisting of undominated strategies. Then, for any bidder i with x * i ≥ 1 we have that: β ti (b −i ) ≤ p(b), because at least x * i ≥ t i non-winning marginal bids in b (including p(b)) become the lowest winning in b −i . This yields β t1 (b −1 ) ≤ p(b) and we obtain (8) from (17). We used our assumption of undominated support essentially only in the analysis that follows (17). If we replace this assumption with the standard no-overbidding assumption, we can continue our analysis from (17), by noticing that β t1 (b −1 ) ≤ β t1 (b −1 ) ≤ β k (b). Thus, (8) can be replaced by: 1≤i≤ t i β ti (b −i ) ≤ SW (b). We can use this latter upper bound in (13), along with i u i (b) ≤ i v i (b) = SW (b), to obtain an upper bound of 4 on the Bayes-Nash Price of Anarchy. Corollary 1 The mixed Bayes-Nash Price of Anarchy of the Uniform Price Auction for nonoverbidding bidders with submodular valuation functions is at most 4. Appendix B: Proof of Lemma 4 Let b be any pure Nash equilibrium configuration in undominated strategies. For simplicity we use x i for x i (b) and β j for β j (b), j = 1, . . . , k. For SW (b) we have: SW (b) = i∈W0(x) v i (x i ) + i∈W1(x) v i (x i ) + i∈W2(x) v i (x i )(18) By definition of W 0 , we can write the first term of (18), i∈W0(x) v i (x i ), as: i∈W0(x) v i (x * i ) + v i (x i ) − v i (x * i ) = i∈W0(x)   v i (x * i ) + xi j=1+x * i m i (j)   ≥ i∈W0(x)   v i (x * i ) + xi j=1+x * i b i (j)  (19) The last inequality is due to the fact that b is an undominated pure Nash equilibrium, thus, Lemma 1 applies. For the third term of (18), we have similarly: i∈W2(x) v i (x i ) = i∈W2(x) xi j=1 m i (j) ≥ i∈W2(x) xi j=1 b i (j)(20) Substituting (19) and (20) in (18), we obtain: SW (b) ≥ i∈W0(x)   v i (x * i ) + xi j=1+x * i b i (j)   + i∈W1(x) v i (x i ) + i∈W2(x) xi j=1 b i (j) Observe that, for every unit missed under b by any bidder i ∈ W(x * ) ∩ W 1 (x), there exists a bidder i ∈ W 0 (x) ∪ W 2 (x) that obtains this unit. If i missed x * i − x i > 0 units in b, there are at least as many bids issued by bidders in W 0 (x) ∪ W 2 (x), that won collectively these units. These bids sum up to at least x * i −xi j=1 β j , i.e. the sum of the x * i − x i lowest winning bids in b. I.e.: i∈W0(x) xi j=1+x * i b i (j) + i∈W2(x) xi j=1 b i (j) ≥ i∈W1(x) x * i −xi j=1 β j(21) Using (21) in the last lower bounding expression of SW (b), we obtain: SW (b) ≥ i∈W0(x) v i (x * i ) + i∈W1(x) v i (x i ) + x * i −xi j=1 β j(22) Finally, we notice that W 0 (x) and W 1 (x) are a partition of W(x * ), thus: SW (b * ) = i∈W0(x) v i (x * i ) + i∈W1(x) v i (x * i )(23) By (22) and (23) we obtain (1). 2 Appendix C: Omitted Proofs from Section 6 Proof of Lemma 6 Fix any player i. The statement trivially holds for j = 0, hence consider j ≥ 1. For any bidding configuration b −i , we let β 1 (b −i ) ≤ · · · ≤ β k (b −i ) denote the winning bids, if i was not present, in non-decreasing order. Fix also the index j (1 ≤ j ≤ k), and using the submodular valuation function v i of i, define the bidding vector m i , b −i ). Obviously s ≤ j. Let p denote the price that he pays for these units. In the case that s = j, p = β j (b −i ), and the statement of the Lemma trivially holds. In the case that s < j, we have that p = max{m i (s + 1), β s (b −i )}. This is because the highest losing bid may be either the next bid of bidder i (i.e., m i (s + 1)), or the highest losing bid in b −i , which is β s (b −i ). This implies that p ≤ β j (b −i ). Thus we can derive the following bound on the utility of bidder i: u i (m [j] i , b −i ) ≥ v i (s) − s · β j (b −i ) ≥ v i (s) − s · β j (b −i ) + j−s r=1 (m i (s + r) − β j (b −i )) = v i (s) + j−s r=1 m i (s + r) − j · β j (b −i ) = v i (j) − j · β j (b −i ) The second inequality above holds because m i (s + r) ≤ m i (s + 1) ≤ β j (b −i ) for all marginal values beyond the s-th one. This completes the proof. Omitted Part of Proof of Theorem 3 Consider the first of the two summands of (10). We explain the derivations below. i v∈V π(v)E w−i\|vi E b∼B (v i ,w −i ) u i (b)(24)= i v−i∈V−i π(v −i \|v i ) vi∈Vi π i (v i ) w−i∈V−i π(w −i \|v i )E b∼B (v i ,w −i ) u i (b) = i vi∈Vi π i (v i ) v−i∈V−i π(v −i \|v i ) w−i∈V−i π(w −i \|v i )E b∼B (v i ,w −i ) u i (b) = i vi∈Vi π i (v i ) w−i∈V−i π(w −i \|v i )E b∼B (v i ,w −i ) u i (b)(25)= v∈V π(v) i E b∼B (v i ,w −i ) u i (b)(26)= v∈V π(v)E b∼B v i u i (b) = E v E b∼B v i u i (b)(27) To obtain the equality following (24), we analyze v∈V π(v) to: v−i∈V−i π(v −i ) vi∈Vi π i (v i ) = (vi,v−i) π(v −i \|v i )π(v i ) For (25), it suffices to move v−i∈V−i π(v −i \|v i ) all the way to the right and observe that it equals 1 by independence of the valuation distributions. Finally, we obtain ((26) by condensing: i vi∈Vi π i (v i ) w−i∈V−i π(w −i \|v i ) into v∈V π(v) and moving i to the right, to sum over the expected valuations of bidders for every valuation profile v ∈ V. Any non-decreasing valuation functionv : ({0} ∪ [k]) → R + with v(0) = 0 can be specified also as a vector of marginal values (m(1), m(2), . . . , m(k)), where m(j) = v(j) − v(j − 1).If v is submodular, then m(1) ≥ m(2) ≥ · · · ≥ m(k). We write m i (·) for the marginal value function of bidder i.The Uniform Price Auction The standard Uniform Price Auction format requires that each bidder i declares his whole valuation curve, by submitting a vector b i of marginal bids, socially optimal welfare is 3, where bidder 1 gets all the units. Consider the pure Nash equilibrium configuration b 1 = (1, 0, 0), b 2 = ( 2 3 , 0, 0) and b 3 = ( 1 2 , 0, 0), for j = 1, 2, 3. None of bidders 2 and 3 have incentive to change their bid, as their utility cannot improve. Should bidder 1 raise b 1 (2) = 0 to b 1 (2) > 1 2 , he would win one more unit additionally to x 1 (b) = 1, but pay x 1 (b ) × 1 2 = 1, thus not improving his utility. If bidder 1 would raise b 1 (2) = 0 to b 1 (2) > 2 3 and b 1 (3) = 0 to b 1 (3) > 1 2 , he would still not improve his utility; it would be u 1 3 4 . 4In the social optimum we obtain welfare 4, by giving all units to bidder 1. For an undominated equilibrium profile, take b 1 = (1, 0, 0, 0), b 2 = ( 1 2 , 0, 0, 0), b 3 = ( 2 3 , 0, 0, 0), b 4 = ( 3 4 , 0, 0, 0). Then, SW (b) = 35/12 and this gives a lower bound of 48/35 for the Price of Anarchy. Lemma 6 6For every bidder i ∈ [n] with submodular valuation v i define the bidding vector m [j] i = (m i (1), m i (2), . . . , m i (j), 0, 0, . . . , 0) . For any conservative bidding profile b −i , and for any number of units j ∈ {0, 1, . . . , k}: Theorem 3 3The Price of Anarchy of Bayes-Nash Equilibria with undominated support in Uniform Price Auctions for bidders with submodular valuation functions is at most 4 − 2 k . Proof. Consider a Bayes-Nash equilibrium B. Fix a bidder i and any valuation profile i = (m i (1), . . . , m i (j), 0, . . . , 0). Assume that bidder i has won s units in the configuration (m [j] This improves over a bound of O(log k) from[20] and over a bound of 4e e−1 , shown subsequently in[32]. It is known that for submodular valuation functions on identical units, the allocation algorithm of the Uniform Price Auction produces an optimal allocation when bidders bid truthfully. This property does not hold in the case of non-identical items, where only a 2-approximation is achieved[17]. Appendix A: Omitted Proofs from Section 4 Proof ofLemma 1Fix any bidder i with a submodular valuation function v i and consider a bidding vector b i = (b i (1), . . . , b i (k)) of i, where b i (1) ≥ b i (2) ≥ ... ≥ b i (k), as required by the bidding rule of the auction. Suppose that for some j, bidder i overbids the marginal value for the j-th unit, i.e., b i (j) > m i (j). We will construct a bidding vector b i and we will show that b i weakly dominates b i . We define b i as follows: b i (r) = b i (r) for any r ≤ j − 1, and b i (r) = m i (r) for every r ≥ j. Note that this is a valid bidding vector for the auction (this holds becauseand the inequality is strict for at least one vector b −i .For a configuration b −i of the other bidders, let p(b) denote the uniform price under b = (b i , b −i ). We start first, with configurations b −i for which j > x i (b). In this case, if bidder i plays according to b i , he will retain at least the same utility, because b i (j) does not grant him any unit and neither does m i (j). Hence, he will keep winning under b i the same number of units and if his j-th bid was the price-setting bid, the price may even decrease and he will be strictly better. Consider now configurations b −i for which j ≤ x i (b). We examine the following subcases:). Then, by playing b i , bidder i will still win the same number of units as before. The price under (b i , b −i ) will either remain the same or may decrease in the case that the price-setting bid was an overbid by bidder i.). In this case, various scenarios may occur depending on the possible configurations for b −i , and on the possible appearance of ties. First, by switching to b i , bidder i will either win the same number of units as before, or he may win less if some tie is resolved against him. In the cases that he wins the same number of units as in b, it is easy to see that the price has either remained the same or it may even have fallen, e.g., if the price-setting bid in b was b i (x i (b) + 1) > m i (x i (b) + 1), which is reduced in b i to m i (x i (b) + 1). Hence bidder i has at least the same utility as before. In the cases where bidder i wins less units, the only possibility is that the price has remained the same and it is only because of ties that i lost some of his previously won units. But then for the units that he lost, their marginal value is the same as the price, hence bidder i simply had zero utility for them in b. Thus he will still have the same utility under b i as before.(III): p(b) > m i (x i (b)). Then in b, i overpays his marginal value for at least one won unit; in this case bidder i can switch to b i and strictly increase his utility. To see this, note that under (b i , b −i ) bidder i will still win the units that give him nonnegative utility (i.e., have marginal value at least p(b)), and he will lose only units that he could not afford anyway.Conclusively, overbidding any marginal value is a weakly dominated strategy for every bidder i.2Proof of Lemma 2By Lemma 1, undominated strategies are conservative w.r.t. marginal bids, thus no bidder may exaggerate his bid for the first unit. Let b i denote such a bidding vector for bidder i, where b i (1) < m i (1). Let b i denote the bidding vector where b i (1) = m i (1) and b i (j) = b i (j), for j = 2, . . . , k. For any configuration b −i due to all other bidders, we show that:, bidder i will maintain his allocation and his utility by increasing b i (1) to m i (1) = b i (1). 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[ "Generalized Efimov effect in one dimension", "Generalized Efimov effect in one dimension" ]	[ "Sergej Moroz \nDepartment of Physics\nUniversity of Colorado\n80309BoulderColoradoUSA\n\nCenter for Theory of Quantum Matter\nUniversity of Colorado\n80309BoulderColoradoUSA\n", "José P D'incao \nDepartment of Physics\nUniversity of Colorado\n80309BoulderColoradoUSA\n\nJILA\nUniversity of Colorado\nNIST\n80309-0440BoulderColoradoUSA\n", "Dmitry S Petrov \nLPTMS\nCNRS\nUniv. Paris-Sud\nUniversité Paris-Saclay\n91405OrsayFrance\n" ]	[ "Department of Physics\nUniversity of Colorado\n80309BoulderColoradoUSA", "Center for Theory of Quantum Matter\nUniversity of Colorado\n80309BoulderColoradoUSA", "Department of Physics\nUniversity of Colorado\n80309BoulderColoradoUSA", "JILA\nUniversity of Colorado\nNIST\n80309-0440BoulderColoradoUSA", "LPTMS\nCNRS\nUniv. Paris-Sud\nUniversité Paris-Saclay\n91405OrsayFrance" ]	[]	We study a one-dimensional quantum problem of two particles interacting with a third one via a scaleinvariant subcritically attractive inverse square potential, which can be realized, for example, in a mixture of dipoles and charges confined to one dimension. We find that above a critical mass ratio, this version of the Calogero problem exhibits the generalized Efimov effect, the emergence of discrete scale invariance manifested by a geometric series of three-body bound states with an accumulation point at zero energy.	10.1103/physrevlett.115.180406	[ "https://arxiv.org/pdf/1506.03856v2.pdf" ]	15,328,024	1506.03856	5617cfca787e6374a7a08102fc1f295fc1aa5d8b	Generalized Efimov effect in one dimension 29 Oct 2015 Sergej Moroz Department of Physics University of Colorado 80309BoulderColoradoUSA Center for Theory of Quantum Matter University of Colorado 80309BoulderColoradoUSA José P D'incao Department of Physics University of Colorado 80309BoulderColoradoUSA JILA University of Colorado NIST 80309-0440BoulderColoradoUSA Dmitry S Petrov LPTMS CNRS Univ. Paris-Sud Université Paris-Saclay 91405OrsayFrance Generalized Efimov effect in one dimension 29 Oct 2015arXiv:1506.03856v2 [cond-mat.other] We study a one-dimensional quantum problem of two particles interacting with a third one via a scaleinvariant subcritically attractive inverse square potential, which can be realized, for example, in a mixture of dipoles and charges confined to one dimension. We find that above a critical mass ratio, this version of the Calogero problem exhibits the generalized Efimov effect, the emergence of discrete scale invariance manifested by a geometric series of three-body bound states with an accumulation point at zero energy. In quantum mechanics three identical bosons in three dimensions interacting resonantly via a short-range two-body potential have an infinite tower of bound states, whose energy spectrum forms a geometric series near the accumulation point at zero energy. This was discovered theoretically by Vitaly Efimov in 1970 [1] and is known today as the Efimov effect. This effect is a beautiful example of few-body universality since it is independent of the detailed form of the interaction potential provided it is tuned to the resonance (i.e., whenever a zero-energy s-wave two-body bound state if formed). The Efimov effect has been extended to systems of distinguishable particles [2][3][4][5][6], liberated from three dimensions [7] and found in other systems [8,9]. During the last decade a number of experiments [10][11][12][13][14] with cold atoms near Feshbach resonances [15] verified various universal aspects related to Efimov physics-the Efimov 4 He trimer has also been recently observed in [16]-and demonstrated the experimental capability to explore fundamental aspects of few-body systems in exotic regimes. From a more general perspective, the most startling feature of the Efimov effect is discrete scale invariance of the three-body problem, manifested in both bound and scattering three-body observables, that originates from continuous scale invariance of the two-body interaction. It thus appears natural to us to generalize the Efimov effect to systems whose two-body interaction is not necessarily short-range and define it as the emergence of discrete scaling symmetry in a threebody problem if the particles attract each other via a two-body scale invariant potential [17]. Motivated by this broader perspective on the Efimov effect, we study a three-body problem with a two-body long-range attractive potential of the form V (r) = − α 2µr 2(1) with µ being the reduced mass and α the dimensionless coupling constant. The potential (1) is scale invariant and, at zero energy or for sufficiently small r, where the energy term can be neglected, in one dimension the two independent solutions of the two-body Schrödinger equation are the powers r 1/2± √ 1/4−α . However, the (inevitable) breakdown of the 1/r 2 law at small distances introduces a length scale b, made explicit by writing the linear combination of the two asymptotic solutions in the form ψ ∼ r b 1/2+ √ 1/4−α − r b 1/2− √ 1/4−α .(2) For the further discussion it is crucial whether α is larger or smaller than 1/4 [4,18,19]. The case α > 1/4 corresponds to the fall of a particle to the center and the discrete scaling is manifest already in the two-body problem. Here, the exponents 1/2 ± 1/4 − α are complex conjugate, the two terms in Eq. (2) should be treated on equal footing, and b becomes an essential parameter, which can never be neglected. By contrast, the case α < 1/4 has two scale-invariant limits b = 0 and b −1 = 0 where, respectively, only the plusbranch r 1/2+ √ 1/4−α or only the minus-branch r 1/2− √ 1/4−α survives in Eq. (2). In practice, these two limits require, respectively, \|b\| ≪ ξ or \|b\| ≫ ξ, where ξ is a typical lengthscale in the problem such as the system size, de Broglie wave length, etc. For instance, in Eq. (2) the minus-branch solution can be neglected if (b/ξ) √ 1−4α ≪ 1. Thus, the plus-branch scaling is realized "automatically" by increasing the typical size of the system, whereas the minus-branch requires a fine tuning of the short-range part of the potential [20][21][22]. Physically, this fine tuning signals the appearance of an additional two-body bound state emerging from the zero-energy threshold which can be realized using, for example, the Feshbach resonance technique [15]. As far as the three-body problem with the two-body interaction (1) is concerned, Calogero solved it in one dimension analytically for three identical particles [23] and found continuous scale invariance for all α < 1/4, which implies the absence of the Efimov effect [24]. In this paper we show that this conclusion does not hold in general for the modified Calogero problem -two identical spinless bosons or fermions interacting with a third particle via the potential (1). In addition to the quantum statistics and the choice b = 0 or b −1 = 0 the modified problem is parametrized by the two continuous dimensionless quantities: α and the mass ratio. Accordingly, we calculate the critical line separating the Efimov and scaleinvariant regions and describe the nature of the three-body bound state spectrum in this parameter space. The three-body Hamiltonian relevant for our problem reads H = − ∂ 2 R1 + ∂ 2 R2 2M − ∂ 2 r 2m + V (r − R 1 ) + V (r − R 2 ),(3) where R 1 and R 2 are the coordinates of two identical particles of mass M and r is the coordinate of the third particle of mass m. The potential V is given by Eq. (1), where µ = mM/(M + m) and α denotes the interspecies dimensionless coupling. A convenient way to solve this problem is obtained using hyperspherical coordinates. First, we introduce the center-ofmass and mass-scaled Jacobi coordinates R CM = mr + M (R 1 + R 2 ) /(2M + m), x = μ/2(2r − R 1 − R 2 ), y = √ 2µ M (R 2 − R 1 ), whereμ = 2mM/(m + 2M ) and µ M = M/2. It is then convenient to define polar (hyperspherical) coordinates x = R cos θ, y = R sin θ with the massscaled hyperradius R = 2 i m i (r i − R CM ) 2 . The interparticle distances in the new coordinates become r − R 1 = R sin(∆ + θ)/ √ 2µ, r − R 2 = R sin(∆ − θ)/ √ 2µ and R 2 − R 1 = R sin θ/ √ M , where ∆ = arctan 1 + 2M/m. Accordingly, after separating the center-of-mass motion, the relative part of the Hamiltonian (3) is written as a twodimensional radial problem H = −∂ 2 R − 1 R ∂ R + 1 R 2 M 2 θ(4) with the hyperangular Schödinger operator M 2 θ = −∂ 2 θ − α sin 2 (∆ + θ) − α sin 2 (∆ − θ) .(5) Two-body scale invariance leads to separability of the threebody problem in hyperspherical coordinates. The relative part of the three-body wave function Ψ(R, θ) can thus be written in the factorized form Ψ(R, θ) = Φ(R)ψ(θ) and the problem separates into two tasks. First, one finds ψ by diagonalizing the operator M 2 θ M 2 θ ψ = −s 2 ψ.(6) Then, Φ(R) is the solution corresponding to the Hamiltonian (4) with M 2 θ substituted by −s 2 . This second task is trivially solved in terms of the Bessel functions J ±is , and the onset of the generalized Efimov effect coincides with the point s 2 = 0: for positive s 2 the system is Efimovian and for negative s 2 it is scale invariant. Thus, the problem of determining the critical mass ratio is equivalent to solving the hyperangular problem (6) and identifying the zero crossing of s 2 as a function of ∆. We will now discuss this procedure. The coincidence angles θ = 0, π (M − M coincidence) and θ = ±∆, π ± ∆ (M − m coincidences) partition the hyperangular circle into six regions (see Fig. 1). Since two particles of mass M are identical, the wave function satisfies ψ(θ) = ψ(−θ) or ψ(θ) = −ψ(−θ), respectively, for bosons or fermions. In addition, the hyperspherical Hamiltonian is symmetric under θ → π − θ and the wave function ψ is either even or odd under this transformation. It is thus sufficient to solve the angular problem only in the domain θ ∈ (0, π/2). Moreover, we will assume that the distinguishable particles are impenetrable. Physically, this is realized by regularizing the inverse square potential (1) with a short-range potential that has a strong repulsive core. Due to the interspecies impenetrability, sectors I and II in Fig. 1 decouple and can be addressed separately. In sector I the hyperangular wave function, ψ(θ), should satisfy the following boundary conditions for θ = 0 [25] ψ = 0 fermions, ψ ′ = 0 bosons(7) and for θ → ∆ − ψ ∼ (∆ − θ) 1/2+ √ 1/4−α plus-branch, ψ ∼ (∆ − θ) 1/2− √ 1/4−α minus-branch.(8) The critical mass ratio is determined by solving Eq. (6) in sector I and is plotted in Fig. 2 (a) and (b) for bosons and fermions, respectively. We found that s 2 is an increasing function of the mass ratio M/m for any choice of α and boundary conditions. In addition, we find no zero-energy (s = 0) solution in sector II that satisfies the proper scale-invariant boundary conditions at the interspecies coincidence point. The wave function is thus zero in sector II, i.e., the probability to find the particle of mass m in between the two identical particles of mass M vanishes. It should be noted that the modified Calogero problem of the type (3) is exactly solvable and scale invariant for the plusbranch under the condition M/m = 1/(1/2 + 1/4 − α) [26][27][28]. The Efimov region corresponds to higher values of M/m and, since the problem is not solvable, we solve it numerically. We use the Numerov method [29] on a logarithmic grid (see Ref. [22]). Nevertheless, we also find approximate analytic solutions for this problem in limiting cases discussed below. For the plus-branch the critical mass ratio diverges at α = 1/8. In fact, for α > 1/8 both branches give rise to the Efimov effect for sufficiently large M/m [30]. Indeed, for M/m → ∞ the angle ∆ = π/2 and the hyperangular potential in Eq. (5) reduces to −2α/ cos 2 θ. One can see that the hyperangular problem becomes Efimovian for 2α > 1/4 independent of the quantum statistics of the heavy particles and the branch choice. This means that the spectrum of M 2 θ is unbound from below with deep bound states localized close to θ = π/2. As a result, a finite π/2 − ∆ is necessary to renormalize this potential and bring the ground state energy −s 2 to zero. Quantitatively, for the plus-branch solution in the vicinity of α = 1/8 we obtain [22] (i) (ii) (iii) (i) (ii) (iii)π 2 − ∆ ≈ N e ± π 2 − 2π √ 8α−1 ,(9) where the upper (lower) sign corresponds to the case of bosons (fermions) and N = 16 exp[−2 − 2 √ 2 − H (−3+ √ 2)/2 ] ≈ 11.887 with H n being the harmonic number. For the minus-branch the spectrum is Efimovian for any 0 < α < 1/4 for M/m ≫ 1 [22]. The less stringent condition for the Efimov effect in this case can be explained by the fact that the minus-branch two-body interaction nearly binds two particles and is, in this sense, more attractive than the plus-branch interaction with the same α. In fact, the hyperangular problem can be solved analytically close to the noninteracting point α = 0. In Ref. [22] we show that for the bosonic case ∆ − π 4 ≈ απ 2(10) and for fermions π 2 − ∆ ≈ απ 4 .(11) The asymptotes (9), (10), and (11) are plotted in Figs. 2 as dashed lines. The identified critical mass ratio is calculated using the wave function ψ(θ) without nodes inside sector I. As one increases the mass ratio, wave functions with increasing number of nodes will give rise to additional towers of Efimov states. Now we describe the qualitative nature of the three-body bound state spectrum.The interaction in Eq. (1) must be regularized at short distances, see [22]. As the short-range potential depth D 0 changes one can tune between a pure plusbranch (b = 0) and minus-branch (b −1 = 0) solutions. The nature of the three-body spectrum will depend on which region in Fig. 2 the system falls into. There are three different regimes: • In the region (i), below the orange curve, there is no Efimov effect for any value of b. • In the region (ii), between the orange and blue critical curves, the spectrum behaves similar to the original Efimov problem [1]. By starting from the plus branch solution with b = 0 and increasing the depth D 0 , threebody bound states emerge one-by-one from the threebody continuum as one approaches b −1 = 0. At the critical point b −1 = 0, where a zero-energy two-body bound state pops up, an infinite tower of Efimov states is formed with the Efimov parameter s − (encoding the geometric factor e 2π/s− for the energy spectrum) which depends on both M/m and α. As one further increases the depth D 0 , the trimers disappear one-by-one into the particle-dimer continuum. • In the region (iii) the spectrum resembles the one appearing in the three-dimensional Efimov problem of particles with unequal scattering lengths [2,31,32]. Now both the plus-and minus-branches support Efimov states characterized by the Efimov parameter s + and s − , respectively, where 0 < s + < s − . The energy spectrum contains an infinite number of threebody bound states close to the zero-energy threshold for any value of b. The interpolation from the plus-to the minus-branch can be understood as follows: Near b = 0 the energy spectrum close to the zero-energy threshold is controlled by the s + parameter. As one approaches b −1 = 0 the virtual dimer state of size of order \|b\| is formed. As a result, the trimers with energies below (above) ǫ d ∼ −1/µb 2 follow the geometric scaling with the Efimov parameter s − (s + ). At resonance b −1 = 0 the geometric spectrum with s − scaling is obtained. When the depth D 0 is increased further, a two-body bound state is formed and three-body states with energies below (above) ǫ d will also follow the geometric scaling with the Efimov parameter s − (s + ) to the point where, when away enough from b −1 = 0, the energy spectrum is again completely controlled by the s + parameter. Can the Efimov effect found in this paper be discovered experimentally? A promising candidate might be a mixture of dipoles and charges that are confined to one dimension. Indeed, the dipole-charge interaction in three dimensions is given by the scale-invariant anisotropic potential V (r) ∼ cos φ/r 2 , where φ is the angle between the direction of the dipole moment and the dipole-charge separation vector. Consider now a system of two identical dipoles of mass M and dipole moment P and a particle of mass m and charge −q confined to a one-dimensional line. Let us also regard the dipoles as dumbbells with fixed dipole moments as illustrated in Fig. 3. This three-body problem is governed by the Hamiltonian H = − 1 2M ∂ 2 R1 + ∂ 2 R2 − 1 2m ∂ 2 r −K e qP 1 (r − R 1 ) 2 + 1 (r − R 2 ) 2 dipole-charge − 2K e P 2 (R 1 − R 2 ) 3 dipole-dipole ,(12) where the Coulomb constant K e = 1/(4πǫ). If we neglect the dipole-dipole term, this Hamiltonian maps on Eq. (3) with α = 2µK e P q (13) and thus gives rise to the Efimov effect provided α < 1/4 and the mass ratio is above the critical value. The presence of the dipole-dipole term introduces a length scale l dd ∼ M K e P 2 ∼ P q and bound states of this size. In the Efimov regime the length l dd provides the high-energy cutoff for the energy spectrum. This cutoff should not affect the (geometric) energy spectrum close to the zero-energy threshold. However, since our problem is one-dimensional, the dipole-dipole interaction effectively fermionizes the dipoles since their wave function is suppressed at R 1 − R 2 ∼ l dd . Thus, the critical mass ratio for the Efimov effect should be read off of Fig. 2 (b) rather than (a) even if the dipoles are identical bosons. As an example, consider two polar molecules interacting with an electron. From Eq. (13) the dipole moment of the polar molecule should satisfy P < P cr = ea 0 /8 ≈ 0.318 Debye, where e is the charge of the electron and a 0 is the Bohr radius. Such a system has a large mass ratio, and, therefore, provided it falls into the region (iii) in Fig. 2 (b), displays Efimov states (without fine-tuning to the minus branch) which can be detected spectroscopically. For a typical mass ratio M/m = 10 5 we find s + > 1 for P > 0.281 Debye. Moreover, Efimov states could also be observed if tuning of the dipole moment P is possible. In that case, near an electron-dipole resonance, dipolar losses should be enhanced every time a new Efimov state is formed. A three-dimensional version of the problem may have better chances to be realized experimentally and should also have interesting quantum-chemistry implications. In that case P cr ≈ 1.63 Debye [33][34][35]. We consider it as a promising project and leave it for future studies. We thank S. Endo, O. Kartavtsev, Y. Nishida, S. Tan for useful suggestions. We are grateful to V. Efimov for discussions concerning the qualitative nature of the energy spectrum. This research was partially supported by the NSF through DMR-1001240. SM is grateful for hospitality to Institute for Nuclear Theory in Seattle, where this work was partially done. JPD acknowledges support from the U. S. National Science Foundation. DSP acknowledges support from the IFRAF Institute. The research leading to these results has received funding from the European Research Council under European Community's Seventh Framework Programme (FR7/2007-2013 Grant Agreement no.341197). 1 SUPPLEMENTAL MATERIAL: GENERALIZED EFIMOV EFFECT IN ONE DIMENSION Square well regularization and resonance Consider two particles with the reduced mass µ interacting in one dimension via the regularized inverse square potenital V (r) = − α 2µr 2 , r > r 0 , V (r) = −D 0 = − γ 2µr 2 0 , 0 < r ≤ r 0 ,(S1) where r 0 > 0, γ > 0. We will also assume V (−r) = V (r), which implies that a wave function should be either even or odd under the reflection r → −r. It should be noted that the regularized potential necessarily supports a parity-even ground state with energy ǫ GS ∼ −1/(µr 2 0 ). This is in agreement with a general theorem that any attractive well supports at least one bound state in one dimension. Here, however, are interested in the zero energy solution. For r > r 0 a general solution of the relative Schrödinger equation at zero energy is given by ψ(r) = A + r r 0 1/2+ √ 1/4−α + A − r r 0 1/2− √ 1/4−α . (S2) On the other hand, for r ≤ r 0 the solution at zero energy is ψ(r) = C sin √ γ r r 0 reflection odd, ψ(r) = C cos √ γ r r 0 reflection even. (S3) By matching the logarithmic derivative at r = r 0 one finds for the reflection odd case X ≡ A + A − = 1/4 − α − 1/2 + √ γ cot √ γ 1/4 − α + 1/2 − √ γ cot √ γ . (S4) For the reflection even case one must replace cot √ γ → − tan √ γ in Eq. (S4). For a generic value of γ, the plusbranch is dominant for r ≫ r 0 since X = 0. Note however that one can fine tune the regularized potential such that the wave function follows the minus-branch for r ≫ r 0 . This happens for X = 0 resulting in the resonance condition 1/4 − α − 1/2 + √ γ cot √ γ = 0 reflection odd, 1/4 − α − 1/2 − √ γ tan √ γ = 0 reflection even.(S5) By increasing the depth D 0 of the short-range potential, the total potential supports more and more bound states. Incidentally, the resonance condition signals the appearance of an additional bound state emerging from the zero-energy threshold. Logarithmic grid and Langer substitution A numerical solution of the Schrödinger equation [∂ 2 θ − v(θ)]ψ = κ 2 ψ (S6) with a potential possessing an inverse square singularity requires a grid which is dense near the singularity allowing to resolve the behavior of the wave function close to it. This can be accomplished by the Langer substitution θ(x) = exp(x), ψ(θ) = exp x 2 y(x) (S7) which transforms Eq. (S6) into ∂ 2 x y(x) + g(x)y(x) = 0 (S8) with g(x) = − exp(2x) κ 2 + v[exp(x)] − 1 4 . (S9) This equation is now solved by the Numerov method on the equidistant x-grid with a lower cutoff x min < 0. Analytic calculation near α = 1/8 Slightly above α = 1/8 for the plus-branch the critical mass ratio is large (∆ → π/2) and the angular problem can be attacked analytically. Here we sketch this calculation. First, it is convenient to redefine the angle to be φ = ∆ − θ. The angular Schrödinger equation (6) to be solved in sector I takes the form − ∂ 2 φ − α sin 2 φ − α sin 2 (φ + 2ǫ) ψ = 0,(S10) where ǫ = π/2 − ∆ ≪ 1. In the regime φ ≫ ǫ this equation simplifies to − ∂ 2 φ − 2α sin 2 φ ψ = 0 (S11) with a general solution ψ =(1 − cos φ) 1 4 × L 1 P √ 1−8α/2 −1/2 (cos φ) + L 2 Q √ 1−8α/2 −1/2 (cos φ) ,(S12) where P m n (x) and Q m n (x) are associated Legendre functions. The quantum statistics condition (7) imposed at φ = ∆ fixes the ratio L 1 /L 2 . In the regime φ ≪ 1 one can expand the denominators in Eq. (S10) and it transforms to − ∂ 2 φ − α φ 2 − α (φ + 2ǫ) 2 ψ = 0 (S13) with a general solution ψ = ǫ x(2 + x) × S 1 P √ 1−4α ( √ 1−8α−1)/2 (1 + x) + S 2 Q √ 1−4α ( √ 1−8α−1)/2 (1 + x) ,(S14) where x = φ/ǫ. The ratio S 1 /S 2 is now determined by applying the plus-branch boundary condition ψ ∼ x 1/2+ √ 1/4−α at x = 0 + [equivalent to Eq. (8)]. In the intermediate regime x ≫ 1 and φ ≪ 1 the wave functions (S12) and (S14) must now be matched. In this region they are given by ψ = B + φ 1 2 (1+ √ 1−8α) + B − φ 1 2 (1− √ 1−8α) ,(S15) where B + /B − = exp(iζ) with a real phase ζ. The phases extracted from the solutions (S12) and (S14) can differ only by the angle 2πn, where n is an integer. The nodeless wave function ψ is found for n = 1. The matching of the phases ζ in the intermediate region with n = 1 gives rise to Eq. (9). Analytic calculation near α = 0 Here we show how the mass ratio for the minus-branch can be determined analytically close to the non-interacting point α = 0. In this case identical bosons and fermions must be treated differently. In the case of identical bosons the critical mass ratio vanishes at α = 0. We can thus start from the analytical solution of the angular equation (6) at ∆ = π/4 ψ ∝ sin 1/2− √ 1/4−α (θ + π 4 ) sin 1/2− √ 1/4−α (θ − π 4 ) (S16) with the positive eigenvalue − s 2 = 2(1 − 2 1/4 − α − 2α) = 4α 2 + O(α 3 ). (S17) Let us now take into account a small deviation of ∆ from π/4 perturbatively. In order to do this we redefine the angle θ = 4∆θ/π such that the configurational space is the fixed interval θ ∈ (0, π/4) independent of ∆. After this redefinition, Eq. (6) becomes − ∂ 2 θ − αρ 2 sin 2 [ρ( π 4 +θ)] − αρ 2 sin 2 [ρ( π 4 −θ)] ψ = −s 2 ρ 2 ψ, (S18) where ρ = 4∆/π > 1. The linear shift of −s 2 (ρ) with respect to ρ − 1 ≪ 1 is now determined by the first-order perturbation formula ∂(−s 2 ρ 2 )/∂ρ = ψ\|v 1 \|ψ / ψ\|ψ ,(S19) where v 1 is the first derivative of the potential in Eq. (S18) with respect to ρ at ρ = 1 and ψ is given by Eq. (S16). We have explicitly v 1 (θ) = − α sec 3 (2θ) × − 3π + 8 cos(2θ) + π cos(4θ) + 16θ sin(2θ) . In fact, the integrals on the right hand side of Eq. (S19) can be calculated for a constant ψ, i.e., setting α = 0 in Eq. (S16). Then, combining the result with Eq. (S17) we finally obtain − s 2 ≈ 4α 2 − 2α(ρ − 1),(S21) from which Eq. (10) follows immediately. For identical fermions the mass ratio diverges at α = 0. Hence one can follow the procedure described in the previous subsection with the wave function (S14) now satisfying the minus-branch boundary condition ψ ∼ x 1/2− √ 1/4−α at x = 0 + . As a result, one finds Eq. (11). Born-Oppenheimer approximation for minus-branch Here we study the minus branch in the regime M ≫ m, where the Born-Oppenheimer (BO) approximation can be used and the emergence of the Efimov effect can be intuitively understood. Within this approximation the three-body wave function is factorized Ψ(R, r) = Φ(R)ψ(r; R), where the function ψ(r; R) corresponds to a bound state of the light particle in the combined potential of two heavy centers located at R 1 = 0 and R 2 = −R. This wave function satisfies − 1 2m ∂ 2 r + α r 2 + α (r + R) 2 H light ψ = ǫψ.(S23) Note that the distance R is the only characteristic length scale in Eq. (S23). Therefore, if a bound state exists at some R, it also exists at any R, and its energy equals ǫ R = −σ/mR 2 , where σ is a positive dimensionless number which depends on α and on the choice of the light-heavy boundary condition. Next, the Born-Oppenheimer approximation assumes that the light particle adiabatically follows displacements of the heavy ones for which ǫ R acts as the effective interaction potential. Thus, the Schrödinger equation for the relative motion of the heavy particles reads − 1 M ∂ 2 R + ǫ GS (R) Φ = EΦ.(S24) The energy spectrum E determines the energy spectrum of the three-body system. The spectrum of Eq. (S24) is Efimovian, E n ∼ e − 2πn s +θ , where s ≈ σM/m. For simplicity, we neglected in Eq. (S24) the diagonal correction to the potential V diag (R) = ψ\|∂ 2 R ψ /M which is subleading and does not modify the argument above. Now we demonstrate that Eq. (S23) actually supports at least one bound state for the minus-branch condition for any 0 < α < 1/4. To this end, we consider the minus branch variational wave-function which equals ψ κ ∝ r 1/2− √ 1/4−α exp(−κr)(S25) for r > 0 and vanishes for r < 0. We find that for any 0 < α < 1/4 and for a sufficiently small variational parameter κ ψ κ \|H light \|ψ κ < 0. This can be seen by evaluating analytically the variational en-ergy (S26) and Taylor expanding it around κ = 0. 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[]	[ "Xavier Allamigeon ", "ANDStéphane Gaubert ", "Ricardo D Katz " ]	[]	[ "2010 Mathematics Subject Classification. 14T05 (Primary) 15A80, 52A01" ]	We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.	10.1016/j.laa.2011.02.004	[ "https://arxiv.org/pdf/1004.2778v2.pdf" ]	14,058,562	1004.2778	00fd5df126119b23ba60833b46805b0bfa59af22	29 Oct 2010 Date: April 15, 2010, revised October 27, 2010. Xavier Allamigeon ANDStéphane Gaubert Ricardo D Katz 2010 Mathematics Subject Classification. 14T05 (Primary) 15A80, 52A01 166029 Oct 2010 Date: April 15, 2010, revised October 27, 2010.arXiv:1004.2778v2 [math.CO] TROPICAL POLAR CONES, HYPERGRAPH TRANSVERSALS, AND MEAN PAYOFF GAMESand phrases Max-plus semiringmax-plus convexitytropical convexitypolyhedrahypergraph transver- salsminimal hitting setsminimal solutions We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone. Introduction The max-plus or tropical analogue of classical convexity has emerged in a number of works. Early contributions come back to Zimmermann [Zim77] and Cuninghame-Green [CG79]. The analogues of cones (thought of as modules over the tropical semiring) have been studied by Litvinov, Maslov, and Shpiz [LMS01] as part of "Idempotent analysis", and by Cohen, Gaubert and Quadrat [CGQ04], motivated by discrete event systems. Relations with abstract convexity have appeared in a further work with Singer [CGQS05], and in the work of Briec and Horvath [BH04]. The interest in the subject has been renewed after the work of Develin and Sturmfels [DS04], who developed a combinatorial approach motivated by tropical geometry. This was at the origin of a number of works of the same authors and of Joswig, Yu, and Block, see [Jos05,JSY07,BY06]. Some recent developments include [BSS07,GK09,Jos09,GM10,AGG10b]. In classical convex analysis, duality techniques play an important role, and, in the light of the current development of tropical convexity, it is natural to ask whether these carry over to the tropical setting. In particular, a polyhedral cone can be classically represented in two different ways, either internally, "by generators," as the set of nonnegative linear combinations of a finite set of vectors, or externally, "by relations," as the solution set of a finite system of homogeneous linear inequalities. If the cone is pointed, among the generating sets of vectors, there turns out to be a unique minimal one (up to a scaling of each vector), which consists of representatives of the extreme rays of the cone. If the cone is of full dimension, among the defining systems of inequalities, there turns out to be a unique minimal one (up to a scaling of each inequality), corresponding to the facets of the cone, or, by duality, to the extreme rays of its polar. Moreover, by Farkas lemma, every (homogeneous, linear) inequality verified by all the vectors of the cone can be obtained by taking a nonnegative linear combination of the "facet defining" inequalities. We may ask for the tropical analogues of these properties. For the internal representation, precisely the same result holds: the tropical analogue of the Minkowski theorem [GK06,BSS07,GK07] shows that a tropical polyhedral cone is generated by its extreme rays, and that every generating set must contain one representative of each extreme ray. The external representation has also been studied in the tropical setting. The analogue of the polar, which consists of the set of inequalities verified by the elements of the cone, was introduced in [GK09]. The polar of a tropical polyhedral cone is in fact a tropical polyhedral cone in a space of double dimension. It has a finite family of extreme rays, which determine a finite family of "extreme" inequalities, having the property that every inequality verified by all the vectors of the cone is a linear combination of the extreme inequalities. Moreover, the set of extreme inequalities is the unique minimal one having this property (up to a scaling of each inequality). In this paper, we pursue the investigation of tropical analogues of classical properties concerning convex duality, and in particular polars. First, we establish (Theorem 4 and Proposition 5 below) a characterization of the extreme rays of the polar in terms of minimal set covers. The latter may be thought of as weighted generalizations of minimal transversals (or hitting sets) in hypergraphs. It follows that enumerating the extreme rays of the polar is at least as hard as the well known problem of enumerating the minimal transversals of an hypergraph (Corollary 7 and Proposition 2). Moreover, by combining this characterization with a result of Elbassioni [Elb08], building on a line of works on minimal transversals and minimal solutions of monotone inequalities by Fredman, Khachiyan, Boros, Elbassioni, Gurvich, and Makino [FK96, BEG + 02, KBEG06], it follows that the set of extreme rays of the polar of a tropical polyhedral cone can be computed in incremental quasi-polynomial time. Next, we answer to a question raised in [GK09], which asks for a tropical analogue of Farkas lemma. Indeed, it was observed there that the statement of the classical Farkas lemma is no longer valid in the tropical world: there are inequalities which can be logically deduced from a finite system of inequalities but which cannot be obtained by taking linear combinations of the inequalities in this system (see Figure 5 below for an example). We show here that there is indeed a tropical analogue of Farkas lemma, if we consider properly its role. The classical result provides a certificate (nonnegative weights, or Lagrange multipliers), which allows one to easily check (by computing a linear combination) that an inequality follows from other inequalities. The tropical analogue, Theorem 18 below, shows that there is still a concise certificate, which is no longer a collection of Lagrange multipliers, but consists of a strategy of a mean payoff game. It also follows from our approach that checking whether an inequality is implied by a finite family of inequalities is polynomial time equivalent to the problem of solving a mean payoff game (the latter was already known to be in NP ∩ co-NP, and the existence of a polynomial time algorithm for this problem is a long standing open question). Theorem 18 relies on a recent work of Akian, Gaubert and Guterman [AGG09a,AGG10a], setting up a correspondence between mean payoff games and external representations of tropical polyhedra, and showing that the value of a mean payoff game is nonnegative if, and only if, the corresponding tropical polyhedron is non-empty. In the present paper, we use similar techniques to show that the decision problem associated with the tropical Farkas lemma is also equivalent to a mean payoff game problem (Corollary 22). We also discuss examples, computing in particular the extreme rays of the polar of the tropical analogues of cyclic polyhedral cones. Whereas in classical algebra, cyclic polyhedral cones have an exponential number of facets (actually, they maximize the numbers of facets among all the cones of the same dimension and with the same number of extreme rays), in the tropical setting their polars turn out to have only a polynomial number of extreme rays, see Proposition 6 below. We finally show that unlike in the classical case, there is no unique minimal set of inequalities defining a given tropical polyhedral cone. Extreme elements of the polar We start by recalling some basic definitions and notation. Tropical or max-plus algebra is the analogue of classical linear algebra developed over the max-plus semiring R max , which is the set R ∪ {−∞} equipped with the addition (a, b) → max(a, b) and the multiplication (a, b) → a + b. To emphasize the semiring structure, we write a ⊕ b := max(a, b), ab := a + b, ¼ := −∞ and ½ := 0. The semiring operations are extended in the usual way to matrices over the max-plus semiring: (A ⊕ B) ij := A ij ⊕ B ij , (AB) ij := ⊕ k A ik B kj and (λA) ij := λA ij for all i, j, where A, B are matrices of compatible sizes and λ ∈ R max . In what follows, we shall denote by G ·i (resp. G i· ) the ith column (resp. row) of the matrix G, and by e i the ith vector of the canonical basis of R n max , i.e. the vector defined by (e i ) j := ½ if j = i and (e i ) j := ¼ otherwise. Several classical concepts and results have their tropical analogues. In particular, the tropical analogues of convex sets were introduced by Zimmermann [Zim77]. Since in the max-plus semiring any scalar is "positive", i.e. for any λ ∈ R max we have λ ¼, it is natural to define the tropical segment joining two points x, y ∈ R n max as the set of points of the form λx ⊕ µy where λ and µ are elements of R max such that λ ⊕ µ = ½. Then, a subset of R n max is said to be a tropical convex set if it contains any tropical segment joining two of its points. Similarly, the tropical cone generated by x, y ∈ R n max is defined as the set of vectors of the form λx ⊕ µy where now λ and µ are arbitrary elements of R max . A tropical (convex) cone is a subset K of R n max which contains any tropical cone generated by two of its vectors. An example of tropical convex set is given on the left hand side of Figure 1: the tropical convex set is the closed gray region together with the horizontal segment joining the point u to it. Three tropical segments in general position are represented in bold. Comparing the shapes of these segments with the shape of the set, one can check geometrically that this set is a tropical convex set. A tropical cone generated by two vectors u, v ∈ R 2 max is represented on the right hand side of Figure 1 by the unbounded gray region. Given a tropical cone K ⊆ R n max , we shall say that a non-identically ¼ vector x ∈ K is extreme in K if x = y ⊕ z with y, z ∈ K implies x = y or x = z. The set of scalar multiples of an extreme vector of K is an extreme ray of K. A tropical cone K is said to be finitely generated (or polyhedral) if it contains a finite set of vectors {y r } r∈ [p] such that every vector x ∈ K can be expressed as a tropical linear combination of these vectors, meaning that x = ⊕ r∈[p] λ r y r for some λ r ∈ R max . Here, and in the sequel, we set [p] := {1, . . . , p} for any p ∈ N. The polar of a subset K ⊆ R n max can be defined [GK09] as (1) K • := (a, b) ∈ (R n max ) 2 \| ax bx, ∀x ∈ K , where xy := ⊕ i∈[n] x i y i for any pair of vectors x, y ∈ R n max . Note that the polar of K is a tropical cone of (R n max ) 2 which represents the set of (tropical) linear inequalities satisfied by the elements of K. In what follows, we shall usually identify an element (a, b) of K • with the corresponding inequality ax bx, which will be called valid for K because it is satisfied by all its elements. In this paper, we are mainly interested in polyhedral cones. Therefore, we shall consider the case in which the tropical cone K is the row space R(G) of some p × n matrix G with entries in R max , i.e. (2) R(G) := x ∈ R n max \| x = ⊕ r∈[p] λ r G r· , λ r ∈ R max for r ∈ [p] , where without loss of generality we assume that G does not have an identically ¼ column. Observe that in this case we have R(G) • = (a, b) ∈ (R n max ) 2 \| Ga Gb , which implies that R(G) • has a finite number of extreme rays, because it can be expressed as the solution set of a two sided (tropical) linear system of equations, namely R(G) • = (a, b) ∈ (R n max ) 2 \| Ga ⊕ Gb = Gb , and the solution sets of such systems are known to be finitely generated tropical cones (see [BH84,Gau92,GP97]). We refer the reader to [GK10, GK09, GK06] for more information. By the separation theorem for closed cones of [Zim77,SS92,CGQS05], it follows that R(G) is characterized by its polar cone, i.e. R(G) = {x ∈ R n max \| ax bx , ∀(a, b) ∈ R(G) • } . This implies that R(G) can be expressed as the solution set of the (finite) set of linear inequalities associated with the extreme rays of R(G) • . More precisely, the extreme rays of R(G) • determine a finite family of linear inequalities defining R(G), which has the property that any valid inequality for R(G) can be expressed as a (tropical) linear combination of the inequalities in this family. We shall say that a linear inequality in the variables x j , j ∈ [n], is trivial if it is of the form x i x i or x i ¼, and that it is of type i if it is of the form x i j∈[n\i] z j x j , where z ∈ R n−1 max and [n \ i] := [n] \ {i} = {1, . . . , n} \ {i} . An inequality satisfied by the elements of a tropical cone is said to be extreme if it corresponds to an extreme vector of the polar of this cone. A system of representatives of the extreme inequalities is any family containing one and only one inequality proportional to every extreme inequality. Proposition 1. Every extreme inequality is either proportional to a trivial inequality or proportional to an inequality of type i, for some i ∈ [n]. Proof. Firstly, note that any vector of the form (¼, e i ), corresponding to the trivial inequality x i ¼, belongs to the polar of every cone. Therefore, these inequalities are clearly extreme and any extreme inequality corresponding to a vector of the form (¼, b), where b ∈ R n max , must be a scalar multiple of one of these inequalities. It can also be checked that the vectors (e i , e i ), corresponding to the trivial inequalities x i x i , are also extreme because vectors of the form (e i , λe i ), where λ < ½, do not belong to the polar of R(G) (recall that we assume that G does not contain an identically ¼ column). Consider now the inequality ax bx corresponding to a vector (a, b) of R(G) • with a = ¼. Since (a ′ ⊕ a ′′ )x bx =⇒ a ′ x bx and a ′′ x bx , it follows that ax bx can only be extreme if a is a scalar multiple of e i , for some i ∈ [n]. Therefore, without loss of generality, we may assume that a = e i . We next show that the inequality e i x bx is extreme only if b = e i or b i = ¼, which in the latter case means that e i x bx is of type i. Assume that e i x bx is not of type i, i.e. that b i = ¼. If b i < ½, then (e i , ⊕ j∈[n\i] b j e j ) ∈ R(G) • and as (e i , b) = (e i , ⊕ j∈[n\i] b j e j ) ⊕ (¼, b i e i ), we conclude that (e i , b) is not extreme. Analogously, if b i > ½, since (e i , b) = (e i , e i ) ⊕ (¼, b), it follows that neither in this case (e i , b) is extreme. Finally, if b i = ½, we have (e i , b) = (e i , e i ) ⊕ (¼, ⊕ j∈[n\i] b j e j ) implying that (e i , b) is extreme only if ⊕ j∈[n\i] b j e j = ¼, i.e. only if b = e i . Definition 1. The ith polar K • i of a subset K ⊆ R n max is the tropical cone (3) K • i := b ∈ R n max \| b i x i ⊕ j∈[n\i] b j x j , ∀x ∈ K . Since b ∈ R n max belongs to the ith polar K • i if, and only if, (b i e i , ⊕ j∈[n\i] b j e j ) ∈ (R n max ) 2 belongs to the polar K • , it follows that the extreme inequalities of type i of K correspond to the extreme rays of K • i associated with extreme vectors b such that b i = ¼. Moreover, as e j ∈ K • i for all j ∈ [n \ i], note that any extreme vector b of K • i with b i = ¼ must be a scalar multiple of one of these vectors of the canonical basis of R n max . Therefore, by Proposition 1, it follows that the study of the extreme inequalities of K reduces to the study of the extreme rays of K • i . In fact, the two underlying enumeration problems are polynomial time equivalent, as shown by the following result. Proposition 2. The enumeration problem for the extreme rays of the polar of a tropical polyhedral cone is polynomial time equivalent to the enumeration problem for the extreme rays of the ith polar of a tropical polyhedral cone. Proof. Let R(G) be a tropical polyhedral cone. As we already showed, the extreme inequalities of type i of R(G) correspond to the extreme rays of R(G) • i associated with extreme vectors b such that b i = ¼. Moreover, as e j ∈ R(G) • i for all j ∈ [n \ i], any extreme vector b of R(G) • i with b i = ¼ must be a scalar multiple of one of these vectors of the canonical basis of R n max . Therefore, by Proposition 1, it follows that the enumeration problem for the extreme rays of R(G) • reduces to the enumeration problem for the extreme rays of R(G) • i for i ∈ [n]. Conversely, given a tropical polyhedral cone R(G), let K ⊆ R n max be the tropical cone generated by (i.e. all the tropical linear combinations of) the rows of G and the vectors e j for all j ∈ [n \ i]. Then, b ∈ R n max belongs to the ith polar R(G) • i of R(G) if, and only if, (b i e i , ⊕ j∈[n\i] b j e j ) ∈ (R n max ) 2 belongs to the polar K • of K. Therefore, it follows that b is an extreme vector of R(G) • i if, and only if, (b i e i , ⊕ j∈[n\i] b j e j ) is an extreme vector of K • . Moreover, any non-trivial extreme inequality of K is of type i. In consequence, the enumeration problem for the extreme rays of the ith polar of R(G) reduces to the enumeration problem for the extreme rays of the polar of K. x 1 x 2 x 3 G 1· G 2· G 3· G 4· G 5· G 6· x 1 x 2 x 3 x 1 x 2 x 3 Example 1. Consider the row space R(G) of the matrix (4) G =         −3 0 0 0 −3 0 0 0 −3 1 0 0 0 1 0 0 0 1         . This tropical polyhedral cone is represented on the left hand side of Figure 2 by the closed region in dark gray together with the line segments joining the points G 1· , G 2· and G 3· to it. This illustration is in barycentric coordinates, meaning that a vector (x 1 , x 2 , x 3 ) of R 3 max is represented by the barycenter with weights (e x 1 , e x 2 , e x 3 ) of the three vertices of a simplex. Then, two vectors that are proportional in the tropical sense are represented by the same point. This is convenient to make two-dimensional pictures of tropical convex cones of dimension three. Moreover, the barycentric representation permits to show vectors with infinite entries: the latter appear at the boundary of the simplex. The extreme rays of the 2nd polar of R(G) correspond to the following inequalities: x 1 ⊕ (−1)x 3 (−1)x 2 x 1 (−3)x 2 x 3 (−3)x 2 (−1)x 1 ⊕ x 3 (−1)x 2 x 1 ¼ x 3 ¼ The tropical cones associated with the four extreme inequalities of type 2 above are represented on the right hand side of Figure 2. According to the proof of Proposition 2, these four non-trivial inequalities precisely define the cone generated by the rows of G and the vectors e 1 and e 3 , as illustrated in the middle part of Figure 2. The extreme vectors of tropical polyhedral cones can be characterized combinatorially in terms of directed hypergraphs (we shall also use undirected hypergraphs in the sequel, so we shall always make it clear whether the hypergraph is directed), see [AGG09b,AGG10b,All09a]. Let us recall that a directed hypergraph is a couple (N, E), where N is a set of nodes and E is a set of directed hyperedges, which are of the form (M, M ′ ) with M, M ′ ⊆ N . When each of the sets M and M ′ has only one element, we say that the hyperedge is an edge. The notion of reachability in directed hypergraphs can be defined recursively: node r is said to be reachable from node h if r = h or there exists a directed hyperedge (M, M ′ ) ∈ E such that r ∈ M ′ and any node in M is reachable from h. As usual, a strongly connected component is a maximal subset of nodes C satisfying the property that every node in C is reachable from any node in C. Given a tropical cone K ⊆ E = {(arg max(b r y), arg max(a r y)) \| r ∈ [p], a r y = b r y = ¼} , where arg max(cx) is the argument of the maximum cx = ⊕ i∈[n] c i x i = max i∈[n] (c i + x i ) for any c, x ∈ R n max . In [All09a],[AGG10b, Theorem 3.7], it was shown that a vector y of a tropical polyhedral cone K is extreme if, and only if, the tangent directed hypergraph H(K, y) of K at y admits a smallest strongly connected component. The term "smallest" refers to the order relation in which a strongly connected component C 1 is smaller than a strongly connected component C 2 if C 1 has a node which is reachable from a node of C 2 . So, this condition means that there is a node which is reachable for all the other nodes in the tangent directed hypergraph H(K, y). We refer the reader to [AGG10b] and to [All09a] for details (note that in the latter reference, the order of strongly connected components is reversed). In the special case of the ith polar of R(G), a simpler characterization of its extreme vectors holds. The following theorem shows that a vector b ∈ R(G) • i with b i = ¼ is extreme if, and only if, the tangent directed hypergraph H(R(G) • i , b) of R(G) • i at b contains a star-like directed subhypergraph which is reduced to edges directed to node i and leaving the other nodes, see Figure 3. Alternative characterizations are given in Theorem 5 of [GK10] and Proposition 5.13 of [All09a]. Theorem 3 (Star-like directed sub-hypergraph). Let b be a vector of the ith polar of the row space of G, i.e. b ∈ R(G) • i , such that b i = ¼.(R(G) • i , b) of R(G) • i at b contains all the edges ({h}, {i}) for h ∈ {j ∈ [n \ i] \| b j = ¼}. Proof. In the first place, assume that H(R(G) • i , b) does not contain the edge ({h}, {i}) for some h ∈ {j ∈ [n \ i] \| b j = ¼}. Then, for each r ∈ [p] such that b i G ri = ⊕ j∈[n\i] b j G rj = ¼, we have arg max(⊕ j∈[n\i] b j G rj ) = {h}. Therefore, there exists λ < b h such that b ′ ∈ R(G) • i , where the vector b ′ ∈ R n max is defined by b ′ h = λ and b ′ j = b j for all j ∈ [n \ h]. Since e h also belongs to R(G) • i and b = b ′ ⊕ b h e h , it follows that b is not extreme. Conversely, assume now that H(R(G) • i , b) contains the edge ({h}, {i}) for each h ∈ [n \ i] such that b h = ¼. Suppose that b = b ′ ⊕ b ′′ , where b ′ , b ′′ ∈ R(G) • i . Then, either we have b i = b ′ i or b i = b ′′ i . Assume, without loss of generality, that b i = b ′ i . We claim that in this case b = b ′ , which proves that b is extreme. By the contrary, suppose that b = b ′ . Then, there exists h ∈ {j ∈ [n \ i] \| b j = ¼} such that b ′ h < b h . Since H(R(G) • i , b) contains the edge ({h}, {i}), there exists r ∈ [p] such that b i G ri = ⊕ j∈[n\i] b j G rj = ¼ and arg max(⊕ j∈[n\i] b j G rj ) = {h}. But as b ′ b, b ′ h < b h and b ′ i = b i , it follows that b ′ i G ri = b i G ri = b h G rh > ⊕ j∈[n\i] b ′ j G rj , contradicting the fact that b ′ ∈ R(G) • i . This proves our claim. Covering by level sets and hypergraph transversals We begin this section by characterizing the extreme inequalities of a tropical polyhedral cone in terms of minimal elements of tropical polyhedra. Recall that a vector z of a set Z ⊆ R n−1 max is a minimal element of Z if z ′ ∈ Z and z ′ z imply z ′ = z. Theorem 4. A system of representatives of the extreme inequalities satisfied by the elements of the row space R(G) consists of the trivial inequalities, together with the inequalities of type i, x i ⊕ j∈[n\i] z j x j where i ∈ [n] , in which the vector of coefficients z is a minimal element of the tropical polyhedron Z i := z = (z j ) j∈[n\i] ∈ R n−1 max \| ⊕ j∈[n\i] z j G ·j G ·i . Proof. By Proposition 1, we only need to show the characterization for extreme inequalities of type i, so let us consider an inequality of type i, x i j∈[n\i] z j x j ,(5) in the polar cone R(G) • of R(G). Since every row of G must satisfy this inequality, we have G ·i j∈[n\i] z j G ·j . Assume first that z is a minimal element of Z i . Let us write Inequality (5) in the form ax bx. If this inequality is not extreme in R(G) • , then, we can write (a, b) = (a ′ , b ′ ) ⊕ (a ′′ , b ′′ ), where (a ′ , b ′ ) and (a ′′ , b ′′ ) belong to R(G) • and both of them differ from (a, b). We deduce from e i = a = a ′ ⊕ a ′′ that either a ′ = e i or a ′′ = e i , say a ′ = e i . Then, the inequality a ′ x b ′ x is of type i, because b ′ i b i = ¼ and so b ′ i = ¼. Moreover, the vector z ′ ∈ R n−1 max defined by z ′ j = b ′ j for j ∈ [n \ i] belongs to Z i . Our assumption that (a, b) = (a ′ , b ′ ) implies that z ′ is (strictly) smaller than z, contradicting the minimality of z. It follows that Inequality (5) is extreme. Conversely, if z is not minimal, since the set of elements in Z i smaller than z is compact, there exists a minimal element z ′ ∈ Z i such that z ′ z. By the definition of Z i , the inequality of type i arising from z ′ belongs to the polar of R(G), and since z ′ z, Inequality (5) can be obtained by summing to the former inequality suitable multiples of the inequalities x j ¼, j ∈ [n\i]. Therefore, we conclude that Inequality (5) is not extreme. For i ∈ [n], let S i := {k ∈ [p] \| G ki = ¼} . Given a scalar λ ∈ R max and j ∈ [n \ i], we define the level set L j (λ) := {k ∈ S i \| λG kj G ki } . Then, for any z ∈ R n−1 max we have z ∈ Z i ⇐⇒ j∈[n\i] z j G ·j G ·i ⇐⇒ S i ⊆ j∈[n\i] L j (z j ) . Since the maps λ → λG kj are non-decreasing, it follows that the family of level sets {L j (λ)} λ∈Rmax is a chain. For each j ∈ [n \ i], it consists of at most p + 1 sets, say ∅ = L 1 j L 2 j · · · L k j j for some k j ∈ [p + 1]. Note that for each level set L r j there exists a minimal λ ∈ R max such that L j (λ) = L r j , which is given by w r j := max G −1 kj G ki \| k ∈ L r j , where the maximum over the empty set is defined to be ¼, and G −1 kj is understood in the tropical sense (i.e., −G kj with the usual notation). We shall say that {L r j j } j∈[n\i] , where r j ∈ [k j ] for j ∈ [n \ i], is a minimal cover of S i if S i ⊆ ∪ j∈[n\i] L r j j but this inclusion is no longer satisfied if some non-empty set L r j j is replaced by L r j −1 j . The following simple observation gives a combinatorial interpretation of the extreme rays of the polar in terms of set covers, which we shall relate to minimal transversals in hypergraphs at the end of this section. Proposition 5. The minimal elements of Z i correspond to the minimal covers of S i by level sets L r j j as j ∈ [n \ i] and r j ∈ [k j ]. Proof. In the first place, assume that z is a minimal element of Z i . For each j ∈ [n \ i], let r j ∈ [k j ] be such that L r j j = L j (z j ). Then, since ⊕ j∈[n\i] z j G ·j G ·i , we have S i ⊆ j∈[n\i] L j (z j ) = j∈[n\i] L r j j , showing that the family of level sets {L r j j } j∈[n\i] is a cover of S i . We claim that this cover must be minimal. Otherwise, in {L r j j } j∈[n\i] we could replace some non-empty set L r h h by L r h −1 h and still obtain a cover of S i . Therefore, if we replaced the component z h of z by w r h −1 h , we would obtain an element of Z i . However, since z h w r h h > w r h −1 h , this would contradict the minimality of z (indeed, as z is minimal, note that we must have z j = w r j j for j ∈ [n \ i]), proving our claim. Conversely, assume that {L r j j } j∈[n\i] is a minimal cover of S i . If we define z ∈ R n−1 max by z j = w r j j for j ∈ [n \ i], since L j (z j ) = L j (w r j j ) = L r j j , it follows that S i ⊆ j∈[n\i] L r j j = j∈[n\i] L j (z j ) , and thus z ∈ Z i . If z was not minimal, we could decrease one of its components, say z h , so that the resulting vector still belongs to Z i . This would mean that in {L r j j } j∈[n\i] we could replace the set L r h h by L r h −1 h and still obtain a cover of S i . However, this contradicts that {L r j j } j∈[n\i] is a minimal cover of S i . We now analyze two examples of tropical cones with p generators in dimension n. The first one shows that the growth of the number of extreme rays of the polar of such a cone cannot be polynomially bounded in p and n. Example 2. Assume that p divides n − 1. Then, we can choose the matrix G in such a way that the polar of the cone R(G) generated by its rows has at least ((n − 1)/p) p extreme rays. Let q = (n − 1)/p and G = [F 1 , . . . , F p , e], where e is the p dimensional tropical unit column vector (with all entries equal to ½), and for k ∈ [p], F k is a p × q matrix such that the maximum in each column is attained on row k and only on this row. An example of such a matrix in which n = 7, p = 3 and q = 2 is: G =   5 4 1 1 1 1 0 1 1 5 4 1 1 0 1 1 1 1 5 4 0   . We obtain a minimal cover of S n = [p] by level sets as follows. For each k ∈ [p], select precisely one index j k in the set of column indices of F k . Set z j k = G −1 kj k . Since the maximum of column j k is attained only on row k, we have L j k (z j k ) = {k}. If we define z j = ¼ for j ∈ {j k \| k ∈ [p]}, it follows that {L j (z j )} j∈[n−1] is a minimal cover of S n by level sets. By Proposition 5, each of these minimal covers yield a minimal point of Z n , and so, an extreme vector of the polar of R(G). Since for each k ∈ [p] there are q ways to choose j k , there is a total number of q p = ((n − 1)/p) p choices. Note that each of these choices leads to a different extreme vector. Example 3 (Cyclic polyhedral cone). Consider the p × n matrix G defined by G ij = t j−1 i , where t 1 < t 2 < · · · < t p are p real numbers. The classical convex cone generated by the rows of G is the cyclic polyhedral cone. Among all the cones in dimension n generated by p vectors, the latter is known to maximize the number of facets, or equivalently, the number of extreme rays of its polar. This result is part of the celebrated McMullen upper bound theorem [McM70], see [Zie98,Mat02] for more background. The cyclic polyhedral cone can be defined in the same way in the tropical case. Thus, the exponentiation is now understood tropically, so that the entries of G are given by G ij = t i × (j − 1), and the tropical cyclic polyhedral cone, P(p, n), is the set of all tropical linear combinations of the rows of G. This cone was first considered by Block and Yu in [BY06], and a dual notion (polars of cyclic polyhedral cones) depending on a sign pattern was studied in [AGK10], see below. The number of extreme rays of the classical polar of the cyclic polyhedral cone is known to be of order p ⌊(n−1)/2⌋ as p tends to infinity, for a fixed n, see for example [Mat02]. This should be opposed to the tropical case, in which the number of extreme rays is polynomially bounded, as shown by the following proposition. Proposition 6. The polar of the tropical cyclic polyhedral cone P(p, n) is generated by 2n + n i=1 ((p − 1)(i − 1)(n − i) + n − 1) = O pn 3 extreme vectors. Proof. In order to prove this proposition, in the first place it is necessary to recall some concepts and results from [AGK10]. A sign pattern for the cyclic polyhedral cone P(p, n) is a p × n matrix (ǫ ij ) whose entries are + and − signs. The polar of the signed cyclic polyhedral cone with sign pattern (ǫ ij ) is defined (see [AGK10, Definition 1]) as the set of vectors x ∈ R n max such that a r x b r x for all r ∈ [p] , where a r j = G rj if ǫ rj is − (resp. b r j = G rj if ǫ rj is +) and a r j = ¼ otherwise (resp. b r j = ¼ otherwise). An oriented lattice path in the sign pattern (ǫ ij ) is a sequence of entries of (ǫ ij ) starting from some top entry (1, i) and ending with some bottom entry (p, j) such that the successive entries are always either immediately at the right or immediately at the bottom of the previous ones. Therefore, such a path is composed of vertical segments oriented downward (vertical segments are supposed to be composed of at least two entries, with exception of the first and last vertical segments which are allowed to be composed of only one entry so that every paths starts and ends with a vertical segment) and horizontal segments oriented to the right (which are supposed to be composed of at                   · + · · · · · · · · · · · · + · · · · · · · · · · · · + · · − · · · · · · · · · · · · + · · · · · · · · · · · · + · · · · · · · · · · · · + − · · · · · · · · · · · · + · · · · · · · · · · · · − · · · + · · · · · · · · · · · · + · · · · · · · · · · · · + · · · · · · · · · · · · − + · ·                                     + + + + + − + + + + + + + + + + + − + + + + + + + + + + + − + + + + + + + + + + + − + + + + + + + + + + + − + + + + + + + + + + + − + + + + + + + + + + + − + + + + + + + + + + + − + + + + + + + + + + + − + + + + + + + + + + + − + + + + + + + + + + + − + + + + + + (v) as soon as a pair (−, +) occurs as the pair of extreme signs of some horizontal segment, the pairs of signs corresponding to all the horizontal segments below this one must also be equal to (−, +).                   Examples of tropically allowed lattice paths are shown in Figure 4, we refer the reader to [AGK10] for more information. The interest in tropically allowed lattice paths is given by Theorem 2 of [AGK10], which shows that the extreme rays of the polar of the signed cyclic polyhedral cone with sign pattern (ǫ ij ) are in one to one correspondence with tropically allowed lattice paths for (ǫ ij ). Consider now the ith polar K • i of P(p, n), which consists of the vectors b ∈ R n max such that b i x i j∈[n\i] b j x j for all x ∈ P(p, n). By the discussion in Section 2, it follows that the extreme rays of the polar of P(p, n), with exception of those associated with the trivial inequalities x i x i , correspond to the extreme rays of the cones K • i , i ∈ [n] . Observe that, if we consider the sign pattern (ǫ ij ) all whose entries are + signs with exception of column i in which they are − signs, then K • i is the polar of the signed cyclic polyhedral cone with sign pattern (ǫ ij ). Given the structure of this sign pattern, its tropically allowed lattice paths (which therefore correspond to the extreme inequalities of P(p, n) associated with the extreme rays of K • i ) can be classified as follows. (i) There are n − 1 vertical tropically allowed paths, corresponding to the trivial inequalities x j ¼ for j ∈ [n \ i]. (ii) There are n − 1 tropically allowed paths with exactly one horizontal segment. If the extreme pair of signs of this segment is (+, −), then, the − sign must be on the last row, and so there are i−1 choices for the column containing the + sign (i.e. the column containing the first vertical segment). If the extreme pair of signs of the horizontal segment is (−, +), then, the − sign must be on the first row and thus there are n − i choices for the column containing the + sign (i.e. the column containing the last vertical segment). This makes a total of n − 1 extreme rays, corresponding to inequalities of the form t i p x j t j p x i for j ∈ [i − 1] and t i 1 x j t j 1 x i for j ∈ [i + 1, n], where for all r ∈ [n], we set [r, n] := {r, r + 1, . . . , n}. (iii) The other tropically allowed paths must consist of two horizontal segments on consecutive rows, which have (+, −) and (−, +) as successive pairs of extreme signs, see Figure 4 for an example. The first row m can take p − 1 values, and for each of these, we must choose a column j in [i − 1] containing the first vertical segment of the path and a column k in [i + 1, n] containing the last vertical segment of the path. The associated extreme ray can be shown (see [AGK10]) to correspond to the inequality t i m+1 t j m x k ⊕ t i m t k m+1 x j t j m t k m+1 x i . Thus, for a fixed i, we have a total of n − 1 + (p − 1)(n − i)(i − 1) extreme inequalities, excluding the trivial inequalities x j ¼, j ∈ [n \ i] (since these inequalities arise several times for different values of i, they must be counted separately). Summing the latter quantity over i ∈ [n], and adding the 2n trivial inequalities x i x i and x i ¼, i ∈ [n], we arrive at the formula given in the proposition. Remark 1. As mentioned earlier, in classical convex geometry, cyclic polyhedral cones are known to maximize the number of extreme rays of the polar, among all cones with p generators in dimension n. By combining Proposition 6 and Example 2, we see that the same is not true in the tropical case. It would be interesting to find the "maximizing model" in this case. We now derive some algorithmic consequences of Theorem 4. The extreme rays of the polar of a tropical polyhedral cone can be computed by the tropical double description method [AGG10b], see also [AGG09b,All09a] for more information. The latter is a general method, which determines the extreme rays of a tropical polyhedral cone defined as the intersection of tropical half-spaces. This method computes a sequence of intermediate polyhedral cones, given by the intersection of successive half-spaces. Its execution time is polynomial in the size of the input and the maximal number of extreme rays of these intermediate cones. However, there are instances in which this number blows up, so that the execution time can be exponential in the size of the input and the output. Theorem 4 will allow us to exploit known algorithms for variants of a classical problem in hypergraph theory: finding all minimal transversals. This leads to an alternative algorithm to compute the polar, which, by comparison with the tropical double description method, has the advantage of running in a time which is quasi-polynomial in the size of the input and the output. However, it should be noted that this alternative algorithm can only be applied to the intersection of tropical half-spaces defined by inequalities of the same type i (the ith polar of a tropical polyhedral cone is given by the intersection of such half-spaces), while the tropical double description method can handle the intersection of tropical half-spaces defined by inequalities of different types. Let us recall that, given a (undirected) hypergraph with set of nodes N and set of hyperedges E (i.e. E is a family of subsets of N ), a transversal or hitting set of this hypergraph is a set T ⊆ N such that M ∩ T = ∅ for all M ∈ E. A transversal T is minimal if no proper subset of T is also a transversal. The minimal transversal problem consists in finding all minimal transversals of a given hypergraph. We next show that the minimal covers arising in Proposition 5 may be thought of as weighted generalizations of hypergraph transversals. Consider a hypergraph with set of nodes [n − 1], and let E = {M 1 , . . . , M p } be its set of hyperedges. We associate with this hypergraph the p × n matrix G = [F, e], where e is the p dimensional tropical unit column vector and F is the p × (n − 1) matrix defined by: F ij = ½ if j ∈ M i and F ij = ¼ otherwise. Then, it can be easily checked that the entries of any minimal element of the tropical polyhedron z ∈ R n−1 max \| ⊕ j∈[n−1] z j G ·j G ·n = z ∈ R n−1 max \| F z e can only take values in the set {½, ¼} and that z ∈ {½, ¼} n−1 is a minimal element of this polyhedron if, and only if, the set {j ∈ [n − 1] \| z j = ¼} is a minimal transversal of the given hypergraph. In other words, the rows of F represent the incidence vectors of hyperedges, and the minimal elements z are the incidence vectors of minimal transversals. Therefore, by Theorem 4, minimal transversals of the given hypergraph correspond to extreme rays of the nth polar of R(G) associated with vectors b such that b n = ¼. We summarize this discussion by the following corollary. Corollary 7. The minimal transversal problem reduces to the computation of the extreme rays of the ith polar of a tropical cone. Fredman and Khachiyan [FK96] showed that the minimal transversal problem can be solved in incremental quasi-polynomial time. This means that given a set S of already computed minimal transversals, the time needed to compute a new minimal transversal or to decide that there are no more minimal transversals is bounded by 2 polylog(m) , where m = k + \|S\| and k is the size of the input. Boros, Elbassioni, Gurvich, Khachiyan and Makino extended this result in [BEG + 02] to the case of systems of monotone linear inequalities, and considered general dualization problems (see also [KBEG06]). Elbassioni showed in [Elb08, Theorem 1], as a consequence of [BEG + 02], that the minimal elements of a set of the form {x ∈ R n max \| Ax b , l x u} , where b, l and u are vectors of R n max and A is a p × n matrix with entries in R max , can also be computed in incremental quasi-polynomial time. (Actually, the setting of [Elb08] concerns "maxtimes" inequalities, but the present setting is equivalent.) Taking A as the matrix whose columns are the columns of G with exception of column i and b as the ith column of G, if we define l h := ¼ and u h := w k h h for h ∈ [n \ i], we conclude from Elbassioni's theorem that the minimal elements of Z i can be computed in incremental quasi-polynomial time. Combining this remark with Theorem 4, we obtain: Corollary 8. The extreme rays of the polar of a tropical polyhedral cone can be computed in incremental quasi-polynomial time. A tropical analogue of Farkas lemma involving mean payoff games The classical Farkas lemma shows that a (homogeneous) linear inequality over the reals can be logically deduced from a finite family of linear inequalities if, and only if, it can be expressed as a nonnegative linear combination of the inequalities in this family. As it was observed in [GK09], the same is not true in the tropical setting (see Figure 5 below). This raises the question of deciding whether Ax Bx =⇒ cx dx (6) for all x ∈ R n max , where A, B are p × n matrices and c, d are row vectors of dimension n, all of them with (effective) entries in R max . Equivalently, given a finite system of (tropical) linear inequalities, we may ask whether one of them is redundant. In recent works [AGG09a, AGG10a], Akian, Gaubert and Guterman showed that checking whether a tropical polyhedral cone is trivial (i.e. reduced to the identically ¼ vector) reduces to solving a mean payoff game problem. We next show that the problem of deciding whether implication (6) holds also reduces to a mean payoff game problem. We refer the reader to [GKK88,ZP96] for more background on these games. In order to perform this reduction to games, it is convenient to establish first some simple preliminary properties. In many applications, the finite entries of the matrices A, B and the vectors c, d are integers. Then, it follows from the next result that the validity of implication (6) does not change if one considers real or integer variables. In the sequel, if H is a subgroup of (R, +), we denote by H max the semiring (H ∪ {−∞}, max, +). Proposition 9. Assume that A, B ∈ H p×n max and c, d ∈ H n max , where H is a subgroup of (R, +). Then, the implication Ax Bx =⇒ cx dx holds for all x ∈ R n max if, and only if, it holds for all x ∈ H n max . Proof. Let K := {x ∈ R n max \| Ax Bx}. The tropical analogue of the Minkowski theorem [GK06, GK07,BSS07] shows that every vector of K is a tropical linear combination of vectors in the extreme rays of K. Hence, the implication is valid for all x ∈ R n max if, and only if, every vector x in an extreme ray of K satisfies cx dx. Since all the vectors in a ray are proportional in the tropical sense, it suffices to check that cx dx holds for a suitably normalized vector of each extreme ray. For instance, we may normalize a vector of a ray by requiring that x j = 0, where j is the first index in [n] such that x j is finite. The normalized vectors of extreme rays will be referred to as extreme generators. Then, the explicit construction of the extreme generators, in the tropical double description algorithm [AGG10b], shows that every finite entry of the extreme generators belongs to the subgroup H of (R, +), because all the operations performed by the algorithm preserve this subgroup. Hence, if the implication holds for all x ∈ H n max , it holds in particular for all the normalized vectors of the extreme rays of K, and so it holds for all x ∈ R n max . We shall need the following technical proposition. M jr 1 + M r 1 r 2 + · · · + M r m−1 k , where M st := max i∈[p], A is ,B it =¼ \|A is − B it \| . In particular, \|y j − y k \| M := (n − 1) max s,t∈[n] M st . Proof. To bound \|y j − y k \|, we shall use the characterization of the extreme vectors of K in terms of tangent directed hypergraphs established in [AGG10b, Theorem 3.7], which was already recalled in Section 2. The tangent directed hypergraph of K at y ∈ R n max is the directed hypergraph E = {({j ∈ [n] \| B rj y j = B r· y}, {j ∈ [n] \| A rj y j = A r· y}) \| r ∈ [p], A r· y = B r· y = ¼} . Here, we shall also use an undirected graph, denoted by G(K, y), with the same set of nodes N and with an edge connecting nodes j and k if there exists an hyperedge (M, M ′ ) ∈ E such that j ∈ M and k ∈ M ′ . In other words, the edge (j, k) belongs to G(K, y) if, and only if, A rk y k = A r· y = B r· y = B rj y j = ¼ for some r ∈ [p]. Recall that the result of [AGG10b] shows that y ∈ K belongs to an extreme ray of K if, and only if, H(K, y) has a smallest strongly connected component. It follows that in particular the underlying undirected graph G(K, y) must be connected. Note that for any edge (j, k) of G(K, y) we have (with the usual notation) A rk + y k = B rj + y j for some r ∈ [p], and so \|y j − y k \| M kj . Consider now any two nodes j, k of N . Since G(K, y) is connected, it must contain an undirected path j, r 1 , . . . , r m−1 , k of length m \|N \| − 1 n − 1 connecting these two nodes, which shows (7). The following immediate corollary shows that when implication (6) does not hold, we can construct a counter example by assigning to the variables values which are not too large. Corollary 11. If the implication Ax Bx =⇒ cx dx does not hold, there is a vector y ∈ R n max that satisfies (7) such that Ay By and cy > dy (a counter example). Proof. If implication (6) does not hold, there is at least one extreme generator y of the tropical polyhedral cone {x ∈ R n max \| Ax Bx} such that cy > dy, and so (7) is valid for y. Remark 2. The previous corollary is related to the "small model property" established by Bezem, Nieuwenhuis, and Rodríguez-Carbonell for the "max-atom" problem. Lemma 1 of [BNRC10] (see also [BNRC08]) deals with a system of inequalities of the form x i k i,r,s + max(x r , x s ), for all (i, r, s) ∈ V , where the set V is given, and every coefficient k i,r,s is a given integer. They show that if this system has a finite integer solution, then, it also has a finite integer solution y such that \|y i − y j \| (i,r,s)∈V \|k i,r,s \|. We now present the reduction to games. Given a scalar λ ∈ R, we shall consider the system of inequalities Ax Bx, λdx cx. Following [AGG09a], with this system we associate a mean payoff game in which there are two players, the maximizer "Max" and the minimizer "Min". This game can be represented by a bipartite digraph G λ with two classes of nodes: the row nodes [p + 1] and the column nodes [n]. For i ∈ [p] and j ∈ [n], we draw an arc with weight B ij from row node i to column node j if B ij ∈ R, and we draw an arc with weight −A ij from column node j to row node i if A ij ∈ R. Similarly, we draw an arc from row node p + 1 to column node j with weight c j if c j ∈ R, and we draw an arc from column node j to row node p + 1 with weight −λ − d j if d j ∈ R. This is illustrated in Example 4 and Figure 6 (Left) below. The mean payoff game associated with this bipartite digraph consists of moving a token along its edges. When the token is at row node i, Player Max must select an arc leaving i and receives the weight of the arc as a payment from Player Min. When the token is at column node j, Player Min must select an arc leaving j, and pays to Player Max the weight of this arc. (We warn the reader that an opposite convention of sign is used in [AGG09a]: it is assumed there that at each step, the player who makes the moves receives the amount indicated on the arc, whereas here, Player Max receives this amount, even when Player Min makes the move.) We shall need the following simple assumption, which guarantees that each player has at least one action available in every node. Assumption 1. For all j ∈ [n], d j ∈ R or there exists i ∈ [p] such that A ij ∈ R. There exists j ∈ [n] such that c j ∈ R and, for all i ∈ [p], there exists j ∈ [n] such that B ij ∈ R. Observe that, since we are interested in studying the implication Ax Bx =⇒ cx dx, we may always assume that the conditions of Assumption 1 hold. Indeed, by adding to the p inequalities Ax Bx the n trivial inequalities x j x j , j ∈ [n], we obtain an equivalent implication in which the first condition of Assumption 1 is satisfied. If for some i ∈ [p] we have B ij = ¼ for all j ∈ [n], then Ax Bx implies x j = ¼ for all j ∈ [n] such that A ij = ¼. Therefore, by eliminating the ith inequality and the variables x j for which A ij = ¼, we obtain a new implication which is equivalent to the original one. By repeating this elimination procedure a finite number of times, we eventually arrive at an equivalent implication (involving a subset of variables) which satisfies the last condition of Assumption 1. Finally, observe that we may always assume that c is not the identically ¼ vector because otherwise the implication trivially holds. Hence, in the sequel, we shall always require the matrices to satisfy Assumption 1 without stating it explicitly. The dynamic programming operator g λ of the game described above is the self-map of R n given by [g λ (x)] j := min min i∈[p],A ij ∈R − A ij + max k∈[n] (B ik + x k ) , −λ − d j + max k∈[n] (c k + x k ) if d j ∈ R, [g λ (x)] j := min i∈[p],A ij ∈R − A ij + max k∈[n] (B ik + x k ) otherwise. Observe that in this section, for more readability, we come back to the usual notation (instead of the tropical one) when dealing with dynamic programming operators of games. The fact that g λ preserves R n follows readily from Assumption 1, which implies that the maxima and minima appearing in the previous expressions only take finite values when x ∈ R n and λ ∈ R. Note also that g λ has a unique continuous extension to R n max , that we will denote by the same symbol g λ (R n max is equipped with the product topology which arises when considering the metric (x, y) → \|e x − e y \| on R max ). Actually, the meaning of the previous expressions giving [g λ (x)] j is unambiguous, even when x ∈ R n max , and this determines the extension. Observe that g λ satisfies x y =⇒ g λ (x) g λ (y), i.e. g λ is order preserving. Moreover, for any scalar µ ∈ R max and x ∈ R n max , we have g λ (µ + x) = µ + g λ (x), so we shall say that g λ commutes with the (usual) addition of a scalar. We denote by ρ(f ) the (non-linear) spectral radius of a continuous order preserving self-map f of R n max that commutes with the addition of a scalar. Recall that ρ(f ) is defined as the maximal scalar µ for which there exists a non-identically ¼ vector x ∈ R n max (non-linear eigenvector) such that f (x) = µ + x. In other words, ρ(f ) is the maximal "additive eigenvalue" of f . We refer the reader to [AGG09a] for more background. Since the map g λ preserves R n , is piecewise affine and sup-norm nonexpansive, the following limit, called cycle time, is known to exist [Koh80]: χ(g λ ) := lim k→∞ g k λ (0) k . (8) Here, g k λ denotes the kth iterate of g λ and 0 is the n dimensional zero vector. Kohlberg actually shows a stronger result, that there is an invariant half-line on which g λ acts by translation. It follows easily from this result that the jth entry of χ(g λ ), denoted by χ j (g λ ), coincides with the value of the game when the initial state is column node j and the payoff of an infinite run is defined as the average payment per turn made by Player Min, as in [LL69]. (The equivalence is detailed in [AGG09a].) A Collatz-Wielandt type formula ([AGG09a, Lemma 2.8], see also [GG04]) shows that ρ(g λ ) = χ(g λ ) := lim k→∞ 1 k max j∈[n] (g k λ (0)) j ,(9) and so ρ(g λ ) = max j∈[n] χ j (g λ )(10) can be interpreted as the value of an associated mean payoff game in which Player Max is allowed to select the initial state j, see Proposition 2.11 of [AGG09a] for details. We shall refer to ρ(g λ ) as the mean payoff (value) of the latter game. The introduction of these mean payoff games is motivated by the following propositions. Proposition 12. The system of inequalities Ax Bx does not imply the scalar inequality cx dx if, and only if, max j∈[n],c j =¼ χ j (g λ ) 0 for some λ > 0. Proof. There exists a vector y such that Ay By and cy > dy if, and only if, there exists a scalar λ > 0 such that the tropical polyhedral cone (11) K λ := {x ∈ R n max \| Ax Bx, λdx cx} contains a vector x for which cx = ¼, i.e. K λ contains a vector x satisfying x j = ¼ for some j ∈ [n] such that c j = ¼. Then, the proposition follows from Theorem 3.2 of [AGG09a], which shows that the tropical polyhedral cone K λ contains a vector x such that x j = ¼ if, and only if, χ j (g λ ) 0, i.e. column node j is a winning initial state for Player Max in the mean payoff game associated with the system of inequalities Ax Bx, λdx cx. The next result considers the case of vectors with only finite entries. Proposition 13. The implication Ax Bx =⇒ cx dx does not hold for all x ∈ R n if, and only if, min j∈[n] χ j (g λ ) 0 for some λ > 0. Proof. The implication Ax Bx =⇒ cx dx does not hold for all x ∈ R n if, and only if, there exists a scalar λ > 0 such that the tropical polyhedral cone K λ defined in (11) contains a finite vector. Then, as in the proof of Proposition 12, the result follows from Theorem 3.2 of [AGG09a] because this theorem shows that K λ contains a finite vector if, and only if, χ j (g λ ) 0 for all j ∈ [n]. The situation in which cx = dx = ¼ for some non-identically ¼ vector x in the tropical cone {x ∈ R n max \| Ax Bx} appears to be degenerate. Hence, in the sequel we shall use the following technical assumption, which, as we shall shortly see, implies no loss of generality. Assumption 2. If x ∈ R n max is such that Ax Bx and cx = dx = ¼, then x is the identically ¼ vector. The following lemma shows that the vector d may always be required to be finite. Then, the previous assumption is trivially satisfied. Lemma 14. Let the constant M be defined as in Proposition 10, and define the vector d ′ ∈ R n by d ′ i := d i if d i = ¼ −M − 1 + min j∈[n],c j =¼ c j otherwise. Then d can be replaced by d ′ without changing the validity of the implication Ax Bx =⇒ cx dx. Proof. Since d d ′ , the implication in which d ′ appears is weaker than the one with d. Assume that the latter implication does not hold. Then, by Corollary 11, there is a vector y such that Ay By and dy < cy, and this vector satisfies (7). Let j denote any index such that c j y j = cy. Using (7) we deduce that d ′ k y k (−M − 1)c j y k < c j y j = cy for any k such that d k = ¼, and so d ′ y < cy, showing that the implication in which d is replaced by d ′ does not hold. Thanks to Assumption 2, the validity of the implication Ax Bx =⇒ cx dx can now be characterized in terms of the spectral radius. Proposition 15. The system of inequalities Ax Bx does not imply the scalar inequality cx dx if, and only if, ρ(g λ ) 0 for some λ > 0. Proof. Since Assumption 2 holds, there is a vector y such that Ay By and cy > dy if, and only if, there exists a scalar λ > 0 such that the tropical polyhedral cone K λ defined in (11) is not trivial (i.e., not reduced to the identically ¼ vector). Then, the conclusion follows from Theorem 3.1 of [AGG09a], which shows that K λ is not trivial if, and only if, the mean payoff game associated with the system of inequalities Ax Bx, λdx cx has at least one winning initial state for Player Max, which by Lemma 2.8 and Proposition 2.11 of the same paper holds if, and only if, the associated dynamic programming operator g λ has spectral radius at least 0. We call the map λ → ρ(g λ ) the spectral function. The idea of considering a parametric spectral radius somehow similar to this one appears in [GS09], where it is used to solve a different problem (two-sided eigenproblem). We next indicate some elementary properties of the spectral function. Lemma 16. The spectral function λ → ρ(g λ ) is non-increasing. Proof. We claim that for all k ∈ N and x ∈ R n , the map λ → g k λ (x) is order reversing from R to R n , meaning that λ µ implies g k µ (x) g k λ (x). We prove this claim by induction. For k = 1, this property is immediate. Assume that our claim holds for k = r. Then, if λ µ, we have g r+1 µ (x) = g µ (g r µ (x)) g λ (g r µ (x)), and since y → g λ (y) is order preserving, using the induction hypothesis, we conclude that g λ (g r µ (x)) g r+1 λ (x), proving our claim. Thus, it follows from (9) that the map λ → ρ(g λ ), which is a pointwise limit of non-increasing functions, is non-increasing. Now we show that the spectral function is piecewise affine, by describing explicitly a complete family of "tangent" affine maps. This description involves the notion of strategy. We call strategy for Player Min a map σ which assigns to each column node j ∈ [n] a row node σ(j) ∈ [p + 1] such that A σ(j)j ∈ R if σ(j) ∈ [p], or d j ∈ R if σ(j) = p + 1. We associate with the strategy σ the map g σ λ defined by: [g σ λ (x)] j := −A σ(j)j + max k∈[n] (B σ(j)k + x k ) if σ(j) ∈ [p] , −λ − d j + max k∈[n] (c k + x k ) if σ(j) = p + 1 . Observe that with the tropical notation, [g σ λ (x)] j = A −1 σ(j)j B σ(j)· x if σ(j) ∈ [p] , λ −1 d −1 j cx if σ(j) = p + 1 , so g σ λ is a tropical linear map. By the definition of g λ , we have the following selection property, which holds for all λ ∈ R, For each x ∈ R n max there exists a strategy σ such that g λ (x) = g σ λ (x) . We shall use the following result, which may be thought of as a variant of the "duality theorem" established in [GG98b] (see also [GG98a]). Lemma 17. Let g : R n max → R n max be a continuous order preserving map that commutes with the addition of a scalar. Assume that g is the pointwise infimum of a family of maps {h σ } σ∈Σ all of which are continuous, order preserving and commute with the addition of a scalar. If for each x ∈ R n max there exists σ ∈ Σ such that g(x) = h σ (x) (selection property), then ρ(g) = min σ∈Σ ρ(h σ ) .(13) Proof. It follows from the characterization (9) that the map g → ρ(g) is non-decreasing. Hence, ρ(g) min σ∈Σ ρ(h σ ). The Collatz-Wielandt formula (see [AGG09a, Lemma 2.8]) shows that the spectral radius of a map f : R n max → R n max which is continuous, order preserving and commutes with the addition of a scalar satisfies the equality: ρ(f ) = inf {µ ∈ R \| ∃y ∈ R n , f (y) µ + y} .(14) Therefore, for any α > 0, there exists a vector y ∈ R n such that g(y) ρ(g) + α + y. Using the selection property, we deduce that h σ (y) = g(y) ρ(g) + α + y for some σ ∈ Σ. Hence, by (14) we have ρ(h σ ) ρ(g) + α. Since this holds for any α > 0, we conclude that min σ∈Σ ρ(h σ ) ρ(g). By the previous lemma, it follows that ρ(g λ ) = min σ ρ(g σ λ ) (15) for all λ ∈ R, where the minimum is taken over the set of all strategies for Player Min. A strategy σ for Player Min defines a "one player sub-game" in which only Player Max has to make choices. This sub-game corresponds to the sub-graph G σ λ of G λ in which for each column node j we delete all the arcs leaving this node except the one going to row node σ(j). Define the length of a circuit in the digraph G σ λ to be the number of column nodes that it contains. Then, it follows from the max-plus spectral theorem (see for example [CG79,BCOQ92]) that ρ(g σ λ ) coincides with the maximal weight-to-length ratio of circuits in the digraph G σ λ , see also [CTGG99] for a discussion adapted to the present setting. The classical Farkas lemma gives a simple "certificate" that a linear inequality over the reals is implied by a finite family of linear inequalities. This certificate consists of nonnegative coefficients (Lagrange multipliers) expressing the given inequality as a linear combination of the inequalities in the family. The following result does the same in the tropical setting. However, the certificate is now of a different nature: the collection of Lagrange multipliers is replaced by a strategy. Theorem 18 (Tropical analogue of Farkas lemma). The implication Ax Bx =⇒ cx dx holds if, and only if, there exists a strategy σ for Player Min such that every circuit in the digraph G σ 0 has nonpositive weight, and if a circuit in this digraph has zero weight, then it passes through row node p + 1. Proof. Observe that for each strategy σ, the map λ → ρ(g σ λ ) is piecewise affine. Actually, it is given by the maximal weight-to-length ratio of the (elementary) circuits in the digraph G σ λ , and the weight of each of these circuits is an affine function of λ. Since there is a finite number of strategies, we deduce from (15) that there exist a strategy σ and a positive numberλ such that ρ(g λ ) = ρ(g σ λ ) , for all λ ∈ [0,λ]. As the spectral function is non-increasing, by Proposition 15 it follows that the implication Ax Bx =⇒ cx dx holds if, and only if, ρ(g λ ) = ρ(g σ λ ) < 0 for all λ ∈ (0,λ]. Now, using the characterization of the spectral radius of the tropical linear map g σ λ as the maximal weight-to-length ratio of circuits in the digraph G σ λ , from ρ(g σ λ ) < 0 for all λ ∈ (0,λ], we deduce that every circuit in G σ 0 must have nonpositive weight. Otherwise, by continuity of λ → ρ(g σ λ ), we would have ρ(g σ λ ) > 0 for some λ > 0, which is nonsense. We also deduce that every circuit of zero weight in G σ 0 (if any) must contain an arc on which the parameter −λ appears, i.e., an arc of weight −d j from some column node j to row node p + 1. Otherwise, by definition of G σ λ , the weight of this circuit would also be zero in G σ λ for every λ > 0, contradicting the fact that ρ(g σ λ ) < 0. This shows that the condition of the theorem is necessary. Conversely, assume that there exists a strategy σ satisfying the condition of the theorem. Then, by the characterization of the spectral radius ρ(g σ λ ) as the maximal weight-to-length ratio of circuits in G σ λ , it follows that ρ(g σ λ ) < 0 for all λ > 0. Since by (15) we have ρ(g λ ) ρ(g σ λ ) < 0 for all λ > 0, from Proposition 15 we conclude that the implication Ax Bx =⇒ cx dx holds. We now state a dual result, in which strategies are used to certify that the implication does not hold. We shall consider a strategy π for Player Max, which is a map from the set of row nodes to the set of column nodes, assigning to each row node i a unique arc leaving it, with destination to some column node π(i). Each such strategy defines a new sub-game, by erasing all arcs leaving row node i but the one going to column node π(i). We denote by G π λ the corresponding sub-graph of G λ . Define now the map g π λ by [g π λ (x)] j := min min i∈[p],A ij ∈R − A ij + B iπ(i) + x π(i) , −λ − d j + c π(p+1) + x π(p+1) if d j ∈ R, [g π λ (x)] j := min i∈[p],A ij ∈R − A ij + B iπ(i) + x π(i) otherwise. Observe that for every strategy π for Player Max, g π λ is a self-map of R n that commutes with the addition of a scalar, and it has a unique (continuous) order preserving extension to R n max , which is also denoted by g π λ . Hence, the definition of the additive spectral radius, ρ, applies to the map g π λ . Since g π λ g λ and for each x ∈ R n max there exists a strategy π such that g λ (x) = g π λ (x), we deduce that ρ(g λ ) = max π ρ(g π λ )(16) for all λ ∈ R. Indeed, we have already noted that the characterization (9) implies that the spectral radius ρ of a map is an order preserving function of this map, and so ρ(g λ ) max π ρ(g π λ ) .(17) To see that the equality is attained, the argument is dual to the proof of Lemma 17 above. If u is an eigenvector of g λ , so that g λ (u) = ρ(g λ ) + u, using the former selection property, we deduce that g π λ (u) = g λ (u) = ρ(g λ ) + u for some strategy π for Player Max, and so, ρ(g λ ) ρ(g π λ ), which implies that the equality holds in (17). By applying (10) to g π λ , we get ρ(g π λ ) = max j∈[n] χ j (g π λ ) .(18) Then, using (16) and (18), we obtain the following immediate consequence of Proposition 15. Proposition 19. The implication Ax Bx =⇒ cx dx does not hold if, and only if, there exists a strategy π for Player Max, a column node j ∈ [n] and a scalar λ > 0 such that χ j (g π λ ) 0. The cycle time χ(g π λ ) has a simple characterization. For each strongly connected component C of the digraph G π λ , let ν C denote the minimal weight-to-length ratio of the circuits in C (the length being defined as the number of column nodes in the circuit). Then, it is known that χ j (g π λ ) = min C ν C (19) where the minimum is taken over all the strongly connected components C to which there is a path in G π λ from column node j (see for instance [CTGG99,§ 1.4] or [CTCG + 98]). Recall that every minimal ratio ν C can be computed in polynomial time by Karp's algorithm. Arguing as in the proof of Theorem 18 and using (19), we arrive at the following result, which expresses Proposition 19 in combinatorial terms. This is somehow dual to Theorem 18. Corollary 20. The implication Ax Bx =⇒ cx dx does not hold if, and only if, there exists a strategy π for Player Max with the following property: in the digraph G π 0 there exists a column node j ∈ [n] such that every circuit reachable from j has nonnegative weight, and if a circuit of zero weight is reachable from j, then it does not pass through row node p + 1. Example 4. Consider the inequalities x 1 ⊕ (−2)x 3 x 2 and x 2 (−3)x 1 ⊕ x 3 . We next apply the previous method to show that these inequalities imply the inequality x 1 ⊕ x 2 x 3 . The tropical cones associated with these inequalities are illustrated in Figure 5 in barycentric coordinates. Observe that in this case, we have A = 0 ¼ −2 ¼ 0 ¼ , B = ¼ 0 ¼ −3 ¼ 0 , c = 0 0 ¼ , and d = ¼ ¼ 0 . x 1 x 2 x 3 x 1 ⊕ (−2)x 3 x 2 x 1 x 2 x 3 x 2 (−3)x 1 ⊕ x 3 x 1 x 2 x 3 x 1 ⊕ (−2)x 3 x 2 x 2 (−3)x 1 ⊕ x 3 x 1 x 2 x 3 x 1 ⊕ x 2 x 3 Figure 5. The final tropical linear inequality follows from the first two ones, although it cannot be obtained from them by tropical linear combinations. The associated bipartite digraph G λ is depicted in Figure 6, where row nodes are represented by squares and column nodes by circles. If we consider the strategy σ for Player Min defined by σ(1) = 1, σ(2) = 2 and σ(3) = 3, it can be checked that all circuits in G σ 0 have nonpositive weight and that any circuit of zero weight passes through row node p + 1 = 3. The latter can also be checked by deleting row node p+1 = 3 from G σ 0 , and the arcs adjacent to it (dotted arcs on Figure 6, middle) because the resulting digraph contains only one circuit and this circuit has negative weight. Therefore, by Theorem 18 we conclude that Ax Bx implies cx dx, as can be seen in Figure 5. Consider now the inequality 1x 1 ⊕ x 2 x 3 instead of x 1 ⊕ x 2 x 3 , so that in Figure 6 the weight of the arc connecting row node p + 1 = 3 with column node 1 is now 1 instead of 0. If we define the strategy π for Player Max by π(1) = 2, π(2) = 3 and π(3) = 1, then all circuits in G π 0 have positive weight. Thus, from Corollary 20, it follows that Ax Bx does not imply cx dx. Observe that x = (0, 0, 0) t satisfies Ax Bx but not cx dx. Remark 3. Let us now restrict our attention to instances in which the entries of the matrices A, B and the vectors c, d belong to Q max . Then, Theorem 18 implies that the problem "does Ax Bx =⇒ cx dx hold?" is in NP. Indeed, any strategy σ for Player Min satisfying the condition of the theorem provides a certificate which can be checked in polynomial time. To see this, it suffices to compute the maximal weight-to-length ratio of circuits in G σ 0 , which can be done by applying Karp's algorithm to every strongly connected component of G σ 0 . To be valid, the certificate requires these maximal weight-to-length ratios to be nonpositive. Moreover, if one of these maximal weight-to-length ratios is zero (indeed, to be valid, only the one corresponding to the strongly connected component containing row node p + 1 could be zero), we must check whether there is a circuit of zero weight in G σ 0 which does not pass through row node p + 1. This can be verified by deleting row node p + 1 and the arcs connected to it from the digraph G σ 0 and computing the maximal weight-to-length ratio of circuits in the resulting digraph. The sub-game G π 0 induced by a strategy π for Player Max (in bold), certifying that for a perturbed vector c (the weight c 1 of the arc connecting row node 3 with column node 1 is now 1), the implication no longer holds. Every circuit has now positive weight. Remark 4. Similarly, Corollary 20 implies that the problem "does Ax Bx =⇒ cx dx hold?" belongs to co-NP. A negative certificate (disqualification) is now a strategy π for Player Max and the validity of such a certificate can still be checked in polynomial time. We next only sketch the argument (which is more involved than in the preceding case), leaving details to the reader. First, apply (19) to compute χ(g π 0 ). This requires calling Karp's algorithm at most n times, and can therefore be done in polynomial time. If χ j (g π 0 ) > 0 for some j ∈ [n], the certificate is valid. If χ j (g π 0 ) < 0 for all j ∈ [n], the certificate is invalid. If none of the previous conditions is satisfied, for each j ∈ [n] such that χ j (g π 0 ) = 0 we proceed as follows. Assume that the strongly connected component C containing row node p + 1 is reachable from j and that ν C = 0 (otherwise, the certificate is valid). Now, consider a potential transformation, which consists, for every arc (i, j) in C, in replacing the weight w ij of this arc by w ′ ij = u i + w ij − u j , where a real number u k (the potential) must be chosen for each row or column node k. Obviously, this transformation does not change the weight of circuits, and a fortiori, the weight-to-length ratios of circuits. It follows from the duality theorem in linear programming that we can find a potential such that w ′ ij 0 for all arcs (i, j) in C, and then the circuits of zero weight in C are precisely the circuits composed of those arcs (i, j) such that w ′ ij = 0. Now, we delete all the arcs but these ones, which yields a sub-graph. If no circuit in this sub-graph passes through row node p + 1, the certificate is valid. Remark 5. In the proof of the tropical Farkas lemma, following the route of [AGG09a], we used techniques of non-linear Perron-Frobenius theory showing that the spectral radius ρ is a "morphism" with respect to the infimum or supremum of families of maps having a selection property, meaning that (15) and (16) hold. One might look for an alternative and more combinatorial proof. Indeed, we may define directly the value of the mean payoff game for every initial state as in [LL69] and then define ρ as the maximum of this value over all the initial states, instead as the limit of the value per time unit of the finite horizon game (9). As pointed out above, the theorem of Kohlberg [Koh80] implies that the two definitions of ρ coincide, in other words, that the value commutes with the limit. Instead of Kohlberg's theorem, one might exploit the combinatorial approach of Gurvich, Karzanov, and Khachiyan [GKK88], which relies on potentials solving certain systems of inequalities. It should be noted that the potential vector returned by their algorithm is of the same nature as an invariant half-line (the basepoint of an half-line determines a potential). We finally point out that Möhring, Skutella, and Stork [MSS04] established an equivalence between mean payoff games and certain scheduling problems with and/or constraints, which turn out to be equivalent to the existence of finite vectors in a tropical polyhedron. Some techniques used in [MSS04] might also yield alternative approaches to the present problems. When the entries of the matrices A, B and the vectors c, d belong to Z max , there turns out to be a simpler characterization. Proposition 21. The implication Ax Bx =⇒ cx dx does not hold if, and only if, ρ(g 1 ) 0. Proof. By Proposition 9, if the implication does not hold, there is a vector y ∈ Z n max such that Ay By and dy < cy. Since the finite entries of d, c and y are integers, we must have λdy cy for λ = 1. It follows that ρ(g 1 ) 0. Conversely, if ρ(g 1 ) 0, the system Ax Bx, λdx cx for λ = 1 has at least one non-identically ¼ solution y ∈ R n max . Then, by Assumption 2, we must have cy > ¼. It follows that dy < cy, showing that the implication does not hold. Recall that for a given λ, ρ(g λ ), which is the value of a mean payoff game, can be computed in pseudo-polynomial time by standard value iteration arguments. See [ZP96] and also [AGG09a, Section 3.2] for a refinement using the Collatz-Wielandt property. The existence of a polynomial time algorithm is an open question. By combining the results of [AGG09a,AGG10a] and Proposition 21, we arrive at the following result, in which the matrices and vectors are still required to have entries in Z max , and payments of games are still integers. Corollary 22. The problem "does a mean payoff game have at least one winning initial state?" (i.e., if g is the dynamic programming operator of a mean payoff game, does ρ(g) 0 hold?) is polynomial time equivalent to the problem of deciding whether the implication Ax Bx =⇒ cx dx holds. Proof. Theorem 3.1 of [AGG09a] shows that checking whether ρ(g) 0 is polynomial time equivalent to deciding whether an associated tropical polyhedral cone {x ∈ R n max \| Ax Bx} is not reduced to the trivial (identically ¼) vector. The latter reduces to checking whether the implication Ax Bx =⇒ cx dx does not hold, where c, d are any pair of vectors such that c j > d j > ¼ for all j ∈ [n]. Conversely, assume we have an oracle allowing us to decide whether ρ(g) 0 for any dynamic programming operator g of a mean payoff game with integer rewards in which n states belong to Player Min. By Proposition 21, it suffices to apply this oracle to the map g 1 to decide whether the implication holds. Remark 6. When ρ(g 0 ) < 0, the unique solution of the system Ax Bx, dx cx is the identically ¼ vector, and vice versa. Thus, the stronger implication Ax Bx, x ≡ ¼ =⇒ cx < dx is characterized by ρ(g 0 ) < 0. Proposition 21 leads to a greedy algorithm to construct non-redundant systems of inequalities defining a tropical polyhedral cone K. With this aim, we apply the following procedure. (i) We start from the extreme rays of the polar of K, which correspond to a finite family of inequalities a j x b j x, j ∈ J. (ii) We check, using the characterization of the proposition (by computing ρ(g 1 ), the value of a mean payoff game), whether any of these inequalities is implied by the other ones. If this is the case, we delete the inequality from the list. In this way, we arrive at a minimal set of inequalities defining K. The next example shows that such a set is not unique: running the previous greedy algorithm by scanning the inequalities in different orders yields incomparable minimal sets of defining inequalities. For instance, if K := P(5, 4) is the tropical cyclic polyhedral cone with five extreme rays in dimension 4, with t i = i for i ∈ , applying the previous algorithm we get the following minimal system of inequalities defining K: −1x 2 x 1 ⊕ −3x 3 −2x 2 x 1 ⊕ −5x 3 −3x 2 x 1 ⊕ −7x 3 −15x 4 x 1 −2x 3 x 2 ⊕ −5x 4 −3x 3 x 2 ⊕ −7x 4 −4x 3 x 2 ⊕ −9x 4 −5x 3 x 2 −5x 4 x 3 x 1 −1x 2 x 1 −2x 3 x 2 −2x 4 −4x 2 x 1 ⊕ −9x 3 −1x 3 x 2 ⊕ −3x 4 It can be checked that by replacing the four inequalities on the last two rows by x 2 −1x 3 x 3 −1x 4 −4x 2 x 1 ⊕ −14x 4 −2x 3 x 1 ⊕ −4x 4 we still get a minimal defining system. The following vectors are certificates that the first system of 14 inequalities above is minimal: each vector satisfies all the inequalities but the one on the same row and column. For the second system of inequalities, the certificates are: (0, 2, 3, 5) (0, 3, 5, 7) (0, 4, 7, 10) (¼, 0, 5, 10) (0, 3, 6, 8) (0, 4, 8, 11) (0, 4, 9, 13) (0, 1, 10, 15) (0, 1, 2, 15) (0, ¼, ¼, ¼) (0, 1, ¼, ¼) (0, 1, 2, ¼) (0, 7, 11, 15) (0, 2, 4, 5) Let us finally give more details on how the previous inequalities were obtained. We know from Proposition 6 that the polar cone of K has precisely 36 extreme rays. Excluding the 8 inequalities x i x i and x i ¼, for i ∈ [4], we have 28 non-trivial inequalities. The latter can be obtained by enumerating the corresponding tropically allowed lattice paths, following the proof of Proposition 6, or using directly the tropical polyhedral library TPLib [All09b]. These 28 inequalities consist of the 18 inequalities listed in the two groups above, together with the following 10 inequalities: 3x 1 x 4 4x 2 5x 1 ⊕ x 4 3x 3 7x 1 ⊕ x 4 4x 3 10x 1 ⊕ x 4 5x 3 13x 1 ⊕ x 4 6x 2 8x 1 ⊕ x 4 8x 2 11x 1 ⊕ x 4 −10x 3 x 1 −10x 4 x 2 −5x 2 x 1 Then, we eliminated successively redundant inequalities, by using the previously mentioned greedy method, in which we used the value iteration algorithm of [ZP96], or rather its variant in [AGG09a], to compute ρ(g 1 ) at each step. Figure 1 . 1Illustration of tropical convexity in R 2 max . (Left): A tropical convex set and three tropical segments in general position. (Right): The tropical cone generated by the vectors u and v. Figure 2 . 2Illustration of the proof of Proposition 2. (Left): The row space R(G) of the matrix G in (4). (Middle): The tropical cone K obtained by adding to the generators of R(G) the vectors e 1 and e 3 of the canonical basis of R 3 max . (Right): The four extreme inequalities of type 2. Figure 3 . 3Tangent directed hypergraph at an extreme vector b of the ith polar of a tropical polyhedral cone. The edges of the star-like directed sub-hypergraph are in black (thin lines). The whole directed hypergraph may contain additional hyperedges pointing to node i, here in gray. R n max defined by p linear inequalities a r x b r x, where {a r } r∈[p] and {b r } r∈[p] are two families of vectors of R n max , the tangent directed hypergraph of K at y ∈ R n max is defined as the directed hypergraph H(K, y) = (N, E) with set of nodes N = {i ∈ [n] \| y i = ¼} and set of hyperedges Then, b belongs to an extreme ray of this ith polar if, and only if, the tangent directed hypergraph H Figure 4 . 4Tropically allowed lattice paths. (Left): For the given tropically allowed lattice path, the signs of the entries indicated by the symbol "·" are irrelevant. (Right): A tropically allowed lattice path with two horizontal segments for the sign pattern associated with ith polar of P(p, n), where here n = 12, p = 11 and i = 6. least two entries). An oriented lattice path is said to be tropically allowed for the sign pattern (ǫ ij ) if the following conditions are satisfied:(i) every sign occurring on the initial vertical segment, except possibly the sign at the bottom of the segment, is positive;(ii) every sign occurring on the final vertical segment, except possibly the sign at the top of the segment, is positive;(iii) every sign occurring in any other vertical segment, except possibly the signs at the top and bottom of this segment, is positive;(iv) for every horizontal segment, the pair of signs consisting of the signs of the leftmost and rightmost positions of the segment is of the form (+, −) or (−, +); Proposition 10 . 10The finite entries of every vector y in an extreme ray of the tropical polyhedral cone K := {x ∈ R n max \| Ax Bx} satisfy: \|y j − y k \| max m n−1, r 1 ,...,r m−1 ∈[n] H (K, y) = (N, E) with set of nodes N = {i ∈ [n] \| y i = ¼} and set of hyperedges Figure 6 . 6Illustration of Theorem 18 and Corollary 20. (Left): The parametric game G λ allowing one to check the implication shown in Figure 5; row nodes are represented by squares (Max plays), column nodes by circles (Min plays); the strategy σ for Player Min certifying the implication is shown in bold. (Middle): The sub-game G σ 0 induced by this strategy: there are no circuits of positive weight, and every circuit of zero weight passes through the special row node p + 1. (Right): [5] in the definition of Example 3, i.e. if K is the set of tropical linear combinations of the rows of the matrix Tropical polyhedra are equivalent to mean payoff games. M Akian, S Gaubert, A Guterman, arXiv:0912.2462v1E-printM. Akian, S. Gaubert, and A. Guterman. Tropical polyhedra are equivalent to mean payoff games. E-print arXiv:0912.2462v1, 2009. Computing the extreme points of tropical polyhedra. X Allamigeon, S Gaubert, Goubault, arXiv:0904.3436E-printX. Allamigeon, S. Gaubert, andÉ. Goubault. Computing the extreme points of tropical polyhedra. 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Postal address:Instituto de Matemática "Beppo Levi", Universidad Nacional de Rosario, Avenida Pellegrini 250, 2000 Rosario, Argentina. E-mail address: [email protected]	[]
[ "DIVISION ALGEBRAS GRADED BY A FINITE GROUP", "DIVISION ALGEBRAS GRADED BY A FINITE GROUP" ]	[ "Eli Aljadeff ", "Darrell Haile ", "Yaakov Karasik " ]	[]	[]	Let k be a field containing an algebraically closed field of characteristic zero. If G is a finite group and D is a division algebra over k, finite dimensional over its center, we can associate to a faithful G-grading on D a normal abelian subgroup H, a positive integer d and an element ofis the Schur multiplier of H. Our main theorem is the converse: Given an extension 1 → H → G → G/H → 1, where H is abelian, a positive integer d, and an element of Hom(M (H), k × ) G , there is a division algebra with center containing k that realizes these data. We apply this result to classify the G-simple algebras over an algebraically closed field of characteristic zero that admit a division algebra form over a field containing an algebraically closed field.	10.1016/j.jalgebra.2021.03.008	[ "https://arxiv.org/pdf/1904.10686v1.pdf" ]	129,945,124	1904.10686	aeb846f7a746fd764ed42656870814e159c67ef8	DIVISION ALGEBRAS GRADED BY A FINITE GROUP 24 Apr 2019 Eli Aljadeff Darrell Haile Yaakov Karasik DIVISION ALGEBRAS GRADED BY A FINITE GROUP 24 Apr 2019 Let k be a field containing an algebraically closed field of characteristic zero. If G is a finite group and D is a division algebra over k, finite dimensional over its center, we can associate to a faithful G-grading on D a normal abelian subgroup H, a positive integer d and an element ofis the Schur multiplier of H. Our main theorem is the converse: Given an extension 1 → H → G → G/H → 1, where H is abelian, a positive integer d, and an element of Hom(M (H), k × ) G , there is a division algebra with center containing k that realizes these data. We apply this result to classify the G-simple algebras over an algebraically closed field of characteristic zero that admit a division algebra form over a field containing an algebraically closed field. introduction If A is a finite dimensional algebra over a field k and G is a finite group, we say A is G-graded if there is a decomposition A = ⊕ g∈G A g of A into a direct sum of k-vector spaces such that A g1 A g2 ⊂ A g1g2 for all g 1 , g 2 ∈ G. We say the grading is faithful if A g = 0 for all g ∈ G. In this paper we are interested in the case where A = D is division algebra over k faithfully graded by a finite group G and where k contains an algebraically closed field of characteristic zero. We begin with a brief description of our results followed by a more detailed discussion: Writing D = ⊕ g∈G D g we see easily that D e is a division algebra of degree d (say) over its center, and k = k · 1 ⊆ D e (However it is not necessarily the case that the center K of D is contained in D e or that K is even a graded subalgebra of D). The group G acts on the center L of D e with kernel H. We obtain in this way an extension 1 → H → G → G/H → 1 and a two-cocycle α with class in H 2 (H, L × ). The condition that k contains an algebraically closed field of characterisitic zero forces H to be abelian. Moreover the class of α is Ginvariant, that is, fixed by the natural action of G on H 2 (H, L × ) and gives rise to a G-invariant element in Hom(M (H), µ), where M (H) is the Schur multiplier of H and µ is the group of roots of unity in k. Our main result (Theorem 1.6 ) is the converse: Given an extension of groups 1 → H → G → G/H → 1 where H is abelian, a positive integer d, and a G-invariant element of Hom(M (H), µ), there is a field k containing an algebraically closed field of characterisitic zero and a division algebra D over k giving rise to the prescribed data. We apply our result (Theorem 1.10) to classify the G-simple algebras over an algebraically closed field Key words and phrases. graded algebras, graded division algebras, G-simple algebras, forms of algebras. The first author was partially supported by the ISRAEL SCIENCE FOUNDATION (grant No. 1516/16). Much of this work was done while the second author was a visitor at Guangtong-Technion Israel Institute of Technology. He would like to thank the institute for its support. of characteristic zero that admit a division algebra form over a field k containing an algebraically closed field. We proceed to the more detailed description. We begin the paper by determining the general structure of gradings by the group G on K-central division algebras (so K contains the base field k where as always k is assumed to contain an algebraically closed field of characteristic zero) and break the analysis into two cases: (1) The case in which the center K is contained in the e-homogeneous component. (2) The general case, in which the center K may not be contained in D e , and may not be G-graded. There are three types of gradings on division algebras. Our first two theorems (for the two cases mentioned above) will show that any grading by a finite group is a combination of these three types: (1) The trivial grading: For D any finite dimensional division algebra and G = {1}. (2) The crossed product grading: If L/K is a Galois extension with G as Galois group, we can construct the skew group algebra L t G which is isomorphic to the group algebra LG as a left vector space over L. We denote its elements by σ∈G l σ u σ , where l σ ∈ L and u σ is a symbol in L t G corresponding to the element σ ∈ G. The product is defined using distributivity and so as to satisfy the condition lu σ yu τ = lσ(y)u στ . It is well known that the algebra L t G is isomorphic to M n (K) where n = ord(G). In particular the skew group algebra structure gives a Ggrading on M n (K). Now we can twist the multiplication by a 2-cocycle of G with coefficients in L * . If α : G×G → L * is such a 2-cocycle we construct the crossed product algebra, denoted by L α t G, which again is isomorphic to LG and L t G as a left vector space over L with the product determined by the formula lu σ yu τ = lσ(y)α(σ, τ )u στ . Cohomologous 2-cocycles in H 2 (G, L * ) yield G-graded isomorphic algebras over K and in particular isomorphic algebras over K. On the other hand, noncohomologous 2-cocycles yield nonisomorphic K-central simple algebras and in particular nonisomorphic G-graded algebras. The crossed product L α t G is K-central simple and is a G-graded twisted form of L t G. It is well known (we provide a proof below) that for any finite group G one can find a G-Galois extension L/K and a cocycle α such that L α t G is a K-central division algebra. Moreover one can find such crossed product division algebras for which the center contains an algebraically closed field of characteristic zero. Note that here the e-component is L, a maximal subfield in L α t G. (3) The case of twisted group algebras: Let H be a finite group and consider twisted group algebras K α H where α is a 2-cocycle H × H → K * , and where the action of H on K * is trivial. Because char(K) = 0 the twisted group algebra is semisimple but in general not simple. It seems to be difficult to determine the finite groups H which admit a 2-cocycle over an aribitrary field K of characteristic zero such that the twisted group algebra K α H is a division algebra. In our case, in which the base field contains an algebraically closed field of characteristic zero, we will see that the group H must be abelian. It is easy to show that for any abelian group H we can find a cocycle α and a field K such that the twisted group algebra is a field and so in particular a finite dimensional division algebra. The case where K α H is central over K is of special interest. These are obtained precisely when H is abelian of the form A × A and the cocycle is a suitable nondegenerate cocycle on H (see the following Remark). Remark 1.1. Classically, a finite group G is said to be of central type if it has a complex representation of degree [G : Z] 1/2 where Z = Z(G) is the center of G. In particular the index of Z in G is an integer square. Since a linear representation of G gives rise to a 2-cocycle α on G/Z with values in C * and a projective representation of G/Z corresponding to α, researchers borrowed that terminology and defined (nonclassically) a group Λ of central type as a group that admits a 2-cocycle α ∈ Z 2 (Λ, C * ) such that the twisted group algebra C α Λ ∼ = M n (C). We say the cocycle α is nondegenerate. In this article we slightly extend the terminology and say a 2- cocycle Z 2 (Λ, K * ) is nondegenerate if the twisted group algebra K α Λ is a K-central simple algebra. We can now state our first structure theorem. Theorem 1.2. Let D be a division algebra, finite dimensional over its center K. Suppose D is faithfully graded by a finite group G. Let D e be the identity component of D and denote by L its center. Assume K is contained in D e and that K contains an algebraically closed field of characteristic zero. Then the following conditions hold: (1) Conjugation on D e by nonzero homogeneous elements x g ∈ D g determines a G-action on L, the center of D e . Furthermore, K = L G and hence if we denote by H the kernel of the action, L/K is a G/H-Galois extension. (2) The group H is abelian and is isomorphic to A × A, for some abelian group A. (3) There exists a 2-cocycle α ∈ Z 2 (H, L * ) such that D H = h∈H D h ∼ = D e ⊗ L L α H. Furthermore, the cocycle α is nondegenerate, that is L α H is an Lcentral simple algebra. The subalgebras L α H and D H have center L. (4) The cohomology class [α] ∈ H 2 (H, L * ) is G-invariant. (5) The algebra D = D G is a generalized G/H-crossed product in the sense of O. Teichmuller over the algebra D H (see [8] and [9]). Its degree is dim L (D e )\|H\|(G : H). Our second structure theorem deals with the general case; we drop the assumption that K, the center of D, is contained in the identity component and do not assume K is G-graded. (1) Conjugation on D e with nonzero homogeneous elements x g ∈ D g determines a G-action on L with K 0 = L G . If we denote by H the kernel of the action, then L/K 0 is a G/H-Galois extension. In the next theorem we determine under what circumstances the center K ( in Theorem 1.3 above) is G-graded. By statement (5) of the theorem, K is a subfield of D H and hence if K is G-graded, it is necessarily H-graded. (2) The algebra D is finite dimensional over K 0 . (3) The group H is abelian. (4) There exists a 2-cocycle α ∈ Z 2 (H, L * ) such that D H = h∈H D h ∼ = D e ⊗ L L α H. Hence, if we denote L 1 = Z(L α H), We continue the notation as in Theorem 1.2. For every h ∈ H we let x h be a representative in L α H (i.e., any nonzero element in (L α H) h ). Let S = {h ∈ H : x h x h ′ = x h ′ x h for all h ′ ∈ H}, that is, S is the set of elements in H such that x s ∈ L 1 , the center of L α H. We claim that in fact L 1 = L α S: It is clear that L α S ⊆ L 1 . Conversely if z = h∈H l h x h lies in L 1 , then for all r ∈ H, x r z = zx r . But H is abelian and x r commutes with the elements of L. It follows that if l h = 0 then h ∈ S. Note that because G acts on L 1 , it acts also on S, so S is a normal subgroup of G. Theorem 1.4. The center K is G graded if and only if S is central in G. The most substantial part of this paper is devoted to proving a converse to the structure theorems. To this end we adopt the following terminology: Let D be a finite dimensional division algebra with center K and suppose D is faithfully graded by a finite group G. By the structure theorems this gives rise to (1) a group extension with abelian kernel Because H is abelian the Schur multiplier can be identified with the wedge prod- 1 → H → G → G/H → 1,uct H ∧ H and if h 1 , h 2 ∈ H, then φ [α] (h 1 ∧ h 2 ) = α(h 1 , h 2 )α(h 2 , h 1 ) −1 . In the twisted group algebra L α H = h∈H Lx h one sees easily that for all h 1 , h 2 ∈ H, the commutator x h1 x h2 x −1 h1 x −1 h2 is a root of unity and we also have φ [α] (h 1 ∧ h 2 ) = x h1 x h2 x −1 h1 x −1 h2 . If we let F denote an algebraically closed field containing K then in fact H 2 (H, F * ) G ∼ = Hom(M (H), µ) G . The class [α] ∈ H 2 (H, L * ) G is nondegenerate if and only if its image in H 2 (H, F * ) G is nondegenerate (because L α H ⊗ L F = F α H). It therefore makes sense to speak of the nondegeneracy of φ [α] ∈ Hom(M (H), µ) G . Letting [β] ∈ H 2 (G/H, H) be the cocycle determined by the group extension given above, we see that any faithfully G-graded finite dimensional division algebra over its center determines uniquely an ordered triple ([β], φ, d) ∈ H 2 (G/H, H) × Hom(M (H), µ) G × N where N = {1, 2, 3, . . . } denotes the set of natural numbers. Definition 1.5. Let G be a finite group and H an abelian normal subgroup. We say the triple ([β], φ, d) ∈ H 2 (G/H, H) × Hom(M (H), µ) G × N is realizable if there exists a finite dimensional division algebra, faithfully G-graded which yields that triple. Here is the main result of the paper. Theorem 1.6. If G is a finite group and H is an abelian normal subgroup, then every ordered triple ([β], φ, d) ∈ H 2 (G/H, H) × Hom(M (H), µ) G × N is realizable. If φ is nondegenerate then the center of every realization will lie in the e-component of its center. We will say that an extension 1 → H → G → G/H → 1 with an abelian kernel is realizable if there exists a G-invariant map φ : M (H) → µ such that the triple ([β], φ, 1) is realizable. Because the trivial map φ is G-invariant we1 → H → G → G/H → 1 with an abelian kernel is realizable by a division algebra of degree [G : H]. Example 1.8. In [5] Cuadra and Etingof give an example of a finite dimensional division algebra graded by Q 8 , the quaternion group of order 8. In their example the center is not graded. In the following we describe the possible grading structure on division algebras by the group Q 8 and by the group D 4 , the dihedral group of order 8. As above D will denote the division algebra, K its center and K 0 its ecenter. We will assume D e = L, so d = 1 (See the remarks at the beginning of section 3). As above the field K 0 is assumed to contain an algebraically closed field of characteristic zero. Let 1 → H → Q 8 → Q 8 /H → 1 (1) If H = {1} we obtain a Q 8 crossed product division algebra of degree 8. H is cyclic). The e-component L is an extension of degree 2 over K 0 and the H-component is a field extension of L of degree 4. The center K is ungraded, an extension of degree 4 over K 0 . The division algebra D is a quaternion algebra over K. This is the example of Cuadra and Etingof. The e-component L is of degree 2 over K 0 and the H-component is a field extension of L of degree 4. The center K is ungraded, an extension of degree 4 over K 0 . The division algebra D is a quaternion algebra over K and because the extension is necessarily split, there is a quaternion algebra (2) If H ∼ = Z 2 the e-component is a field L which is a Z 2 × Z 2 Galois extension of the center K. The H-component D H is a field, and in fact K = D H , so the center is graded. The division algebra D is a Z 2 × Z 2 crossed product over K. (3) If H ∼ = Z 4 , the group S of Theorem 1.4 equals H (becauseNow let 1 → H → D 4 → D 4 /H → 1 (1) If H = {1} we obtain a D 4 crossed product division algebra of degree 8. (2) If H ∼ = Z 2 the e-component L is a Z 2 × Z 2 Galois extension of the center K = K 0 . The H-component D H is a field, and in fact K = D H , so the center is graded. The division algebra D is a Z 2 × Z 2 crossed product over K . Moreover there is a K 0 -subalgebra D 0 , also a Z 2 × Z 2 crossed product such that D ∼ = D 0 ⊗ K0 K. (3) If H ∼ = Z 2 × Z 2 and the map φ is trivial, then the e-component is L, an extension of degree 4 over K 0 . The subgroup S equals H and the division algebra D is a quaternion algebra over a nongraded center K which is of degree 2 over K 0 . Because the sequence is split, D is isomorphic to D 0 ⊗ K0 K, where D 0 is a quaternion algebra over K 0 . (4) If H ∼ = Z 2 × Z 2D 0 over K 0 such that D ∼ = D 0 ⊗ K0 K. Next we address the following problem: What are the G-simple algebras over an algebraically closed field of characteristic zero that admit a G-graded twisted form division algebra? Let D be a finite dimensional division algebra over its center K. Let K 0 = K ∩D e as above. Suppose K 0 contains an algebraically closed field of characteristic zero. Extension of scalars D E = D ⊗ K0 E where E is any algebraically closed field containing K 0 yields a finite dimensional G-simple algebra in which E will be the central homogeneous elements of degree e. Such algebras where classified by means of elementary and fine gradings by Bahturin,Sehgal and Zaicev [3]. Here is their result: Theorem 1.9. Let A be a finite dimensional G−simple algebra over an algebraically closed field F of characteristic zero. There is a subgroup H of G, a two cocycle c ∈ Z 2 (H, F * ) (the H−action on F * is trivial) and g = (g 1 , . . . , g m ) ∈ G (m) , such that A ∼ = F c H ⊗ M m (F ), where (by this identification) the G−grading on A is given by A g = span F u h ⊗ e i,j \|g = g −1 i hg j . Unlike the ungraded case, namely matrix algebras over E, not every finite dimensional G-simple algebra over E admits a G-graded twisted form division algebra and even not a G-graded division algebra twisted form (see [2], Theorems 1.8 and 1.12). We have the following result. (1) H is abelian (2) Every coset of H is represented in the elementary grading. Moreover the number of representatives is equal for all cosets. (3) H is normal in G (4) The cohomology class [α] ∈ H 2 (H, E * ) is G invariant where the action of G on E is trivial. proof of the structure theorems In this section we prove Theorems 1.2 − 1.4. We start with Theorem 1.2. Proof. Let D be a finite dimensional algebra over its center K and suppose D is graded by a finite group G. We are assuming the center K is contained in D e , the identity component of D. = u h θ −1 h centralizes D e . Let h and h ′ in H. Clearly x h x h ′ = α(h, h ′ )x hh ′ where α(h, h ′ ) ∈ D * e , but because x h and x h ′ centralize D e , we have α(h, h ′ ) ∈ L * . It follows that α : H × H → L * is a 2-cocycle in Z 2 (H, L * ). Finally, because the set {x h } h∈H is linearly independent over D e (and in particular over L) we obtain D H = D e ⊗ L L α H, as desired. Next we will show L is the center of the twisted group algebra L α H: If L 1 = Z(L α H) = Z(D H ), conjugation by nonzero homogeneous elements of D induces an action of G on L 1 with fixed field K. But H acts trivially on L 1 and so L G/H 1 = K. Because L ⊆ L 1 and the action of G/H is faithful, we obtain L = L 1 , as desired. It follows that the 2-cocycle α is nondegenerate and H is of central type. We show next that H is abelian: Because the group Γ of trivial units in L α H, that is the group of elements of the form, L * x h , h ∈ H, is center by finite, it follows from a theorem of Schur that the commutator subgroup of Γ is finite. That means the commutator subgroup consists of roots of unity, which in our case are in the base field k. It follows that if h 1 , h 2 ∈ H, then x h1 x h2 x −1 h1 x −1 h2 lies in k. But x h1 x h2 x −1 h1 x −1 h2 ∈ Lx h1h2h −1 1 h −1 2 , so x h1h2h −1 1 h −1 2 ∈ L which is in the e-component. Hence the commutator h 1 h 2 h −1 1 h −1 2 is trivial. Next we will show that the class [α] ∈ H 2 (H, L * ) is G invariant, where the action of G on the cohomology is induced by the given action on H and L. Let y g be a nonzero homogeneous element of degree g ∈ G. If h ∈ H, then y g x h y −1 g lies in D H and commutes with D e . It follows that conjugation by y g stabilizes L α H. We need to show the cocycle g(α) determined by g(α)(h, h ′ ) = y −1 g α(ghg −1 , gh ′ g −1 )y g is cohomologous to α. Write x h x h ′ = α(h, h ′ )x hh ′ . Conjugating both sides by y −1 g we get y −1 g x h y g y −1 g x h ′ y g = y −1 g α(h, h ′ )y g y −1 g x hh ′ y g Because y −1 g x h y g ∈ D g −1 hg and centralizes D e , we have y −1 g x h y g = γ g −1 hg x g −1 hg where γ g −1 hg ∈ L * . Note that the elements of D e centralize x h , so the value γ g −1 hg does not depend on the choice of y g ∈ D g = D e y g . Thus the equation above yields γ g −1 hg x g −1 hg γ g −1 h ′ g x g −1 h ′ g = y −1 g α(h, h ′ )y g γ g −1 hh ′ g x g −1 hh ′ g and so γ g −1 hg γ g −1 hg g −1 h ′ g α(g −1 hg, g −1 h ′ g)x g −1 hh ′ g = y −1 g α(h, h ′ )y g γ g −1 hh ′ g x g −1 hh ′ g . We conclude that γ g −1 hg γ g −1 hg g −1 h ′ g α(g −1 hg, g −1 h ′ g) = y −1 g α(h, h ′ )y g γ g −1 hh ′ g , showing the cocycles are cohomologous. It is now clear that the algebra D is a crossed product of G/H with coefficients in D H . It is straightforward to check that its degree is \|H\|(G : H). We proceed to prove Theorem 1.3, that is, we drop the assumption that the center is contained in the e-component. Proof. We have D = D e ⊕ D g2 ⊕ · · · ⊕ D gn and L = Z(D e ). As above, the group G acts on L and if H denotes the kernel of the action. then K 0 = L G/H . We want to show, as above, that we can find representatives x h , h ∈ H which centralize D e so that we have D H ∼ = D e ⊗ L L α H, for some α ∈ Z 2 (H, L * ), L 1 = Z(L α H) = Z(D H ). In order to apply Skolem Noether we need to know D e is finite dimensional over its center L. We are given that D is finite dimensional over K. Claim: the center K is contained in DH , whereH is the subgroup of H consisting on all elements g ∈ G such that conjugation with u g ∈ D g induces an inner automorphism of D e . Note that in that case there exists (as above) an element x g ∈ D g that centralizes D e . To prove the claim let z ∈ K. We have z = α e u e + α g2 u g2 + · · · + α gn u gn We show first α g = 0 if g ∈ H. Conjugation by elements of L centralizes z and preserves all homogeneous components. Since for every g we can find l g which does not centralize u g , the claim follows. Suppose now conjugation by u h does not give an inner automorphism on D e . Need to show α h = 0. Conjugation with any nonzero elements of D e , fixes z and so it must centralize α h u h . This shows conjugation of D e by α −1 h and u h determine the same action and hence the action of u h is inner if α is nonzero. This prove that K is contained in DH . But, since for every h ∈H there is x h homogeneous of degree h which centralizes D e we see that their product also centralizes D e and hence the algebra generated by x h , h ∈H over L is isomorphic to a twisted group algebra L αH where α :H ×H → L * is a 2cocycle. This shows D e is finite dimensional over L (because it is finite dimensional over K). We conclude that D e is finite dimensional over K 0 = L ∩ K. Note that in fact H =H, because by Skolem Noether, conjugation by any homogeneous element of D H is inner. The proof that H is abelian and that the cocycle α is G-invariant is the same as in Theorem 1.2. Let L 1 = Z(L α H). We have D H ∼ = D e ⊗ L L α H and hence L 1 = Z(D H ). Clearly, conjugation by homogeneous elements induces an action of G on L 1 whose kernel contains H. But the kernel must be equal to H since L is contained L 1 . Let S = {h ∈ H : x h x h ′ = x h ′ x h for all h ′ ∈ H}, that is, S is the set of elements in H such that x s ∈ L 1 , the center of L α H. We claim that in fact L 1 = L α S: It is clear that L α S ⊆ L 1 . Conversely if z = h∈H l h x h lies in L 1 , then for all r ∈ H, x r z = zx r . But H is abelian and x r commutes with the elements of L. It follows that if l h = 0 then h ∈ S. Because L 1 = L α S is the center of L α H the group H/S must be of central type, that is, isomorphic to A × A for some abelian group A. As in the previous theorem we now have that D G is a crossed product of G/H with D H . It is straightforward to check that its degree is (H : S)(G : H). We proceed to Theorem 1.4. Proof. The set up is as in Theorem 1.3. Recall the field L 1 = Z(L α H) where H is abelian, S is the subgroup of H whose representatives x s commute with x h for all h ∈ H. We have seen that L 1 = L α S and that S is normal in G. Let S 1 = S ∩ Z(G). Clearly L 2 = L α S 1 is contained in L 1 and normalized by G. Suppose S 1 = S, that is S is central in G. We want to show the center K is Hgraded and in particular G-graded. Indeed, if z ∈ k, z = h∈S γ s x s , conjugation by nonzero homogeneous elements y g acts on one hand trivially on z and on the other hand by multiplication of each x s by a nonzero scalar. Since the elements x s are linearly independent the result follows. Suppose now S is not central in G, that is, S 1 is a proper subgroup of S. We want to show the center K is not graded. Will show there is an element z = s∈S γsx s ∈ K where the elements x s are not central. Take s 0 ∈ S \ S 1 . Clearly x s0 is not in K. For every g ∈ G we choose nonzero y g ∈ D g , where as usual, if h ∈ H, then y h is chosen to commute with D e . Consider the element z = g∈G y g x s0 y −1 g . We show z is central. Since x s0 is in L 1 and conjugation by y g normalizes L 1 we have it is central in D H . It is sufficient to show z is invariant under conjugation by y g . But this is clear since conjugation of the sum by y g permutes the components modulo nonzero elements in D e which clearly centralize x s0 . It remains to show z = 0. Write g(x s0 ) = y g x s0 y −1 g = β g x g(x0) where β : G → L * . Note that for all h ∈ H β(h) = 1 because s 0 ∈ S. We claim β is a 1-cocycle. Indeed, g 1 g 2 (x s0 ) = g 1 (g 2 (x s0 )) The left hand side yields β(g 1 g 2 )x g1g2(s0) whereas the right hand side yields g 1 (β(g 2 )x g2(s0) ) = β(g 2 ) g1 β(g 1 )x g1g2(s0) and the result follows. Because β(H) = 1 we get an induced 1-cocycle ( which we will also call β) with class in H 1 (G/H, L × ). Applying Hilbert's Theorem 90 there is t ∈ L * with β(g) = g(t)t −1 . We can therefore write g(x s0 ) = g(t)t −1 x g(x0) and so g(tx s0 ) = t −1 x g(x0) Replacing the element x g(x0) by tx g(x0) and letting C G (s 0 ) denote the centralizer of s 0 in G, we see that z = \|C G (s 0 )\|x s0 + g ∈CG(s0) y g x s0 y −1 g which is nonzero. Note that here we have used the fact that the characteristic is zero. This completes the proof of the theorem. construction of division algebras graded by G In this section we prove that given a group extension 1 → H → G → G/H → 1 where H is abelian, a positive integer d, and a map φ : M (H) → µ which is Ginvariant there is a finite dimensional division algebra over its center, faithfully G-graded which realizes the given data. Our constructions will produce graded division algebras D in which d, which is the degree of D e , equals 1, that is, in which D e is a field. It is straightforward to pass from this case to the case of arbritrary d: Let L denote the field D e . Adjoin new variables a, b to k and let E = (a, b) d be the symbol algebra of degree d over k(a, b) determined by a and b. Then E is a division algebra and we can form D = D ⊗ K0 (E ⊗ k(a,b) K 0 ), where K 0 is the e-center of D and E ⊗ k(a,b) K 0 has the trivial grading. ThenD is a division algebra and has the obvious G-grading with D e = E ⊗ k(a,b) L. So the degree ofD e is d, as desired. Prior to our general treatment, we shall discuss two special cases: in case (1) we assume H = {1}, and in case (2) we assume G = H. The case where H is trivial is well known. We recall here a proof which is attributed to S. Rosset and K. Brown. Their idea will appear in the proof of the general case. Let 1 → R → F → G → 1 be a presentation of the finite group G where the groups F and R are finitely generated free. Taking quotient groups modulo the commutator subgroup [R, R] we obtain β : 1 → N = R/[R, R] → Γ = F/[R, R] → G → 1. We want the action of G on N in this extension to be faithful, that is, we want the centralizer of N in Γ to be N itself. This can be arranged as follows: Let A be the free abelian group on n = \|G\| generators and let G act on A by permuting these generators. We obtain the split exact sequence 1 → A → A ⋊ G → G → 1. We take our group F to be free on 2n generators y g , x g for all g ∈ G and map F to G by sending each y g to the identity and each x g to g. The group F maps to A ⋊ G by sending the y ′ g s to generators of A and each x g to g. The resulting extension denoted above by β maps to this split exact sequence and a simple diagram chase then shows that G acts faithfully on N . By a theorem of Higman ([6], Theorem 2) Γ = F/[R, R] is torsion free. Because Γ is abelian by finite, it follows from a theorem of Brown that the group algebra kΓ is a domain (See [4], Cor. 2). Furthermore, it follows from Goldie's theorem that kΓ as a classical ring of quotients which is actually a division algebra D. Clearly, since N is of finite index in Γ, the division algebra D is obtained by localizing kΓ at S = kN \ {0}. Because the action of G on N is faithful, it is faithful on kN and hence also on the field E = F rac(kN ). It is clear now that S −1 kΓ is a G-crossed product of the form (E/E G , G,β) whereβ is the cohomology class in H 2 (G, E * ) induced by the extension β above. We next consider case (2) in which G = H is abelian and φ is any map φ : M (H) → µ ⊂ k * . Note that in this case every cohomology class is invariant. Let 1 → R → F H → H → 1 be a presentation of H. We take quotients modulo [R, F H ] and obtain the central extension 1 → R/[R, F H ] → F H /[R, F H ] → H → 1. The group R/[R, F H ] is finitely generated abelian and so is the direct product of a finite group U and a finitely generated torsion free group T ∼ = Z r . We claim (and it is well known) that the torsion part is precisely M (H), the Schur multiplier of H. Let Γ H = F H /[R, F H ]. The group U is characteristic in R/[R, F H ] and so normal in Γ H . We therefore have the extension β : 1 → U → Γ H → Γ H /U → 1 which is easily seen to be the inflation of the following extensionβ: β : 1 → U → Γ H /T → H → 1 We claim Γ H /U is torsion free: Indeed, Γ H /U = F H /[R, F H ]/(R ∩ [F H , F H ])/[R, F H ] ∼ = F H /(R ∩ [F H , F H ]) = F H /[F H , F H ] because H is abelian. The claim follows. The extension β yields a crossed product kU β Γ H /U . Composing the map φ, whose image is in µ, with the 2-cocycle β we obtain a cocycle which represents a class in H 2 (Γ H /U, µ) and which we denote by φ(β). Thus we obtain the twisted group algebra k φ(β) Γ H /U . Because Γ H /U is torsion free and abelian by finite, the theorem of Moody (See for instance [7], Lemma, 37.8) implies that this twisted group algebra is an Ore domain. Because β is the inflation ofβ, the elements of T in k φ(β) Γ H /U are central and hence, by inverting the elements of T , we obtain the twisted group division algebra k(T ) φ(β) H over the field k(T ). Letting k(T ) φ(β) H = h∈H k(T )x h we compute easily that for all h 1 , h 2 ∈ H, the commutator x h1 x h2 x −1 h1 x −1 h2 = φ(h 1 ∧ h 2 ) and so, by the comments preceding Definition 1.5, this division algebra realizes the given data. We know turn to proof of the general case. Let 1 → R → F G → G → 1 be a presentation of the finite group G (We will be more precise concerning the choice of F G below). We denote the map F G → G by π. For the subgroup H we obtain the following induced extension, where F H = π −1 (H): 1 → R → F H → H → 1 Taking quotients modulo [R, F H ] in these two extensions we obtain 1 → R/[R, F H ] → F G /[R, F H ] → G → 1 and 1 → R/[R, F H ] → F H /[R, F H ] → H → 1. As in case (2) We have the extension: β : 1 → U → Γ G → Γ G /U → 1. Because φ : M (H) → µ is G invariant, the map φ • β : Γ G /U × Γ G /U → µ is a 2-cocycle. Therefore we can form the twisted group algebra k φ•β Γ G /U . The group Γ G /U is abelian by finite and we have seen that it is torsion free. Again as in case (2) we infer that k φ•β Γ G /U is an Ore domain. We claim the ring of quotients of k φ•β Γ G /U is a G graded division algebra satisfying the desired conditions. What do we have to prove? Because it is an Ore domain the ring of quotients is a division algebra which we denote by D. So we need to prove that D is finite dimensional over its center, G-graded and the grading gives rise to the given group extension 1 → H → G → G/H → 1 where H is abelian, and the given G-invariant map φ : M (H) → µ. We first consider the subalgebra k φ•β Γ H /U . This is the same as the dvision algebra obtained in case (2) and as we saw there the cocycle β restricted to Γ H /U is inflated from the extension 1 → U → Γ H /T → H → 1 Therefore the elements of T in k φ(β) Γ H /U are central and hence, by inverting the elements of T , we obtain the twisted group division algebra k(T ) φ(β) H over the field k(T ). Also as in case (2) the cocycle on H maps to φ, as desired. From the extension 1 → T → Γ G /U → G → 1 we see that if we set D e = k(T ) and choose representatives x g in Γ G /U for the elements g in G, then D = g∈G D e x g is faithfully G-graded. We are then left with showing that H is exactly the kernel of the action of G on k(T ). For that we need to be more precise about our choice of F G : What is required is that in the sequence 1 → T → F G /U → G → 1 the kernel of the action of G on T is exactly H (We already know the kernel contains H). We proceed as in case (1): Let A be the free abelian group on m = \|G/H\| generators and let G act on A by permuting these coset generators. We obtain the split exact sequence 1 → A → A ⋊ G → G → 1. We take our group F G to be free on m + n generators y g1 , y g2 , . . . , y gm , where g 1 , g 2 , . . . , g m are distinct coset representatives of H in G union the generators x g for all g ∈ G. We map F G to G by sending each y gi to the identity and each x g to g. We map F G to A ⋊ G by sending the y ′ gi s to generators of A and each x g to g. With this choice of presentation for G a straightforward calculation shows that we have a induced map of extensions from 1 → R/[R, F H ] → F G /[R, F H ] → G → 1 to 1 → A → A ⋊ G → G → 1. Because U is torsion and A is torsion free we get an induced map of extensions from 1 → T → Γ G /U → G → 1 to 1 → A → A ⋊ G → G → 1. A simple diagram chase now shows that the kernel of the action of G on T is exactly H, as desired. twisted forms of finite dimensional G-simple algebras In this section we prove Theorem 1.10. Proof. As we have seen above, any finite dimensional division algebra over its center K, faithfully graded by a finite group G, is finite dimensional over the e-center K 0 and hence if we extend scalars over K 0 to its algebraic closure F we obtain a finite dimensional algebra, G-simple over F . These algebras were characterised by Bahturin, Sehgal and Zaicev (see Theorem 1.9) in terms of fine and elementary grading. More precisely given a G-simple algebra A finite dimensional over its e-center F , where F is an algebraically closed field of characteristic zero, the Ggrading is given by a presentation P A = (H, (g 1 , . . . , g s ), α) where H is a subgroup of G, (g 1 , . . . , g s ) is an s-tuple in G and α an element in H 2 (H, F * ). By [1], Lemma 1.3, we may replace the elements in the s-tuple by right H-cosets representatives. Moreover by permuting elements of the s-tuple we may assume equal representatives are adjacent to each other. We denote the normalized tuple by Θ. We let n i denote the multiplicity (possibly zero) of the ith coset in Θ, hence s = n 1 + · · · + n r where r = [G : H]. Now, we are interested in G-simple algebras over F which are obtained by scalar extensions from finite dimensional division algebras which are G-graded. Since these are in particular G-graded division algebras, we may apply [2], Theorems 1.8 and 5.3, which say that a finite dimensional G-simple algebra admits a G-graded division algebra form B whose e-center contains an algebraically closed field, if and only if the G-grading on A is given by a presentation P A = (H, (g 1 , . . . , g s ), α) where H normal in G, all cosets representatives are equally represented (possibly zero, i.e if G = H) and the class α is G-invariant. We need to prove that such G-simple algebras A but with H-abelian are precisely the G-simple algebras which admit a G-graded form which is an ungraded division algebra. This will follow rather easily from the sections above. Indeed, we have seen that a division algebra D that is G-graded, finite dimensional over its center K (and hence finite dimensional over its e-center K 0 ) gives rise to a group extension with abelian kernel 1 → H → G → G/H → 1, a G-invariant map φ : M (H) → µ (where the action of G on µ is trivial and the action on M (H) is induced by conjugation) and an integer d, the degree of the division algebra D e . We claim that extending scalars of D over the e-center K 0 yields a finite dimensional G-simple algebra A with presentation P A = (H, (g 1 , . . . , g s ), α φ ), namely the same group H and α φ is the cohomology class in H 2 (H, F * ) which corresponds to the given map φ by means of the Universal Coefficient Theorem. This is basically clear. We will show only that the cocycle α in P A is indeed α φ . By the previous remarks we see that extension of scalars to F yields a 2-cocycle of H with values in the units of F ⊗ K0 L = F × · · · × F ([G : H]-times). Now, by the Universal Coefficient Theorem a cocycle with values in F * (algebraically closed) is cohomologous to a cocycle whose values are roots of unity and because, by assumption, these are contained in the e-center, adding the fact that α is G-invariant, the values of α are in fact diagonally embedded in F × · · · × F . This proves our claim. In fact we see that a G-grading on a division algebra D, realizes the data (group extension with abelian kernel, G-invariant map φ and integer d) if and only if extending scalars to the algebraic closure of K e yields a G-simple algebra with presentation P A = (H, (g 1 , . . . , g s ), α) as in [2], Theorem 1.8, with the extra condition that H is abelian. Note that the integer d is the multiplicity of the cosets representatives in the s-tuple (g 1 , . . . , g s ). Finally, since any triple (extension, map, integer) can be realized by a G-grading on a finite dimensional division algebra the same holds for the G-simple algebra A. Theorem 1 . 3 . 13Let D be a division algebra, finite dimensional over its center K. Suppose D is faithfully graded by a finite group G. Let D e be the identity component of D and denote by L its center. Let K 0 = K ∩ L denote the central e-homogeneous elements of D. Assume K 0 contains an algebraically closed field of characteristic zero. With these conditions the following hold: then L 1 is also the center of the division algebra D H . (5) Conjugation on D H by nonzero homogeneous elements induces an action of G on L 1 with fixed field K, the center of D. Furthermore, the extension L 1 /K is G/H-Galois. (6) The cohomology class [α] ∈ H 2 (H, L * ) is G invariant. If we let S = {s ∈ H\|x s ∈ L 1 } (= {s ∈ H\|α(s, h) = α(h, s) for all h ∈ H}) , then S is a subgroup of H and L 1 = L α S. The group H/S is of central type, hence isomorphic to A × A for some abelian group A. (7) The division algebra D = D G is a G/H crossed product over the algebra D H . Its degree is (H : S)(G : H). class [α] ∈ H 2 (H, L * ) G , and (3) a positive integer d, the degree of D e over its center. Let M (H) denote the Schur multiplier of H and let φ [α] ∈ Hom(M (H), L * ) be the map corresponding to [α] by means of the Universal Coefficients Theorem. By naturality, the map φ [α] is G-invariant. Furthermore, because the values of φ [α] are roots of unity and these are assumed to be in K 0 , we have φ [α] ∈ Hom(M (H), µ) G , where µ denotes the group of roots of unity in K 0 (the action of G on µ is trivial). and the map φ is nontrivial, then φ is the only nontrivial element of Hom(M (H), µ) and so φ is G-invariant. The e-component L contains the center K = K 0 and L has degree 2 over K. The division algebra D has degree 4 over K. The H-component of D is a quaternion algebra with center L. Because the sequence is split, D ∼ = Q 1 ⊗ K Q 2 is a tensor product of two quaternion algebras over the center K. (5) If H ∼ = Z 4 , the group S of Theorem 1.4 equals H (because H is cyclic). Theorem 1. 10 . 10Let A be a finite dimensional G-simple algebra over an algebraically closed field E of characteristic zero. Then A admits a G-graded twisted form division algebra over a field K 0 which contains an algebraically closed field of characteristic zero if and only if the following hold: Let L = Z(D e ). Conjugation on D e with nonzero homogeneous elements induces an action of G on L and we denote by H the kernel of this action. Clearly K = L G = L G/H . We claim that D H ∼ = D e ⊗ L L α H for some α ∈ Z 2 (H, L * ): For all h ∈ H, let u h be a nonzero homogeneous element of degree h. Because h acts trivially on L, and D e is finite dimensional over L, conjugation by u h determines an inner automorphism of D e and hence, by the Skolem-Noether Theorem, there is a (nonzero) element θ h in D e such that x h Indeed, by the Hopf formula the Schur multiplier is given by (R∩[F H , F H ])/[R, F H ] so it is necessary and sufficient to show that R/[R, F ]//(R ∩ [F H , F H ])/[R, F H ] ∼ = R/(R ∩ [F H , F H ]) is torsion free. But R/(R ∩ [F H , F H ]) is isomorphic to R[F H , F H ]/[F H , F H ], a subgroup of the torsion free group F H /[F H , F H ]. This proves the claim. , the group R/[R, F H ] is the direct product of a finite group U = M (H) and a finitely generated torsion free group T ∼ = Z r . Moreover U is characteristic in R/[R, F H ] and hence normal in both Γ H = F H /[R, F H ] and Γ G = F G /[R, F H ]. The group Γ G /U is torsion free: We have Γ G /U ∼ = F G /[R, F H ]/(R ∩ [F H , F H ])/[R, F H ] ∼ = F G /[F H , F H ] because H is abelian, and F G /[F H , F H ] is torsion free by Higman's theorem. have the following corollary (see Cuadra and Etingof result ([5] Theorem 2.1)).Corollary 1.7. Every extension Simple G-graded algebras and their polynomial identities. E Aljadeff, D Haile, Trans. Amer. Math. Soc. 36613E. Aljadeff and D. Haile, Simple G-graded algebras and their polynomial identities, Trans. Amer. Math. Soc. 366 (2014), 1749-1771. 13 Verbally prime T-ideals and graded division algebras. E Aljadeff, Y Karasik, Adv. Math. 33214E. Aljadeff and Y. Karasik, Verbally prime T-ideals and graded division algebras, Adv. Math. 332 (2018), 142-175. 7, 13, 14 Finite-dimensional simple graded algebras. Y Bahturin, M Zaicev, S K Sehgal, Sb. Math. 1997Mat. Sb.Y. Bahturin, M. Zaicev, and S. K. Sehgal, Finite-dimensional simple graded algebras. (Russian), Mat. Sb., 199(7), (2008), 21-40; translation, Sb. Math., 199(7), (2008), 965-983. 6 On zero divisors in group rings. K A Brown, Bull. London Math. Soc. 83K. A. Brown, On zero divisors in group rings, Bull. London Math. Soc. 8 (1976), no. 3, 251-256. 11 Finite dimensional Hopf actions on central division algebras. J Cuadra, P Etingof, Int. Math. Res. Not. IMRN. 5J. Cuadra and P. Etingof, Finite dimensional Hopf actions on central division algebras, Int. Math. Res. Not. IMRN no. 5 (2017) 1562-1577. 5 Finite groups having isomorphic images in every finite group of which they are homomorphic images. G Higman, The Quaterly Journal of Mathematics. 26G. Higman, Finite groups having isomorphic images in every finite group of which they are homomorphic images, The Quaterly Journal of Mathematics 2, no. 6 (1955) 250-254. 11 Infinite crossed products. D S Passman, Pure and Applied Mathematics. 13512Academic Press, IncD. S. Passman, Infinite crossed products, Pure and Applied Mathematics, 135. Academic Press, Inc., Boston, MA, 1989. 12 Uber die sogenannte nichtkommutative Galoissche Theorie und die Relation 3. O Teichmuller, O. Teichmuller, Uber die sogenannte nichtkommutative Galoissche Theorie und die Relation 3 Generalized crossed products. J.-P Tignol, Seminaire Mathematique (nouvelle serie). Louvain-la-Neuve; Belgium106Universite Catholique de LouvainJ.-P. Tignol, Generalized crossed products, Seminaire Mathematique (nouvelle serie) 106 (1987), Universite Catholique de Louvain, Louvain-la-Neuve, Belgium. 3 Germany. E-mail address: aljadeff 'at' technion.ac.il (E. Aljadeff), E-mail address: haile 'at. * Mathematisches Institut, Giessen, GiessenJustus-Liebig-Univeristätindiana.edu (D. Haile* Mathematisches Institut, Justus-Liebig-Univeristät, Giessen, Giessen, Ger- many. E-mail address: aljadeff 'at' technion.ac.il (E. Aljadeff), E-mail address: haile 'at' indiana.edu (D. Haile). E-mail address: Igor.Karasik 'at' math.uni-giessen. de (Y. KarasikE-mail address: Igor.Karasik 'at' math.uni-giessen.de (Y. Karasik),	[]
[ "The decay of multiscale signals -deterministic model of the Burgers turbulence", "The decay of multiscale signals -deterministic model of the Burgers turbulence" ]	[ "S N Gurbatov \nRadiophysics Dept\nUniversity of Nizhny\nNovgorod 23, Gagarin Ave603600Nizhny NovgorodRussia\n\nObservatoire de la Cote d'Azur, Laboratorie G.D. Cassini\nUniversity of Nizhny Novgorod\nB.P. 229, 06304, Nice Cedex 4, 23, Gagarin Ave603600Nizhny NovgorodFrance., Russia\n", "A V Troussov [email protected] \nJoint Institute of Physics of the Earth RAS\nMolodezhnaya Str., 3117296MoscowGC RASRussia\n\nObservatoire de la Cote d'Azur, Laboratorie G.D. Cassini\nUniversity of Nizhny Novgorod\nB.P. 229, 06304, Nice Cedex 4, 23, Gagarin Ave603600Nizhny NovgorodFrance., Russia\n", "Sergey N Gurbatov ", "Radiophysics Dept " ]	[ "Radiophysics Dept\nUniversity of Nizhny\nNovgorod 23, Gagarin Ave603600Nizhny NovgorodRussia", "Observatoire de la Cote d'Azur, Laboratorie G.D. Cassini\nUniversity of Nizhny Novgorod\nB.P. 229, 06304, Nice Cedex 4, 23, Gagarin Ave603600Nizhny NovgorodFrance., Russia", "Joint Institute of Physics of the Earth RAS\nMolodezhnaya Str., 3117296MoscowGC RASRussia", "Observatoire de la Cote d'Azur, Laboratorie G.D. Cassini\nUniversity of Nizhny Novgorod\nB.P. 229, 06304, Nice Cedex 4, 23, Gagarin Ave603600Nizhny NovgorodFrance., Russia" ]	[]	This work is devoted to the study of the decay of multiscale deterministic solutions of the unforced Burgers' equation in the limit of vanishing viscosity.It is well known that Burgers turbulence with a power law energy spectrum E 0 (k) ∼ \|k\| n has a self-similar regime of evolution. For n < 1 this regime is characterised by an integral scale L(t) ∼ t 2/(3+n) , which increases with the time due to the multiple mergings of the shocks, and therefore, the energy of a random wave decays more slowly than the energy of a periodic signal.In this paper a deterministic model of turbulence-like evolution is considered. We construct the initial perturbation as a piecewise linear analog of the Weierstrass function. The wavenumbers of this function form a "Weierstrass spectrum", which accumulates at the origin in geometric progression. "Reverse" sawtooth functions with negative initial slope are used in this series as basic functions, while their amplitudes are chosen by the condition that the distribution of energy over exponential intervals of wavenumbers is the same as for the continuous spectrum in Burgers turbulence. Combining these two ideas allows us to obtain an exact analytical solution for the velocity field. We also notice that such multiscale waves may be constructed for multidimensional Burgers' equation.This solution has scaling exponent h = −(1 + n)/2 and its evolution in time is selfsimilar with logarithmic periodicity and with the same average law L(t) as for Burgers turbulence. Shocklines form self-similar regular tree-like structures. This model also describes important properties of the Burgers turbulence such as the self-preservation of the evolution of large scale structures in the presence of small scales perturbations.PACS 43.25.Cb,47.27.Eq.	10.1016/s0167-2789(00)00090-7	[ "https://arxiv.org/pdf/physics/0002043v1.pdf" ]	14,354,154	physics/0002043	9048ff96af8aae79af4d3fe7364f1e7800afc09c	The decay of multiscale signals -deterministic model of the Burgers turbulence 22 Feb 2000 October 29, 2018 S N Gurbatov Radiophysics Dept University of Nizhny Novgorod 23, Gagarin Ave603600Nizhny NovgorodRussia Observatoire de la Cote d'Azur, Laboratorie G.D. Cassini University of Nizhny Novgorod B.P. 229, 06304, Nice Cedex 4, 23, Gagarin Ave603600Nizhny NovgorodFrance., Russia A V Troussov [email protected] Joint Institute of Physics of the Earth RAS Molodezhnaya Str., 3117296MoscowGC RASRussia Observatoire de la Cote d'Azur, Laboratorie G.D. Cassini University of Nizhny Novgorod B.P. 229, 06304, Nice Cedex 4, 23, Gagarin Ave603600Nizhny NovgorodFrance., Russia Sergey N Gurbatov Radiophysics Dept The decay of multiscale signals -deterministic model of the Burgers turbulence 22 Feb 2000 October 29, 2018arXiv:physics/0002043v1 [physics.flu-dyn] (Permanent address) (Permanent address)Burgers' equation; Burgers turbulence This work is devoted to the study of the decay of multiscale deterministic solutions of the unforced Burgers' equation in the limit of vanishing viscosity.It is well known that Burgers turbulence with a power law energy spectrum E 0 (k) ∼ \|k\| n has a self-similar regime of evolution. For n < 1 this regime is characterised by an integral scale L(t) ∼ t 2/(3+n) , which increases with the time due to the multiple mergings of the shocks, and therefore, the energy of a random wave decays more slowly than the energy of a periodic signal.In this paper a deterministic model of turbulence-like evolution is considered. We construct the initial perturbation as a piecewise linear analog of the Weierstrass function. The wavenumbers of this function form a "Weierstrass spectrum", which accumulates at the origin in geometric progression. "Reverse" sawtooth functions with negative initial slope are used in this series as basic functions, while their amplitudes are chosen by the condition that the distribution of energy over exponential intervals of wavenumbers is the same as for the continuous spectrum in Burgers turbulence. Combining these two ideas allows us to obtain an exact analytical solution for the velocity field. We also notice that such multiscale waves may be constructed for multidimensional Burgers' equation.This solution has scaling exponent h = −(1 + n)/2 and its evolution in time is selfsimilar with logarithmic periodicity and with the same average law L(t) as for Burgers turbulence. Shocklines form self-similar regular tree-like structures. This model also describes important properties of the Burgers turbulence such as the self-preservation of the evolution of large scale structures in the presence of small scales perturbations.PACS 43.25.Cb,47.27.Eq. Introduction The nonlinear diffusion equation ∂v ∂t + v ∂v ∂x = ν ∂ 2 v ∂x 2 ; v(x, t = 0) = v 0 (x). (1.1) was originally introduced by J.M.Burgers in [6] (1939) as a model for hydrodynamical turbulence. Burgers' equation (1.1) describes two fundamental effects characteristic of any turbulence [10]: the nonlinear redistribution of energy over the spectrum and the action of viscosity in small scales. Burgers' equation was used later to describe a large class of physical systems in which the nonlinearity is fairly weak (quadratic) and the dispersion is negligible compared to the linear damping [29]. The most important example of such waves are acoustical waves with finite amplitude [24]. Another class of problems, arising, e.g., in surface growth, also leads to Burgers' equation [5], [8], [30]. The three dimensional form of (1.1) has been used in cosmology to describe the formation of large scale structures of the Universe at a nonlinear stage of gravitational instability ( see e.g. [15], [25], [13], [28] ). In the physically important case of large Reynolds number, the action of viscosity is significant only in the small regions with high gradient of the velocity field. In the limit ν → 0, the solution of Burgers' equation has the following form (see [18], [7], [13]): v(x, t) = x − y(x, t) t , (1.2) where y(x, t) is the coordinate of the maximum of the function G(x, y, t) = Ψ 0 (y) − (x − y) 2 2t , v 0 (x) = − ∂Ψ 0 (x) ∂x . (1.3) Strong interaction between coherent harmonics leads to the appearance of local selfsimilar structures in Burgers' equation. A periodic initial perturbation with zero mean velocity is transformed asymptotically into a sawtooth wave with gradient ∂ x v = 1/t and with the same period l 0 . It is important that at this stage the amplitude a(t) = l 0 /t and the energy density σ 2 (t) ≃ l 2 0 /12t 2 do not depend on the initial amplitude. An initial one-signed pulse with the area m > 0, localised at t = 0 in the neighbourhood of the point x = 0, also has asymptotically a universal form: it transforms into a triangular pulse with the gradient ∂ x v = 1/t and increasing coordinate of the shock x s ≈ (2mt) 1/2 . Due to the increase of the integral scale the amplitude of such a pulse a(t) = x s (t)/t ∼ m 1/2 t −1/2 and its energy will decrease more slowly than for a periodic signal, like t −1/2 . Continuous random initial fields are also transformed into sequences of regions with the same gradient ∂ x v = 1/t, but with random locations of the shocks separating them. Due to the multiple merging of the shocks the statistical properties of such random fields are also self-similar and may be characterised by the integral scale of the turbulence L(t). The merging of the shocks leads to an increase of the integral scale L(t), and because of this the energy σ 2 (t) ∼ L 2 (t)/t 2 (1.4) of a random wave decreases more slowly than the energy of periodic signals. The type of turbulence evolution is determined by the behaviour of the large scale part of the initial energy spectrum E 0 (k) = α 2 k n b 0 (k); (1.5) E 0 (k) = 1 2π v 0 (x), v 0 (x + z) e ikz dz. (1.6) Here b 0 (k) is a function which falls off rapidly for k > k 0 ∼ l 0 , and b 0 (0) = 1. For n > 1 the law of enrgy decay strongly depends on the statistical properties of the initial field (see e.g. [30] and references therein). For the initial Gaussian perturbation the integral scale L(t) ∼ t 1/2 times logarithmic correction obtains and is determined by two integral characteristics of the initial spectrum: the variances of the initial potential Ψ 0 and the velocity v 0 (x) [20], [9], [13], [16]. For n < 1 the structure function of the initial potential increases as a power law in space. Then the initial potential field is Brownian, or fractional Brownian motion, and some scaling may be used [7], [20], [13], [26], [3], [22], [23]. In this case the turbulence is also self-similar and the integral scale L(t) increases as L(t) = (αt) 2/(3+n) . (1.7) The energy of the turbulence is derived from (1.4): σ 2 (t) ∼ t −p , p = 2(n + 1) n + 3 . (1.8) The difference between these two cases ( n < 1 and n > 1 ) is connected to the process of parametric generation of low frequency component of the spectrum. For the case n < 1 the newly generated low frequency components are relatively small and we have the conservation of large scale part of the spectrum: E(k, t) = E 0 (k) = α 2 k n , f or k << 1/L(t). (1.9) Thus, the laws of turbulent decay are more complex than for simple signals, which can be attributed to multiple merging of the shocks. In [12] a model of a regular fractal signal with decay lower than for single one-signed pulse was introduced. The initial signal v 0 (x) was constructed as a sequence of one-signed pulses whose positions form a Cantor set with capacity (fractal dimension) D = ln N/ ln β, where N p is the number of pulses in the scale L p ≃ L 1 Nβ p−1 , 0 < D < 1. Multiple merging makes the decay of the wave slower and the general behaviour of the energy decay may be approximated by the power law with the exponent in (1.8): p = 1 − D 2 − D , 0 < p < 1. (1.10) The evolution proves to be self-similar in successive time periods (t i , t i−1 ) and (t i+1 , t i+2 ), where t i+1 /t i = β 2 /N. This shows log-periodical self-similarity of the field evolution. Linear and non-linear decay of fractal and spiral fields given by the sequences of regular pulses was also investigated in [1]. It was shown that the power law (1.8) with the exponent given by the formula (1.10) holds also true for homogeneous fractal pulse signal with capacity D. Another model of a multiscale signal, which has the same general behaviour on the external scale L(t) (1.7), and energy of the Burgers turbulence, was also discussed in [12]. It was assumed therein that the initial signal is a discrete set of modes -the spatial harmonics v 0 (x) = ∞ p=0 a p sin(k p x + ϕ p ), (1.11) with wavenumbers k p and amplitudes a p given in terms of a parameter ǫ by k p = k 0 ǫ p , a p = a 0 ǫ −hp , h = −( n + 1 2 ), a 0 = αk (n+1)/2 0 . (1.12) Amplitudes a p and the scaling exponent h are chosen from the condition that the mean energy of harmonics in the interval ∆ p = k p − k p+1 be identical to that corresponding to the spectral density (1.5): a 2 p = E(k p )∆ p . For ǫ << 1 and n > −1 the harmonics are spread over the spatial spectrum and accumulate at the point k = 0 with decreasing amplitude. The main approach in this model was that the energy of the wave is the sum of energies of independent modes. The approach is nontrivial, but nevertheless leads to the same laws of the integral scale L(t) (1.7) and the energy decay (1.8) as in the case of continuous spectrum (1.5). Let us point out that representation of the field given by the formula (1.11) is similar to shell models, which were introduced as useful models addressing the problem of analogous scaling in fully developed turbulence (see, e.g., [11], [19] and references in there). In present paper we consider the evolution of a regular signal whose behaviour in general is similar to the evolution of the Burgers turbulence with continuous spectrum (1.5). The main difference,as compared to the model discussed above, is that we construct the exact solution of Burgers' equation using as an initial mode the "reverse" sawtooth wave. The frequency ratio in our model is ǫ = 1/N, where N is an integer and N ≥ 2. These perturbation are similar to the well known Weierstrass and Weierstrass-Mandelbrot fractal functions (see [21], [4]). For the analysis of Burgers' equation it is convenient to use a mechanical interpretation. There is a one-to-one correspondence between the solution of Burgers' equation and the dynamic of a gas of inelastically interacting particles [13], [14]. Let us take a one-dimensional particle flux with a contact interaction: as long as the particles do not run into each other they move with constant velocity. In the collision they stick together, forming a deltafunction singularity in the matter density. This leads to the appearance of gas of two species: a hydrodynamical flux of the "light" initial particles, and a gas of "heavy" particles arising in the adhesion process of light particles. The evolution of the particle velocity field will be described by the solution of Burgers' equation if we assume that the initial density of the light particles is ρ 0 = const, the velocity of particles is equal to the initial velocity in (1.1), and that the collision of the particles conserve their mass and momentum. This analogy permits construction of a very fast (linear time) algorithm of solution of Burgers' equation [27]. In our case, for the initial reverse sawtooth wave, all the matter turns into heavy particles at the same moment of time. Thus, after this time the evolution of the Burgers turbulence is fully determined by the motion of heavy particles, whose positions are positions of shocks, and masses are equal to ∆v · t, where ∆v are the amplitudes of the shocks. The paper is organized as follows. In Section 2 we consider the evolution and interaction of "reverse" sawtooth modes. In section 3 we consider the interaction of small scale mode with large scale structures. In Section 4 we investigate the properties of the sawtooth Weierstrass-Mandelbrot fractal function. In section 5 we show that deterministic model has logarithmic periodic self-similarity. We also discuss here the multi-dimensional generalization of this model. Section 5 presents concluding remarks. Evolution and interaction of "reverse" sawtooth modes in Burgers' equation Let us introduce the pth "reverse" saw tooth mode as v (p) (x, 0) = a p A(k p x + ϕ p ) (2.1) Here a p , k p are amplitude and wavenumber of the mode, −ϕ p is its phase. The function A(x) is 2π periodic function, given on its first period by the following expression A(x) = π − x, x ∈ [0, 2π[ (2.2) The set of "reverse sawtooth" functions is not orthogonal, but nevertheless we will introduce a set of modes satisfying equation (1.12), whose wavenumbers and amplitudes satisfy the same relations as the sinusoidal modes of [12], i.e. relation (1.11). We introduce the term "reverse sawtooth" because this signal is a sawtooth with teeth facing to the right, but the term "sawtooth" by itself is widely used in Burgers turbulence literature to refer to the late stage of the evolution of the wave profile a sequence of sawteeth with positive slope 1/t. The solution of Burgers' equation with a linear velocity profile v 0 = −γ(x − x + ) is well-known (see, e.g., [13] ): v(x, t) = −γ(x − x + ) 1 − γt (2.3) The value γ −1 has the dimension of time, and for γ > 0, at the finite time t = γ −1 the gradient ∂ x v becames infinite. For γ < 0 the gradient becomes equal to ∂ x v = t −1 , independent of the value of γ at times t ≫ \|γ\| −1 . Thus, we have from (1.12), (2.1),(2.2), that the evolution of the pth mode is characterised by the nonlinear time t p = γ −1 p = a p k p −1 = t 0 /(ǫ (n+3)/2 ) p , t 0 = 1/αk n+3/2 0 (2.4) Based on solution (2.3 ) it is easy to see that for the "first" period (if ϕ p = 0) the evolution of the pth reverse mode at the initial stage (t < t p ) may be described as v (p) (x, t) =        x/t, if 0 < x < t tp π kp ; 1 tp ( π kp − x)/ 1 1−t/tp , if \| π kp − x\| < t tp π kp ; (x − π kp )/t, if t tp π kp < x − π kp < π kp . (2.5) On the other hand, at time t = t p , the mode transforms into a "direct" sawtooth wave with slope ∂ x v = 1/t , independent of the amplitude and wavenumber of the mode: v (p) (x, t) = x/t, if 0 < x < π kp ; (x − π kp )/t, if π kp < x < 2π kp . (2.6) The density of energy σ 2 (t) = v 2 (x, t) L , where L denotes averaging over the period, is conserved before t < t p , and decreases like (k p t) −2 after t > t p . Consider now the evolution of a gas of sticky particles in the case of independent evolution of the pth mode. In the general case, the density of the gas is calculated by using the Jacobian of the transformation from Lagrangian to Eulerian coordinates and may be written in the form ( see, e.g., [13] ) ρ(x, t) = ρ 0 (1 − t∂ x v(x, t)) (2.7) Then it is obvious that at the initial stage ρ(x, t) = ρ 0 1 1 − t/t p , \| π k p − x\| < t t p π k p (2.8) while ρ is zero outside this interval in each period. At time t = t p all the light particles in each period collide into a single heavy particle with mass m p = ρ 0 L p = ρ 0 2π k p ,(2.9) and the heavy particles have positions x p,l = π k p − ϕ p k p + 2π k p l; l = 0, ±1, ±2, ... (2.10) equal to the zero positions of the initial pth mode. The process of light merging particles and the evolution of the velocity is shown in Fig.1. Consider now the joint evolution of two successive modes: pth and (p + 1)th. From (2.4) one can see that the ratio of nonlinear times of the successive modes is t p+1 t p = ǫ −(n+3)/2 ≡ ǫ 1−h , (2.11) which does not depend on p and increases if the exponent n is greater than −3. The gradient of the initial field v (p) (x) + v (p+1) (x) is −(γ p + γ p+1 ), so the effective nonlinear time for such a sum is t p,ef f = t p,p+1 = 1 γ p + γ p+1 = t p 1 + t p /t p+1 ,(2.12) Because all parts of the initial perturbation have the same slope, all light particles will collide at the same time t = t p,p+1 . The mass of heavy particles after merging are m p,i = ρ 0 ∆ i,i+1 , where ∆ i,i+1 is the distance between adjacent shocks in the initial perturbation. The ratio of periods of two adjacent modes is L p+1 /L p = k p /k p+1 = ǫ −1 . If N = ǫ −1 is an integer larger than 1, there will be N + 1 heavy particles on the period of the larger scale mode (p + 1)th: (N − 1) with the mass m p = ρ 0 L p (2.9), and two particles with total mass equal to m p . These two particles only exist when the shock of the (p + 1)th mode is located in the interval between shocks of the pth mode. For simplicity, we will consider the case where the spatial relations k p+1 ϕ p+1 = k p ϕ p + 2πr/N, (2.13) between the phases of successive modes, hold. In this case the discontinuities of the (p+1)th mode do not produce new shocks in the total perturbation v (p) (x)+v (p+1) (x). Thus, at times t larger than t > t r,ef f , the masses of all heavy particles will be the same, as would be the case without the large scale modes (2.9). The positions of these heavy particles at time t = t p,ef f are X (p,l) (t p,ef f ) = x p,l + v (p+1) (x p,l )t p,ef f ,(2.14) where x p,l are the zero positions of the pth mode (2.10). The velocity of this particle is equal to v (p+1) (x p,l ). Equation (2.14) is obvious if we use the trivial equality v (p) (x p,l )+v (p+1) (x p,l ) = v (p+1) (x p,l ) , and also note that the position of the heavy particle x p,l (t p,ef f ) is equal to the position at the same time of all light particles with initial coordinate x = x p,l . From (2.14) we immediately have that after time t p,ef f the positions of the particles are X (p,l) (t) = x (p,l) + v (p+1) (x p,l )t. (2.15) The difference between the coordinates of the adjacent particles X p,l (t) and X p,l+1 (t) decreases with time, proportionally to the gradient of v (p+1) (x): X (p,l+1) (t) − X p,l (t) = (x p,l+1 − x p,l ) − t ∂v (p+1) (x) ∂x (x p,l+1 − x p,l ) ≡ (x p,l+1 − x p,l )(1 − t/t p+1 ). (2.16) These particles collide at time t = t p+1 (2.4) and the newly created heavy particles will have masses m (p+1) = ρ 0 L p+1 (2.17) and positions x p+1,l = π k p+1 − ϕ p+1 k p + 2πl k p+1 ; l = 0, ±1, ±2, ... (2.18) The velocity of these particles is zero. Thus, at times t larger then t p+1 , the evolution of the initial perturbation v 0 (x) = v (p) (x)+ v (p+1) (x) will be the same as the evolution of only the large scale mode v (p+1) (x). The process of particles merging and the evolution of the velocity for the sum of two successive modes with the periods ratio N = 2 are shown in Fig. 2. By recurrence, it is evident that for finite number of modes v( 3 Interaction of small scale "reverse" sawtooth mode with large scale structures x) = v (p) (x) + v (p+1) (x) + ... + v (M ) (x) Let us now consider the interaction of the pth mode with an infinite series of larger scale modes W p (x) = ∞ r=p+1 v r (x) ≡ ∞ r=p+1 a r A(k r x + ϕ r ),(3.1) assuming that the phases of the modes satisfy the relations ( 2.13 ) and that k r = k 0 ǫ r , a r = a 0 ǫ −hr , h = −(n + 1)/2. From (3.1) and (2.4) we have for the gradient of the initial perturbation v 0 (x) = v p (x) + W p (x) ∂ x v 0 (x) = ∂ x v p (x) + ∂ x W p (x) = ∞ r=p γ p = = γ 0 ∞ r=p (ǫ (n+3) 2 ) r = γ 0 ǫ (n+3) 2 p 1 1−ǫ (n+3) 2 ,(3.2) the condition n > −3 (h < 1) being necessary for the series to converge. From (3.2), we have for the effective time of nonlinearity of pth modẽ t p = 1/∂ x v 0 (x) = t p (1 − ǫ (n+3) 2 ),(3.3) with the original t p determined by the equation (2.4). Thus, after the time of collisiont p , heavy particles with mass m p = ρ 0 L p (3.2) appear. The coordinates of these particles will be determined by an equation similar to (2.15) x p,l (t) = x p,l (t) + W p (x p,l )t, (3.4) with the velocity of particles determined by the function W p (x) (see 3.1), which is a sum of all larger modes, and x p,l are the coordinates of the zeros of the pth modes. The difference between the coordinates of adjacent particles x p,l and x p,l will then decrease with time like (1 − t/t p+1 ), wheret p+1 = t p+1 (1 − ǫ (n+3)/2 ) is the inverse of the gradient of the function W p (x), see (3.1) and (3.2). Thus, the time of particle collision for this generation will be described by equation (3.3) with p = p + 1, and the new masses will be determined by the period of the (p + 1)th mode -see equation (2.17). The extrapolation of this particle merging process to the next generations is evident by recurrence. The qth collision of heavy particles takes place at timet p+q = t p+q (1 − ǫ n+3 2 ), the masses of these particles at this time are determined by the period of the (p + q)th mode m p+q = ρ 0 L p+q = 2πρ 0 /k p+q (2.9), (2.17). In the time interval t ∈ [t p+q ,t p+q+1 ], t p+q+1 t p+q = ǫ − (n+3) 2 = N n+3 2 ; (3.5) the coordinates of particles will be determined by the equation (3.4) with p = p + q. Here W p+q (x) is the sum of the velocities of all larger modes with r > p + q, and x p+q,l are the zeros of (p + q)-th mode. It is important to note that at time t >t p+q the evolution of the particles is solely determined by the modes with r ≥ p + q. It means, that at times t >t p+q the position of the particles does not depend on the presence in the initial condition of the small scale modes with r < p + q. Thus, two processes with different initial velocities:ṽ 0 (x), the field with small scales, and v 0 (x), the field without small scales modes: v 0 (x) = W p+q−1 (x);ṽ 0 (x) = W p−1 (x) (3.6) will have the same evolution after t >t p+q−1 . Even if p → −∞ (when modes with very small scales L p ∼ ǫ −p = N p and very large amplitudes a p ∼ a 0 (ǫ (n+1)/2 ) p = a 0 (N (n+1)/2 ) −p ) are present in the initial perturbation) the multiple merging of the particles will lead to the independence of the evolution of large scale modes with respect to the small scale modes. This effect is similar to the self-preservation of large scale structures in Burgers turbulence [2], [17]. When the initial field v 0 (x) is noise, the highly nonlinear structures continuously interact and due to the merging of shocks, their characteristic scale L(t) constantly increases. The presence of small scale noise perturbation v h (x) results in additional fluctuations in the shock coordinates ∆x k (t), and these fluctuations increase in strength with the passage of time. Thus, the final result of the evolution of the field is determined by the competition of two factors, the increase in the external scale L(t) of the structures and the increase in the strength ∆x k (t) of shock coordinates fluctuations, the later being related to the perturbation v h (x). In a turbulence, having power index n < 1 (1.5), multiple merging of shocks leads to self-preservation of the large scale structures independently of the presence of small scale components. For the model signal this effect appears for arbitrary n due to the special choice of wavenumbers and phases of interacting modes. It was stressed in the introduction that the solution of Burgers' equation has a one-to-one correspondence with the dynamics of the gas of inelastically interacting particles ( [13]). The stage when all light particles collide, forming heavy particles, corresponds to the solution of Burgers' equation with a well-defined slope ∂ x v = 1/t. In this case the profile of the field v(x, t) is fully determined by the coordinates and amplitudes of the shocks. Their coordinates X s (t) are equal to the coordinates of heavy particles, their velocity v s (t) = dX s (t) dt = (v s (x s − 0, t) + v s (x s + 0, t))/2 (3.7) is equal to the velocity of the particles, and the amplitude of the shock ∆v s (x) = (v(x s − 0, t) − v(x s + 0, t)) = m/t (3.8) is determined by the mass of the particle (ρ 0 ≡ 1) (see, e.g., [13]). Thus, the investigation of the motion of heavy particles permits to fully reconstruct the properties of the velocity field v(x, t) of Burgers' equation. The sawtooth Weierstrass-Mandelbrot fractal function It was shown in the previous section that the evolution of the particles (shocks) is determined by the function W p (x) (3.1). The basis functions of W p (x) are the reverse sawtooth periodic functions with wavenumbers k r = k 0 ǫ r and amplitudes a r = a 0 ǫ −hr , h = −(n + 1)/2 satisfy relations (1.12). Wavenumbers form a geometrical progression like in the Weierstrass function (see [4]) and accumulate at the origin k = 0. In the original Weierstrass function, the situation was the opposite with increasing frequencies, but nevertheless the function W p (x) has many properties of Weierstrass function and of its generalisation -the Weierstrass-Mandelbrot function (see [21], [4]). We consider here a deterministic function W p (x) with the special phase relation ϕ p = (2πk/N)p (k = 1, 2, ..., N; N = 1/ǫ ), thus, the discontinuities in the largest modes r > p + 1 coincide with some of the discontinuities of the smaller mode r = p + 1. The function W p (x) is continuous in the intervals 2π/k p+1 = 2π/(k 0 ǫ p+1 ) with the same slope in each interval. The inverse value of this slopet p+1 t p+1 = t p+1 (1 − ǫ n+3 2 ); t p+1 = t 0 (ǫ n+3 2 ) p+1 (4.1) is proportional to the nonlinear time t p+1 of the smallest mode. Of course, we need n > −3, so that the convergence of (3.2) is assured and the inequalities t p+1 > t p hold. The amplitudes of the modes are proportional to ǫ (n+1)/2 and for n > −1 the function W p (x) is bounded W p (x) ≤ ∞ r=p+1 a r = a 0 (ǫ n+1 2 ) p+1 1 1 − ǫ n+1 2 . (4.2) Thus, for finite p the energy of W p (x) is also finite. For the case of the phase relation introduced above, the functions W p (x) also have scaling properties, so that for instance for k = 0, we have W p (x) = ǫ −hp W 0 (ǫ p x); W p (ǫ m x) = ǫ hm W m+p (x). (4.3) The case −1 < n < 1 is similar to the initial conditions with generalized white noise in Burgers turbulence. The energy of the initial signal in such turbulence is determined by the largest cutoff wavenumber, so in our model by the smallest scale p. If p → −∞ the energy of the model signal (as the energy of white noise) will tend to infinity. But from the considerations in the previous section we have that at the finite time t all the modes with t p < t have finite energy ∼ L 2 p /t 2 due to the nonlinear dissipation, so that the whole energy of the turbulence is also finite. Thus, even in the case of "divergent" initial conditions (p → −∞), we will have a "convergent" solution for any time t > 0. The case n < −1 is similar to having fractional Brownian motion initial condition in Burgers turbulence. In this case, the series (4.2) diverges and the initial signal W p (x) is unbounded. But for Burgers turbulence ( for the process of particles motion and collisions ) only relative velocity of the particles matters. So we can use the same regularisation procedure with W p (x). Such a procedure was done with the Weierstrass function in [21]. In our case, taking ϕ p ≡ 0, we can introduce the function W ∞ p (x) = W p (x) − W p (0), according to [21], which is finite in all finite spatial intervals. The other way to get a bounded function is to use special phase relations for the modes. Self-similarity properties of deterministic model in one and two dimension Here we summarise the properties of the evolution of the multiscale deterministic signal using some additional information about scaling characteristics of W p (x), and compare them with the properties of the Burgers turbulence. Let us consider the evolution of the multiscale signal v 0 (x) = v p (x) + W p (x). (5.1) It was shown that at times t for which t >t p , heavy particles with mass M p = ρ 0 L p appear and their coordinates are determined by the relation (3.5). These particles collide at timẽ t p+1 , (t p+1 /t p = N (n+3) 2 , N = ǫ −1 ), and new particles with masses m p+1 = ρ 0 L p+1 = m p N appear. Their motion will be determined by the same law (3.5) with substitution p → p + 1. Using the scaling properties of W p (x) (4.3) we have, that the motion of the particles in this interval will be similar to the motion of the particles in the interval [t p ,t p+1 ] if we rescale the time t/t p ⇒ t/t p+1 . Since the ratio t p+1 /t p does not depend on p, one can speak about the logarithmic periodic self-similarity of the motion of the particles. This means that at arbitrary interval [t q ,t q+1 ] the motion of the particles will be similar to the motion of the particles in the intervalt p ,t p+1 , by the scaling factor x p /x q = ǫ p−q in space, and the scaling function t p /t q = (ǫ − n+3 2 ) p−q in time. The coordinates and masses of the particles fully determine the velocity field, and so the solution of Burgers' equation is also logarithmic periodic self-similar. With each collision, the mass M(t) of the particles increases N = 1/ǫ times. The time interval between the two successive collisions increases as t p+1 /t p = N n+3 2 . Thus, by the approximation of piecewise constant function m(t) by the power law m(t) ≃ m 0 (t/t 0 ) (n+3)/2 . (5.2) we obtain the same result as for the Burgers turbulence. In our case, m is proportional to the period of the smallest mode at time t, and is analogous to the integral scale in Burgers turbulence. In the case n > −1 we can also estimate the energy decay of the model signal. For n > −1 and ǫ ≪ 1 the main energy of the signal at time t is in the smallest mode and is proportional to L 2 (t)/t 2 . Thus, we have here again the same law for the energy decay as for Burgers turbulence. The numerical simulation based on the algorithm [27] was done to illustrate the process of particles merging and velocity field evolution. The trajectories of the particles and profile of the field at different times are plotted for the initial "white noise" signal (n = 0, h = −1/2) in Fig.3, and for the initial "Brownian" motion (n = −2, h = 1/2) in Fig.4. Ten modes with the ratio of successive wavenumbers ǫ = 1/N = 1/2 were used. The plots show the initial stage of the evolution in some relatively small region where the finiteness of the number of modes is not significant. In Fig. 3 one can see that for n = 0 the initial "sawtooth" multiscale function oscillates near v = 0 like a "white noise" with finite variance. After the collision of light particles, when the reverse sawtooth function transforms into a sawtooth wave with positive gradient ∂ x v = 1/t, the structure of the signal is relatively simple, and even for N = 2 the main energy remains in the mode with smallest wavenumber. In the case n = −2 the initial profile has a large deviation behaviour which is typical for Brownian motion functions. After merging of light particles, the sawtooth profile has a set of small shocks with different amplitudes, which is also similar to the properties of Brownian signal in the Burgers turbulence [28]. In Figs. 3(c) and 4(c) the velocity field at three successive merger times t * /t * * = N (n+3)/2 are plotted. These figures show the logarithmic periodic self-similarity of the evolution of multiscale signals. We notice now that such multiscale waves may be constructed for multidimensional Burgers' equation. Let us assume that the initial vector field V p (x) is an infinite series of "reverse" modes v r (x): V p (x) = ∞ r=p v r (x) ,(5.3) In the two dimensional case, the rth "reverse" mode may be composed of piecewise linear functions defined on a system of regular triangles of size L r covering the plane. We consider here the special case when the ratio of the scales of two adjacent modes is L r+1 /L r = ǫ −1 = N = 2. We assume also the special symmetry and phase relation between the different modes. In our case one big triangle is divided into four smaller triangles with vertices located at midpoints of its sides ( see Fig.5 ). We assume that inside each triangle in the rth mode the velocity has a linear profile v r (x) = −γ r (x − x + ), where x + is the coordinate of the center of the triangle. The solution of the multidimensional Burgers equation for such initial perturbation v (x, t) = −γ (x − x + ) 1 − γt (5.4) is now valid inside the triangle of size L r (t) = L r (1 − tγ r ) . The value γ −1 r has the dimension of time, and at the finite time t r = γ −1 r the velocity gradient becomes infinite. On the other hand, at time t = t r , the mode transforms into a "direct" sawtooth wave with the universal behaviour inside the new set of triangles and with the gradient 1/t independent of the amplitude and wavenumber of the mode: v (x, t) = x − x c t , (5.5) where x c is now the center of the triangle, coinciding with the top of the initial triangular set. Consider now the evolution of a gas of sticky particles in the case of independent evolution of the rth mode. Then, it is obvious that at the initial stage, inside the "collapsing" triangle of size L r (t) = L r (1 − tγ r ) the density increases as ρ(x, t) = ρ 0 1 (1 − t/t p ) 2 , (5.6) while ρ is zero outside "collapsing" triangular in each cell. At time t = t r , all the light particles in each cell collide into a single heavy particle with mass m r = ρ 0 L 2 r √ 3/4,(5.7) and the heavy particles' positions are equal to the center of the initial triangle x + . We assume also that the evolution of the rth mode is characterised by a nonlinear time t r the same as in the one dimension case (2.4): t r = γ −1 r = t 0 /(2 −(n+3)/2 ) r . (5.8) Let us now consider the evolution of the vector field V p (x) (5.3) which is an infinite series of "reverse" modes. The evolution of the vector field is very similar to the evolution of the scalar field (3.1). For the gradient of the initial perturbation V p (x) we have the same relation (3.2) as in 1D. The effective time of nonlinearity of the smallest pth mode in the vector field (5.3), in presence of all large scale modes, is determined by the equation (3.3). Thus, after the time of collisiont p , heavy particles with mass m p (5.7) appear. Velocities of these particles will be determined by the function V p+1 (x) (5.3), which is a sum of all larger modes, but the number of particle collisions is determined by the next p + 1 mode. At timẽ t p+1 we have a collision of four heavy particles. The extrapolation of this particle merger process to next generations is evident by recurrence. The qth collision of heavy particles takes place at timet p+q ), (5.8) the masses of these particles at this time are determined by the scale of the (p + q)th mode m p+q = ρ 0 L 2 p+q √ 3/4 (5.7). Here also one can speak about the logarithmic periodic self-similarity of the motion of the particles. This means that at arbitrary interval [t q ,t q+1 ] the motion of the particles will be similar to the motion of the particles in the intervalt p ,t p+1 , by the scaling factor x p /x q = 2 (q−p) in space, and the scaling function t p /t q = (2 n+3 2 ) (p−q) in time. We used computer simulation for studying two dimensional case; results of the simulation were generated in so called VRML (Virtual Reality Modeling Language), which enables to handle three dimensional figure in different projections. Fig. 6 presents snapshots of this modeling. On the Fig. 6(a) one can see that particles formed by small triangles move towards the center of an embracing triangle; this center in its turn, moves towards the center of the next bigger triangle in the hierarchy; e.t.c. Fig.6(b) gives the side-view of this process. Conclusion In conclusion, we would like to point out that the evolution of the multiscale signal with the Weierstrass spectrum simulates properties of Burgers turbulence such as self-similarity, conservation of large scale structures and has the same laws of the energy decay and integral scale. The difference between the deterministic model and Burgers turbulence is that here we have the exact solution for the evolution of multiscale signals and these properties are not stochastic but deterministic. The evolution of the multiscale signal is exactly self-similar in logarithmically spaced time intervals. The evolution of the large scale modes is completely independent of the small scales modes, even if these have very large amplitudes. These properties take place for Burgers turbulence in the stochastical sense and, moreover, for a signal with cutoff frequencies of small scales, only asymptotically. Of course, these properties of the multiscale signal are determined by the special form of modes (reverse sawtooth function), the special relations between wavenumbers of modes (k r+1 = k r /N, where N is an integer) and their phase relations. On the other hand, these model signals do not reflect such properties of Burgers turbulence as qualitative difference in the behaviour of the turbulence for n < 1 and n > 1 in the power spectrum (1.9) due to the process of generation of large scale components in the spectrum. For the deterministic model this process is not present due to the special relation of wavenumbers. Let as now move to the mechanical interpretation of solution of the Burgers equation. For the initial reverse sawtooth wave, all the matter turns into heavy particles at the same moment of time. Thus, after this time, the evolution of Burgers turbulence is fully determined by the motion of heavy particles. The trajectories of heavy particles form regular tree-like structure on the plane (X, t), see Figs. 3 and 4. The properties of this structure depend on the parameters of our model. The integer ǫ −1 = N is the number of trajectories which intersect at one point and form a new branch of our structure. For ǫ −1 = N = 2 we, thus, obtain binary tree structure. Changing of the parameter h stretches or contracts the structure in the t direction. One can say that our structure is the plane representation of the N−tree; the root of our tree is located at t = +∞. This tree is similar to the flattened fractal model of botanical umbrella tree (see [21]). If we take some node (X, T ) of this structure as a root, and consider the trajectories of all heavy particles, which will merge at the moment of time T at this point (X, T ) we shall also obtain an n-tree. The whole tree seems self-similar, because every branch plus the branches it carries is a reduced scale version of the whole. We also notice that such multiscale waves may be constructed for multidimensional Burgers' equation. Figure captions the evolution of the field after t M will be the same as the evolution of only the largest mode v (M ) (x). The reason for this, is of course, the special relation between the phases ϕ p and the wavenumbers k p of all interacting modes: k p = k 0 /N p ( see equation (1.12) ). For integer N the minimal value of any combination of these wavenumbers is equal to the largest mode wavenumber k M = k 0 /N M . So the nonlinear interaction does not produce new components at frequencies less than k M . Figure 1 : 1Evolution of one-mode pulse. (a) particle trajectories; (b) evolution of the initial velocity field given in the same spatio-temporal scale. Bold lines on the time axis denote moments at which the profiles of the velocity are plotted. Figure 2 : 2Evolution and interaction of two modes. (a) particle trajectories; (b) evolution of the initial velocity field given in the same spatio-temporal scale. Bold points on the time axis denote moments at which the profiles of the velocity are plotted. Figure 3 : 3Evolution of the multiscale fractal signal with n = 0 (h = −1/2), corresponding to "white noise" signal. (a) particle trajectories; (b) evolution of the initial velocity field; (b) velocity field taken at the initial moment of time and then at three successive time moments of self-similarity. Figure 4 : 4Evolution of the multiscale fractal signal with n = −2 (h = 1/2), corresponding to "Brownian motion" signal. (a) particle trajectories; (b) evolution of the initial velocity field; (b) velocity field taken at the initial moment of time and then at three successive time moments of self-similarity. Figure 5 : 5Plane construction for two dimensional case. The hierarchy of triangles, used for the construction of the multiscale signal, with four layers shown. The initial signal is constructed as a series of signals piece-wise linear on triangles. Figure 6 : 6Particle trajectories for the multiscale fractal signal in two dimensional case presented in spatio-time three dimensional space; the width of particle trajectory reflects its mass: (a) top view; (b) side-view. AcknowledgementsThe authors are grateful to U. Frisch for useful discussions and for his hospitality at the Observatoire de la Côte d'Azur, to G.M. Molchan J R Angilella, J C Vassilicos, Spectral, diffusive and convective proprties of fractal and spiral fields. 124J.R. Angilella, J.C. 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[ "Optimal Energy Allocation For Delay-Constrained Traffic Over Fading Multiple Access Channels", "Optimal Energy Allocation For Delay-Constrained Traffic Over Fading Multiple Access Channels" ]	[ "Antonious M Girgis ", "Amr El-Keyi ", "Mohammed Nafie [email protected] \nElectronics and Communications Dept\nFaculty of Engineering\nCairo University\nGizaEgypt\n", "\nWireless Intelligent Networks Center (WINC)\nNile University\nCairoEgypt\n" ]	[ "Electronics and Communications Dept\nFaculty of Engineering\nCairo University\nGizaEgypt", "Wireless Intelligent Networks Center (WINC)\nNile University\nCairoEgypt" ]	[]	In this paper, we consider a multiple-access fading channel where N users transmit to a single base station (BS) within a limited number of time slots. We assume that each user has a fixed amount of energy available to be consumed over the transmission window. We derive the optimal energy allocation policy for each user that maximizes the total system throughput under two different assumptions on the channel state information. First, we consider the offline allocation problem where the channel states are known a priori before transmission. We solve a convex optimization problem to maximize the sum-throughput under energy and delay constraints. Next, we consider the online allocation problem, where the channels are causally known to the BS and obtain the optimal energy allocation via dynamic programming when the number of users is small. We also develop a suboptimal resource allocation algorithm whose performance is close to the optimal one. Numerical results are presented showing the superiority of the proposed algorithms over baseline algorithms in various scenarios.	10.1109/glocom.2016.7842084	[ "https://arxiv.org/pdf/1607.00711v1.pdf" ]	3,253,701	1607.00711	eecb51db7bbe947cee0cca5081f2586e47154ed8	Optimal Energy Allocation For Delay-Constrained Traffic Over Fading Multiple Access Channels Antonious M Girgis Amr El-Keyi Mohammed Nafie [email protected] Electronics and Communications Dept Faculty of Engineering Cairo University GizaEgypt Wireless Intelligent Networks Center (WINC) Nile University CairoEgypt Optimal Energy Allocation For Delay-Constrained Traffic Over Fading Multiple Access Channels Index Terms-Resource allocationmultiple-access channelsfadingdynamic programming In this paper, we consider a multiple-access fading channel where N users transmit to a single base station (BS) within a limited number of time slots. We assume that each user has a fixed amount of energy available to be consumed over the transmission window. We derive the optimal energy allocation policy for each user that maximizes the total system throughput under two different assumptions on the channel state information. First, we consider the offline allocation problem where the channel states are known a priori before transmission. We solve a convex optimization problem to maximize the sum-throughput under energy and delay constraints. Next, we consider the online allocation problem, where the channels are causally known to the BS and obtain the optimal energy allocation via dynamic programming when the number of users is small. We also develop a suboptimal resource allocation algorithm whose performance is close to the optimal one. Numerical results are presented showing the superiority of the proposed algorithms over baseline algorithms in various scenarios. Abstract-In this paper, we consider a multiple-access fading channel where N users transmit to a single base station (BS) within a limited number of time slots. We assume that each user has a fixed amount of energy available to be consumed over the transmission window. We derive the optimal energy allocation policy for each user that maximizes the total system throughput under two different assumptions on the channel state information. First, we consider the offline allocation problem where the channel states are known a priori before transmission. We solve a convex optimization problem to maximize the sum-throughput under energy and delay constraints. Next, we consider the online allocation problem, where the channels are causally known to the BS and obtain the optimal energy allocation via dynamic programming when the number of users is small. We also develop a suboptimal resource allocation algorithm whose performance is close to the optimal one. Numerical results are presented showing the superiority of the proposed algorithms over baseline algorithms in various scenarios. Index Terms-Resource allocation, multiple-access channels, fading, dynamic programming. I. INTRODUCTION Wireless communication channels are characterized by their time-varying fading nature that has significant effect on the performance of wireless networks. Various algorithms have been proposed to design efficient resource allocation schemes that optimize the system performance over fading channels, e.g., by minimizing the transmission power, minimizing the delay, or maximizing the system throughput. Resource allocation over fading channels has been studied for pointto point communication in different contexts, e.g., [1]- [5]. In [1] the expected Shannon capacity for fading channels was obtained when the channel state information (CSI) is known causally at the transmitter and the receiver. Furthermore, it has been shown that the "water-filling" algorithm achieves the maximum expected capacity. The authors of [5] considered the problem of minimizing the expected energy to transmit a single packet over a fading channel subject to a hard deadline. In [2]and [3], a dynamic program formulation was proposed to maximize a general throughput function under constraints on the delay and the amount of energy available at the transmitter. In [4], the work of [2] was extended to energy harvesting systems where the transmitter has causal CSI. The capacity region of the multiple access channel (MAC) has been studied in various settings, see for example [6]- [11]. In [6], the capacity region of the Gaussian multipleinput multiple-output (MIMO) MAC was characterized. The authors of [6] proposed an iterative water-filling algorithm to obtain the optimal transmit covariance matrices of the users that maximize the weighted sum capacity. In [7] the capacity region of the fading MAC was characterized by Tse and Hanly. Furthermore, the power allocation policy that maximizes the long-term achievable rates subject to average power constraints for each user was introduced. In [8], Hanly and Tse introduced an information-theoretic characterization of the capacity region of the fading MAC with delay constraints. In addition, they provided the optimal power allocation policy that achieves the delay-limited capacity. In [12], Wang developed the optimal energy allocation strategy for the fading MAC with energy harvesting nodes by assuming that the CSI is non-causally known before the beginning of transmission. In [13], the capacity region of the fading MAC with power constraint on each codeword was investigated. However, the authors of [13] focused their work on the low signal-to-noise ratio (SNR) regime where they showed that the one-shot power allocation policy is asymptotically optimal. In this paper, we consider a system composed of multiple users transmitting to a single base station (BS) over a fading MAC. The transmission occurs over a limited time duration in which each user has a fixed amount of energy. Some motivating scenarios and applications for this system model are introduced in [2], [3], [8], e.g., satellites, remote sensors, and cellular phones with limited amount of energy transmitting delay-sensitive data to a single receiver. We develop energy allocation strategies to maximize the expected sum-throughput of the fading MAC subject to hard deadline and energy constrains. First, we consider the offline allocation problem in which the channel states are known a priori to the BS. We show that the optimal solution of this problem can be obtained via the iterative water filling algorithm. Next, a dynamic program formulation is introduced to obtain the optimal online allocation policy when only causal CSI is available at the BS. Since the computational complexity of the optimal online policy increases exponentially with the number of users, we develop a suboptimal solution for the online allocation problem by exploiting the proposed offline allocation policy. Moreover, we investigate numerically the performance of the proposed policies and compare them with the equal-energy allocation and the one-shot energy allocation policy of [13]. The rest of the paper is organized as follows. In Sec-tion II, we present the system model and formulate the maximum sum-throughput optimization problem. The offline energy allocation is introduced in Section III. We study the online allocation in Section IV, where dynamic programming is utilized to obtain the optimal policy and a suboptimal policy with reduced computational complexity is proposed. In Section V, we present our numerical results and compare the performance of different policies in various scenarios. Finally, we conclude the paper in Section VI. II. SYSTEM MODEL We consider a discrete-time MAC as shown in Fig. 1, where N users communicate with a single BS in a slotted wireless network. We assume a flat-fading channel model in which the channel gain of each user is constant over the duration of the time slot and changes independently from time slot to another according to a known continuous distribution. Thus, the received signal by the BS at time slot t is given by y t = N i=1 h (i) t x (i) t + n t(1) where n t is a zero-mean white Gaussian noise with variance σ 2 , and x (i) t is the transmitted signal of user i at time slot t. The channel gain between the ith user and the BS at time slot t is denoted by h (i) t , where the channel gains of each user h (i) t , i ∈ {1, · · · , N }, are independent identically distributed with the cumulative distribution function (CDF) F (i) H (x) . Let E i denote the maximum amount of energy that can be expended by user i during T time slots, where T denotes the transmission window in which each user must transmit his data. Let N = {1, · · · , N } denote the set of users communicating with the BS, and T = {1, · · · , T } denote the set of the time slots during which communication occurs. Our goal is to maximize the sum-throughput of the MAC over the transmission window under constraints on the available energy for each user. Let e (i) t denote the consumed energy by the ith user at time slot t. Hence, the maximum achievable sum-throughput of the MAC at time slot t, when the channel gains of all users at time slot t are known, is given by [14] R (e t , h t ) = τ W log 2 1 + 1 τ N o N i=1 h (i) t e (i) t(2) where W and τ are the channel bandwidth, and the time slot duration, respectively, and N o = W σ 2 is the noise power in watts. 1 In (2), h t = h (1) t , · · · , h (N ) t and e t = e (1) t , · · · , e (N ) t are the channel gains vector and the consumed energy vector of all users at time slot t, respectively. Let E (i) 1 t h   N t h t n 1 User N User T E , 1 T E N , -user MAC Channel BS t y   1 t x N   N t x Fig. 1: System model where the initial state of the energy queue is E (i) 1 = E i . In addition, the energy vector E t = E (1) t , · · · , E (N ) t represents the energy levels of all users at time slot t ∈ T . We aim to get the energy allocation policy for each user i ∈ N to maximize the expected sum-throughput of the MAC over a deadline of T slots. Towards this objective, we formulate the following optimization problem: max e1,··· ,e T E T t=1 R (e t , h t ) s.t. T t=1 e (i) t = E i i ∈ N e t 0 t ∈ T(4) where 0 denotes a row vector whose elements are equal to zero, E denotes the expectation with respect to the channel vectors h t , t ∈ T , and the maximization is over all feasible energy allocation policies. In the following sections, we first study the offline allocation policy in which the channel gains of all users are known a priori for T time slots. Next, we study different online allocation policies that maximize the expected sum-throughput of the MAC when only causal CSI is available. III. OFFLINE ENERGY ALLOCATION In this section, we introduce the optimal offline energy allocation policy when the channel vectors h t , t ∈ T , are noncausally known to the BS and the users at the beginning of the transmission. Since the channel vectors h t , t ∈ T are a priori known, the optimization problem (4) can be reformulated as a deterministic optimization problem max e1,··· ,e T T t=1 R (e t , h t )(5) subject to the same constraints of the optimization problem (4). Theorem 1. The optimal offline transmission policy for the users is obtained by solving the following equations e (i) t = γ (i) o − γ (i) t + ∀i ∈ N , t ∈ T (6) T t=1 e (i) t = E i ∀i ∈ N (7) γ (i) t = τ N o + n =i h (n) t e (n) t h (i) t (8) where γ (i) o is a threshold value obtained by substituting from (6) into (7), and (x) + = max (0, x). Proof: Refer to the Appendix. In the single user case, i.e., N = 1, the optimal offline policy in Theorem 1 is the conventional water-filling algorithm [1], where the noise to the channel gain ratio at each time slot t determines the amount of energy allocated to the time slot t. In case of the multiple users, i.e., N > 1, we note that the energy allocation policy of the ith user for a given energy allocation of the other users e (i) = e (i) 1 , · · · , e (i) T , is also obtained via the water-filling algorithm. However, in this case, the interference signals of the other users n =i h for i = 1 to N do 4: Let γ (i) t = τ σ 2 + n =i h (n) t e (n) t h (i) t , ∀ t ∈ T 5: e (i) t = γ (i) o − γ (i) t + , ∀ t ∈ T 6: T t=1 e (i) t = E i 7: end for 8: end for Note that a closed-form expression for the optimal solution introduced in Theorem 1 can not be found. Nevertheless, the optimal solution can be obtained by applying the iterative water filling algorithm (IWF) described in Algorithm 1 to iteratively solve equations (6)- (8) where L max is the maximum number of iterations. In each iteration, the IWF algorithm successively updates the optimal energy allocation of each user using the water-filling algorithm while assuming that the allocation policy of the other users are fixed. Hence, at each iteration the algorithm tries to maximize the objective function of the problem (5) by adapting the energy allocation of a single user while considering the signals of the other users n =i h (n) t e (n) t as noise. Since the objective function is monotonically increasing in the energy allocation policy of each user (i) , the objective function cannot decrease after any iteration. As a result, the IWF solution approaches the optimal solution of problem (5) as the number of iterations L max increases where L max determines the error tolerance. The IWF algorithm was applied in [6] to find the optimal transmit covariance matrices of the users that achieve the boundary of the Gaussian MIMO-MAC capacity. In a similar manner to [6], we can assume the channel gains of the ith user over the time window (h (i) 1 , · · · , h (i) T ) as effective channel gains of T transmit antennas of the ith user. Therefore the results of the IWF algorithm obtained in [6] can be applied here. Theorem 2. For a finite number of iterations, the IWF algorithm described in Algorithm 1 converges to the optimal allocation policy which is the solution of the optimization problem in (5). Furthermore, the IWF algorithm achieves a sum-throughput lower than the optimal within (N −1)T 2 nats after a single iteration. Proof: See Theorem 4 and Theorem 5 in [6]. IV. ONLINE ENERGY ALLOCATION In this section, we assume that the channel vector h t is causally known to the BS and the users at the beginning of time slot t while future channel states are not known. Let X t = (E t , h t ) denote the state of the system which is comprised of the channel gains and the energy levels of all users at time slot t. We aim to obtain the energy allocation policy G * = [e * 1 (X 1 ) , · · · , e * T (X T )] that maximizes the expected sum-throughput of the MAC within a duration of T slots by sequentially solving the optimization problem in (4). The optimal energy allocation policy G * can be obtained by formulating the optimization problem in (4) as a finite horizon dynamic program (DP) that can be described by the following two equations U T (E T , h T ) = R (E T , h T ) (9a) U t (E t , h t ) = max 0 et Et R (e t , h t ) + U t+1 (E t − e t ) ∀1 ≤ t < T (9b) where U t+1 (E) = E {U t+1 (E, h)}. The equations in (9) are Bellman's equations of the finite horizon DP [15], where U t+1 (E) is the maximum expected sum-throughput that can be obtained during the remaining T − t slots given that the energy levels of all users is E. Note that the optimal policy is a vector of functions mapping the current state of the system (the channel gains and the energy levels) to an amount of energy determined for each user. In (9a) the users transmit with all available energy E T to maximize the total sum-throughput at the last time slot T . On the other hand, for time slots t = 1, · · · , T −1 there is a tradeoff between the current reward R (e t , h t ) and the expected future reward U t+1 (E t − e t ). Hence, the optimal energy allocated for each user at time slot t, is determined by maximizing the current throughput plus the expected future throughput. A. The optimal policy The optimal allocation policy G * is obtained recursively by solving Bellman's equation in (9) at each time slot t ∈ T , where U t+1 (E) is computed backwards in time. However, we can not get a closed-form expression for the expected reward function U t+1 (E) even in the single user case. Therefore, U t+1 (E) is computed numerically using the discretization method [15]. B. Suboptimal policy The computational complexity required to solve (9) numerically grows exponentially with the number of users [15]. In order to alleviate this problem, the one-shot energy allocation policy was introduced in [3] and [13] to solve (9) efficiently. The one-shot energy allocation policy arises from the linear approximation of the throughput function, i.e., R (e t , h t ) ≈ 1 σ 2 N i=1 h (i) t e (i) t .(10) Note that the linear approximation is acceptable in the wideband regime, i.e., when W → ∞, and/or when all users transmit at low SNR, where the transmit SNR of the ith user is given by SNR i = Eih (i) o τ No , h (i) o = ∞ 0 xdF (i) H (x). Hence, Bellman's equations can be restated as follows U T (E T , h T ) = N i=1 h (i) T E (i) T (11a) U t (E t , h t ) = max 0 et Et N i=1 h (i) t e (i) t +Ũ t+1 (E t − e t ) , 1 ≤ t < T (11b) By applying the DP recursion backward in time from the time slot t = T to the time slot t = 1, the expected reward functioñ U t (E) for t ∈ T can be computed as follows U t (E) = N i=1 E (i) ν (i) t(12) Furthermore, the one-shot energy allocation policy which solves (11) is given by e (i) t = E (i) t if h (i) t > ν (i) t+1 0 if h (i) t ≤ ν (i) t+1 , ∀ i ∈ N(13) where ν (i) t = ν (i) t+1 + ∞ ν (i) t+1 xdF (i) H (x), and ν (i) T = h (i) o . The one-shot energy allocation policy allocates the available energy for the ith user E i to the earliest time slot t ∈ T that has a channel gain h (i) t > ν (i) t+1 . We refer to [13] for more insights and details. Next, we develop a low-complexity suboptimal solution to solve the recursive DP introduced in (9). We show through numerical simulations that the performance of the suboptimal algorithm is close to that of the optimal policy when the number of users is small. The suboptimal solution is obtained by applying the certainty equivalent controller (CEC) scheme (see Chapter 6 in [15]), in which the following three steps are applied at each time slot t: 1) Certainty step: We replace all uncertain variables with their means. Hence we assume that the future channel gain of each user over the remaining T − t slots is equal to its mean, i.e., h k = h o for k = t + 1, · · · , T where h o = h (1) o , · · · , h (N ) o , h (i) o = ∞ 0 xdF (i) H (x). 2) Optimization step: After the certainty step, the recursive optimization problem in (9) at time slot t can be reformulated as the following deterministic optimization max et,··· ,ẽ T R (ẽ t , h t ) + T k=t+1 R (ẽ k , h o ) s.t. T k=tẽ (i) k ≤ E (i) t , i ∈ Ñ e k 0, k = t, · · · , T(14) where the solution of the optimization problem in (14) is obtained in a similar way to the offline allocation problem introduced in Section III by applying the IWF algorithm in Algorithm 1 over T − t + 1 slots with an amount of energy available at each user E (i) t for i ∈ N . 3) Allocation step: We set e t =ẽ t and compute the energy levels of all users at t + 1 using Equation 3. Then, we go to the next time slot t + 1. V. NUMERICAL RESULTS In this section, we numerically evaluate the performance of various energy allocation policies introduced throughout the paper. For comparison, we consider a simple energy allocation policy namely the equal-energy allocation, where each user allocates an equal amount of energy for each time slot of the transmission window regardless the effect of the channel t = E i T , ∀i ∈ N , ∀t ∈ T(15) Notice that this policy is optimal in case of time-invariant channels, where the channel gain of each user is constant over the deadline. For simplicity, we consider a symmetrical case, where all users are equipped with an equal amount of energy, i.e., E i = E, ∀i ∈ N , and the channel gains of all users are i.i.d., where the channel gains are generated according to the exponential distribution with parameter λ = 1, i.e., F (i) H (x) = 1 − e −x , ∀i ∈ N . Also, we consider the following parameters: the bandwidth W = 1 MHz, the noise power N o = 1 watts, and the slot length τ = 1 seconds, and hence, the transmit SNR of each user SNR i = E, ∀i ∈ N . We use the performance of the offline allocation policy as an upper bound on the performance of online policies. In the following figures, the performance of the optimal offline, suboptimal, one-shot, and equal-energy policies are obtained by averaging over 10 4 randomly generated channel realizations, while the performance of the optimal online policy is obtained by using the discretization method [15]. Figs 2 and 3 show the average sum-throughput of the MAC versus the transmit SNR of each user for a system composed of N = 2 users and transmission window length equal to T = 5 time slots. Fig. 2 focuses on the low SNR regime where the SNR is varied from −30 dB to 0 dB. It is clear that the performance of the proposed suboptimal and the one-shot policies is close to the optimal one, although, the proposed suboptimal policy performs better when the SNR approaches 0 dB. Moreover, the equal-energy allocation policy has the worst performance. In Fig. 3, the SNR varies from 0 dB to 20 dB to investigate the performance of the different policies in the medium and high SNR regimes. We can see from this figure that the one-shot policy deviates from the optimal solution, since the linear approximation of the throughput function is no longer valid at high SNR. However, the performance of the proposed suboptimal policy is still very close to that of the optimal solution. Next, we investigate the effect of the number of users N on the performance of different policies. Fig. 4a and Fig. 4b show the average sum throughput of the system (for T = 5 slots) at SNR= −10 dB and SNR=10 dB, respectively. We can see from Fig. 4a that both the proposed suboptimal policy and the one-shot policy almost have the same performance for any number of users in the low SNR regime. When the number of users is much larger than the time slots of the transmission window, i.e., N T , each time slot of the transmission widow would be shared with a lot of users. In other words, each user would suffer from high interference signals at each time slot of the transmission window. Therefore the best choice is to allocate the available energy of each user to a single time slot of the transmission window that has a favorable channel gain. Hence Fig. 4b shows that the one-shot policy converges to the proposed suboptimal policy in the high SNR regime for N T . However, the equal-energy allocation policy has better performance than the one-shot policy when the number of users is small. On the other hand, Fig. 4a and Fig. 4b show that the gap between the equal-energy allocation policy and the suboptimal policy increases as the number of users increases since the competition on the available resources (the time slots of the transmission window) increases as the number of users increases. VI. CONCLUSION In this paper, we have proposed energy allocation strategies for the N -user fading MAC with delay and energy constraints under two different assumptions on the channel states information. In the offline allocation, a convex optimization problem is formulated with the objective of maximizing the sum-throughput of the fading MAC within the transmission window where the optimal solution is obtained by applying the iterative water filling algorithm. In the online allocation, the problem is formulated via dynamic programming, and the optimal solution is obtained numerically by using the discretization method when the number of users is small. In addition, we have proposed a suboptimal solution with reduced computational complexity that can be used when the number of users is large. Numerical results have been provided to show the superiority of the proposed algorithms compared to the equal-energy allocation and the one-shot allocation algorithms. APPENDIX The optimal offline transmission policy is obtained by solving the optimization problem (5). Since the objective function of (5) is the sum of concave functions R (e, h), and the constrains are affine functions, then the optimization problem (5) is a convex optimization problem that can be solved using Lagrange method. The Lagrangian is given by 16) where µ (i) is the Lagrange multiplier associated with the ith equality constraint in (4), and λ (i) t is the Lagrange multiplier associated with the ith inequality constraint in (4). Slaters condition is satisfied for this problem, and hence, the Karush-Kuhn-Tucker (KKT) conditions are necessary and sufficient for optimality [16]. The KKT conditions are given by L = T t=1 R (e t , h t )− N i=1 µ (i) T t=1 e (i) t − E i − N i=1 T t=1 λ (i) t e (i) t( time slot t are considered as noise. Hence, the energy allocation policy of the ith user is significantly affected by the energy allocation policy of the other users, where the allocated energy for the ith user in time slot t depends on γ (i) t which represents the ratio between the interference-plus-noise power and the channel gain of the ith user at time slot t.Algorithm 1 Iterative water-filling (IWF) algorithm 1: Initialization: e t = 0, ∀ t ∈ T 2: for l = 1 to L max do 3: Fig. 2 : 2Average sum-throughput in Mbits in the low SNR regime for T = 5 and N = 2. Fig. 3 : 3Average sum-throughput in Mbits in the high SNR regime for T = 5 and N = 2 fading and the allocation policy of the other users, i.e., e (i) Fig. 4 : 4Average sum-throughput of the MAC versus the number of users N for SNR = −10 dB and SNR = 10 dB t be the available energy for user i at time slot t. Thus, the evolution of the energy queue of the ith user is given byE (i) t+1 = E (i) t − e (i) t t = 1, · · · , T − 1(3)1 Note that the successive cancellation decoding strategy is the optimal decoding scheme that achieves the maximum sum-throughput of the MAC[8] t e (k) t − µ (i) − λ (i) t ∀i ∈ N , t ∈ T (17a) λ (i) t e (i) t = 0, λ (i) t ≥ 0, e (i) t ≥ 0, ∀i ∈ N , t ∈ T (17b) T t=1 e (i) t = E i , ∀i ∈ N (17c)Solving the above KKT conditions yields (6)-(8). Capacity of fading channels with channel side information. A J Goldsmith, P P Varaiya, IEEE Transactions on Information Theory. 436A. J. Goldsmith and P. P. 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[ "KFHE-HOMER: Kalman Filter-based Heuristic Ensemble of HOMER for Multi-Label Classification", "KFHE-HOMER: Kalman Filter-based Heuristic Ensemble of HOMER for Multi-Label Classification" ]	[ "Arjun Pakrashi [email protected] \nInsight Centre for Data\nAnalytics University College Dublin Dublin\nIreland\n", "Brian Mac Namee \nInsight Centre for Data\nAnalytics University College Dublin Dublin\nIreland\n" ]	[ "Insight Centre for Data\nAnalytics University College Dublin Dublin\nIreland", "Insight Centre for Data\nAnalytics University College Dublin Dublin\nIreland" ]	[]	Multi-label classification allows a datapoint to be labelled with more than one class at the same time. Ensemble methods generally perform much better than single classifiers. Except bagging style ensembles like ECC, RAkEL, in multi-label classification, other ensemble methods have not been explored much. KFHE (Kalman Filter-based Heuristic Ensemble), is a recent ensemble method which uses the Kalman filter to combine several models. KFHE views the final ensemble to be learned as a state to be estimated which it estimates using multiple noisy "measurements". These "measurements" are essentially component classifiers trained under different settings. This work extends KFHE to multi-label domain by proposing KFHE-HOMER which enhances the performance of HOMER using the KFHE framework. KFHE-HOMER sequentially trains multiple HOMER classifiers using weighted training datapoints and random hyperparameters. These models are considered as measurements and their related error as the uncertainty of the measurements. Then the Kalman filter framework is used to combine these measurements to get a more accurate estimate. The method was tested on 10 multi-label datasets and compared with other multi-label classification algorithms. Results show that KFHE-HOMER performs consistently better than similar multi-label ensemble methods.	null	[ "https://arxiv.org/pdf/1904.10552v1.pdf" ]	129,945,049	1904.10552	76146c95fa8c0805662b787ef8e70dcbc7ad337e	KFHE-HOMER: Kalman Filter-based Heuristic Ensemble of HOMER for Multi-Label Classification Arjun Pakrashi [email protected] Insight Centre for Data Analytics University College Dublin Dublin Ireland Brian Mac Namee Insight Centre for Data Analytics University College Dublin Dublin Ireland KFHE-HOMER: Kalman Filter-based Heuristic Ensemble of HOMER for Multi-Label Classification Multi-label classification allows a datapoint to be labelled with more than one class at the same time. Ensemble methods generally perform much better than single classifiers. Except bagging style ensembles like ECC, RAkEL, in multi-label classification, other ensemble methods have not been explored much. KFHE (Kalman Filter-based Heuristic Ensemble), is a recent ensemble method which uses the Kalman filter to combine several models. KFHE views the final ensemble to be learned as a state to be estimated which it estimates using multiple noisy "measurements". These "measurements" are essentially component classifiers trained under different settings. This work extends KFHE to multi-label domain by proposing KFHE-HOMER which enhances the performance of HOMER using the KFHE framework. KFHE-HOMER sequentially trains multiple HOMER classifiers using weighted training datapoints and random hyperparameters. These models are considered as measurements and their related error as the uncertainty of the measurements. Then the Kalman filter framework is used to combine these measurements to get a more accurate estimate. The method was tested on 10 multi-label datasets and compared with other multi-label classification algorithms. Results show that KFHE-HOMER performs consistently better than similar multi-label ensemble methods. Introduction In multi-label classification problems a datapoint can be assigned to more than one class, or label, simultaneously [9]. For example, an image can be classified as containing multiple different objects, or a document can be labelled with more than one topic. In multi-class classification problems, objects can only belong to a single class, which makes multi-label classification a more general classification approach. Multi-label classification algorithms either break the multi-label problem down into smaller multi-class classification problems-for example ensemble classifier chains [16] known as problem transformation methods-or modify multi-class algorithms to directly train on multi-label datasets-for example BPMLL [25] known as algorithm adaptation methods. Ensemble classification methods train multiple component classifiers and aggregate them. Generally, ensemble methods perform better than the single component classifiers [10]. In multi-label classification literature, several methods are proposed which combine multiple multi-label models to form an ensemble. Ensemble classifier chains (ECC) [16] trains a classifier chain (CC) [16] model at each iteration with a subset of the training dataset and a random chain order of the labels and then combines the results. Random k-Label sets (RAkEL) [21] first partitions the labels into random overlapping subsets, and for each subset trains a label powerset classifier [1] and aggregates the results. Ensemble of binary relevance (EBR) [16] simply bags multiple binary relevance models [1]. Ensemble of pruned set (EPS) [15] is a bagged version of pruned set method, which is similar to label powerset, but removes the infrequently occurring labelsets (unique combination of labels) and focuses on the more important label relationships. AdaBoost.M1 [18] uses boosting in multi-label context. Hierarchy of multi-label classifiers (HOMER) [22] partitions the dataset hierarchically in a tree format, where each node in the hierarchy only predicts if a datapoint has at least one relevant of a group. Then it further splits the group of labels into smaller groups to form child nodes. The prediction task starts from the root, and at any given internal node v, the datapoint is passed down to a child based on the prediction of the model at v. The final label predictions are done at the leaves. Although, except AdaBoost.M1 most of these methods essentially combines multiple multi-label classification methods using simple bagging. AdaBoost.M1 performs boosting, but the performance was previously found to be poor [12]. Kalman filter-based Heuristic Ensemble (KFHE) [14] is an multi-class ensemble algorithm which takes a new approach. Unlike the existing boosting or bagging methods, KFHE imagines the ensemble to be trained as a hypothesis to be estimated in the hypothesis space. Then it takes noisy measurements in the hypothesis space, which is essentially trained classifiers, which it then combines using a Kalman filter. In effect, KFHE behaves like a combination of boosting and bagging. As ensembles generally outperforms the individual classifiers and the multilabel literature lacks non-bagging style ensemble classifiers, this leaves an opportunity to improve a multi-label classification using the KFHE framework. The main contributions of this work are: -To propose a multi-label ensemble classification method, KFHE-HOMER, which combines multiple HOMER classifier models using the KFHE framework. -Extensive experiment demonstrating the effectiveness of KFHE-HOMER. -Introduction of E-HOMER, a simple bagged ensemble version of HOMER. The remainder of the paper is structured as follows. Section 2 presents the relevant aspects of HOMER and KFHE. Section 3 introduces the proposed method KFHE-HOMER and E-HOMER. The experimental setup is described in Section 4 and the results are discussed in detail in Section 5. Finally, Section 6 discusses future works and concludes the paper. Related Work In this section, first the HOMER method will be discussed, followed by a basic introduction of Kalman filter required to describe KFHE. Next the KFHE method will be briefly explained. HOMER Hierarchy of multi-label classifiers (HOMER) [22] is a problem transformation method which divides the multi-label dataset into smaller subsets, but hierarchically. This method divides the dataset based on the labels, but establishes a hierarchical relationship between the partitions. It starts with the entire dataset D(x i , l i )∀ 1≤i≤n at the root node of the hierarchy which has all the labels L = {λ 1 , . . . , λ q }. Here x i is the datapoint, l i = [l i1 , . . . , l iq ] is the vector or labels assigned to the corresponding datapoint. If a label λ q is applicable to x i then l iq = 1, otherwise l iq = 0. Every leaf node has one associated label λ l . Any internal node v have only a subset of labels L v ⊂ L associated with itself, which is a union of the label subsets associated to its children. Therefore, L v = ∪L c , c ∈ children(v), where children(v) indicates the children of the node v in the hierarchy. Each node only keeps the datapoints which have at least one of the associated labels of the node applicable to them. Therefore, the dataset in node v is D v = {(x j , l j )\|∀(x j , l j ) ∈ parent(v), ∃ k λ q ∈ L v ∧ l jq = 1}. In each internal node v, the original labels of the datapoints are not stored, instead a meta-labels are generated and stored. At node v if the labels are partitioned into k partitions, then k meta-labels are created, where the meta label µ i represents the union of the labels in the ith partition. A datapoint is labelled with µ i if the datapoint is associated to any of the labels in the ith partition. The meta labels are generated as D v = {(x j , l j )\|l j = ∪ c∈children(v) µ c }. The root node and all the internal nodes v will have a trained model h v associated to it, which is trained using the dataset D v with the target being the meta labels associated with each datapoint in the child nodes. The utility of the meta labels is to indicate which branch or branches in the hierarchy has to be followed to the root for the prediction of the labels. A new datapoint d starts from the root of the hierarchy travels one or more paths from root to leaf. All the labels represented by the leaves which are encountered by the datapoint d are taken as the prediction. The partition of the labels at each level of the hierarchy is done in such a way that the similar labels stay together, this is to decrease the number of paths traveled from root to leaf. To keep the similar labels together, at each level a clustering algorithm on the label assignments of D v is performed on each node to partition the label subset L v into k disjoint label subsets for the node v's children. A variation of k-Means algorithm, named the balanced k-Means is proposed, which attempts to balance the cluster sizes. Static State Estimation using Kalman filter The discrete Kalman filter is a mathematical framework to estimate an unobservable state of a linear stochastic discrete time controlled process through noisy measurements [6]. In this section a very brief and intuitive overview of the Kalman filter to estimate a static one dimensional state will be given. Let there be a state, y, of a linear stochastic system has to be estimated, where y cannot be observed directly. The state of the system can be estimated in two ways. Firstly, given an estimate of the stateŷ t−1 with a related variance p t−1 at time step (t − 1), a linear model is used to make an a priori state estimateŷ − t . The variance related toŷ − t is also updated to p − t . This variance can be imagined as the uncertainty of state. This is known as the time update step. Secondly, an external sensor can be used to get an estimate through a measurement, z t , of the state with a related variance r t , which can also be seen as the uncertainty of the measurement. Given these two noisy state estimates, the a priori estimate,ŷ − t , its related variance p − t , and the measurement z t , its related variance r t , the Kalman filter combines them optimally to get an a posteriori state estimate,ŷ t , which potentially has a lower uncertainty than the previous two. This is known as the measurement update step. The Kalman filter iterates through the time update and the measurement update steps. At iteration t, the a priori estimate is used in the measurement update step to get an a posteriori estimate, which is fed back to the time update in the next iteration as the a priori estimate. If the state to be estimated is assumed to be static then the time update step is considered to be non-existent, hence it becomesŷ − t =ŷ t−1 and p t − = p t−1 . This kind of scenario can occur in cases when, say, the voltage level of a DC battery or the altitude of a cruising aircraft is being estimated. In both of the cases, the DC voltage and the altitude of the aircraft is supposed to be constant, but unknown. In such cases, the measurement of the static state from a noisy sensor is repeatedly combined using the measurement update step. After consideringŷ − t =ŷ t−1 and p − t = p t−1 the measurement update steps are as followsŷ t =ŷ t−1 + k t (z t −ŷ t−1 ) (1) k t = p t−1 /(p t−1 + r t ) (2) p t = (1 − k t )p t−1(3) Here z t , the measurement, can be an external source or sensor (voltage or altitude sensor), r t is the related measurement variance indicating the uncertainty of the estimate. The k t is the Kalman gain, which optimally combines the a priori estimate and the measurement. A complete and detailed explanation of Kalman filters can be found in [6,23]. KFHE Fig. 1: The high level interactions between kf-m and kf-w [14] Kalman filter-based heuristic ensemble (KFHE) [14] is a multi-class ensemble algorithm proposed by Pakrashi and Mac Namee, which takes a different approach by viewing the ideal hypothesis for a specific classification problem as the static state to be estimated in the hypothesis space [5]. A classifier training task can be seen as searching for a hypothesis for a given problem in the hypothesis space, where a training algorithm navigates through the hypothesis space toward the ideal hypothesis 1 . For a specific learning problem the target of a learning algorithm is to learn the ideal hypothesis, which can be assumed to be stationary within the hypothesis space. In KFHE, estimation of this stationary hypothesis is modelled as a static state estimation problem and is then estimated by multiple noisy measurements as explained above. Two Kalman filters interact with each other in KFHE. The Kalman filter which estimates the ideal hypothesis is called the model Kalman filter, abbreviated as kf-m. The kf-m estimates the final model or hypothesis by combining multiple noisy measurements. The measurement in this case is defined as z (y) t = (h t (D) +ŷ t−1 )/2(4) Where h t = H(D,ŵ t−1 ) is a classifier model trained using algorithm H (decision tree, SVM, etc.) trained using different weights updated in the previous iteration,ŵ t−1 , assigned to different datapoints. A datapoint is weighted more if it was misclassified previously. Although, unlike AdaBoost [8], the weight for Table 1: Intermediate representation of a state for KFHE and KFHE-HOMER. A trained model is represented using the prediction scores of a given set of datapoints. This representation is used withŷ t , z t and h t (D). c 1 c 2 c 3 x 1 0.10 0.89 0.01 x 2 0.08 0.27 0.65 . . . . . . . . . . . . x n 0.77 0.20 0.03 the datapoins are determined by the weight Kalman filter or kf-w. The intuition of averaging the h t andŷ t is to estimate the impact h t will be induced on y t . Although, other measurement heuristics can be used, Eq. (4) will be used in this work. Note that, the ensemble model h t cannot directly be used with the equations in Section 2.2, therefore an intermediate proxy representation is used for the states in kf-m. The intermediate representation of a trained model is the labelwise prediction scores of a given dataset by the model of the corresponding state, as shown in. Therefore, the intermediate representation of a model (individual or ensemble) would be the predictionŷ t as shown in Figure 1. All arithmetic operations on this representation are done element wise, as it represents one single state as in Section 2.2. This representation is used forŷ t and z t . In the final estimated state the class assignment is done by taking the class with the highest score. The kf-w estimatesŵ t , which is a vector of weights to be used by the measurement step of kf-m. kf-w is identical to kf-m, but it estimates the weights. The measurement for the kf-w, z (w) t is a function f of weighted per datapoint error: z (w) t = [z (w) ti \|z (w) ti =ŵ ti × f (class(c i ) = class(z (y) ti )) 1 ≤ i ≤ n](5) Here, class indicates the class assignment of the prediction and c i indicates the class assigned to the datapoint x i . There are two variants discussed in [14] where f can be either the exponential function or the linear function in Eq. (5). The use of exponential function is similar to AdaBoost, but the final combination inŵ t is done using the kf-w filter in the measurement update step using equation similar to Eq. (1) The related noise is set same as the measurement noise taken in kf-m, which makes an assumption that weight measurement will induce an error no more than what the kf-m had in the previous iteration. The training step stores the component classifiers h t and the Kalman gains k The setting of the measurements and the related errors are the heuristic components of the method, which are set by making assumptions. An overall interaction of the kf-m and kf-w is shown in Figure 1. The superscript (y) indicates that the variables are related to kf-m, and the superscript (w) indicates these variables are related to kf-w.ŷ is the state estimate by kf-m, andŵ t is estimated by kf-w. The detailed derivation and explanation of KFHE can be found in [14]. KFHE-HOMER In this work, KFHE-HOMER extends KFHE to multi-label domain by combining multiple HOMER models. Also, a bagged ensemble version of HOMER, named E-HOMER, is introduced. HOMER is a method which has competitive performance [11,13] also with relatively faster training times. As explained in Section 2.2, there are two components of KFHE, the kf-m which estimates the hypothesis, and the kf-w, which computes the weights of the training datapoints during each measurement. To make KFHE work in a multi-label setting, the measurements of the kf-m and kf-w steps were adapted in this work. For KFHE-HOMER, the measurement at each step is considered as the average of a trained HOMER classifier and the previous estimate of the ensemble as shown in Eq. (4). The related measurement uncertainty r (y) t is considered as the hamming loss (hloss) [26] of the trained model. Each HOMER model at every step is trained on different weights,ŵ t , assigned to different datapoints, where the weights are determined by the kf-w component. The kf-w estimates one single vector of weightsŵ t using which a sampling with replacement of the training dataset is done. The measurement z (w) t for kf-w is taken as per-datapoint weighted hamming loss can be defined as follows z (w) t = [z (w) ti \|z (w) ti =ŵ ti × exp(hloss(x i , l i )) 1 ≤ i ≤ n](6) Eq. (6) is similar to Eq. (5) but uses hamming loss and exponential function to highlight misclassified datapoint in the measurement which will later be used by the measurement update step to get the weightsŵ t to be used in kf-m in the next iteration. In this case the related uncertainty is kept as in KFHE. The model h t in this case is a trained HOMER classifier model H(D,ŵ t−1 , C, k, φ). To weight the datapoints for training, the HOMER classifiers are trained using samples using the distributionŵ t−1 , the last updated weights, here the sample size is twice the size is 2n, or twice the size of orignial number of datapoints. Also, while training, the clustering algorithm C used by HOMER is randomly selected from {random, k-means, balanced k-means}, the number of cluster k is randomly selected too. Also, the kernel φ of the underlying SVM used by HOMER, is also selected randomly. Next the measurement is done using Eq. (4). The intuition of averaging is to estimate how much impact h t will be induced onŷ t−1 . Given the different HOMER models trained using different hyperparameters, many of them may lead to a poor measurement. The KFHE framework combines the measurements based on the measurement errors. If the measurement uncertainty r (y) t is higher than the uncertainty of the ensemble found upto tth iteration p (y) t , then the measurement is weighted less and the Kalman gain is lower than 0.5, and when the measurement error is lower the measurement is incorporated more, as a result of the Kalman gain being greater than 0.5. Therefore, based on this property, the HOMER models which have a poor performance will have a much less impact on the entire ensemble, whereas a more accurate HOMER model will have more impact on the entire ensemble. A simple bagged version of HOMER, E-HOMER, is also introduced and compared with KFHE-HOMER. In E-HOMER, at each iteration a HOMER classifier is trained on a bootstrap sample of the dataset of size 2n along with random HOMER cluster type, random number of clsuters and selecting the type of underlying SVM kernel randomly, as done in the case of KFHE-HOMER. The values ofŷ 0 , p (y) 0 andŵ 0 , p (w) 0 has to be initialised.ŷ 0 is initialised using a single HOMER classifier model h 0 . The value of p (y) 0 is set to 1 indicating maximum uncertainty. Equal weights to every point is given inŵ 0 and p (w) 0 is also initialised with 1. The KFHE-HOMER training algorithm is given in Algorithm 1. Here, all the lowercase symbols corresponds to symbols in Eq. (1), (2) and (3). The superscript (y) and (w) indicates that the corresponding variables are related to kf-m and kf-w respectively. In line 5-8 the different hyperparameter of HOMER is selected randomly. Next, the HOMER model is trained in line 9, and the measurement is done in line 10. Line 12 computes the Kalman gain k The prediction algorithm is same as in KFHE and is shown in Algorithm 2. Here the trained models and the Kalman gain values learned during the training along with the new datapoint is given. Using the models in line 5 the Kalman gain is repeatedly used to combine the measurements in line 4. After T iterations the predicted labels for the new datapoint d, the estimateŷ (y) T is returned. To find the label assignments, these scores are thresholded at 0.5. Experiment To evaluate the effectiveness of KFHE-HOMER, experiments were performed on ten well-known multi-label benchmark datasets listed in Table 2. In Table 2 the different properties of the multi-label datasets are summarised. Instances, Inputs and Labels are the number of datapoints, the dimension of the datapoints and the number of labels, respectively. Labelsets indicates the number of unique combinations of labels. Cardinality measures the average number of labels assigned to each datapoint and MeanIR [2] indicates the degree of imbalance of the labels, where higher values indicate higher imbalance. Algorithm 1 KFHE-HOMER training 1: procedure train(D = {(x i , l i )\|1 ≤ i ≤ n}, T ) 2: p (w) 0 = 1,ŵ 0 = [1/n, . . . , 1/n] 3: h t = H(D,ŵ 0 , C, k, φ),ŷ 0 = h 0 (D) 4: t = 1 5: for t ≤ T do 6: kf-m Section 7: Randomly select C, k and φ 8: C ∈ {k-means, balanced k-means, random}, 9: k ∈ {2, . . . , L }, 10: φ ∈ {linear , radial } 11: h t = H(D,ŵ t−1 , C, k, φ) 12: z (y) t = (h t (D) +ŷ t−1 )/2 Measurement 13: r (y) t = hloss(D, z (y) t ) 14: k (y) t = p (y) t−1 /(p (y) t−1 + r (y) t ) Kalman gain 15:ŷ t =ŷ t−1 + k (y) t (z (y) t −ŷ t−1 ) Measurement update 16: p (y) t = (1 − k (y) t )p (y) t−1 17: kf-w Section 18: z (w) t = [z (w) ti \|z (w) ti =ŵ ti × exp(hloss(x i , l i )) 1 ≤ i ≤ n] 19: r (w) t = r (m) t 20: k (w) t = p (w) t−1 /(p (w) t−1 + r (w) t ) Kalman gain 21:ŵ t =ŵ t−1 + k (w) t (z (w) t −ŵ t−1 ) Measurement update 22: p (w) t = (1 − k (w) t )p (w) t1: procedure predict(d, {h t , k (y) t \|∀ 1≤t≤T }, T ) 2:ŷ (y) 0 = h 0 (x), t = 1 3: for t ≤ T do 4:ẑ (y) t = (h t (d) +ŷ t−1 )/2 Measurement 5:ŷ t =ŷ t−1 + k (y) t (z (y) t −ŷ t−1 ) Measurement update 6: t = t + 1 7: end for 8: return (ŷ (y) T ) 9: end procedure The label-based macro-averaged F-Score [26] was used to measure the performance of models in these evaluations. This was chosen over Hamming loss, which has been used in several previous studies (e.g. [3,20,24]). This is because with the highly imbalanced multi-label datasets used in this study (see the high MeanIR scores for several datasets in Table 2) when Hamming loss is used the majority classes may overwhelm the performance of the minority class. The label-based macro-averaged F-Score is defined as follows F = 1 q q l=1 2 × P recision l × Recall l P recision l + Recall l For all evaluations 2 times 5 crossvalidation experiments were performed. The iterative stratification method [19] was used to generate the folds in the crossvalidation experiment. Performance of KFHE-HOMER was compared with ECC, a state-of-the-art ensemble based multi-label classifier, and E-HOMER, a bagging-based ensemble using HOMER as the base model. Also, individual CC model, HOMER-K (using k-means clustering) and HOMER-B (using balanced clustering) models were included to understand how much the ensembles led to improved performance over single base models. The cluster size for HOMER was selected using the best values found in the benchmark experiments in [13]. The HOMER and CC models used support vector machines (SVM) as their underlying learner, as they have proved to perform very well [11,13]. At each iteration of E-HOMER and KFHE-HOMER the type of HOMER clustering C was selected randomly from {balanced k-means, k-means, random}. The number of clusters k was selected randomly from the range k ∈ {2, . . . , L }. The kernel types φ for each of the base SVM models used by the component HOMER models were also selected randomly, from {linear, radial}. For ECC and E-HOMER the bootstrap sample was selected to be twice the size of the training dataset, to keep it consistent with KFHE-HOMER. For ECC, E-HOMER and KFHE-HOMER 100 component classifiers were trained. Table 3: Experimental results. Values in cells are mean label-based macroaveraged F-Scores from the crossvalidation experiments, and their standard deviations. The rank of each score for a dataset across the algorithms compared is shown in parenthesis. The last row shows the average rank of each algorithms. The "⇓", "↓" and " " symbols indicate the significance level at which the performance of an algorithm is shown to be worse than the performance of KFHE-HOMER for a dataset, "⇓" = 0.01, "↓" = 0.05 and " " = 0.1. The " "symbol indicates that the corresponding method is better than KFHE-HOMER at a significance level of 0.1. Therefore, the experimental environment were kept identical for all ensemble methods for a fair comparison. KFHE-HOMER and E-HOMER is implemented in R 2 , and the utiml library [17] is used for the multi-label classifiers. Table 3 shows the results of the experiments performed. The columns indicate the algorithms and the rows indicate the datasets. In each cell, the mean and standard deviation label-based macro-averaged F-Score (higher values are better) across the crossvalidation performed are shown. The values in the parenthesis indicate the relative ranking (lower values are better) of the algorithm with respect to the corresponding dataset. The last row of Table 3 indicates the overall average ranks of the algorithms compared. Table 3 shows that KFHE-HOMER attains the best average rank of 1.1. KFHE-HOMER attained the top rank for all the datasets, except for llog where KFHE-HOMER got the second rank. E-HOMER attained the second best overall average rank of 2.7, whereas ECC attained the third best overall average rank of 3.5. The classifiers, CC, HOMER-B and HOMER-K attained average ranks of 3.8, 4.7 and 5.2 respectively. This shows that for each case the ensemble methods have consistently performed better than the component classifiers. The E-HOMER method has performed better in almost all cases compared to a single HOMER model, as well as have performed better than ECC, which demonstrates the effectiveness of ensembling the HOMER method. The difference between E-HOMER and KFHE-HOMER is the aggregation method of the component HOMER classifier models and KFHE-HOMER has performed better than E-HOMER in all the cases. This demonstrates that KFHE-HOMER performs better than a bagged version of HOMER. Results To understand if KFHE-HOMER did attain significantly different (better or worse) results than the other methods per dataset, a two-tailed paired Wilcoxon's signed rank sum test [4] was performed over the folds of each crossvalidation experiment. KFHE-HOMER was set as the control method and compared to the other methods. The different significance levels at which the differences were found are indicated in the table. The symbols "⇓", "↓" and " " beside the values in Table 3 indicates that the method was significantly worse than KFHE-HOMER at a significance level of 0.01, 0.05 and 0.1, respectively. " " symbol indicates that a method was significantly better than KFHE-HOMER with a significance level of 0.1. From Table 3 it is clear that for almost all the datasets, KFHE-HOMER was significantly better than both the single HOMER-K and HOMER-B models. Also, KFHE-HOMER is significantly better than CC for most of the datasets. Interestingly, CC was better than KFHE-HOMER in the case of llog dataset. To further analyse the overall difference of the methods over the different datasets, a multiple classifier comparison was performed following the recommendations of García et al. [7]. A post-hoc Friedman aligned rank test was performed with the Finner p-value correction. The results of this evaluation is summarised in Figure 2, where the scale indicates the average ranks and if the methods are not connected with a horizontal line then they are significantly different over different datasets with a significance level of 0.05. This shows that KFHE-HOMER was significantly better than all the methods except E-HOMER in which case the null hypothesis of Friedman aligned rank test could not be rejected with a significance level of 0.05. Although, KFHE-HOMER attained better ranks in all the datasets. An overall pairwise table of the p-values of the post-hoc Friedman test is shown in Table 4. The lower diagonal of Table 4 a value in a cell is the p-values of the post-hoc Friedman aligned rank test with the Finner p-value correction for of the corresponding pair of algorithms in the rows and columns. Also, in the upper diagonal of the Table 4 each cell has the win/lost/tie count of the algorithm in the corresponding row, over the algorithms in the corresponding column. To summarise, the results indicate that -KFHE-HOMER was able to perform consistently better than its component classifiers. -The aggregation method of KFHE-HOMER using the Kalman filter is more effective than E-HOMER, a simple bagged version of HOMER. -KFHE-HOMER performed significantly better than ECC. Conclusion and Future Work This work introduces a multi-label classification method, KFHE-HOMER, by extending the Kalman filter-based Heuristic Ensemble (KFHE). The method KFHE views the ensemble classifier model to be trained as a state to be estimated in the hypothesis space. Then the state is estimated using a Kalman filter by using multiple noisy measurements, where each measurement is a trained classifier and the noise is its related classification error. In KFHE-HOMER, the KFHE framework to aggregate multiple HOMER models. The method combines multiple HOMER models trained on weighted samples and different hyperparameter settings. The KFHE framework combines these models based on the classification error of the HOMER models. Experiments showed that KFHE-HOMER performed consistently the best compared to ECC and other state-of-the-art multi-label classifiers as HOMER and classifier chains. Also, KFHE-HOMER performed better than a simple ensemble of bagged HOMER named E-HOMER demonstrating the effectiveness of the KFHE framework. The KFHE framework can be utilised with other methods in future along with improvements to the KFHE methods. Some research directions for future can be as follows: -A per-label KFHE can be studied, where instead of the combination of multiple labels using one Kalman gain, per-label Kalman gains will be maintained. -The present algorithm does not have a time update step, which can introduced and studied. -To stop the method converging to fast, process noise or a slowdown mechanism can be introduced, which may improve performance in some cases where the Kalman gain becomes 1. -Instead of one singe measurement per iteration, multiple measurements can be performed and combined to achieve a better performance. This research was supported by Science Foundation Ireland (SFI) under Grant Number SFI/12/RC/2289. When a new datapoint is encountered during the prediction step the equation Eq. (1) is repeatedly used using the component classifiers h t and the Kalman gains k (y) t found during the training stage. kf-m and line 13 computes the proxy representation of the ensembleŷ t , which indicates the KFHE-HOMER ensemble predictions on the training dataset. The kf-w steps are similar and are performed from line 16-20. The process runs until a maximum number of ensemble iterations T . 6387 ± 0.04 (5) ⇓ 0.5957 ± 0.05 (6) ⇓ 0.6681 ± 0.04 (3) 0.6484 ± 0.03 (4) ⇓ yeast 0.4899 ± 0.01 (1) 0.4400 ± 0.04 (4) ⇓ 0.4403 ± 0.02 (3) ⇓ 0.4565 ± 0.01 (2) ⇓ 0.3812 ± 0.02 (5) ⇓ 0.3524 ± 0.02 (6) ⇓ scene 0.8090 ± 0.02 (1) 0.8008 ± 0.02 (2) 0.7806 ± 0.02 (4) ⇓ 0.7818 ± 0.02 (3) ↓ 0.7777 ± 0.02 (5) ⇓ 0.2011 ± 0.03 (6) ⇓ emotions 0.7046 ± 0.03 (1) 0.6891 ± 0.03 (3) 0.6731 ± 0.03 (5) ↓ 0.6595 ± 0.04 (6) ↓ 0.6964 ± 0.1985 ± 0.02 (5) ⇓ 0.1888 ± 0.01 (6) ⇓ 0.2123 ± 0.02 (4) ⇓ birds 0.3928 ± 0.05 (1) 0.3834 ± 0.04 (2) 0.3463 ± 0.06 (3) ↓ 0.3297 ± 0.04 (4) ↓ 0.3256 ± 0.05 (5) ↓ 0.3203 ± 0.03 (6) ⇓ genbase 0.9402 ± 0.03 (1) 0.8911 ± 0.03 (4) ⇓ 0.9293 ± 0.02 (2) 0.9245 ± 0.02 (3) 0.7534 ± 0.06 (6) ⇓ 0.8810 ± 0.03 (5) ⇓ cal500 0.1458 ± 0.01 (1) 0.1455 ± 0.01 (2) 0.1310 ± 0.01 (3) ↓ 0.0859 ± 0.01 (4) ⇓ 0.0331 ± 0.01 (6) ⇓ 0.0642 ± 0. Fig. 2 : 2Rank plot of post-hoc Friedman rank sum test with Finner p-value correction. The scale indicates the average ranks. The methods which are not connected with the horizontal lines are significantly different with a significance level of 0.05. Finner p-value correction. * α = 0.1, ** α = 0.05 and * Table 2 : 2Multi-label datasets Dataset Instances Inputs Labels Labelsets Cardinality MeanIRflags 194 26 7 24 3.392 2.255 yeast 2417 103 14 77 4.237 7.197 scene 2407 294 6 3 1.074 1.254 emotions 593 72 6 4 1.869 1.478 medical 978 1449 45 33 1.245 89.501 enron 1702 1001 53 573 3.378 73.953 birds 322 260 20 55 1.503 13.004 genbase 662 1186 27 10 1.252 37.315 cal500 502 68 174 502 26.044 20.578 llog 1460 1004 75 189 1.180 39.267 Table 4 : 4Upper diagonal: win/lose/tie. Lower diagonal: Results of the Friedman aligned rank test with In this text hypothesis and classifier model are used interchangably. A version of KFHE-HOMER and E-HOMER is available at: https://github.com/phoxis/kfhe-homer Learning multi-label scene classification. 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[ "A Compact Circumstellar Shell as the Source of High-velocity Features in SN 2011fe", "A Compact Circumstellar Shell as the Source of High-velocity Features in SN 2011fe" ]	[ "Brian W Mulligan [email protected] \nThe University of Texas at Austin\nThe University of Texas at Austin\n\n", "J Craig Wheeler \nThe University of Texas at Austin\nThe University of Texas at Austin\n\n" ]	[ "The University of Texas at Austin\nThe University of Texas at Austin\n", "The University of Texas at Austin\nThe University of Texas at Austin\n" ]	[]	High-velocity features (HVF), especially of Ca II, are frequently seen in Type Ia supernovae observed prior to B-band maximum (Bmax). These HVF start at more than 25, 000 km s −1 in the days after first light, and slow to about 18, 000 km s −1 near Bmax. To recreate the Ca II near-infrared triplet (CaNIR) HVF in SN 2011fe, we consider the interaction between a Type Ia supernova and a compact circumstellar shell, employing a hydrodynamic 1-D simulation using FLASH. We generate synthetic spectra from the hydrodynamic results using syn++. We show that the CaNIR HVF and its velocity evolution is better explained by a supernova model interacting with a shell than a model without a shell, and briefly discuss the implications for progenitor models.	10.1093/mnras/sty027	[ "https://arxiv.org/pdf/1505.05145v2.pdf" ]	73,640,723	1505.05145	7b33f37478e4c11774134902677c069c0de69507	A Compact Circumstellar Shell as the Source of High-velocity Features in SN 2011fe 19 May 2015 May 21, 2015 Brian W Mulligan [email protected] The University of Texas at Austin The University of Texas at Austin J Craig Wheeler The University of Texas at Austin The University of Texas at Austin A Compact Circumstellar Shell as the Source of High-velocity Features in SN 2011fe 19 May 2015 May 21, 2015Subject headings: supernovae: general -supernovae: individual (SN 2011fe) -line: formation -line: profiles High-velocity features (HVF), especially of Ca II, are frequently seen in Type Ia supernovae observed prior to B-band maximum (Bmax). These HVF start at more than 25, 000 km s −1 in the days after first light, and slow to about 18, 000 km s −1 near Bmax. To recreate the Ca II near-infrared triplet (CaNIR) HVF in SN 2011fe, we consider the interaction between a Type Ia supernova and a compact circumstellar shell, employing a hydrodynamic 1-D simulation using FLASH. We generate synthetic spectra from the hydrodynamic results using syn++. We show that the CaNIR HVF and its velocity evolution is better explained by a supernova model interacting with a shell than a model without a shell, and briefly discuss the implications for progenitor models. Introduction Type Ia supernovae (SN Ia) provide a fundamental tool for our understanding of the history of the universe. SN Ia are 'standardizable candles' used to explore the expansion of the universe as well as the chemical enrichment of galaxies (Riess et al. 1998;Perlmutter et al. 1999;Tsujimoto & Shigeyama 2012). The configuration of the progenitor system and the cause of the explosion remain elusive. Observations of SN 2011fe within the first day after the explosion have shown that any optically thick material was within 0.1R ⊙ of the exploding star prior to the explosion Piro & Nakar 2014). Spectroscopy of SN Ia in the first days and weeks after the explosion also reveal high-velocity features (HVF) in Ca II, Si II, and other ions (Hatano et al. 1999;Parrent et al. 2012;Marion et al. 2013). The HVF has a velocity ∼ > 25, 000 km s −1 at 15-18 days before B-band maximum (Bmax) and slows to a plateau of about 18, 000 by Bmax. The photosphere also starts at high velocity and moves to lower velocities over the same interval, but the HVF consistently remain ∼ > 7, 000 kms −1 faster than the photospheric velocity features (PVF) (Marion et al. 2013;Maguire et al. 2014;Silverman et al. 2015). Ca II HVF appear in over 90% of normal SN Ia (Mazzali et al. 2005;Childress et al. 2014;Maguire et al. 2014;Silverman et al. 2015) and show significant polarization (Wang et al. 2003), indicating that the material has a high covering factor and is asymmetric. The source of the high-velocity material will give insight to the nature of the progenitor system or mechanism by which the explosion is initiated; understanding both of these is necessary to control the systematics in the use of SN Ia as cosmological probes. Previous suggestions for the source of the high-velocity material include plumes of partially burned ejecta (Wang et al. 2003), buoyant bubbles due to a gravitationally confined detonation (Kasen & Plewa 2005), or interaction with a circumstellar medium (CSM) with a total mass of high-velocity material of about 0.02 M ⊙ and solar abundance (Gerardy et al. 2004;Quimby et al. 2006). These models consider HVF velocity at the time near or after Bmax, when the HVF are in the asymptotic phase and are fading. It is necessary to explain the full evolution of the velocity and strength throughout the period the HVF is detectable in order to understand the source. In this letter, we consider the interaction between a circumstellar shell (CSS) of mass 0.005M ⊙ located within 0.3 R ⊙ of the center of the explosion and compare this model to a supernova with no CSS sources of the HVF. We concentrate on the Ca II near-infrared triplet (CaNIR) in SN 2011fe between −16.5 d and +2.9 d. In §2 we describe the hydrodynamic methods used, the SN model ( §2.1), and the CSS ( §2.2). In §3 we discuss the methods used for generating synthetic spectra from the hydrodynamic results. We present the results in §4, and summarize our conclusions in §5. Hydrodynamics We use FLASH 4.1 (Fryxell et al. 2000) with multi-pole gravity, 1-D spherical geometry, the Helmholtz equation of state (EOS) (Timmes & Arnett 1999) for the supernova-only model and the gamma law EOS for the model including the CSS. The simulation volume has the supernova at the center and a radius of 10 12 cm, with a minimum resolution of 4.2 × 10 6 cm per zone. The multispecies unit is used to track H, 3 He, 4 He, 12 C, 16 O, 24 Mg, 32 Si, and 56 Ni. The model is evolved to about 50s after the explosion at which time the the shock has fully propagated through the shell (when it is included) and both the ejecta and shell are in or near the free expansion phase. The choice of end point and EOS for a given model is governed by the density floor in the Helmholtz EOS and the desire to have similar end times for each model. Explosion Model The explosion models are those of Gamezo et al. (2005) delayed-detonation models b and c. We maintain the resolution of the original model, which has a maximum radius of 5.4 × 10 8 cm resolved into 128 3 zones (4.2 × 10 6 cm per zone). We use spherical averaging to reduce the model from 3-D to 1-D. The model provides C, O, Mg group, Si group, and Fe group abundances for each zone. Exterior to the ejecta we apply a CSM with density 10 −9 g cm −3 and temperature of 20, 000 K. Circumstellar shell model We have explored a range in masses and density distributions for the shell. These results that will be reported in a forthcoming paper (Mulligan & Wheeler in preparation). For this work, we use a shell with inner radius 7 × 10 8 cm and outer radius 2 × 10 10 cm (0.3 R ⊙ ), a total mass of 0.005 M ⊙ with a density profile that decreases linearly outward, and a temperature of 30, 000 K. Exterior to the shell, the CSM parameters are applied as described in the previous subsection. For the hydrodynamic simulation, we assume a hydrogenic shell with solar abundances. Total He abundance in the shell is taken to be solar, but we enhance the 3 He by 10% to act as a tracer. Spectral synthesis We use the result of the hydrodynamic simulation to generate an optical depth profile as a function of velocity. The line optical depth for an individual zone is computed using the Sobolev approximation then binned with zones having a similar velocity. We assume excitation temperature is constant over the line-generating regions in the ejecta and shell. We define scaling factors C E and C S to account for the fraction of Ca II relative to all Si group elements in the ejecta and Ca II abundance by mass within the shell, respectively. We use syn++ (Thomas et al. 2011), modified to allow an arbitrary optical depth as a function of velocity, to generate the synthetic spectra. When generating the spectra, the time after explosion, photospheric temperature and velocity, and the abundance scale factors C E and C S are free parameters. The photospheric velocity and temperature set the minimum velocity of the photospheric line-generating region and the shape of the continuum, respectively. We adopt a lower limit to the temperature of the photosphere of 8, 000 K. The effects of the excitation temperature, ion mass fraction, and time after explosion are degenerate. We fix the excitation temperature to 10, 000K and the time after explosion to 1 day and vary only the ion mass fraction to represent the value of these combined parameters. The abundance scale factors are limited such that the total optical depth stays below 10 4 to prevent unobserved Ca II features from appearing in the synthetic spectra, e.g. a λ5020 doublet. We fit the model spectra to the observed spectra SN 2011fe from Parrent et al. (2012) N s F s λ dλ = λu λ l N o F o λ dλ = 0.5,(1) where F λ is the specific flux density from either our synthetic spectra (F s λ ) or observed spectra (F o λ ). We have chosen the bounds to be λ l = 4500Å and λ u = 9000Å to cover the majority of the optical spectrum. A multi-variable simplex fitting routine is used to minimize the variance σ 2 = (N s F s λ −N o F o λ ) 2 over the range 7750Å and 8400Å, which covers most of the CaNIR absorption feature. The CaNIR feature is chosen because it is the most conspicuous HVF, and it allows a fit without concern of blending with Si, as may occur in the Ca H&K feature (Maguire et al. 2012;Foley 2013;Silverman et al. 2015). Details of the dynamic models and line-fitting procedure will be given in Mulligan & Wheeler (in preparation). Results and discussion Synthetic spectra for the models are shown in Figures 1 and 2 for the case of the supernova-only model (SN-O) and supernova with a shell of mass 0.005 M ⊙ (SN+S), respectively. These figures also present the observed spectra of SN 2011fe from Parrent et al. (2012). The photospheric velocity and temperature, abundance scaling factors (C S and C E ), and variance of the fit (σ) are listed in Table 1. The photosphere velocity and temperature are based only on the fit to the CaNIR feature and are therefore not strong indicators of the actual photospheric parameters; fitting of additional absorption features will give more reliable values. In the discussion below, phase is based on Bmax on JD 2,455,814.4 (Vinkó et al. 2012). The SN+S model tends to produce a better fit to the width of the CaNIR feature over the entire period studied, although the variance usually differs by only about 0.1 between the SN-O and SN+S models. The largest exception is for the earliest available spectrum (−16.54 d), when the SN+S model is a better fit by about 0.5 dex than the SN-O model. This is also apparent by visually inspecting the resulting spectra: the SN-O model leaves gaps both red-ward and blue-ward of the feature minimum. Part of the reason the earliest fit of the SN+S model is better than the remainder of the fits is the success in capturing the whole of the P Cygni peak; with the exception of -10.54d, there is excess flux at the peak in the observations that the models do not reproduce. This is due to the choice of a fixed upper limit for fitting at 8400Å, which does capture the peak on −16.54 d, but is blue-ward of the peak at all later times. From visual inspection, the fit tends to result in the flux near 8400Å matching the observed value but with an incorrect slope. A dynamically chosen upper wavelength limit for the fit would improve this result; this will be incorporated in future work. At the earliest times in the SN+S model, the photosphere lies in the shell rather than in the ejecta. The contact discontinuity between ejecta and shell material lies at 20, 880 km s −1 . On −16.54 d our fit finds a photosphere velocity of 21, 660 km s −1 , which is within the shell; one day later the photosphere lies at 20, 820 km s −1 , just inside the ejecta. This results in no measurable PVF in our model in these first two epochs. By −13.21 d and later, the photosphere has receded into or below the freshly synthesized Ca, giving distinct PVF and HVF due to the ejecta and shell, respectively. At −7.27 d and later, the shell has become optically thin to CaNIR and the HVF due to the shell becomes very weak. There is a feature near 8000Å in both the observed and the SN+S model spectra that is miss-ing from the SN-O model spectra. This feature is due exclusively to absorption within the ejecta in the SN+S model. The lack of such a feature in the SN-O model suggests that it is a result of the interaction between the ejecta and shell. We have tested simplified models of a density enhancement at the reverse shock in the ejecta that do not immediately account for this interaction feature (IF). We will further explore the physical origin of the IF in our upcoming paper, Mulligan & Wheeler (in preparation). Velocity evolution Generating spectra with C S or C E set to a small value (10 −20 ) allows us to individually measure the effect of the material in the shell and the ejecta. We flatten the ejecta-only and shellonly spectra using a continuum from the photosphere temperature and velocity, then identify the point of minimum flux of the CaNIR feature for the shell, which we refer to as the HVF velocity, and ejecta, which we refer to as the PVF velocity. We report these velocities in Figure 3, with the SN 2011fe PVF and HVF velocities from Parrent et al. (2012) and Silverman et al. (2015) shown for reference. The IF is identified by visual inspection of the ejecta-only spectra generated from the SN+S model. At times before −7.27 d, where there are multiple small features blue-ward of the CaNIR minimum, we chose the feature that had a wavelength range similar to the IF on the successive epoch. We find a local minimum in the flux at the IF and report that velocity in Figure 3. Comparing the HVF, PVF and IF velocities in this work to those of Parrent et al. (2012) and Silverman et al. (2015) requires recognition of the differing methods for velocity measurement. Parrent et al. (2012) report the v min parameter they use for a SYNAPPS fit, which is equivalent to our photospheric velocity for the ejecta, but we have no equivalent for the shell other than the velocity of the contact discontinuity. Silverman et al. (2015) use a central velocity from a Gaussian profile line-fitting routine, which should be similar to our measured velocity, though some difference can be expected because the bottom of the CaNIR feature is not smooth. At the earliest times, the photosphere is in the shell so the SN+S model has only a HVF and no measurable PVF, although a PVF is identified by both Parrent et al. (2012) and Silverman et al. (2015). The use of the v min parameter by Parrent et al. (2012) can falsely suggest the presence of a PVF that is absent due to very low effective optical depths. The method for identifying PVF and HVF used by Silverman et al. (2015) and others may artificially require a two component fit because of the non-Gaussianity of the spectral feature as well as the choice of the P Cygni peak as the basis for the red-ward continuum. The similarity of the velocities produced by the SYNAPPS and Gaussian-fitting methods at the early times cannot be discounted, however our model results call for deeper consideration of the techniques used to characterize the early data. The SN-O model produces a PVF with a velocity that is higher than the PVF velocity identified by Parrent et al. (2012) and Silverman et al. (2015) by about 5, 000 kms −1 at −10.17d and earlier, but is too slow to explain the HVF. The PVF velocity of the SN+S model agrees with that of Parrent et al. (2012) and Silverman et al. (2015) on the first day that it is measurable (−13.21 d), but remains at about the same velocity over the following six days. At the latest times, it is again in very good agreement with Parrent et al. (2012) and Silverman et al. (2015). To explain the HVF in Parrent et al. (2012) and Silverman et al. (2015), we suggest that the observationally-identified HVF is generated by the shell at the earliest times (−10.54d and earlier) and by the IF at later times. Our HVF velocity is about 3, 000 km s −1 slower than their HVF velocities during the shell-dominated phase; a slightly lower mass shell may explain the discrepancy, or it may be a result of the differing velocity measurement methods. We will further explore the effect of the mass of the shell in our upcoming paper Mulligan & Wheeler (in preparation). Implications for progenitor models Our results disfavor progenitor models that predict, or are consistent with, very little CSM. Models in this category could be spin-up / spindown models in which mass transfer has long since ceased (Di Stefano et al. 2011;Di Stefano & Kilic 2012;Justham 2011) or models in which isolated white dwarfs explode by pycnonuclear reactions (Chiosi et al. 2015). In the absence of a CSM, (Vinkó et al. 2012). c The photosphere is at a velocity that is interior to all of the freshly-synthesized Ca and thus the velocity is not well constrained by this fit. The high temperature is a result of the low velocity. these models would not produce HVF by the mechanism modeled here. An important category of SN Ia explosion models involve edge-lit double detonations whereby the explosion is triggered in a sub-Chandrasekhar white dwarf by the detonation of a thin layer of He on the outside of a C/O core. The required mass of He is estimated to range from ∼ 0.05 to ∼ 0.1M ⊙ (Fink et al. 2010;Woosley & Kasen 2011), although the mass of He might be considerably less if the He is enriched with C/O (Shen & Moore 2014). In the context of the current models, we note that the high-velocity ejection of such a He shell is unlikely to produce an observed HVF because the velocity will be too high and the optical depth too low. Some progenitor model components can also be discounted as sources of the CSS. A thin accretion disk with a small solid angle would not account for the ubiquity of the HVF and such disks are typically much less massive than the shell that we have invoked. A steady-state wind would result in excess optical light, which is not observed (Gerardy et al. 2004), and would contain substantially less mass than does the shell presented here. The CSS may be a result of a thick disk or other phenomenon. Investigation into the composition of the CSS will provide insight into these sources. Conclusion We have compared the synthetic spectra from a model with only supernova ejecta (SN-O) and a model with the supernova ejecta colliding with a circumstellar shell of mass 0.005M ⊙ (SN+S) to the observed Ca II near-infrared triplet (CaNIR) feature in SN 2011fe (Parrent et al. 2012). The SN+S model generally produces a better fit to the width of the feature than the SN-O model. The SN+S model spectra and observations of SN 2011fe have a sub-feature near 8000Å at late times which is absent in the SN-O model spectra. This feature, which we dub the interaction feature (IF), is a result of the interaction between the ejecta and the shell in our model. The combination of the shell-HVF and IF in the SN+S model more consistently reproduces the observed HVF velocity evolution than does the SN-O model. The PVF velocity evolution in the SN+S model matches that of Parrent et al. (2012) and Silverman et al. (2015) when it first appears in this model and again near Bmax, but is about 2, 000 km s −1 fast at intermediate times. The SN-O model fails to account for either the HVF or the PVF velocity at any time before about −8d, and accounts for only the PVF velocity after this epoch. We have made a plausible case that a shell of mass ∼ 0.005 M ⊙ accounts for the common presence of HVF in SN Ia. Discrepancies between the spectra generated by our models and the observed SN 2011fe features may be improved by using a more finely tuned mass or structure of the shell. Fitting of additional absorption features is necessary to place constraints on the physical quantities of the shell, such as the geometry, mass, and composition. These quantities will give us insight into possible origins of the shell and the cause of Type Ia supernovae. Special thanks to Jeffrey Silverman, Howie Marion, and Jozsef Vinkó for many discussions about high-velocity features and the fits shown in this paper. Thanks to Vadim Gamezo for providing the source model of the post-explosion supernova. This work was supported in part by NSF grant AST-1109801. The Texas Advanced Computing Center (TACC) at the University of Texas at Austin has provided HPC resources that have contributed to the research results reported within this letter. (Parrent et al. 2012), the generated spectrum from the model is shown in solid red, a spectrum showing the effects of only the shell is shown in dashed blue, and that of only the supernova ejecta is shown in dashed red. At early and late times the ejecta and shell component of the spectrum, respectively, is shown as a continuum due to high photospheric velocity (early times) and thinning of the material (late times). The gray shaded regions at −13.21 d and later represent the approximate range of the IF in the SN+S model spectra. Fig. 1 .Fig. 2 . 12-Results of fitting the CaNIR feature in the SN-O model to SN 2011fe. SN 2011fe spectra are shown in black (Parrent et al. 2012) and the generated spectrum from the model is shown in red. The gray shaded regions at −13.21 d and later represent the approximate range of the IF in the SN+S model spectra; there is no equivalent feature in the SN-O model spectra in these regions. -Results of fitting the CaNIR feature in the SN+S model to SN 2011fe. SN 2011fe spectra are shown in black Fig. 3 . 3-Evolution of the Ca II near IR triplet feature velocities for the SN-O model (blue, dash-dot, ), the SN+S model (dashed, PVF: red, ; HVF: yellow, ; IF: magenta, ), and the Ca II PVF and HVF velocities reported by Parrent et al. (2012) (×, PVF: green; HVF: brown) and Silverman et al. (2015) (+, PVF: cyan; HVF: dark brown). The Parrent et al. data stop at −10 d, but the Silverman et al. HVF velocities extend to Bmax where they correspond to the IF velocity in the SN+S model. over the period −16.54 d to +2.87 d. For fitting purposes, normalization factors N s and N o are computed such thatλu λ l Table 1 1Parameters and variance of fitPhotosphere Photosphere velocity temperature Model MJD a Phase b (1000 km s −1 ) (1000 K) log C E log C S log σ SN-O 797.66 -16.54 24.80 8.42 1.98 . . . -1.73 798.76 -16.24 22.26 8.00 1.14 . . . -1.60 800.69 -13.21 17.97 8.00 0.43 . . . -1.89 801.24 -12.66 19.58 11.76 0.06 . . . -2.14 803.36 -10.54 19.85 9.14 -0.30 . . . -2.31 803.73 -10.17 19.30 8.00 -0.81 . . . -1.93 806.63 -7.27 11.95 8.71 -1.90 . . . -2.53 811.73 -2.17 9.39 c 19.55 c -2.17 . . . -3.16 816.77 2.87 5.97 c 22.52 c -2.12 . . . -2.76 SN-S 797.66 -16.54 21.66 8.01 . . . 0.41 -2.32 798.76 -16.24 20.82 8.00 . . . 0.08 -1.65 800.69 -13.21 18.35 8.03 0.12 -0.27 -1.72 801.24 -12.66 16.98 10.19 -0.45 -0.60 -1.82 803.36 -10.54 18.59 8.18 0.09 -0.91 -2.21 803.73 -10.17 17.88 8.00 -1.33 -1.63 -1.81 806.63 -7.27 19.09 8.47 -0.29 -2.72 -2.43 811.73 -2.17 7.34 c 18.90 c -2.39 -3.05 -3.19 816.77 2.87 7.97 c 22.81 c -2.18 -4.11 -2.79 a Relative to JD 2,455,000.5. b Relative to B-band maximum on MJD 813.90 . 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[ "Deep Quality Estimation: Creating Surrogate Models for Human Quality Ratings", "Deep Quality Estimation: Creating Surrogate Models for Human Quality Ratings" ]	[ "Florian Kofler \nDepartment of Informatics\nTechnical University Munich\nGermany\n\nTranslaTUM -Central Institute for Translational Cancer Research\nTechnical University of Munich\nGermany\n\nDepartment of Diagnostic and Interventional Neuroradiology\nSchool of Medicine\nKlinikum rechts der Isar\nTechnical University of Munich\nGermany\n\nHelmholtz AI\nHelmholtz Zentrum München\nGermany\n", "Ivan Ezhov \nDepartment of Informatics\nTechnical University Munich\nGermany\n\nTranslaTUM -Central Institute for Translational Cancer Research\nTechnical University of Munich\nGermany\n", "Lucas Fidon \nSchool of Biomedical Engineering & Imaging Sciences\nKing's College London\nUnited Kingdom\n", "Izabela Horvath \nDepartment of Informatics\nTechnical University Munich\nGermany\n\nInsitute for Tissue Engineering and Regenerative Medicine\nHelmholtz Institute Munich (iTERM)\nGermany\n", "Ezequiel De La Rosa \nDepartment of Informatics\nTechnical University Munich\nGermany\n\nicometrix\nLeuvenBelgium\n", "John Lamaster \nDepartment of Informatics\nTechnical University Munich\nGermany\n\nDepartment of Quantitative Biomedicine\nUniversity of Zurich\nSwitzerland\n", "Hongwei Li \nDepartment of Informatics\nTechnical University Munich\nGermany\n\nDepartment of Quantitative Biomedicine\nUniversity of Zurich\nSwitzerland\n", "Tom Finck \nInsitute for Tissue Engineering and Regenerative Medicine\nHelmholtz Institute Munich (iTERM)\nGermany\n\nImperial College London\nSouth KensingtonLondonUnited Kingdom\n", "Suprosanna Shit \nDepartment of Informatics\nTechnical University Munich\nGermany\n\nTranslaTUM -Central Institute for Translational Cancer Research\nTechnical University of Munich\nGermany\n", "Johannes Paetzold \nDepartment of Informatics\nTechnical University Munich\nGermany\n\nTranslaTUM -Central Institute for Translational Cancer Research\nTechnical University of Munich\nGermany\n\nInsitute for Tissue Engineering and Regenerative Medicine\nHelmholtz Institute Munich (iTERM)\nGermany\n\nImperial College London\nSouth KensingtonLondonUnited Kingdom\n", "Spyridon Bakas \nCenter for Biomedical Image Computing and Analytics (CBICA)\nUniversity of Pennsylvania\nPhiladelphiaPAUSA\n\nDepartment of Radiology\nPerelman School of Medicine\nUniversity of Pennsylvania\nPhiladelphiaPAUSA\n\nDepartment of Pathology and Laboratory Medicine\nPerelman School of Medicine\nUniversity of Pennsylvania\nPhiladelphiaPAUSA\n", "Marie Piraud \nHelmholtz AI\nHelmholtz Zentrum München\nGermany\n", "Jan Kirschke \nDepartment of Diagnostic and Interventional Neuroradiology\nSchool of Medicine\nKlinikum rechts der Isar\nTechnical University of Munich\nGermany\n", "Tom Vercauteren \nSchool of Biomedical Engineering & Imaging Sciences\nKing's College London\nUnited Kingdom\n", "Claus Zimmer \nDepartment of Diagnostic and Interventional Neuroradiology\nSchool of Medicine\nKlinikum rechts der Isar\nTechnical University of Munich\nGermany\n", "Benedikt Wiestler \nDepartment of Diagnostic and Interventional Neuroradiology\nSchool of Medicine\nKlinikum rechts der Isar\nTechnical University of Munich\nGermany\n", "Bjoern Menze \nDepartment of Informatics\nTechnical University Munich\nGermany\n\nDepartment of Quantitative Biomedicine\nUniversity of Zurich\nSwitzerland\n" ]	[ "Department of Informatics\nTechnical University Munich\nGermany", "TranslaTUM -Central Institute for Translational Cancer Research\nTechnical University of Munich\nGermany", "Department of Diagnostic and Interventional Neuroradiology\nSchool of Medicine\nKlinikum rechts der Isar\nTechnical University of Munich\nGermany", "Helmholtz AI\nHelmholtz Zentrum München\nGermany", "Department of Informatics\nTechnical University Munich\nGermany", "TranslaTUM -Central Institute for Translational Cancer Research\nTechnical University of Munich\nGermany", "School of Biomedical Engineering & Imaging Sciences\nKing's College London\nUnited Kingdom", "Department of Informatics\nTechnical University Munich\nGermany", "Insitute for Tissue Engineering and Regenerative Medicine\nHelmholtz Institute Munich (iTERM)\nGermany", "Department of Informatics\nTechnical University Munich\nGermany", "icometrix\nLeuvenBelgium", "Department of Informatics\nTechnical University Munich\nGermany", "Department of Quantitative Biomedicine\nUniversity of Zurich\nSwitzerland", "Department of Informatics\nTechnical University Munich\nGermany", "Department of Quantitative Biomedicine\nUniversity of Zurich\nSwitzerland", "Insitute for Tissue Engineering and Regenerative Medicine\nHelmholtz Institute Munich (iTERM)\nGermany", "Imperial College London\nSouth KensingtonLondonUnited Kingdom", "Department of Informatics\nTechnical University Munich\nGermany", "TranslaTUM -Central Institute for Translational Cancer Research\nTechnical University of Munich\nGermany", "Department of Informatics\nTechnical University Munich\nGermany", "TranslaTUM -Central Institute for Translational Cancer Research\nTechnical University of Munich\nGermany", "Insitute for Tissue Engineering and Regenerative Medicine\nHelmholtz Institute Munich (iTERM)\nGermany", "Imperial College London\nSouth KensingtonLondonUnited Kingdom", "Center for Biomedical Image Computing and Analytics (CBICA)\nUniversity of Pennsylvania\nPhiladelphiaPAUSA", "Department of Radiology\nPerelman School of Medicine\nUniversity of Pennsylvania\nPhiladelphiaPAUSA", "Department of Pathology and Laboratory Medicine\nPerelman School of Medicine\nUniversity of Pennsylvania\nPhiladelphiaPAUSA", "Helmholtz AI\nHelmholtz Zentrum München\nGermany", "Department of Diagnostic and Interventional Neuroradiology\nSchool of Medicine\nKlinikum rechts der Isar\nTechnical University of Munich\nGermany", "School of Biomedical Engineering & Imaging Sciences\nKing's College London\nUnited Kingdom", "Department of Diagnostic and Interventional Neuroradiology\nSchool of Medicine\nKlinikum rechts der Isar\nTechnical University of Munich\nGermany", "Department of Diagnostic and Interventional Neuroradiology\nSchool of Medicine\nKlinikum rechts der Isar\nTechnical University of Munich\nGermany", "Department of Informatics\nTechnical University Munich\nGermany", "Department of Quantitative Biomedicine\nUniversity of Zurich\nSwitzerland" ]	[]	Human ratings are abstract representations of segmentation quality. To approximate human quality ratings on scarce expert data, we train surrogate quality estimation models. We evaluate on a complex multi-class segmentation problem, specifically glioma segmentation, following the BraTS annotation protocol. The training data features quality ratings from 15 expert neuroradiologists on a scale ranging from 1 to 6 stars for various computer-generated and manual 3D annotations. Even though the networks operate on 2D images and with scarce training data, we can approximate segmentation quality within a margin of error comparable to human intra-rater reliability. Segmentation quality prediction has broad applications. While an understanding of segmentation quality is imperative for successful clinical translation of automatic segmentation quality algorithms, it can play an essential role in training new segmentation models. Due to the split-second inference times, it can be directly applied within a loss function or as a fully-automatic dataset curation mechanism in a federated learning setting. contributed equally as senior authors arXiv:2205.10355v2 [cs.CV] 30 Aug 2022 2 F. Kofler et al.	10.48550/arxiv.2205.10355	[ "https://export.arxiv.org/pdf/2205.10355v2.pdf" ]	248,987,490	2205.10355	ef90cbdf1fc5a441bf7e2ea680e8a5275fe1a611	Deep Quality Estimation: Creating Surrogate Models for Human Quality Ratings Florian Kofler Department of Informatics Technical University Munich Germany TranslaTUM -Central Institute for Translational Cancer Research Technical University of Munich Germany Department of Diagnostic and Interventional Neuroradiology School of Medicine Klinikum rechts der Isar Technical University of Munich Germany Helmholtz AI Helmholtz Zentrum München Germany Ivan Ezhov Department of Informatics Technical University Munich Germany TranslaTUM -Central Institute for Translational Cancer Research Technical University of Munich Germany Lucas Fidon School of Biomedical Engineering & Imaging Sciences King's College London United Kingdom Izabela Horvath Department of Informatics Technical University Munich Germany Insitute for Tissue Engineering and Regenerative Medicine Helmholtz Institute Munich (iTERM) Germany Ezequiel De La Rosa Department of Informatics Technical University Munich Germany icometrix LeuvenBelgium John Lamaster Department of Informatics Technical University Munich Germany Department of Quantitative Biomedicine University of Zurich Switzerland Hongwei Li Department of Informatics Technical University Munich Germany Department of Quantitative Biomedicine University of Zurich Switzerland Tom Finck Insitute for Tissue Engineering and Regenerative Medicine Helmholtz Institute Munich (iTERM) Germany Imperial College London South KensingtonLondonUnited Kingdom Suprosanna Shit Department of Informatics Technical University Munich Germany TranslaTUM -Central Institute for Translational Cancer Research Technical University of Munich Germany Johannes Paetzold Department of Informatics Technical University Munich Germany TranslaTUM -Central Institute for Translational Cancer Research Technical University of Munich Germany Insitute for Tissue Engineering and Regenerative Medicine Helmholtz Institute Munich (iTERM) Germany Imperial College London South KensingtonLondonUnited Kingdom Spyridon Bakas Center for Biomedical Image Computing and Analytics (CBICA) University of Pennsylvania PhiladelphiaPAUSA Department of Radiology Perelman School of Medicine University of Pennsylvania PhiladelphiaPAUSA Department of Pathology and Laboratory Medicine Perelman School of Medicine University of Pennsylvania PhiladelphiaPAUSA Marie Piraud Helmholtz AI Helmholtz Zentrum München Germany Jan Kirschke Department of Diagnostic and Interventional Neuroradiology School of Medicine Klinikum rechts der Isar Technical University of Munich Germany Tom Vercauteren School of Biomedical Engineering & Imaging Sciences King's College London United Kingdom Claus Zimmer Department of Diagnostic and Interventional Neuroradiology School of Medicine Klinikum rechts der Isar Technical University of Munich Germany Benedikt Wiestler Department of Diagnostic and Interventional Neuroradiology School of Medicine Klinikum rechts der Isar Technical University of Munich Germany Bjoern Menze Department of Informatics Technical University Munich Germany Department of Quantitative Biomedicine University of Zurich Switzerland Deep Quality Estimation: Creating Surrogate Models for Human Quality Ratings automatic quality control · quality estimation · segmenta- tion quality metrics · glioma · BraTS Human ratings are abstract representations of segmentation quality. To approximate human quality ratings on scarce expert data, we train surrogate quality estimation models. We evaluate on a complex multi-class segmentation problem, specifically glioma segmentation, following the BraTS annotation protocol. The training data features quality ratings from 15 expert neuroradiologists on a scale ranging from 1 to 6 stars for various computer-generated and manual 3D annotations. Even though the networks operate on 2D images and with scarce training data, we can approximate segmentation quality within a margin of error comparable to human intra-rater reliability. Segmentation quality prediction has broad applications. While an understanding of segmentation quality is imperative for successful clinical translation of automatic segmentation quality algorithms, it can play an essential role in training new segmentation models. Due to the split-second inference times, it can be directly applied within a loss function or as a fully-automatic dataset curation mechanism in a federated learning setting. contributed equally as senior authors arXiv:2205.10355v2 [cs.CV] 30 Aug 2022 2 F. Kofler et al. Introduction Large and long-standing community challenges, such as BraTS [2], have created a multitude of fully-automatic segmentation algorithms over the years. To fully exploit the potential of these task-specific algorithms, be it for clinical or scientific purposes, it is essential to understand the quality of their predictions and account for segmentation failures. 13 As individual segmentation quality metrics are only able to cover isolated aspects of segmentation quality, most segmentation challenges evaluate on a combination of metrics [15]. Human expert quality ratings have become a prominent tool to complement conventional analysis, among others [14,17,20] for segmentation outputs [13]. In contrast, to narrowly defined quality metrics, these more holistic measures capture various quality aspects. However, they are prohibitively expensive regarding data acquisition times and financials to be deployed regularly. Related work: Therefore, previous research tried to approximate segmentation performance by other means. Reverse classification accuracy (RCA) has been proposed to evaluate segmentation accuracy for applications of multi-organ segmentation in magnetic resonance imaging (MRI) [21] and cardiac MRI segmentation [19]. The authors used the maximum predicted segmentation quality metric estimated by a multi-atlas-based registration method as a proxy to estimate the true quality metric. However, the application of RCA is so far limited to organ segmentation. Further, the analysis is restricted to established segmentation quality metrics such as DSC and Hausdorff distance. It is unclear how it would generalize to expert scoring and to lesion segmentation. A method to estimate the Dice score of an ensemble of convolutional neural networks (CNNs) for segmentation has also been proposed [7]. They propose to train a linear classifier regression model to predict the Dice score for every CNN in the ensemble. Here, the Dice scores for every pair of segmentation predictions of the models in the ensemble serve as input for the regressor. Further, Audelan et al. proposed an unsupervised learning method for predicting DSC using Bayesian learning [1]. The above methods are coupled with segmentation algorithms for the estimation of segmentation quality metrics. Unlike this, Fournel et al. predict the segmentation quality metric directly from the input image and segmentation map using a CNN [5]. Similarly, it is possible to predict an ensemble's segmentation performance from discord between individual segmentation maps, even when ignoring the image data [12]. The BraTS challenge [16] features the multi-class segmentation problem of glioma segmentation. Distinguishing between enhancing tumor, necrosis, and edema is a complex subtask scattered over multiple imaging modalities. It is evaluated using Sørensen-Dice coefficient (DSC) and Hausdorff distance (HD) for the whole tumor, tumor core and enhancing tumor channels. Previous research revealed that BraTS tumor segmentation algorithms typically perform well when monitoring established segmentation quality ratings such as DSC, HD, and others. However, when they fail, they fail spectacularly [4,2,11,12,16]. These findings reflecting multiple established segmentation quality metrics are supported by surveys with expert neuroradiologists [13]. Here, the multi-faceted concept of segmentation quality was condensed to a single expert quality rating. To this end, BraTS glioma segmentation is a good candidate for studying how a holistic rating can complement or replace them. Contribution: In contrast to the above-mentioned methods, which predict narrowly defined quality metrics such as DSC or HD, we focus on the approximation of more holistic expert neuroradiologists' ratings. We build surrogate regression models for these abstract human quality ratings to estimate the segmentation quality of MICCAI BraTS segmentation algorithms. A sophisticated augmentation pipeline compensates for the scarce 2D training data available. Despite these obstacles, our model manages to create robust estimates for 3D segmentation quality on an internal and external test set. While our models are agnostic to the segmentation method and are even compatible with 2D manual segmentations, split-second inference times enable broad downstream applications in scientific and clinical practice. Methods: Network Training We train multiple regression networks to approximate the human quality ratings. Segmentation quality rating: We use human quality ratings provided by Kofler et al. [13]. In this study, expert radiologists rated the quality of glioma segmentations in two experiments. In the first experiment, 15 neuroradiologists rated the segmentations' center of mass for 25 exams with four different segmentations from axial, sagittal, and coronal views, resulting in 300 trials. The experiment featured one manual and three computer-generated segmentations. In the second experiment, three neuroradiologists rated another 50 exams with one manual and five computer-generated segmentations only on axial views, again resulting in 300 trials. The rating scale ranges from 1 star for very bad to 6 stars for very good segmentations. Network input and output: To predict the above-mentioned quality rating, the network receives the four MR modalities, namely T1, T1c, T2, and FLAIR. The tumor segmentations are either supplied in a single label channel encoding the different tumor tissues or with three label channels following BraTS annota-tion concepts, as illustrated by Figure 1. We try this style of encoding the tumor segmentations as this approach has been proven successful for training BraTS segmentation algorithms [2]. Training constants: The dataset is randomly split into 80 percent (60) of the 75 exams for training and 20 percent (15) for testing. Batch size is kept constant at 80 and learning rate at 1e-3. To compensate for the scarce training data, we employ a heavy augmentation pipeline featuring Gaussian noise, flips, and random affine plus elastic transformations. Additionally, we use batchgenerators [9] to augment with contrast, brightness, gamma, low resolution, and rician noise. Further, we simulate MR artifacts with TorchIO [18], specifically motion, ghosting, spikes, and bias fields. We employ a Mean Square Error (MSE) loss for all training runs to especially penalize far-off predictions. The human quality ratings serve as reference annotations for the loss computations. Due to the scarcity of training data, we do not conduct model selection and use the last checkpoint after 500 epochs of training across all training runs. Training variations: Between training runs we experiment with three optimizers, namely Ranger21 [22], AdamW, and SGD with a momentum of 0.95. We use DenseNet121 and DenseNet201 [8] to investigate whether the performance prof-its from more trainable parameters. Moreover, we try a channel-wise min/max, and a nn-Unet [9] inspired percentile-based normalization using the 0.5 and 99.5 percentiles for minimum and maximum, respectively. Hardware: All computations take place on a small workstation with an 8-core Intel(R) Xeon(R) W-2123 CPU @ 3.60GHz with 256GB RAM and a NVIDIA Quadro P5000 GPU. Computation times: A training run takes approximately two hours. The above machine infers four 3D exams per second, including mass computation. Without the extraction of 2D slices, the inference performance increases to 25 per second. Note that the implementation is not fully-optimized for computation time, as the split-second inference times are in no way impeding our purposes. Memory consumption: With the batch size of 80 we use most of the 16gb CUDA memory of the NVIDIA Quadro P5000 GPU. Evaluation Experiments We conduct two experiments to evaluate the performance of our models. In the first experiment we identify and evaluate our best performing model. In the second experiment we validate its generalization capabilities on an external test set. In both experiments, we generate network predictions for the axial, coronal, and sagittal views and compute a mean rating for each exam. Internal evaluation experiment To investigate which of the six hyperparameter combinations works best by evaluating on an internal test set. Dataset: We use the previously held-back 20 percent of the training data for evaluation, as described in Section 2. Procedure: We run inference on the test set for each model. Following, we evaluate using established regression metrics, namely mean absolute error (MAE), root mean square error (RMSE). We further compute Pearson r (r) to measure the linear correlation between network predictions and ground truth labels. Results: Table 1 illustrates the performance differences between training runs. We observe that the simple DenseNet121 trained with percentile-based normalization, Ranger21 optimizer, and BraTS label encoding approximates the human rating best. Figure 2 visualizes the model's predictions compared to the averaged human star ratings. According to the scatter plot, illustrated in Figure 2, the model performs more accurately for better segmentations. This is also reflected by a Bland-Altman plot, see Figure 3. It is important to note that the variance in human quality assessment also increases for lower quality segmentations, see 3. Bland-Altman plot: Network predictions vs. human star rating. The model reveals higher prediction accuracy for better-quality segmentations. This is not surprising given that human raters also display higher agreement for such cases and that these are better represented in the training data, compare External evaluation experiment To better understand our model's generalizability, we further evaluate an external dataset. Data set: The dataset features manual annotations for 68 exams generated by two expert radiologists in consensus voting. It includes 15 high-grade glioma (GBM) and 13 low-grade glioma (LGG) from University Hospital rechts der Isar. Furthermore, 25 GBM and 15 LGG from the publicly available Rembrandt dataset [6] are added to the analysis. We obtain five segmentations from BraTS algorithms and four fusions from BraTS Toolkit [11]. This way, we have a total of 612 segmentations to evaluate. Procedure: We select the best model obtained from the first experiment. We feed 2D views of the 3D augmentations' center of mass to the network to obtain an axial, sagittal, and coronal quality rating. As there are no human quality ratings for this dataset, we measure segmentation performance using established quality metrics, namely Sørensen-Dice coefficient (DSC) and surface Dice coefficient (SDSC). Results: We find a strong correlation between the quality ratings predicted by the network and DSC, see Figure 5. We observe a Pearson r of 0.75 for the axial, 0.76 for the coronal, and 0.77 for the sagittal view, while the averaged rating across views has a 0.79 correlation. This is supported by a high correlation between the mean rating and SDSC (Pearson r: 0.85), suggesting that the model generalizes well to the external data set. Fig. 4. Network predictions vs. minimum, mean, and maximum human rating. For lower quality segmentations, human raters disagree more. In most cases, the network's predictions are in the range or close to the human ratings. As already visible in Figure 3, the network tends to overestimate the quality of bad segmentations. However, they are still assigned systematically lower scores. In practice, this difference is sufficient to distinguish between good and bad segmentations by employing a simple thresholding operation. Discussion We demonstrate that a simple DenseNet121 is able to serve as a surrogate model for the abstract human segmentation quality rating under a scarce training data regime. Notably, the mean absolute error deviation is lower than the difference Kofler et al. [13] reported for individual human raters from the mean human rating. Apparently, the 3D segmentation quality is sufficiently encoded in the 2D center of mass slices. Our experiments show that the choice of hyperparameters is not critical as all networks reach solid performance. Expert radiologists are among the highest-paid doctors and are notoriously hard to come by. Given that the inference of the approximated quality rating only takes split seconds, there are broad potential applications for deep quality estimation (DQE) networks: One obvious application for DQE is quality monitoring during inference. Even though fully-automatic glioma segmentation algorithms, on average, tend to outperform human annotators [13], they sometimes fail spectacularly. For successful clinical translation of such algorithms, detection and mitigation of failure cases is imperative. Another potential use case for DQE is dat aset curation. Data set curation is an important aspect of model training, as broken ground truth labels can destroy model performance. As we demonstrate in the evaluation experiments, DQE can differentiate between trustworthy and broken ground truth cases in a fully-automatic fashion. This property becomes especially valuable in a federated learning setting, where researchers have no access to the ground truth labels. DQE allows training models only on trustworthy exams by applying a simple thresholding operation on the estimated quality score. Limitations: It is unclear how well our approach generalizes to other (segmentation) tasks and quality metrics. Taking into account that glioma segmentation is a complex multi-class segmentation problem, the scarce training data, the 3D to 2D translation, and the abstract nature of the human-generated quality judgments, we believe there is a slight reason for optimism. The proposed model performs better for predicting high-quality segmentations. This is perhaps not surprising given that humans agree more on the quality of such cases and that these are better represented in the training data, as visible in Figures 2 to 4. Nevertheless, in practice, simple thresholding of the predicted star ratings can sufficiently distinguish segmentation qualities. Outlook: As we demonstrated, DQE can approximate non-differentiable quality metrics, such as the abstract human segmentation quality rating, with a differentiable CNN. This promises the possibility of training new (segmentation) networks with surrogates of non-differentiable quality metrics by using DQE within the loss function. 14 Future research should address these open questions. Fig. 1 . 1Example inputs for the CNN training -axial center of mass slices of a glioma exam. Besides the illustrated T1, T1c, T2, and FLAIR MR sequences, the tumor segmentations are supplied to the network. The different tumor tissue types are visualized in colors: Red: necrosis; Yellow: enhancing tumor ; Green: edema. For BraTS label encoding, the enhancing tumor is encoded in a binary label channel, while a second tumor core channel is formed by combining enhancing tumor and necrosis and a third whole tumor channel is calculated by combining all three tumor tissue labels. Software: All computations happen with NVIDIA Driver v470.103.01, CUDA v11.4 with PyTorch 1.9.0. The networks are implemented via MONAI [3] 0.6.0. Segmentation metrics are computed with pymia 0.3.1 [10] and regression metrics via scikit-learn 0.24.2. Figure 4 . 4 Fig. Fig. 3. Bland-Altman plot: Network predictions vs. human star rating. The model reveals higher prediction accuracy for better-quality segmentations. This is not surprising given that human raters also display higher agreement for such cases and that these are better represented in the training data, compare Figure 4. Figure 4 . 4Fig. 3. Bland-Altman plot: Network predictions vs. human star rating. The model reveals higher prediction accuracy for better-quality segmentations. This is not surprising given that human raters also display higher agreement for such cases and that these are better represented in the training data, compare Figure 4. Fig. 5 . 5Scatter plot: Predicted star rating averaged across views vs. DSC for 612 segmentations. Even though we observe moderate heteroscedasticity, a linear model is able to describe the data well, as illustrated by the dotted pink line. We observe a Pearson r of 0.79 between the variables, meaning our model can predict the segmentation quality as measured by DSC quite well. Again we observe more accurate predictions for better segmentations. Table 1 . 1Training results for different training parameters. We report mean absolute error (MAE), root mean square error (RMSE), and Pearson r (r). The selected model is highlighted in pink.architecture optimizer normalization labels MAE RMSE r DenseNet121 SGD percentile tissue 0.65 0.88 0.64 DenseNet121 AdamW percentile tissue 0.59 0.80 0.62 DenseNet121 Ranger21 percentile tissue 0.60 0.83 0.58 DenseNet201 Ranger21 percentile tissue 0.59 0.80 0.63 DenseNet121 Ranger21 min.max BraTS 0.57 0.82 0.61 DenseNet121 Ranger21 percentile BraTS 0.51 0.80 0.66 Models often output uncertainty levels to account for this. However, we believe the judgment of external entities is more trustworthy for quality assurance purposes. Remarkably, while this separation of concerns is a well-established practice in other fields, it remains largely ignored in machine learning. For instance, imagine a world where aircraft pilots could self-certify their ability to fly. A big advantage here (compared to, e.g., GAN training) is the possibility to train the networks sequentially and thereby stabilize the training process. 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[ "Safe Learning of Quadrotor Dynamics Using Barrier Certificates ", "Safe Learning of Quadrotor Dynamics Using Barrier Certificates " ]	[ "Li Wang ", "Evangelos A Theodorou ", "Magnus Egerstedt " ]	[]	[]	To effectively control complex dynamical systems, accurate nonlinear models are typically needed. However, these models are not always known. In this paper, we present a datadriven approach based on Gaussian processes that learns models of quadrotors operating in partially unknown environments. What makes this challenging is that if the learning process is not carefully controlled, the system will go unstable, i.e., the quadcopter will crash. To this end, barrier certificates are employed for safe learning. The barrier certificates establish a non-conservative forward invariant safe region, in which high probability safety guarantees are provided based on the statistics of the Gaussian Process. A learning controller is designed to efficiently explore those uncertain states and expand the barrier certified safe region based on an adaptive sampling scheme. In addition, a recursive Gaussian Process prediction method is developed to learn the complex quadrotor dynamics in real-time. Simulation results are provided to demonstrate the effectiveness of the proposed approach.	10.1109/icra.2018.8460471	[ "https://arxiv.org/pdf/1710.05472v1.pdf" ]	35,948,052	1710.05472	85668dd2acfdec9b8ae01986e75fd6d0145b4d73	Safe Learning of Quadrotor Dynamics Using Barrier Certificates * 16 Oct 2017 Li Wang Evangelos A Theodorou Magnus Egerstedt Safe Learning of Quadrotor Dynamics Using Barrier Certificates * 16 Oct 2017 To effectively control complex dynamical systems, accurate nonlinear models are typically needed. However, these models are not always known. In this paper, we present a datadriven approach based on Gaussian processes that learns models of quadrotors operating in partially unknown environments. What makes this challenging is that if the learning process is not carefully controlled, the system will go unstable, i.e., the quadcopter will crash. To this end, barrier certificates are employed for safe learning. The barrier certificates establish a non-conservative forward invariant safe region, in which high probability safety guarantees are provided based on the statistics of the Gaussian Process. A learning controller is designed to efficiently explore those uncertain states and expand the barrier certified safe region based on an adaptive sampling scheme. In addition, a recursive Gaussian Process prediction method is developed to learn the complex quadrotor dynamics in real-time. Simulation results are provided to demonstrate the effectiveness of the proposed approach. I. INTRODUCTION Safety is crucial to many physical control dynamical systems, such as autonomous vehicles, industrial robots, chemical reactors, and air-traffic control systems [11], [2], [5]. If the system reaches certain unsafe states or even fails, both the operator and the controlled plant might be put in serious danger. The existence of model inaccuracies and unknown disturbances create an even greater challenge to the design of safe controllers for these systems. Tools such as robust control and adaptive control methods have been developed in classic control theory to ensure the safety and stability of the system, see [9], [4] and the references therein. Meanwhile, machine learning based control approaches are becoming increasingly popular as a way to deal with inaccurate models [8], [17], due to their abilities to infer unknown models from data and actively improve the performance of the controller with the learned model. In contrast to classic control methods, learning based control approaches require only limited expert knowledge and fewer assumptions about the system [22]. However, there always exists an inherent trade-off between safety and performance in these methods [3]. Data-driven learning approaches rarely provides safety guarantees, which limits their applicability to real-world safety critical control dynamical systems [22]. The objective of this paper is to construct high probability safety guarantees for Gaussian Process (GP) based learning approaches using barrier certificates. In order to promote the application of learning based control methods in safety-critical systems, a number of safe learning approaches have been proposed in the literature. Among these methods, the use of learning Control Lyapunov Functions (CLF) is shown to be a promising approach. A learning from demonstration method was developed in [13] to search for a CLF from several demonstrations, and the learned CLF was used to stabilize the system. But the learned controller did not consider actuator limits and other safe operation constraints. [21] introduced a verifier to explicitly validate the learned CLF. However, when the model of the system is inaccurate, the verifier needs to check an infinite number of inequalities throughout the state space, which is computationally difficult [12]. [6] seeks to learn CLF and maximize the safe operation region for the system with GP model. High probability safety guarantees are provided based on Lyapunov stability and GP statistics. In addition, a reachability-based safe learning approach was presented in [1] to reduce the conservativeness of reachability analysis by learning the disturbance from data. In contrast to the aforementioned methods, this paper interprets the safe operation region as general invariant sets established with barrier certificates, which permits a much richer set of safe control options, rather than Lyapunov sublevel sets. The barrier certificates formally define a forward invariant safe region, where all system trajectories starting in this region remains in this region for all time [18], [26], [2]. With the barrier certificates, the safety of the system can be certified without explicitly computing the forward reachable set [19]. Barrier certificates were successfully applied to many safety critical dynamical systems, such as adaptive cruise control [2], bipedal walking [15], quadrotor control [25], and swarm robotics [23], [24]. In this paper, we construct a safe operation region with barrier certificates, and gradually expand the certified safe region as the uncertainty of the system reduces. The unknown dynamics of the system is represented with a GP model, which provides both the mean and variance of the prediction. Using the statistics of GP model, a high probability safety guarantee of the system with inaccurate model is provided. The search for maximum volume barrier certificates involves the validation of an infinite number of inequality constraints, which is computationally expensive. Inspired by the discrete sampling technique used in [5], we design an adaptive sampling algorithm to significantly reduce the computation intensity, i.e., the more certain regions in the state space are sampled less without loss of safety guarantees. In addition, a recursive learning strategy based on GP is designed to learn the complex 3D nonlinear quadrotor dy-namics online. The learned dynamical model of the quadrotor is then incorporated into a differential flatness based flight controller to improve the trajectory tracking performance. The main contributions of this paper are threefold. First, a safe learning strategy is developed based on barrier certificates, which admits a rich set of learning control options. Second, an adaptive sampling algorithm is proposed to significantly reduce the computation intensity of the learning process. Third, an recursive learning strategy based on GP is presented to learn the complex 3D nonlinear quadrotor dynamics online. The rest of this paper is organized as follows. The preliminaries of barrier certificates and GP are briefly revisited in Section II. A safe learning strategy based on barrier certificates is presented in Section III. Section IV contains a real-time learning algorithm for 3D quadrotor dynamics based on GP. Simulation results are provided in Section V, and the paper is ended by conclusions in Section VI. II. PRELIMINARIES OF BARRIER CERTIFICATES AND GAUSSIAN PROCESS Preliminary results regarding the two fundamental tools, i.e., barrier certificates and Gaussian Process, used to formulate the safe learning strategy are presented in this section. A. Barrier Certificates and Set Invariance Consider a control affine dynamical systeṁ x = f (x) + g(x)u,(1) where x ∈ X ⊆ R n and u ∈ U ⊆ R m are the state and control of the system, f : R n → R n and g : R n → R m are Lipschitz continuous. Let the safe set of the system be encoded as the superlevel set of a smooth function h : R n → R, C = {x ∈ R n \| h(x) ≥ 0}.(2) The function h(x) is termed a Control Barrier Function (CBF), if there exists an extended class-κ function (κ(0) = 0 and strictly increasing) such that sup u∈U ∂ h ∂ x f (x) + ∂ h ∂ x g(x)u + κ(h(x)) ≥ 0, for all x ∈ E with C ⊆ E . Given a CBF, the barrier certified safe control space S(x) is defined as S(x) = u ∈ U \| ∂ h ∂ x f (x) + ∂ h ∂ x g(x)u + κ(h(x)) ≥ 0 , x ∈ E . With barrier certificates, the invariance property of C is established with the following theorem, Theorem [26]: Given a set C ⊂ R n defined by (2) and a CBF h defined on E , with C ⊆ E ⊂ R n , any Lipschitz continuous controller u : E → R such that u ∈ S(x) for the system (1) renders the set C forward invariant. This type of barrier certificates expands the certified safe control space significantly by allowing h(x) to decrease within C as opposed to strictly increasing [26], [2]. Compared with Lyapunov sublevel set based safe region, barrier certificates provide a more permissive notion of safety. As a result, barrier certificates based safe learning controllers have more freedom to efficiently explore those unknown states. This fact can be illustrated with the following example. Example 1: Consider an autonomous dynamical system ẋ 1 x 2 = x 2 + 0.8x 2 2 −x 1 − x 2 + x 2 1 x 2 ,(3) the safe region of this system is estimated with both the Lyapunov sublevel set and barrier certificates. Since (3) is a polynominal system, the safe sets can be computed directly with Sum-of-Squares programs using YALMIP [14] and SMRSOFT [7] solvers. Both the Lyapunov function and barrier certificates are limited to second order polynomials. The safe region estimated with the optimal polynomial Lyapunov function is From Fig. 1, it can be observed that the barrier certified safe region A 2 is much larger than the Lyapunov based safe region A 1 . Consequently, safe learning controller based on barrier certificates are allowed to explore more states of the system. In this paper, we will leverage the non-conservative safety guarantee of barrier certificates to allow a much richer set of safe learning control options. A 1 = {x \| V * (x) ≤ 1}, where V * (x) = 1.343x 2 1 + 0.5155x 1 x 2 + 1.152x 2 2 . The safe region estimated with barrier certificates is A 2 = {x \| h * (x) ≥ 0}, where h * (x) = 1 − 0.4254x 1 − 0.3248x 2 − 0.7549x 2 2 − 0.8616x 2 1 − 0.2846x 1 x 2 . -2 -1 0 1 2 x 1 -2 -1 0 1 2 x 2 A 2 (h * (x) = 0) A 1 (V * (x) = 1) B. Gaussian Processes A GP is a nonparametric regression method that can capture complex unknown functions [20]. With a GP, every point in the state space is associated with a normally distributed random variable, which allows us to derive high probability statements about the system. Adding some unknown dynamics d(x) to the original class of control-affine systems (1), we now consider a system with partially unknown dynamics in this paper, i.e., x = f (x) + g(x)u + d(x),(4) where x ∈ X ⊆ R n and u ∈ U ⊆ R m are the state and control of the system. Although the proposed method applies to general dynamical systems, here we restrict our attention to the class of systems that can be addressed with existing computation tools. It is also assumed that d(x) is Lipschitz continuous. This assumption is necessary, because we want to generalize the learned dynamics to states that are not explored before. Since the unmodeled dynamics d(x) is n dimensional, each dimension is approximated with a GP model G P(0, k(x, x ′ )) with a prior mean of zero and a covariance function of k(x, x ′ ), where k(x, x ′ ) is the kernel function to measure the similarity between any two states x, x ′ ∈ X . In order to make GP inferences on the unknown dynamics, we need to get measurements of d(x). This measurementd(x) is obtained indirectly by subtracting the inaccurate model prediction [ f (x) + g(x)u] from the noisy measurement of the system dynamics [ẋ + N (0, σ 2 n )]. Since any finite number of data points form a multivariate normal distribution, we can obtain the posterior distribution of d(x * ) at any query state x * ∈ X by conditioning on the past measurements [20]. Given a collection of w measurements y w = [d(x 1 ),d(x 2 ), ...,d(x w )] T , the mean m(x * ) and variance σ 2 (x * ) of d(x * ) at the query state x * are m(x * ) = k T * (K + σ 2 n I) −1 y w , (5) σ 2 (x * ) = k(x * , x * ) − k T * (K + σ 2 n I) −1 k * ,(6) where ⌊K⌋ (i, j) = k(x i , x j ) is the kernel matrix, and k * = [k(x 1 , x * ), k(x 2 , x * ), ..., k(x w , x * )] T . With the learned system dynamics based on GP, a high probability confidence interval of the unmodeled dynamics d(x) can be established as D(x) = {d \| m(x) − k δ σ (x) ≤ d ≤ m(x) + k δ σ (x)}, (7) where k δ is a design parameter to get (1 − δ ) confidence, δ ∈ (0, 1). For instance, 95.5% and 99.7% confidence are achieved at k δ = 2 and k δ = 3, respectively. III. SAFE LEARNING WITH BARRIER CERTIFICATES In order to ensure that the learning based controller never enters the unsafe region, we will learn barrier certificates for the system and use the learned certificates to regulate the controller. As discussed in Section II, the barrier certificates certify a safe region that is forward invariant. We can first start with an conservative barrier certificate with certified safe region C 0 (x), then gradually expand this certified safe region with the collected data until it stops growing. This incremental learning process is visualized in Fig. 2. The green region C 0 and the yellow regions C n are the initial and final barrier certified safe regions, respectively. The barrier certified safe region gradually grows as more and more data points are sampled in the state space. More concretely, the goal of the learning process is to maximize the volume of the barrier certified safe region C by adjusting h(x), i.e., max h(x) vol(C ) s.t. max u∈U min d∈D (x) ∂ h ∂ x ( f (x) + g(x)u + d) + γh(x) ≥ 0, ∀x ∈ C . Since u and d are independent from each other, we can rewrite this optimization problem into max h(x) vol(C ) s.t. max u∈U ∂ h k ∂ x g(x)u + min d∈D (x) ∂ h ∂ x d + ∂ h ∂ x f (x) + γh(x) ≥ 0, ∀x ∈ C(8) Using the high confidence interval D(x) in (7), the barrier certificates constraint can be considered as max h(x) vol(C ) s.t. max u∈U ∂ h ∂ x g(x)u + ∂ h ∂ x m(x) − k δ ∂ h ∂ x σ (x) + ∂ h ∂ x f (x) + γh(x) ≥ 0, ∀x ∈ C .(9) When more data points are collected about the system dynamics, the uncertainty σ (x) will gradually decrease. As a result, more states will satisfy the barrier certificates constraint. The goal of the exploration task is to actively collect data to reduce σ (x) and maximize the volume of C . It should be pointed out that the barrier certified region maximization problem (9) is a non-convex, infinite dimensional optimization problem, which is intractable to solve in practice. We will make two simplifications to make it solvable, namely by employing adaptive sampling of the state space and parameterization of the shape of C . A. Adaptive Sampling of the State Space Due to the Lipschitz continuity of the system dynamics, the safety of the system in X can be evaluated by only sampling a finite number of points in X . Inspired by [5], we will show that we can adaptively sample the state space without losing safety guarantees. Similar to Lemma 4 in [5], it can be shown that h(x) andḣ(x) are Lipschitz continuous in x with Lipschitz constants L h and Lḣ, respectively. Let X τ ⊂ X be a discretization of the state space X . The closest point in X τ to x ∈ X is denoted as [x] τ , where x − [x] τ ≤ τ 2 . Lemma 3.1: If the following condition holds for all x ∈ X τ , max u∈U ∂ h ∂ x g(x)u + ∂ h ∂ x m(x) − k δ ∂ h ∂ x σ (x) + ∂ h ∂ x f (x) + γh(x) ≥ (Lḣ + γL h )τ,(10) then the safety barrier constraint max u∈U min d∈D (x) ∂ h ∂ x ( f (x) + g(x)u + d) + γh(x) ≥ 0(11) is satisfied for all x ∈ X with probability (1 − δ ), δ ∈ (0, 1). Proof: With the definition of the high confidence interval D(x), (10) can be rewritten as max u∈U min d∈D (x) ∂ h ∂ x ( f (x) + g(x)u + d) + γh(x) ≥ (Lḣ + γL h )τ, with a probability of (1 − δ ), for all x ∈ X τ . This is equivalent toḣ (x) + γh(x) ≥ (Lḣ + γL h )τ, for all x ∈ X τ . Because of the Lipschitz continuity of h(x) andḣ(x), we have for any x ∈ X , h(x) + γh(x) ≥ (ḣ([x] τ ) − Lḣτ) + γ(h([x] τ ) − L h τ) ≥ 0. This means that the safety barrier constraint is satisfied for any x ∈ X , if (10) holds for all x ∈ X τ . With the discretization of the state space, we only need to sample a finite number of points to validate the barrier certificates. However, the number of required sampling points is still very large. The following adaptive sampling strategy further reduces the number of sampling points required. Proposition 3.2: If the following condition is satisfied at x ∈ X , max u∈U ∂ h ∂ x g(x)u + ∂ h ∂ x m(x) − k δ ∂ h ∂ x σ (x) + ∂ h ∂ x f (x) + γh(x) ≥ (Lḣ + γL h )k τ τ,(12) with k τ ≥ 0, then the safety barrier constraint (11) is satisfied for all y ∈ X such that x − y ≤ k τ τ. Proof: The proof is similar to lemma 3.1. Leveraging the Lipschitz continuity of the barrier certificates, we can adaptively sample the state space without losing safety guarantees. Sparse sampling is performed at places with large safety margin, while dense sampling is only required at places with small safety margin. B. Parameterization of the Barrier Certificates Because maximizing the volume of C is a non-convex problem in general, we can parameterize the barrier certificate h µ (x) with µ to simplify the optimization problem. For example, h µ (x) can be formulated as 1 − Z(x) T µZ(x), where Z(x) is the vector of monomials, and µ is a positive semidefinite matrix. Then maximizing vol(C ) is equivalent to minimize the trace of µ. Further simplification can be made to fix the shape of C (by optimizing only with the known dynamics) and enlarge the level set of barrier certificates. With the shape parameterization and adaptive sampling technique, the barrier certificate maximization problem (9) can be written as max µ vol(C ) s.t. max u∈U ∂ h µ ∂ x g(x)u + ∂ h µ ∂ x m(x) − k δ ∂ h µ ∂ x σ (x) + ∂ h µ ∂ x f (x) + γh µ (x) ≥ (Lḣ + γL h )τ, ∀x ∈ C ∩ X τ .(13) In order to increase the learning efficiency during the exploration phase, the most uncertain state in C is sampled, x next = argmax x∈C ∩X τ σ (x).(14) It is assumed that a nominal exploration controllerû can always be designed to drive the system from the current state x to x next , i.e.,û = GoTo(x, x next ). Then the safety barrier certificates are enforced through a QP-based controller to "rectify" the nominal control such that the system is always safe, u * = argmin u∈U J(u) = u −û 2 s.t. ∂ h ∂ x g(x)u + ∂ h ∂ x m(x) − k δ ∂ h ∂ x σ (x) + ∂ h ∂ x f (x) + γh(x) ≥ 0.(15) Therefore, the actual exploration controller u * tries to stay as close as possible to the desired controllerû, while always honoring the safety requirements. The exploration phase ends when the safe region C does not grow any more. The learned maximum barrier certificates can be further used to regulate other control tasks the system want to achieve. C. Overview of the Safe Learning Algorithm An overview of the barrier certificates based safe learning algorithm is provided in Algorithm 1. At the beginning, a conservative barrier certified safe region C 0 is provided. The most uncertain state x next is computed based on the current GP model. Then, the QP based controller (15) is used to ensure that the system is driven to x next without ever leaving C n . After updating the GP model with the sampled data at x next , the barrier certificate optimization problem (13) is solved. The adaptive sampling technique (12) is adopted here to reduce the number of states to be sampled. This process is repeated until the safe region C n stops growing. Algorithm 1 Barrier Certificates based Safe Learning Input: Initial safe set C 0 ⊆ X , GP model G P(0, k(x, x ′ )), discretization X τ , tolerance ε Output: Final safe set C n Initialization : n = 0, x = x 0 1: repeat 2: n = n + 1 3: Find x next with (14) 4: Design nominal controllerû = GoTo(x, x next ) 5: Rectifyû with (15) and drive to x next 6: Sample x next , update GP 7: Expand vol(C n ) with (13) and adaptive sampling (12) 8: until vol(C n )-vol(C n−1 )≤ ε 9: return C n IV. ONLINE LEARNING OF QUADROTOR DYNAMICS The safe learning approach developed in Section III relies on a learning controller that drives the system to explore interested states. The challenge of designing this learning controller is that the 3D quadrotor system considered in this paper is highly nonlinear and unstable. In this section, we will present a recursive learning controller based on GP to learn the complex quadrotor dynamics online. A. Differential Flatness of 3D Quadrotor Dynamics The quadrotor is a well-modelled dynamical system with forces and torques generated by four propellers and gravity [27]. The relevant coordinate frames and Euler angles (roll φ , pitch θ , and yaw ψ) are illustrated in Fig. 3. The world, body, and intermediate frames (after yaw angle rotation) are denoted by the subscripts w, b, and c, respectively. where sθ and cθ stand for sin θ and cos θ , respectively. Here, we adopt the quadrotor model used in [10] to describe the nonlinear quadrotor dynamics,         r = gz w + 1 m Rz w f z ,   φ θ ψ    =    1 sφtθ cφtθ 0 cφ −sφ 0 sφ scθ cφ scθ    ω,(16) where z w = [0 0 1] T , and r = [x, y, z] T , m, and g are the position of the center of mass, the mass, and the gravitational acceleration of the quadrotor, respectively. tθ and scθ are short for tan θ and sec θ . The control inputs of the quadrotor are the body rotational rates (ω = [ω x , ω y , ω z ] T ) and the thrust ( f ). It is assumed that the body rotational rates of quadrotor are directly controllable through the fast response onboard controller, due to the small rotational inertia and high torque features of quadrotors [10]. Similar to [27], the dynamics in (16) η T ]. With the differential flatness property, quadrotor trajectory planning can be simplified as smooth parametric curves. Given a desired trajectory η d (t) ∈ C 3 that is three times differentiable, the feed forward control u FF = [ f FF , ω T FF ] can be derived by inverting the dynamics in (16),          f F F = −m r d − gz w , ω FF =    1 0 −sθ d 0 cφ d sφ d cθ d 0 −sφ d cφ d cθ d      φ ḋ θ ḋ ψ d    where θ d = atan2(β a , β b ), φ d = atan2(β c , β 2 a + β 2 b ), β a = −ẍ d cos ψ d −ÿ d sin ψ d , β b = −z d + g, and β c = −ẍ d sin ψ d + y d cos ψ d . Differential flatness only gives the feed forward control u FF . In addition, the unknown model error and tracking error need to be handled by a feedback control u FB . The actual control applied to the quadrotor is u = u FF + u FB , where          f FB = K p < Rz w , r d − r > +K d < Rz w ,ṙ d −ṙ >, ω FB = K p    φ d − φ θ d − θ ψ d − ψ    + K d   φ d −φ θ d −θ ψ d −ψ    +K p    y d − y x − x d 0    Note that with an inaccurate model, a high-gain feedback controller is needed to counteract both the model error and disturbances. As a better model is learned over time, only a low-gain feedback controller is needed with an improved tracking performance [16]. B. Learning based Control Using Gaussian Process The previous section deals with precise quadrotor models. But it is often difficult to acquire accurate parameters for quadrotor systems. In addition, the model (16) neglects the uncertain effects of damping, drag force, and wind disturbances. Here, we will use GP models to learn the unmodeled dynamics. The unmodeled dynamics can be captured with six GPs along each dimension in the state space, i.e.,                   r = gz w + 1 m Rz w f z +    G P 1 (0, k(q, q ′ )) G P 2 (0, k(q, q ′ )) G P 3 (0, k(q, q ′ ))    ,   φ θ ψ    =    1 sφtθ cφtθ 0 cφ −sφ 0 sφ scθ cφ scθ    ω +    G P 4 (0, k(q, q ′ )) G P 5 (0, k(q, q ′ )) G P 6 (0, k(q, q ′ ))    , where the input to the GPs is q = [r T ,ṙ T , θ , φ , ψ] T , and the observations for the GPs are s = [r T ,φ ,θ ,ψ] T , respectively. At a new query point q * , the mean m i (q * ) and variance σ 2 i (q * ) of the unknown dynamics can be inferred with (5). Based on the learned dynamics, a differential flatness based feed forward controller can be derived as,          f F F = −m r d − [m 1 (q), m 2 (q), m 3 (q)] T − gz w , ω FF =    1 0 −sθ d 0 cφ d sφ d cθ d 0 −sφ d cφ d cθ d      φ d − m 4 (q) θ d − m 5 (q) ψ d − m 6 (q)    , where θ d = atan2(β a ,β b ), φ d = atan2(β c , β 2 a +β 2 b ), β a = −(ẍ d − m 1 (q)) cos ψ d − (ÿ d − m 2 (q)) sin ψ d , β b = −(z d − m 3 (q)) + g, andβ c = −(ẍ d − m 1 (q)) sin ψ d + (ÿ d − m 2 (q)) cos ψ d . C. Recursive Online GP Learning One issue with the GP regression is that the time complexity of GP inference is O (N 3 ), where N is the number of data points. The majority of the time is used to compute the inverse of the kernel matrix K. While various approximation methods can be used to reduce the GP inference time, it is still challenging to perform online GP inference for complex dynamically systems like quadrotor. Here, we propose a recursive online GP Learning method to compute the exact GP inference. As the quadrotor moves forward, we will actively add multiple relevant data points into the kernel matrix at each time step. At the same time, the data points that contribute the least to the inference are deleted. The recursive data addition and deletion operations are described as following. 1) Adding Multiple New Data to the Kernel Matrix: Let the kernel matrix at the ith time step be K i , we can save the matrix inverse result from the previous step as L i = (K i + σ 2 n I) −1 . Denote the number of new data to be added as M. With the new data y i+1 and kernal vector k i+1 , we have L i+1 = L −1 i k i+1 k T i+1 c i+1 + σ 2 n I −1 = L i + L i k i+1 (c i+1 + σ 2 n I − k T i+1 L i k i+1 ) −1 k T i+1 L i −(c i+1 + σ 2 n I − k T i+1 L i k i+1 ) −1 k T i+1 L i L i k i+1 (c i+1 + σ 2 n I − k T i+1 L i k i+1 ) −1 (c i+1 + σ 2 n I − k T i+1 L i k i+1 ) . Notice that inversion operation only needs to be performed on a M × M matrix rather than a large N × N matrix. 2) Deleting Multiple Old Data from the Kernel Matrix: After deleting M data points from the old Kernel matrix inversion L i = (K i + σ 2 n I) −1 , the new inverse of the kernel matrix becomesL i = (K i + σ 2 n I) −1 . First, the data to be deleted is permuted to the bottom of the kernel matrix with a permutation matrix P π , where π : N → N is a permutation of N elements. The permuted kernel matrix is K p i = P π K i P T π , which can be written into a block matrix form, K P i = K i E i E T i F i , where E i , F i are the known parts to be deleted. Similarly, L P i = P π L i P T π = L −1 i E i E T i F i + σ 2 n I −1 . Since L P i is known, it can be written into block matrix form with the same block dimensions with (IV-C.2), L P i = A i B i B T i C i . With the block matrix inversion rule,L i can be recovered as L i = A i − B i C −1 i B T i , which means to perform the deletion operation, the only matrix inverse required is C −1 i ∈ R M×M . With the recursive data addition and deletion method, the GP inference can be obtained efficiently online. V. SIMULATION RESULTS The GP based learning algorithm is validated on a simulated quadrotor model in two examples, i.e., online learning of quadrotor dynamics and learning safety barrier certificates. In the simulation, the actual weight of the quadrotor is 1.4 times the weight used in the computation. In addition, an unknown constant wind of 0.1g is applied in the environment as illustrated in Fig. 4. Since the standard fixed pitch quadrotor cannot generate reverse thrust, the thrust control is limited to f z ∈ [−1.8mg, 0]. This simulation setup is very challenging, because the learning based quadrotor controller needs to deal with very inaccurate model and limited thrust. A. Online Learning of Quadrotor Dynamics In the first example, the quadrotor is commanded to track a nominal trajectory (illustrated in Fig. 4) using a differential flatness based controller with the given inaccurate model. A PD controller is wrapped around to stabilize the quadrotor. During the simulation, the quadrotor is intentionally pushed to unknown regions that has not been explored before. This will help us evaluate the scalability of the algorithm. The desired trajectory of the quadrotor is given aŝ η = [r(t) T ,ψ(t)] ∈ C 3 , while the actual trajectory is η = [r(t) T , ψ(t)]. In practice, the actual trajectory might deviate significantly from the desired trajectory when the model is r i = ... r i − K · [(r i −r i ), (ṙ i −ṙ i ), (r i −r i )] T . In the simulation, the sample size of the recursive GP model is fixed at 300 data points. At each time step, the most irrelevant data point is thrown away, and the most relevant data point is added to the GP model. The data relevance is decided by the kernel function k(q, q * ), where q = [r T ,ṙ T , θ , φ , ψ] T . It can observed that the tracking error of the learning based controller is significantly smaller than the tracking error without GP inference, as shown in Fig. 5. With the recursive learning strategy, it is demonstrated in Fig. 6 that the GP inference time is always kept below 20ms. Thus, the recursive GP inference method is very suitable for online learning of quadrotor dynamics. By pushing the quadrotor to unexplored regions, we can found that learning with q ′ = [ṙ T , θ , φ , ψ] T yields much better scalability than learning with q = [r T ,ṙ T , θ , φ , ψ] T . The reason might be the position r is not as important as other features in the current simulation setup. B. Learning Safety Barrier Certificates In this example, the motion of the quadrotor is constrained within an ellipsoid safe region, i.e., The quadrotor is controlled to fly back and forth on a vertical path inside the ellipsoid. The goal is to learn how aggressively the quadrotor can fly in the z direction with an inaccurate model and limited thrust. x The barrier certificates are parameterized as h µ (r) = 1 − (z + 0.8) 2 0.36 − µż 2 − x 2 0.16 − y 2 0.16 −ẋ 2 0.25 −ẏ 2 0.25 ≥ 0, where µ is the barrier parameter to regulate how fast the quadrotor can fly in the z direction. Small values of µ correspond to large admissible speedż, which means more aggressive flight behavior. Thus, the objective of the learning process is to minimize µ with the collected data. To reduce the number of required sample points, the adaptive sampling strategy developed in Section III-A was adopted. An illustrative example of the adaptive sampling strategy is given in Fig. 7. It can be observed that places closer to the boundary of the safe region (z = −1.2 and z = −0.2) are sampled much denser than the place closer to the center of the safe region (z = 0). Furthermore, downward speed (ż > 0) is sampled much denser than the upward speed (ż < 0). This might be caused by the lack of reverse thrust to counter the unmodeled dynamics. A conservative barrier certificate (µ = 6.3) is provided at the beginning of the learning process. Then, the quadrotor gradually explores the safe region C 0 and expands it to C n (µ = 0.6), as illustrated in Fig. 8. The nominal exploration controller is always regulated by the barrier certificates using the QP-based controller in (15). During the learning process, the quadrotor never leaves the barrier certified safe region. VI. CONCLUSIONS A safe learning algorithm based on barrier certificates was developed in this paper. The learning controller is regulated by the barrier certificates, such that the system never enters the unsafe region. The unmodel dynamics of the system was approximated with a Gaussian Process, from which a high probability safety guarantee for the dynamical system was derived. The barrier certified safe region is gradually expanded as the uncertainty of the system dynamics is reduced with more data. This safe learning technique was applied on a quadrotor system with 3D nonlinear dynamics. The computation time of this learning method is reduced significantly with an adaptive sampling strategy and a recursive The region enclosed by the solid green ellipse C 1 is the current safe region, while the region enclosed by the dashed red ellipse C 2 is the optimized next safe region. The green cross markers and red asterisk markers are the data points already sampled and to be sampled, respectively. The red circles centered at those sample points are the confident safe regions based on (12). All the unexplored region between C 1 and C 2 are covered by the circular confident safe region. GP inference method. Simulation results demonstrated the effectiveness of the proposed method. * The work by the first and third authors was sponsored by Grant No. 1544332 from the U.S. National Science Foundation. † Li Wang and Magnus Egerstedt are with the School of Electrical and Computer Engineering, Evangelos A. Theodorou is with the School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA. Email: {liwang, magnus, evangelos.theodorou}@gatech.edu Fig. 1 : 1Estimates of safe regions for system(3). The regions enclosed by the dashed red ellipse and solid green ellipse are estimated safe regions with optimal polynomial Lyapunov function V * (x) and barrier certificates h * (x), respectively. Fig. 2 : 2Incremental learning of the barrier certificates. Fig. 3 : 3Quadrotor coordinate frames. The Euler angles are defined with the ZY X convention. Hence, the rotation matrix from the body frame to the world frame can be written as R =   cθ cψ sφ sθ cψ − cφ sψ cφ sθ cψ + sφ sψ cθ sψ sφ sθ sψ + cφ cψ cφ sθ sψ − sφ cψ −sθ sφ cθ cφ cθ   , is differentially flat with the flat output chosen as η = [r T , ψ T ] T . The full state q = [r T ,ṙ T , θ , φ , ψ] T and control u = [ f , ω T ] T can be represented as an algebraic function of [η T ,η T ,η T , ... Fig. 4 : 4A simulated quadrotor flies in an unknown wind field with an inaccurate model. very inaccurate. To track the desired trajectory, the nominal trajectory is designed with a pole placement controller, ... Fig. 5 : 5Tracking error of the differential flatness based flight controller with and without GP inference. Fig. 6 : 6Recursive GP inference time per iteration. Fig. 7 : 7Adaptive sampling of the state space. Fig. 8 : 8Initial and final barrier certificates. 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[ "TESS-Keck Survey IX: Masses of Three Sub-Neptunes Orbiting HD 191939 and the Discovery of a Warm Jovian Plus a Distant Sub-Stellar Companion", "TESS-Keck Survey IX: Masses of Three Sub-Neptunes Orbiting HD 191939 and the Discovery of a Warm Jovian Plus a Distant Sub-Stellar Companion" ]	[ "Jack Lubin \nDepartment of Physics & Astronomy\nUniversity of California Irvine\n92697IrvineCAUSA\n", "Judah Van Zandt \nDepartment of Physics & Astronomy\nUniversity of California Los Angeles\n90095Los AngelesCAUSA\n", "Rae Holcomb \nDepartment of Physics & Astronomy\nUniversity of California Irvine\n92697IrvineCAUSA\n", "Lauren M Weiss \nInstitute for Astronomy\nUniversity of Hawai'i\n2680 Woodlawn Drive96822HonoluluHIUSA\n", "Erik A Petigura \nDepartment of Physics & Astronomy\nUniversity of California Los Angeles\n90095Los AngelesCAUSA\n", "Paul Robertson \nDepartment of Physics & Astronomy\nUniversity of California Irvine\n92697IrvineCAUSA\n", "Joseph M Akana Murphy \nDepartment of Astronomy and Astrophysics\nUniversity of California\n95060Santa CruzCAUSA\n", "Nicholas Scarsdale \nDepartment of Astronomy and Astrophysics\nUniversity of California\n95060Santa CruzCAUSA\n", "Konstantin Batygin \nDivision of Geological and Planetary Sciences\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n", "Alex S Polanski \nDepartment of Physics & Astronomy\nUniversity of Kansas\n1251 Wescoe Hall Dr1082, 66045Malott, LawrenceKSUSA\n", "Natalie M Batalha \nDepartment of Astronomy and Astrophysics\nUniversity of California\n95060Santa CruzCAUSA\n", "Ian J M Crossfield \nDepartment of Physics & Astronomy\nUniversity of Kansas\n1251 Wescoe Hall Dr1082, 66045Malott, LawrenceKSUSA\n", "Courtney Dressing \nDepartment of Astronomy\nUniversity of California Berkeley\n94720BerkeleyCAUSA\n", "Benjamin Fulton \nNASA Exoplanet Science Institute/Caltech-IPAC\n1200 E California Blvd314-6, 91125PasadenaMC, CAUSA\n", "Andrew W Howard \nDepartment of Astronomy\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n", "Daniel Huber \nInstitute for Astronomy\nUniversity of Hawai'i\n2680 Woodlawn Drive96822HonoluluHIUSA\n", "Howard Isaacson \nDepartment of Astronomy\nUniversity of California Berkeley\n94720BerkeleyCAUSA\n\nCentre for Astrophysics\nUniversity of Southern Queensland\nToowoombaQLDAustralia\n", "Stephen R Kane \nDepartment of Earth and Planetary Sciences\nUniversity of California\n92521RiversideCAUSA\n", "Arpita Roy \nSpace Telescope Science Institute\n3700 San Martin Drive21218BaltimoreMDUSA\n\nDepartment of Physics and Astronomy\nJohns Hopkins University\n3400 N Charles St21218BaltimoreMDUSA\n", "Corey Beard \nDepartment of Physics & Astronomy\nUniversity of California Irvine\n92697IrvineCAUSA\n", "Sarah Blunt \nDepartment of Astronomy\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n", "Ashley Chontos \nInstitute for Astronomy\nUniversity of Hawai'i\n2680 Woodlawn Drive96822HonoluluHIUSA\n", "Fei Dai \nDivision of Geological and Planetary Sciences\n1200 E California Blvd91125PasadenaCAUSA\n", "Paul A Dalba \nDepartment of Earth and Planetary Sciences\nUniversity of California\n92521RiversideCAUSA\n", "Kaz Gary \nDepartment of Physics & Astronomy\nUniversity of Kansas\n1251 Wescoe Hall Dr1082, 66045Malott, LawrenceKSUSA\n", "Steven Giacalone \nDepartment of Astronomy\nUniversity of California Berkeley\n94720BerkeleyCAUSA\n", "Michelle L Hill \nDepartment of Earth and Planetary Sciences\nUniversity of California\n92521RiversideCAUSA\n", "Andrew Mayo \nDepartment of Astronomy\nUniversity of California Berkeley\n94720BerkeleyCAUSA\n", "Teo Močnik \nGemini Observatory/NSF's NOIRLab\n670 N. A'ohoku Place96720HiloHIUSA\n", "Molly R Kosiarek \nDepartment of Astronomy and Astrophysics\nUniversity of California\n95064Santa CruzCAUSA\n", "Malena Rice \nDepartment of Astronomy\nYale University\n06511New HavenCTUSA\n", "Ryan A Rubenzahl \nDepartment of Astronomy\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n", "David W Latham \nCenter for Astrophysics \|\nHarvard & Smithsonian\n60 Garden Street02138CambridgeMassachusettsUSA\n", "S Seager \nDepartment of Physics\nKavli Institute for Astrophysics and Space Research\nMassachusetts Institute of Technology\n02139CambridgeMAUSA\n\nDepartment of Earth, Atmospheric and Planetary Sciences\nMassachusetts Institute of Technology\n02139CambridgeMAUSA\n\nDepartment of Aeronautics and Astronautics\nMIT\n77 Massachusetts Avenue02139CambridgeMAUSA\n", "Joshua N Winn \nDepartment of Astrophysical Sciences\nPrinceton University\n4 Ivy Lane08544PrincetonNJUSA\n" ]	[ "Department of Physics & Astronomy\nUniversity of California Irvine\n92697IrvineCAUSA", "Department of Physics & Astronomy\nUniversity of California Los Angeles\n90095Los AngelesCAUSA", "Department of Physics & Astronomy\nUniversity of California Irvine\n92697IrvineCAUSA", "Institute for Astronomy\nUniversity of Hawai'i\n2680 Woodlawn Drive96822HonoluluHIUSA", "Department of Physics & Astronomy\nUniversity of California Los Angeles\n90095Los AngelesCAUSA", "Department of Physics & Astronomy\nUniversity of California Irvine\n92697IrvineCAUSA", "Department of Astronomy and Astrophysics\nUniversity of California\n95060Santa CruzCAUSA", "Department of Astronomy and Astrophysics\nUniversity of California\n95060Santa CruzCAUSA", "Division of Geological and Planetary Sciences\nCalifornia Institute of Technology\n91125PasadenaCAUSA", "Department of Physics & Astronomy\nUniversity of Kansas\n1251 Wescoe Hall Dr1082, 66045Malott, LawrenceKSUSA", "Department of Astronomy and Astrophysics\nUniversity of California\n95060Santa CruzCAUSA", "Department of Physics & Astronomy\nUniversity of Kansas\n1251 Wescoe Hall Dr1082, 66045Malott, LawrenceKSUSA", "Department of Astronomy\nUniversity of California Berkeley\n94720BerkeleyCAUSA", "NASA Exoplanet Science Institute/Caltech-IPAC\n1200 E California Blvd314-6, 91125PasadenaMC, CAUSA", "Department of Astronomy\nCalifornia Institute of Technology\n91125PasadenaCAUSA", "Institute for Astronomy\nUniversity of Hawai'i\n2680 Woodlawn Drive96822HonoluluHIUSA", "Department of Astronomy\nUniversity of California Berkeley\n94720BerkeleyCAUSA", "Centre for Astrophysics\nUniversity of Southern Queensland\nToowoombaQLDAustralia", "Department of Earth and Planetary Sciences\nUniversity of California\n92521RiversideCAUSA", "Space Telescope Science Institute\n3700 San Martin Drive21218BaltimoreMDUSA", "Department of Physics and Astronomy\nJohns Hopkins University\n3400 N Charles St21218BaltimoreMDUSA", "Department of Physics & Astronomy\nUniversity of California Irvine\n92697IrvineCAUSA", "Department of Astronomy\nCalifornia Institute of Technology\n91125PasadenaCAUSA", "Institute for Astronomy\nUniversity of Hawai'i\n2680 Woodlawn Drive96822HonoluluHIUSA", "Division of Geological and Planetary Sciences\n1200 E California Blvd91125PasadenaCAUSA", "Department of Earth and Planetary Sciences\nUniversity of California\n92521RiversideCAUSA", "Department of Physics & Astronomy\nUniversity of Kansas\n1251 Wescoe Hall Dr1082, 66045Malott, LawrenceKSUSA", "Department of Astronomy\nUniversity of California Berkeley\n94720BerkeleyCAUSA", "Department of Earth and Planetary Sciences\nUniversity of California\n92521RiversideCAUSA", "Department of Astronomy\nUniversity of California Berkeley\n94720BerkeleyCAUSA", "Gemini Observatory/NSF's NOIRLab\n670 N. A'ohoku Place96720HiloHIUSA", "Department of Astronomy and Astrophysics\nUniversity of California\n95064Santa CruzCAUSA", "Department of Astronomy\nYale University\n06511New HavenCTUSA", "Department of Astronomy\nCalifornia Institute of Technology\n91125PasadenaCAUSA", "Center for Astrophysics \|\nHarvard & Smithsonian\n60 Garden Street02138CambridgeMassachusettsUSA", "Department of Physics\nKavli Institute for Astrophysics and Space Research\nMassachusetts Institute of Technology\n02139CambridgeMAUSA", "Department of Earth, Atmospheric and Planetary Sciences\nMassachusetts Institute of Technology\n02139CambridgeMAUSA", "Department of Aeronautics and Astronautics\nMIT\n77 Massachusetts Avenue02139CambridgeMAUSA", "Department of Astrophysical Sciences\nPrinceton University\n4 Ivy Lane08544PrincetonNJUSA" ]	[]	Exoplanet systems with multiple transiting planets are natural laboratories for testing planetary astrophysics. One such system is HD 191939 (TOI-1339), a bright (V=9) and Sun-like (G9V) star, which TESS found to host three transiting planets (b, c, and d). The planets have periods of 9, 29, and 38 days each with similar sizes from 3 to 3.4 R ⊕ . To further characterize the system, we measured the radial velocity (RV) of HD 191939 over 415 days with Keck/HIRES and APF/Levy. We find that M b = 10.4 ± 0.9M ⊕ and M c = 7.2 ± 1.4M ⊕ , which are low compared to most known planets of comparable radii. The RVs yield only an upper-limit on M d (<5.8 M ⊕ at 2σ). The RVs further reveal a fourth planet (e) with a minimum mass of 0.34 ± 0.01 M Jup and an orbital period of 101.4 ± 0.4 days. Despite its non-transiting geometry, secular interactions between planet e and the inner transiting planets indicate that planet e is coplanar with the transiting planets (∆i < 10 • ). We identify a second high mass planet (f) with 95% confidence intervals on mass between 2-11 M Jup and period between 1700-7200 days, based on a joint analysis of RVs and astrometry from Gaia and Hipparcos. As a bright star hosting multiple planets with well-measured masses, HD 191939 presents many options for comparative planetary astronomy including characterization with JWST.	10.3847/1538-3881/ac3d38	[ "https://arxiv.org/pdf/2108.02208v2.pdf" ]	236,924,440	2108.02208	f8af42daea85c7bc0220e566e22faf880f766122	TESS-Keck Survey IX: Masses of Three Sub-Neptunes Orbiting HD 191939 and the Discovery of a Warm Jovian Plus a Distant Sub-Stellar Companion 4 Jan 2022 Jack Lubin Department of Physics & Astronomy University of California Irvine 92697IrvineCAUSA Judah Van Zandt Department of Physics & Astronomy University of California Los Angeles 90095Los AngelesCAUSA Rae Holcomb Department of Physics & Astronomy University of California Irvine 92697IrvineCAUSA Lauren M Weiss Institute for Astronomy University of Hawai'i 2680 Woodlawn Drive96822HonoluluHIUSA Erik A Petigura Department of Physics & Astronomy University of California Los Angeles 90095Los AngelesCAUSA Paul Robertson Department of Physics & Astronomy University of California Irvine 92697IrvineCAUSA Joseph M Akana Murphy Department of Astronomy and Astrophysics University of California 95060Santa CruzCAUSA Nicholas Scarsdale Department of Astronomy and Astrophysics University of California 95060Santa CruzCAUSA Konstantin Batygin Division of Geological and Planetary Sciences California Institute of Technology 91125PasadenaCAUSA Alex S Polanski Department of Physics & Astronomy University of Kansas 1251 Wescoe Hall Dr1082, 66045Malott, LawrenceKSUSA Natalie M Batalha Department of Astronomy and Astrophysics University of California 95060Santa CruzCAUSA Ian J M Crossfield Department of Physics & Astronomy University of Kansas 1251 Wescoe Hall Dr1082, 66045Malott, LawrenceKSUSA Courtney Dressing Department of Astronomy University of California Berkeley 94720BerkeleyCAUSA Benjamin Fulton NASA Exoplanet Science Institute/Caltech-IPAC 1200 E California Blvd314-6, 91125PasadenaMC, CAUSA Andrew W Howard Department of Astronomy California Institute of Technology 91125PasadenaCAUSA Daniel Huber Institute for Astronomy University of Hawai'i 2680 Woodlawn Drive96822HonoluluHIUSA Howard Isaacson Department of Astronomy University of California Berkeley 94720BerkeleyCAUSA Centre for Astrophysics University of Southern Queensland ToowoombaQLDAustralia Stephen R Kane Department of Earth and Planetary Sciences University of California 92521RiversideCAUSA Arpita Roy Space Telescope Science Institute 3700 San Martin Drive21218BaltimoreMDUSA Department of Physics and Astronomy Johns Hopkins University 3400 N Charles St21218BaltimoreMDUSA Corey Beard Department of Physics & Astronomy University of California Irvine 92697IrvineCAUSA Sarah Blunt Department of Astronomy California Institute of Technology 91125PasadenaCAUSA Ashley Chontos Institute for Astronomy University of Hawai'i 2680 Woodlawn Drive96822HonoluluHIUSA Fei Dai Division of Geological and Planetary Sciences 1200 E California Blvd91125PasadenaCAUSA Paul A Dalba Department of Earth and Planetary Sciences University of California 92521RiversideCAUSA Kaz Gary Department of Physics & Astronomy University of Kansas 1251 Wescoe Hall Dr1082, 66045Malott, LawrenceKSUSA Steven Giacalone Department of Astronomy University of California Berkeley 94720BerkeleyCAUSA Michelle L Hill Department of Earth and Planetary Sciences University of California 92521RiversideCAUSA Andrew Mayo Department of Astronomy University of California Berkeley 94720BerkeleyCAUSA Teo Močnik Gemini Observatory/NSF's NOIRLab 670 N. A'ohoku Place96720HiloHIUSA Molly R Kosiarek Department of Astronomy and Astrophysics University of California 95064Santa CruzCAUSA Malena Rice Department of Astronomy Yale University 06511New HavenCTUSA Ryan A Rubenzahl Department of Astronomy California Institute of Technology 91125PasadenaCAUSA David W Latham Center for Astrophysics \| Harvard & Smithsonian 60 Garden Street02138CambridgeMassachusettsUSA S Seager Department of Physics Kavli Institute for Astrophysics and Space Research Massachusetts Institute of Technology 02139CambridgeMAUSA Department of Earth, Atmospheric and Planetary Sciences Massachusetts Institute of Technology 02139CambridgeMAUSA Department of Aeronautics and Astronautics MIT 77 Massachusetts Avenue02139CambridgeMAUSA Joshua N Winn Department of Astrophysical Sciences Princeton University 4 Ivy Lane08544PrincetonNJUSA TESS-Keck Survey IX: Masses of Three Sub-Neptunes Orbiting HD 191939 and the Discovery of a Warm Jovian Plus a Distant Sub-Stellar Companion 4 Jan 2022Draft version January 5, 2022 Typeset using L A T E X twocolumn style in AASTeX63 2HD 191939TESSKeck HIRESMulti-Planet Exoplanet systems with multiple transiting planets are natural laboratories for testing planetary astrophysics. One such system is HD 191939 (TOI-1339), a bright (V=9) and Sun-like (G9V) star, which TESS found to host three transiting planets (b, c, and d). The planets have periods of 9, 29, and 38 days each with similar sizes from 3 to 3.4 R ⊕ . To further characterize the system, we measured the radial velocity (RV) of HD 191939 over 415 days with Keck/HIRES and APF/Levy. We find that M b = 10.4 ± 0.9M ⊕ and M c = 7.2 ± 1.4M ⊕ , which are low compared to most known planets of comparable radii. The RVs yield only an upper-limit on M d (<5.8 M ⊕ at 2σ). The RVs further reveal a fourth planet (e) with a minimum mass of 0.34 ± 0.01 M Jup and an orbital period of 101.4 ± 0.4 days. Despite its non-transiting geometry, secular interactions between planet e and the inner transiting planets indicate that planet e is coplanar with the transiting planets (∆i < 10 • ). We identify a second high mass planet (f) with 95% confidence intervals on mass between 2-11 M Jup and period between 1700-7200 days, based on a joint analysis of RVs and astrometry from Gaia and Hipparcos. As a bright star hosting multiple planets with well-measured masses, HD 191939 presents many options for comparative planetary astronomy including characterization with JWST. INTRODUCTION Bright systems with multiple planets are valuable to the exoplanet community. They are amenable to precise RV monitoring and are natural laboratories of planetary astrophysics. With multiple planets forming from the same protoplanetary disk, such systems allow for comparative exoplanetology investigations, as we can assume a similar history of formation conditions for each planet. NASA's Transiting Exoplanet Survey Satellite (TESS ; Ricker et al. 2015) is an all-sky photometric survey searching for planets around the brightest stars, and its discoveries continue to deliver new planetary systems for detailed investigation. Due to the 28-day per sector survey strategy, TESS is finding many exoplanets in short period orbits (<14 days). A primary science goal of the TESS mission is to measure the masses of 50 planets smaller than 4 Earth radii. The TESS-Keck Survey (TKS) is a collaboration among astronomers at Keck partner institutions to combine efforts and telescope time to meet and exceed this science goal (see TKS-0 (Chontos et al. 2021), TKS-I (Dalba et al. 2020), TKS-II , TKS-III (Dai et al. 2020), TKS-IV (Rubenzahl et al. 2021)). Our survey is further concerned with the formation, evolution, and dynamics of various types of exoplanetary systems. Three of TKS's main goals are characterizing systems with multiple planets, those with possible distant giant planets, and those that show promise for high-quality atmospheric characterization (Chontos et al. 2021) HD 191939 is a solar-like star (G9V) that hosts a multi planet system which addresses most of our areas of interest. TESS observed the star for 252 days in 9 nonconsecutive sectors during its primary mission, allowing for a long baseline (326 days) of photometry and enabling discovery of longer-period planets. Badenas-Agusti et al. (2020) have already announced three transiting planets, two of which would not have been discovered without multiple sectors of coverage. This work includes the first mass measurements of the transiting planets and we have uncovered an additional Jovian planet as well as a high mass planet. In all, this system has a wide diversity of planet masses and periods. We find that the transiting planets of HD 191939 fall into some of the patterns uncovered by statistics papers on the Kepler planets. They have nearly identical radii, * NSF Graduate Research Fellow † NSF Astronomy and Astrophysics Postdoctoral Fellow as is typical of the Kepler planets (Weiss et al. 2018), yet their spacing is irregular. They have similar masses, consistent with the pattern found in Millholland et al. (2017), but the planets have low masses for their sizes (Weiss & Marcy 2014), implying lower than average densities. The masses we present are some of the most precise mass measurements of small transiting planets in a multiplanet system (2 of 3 with 5σ mass or better). In this paper, we describe our data sources ( §2) and analyze the system properties, describing the host star properties ( §3.1) as well as our RV model ( §3.2) and photometry model ( §3.3). Next, we describe the densities and compositions of the transiting planets ( §4). We then explore the system dynamics in detail, including constraining the properties of planet f with new techniques ( §5), placing limits on the inclination of planet e ( §6), and describing the resonant interactions of planets c and d ( §7). We then quantify the possibility of additional planets in the system ( §8) before investigating follow up opportunities for HD 191939 by examining the system's atmospheric and . We present our conclusions in §10. OBSERVATIONS TESS Photometry Due to the star's high northern declination, TESS observed HD 191939 for a total of 9 sectors in Cycle 2. Data were obtained with a 2-minute cadence during sectors 15-19, 21-22, and 24-25, spanning a total baseline of 326 days from 2019-07-18 to 2020-06-08, though the star was not observed for the entirety of this time (Stassun et al. 2018). We downloaded data processed through the Science Processing Operations Center (SPOC) pipeline through the Mikulski Archive for Space Telescopes (MAST), and used the Pre-search Data Conditioning (PDC) light curves for our analysis. (Jenkins et al. 2016). Radial Velocities We acquired 73 RV observations with Keck/HIRES at the W.M. Keck Observatory on Maunakea, Hawaii between November 2019 and December 2020, see Table 3. We reduced the spectra in the standard procedure of the California Planet Search (Howard et al. 2010). We used a high SNR template from Keck/HIRES to generate a deconvolved stellar spectral template (DSST). We took all RV observations with a warm iodine cell in the light path for wavelength calibration (Valenti et al. 1995;Butler et al. 1996) with median SNR of ∼216 per pixel at the iodine wavelength region of ∼500 nm. We also acquired 104 RV observations with the Automated Planet Finder telescope (APF) (Vogt et al. 2014) at Lick Observatory in California between December 2019 and December 2020. At the beginning of the baseline, we observed twice per night and binned the two observations. After February 2020, we changed our observing strategy to obtain one spectrum per night due to time constraints within our survey. We used the same Keck/HIRES template to calculate the APF RVs because it produced a higher-quality DSST than the APF template. The median SNR for APF observations was ∼76 per pixel at the iodine wavelength region of ∼500 nm. To maintain only high quality data points, we removed all (7) RVs from the APF time series that had SNR < 31, equivalent to an RV error of 9 m/s. We also removed 1 APF observation that was taken within 5 minutes of 12 • twilight in the morning. SYSTEM PROPERTIES Host Star We analyzed our iodine-free HIRES spectrum with the SpecMatch-Syn code to derive the T eff , log g, and metallicity [Fe/H] of the host star, and we list our results in Table 1. We then derived stellar mass, radius, and age according to the approach described in . We incorporated Gaia DR2 parallaxes (Gaia Collaboration et al. 2018), 2MASS apparent K magnitude, and the MIST models (Choi et al. 2016) using the isoclassify package (Huber et al. 2017;Berger et al. 2020). Following Tayar et al. (2020), we inflated the error bar on the stellar mass measurement by adding a systematic error term of 0.03 M in quadrature. Given the limited spread in the HR diagram at HD 191939's T eff (5348 ± 100 K, G9V), isochrone ages have large uncertainties. However, they indicate this star is older than 8.7 Gyr (2σ confidence). We determined the abundances for 15 individual elements using KeckSpec (Rice & Brewer 2020) finding the composition of HD 191939 is generally sub-solar for most elements. We determine the Mg/Si ratio to be consistent with both the solar value and most local stars (Brewer & Fischer 2017). C/O, however, is found to be 0.34 ± 0.09; 2σ lower than the solar value implying the assumption of solar abundances for these elements may not be applicable to stellar atmospheric models. We obtain [Y/Mg] = −0.08 ± 0.1 and use this with the abundance-age relation of Nissen et al. (2020) which gives an age estimate of 7 ± 3 Gyr. Although this is consistent with the lower bound obtained from an isochronal fit, HD 191939 is 400K cooler than the Sun-like stars used for this relation and should be treated with caution. HD 191939 is a chromospherically inactive star with log R HK = -5.11 ± 0.05. We computed the Ca II H&K index (S HK ) as described in Isaacson & Fischer (2010) for both our Keck/HIRES and APF time series, see Table 3. We find no significant correlations between the S HK values and RVs. Additionally, we find no statistically significant periodicities in the S HK time series. RV Model Soon after beginning our RV observations, we saw evidence of an additional planet beyond those identified by TESS , including observations which showed a ∼ 40 m/s change in the RV, consistent with a massive planet. Continued observations further traced a large amplitude periodicity near ∼ 100 days. The Generalized Lomb-Scargle (GLS) periodogram (Zechmeister & Kürster 2009) of the RVs is dominated by this signal which we attribute to a fourth planet (e) (Figure 1, top panel). To discern the architecture of the system, we performed a model comparison analysis. Using RadVel ), we tested a variety of RV models: 3-5 total planets and either allowing eccentricity to vary for each or fixing it to zero, as well as allowing or prohibiting trend and/or curvature terms. We used Markov Chain Monte Carlo (MCMC) to explore the parame- 1.7 ± 0.2 APF Zeropoint, γAPF (ms −1 ) −8.01 APF Jitter, σAPF (ms −1 ) 3.7 ± 0.6 Trend,γ (ms −1 d −1 ) 0.114 ± 0.006 Curve,γ (ms −1 d −2 ) (−6 ± 2) × 10 −5 * Equilibrium Temperatures assume zero bond albedo ter space and estimate uncertainties; all planet models discussed here converged by the default RadVel criteria unless otherwise stated. In all models, we fixed the periods and times of conjunction of the transiting planets to the values found from our TESS photometry model. We set uniform priors on the Doppler amplitudes (−∞, ∞), allowing negative values for all planets to avoid biasing the masses to higher values. We set a uniform prior from 1 to 1000 days on the period of planet e and a uniform prior on its time of conjunction (2459000, 2459100) BJD. For both instruments, we set a prior on the instrumental jitter as uniform (0, 10) m/s. Lastly we set a prior on the trend term uniform from (-1, 1) m/s/d and on the curvature term uniform (-0.1, 0.1) m/s/d 2 . Our preferred RV model has an Akaike Information Criterion (AIC) (Akaike 1974) of 858 and contains four planets on circular orbits, as well as both a trend and curvature term, which models a subset of a sinusoid as a quadratic to represent a 5th body in the system. The closest neighboring model, in terms of AIC, is one with 4 planets plus a trend but no curvature term (AIC = 866). Our preferred model is very strongly preferred over a model with 3 transiting planets (did not converge, AIC = 1332) and a 4-planet model with no trend and no curvature (AIC = 1202). Our full RV timeseries can be seen in Figure 2, and phase-folded RV time series for each planet can be seen in Figure 3. The orbital parameters and masses of all planets can be found in Table 1. We searched the residuals of our preferred model for additional planets but found no statistically significant signals. Photometry Model We pre-whitened the TESS photometry by using a Gaussian process model to subtract out low amplitude stellar and instrumental variability from the light curve. We then performed a blind transit search using Transit Least Squares (TLS; Hippke & Heller 2019). This recovered three transiting planets with period, depth, and duration values and errors consistent with the previously published values in Badenas-Agusti et al. (2020). We also performed a more targeted search with TLS for transits of planet e but found no evidence of any such events. We then modeled the transits of planets b, c, and d with the Exoplanet package (Foreman-Mackey et al. 2020b) and re-derived planet parameters using our updated stellar parameters and the TLS output as priors (Table 1). To calculate this model, we assumed circular orbits and fit for seventeen parameters: (1.) Orbital periods, with Gaussian priors informed by our TLS search values, (2.) Times of inferior conjunction, with Gaussian priors informed by our TLS search, (3.) Planet-to-star radius ratios, with a log-uniform prior from 0.01 to 0.1, (4.) Im-pact parameters, with a uniform prior from 0 to 1, (5.) Stellar radius, with Gaussian priors defined by the updated stellar parameters, (6.) Stellar mass, with Gaussian priors defined by the updated stellar parameters, (7.) Quadratic limb darkening parameters calculated using Python Limb Darkening Toolkit (Parviainen & Aigrain 2015), and (8.) A white noise scaling term for the TESS light curve. Exoplanet implements MCMC algorithm, which we ran with 2000 iterations and a 500 step burn in and found that all chains converged. Additionally, we derived transit midpoints for each transit of each planet which is discussed further in §7 (see Appendix, Table 2). Figure 4 shows our modeled phasefolded light curves for each planet and Figure 5 shows the full reduced TESS light curve of the star with transits color coded. Given our weak detection of planet d in the RV time series, we returned to the photometry to confirm the period and transit times. Due to the positioning of data gaps in the light curve, there are 4 "odd" transits and 1 "even" transit of planet d. We considered the possibility that the single even transit, occurring at 2458781.89 BJD, comes from a different source than the four odd transits. In such a scenario, the orbital period of planet d would double to 76.7 days. However, comparing between the transits, including matching depths, durations, and ingress/egress shapes, we found no inconsistencies between the single transit and the other four. Furthermore, we find no evidence for a ∼76 day periodicity in our RV time series. Thus we conclude that all five transits do originate from a single planet with an orbital period of 38.4 days. COMPOSITION OF TRANSITING PLANETS How do the transiting planets in this system compare to other known transiting planets? We find planet b imparts a Doppler semi-amplitude of 3.8 ± 0.3 ms −1 , corresponding to a mass of 10.4±0.9M ⊕ ; plant c imparts 1.8±0.3 ms −1 , corresponding to 7.2±1.4M ⊕ ; and planet d imparts 0.6 ± 0.3 ms −1 , corresponding to 2.8 ± 1.5M ⊕ . The placement of the three transiting planets on a massradius diagram reveals that they exist at the periphery of the known planet population ( Figure 6). Planet b fits more consistently with previously known planets, while planet d is a low-mass outlier. The relatively low masses for their radii implies small densities. We find planet b has a bulk density of 1.5±0.2 g/cc, planet c has 1.4±0.3 g/cc, and planet d has 0.5 ± 0.3 g/cc. Fulton et al. (2017) and Van Eylen et al. (2018) described the radius gap as a region of radius phase space from 1.5-2.0 R ⊕ where relatively few planets are found. Studies have explained this gap as most likely due to a transitional phase between planets with and without extended H/He envelopes, which may be due to photoevaporation (Lopez & Fortney 2014;Owen & Wu 2017). Given that all three transiting planets in the HD 191939 system have radii above the gap, it is likely that the best description of their compositions is that of a volatile rich envelope surrounding a rocky core (Weiss & Marcy 2014;Rogers 2015;Fulton et al. 2017). Employing Smint (Piaulet 2020), which interpolates the model grids from Lopez & Fortney (2014) and Zeng et al. (2016) and samples posterior space with MCMC, we explored the possible fractions of H/He by mass for the three transiting planets assuming a dry, Earth-like, rock-iron core. Using a flat prior for the age from 9 to 13 Gyr, we find H/He envelopes of 6.5 ± 0.5% for planet b, 5.7 ± 0.6% for planet c, and 6.4 ± 0.5% for planet d. From our RV model, we place a 2σ upper limit on planet d's mass at 5.8 M ⊕ . This corresponds to 2σ upper limit on planet d's density of 1.1 g/cc. While this density upper limit places it within the range of planets b and c, the potential low density for planet d is noteworthy. In the literature, there is a population of low density planets: the Kepler-51 system (Masuda 2014), Kepler-79d (Jontof-Hutter et al. 2014), and Kepler-87c (Ofir et al. 2014), which are collectively described as "superpuffs" for their inflated radii (4-8 R ⊕ ) and low masses (2-5 M ⊕ ), which implies densities of ∼0.1 g/cc. While HD 191939 d is not a super-puff since its radius is smaller (only 3 R ⊕ ), it does share a notable characteristic with the super-puffs: they all exist in or near resonance with another planet in their systems. The super-puff planets may have low masses for their sizes as part of a selection bias: the planet masses are derived from transit timing variation (TTV) interactions, which are most prominent for planets in or near a resonance chain. HD 191939 d's potential low density, combined with its placement as the outer member of a near 4:3 resonance with planet c (see §7 for more detail), draws some comparison to the super-puffs and brings forward questions on its possible formation history. Two different mechanisms have been proposed for explaining the prevalence of highly inflated plants in or near resonance. Lee & Chiang (2016) showed super-puff planets most easily gain their extended atmospheres in dust-free environments at distances beyond 1 AU before migrating inwards. As part of this migration, they are more likely to form the outer companion of a resonance chain with another interior planet in the system. Under this formation scenario, planet d would likely contain a large fraction of water, a composition which we do not explore in this paper. Millholland (2019) describes how super-puffs that exist just wide of resonance with another planet are thought to have preferentially high obliquities, which could drive heat dissipation through obliquity tides resulting in inflated planet radii. HD 191939 d represents a unique opportunity to study a possible low density planet and to test the above theories for two reasons. The mass measurement we provide comes from the RV method rather than TTVs. The location in the system interior to the Jovian planet e can provide dynamical constraints for any potential migration history. Of the super-puffs listed above, only Kepler-79d has a confirmed planet exterior to its orbit in the system, and this planet is another sub-Neptune. The relatively small masses, low densities, and high equilibrium temperatures of these planets might combine to drive atmospheric escape on some or all of the three inner planets. By the Jeans escape mechanism, to first order approximation a gas will eventually com-pletely escape if its thermal velocity exceeds one sixth the planet's escape velocity. Planet b's temperature is likely high enough to allow the steady escape of atomic and molecular hydrogen. Fixing each planet's radius to the median values of our photometry model, we calculated whether molecular hydrogen would escape each planet for a grid of every combination of planet mass and equilibrium temperature out to 3σ of each value. We find that molecular hydrogen escapes planet b in 84% of combinations, 52% for planet c, and 94% for planet d. Following the same procedure, planet d's small mass means it may not even be able to retain helium as 47% of combinations allow this gas to escape. If any of these planets are experiencing atmospheric escape, transmission spectroscopy with JWST might show evidence. PLANET F CONSTRAINTS What is the nature of the 5th planet in the system? Our RV analysis favors both a trend and curvature in the residuals of the preferred 4-planet model, suggesting a 5th planet with an orbital period much longer than our 415-day observing baseline. The presence of this planet can be further constrained by the change in HD191939's proper motion over a period of 24 years. Using these independent data sets, we can place constraints on the mass and semi-major axis of planet f. We derived these constraints using a novel method which compares model orbits using just 3 free parameters. We quantify long-period signals in the RV residuals through trend (γ) and curvature (γ) terms; and astrometric motion through ∆µ, the difference in proper motions at two epochs. We generated a set of randomlysampled orbits and computed these three parameters for each. A high-likelihood orbital model is one that reproduces the true values ofγ,γ, and ∆µ. To produce a set of model orbits, we first defined our search range for both mass and semi-major axis. We started with τ min , the lower bound on orbital period. Planet f produced only a small detected curvature over our observing baseline, a feature that we estimate would require an orbital period 4 times the baseline. This yielded a lower semi-major axis limit of 2.6 AU. We limited our search to semi-major axes within 50 AU. We used a similar argument to obtain a lower bound on M p . We took the maximum ∆RV from the residuals of fitting for planets b-e and set it equal to the semi-amplitude of a planet with a period of τ min , again assuming a circular orbit. From this amplitude, we calculated a minimum mass of 2.05 M J . We chose 200 M J as the upper limit of our mass search, reasoning that more massive objects would be luminous enough to detect in high-contrast imaging. We marginalized over four additional orbital parameters: inclination i, eccentricity e, argument of periastron ω, and mean anomaly M . In total we drew 10 8 random samples from this 6-dimensional parameter space using the following prior distributions: • log a 1 AU ∼ U(2.62, 50) • log Mp 1 MJ ∼ U(2.05, 200) • cos(i) ∼ U(0, 1) • ω ∼ U(0, 2π) • M ∼ U(0, 2π) • e ∼ B(0.867, 3.03) where B is the two-parameter Kipping (2013) beta distribution for e. We used the same samples to generate both the RV curves and the astrometric proper motions. To impose RV constraints, we computed for each sample the first (γ) and second (γ) time derivatives of the stellar radial velocity. We began by differentiating the true anomaly ν: ν = 2 tan −1 1 + e 1 − e tan E 2 (1) ν = 2π √ 1 − e 2 τ (1 − e cos(E)) 2 ,(2) where τ is the orbital period calculated from Kepler's Third Law and E is the eccentric anomaly, which we obtained by numerically solving Kepler's equation: M = E − e sin E.(3) The second derivative of ν is also needed to computeγ: ν = −ν 2 2e sin(E) √ 1 − e 2(4) With the derivatives of ν, we can write the equations forγ andγ. We start with the RV value itself, γ: γ = K [e cos(ω) + cos(ν + ω)] ,(5) where K = G 1 − e 2 M p sin i a(M p + M ) . The derivatives of γ are: γ = −K [ν sin(ν + ω)](7) andγ = −K ν 2 cos(ν + ω) +ν sin(ν + ω) . We evaluated the sample likelihood according to P (γ,γ\|γ m ,γ m ) ∝ exp − (γ −γ m ) 2 2σ 2 γ + (γ −γ m ) 2 2σ 2 γ .(9) To obtain the 2D a-M p joint posterior, we marginalized over {e, i, ω, M }. The results from the RV only constraints can be seen in Figure 7 in green with 1-and 2-σ contours. We next incorporated astrometry to further constrain the characteristics of the fifth planet. Brandt (2021) aligned the reference frames of Hipparcos ( Hip 1997) and Gaia EDR3 (Lindegren et al. 2020) to produce a self-consistent catalog of stellar proper motions measured at epochs 1991.25 and 2015.5. Brandt reported the proper motion based on the difference in position between these epochs. The Gaia and position-derived proper motions, µ G = (150.19 ± 0.02, −63.99 ± 0.02) mas/yr and µ HG = (150.31±0.03, −63.94±0.03) mas/yr, were the most precise, and indicated a change in proper motion ∆µ = \| µ G − µ HG \| of 0.13 ± 0.03 mas/yr over the 24 years separating the two epochs. Using the same orbit models as in the RV analysis, we first computed the average proper motion vector in the Gaia EDR3 epoch. We also used the change in astrometric position between the Gaia and Hipparcos epochs to obtain an average proper motion over the 24 year baseline. We then computed the magnitude of the difference vector ∆µ m and evaluated the likelihood via P (∆µ\|∆µ m ) ∝ exp − (∆µ − ∆µ m ) 2 2σ 2 ∆µ .(10) The detected proper motion difference rules out high mass models that were permitted by our RV-only analysis. The blue region of Figure 7 shows the range of a-M p values that are allowed by astrometry at the 1 and 2σ levels. Because the RV and astrometric data sets are independent, we may evaluate the joint RV-astrometry likelihood by multiplying Equations 9 and 10. Figure 7 shows in red the region of a-M p space that is allowed by both the RV and astrometric constraints. We find at 95% confidence that planet f has a mass of 2-11 M J and orbits at a distance of 2.6-7.0 AU. Throughout this paper we refer to this companion as a "planet" because these current mass constraints place it most likely below the generally accepted upper mass limit for planets of ∼ 13 M J ; but we caution that the high-mass tail of the probability distribution includes objects that would typically be characterized as brown dwarfs. Such high mass objects on the planet-brown dwarf boundary are thought to form by one of two general formation pathways: core accretion (Pollack et al. 1996) or gravitational instability (Boss 1997). Core accretion is more successful at producing low mass objects and is the most plausible formation channel for planets b through e. Schlaufman (2018) showed a transition point in formation mechanism at 10 M J , which may represent a mass upper limit for objects formed via core-accretion. Therefore, more massive objects more likely formed via gravitational instability and are therefore not planets. If planet f is at the upper end of its mass range, gravitational instability becomes a plausible pathway. This raises the possibility that both mechanisms were active in the HD 191939 system. We advocate for continued Doppler/astrometric monitoring of the HD 191939 system to fully resolve this companion's orbit and measure its mass more precisely to identify which formation channel is more likely. PLANET E IS NEARLY COPLANAR What is the inclination of planet e? Given the emergence of planet e in our RV data, we searched the TESS photometry for evidence of its transit. We would expect this 0.34 ± 0.01 M J / sin i Jovian planet to have a radius of ∼ 1 R J , implying a transit depth on the order of 1%. At a 101 day orbital period, assuming zero eccentricity and an edge-on orbit, we expect the duration of its transit to be ∼ 8 hours. Such a transit event should be obvious in the data by visual inspection. We do not see planet e's transit (see Figure 5). Within the error bars of our period and time of conjunction for planet e, it is possible that TESS missed the transits of planet e by unlucky timing. Still, the most likely explanation for the missing transits is that the planet is non-transiting. We did not search for a transit of planet f because its transit event should be a similar depth but even longer than planet e's and it was not near its expected time of conjunction at the time of TESS 's observations. Assuming planet e is non-transiting and has a radius of 1 R J , we place an upper limit on the inclination at 89.5 • . To place a lower limit, we explored the dynamics of the system with Laplace-Lagrange secular perturbation theory (Marquis de Laplace 1825). Following the methods in Murray & Dermott (2010), we analytically derived equations for the time dependence of the inclination for each of the planets in the system. We chose to ignore effects from planet f. Due to planet f's large semi-major axis relative to the other 4 planets, the inner 4 will move together under its influence. Additionally, any of effects from planet f will play out over much longer timescales than we are interested in (∼2 orders of magnitude longer). For the four planets in question, we used the median values for mass and semi-major axis from Table 1. Within the Laplace-Lagrange framework, eccentricity and inclination become decoupled; for simplicity and consistency with our preferred RV model, we assumed circular orbits. The Laplace-Lagrange secular perturbation theory is built on the foundation of the disturbing function, where I is the inclination, j and k are planet indices that run from 0 to N with N being the number of planets in the system: where R j is the disturbing function ∂I j ∂t = − 1 n j a 2 j I j ∂R j ∂Ω j ,(11)R j = n j a 2 j 1 2 B jj I 2 j + B jk I j I k cos(Ω j − Ω k )(12) and B jk = 1 4 G(M * + m j ) a 3 j 1 2 m k M * + m j α jk α jk b (1) 3 2 (α jk ),(13) and n j = G(M * + m j ) a 3 j(14) where B jk = −B jj . Terms α jk and α jk are constants determined by semi-major axis ratios of the jth and kth planets, b 3 2 (α jk ) is a definite integral also dependent on semi-major axes (Murray & Dermott 2010), and Ω is the longitude of ascending node. From the disturbing function we constructed the B matrix. The eigenvalues of the B matrix, f k , represent the periodicity of the oscillations of the planets' inclination and the eigenvectors (which are unscaled and must be normalized) along with the initial conditions of the system's configuration represent the amplitude of the oscillations. In the normalization process we calculated both a scaling factor and a phase angle for the oscillation periodicity of each planet, γ k . This is accomplished by implementing the initial conditions at t = 0 (both I o and Ω o ) to generate a set of set of 2N equations from which we can solve for N scaling factors and N phase angles. With these scaling factors in hand, the final amplitudes of the oscillations, V jk , are determined. Then we calculated the inclinations of each planet at a given time: I j = (p 2 j + q 2 j ) 1 2 ,(15) where p j and q j are parameterized variables: p j = N planets k=0 V jk sin(f k t + γ k ),(16)q j = N planets k=0 V jk cos(f k t + γ k ).(17) Within this framework, we derived I j (t) for each planets j ∈ {b, c, d, e} for various initial configurations of the system. For each configuration, planets b, c, and d were initialized at 0 • , corresponding to placing all three on the same plane. Note that the plane from which we are measuring inclinations is 90 • transposed from the conventional plane of reference for inclinations, the sky plane. For ease of reference, we call this plane the LL-Plane. We also initialized all four planets' longitude of ascending node, Ω, to the same value, arbitrarily 0 • . We tested various trials where Ω e was initialized at different values between 0 • -360 • and found it had little to no affect on the outcome of our experiment. In each configuration we set the starting inclination for planet e to different values, stepping in 0.5 • intervals from 0 • to 12.0 • . We computed I j (t) for an 8,000 year span, roughly double the longest eigenfrequency. For every year in a configuration, we computed the mutual inclination of the three planets: cos I xy = cos I x cos I y + sin I x sin I y cos(Ω x − Ω y ), (Carter et al. 2012). We determined a maximum limiting angle for mutual transiting of the inner 3 planets by geometric reasoning. We calculated the minimum transiting inclinations for both the innermost and second innermost planets, by i min ≈ R * a . Then the sum of these two angles is the limiting angle. This corresponds to placing the innermost and second innermost planets at the opposite limbs of the star. For a given timestamp, if the mutual inclinations of all pairs of planets are less than the limiting angle, then all planets transit together at that timestamp. Figure 8 shows the I j (t) curves for two examples from our trials as well as the results of all trials. For each trial of planet e's starting inclination, we computed the percent of timestamps within the 8000 year time span during which all three of the inner planets transited with respect to an arbitrary line of sight. As expected, the farther from the LL-Plane that we start planet e's inclination, the smaller the percent of the timestamps during which all three inner planets will transit. There is a range of starting inclinations for which we would expect all three inner planets to transit 100% of the timestamps, from 0 • to 2.0 • in the LL-Plane. We nominally rule out inclinations less than 0.5 • based on the absence of a transit for planet e, although this limit does not take into account the uncertainty in planet e's radius and the simplification that all three inner planets start at 0 • . In sample tests where we included planet f with mass and semi-major axis values drawn from results in §5, we find the results to be similar. Including planet f, the value for the percentage of timestamps where the inner 3 planets are all transiting for any given inclination of planet e is within 5% of the value as when we exclude planet f. Above 2.0 • in the LL-plane, the percentage of timestamps where all three are transiting together falls sharply and then decreases asymptotically towards 0%. From these results, we conservatively place a upper limit on the planet e's mutual inclination at 10 • . This angle corresponds to a lower limit for absolute inclination of 80 • in the conventional sky-plane frame of reference. For starting inclinations above 10 • , the amplitudes of the planets' oscillations in inclination space become large enough that it is rare for all three to transit together from an arbitrary line of sight: < 10% of the timestamps tested. Mutual inclinations of planet e larger than 10 • are viable solutions. However, in those scenarios, the decreasingly short windows in time where all three planets transit make Earth observers increasingly lucky to have caught the system at one of these rare moments in its dynamical periodicity. This investigation suggests that planet e is likely to be nearly coplanar with the three transiting planets. TTVS AND MMR Planets c and d have orbital periods very near to 4:3 mean-motion resonance (MMR). But do they indeed reside in MMR? We explored this possibility and the implications which follow. In general, planets which reside in MMR are characterized by period ratios of P 2 P 1 = j j − 1 ,(19) where j is an integer and subscripts 1 and 2 denote the inner and outer planet of the pair, respectively. We quantify the "proximity" to MMR by ∆ 12 = P 2 P 1 j − 1 j − 1,(20) following Lithwick et al. (2012). Applying this formula to planets c and d, ∆ cd = 0.6432 ± 0.0001%. Following Batygin & Adams (2017), the resonant bandwidth can be approximated as: \|χ\| < ∼ 5 j − 1 j 2/3 M 1 + M 2 M * 2/3 .(21) For planets c and d, χ cd = 0.662 ± 0.001%. Because ∆ < \|χ\|, we cannot rule out that the two planets are librating in MMR. Under the assumption that planets c and d are close to but not in MMR, we calculated the period and amplitude of TTV oscillations of the pair following Lithwick et al. (2012). TTV oscillations will be oppositely-phased sinusoids, each at a period designated as the super period (SP): P SP = P 2 j \|∆\| ,(22) with amplitudes T T V 1 = P 1 ( m2 M * ) πj 2/3 (j − 1) 1/3 ∆ −f − 3Z 2∆ ,(23) and T T V 2 = P 2 ( m1 M * ) πj∆ −g − 3Z 2∆ ,(24) where f and g are constants associated with the MMR ratio, in this case 4:3, and Z is a linear combination of the free eccentricities of the two planets. We calculated the super period of planets c and d to be 1490 ± 10 days. In the circular orbit limit, Z = 0 and the amplitudes of planet c and d's TTV oscillations are 15.5 ± 9.1 minutes and 59.2 ± 13.8 minutes, respectively. If the phase of the oscillations is near an inflection point, Planet d's oscillation would be large enough that it could be detected even though TESS has only sampled about a fifth of the super period. To further investigate, we calculated the TTV associated with each transit event. We generated model transits offset from the expected transit time by between ±60 minutes and calculated the chi-squared (χ 2 ) fit of these model transits to the light curve. We adopted the offset that minimized the χ 2 statistic as the value of the TTV. The 1σ error bars are calculated from the offset where the χ 2 increased from its minimum value by 1.0. We performed this process for each transit of each planet. Figure 9 as well as Table 2 shows all of the TTVs for each planet. Planet d's 5 transits cover ∼230 days of time, or about 15% of the super period. Its TTVs do not show a trend. Planet c's transits similarly span only ∼230d. Due to TESS 's observing strategy, planet c transited just hours before sector 24 observations and hours before and after sector 25 observations, at times when the star was not visible to TESS . It is noteworthy that the two planets behave similarly in that when one is late, the corresponding transit of the other is similarly late and vice versa for early transits. Planet b's TTVs are consistent with zero, showing no trend or significant sinusoidal variation. These results can be interpreted in two ways. First, and most likely, TESS has not sampled enough of the 1500 day super period to make a conclusive finding. Alternatively, we could be sampling TTVs very near the maximum or minimum of the TTV signal's phase, so the ∆TTV over the baseline is too small for a significant detection. TESS 's extended mission cycle 4 will shed more light onto these three possibilities. Figure 9. TTVs of the transiting planets over the duration of the TESS photometry. We do not detect significant TTVs for any of the transiting planets over the observing baseline. GAP COMPLEXITY Could there be an additional planet hiding in the gap between planets b and c? With planets c and d very near MMR, it is noticeable that there are not more pairs of planets also spaced in near resonant orbits. Following the peas-in-a-pod architecture where multi-planet systems show similarly sized planets in regular orbital distance spacing, we might expect more than just one pair in this system to exhibit near-resonance, especially considering that the transiting planets have very similar radii (Leleu et al. 2021). In the residuals of our GLS periodogram (Figure 1), there is a noticeable peak between planets b and c at 17.7 days. A planet at this period would be particularly interesting as it would be near 2:1 resonance with planet b and 8:5 resonance with planet c. A planet at this period would also fill the gap in log Period space of this system well. Given that we have a strong RV detection of planet c, any additional planet in this gap between planets b and c must be less massive than planet c and inclined. When we add a fit for a 17.7d planet in our preferred model, we find a 2σ upper limit to its mass to be 6 M ⊕ . In order to be non-transiting, its inclination must be at least 2 • from the LL plane. We followed the methods in Gilbert & Fabrycky (2020) to calculate the Gap Complexity, C, for the HD 191939 system. C describes the deviation from uniform planet spacing in a system. C = 0 indicates uniform spacing in log Period space, while as C → 1 the less uniform the spacing. For Kepler systems, C peaks at 0 with the majority (∼75%) of systems having C < 0.2. Systems with larger C values are more likely to have additional planets hiding in the gaps between known planets. We calculate C HD191939 = 0.846 considering the transiting planets only, as planet e does not fall into the peas-in-a-pod configuration. We interpret the high value of C to mean that there is a significant gap, which Figure 10. All multi-planet systems with 5σ masses and radii for small planets (Rp < 10R⊕, Mp < 100M⊕) with TSMs > 20. Planets are plotted by mass and arranged vertically in order of host star effective temperature (hotter at the top). HD 191939 b and c have TSM values that are individually among the best in the sub-Neptune population, and are unique in having the same host star. Due to Planet d's weak mass measurement, it appears in this plot unfilled. HD 191939 is the only system to date with multiple planets with TSMs greater than 100 that also does not saturate JWST . . could be the site of an additional planet. When we include a hypothetical planet on a 17.7 day period with the known transiting planets, we calculate C HD191939 = 0.18. This value is consistent with the findings of Gilbert & Fabrycky (2020) for the general pattern of multi-planet system configurations. Adding a 17.7 day planet to our preferred model does not improve the likelihood enough to justify the extra three parameters. Nevertheless, this planet candidate is interesting and deserves continued attention with additional RV observations. FOLLOW UP PROSPECTS How well suited is this system for further follow up? We identified HD 191939 as a key TKS target for atmospheric follow up with the target selection algorithm described in Scarsdale et al. (in prep). As a bright (J = 7.6 mag) multi-planet system, space-based spectroscopic observations offer a unique opportunity for studies in planet formation and evolution. We use the Transmission Spectroscopy Metric (TSM; Kempton et al. 2018) to quantify the expected signal-tonoise ratio of JWST -NIRISS observations for the transiting planets: TSM p = S × R 3 p T eq M p R 2 * × 10 −0.2m J ,(25) where S is a dimensionless normalization constant, equal to 1.28 for planets 2.75 < R p < 4.0 R ⊕ . The TSM is a proxy for the expected SNR from a 10-hour observing program with JWST -NIRISS assuming a cloudfree, solar-metallicity, H 2 -dominated atmosphere. For reference, HD 3167 c, a sub-Neptune orbiting an early-K dwarf with a recent water vapor detection from five HST -WFC3 transits (Mikal-Evans et al. 2021), has a TSM of about 100. Using the derived planet parameters from Table 1, we find HD 191939 b has a TSM of 151 ± 18, which places it in the top quartile of targets in the 2.75 < R p < 4.0 R ⊕ range from the statistical sample in Kempton et al. (2018). HD 191939 c has a TSM of 106 ± 24, placing it in the third quartile from the top of TSM values for planets between 2.75 and 4.0 R ⊕ . We place a lower limit on the TSM of planet d, finding TSM d > 72 at 2-σ confidence. For the transit durations reported in Table 1, our TSM values scale to an expected single-transit SNR with JWST-NIRISS of 84 ± 10, 71 ± 16, and > 53 for planets b, c, and d respectively, where the lower limit for planet d represents 2-σ confidence. We used PandExo (Batalha et al. 2017) to estimate the nominal heights of molecular features in a singletransit JWST -NIRISS transmission spectrum for planet b, assuming a cloud-free, solar-metallicity atmosphere. In this ideal case we find feature heights of ∼100-300 ppm between 1 and 5 µm. In reality, clouds and/or enhanced atmospheric metallicity will probably reduce these amplitudes by a factor of three or more (Wakeford et al. 2019). Additionally a sub-Solar C/O ratio, which may be implied from the host star's abundance measurements, also disagrees with the ideal case of a solar-metallicity composition and would produce spectra dominated by CO, H 2 O, and CO 2 . A spin-orbit measurement for this system would be particularly informative to planetary formation theories. Only 8 systems with three or more planets have had their sky-projected obliquity angles, λ, measured. In the HD 191939 system, the three inner planets all lie in nearly the same orbital plane, while we have shown that the giant planet should lie close to this plane. If they are misaligned with respect to the stellar spin axis, that could inform the dynamical history of the system and the roles that planets e and f have played in shaping the system. However, the low v sin i (see Table 1) of the host star might be prohibitive to a Rossiter-McLaughlin (RM;Rossiter 1924;McLaughlin 1924;Gaudi & Winn 2007) measurement of even the largest expected signal from planet b. A simulation using arome (Boué et al. 2013) finds that for v sin i = 1 km/s and λ = 0 • , planet b's expected RM amplitude is 1.5 m/s. HD 191939 will be observed again by TESS in Cycle 4. Nominal dates for observations include 6 sectors of additional coverage: 41, 48, 49, 51, 52, and 55. These observations will extend the total baseline of photometry observations to 2022-09-01 for a a total of 1142 days, about 76% of the super period between planets c and d. CONCLUSIONS The overall architecture of the HD 191939 system − multiple small planets, then a warm Saturn, followed by a high mass planet − seemingly stands alone among known systems. Sub-Neptunes are near ubiquitous (Howard et al. 2012;Petigura et al. 2013), but the a priori occurrence rate for warm sub-Jovians (30 − 300 M ⊕ at 0.1−1.0 AU) is much smaller at ∼3%, and similarly at ∼5% for cold super-Jovians (300 − 6000 M ⊕ at 3−10 AU) (Fulton et al. 2021). We cannot simply multiply together these occurrence rates to discern how rare it is for such a system like HD 191939 to exist, as Weiss et al. (2018) found that adjacent planets tend to have similar sizes, and some studies have found a relationship between sub-Neptune occurrence and giant planet occurrence (Zhu & Wu 2018;Bryan et al. 2019) We searched the literature for analog systems by performing cuts on the known population for systems with 4 planets, with three sub-Neptunes (M p < 25M ⊕ ) interior to a warm Saturn (50M ⊕ < M p < 300M ⊕ , with orbital period of 50−360 days) and a long period high mass planet. However, there are a few systems that stand out as notable. Mills et al. (2019) describe three systems, Kepler-65, Kepler-68, and Kepler-25 with high mass outer planets. Kepler-65 has a tight inner system of three sub-Neptunes and a 0.28 M J planet with an orbital period of 258 days, similar to the inner system of HD 191939, but there is no evidence for a trend over a ∼2000 day baseline. Kepler-25 is similar in having two inner sub-Neptunes in/near resonance (2:1) and a Saturn mass planet at just over a 100 day orbit; but again, no evidence for a long period companion represented by trend over its ∼3000 day observing baseline. Kepler-68 may represent the most similar system to HD 191939. It has an inner system of of two sub-Neptunes, then a Jovian with an orbital period of 634 days, and then strong evidence for curvature in the residuals. Mills et al. (2019) attribute this curvature to an object with a period much longer than the ∼3000 day baseline and place a lower limit of 0.6 M J , but no upper limit. Lastly, Kepler-129 (Zhang et al. 2021) bears resemblance to HD 191939 in having two inner planets at < 45 M ⊕ and a high mass Jovian (8.3 M J ) on ∼7 year orbit. Zhang et al. (2021) also discusses the perturbations of inclinations of the inner transiting planets due to the long period Jovian. Each of these systems has pieces of the HD 191939 system, but none have the full architecture. Bright, multi-planet systems are invaluable to the exoplanet community due to their enhanced follow up opportunities and comparative planet prospects. With photometry from TESS and RV data from both Keck/HIRES and the APF, we have characterized the HD 191939 system: 3 transiting sub-Neptune planets, a fourth Jovian, and 5th high mass planet. We have measured the planets' masses, as well as their radii and densities where applicable. Because of our strong mass measurements of 3 of the 4 inner planets (>5σ), we are able to explore and further investigate many aspects of the system to answer more detailed questions about the system. Our main conclusions are as follows: • The bulk densities of the transiting planets are ρ b = 1.5 ± 0.2 g/cc, ρ c = 1.4 ± 0.3 g/cc, and ρ d = 0.5 ± 0.3 g/cc. We find the compositions of the planets are best explained by extended H/He atmospheres. • By new technique for constraining the mass and period of distant companions using both RV and astrometric data sets, we find planet f to be between 2-11 M J on a 1700-7200 day orbital period at 95% confidence. • Through a dynamical analysis using Laplace-Lagrange secular perturbation theory, we constrain the inclination of the non-transiting planet e. We find it most likely orbits within a plane less than 10 • from the plane roughly shared by the three transiting planets. • By investigation into the potential mean motion resonance of planets c and d, we predict their TTV amplitudes to be 15.5 ± 9.1 minutes and 59.2 ± 13.8 minutes, respectively over a super period of 1490 ± 10 days. However, we find no evidence for significant TTVs over the short observing baseline (326 days) compared to the super period of the interaction (1500 days). • We analyze of the RV residuals and Gap Complexity of the system to investigate the potential for additional planets in the system, identifying a possible planet candidate at 17.7 days which deserves continued attention. • We evaluate the transiting planets' prospects for atmospheric characterization through transmission spectroscopy with JWST . HD 191939 is the only system that does not saturate JWST-NIRISS where two planets both have TSMs greater than 100, making it an excellent candidate for comparative atmospheric studies. With its three transiting mini-Neptunes, one nontransiting Jovian planet, and distant high mass planet surrounding a bright, nearby host star, HD 191939 provides a rich natural laboratory for detailed atmospheric characterization and dynamical studies. Figure 1 . 1GLS periodograms of the combined time series of Keck/HIRES and APF data. The two data sets were first filtered by removing instrumental offsets as well as the trend and curvature according to the best fit parameters from our preferred model. In each descending panel, we have removed one planet at a time. The bottom panel shows the window function of the time series. Figure 2 . 2a) Our complete RV time series with our preferred model (blue) as well as b) residuals including trend and curvature. Data collected from Keck/HIRES are shown as black circles while data from the APF are shown by green diamonds. Figure 3 . 3The phase folded RV time series for each planet with periods less than our baseline. Red circles are bins of size 0.08 phase. Figure 4 . 4Phase folded light curves for each of the transiting planets with our best fit model overlaid and residuals below. Figure 5 . 5TESS photometry from sectors15-19, 21, 22, 24, and 25 highlighting the transits of the three sub-Neptunes which are indicated by color-coded arrows. Our RV model's predicted transit midpoint times for planet e are shown by vertical dashed red lines along with 3σ error windows as light red shaded regions. The predicted ∼ 8 hour transit duration (for a central transit) is shown by dark red shading. An additional transit window occurred in Sector 23 when the star was not visible in any TESS cameras. Figure 6 . 6A Mass-Radius diagram highlighting the HD 191939 transiting planets. Larger marker sizes correspond to more precise mass measurements, excluding the HD 191939 planets. Planet d's marker represents the 2σ upper limit, and its arrow points back to the median value. Grey points are from the NASA Exoplanet Archive as of 2021-07-01, with cuts to include only 2σ masses or better. Figure 7 . 7Constraints on the mass and semi-major axis of planet f. The green region shows values that are consistent with the measured RV trend and curvature. The blue region shows values that are consistent with the Hipparcos/Gaia astrometry. The red region shows the values consistent with both RV and astrometry. Dark and light regions indicate the 1 and 2σ confidence intervals, respectively. Planet f is likely between 2-11 MJ , orbiting between 2.6-7.0 AU. Figure 8 . 8Details on the Laplace-Lagrange analysis. Left: The inclination curves for each planet when planet e is given a starting value of Ie = 0.5 • vs Ie = 6.0 • . When the mutual inclination of the three is small enough for all three to transit together, the line is opaque. Right: The percent of time during which the inner three planets transit depends on the inclination of planet e. The vertical black dashed line indicates the nominal maximum inclination for which we would expect planet e to still transit. The horizontal red dashed indicates the 10% threshold for our conservative estimate on the upper limit to the giant planet's inclination. Facilities Automated Planet Finder (Levy), Keck I (HIRES), TESS Software: Astropy (Astropy Collaboration et al. 2013), corner.py (Foreman-Mackey 2016), emcee (Foreman-Mackey et al. 2013), isoclassify (Huber et al. 2017), Jupyter (Kluyver et al. 2016), KeckSpec (Rice & Brewer 2020) matplotlib (Hunter 2007), numpy (Van Der Walt et al. 2011), pandas (McKinney 2010), Python Limb Darkening Toolkit (Parviainen & Aigrain 2015) RadVel (Fulton et al. 2018), Smint (Piaulet 2020) SpecMatch-Syn (Petigura et al. 2017) Transit Least Squares (Hippke & Heller 2019) exoplanet (Foreman-Mackey et al. 2020a) and its dependencies (Agol et al. 2019; Astropy Collaboration et al. 2018; Espinoza 2018; Luger et al. 2019; Salvatier et al. 2016; Theano Development Team 2016) 12. APPENDIX Table 1 . 1System ParametersStellar Parameters Table 2 . 2Transit Mid-times Planet Epoch # Mid-time (BJD) Error (BJD)b 1 2458715.3552 0.0023 b 3 2458733.1156 0.0028 b 4 2458741.9962 0.0023 b 6 2458759.7587 0.0024 b 7 2458768.6376 0.0029 b 9 2458786.3987 0.0021 b 10 2458795.2825 0.0026 b 11 2458804.1602 0.0026 b 12 2458813.0424 0.0041 b 13 2458821.9181 0.0023 b 14 2458830.8014 0.0029 b 15 2458839.6806 0.0018 b 19 2458875.2017 0.0018 b 21 2458892.9626 0.0020 b 22 2458901.8430 0.0020 b 23 2458910.7222 0.0023 b 24 2458919.6017 0.0024 b 29 2458964.0028 0.0036 b 30 2458972.8837 0.0021 b 31 2458981.7649 0.0021 b 32 2458990.6450 0.0020 b 33 2458999.5248 0.0029 b 34 2459008.4056 0.0023 c 1 2458726.0546 0.0033 c 2 2458754.6340 0.0028 c 3 2458783.2116 0.0031 c 4 2458811.7992 0.0031 c 5 2458840.3758 0.0029 c 7 2458897.5359 0.0038 c 8 2458926.1168 0.0031 d 1 2458743.5531 0.0038 d 2 2458781.9029 0.0029 d 3 2458820.2645 0.0031 d 5 2458896.9613 0.0038 d 6 2458973.6648 0.0031 Table 3 . 3Radial Velocity Time Series The full data set in a machine readable format is available online.BJD RV (m/s) RV err (m/s) S-Value S-Value err Instrument 2458795.832 -20.961 1.225 0.146 0.001 HIRES 2458802.800 -9.760 1.291 0.146 0.001 HIRES 2458815.779 -14.967 1.241 0.142 0.001 HIRES 2458834.647 1.901 3.891 0.141 0.002 APF 2458834.661 8.670 3.813 0.145 0.002 APF 2458837.734 -6.977 4.347 0.176 0.002 APF ACKNOWLEDGMENTSWe thank the anonymous referee for their insightful and thorough comments. We are grateful to Tim Brandt for his insight and contributions to the methods of §5. We thank the time assignment committees of the University of California, the California Institute of Technology, NASA, and the University of Hawai'i for supporting the TESS-Keck Survey with observing time at Keck Observatory and on the Automated Planet Finder. We thank NASA for funding associated with our Key Strategic Mission Support project. We gratefully acknowledge the efforts and dedication of the Keck Observatory staff for support of HIRES and remote observing. We recognize and acknowledge the cultural role and reverence that the summit of Maunakea has within the indigenous Hawaiian community. We are deeply grateful to have the opportunity to conduct observations from this mountain. We thank Ken and Gloria Levy, who supported the construction of the Levy Spectrometer on the Automated Planet Finder. We thank the University of California and Google for supporting Lick Observatory and the UCO staff for their dedicated work scheduling and operating the telescopes of Lick Observatory. This paper is based on data collected by the TESS mission. Funding for the TESS mission is provided by the NASA Explorer Program. We acknowledge the use of public TESS data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. This paper includes data collected by the TESS mission that are publicly available from the Mikulski Archive for Space Telescopes (MAST). Astrometric and photometric star catalogues derived from the ESA HIPPARCOS Space Astrometry Mission. ESA Special Publication. 1200The HIPPARCOS and TYCHO catalogues, ESA Special Publication, Vol. 1200, The HIPPARCOS and TYCHO catalogues. Astrometric and photometric star catalogues derived from the ESA HIPPARCOS Space Astrometry Mission . 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[ "The α ′ Expansion On A Compact Manifold Of Exceptional Holonomy", "The α ′ Expansion On A Compact Manifold Of Exceptional Holonomy" ]	[ "Katrin Becker \nGeorge P. and Cynthia W\nMitchell Institute for Fundamental Physics and Astronomy\nTexas A& M University\n77843-4242College StationTXUSA\n", "Daniel Robbins \nInstitute for Theoretical Physics\nUniversity of Amsterdam\n94485, 1090 GLPostbus, AmsterdamThe Netherlands\n", "Edward Witten \nSchool of Natural Sciences\nInstitute for Advanced Study\nDepartment of Physics\nUniversity of Washington\nEinstein Drive08540, 98195Princeton, SeattleNJ, WashingtonUSA, USA\n" ]	[ "George P. and Cynthia W\nMitchell Institute for Fundamental Physics and Astronomy\nTexas A& M University\n77843-4242College StationTXUSA", "Institute for Theoretical Physics\nUniversity of Amsterdam\n94485, 1090 GLPostbus, AmsterdamThe Netherlands", "School of Natural Sciences\nInstitute for Advanced Study\nDepartment of Physics\nUniversity of Washington\nEinstein Drive08540, 98195Princeton, SeattleNJ, WashingtonUSA, USA" ]	[]	In the approximation corresponding to the classical Einstein equations, which is valid at large radius, string theory compactification on a compact manifold M of G 2 or Spin(7)holonomy gives a supersymmetric vacuum in three or two dimensions. Do α ′ corrections to the Einstein equations disturb this statement? Explicitly analyzing the leading correction, we show that the metric of M can be adjusted to maintain supersymmetry. Beyond leading order, a general argument based on low energy effective field theory in spacetime implies that this is true exactly (not just to all finite orders in α ′ ). A more elaborate field theory argument that includes the massive Kaluza-Klein modes matches the structure found in explicit calculations. In M-theory compactification on a manifold M of G 2 or Spin(7) holonomy, similar results hold to all orders in the inverse radius of M -but not exactly. The classical moduli space of G 2 metrics on a manifold M is known to be locally a Lagrangian submanifold of H 3 (M, R) ⊕ H 4 (M, R). We show that this remains valid to all orders in the α ′ or inverse radius expansion.	10.1007/jhep06(2014)051	[ "https://arxiv.org/pdf/1404.2460v1.pdf" ]	53,526,085	1404.2460	95912373eae96d1210f20bab127ed6eff14c51bb	The α ′ Expansion On A Compact Manifold Of Exceptional Holonomy 9 Apr 2014 Katrin Becker George P. and Cynthia W Mitchell Institute for Fundamental Physics and Astronomy Texas A& M University 77843-4242College StationTXUSA Daniel Robbins Institute for Theoretical Physics University of Amsterdam 94485, 1090 GLPostbus, AmsterdamThe Netherlands Edward Witten School of Natural Sciences Institute for Advanced Study Department of Physics University of Washington Einstein Drive08540, 98195Princeton, SeattleNJ, WashingtonUSA, USA The α ′ Expansion On A Compact Manifold Of Exceptional Holonomy 9 Apr 2014 In the approximation corresponding to the classical Einstein equations, which is valid at large radius, string theory compactification on a compact manifold M of G 2 or Spin(7)holonomy gives a supersymmetric vacuum in three or two dimensions. Do α ′ corrections to the Einstein equations disturb this statement? Explicitly analyzing the leading correction, we show that the metric of M can be adjusted to maintain supersymmetry. Beyond leading order, a general argument based on low energy effective field theory in spacetime implies that this is true exactly (not just to all finite orders in α ′ ). A more elaborate field theory argument that includes the massive Kaluza-Klein modes matches the structure found in explicit calculations. In M-theory compactification on a manifold M of G 2 or Spin(7) holonomy, similar results hold to all orders in the inverse radius of M -but not exactly. The classical moduli space of G 2 metrics on a manifold M is known to be locally a Lagrangian submanifold of H 3 (M, R) ⊕ H 4 (M, R). We show that this remains valid to all orders in the α ′ or inverse radius expansion. Let M be a compact seven-manifold of G 2 holonomy. Compactification on M gives a classical solution of ten-dimensional supergravity, with unbroken supersymmetry in three dimensions. Is the analogous statement true in string theory, allowing for α ′ corrections? Differently put, corresponding to a classical solution of Einstein's equations with G 2 holonomy, is there a superconformally invariant two-dimensional σ-model with target M? To be more precise, we consider the question in the context of Type II superstring theory (or somewhat similarly, the heterotic string with the spin connection embedded in the gauge group). Then the question is whether there is a family of σ-models (without torsion) with target M and (1, 1) superconformal symmetry, depending on the moduli of the classical G 2 metric. The question arises because [1] the α ′ expansion in σ-model perturbation theory generates corrections to the Einstein equations and to the conditions for supersymmetry. A "yes" answer means that the classical metric of G 2 holonomy can be modified to compensate for these corrections. This natural question appears not to have been addressed in the literature. A somewhat different question has been answered [2]: if such a σ-model exists, what is its chiral algebra? The answer involves an interesting extension of the N = 1 superconformal algebra in two dimensions. Also, the leading α ′ correction has been analyzed in some examples [3]. Explicit formulas were found showing that the leading correction does not destroy spacetime supersymmetry in these examples. 1 In the context of compactification on a six-dimensional Calabi-Yau manifold X, there is a superficially similar question: can a classical metric of SU(3) holonomy be corrected to compensate for the modifications of Einstein's equations that arise in σ-model perturbation theory and so to maintain spacetime supersymmetry? This question has a nice answer from the point of view of the two-dimensional σ-model. One uses the fact that (in a σ-model without torsion 2 ) the condition for (2, 2) worldsheet supersymmetry, without assuming conformal invariance, is that the target space X should be Kahler. Moreover, it is possible to regularize the σ-model preserving (2,2) supersymmetry. Hence in analyzing the renormalization group flow on the metric of X that is induced by σ-model corrections, one can consider only flows in the space of Kahler metrics. At one-loop order, one meets the classical Einstein equations, and this is where the Calabi-Yau condition comes in. What about the higher order corrections? They actually give a flow in the space of Kahler potentials (that is, a flow that keeps fixed the Kahler class of the target space metric). Once one knows this, one can easily argue [5] that the metric of X can be corrected order by order in σ-model perturbation theory (and even exactly) to maintain superconformal invariance and therefore spacetime supersymmetry. 3 Unfortunately, we have not been able to generalize this argument for a manifold M of G 2 holonomy. Roughly, to do this one would want a global or "off-shell" version of the chiral algebra that was described in [2]. In other words, one would want to identify part of this structure (analogous to global (2,2) supersymmetry in the Calabi-Yau case) that can be preserved in the presence of a suitable regulator, so that it is valid in σmodel perturbation theory. For example, it might be that with some regularization, the renormalization group flow of the σ-model with target M takes place only in the space of metrics that can be derived from a (not necessarily torsion-free 4 ) G 2 -structure derived from a closed three-form φ with a fixed cohomology class. This would be analogous to the fact that in the Calabi-Yau case, the flow (with a regularization that preserves global (2, 2) supersymmetry) takes place only in the space of Kahler forms with a fixed 1 The examples in question are explicit, complete but non-compact manifolds of G 2 holonomy. In the present paper, we phrase our statements in terms of compact manifolds of exceptional holonomy to avoid some analytical details, but one expects our main results to carry over to a large class of complete but not compact examples. 2 There are more general (2, 2) models with torsion [4], but we need not consider them here. 3 As a statement about worldsheet superconformal invariance, this argument actually works for Calabi-Yau n-folds for any n. 4 A G 2 -structure on a seven-manifold M is a three-form φ that obeys mild inequalities which ensure that the formula (A.9) does define a Riemannian metric. The G 2 -structure is said to be torsion-free (and in this case M is called a G 2 manifold) if the metric has G 2 holonomy. cohomology class. Given such a statement, perhaps one could imitate the argument in [5]. But we have been unable to find such a statement. In the Calabi-Yau case, an alternative argument uses elementary properties of the spacetime effective action to show that α ′ corrections do not destroy spacetime supersymmetry. A version of the argument 5 appropriate to the heterotic string, in which Calabi-Yau compactification preserves N = 1 supersymmetry in spacetime, proceeds as follows [6]. To disturb N = 1 supersymmetry, one must generate a correction either to the spacetime superpotential or to the Fayet-Iliopoulos (FI) D-terms. Simple arguments based on scaling and holomorphy show that at string tree level, α ′ corrections cannot generate either of these effects. 6 Similar reasoning applies to Type II superstring theory on a Calabi-Yau manifold. In this case, one has N = 2 supersymmetry in spacetime, which can only be disturbed by FI terms (there is no analog of the superpotential), and after disposing of these terms, one learns that spacetime supersymmetry is unbroken in the full quantum string theory. In Type II superstring theory on a manifold of G 2 holonomy, a similar argument based on holomorphy of the spacetime effective action shows that σ-model corrections cannot disturb spacetime supersymmetry. Such a model has N = 2 supersymmetry in three-dimensional spacetime, which is similar to N = 1 in four dimensions, and a priori spacetime supersymmetry might be disturbed by a correction to the superpotential or by FI D-terms. Holomorphy ensures that α ′ corrections cannot correct the spacetime superpotential: the superpotential is a function of chiral superfields whose imaginary parts are RR fields or axion-like modes from the NS-NS sector, all of which decouple at zero momentum in σ-model perturbation theory. And since the gauge fields of Type II superstring theory on a G 2 manifold arise from the RR sector, their FI terms would actually violate a symmetry of σ-model perturbation theory (the symmetry (−1) F L that counts left-moving fermions mod 2). The reason for the present paper is that although we consider the argument in the last paragraph to be satisfactory, we wanted to understand what happens more explicitly. We begin in section 2.1 by considering the first non-trivial α ′ correction to the Einstein equations and the conditions for unbroken supersymmetry in Type II compactification on a G 2 manifold. We show explicitly that the metric can always be corrected to maintain spacetime supersymmetry in this order. We find that the same will be true in higher orders if a certain four-form α and five-form β (which can be computed order by order in α ′ ) are always exact. We do not know how to show this directly from σ-model considerations. However, we show that the spacetime arguments mentioned in the last paragraph amount to predicting the exactness of a certain four-form and five-form. We expect these to coincide with the α and β that come from σ-model perturbation theory. An obvious question is to consider instead compactification on an eight-manifold of Spin(7) holonomy. In section 3, we show explicitly that once again in this case, the leading α ′ correction to the supersymmetry transformations can be compensated by adjusting the metric to maintain spacetime supersymmetry. Simple arguments based on the spacetime effective action predict that this result must persist in higher orders. For example, Type 5 This argument is not restricted to the case of embedding the spin connection in the gauge group; it applies to the larger class of supersymmetric heterotic string compactifications described in [6]. 6 For similar reasons, string loop corrections cannot generate a superpotential [7,8], although they can generate the FI terms at one-loop order [9]. Nonperturbatively in the string coupling, spacetime instantons can and typically do generate a nonperturbative superpotential in these models. Nonperturbatively in α ′ , the same is true of worldsheet instantons in the general class of models mentioned in footnote 5, though not in models constructed via the standard embedding of the spin connection in the gauge group. IIA compactification on a Spin(7) manifold gives a model with (1, 1) supersymmetry in two dimensions. In this model, the symmetry (−1) F L acts as a discrete R-symmetry that -when combined with the decoupling of RR fields of zero momentum in σ-model perturbation theory -ensures that α ′ corrections cannot break spacetime supersymmetry. We elaborate on such arguments in section 3.3. These questions have some obvious further generalizations. One can consider Mtheory compactifications to four or three dimensions on a G 2 or Spin(7) manifold M. The analog of the α ′ expansion is an expansion in powers of 1/r, with r the radius of M. In each of these cases, supersymmetry is maintained to all finite orders 7 in 1/r. On a G 2 manifold, holomorphy of the superpotential together with decoupling of the C-field at zero momentum leads to essentially the same argument as in the case of Type II superstring theory on a G 2 manifold. On a Spin(7) manifold, as explained in section 3, one makes much the same argument as in the Type IIA case, using a reflection symmetry in the non-compact directions instead of its string theory reduction, which is (−1) F L . Finally, one can ask about the heterotic string on a manifold of G 2 or Spin(7) holonomy. There are in the supergravity limit many supersymmetric compactifications that are not obtained by simply embedding the spin connection in the gauge group in the usual fashion. 8 Replacing Type II superstrings by the heterotic string reduces the spacetime supersymmetry, and an argument based on holomorphy is not available. Moreover, there is no useful R-symmetry. So we expect that in this class of heterotic string compactifications, α ′ corrections do spoil spacetime supersymmetry. G Holonomy Manifolds Preliminaries We consider Type II (either IIA or IIB) string theory in a spacetime of the form R 2,1 ×M, where M is a compact seven-dimensional manifold. We will not turn on fluxes 9 , and we wish to preserve a minimal amount of supersymmetry in three dimensions. The condition for finding a supersymmetry generator leaving the vacuum invariant imposes strong constraints on spacetime. 10 For example, within supergravity it requires the existence of a covariantly constant spinor η on M. This implies that M is Ricci-flat, the three-form φ abc = iη T Γ abc η, (2.1) and the dual four-form ψ = ⋆φ are covariantly constant, and the holonomy group of M is G 2 . Such a spacetime solves the Einstein equations, and moreover the string theory beta-functions vanish on such a solution, up to three-loop order. However, the four-loop correction to the beta-function for the metric does not vanish, in general, for a G 2 -holonomy space. We would like to show that, as in the Calabi-Yau 7 In the G 2 case, nonperturbatively in 1/r, instantons derived from wrapped M2-branes generate a spacetime superpotential [18] that triggers supersymmetry breaking. 8 Instead, the gauge field obeys an appropriate equation associated to spacetime supersymmetry. On a G 2 manifold, this equation is π 7 (F ) = 0, where F is the Yang-Mills field strength and π 7 is the projector onto two-forms that transform in the 7 of G 2 . On a Spin(7) manifold, the equation is again π 7 (F ) = 0, or alternatively ⋆F + Φ ∧ F = 0, where ⋆ is the Hodge star and Φ is the covariantly constant four-form, suitably normalized. These are the analogs of the hermitian Yang-Mills (or Donaldson-Uhlenbeck-Yau) equation that was employed in [6]. 9 For some analysis including G-flux at the classical level, see [10]. 10 In what follows, indices M, N, . . . , µ, ν, . . . and a, b, . . . are tangent to the ten-, three-and sevendimensional spaces, respectively. case, we can always find a globally-defined α ′ -dependent metric, which is close to the Ricci-flat metric, and which solves the equations to all orders in α ′ . Locally, the moduli of the G 2 metric on M are given by the cohomology class of the three-form φ. This cohomology class takes values in a certain cone C ⊂ H 3 (M, R), which in general is not well-understood. The analysis of this paper holds for any choice of the cohomology class of φ, within the cone C. Leading Order Correction Since in ten-dimensions the supersymmetry algebra only closes on-shell, a correction to the equations of motion will, in general, lead to corrections to the supersymmetry variations. To parametrize these corrections, it is useful to recall that in seven dimensions, the spinor representation is a real representation of dimension 8. Suppose we have a nowhere-vanishing spinor field η on M, which we can normalize so that η T η = 1. Let Γ a , a = 1, . . . , 7 be the Dirac matrices, obeying {Γ a , Γ b } = δ ab . One cannot choose the Γ a to be real, but one can choose them to be purely imaginary (so that the SO(7) generators Γ ab = 1 2 [Γ a , Γ b ] are real) and we will do so. Any other real spinor ψ can then be expanded in the basis {η, iΓ a η}, a = 1, . . . , 7. Explicitly ψ = η η T ψ + Γ a η η T Γ a ψ . (2.2) In particular, by expanding in this basis, we can encode any possible corrections to the transformation law of the ten-dimensional gravitino under spacetime supersymmetry in terms of two tensors A a and B a b : δψ a = ζ ⊗ D a η + A a η + iB b a Γ b η . (2.3) Here ζ is a three-dimensional spinor, which we can take to be constant (in looking for Lorentz-invariant solutions in three dimensions). To order α ′3 , the corrections to the supersymmetry transformation were presented in ref. [3]: A a = 0, B a b = c 2 α ′3 φ acd ∇ c Z db , (2.4) where Z ab is the symmetric tensor built out of three Riemann tensors Z ab = 1 32g ǫ ac 1 ···c 6 ǫ bd 1 ···d 6 R c 1 c 2 d 1 d 2 R c 3 c 4 d 3 d 4 R c 5 c 6 d 5 d 6 , (2.5) c is a constant, and g = det g ab . Given the α ′ -corrected supersymmetry transformations, next we construct a supersymmetric solution perturbatively in α ′ . Henceforth we label quantities of the corrected space with primes while unprimed quantities are unperturbed. So the α ′ -corrected internal space is denoted by M ′ . To preserve supersymmetry we must find a globally-defined spinor η ′ on M ′ which equals η to leading order in α ′ and obeys the appropriate α ′deformed equation. Since A, B are already O(α ′3 ), the vanishing of the variation (2.3) to order α ′3 becomes D ′ a η ′ + A a η + iB b a Γ b η = 0. (2.6) If we are given such an η ′ , even just defined locally, it is possible to construct a tensor φ ′ abc = iη ′T Γ ′ abc η ′ ,(2.7) (defining what is called a G 2 -structure that may have torsion) and an associated metric 11 g ′ ab . We also have ψ ′ abcd = η ′T Γ ′ abcd η ′ , (2.8) or equivalently ψ ′ = ⋆ ′ φ ′ , where the Hodge star is constructed from g ′ ab . Eq. (2.6) can be converted into conditions on φ ′ and ψ ′ by taking derivatives of eqns. (2.7) and (2.8). We define the four-form α and five-form β α = dφ ′ , β = dψ ′ ,(2.9) which are explicitly given by α abcd = 8A [a φ bcd] − 8B e [a ψ bcd]e , β abcde = 10A [a ψ bcde] − 40B [ab φ cde] . (2.10) This establishes a connection between A a and B a b and failure of φ ′ and ψ ′ to be closed (and thus to the torsion forms defined in proposition 1 of ref. [11]). With a view to the generalization beyond order α ′3 , we have written the correction in eqn. (2.6) in terms of A a and B a b . So far these tensors could be anything. However, from eqs. (2.9) and (2.10) we see that a necessary condition on A a and B a b for the existence of a G 2 -structure space M ′ close to M is that the forms α and β are exact. As we show next this condition is also sufficient. To order α ′3 , we can use the explicit expressions above to show that α = dχ and β = dξ, with χ abc = −cφ abc Z + 3cφ d [ab Z c]d , ξ abcd = −4cψ e [abc Z d]e ,(2.11) where Z = Z a a . Thus α and β are exact to order α ′3 . Existence Of A Solution Our task is now to find a globally-defined G 2 -structure φ ′ (and its associated metric g ′ and four-form ψ ′ ) which is close to φ, i.e. φ ′ = φ + δφ,(2.12) and solves the equations dφ ′ = α, dψ ′ = β,(2.13) for given exact forms α = dχ and β = dξ. We can satisfy the first equation by 12 φ ′ = φ + χ + db,(2.14) for some two-form b. What about the second one? To leading order in fluctuations the dual four-form ψ ′ = ⋆ ′ φ ′ satisfies dψ ′ = d ⋆ 4 3 π 1 + π 7 − π 27 (χ + db) . (2.15) Here π 1 , π 7 , π 27 are the projections of three-forms onto ∧ 3 1 , ∧ 3 7 and ∧ 3 27 respectively (these are the subspaces of ∧ 3 that transform in the indicated representations of G 2 ; see the appendix for concrete expressions). Deformations of G 2 -structures have been studied in the literature (see, for example, [12,13,14]), and the tools developed in that context are useful in what follows. All we need to know to derive eqn. (2.15) is described in the appendix. Besides the explicit δφ appearing in φ ′ , there is also implicit δφ dependence in ⋆ ′ since the metric is a functional of φ ′ . Taking this fact into account one easily derives eqn. (2.15). We get from (2.15) an equation for b, which can be written as d † π 27 − π 7 − 4 3 π 1 db = d † ρ, ρ = − ⋆ ξ − π 27 − π 7 − 4 3 π 1 χ. (2.16) Here d † = − ⋆d⋆ in acting on three-forms in seven dimensions. To proceed further, we decompose b into irreducible representations of G 2 . The space of two-forms decomposes as ∧ 2 = ∧ 2 7 ⊕ ∧ 2 14 , where 7 and 14 are the representations of G 2 of the indicated dimension. Actually, the component of b in ∧ 2 7 does not contribute to the right hand side of eqn. (2.15), as is shown in eqn. (A.31) of the appendix. Hence we can assume that b ∈ ∧ 2 14 . The reason that the part of b in ∧ 2 7 does not contribute to the equation for unbroken supersymmetry is that for b ∈ ∧ 2 7 , db is the change in φ generated by an infinitesimal diffeomorphism. Indeed, a general section of ∧ 2 7 is b ab = φ ab c v c for some vector field v c , and the corresponding change in the metric is δg ab = 2∇ (a v b) , which is the first-order change in the metric generated by v. It is also true that d † ρ ∈ ∧ 2 14 . This elementary but somewhat tricky fact is explained in the appendix (see also eqn. (2.51) for another point of view). We want to show that equation (2.16) has a solution for b. To do this, first let ∆ = d † d + dd † be the Hodge-de Rham Laplacian, and observe that it is possible to solve the equations ∆b = d † ρ, d † b = 0. (2.17) Indeed, standard Hodge theory says that a two-form b obeying these equations always exists for any three-form ρ on any compact manifold, since d † ρ is orthogonal to the harmonic two-forms. Moreover, on a G 2 manifold, ∆ preserves the decomposition ∧ 2 = ∧ 2 14 ⊕ ∧ 2 7 , so a solution exists with b ∈ ∧ 2 14 . We can write the first equation in (2.17) as d † (π 27 + π 7 + π 1 ) db = d † ρ, (2.18) since ∧ 3 = ∧ 3 27 ⊕ ∧ 3 7 ⊕ ∧ 3 1 . But actually, the π 1 and π 7 terms do not contribute in eqn. (2.18). π 1 does not contribute because for any b ∈ ∧ 2 14 , π 1 db = 0; this is because the representation 1 of G 2 does not appear in the decomposition of 7 ⊗ 14, so it does not appear in the first derivatives of b. Also, if d † b = 0, one has π 7 (db) = 0; this is because the 7 of G 2 appears only once in the decomposition of 7 ⊗ 14, so there is essentially only one way to form linear combinations of the first derivatives of b transforming as the 7 of G 2 , and hence π 7 (db) is linear in d † b ∈ ∧ 1 = ∧ 1 7 . Since the π 1 and π 7 terms do not contribute, eqn. (2.18) is equivalent to eqn. (2.16), which we wished to solve. All Orders In α ′ Next we describe the generalization to arbitrary orders in α ′ . We are given supersymmetry transformations δψ a = ζ ⊗ D ′ a η ′ + A a η ′ + iB a b Γ ′ b η ′ . (2.19) Here A, B are functionals of φ ′ : A a = A a [φ ′ ], B a b = B a b [φ ′ ]. (2.20) This time we include all orders in the derivative expansion. So B[φ ′ ] = ∞ n=3 B n [φ ′ ], (2.21) where each B n [φ ′ ] is a local covariant expression (or functional) constructed out of 13 φ ′ and its associated metric g ′ , Riemann tensor R ′ , and covariant derivatives ∇ ′ , and where B n [φ ′ ] contains 2n + 1 explicit derivatives (with each Riemann tensor counting two and each covariant derivative counting one). There is a similar expansion for A a . We can then construct φ ′ and its dual ψ ′ = ⋆ ′ φ ′ φ ′ = η ′T Γ ′ abc η ′ , ψ ′ = η ′T Γ ′ abcd η ′ ,(2.22) and use the condition for unbroken supersymmetry to compute dφ ′ = α[φ ′ ], dψ ′ = β[φ ′ ]. (2.23) Explicitly α abcd = 8A [a φ ′ bcd] − 8B e [a ψ ′ bcd]e , β abcde = 10A [a ψ ′ bcde] − 40B [ab φ ′ cde] . (2.24) Next we wish to construct the supersymmetric background perturbatively in α ′ . We denote the order n term of φ ′ by φ n . We proceed by induction. In our previous analysis, we constructed φ 3 explicitly. We assume the G 2 -structure is known up to order n − 1 in α ′ . We denote this G 2 -structure byφ = φ + n−1 k=3 φ k ,(2.25) and we construct the order n contribution to φ ′ as a small perturbation aroundφ φ ′ =φ + δφ, (2.26) 13 In the α ′ or 1/r expansion, the effective action and (therefore) the supersymmetry transformations are constructed in terms of g ′ without reference to φ ′ , so actually B can be constructed from the metric g ′ and its derivatives, without reference to φ ′ . (Note that g ′ can be expressed in terms of φ ′ via (A.9), but not the other way around. In the tangent space at any point, g ′ has SO(7) symmetry and φ ′ reduces the symmetry to G 2 .) To leading order, the expression for B in terms of the metric was given in [3]. where δφ = φ n . This deformation ofφ leads to a deformation of ψ ′ ψ ′ =ψ + δψ, (2.27) whereψ =⋆φ and δψ =⋆ 4 3π 1 +π 7 −π 27 δφ, (2.28) where⋆ andπ are the Hodge dual and projection operators with respect to the G 2structureφ. We will label the term of order n in α ′ of any quantity by \| n and from the above we see that the order n of ψ ′ is ψ ′ \| n =⋆φ \| n + ⋆ 4 3 π 1 + π 7 − π 27 φ n ,(2.29) Note thatφ includes terms only up to order n − 1 but⋆ andψ can, in general, receive contributions of any order since they are non-linear functionals ofφ. Let us suppose that the dependence of A a and B a b on φ ′ is such that order by order in α ′ the forms α and β are exact. (In section 2.4, we interpret this statement in terms of effective field theory.) Then there exist globally-defined χ n and ξ n with α[φ ′ ] \| n = dχ n , β[φ ′ ] \| n = dξ n . (2.30) Note that since α and β are already order (α ′ ) 3 explicitly, we can view χ n and ξ n as being functionals ofφ; they do not depend on φ n . By setting φ n = χ n + db n , (2.31) we solve the part of the first equation in eqn. (2.23) that is of the order n in α ′ . The second equation in (2.23) turns into a partial differential equation for b n . Indeed, we take the exterior derivative of (2.29) and set dψ ′ \| n = dξ n , so dξ n = d(⋆φ) \| n +d ⋆ 4 3 π 1 + π 7 − π 27 (χ n + db n ). (2.32) This can be recast in the form ∆b n = d † ρ n , d † b n = 0, (2.33) with ⋆ρ n = ξ n − ⋆ 4 3 π 1 + π 7 − π 27 χ n −⋆φ \| n . (2.34) In complete analogy to the leading order case, any piece b n ∈ ∧ 2 7 drops out of eqn. (2.33). Taking b n ∈ ∧ 2 14 , we obtain eqn. (2.33) after choosing the gauge d † b n = 0. The source is co-exact, and satisfies π 7 (d † ρ n ) = 0 (this is shown in Appendix A.1.5), and the remaining steps follow word by word the reasoning we used to leading order in α ′ . Interpretation In Three-or Four-Dimensional Field Theory Superfields We aim here to interpret the above results in the effective field theory that arises by compactification of Type IIA or Type IIB superstring theory on a G 2 manifold M. This is a theory with N = 2 supersymmetry (four supercharges) in three dimensions. In studying this theory, we will go beyond low energy effective field theory and include Kaluza-Klein harmonics in a way that preserves three-dimensional supersymmetry. We should warn the reader that more work is needed to fully justify the way we do this. Our analysis is somewhat speculative. We will make the analysis for Type IIA and leave the analogous story for Type IIB for the future. Rather than the α ′ expansion of Type IIA on R 3 × M, we can in a similar way study the 1/r expansion of M-theory on R 4 × M (here r is the radius of M). It is not completely trivial that the analysis is the same for Type IIA and for M-theory, because in general N = 2 theories in three dimensions, there can be supersymmetric interactions (Chern-Simons couplings of vector multiplets, for instance) that do not arise by classical dimensional reduction from four dimensions. If these were important, the Type IIA and M-theory analyses would be essentially different. However, the supersymmetric interactions that will be relevant are the most basic ones related to superpotentials and Kahler potentials for chiral superfields, and these are possible in four dimensions. Hence at the general level of the following discussion, the constraints on the α ′ expansion of Type IIA are the same as those on the 1/r expansion of M-theory and the structure we will find applies to each. Our analysis is limited to finite orders in α ′ or 1/r because we assume locality along M, which nonperturbatively in α ′ is violated by worldsheet instantons (Type IIA), and nonperturbatively in 1/r is violated by M2-brane instantons (M-theory). What happens nonperturbatively in 1/r is quite different from what happens nonperturbatively in α ′ . In M-theory, the M2-brane instantons violate a certain shift symmetry (adding a harmonic form to the C-field) and generate a spacetime superpotential [18] that destabilizes the R 4 × M compactification. In Type IIA, the worldsheet instantons respect the relevant shift symmetry 14 and a simple argument given in the introduction shows that, to all finite orders in the string coupling constant g st , the R 3 × M compactification remains supersymmetric. 15 In particular, setting g st = 0, one expects an exact superconformal field theory describing G 2 compactification. The more detailed analysis we give here, which aims to explain what we have found in sections 2.1-2.3, assumes locality along M and so is valid only to all finite orders in α ′ . Exploiting three-or four-dimensional supersymmetry as well as locality along M, we will try to describe Type IIA or M-theory compactification on M in terms of three-or four-dimensional superfields that are also functions, or forms, along M. This part of the analysis will be more transparent in the M-theory language. Once we have identified the relevant superfields, we will phrase our discussion in terms of the α ′ (rather than 1/r) expansion. The bosonic fields of eleven-dimensional supergravity consist of the metric tensor 16 g M N and the three-form field C M N P . In compactification on M, these fields will give the propagating bosonic modes of four-dimensional supermultiplets. We can see what these supermultiplets must be as follows: (1) The part g µν (x; y) of the metric tensor (here x is a coordinate along R 4 and y along M) gives fields of spin 2 or 0 on R 4 that are scalar functions on M. Clearly, the Kaluza-Klein expansion of M-theory on R 4 × M contains massive spin 2 supermultiplets. We will not try to understand these multiplets here, though this will have to be part of a full understanding. (2) There are two sources of spin 1 fields along R 4 that will be the bosonic parts of vector multiplets. From the C-field, we get the components C µab (x; y), which we interpret as the bosonic part of a vector multiplet V ab that is a two-form along M (and a vector multiplet in R 4 ). Similarly, the components g µ a of the eleven-dimensional metric give vector multiplets V a that comprise a vector field along M. (3) Taking advantage of the fact that M-theory on R 4 × M is invariant under a reflection of R 4 combined with a sign change of C, spin zero fields in this theory can be classified as scalars or pseudoscalars. Pseudoscalars come from the part C abc (x; y) of the C-field, which gives us a pseudoscalar field on R 4 that is a three-form on M. Scalars come from the part g ab of the metric and also from the part C µνa of the C-field (here one must recall that a two-form on R 4 , such as C µνa , is dual to a field of spin 0). We expect all these fields to combine to the propagating modes of a field C abc that will be a chiral superfield on R 4 and a three-form on M. The bottom component of C abc is a complex field of spin 0 that we will call C abc . We expand C abc (x; y) = φ abc (x; y) + iC abc (x; y), where the imaginary part is the pseudoscalar field C abc , and the real part φ abc is constructed from g ab and (the dual of) C µνa . Concerning this last point, we note that in expansion around a metric of G 2 holonomy, a three-form C abc transforms as 1 ⊕ 7 ⊕ 27, while a perturbation in the metric g ab transforms as 1 ⊕ 27, and the dual of 17 C µνa transforms as 7. So the pieces are there for the scalar and pseudoscalar fields to combine properly into the complex three-form C abc . The reason that we denote the real part of C abc as φ is that in the classical limit (1/r → 0 or α ′ → 0), expanding around a metric of G 2 holonomy, φ will coincide with the covariantly constant three-form φ associated to the G 2 metric. Thus φ will be an α ′ -corrected version of φ. φ will be the analog in our present analysis of the α ′ -corrected three-form that was called φ ′ in section 2.2. The relation between φ ′ and φ will be the subject of section 2.4.4. We can use the chiral multiplets and vector multiplets that have just been introduced to describe M-theory on R 4 × M, to all finite orders in 1/r, in a way that exhibits supersymmetry (and locality) along R 4 and locality along M. The same set of superfields can be used to similarly describe Type IIA on R 3 × M. Rather than always repeating ourselves, we formulate the following statements in terms of Type IIA. In what follows, the goal will be to describe possible supersymmetric vacua. In such a vacuum, the fields are independent of x. So we can drop the dependence of the fields on x (that is, on R 3 or R 4 ) and concentrate on the dependence on y. Conditions For Unbroken Supersymmetry In general, in a globally supersymmetric four-dimensional theory of chiral multiplets Φ and vector multiplets V ζ , or a three-dimensional theory of such multiplets with no Chern-Simons couplings for the vector multiplets, 18 the condition for unbroken supersymmetry is δW = 0 = D ζ , where δW is the variation of the spacetime superpotential W (Φ), and D ζ are the auxiliary fields in the vector multiplets V ζ . After coupling to supergravity, to get unbroken supersymmetry in Minkowski spacetime, one additionally needs W = 0. In the present context, we can easily make explicit the conditions W = δW = 0. W will have to be a holomorphic function of the three-form field C. It must be constructed without use of a metric on M (since the metric is a function of the real part of C and thus is a non-holomorphic function of C). So up to a constant multiple, the superpotential must be W = M C ∧ dC. (2.35) The condition δW = 0 is thus simply dC = 0. (2.36) In more detail, this is G = 0,(2.37) where G = dC is the field strength of the three-form field C (or more precisely the part of this field strength that is a four-form along M), and d φ = 0. (2.38) Thus in contrast to section 2.2 where the α ′ -corrected three-form φ ′ did not obey dφ ′ = 0, here there will be no α ′ correction to the statement d φ = 0. The further condition W = 0 is automatic if C is a globally-defined three-form (for then dC = 0 implies that C∧dC = 0); sinceφ is certainly globally-defined, this is true if the C-field is topologically trivial. To learn more, we will have to impose the second condition for unbroken supersymmetry, which is the vanishing of the auxiliary fields D ζ . As explained above, the theory of interest has two kinds of vector multiplets, namely a two-form V ab and a vector field V a . The symmetry gauged by V a is the group of diffeomorphisms of M. On the other hand, the two-form V ab gauges the group of C-field gauge transformations C → C + dΛ, φ → φ, where Λ is an arbitrary two-form on M. Thus the transformation is δC = idΛ. (2.39) Clearly, V ab is odd under C → −C, while V a is even. It turns out that the corresponding auxiliary field D V and D V transform oppositely to V and V (this is because the Kahler form that we analyze below is odd under C → −C), so D V is even and D V is odd. D V and D V are gauge-invariant local functionals of C, so they are really depend on C only through G = dC. Eqn. (2.37) tells us that G = 0 in a supersymmetric vacuum, and once we set G = 0, D V will automatically vanish, since it is odd in G. So the additional condition for unbroken supersymmetry that we are looking for will be D V = 0. In general, consider any theory with four supercharges with chiral multiplets C = C + . . . that parametrize a Kahler manifold X , and with vector multiplets V ζ generating a group G of symmetries of X . On-shell, the corresponding auxiliary fields D ζ are functions D ζ (C, C) on X and together these functions comprise the "moment map" for the action of G on X . In general, these functions transform in the representation of G that is dual to the adjoint representation. In the present context, we take G to be the group of C-field gauge transformations, so a generator of G is a two-form Λ, as in eqn. (2.39). Hence, the auxiliary field will be a five-form D(C, C) which, order by order in the α ′ expansion, will be constructed locally from C, C, and their derivatives. But actually, the group G does not act faithfully on C: the C-field gauge transformation generated by a closed two-form Λ is trivial. Hence D(C, C) takes values in the dual to the space of two-forms modulo closed ones. In seven dimensions, the dual of two-forms mod closed ones is the space of exact five-forms, and therefore D(C, C) will be exact, as we find explicitly below. In our problem, the Kahler manifold X parametrized by the C's is simply the space of complex-valued three-forms on M. We must discuss the Kahler metric on X , since in general the moment map depends on the Kahler metric. The Kahler metric ds 2 is determined in the usual fashion by a Kahler potential K(C, C): ds 2 = M ×M δC(y) ⊗ δC(y ′ ) δ 2 K δC(y)δC(y) . (2.40) (We will write δ for a variation on the infinite-dimensional space X , and d for the exterior derivative on M.) Though we have written the metric as an integral over M × M, this integral actually collapses, order by order in the α ′ expansion, to an integral over a single copy of M. The reason for this is that the functional K is local order by order in α ′ (that is, it is the integral over M of a local function of C, φ, and their derivatives up to a finite order), so that the second variation δ 2 K/δC(y)δC(y ′ ) is a sum of terms proportional to δ(y, y ′ ) and its derivatives. This ensures that the integral on the right hand side of (2.40) collapses, order by order, to a local integral over M. Type IIA superstring theory has a symmetry (−1) F L that ensures that K is an even function of C. Also, the Kahler metric of X does not depend on the orientation 19 of M, since Type IIA superstring theory on R 3 × M is invariant under simultaneous reversal of R 3 and M. The Kahler potential K of X is not uniquely determined, since one is free to make a Kahler transformation K → K + f + f , where f is a holomorphic function on X . But since the Kahler metric of X is gauge-invariant, K can be chosen to be gaugeinvariant and thus to depend on C only via G = dC. (The fact that K does not depend on the orientation of M excludes a term M C ∧ dC in K.) The fact that K depends on C only via G = dC will ensure that the C-dependent terms in K do not contribute to the functions D(C, C). To explain this, we will expand K in powers of G, keeping the quadratic term (there is no linear term as K is even in C). It will be clear from the derivation that terms in K that are higher than quadratic in G will not contribute to D(C, C) at G = 0, and actually we will see that also the quadratic terms do not contribute. Thus we write K = 4K 0 ( φ) + M ×M G(y) ∧ G(y ′ ) ∧ K 1 (y, y ′ ) + . . . . (2.41) (The factor of 4 is for later convenience.) Here K 1 (y, y ′ ) is a three-form in each variable. As in (2.40), we write the second term here as an integral over M × M, but order by order in α ′ , it actually collapses to an integral over a single copy of M, since K 1 (y, y ′ ) is proportional to δ(y, y ′ ) and its derivatives up to a finite order. The Kahler metric derived from (2.41) is ds 2 = M ×M δC(y) ⊗ δC(y ′ ) δ 2 K 0 δ φ(y)δ φ(y ′ ) + 1 2 M ×M dδC(y) ∧ dδC(y ′ ) ∧ K 1 (y, y ′ ) + O(G). (2.42) We omit terms proportional to G as we will be setting G = 0. The corresponding Kahler form is ω = M ×M δ φ(y) ∧ δC(y ′ ) δ 2 K 0 δ φ(y)δ φ(y ′ ) + 1 2 M ×M dδ φ(y) ∧ dδC(y ′ ) ∧ K 1 (y, y ′ ). (2.43) In general, given a vector field V ζ acting on a Kahler manifold X with Kahler form ω, the corresponding D-auxiliary field D ζ is characterized (up to an additive constant which is known as the Fayet-Ilioupoulos D-term) by the relation δD ζ = ι V ζ ω, where ι V ζ is the operation of contraction with respect to V ζ . To implement this in the present context, we let V Λ be the vector field that generates the C-field gauge transformation δC = dΛ, for some two-form Λ. The contraction ι V Λ ω is evaluated by replacing δC in the formula for ω by −dΛ (there is a minus sign because ι V Λ anticommutes with δ φ). When we do this, the K 1 term does not contribute because dδC(y ′ ) is replaced by d 2 Λ(y ′ ) = 0. So ι V Λ ω = − M ×M δ φ(y) ∧ dΛ(y ′ ) ∧ δ 2 K 0 δ φ(y)δ φ(y ′ ) . (2.44) We are now supposed to set this equal to δD Λ . Clearly the desired relation δD Λ = ι V Λ ω is satisfied by 20 D Λ = M Λ ∧ d δK 0 δ φ . (2.45) The condition that D Λ = 0 for all Λ is that D = 0, where D = d ψ, (2.46) with ψ = δK 0 δ φ . (2.47) Clearly, the five-form D is always exact, even when it does not vanish. We expect that D corresponds to the exact five-form β that was found in sections 2.2 and 2.3. The conditions for unbroken supersymmetry are δW = D = 0, or in other words d φ = d ψ = 0. In the classical limit α ′ → 0, K 0 is [18,19,20] a multiple of log V (M), where 21 V (M) is the volume of M (computed using the metric (A.9), where in the classical limit, we need not distinguish φ from φ). With K 0 a constant multiple of log V (M), ψ as defined in (2.47) is a multiple of ψ = ⋆φ, where ⋆ is the Hodge star defined using the metric (A.9). So the conditions δW = D = 0 give the expected results 0 = dφ = d ⋆ φ, which characterize G 2 holonomy at the classical level. In our framework, K 0 and therefore the 20 Diffeomorphism invariance does not allow us to add an FI term to this formula. An FI term would be a contribution to D Λ that is constant, that is independent of φ. But there is no diffeomorphism-invariant way to construct a five-form contribution to D Λ if it is not allowed to depend on the only field in the problem, namely φ. 21 See also [21] for properties of this volume functional. function ψ( φ) will receive corrections order by order in α ′ , and these give the corrections to G 2 holonomy. As for what metric should be used to describe M, taking into account the α ′ corrections, this question does not have a unique answer. One can simply use eqn. (A.9) to define a metric g on M, using φ instead of φ. With α ′ corrections included, this metric will not have G 2 holonomy, since although there is a closed four-form ψ, it is not simply ⋆ φ φ (where ⋆ φ is defined using the metric g). Since this metric does not have G 2 holonomy, it is not really distinguished. One can just as well modify eqn. (A.9) by adding α ′ corrections in the relation between g and φ. Diffeomorphism Invariance And Expansion In Powers Of α ′ In this analysis, we have made no use of the second set of vector multiplets V a , which gauge the diffeomorphisms of M. Their D auxiliary fields trivially vanish once we set G = 0, so the vanishing of these fields gives no additional constraint. However, the existence of this gauge symmetry means that the functional K 0 ( φ) is diffeomorphisminvariant. Let us determine the implications of this. Let v a be a vector field on M that generates an infinitesimal diffeomorphism. The transformation of a general three-form φ generated by this vector field is δ φ = (ι v d + dι v ) φ, where ι v is the operation of contraction with v. If we restrict ourselves to the case that d φ = 0, then δ φ = dι v φ. (2.48) Diffeomorphism invariance of K 0 means that (if d φ = 0) K 0 is invariant under this transformation, so 0 = M dι v φ ∧ δK 0 δφ = M dι v φ ∧ ψ. (2.49) After integrating by parts, and recalling that d ψ is the exact five-form D, we learn that for any vector field v on M, 0 = W ι v φ ∧ D. (2.50) Now recall that any φ that obeys some mild inequalities such that the formula (A.9) defines a Riemannian metric g gives a reduction of the structure group of the tangent bundle of M to G 2 ; this is known as a G 2 -structure on M. In particular, if φ arises in α ′ perturbation theory starting with the covariantly constant three-form φ of a classical metric of G 2 holonomy, then certainly the relevant inequalities are obeyed and φ does define a G 2 -structure. This means that it determines a decomposition of the space of two-forms on M as ∧ 2 = ∧ 2 7 ⊕ ∧ 2 14 , with corresponding projection operators π 7 and π 14 . Since ι v φ is an arbitrary section of ∧ 2 7 , eqn. (2.50) is equivalent to the identity π 7 (D) = 0 (2.51) which holds whenver d φ = 0. Let us now make a few remarks about solving the equations d φ = d ψ = 0 in an expansion in powers of α ′ . K 0 has an expansion in powers of α ′ with the leading term being the classical expression K 0,cl and the first correction being of order (α ′ ) 3 : K 0 = K 0,cl + ∞ n=3 (α ′ ) n K 0,n . (2.52) Correspondingly we expand φ in a series in α ′ . The classical term is the covariantly constant three-form φ of a classical metric of G 2 holonomy. The corrections must preserve the fact that d φ = 0, and (for the same reason as in footnote 12) we may as well assume that the corrections do not change the cohomology class of φ. Therefore we assume that the series takes the form φ = φ + ∞ n=3 (α ′ ) n db n ,(2.53) with two-forms b n . In this expansion, we can assume that b n ∈ ∧ 2 14 for the same reason as in section 2.2.1. Indeed, we are only interested in the solution of the equations d φ = d ψ = 0 up to diffeomorphism. In each order in the expansion, an infinitesimal diffeomorphism (with a generator of order (α ′ ) n ) can shift π 7 (b n ) in an arbitrary fashion, and therefore in solving the equations we can assume that π 7 (b n ) = 0. The equation d ψ = 0, where ψ = δK 0 /δ φ and K 0 has the expansion (2.52), can then be expanded in powers of α ′ . In order (α ′ ) n , with n ≥ 3, we get a linear equation for b n that has the structure explored in sections 2.2.1-2.3. This equation has an essentially unique solution for the same reasons as described there. The Relation Between The Two Expansions The expansion that we have just described is very similar to the expansion that we described starting in section 2.2, but there is one notable difference. In section 2.2, the quantum-corrected G 2 -structure was described by a three-form φ ′ that did not obey dφ ′ = 0; see eqns. (2.9) and (2.10) for explicit formulas in lowest order. By contrast, in the present analysis, the equation d φ = 0 is exact and only the equation d ⋆ φ = 0 receives α ′ corrections. Reexamining the formulas of section 2.2, we see that at least in leading order, although dφ ′ = 0, one has d(φ ′ − χ) = 0 where χ is a locally-defined function of φ ′ (this is shown in eqn. (2.11), where to the given order, φ can be replaced by φ ′ ). Hence, although the three-form φ ′ that determines the α ′ -corrected metric is not closed, it fails to be closed by a term that could be removed by a local change of variables, that is by the addition to φ ′ (y) at a point y ∈ M of a function of φ ′ (y) and its derivatives up to finite order. It is logical to think that φ ′ − χ in section 2.2 corresponds to φ in our present analysis. We suggest that the relation between φ ′ and φ can be regarded as the relation between two different regularizations of the supersymmetric σ-model with target M. In section 2.2, we used formulas that arise if one computes the effective action of ten-dimensional superstring theory on a general ten-manifold Z and then specializes to Z = R 3 ×M. These formulas have ten-dimensional Poincaré covariance and (order by order in α ′ ) locality, but they do not have manifest spacetime supersymmetry. The standard regularizations of the σ-model preserve Poincaré covariance and locality and lead to such formulas. We will call such a regularization a 10-dimensional regularization. By contrast, in section 2.4.1, we asked for a formalism that preserves three-dimensional covariance and locality and supersymmetry and seven-dimensional covariance and locality, but we did not assume ten-dimensional covariance. If there is a regularization of the σ-model compatible with these requirements, it is a different one than is customarily used in ten dimensions. In such a regularization, one would expect φ (since it is part of the superfield C) to be the natural variable, rather than φ ′ . We will call this a 3 × 7-dimensional supersymmetric regularization. Two reasonable regularizations differ, order by order in perturbation theory, by a local change of variables. Thus, the variable φ used in a hypothetical 3 × 7-dimensional supersymmetric regularization would be expected to differ by a local change of variables from the variable φ ′ used in a standard 10-dimensional regularization. This is the structure that we have found explicitly in the first non-trivial order, and this encourages us to think that a 3 × 7-dimensional supersymmetric regularization might exist, though we do not know how to construct one. The significance might be as follows. As we recalled in the introduction, in the case of compactification of Type II superstring theory on a Calabi-Yau manifold, there is a nice σ-model argument [5] showing that supersymmetry is unbroken order by order in α ′ . We believe that a 3×7-dimensional supersymmetric regularization, if it exists, might lead to a similar argument for G 2 manifolds. All this may have an analog for the problem we study in section 3: a 2 × 8-dimensional supersymmetric regularization might provide a good framework for understanding α ′ corrections to compactification on an eight-manifold of Spin(7) holonomy. Some Properties Of The Moduli Space Here we will describe an interesting consequence of the relation (2.47) between ψ and φ. This consequence can be deduced with no knowledge of the functional K 0 except its classical limit for α ′ → 0, which ensures that the moduli space of superconformal σmodels with target M goes over for α ′ → 0 so the corresponding classical moduli space of G 2 metrics. Since φ is closed, its cohomology class [ φ] defines an element of H 3 (M, R). Likewise, ψ has a cohomology class [ψ] ∈ H 4 (M, R). In the classical limit, these go over to the cohomology classes of the familiar covariantly constant forms φ and ψ = ⋆φ. Let M 0 be the moduli space of G 2 metrics on M, modulo topologically trivial diffeomorphisms. (This is a rough analog of the Teichmuller space of a Riemann surface.) The classical moduli space M 0 is conical (since a G 2 metric can be rescaled by a positive constant). The corresponding moduli space M of superconformally invariant σ-models with target M is not conical. However, the fact that a classical G 2 metric can be deformed order by order in α ′ to give a superconformally-invariant σ-model, and that this deformation is unique up to a reparametrization of the variables in the σ-model, ensures that M looks like M 0 near infinity. This ensures that M inherits some properties of M 0 (at least in the large volume region), so we will first state some general properties of M 0 . We R) is only valid to all finite orders in α ′ , but we expect a natural such map to exist in general. Perhaps it can be defined in superconformal field theory. We expect the map ̺ to be everywhere an immersion of a middle-dimensional submanifold, but the assertion that ̺(M) can be parametrized locally by [ φ] may be valid only near large volume.) Now we come to a more delicate statement that does depend on the relation (2.47). For α ′ = 0, this statement is Proposition 10.4.5 in [16]. For a closed three-form φ and closed four-form ψ, the integral M ψ ∧ φ depends only on the cohomology classes [ φ] and [ ψ]. We can use this integral to define a symplectic form ̟ on Q = H 3 (M, R)⊕H 4 (M, R): ̟ = M δ φ ∧ δ ψ. (2.54) The claim is that ̺(M) is a Lagrangian submanifold of Q (which means that ̟ vanishes when restricted to ̺(M)). We will explain this argument very explicitly. We pick a basis of H 3 (M, R) and a dual basis of H 4 (M, R) and let φ λ and ψ λ be the components of [ φ] and [ ψ] with respect to these bases. (For legibility of the following formulas, we prefer to write φ λ and ψ λ rather than [ φ] λ and [ ψ] λ .) In these variables, ̟ = λ d φ λ ∧ d ψ λ . (2.55) (After reducing to the finite set of variables φ λ and ψ λ , we write the exterior derivative as d rather than δ.) Once we restrict to M, φ is uniquely determined by is cohomology class [ φ], so K 0 ( φ) can be regarded as a function of [ φ] and hence of its components φ λ , and eqn. (2.47) tells us that along M, ψ λ = ∂K 0 ∂ φ λ . (2.56) It follows that d ψ λ = ν d φ ν ∂ 2 K 0 ∂ φ λ ∂ φ ν ,(2.57) and hence when restricted to M, ̟ = λ,ν d φ λ ∧ d φ ν ∂ 2 K 0 ∂ φ λ ∂ φ ν = 0, (2.58) where we use the fact that ∂ 2 K 0 /∂ φ λ ∂ φ ν is symmetric in λ and ν. This shows that ̺(M) is Lagrangian in Q. We expect this conclusion to be valid exactly, not just to all finite orders in α ′ . The claim that K 0 is a local functional of φ is valid only to all finite orders in α ′ , but in the exact theory, we expect that there is a Kahler potential on the space X that can be used as input to this analysis, leading to the same conclusion. Spin(7) Holonomy Manifolds In this section, we consider Type II superstring theory on a spacetime of the form R 1,1 ×M, where M is a compact eight-dimensional manifold. At the classical level, the condition for such a compactification to preserve supersymmetry on R 1,1 , in the absence of fluxes, 22 is that the holonomy group of M should be Spin(7) (or a subgroup thereof). If the holonomy group is precisely Spin (7), which is the case we will concentrate on, then compactification on M preserves (1, 1) supersymmetry on R 1,1 (Type IIA) or (0, 2) supersymmetry (Type IIB). Our goal is to investigate α ′ corrections to such compactifications. An eight-manifold of Spin(7) holonomy admits a covariantly constant spinor field η of negative chirality, which we normalize (up to sign) by setting η T η = 1. Spin (7) and G 2 geometries share many common features when described in terms of the covariantly constant spinor so we will be brief and collect technical details in the appendix. We will have to explain, however, a few key differences between the two cases. For any η of negative chirality, the four-form Φ abcd = η T Γ abcd η, (3.1) is anti-selfdual. If η is covariantly constant, then Φ also covariantly constant and in particular is closed and co-closed, dΦ = d ⋆ Φ = 0. Leading Order Correction As in the G 2 -holonomy case, to describe corrections of order α ′3 to the supersymmetry transformations, we expand in a basis of real, antichiral spinors. The negative chirality spinors of SO(8) decompose under Spin(7) as 1 ⊕ 7, and the two-forms transform as 28 = 7 ⊕ 21. So we can take a basis of negative chirality spinors given by η and c ab Γ ab η, where c ab is an antisymmetric tensor transforming in the 7. Hence the condition to have an unbroken supersymmetry with α ′3 corrections included must take the form of the existence of a spinor η ′ = η + O((α ′ ) 3 ) satisfying D ′ a η ′ = A a η + C bc a Γ bc η, (3.2) where A a and C bc a are real tensors on M, proportional to (α ′ ) 3 , that are locally constructed from Φ. Note that C bc a transforms in the 8 ⊗ 7 ∼ = 8 ⊕ 48 of Spin(7). Eqn. (3.2) is readily seen to imply that the one-form A = A a dy a is A = 1 2 d log(η ′T η ′ ), and hence if we rescale η ′ so that η ′T η ′ = 1 (this may not be the natural normalization for other purposes), it obeys eqn. (3.2) with A a = 0. Actually, the standard formulas for the (α ′ ) 3 correction do have A a = 0. We define a corrected four-form Φ ′ abcd = η ′T Γ ′ abcd η ′ . (3.3) The α ′ -corrected condition (3.2) for unbroken supersymmetry is equivalent to the condition dΦ ′ = γ (3.4) for Φ ′ , where we define γ abcde = −80C f g [a g b\|f \| Φ cde]g . (3.5) Note that both the 8 and 48 pieces of C appear in γ. These are known as the torsion classes of the Spin(7)-structure, and their data is equivalent to that contained in C bc a . At leading order, from [15] we have C a bc = −c α ′3 4 Φ qrs a ∇ q Z bc rs . (3.6) Here c is a constant, and Z abcd is given by Z abcd = 1 64g ǫ abe 1 ···e 6 ǫ cdf 1 ···f 6 R e 1 e 2 f 1 f 2 R e 3 e 4 f 3 f 4 R e 5 e 6 f 5 f 6 . (3.7) From here we find γ = dχ with χ abcd = 8cΦ e [abc Z d]e − 12cΦ ef [ab Z cd]ef . (3.8) So γ is exact. As we describe next, this cohomology condition on γ is necessary and sufficient for the existence of a Spin (7)-structure on M that is close to the classical one and obeys the α ′ -corrected condition for supersymmetry, to this order. Before trying to solve the eqn. (3.4) for unbroken supersymmetry, we must understand an important difference between a G 2 -structure on a seven-manifold and a Spin(7)structure on an eight-manifold. At a given point p in a seven-manifold, any three-form φ that obeys some inequalities is invariant under a G 2 subgroup of the group GL(7, R) that acts on the tangent space at p. (These inequalities amount to saying that a metric can be defined by the formula of eqn. (A.9).) If φ obeys the relevant inequalities, we say it defines a G 2 -structure at p. A three-form φ on a seven-manifold M that obeys the relevant conditions everywhere on M is said to define a G 2 -structure on M; if φ has this property, then any three-form φ ′ that is sufficiently close to φ does as well. Matters are different in one dimension more. At a given point p in an eight-manifold M, a generic four-form Φ ′ is not invariant under a Spin(7) subgroup of the group GL(8, R) that acts on the tangent space at p. Φ ′ has Spin(7) symmetry if and only if there is an orientation-preserving element of GL(8, R) that maps Φ ′ to a standard fiducial four-form Φ 0 . Following ref. [16], section 10.5, we write A p M for the space of four-forms that obey this condition at a point p ∈ M, and AM for the space of four-forms that obey the condition for all p ∈ M. From this description, clearly GL(8, R) acts transitively on A p M, with the stabilizer group of a point being Spin (7). So there is a one-to-one correspondence between A p M and the coset GL(8, R)/Spin (7). Counting dimensions, we have If Φ defines a Spin(7)-structure, then a deformation of Φ in the 1 represents a rescaling of Φ, preserving in each tangent space the Spin(7) subgroup of GL(8, R) that leaves Φ fixed. A deformation in the 7 ⊕ 35 represents a rotation of that Spin(7) subgroup inside GL(8, R). But a deformation of Φ that is in the 27 cannot be interpreted as a deformation of the Spin(7)-structure. Thus the tangent space to A p M at the point corresponding to Φ decomposes as 1 ⊕ 7 ⊕ 35, and the condition that a deformation δΦ of Φ represents a deformation of the Spin(7)-structure of M is that π 27 (δΦ) = 0 (3.10) or equivalently δΦ ∈ ∧ 4 1 ⊕ ∧ 4 7 ⊕ ∧ 4 35 . (3.11) Explicitly, the meaning of the condition (3.11) on δΦ is the following. Comparing the definition of the unperturbed four-form Φ in eqn. (3.1) to the definition of the perturbed four-form Φ ′ in eqn. (3.3), we see that the perturbation Φ ′ has two sources: (i) the perturbation from η to η ′ ; and (ii) the perturbation in the gamma matrices that appear in Γ ′ abcd in eqn. (3.3). Since we have set η ′T η ′ = 1, the perturbation η ′ − η transforms in the 7. The perturbation in the gamma matrices arise because of a perturbation in the metric g of M. The metric is a symmetric second rank tensor g ab ; perturbations of g ab about a metric of Spin(7) holonomy transform as 1 ⊕ 35. Altogether, then, first order perturbations of Φ should transform as 1 ⊕ 7 ⊕ 35, with no contribution transforming as 27. Now we expand Φ ′ = Φ + δΦ, (3.12) and try to pick δΦ to satisfy d(δΦ) = γ and also π 27 (δΦ) = 0. If π 27 (χ) = 0, we would simply take Φ ′ = Φ + χ, and this would give the deformed Spin(7)-structure. However, π 27 (χ) = 0 and therefore a more elaborate discussion is required. We will try Φ ′ = Φ + (1 − π 27 ) (χ + dc) ,(3.13) for some three-form c. By construction we have π 27 (Φ ′ ) = 0, so it remains only to show that we can find some globally-defined c such that dΦ ′ = γ, or dπ 27 dc = −dπ 27 χ. (3.14) The decomposition of the space of three-forms under Spin (7) is ∧ 3 = ∧ 3 8 ⊕ ∧ 3 48 . A c ∈ ∧ 3 8 corresponds to the change in Φ induced by an infinitesimal diffeomorphism. It does not contribute to eqn. (3.14) since 8 ⊗ 8 ∼ = 1 ⊕ 7 ⊕ 21 ⊕ 35, so that π 27 dc = 0 for c ∈ ∧ 3 8 . So we restrict to c ∈ ∧ 3 48 . Similarly the source −dπ 27 χ is also in ∧ 5 48 (dually to the decomposition of ∧ 3 , one has ∧ 5 = ∧ 5 8 ⊕ ∧ 5 48 ; since 8 does not appear in the decomposition of 8 ⊗ 27, one has dπ 27 χ ∈ ∧ 5 48 ). Because of the decomposition ∧ 4 = ∧ 4 1 ⊕ ∧ 4 7 ⊕ ∧ 4 27 ⊕ ∧ 4 35 , we have on four-forms 1 = π 1 + π 7 + π 27 + π 35 , so d (π 1 + π 7 + π 27 + π 35 ) dc = 0. (3.15) Hence the equation (3.14) that we are trying to solve is equivalent to d (π 1 + π 7 − π 27 + π 35 ) dc = 2dπ 27 χ. (3.16) For c ∈ ∧ 3 48 , π 1 (dc) = 0, since 1 does not appear in the decomposition of 8 ⊗ 48. So we can reverse the sign of the π 1 term in eqn. (3.16). We will show that we can solve eqn. (3.16) while also requiring that c is coclosed, d † c = 0. If this condition is satisfied, then π 7 (dc) = 0; indeed, 7 occurs only once in the decomposition of 8 ⊗ 48, so it occurs only once in the first derivatives of c, and this contribution is a multiple of d † c. So if d † c = 0, we can reverse the sign of the π 7 (dc) term in eqn. (3.16). At this point, we use the fact that (since ∧ 4 + = 35 and ∧ 4 − = 1 ⊕ 7 ⊕ 27) the Hodge ⋆ operator on four-forms is ⋆ = −π 1 − π 7 − π 27 + π 35 . Given this, the equations that we have to satisfy can be written d † dc = −2d † π 27 χ, d † c = 0, or equivalently ∆c = −2d † π 27 χ, d † c = 0, (3.17) where ∆ = d † d+dd † is the Hodge-de Rham Laplacian. By the general theory of the Hodgede Rham Laplacian, since the source is orthogonal to the kernel of ∆, these equations have a solution (unique up to the possibility of adding a harmonic three-form to c), and the solution can be taken to lie in ∧ 3 48 because the Hodge-de Rham Laplacian respects the decomposition ∧ 3 = ∧ 3 8 ⊕ ∧ 3 48 and the source lies in ∧ 3 48 . All Orders In α ′ To extend this result to all orders in α ′ , we proceed as in section 2.2. We expand Φ ′ = Φ + ∞ k=3 (α ′ ) k Φ k . (3.18) As in the G 2 case, a solution will only exist, in a given order in the expansion, if a certain closed form is actually exact. In this case, in order (α ′ ) k , the condition that we will have to satisfy is dΦ k = γ[Φ ′ ]\| k , where γ[Φ ′ ]\|dΦ k = dχ k ,(3.19) with also a constraint that ensures that Φ ′ ∈ AM. If the constraint were simply that π 27 Φ k = 0, then the problem of finding Φ k would be isomorphic to the problem already solved in leading order in section 3.1, with a different source on the right hand side. Actually, AM is a nonlinear space and the constraint Φ ′ ∈ AM is nonlinear in Φ ′ . As a result the appropriate condition on Φ k is not that π 27 Φ k = 0, but that π 27 (Φ k ) is a certain nonlinear function Θ k of the Φ n , n < k. (3.20) and now the problem is indeed isomorphic to the one that we have already studied. After writing Φ k = Φ ′ k + Θ k where π 27 (Φ ′ k ) = 0, eqn. (3.19) becomes dΦ ′ k = d(χ k − Θ k ), Interpretation In Two-or Three-Dimensional Field Theory Here we will rather briefly interpret the results of sections 3.1 and 3.2 in the language of supersymmetric field theory in two or three dimensions. As in our discussion of the G 2 case in section 2.4, it will take more work to fully justify our proposal. We can consider Type IIA superstring theory compactified to two dimensions on a Spin(7) manifold M, giving a two-dimensional theory with (1, 1) supersymmetry, or Type IIB compactified on M, giving a two-dimensional theory with (0, 2) supersymmetry. As in section 2.4, we will focus on the Type IIA case, and we also observe that Type IIA compactification on M to two-dimensions is similar to M-theory compactification on M to three-dimensions, with the α ′ expansion replaced by the 1/r expansion. In M-theory, compactification on a Spin(7) manifold gives a three-dimensional theory with N = 1 supersymmetry (two supercharges). Our discussion applies to each of these cases. Two-dimensional theories with (1, 1) supersymmetry and three-dimensional theories with N = 1 supersymmetric can be conveniently formulated in terms of a superspace with two or three bosonic coordinates x µ and two fermionic coordinates θ α , α = 1, 2. The superfields that will be important in our analysis are unconstrained functions Λ(x µ , θ α ) on superspace (these are often called scalar superfields, but we will be more precise in our terminology below). For the purposes of finding supersymmetric vacuum states, the most important supersymmetric interaction is the superpotential. This is a function W (Λ) such that the condition for a supersymmetric vacuum is dW = 0. In either Type IIA or M-theory compactification on a Spin(7) manifold, there is a discrete symmetry τ under which the superpotential is odd. In Type IIA, one can take τ to be the operation (−1) F L that reverses the sign of left-moving worldsheet fermions. In M-theory, one can take τ to be a reflection of one of the uncompactified directions accompanied by a sign change of the three-form field C. There are therefore two kinds of superfield: we call Λ a scalar superfield if its bottom component is even under τ , and a pseudoscalar superfield if its bottom component is odd under τ . We will only analyze supersymmetric vacua that are τ -invariant (this means that we omit vacua with fluxes of the field G = dC, as have been studied in [22]). Let S I be the scalar superfields and T J the pseudoscalar superfields. Since the superpotential is odd under τ , it can be expanded in powers of the T 's with only odd order terms appearing: W = I T I F I (S J ) + O(T 3 ). (3.21) Here the F I are in general completely arbitrary functions of the S J . In a globally supersymmetric theory, the τ -invariant supersymmetric states correspond to solutions of dW = 0 with also T = 0. Clearly the necessary condition is simply that F I (S J ) = 0 (3.22) for all I. After coupling to supergravity, one also wants W = 0 to get a supersymmetric vacuum in Minkowski spacetime; clearly in a theory of this kind, this is an immediate consequence of setting T = 0. In M-theory on R 3 ×M (or similarly in Type IIA on R 2 ×M), the obvious pseudoscalar fields are obtained by taking all indices of the three-form field C to be tangent to M. This gives us a field C abc (x, y), which as in section 2.4 we regard as a pseudoscalar field on R 3 that is also a three-form on M. We expect C abc (x, y) to be the bottom component of a superfield that we will, for brevity, also call C abc . We will assume that to describe the theory in a way that has manifest covariance, locality and supersymmetry along R 3 or R 2 and manifest covariance and locality along M, we should also introduce a scalar superfield 23Φ that is a four-form on M constrained to take values in AM. Because of the usual gauge-invariance C → C + dΛ for a two-form Λ on M, the superpotential W depends on C only through its field strength G = dC, where here we consider only the part of G that is a four-form on M (and a pseudoscalar function on R 3 ). We expand the superpotential in powers of G; as in the last paragraph only odd powers appear and only the linear term is important for understanding supersymmetric vacua at G = 0. Thus the general form of the superpotential is W = M G ∧ P (Φ) + O(G 3 ) (3.23) where to any finite order in α ′ , P is a local functional ofΦ. The condition for a critical point at G = 0 is simply dP (Φ) = 0. (3.24) 23 How isΦ constructed in terms of the usual degrees of freedom of supergravity? As explained in section 3.1, a perturbation inΦ can be decomposed under Spin(7) as 1 ⊕ 7 ⊕ 35, while a perturbation in the metric g of M can be decomposed as 1 ⊕ 35. What is the proper interpretation of the 7 contribution inΦ? In section 2.4.1, it was suggested that the answer to the analogous question for G 2 (in the M-theory case) arises from scalar fields obtained by dualizing C µνa . For Spin (7) an analog is to consider scalar fields obtained by dualizing C µab , which transforms as 7 ⊕ 21. The 7 would hypothetically complete the construction ofΦ and the 21 would represent additional fields that would be included in a more complete description. In the classical limit α ′ → 0, we must have simply P (Φ) =Φ, so that eqn. (3.24) reduces to the classical condition dΦ = 0 which (with the constraint thatΦ is valued in AM) is equivalent to Spin(7) holonomy. But now we can easily go to higher orders. When we expandΦ = Φ + ∞ k=3 (α ′ ) k Φ k ,(3.dΦ k = dχ k (3.26) where χ k is a nonlinear function of the Φ n for n < k. There will also be a constraint that determines π 27 (Φ k ) in terms of the previous Φ n 's. As we have seen, it is always possible to solve conditions of this form. The main point is that the supersymmetric structure implies that the five-forms that were called γ[Φ ′ ]\| k in section 3.2 are always exact. In fact, they are always dχ k , where χ k is a local function of the Φ n 's with n < k. A Background On Manifolds Of Exceptional Holonomy In this appendix we review some facts which have been used in the main body of the paper. These facts are collected for the convenience of the reader. Before getting into details on G 2 or Spin (7), recall that the standard inner product of forms is defined by χ, ξ = 1 p! d d x √ gχ a 1 ···ap ξ a 1 ···ap . (A.1) This inner product respects the decomposition of forms into G 2 and Spin (7) where the Levi-Civita symbol ε a 1 ...a 7 is a tensor. Define φ abc = iη T Γ abc η, ψ abcd = 1 3! ε abcdklm φ klm = η T Γ abcd η. (A.6) Any real spinor can be expanded in the above basis. In particular Γ ab η = −iφ c ab Γ c η, iΓ abc η = φ abc η − iψ k abc Γ k η. (A.7) Using (A.4) one derives some useful identities together with additional identities obtained by contraction. φ ab k φ cdk = 2g a[c g d]b − ψ abcd , ψ abck φ dek = 6δ [ A.1.2 Metric Given a G 2 -structure φ the above relations can be used to obtain g ab = − 1 144 ǫ ijklmnp φ aij φ bkl φ mnp . (A.9) Given a G 2 -structure φ this can be used as the defining equation for the metric g ab . Indeed, the epsilon tensor takes values 24 ±1/ √ g and so taking the determinant of eqn.(A.9) we can solve for g in terms of φ and this in turn lets us write the metric g ab in terms of φ only. We can then consider φ to be the fundamental object from which the metric, Riemann tensor, covariant derivatives and ψ = ⋆φ are obtained. The deformed structure φ ′ = φ+δφ will give rise to deformations in the metric, g ′ = g+δg, and the four-form, ψ ′ = ψ + δψ. Plugging these into the contraction φ a cd φ bcd = 6g ab , (A. 13) one can derive that to first order in the deformation δφ we have δg ab = − 1 18 g ab φ cde δφ cde + 1 2 φ cd (a δφ b)cd , (A. 14) and using this metric to construct the Hodge star we get δψ abcd = − 1 9 ψ abcd φ ef g δφ ef g − 1 3 φ [abc ψ ef g d] δφ ef g − 6φ e [ab δφ cd]e . (A. 15) In both of these expressions, indices are raised with the undeformed metric g ab . In terms of the pieces of δφ which transform in different representations of G 2 (the copy of G 2 which leaves the original φ invariant), we can rewrite these first order deformations as δg ab = φ cd A.1.5 The Torsion Forms Given a G 2 -structure φ ′ , its exterior derivative dφ ′ is a four-form and the exterior derivative of the dual dψ ′ = d⋆ ′ φ ′ is a five-form. These forms can be decomposed into irreducible representations of G 2 . Following proposition 1 of ref. [11], we write dφ ′ = τ 0 ψ ′ + 3τ 1 ∧ φ ′ + ⋆ ′ τ 3 , dψ ′ = 4τ 1 ∧ ψ ′ + ⋆ ′ τ 2 , (A.18) where τ 3 ∈ ∧ 3 27 and τ 2 ∈ ∧ 2 14 . It was mentioned in ref. [11] that the projections of dφ ′ and dψ ′ onto ∧ 4 7 and ∧ 5 7 are closely related and in particularτ 1 = τ 1 . The basic reason is that for any given G 2 -structure the following identity holds (eqn. (3.8) of ref. [11]) (dψ ′ ) abcde ψ ′ bcde = 4(dφ ′ ) abcd φ ′ bcd ; (A. 19) this can be shown using the definition ψ ′ = ⋆ ′ φ ′ . Here we note a consequence of this result which we use in the main body of the paper. If we consider a perturbation about a G 2 holonomy space with G 2 -structure φ by setting φ ′ = φ + δφ, then to first order in fluctuations eqn. (A. 19) becomes (dψ ′ ) abcde ψ bcde = 4(dφ ′ ) abcd φ bcd = d ⋆ 4 3 π 1 + π 7 − π 27 δφ abcde ψ bcde . (A.20) In section 2.2.1, to first order in α ′ , we solve the conditions dφ ′ = dχ and dψ ′ = dξ for globally-defined χ and ξ. In this case eqn. (A.20) implies (dξ) abcde ψ bcde = d ⋆ 4 3 π 1 + π 7 − π 27 χ abcde ψ bcde . (A. 21) This is equivalent to the vanishing of the ∧ 2 7 projection of d † ρ, which is defined in eqn. (2.16). A similar argument can be used to show that d † ρ n ∈ ∧ 2 14 in section 2.3. In this case we take φ ′ = φ + n−1 i=3 φ i + φ n =φ + δφ. (A.22) Soφ is the G 2 -structure up to order n − 1 in α ′ . We consider a small perturbation of order n around this G 2 -structure δφ = φ n . Then ψ ′ =ψ + δψ, (A.23) whereψ =⋆φ. Note thatφ is at most of order n while ψ ′ could, in principle, receive contributions to all orders in α ′ since it it a non-linear functional ofφ. So ψ ′ \| n =⋆φ \| n +δψ \| n . Now, locally we can always solve (2.19) to find η ′ such that the associated forms φ ′ and ψ ′ satisfy (2.30), but a priori we may not be able to extend η ′ to a global solution. Note that the local solution for φ ′ can always be written in the form χ n + db n , where now b n may not be globally defined. By substituting this local solution for ψ ′ above, we obtain which is equivalent to the vanishes of the 7 part of the source d † ρ n . Finally, since b n can be shown to drop out of this relation completely, the result will hold true for any valid solution. A.1.6 Some Useful Identities If M is a G 2 holonomy manifold, a few useful properties can be derived. Defining (L · λ) abc = ψ d abc λ d , λ ∈ ∧ 1 , (A.28) we have d † L · λ = (2π 7 − π 14 ) dλ. (A.29) For any two-forms π 7 (db) = − 1 4 L · d † b, b ∈ ∧ 2 14 , (A.30) and d † 4 3 π 1 + π 7 − π 27 dα = 0, α ∈ ∧ 2 7 . (A.31) Another useful identity is π 1 (db) = 0, ∀b ∈ ∧ 2 14 . (A.32) A.2 Spin (7) A. Spinor Conventions For an eight-manifold with Spin(7)-structure, we have a nowhere-vanishing real spinor η. We will choose conventions in which η is antichiral, Γ 9 η = −η,(A. 18 3 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Leading Order Correction . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 Existence Of A Solution . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 All Orders In α ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Interpretation In Three-or Four-Dimensional Field Theory . . . . . . . . 11 2.4.1 Superfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 11 2.4.2 Conditions For Unbroken Supersymmetry . . . . . . . . . . . . . 12 2.4.3 Diffeomorphism Invariance And Expansion In Powers Of α ′ . . . . 16 2.4.4 The Relation Between The Two Expansions . . . . . . . . . . . . 17 2.4.5 Some Properties Of The Moduli Space . . . . . . . . . . . . . . . Spin(7) Holonomy Manifolds 19 3.1 Leading Order Correction . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 All Orders In α ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Interpretation In Two-or Three-Dimensional Field Theory . . . . . . . . 23 A Background On Manifolds Of Exceptional Holonomy 25 A.1 G 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 A.1.1 Spinor Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . 25 A.1.2 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 A.1.3 Decomposition Of Differential Forms Into Irreducible Representations Of G 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 A.1.4 Deformations Of G 2 -Structures . . . . . . . . . . . . . . . . . . . 27 A.1.5 The Torsion Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 27 A.1.6 Some Useful Identities . . . . . . . . . . . . . . . . . . . . . . . . 29 A.2 Spin(7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 A.2.1 Spinor Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . 29 A.2.2 Decomposition Of Differential Forms Into Irreducible Representations Of Spin(7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 A.2.3 Deformations Of Spin(7)-Structures . . . . . . . . . . . . . . . . . 30 A.2.4 The L Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . define a map ̺ : M → Q = H 3 (M, R) ⊕ H 4 (M, R) that takes a point in M to the point [ φ] ⊕ [ ψ] ∈ Q. This definition makes sense both in the classical theory at α ′ = 0 and (at least to all orders in α ′ ) in the quantum-corrected theory. A basic result about classical G 2 manifolds is that if we set α ′ = 0, the map ̺ is locally an embedding. Moreover, in the classical theory, ̺(M) is middle-dimensional in Q and can be parametrized locally by [ φ] (with [ ψ] regarded as a function of [ φ]), as stated in Theorem 10.4.4 of [16]. All these statements, which are basic facts in the theory of classical G 2 manifolds, are stable under arbitrary small perturbations of the map ̺ : M → H 3 (M, R) × H 4 (M, R). So since these statements hold for the classical moduli space M 0 , and given that M is asymptotic to M 0 at infinity, they also hold for M, at least for sufficiently large volume. (The definition we have given of the map ̺ : M → H 3 (M, R) ⊕ H 4 (M, p M is not an open subset of the space ∧ 4 of all four-forms at p; it is of codimension 70 − 43 = 27. Under a Spin(7) subgroup of GL(8, R), ∧ 4 decomposes as 1 ⊕ 7 ⊕ 35 ⊕ 27. 25) the equation (3.24) will give, in each order, an equation of the general form representations, so for distinct irreducible representations r and s π r (χ), π s (ξ) = 0.(A.2)We use d † = (−1) p(d+1−p) ⋆ d⋆ for the adjoint of the exterior derivative acting on p-forms.With this definition, χ, dξ = d † χ, ξ . use a basis in which the gamma matrices Γ a are purely imaginary antisymmetric matrices satisfying {Γ a , Γ b } = 2g ab . The Clifford algebra is spanned by real symmetric matrices {1, iΓ abc } and real anti-symmetric matrices {iΓ a , Γ ab }. The eight spinors {η, iΓ a η}, are a basis. The completeness of this basis says Γ a ηη T Γ a + ηη T = 1. a 1 ,...,a 7 ε a 1 ...a 7 Γ a 1 ...a 7 = −i, (A.5) 1 (δφ) + ⋆π 7 (δφ) − ⋆π 27 (δφ). (A.17) then expand eqn. (A.19) aboutφ and use the fact that (A.19) is valid with primed quantities substituted by tilde quantities we derive [d(δψ)] abcde ψ bcde \| n = 4 [d(δφ)] abcd φ bcd \| n , (A.25) at order n in α ′ . According to (A.24) d(δψ) \| n = dψ ′ \| n −d(⋆φ) \| n . (A.26) dξ n − d(⋆φ) \| n abcde ψ bcde = 4 (dχ n ) abcd φ bcd , (A.27) abc φ def β def . (A.12) A.1.4 Deformations Of G 2 -Structures⊲ for ∧ 3 ∼ = ∧ 3 1 ⊕ ∧ 3 7 ⊕ ∧ 3 27 (π 1 β) abc = 1 42 φ abc φ def β def , (π 7 β) abc = 1 4 β abc − 3 8 ψ de [ab β c]de − 1 24 φ abc φ def β def , (π 27 β) abc = 3 4 β abc + 3 8 ψ de [ab β c]de + 1 56 φ and Γ i are pure imaginary antisymmetric 16 × 16 gamma matrices for SO(8). We can normalize η so that η T η = 1, (A.35) and we also have propertiesΓ a ηη T Γ a = Π + , ηη T − 1 8 Γ ab ηη T Γ ab = Π − , (A.36) Define Φ abcd = η T Γ abcd η. abcdef gh Γ ef gh = Γ 9 Γ abcd , (A.39)we have ⋆Φ = −Φ. We can also derive Φ abcg Φ def g = 6δ33) where Γ 9 = 1 √ g Γ 1 Γ 2 · · · Γ 8 , (A.34) where Π ± = 1 2 (1 16×16 ± Γ 9 ) . (A.37) (A.38) Note that since √ g 4! ǫ [a [d δ b e δ c] f ] − 9δ [a [d Φ bc] ef ] , (A.40) and its contractions. Here g ′ is defined in terms of φ ′ by the formula (A.9).12 The possibility of adding to φ ′ a closed but not exact three-form is not really interesting here, because this could be absorbed in shifting the cohomology class of the starting three-form φ. As remarked at the end of section 2.1, this cohomology class is arbitrary (within a certain cone). So the form given in (2.14) is essentially the most general. They violate a different shift symmetry involving the B-field of the NS sector.15 Nonperturbatively in g st , this stability is lost because of D2-brane instantons, which are analogs of M2-brane instantons in M-theory.16 See footnote 10 for index conventions. The dual of C µνa is even under C → −C accompanied by a reflection of one coordinate in R 4 , sô φ abc is even under this symmetry while C abc is odd. Hence the Kahler potential that we introduce later is invariant underφ →φ, C → −C. In the present context, the symmetry of Type IIA on R 3 × M under a reflection of R 3 (together with a reversal of orientation of the string worldsheet) ensures that Chern-Simons couplings are not possible for the vector multiplets V ab and V a . As explained in footnote 24 of the appendix, a G 2 manifold does not have a preferred orientation. For some results on the case with G-flux included at the classical level (in the M-theory context and with M restricted to have holonomy SU (4) ⊂ Spin(7)), see[22]. See[23] for an analysis of the analogous conditions for Spin(7). Given a choice of φ, the sign of the epsilon tensor or equivalently the orientation of M is determined to make the metric defined in (A.9) positive. However, without changing the metric, we could reverse the sign of φ and also the orientation of M . Thus a G 2 manifold does not have a preferred orientation. AcknowledgementA.1.3 Decomposition Of Differential Forms Into Irreducible RepresentationsOf G 2Using the fact that the tangent and cotangent spaces at points of M transform as the fundamental seven-dimensional representation of G 2 , one can derive the transformations of p-forms (living in ∧ p ∼ = ∧ p (T * M)),(A.10)We also have ∧ 7−p ∼ = ∧ p . We list the projections(π 14 α) ab = 2 3 α ab + 1 6 ψ cd ab α cd .(A.11)A.2.2 Decomposition Of Differential Forms Into Irreducible RepresentationsOf Spin(7)Under Spin(7), the spaces of differential forms decompose asand ∧ 8−n ∼ = ∧ n . We list the projectionsThe space of four-forms decomposes into self-dual forms ∧ 4 + and anti-self-dual formsA.2.3 Deformations Of Spin(7)-StructuresThe metric of a Spin(7)-structure has been derived in ref.[17]. Here we require the result for the deformations of the metric. The deformed structure Φ ′ = Φ + δΦ will give rise to a metric deformation With this metric, (A.40) continues to hold (at leading order) with Φ replaced by Φ ′ and g by g ′ .A.2.4 The L OperatorOn a Spin(7)On forms in irreducible representations of G 2 , L does not change the representation. Moreover, L is invertible on its image. This leads to the isomorphismsThe kernel isThere are potentially three ways of constructing a (p + 1)-form by differentiating a pform ω; we can make dω, d † Lω, or Ld † ω. If ω transforms in an irreducible representation r, then each of these must transform in representations contained in the product 8 ⊗ r. If some given irreducible representation s only occurs once in the decomposition of 8 ⊗ r, then it means that the forms π s (dω), π s (d † Lω), and π s (Ld † ω) must all be proportional to each other. We use this to derive some useful identities: ⊲ similar relations can be derived for higher degree forms. The only other facts we will need are that π 27 (dω) = 0 for ω ∈ ∧ 3 8 and for ω ∈ ∧ 3 48 , we have π 1 (dω) = 0 and Ld † ω = −4π 7 (dω).(A.55)Starting with the standard Hodge decomposition, the above identities can be used to derive the following decompositions ⊲ for any α ∈ ∧2 21, there exists a one-form σ and a co-closed three-form ρ ∈ ∧ 2 21 such that α = π 21 (dσ) + ρ.(A.56)⊲ for any ξ ∈ ∧ 3 48 , there exists a two-form µ and co-closed three-form ν ∈ ∧ 3 48 such that ξ = π 48 (dµ) + ν.(A.57) Four Loop Beta Function For The N = 1 and N = 2 Nonlinear Sigma Model In Two Dimensions. M T Grisaru, A E M Van De Ven, D Zanon, Phys.Lett. 173423M. T. Grisaru, A. E. M. van de Ven, and D. Zanon, "Four Loop Beta Function For The N = 1 and N = 2 Nonlinear Sigma Model In Two Dimensions," Phys.Lett. B173 (1986) 423. Superstrings And Manifolds Of Exceptional Holonomy. S Shatashvili, C Vafa, hep-th/9407025Selecta Math. 1S. Shatashvili and C. Vafa, "Superstrings And Manifolds Of Exceptional Holonomy," Selecta Math. 1 (1995) 347, hep-th/9407025. Supersymmetric Deformations of G 2 Manifolds from Higher Order Corrections to String and M Theory. H Lu, C N Pope, K S Stelle, P K Townsend, hep-th/0312002JHEP. 041019H. Lu, C. N. Pope, K. S. Stelle and P. K. Townsend, "Supersymmetric Deformations of G 2 Manifolds from Higher Order Corrections to String and M Theory," JHEP 0410, 019 (2004), hep-th/0312002. Twisted Multiplets And New Supersymmetric Nonlinear Sigma Models. S J GatesJr, C Hull, M Rocek, Nucl. Phys. 248157S. J. Gates, Jr., C. Hull, and M. Rocek, "Twisted Multiplets And New Supersymmetric Nonlinear Sigma Models," Nucl. Phys. B248 (1984) 157. Conformal Invariance Of Supersymmetric σ Models On Calabi-Yau Manifolds. D Nemeschansky, A Sen, Phys. Lett. 178365D. Nemeschansky and A. Sen, "Conformal Invariance Of Supersymmetric σ Models On Calabi-Yau Manifolds," Phys. Lett. B178 (1986) 365. New Issues In Manifolds Of SU(3) Holonomy. E Witten, Nucl. Phys. 26879E. Witten, "New Issues In Manifolds Of SU(3) Holonomy," Nucl. Phys. B268 (1986) 79. Is The Superstring Weakly Coupled?. M Dine, N Seiberg, Phys. Lett. 162299M. Dine and N. Seiberg, "Is The Superstring Weakly Coupled?" Phys. Lett. B162 (1985) 299. Nonrenormalization Theorems In Superstring Theory. M Dine, N Seiberg, Phys. Rev. Lett. 57M. Dine and N. Seiberg, "Nonrenormalization Theorems In Superstring Theory," Phys. Rev. Lett. 57 (1986) 2625-8. Fayet-Iliopoulos Terms In String Theory. M Dine, N Seiberg, E Witten, Nucl. Phys. 289589M. Dine, N. Seiberg, and E. Witten, "Fayet-Iliopoulos Terms In String Theory," Nucl. Phys. B289 (1987) 589. Supersymmetric M theory compactifications with fluxes on seven-manifolds and G structures. P Kaste, R Minasian, A Tomasiello, hep-th/0303127JHEP. 03074P. Kaste, R. Minasian and A. Tomasiello, "Supersymmetric M theory compactifications with fluxes on seven-manifolds and G structures," JHEP 0307, 004 (2003), hep-th/0303127. Some Remarks on G 2 -Structures. R L Bryant, arXiv:math/0305124R. L. Bryant,"Some Remarks on G 2 -Structures", arXiv:math/0305124. Flows of G 2 -Structures, I. S Karigiannis, arXiv:math/0702077S. Karigiannis, "Flows of G 2 -Structures, I", arXiv:math/0702077. Local Geometry of the G 2 Moduli Space. S Grigorian, S. -T Yau, arXiv:0802.0723Commun. Math. Phys. 287459S. Grigorian and S. -T. Yau, "Local Geometry of the G 2 Moduli Space", Commun. Math. Phys. 287, 459 (2009), arXiv:0802.0723. Moduli Spaces of G 2 Manifolds. S Grigorian, arXiv:0911.2185Rev. Math. Phys. 221061S. Grigorian, "Moduli Spaces of G 2 Manifolds", Rev. Math. Phys. 22, 1061 (2010), arXiv:0911.2185. String and M-Theory Deformations of Manifolds with Special Holonomy. H Lu, C N Pope, K S Stelle, P K Townsend, hep-th/0410176JHEP. 050775H. Lu, C. N. Pope, K. S. Stelle and P. K. Townsend, "String and M-Theory Deformations of Manifolds with Special Holonomy," JHEP 0507, 075 (2005), hep-th/0410176. Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs series by. D Joyce, Oxford University PressD. Joyce, Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs series by Oxford University Press. Deformations of G 2 and Spin(7) Structures on Manifolds. S Karigiannis, arXiv:math/0301218Canadian Journal of Mathematics. 57S. Karigiannis, "Deformations of G 2 and Spin(7) Structures on Manifolds", Canadian Journal of Mathematics 57, 1012-1055 (2005), arXiv:math/0301218. Superpotentials and Membrane Instantons. J A Harvey, G Moore, hep-th/9907026J. A. Harvey and G. Moore, "Superpotentials and Membrane Instantons," hep-th/9907026. Moduli Spaces And Brane Solitons for M-theory Compactications on Holonomy G 2 Manifolds. J Gutowski, G Papadopoulos, hep-th/0104105Nucl. Phys. 615J. Gutowski and G. Papadopoulos, "Moduli Spaces And Brane Solitons for M-theory Compactications on Holonomy G 2 Manifolds," Nucl. Phys. B615 (2001) 237-265, hep-th/0104105. C Beasley, E Witten, hep-th/0203061A Note On Fluxes And Superpotentials In M-theory Compactifications On Manifolds of G 2 Holonomy. C. Beasley and E. Witten, "A Note On Fluxes And Superpotentials In M-theory Compactifications On Manifolds of G 2 Holonomy," hep-th/0203061. N Hitchin, arXiv:math/0010054The Geometry Of Three-Forms In Six And Seven Dimensions. N. Hitchin, "The Geometry Of Three-Forms In Six And Seven Dimensions," arXiv:math/0010054. K Becker, M Becker, hep-th/0107044Supersymmetry Breaking, M-Theory, and Fluxes. 38K. Becker and M. Becker, "Supersymmetry Breaking, M-Theory, and Fluxes," JHEP 0107 (2001) 038, hep-th/0107044. A Note on compactifications on spin(7) -holonomy manifolds. K Becker, hep-th/0011114JHEP. 01053K. 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[ "Statistical investigation of relationship between spread of coronavirus disease (COVID- 19) and environmental factors based on study of four mostly affected places of China and five mostly affected places of Italy", "Statistical investigation of relationship between spread of coronavirus disease (COVID- 19) and environmental factors based on study of four mostly affected places of China and five mostly affected places of Italy" ]	[ "Soumyabrata Bhattacharjee [email protected] \nThe Assam Royal Global University\nGuwahati -35, AssamIndia\n" ]	[ "The Assam Royal Global University\nGuwahati -35, AssamIndia" ]	[]	COVID-19 is a new type of coronavirus disease which is caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). It originated in China in the month of December 2019 and quickly started to spread within the country. On 31 st December 2019, it was first reported to country office of World Health Organization (WHO) in China. Since then,it has spread to most of the countries around the globe. However, there has been a recent rise in trend in believing that it would go away during summer days, which has not yet been properly investigated. In this paper, relationship of daily number of confirmed cases of COVID-19 with three environmental factors, viz. maximum relative humidity (RHmax), maximum temperature (Tmax) and highest wind speed (WSmax), considering the incubation period, have been investigated statistically, for four of the most affected places of China, viz.Beijing, Chongqing, Shanghai, Wuhan and five of the most affected places of Italy, viz.Bergamo, Cremona, Lodi, Milano. It has been found that the relationship with maximum relative humidity and highest wind is mostly negligible, whereas relationship with maximum temperature is ranging between negligible to moderate.	null	[ "https://arxiv.org/pdf/2003.11277v1.pdf" ]	214,641,369	2003.11277	4d30f0a58b91a1e7ff4d97f64cf960acd813007e	Statistical investigation of relationship between spread of coronavirus disease (COVID- 19) and environmental factors based on study of four mostly affected places of China and five mostly affected places of Italy Soumyabrata Bhattacharjee [email protected] The Assam Royal Global University Guwahati -35, AssamIndia Statistical investigation of relationship between spread of coronavirus disease (COVID- 19) and environmental factors based on study of four mostly affected places of China and five mostly affected places of Italy CoronavirusCOVID-19Environmental factors COVID-19 is a new type of coronavirus disease which is caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). It originated in China in the month of December 2019 and quickly started to spread within the country. On 31 st December 2019, it was first reported to country office of World Health Organization (WHO) in China. Since then,it has spread to most of the countries around the globe. However, there has been a recent rise in trend in believing that it would go away during summer days, which has not yet been properly investigated. In this paper, relationship of daily number of confirmed cases of COVID-19 with three environmental factors, viz. maximum relative humidity (RHmax), maximum temperature (Tmax) and highest wind speed (WSmax), considering the incubation period, have been investigated statistically, for four of the most affected places of China, viz.Beijing, Chongqing, Shanghai, Wuhan and five of the most affected places of Italy, viz.Bergamo, Cremona, Lodi, Milano. It has been found that the relationship with maximum relative humidity and highest wind is mostly negligible, whereas relationship with maximum temperature is ranging between negligible to moderate. affecting life of more than 8000 people in 26 different countries [1]. In 2012, another disease called called Middle East Respiratory Syndrome (MERS), caused by MERS coronavirus (MERS-CoV), started from Saudi Arabia, became an epidemic and spread to 27 countries [2]. In the later part of 2019, an outbreak of pneumonia of unknown cause started appearing in Wuhan, China and was reported to the Country Office of WHO in China on 31 st December 2019. On 30 th January 2020, it was declared as Public Health Emergency of International Concern. On 11 th February 2020, WHO named the disease as coronavirus disease and the virus causing the disease as severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) [3] . On 11 th March 2020, WHO declared COVID-19 as pandemic [4] and by 16 th March 2020, the virus has spread to 156 countries/regions [5] . From 19 th January 2020 onwards, there has been hearsay about relationship between coronavirus and summer [6] . However, there has been a sharp rise in interest shown by people all around the world, in the relationship, as shown by Google Trend [6] , after President of the United States of America, who is currently one of the most influential person in the world [7] , tweeted that the virus might be gone with warmer weather [8] . If such claims are not properly investigated then it might end up being rumour and ultimately hinder the disease control process [9] , which has not yet been done widely. One of the terms to describe severity of any infectious disease is effective reproduction number (R) [10]. Wang et al., tried to develop an equation to estimate the value of R with respect to the value of Temperature (T) and relative humidity (RH) [11], but they developed the equation based on data only from Chinese cities and applied that for places outside China. However, the environment and their effect are different at different places of the globe and the effect of incubation period has also not been considered while developing the equation. Incubation period of COVID-19 has been investigated by Baum et al. and it has been found that the median incubation period is approximately 5 days [12]. In the current paper, the relationship of daily number of confirmed COVID-19 cases, which has been assumed to be reflecting how contagious the disease is, from four of the most affected places of China and five of the most affected places of Italy has been investigated with RHmax, Tmax and WSmax. The environmental factors, that have been considered, are that of 5 days back from the date of reporting the case, to consider the effect of incubation period. Methodology: Since 21 st January, WHO has been publishing daily situation report on COVID- 19 [13]. The data for the Chinese places has been collected from these reports and has been summarised in Table 1. All the weather data, reported in Table 1 Once the data has been collected, then to find out how well the daily number of newly confirmed cases of COVID-19 relates to RHmax, Tmax and WSmax, at first the daily number of cases is plotted against each one of the considered environmental factor and visually inspected, then Pearson's correlation coefficient (r) for each such pair has been calculated [16] using MATLAB (R2019b). After that, the each of the values are interpreted according to the rule of thumb mentioned by Mukaka [17]. To further strengthen the finding, hypothesis test has also been done on them using 95% confidence interval. Null hypothesis, for each such pair, being that the environmental factors do not influence the spread of disease. But, while testing hypothesis, the results are not interpreted depending only on the p-values, as cautioned by Wasserstein et al. [18]. Instead Bayes Factor (BF), for each one of them is calculated using the expression − 1 ln( ) [19] and then the relationship is once again interpreted based on the value of BF as mentioned in the classification scheme by Jamil et al. [20]. Results and discussion: From figure 1 -27, it can be seen that the influence of environmental factors is neither that strong nor that can be outrightly rejected. The value and interpretation of Pearson's correlation coefficient and Bayes Factor is listed in Table 3. Amidst the commotion, a belief is getting popular that the virus would die its own death with the arrival of summer season, but in the current paper, it has been found that the relationship between the effectiveness of virus and different environmental factors is not that strong. Hence, it can be concluded that the virus shows no sign as of now, to become dormant during summer days. The current piece of work is based on preliminary data that's available. A better relation can be predicted when more data become available. Figure 1 :Figure 2 :Figure 3 :Figure 5 :Figure 7 :Figure 8 :Figure 9 :Figure 10 :Figure 11 :Figure 13 :Figure 14 :Figure 15 :Figure 17 :Figure 19 :Figure 20 :Figure 21 :Figure 23 :Figure 27 : 12357891011131415171920212327Effect of maximum relative humidity in Beijing Effect of maximum relative humidity in Chongqing Effect of maximum relative humidity in Shanghai Figure 4: Effect of maximum relative humidity in Wuhan Effect of maximum relative humidity in Bergamo Figure 6: Effect of maximum relative humidity in Brescia Effect of maximum relative humidity in Cremona Effect of maximum relative humidity in Lodi Effect of maximum relative humidity in Milano Effect of maximum temperature in Beijing Effect of maximum temperature in Chongqing Figure 12: Effect of maximum temperature in Shanghai Effect of maximum temperature in Wuhan Effect of maximum temperature in Bergamo Effect of maximum temperature in Brescia Figure 16: Effect of maximum temperature in Cremona Effect of maximum temperature in Lodi Figure 18: Effect of maximum temperature in Milano Effect of maximum wind speed in Beijing Effect of maximum wind speed in Congqing Effect of maximum wind speed in Shanghai Figure 22: Effect of maximum wind speed in Wuhan Effect of maximum wind speed in Bergamo Figure 24: Effect of maximum wind speed in Brescia Effect of maximum wind speed in Milano Table 1 : 1Details of considered places from ChinaSimilarly, the data for Italy has been collected from the official GitHub repository of Department of Civil Protection, Italy [15] and the summary is presented inTable 2along with weather data which is obtained from Weather Underground[14].Date Place Number of additional confirmed cases in last 24 hours RHmax (%) Tmax (°F) WSmax (mph) 20-01-2020 Beijing 3 65 35 4 21-01-2020 Beijing 0 74 36 4 22-01-2020 Beijing 5 75 37 4 02-02-2020 Beijing 27 75 41 4 Table 2 : 2Details of considered places from ItalyDate Place Number of additional confirmed cases in last 24 hours RHmax (%) Tmax (°F) WSmax (mph) 24-02-2020 Bergamo 0 93 57 17 25-02-2020 Bergamo 18 70 55 8 26-02-2020 Bergamo 2 75 55 7 27-02-2020 Bergamo 52 81 55 8 28-02-2020 Bergamo 31 87 61 14 29-02-2020 Bergamo 7 93 61 10 01-03-2020 Bergamo 99 93 57 9 02-03-2020 Bergamo 34 100 55 24 03-03-2020 Bergamo 129 57 52 9 04-03-2020 Bergamo 51 65 57 21 05-03-2020 Bergamo 114 87 52 8 06-03-2020 Bergamo 86 100 46 9 07-03-2020 Bergamo 138 100 48 13 08-03-2020 Bergamo 236 100 54 10 09-03-2020 Bergamo 248 87 54 8 10-03-2020 Bergamo 227 100 50 9 11-03-2020 Bergamo 343 100 48 8 12-03-2020 Bergamo 321 93 59 9 13-03-2020 Bergamo 232 76 55 8 24-02-2020 Brescia 0 93 57 17 Table 3 : 3Values and interpretation of statistical indicators Conclusion: COVID-19 started in the month of December 2019, from China and is rapidly spreading to different countries of the world. Millions of people have already been infected by the virus SARS-CoV-2.Country Place Statistical Indicator Highest RH on that day HIGHEST TEMP on that day (FAHRENHEIT) HIGHEST WIND SPEED on that day (mph) China Beijing r 0.054 Negligible -0.2335 Negligible -0.0388 Negligible BF 1.575587437 Negligible 1.429067674 Negligible 2.064372541 Negligible Chongqing r 0.0344 Negligible -0.3925 Low negative -0.1919 Negligible BF 2.290063003 Negligible 9.817365187 Moderate 1.129149822 Negligible Shanghai r -0.0497 Negligible -0.325 Low negative -0.149 Negligible BF 1.682343457 Negligible 3.547988527 Moderate 1.00598094 Negligible Wuhan r 0.0282 Negligible 0.0421 Negligible -0.1338 Negligible Acknowledgement: Declarations of interest: noneBibliography: WHO \| SARS (Severe Acute Respiratory Syndrome). 16"WHO \| SARS (Severe Acute Respiratory Syndrome)." [Online]. Available: https://www.who.int/ith/diseases/sars/en/. [Accessed: 16-Mar-2020]. 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[ "Charge-spin current conversion in high quality epitaxial Fe/Pt systems: isotropic spin Hall angle along different in-plane crystalline directions", "Charge-spin current conversion in high quality epitaxial Fe/Pt systems: isotropic spin Hall angle along different in-plane crystalline directions" ]	[ "C Guillemard \nUMR 7198\nUniversité de Lorraine\nCNRS\nInstitut Jean Lamour\nF-54011NancyFrance\n", "S Petit-Watelot \nUMR 7198\nUniversité de Lorraine\nCNRS\nInstitut Jean Lamour\nF-54011NancyFrance\n", "S Andrieu \nUMR 7198\nUniversité de Lorraine\nCNRS\nInstitut Jean Lamour\nF-54011NancyFrance\n", "J.-C Rojas-Sánchez \nUMR 7198\nUniversité de Lorraine\nCNRS\nInstitut Jean Lamour\nF-54011NancyFrance\n" ]	[ "UMR 7198\nUniversité de Lorraine\nCNRS\nInstitut Jean Lamour\nF-54011NancyFrance", "UMR 7198\nUniversité de Lorraine\nCNRS\nInstitut Jean Lamour\nF-54011NancyFrance", "UMR 7198\nUniversité de Lorraine\nCNRS\nInstitut Jean Lamour\nF-54011NancyFrance", "UMR 7198\nUniversité de Lorraine\nCNRS\nInstitut Jean Lamour\nF-54011NancyFrance" ]	[]	We report the growth of MgO[001]//Fe(6 nm)/MgO(7 nm) and MgO[001]//Fe(6 nm)/Pt(6 nm) by molecular beam epitaxy and show that the full characterization by spin-orbit ferromagnetic resonance (SO-FMR) allows the determination of magnetic anisotropies as by classical FMR-only studies. The spin mixing conductance of epitaxial Fe/Pt interface was measured to be 19 ffect	10.1063/1.5079236	[ "https://arxiv.org/pdf/1810.08716v2.pdf" ]	53,450,121	1810.08716	acbf267b633a94721a13b364cc9001be40ee9012	Charge-spin current conversion in high quality epitaxial Fe/Pt systems: isotropic spin Hall angle along different in-plane crystalline directions C Guillemard UMR 7198 Université de Lorraine CNRS Institut Jean Lamour F-54011NancyFrance S Petit-Watelot UMR 7198 Université de Lorraine CNRS Institut Jean Lamour F-54011NancyFrance S Andrieu UMR 7198 Université de Lorraine CNRS Institut Jean Lamour F-54011NancyFrance J.-C Rojas-Sánchez UMR 7198 Université de Lorraine CNRS Institut Jean Lamour F-54011NancyFrance Charge-spin current conversion in high quality epitaxial Fe/Pt systems: isotropic spin Hall angle along different in-plane crystalline directions 1 We report the growth of MgO[001]//Fe(6 nm)/MgO(7 nm) and MgO[001]//Fe(6 nm)/Pt(6 nm) by molecular beam epitaxy and show that the full characterization by spin-orbit ferromagnetic resonance (SO-FMR) allows the determination of magnetic anisotropies as by classical FMR-only studies. The spin mixing conductance of epitaxial Fe/Pt interface was measured to be 19 ffect . We address about the proper conditions to determine those relevant spintronics parameters. TEXT The conversion of spin current into charge current and vice versa play key roles in new research efforts in spintronics and related applications. This interconversion can be achieved without any external magnetic field or magnetic material in 3-Dimensional systems that exhibit strong spin-orbit coupling [1], namely the spin Hall effect (SHE) [2][3][4]. The quantification of the efficiency of such interconversion is called the spin Hall angle (SHA). The spin Hall angle determination is thus relevant to find out new materials for applications like in magnetic memories because it will allow reducing power consumption. Large spin Hall angles have been found in heavy metals like Pt [5][6][7][8], Ta [9] and W [10,11], and alloys like CuBi [12], AuW and AuTa [13,14]. So far the quantification has been evaluated mostly on sputtered polycrystalline samples. In a heavy metal or alloy layer which is in contact with a layer of a different material a injected spin current might decrease through the interface due to interfacial interactions and the spin Hall angle becomes and effective spin Hall angle ffect SHE e  [8,15]. There are several techniques available to evaluate the effective spin Hall angle, or effective spin orbit torque (SOT), like spin pumping ferromagnetic resonance [7,[16][17][18][19], spin-orbit ferromagnetic resonance [6,[20][21][22][23], no-local injection in lateral spin valve, current-induced magnetization switching, harmonic measurements, spin Hall magnetoresistance and so on. In all these experiments, in the measurements as well as in the analysis, there are many details and approximations that are often overlooked. In this paper we focus on spin-orbit ferromagnetic resonance (SO-FMR), technique to study epitaxial samples. Sample growth Epitaxial s//Fe(6 nm)/Pt (5 nm [24]. In the following we will refer only to the MgO crystalline axes. FMR on epitaxial s//Fe/MgO Small pieces of about few mm 2 were extracted from the center region of s//Fe/MgO sample to avoid edge deposition issues and measured by FMR. A grounded coplanar wave guide (GCPW) was used as shown in the Fig. 2a. Hence the radio-frequency magnetic field was maintained transversal (along y direction) to the dc magnetic field H dc (along x). DC field was applied parallel to the film plane in the present study. The frequency and input power of the microwave source was fixed and H dc was swept around the resonance field as shown in Fig. 2b. The transmitted power is detected using a power diode detector and lock-in technique. The amplitude of the rf power was modulated at f LO (= 433 Hz) and the output signal of the diode is measured at f LO . A typical FMR spectrum is shown in the Fig. 2b when H dc is applied parallel to the [100]MgO crystalline axis. The observed symmetrical Lorentz curve verifies that, in this geometry, the absorbed microwave power is proportional to the imaginary part of the magnetic susceptibility  yy " with practically no losses due to dispersion in the GCPW. We can identify the resonance field H res , and the linewidth H (half width at half maximum). Broadband frequency dependence was studied when H dc is applied along different crystalline directions. The fdependence of the resonance field and linewidth was analyzed and plotted in It is worth to note that every 90 degrees we recover the same relationship without any shift (vertical neither horizontal). This means the lack of additional in-plane uniaxial anisotropy, which is an experimental verification of a high quality sample. After minimizing numerically the free density energy of the system we get the equilibrium position of the magnetization and the resonance condition dispersion following the results of Smit and Beljers [25], and Suhl [26], which formalism has been developed in detail elsewhere [27]. Thus the effective magnetic saturation M eff is 1450 emu/cm 3 = 1450 kA/m, and the cubic magnetocrystalline field  0 H cub is -48 mT. We observe that the magnetic damping constant is the same along different crystalline directions with a value of Fe 0.0041 0.0004  . It is worth to note here that the frequency-independent contribution due to inhomogeneity is quite low,  0 H 0  0.4 mT, is another verification of high quality epitaxial growth. SO-FMR on epitaxial s//Fe/Pt In order to evaluate the charge-spin current conversion spin-orbit ferromagnetic resonance (SO-FMR), was used. This is also known as spin transfer FMR or spin diode FMR, which is somehow the reciprocal dynamic effect of spin pumping FMR. We inject directly the microwave frequency charge current in the s//Fe/Pt slabs which is converted into spin current inside the Pt layer due to the SHE. Therefore an oscillating spin current is injected from Pt into Fe layer inducing precession of its magnetization. This, in turn, leads to an oscillatory radiofrequency resistance which mixed with the rf current allows, at the resonance condition, dc voltage detection across the slab using a bias tee [6,[20][21][22][23]. The dc voltage is picked up at 45° of the applied H dc (see inset in Figure 3). The dc voltage is composed of a mix between a symmetrical Lorentzian function and an antisymmetrical one around the resonance field H res . The amplitude of each contribution is V sym and V anti . We use the following general function to fit the mixed voltage measured considering also an offset V offset :            (1) The slabs for SO-FMR were patterned by standard UV lithography and have lateral sizes of 20 m  90 m. Ti/Au electrodes were deposited by lift off technique. In the inset of figure 3(a) is shown a picture of one device along with grounded-signal-grounded, GSG, rf contacts. Figure 3 shows raw data of the dc voltage measured when H dc is applied along different crystalline axes. After the analysis of this broadband frequency dependence results we show that by SO-FMR we can also account for the in-plane magnetic anisotropies, i. which is lower than previously reported for FM/Pt systems, even in epitaxial Fe/Pt samples studied by spin pumping [29,30]. Critical for this result is that the Fe layer is exactly the same for both reference Fe/MgO and Fe/Pt bilayer which is crucial especially in epitaxial samples where damping is much more sensitive to any defect (which is not the case in polycrystalline samples). Future work will include measuring the reference damping with the same SO-FMR method using a structure like Fe/Cu/MgO and thickness dependence. In the simplest model, the quantification of the effective spin Hall angle ffect SHE e  is proportional to the ratio of the symmetrical Lorentzian voltage over the anti-Lorentzian one, V symm /V anti . When the resonance field is large enough to consider uniform precession of the magnetization we can use the simplest model to calculate the effective spin Hall angle [6,21- where we have highlighted the condition of uniform precession of the magnetization ( res sat HH  ), with H sat being the saturation field. In the in-plane hard axes, the field necessary to get the saturation state is above 0.1 T, obtained after M(H) cycles in PPMS-VSM (not shown). The saturation magnetization is M s = 1675 emu/cm 3 , close to the bulk value (1714 emu/ cm 3 ) [31]. We can calculate the four-fold magnetocrystalline anisotropy ( cub cub s /2 K H M  ). It results in our Fe film that 5 cub 3.9 10 K  erg/ cm 3 , below the bulk value of Fe ( 5 4.8 10 erg/ cm 3 ) [31]. As shown in Figure 4(c) and using eq. (3), we find that for Pt in Fe/Pt system ffect SHE 0.051 0.008 e  , which is similar that previous effectives values reported [6,8]. Moreover, within the experimental resolution, we have found similar values when H is applied along different crystalline directions once it is reached the condition: res sat HH  as displayed in figure 4(c). Additionally, we have grown s//Fe(6)/Pt(t) with t between 1 and 10 nm, and then patterned in double Hall bar to measure the electrical resistance. From the Ptthickness dependence of such results we have estimated the resistivity of Pt layer whose value is Pt 15.6 0.5 μΩ.cm   along different in-plane crystalline directions. It has been shown that Elliot-Yafet mechanism dominates the conduction electron spin relaxation in Pt [15,32,33] where the product Pt sf l   is a constant. Experimental values of such a product [8,32] [8]. Some other systems that showed isotropic results despite in-plane anisotropies can be found in the case of in-plane exchange bias and spin-orbit torque in AFM/FM [34], or isotropic damping in epitaxial Fe [35], for similar level of thicknesses. However it just has been shown that for ultrathin Fe layer (0.8 nm) we can detect anisotropic magnetic damping [35], as predicted theoretically for epitaxial magnetic layers [36]. Ultrathin magnetic layers are out of the scope of the present study but it is a very interesting perspective to enhance this study especially with epitaxial Heusler alloys showing ultralow damping [37]. We observe that for f > 10 GHz (>15 GHz) along the hard (easy) axes it reaches a constant value (as expected it does not depend on the frequency). The resonance field for such a frequencies are above 0.1 T, i.e., well above saturation field so uniform precession of magnetization is reached. In summary, we have performed FMR and SO-FMR in thin highly epitaxial, Fe and Fe/Pt samples respectively. From both methods we show that one can access detailed magnetic anisotropies. Particularly we have accounted for the cubic magnetocrystalline anisotropy of Fe. Using exactly the same Fe bottom layer, a low value of spin mixing conductance in Fe/Pt was measured. We show that considering uniform precession of the magnetization (resonance field above saturation field), the effective spin Hall angle can be determined in a reliable way for FM/Pt systems, and, it is independent of the microwave frequency. Furthermore, we show that the magnetic Gilbert damping constant as well as the effective spin Hall angle are isotropic in epitaxial system for the level of thickness in the present study, Fe(6 nm)/Pt(5 nm). Our results also highlight the importance of taking care of the proper conditions to the estimation of the effective charge-spin current conversion efficiency or spin Hall angle, and the effective spin mixing conductance in epitaxial FM/HM systems. Those parameters and their proper determination are relevant for spintronics applications. -plane crystalline directions. It was found that ffect SHE e  is the same in all directions. When taking into account high enough excitation frequencies to achieve uniform precession of magnetization, the effective spin hall angle for epitaxial Pt in Fe/Pt is Fig. 1 : 1(color online) (a) Schematic of the full stack that has been grown by MBE. After the deposition of the Fe layer, half of the sample was cover to deposit 7 nm of MgO or 5 nm of Pt. b) RHEED pattern shown the good quality and 2D grown of Fe as well as the Pt layers. c) The cubic Fe cell grown rotated by 45° on top of the cubic cell of MgO. Fig. 2 ( 2c) and (d), respectively. We can observe in the Fig. 2(c) the typical relationship dispersion f vs. H in a system with cubic magnetocrystalline anisotropy where the in-plane easy axes are along the [110]MgO directions and the in-plane hard axes are along [100]MgO ones. Figure 2 : 2(a) Picture of the grounded coplanar wave guide GCPW used along with GSG radiofrequency contacts. The sample position, the direction of the dc magnetic field, H dc , and the direction of the rf field, h rf , in the middle of the signal line are also indicated. (b) Typical Figure 3 . 3Raw data of the SO-FMR scans. (a) H dc is applied parallel to the [010]MgO crystalline direction (scans shown only for f  6 GHz). The inset is a picture of a device along with Ti/Au electrodes and GSG rf contacts. (b) H dc is applied parallel to the [110]MgO crystalline direction. H dc is always applied at 45° of the slab. Figure 4 . 4Results of the SO-FMR study. (a) Dispersion relationship and magnetic anisotropies determination. (b) Linewidth and damping constant determination. (c) Determination of effective spin Hall angle. ) we can calculate the effective spin mixing conductance ff e g  following standard spin pumping theory[8,13,28]:e, the easy (hard) axes when H dc is parallel to [110]MgO ([100]MgO). That is summarized in figure 4(a) where we can identify similar results of only FMR as observed in Fig. 2(c). This means that despite the pick-up voltage is at 45° of the applied field in the SO-FMR experiment, what matters is the direction along which is applied H dc with respect to the crystalline axis of the samples. So far in our knowledge, this is the first time that it is shown such results in a system with cubic anisotropy. The magnetic damping in Fe/Pt results Fe/Pt 0.0065 0.0004   as shown in figure 4(b). With the results from the Fe reference layer ( Fe 0.0041 0.0004    eff Fe eff Fe/Pt Fe B 4 Mt g g      (2) Fe  is quite large. To compute ff e g  we use the difference of the magnetic damping Fe/Pt Fe ()   which gives the contribution due to spin pumping only since the Fe layer is a true reference sample as described above. 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[ "Coherent and semiclassical states of a charged particle in electromagnetic fields", "Coherent and semiclassical states of a charged particle in electromagnetic fields" ]	[ "A S Pereira \nDepartment of Higher Mathematics and Mathematical Physics\nInstitute of Physics and Technology\nNational Research Tomsk Polytechnic University\nTomskRussia\n" ]	[ "Department of Higher Mathematics and Mathematical Physics\nInstitute of Physics and Technology\nNational Research Tomsk Polytechnic University\nTomskRussia" ]	[]	In the present article we extend our study (BJP 45 (2015) 369) of generalized coherent states (GCS) of a one-dimensional particle considering such important physical system as a 3-dimensional charged particle in electric and magnetic fields. Constructing GCS in many-dimensional case, we meet nontrivial technical complications that make the consideration nontrivial and instructive.The GCS of a system under consideration are constructed. We study properties of these GCS such as completeness relations, minimization of uncertainty relations and so on. We point out which family of the obtained GCS of a charged particle in magnetic field is related with the CS constructed first by Malkin and Man'ko. We obtain conditions under which some of the GCS can be considered as semiclassical states. * Electronic address: [email protected] arXiv:1711.07406v2 [quant-ph]	10.1007/s13538-018-0562-z	[ "https://arxiv.org/pdf/1711.07406v2.pdf" ]	119,348,433	1711.07406	1708646e070b34461063cf24885e165f2af4d715	Coherent and semiclassical states of a charged particle in electromagnetic fields 21 Nov 2017 A S Pereira Department of Higher Mathematics and Mathematical Physics Institute of Physics and Technology National Research Tomsk Polytechnic University TomskRussia Coherent and semiclassical states of a charged particle in electromagnetic fields 21 Nov 2017 In the present article we extend our study (BJP 45 (2015) 369) of generalized coherent states (GCS) of a one-dimensional particle considering such important physical system as a 3-dimensional charged particle in electric and magnetic fields. Constructing GCS in many-dimensional case, we meet nontrivial technical complications that make the consideration nontrivial and instructive.The GCS of a system under consideration are constructed. We study properties of these GCS such as completeness relations, minimization of uncertainty relations and so on. We point out which family of the obtained GCS of a charged particle in magnetic field is related with the CS constructed first by Malkin and Man'ko. We obtain conditions under which some of the GCS can be considered as semiclassical states. * Electronic address: [email protected] arXiv:1711.07406v2 [quant-ph] I. INTRODUCTION Coherent states (CS) play an important role in modern quantum theory as states that provide a natural relation between quantum mechanical and classical descriptions. They have a number of useful properties and as a consequence a wide range of applications, e.g., in semiclassical description of quantum systems, in quantization theory, in condensed matter physics, in quantum computations, and so on, see e.g., [1][2][3][4][5][6][7]. Starting by the works [8,9] CS are defined as eigenvectors of some annihilation operators that are at the same time integrals of motion. Of course such defined CS have to satisfy the corresponding Schrödinger equation. In the frame of such a definition one can in principle construct CS for a general quadratic system. In the article [10] we, following these scheme, constructed different families of CS (GCS) for one-dimensional systems with general time-dependent quadratic Hamiltonian, see too [11,12]. In the article [12], we have demonstrated that the GCS of a free particle can be treated as quantum states that describe a semiclassical motion, whereas there exist GCS that describe pure quantum motion. In the present article we extend our study beyond one-dimensional systems, considering such important physical system as a 3-dimensional charged particle in electric and magnetic fields. Considering many-dimensional systems we meet nontrivial technical complications that makes the consideration nontrivial and instructive. We discuss properties of the constructed GCS such as completeness relations, minimization of uncertainty relations and so on. As in the one-dimensional case, we succeeded to find conditions that allow one to attribute the GCS either to the class of semiclassical quantum states or to purely quantum states. This article is organized as follows. In Section 2, we outlook classical and quantum descriptions of a charged particle in parallel electric and magnetic fields and obtain for such a system integrals of motion which are creation and annihilation operators. In section 3, we construct GCS that satisfies the Schrödinger equation and calculate the mean values, standard deviations and uncertainty relations, as well as we discuss properties of the constructed GCS such as completeness relations, minimization of uncertainty relations, and so on. We also demonstrated that the GCS of a charged particle in magnetic field is related with the CS constructed first by Malkin and Man'ko. In section 4, we obtain conditions under the electric and magnetic field such that the GCS can be considered as semiclassical states or purely quantum states. II. CHARGED PARTICLE IN PARALLEL ELECTRIC AND MAGNETIC FIELDS A. Classical and quantum equations of motion Consider a charged particle 1 , with total charge e, moving in a 3-dimensional Euclidean space, H = P 2 2m + eA 0 , P = p − e c A,(1) interacting with constant, uniform and parallel electric E and magnetic B fields E = (0, 0, E) , B = (0, 0, B) , A 0 = −zE sin 2 α, A = 1 2 −By, Bx, −2ctE cos 2 α , α ∈ [0, π/2] .(2) The corresponding Hamiltonian can be written in the following form H = H xy + H z , H xy = p 2 ⊥ 2m + mω 2 r 2 ⊥ 8 − ω 2 L, H z = p 2 z 2m − mξz sin 2 α + ξp z t cos 2 α + mξ 2 2 t 2 cos 4 α, L = xp y − yp x , ω = eB mc , ξ = eE m .(3) Here p is the canonical momenta conjugated to the coordinates r, ω is the cyclotron frequency, and ξ denotes the acceleration along the z-axis. The velocity v is related to the canonical variables as mv = P. The division of H into two separate parts, H xy and H z , indicates implicitly the independence between particle's motion on the xy-plane (hereafter referred as xy-motion) from the motion along the z-direction (hereafter referred as z-motion). This fact is explicitly [1] Throughout the text, e denotes the algebraic electric charge (e = − \|e\| for an electron), bold letters represent vectors, e. g., r = (x, y, z), latin indices are j, k, ... = 1, 2, 3 and greek indices are β, η, ... = 1, 2. confirmed by the structure of canonical equations of motion 2 v x (t) =ẋ (t) = ∂H ∂p x = p x m + ω 2 y,ṗ x (t) = − ∂H ∂x = ω 2 p y − mω 2 4 x, v y (t) =ẏ (t) = ∂H ∂p y = p y m − ω 2 x,ṗ y (t) = − ∂H ∂y = − ω 2 p x − mω 2 4 y, v z (t) =ż (t) = ∂H ∂p z = p z m + ξt cos 2 α,ṗ z (t) = − ∂H ∂z = mξ sin 2 α.(4) The general solution for the z-motion follows from the third line of Eq. (4), z (t) = z 0 + v 0 z t + 1 2 ξt 2 , p z (t) = m v 0 z + ξt sin 2 α , z 0 = z (0) , v 0 z = v z (0) = p 0 z m , p 0 z = p z (0) ,(5) corresponds to an accelerated motion along the z-direction. As for the xy-motion, it can be presented in two equivalent ways. The first one is given in terms of the initial Cauchy data (r ⊥ (0) = r 0⊥ , v ⊥ (0) = v 0 ⊥ ), x (t) = x 0 + v 0 y 1 − cos (ωt) ω + v 0 x sin (ωt) ω , p x (t) m = v 0 x cos (ωt) + v 0 y sin (ωt) + v 0 x − ωy 0 2 , y (t) = y 0 − v 0 x 1 − cos (ωt) ω + v 0 y sin (ωt) ω , p y (t) m = v 0 y cos (ωt) − v 0 x sin (ωt) + v 0 y + ωx 0 2 ,(6) which can be equivalently given in terms of the perpendicular velocity components, x (t) = x 0 − v y (t) − v 0 y ω , y (t) = y 0 + v x (t) − v 0 x ω , p x (t) = m v x (t) + v 0 x − ωy 0 2 , p y (t) = m v y (t) + v 0 y + ωx 0 2 , v x (t) = v 0 y sin (ωt) + v 0 x cos (ωt) , v y (t) = v 0 y cos (ωt) − v 0 x sin (ωt) .(7) In a second representation, the xy-motion is parametrized in terms of the coordinates of the center of the orbit (x c , y c ), its radius R and the initial phase θ 0 instead the initial Cauchy data, x (t) = x c + R cos θ, y (t) = y c + R sin θ, θ = ωt + θ 0 , x c = R c cos θ c , y c = R c sin θ c , R 2 c = x 2 c + y 2 c , R 2 = (x − x c ) 2 + (y − y c ) 2 ,(8) [2] Quantities associated with the xy-motion are labelled with the symbol "⊥" (e. g., r ⊥ = (x, y)) while quantities associated with the z-motion are labelled with the symbol " " (e. g., r = z). as illustrated in Fig. 1. In this representation, the distance from the origin r (t) can be expressed in terms of R, R c and the relative angle θ − θ c as r 2 = x 2 + y 2 = R 2 + R 2 c + 2RR c cos (θ − θ c ) ,(9) whose maximum r max = R + R c and minimum r min = \|R − R c \| correspond to configurations in which θ = θ c and θ = θ c + π, respectively. From the perpendicular velocities v ⊥ (t) = (ẋ (t) ,ẏ (t)),ẋ (t) = v x (t) = −Rω sin θ = −ω (y − y c ) ,y (t) = v y (t) = Rω cos θ = ω (x − x c ) ,(10) one can readily see that both sets of solutions can be mapped to each other through the following relations between the initial conditions x 0 , y 0 , v 0 x , v 0 y and the integration constants (x c , y c , θ 0 , R) as x 0 = x c + R cos θ 0 , y 0 = y c + R sin θ 0 , v 0 x = −Rω sin θ 0 , v 0 y = Rω cos θ 0 .(11) Eqs. (8) and (10) may be written as follows x (t) = x c + 1 ω 2E ⊥ m cos θ, y (t) = y c + 1 ω 2E ⊥ m sin θ, v x (t) = − 2E ⊥ m sin θ, v y (t) = 2E ⊥ m cos θ,(12) where the perpendicular energy E ⊥ is a conserved quantity, E ⊥ = m v 2 x + v 2 y 2 = m (v 0 x ) 2 + v 0 y 2 2 = mR 2 ω 2 2 .(13) It is worth noting that in the limit of zero magnetic field, the xy-motion tends to a uniform motion according to Eq. (6), while diverges according to Eqs. (12), due to the ω −1 singularity. At a first sight, this might be source of discrepancy but, to solve the Cauchy problem, one has to specify the initial coordinates x 0⊥ and velocities v 0 ⊥ . This means that the integration constants x c , y c and θ 0 must be expressed in terms of the latter constants. In fact such, solving Eqs. (11) for x c , y c , and θ 0 and substituting into Eqs. (12), the divergence is eliminated. The quantum nonrelativistic motion of the system under consideration is described by the corresponding Schrödinger equation i ∂ t Ψ (r, t) =ĤΨ (r, t) ,Ĥ =Ĥ xy +Ĥ z , ∂ t = ∂ ∂t , H xy =p 2 ⊥ 2m + mω 2 8 r 2 ⊥ − ω 2L ,L = xp y − yp x ,p = −i ∇, H z =p 2 z 2m − mξz sin 2 α + ξp z t cos 2 α + mξ 2 2 t 2 cos 4 α.(14) Introducing dimensionless variables q,π, and τ as q = l −1 r , τ = ml 2 t,π = l p,Π = l P →v = lmΠ , Ω = ml 2 ω, Ξ = m 2 l 3 2 ξ, q k ,Π j = [q k ,π j ] = iδ kj ,(15) we pass to an equivalent Schrödinger equation SΦ (q, τ ) = 0,Ŝ = ∂ τ + i H ⊥ + H , Φ (q, τ ) = √ l 3 Ψ lq, ml 2 τ , H ⊥ =π 2 ⊥ 2 + Ω 2 q 2 ⊥ 8 − Ω 2L ,L = q 1π2 − q 2π1 ,Ĥ xy = 2 ml 2 H ⊥ , H =π 2 3 2 − Ξq 3 sin 2 α + Ξτπ 3 cos 2 α + Ξ 2 τ 2 2 cos 4 α,Ĥ z = 2 ml 2 H .(16) B. Special sets of integrals of motion Following well-known method [8][9][10], we are going to find special setsÂ j (τ ) of quantum integrals of motion in the problem under consideration. They have to be linear combinations of basic operators q,π and be, at the same time, annihilation and creation operators 3 , A j (τ ) = f jk (τ ) q k + ig jk (τ )π k √ 2 + ϕ j (τ ) ,(17)Â j (τ ) ,Â † k (τ ) = δ jk , Â j (τ ) ,Â k (τ ) = Â † j (τ ) ,Â † k (τ ) = 0, ∀τ.(18) Here, the coefficients f jk (τ ), g jk (τ ) and ϕ j (τ ) are some time-dependent functions. One can easily see that bothÂ j (τ ) andÂ † j (τ ) are integrals of motion if d τÂj (τ ) = Ŝ ,Â j (τ ) = 0.(19) Since the xy-motion is independent from the z-motion, the functions f 3k (τ ) and g 3k (τ ) can be chosen as f 3k (τ ) = f 3 (τ ) δ 3k , g 3k (τ ) = g 3 (τ ) δ 3k ,(20) which means that the operatorÂ 3 (τ ) depends on the operators q 3 andπ 3 only. Under such a condition the unknown functions f jk (τ ), g jk (τ ) and ϕ j (τ ) must obey the following equationṡ f β1 (τ ) = Ω 2 f β2 (τ ) + iΩ 2 4 g β1 (τ ) ,ġ β1 (τ ) = if β1 (τ ) + Ω 2 g β2 (τ ) ,φ β (τ ) = 0, f β2 (τ ) = − Ω 2 f β1 (τ ) + iΩ 2 4 g β2 (τ ) ,ġ β2 (τ ) = if β2 (τ ) − Ω 2 g β1 (τ ) , f 3 (τ ) = 0,ġ (τ ) = if 3 (τ ) ,φ 3 (τ ) = Ξ g 3 (τ ) sin 2 α − iτ f 3 (τ ) cos 2 α i √ 2 .(21) [3] The summation convention over repeated scripts is assumed throughout, unless otherwise explicitly stated. In addition, it follows from Eqs. (18) and (19) that functions f jk (τ ) and g jk (τ ) are subjected to the following conditions f jk (τ ) g * k k (τ ) + f * k k (τ ) g jk (τ ) = f jk (0) g * k k (0) + f * k k (0) g jk (0) = 2δ jk , f jk (τ ) g k k (τ ) − f k k (τ ) g jk (τ ) = f jk (0) g k k (0) − f k k (0) g jk (0) = 0.(22) The general solution of (21) has the form f β1 (τ ) = iΩ 2 c β1 − ib β2 1 + cos (Ωτ ) 2 − ib β1 2 sin (Ωτ ) , ϕ β (τ ) = ϕ β (0) , f β2 (τ ) = − iΩ 2 c β2 − ib β1 1 + cos (Ωτ ) 2 + ib β2 2 sin (Ωτ ) , g β1 (τ ) = c β2 + b β1 1 − cos (Ωτ ) Ω + b β2 sin (Ωτ ) Ω , g β2 (τ ) = c β1 − b β2 1 − cos (Ωτ ) Ω + b β1 sin (Ωτ ) Ω , f 3 (τ ) = f 0 , g 3 (τ ) = g 0 + if 0 τ, ϕ 3 (τ ) = − f 0 τ 2 + ig 3 (τ ) sin 2 α Ξτ √ 2 ,(23) where c βη , b βη , f 0 and g 0 are some constants. Without any loss of generality, we can set ϕ j (0) = 0. It follows from (19) that a common basis forŜ andÂ j (τ ) can be found in the form S \|ζ, τ = λ ζ (τ ) \|ζ, τ ,(24)A j (τ ) \|ζ, τ = ζ j \|ζ, τ ,(25) wherein ζ = (ζ 1 , ζ 2 , ζ 3 ) are complex numbers and λ ζ (τ ) is an arbitrary time-dependent function. III. GENERALIZED COHERENT STATES We define the GCS as solutions of the Schrödinger equation that are eigenstates of the operatorsÂ j (τ ) with eigenvalues ζ j = ζ ⊥ , ζ 3 , ζ ⊥ = ζ 1 , ζ 2 . One can represent the GCS as Φ ζ (q, τ ) = q\|ζ, τ = Φ ζ ⊥ (q ⊥ , τ ) Φ ζ 3 (q 3 , τ ) .(26) Then, to obey three conditions (25), the functions Φ ζ ⊥ (q ⊥ , τ ) and Φ ζ 3 (q 3 , τ ) have to satisfy the following equationŝ A β Φ ζ ⊥ (q ⊥ , τ ) = f β1 q 1 + f β2 q 2 + g β1 ∂ q 1 + g β2 ∂ q 2 √ 2 Φ ζ ⊥ (q ⊥ , τ ) = ζ β Φ ζ ⊥ (q ⊥ , τ ) ,(27)A 3 Φ ζ 3 (q 3 , τ ) = f 0 q 3 + g 3 ∂ q 3 √ 2 + ϕ 3 Φ ζ 3 (q 3 , τ ) = ζ 3 Φ ζ 3 (q 3 , τ ) .(28) The general solution of Eq. (28) reads Φ ζ 3 (q 3 , τ ) = exp − f 0 g 3 q 2 3 2 + √ 2 ζ 3 − ϕ 3 g 3 q 3 + iφ 3 (τ ) ,(29) where φ 3 (τ ) is an arbitrary function of τ . The general solution of the set (27), see Appendix A, has the form Φ ζ ⊥ (q ⊥ , τ ) = exp − G 1 q 2 1 + G 2 q 2 2 2 + F q 1 q 2 + √ 2 (Q 1 q 1 − Q 2 q 2 ) + iφ ⊥ (τ ) ,(30) where G 1 = f 11 g 22 − f 21 g 12 g 11 g 22 − g 12 g 21 , G 2 = f 22 g 11 − f 12 g 21 g 11 g 22 − g 12 g 21 , Q 1 = g 22 ζ 1 − g 12 ζ 2 g 11 g 22 − g 12 g 21 , Q 2 = g 21 ζ 1 − g 11 ζ 2 g 11 g 22 − g 12 g 21 , F = f 11 g 21 − f 21 g 11 g 11 g 22 − g 12 g 21 ,(31) and φ ⊥ (τ ) is an arbitrary time-dependent function. Thus, Φ ζ (q, τ ) = Φ ζ ⊥ (q ⊥ , τ ) Φ ζ 3 (q 3 , τ ) = exp − G 1 q 2 1 + G 2 q 2 2 2 + F q 1 q 2 + √ 2 (Q 1 q 1 − Q 2 q 2 ) − f 0 g 3 q 2 3 2 + √ 2 ζ 3 − ϕ 3 g 3 q 3 + iφ (τ ) ,(32) being φ (τ ) ≡ φ 3 (τ ) + φ ⊥ (τ ) some function of τ , such that Φ ζ (q, τ ) satisfies the Schrödinger equation (16). This means that the function φ (τ ) must be given by φ (τ ) =φ 1 (τ ) + i 2 ln g 3 − i ln C, φ 1 (τ ) = Q 2 1 + Q 2 2 − G 1 + G 2 2 + ζ 3 − ϕ 3 g 3 + iΞτ √ 2 cos 2 α 2 dτ,(33) where C is a normalization constant, which can be found from the normalization condition. The normalized function Φ ζ (q, τ ) reads: Φ ζ (q, τ ) = Re G 1 Re G 2 − Re 2 F π 3 g 2 3 1/4 exp i Reφ 1 (τ ) − \|g 3 \| 2 Re 2 ζ 3 − ϕ 3 g 3 − Re G 2 Re 2 Q 1 − 2 Re Q 1 Re Q 2 Re F + Re G 1 Re 2 Q 2 Re G 1 Re G 2 − Re 2 F − G 1 q 2 1 + G 2 q 2 2 2 − f 0 g 3 q 2 3 2 + F q 1 q 2 + √ 2 (Q 1 q 1 − Q 2 q 2 ) + √ 2 ζ 3 − ϕ 3 g 3 q 3 .(34) The family of states Φ ζ (q, τ ) is parametrized by constants c βη , b βη , f 0 and g 0 . It is possible to see from Eqs. (21) that the following relations hold g 12 (τ ) = ig 11 (τ ) ⇐⇒ f 12 (τ ) = if 11 (τ ) & g 21 (τ ) = ig 22 (τ ) ⇐⇒ f 21 (τ ) = if 22 (τ ) .(35) They imply, in turn, relations c 12 = −ic 11 , c 21 = −ic 22 , b 12 = −ib 11 , b 21 = −ib 22 ,(36) and f 11 (τ ) = iΩ 2 c 11 − b 11 1 + e iΩτ 2 , f 22 (τ ) = − iΩ 2 c 22 − b 22 1 + e −iΩτ 2 , g 11 (τ ) = −ic 11 + b 11 1 − e iΩτ Ω , g 22 (τ ) = −ic 22 − b 22 1 − e −iΩτ Ω .(37) In addition, it follows from Eqs. (22) that 2 Re (f 11 g * 11 ) = 2 Im (b 11 c * 11 ) − Ω \|c 11 \| 2 = 1, Re (f 0 g * 3 ) = Re (f 0 g * 0 ) = 1, 2 Re (f 22 g * 22 ) = 2 Im (b 22 c * 22 ) + Ω \|c 22 \| 2 = 1, f 11 g 22 − f 22 g 11 = ib 22 c 11 − (ib 11 + Ωc 11 ) c 22 = 0.(38) On this stage, the GCS Φ ζ (q, τ ) can be rewritten as follows Φ ζ (q, τ ) = 1 π 3/4 2g 2 11 g 3 exp i Reφ 2 (τ ) − 2 \|g 11 \| 2 Re 2 Q 1 + Re 2 Q 2 − \|g 3 \| 2 Re 2 ζ 3 − ϕ 3 g 3 + iΩτ 2 − f 11 g 11 q 2 1 + q 2 2 2 − f 0 g 3 q 2 3 2 + √ 2 ζ 3 − ϕ 3 g 3 q 3 + √ 2 (Q 1 q 1 − Q 2 q 2 ) ,(39)whereφ 2 (τ ) = Q 2 1 + Q 2 2 + ζ 3 − ϕ 3 g 3 + iΞτ √ 2 cos 2 α 2 dτ, Q 1 = g 22 ζ 1 − ig 11 ζ 2 2g 11 g 22 , Q 2 = ig 22 ζ 1 − g 11 ζ 2 2g 11 g 22 .(40) It follows from Eqs. (17), (25) and (35) that ζ 1 = f 11 (τ ) [q 1 (τ ) + iq 2 (τ )] + ig 11 (τ ) [π 1 (τ ) + iπ 2 (τ )] √ 2 , ζ 2 = if 22 (τ ) [q 1 (τ ) − iq 2 (τ )] − g 22 (τ ) [π 1 (τ ) − iπ 2 (τ )] √ 2 , ζ 3 = f 0 q 3 (τ ) + ig 3 (τ ) π 3 (τ ) √ 2 + ϕ 3 (τ ) ,(41) wherein the mean values q j (τ ) = ζ, τ \|q j \| ζ, τ and π j (τ ) = ζ, τ \|π j \| ζ, τ obey the classical equations of motion (4) written in dimensionless variables, π 1 (τ ) = Ω 2 π 2 (τ ) − Ω 2 q 1 (τ ) 4 ,π 2 (τ ) = − Ω 2 π 1 (τ ) − Ω 2 q 2 (τ ) 4 , q 1 (τ ) = Ω 2 q 2 (τ ) + π 1 (τ ) ,q 2 (τ ) = − Ω 2 q 1 (τ ) + π 2 (τ ) , π 3 (τ ) = Ξ sin 2 α,q 3 (τ ) = π 3 (τ ) + Ξτ cos 2 α.(42) In terms of classical trajectories q j (τ ) and π j (τ ), the functions Φ ζ (q, τ ) take the form Φ ζ (q, τ ) = 1 π 3/4 g 11 √ 2g 3 exp − f 11 g 11 [q 1 − q 1 (τ )] 2 + [q 2 − q 2 (τ )] 2 2 − f 0 g 3 [q 3 − q 3 (τ )] 2 2 − π 1 (τ ) [2q 1 − q 1 (τ )] + π 2 (τ ) [2q 2 − q 2 (τ )] + π 3 (τ ) [2q 3 − q 3 (τ )] 2i + iΩτ 2 + iφ 3 , (43) whereφ 3 (τ ) =φ * 3 (τ ) = Ξ q 3 (τ ) sin 2 α − τ (π 3 (τ ) + Ξτ cos 2 α) cos 2 α 2 dτ.(44) Let us consider a different representation for the GCS. Using the operatorsÂ j (τ ) and A † j (τ ), we can construct a Fock space \|n, τ , n = (n 1 , n 2 , n 3 ), n j = 0, 1, 2, ..., at any time instant τ , \|n, τ = Â † 1 (τ ) n 1 Â † 2 (τ ) n 2 Â † 3 (τ ) n 3 √ n 1 !n 2 !n 3 ! \|0, τ ,Â j (τ ) \|0, τ = 0, m, τ \|n, τ = δ m j n j , ∞ n 1 ,n 2 ,n 3 =0 \|n, τ n, τ \| = 1. Then, \|ζ, τ = exp − \|ζ\| 2 2 ∞ n 1 ,n 2 ,n 3 =0 ζ n 1 1 ζ n 2 2 ζ n 3 3 √ n 1 !n 2 !n 3 ! \|n, τ =D (ζ, τ ) \|0, τ ,(46) where the displacement operatorD (ζ, τ ) readŝ D (ζ, τ ) ≡ e − \|ζ\| 2 2 e ζ jÂ † j (τ ) e −ζ * jÂ j (τ ) = e ζ jÂ † j (τ )−ζ * jÂ j (τ ) .(47) Using the completeness property of the states \|n, τ , we can find their overlapping and prove a completeness relation for the GCS, ζ , τ \|ζ, τ = exp (ζ ) * · ζ − \|ζ \| 2 + \|ζ\| 2 2 , 1 π 3 \|ζ, τ ζ, τ \| d 2 ζ = 1, d 2 ζ = d 2 ζ 1 d 2 ζ 2 d 2 ζ 3 , d 2 ζ j = d Re ζ j d Im ζ j , ∀τ.(48) The states \|n, τ (45) in q-representation Φ n (q, τ ) = q\|n, τ have the form (see appendix B) Φ n (q, τ ) = 1 √ n 1 !n 2 !n 3 ! g * 3 2g 3 n 3 2 H n 3 q 3 + √ 2 Re (ϕ 3 g * 3 ) \|g 3 \| × g * 11 g 11 n 1 g * 22 ig * 11 n 2 H n 1 ,n 2 q 1 − iq 2 √ 2g * 11 , q 1 + iq 2 √ 2g 11 Φ 0 (q, τ ) .(49) Using the relations exp (µZ + νZ * − µν) = ∞ m 1 ,m 2 =0 H m 1 ,m 2 (Z, Z * ) µ m 1 ν m 2 m 1 !m 2 ! , exp 2κh − h 2 = ∞ m=0 H m (κ) m! h m ,(50) we can calculate the sum in Eq. (46) and rewrite the CS as follows Φ ζ (q, τ ) = Φ 0 (q, τ ) exp √ 2 q 3 + √ 2 Re (ϕ 3 g * 3 ) g 3 ζ 3 − ζ 2 3 g * 3 2g 3 + ζ 1 q 1 − iq 2 √ 2g 11 + ζ 2 g * 22 \|g 11 \| 2 q 1 + iq 2 i √ 2 − g * 22 ig 11 ζ 1 ζ 2 − \|ζ\| 2 2 ,(51) where the vacuum state Φ 0 (q, τ ) reads Φ 0 (q, τ ) = 1 π 3/4 g 11 √ 2g 3 exp i Reφ 4 (τ ) − \|g 3 \| 2 Re 2 ϕ 3 g 3 + iΩτ 2 − f 11 g 11 q 2 1 + q 2 2 2 − f 0 g 3 q 2 3 2 − √ 2 ϕ 3 g 3 q 3 ,φ 4 (τ ) = iΞτ √ 2 cos 2 α − ϕ 3 g 3 2 dτ.(52) Using the relations \|f 11 (τ )\| = \|f 22 (τ )\| , \|g 11 (τ )\| = \|g 22 (τ )\| ,(53) one can easily verify that the states (51) coincide with ones (39). A. Standard deviations and uncertainty relations We recall that the standard deviation σ χ (τ ) of a some physical quantity χ in the states \|ζ, τ is calculated via the corresponding operatorχ as follows σ χ (τ ) ≡ ζ, τ \| (χ − ζ, τ \|χ\|ζ, τ ) 2 \|ζ, τ = χ 2 (τ ) − (χ (τ )) 2 , (χ (τ )) 2 ≡ ( ζ, τ \|χ\|ζ, τ ) 2 , χ 2 (τ ) ≡ ζ, τ \|χ 2 \|ζ, τ .(54) Below, we calculate standard deviations of some physical quantities of the problem under discussion. To this end, it is convenient to express the operators q j andπ j in terms of the annihilation and creation operatorsÂ j (τ ) andÂ † j (τ ). It follows from (17), (18) and (35) that q 1 = g * 11Â 1 + g 11Â † 1 − i g * 22Â 2 − g 22Â † 2 √ 2 ,π 1 = f * 11Â 1 − f 11Â † 1 − i f * 22Â 2 + f 22Â † 2 i √ 2 , q 2 = g * 22Â 2 + g 22Â † 2 − i g * 11Â 1 − g 11Â † 1 √ 2 ,π 2 = f * 22Â 2 − f 22Â † 2 − i f * 11Â 1 + f 11Â † 1 i √ 2 , q 3 = g * 3Â 3 + g 3Â † 3 − 2 Re (g * 3 ϕ 3 ) √ 2 ,π 3 = f * 0Â 3 − f 0Â † 3 − 2i Im (f * 0 ϕ 3 ) i √ 2 .(55) Then, using Eq. (25) we can easily to find q 1 (τ ) = √ 2 [Im (g * 22 ζ 2 ) + Re (g * 11 ζ 1 )] = q 0 1 − Π 2 (τ ) − Π 0 2 Ω , q 2 (τ ) = √ 2 [Im (g * 11 ζ 1 ) + Re (g * 22 ζ 2 )] = q 0 2 + Π 1 (τ ) − Π 0 1 Ω , π 1 (τ ) = √ 2 [Im (f * 11 ζ 1 ) − Re (f * 22 ζ 2 )] = Π 1 (τ ) + Π 0 1 − Ωq 0 2 2 , π 2 (τ ) = √ 2 [Im (f * 22 ζ 2 ) − Re (f * 11 ζ 1 )] = Π 2 (τ ) + Π 0 2 + Ωq 0 1 2 , q 3 (τ ) = √ 2 Re [g * 3 (ζ 3 − ϕ 3 )] = q 0 + Π 0 3 τ + Ξ 2 τ 2 , π 3 (τ ) = √ 2 Im [f * 0 (ζ 3 − ϕ 3 )] = Π 0 3 + Ξτ sin 2 α,(56) where Π 1 (τ ) = Π 0 1 cos (Ωτ ) + Π 0 2 sin (Ωτ ) , Π 2 (τ ) = Π 0 2 cos (Ωτ ) − Π 0 1 sin (Ωτ ) , q 0 1 = √ 2 [Re (c * 22 ζ 2 ) − Im (c * 11 ζ 1 )] , Π 0 2 = √ 2 [Re (b * 11 ζ 1 ) − Im (b * 22 ζ 2 )] , q 0 2 = √ 2 [Re (c * 11 ζ 1 ) − Im (c * 22 ζ 2 )] , Π 0 1 = √ 2 [Re (b * 22 ζ 2 ) − Im (b * 11 ζ 1 )] , q 0 = √ 2 Re (g * 3 ζ 3 ) , Π 0 3 = √ 2 Im (f * 0 ζ 3 ) .(57) Mean values of the operators q 2 j andπ 2 j in the states \|ζ, τ are given by q 2 β (τ ) = (q β (τ )) 2 + \|g 11 \| 2 + \|g 22 \| 2 2 , π 2 β (τ ) = (π β (τ )) 2 + \|f 11 \| 2 + \|f 22 \| 2 2 , q 2 3 (τ ) = (q 3 (τ )) 2 + \|g 3 \| 2 2 , π 2 3 (τ ) = (π 3 (τ )) 2 + \|f 0 \| 2 2 . (58) σ q β (τ ) ≡ σ q (τ ) = \|g 11 \| 2 + \|g 22 \| 2 2 , σ π β (τ ) ≡ σ π (τ ) = \|f 11 \| 2 + \|f 22 \| 2 2 , σ q 3 (τ ) = \|g 3 \| √ 2 , σ π 3 (τ ) = σ π 3 (0) = σ π 3 = \|f 0 \| √ 2 .(59) By mean from (38) and (59), the constants c ββ , b ββ , f 0 and g 0 can be related to the initial standard deviations σ q j (0) ≡ σ q j and σ π j (0) ≡ σ π j in the form \|c 11 \| = \|c 22 \| = σ q , \|b 11 \| = σ π 1 + 2Ω + Ω 2 σ 2 q 4σ 2 π , \|b 22 \| = σ π 1 − 2Ω − Ω 2 σ 2 q 4σ 2 π , Im (c 11 b * 11 + c * 22 b 22 ) = −Ωσ 2 q , Re (c 11 b * 11 + c * 22 b 22 ) = 4σ 2 q σ 2 π − 1, cos (arg g 0 − arg f 0 ) = 1 2σ π 3 σ q 3 , Im (f * 0 g 0 ) = 4σ 2 q 3 σ 2 π 3 − 1.(60) Taking the relations (37) and (60) into account, we can write (59) as follows σ q (τ ) = σ q cos (Ωτ ) + 4σ 2 π + Ω 2 σ 2 q 2σ 2 q 1 − cos (Ωτ ) Ω 2 + 4σ 2 q σ 2 π − 1 sin (Ωτ ) Ωσ 2 q , σ π (τ ) = σ π 1 + Ω 2 σ 2 q 4σ 2 π − 1 1 − cos (Ωτ ) 2 − Ω 4σ 2 π 4σ 2 q σ 2 π − 1 sin (Ωτ ), σ q 3 (τ ) = σ q 3 1 + 4σ 2 π 3 − 1 σ 2 q 3 τ + σ 2 π 3 σ 2 q 3 τ 2 .(61) One can see that σ q (τ ), σ π (τ ) and σ q 3 (τ ) are real functions if σ q σ π ≥ 1 2 , σ q 3 σ π 3 ≥ 1 2 ,(62) which correspond the Heisenberg uncertainty relation in τ = 0. In what follows, we consider the conditions σ π = 1 2σ q , σ π 3 = 1 2σ q 3 .(63) Thus, the Heisenberg uncertainty relation is minimal at τ = 0. For any τ , we have that σ q (τ ) σ π (τ ) = 1 2 1 + 1 − 2 − Ω 2 σ 4 q Ω 2 σ 4 q 4Ω 2 σ 4 q sin 2 (Ωτ ) ≥ 1 2 , σ q 3 (τ ) σ π 3 = 1 2 1 + τ 2 4σ 4 q 3 ≥ 1 2 .(64)σ q = 1 √ Ω .(65) One can see that the product σ q (τ ) σ π (τ ) is limited from above for any τ , 1 2 ≤ σ q (τ ) σ π (τ ) ≤ 1 4 2 + Ω 2 σ 4 q Ω 2 σ 4 q + 1 Ω 2 σ 4 q .(66) To obtain the Robertson-Schrödinger relation [13,14] we need calculate the covariance σ χκ (τ ) = (χ − χ ) (κ − κ ) + (κ − κ ) (χ − χ ) 2 .(67) Then, σ q β π β (τ ) = π β q β − π β q β + i 2 = f * 11 g 11 + f * 22 g 22 − 1 2i , σ q 3 π 3 (τ ) = π 3 q 3 − π 3 q 3 + i 2 = f * 0 g 3 − 1 2i .(68) Thus, we have that σ 2 q j (τ ) σ 2 π j (τ ) − σ 2 q j π j (τ ) = 1 4 .(69) The Robertson-Schrödinger uncertainty relations are minimized for any τ , this means that the GCS are squeezed states, see e.g., [15]. Now, let us study mean values and standard deviations of the velocities, Π 1 =π 1 + Ω 2 q 2 ,Π 2 =π 2 − Ω 2 q 1 ,Π 3 =π 3 + Ξτ cos 2 α.(70) It follows from (56) that Π 1 (τ ) = π 1 (τ ) + Ω 2 q 2 (τ ) , Π 2 (τ ) = π 2 (τ ) − Ω 2 q 1 (τ ) , Π 3 (τ ) = π 3 (τ ) + Ξτ cos 2 α. (71) Using relations (56), (58) and (59), we can calculate mean values of the operatorsΠ 2 1 ,Π 2 2 andΠ 2 3 , Π 2 β (τ ) = (Π β (τ )) 2 + σ 2 π (τ ) + Ω 2 σ 2 q (τ ) 4 = (Π β (τ )) 2 + 1 + Ω 2 σ 4 q 4σ 2 q , Π 2 3 (τ ) = (Π 3 (τ )) 2 + \|f 0 \| 2 2 . (72) σ Π β (τ ) = 1 + Ω 2 σ 4 q 2σ q ≡ σ Π , σ Π 3 (τ ) = \|f 0 \| √ 2 ≡ σ π 3 .(73) Taking into account relation (63), we see that in (43) exist families of GCS which differ one from another by values of the parameters σ = σ q ,σ q 3 , Φ σ (q, τ ) = exp − f σq 11 (τ ) g σq 11 (τ ) [q 1 −q 1 (τ )] 2 +[q 2 −q 2 (τ )] 2 2 − [q 3 −q 3 (τ )] 2 2(2σ 2 q 3 +iτ ) + iφ 3 + iΩτ 2 g σq 11 (τ ) √ 2 3 π 3 σ q 3 + iτ 2σq 3 − π 1 (τ ) [2q 1 − q 1 (τ )] + π 2 (τ ) [2q 2 − q 2 (τ )] + π 3 (τ ) [2q 3 − q 3 (τ )] 2i ,(74) where f σq 11 (τ ) = iΩσ q 2 + 1 + Ωσ 2 q 1 + e iΩτ 4iσ q , g σq 11 (τ ) = −iσ q − 1 + Ωσ 2 q 1 − e iΩτ 2iΩσ q .(75) Setting electric E and magnetic B fields to zero in Eq. (74), we obtain the GCS for the free 3-dimensional particle, Φ ζ (q, τ ) = exp − [q 1 −q 1 (τ )] 2 2(2σ 2 q +iτ ) − [q 2 −q 2 (τ )] 2 2(2σ 2 q +iτ ) − [q 3 −q 3 (τ )] 2 2(2σ 2 q 3 +iτ ) √ 2 3 π 3 σ q + iτ 2σq 2 σ q 3 + iτ 2σq 3 − π 1 [2q 1 − q 1 (τ )] + π 2 [2q 2 − q 2 (τ )] + π 3 [2q 3 − q 3 (τ )] 2i ,(76) where q j (τ ) = q 0 j + Π 0 j τ and π j = Π 0 j . The function (76) is a product of three one-dimensional GCS that depend on q 1 , q 2 and q 3 respectively. These one-dimensional GCS coincide with ones obtained and studied in our previous publications [12]. B. CS of a charged particle in constant magnetic field Quantum nonrelativistic motion of a charged particle in a constant magnetic field was studied in a number of articles, see e.g., [16][17][18][19][20]. CS of such a system were obtained first by Malkin and Man'ko [21], see too [22][23][24][25][26]. The states (51) for zero electric field are GCS for such a system. To compare the latter states with ones by Malkin and Man'ko one has to projet them on the xy-plane and to use the relation (65). In Ref. [21], the CS ofĤ xy (14) are given by ( = 1) Φ αβ (ε) = mω 2π exp − \|ε\| 2 + √ 2βε + i √ 2αε * − iαβ − \|α\| 2 + \|β\| 2 2 ,(77) where ε = √ mω x + iy 2 ,â = − i √ 2 ε + ∂ ∂ε * ,b = 1 √ 2 ε * + ∂ ∂ε âΦ αβ = αΦ αβ ,bΦ αβ = βΦ αβ .(78) Being rewritten in term of the operatorsâ andb the HamiltonianĤ xy takes the form H xy = ω â †â + 1 2 .(79) Then, the time evolution of Φ αβ (ε) is given by means of α (t) = αe −iωt as follows Φ αβ (ε, t) = exp −iĤ xy t Φ αβ (ε) = mω 2π exp − \|ε\| 2 + √ 2 iαe −iωt ε * + βε − iαe −iωt β − \|α\| 2 + \|β\| 2 2 − iωt 2 .(80) Now, let us consider the states (51) in dimensional variables (15), with = 1, on the plane xy as follows Ψ σ ⊥ (µ, t) = 1 √ πg σ ⊥ 11 exp − f σ ⊥ 11 g σ ⊥ 11 \|µ\| 2 2 + 1 √ 2 ζ 1 µ * g σq 11 + ζ 2 µ ig 22 − g * σ ⊥ 22 ig σ ⊥ 11 ζ 1 ζ 2 − \|ζ 1 \| 2 + \|ζ 2 \| 2 2 + iωt 2 ,(81) where f σ ⊥ 11 (t) = (1 + mωσ 2 ⊥ ) (1 + e iωt ) − 2mωσ 2 ⊥ 4iσ ⊥ , f σ ⊥ 22 (t) = 2mωσ 2 ⊥ + (1 − mωσ 2 ⊥ ) (1 + e −iωt ) 4iσ ⊥ , g σ ⊥ 11 (t) = 2mωσ 2 ⊥ − (1 + mωσ 2 ⊥ ) (1 − e iωt ) 2imωσ ⊥ , g σ ⊥ 22 (t) = 2mωσ 2 ⊥ + (1 − mωσ 2 ⊥ ) (1 − e −iωt ) 2imωσ ⊥ , µ = x + iy, σ q = l −1 σ ⊥ , Φ ζ ⊥ (q ⊥ , τ ) = lΨ σ ⊥ (µ, t) .(82) Using the condition (65) as follows σ ⊥ = 1 √ mω . (83) f σ ⊥ 11 (t) = le iωt √ mω 2i , g σ ⊥ 11 (t) = e iωt il √ mω , f σ ⊥ 22 = l √ mω 2i , g σ ⊥ 22 = 1 il √ mω .(84) Thus, the states (81), Ψ σ ⊥ (µ, t) −→ Ψ (µ, t), take the form Ψ (µ, t) = mω π exp − mω \|µ\| 2 4 + mω 2 iζ 1 µ * e iωt + ζ 2 µ −ie −iωt ζ 1 ζ 2 − \|ζ 1 \| 2 + \|ζ 2 \| 2 2 − iωt 2 .(85) Using the relations µ = 2 √ mω ε, ζ 1 = α, ζ 2 = β,(86) we can show that Ψ (µ, t) = Φ αβ (ε, t) .(87) Thus, we find that the GCS (51) coincides with the CS of Malkin and Man'ko given by the condition (83). IV. GCS AS SEMICLASSICAL STATES In general case, the GCS cannot be considered as semiclassical states (SS) because the standard deviations can grow over the time or by means of any other parameter of the system. The study of the SS have attracted attention of theoretical physicists and chemical, see e.g., [27][28][29][30][31][32][33][34][35][36]. We have that the GCS can considered as SS if the conditions below are satisfied: I -Position and velocity mean value must propagate along the classical trajectories. II -Position and velocity standard deviation must have a short interval of variation. In our previous publication [12], we find that SS of a free particle is given by condition v 2mσ x ,(88) wherein v is the velocity and σ x is the position standard deviation. This means that the velocity mean value must be much larger than its corresponding standard deviation. Hence, the variation of the position mean value is much larger than the variation of its corresponding standard deviation. In a certain sense, this implies that the mean values propagate with a constant standard deviation, which is consistent with the conditions I and II listed above. Of course that in a long instant time we will not have SS anymore. First, let us study the conditions of SS on the z-motion. To this, we return to the initial dimensional variables and rewrite the Eqs. (56) (61), (71) and (73) in these variables, as follows z (t) = z 0 + v 0 z t + ξ We can see that the condition of SS is not satisfied only in Case I because the standard deviation of the velocity is a function of the magnetic field. The Case IV presents the best description of the SS because the position and velocity mean values presented simultaneously the lowest value in their measurements. In addition, it admit the free particle limit. Taking into account the limit E = B = 0, we find the conditions of SS of a free particle in 3-dimension from (89) and (103) In this work, we obtain the classical equations of motion which are reduced to the free particle case in the limit of null field by mean of the initial Cauchy data. We find the corresponding GCS that are solutions of the Schrödinger equation, parametrized by the standard deviations σ x , σ y and σ z at the initial time instant. These states are squeezed states which present potential application in optical. In addition, we can obtain the GCS of a free particle taking into account the limit of null field, such result it is important to the scattering process. We obtain conditions under which the GCS can be considered as SS and these conditions hold on the limit of null field. Consider the Eq. (28) with β = 2, (g 21 ∂ q 1 + g 22 ∂ q 2 ) Φ ζ ⊥ (q ⊥ , τ ) = √ 2ζ 2 − f 21 q 1 − f 22 q 2 Φ ζ ⊥ (q ⊥ , τ ) .(A3) Comparing the Eqs. (A3) and (A2) we have only one possibility to make the following identifications w (x, y) = Φ ζ ⊥ (q ⊥ , τ ) , a = g 21 , b = g 22 , q 1 = x, q 2 = y. But, for the functions f (x) −→ f (q 1 ) and g (y) −→ g (q 2 ) there is more than one possibility, for example, I. f (q 1 ) = −f 21 q 1 , g (q 2 ) = √ 2ζ 2 − f 22 q 2 , II. f (q 1 ) = √ 2ζ 2 − f 21 q 1 , g (q 2 ) = −f 22 q 2 , III. f (q 1 ) = ζ 2 √ 2 − f 21 q 1 , g (q 2 ) = ζ 2 √ 2 − f 22 q 2 .(A5) Any of the possibilities listed above leads to the same solution of the Eqs. (27). Let us consider the situation III. Thus, the general solution of the Eq. (A3) takes the form Φ ζ ⊥ (q ⊥ , τ ) = Υ (u) exp ζ 2 q 1 √ 2g 21 + ζ 2 q 2 √ 2g 22 − f 21 g 21 q 2 1 2 − f 22 g 22 q 2 2 2 , u = g 22 q 1 − g 21 q 2 .(A6) At the same time, the solution (A6) must be satisfy the Eq. (27) with β = 1, (f 11 q 1 + f 12 q 2 + g 11 ∂ q 1 + g 12 ∂ q 2 ) Φ ζ ⊥ (q ⊥ , τ ) = √ 2ζ 1 Φ ζ ⊥ (q ⊥ , τ ) . Using the relation f 11 g 21 − f 21 g 11 = f 22 g 12 − f 12 g 22 and the transformations ∂ q 1 = g 22 ∂ u + g * 22 ∂ u * , ∂ q 2 = −g 21 ∂ u − g * 21 ∂ u * .(A8) We find that (A6) is a solution of the Eq. (A7) if the function Υ (u) satisfies the following equation ∂ u + F g 21 g 22 u + F 0 √ 2g 21 g 22 Υ (u) = 0, F = f 11 g 21 − f 21 g 11 g 11 g 22 − g 12 g 21 , F 0 = (g 11 g 22 + g 12 g 21 ) ζ 2 − 2g 21 g 22 ζ 1 g 11 g 22 − g 12 g 21 , whose general solution reads Υ (u) = exp − F g 21 g 22 u 2 2 − F 0 √ 2g 21 g 22 u + iφ ⊥ ,(A10) where φ ⊥ (τ ) is an arbitrary time-dependent function. FIG. 1 : 1Particle's classical circular motion along the xy-plane. The 3D trajectory correspond to a helix, with a circular xy-motion and accelerated z-motion, whose direction point out of the picture, aligned with the external electromagnetic fields. Consider the inequality between the velocity v z (τ ) and its corresponding standard deviationOne can see that at any time instant t the inequality holds true provided that the conditionThe condition (90) implies that the variation of z (t) is much larger than the variation of σ z (t). Thus, we can writeThe electric field allows to have SS when the condition (90) is no longer true. However, this is possible after an instant of time, given byWe can see that much larger are the parameters ξ, v 0 z is smaller the time required for the conditions of SS to be satisfied. Then, it follows from (91) thatOn the xy-plane, the standard deviation of the position σ ⊥ (t) and velocity σ v ⊥ are given bybeyond those discussed for z-motion. One can see that the standard deviations depends on the magnetic field B. Thus, we must analysis the following cases:the velocity standard deviation becomes much large and the position standard deviation becomes the smallest possible.the position standard deviation becomes a constantand the velocity standard deviation is given bywe have that the velocity standard deviation assumes its lowest valueBut, the position standard deviation begins to grow over timeThe simultaneous measurements in the position and velocity have the standard deviation with the lowest value possible. In addition, we must impose the condition below, such that in the limit of null field the conditions of SS holds true v 0Appendix A: Solution from the set(27)To solve equations from the set(27), we are going to use the fact (see Ref.[37]) thatwith an arbitrary function Υ (z) is the general solution of the equationAppendix B: Calculating the states Φ n (q, τ )The number states Φ n (q, τ ) = q\|n, τ we find from (45) as followswhere the vacuum state Φ 0 (q, τ ) is given in (52).The creation operatorsÂ † j (τ ) we can write as followŝwherein the functions Φ ⊥ and Φ 3 are given byUsing the relations, see[38][39][40][41], below H n,m (Z, Z * ) = (−1) n+m e ZZ * ∂ n Z * ∂ m Z e −ZZ * , Z ∈ C,where H n (x) are the Hermite polynomials and H n,m (Z, Z * ) are the Hermite polynomials of two variables, we find thatThen, the sum (B1) takes the form Φ n (q, τ ) = 1 √ n 1 !n 2 !n 3 ! g * H n 1 ,n 2 q 1 − iq 2 √ 2g *11, q 1 + iq 2 √ 2g 11 Φ 0 (q, τ ) . 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Y Xu, Complex versus real orthogonal polynomials of two variables Integral Transforms and Special Functions. 26134Y. Xu 2015 Complex versus real orthogonal polynomials of two variables Integral Transforms and Special Functions, 26:2, 134. I S Gradshteyn, I M Ryzhik, Table of integrals, series, and products. OxfordElsevier IncI. S. Gradshteyn and I. M. Ryzhik 2007 Table of integrals, series, and products (Elsevier Inc, Oxford).	[]
[ "Quark Phase Transition Parameters and δ-Meson Field in RMF Theory", "Quark Phase Transition Parameters and δ-Meson Field in RMF Theory" ]	[ "G B Alaverdyan \nYerevan State University\nA.Manoogyan str.10025YerevanArmenia\n" ]	[ "Yerevan State University\nA.Manoogyan str.10025YerevanArmenia" ]	[]	The deconfinement phase transition from hadronic matter to quark matter in the interior of compact stars is investigated. The hadronic phase is described in the framework of relativistic meanfield (RMF) theory, when also the scalar-isovector δ-meson effective field is taken into account. To describe a quark phase the MIT bag model is used. The changes of the mixed phase threshold parameters caused by the presence of δ-meson field are investigated. 15 interaction term and contribution of the δ-meson field, respectively. This Lagrangian density 16 (1) contains the meson-nucleon coupling constants, g σ , g ω , g ρ , g δ and also parameters of σ-17 field self-interacting terms, b and c. In our calculations we take a δ = (g δ /m δ ) 2 = 2.5 fm 2 18 for the δ coupling constant, as in [5]. Also we use m N = 938.93 MeV for the bare nucleon 19 mass, m * N = 0.78 m N for the nucleon effective mass, n 0 = 0.153 fm −3 for the baryon number 20 density at saturation, f 0 = −16.3 MeV for the binding energy per baryon, K = 300 MeV for 21 the incompressibility modulus, and E (0) sym = 32.5 MeV for the asymmetry energy. Five other 22	null	[ "https://arxiv.org/pdf/0907.4065v1.pdf" ]	119,236,959	0907.4065	013b90a4e1fc561eeffec4a43546ffb60470eed9	Quark Phase Transition Parameters and δ-Meson Field in RMF Theory 23 Jul 2009 July 23, 2009 G B Alaverdyan Yerevan State University A.Manoogyan str.10025YerevanArmenia Quark Phase Transition Parameters and δ-Meson Field in RMF Theory 23 Jul 2009 July 23, 2009Preprint submitted to Nuclear Physics AarXiv:0907.4065v1 [nucl-th] The deconfinement phase transition from hadronic matter to quark matter in the interior of compact stars is investigated. The hadronic phase is described in the framework of relativistic meanfield (RMF) theory, when also the scalar-isovector δ-meson effective field is taken into account. To describe a quark phase the MIT bag model is used. The changes of the mixed phase threshold parameters caused by the presence of δ-meson field are investigated. 15 interaction term and contribution of the δ-meson field, respectively. This Lagrangian density 16 (1) contains the meson-nucleon coupling constants, g σ , g ω , g ρ , g δ and also parameters of σ-17 field self-interacting terms, b and c. In our calculations we take a δ = (g δ /m δ ) 2 = 2.5 fm 2 18 for the δ coupling constant, as in [5]. Also we use m N = 938.93 MeV for the bare nucleon 19 mass, m * N = 0.78 m N for the nucleon effective mass, n 0 = 0.153 fm −3 for the baryon number 20 density at saturation, f 0 = −16.3 MeV for the binding energy per baryon, K = 300 MeV for 21 the incompressibility modulus, and E (0) sym = 32.5 MeV for the asymmetry energy. Five other 22 Deconfinement phase transition parameters 1 The modern concept of hadron-quark phase transition is based on the feature of that tran-2 sition, that is the presence of two conserved quantities in this transition: baryon number and 3 electric charge [1]. It is known, that depending on the value of surface tension, σ s , the phase 4 transition of nuclear matter into quark matter can occur in two scenarios [2]: ordinary first order 5 phase transition with a density jump (Maxwell construction), or formation of a mixed hadron-6 quark matter with a continuous variation of pressure and density [1]. Uncertainty of the surface 7 tension values does not allow to determine the phase transition scenario, taking place in realty. 8 In our recent paper [3] in the assumption that the transition to quark matter is a usual first-order 9 phase transition, described by Maxwell construction, we have shown that the presence of the 10 δ-meson field leads to the decrease of transition pressure P 0 , of baryon number densities n N and 11 n Q . In this article we investigate the deconfinement phase transition, when the transition proceeds 12 through a mixed phase. 13 For description of hadronic phase we use the relativistic Lagrangian density of many-particle system consisting of nucleons, p, n, and exchanged mesons σ, ω, ρ, δ: L σωρδ (σ(x), ω µ (x), ρ µ (x), δ(x)) = L σωρ (σ(x), ω µ (x), ρ µ (x)) − U(σ(x)) + L δ ( δ(x)),(1) where L σωρ is the linear part of relativistic Lagrangian density without δ-meson field and to determine the re-denoted mean-fields, σ ≡ g σσ , ω ≡ g ωω0 , δ ≡ g δδ (3) , and ρ ≡ g ρρ0 [4], U(σ) = 14 b 3 m N (g σ σ) 3 + c 4 (g σ σ) 4 and L δ ( δ) = g δψN τ N δψ N + 1 2 ∂ µ δ∂ µ δ − m δ δ 2 are the σ-meson self- constants, a i = (g i /m i ) 2 (i = σ, (3) , 27 depending on baryon number density n and asymmetry parameter α = (n n − n p )/n. The standard 28 QHD procedure allows to obtain expressions for energy density ε(n, α) and pressure P(n, α). 29 The results of our analysis show that the scalar -isovector δ-meson field inclusion increases the 30 value of the energy per nucleon. This change is strengthened with the increase of the nuclear 31 matter asymmetry parameter, α. The δ-field inclusion leads to the increase of the EOS stiffness 32 of nuclear matter due to the splitting of proton and neutron effective masses, and also due to the 33 increase of asymmetry energy (for details see Ref. [6]). 34 To describe the quark phase the MIT bag model is used, in which the interactions between Table 1 represents the parameter sets of the mixed phase both with and without δ-meson field. 39 It is shown that the presence of δ-field alters threshold characteristics of the mixed phase. The ω, ρ), b and c, then can be numerically determined: 10 −2 fm −1 , c = 1.319 · 10 −2 . If we neglect the δ channel, then a δ = 0 and a ρ = 4.794 25 fm 2 . The knowledge of the model parameters makes it possible to solve the set of four equations 26 35 u 35, d, s quarks inside the bag are taken in a one-gluon exchange approximation [7]. We choose 36 m u = 5 MeV, m d = 7 MeV and m s = 150 MeV for quark masses, B = 60 MeV/fm 3 for bag 37 parameter and α s = 0.5 for the strong interaction constant. lower threshold parameters, n N , ε N , P N , are increased, meanwhile the upper ones, n Q , ε Q , P Q , 41 are slowly decreased. For EOS, used in this study, the central pressure of the maximum mass 42 neutron stars is less than the mixed phase upper threshold P Q . Thus, the corresponding hybrid 43 stars do not contain pure strange quark matter core. 44 H. Heiselberg and M. Hjorth-Jensen, (1999), arXiv: nucl-th/9902033. 48 [3] G.B. Alaverdyan, Gravitation & Cosmology 15 (2009) 5, arXiv: 0902.0050 [astro-ph.SR]. 49 [4] N. K. Glendenning, Compact Stars, Springer (2000). 50 [5] B. Liu, V. Greco, V. Baran, M. Colonna, and M. Di Toro, Phys. Rev. C65 (2002) 045201. V. Greco, M. Colonna, Table 1 : 1The Mixed phase parameters with (σωρδ) and without (σωρ) δ-meson field. n N n Q P N P Q ε N ε Q fm −3 fm −3 MeV/fm 3 MeV/fm 3 MeV/fm 3 MeV/fm 3 σωρ 0.072 1.083 67.728 1280.889 0.336 327.747 σωρδ 0.077 1.083 72.793 1280.884 0.434 327.745 38 . N K Glendenning, Phys. Rev. 461274N.K. Glendenning, Phys. Rev. D46 (1992) 1274. . H Heiselberg, C J Pethick, E S Staubo, Phys. Rev. Lett. 701355H. Heiselberg, C.J. Pethick, and E.S. Staubo, Phys. Rev. Lett. 70 (1993) 1355. . G B Alaverdyan, Astrophysics. 52132G. B. Alaverdyan, Astrophysics 52 (2009) 132. . E Farhi, R L Jaffe, Phys. Rev. 302379E. Farhi and R. L. Jaffe, Phys. Rev. D30 (1984) 2379.	[]
[ "JCAP05(2015)034 ournal of Cosmology and Astroparticle Physics An IOP and SISSA journal J Super-inflation and generation of first order vector perturbations in ELKO", "JCAP05(2015)034 ournal of Cosmology and Astroparticle Physics An IOP and SISSA journal J Super-inflation and generation of first order vector perturbations in ELKO" ]	[ "Abhishek Basak [email protected] \nSchool of Physics\nIndian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM)\n695016TrivandrumIndia\n", "S Shankaranarayanan \nSchool of Physics\nIndian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM)\n695016TrivandrumIndia\n" ]	[ "School of Physics\nIndian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM)\n695016TrivandrumIndia", "School of Physics\nIndian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM)\n695016TrivandrumIndia" ]	[]	In this work we construct a model where the first order vector perturbations can be generated during inflationary expansion. For the non-standard spinors, known as ELKO, we show that the components of the first order perturbed energy-momentum tensor of the ELKO is non-zero for pure vector part of the metric perturbation (B i ). We show that vector perturbations do not decay in the super-Hubble scale and for a specific super-inflation background model we show that the vector perturbations are nearly scale invariant, while its amplitude is smaller than the primordial scalar perturbations. We also comment on the generation of vorticity.	10.1088/1475-7516/2015/05/034	null	119,181,181	1410.5768	3f1b1e63b503feb51dd5825d9adaf7ff17b07de9	JCAP05(2015)034 ournal of Cosmology and Astroparticle Physics An IOP and SISSA journal J Super-inflation and generation of first order vector perturbations in ELKO Published May 20, 2015 Abhishek Basak [email protected] School of Physics Indian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM) 695016TrivandrumIndia S Shankaranarayanan School of Physics Indian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM) 695016TrivandrumIndia JCAP05(2015)034 ournal of Cosmology and Astroparticle Physics An IOP and SISSA journal J Super-inflation and generation of first order vector perturbations in ELKO Published May 20, 201510.1088/1475-7516/2015/05/034Received October 29, 2014 Revised February 26, 2015 Accepted April 30, 2015ArXiv ePrint: 1410.5768 Article funded by SCOAP 3 . Content from this work may be used under the terms of the Creative Commons Attribution 3.0 License. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.inflationcosmology of theories beyond the SMcosmological perturbation theory In this work we construct a model where the first order vector perturbations can be generated during inflationary expansion. For the non-standard spinors, known as ELKO, we show that the components of the first order perturbed energy-momentum tensor of the ELKO is non-zero for pure vector part of the metric perturbation (B i ). We show that vector perturbations do not decay in the super-Hubble scale and for a specific super-inflation background model we show that the vector perturbations are nearly scale invariant, while its amplitude is smaller than the primordial scalar perturbations. We also comment on the generation of vorticity. Introduction Inflationary paradigm has been highly successful in explaining the observed Universe. In the standard inflationary scenario, inflation is driven by a slowly rolling scalar field through its potential. The cosmological perturbation theory during inflation has predicted the CMB observations quite successfully, for example density perturbation or scalar perturbations in the first order [1,2]. However, generation of primordial seed magnetic field is still unresolved. There are various mechanisms to generate primordial magnetic field [3][4][5]. One mechanism for the generation of primordial magnetic field is to generate vorticity during inflation which can be sourced by primordial vector modes. However, it has been observed that the first order perturbation theory can not generate growing vector perturbations. The reason is easily understood as the non-diagonal component of perturbed energy-momentum tensor do not contain any vector modes. In the absence of anisotropic stress, the vector modes decay quickly as the Universe expands [1,6]. To avoid this problem of first order vector modes attempts have been made to generate vector modes in the collapsing Universe during the contracting phase of the cyclic models of the Universe [7]. It has been shown that during this contracting phase the vector modes indeed grow, but this growth cannot be stopped which may finally lead to breakdown of the perturbation theory [8]. Unlike the first order perturbation, it has been observed that in the case of second order perturbation theory the vector modes can be sourced and vorticity can be generated [9][10][11][12] even in the standard scalar field driven inflationary theory. In this work we consider the inflationary scenario driven by non-standard spinors also known as ELKO [13][14][15][16]. These kind of spinors have mass dimension one and follows Klein-Gordon equation instead of Dirac equation. It has been shown that this kind of spinors can drive inflation and lead to scalar power spectrum consistent with the observed data [17][18][19]. Here we show explicitly that ELKO driven inflation can generate growing vector modes even in the first order. We show that, unlike the scalar field inflation, the non-diagonal components of the stress-tensor (specifically (η, i) component, where η is the conformal time defined later and i the spatial index) is non-zero corresponding to the pure vector modes of JCAP05(2015)034 the metric perturbation. Rewriting the non-diagonal components, we show that the vector perturbations, like the scalar perturbations, satisfy second order differential equation. In order to make a definite prediction, we assume that the background field leads to superinflation, i.e.Ḣ > 0 [20][21][22][23], where H is the Hubble parameter. The super-inflationary phase requires new physics which, in our case, is provided by ELKO. We have shown that the vector modes can be frozen in the super-Hubble scale and is scale invariant. However, the amplitude is small compared to the scalar perturbation. The suppression factor of the amplitudes of the vector modes is exp − 9 8 ∆N , where ∆N is the number of e-foldings necessary for superinflation. As in the case of ELKO driven inflationary theories, the spectral index of the scalar modes of perturbations depends on the slow-roll parameter ǫ = −Ḣ H 2 , super-inflationary phase (H > 0) may produce blue tilt in the spectral index of the scalar perturbations [19]. Hence, the super-inflationary phase precedes the standard inflationary phase. In order for the scalar perturbations from the ELKOS to be consistent with the CMB observations, the number of e-foldings in the super-inflation phase can only be of the order unity. After the end of super-inflationary phase the standard inflationary phase with (Ḣ < 0) starts. In section 2, we give the definitions of ELKO Lagrangian and the energy-momentum tensor. In this section we further calculate the non-diagonal component (specifically, (η, i)) of the perturbed energy-momentum tensor in the linear order. In section 3 we use the linear order perturbed Einstein equation and using the perturbed energy-momentum tensor we calculate the evolution equation of the pure vector modes B i . At super-Hubble scale, it is shown that the vector modes can be frozen in time. In section 4 we give the approximate background scaling solution, consistent with the super-inflation. Section 5 contains the solutions of the vector modes in the sub-Hubble and super-Hubble scale. Finally, in section 6 we end with conclusions and comment on the generation of vorticity in the first order. Linear order perturbed ELKO energy-momentum tensor In this work we are interested in the inflationary theory based on non-standard spinor known as ELKO. ELKO are the eigenspinors of charge conjugation operator. This kind of spinors are non-standard because, unlike classical Dirac spinors in case of ELKO spinors (CP T ) −2 = −I. Other major difference between classical Dirac spinors and ELKO is, Dirac spinors have mass dimension 3 2 whereas ELKO spinors have mass dimension one. Therefore, these kind of spinors follow second order Klein-Gordon equation instead of Dirac equation which is first order in time. The energy-momentum tensor of ELKO can be written as [18]: T µν = ¬ λ ← − ∇ (µ − → ∇ ν) λ − g µν L + F µν ,(2.1) where λ and ¬ λ are ELKO and its dual, respectively and µ is the space-time index. The dual of ELKO is defined in the same spirit as is done in case of Dirac spinors (Dirac adjoint in case of Dirac spinors,ψ = ψ † γ 0 , γ 0 is the 0-th component of Dirac gamma matrix), such that ( ¬ λλ) becomes a space-time scalar. For a detailed discussion on the construction of ELKO and its dual, one can look into [18]. The ELKO Lagrangian (L) is L = 1 2 g µν ¬ λ ← − ∇ (µ − → ∇ ν) λ − V ¬ λλ (2.2) The exact form of the potential is arbitrary at this stage as it will be clear in the following sections that it does not not enter in the equation of motion of the vector modes directly. -2 - JCAP05(2015)034 The covariant derivative on ELKO and its dual are defined as: ¬ λ ← − ∇ ≡ ∂ µ ¬ λ + ¬ λΓ µ , − → ∇λ ≡ ∂ µ λ − Γ µ λ. (2.3) Where Γ µ is the spin-connection appearing because of propagation of spinors in curved spacetime. The expression of Γ µ is given by Γ µ = i 4 ω ab µ f ab , (2.4) where a, b are the spinor indices. f ab = i 2 γ a , γ b is the generator of the Lorentz group and γ a is the Dirac gamma matrix. ω ab µ is defined as ω ab µ = e a α e αb ;µ . Here (; µ) denotes the covariant derivative with respect to µ. e µ a is the vierbien. The vierbiens are related to the space-time metric g µν by the following relation: e µ a e ν b η ab = g µν , (2.5) where η ab = diag. (1, −1, −1, −1) is the Minkowski metric. F µν is the additional term that comes from the variation of spin-connection, Γ µ , with respect to the metric g µν . We work in the conformal time where the background metric is given as: g (0) µν = a 2 (η) O O −a 2 (η) δ ij . (2.6) Here η is the conformal time defined as dη = dt/a, t is the cosmic time and a is the scale factor. The expression of F µν is given as F µν = 1 2 ∇ ρ J µνρ , (2.7) where the expression of J µνρ is: J µνρ = − i 2 ¬ λ ← − ∇ (µ f ν)ρ λ + ¬ λf ρ(µ − → ∇ ν) λ . (2.8) f µν is given as f µν = e µ a e ν b f ab . We prefer to work with the following ansatz for the back ground ELKO and its dual λ = ϕ (η) ξ, ¬ λ = ϕ (η) ¬ ξ, (2.9) where ϕ is real, ξ and ¬ ξ are the two constant matrices with the property ¬ ξξ = I, (2.10) such that ¬ λλ = ϕ (η) 2 . In the above ansatz ϕ (η) is a background scalar quantity dependent on time only. The advantages of using the above ansatz are: (i) The components of the energy-momentum tensor can be written in terms of one scalar field (ϕ) instead of two spinors and (ii) it can be ensured that the theory does not have any negative energy or ghost modes [18]. Using (2.9) and (2.10) one can show that the (η, η) component of the background energy-momentum tensor, which will be used later, becomes: T ηη(0) = a −4 1 2 ϕ ′2 + 3 8 H 2 ϕ 2 + a 2 V ,(2. JCAP05(2015)034 where ′ denotes derivative with respect to η and H is the Hubble parameter defined as H = a ′ /a. The perturbed Einstein equation is given by: δG µ ν = 8πGδT µ ν , (2.12) For the vector modes in the metric perturbation we choose the Newtonian gauge where the components of metric perturbation are given below: δg ηi = a 2 B i , (2.13) where 'i' is the spatial index -x, y and z. All other components of the metric perturbation are equal to zero. Here B i is the divergence less vector mode i.e., ∂ i B i = 0. The (η, i) component of the Einstein equation (2.12) is given by, 1 2 a −2 ∆B i = 8πGδT η i , (2.14) where ∆ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 . It is important to note that in the standard scalar field driven inflation the right hand side of the equation (2.14) vanishes. Therefore, in case of the standard scalar field theory the solution of the vector modes B i are identically zero in all scales. But here we will show that unlike the standard canonical scalar field theory, in case of ELKO the expression of δT η i has the vector modes B i . To calculate the perturbed energy momentum tensor we use the following ansatz for the perturbed ELKO and its dual: δλ = δϕ (η, x) ξ, δ ¬ λ = δϕ (η, x) ¬ ξ. (2.15) The expression of energy-momentum tensor (2.1) shows that it is a sum of three components: T 1 µν = ¬ λ ← − ∇ (µ − → ∇ ν) λ , T 2 µν = g µν L and T 3 µν = F µν . Here we show the (η, x) component of the three terms of the perturbed energy-momentum tensor. The other components ((η, y) and (η, z)) can be written accordingly. The expression of δT 1 ηx , δT 2 ηx and δT 3 ηx can be written as follows: δT 1 ηx = a −4 B x ϕ ′2 − ϕ ′ δϕ ,x − 1 8 HB ′ x ϕ 2 , (2.16) δT 2 ηx = a −4 1 2 B x ϕ ′2 + 3 8 H 2 B x ϕ 2 − a 2 B x V , (2.17) δT 3 ηx = a −4 3 4 H 2 B x ϕ 2 − 1 16 B ′′ x ϕ 2 − 1 8 B ′ x ϕϕ ′ + 1 4 Hϕδϕ ,x + 1 8 ∆B x ϕ 2 . (2.18) In the above expression ( ,x ) denotes the partial derivative with respect to x. Therefore, one can generalize (η, i) component of the energy-momentum tensor for vector perturbation as: δT ηi = a −4 1 2 B i ϕ ′2 − ϕ ′ δϕ ,i − 1 8 HB ′ i ϕ 2 + 3 8 H 2 B i ϕ 2 + a 2 B i V − 1 16 B ′′ i ϕ 2 − 1 8 B ′ i ϕϕ ′ + 1 4 Hϕδϕ ,i + 1 8 ∆B i ϕ 2 (2.19) JCAP05(2015)034 The expression of energy-momentum tensor in the mixed form can be calculated using the following relation: δT µ ν = δT µσ g (0) σν + T µσ(0) δg σν . (2.20) Therefore, using the expression (2.11) (η, i) component of the energy-momentum tensor in the mixed form becomes: δT η i = a −2 1 8 HB ′ i ϕ 2 + 1 16 B ′′ i ϕ 2 + 1 8 B ′ i ϕϕ ′ + ϕ ′ − 1 4 Hϕ δϕ ,i − 1 8 ∆B i ϕ 2 (2.21) 3 Evolution equation of B i Substituting equation (2.21) in the equation (2.14) the expression of the Einstein equation can be written as: ∆B i 1 + 1 4 ϕ 2 M 2 pl = 1 M 2 pl 1 4 HB ′ i ϕ 2 + 1 8 B ′′ i ϕ 2 + 1 4 B ′ i ϕϕ ′ + 2 ϕ ′ − 1 4 Hϕ δϕ ,i ,(3.1) where M pl = 1 √ 8πG is the reduced Planck Mass. The above equation can be rewritten as B ′′ i + 2 ϕ ′ ϕ + H B ′ i − 2 + 8M 2 pl ϕ 2 ∆B i + 16 ϕ ′ − 1 4 Hϕ δϕ ,i ϕ 2 = 0, (3.2) In the Fourier mode the equation (3.1) can be written as B ′′ i + A 1 B ′ i + A 2 k 2 B i − 16ik i ϕ ′ − 1 4 Hϕ δϕ ϕ 2 = 0,(3.3) where, A 1 = 2 ϕ ′ ϕ + H and A 2 = 2 + B ′′ i + A 1 B ′ i − A 2 ∆B i = 0. (3.5) The above equation looks very similar to the evolution equation of scalar perturbations during inflation. In the next section we show that the condition (3.4) leads to consistent background evolution. Background scaling solution Before proceeding with the power-spectrum calculation, we show that the condition (3.4) leads to a consistent background evolution and that it leads to Super-inflation (Ḣ > 0). The Klein-Gordon equation of the background field ϕ is given as [18] ϕ ′′ + 2Hϕ ′ − 3 4 H 2 ϕ + a 2 V ,ϕ = 0, (4.1) -5 - JCAP05(2015)034 where ( ,ϕ ) denotes the derivative with respect to the background field ϕ. The modified Friedmann equations are given as follows: H 2 = 1 1 −F 1 3M 2 pl ϕ ′2 2 + a 2 V , (4.2) H ′ = 1 1 −F 1 3M 2 pl a 2 V − ϕ ′2 + HF ′ , (4.3) whereF = ϕ 2 8M 2 P l . To find the background scaling solutions for power-law type of potential we choose the following forms of scale factor, background field and potential: a (η) = A (−η) −q , ϕ (η) = ϕ 0 (−η) p , V = V 0 ϕ β ,(4.4) where A, ϕ 0 and V 0 are some arbitrary constants which can be expressed in terms of the exponents −q, p and β using the three background equations. Keeping in mind the condition that ϕ ′ = 1 4 Hϕ one can easily write p = −nq, where n = 1 4 . The equations (4.2) and (4.3) are qualitatively same. Substituting (4.4) in the background equations (4.1) and (4.2) one find the following relations between β and q respectively q = 2/ (2 + nβ − 2n) , q = 2/ (2 + nβ) . where H = H/a. From the above expression one can see that whenφ Hϕ > 1 2 one gets the standard inflationary theory withḢ < 0 and whenφ Hϕ < 1 2 one gets the super-inflationary theory withḢ > 0. Therefore, using the condition (3.4) the equation (4.7) tells us thatḢ > 0. One can also show that for 0 < q < 1, the expression of the scale factor (a (η) = A (−η) −q ) gives usḢ > 0. So, under the condition 0 < q < 1 one gets positive acceleration and at the same time one can see that the scale factor grows in cosmic time. This phase of evolution is known as super-inflation [20][21][22]. Solutions in the sub-Hubble and super-Hubble scale Following [24,25] the general solution of (3.5) can be written as: B i = ε ir k b r k B (η, k) e i k. x + b † r k B * (η, k) e −i k. x d 3 k (2π) 3/2 . (5.1) b r and b † r are the annihilation and creation operators respectively. Here ε µr ( k) = (0, ε r ( k)) is the polarisation vector and r = 1, 2, 3. ε 1 ( k), ε 2 ( k) are the mutually orthogonal unit vectors also orthogonal to k and ε 3 ( k) is the unit vector along the direction of k. Here we have defined the polarisation vectors slightly differently than in the references [24,25] -6 - JCAP05(2015)034 (check appendix (A) for discussion). The advantage of doing so is that, instead of redefining the scalar (B = aB) [24,25], in Fourier mode one can now directly write the evolution equation (3.5) in terms of the scalar quantity B (η, k) by replacing ∆ with −k 2 , B ′′ + A 1 B ′ + A 2 k 2 B = 0. (5.2) One can see that the coefficient of k, √ A 2 , in equation (5.2) is not a constant in time. One can remove the time dependent coefficient of k 2 by redefining the time parameter as dη = √ A 2 dη [26]. Under this change of the time variable, equation (3.5) can be written as B ,ηη +Ã 1 B ,η + k 2 B = 0, (5.3) whereÃ 1 = 1 2 A 2,η A 2 + A 1 √ A 2 . Here, in terms ofη, A 1 becomes A 1 (η) = 2 √ A 2 ϕ ,η ϕ +H , whereH = a ,η /a. One can eliminate B ,η term from equation (5.3) by redefining B (η, k) = B (η, k) f (η). Substituting this form of B in equation (5.3) one can eliminate B ,η term by setting its coefficient as zero, which finally gives us f = exp − 1 2 Ã 1 dη , (5.4) B ,ηη + k 2 − Ã 1,η 2 +Ã 2 1 4 B = 0. (5.5) From the equation (4.2) one can see that ϕ 2 < 8M 2 P l , as H can not be imaginary. Thus, when 8M 2 P l ϕ 2 is much larger than 2, one can write Using (4.4) and the expression of √ A 2 one can write the expression ofη in terms of η as: A 2,η A 2 ≈ −2 ϕ ,−η = (−η) 1−2p 1−2p . Therefore, one can see that η → −∞ =⇒η → −∞ and η → 0 =⇒η → 0, when p ≪ 1. Thus, the expression of the scale factor in terms ofη becomes a (η) = (−η) −m , where m = q 1−2p . Therefore, in this case also 0 < m ≪ 1. Finally, using the expression of a(η) the evolution equation (5.6) can be expressed in terms of Bessel differential equation: B ,ηη + k 2 − 1 η 2 ν 2 − 1 4 B = 0,(5.7) where ν 2 = 1 4 1 + 153 16 + 9 2ǫ m 2 . Whereǫ = 1 −H ,η H is the slow-roll parameter in the standard inflationary scenario. However, it should be noted that in the super-inflationary phaseǫ can be ∼ 1 or large. In the super-inflationary phase one can achieve acceleration withoutǫ requiring to be smaller than one. JCAP05(2015)034 As ν is positive, the solution of the equation can be expressed in terms of the Hankel function of the first and second kind B k = −η a 1 (k) H (1) ν (x) + a 2 (k) H (2) ν (x) , (5.8) where x = −kη. In the limit of x ≫ 1, the properties of the Hankel functions are following H (1) ν (x ≫ 1) ≈ 2 πx e ix−i π 2 (ν+ 1 2 ) , H (2) ν (x ≫ 1) ≈ 2 πx e −ix+i π 2 (ν+ 1 2 ) . (5.9) Following the methods used in case of standard inflationary theory, to match with the plane wave solution in the sub-Hubble scale -B k ∼ e ix √ k -one can set a 2 = 0 and a 1 = √ π 2 e i π 2 (ν+ 1 2 ) . The property of the Hankel function H (1) ν (x), in the limit of x ≪ 1 becomes H (1) ν (x ≪ 1) ≈ 2 π e −i π 2 2 (ν− 3 2 ) Γ (ν) Γ 3 2 x −ν . (5.10) Thus, the solution of B k in the super-Hubble scale can be written as \| B k \|∼ −ηx −ν = k √ k 3 (k) 1 2 −ν (−η) 1 2 −ν . (5.11) Using the fact that the solutions nearly remains unchanged after Hubble crossing, one can use k =H during Hubble crossing and the expression (5.11) can be rewritten as \| B k \|∼H √ k 3 (k) 1 2 −ν (−η) 1 2 −ν . (5.12) Finally, the expression of (B) can be written as: \| B k \|∼ e − 9 8 ∆NH √ k 3 (k) 1 2 −ν (−η) 1 2 −ν . (5.13) From the expression of ν one can identify the spectral index n V for vector modes as: n V = 153 16 + 9 2ǫ m 2 . (5.14) For a small value of m one can approximate ν ∼ 1 2 . Therefore, from equation (5.13) one can see that in the super-Hubble scale the vector modes will be nearly frozen and nearly scale independent similar to the scalar perturbations. It is important to note that there are no observational evidence for scale independent vector perturbation yet. However, superinflationary phase naturally provides the scale invariance. One can identify that the term similar toH √ k 3 also appears in the amplitude of the scalar perturbations, for example one can see reference [27]. But in case of vector modes the amplitude is suppressed by the factor (e − 9 8 ∆N ) compared to the scalar modes. Here, ∆N is the number of e-foldings required for super-inflation. The super-inflationary phase requires new physics which in our case is provided by ELKO. Hence, the super-inflationary phase precedes the standard inflationary phase and the number of e-folding is ∼ O (1). The observation of the vector modes in the CMB polarization can restrict the number of e-foldings of super-inflation. In this work we have calculated the perturbed energy-momentum tensor of the ELKO. It has been shown that unlike the standard scalar field case, the non-diagonal component (specifically, (η, i)) of the stress-energy tensor is non-zero. The same component of the Einstein equation gives us the evolution equation of the vector modes (B i ) which looks similar to the scalar perturbation equation. We have analytically obtained the vector perturbations for the background evolution satisfied by ϕ ′ = 1 4 ϕH . We have shown that this condition leads to super-inflation (Ḣ > 0). We have shown explicitly that the vector perturbations are nearly scale invariant and frozen in the super-Hubble scales. However, the amplitude of these perturbations are smaller compared to the scalar perturbation. In the super-Hubble scale one can look at the behaviour of the solution of (3.3) by setting k → 0. As k = k 2 x + k 2 y + k 2 z , each k i will also be small, i.e., k i → 0. As, so far we have not observed any vector modes, one can presume that they have smaller amplitude than the scalar perturbation δϕ/ϕ. Therefore, in general, one can not ignore the last term in (3.3) in the super-Hubble scale. However, as δϕ/ϕ is restricted by the observation to be very small, we presume that it is always possible to find a suitable k i for which the last term in (3.3) can also be smaller than the first two terms. Under such conditions, in the super-Hubble scale, equation (3.3) can be simplified as B ′′ i + A 1 B ′ i = 0,(6.1) which tells us that for positive value of A 1 , the solution for B i is frozen in the super-Hubble scale for a given initial condition, similar to the scalar perturbation. However, the initial condition is given by the solutions in the sub-Hubble scale, which can bring the k dependence in the full solutions of (3.3). In this work we have used the condition ϕ ′ = 1 4 ϕH which gives us super-inflation. However, one can understand from equation (6.1) that during standard inflation, even when the above condition is violated, the vector modes will be nearly constant in time in the super-Hubble scale. This needs further investigation. As, usually the vorticity in the perfect fluid follow nearly the similar equation as the vector modes (in this case it will be second order differential equation in time with some additional terms), hence, the vorticity can also be generated in the first order perturbation theory. With the generated vorticity one can also look into the production of large-scale primordial magnetic field. Once the Planck polarization results are published, the consequences of this kind of models of vector perturbations can be verified. For the observational possibilities one can look into the ref. [28]. We know that most of the inflationary models with standard scalar fields can not produce pure vector modes in the first order. Apart from producing scalar perturbations consistent with observations, the inflationary theory driven by non-standard spinors like ELKO can also produce pure vector modes in the first order. Therefore, the observation of vector modes in the CMB polarization can make this kind of non-standard spinors a potential candidate for inflaton. ϕ 2 . 2The Fourier modes are related to the partial derivatives as ∂ i ≡ −ik i . The last term of the above equation acts as a source term. For simplicity, one can set the last term in eq. (3.3) to vanish, i.e., condition, the evolution equation(3.2) simplifies to background equation (4.3) in cosmic time gives us: H 1 − πGϕ 2 = −4πGφ 2 + 2πGHϕφ, (4.7) η ϕ . Therefore, under this condition the expression ofÃ 1 becomesÃ 1 ≈ ϕ ,η ϕ + 2H. Using the condition (3.4) (which, in terms of η, remains unchanged) the factor f becomes f = exp − 9 8 ∆N . Here ∆N ≈ H dη = Hdη denotes the number of e-folding required for super-inflation. This factor acts as a suppressing factor. The larger the number of e-folding, the larger is the suppressing factor. Using the expression ofÃ 1 and the condition (3.4) in terms ofη equation (5.5) can be expressed in terms of H as B ,ηη + k 11) AcknowledgmentsWe thank T. R. Seshadri and Kandaswamy Subramanian for useful discussions. The work is supported by Max Planck-India Partner Group on Gravity and Cosmology. SS is partially supported by Ramanujan Fellowship of DST, India.-9 -JCAP05(2015)034A Polarisation vector in the curved space-time Following the formalism in the quantum field theory (for example, one can look at[29]) in flat Minkowski space-time, the orthonormality and completeness relations of the polarisation vectors ε µ r k can be written asHere ε rµ ( k) = η µν ε ν r ( k). The above formalism is used in the quantisation of the electromagnetic field (A µ ) in the Minkowski space-time. One can choose the polarisation vector as ε µ r ( k) = (0, ε r ( k)), where r = 1, . . . , 3. ε 1 ( k), ε 2 ( k) are the mutually orthogonal unit vectors also orthogonal to k. ε 3 ( k) is the unit vector along the direction of k. Therefore, ε 1 ( k), ε 2 ( k) are also orthogonal to ε 3 ( k).To generalise the expressions (A.1), (A.2) and (A.3) in the curved space-time, in reference[24,25]the authors have chosen to multiply 1/a with all components of the polarisation vector ε µ r ( k). Then one can replace η µν with g µν in (A.2) and the expression (A.3) remains unchanged. This formalism gives correct solutions of the electro-magnetic vector potential A i as one can always absorb the scale factor in the scalar term which appears in the Fourier decomposition of A i . 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[ "DATA-DRIVEN TIGHT FRAME FOR CRYO-EM IMAGE DENOISING AND CONFORMATIONAL CLASSIFICATION", "DATA-DRIVEN TIGHT FRAME FOR CRYO-EM IMAGE DENOISING AND CONFORMATIONAL CLASSIFICATION" ]	[ "Yin Xian \nDepartment of Mathematics\nHong Kong University of Science and Technology\n\n", "Hanlin Gu \nDepartment of Mathematics\nHong Kong University of Science and Technology\n\n", "Wei Wang \nDepartment of Chemistry\nDepartment of Applied Mathematics\nHong Kong University of Science and Technology\nBeijing Computational Science Research Center\n\n", "Xuhui Huang \nDepartment of Chemistry\nDepartment of Applied Mathematics\nHong Kong University of Science and Technology\nBeijing Computational Science Research Center\n\n", "Yuan Yao \nDepartment of Mathematics\nHong Kong University of Science and Technology\n\n", "Yang Wang \nDepartment of Mathematics\nHong Kong University of Science and Technology\n\n", "Jian-Feng Cai \nDepartment of Mathematics\nHong Kong University of Science and Technology\n\n" ]	[ "Department of Mathematics\nHong Kong University of Science and Technology\n", "Department of Mathematics\nHong Kong University of Science and Technology\n", "Department of Chemistry\nDepartment of Applied Mathematics\nHong Kong University of Science and Technology\nBeijing Computational Science Research Center\n", "Department of Chemistry\nDepartment of Applied Mathematics\nHong Kong University of Science and Technology\nBeijing Computational Science Research Center\n", "Department of Mathematics\nHong Kong University of Science and Technology\n", "Department of Mathematics\nHong Kong University of Science and Technology\n", "Department of Mathematics\nHong Kong University of Science and Technology\n" ]	[]	The cryo-electron microscope (cryo-EM) is increasingly popular these years. It helps to uncover the biological structures and functions of macromolecules. In this paper, we address image denoising problem in cryo-EM. Denoising the cryo-EM images can help to distinguish different molecular conformations and improve three dimensional reconstruction resolution. We introduce the use of data-driven tight frame (DDTF) algorithm for cryo-EM image denoising. The DDTF algorithm is closely related to the dictionary learning. The advantage of DDTF algorithm is that it is computationally efficient, and can well identify the texture and shape of images without using large data samples. Experimental results on cryo-EM image denoising and conformational classification demonstrate the power of DDTF algorithm for cryo-EM image denoising and classification.	10.1109/globalsip.2018.8646614	[ "https://arxiv.org/pdf/1810.08829v2.pdf" ]	67,874,019	1810.08829	339a11c12605457db9afd8b7359c662957a1da08	DATA-DRIVEN TIGHT FRAME FOR CRYO-EM IMAGE DENOISING AND CONFORMATIONAL CLASSIFICATION Yin Xian Department of Mathematics Hong Kong University of Science and Technology Hanlin Gu Department of Mathematics Hong Kong University of Science and Technology Wei Wang Department of Chemistry Department of Applied Mathematics Hong Kong University of Science and Technology Beijing Computational Science Research Center Xuhui Huang Department of Chemistry Department of Applied Mathematics Hong Kong University of Science and Technology Beijing Computational Science Research Center Yuan Yao Department of Mathematics Hong Kong University of Science and Technology Yang Wang Department of Mathematics Hong Kong University of Science and Technology Jian-Feng Cai Department of Mathematics Hong Kong University of Science and Technology DATA-DRIVEN TIGHT FRAME FOR CRYO-EM IMAGE DENOISING AND CONFORMATIONAL CLASSIFICATION Index Terms-Cryo-EM imagesimage denoisingcon- formational classificationdata-driven tight frame The cryo-electron microscope (cryo-EM) is increasingly popular these years. It helps to uncover the biological structures and functions of macromolecules. In this paper, we address image denoising problem in cryo-EM. Denoising the cryo-EM images can help to distinguish different molecular conformations and improve three dimensional reconstruction resolution. We introduce the use of data-driven tight frame (DDTF) algorithm for cryo-EM image denoising. The DDTF algorithm is closely related to the dictionary learning. The advantage of DDTF algorithm is that it is computationally efficient, and can well identify the texture and shape of images without using large data samples. Experimental results on cryo-EM image denoising and conformational classification demonstrate the power of DDTF algorithm for cryo-EM image denoising and classification. INTRODUCTION The cryo-electron microscope (cryo-EM) has been established as one of the fundamental techniques in structural biology. It can help to understand the macromolecules' structure, the arrangement of the atoms, and the biological mechanism of proteins [1,2]. Unlike X-ray crystallography, cryo-EM does not require crystallization. Crystallization may change the conformation of the macromolecules, and many proteins and viruses are resistant to it [3]. Cryo-EM is advantageous over Nuclear Magnetic Resonance spectroscopy (NMR) in solving macromolecules in native state. However, because of the limitation of resolution, cryo-EM was not popular in the past. Recent revolutionary advancement in detectors and softwares have improved the resolution of cryo-EM to atomic scale, and it is significantly popular these years [3]. The Nobel Prize in Chemistry in 2017 was awarded for work that developed cryo-EM. The cryo-EM images are created by the electron microscope that provides a top view of the molecules that are frozen Yin Xian and Hanlin Gu have equivalent contributions. in a thin layer of vitreous ice. The created image is called micrograph [4]. Image processing is crucial and it helps remove bad image samples, and facilitates orientation estimation, 3D inversion, 3D reconstruction and conformational classification [5]. The challenge of cryo-EM images processing is that the images are highly influenced by noise [6,7]. The point spread function of the microscope also blurs the images. The noise comes from various sources, and the type and level of noise in the dataset are unknown. The noise will obscure the conformational difference of molecules. It will also obscure projection of the same molecular structure in different viewing directions. In this paper, we address the image denoising problem in cryo-EM, and evaluate the effect of noise reduction in 2D conformational classification. A lot of methods have been proposed to remove noise in cryo-EM images. Singer and his group have designed a toolbox ASPIRE, and proposed Covariance Wiener Filtering (CWF) [8] for image denoising. CWF needs large samples of data in order to estimate the covariance matrix correctly, and have good denoising effect. They also proposed class averaging method, such as vector diffusion map [9] for image denoising. These methods operate on Fourier domain. Other than that, the non-local mean method [10] has also been applied for cryo-EM image denoising. In this paper, we propose to use the multi-image datadriven tight frame (DDTF) [11,12] for cryo-EM image denoising. The DDTF method is inspired by the wavelet tight frame method [13] and the K-SVD method [14]. It uses learned filters to form a tight frame. The Unitary Extension Principle (UEP) condition [15] can be used to construct tight frames. However, it is not easy to satisfy the UEP condition. The DDTF algorithm relaxes the UEP condition, and generate filters with orthogonality. In the image patch space, the generated filters form an orthogonal dictionary. The K-SVD method needs a highly redundant dictionary to obtain a sparse code. The DDTF simplifies the process to obtain the filters coefficients, and reduce the computational cost compared with K-SVD. The use of data-driven tight frame can also better represents images with rich textures compared with standard wavelet methods and PDE based methods [11]. We further use the denoised images for conformational classification. Experimental results show that DDTF outperform BM3D and KSVD in image denoising and classification. It demonstrates the power of DDTF method for cryo-EM image processing. BACKGROUND The problem of cryo-EM image formation model is: [16]: G(u) = C * Y (u) + Z(u)(1) where Y be the clean ideal image, and Z be the additive noise. Let C be the point spread function of the microscope. G is the measure image in real space, and * is the convolution operator. The Fourier transform of the point spread function is the Contrast Transfer Function (CTF). In order to obtain an image close to the original image, the level of noise has to reduce, and the point spread function effect needs to be estimated. In this paper, we are focusing on reducing the noise level of the cryo-EM images. When the point spread function is known, the estimated image can be obtained by deconvolving the denoised images with the point spread function. Singer and his group have developed Covariance Wiener Filtering (CWF) for cryo-EM image denoising. The procedure of this method is to first estimate the covariance matrix of the clean images from the noisy images, and then apply the traditional wiener filtering method, with the use of the estimated covariance matrix, for denoising [8]. The drawback of this method is that it needs a large number of images to accurately estimate the covariance matrix. When the covariance matrix is not accurately estimated, the performance of this method is not good. In this paper, we are seeking to denoise a single cryo-EM image well when the number of images is limited. In the image processing area, the Block-matching and 3D filtering (BM3D) method [17] is considered as an effective baseline. It groups similar and nonlocal image patches into a 3D array and filters the 3D array. The image patches are then put back to the original positions and reweighed to form a denoised image. BM3D works particularly well for images with self-similarities. K-SVD [14] is a dictionary learning method. K-SVD embeds the local overcompleted dictionary into a global Bayesian estimator. For a given noisy image G, the formula for K-SVD image denoising is to solve: α = arg min α \|\|Dα − G\|\| 2 2 + µ\|\|α\|\| 0 (2) where D is the dictionary, and α is the sparse code. The denoised image is given byŶ = Dα. The algorithm introduces the idea of updating image representation adaptively, and iteratively updates the sparse coding step and the dictionary up-date step. Because the dictionary in K-SVD is unstructured, the computational cost of this method is heavy. MULTI-IMAGE DATA-DRIVEN TIGHT FRAME The data-driven tight frame (DDTF) is proposed based on the wavelet tight frame method and the K-SVD method. Compared with the wavelet tight frame, as well as the ridgelet, curvelet and shearlet tight frame methods, the data adpative tight frame method is effective to process natural images that are rich with texture [11]. DDTF process: Given an image G of size m × n, let W be the analysis operator, its adjoint W T is a synthesis operator defined by filters {a i } r i=1 : W T = [S a1 , S a2 , · · · , S ar ], where S a is a linear convolution operator: [11] [S a v](n) = [a * v](n) = k∈Z a(n − k)v(k), where v and a are in l 2 space. S a is of size L × L, where L = m × n. The column of W T forms a tight frame, and W T W = I L .(3) I L is an identity matrix of size L. A tight frame W T can be constructed by the minimization [11]: min α,{ai} r i=1 \|\|α − W (a 1 , a 2 , · · · , a r )G\|\| 2 2 + λ 2 \|\|α\|\| 0 The filter coefficients {a i } r i=1 and the sparse frame coefficients α can be solved iteratively. Specifically, given initial filter {a (0) i } r i=1 , at the step k, k = 1, 2, · · · , M , we have, α (k) := min α \|\|α − W (a (k) 1 , a (k) 2 , · · · , a (k) r )G\|\| 2 2 + λ 2 \|\|α\|\| 0 (5) {a (k+1) i } r i=1 := min {ai} r i=1 \|\|α k − W (a 1 , a 2 , · · · , a r )G\|\| 2 2(6) For eq. (5), the filter coefficients {a i } r i=1 are given, and α is update. For eq. (6), the sparse frame coefficient α is given, and {a i } r i=1 is update. It can be proved that α * = T µ (W G) is a unique solution for eq. (5), where T µ is a hard thresholding operator. Let A = [â 1 ,â 2 , · · · ,â r ], whereâ is a vectorized form of a 2D filter a. The unique solution of eq. (6) can be obtained by A * = 1 r QP T . P and Q satisfies the singular value decomposition ofḠᾱ, that isᾱḠ = P DQ T .Ḡ = [g 1 , · · · , g n ], where g i is the i-th vectorized image patches of G. andᾱ is the corresponding sparse frame coefficient matrix [11]. We can get the filter coefficients {a (0) i } r i=1 by A * . The denoised image can be obtained by G * = W T (T µ (W G)).(7) Connection with dictionary learning: This subsection shows the DDTF in the image patch space is essentially dictionary learning. Let k 1 × k 2 as the image patch size. For an image G, let X = [x 1 , x 2 , · · · , x mn ] is the image patch for G. x j is the j-th vectorized patch in G. Considering a sparse approximation for the dataset X. the frame operator W becomes an orthogonal dictionary. The computation of the algorithm can be further accelerated. Since every signal has both high frequency and low frequency components, represent the orthogonal dictionary as W = [A 1 , A 2 ], where A 1 is a predefined low pass filter, and A 2 is the learned hight pass filters. Eq. (5) and eq. (6) then becomes [12] min A2,α \|\|X − [A 1 , A 2 ]α\|\| 2 2 + λ 2 \|\|α\|\| 0 (8) min A2 \|\|X − (A 1 α A1 + A 2 α A2 )\|\| 2 2(9) where α = [α A1 , α A2 ] denotes the codes associated with A 1 and A 2 . The orthogonality constraint of W, according to eq. (3), becomes A T 2 A 2 = I r ; A T 1 A 2 = 0.(10) The procedure to update the A 2 and α is similar to the tight frame case. The denoise dimage patches can be also obtained by the formula X * = W T µ (W T X),(11) where W = [A 1 , A * 2 ] in this case. Re-synthesizing the image patches, we can obtain the denoised image. Multi-image denoising: For multi-image denoising, W is learned from multiple images. Given N input images {G i } N i=1 of size m × n, take k 1 × k 2 as the image patch size as before, and concatenate the image patches together to form the input: X = [X 1 , X 2 , · · · , X N ] ∈ R k1k2×N mn . where X i = [x i,1 , x i,2 , · · · , x i,mn ] is the image patch for the i-th image G i . Apply eq. (5) and eq. (6) to update and obtain the sparse frame coefficient α and filter coefficients {a i } r i=1 . Apply eq. (11) to reduce the noise level of the image patches, and re-synthesize to obtain the denosied images. The advantage of using multiple images is that using multiple images can better capture the distribution of noise information and perform denoising. The position of the macromolecules is not always at the center of the cryo-EM images. Constructing W from mutiple images can better eliminate the background noise. EXPERIMENTAL RESULTS Datasets We test the efficiency of the denoising algorithms on the structurally heterogeneous synthetic dataset. The dataset is generated based on five representative atomistic structures of Thermus aquaticus RNA polymerase. It is obtained from the snapshots of molecular dynamics simulations. The images are generated by two dimensional projections of a three dimensional model of RNA polymerase. The size of the images are 128 × 128. We evaluate the effectiveness of the noise reduction algorithms according to the Peak Signal to Noise Ratio (PSNR), and conformational classification error rate. The point spread function is set to be a Delta function in this case. Additive Gaussian noise is added to the clean images. The clean images are obtained from the voxelization and 2D projection of the atomic structure of the molecules. It is prepared by the Xmipp package [18]. The noisy datasets are prepared at different level of signal to noise ratio (SNR). The SNR is defined by Es σ 2 , where E s is the power of the signal, and σ 2 is the variance of the noise. Denoised results We first of all evaluate the CWF algorithm on the noisy images when SNR is 0.1. The result is shown in Figure 1 (c). Compared with the results shown in [8], which uses 10,000 images to estimate the mean and covariance matrix for denoising, we use 2031 images for experiment given the limitation of computational resource. According to the results, CWF fails to capture the shape and content of the polymerase, and is not suitable for the conformational classification in this case. It is possible that the estimated covariance matrix is not accurately estimated and leads to not desirable denoising result. We evaluate DDTF, BM3D and K-SVD on the dataset. The levels of SNR are 0.8, 0.4, 0.2, 0.1, 0.05 and 0.01. There are 2031 images for each SNR level. The pixel value of each image is in the range of 0 to 255. In the experiments, the image patch size of DDTF, BM3D and K-SVD are 16. We use 20 images to obtain the filter coefficients and sparse code of the multi-image DDTF. The initial filters are generated by a discrete cosine function. For K-SVD, we use random initial dictionary for initialization. Figure 1 shows the noise reduction effect of of each algorithms when SNR is equal to 0.1. Table 1 The PSNR measures the ratio of the maximum possible power of a signal and the power of noise that affects the fidelity of signal representation. It is generally used to show the image reconstruction quality. According to the results shown in Table 1, DDTF performs better than BM3D and K-SVD in PSNR. From Figure 1, the multi-image DDTF better preserves the shape, and has less artifacts compared with other methods. Since the cryo-EM images have little self-similarity pattern, the BM3D method, which uses nonlocal information of the images for denoising, does not perform well. Conformational classification results After image denoising, we classify the "clamp-open" structure and "clamp-close" structure of the RNA polymerase among the images. Classification helps us to solve the relative population distribution of stable conformations of macromolecules. The illustration of these two structures are shown in Figure 2. The numbers of "clamp-open" structure and "clamp-close" structure images are 420 and 429 respectively. We compared the denoised image with DDTF, BM3D and K-SVD methods, and noisy image without denoising as inputs for the conformational classification. We perform a brute force classification of images by pixel information. We select 31 template images from landmarks around the "North Pole" of the sphere for each conformation. Because the molecules can be rotated at random angels, we rotate and reflect the template images, and the Euclidean distance is calculated between the tested images and the rotated and reflected templates. Classification is performed based on these distances. When the distance of the noisy image is close to the "clampopen" template, the image is classified as "clamp-open" class. Similarly, when the noisy image is close to "clamp-close" template, the image is in the "clamp-close" class. The classification results are shown in Table 2. According to the results, we can see that DDTF performs better than BM3D and K-SVD in classification error rate. The lower the SNR, the better DDTF compared with BM3D and KSVD. DDTF can well capture the shape and texture of images. It helps to distinguish different molecules conformation. CONCLUSION In this paper, we applied DDTF method for cryo-EM image denoising and conformational classification. The denoising effect of data-driven tight frame is better than K-SVD and BM3D. It also improves conformational classification accuracy over other algorithms. Our research demonstrates that data-driven tight frame is an effective algorithm for cryo-EM image processing. Fig. 1 . 1shows the average PSNR. For an image x of size L × M , the PSNR of its estimated imagex is defined by P SN R(x,x) = 10 log 10 x(i, j) − x(i, j)) 2. Noise reduction of different algorithms when SNR=0.1 Fig. 2 . 2Illustration of conformations of RNA polymerase Table 1 . 1PSNR(dB) SNR DDTF BM3D K-SVD 0.8 45.2005±1.51 40.5436±1.57 37.8226±0.72 0.4 42.4517±1.61 36.7738±1.50 35.6769±0.66 0.2 41.3008±1.66 33.6792±1.34 33.8494±0.60 0.1 38.9199±1.69 31.7994±1.07 32.4848±0.53 0.05 36.6697±1.46 31.1473±0.85 31.6649±0.52 0.01 33.2232±1.35 31.0944±0.77 31.0573±0.60 Table 2 . 2Classification Error Rate (%)SNR DDTF BM3D K-SVD Noisy Image 0.8 0 0 0 0.59 0.4 0 0.35 0 22.85 0.2 0.35 0.12 0.58 48.00 0.1 0.47 10.95 14.72 50 0.05 2 36.04 44.76 50 The development of cryo-EM into a mainstream structural biology technique. E Nogales, Nature methods. 13124E. Nogales, "The development of cryo-EM into a main- stream structural biology technique," Nature methods, vol. 13, no. 1, pp. 24, 2015. A primer to single-particle cryo-electron microscopy. Y Cheng, N Grigorieff, P Penczek, T Walz, Cell. 1613Y. Cheng, N. Grigorieff, P. Penczek, and T. 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[ "Hydrodynamics of topological Dirac semi-metals with chiral and Z 2 anomalies", "Hydrodynamics of topological Dirac semi-metals with chiral and Z 2 anomalies" ]	[ "Marek Rogatko [email protected]@tytan.umcs.lublin.pl \nInstitute of Physics Maria Curie\nSkłodowska University\npl. Marii Curie-Skłodowskiej 120-031LublinPoland\n", "Karol I Wysokinski \nInstitute of Physics Maria Curie\nSkłodowska University\npl. Marii Curie-Skłodowskiej 120-031LublinPoland\n" ]	[ "Institute of Physics Maria Curie\nSkłodowska University\npl. Marii Curie-Skłodowskiej 120-031LublinPoland", "Institute of Physics Maria Curie\nSkłodowska University\npl. Marii Curie-Skłodowskiej 120-031LublinPoland" ]	[]	We consider the hydrodynamical model of topological Dirac semi-metal possessing two Dirac nodes separated in momentum space along a rotation axis. It has been argued that the system in question, except the chiral anomaly, is endowed with the other one Z 2 . In order to model such a system we introduce two U (1)-gauge fields. The presence of the additional Z 2 anomaly leads to the non-trivial modifications of hydrodynamical equations and to the appearance of new kinetic coefficients bounded with the vorticity and the magnetic parts of Maxwell and auxiliary U (1)-gauge fields.	10.1007/jhep09(2018)136	[ "https://arxiv.org/pdf/1804.02202v2.pdf" ]	56,271,709	1804.02202	7ed7ff7bc518b901c6d95da5fc37c5aa4c1f8dd5	Hydrodynamics of topological Dirac semi-metals with chiral and Z 2 anomalies 6 Apr 2018 Marek Rogatko [email protected]@tytan.umcs.lublin.pl Institute of Physics Maria Curie Skłodowska University pl. Marii Curie-Skłodowskiej 120-031LublinPoland Karol I Wysokinski Institute of Physics Maria Curie Skłodowska University pl. Marii Curie-Skłodowskiej 120-031LublinPoland Hydrodynamics of topological Dirac semi-metals with chiral and Z 2 anomalies 6 Apr 2018Prepared for submission to JHEParXiv:1804.02202v1 [hep-th]Gauge-gravity correspondenceHolography and condensed matter physics (AdS/CMT)Black Holes We consider the hydrodynamical model of topological Dirac semi-metal possessing two Dirac nodes separated in momentum space along a rotation axis. It has been argued that the system in question, except the chiral anomaly, is endowed with the other one Z 2 . In order to model such a system we introduce two U (1)-gauge fields. The presence of the additional Z 2 anomaly leads to the non-trivial modifications of hydrodynamical equations and to the appearance of new kinetic coefficients bounded with the vorticity and the magnetic parts of Maxwell and auxiliary U (1)-gauge fields. Introduction The possibility of the hydrodynamic approach to transport relies on the fact that strong interactions of the constituent particles which, at low energies and long length -scales, move like a fluid can be described with only a few collective or slowly varying variables. These include the local velocity v(x), temperature T (x) and chemical potentials µ a (x) related to all conserved charges (their densities are denoted by ρ a (x)). The hydrodynamics of the relativistic fluid has been developed by Landau [1] and others [2] and generalized to take relativistic triangle anomalies [3,4] into account. A purely hydrodynamic derivation of the anomaly effects, considering the first order in derivation expansion was presented in [5]. The idea was to examine the local entropy production rate in the presence of anomalies and impose the positivity constraint stemming from the second law of thermodynamics. It was shown that the contributions from the anomaly to the entropy production were locally unbounded and might potentially violate the second law of thermodynamics, so the proper generalizations were necessary. In turn, these facts lead to a set of differential equations for the novel transport coefficients connected with the anomaly. Further, this idea was implemented to the case of anomalous superfluids [6]- [8] and non-abelian symmetry [9,10]. On the other hand, the chiral magnetic anomaly, i.e., anomaly induced phenomenon of electric charge separation along the axis of the applied magnetic field in the presence of fluctuating topological charge was widely studied [11]- [16]. The aforementioned phenomenon have attracted a lot of attention due to the possible explanation of an experimentally observed charge asymmetry in heavy ion collisions and provided explanation for the observed decay of neutral pion into photons. The anomalies have been predicted [17] and later found [18,19] to play an important part in the description of electrons in solids. The necessity of relativistic description of electrons in solids may appear superficial, as the velocity of electrons in solids typically equals a small fraction of the light velocity. However, the spectrum of electrons in many materials and close to some special points in the Brillouin zone, has a relativistic form characteristic for massless particles. Such Diraclike massless nature of spectrum is protected by symmetries and has been spotted in the two -dimensional graphene [20] and at the surfaces of the crystalline topological insulators [21]. The Dirac -like spectrum is predicted and observed in the three -dimensional materials known as Dirac or Weyl semi-metals [22][23][24][25][26][27][28]. The transport properties of graphene with the Dirac point at the Fermi energy have been proposed to follow the hydrodynamic description [29]. Later measurements confirmed the hydrodynamic behavior of electrons in graphene [30] and in three -dimensional systems [27,28,[31][32][33]. All this makes the relativistic hydrodynamic approach to electrons in condensed matter a timely and important issue. Moreover, the recent experimental works provide clear evidences that chiral anomaly is observed in condensed matter systems. Namely, it was spotted in Dirac semi-metal Na 3 Bi [34], ZrTe 5 [35], as well as, in Weyl semi-metal TaAs and NbP [36]- [38]. The mentioned two classes of Dirac semi-metals (DSM) have acquired attention in the contemporary investigations. In the first one the Dirac points appear at the time reversal invariant momenta in the first Brillouin zone, while in the other the Dirac points take place in pairs and are separated in momentum space along a rotational axis [39,40]. It turns out that the experimentally found examples of DSM belong mainly to the second class of the aforementioned materials. The Dirac points in the second class of aforementioned semi-metals are endowed with a non-trivial Z 2 topological invariant protecting the nodes and leading to the presence of Fermi arc surface states [41]- [44]. The novel charge, in a close analogy to the chiral one, is also not conserved under the action of external fields. The non-conservation of the novel anomalous charge has been argued to have an effect on transport characteristics of materials [45]. Thus the recent studies of three dimensional condensed matter systems open the doors to symmetries not spotted in other relativistic objects, making the subject even more intriguing. The main motivation behind our considerations is a natural question about the possible influence of Z 2 topological charge on transport characteristics of the studied materials. The aim of the present work is to generalize relativistic hydrodynamics including the chiral anomaly [3,4] and the additional anomaly, which we call Z 2 anomaly after the paper [45]. The two anomalous charges in the considered theory require the existence of the two conjugate to them chemical potentials (µ and µ d ). At the equilibrium both chemical potentials take zero values. Accordingly we also introduce two U (1)-gauge fields, one being the standard Maxwell field coupled to the chiral anomalous charge and other coupled to the Z 2 topological charge. The derived set of hydrodynamic equations generalizes those previously found [5] and extensively discussed [46,47] in the literature. The organization of the paper is as follows. In the next section 2 we present the calculations leading to generalization of the relativistic hydrodynamic equations [5] in such a way that they take into account two anomalous charges, responsible for chiral and Z 2 anomalies. In section 3 we conclude with the discussion of the main results and possible modifications of the transport characteristics of materials. Hydrodynamical model In this section we examine the hydrodynamical model of topological Dirac semi-metal in which two Dirac nodes, protected by rotational symmetry, are separated in momentum space along a rotation axis. It has been argued that the aforementioned system constitutes a source of the additional Z 2 anomaly, except the chiral one, which leads to the nonconservation of the corresponding anomalous Z 2 topological charge [45]. In order to model such a system we consider anomalous charges connected with two U (1)-gauge fields. One of them is the ordinary Maxwell gauge field, the other is the additional one connected with the Z 2 anomalous charge. The hydrodynamical equations of motion in the presence of Z 2 and chiral anomalies are provided by ∂ α T αβ (F, B) = F βα j α (F ) + B βα j α (B), (2.1) ∂ α j α (F ) = C 1 E α B α + C 2ẼαB α , (2.2) ∂ α j α (B) = C 3Ẽα B α + C 4 E αB α , (2.3) where C i , i = 1, . . . , 4 denote the constants which determine the adequate anomalies. The electric and magnetic components of the two gauge fields, in the fluid rest frame, are written respectively as E α = F αβ u β , B α = 1 2 ǫ αβρδ u β F ρδ , (2.4) E α = B αβ u βB α = 1 2 ǫ αβρδ u β B ρδ . (2.5) F αβ = 2∂ [α A β] stands for the ordinary Maxwell field strength tensor, while the second U (1)- gauge field B αβ is given by B αβ = 2∂ [α B β] . On the other hand, j α (F ), j α (B) represent the adequate currents connected with the gauge fields. The relation (2.2) describes the modifications of the anomalous chiral charge conservation law when the external magnetic and electric fields parallel to each other are applied to the system, while the equation (2.3) expresses the changes of the anomalous Z 2 charge conservation law. The energy momentum tensor and the currents needed for the hydrodynamic description of the relativistic fluid are given by [1,5] T αβ = ǫ + p u α u β + pg αβ + τ αβ , (2.6) j α (F ) = ρ u α + V α F , (2.7) j α (B) = ρ d u α + V α B , (2.8) where ǫ is the energy per unit volume, p the pressure of the fluid, ρ, ρ d are the U (1) charge densities, while τ αβ and V α F (B) depict higher order corrections in velocity gradients and correspond to the dissipative effects in the fluid. In the rest frame of the fluid element, there are no dissipative forces and u α τ αβ = 0 and u α V α F = u α V α B = 0. The four-vector u α , with the normalization u α u α = −1, describes the flow of the considered fluid. Using the thermodynamical relations ǫ + p = T s + µ ρ + µ d ρ d , dp = s dT + ρ dµ + ρ d dµ d , (2.9) where s is the entropy per unit volume, the explicit expression for energy-momentum tensor and u β ∂ α T αβ , as well as, the expressions for ∂ α j α (F ) and ∂ α j α (B), one arrives at the following relation: ∂ α su α − µ T V α F − µ d T V α B = − 1 T τ αβ ∂ α u β (2.10) − V α F ∂ α µ T − E α T − V α B ∂ α µ d T −Ẽ α T − µ T C 1 E α B α + C 2ẼαB α − µ d T C 3Ẽα B α + C 4 E αB α . As was pointed out in [5], if we did not take into account the influence of the anomalies (i.e., C i = 0), and supposed the positivity of the conductivities σ F > 0 (σ B > 0) and viscosity parameters η and ζ [1] entering the formula for τ αβ , the right-hand side of (2.10) would be positive for the following relations: V α F = −σ F T P αβ ∂ β µ T + σ F E α , (2.11) V α B = −σ B T P αβ ∂ β µ d T + σ BẼ α . (2.12) Thus, the equation (2.10) can be interpreted as describing the entropy production. Its righthand side is greater or equal to zero, as required by the second law of thermodynamics. The presence of anomalies changes the situation drastically. The terms with C i = 0 can have either sign and, when negative, can even overcome the rest of the terms appearing in the equation (2.10) and thus spoil the positivity of entropy production. Therefore, the entropy flux s α , as well as, all the dissipative terms contributing to the transport current have to be modified. The most general modification of the entropy current, which comprises standard dissipation terms, vorticity ω α = (1/2)ǫ αβρδ u β ∂ ρ u δ and the terms proportional to the magnetic components of the two U (1)-gauge fields are taken in the form s α = su α − µ T V α F − µ d T V α B + D ω α + D B B α + DBB α . (2.13) The dissipative contribution to the U (1)-gauge field currents are also modified by new transport coefficients ξ, ξ B , ξ d and ξB V α F = −σ F T P αβ ∂ β µ T + σ F E α + ξ ω α + ξ B B α ,(2. 14) V α B = −σ B T P αβ ∂ β µ d T + σ BẼ α + ξ d ω α + ξBB α . (2.15) The symbol P αβ = g αβ + u α u β stands for the projector orthogonal to the four-velocity u α , and the unknown functions ξ, ξ d , ξ B , ξB, D, D B , DB depend on T and µ, µ d . Our aim is to find the general formula for these new transport coefficients induced by the quantum anomalies. Assuming that all the anomaly coefficients C i = 0 and repeating the standard algebraic manipulations [1] required by the positivity proof of ∂ α s α , one gets the equation containing on its right-hand side the following additional terms ∂ α (Dω α + D B B α + DBB α ) − (ξω α − ξ B B α ) ∂ α µ T − E α T (2.16) −(ξ d ω α − ξBB α ) ∂ α µ d T −Ẽ α T − µ T (C 1 E α B α + C 2ẼαB α ) − µ d T (C 3Ẽα B α + C 4 E αB α ). In order to satisfy the requirement of the entropy current positivity, the above terms are demanded to vanish [5]. To proceed we relate the derivatives of the vorticity ∂ α ω α to the vorticity ω α itself and similarly, the ∂ α B α is related to B α . For our hydrodynamics (linear in the derivatives of velocity) it is enough to find the required relations for the ideal fluid. They may be achieved by projecting the underlying equations of motion (2.6)-(2.8) of the hydrodynamical model along two orthogonal directions. Namely, along u α and P α β = δ α β + u α u β . As a result we achieve the following relations for the ideal hydrodynamics ( i.e., with τ αβ = 0, V α F = V α B = 0) ∂ α ω α = 2ω α ǫ + p − ∂ α p + F αβ j β (F ) + B αβ j β (B) , (2.17) ∂ α B α = −2ω α E α + B α ǫ + p − ∂ α p + F αβ j β (F ) + B αβ j β (B) , (2.18) ∂ αB α = −2ω αẼ α +B α ǫ + p − ∂ α p + F αβ j β (F ) + B αβ j β (B, B α ,B α , ω α E α , ω αẼ α , E α B α ,Ẽ αB α ,Ẽ α B α , E αB α ,. The condition ∂ α s α ≥ 0 demands vanishing all factors multiplying the above terms. It eventuates in the following differential equations ∂ α D − 2 ∂ α p ǫ + p D − ξ ∂ α µ T − ξ d ∂ α µ d T = 0, (2.20) ∂ α D B − ∂ α p ǫ + p D B − ξ B ∂ α µ T = 0, (2.21) ∂ α DB − ∂ α p ǫ + p DB − ξB ∂ α µ d T = 0,(2.22) and the additional conditions and require vanishing of the coefficients multiplying ∂ α p, ∂ αμ and ∂ αμd , which can be considered as having arbitrary values at the initial time slice [5]. This leads to three sets of the differential equations. The first defines the parameter D(p,μ,μ d ) 2D ρ ǫ + p − 2D B + 1 T ξ = 0, (2.23) 2D ρ d ǫ + p − 2DB + 1 T ξ d = 0, (2.24) ρ D B ǫ + p + ξ B T − µ C 1 T = 0, (2.25) ρ d DB ǫ + p + ξB T − µ C 2 T = 0, (2.26) ρ DB ǫ + p − µ d C 4 T = 0, (2.27) ρ d D ǫ + p − µ d C 3 T = 0.∂ α D = ∂D ∂p μ,μ d ∂ α p + ∂D ∂μ p,μ d ∂ αμ + ∂D ∂μ d p,μ ∂ αμd , (2.29) ∂ α D B = ∂D B ∂p μ,μ d ∂ α p + ∂D B ∂μ p,μ d ∂ αμ + ∂D B ∂μ d p,μ ∂ αμd , (2.30) ∂ α DB = ∂DB ∂p μ,μ d ∂ α p + ∂DB ∂μ p,μ d ∂ αμ + ∂DB ∂μ d p,μ ∂ αμd ,(2.∂D ∂p μ,μ d − 2D ǫ + p = 0, (2.33) ∂D ∂μ p,μ d − ξ = 0, (2.34) ∂D ∂μ d p,μ − ξ d = 0, (2.35) while the next two give the dependence of the partial derivatives of D B (p,μ,μ d ) ∂D B ∂p μ,μ d − D B ǫ + p = 0, (2.36) ∂D B ∂μ p,μ d − ξ B = 0, (2.37) ∂D B ∂μ d p,μ = 0, (2.38) and DB(p,μ,μ d ) ∂DB ∂p μ d ,μ − DB ǫ + p = 0, (2.39) ∂DB ∂μ p,μ d = 0, (2.40) ∂DB ∂μ d p,μ − ξB = 0. (2.41) Using the Gibbs-Duhem thermodynamic relations (2.9) we can arrive at the expression dp = ǫ + p T dT + ρT dμ + ρ d T dμ d (2.42) which in turn can be easily cast into dT = T ǫ + p dp − ρT 2 ǫ + p dμ − ρ d T 2 ǫ + p dμ d . (2.43) This provides the relations as follows: ∂T ∂p μ,μ d = T ǫ + p , ∂T ∂μ p,μ d = − ρ T 2 ǫ + p , ∂T ∂μ d p,μ = − ρ d T 2 ǫ + p . (2.44) By virtue of (2.44) the first equations from each of the sets of the relations (2.33), (2.36) and (2.39), can be immediately integrated. The results yields D = T 2 d(μ,μ d ), D B = T d B (μ,μ d ), DB = T dB(μ,μ d ), (2.45) where d i = d(μ,μ d ), d B (μ,μ d ), dB(μ,μ d ) are the new functions, which do not depend on temperature T . Thus it is more convenient to treat D i as functions of temperature T , and chemical potentialsμ andμ d . To this end we assume the following dependence of the temperature T = T (p,μ,μ d ) and use the relation ∂D i (T,μ,μ d ) ∂μ p,μ d = ∂D i (T,μ,μ d ) ∂μ T,μ d + ∂D i (T,μ,μ d ) ∂T μ,μ d ∂T ∂μ p,μ d . (2.46) The formula similar to (2.46) for the derivative with respect toμ d is supposed. This leads to the system of differential equations provided by T ∂D ∂T μ,μ d − 2D = 0, (2.47) ∂D ∂μ T,μ d − ρ T 2 ǫ + p ∂D ∂T μ,μ d − ξ = 0, (2.48) ∂D ∂μ d T,μ − ρ d T 2 ǫ + p ∂D ∂T μ,μ d − ξ d = 0,(2.49) and for D B one gets T ∂D B ∂T μ,μ d − D B = 0, (2.50) ∂D B ∂μ T,μ d − ρ T 2 ǫ + p ∂D B ∂T μ.μ d − ξ B = 0, (2.51) ∂D B ∂μ d T,μ − ρ d T 2 ǫ + p ∂D ∂T μ,μ d = 0. (2.52) Consequently, one obtains the similar equations for DB T ∂DB ∂T μ,μ d − DB = 0, (2.53) ∂DB ∂μ T,μ d − ρ T 2 ǫ + p ∂DB ∂T μ,μ d = 0, (2.54) ∂DB ∂μ d T,μ − ρ d T 2 ǫ + p ∂DB ∂T μ,μ d − ξB = 0. (2.55) To proceed, we shall replace the derivatives of the type ∂D i , respectively. Consequently, we obtain the three sets of partial differential equations ∂D ∂μ T,μ d = 2ρT ǫ + p D + ξ = 2T D B , (2.56) ∂D ∂μ d T,μ = 2ρ d T ǫ + p D + ξ d = 2T DB,(2.57) where the second equalities follow from the equations (2.23) and (2.24) ∂D B ∂μ T,μ d = ρT ǫ + p D B + ξ B = C 1 Tμ, (2.58) ∂D B ∂μ d T,μ = ρ d T ǫ + p D B = C 3 Tμ d . (2.59) In the above derivations we use the relations (2.25) and (2.28). The last set of the equations can be easily achieved by incorporating (2.27) and (2.26). Namely, one has ∂DB ∂μ T,μ d = ρT ǫ + p DB = C 4 Tμ d , (2.60) ∂DB ∂μ d T,μ = ρ d T ǫ + p DB + ξB = C 2 Tμ. (2.61) On this account, it is customary to write the solutions of the aforementioned sets of the partial differential equations as follows: D B = 1 2 C 1 Tμ 2 + 1 2 C 3 Tμ 2 d + γ 1 (2.62) DB = 1 2 (C 2 + C 4 )Tμμ d + γ 2 (2.63) D = 1 3 C 1 T 2μ3 + 1 2 (C 2 + C 4 )T 2μμ2 d + γ 1μ + γ 2μd + γ 3 ,(2.64) where γ i , i = 1, 2, 3 are integration constants. Contrary to some claims in the literature, these constants are required to vanish, on the account of the relation (2.45). Consequently, one can readily get the following expressions for the four novel kinetic coefficients: ξ = C 1 µ 2 1 − 2 3 ρµ ǫ + p + µ 2 d C 3 − (C 2 + C 4 ) ρµ ǫ + p (2.65) ξ d = − 2 3 C 1 ρµ 3 ǫ + p + (C 2 + C 4 )µµ d 1 − ρ d µ d ǫ + p (2.66) ξ B = C 1 µ 1 − 1 2 ρµ ǫ + p − 1 2 C 3 ρµ 2 d ǫ + p , (2.67) ξB = 1 2 (C 2 + C 4 )µ 1 − ρµ d ǫ + p . (2.68) In the derivation of the above equations we have made use of the continuity of the partial differentials of DB, which provides the equality C 2 = C 4 . Equations (2.65)-(2.68) constitute the main results of the paper. They provide the generalization and in the appropriate limit reduce to those obtained earlier [5]. As the very nontrivial result we remark the fact, novel in comparison to the paper [5], that the kinetic coefficient ξ d is induced by the parameter C 1 = C considered in that work. The extra parameter C 3 introduced here, modifies the values of kinetic coefficients previously found in systems with the triangle anomaly. In the next section, we shall discuss the application of the theory in question to Weyl semi-metals with the two discussed anomalies. 3 Application to Dirac semi-metals with Z 2 topological charge As was mentioned in the introduction most of the known Dirac semi-metals, in particular Na 3 Bi or Cd 2 As 3 , possess a chiral anomaly and two Dirac nodes, each carrying topological Z 2 charge. In these materials two Dirac nodes are protected by rotational symmetry of the crystal. The two anomalies show up in our results as two different chemical potentials: µ corresponds to the chiral anomaly and its change results in the appearance of the chiral currents while µ d decides about the position in energy of the two Dirac nodes. In the presence of a magnetic field parallel to an electric field the corresponding currents are not conserved. The current related to Z 2 anomaly is a spin current, at least so, when the spin is approximately conserved [45]. Due to this interpretation of the Z 2 bound current, one expects that spin -related magnetic fieldB α vanishes, what is equivalent to disappearing of C 2 and C 4 . Assuming C 2 = 0 and C 4 = 0, one immediately observes the disappearance of the ξB kinetic coefficient. Accordingly with the above claim, the other coefficients imply ξ = C 1 µ 2 1 − 2 3 ρµ ǫ + p + C 3 µ 2 d , (3.1) ξ d = − 2 3 C 1 ρµ 3 ǫ + p , (3.2) ξ B = C 1 µ 1 − 1 2 ρµ ǫ + p − 1 2 C 3 ρµ 2 d ǫ + p . (3.3) It is worth pointing out that even in the presence of theẼ α field, the spin conductivity ξ d (and possibly the spin Hall effect) is not affected by it. However, the parameter C 3 modifies the kinetic coefficients related to the chiral anomaly. These findings cosound with the recent kinetic calculations [45], where the authors have noted that the Z 2 anomaly affects magneto-transport properties of Dirac semi-metals. The observational manifestation of the Z 2 anomaly found earlier is connected with the reduction of the diagonal resistivity due to the spin Hall effect and the narrowing of the angular dependence of the magnetoresistance. The detailed analysis of the magneto-conductivity and magneto-resistivity of the Weyl semi-metals based on the presented hydrodynamic approach [48] will be presented in the future publication. Summary and conclusions We have examined the generalized equations of relativistic hydrodynamics allowing the description of electrons in condensed matter systems with linear spectrum and the two different types of anomalies. One of them is the well known chiral anomaly, while the other one, authorizes the anomaly observed in one class of Dirac semi-metal characterized by two Dirac nodes separated in momentum space and lying on the axis of rotation. With the Z 2 anomaly the corresponding charge density ρ d is connected. Its existence forces the non-trivial generalization of the relativistic hydrodynamics. We have found that the additional kinetic parameters, bounded with two different anomalous charges and required by the second law of thermodynamics and positiveness of the entropy production during the flow of electron fluid, enter the hydrodynamic equations in the similar manner. They are a source of the additional kinetic coefficients called earlier magnetic conductivities. In fact these are spin and spin Hall conductivities [45]. Their appearance in the hydrodynamic equations can be traced back to the necessity of adding dissipative terms proportional to the vorticity and magnetic components of the two U (1)gauge fields. Up to the first order in the velocity gradients, they constitute the important component in the proper description of the relativistic fluid. The lack of the magnetic field acting on spin degrees of freedomB α , connected with the discussed Z 2 anomaly, present in topological second type Weyl semi-metals, forces us to put C 2 = C 4 = 0. Interestingly, the very existence of the Z 2 anomaly induces the kinetic coefficient ξ d connected with the Z 2 related conductivity. We argue that this conductivity is bounded with spin conductivity and spin Hall effect in the kinetic approach to the problem in question. The fieldẼ α , in the presence of the parallel to it magnetic field B α , changes via the parameter C 3 , the kinetic coefficients ξ and ξ B related to the chiral anomaly. 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[ "In situ single-shot diffractive fluence mapping for X-ray free-electron laser pulses", "In situ single-shot diffractive fluence mapping for X-ray free-electron laser pulses" ]	[ "Michael Schneider [email protected] \nMax-Born-Institut Berlin\nMax-Born-Straße 2a12489BerlinGermany\n", "Christian M Günther \nMax-Born-Institut Berlin\nMax-Born-Straße 2a12489BerlinGermany\n\nInstitut für Optik und Atomare Physik\nTechnische Universität Berlin\nHardenbergstraße 36a10623BerlinGermany\n", "Bastian Pfau \nMax-Born-Institut Berlin\nMax-Born-Straße 2a12489BerlinGermany\n", "Flavio Capotondi \nElettra Sincrotrone Trieste S.C.p.A\nStrada Statale 14, KM 163.534012BasovizzaTSItaly\n", "Michele Manfredda \nElettra Sincrotrone Trieste S.C.p.A\nStrada Statale 14, KM 163.534012BasovizzaTSItaly\n", "Marco Zangrando \nElettra Sincrotrone Trieste S.C.p.A\nStrada Statale 14, KM 163.534012BasovizzaTSItaly\n\nIstituto Officina dei Materiali\nConsiglio Nazionale delle Ricerche\n34149BasovizzaTSItaly\n", "Nicola Mahne \nElettra Sincrotrone Trieste S.C.p.A\nStrada Statale 14, KM 163.534012BasovizzaTSItaly\n\nIstituto Officina dei Materiali\nConsiglio Nazionale delle Ricerche\n34149BasovizzaTSItaly\n", "Lorenzo Raimondi \nElettra Sincrotrone Trieste S.C.p.A\nStrada Statale 14, KM 163.534012BasovizzaTSItaly\n", "Emanuele Pedersoli \nElettra Sincrotrone Trieste S.C.p.A\nStrada Statale 14, KM 163.534012BasovizzaTSItaly\n", "Denys Naumenko \nElettra Sincrotrone Trieste S.C.p.A\nStrada Statale 14, KM 163.534012BasovizzaTSItaly\n", "Stefan Eisebitt tos.e.email:[email protected] \nMax-Born-Institut Berlin\nMax-Born-Straße 2a12489BerlinGermany\n\nInstitut für Optik und Atomare Physik\nTechnische Universität Berlin\nHardenbergstraße 36a10623BerlinGermany\n" ]	[ "Max-Born-Institut Berlin\nMax-Born-Straße 2a12489BerlinGermany", "Max-Born-Institut Berlin\nMax-Born-Straße 2a12489BerlinGermany", "Institut für Optik und Atomare Physik\nTechnische Universität Berlin\nHardenbergstraße 36a10623BerlinGermany", "Max-Born-Institut Berlin\nMax-Born-Straße 2a12489BerlinGermany", "Elettra Sincrotrone Trieste S.C.p.A\nStrada Statale 14, KM 163.534012BasovizzaTSItaly", "Elettra Sincrotrone Trieste S.C.p.A\nStrada Statale 14, KM 163.534012BasovizzaTSItaly", "Elettra Sincrotrone Trieste S.C.p.A\nStrada Statale 14, KM 163.534012BasovizzaTSItaly", "Istituto Officina dei Materiali\nConsiglio Nazionale delle Ricerche\n34149BasovizzaTSItaly", "Elettra Sincrotrone Trieste S.C.p.A\nStrada Statale 14, KM 163.534012BasovizzaTSItaly", "Istituto Officina dei Materiali\nConsiglio Nazionale delle Ricerche\n34149BasovizzaTSItaly", "Elettra Sincrotrone Trieste S.C.p.A\nStrada Statale 14, KM 163.534012BasovizzaTSItaly", "Elettra Sincrotrone Trieste S.C.p.A\nStrada Statale 14, KM 163.534012BasovizzaTSItaly", "Elettra Sincrotrone Trieste S.C.p.A\nStrada Statale 14, KM 163.534012BasovizzaTSItaly", "Max-Born-Institut Berlin\nMax-Born-Straße 2a12489BerlinGermany", "Institut für Optik und Atomare Physik\nTechnische Universität Berlin\nHardenbergstraße 36a10623BerlinGermany" ]	[]	Free-electron lasers (FELs) in the extreme ultraviolet (XUV) and X-ray regime opened up the possibility for experiments at high power densities, in particular allowing for fluencedependent absorption and scattering experiments to reveal non-linear light-matter interactions at ever shorter wavelengths. Findings of such non-linear effects are met with tremendous interest, but prove difficult to understand and model due to the inherent shot-toshot fluctuations in photon intensity and the often structured, non-Gaussian spatial intensity profile of a focused FEL beam. Presently, the focused beam is characterized and optimized separately from the actual experiment. Here, we present the simultaneous measurement of XUV diffraction signals from solid samples in tandem with the corresponding single-shot spatial fluence distribution on the actual sample. Our in situ characterization scheme enables direct monitoring of the sample illumination, providing a basis to optimize and quantitatively understand FEL experiments.	10.1038/s41467-017-02567-0	null	19,319,088	1705.03814	20bf7283d354c9f1eb8c858dff3bcbd4cda5d527	In situ single-shot diffractive fluence mapping for X-ray free-electron laser pulses Michael Schneider [email protected] Max-Born-Institut Berlin Max-Born-Straße 2a12489BerlinGermany Christian M Günther Max-Born-Institut Berlin Max-Born-Straße 2a12489BerlinGermany Institut für Optik und Atomare Physik Technische Universität Berlin Hardenbergstraße 36a10623BerlinGermany Bastian Pfau Max-Born-Institut Berlin Max-Born-Straße 2a12489BerlinGermany Flavio Capotondi Elettra Sincrotrone Trieste S.C.p.A Strada Statale 14, KM 163.534012BasovizzaTSItaly Michele Manfredda Elettra Sincrotrone Trieste S.C.p.A Strada Statale 14, KM 163.534012BasovizzaTSItaly Marco Zangrando Elettra Sincrotrone Trieste S.C.p.A Strada Statale 14, KM 163.534012BasovizzaTSItaly Istituto Officina dei Materiali Consiglio Nazionale delle Ricerche 34149BasovizzaTSItaly Nicola Mahne Elettra Sincrotrone Trieste S.C.p.A Strada Statale 14, KM 163.534012BasovizzaTSItaly Istituto Officina dei Materiali Consiglio Nazionale delle Ricerche 34149BasovizzaTSItaly Lorenzo Raimondi Elettra Sincrotrone Trieste S.C.p.A Strada Statale 14, KM 163.534012BasovizzaTSItaly Emanuele Pedersoli Elettra Sincrotrone Trieste S.C.p.A Strada Statale 14, KM 163.534012BasovizzaTSItaly Denys Naumenko Elettra Sincrotrone Trieste S.C.p.A Strada Statale 14, KM 163.534012BasovizzaTSItaly Stefan Eisebitt tos.e.email:[email protected] Max-Born-Institut Berlin Max-Born-Straße 2a12489BerlinGermany Institut für Optik und Atomare Physik Technische Universität Berlin Hardenbergstraße 36a10623BerlinGermany In situ single-shot diffractive fluence mapping for X-ray free-electron laser pulses 10.1038/s41467-017-02567-0ARTICLE OPEN Correspondence and requests for materials should be addressed to M.S. NATURE COMMUNICATIONS \| (2018) 9:214 1 Free-electron lasers (FELs) in the extreme ultraviolet (XUV) and X-ray regime opened up the possibility for experiments at high power densities, in particular allowing for fluencedependent absorption and scattering experiments to reveal non-linear light-matter interactions at ever shorter wavelengths. Findings of such non-linear effects are met with tremendous interest, but prove difficult to understand and model due to the inherent shot-toshot fluctuations in photon intensity and the often structured, non-Gaussian spatial intensity profile of a focused FEL beam. Presently, the focused beam is characterized and optimized separately from the actual experiment. Here, we present the simultaneous measurement of XUV diffraction signals from solid samples in tandem with the corresponding single-shot spatial fluence distribution on the actual sample. Our in situ characterization scheme enables direct monitoring of the sample illumination, providing a basis to optimize and quantitatively understand FEL experiments. T he development of free-electron lasers for the extreme ultraviolet (XUV) and X-ray regime has been one of the major leaps in photon-based science in the last few decades. It enabled key advances in the study of ultrafast dynamics of excitations in matter with a unique combination of coherent femtosecond pulses, and ultrahigh fluences up to several J cm −2 for XUV radiation, and several kJ cm −2 in the hard X-ray regime 1,2 . In recent years, observations of non-linear effects in solids such as wave mixing 3,4 , stimulated emission [5][6][7] , and absorption saturation 8 have been reported. Conducting such experiments requires sophisticated control of the sample illumination. This includes the in situ control of the focus position and, possibly, the precise alignment of several free-electron laser (FEL) beams. After the experiment, an exact knowledge of the number of photons per unit time and area on the sample is crucial to interpret the measurements. A number of well-established techniques exist to estimate these pivotal parameters. Gas monitor detectors are able to measure the total photon number in a single, few-femtosecond pulse 9 , but cannot account for the intensity distribution within the focal spot on the sample. This distribution is typically measured separately from the actual experiment using wave-front sensing [10][11][12] , ablative imprints 13,14 , or by detecting the transmitted intensity through a small aperture or behind a sharp knife-edge scanned across the beam in the sample plane 2,15 . These approaches are highly invasive and cannot be performed in tandem with the majority of FEL experiments. They are in particular incompatible with all scattering experiments in the forward direction and cannot account for the finite acceptance of a sample smaller than the beam size or for the beam position on a larger and potentially inhomogeneous sample. This leads to significant uncertainties, especially in diffract-and-destroy experiments, where a new sample is aligned after every single shot [16][17][18] . In this work, we demonstrate the simultaneous measurement of the spatial fluence distribution on the sample in conjunction with the diffraction signal from a solid sample in a single-shot XUV FEL experiment. Our measurement scheme is derived from work on monolithically integrated gratings on carrier membranes 19 and from the theoretical treatment of zone-plate diffraction under off-axis illumination 20 . Via our integrated diffraction monitor design, we are able to map the incident photon distribution on the sample to the detector plane. There, the illumination is recorded simultaneously with the sample's scattering signal. This allows a precise control of the sample position during the experiment and yields reliable information on the sample illumination that is crucial for the interpretation of the data recorded. Results Role of the fluence estimate in non-linear experiments. We demonstrate the importance of a precise fluence measurement for the interpretation of non-linear effects in Fig. 1. Here, we simulate a strongly focused, non-Gaussian FEL beam and consider three different estimates of the spatial fluence distribution f in the sample plane (ξ, η). The distributions represent, respectively, an accurate measurement (Fig. 1a), a blurred, low-resolution estimate as is typically the result of an aperture scan (Fig. 1b) and a constant estimate, where the shot energy is distributed uniformly over a certain area (Fig. 1c). The fluence histograms vary drastically for the different estimates, as shown in Fig. 1d. For each fluence distribution, we calculate the signal levels assuming a linear, power-law, or saturating fluence dependency ( Fig. 1e-g, respectively). These non-linear relations occur for example in two-photon absorption or saturable absorption experiments. It is evident that, except for the linear case, an inaccurate fluence assumption obfuscates some or all of the characteristic parameters of the effect under study. Thus, a correct interpretation of fluence-dependent measurements will only be possible with an accurate, in situ characterization of the incident photon distribution on the sample on a shot-by-shot basis. Grating design. We consider an FEL diffraction experiment that detects scattered radiation as a function of momentum transfer q on a two-dimensional pixelated detector, as shown in Fig. 2. Note that this geometry also includes standard spectroscopy of the photon beam, where the beam at a selected momentum transfer (such as q = 0) is detected as a function of wavelength. In material and life sciences, thin membranes of Si 3 N 4 , Si, or polymers are commonly used to administer samples to the FEL beam. We equip these membranes with a grating structure, that gives rise to an additional scattering signal at a selected detector position 19 . The key idea of this work is to design the gratings such that each point on the sample surface diffracts the incoming light to a separate position on the detector while preserving the spatial relationship of the originating sample points. Figure 3 sketches the basic idea of our concept as a step-wise evolution from regular, to segmented, and finally to the spatially resolving gratings we discuss here. We start with the following sinusoidal transmission function for a regular grating 21 : tðξ; ηÞ ¼ 1 2 þ 1 2 cos 2π p ξ cosðφÞ þ η sinðφÞ ð Þ :ð1Þ Here, ξ and η are spatial coordinates in the sample plane, while p and φ are the grating period and orientation angle, respectively. They are given by the position (x, y, z det ) of the grating's first diffraction order in the detector plane: φ ¼ arctan y=x ð Þ; ð2Þ p ¼ λ sin arctan ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi x 2 þ y 2 p =z det À Á À Á :ð3Þ We turn φ and p into functions of the sample coordinates by setting x ¼ x 0 þ mξ;ð4Þy ¼ y 0 þ mη:ð5Þ When inserted into Eq. (1), the result is a grating, the pitch and orientation of which varies continuously and that diffracts an image of its own illumination function, magnified by the dimensionless parameter m and centered at (x 0 , y 0 ), to the detector. We note that the thereby obtained structures constitute segments of Fresnel zone plates (Supplementary Note 1). Since Eqs (2) and (3) relate to far-field diffraction, the mapping is only valid if the detector is sufficiently far away from the sample to be in the Fraunhofer regime 21 . Specifically, this requires z det >2 w 2 0 λ ;ð6Þ where w 0 is the beam's waist size on the sample. q = 0 p ( , ) ( , ) a b Fig. 2 Experiment geometry and sample design. a An optical system focuses the incoming beam (red lines) onto the sample. Downstream, a 2D pixelated detector records the scattered radiation (red cone). b Enlarged view of the sample and the scattering geometry. The sample bears a suitably tailored, continuously varying grating with local periodicity p(ξ, η) and local orientation angle φ(ξ, η), where ξ and η are the coordinates in the sample plane. Incident light is diffracted away from the undeflected beam (q = 0) with a momentum transfer of ±q(p, φ) according to the local grating parameters. We design the grating such that it maps an enlarged image of the incident illumination, centered around ±(x 0 , y 0 ) in the detector plane at distance z det. . a A regular grating diffracts incoming light to two symmetric points in Fourier-space d and hence reveals no spatial information on the illumination function. b A two-by-two segmented grating yields two symmetric sets of four diffraction spots e. The intensity of each spot is proportional to the illumination of the corresponding sample quadrant. This constitutes the most basic form of a spatially resolving beam profile monitor based on an integrated grating. c A grating with suitably varying period and orientation forms a magnified image of its own illumination in Fourier-space f. The colors in the real-space images indicate the local grating period and mark the corresponding points in the diffraction images In simulations with Gaussian beams, we observe that the following condition must simultaneously be satisfied: z det m <2π w 2 0 λ :ð7Þ This relation enforces that the illumination does not change drastically within a small number of grating periods, which would lead to errors in the diffracted fluence maps (see Supplementary Notes 1 and 2 for a detailed discussion). FEL experiment. We present a single-shot XUV FEL diffraction pattern obtained in this fashion in Fig. 4a. The whole diffraction pattern consists of the ring-shaped primary sample signal (Methods) and the grating's positive and negative diffraction orders. Note that this particular experiment investigates the nonlinear breakdown of the primary sample signal at high XUV fluences 22 . Under these special circumstances, the measured intensities of both signal contributions are not well matched. The extracted map (Fig. 4b) reveals a complex focal spot with a bright central area and several side lobes of considerable intensity. Its brightest feature has a full width at half maximum (FWHM) of 3.9 μm and aggregates 58% of the shot energy. The recorded FEL single shot has a moderate pulse energy of 11 μJ. This corresponds to a peak fluence of 40 J cm −2 , which is well within the regime of previously reported non-linear XUV light-matter interactions 5,8,23 . Additionally, the beam's smallest features (1.9 µm FWHM) are clearly resolved by our grating monitor. The whole spatial distribution is in very good agreement with the independently obtained data for a different single shot via a Hartmann wave-front sensor measurement, shown in Fig. 4c. Both measurements are furthermore consistent with the atomic force microscopy image of a single-shot damage crater in the sample substrate, shown in Fig. 4d. The line scans in Fig. 4e further demonstrate that the relative intensities measured by our grating monitors agree well with the wave-front sensor and damage crater data. Deviations between the three independent measurements are due to uncertainties associated with each technique (e.g., melting and redeposition in the damage crater, choice of numerical propagation parameters for wave-front sensor image, wavelength, and distance-scaling of the diffraction image) and the fact that they originate from different FEL shots. Figure 5 displays a series of spatial fluence distributions, recorded at various positions along the beam propagation axis. In order to record the diffraction signals (Fig. 5a) inherent to the FEL source-we accumulate 2000 strongly attenuated shots per image. Thus, a perfect agreement with the wave-front sensor data (Fig. 5b) cannot be expected since the latter is extrapolated from a single-shot measurement. In particular, the accumulated images are blurred due to spatial jitter, that is, small, random changes of the beam pointing on a shot-to-shot basis. We are able to model the blurred, accumulated image at z = 0 (Fig. 5a, center) by applying a Gaussian filter to a destructive single-shot diffraction image (not shown), recorded at the same position. From this, we estimate the amount of spatial jitter in this particular case to be 3.5 μm (FWHM) in the horizontal, and 1.9 μm (FWHM) in the vertical direction. This is a significant fraction of the brightest feature's spatial extent of 3.9 μm. It is obvious, that-in addition to mapping the internal structure of the focus-the access to the spatial jitter of the beam on the sample on a single-shot basis is extremely valuable. This applies particularly to laterally inhomogeneous samples with spatially varying material composition, including particles sparsely dispersed on a membrane. Throughout the series, grating monitor and wave-front sensor data generally agree well. As discussed, the principal cause for the differences to the single-shot wave-front measurement is spatial jitter of the FEL beam. Note that the fluence distribution changes substantially within a few millimeters along the beam axis. This is due to the finite size of the optical elements that act as limiting apertures and introduce diffraction artifacts into the beam. We remark that our in situ fluence-mapping approach can easily be used to track and optimize the sample position with respect to a focus both in the transverse direction as well as along the beam axis, e.g., when aligning samples for an experiment. Discussion The attainable spatial resolution of our approach is closely linked to the manufacturing process. We use focused ion beam (FIB) milling to directly pattern the grating structures into the sample membrane (see Methods section for details on the manufacturing process). This fast and flexible method is capable of producing structure sizes down to a few tens of nanometers with low aspect ratios. As a rule of thumb, the necessary structure size (i.e., the grating half-pitch) to map a particular focal spot is about onetenth of the spot size. Note that Eq. (7) enforces this condition (Supplementary Note 2 and Supplementary Fig. 1). Given the milling resolution of the FIB process, our approach is directly capable of mapping sub-micrometer focal spots in the XUV and soft X-ray regime. Moving toward even shorter wavelengths and into the hard Xray regime, the absorption contrast for most materials-and thereby the grating efficiency-diminishes. This necessitates higher aspect ratios for the grating structures and thus a more challenging manufacturing process. Furthermore, the achievable focal spots of the FEL beam are considerably smaller, extending into the sub-100 nm region 2,15 . Consequentially, smaller grating structures are necessary to satisfy Eq. (7). Due to their close relationship to our grating monitors, it seems reasonable to consider recent progresses in hard X-ray zone-plate manufacturing for the discussion of these issues. Zone plates with 15 nm outer zone width have successfully been manufactured for hard X-ray radiation and are used in experiments with 9 keV photon energy 24 . With such manufacturing capabilities, it is feasible to directly transfer our concept to the hard X-ray regime and focal spot sizes on the order of 100 nm. Additionally, our concept does not require high diffraction efficiencies and aspect ratios can accordingly be smaller than for a zone plate. This makes it possible to utilize even smaller structures, and achieve sub-100 nm resolution. However, the complex manufacturing process might-at the current technological state-prohibit the time-and cost-efficient fabrication of a large number of samples for destructive studies. The actual limit will of course depend on the specific experiment, including photon energy, sample size and thickness, available detector space, and the experiment geometry. The low absorption of hard X-ray radiation in most of the conceivable grating materials is greatly beneficial when transparent beam monitors are considered. In such cases, our grating concept could provide permanent and reliable in situ feedback of the beam position and spatial structure at critical beam-line positions, such as split-and-delay units 25,26 or intermediate focus stages 15 . This is further encouraged by the fact that spatial constraints in the experimental setup apply to a much lesser degree since high-vacuum conditions are usually not required. We note that no computational treatment of the measured fluence distributions is necessary. This makes the concept suitable for live monitoring, even at very high repetition rates. Our fluence-mapping approach is a unique tool for true in situ, single-shot-capable monitoring of the fine structure of the sample illumination in transmission-type scattering experiments. It is, to our knowledge, the only approach that allows for a simultaneous, non-invasive mapping of the fluence distribution on the sample together with a scattering signal of interest. The approach provides an instantaneous online signal, which can be interpreted without any further computation, and can thus be used as instant feedback to align the upstream optical system. In the study of fluence-dependent phenomena, it provides crucial information for the correct interpretation of the data. The position and magnification of the photon-fluence map on the detector is, within the discussed constraints and the limits of the particular manufacturing process, freely selectable. Furthermore, the derivation of the grating formula can easily be adapted for diffraction experiments in reflection geometry. This makes our fluencemapping approach compatible with a large variety of experiments and samples. Given these features, we expect this approach to become a valuable tool for alignment and optimization, and in particular for the study of non-linear light-matter interaction in the XUV and X-ray regime. Methods Diffraction experiment. We perform small-angle scattering in transmission geometry at the FERMI@Elettra FEL source, using the DiProI end-station 27 . Focusing is provided via a bendable Kirkpatrik-Baetz optics 28 . A Princeton Instruments PI-MTE in-vacuum charge-coupled device (CCD) camera (2048 × 2048 pixel, 13.5 μm edge length) detects the scattered radiation 75-150 mm downstream of the sample. For the data in Fig. 4a, the sample-CCD distance is 75 mm. A cross-shaped beamstop in front of the camera blocks the intense radiation in the forward direction and potential membrane edge scattering. We subtract dark images to flatten the image background. In the single-shot image, we manually remove a linear background that is due to a read-out artifact in the CCD wells that are read after the highest intensity occurs and de-noise the image by applying a Gaussian filter with 2 px width. The incident X-ray pulses are tuned to a wavelength of 20.8 nm (59.6 eV) with single-shot pulse energies ranging from 0.5 to 60 μJ. For accumulating measurements at a repetition rate of 10 Hz, solid-state filters and a gas absorber reduce the pulse energy to a range between 20 and 80 nJ. The Hartmann WFSmanufactured by Imagine Optique-is equipped with a 72 × 72 pinhole grid (pinhole diameter 60 µm, pitch 180 µm) and has a nominal accuracy of λ/100. Sample fabrication. The samples consist of Si 3 N 4 membranes of 30 nm thickness with 30-200 μm edge length. For the purpose of other experiments, a magnetic Co/ Pt multi-layer is deposited on the membranes by DC magnetron sputtering. In the experiments reported here, this sample layer is in a labyrinth-like domain state with magnetization vectors parallel or anti-parallel to the FEL beam axis. At the selected photon wavelength, these domains give rise to a ring-shaped scattering signal on the CCD detector via the X-ray magnetic circular dichroism effect. We use a focused Ga + ion beam (FIB) to mill gratings directly into the Si 3 N 4 membrane at 30 kV acceleration voltage and 93 pA beam current. For the milling process, the gratings are generated on a grid of 3500 × 3500 points with x 0 = y 0 = 9.5 mm, λ = 20.8 nm, z = 150 mm, and m = 80. In this particular case, milling the 35 × 35 μm grating with a nominal topographic amplitude of 2 nm takes 60 s. Data availability. The data that support the findings of this study are available from the corresponding authors upon reasonable request. Received: 24 April 2017 Accepted: 11 December 2017 Fig. 1 Fig. 3 13Examples of spatial fluence distribution estimates. Simulated two-dimensional maps of the spatial fluence distribution in the sample plane (f(ξ, η), representing a an exact measurement, b a blurred, low-resolution measurement, and c a constant estimate. All maps sum to the same overall shot energy E shot . d Fluence histograms of the three maps in a-c. e-g Simulated scattering signal I(f) for increasing overall shot energy in the three fluence distributions under the assumption of, respectively, linear, quadratic, and saturating fluence dependency. Note the double logarithmic scale in f Schematic evolution of our grating design from regular gratings. The images show the real-space structures a-c and respective diffraction pattern df Fig. 4 Fig. 5 45without risk of destroying the sample with a particularly high-powered single shot-as might occur due to random intensity fluctuations Single-shot spatial fluence distribution and primary sample signal. a Logarithmic false-color image of a single-shot diffraction pattern with sample signal (ring-shaped feature) and spatial fluence distributions. b Crop of the spatial fluence map from a on a linear intensity scale. c Spatial fluence distribution in the sample plane on a linear scale, extrapolated from a Hartmann wave-front sensor (WFS) measurement of a different FEL single-shot. d Atomic force micrograph (AFM) of a single-shot damage crater in the sample substrate. All scale bars correspond to 10 μm. e Line profiles along the indicated lines in b-d. To extract an intensity profile from d, we assume that the absorption in the substrate material follows a Beer-Lambert law. Neglecting thermal melting and redeposition, the damage crater topography then represents the surface of constant intensity at which the incident fluence is attenuated below the ablation threshold13 Grating a WFS b z =−8 mm z = −6 mm z= −4 mm z = −Spatial fluence distributions along the beam propagation axis. a Multi-shot image of the grating's positive diffraction order. The fluence distributions appear blurred due to spatial jitter. b Spatial fluence distribution calculated from a single-shot wave-front sensor measurement. The position z of the sample plane is given relative to the nominal focus position. Note that all images are individually normalized. The peak intensity drops rapidly when the sample moves out of the focus position. Scale bars correspond to 10 μm NATURE COMMUNICATIONS \| DOI: 10.1038/s41467-017-02567-0 NATURE COMMUNICATIONS \| (2018) 9:214 \| DOI: 10.1038/s41467-017-02567-0 \| www.nature.com/naturecommunications © The Author(s) 2018 AcknowledgementsWe thank the FERMI accelerator and laser teams for their great support during the experiments.Additional informationSupplementary Information accompanies this paper at https://doi.org/10.1038/s41467-017-02567-0.Competing interests: The authors declare no competing financial interests.Reprints and permission information is available online at http://npg.nature.com/ reprintsandpermissions/ Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 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[ "Integrated-light analyses vs. colour-magnitude diagrams -II. Leo A, an extremely young dwarf in the Local Group", "Integrated-light analyses vs. colour-magnitude diagrams -II. Leo A, an extremely young dwarf in the Local Group" ]	[ "T Ruiz-Lara [email protected] \nInstituto de Astrofísica de Canarias\nCalle Vía Láctea s/nE-38205La Laguna, TenerifeSpain\n\nDepartamento de Astrofísica\nUniversidad de La Laguna, E-38200 La LagunaTenerifeSpain\n", "C Gallart \nInstituto de Astrofísica de Canarias\nCalle Vía Láctea s/nE-38205La Laguna, TenerifeSpain\n\nDepartamento de Astrofísica\nUniversidad de La Laguna, E-38200 La LagunaTenerifeSpain\n", "M Beasley \nInstituto de Astrofísica de Canarias\nCalle Vía Láctea s/nE-38205La Laguna, TenerifeSpain\n\nDepartamento de Astrofísica\nUniversidad de La Laguna, E-38200 La LagunaTenerifeSpain\n", "M Monelli \nInstituto de Astrofísica de Canarias\nCalle Vía Láctea s/nE-38205La Laguna, TenerifeSpain\n\nDepartamento de Astrofísica\nUniversidad de La Laguna, E-38200 La LagunaTenerifeSpain\n", "E J Bernard \nUniversité Côte d'Azur, OCA\nCNRS\nLagrangeFrance\n", "G Battaglia \nInstituto de Astrofísica de Canarias\nCalle Vía Láctea s/nE-38205La Laguna, TenerifeSpain\n\nDepartamento de Astrofísica\nUniversidad de La Laguna, E-38200 La LagunaTenerifeSpain\n", "P Sánchez-Blázquez \nDepartamento de Física Teórica\nUniversidad Autónoma de Madrid\nE-28049CantoblancoSpain\n", "E Florido \nDepartamento de Física Teórica y del Cosmos\nUniversidad de Granada\nCampus de Fuentenueva, E18071GranadaSpain\n\nInstituto Carlos I de Física Teórica y computacional, Universidad de Granada\nE-18071GranadaSpain\n", "I Pérez \nDepartamento de Física Teórica y del Cosmos\nUniversidad de Granada\nCampus de Fuentenueva, E18071GranadaSpain\n\nInstituto Carlos I de Física Teórica y computacional, Universidad de Granada\nE-18071GranadaSpain\n", "I Martín-Navarro \nUniversity of California Observatories\n1156 High Street95064Santa CruzCAUSA\n\nMax-Planck Institut für Astronomie\nKonigstuhl 17D-69117HeidelbergGermany\n" ]	[ "Instituto de Astrofísica de Canarias\nCalle Vía Láctea s/nE-38205La Laguna, TenerifeSpain", "Departamento de Astrofísica\nUniversidad de La Laguna, E-38200 La LagunaTenerifeSpain", "Instituto de Astrofísica de Canarias\nCalle Vía Láctea s/nE-38205La Laguna, TenerifeSpain", "Departamento de Astrofísica\nUniversidad de La Laguna, E-38200 La LagunaTenerifeSpain", "Instituto de Astrofísica de Canarias\nCalle Vía Láctea s/nE-38205La Laguna, TenerifeSpain", "Departamento de Astrofísica\nUniversidad de La Laguna, E-38200 La LagunaTenerifeSpain", "Instituto de Astrofísica de Canarias\nCalle Vía Láctea s/nE-38205La Laguna, TenerifeSpain", "Departamento de Astrofísica\nUniversidad de La Laguna, E-38200 La LagunaTenerifeSpain", "Université Côte d'Azur, OCA\nCNRS\nLagrangeFrance", "Instituto de Astrofísica de Canarias\nCalle Vía Láctea s/nE-38205La Laguna, TenerifeSpain", "Departamento de Astrofísica\nUniversidad de La Laguna, E-38200 La LagunaTenerifeSpain", "Departamento de Física Teórica\nUniversidad Autónoma de Madrid\nE-28049CantoblancoSpain", "Departamento de Física Teórica y del Cosmos\nUniversidad de Granada\nCampus de Fuentenueva, E18071GranadaSpain", "Instituto Carlos I de Física Teórica y computacional, Universidad de Granada\nE-18071GranadaSpain", "Departamento de Física Teórica y del Cosmos\nUniversidad de Granada\nCampus de Fuentenueva, E18071GranadaSpain", "Instituto Carlos I de Física Teórica y computacional, Universidad de Granada\nE-18071GranadaSpain", "University of California Observatories\n1156 High Street95064Santa CruzCAUSA", "Max-Planck Institut für Astronomie\nKonigstuhl 17D-69117HeidelbergGermany" ]	[]	Context. Most of our knowledge on the stellar component of galaxies is based on the analysis of distant systems and comes from integrated light data. It is important to test whether the results of the star formation histories (SFH) obtained with standard fullspectrum fitting methods are in agreement with those obtained through colour-magnitude diagram (CMD) fitting (usually considered the most reliable approach). Aims. We compare SFHs recovered from both techniques in Leo A, a Local Group dwarf galaxy whose majority of stars formed during the last 8 Gyrs. This complements our previous findings in a field in the Large Magellanic Cloud bar, where star formation has been on-going since early epochs though at varying rates. Methods. We have used GTC/OSIRIS in long-slit mode to obtain a high-quality integrated light spectrum by scanning a selected region within Leo A, for which a CMD reaching the old main mequence turn-off (oMSTO) is available from HST. We compared the SFH obtained from the two datasets, using state-of-art methods of integrated light (STECKMAP) and resolved stellar population analysis. In the case of the CMD, we computed the SFH both from a deep CMD (observed with HST/ACS), and from a shallower one (archival data from HST/WFPC2). Results. The agreement between the SFHs recovered from the oMSTO CMD and from full spectrum fitting is remarkable, particularly regarding the time evolution of the star formation rate. The overall extremely low metallicity of Leo A is recovered up to the last 2 Gyrs, when some discrepancies appear. A relatively high metallicity found for the youngest stars from the integrated data is a recurring feature that might indicate that the current models or synthesis codes should be revised, but that can be significantly mitigated using a more restrictive metallicity range. We thoroughly inspect the robustness of both approaches separately, finding that the subtle differences between them are inherent to the methods themselves. The SFH recovered from the shallow CMD also presents differences with the other two. Conclusions. Modern full-spectral fitting codes are able to recover both average constant SFHs (LMC case) and SFHs with a dominant fraction of young stellar populations. The analysis of high S/N spectra seems to provide more reliable SFH estimates than that of CMDs not reaching the oMSTO. The comparison presented in this paper needs to be repeated for predominantly old systems, thus assessing the performance of full-spectrum fitting for a full range of SFHs.	10.1051/0004-6361/201732398	[ "https://arxiv.org/pdf/1805.04323v2.pdf" ]	56,444,925	1805.04323	83ea06a85b1f97945a9124b74b8a2b0216cd9f00	Integrated-light analyses vs. colour-magnitude diagrams -II. Leo A, an extremely young dwarf in the Local Group July 25, 2018 T Ruiz-Lara [email protected] Instituto de Astrofísica de Canarias Calle Vía Láctea s/nE-38205La Laguna, TenerifeSpain Departamento de Astrofísica Universidad de La Laguna, E-38200 La LagunaTenerifeSpain C Gallart Instituto de Astrofísica de Canarias Calle Vía Láctea s/nE-38205La Laguna, TenerifeSpain Departamento de Astrofísica Universidad de La Laguna, E-38200 La LagunaTenerifeSpain M Beasley Instituto de Astrofísica de Canarias Calle Vía Láctea s/nE-38205La Laguna, TenerifeSpain Departamento de Astrofísica Universidad de La Laguna, E-38200 La LagunaTenerifeSpain M Monelli Instituto de Astrofísica de Canarias Calle Vía Láctea s/nE-38205La Laguna, TenerifeSpain Departamento de Astrofísica Universidad de La Laguna, E-38200 La LagunaTenerifeSpain E J Bernard Université Côte d'Azur, OCA CNRS LagrangeFrance G Battaglia Instituto de Astrofísica de Canarias Calle Vía Láctea s/nE-38205La Laguna, TenerifeSpain Departamento de Astrofísica Universidad de La Laguna, E-38200 La LagunaTenerifeSpain P Sánchez-Blázquez Departamento de Física Teórica Universidad Autónoma de Madrid E-28049CantoblancoSpain E Florido Departamento de Física Teórica y del Cosmos Universidad de Granada Campus de Fuentenueva, E18071GranadaSpain Instituto Carlos I de Física Teórica y computacional, Universidad de Granada E-18071GranadaSpain I Pérez Departamento de Física Teórica y del Cosmos Universidad de Granada Campus de Fuentenueva, E18071GranadaSpain Instituto Carlos I de Física Teórica y computacional, Universidad de Granada E-18071GranadaSpain I Martín-Navarro University of California Observatories 1156 High Street95064Santa CruzCAUSA Max-Planck Institut für Astronomie Konigstuhl 17D-69117HeidelbergGermany Integrated-light analyses vs. colour-magnitude diagrams -II. Leo A, an extremely young dwarf in the Local Group July 25, 2018Received -; accepted -Astronomy & Astrophysics manuscript no. LeoA_accepted c ESO 2018galaxies: stellar content -galaxies: dwarf -galaxies: Local Group -techniques: spectroscopy Context. Most of our knowledge on the stellar component of galaxies is based on the analysis of distant systems and comes from integrated light data. It is important to test whether the results of the star formation histories (SFH) obtained with standard fullspectrum fitting methods are in agreement with those obtained through colour-magnitude diagram (CMD) fitting (usually considered the most reliable approach). Aims. We compare SFHs recovered from both techniques in Leo A, a Local Group dwarf galaxy whose majority of stars formed during the last 8 Gyrs. This complements our previous findings in a field in the Large Magellanic Cloud bar, where star formation has been on-going since early epochs though at varying rates. Methods. We have used GTC/OSIRIS in long-slit mode to obtain a high-quality integrated light spectrum by scanning a selected region within Leo A, for which a CMD reaching the old main mequence turn-off (oMSTO) is available from HST. We compared the SFH obtained from the two datasets, using state-of-art methods of integrated light (STECKMAP) and resolved stellar population analysis. In the case of the CMD, we computed the SFH both from a deep CMD (observed with HST/ACS), and from a shallower one (archival data from HST/WFPC2). Results. The agreement between the SFHs recovered from the oMSTO CMD and from full spectrum fitting is remarkable, particularly regarding the time evolution of the star formation rate. The overall extremely low metallicity of Leo A is recovered up to the last 2 Gyrs, when some discrepancies appear. A relatively high metallicity found for the youngest stars from the integrated data is a recurring feature that might indicate that the current models or synthesis codes should be revised, but that can be significantly mitigated using a more restrictive metallicity range. We thoroughly inspect the robustness of both approaches separately, finding that the subtle differences between them are inherent to the methods themselves. The SFH recovered from the shallow CMD also presents differences with the other two. Conclusions. Modern full-spectral fitting codes are able to recover both average constant SFHs (LMC case) and SFHs with a dominant fraction of young stellar populations. The analysis of high S/N spectra seems to provide more reliable SFH estimates than that of CMDs not reaching the oMSTO. The comparison presented in this paper needs to be repeated for predominantly old systems, thus assessing the performance of full-spectrum fitting for a full range of SFHs. Introduction Stars are one of the main constituents of the baryonic component of galaxies and, as a consequence, the characterisation of how the rate of star formation and the stellar chemical composition vary as a function of time is key to understanding their evolution. The determination of these so called Star Formation Histories (SFH) of galaxies (e.g. Searle et al. 1973;Gallagher et al. 1984;Hodge 1989;Madau et al. 1996;Kauffmann et al. 2003;Tolstoy et al. 2009;Gallart et al. 2015) can help us to constrain their cosmological assembly history and trace back past events in their evolution. However, the derivation of SFHs based on the distribution of stars in a deep Colour-Magnitude Dia-gram (CMD) reaching the oldest main sequence turn-off (oM-STO, Bertelli et al. 1992;Gallart et al. 1999;Hernandez et al. 1999;Holtzman et al. 1999;Olsen 1999;Dolphin 2002;Dolphin et al. 2002;Cole et al. 2007;Aparicio & Hidalgo 2009;Cignoni & Tosi 2010), often considered the more reliable approach, is only applicable to the few dozens of nearby systems found in and around the Local Group (within distances up to ∼ 1 Mpc, McConnachie 2012). Thus, the study of the stellar content of most galaxies relies on the information that we can obtain from integrated light (MacArthur et al. 2004;Cid Fernandes et al. 2007;Ocvirk 2010;Sánchez-Blázquez et al. 2011;Pérez et al. 2013 Gallazzi et al. 2014;Sánchez-Blázquez et al. 2014;Beasley et al. 2015;González Delgado et al. 2015;Ruiz-Lara et al. 2016;Zibetti et al. 2017). In the last few decades, there has been an enormous effort to improve the recovery of reliable SFHs from integrated spectroscopic data. Advances in the modeling of stellar populations (e.g. Bruzual & Charlot 2003;Lee & Worthey 2005;Schiavon 2007;Conroy et al. 2009;Vazdekis et al. 2010Vazdekis et al. , 2016 based on the improvement of stellar libraries (e.g. Prugniel & Soubiran 2001;Le Borgne et al. 2003;Valdes et al. 2004;Sánchez-Blázquez et al. 2006;Prugniel et al. 2007), isochrones and evolutionary tracks (e.g. Girardi et al. 2000;Pietrinferni et al. 2004;Bressan et al. 2012;Pietrinferni et al. 2013); studies to properly characterise the shape of the stellar Initial Mass Function (IMF; Salpeter 1955; Vazdekis et al. 1996;Kroupa 2001;Weidner et al. 2013b,a;Peacock et al. 2014) and the effect that these shapes have on the observed stellar populations (Martín-Navarro et al. 2015); and the development of new inversion codes to recover the star formation history from the integrated spectra of galaxies (Cid Fernandes et al. 2005;Ocvirk et al. 2006b,a;Koleva et al. 2009;Sánchez et al. 2016b,a) have substantially increased our capability to obtain the SFHs of external galaxies. However, astronomers are still assessing to what extent we can rely on the outcome of the analysis of integrated spectroscopic data. A good number of works have tested the consistency between the SFHs, average ages and metallicities derived using integrated information and those using other methods such as the analysis of CMDs. However, most of these studies were focused on single stellar populations such as stellar clusters (e.g. Gibson et al. 1999;Beasley et al. 2002;de Grijs & Anders 2006;Santos et al. 2006;Wolf et al. 2007;Mendel et al. 2007;González Delgado & Cid Fernandes 2010;Ocvirk 2010;Barber et al. 2014;Kuncarayakti et al. 2016), and just a few analysed more complex systems such as dwarf galaxies (Makarova et al. 2010;García-Benito & Pérez-Montero 2012) with available (albeit shallow) CMDs. Nevertheless, to fully assess the reliability of the information recovered from integrated light data, it is crucial to check this consistency analysing the most complex systems where this comparison can be properly done, and for which CMDs reaching the oMSTO can be obtained, covering a wide range of SFHs and metal enrichments to test the limitations of the methods. In Ruiz-Lara et al. (2015, hereafter Paper I), we started a project aimed at testing the performance of some of the most used full-spectrum fitting codes at recovering complex SFHs. As a case example, we derived the SFH of a region within the bar of the Large Magellanic Cloud (LMC) for which a CMD reaching the oMSTO and high quality spectroscopic data was available (see also Alloin et al. 2002;Lilly & Fritze-v. Alvensleben 2005, for preliminary studies using the same dataset). Previous studies have found different star forming episodes since the formation of the LMC bar and until the present time, with some periods of low star formation at intermediate ages (Holtzman et al. 1999;Olsen 1999;Smecker-Hane et al. 2002;Weisz et al. 2013;Monteagudo et al. 2018), accompanied by a continuous chemical enrichment especially concentrated at early times and during the last few gigayears (Carrera et al. 2008;Harris & Zaritsky 2009;Weisz et al. 2013;Monteagudo et al. 2018). We compared the SFHs recovered from the CMD using the IAC-star/MinnIAC/IAC-pop set of routines (Aparicio & Gallart 2004;Aparicio & Hidalgo 2009;Hidalgo et al. 2011;Monelli et al. 2010) with those recovered from the spectrum applying three modern full-spectrum fitting codes, namely STECKMAP (Ocvirk et al. 2006a,b), STARLIGHT (Cid Fernandes et al. 2005), and ULySS (Koleva et al. 2009). STECKMAP gave the best agreement with the CMD results. The only way of obtaining comparable results using STARLIGHT or ULySS was by using complex stellar populations (considering continuous star formation over an extended period of time), instead of simple ones, as input stellar models. This last approach has subsequently been adopted in some works (e.g. González Delgado et al. 2017). Although the agreement between SFHs derived from the CMD and from the spectrum was reassuring, the SFH of the LMC bar is just one case among the wide variety of behaviors exhibited by galactic systems. The natural continuation of such work is the analysis of other stellar systems sampling a wide range of evolutionary histories. One of the most striking dwarf galaxies in the Local Group is Leo A, a dwarf irregular for which recent works suggest that could have been almost purely gaseous during the epoch of giant galaxy assembly (z ∼ 2, Cole et al. 2007). Leo A is an isolated galaxy with a well-reported population of young and massive stars (Demers et al. 1984;Tolstoy 1996). Using an HST CMD reaching the oMSTO, Cole et al. (2007) (see also Gallart et al. 2015) found that around 80% of the star formation in Leo A occurred within the last 8 Gyr of evolution, with a peak on the Star Formation Rate (SFR) between 1 to 3 Gyr ago and an almost constant low metallicity (Z 1 ∼ 0.0008 +0.0005 −0.0003 ). The discovery of RR Lyrae variables in Leo A support the presence of a modest amount of old stars (older than 10 Gyr) coexisting with the dominant young component . The characteristics of the Leo A SFH, in comparison with the more extended SFH of the LMC bar (analysed in Paper I), make of this galaxy a key example to continue testing the performance of full-spectrum fitting techniques in systems with different stellar compositions. The results of this comparison will allow us to check, in particular, the ability of the integrated techniques to single out different fractions of old population in systems with abundant young and intermediate-age population. In this work we obtain and compare the SFHs computed using deep CMDs and GTC/OSIRIS long slit spectroscopic data of a region within Leo A as an example of a predominantly young galactic system. Section 2 describes the observations and the data reduction of the spectroscopic and photometric data used in this study. The determination of the SFH from those datasets is presented in Sect. 3. The discussion and main conclusions are outlined in Sect. 4 and 5. Observations and data reduction In this project we compare the SFH of Leo A derived with two of the main approaches to study the stellar content of galaxies: fullspectrum fitting techniques applied to high-quality spectra and the modelling of CMDs reaching the oMSTO. Leo A is one of the few close dwarf galaxies for which old main sequence stars can be resolved via HST photometry allowing for the observation of a deep CMD. At the same time, despite Leo A's low surface brightness, the light collecting power of the new generation of giant telescopes makes possible to obtain a high-quality integrated spectrum. To derive the SFH following both approaches we carefully selected an extended region in this galaxy for which 1 Throughout the paper we use Z to denote the metallicity of the stellar component of Leo A (unless expressed otherwise). The following equations can be used to transform from Z to [M/H] HST/ACS data are available, avoiding as much background and foreground objects as possible. We scanned the selected region using GTC/OSIRIS in its long slit configuration (see Fig. 1) to obtain the integrated spectrum. In this section we give all the details concerning the analysed data. Integrated light spectrum The region within Leo A selected to obtain the integrated spectrum was observed using the OSIRIS imager and spectrograph 2 mounted at the Gran Telescopio Canarias (GTC) in the Observatorio del Roque de los Muchachos, La Palma. The combination of the OSIRIS instrument in its long slit configuration and the light collecting capability of a 10m-class telescope such as GTC allows for the acquisition of high-quality spectra for objects of low surface brightness such as Leo A (µ V ∼ 24.8 mag/arcsec 2 , averaged over one effective radius, McConnachie 2012). The observations were carried out as part of the GTC94-15B_000 program on December 2015, January 2016, and February 2016 using the R1000B grism and a slit width of 1.2" that allows for a nominal wavelength coverage from 3630 to 7500 Å and spectral resolution of ∼ 11 Å (FWHM). This particular configuration was chosen as a compromise between the wavelength range covering the blue part, the amount of light gathered from the faint target (slit width), and the spectral resolution. However, due to peculiarities in this particular program the definitive useful wavelength range was restricted from 4000 to 5000 Å (see below). The final scanned region was determined by 16 different slit positions located side by side on the sky, aligned with the semimajor axis of Leo A (position angle of -76.1 o ) covering a total area of 19.2"× ∼ 7'. Two separate exposures of 1800 seconds each were taken at each slit position (16 hours on target). The size of the Leo A scanned field has been determined taking into account some crucial considerations: i) the total area has to contain enough stars to reliably determine the SFH from the CMD and to avoid under-sampling of some minority stellar populations in the field; ii) the integration area contains enough light as to obtain a high-quality integrated spectrum by summing all spectra within Leo A (S/N/pixel ∼ 60 in this case), and iii) to minimise foreground and background objects not belonging to Leo A. An inaccurate sampling of the stellar content (such as including few young stars dominating the final spectrum but with little mass contribution) may bias the reconstructed SFH. We analysed this problem in Paper I, and concluded that sampling is not a problem when integrating the light from fields that contain enough stars to reliably determine the SFH from CMD data. Hidalgo et al. (2011) concluded that 15000 stars down to the oMSTO in the CMD of a stellar system should be sufficient to accurately determine its SFH. For a system with a considerable young population as Leo A, it is essential to have such a large number of stars in the CMD in order to have a reliable determination of the young SFH. With the 16 observed slit positions, we could analyse an area in common with the HST/ACS data of 19.2"×202" (the second number being the width of the ACS field of view) which contains 16253 stars in the CMD (see Sect. 2.2). Classical reduction procedures such as bias subtraction, flatfielding, wavelength calibration, cosmic rays removal (L.A. Cosmic, van Dokkum 2001) and sky subtraction were performed separately in each of the two CCDs composing the OSIRIS field of view using an IDL/python-based reduction pipeline de-signed to reduce OSIRIS long slit spectroscopic data. In particular, the sky subtraction is an essential step when dealing with low-surface brightness targets and deserves special consideration. The typical sky brightness in La Palma is µ V ∼ 21.9 mag/arcsec 2 (Benn & Ellison 1998), i.e. three magnitudes brighter than Leo A (µ V ∼ 24.8 mag/arcsec 2 ). In addition, the relatively low spectral resolution of our data further complicates the proper recovery of the shape of the sky lines. As a consequence, we have decided to apply a method that optimizes the usage of the data, the Kelson's sky subtraction algorithm (Kelson 2003). This technique relies on the knowledge of the CCD distortions and the curvature of the spectral features to obtain a characteristic sky spectrum from carefully selected pixels with sky information (avoiding bright objects as well as Leo A contamination). We noted that some pixels close to the edge of the CCDs were affected by some illumination issue that was also visible in the exposures of the spectrophotometric standards. We corrected this issue by fitting a smooth, low-order polynomial as a function of pixel position in order to raise the affected sky pixels to the "real" sky level. After that, a light profile along the slit clearly showed the extension of Leo A, its light distribution, and two nearly flat regions at both sides of the object. We applied the Kelson's algorithm to a subset of the pixels in those sky regions (one for each part of the galaxy) to properly subtract the sky on our target. We performed several sky subtraction tests modifying the input parameters for the sky spectrum computation as well as changing the sky regions to test possible effects of this choice. We also tried different sky subtraction approaches. While most of the features in the blue part of the spectrum remain largely unaltered, the sky subtraction towards the longest wavelengths (5500 Å redwards) was very unstable. In addition, some clear residuals were left in the red end of the spectrum as a consequence of the large amount of sky spectral features and sky flux. For this reason, and in order to avoid the effect of an inaccurate sky subtraction, we have decided to restrict our analysis to wavelengths bluer than 5500 Å. Figure 2 shows a comparison of the scanned region as seen with the HST (top two rows) and a reconstruction from the 16 slit positions observed with OSIRIS (third row). Despite the difference in spatial resolution some patterns can be identified in both images. The final integrated spectrum is obtained as a mean of all the pixels within the ACS field of view avoiding saturated stars, background galaxies, and bright stars that might belong to our Milky Way rather than to Leo A (see Sect. 2.2). We will name this spectral extraction, extraction A. For this first extraction, the Milky Way candidates are selected based on their position on the observed CMD as bright stars suspected of not belonging to Leo A (see Sect. 3.2). The low spectral resolution of the OSIRIS data hampers a proper characterisation of the membership of those bright stars using their radial velocities. As a consequence, we decided to perform a second extraction including those Milky Way candidates to further investigate the effect of these bright stars in the final SFH reconstruction. The effect of including or not these few stars is negligible in the CMD analysis, but their light could potentially affect the observed integrated spectrum (see fifth row, extraction B). We must bear in mind that this Milky Way contamination is an issue naturally found in this test due to the large area covered, but negligible in the analysis of external galaxies, typically subtending a much smaller area in the sky, where the tested codes are meant to be used. Analysed region and masking procedure. Top two rows: Highspatial resolution HST image of the scanned region within the ACS (area covered is 19.2"×202"). The next three rows show pseudoimages reconstructed from the 16 slit positions observed with OSIRIS in Leo A. Third row: The entire extension of the observed field ( 19.2"×7'). Fourth row: The region from which the integrated spectrum is obtained; in this case we mask background galaxies, saturated stars, and Milky Way stars candidates (extraction A). Fifth row: The region from which the integrated spectrum is obtained; in this case we mask just foreground galaxies and saturated stars (extraction B). The final extracted spectrum is shown in Fig. 3. A visual inspection shows another issue related with the low surface brightness of the scanned region. At ∼ 4760 Å and 5080 Å the integrated spectrum (regardless of the extraction procedure) displays two "bumps" that, in principle, do not seem to be linked to any physical feature (of stellar, gaseous, or molecular origin). A careful inspection of the sky-subtracted frames (for all slit positions) suggests that every pixel belonging to extremely to lowest surface brightness regions of the galaxy display light enhancements around such wavelengths for all the slit positions. In addition, we found this issue even in the spectrophotometric standard frames, in those pixels exposed to extremely low amounts of light (especially in the so called first CCD, CCD1). However, there is no hint of such light increases in regions where stars or background galaxies are located, i.e. in those pixels with high signal. Since these "bumps" are not physically connected with the Leo A content, we have decided to avoid those wavelengths affected by these unreal features for the SFH recovery. In addition, the extremely low surface brightness of the scanned region along with the low sensitivity of the OSIRIS instrument using the R1000B grism below 4000 Å further restrict the spectral range useful for our purposes. Despite all the described difficulties, the observed spectrum is of exceptional quality from ∼ 4000 Å to ∼ 5000 Å, with the exception of a region around the "bump" prior to Hβ. Although a better spectral sampling would be ideal for the SFH recovery from spectroscopic data, this final useful range contains some of the most important Hydrogen Balmer lines (Hβ, Hγ, and Hδ, prominent absorption lines in the figure), as well as some iron and calcium absorption features. Taking into account the goal of this work, the limited wavelength range under analysis can be interpreted as an extra challenge for testing modern full-spectrum fitting techniques rather than a caveat of this analysis. The final S/N of the integrated spectrum is around 60 (per pixel, ∼ 100 per Å) with a measured resolution of around 10.8 Å (FWHM). A visual inspection of the observed spectrum suggest the presence of a young stellar population in Leo A (deep Balmer lines) Fig. 3. Example of a typical STECKMAP fit to the Leo A integrated spectrum. We represent the observed spectrum with a blue solid line, the best STECKMAP fit with a red solid line, the limits of the fit with vertical red-dashed lines, and the masked region with shade. with some emission that can be clearly seen filling in the three Balmer lines covered by our data. Resolved stellar populations The CMD analysed in this work was obtained from Leo A HST/ACS observations collected from 26 th December 2005 to 8 th January 2006. These data were used in Cole et al. (2007) to obtain for the first time the Leo A SFH from a deep CMD reaching the oMSTO, and later on to search for variable stars (Bernard et al. 2013). Sixteen HST orbits were devoted to obtain precise photometry in two different bands (F475W and F814W). In each of the 16 orbits two exposures of ∼1200s were taken per band, accounting for a total of 19200 and 19520 seconds of integration time in the F475W and F814W bands, respectively. The photometry of the individual stars was obtained with the DAOPHOT/ALLFRAME set of routines (Stetson 1994) to the nondrizzled HST/ACS images. The completeness and photometry errors were characterized through artificial star tests. A total of ∼ 8×10 5 stars were used (see Monelli et al. 2010, for details on the procedure used to obtain the photometry and the artificial stars test). Although these observations comprised photometric information for some 9.5×10 4 stars in total, in this work we are mainly interested in the area scanned with GTC/OSIRIS. The (M F814W , M F475W -M F814W ) CMD of the scanned region within Leo A (see Fig. 1) is presented in Fig. 4. In total, it comprises 16253 stars with precise and accurate photometric magnitude measurements. We transform from apparent to absolute magnitudes each observed star assuming a distance modulus of 24.48 mag (determined in Bernard et al. 2013, using RRLyrae stars) and a galactic extinction of A F475W = 0.068 mag and A F814W = 0.032 mag (Schlafly & Finkbeiner 2011). The photometry reaches down to 3.9 and 3.5 absolute magnitude (F475W and F814W, respectively), which corresponds to an apparent magnitude of 28.6 and 27.9, respectively with a 50% of completeness. A quick visual inspection of Fig. 4 already shows the presence of a conspicuous young stellar population. The main aspects that stand out are the prominent and bright main se- quence (MS) up to M F814W ∼ −2, the vertically extended red clump (M F814W ∼ −1.0), and the well defined and narrow red giant branch (RGB), suggesting a low spread in metallicity. The oMSTO is located around M F814W ∼ 2.5, well above our 50% completeness level. In addition, there are some stars that, considering their position in the CMD, are suspected of not belonging to Leo A but to the MW (red points). Extractions A and B of the integrated spectrum include and exclude these suspected foreground stars (see Sect. 2.1). Determination of the Leo A Star Formation History In this paper we expand the comparison between the SFH derived from resolved and unresolved stellar populations presented in Paper I (focused on a region in the LMC bar) by analysing data from Leo A, a complex system dominated by a young and intermediate-age stellar component (Cole et al. 2007). As in Paper I we compare the results of the CDM-fitting analysis to the results obtained from integrated spectra. In that paper we analysed different inversion codes and identified some of the main advantages and disadvantages of each one. In this paper, we focus in the performance of a single code, STECKMAP. Integrated spectrum analysis The strategy to obtain the SFH from integrated spectra relies on the comparison between observed data and a set of modelled stellar spectra with some given prescriptions (IMF, SFH, etc.). The main differences among the variety of inversion codes are the way of treating the original data (e.g. polynomial fitting or not) and the way they compare observations with models (min-imisation algorithms). In this work, the derivation of the SFH from the Leo A integrated spectrum is based on the combination of three well-known analysis codes: pPXF (penalised pixel fitting code, Cappellari & Emsellem 2004;Cappellari et al. 2011); GANDALF (Gas AND Absorption Line Fitting, Sarzi et al. 2006;Falcón-Barroso et al. 2006); and, STECKMAP (STEllar Content and Kinematics via Maximum A Posteriori likelihood, Ocvirk et al. 2006a,b). Although STECKMAP is the code that we use to ultimately recover the SFH for a given input spectrum, the other two codes play an essential role in this methodology. First, pPXF recovers the best combination of the MILES 3 stellar population models (Vazdekis et al. 2016) convolved with a given line-ofsight velocity distribution (LOSVD) to determine the optimal stellar velocity and velocity dispersion while masking areas affected by gaseous emission. Second, we make use of GANDALF to include in the fit, apart from the already mentioned stellar populations models, additional Gaussians to deal with the emission lines and to take into account their possible contamination filling absorption features. Once the gaseous emission is modelled, we subtract from the observed spectrum those emission lines detected with a S/N above 3. The outcome of these two steps are the stellar kinematics and a pure absorption spectrum from our original data. It is in the third step when we are able to properly recover the stellar content shaping the observed spectrum by applying STECKMAP to the emission-cleaned spectroscopic data. This code, based on a Bayesian minimization method, is able to recover the combination of stellar model templates that best resemble our observed spectrum (see Fig. 3), and thus, its SFH (SFR as a function of time and Age-Metallicity Relation, AMR). Although STECKMAP does not rely on any a priori shape of the solution, it prefers smooth solutions over discontinuous ones (regularization). The smoothness of the final solution is determined by the user via the smoothing parameters: µ x , µ Z and µ v for the SFR, AMR and LOSVD functions, respectively. In this final step we fix the stellar kinematics to those values computed using pPXF to avoid the well-known degeneracy that arises when fitting simultaneously the velocity dispersion and the stellar metallicity . Errors in the SFH from the STECKMAP analysis are computed via 25 Monte Carlo (MC) simulations. This methodology has been extensively used to recover the stellar content in external galaxies Seidel et al. 2015;Pérez et al. 2017). For a thorough description of the method, the codes, and the different input parameters we refer the reader to Paper I and Ruiz-Lara et al. (2017), along with the papers where each code is presented and described. The final shape of the SFH recovered by STECKMAP might depend on the input parameters, i.e. the wavelength range under analysis, the model stellar templates, and the smoothing parameters. In this work, the wavelength range is limited by our spectroscopic data. Regarding the stellar templates, in Paper I we already demonstrated that the choice of models has a second order effect on the SFH recovery, affecting mainly the AMR. However, the selection of a reasonable set of smoothing parameters that improves the fit is a key test to be done before studying a new set of data with STECKMAP. Due to the peculiarities of this particular experiment (scan of a wide region of a Local Group galaxy), another aspect to take into account is the possible foreground contamination from our own Galaxy. It might affect the shape of the observed spectrum and thus, that of the recovered SFH (extractions A and B, see Sect. 2.1). For the sake of clarity, we will focus on the solution from extraction A (which we consider the safest and more reasonable option), using the new set of MILES models 4 with the BaSTI (Pietrinferni et al. 2004) isochrones (ages up to 13.5 Gyr, Vazdekis et al. 2016) to directly compare to the CMD analysis (see Sect. 3.2), and the parameters that maximise the quality of the fit (µ x = 0.1 and µ Z = 100). In Appendix A we show that the solutions are very robust and mostly insensitive to changes on these parameters. Figure 5 shows the normalised SFR (top panel), the AMR (middle panel) and the cumulative mass fraction (bottom panel) from STECKMAP (red lines and shaded areas). The normalised SFR and the cumulative mass fractions clearly show the youth of Leo A, with only around 16% of its mass formed more than 7 Gyr ago. Stars younger than 7 Gyr old become gradually more abundant until the SFR reaches a peak at around 2.3 Gyr. After a sharp decrement in the SFR with a minimum at ∼ 1 Gyr ago, the SFR increases again to reach another maximum a few hundreds of million years ago, when the SFR drops again. This SFR is basically in agreement with other works claiming a nonnegligible presence of old stellar populations but with the bulk of the Leo A population being younger than 8 Gyr Cole et al. 2007;Bernard et al. 2013;Gallart et al. 2015). The AMR has a constant value (Z = 0.0007) from 10 Gyr to 1 Gyr ago. For older populations, the recovered metallicities is around Z = 0.0009 (with large errors, consistent with the 1-10 Gyr range), while a higher metallicity (up to Z = 0.015) is inferred for recently formed stars. Several studies have also found such low metallicity in the stellar component of Leo A; e.g. Z ∼ 0.00053 from Kirby et al. (2013). Kirby et al. (2017) Using this low value of the oxygen abundance as a proxy of the metallicity of the very young stars, we can conclude that the high stellar metallicity we are finding for the youngest stars using STECKMAP seem to be in contradiction with the current gas oxygen abundance in Leo A. CMD analysis In this section, we derive the SFH of the scanned region of Leo A through the fitting of a deep CMD obtained from the ACS data (see Sect. 2.2). We use an updated python code (Bernard et al. 2015a) that closely resembles the procedures adopted in Paper I with the IAC-star/MinnIAC/IAC-pop set of routines. This 4 We do not apply any pre-selection on what sub-set of the MILES models to use in the fitting procedure as it is customary in most of the SFH derivations from spectra of integrated light, we use them all (except the oldest one, 14.0 Gyr). As a consequence, the age and metallicity limits as well as their sampling are determined by the models themselves (see Vazdekis et al. 2016 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.60, 0.70, 0.80, 0.90, 1.0, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5, 2.75, 3.0, 3.25, 3.5, 3.75, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 9.5, 10.0000, 10.5, 11.0, 11.5, 12.0, 12.5, 13.0, 13.5] × [0.0001, 0.0003, 0.0006, 0.001, 0.002, 0.004, 0.0084, 0.010, 0.022, 0.027, 0.034, 0.047] * STECKMAP resamples these age bins when providing the solution by dividing logarithmically the age range. method relies on the quantitative comparison, via minimization of the Poissonian equivalent to the χ 2 statistics (e.g. Dolphin et al. 2002), of the distribution of stars on an observed CMD with that of a combination of synthetic CMDs corresponding to simple stellar populations where observational effects have been simulated. The optimal solution is the linear combination of these simple populations best reproducing the observed CMD. We need to provide the code with a number of age and metallicity bins defining these theoretical simple populations as well as a series of regions within the CMD ("bundles") that are used in the fitting procedure to maximise the information from the observed CMD. Each "bundle" is divided in boxes of different size that will determine the relative importance of each "bundle" in the fit. Small boxes are preferred in regions where stars in well-known evolutionary phases abound as the larger the number of boxes in a given "bundle", the larger the importance of it in the fit. The SFH derivation is performed in the same way as in Bernard et al. (2015a) by counting the number of stars in each box in the observed and in the synthetic diagrams. In this particular case, we extracted the simple stellar populations from a "mother", synthetic CMD (Aparicio & Hidalgo 2009) composed of 5×10 7 stars computed using the BaSTI stellar evolution library 5 (Pietrinferni et al. 2004) following a constant SFR at all ages (0.03-13.5 Gyr) and a flat metallicity distribution from Z = 0.0001 to Z = 0.01. We note here that both analyses (integrated light spectrum and CMD) use the same set of isochrones. We assume a Kroupa IMF (Kroupa 2001), a binary fraction of 40% (β = 0.4) and a minimum mass ratio q = 0.5. We adopted this wide metallicity range rather than a more restrictive one (including just metallicities reported in the literature, e.g. Cole et al. 2007;Kirby et al. 2017) to minimise the number of assumptions that might affect the final solution. In order to take into account observational errors, this synthetic CMD was dispersed according to the artificial star tests (see Sect. 2.2) and taking into account the spatial distribution of Leo A stars. We define the different simple stellar populations according to the following age and metallicity bins, defined with similar criteria as in other SFH works using resolved CMDs ( The adopted "bundle" strategy for this analysis (see Fig. 4) uses the smallest boxes in the MS and sub-giant region ("bundle" 1), red clump (2), and bright MS (3). The RGB (4) and blue super-giant region (5) has the largest boxes along with "bundles" 6 and 7. Modifications in the input parameters for the fit, especially the "bundle" strategy or the way to deal with binaries, might affect the recovered SFH. In Appendix A we assess how different input configurations modify the solution. Given the similarities among the SFHs derived from the different tests reported in the Appendix, we stick to the configuration that makes use of the largest number of stars from the observed CMD while still modulating the importance of different regions of the CMD in the fit by the use of the "bundle" strategy. Due to possible uncertainties in the determination of the reddening and the distance modulus as well as uncertainties in the photometric or model calibration; different solutions are computed by applying small shifts in colour and magnitude to the observed CMD. In total, a solution is calculated in each point of a grid of 25 positions within ±0.06 mag and ±0.15 in colour and magnitude, respectively, with a further refining of the grid around the position where the minimum χ 2 is obtained. In this step each solution is computed once. After the minimum position is determined, we run 20 different solutions by i) slightly shifting the boxes and, ii) modifying the age and metallicity bins for the simple populations. The final solution is the average of these 20 solutions at the position where the minimum χ 2 was found ([∆(colour) min , ∆(Mag) min ]). Uncertainties on the SFRs were computed as described in Hidalgo et al. (2011). They are assumed to be a combination in quadrature of the uncertainties due to the effect of binning in the colour-magnitude and age-metallicity planes (from the dispersion of 20 solutions obtained after shifting the bin limits), and those due to the effect of statistical sampling in the observed CMD (the dispersion of 5 We used an adapted version of the CMD code available at the BaSTI webpage (http://basti.oa-teramo.inaf.it/index. html), made available to us by S. Cassisi 6 These age and metallicity bins do not match completely the values used in the integrated spectrum analysis on purpose. We have preferred to stick to customary procedures dealing with spectrum fitting and CMD fitting techniques seeking for a proper comparison between methods. A&A proofs: manuscript no. LeoA_accepted 20 solutions obtained after resampling the observed CMD following Poissonian statistics). For a more detailed explanation of the methodology we refer the reader to Monelli et al. (2010), Bernard et al. (2015a) and references therein. The application of the above-described method to the CMD of the scanned region within the HST/ACS data for Leo A gave a best solution (minimum χ 2 ) that was found in [∆(colour) min , ∆(Mag) min ] = [0.07,-0.05] with a χ 2 of 1.3. The recovered SFH is shown in Fig. 5 (blue colours). According to this solution, during the first 6 Gyr of evolution, Leo A formed approximately 11% of its total mass. Afterwards, a period of star formation activity follows with three main periods of enhanced star formation with a maximum at 5.8 (lasting 3.0 Gyrs), 3.0 (extending over 1.8 Gyr), and 0.4 (active during the last 1.6 Gyr of evolution) Gyrs ago. This recovered time evolution of the SFR is in general agreement with previous works (Cole et al. 2007;Gallart et al. 2015). Regarding the AMR, we found a rather constant and low metallicity value of 0.0005 for stars with ages from 2.5 to 10.5 Gyr with younger stars displaying higher values. As expected, given the low fraction of stars older than 10.5 Gyr, they display the larger uncertainties. On the other hand, stars younger than 2.5 present a smooth increase of their metallicities with the newly born stars having metallicities up to 0.0075. Although to a lesser degree than that found in the STECKMAP AMR recovery, the computed metallicity for the youngest stars is above, not only previous measurements in Leo A stars (Cole et al. 2007;Kirby et al. 2013Kirby et al. , 2017, but also that expected from measurement of the gaseous component (van Zee et al. 2006). However, we must bear in mind at this point that we cannot directly compare this SFH with previous determinations as in this case we are just analysing a specific region of Leo A defined by the scanned region (see Sect. 2.1). Discussion In this section we assess the reliability of SFHs recovered via the analysis of high S/N spectra. First of all, we compare the outcome of the analysis of the Leo A integrated spectrum and deep CMD (Sects. 2 and 3). Afterwards, we test the accuracy on the SFH recovery from the shallow CMD as compared to that from integrated spectra. Figure 5 shows the recovered SFHs after applying typical techniques to analyse integrated spectra and deep CMDs. The similarities among both recovered SFHs are remarkable, both in terms of the overall behaviour of the SFR as a function of time and the shape of the AMR. The SFR is very low from the oldest ages up to around 7 Gyr ago, when it rose until ∼ 3 Gyr ago. A peak of star formation is found from both approaches at ∼ 2.8 Gyr ago. After this peak, the SFR decreases till a minimum value around 1 Gyr ago, followed by a final increase up to the present time. Although the recovered SFH (top panel of Fig. 5) in both analyses might differ in the details, the overall trend is remarkably similar (see bottom panel, cumulative mass fraction). It is in the AMR where some discrepancies are found. The recovered AMR is fairly constant at a value of around 0.0006 (CMD analysis) and 0.0007 (spectral analysis). Slightly higher metallicity values are inferred for young (younger than 2.5 Gyr) and old (older than 10.5 Gyr) stars than for the intermediate-age stellar populations. The behaviour at the oldest end is affected by larger uncertainties, as can be expected since very few stars/light from an old stellar population is available to constrain the solution. Comparing integrated spectrum and CMD analysis However, while the AMR recovered through CMD fitting increases from Z = 0.0006, 2.5 Gyr ago, to Z = 0.006, present day, the behaviour from the study of the integrated spectra is more abrupt, with the enrichment starting 0.5 Gyr ago and reaching up to Z = 0.015 with some wiggles at the youngest ages. Although these discrepancies are observed and should be noted, the derived metallicities are consistent within the errors of each method. In Paper I we already reported that the results from STECKMAP and CMD fitting were more in agreement when considering the SFR than the AMR. In that case, the metallicities recovered from the spectral analysis were generally higher than those from the CMD analysis, with the largest discrepancies found at the youngest ages (younger than 1 Gyr), where STECKMAP needed a large enrichment in order to properly fit the observed spectrum. Then, we speculated with the possibility that this enrichment was an artifact due to the low SFR at young ages (lack of young stars in the analysed region) or the effect of the smoothing penalty function in the AMR (large value of µ Z ). In Leo A, where a prominent young component dominates and this issue is found regardless of the smoothing parameters (see Appendix A), this is not a likely explanation. This issue might be related with the analysis of the integrated light or the intrinsic difficulty of current stellar models to discriminate different metallicities at young ages. The amount of metals in a stellar system is imprinted in its integrated spectrum on the shape of characteristic absorption spectral features. However, metal-poor and young stars tend to be hotter, which generally weakens the strength of the observed metal-lines (causing the disappearance of some). This can lead to the observation of spectra characterised by strong Balmer lines and blue, featureless continuum, not easily attributable either to a young age or a low metallicity (i.e. age-metallicity degeneracy). As a consequence, the recovery of the correct metallicity, especially for young ages, is affected by larger uncertainties. Apart from the intrinsic modeling issues, another possible cause for the observed differences in the AMR at young ages is the different metallicity ranges of the models employed in both approaches. While in the case of the CMD we restricted the models to vary between metallicity values of 0.0001 and 0.01 (for reasons previously explained), in the integrated light analysis we use the entire extend of the MILES models (0.0001 to 0.047). We proceed this way in order to apply both methods as independiently as possible and following the customary procedures. In the case of the analysis of the CMD, constraints on the metallicities of the system under analysis can be established either from the chemistry of individual stars (Kirby et al. 2017) or by overplotting to the observed CMD isochrones of different ages and metallicities. However, nothing of that sort can be done in the case of the spectral analysis of external galaxies. In order to investigate the effect that a more restrictive metallicity range might have on the recovered solution, we have computed a different solution using only those MILES models with metallicities between 0.0001 and 0.01. Similar conclusions can be outlined regarding the shape of the temporal evolution of the SFR as before. However, in the case of the recovered AMR we can conclude that, although the maximum metallicity is now consistent with the CMD analysis (0.008), the shape is still different, implying that the chosen metallicity range cannot completely solve the problem, although it can alleviate it considerably. If we focus on the details instead of on the overall shape of the SFH, we can find some additional discrepancies. However, at such levels of detail (of the order of time-scales of 1 Gyr), we are dealing with uncertainties within the solutions themselves (see Appendix A for an assessment on the robustness and reliability of the two derived SFHs individually). The main goal of this project (started in Paper I with the analysis of a field in the LMC bar) is the examination of the performance of spectral analysis at recovering the stellar content of complex systems. Leo A is the second of such systems in which the recovered SFHs from the analysis of an integrated light spectrum and the corresponding CMD agree. As a consequence, we can conclude that both approaches produce similar solutions in systems with predominantly young (Leo A) or approximately constant (LMC bar) star formation. In addition, they are able to discriminate different fractions of young and old stellar populations. However, we still need to assess the consistency in systems that can be considered predominantly old or dominated by intermediate-age populations. As a natural continuation of this project, we plan to repeat this comparison in other Local Group systems to consider the whole range of SFHs and chemical enrichment histories representative of galactic environments. On the reliability of SFHs determined from shallow CMDs The information in the CMD of a resolved stellar system is generally regarded as the most direct way to obtain reliable information on its SFH, since stars in different evolutionary stages can be singled-out. This has led to important allocations of ground-based and HST time to obtain this kind of observations (e.g. SMASH, Nidever et al. 2017;LCID, Bernard et al. 2008;ANGST, Dalcanton et al. 2009;PHAT, Hildebrandt et al. 2010). However, there is some debate and research (Weisz et al. 2014;Bernard et al. 2015b) as to what extent the SFH of a stellar system, extended to its whole lifetime, can be reliably determined using a CMD which does not reach the oMSTO, but a shallower CMD reaching the magnitude level of the horizontal branch. This implies that the information on the old population is only provided by stars in advanced evolutionary stages such as the RGB, asymptotic giant branch, and horizontal branch/red clump. For systems with recent star formation, such diagrams provide information from MS stars younger than ∼ 1 Gyr only. Apart from that, stellar evolution models for stars in advanced evolutionary stages are affected, in principle, by larger uncertainties. Additionally, stars of different ages and metallicites are tighly packed in the CMD, where their positions are affected by important degeneracies (e.g. the well known age-metallicity degeneracy on the RGB). In this work and in Paper I we have found that, for a young system such as Leo A or the LMC bar, an integrated spectrum leads to a SFH that is in good agreement with that derived with a CMD reaching the oMSTO. In this section we will try to answer in the case of Leo A, whether a shallow CMD (where some stellar phases are missing) can also recover SFHs in close agreement with the one obtained from the deepest CMDs. With this aim, we downloaded the photometry and artificial star tests corresponding to HST/WFPC2 observations from the HST Local Group Stellar Photometry Archive (Holtzman et al. 2006), maintained by J. Holtzmann 7 . In particular, we used the WFPC2 pointing covering most of the galaxy (u2x501 field, 1800 seconds in the F555W and F814W HST filters) and originally observed within the GO program 5915 (P.I. Evan Skillman). These observations are much shallower than the ones analysed previously from the HST/ACS data, reaching down to ap- Fig. 6. Comparison of the SFHs recovered for Leo A from the analysis of a deep CMD (blue, ACS/HST data) and a shallow one (red, WFPC2/HST data). Panels are as those in Fig. 5. Inset panels focus on the SFR and AMR at young ages. parent magnitudes of 26.5 and 25.6 (F555W and F814W) with a 50% of completeness. In Fig. 6 we show the recovered SFHs from both CMDs. Although the shape of the AMR is fairly well recovered (always within errors), the time evolution of the SFR displays some discrepancies, especially at old ages. Our analysis of a shallow CMD gives similar results as those published previously using similar shallow CMDs (Tolstoy et al. 1998;Orban et al. 2008), i.e. Leo A is predominantly young. However, the analysis of a shallow CMD routinelly finds a non negligible amount of old population which we do not find neither with the analysis of higher-quality, deep CMDs nor from spectroscopic data (see Fig. 5). It is worth noting that, although integrated light analysis are also affected by the previously mentioned modeling issues, they present some advantages allowing us to understand why its analysis compares so well with the analysis of deep CMDs. Optical integrated spectra do contain light from all stellar stages (including the oMSTO) and thus, the analysis of integrated light is, in principle, sensitive to all stellar ages. However, the contribution to the integrated spectrum from stellar populations at different stages is highly dependent upon the wavelength range. At bluer wavelengths (B-band, approximately the wavelength range analysed in this work), even for old systems ∼ 49% of the light is coming from MS and subgiant stars with different contributions from other phases (9% Asymptotic Giant Branch, 29% RGB, 13% Horizontal Branch, according to the Vazdekis et al. 2016 models). As we go to redder wavelengths, RGB stars become the main contributor to the observed spectrum, losing considerably age sensitivity. This fact, added to the amount of absorption features in the blue end of the optical spectrum, makes the analysed wavelength range ideal for our purposes. Regarding the recovery of the metallicity via integrated light, we must be careful, especially in the case of the metallicity of young stars due to the age-metallicity degeneracy (as already commented before, see Sect. 4.1). In view of these results, we can claim that, at least for young systems such as Leo A, the analysis of high S/N integrated spectra provides SFHs that are closer to those recovered by the analysis of a deep CMD than those from a shallow CMD. A more rigorous study analysing a larger number of systems is necessary to assess what is the optimal type of data needed to study stellar objects near the edge and beyond the LG. Conclusions In this paper, we apply two very distinct approaches to characterize the SFH of the central region of Leo A, a dwarf irregular galaxy in the Local Group. On one hand, we scan the central part of Leo A with the GTC/OSIRIS long-slit mode in order to obtain an integrated spectrum that has been analysed with STECKMAP. On the other hand, we analyse an HST CMD reaching the oM-STO, following commonly used CMD fitting methods. Both approaches give remarkably consistent results in terms of the time evolution of the SFR in Leo A as well as its overall chemical enrichment (AMR), If we focus on the details (time-scales of less than 1 Gyr), there are some differences between both approaches that can be attributed to uncertainties inherent to each method, and thus, details from both recovered SFHs have to be taken with caution. Slight differences are also found in the recovered metallicity of young stars, differences that can be highly minimised restricting the metallicity range of the used stellar models. This general agreement is reassuring, especially considering that most of what we currently know about galaxy formation and evolution comes from studies of external systems where individual stars cannot be resolved. Together with the conclusions in Ruiz-Lara et al. (2015), the results presented in this paper support the use of high-quality spectroscopic data collected at ground-based facilities to study the SFHs of external as well as nearby, semiresolved systems for which a CMD reaching the oMSTO cannot be obtained. Through the results of the analysis of a shallow CMD (reaching just below the horizontal branch) we provide a hint that a high S/N spectrum like the one used in this work may be preferred to a shallow CMD (which are very expensive to obtain for systems beyond the LG), at least in the case of young stellar systems such as Leo A. Table A.1 summarises the different runs performed in order to obtain the final SFH from the integrated spectrum. These runs focus on the effect of the choice of the smoothing parameters and the possible influence of bright foreground stars (Milky Way candidates). The solution that has been compared with the CMD approach was the one that minimised the residuals in the spectral fit (rms, run 6) from extraction A. In order to investigate to what extent different smoothing parameters (µ x and µ Z ) affect the recovery of the SFH, we have tested 11 different sets of parameters sampling a reasonable range of values (see Table A.1). In Fig. A.1 we compare the shape of the solution used throughout this paper (extraction A and minimum rms, red) with the envelope of the 11 solutions (including errors) using this input configuration (extraction A) but considering all the different smoothing parameter sets. We must highlight that, in spite of some differences, every recovered SFH presents a common pattern: a near absence of stars older than 7 Gyr, two peaks of stars with ages 2.3 and 0.1 Gyr, an almost flat AMR with values of Z ∼ 0.0007 and a chemical enrichment towards young stellar populations. This enrichment might reach up to solar values in some runs. The small differences in the rms values found among runs show that the choice of parameters is not a crucial factor in this case, and proves the robustness of the STECKMAP fits in analysing the Leo A spectrum. Both aspects suggest that small, secondary peaks, drops and trends in the recovered SFH, apart from the above outlined general behaviour, are not sufficently robust as to be claimed as real, and might be artifacts of the particular set of input parameters. For the sake of completeness, we also assess the effect that bright Milky Way candidate stars might have on the SFH recovery. Although this is not an issue affecting normal studies of stellar populations in external systems, it is an aspect that might affect the conclusions of the current work. In Sect. 2 we inspected the observed CMD pinpointing bright stars that might not belong to Leo A but could have a potential effect on the shape of the observed spectrum and thus, on the recovered SFH. Figure A.2 compares the envelopes of all the solutions (including errors) for the two different extractions performed in Sect. 2.1 (see also Fig. 2) as well as the best solutions in each set of runs (solid lines). Although the light coming from these bright, MW candidates slightly modifies the shape of the spectrum, the light coming from Leo A dominates in the SFH recovery and both extractions lead to similar SFHs. The main differences are spotted at old ages (10 to 12 Gyr ago) and at very young ages. These minor discrepancies are somehow expected, as the position in the observed CMD of most of these stars suggests that they might be young if at Leo A distance. As a consequence, extraction B Table A.1). Panels are as those in the top and middle panels of Fig. 5. In red we show the best solution (minimum rms) for the extraction A (run 6). Embedded in the grey shaded area are runs 1 to 11 using extraction A, including errors. (which includes the light from these stars) presents a slightly larger fraction of stars younger than 1 Gyr in detriment of star with ages around 10 to 12 Gyr. Despite these small differences, throughout the paper we have decided to stick our analysis to what we consider the safer and more standard extraction, extraction A. Appendix A.2: Resolved stellar populations As in the case of the SFH recovery from integrated data, there is a degree of uncertainty in the results from the analysis of the Leo A observed CMD due to the choice of input parameters (see Table A.2). In this appendix we focus on the effect of the two main aspects that might change the recovered SFH and thus, our conclusions (see Table A.2), namely the "bundle" strategy (highly human-dependant) and the way of treating binaries in the computation of the synthetic CMD (Belloni et al. 2017). Although the purpose of this work is not a complete study of the SFH of Leo A, in these robustness tests we have decided to analyse the entire Leo A CMD with around 95000 stars (not only the stars within the scanned region). The larger number of stars in comparison with the scanned region allowed us to better discriminate the possible effect of the input parameters under analysis as larger errors are obtained as we reduce the number of stars. Figure A.3 shows the three different "bundle" strategies used in these tests. There are some areas within the CMD that are populated by stars in evolutionary phases possibly less well described by theory. As a consequence, taking into account that the Table A.1. Set of input parameters for the STECKMAP runs to recover the SFH from the integrated spectrum. The main parameters that might affect the shape of the recovered SFH are the smoothing parameters (µ x and µ Z ) and the extraction mask (extraction A or B). With these runs we cover all reasonable values of smoothness in the solution. We show as well the rms, computed as the mean values of the absolute differences between the data and the fit as a proxy for the quality of each fit. The solutions with the lowest rms are marked with a *. Test "Bundle" strategy β q A All CMD in 7 "bundles" 0.4 0.5 B A except RC and RGB 0.4 0.5 C All CMD in 1 single "bundle" 0.4 0.5 D Same as A 0.7 0.1 Table A.1), are embedded within each shaded area. recovery of the SFH relies on the comparison of observed CMDs with synthetic ones based on stellar evolution, one might think that the addition of these regions in the analysis might affect the recovered results. Test A analyses all regions with stars within the CMD in 7 different boxes including the MS, sub-giant, Red Clump (RC) and RGB stars with different boxes sizes. Test B follows the previous strategy but without the RC and RGB "bundles". Test C is intended as an experiment to further assess the effect of the "bundle" definition, which is rather subjective, by analysing the CMD with a single "bundle" and a fine grid containing all the stars. The outcome of all these tests are shown in Fig. A.4. All solutions are consistent with a system with very low star formation until ∼ 8 Gyr ago, when the SFR starts to rise to a period of higher activity between 6 and 2 Gyr ago, followed by a slight decrease followed by a recent rise. The behaviour of the AMR is also very similar among tests: a flat AMR is found with the exceptions of a recent chemical enrichment and larger uncertainties at old ages, where the number of stars in which the solution is based is minimal. As the cumulative mass fraction shows, all these discrepancies are mere subtleties, and thus, the effect of the chosen "bundle" strategy does not affect the overall shape of the recovered SFH, but only the details. In fact, all these recovered SFHs are in agreement with previous works using the same data. The different choices of input parameters also have a small effect on the properties of the best-fit CMD. Figure A.5 shows the metallicity distribution of a subset of RGB stars (matching the absolute magnitudes covered by Kirby et al. 2017) in the three best-fit CMDs (for tests A, B, and C; blue, red and green, respectively) in comparison with that presented in Kirby et al. (2017) from the spectroscopic chemical composition of 113 individual RGB stars (black). The resemblance between the four distributions is remarkable, although some differences can be highlighted. Tests A and C are those that present the higher sim-Article number, page 13 of 15 A&A proofs: manuscript no. LeoA_accepted . Different "bundle" strategies adopted to test the robustness of the recovery of the SFH through CMD fitting. A vs. B allows us to tackle the effect of adding the RGB or the RC in the CMD fit. Test C allows us to check the effect of considering all the CMD at once in a single "bundle" with small boxes. ilarities as compared with the Kirby et al. (2017) distribution, while test B results in a broader [Fe/H] distribution. This fact indicates that adding as many evolutionary phases as possible in our analysis improves the reliability of the recovered SFHs. In particular, if we include the observed RGB in the fit, the characteristics of the modeled RGB resembles those of the observed one (metallicity distribution) even though this stellar evolution phase is affected, in principle, by larger uncertainties. This result, in fact, may provide an indication of the reliability of current, up-to-date stellar evolution models such as those of the BaSTI library, even in advanced evolutionary stages. Another parameter that affects the distribution of stars in the CMDs is stellar multiplicity, as stars are typically born in pairs and even in groups. However, the problem of how to model the binary population is far from trivial (Kroupa & Petr-Gotzens 2011;Kroupa et al. 2013;Belloni et al. 2017). As a consequence, current codes to create synthetic CMDs are based on simple parametrisation in which properties as the binary fraction (β) or the binary mass ratio (q) are essential. Figure A.6 compares the SFH recovered by comparing the Leo A observed CMD with synthetic CMDs with different β and q values (see Table A.2). As in the previous set of tests, the recovered SFH is fully consistent in both cases. Only subtle discrepancies appear among tests and slightly younger populations are found (see bottom panel of Fig. A.6) as the amount of binaries increase (see also Monelli et al. 2010). This suggests that the overall shape of the SFH presents little dependence on this particular choice. Based on this analysis, we conclude that all the tests produce consistent solutions from the study of the Leo A CMD reaching the oMSTO, with no obvious "best" solution. Small discrepancies inherent to the method are found with different input parameters at short time-scales (of the order of 1 Gyr). In addition, this analysis opens the door of an alternative and more objective "bundle" definition in which just one "bundle" is considered. To sum up, we can conclude that both approaches present their own typical uncertainties (usually subtle) and that the choice of input parameters within reasonable limits has a small impact. To what extent we can rely on these subtle details is an aspect to be improved within each technique in the coming years. Comparison of the SFHs recovered from the analysis of the Leo A CMD using different "bundle" strategies (see Fig. A.3). In blue, red and green we show the SFH (and the errorrs) recovered using "bundle" strategies A, B and C, respectively. Panels are as those in Fig. 5. Inset panels focus on the SFR as well as AMR at young ages. Comparison of the SFHs recovered from the analysis of the Leo A CMD with different synthetic CMDs treating binaries with different recipes ("bundle" strategy as in test A). Panels are as those in Fig. 5. Inset panels focus on the SFR as well as AMR at young ages. Fig. 1 . 1Observational layout: SDSS r-band image(Abazajian et al. 2009) of Leo A with the fields analysed in this works superimposed. The blue shaded area corresponds to the HST/ACS field. The red shaded area represents the scanned region observed with GTC/OSIRIS. The area of overlap between the two datasets is the comparison area from where both SFHs are extracted. Fig. 2 . 2Fig. 2. Analysed region and masking procedure. Top two rows: Highspatial resolution HST image of the scanned region within the ACS (area covered is 19.2"×202"). The next three rows show pseudoimages reconstructed from the 16 slit positions observed with OSIRIS in Leo A. Third row: The entire extension of the observed field ( 19.2"×7'). Fourth row: The region from which the integrated spectrum is obtained; in this case we mask background galaxies, saturated stars, and Milky Way stars candidates (extraction A). Fifth row: The region from which the integrated spectrum is obtained; in this case we mask just foreground galaxies and saturated stars (extraction B). Fig. 4 . 4(M F814W , M F475W -M F814W ) CMD based on the ACS data. Red points depict the Milky Way star candidates that differentiate extraction A from extraction B (see Sect. 2.1). The 7 polygons show the "bundles". The inset table shows the size of the boxes within "bundles". measured the metalliticy of 113 RGB stars finding metallicities between 0.0001 and 0.002, with only 2 and 6 stars with metallicities below and above this range. On the other hand, van Zee et al. (2006) measured a very low oxygen abundance in the warm gas phase in Leo A (12+log(O/H) = 7.38 ± 0.1 corresponding to 5 % of the solar oxygen content assuming 12+log(O/H) = 8.69 ± 0.05 for the Sun; Asplund et al. 2009). Fig. 5 . 5Comparison between the Leo A SFH from the CMD (blue) and the integrated spectrum using STECKMAP (red). Top panel: Normalised SFR: the area under each curve is 1. Middle panel: Age-Metallicity relation. Bottom panel: Cumulative mass fraction. See text for details on the computation of the uncertainties. Inset panels focus on the SFR and AMR at young ages. Fig. A. 1 . 1Comparison of the SFHs recovered from the analysis of the Leo A integrated spectrum using different smoothing parameters (see Fig. A. 2 . 2Comparison of the SFHs recovered from the analysis of the Leo A integrated spectrum using extraction A (blue) or B (red). Panels are as those in the top and middle panels ofFig. 5. All the different solutions from the different runs, including errors (see Fig Fig. A.3. Different "bundle" strategies adopted to test the robustness of the recovery of the SFH through CMD fitting. A vs. B allows us to tackle the effect of adding the RGB or the RC in the CMD fit. Test C allows us to check the effect of considering all the CMD at once in a single "bundle" with small boxes. Fig Fig. A.4. Comparison of the SFHs recovered from the analysis of the Leo A CMD using different "bundle" strategies (see Fig. A.3). In blue, red and green we show the SFH (and the errorrs) recovered using "bundle" strategies A, B and C, respectively. Panels are as those in Fig. 5. Inset panels focus on the SFR as well as AMR at young ages. Fig. A. 5 . 5Modelled and observed RGB stars metallicity distributions. We compare the metallicity distributions of the RGB stars from the best-model CMD for tests A, B and C (blue, red and green) with that presented inKirby et al. (2017) determined through the spectroscopic analysis of individual stars (black, 113 stars). The distributions are built as the sum of individual gaussians for each star. Fig . A.6. ; Cid Fernandes Article number, page 1 of 15 arXiv:1805.04323v2 [astro-ph.GA] 24 Jul 2018A&A proofs: manuscript no. LeoA_accepted et al. 2013; or [Fe/H] (Vazdekis et al. 2015): [M/H] = [Fe/H] + A × [Mg/Fe]; A ∼ 0.75 (1) [M/H] log 10 (Z/Z ) (2) Article number, page 2 of 15 T. Ruiz-Lara et al.: Leo A, an extremely young dwarf in the Local Group Table A . A2. Set of input parameters for the tests to recover the SFH from the Leo A observed CMD. The main parameters that might affect the shape of the recovered SFH are the "bundle" strategy and the way of dealing with binaries during the synthetic CMD computation. For a graphical description of the different "bundle" strategies seeFig. A.3. Complete information regarding the OSIRIS instrument can be found at http://www.gtc.iac.es/instruments/osiris/ The MILES models are publicly available at http://miles.iac. es and are based on the MILES empirical library of stellar spectra(Sánchez-Blázquez et al. 2006;Falcón-Barroso et al. 2011) http://astronomy.nmsu.edu/holtz/archival/html/lg. html Acknowledgements. We thank the anonymous referee for the careful reading of the manuscript and the very useful comments that have improved considerably this work. We also thank Alexandre Vazdekis and Jesús Falcón-Barroso for useful discussions. This research has been mainly supported by the Spanish Ministry of Economy and Competitiveness (MINECO) under the grants AYA2014-56795-P and AYA2016-77237-C3-1-P. EJB acknowledges support from the CNES postdoctoral fellowship program. GB gratefully acknowledges financial support by the Spanish Ministry of Economy and Competitiveness (MINECO) under the Ramon y Cajal Programme (RYC-2012-11537) and the grant AYA2014-56795-P. PSB thanks the support under the grant AYA2013-48226-C3-3-P (MINECO). EFN and IP thanks the support received via grants AYA 2014-53506-P (MINECO) and FQM-108 (Junta de Andalucía). IMN acknowledges funding from the Marie Skłodowska-Curie Individual Fellowship 702607, and from grant AYA2013-48226-C3-1-P from the Spanish Ministry of Economy and Competitiveness (MINECO). This research has been based on ob-servations made with the Gran Telescopio Canarias (GTC), installed at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, in the island of La Palma as well as based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with HST program 10590 and GTC program GTC94-15B_000. This research makes use of python (http://www.python.org); Matplotlib(Hunter 2007), a suite of open-source python modules that provide a framework for creating scientific plots; and Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2013).Appendix A: Robustness of the solutionsIn the main body of this paper we have focused our comparison between the performances of full-spectral fitting and CMDbased techniques in a one-to-one comparison. This means that, based on standard criteria, we have chosen a particular set of input parameters to compute the final solution from each approach. However, the details of this solution are somewhat dependent on the set of input parameters. In this appendix we assess how different input parameters affect the recovery of the SFH. 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[ "Rectification and One-Way Street for the Energy Current in Boundary-Driven Asymmetric Quantum Spin Chains", "Rectification and One-Way Street for the Energy Current in Boundary-Driven Asymmetric Quantum Spin Chains" ]	[ "Emmanuel Pereira \nDepartamento de Física-Instituto de Ciências Exatas\nUniversidade Federal de Minas Gerais\n30.161-970 Belo Horizonte MG702CPBrazil\n" ]	[ "Departamento de Física-Instituto de Ciências Exatas\nUniversidade Federal de Minas Gerais\n30.161-970 Belo Horizonte MG702CPBrazil" ]	[]	Motivated by the demand of efficient quantum devices to engineer the energy transport, we analyze some inhomogeneous quantum spin systems, including the XXZ chains, with magnetization baths at the ends. Aimed at finding general properties, we study the effects of suitable transformations on the boundary-driven Lindblad master equation associated to the dynamics of the systems. For asymmetric models with target polarization at the edges or twisted XY boundary gradients, we show properties of the steady state which establish features of the energy current, irrespective of the system size and of the regime of transport. We show the ubiquitous occurrence of energy rectification and, more interestingly, of an unusual phenomenon: in the absence of external magnetic field, there is an one-way street for the energy current, i.e., the direction of the energy current does not change as we invert the magnetization baths at the boundaries. Given the extensiveness of the procedures, which essentially involve properties of the Lindblad master equation, our results certainly follow for other interactions and other boundary conditions. Moreover, our results indicate graded spin chains as genuine quantum rectifiers. PACS numbers: 05.70.Ln, 05.60.Gg, 75.10.PqThe derivation of the laws of transport from the underlying microscopic models is one of the purposes of nonequilibrium statistical physics. Many works are devoted to this subject, and, nowadays, besides the study of fundamental questions, such as the onset of Fourier's law, a bedrock of heat transport theory, we observe a considerable effort which aims at understanding the mechanisms of manipulation and control of the heat or energy current [1], including experimental works[3]. These studies are mainly inspired by the amazing progress of modern electronics due to the invention of electrical diodes, transistors and other nonlinear solid-state devices. The current lacking of an efficient analog of diode, i.e., the lacking of a device which clearly has a preferential direction for the energy flow [2], prevents the advance of phononics, the counterpart of electronics dedicated to the manipulation of the energy current, and makes several authors to devote their investigations to the mechanism of rectification.Given the present ambient of miniaturization, the possibility of quantum effects and the exiguity of quantum results in this specific area, we propose to investigate the mechanism of energy transport and rectification in genuine quantum models. In the present work, we consider the analysis of the energy (and spin) currents in some open quantum spin chains, including versions of the XXZ models. We emphasize that the detailed investigation of these systems, such as the XXZ chain, the archetypal model of open quantum system, besides being profitable for the understanding of quantum effects on the mechanism of the energy current, it also involves issues which * Electronic address: [email protected] interest to many areas, such as optics and cold-atoms, condensed matter, quantum information, etc.[4].We analyze some one-dimensional spin models driven out equilibrium by magnetization reservoirs coupled to the boundary spins, i.e., by the presence of pumping applied at the edges. More precisely, we analyze some spin models with dynamics given by boundary-driven Lindblad equations. We remark that such a choice of reservoirs does not precisely describe a quantum system passively coupled to heat baths [5] (problem to be treated in future works). But it is worth stressing that these specific quantum spin chains, e.g., the XXZ versions with magnetization pumping, are also of great theoretical interest[6]. Moreover, these models can also be experimentally realized due to the advance of nanotechnology, which allows us to manipulate different materials, even those with few elements, and to engineer different quantum reservoirs and specific designs for the coupling between systems and reservoirs[21].Inspired, in some way, by the work of Popkov and Livi [8], our strategy is the following: instead of performing extensive and intricate computations to determine the steady density matrix, we simply study the action on the Lindblad master equation (LME) of some properly chosen operators, related to the inversion of the baths, i.e., to the permutation between the bath linked to the first site and the bath linked to the last site. And so, we determine the effects of such transformations on the stationary density matrix, as well as on the energy and spin currents. As outcome, we reveal properties, independent of the system size and regime of transport, which lead to the occurrence of energy rectification and, more interestingly, to a particular phenomenon: in the absence of magnetic field, for the inhomogeneous, asymmetric (e.g., graded) XXZ model with target polarization at the edges	10.1103/physreve.95.030104	[ "https://arxiv.org/pdf/1703.05567v1.pdf" ]	2,117,856	1703.05567	861d6013403cf80100471884beab724abe74ce5e	Rectification and One-Way Street for the Energy Current in Boundary-Driven Asymmetric Quantum Spin Chains 16 Mar 2017 (Dated: October 1, 2018) Emmanuel Pereira Departamento de Física-Instituto de Ciências Exatas Universidade Federal de Minas Gerais 30.161-970 Belo Horizonte MG702CPBrazil Rectification and One-Way Street for the Energy Current in Boundary-Driven Asymmetric Quantum Spin Chains 16 Mar 2017 (Dated: October 1, 2018) Motivated by the demand of efficient quantum devices to engineer the energy transport, we analyze some inhomogeneous quantum spin systems, including the XXZ chains, with magnetization baths at the ends. Aimed at finding general properties, we study the effects of suitable transformations on the boundary-driven Lindblad master equation associated to the dynamics of the systems. For asymmetric models with target polarization at the edges or twisted XY boundary gradients, we show properties of the steady state which establish features of the energy current, irrespective of the system size and of the regime of transport. We show the ubiquitous occurrence of energy rectification and, more interestingly, of an unusual phenomenon: in the absence of external magnetic field, there is an one-way street for the energy current, i.e., the direction of the energy current does not change as we invert the magnetization baths at the boundaries. Given the extensiveness of the procedures, which essentially involve properties of the Lindblad master equation, our results certainly follow for other interactions and other boundary conditions. Moreover, our results indicate graded spin chains as genuine quantum rectifiers. PACS numbers: 05.70.Ln, 05.60.Gg, 75.10.PqThe derivation of the laws of transport from the underlying microscopic models is one of the purposes of nonequilibrium statistical physics. Many works are devoted to this subject, and, nowadays, besides the study of fundamental questions, such as the onset of Fourier's law, a bedrock of heat transport theory, we observe a considerable effort which aims at understanding the mechanisms of manipulation and control of the heat or energy current [1], including experimental works[3]. These studies are mainly inspired by the amazing progress of modern electronics due to the invention of electrical diodes, transistors and other nonlinear solid-state devices. The current lacking of an efficient analog of diode, i.e., the lacking of a device which clearly has a preferential direction for the energy flow [2], prevents the advance of phononics, the counterpart of electronics dedicated to the manipulation of the energy current, and makes several authors to devote their investigations to the mechanism of rectification.Given the present ambient of miniaturization, the possibility of quantum effects and the exiguity of quantum results in this specific area, we propose to investigate the mechanism of energy transport and rectification in genuine quantum models. In the present work, we consider the analysis of the energy (and spin) currents in some open quantum spin chains, including versions of the XXZ models. We emphasize that the detailed investigation of these systems, such as the XXZ chain, the archetypal model of open quantum system, besides being profitable for the understanding of quantum effects on the mechanism of the energy current, it also involves issues which * Electronic address: [email protected] interest to many areas, such as optics and cold-atoms, condensed matter, quantum information, etc.[4].We analyze some one-dimensional spin models driven out equilibrium by magnetization reservoirs coupled to the boundary spins, i.e., by the presence of pumping applied at the edges. More precisely, we analyze some spin models with dynamics given by boundary-driven Lindblad equations. We remark that such a choice of reservoirs does not precisely describe a quantum system passively coupled to heat baths [5] (problem to be treated in future works). But it is worth stressing that these specific quantum spin chains, e.g., the XXZ versions with magnetization pumping, are also of great theoretical interest[6]. Moreover, these models can also be experimentally realized due to the advance of nanotechnology, which allows us to manipulate different materials, even those with few elements, and to engineer different quantum reservoirs and specific designs for the coupling between systems and reservoirs[21].Inspired, in some way, by the work of Popkov and Livi [8], our strategy is the following: instead of performing extensive and intricate computations to determine the steady density matrix, we simply study the action on the Lindblad master equation (LME) of some properly chosen operators, related to the inversion of the baths, i.e., to the permutation between the bath linked to the first site and the bath linked to the last site. And so, we determine the effects of such transformations on the stationary density matrix, as well as on the energy and spin currents. As outcome, we reveal properties, independent of the system size and regime of transport, which lead to the occurrence of energy rectification and, more interestingly, to a particular phenomenon: in the absence of magnetic field, for the inhomogeneous, asymmetric (e.g., graded) XXZ model with target polarization at the edges Motivated by the demand of efficient quantum devices to engineer the energy transport, we analyze some inhomogeneous quantum spin systems, including the XXZ chains, with magnetization baths at the ends. Aimed at finding general properties, we study the effects of suitable transformations on the boundary-driven Lindblad master equation associated to the dynamics of the systems. For asymmetric models with target polarization at the edges or twisted XY boundary gradients, we show properties of the steady state which establish features of the energy current, irrespective of the system size and of the regime of transport. We show the ubiquitous occurrence of energy rectification and, more interestingly, of an unusual phenomenon: in the absence of external magnetic field, there is an one-way street for the energy current, i.e., the direction of the energy current does not change as we invert the magnetization baths at the boundaries. Given the extensiveness of the procedures, which essentially involve properties of the Lindblad master equation, our results certainly follow for other interactions and other boundary conditions. Moreover, our results indicate graded spin chains as genuine quantum rectifiers. The derivation of the laws of transport from the underlying microscopic models is one of the purposes of nonequilibrium statistical physics. Many works are devoted to this subject, and, nowadays, besides the study of fundamental questions, such as the onset of Fourier's law, a bedrock of heat transport theory, we observe a considerable effort which aims at understanding the mechanisms of manipulation and control of the heat or energy current [1], including experimental works [3]. These studies are mainly inspired by the amazing progress of modern electronics due to the invention of electrical diodes, transistors and other nonlinear solid-state devices. The current lacking of an efficient analog of diode, i.e., the lacking of a device which clearly has a preferential direction for the energy flow [2], prevents the advance of phononics, the counterpart of electronics dedicated to the manipulation of the energy current, and makes several authors to devote their investigations to the mechanism of rectification. Given the present ambient of miniaturization, the possibility of quantum effects and the exiguity of quantum results in this specific area, we propose to investigate the mechanism of energy transport and rectification in genuine quantum models. In the present work, we consider the analysis of the energy (and spin) currents in some open quantum spin chains, including versions of the XXZ models. We emphasize that the detailed investigation of these systems, such as the XXZ chain, the archetypal model of open quantum system, besides being profitable for the understanding of quantum effects on the mechanism of the energy current, it also involves issues which * Electronic address: [email protected] interest to many areas, such as optics and cold-atoms, condensed matter, quantum information, etc. [4]. We analyze some one-dimensional spin models driven out equilibrium by magnetization reservoirs coupled to the boundary spins, i.e., by the presence of pumping applied at the edges. More precisely, we analyze some spin models with dynamics given by boundary-driven Lindblad equations. We remark that such a choice of reservoirs does not precisely describe a quantum system passively coupled to heat baths [5] (problem to be treated in future works). But it is worth stressing that these specific quantum spin chains, e.g., the XXZ versions with magnetization pumping, are also of great theoretical interest [6]. Moreover, these models can also be experimentally realized due to the advance of nanotechnology, which allows us to manipulate different materials, even those with few elements, and to engineer different quantum reservoirs and specific designs for the coupling between systems and reservoirs [21]. Inspired, in some way, by the work of Popkov and Livi [8], our strategy is the following: instead of performing extensive and intricate computations to determine the steady density matrix, we simply study the action on the Lindblad master equation (LME) of some properly chosen operators, related to the inversion of the baths, i.e., to the permutation between the bath linked to the first site and the bath linked to the last site. And so, we determine the effects of such transformations on the stationary density matrix, as well as on the energy and spin currents. As outcome, we reveal properties, independent of the system size and regime of transport, which lead to the occurrence of energy rectification and, more interestingly, to a particular phenomenon: in the absence of magnetic field, for the inhomogeneous, asymmetric (e.g., graded) XXZ model with target polarization at the edges or twisted XY boundary gradients, and certainly other boundary conditions and different interactions, there is an one-way street for the energy current. Namely, the direction of the energy current is determined by the asymmetry in the chain, not by the magnetization baths. In other words, not only the magnitude, but also the direction of the energy flow does not change as we invert the magnetization baths at the boundaries. We must stress that the existence of asymmetry in the structure of a given model is absolutely not a guarantee for the existence of energy rectification. The ingredients for asymmetry in the energy flow, i.e., for the occurrence of rectification and related effects, are not trivial. In order to make transparent that it is indeed an intricate problem, before describing our investigations on the spin chains, we briefly revisit the classical chain of oscillators, in which this problem of rectification has been already investigated in details. Since the pioneering works of Debye and Peierls, the prototype model for heat conduction in insulating solids is given by chains of anharmonic oscillators. Consequently, such systems have been recurrently and exhaustively studied [9], in particular, in reference to the energy rectification [10]. For transparency, we will succinctly analyze the phenomenon in a minimal chain of 3 sites. We repeat that a more detailed study in larger systems is already known [11]. As well known, Fourier's law holds in many systems: here, let us focus on chains of oscillators with nearest neighbor harmonic interparticle interactions and anharmonic onsite potentials [11]. In such cases, we usually have a thermal conductivity depending on the local temperature and other parameters. To investigate rectification in this chain of oscillators, we start from the expression for local Fourier's law derived for such systems in previous works. For inhomogeneous chains [11], the Fourier's law in any part of the chain, i.e., the heat flow from site j to j + 1, F j,j+1 , is given by F j,j+1 = −κ j,j+1 (∇T ) j = −1 c j T α j + c j+1 T α j+1 (T j+1 − T j ) ,(1) where T j is the local temperature, α ≥ 0, and c j depends on local parameters (particle mass, etc.). In a graded system, we have c j−1 > c j > c j+1 or c j−1 < c j < c j+1 . Hence, for a chain of 3 sites, F 1,2 = −κ 1,2 (T 2 − T 1 ) , F 2,3 = −κ 2,3 (T 3 − T 2 ) . In the steady state, where F = F 1,2 = F 2,3 , we have F = −1 c 1 T α 1 + c 2 T α 2 (T 2 − T 1 ) = −1 c 2 T α 2 + c 3 T α 3 (T 3 − T 2 ). And summing up the 2 parts of the equation above, we get F = −1 c 1 T α 1 + 2c 2 T α 2 + c 3 T α 3 (T 3 − T 1 ) . To determine the heat flow, we recall that the temperatures T 1 and T 3 at the edges are given a priori, and so, T 2 is computed by using the equations above. The computation becomes very simple if we still assume a very small gradient of temperature in the chain, namely, T 1 = T + a 1 ε and T 3 = T + a 3 ε (again, given T , a 1 , a 3 and ε, which is small). Hence, considering the result only up to O(ε), algebraic manipulations give us a 2 = 1 2c 2 + c 1 + c 3 [c 1 a 3 + c 2 (a 1 + a 3 ) + c 3 a 1 ] . And so, the temperature T 2 and the heat flow F are determined. Note that we may write F = −κ(T 3 − T 1 ) ⇒ κ = 1/ (c 1 T α 1 + 2c 2 T α 2 + c 3 T α 3 ) . Carrying out the computation for the inverted system, i.e., for the chain in which a ′ 1 = a 3 and a ′ 3 = a 1 , we obtain a ′ 2 , which will be given by the expression for a 2 above but with the replacement a 1 ↔ a 3 , a ′ 2 = 1 2c 2 + c 1 + c 3 [c 1 a 1 + c 2 (a 1 + a 3 ) + c 3 a 3 ] . Moreover, we obtain the inverted flow F ′ = −κ ′ (T 1 −T 3 ), where κ ′ is such that 1 κ − 1 κ ′ = αεT α−1 (c 1 −c 3 )(a 1 −a 3 ) c 1 + c 3 2c 2 + c 1 + c 3 .(2) Despite the simplicity of this minimal model, important information can be extracted from the equations above. For inhomogeneous (graded) chains and generic temperatures at the edges, an interesting feature is that the temperature profile T ′ , for the system with inverted baths (T ′ 1 = T 3 and T ′ 3 = T 1 ), is different from that obtained by inverting the original temperature profile T . That is, if we simply invert the original profile, we get a ′ 1 = a 3 , a ′ 3 = a 1 and a ′ 2 = a 2 , which is not the profile for the inverted chain. And it follows even for the specific case of α = 0, that describes a system in which the thermal conductivity does not depend on temperature. By the other side, rectification holds only if α = 0, see Eq. (2). In other words, if we invert an asymmetric chain of oscillators between two baths, even if we observe a different temperature profile, i.e., a final profile which is not the inversion of the original one, the occurrence of rectification is not mandatory. In short, asymmetry in a chain is not a synonym of rectification, i.e., it is not a synonym of asymmetry in the energy current [12,13]. A further comment is relevant. It is rigorously proved [14] that rectification is absent in any asymmetric version of the harmonic chain of oscillators with self-consistent inner stochastic baths -these inner baths mimic the anharmonic potentials absent in the initial Hamiltonian. Such a model obeys the Fourier's law as described above, with α = 0. In short, the previous derived result is not due to the size of the chain, nor due to the approximation in the temperature gradient. Now, we introduce the spin systems. We consider here the one-dimensional quantum spin model, with Hamiltonian (for = 1) H = N −1 i=1 α i,i+1 σ x i σ x i+1 + α ′ i,i+1 σ y i σ y i+1 + ∆ i,i+1 σ z i σ z i+1 + N i=1 B i σ z i ,(3) where σ β i (β = x, y, z) are the Pauli matrices and B i is the external magnetic field acting on site (particle) i. In fact, we can consider several other H: the main restriction is the invariance of H under the transformations to be presented. Anyway, for simplicity, we will restrict the analysis to the simple case of the XXZ version with α i,i+1 = α ′ i,i+1 = α. The time evolution of the system density matrix ρ is given by a Lindblad quantum master equation [15] dρ dt = i[ρ, H] + L(ρ) .(4) The dissipator L(ρ) describes the coupling with the baths, and it is given by L(ρ) = L L (ρ) + L R (ρ) , L L,R (ρ) = s=± L s ρL † s − 1 2 L † s L s , ρ .(5) For L L , in the case of a XXZ chain with target σ z polarization at the edges, we have L ± = γ 2 (1 ± f L )σ ± 1 ,(6) and similarly for L R , but with σ ± N and f R replacing σ ± 1 and f L . In the previous expressions, {·, ·} denotes the anticommutator; σ ± j are the spin creation and annihilation operators σ ± j = (σ x j ± iσ y j )/2 ; γ is the coupling strength to the spin baths; f L and f R give the driving strength, and are related to the polarization of extra spin at the boundaries: f L = σ z 0 and f R = σ z N +1 . The expressions for spin and energy currents follow from the LME and continuity equations, see e.g. Ref. [16] for detailed derivations. At site j, inside the chain, for the magnetization current J j and energy current F j , we have J j = 2α σ x j σ y j+1 − σ y j σ x j+1 , F j = F XXZ j + F B j , F XXZ j = 2α α σ y j−1 σ z j σ x j+1 − σ x j−1 σ z j σ y j+1 +∆ j−1,j σ z j−1 σ x j σ y j+1 − σ z j−1 σ y j σ x j+1 +∆ j,j+1 σ x j−1 σ y j σ z j+1 − σ y j−1 σ x j σ z j+1 , F B j = 1 2 B j J j−1 + J j .(7) To carry out the study, we take f = f L = −f R . And so, within such a choice, the inversion of the baths at the edges (f L ↔ f R ) is given by the change in the sign of f . Analyzing the dissipator L(ρ), we note that it does not modifies if we change the sign of f and, at the same time, make the replacements σ + 1 ↔ σ − 1 and σ + N ↔ σ − N . Hence, in the case of absence of the external magnetic field, B ≡ 0, we investigate the effects of the following transformation [the idea is to find a suitable transformation, related to the inversion, i.e., permutation of the baths] U = σ x 1 ⊗ σ x 2 ⊗ . . . ⊗ σ x N , U † = U −1 .(8) From the LME we have d dt U −1 ρU = iU −1 ρHU − iU −1 HρU + U −1 LU +iU −1 ρU U −1 HU − iU −1 HU U −1 ρU + U −1 LU . But, U −1 HU = H (for B = 0), and U −1 L(f )U = L(−f ). That is, if ρ is a solution of the LME, then U −1 ρU is a solution of the LME with −f , which means, it is a solution of the system with inverted baths. Recalling the uniqueness of the stationary solution of these LME [8,17,18], we can say that if ρ is the steady solution of the LME, then U −1 ρU is the steady solution of the LME with −f , i.e., a solution of the system with inverted baths. In relation to the effect of U on the currents, we have U −1 F XXZ j U = F XXZ j ,(9) see Eqs. (7). Consequently, F XXZ j ≡ tr(ρF XXZ j ) = tr(ρU −1 F XXZ j U ) = tr(U −1 ρU F XXZ j ) = tr(ρ(−f )F XXZ j ) . Precisely, F XXZ j = F XXZ j ib , where ib means the system with inverted baths. In other words, F XXZ j (f ) = F XXZ j (−f ) , i.e., F XXZ j is an even function of f . The implications of such property are evident. First, for the case of a homogeneous chain, it proves the vanishing of F XXZ j , as described in Ref. [8], there with the use of the left-right reflexion operation. To show the vanishing here, note that in the homogeneous chain, there is nothing which may indicate the direction for the energy flow -it must be given by the baths. And so, if we invert the baths, the direction of the energy flow must invert. As it does not happen due to the property above, we must conclude that F XXZ j vanishes in the homogeneous system. And, for the case of a graded chain (note that, in such case, the reflexion operation does not work anymore), in any situation which allows an energy current, its direction is determined by some structure in the chain, not by the baths. It means that we have a oneway street for the energy flow, say, a "complete, integral rectification". And, for the magnetization current J j , U −1 J j U = −J j ,(10) see Eq. (7). Then, J j ≡ tr(ρJ j ) = −tr(ρU −1 J j U ) = −tr(U −1 ρU J j ) = −tr(ρ(−f )J j ) . In short, J j (f ) = − J j (−f ) , namely, J j is an odd function of f . It means that the direction of the magnetization flow is given by the baths: if we invert the baths, the direction of the flow is inverted. Note also that rectification is absent: the magnitude of the spin current is preserved. The extension of our findings for the system in the presence of a homogeneous magnetic field B is immediate. Indeed, a homogeneous B does not affect expectation values of spin-conserving operators (more comments and details in Ref. [16] and references there in), and so, both F XXZ j and J j do not change. Now, the total energy current is given by, see Eqs. (7), F j = F XXZ j + B J j ,(11) namely, it is a sum of an even function of f with another odd function of f . Consequently, unless vanishing, F (f ) = F (−f ) , that is, the occurrence of energy rectification is transparent. Note also that the presence of a large magnetic field B decreases the rectification power. It is important to report that, in a recent work [19], considering the investigation of energy transport in the open quantum XXZ chain, we perform analytical computations for the energy and spin currents by finding the stationary density matrix of the associated LME. Precise algebraic formulas are obtained for a minimal chain with 3 sites (the restriction to small systems was due to technical difficulties), and numerical computations extend some results to systems up to 8 sites. An exact and huge expression for F is derived in Ref. [19] for the minimal chain. For clearness, we write below the dominant terms in a expansion in powers of f , the driving strength, and of δ, the asymmetry parameter, defined ahead. For the Hamiltonian (3) of the graded chain with 3 sites, we take α i,i+1 = α ′ i,i+1 = 1, ∆ 1,2 = ∆ − δ, ∆ 2,3 = ∆ + δ, B i = B ; and so, we have F = Bf 912 969 + 48∆ 2 +f 2 δ 32(20224∆ 4 + 64256∆ 2 − 1083) (51 + 16∆ 2 )(323 + 16∆ 2 ) 2 . The formula above for the energy current, in the minimal chain, shows that it is nonzero, for f = 0, even if B = 0. Moreover, again for B = 0, the energy current does not change as we invert the baths. These results follow also in the exact formula (beyond O(f 2 )). Our analysis, carried out in the present work by means of a completely different approach, shows that the energy rectification and the one-way street for energy current as B = 0 are not only an accident in a very small chain, but they are features of these graded XXZ systems and many other spin models with the same symmetries. Another relevant version of the XXZ model is given by the chain coupled to boundary baths which tend to polarize the spins at the ends along different directions [17]. Now, beginning with B = 0, we analyze the version in which the dissipators in the LME are given by L L (ρ) = − 1 2 2 m=1 ρ, W † m W m + 2 m=1 W m ρW † m , L R (ρ) = − 1 2 2 m=1 ρ, V † m V m + 2 m=1 V m ρV † m ,(12) with W 1 = 1 − k 2 (σ z 1 + iσ x 1 ) , W 2 = 1 + k 2 (σ z 1 − iσ x 1 ) , V 1 = 1 + k ′ 2 (σ y N + iσ z N ) , V 2 = 1 − k ′ 2 (σ y N − iσ z N ) , where we will take −1 ≤ k ≤ 1, and, in our case, k ′ = −k. To carry out the analysis, we define the operator σ r , where σ r ≡ 0 1 i 0 ⇒ (σ r ) † = (σ r ) −1 = 0 −i 1 0 . [Of course, in order to obtain a suitable transformation σ r , we have performed a detailed algebraic study involving the effects of generic transformations on the terms of the LME.] It follows that σ r σ x (σ r ) † = σ y , σ r σ y (σ r ) † = σ x , σ r σ z (σ r ) † = −σ z . Discarding the indices 1 and N in σ x , σ y , σ z written in W 1 , W 2 , V 1 and V 2 , we have σ r W 1 (σ r ) † = iV 1 , σ r W 2 (σ r ) † = −iV 2 , σ r V 1 (σ r ) † = −iW 1 , σ r V 2 (σ r ) † = iW 2 , etc. Thus, for the analysis of (a)symmetries in the LME, we define U † = σ r 1 ⊗ σ r 2 ⊗ . . . ⊗ σ r N .(13) And we have that, if ρ is a solution of the LME, then U † ρU is a solution of the LME, but with inverted baths. Turning to the currents, we have U † F XXZ j U = F XXZ j , U † J j U = −J j . And so, it follows an analysis similar to the previous one, presented for the case of target spins at the edges of the chain: F XXZ j = F XXZ j ib , J j = − J j ib , etc. Consequently, we find similar phenomena: one-way street and rectification. To conclude, we make some remarks. In Ref. [8], the authors analyze homogeneous XXZ driven spin chains by considering symmetries of the LME. They show that different pumping applied at the edges, irrespective of the system size or the regime of transport, can switch on and off the spin and/or energy currents in the stationary state. In the present work, motivated by the investigation of the energy rectification phenomenon (evidently, in asymmetric models), we also study the behavior of the LME under suitable transformations. We show that, again, irrespective of the system size or the regime of transport, in inhomogeneous, asymmetric spin chains we can find energy rectification or the emergence of an one-way street for the energy current, even in a chain with no energy current in the homogeneous case. Some interesting recent results concerning the rectification of the spin current by direct computation in the steady state are presented in Ref. [20]. There, the authors consider XXZ chains with target σ z polarization at the edges, and in the presence of an inhomogeneous external magnetic field. Another interesting and recent result concerning transport in spin chains is presented in Ref. [21], where the authors find diffusive and subdiffusive high temperature spin transport in a disordered Heisenberg chain in the ergodic regime. A further comment on the interest of asymmetric models is pertinent. 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[ "Modelling and Simulation of Scheduling Policies Implemented in Ethernet Switch by Using Coloured Petri Nets", "Modelling and Simulation of Scheduling Policies Implemented in Ethernet Switch by Using Coloured Petri Nets" ]	[ "B Brahimi \nCentre de Recherche en Automatique de Nancy\nFaculté des Sciences\nUMR-CNRS 7039\nBP 23954506VANDOEUVRE LES NANCY Cedex\n", "C Aubrun \nCentre de Recherche en Automatique de Nancy\nFaculté des Sciences\nUMR-CNRS 7039\nBP 23954506VANDOEUVRE LES NANCY Cedex\n", "E Rondeau \nCentre de Recherche en Automatique de Nancy\nFaculté des Sciences\nUMR-CNRS 7039\nBP 23954506VANDOEUVRE LES NANCY Cedex\n" ]	[ "Centre de Recherche en Automatique de Nancy\nFaculté des Sciences\nUMR-CNRS 7039\nBP 23954506VANDOEUVRE LES NANCY Cedex", "Centre de Recherche en Automatique de Nancy\nFaculté des Sciences\nUMR-CNRS 7039\nBP 23954506VANDOEUVRE LES NANCY Cedex", "Centre de Recherche en Automatique de Nancy\nFaculté des Sciences\nUMR-CNRS 7039\nBP 23954506VANDOEUVRE LES NANCY Cedex" ]	[]	The objective of this paper is to propose models enabling to study the behaviour of Ethernet switch for Networked Control Systems. Two scheduler policies are analyzed: the static priority and the WRR (Weighted Round Robin). The modelling work is based on Coloured Petri Nets. A temporal validation step based on the simulation of these modelling, shows that the obtained results are near to the expected behaviour of these scheduler policies.	10.1109/etfa.2006.355373	[ "https://arxiv.org/pdf/cs/0609150v1.pdf" ]	6,089,006	cs/0609150	bfb0086270386ddf2473c7c19be48fd7f93c8476	Modelling and Simulation of Scheduling Policies Implemented in Ethernet Switch by Using Coloured Petri Nets B Brahimi Centre de Recherche en Automatique de Nancy Faculté des Sciences UMR-CNRS 7039 BP 23954506VANDOEUVRE LES NANCY Cedex C Aubrun Centre de Recherche en Automatique de Nancy Faculté des Sciences UMR-CNRS 7039 BP 23954506VANDOEUVRE LES NANCY Cedex E Rondeau Centre de Recherche en Automatique de Nancy Faculté des Sciences UMR-CNRS 7039 BP 23954506VANDOEUVRE LES NANCY Cedex Modelling and Simulation of Scheduling Policies Implemented in Ethernet Switch by Using Coloured Petri Nets The objective of this paper is to propose models enabling to study the behaviour of Ethernet switch for Networked Control Systems. Two scheduler policies are analyzed: the static priority and the WRR (Weighted Round Robin). The modelling work is based on Coloured Petri Nets. A temporal validation step based on the simulation of these modelling, shows that the obtained results are near to the expected behaviour of these scheduler policies. I. INTRODUCTION Since the 1980s, a great deal of research has focused on the problem of distributed control over networks and the so-called Networked Control System (NCS) has developed. The source of this enthusiasm can be traced to the many advantages gained by eliminating the restrictions of traditional point-to-point control architectures. As an alternative to the point-to-point architecture, the NCS offers more flexibility for reconfiguration (e.g. to achieve fault-tolerance). Research on fault-tolerance aspects of NCS is at a very early stage of development and as such is a new requirement. Today most studies on fault-tolerance only include the effects of faults at either component or local controller levels. The autonomous fault-tolerant NCS is a distributed system involving fault diagnosis and control at various local to global levels of system embedding. Implementation of the subsequent concepts can be achieved by using the technologies of wireless networks, embedded systems, nomad components, electronics tags, etc... Emphasis will be also placed on algorithms and procedures that facilitate the detection and isolation, at an early stage, of anomalies (variances or irregularities in the networks and/or in the system) and to switch to the fault-tolerant control strategy. In particular, no significant theory in fault management and autonomous operating conditions exists, and only a few tools are available. Considerable efforts are still needed to make the range of theoretical results or methods in the control field applicable to networked systems. Integrated solution offering a synergy between communications, and computation and control, representing a new area of study for fault diagnosis and fault-tolerant control has to be developed. In this context the aim of the NeCST (Networked Control System Tolerant to faults) project is to explore research opportunities in the direction of distributed control systems in order to enhance the performance of diagnostics and fault tolerant control systems. This leads to an improvement in the intensive use of NeCS technologies for the reactivity; autonomy and monitoring of large scale systems. The systems under consideration in the framework of this project can be considered as a distributed network of nodes operating under highly decentralised control, but unified in accomplishing complex system-wide goals. One of the key factors in designing such a complex system is that both the physical subsystem and the control part have to be designed together in an integrated manner. The figure 1 shows the global approach of the NeCST project. The networked control system is decomposed in two parts: the network and the (industrial) application, which has to be studied in closed way. The default detection is achieved in comparing the performance indicators provided by the real system (measured values) and by its model (estimated values). For network point of view, the communication information such as loss bandwidth, delay, and jitter can be monitored by using SNMP protocol and in a same time can be calculated from stochastic models or deterministic approaches. When the measured values and estimated values are different, it corresponds to one or more dysfunctions. The diagnostic has to enable to compensate the problem by using different mechanisms according to the importance of the fault: -The accommodation is used in line and consists in only changing software parameters (traffic priority, scheduling policy,…) -The reconfiguration temporarily stops the process for defining for example a new physical architecture in terms of network cabling, embedded components deployment,… -The degradation modifies the application parameters, interrupts non-vital tasks in order to maintain the main functions of the process. In the NeCST project, the communication system studied is the switched Ethernet architecture because it is more and more used in the Networked Control Systems. In this context, different works have been already investigated on the conceptual model of NeCST ( figure 1). Firstly, the network calculus theory is used to model the switched Ethernet architecture and to estimate the bounded end-to-end delays [3]. Secondly, the IEEE 1588 protocol (Precision Time Protocol) is implemented to synchronise the clocks of the different distributed devices. This enables to easily measure the delays. Thus when the measured delay is upper to the estimated bounded delay, it means that the network model does not correspond to the real network configuration and a fault is then detected (but not isolated). In previous paper [1], accommodation mechanisms have been studied at application level. In this case, the evolution of network delays (estimated or measured) is compensated with the parameters of control models such as smith predictor model or robust control theory. The problem is when the control model is not able to ensure the stability of the system due to a delay more important than the one originally estimated in the design of the NCS. In this case, a second level of compensating has to be considered and has to be applied on the communication system. It is the goal of this paper. The objective is to analyse several compensation mechanisms implemented inside the network devices (Ethernet switch) in order to optimise the traffic according to the applicative importance of the messages. It corresponds to the adjustment between the quality of service offered by the network and the quality of control required by the application. Some characteristics of Ethernet such as the nondeterministic access protocol and the propagation time of information in the network, raises questions on the capacity of these media to guarantee properties of determinism and reactivity of the control. Many works [2,3,4] showed that the use of switched Ethernet can be a solution to this problem. Protocols development [5] and new services offered by these technologies [6] gave place to many scientific works. This paper addresses the problem of performances evaluation of the internal architecture of an Ethernet switch. When the Ethernet switch performances are disturbed by an external event such as instrument failure or additional load of the incoming traffic, transmission delay may increase. The internal Ethernet switch architecture implements many complex mechanisms witch enable to ensure the exchanges of information between the nodes of application. Thus the communication behaviour representing all this complexity has to be modelled, verified and evaluated by using formal models. The formalization of the communication architecture requires models presenting the following attributes [7]: -Expression of parallelism, synchronization, the interaction, resource sharing; -Expression of the temporal (and stochastic) characteristics associated to the mechanisms; -Possibility of qualitative analysis (checking of the logic of the non-temporal and/or temporal mechanisms); -Possibility of quantitative analysis (performance evaluation and/or reliability). The approach presented in the paper is based on the well known Petri nets formalism. This formalism offers a framework well adapted and progressive for the representation and the analysis of the communications systems. The model is obtained from the temporal Petri nets which are well adapted to the formulation of the problem studied is this paper. In adding, this work presents a generic and modular approach which requires the use of "colouring" and hierarchy of the model. The temporal analysis of the hierarchical coloured temporal Petri net is obtained by simulation with CPNTools. This software developed by the Aarhus university to Denmark [8]. In the first section, the different components of the switch Ethernet are introduced. The second section is devoted to the explanation of the model structure. In this section, simulation results are presented for the following type of scheduler: -Static Priority, -Weighted Round Robin (WRR) The first one tends to generate famine for the weak priority packets, and the second one has to solve this problem. In the third section, the confrontation of the obtained models under a congestion situation of the switch is carried out. The results of simulation are discussed. Finally conclusions and perspectives are presented. II. ETHERNET SWITCH MODELLING A. Introduction A Switch is a complex system which includes different mechanisms and technologies. [9] and [10] decompose the switching architecture in three main functional components: -The queuing models refers to the buffering and the congestion mechanisms implemented in the switch, -The switching algorithm implementation refers to the decision making process within the switch (how and where a switching decision is made), -The switching fabric is the path that data take to move from one port to another. There are different ways to build up the switch architecture with each of these components. In this deliverable, only one specific switching architecture is considered: Cisco Catalyst 2900 XL. Its modelling is shown at the figure 1 and uses the elementary components defined below: One multiplexer and one queue to represent a switch using a shared memory (1&2), -One demultiplexer to model the switching step (3), -One demultiplexer for each output port (4) -As much buffer as defined priorities (up to 8) for each output port (6), -And one multiplexer for each output port defining the bandwidth used (7). The figure 2 shows the model of a switch which manages two priorities on the frames. Coloured Petri Nets (CPN), proposed by K.Jensen [11], is an extended version of classical Petri Net. In addition to places, transitions and tokens, the concepts of colours, guards and expressions are introduced so that computed data values can be carried by the tokens. This concept enables to introduce information (simple or complex) into the tokens. Moreover, the model of switch based on the CPN design tool is more compact. A Coloured Petri net is graphical oriented language for design, specification, simulation and verification of systems. It is in particular well-suited for systems in which communication, synchronisation and resource sharing is crucial problem. CPN has an intuitive, graphical representation that is make its use easy. Typical examples of application areas are communication protocols, distributed systems, automated production systems and others. Moreover, many works emerged to show the advantages of this kind of modelling [12,13,14,15,16]. Fig. 3. Tree structure of the model The model of the switch is built following a hierarchical and modular architecture. The structure of the model is represented in figure 3. Each node of the tree structure is a sub-model which corresponds to a substitution transition of the initial model. The figure 4, shows the root of the hierarchical representation of the architecture of the model. The source of traffic whose activity is modelled by the transition traffic source generates packets to the switch by producing tokens at the Ptr1 place. The switch transmits the packets generated by traffic source towards the consumer via the places Ptr2 and Ptr2 '. The places play the role of inputs/outputs for sub-models, the places Pbp1 and Pbp2 represents the availability of the output: when those are free, Pbp1 and Pbp2 are marked. C. Source modelling The colours associated to the tokens which represent the Ethernet packets are triplets: source equipment INP, destination equipment OUTP, and level of priority represented by PRIO. We have three levels of priorities for our model, obviously, other attributes can be added. Each triplet supports temporal stamp. Colorset packet = product INPOUTPPRIO timed. The colour ord (see the label (i,g) in the figure 5)also called variable inherits the colorset packet. This variable models the packets which cross the different sub-models of figure 4. (traffic_source, swicth and consumers). The packets are generated by the traffic source represented by the figure 5. It is a periodic source of period d (the period is equal to 5 in this figure). This period enables to model the sampling period of sensors and actuators. D. Ethernet switch modelling The packets thus generated by the traffic source to destination of the switch cross the different sub-modules constituting the Ethernet switch. The sub-model of the substitution transition of the switch is represented by the figure 6. Firstly, the packets cross the FIFO queue. Secondly, they cross the demultiplexer. The goal is to route the packets to the output port (defined by attribute OUTP). And according to the priority which is associated to the packets (defined by attribute PRIO), the packets are stored in the FIFO queue (there is as many queue as of priorities). Finally the scheduling policy defines in the scheduler, processes the different packets. FIFO model is represented in the figure 7. This algorithm processes the packets in the order of their arrival. The main advantage of this algorithm is its simplicity of implementation. On the other hand, it does not make any distinction between flows, i.e. it offers only one level of service (it is not adapted to guarantee the quality of service). To model a FIFO queue with CPNTools, we use the concatenation primitive . The set of the demultiplexers is represented by the model described in the figure 8. Transition DMULX models the first demultiplexing according to the destination attribute associated with each packet,. In the figure 8, two destinations are modelled O1, O2 (which correspond to output 1 and 2, respectively). Technically, this demultiplexing is defined by crossing conditions (or not) associated to the output arcs of the transition DMULX. The function of the transitions DMULX1 and DMULX2 is to route the packets according to their priorities. In this study, three priority levels are defined: H: high level of priority; M: average priority and b: low priority. In the same manner that the transition DMULX, the routing is carried out by conditions associated to the output arcs of the two transitions. Then, the FIFO queues which represent the different priority levels, receive the packets coming from the demultiplexers (DMULX1 and DMULX2). Finally, the packets stored in the FIFO queues are processed according to their priority, and also in taking into account the selected scheduling policies. In the next section, we model the static priority policy and Weighted Round Robin (WRR). E. Static priority modelling The model of the figure 9 describes the behaviour of a static priority scheduling. The type of packets is classified in three groups: -Packets of flows which have a priority higher than the other packets. These packets are called packets H and are modelled by tokens in the place Ptr4' 1. -The packets which belong to flows of priority lower than the latter one. They are modelled by tokens in the place Ptr4' 2. -The packets which belong to flows having the lowest priority. Fig. 9. Priority static scheduler model A packet with lowest priorities can be delayed by the other packets due to the non-pre-emptive characteristics of this kind of packet scheduling algorithms. This situation occurs when a higher priority packet arrives during the transmission of the packet with a low priority. Even if the packet has a higher priority it has to wait until the low priority packet is fully transmitted. The interest of the static priority algorithms is that it is easy to implement. But it manages the bandwidth in iniquity way. The consequence is the low priority packets could be not transmitted (famine problem) and could generated perturbations on the controlled systems. To eliminate the iniquity situation, the Fair Queuing algorithm is studied in the next section. F. Weighted Round Robin modelling The Fair Queuing (FQ) algorithm enables to share the bandwidth equitably between several flows. A weight is associated to each flow, defining the amount of bandwidth which must be allocated to it. The implementation of such algorithm is very complex. We propose to use a similar algorithm called the Round-Robin (RR) algorithms and easier to implement. RR Algorithms assign a queue to each flow. In a cycle way, the server visits the queues recurrently to process the packets on stand-by as service. The duration of the service is fixed by the service policy defined by RR algorithm. At the end of service allocated time, the server visits the following queue considered in the cycle. And, the same procedure starts again. The most known RR algorithms in practice are: -Weighted Round Robin (WRR) -Deficit Round Robin (DRR). The WRR has the capability to deal with the problem of iniquity which is the main problem addressed in this paper. WRR assigns a weight of Wi to each queue i which fixes the service quantity. For each visit of queue i, the server transmits the packets which are stored until either the queue is empty or the number of the served packets reaches wi. The figure 10 represents WRR algorithm for three priority levels. When the server visits the queue represented by Ptr4' 1 (place of reception of the high priority packets), the place of the type "E" get marked. For each visit of queue the number of packet transmission is limited to wi. The place of the "wfi" type counts the number of packets which have been served during the current visit of the server. This place is put at wi on each arrival of the server to the queue. Its marking is then decremented when each packet is transmitted (i.e. following the transition TH1 firing). When the server leaves the queue this place is set to zero by the transition firing which has a guard of [Lw<>nil] The server leaves the queue when the place Ptr4' 1 becomes empty, and/or when Wf (allocated time) is reached. When the service of this queue is finished, the server processes the next queue in the cycle (represented by the place Ptr4' 1 which receives the mean priority packets). If this place is empty, the server waits for new incoming packets in the buffer (Ptr4' 1). This step is necessary when the queues are empty to avoid that the transitions are always fired. In this case the graph is not bounded. The PN is as many duplicated once as there is various priority levels. The advantage the coloured Petri nets is to reduce the size of the initial PN by folding. The figure 10 represents the CPN without folding. G. Consumer modelling The packets are transmitted under the condition that the transmission line is free (the places Pbp1, Pbp2 must be marked).Then the packets are consumed by periodic consumers as shown in the figure 11. As we consider an Ethernet switch with two output ports, two consumers are modelled. Place C1 represents a counter of the consumed packets for the different priorities. The counter is incremented if the condition associated to the input arc of this place is verified. H. Conclusion The general model of an Ethernet switch is obtained by assembling the different sub-models described in this section. For example, the figure 12 (see the last page) describes a hierarchical model of an Ethernet switch implementing the static priority scheduler. II. SIMULATION OF ETHERNET SWITCH MODELS A. Introduction The objective of this section is to test 2-ports Ethernet switch model and to analyse the behaviour of the two schedulers: static priority and WRR. We have defined a scenario which collects six periodic source traffic and six periodic consumers. In order to simulate the model (see the figure 12), the six sources of periodic traffic (period d = 5 time unit) are folded in only one. Three periodic consumers (period d = 5 t.u) are associated with the first output port of switch. And three other periodic consumers with the same period are attached to the second one. For each output port the three periodic consumers are folded in only one. B. Static priority scheduler simulation The simulation of the model for 10.000 simulation steps and 565 t.u, give the results shown in the figure 13. it shows that 98,24 % of the generated packets of high priority are consumed, and only 0,87% of the mean priority packets are consumed and no packet of low priority is consumed. The end to end delay of high priority packets is 10 t.u and the standard deviation is nil. We can carry out a quantitative analysis according to the results obtained. The static priority scheduling processes immediately the high priority packet, at the expense of other lower priorities packets. The advantage of the static priority algorithm is to be easy to implement. Nevertheless, it shares the bandwidth in iniquity way. The static priority algorithm generates famine for the consumers which are waiting for the mean and low priority packets. The consequence of this problem could destabilize the process. Then, this model is simulated in the case where the switch is in a congestion state (figure 14). For this, for each source 2 packets with a period equal to 5 t.u are added on the previous scenario. The obtained results show that the packets are consumed with a delay (for example: we can observe in the figure 14 a delay upper to 200 t.u). In the worst case, some packets are not consumed such as packets 35, 36… The average value of the end-to-end delay is of 50,38 t.u. The standard deviation is equal to 34,24 t.u and generates jitters on the network. The packets with other priority are not served. C. WRR simulation The model was simulated for 10000 simulation steps and 565 t.u. At first we analyse the previous scenario without congestion in taking into account the WRR algorithm. With the same parameters attributed to the traffic sources and to the consumers. WRR algorithm has three queues corresponding to each priority. A weight is attached to a priority as follows: W1 (60%), W2 (30%), W3 (10%). The percentages represent the weights assigned to the high, mean, low priority packets, respectively. The figure 15 represents the results obtained with this method. The results show that 59,65 % of packets generated with high priority are consumed, and 28,94% of the generated of mean priority packet are consumed and 10,52% for the low priority packets. The end to end delay of the high priority packet is 11,70 t.u and the standard deviation is 2,64 t.u. For the mean priority packet, the end to end delay is 10 t.u and a standard deviation is nil. Finally, for the low priority packets the end to end delay is 9,44 t.u and the standard deviation is 1,66 t.u. The simulation results clearly show equity (fairness) of consumption between the various packets, according to the predefined weights. The main advantage of this algorithm is to avoid the famine phenomenon. The interest of WRR algorithm in the context of NCS is to be able to adjust the weights in considering the application constraints to avoid process instability. Secondly, we analyse the previous scenario with congestion. The figure 16 shows the switch is overloaded. Moreover, only 28,98% of the generated with high priority packets are consumed, 15,94% of mean priority packets and 5,79% of low priority packet are consumed. The end to end delays are 9,54 t.u, 10 t.u and 10 t.u, respectively. The standard deviation are 1,50 t.u, 0 t.u and 0 t.u for the high, mean and low priority packets, respectively. The interests of WRR algorithm in the congestion state are to minimise the end to end delay and the standard deviation in regard to the results obtained from the static priority algorithm. This paper proposes a general model to represent the behaviour of an Ethernet switch by using CPN. Two schedulers have been modelled and evaluated showing the interest to implement a WRR policy which solves the iniquity problem. In the future work, the goal of our research is to propose algorithms which enable to adjust the weights defined in the WRR algorithm in considering the time constraints of process. The benefit of this approach is to avoid instability situation in combining the quality of service offered by the network and the quality of control defined by the application. These algorithms will be embedded in each network devices to be able to compensate dynamically the impact of the delay on the networked control system. Finally, we are studying other scheduling policies such as the Earliest Deadline First (EDF) which is interesting to dynamically control the network [17]. Fig. 1 . 1Conceptual model of NeCST Fig. 2 . 2IEEE 802.1 P/Q switch model B. Ethernet communication modelling by using Coloured Petri Nets Fig. 4 . 4Root level of the model Fig. 5 . 5Periodic Fig. 6 . 6Ethernet switch modelThe output packets of the switch are consumed by a periodic consumer. Fig. 7 . 7FIFO queue model Fig. 8 . 8Demultiplexer model Fig. 10 . 10WRR model Fig. 11 . 11Periodic consumer model Fig. 14 . 14Static priority scheduler simulation with congestion Fig. 16 . 16WRR simulation with congestion III. Conclusion ACKNOWLEDGMENTThe authors wish to acknowledge the funding support for this research under the European Union 6th Framework Program contract n° IST -2004-004303 Network Controles Systems Tolerant to faults (NeCST). Control compensation based on upper bound delay in networked control systems. Jp, N Georges, E Vatanski, C Rondeau, S L Aubrun, Jämsä-Jounela, 17th International Symposium on Mathematical Theory of Networks Systems. KyotoJP. Georges, N. Vatanski, E. Rondeau, C. Aubrun, SL. Jämsä-Jounela "Control compensation based on upper bound delay in networked control systems", 17th International Symposium on Mathematical Theory of Networks Systems, Kyoto, July 2006 Performance evaluation of switched Ethernet for realtime industrial communications. K C Lee, S Lee, Computer Standards & Interfaces. 24K.C.Lee et S.Lee. Performance evaluation of switched Ethernet for real- time industrial communications. Computer Standards & Interfaces 24 (2002) 411-423. Conforting the performances of switched Ethernet network with industrial constraints by using the network calculus. J-P Georges, T Divoux, Et E Rondeau, International Journal of Communication systems. 18J-P.Georges, T.Divoux, et E.Rondeau. Conforting the performances of switched Ethernet network with industrial constraints by using the network calculus. International Journal of Communication systems 2005, 18:877-903. Analyse of switched Ethernet networks with different topologies used in automation systems. S Rüping, E Vonnhame, J Jasperneite, Proc. Of fieldbus technology conference (FeT'99). Of fieldbus technology conference (FeT'99)MagdeburgSpringer-VerlagS. Rüping, E. Vonnhame, and J.Jasperneite. Analyse of switched Ethernet networks with different topologies used in automation systems. Proc. Of fieldbus technology conference (FeT'99), 351-358, Magdeburg, Springer- Verlag, 1999. DP-Ethernet: the profibus DP protocol implemented on Ethernet. S Vitturi, Computer Communications. 26S.Vitturi. DP-Ethernet: the profibus DP protocol implemented on Ethernet. Computer Communications, 26:1095-1104, 2003. Sensor integration in industrial environment: from fieldbus to web sensors. A Flammini, P Ferrari, E Sisinni, D Marioli, A Taroni, Computer Standard & Interfaces. 252A.Flammini, P. Ferrari, E. Sisinni, D. Marioli and A.Taroni. Sensor integration in industrial environment: from fieldbus to web sensors. Computer Standard & Interfaces, 25(2): 183-194, 2003. Les réseaux de Petri étendus et méthodologie pour l'analyse de performances. In the book 'Méthodes exactes d'analyse de performance des réseaux. G Juanole, M Diaz, F Vernadat, IC2 Réseaux et Télécoms (Information -Commande -Communication), Edition Hermès science. G.Juanole, M.Diaz and F.Vernadat. Les réseaux de Petri étendus et méthodologie pour l'analyse de performances. In the book 'Méthodes exactes d'analyse de performance des réseaux'. IC2 Réseaux et Télécoms (Information -Commande -Communication), Edition Hermès science, 2004. CPN Tools for editing, simulating, and analysing Coloured Petri Net. A V Ratzer, L Wells, H M Larsen, M Laursen, J F Qvortrup, M S Stissing, M Westergaard, S Christensen, K Jensen, LNC. 2679A.V. Ratzer, L.Wells, H.M.Larsen, M.Laursen, J.F.Qvortrup, M.S.Stissing, M. Westergaard, S. Christensen, K.Jensen. CPN Tools for editing, simulating, and analysing Coloured Petri Net. LNC, 2679:450- 462, 2003. A guide to LAN switch architectures and performance. M Phipps, packet Cisco Systems user magazine. Cisco Systems4M.Phipps. A guide to LAN switch architectures and performance. In: packet Cisco Systems user magazine. Vol.4 pp.54-57. Cisco Systems.1999. Switch book. R Seifert, John Wiley & sonsR. Seifert. Switch book. John Wiley & sons 2000. Couloured Petri Nets. K Jensen, Springer-Verlag1K.Jensen. Couloured Petri Nets. Volume 1-2. Springer-Verlag, 1992. Specification and validation of an Edge router discovery protocol for mobile Ad Hoc networks. L M Kristensen, K Jensen, Springer-VerlagBerlin HeidelbergL.M.Kristensen et K.Jensen. Specification and validation of an Edge router discovery protocol for mobile Ad Hoc networks. Springer-Verlag Berlin Heidelberg 2004. Coloured Petri Nets based modelling and simulation of the static and dynamic allocation policies of the asynchronous bandwidth in the fieldbus protocol. A B Mnaouer, T Sekiguchi, Y Fujii, T Ito, H Tanaka, Lecture Notes in Computer Science. J. Billington, M. Diaz, G. Rozenberg1605Springer-VerlagApplication of Petri Nets to Communication NetworksA.B. Mnaouer, T. Sekiguchi, Y. Fujii, T. Ito et H. Tanaka. Coloured Petri Nets based modelling and simulation of the static and dynamic allocation policies of the asynchronous bandwidth in the fieldbus protocol. In: J. Billington, M. Diaz, G. Rozenberg (eds.): Application of Petri Nets to Communication Networks, Lecture Notes in Computer Science Vol. 1605, Springer-Verlag 1999, 93-130. Experience with modelling TCP's connection management procedures with CPNs. B Han, J Billington, Proceedings of the Fifth Workshop and Tutorial on Practical Use of Coloured Petri Nets and the CPN Tools. K. Jensenthe Fifth Workshop and Tutorial on Practical Use of Coloured Petri Nets and the CPN ToolsDepartment of Computer Science, University of AarhusB. Han et J. Billington. Experience with modelling TCP's connection management procedures with CPNs. In: K. Jensen (ed.): Proceedings of the Fifth Workshop and Tutorial on Practical Use of Coloured Petri Nets and the CPN Tools, October 2004, Department of Computer Science, University of Aarhus, 57-76. An Evaluation of network response time using a Coloured Petri Net model of switched LAN. D A Zaitsev, Proceedings of the Fifth Workshop and Tutorial on Practical Use of Coloured Petri Nets and the CPN Tools. K. Jensenthe Fifth Workshop and Tutorial on Practical Use of Coloured Petri Nets and the CPN ToolsDepartment of Computer Science, University of AarhusD.A. Zaitsev. An Evaluation of network response time using a Coloured Petri Net model of switched LAN. In: K. Jensen (ed.): Proceedings of the Fifth Workshop and Tutorial on Practical Use of Coloured Petri Nets and the CPN Tools, October 2004, Department of Computer Science, University of Aarhus, 157-166. Evaluation of realtime capabilities of Ethernet-based automation systems using formal verification and simulation. Ecole d'été Temps Réel. G Marsal, D Witsch, B Denis, J.-M Faure, G Frey, NancyG.Marsal, D.Witsch, B.Denis, J.-M.Faure et G.Frey. Evaluation of real- time capabilities of Ethernet-based automation systems using formal verification and simulation. Ecole d'été Temps Réel , Nancy, Septembre. ETR'2005. Real time communication for industrial embedded systems using Switched Ethernet. H Hoang, Proceedings of the 18 th International Parallel and Distributed Processing Symposium (IPDPS'4). the 18 th International Parallel and Distributed Processing Symposium (IPDPS'4)H. Hoang. Real time communication for industrial embedded systems using Switched Ethernet. Proceedings of the 18 th International Parallel and Distributed Processing Symposium (IPDPS'4) , 2004. Hierarchical model of Ethernet switch. Fig, Fig. 12. Hierarchical model of Ethernet switch	[]
[ "The Dynamical Evolution of Black Hole-Neutron Star Binaries in General Relativity: Simulations of Tidal Disruption", "The Dynamical Evolution of Black Hole-Neutron Star Binaries in General Relativity: Simulations of Tidal Disruption" ]	[ "Joshua A Faber \nDepartment of Physics\nUniversity of Illinois at Urbana-Champaign\n61801UrbanaIL\n", "Thomas W Baumgarte \nDepartment of Physics and Astronomy\nBowdoin College\n04011BrunswickME\n", "Stuart L Shapiro \nDepartment of Physics\nUniversity of Illinois at Urbana-Champaign\n61801UrbanaIL\n", "Keisuke Taniguchi \nDepartment of Physics\nUniversity of Illinois at Urbana-Champaign\n61801UrbanaIL\n", "Frederic A Rasio \nDepartment of Physics and Astronomy\nNorthwestern University\n60208EvanstonIL\n" ]	[ "Department of Physics\nUniversity of Illinois at Urbana-Champaign\n61801UrbanaIL", "Department of Physics and Astronomy\nBowdoin College\n04011BrunswickME", "Department of Physics\nUniversity of Illinois at Urbana-Champaign\n61801UrbanaIL", "Department of Physics\nUniversity of Illinois at Urbana-Champaign\n61801UrbanaIL", "Department of Physics and Astronomy\nNorthwestern University\n60208EvanstonIL" ]	[]	We calculate the first dynamical evolutions of merging black hole-neutron star binaries that construct the combined black hole-neutron star spacetime in a general relativistic framework. We treat the metric in the conformal flatness approximation, and assume that the black hole mass is sufficiently large compared to that of the neutron star so that the black hole remains fixed in space. Using a spheroidal spectral methods solver, we solve the resulting field equations for a neutron star orbiting a Schwarzschild black hole. The matter is evolved using a relativistic, Lagrangian, smoothed particle hydrodynamics (SPH) treatment. We take as our initial data recent quasiequilibrium models for synchronized neutron star polytropes generated as solutions of the conformal thin-sandwich (CTS) decomposition of the Einstein field equations. We are able to construct from these models relaxed SPH configurations whose profiles show good agreement with CTS solutions. Our adiabatic evolution calculations for neutron stars with low-compactness show that mass transfer, when it begins while the neutron star orbit is still outside the innermost stable circular orbit, is more unstable than is typically predicted by analytical formalisms. This dynamical mass loss is found to be the driving force in determining the subsequent evolution of the binary orbit and the neutron star, which typically disrupts completely within a few orbital periods. The majority of the mass transferred onto the black hole is accreted promptly; a significant fraction (∼ 30%) of the mass is shed outward as well, some of which will become gravitationally unbound and ejected completely from the system. The remaining portion forms an accretion disk around the black hole, and could provide the energy source for short-duration gamma ray bursts.	10.1103/physrevd.73.024012	[ "https://arxiv.org/pdf/astro-ph/0511366v1.pdf" ]	5,564,770	astro-ph/0511366	af7940e90686d58d7d8df2e5a32137b529b874ed	The Dynamical Evolution of Black Hole-Neutron Star Binaries in General Relativity: Simulations of Tidal Disruption 11 Nov 2005 Joshua A Faber Department of Physics University of Illinois at Urbana-Champaign 61801UrbanaIL Thomas W Baumgarte Department of Physics and Astronomy Bowdoin College 04011BrunswickME Stuart L Shapiro Department of Physics University of Illinois at Urbana-Champaign 61801UrbanaIL Keisuke Taniguchi Department of Physics University of Illinois at Urbana-Champaign 61801UrbanaIL Frederic A Rasio Department of Physics and Astronomy Northwestern University 60208EvanstonIL The Dynamical Evolution of Black Hole-Neutron Star Binaries in General Relativity: Simulations of Tidal Disruption 11 Nov 2005(Dated: October 27, 2018)numbers: 0430Db0425Dm4711+j9585Sz We calculate the first dynamical evolutions of merging black hole-neutron star binaries that construct the combined black hole-neutron star spacetime in a general relativistic framework. We treat the metric in the conformal flatness approximation, and assume that the black hole mass is sufficiently large compared to that of the neutron star so that the black hole remains fixed in space. Using a spheroidal spectral methods solver, we solve the resulting field equations for a neutron star orbiting a Schwarzschild black hole. The matter is evolved using a relativistic, Lagrangian, smoothed particle hydrodynamics (SPH) treatment. We take as our initial data recent quasiequilibrium models for synchronized neutron star polytropes generated as solutions of the conformal thin-sandwich (CTS) decomposition of the Einstein field equations. We are able to construct from these models relaxed SPH configurations whose profiles show good agreement with CTS solutions. Our adiabatic evolution calculations for neutron stars with low-compactness show that mass transfer, when it begins while the neutron star orbit is still outside the innermost stable circular orbit, is more unstable than is typically predicted by analytical formalisms. This dynamical mass loss is found to be the driving force in determining the subsequent evolution of the binary orbit and the neutron star, which typically disrupts completely within a few orbital periods. The majority of the mass transferred onto the black hole is accreted promptly; a significant fraction (∼ 30%) of the mass is shed outward as well, some of which will become gravitationally unbound and ejected completely from the system. The remaining portion forms an accretion disk around the black hole, and could provide the energy source for short-duration gamma ray bursts. I. INTRODUCTION The infall of compact objects into black holes (BHs) is of considerable interest in many branches of astrophysics. In particular, many of the arguments that can be made about coalescing neutron star-neutron star (NSNS) binaries also apply to coalescing black hole-neutron star (BHNS) binaries. Both are strong candidates for the central engines of short-duration gamma ray bursts (GRBs), since the merger timescale following tidal disruption is comparable to the GRB duration and the gravitational binding energies provide the characteristic energy scales inferred by observers [1,2]. It is possible that any ejected matter may contribute significantly to the r-process elemental abundance of the universe [3,4,5]. Additionally, they are expected to be among the most important sources of gravitational waves (GWs) that can be detected by both terrestrial laser interferometers such as LIGO [6], VIRGO [7], GEO [8], and TAMA [9], as well as the proposed space-based interferometer LISA [10]. The key difference between the sources that can be observed with LIGO (and comparable detectors) and LISA is the characteristic frequency of the GW emission: LISA's characteristic frequency range falls within 10 −4 − 10 −1 Hz, whereas LIGO operates between 10 − 500 Hz. Because of this, LIGO is most sensitive to the mergers of stellar-mass BHs, whereas LISA will observe more massive merging systems that involve either intermediate mass BHs (IMBHs), M BH = 10 2 − 10 4 M ⊙ , or supermassive BHs (SMBHs), M BH > 10 5 M ⊙ . The formation history leading to these encounters is likely to involve completely different processes. Compact binaries with stellar-mass BHs are likely to be formed through typical stellar binary evolution, at rates that depend on parameters such as the binary mass ratio distribution, common-envelope efficiency, and the physics of supernova kicks, all of which remain somewhat uncertain (see [11,12,13] and references therein for a thorough review). The mass distribution of BHs in such systems is poorly constrained, as none have been observed to date, but may vary widely, spanning a range 2M ⊙ < M BH < 25M ⊙ [14]. For sufficiently tight binaries, merger will occur within a Hubble time. In these cases, the dissipative effects of gravitational radiation will cause the orbit to circularize as the binary separation shrinks, so that the eccentricity of the orbit is expected to be almost zero by the time the binary enters the LIGO band. Whether or not the compact object is tidally dis-rupted by its BH companion, as well as where this would occur in the latter case with respect to the Innermost Stable Circular Orbit (ISCO), depends on both the compaction of the compact object and the mass ratio (see Section II). This simple picture does not apply to compact objects orbiting BHs with considerably higher mass. Both IMBHs and SMBHs are expected to reside within stellar clusters, whose dynamics will be determined by both stellar-BH and stellar-stellar gravitational encounters (scattering). Some stars will typically be scattered, either strongly or weakly, into the "loss cone", i.e., the volume of phase space encompassing orbits with sufficiently small periastrons that the star will be tidally disrupted before being kicked into another orbit by future encounters (see [15] for a review of the original derivations, and [16,17] for more recent work). As a result, most objects that enter the loss cone do so at very high eccentricity, with periastron distances of 5−50 Schwarzschild radii [18,19]. In many cases, these systems will approach the BH with eccentricities e > ∼ 0.1 [20]. GW detections from coalescence with higher mass BHs may yield very little information about the physics of NS matter, for the case of a NS falling into an IMBH, or any compact object (BH, NS, or white dwarf) falling into an SMBH with M > ∼ 10 6 M ⊙ . These objects should plunge through the ISCO of the BH intact, since the tidaldisruption radius lies within the ISCO, and will likely be swallowed whole by the BH. For the opposite case, applicable to white dwarfs (WD) falling into IMBHs (and NSs into stellar-mass BHs), tidal disruption will occur outside the ISCO, a process we describe in detail in Sec. II. For the vast majority of its lifetime, a stellar-mass compact object binary will inspiral very slowly, such that it can be described by a point-mass, post-Newtonian (PN) treatment. PN formalisms for the adiabatic inspiral epoch are now completely determined up to 3.5PN order [21], and include lowest-order spin-orbit and spinspin terms [22,23]. Once finite-size and tidal effects become important at close separation, it becomes necessary to solve the fully nonlinear Einstein field equations. Quasi-equilibrium binary configurations in circular orbits have been calculated in GR for NSNS [24,25,26,27,28,29,30], BHNS (see [31], hereafter BSS; [32], hereafter TBFS, and references therein), and BHBH binaries ( [33,34,35,36]; for a thorough review of the topic and references, see [37,38]). Details of the transition from slow inspiral to rapid plunge, and deviations from the point-mass energy versus frequency relation found in quasi-equilibrium sequences, may yield important information about the physical parameters of the NS equation of state (EOS; see, e.g., [24,39,40]). It has been suggested [41] that a combination of 10 − 50 broadband and narrowband observations of NSNS mergers might be able to constrain the NS radius to within a few percent. We will show below that BHNS mergers may be just as interesting, but it is likely that the interpretation of physical features in the GW signal will be significantly more complicated, since differences between stable and unstable modes of mass transfer may lead to radically different scenarios. Eventually, for those systems in which the tidal limit is reached before the ISCO, mass transfer onto the BH will begin. This process is fundamentally dynamic in nature, and can only be modeled accurately by relativistic, three-dimensional hydrodynamic calculations. Attempts have been made to model these systems analytically, but as we will show below, the conclusions rely on a number of unphysical assumptions. The earliest work describing mass transfer in detail for compact object binary mergers [42] assumed that mass transfer in NSNS binaries would conserve both mass and orbital angular momentum, and that both NSs would remain on a quasi-circular orbit in corotation during the evolution; a similar set of assumptions was used to describe BHNS binaries as a possible source of gamma-ray bursts [43]. A more complex treatment developed in [44] drops the assumption of circularity, since it is not seen to hold in numerical calculations (e.g., [4]). Still, their model for the evolution of BHNS systems undergoing mass transfer depends on a number of ad hoc assumptions that need to be tested by dynamical calculations in order to be proven valid. Beyond uncertainties about the form of the lateinspiral GW signal produced by a BHNS merger, there remains the question of the event rate, which remains uncertain given the complete lack of detection of such systems to date. Still, it is possible to estimate the likely merger rate using population synthesis models, which can be calibrated against the observed galactic NSNS binary population and supernova rates. Recent estimates predict an advanced LIGO annual detection rate of anywhere from a few mergers [13] up to potentially several hundred [12]. Should a BHNS binary merger be observed, it might reveal a great deal about the physics of matter at nuclear densities. In particular, the onset of mass transfer would yield a clear indication about the NS radius, and, as we will explain in detail below, the stability of the mass transfer would yield important information as to the nuclear EOS. Whereas for NSNS binaries the characteristic frequencies of GW emission during the merger and formation of a remnant (either a hypermassive NS or BH) will typically occur at frequencies outside the peak sensitivity of even an advanced LIGO detector, the same is not true for BHNS binaries. Since the frequency at the onset of instability scales roughly inversely with the total binary mass, we expect stellar-mass BHNS mergers to occur at characteristic frequencies at which LIGO will be most sensitive, ∼ 100 − 500 Hz. If the GW signal from a merger was observed to be coincident with a short-duration gamma-ray burst, we could potentially determine their distance, luminosity, and characteristic beaming angle [45]. A detailed theoretical understanding of these systems is now more urgent than ever, in light of the recent localizations of short GRB afterglows [46,47,48,49,50,51], the first ever for these systems (many long-duration GRBs have been localized, but are believed to be the result of collapsing stars, not merging compact binaries). Unfortunately, the current state-of-the-art for hydrodynamic calculations of BHNS inspiral and merger is far behind that for NSNS mergers. Calculations of the latter have been performed using a variety of Newtonian, PN, and relativistic gravitational formalisms (see, [38,52] for thorough reviews, and [53], hereafter FGR, for a more recent summary). Many calculations have now been performed in either the conformal flatness (CF) approximation to general relativity (GR) [53,54], or in full GR [55,56,57,58]. These GR calculations now include sophisticated treatments of the NS EOS and physically appropriate initial spins ( [58]; NSs are expected to be nearly irrotational in the inertial frame, since the viscous timescale is much longer than the inspiral timescale, see [59,60] and Sec. II below). The key difficulty that must be overcome to perform simulations of relativistic BHNS mergers is the same one that arises in the study of BHBH binaries; the presence of a spacetime singularity inside the black hole. To avoid encountering the singularity in a numerical simulation, the BH interior is excised from the computational grid in most current applications. This is justified by the fact that no information can propagate from the BH interior to the exterior, so the exterior can be evolved independently of the interior. While progress has been reported, especially very recently [61], black hole evolution calculations have been plagued by numerical instabilities. In some ways, BHNS mergers are even more difficult to evolve consistently, since both the singular behavior of the BH as well as the hydrodynamic nature of the NS must be confronted. Whereas the BHBH problem involves a pure vacuum solution of the Einstein field equations, the NS must always be evolved in such a way that the relativistic fluid is treated properly. As a result of these difficulties, all hydrodynamic calculations performed to date of stellar-mass BHNS mergers have used Newtonian or quasi-Newtonian gravitational treatments [1,5,62,63,64,65,66]. Needless to say, binaries containing a BH can be evolved accurately only by using relativistic hydrodynamics in a relativistic spacetime. We emphasize here that this applies both to the tidal field created by the BH, as well as the self-gravity of the NS. Previous Newtonian calculations have in some cases [64,66] used an approximate black hole potential, suggested in [67], that creates an ISCO at 6M BH , but no single static potential can generate the full set of relativistic forces experienced by matter in the strongfield regime. Calculations employing a fixed background BH metric have typically been performed for stars undergoing a tidal interaction with a massive BH, rather than a stellar-mass BH, with relativistic dynamical terms but a Newtonian treatment of the self-gravity. The secondary, in fact, is often assumed to be a white dwarf or main-sequence star. These models include SPH treatments without self-gravity [68], and both PPM [69] and spectral method [70] treatments with Newtonian selfgravity. More recently, SPH techniques have been devised that evolve the NS matter in the background metric of a stellar-mass BH, using Newtonian-order correction to model the NS self-gravity, for both SPH [71,72] and characteristic gravity [73]. This approach is appropriate for describing main-sequence stars or white dwarfs. However, since the tidal disruption is a result of a competition between the black hole tidal force and stellar self-gravity, this approach is not sufficient to describe BHNS binaries accurately. Modeling tidal disruption in BHNS binaries requires a relativistic treatment of both the black hole and the neutron star. Here, we will make use of the CF approximation to GR, introduced by Isenberg [74] and Wilson and collaborators [75]. The CF approximation amounts to assuming that the spatial metric remains conformally flat, so that the gravitational fields can be found by solving the constraint equations of GR, decomposed in the conformal thinsandwich (CTS) decomposition [76], alone. The CTS formalism has been used in numerous applications to construct initial data describing both NSNS and BHNS binaries in quasiequilibrium [24,25,26,27,28,29,30,31,32]. For these initial data the choice of a conformal background metric is completely consistent with Einstein's initial value (constraint) field equations, although different choices may describe the astrophysical situation at hand more or less accurately. The situation is different for dynamical simulations in the CF approximation (e.g. [53,54,75,77,78]), since the assumption that the spatial metric remains conformally flat is no longer strictly consistent with Einstein's field evolution equations. For many applications, however, CF provides an excellent approximation. For spherically symmetric configurations, as an example, the CF approach is exact, and for many other applications the error has been shown to be in the order of at most a few percent (see, e.g., [79]). It is particularly useful for exploring dynamical behavior, e.g., collapse or tidal break-up, which occurs on dynamical timescales and is unaltered by secular effects like gravitational radiation-reaction. In this paper we present the dynamical extension of BSS, who calculated the first relativistic, quasiequilibrium BHNS sequences as solutions of the CTS decomposition of the Einstein field equations. Modifying their code to treat the metric for the Schwarzschild BH in isotropic (CF) coordinates, rather than the Kerr-Schild coordinates reported in BSS, we take their corotating quasiequilibrium configurations as initial data. As in BSS, we assume an extreme mass ratio, M BH ≫ M NS , which allows us to hold the BH position fixed and restrict the computational grid to a neighborhood of the NS, thereby avoiding complications arising in the BH interior. We also assume a polytropic equation of state for the neutron star, as well as synchronous rotation. The resulting dynamical calculations are the first of their kind to solve the CTS field equations for the spacetime around the NS self-consistently by treating both the NS and BH relativistically. They allow us to study details of the dynamical mass-transfer process, particularly its stability. The CF approximation holds a stable equilibrium configuration constructed in the CTS formalism in strict dynamical equilibrium. Our calculation is a prototype of more detailed general relativistic calculations we hope to provide in the future that will involve irrotational NS models with more realistic EOSs and compactions, arbitrary mass ratios, and a fully self-consistent treatment of the spatial metric. In the CF approximation, gravitational radiation reaction must be added in by hand in order to drive the system toward merger. While it is the secular energy losses to gravitational radiation that initially drive the binary system toward the point of tidal disruption, they play a much reduced role in the dynamics thereafter. Indeed, while secular forces determine the path the binary takes prior to merger, the merger itself is a fundamentally dynamical process, as we discuss in great detail below. Our work is organized as follows. In Sec. II we discuss the important physical scales that define our problem, and present a detailed treatment of the traditional picture for determining the stability of mass transfer. We then discuss the limitations of this model, and explain why it may not be applicable for BHNS mergers. In Sec. III we describe our numerical methods, including the details of both our implementation of the CF field equations as well as our use of smoothed particle hydrodynamics (SPH) techniques to evolve the fluid configuration. In Sec. IV we compare our relaxed initial data to previous quasiequilibrium models, and find that we can construct configurations that satisfy the field equations to high accuracy while reproducing previous results. In Sec. V we present our simulations of merging binaries for different models of the NS polytropic EOS. Finally, in Sec. VI we discuss our results in the context of GW astrophysics, and describe our plans for further calculations. II. PHYSICAL OVERVIEW The evolution of binaries containing NSs is a fully relativistic problem, since lowest-order PN approximations break down in the strong gravitational fields present during late stages of the merger. However, we can use information from Newtonian and quasi-Newtonian calculations to estimate the various timescales and physical regimes we expect to encounter. Thus, we first classify the relevant physical scales we expect to encounter in our study of BHNS binary evolution, and then generalize the standard model for stable, binary mass transfer to relativistic stars. In doing so, we will explain why this model, variants of which have been used previously to describe the evolution of compact binaries, is unlikely to apply to BHNS mergers. Our simple mass-transfer model does have physical relevance, as it can apply to the case of a WD inspiraling in a nearly circular fashion toward an IMBH in a globular cluster. Such a star will begin transferring mass long before reaching the ISCO. However, since WDs are typically kicked into highly eccentric orbits prior to interactions with the BH, the orbit may not have time to circularize fully before the onset of mass transfer. In such cases, the binary evolution will be more complicated than the scenario we consider here; it has been studied before by several groups (see [16,80], and references therein). A. Units, Timescales and Characteristic Lengths The four most important timescales characterizing the problem at hand are the NS dynamical timescale t D , the viscous timescale t v , the orbital timescale T , and the GW radiation-reaction timescale t GW . Throughout this paper, we set G = c = 1. The BH and NS masses can be written in terms of the initial mass ratio q as M BH = q −1 M NS or equivalently M NS = qM BH , and the NS radius R NS = C −1 M NS = qC −1 M BH , where C ≡ M NS /R NS is the compactness parameter. The dynamical timescale of the NS is given by t D ≡ 2R 3 NS M NS = 2 1/2 C −1.5 M NS = 2 1/2 qC −1.5 M BH . (1) We wish to compare this with the orbital and radiationreaction timescales at the radius where Roche-lobe overflow will begin. To estimate this radius, we will use the approximate form proposed in [81], R r = 0.46a q 1 + q 1/3 ,(2) which gives the (volume-averaged) Roche-lobe radius as a function of the mass ratio and binary separation a, for a binary treated as a pair of point masses in the corotating frame. This definition differs from the original definition of the Roche lobe, which was defined for incompressible matter (point masses are in effect infinitely compressible), but the physical scalings are the same; the coefficient becomes 0.41 for incompressible matter instead of 0.46. The Roche-lobe radius is equal to the NS radius at a separation a R = 2.17q −1/3 (1 + q) 1/3 C −1 M NS = 2.17q 2/3 (1 + q) 1/3 C −1 M BH ,(3) at which point the (Keplerian) orbital period is T ≡ 2π a 3 R M T = 20.1(1 + q) 1/2 C −1.5 M NS = 20.1q(1 + q) 1/2 C −1.5 M BH ∼ 14t D ,(4) where M T ≡ M NS + M BH is the total binary mass, and the last relation holds for q ≪ 1, or equivalently, M T ≃ M BH . We note that in this limit, the orbital period is a fixed multiple of the NS dynamical timescale, regardless of the properties of the NS. For a point-mass binary on a circular orbit, lowest order radiation reaction predicts that the binary will inspiral, losing energy and angular momentum, on a characteristic timescale t GW given by t GW = ȧ a = Ė E = J 2J = 5 64 a 4 M NS M BH M T = 4τ,(5) where τ is the coalescence time, i.e., the remaining time until the point-mass binary would reach a = 0. At the Roche-lobe separation, assuming q ≪ 1, we have t GW = 1.73q 2/3 C −4 M NS = 1.73q 5/3 C −4 M BH = 1.23q 2/3 C −5/2 t D . (6) In general, the radiation-reaction timescale will be at least an order of magnitude longer than the dynamical timescale for any binary which begins mass transfer outside the ISCO radius. The orbital period T , however, may become similar to the coalescence time t GW , indicating that the infall becomes quite rapid. In Fig. 1, we show the regions in parameter space for which the critical separation for Roche-lobe overflow lies within or outside of the ISCO, for a wide variety of compact object-BH binaries. Dashed vertical lines correspond to the approximate compactness of either a WD or a NS, whereas dotted horizontal lines show the approximate mass ratios to be expected for a 10M ⊙ stellar-mass BH, an IMBH, or a SMBH. Systems above the critical curve reach the tidal limit before the ISCO, and are likely to transfer mass onto the BH. For those sufficiently below the curve, we expect that the compact object will pass through the ISCO intact, and plunge onto the BH relatively intact. We note however, that this simple picture may very well be altered by a number of more complicated effects. Recently, Miller [82] has argued that even if systems are expected to reach the mass-shedding limit prior to crossing the ISCO, in many cases they will have already begun to plunge. Indeed, since the binary energy as a function of separation flattens out significantly near the ISCO, t GW , which is already nearly of order T , will systematically underestimate the infall velocity (a similar argument was used in [40] to argue that the GW energy spectrum produced in NSNS mergers declines dramatically near the ISCO). Because of this, the ISCO may systematically underestimate the binary separation at which prompt merger becomes inevitable. On the other hand, describing the "plunge" of an extended object like a NS may provide a misleading picture of the dynamical merger in cases where the tidal disruption occurs inside the ISCO but outside the horizon. While it is certain that some matter, likely a significant fraction of the NS mass, will plunge inward directly onto the BH, this may liberate a great deal of angular momentum into the outer parts of the NS [83]. As a result, some fraction of the mass may survive the plunge, at least initially, in the form of a "mini-NS", which will escape outside the ISCO on an elliptical orbit. Needless to say, only dynamical calculations will clarify the role played by these competing effects. The final timescale we must consider is the viscous damping timescale t Vis , which we expect to play a crucial role in determining the fate of the binary once mass transfer commences. In the limit that the viscous timescale is extremely short (high viscosity), we expect two important phenomena to occur. First, tidal dissipation can synchronize the binary so that the secondary is corotating upon the onset of mass transfer. From [59], we note that a binary will be synchronized by the time mass transfer begins only if β ≡ R NS t Vis > ∼ 60(1 + q) 5/3 q −2/3 C 3 ,(7) t Vis < ∼ 1 60 (1 + q) −5/3 q 2/3 C −4 M NS < ∼ 1 60 (1 + q) −5/3 q 5/3 C −4 M BH ,(8) where β is the ratio of the light crossing time of the secondary to its viscous timescale, as defined by [59]. If we follow [84] and assume that turbulent viscosity is the primary damping mechanism, we can define α Vis , the turbulent viscosity parameter, so that α Vis ≡ t D t Vis .(9) We see that synchronization will occur if α Vis > ∼ 60(1 + q) 5/3 q −2/3 C 5/2 .(10) On Fig. 1, we show curves marking the critical mass ratiocompactness dependence for β = 1, which we define as the "causal limit", as well as for α Vis = 0.1, which is the maximum plausible value for turbulent viscosity in physical systems of interest [85]. Configurations to the left of the curve can synchronize before merger; this includes essentially all mergers where the secondary is either an MS star or a WD. NS mergers, on the other hand, will be irrotational in general, especially when the primary is a BH, since the required viscosity to synchronize the NS increases as the primary mass increases [59,60]. Viscosity should also play a role after mass transfer starts, as we will discuss in detail below. B. The stability of mass transfer Once the secondary fills its Roche lobe, it will begin to transfer mass onto its companion. Such a process can be either stable or unstable, depending on its response to mass loss. If the volume of the Roche lobe shrinks faster than (or expands slower than) the stellar radius, the process is unstable, and the star will typically be disrupted violently. On the other hand, if a small amount of mass loss causes the star to shrink back within the Roche lobe, it is possible for the mass loss to temporarily cease, or at the very least settle down to a much smaller equilibrium level, whose value can be determined based on the assumptions made about conservation of mass and angular momentum, as we discuss below. We first note that models of stable mass transfer typically assume that the binary orbit remains quasicircular, which in turn is only possible if the viscous timescale is short relative to the orbital and GW timescales. Maintaining a circular orbit requires that the orbital energy evolve according to a fixed relation in terms of the orbital angular momentum and the mass of the secondary, but there is no reason to assume that such a relation should hold a priori. Indeed, it is viscous dissipation that drives the orbit toward circularity, by converting excess orbital energy into other forms. Furthermore, when the viscous timescale is long, the mass-transfer rate can grow extremely rapidly, since the inner Lagrange point travels into the secondary at roughly v in R N S /a R , unbinding progressively denser material from the NS. If this leads to an unstable runaway, it is the mass loss that drives the orbital evolution, and we expect to find the development of an orbital eccentricity. This violates the typical assumptions made in conservative mass-transfer models, which assume that mass loss is steady, and slow enough that the orbit can remain circular as mass is lost. The early attempt to follow the mass transfer process in detail for NSNS binaries was provided by [42], who modeled the heavier NS as a point mass, and the lighter secondary using an EOS that yields a nearly flat massradius relation down to M NS ∼ 0.3M ⊙ , below which the NS begins to expand rapidly with further decreasing mass. Rather than assume conservative mass transfer, they parameterized the possible loss of both mass and angular momentum from the system, finding that the former has very little effect on their results. In their model, mass transfer leads to a widening of the binary orbit, under the condition that the NS radius must equal the radius of its Roche lobe. As mentioned above, this will only hold for systems in which t Vis ≪ t GW . Over time, the mass loss rate and GW luminosity decrease rapidly from their large initial values at closest approach (as does the rate of neutrino production as the NS matter decompresses during the transfer), until eventually the low-mass NS begins to expand rapidly [86] and unstable mass transfer begins. Many of these ideas were revisited for a discussion of BHNS binaries in [43], in light of the optical identification of GRB counterparts at cosmological distances. Assuming a Newtonian n = 1.5 polytropic EOS and fully conservative mass transfer, they find that the initial masstransfer rate between a 1.4M ⊙ NS and a BH with mass M BH = 3 − 5M ⊙ will occur at a rate of ∼ 100M ⊙ s −1 for approximately 1 ms before decaying away according to the approximate power-law relationṀ NS ∝ t −14/11 , which corresponds to M NS (t) ∝ t −3/11 . As in [42], they assume that the process will terminate when the NS reaches a critical minimum mass and begins to expand unstably. The most recent treatment of BHNS coalescence makes a completely different set of assumptions about the dynamics during mass transfer. Based on the Newtonian BHNS numerical calculations of Rosswog, Speith, and Wynn [5], Davies, Levan, and King [44] assume that the rapid timescale for mass transfer will violate the assumption of circular orbits, which underlies the typical conservative, quasiequilibrium mass-transfer formulation. Instead, they make the following assumptions: 3. Half of the angular momentum lost to the transferred mass will return to the NS, placing it on an eccentric orbit that will typically not lead to overflow during the next periastron passage. This model does reproduce well the extremely high mass loss rates initially seen during Newtonian numerical calculations of BHNS mergers [1,5,63], but the assumptions adopted are somewhat ad hoc. In particular, transferring angular momentum back to the NS without adjusting its mass causes a discontinuous evolution of the binary orbit. In some cases, the NS will find itself on an orbit whose periastron is outside the mass-shedding limit, leading to a period of stable evolution until GW dissipation forces the orbit to decay inward again back to the onset of mass transfer. In contrast, we find below that mass transfer can be quenched temporarily, but from this point on the NS follows an elliptical trajectory that will take it back within the mass-shedding limit prior to the next periastron passage. In Appendix A, we derive a semi-analytic formulation for conservative mass transfer onto a BH, modeling secondaries either by a Newtonian or a relativistic polytrope. We recover the scaling relations found in [42,43], and generalize them for arbitrary polytropic indices. Although these relations are unlikely to hold for merging BHNS systems, as shown in Fig. 1, the Newtonian results can be applied to merging WDs as well as main sequence stars undergoing mass transfer. We note that there are semi-analytic formalisms for describing non-conservative mass transfer as well (see, e.g. [87] for a formalism involving mass transfer from a main sequence star onto a companion), and that these have been useful in describing WDBH mergers [88], but that the actual NS tidal disruption process is sufficiently dynamic that essentially all analytic treatments break down. The theory of accretion disk dynamics presents several interesting connections to that of merging binaries, since questions about the stability of mass transfer appear as well (see, e.g., [89] and references therein). One key difference between the models is the typical radial angular momentum distribution; parameterizing the tangential velocity profile as v t (r) ∝ r α , irrotational NS have a nearly flat velocity profile, α ∼ 0, and corotating NS a flat angular velocity profile, α = 1, both larger than the Keplerian value α K = −0.5. Moreover, NS differ greatly from disks because of their infall velocity when they pass through the ISCO, and their strong self-gravity. While angular momentum distributions with larger values of α help to stabilize mass transfer in disks, stronger self-gravity destabilizes mass transfer [89]. Thus, it is hard to generalize across the classes, although we note that irrotational NS should, if anything, be more prone to unstable mass transfer, as the NS loses more angular momentum per unit mass lost from its inner edge. III. NUMERICAL TECHNIQUES To compute the dynamical evolution of a BHNS binary, we fix the position of the BH and assume that the surrounding spacetime metric takes the form appropriate to a nonspinning Schwarzschild BH. The approximation of a fixed BH position is correct in the limit that M BH ≫ M NS . Here, we will study binaries with mass ratios q ≡ M NS /M BH = 0.1, which is presumably within the range of values for which the approximation of an extreme mass ratio is valid. To calculate gravitational forces and evolve the fluid configuration, we will work within the CF formalism, which we explain in more detail in Section III B below. We assume that the spatial metric is remains conformally flat, so that it can be written in the form ds 2 = (−α 2 + β k β k )dt 2 + 2β i dx i dt + ψ 4 δ ij dx i dx j ,(11) where α and β i are the lapse function and shift vector, respectively. Under this assumption we only need to solve the 3+1 constraint equations for ψ, α and β i to determine the metric. Our initial configuration places the NS in a corotating initial configuration. Irrotational configurations, which are more realistic astrophysically, will be treated in a later publication. We model the NSs as relativistic polytropes, and assume adiabatic evolution, which we describe in detail in Sec. III A. The code we use both to relax and evolve BHNS binaries is similar to that introduced in FGR [53] for evolving NSNS binaries. We solve the five linked non-linear field equations of the CF formalism, Eqs. (24), (25), and (26) below, using the LORENE libraries, publicly available at http://lorene.obspm.fr. These Poisson-like equations are solved using spectral methods, decomposing the fields and their sources in a set of radially distinct domains into radial and angular expansions. Dynamical evolution is treated through SPH discretization. Many aspects of the code were discussed in detail in FGR, so we concentrate instead on the changes and new features introduced to evolve BHNS binaries. Roughly speaking, we have made three significant changes to the code to admit the presence of a BH in the binary. First, the asymptotic Schwarzschild BH contribution to the spacetime metric is held fixed, allowing us to solve the field equations describing the selfgravity of the NS in a fully consistent way. Second, as discussed below, we solve Poisson-like elliptic equations for ψ and (αψ), as in BSS and elsewhere, rather than for ν ≡ ln α and β ≡ ln(αψ 2 ), as in FGR and related treatments (TBFS denotes the latter quantity "σ"). Third, we restrict the spatial domain of our spectral methods field solver to a finite radius centered on the NS, as was done in BSS and TBFS, which allows us to avoid problems near the BH. Indeed, our computational domain is chosen so as not to overlap the event horizon at any time. As a result, we do not make use of the asymptotic boundary conditions typically used by LORENE-based codes, which can be extended to spatial infinity through the proper coordinate transformations [91]. The use of a restricted spatial domain has been introduced before, in the context of domains with ingoing and outgoing GWs [92], but with a set of BC's that are not appropriate to the (elliptic) problem at hand. Instead, as we describe below, we have introduced a multipole expansion BC, used here and in TBFS, which should be more accurate than the lowest-order power-law falloff conditions traditionally used in grid-based calculations. Below, we first summarize the relevant equations that comprise relativistic hydrodynamics (Sec. III A) and the CF formalism (Sec. III B), introduce the "split" equations which factor out the BH contributions to the spacetime in Sec. III C, describe our new approach for introducing a multipole BC in Sec. III D, and finally describe how this affects the evaluation of various quantities in the SPH evolution equations Sec. III E. A. Relativistic Hydrodynamics We assume that the matter can be described as a perfect fluid so that the stress-energy tensor takes the form T µν = ρ 0 (1 + ε + P ρ 0 )u µ u ν + P g µν ,(12) where ρ 0 , ε, P , and u µ denote the rest mass density, specific internal energy, pressure, and 4-velocity, respectively. We will describe the NS by a relativistic polytropic EOS that evolves adiabatically with index Γ. Hence, the pressure obeys the relation P = (Γ − 1)ρ 0 ε,(13) and initially satisfies P = κρ Γ 0 ,(14) where κ is a constant. As discussed in BSS, we can scale away dimensional units by setting κ = 1 (see their Sec. IIIc). The Lagrangian continuity equation (FGR, [54]) can be written as dρ * dt + ρ * ∂ i v i = 0(15) where we define the conserved density ρ * ≡ αu 0 ψ 6 ρ 0 = γ n ψ 6 ρ 0 ,(16) and the coordinate velocity v i = u i u 0 = −β i + u i u 0 ψ 4 ,(17) and introduce the Lorentz factor for the fluid γ n ≡ αu 0 . Lagrangian time derivatives are related to Eulerian partial time derivatives through the familiar relation d/dt ≡ ∂/∂t + v i ∂ i . To determine the Lorentz factor, we solve the normalization condition for the 4-velocity, γ n 2 = (αu 0 ) 2 = 1+ u i u i ψ 4 = 1+ũ iũi ψ 4 1 + Γκρ Γ−1 * (γ n ψ 6 ) Γ−1 −2 , (18) implicitly. The Euler equation can be written dũ i dt = − αψ 6 ρ * ∂ i P − αhu 0 ∂ i α +ũ j ∂ i β j + 2hα(γ 2 n − 1) γ n ψ ∂ i ψ,(19) where the specific momentum is defined bỹ u i ≡ hu i ,(20) and the specific enthalpy h by h ≡ 1 + Γε.(21) Finally, the energy equation takes the form de * dt + e * ∂ i v i = 0,(22) where e * = γ n ψ 6 (ρ 0 ε Γ−1 ) 1/Γ . For an adiabatic evolution without shock heating, the energy equation is satisfied automatically by adopting Eq. (14). To account for shocks, we included an artificial viscosity prescription composed of both linear and quadratic terms (the relativistic analogue of the form introduced in [90], similar to that found in [54]). We found no evidence for significant shocks within the body of the NS, as only the matter in the mass transfer stream directed toward the BH showed signs of significant heating very near the BH. Using the value of κ ≡ P/ρ Γ 0 ≡ (Γ − 1)ε/ρ Γ−1 0 as a measure, a quantity that remains constant during an adiabatic evolution, we found variation of no more than 5% within the body of the NS. This is hardly a surprise, as there is no physical mechanism such as a collision to cause significant shocking within the bulk of the NS. Shock heating will be important for understanding the evolution of the initially low-density accretion stream that falls toward the BH, especially near the event horizon. In this region, the heating can be substantial, but it seems not to introduce significant feedback on the NS remnant. Given these results, we replace the energy equation, Eq. (22), with its adiabatic solution, Eq. (14), throughout the calculations described here. In future calculations, where shocks may be more important, we will restore the full evolution of the energy equation with an artificial viscosity prescription and allow for shocks everywhere. This will be especially important for irrotational NS calculations, since the matter transferring through the inner Lagrange point has significantly greater angular momentum than in the irrotational case, and a great deal of it will likely forming a disk rather than accreting promptly. B. The CF formalism In the CF formalism [74,75] we assume that the spatial metric is not only conformally flat initially, but that it remains conformally flat. In particular, for the 3-metric we approximate ∂ tγij = 0 so that in rectangular coordinatesγ ij = δ ij at all times. Strictly speaking, this is inconsistent with Einstein's evolution equations, but is often a very good approximation, particularly on dynamical timescales when secular motion due to radiationreaction is not important. Under this approximation the evolution equation for the spatial metric yields a relation between the extrinsic curvature and the shift, K ij ≡ ψ 4 2α δ il ∂ j β l + δ jl ∂ i β l − 2 3 δ ij ∂ l β l .(23) Inserting this expression into the momentum constraint yields an equation for the shift β i ∇ 2 β i + 1 3 ∂ i (∂ j β j ) = 16παψ 4 (E + P )U i +2αψ 4 K ij ∇ j (ln[α/ψ 6 ]) ≡ S i β , (24) where ∇ 2 is the flat space Laplacian. The Hamiltonian constraint is an equation for the conformal factor ψ ∇ 2 ψ = −2πψ 5 E − 1 8 ψ 5 K ij K ij ≡ S ψ .(25) To derive an equation for the lapse α, the remaining undetermined function in the metric, Eq. (11), we choose maximal slicing K ≡ γ ij K ij = 0 at all times, which implies ∂ t K = 0. This choice can be combined with the evolution equation for the extrinsic curvature, which then yields ∇ 2 (αψ) = 2παψ 5 (E + 2S) + 7 8 αψ 5 K ij K ij ≡ S αψ . (26) In the above equations the matter sources E, S and U i are projections of the stress-energy tensor T µν and can be expressed as E = ρ 0 hγ 2 n − P,(27)S = 3P + γ 2 n − 1 γ n (E + P ),(28)U i =ũ i γ n hψ 4 .(29) In practice, it is easier to decompose the three coupled equations for the shift, Eq. (24), into four decoupled Poisson equations. To do so, we follow [93,94] and define β i = 4B i − 1 2 [∂ i (χ + B k x k )] = 7B i − ∂ i χ − x k ∂ i B k 2 ,(30) and solve the set ∇ 2 B i = S i β 4 ,(31)∇ 2 χ = − S i β x i 4 .(32) These Poisson-like equations, found in [94] and elsewhere, are exactly equivalent to those found in FGR for ν ≡ ln α and β ≡ ln(αψ 2 ), and share the same asymptotic fall-off behavior, but have radically different properties near the horizon of the BH, where the lapse function goes to zero. This causes divergences in the values of ν and β, whereas α and ψ remain finite and easy to deal with in a numerical treatment. Our chosen variables also exhibit a slightly different behavior when we split them into additive pieces contributed largely by the NS and BH, i.e., the contributions from the NS and BH to ψ and αψ are additive, whereas the logarithmic dependence of the "ν − β" set means that the two contributions are combined multiplicatively. As several different sets of notation have now been introduced into the literature to define equivalent quantities in the CF formalism, we present alternate notations used in a selection of other works in Table I. C. BHNS binaries The CF approximation is exact for spherically symmetric configurations, reproducing the TOV equation for fluid configurations as well as the Schwarzschild solution for a stationary, non-spinning black hole. In isotropic coordinates, such a solution is given by ds 2 = − 1 − M BH /2r 1 + M BH /2r 2 dt 2 + 1 + M BH 2r 4 δ ij dx i dx j . (33) From this metric we identify the BH lapse and conformal factors as α BH = 1 − M BH /2r 1 + M BH /2r ,(34)ψ BH = 1 + M BH 2r .(35) The BH contribution to the shift (β i ) BH , vanishes in isotropic coordinates (unlike in the Kerr-Schild coordinates used by BSS). To convert this line element to the more familiar Schwarzschild (areal) coordinates, with ds 2 = − 1 − 2M BH r dt 2 + 1 − 2M BH r −1 dr 2 +r 2 dΩ 2 ,(36) one makes the coordinate transformatioñ r = 1 + M BH 2r 2 r,(37)r = 1 2 r − M BH + r(r − 2M BH ) .(38) Note that this implies that the Schwarzschild radius and ISCO radius take the values [94] Lapse r = 2 M BH ↔ r = 0.5 M BH ,(39)r = 6 M BH ↔ r = 4.949 M BH .(40)α N N α α α Shift βi −Ni −Ni βi βi βi Conformal Factor ψ √ A √ A ψ φ ψ Rest Density ρ * ρ * ΓnA 3 ρ ρ * Dφ 6 ρ * Lorentz Factor γn γn Γn αu 0 W αu 0 Velocity v i v i N U i + N i v i V i v i Specific Momentumũiũi wiũi Si/(Dφ 6 )ũi Enthalpy h h h w h 1 + Γǫ At asymptotically large distances,r = r + M BH . As discussed at length in [27], it is useful to "split" the field equations when dealing with binaries, so that the terms on the RHS of each equation are concentrated on either component of the binary. Here, the method is slightly different. Since the BH solution is an exact solution of the vacuum field equations, it can be subtracted out of the full metric field equations to yield the CF solution for the largely-NS contribution to the fields. Defining N ≡ αψ, and accordingly, N BH ≡ α BH ψ BH = 1 − M BH /2r, we split the fields such that ψ = ψ BH + ψ NS ,(41)α = α BH + α NS ,(42)N ≡ αψ = N BH + N NS (43) ⇒ N NS = α NS ψ NS + α BH ψ NS + α NS ψ BH . The NS piece of the field equations, Eqs. (25) and (26), can be expressed as ∇ 2 ψ NS = −2π(ψ BH + ψ NS ) 5 E − 1 8 (ψ BH + ψ NS ) 5 (K ij ) NS (K ij ) NS ,(44)∇ 2 N NS = 2π(N BH + N NS )(ψ BH + ψ NS ) 4 (E + 2S) + 7 8 (N BH + N NS )(ψ BH + ψ NS ) 4 (K ij ) NS (K ij ) NS .(45) The BH contributes to the shift vector, Eq. (24) only through the lapse function and conformal factor, since the black hole contribution to the shift vanishes in isotropic coordinates. D. Multipole Boundary Conditions The LORENE-based field solver we use decomposes the angular dependence of all scalar, vector, and tensor quantities into spherical harmonics (the radial decomposition into Chebyshev polynomials is described in detail in [91]). For configurations in which the outermost boundary extends to spatial infinity, the outer boundary condition can be set exactly to zero for any field which satisfies a power-law falloff. For a BHNS binary, however, that is not an option, since we encounter numerical difficulties when the computational domain overlaps the BH singularity. Instead, we must impose an approximate BC for each field on the outermost (spherical) boundary, which lies at a finite radius, as shown in Fig. (2). This outermost boundary is chosen so that it never overlaps the BH event horizon. Any Poisson-like equation ∇ 2 Φ = ρ with compact support has an exterior solution given by Φ(r, θ, φ) = ∞ l=0 l m=−l R 0 ρr l Y * lm (θ, φ)d 3 r ×Y lm (θ, φ)r −(l+1) ≡ ∞ l=0 l m=−l ρ lm Y lm (θ, φ)r −(l+1) ,(46) where we define the multipole moments of the source term ρ lm [see Eq. (4.2) of [95]]. This solution established the boundary conditions for our outermost computational domain, and can be matched to the interior solutions to yield the field solution everywhere in space. Here, the source terms of the Poisson-like Eqs. (24), (25), and (26) are not compact, but instead satisfy rather steep power-law falloffs, allowing us to use the same formalism while introducing only small errors. Noting that the matter configurations are equatorially symmetric, we only sum over multipoles with the same equatorial symmetry as the particular field (i.e., l + m even for ψ, α, β x , β y , and χ; l + m odd for β z ). Rather than evaluate the real field source integrals against the complex spherical harmonics Y lm , we evaluate both the multipole moments and the resulting expansions against the real and imaginary parts of the spherical harmonics Y lm with m ≥ 0 (noting that Y l0 are purely real and that Y l,−m = (−1) m Y * lm ). Finally, we truncate the expansion at a predetermined value l = l max , where throughout this paper we use l max = 4, or hexadecapole order. Thus, we assume ρ lm = 0 for all terms with l > l max when we define the BC's for our field equations. This is done for two reasons. First, the multipole coefficients fall off steeply at high l, so that the higher order multipole make ever smaller contributions to the field at large separation. Second, including higher order multipoles can lead to purely numerical instabilities in the field solvers for a finite set of Chebyshev polynomials, since the rapid oscillations with respect to angle can lead to large gradients in derivative-based quantities. We find that that a multipole treatment can lead to significantly higher accuracy for our boundary solution, at the cost of some computational efficiency. To avoid numerical instabilities arising from quickly growing higher-order multipoles, we employ underrelaxation during each iteration, updating each field such that ψ new = (1 − λ)ψ old + λψ new , where ψ old is the field value from the previous iteration, andψ new is the new solution found from solving the elliptic equation. We find good stability and efficiency by setting λ = 0.5 initially, and increasing the value to λ = 0.5 − 0.05 log(∆β y ) with each iteration, where ∆β y is the maximum relative change in the y-component of the shift vector from iteration. The iteration loop terminates when ∆β y < 10 −9 , at which point λ ≃ 0.95, representing very weak underrelaxation. Our multipole BC's allow us to calculate the forces on particles that fall outside the computational domain directly, since the multipole expansion for the metric is valid throughout space. Indeed, for such particles, we calculate the BH contribution to the lapse and conformal factor from Eqs. (34) and (35), the NS contribution from the multipole expansions given by Eq. (46), and the gradients of the NS contribution from ∂Φ ∂x i = ∂ i l,m r l ρ lm Y lm r 2l+1 = l,m ρ lm ∂ i (r l Y lm ) r 2l+1 − (2l + 1)x i r l Y lm r 2l+3 ,(47) noting that ∂r/∂x i = x i /r. This works directly for the lapse and conformal factor, but the shift vector is slightly more complicated. Recall that we have solved elliptic equations not for β i , but for B i and χ, as defined in Eq. (30), and thus only know the multipole decomposition of the latter quantities. In terms of these, the gradient of the shift is given by ∂ j β i = 7∂ j B i − ∂ i B j − ∂ i ∂ j χ − x k ∂ i ∂ j B k 2 ,(48) where we also evaluate terms of the form ∂ 2 Φ ∂x i ∂x j = l,m ρ lm ∂ i ∂ j (r l Y lm ) r 2l+1 − 2l + 1 r 2l+3 x i ∂ j (r l Y lm ) + x j ∂ i (r l Y lm ) +δ ij r l Y lm + (2l + 1)(2l + 3)x i x j r l Y lm r 2l+5 .(49) Since the lapse goes to zero at the horizon, particles approaching it become frozen in proper time, and cannot penetrate within. The approach we use has several advantages over the leading order power-law falloff BC's typically used in BSS and other grid-based field calculations (e.g., [56]). First, we lose less information about the source terms by extending to higher order multipoles ( [57] include dipole order falloff terms for the lapse and conformal factor in full GR, while [54] include quadrupole-order terms for these in CF gravity). Moreover, we avoid a problem associated with symmetries present in our quasi-equilibrium initial conditions which are broken during the dynamical evolution. In particular, as we show in Appendix B, our quasiequilibrium configurations can be shown to have a vanishing monopole contribution to β x , and vanishing monopole and dipole contributions to β z . Once the binary becomes tidally disrupted, however, we expect the monopole contribution to β x and the dipole contribution to β z to grow in magnitude (β z may never have a monopole contribution, since equatorial symmetry holds for dynamical configurations as well). While these terms are growing, we are faced with a situation where the leading-order term may very well not be the largest magnitude multipole contribution on the boundary. Defining a global power-law falloff index to fit the boundary condition. as in previous treatments, is impossible even when the two lowest-order moments are known, since the index varies with angle. Instead, our multipole summation handles this situation naturally, calculating all low-order moments accurately. E. SPH Discretization, Computational Domains, and Timestepping Many of the methods used to perform an SPH discretization of the CF hydrodynamic and field equations are discussed in FGR, so here we summarize briefly the fundamental aspects of the SPH treatment and the new features present in our BHNS code. The neighbor finding algorithms used in our code are based on routines from StarCrash, a publicly available, extensively documented Newtonian SPH code, which can be found online at www.astro.northwestern.edu/theory/StarCrash. Motivated by the form of the Lagrangian continuity equation (15), we define the mass m a of each particle, fixed in time, in terms of the conserved density ρ * , such that (ρ * ) a = b m b W ab ,(50) where W ab is the C 2 , piecewise, "W 4 " smoothing kernel function for a pair of particles introduced by [96], and used in FGR and elsewhere. For each particle, we define a smoothing length h a , and compute all sums over particles that lie within a sphere of radius 2h a surrounding each particle (we calculate all SPH quantities using a "gather-scatter" technique, as described in FGR and the StarCrash documentation). Smoothing lengths are updated using underrelaxation in order to maintain a roughly constant number of neighbors for each particle, set at the beginning of each run. Each particle is advanced through space with a velocity v i = dx i /dt, which we evaluate with a second-order accurate leapfrog evolution scheme, calculating forces from the Euler equation (19) at the half-timestep. Since the calculation is adiabatic, the energy equation, Eq. (22) is automatically satisfied when we use the adiabatic EOS, Eq. (14). A typical timestep in our evolution scheme, started with particle velocities evaluated half a timestep in advance of the particle positions, involves a number of computational elements. First, we advance all particles a full timestep, and re-evaluate the particle neighbor lists and the SPH expressions for the density of each. We then use the SPH form for the density at each particle position to define the computational domains used by the Lorene field solver, shown schematically in Fig. 2. To do so, we calculate the position of the NS center of mass from all particles having a density (ρ * ) a > ρ crit , where ρ crit is a critical value chosen to encompass the vast majority of particles at the beginning of a run. Next, we calculate the surface of the innermost computational domain R 1 (θ, φ), as the smallest triaxial ellipsoid, centered on the NS center of mass, that contains all particles that lie at greater radii from the BH than the NS center of mass, treating the particles as spheres of radius 2h a . This is very similar to the technique described in FGR, except that there we included all particles that passed the density cut, regardless of which side of the NS they fell within. Here, however, the dynamics of the mass transfer are different. In equal-mass NSNS binaries, the NS only begin to disrupt at very close separations, never deviating particularly far from an ellipsoidal configuration up to the point of merger. Here, mass transfer is initially one-sided toward the BH, and the outer half of the NS remains virtually intact while the inner half becomes deformed by the tidal gravity of the BH. We find that our field solver performs best if we define our elliptical domain based on the profile from the outer half of the NS, as it can handle without difficulty field sources located outside the innermost domain, but produces numerical errors if the density field of the NS drops to zero within the innermost domain. (see the second panel of Fig. 2). The two outer domains, which have the topology of spherical shells, are defined initially such that their outer boundaries are spheres at radii equal to twice and three times that of the maximum extent of the innermost shell, i.e. R 2 (θ, φ) = 2 × max(R 1 ); R 3 (θ, φ) = 3 × max[R 1 ]. Over time, we hold the outermost boundary fixed at this radius, and adjust that of the second domain to be the geometric mean of the outer radius and the maximum value from the inner domain, i.e., R 2 = 0.5 * (R 3 + max[R 1 ]) (compare the two panels of Fig. 2). Once the computational domains are defined, we use the techniques of [91] to define a set of "collocation (24). Initially, a triaxial ellipsoid with surface R1(θ, φ) and origin at the NS center of mass is fitted around all SPH particles for which (ρ * )i > ρcrit (top panel). Two annular "shell-like" domains with spherical outer boundaries are also laid down, with radii R2 and R3, twice and three times the maximum value of R1. During the evolution (bottom panel), we use the same procedure to fit R1(θ, φ), keep the value of R3 fixed, and calculate R2 as the mean of R3 and the maximum of R1. Some particles first leave the innermost domain, and then the entire computational volume, particularly those accreted by the black hole, shown as a circle of radius 0.5 MBH centered at the origin. points" at which we compute the local SPH expression for ρ * ,ũ, and P * ≡ κρ Γ * , noting the latter remains equivalent to Eq. (13), for adiabatic evolution and polytropic initial data. From these, we calculate all other hydrodynamic quantities using the Lorene library routines, and solve the field equations iteratively. After every iteration of the field solver, all hydro quantities are updated to reflect the new fields. Once a convergent solution is found, we must export back all relevant matter and field terms from the spectral decomposition to the particle positions. For particles in the innermost domain, we evaluate most hydrodynamical terms directly from the spectral decomposition. Thus, denoting by "SB" those terms evaluated in the spectral basis and "SPH" those quantities defined only on a particle-by-particle basis, we calculate the Euler equation as dũ i dt in = [αψ 6 ] SB ∂ i P ρ * SPH − αhu 0 ∂ i α + 2hα(γ 2 n − 1) γ n ψ ∂ i ψ SB + ∂ i β j SB [ũ j ] SPH .(51) This approach works in the outermost domains for extrinsic quantities like ρ * that go to zero smoothly at the surface of the NS matter, but fails for intrinsic quantities that have discontinuities there, , e.g., u 0 and γ n , since the Chebyshev radial decomposition cannot describe discontinuous functions. Instead, we evaluate hydrodynamic terms for particles in these domains on a particle-byparticle basis, and evaluate field quantities and derivatives through the spectral decomposition, dũ i dt out = [αψ 6 ] SB ∂ i P ρ * SPH −[α∂ i α] SB [hu 0 ] SPH + α ψ ∂ i ψ SB 2h(γ 2 n − 1) γ n SPH + ∂ i β j SB [ũ j ] SPH .(52) After calculating the forces for the RHS of the Euler equation, we advance the velocities from their original half-timestep value forward to a half-timestep ahead of the positions, and then resolve the field equations to determine the velocity v i using the same approach described above for particles based upon their computational domain, v i = 1 ψ 4 u 0 SB [ũ i ] SPH − [β i ] SB ,(53) in the innermost domain, with u 0 evaluated via SPH instead for the outer ones. IV. EQUILIBRIUM MODELS The first step in evolving BHNS binaries is the construction of accurate initial data. In our approach, this requires not only determining the fields and hydrodynamic quantities within and surrounding the NS, but also the construction of a relaxed SPH discretization configuration describing the NS itself. We take as our starting point data constructed from the grid-based equilibrium scheme described in BSS. We modified the scheme of BSS to allow for a conformal background metric corresponding to a Schwarzschild BH in isotropic coordinates, which can be adopted more easily for the CF approximation used here, rather than the Kerr-Schild background used in BSS (see also TBFS). To construct an SPH particle decomposition, we first lay down a hexagonal close-packed lattice of SPH particles over the Cartesian coordinate volume where the density of the star is positive. Tentative particle masses are assigned to be proportional to the density ρ * , normalized to match the proper NS mass. Next, we calculate the SPH value for the density of each particle, and adjust the masses and smoothing lengths of each particle until each has approximately the correct number of neighbors as well as the correct density, to within ∼ 2%. While the resulting configuration could serve as acceptable initial data, we can do better by evolving the configuration in the corotating frame with drag forces applied, to damp away spurious deviations from true equilibrium. This also allows us to relax to quasiequilibrium initial models with binary separations differing by up to ∼ 20% in either direction using the same initial data from BSS. Of course, the new field solution will be different, reflecting the change in magnitude of the tidal terms, but we have found that after approximately 1000 timesteps of relaxed evolution, the overall level of spurious motion is equivalent. We used two grid-base datasets to generate our initial data. For configuration A, the NS is modeled as a relativistic Γ = 1.5 polytrope, of compaction M NS /R NS = 0.042 (or equivalently, massM NS = 0.05 orbiting a BH of massM BH = 10M NS at a separation a 0 = 11.8M BH . Configuration B features a NS with the same compaction but a stiffer EOS, Γ = 2 and a 0 = 11.1M BH (whereM is the dimensionless mass defined in Sec. IIIC of BSS). Note that in these units the maximum compaction of an isolated NS is 0.214 and 0.074 for adiabatic indices Γ = 2 and Γ = 1.5, respectively. To convert from the units of BSS to those used here, many quantities must be linearly rescaled. In particular, for configuration A,r = 10.5, and M BH = 4.72. Thus all distances should be multiplied by a factorr/M BH = 2.2 to convert from the "hatted" units of BSS to those here expressed in terms of the BH mass. Similarly, for configuration B,r = 1.32 and M BH = 0.5, so the rescaling factor is 2.64. Both NS models are undercompact compared to the expected physical NS parameters. Since our method assumes an extreme mass ratio, we are limited to low compactness NS models in order to study cases where tidal disruption occurs outside the ISCO, as can be seen from Fig. 1. Thus, while our configurations do not exactly represent physical parameters expected to be found in BHNS binaries, they serve as an analogue to binaries containing lower-mass BHs and more compact NS that will have comparable tidal-disruption radii located outside the ISCO. In a future work, we will treat more physically realistic NS compactnesses, as well as NS spins, including cases for which the tidal-disruption radius is within the ISCO. Below we describe the technique by which we generate our relaxed SPH initial conditions in Sec. IV A, and then show the comparison between our resulting models and the grid-based data in Sec. IV B. A. Relaxation of Initial Data When preparing a fluid configuration to be evolved using SPH, it is generally necessary to use some form of relaxation first. Otherwise, numerical deviations from equilibrium present in the discretized initial configuration will drive the dynamics, leading to a variety of spurious effects. Relaxation is easiest to perform for configurations in which the matter will be stationary in some reference frame, such as a corotating system, since the spurious component of each particle's velocity can be easily identified and damped away by a drag term in the force equation. This statement holds equally true for Newtonian and relativistic formalisms, although the latter require a slightly more complicated numerical treatment, for reasons we discuss below, primarily due to the presence of velocity-dependent forces as well as a more complicated set of variables used to define the equations of motion. In order to derive the proper equations for a relaxation scheme in a relativistic setting, it is useful to start with a brief review of how the process works in Newtonian physics, and then generalize to the appropriate relativistic equation. In what follows, we define our coordinates such that the x-axis corresponds to the line connecting the centers of mass of the two objects, the y-direction to their orbital velocity, and the z-direction to the binary's angular velocity. In Newtonian physics, we have a set of inertial frame evolution equations d r dt = v,(54)d v dt = a,(55) where the RHS of each are known functions used to define an initial condition. For the case of a corotating equilibrium binary configuration, these quantities take the form v eq = Ω × r, a eq = −Ω 2 r cyl ,(56) where we use the subscript "eq" to indicate the relaxed value, and where r cyl is the "cylindrical" radius. To evolve the fluid during the relaxation, we shift to the frame in which the matter is stationary. Thus we define V ≡ v − v eq = v − Ω × r,(58) so that V = 0 for equilibrium configurations, and evolve d x/dt = V . We determine Ω as an eigenvalue from the condition that the binary center-of-mass separation is already known, by summing over all of our SPH particles. Based on symmetry considerations, only the xcomponent of the equation yields a non-trivial result: m i d V dt x i = 0 = i m i a x i − i m i ( a eq ) x i = i m i a x i + i m i Ω 2 x i ,(59) which implies Ω = − i m i a x i i m i x i .(60) We add to the force equation a linear drag term with some characteristic timescale τ in order to damp away the spurious motion in of the initial condition. The relaxation timescale should generally be approximately equal to the dynamical time of the system. Thus, we evolve d V dt = ( d v dt + Ω 2 r cyl ) + a drag = a + Ω 2 r cyl − V τ .(61) As we approach equilibrium, both the term in parentheses as well as the drag term separately approach zero. The relativistic case is slightly more complicated, but we can derive analogous relativistic expressions for all of our Newtonian ones. Our equations of motion now take the form d x dt = v,(62)dũ dt = a,(63) where the velocity variables are related by Eq. (17), v = u/(ψ 4 u 0 ) − β. In a corotating frame in equilibrium, we know that d(ψ 4 u 0 )/dt = 0, and will treat the term as a constant. The shift vector has the time-dependence of the other vector quantities. For a corotating configuration, we have v eq = Ω × r, and the slightly more complicated Euler equation a eq = d dt [ψ 4 u 0 ( v + β)] = ψ 4 u 0 [Ω × ( v eq + β)] = ψ 4 u 0 [−Ω 2 r cyl + Ω × β].(65) The matter will be propagated on trajectories with velocity V = v − v eq , just as before. Now, however, we have the condition V = v − v eq =ũ ψ 4 u 0 − β − Ω × r,(66) whose time derivative in equilibrium must satisfy the condition a eq /(ψ 4 u 0 ) − Ω × β + Ω 2 r cyl = 0. In the particle decomposition, we find m i dV x dt i = 0 = i m i a x i (ψ 4 u 0 ) i + i m i Ωβ y i + i m i Ω 2 x i = i m i a x i (ψ 4 u 0 ) i +Ω i m i β y i + Ω 2 i m i x i ,(67) which can be solved for Ω. To the force equation, we also add a linear drag term with a characteristic timescale τ , but the drag term must forceũ toward its equilibrium valueũ eq = ψ 4 u 0 (Ω× r+ β), rather than toward zero, so that dũ dt = a − a eq −ũ −ũ eq τ = a + ψ 4 u 0 (Ω 2 r cyl − Ω × β) −ũ − ψ 4 u 0 (Ω × r + β) τ .(68) This equation should, barring any physical instabilities, damp away spurious motion and produce a corotating equilibrium configuration. B. Comparison with Other Quasi-Equilibrium Sequences Our grid-based models, based on the scheme described in BSS but with isotropic background coordinates, were constructed using 48 × 48 × 24 grids, with outer boundaries placedx = ±3,ȳ = ±3,z = 3 for the Γ = 2 case, andx = ±2,ȳ = ±2,z = 2 for the Γ = 1.5 case. SPH configurations were generated corresponding to these binary separations, as well as wider and narrower binary separations constructed by translating the NS to the appropriate position. The number of particles used to construct the SPH configurations were n p = 103, 953 and n p = 77, 908 for the Γ = 2 and Γ = 1.5 EOS, respectively. For the Γ = 1.5 EOS configuration only particles of mass m i > 10 −4 m i;max were accepted, where m i;max is the maximum mass of any SPH particle present. This mass cut is useful for eliminating an outermost layer of negligible total mass, which is typically blown off the surface of the NS anyway by even a tiny amount of spurious motion resulting from deviations from pure equilibrium in the initial condition. Each SPH configuration was relaxed for 1000 timesteps, which corresponds to ∼ 10t D , a sufficient time given that the initial models were rather relaxed to begin with. Parameters for our grid-based models and the final SPH configurations at the end of relaxation are listed in Table II. Models for Γ = 1.5 are labeled A1 -A7, while models for Γ = 2 are labeled B1 -B4. As a check, we compare the value of the period determined during the SPH relaxation process T , to the exact relativistic Kepler relation for a point mass about a BH, T N ≡ r 3 /M BH , and find very good agreement. Here the binary separation is measured in areal coordinates, whose relation to CF coordinates is given by (37). We note that configurations A1-A3 do not settle down completely during relaxation. In each case, the central density of the NS dropped monotonically as the configuration expanded in the x-direction, indicating that mass transfer would eventually begin even with drag forces applied. In general, we find very good agreement between the grid-based initial data and our SPH configurations for stable configurations. In Fig. 3 and 4 we show a comparison between the field values and densities from our SPH configuration and the grid-based data along the xaxis, for configurations A5 and B3. In both cases the relevant fields agree to generally within about 1 − 2%. The only exception is configuration A5, where we find some disagreement between the two methods on the half of the NS facing the BH. The discrepancy is primarily due to the different BC's: the grid based data imposes a 1/r power-law falloff condition on a cube whose inner edge is located atx = −2, whereas the multipole solution used for the SPH configuration is imposed on a spherical boundary atr = 4.5. Thus, the spectral methods solution integrates over a much larger volume of space, and allows for higher-order terms in the field solution at the boundary, which are not insignificant at a distance corresponding to a few NS radii. The small disagreement in the density profile is in part an SPH effect: SPH typically smooths out the density field over each particle's smoothing length. Since our particles are initially equally spaced along a lattice, this length is ∼ δx = 0.05, and we cannot fully resolve the sharp density peak at the NS center. We can formulate an independent check on the selfconsistency of our initial data by checking how well they satisfy the integrated Euler equation, h u 0 = constant.(69) This condition is typically used to generate initial data in grid-based calculations, e.g., Eq. (42) of BSS and Eq. (28) of TBFS, but appears nowhere in our relaxation scheme. As we show in Appendix C, it can be derived as a consequence of a relaxed initial configuration for which the RHS of the Euler equation (19) is zero. In Fig. 5 we show on a particle by particle basis the value of h/u 0 , for configurations A5 and B3, plotted for clarity against the particle's radial coordinate position outward from the NS center of mass. In both cases, we find the standard deviation from the mean is < 0.01%, and the maximum discrepancy < 0.1%. V. EVOLUTION OF BHNS BINARIES From our relaxation results, it appears that the tidal disruption limit for the adopted choices of NS EOS would occur at binary separations a 0 /M BH ∼ 11.0, in line with the predictions of Eq. (2). For models with smaller binary separations, we were unable to find a convergent solution for the configuration with a stable central density maximum. These results can be confirmed through dynamical calculations in the strict CF formalism which ignore all energy and angular momentum losses from gravitational radiation-reaction. From our discussion in Sec. II B, we might expect the possibility of qualitatively different behavior for NSs with the stiffer vs. softer EOS evolved from an initial configuration near the stability limit. For the softer EOS, we expect unstable mass transfer: the NS should disrupt completely once mass transfer begins. For the stiffer EOS, we might expect stable mass transfer in the strongly viscous regime. However, as there is no dissipative mechanism, such as viscosity, powerful enough to circularize the orbit after the onset of mass transfer, the picture is considerably more complicated. All models A (for Γ = 1.5), or all models B (for Γ = 2), describe the same physical binary system, at different separations representing different moments in its evolution. Clearly, for each binary there is only one correct inspiral history. In our separate runs we pick up this history at different points (approximating the orbit as circular), which helps to locate the onset of tidal disruption and analyze the system's dynamical evolution. Ideally, we should start an evolution calculation at some large separation and evolve it forward to complete coalescence, since the assumption of quasicircularity is violated in an ever increasing fashion by inspiraling systems. However, even when we augment our CF equations with a radiation-reaction potential to drive the secular inspiral, we have neither the time nor the numerical stability to calculate an evolution for an indefinite time. Thus, our models started from different initial separations represent a series of approximations to the true binary evolution, which illustrate the dynamics of tidal break-up, and should not be taken as physically distinct evolutionary paths. GW energy and angular momentum losses drive the BHNS binary toward coalescence. However, the GW timescale is much slower than the dynamical timescale, so t GW ≪ t D , we expect that radiation-reaction losses will cease to play an important role in the hydrodynamical evolution once phenomena associated with the dynamical timescale, such as tidal break-up and mass loss, begin. Nevertheless, our evolution calculations started from outside the stability limit and including the effects of GW radiation-reaction yield our best models for the As a soft NS EOS, we choose a polytropic model with adiabatic index Γ = 1.5 (or equivalently, polytropic index n = 2). As we see from Sec. A 2, a NS of compactness C = 0.042 is expected to undergo unstable mass transfer in the high viscosity limit regardless of the binary mass ratio (and thus in the inviscid limit as well). To test out how well this statement applies in the inviscid limit, which applies to our calculations (see Tables V and VI of [97] for numerical estimates of the viscosity present in a lower-resolution implementation of our current SPH scheme) as well as to physically reasonable NS, we evolve binary BHNS configurations from a number of initial separations. This also allows us to estimate the critical separation marking the onset of mass transfer, which according to Eq. (2) should be at a R = 11M BH . In Fig. 6, we show the evolution of run A5, at an initial time t = 0, corresponding to the initial relaxed configuration, as well as t/P = 1, 2, and 2.5. The NS revolves clockwise around the BH, which is fixed at the origin. The event horizon, located at R BH = 0.5M BH , is shown as a circle. In the first plot, the NS has essentially filled its Roche lobe, and has primary axis ratios a 2 /a 1 ∼ a 3 /a 1 = 0.86. We note that the figure shows particle locations, rather than a surface density representation. In fact, particles near the edge of the NS have a density four orders of magnitude lower than in the NS center. After a full orbit, we see in the second panel of Fig. 6 that the NS has begun to shed a small amount of mass, which indicates that it is near the mass-shedding limit, not necessarily past it. Our initial SPH configuration is relaxed to the point where the spurious motion resulting from deviations from equilibrium is small, but not zero. Given that the dynamical timescale of the extremely low-density outer layers of the NS is so long, and the SPH particle masses so small (roughly proportional to the density), even a tiny error in the initial data may result in significant spurious velocities in these layers over time. For an isolated NS, these particles will remain bound, but the same is not true for a NS in a binary. Here, particles that escape the NS surface will often travel outside the Roche lobe and be lost from the NS. As a result, the NS will lose a very small amount of mass and angular momentum. By the third panel of Fig. 6, the NS has expanded to the point that matter is now lost through both the inner and outer Lagrange points. This leads to the formation of a stream of matter thrown out into an extended halo around the binary system, most of which remains bound to the BH. Meanwhile, there remains a mass stream of material accreting directly onto the BH. We believe that the relativistic nature of the BH gravitational potential plays an important role in the dynamics of this accretion process. The matter streaming through the Lagrange point passes sufficiently close to the BH to fall well within the ISCO on its first passage. As a result, most of it accretes directly onto the BH, rather than forming a disk. The instability of orbits near the BH is likely to play an important role in suppressing the formation of an accretion disk. We note, however, that our assumption of an extreme mass ratio may bias the evolution towards prompt accretion (as does the assumption of initial synchronization), since the BH is a fixed target, rather than one orbiting the binary center of mass itself. Finally, in the last panel, the NS is nearing a state of complete disruption, and will continue to do so until we can no longer locate a gravitationally bound object. A similar pattern holds for all runs we performed with the same soft EOS, regardless of the binary separation. Due to the nature of the instability, all models we calculated led to the eventual tidal disruption of the NS, since there is no stabilizing mechanism to suppress mass loss once mass transfer begins, no matter how small the mass-transfer rate. It does take longer for the NS to be disrupted in configurations placed at a greater initial binary separation, however. In the bottom plot of Fig. 7, we show the evolution of the mass loss over time for all of the configurations using the soft NS EOS, defined as the total mass that can be found outside the innermost computational domain at any given time. We see that in all cases mass loss is an unstable process, occurring at a rate that grows extremely rapidly until the major-FIG. 6: SPH particle configurations at times t = 0, 286, 566, and 693, projected into the orbital plane, for run A5 (for which the orbital period is T = 287 MBH). These configurations correspond to the initial configuration, and after 1, 2, and 2.5 full orbits, respectively. The BH is shown as a circle at a radius r = 0.5MBH. We see that once mass transfer begins, the NS begins to expand, forming a low-density single-armed spiral pattern. Note that particles have different masses, and that those in the center of the NS are in most cases significantly more massive than those originally in the outer layers. ity of its original mass is no longer bound to the NS. In the case started from the largest separation, slightly over half of the NS mass is accreted directly onto the BH or into orbits that lie within the ISCO, while half is lost outward to form the "spiral arm" pattern seen in Fig. 6. This pattern is to be expected, as the primary response of the NS to losing mass is expansion. The inner half of the NS is pushed inward toward the BH, while the outer half expands outward. For a synchronized configuration, we know that the average specific angular momentum of the NS is generally greater than the specific angular momentum near the center-of-mass of the NS, since j ≡ v × r ∼ Ωr 2 is weighted more heavily by matter on the outside of the NS. In response to the expansion of the NS, we expect the orbit to tighten, which is confirmed by the numerical results. In Fig. 7, we show the binary separation over time for the runs with a soft NS EOS. We see that the onset of mass transfer leads to a rapid decrease in the binary separation, prompting the explosive mass loss. Only in the late stages does the remnant of the NS begin to move back outward, but by then tidal disruption is inevitable. As the initial separation is decreased, the qualitative behavior of the system stays essentially the same, but Table II. Here, particles are defined as "lost" when they lie outside the innermost computational domain around the NS. In all cases, mass loss never quenches once it begins, eventually leading to the complete disruption of the NS. more of the NS mass ends up being transferred inward toward the BH. In Fig. 8, we show the amount of mass lost inward (top panels) and outward (bottom panels) for all of the runs we performed with the soft EOS. For configurations near the stability limit, we find almost twice as much mass falls inward toward the BH as is expelled outward into the spiral arm. As these calculations were performed without radiation-reaction effects, we expect that including them would tip the balance of the mass transfer further toward mass accretion onto the BH, since the binary orbit will be driven by radiation to smaller separations than we find here. Unlike the situation found in NSNS binaries, for which > 99% of the matter typically remains bound to the system, we find that a significant amount of matter is unbound from the system during all of our calculations with the soft EOS, representing in each run between 3 − 5% of the original mass of the NS. This fraction is much larger than that typically found in relativistic calculations of synchronized binary NS systems [53,57], but roughly consistent with previous results from Newtonian BHNS calculations [5,64]. However, we note that this fraction is almost certainly an overestimate, perhaps greatly so: irrotational configurations suppress the amount of mass that will become unbound (see, e.g., [98] or FGR for a similar argument with respect to NSNS mergers) . The Fig. 7. In general, the smaller the initial separation, the more mass that falls inward toward the BH. For the runs started from a larger separation, much of the initial mass loss results from spurious numerical of low mass particles out of the Roche lobe, leading to nearly equal flows directed inward and outward. In contrast, the mass loss for the closer cases is in exactly the form expected for Roche-lobe overflow through the inner Lagrange point. total angular momentum of an irrotational configuration is less than that of a synchronized configuration, and the decrease in specific angular momentum is largest at the outer edge of the NS. Thus, we expect that while some mass may still be unbound when the NS is initially irrotational, the amount may be significantly less. B. BHNS mergers with a stiffer EOS: Γ = 2 To study a configuration that would be predicted to undergo stable mass transfer in the classical conservative quasiequilibrium scenario, described in Sec. II B, we model the NS with a stiffer, Γ = 2 polytropic EOS, which has been used in numerous studies as a first approximation to a stiff NS EOS. We note that in Newtonian physics, this is the critical polytropic index for which an isolated NS has a radius independent of its mass. In relativistic gravity, self-gravity effects cause the NS radius to increase as the NS mass decreases. Evolution without GW radiation-reaction In Fig. 9 we show the binary separation (top panels) and NS mass fraction lost (bottom panels) as a function of time for runs B1-B4. We find three qualitatively different behaviors over time, which we will describe in turn. We note first that the tidal limit separation is not drastically different for this EOS compared to the softer EOS. Both cases show very good agreement with the classical Roche limit result, as we would expect. FIG. 9: Binary separation (top panel) and total mass lost from the NS (bottom panel) as a function of time for the runs calculated using the stiff NS EOS without GW radiationreaction, runs B1-B4, which have Γ = 2. Conventions are as in Fig. 7. For run B1, we see periodic mass loss, as the NS gets kicked into an elliptical orbit, losing mass during every periastron passage. Runs B2 and B3 are similar to those with the softer EOS, as the NS gets disrupted during the first mass loss phase. Run B4 is essentially stable, only showing the long-term effects of numerical diffusion of particles. Run B4 is completely stable for approximately three orbits. Eventually, the NS begins to lose a small amount of mass, which gives us an estimate for the length of time over which the code can reliably maintain an equilibrium configuration in the absence of relaxation (t/T ∼ 3). We note from Fig. 10 that the mass loss from the NS is almost evenly divided between a component moving toward the BH and the component directed away, indicating that the mass loss is a numerical artifact caused by particles near the edge of the NS diffusing outside the Roche lobe over time. Runs B2 and B3 are started from a binary separation approximately equal to the mass-shedding limit. In both cases, the NS makes approximately one orbit after mass Fig. 9. Conventions are as in Fig. 8. Approximately 50% more mass is lost inward for runs B2 and B3, started from near the stability limit, whereas for run B1, marked by an elliptical orbit and periodic mass transfer, nearly 4 times much mass is lost toward the BH. transfer begins without any appreciable change in the binary separation. By this point, approximately 10% of the NS mass is stripped away, and the binary begins to move outward. As we saw in the case of the softer EOS, the mass loss is unstable and the mass-transfer rate grows rapidly. In both cases, the NS is tidally disrupted before the mass transfer halts. By the time the NS is disrupted, approximately 60% of the original mass has fallen inward toward the BH, as we see from Fig. 10. Of the matter shed outward, approximately 0.075M NS is unbound from the system. This result fits in with the general picture derived from NSNS binaries, in which mass shedding outward is more efficient for stiffer EOS (see, e.g., [99] and references therein). The evolution of the NSs in runs B2 and B3 started from initial conditions differing only in their binary separations (run B3 starts from 0.5% further out), which are nearly equal to the mass-shedding limit. We find that the closer the NS is when mass transfer begins, the more mass is lost inward toward the BH relative to that lost outward, until the NS expands to the point that it greatly overfills its Roche lobe and mass loss becomes more isotropic around the axis describing the NS velocity. These observations go a great deal toward clarifying the evolution seen in run B1, the only case in which the mass transfer was periodic, rather than continuous. Here the initial configuration places the NS within the massshedding limit, as GW losses would be expected to do for physical BHNS systems. Indeed,runs B3a and B3b, described in Sec. V B 2, which include the effects of radiation reaction, are driven to approximately this binary separation by GW radiation-reaction energy and angular momentum losses. When mass transfer begins, it occurs only toward the BH, which causes the NS to move outward. This expansion in the orbit, which happens with no outwardly directed stream to counter the outward acceleration, occurs sufficiently fast that the Roche-lobe expansion is enough to quench mass transfer. At this point, the NS is on a continuously expanding eccentric orbit (e ∼ 0.1), whose apocenter lies outside the Roche limit and pericenter within it. As the NS crosses over the new mass-shedding limit, mass transfer begins again, pushing the NS onto an even wider eccentric orbit, similar to the pattern seen in [5] for NS with a stiffer EOS. This scenario is ostensibly similar to that proposed by [44], but with one crucial difference. In both cases, mass transfer forces the NS onto a highly elliptical orbit. Here, however, mass transfer occurs during every periastron passage, whereas in their model the NS can be kicked into such a widely separated orbit that GW radiation-reaction must drive the NS back toward the mass-shedding point. We believe that the assumptions that go into the latter model lead to this unphysical result. In [44], it is assumed that the NS loses a specified amount of mass during masstransfer events but recovers half of its angular momentum during the next half orbital period. This can lead to a discontinuous evolution of the binary separation. Here, we see that once mass transfer ceases, the NS will follow a nearly unperturbed elliptical orbit, with essentially no change in its orbital angular momentum. Mass transfer must resume when the orbit crosses this same point as it approaches pericenter during the next passage. Mergers including GW radiation-reaction While evolution calculations lacking GW radiationreaction terms can be useful for studying the processes that control the moment to moment dynamical evolution of the system, we know that the effects of radiationreaction must play an important role in the secular dynamics of the system. Indeed, given the potentially unstable nature of mass loss, we expect the inwardly directed component of the NS velocity to be critical. Since the mass transfer leads not to an instantaneous change in the NS velocity, but rather in its acceleration, there will be a time period immediately after the onset of mass transfer during which the NS loses mass while falling further inward. All the while, the inner Lagrange point will move further within the NS, regardless as to how its radius adjusts on the dynamical timescale. To model radiation-reaction, we add a damping force to the material, representing the lowest-order quadrupole contribution to the radiation reaction potential. Thus, we add to the RHS of Eq. (19) an acceleration term of the form a i:reac = N 2 hu 0 ∂ i χ,(70) where χ is a quadrupole radiation-reaction potential (see Sec. 36.1 of [100]), defined here as χ = 1 5 x k x l Q [5] kl ,(71) in terms of the fifth time derivative of the quadrupole moment, Q kl = ST F ρ ADM x k x l d 3 x .(72) Here ρ ADM is the quantity that can be integrated to give the matter contribution to the ADM mass ρ ADM ≡ ψ 5 E, which appears in the field equation for ψ, Eq. (25). Following the argument found in Sec. IIIa of FGR, we evaluate the fifth time derivative of the quadrupole moment in terms of the expression for the first time derivative in the limit that it is dominated by terms representing orbital motion and not changes in the density with time (i.e., dρ ADM /dt is negligible), (Q 1 ) kl = ST F ρ ADM (x k v l + x l v k )d 3 x ,(73) where "STF" means the symmetric trace-free component of the tensor. We then assume that further time derivatives result purely from the orbital motion, and approximate Q [5] kl ≈ 16ω 4Q kl ,(74) where the angular velocity ω is taken as the ratio of the angular momentum to the moment of inertia, ω = a m a (x a (v y ) a − y a (v x ) a ) a m a (x 2 a + y 2 a ) .(75) While the resulting radiation-reaction force will differ slightly from the true quadrupole expression found by taking the exact time derivatives, it is sufficiently accurate for our purposes here, generally within 10%. It is important to remember that the radiation-reaction force drives the binary inward on a secular timescale t GW , but plays almost no role once effects occurring on the much more rapid dynamical timescale become dominant. To test how radiation-reaction effects change the scenario described above, we calculated two runs that included radiation-reaction terms. Both took as initial data the configuration used also for run B3, using the stiffer Γ = 2 NS EOS, placed nearly at the stability limit. For run B3a, we added a radiation-reaction acceleration term described by Eq. (70); for run B3b, we doubled the magnitude of this force. The evolution of the former, run B3a, is shown in Fig 11. Mass transfer occurs in a welldefined stream for the first two orbital periods, until the FIG. 11: SPH particle configurations at times t/MBH = 0, 261, 514, and 767, projected into the orbital plane, for run B3a, which includes the dissipative effects of radiation-reaction (for this configuration, T = 265MBH). In terms of the binary orbit, these correspond to the initial configuration and 1, 2, and 3 full orbits, respectively. Conventions are as in Fig. 6. Here, we see that radiation-reaction initially drives only an inwardly directed mass-transfer stream onto the BH, until somewhere after t/MBH = 700, when the expansion of the NS becomes unstable and tidal disruption occurs. expansion of the NS eventually drives the rapid tidal disruption of the NS. In Fig. 12, we show the evolution of the binary separation (top panels) and NS mass loss (bottom panels) for runs B3a and B3b. We find that the stronger the radiation-reaction losses, the greater the initial mass loss from the NS, since the first passage within the massshedding limit takes it closer to the BH. This in turn drives the NS back outward, quenching the mass loss temporarily until the next periastron passage, after which the NS tidally disrupts. From Fig. 13, we see that stronger radiation-reaction losses favors a larger amount of mass lost inward onto the BH (and thus less outward into a disk), as one would expect. Thus, we conclude that the inclusion of GW radiation-reaction terms have the effect of increasing the chance that some fraction of the original NS mass will remain bound after an initial phase of mass loss, since the NS rebounds more sharply outward than for cases in which radiation-reaction losses are ignored. Tidal disruption, while still seemingly inevitable, also occurs at a greater distance from the BH. In the bottom panel of Fig. 14, we show the gravity wave signal produced in run B3a, in both polarizations. Fig. 7. The only difference between the two runs is the magnitude of the radiation-reaction effects; we use the quadrupole order form in run B3a, and double its strength for run B3b. We find that doubling the radiative drag force forces the binary to a smaller separation, leading to a larger initial burst of mass loss and a more rapid increase in the binary separation. As a result, the system tidally disrupts slower than the case shown in run B3a. The two components are defined by the familiar relations D o h + =Q xx −Q yy ,(76)D o h × = 2Q xy ,(77) where D o is the distance from the observer to the binary. The corresponding angular frequency of the GW signal, approximately equal to twice the orbital angular frequency, is shown in the top panel of the figure. Prior to disruption, the GW waveform takes the classic pointmass form, with a steadily but extremely slowly increasing amplitude and frequency (relativistic and finite-size affects cause minor deviations from the point-mass form; see [40] for a discussion). Once the mass transfer begins, however, the frequency reaches a maximum and begins to decrease quickly, as does the amplitude. In general, if the NS does transfer sufficient mass to survive the initial infall, we expect this decrease in frequency and amplitude until the signal can no longer be reliably observed. Should the orbit be elliptical, as we found for run B1, this will show up in the GW waveform as well. As matter accretes onto the BH, it will excite quasinormal ringing modes that could in theory be visible in Fig. 9. Conventions are as in Fig. 8. Initially, in run B3b with twice the physical GW radiation-reaction force applied, all mass lost in run B3b is directed inward. This leads to a rapid increase in the binary separation and slows the growth of the mass-transfer rate. In contrast, in run B3a, mass loss is more evenly balanced between inwardly and outwardly directed flows, and the NS disrupts more quickly. the gravitational wave signal. Unfortunately, our numerical approach limits our ability to determine the gravitational wave signal we expect from these modes. Indeed, such a calculation would require a dynamical treatment of the spacetime very near the BH, whereas in our calculation, the key physics for ringdown occurs in the asymptotic region located outside our computational domain, where the BH remains stationary. To evaluate quasinormal mode ringing, it will be necessary to relax the CF approximation, in which no gravitational radiation is generated, and to evolve the fields everywhere. Additionally, the use of a lapse function that penetrates the horizon will be crucial for studying the problem selfconsistently. We plan to study these issues in future work. VI. SUMMARY AND DISCUSSION We have performed relativistic calculations of BHNS mergers in CF gravitation, in the limit that the BH is much more massive than the NS. These calculations mark the first time that a relativistic treatment has been applied both to the self-gravity of the NS as well as the BH spacetime. FIG. 14: Gravity wave emission angular frequency (top panel) and waveform in both polarizations, h+ (solid curve) and h× (dashed curve), defined by Eqs. (76) and (77), for the merger calculated in run B3b. Initially, we see conclusion of the standard "chirp" signal, in which the binary separation decreases while the GW amplitude and frequency increases, all of which happens extremely gradually on the secular GW timescale. This lasts until the onset of mass transfer, at which point we encounter a much more rapid "reverse chirp", as the GW amplitude and frequency rapidly decrease while the NS is tidally disrupted. For systems studied here in which the onset of tidal disruption occurs outside the ISCO, previously proposed analytical models do not properly describe all the complex tidal phenomena that pertain to the evolution. In all the runs we calculated, mass transfer plays a leading role in determining the dynamics of the system. In general, mass transfer in a stream directed toward the BH causes the orbital separation to increase, in many cases quite dramatically. Mass transfer also causes the NS radius to expand on the dynamical timescale, with relativistic selfgravity effects leading to a more rapid expansion than is seen in Newtonian gravity for a given NS EOS. As a result, mass transfer is significantly more unstable in relativistic gravity. We conclude immediately that previous models of mass transfer in compact object binaries that assume the orbit remains quasi-circular [42,43] are not applicable here. Furthermore, the model put forward in [44] also seems to be insufficient, in that the orbital parameters evolve discontinuously from one orbit to the next (as a result of angular momentum being added to the NS while its mass is held fixed). We find instead that if some remnant of the NS survives the initial burst of mass loss, it can end up on an elliptical orbit that takes it back outside its mass-shedding separation. Dur-ing every successive periastron passage, however, more mass will be lost, eventually leading to the complete disruption of the NS. As the NS expands during mass loss, it eventually loses mass outward as well as inward, so long as the plunge does not take it too far within the ISCO, leading to a prompt plunge onto the BH. While the majority of matter released through the outer Lagrange point remains bound to the BH in the former case, a significant fraction is ejected with sufficient velocity to become unbound from the system completely, approximately 5 − 7% for the Γ = 2 EOS we considered. This mass loss also limits the radial expansion of the orbit, and as a result we find in many of our calculations that mass transfer is never quenched once it begins. Even though the NS moves outward, it persists in configurations for which the Roche lobe lies within the star, and mass loss continues until the NS is completely disrupted. We plan to improve our simulations and relax several approximations in the near future. In particular we plan to adopt the astrophysically more realistic irrotational initial configuration of TBFS instead of the corotating configurations used here. Relativistic NSNS [53,55] and Newtonian BHNS [5] calculations have shown that for these cases the amount of matter ejected from the binary system is expected to be significantly smaller, since the material on the outer edge of the NS has significantly less specific angular momentum. Thus, while we expect that BHNS mergers will expel more material than in the case of NSNSs, in which the binary components are more comparable in mass, we would assume that the ejected fractions found here are overestimates. Calculating mergers using irrotational NSs should also increase the probability that escaping matter forms an accretion disk around the BH, even if that disk is shortlived (see, e.g., [83]). We typically found in our calculations here that most of the matter transferred toward the BH ends up accreting onto it directly, since the specific angular momentum is not sufficient to create a disk. Even though an irrotational NS has less total angular momentum than a corotating one, the matter on the inner edge has a higher specific angular momentum, and is more likely to orbit around the BH rather than infall directly. In addition, prompt accretion of matter may also be inhibited slightly by a moving BH orbiting the binary center of mass, whereas in our assumption of an extreme mass ratio, the BH position is fixed. In our future work, we will test out how shock heating in the accretion disk affects its evolution, and determine if there are cases in which feedback onto the NS will affect its future evolution as well. By including a relativistic artificial viscosity treatment, we will follow the thermodynamic evolution of the disk, as well as that of the NS and any outward mass loss. Of course, to investigate BHNS mergers fully and accurately, we will ultimately have to abandon the assumptions underlying the CF metric as well. While our description of an isolated Schwarzschild BH is exact, the BH lapse goes to zero just outside the event horizon. This causes matter to "pile up" around the BH, since the proper time ceases to advance in this region. While this poses no difficulty for determining the fate of the material, which will clearly be accreted, it does act as a computationally challenging ridge of growing mass concentration as material is transferred continuously toward the BH. These calculations adopted "undercompact" NSs because we are interested in systems that disrupt outside the ISCO while restricting our attention to the case of extreme mass ratios. By performing dynamical evolutions using more compact configurations, we will investigate the transition to the opposite case, in which plunge begins while the NS is still completely bound. Based upon our results, we expect that that in some cases, the mass-transfer rate may prove sufficient to kick the core of the NS back out to a wider, highly elliptical orbit. The phase space for which this will occur, in terms of the NS EOS and binary mass ratio, remains poorly understood, and may not conform to simple analytic estimates, which have difficulty describing the unstable processes that characterize the merger. In [5], it was found for a quasi-Newtonian potential that for a mass ratio q = 0.1, a NS described by a Γ = 2 EOS was disrupted during the initial passage, whereas one with a Γ = 3 EOS led to a punctuated mass-transfer scenario similar to the one we describe for run B1 above in Sec. V B 2. Probing the assumed forms of the NS EOS that lead to complete disruption versus survival of a remnant NS core may prove to be a crucial diagnostic tool for determining the true physical NS EOS. In many prior works, it has been assumed that so long as a ISCO < a Roche , the NS will necessarily be tidally disrupted, with some of its mass deposited into a disk around the BH. According to [82], this may not be true. Instead, the plunge may start well outside of the ISCO, since, based on the properties of quasiequilibrium models, the inspiral time scale may approach the orbital time scale already well outside of the ISCO (so that the transition through the ISCO is nearly dynamical rather than adiabatic). It is unclear exactly what role the spin of the BH will play in this process, though it appears that a prompt merger is more likely for a Schwarzschild BH than a spinning Kerr BH [71], where the latter case is favored by binary evolution calculations [101,102]. Two different factors can mitigate a prompt merger, however, and will need to be investigated numerically in more detail. First, the onset of a "plunge" also marks the point at which we assume the quasi-equilibrium formulation to break down. From that point onward, the NS will no longer follow the trajectory predicted by quasiequilibrium results, and may therefore not plunge as fast as quasiequilibrium models would predict (in fact, these predictions provide completely unphysical overestimates of the infall velocity at the ISCO itself). Perhaps more importantly, there is no guarantee that a plunge phase will lead to the entire NS being swallowed by the BH [83]. Angular momentum can be transferred outward within the NS on something approximating the dynamical timescale, and it is possible that some fraction of the mass, perhaps a significant fraction, will survive the initial plunge phase. Using the techniques developed here, and initial configurations taken from TBFS, we will study how varying the NS spin and compaction affect the final fate of the NS. It will be necessary to perform relativistic merger calculations in full generality, in order to drop the assumption that the binary mass ratio is extreme. We are currently constructing such quasiequilibrium data, which will then serve as initial data for dynamical simulations [103]. We expect that these dynamical simulations will be plagued by the same difficulties encountered in dynamical simulations of BHBH binaries, and therefore anticipate that this will be a very challenging project. Until that point, however, a great deal can be accomplished. First and foremost would be to identify the boundaries in phase space that separate qualitatively different phenomena that can occur during a BHNS merger. The most obvious categories would be prompt merger, prompt tidal disruption leading to an accretion disk, or a period of periodic mass-transfer bursts, if the latter does occur at all. The former distinction should prove useful for understanding any potential X-ray/gamma-ray emission from these systems, and aid in our understanding of short gamma-ray bursts like the recently observed GRB 050509b [46,47], GRB 050709 [48,49], GRB 050724 [50], and GRB 050813 [51]. Below, we derive a formalism that can be used to describe conservative quasiequilibrium mass transfer, for those cases where the viscosity is high, and the binary orbit remains circular during the mass-transfer process. For typical (high-mass) NSs, this regime does not apply ( [59,60]; see also Fig. 1 above), but can apply to lowmass NSs, WDs, and main sequence stars orbiting BHs. We nevertheless refer to the star as a NS below. Newtonian polytropes In Newtonian physics, the mass-radius relationship is given as a function of the polytropic index n by R NS ∝ M (1−n)/(3−n) NS . (A1) For a given value of the parameter κ in P = κρ (1+1/n) 0 , the familiar result is that NSs with n = 1 have a uniform radius independent of their mass. NSs with n < 1 have radii which decrease as the mass decreases, whereas those with n > 1 increase in size as they lose mass. We see that if the NS loses mass at a rateṁ ≡ −Ṁ NS (implyinġ m > 0 for the case of interest), the change in the radius is given byṘ NS R NS = n − 1 3 − nṁ M NS .(A2) Once mass transfer begins, the NS will shed mass, and in doing so, lose both energy and angular momentum. This will affect the binary orbit, in a way which depends on both the magnitude and fate of the angular momentum of the ejected material. For conservative mass transfer, the orbital angular momentum J and the total binary mass M T ≡ M NS + M BH = (1 + q)M NS /q are conserved globally, as mass is transferred from the NS to the BH. We assume that the orbit remains circular. Since q = M NS /(M T − M NS ), we see thatq q = −(1 + q)ṁ M NS .(A3) From the angular momentum of a circular orbit, J = M NS M BH Ga M T ,(A4) we find that a = M T J 2 G [M NS (M T − M NS )] −2 = J 2 G M −3 NS q(1 + q), (A5) a/a = 2(1 − q)ṁ M NS .(A6) The Roche-lobe radius R r changes during the process as well. Taking the logarithmic time derivative of Eq. (2) and combining with Eq. (A6), we finḋ R r R r =ȧ a +q 3q(1 + q) =ȧ a −ṁ 3M NS = 5 − 6q 3ṁ M NS .(A7) Combining this final expression with Eq. (A2), we see that mass transfer will be unstable ifṘ NS /R NS >Ṙ r /R r , or equivalently, n > 9 − 9q 4 − 3q , (A8) q > 9 − 4n 9 − 3n ,(A9) or in terms of the adiabatic index Γ ≡ 1 + 1/n, mass transfer is unstable if Γ < 13 − 12q 9 − 9q , (A10) q > 9Γ − 13 9Γ − 12 .(A11) In particular, the critical mass ratio for unstable mass transfer for some polytropic indices commonly used in numerical calculations are n = 1/2 (Γ = 3) : q > 14/15, (A12) n = 1 (Γ = 2) : q > 5/6, (A13) n = 3/2 (Γ = 5/3) : q > 2/3. (A14) As a general rule, for polytropic indices n < 3/2, mass transfer is stable only for systems with components of similar mass. For n > 3/2, the critical mass ratio drops quickly, down to the limiting case n = 9/4, the softest polytropic EOS for which stable mass transfer can ever occur, at which point q = 0. Relativistic polytropes Relativistic polytropes are not self-similar for a fixed value of the polytropic index; as the mass decreases, nonlinear gravitational effects become weaker, and the star's scale-free density profile grows in size. It is straightforward to incorporate this into our discussion of mass transfer. We recast the mass-radius relationship in the form R NS ≡ ξM (1−n)/(3−n) NS f (C),(A15) where ξ sets the physical scale of the mass-radius relation, C ≡ M/R is the compactness of a spherical star, and f (C) accounts for the relativistic corrections to the massradius relation. Since relativistic corrections effectively increase the strength of gravity, f (C) must be a monotonically decreasing function of compactness. As C → 0, ξ approaches the proper Newtonian value, ξ = ξ N , and f (0) → 1. Taking a logarithmic derivative of Eq. (A15) shows us thatṘ NS R NS =ḟ f + n − 1 3 − nṁ M NS .(A16) But we knoẇ f f = 1 f ∂f ∂CĊ = 1 f ∂f ∂C Ṁ NS R NS − MṘ NS R 2 NS = − C f ∂f ∂C ṁ M NS +Ṙ NS R NS ,(A17) so we conclude thaṫ R NS R NS = n − 1 3 − n − 2 3 − n C f ∂f ∂C 1 + C f ∂f ∂C −1 ṁ M NS . (A18) From Eq. (A8), we see that the critical mass ratio for instability becomes q c = 9 − 4n 9 − 3n + 1 3 − n C f ∂f ∂C 1 + C f ∂f ∂C −1 ,(A19) where the second term is always negative, indicating that more compact configurations are more unstable against mass transfer. In Figure 15, we show the critical compactness values separating stable and unstable mass transfer for relativistic polytropes, as a function of the binary mass ratio and the polytropic (bottom panels) and adiabatic (top panels) indices. As a general rule, as the compactness of the NS increases, mass transfer is more likely to be unstable. The Newtonian curves (C = 0) have an analytic form given by Eqs. (A8) and (A10). Also shown are the two models we evolve dynamically in this paper, both with C = 0.042. The case with n = 1(Γ = 2), shown as a triangle, would be expected to demonstrate stable mass transfer in the highly viscous regime, but as we show in Sec. V B, the true situation is nowhere near this simple for typical NSs. The case with n = 2(Γ = 1.5), shown as a square, is expected to lead to unstable mass transfer. Representing the same results against compactness, rather than mass ratio, shows how little parameter space is available for unstable mass transfer in the viscous limit. In Figure 16, we show the critical binary mass ratio for stable or unstable mass transfer as a function of the NS compactness and polytropic (bottom) or adiabatic (top) index. We see that for a given compactness, there is a rather limited range of polytropic indices which can produce unstable mass transfer. The heavy lines show the maximum possible compactness allowed for a given polytropic EOS; no model to the right of those curves can be constructed. The models that we evolve dynamically are shown as well. As noted in Sec. V B, many models expected to undergo stable mass transfer in the viscous limit are extremely unstable during mass transfer when viscosity is not a dominant driver of the evolution [59,60], so the true parameter space for instability is actually much larger for typical NSs than analytic models would otherwise predict. Time evolution We describe here a crude treatment of the dynamics of mass transfer in the presence of gravitational radiationreaction losses, using a number of simplifying assumptions. Most important among these will be treating the problem in the quasi-equilibrium regime. We will study only conservative mass transfer, in the highly viscous limit. The evolution of the binary separation can be derived from the expression for the total angular momentum for a circular orbit, Eq. (A4). Making no assumptions about the evolution of the system angular momentum, we finḋ J J =Ṁ NS M NS +Ṁ BH M BH +ȧ 2a = (q − 1)ṁ M NS +ȧ 2a . (A20) Setting the RHS equal to zero gives the standard expression for conservative mass transfer, Eq. (A6). For the case of a relativistic binary, we know angular momentum will not be conserved. Instead, angular momentum is radiated away from the system in gravitational waves. As we are assuming circular orbits, we will make use of the standard angular momentum loss rate, J J = − 32 5 M NS M BH M T a 4 .(A21) From our two angular momentum relations, we can derive a relation linking the mass loss rate to the change in binary separation, and determine the time-averaged mass loss rate. To do so, we will make a very simple assumption: that mass loss takes place at the location where we predict Roche-lobe overflow to occur. Solving Eq. (A7) for the binary separation and assum- Plugging this result into Eq. (A20), we finḋ J J = − 9 − 4n 9 − 3n − q + 1 3 − n C f ∂f ∂C 1 + C f ∂f ∂C −1 ṁ M NS . (A23) Re-expressing the radiation-reaction angular momentum loss rate, Eq. (A21), for a binary located at the Rochelobe overflow point, we finḋ We are now in a position to analyze the mass-transfer process from its onset until the final fate of the binary. To do so, we will make a few simplifying assumptions. First, we assume the NS is much less massive than the BH, such that M BH ∼ M T and q ≪ 1. Also, we will ignore the relativistic changes to the Newtonian powerlaw mass-radius relation, which has the effect of setting f = 1 uniformly (and thus ∂f /∂C = 0). In general, relativistic polytropes expand more than their Newtonian analogues during mass loss, which increases the amount of orbital expansion seen for a given amount of mass loss. This statement would imply that a given amount of mass loss leads to a greater loss of angular momentum, or conversely, that for a fixed angular momentum loss rate we would see a slight suppression of the mass loss rate. Under these assumptions, we finḋ − 9−3n 9+5n ,(A26) with asymptotic behavior M NS (t) ∝ t − 9−3n 9+5n ,(A27)m ∝ t − 18+2n 9+5n .(A28) Considering some familiar polytropic EOS, we find: Eq. (A29) recovers the special value found by [43], in the limit q ≪ 1. We note that the typical rate found here satisfiesṀ NS ∼ M NS /t GW 0 [cf., Eq. (A25)], while our dynamical simulations for cases in which viscosity is not important showṀ NS ≫ M NS /t GW 0 . APPENDIX B: SYMMETRIES When constructing quasi-equilibrium configurations in either Newtonian or relativistic gravity, it is important to take note of the various symmetries present in the relevant equations. These symmetries can either be enforced numerically, to save computational resources, or be used as a check to make sure that a numerical code is producing physically valid results. For quasi-equilibrium binary BHNS systems, virtually all gravitational formalisms will produce a configuration that is equatorially symmetric so long as the NS spin axes are parallel to the orbital angular momentum axis, regardless of whether the NS is corotating, counterrotating, or irrotational. If we fix the z-axis to be parallel to the various angular momenta mentioned above, we find that the transformation z → −z, with a corresponding reflection for all vector and tensor quantities, leaves the hydrostatic equations and the metric invariant, and is compatible with the velocity field of the initial configuration as well. For quasi-equilibrium configurations evaluated using full GR in the Kerr-Schild metric, this is the only symmetry plane of note present in the initial data. In our calculations, this symmetry is enforced at the code level: each SPH particle is treated as if it were a pair of particles, each of half the total mass, with one copy lying above the equatorial plane and one below. All spectral decompositions include this symmetry as well, setting to zero all spherical harmonics that are incompatible with a fully symmetric description. We show here that there is an additional symmetry plane for the case of equilibrium BHNS binaries in conformally flat gravity, because the formalism is timesymmetric. Directions are defined such that the axis of separation between the BH and NS lies along the xaxis, and the orbital angular momentum points in the z-direction. Thus, the y-direction represents the direction of motion for each object. Reversing the time direction requires us to perform two operations in order to maintain invariance. First, we must invert all vector quantities, most notably the velocity and the shift vector. Second, we must perform an inversion in the y-direction, to account for the inversion of our angular coordinate φ → −φ. We find the following relations are compatible with the initial velocity field, and leave our hydrostatic equations invariant: Scalars : f (x, y, z, t) = f (x, −y, z, −t), (B1) Vectors : [v x , v y , v z ](x, y, z, t) = − [v x , −v y , v z ](x, −y, z, −t). (B2) Evaluating the expressions above at t = 0 yields the symmetries in the y-direction for our initial data. This symmetry can be extended to tensor quantities as well: two index tensor elements satisfy the relation T ij (x, −y, z) = (−1) 1+Ny T ij (x, y, z), where N y is the number of "y" indices present for a given element. The additional factor of −1 is necessary to describe the inversion properties under time symmetry. Equatorial symmetry can be described in the same language, T ijk... (x, y, −z) = (−1) Nz T ijk... (x, y, z). It is straightforward to check that these symmetries are compatible with all equations governing the construction of quasi-equilibrium NS binaries in CF gravity. To do so, we note that any tensorial operation on fields satisfying these symmetries will maintain these symmetries, including gradients and inner products. First, we note that the matter sources in the equations for the conformal factor and lapse function, Eqs. (25) and (26), E, P , and S, as scalars, are symmetric in both y and z. The same pattern follows for scalar densities and quantities like u 0 . As an immediate consequence, the lapse and conformal factor share the same symmetries, since the only other source terms involve K ij K ij , itself a scalar. The equilibrium velocity field is v = Ω × r = [−Ωy, Ωx, 0], which satisfies [v x , v y , v z ](x, −y, z) = −[v x , −v y , v z ](x, y, z).(B3) Equatorial symmetry is satisfied trivially. In the CF case, the BH component of the shift vector is zero, and also satisfies the symmetry relations trivially. Since the shift equation, Eq. (24) depends only on the velocity field and other tensors, we can extend the symmetry to describe all quantities present in the equation. This statement cannot be made for the Kerr-Schild metric, and serves as the simplest example of how time-symmetry fails. For the Kerr-Schild case, β i BH−KS ∝ x i → [β x , β y , β z ](x, −y, z) = [β x , −β y , β z ](x, y, z).(B4) Thus, the BH contribution to the shift satisfies a different symmetry relation than the initial velocity field, and there is no global symmetry present. The physical reasons underlying this fact are discussed in more detail in Appendix E of [104]. APPENDIX C: PRESERVING STATIONARY EQUILIBRIUM DURING CF EVOLUTION We show here that the CF evolution equations maintain strict stationary equilibrium for initial data satisfying both the thin-sandwich equations and the integrated Euler equation. That is, if we have an initially uniformly rotating matter configuration that satisfies the condition h/u 0 = C, where C is constant throughout space at t = 0, and if the gravitational field satisfies the CTS equations, Eqs. (24), (25), and (26), the time derivatives of all quantities go to zero. Note that under these assumptions, this statement applies to both Eulerian and Lagrangian time derivatives, related by the expression d/dt = ∂/∂t + v j ∂/∂x j since the difference term goes to zero when v j = 0, which applies in the corotating frame. First, we note that the continuity equation, Eq. (15), and energy equation, Eq. (22), are trivially conserved when v i = 0, implying that ρ * and e * are conserved automatically. The only evolution equation that requires a more thorough look is the Euler equation, Eq. (19), which we reexpress for convenience in the equivalent form dũ i dt = − ∂ i P ρ * u 0 − αhu 0 ∂ i α +ũ j ∂ i β j + 2ũ kũk hu 0 ψ 5 ∂ i ψ. (C1) We can show that the RHS of this expression is zero by starting from the integrated Euler equation, which implies that ∂ i h u 0 − h∂ i u 0 (u 0 ) 2 = 0,(C2) From the relativistic Gibbs-Duhem relation, we know that [27] ∂ i h u 0 = ∂ i P ρ 0 u 0 . The gradient of u 0 is determined from the normalization u a u a = u 0 u 0 = −1, where we make use of the fact that u i = 0. From this, we conclude u 0 = (α 2 − ψ 4 δ ij β i β j ) −1/2 ,(C4) and differentiating, ∂ i u 0 = −(u 0 ) 3 (α∂ i α − ψ 4 δ jk β j ∂ i β k − 2ψ 3 δ jk β j β k ∂ i ψ). (C5) Inserting the expression u i = g 0i u 0 = −ψ 4 u 0 δ ij β j for a configuration with u i = 0, we see that h∂ i u 0 (u 0 ) 2 = − αhu 0 ∂ i α +ũ k ∂ i β k − 2ũ kũk hu 0 ψ 5 ∂ i ψ , (C6) and combining terms, 0 = ∂ i h u 0 = ∂ i h u 0 − h∂ i u 0 (u 0 ) 2 = ∂ i P ρ 0 u 0 + αhu 0 ∂ i α +ũ k ∂ i β k − 2ũ kũk hu 0 ψ 5 ∂ i ψ = − dũ i dt = 0.(C7) Thus, the RHS of the Euler equation, Eq. (19), is zero under these assumptions, andũ i is also invariant. We now consider the field equations for the CTS initial data. The fields are described by a set of linked elliptic equations, Eqs. (24), (25), and (26), whose source terms involve the fields themselves, as well as three quantities: E = (αu 0 ) 2 [ ΓP Γ − 1 + ρ 0 ] − P,(C8)U i = δ ij u j ψ 4 γ n (1 + Γǫ) ,(C9)S = 3P + ψ 4 (E + P )δ ij U i U j ,(C10) where ρ 0 , ǫ, and P are the standard relativistic mass density, internal energy density, and pressure, respectively. The Lorentz factor γ n ≡ αu 0 , can be solved implicitly, from Eq. (18). Coupled with our field equations, we have a set of six completely linked equations for six variables (γ n , α, ψ, β i ) and our conserved matter quantities (ρ * , e * ,ũ i ). So long as a unique solution exists for our choice of matter variables, we know that this solution will remain invariant so long as we choose our elliptic equation boundary conditions to be invariant. Thus, the RHS of all our evolution equations will remain zero, and the matter configuration will remain in equilibrium. FIG. 1 : 1The critical mass ratio q = M * /MBH as a function of the secondary's compaction C = M * /R * , for which mass transfer begins at the ISCO, taken here as a = 6MBH (solid line). For systems above the curve, mass transfer begins while the orbit is stable; for those below, the secondary may plunge into the BH before being tidally disrupted. Dashed vertical curves show characteristic compactions of a WD or NS; dotted horizontal curves show typical mass ratios for a 1M⊙ compact object orbiting a 10M⊙ stellar-mass BH, a 10 3 M⊙ IMBH, or a 10 6 M⊙ SMBH. Dot-dashed curves show where β, the ratio of the light crossing time the viscous timescale of the NS, equals unity, and where αVis, the nondimensional turbulent viscosity assumes its largest reasonable value for the turbulent viscosity [See Eqs.(7)and(9)]. Only configurations to the left of these curves can synchronize prior to merger. FIG. 2 : 2A schematic representation of the spectral methods computational domains used to solve the NS components of the field equations, Eqs. (25)- FIG. 3 : 3From top to bottom, the values of ρ * , β y , ψ, and α along the x-axis for the SPH (solid line) and grid-based (dashed line) data representing configuration B3, a NS with a Γ = 2 polytropic EOS and an initial binary separation a0/MBH = 11.1. The agreement is generally to within 1 − 2%. FIG. 4 : 4The values of ρ * , β y , ψ, and α along the x-axis for the SPH (solid line) and grid-based (dashed line) data representing configuration A5, a NS with a Γ = 1.5 polytropic EOS and an initial binary separation a0/MBH = 11.8. Conventions are the same asFig. 3. FIG. 5 : 5Value of the SPH expression for the integrated Euler equation constant h/u 0 as a function of the particle's radius from the NS center of mass for configurations B3 (top) and A5 (bottom), featuring a NS EOS with Γ = 2 and Γ = 1.5, respectively. Note that the proper value differs between the two cases. The standard deviation in both cases is < 0.01%, with maximum variation < 0.1%.physical evolution of the systems in question.A. BHNS mergers with a soft EOS: Γ = 1.5 FIG. 7 : 7Binary separation (top panel) and total mass lost from the NS (bottom panel) as a function of time for all runs calculated using the soft NS EOS, runs A1-A7, which have Γ = 1.5. The properties of the initial configurations, which differ only in their binary separation, are shown in FIG. 8 : 8Total mass lost inward toward the BH (top panel) and outward away from the BH (bottom panel) as a function of time, for the runs shown in FIG. 10 : 10Total mass lost inward toward the BH (top panel) and outward away from the BH (bottom panel) as a function of time, for the runs shown in FIG. 12 : 12Binary separation (top panel) and total mass lost from the NS (bottom panel) as a function of time for the runs calculated using the stiff NS EOS and dissipative GW radiation-reaction effects, runs B3a and B3b, which have Γ = 2. Conventions are as in FIG. 13 : 13Total mass lost inward toward the BH (top panel) and outward away from the BH (bottom panel) as a function of time, for the runs shown in FIG. 15 : 15Critical compactness for determining the stability of mass transfer from a relativistic polytropic NS companion as a function of the binary mass ratio, defined in terms of the polytropic index n (bottom), or the adiabatic index Γ ≡ 1 + 1/n. Mass transfer is unstable to the right of the curve, stable to the left. The square and triangle represent the positions of the two models we are evolving dynamically. FIG. 16 : 16Critical mass ratio for determining the stability of mass transfer from a relativistic polytropic NS companion as a function of the NS compactness, defined in terms of the polytropic index n (bottom), or the adiabatic index Γ ≡ 1 + 1/n. Mass transfer is unstable to the right of the curve, stable to the left. The heavy solid lines depict the maximum possible compaction for a given polytropic EOS. The square and triangle represent the positions of the two models we are evolving dynamically.ing R r = R NS , we can insert Eq. (A18) and finḋ = −143ξ −4 f (C) −4 (M T − M NS )M NS . (A24) : M ∝ t −3/11 ;ṁ ∝ t −14/11 , (A29) n = 1.0 (Γ = 2) : M ∝ t −3/7 ;ṁ ∝ t −10/7 , (A30) n = 0.5 (Γ = 3) : M ∝ t −15/23 ;ṁ ∝ t −38/23 , (A31) n = 0 (Γ → ∞) : M ∝ t −1 ;ṁ ∝ t −2 . TABLE I : IA comparison of our notation for various relativistic quantities to that found in a selection of previous works using the CF formalism:[27,53,54,75,94]. For those cases where no unique terminology was defined, we give the simplest equivalent algebraic form.QuantityHere FGR[53] Gourgoulhon[27] Oechslin[54] Wilson[75] Shibata TABLE II : IIParameters for our relaxed initial models. TN is the exact relativistic Keplerian period for a point mass about a BH, defined by Eq.(37).Run a/MBH ΩMBH T /MBH TN/MBHΓ = 1.5, q = 0.1, C = 0.042 A1 10.438 0.0258 243.8 243.8 A2 10.745 0.0247 254.0 253.7 A3 11.256 0.0232 270.6 270.3 A4 11.513 0.0225 279.1 278.8 A5 11.767 0.0218 287.5 287.3 A6 12.027 0.0212 296.5 296.1 A7 12.791 0.0194 323.1 322.5 Γ = 2, q = 0.1, C = 0.042 B1 10.539 0.0255 246.4 247.0 B2 10.961 0.0242 260.2 260.7 B3 11.093 0.0238 264.6 265.0 B4 11.648 0.0222 283.0 283.3 . During this time period, the NS, treated as a uniform density sphere, will lose mass from a shell whose depth is a distance equivalent to the infall rate from the beginning of the mass-transfer rate multiplied by half an orbital period. AcknowledgmentsJAF is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-0401533. TWB gratefully acknowledges support from the J. S. Guggenheim Memorial Foundation. 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[ "STRUCTURALLY STABLE NON-DEGENERATE SINGULARITIES OF INTEGRABLE SYSTEMS", "STRUCTURALLY STABLE NON-DEGENERATE SINGULARITIES OF INTEGRABLE SYSTEMS" ]	[ "E A Kudryavtseva ", "A A Oshemkov " ]	[]	[]	In this paper, we study singularities of the Lagrangian fibration given by a completely integrable system. We prove that a non-degenerate singular fibre satisfying the so-called connectedness condition is structurally stable under (small enough) real-analytic integrable perturbations of the system. In other words, the topology of the fibration in a neighbourhood of such a fibre is preserved after any such perturbation. As an illustration, we show that a simple saddle-saddle singularity of the Kovalevskaya top is structurally stable under real-analytic integrable perturbations, but structurally unstable under C ∞ -smooth integrable perturbations.MSC: 37J35, 37J39, 53D20, 70E40Consider the (local) Hamiltonian R n -action on M generated by the momentum map F , i.e. by the (local) flows of the vector fields X f i . Orbits of this (local) action will be called simply orbits. Note that all fibers are invariant under the (local) flows of the vector fields X f i , thus each fiber is a union of Affiliations: 2 KUDRYAVTSEVA, OSHEMKOV But, by the assumption, L is non-degenerate or satisfies the condition (ii) from Theorem 1.12, therefore the k e + k h + k f symplectic submanifolds K i ∩ U (O) are pairwise transversal and have symplectic pairwise intersections at m ∈ O. It follows from Lemma B.1 (a) that there exists a real-analytic symplectomorphism Φ : (U (m ), ω) → (R 2n , ω can ) such that the n − r = k e + k h + 2k f functions coincide with the quadratic functions h i of elliptic, hyperbolic and focus-focus types, resp., while the remaining r functions J i • F • Φ −1 (whose differentials are automatically linearly independent at m , since r = rank d(J • F )(m )) are linear functions λ 1 , . . . , λ r .Thus O is non-degenerate of Williamson type (k e , k h , k f ) and rank r , and the map J • F is a Vey momentum map at m . In fact, we have even more: it is a Vey momentum map at O (Definition 1.5),	10.1134/s106192082201006x	[ "https://arxiv.org/pdf/2112.00130v2.pdf" ]	244,773,182	2112.00130	f47ff62fe61fde266327e074aa6f494bc2d27d7c	STRUCTURALLY STABLE NON-DEGENERATE SINGULARITIES OF INTEGRABLE SYSTEMS E A Kudryavtseva A A Oshemkov STRUCTURALLY STABLE NON-DEGENERATE SINGULARITIES OF INTEGRABLE SYSTEMS In this paper, we study singularities of the Lagrangian fibration given by a completely integrable system. We prove that a non-degenerate singular fibre satisfying the so-called connectedness condition is structurally stable under (small enough) real-analytic integrable perturbations of the system. In other words, the topology of the fibration in a neighbourhood of such a fibre is preserved after any such perturbation. As an illustration, we show that a simple saddle-saddle singularity of the Kovalevskaya top is structurally stable under real-analytic integrable perturbations, but structurally unstable under C ∞ -smooth integrable perturbations.MSC: 37J35, 37J39, 53D20, 70E40Consider the (local) Hamiltonian R n -action on M generated by the momentum map F , i.e. by the (local) flows of the vector fields X f i . Orbits of this (local) action will be called simply orbits. Note that all fibers are invariant under the (local) flows of the vector fields X f i , thus each fiber is a union of Affiliations: 2 KUDRYAVTSEVA, OSHEMKOV But, by the assumption, L is non-degenerate or satisfies the condition (ii) from Theorem 1.12, therefore the k e + k h + k f symplectic submanifolds K i ∩ U (O) are pairwise transversal and have symplectic pairwise intersections at m ∈ O. It follows from Lemma B.1 (a) that there exists a real-analytic symplectomorphism Φ : (U (m ), ω) → (R 2n , ω can ) such that the n − r = k e + k h + 2k f functions coincide with the quadratic functions h i of elliptic, hyperbolic and focus-focus types, resp., while the remaining r functions J i • F • Φ −1 (whose differentials are automatically linearly independent at m , since r = rank d(J • F )(m )) are linear functions λ 1 , . . . , λ r .Thus O is non-degenerate of Williamson type (k e , k h , k f ) and rank r , and the map J • F is a Vey momentum map at m . In fact, we have even more: it is a Vey momentum map at O (Definition 1.5), Introduction In this work, we study singularities of integrable systems. Recall that an integrable system is specified by a triple (M 2n , ω, F ), where (M, ω) is a symplectic 2n-manifold and F = (f 1 , . . . , f n ) : M → R n is a momentum map, consisting of n almost everywhere independent functions f i that pairwise Poisson commute: ω(X f i , X f j ) = 0 for all i, j = 1, . . . , n, where X f i is defined by the rule ω(·, X f i ) = df i . The momentum map F naturally gives rise to a (singular) Lagrangian fibration on M whose fibers are connected components of the common level sets F −1 (a), a ∈ R n . One can also write this fibration as the quotient map F : M → B, where B is the set of connected components of F −1 (a), a ∈ R n , equipped with the quotient topology [15]. The space B is usually referred to as the bifurcation complex (or the unfolded momentum domain) of the system. Definition 1. 1. Two integrable systems (M i , ω i , F i ), i = 1, 2, will be called equivalent (resp. symplectically equivalent) if there exists a homeomorphism (resp. symplectomorphism) Φ : M 1 → M 2 and a homeomorphism φ : B 1 → B 2 such that φ • F 1 = F 2 • Φ. The systems will be called equivalent in a strong sense or left-right equivalent (resp. symplectically equivalent in a strong sense) if there exists a homeomorphism (resp. symplectomorphism) Φ : M 1 → M 2 and a diffeomorphism J : U 1 → U 2 such that J • F 1 = F 2 • Φ, for some neighbourhoods U i of F i (M i ) in R n , i = 1, 2. orbits. If the map F is proper then the flows of the vector fields X f i are complete, and we have a usual (well-defined) R n -action. By a singularity of an integrable system, we will mean (following [48,49,50,53,7,18,6]) the fibration germ at either a singular orbit (or its subset) or a singular fiber called local and semilocal singularities, resp. We note that, in literature, there is also the word "semiglobal" instead of our "semilocal", see e.g. [45]. We recall that a point m 0 ∈ M is called a singular point of this fibration if rank dF (m 0 ) < n. An orbit is called singular if it contains a singular point (so, all its points are singular). A fiber is called singular if it contains at least one singular point; the minimal rank of singular points belonging to this fiber is called rank of the fiber. Topology and geometry of integrable Hamiltonian systems with non-degenerate singularities have been studied from local [44,33,14,19,48,49,1,38,54], semilocal [15,34,48,49,2,36,27,28] and global viewpoints [15,2,4,54]. N.T. Zung developed a semilocal topological classification of nondegenerate singularities [48,49], and reduced a global topological (resp. symplectic) classification to rough topological (resp. symplectic) classification for "generic" integrable systems with singularities [53,54]. 1. 1. Structurally stable singularities. Our central object will be structurally stable singularities. Informally speaking, a singularity is called structurally stable if the topology of the fibration is preserved after any (small enough) real-analytic integrable perturbations of the system. Let us proceed with precise formulations. In this paper, we assume that the manifold M , the symplectic structure ω and the momentum map F are real-analytic. In the following definition, 0 denotes the C 0 -norm on the space of real-analytic pairs (ω C , F C ) on U C 0 . Here U C 0 denotes a (small) open complexification of a neighbourhood U 0 , while ω C , F C are holomorphic extensions of ω, F to U C 0 . Definition 1.2 ([30, Def. 4.1]). A compact subset K of a singular orbit or fiber (and the singularity at K) of an integrable system (M, ω, F ) will be called structurally stable (resp. symplectically structurally stable) if K has a neighbourhood U 0 and its (small) open complexification U C 0 such that, for any smaller neighbourhood U 1 with a compact closure U 1 ⊂ U 0 , there exists ε > 0 satisfying the following condition: for any real-analytic integrable perturbation (U 0 ,ω,F ) of (U 0 , ω\| U 0 , F \| U 0 ) such that ω C − ω C 0 + F C − F C 0 < ε, the integrable systems (U, ω\| U , F \| U ) and (Ũ ,ω\|Ũ ,F \|Ũ ) are equivalent (resp. symplectically equivalent), cf. Definition 1.1, for some neighbourhoods U,Ũ ⊆ U 0 containing U 1 . 1 In a similar way, structural stability is defined for an arbitrary compact subset K of M . If the integrable systems (U, ω\| U , F \| U ) and (Ũ ,ω\|Ũ ,F \|Ũ ) are equivalent (resp. symplectically equivalent) in a strong sense, the singularity will be called structurally stable (resp. symplectically structurally stable) in a strong sense. In a similar way, one defines structural stability under integrable perturbations of some class, e.g. the classes of C ∞ perturbations, G-symmetry-preserving perturbations (where G is a symmetry group of the singularity), parametric perturbations (analytically or smoothly depending on a small parameter) etc. A Morse critical point of a smooth function on a surface can be viewed as a simplest singularity of integrable Hamiltonian systems with 1 d.f. It is well known that, due to the Morse lemma, Morse critical 1 The notion of structural stability is known for C 1 vector fields (or flows) that are defined on a compact domain U 0 and satisfy a transversality condition on the boundary of U 0 , in which case one has U 1 = U =Ũ = U 0 . However, for fibration germs, the domain U 0 is unfixed, and a transversality condition on ∂U 0 is often not fulfilled. We overcome these difficulties by using U 1 , U, Ũ . points are structurally stable, moreover they are symplectically structurally stable [9]. Non-degenerate singularities (cf. Sec. 1.2) are natural generalization of Morse critical points, and locally they are direct products of elliptic, hyperbolic, focus-focus and regular components (Theorems 1.3, 1.4). As we noted above, non-degenerate singularities have been extensively studied. Nevertheless, some questions on structural stability of semilocal non-degenerate singularities remained open until now, and we give solutions to them in this paper in the real-analytic case (Theorems 1.12, 1. 16 Below we mention some known results on structural stability of singularities: 1) Infinitesimal stability (i.e. stability under infinitesimal integrable deformations of the system [17,Def. 8]) was studied for 2-degrees of freedom integrable systems, namely: non-degenerate rank-0 and rank-1 singular points and a rank-1 parabolic singular point are infinitesimally stable [17,Def. 9, Theorems 2 and 3]. 2) Structural stability was proved [21,Proposition 3.6] for focus fibres of any dimension, satisfying connectedness condition (iii, iv) (or (v, vi)) of Theorem 1.12 (such singularities were called irreducible in [21]). 3) Structural stability under "component-wise" C ∞ integrable perturbations was proved [41] for saddle-saddle fibers satisfying connectedness condition (v, vi) of Theorem 1. 12. 4) Symplectic structural stability in a strong sense of non-degenerate compact orbits is known in real-analytic case [30, Example 4.2 (A)] (see also Theorem 2.1). 5) In contrast to elliptic singularities (which are symplectically structurally stable due to the Eliasson Theorem 1.3), simple semilocal singularities of hyperbolic and focus-focus types are symplectically structurally unstable. This follows from the presence of "moduli" in their symplectic classifications [11,45]. Moreover, "moduli" also appear even in smooth classification for some classes of focus singularities of arbitrary dimension [5], which therefore are smoothly (and, hence, symplectically) structurally unstable. 6) For a parabolic singular point (cf. I in Fig. 2 (b)), structural stability and C ∞ -smooth structural stability follow from [33]. In the analytic case, symplectic structural stability of a parabolic point follows from [43,Theorem 3] (note that a parabolic point is infinitesimally non-degenerate, see [47,Theorem 5.25] for a proof). 7) Parabolic orbits and cuspidal tori are structurally stable due to [33], moreover they are C ∞smoothly structurally stable due to [31]. However they are not symplectically structurally stable, because of the presence of "moduli" in their symplectic classifications (see [6] for real-analytic case, [32] for the smooth and real-analytic cases). 8) Structural stability under integrable perturbations preserving a Hamiltonian (S 1 ) n−1 -action (for ndegree of freedom integrable systems) was proved for many degenerate local singularities, e.g., parabolic orbits with resonances [22] (which are smoothly structurally stable when the resonance order is different from 4 [18,30]), their parametric bifurcations [18], periodic integrable Hamiltonian Hopf bifurcation [42,18] and its hyperbolic analogue [36,Sec. 2], periodic integrable Hamiltonian Hopf bifurcations with resonances and their parametric bifurcations [12], normally-elliptic parabolic orbits [8], normallyhyperbolic parabolic orbits etc. The above F -preserving Hamiltonian (S 1 ) n−1 -action is generated by n−1 functions, some of which are real-analytic functions multiplied with √ −1 (in the real-analytic case) [30,Example 3.12]. It is conjectured in [30,Example 4.2 (B)] that, using "hidden" torus actions, one can prove structural stability in a strong sense of the singular orbits mentioned above in real-analytic case. 9) Structural stability under real-analytic parametric integrable perturbations can be proved for the singularities mentioned in item 7 from above, via the convergence of the Birkhoff normal form [55] and its analytic dependence on the perturbation parameters. In this paper, we prove (Theorem 1.16) that a non-degenerate semilocal singularity is structurally stable under real-analytic integrable perturbations, provided that it satisfies the connectedness condition (Definition 1.6). We also give several criteria (Theorem 1.12 and Corollary 1.13) for a semilocal singularity to satisfy the required assumptions (connectedness condition and non-degeneracy). As an illustration, we show that a saddle-saddle singularity of the Kovalevskaya top (and an arrangement of semilocal singularities containing this singularity) is structurally stable under real-analytic integrable perturbations, but structurally unstable under C ∞ -smooth integrable perturbations (Example 4.1). A singular point m 0 of rank 0 is called non-degenerate (cf. e.g. [14,10,35,49,2]) if the linearizations A j of the Hamiltonian vector felds X f j at the singular point span a Cartan subalgebra of the Lie algebra of the Lie group Sp(T m 0 M, ω\| m 0 ) Sp(2n, R), i.e. the operators A 1 , . . . , A n span an n-dimensional commutative subalgebra and there exists a linear combination A = n j=1 c j A j , c j ∈ R, having a simple spectrum: \| Spec A\| = 2n. A singular point m 0 of rank r is called non-degenerate (cf. e.g. [10]) if the rank-0 singular point of the corresponding reduced integrable Hamiltonian system with n − r degrees of freedom (obtained by local symplectic reduction under the action of f 1 , . . . , f r such that df 1 ∧ · · · ∧ df r \| m 0 = 0) is non-degenerate. A singular orbit (respectively, fiber) is called non-degenerate if each singular point contained in this orbit (fiber) is non-degenerate. Notice that, for an orbit, this condition holds automatically if at least one of its points is non-degenerate, but for a fiber it is not the case. The following assertion (known as Eliasson's theorem) is formulated for reader's interest, but it is not used any further in this article. Its proof is known for non-degenerate corank 1, elliptic, and focus-focus corank 2 singularities [9,14,46,38] (more specifically, the rank 0 case was treated in [9,14,46], resp., and the general case follows from rank 0 case due to [38,Corollary 3.5]). It is not clear whether there exists in the literature a complete proof of this assertion in the general case for singularities of all types. For each non-degenerate singular point m 0 ∈ M , the fibration is locally symplectically equivalent to the direct product of a regular fibration and several copies of elliptic, hyperbolic and focus-focus singularities, i.e., to a canonical system (1) h s = λ s for 1 ≤ s ≤ r, h r+j = 1 2 (x 2 j + y 2 j ) for 1 ≤ j ≤ k e , h r+j = x j y j for k e + 1 ≤ j ≤ k e + k h , h r+j = x j y j + x j+1 y j+1 and h r+j+1 = x j+1 y j − y j+1 x j for j = k e + k h + 2i − 1, 1 ≤ i ≤ k f , (2) ω can = r s=1 dλ s ∧ dϕ s + n−r j=1 dx j ∧ dy j . Thus, the canonical momentum map is defined by regular components h j (1 ≤ j ≤ r), elliptic components h r+j (1 ≤ j ≤ k e ), hyperbolic components h r+ke+j (1 ≤ j ≤ k h ) and focus-focus pairs of components h r+ke+k h +2j−1 , h r+ke+k h +2j (1 ≤ j ≤ k f ). We say that the singular point m 0 has Williamson type (k e , k h , k f ) [49, Def. 2.3]. Notice that r + k e + k h + 2k f = n and r is the rank of m 0 . In real-analytic case, Theorem 1.3 admits the following strengthening. (O) = {0} r × R ro × (S 1 ) rc × {(0, 0)} n−r . Here V = D r × R ro × (S 1 ) rc × (D 2 ) n−r with the coordinates (λ s ) r s=1 , (ϕ s ) r s=1 and (x j , y j ) n−r j=1 ; Γ is a finite group (called the twisting group at O) that acts on V freely and component-wise; the action of Γ on (S 1 ) rc is by translations, its action on each hyperbolic disk D 2 is by multiplications by ±1, its action on the remaining components is trivial (i.e. on D r , on R ro , on each elliptic disk D 2 and on each focus-focus polydisk D 2 × D 2 ), and its action on (D 2 ) n−r is effective. Thus, due to Theorem 1.4, in analytic case, the momentum map itself is conjugated (left-right equivalent) to the canonical one. In non-analytic case, the fibrations are the same, but the momentum maps are not necessarily conjugated, even in the case of a point. Examples of such situations (so-called splittable singularities) can be found in [2, Fig. 1.9, 1.10, 9.63 and comments to and after them], [7, Sec. 5.3]. This is why a description of structurally stable singularities in the smooth case is more difficult than in analytic case. In this paper, we consider analytic case only. Definition 1. 5. The composition J •F from Theorem 1.4 (a) (resp. (b)) will be called a Vey momentum map at the point m 0 (resp. at the orbit O). We note that the non-regular components of the Vey momentum map at a rank-r point (after mutiplying some of them by √ −1) generate a (n − r)-torus action near the point, which shows that these components are well-defined up to additive constants, but these constants can be uniquely chosen in order to make the values of the generating functions equal 0 at fixed points of the action. A proof of Theorem 1.4 is given in App. B. Theorem 1.4 (a) was proved by J. Vey [44], and its equivariant generalization was proved by the first author [30, Lemma 6.2]. For a compact orbit O, Theorem 1.4 (b) was proved in [38, Theorem 2.1] for C ∞ and real-analytic cases (see also [30,Example 4.2 (A)] for real-analytic case), and its equivariant generalization was proved [38,Theorem 4.3] for C ∞ and real-analytic cases. 1. 3. The connectedness condition. Let L be a compact singular fiber (perhaps degenerate). In this subsection, we will assume that L is almost non-degenerate in the following sense: L consists of finitely many orbits, moreover if an orbit O 1 is contained in the boundary of an orbit O ⊂ L then rank of O 1 is less than rank of O. It is clear that if L is non-degenerate then it is almost non-degenerate. Definition 1. 6. We say that the singular fiber L (and the semilocal singularity at L) of rank r satisfies the connectedness condition if it contains a non-degenerate rank-r orbit O 0 ⊂ L such that each of the k h + k f subsets K i := {m ∈ U (L) \| d(J i • F )(m) = 0}, i ∈ {r + k e + a} k h a=1 ∪ {r + k e + k h + 2b} k f b=1 , is connected and contains any compact orbit O ⊂ L. Here U (L) denotes a small neighbourhood of L, (k e , k h , k f ) is Williamson type of a point m 0 ∈ O 0 , and J • F is a Vey momentum map (Definition 1.5) at m 0 (we note that J depends on the choice of the orbit O 0 ). In practice, for verifying that K i is connected, it is enough to check that, for each compact orbit O ⊂ L, K i contains a path γ i joining O to O 0 . Note that K i = {m ∈ U (L) \| df i (m) = 0}, provided that the bifurcation diagram of F \| U (O 0 ) has a standard form (cf. Remark 1.11). Denote by K O 0 i the connected component of K i containing O 0 . Remark 1. 7. We could require in Definition 1.6 that O is also contained in the subsets K i for elliptic and/or focus-focus components 0 . But this requirement automatically holds, since K i ⊇ L for elliptic components h i (see [49,Proposition 2.6 h i , i ∈ {r + a} ke a=1 ∪ {r + k e + k h + 2b − 1} k f b=1 , of the Vey momentum map J • F at m 0 ∈ O]), and K i ⊇ K O 0 i+1 for focus-focus pairs of components h i , h i+1 . Remark 1.8. Observe that K O 0 j is a symplectic Bott critical submanifold of the function J j • F (j > r), since K O 0 j (resp., its complexification (K O 0 j ) C ) is the fixed point set of the Hamiltonian S 1 -action generated by the function J j • F (resp., iJ C j • F C ) near K O 0 j , due to Lemma A.1 (a, b). Obviously, all points of K j are singular for the momentum map F \| U (L) . Thus, the singular point set of F \| U (L) contains i K i ⊇ i K O 0 i . If L is non-degenerate, the connectedness condition simply means that the singular point set of F \| U (L) is exhausted by i K O 0 i and K O 0 i = K i . Then the submanifold K i consists of points of rank ≤ 1 2 dim K i , and it is the closure of its open subset K i \ i =i K i consisting of non-degenerate rank-1 2 dim K i singular points. Definition 1.9. The singular values set of the momentum map F is called the bifurcation diagram of the momentum map. Definition 1.10 ([49, Def. 6.3], [2, Def. 9.7], [7]). We say that the singular fiber L (and the semilocal singularity at L) satisfies the non-splitting condition 2 if L is non-degenerate and the bifurcation diagram of the momentum map F restricted to a small neighbourhood U (L) of L coincides with the bifurcation diagram of the momentum map F restricted to a small neighbourhood U (O) of any compact orbit O ⊆ L. Remark 1. 11. Suppose that L satisfies the non-splitting condition (Definition 1.10). Then all compact orbits in L have the same (minimal for L) rank r and the same Williamson type (k e , k h , k f ), which will be called Williamson type of L. Moreover any orbit in L is diffeomorphic to R a+b × (S 1 ) r+b and has Williamson type (k e , k h − a, k f − b), for some a, b ∈ Z + [49, Propositions 2.6 and 3 .5]. Therefore (after replacing F by a Vey momentum map J • F at a compact orbit O 0 ⊂ L), we can assume that the bifurcation diagram of F \| U (L) has a standard form. Examples of non-degenerate semilocal singularities which do not satisfy the non-splitting condition (and, hence, the connectedness condition, cf. Corollary 1.13) can be found in [2, Fig. 9.60-9.63 and Comments to them], [29, Lemma 3, Remark 4], [25]. Theorem 1. 12. Suppose that L is a compact, almost non-degenerate singular fiber. If (i) the fiber L satisfies the connectedness condition (Definition 1.6), then the non-degeneracy of L is equivalent to the following: moreover (i, ii) implies the following: (ii) the k e + k h + k f submanifolds K i , i ∈ {r + a} ke+k h a=1 ∪ {r + k e + k h + 2b} k f b=1 , (iii) the fiber L satisfies the non-splitting condition (Definition 1.10). If L satisfies the non-splitting condition (iii), then the connectedness condition (i) is equivalent to the following: (iv) K i is connected for all i ∈ {r + k e + a} k h a=1 ∪ {r + k e + k h + 2b} k f b=1 (or, equivalently, for all i = r + 1, . . . , n, cf. Remark 1.7). If the singularity at L is of almost-direct-product type topologically (cf. [49,Def. 7.2] or [7]), i.e. (v) L is non-degenerate and its small neighbourhood is equivalent to the quotient (V 1 ×· · ·×V n−k f )/Γ of the direct product of r regular, k e elliptic, k h hyperbolic and k f focus-focus semilocal singularities by a free component-wise action of a finite group Γ, then the connectedness condition (i) is equivalent to the following: (vi) the action of Γ on each component V i is transitive on the set of its singular points. Corollary 1.13 (Criteria for connectedness condition and non-degeneracy). For any compact, almost non-degenerate singular fiber L, the following conditions are equivalent: • L is non-degenerate and satisfies the connectedness condition, • (i) and (ii); • (iii) and (iv); • (v) and (vi), where (i)-(vi) are the conditions from Theorem 1. 12. In Theorem 1.12 and in what follows, the simultaneous fulfillment of the conditions (i) and (ii) is denoted by (i, ii), and similarly for (iii, iv) etc. We note that, for saddle-saddle singularities, (iii, iv) simply means that both components of the l-type [3,37,2] of the singularity are connected. Also note that (iii) in (iii, iv) is essential, see an example in [2, Comment to Fig. 9.60]. In fact, (v, vi) appears in [41] for saddle-saddle singularities. The equivalence of (iii, iv) and (v, vi) for rank-0 focus singularities is proved in [21,Proposition 3.4]. Remark 1.14. In Theorem 1.12, the following implications are easy: (i, iii)=⇒(ii, iv), (iii, iv)=⇒(i) and (i, v)⇐⇒(v, vi). Thus, for proving Theorem 1.12, it is enough to prove the implications (i, ii)=⇒(iii) and (i, non-degeneracy of L)=⇒(ii). Proofs of Theorem 1.12 and Corollary 1.13 are given in App. A. By complexity of a compact fiber L, we will mean the number of compact orbits contained in this fiber (actually, by Remark 1.11, compact orbits in L coincide with orbits of the minimal rank, provided that the non-splitting condition holds). Example 1. 15. A topological classification of semilocal non-degenerate singularities satisfying the nonsplitting condition and having complexity ≤ c is known for the following Williamson types (k e , k h , k f ) (cf. Remark 1.11): For example, singularities of the following topological types satisfy the connectedness condition: A, B, A * , F 1 , four saddle-saddle [34] and two saddle-focus [36] singularities of complexity 1 • rank-0 saddle type, k e = k f = 0, with (k h , c) = (2, 2) [34,(3) B × B, (B × C 2 )/Z 2 , (B × D 1 )/Z 2 , (C 2 × C 2 )/(Z 2 + Z 2 ), B × F 1 , (B × F 2 )/Z 2 , eleven saddle-saddle singularities of complexity 2 [41] (4) (D 1 × D 1 )/Z 2 , (P 4 × P 4 )/D 4 , (C 2 × C 2 )/Z 2 , (C 1 × I 1 )/Z 4 , (K 3 × K 3 )/(Z 4 + Z 2 ), (C 1 × J 1 )/Z 4 , (C 1 × K 3 )/Z 4 , (C 1 × P 4 )/Z 4 , (D 1 × C 2 )/Z 2 , (C 2 × P 4 )/(Z 2 + Z 2 ) ( where the latter case corresponds to two different types of singularities), three saddle-focus [26] and one 4 d.f. focus (cf. [21,Proposition 3.5], [20,Sec. 8]) singularities of complexity 2 (5) (D 1 × F 2 )/Z 2 , (C 1 × F 4 )/Z 4 , (C 2 × F 2 )/Z 2 , (F 2 × F 2 )/Z 2 , and their almost-direct products with each other and with a regular component W reg = D 1 × S 1 (we note that an almost-direct product of singularities satisfying the connectedness condition, obviously, also satisfies it). Here A * := (B × W reg )/Z 2 . By abusing notations, we will preserve the notation V for a singularity of the topological type V × W reg . Some elementary semilocal singularities are shown in Fig. 1. (3)- (5), as well as their almostdirect products with each other and with regular components, are structurally stable in a strong sense under integrable real-analytic perturbations. A B C 1 C 2 D 1 I 1 J 1 K 3 P 4 F k This theorem immediately implies the following. A proof of Theorem 1.16 is given in Sec. 3. Theorem 1.16 for rank-0 focus singularities satisfying connectedness condition (iii, iv) (or (v, vi)) of Theorem 1.12 was proved in [21, Proposition 3.6]. Symplectic structural stability of local singularities Theorem 2.1 (Local symplectic structural stability). Suppose (M, ω, F ) is a real-analytic integrable system and we have one of the following situations: (a) K is a non-degenerate singular point of the system, (b) K is a compact subset of a non-degenerate singular orbit O, and the flows of the vector fields X f 1 , . . . , X fn are complete on O. Then the singularity at K (local singularity) is symplectically structurally stable in a strong sense (Definition 1. 2) under real-analytic integrable perturbations (but not necessarily under C ∞ integrable perturbations). In Theorem 2.1, we do not assume that the orbit O ⊇ K is compact. If O is compact, we can assume that K = O. If O is non-compact, we can assume, e.g., that K is a torus K 0 defined in the proof below. In Theorem 2.1, we do not consider the case of a non-compact K (e.g. when O is a non-compact singular orbit and K = O), because even the notion of structural stability in Definition 1.2 is given only for compact subsets K. Step 1. By Theorem 1.4, each local non-degenerate singularity (a fibration germ at a point, or at an orbit) can be reduced to a normal form by a local symplectomorphism Φ and a local diffeomorphism J. Let us prove persistence of Φ and J under (small) real-analytic integrable perturbations. Without loss of generality, we can and will assume that In this way, we see that the (S 1 ) n−ro -action and its normalization at the r c -torus K 0 ⊆ O in Theorem 1.4 (b) are persistent and rigid (resp.) under (small) integrable real-analytic perturbations. In detail, if the "perturbation" is ε-small, we can construct neighbourhoods U,Ũ ⊆ U (O) C of K 0 in M C each of which contains U 1 , a "perturbed" (S 1 ) n−ro -action on the neighbourhoodŨ , a "perturbed" real-analytic Vey momentum mapJ •F onŨ ∩ M and a "perturbed" real-analytic symplectomorphism Φ : (Ũ ∩ M,ω) → (V /Γ, ω can ) such thatΦ(Ũ ∩ M ) = Φ(U ∩ M ) andJ •F •Φ −1 has a canonical form (1). Moreover, the "perturbed" change (Φ,J) is O(ε)-close to the "unperturbed" change (Φ, J). K 0 ⊆ K ⊆ K 1 , where K 0 is an (S 1 ) rc -orbit, and K 1 is the union of r c -tori φ t 1 J 1 •F • · · · • φ We can extend the symplectomorphismΦ to the "perturbed" neighbourhoodŨ (K) := . . , h n ) on the wholeŨ (K), as required. We also haveΦ(Ũ (K)) = Φ(U (K)) for the similar "unperturbed" neighbourhood U (K) := t∈B 0,C φ t 1 J 1 •F • · · ·•φ tr õ Jr o •F (Ũ ∩M ) of K 1 using the "perturbed" Hamiltonian R ro -action generated byJ 1 •F , . . . ,J ro •F . ThusJ •F •Φ −1 = (h 1 , .t∈B 0,C φ t 1 J 1 •F • · · · • φ tr o Jr o •F (U ∩ M ) of K 1 ⊇ K. Step 2. Thus F \| U (K) andF \|Ũ (K) are conjugated via the symplectomorphismΦ −1 • Φ : U (K) →Ũ (K) and the diffeomorphismJ −1 • J : W →W : F = (J −1 • J) • F • (Φ −1 • Φ) −1 , which yields symplectic structural stability of the singularity at K in a strong sense (Definition 1.2). Definition 2.2. The compositionJ •F from the proof of Theorem 2.1 (a) (resp. (b)) will be called a perturbed Vey momentum map near the point m 0 (resp. near the orbit O). Structural stability of semilocal singularities In this section, we formulate Principle Lemma and derive Theorem 1. 16 Thus, Principle Lemma 3.1 shows that, in order to construct Vey momentum maps for all orbits in L, it is enough to construct it just for one compact orbit O 0 in L. Actually, Principle Lemma is very useful for symplectic classification of singularities under consideration, as we will show in the next work. We remark that it is not hard to prove that, if the properties (a) and (b) hold for all compact orbits in L, then they hold for all orbits in L (including non-compact ones), but proving the properties (a) and (b) for all compact orbits in L is more difficult and requires the assumptions on the fiber (namely, non-degeneracy and fulfillment of the connectedness condition). A proof of Principle Lemma is given in App. A. This lemma was proved in [21, proof of Theorem 4.1] for focus singularities satisfying connectedness condition (iii, iv) (or (v, vi)) of Theorem 1.16; it was used for proving that the equivalence and C ∞ -smooth equivalence actually coincide for such semilocal singularities [21, Theorem 4.1]. 3.1. Proof of Theorem 1. 16. Let U (O 0 ) and U (L) be small neighbourhoods of O 0 and L, resp. For proving Theorem 1. 16, it suffices to show that, if an integrable perturbation is small, then there exist a neighbourhoodŨ (L) of L close to U (L) and a (perhaps, non-analytic) homeomorphism Ψ :Ũ (L) → U (L) close to the identity such thatF =J −1 • J • F • Ψ. Step 1. Observe that the image and the "unperturbed" bifurcation diagram of a Vey momentum map J • F at a compact orbit O 1 ⊆ L are standard and are completely determined by the Williamson type of O 1 . By Theorem 2.1, every compact orbit is structurally stable in a strong sense under integrable real-analytic perturbations, thus its Williamson type is preserved under such perturbations. Thus, Principle Lemma 3.1 implies that the "unperturbed" bifurcation diagram of J • F \| U (L) is the same as the "unperturbed" bifurcation diagram of J • F \| U (O 1 ) , and the same as the "perturbed" bifurcation diagram ofJ •F \|Ũ (L) . In particular, the perturbed semilocal singularity atL :=F −1 (J −1 (0)) satisfies the non-splitting condition (Definition 1.10). Without loss of generality, we can and will assume that J andJ coincide with the identity. Due to Principle Lemma 3.1, for each singular orbit O ⊂ L, the perturbed fiberL contains a singular orbitÕ close to O and having the same rank, Williamson type and local bifurcation diagram as those of O. Step 2. Due to the Zung topological classification [49,Theorem 7.3] of non-degenerate semilocal singularities satisfying the non-splitting condition, the semilocal singularity at L is equivalent to the almost-direct product of several semilocal singularities of the following types: regular, elliptic, hyperbolic and focus-focus ones. The proof of [49,Theorem 7.3] uses the l-type of the singularity at L, which is the (unordered) collec- [37]. A key ingredient of [49, proof of Theorem 7.3] is to show that the Cl-type of the singularity at L is isomorphic to the Cl-type of an almost-direct-product singularity. tion (V 1 , . . . , V ke+k h +k f ) of symplectic foliated (2r + 2)-or (2r + 4)-submanifolds V i =   ke+k h a=1 a =i K r+a   ∩   k f b=1 ke+k h +b =i K r+ke+k h +2b   ⊂ U (L) with singular fibres K i = V i ∩ L (if r = 0, then (V i , K i ) are so- One can deduce from Step 1 that the semilocal singularities at L andL have naturally isomorphic Cl-types. Applying the same arguments 3 as in [49, proof of Theorem 7.3], one obtains that these singularities are equivalent in a strong sense, as required. Structurally stable non-degenerate singularities of the Kovalevskaya top The base of the Liouville fibration is called the bifurcation complex; it was introduced by A.T. Fomenko (the cell-complex K in [15,Sec. 5], the affine variety A in [13] called the unfolded momentum domain); it is a (branched) covering of the momentum domain F (M ) [13]. As pointed out by V.I. Arnold, it is interesting to investigate singularities of the bifurcation complex. S.P. Novikov stated the following important question: in which form does the bifurcation complex, as a topological invariant of the Liouville fibration, "feels" algebraicity of the momentum map F . g = 0, (b) 0 < g 2 < 1, (c) 1 < g 2 < 8/(3 √ 3), (d) 8/(3 √ 3) < g 2 < 2, (e) g 2 > 2.(R 1 ,R 2 ,R 3 ,S 1 ,S 2 ,S 3 ) with the Poisson structure (6) {S i , S j } = ε ijk S k , {R i , R j } = 0, {S i , R j } = {R i , S j } = ε ijk R k , where {i, j, k} = {1, 2, 3}, and ε ijk is either the sign of the permutation (i, j, k) if all the i, j, k are different, or 0 otherwise. If the principle moments of inertia (I 1 , I 2 , I 3 ) of the rigid body are proportional to (2, 2, 1) and the center of masses lies in the plane perpendicular to the symmetry axis, the rigid body is called the Kovalevskaya top. The corresponding Hamiltonian system is integrable, with first integrals H = 1 2 S 2 1 + S 2 2 + 2S 2 3 + R 1 , f 1 = R 2 1 + R 2 2 + R 2 3 , f 2 = S 1 R 1 + S 2 R 2 + S 3 R 3 , K = S 2 1 2 − S 2 2 2 − R 1 2 + (S 1 S 2 − R 2 ) 2 (here H is the Hamilton function, f i are Casimir functions of the Poisson bracket). Consider the restriction of this Kovalevskaya's system to a symplectic leaf M 4 g = {f 1 = 1, f 2 = g}, where g ∈ R is the "area constant" regarded as a parameter of the system. The bifurcation diagrams (together with the bifurcation complexes) of the momentum maps F g = (H, K)\| M 4 g : M 4 g → R 2 are shown in [24] and in Fig. 2. All local non-degenerate singularities are symplectically structurally stable in a strong sense under real-analytic integrable perturbations, due to Theorem 2.1. Let us describe them. Every open arc of the bifurcation diagram corresponds to one or two 1-parameter families of rank-1 singular orbits of elliptic or hyperbolic types. The vertices IV and V correspond to rank-0 singular points of hyperbolic-hyperbolic (if 1 = g 2 < 2), hyperbolic-elliptic (if g 2 > 2) and elliptic-elliptic types, respectively; these points form the fixed point set of the Z 2 -action on M 4 g given by the canonical involution (R 1 , R 2 , R 3 , S 1 , S 2 , S 3 ) → (R 1 , −R 2 , −R 3 , S 1 , −S 2 , −S 3 ) of the Kovalevskaya system. All semilocal non-degenerate rank-0 singularities also happen to be structurally stable in a strong sense under real-analytic integrable perturbations (although not all under C ∞ integrable perturbations), but some non-degenerate rank-1 singularities are not: • Due to [24] or Corollary 1.17, all open arcs of the bifurcation diagram (Fig. 2), except for C 2 , correspond to 1-parameter families of structurally stable in a strong sense (under real-analytic integrable perturbations) semilocal rank-1 singularities A, B, A * (Fig. 1). However the semilocal rank-1 singularities corresponding to points of the open arc C 2 (Fig. 2, 3) are structurally unstable under C ∞ integrable perturbations (this easily follows from the existence of a system-preserving free Hamiltonian S 1 -action near such a semilocal singularity [49,Theorem 4.1 Fig. 3. • Due to Corollary 1.17, the vertices IV and V correspond to structurally stable in a strong sense (under real-analytic integrable perturbations) semilocal rank-0 singularities. However the semilocal rank-0 singularity at IV is structurally unstable under C ∞ integrable perturbations if g 2 < 1 (since the singular fiber IV is contained in the closure of the family of singular fibers C 2 , which are structurally unstable under C ∞ integrable perturbations by above). ]). A related perturbation is shown in Note that the semilocal singularities corresponding to C 2 (and probably to IV , topologically (B×C 2 )/Z 2 , for g 2 < 1) are structurally stable under Z 2 -symmetry-preserving C ∞ integrable perturbations, since C 2 /Z 2 = B is structurally stable. Figure 3. Semilocal singularity C 2 and its perturbed fibration (with notations due to [2]). White circles transform to gray ones through a singular fiber, and after that gray circles transform to black ones. = C 2 = = B − B = The arrangement of semilocal singularities corresponding to any closed arc contained in the half-open arc C 2 ∪ {IV } and containing the point IV for g 2 < 1 is structurally stable in a strong sense under real-analytic integrable perturbations (even under Z 2 -symmetry-breaking perturbations, although not under C ∞ integrable perturbations, by above) and satisfies the assertion of Principle Lemma 3. 1. This can be proved using structural stability of IV under real-analytic integrable perturbations (see above) and Principle Lemma 3.1 for IV , by noticing that the assertion of Principle Lemma can be extended from IV to C 2 ∪ {IV } by analytic continuation. Since O 0 is non-degenerate and compact, it is symplectically structurally stable in a strong sense by Theorem 2.1. Thus, for a perturbed system, we have similar objectsŨ (O 0 ),Φ andJ close to U (O 0 ), Φ and J resp., whereΦ :Ũ (O 0 ) → R 2n is a perturbed local symplectomorphism, andJ :W → R n a perturbed diffeomorphism (as in the proof of Theorem 2.1) near the compact rank-r orbit O 0 . In particular, we have a perturbed Vey momentum mapJ •F near O 0 (Definition 2.2). Denote K i ∩ L by K i (for r + 1 ≤ i ≤ n). Due to Remark 1.14, for proving Theorem 1.12, it is enough to prove the implications (i, ii)=⇒(iii) and (i, non-degeneracy of L)=⇒(ii). Principle Lemma 3.1 and Theorem 1.12 readily follow from the following lemma. Lemma A. 1. Under the above assumptions, the following holds. (a) The functions J s • F, 1 ≤ s ≤ r + k e , J r+ke+k h +2j • F, 1 ≤ j ≤ k f (corresponding to regular and elliptic components and elliptic parts of focus-focus components at O 0 , cf. (7)) generate a Hamiltonian (S 1 ) r+ke+k f -action near the fiber L w.r.t. ω. The fiber L is fixed under the (S 1 ) ke -subaction of this action; each K r+i is a Bott critical points set of the function J r+i •F , 1 ≤ i ≤ k e . The "perturbed" functions (8)J s •F , 1 ≤ s ≤ r + k e ,J r+ke+k h +2j •F , 1 ≤ j ≤ k f , generate a Hamiltonian (S 1 ) r+ke+k f -action near the fiber L w.r.t.ω. (b) If K j corresponds to a hyperbolic component h j in (7), then K j is a Bott critical points set of the function J j • F , the function iJ C j • F C generates a Hamiltonian S 1 -action near K j w.r.t. ω C , and the function iJ C j •F C generates a Hamiltonian S 1 -action near K j w.r.t.ω C . If K j corresponds to focus-focus components h j−1 , h j in (7), then K j is a Bott critical point set of each function J j−1 • F and J j • F , the functions iJ C j−1 • F C and J C j • F C generate a Hamiltonian (S 1 ) 2 -action near K j w.r.t. ω C , and the functions iJ C j−1 •F C andJ C j •F C generate a Hamiltonian (S 1 ) 2 -action near K j w.r.t.ω C . (c) Suppose that L is almost non-degenerate (see Subsec. 1.3), and L is either non-degenerate or satisfies the condition (ii) from Theorem 1. 12. Suppose that an orbit O ⊂ L has rank r and lies in k h of the k h subsets K j corresponding to hyperbolic components h j and in k f of the k f subsets K j corresponding to focus-focus components h j−1 , h j in (7). Then r + k e + k h + 2k f = n; O is nondegenerate of Williamson type (k e , k h , k f ) and rank r = r + k h − k h + 2(k f − k f ) ≥ r. In particular, the (S 1 ) r -action generated by J 1 • F, . . Proof. (a) Observe that the functions (8) from the Vey presentation generate a Hamiltonian (S 1 ) r+ke+k faction onŨ (O 0 ) w.r.t.ω, moreover the (S 1 ) r -subaction is locally-free onŨ (O 0 ), and O 0 is fixed under the unperturbed (S 1 ) ke+k f -subaction. Since L is compact, the time-2π map of the flow of each vector field XJ s•F , 1 ≤ s ≤ r + k e or s = r + k e + k h + 2j, 1 ≤ j ≤ k f , is well-defined on a neighbourhood of L. Since this map is the identity on a U (O 0 ), and L is connected, it follows by uniqueness of analytic continuation that it is the identity on a neighbourhood of L. Thus, the functions (8) generate a Hamiltonian (S 1 ) r+ke+k f -action on some neighbourhood of L. Due to [49,Proposition 2.6], the fiber L is fixed under the unperturbed (S 1 ) ke -subaction. (b) Suppose that h j is a hyperbolic component in (7). It follows from the Vey presentation (1), (2), (7) that the function iJ C j • F C generates a Hamiltonian S 1 -action on some neighbourhood U (O 0 ) C of O 0 in M C w.r.t. ω. Similarly, the perturbed function iJ C j •F C generates a Hamiltonian S 1 -action on some neighbourhoodŨ (O 0 ) C w.r.t.ω. By definition of K j , we have O 0 ⊆ K j ⊆ L and d(J j • F ) = 0 at each point of K j . Therefore, O 0 is fixed under the S 1 -action generated by iJ C j • F C , and the time-2π map of the flows of the vector fields X iJ j •F and X iJ j •F are well-defined on some neighbourhood of K j in M C . Since this time-2π maps are the identity on U (O 0 ) C and K j is connected, it follows by uniqueness of analytic continuation that these time-2π maps are the identity on some neighbourhood of K j . Thus, the functions iJ j • F and iJ j •F generate Hamiltonian S 1 -actions on some neighbourhoods U (K j ) C andŨ (K j ) C of K j . We also showed that (K j ) C is the fixed points set on U (K j ) of the unperturbed S 1 -action, whence K j is a symplectic submanifold and it is a Bott critical points set of J j • F . If h j−1 , h j are focus-focus components in (7), then similar arguments show that • the functions iJ C j−1 • F C , J C j • F C generate a Hamiltonian (S 1 ) 2 -action on some neighbourhood U (K j ) C of K j in M C w.r.t. ω C , • (K j ) C is the fixed points set of this (S 1 ) 2 -action, • the perturbed functions iJ C j−1 •F C ,J C j •F C generate a Hamiltonian (S 1 ) 2 -action on some neigh- bourhoodŨ (K j ) C of K j w.r.t.ω C . (c) By (a), the "regular" and the "elliptic" Vey functions J s •F , 1 ≤ s ≤ r+k e , generate a Hamiltonian (S 1 ) r+ke -action on some neighbourhood of L, and the (S 1 ) ke -subaction is fixed on L. By (b), we have k h "hyperbolic" functions (9) iJ C j • F C , 1 ≤ j ≤ k h , and 2k f "focus-focus" pairs of functions Let us first show that r + k e + k h + 2k f = n and O is non-degenerate of Williamson type (k e , k h , k f ). Choose a point m ∈ O. Consider two cases. (10) iJ C j −1 • F C , J C j • F C , k h + 1 ≤ j ≤ k h + k f ,r+i • F , 1 ≤ i ≤ k e , iJ C r+ke+j • F C , 1 ≤ j ≤ k h , and iJ C r+ke+k h +2j−1 • F C , J C r+ke+k h +2j • F C , 1 ≤ j ≤ k f . Therefore r = rank dF (m ) = rank d(J • F )(m ) ≤ r = rank dF (m 0 ) . But m 0 has minimal rank on L by connectedness condition. Therefore r = r, thus the functions J s • F , 1 ≤ s ≤ r, generate a locally-free (S 1 ) r -action on O. Thus m is a rank-r point of the Hamiltonian (S 1 ) n -action generated by the above functions (having the form J s • F , iJ j • F ). By [30,Theorem 3.10], there exists a real-analytic symplectomorphism Φ : (U (O), ω) → (V /Γ , ω can ) such that J s • F • Φ −1 = h s , 1 ≤ s ≤ r, the regular components h 1 , . . . , h r on V , while J i • F • Φ −1 = n j=r+1 k ij h j , r + 1 ≤ i ≤ n, for some integers k ij and n − r quadratic functions h i of elliptic, hyperbolic and focus-focus types (the number of components h i of each type is not necessary k e , k h , k f as in (7)). As we showed above, each K C i (i > r) is a fixed point set of the corresponding S 1 -subaction (resp. (S 1 ) 2 -subaction) of the (S 1 ) n−r -action on U (O) C , and the type of this subaction is given by h i (resp. h i−1 , h i ) in (7). Since, by assumption, L either is non-degenerate or satisfies the condition (ii) from Theorem 1.12, we conclude that J i • F • Φ −1 = h i , r + 1 ≤ i ≤ n, after changing Φ if necessarily. In particular, O is non-degenerate and has Williamson type (k e , k h , k f ) and rank r = r, moreover J • F is a Vey momentum map at O. Case 2: O is noncompact. Thus its closure O contains a compact orbit O 1 ⊂ L (because L is almost non-degenerate). By Case 1, O 1 is non-degenerate of rank r and Williamson type (k e , k h , k f ), moreover O 1 lies in each K i , r + 1 ≤ i ≤ n, and there exists a real-analytic symplectomorphism Φ 1 : (U (O 1 ), ω) → (V /Γ 1 , ω can ) such that J i • F • Φ −1 1 = h i , 1 ≤ i ≤ n. Since O 1 is non-degenerate and O 1 ⊂ O, we conclude that O is non-degenerate too, moreover (by Remark 1.11) it has Williamson type (k e , k h − a, k f − b) and is diffeomorphic to R a+b × (S 1 ) r+b , for some a, b ∈ Z + . Since O 1 lies in each K i , r + 1 ≤ i ≤ n, it follows that k h = k h − a and k f = k f − b. Thus r = r + a + 2b and r + k e + k h + 2k f = r + k e + k h + 2k f = n, as required. We also obtain that the functions J i • F , 1 ≤ i ≤ r, generate a locally-free (S 1 ) r -action on O. It remains to show that contained in k e + k h + k f subsets K C i , i ∈ {r + 1, . . . , r + k e } ∪ { j } k h j=1 (resp. i ∈ { j } k h +k f j=k h +1 ), see (9),(10) , each of which is a fixed point set of the corresponding S 1 -subaction (resp. (S 1 ) 2 -subaction) of the (S 1 ) n−r -action on U (O) C , and the type of this subaction is given by h i (resp. h i−1 , h i ) in (7). since, due to (a), the remaining r = r + k h − k h + 2k f − 2k f functions (namely, the functions J s • F , 1 ≤ s ≤ r, and the remaining k h − k h hyperbolic functions J j • F and k f − k f focus-focus pairs of functions J j−1 • F, J j • F ) generate an R k h −k h +k f −k f × (S 1 ) r+k f −k f -action near L, which is locally-free near O, and we can use this action for extending the local symplectomorphism Φ to a neighbourhood of the cylinder O ≈ R k h −k h +k f −k f × (S 1 ) r+k f −k f , as in the proof of Theorem 1.4 (b). Due to (a) and (b) , the n − r = k e + k h + 2k f perturbed functionsJ r+i •F , 1 ≤ i ≤ k e , iJ j •F , 1 ≤ j ≤ k h , and iJ j −1 •F ,J j •F , k h + 1 ≤ j ≤ k h + k f , generate a "perturbed" Hamiltonian (S 1 ) n−r -action near O. Since the mapJ •F is close to J • F ,O ≈ R k h −k h +k f −k f × (S 1 ) r+k f −k f using the perturbed locally-free R k h −k h +k f −k f × (S 1 ) r+k f −k f -action generated byJ s •F , 1 ≤ s ≤ r, and the remaining k h − k h hyperbolic functionsJ j •F and k f − k f focus-focus pairs of functionsJ j−1 •F ,J j •F . Proof of Corollary 1. 13. We have to prove the equivalence of four conditions. It follows from Theorem 1.12 that all of these conditions except for the last one are pairwise equivalent, and the last one implies the previous ones. Moreover the last one follows from the previous one, provided that (iii) implies (v). It is left to note that the latter implication is the Zung topological classification [49,Theorem 7.3]. Appendix B. Local normal form and its rigidity Here we give a proof of Theorem 1.4 using the following lemma, which we also use (in Sec. 2 (11) J r+1 • F, . . . , J r+ke • F, iJ r+ke+1 • F, . . . , iJ r+ke+k h • F, iJ r+ke+k h +1 • F, J r+ke+k h +2 • F, . . . , iJ n−1 • F, J n • F, for some real-analytic functions J j : W → C with J j (z) = z j + O(\|z\| 2 ) as z → 0, r + 1 ≤ j ≤ n. (b) The above (S 1 ) n−r -action and its normalization are persistent and rigid (resp.) under real-analytic integrable perturbations in the following sense. Suppose we are given a neighbourhood U 1 of m 0 in M C and a neighbourhood W 1 of the origin in C n having compact closures U 1 ⊂ U and W 1 ⊂ W , and an integer k ∈ Z + . Then there exists ε > 0 such that, for any ("perturbed") real-analytic integrable Hamiltonian system (U ∩ M,ω,F ) whose holomorphic extension to U is ε−close to (U, ω C , F C ) in C 0 -norm, the following properties hold. On some neighbourhoodŨ ⊇ U 1 , there exists a uniqueF C -preserving Hamiltonian (w.r.t. the "perturbed" symplectic structureω) (S 1 ) n−r -action generated by functions (12)J r+1 •F , . . . ,J r+ke •F , iJ r+ke+1 •F , . . . , iJ r+ke+k h •F , iJ r+ke+k h +1 •F ,J r+ke+k h +2 •F , . . . , iJ n−1 •F ,J n •F , whereJ j (z 1 , . . . , z n ), r + 1 ≤ j ≤ n, are real-analytic functions on some neighbourhoodW ⊇ W 1 that are O(ε)−close to J j (z 1 , . . . , z n ) in C k -norm. There exists a real-analytic symplectomorphism Φ : . . , h n ), whereJ s (z) = z s for 1 ≤ s ≤ r,J = (J 1 , . . . ,J n ) :W → R n . If the system depends on a local parameter (i.e. we have a local family of systems), moreover its holomorphic extension to M C depends smoothly (resp., analytically) on that parameter, then J and Φ can also be chosen to depend smoothly (resp., analytically) on that parameter. (Ũ ∩ M,ω) → (R 2n , ω can ) whose holomorphic extension toŨ is O(ε)−close to Φ C in C k -norm such thatJ •F •Φ −1 = (h 1 , . Proof. (a) We divide the proof into two steps. Step 1. We can extend the functions f 1 , . . . , f r to a system of local canonical coordinates Φ = (λ, ϕ, x, y) = (λ 1 , ϕ 1 , . . . , λ r , ϕ r , x 1 , y 1 , . . . , x n−r , y n−r ) : U → R 2n on a small neighbourhood U of m 0 such that f s = λ s for 1 ≤ s ≤ r, Φ(m 0 ) = 0, dΦ(m 0 ) = dΦ(m 0 ) and ω\| U = Φ * ω can (Darboux coordinates), see (2). Consider two cases. Case 1: r = 0. Consider the 1-parameter family of "rescaling" coordinate systems Φ ε = (x , y ) such that x = εx , y = εy , where ε > 0 is a small parameter. Without loss of generality, we can and will assume that F (m 0 ) = 0. By Hadamard's lemma, f j • Φ −1 ε (x , y ) = f j • Φ −1 (εx , εy ) = ε 2 f j (x , y , ε) for some real-analytic functions f j (x , y , ε). Clearly, ω = ε 2 Φ * ε (dx ∧dy ) where dx ∧dy := n j=1 dx j ∧dy j . Choose a small ε 0 > 0 such that Φ(U ) ⊇ B 0,ε 0 := {w ∈ R 2n \| \|w\| < ε 0 }. Denote U ε := Φ −1 (B 0,ε ) for 0 < ε ≤ ε 0 . Thus the rescaling diffeomorphism Φ ε : U ε → B 0,1 transforms the integrable Hamiltonian system x , y , ε), . . . , f n (x , y , ε)). (13) (U ε , ε −2 ω, ε −2 F ) to the integrable system (14) (B 0,1 , dx ∧ dy , F ), where F • Φ −1 ε = ε 2 F (x , y , ε), F (x , y , ε) := (f 1 ( Observe that the "unperturbed" system (i.e. (14) with ε = 0) (15) (B 0,1 , dx ∧ dy , F \| ε=0 = (h 1 , . . . , h n )) coincides with the linearization of the original system at m 0 , which has the canonical form by assumption of the lemma. Thus, on a small neighbourhood U = U ε of m 0 , our system (13) can be viewed as a system (14) obtained from the (canonical) "unperturbed" system (15) (11) where J j = J j (z 1 , . . . , z n ) are real-analytic functions such that J j (z 1 , . . . , z n ) = z j + O(\|z\| 2 ), 1 ≤ j ≤ n. 4 By [30, Theorem 3.10(a) or Lemma 6.2(a)], the latter (S 1 ) n -action is linearizable at m 0 ([30, Def. 3.1, 3.7]). In other words, there exists a real-analytic symplectomorphismΦ : (U, ω) → (R 2n , ω can ) sending the point m 0 to the origin, with dΦ(m 0 ) = dΦ(m 0 ), and transforming the momentum map J • F to a collection of quadratic functions on V = Φ(Û ), which does not depend on ε and, hence, coincides with (h 1 , . . . , h n ) from (15). Thus J andΦ have the required properties. Case 2: r > 0. One performs a local Hamiltonian reduction and reduces the problem to an rparameter family of integrable systems with n − r degrees of freedom, with a non-degenerate rank-0 point m 0 . This can be done by the same arguments as in the case of a compact orbit O (see [38,Sec. 4] or [30,Sec. 7]). In detail: on a small neighbourhood U 0 of m 0 , we can extend the functions f 1 , . . . , f r to a system of local canonical coordinates Φ 0 = (λ, ϕ, x, y) = (λ 1 , ϕ 1 , . . . , λ r , ϕ r , x 1 , y 1 , . . . , x n−r , y n−r ) : U 0 → R 2n such that f s = λ s for 1 ≤ s ≤ r, Φ 0 (m 0 ) = 0, and ω\| U = Φ * ω can (Darboux coordinates), see (2). Take a local disk P = {ϕ = 0} of dimension 2n − r that intersects the local orbit O through m 0 transversally at m 0 . Then the local disk {λ 1 = const, . . . , λ r = const} ∩ P near m 0 has an induced symplectic structure and induced functions f r+1 , . . . , f n that pairwise Poisson commute. Applying the case of a rank-0 point and parameters λ 1 , . . . , λ r , which is a parametric extension of Case 1 (such an extension is valid due to the parametric extensions [30, Theorems 2.2(b) and 3.10(b) or Lemma 6.2(b)] of [30, Theorems 2.2(a) and 3.10(a) or Lemma 6.2(a)]), we can define an F -preserving Hamiltonian (S 1 ) n−r -action on P C and local functionsx 1 ,ŷ 1 , . . . ,x n−r ,ŷ n−r on P , such that they form a local symplectic coordinate system on each local disk {λ 1 = const, . . . , λ r = const} ∩ P , with respect to which the Hamiltonian (S 1 ) n−r -action is linear and does not depend on the values of λ 1 , . . . , λ r . Moreover we havex(m 0 ) =ŷ(m 0 ) = 0, the local coordinates (x,ŷ) on the local disk {λ = 0} ∩ P have the same linearization at m 0 as (x, y). We extendx 1 ,ŷ 1 , . . . ,x n−r ,ŷ n−r to functions on U by making them invariant under the local Hamiltonian flows of X f 1 , . . . , X fr . Since dω = 0, it follows [38,Lemma 4.2] that the symplectic structure ω on U has the form ω = r s=1 dλ s ∧ d(ϕ s + g s ) + n−r j=1 dx j ∧ dŷ j , for some real-analytic functions g s on a neighbourhood of m 0 in U , that are invariant under the local Hamiltonian flows of X f 1 , . . . , X fr . Defineλ s := λ s = f s ,φ s := ϕ s + g s , and J s (z 1 , . . . , z n ) = z s for 1 ≤ s ≤ r. Then with respect to the coordinate systemΦ = (λ,φ,x,ŷ) on U , the symplectic form ω on U has the standard form and the 4 Indeed: J(F (Φ −1 (x, y))) = J(F (Φ −1 ε (x , y ))) = J(ε 2 F (x , y , ε)) = ε 2 J (F (x , y , ε), ε) for some real-analytic map J (z, ε) such that J(ε 2 z) = ε 2 J (z, ε). Hence the rescaling diffeomorphism Φ C ε conjugates the (S 1 ) n -action (U C , ε −2 ω C , ε −2 J C • F C ) with the (S 1 ) n -action (B C 0,1 , (dx ∧ dy ) C , J C • F C ). Hence the linearization of X Jj •F at m 0 is dΦ ε (m 0 )-conjugated with the linearization of X J j •F at 0 for any ε > 0. But the quadratic part of F at x = y = 0 does not depend on ε and coincides with F \| ε=0 = (h 1 , . . . , h n ), see (15). This implies that J j (z 1 , . . . , z n ) = z j + O(\|z\| 2 ) and, hence, J j (z 1 , . . . , z n ) = z j + O(\|z\| 2 ), 1 ≤ j ≤ n. Hamiltonian (S 1 ) n−r -action on P C is linear and does not depend on λ. This implies that ω =Φ * ω can and the functions . . , J n−1 • F •Φ −1 , iJ n • F •Φ −1 generating this linear Hamiltonian (S 1 ) n−r -action are quadratic functions in x j , y j and do not depend on λ. Clearly, the functions J 1 • F •Φ −1 , . . . , J n • F •Φ −1 have the canonical form (1), and by construction J s (z 1 , . . . , z n ) = z s for 1 ≤ s ≤ r, J s (z 1 , . . . , z n ) = z s + O(\|z\| 2 ) for r + 1 ≤ s ≤ n. J r+1 • F •Φ −1 , . . . , J r+ke • F •Φ −1 , iJ r+ke+1 • F •Φ −1 , . . . , iJ r+ke+k h • F •Φ −1 , J r+ke+k h +1 • F •Φ −1 , iJ r+ke+k h +2 • F •Φ −1 , . Step 2. It remains to prove the last assertion of (a). We will prove it for r = 0 (the case r > 0 can be reduced to the case r = 0 by a local Hamiltonian reduction, as in Step 1). Suppose Φ, Φ are two local symplectomorphisms at m 0 bringing J • F to the canonical form. Then Ψ := (Φ −1 • Φ ) 2 is a F -preserving real-analytic symplectomorphism of a neighbourhood of m 0 to M fixing m 0 and being homotopic to the identity in the space of F -preserving homeomorphisms. Take a regular point m 1 ∈ L C close to m 0 . Consider the rescaling diffeomorphism Φ ε : U ε → B 0,1 from Step 1. By Hadamard's lemma, the map Φ • Ψ • Φ −1 ε : B 0,1 → B 0,1 has the form εΨ ε , where Ψ ε : B 0,1 → B 0,1 is a 1-parameter family of real-analytic maps in x , y , ε. Clearly, Ψ ε preserves J (F (x , y , ε), ε) and dx ∧ dy . One checks that the unperturbed map Ψ 0 is linear and coincides with d(Φ • Ψ • Φ −1 )(m 0 ). Take a point m = (x , y ) ∈ B C 0,1 which is a regular point of the singular fiber of the unperturbed system (15). Without loss of generality, m is fixed under the unperturbed linear map Ψ 0 (this can be achieved by replacing Ψ with its composition with the time-1 map of the Hamiltonian flow generated by a linear combination of J j • F , 1 ≤ j ≤ n). We can extend the functions λ j = J j C • F C to a local system of canonical holomorphic coordinates λ j , µ j near m (Darboux coordinates) depending analytically on ε such that µ j (m ) = 0. Since m is fixed under the unperturbed map Ψ 0 , it follows that, in these coordinates the perturbed map Ψ C ε on a neighbourhoodŨ (m ) C of the point m has the form (λ, µ) → (λ, µ + ∂S C ε ∂λ ), for some real-analytic function S ε = S ε (λ) on a neighbourhood of the origin such that S 0 (0) = 0 and ∂S 0 (0) ∂λ j = 0. Thus, onŨ (m ) C , the map Ψ C ε coincides with the time-1 map of the Hamiltonian flow generated by S C ε (J j C (F C (x , y , ε), ε)). Choose a point m ε = (x ε , y ε ) ∈Ũ (m ) C with λ j (m ) = µ j (m ) = 0. Thus, Ψ C coincides with the time-1 map of the Hamiltonian flow generated by S • J • F , where S(z) = ε 2 S ε (z/ε 2 ) on a small neighbourhood of the point m ε := Φ −1 (εm ε ) ∈ L C . Since the analytic symplectomorphisms Ψ C and (φ 1 S•J•F ) C are well-defined on some neighbourhood U of m 0 in M C and coincide with each other on a neighbourhood of the path m u ∈ L C ∩ U , 0 ≤ u ≤ ε, by analytic continuation they must coincide on the whole U . This yields Lemma B.1 (a). (b) On a neighbourhoodŨ of m 0 close to U , we can extend the "perturbed" functionsf 1 , . . . ,f r to a "perturbed" system of local canonical coordinatesΦ = (λ,φ,x,ỹ) :Ũ → R 2n such thatf s =λ s for 1 ≤ s ≤ r andω\|Ũ =Φ * ω can (perturbed Darboux coordinates), see (2). We obtain a "perturbed" r-parameter family of integrable systems with n − r degrees of freedom, with parametersλ 1 , . . . ,λ r . Since the "unperturbed" system with zero values of the parameters (λ 1 = · · · = λ r = 0) admits a non-degenerate rank-0 point m 0 , we can derive the assertion (b) from Case 1 of (a) similarly to deriving Case 2 of (a), by applying to the "perturbed" system the "perturbative" Theorem 1 . 3 ( 13Smooth local normal form). Theorem 1 . 4 ( 14Real-analytic local normal form). In real-analytic case, for each non-degenerate singular point m 0 ∈ M , (a) There exists a neighbourhood U of m 0 , in which the system is symplectically equivalent in a strong sense to (1), (2), i.e. there exist a real-analytic symplectomorphismΦ = (λ 1 , ϕ 1 , . . . , λ r , ϕ r , x 1 , y 1 , . . . , x n−r , y n−r ) : (U, ω) → (R 2n , ω can )and a real-analytic diffeomorphism germ J = (J 1 , . . . , J n ) : (R n , F (m 0 )) → (R n , 0) such that Φ(m 0 ) = 0 and the map J • F • Φ −1 = (h 1 , . . . , h n ) has a canonical form (1). (b) If the flows of X f 1 , . . . , X fn are complete on the orbit O of m 0 , then this orbit is diffeomorphic to a cylinder R ro × (S 1 ) rc with r o + r c = r, and there exist a neighbourhood U (O) of O, a real-analytic symplectomorphism Φ : (U (O), ω) → (V /Γ, ω can ) and a real-analytic diffeomorphism germ J = (J 1 , . . . , J n ) : (R n , F (m 0 )) → (R n , 0) such that the map J • F • Φ −1 = (h 1 , . . . , h n ) has a canonical form (1) and Φ from Definition 1.6 and Remark 1.8 are pairwise transversal at every compact orbit O ⊆ L and have symplectic pairwise intersections at O; 3, 37] (43 topological types of saddle-saddle singularities, where B × B and (B × C 2 )/Z 2 appear in the Kovalevskaya top, (B × D 1 )/Z 2 appears in the Goryachev-Chaplygin-Sretenskii case, (C 2 × C 2 )/Z 2 occurs in the Clebsch case), with (k h , c) = (3, 1) [23], [2, Theorem 9.13] (32 topological types of saddle-saddlesaddle singularities), and with any (k h , c) [39, 40]; • rank-r center-focus type, k h = 0, with any (r, k e , k f , c) [20] (e.g. F 1 appears in the Lagrange case, F 2 occurs in the Clebsch case); • rank-0 saddle-focus type, k e = 0, with (k h , k f , c) = (1, 1, 1) [36] and with (k h , k f , c) = (1, 1, 3) [26] (11 topological types of saddle-focus singularities of complexity 2, and 21 topological types of saddle-focus singularities of complexity 3). Figure 1 . 1Elementary semilocal singularities: elliptic, some hyperbolic and focus-focus singularities. Corollary1.17 (Structural stability of simple singularities). Suppose L is a compact non-degenerate singular fiber containing a unique compact orbit O 0 . Then the semilocal singularity at the fiber L is structurally stable in a strong sense under real-analytic integrable perturbations (but not necessarily under C ∞ integrable perturbations, see Example 4.1 for the saddle-saddle singularity (B × C 2 )/Z 2 ). ( a ) aSuppose m 0 ∈ M is a non-degenerate rank-r point and K = {m 0 }. Due to LemmaB.1 (b), the local diffeomorphisms Φ and J bringing the local momentum map at m 0 to the normal form (1),(2) are persistent under real-analytic integrable perturbations.(b) Suppose O ⊆ L is a non-degenerate rank-r orbit, and K ⊆ O its compact subset. By Theorem 1.4 (b), there exist a neighbourhood U (O) of O in M , a symplectomorphism Φ : (U (O), ω) → (V /Γ, ω can ) and a diffeomorphism J = (J 1 , . . . , J n ) : W → R n such that J • F • Φ −1 has a canonical form(1), where V = D r ×R ro ×(S 1 ) rc ×(D 2 ) n−r with the standard symplectic form(2), and W ⊂ R n is a neighbourhood of F (O) in R n . Since we can take a smaller neighbourhood if necessary, we can assume that we have the Hamiltonian (S 1 ) n−ro -action on a neighbourhoodU (O) C ⊃ U (O) of O in M C generated by n − r o functions having the form J C r • F C and iJ C j • F C .Moreover, the orbit O is fixed under the (S 1 ) n−rsubaction, and the (S 1 ) rc -subaction is locally-free onO. We want to show that Φ and J are persistent on a neighbourhood of K under integrable real-analytic perturbations. Jr o •F (K 0 ) forming a r o -parameter family with parameters t = (t 1 , . . . , t ro ) ∈ B 0,C := {u ∈ R ro \| \|u\| < C}, for some fixed real value C > 0 (here φ tf denotes the Hamiltonian flow generated by a function f ). Choose a neighbourhood U 1 of K in M C having a compact closure U 1 ⊂ U (O) C . Now, we can follow the same arguments as in our proof of Lemma B.1 (b) and Theorem 1.4 (b) (see App. B). from it. Let L ⊂ M be a compact non-degenerate fiber satisfying the connectedness condition. Let O 0 ⊆ L be an orbit of minimal rank in L satisfying the properties from Definition 1.6 of the connectedness condition. Due to Theorem 1.4 (b), we can define a Vey momentum map J • F at O 0 (Definition 1.5).Observe that O 0 is compact (otherwise its boundary contains a compact orbit of smaller rank, because L is almost non-degenerate, see Subsec.1.3), thus it is symplectically structurally stable in a strong sense by Theorem 2.1, so we can define a perturbed Vey momentum mapJ •F near O 0 (Definition 2.2).Principle Lemma 3.1. Under the above assumptions, every orbit O ⊆ L has the following properties. The Vey momentum map J • F at the compact orbit O 0 can serve as a Vey momentum map at the orbit O (Definition 1.5), with the same regular, elliptic, hyperbolic and focus-focus components apart from some of the hyperbolic and/or focus-focus components at O 0 which are regular components at O. If O is compact then it has the same rank and the same Williamson type as O 0 . ( b ) bThe perturbed Vey momentum mapJ •F near O 0 (Definition 2.2) can serve as a perturbed Vey momentum map near O. called "atoms", may be non-connected). Connected components of K i \ ∪(compact orbits) called primitive orbits can be "moved" along each other[49, proof of Theorem 7.3], provided that the given two primitive orbitsO 1 and O 2 lie in different V i and in the closure of an orbit O ⊂ L of dimension dim O = dim O 1 + dim O 2 − r, where ∂O = O 1 + O 2 − O 1 − O 2 algebraically. Here, by moving O 1 along O 2 ,one get a primitive orbit O 1 , which lies in the same V i as O 1 . By moving O 2 along O 1 , one get a primitive orbit O 2 , which lies in the same V j as O 2 . Let C ⊂ U (L) be the union of all V i and of all orbits O ⊂ L corresponding to the moves of primitive orbits (see above). Let the tuple (C, V 1 , . . . , V ke+k h +k f ), equipped with inclusions V i → C and some orientations, be called the Cl-type of the singularity at L [2]. In the saddle-saddle case (r = k e = k f = 0 and k h = 2), two singularities are equivalent if and only if their Cl-types are isomorphic Figure 2 . 2Bifurcation diagram and singularity types for the Kovalevskaya top with (a) Example 4. 1 ( 1Kovalevskaya's top). Consider the motion of a rigid body with a fixed point in a gravity field. The dynamical system is a Hamiltonian system on e(3) * which is R 6 Appendix A. Proof of Principle Lemma 3.1, Theorem 1.12 and Corollary 1.13 Let L ⊂ M be a compact rank-r fiber satisfying the connectedness condition (Definition 1.6), and let O 0 ⊂ L be the corresponding non-degenerate rank-r orbit. Suppose Williamson type of O 0 is (k e , k h , k f ). By Theorem 1.4 (b), there exist a neighbourhood U (O 0 ) of O 0 , a local symplectomorphism Φ : U (O 0 ) → R 2n at O 0 and a real-analytic diffeomorphism J = (J 1 , . . . , J n ) : W → R n such that (7) J • F • Φ −1 = (h 1 , . . . , h n ) (cf. (1), (2)), so J • F is a Vey momentum map at the orbit O 0 (Definition 1.5). . , J r • F (corresponding to regular components at O 0 ) is locally free on O (and, hence, on L). Furthermore, the map J • F is a Vey momentum map at O (Definition 1.5) w.r.t. ω with the same regular, elliptic, hyperbolic and focus-focus components apart from k h − k h hyperbolic and 2(k f − k f ) focus-focus components at O 0 which are regular components at O; the map J •F is a perturbed Vey momentum map near O (Definition 2.2) w.r.t.ω. a Hamiltonian (S 1 ) k h +2k f -action on some neighbourhood U (O) C of O, and this action is fixed on O. Case 1 : 1O is compact. Thus k h = k h and k f = k f by the connectedness condition. Thus, m is a fixed point of the Hamiltonian (S 1 ) n−r -action on U (O) C generated by the functions J J • F is a Vey momentum map at O (Definition 1.5) w.r.t. ω, andJ •F is a perturbed Vey momentum map near O (Definition 2.2) w.r.t.ω. On one hand, by (a) and (b), O is fixed under the Hamiltonian (S 1 ) n−r -action on U (O) C . On the other hand, as we showed above, O is and App. A) in the proofs of Theorems 2.1, 1.12 and Principle Lemma 3.1. Lemma B.1. Suppose m 0 ∈ M is a singular rank-r point of a real-analytic integrable system (M, ω, F ). Suppose the first differentials of the functions f r+1 , . . . , f n at m 0 vanish and, in some canonical chart Φ 0 : (U 0 , ω) → (R 2n , ω can ) with Φ 0 (m 0 ) = 0, the second differentials of f r+1 • Φ −1 0 , . . . , f n • Φ −1 0 at 0 coincide with the second differentials of h r+1 , . . . , h n in (1), and d(f s • Φ −1 0 )(0) = dλ s for 1 ≤ s ≤ r. Then There exist a neighbourhood U of m 0 in M C , a neighbourhood W ⊇ F C (U ) of F (m 0 ) in C n , and a unique Hamiltonian (S 1 ) n−r -action on U generated by the functions There exists a real-analytic symplectomorphism Φ :(U ∩ M, ω) → (R 2n , ω can ) such that Φ(m 0 ) = 0, dΦ(m 0 ) = dΦ 0 (m 0 ), and J • F • Φ −1 = (h 1 , . . . , h n ), where J s (z) := z s for 1 ≤ s ≤ r, J = (J 1 , . . . , J n ) : W → C n .For a fixed J, any two such symplectomorphisms Φ, Φ near m 0 are related by (Φ −1 • Φ ) 2 = φ 1 S•J•F , for some real-analytic function S = S(z 1 , . . . , z n ), where φ t f denotes the Hamiltonian flow generated by the function f . and 2.1, Corollary 1.13, Examples 1.15 and 4.1). 1.2. Non-degenerate singularities: local symplectic normal form. Sufficient conditions for structural stability of a singularity are given in Theorems 1.16 and 2.1. For their formulation, let us recall the notion of a non-degenerate singularity. 1.4. Main result. Theorem 1.16 (Semilocal structural stability test). Suppose L is a compact non-degenerate singular fiber satisfying the connectedness condition (cf. Definition 1.6 and Corollary1.13). Then the semilocal singularity at the fiber L is structurally stable in a strong sense (Definition 1.2) under real-analytic integrable perturbations (but not necessarily under C ∞ integrable perturbations).Thus, by Theorem1.12 (v, vi) and Theorem1.16, all singularities which is a Vey momentum map at O by above, it follows from Lemma B.1 (b) thatJ •F is a perturbed Vey momentum map near m (Definition 2.2). In fact, we have even more: it is a perturbed Vey momentum map near O, since we can extend the corresponding local symplectomorphismΦ to a neighbourhood of the cylinder by O(ε)-small integrable perturbation. Using this and [30, Lemma 2.3], one can show that, for each S 1 -subaction of the above (S 1 ) n -action on (T m 0 M ) C (see Step 1), there exists a point m 1 ⊂ L C satisfying the conditions (i)-(iii) of [30, Theorem 2.2(a)]. By [30, Theorem 2.2(a)], on a small open complexification U C of U = U ε , there exists a Fpreserving Hamiltonian (S 1 ) n -action generated by some functions Such singularities are also called stable [16, Sec. 2.10] or topologically stable [49, Def. 4.5, 5.3, 6.3]. Instead of arguments from[49], we can apply Principle Lemma 3.1 for obtaining another proof of the fact that the singularities at L andL are equivalent in a strong sense. This yields Theorem 1.4.The authors are grateful to Alexey Bolsinov for helpful comments on a preliminary version of the paper and to Anton Izosimov for informing us about his results on structural stability of focus singularities. The work on semilocal singularities (Theorems 1.12 Theorem 2.2(b) and 3.10(b) or Lemma 6.2(b)] of [30, Theorem 2.2(a) and 3.10(a) or. 30[30, Theorem 2.2(b) and 3.10(b) or Lemma 6.2(b)] of [30, Theorem 2.2(a) and 3.10(a) or This yields Lemma B.1 (b). This yields Lemma B.1 (b). (a) We want to bring our system to a canonical form. B.1. Proof of Theorem 1.4.. This can be done using [44]. Let us give another proof based on Lemma B.1 (a) (which we proved using [30])B.1. Proof of Theorem 1.4. (a) We want to bring our system to a canonical form (1), (2) on a small neighbourhood U of the point m 0 . This can be done using [44]. Let us give another proof based on Lemma B.1 (a) (which we proved using [30]). After replacing f 1 , . . . , f n by their linear combinations, we can assume that df j (m 0 ) = 0 for each. After replacing f 1 , . . . , f n by their linear combinations, we can assume that df j (m 0 ) = 0 for each f r to a system of local canonical coordinates Φ 0 = (λ, ϕ, x, y) = (λ 1 , ϕ 1. . ; . , ϕ r , x 1 , y 1 , . . . , x n−r , y n−r ) : U → R 2n such that can (Darboux coordinates), seeOn a small neighbourhood U of m 0 , we can extend the functions f 1 , . . . , f r to a system of local canonical coordinates Φ 0 = (λ, ϕ, x, y) = (λ 1 , ϕ 1 , . . . , λ r , ϕ r , x 1 , y 1 , . . . , x n−r , y n−r ) : U → R 2n such that can (Darboux coordinates), see (2). f n by their linear combinations, and applying to x, y a linear canonical transformation if necessary) the second differentials of f r+1 • Φ −1 0 \| λ=0. It follows from the Williamson theorem that (after replacing f r+1 , . . .It follows from the Williamson theorem that (after replacing f r+1 , . . . , f n by their linear combinations, and applying to x, y a linear canonical transformation if necessary) the second differentials of f r+1 • Φ −1 0 \| λ=0 , . . . , f n • Φ −1 \| λ=0 at the origin have a canonical form, i.e. coincide with the second differentials of h r+1 \| λ=0 , . . . , h n \| λ=0 in (1). In particular, the linearizations at the point m 0 of the restrictions of the Hamiltonian vector fields generated by (11) to {λ = 0} have 2π-periodic flows on. 1there exist J and Φ with required properties\| λ=0 at the origin have a canonical form, i.e. coincide with the second differentials of h r+1 \| λ=0 , . . . , h n \| λ=0 in (1). In particular, the linearizations at the point m 0 of the restrictions of the Hamiltonian vector fields generated by (11) to {λ = 0} have 2π-periodic flows on (T m 0 M ) C . Due to Lemma B.1 (a), there exist J and Φ with required properties. Since the flows of all X f i are complete on O, it is diffeomorphic to a cylinder R ro × (S 1 ) rc , where r o and r c are degree of openness and degree of closedness of O. Suppose O is a rank-r orbit. respectively [49, Def. 3.4], r = r o + r cSuppose O is a rank-r orbit, m 0 ∈ O. Since the flows of all X f i are complete on O, it is diffeomorphic to a cylinder R ro × (S 1 ) rc , where r o and r c are degree of openness and degree of closedness of O, respectively [49, Def. 3.4], r = r o + r c . In terminology of [30, Def. 3.7], this means that the integer (n − r) × (n − r)-matrix p j (whose columns are "extended" elliptic and hyperbolic resonances of the singularity) is the unity matrix: p j = δ j (we can achieve this, since our matrix p j is a non-degenerate square matrix, and we are allowed to replace the functions J j • F by their linear combinations forming a non-degenerate matrix). We can manage that the action of Γ is trivial on each elliptic disk D 2 and on each focus-focus polydisk D 2 ×D 2 , because the twisting resonances are well-defined only up to adding any linear combinations of the "extended" elliptic resonances [30, Remark 3.11(C)]. The action of Γ on (D 2 ) n−r is effective, since otherwise the above (S 1 ) rc -action is non-effective. Case 2: r o > 0. We deduce this case from a parametric version of Case 1 (similarly to the proof of Lemma B.1 (a), Step 1, Case 2) by considering the corresponding reduced integrable Hamiltonian system with n−r o degrees of freedom (obtained by local symplectic reduction under the local Hamiltonian action generated by f 1 , . . . , f ro ). In this way, we see from Case 1 and. . . • F, , . J R )/∂(z Ro+1, . V /Γ, Ω Can, there exists a locally-free Fpreserving Hamiltonian (S 1 ) rc -action on a neighbourhood U (O) of O. This (S 1 ) rc -action is generated by functions of the form J ro+1. , z r ) = 0 and df 1 ∧ · · · ∧ df r = 0 at some (and hence each) point of O. Without loss of generality, this (S 1 ) rc -action is effective. Besides, we can extend to U (O) C the Hamiltonian (S 1 ) n−r -action on U C generated by J C j •F C , r +1 ≤ j ≤ n, constructed in (a). 30, Theorem 3.10(b)] that the system (U (O), ω, J •F ) is symplectomorphic to a neighbourhood of the cylinder {0} r ×R ro ×(S 1 ) rc ×{(0, 0)} n−r in the linear model (V /Γ, ω can , (h 1 , . . . , h n )) having the form (1), (2), as required. ReferencesBy [19] or [51, 55] (or [49, Theorem 6.1] in the C ∞ case with a proper F ), there exists a locally-free F - preserving Hamiltonian (S 1 ) rc -action on a neighbourhood U (O) of O. This (S 1 ) rc -action is generated by functions of the form J ro+1 • F, . . . , J r • F for some real-analytic functions J s (z 1 , . . . , z n ), r o + 1 ≤ s ≤ r. Without loss of generality, ∂(J ro+1 , . . . , J r )/∂(z ro+1 , . . . , z r ) = 0 and df 1 ∧ · · · ∧ df r = 0 at some (and hence each) point of O. Without loss of generality, this (S 1 ) rc -action is effective. Besides, we can extend to U (O) C the Hamiltonian (S 1 ) n−r -action on U C generated by J C j •F C , r +1 ≤ j ≤ n, constructed in (a). The above (S 1 ) rc -action and (S 1 ) n−r -action give rise to the Hamiltonian (S 1 ) n−ro -action on U (O) C generated by J C j • F C , r o + 1 ≤ j ≤ n. Put J s (z 1 , . . . , z n ) := z s , 1 ≤ s ≤ r o . Consider two cases. Case 1: r o = 0, thus the orbit O is compact. By [30, Theorem 3.10(a)], the above (S 1 ) n−ro -action is symplectomorphic to a linear model, thus the system (U (O), ω, J • F ) is symplectomorphic to a linear model (V /Γ, ω can , (h 1 , . . . , h n )) [30, Def. 3.7] having the form (1), (2). In terminology of [30, Def. 3.7], this means that the integer (n − r) × (n − r)-matrix p j (whose columns are "extended" elliptic and hyperbolic resonances of the singularity) is the unity matrix: p j = δ j (we can achieve this, since our matrix p j is a non-degenerate square matrix, and we are allowed to replace the functions J j • F by their linear combinations forming a non-degenerate matrix). We can manage that the action of Γ is trivial on each elliptic disk D 2 and on each focus-focus polydisk D 2 ×D 2 , because the twisting resonances are well-defined only up to adding any linear combinations of the "extended" elliptic resonances [30, Remark 3.11(C)]. The action of Γ on (D 2 ) n−r is effective, since otherwise the above (S 1 ) rc -action is non-effective. Case 2: r o > 0. 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[ "SPARQLing Database Queries from Intermediate Question Decompositions", "SPARQLing Database Queries from Intermediate Question Decompositions" ]	[ "Irina Saparina \nHSE University\nYandex, MoscowRussia\n", "Anton Osokin \nHSE University\nYandex, MoscowRussia\n" ]	[ "HSE University\nYandex, MoscowRussia", "HSE University\nYandex, MoscowRussia" ]	[ "Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing" ]	To translate natural language questions into executable database queries, most approaches rely on a fully annotated training set. Annotating a large dataset with queries is difficult as it requires query-language expertise. We reduce this burden using grounded in databases intermediate question representations. These representations are simpler to collect and were originally crowdsourced within the Break dataset (Wolfson et al., 2020). Our pipeline consists of two parts: a neural semantic parser that converts natural language questions into the intermediate representations and a non-trainable transpiler to the SPARQL query language (a standard language for accessing knowledge graphs and semantic web). We chose SPARQL because its queries are structurally closer to our intermediate representations (compared to SQL). We observe that the execution accuracy of queries constructed by our model on the challenging Spider dataset is comparable with the state-of-the-art text-to-SQL methods trained with annotated SQL queries. Our code and data are publicly available. 1 Question: Which teachers work in NY? Show the names in alphabetical order. Database: teacher Name school Lucy Wong Joseph Huts Executable SPARQL query S_ID 1 1 ID State Name 1 NY NYU 2 CA Stanford QDMR-to-SPARQL translator RAT-encoder + AST-decoder QDMR with grounded arguments:Figure 1: Overall map of our approach: we feed a question and a database schema into the encoder-decoder model to obtain the grounded QDMR. The grounded QDMR is then fed into our QDMR-to-SPARQL translator to obtain an executable SPARQL query. The generated query is executed on the database in the RDF format.	10.18653/v1/2021.emnlp-main.708	[ "https://www.aclanthology.org/2021.emnlp-main.708.pdf" ]	237,491,955	2109.06162	5d26525c33b0a3ca97284f6cf304ff9de5e90c92	SPARQLing Database Queries from Intermediate Question Decompositions Association for Computational LinguisticsCopyright Association for Computational LinguisticsNovember 7-11, 2021. 2021 Irina Saparina HSE University Yandex, MoscowRussia Anton Osokin HSE University Yandex, MoscowRussia SPARQLing Database Queries from Intermediate Question Decompositions Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing the 2021 Conference on Empirical Methods in Natural Language ProcessingAssociation for Computational LinguisticsNovember 7-11, 2021. 20218984 To translate natural language questions into executable database queries, most approaches rely on a fully annotated training set. Annotating a large dataset with queries is difficult as it requires query-language expertise. We reduce this burden using grounded in databases intermediate question representations. These representations are simpler to collect and were originally crowdsourced within the Break dataset (Wolfson et al., 2020). Our pipeline consists of two parts: a neural semantic parser that converts natural language questions into the intermediate representations and a non-trainable transpiler to the SPARQL query language (a standard language for accessing knowledge graphs and semantic web). We chose SPARQL because its queries are structurally closer to our intermediate representations (compared to SQL). We observe that the execution accuracy of queries constructed by our model on the challenging Spider dataset is comparable with the state-of-the-art text-to-SQL methods trained with annotated SQL queries. Our code and data are publicly available. 1 Question: Which teachers work in NY? Show the names in alphabetical order. Database: teacher Name school Lucy Wong Joseph Huts Executable SPARQL query S_ID 1 1 ID State Name 1 NY NYU 2 CA Stanford QDMR-to-SPARQL translator RAT-encoder + AST-decoder QDMR with grounded arguments:Figure 1: Overall map of our approach: we feed a question and a database schema into the encoder-decoder model to obtain the grounded QDMR. The grounded QDMR is then fed into our QDMR-to-SPARQL translator to obtain an executable SPARQL query. The generated query is executed on the database in the RDF format. Introduction The difficulty of collecting and annotating datasets for the task of translating a natural language question to an executable database query is a significant obstacle to the progress of the technology. The most popular multi-database text-to-SQL dataset, Spider (Yu et al., 2018), has 10K questions, which is smaller compared to question answering datasets of other types: the DROP dataset with text paragraphs has 97K questions (Dua et al., 2019) and the GQA dataset with images has 22M questions (Hudson and Manning, 2019). The Spider dataset was created by 11 Yale students proficient in SQL, and it is difficult to scale such a process up. Recently, Wolfson et al. (2020) proposed the Question Decomposition Meaning Representation, QDMR, which is a way to decompose a question into a list of "atomic" steps representing an algorithm for answering the question. Importantly, they developed a crowdsourcing pipeline to annotate QDMRs and showed that it can be used at scale: they collected 83K QDMRs for questions (all in English) coming from different datasets (including Spider) and released them in the Break dataset. QDMRs resemble database queries but are not connected to any execution engine and cannot be run directly. Moreover, QDMRs were collected when looking only at questions and thus have no information about the database structure. Entities mentioned in QDMR steps usually have counterparts in the corresponding database but do not have links to them (grounding). In this paper, we build a system for translating a natural language question first into QDMR and then into an executable query. We use modified QDMRs, where the entities described with text are replaced with their database groundings. Our system consists of two translators: a neural network for text-to-QDMR and a non-trainable QDMR-to-SPARQL transpiler. See Figure 1, for an illustration of our system. In the text-to-QDMR part, we use an encoderdecoder model. Our encoder is inspired by RATtransformer and uses BERT (Devlin et al., 2019) or GraPPa (Yu et al., 2021). Our decoder is a syntax-guided network (Yin and Neubig, 2017) designed for our version of the QDMR grammar. We trained this model with full supervision, for which we automatically grounded QDMRs for a subset of Spider questions. In the second part of the system, our goal was to translate grounded QDMRs into one of the existing query languages to benefit from the efficiency of database software. The most natural choice would be to use SQL, but designing such a translator 1 Figure 2: Database from Figure 1 converted to the RDF format (the RDF graph). The red nodes correspond to the values from teacher table, the green ones -to the values from school table. Arcs correspond to the relations between primary key and other values of the same row (arc:tbl:col) and along the foreign keys (arc:t_src:c_src:t_tgt:c_tgt). was difficult due to structural differences between QDMR and SQL. Instead, we implement a translator from QDMR to SPARQL, 2 which is a query language for databases in the Resource Description Framework (RDF) format (Prud'hommeaux and Seaborne, 2008;Harris and Seaborne, 2013). SPARQL is a standard made by the World Wide Web Consortium and is recognized as one of the key technologies of the semantic web. See Figure 2 for an example of the RDF database. We evaluated our system with the execution accuracy metric on the Spider dataset (splits by Wolfson et al., 2020) and compared it with two strong baselines: text-to-SQL systems BRIDGE (Lin et al., 2020) and SmBoP (Rubin and Berant, 2021) from the top of the Spider leaderboard. On the cleaned-up validation set, our system outperforms both baselines. On the test set with original annotation, our system is in-between the baselines. Additionally, we experimented with training our models on extra data: items from Break without schema but with QDMRs. This teaser experiment showed potential for further improvements. This paper is organized as follows. Sections 2 and 3 present two main parts of our system. Section 4 contains the experimental setup, Section 5our results. We review related works in Section 6 and conclude in Section 7. 2 QDMR-to-SPARQL translator 2.1 QDMR logical forms Question Decomposition Meaning Representation (QDMR) introduced by Wolfson et al. (2020) is an intermediate format between a question in a natural language (tested in English) and an executable query in some formal query language. QDMR is a sequence of steps, and each step corresponds to a separate logical unit of the question (see Table 1). A QDMR step can refer to one of the previous steps, allowing one to organize the steps into a graph. We work with QDMR logical forms (LF), which can be automatically obtained from the text-based QMDRs, e.g., with the rule-based method of Wolfson et al. (2020). Steps of a logical form are derived from the corresponding steps of QDMR. Each step of LF includes an operator and its arguments. We show some operators in Table 2 Grounding QDMRs in databases QDMR logical forms are similar to the programmed queries but are not connected to any execution engine and cannot be executed directly. To execute these LFs using knowledge from a database, one needs to associate their arguments with the entities of the database: tables, columns, values. We refer to this association as grounding and provide the details below. Arguments of LF operators can be of different types (see Table 2 and Appendix A) and some types require groundings. Type ref indicates a reference to one of the existing LF steps. Type text corresponds to a text argument that needs to be grounded to a table, column or value in the database. Type choice corresponds to the choice among a closed list of possible options, and type bool corresponds to the True/False choice. There are also a few edge cases that require special processing. First, the value argument of the COMPARATIVE operator can be either ref or text. Second, the operator argument of AGGREGATE/GROUP can actually be grounded to a column. We introduced this exception because a database can contain only the aggregated information without information about individual instances. As the QDMR annotation is built without looking at the database it cannot distinguish the two cases. In the example of Table 1, if the database has a column num_teachers in the table school we would need to ground count to the column num_teachers. We describe our procedure for annotating LF arguments with groundings in Section 4.2. Executable queries in SPARQL To convert a QDMR LF with grounded step arguments into an actually executable query, it is beneficial to translate QDMR into one of the existing query languages to use an existing efficient implementation at the test time. In this paper, we translate QDMR queries into SPARQL, a language for querying databases in the graph-based RDF format (Prud'hommeaux and Seaborne, 2008;Harris and Seaborne, 2013). Next, we briefly overview the RDF database format and SPARQL and then describe our algorithm for translating grounded LFs into SPARQL queries. RDF format. In RDF, data is stored as a directed multi-graph, where the nodes correspond to the data elements and the arcs correspond to relations. RDF-graphs are usually defined by sets of subject-predicate-object triples, where each triple defines an arc: the subject is the source node, the predicate is the type of relation and the object is the target node. Relational data to RDF. To evaluate our approach on the Spider dataset containing relational databases (in the SQLite format), we convert relational databases to the RDF format. The conversion is inspired by the specification of Arenas et al. (2012). For each table row of the relational database, we add to the RDF graph a set of triples corresponding to each column. For the primary key column 4 key of a table tbl, we create a triple with the self-link arc:tbl:key pointing from the key element to itself. For any other column col in the table tbl, we create a triple with the separate edge type arc:tbl:col, which connects the primary key element of a row to the corresponding element in col. For each foreign key of the database, we create an arc type arc:t_src:c_src:t_tgt:c_tgt (here the target column c_tgt has to be a key). Then we add to the RDF graph the triples with these foreignkey relations. See Figure 2 with an example of the RDF graph for the database of Figure 1. SPARQL. In a nutshell, a SPARQL query is a set of triple patterns where some elements are replaced with variables. The execution happens by searching the RDF graph for subgraphs that match the patterns. to the RDF graph of Figure 2 searches for pairs of nodes that are connected with arcs of type arc:school:State. Entries starting with symbol ? represent variables. See Figure 1 for an example of a more complicated query. SPARQL also supports subqueries and aggregators, the GROUP, SORT, UNION, MINUS keywords, etc. See, e.g., the Wikidata SPARQL tutorial 5 for a detailed overview of SPARQL features. Translating grounded QDMR to SPARQL. We implemented a translator from a grounded QDMR LF into SPARQL. Note that LFs do not have a formal specification defining the execution, so our translator fills in the formal meaning. Our translator recursively constructs graph patterns that contain a result of LF steps. When processing a step, the method first constructs one or several patterns for the step arguments and then connects them into another pattern. At the beginning of the process, we request the method to construct the pattern containing the last QDMR step, which corresponds to the query output. We provide the details of our translator in Appendix A. Text-to-QDMR parser In this section, we describe our approach to generating a grounded QDMR LF from a given question and a database schema. Our encoder consists of BERT-like pretrained embeddings (Devlin et al., 2019;Yu et al., 2021) and a relation-aware transformer . Our decoder is an LSTM model that generates an abstract syntax tree in the depth-first traversal order (Yin and Neubig, 2017). Encoder In our task, the input is a sequence of question tokens and a set of database entities eligible for grounding: tables, columns, and extracted values. To choose values from a database, we use string matching between question tokens and database values (see Appendix B). Additionally, we extract numbers and dates from the question that can be valid comparative values not from the database. To avoid ambiguity of the encoding, we combine the multiple identical values from different columns into one. (Yu et al., 2021). The obtained representations are fed into the relationaware transformer, RAT . RAT module. RAT ) is based on relation-aware self-attention layer (Shaw et al., 2018) . Unlike the standard self-attention in the transformer model (Vaswani et al., 2017), this layer explicitly adds embeddings r ij that encode relations between two inputs x i , x j . The RAT self-attention weights are α ij = softmax x i W Q (x j W K +r ij ) √ d , where W K , W Q , d are the standard self-attention parameters. The relations between the columns and tables come from the schema structure, e.g., the tableprimary key and foreign key relations. We also have relations based on matches: question -table and question -column matches based on the ngram comparison (Guo et al., 2019) and questionvalue matches from our value extracting procedure. Decoder The decoder is a recurrent model with LSTM cells that generates an abstract syntax tree (AST) in the depth-first traversal order (Yin and Neubig, 2017). At each prediction, the decoder selects one of the allowed outputs, the list of allowed outputs is defined by our QDMR grammar (see Appendix C). The output can be the grammar rule (transition to a new node in AST), the grounding choice or the previous step number (leaf nodes in AST). To predict grammar rules, we use the same modules as in the RAT-SQL model . The decoder predicts comparator, aggregator and sort directions using the output of MLP. For table, column or value grounding, we use the pointer network attention mechanism (Vinyals et al., 2015). To predict a reference to a previous QDMR step, we use an MLP with a mask in the output softmax. To avoid incorrect QDMR output, we use several restrictions in the decoding process. Most of them are in the prediction of comparative arguments, e.g., we check type consistency (see Appendix D). Training We follow the RAT-SQL training procedure in the main aspects. We use the standard teacher forcing approach for autoregressive models. We found that an additional alignment loss proposed for RAT-SQL did not lead to any improvements in our case, so we trained the models with the cross-entropy loss with label smoothing. See Appendix B for implementation details. Augmentations. We randomly permute tables, columns and values when training. We experimented with a random choice of QDMR graph linearization at training but did not observe performance improvements. We also tried to randomly select one of the multiple available QDMR groundings, but it did not help as well. Experiment setup 4.1 Data For training and evaluation, we use the part of the Break dataset that corresponds to the Spider dataset. 6 Data includes questions and databases from Spider, QDMR logical forms from Break and groundings that we collected. Automatic grounding annotation is challenging, but we are able to annotate with target groundings more than 60% of the Break data (see Section 4.2). Our splits are based on the Break splits but take into account the grounding annotation. The Break dataset does not include the Spider test, as it is hidden, while the 6 The Break dataset also contains QDMRs for other textto-SQL datasets, e.g., single-database ATIS and GeoQuery. Comparison in the regime of fine-tuning on a specific database is also interesting, but baseline and our codebases failed due to the limitations of the SQL parsers (coming from Spider). This issue might be resolved by switching to a different SQL parser but it appeared technically infeasible at the time of writing. Dataset Train Dev Test Break dev and test are the halves of the Spider dev. The gold QDMR and grounding annotation on the Break test is also hidden. The overall dataset statistics are shown in Table 3. We fixed typos and annotation errors in some train and dev examples. We also corrected some databases on train and dev: we deleted trailing white spaces in values (led to mismatches between SQL query and database) and added missing foreign keys (necessary for our SPARQL generator) based on the procedure of Lin et al. (2020). We kept the test questions and SQL queries unchanged from the original Spider dataset, which implied that some dataset errors could degrade comparisons of SQL and SPARQL results. Annotating Groundings for LFs We process LFs from the Break dataset in several stages. At the first stage, we iterate over all the operators and make their arguments compatible with our specification (see Table 2). At the second stage, we collect candidate groundings for each argument that requires grounding. At this stage, we use all available sources of information: text-based similarity between the text argument and the names of the database entities, the corresponding SQL query from Spider, explicit linking between the question tokens and the elements of the schema released by Lei et al. (2020). Importantly, we can match the output of LF to the output of the SQL query and propagate groundings inside LF, which allows to obtain many high-quality groundings. At the third stage, we use collected candidate groundings and group them in all possible ways to obtain candidate LFs with all arguments grounded. Then, for each candidate LF, we run our QDMRto-SPARQL translator and execute the obtained query. We accept the candidate if there are no failures in the pipeline and the result of the SPARQL query equals the result of the SQL one. Finally, we included the question in the dataset if we had accepted at least one grounded LF. Note that we can accept several versions of grounding for each question. We cannot figure out which one is better at this point, so we can either pick one randomly or use all of them at training. Evaluation Metric For evaluation on the Spider dataset, most textto-SQL methods use the metric called exact set matching without values. This metric compares only some parts of SQL queries, e.g., values in conditions are not used, and sometimes incomplete non-executable queries can achieve high metric values. As our approach does not produce any SQL query at all, this metric is not applicable. Instead, we use an execution-based metric, a.k.a. execution accuracy. This metric compares the results produced by the execution of queries (allowing arbitrary permutation on the output columns). Recently, the Spider leaderboard started supporting this metric, but submitting directly to the leaderboard is still not possible for us because the exposed interface requires SQL queries. We modify the Spider execution accuracy evaluation in such a way that it can support any query language that can be executed and provide results. When comparing the results of SPARQL to the results of SQL, we faced several challenges: • the order of output columns in SQL does not match the order in the question; • in Spider, when selecting relations w.r.t. argmin or argmax there is no consistent policy whether to pick all the rows satisfying the constraints or only one of them; • the order of rows in the output of SQL is stable, but the order of rows in the output of SPARQL varies depending on minor launching conditions; • in SPARQL, sorting is unstable (can arbitrarily change elements with equal sorting key values), but SQL sorting is stable; The first two points can make SQL-to-SQL comparisons invalid as well, and the others affect only SQL-to-SPARQL comparisons. To resolve these issues, we implemented the metric supporting SQL-to-SQL, SQL-to-SPARQL, SPARQL-to-SPARQL comparisons with the following properties: • we reorder the columns of the outputs based on the columns the output values come from. If the matching fails, we try to compare output tables with the given order of columns; • if one of the outputs is from an SQL query ending with "ORDER BY···LIMIT 1", we check that the produced one row is contained in another output; • if one of the outputs has done unstable sorting, we allow it to provide a key w.r.t. which the sorting was done and try to match the order of the rows in another output by swapping the rows with identical sorting-key values; • before comparison, we extract the column types from both outputs and convert each value to the standardized representation. Results Comparison with text-to-SQL methods First, we compare our approach to state-of-the-art text-to-SQL methods (that generate full executable queries) BRIDGE (Lin et al., 2020) and SmBoP (Rubin and Berant, 2021), both from the top of the Spider leaderboard. See Table 4 for the results. As our training data includes only 50% of the original Spider train, we add to the comparison BRIDGE and SmBoP models trained on the same data subset. We use the official implementations of both models. All models are trained together with finetuning pretrained contextualized representations: BRIDGE encoder uses BERT, SmBoP encoder uses GraPPa, our model has both BERT and GraPPa versions. We choose the final model of each training run of our system based on the best dev result from the last 10 checkpoints with the step of 1000 iterations. For BRIDGE and SmBoP, we used the procedures provided in the official implementations (they similarly look at the same dev set). The estimated std of our model is 0.9 on the dev set (estimated via retraining our BERT-based model with 5 different random seeds). On the development set, our models achieve better execution accuracy than text-to-SQL parsers even trained on full Spider data. On the test set, our models outperform BRIDGE but not SmBoP when trained on the same amount. See Table 6 for qualitative results of our GraPPa-based model. We did not include the results of RAT-SQL in Table 4, because this model was trained to optimize exact set matching without values, so the model output contains placeholders instead of values. The model trained on full Spider reproduces the exact matching scores shown by but gives only 40.2% execution accuracy on dev and 39.9% on test. Correct predictions mostly came from correct SQL queries without values. We also tried the available feature of value prediction in the official implementation of RAT-SQL and obtained better execution accuracy scores (48.5% on dev and 46.4% on test), but they were still very low. Additional training data from Break The Break dataset contains QDMR annotations for several question answering datasets, so we tried to enrich training on Spider with QDMRs from other parts of Break. Table 5 shows the execution accuracy on our dev and test in these settings. Adding training data for both versions of the model leads to performance improvement on the test set, but slightly decreases the dev set results. When training with the data from other parts of Break, we simply assume that the schema is empty and use all the textual QDMR arguments as values. More careful exploration of additional QDMR data is left for future work. in both models decreases the execution accuracy. Next, we tested different configurations of RATencoder: • without relations that come from the schema structure (e.g., the table -primary key and foreign key relations); • with the small number of default relations: without distinguishing table, column or value, because these elements are considered as elements of one unified grounding type; • the regular transformer instead of RAT. Ablation study The model without schema relations lost 11% on dev, which shows that encoding schema with RATencoder is an important part of the model. This also limits the use of additional data from Break, where schemas do not exist. The variety of relations in RAT-encoder is also important, as RAT itself. Our findings are consistent with the ablations of . Related Work Text-to-SQL. The community has recently made significant progress and moved from fixed-schema datasets like ATIS or GeoQuery (Popescu et al., 2003;Iyer et al., 2017) to the WikiSQL or Overnight datasets with multiple single-table schemas (Wang et al., 2015;Zhong et al., 2017) and then to the Spider dataset with multiple multi-table multi-domain schemas (Yu et al., 2018). Since the release of Spider, the accuracy has moved up from around 10% to 70%. Most recent systems are structured as encoderdecoder networks. Encoders typically consist of a module fine-tuned from a pretrained language model like BERT (Devlin et al., 2019) and a module for incorporating the schema structure. Guo et al. (2019); Zhong et al. (2020);Lin et al. (2020) represented schemas as token sequences, Bogin et al. (2019a,b) used graph neural networks and used relation-aware transformer, RAT, to encode a graph constructed from an input schema. In this paper, we use the RAT module to encode the schema but enlarge the encoded graph by adding value candidates as nodes. Decoders are typically based on a grammar representing a subset of SQL and produce output tokens in the depth-first traversal order of an abstract syntax tree, AST, following Yin and Neubig (2017). A popular choice for such a grammar is to use SemQL of Guo et al. (2019) or to use a lighter grammar with more intensive consistency checks inside beam search like in BRIDGE (Lin et al., 2020). Recently, Rubin and Berant (2021) proposed a different approach to decoding based on bottom-up generating of sub-trees on top of the relational algebra of SQL. In our paper, we follow the standard AST-based approach but for the grammar describing grounded QDMRs. We also use some consistency checks and the decoding time to prevent some easily avoidable inconsistencies. There is also a line of work on weaklysupervised learning of text-to-SQL semantic parsers, where SQL queries or logical forms for the training set are not available at all. Some works (Min et al., 2019;Wang et al., 2019;Agarwal et al., 2019;Liang et al., 2018) reported results on the WikiSQL dataset, worked on GeoQuery and Overnight datasets. We are not aware of any works reporting weakly-supervised results on the multi-table Spider dataset. Pretraining on text and tables. One possible direction inspired by the success of pretraining language models on large text corpora is to pretrain model on data with semantically connected text and tables. Yin et al. (2020, TaBERT) (2021, GAP) used synthetic data generated by the models for SQL-to-text and table-to-text auxiliary tasks. In this paper, we do not pretrain such models but experiment with GraPPa as the input encoder. QDMR. Together with the Break dataset, Wolfson et al. (2020) created a task of predicting QDMRs given questions in English. As a baseline, they created a seq2seq model enhanced with a copy mechanism of Gu et al. (2016). Recently, Hasson and Berant (2021) built a QDMR parser that is based on dependency graphs and uses RAT modules. Differently from this line of work, we use a modified version of QDMRs, and our models never actually predict QDMR arguments as text but always directly their groundings. SPARQL. SPARQL was used in several lines of work on semantic parsing for querying knowledge bases. The SEMPRE system of Berant et al. (2013) relied on SPARQL to execute logical forms on the Freebase knowledge base. Yih et al. (2016) and Talmor and Berant (2018) created the We-bQuestions and ComplexWebQuestions datasets, respecively, where annotations were provided in the form of SPARQL queries. A series of challenges on Question Answering over Linked Data Challenge, QALD (Lopez et al., 2013), and the LC-QuAD datasets (Trivedi et al., 2017;Dubey et al., 2019) targeted the generation of SPARQL queries directly. Our paper is different from these lines of work as we rely on supervision via QDMRs and not SPARQL directly. There also exist several lines of works on converting queries from/to SPARQL, and the problems are difficult. See, e.g., the works of Michel et al. Conclusion In this paper, we proposed a way to use the recent QDMR format (Wolfson et al., 2020) as annotation for generating executable queries to databases given a question in a natural language. Using QDMRs is beneficial because they can be collected through crowdsourcing potentially easier than correct database queries. Our system consists of two main parts. First, we have a learned text-to-QDMR translator that we built on top of the recent RAT-SQL system and trained on an annotated with QDMRs part of the Spider dataset. Second, we have a non-trainable QDMR-to-SPARQL translator, which generates queries executable on databases in the RDF format. We evaluated our system on the Spider dataset and showed it to perform on par with the modern textto-SQL methods (BRIDGE and SmBoP) trained with full supervision in the form of SQL queries. We also showed that additional QDMR annotations for questions not aligned with any databases could further improve the performance. The improvement shows great potential for future work. Supplementary Material (Appendix) SPARQLing Database Queries from Intermediate Question Decompositions A QDMR-to-SPARQL translator Table 8 contains the full list of QDMR operators used in our paper. Algorithm 1 sketches the QDMR-to-SPARQL translator. It is a recursive procedure that creates SPARQL queries for all QDMR LF steps. At its core, it constructs one or several patterns for the step arguments and then connects them into another pattern in a way specific to the LF operator of the current step. Importantly, the patterns for LF operators can be of two types: inner (inline) and full. An inner pattern represents the internal part of a query that needs to be placed inside the curly brackets {...}. A full pattern corresponds to a full query that can be executed directly (starts with the SELECT keyword). An inner pattern can be converted to full by using the SELECT <output vars> WHERE {<inner>} construction. The full pattern can be converted to inner by creating a subquery via {<full>} (here, the output variables of <full> pattern become available in the scope where the subquery is created). Different LF operators require and produce different patterns: inner of full. Next, we specify a pattern for each LF operator. The SELECT operator adds the grounded object to the context: a self link for a table, a link for a column, a link with a filtering condition for a value. The PROJECT operator creates a context for the argument and does the same as SELECT. To connect instances from different columns, we use the breadth-first search to find the shortest path in the undirected graph where all the columns of all tables represent nodes and edges appear between the primary key of each table and all other columns of the same table, and along with the foreign links. The COMPARATIVE operator first creates an inner <pattern> for its arguments and then adds a filtering condition from the l.h.s. values <filter_var>, the operation <comparator> and the r.h.s. value <value>: <pattern> FILTER(<filter_var><comparator><value>). C ← convert C to inline/full 18: return C The AGGREGATE operator computes the aggregator <agg_op> from a set of values. This operator takes the inner pattern <pattern> as input (with <var> correspondings to the set of values to aggregate) and produces the full query with the output variable <output_var> as the output: SELECT (<agg_op>(<var>) as <output_var>) WHERE { <pattern> } The SUPERLATIVE operator filters the instances such that some related attribute has the min/max value. The operator first computes the min/max value with a built-in AGGREGATE operator then filters (similar to COMPARATIVE) the patterns based on the computed value: {SELECT (<agg_op>(<var>) AS <minmax_var>) WHERE { <pattern_inner> } } <pattern_outer> FILTER(<query_outer_var>=<minmax_var>). The SUPERLATIVE operator requires two inner patterns as input <pattern_inner>, <pattern_outer> and makes an inner pattern as the output. The GROUP operator groups the values <var> by the equal values of the related attribute <index_var>: SELECT The aggregation is done with the operator <agg_op>. The input pattern <pattern> is inner, and the output is the full pattern with the output variable <output_var>. The UNION operator can actually correspond to several operators: horizontal union, vertical union, union of aggregators, union after group. By horizontal union, we mean the union of two or more related variables from the same pattern. These variables have to correspond to different database columns. By vertical union, we mean the union of two or more variables corresponding to the same column but coming from different patterns. This case is implemented with the UNION keyword from SPARQL using the following construction: { <pattern1> } UNION { <pattern2> } The union-after-group case is a special but common situation when arguments contain the result of the GROUP operator and the index variable of the same operator. We implement this case similar to the pattern of the GROUP operator but with several variables in the output. The union of aggregators is another common special case when the arguments of the UNION contain several aggregators from the same pattern. We simply output these several aggregators by concatenating them after the SPARQL SELECT keyword. The INTERSECT operator effectively consists in sequentially applying two COMPARATIVE operators that do not have explicit comparisons as arguments. The DISCARD operator is based of the pattern very similar to the vertical union: { <pattern1> } MINUS { <pattern2> } The SORT operator consists in adding the ORDER BY keyword at the end of the full pattern: B Implementation details We implemented our model on the top of the RAT-SQL code 7 built with Pytorch (Paszke et al., 2019). We use pretrained BERT and GraPPa from the Transformers library (Wolf et al., 2020). To support SPARQL queries and RDF databases, we used two libraries: RDFLib 8 and the open-source version of the Virtuoso system. 9 RDFLib was much easier to install (a python package), but Virtuoso allowed to run SPARQL queries on pre-loaded databases much faster. To choose relevant values from a database, we tokenized question and all unique database values using the Stanford CoreNLP library (Manning et al., 2014), filtered tokens using NLTK 10 English stopwords, and then picked top-25 values with higher similarity scores calculated as follows: • for a numeric value, we gave the maximum score if it exactly matched with some question token, otherwise, we gave the minimum score; • for other tokens, we gave the maximum score if the value and question stems were the same (we used the Porter and Snowball stemmers from NLTK), otherwise, we calculated similarity score based on the longest continuous matching subsequence (we used Python Se-quenceMatcher class). For the neural network architecture and training, we used the same hyperparameters as RAT-SQL : 8 RAT layers, each with 8 heads and the hidden dimension of 256, 1024 and 512 in self-attention, position-wise feedforward network and decoder LSTM, respectively. We trained the model with the Adam optimizer (Kingma and Ba, 2014) and polynomial decay scheduler used by . The batch size was 24, the overall number of iterations was 81000 for all models. The training time on 4 NVIDIA V100 GPUs was approximately 24 hours. Following Huang et al. (2018); Zhang et al. (2019); Wang et al. (2020), the input tokens of four types (question, table, column and value) are interleaved with [SEP], combined into a sequence and encoded: we experiment with BERT (Devlin et al., 2019) and GraPPa (2019); Abatal et al. (2019) and references therein. and provide the full list in Appendix A. 3 For each state, how many teachers are there? #1 return states #2 return teachers in #1 #3 return number of #2 for each #1 #4 return #1 and #3Question: QDMR (Break) QDMR logical form (Break) #1 SELECT[states] #2 PROJECT[teachers in #REF, #1] #3 GROUP[count, #2, #1] #4 UNION[#1, #3] grounded QDMR (ours) #1 SELECT[School.State] #2 PROJECT[teacher, #1] #3 GROUP[count, #2, #1] #4 UNION[#1, #3] Table 1 : 1Examples of different QDMR formats: textual QDMR, QDMR logical form (from Break) and our version of QDMR with grounded arguments. Table 2 : 2QDMR operators and their arguments with types. See Appendix A for the full version -Table 8. Table 3 : 3Dataset statistics for the original Spider, part of Spider with QDMR annotations from Break and part of Spider with QDMRs and groundings. Break dev and test are splits of original Spider dev. Break test is hidden, so we do not have annotation for this part. Table 4 : 4Execution accuracy of our model compared to state-of-the-art text-to-SQL methods on our develop- ment and test sets. Train Pretrain Augs Dev Test subset BERT + 79.3 60.8 full Break BERT - 78.4 61.8 full Break BERT + 78.9 61.8 subset GraPPa + 80.4 62.0 full Break GraPPa - 74.6 62.6 full Break GraPPa + 73.9 61.4 Table 5 : 5Execution accuracy of our model trained on only the Spider subset of Break compared to using additional data from Break (on our development and test sets). Table 7 7presents results of ablations on the development set. First, note that disabling augmentations Q: How many concerts are there in year 2014 or 2015? SQL: SELECT count( * ) FROM concert WHERE YEAR = 2014 OR YEAR = 2015 What is the year that had the most concerts? SELECT year FROM concert GROUP BY year ORDER BY count( * ) DESC LIMIT 1 #1 SELECT[concert.Year] #2 PROJECT[concert, #1] #3 GROUP[count, #2, #1] #4 SUPERLATIVE[max, #1, #3]What are the names of the stadiums without any concerts?What are the number of concerts that occurred in the stadium with the largest capacity?Ours #1 SELECT[concert] #2 PROJECT[concert.Year, #1] #3 COMPARATIVE[#1,#2,=2014] #4 COMPARATIVE[#1,#2,=2015] #5 UNION[#3, #4] #6 AGGREGATE[count, #5] Q: Show location and name for all stadiums with a capacity between 5000 and 10000. SQL: SELECT location, name FROM stadium WHERE capacity BETWEEN 5000 AND 10000 Ours #1 SELECT[stadium] #2 PROJECT[stadium.Capacity, #1] #3 COMPARATIVE[#1,#2, ≥5000] #4 COMPARATIVE[#1,#2, ≤10000] #5 INTERSECTION[#1, #3, #4] #6 PROJECT[stadium.Location, #5] #7 PROJECT[stadium.Name, #5] #8 UNION[#6, #7] Q: SQL: Ours Q: SQL: SELECT name FROM stadium WHERE stadium_id NOT IN (SELECT stadium_id FROM concert) Ours #1 SELECT[stadium] #2 COMPARATIVE[#1, #1, concert] #3 DISCARD[#1, #2] #4 PROJECT[stadium.Name, #3] Q: SQL: SELECT count( * ) FROM concert WHERE stadium_id =( SELECT stadium_id FROM stadium ORDER BY capacity DESC LIMIT 1) Ours #1 SELECT[stadium] #2 PROJECT[stadium.Capacity, #1] #3 SUPERLATIVE[max, #1, #2] #4 PROJECT[concert, #3] #5 AGGREGATE[count, #4] Q: What is the average and maximum capacities for all stadiums? SQL: SELECT avg(capacity), max(capacity) FROM stadium Ours #1 SELECT[stadium] #2 PROJECT[stadium.Average, #1] #3 AGGREGATE[avg, #2] #4 AGGREGATE[max, #2] #5 UNION[#3, #4] Table 6 : 6Qualitative results of our GraPPa-based model. and denote correct and incorrect execution results respectively.Model Pretrain Dev Base BERT 79.3 -w/o augmentations BERT 75.7 -w/o schema relations BERT 68.1 -with default relations BERT 65.4 -w/o relation-aware layers BERT 51.0 Base GraPPa 80.4 -w/o augmentations GraPPa 75.7 Table 7 : 7Execution accuracy for our ablation study. and Herzig et al. (2020, TaPas) used text-table pairs extracted from sources like Wikipedia for pretraining. Yu et al. (2021, GraPPa) used synthetic question-SQL pairs. Deng et al. (2021, STRUG) used the table-totext dataset of Parikh et al. (2020, ToTTo). Shi et al. (<agg_op>(<var>) AS <output_var>) WHERE { <pattern> } GROUP BY <index_var> UNION ref1, ref2, etc. ref Get the union of ref1, ref2, etc. Order instances of subject attr ref such that related attr direction choice is ordered in asc/desc directionOperator Arguments Type Description SELECT subject text Select subject distinct bool (possibly distinct values) PROJECT projection text Select projection subject ref related to subject distinct bool (possibly distinct values) COMPARATIVE subject ref Select subject such that attr ref related attr compares comparator choice (using =, =, >, <, ≥, ≤, like) value text/ref to value distinct bool (possibly distinct values) SUPERLATIVE subject ref Select subject such that attr ref related attr has operator choice max/min values AGGREGATE subject ref Compute max/min/sum/count operator choice of subject GROUP subject ref Group instances of subject attr ref such that attr has same values operator choice (aggregate with max/min/sum/count) INTERSECT subject ref Get instances of subject attr1, attr2 ref related to both attr1 and attr2 DISCARD subject ref Get instances of subject minus ref excluding instances of minus SORT subject ref Table 8 : 8QDMR operators, their arguments, types of the arguments. The full version ofTable 2. https://github.com/yandex-research/ sparqling-queries SPARQL is a recursive acronym for SPARQL Protocol and RDF Query Language. Differently fromWolfson et al. (2020) we merged the operation FILTER into COMPARATIVE due to their similarity and excluded ARITHMETIC, BOOLEAN and undocumented COMPARISON because they are extremely rare in the Spider part of Break. For simplicity, we assume that each table has a singlecolumn primary key (otherwise, we add a new ID column). https://www.wikidata.org/wiki/ Wikidata:SPARQL_tutorial https://github.com/microsoft/rat-sql 8 https://github.com/RDFLib/rdflib AcknowledgementsThis research was supported in part through computational resources of HPC facilities at HSE University(Kostenetskiy et al., 2021).D Restrictions in the decoding processThe decoding process at the inference stage is sequential, and at each step, there is a set of eligible choices. 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[ "https://github.com/yandex-research/", "https://github.com/microsoft/rat-sql", "https://github.com/RDFLib/rdflib" ]
[ "Matroids, Antimatroids, and the Generalized External Order", "Matroids, Antimatroids, and the Generalized External Order" ]	[ "Bryan R Gillespie \nDepartment of Mathematics\nUniversity of California\nBerkeley\n" ]	[ "Department of Mathematics\nUniversity of California\nBerkeley" ]	[]	Las Vergnas's active orders are a collection of partial orders on the bases of a matroid which are derived from the classical notion of matroid activity. In this paper, we construct a generalization of Las Vergnas's external order which is defined on the independence complex of a matroid. We show that this poset is a refinement of the geometric lattice of flats of the matroid, and has the structure of a supersolvable joindistributive lattice. We uniquely characterize the lattices which are isomorphic to the external order of a matroid, and we explore a correspondence between matroid and antimatroid minors which arises from the poset construction. * ordered matroid, we define for each independent set I a set of externally passive elements, EP M (I), using a general definition given in[15]. The external order can be generalized to independent sets by the following.Definition. If I and J are independent sets of the ordered matroid M, then we define the generalized external order ≤ ext byBy [15], Proposition 3.1, this is equivalent to Las Vergnas's ordering in the case where I and J are two bases. For a variety of technical reasons, throughout this exposition we will instead work with the reverse of this ordering:J) Whenever we refer to the "external order" in this work, we will be referring to this reversed order unless otherwise noted. We use distinct notation for these two orders to reduce ambiguity, particularly because there are other contexts in which Las Vergnas's original ordering convention fits more naturally with existing literature.By associating each independent set with its corresponding set of externally passive elements, we define a set system F ext := {EP(I) : I ∈ I(M)}, and show:	null	[ "https://arxiv.org/pdf/1712.04080v1.pdf" ]	119,158,573	1712.04080	cd0044d8d31d4df33bd2581db77f3805eed8b2e1	Matroids, Antimatroids, and the Generalized External Order 12 Dec 2017 December 13, 2017 Bryan R Gillespie Department of Mathematics University of California Berkeley Matroids, Antimatroids, and the Generalized External Order 12 Dec 2017 December 13, 2017 Las Vergnas's active orders are a collection of partial orders on the bases of a matroid which are derived from the classical notion of matroid activity. In this paper, we construct a generalization of Las Vergnas's external order which is defined on the independence complex of a matroid. We show that this poset is a refinement of the geometric lattice of flats of the matroid, and has the structure of a supersolvable joindistributive lattice. We uniquely characterize the lattices which are isomorphic to the external order of a matroid, and we explore a correspondence between matroid and antimatroid minors which arises from the poset construction. * ordered matroid, we define for each independent set I a set of externally passive elements, EP M (I), using a general definition given in[15]. The external order can be generalized to independent sets by the following.Definition. If I and J are independent sets of the ordered matroid M, then we define the generalized external order ≤ ext byBy [15], Proposition 3.1, this is equivalent to Las Vergnas's ordering in the case where I and J are two bases. For a variety of technical reasons, throughout this exposition we will instead work with the reverse of this ordering:J) Whenever we refer to the "external order" in this work, we will be referring to this reversed order unless otherwise noted. We use distinct notation for these two orders to reduce ambiguity, particularly because there are other contexts in which Las Vergnas's original ordering convention fits more naturally with existing literature.By associating each independent set with its corresponding set of externally passive elements, we define a set system F ext := {EP(I) : I ∈ I(M)}, and show: Introduction The classical notion of matroid activity plays an important role in understanding fundamental properties of a matroid, including the h-vector of its independence complex and the matroid Tutte polynomial. In 2001, Michel Las Vergnas introduced another structure derived from matroid activity, a collection of partial orders on the bases of a matroid which he called the active orders [15]. These orders elegantly connect matroid activity to a system of basis exchange operations, and are closely related to the broken circuit complex and the Orlik-Solomon algebra of a matroid. In [1] and [2], the combinatorial structure of these active orders arises in relation to the initial ideal of certain projective varieties derived from affine linear spaces. In the theory of zonotopal algebra (see for instance [12], [3] and [16]), the active orders connect with a class of combinatorial objects called forward exchange matroids, where the bases associated with a forward exchange matroid satisfy axioms which are equivalent to their forming an order ideal in the external order. The primary purpose of the present work is to define a generalization of Las Vergnas's external order which extends the order to the independent sets of a matroid. If M is an Theorem 1. If M is an ordered matroid, then the set system F ext of externally passive sets of M is an antimatroid. An antimatroid is a special class of greedoid which appears particularly in connection with convexity theory. Specifically, associated with any antimatroid is a convex closure operator, a closure operator on the ground set which combinatorially abstracts the operation of taking a convex hull, in the same way that a matroid closure operator abstracts the operation of taking a linear span. The convex closure operator on an ordered matroid derived from F ext in particular bears a strong similarity to the convex closure operator for oriented matroids, which were first explored by Las Vergnas in [14]. In a 1985 survey paper [10], American mathematicians Paul Edelman and Robert Jamison noted: The authors have previously referred to these objects by the cacophonous name of 'antimatroids'. We hope there is time to rectify this and that Gresham's Law does not apply to mathematical nomenclature. In the intervening 30 years, the name nevertheless appears to have become ensconced in the mathematical literature. However, in light of our Theorem 1 and other structural results of antimatroids, the name is perhaps not so poorly chosen, as the generalized external order provides an explicit connection between antimatroids and their combinatorial namesake. The characterization of F ext as an antimatroid further allows us to connect the external order with the large existing literature on lattice theory. The feasible sets of an antimatroid have a highly structured inclusion ordering called a join-distributive lattice 1 , which is thus inherited by the generalized external order. Moreover, the poset is in fact a refinement of the geometric lattice of flats associated with the matroid M, obtained by suitably combining copies of Las Vergnas's original external order for the different flats of M. Las Vergnas's original construction required the inclusion of an additional zero element (the ' * ' in Figure 1) in order to form a proper lattice structure. In the generalized order, bases whose meet in the original order would have been the extra zero element instead are joined at an independent set of lower rank. The fact that the external order comes from an antimatroid allows us to describe features of the lattice structure combinatorially. In addition, using results of Gordon and McMahon [11] for general greedoids, we are able to further derive the following explicit partition of the boolean lattice. This partition bears a resemblance to the well-known partition of Crapo, described in [6], and in fact it can be shown that this partition is a proper refinement of Crapo's. Another main purpose of this exposition is to discuss the way in which the external orders fit into the context of antimatroids and join-distributive lattices. To refine our understanding, we characterize a proper subclass of the join-distributive lattices which we call matroidal join-distributive lattices. Definition. Given a lattice L and an element x ∈ L, let r c (x) denote the number of elements in L which cover x. A join-distributive lattice L is called matroidal if r c is decreasing in L, and it satisfies the semimodular inequality r c (x ∧ y) + r c (x ∨ y) ≤ r c (x) + r c (y) For an element x of a join-distributive lattice L, one can associate a set I(x) called the independent set corresponding with x. If L is the external order lattice of an ordered matroid M, then the I operator recovers the matroid independent sets of M. Even for an arbitrary join-distributive lattice, the collection of independent sets is closed under taking subsets, and thus forms a simplicial complex. This join-distributive independence complex in fact provides an alternate characterization of matroidal join-distributive lattices. Theorem 2. A join-distributive lattice L is matroidal if and only if its independent sets are those of a matroid. In particular, this shows that the external order of an ordered matroid is a matroidal joindistributive lattice. This result goes a long way towards understanding where the external order sits among all join-distributive lattices, but surprisingly, there are matroidal joindistributive lattices which are not an external order. If we denote the class of join-distributive lattices by JD, the class of matroidal join-distributive lattices by MJD, and the class of lattices derived from the external order by EO, then EO MJD JD Figure 3 in Section 5.2 gives an example of a lattice in MJD but not EO, and Figure 2 in Section 2.3 gives an example of a lattice in JD but not MJD. A further refinement is necessary to precisely classify the lattices isomorphic to an external order, and that refinement comes from the notion of edge lexicographic or ELshellability. A graded poset P is EL-shellable if its Hasse diagram admits a labeling of its edges by integers which satisfies certain lexicographic comparability conditions on unrefinable chains. EL-shellability of a graded poset implies shellability of its order complex, and the notion has been widely studied for different classes of posets. The external order is EL-shellable, and in fact it satisfies a stronger property called S n EL-shellability. We study how S n EL-shellability relates to antimatroids, and we show that Theorem 3. A finite lattice L is isomorphic to the external order ≤ * ext of an ordered matroid if and only if it is join-distributive, matroidal, and S n EL-shellable. McNamara introduced S n EL-shellability in [17] as a way to characterize the supersolvable lattices of Stanley [19], and in particular, he proved that the two properties are equivalent. This implies that one may replace "S n EL-shellable" with "supersolvable" in the above classification of the external order. The remainder of the document is structured as follows. Section 2 gives a brief overview of necessary background material in the areas of matroid theory, antimatroid theory, and the theory of join-distributive lattices. Section 3 develops additional technical results relating feasible and independent sets of join-distributive lattices. Section 4 constructs the generalized external order and explores its structure and connections with greedoid theory. Section 5 then characterizes matroidal join-distributive lattices and relates them to S n EL-shellability, and Section 6 relates the deletion and contraction operations of matroids and antimatroids. Background We will be studying the relations between several objects in the areas of lattice theory and discrete geometry, for which significant theory has been developed. We provide a brief review of relevant background here, and refer the reader to standard sources for additional details. For general matroid notions, Oxley [18] is comprehensive, and for concepts related to matroid activity, Björner [6] gives a concise overview. For the topics of greedoids and antimatroids, our primary references are Björner and Ziegler's survey [7], as well as the book [13] by Korte, Lovász and Schrader. General lattice theory is developed in detail in Stanley [20], Chapter 3, and the literature on join-distributive lattices is discussed in some detail in the introduction of Czédli [8]. Matroids To begin, we define matroids, a combinatorial object which generalizes both the concept of linear independence of vectors in a vector space, and the concept of cycle-freeness of edge sets in a graph. The basic object of interest is the set system. Definition 2.1. If E is a finite set, a set system is a pair (E, F ) where F is a nonempty collection of subsets of E. We will sometimes refer to F as a set system when we don't need to emphasize the ground set. A common notation in the study of finite set systems is to use a string of lower-case characters or numbers to refer to a small finite set. For instance, if a, b ∈ E are elements of a ground set, then the string ab denotes the set {a, b}. If A ⊆ E, then A ∪ ab denotes the set A ∪ {a, b}. In practice this notation enhances rather than confounds communication, so we will adopt it in the present work when the meaning is clear from the context. We can now define matroids in terms of their collections of "independent sets" as follows. Definition 2.2. A set system M = (E, I) is called a matroid if • If I ∈ I and J ⊆ I, then J ∈ I; and • For I, J ∈ I, if \|I\| > \|J\|, then there is an element x ∈ I such that J ∪ x ∈ I. A set in I is called an independent set of the matroid M. The first property above is called the hereditary property for a set system, and the second is called the matroid independence exchange axiom. The independence axioms for matroids are one of many different equivalent definitions of matroids frequently called "cryptomorphisms". Among the classical cryptomorphisms are axiom systems for bases, circuits, rank functions, closure operators, and the greedy algorithm. A fluent understanding of the definitions of these concepts and the relations between them will be helpful in the remainder of this work, and is explored in detail in [18] Chapter 1. A pair of constructions which will be used frequently are the basic circuit and basic bond. A classical characterization of the basic circuit and basic bond is given by the following lemma. Lemma 2.4. Let M be a matroid with a basis B, and let b ∈ B and x / ∈ B. Then the following are equivalent: • b ∈ ci(B, x) • x ∈ bo(B, b) • B \ b ∪ x is a basis of M For notational convenience, we extend the definition of basic circuits and basic cocircuits in the following way. For elements outside of F , neither of these expressions are defined. A concept of fundamental importance in the remainder of this work is the notion of matroid activity. Definition 2.6. An ordered matroid is a matroid M = (E, I) along with a total order ≤ on the ground set E. We will frequently refer to M as an ordered matroid without specifying the order when no ambiguity arises. Definition 2.7. Let M = (E, I) be an ordered matroid, and let B be a basis of M. For x ∈ E \ B, we call x externally active with respect to B if x is the minimum element of the basic circuit ci(B, x), and externally passive otherwise. For b ∈ B, we call b internally active with respect to B if b is the minimum element of the basic cocircuit bo(B, b), and internally passive otherwise. We denote the sets of externally active and externally passive elements with respect to a basis B by EA M (B) and EP M (B), and the sets of internally active and internally passive elements by IA M (B) and IP M (B). Note in particular that the internal and external activities are dual notions. If M * is the dual matroid of M, then EA M (B) = IA M * (E \ B), and similarly for the other sets. Historically, the most important property of the notions of matroid activity is that they generate an important algebraic invariant of matroids called the Tutte polynomial. Proposition 2.8. Given an ordered matroid M, the Tutte polynomial of M is given by T M (x, y) = B∈B(M ) x \|IA(B)\| y \|EA(B)\| and is independent of the ordering of M. The Tutte polynomial is what is called the universal Tutte-Grothendieck invariant for the class of all matroids, and in particular it encodes a breadth of combinatorial data corresponding to a matroid. Antimatroids Greedoids are a generalization of matroids which capture the structure necessary for the matroid greedy algorithm to apply. The generalization gives rise to a rich hierarchy of subclasses, including matroids, which are outlined in exquisite detail in [7], Figure 8.5. Definition 2.9. A set sytem G = (E, F ) is called a greedoid if • For every non-empty X ∈ F , there is an x ∈ X such that X \ x ∈ F ; and • For X, Y ∈ F , if \|X\| > \|Y \|, then there is an element x ∈ X such that Y ∪ x ∈ F . A set in F is called a feasible set of the greedoid E. The first property above is a weakening of the matroid hereditary property called accessibility, and the second property above is exactly the matroid independence exchange axiom, which we sometimes will call the greedoid exchange axiom for clarity. For our discussion, the most important subclass of greedoids aside from matroids is the antimatroids, defined by: Definition 2.10. A set system (E, F ) is called an antimatroid if • F is a greedoid; and • if X ⊆ Y are sets in F and a ∈ E \ Y with X ∪ a ∈ F , then Y ∪ a ∈ F . The second property in this definition is called the interval property without upper bounds. Antimatroids as set system of feasible sets can be formulated in a variety of equivalent manners, and we will state for reference several of these which will also be useful. . If F is a set system, then the following conditions are equivalent. • F is an antimatroid; • F is accessible, and closed under taking unions; and • ∅ ∈ F , and F satisfies the exchange axiom that if X, Y are sets in F such that X Y , then there is an element x ∈ X \ Y such that Y ∪ x ∈ F . Before moving on to some of the essential characteristics of these objects, we offer a remark concerning the name "antimatroid". Independent Sets, Circuits and Cocircuits As with matroids, the theory of antimatroids admits a number of cryptomorphic definitions, which include a theory of rooted circuits and a dual theory of rooted cocircuits. For more details, see [7] Section 8.7.C as well as [13] Section 3.3. Definition 2.12. If (E, F ) is a set system and A ⊆ E, define the trace F : A by F : A := {X ∩ A : X ∈ F } If F is a greedoid, then A ⊆ E is called free or independent if F : A = 2 A . If A is not independent, it is called dependent. Definition 2.13. If (E, F ) is a set system and A ∈ F , then the feasible extensions of A are the elements of Γ(A) := {x ∈ E \ A : A ∪ x ∈ F} The following lemma relates freeness to feasible extensions, and follows directly from Lemma 3.1 of [13]. Of particular note is that the collection of independent sets of an antimatroid is closed under taking subsets, and thus forms a simplicial complex as a set system. We will discuss more properties of independent sets and their relationship with feasible sets of an antimatroid in Section 3. The cryptomorphisms of rooted circuits and rooted cocircuits are presented in terms of rooted sets: Definition 2.15. If A is a set and a ∈ A, then the pair (A, a) is called a rooted set with root a. In this case, we may equivalently refer to A as a rooted set if the root is clear from context. Now we can define the circuits of an antimatroid. In particular, the following holds for circuits of an antimatroid. Proposition 2.17 ([7]). If (E, F ) is an antimatroid and C ⊆ E, then there is a unique element a ∈ C such that F : C = 2 C \ {{a}}. We call the rooted set (C, a) a rooted circuit of F . Let C(F ) denote the collection of rooted circuits of an antimatroid F . Rooted circuits give a cryptomorphism for antimatroids due to the following fundamental result. That is, an antimatroid is fully determined by its collection of rooted circuits. Further, we can axiomatize the rooted families which give rise to an antimatroid. (CI1) If (C 1 , a) ∈ C, then there is no rooted set (C 2 , a) ∈ C with C 2 C 1 . (CI2) If (C 1 , a 1 ), (C 2 , a 2 ) ∈ C and a 1 ∈ C 2 \ a 2 , then there is a rooted set (C 3 , a 2 ) ∈ C with C 3 ⊆ C 1 ∪ C 2 \ a 1 . Björner and Ziegler noted that these axioms bear a curious resemblance to the circuit axioms for matroids, and we will see in Section 4 that this resemblance is not superficial. A second cryptomorphism for antimatroids is their rooted cocircuits, which form a certain type of dual to their rooted circuits. Definition 2.20. If (E, F ) is an antimatroid and F ∈ F , then an element a ∈ F is called an endpoint of F if F \ a ∈ F . If F ∈ F has a single endpoint a, then we call F a cocircuit, and we call the rooted set (F, a) a rooted cocircuit of F . Equivalently, (F, a) is a rooted cocircuit iff F ∈ F is minimal containing a. We denote by C * (F ) the collection of rooted cocircuits of an antimatroid F . In many places in the literature, antimatroid cocircuits are also called paths, but we use the name cocircuit to emphasize their duality with antimatroid circuits. The descriptive power of these rooted sets is exemplified by the following lemma. In particular, this shows that the cocircuits of an antimatroid also uniquely determine the feasible sets. As with circuits, there is also an axiomatic characterization of the set systems which form the collection of rooted cocircuits of an antimatroid. (CC2) If (D 1 , a 1 ) ∈ C * and a 2 ∈ D 1 \ a 1 , then there is a rooted set (D 2 , a 2 ) ∈ C * with D 2 ⊆ D 1 \ a 1 . Since rooted circuits and rooted cocircuits suffice to specify an antimatroid, when convenient we will sometimes denote an antimatroid using these rooted set systems, as a pair (E, C) or (E, C * ). Finally, we describe the duality which relates the circuits and cocircuits of an antimatroid. Definition 2.23. If E is a finite set and U is a collection of subsets of E, then U is called a clutter if no set in U is contained in another. If U is a clutter, then the blocker of U, denoted B(U) is the collection of minimal subsets B(U) := min {V ⊆ E : V ∩ U is nonempty for each U ∈ U } A basic result of blockers is that the operation of taking blockers is an involution on clutters. In particular, this involution provides the essential connection between antimatroid circuits and cocircuits. Proposition 2.26. Let (E, F ) be an antimatroid with circuits and cocircuits C and C * respectively. Then for each x ∈ E, we have that C x and C * x are clutters, and C * x is the blocker of C x and vice versa. Minors Finally, we will recall two notions of minors which may be defined respectively for greedoids and for antimatroids. First, we give the standard definitions of deletion and contraction for general greedoids. Definition 2.27. If G = (E, F ) is a greedoid and A ⊆ E, then the greedoid deletion G \ A is the set system (E \ A, F \ A), where F \ A = {F ⊆ E \ A : F ∈ F} The greedoid contraction G / A is the set system (E \ A, F / A) where F / A = {F ⊆ E \ A : F ∪ A ∈ F } A greedoid deletion G \ A is always a greedoid, while in general a greedoid contraction G / A is a greedoid only when A is feasible, as otherwise ∅ is not included in the resulting set system. A greedoid minor is a deletion of a contraction of a greedoid. Aside from the limitation that the contracting set is feasible, greedoid minors behave like matroid minors in that the deletion and contraction operations commute with themselves and each other. We provide these definitions for arbitrary greedoids primarily for background and context. For antimatroids in particular, there is an alternate formulation of minors based on rooted circuits which will be central to the discussion in Section 6. Definition 2.28. If A = (E, C) is an antimatroid with rooted circuits C and A ⊆ E, then the antimatroid deletion A \ A is the pair (E \ A, C \ A) where C \ A = {(C, x) : (C, x) ∈ C, C ∩ S = ∅} The antimatroid contraction A / A is the pair (E \ A, C / A) where C / A = min {(C \ S, x) : (C, x) ∈ C, x / ∈ S} and where min R for a collection R of rooted sets denotes the subcollection of those which are (non-strictly) minimal under inclusion as non-rooted sets. In particular, these deletion and contraction operations produce antimatroids, and also behave like matroid minors. Proposition 2.29 ([9], Propositions 12 and 14). If A = (E, F ) is an antimatroid and A ⊆ E, then A / A and A \ A are antimatroids. If A, B ⊆ E are disjoint, then • (A \ A) \ B = (A \ B) \ A • (A \ A) / B = (A / B) \ A • (A / A) / B = (A / B) / A An antimatroid minor may then be defined as a deletion of a contraction of an antimatroid. Although not immediately obvious from the circuit definition, these operations may also be characterized in the following way in terms of antimatroid feasible sets. Proposition 2.30. If (E, F ) is an antimatroid and A ⊆ E, then • F \ A is given by the trace F : (E \ A) • F / A is given by the greedoid deletion F /A = {F ∈ F : F ∩ A = ∅} Antimatroid deletion by a set A can in general be thought of as collapsing the edges of the antimatroid Hasse diagram whose labels for the natural edge labeling (see Definition 3.2) are elements of A. Join-distributive Lattices Finally, we review background on the class of posets called join-distributive lattices, which fundamentally connect antimatroids with lattice theory. Beyond standard notions of lattice theory, we require the following definitions, which follow the exposition of [8]. Definition 2.31. A lattice L is called semimodular or upper semimodular if for all x, y ∈ L, if x ⋗ x ∧ y, then x ∨ y ⋗ y. Definition 2.32. A lattice L is called meet semidistributive if it satisfies the meet semidistributive law, that for all x, y ∈ L and for any z ∈ L, if x ∧ z = y ∧ z, then the common value of these meets is (x ∨ y) ∧ z. Definition 2.33. Given a lattice L, an element x ∈ L is called meet-irreducible if it is covered by exactly one element of L, and is called join-irreducible if it covers exactly one element of L. We denote the set of meet-irreducibles of L by MI(L), and the set of join-irreducibles of L by JI(L). Definition 2.34. Given a lattice L and an element x ∈ L, an irredundant meet decom- position of x is a representation x = Y with Y ⊆ MI(L) such that x = Y ′ for any proper subset Y ′ of Y . The lattice L is said to have unique meet-irreducible decompositions if each x ∈ L has a unique irredundant meet-decomposition. Definition 2.35. If x ∈ L is a member of a locally finite lattice, let j(x) denote the join of all elements covering x. Using this terminology, we can define join-distributive lattices and give several equivalent formulations which will be variously useful for our discussion. The most important property of join-distributive lattices for our purposes is a remarkable correspondence with antimatroids, very similar to the correspondence of Birkhoff's representation theorem for finite distributive lattices. The primary consequence of this correspondence is that join-distributive lattices are essentially equivalent to antimatroids: T gives a one-to-one correspondence between joindistributive lattices and antimatroids F which have no loops, or equivalently, for which the ground set E is covered by the feasible sets of F . x → S x , where S x ∈ F is the unique meet irreducible in L(F ) covered by (S x ∪ x) ∈ F . This bijection of ground sets induces a canonical isomorphism between F and T (L(F )). In general, we will allow for antimatroids with loops. This introduces a slight ambiguity in the equivalence between antimatroids and join-distributive lattices, as an antimatroid with loops has the same feasible sets and associated join-distributive lattice as a corresponding antimatroid with loops removed. This should not cause confusion in practice, however, so we will often refer to general antimatroids and join-distributive lattices interchangeably, keeping this subtlety in mind. Feasible and Independent Sets of Join-distributive Lattices Before moving to the main new results of this work, we will develop some additional theory in the realm of antimatroids and join-distributive lattices which will be useful later. Our aim is to explore the robust connections between the independent sets and the feasible sets of an antimatroid, so we will work in the equivalent context of join-distributive lattices, which provide a more symmetric way to represent these set systems. To begin, we give some notation to describe covering relations and independent sets in join-distributive lattices. J(x) = {e(w, x) : w ∈ L, (w, x) ∈ Cov(L)} I(x) is the independent set associated to x, and is equal to the independent set of feasible extensions of T (x) in the antimatroid corresponding to L. We adopt the following additional notation. Of particular importance is the following: Proof. If x ≤ y, then T (x) ⊆ T (y). If a ∈ I(y) ∩ T (x), then a is a member of both I(y) and T (y), contradicting disjointness. Otherwise, x ∨ y > y. In particular, there is a covering element y a for some a ∈ I(y) such that T (y a ) = T (y) ∪ a, and y a ≤ x ∨ y. Thus a ∈ T (y a ) ⊆ T (x ∨ y) = T (x) ∪ T (y), so because a / ∈ T (y) we conclude that a ∈ T (x) ∩ I(y). Corollary 3.6. The map I : L → I(L) is one-to-one. Proof. If x, y ∈ L satisfy I(x) = I(y), then T (x) ∩ I(y) = T (y) ∩ I(x) = ∅, so x ≤ y and y ≤ x. In particular, an element of a join-distributive lattice is uniquely identified with its corresponding independent set. In fact, this property characterizes the antimatroids among all greedoids. Γ : A → {x ∈ E \ A : A ∪ x ∈ F } is one-to-one. Proof. The forward direction is just restating Corollary 3.6 in the context of antimatroids. So suppose that F is a greedoid and the map Γ is one-to-one. To see that F is an antimatroid, we prove that it satisfies the interval property without upper bounds. As a base case, suppose A, B ∈ F with B = A ∪ x for some x / ∈ A. If A ∪ y ∈ F for some y / ∈ B, we want to show that B ∪ y ∈ F as well. Suppose this is not the case, so that B ∪ y = A ∪ xy / ∈ F . Then we will show that A ∪ x and A ∪ y are mapped to the same set under Γ. To this end, suppose that z ∈ Γ(A ∪ x) for some z, so that A ∪ xz ∈ F . Then in particular, \|A ∪ y\| < \|A ∪ xz\|, so by the greedoid exchange axiom we know there is an element w ∈ (A ∪ xz) \ (A ∪ y) = xz such that A ∪ yw ∈ F . However, by assumption we know that A ∪ xy / ∈ F , so we must have w = z. Then A ∪ yz ∈ F , so z ∈ Γ(A ∪ y). This implies that Γ(A ∪ x) ⊆ Γ(A ∪ y). A symmetric argument proves the reverse inclusion, so we see that Γ maps the two sets to the same independent set, a contradiction. We conclude that in this context, B ∪ y = A ∪ xy ∈ F . In general if A, B ∈ F with A B, then by repeatedly applying the greedoid exchange axiom, there is a sequence of covering sets A i ∈ F with A = A 0 A 1 · · · A k = B where A i+1 = A i ∪ x i for some x i ∈ E. The interval property without upper bounds follows by inducting on the length of this chain using the previous base case. As mentioned previously, the independent sets of an antimatroid are closed under taking subsets, and so form a simplicial complex. In terms of the lattice structure of L, we get a stronger fact, that the inclusion order on the complex embeds in L in the following way. If A ∈ I(L), let x A denote the corresponding lattice element I −1 (A). Lemma 3.8. If J is an independent set of a join-distributive lattice L, and I ⊆ J, then I is independent, and x I ≥ x J . Proof. If I J, there is a lattice element x J ′ ⋗ x J such that I ⊆ J ′ . This follows because if a ∈ J \I, then by definition of independent sets, there is a covering element x J ′ ⋗x J such that T (x J ′ ) \ T (x J ) = {a}. In particular, because L is join-distributive, the interval [x J , j(x J )] is boolean, and so j(x J ′ ) ≥ j(x J ). Noting that for any x ∈ L the relation I(x) = T (j(x)) \ T (x) holds, we have J ′ = T (j(x J ′ )) \ T (x J ′ ) ⊇ T (j(x J )) \ (T (x J ) ∪ a) = J \ a ⊇ I Since L is of finite length, repeated applications of the above must terminate, producing a saturated chain whose greatest element is x K for an independent set K satisfying K ⊇ I but not K I. Hence I = K is independent, and x I ≥ x J . We now state and prove some additional lemmas concerning independent sets of joindistributive lattices which will be useful in later sections. Lemma 3.9. If x ≤ y in a join-distributive lattice L, then I(x) ⊆ I(y) ∪ T (y). Proof. Suppose that a ∈ I(x), and a / ∈ T (y). Then there is an element x a ⋗ x such that T (x a ) = T (x) ∪ a, and by the antimatroid interval property without upper bounds, there must be an element y a ∈ L such that T (y a ) = T (y) ∪ a, and so we have y a ⋗ y. We conclude that a ∈ I(y). x I ∧x J = x K for K independent, then K ⊆ I ∪ J. Proof. Let a ∈ K, and suppose that a / ∈ I ∪J. Since x K ≤ x I , x J , we know that a ∈ I ∪T (x I ) and a ∈ J ∪T (x J ). Thus since a is in neither I nor J, we can conclude that a ∈ T (x I )∩T (x J ). However, since a ∈ K, there exists K ′ independent such that T ( x K ′ ) = T (x K ) ∪ a. Since T (x K ) ⊆ T (I) ∩ T (J) and a ∈ T (I) ∩ T (J), we have that T (x K ′ ) ⊆ T (I) ∩ T (J). We see now that x K ⋖ x K ′ ≤ x I , x J , and this contradicts the claim that x K is the meet of x I and x J . If A ⊆ MI(L), let x A denote the meet of all elements x I for I ⊆ A independent. The element x A ∈ L is equal to x K for some independent set K, and by induction on Lemma 3.10, we have that K ⊆ A. Let I(A) denote this independent set, and note that if A is itself independent, then I(A) = A by Lemma 3.8. Lemma 3.11. If A, B ⊆ MI(L), then x A ∨ x B ≤ x A∩B , and x A ∧ x B ≤ x A∪B . Proof. For the first inequality, let I be independent with I ⊆ A ∩ B, and note that I ⊆ A and I ⊆ B, so x A ≤ x I and x B ≤ x I . In particular, x A ∨ x B ≤ x I , so since this holds for arbitrary I ⊆ A ∩ B, it is also true for the meet of all such elements, hence x A ∨ x B ≤ x A∩B . For the second inequality, let I be independent with I ⊆ A ∪ B, and let I 1 = I ∩ A, and I 2 = I ∩ B. By Lemma 3.8 both I 1 and I 2 are independent, and they satisfy x I 1 , x I 2 ≥ x I . Thus x I 1 ∧ x I 2 ≥ x I , and in fact we will see that x I 1 ∧ x I 2 = x I . If K is independent with x K = x I 1 ∧x I 2 , then by Lemma 3.10, we have that K ⊆ I 1 ∪I 2 = I. For a ∈ I, suppose without loss of generality that a ∈ I 1 . In particular, a / ∈ T (x I 1 ), and this implies a / ∈ T (x K ) because x I 1 ≥ x K . But by Lemma 3.9, since x K ≥ x I , we have that I ⊆ K ∪ T (x K ) , and so we conclude that a ∈ K. Since a ∈ I was arbitrary, we thus have I ⊆ K, so the two sets are equal. Finally, note that since x I 1 ∧ x I 2 = x I and I 1 ⊆ A, I 2 ⊆ B, we have that x A ∧ x B ≤ x I . Since I was chosen arbitrarily in A ∪ B, we conclude x A ∧ x B ≤ x A∪B . Extending Las Vergnas's External Order In [15], Michel Las Vergnas defined partial orderings on the bases of an ordered matroid which are derived from the notion of matroid activity. His external order, defined in terms of matroid external activity, is the starting point for the remainder of this work. B 1 ≤ ext B 2 iff EP(B 1 ) ⊇ EP(B 2 ) The poset obtained by this definition depends on the ordering associated with M, but has some suggestive properties, summarized in the following. 1 ⋖ B 2 in P iff B 2 = B 1 \ b ∪ a, where b ∈ B 1 , and a is the maximal element of bo(B 1 , b) externally active with respect to B 1 . In this case, EP(B 2 ) = EP(B 1 ) ∪ b • L is a lattice with combinatorially defined meet and join operators A dual order, the internal order, can be derived from the external order on the dual ordered matroid M * , and has analogous properties. The Generalized External Order In the same paper, Las Vergnas defined a generalized notion of matroid activity which will be the key to generalizing the external order. In particular, the above definition reduces to the classical definition of matroid activity when A is chosen to be a basis of M. One of the primary properties of external activity that allows the construction of the external lattice on bases is the fact that the map B → EP(B) is one-to-one. This characteristic fails spectacularly for the generalized definition of external activity. However, when we restrict our attention to independent sets, the situation is better. Proof. Suppose that x ∈ Act M \| F (I). Then x ∈ F , and there is a circuit C of M\| F such that x ∈ C ⊆ I ∪ x and x is the smallest element of C. However, the circuits of M\| F are just the circuits of M which are contained in F , so in particular we have that C is also a circuit of M, which shows that x ∈ Act M (I). Now suppose that x ∈ Act M (I). Then there is a circuit C of M such that x ∈ C ⊆ I ∪ x and x is the smallest element of C. In particular, we have that C \ x is an independent subset of F . If x / ∈ F , then we would have x / ∈ span(C \ x), which would imply that C = (C \ x) ∪ x is independent, a contradiction. Thus it must be the case that x ∈ F . This means that C ⊆ I ∪ x ⊆ F , so this implies that C is also a circuit of M\| F . Since C still satisfies the conditions required by the definition of activity in M\| F , we conclude that x ∈ Act M \| F (I). In particular, we have the following. With this result in mind, we extend Las Vergnas's external order to the independent sets of an ordered matroid. Definition 4.7. Let M be an ordered matroid. Then the external order on the independent sets of M is defined by: I 1 ≤ ext I 2 iff EP(I 1 ) ⊇ EP(I 2 ) In particular, because EP restricted to the bases of M is the same as the classical definition used by Las Vergnas, the original external order on the bases of M appears as a subposet of this generalization. As noted in the introduction, for technical convenience we will work with the reverse of this order, I 1 ≤ * ext I 2 iff EP(I 1 ) ⊆ EP(I 2 ) Whenever we refer to the external order, we will be referring to the reversed order ≤ * ext unless otherwise noted. To understand the properties of the generalized external order, we will relate the notion of matroid external activity to an analogous notion for antimatroids, as follows. We first note that the rooted circuits of an antimatroid can be thought of as minimal obstructions to extending feasible sets. On the other hand, if each rooted circuit (C, a) has nonempty intersection with T (x), then the intersection of C with T (x) ∪ a is not equal to the singleton set {a}, and so by Proposition 2.18, we have that T (x) ∪ a ∈ F , so a ∈ I(x). A consequence of this fact is that the rooted circuits of an antimatroid allow us to recover the feasible set associated to a given independent set without reference to any other global structure of the antimatroid. Proof. Let T 0 (x) denote the set in the right side of the equality, and let a be an arbitrary element in E \ I(x). If a / ∈ T 0 (x), then there is a rooted circuit (C, a) ∈ C such that C ⊆ I(x) ∪ a. But then C \ a ⊆ I(x), so C ∩ T (x) is either {a} if a ∈ T (x) or empty if a / ∈ T (x). By Proposition 2.18, since T (x) ∈ F , we see that C ∩ T (x) = {a}, so we conclude that in this case, a / ∈ T (x). Thus T (x) ⊆ T 0 (x). Now suppose that a ∈ T 0 (x). If I(x) ∪ a is independent, say I(y) = I(x) ∪ a, then by Lemma 3.8 we know that x ≥ y, so by Lemma 3.9, I(y) ⊆ I(x) ∪ T (x), and thus a ∈ T (x). If I(x) ∪ a is not independent, it contains a rooted circuit (C, b) ∈ C. Since any subset of I(x) is independent and thus not a circuit, we must have that a ∈ C. However, a cannot be the root of C because in this case C ⊆ I(x) ∪ a violates the fact that a ∈ T 0 (x). However, if b = a then b ∈ I(x), so by Lemma 4.8 we have that C ∩ T (x) is nonempty. Since all elements of C aside from a are in I(x) which is disjoint from T (x), we conclude that a ∈ T (x). Thus T 0 (x) ⊆ T (x) as well. In light of this lemma, it makes sense to define the external activity in an antimatroid as follows. Definition 4.10. Let (E, F ) be an antimatroid with rooted circuits C, and let I be an independent set. Then for a ∈ E \ I, we say that a is externally active with respect to I if there exists a rooted circuit (C, a) ∈ C such that C ⊆ I ∪ a. Otherwise we say that a is externally passive. We denote the active elements of F by EA F (I), and the passive elements by EP F (I), where the subscripts may be omitted if there is no risk of ambiguity. If L is a join-distributive lattice, then EA L (x) and EP L (x) denote the active and passive elements of I(x) in the associated antimatroid F (L). In particular, for x ∈ L a join-distributive lattice, Lemma 4.9 shows that T (x) is the set of externally passive elements of I(x). We can now connect the external order with the theory of antimatroids. Proof. For axiom (CI1), note that if (C 1 , a) and (C 2 , a) are in C, then C 1 and C 2 are circuits of M, and thus C 1 is not a proper subset of C 2 by properties of matroid circuits. For axiom (CI2), suppose (C 1 , a 1 ), (C 2 , a 2 ) ∈ C with a 1 ∈ C 2 \ a 2 . By definition of C we know that a 1 = min(C 1 ) and a 2 = min(C 2 ), so in particular we know that a 1 > a 2 , and a 2 / ∈ C 1 . Note that matroid circuits satisfy the following strong elimination axiom: If C 1 , C 2 are circuits with a 1 ∈ C 1 ∩ C 2 and a 2 ∈ C 2 \ C 1 , then there is a circuit C 3 ⊆ (C 1 ∪ C 2 ) \ a 1 which contains a 2 . Applying this elimination axiom to our present circuits, we obtain a matroid circuit C 3 ⊆ (C 1 ∪ C 2 ) \ a 1 with a 2 ∈ C 3 . a 2 is minimal in C 1 ∪ C 2 , so this implies that a 2 = min(C 3 ), and (C 3 , a 2 ) ∈ C. Thus C satisfies axiom (CI2) as well. In particular, we can see that EP L (x) = EP M (I(x)) for each x ∈ L, and so the feasible set of F associated with each independent set I(x) is given by the set of (matroid) externally passive elements of I(x). Thus the feasible sets of F are exactly the sets in F ext (M), as we wished to show. A further consequence of this argument is that the independent set associated with each feasible set EP M (I) in F ext (M) is in fact I. Following from this correspondence with antimatroids, we may apply Proposition 2.39 to obtain the following. Corollary 4.13. If M = (E, I) is an ordered matroid, then the external order ≤ * ext on I is a join-distributive lattice. Meet-irreducible sets in the lattice correspond with the non-loops of E, and joins correspond to taking unions of externally passive sets. Combinatorial Structure Using the antimatroid structure of the generalized external order, we are able to prove a variety of properties of the poset, many of which generalize the properties enjoyed by the classical order on matroid bases. In the following, M = (E, I) denotes an ordered matroid. I, a)). • If ch(I, a) is empty, J a = I \ a. For each a ∈ I, we have EP(J a ) = EP(I) ∪ a, and thus the sets J a are the independent sets covering I in the external order. Proof. Let a ∈ I, and denote F = span(I), I 0 = I \ a, and F 0 = span(I 0 ). From Lemma 4.15 we know that there exists an independent set J such that EP(J) = EP(I) ∪ a. Since E \ F ⊆ EP(I) and EP(J) ∩ J = ∅, we have that J ⊆ F . Using the antimatroid interval property without upper bounds, with the fact that independent sets are the sets of antimatroid feasible extensions, we know that I 0 ⊆ J. Thus since J is independent and contained in F , either J = I 0 , or J = I 0 ∪ b for some b ∈ bo(I, a). In the latter case, since b ∈ J, b / ∈ EP(J) = EP(I) ∪ a, so this implies that b is an element of the active chain ch(I, a). If ch(I, a) is empty, then we must be in the first case above, so J = I \ a = J a as desired. If ch(I, a) is nonempty, let c be its maximal element, which in particular is in F \ F 0 , and is not in EP(I) ∪ a. On one hand, suppose that J = I 0 . Then F \ span(J) = F \ F 0 ⊆ EP(J), so c ∈ EP(J), and this implies that EP(J) = EP(I) ∪ a, a contradiction. On the other hand, suppose that J = I 0 ∪ c ′ for some c ′ ∈ ch(I, a), c ′ < c. Then because c / ∈ F 0 , we must have ci(J ′ , c) I 0 ∪ c, so c ′ ∈ ci(J ′ , c). This implies that c is externally passive since c ′ < c, so again EP(J) = EP(I) ∪ a. Since there is only one remaining possibility for J, we conclude that J = I \a∪c = J a . The downward covering relations are somewhat more complicated to describe in general, but a particular covering always exists. If x ∈ span(I), then let y = min(ci(I, x)), and let J = I \ y ∪ x. Then ci(J, y) = ci(I, x), so since y < x, we have that y is externally active with respect to J, and in particular is contained in the active chain ch(J, x). In fact, we can show that y = max(ch(J, x)). If this were not the case, then there is an element z > y with z ∈ EA(J) ∩ bo(J, x). Then z ∈ bo(J, x) = bo(I, y), which means that x ∈ ci(J, z) and y ∈ ci(I, z). Since z ∈ EA(J), we have z < x, and since z > y we have that z ∈ EP(I). This contradicts the assumption that x was minimal in EP(I). We conclude that y = max(ch(J, x)), so again by Proposition 4.17, we have that EP(J) = EP(I) \ x. Proof. This follows because ch(I, x) consists only of elements smaller than x, so any covering relation corresponds with either a replacement of an element with a smaller one, or with removal of an element entirely. We can give explicit combinatorial formulations for the meet and join of independent sets in the external order. If I > * ext B, then there is an independent set J ≤ * ext I which covers B, so that EP(J) = EP(B)∪x for some x ∈ E. However, such a J exists exactly when x ∈ B, so since B ⊆ E \A, we have x / ∈ A. Thus EP(I) \ A is nonempty. As a further consequence, we obtain the following partition of the boolean lattice into boolean subintervals. This partition resembles the classic partition of Crapo (see for instance [6]), and in fact, it can be shown that this partition is a refinement of Crapo's. Gordon and McMahon [11] mention that the existence of such a partition is implied by their Theorem 2.5 applied to matroid independent sets, and this explicit form can be proved by first generalizing the idea of their Proposition 2.6 to external activity for arbitrary independent sets. Interestingly, an independent proof is obtained by instead applying Theorem 2.5 to the antimatroid F ext (M). This gives the interval partition [EP(I), E \ I] for I independent and the desired interval partition is obtained from this by taking set complements. The details of these proofs are omitted. Finally, we note that the external order is a refinement of the geometric lattice of flats of the associated matroid. Note in particular that the classical ordering convention ≤ ext which is consistent with Las Vergnas's original definition then gives an order preserving surjection onto the geometric lattice of flats of a matroid. This is a significant reason why in some contexts the classical order convention, rather than the reverse, may be more convenient. Lattice Theory of the Extended Order With the external order identified as a join-distributive lattice, a natural question which arises is to classify the lattices this construction produces. To do so, we will need to incorporate two main ideas. First, we will define the subclass of matroidal join-distributive lattices which characterizes the join-distributive lattices whose independent are those of a matroid. Second, we will identify a property, S n EL-shellability, which ensures a certain order consistency condition for the roots of circuits. We will see in Theorem 3 that these two lattice-theoretic properties, which are satisfied by the external order, are in fact enough to characterize the lattices isomorphic to the external order of an ordered matroid. Matroidal Join-distributive Lattices The most apparent connection between the external order and the underlying ordered matroid is in the equality of the matroid and antimatroid independent sets. We now define the class of matroidal join-distributive lattices to further explore this connection. Recall that for arbitrary A ⊆ E, we denote by x A the meet of the elements I A = {x I : I ⊆ A is independent} In general, x A is equal to a minimal element x I with I ⊆ A independent, and since r c is decreasing in L, covering rank is maximized in I A by x I . This means that I is a maximal size independent subset of A, so we conclude that r(A) = r c (x A ). Now for A, B ⊆ E, by Lemma 3.11 we know x A ∧ x B ≤ x A∪B and x A ∨ x B ≤ x A∩B . Thus with the semimodular inequality for r c and because r c is a decreasing function, we have r(A ∪ B) + r(A ∩ B) = r c (x A∪B ) + r c (x A∩B ) ≤ r c (x A ∧ x B ) + r c (x A ∨ x B ) ≤ r c (x A ) + r c (x B ) = r(A) + r(B) Thus r satisfies the semimodular inequality. Finally, note that if A is independent, then r(A) = \|A\|, and if A is not independent, then the independent subsets of A are proper, so r(A) < \|A\|. Thus the sets A ∈ I are exactly the subsets of E for which r(A) = \|A\|, and so I is the set of independent sets of the matroid with rank function r. With a little more work, we can also prove the converse of this statement: a joindistributive lattice whose independent sets form a matroid is itself matroidal. To this end, a few additional lemmas will be useful. Definition 5.4. Let L be a join distributive lattice whose independent sets are the independent sets of a matroid. Then for x ∈ L, let F x denote the matroid flat cl(T (x) c ). Lemma 5.5. If L is a join-distributive lattice whose independent sets are the independent sets of a matroid M, then for any x ∈ L, the independent set I(x) is a basis of F x . In particular, r c (x) = \|I(x)\| = r(F x ). Proof. Since I(x) ⊆ T (x) c , we have I(x) ⊆ F x for any x, so suppose there is an x ∈ L such that I(x) doesn't span F x . In particular, by properties of matroids there is an element a ∈ T (x) c \ I(x) such that I(x) ∪ a is independent in M, and since I(L) = I(M), there is an element y ∈ L with I(y) = I(x) ∪ a. By Lemma 3.8, we have y < x, and by Lemma 3.9, this means that I(y) ⊆ I(x) ∪ T (x). However, this is a contradiction since a ∈ T (x) c \ I(x). Lemma 5.6. Let L be a join-distributive lattice whose independent sets are the independent sets of a matroid M. If x, y ∈ L satisfy I(x) ⊇ I(y), then the elements of T (y) \ T (x) lie outside of F y . Proof. If x = y this is vacuously true, so suppose x = y. By Lemma 3.8, we have x < y, so there is a sequence of elements x = z 0 ⋖ z 1 ⋖ · · · ⋖ z k = y with edge labels a i = e(z i−1 , z i ). In particular, T (y) \ T (x) = {a 1 , . . . , a k }. For each i, a i ∈ I(z i−1 ). If a i were in I(y) for some i, then we would have a i ∈ T (z i ) ⊆ T (y), so in particular this contradicts disjointness of T (y) and I(y). By induction using Lemma 3.9, we see that I(z i ) ⊇ I(y) for each i. Thus the sets I(y) ∪ a i ⊆ I(z i−1 ) are independent, and a i / ∈ cl(I(y)) for each i. The conclusion follows from Lemma 5.5. Lemma 5.7. Let L be a join-distributive lattice whose independent sets are the independent sets of a matroid M. If x, y ∈ L, then • F x∨y ⊆ F x ∩ F y • F x∧y = cl(F x ∪ F y ) Proof. For the first relation, note that T (x ∨ y) = T (x) ∪ T (y), so F x∨y = cl((T (x) ∪ T (y)) c ) = cl(T (x) c ∩ T (y) c ) ⊆ cl(T (x) c ) ∩ cl(T (y) c ) = F x ∩ F y For the second, begin by noticing that T (x ∧ y) ⊆ T (x) ∩ T (y), so F x∧y = cl(T (x ∧ y) c ) ⊇ cl((T (x) ∩ T (y)) c ) = cl(T (x) c ∪ T (y) c ) = cl(F x ∪ F y ) Let G x∧y := cl(F x ∪ F y ), and suppose the containment F x∧y ⊇ G x∧y is proper. Then since I(x ∧ y) is a basis for F x∧y , we have I(x ∧ y) \ G x∧y is nonempty, containing an element a. Then there exists z ∈ L with I(z) = I(x ∧ y) \ a, and by Lemma 3.8, we have z > x ∧ y. Since a lies outside of G x∧y , we know that I(z) = I(x ∧ y) \ a has span F z ⊇ G x∧y , so in particular F z contains both F x and F y . By Lemma 5.6, since I(x ∧ y) ⊇ I(z), we know that T (z) \ T (x ∧ y) contains only elements outside of F z . However, since F x , F y ⊆ F z , we have T (z) \ T (x ∧ y) ⊆ F c z ⊆ F c x ∩ F c y ⊆ T (x) ∩ T (y) Noting that T (x ∧ y) ⊆ T (x) ∩ T (y), we further conclude that T (z) ⊆ T (x) ∩ T (y), and thus z ≤ x ∧ y. This contradicts z > x ∧ y, so we see that the inclusion F x∧y ⊇ G x∧y must be equality as desired. Finally, we can prove the converse to Proposition 5.3. Proposition 5.8. Let L be a join-distributive lattice. If I(L) is the collection of independent sets of a matroid, then L is matroidal. Proof. Suppose that x ≤ y in L, so that T (x) ⊆ T (y). Then in particular, F x = cl(T (x) c ) ⊇ cl(T (y) c ) = F y , so r c (x) = r(F x ) ≥ r(F y ) = r c (y) and thus r c is decreasing. To prove that r c satisfies the semimodular inequality, we appeal to the corresponding inequality for matroid rank functions. Using Lemmas 5.5 and 5.7, we have r c (x ∧ y) + r c (x ∨ y) = r(F x∧y ) + r(F x∨y ) ≤ r(cl(F x ∪ F y )) + r(F x ∩ F y ) = r(F x ∪ F y ) + r(F x ∩ F y ) ≤ r(F x ) + r(F y ) = r c (x) + r c (y) Gathering the above results, we have proven the following. Theorem 2. A join-distributive lattice L is matroidal if and only if I(L) is the collection of independent sets of a matroid. It is clear from this result that the generalized external order for an ordered matroid M gives a matroidal join-distributive lattice. A natural question to address, then, is whether all matroidal join-distributive lattices arise as the external order for some ordering of their underlying matroid. In fact, this question can be answered in the negative, as the following counterexample demonstrates. Example. Consider the antimatroid on ground set E = {a, b, c, d} whose feasible sets are F = {∅, d, c, bd, cd, ac, abd, bcd, acd, abc}. The Hasse diagram for the corresponding joindistributive lattice appears in Figure 3. In particular, the collection of independent sets of this antimatroid is the uniform matroid U 2 4 of rank 2 on 4 elements. Suppose this were the external order with respect to some total ordering < on E. In this case, we observe that • a is active with respect to I = bc, so a is smallest in the basic circuit ci(bc, a) = abc • b is active with respect to I = ad, so b is smallest in the basic circuit ci(ad, b) = abd But this implies that both a < b and b < a, a contradiction. Thus this matroidal joindistributive lattice cannot come from a total ordering on the ground set E. The External Order and S n EL-labelings To bridge the gap between matroidal join-distributive lattices and the external order, we will need one more key notion, a combinatorial construction on a graded poset called an S n EL-labeling, or snelling. Definition 5.9. If P is a finite poset, then a map λ : Cov(P ) → Z on the covering pairs of P is called an edge labeling of P . If m is an unrefinable chain x 0 ⋖ x 1 ⋖ · · · ⋖ x k in P , then the sequence λ(m) = (λ(x 0 , x 1 ), λ(x 1 , x 2 ), . . . , λ(x k−1 , x k )) is called the label sequence of m, and an unrefinable chain m is called increasing if λ(m) is increasing. The existence of an EL-labeling on a poset P in particular implies that the order complex of P is shellable, and this is the application for which the notion was introduced by Björner in [5]. In particular, a poset which admits an EL-labeling is called EL-shellable. EL-labelings are not sufficiently rigid to capture the combinatorial property we are trying to isolate, but the following strengthening, first introduced by McNamara in [17], couples well with the set system structure of antimatroids. Definition 5.11. An EL-labeling on a finite graded poset P is called an S n EL-labeling or snelling if the label sequence λ(m) of any maximal chain in P is additionally a permutation of the integers 1 to n. A poset which admits an S n EL-labeling is called S n EL-shellable. We proceed to relate S n EL-labelings of join-distributive lattices to the following useful property for antimatroid circuits. Definition 5.12. If (E, F ) is an antimatroid with rooted circuits C, we say that F is confluent if there is an ordering ≤ on the elements of E such that the root of any rooted circuit C ∈ C is given by x = max ≤ (C). We call such an ordering a confluent ordering for F . Similarly, a join-distributive lattice is called confluent if its corresponding antimatroid is confluent. This definition captures the essential structure that distinguishes the external order from other matroidal join-distributive lattices. A useful consequence of confluence is that comparable feasible sets in a confluent antimatroid have lex comparable independent sets in the following sense. Lemma 5.13. In a confluent join-distributive lattice L, if x, y ∈ L satisfy x ≤ y, then I(x) ≤ I(y) in lex ordering, where prefixes of a word S are considered larger than S. Proof. If x ⋖ y, then T (y) = T (x) ∪ a for some a ∈ E = MI(L), and in particular a ∈ I(x). By Lemma 3.9, I(x) \ a ⊆ I(y). Since I(y) is the set of elements in E \ T (y) which are not the root of a circuit disjoint from T (y), any new elements in I(y) \ I(x) are elements b which are the root of a circuit (C, b) with a ∈ C. Since the ordering on E is confluent, the root b is maximal in C, so b > a. This shows that I(y) consists of the elements in I(x) \ a plus a (potentially empty) set of elements S all of which are larger than a. The ordering I(x) < I(y) follows, and the general fact for y not covering x follows by induction on the length of a maximal chain between x and y. The main structural result of this section is Proposition 5.15, which is similar to the work of Armstrong in [4] characterizing supersolvable matroids. In fact, our result can be derived from Armstrong's Theorem 2.13, which lists several conditions which are equivalent to S n EL-shellability of a join-distributive lattice. Our result in particular shows that the condition "(E, F ) is a confluent antimatroid" is also equivalent to the conditions listed in Armstrong's theorem. We provide an independent proof of Proposition 5.15 for the reader's convenience. The proof has the particular advantage of more directly relating S n EL-labelings with the natural labelings of antimatroids without needing to pass through the theory of supersolvable lattices. We begin by proving the following lemma. Lemma 5.14. Let L be a join-distributive lattice. Then any S n EL-labeling of L is equivalent to the natural edge labeling of L for some ordering of its labels. Proof. Let ǫ : Cov(L) → [n] be an S n EL-labeling of L, and let e : Cov(L) → MI(L) denote the natural edge labeling of L. First we prove that for any diamond of elements x, y, x ′ , y ′ ∈ L as below, we have that ǫ(x, x ′ ) = ǫ(y, y ′ ). x x ′ y y ′ To see this, suppose that m is a maximal chain of L which includes the covering relations x ⋖ x ′ ⋖ y ′ , and let m ′ be the maximal chain of L which is identical to m except that it replaces the covering relations x ⋖ x ′ ⋖ y ′ with the relations x ⋖ y ⋖ y ′ . Then the edge labels of m and m ′ form permutations of [n], and the edge labels below x and above y ′ in each chain are identical. In particular, since both are permutations, the sets of labels {ǫ(x, x ′ ), ǫ(x ′ , y ′ )} and {ǫ(x, y), ǫ(y, y ′ )} are the same, say {a, b} with a < b. Since ǫ is an S n EL-labeling, exactly one chain in the interval [x, y ′ ] is in increasing order, which means that ǫ gives one of the two labelings: x x ′ y y ′ a b a b x x ′ y y ′ b a b a In either case, ǫ(x, x ′ ) = ǫ(y, y ′ ), as we wished to show. Now let x, x ′ ∈ L be a covering pair, x ⋖ x ′ , let y ∈ MI(L) be the edge label e(x, x ′ ), and let y ′ be the unique element covering y in L. We will show that in this case, ǫ(x, x ′ ) = ǫ(y, y ′ ). To see this, note that x ≤ y, and let m be a maximal chain between x and y, given by x = z 0 ⋖ z 1 ⋖ · · · ⋖ z k = y. If k = 0, then x = y and the desired relation holds trivially. Otherwise, by the interval property without upper bounds, there exist elements z ′ i ⋗ z i with e(z i , z ′ i ) = y, and we observe a parallel chain m ′ given by x ′ = z ′ 0 ⋖ z ′ 1 ⋖ · · · ⋖ z ′ k = y ′ . Then each pair of coverings z i ⋖ z ′ i and z i+1 ⋖ z ′ i+1 form a diamond of elements as in the previous argument, and so ǫ( z i , z ′ i ) = ǫ(z i+1 , z ′ i+1 ) for each i. This shows that ǫ(x, x ′ ) = ǫ(y, y ′ ). Finally, let m now denote the unique increasing maximal chain of L in the labeling ǫ, given by 0 = x 0 ⋖ x 1 ⋖ · · · ⋖ x n = 1. In particular, since the labels of m are an increasing permutation of [n], we have that ǫ(x i−1 , x i ) = i for each i. Then each covering in this chain corresponds with the meet irreducible y i = e(x i−1 , x i ), which is covered by a unique element y ′ i . By the above argument, ǫ(y i , y ′ i ) = ǫ(x i , x ′ i ) = i as well. In particular, this implies that for any covering relation x ⋖ x ′ in L, the label ǫ(x, x ′ ) is given by the label ǫ(y i , y ′ i ) = i, where e(x, x ′ ) = y i . Thus ǫ(x, x ′ ) = ϕ(e(x, x ′ )) for the bijection ϕ : MI(L) → [n] given by y i → i, and we see that ǫ is equivalent to e under the ordering induced by ϕ. Applying this lemma, we can demonstrate the equivalence of confluence and S n ELshellability for join-distributive lattices. We will prove in two parts the following: Proposition 5.15. A join-distributive lattice is S n EL-shellable if and only if it is confluent. Lemma 5.16. If L is a confluent antimatroid, then the natural edge labeling of L is an S n EL-labeling for any confluent ordering. Proof. Fix a confluent ordering of E = MI(L), and as usual, let e : Cov(L) → E denote the natural edge labeling of L. The fact that the sequence of labels of any maximal chain gives a permutation of E is clear from the fact that the union of the edge labels of a maximal chain is equal to E = T (1). Thus it is sufficient to show that every interval [x, y] has a unique increasing maximal chain. Further, since the edge labels of any maximal chain in [x, y] are a permutation of T (y) \ T (x) and determine the chain uniquely, it is enough to prove that there is a chain whose edge labels are the increasing sequence of the elements of T (y) \ T (x). For this, we proceed by induction on the size of T (y) \ T (x). If x = y, then the empty chain is sufficient, so suppose that x < y, and let a = min(T (y) \ T (x)). For any z ∈ [x, y], we have that I(z) is lex greater than or equal to I(x) in the sense of Lemma 5.13. Further, if we denote J = I(x) \ T (y), then we have J ⊆ I(z) by the antimatroid interval property without upper bounds. Thus the smallest element of lexicographic divergence between I(x) and I(z) must be an element b of I(x) ∩ T (y) which is contained in I(x) but not in I(z). In particular we have b ∈ T (y) \ T (x). Since a is smallest in T (y) \ T (x), if a / ∈ I(x), then the smallest element of divergence between I(x) and I(z) is larger than a, so a / ∈ I(z). However, this holds for any z ∈ [x, y], so if it were the case that a / ∈ I(x), then we would conclude that there are no edges in [x, y] labeled by a, which would imply that a / ∈ T (y), a contradiction. Thus we must have a ∈ I(x). In particular, this means that there is an element x ′ covering x such that T (x ′ ) = T (x)∪a, and by induction, there is a unique increasing chain in the interval [x ′ , y], whose labels are the increasing permutation of the elements in T (y) \ (T (x) ∪ a). Appending this chain to the covering relation x ⋖ x ′ gives an increasing chain in [x, y], and completes the proof. Lemma 5.17. If L is a non-confluent join-distributive lattice, then L is not S n EL-shellable. Proof. Let (E, F ) be the associated antimatroid of L, and suppose that L is non-confluent. Then for any ordering of E, there is a rooted circuit C whose root is not maximal in C. Suppose that nevertheless, L is S n EL-shellable. By Lemma 5.14, an S n EL-labeling corresponds with the natural labeling e : Cov → E for some ordering of E. With respect to that ordering, there is a rooted circuit (C, a) of F such that a = max(C). Let b = max(C). By Proposition 2.26, the stem C \ a of C is in the blocker for the clutter of stems C * a = {D \ a : (D, a) an antimatroid cocircuit of F} In particular, since a blocker consists of the minimal sets intersecting each set in a clutter, we have that (C \ a) \ b is not in the blocker of C * a , and so some antimatroid cocircuit (D, a) must include b in its stem D \ a. In particular, D is feasible and corresponds with a join-irreducible element of L where the single feasible set covered by D is D \ a. If x ∈ L satisfies T (x) = D, then any chain m given by 0 = z 0 ⋖ z 1 ⋖ · · · ⋖ z k = x has edge labels which are a permutation of the elements of D. Further, since the only feasible set covered by D is D \ a, we have that e(z k−1 , z k ) = a. This implies that a comes after b in the sequence of edge labels of m, and so m is not an increasing chain. This contradicts the fact that in an S n EL-labeling, any interval must have a unique increasing maximal chain. We conclude that no S n EL-labeling exists, and so a join-distributive lattice which is non-confluent is not S n EL-shellable. Finally, Proposition 5.15 allows us to classify the matroidal join-distributive lattices which are the external order for a matroid. Specifically, it is immediate that a matroidal joindistributive lattice L is the external order of an ordered matroid iff it is confluent, in which case the underlying matroid may be ordered by the reverse of any confluent ordering of L. Thus we immediately conclude Corollary 5.18. A matroidal join-distributive lattice L with corresponding matroid M is the external order for some ordering of M if and only if L is S n EL-shellable. Aggregating our results to this point, we can now state a complete characterization of lattices corresponding with the external order of an ordered matroid. Theorem 3. A finite lattice L is isomorphic to the external order ≤ * ext of an ordered matroid if and only if it is join-distributive, matroidal, and S n EL-shellable. Deletion and Contraction We continue by exploring a correspondence between the deletion and contraction operations of matroids and antimatroids which is introduced by the external order construction. In the following, let (E, F ) denote an antimatroid, and unless otherwise noted, for A ⊆ E let F \ A and F / A denote antimatroid deletion and contraction, as defined in Section 2.2.2. Definition 6.1. We call an element a ∈ E an extending element of F if a is the root of any circuit of F which contains it. We say that A ⊆ E is an extending set of F if there is an ordering A = {a 1 , . . . , a k } such that a i is an extending element of F \ {a 1 , . . . , a i−1 } for each i. It is not hard to show that an antimatroid (E, F ) is confluent (cf. Section 5.2) if and only if E is an extending set. The following lemma relates antimatroid deletion with the standard greedoid deletion and contraction operations. Lemma 6.2. If A ∈ F is a feasible set, then the antimatroid deletion F \ A is equal to the greedoid contraction F / A. If A is an extending set of F , then the antimatroid deletion F \ A is equal to the greedoid deletion F \ A. The first part of this lemma is discussed in [9], Section 4, but we will prove both parts here for completeness. Proof. Because antimatroid and greedoid minors satisfy the usual commutativity properties of minors, in each case it is sufficient to prove the lemma when A = {a} is a singleton set. If A = {a} is a feasible set, then F \ A = {F \ a : F ∈ F}. On the other hand, the greedoid contraction by {a} consists of all sets G ⊆ E such that G ∪ a ∈ F . In particular, the feasible sets F containing a correspond with the feasible sets G = F \ a in the greedoid contraction, so any feasible set in the greedoid contraction is also feasible in the antimatroid deletion. The remaining feasible sets in the antimatroid deletion are sets F ∈ F with a / ∈ F . For these sets, note that because ∅ ⊆ F and ∅ may be extended to {a}, we see by the antimatroid interval property without upper bounds that F ∪ a ∈ F as well. Thus F = F \ a = (F ∪ a) \ a is also feasible in the greedoid contraction. Now suppose A = {a} where a is an extending element of F . One consequence of being an extending element is that for any feasible set F , if a ∈ F , then F \ a is feasible. To see this, let F ∈ F be a feasible set containing a, and suppose that F \ a / ∈ F . Then there exists a rooted circuit (C, x) such that (F \ a) ∩ C = {x}, and in particular, we have that the root x is not equal to a. Because a is an extending element, we conclude that a / ∈ C. However, this means that F ∩ C = {x} as well, so we conclude that F is not feasible, a contradiction. From this we see that the antimatroid deletion F \ A = {F \ a : F ∈ F} is given by the feasible sets of F which don't contain a. This is exactly the greedoid deletion by {a}. Note that for A feasible, it follows directly that F \ A corresponds with the the interval [A, E] in F via the map F → F ⊔ A. For A extending, it follows that E \ A is feasible, and F \ A is equal to the interval [∅, E \ A] in F . We now show that matroid and antimatroid deletion are in exact correspondence for matroidal antimatroids. Proof. Recall that the circuits of an antimatroid are the minimal non-independent sets, so an antimatroid is matroidal with associated matroid M iff its circuits are the circuits of M. Now let C denote the collection of rooted circuits of F . Then the circuits of F \ a are given by C \ a = {C ∈ C : C ∩ {a} = ∅} Forgetting the roots, these are exactly the circuits of M \ a, so we conclude that F \ a is matroidal with associated matroid M \ a. Remembering the roots, if M is ordered then F = F ext (M) iff every circuit C of F has root x = min(C). This property is preserved by restricting to a subset of the circuits, so we see that if F = F ext (M), then F \ a = F ext (M \ a). Antimatroid contractions do not behave as nicely as deletions with respect to matroid structure -in many cases, contraction does not even preserve the property of being matroidal! However, for certain contraction sets the situation is still favorable. Proposition 6.4. Suppose that F is matroidal with associated matroid M. • For A feasible, the antimatroid contraction F / A is matroidal with associated matroid M ′ = M / A. • For A extending, the antimatroid contraction F / A is matroidal with associated matroid M ′ = M \ A. For either case, if F = F ext (M) for an ordered matroid M, then F / A = F ext (M ′ ), where the order on M ′ is induced by the order on M. Proof. As in Lemma 6.2, it is sufficient to prove these cases when A = {a} is a singleton set because of commutativity properties of minors. If A = {a} is a feasible set, then A ∩ C = {a} for any rooted circuit C, and so a is never the root of a circuit of F . In particular, this means that C(F / a) = min {(C \ a, x) : (C, x) ∈ C(F )} The circuits of M / a are exactly the underlying sets of the rooted circuits of F / a, so we conclude that F / a is matroidal with associated matroid M / a. If M is ordered and F = F ext (M), then any rooted circuit (C ′ , x) of F / a corresponds with a rooted circuit (C, x) of F , where C ′ = C \ a. Since F = F ext (M), we have x = min(C), and since x = a, we have also that x = min(C ′ ), so the root of each circuit of F / a is the minimal element of the circuit. This implies that F / a = F ext (M / a) for the induced order on M / a. If A = {a} for a an extending element of F , then a is the root of any circuit containing it. In particular this means that Thus in this case, F / a = F \ a, and the result follows from Proposition 6.3. Although antimatroid contraction doesn't preserve matroid structure for arbitrary contraction sets, if F is the external order for an ordered matroid, the resulting set system is related nicely to the external orders for the corresponding matroid deletion and contraction. We start with two lemmas, one due to Dietrich, and the other a short technical lemma on matroid deletions. Lemma 6.5 ([9], Lemma 13). If (C, x) ∈ C(F ) and A ⊆ E with x / ∈ A, then there exists a rooted circuit (C ′ , x) ∈ C(F / A) with C ′ ⊆ C \ A. Lemma 6.6. Let M be a matroid on ground set E, and let A ⊆ E. If C ∈ C(M), then for each x ∈ C \ A, there exists C ′ ∈ C(M / A) with C ′ ⊆ C and x ∈ C ′ . Proof. We induct on the size of A. If A = ∅, then the lemma holds trivially. Now suppose that \|A\| ≥ 1, and let a ∈ A. We will apply a result from [18] Exercise 3.1.3, which states that • If a ∈ C, then either a is a loop or C \ a is a circuit of M / a • If a / ∈ C, then C is a union of circuits of M / a Let C ∈ C, assume without loss of generality that C \ A is nonempty, and let x ∈ C \ A. Suppose first that a ∈ C. If a were a loop, this would imply C = {a}, which contradicts our assumption that C \ A is nonempty. By the above, we now have that C \ a is a circuit of M / a. In particular, x ∈ (C \ a) \ (A \ a), so by induction there exists a circuit C ′ of M / A = (M / a) /(A \ a) such that C ′ ⊆ C \ a ⊆ C and x ∈ C ′ . Thus the lemma holds. Now suppose that a / ∈ C. Then C is a union of circuits of M / a, so in particular there is a circuit C ′ ∈ C(M / a) with C ′ ⊆ C and x ∈ C ′ . Inductively there exists a circuit C ′′ of M / A = (M / a) /(A \ a) such that x ∈ C ′′ and C ′′ ⊆ C ′ ⊆ C. This completes the proof. Using these lemmas, we prove the following. Proof. We begin with the left inclusion. Suppose that F ⊆ E \ A is not feasible in F / A, so that there exists a rooted circuit (C, x) of F / A such that F ∩ C = x. Then in particular, C = C 0 \ A for a rooted circuit (C 0 , x) ∈ C(F ) with x / ∈ A. Since F = F ext (M), the set C 0 is a circuit of M, and x = min(C 0 ). By Lemma 6.6, there exists a circuit C ′ ∈ C(M / A) with C ′ ⊆ C 0 \ A = C and x ∈ C ′ . Since x = min(C 0 ), we also have x = min(C ′ ), so (C ′ , x) is a rooted circuit of F ext (M / A). In particular we see that C ′ ∩ F = {x}, so we conclude that F is also not feasible in F ext (M / A). For the right inclusion, suppose that F ⊆ E \ A is not feasible in F ext (M \ A), so that there exists a rooted circuit (C, x) of F ext (M \ A) with C disjoint from A and F ∩ C = x. Then (C, x) ∈ C(F ), and by Lemma 6.5, there is a circuit (C ′ , x) ∈ C(F / A) with C ′ ⊆ C \ A = C. In particular, F ∩ C ′ = x, so we conclude that F is also not feasible in F / A. Further Work We conclude by discussing some potentially fruitful directions for future study. One common thread which was encountered when investigating the external order is the incongruity between matroids and antimatroids in the area of duality. Matroids admit a classical involutive duality operator M → M * , where M * is defined on the same ground set as M and the bases of M * are the complements of the bases of M. On the other hand, in [9], Dietrich proves that for antimatroids, no involutive duality operator exists which satisfies certain desirable properties with respect to antimatroid deletion and contraction. Given that the external order provides a correspondence between ordered matroids and a subclass of antimatroids, it would be interesting if matroid duality could be lifted to at least a subclass of antimatroids. Question 1. Is there a natural notion of duality for matroidal antimatroids which corresponds with matroid duality of independence complexes? A positive answer to this question would likely provide the basis for a generalized notion of internal and external matroid activity derived from antimatroid circuit roots rather than a total order on the ground set, and one might study whether such a generalization exhibits the standard behaviors and properties of matroid activity. If such a generalization exists, it would be interesting if it could be related to the decision trees of [11]. Another notion which arises from the external order is the idea that we can isolate certain distinguished local exchange moves between the independent sets of a matroid, which correspond with the covering relations of the external order (cf. Proposition 4.17). Due to the rooted circuit structure of antimatroids, and in particular by Lemma 4.8, similar exchange structure can be isolated for the independence complexes of arbitrary antimatroids. Question 2. Are there other natural classes of simplicial complexes that can be represented as the independence complexes of suitable antimatroids? What exchange structure emerges from such representations, and what does this reveal about the structure of these complexes? The independence complexes of antimatroids seem to be a rather broad class. One place to start could be to determine whether there exist simplicial complexes which cannot be realized as an antimatroid independence complex. Figure 1 comparesFigure 1 : 11Las Vergnas's external order with the generalized order for the linear matroid represented by the columns of the matrix Las Vergnas's external order ≤ ext on bases B, the generalized order ≤ * ext on independent sets I, and the corresponding externally passive sets EP X (I). Note that Las Vergnas's order embeds in the generalized order (in bold) in reversed orientation. Proposition 4. 23 . 23If M is an ordered matroid with ground set E, then the intervals [I, I ∪ EA(I)] for I independent form a partition of the boolean lattice 2 E . Definition 2. 3 . 3Let M = (E, I) be a matroid, and let B be a basis of M. For x / ∈ B, define ci M (B, x) the basic circuit of x in B to be the unique circuit contained in B ∪ x. Dually, for b ∈ B define bo M (B, b) the basic cocircuit or basic bond of b in B to be the unique cocircuit contained in (E \ B) ∪ b. Definition 2 . 5 . 25Let M be a matroid, let I ∈ I(M), and denote F = span(I). For x ∈ F \ I, define ci(I, x) = ci M \| F (I, x) and for y ∈ I, define bo(I, y) = bo M \| F (I, y) Proposition 2.11 ([7], Proposition 8.2.7) Lemma 2 . 14 . 214If (E, F ) is an antimatroid, then X ⊆ E is independent if and only if it is equal to the feasible extensions Γ(A) of some feasible set A ∈ F . Definition 2 . 216. A circuit of an antimatroid (E, F ) is a minimal dependent subset of E. Let (E, F ) be an antimatroid and A ⊆ E. Then A is feasible if an only if C ∩ A = {a} for every rooted circuit (C, a). Proposition 2 . 219 ([7],Theorem 8.7.12). Let C be a family of rooted subsets of a finite set E. Then C is the family of rooted circuits of an antimatroid if an only if the following two axioms are satisfied: Lemma 2.21 ([13],Lemma 3.12). If (E, F ) is an antimatroid and A ⊆ E, then A is feasible if and only if it is a union of cocircuits. If A has k endpoints {a 1 , . . . , a k }, then A is a union of k cocircuits {A 1 , . . . , A k }, where the root of each A i is a i . Proposition 2 . 22 . 222Let C * ⊆ {(D, a) : D ⊆ E, a ∈ D} be a family of rooted subsets of a finite set E. Then C * is the family of rooted cocircuits of an antimatroid (E, F ) if an only if the following two axioms are satisfied: (CC1) If (D 1 , a) ∈ C * , then there is no rooted set (D 2 , a) ∈ C * with D 2 D 1 . Lemma 2 . 24 . 224For any clutter U, the blocker V = B(U) is a clutter, and B(V) = U. Definition 2 . 25 . 225If A is a collection of rooted subsets of a ground set E and x ∈ E, let A x denote the collection of sets {A \ x : (A, x) ∈ A}. Definition 2 . 236. A finite lattice is called join distributive if it is semimodular and meetsemidistributive.Proposition 2.37 ([8], Proposition 2.1). For a finite lattice L, the following are equivalent. 1. L is join-distributive 2. L has unique meet-irreducible decompositions 3. For each x ∈ L, the interval [x, j(x)] is a boolean lattice 4. The length of each maximal chain in L is equal to \|MI(L)\|. Definition 2.38 ([7]). Given a finite join-distributive lattice L, let F (L) denote the set system which is the image of the map T : L → 2 MI(L) given byT : x → {y ∈ MI(L) : y x}Proposition 2.39 ([7], Theorem 8.7.6). T is a poset isomorphism from L to F (L) ordered by inclusion, and joins in L correspond to unions in F (L). F (L) is an antimatroid with ground set MI(L), and the poset F of feasible sets of any antimatroid, ordered by inclusion, forms a join-distributive lattice. Figure 2 2demonstrates the application of this map to produce an antimatroid from a join-distributive lattice. Figure 2 : 2The T map applied to a join-distributive lattice with labeled meet irreducibles Explicitly, if F is an antimatroid with ground set E = F ∈F F , let L(F ) denote the joindistributive lattice formed by the feasible sets of F under set inclusion. Then the elements of E are in bijection with the meet irreducibles of L(F ) by the map Definition 3. 1 . 1For a poset P , let Cov(P ) ⊆ P × P denote the covering pairs (x, y), with x ⋖ y. Definition 3 . 2 . 32Let L be a join-distributive lattice. Recall from Definition 2.38 the map T : L → 2 MI(L) which maps L to its associated antimatroid, and let e : Cov(L) → MI(L) denote the natural edge labeling, given by e : (x, y) → T (y) \ T (x). Such set differences are singletons, hence the map is well-defined into MI(L). Definition 3 . 3 . 33If x ∈ L is an element of a join-distributive lattice, let I(x) denote the set of elements I(x) = {e(x, y) : y ∈ L, (x, y) ∈ Cov(L)} and let J(x) denote the set of elements Definition 3. 4 . 4If L is a join-distributive lattice, • Let F (L) = (MI(L), {T (x) : x ∈ L}) denote the (loopless) antimatroid associated with L • Let I(L) = {I(x) : x ∈ L} denote collection of independent sets of L • Let C(L) denote the collection of rooted circuits of F (L), which we interchangeably refer to as the rooted circuits of L Notice that I(x) is disjoint from T (x), and J(x) is a subset of T (x). The meet-irreducible elements x ∈ MI(L) are characterized by the condition \|I(x)\| = 1, in which case I(x) = {x}. The join-irreducible elements y ∈ JI(L) are characterized by the condition \|J(x)\| = 1, and in particular correspond with the rooted cocircuits of F (L). Lemma 3. 5 . 5For x, y ∈ L elements of a join-distributive lattice, T (x) has empty intersection with I(y) if and only if x ≤ y. Proposition 3.7. A greedoid (E, F ) is an antimatroid if and only if the feasible extension operator Lemma 3 . 10 . 310If I, J are independent sets of a join-distributive lattice L, then if Definition 4. 1 1([15]). Let M be an ordered matroid. Then Las Vergnas's external order on the set of bases of M is defined by: Proposition 4. 2 . 2Let M = (E, I) be an ordered matroid, and let P = (B(M), ≤ * ext ) be the external order on the bases of M. Let L denote the poset P with an additional minimal element 0 added to the ground set. Then • P is a graded poset, graded by \|EP(B)\| • Two bases B 1 and B 2 satisfy a covering relation B Definition 4. 3 . 3Let M = (E, I) be an ordered matroid, and let A ⊆ E. Then we say thatx ∈ E is M-active with respect to A if there is a circuit C of M with x ∈ C ⊆ A ∪ x such that x isthe smallest element of C. We denote the set of such M-active elements by Act M (A), and define 1. EA M (A) := Act M (A) \ A 2. EP M (A) := (E \ A) \ EA M (A) 3. IA M (A) := Act M * (E \ A) ∩ A 4. IP M (A) := A \ IA M (A) Lemma 4. 4 . 4Let M = (E, I) be an ordered matroid, and let I ∈ I. Then if F is the flat spanned by I, we have Act M (I) = Act M \| F (I) Corollary 4. 5 . 5If M = (E, I) is an ordered matroid and I ∈ I with F = span(I), then EP M (I) = EP M \| F (I) ∪ (E \ F ) and in particular F = span(E \ EP M (I))Proof. The first equality follows directly from the above lemma, noting that Act M (I) = Act M \| F (I) ⊆ F . The second equality follows becauseI ⊆ E \ EP M (I) ⊆ F Corollary 4.6. If M is an ordered matroid, then the map EP M : I → 2 E is one-to-one.Proof. From previous theory we know that EP is one-to-one when restricted to the bases of a matroid. Now let I, J be distinct independent sets of M, with F I = span(I) andF J = span(J). If F I = F J , then by the above lemma, span(E \ EP M (I)) = F I = F J = span(E \ EP M (J)) Thus in this case the two passive sets cannot be equal. If F I = F J , call this common spanning flat F . Then I and J are distinct bases of the restriction matroid M\| F . This gives that EP M \| F (I) = EP M \| F (J), so EP M (I) = EP M \| F (I) ∪ (E \ F ) = EP M \| F (J) ∪ (E \ F ) = EP M (J) because the unions with (E \ F ) are disjoint unions. Lemma 4. 8 . 8Let (E, F ) be an antimatroid with associated join-distributive lattice L, let x ∈ L, and let a ∈ E \ T (x). Then a ∈ I(x) if and only if each rooted circuit (C, a) of F has nonempty intersection with T (x).Proof. If a ∈ I(x), then T (x) ∪ a is a feasible set. If a rooted circuit (C, a) is disjoint from T (x), then the intersection of C with T (x) ∪ a is equal to the singleton set {a}. However, this violates the definition from Proposition 2.17 of the root of a rooted circuit. Lemma 4. 9 . 9Let (E, F ) be an antimatroid with associated join-distributive lattice L, and let x ∈ L. Then T (x) = {a ∈ E \ I(x) : C I(x) ∪ a for any (C, a) ∈ C(F )} Proposition 4. 11 . 11If M is an ordered matroid, then the collection of rooted sets C = C ext (M) := {(C, min(C)) : C ∈ C(M)} satisfies the axioms of rooted antimatroid circuits. Proposition 4 . 411 allows us to conclude the following structural characterization of the generalized external order. Definition 4 . 12 . 412If M is an ordered matroid, let F ext = F ext (M) := {EP M (I) : I ∈ I(M)} Theorem 1. If M is an ordered matroid, then F ext (M) is the collection of feasible sets of the antimatroid with rooted circuits C ext (M). Proof. Denote M = (E, I). By Proposition 4.11, C ext (M) forms the rooted circuits of an antimatroid (E, F ). Let L be the associated join-distributive lattice. By definition of antimatroid circuits as minimal dependent sets, we have that I(L) = I so that the sets I(x), x ∈ L are in correspondence with the matroid independent sets of M. By Lemma 4.9, any element x ∈ L has T (x) = EP L (x) = {a ∈ E \ I(x) : C I(x) ∪ a for any (C, a) ∈ C ext (M)} Lemma 4 . 14 . 414The following basic properties hold for independent sets and externally passive sets in M.1. If I, J ∈ I, then I ≤ * ext J if and only if EP(I) ∩ J = ∅ 2. If I, J ∈ I and J ⊇ I, then J ≤ * ext I 3. If I, J ∈ I, then I ∧ J ⊆ I ∪ J Proof. The three parts are restatements of Lemmas 3.5, 3.8 and 3.10 respectively in the context of the generalized external order. Lemma 4 . 15 . 415If M = (E, I) is an ordered matroid, I ∈ I, and a ∈ E \ EP(I), the set EP(I) ∪ a is the set of externally passive elements of some independent set iff a ∈ I.Proof. Let L be the join-distributive lattice associated with the antimatroid F ext (M), and let x ∈ L be the element with I(x) = I. Then I(x) is the set of feasible extensions of T (x) = EP(I), so EP(I) ∪ a is feasible in F ext (M) iff a ∈ I(x) = I. The result follows because the feasible sets are exactly the sets of externally passive elements.We now characterize the covering relations in the external order. Definition 4. 16 . 16For an ordered matroid M, if I us independent and a ∈ I, define the active chain of a in I to be the set ch(I, a) = EA M (I) ∩ bo(I, a) Proposition 4.17. Let M be an ordered matroid, and let I ∈ I(M). Then for each a ∈ I, define the independent set J a by • If ch(I, a) is nonempty, J a = I \ a ∪ max(ch( Lemma 4 . 18 . 418Let M be an ordered matroid. If I is independent and x = min(EP(I)), then there is an independent set J such that EP(J) = EP(I) \ x. Proof. If x / ∈ span(I), then let J = I ∪ x. Then the active chain ch(J, x) is empty, so from Proposition 4.17, EP(J) = EP(I) \ x. Corollary 4. 19 . 19If M = (E, I) is an ordered matroid and I, J ∈ I satisfy I ≤ * ext J, then I is lexicographically greater than or equal to J, where prefixes are considered small. Lemma 4 . 20 . 420If A ⊆ E, then the lex maximal basis B of M \ A satisfies EP(B) ⊆ A. If I > * ext B for some independent set I, then EP(I) \ A is nonempty. Proof. Suppose x ∈ EP(B) \ A. Then the element y = min(ci(B, x)) is an element of B, and the basis B ′ = B \ y ∪ x gives a basis in M \ A which is lex greater than B, a contradiction. Thus EP(B) ⊆ A. Proposition 4. 21 . 21The minimum element of the external order is the lex maximal basis of M, and the maximum element of the external order is the empty set. If I, J ∈ I, then meets and joins in the external order are described by• I ∧ J is the lex maximal basis of M \ (EP(I) ∩ EP(J)) • I ∨ J is the lex maximal basis of M \ (EP(I) ∪ EP(J)) Proof. The proof is by repeated application of Lemma 4.20. The lex maximal basis B of M = M \ ∅ has EP(B) ⊆ ∅, so B is the minimum element in the external order. Likewise, EP(∅) is the ground set of M minus any loops (which are never externally passive), so ∅ is the maximum element.To characterize meets, let K be the lex maximal basis of M \ (EP(I) ∩ EP(J)). Then EP(K) ⊆ EP(I) ∩ EP(J), so we have that K ≤ * ext I ∧ J. Further, if K ′ ≥ * ext K, then EP(K ′ ) contains an element outside of EP(I) ∩ EP(J), which shows that K ′ is not less than one of I or J. Since K ≤ * ext I, J and no larger independent set is, we conclude that K = I ∧ J. To characterize joins, let K be the lex maximal basis of M \ (EP(I) ∪ EP(J)), so that EP(K) ⊆ EP(I) ∪ EP(J). By properties of antimatroids, EP(I ∨ J) = EP(I) ∪ EP(J), so in particular, we have K ≤ * ext I ∨ J. If this relation is not equality however, we note that EP(I ∨ J) contains an element outside of EP(I) ∪ EP(J), which is a contradiction. Thus we must have equality, so K = I ∨ J.From this we also conclude Corollary 4.22. I is the lex maximal basis of M \ EP(I) for any independent set I. Proposition 4. 23 . 23If M is an ordered matroid with ground set E, then the intervals [I, I ∪ EA(I)] for I independent form a partition of the boolean lattice 2 E . Proposition 4 . 24 . 424The natural map from the external order ≤ * ext on M to the geometric lattice of flats of M given by I → span(I) is surjective and monotone decreasing. In particular, the external order on M is a refinement of the geometric lattice of flats of M.Proof. Suppose I and J are independent with I ≤ * ext J. In particular, EP(I) contains all elements outside of span(I), and by Lemma 4.14, we also have EP(I) ∩ J = ∅. Thus J ⊆ span(I), so we conclude span(J) ⊆ span(I). Definition 5 . 1 . 51If L is a join-distributive lattice, define the covering rank function r c of L by r c : x → \|I(x)\| counting the number of elements in L which cover x.Definition 5.2. We call a join-distributive lattice L matroidal if the covering rank function r c is decreasing, and satisfies the semimodular inequalityr c (x ∧ y) + r c (x ∨ y) ≤ r c (x) + r c (y) Proposition 5.3.If L is a matroidal join-distributive lattice, then I(L) is the collection of independent sets of a matroid with ground set MI(L).Proof. For notational convenience, let I = I(L) and let E = MI(L). We will show that the function r : 2 E → Z ≥0 defined byr(A) = max {\|I\| : I ∈ I, I ⊆ A}is a matroid rank function on 2 E whose independent sets are I.Both the fact that 0 ≤ r(A) ≤ \|A\| for any subset A and that r(A) ≤ r(B) for subsets A ⊆ B ⊆ E are clear from the definition of r. Thus all that remains is to prove the semimodular inequality r(A ∪ B) + r(A ∩ B) ≤ r(A) + r(B) for any subsets A, B ⊆ E. Figure 3 : 3Feasible sets of F with edge labels, and corresponding independent sets Definition 5 . 10 . 0 • 5100If P is a finite graded poset, then an edge labeling λ is called an edge lexicographic or EL-labeling if • Any interval [x, y] ⊆ P has a unique increasing maximal chain m Any other maximal chain in [x, y] has edge labels which are lex greater than the edge labels of m 0 Proposition 6 . 3 . 63Suppose that F is matroidal with associated matroid M. Then for A ⊆ E, the antimatroid deletion F \ A is matroidal with associated matroid M \ A. If F = F ext (M) for an ordered matroid M, then F \ A = F ext (M \ A), where the order on M \ A is induced by the order on M. C ( F (/ a) = min {(C \ a, x) : (C, x) ∈ C(F ), x = a} = {(C, x) : (C, x) ∈ C(F ), a / ∈ C} = C(F \ a) Proposition 6 . 7 . 67Let M be an ordered matroid with ground set E, and suppose F = F ext (M) is the external order for M. 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[ "Perturbative QCD Correction to the B → π Transition Form Factor", "Perturbative QCD Correction to the B → π Transition Form Factor" ]	[ "A Khodjamirian \nInstitut für Theoretische Physik\nUniversität Würzburg\nD-97074WürzburgGermany\n\nOn leave from Yerevan Physics Institute\n375036YerevanArmenia\n", "R Rückl \nInstitut für Theoretische Physik\nUniversität Würzburg\nD-97074WürzburgGermany\n\nMax-Planck-Institut für Physik\nWerner-Heisenberg-InstitutD-80805MünchenGermany\n", "S Weinzierl \nService de Physique Théorique\nCentre d'Etudes de SaclayF-91191Gif-sur-Yvette CedexFrance\n", "O Yakovlev \nInstitut für Theoretische Physik\nUniversität Würzburg\nD-97074WürzburgGermany\n\nOn leave from Budker Institute of Nuclear Physics (BINP)\n630090NovosibirskRussia\n" ]	[ "Institut für Theoretische Physik\nUniversität Würzburg\nD-97074WürzburgGermany", "On leave from Yerevan Physics Institute\n375036YerevanArmenia", "Institut für Theoretische Physik\nUniversität Würzburg\nD-97074WürzburgGermany", "Max-Planck-Institut für Physik\nWerner-Heisenberg-InstitutD-80805MünchenGermany", "Service de Physique Théorique\nCentre d'Etudes de SaclayF-91191Gif-sur-Yvette CedexFrance", "Institut für Theoretische Physik\nUniversität Würzburg\nD-97074WürzburgGermany", "On leave from Budker Institute of Nuclear Physics (BINP)\n630090NovosibirskRussia" ]	[]	We report on the perturbative O(α s ) correction to the light-cone QCD sum rule for the B → π transition form factor f + . The correction to the product f B f + in leading twist approximation is found to be about 30%, that is similar in magnitude to the corresponding O(α s ) correction in the two-point sum rule for f B . The resulting cancellation of large QCD corrections in f + eliminates one important uncertainty in the sum-rule prediction for this form factor.	10.1016/s0370-2693(97)00936-2	[ "https://arxiv.org/pdf/hep-ph/9706303v1.pdf" ]	18,195,894	hep-ph/9706303	7fbc1a46bc7ff2dab64e9d93dcd3b3e51fccd07e	Perturbative QCD Correction to the B → π Transition Form Factor Jun 1997 A Khodjamirian Institut für Theoretische Physik Universität Würzburg D-97074WürzburgGermany On leave from Yerevan Physics Institute 375036YerevanArmenia R Rückl Institut für Theoretische Physik Universität Würzburg D-97074WürzburgGermany Max-Planck-Institut für Physik Werner-Heisenberg-InstitutD-80805MünchenGermany S Weinzierl Service de Physique Théorique Centre d'Etudes de SaclayF-91191Gif-sur-Yvette CedexFrance O Yakovlev Institut für Theoretische Physik Universität Würzburg D-97074WürzburgGermany On leave from Budker Institute of Nuclear Physics (BINP) 630090NovosibirskRussia Perturbative QCD Correction to the B → π Transition Form Factor Jun 1997arXiv:hep-ph/9706303v1 9 We report on the perturbative O(α s ) correction to the light-cone QCD sum rule for the B → π transition form factor f + . The correction to the product f B f + in leading twist approximation is found to be about 30%, that is similar in magnitude to the corresponding O(α s ) correction in the two-point sum rule for f B . The resulting cancellation of large QCD corrections in f + eliminates one important uncertainty in the sum-rule prediction for this form factor. 1. The semileptonic decay B → πlν l is one of the most important reactions for the determination of the CKM parameter V ub . However, in order to extract V ub from data one needs an accurate theoretical calculation of the hadronic matrix element π(q)\|ūγ µ b\|B(p + q) = 2f + (p 2 )q µ + (f + (p 2 ) + f − (p 2 ))p µ , where p + q, q and p denote the B and π four-momenta and the momentum transfer, respectively, and f ± are two independent form factors. A very reliable approach to calculate f ± in the framework of QCD is provided by the operator product expansion (OPE) on the light-cone [1,2,3] in combination with QCD sum rule techniques. The sum rule for the form factor f + (p 2 ) has been obtained in [4,5] taking into account all twist 2, 3 and 4 operators, while f − (p 2 ) is derived in [6]. The most important missing elements of these calculations are the perturbative QCD corrections to the correlation function leading to (1). Here we report on a calculation of the O(α s ) correction to f + which eliminates one of the main uncertainties in the existing sum rule results. The calculation has several aspects which are worth pointing out. Firstly, the sum rule is actually derived for the product f B f + , f B being the B meson decay constant defined by B\|biγ 5 d\|0 = m 2 B f B /m b .(2) The form factor f + itself is then obtained by dividing out f B taking the value determined from the corresponding two-point QCD sum rule. In previous estimates, the O(α s ) correction to f B was thereby ignored for consistency because of the lack of the O(α s ) correction to f B f + . Our calculation now allows to take into account the correction to f B which is known to be sizeable. Secondly, knowing the O(α s ) corrections, also the heavy quark mass entering the sum rule can be properly defined. Thirdly, perturbative corrections to exclusive amplitudes involving light-cone wave functions have so far been studied only for massless quarks. For example, in [7,8,9] the amplitude of the pion transition to two virtual photons was calculated to O(α s ). The calculation for a finite quark mass is new and will have numerous applications. The main result of our work is the following. The O(α s ) correction to the light-cone sum rule for the product f B f + calculated in the leading twist approximation is about 30% and positive. Since the O(α s ) correction to f B is similar in size and of the same sign, the large QCD corrections cancel in f + making the prediction of the form factor very reliable, at least from the point of view of perturbative QCD. In this letter, we outline our calculation, present the final analytical results, and give first numerical estimates. Technical details, a thorough numerical analysis, and further applications will be presented elsewhere. can be calculated in the region (p + q) 2 < 0 and p 2 < m 2 b − O(1GeV 2 ) using OPE near the light-cone, i.e. at x 2 ≃ 0. In (3), we have multiplied the pseudoscalar current by the b-quark mass in order to assure renormalization-group invariance of the correlation function. After contracting the b-quark fields in (3), F µ is expressed in terms of bilocal matrix elements of increasing twist. In the present calculation we focus on the leading twist 2 contribution which enters through the following matrix element: (4) where the ellipses stand for terms of higher twist. The path-ordered gluon operator ensures gauge invariance. In the light-cone gauge, x · A = 0, adopted here as usual this operator is unity. The distribution function ϕ π (u) is known as the twist 2 light-cone wave function of the pion [1,2,3]. π(q) ū(x)γ µ γ 5 P exp   ig s 1 0 dα x · A(αx)   d(0) 0 = −iq µ f π 1 0 duϕ π (u)e iuq·x + ... , Comparison of (1) and (3) shows that in order to calculate f + one only has to deal with the invariant amplitude F in (3). With (4), F can be written as a convolution of a hard amplitude T (p 2 , (p + q) 2 , u) calculable within perturbation theory, with the pion wave function ϕ π (u) containing the long-distance effects: F (p 2 , (p + q) 2 ) = −f π 1 0 duϕ π (u)T (p 2 , (p + q) 2 , u).(5) In zeroth order in α s , the hard amplitude represented graphically in Fig. 1a reads T 0 (p 2 , (p + q) 2 , u) = m 2 b p 2 (1 − u) + (p + q) 2 u − m 2 b .(6) At fixed p 2 < m 2 b , F is an analytic function in the complex (p + q) 2 -plane, with a cut along the real axis starting from (p + q) 2 = m 2 b . One can therefore write a dispersion relation F (p 2 , (p + q) 2 ) = ∞ m 2 b ρ QCD (p 2 , s)ds s − (p + q) 2 .(7) Equating the QCD result obtained with ρ QCD = 1 π ImF and the hadronic representation of F following from (7) with the spectral density ρ(p 2 , s) = δ(s − m 2 B )2m 2 B f B f + (p 2 ) + ρ QCD (p 2 , s)Θ(s − s 0 )(8) yields the desired relation between f + and the invariant function F . In (8), the first term stems from the B ground state, whereas the second term represents the contributions from the higher resonances and the continuum in the B-meson channel above the threshold s 0 . Invoking quark-hadron duality the latter is replaced by the spectral density ρ QCD . The sum rule finally follows from the above after Borel transformation in (p + q) 2 : f B f + (p 2 ) = 1 2m 2 B s 0 m 2 b ρ QCD (p 2 , s)e m 2 B −s M 2 ds ,(9) where ρ QCD (p 2 , s) = − f π π 1 0 duϕ π (u)ImT (p 2 , s, u) .(10) With the zeroth order approximation (6), one easily obtains ImT 0 (p 2 , s, u) = −πδ(1 − p 2 m 2 b (1 − u) − s m 2 b u).(11) Substitution of (10) and (11) in (9) and integration over s reproduce the leading twist 2 contribution to the light-cone sum rule given in [4]. In this approximation the evolution of ϕ π is taken into account in the leading order (LO). In order to go to the next-to-leading order (NLO), one has to calculate the O(α s ) correction to ImT and use the NLO-evolution of ϕ π . This problem is solved below. 3. The first step is to calculate the O(α s ) correction to the hard amplitude T which we write as T (r 1 , r 2 , u) = T 0 (r 1 , r 2 , u) + α s C F 4π T 1 (r 1 , r 2 , u) ,(12) introducing convenient dimensionless variables r 1 = p 2 /m 2 b and r 2 = (p + q) 2 /m 2 b . The zeroth order amplitude T 0 is given in (6). In Figs. 1b -1g we show the Feynman diagrams determining the first order amplitude T 1 . The calculation is performed in general covariant gauge in order to have a possibility to check the gauge invariance of the result. Both the ultraviolet (UV) and infrared divergences are regularized by dimensional regularization and renormalized in the MS scheme with totally anticommuting γ 5 . This choice is motivated by the fact that the same scheme is used in the calculation of the NLO evolution kernel of the wave function ϕ π (u) [10]. From the diagrams depicted in Fig. 1 we find T 1 (r 1 , r 2 , u) = 3(1 + ρ) (1 − ρ) 2 ∆ − ln m 2 b µ 2 + 1 − 2 1 − ρ 2G (ρ) −G (r 1 ) −G (r 2 ) + 2 (r 1 − r 2 ) 2 1 − r 2 u G (ρ) −G (r 1 ) + 1 − r 1 1 − u G (ρ) −G (r 2 ) + ρ + (1 − ρ) ln (1 − ρ) ρ 2 − 2 1 − ρ (1 − r 2 ) ln (1 − r 2 ) r 2 + 3 − ρ (1 − ρ) 2 − 2 (1 − u)(r 1 − r 2 ) (1 − ρ) ln (1 − ρ) ρ − (1 − r 2 ) ln (1 − r 2 ) r 2 (13) with ∆ = 2 4 − d − γ E + ln(4π), ρ = r 1 + u(r 2 − r 1 ),(14)G (ρ) = Li 2 (ρ) + ln 2 (1 − ρ) − ln(1 − ρ) ∆ − ln m 2 b µ 2 + 1 , Li 2 (x) = − x 0 dt t ln(1−t) being the Spence function. The UV renormalization scale and the factorization scale of the collinear (COL) divergences are taken to be equal and denoted by µ. In order to trace the origin of the various divergent terms we have performed additional explicit calculations. In particular, we have used mass regularization by giving the light quarks a small but finite mass, and momentum regularization keeping the light quarks off mass shell. In this way, we have unambiguously separated the COL-divergent terms from the UV-divergent terms. The latter add up to T U V 1 (r 1 , r 2 , u) = 6ρ (1 − ρ) 2 ∆.(15) The correlation function (3) involves the unrenormalized quark currents J 5 =bγ 5 d and J µ =ūγ µ b as well as the bare b-quark mass m b . As usual, we define the corresponding renormalized quantities by J 5 → Z 5 J 5 , J µ → Z V J µ , m b → Z mmb . In the MS-scheme, the renormalization constants are given by Z 5 = 1 + 3∆ α S C F 4π , Z V = 1, Z m = 1 − 3∆ α S C F 4π .(16) We see that the overall renormalization factor of (3) is Z m Z 5 Z V = 1. The UV-renormalized hard amplitude T then follows from the unrenormalized result (12) just by reexpressing the unrenormalized mass m b through the renormalized massm b . As a result, an additional O(α s ) contribution to T emerges which exactly cancels the term T U V 1 given in (15). In addition to the UV-divergent terms, the function T 1 contains the COL-divergent terms: T COL 1 (r 1 , r 2 , u) = −∆T 0 (u) 3 − 2 ln 1 − r 2 1 − r 1 (1 − r 1 )(1 − r 2 ) − u(1 − r 1 )(r 2 − r 1 ) u(r 2 − r 1 ) 2 (17) −2 ln 1 − ρ 1 − r 2 (1 − r 1 )(1 − r 2 ) − u(1 − u)(r 2 − r 1 ) 2 u(1 − u)(r 2 − r 1 ) 2 . It is straightforward to check that T COL 1 can be written in the form T COL 1 (r 1 , r 2 , u) = −∆ 1 2 1 0 dwV (w, u)T 0 (r 1 , r 2 , w) ,(18) where V (w, u) is the kernel of the Brodsky-Lepage evolution equation [3] of the light-cone wave function ϕ π (u) introduced in (4): dϕ π (u, µ)/d ln µ = 1 0 dωV (u, ω)ϕ π (ω, µ)(19) with V (w, u) = α s (µ)C F π 1 − w 1 − u 1 + 1 w − u Θ(w − u) + w u 1 + 1 u − w Θ(u − w) + .(20) The operation + is defined by V (w, u) + = V (w, u) − δ(w − u) 1 0 V (v, u)dv.(21) The appearance of COL-divergent terms in the hard amplitude T in the form (18) reflects the factorization of the correlation function into a wave function and a hard amplitude [2,8,9]. For the factorization scheme we have again adopted the MS-scheme, i.e. we have subtracted the terms in the UV-renormalized hard scattering amplitude proportional to ∆. These are the terms absorbed in the definition of the scale-dependent wave function. The remaining µ-dependences of the hard scattering amplitude and of the wave function compensate each other. Up to now we have worked in the MS-scheme. However, the MS quark mass depends explicitly on the renormalization scale µ and implicitly on the renormalization prescription. A renormalization-scheme-independent definition of the quark mass within QCD perturbation theory is given by the pole mass which we denote m * b . Since we intend to use the set of parameters (m * b , f B , s 0 ) determined self-consistently from an independent analysis of the two-point sum rule for f B [11] it is convenient to replacem b by m * b also in the sum rule for f B f + using the well-known one-loop relation: m b = m * b 1 + α S C F 4π −4 + 3 ln m * 2 b µ 2 .(22) To O(α s ), this replacement adds the term duϕ π (u, µ)T (r 1 , r 2 , u, µ) , where the renormalized hard amplitude is given by T (r 1 , r 2 , u, µ) = 1 ρ − 1 + α s (µ)C F 4π 1 ρ − 1 (−4 + 3 ln m * 2 b µ 2 ) + 2 ρ − 1 [2G (ρ) − G (r 1 ) − G (r 2 )] + 2 (r 1 − r 2 ) 2 1 − r 2 u [G (ρ) − G (r 1 )] + 1 − r 1 1 − u [G (ρ) − G (r 2 )] + ρ + (1 − ρ) ln (1 − ρ) ρ 2 + 2 ρ − 1 (1 − r 2 ) ln (1 − r 2 ) r 2 − 2 ρ − 1 − 2 (1 − u)(r 1 − r 2 ) (1 − ρ) ln (1 − ρ) ρ − (1 − r 2 ) ln (1 − r 2 ) r 2 .(25) Here G(ρ) =G(ρ)\| ∆=0 , and ϕ π (u, µ) is the pion wave function evolved to the scale µ in NLO. To proceed further according to (7) and (10) we calculate the imaginary part of the hard scattering amplitude (25) for r 2 > 1 and r 1 < 1: − 1 π ImT (r 1 , r 2 , u, µ) = δ(1 − ρ) + α s (µ)C F 4π δ(1 − ρ) π 2 − 6 + 3 ln m * 2 b µ 2 − 2Li 2 (r 1 ) +2Li 2 (1 − r 2 ) −2 ln r 2 − 1 1 − r 1 2 + 2 ln r 2 + 1 − r 2 r 2 (2 ln(r 2 − 1) − ln(1 − r 1 )) +θ(ρ − 1) 8 ln(ρ − 1) ρ − 1 + + 2 ln r 2 + 1 r 2 − 2 − 2 ln(r 2 − 1) + ln m * 2 b µ 2 1 ρ − 1 + −2 r 2 − 1 (r 1 − r 2 )(ρ − r 1 ) ln ρ − 2 ln(ρ − 1) + 1 − ln m * 2 b µ 2 +2 1 − r 1 (r 1 − r 2 )(r 2 − ρ) ln ρ r 2 − 2 ln ρ − 1 r 2 − 1 −4 ln ρ ρ − 1 + 2 1 r 2 − ρ 1 ρ − 1 r 2 + 1 ρ 2 − 1 ρ +θ(1 − ρ) 2 ln r 2 + 1 r 2 − 2 ln(r 2 − 1) − ln m * 2 b µ 2 1 ρ − 1 + − 2 1 − r 1 (r 1 − r 2 )(r 2 − ρ) ln r 2 + 1 − 2 ln(r 2 − 1) − ln m * 2 b µ 2 −2 1 r 2 − ρ 1 − r 2 r 2(26) Here, the operation + is defined by dρf (ρ) 1 1 − ρ + = dρ (f (ρ) − f (1)) 1 1 − ρ .(27) This prescription takes care of the spurious infrared divergencies which one encounters by taking the imaginary part of (24). Substituting (26) and (10) to (9) one obtains the desired sum rule in O(α s ) for the form factor f + in the leading twist 2 approximation: f B f + (p 2 ) = − f π 2πm 2 B s 0 m * 2 b ds 1 0 du ϕ π (u, µ)ImT p 2 m 2 * b , s m * 2 b , u, µ e m 2 B −s M 2 ds .(28) The subleading twist 3 and 4 contributions are presently known only in zeroth order in α s [4,6]. They will be taken into account in the numerical analysis. 4. The second step is to determine the decay constant f B and the pion wave function ϕ π (u, µ) in NLO. For that purpose we have analyzed the two-point sum rule for f B obtained from the renormalization-group-invariant correlation function [11]. For the running coupling constant we use the two-loop expression with N f = 4 and Λ (4) = 234 MeV [12] corresponding to α s (M Z ) = 0.112. For µ 2 we take the value µ 2 B = m 2 B − m * 2 b corresponding to the average virtuality of the correlation function which in turn is given by the Borel mass parameter M 2 . With this choice the following correlated results are extracted from the two-point sum rule: m 2 b 0 \| T {J + 5 (x)J 5 (0)} \| 0 in O(α s )f B = 180 ± 30 MeV m * b = 4.7 ∓ 0.1 GeV, s 0 = 35 ± 2 GeV 2 .(29) In the following, we adopt the central values in the above intervals. Note that without O(α s ) correction one obtains f B = 140 ± 30 MeV. The remaining parameters entering (28) are directly measured: m B = 5.279 GeV and f π = 132 MeV. The wave function ϕ π can be expanded in terms of Gegenbauer polynomials C 3/2 n (2u − 1). Arguments based on conformal spin expansion [13] allows one to neglect higher terms in this expansion. We adopt the ansatz suggested in [14]: ϕ π (u, µ 0 ) = Ψ 0 (u) + a 2 (µ 0 )Ψ 2 (u) + a 4 (µ 0 )Ψ 4 (u),(30) where Ψ n (u) = 6u(1 −u)C 3/2 n (2u −1). The asymptotic wave function ϕ π (u) = 6u(1 −u) is unambigously fixed [3]. The terms n > 0 describe nonasymptotic corrections. The coefficients a 2 (µ 0 ) = 2/3 and a 4 (µ 0 ) = 0.43 at the scale µ 0 = 500 MeV have been extracted [14] from a two-point QCD sum rule for the moments of ϕ π (u) [1]. In NLO, the evolution of the wave function is given by [9]: ϕ π (u, µ) = n a n (µ 0 ) exp   − αs(µ) αs(µ 0 ) dα γ n (α) β(α)      Ψ n (u) + α s (µ) 4π k>n d k n (µ)Ψ k (u)   (31) with a 0 = 1. The coefficients d k n (µ) are due to mixing effects, induced by the fact that the polynomials Ψ n (u) are the eigenfunctions of the LO, but not of the NLO evolution kernel. The QCD beta-function β [12] and the anomalous dimension γ n of the n-th moment a n (µ) of the wave function have to be taken in NLO. Explicitly [15], γ n = α s 4π γ n 0 + α s 4π 2 γ n 1(32) with γ 0 0 = 0, γ 0 1 = 0 , γ 2 0 = 100 9 , γ 2 1 = 34450 243 − 830 81 N F , γ 4 0 = 728 45 , γ 4 1 = 662846 3375 − 31132 2025 N F .(33) The NLO mixing coefficients are [9,10] d k n (µ) = M nk γ k 0 − γ n 0 − 2β 0     1 − α s (µ) α s (µ 0 ) γ k 0 −γ n 0 −2β 0 2β0     ,(34) With the above input and (31) we find a 2 (µ B ) = 0.218 and a 4 (µ B ) = 0.084. Now, we are ready to exploit the sum rule (28) numerically. In Fig. 2, the product f B f + (0) is plotted as a function of the Borel parameter M 2 . The O(α s ) correction turns out to be large, between 30% and 35% , and stable under variation of M 2 . More specifically, in the interval M 2 = 8 ÷ 12 GeV 2 we obtain in LO f B f + (0) = 0.0229 ÷ 0.0224 GeV, f + (0) = 0.163 ÷ 0.160,(36) and in NLO f B f + (0) = 0.0306 ÷ 0.0295 GeV, f + (0) = 0.170 ÷ 0.164(37) where f B = 140 MeV and 180 MeV has been used, respectively. Note the almost complete cancellation of the NLO correction in f + . Furthermore, Fig. 3 shows the momentum dependence of the form factor f + (p 2 ) in the region 0 < p 2 < 15 ÷ 17 GeV 2 for M 2 = 10 GeV 2 , where the sum rule (28) is expected to be valid. Finally, it is interesting to compare the µ dependence in LO and NLO. This is done in Fig. 4. The very mild µ -dependence in LO only results from the evolution of the wave function. In NLO, the µ-dependence is stronger than in LO but similar to the µ-dependence of f B . As a result, the residual scale dependence of f + is again mild. The above results refer to the leading twist 2 approximation. If one adds the LO twist 3 and 4 contributions, one obtains at p 2 = 0 f + (0) = 0.27 .(38) This value should be compared with the LO estimate f + (0) = 0.30 obtained in [4,6]. 5. In this paper, we have presented the perturbative QCD correction in O(α s ) to the leading twist 2 approximation of the light-cone sum rule for the B → π form factor f + . Both UV and collinear divergences are handled by dimensional regularization and MS renormalization. The collinear divergences in the hard amplitude are factorized and absorbed in the evolution of the light-cone wave function. Numerically, the O(α S ) correction to the product f B f + amounts to about 30%. We have shown that this large correction is almost completely compensated by the corresponding correction to the twopoint sum rule for f B . The remaining O(α s ) effect on f + is therefore small. This finding improves the accuracy and reliability of the light-cone sum rule estimate substantially. Furthermore, we have shown that the O(α s ) correction to the sum rule for f B f + depends only very weakly on the momentum transfer. This observation, together with the above-mentioned compensation strongly suggests that the dominant O(α s ) effect in the correlation function (3) comes from the γ 5 vertex (see Fig. 1c) which is also present in the two-point correlation function for f B . Thus, our calculation strongly supports the conjecture [4,5,16] that the perturbative correction may drop out in the ratio f B f + /f B . Recently, an estimate of the perturbative correction to the B → π form factor was obtained [17] in a different approach combining the constituent quark model for B and π with light-cone wave functions. Although the results agree qualitatively it is difficult to directly compare our result with this model-dependent calculation. A more detailed account of our calculation as well as applications to various exclusive B and D decays will be published elsewhere. renormalized amplitude T 1 . The final result for the invariant function (5) then reads F (r 1 , r 2 ) = −f π 1 0 where the numerical values of the first few elements of the matrix M nk are M 02 = −11.2 + 1.73N F , M 04 = −1.41 + 0.565N F , M 24 = −22.0 + 1.65N F . Figure 1 :Figure 2 :Figure 3 :Figure 4 : 1234Feynman diagrams contributing to the correlation function (3): (a) zeroth order in α s , (b-g) first order in α s . Light-cone sum rule estimate for f B f + (0) in leading twist 2 approximation as a function of the Borel parameter M 2 : NLO (solid ) in comparison to LO (dashed). Momentum dependence of the form factor f + (p 2 ) in leading twist 2 approximation: LO (dashed) in comparison to NLO (solid). Scale dependence of the light-cone sum rule estimate of f B f + (0) in leading twist 2 approximation: NLO (solid) in comparison to LO (dotted). Acknowledgements.We are grateful to A. Ali, V. Braun, A. Grozin and A. Vainshtein for useful discussions. 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[ "Lattice rules for nonperiodic smooth integrands", "Lattice rules for nonperiodic smooth integrands" ]	[ "Josef Dick ", "Dirk Nuyens ", "Friedrich Pillichshammer " ]	[]	[]	The aim of this paper is to show that one can achieve convergence rates of N −α+δ for α > 1/2 (and for δ > 0 arbitrarily small) for nonperiodic α-smooth cosine series using lattice rules without random shifting. The smoothness of the functions can be measured by the decay rate of the cosine coefficients. For a specific choice of the parameters the cosine series space coincides with the unanchored Sobolev space of smoothness 1.We study the embeddings of various reproducing kernel Hilbert spaces and numerical integration in the cosine series function space and show that by applying the so-called tent transformation to a lattice rule one can achieve the (almost) optimal rate of convergence of the integration error. The same holds true for symmetrized lattice rules for the tensor product of the direct sum of the Korobov space and cosine series space, but with a stronger dependence on the dimension in this case.	10.1007/s00211-013-0566-0	[ "https://arxiv.org/pdf/1211.3799v1.pdf" ]	6,308,436	1211.3799	c52fb3af9e0fbbcc430024419c6a174dfb33708a	Lattice rules for nonperiodic smooth integrands May 5, 2014 Josef Dick Dirk Nuyens Friedrich Pillichshammer Lattice rules for nonperiodic smooth integrands May 5, 2014 The aim of this paper is to show that one can achieve convergence rates of N −α+δ for α > 1/2 (and for δ > 0 arbitrarily small) for nonperiodic α-smooth cosine series using lattice rules without random shifting. The smoothness of the functions can be measured by the decay rate of the cosine coefficients. For a specific choice of the parameters the cosine series space coincides with the unanchored Sobolev space of smoothness 1.We study the embeddings of various reproducing kernel Hilbert spaces and numerical integration in the cosine series function space and show that by applying the so-called tent transformation to a lattice rule one can achieve the (almost) optimal rate of convergence of the integration error. The same holds true for symmetrized lattice rules for the tensor product of the direct sum of the Korobov space and cosine series space, but with a stronger dependence on the dimension in this case. Introduction Quasi-Monte Carlo (QMC) rules are equal weight quadrature rules 1 N N −1 n=0 f (x n ) which can be used to approximate integrals of the form [0,1] s f (x) dx, see [10,23,28] for more information. In QMC rules, the quadrature points {x 0 , x 1 , . . . , x N −1 } are chosen according to some deterministic algorithm. One can show that the convergence rate of the integration error 1 There are several deterministic construction methods for the quadrature points. One such method yields so-called digital nets. These yield a convergence of the integration error of N −1 (log N ) s for functions of bounded variation [10,23]. Higher order digital nets, using an interlacing factor of d, yield a convergence rate of order N − min(α,d) (log N ) sα for integrands with square integrable partial mixed derivatives of order α in each variable [8,9]. An alternative to digital nets are lattice rules [23,28]. In this case, for a given positive integer N one chooses a vector g ∈ {1, . . . , N − 1} s called the generating vector and defines the quadrature points by ng N for 0 ≤ n < N. Here, for a real number x, the braces {x} denote the fractional part, i.e., modulo 1. For vectors, the fractional part is taken component wise. It is well known that there are generating vectors for lattice rules for which the integration error converges with order N −α+δ (δ > 0) for smoothness α ≥ 1, but with the restriction that f and its partial derivatives up to order α − 1 in each variable have to be periodic [19,Theorem 18,p. 120] and [29,30]. Fast computer search algorithms for such vectors are known from [25,26]. Hence, in order to be able to benefit from the fast rate of convergence, one needs to apply a transformation which makes the integrand (and its partial derivatives) periodic. This can cause some problems though and is not always recommended [19,22]. Since arbitrarily high rates of convergence can be obtained using digital nets for nonperiodic functions, the question arises whether this is also possible for lattice rules. Until now, lattice rules achieve a convergence rate of at most N −1+δ (δ > 0) for nonperiodic integrands (via estimates of the star-discrepancy [23]). If one applies the so-called tent transformation and a random shift, this rate of convergence can be improved to N −2+δ for any δ > 0 for the worst-case error in an unanchored Sobolev space of smoothness 2, see [15]. In this paper we present quadrature rules which achieve a convergence rate of order N −α+δ , δ > 0, for nonperiodic functions with smoothness α > 1/2. The way we measure smoothness in this paper is slightly different from the setting used in, for instance, higher order digital nets [8,9] though. We consider functions f which can be represented by a cosine series. Note that every continuous function f ∈ L 2 ([0, 1]) can be represented by a cosine series (see [18,Theorem 1] for this basis over [−1, 1]). This follows from the fact that the functions 1, √ 2 cos(πx), √ 2 cos(π2x), √ 2 cos(π3x), . . . are L 2 -orthogonal and complete. As mentioned above, the "smoothness" of the cosine series in our context is measured by the rate of decay of the cosine coefficients. To illustrate this, consider a one-dimensional function f : [0, 1] → R given by its cosine series f (x) = f cos (0) + ∞ k=1 f cos (k) √ 2 cos(πkx) = f (0) + ∞ k=1 f cos (2k) √ 2 cos(π2kx) + (1) The sum over the even frequencies is a 1-periodic function f per over [0,1]. If the coefficients f (h) decay with order h −α then f per is α-times differentiable in the classical sense. However, this does not apply to the sum over the odd coefficients. For instance, the cosine series for x → x − 1 2 is given by − 4 π 2 ∞ k=1 k odd 1 k 2 cos(πkx)(2) and hence the odd coefficients converge with order k −2 only, although x is infinitely times differentiable. Below we introduce a reproducing kernel Hilbert space based on cosine series, with the smoothness measured by the decay rate of the cosine coefficients. Although the smoothness of a cosine series measured by the differentiability of the series can be larger than the decay rate of the cosine coefficients suggests, the opposite can not happen. That is, we show that the reproducing kernel Hilbert space based on cosine series is embedded in the unanchored Sobolev space with the same value of the smoothness parameter. The case of smoothness 1 provides an exception, since there the cosine series space and the unanchored Sobolev space coincide. Various reproducing kernel Hilbert spaces are introduced in Section 2 and their embeddings are studied in Section 3. In this paper we present two methods which allow us to achieve a higher convergence rate for smoother nonperiodic functions using lattice rules, namely: 1. application of the tent transformation to the integration nodes; 2. symmetrization of the integration nodes. The tent transformation, φ : [0, 1] → [0, 1], φ(x) := 1 − \|2x − 1\| is a Lebesgue measure preserving function. The idea of using this transformation in conjunction with a random shift for integration based on lattice rules comes from Hickernell [15] and was also used in [6] for digital nets. In contrast to these works, here we do not rely on a random element in our quadrature rules. We show that a tent transformed lattice rule achieves an integration error of order N −α+δ , for any δ > 0, for functions belonging to a certain reproducing kernel Hilbert space of cosine series with smoothness parameter α. This result follows by showing that the worst-case error in the cosine series space for a tent-transformed lattice rule is the same as the worst-case error in a Korobov space of smooth periodic functions using lattice rules. Thus all the results for integration in Korobov spaces using lattice rules [7,12,20,25,26] also apply for integration in the cosine series space using tent-transformed lattice rules. In particular for smoothness 1, this yields deterministic point sets for numerical integration in unanchored Sobolev spaces with the same tractability properties as for numerical integration in the Korobov space. Furthermore, we also use symmetrized lattice rules. We show that these rules achieve the optimal order of convergence for integration of sums of products of cosine series and Fourier series. We apply the transformation x → 1 − x to each possible set of coordinates separately, so that if we start off with N points we get O(2 s−1 N ) points (see Section 4.2). This symmetrization approach is also mentioned in [19,28,34] and is one of the symmetry groups applied in the construction of cubature formulae, see, e.g., [4,13]. We prove that a lattice rule symmetrized this way achieves an integration error of order N −α+δ , δ > 0, for functions belonging to a certain reproducing kernel Hilbert space of cosine series and Fourier series with smoothness parameter α. The advantage of symmetrized lattice rules is that functions of the form cos(π(2k −1)x), where k is a nonnegative integer, are integrated exactly and hence only the smoothness of the periodic part determines the convergence rate. Note that the decay rate of the cosine series coefficients of the periodic part and of the Fourier series coefficients part coincides with the classical smoothness. Thus the problem with functions where the smoothness in terms of differentiability differs from the rate of decay of the cosine series is overcome using symmetrization. For instance, the function x → x is integrated exactly using symmetrization. However, a disadvantage of the symmetrization compared to the tent transformation is that the number of function evaluations grows exponential in the dimension and therefore symmetrization is only useful in smaller dimensions. For both methods, the rates of convergence we obtain are essentially optimal by an adaption of the lower bound of Bakhvalov [3], which is presented in Section 4.3. In the next section we introduce four reproducing kernel Hilbert spaces, the unanchored Sobolev space, the Korobov space, the cosine series space and the sum of the cosine and Korobov space. Since the unanchored Sobolev space and the Korobov space are frequently studied in the literature, we study the relations among these four spaces in Section 3 to put our results into context. It is shown that the Korobov space and the cosine series space differ, but both are embedded in the sum of the cosine series and Korobov space, which is itself embedded in the unanchored Sobolev space. In Section 4 we study numerical integration in the cosine series space using tent-transformed lattice rules and numerical integration in the sum of the cosine series and Korobov space using symmetrized lattice rules. Numerical results are presented in Section 5 and a conclusion is presented in Section 6. We write Z for the set of integers, N := {1, 2, . . .} for the set of positive integers and N 0 := {0, 1, 2, . . .} for the set of nonnegative integers. We also write R s + := {x ∈ R : x > 0} s . Furthermore, for s ∈ N we write [s] := {1, . . . , s}. Reproducing kernel Hilbert spaces In this section we introduce several reproducing kernel Hilbert spaces [2]. For a reproducing kernel K : [0, 1] × [0, 1] → R we denote by H(K) the corresponding reproducing kernel Hilbert space with inner product ·, · K and corresponding norm f K = f, f K . For any y ∈ [0, 1] we have K(·, y) ∈ H(K) and we have the reproducing property f (y) = f, K(·, y) for all y ∈ [0, 1] and f ∈ H(K). Further, the function K is symmetric in its arguments and positive semi-definite. For higher dimensions s > 1 we consider tensor product spaces. The reproducing kernel is in this case given by K s (x, y) = s j=1 K(x j , y j ), where x = (x 1 , . . . , x s ) and y = (y 1 , . . . , y s ). Again, the corresponding reproducing kernel Hilbert space is denoted by H(K s ), the corresponding inner product is denoted by ·, · Ks and the corresponding norm by f Ks = f, f Ks . For further information on reproducing kernel Hilbert spaces we refer to [2] (or [10,Chapter 2] for reproducing kernel Hilbert spaces in the context of numerical integration). The unanchored Sobolev space The unanchored Sobolev space is a reproducing kernel Hilbert space H(K sob 1,γ ) with reproducing kernel K sob 1,γ : [0, 1] × [0, 1] → R given by K sob 1,γ (x, y) := 1 + γB 1 (x)B 1 (y) + γ B 2 (\|x − y\|) 2 , where B 1 (z) = z − 1/2 and B 2 (z) = z 2 − z + 1/6 are Bernoulli polynomials and γ > 0 is a real number. The inner product in this space is given by f, g K sob 1,γ = 1 0 f (x) dx 1 0 g(x) dx + 1 γ 1 0 f (x)g (x) dx. In higher dimensions s > 1 we consider the tensor product space H(K sob 1,γ 1 )⊗· · ·⊗H(K sob 1,γs ). The reproducing kernel is in this case given by K sob 1,γ,s (x, y) := s j=1 K sob 1,γ j (x j , y j ), where x = (x 1 , . . . , x s ), y = (y 1 , . . . , y s ) and γ = (γ 1 , . . . , γ s ) ∈ R s + . The reproducing kernel Hilbert space H(K sob 1,γ ) can be generalized to higher order smoothness. For α ∈ N, consider the reproducing kernel K sob α,γ : [0, 1] × [0, 1] → R given by K sob α,γ (x, y) := 1 + γ α τ =1 B τ (x)B τ (y) (τ !) 2 − (−1) α γ B 2α (\|x − y\|) (2α)! , where B τ is the Bernoulli polynomial of order τ . The inner product for this space is given by f, g K sob α,γ = 1 0 f (x) dx 1 0 g(x) dx + 1 γ α−1 τ =1 1 0 f (τ ) (x) dx 1 0 g (τ ) (x) dx + 1 γ 1 0 f (α) (x) g (α) (x) dx. To obtain reproducing kernel Hilbert spaces for the domain [0, 1] s , we consider again the tensor product of the one-dimensional spaces. This space has the reproducing kernel K sob α,γ,s (x, y) := s j=1 K sob α,γ j (x j , y j ). Quasi-Monte Carlo rules in H(K sob α,γ,s ) which yield the optimal rate of convergence of order N −α+δ for any δ > 0 where studied in [9]. The Korobov space For α > 1/2, h ∈ Z and γ > 0 we define r α,γ (h) := 1 if h = 0, γ\|h\| −2α if h = 0.(3) For h = (h 1 , . . . , h s ) ∈ Z s and γ = (γ 1 , . . . , γ s ) ∈ R s + , we set r α,γ,s (h) := s j=1 r α,γ j (h j ). The Korobov space is a reproducing kernel Hilbert space of Fourier series. The reproducing kernel for this space is given by K kor α,γ (x, y) := h∈Z r α,γ (h) e 2πih(x−y) , and in higher dimensions s > 1 by K kor α,γ,s (x, y) := s j=1 K kor α,γ j (x j , y j ) = h∈Z s r α,γ,s (h) e 2πih·(x−y) , where the "·" denotes the usual inner product in R s . Let the Fourier coefficient for a function f : [0, 1] s → R be given by f (h) := [0,1] s f (x) e −2πih·x dx. Then the inner product in the reproducing kernel Hilbert space H(K kor α,γ,s ) is given by f, g K kor α,γ,s = h∈Z s f (h) g(h) r −1 α,γ,s (h). The corresponding norm is defined by f K kor α,γ,s = f, f K kor α,γ,s . The half-period cosine space The half-period cosine space is a reproducing kernel Hilbert space of (half-period) cosine series with reproducing kernel K cos α,γ (x, y) := 1 + ∞ k=1 r α,γ (k) √ 2 cos(πkx) √ 2 cos(πky), where r α,γ is defined as in (3) and where α > 1/2 and γ > 0. The inner product is given by f, g K cos α,γ = ∞ k=0 f cos (k) g cos (k) r −1 α,γ (k), where f cos and g cos are the cosine coefficients of f and g, respectively, as defined in (1). We can generalize the reproducing kernel Hilbert space H(K cos α,γ ) to the domain [0, 1] s by setting K cos α,γ,s (x, y) := s j=1 K cos α,γ j (x j , y j ), where γ = (γ 1 , . . . , γ s ) ∈ R s + and x = (x 1 , . . . , x s ), y = (y 1 , . . . , y s ) ∈ [0, 1] s . The inner product is then given by f, g K cos α,γ,s = k∈N s 0 f cos (k) g cos (k) r −1 α,γ,s (k), where the multi-dimensional cosine coefficients for a function f : [0, 1] s → R are given by f cos (k) := [0,1] s f (x) 2 \|k\| 0 /2 s j=1 cos(πk j x j ) dx, where for k = (k 1 , . . . , k s ) ∈ N s 0 we define \|k\| 0 := \|{j ∈ [s] : k j = 0}\| to be the number of nonzero components in k. The corresponding norm is defined by f K cos α,γ,s = f, f K cos α,γ,s , in particular we have f 2 K cos α,γ,s = k∈N s 0 \| f cos (k)\| 2 r α,γ,s (k) = h∈Z s 2 −\|h\| 0 \| f cos (\|h\|)\| 2 r α,γ,s (h) , where \|h\| = (\|h 1 \|, . . . , \|h d \|). The sum of the Korobov space and the half-period cosine space In this section we introduce the kernel K kor+cos α,γ (x, y) := 1 2 K kor α,γ (x, y) + K cos α,γ (x, y) = 1 2 h∈Z r α,γ (h) e 2πih(x−y) + 1 2 + ∞ k=1 r α,γ (k) cos(πkx) cos(πky) = 1 + γ ∞ k=1 k −2α (cos(2πk(x − y)) + cos(πkx) cos(πky)) . The space resulting from the sum of kernels is studied in [2, Part I, Section 6]. The norm in the reproducing kernel Hilbert space H(K kor+cos α,γ ) is then defined by f 2 K kor+cos α,γ = min f =f kor +fcos 2 f kor 2 K kor α,γ + f cos 2 K cos α,γ , where the minimum is taken over all functions f kor ∈ H(K kor α,γ ) and f cos ∈ H(K cos α,γ ) such that f = f kor + f cos . For dimensions s > 1 we define the reproducing kernel by K kor+cos α,γ,s (x, y) := s j=1 1 2 K kor α,γ j (x j , y j ) + K cos α,γ j (x j , y j ) . Thus the space H(K kor+cos α,γ,s ) is the tensor product H(K kor+cos α,γ 1 ) ⊗ · · · ⊗ H(K kor+cos α,γs ). For u ⊆ [s] we define K kor+cos α,γ,s,u (x, y) := 2 −s j∈u K kor α,γ j (x j , y j ) j∈[s]\u K cos α,γ j (x j , y j ), where as usual an empty product is considered to be one. This is a reproducing kernel for the space H(K kor+cos α,γ,s,u ) =   j∈u H( 1 2 K kor α,γ j )   ⊗   j∈[s]\u H( 1 2 K cos α,γ j )   with the inner product f, g K kor+cos α,γ,s,u = 2 s hu∈Z \|u\| k [s]\u ∈N s−\|u\| 0 f u,kor+cos (h u , k [s]\u ) g u,kor+cos (h u , k [s]\u ) r −1 α,γ,s (h u , k [s]\u ), where f u,kor+cos (h u , k [s]\u ) := [0,1] s f (x) ( √ 2) \|k [s]\u \| 0 j∈u e −2πih j x j j∈[s]\u cos(πk j x j ) dx. Clearly we have that K kor+cos α,γ,s (x, y) = u⊆[s] K kor+cos α,γ,s,u (x, y). Embeddings In this section we investigate the relationships between the spaces introduced above. For two reproducing kernel Hilbert spaces H(K 1 ) and H(K 2 ) we say that H(K 1 ) is continuously embedded in H(K 2 ) if H(K 1 ) ⊆ H(K 2 ) and if f K 2 ≤ C f K 1 for all f ∈ H(K 1 ) for some constant C > 0 independent of f . We write H(K 1 ) → H(K 2 ) in this case. On the other hand it is possible that H(K 1 ) is not a subset of H(K 2 ), i.e., there is a function in H(K 1 ) which is not in H(K 2 ). In this case we write H(K 1 ) ⊂ H(K 2 ). If H(K 1 ) → H(K 2 ) and H(K 2 ) → H(K 1 ) we write H(K 1 ) H(K 2 ). The Korobov space and the unanchored Sobolev space It is well known, see, e.g., [24,Appendix A], that the Korobov space is continuously embedded in the unanchored Sobolev space H(K kor α,γ,s ) → H(K sob α,γ,s ). Conversely, as is also well known, for instance the function g : [0, 1] s → R, g(x) = x 1 is in H(K sob α,γ,s ) for all α ∈ N, but not in H(K kor α,γ,s ), since g is not periodic. Thus H(K sob α,γ,s ) ⊂ H(K kor α,γ,s ). We note that for f ∈ H(K kor α,γ,s ), α ∈ N, we have that f K sob α,γ,s = f K sob α,γ(2π) −2α ,s where γ(2π) −2α denotes the rescaled sequence (γ 1 (2π) −2α , . . . , γ s (2π) −2α ). The half-period cosine space and the unanchored Sobolev space We now consider the half-period cosine space and the unanchored Sobolev space. For α = 1 there is a peculiarity. Lemma 1. We have K sob 1,γ (x, y) = 1 + γ π 2 ∞ k=1 1 k 2 2 cos(kπx) cos(kπy) = K cos 1,γπ −2 (x, y). For completeness we include a short proof. Proof. In the following we calculate the cosine coefficients of K sob 1,γ . It is easy to check that we have 1 0 1 0 K sob 1,γ (x, y) cos(πnx) cos(πmy) dx dy = 0 for (n, m) ∈ {(0, k) : k > 0} ∪ {(k, 0) : k > 0}. Further we have 1 0 1 0 K sob 1,γ (x, y) dx dy = 1. We have B 2 (\|x − y\|) =B 2 ({x − y}) = 1 2π 2 k∈Z\{0} e 2πik(x−y) k 2 = ∞ k=1 cos(2πkx) cos(2πky) π 2 k 2 + ∞ k=1 sin(2πkx) sin(2πky) π 2 k 2 . Using (2) we obtain (x − 1 2 )(y − 1 2 ) + B 2 (\|x − y\|) 2 = ∞ k,l=1 16 π 4 (2k − 1) 2 (2l − 1) 2 cos(π(2k − 1)x) cos(π(2l − 1)y) + ∞ k=1 cos(2πkx) cos(2πky) 2π 2 k 2 + ∞ k=1 sin(2πkx) sin(2πky) 2π 2 k 2 . This immediately implies that if m is even and n is odd, or m is odd and n is even, or m, n are even with m = n. If m = n = k for even k > 0, we obtain 1 0 1 0 K sob 1,γ (x, y) √ 2 cos(πkx) √ 2 cos(πky) dx dy = γ π 2 k 2 . Now let m, n > 0 be odd. We have 1 0 sin(2πkx) cos(πmx) dx = 2k(1−(−1) m ) π(4k 2 −m 2 ) and therefore 1 0 1 0 K sob 1,γ (x, y) √ 2 cos(πmx) √ 2 cos(πny) dx dy = 8 π 4 m 2 n 2 + 16 π 4 ∞ k=1 1 (4k 2 − m 2 )(4k 2 − n 2 ) . For m = n we have ∞ k=1 1 (4k 2 −m 2 )(4k 2 −n 2 ) = − 1 m 2 n 2 and further ∞ k=1 1 (4k 2 −m 2 ) 2 = π 2 m 2 −8 16m 4 . Thus we obtain 1 0 1 0 K sob 1,γ (x, y) √ 2 cos(πmx) √ 2 cos(πny) dx dy = 0 if m = n, γ (πm) 2 if m = n. Note that the cosine series for K sob 1,γ converges absolutely. Since the function K sob 1,γ is continuous, the cosine series converges to the function pointwise. This completes the proof. The above lemma and Mercer's theorem also yield the eigenfunctions of the operator T (g)(y) = 1 0 K sob 1,γ (x, y)g(x) dx. These are 1, √ 2 cos(πx), √ 2 cos(2πx), √ 2 cos(3πx), . . . and the corresponding eigenvalues are 1, π −2 , (π2) −2 , (π3) −2 , . . .. Remark 1. An analoguous result for a slightly different reproducing kernel Hilbert space was established in [33]. For this space one obtains the same set of eigenfunctions. For the anchored Sobolev space the eigenfunctions are slightly different and have been found in [32]. We thus find H(K sob 1,γ ) = H(K cos 1,γπ −2 ) and f K sob 1,γ = f K cos 1,γπ −2 for all f ∈ H(K sob 1,γ ). The same also applies for the higher dimensional tensor product space H(K sob 1,γ,s ) = H(K cos 1,γπ −2 ,s ) and f K sob 1,γ,s = f K cos 1,γπ −2 ,s for all f ∈ H(K sob 1,γ,s ), where γπ −2 denotes the sequence (γ 1 π −2 , . . . , γ s π −2 ). Thus H(K sob 1,γ,s ) H(K cos 1,γ,s ). Now consider α > 1. The function x → x belongs to H(K sob α,γ ) for all α ∈ N. On the other hand we have x = 1 2 − 4 π 2 ∞ k=1 k odd cos(πkx) k 2 . Hence the function x → x is not in H(K cos α,γ ) for α ≥ 3/2 and therefore H(K sob α,γ ) ⊂ H(K cos α,γ ) for α ∈ N, α ≥ 2. Conversely, let α ∈ N, α ≥ 2. Let f ∈ H(K cos α,γ ) be given by f (x) = f cos (0) + ∞ k=1 f cos (k) √ 2 cos(πkx) with f 2 K cos α,γ = \| f cos (0)\| 2 + 1 γ ∞ k=1 \| f cos (k)\| 2 \|k\| 2α < ∞. Then for 1 ≤ τ ≤ α we have f (τ ) (x) = ∞ k=1 f cos (k)(−1) τ /2 (kπ) τ √ 2 φ τ (πkx), where φ τ (z) = cos(z) for τ even and φ τ (z) = sin(z) for τ odd and where for a real number x, x denotes the smallest integer bigger or equal to x. Thus 1 γ 1 0 f (τ ) (x) dx 2 ≤ 1 γ 1 0 \|f (τ ) (x)\| 2 dx = π 2τ γ ∞ k=1 \| f cos (k)\| 2 k 2τ ≤ π 2τ f 2 K cos α,γ . Thus f K sob α,γ ≤ α τ =0 π 2τ 1/2 f K cos α,γ = π 2(α+1) − 1 π 2 − 1 1/2 f K cos α,γ . Thus we have H(K cos α,γ ) → H(K sob α,γ ). This result can be generalized to the tensor product space, thus H(K cos α,γ,s ) → H(K sob α,γ,s ). The half-period cosine space and the Korobov space For α = 1 the embedding results for the half-period cosine space and the Korobov space follow from the previous two subsections. Let now α > 1. Let f (x) = sin(2πx). Then f ∈ H(K kor α,γ ) for all α > 1/2. On the other hand we have for k ∈ N f cos (k) = 1 0 sin(2πx) √ 2 cos(πkx) dx = 4 √ 2 π(4−k 2 ) if k is odd, 0 otherwise. Thus f 2 K cos α,γ = 1 γ ∞ k=1 k odd \| f cos (k)\| 2 k 2α = 1 γ 32 π 2 ∞ k=1 (2k − 1) 2α ((2k − 1) 2 − 4) 2 . Thus we have f K cos α,γ = ∞ for α ≥ 3/2. Thus H(K kor α,γ ) ⊂ H(K cos α,γ ) for α ≥ 3/2. Conversely, let now f (x) = cos(πx). Then f ∈ H(K cos α,γ ) for all α > 1/2. On the other hand we have for h ∈ Z f kor (h) = 1 0 cos(πx) e −2πihx dx = 4ih π(1 − 4h 2 ) . For any α ≥ 1/2 we have f 2 K kor α,γ = 1 γ 16 π 2 h∈Z\{0} \|h\| 2α h 2 (4h 2 − 1) 2 = ∞, which implies that f / ∈ H(K kor α,γ ). As we need α > 1/2 we therefore have H(K cos α,γ ) ⊂ H(K kor α,γ ). Embeddings of the sum of the Korobov and half-period cosine space Since K kor+cos α,γ = 1 2 K kor α,γ + 1 2 K cos α, Summary of embeddings We summarize the obtained embedding results in the following theorem. Numerical integration We now study the worst-case error for QMC integration. As quality measure for the QMC algorithm we use the worst-case integration error. Let P = {x 0 , . . . , x N −1 } and let H(K) be an arbitrary reproducing kernel Hilbert space with reproducing kernel K and norm · K . Then the worst-case error for QMC integration in H(K) using the point set P is defined as e(H(K); P ) := sup f ∈H(K) f K ≤1 [0,1] s f (x) dx − 1 N N −1 n=0 f (x n ) , see, e.g., [10,24] for a general reference. We use the following formula for the square worst-case error (see [10,Proposition 2.11] or [14]): e 2 (H(K); P ) = [0,1] 2s K(x, y) dx dy − 2 N N −1 n=0 [0,1] s K(x, x n ) dx + 1 N 2 N −1 n,n =0 K(x n , x n ). Integration in the Sobolev space H(K sob α,γ,s ) has been considered in [9,10] and integration in the Korobov space has been studied for instance in [7,8,17,19,20,25,26], as well as other papers. In this paper we study numerical integration in the half-period cosine space and in the sum of the Korobov space and the half-period cosine space. For the former space we use tent-transformed lattice rules and for the latter one we use symmetrized lattice rules. Numerical integration in the half-period cosine space We now study numerical integration in the half-period cosine space using tent-transformed lattice rules. For a nonnegative real number x we denote the fractional part of x by {x} = x − x . For a vector x of nonnegative real numbers, the expression {x} denotes the vector of fractional parts. A lattice point set with N ≥ 2 points and generating vector g ∈ {1, . . . , N − 1} s is given by P (g, N ) := ng N : 0 ≤ n < N .(4) For x ∈ [0, 1] we define the tent-transformation by φ(x) = 1 − \|2x − 1\| and for vectors we apply the function φ component-wise. The tent-transformed lattice point set is now given by P φ (g, N ) := φ ng N : 0 ≤ n < N . We call a lattice rule which is based on P φ (g, N ) a tent-transformed lattice rule. The following theorem gives a useful formula for the worst-case integration error in H(K cos α,γ,s ) of tent-transformed lattice rules. Theorem 2. The squared worst-case error for QMC integration in the half-period cosine space H(K cos α,γ,s ) using a tent-transformed lattice rule is given by e 2 (H(K cos α,γ,s ); P φ (g, N )) = h∈L ⊥ \{0} r α,γ,s (h), where L ⊥ := {h ∈ Z s : h · g ≡ 0 (mod N )} is the dual lattice. Proof. Let f ∈ H(K cos α,γ,s ) with f K cos α,γ,s < ∞ and with expansion f (x) = k∈N s 0 f cos (k)( √ 2) \|k\| 0 s j=1 cos(πk i x i ).(5) For any k ∈ N 0 we have cos(πkφ(x)) = cos(2πkx) for all x ∈ [0, 1], and hence f φ ng N = k∈N s 0 ( √ 2) \|k\| 0 f cos (k) s j=1 cos πk j φ ng j N = k∈N s 0 ( √ 2) \|k\| 0 f cos (k) s j=1 cos 2πk j ng j N = h∈Z s ( √ 2) −\|h\| 0 f cos (\|h\|) e 2πin(h·g)/N . Therefore we obtain 1 N N −1 n=0 f φ ng N − [0,1] s f (x) dx = 0 =h∈Z s ( √ 2) −\|h\| 0 f cos (\|h\|) 1 N N −1 n=0 e 2πi n(h·g)/N . The sum in the braces is a character sum over the group Z/N Z which is one if h · g is a multiple of N and zero otherwise. From this we get 1 N N −1 n=0 f φ ng N − [0,1] s f (x) dx = h∈L ⊥ \{0} ( √ 2) −\|h\| 0 f cos (\|h\|).(6) From this formula and an application of the Cauchy-Schwarz inequality we obtain 1 N N −1 n=0 f φ ng N − [0,1] s f (x) dx = h∈L ⊥ \{0} r α,γ,s (h) 1/2 ( √ 2) −\|h\| 0 f cos (\|h\|) r α,γ,s (h) 1/2 ≤   h∈L ⊥ \{0} r α,γ,s (h)   1/2 h∈Z s 2 −\|h\| 0 \| f cos (\|h\|)\| 2 r α,γ,s (h) 1/2 =   h∈L ⊥ \{0} r α,γ,s (h)   1/2 f K cos α,γ,s . Here we obtain equality by (6) for the function with cosine series coefficients given by f cos (k) = ( √ 2) \|k\| 0 r α,γ,s (k) for k ∈ {\|h\| : h ∈ L ⊥ \ {0}} 0 otherwise. Hence the result follows by the definition of the worst-case error. The above result shows that the square worst-case error of tent-transformed lattice rules for numerical integration in the cosine space coincides with the square worst-case error for numerical integration in a Korobov space using the same lattice rules but without applying the tent-transformation, that is, N )). Thus all the results which hold for the worst-case error in the Korobov space using lattice rules, also hold for the worst-case error in the half-period cosine space using tenttransformed lattice rules. This applies for instance to the component-by-component construction [7,20,30,29] and fast component-by-component construction [25,26], general weights [12,21] and extensible lattice rules [5,11,16,17]. N )) ≤ C α,γ,s,τ (N − 1) −τ /2 , for all 1 ≤ τ < 2α, where the constant C α,γ,s,τ > 0 is given by e 2 (H(K kor α,γ,s ); P (g, N )) = −1 + h∈Z s r α,γ,s (h) 1 N N −1 n=0 e 2πinh·g/N 2 = h∈L ⊥ \{0} r α,γ,s (h) = e 2 (H(K cos α,γ,s ); P φ (g,C α,γ,s,τ =   ∅ =u⊆[s] γ 1/τ u (2ζ(2α/τ )) \|u\|   τ /2 = −1 + s j=1 (1 + 2ζ(2α/τ )γ 1/τ j ) τ /2 . Note that for certain choices of weights γ, the upper bound can be made independent of the dimension s and we then obtain (strong) tractability results. See [7,12,20,24] for a discussion of tractability results which apply in this context. Since the cosine series space H(K cos 1,γ,s ) and the unanchored Sobolev space H(K sob 1,γ,s ) coincide, we also get the following result. e(H(K sob 1,γ,s ), P φ (g, N )) ≤ C 1,γ,s,τ (N − 1) −τ /2 , for all 1 ≤ τ < 2, where C 1,γ,s,τ =   ∅ =u⊆[s] γ 1/τ u (2ζ(2/τ )) \|u\|   τ /2 = −1 + s j=1 (1 + 2ζ(2/τ )γ 1/τ j ) τ /2 . Remark 2. An analoguous result can also be obtained for the space considered in [33], since the eigenfunctions are the same as for H(K sob 1,γ ). This result is a deterministic version of the main results in [7,20], where a random shift was required to achieve this bound. The tractability results of [7,12,20] also apply. Numerical integration in the Korobov plus half-period cosine space We now study numerical integration in the space H(K kor+cos α,γ,s ) using symmetrized lattice rules. Let x = (x 1 , . . . , x s ) and let u ⊆ [s]. Then let sym u (x) denote the vector whose jth coordinate is x j if j ∈ u and 1 − x j otherwise, i.e., sym u (x) = (y 1 , . . . , y s ) with y j = 1 − x j if j ∈ u, x j if j ∈ u. For a lattice point set as in (4) let P sym (g, N ) := sym u ng N : 0 ≤ n < N, u ⊆ [s] . We call a lattice rule which is based on P sym (g, N ) a symmetrized lattice rule. Note that P sym (g, N ) consists of O(2 s−1 N ) elements as we show next. Proof. The argument comes from [34]. We have the following symmetry kg j ≡ N − (N − k)g j (mod N ), for all 0 ≤ k < N and j = 1, . . . , s, which corresponds exactly to x k,j = 1 − x N −k,j . This means we only have to evaluate and symmetrize the points for 0 ≤ k ≤ N/2. 1. For 0 < k < N/2 we have 2 s (N − 1)/2 points if 2 N and 2 s (N/2 − 1) if 2 \| N . 2. For k = 0 symmetrization returns all 2 s corner points. 3. If 2 \| N then for k = N/2 we have x N/2 = ( 1 2 , . . . , 1 2 ) and thus no symmetrization is needed. Counting the number of function evaluations now gives the result from above. The analysis used in the previous proof can also be used in an implementation where further stream lining can be done by noticing that x → 1 − x is its own inverse and thus the 2 s symmetric points for 0 < k < N/2 can be constructed easily in gray code ordering. Nevertheless, we increase the number of points by a factor of 2 s−1 and so this is only feasible for moderate dimensions. For the derivations below we symmetrize all N points for ease of notation. The following theorem gives a useful formula for the worst-case integration error in H(K kor+cos α,γ,s ) of symmetrized lattice rules. Theorem 3. The squared worst-case error for QMC integration in the sum of the halfperiod cosine space and the Korobov space H(K kor+cos α,γ,s ) using a symmetrized lattice rule is given by e 2 (H(K kor+cos α,γ,s ); P sym (g, N )) = h∈L ⊥ \{0} r α,γ,s (h), where L ⊥ := {h ∈ Z s : h · g ≡ 0 (mod N )} is the dual lattice. Proof. f v (x) = hv∈Z \|v\| k [s]\v ∈N s−\|v\| 0 f v,kor+cos (h v , k [s]\v ) ( √ 2) \|k [s]\v \| 0 j∈v e 2πih j x j j∈[s]\v cos(πk j x j ) such that f 2 K kor+cos α,γ,s = v⊆[s] f v 2 K kor+cos α,γ,s,v . Note that for k ∈ Z and x ∈ R, cos(πkx) + cos(πk(1 − x)) = 2 cos(πkx) = e πikx + e −πikx if k is even, 0 if k is odd. For given v ⊆ [s] we therefore have 1 2 s N N −1 n=0 u⊆[s] f v sym u ng N = 1 2 s N N −1 n=0 hv∈Z \|v\| k [s]\v ∈N s−\|v\| 0 ( √ 2) \|k [s]\v \| 0 f v,kor+cos (h v , k [s]\v ) × j∈v e 2πih j ng j /N + e −2πih j ng j /N j∈[s]\v (cos(πk j {ng j /N }) + cos(πk j (1 − {ng j /N }))) = 1 2 s N N −1 n=0 hv∈Z \|v\| k [s]\v ∈N s−\|v\| 0 ( √ 2) \|k [s]\v \| 0 f v,kor+cos (h v , 2k [s]\v ) × j∈v e 2πih j ng j /N + e −2πih j ng j /N j∈[s]\v e 2πik j ng j /N + e −2πik j ng j /N = 1 2 s N N −1 n=0 h∈Z s 2 \|v\| ( √ 2) \|h [s]\v \| 0 2 s−\|v\|−\|h [s]\v \| 0 f v,kor+cos (h v , 2\|h [s]\v \|)e 2πinh·g/N = h∈Z s ( √ 2) −\|h [s]\v \| 0 f v,kor+cos (h v , 2\|h [s]\v \|) 1 N N −1 n=0 e 2πinh·g/N . The sum in the braces is a character sum over the group Z/N Z which is one if h · g is a multiple of N and zero otherwise. From this we get 1 2 s N N −1 n=0 u⊆[s] f v sym u ng N − [0,1] s f v (x) dx = h∈L ⊥ \{0} ( √ 2) −\|h [s]\v \| 0 f v,kor+cos (h v , 2\|h [s]\v \|). (7) Thus we find 1 2 s N N −1 n=0 u⊆[s] f sym u ng N − [0,1] s f (x) dx ≤ v⊆[s] h∈L ⊥ \{0} ( √ 2) −\|h [s]\v \| 0 \| f v,kor+cos (h v , 2\|h [s]\v \|)\| = v⊆[s] h∈L ⊥ \{0} r α,γ,s (h) 1/2 ( √ 2) −\|h [s]\v \| 0 \| f v,kor+cos (h v , 2\|h [s]\v \|)\| r α,γ,s (h) 1/2 ≤   v⊆[s] h∈L ⊥ \{0} r α,γ,s (h)   1/2   v⊆[s] h∈L ⊥ \{0} 2 −\|h [s]\v \| 0 \| f v,kor+cos (h v , 2\|h [s]\v \|)\| 2 r α,γ,s (h)   1/2 ≤   2 s h∈L ⊥ \{0} r α,γ,s (h)   1/2    1 2 s v⊆[s] 2 s hv∈Z \|v\| k [s]\v ∈N s−\|v\| 0 \| f v,kor+cos (h v , k [s]\v )\| 2 r α,γ,s (h v , k [s]\v )    1/2 ≤   2 s h∈L ⊥ \{0} r α,γ,s (h)   1/2   1 2 s v⊆[s] f 2 K kor+cos α,γ,s,v   1/2 ≤   h∈L ⊥ \{0} r α,γ,s (h)   1/2 f K kor+cos α,γ,s . Thus the result follows. Again we can relate the worst-case error e 2 (H(K kor+cos α,γ,s ); P sym (g, N )) to the worst-case error in a Korobov space. We have e(H(K kor+cos α,γ,s ); P sym (g, N )) = e(H(K kor α,γ,s ); P (g, N )). Thus the results for integration in the sum of the half-period cosine space and the Korobov space using symmetrized lattice rules are the same as in a Korobov space using lattice rules. In particular, the component-by-component algorithm can be used [7,20,30,29] and also its fast version [25,26], general weights [12,21] and extensible lattice rules [5,11,16,17]. N )) ≤ C α,γ,s,τ (N − 1) −τ /2 , for all 1 ≤ τ < 2α, where the constant C α,γ,s,τ > 0 is given by C α,γ,s,τ =   ∅ =u⊆[s] γ 1/τ u (2ζ(2α/τ )) \|u\|   τ /2 = −1 + s j=1 (1 + 2ζ(2α/τ )γ 1/τ j ) τ /2 . Due to the fact that the number of points of P sym (g, N ) is M = O(2 s−1 N ) we do not get tractability results. Notice that in terms of the number of points one gets (N −1) −τ /2 ≈ 2 (s−1)τ /2 M −τ /2 , which also implies a strong dependence on the dimension. A consequence of the symmetrization procedure is that all functions of the form are integrated exactly. This is because all the odd frequencies in a cosine series are integrated exactly. Specifically for the half-period cosine space we can state the following result. Corollary 4. The squared worst-case error for QMC integration in the half-period cosine space H(K cos α,γ,s ) using a symmetrized lattice rule is given by e 2 (H(K cos α,γ,s ); P sym (g, N )) = h∈L ⊥ \{0} r α,γ,s (2h), where L ⊥ := {h ∈ Z s : h · g ≡ 0 (mod N )} is the dual lattice. Proof. Similar to the proof of Theorem 3 with v = ∅. A lower bound on the worst-case error We prove the following lower bound for integration in the half-period cosine space. Let P = {x 0 , . . . , x N −1 } ⊆ [0, 1] s be an N element point set and let w = (w 0 , . . . , w N −1 ) be an arbitrary real tuple. Let e(H(K cos α,γ,s ); P ; w) = sup We have e 2 (H(K cos α,γ,s ); P ; w) f ∈H(K cos α,γ,s ) f K cos α,γ,s ≤1 [0,1] s f (x) dx − N −1 n=0 w n f (x n ) .= [0,1] 2s K(x, y) dx dy − 2 N −1 n=0 w n [0,1] s K(x, x n ) dx + N −1 n,n =0 w n w n K(x n , x n ) = (1 − β) 2 + k∈N s 0 \{0} r α,γ,s (k) 2 \|k\| 0 N −1 n=0 w n s j=1 cos(πk j x j,n ) 2 where x n = (x 1,n , . . . , x s,n ). For m = (m 1 , . . . , m s ) ∈ N s 0 and \|m\| := m 1 +· · ·+m s we will now construct a function G(y) := \|m\|=t F m (y), parametrized by the points x n and weights w n of the arbitrary cubature rule, to obtain a lower bound on the worst-case error. For this we will pick its cosine coefficients to be bounded above by r α,γ,s (k). Let the integer t be chosen such that 2N ≤ 2 t < 4N. Let a := α + 1 and let f : R → R be the a-times differentiable function f (x) := x a+1 (1 − x) a+1 for 0 < x < 1, 0 otherwise.(8) Note that f (x) > 0 for 0 < x < 1 and supp(f (τ ) ) = (0, 1) for all 0 ≤ τ ≤ a. I(f ) = B(a + 2, a + 2) = ((a + 1)!) 2 (2a + 3)! , where B denotes the beta function. For k = 0 we have f m,cos (k) = 1 0 f (2 m+2 x) √ 2 cos(πkx) dx = 1 √ 2 2 −m−2 0 f (2 m+2 x) e πikx + e −πikx dx = 1 2 m+2 √ 2 1 0 f (y) e 2πik2 −m−3 y + e −2πik2 −m−3 y dy = f (k2 −m−3 ) + f (−k2 −m−3 ) 2 m+2 √ 2 , where f (h) = 1 0 f (x) e −2πihx dx denotes the Fourier transform of f . Since, by definition, f (τ ) (0) = f (τ ) (1) = 0 for all 0 ≤ τ ≤ a, and f is a-times differentiable, repeated integration by parts shows that for any m ∈ N 0 we have \| f (k2 −m−3 )\| ≤ C a min(1, (k2 −m−3 ) −a ), where the constant C a > 0 depends only on a (and f ). Thus we have \| f m,cos (k)\| ≤ C a 2 −m−3/2 min(1, (k2 −m−3 ) −a ) ≤ C a 2 −m min(1, 2 am r a/2,1 (k)). This bound even holds for f m,cos (0) if C a is large enough. For the multivariate case we have the bound \| f m,cos (k)\| ≤ C(a, s) s j=1 2 −m j min(1, 2 am j r a/2,1 (k j )) = C(a, s) 2 (α−1)\|m\| s j=1 2 −αm j min(1, 2 am j r a/2,1 (k j )). By summing \| f m,cos (k)\| 2 over all choices of m where \|m\| = t we obtain m∈N s 0 \|m\|=t \| f m,cos (k)\| 2 ≤ 2 2(α−1)t C 2 (a, s) m∈N s 0 \|m\|=t s j=1 2 −2αm j min(1, 2 2am j r a,1 (k j )) ≤ 2 2(α−1)t C 2 (a, s) s j=1 ∞ m=0 2 −2αm min(1, 2 2am r a,1 (k j )). The last sum can now be estimated by ∞ m=0 2 −2αm min(1, 2 2am r a,1 (k j )) = 0≤m≤(log 2 r −1 a,1 (k j ))/2a 2 2(a−α)m r a,1 (k j ) + m>(log 2 r −1 a,1 (k j ))/2a 2 −2αm ≤ r −1 a−α,1 (k j )2 2(a−α) − 1 2 2(a−α) − 1 r a,1 (k j ) + r α,1 (k j )2 2α 2 2α − 1 ≤ r α,1 (k j ) 1 + 2 2α 2 2α − 1 ≤ 3 r α,1 (k j ). Thus, since 2N ≤ 2 t < 4N , we have r α,γ,s (k) ≥ C 0 (a, γ, s)2 −2(α−1)t m∈N s 0 \|m\|=t \| f m,cos (k)\| 2 ≥ C 1 (a, γ, s) 2 2t N 2α m∈N s 0 \|m\|=t \| f m,cos (k)\| 2 . Now for x = (x 1 , . . . , x s ) and y = (y 1 , . . . , y s ) and for u ⊆ [s] define x (±) u y = (z 1 , . . . , z s ), Denote with B c m,u the complement with respect to [0, 1] s . Then for λ s the s-dimensional Lebesgue measure we have λ s (supp(F m,u )) = λ s (B c m,u ). Since supp(f m (x n (±) u y)) as a function of y is contained in the interval j∈u (−x j,n , −x j,n + 2 −m j −2 ) j∈[s]\u (x j,n − 2 −m j −2 , x j,n ) we have where z j = x j + y j if j ∈ u and z j = x j − y j if j ∈ u. Define the functionssupp(F m,u ) ⊆ N −1 n=0 j∈u (−x j,n , −x j,n + 2 −m j −2 ) j∈[s]\u (x j,n − 2 −m j −2 , x j,n ). Thus λ s (supp(F m,u )) = λ s (B c m,u ) ≤ N 2 −\|m\|−2s . Now, for all m satisfying \|m\| = t we obtain λ s (B m ) = 1 − λ s (B c m ) = 1 − λ s   u⊆[s] B c m,u   ≥ 1 − u⊆[s] N 2 −\|m\|−2s = 1 − N 2 \|m\|+s > 1/2, since 2N ≤ 2 t < 4N . We can expand F m (y) − [0,1] s F m (y) dy in terms of the coefficients f m,cos (k): 1 2 s u⊆[s] N −1 n=0 w n f m (x n (±) u y) − f m,cos (0)β = N −1 n=0 w n k∈N s 0 \{0} f m,cos (k) ( √ 2) \|k\| 0 2 s u⊆[s] j∈u cos(πk j (x j,n + y j )) j∈[s]\u cos(πk j (x j,n − y j )) = N −1 n=0 w n k∈N s 0 \{0} f m,cos (k)( √ 2) \|k\| 0 s j=1 cos(πk j (x j,n − y j )) + cos(πk j (x j,n + y j )) 2 Thus, by definition of B m , we have λ s (B m )\| f m,cos (0)\| 2 β 2 = Bm   1 2 s u⊆[s] N −1 n=0 w n f m (x n (±) u y) − f m,cos (0)β   2 dy ≤ [0,1] s   1 2 s u⊆[s] N −1 n=0 w n f m (x n (±) u y) − f m,cos (0)β   2 dy = k∈N s 0 \{0} \| f m,cos (k)\| 2 N −1 n=0 w n s j=1 cos(πk j x j,n ) 2 . We are now ready to piece this all together to obtain cos(πk j x j,n ) 2 ≥ (1 − β) 2 + C 1 (a, γ, s) 2 2t N 2α m∈N s 0 \|m\|=t k∈N s 0 \{0} \| f m,cos (k)\| 2 2 \|k\| 0 N −1 n=0 w n s j=1 cos(πk j x j,n ) 2 ≥ (1 − β) 2 + C 1 (a, γ, s) 2 2t N 2α m∈N s 0 \|m\|=t λ s (B m )\| f m,cos (0)\| 2 β 2 ≥ (1 − β) 2 + C 2 (a, γ, s)β 2 (I(f )) 2s 2 2t N 2α 2 −2t−4s m∈N s 0 \|m\|=t 1 ≥ (1 − β) 2 + C 3 (α, γ, s)β 2 N −2α t + s − 1 s − 1 . Set A := C 3 (a, γ, s)N −2α t+s−1 s−1 . Then the last expression can be written as (1−β) 2 +Aβ 2 , which satisfies e 2 (H(K cos α,γ,s ); P ; w) ≥ (1 − β) 2 + Aβ 2 ≥ min(1, A) 2 ≥ C 4 (α, γ, s)N −2α t + s − 1 s − 1 , which implies the result, since t ≥ log 2 (N ). Numerical results In this section we show some numerical examples of applying lattice rules, tent-transformed lattice rules and symmetrized lattice rules to some test functions. For this we use the lattice sequence from [16] which was constructed to give 3rd order convergence in a Korobov space. It is a 10-dimensional base 2 sequence with a maximum of 2 20 points and is comprised of embedded lattice rules with sizes 2 m for m = 0, . . . , 20. This lattice sequence was also used in [27] for some higher order convergence tests. We report on two test functions to show some effects: g s,w (x) := s j=1 1 + w j 21 −10 + 42x 2 j − 42x 5 j + 21x 6 j , h s,w (x) := s j=1 1 + w j 8 31 − 84x 2 j + 8x 3 j + 70x 4 j − 28x 6 j + 8x 7 j − 16 cos(1) − 16 sin(x j ) . Both functions integrate to 1 over [0, 1] s . The parameter w acts like a product weight w j . All tests use 2 20 (plus 1 for the symmetrized rule) function evaluations for their final result. In Figure 1 we report the actual running time in microseconds of optimized C++ code (to accommodate for the difference in which integration nodes are constructed). Every mark represents an approximation with 2 m function evaluations or 2 s−1 2 m + 1 for the symmetrized rule. From this it can be seen that all three methods approximately have the same cost in this implementation, but in general this is dependent on the relative differences in the time for generating lattice sequence points, symmetrization and function evaluation. For completeness we note that the tests were run on a 1.8 GHz Intel Core i7 and compiled with the clang++ 3.1 C++ compiler. Tests on an older 2 GHz Intel Xeon compiled with the g++ 4.8 compiler gave similar looking results (but slower). The C++ source code and the raw data (for more test functions than shown here) can be downloaded from the KU Leuven Lirias document repository. The function g s,w (x) is a product of 1 + w j (B 6 (x j ) + E 5 (x j )), where B 6 is the degree 6 Bernoulli polynomial and E 5 is the degree 5 Euler polynomial. From their Fourier expansions [1] follows that we expect 3rd order convergence for both the tent-transformed and symmetrized lattice sequence, while we only expect 1st order convergence for the standard lattice sequence. We show some results in the left hand column of Figure 1 for s = 8. When the weight parameter is close to 1 (w = 0.9 in the middle panel) we notice that the performance of the symmetrized and the tent-transformed rules is similar. If w is much smaller (w = 0.1 in the top panel) then the tent-transformed rule wins, while if w is larger (w = 2 in the bottom panel) then the symmetrized rule wins; these effects are more pronounced when the dimension gets larger. The function h s,w (x) is a more arbitrary function (including sin(x j ) and E 7 (x j )) for which its cosine expansion converges only like k −2 . We thus expect 1st order convergence for the tent-transformed and symmetrized methods. For s = 10 the behavior is illustrated in the right hand column of Figure 1. Here again it can be seen that small values of w give the advantage to the tent-transformed lattice rule, see the top panel where w = 0.1. However, already for w = 0.9 the symmetrized rule outperforms both other methods. The bottom panel shows the case w = 1 where it can be seen that the number of points needed to reduce the error below 1 is already getting quite high. In such a case there is an exponential dependence on the number of dimensions and the symmetrized rule is then the preferred choice if the dimension is small enough (this can also be seen in the left bottom panel which has w = 2 for g s,w (x) in s = 8 dimensions). We remark that the graphs, where one replaces the time in the abscissa by the number of points N in Figure 1, are very similar to the ones shown in Figure 1. Conclusion In this paper we have shown that it is possible to obtain a convergence rate of N −α+δ for any δ > 0 for sufficiently smooth integrands for lattice-type rules also for nonperiodic functions. Previously, the only QMC rules with such properties where higher order digital nets [9]. However, the function spaces we consider here are smaller than the usual smooth Sobolev spaces used for higher order digital nets. Since for smoothness 1 the unanchored Sobolev space and the half-period cosine space coincide, we obtain that tent-transformed lattice rules achieve the same convergence behavior and tractability results as lattice rules in Korobov spaces. In contrast to previous results this technique does not need randomization. f (x n ) − [0,1] s f (x) dx depends on the smoothness of the integrand and some property of the quadrature points. ∞ k=1 ff k=1cos (2k − 1) √ 2 cos(π(2k − 1)x) =: c + f per (x) + f nonper (x),where f cos (k) := (x) √ 2 cos(πkx) dx for k ∈ N. (πmx) cos(πny) dx dy = 0 Corollary 1 . 1Using the fast component-by-component algorithm one can obtain a generating vector g ∈ {1, . . . , N − 1} s such that e(H(K cos α,γ,s ); P φ (g, Corollary 2 . 2Using the fast component-by-component algorithm one can obtain a generating vector g ∈ {1, . . . , N − 1} s such that Lemma 2 . 2The number of nodes in the symmetrized lattice rule P sym (g, N ) is given by 2 s−1 (N + 1) if 2 N and 2 s−1 N + 1 if 2 \| N . Corollary 3 . 3Using the fast component-by-component algorithm one can obtain a generating vector g ∈ {1, . . . , N − 1} s such that e(H(K kor+cos α,γ,s ); P sym (g, k 1 , 1...,ks∈N k 1 ,...,ks odd b k 1 ,...,ks s j=1 cos(πk j x j ) for all b k 1 ,...,ks ∈ R are integrated exactly. Likewise, all polynomials of the form k 1 ,...,ks∈N k 1 ,...,ks odd a k 1 ,...,ks s j=1 (x j − 1/2) k j for all a k 1 ,...,ks ∈ R Theorem 4 . 4For P an arbitrary N -element point set in [0, 1] s and w= (w 0 , . . . , w N −1 ) ∈ R N we have e(H(K cos α,γ,s ); P ; w) ≥ C(α, γ, s) (log N ) (s−1)/2 N α where C(α, γ,s) > 0 depends on α, γ and s, but not on N and w. Proof. We follow the proof of Temlyakov [31, Lemma 3.1]. Let β := N −1 n=0 w n . If β = 0 then e(H(K cos α,γ,s ); P ; w) ≥ 1, since for f = 1 the integration error is 1. In this case the result holds trivially. Thus we can assume now that β = 0. For m ∈ N 0 let f m (x) := f (2 m+2 x) and for m = (m 1 , . . . , m s ) ∈ N s 0 and x = (x 1 , . . . , x s ) ∈ R s let (y) dy. For f given by(8) we obtain F m,u ( ,u := {y ∈ [0, 1] s : F m,u (y) = 0} , B m := {y ∈ [0, 1] s : F m,u (y) = 0 for all u ⊆ [s]} = u⊆[s] B m,u . (πk j x j,n ) cos(πk j y j )πk j y j ). e 2 ( 2H(K cos α,γ,s ); P ; w) = (1 − β) 2 + k∈N s 0 \{0}r α,γ,s (k) 2 \|k\| 0 Figure 1 : 1Left column: function g s,w (x) in 8 dimensions for w = 0.1, w = 0.9 and w = 2. Right column: function h s,w (x) in 10 dimensions for w = 0.1, w = 0.9 and w = 1. Theorem 1 . 1For α = 1 we haveH(K kor 1,γ,s ) → H(K sob 1,γ,s ) H(K cos 1,γ,s ) H(K kor+cos 1,γ,s ) and H(K sob 1,γ,s ), H(K cos 1,γ,s ), H(K kor+cos 1,γ,s ) ⊂ H(K kor 1,γ,s ). For α ∈ N with α > 1 we have H(K kor α,γ,s ), H(K cos α,γ,s ) → H(K kor+cos α,γ,s ) → H(K sob α,γ,s ) and for α ∈ R we have H(K cos α,γ,s ) ⊂ H(K kor α,γ,s ), H(K kor+cos α,γ,s ) ⊂ H(K kor α,γ,s ), H(K kor+cos α,γ,s ) ⊂ H(K cos α,γ,s ) for α ≥ 3/2, H(K kor α,γ,s ) ⊂ H(K cos α,γ,s ) for α ≥ 3/2. We do not know whether H(K sob α,γ,s ) differs from H(K kor+cos α,γ,s ) for α > 1. AcknowledgementsJ.D. is supported by an Australian Research Council Queen Elizabeth II fellowship. 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[ "The width of Herschel filaments varies with distance", "The width of Herschel filaments varies with distance" ]	[ "G V Panopoulou \nCalifornia Institute of Technology\nMail Code 350-17, 1200 E. California Blvd91125PasadenaCAUSA\n", "S E Clark \nDepartment of Physics\nStanford University\n94305StanfordCaliforniaUSA\n\nKavli Institute for Particle Astrophysics & Cosmology\nStanford University\nP. O. Box 245094305StanfordCAUSA\n", "A Hacar \nDepartment of Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria\n", "F Heitsch \nDepartment of Physics & Astronomy\nUniversity of North\nCarolina -Chapel Hill, Chapel Hill27599North CarolinaUSA\n", "J Kainulainen \nDepartment of Space, Earth and Environment\nChalmers University of Technology\nSE-412 93GothenburgSweden\n", "E Ntormousi \nScuola Normale Superiore di Pisa\nPiazza dei Cavalieri, 756126PisaItaly\n", "D Seifried \nI. Physical Institute\nUniversity of Cologne\nZülpicher Str. 7750937CologneGermany\n", "R J Smith \nJodrell Bank Centre for Astrophysics\nDepartment of Physics and Astronomy\nUniversity of Manchester\nOxford RoadM13 9PLManchesterUK\n" ]	[ "California Institute of Technology\nMail Code 350-17, 1200 E. California Blvd91125PasadenaCAUSA", "Department of Physics\nStanford University\n94305StanfordCaliforniaUSA", "Kavli Institute for Particle Astrophysics & Cosmology\nStanford University\nP. O. Box 245094305StanfordCAUSA", "Department of Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria", "Department of Physics & Astronomy\nUniversity of North\nCarolina -Chapel Hill, Chapel Hill27599North CarolinaUSA", "Department of Space, Earth and Environment\nChalmers University of Technology\nSE-412 93GothenburgSweden", "Scuola Normale Superiore di Pisa\nPiazza dei Cavalieri, 756126PisaItaly", "I. Physical Institute\nUniversity of Cologne\nZülpicher Str. 7750937CologneGermany", "Jodrell Bank Centre for Astrophysics\nDepartment of Physics and Astronomy\nUniversity of Manchester\nOxford RoadM13 9PLManchesterUK" ]	[]	Context. Filamentary structures in nearby molecular clouds have been found to exhibit a characteristic width of 0.1 pc, as observed in dust emission. Understanding the origin of this universal width has become a topic of central importance in the study of molecular cloud structure and the early stages of star formation. Aims. We investigate how the recovered widths of filaments depend on the distance from the observer by using previously published results from the Herschel Gould Belt Survey. Methods. We obtained updated estimates on the distances to nearby molecular clouds observed with Herschel by using recent results based on 3D dust extinction mapping and Gaia. We examined the widths of filaments from individual clouds separately, as opposed to treating them as a single population. We used these per-cloud filament widths to search for signs of variation amongst the clouds of the previously published study. Results. We find a significant dependence of the mean per-cloud filament width with distance. The distribution of mean filament widths for nearby clouds is incompatible with that of farther away clouds. The mean per-cloud widths scale with distance approximately as 4-5 times the beam size. We examine the effects of resolution by performing a convergence study of a filament profile in the Herschel image of the Taurus Molecular Cloud. We find that resolution can severely affect the shapes of radial profiles over the observed range of distances. Conclusions. We conclude that the data are inconsistent with 0.1 pc being the universal characteristic width of filaments.	10.1051/0004-6361/202142281	[ "https://export.arxiv.org/pdf/2111.08125v3.pdf" ]	244,130,063	2111.08125	dc9f8f3eff4442658d3e264909603162b5811d2b	The width of Herschel filaments varies with distance August 9, 2022 August 9, 2022 G V Panopoulou California Institute of Technology Mail Code 350-17, 1200 E. California Blvd91125PasadenaCAUSA S E Clark Department of Physics Stanford University 94305StanfordCaliforniaUSA Kavli Institute for Particle Astrophysics & Cosmology Stanford University P. O. Box 245094305StanfordCAUSA A Hacar Department of Astrophysics University of Vienna Türkenschanzstrasse 171180ViennaAustria F Heitsch Department of Physics & Astronomy University of North Carolina -Chapel Hill, Chapel Hill27599North CarolinaUSA J Kainulainen Department of Space, Earth and Environment Chalmers University of Technology SE-412 93GothenburgSweden E Ntormousi Scuola Normale Superiore di Pisa Piazza dei Cavalieri, 756126PisaItaly D Seifried I. Physical Institute University of Cologne Zülpicher Str. 7750937CologneGermany R J Smith Jodrell Bank Centre for Astrophysics Department of Physics and Astronomy University of Manchester Oxford RoadM13 9PLManchesterUK The width of Herschel filaments varies with distance August 9, 2022 August 9, 2022Astronomy & Astrophysics manuscript no. main Letter to the EditorISM: clouds -ISM: structure -stars: formation -Galaxy: local interstellar matter Context. Filamentary structures in nearby molecular clouds have been found to exhibit a characteristic width of 0.1 pc, as observed in dust emission. Understanding the origin of this universal width has become a topic of central importance in the study of molecular cloud structure and the early stages of star formation. Aims. We investigate how the recovered widths of filaments depend on the distance from the observer by using previously published results from the Herschel Gould Belt Survey. Methods. We obtained updated estimates on the distances to nearby molecular clouds observed with Herschel by using recent results based on 3D dust extinction mapping and Gaia. We examined the widths of filaments from individual clouds separately, as opposed to treating them as a single population. We used these per-cloud filament widths to search for signs of variation amongst the clouds of the previously published study. Results. We find a significant dependence of the mean per-cloud filament width with distance. The distribution of mean filament widths for nearby clouds is incompatible with that of farther away clouds. The mean per-cloud widths scale with distance approximately as 4-5 times the beam size. We examine the effects of resolution by performing a convergence study of a filament profile in the Herschel image of the Taurus Molecular Cloud. We find that resolution can severely affect the shapes of radial profiles over the observed range of distances. Conclusions. We conclude that the data are inconsistent with 0.1 pc being the universal characteristic width of filaments. Introduction The formation of stars appears to occur preferentially within filamentary structures (André et al. 2014;Könyves et al. 2015). Special attention has been given to one morphological property of these structures: their width. In models of idealized hydrostatic cylinders, the radius is related to the stability of the structure (Ostriker 1964). In particular, the radius of the column density profile is expected to scale inversely with the column density following the thermal Jeans length. Arzoumanian et al. (2011) analyze a large sample of filaments in Herschel dust continuum images and show that the observed widths of filaments are almost independent of the column density and are uncorrelated with the Jeans length -contrary to theoretical expectations. Despite the wide range of filament column densities, filament widths are found to follow a narrow distribution, which peaks at ∼ 0.1 pc with a spread of only a factor of 2 (Arzoumanian et al. 2011). This surprising finding has led to the proposition that filaments show a characteristic width -one that is universal among Hubble fellow clouds with drastically different properties (e.g., star formation rate, mean column density, Arzoumanian et al. 2011;André et al. 2014). Recently, Arzoumanian et al. (2019) extended the analysis of filament widths to a much larger sample of filaments in the Herschel Gould Belt Survey (HGBS), finding results in agreement with their earlier study. Multiple theoretical models have been proposed to explain the observed distribution of widths and the apparent independence with the column density (Fischera & Martin 2012a,b;Hennebelle 2013;Hennebelle & André 2013;Federrath 2016;Auddy et al. 2016;Federrath et al. 2021;. To date, no model has been able to reproduce the properties of the distribution over the wide range of filament column densities in the sample of Arzoumanian et al. (2019). The presence of a characteristic width has been called into question from several investigations. Panopoulou et al. (2017) show that commonly adopted choices in the analysis of filament radial profiles lead to significant biases in the resulting width distribution. First, the width of Herschel filaments was originally determined from the full-width-at-half-maximum (FWHM) of a Article number, page 1 of 11 arXiv:2111.08125v3 [astro-ph.GA] 8 Aug 2022 A&A proofs: manuscript no. main single radial profile: one that results from averaging the contribution of equidistant points at each radius along the filament spine. The reported distribution of widths is thus a distribution of the mean filament widths, leading to an artificially narrow spread as a result of the central limit theorem. When considering widths measured at all points along a filament's crest, broader distributions are invariably found (with a spread 2-3 times that of the crest-averaged distribution, Panopoulou et al. 2017;Arzoumanian et al. 2019;Suri et al. 2019). Second, the determination of the filament FWHM has a strong dependence on the choice of the maximum radial distance within which the fit is performed (Smith et al. 2014). Ossenkopf-Okada & Stepanov (2019) perform an independent analysis of Herschel data using a wavelet decomposition and do not find signs of a characteristic width common to all of the clouds in their study. Recently, Louvet et al. (2021) investigated the effect of the telescope beam size on the core mass function (CMF), finding that both the peak of the CMF and the radial extent of filaments are dependent on the resolution. Juvela et al. (2012a) study how telescope resolution can affect the properties of recovered filament profiles by employing magneto-hydrodynamical (MHD) simulations and radiative transfer post-processing. The resulting filament widths are mildly affected (10% level) unless the structures are placed at distances 400 pc (beyond which the structures become unresolved), they are affected by background confusion, or they have complex dust opacity (in which case biases of ∼40% are found). Observations treating the effect of varying dust optical properties have also found slight variations in filament widths compared to the case of the assumed simple opacity on the order of 60% (Howard et al. 2019). The level of bias caused by resolution on filament widths in the aforementioned works is modeldependent; for example, the 0.01-pc-wide filaments simulated by Seifried et al. (2017) have observed widths a few times wider than their true value. In this Letter, we revisit the original data that support the presence of a characteristic width of 0.1 pc. Using the most recent developments in the determination of molecular cloud distances based on Gaia, we revised the estimates of filament widths published from the HGBS survey. We demonstrate that the mean filament width increases as a function of distance (Sect. 3). This trend refers to the ensemble average of widths over the population of filaments in a cloud. We investigate whether the trend could be related to telescope resolution though a convergence study of a single filament profile (Sect. 4). We discuss our findings and conclude in Sect. 5. Arzoumanian et al. (2019) identified filaments on column density maps from Herschel for eight clouds in the HGBS. We briefly summarize the salient points in their analysis leading to the determination of filament widths. Data Literature measurements of filament widths In each image, the crest of filamentary structures (skeleton) was obtained by using the DisPerSe algorithm (Sousbie 2011). For each filament, a single radial column density profile was created by taking the median of all points that are equidistant from the crest along the length of the filament (we refer to this as the crest-averaged profile). At some distance from the crest, the profile flattens and merges with the background. The radius at which this happens is denoted as r out . The width of the filament profile within r out of the crest was measured in two ways. First, the au-thors found the radius where the profile drops to half-maximum of the crest-averaged profile after background subtraction (halfradius, hr, in their notation). The width is defined as the halfdiameter hd = 2hr. Second, a Gaussian function plus background was fit within 1.5 hr, and the width of the filament is the resulting FWHM of the Gaussian. For each cloud, the distribution of filament widths was constructed. Arzoumanian et al. (2019) calculated a "deconvolved" width, or half-diameter, as follows: hd dec = √ hd 2 − HPBW 2 (HPBW is the telescope halfpower-beam-width). In this paper we use the nonparametric estimation of filament width, reported as the "deconvolved" half-diameter, hd dec in Table 3 of Arzoumanian et al. (2019). In our notation, we define hd dec to be FWHM dec . When necessary, we convert the "deconvolved" width, FWHM dec , to the observed width, FWHM obs , following Arzoumanian et al. (2019): FWHM obs = FWHM 2 dec + HPBW 2 ,(1) where HPBW, the telescope half-power-beam-width, is equal to 18.2 . We stress that calculating FWHM dec as in Arzoumanian et al. (2019) does not accurately correct for the convolution with the beam, as we show in Appendix B, but we chose to use FWHM dec to facilitate comparison with their work. In addition to the hd dec , we also used the values of 2r out as well as the spread of the per-cloud distribution of hd dec -which we denote as σ(FWHM dec )-as provided in their table 3. Cloud distances We used the latest 3D dust extinction maps based on Gaia for the determination of distances to clouds in the Arzoumanian et al. (2019) sample. Zucker et al. (2020) have provided highly accurate distance measurements (to within ∼ 5%) for a subset of the clouds in this sample, namely: IC 5146, Orion B, Taurus L1495/B213, and Ophiuchus. While they also provide estimates for the Aquila Rift, the Pipe Nebula, and the Polaris Flare, these are based on sightlines passing outside the area covered by Herschel. We therefore reanalyzed data from 3D dust extinction toward these three clouds, as well as Musca, which does not have a recent distance estimate in the literature, to determine the distance to the filamentary structures seen in the Herschel images. For this, we used the Leike et al. (2020) 3D dust map which provides the highest distance resolution among the existing maps within the Solar neighborhood, as described in Appendix A. Distances to the Polaris Flare and IC 5146 are the most discrepant between the updated measurements and the default values adopted in Arzoumanian et al. (2019). The results are summarized in Dependence of filament widths on distance Using the new cloud distances, d new , we rescaled the per-cloud mean filament widths from Arzoumanian et al. (2019) (from their table 3, see Section 2) to obtain revised estimates of FWHM dec for each cloud. The same operation was performed to the spread of the distribution of widths for each cloud. . A colored symbol shows the ensemble average width of filaments within a cloud. The vertical lines span the ±1σ range of the distribution of the filament widths of that cloud. Two points mark the full range of distances to a cloud, when such measurements are available. A dotted line shows the Herschel beam size. The background shows a distribution of the filament widths for each cloud, drawn from a Gaussian distribution with the number of measurements per cloud reported in the original study (see text). We only show a realization for the nearest distance limit of each cloud for clarity. and error-bars denote the standard deviation of the distribution of filament widths in each cloud. The mean filament width systematically increases by a factor of ∼ 4 when comparing the nearest and farthest clouds. For each cloud we use the full range of likely distances, as obtained in Appendix A, to demonstrate the full error budget. We find a similar trend in the original data as published in Arzoumanian et al. (2019) (which show a factor of ∼ 2 increase). For example, using the values reported in table 3 of Arzoumanian et al. (2019), filament widths in IC 5146 (with a mean of 0.13 pc and σ of 0.04 pc) are more than one standard deviation larger than those in Taurus (with a mean of 0.06 pc and a spread of 0.02 pc). We have verified that the trend is also seen in the widths obtained by Gaussian fits provided in table 3 of Arzoumanian et al. (2019). Clouds at distances larger than 300 pc (Polaris, Orion, IC5146) clearly show higher values for the mean width. If one were to combine filament widths from all clouds, as in Arzoumanian et al. (2019), the resulting distribution would have a mean of ∼ 0.1 pc. This is noted in their section 4.1 when alternative distances are considered. However, it is clear from Fig. 1 that the clouds that are farther than ∼300 pc pull the mean of the distribution toward higher values. In particular, the 234 filaments of Orion B constitute 40% of the total number of filaments in the sample and significantly contribute to the larger values around 0.1 pc. To demonstrate the level of discrepancy, we compared the distributions of the two clouds with the largest number of filaments (Orion B and Taurus, with 234 and 110 filaments, re-spectively) via a two-sided Kolmogorov-Smirnov (K-S) test. We used the reported mean width, the ±1σ standard deviation, and the number of identified filaments, N fil , to draw mock filament widths from a normal distribution. Fig. 1 shows one random realization of filament widths drawn as described above for all clouds (background grayscale), using the same number of filaments per cloud as the original paper. The K-S test rejects the hypothesis that the distribution of widths from filaments in Orion is drawn from the same distribution as that of Taurus, with a pvalue < 10 −40 . This is true for both the original widths as well as the rescaled values after updating cloud distances. Finally, we also examined the spread of the per-cloud distribution of filament widths. If all clouds had the same intrinsic spread of filament widths, we would expect to find the reported σ of the distribution of FWHM dec σ(FWHM dec ) to be independent of distance 1 . Contrary to that expectation, the per-cloud spread σ(FWHM dec ) are correlated with the cloud distance, and they are found to increase from 0.026 for the nearest cloud to 0.09 pc for the farthest cloud (Pearson correlation coefficient of 0.94 and a p-value of 4 ×10 −4 ). It is unlikely that the different sample sizes are driving this scaling, as the σ(FWHM dec ) are uncorrelated with the number of filaments per cloud, N fil . In summary, we have found that the mean and the spread of the per-cloud distribution of filament widths significantly depend on distance from the observer. A K-S test rejects the null hypothesis that the distribution of widths in nearby clouds is consistent with that of farther away clouds. These findings contradict the interpretation of the mean width of 0.1 pc, averaged over all clouds, as being representative of the whole filament sample (i.e., universal). Resolution may strongly affect filament profiles The apparent increase of the mean filament widths as a function of distance ( Fig. 1) suggests that filament profiles are not resolved. Yet, filament FWHM dec are several times larger than the beam size of 18.2 (as can be seen by comparing the data with the line of Fig. 1). To understand this apparent contradiction, we investigated the effect of the resolution on the profile of a filament in the nearby Taurus molecular cloud. We tested the following hypothesis: Is resolution, in principle, capable of producing as significant a rise of filament width with distance as observed? To answer this question, we performed a simple experiment: we progressively reduced the resolution of the map of the Taurus main filament, effectively "observing" it with angular resolution corresponding to the original Herschel beam at larger distances (i.e., we performed a convergence test as in the CMF study of Louvet et al. 2021). We then measured the FWHM of the filament at these different resolutions. As we are interested in understanding the effect of the beam size, we used the observed FWHM obs , not the "deconvolved" FWHM dec (Eq. 1), as well as the updated distance estimates, d new (Table A.1). The measurement of FWHM obs involves two main operations: (a) convolution with the telescope beam and (b) truncation of the profile at radii larger than r out (Section 2). First, we numerically convolved the column density image of Taurus to achieve resolutions equivalent to 0.023 pc, 0.037 pc, and 0.067 pc (corresponding to a beam size of 18.2 at distances of 260 pc, 423 pc, and 762 pc). Using the same filament skeleton for all images, we constructed the filament's radial profile, determined Comparison between the observed dependence of FWHM obs on the distance (gray data points, symbols as in Fig. 1) and that of a radial profile of a filament in Taurus, after reducing the angular resolution of the Herschel map to correspond to the physical resolution of a 18.2 beam at the observed cloud distances (red diamonds). Dotted lines mark 1, 2, 4, and 5 times the beam size. r out , and measured FWHM obs . We refer the reader to Appendix B for more details. In Fig. 2 we compare the FWHM obs obtained for this resolution study with the per-cloud mean FWHM obs , as a function of cloud distance. The effect of resolution dramatically changes the FWHM obs of the Taurus filament profile (red diamonds), with values increasing by a factor of 10 from the original image (140 pc) to the largest distance of 762 pc. While the per-cloud mean FWHM obs do rise with distance, this rise is shallower than that of the single profile of the resolution study. We note that the reduction of angular resolution is not necessarily equivalent to placing the same cloud at different distances. However, this convergence study allows us to examine whether all filaments observed with the same resolution in physical units have the same width of 0.1 pc. This does not seem to be the case: if the Taurus filament had been observed with a physical resolution of 0.067 pc (as filaments in IC 5146) instead of 0.012 pc, it would appear to have a width of 0.99 pc compared to the mean width of 0.24 pc of filaments in IC 5146. We have thus shown that resolution can produce an increase in the FWHM obs of a single filament by a factor of ∼ 10, which is much larger than the increase in the per-cloud mean FWHM obs (factor of 4). The observed rise in the per-cloud mean FWHM obs with distance could, at least partially, be explained by resolution. The remarkably linear dependence of FWHM obs on the distance of ∼ 4 × beam at large distances would suggest that indeed filaments are not resolved. Resolution and beam confusion can lead to the detection of structures of a few times the beam size; for example, cloud diameters were found to be 3 times the beam size in Verschuur (1993) as a result of the hierarchical nature of the medium. Further work is needed to reliably disentangle possible effects of resolution from a possible intrinsic variation of filament widths. The mean width of the entire cloud sample of 0.1 pc may not be robust to changes in resolution. Verifying or disproving this, however, is beyond the scope of the present work. Conclusions We have revisited the observational results of Arzoumanian et al. (2019) on filament widths in the HGBS. By examining the data of different clouds separately, we have unveiled a trend that has been hidden in the data: a dependence of the per-cloud mean filament widths on the distance from the observer. While the ensemble average widths over all clouds is 0.1 pc, this is not a representative statistic. The distribution of mean filament widths for nearby clouds is incompatible with that of farther away clouds. In addition, the spread of per-cloud filament widths depends on distance. The data thus contradict a universal width of 0.1 pc with a spread of ∼ 2 for all clouds, as originally inferred by Arzoumanian et al. (2011). The scaling of FWHM with distance is reminiscent of that found in other Herschel surveys (Schisano et al. 2014;Rivera-Ingraham et al. 2016), recognised as the effect of resolution. Even though the observed filament widths are multiple times the beam size, we have demonstrated that the effect of resolution on the shape of an observed filament profile is not negligible. We considered the case of the profile of a filament in the Taurus molecular cloud. We performed a resolution study by convolving the column density image of Taurus with varying beam sizes. The resolution has a dramatic effect on the recovered filament width: the width increases with distance more steeply than the per-cloud mean widths. In combination with the almost linear scaling of mean filament widths with distance (4-5 times the beam size), our results strongly suggest that resolution biases the measurement of filament widths. Our results are in agreement with a growing body of evidence showing that filament radial profiles exhibit more complexity than can be captured by the picture of a "characteristic" width. Filaments show substructure when observed with spectral line tracers (Hacar et al. 2013) and can appear significantly narrower when observed at a higher resolution than Herschel (e.g., with interferometers targeting spectral lines of dense gas tracers, Fernández-López et al. 2014;Hacar et al. 2018;Monsch et al. 2018). Filament radial profiles can vary by factors of 2-10 from one end of the structure to the other (Juvela et al. 2012b;Suri et al. 2019), and averaging is shown to bias the shape of the resulting profile . By understanding the inherent biases in characterizing filament profiles, we can correct for them. Assuming a Gaussian deconvolution (Eq. 1) does not accurately recover the intrinsic FWHM (Appendix B). Beam effects should be mitigated by forward modeling (e.g., Juvela et al. 2012a;Smith et al. 2014;Federrath 2016;Seifried et al. 2017) or analytical functions that explicitly take these into account (Fischera & Martin 2012a). Tools that go beyond the 1D description of a filament profile A&A proofs: manuscript no. main Appendix A: Cloud distances We used the latest results from 3D dust extinction mapping to update the distance estimates of clouds in the sample of Arzoumanian et al. (2019). Zucker et al. (2020) provide distance measurements for a large list of nearby molecular clouds. They combined stellar photometry with the Gaia Data-Release-2 stellar parallax to model stellar extinction as a function of distance in discrete sightlines toward these clouds. For each cloud we considered all sightlines (from their table A.1) that fall within the footprint of the Herschel maps (Fig. A.1). We combined the systematic and statistical uncertainties for each measurement and compared the ±1σ ranges among all sightlines. We considered the lower limit of the cloud distance as the minimum of these bounds, and similarly for the upper limit. We thus obtained a lower and upper limit for the distance to four clouds in the sample: IC 5146, Orion B, Taurus, and Ophiuchus. For the remaining four clouds in the sample (Aquila, Musca, Polaris, and Pipe), there are no suitable distance estimates from Zucker et al. (2020), that is to say either no sightline passes within the Herschel footprint, or the cloud was not studied at all, as for Musca. We therefore analyzed 3D dust maps to calculate distance limits. We used the Leike et al. (2020) map which offers the best distance resolution among available maps (∼ 1 pc). First, we selected lines of sight that overlap with filaments as identified in Arzoumanian et al. (2019) (shown in Fig. A.2). Using the dustmaps python package, we queried the Leike et al. (2020) map to obtain the differential optical depth (optical depth per parsec) for each line of sight. We converted the differential optical depth to differential G−band extinction, δA G , following Panopoulou et al. (2021). For all sightlines, the differential optical depth shows a prominent peak at a certain distance (Fig. A.3). We found the distance of the δA G peak for each sightline. The minimum and maximum peak locations for each cloud are considered to be the lower and upper limits for its distance. Distance measurements for the sightlines toward all clouds in the sample are given in Table A.1. For measurements from Zucker et al. (2020), the quoted distance uncertainties include the full statistical and systematic uncertainty as provided in their table A.1. For measurements from this work, we quote the uncertainty determined by the chosen distance binning of the 3D dust map (5 pc). In the following, we compare the updated distance estimates with literature values used in Arzoumanian et al. (2019). IC5146 The distance range for this cloud is [686,833] pc by combining the results of Zucker et al. (2020), as described above. The original distance adopted by Arzoumanian et al. (2011) of 460 +40 −60 pc was derived by Lada et al. (1999) using star counts. An alternative distance of 950± 80 pc was also considered (appendix A of Arzoumanian et al. 2011), which was derived by stellar photometry of late B-type members of the IC 5146 cluster (Harvey et al. 2008). Orion B The distance range for this cloud is [386,474] pc (from Zucker et al. 2020). This is consistent with the distance of 400 pc adopted in Arzoumanian et al. (2019) (obtained from Gibb 2008). Polaris Flare Our distance estimate for the Polaris Flare is [350,360] pc. This is significantly discrepant from the distance of 150 pc assumed in Arzoumanian et al. (2019). Despite this being a commonly adopted distance (e.g., Bensch et al. 2003;Ward-Thompson et al. 2010), it is nevertheless incorrect, and can be traced back to Zagury et al. (1999), who assumed that the cloud is foreground to Polaris (the star) (see Schlafly et al. 2014, for a detailed literature review). Polaris (the star) exhibits some extinction and polarization, leading Panopoulou et al. (2016) to also erroneously conclude that the cloud is in front of the North star. However the bulk of extinction clearly arises at 350 pc as seen in Fig. A.3 (and also Schlafly et al. 2014). Pipe Extinction toward the Pipe nebula is found to lie within [150,155] pc. This is consistent with the previously adopted distance of 145±16 pc, based on linear polarimetry and Hipparcos distances (Alves & Franco 2007). Aquila Our distance limits to Aquila are [250,260] pc, consistent with the distance of 260±37 pc, which was based on stellar extinctions in the general area of Serpens (Straižys et al. 1996). Musca Our determination of the distance to Musca is 170 pc; there is no variation among the selected sightlines. The previously adopted distance was 200 pc, selected by Cox et al to be "in between" the estimates by Franco (1991) and Knude & Hog (1998) (stellar extinctions toward the general region of the Chamaeleon clouds). Taurus L1495 The distance limits from Zucker et al. (2020) are [120,140] pc, which is consistent with the previously adopted distance of 140 pc (e.g., based on stellar photometry from Kenyon et al. 1994). Ophiuchus The distance limits for Ophiuchus of [131,145] pc are consistent with the previously adopted distance of 140 pc (in turn based on highly accurate distance measurements toward parts of the cloud of 138±3 pc and 144±1 pc from Very Long Baseline Interferom- the final resolution of the image (beam size) has a physical size of 0.023 pc, 0.037 pc, and 0.067 pc, corresponding to the Herschel beam of 18.2 observed at distances of 260 pc, 423 pc, and 762 pc. More specifically, we calculated the kernel standard deviation as: σ kernel = beam 2 final − beam 2 initial /(2 √ 2 ln 2), where beam final is the desired resolution of the image, beam initial is the Herschel beam, and all quantities are measured in units of pixel on the image. We used the publicly available skeleton of the Taurus Herschel map, which was produced using DisPerSe -see Arzoumanian et al. (2019). We input this skeleton to the radfil Python package (Zucker & Chen 2018) and extracted the median radial profile along the filament crest (shown in Fig. B.1). We focused on a single filament to isolate the effect of resolution from other effects such as averaging over the filament population. The median radial profile obtained from the images "observed" with lower angular resolution is shown in Fig. B.2. By changing the resolution, we see a reduction in the peak amplitude of the profile, while the overall shape remains similar in angular units (Fig. B.2, left panel). When comparing the profiles in physical units, the profiles become drastically broader for larger beam sizes (lower resolution). For each profile, we measured the FWHM similarly to Arzoumanian et al. (2019), as follows. We found the outer truncation radius, r out , where the derivative of the profile, dNH 2 /dr, be-comes consistent with zero over a range of distances. The derivative and corresponding r out are shown in Fig. B.2 (bottom right panel). The values of r out are 0.25 pc, 0.5 pc, 0.95 pc, and 1.5 pc for the profiles at 140 pc, 260 pc, 423 pc, and 762 pc. We then calculated the mean column density of the profile at all radial distances beyond r out . We subtracted this mean value (background) from the profile and then set all negative values to zero. We finally found the half-radius and multiplied by 2 to obtain FWHM obs . The results are shown in Fig. 2 and discussed in the main text. Previous works have indicated that filament widths have a dependence on the choice of the truncation radius (Smith et al. 2014;Panopoulou et al. 2017). To ensure a fair comparison with Arzoumanian et al. (2019), we compared our recovered r out with the reported per-cloud average r out from their work. We rescaled the values of 2 r out (Arzoumanian et al. 2019, table 3) to the new cloud distances and plotted them in Fig. B.3. The values were found to scale approximately linearly with distance. We fit a linear regression to the data using the lower distance limits and obtained the following: 2 r out = 0.00112 d new + 0.072 pc. (B.1) Fitting the lower or upper limits of d new yields essentially identical results. Dividing by 2, we obtained the slope and intercept for obtaining the truncation radius r out at a given distance. We checked that using the mean r out for a given distance from eq. B.1 produces essentially identical FWHM obs with respect to those Herschel column density maps of clouds for which we found the distance using the 3D dust extinction map of Leike et al. (2020). Red circles mark the sightlines used to derive distances. that used the profile-flattening criterion for the example filament in our resolution study. The one exception is for the largest distance considered, where the Taurus profile differs by 40% for the two choices of r out . The effect of the beam size has previously been treated as a simple convolution of Gaussians (Eq. 1). However, Eq. 1 should not be used to deconvolve any arbitrary functional form of a profile from a Gaussian beam. This can be readily understood as a consequence of the Fourier properties of a Gaussian function and the convolution theorem. Indeed, as noted in Zucker & Chen (2018), Eq. 1 does not have the desired effect of correcting for beam convolution. For the example profiles shown in Fig. B.2, the "deconvolved" FWHM dec from Eq. 1 can differ by up to a factor of 10 from the initial FWHM obs of the profile at 140 pc. A simple beam deconvolution does not recover intrinsic properties of the profile. Fig. 1 Fig. 1 . 11shows the mean "deconvolved" filament width, FWHM dec , as a function of d new , for each cloud in theArzoumanian et al. (2019) sample separately. Each data point represents an ensemble average over the set of filaments identified in a cloud by their study, Mean "deconvolved" filament width for each cloud analyzed byArzoumanian et al. (2019), as a function of the updated cloud distance (d new ) Fig. 2. Comparison between the observed dependence of FWHM obs on the distance (gray data points, symbols as in Fig. 1) and that of a radial profile of a filament in Taurus, after reducing the angular resolution of the Herschel map to correspond to the physical resolution of a 18.2 beam at the observed cloud distances (red diamonds). Dotted lines mark 1, 2, 4, and 5 times the beam size. (e.g., wavelet decompositionRobitaille et al. 2014;Ossenkopf- Okada & Stepanov 2019;Robitaille et al. 2019) can allow for a deeper, more nuanced understanding of the nature of interstellar filaments and their environment (e.g., as a hierarchical medium that is only limited by the beam size,Robitaille et al. 2020). Fig. A. 1 . 1Herschel column density maps of clouds with distance measurements fromZucker et al. (2020). Red circles mark the sightlines with measured distance. Fig. A.2. Herschel column density maps of clouds for which we found the distance using the 3D dust extinction map of Leike et al. (2020). Red circles mark the sightlines used to derive distances. Fig. A. 3 . 3Distance determination for the Pipe Nebula, Musca, Polaris Flare, and Aquila Rift clouds. Each subpanel shows the differential G − band extinction per parsec as a function of distance for different sightlines within the Herschel map of each cloud (transparent black lines). Old distance estimates (adopted inArzoumanian et al. 2019) are shown as a vertical dashed line, while the new distance measurement for each line of sight is shown as a red vertical segment at the top of each panel.Article number, page 9 of 11 Fig. B. 2 .Fig. B. 3 . 23Median radial profile of a filament in Taurus (Fig. B.1) for different choices of resolution, corresponding to the Herschel beam of 18.2 at distances of 140 pc, blue; 260 pc, orange; 423 pc, green; and 762 pc, red. Top left: Profile comparison in angular units. Top right: Profile comparison in physical units. Bottom left: Profiles after background subtraction (with logarithmic horizontal axis for better visualization). The vertical dashed lines mark the r out defined in the bottom right panel. Bottom right: Derivative of radial profile. The radial distance where the derivative flattens determines r out (vertical dashed lines), following Arzoumanian et al. (2019). Outer truncation diameter as a function of (updated) cloud distance. The line is a linear fit to the data points choosing all lower distance limits. A fit to all upper distance limits yields similar results. Symbols as in Fig. 1. Table A . A1. Throughout the text, updated dis- tance estimates are denoted as d new , while those used in Arzou- manian et al. (2019) are denoted as d old . Table A A.1. Summary of sightlines used to determine distance limits to clouds. Galactic coordinates are provided, as well as the distance estimates for each sightline. References for the new distance limits: (1)Zucker et al. (2020), (2) this work.100 200 300 400 0.0 0.1 0.2 A G (mag/pc) Pipe Old distance New distances 100 200 300 400 0.00 0.02 0.04 0.06 A G (mag/pc) Musca 100 200 300 400 0.00 0.02 0.04 0.06 A G (mag/pc) Polaris 100 200 300 400 Distance (pc) 0.00 0.02 0.04 0.06 A G (mag/pc) Aquila A&A proofs: manuscript no. main 64.90°64.80°64.70°64.60°64.50°2Fig. B.1. Filament in Taurus observed with different beam sizes. The filament crest (obtained from the HGBS archive) is marked with a red line. The images were obtained by convolving the original Herschel column density image of Taurus (left panel) with a Gaussian kernel to achieve effective resolutions corresponding distances of 260 -762 pc. The right panel shows the map for 423 pc.7.50°2 7.40°2 7.30°2 7.20°R .A. Dec. 140 pc 423 pc 0.0 0.5 1.0 NH 2 (cm 2 ) ×10 22 200 0 200 Angular distance [arcsec] 0.00 0.25 0.50 0.75 1.00 1.25 1.50 NH 2 (cm 2 ) 1e22 140 pc 260 pc 423 pc 762 pc 1 0 1 Radial distance [pc] 0.00 0.25 0.50 0.75 1.00 1.25 1.50 NH 2 (cm 2 ) 1e22 10 2 10 1 10 0 Radial distance [pc] 0.00 0.25 0.50 0.75 1.00 1.25 NH 2 (cm 2 ) 1e22 10 2 10 1 10 0 Radial distance [pc] This expectation holds if the intrinsic spread is measurable with Herschel, e.g., not limited by resolution.Article number, page 3 of 11 A&A proofs: manuscript no. main Article number, page 8 of 11 Panopoulou et al.: Herschel filament widths vary with distance Appendix B: A case study in Taurus on the effects of beam convolutionWe investigated the effect of resolution on a radial profile from the Taurus B211/B213 filament. First, we convolved the column density map of Taurus with Gaussian kernels of different sizes to simulate the effect of observing the same cloud at lower resolution. 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[ "Surjectivity of cycle maps for singular varieties", "Surjectivity of cycle maps for singular varieties" ]	[ "Robert Laterveer " ]	[]	[]	A theorem of Jannsen asserts that if a smooth projective variety has injective cycle class maps, it has surjective cycle class maps. The object of this note is to present a version of Jannsen's theorem for singular quasi-projective varieties.	10.1007/s10711-015-0080-x	[ "https://arxiv.org/pdf/1507.04483v1.pdf" ]	73,548,468	1507.04483	c81669508415a4afc7aaa86cc2ad50c0ff9361e8	Surjectivity of cycle maps for singular varieties Robert Laterveer Surjectivity of cycle maps for singular varieties arXiv:1507.04483v1 [math.AG] 16 Jul 2015 manuscript No. (will be inserted by the editor)Algebraic cycles · Chow groups · Pure motives · Singular varieties Mathematics Subject Classification (2010) 14C15 · 14C25 · 14C30 A theorem of Jannsen asserts that if a smooth projective variety has injective cycle class maps, it has surjective cycle class maps. The object of this note is to present a version of Jannsen's theorem for singular quasi-projective varieties. Introduction Let X be a smooth complex projective variety. The cycle class maps cl i : A i X Q → H 2i (X, Q) from Chow groups to singular cohomology have given rise to some of the most profound and fascinating conjectures in algebraic geometry: the Hodge conjecture (concerning the image of cl i ), and the Bloch-Beilinson conjectures (concerning the structure of the kernel of cl i ). Since Mumford's work [19], it is well-known that if the Chow groups A i X Q are "small" (in the sense of being supported on some subvariety), then also the singular cohomology groups are small (in the sense that they are supported on some subvariety). The following result can be seen as an extreme instance of this general principle: Theorem 1 (Jannsen [12]) Suppose X is a smooth projective variety, such that cl i : A i X Q → H 2i (X, Q) is injective for all i. Then there is an isomorphism This result can be proven using the Bloch-Srinivas method of decomposing the diagonal, and the formalism of correspondences [4], [26]. i≥0 cl i : i≥0 A i X Q ∼ = → j≥0 H j (X, Q) . In this note, we look at the following question: Question 1 If X is only quasi-projective, and/or singular, in what sense is theorem 1 still true ? This question has been treated by Lewis [18, Corollary 0.3], assuming a generalized version of the generalized Hodge conjecture holds. In this note, by contrast, we wanted to see how far we could get unconditionally. Our main result gives a version of theorem 1, provided the singular locus of X is not too large: Theorem 2 Let X be a quasi-projective variety of dimension n, and suppose there exists a compactification of X with singular locus of dimension ≤ n+1 3 . Suppose all cycle class maps are injective. Then the cl i induce an isomorphism i≥0 : i≥0 A i X Q ∼ = → ℓ≥0 W −ℓ H ℓ (X, Q) . That is, W −ℓ H ℓ X = 0 if ℓ is odd; Im cl i if ℓ = 2i. To prove this result, we adapt Jannsen's original method (i.e. the decomposition of the diagonal, plus the formalism of correspondences) to the singular and quasi-projective case. The decomposition of the diagonal goes through unchanged, except that in the quasi-projective case the boundary of a compactification appears in the decomposition (this is lemma 1). As to correspondences: suppose X is projective (not necessarily smooth) of dimension n. Then a correspondence, i.e. a cycle C ∈ A n (X × X) Q , induces an action C * : H i (X, Q) → H 2n−i (X, Q) in a natural way (using the cap-product). Since ∆ * is just the canonical map (capping with the class of X), and since we can control the image of this canonical map in favourable cases (lemma 4), we can conclude by looking at the action of the components occuring in the decomposition of ∆. We raise several questions in the course of this note. Indeed, in several respects even the smooth quasiprojective case is far from being as clear-cut and well-understood as the smooth projective case; a relation with "Voisin's standard conjecture" [25] naturally appears (remark 3). Convention In this note, the word variety refers to a quasi-projective algebraic variety over C. The Bloch-Srinivas argument Definition 1 We will use A i to denote the Chow group of i-dimensional cycles, and A i to denote the Fulton-MacPherson operational Chow cohomology [7]. By construction, A * acts on Chow groups A * ; in particular, for any projective X there are natural maps A i X → Hom A i X, Z , b → deg(b ∩ −) . We recall [21] there are functorial cycle class maps cl i : A i X → Gr W 2i H 2i (X, Q) , where W is Deligne's weight filtration. Definition 2 We will use the notation A i c for "compactly supported operational Chow cohomology". This is defined as follows: for any quasi-projective X, let X ⊂X be a compactification with boundary D. Define A i c X by the exact sequence 0 → A i c X → A iX → A i D . This is independent of choice ofX [8] (actually, A i c X is what is denoted R 0 A i X in [8]) . Moreover A i c is a contravariant functor for arbitrary morphisms [8]. There are natural maps A i c X Q → Hom A i X Q , Q , defined by the above exact sequence. There are also functorial cycle class maps cl i : A i c X → Gr W 2i H 2i c (X, Q) , again defined by the above exact sequence. Definition 3 Let X be a quasi-projective variety. Following Voisin [25], [26], we say that X has trivial Chow groups if the cycle class maps cl i : A i X Q → H 2i (X, Q) are injective for all i. Definition 4 Let X be a quasi-projective variety. We say that Niveau A i X Q ≤ r if there exists a closed (i+r)-dimensional subvariety Y ⊂ X such that A i (X \ Y ) Q = 0. The key to the whole argument is the following decomposition lemma. This is the Bloch-Srinivas argument [4]; in his book, Bloch attributes this argument to Colliot-Thélène [1, appendix to lecture 1]. Lemma 1 LetX be a projective variety of dimension n, and X ⊂X the complement of a closed subvariety D. Suppose Niveau A i X Q ≤ r for all i . Then there is a decomposition of the diagonal ∆ = ∆ 0 + ∆ 1 + · · · + ∆ n + Γ ∈ A n (X ×X) Q , where ∆ j is supported on V j × W j , and V j ⊂X is of dimension j + r, W j ⊂X is of dimension n − j, and Γ is supported onX × D. Proof This is an application of the Bloch-Srinivas method [4]. We use the following two well-known lemmas: Lemma 2 Let X and Z be quasi-projective varieties, and suppose Z is irreducible of dimension n. Then for any i A i (X k(Z) ) ∼ = lim − → A i+n (X × U ) , where the limit is taken over opens U ⊂ Z. Proof This is usually stated for smooth projective varieties [1, appendix to Lecture 1]. If one is brave, one goes checking in Quillen's work to see that the proof given in loc. cit. for the smooth case still goes on for singular varieties. Alternatively, take a resolution of singularities and reduce to the smooth case using the "descent" exact sequences, and the fact that lim − → is an exact functor. Lemma 3 Let X be a quasi-projective variety defined over a field k, and let k ⊂ K be a field extension. Then A i (X k ) Q → A i (X K ) Q is injective. Proof This is usually stated for smooth varieties [1, appendix to Lecture 1], but the same argument works in general: use lemma 2 to reduce to the case of a finite extension. For a finite extension, take a resolution of singularities; for smooth varieties, the existence of the norm implies the extension map is a split injection; by descent, the same is true for singular varieties. Now we proceed with the proof of lemma 1. We can reduce to some subfield k ⊂ C which is finitely generated over its prime subfield. Consider the restriction ∆ ∈ A n (X ×X) Q → A n (X ×X) Q → A 0 (X k(X) ) Q . The last group is supported in dimension r, so we get a rational equivalence ∆ = ∆ 0 + ∆ 1 + Γ 1 ∈ A n (X ×X) Q , where ∆ 0 is supported on V 0 ×X, where V 0 has dimension r, and ∆ 1 is supported onX × W 1 for some divisor W 1 , and Γ 1 is supported on D ×X. Applying the same process to ∆ 1 and continuing inductively, we end up with a decomposition ∆ = ∆ 0 + ∆ 1 + · · · + ∆ n + Γ ′ ∈ A n (X ×X) Q , where the ∆ j are as desired, but Γ ′ is supported on D ×X. Taking the transpose and renumbering, we end up with a decomposition as desired. Remark 1 In case X is smooth projective, lemma 1 was proven in [15], inspired by [4] and [20]. The smooth projective case The following is well-known. Proposition 1 (Jannsen [12], Kimura, Vial) Let X be a smooth projective variety of dimension n. The following are equivalent: (i) The groups A alg i X Q are 0 for i < n 2 ; (ii) X has trivial Chow groups; (iii) Niveau(A i X Q ) ≤ 0 for all i; ( iv) The cycle class maps induce a ring isomorphism A * X Q ∼ = → H * (X, Q) ; (v) For any variety Z, and for any i, the product map induces an isomorphism l+m=i A l X Q ⊗ A m Z Q ∼ = → A i (X × Z) Q ; (vi) For any variety Z, and for any i, j, the product map induces an isomorphism l+m=i A l X Q ⊗ A m (Z, j) Q ∼ = → A i (X × Z, j) Q of higher Chow groups [2]. Proof We will recall the proof, as a warm-up for what follows. To see that (i)⇒(ii), we work inductively. First, the hypothesis A alg 0 X Q = A hom 0 X Q = 0 implies a decomposition of the diagonal ∆ = X × x + Γ 1 ∈ A n (X × X) Q , with Γ 1 supported on D × X, for some divisor D. Considering the action of ∆ on Griff n−1 X Q , we find Griff n−1 X Q = 0 . (Indeed, the action factors over Griff n−1 ( D) Q = 0, for some desingularisation D.) Taken together with the hypothesis A n−1 alg X Q = 0, we find that A hom 1 X Q = 0, and we continue likewise. After n 2 steps, we end up with a decomposition ∆ = ∆ 0 + ∆ 1 + · · · + ∆ ⌊ n 2 ⌋ + Γ ∈ A n (X × X) Q , where ∆ j comes from A j X Q ⊗ A n−j X Q , and Γ is supported on V × X with dim V ≤ n 2 . We can apply this decomposition to A hom i X Q to check that (ii) holds. (Indeed, for i ≥ n 2 , the component Γ does not act on A hom i X Q for dimension reasons; neither do the ∆ j act.) To see that (ii)⇒(iii), remark that the cohomology groups H 2i (X, Q) are finite-dimensional Q-vector spaces. To get the implication (iii)⇒(iv), let the decomposition of the diagonal act on the kernel and cokernel to see that both vanish. Now, let's prove the implication (iv)⇒(v). Let S ⊂ Z denote the singular locus, and let Z → Z denote a resolution of singularities with exceptional divisor E. There is a commutative diagram with exact rows → A i (X × E) Q → A i (X × Z) Q ⊕ A i (X × S) Q → A i (X × Z) Q → 0 ↑ ↑ ↑ → l+m=i A l X Q ⊗ A m E Q → l+m=i A l X Q ⊗ (A m ( Z) ⊕ A m S) Q → l+m=i A l X Q ⊗ A m Z Q → 0 By noetherian induction, we are thus reduced to the case where Z is smooth. Writing out a similar diagram for a compactification, we reduce to the case where Z is smooth and projective. Let's suppose now Z is smooth projective, say of dimension d. We first prove surjectivity of the product map: Take c an element of A i (X × Z) Q . We may suppose everything (X, Z, c and the subvarieties supporting the A i X Q ) is defined over a field k ⊂ C finitely generated over its prime subfield. Consider what happens to c under the restriction c ∈ A i (X × Z) Q → A i−d (X k(Z) ) Q = lim − → A i (X × U ) Q , where the limit is taken over opens U ⊂ Z, and the equality is established in lemma 2. Since k(Z) ⊂ C, lemma 3 implies that A i−d (X k(Z) ) Q → A i−d (X C ) Q is injective, so that Niveau A i−d (X k(Z) ) Q ≤ 0 . It follows that the cycle c can be written c = b × Z + c ′ ∈ A i (X × Z) Q , where b ∈ A i−n X Q and c ′ supported on X × Z ′ , for Z ′ ⊂ Z some divisor. By induction, the statement is true for X × Z ′ , and so we find that c is a sum of product cycles as desired. Next, we prove the product map is injective. So let p : l+m=i A l X Q ⊗ A m Z Q → A i (X × Z) Q denote the product map, and let a be an element in Ker p. We write a = l+m=i a l,m ∈ l+m=i A l X Q ⊗ A m Z Q , and we let L be the maximum l for which a l,m = 0. Hypothesis (iv) implies that A L X Q is finite-dimensional, and that there is a perfect pairing A L X Q × A n−L X Q → Q . Let b 1 , . . . , b r be a basis of A L X Q , and let b ∨ 1 , . . . , b ∨ r denote the dual basis of A n−L X Q . We write a L,m = j c j b j ⊗ d j ∈ A L X Q ⊗ A m Z Q . Since by hypothesis, p(a) = 0, we have p(a L,m ) · (b ∨ j × Z) = p(a) · (b ∨ j × Z) = 0 ∈ A m (X × Z) Q . But p(a L,m ) · (b ∨ j × Z) = c j (point) × d j projects to c j d j ∈ A m (Z) Q under projection to the second factor, so we find c j d j = 0 ∈ A m (Z) Q ∀j . But this means a L,m = 0; contradiction. To get that (v) implies (i): taking Z = X, one obtains a complete decomposition of the diagonal of X; having the diagonal act on A alg i X Q , one obtains the required vanishing. It remains to establish an equivalence with (vi): using a commutative diagram extending the above diagram to the left (this exists, thanks to the notorious moving lemma for higher Chow groups [3], [17]), one is again reduced to the case Z smooth projective. Now we use a result of Kimura [14] and Vial [23, Theorem 5], which states that (ii) is equivalent to the fact that the Chow motive of X is a sum of twisted Lefschetz motives. Hence the motive of X × Z is a sum of twists of the motive of Z; as higher Chow groups only depend on the Chow motive, this implies (vi). Finally, (vi) ⇒ (v) is obvious, and we are done. The smooth quasi-projective case In case X is a smooth quasi-projective variety, the situation is not as well-understood as in the smooth projective case; the equivalences of proposition 1 become difficult open problems. For instance, we raise the following questions: Question 2 Does the implication (i)⇒(ii) of proposition 1 still hold for X smooth quasi-projective ? Here is what we can prove unconditionally: Proposition 2 Let X be a smooth quasi-projective variety with trivial Chow groups. Then the cycle class maps induce isomorphisms i A i X Q ∼ = → ℓ W −ℓ H ℓ (X, Q) . That is, W −ℓ H ℓ (X, Q) = 0 if ℓ is odd; Im cl i if ℓ = 2i. Since "having trivial Chow groups" obviously implies that the niveau of all Chow groups is ≤ 0, proposition 2 follows from the following more general proposition: Proposition 3 Let X be a smooth quasi-projective variety of dimension n. Suppose Niveau(A i X Q ) ≤ 0 for all i . Then W −ℓ H ℓ (X, Q) = 0 if ℓ is odd; Im cl i if ℓ = 2i. Moreover, cl i : A i c X Q → Gr W 2i H 2i c (X, Q) is injective. Proof Let τ : X ⊂X be a smooth compactification, with boundary D. From lemma 1, we obtain a decomposition of the diagonal ofX ∆ = ∆ 0 + ∆ 1 + · · · + ∆ n + Γ ∈ A n (X ×X) Q , where ∆ j is supported on V j × W j , and V j (resp. W j ) is of dimension j (resp. n − j), and Γ is supported on X × D. Let a ∈ Gr k F W −ℓ H ℓ (X, C), with ℓ = −2k. Letā ∈ Gr k F H ℓ (X, C) be an element restricting to a. The action of ∆ j onā is 0 for dimension reasons. (Indeed, let V j and W j denote resolutions of singularities. Then the action of ∆ j factors over H k+n,n−ℓ−k ( V j , C) and H k+n−j,n−ℓ−k−j ( W j , C) , and one of these groups is 0 for dimension reasons.) It follows that a = τ * ā = τ * (∆ 0 ) * (ā) + · · · + (∆ n ) * (ā) = 0 ∈ Gr k F W −ℓ H ℓ (X, C) , so Gr k F W −ℓ H ℓ (X, C) for all ℓ = −2k. In particular, for ℓ odd we find that W −ℓ H ℓ (X, C) = j Gr j F W −ℓ H ℓ (X, C) = 0 . Next, let a ∈ W −2i H 2i (X, Q) ∩ F −i . Using the polarisation on H 2i (X, Q) (cf. [26, ]), one finds there exists a Hodge classā ∈ H 2i (X, Q) which restricts to a (i.e. τ * ā = a). For j = n − i, the action of ∆ j onā is 0 for dimension reasons. ( This is similar to the prior parenthesis: the action of ∆ j factors over H n−i,n−i ( V j , Q) and H n−i−j,n−i−j ( W j , Q) , and one of these groups is 0 for dimension reasons.) For j = n − i, we have that (∆ n−i ) * ā ⊂ Im H 2i (W n−i ) → H 2iX ⊂ Imcl i . It follows that a = τ * ā = τ * (∆ n−i ) * ā + Γ * ā = τ * (∆ n−i ) * ā ⊂ Imcl i . It remains to prove the statement for cl i . Taking the transpose of all elements involved, we may suppose we have a decomposition ∆ = ∆ 0 + ∆ 1 + · · · + ∆ n + Γ ∈ A n (X ×X) Q , where ∆ j are as before, but Γ is now supported on D ×X. Let a ∈ A i c X Q , and letā be the image of a in A iX Q . The restriction ofā to D is 0 (i.e., if ψ : D →X denotes the inclusion, we have ψ * (ā) = 0 ∈ A i (D, Q)), hence Γ * ā = 0. Just as above, the correspondence ∆ j does not act onā except for j = i, henceā = (∆ i ) * ā . Suppose nowā ∈ A i homXQ . The action of ∆ i on A i homXQ factors over A 0 hom ( W i ) Q , which is 0; it follows thatā = 0, whence a = 0. The singular case In this section, we consider quasi-projective (possibly singular) varieties X. We prove our main result as promised in the introduction; this is a version of Jannsen's theorem for varieties whose singular locus is not too large: Theorem 3 Let X be a quasi-projective variety of dimension n, and suppose there is a compactification of X with singular locus of dimension ≤ n+1 3 . Suppose X has trivial Chow groups. Then cycle class maps induce an isomorphism i A i X Q ∼ = → ℓ W −ℓ H ℓ (X, Q) . That is, W −ℓ H ℓ (X, Q) = 0 if ℓ is odd; Im cl i if ℓ = 2i. This follows from the following more precise version: Theorem 4 Let X be a quasi-projective variety of dimension n, and suppose a compactification of X has singular locus of dimension ≤ s. Suppose Niveau(A i X Q ) ≤ 0 for all i . Then W −ℓ H ℓ (X, Q) = 0 if ℓ is odd; Im cl i if ℓ = 2i, provided ℓ ∈ [0, n − s] ∪ [2s, 2n]. Moreover, cl i : A i c X Q → Gr W 2i H 2i c (X, Q) is injective in the range i > s. Proof Let τ : X →X denote the given compactification, with boundary D =X \ X. Applying lemma 1, we find a decomposition of the diagonal ofX of the form ∆ = ∆ 0 + ∆ 1 + · · · + ∆ n ∈ A n (X ×X) Q , where ∆ j is supported on V j × W j , and V j ⊂X has dimension j and W j has dimension n − j. We can view ∆ (and the ∆ j ) as a correspondence ∆ * : H i (X, Q) → W i−2n H 2n−i (X, Q) , where forā ∈ H i (X, Q), we define ∆ * (ā) := (π 2 ) * (π * 1ā ) ∩ ∆ ∈ W i−2n H 2n−i (X, Q) (here π 1 resp. π 2 denotes projection on the first resp. second factor). It is easily checked that ∆ * (ā) =ā ∩ [X] ∈ H 2n−i (X, Q) . (Indeed, let f : X →X be a resolution of singularities, with projections π 1 , π 2 from X × X to the two factors. Let ∆ denote the diagonal of X, so that ∆ = (f × f ) * ∆. Then ∆ * (ā) = (π 2 ) * π * 1ā ∩ ∆ = (π 2 ) * (f × f ) * (f × f ) * π * 1ā ∩ ∆ = f * ( π 2 ) * π * 1 f * ā ∩ ∆ = f * ∆ * (f * ā ) = f * (f * ā ∩ [ X]) =ā ∩ [X].) Case 1: ℓ ≤ n − s. Let a ∈ Gr p F W −ℓ H ℓ (X, C), and letā ∈ Gr p F W −ℓ H ℓ (X, C) be an element restricting to a. According to lemma 4 below, we can find b ∈ Gr W 2n−ℓ H 2n−ℓ (X, C) such that a = b ∩ [X] ∈ W −ℓ H ℓ (X, C) . The "Poincaré duality" map from H 2i to H 2n−i , being a map of Hodge structures, is strictly compatible with the Hodge filtration, so we may suppose b ∈ Gr p+n F . Note that we have a = τ * ā = τ * (∆ 0 ) * b + · · · + (∆ n ) * b ∈ Gr p F W −ℓ H ℓ (X, C) . On the other hand, for dimension reasons we have (∆ j ) * Gr p+n F GrH 2n−ℓ (X, C) = 0 unless 2n − ℓ = 2j = 2(p + n) . It follows that a = 0 if ℓ = −2p; in particular W −ℓ H ℓ (X, Q) = 0 for ℓ odd. Next, let ℓ = 2i and consider a ∈ W −2i H 2i (X, Q). Using lemma 4, we can find again b ∈ H 2n−2i (X, Q) such that a = τ * b ∩ [X] = τ * (∆ 0 ) * b + · · · + (∆ n ) * b ∈ H 2i (X, Q) . But for reasons of dimension, (∆ j ) * b = 0 for j = n − i , and clearly (∆ n−i ) * b ∈ Im cl i . Case 2: ℓ ≥ 2s. Let S ⊂ X denote the singular locus, and U = X \ S the non-singular locus. Then obviously Niveau A i (U ) Q ≤ 0 for all i . This implies (by proposition 3 above) that cl i : A i U Q → W −2i H 2i (U, Q) is surjective for all i, and W −ℓ H ℓ (U, Q) = 0 for ℓ odd. But the map W −ℓ H ℓ (X, Q) → W −ℓ H ℓ (U, Q) is an isomorphism for ℓ > 2s, so W −ℓ H ℓ (X, Q) = 0 if ℓ > 2s is odd . Restriction induces a surjection W −2i H 2i (X, Q) → W −2i H 2i (U, Q) for reasons of weight; this fits into a commutative diagram with exact rows A i S Q → A i X Q → A i U Q → 0 ↓ cl i ↓ cl i ↓ cl i W −2i H 2i (S, Q) → W −2i H 2i (X, Q) → W −2i H 2i (U, Q) → 0 . The right vertical arrow is surjective, as we just noted, and the left vertical arrow is an isomorphism for i ≥ s. The "Moreover" part follows from the commutative diagram A i c U Q → A i c X ↓ ↓ Gr W 2i H 2i c (U, Q) → Gr W 2i H 2i c (X, Q) The horizontal maps are isomorphisms since i > s; the left vertical arrow is injective by proposition 3. Lemma 4 Let X be a projective variety of dimension n, and with singular locus of dimension ≤ s. Then the natural map Gr W j H j (X, Q) → W j−2n H 2n−j (X, Q) is injective for j ≤ n − s, and surjective for j ≥ n + s. Proof Let IH j X denote middle-perversity intersection homology with rational coefficients. It follows from work of Durfee [5] that IH j X = Gr W j H j (X, Q), j ≥ n + s; W j−2n H 2n−j (X, Q), j ≤ n − s . It is well-known [9], [10] that the "Poincaré duality" map factors as Gr W j H j (X, Q) → IH j X → W j−2n H 2n−j (X, Q) . Moreover, it is known by work of Weber [27] (cf. also [11]) that the first arrow is injective, and the second arrow surjective. Remark 4 It seems likely theorem 4 is true without any condition on the singular locus. This is proven by Lewis [18, Corollary (0.2)], under the assumption of (a generalized version of) the generalized Hodge conjecture. cl i : A i X Q → W −2i H 2i (X, Q) , cl i : A i X Q → Gr W 2i H 2i (X, Q) are isomorphisms for all i. (The first isomorphism is [21, Theorem 3]; the second isomorphism is obtained by combining the first isomorphism with [21, Theorem 2].) The argument in the proof of theorem 4 suggests the following question: Question 4 Let X be any quasi-projective variety with Niveau(A i X Q ) ≤ 0 for all i. Is it true that the natural map Im A * X Q → A * X Q → Im H * (X, Q) → H * (X, Q) is an isomorphism ? A partial answer is given by the following result: for X projective, the right-hand side is generated by algebraic cycles. Proposition 4 Let X be a projective variety of dimension n, and suppose Niveau(A i X Q ) ≤ 0 ∀i. Then Im H * (X, Q) → H * (X, Q) is generated by algebraic cycles. That is, Im H ℓ (X, Q) → H 2n−ℓ (X, Q) = 0 if ℓ is odd ; ⊂ Im cl n−i if ℓ = 2i . Proof First, let's suppose ℓ is odd. Then the vanishing of Im H ℓ (X, Q) → H 2n−ℓ (X, Q) follows from the following lemma: Lemma 5 Set-up as in the proposition. Then Gr p−n F Im H ℓ (X, C) → H 2n−ℓ (X, C) = 0 for ℓ = 2p . Proof (of the lemma) By strict compatibility of the Hodge filtration, the group in the statement of the lemma is the same as Im Gr p F H ℓ (X, C) → Gr p−n F H 2n−ℓ (X, C) . This is the same as ∆ * Gr p F H ℓ (X, C) , where correspondences act as defined in the proof of proposition 4. Now we apply the decomposition of ∆ given by lemma 1. The action of the component ∆ j factors as follows: · · · ↑ ↓ Gr p F H ℓ ( V j , C) Gr p−n F H 2n−ℓ ( W j , C) ↑ ↓ Gr p F H ℓ (X, C) (∆ j ) * → Gr p−n F H 2n−ℓ (X, C) Since dim V j = j, the upper left group vanishes for p > j; likewise, since dim W j = n − j, the upper right group vanishes for p < j. It follows that the only non-trivial action is for p = j. But the group Gr j F H ℓ ( V j , C) vanishes unless ℓ = 2j. It remains to treat the case ℓ = 2i. Let a ∈ H 2i (X, Q). From the proof of the above lemma, we find that a ∩ [X] = ∆ * a = (∆ i ) * a ∈ H 2n−2i (X, Q) (indeed, for j = i, the action (∆ j ) * H 2i (X, C) = 0 ∈ H 2n−2i (X, C) , since it is 0 on each Gr p F . But H 2i (X, Q) (∆ j ) * → H 2n−2i (X, Q) ↓ ↓ H 2i (X, C) (∆ j ) * → H 2n−2i (X, C) commutes, so that also (∆ j ) * H 2i (X, Q) = 0 for j = i.) But (∆ i ) * H 2i X ⊂ Im H 2n−2i (W i , Q)) → H 2n−2i (X, Q) ⊂ Im cl n−i . Remark 6 A result analogous to Jannsen's theorem is proven by Esnault-Levine [6]. They prove that if X is smooth projective such that all cycle class maps into Deligne cohomology are injective, these cycle class maps are surjective (this is reproven, and rendered more precise, in [23, Theorem 4].) Lewis extends this to singular and quasi-projective varieties, again assuming (a generalized version of) the generalized Hodge conjecture [18, Corollary (0.3)]. It would be interesting to try whether the approach of the present note can be applied to this problem; I haven't looked into this. Remark 7 The argument of the proof of theorem 4 can also be used to obtain a new version of Mumford's theorem for singular varieties, plus a verification of the Hodge conjecture for certain singular varieties [16]. ( In particular, H p,q (X, C) = 0 for all p = q.) Remark 2 2Jannsen proved that properties (ii), (iii) and (iv) in proposition 1 are equivalent[12]. The fact that it suffices to consider algebraically trivial cycles (i.e. point (i) in proposition 1) is a particular instance of a more general phenomenon, discovered by Vial: if a morphism of Chow motives f :N → M , with N finite-dimensional, induces a surjection A alg * (N ) Q → A alg * (M ) Q , then also A hom * (N ) Q → A hom * (M ) Q is surjective [24,Theorem 7]. The above manifestation is just the case where N is a Lefschetz motive.Property (v) is studied in depth in[22], where it is called the "Chow-Künneth property". Notably, [22, Theorem 4.1] generalizes the result of Kimura and Vial evoked in the above proof. Question 3 3Does the implication (iii)⇒(ii) of proposition 1 still hold for X smooth quasi-projective ? Remark 3 For both questions, the answer is positive provided the "Voisin standard conjecture" [25, Conjecture 0.6],[26] is true. Moreover, a positive answer to either question would imply the following result: if X is smooth quasi-projective with trivial Chow groups, then any open U ⊂ X has trivial Chow groups. As shown in[25], this result (i.e. that "having trivial Chow groups" transfers from a variety to its open subsets) would be a consequence of the truth of the Voisin standard conjecture. Remark 5 Linear varieties (in the sense of [21]) form a subclass of the class of varieties with trivial Chow groups. For a projective (possibly singular) linear variety X, Totaro has shown [21] that Acknowledgements This note was stimulated by the Strasbourg "groupe de travail" based on the monograph[26]. It is a pleasure to thank the participants of this groupe de travail for a very pleasant atmosphere and stimulating interactions. Algebraic cycles and higher K-theory. S Bloch, Lectures on algebraic cycles. 2. S. BlochDuke Univ. Press61Advances in MathS. Bloch, Lectures on algebraic cycles, Duke Univ. Press 1980, 2. S. Bloch, Algebraic cycles and higher K-theory, Advances in Math. vol. 61 (1986), 267-304, The moving lemma for higher Chow groups. S Bloch, J. Alg. Geom. 3S. Bloch, The moving lemma for higher Chow groups, J. Alg. Geom. 3 (1994), 537-568, S Bloch, V Srinivas, Remarks on correspondences and algebraic cycles. 105S. Bloch and V. Srinivas, Remarks on correspondences and algebraic cycles, American Journal of Mathematics Vol. 105, No 5 (1983), 1235-1253, Intersection homology Betti numbers. 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[ "Distributed Space Time Coding for Wireless Two-way Relaying", "Distributed Space Time Coding for Wireless Two-way Relaying" ]	[ "Vijayvaradharaj T Muralidharan \nDept. of Electrical Communication Engineering\nIndian Institute of Science\nBangalore-560012India\n", "Senior Member, IEEEB Sundar Rajan \nDept. of Electrical Communication Engineering\nIndian Institute of Science\nBangalore-560012India\n" ]	[ "Dept. of Electrical Communication Engineering\nIndian Institute of Science\nBangalore-560012India", "Dept. of Electrical Communication Engineering\nIndian Institute of Science\nBangalore-560012India" ]	[]	We consider the wireless two-way relay channel, in which two-way data transfer takes place between the end nodes with the help of a relay. For the Denoise-And-Forward (DNF) protocol, it was shown by Koike-Akino et. al. that adaptively changing the network coding map used at the relay greatly reduces the impact of Multiple Access interference at the relay. The harmful effect of the deep channel fade conditions can be effectively mitigated by proper choice of these network coding maps at the relay. Alternatively, in this paper we propose a Distributed Space Time Coding (DSTC) scheme, which effectively removes most of the deep fade channel conditions at the transmitting nodes itself without any CSIT and without any need to adaptively change the network coding map used at the relay. It is shown that the deep fades occur when the channel fade coefficient vector falls in a finite number of vector subspaces of C 2 , which are referred to as the singular fade subspaces. DSTC design criterion referred to as the singularity minimization criterion under which the number of such vector subspaces are minimized is obtained. Also, a criterion to maximize the coding gain of the DSTC is obtained. Explicit low decoding complexity DSTC designs which satisfy the singularity minimization criterion and maximize the coding gain for QAM and PSK signal sets are provided. Simulation results show that at high Signal to Noise Ratio, the DSTC scheme provides large gains when compared to the conventional Exclusive OR network code and performs slightly better than the adaptive network coding scheme proposed by Koike-Akino et. al.I. BACKGROUND AND PRELIMINARIESA. BackgroundWe consider the two-way wireless relaying scenario shown inFig.1. Two-way data transfer takes place between the nodes A and B with the help of the relay R. It is assumed that all the three nodes operate in The authors are with the DRAFT 2 half-duplex mode, i.e., they cannot transmit and receive simultaneously in the same frequency band. The idea of physical layer network coding for the two way relay channel was first introduced in [1], where the multiple access interference occurring at the relay was exploited so that the communication between the end nodes can be done using a two phase protocol. A protocol called Denoise-And-Forward (DNF) was proposed in[2], which consists of the following two phases: the multiple access (MA) phase (Fig. 1(a)), during which A and B simultaneously transmit to R and the broadcast (BC) phase (Fig. 1(b)) during which R transmits to A and B. Network coding map, which is also referred to as the denoising map, is chosen at R in such a way that A (B) can decode the messages of B (A), given that A (B) knows its own messages. During the MA phase, the transmissions from the end nodes were allowed to interfere at R, but the harmful effect of this interference was mitigated by a proper choice of the network coding map used at R. Information theoretic studies for the physical layer network coding scenario were reported in [3], [4]. A differential modulation scheme with analog network coding for bi-directional relaying was proposed in [5]. The design principles governing the choice of modulation schemes to be used at the nodes for uncoded transmission were studied in [6]. An extension for the case when the nodes use convolutional codes was done in [7]. A multi-level coding scheme for the two-way relaying scenario was proposed in [8]. Power allocation strategies and lattice based coding schemes for bi-directional relaying were proposed in [9]. It was observed in [6] that the network coding map used at the relay needs to be changed adaptively according to the channel fade coefficients, in order to minimize the impact of the Multiple Access Interference (MAI). A computer search algorithm called the Closest-Neighbour Clustering (CNC) algorithm was proposed in [6] to obtain the adaptive network coding maps resulting in the best distance profile at R. An adaptive network coding scheme for MIMO two-way relaying based on the CNC algorithm was proposed in [10]. An alternative procedure to obtain the adaptive network coding maps, based on the removal of deep channel fade conditions using Latin Squares was proposed in [11]. A quantization of the set of all possible channel realizations based on the network code used was obtained analytically in [12]. An extension of the adaptive network coding scheme for MIMO two-way relaying using Latin Rectangles was made in [13]. As an alternative to the adaptive network coding schemes in [6] and [11]-[12], in this paper, we propose a Distributed Space Time Coding (DSTC) scheme, which mitigates the effect of MAI to the fullest extent possible at the transmitting nodes itself without any CSIT. For the proposed DSTC scheme the network coding map used at R need not be changed adaptively according to channel conditions which reduces the complexity at R to a great extent and also eliminates the need for overhead bits from R to May 5, 2014 DRAFT 3 A and B to indicate the choice of the network coding map. A distributed space time coding scheme for a wireless two-way relay network with multiple relay nodes was proposed in [14], in which the DSTC was constructed at the relay nodes. In the proposed scheme, the DSTC is constructed at the end nodes A and B. B. Signal Model Throughout, a quasi-static fading scenario is assumed with the Channel State Information (CSI) available only at the receivers. Let h A and h B denote the fade coefficients associated with A-R and B-R links and h ′ A and h ′ B denote the fade coefficients associated with R-A and R-B links. All the fading coefficients are assumed to follow Rician distribution. Let S denote the unit energy M = 2 λ point constellation used at the end nodes. Let µ : F λ 2 → S denote the mapping from bits to complex symbols used at A and B. 1) Denoise-And-Forward (DNF) protocol: In the sequel, we briefly describe the adaptive network coding schemes based on the DNF protocol proposed in [6], [11] -[12]. Throughout the paper, by DNF protocol, we refer to the schemes proposed in [6] and [11]-[12].In the DNF protocol, transmission occurs in two phases: Multiple Access (MA) phase during which A and B simultaneously transmit to R and Broadcast (BC) phase during which R transmits to A and B.MA Phase: Let x A = µ(s A ), x B = µ(s B ) ∈ S denote the complex symbols transmitted by A and B respectively, where s A , s B ∈ F λ 2 . The received signal at R is given by,The additive noise z R is assumed to be CN (0, σ 2 ), where CN (0, σ 2 ) denotes the circularly symmetric complex Gaussian random variable with mean zero and variance σ 2 .BC Phase: Let (x A ,x B ) ∈ S 2 denote the Maximum Likelihood (ML) estimate of (x A , x B ) at R based on the received complex number y R . Depending on the value of h A and h B , R chooses a many-to-one map M hA,hB : S 2 → S ′ , where S ′ is the signal set (of size between M and M 2 ) used by R during the BC phase.In order to ensure that A (B) is able to decode B's (A's) message, the map M hA,hB should satisfy the exclusive law [6], i.e.,	10.1109/tsp.2012.2231677	[ "https://arxiv.org/pdf/1203.3128v3.pdf" ]	2,329,083	1203.3128	84c74c2212ffa09ef16e69b4d3cc69213c9211ee	Distributed Space Time Coding for Wireless Two-way Relaying May 2012 May 5, 2014 Vijayvaradharaj T Muralidharan Dept. of Electrical Communication Engineering Indian Institute of Science Bangalore-560012India Senior Member, IEEEB Sundar Rajan Dept. of Electrical Communication Engineering Indian Institute of Science Bangalore-560012India Distributed Space Time Coding for Wireless Two-way Relaying May 2012 May 5, 20141 DRAFT We consider the wireless two-way relay channel, in which two-way data transfer takes place between the end nodes with the help of a relay. For the Denoise-And-Forward (DNF) protocol, it was shown by Koike-Akino et. al. that adaptively changing the network coding map used at the relay greatly reduces the impact of Multiple Access interference at the relay. The harmful effect of the deep channel fade conditions can be effectively mitigated by proper choice of these network coding maps at the relay. Alternatively, in this paper we propose a Distributed Space Time Coding (DSTC) scheme, which effectively removes most of the deep fade channel conditions at the transmitting nodes itself without any CSIT and without any need to adaptively change the network coding map used at the relay. It is shown that the deep fades occur when the channel fade coefficient vector falls in a finite number of vector subspaces of C 2 , which are referred to as the singular fade subspaces. DSTC design criterion referred to as the singularity minimization criterion under which the number of such vector subspaces are minimized is obtained. Also, a criterion to maximize the coding gain of the DSTC is obtained. Explicit low decoding complexity DSTC designs which satisfy the singularity minimization criterion and maximize the coding gain for QAM and PSK signal sets are provided. Simulation results show that at high Signal to Noise Ratio, the DSTC scheme provides large gains when compared to the conventional Exclusive OR network code and performs slightly better than the adaptive network coding scheme proposed by Koike-Akino et. al.I. BACKGROUND AND PRELIMINARIESA. BackgroundWe consider the two-way wireless relaying scenario shown inFig.1. Two-way data transfer takes place between the nodes A and B with the help of the relay R. It is assumed that all the three nodes operate in The authors are with the DRAFT 2 half-duplex mode, i.e., they cannot transmit and receive simultaneously in the same frequency band. The idea of physical layer network coding for the two way relay channel was first introduced in [1], where the multiple access interference occurring at the relay was exploited so that the communication between the end nodes can be done using a two phase protocol. A protocol called Denoise-And-Forward (DNF) was proposed in[2], which consists of the following two phases: the multiple access (MA) phase (Fig. 1(a)), during which A and B simultaneously transmit to R and the broadcast (BC) phase (Fig. 1(b)) during which R transmits to A and B. Network coding map, which is also referred to as the denoising map, is chosen at R in such a way that A (B) can decode the messages of B (A), given that A (B) knows its own messages. During the MA phase, the transmissions from the end nodes were allowed to interfere at R, but the harmful effect of this interference was mitigated by a proper choice of the network coding map used at R. Information theoretic studies for the physical layer network coding scenario were reported in [3], [4]. A differential modulation scheme with analog network coding for bi-directional relaying was proposed in [5]. The design principles governing the choice of modulation schemes to be used at the nodes for uncoded transmission were studied in [6]. An extension for the case when the nodes use convolutional codes was done in [7]. A multi-level coding scheme for the two-way relaying scenario was proposed in [8]. Power allocation strategies and lattice based coding schemes for bi-directional relaying were proposed in [9]. It was observed in [6] that the network coding map used at the relay needs to be changed adaptively according to the channel fade coefficients, in order to minimize the impact of the Multiple Access Interference (MAI). A computer search algorithm called the Closest-Neighbour Clustering (CNC) algorithm was proposed in [6] to obtain the adaptive network coding maps resulting in the best distance profile at R. An adaptive network coding scheme for MIMO two-way relaying based on the CNC algorithm was proposed in [10]. An alternative procedure to obtain the adaptive network coding maps, based on the removal of deep channel fade conditions using Latin Squares was proposed in [11]. A quantization of the set of all possible channel realizations based on the network code used was obtained analytically in [12]. An extension of the adaptive network coding scheme for MIMO two-way relaying using Latin Rectangles was made in [13]. As an alternative to the adaptive network coding schemes in [6] and [11]-[12], in this paper, we propose a Distributed Space Time Coding (DSTC) scheme, which mitigates the effect of MAI to the fullest extent possible at the transmitting nodes itself without any CSIT. For the proposed DSTC scheme the network coding map used at R need not be changed adaptively according to channel conditions which reduces the complexity at R to a great extent and also eliminates the need for overhead bits from R to May 5, 2014 DRAFT 3 A and B to indicate the choice of the network coding map. A distributed space time coding scheme for a wireless two-way relay network with multiple relay nodes was proposed in [14], in which the DSTC was constructed at the relay nodes. In the proposed scheme, the DSTC is constructed at the end nodes A and B. B. Signal Model Throughout, a quasi-static fading scenario is assumed with the Channel State Information (CSI) available only at the receivers. Let h A and h B denote the fade coefficients associated with A-R and B-R links and h ′ A and h ′ B denote the fade coefficients associated with R-A and R-B links. All the fading coefficients are assumed to follow Rician distribution. Let S denote the unit energy M = 2 λ point constellation used at the end nodes. Let µ : F λ 2 → S denote the mapping from bits to complex symbols used at A and B. 1) Denoise-And-Forward (DNF) protocol: In the sequel, we briefly describe the adaptive network coding schemes based on the DNF protocol proposed in [6], [11] -[12]. Throughout the paper, by DNF protocol, we refer to the schemes proposed in [6] and [11]-[12].In the DNF protocol, transmission occurs in two phases: Multiple Access (MA) phase during which A and B simultaneously transmit to R and Broadcast (BC) phase during which R transmits to A and B.MA Phase: Let x A = µ(s A ), x B = µ(s B ) ∈ S denote the complex symbols transmitted by A and B respectively, where s A , s B ∈ F λ 2 . The received signal at R is given by,The additive noise z R is assumed to be CN (0, σ 2 ), where CN (0, σ 2 ) denotes the circularly symmetric complex Gaussian random variable with mean zero and variance σ 2 .BC Phase: Let (x A ,x B ) ∈ S 2 denote the Maximum Likelihood (ML) estimate of (x A , x B ) at R based on the received complex number y R . Depending on the value of h A and h B , R chooses a many-to-one map M hA,hB : S 2 → S ′ , where S ′ is the signal set (of size between M and M 2 ) used by R during the BC phase.In order to ensure that A (B) is able to decode B's (A's) message, the map M hA,hB should satisfy the exclusive law [6], i.e., The CNC algorithm proposed in [6] obtains the map M hA,hB which results in the best distance profile during the MA phase at R. The CNC algorithm is run for all possible channel realizations and a partition of the set of all channel realizations is obtained depending on the chosen network coding map. For a given channel realization, the choice of the network coding map is indicated to A and B using overhead bits. During the BC phase R transmits x R = M hA,hB (x A ,x B ) ∈ S ′ . The received signals at A and B during the BC phase are respectively given by, y A = h ′ A x R + z A , y B = h ′ B x R + z B , where z A and z B are independent and CN (0, σ 2 ). Since the map M hA,hB satisfies the exclusive law and A (B) knows its own message x A (x B ), it can decode x B (x A ) by decoding x R . The CNC algorithm optimizes the entire distance profile instead of maximizing only the minimum distance. In some cases, this results in the use of signal sets with a larger cardinality during the BC phase. To solve this problem, an algorithm called the Nearest Neighbour Clustering (NNC) algorithm was proposed in [6] which maximizes the minimum distance alone, instead of optimizing the entire distance profile. The choice of the network coding map obtained depends only on the ratio hB hA and not the individual values of h A and h B [6]. In [11], the values of hB hA for which deep channel conditions occur were identified and network coding maps which remove the harmful effect of these deep channel conditions were obtained by the completion of partially filled Latin Squares. 2) The Proposed DSTC Scheme: For the proposed DSTC scheme, transmission occurs in four phases: Two MA phases during which A and B simultaneously transmit to R followed by two BC phases during which R transmits to A and B. Two independent complex symbols each from A to B and B to A get exchanged at the end of the four phases and hence the information rate in bits per channel use for the proposed scheme is same as that of the DNF protocol. MA Phases: Let x A1 = µ(s A1 ), x A2 = µ(s A2 ) ∈ S denote two independent complex symbols A wants to communicate to B. Similarly, B wants to communicate two independent complex symbols x B1 = µ(s B1 ), x B2 = µ(s B2 ) ∈ S to A. During the i th MA phase i ∈ {1, 2}, A transmits f i A (x A1 , x A2 ) ∈ C, a function of x A1 and x A2 , and similarly B transmits f i B (x B1 , x B2 ) ∈ C, a function of x B1 and x B2 . The received signal at R during the two MA phases can be written as, y R = [y R1 y R2 ] = [h A h B ]   f 1 A (x A1 , x A2 ) f 2 A (x A1 , x A2 ) f 1 B (x B1 , x B2 ) f 2 B (x B1 , x B2 )   + z R1 z R2 , where y Ri denotes the received signal at R during the i th MA phase, z R1 and z R2 are independent and CN (0, σ 2 ). Let x A = [x A1 x A2 ] and x B = [x B1 x B2 ]. The matrix, C(x A , x B ) =   f 1 A (x A1 , x A2 ) f 2 A (x A1 , x A2 ) f 1 B (x B1 , x B2 ) f 2 B (x B1 , x B2 )  (1) represents a DSTC codeword matrix. Note that in the DSTC codeword matrix, x A1 and x A2 can occur only in the first row and, x B1 and x B2 can occur only in the second row. In this way the DSTC differs from space time codes for the conventional 2×1 multiple antenna system with two collocated antennas at the transmitter in which the complex symbols can occupy any entry in the codeword matrix. For a complex number x, let x R and x I denote the real and imaginary parts of x. Definition 1: A DSTC is said to be linear if the entries of the first row of the codeword matrices are complex linear combinations of x R A1 , x I A1 , x R A2 , x I A2 and the entries of the second row are complex linear combinations of x R B1 , x I B1 , x R B2 , x I B2 . Any codeword matrix C(x A , x B ) of a linear DSTC can be written as, C(x A , x B ) = i=1,2 W R Ai x R Ai + W I Ai x I Ai + W R Bi x R Bi + W I Bi x I Bi .(2) The matrices W R Ai , W I Ai , W R Bi and W I Bi are referred to as the weight matrices of the DSTC. Note that the entries of the second (first) row are zeros in the matrices W R Ai and W I Ai (W R Bi and W I Bi ). Definition 2: A linear DSTC is said to be over the signal set S if the entries of the first (second) row of the codeword matrices are complex linear combinations of x A1 and x A2 (x B1 and x B2 ), where x A1 , x A2 , x B1 and x B2 belong to the signal set S. For a linear DSTC over S, codeword matrix C(x A , x B ) is of the form C(x A , x B ) =   x A M A x B M B   , where M A and M B are 2 × 2 complex matrices referred to as the generator matrices at node A and B respectively. Throughout the paper, we consider only linear DSTCs over a signal set S. BC Phases: Let (ŝ A1 ,ŝ A2 ,ŝ B1 ,ŝ B2 ) denote the maximum likelihood estimate of (s A1 , s A2 , s B1 , s B2 ) at R. The relay R transmits x R1 = µ(ŝ A1 ⊕ŝ B1 ) and x R2 = µ(ŝ A2 ⊕ŝ B2 ) during the first and second BC phases respectively, where ⊕ denotes the bit-wise XOR operation. The received signals at the end nodes during the two BC phases are given by, y Ai = h ′ A x Ri + z Ai and y Bi = h ′ B x Ri + z Bi , where i ∈ {1, 2}.s Bi (s Ai ) i ∈ {1, 2}, by decoding x Ri . Note that for the proposed DSTC scheme the signal set used during the BC phase is of the minimum cardinality 2 λ (the cardinality of the signal set should be at least 2 λ to convey λ information bits). In contrast, for the scheme proposed in [6], depending on channel conditions unconventional signal sets with cardinality greater than the minimum cardinality are required. Minimum cardinality signal set is used during the BC phase and throughout the paper the focus is on optimizing the performance during the MA phase. Some of the advantages of the proposed DSTC scheme over the schemes proposed in [6], [11]- [12] are summarized below: • Unlike the schemes proposed in [6], [11]- [12], for the proposed DSTC scheme, the network coding map used at R need not be changed adaptively according to channel conditions. Any network coding map satisfying the exclusive law will give the same performance and for simplicity, the conventional bit-wise Exclusive OR (XOR) map itself can be used. This reduces the complexity at R to a great extent and also eliminates the need for overhead bits from R to A and B to indicate the choice of the network coding map. • For the scheme proposed in [6], for certain channel conditions the adaptive network coding map necessitates the use of unconventional signal sets with cardinality greater than the minimum cardinality required during the BC phase, which results in a degradation in performance. For the proposed scheme, the relay always uses a conventional signal set with minimum cardinality. • The adaptive network coding maps were obtained in [6], by exhaustive computer search. For the proposed scheme no such computer search is required, since the same network code is used irrespective of channel conditions. The contributions and organization of the paper are as follows: • For a classical n t × n r MIMO system with collocated antennas, deep channel fade conditions occur when the channel fade coefficient vector belongs to a finite number of vector subspaces of C nt referred to as the singular fade subspaces. The way in which transmit diversity schemes (space time codes) remove the harmful effect of these singular fade subspaces is discussed. The connection between the dimension of these singular fade subspaces and the transmit diversity order is explained (Section II). • The MAC phase of the DNF protocol for the two-way relaying scenario can be viewed as a virtual May 5, 2014 DRAFT 2 × 1 MISO system. The singular fade subspaces for the classical 2 × 1 MISO system, are singular fade subspaces for the two-way relaying scenario as well. The connection between dimension of these singular fade subspaces and the diversity order for the adaptive network coding schemes proposed in [6] and [11]- [12] is discussed (Section III A). • The singular fade subspaces for the proposed DSTC scheme are identified. • Simulation results presented in Section V show that at high SNR, the DSTC scheme provides large gains when compared to the conventional XOR network code based on the DNF protocol and performs slightly better than the adaptive network coding scheme proposed in [6]. II. THE NOTION OF SINGULAR FADE SUBSPACES FOR THE COLLOCATED MIMO SYSTEM In this section, to explain the notion of singular fade subspaces, we digress from the two-way relaying scenario and focus on the classical MIMO system with collocated antennas. Consider the classical MIMO system with n t transmit antennas at the transmitter Tx and n r receive antennas at the receiver Rx, with H being the n r × n t complex fade coefficient matrix. The entries of the matrix H are assumed to be i.i.d. and Rician distributed. A. Singular Fade Subspaces for the Collocated MIMO system with Spatial Multiplexing Consider the spatial multiplexing of independent complex symbols at Tx, i.e., the received complex vector at Rx is given by y = Hx + z, where x is the transmitted message vector of length n t whose components independently take values from the signal set S and z is CN (0, σ 2 I nt ). Let h k , 1 ≤ k ≤ n r , denote the k th row of H. Since H∆x 2 = nr k=1 \|h k ∆x\| 2 , for the minimum distance of the effective constellation S Rx (H) to be zero, all the vectors h T k , 1 ≤ k ≤ n r , should fall in a vector subspace of the form ∆x ⊥ for some ∆x ∈ ∆S nt . In other words, for H∆x to be zero, the row space of H should be a subspace of the vector subspace of C nt of the form ∆x ⊥ for some ∆x ∈ ∆S nt . The vector subspaces of the form ∆x ⊥ are referred to as the singular fade subspaces. are given by,      0 1   ,   1 0   ,   1 1   ,   1 −1   ,   1 j   ,   1 −j   ,   1 1 + j   ,   1 −1 + j   ,   1 1 − j   ,   1 −1 − j   ,   1 0.5 + 0.5j   ,   1 −0.5 + 0.5j   ,   1 0.5 − 0.5j   ,   1 −0.5 − 0.5j      .(3) The fade coefficient matrix (which is a row vector for this example) is The dimension of the singular fade subspace ∆x ⊥ , and the transmit diversity order of the pair- wise error event (x → x ′ ), are inherently connected, where ∆x = x − x ′ and x, x ′ ∈ S nt . With spatial multiplexing, the transmit diversity order of the pair-wise error event ( x → x ′ ) is 1 while dim( ∆x ⊥ ) = n t − 1. It is the presence of these n t − 1 dimensional singular fade subspaces that results in a transmit diversity order of 1. The receive diversity order n r comes due to the fact that for a fade coefficient matrix to be a deep fade matrix, all the n r rows of the fade coefficient matrix should belong to the same singular fade subspace. The use of full diversity space times space time codes results in the maximum transmit diversity order n t . In the next subsection, the connection between the singular fade subspaces of space time codes and May 5, 2014 DRAFT transmit diversity order will be established. B. Singular Fade Subspaces for the Collocated MIMO system with Space Time Coding Consider the case when Tx uses a space time code C of size n t × T, where T ≥ n t . Let C(x) denote a codeword matrix of the space time code, where x ∈ S K , where K denotes the number of independent complex symbols transmitted. Similar to the spatial multiplexing case, the effective constellation at Rx which is a subset of C nr×T can be defined. It is easy to verify that the minimum distance of the effective constellation at Rx becomes zero when Row(H) is a subspace of the vector subspace Col ⊥ (C (∆x)) , for some ∆x ∈ ∆S K . Note that Col ⊥ (C (∆x)) denotes the vector subspace {u : u T v = 0, ∀v ∈ Col (C (∆x))}. The vector subspaces Col ⊥ (C (∆x)) are the singular fade subspaces for the n t transmit antenna system with the space time code C. The dimension of the singular fade subspace Col ⊥ (C (∆x)) is equal to n t − rank(C (∆x)), while the transmit diversity order for the pair-wise error event (x → x ′ ), x, x ′ ∈ S K equals rank(C (∆x)) [15], where ∆x = x − x ′ . With every pair-wise error event (x → x ′ ), we can associate a singular fade subspace Col ⊥ (C (∆x)) . Among all the pair-wise error events, those error events for which the codeword difference matrix has the least rank determine the overall system transmit diversity order. Equivalently, among all the pair-wise error events, those error events for which the associated singular fade subspace has the largest dimension will dominate the overall error probability. This is expected since among all the singular fade subspaces, the probability that Row(H) falls in the neighbourhood of a subspace of the singular fade subspace, will be the largest for those singular fade subspaces which have the largest dimension. If the space time code is such that C (∆x) is full rank for all ∆x = 0 K , all the singular fade subspaces Col ⊥ (C (∆x)) collapse to be the zero-dimensional trivial singular fade subspace   0 0   , thereby ensuring a transmit diversity order of n t for all the pair-wise error events. Example 3: Consider the 2 × 1 MISO system with Alamouti space time code whose design matrix is given by   x1 x2 −x * 2 x * 1   . Since the design matrix is full rank for all choices of x 1 and x 2 , the column space of the codeword difference matrix is always C 2 and hence all the singular fade subspaces collapse to be the zero dimensional trivial singular fade subspace   0 0   . Equivalently, all the pair-wise error May 5, 2014 DRAFT events (x 1 , x 2 ) → (x ′ 1 , x ′ 2 ) have a transmit diversity order 2. The full rank Alamouti space-time code removed the effect of all the vector subspaces which were non-trivial singular fade subspaces for the spatial multiplexing system, thereby increasing the diversity order of all the pair-wise error events from 1 to 2. Example 4: For a 2 a × 2 a Generalized Linear Complex Orthogonal Design (GCOD) [16], the design matrix G 2 a (x 1 , x 2 , . . . x a+1 ) constructed iteratively is given by,   G 2 a−1 (x 1 , x 2 , . . . x a ) x a+1 I 2 a−1 −x * a+1 I 2 a−1 G H 2 a−1 (x 1 , x 2 , . . . x a )   . The codeword difference matrix for the GCOD is full rank for any signal set. Hence, irrespective of the signal set, the trivial singular fade subspace 0 2 a is the only singular fade subspace for the GCOD. Example 5: Consider the 4×4 Quasi-Orthogonal Design (QOD) [17], whose codeword matrix is given by         x 1 −x * 2 −x * 3 x 4 x 2 x * 1 −x * 4 −x 3 x 3 −x * 4 x * 1 −x 2 x 4 x * 3 x * 2 x 1         . Let ∆x i = x i − x ′ i . Irrespective  x R 1 + jx I 2 0 0 x R 2 + jx I 1   , where x 1 , x 2 ∈ {±1, ±j}. Let ∆x i = x i − x ′ i . The codeword difference matrix is not full rank in the following two cases: Case 1: ∆x R 1 = ∆x I 2 = 0 and at least one out of ∆x I 1 and ∆x R 2 is non-zero. For this case, the singular May 5, 2014 DRAFT fade subspace is given by   1 0   . Case 2: ∆x I 1 = ∆x R 2 = 0 and at least one out of ∆x R 1 and ∆x I 2 is non-zero. For this case, the singular fade subspace is given by   0 1   . Hence, there exists the following two non-trivial singular fade subspaces:   0 1   and   1 0   . However, when the signal set is e jθ {±1, ±j}, where θ is not a multiple of π 4 , the 2 × 2 CIOD offers full transmit diversity. Equivalently, there are no singular fade subspaces other than the trivial singular fade subspace for the 2 × 2 CIOD with the signal set e jθ {±1, ±j}, when θ is not a multiple of π 4 . Example 7: Consider the 4 × 4 CIOD [18] whose codeword matrices are of the form The determinant of the codeword difference matrix for this STC is given by,         x R 1 + jx I 3 x R 2 + jx I 4 0 0 −x R 2 + jx I 4 x R 1 − jx I 3 0 0 0 0 x R 3 + jx I 1 x R 2 + jx I 4 0 0 −x R 4 + jx I 2 x R 3 − jx I 1         , where x 1 , x 2 , x 3 , x 4 ∈ {±1,\|∆x R 1 \| 2 + \|∆x I 3 \| 2 + \|∆x R 2 \| 2 + \|∆x I 4 \| 2 \|∆x R 3 \| 2 + \|∆x I 1 \| 2 + \|∆x R 4 \| 2 + \|∆x I 2 \| 2 . Hence the code-word difference matrix is not full rank in the following two cases: In general, the 2 × 2 CIOD given in Example 6 and the 4 × 4 CIOD given in Example 7, offer full diversity for those signal sets for which the Co-ordinate Product Distance (CPD) 1 is non-zero [18]. Case 1: ∆x R 1 = ∆x I 3 = ∆x R 2 = ∆x I 4 = 0 and at least one out of ∆x I 1 , ∆x R 3 , ∆x I 2 , ∆x R 4 is non-zero. Equivalently, there are non-trivial singular fade subspaces for the 2 × 2 and 4 × 4 CIOD, for signal sets whose CPD is non-zero. In fact, this is true for any Generalized Co-ordinate Interleaved Orthogonal Design (GCIOD), as illustrated in the next example. Example 8: Consider the 2 a × 2 a Generalized Co-ordinate Interleaved Orthogonal Design (GCIOD) [18] whose codeword design matrix is given by,   G 2 a−1 (x1, . . . ,xa) 0 0 G 2 a−1 (xa+1, . . . ,x2a)   . The complex numberx i = x R i + jx I (i+a)2a , where (r) s denotes r modulo s and G 2 a−1 (x 1 , . . . , x a ) is the codeword matrix of the GCOD [16] of size 2 a−1 . The determinant of the codeword difference matrix is given by a i=1 (\|∆x R i \| 2 + \|∆x I (a+i) 2a \| 2 a i=1 (\|∆x I i \| 2 + \|∆x R (a+i) 2a \| 2 . The determinant is non-zero for those signal sets for which the CPD is non-zero and there are no non-trivial singular fade subspaces. For those signal sets for which the CPD is zero, the determinant becomes zero under the following two cases: Case 1: ∆x I i = ∆x R( III. SINGULAR FADE SUBSPACES FOR THE TWO-WAY RELAYING SCENARIO In the previous subsection, the notion of singular fade subspaces was introduced and its connection to the transmit diversity order of the MIMO system with collocated antennas was established. Since the MA phase of the two-way relaying scenario can be viewed as a virtual 2 × 1 MISO system, there exists singular fade subspaces for this case as well. In Subsection III-A, the singular fade subspaces for the two-way relaying scenario are identified. The reason why the adaptive network coding schemes based on the DNF protocol proposed in [6] and [11]- [12] mitigate the effect of these harmful singular fade subspaces is discussed. In Subsection III-B, it is shown that minimizing the harmful effect of these singular fade subspaces can also be achieved by a proper choice of the DSTC, without any need to adaptively change the network code at R according to channel conditions. A. Singular Fade Subspaces for the DNF Protocol Let ∆x A = x A − x ′ A and ∆x B = x B − x ′ B ∈ ∆S. From the discussion in Section II, it follows that the singular fade subspaces for the DNF protocol are of the form   ∆xA ∆xB   ⊥ =   1 −∆x A ∆x B   . The ratio −∆xA ∆xB determines all the singular fade subspaces for the DNF protocol. In [11]- [12], the ratio −∆xA ∆xB was called the singular fade state. As mentioned earlier in Section II, dim     1 −∆x A ∆x B     and the diversity order for the pair-wise error event that a pair (x A , x B ) is wrongly decoded at R as (x ′ A , x ′ B ) (denoted as (x A , x B ) → (x ′ A , x ′ B ) ) are inherently connected. The diversity order for the error event ( To sum up, in the DNF protocol, the transmissions from the nodes A and B are allowed to interfere at R and the effect of MAI is effectively mitigated by adaptively changing the network coding map, thereby removing the harmful effect of all the removable singular fade subspaces. x A , x B ) → (x ′ A , x ′ B ) is equal to rank([∆x A ∆x B ]) = 1 while dim     1 −∆x A ∆x B     = 2 − rank([∆xA ∆xB]) = 1. Let S R (h A , h B ) = {h AxA + h BxB :x A ,x B ∈ S} B. Singular Fade Subspaces for the DSTC Scheme Let ∆x A = x A −x ′ A and ∆x B = x B −x ′ B ∈ ∆S 2 . Then C(∆x A , ∆x B ) = C(x A , x B )−C(x ′ A , x ′ B ) denotes a codeword difference matrix of the DSTC, where C(x A , x B ) is the codeword matrix of the DSTC defined in (1). From the discussion in Section II, it follows that the singular fade spaces for the proposed DSTC scheme are of the form Col ⊥ (C (∆x A , ∆x B )) . Consider the singular fade subspaces of the form Col ⊥ (C (0 2 , ∆x B )) and Col ⊥ (C (∆x A , 0 2 )) , where ∆x A , ∆x B = 0 2 . The first row of the matrix C (0 2 , ∆x B ) has both the entries to be zero. Hence, Col (C (0 2 , ∆x B )) = Hence for a DSTC which is singularity minimal, the only error events which result in diversity order 1 (x A , x B ) → C(x ′ A , x ′ B ), x A = x ′ A , x B = x ′ B ,are of the form C(x A , x B ) → C(x A , x ′ B ), x ′ B = x B and C(x A , x B ) → C(x ′ A , x B ), x ′ A = x A . Hence, the overall coding gain is equal to minimum among all the non-zero singular values of the codeword difference matrices which are of the form C(0 2 , ∆x B ) and C(∆x A , 0 2 ) [15]. Note that the matrices C. A Construction of Singularity Minimal DSTCs for Algebraic Signal Sets A signal set is said to be algebraic if all the signal points of the signal set are algebraic numbers over . Proposition 1: The class of DSTCs whose codeword design matrices are of the form given above are singularity minimal for all algebraic signal sets. Proof: The proof is as follows: For ∆x A = 0 2 and ∆x B = 0 2 , at least one of the two components of ∆x A as well as ∆x B should be non-zero. Hence, (∆x A1 +e j ∆x A2 ) = 0 and (∆x B1 +e j ∆x B2 ) = 0, since e j is transcendental 3 whereas ∆x A1 , ∆x A2 , ∆x B1 and ∆x B2 are algebraic over Q. The codeword difference matrix C(∆x A , ∆x B ) is full rank for all ∆x A = 0 and ∆x B = 0, since det(C(∆x A , ∆x B )) = (ad − bc)(∆x A1 + e j ∆x A2 )(∆x B1 + e j ∆x B2 ) = 0. The DSTC codeword matrix for this case is given by, C(x A , x B ) = 1 √ 2 (xA 1 + e j xA 2 ) (xA 1 + e j xA 2 ) −(xB 1 + e j xB 2 ) (xB 1 + e j xB 2 ) . The scaling factor of 1 √ 2 is to ensure unit average energy per symbol per time slot. It can be verified that the coding gain for this DSTC is approximately 0.6877, same as that of the DSTC given in Example 10. The coding gain of the DSTCs given in Examples 10 and 11 is approximately 0.6877, which is less than the minimum distance of the unit energy 4-PSK signal set, which is Lemma 1: For singularity minimal DSTC over S, where S is a square QAM or 2 λ -PSK signal set, the coding gain is upper bounded by the minimum distance of the signal set S. Proof: See Appendix A. In the following subsection, the condition under which the upper-bound given in the previous lemma is satisfied with equality is identified and explicit construction of DSTCs are provided. A. Constructions of Singularity Minimal, Coding Gain Maximal DSTCs over QAM and PSK signal sets The following proposition states that for DSTCs over S, choosing M A and M B to be unitary matrices ensures that the upper-bound on the coding gain is satisfied with equality, for QAM and PSK signal sets. The coding gain of the DSTC is the minimum among all the non-zero singular values of the codeword difference matrices which are of the form C(0 2 , ∆x B ) and C(∆x A , 0 2 ), i.e., the coding gain is equal to min      min ∆x A ∈∆S 2 , ∆x A =0 2 ∆xAMA , min ∆x B ∈∆S 2 , ∆x B =0 2 ∆xBMB      , which is equal to d min (S). In the following examples, constructions of singularity minimal DSTCs whose generator matrices are unitary are provided. where φ = 1+ √ 5 2 ,φ = 1− √ 5 2 , α = 1 + j − jφ andᾱ = 1 + j − jφ. The DSTC codeword matrix is of the form C(x A , x B ) = xAMA xBMB . The codeword difference matrix C(∆x A , ∆x B ) is full rank for all ∆x A = 0 and ∆x B = 0, when the signal points belong to Z[j] [20]. Hence the DSTC is singularity minimal for all signal sets whose signal points belong to Z[j]. Also, since M A and M B are unitary, for square QAM signal set, the DSTC maximizes the coding gain. Note 2: The DSTC given in Construction 1 was constructed in [20] towards satisfying the design criterion formulated in [21] for the two-user non-cooperative Multiple Access Channel (MAC). In [20], the DSTC given in the above example was shown to be DMT optimal for two-user MAC. For a complex number a, let Q(a) denote the smallest field containing Q and a. It is shown in Lemma 2 below that choosing θ = π 4 ensures singularity minimality for signal sets (for example QAM) whose signal points belong to Q(j) and choosing θ = π 2 λ ensures singularity minimality for signal sets (for example 2 λ -PSK) whose signal points belong to Q(e j 2π 2 λ ). Also, since M A and M B are unitary, this DSTC maximizes the coding gain, for square QAM and 2 λ -PSK signal sets. The advantage of this construction over Construction 1 is that encoding at node A is simple, since it does not involve any linear combination of x A1 and x A2 . Lemma 2: For the DSTC given in construction 2, choosing θ = π 4 ensures singularity minimality for signal sets whose points belong to Q(j) and choosing θ = π 2 λ ensures singularity minimality for signal sets whose signal points belong to Q(e j 2π 2 λ ). Proof: The proof is given for the case when the signal points belong to Q(j). The proof for the case when the signal points belong to Q(e j 2π 2 λ ) is exactly similar and is omitted. Let ∆x Ai = x Ai − x ′ Ai and ∆x Bi = x Bi − x ′ Bi , where x Ai , x ′ Ai , x Bi , x ′ Bi ∈ S ⊂ Q(j) . and i ∈ {1, 2}. To prove singularity minimality, it needs to be shown that when at least one out of ∆x A1 and ∆x A2 (∆x B1 and ∆x B2 ) is non-zero, the codeword difference matrix is full rank. The ratios ∆xB 1 ∆xB 2 and − ∆xB 2 ∆xB 1 belong to Q(j) while tan φ g = √ 5 does not belong to Q(j). Hence, ∆x B1 cos φ g + ∆x B2 sin φ g = 0 and −∆x B1 sin φ g + ∆x B2 cos φ g = 0. Since sin φ g = √ 5 √ 6 and cos φ g = 1 √ 6 , ∆x A2 (∆x B1 cos φ g + ∆x B2 sin φ g ) and ∆x A1 (−∆x B1 sin φ g + ∆x B2 cos φ g ) belong to Q(j, √ 5, √ 6), where Q(j, √ 5,√ 6) denotes the smallest filed containing Q, j, √ 5 and √ 6. The determinant of the codeword difference matrix is given by, ∆x A1 e j π 4 (−∆x B1 sin φ g + ∆x B2 cos φ g ) − ∆x A2 (∆x B1 cos φ g + ∆x B2 sin φ g ). The determinant is non-zero since the ratio ∆xA 2 (∆xB 1 cos φg+∆xB 2 sin φg) ∆xA 1 (−∆xB 1 sin φg+∆xB 2 cos φg) belongs to Q(j, √ 5, √ 6), while e j π 4 does not belong to Q(j, The following proposition states that when conditional ML decoding [22], [23] is employed, the decoding complexity of the DSTCs constructed in the previous section for which the generator matrices √ 5,√ 6) V. SIMULATION RESULTS All the simulation results presented are for the case when the end nodes use 4-PSK signal set. By 'DSTC 1' and 'DSTC 2' we refer to the DSTCs given in Construction 1 and Construction 2 respectively. As a reference scheme, we consider the scheme in which XOR network code is used irrespective of channel conditions and no DSTC is employed, which is referred as 'XOR N/W code'. Assuming unit noise variances at all the nodes, the average energies of the transmissions at the nodes, which are assumed to be equal, is defined to be the Signal to Noise Ratio (SNR). The proposed DSTC scheme is also compared with the adaptive network coding schemes proposed in [6] and [11]- [12]. Since for 4-PSK signal set, the adaptive network coding scheme based on the Nearest Neighbour Clustering (NNC) algorithm proposed in [6] and the scheme based on Latin Squares proposed in [11]- [12] turn out to be the same, without distinguishing them we refer to both as 'adaptive N/W code'. Fig. 2 shows the SNR vs BER performance for different schemes for the case when all the fading coefficients are i.i.d. and Rayleigh distributed. In Fig. 3 and Fig. 4 similar plots are shown for a Rician fading scenario with Rician factors 5 of 0 dB and 5 dB respectively. From Fig. 2 To prove the proposition, we adopt a procedure similar to the one used in [23]. Letỹ R = [y R R1 y I R1 y R R2 y I R2 ] T ,x = [x R A1 x I A1 x R A2 x I A2 x R B1 x I B1 x R B2 x I B2 ] T andz R = [z R R1 z I R1 z R R2 z I R2 ] T . The vectorỹ R can be written asỹ R = H eqx +z R , where H eq is a 4 × 8 real matrix whose entries are functions of h A and h B , determined by the DSTC. Using QR decomposition, the matrix H eq can be decomposed as H eq = QR, where Q ∈ R 4×4 is a orthogonal matrix and R ∈ R 4×8 can be written as [R 1 R 2 ], with R 1 , R 2 ∈ R 4×4 , R 1 being an upper-triangular matrix. The joint ML decoding metric at R is given by 7 . Hence the matrix R is of the form given below. 6 Two matrices M1 and M2 are said to be Hurwitz-Radon orthogonal if M1M H 2 + M2M H 1 = 0. 7 Theorem 2 in [23] proves only the 'if' part. However, following an approach similar to the proof given in [23], it is easy to show that the weight matrices of the DSTC corresponding to the symbolsxi andxj need to be Hurwitz-Radon orthogonal, for the (i, j) th entry of R (i ≤ j) to be zero for all realizations of hA and hB, and hence the 'only if' part also holds. Note that * denotes possible non-zero entries. The claim is that all the entries denoted by * are nonzeros. It is clear that all the diagonal entries are non-zeros. For the (1, 5) th entry in (5) to be a zero, ỹ R − H eqx = Q Tỹ R − Rx = y ′ R − Rx , where y ′ R = Q Tỹ R .W R A1 W R B1 H + W R B1 W R A1 H = 0 uA 1 u H B 1 uB 1 u H A 1 0 = 0, which implies that u A1 and u B1 are orthogonal vectors. Then the vector u B1 should belong to the one-dimensional subspace which is orthogonal to u A1 . Since u A2 also belongs to this one-dimensional subspace and both u B1 as well as u A2 are of unit norm, u B1 = e jθ u A2 , for some angle θ. In that case, the DSTC codeword difference matrix is of the form ∆xA 1 uA 1 + ∆xA 2 uA 2 ∆xB 1 e jθ uA 2 + ∆xB 2 uB 2 , which is not full rank when ∆x A2 , ∆x B1 = 0, ∆x A1 = ∆x B2 = 0 and hence the singularity minimization criterion is violated. Hence, (1, 5) th entry shown by * in (5) is non-zero. By a similar argument, it can be shown that the other non-diagonal entries denoted by * in (5) are non-zeros. From the matrix R given in (5), it can be seen that conditioning on the variables x B1 and x B2 , the symbols x A1 and x A2 can be decoded independently [23]. Since the total number of choices for x B1 and x B2 is M 2 and independently decoding x A1 and x A2 requires 2M computations, the decoding involves For square QAM signal sets, the decoding complexity can be further reduced, since the real and imaginary parts independently take values. From (5), it can be seen that conditioning on x B1 and x B2 , the real and imaginary parts of x A1 as well as x A2 can be decoded independently. Since decoding the real and imaginary points of a signal point in QAM signal set is of constant complexity independent of M (decoding can be done by rounding off to the nearest integer [23]), the ML decoding complexity is O(M 2 ) for square QAM signal sets. This completes the proof. Notations: The complex number √ −1 is denoted by j. The set of integers, Gaussian integers, rational, real and complex numbers are respectively denoted as Z, Z[j], Q, R and C. All the vector spaces and vector subspaces considered in this paper are over the complex field C, unless explicitly mentioned otherwise. Throughout, vectors are denoted by bold lower case letters and matrices are denoted by bold capital letters. Let CN (0, σ 2 I n ) denote the circularly symmetric complex Gaussian random vector with zero mean and covariance matrix σ 2 I n , where I n denotes the n × n identity matrix. Let c 1 , c 2 , . . . c L denote the vector subspace over C spanned by the complex vectors c 1 , c 2 , . . . c L . For a matrix A, A T and A H denotes its transpose and conjugate transpose respectively. For a vector subspace V of a vector space, V ⊥ denotes the vector subspace {x : x T v = 0, ∀v ∈ V } and dim(V ) denotes the dimension of V. The all zero vector of length n is denoted by 0 n . For a square matrix A, let rank(A) denote its rank and let det(A) denote its determinant. For a complex number x, x R and x I denote the real and imaginary May 5, 2014 DRAFT parts of x, x * denotes its conjugate and \|x\| denotes its absolute value. For a vector v, v denotes its Euclidean norm. For a matrix A, Row(A) and Col(A) respectively denote the row space and column space of A. E(X) denotes the expectation of X. Let S Rx (H) ⊂ C nr denote the effective signal set at Rx, i.e., S Rx (H) = {Hx : x ∈ S nt }. Let ∆S denote the difference constellation of the signal set S, i.e., ∆S = {s − s ′ : s, s ′ ∈ S}. The distances between two points in the effective constellation S Rx (H) are of the form H∆x , where ∆x = 0 nt , ∆x ∈ ∆S nt . Definition 3: For an n t × n R MIMO system, the channel fade coefficient matrix H is said to be a deep fade matrix if the minimum distance of the effective constellation S Rx (H) is zero. The row space of a deep fade matrix is said to be a deep fade space. FormallyExample 1 : 1, a singular fade subspace can be defined as follows: Definition 4: A vector subspace V of C nt is said to be a singular fade subspace if all the vector subspaces of V are deep fade spaces. always a singular fade subspace referred to as the trivial singular fade subspace. Consider the 2 × 1 MISO system with spatial multiplexing with 4-PSK signal set S = {±1, ±j}. The difference constellation of 4 PSK signal set has 9 points ∆S = {0, ±2, ±2j, ±1 ± j}. For this case, the set of fourteen singular fade subspaces, which are of the form ∆x ⊥ , where ∆x ∈ ∆S 2 a deep fade matrix (vector) if the row space of the fade coefficient vector is a subspace of one of these 14 singular fade subspaces, i.e., the fade coefficient vector should belong to one of these 14 vector subspaces. For example, is a deep fade matrix. Note that the singular fade subspaces depend only on the number of transmit antennas n t and the signal set S. They are independent of the number of receive antennas n r , as illustrated in the following example. Example 2: Consider the 2 × 2 MIMO system with 4-PSK signal set S = {±1, ±j}. The set of 14 singular fade subspaces for this case is the same as that of 2 × 1 MISO system given in (3). For a fade coefficient matrix to be a deep fade matrix, both the rows should belong to one of these Note 1 : 1Even though the probability that Row(H) is a subspace of one of the singular fade subspaces is zero, with a non-zero probability Row(H) falls in the neighbourhood of a subspace of one of the singular fade subspaces, which results in low values of the minimum distance of the effective constellation. . of the signal set used, the minimum rank of the codeword difference matrix for the 4 × 4 QOD is 2. For example, when ∆x 1 = ∆x 4 = ∆s 1 and ∆x 2 = −∆x 3 = ∆s 2 , the rank of the codeword difference matrix is 2. Equivalently, there exists a non-trivial singular fade subspace, ∆s1 ∆s2 −∆s2 ∆s1 Note that the 2 ×2 Alamouti code removes the effect of the harmful non-trivial singular fade subspaces for any signal set. On the other hand, for the 4 × 4 QOD there exists non-trivial singular fade subspaces for any signal set. In general, a space time code can offer full transmit diversity for some but not all signal sets. In other words, for some signal set, a space time code might have only the trivial singular fade subspace, while for some other signal set, the same space time code might have non-trivial singular fade subspaces. For a space time code which does not offer full transmit diversity for a signal set, there would exist non-trivial singular fade subspaces. These are illustrated in the following example. Example 6 : 6Consider the 2 × 2 Co-ordinate Interleaved Orthogonal Design (CIOD)[18] whose codeword matrices are of the form ±j}. For the 4-PSK signal set considered, this STC does not offer full transmit diversity and there are pair-wise error events which have a transmit diversity order less than 2. .. For this case, the first two columns of the codeword difference matrices are zeros. The column span of the codeword difference matrix is 0Case 2: ∆x I 1 = ∆x R 3 = ∆x I 2 = ∆x R 4 = 0 and at least one out of ∆x I 1 , ∆x Q 3 , ∆x I 2 , ∆x Q 4 is nonzero.Similar to Case 1, it can be shown that the singular fade subspace for this case is given by Hence, for the 4 × 4 CIOD, with 4-PSK signal set x 1 , x 2 , x 3 , x 4 ∈ {±1, ±j}, there exists two nontrivial singular singular fade subspaces. Similar to the 2 × 2 CIOD, when the signal set is a rotated 4-PSK signal set, e jθ {±1, ±j}, where θ is not a multiple of π 4 , the 4 × 4 CIOD offers full transmit diversity and there are no non-trivial singular fade subspaces. a+i)2a = 0, ∀i ∈ {1, . . . , a} and at least one of the elements of the set {∆x R i , ∆x I (a+i) 2a , 1 ≤ i ≤ a} is non-zero. It can be verified that the singular fade subspace for this case is given by e 1 , e 2 , . . . , e 2 a−1 , where e i denotes the 2 a length vector whose i th component is one and all other components are zeros. Case2: ∆x R i = ∆x I (a+i)2a = 0, ∀i ∈ {1, . . . , a} and at least one of the elements of the set {∆x I i , ∆x R (a+i) 2a ,1 ≤ i ≤ a} is non-zero. For this case, the singular fade subspace is given by e 2 a−1 +1 , . . . , e 2 a . [h A h B ] (not necessarily in the neighbourhood of singular fade subspaces), the CNC algorithm chooses the network coding map which results in the best distance profile at R by appropriate clustering of the signal points. The scheme proposed in [11]-[12] avoids distance shortening in the neighbourhood of singular fade subspaces by proper choice of clustering for only the singular fade subspaces and not for every realization of the channel fade coefficients. Consider the two singular fade subspaces: The distance shortening which occurs in the neighbourhood of these singular fade subspaces is unavoidable, since the pairs (x A , x B ) and (x A , x ′ B ) (and also the pairs (x A , x B ) and (x ′ A , x B )) which result in these singular fade subspaces cannot be clustered together without violating the exclusive law. Such singular fade subspaces are referred as the non-removable singular fade subspaces. The dimension of these singular May 5, 2014 DRAFT fade subspaces is one or equivalently, the error events (x A , x B ) → (x A , x ′ B ) and (x A , x B ) → (x ′ A , x B ) always result in diversity order one. The singular fade subspaces other than the non-removable singular fade subspaces are referred as the removable singular fade subspaces.The removable singular fade subspaces are of the form the signal set S used. The non-removable singular fade subspaces are independent of the signal set used. Owing to the presence of non-removable singular fade subspaces, the overall diversity order of the DNF protocol cannot exceed one.From the discussion above, it is clear that there are two classes of singular fade subspaces: removable and non-removable. The non-removable singular fade spaces are created by the channel and is independent of the signal set used. Whatever may be the choice of the network code, the harmful effects of these non-removable singular fade subspaces cannot be mitigated. The harmful effect of the removable singular fade subspaces, which are created by the signal set, can be removed by a proper choice of the adaptive network coding map at R, as in[6] and[11]-[12]. . singular fade subspace Col ⊥ (C (0 2 , ∆x B )By a similar reasoning, Col ⊥ (C (∆x A , 0 2If the DSTC codeword matrices are such that rank(C(∆x A , ∆x B )) = 2, ∀∆x A = 0 2 and ∆x B = 0 2 , all the singular fade subspaces Col ⊥ (C (∆x A , ∆x B )) collapse to be the trivial singular fade subspace 0 2 . Equivalently, all the pair-wise error events C have diversity order 2. Hence, for a properly chosen DSTC, other than the trivial singular fade subspace, the singular fade subspaces are only the two non-removable singular fade subspaces, while for the DNF protocol, in addition, we had the removable singular fade subspaces. In this way, by a proper choice of DSTC, the occurrence of the removable singular fade subspaces is avoided at the transmitting nodes itself, without any CSIT.Hence, we have the following design criterion referred as the singularity minimization criterion for DSTCs for two-way relaying: The DSTC codeword difference matrices C(∆x A , ∆x B ) need to be full rank for all ∆x A = 0 2 and ∆x B = 0 2 , to minimize the number of singular fade subspaces. DSTCs satisfying the above criterion are referred as the singularity minimal DSTCs. C( 0 2Example 9 : 09, ∆x B ) and C(∆x A , 0 2 ) are of rank 1 and have only one non-zero singular value. We have the following coding gain criteria for singularity minimal DSTCs: the minimum among all the non-zero singular values of the codeword difference matrices which are of the form C(0 2 , ∆x B ) and C(∆x A , 0 2 ) needs to be maximized. Consider This DSTC is nothing but the scheme where A and B transmit in separate time slots, making sure that their transmissions do not interfere at the relay. Even though this DSTC avoids all the removable singular fade subspaces, the end-to-end rate in complex symbols per channel use is less than that of the DNF protocol. Q 2 . 2All the commonly used signal sets like QAM and PSK are algebraic signal sets. In this subsection, a class of DSTCs which are singularity minimal for algebraic signal sets is provided. rank complex matrix. Consider the class of DSTCs whose codeword matrices are of the formC(x A , x B ) = a(xA 1 + e j xA 2 ) b(xA 1 + e j xA 2 )c(xB 1 + e j xB 2 ) d(xB 1 + e j xB 2 ) Example 10 :Example 11 : 1011Consider Let 4-PSK be the signal set used at A and B. The DSTC codeword matrix for this case is given by, C(xA, xB) 1 + e j xB 2 )   . A and B are made to transmit in two different time slots which results in low decoding complexity at R, since A's and B's transmissions can be decoded independently. It can be verified that the coding gain for this DSTC is approximately 0.6877. Consider Let 4-PSK be the signal set used at A and B. √ 2 . 2In the next section, it is shown that for DSTCs over square QAM and 2 λ -PSK signal sets, the coding gain is upper bounded by the minimum distance of the signal set and explicit DSTC constructions which achieve this bound with equality are provided.IV. SINGULARITY MINIMAL, CODING GAIN MAXIMAL DSTCS OVER QAM AND PSK SIGNAL SETS In this section, it is shown that the coding gain of the DSTCs over square QAM and 2 λ -PSK signal sets are upper-bounded by the minimum distance of the signal set. In Subsection IV-A, a condition under which a singularity minimal DSTC over square QAM and 2 λ -PSK signal set meets the upper bound with equality is obtained and explicit constructions of DSTCs are provided. In Subsection IV-B, the constructed DSTC's are shown to be fast ML decodable, i.e., the ML decoding complexity of the constructed DSTCs is shown to be less than the brute-force decoding complexity which is O(M 4 ). Note that the generator matrices M A and M B at A and B should be such that the average energy per time slot is unity, i.e., E( x A M A 2 ) ≤ 2 and E( x B M B 2 ) ≤ 2. Proposition 2 : 2For singularity minimal DSTCs over square QAM or 2 λ -PSK signal sets, the coding gain is maximized when the generator matrices M A and M B at A and B are unitary matrices. Proof: When M A and M B are unitary matrices, ∆x A M A = ∆x A and also ∆x B M B = ∆x B . Hence, min ∆xA∈∆S 2 , ∆xA =02 ∆x A M A = min ∆xA 1 ∈∆S,∆x A 1 =0 \|∆x A1 \| = d min (S) and similarly min ∆xB∈∆S 2 , ∆xB =02 ∆x B M B = d min (S), where d min (S) denotes the minimum distance of S. Construction 1 : 1Consider the DSTC over S for which MA = Construction 2 : 2Consider the DSTC for which MA = I2 and M B =  cos φ g − sin φ g e jθ sin φ g cos φ g e jθ   , where φ g = tan −1 √ 5. The DSTC codeword matrix C(x A , x B )is given by, cos φg + xB 2 sin φg e jθ (−xB 1 sin φg + xB 2 cos φg)   . M A and M B are unitary is O(M 3 ) for any arbitrary signal set and is O(M 2 ) for square QAM signal set. Note that the brute force decoding complexity is O(M 4 ).Proposition 3: When the generator matrices of the singularity minimal DSTC over S are unitary, the decoding complexity using conditional ML decoding is O(M 3 ) when the signal set S is arbitrary and is O(M 2 ) when the signal set S is square QAM. Proof: See Appendix B.Compared with the DNF protocol, the decoding complexity is more for singularity minimal coding gain maximal DSTCs over S. For the DNF protocol, the decoding complexity is O(M 2 ) for non square QAM signal sets while it is O(M ) for square QAM signal set 4 . As indicated by the simulation results in the next section, the proposed DSTC offers slightly better performance than the adaptive network coding scheme and eliminates the need for adaptive switching of network coding maps at R. But this comes at the cost of increased decoding complexity at R. a singularity minimal DSTC over S, let the generator matrices be M A = U A and M B = U B , where U A and U B are unitary matrices. Let u Ai and u Bi denote the i th rows of U A and U B respectively. Then the weight matrices of the DSTC defined in (2) are given by, W R Ai = jW 2M 3 3computations and hence the decoding complexity at R is O(M 3 ). Fig. 1. Wireless two-way relaying The goal of minimizing the number of singular fade subspaces results in a new design criterion referred as the singularity minimization criterion for DSTCs. It is shown that for a properly chosen DSTC, most of the vector subspaces which were singular fade subspaces for the DNF protocol, are no longer singular fade subspaces for the DSTC scheme. Also, a criterion to maximize the coding gain of the proposed DSTC scheme is obtained (Section III B). It is shown that for DSTCs which are over S, where S is a square QAM or 2 λ -PSK signal set, the coding gain is maximized when the generator matrices M A and M B at nodes A and B are unitary matrices. Explicit construction of DSTCs over QAM and PSK signal sets which satisfy the singularity minimization criterion and maximize the coding gain are provided. It is shown that for all DSTCs over S with unitary generator matrices M A and M B , the ML decoding complexity at• R is O(M 3 ) for any arbitrary signal set and is O(M 2 ) for square QAM signal sets. Note that the brute force ML decoding complexity is O(M 4 ) (Section IV). denote the effective constellation at R. Letd min (h A , h B ) denote the minimum distance of S R (h A , h B ). When [h A h B ] Tfalls in one of the singular fade subspaces, d min (h A , h B ) becomes zero. Even though the probability that the vector [h A h B ] T belongs to a singular fade subspace is zero, d min (h A , h B ) is greatly reduced when [h A h B ] T falls close to a singular fade subspace, a phenomenon referred as distance shortening. For ∆x A = 0 and ∆x B = 0, the CNC algorithm [6] avoids the distance shortening occurring in the neighbourhood of a singular fadesubspace   ∆xA ∆xB   ⊥ , by ensuring that µ hA,hB (x A , x B ) = µ hA,hB (x ′ A , x ′ B ), i.e., R does not distinguish the pairs (x A , x B ) and (x ′ A , x ′ B ), which are said to be clustered together. In fact, for every realization of . B . BDecoding Complexity of Singularity Minimal, Maximal Coding Gain DSTCs over SAfter the two MA phases, R jointly decodes for the two message vectors x A and x B of A and B respectively. In general, the complexity of this joint ML decoding at R is O(M 4 ), where M is the cardinality of the signal set S. The choice of the generator matrices M A and M B being unitary not only maximizes the coding gain for QAM and PSK signal sets, but also results in a reduced decoding complexity at R. -4, it can be seen that the diversity order is one for all the schemes. Also, it can be seen that at high SNR, both 'DSTC 1' as well as 'DSTC 2' offer nearly the same performance and they perform better than the 'XOR N/W code' as well as the 'adaptive N/W code'. For a Rayleigh fading scenario, at high SNR, the DSTCs offer a gain of 2 dB over 'XOR N/W code' while the 'adaptiveN/W code' offers a gain of about 0.5 dB over the 'XOR N/W code'. For a Rician factor of 0 dB, at high SNR, the DSTCs offer a gain of 2 dB over 'XOR N/W code' while the 'adaptive N/W code' offers a gain of about 1.2 dB over the 'XOR N/W code'. For a Rician factor of 5 dB, at high SNR, the DSTCs offer a gain of 5.5 dB over 'XOR N/W code' while the 'adaptive N/W code' offers a gain of about 4 dB over the 'XOR N/W code'. The reason why the DSTC based scheme performs better than the adaptive N/W coding scheme is as follows: during the BC phase always a 4 point signal set is used for the DSTC based scheme, while depending on channel conditions 4 point or 5 point signal set is used for the adaptive network coding scheme [6], [11]. VI. DISCUSSION A DSTC scheme was proposed for the two-way relaying scenario. It was shown that deep channel fades occur when the channel fade coefficient vector falls in a finite number of vector subspaces called the singular fade subspaces. The connection between the dimension of these vector subspaces and the transmit diversity order was established. Design criterion to minimize the number of singular fade subspaces for the DSTC scheme and maximize the coding gain were obtained. Explicit low decoding complexity constructions of DSTCs were provided. The problem of constructing singularity minimal DSTCs with decoding complexity same as that of the DNF protocol, without sacrificing the coding gain, remains open. Extending the DSTC scheme for two-way relaying with multiple antennas and multi-way relaying are possible directions for future work. This work was supported partly by the DRDO-IISc program on Advanced Research in Mathematical Engineering through a research grant as well as the INAE Chair Professorship grant to B. S. Rajan. Rician factor is the power ratio between the line of sight and scattered components.∆x A M A ≤ d min (S). Similarly, it can be shown that min ∆xA∈∆S 2 , ∆xB =02 ∆x B M B is also upper-bounded by d min (S). Hence, the coding gain of the DSTC over square QAM or 2 λ -PSK signal set is upper-bounded by d min (S). This completes the proof.ACKNOWLEDGEMENT 5 Hence, min ∆xB∈∆S 2 , ∆xB =02 APPENDIX B PROOF OF PROPOSITION 3 A1 H = O 2 , where O 2 denotes the 2 × 2 null matrix. Also, W R A1 W R A2 H = O 2 , since u A1 and u A2 are orthogonal vectors. Hence, A1 H = O 2 . Similarly, using the fact that U A and U B are unitary matrices, it can be shown that the following pairs of matrices are also Hurwitz-Radon orthogonal 6 : {W R Letx i denote the i th component of the vectorx. The i th and j th columns of H eq are orthogonal and hence the (i, j) th entry of R (i ≤ j) is zero for all realizations of h A and h B , if and only if the weight matrices of the DSTC corresponding to the symbolsx i andx j are Hurwitz-Radon orthogonal (follows from Theorem 2, [23]). Hence, W R A1 W I A1 H + W I A1 W R W R A1 W R A2 H + W R A2 W R A1 , W I A2 }, {W I A1 , W R A2 }, {W I A1 , W I A2 }, {W R A2 , W I A2 }, {W R B1 , W I B1 }, {W R B1 , W R B2 }, {W R B1 , W I B2 }, {W I B1 , W R B2 }, {W I B1 , W I B2 }, {W R B2 , W I B2 }. Fig. 2. SNR vs BER for different schemes for 4-PSK signal set for a Rayleigh fading scenario.Fig. 3. SNR vs BER for different schemes for 4-PSK signal set for a Rician fading scenario with a Rician factor 0 dB. Fig. 4. SNR vs BER for different schemes for 4-PSK signal set for a Rician fading scenario with a Rician factor 5 dB.SNR in dB BER XOR N/W Code Adaptive N/W Code DSTC 1 DSTC 2 May 5, 2014 DRAFT 5 10 15 20 25 30 35 40 10 −3 10 −2 10 −1 SNR in dB BER XOR N/W Code Adaptive N/W Code DSTC 1 DSTC 2 5 10 15 20 25 30 35 10 −3 10 −2 10 −1 SNR in dB BER XOR N/W Code Adaptive N/W Code DSTC 1 DSTC 2 The CPD between two complex numbers x and y is defined to be \|x R − y R \|\|x I − y I \|. The CPD of a signal set is defined to minimum among all CPDs between pairs of points in the signal set[18]. May 5, 2014DRAFT A number is said to be algebraic over Q if there exists a polynomial with coefficients from Q of which the number is a root. If there does not exist a polynomial with coefficients from Q of which the number is a root, the number is said to be transcendental[19]. By Lindemann-Weierstrass theorem[19], e jq is transcendental for all q ∈ Q.May 5, 2014 DRAFT For the DNF protocol, with QAM signal set, conditioning on xA, xB can be decoded with constant decoding complexity by rounding off to the nearest integer, which results in an overall decoding complexity of O(M ). A1 ) = 0 for square QAM and 2 λ -PSK signal sets. Since E( x A M A 2 ) ≤ 2, we have \|a 11 \| 2 + \|a 12 \| 2 + \|a 21 \| 2 + \|a22The coding gain of the DSTC is the minimum among all the non-zero singular values of the codeword difference matrices which are of the form C(0 2 , ∆x B ) and C(∆x A , 0 2 ), i.e., the coding gain is equalLet d min (S) denote the minimum distance of the signal set S.Hence, we have, minSince U A is unitary, \|u A11 \| 2 = 1 − \|u A12 \| 2 . For a given λ A1 and λ A2 , the upper-bound in(4)is maximized over all \|u A11 \| 2 when the two terms inside min are equal, i.e., \|u A11 \| 2 λ A1 + \|u A12 \| 2 λ A2 = \|u A11 \| 2 λ A2 + \|u A12 \| 2 λ A1 , for which \|u A11 \| 2 = 1 2 and this maximum value is equal to d 2 min (S) (λA 1 +λA 2 ) 2 .Since, λ A1 + λ A2 ≤ 2, the maximum value of the upper-bound in (4) is less than or equal to d 2 min (S). Hot topic: Physical-layer Network Coding. S Zhang, S C Liew, P P Lam, ACM MobiCom '06. S. Zhang, S. C. Liew and P. P. Lam, "Hot topic: Physical-layer Network Coding", ACM MobiCom '06, pp. 358-365, Sept. 2006. The AntiPackets Can Increase the Achievable Throughput of a Wireless MultiHop Network. P Popovski, H Yomo, IEEE ICC. P. Popovski and H. Yomo, "The AntiPackets Can Increase the Achievable Throughput of a Wireless MultiHop Network", IEEE ICC 2006, Istanbul, Turkey, June 2006. Performance Bounds for Bidirectional Coded Cooperation Protocols. 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[ "A FEW REMARKS ON PIMSNER-POPA BASES AND REGULAR SUBFACTORS OF DEPTH 2", "A FEW REMARKS ON PIMSNER-POPA BASES AND REGULAR SUBFACTORS OF DEPTH 2" ]	[ "\nSchool of Physical Sciences\nChennai Mathematical Institute\nChennaiINDIA\n", "\nJawaharlal Nehru University\nNew DelhiINDIA\n" ]	[ "School of Physical Sciences\nChennai Mathematical Institute\nChennaiINDIA", "Jawaharlal Nehru University\nNew DelhiINDIA" ]	[]	We prove that a finite index regular inclusion of II 1 -factors with commutative first relative commutant is always a crossed product subfactor with respect to a minimal action of a biconnected weak Kac algebra. Prior to this, we prove that every finite index inclusion of II 1factors which is of depth 2 and has simple first relative commutant (respectively, is regular and has commutative or simple first relative commutant) admits a two-sided Pimsner-Popa basis (respectively, a unitary orthonormal basis).	10.1017/s0017089521000379	[ "https://arxiv.org/pdf/2102.01462v3.pdf" ]	231,749,512	2102.01462	20352617bd6d524ebd87adc99e0808cc0c216410	A FEW REMARKS ON PIMSNER-POPA BASES AND REGULAR SUBFACTORS OF DEPTH 2 24 Dec 2021 School of Physical Sciences Chennai Mathematical Institute ChennaiINDIA Jawaharlal Nehru University New DelhiINDIA A FEW REMARKS ON PIMSNER-POPA BASES AND REGULAR SUBFACTORS OF DEPTH 2 24 Dec 2021KESHAB CHANDRA BAKSHI AND VED PRAKASH GUPTA In memory of Vaughan Jones, a true pioneer! The first named author was supported through a DST INSPIRE faculty grant (reference no. DST/INSPIRE/04/2019/002754). 1 We prove that a finite index regular inclusion of II 1 -factors with commutative first relative commutant is always a crossed product subfactor with respect to a minimal action of a biconnected weak Kac algebra. Prior to this, we prove that every finite index inclusion of II 1factors which is of depth 2 and has simple first relative commutant (respectively, is regular and has commutative or simple first relative commutant) admits a two-sided Pimsner-Popa basis (respectively, a unitary orthonormal basis). Introduction Right from the early days of the evolution of the theory of operator algebras, the methods of crossed product constructions and fixed point subalgebras with respect to actions by various algebraic objects on operator algebras have served extremely well to provide numerous examples with specific properties as well as to be considered as suitable candidates for structure results under certain given hypotheses. One of the first such structure results (thanks to Ocneanu, Jones, Sutherland, Popa, Kosaki and Hong) states that every irreducible regular inclusion of factors of type II 1 with finite Jones index is a group subfactor of the form N ⊂ N ⋊G, with respect to an outer action of a finite group G on N . In particular, every such subfactor has depth 2. Further, it has also been established (in a series of papers by Ocneanu, David, Szymański and Nikshych-Vainerman) that every finite index inclusion of type II 1 factors of depth 2 is of the form N ⊂ N ⋊ H, with respect to a minimal action of some biconnected weak Hopf C * -algebra H -see [16,12,7,23,13]. More recently, Popa, Shlyakhtenko and Vaes, in [22], among various interesting results, classified regular subalgebras B of the hyperfinite II 1 -factor R with B ′ ∩ R = Z(B). However, they do not provide any structure for non-irreducible regular inclusions of factors of type II 1 . This short note is a first naive attempt in this direction, in which we prove the following: Theorem 4.6 Let N ⊂ M be a finite index regular inclusion of II 1 -factors with commutative relative commutant N ′ ∩ M . Then, there exists a biconnected weak Kac algebra K and a minimal action of K on N such that N ⊂ M is isomorphic to N ⊂ N ⋊ K. It must be mentioned here that Ceccherini-Silberstein (in [6]) claimed to have proved that every finite index regular subfactor is a crossed product subfactor with respect to an outer action of a finite dimensional Hopf C * -algebra. However, his assertion is incorrect and there is an obvious oversight in his proof as is pointed out in Remark 4.1. Theorem 4.6 is achieved by first proving that any finite index regular inclusion of II 1 -factors with commutative first relative commutant has depth 2 and then an appropriate application of Nikshych-Vainerman's characterization of depth 2 subfactors yields the desired structure. In order to take care of the first part, we utilize the notion of unitary orthonormal basis by Popa to show (in Theorem 4.3) that any regular subfactor with simple or commutative relative commutant is of depth at most 2. It fits well to mention here that, in fact, Popa had recently asked (in [21]) whether every integer index irreducible inclusion of II 1 -factors admits a unitary orthonormal basis or not. It seems to be a difficult question to answer in full generality. In fact, the question can be asked for non-irreducible inclusions as well, and we provide a partial answer in: Then, the second part of Theorem 4.6 is taken care of by a suitable application of the notion of two-sided basis for inclusions of finite von Neumann algebras. In fact, somewhat related to Popa's question, and equally fundamental in nature, is the question related to the existence of a two-sided Pimsner-Popa basis for any extremal inclusion of II 1 -factors, which was asked by Vaughan Jones around a decade back at various places. This question too has tasted too little success. In [2], we had shown that every finite index regular inclusion of II 1 -factors admits a two-sided Pimsner-Popa basis and we have suitably adopted the idea of its proof in proving Theorem 3.21. We move one more step closer towards answering Jones' question by proving the following: Furthermore, M also admits a two-sided basis over N . The flow of the article is in the reverse order in the sense that, after some preliminaries in Section 2, we first make an attempt to partially answer Jones' question regarding existence of two-sided basis in the first half of Section 3 and then move towards Popa's question regarding existence of unitary orthonormal basis in the second half of the same section. Finally, in Section 4, we establish that any regular subfactor with commutative first relative commutant is given by crossed product by a weak Kac algebra. Preliminaries Since there are slightly varying (though equivalent) definitions available in literature, in order to avoid any possible confusion, we quickly recall the definition that we shall be using here. For further details, we refer the reader to [5,15,13,14] and the references therein. Definition 2.1. [15,5] (1) A weak bialgebra is a quintuple (A, m, η, ∆, ε) so that (A, m, η) is an algebra, (A, ∆, ε) is a coalgebra and the tuple satisfies the following compatibility conditions between algebra and coalgebra structures: (a) ∆ is an algebra homomorphism. (b) ε(xyz) = ε(xy 1 )ε(y 2 z) and ε(xyz) = ε(xy 2 )ε(y 1 z) for all x, y, z ∈ A. (c) ∆ 2 (1) = ∆(1) ⊗ 1 1 ⊗ ∆(1) = 1 ⊗ ∆(1) ∆(1) ⊗ 1 .(2) A weak Hopf algebra (or a quantum groupoid ) is a weak bialgebra (A, m, η, ∆, ε) along with a k-linear map S : A → A, called an antipode, satisfying the following antipode axioms: (a) x 1 S(x 2 ) = ε(1 1 x)1 2 , (b) S(x 1 )x 2 = 1 1 ε(x1 2 ) and (c) S(x 1 )x 2 S(x 3 ) = S(x).(3) A weak Hopf algebra (A, m, η, ∆, ε) is said to be a weak Hopf C * -algebra if A is a finite dimensional C * -algebra and the comultiplication map is * -preserving, i.e., ∆(x * ) = ∆(x) * . As in the preceding definition, throughout this paper, we shall use the Sweedler's notation, i.e., ∆(x) = x (1) ⊗ x (2) and (∆ ⊗ Id)∆(x) = x (1) ⊗ x (2) ⊗ x (3) = (Id ⊗ ∆)∆(x) for all x ∈ A. Definition 2.2. [5] A weak Kac algebra is a weak Hopf C * -algebra (A, m, η, ∆, ε, S) such that S 2 = Id A and S is * -preserving. Remark 2.3. ( 1) A weak Hopf algebra is a Hopf algebra if and only if the comultiplication is unit-preserving if and only if the counit is a homomorphism of algebras. In particular, every Kac algebra is a weak Kac algebra. (2) The dual of a weak Kac algebra also admits a canonical weak Kac algebra. Example 2.4. Given a finite groupoid G, the associated groupoid algebra C[G] inherits a canonical weak Kac algebra structure with respect to the comultiplication ∆, the counit ε and the antipode S satisfying ∆(g) = g ⊗ g, ε(g) = 1, S(g) = g −1 for g ∈ G. It is easily seen that C[G] (resp., C[G] * ) is a cocommutative (resp., commutative) weak Kac algebra. And, conversely, it was proved by Yamanouchi that for every cocommutative weak Kac algebra H there exists a finite groupoid G such that H is isomorphic to C[G]. Given any weak Kac algebra A, the target (resp., source) counital map ε t (resp., ε s ) on A, is given by ε t (x) = ε(1 (1) x)1 (2) resp., ε s (x) = 1 (1) ε(x1 (2) ) for x ∈ A, where ∆(1) = 1 (1) ⊗ 1 (2) in Sweedler's notation. These maps are idempotent, i.e., ε t • ε t = ε t , ε s • ε s = ε s , and their images are unital C * -subalgebras (called the Cartan subalgebras) of A: A t := {x ∈ A : ε t (x) = x} and A s := {x ∈ A : ε s (x) = x}. A is said to be connected if the inclusion A t ⊂ A is connected (see [8] for definition). And, A is said to be biconnected if both A and its dual are connected. Remark 2.5. Given a finite groupoid G, the groupoid algebra C[G] is biconnected if and only if G is a group. Crossed product construction. We now briefly recall the notion of the crossed product construction via an action of a weak Hopf C * -algebra, as in [15] (also see [14,13]). Definition 2.6. (1) By a (left) action of a weak Hopf C * -algebra A on a von Neumann algebra M , we mean a linear map A ⊗ M ∋ a ⊗ x → (a ⊲ x) ∈ M which defines a (left) module structure on M and satisfies the conditions (a) a ⊲ xy = (a (1) ⊲ x)(a (2) ⊲ y), (b) (a ⊲ x) * = S(a) * ⊲ x * , and (c) a ⊲ 1 = ε t (a) ⊲ 1 and a ⊲ 1 = 0 iff ε t (a) = 0 for a ∈ A, x, y ∈ M . (2) Under such a (left) action, the crossed product algebra M ⋊ A is defined as follows: As a C-vector space it is the relative tensor product M ⊗ At A, where A (resp., M ) admits a canonical left (resp., right) A t -module structure so that x(z ⊲ 1) ⊗ a ∼ x ⊗ za, for all x ∈ M, a ∈ A, z ∈ A t . For each (a, x) ∈ A × M , [x ⊗ a] denotes the class of the element x ⊗ a and a natural * -algebra structure on M ⊗ At A is given by: 2) ], for all x, y ∈ M and a, b ∈ A. [x ⊗ a][y ⊗ b] = [x(a (1) ⊲ y) ⊗ a (2) b], [x ⊗ a] * = [(a * (1) ⊲ x * ) ⊗ a ((3) The action is said to be minimal if A ′ ∩ (M ⋊ A) = A s . Remark 2.7. [15,13,14] (1) M ⋊ A can be realized as a von Neumann algebra. (2) If M is a II 1 -factor and A is a weak Hopf C -algebra acting minimally on M , then M ⋊ A is also a II 1 -factor. Our interest in actions of weak Hopf C * -algebras stems from the following beautiful characterization of depth 2 subfactors by Nikshych and Vainerman. Before stating them, it would be appropriate to recall the following definition. Definition 2.8. Consider a finite index inclusion N ⊂ M of II 1 -factors and suppose N ⊂ M ⊂ M 1 ⊂ · · · ⊂ M k ⊂ · · · is its tower of Jones' basic construction. Then, the inclusion N ⊂ M is said to have finite depth if there exists a k such that N ′ ∩ M k−2 ⊂ N ′ ∩ M k−1 ⊂ N ′ ∩ M k is an instance of basic construction. The least such k is defined as the depth of the inclusion. We urge the reader to see [8] for various other equivalent formulations of the notion of depth. For any finite index irreducible inclusion N ⊂ M of II 1 -factors, i.e., N ′ ∩ M = C, it was announced by Ocneanu (in [16]) and proved later, separately, by Szymański, David and Longosee [23,7,12] Pimsner-Popa Bases Let N ⊂ M be a unital inclusion of von Neumann algebras equipped with a faithful normal conditional expectation E from M onto N . Then, a finite set B := {λ 1 , . . . , λ n } ⊂ M is called a left (resp., right) Pimsner-Popa basis for M over N via E if every x ∈ M can be expressed as x = n i=1 E(xλ * i )λ i (resp., x = n j=1 λ j E(λ * j x). Further, such a basis {λ i } is said to be orthonormal if E(λ i λ * j ) = δ i,j for all i, j. And, a collection B is said to be a two-sided basis if it is simultaneously a left and a right Pimsner-Popa basis. In this article, when we do not use the adjectives left or right, by a basis we shall always mean a right Pimsner-Popa basis (and not a two-sided basis). Two-sided basis. About a decade back, Vaughan Jones asked the following question at various places 1 . (1) If N ⊂ M is a regular irreducible subfactor of type II 1 of finite index, then (from some works of Ocneanu, Jones, Sutherland, Popa, Kosaki ) it is a well-known fact that it is isomorphic to N ⊂ N ⋊ G, for some outer action of a finite group G on N -see [9], for a precise statement. In particular, M has a two-sided basis over N . (2) In [2], we could drop the irreducibility condition and showed, without depending upon any structure result, that every finite index regular subfactor N ⊂ M of type II 1 admits a two-sided basis. A little thought should convince the reader that the two-sided basis we constructed in [2] is in fact orthonormal. A comment pertaining to an application of the notion of two-sided basis fits in well here: [2]) by exhibiting that, for any finite index regular subfactor N ⊂ M of type II 1 , its index is given explicitly by Remark 3.5.(1) [M : N ] = \|G\| dim(N ′ ∩ M ), where G denotes the generalized Weyl group of the inclusion N ⊂ M , which is defined as the quotient group NM (N ) U (N )U (N ′ ∩M) . Depending upon the structure result of irreducible depth 2 subfactors by Szymański and a result by Kac which determines when a Kac algebra is a group algebra, Nikshych and Vainermann deduced (in [13,Corollary 4.19]) that a depth 2 subfactor of type II 1 with prime index p is necessarily a group subfactor with respect to an outer action of the cyclic group Z/pZ. Interestingly, it turns out that the formula in Equation (1) has the following analogous consequence. Proposition 3.6. Let N ⊂ M be a finite index regular inclusion of II 1 -factors. If [M : N ] = p is prime, then N ⊂ M is irreducible. In particular, the cyclic group G := Z/pZ acts outerly on N and N ⊂ M is isomorphic to N ⊂ N ⋊ G. Proof. Suppose, on contrary, that N ⊂ M is not irreducible. Then, from Equation (1) Adding to the list, we shall provide, in the next section, an yet another application of the notion of two-sided basis for regular inclusions. One more step towards Jones' question. Note that, any irreducible regular factorial inclusion of type II 1 , being isomorphic to a crossed product subfactor by a group, must be of depth 2 (see [8] or Definition 2.8 for definition). Thus, it is natural to ask the following question: We do not know the answer yet in this generality. However, we provide a partial answer in Theorem 3.14, for which we require some preparation. Proof. Suppose {λ i : i ∈ I} is a right basis for K/N . Then, i λ i e K N λ * i = 1, where e K N denotes the Jones projection corresponding to the inclusion N ⊂ K. Let Ω denote the canonical cyclic vector for L 2 (M). Then, for any x ∈ L and y ∈ K, we have i λ i e M L λ * i (yxΩ) = i λ i E M L (λ * i yx)Ω = i λ i E M L (λ * i y)xΩ = i λ i E K N (λ * i y)xΩ [by commuting square condition] = yxΩ. As the commuting square is non-degenerate, we have span LK SOT = M = span KL SOT . In particular, [span LK]Ω · 2 = L 2 (M) = [span KL]Ω · 2 . Therefore, we conclude that i λ i e M L λ * i = 1 and the proof is complete. ✷ Some specific conditions guarantee non-degeneracy of some commuting squares. [2] Let A be a finite dimensional C * -algebra and tr be a faithful tracial state on A. Then, A has a two-sided orthonormal basis over C with respect to tr. The following interesting observation is a folklore. Proposition 3.12. Let N ⊂ M be a finite index depth 2 subfactor of type II 1 and M ⊃ N ⊃ N −1 ⊃ N −2 ⊃ · · · ⊃ N −k ⊃ · · · be a tunnel construction for N ⊂ M . Then, N −2k ⊂ N −2k+1 has depth 2 for all k ≥ 1. Moreover, M 2k−1 ⊂ M 2k is also of depth 2 for all k ≥ 1. Proof. For the tunnel part, it suffices to show that N −2 ⊂ N −1 has depth 2. Let Γ and Ω denote the inclusion matrices for the inclusions ( N ′ −2 ∩ N −1 ⊂ N ′ −2 ∩ N ) and (N ′ −2 ∩ N ⊂ N ′ −2 ∩ M ),< [N −1 : N −2 ] = Ω 2 . Consider the Jones' basic construction tower N −2 ⊂ N −1 ⊂ N ⊂ M ⊂ M 1 ⊂ M 2 ⊂ M 3 · · · ⊂ M k ⊂ · · · . By [4, Theorem 2.13], there exists a * -isomorphism (the shift operator) ϕ : N ′ −2 ∩ M → N ′ ∩ M 2 such that ϕ(N ′ −2 ∩ N ) = N ′ ∩ M 1 and ϕ(N ′ −2 ∩ N −1 ) = N ′ ∩ M . Thus, the truncated towers [N ′ −2 ∩ N −1 ⊂ N ′ −2 ∩ N ⊂ N ′ −2 ∩ M ] and [N ′ ∩ M ⊂ N ′ ∩ M 1 ⊂ N ′ ∩ M 2 ]⊃ N ⊃ N −1 ⊃ N −2 ⊃ · · · ⊃ N −k ⊃ · · · of N ⊂ M . Proof. It suffices to show that M ⊂ M 1 is of depth 2. Fix a 2-step downward basic construction N −2 ⊂ N −1 ⊂ N of N ⊂ M . Then, by the preceding proposition, N −2 ⊂ N −1 is also of depth 2. So, by [13], there exists a biconnected weak Hopf C * -algebra H with a minimal action on N such that (N H ⊂ N ) ∼ = (N −1 ⊂ N ). Thus, N −1 ⊂ N is also of depth 2, by [5] (also see [14,Section 8.1]). Thus, by Proposition 3.12 again, M ⊂ M 1 is also of depth 2. ✷ We are now all set for the theorem of this subsection. Proof. Although some of the arguments below are well-known (see [20]), we provide sufficient details for the sake of self-containment and convenience of the reader. Step I: Any (left/right) basis for M ′ ∩ M 2 over M ′ ∩ M 1 is also a (left/right) basis for M 2 over M 1 . Note that, by Lemma 3.9, it suffices to show that the quadruple M 1 ⊂ M 2 ∪ ∪ M ′ ∩ M 1 ⊂ M ′ ∩ M 2 is a non-degenerate commuting square. Towards this direction, first, recall that the quadruple G 1 := M ⊂ M 1 ∪ ∪ N ′ ∩ M ⊂ N ′ ∩ M 1 is a commuting square -see, for instance, [8,Proposition 4.2.7], wherein the bottom inclusion is connected. Let Λ denote the inclusion matrix for the inclusion N ′ ∩ M ⊂ N ′ ∩ M 1 . Since N ⊂ M is of depth 2,G 2 := M 1 ⊂ M 2 ∪ ∪ N ′ ∩ M 1 ⊂ N ′ ∩ MG 3 := N ′ ∩ M 1 Λ T ⊂ N ′ ∩ M 2 ∪ ∪ M ′ ∩ M 1 Γ ⊂ M ′ ∩ M 2 is also non-degenerate, by Lemma 3.10. In particular, concatenating G 2 and G 3 , we deduce from [20, §1.1.5] that the quadruple M 1 ⊂ M 2 ∪ ∪ M ′ ∩ M 1 ⊂ M ′ ∩ M 2 is a non-degenerate commuting square. Step II: M ′ ∩ M 2 has a two-sided orthonormal basis over M ′ ∩ M 1 . We assert that M ′ ∩ M 1 ⊂ M ′ ∩ M 2 is isomorphic to M ′ ∩ M 1 ⊂ (M ′ ∩ M 1 ) ⊗ Q for some unital * -subalgebra Q of (M ′ ∩ M 1 ) ′ ∩ (M ′ ∩ M 3 ) . Once this is established, we can then readily deduce from Lemma 3.11 that M ′ ∩ M 2 has a two-sided orthonormal basis over M ′ ∩ M 1 . Since N ′ ∩ M ∋ x → Jx * J ∈ M ′ ∩ M 1 is an anti-isomorphism and N ′ ∩ M is simple, so is M ′ ∩ M 1 . Again, since M ⊂ M 1 is also of depth 2, the tower M ′ ∩ M 1 ⊂ M ′ ∩ M 2 ⊂ M ′ ∩ M 3 is an instance of basic construction. So, M ′ ∩ M 3 is also simple. Thus, it follows from [8, Lemma 2.2.2] that (M ′ ∩ M 1 ) ′ ∩ (M ′ ∩ M 3 ) is simple and that M ′ ∩ M 3 ∼ = (M ′ ∩ M 1 ) ⊗ (M ′ ∩ M 1 ) ′ ∩ (M ′ ∩ M 3 ) . Suppose that M ′ ∩ M 1 ∼ = M n (C) and that M ′ ∩ M 3 ∼ = M n (C) ⊗ M k (C). Denote the intermediate subalgebra corresponding to M ′ ∩ M 2 by P . It is well-known that P is of the form M n (C) ⊗ Q, where Q is some unital * -subalgebra of M k (C). We provide the details for the convenience of the reader. By [8, Proposition 4.2.7] again, the quadruple G 4 := P ⊂ M n (C) ⊗ M k (C) ∪ ∪ M n (C) ⊗ 1 ′ ∩ P ⊂ 1 ⊗ M k (C). is also a commuting square. Note that, there exists a unital * -subalgebra Q of M k (C) such that M n (C) ⊗ 1 ′ ∩ P = 1 ⊗ Q. Clearly, M n (C) ⊗ Q ⊆ P . To see the reverse inclusion, consider x = i a i ⊗ b i ∈ P ⊂ M n (C) ⊗ M k (C). Then, we have x = (a i ⊗ 1)E P (1 ⊗ b i ). Since G 4 is a commuting square, we immediately see that E P (1 ⊗ b i ) ∈ 1 ⊗ Q and hence x ∈ M n (C) ⊗ Q. In conclusion, we have P = M n (C) ⊗ Q, as was asserted. Thus, from Steps I and II, we deduce that M 2 has a two-sided orthonormal basis over M 1 . Finally, fix any 2-step downward basic construction N −2 ⊂ N −1 ⊂ N ⊂ M for N ⊂ M . Then, by Proposition 3.12, N −2 ⊂ N −1 also has depth 2. Further, as seen in Proposition 3.12, N ′ −2 ∩ N −1 ∼ = N ′ ∩ M is simple. Hence, we readily deduce from the preceding discussion that M must admit a two-sided orthonormal basis over N . ✷ As an immediate consequence we deduce the following. Remark 3.16. Note that a subfactor as in Theorem 3.14 need not be regular. For instance, any Kac algebra K, which is not a group algebra, acts outerly on the hyperfinite factor R and yields a non-regular irreducible depth 2 subfactor. Unitary orthonormal basis. We now move towards unitary orthonormal bases and touch upon another fundamental question asked recently by Sorin Popa in [21, § 3.5]. Question 3.17 (Sorin Popa). Does there always exist an orthonormal basis consisting of n many unitaries for an integer index (= n) irreducible inclusion of II 1 -factors? Example 3.18. Let H be a subgroup of a finite group G. If G acts outerly on a II 1 -factor N , then N ⋊ G has a unitary orthonormal basis over N ⋊ H. In particular, every irreducible regular subfactor, being isomorphic to a group subfactor, admits a unitary orthonormal basis. In view of the preceding example, it is natural to ask whether we can drop the irreducibility condition or not. The following remarks fall in place here: Remark 3.19. ( [6]. In fact, he asserted (in [ As a partial progress in the resolution of this conjecture, we prove the following: We will need the following couple of results to achieve this. Recall that, for a unital inclusion B ⊂ A of finite dimensional C * -algebras with inclusion matrix Λ, a tracial state tr on A is said to be a Markov trace for 1) The question of existence of unitary orthonormal basis for a general finite index regular subfactor which is not necessarily irreducible (thus modifying Question 3.17) was discussed by Ceccherini-Silberstein B ⊂ A if Λ t Λt = Λ 2t , wheret denotes the trace vector of the tracial state tr. For more on Markov trace, see [8,11]. Lemma 3.22. Let A := C ⊕ C ⊕ · · · ⊕ C (n-copies). (1) The Markov trace tr : A → C for the unital inclusion C ⊂ A is given by tr ((z 1 , . . . , z n )) = 1 n i z i . i , i.e., whose i-th column constitutes of the complex entries in λ i . Then, \|u ij \| = 1 √ n for all 1 ≤ i, j ≤ n and (2) There exists a unitary orthonormal basis for A over C with respect to the Markov trace if and only if there exists a unitary matrix U = [u ij ] ∈ M n (C) such that \|u ij \| = 1 √ n for all 1 ≤ i, j ≤ n.U * U =      tr(λ * 1 λ 1 ) tr(λ * 1 λ 2 ) · · · tr(λ * 1 λ n ) tr(λ * 2 λ 1 ) tr(λ * 2 λ 2 ) · · · tr(λ * 2 λ n ) . . . . . . . . . . . . tr(λ * n λ 1 ) tr(λ * n λ 2 ) · · · tr(λ * n λ n )      = I n . (⇐) Let U = [u ij ] ∈ U (n) be such that \|u ij \| = 1 √ n for all 1 ≤ i, j ≤ n. Consider λ i := √ n (u 1i , u 2i , . . . , u ni ) ∈ A, i = 1, 2, . . . , n. Since \|u ij \| = 1 √ n for all 1 ≤ i, j ≤ n, it follows that λ * i λ i = (1, 1, . . . , 1) for all 1 ≤ i ≤ n, i.e., each λ i is a unitary in A. Further, note that I n = U * U = [u ij ] * [u ij ] =      tr(λ * 1 λ 1 ) tr(λ * 1 λ 2 ) · · · tr(λ * 1 λ n ) tr(λ * 2 λ 1 ) tr(λ * 2 λ 2 ) · · · tr(λ * 2 λ n ) . . . . . . . . . . . . tr(λ * n λ 1 ) tr(λ * n λ 2 ) · · · tr(λ * n λ n )      . Hence, tr(λ * i λ j ) = δ i,j for all 1 ≤ i, j ≤ n, which implies that {λ 1 , . . . , λ n } forms a unitary orthonormal basis for A over C with respect to above tracial state. ✷ Recall that a unitary error basis for a matrix algebra M n (C) is a Hamel basis that is orthogonal with respect to the inner product induced by the canonical trace of M n (C). Little is known about their structure. There are two popular methods of construction of unitary error bases. One is algebraic in nature (due to Knill) and the other combinatorial (due to Werner). Proposition 3.23. Let A be a finite dimensional C * -algebra which is either simple or commutative. Then, A/C has a unitary orthonormal basis with respect to the Markov trace for the unital inclusion C ⊂ A. Proof. Suppose first that A = M n (C) for some n ≥ 2. The existence of a unitary orthonormal basis follows from the known construction of a unitary error basis. We include the details for the reader's convenience. We first recall such a basis for n = 2 (because of its importance and popularity in quantum information theory). The Pauli spin matrices (unitary error bases in dimension 2) are defined as follows: σ x = 0 1 1 0 , σ y = 0 −i i 0 , σ z = 1 0 0 −1 . It is an amazing fact that the set {I 2 , σ x , σ y , σ z } forms an orthonormal basis consisting of unitaries for M 2 (C). For higher dimensions, consider the following two important matrices due to Sylvester and Weyl: U =        1 0 0 · · · 0 0 ω 0 · · · 0 0 0 ω 2 · · · 0 . . . . . . . . . . . . . . . 0 0 0 · · · ω n−1        and V =          0 0 0 · · · 0 1 1 0 0 · · · 0 0 0 1 0 · · · 0 0 0 0 1 · · · 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 · · · 1 0          , where ω := e −2πi/n (a primitive root of unity). Then, it is known that the set {U i V j : 1 ≤ i, j ≤ n} forms a unitary error basis (in fact, a nice error basis) for M n (C). These matrices also appeared in a work of Popa ([18]) (see also [6]). This proves that A/C has unitary orthonormal basis whenever A is simple. Next, let A be isomorphic to C ⊕ C ⊕ · · · ⊕ C (n-copies). Now, for a primitive root of unity ω as above, consider the well-known unitary DFT matrix U := 1 √ n          1 1 1 1 · · · 1 1 ω ω 2 ω 3 · · · ω n−1 1 ω 2 ω 4 ω 6 · · · ω 2(n−1) 1 ω 3 ω 6 ω 9 · · · ω 3(n−1) . . . . . . . . . . . . . . . . . . 1 ω n−1 ω 2(n−1) ω 3(n−1) · · · ω (−1)(n−1)          . Clearly, each entry of U has modulus 1/ √ n. So, by proposition 3.23, there exists a unitary orthonormal basis for A/C with respect to the Markov trace. ✷ The preceding observation will prove to be very crucial in the next section. Thus, it seems it is worthwhile to investigate in detail the existence of unitary basis of the finite dimensional inclusions B ⊂ A, which, in turn, may prove to be useful in answering the question of Popa for hyperfinite irreducible subfactors. On the other hand, since N ⊂ M is regular, from [2, Proposition 3.7], we know that M/R also has a unitary orthonormal basis, say, {v j : j ∈ J}. We assert that {v j u i : i ∈ I, j ∈ J} is a unitary orthonormal basis for M/N . It is easy to see that {v j u i : i ∈ I and j ∈ J} is a Pimsner-Popa basis for M/N . Also, E M N (u * i v * j v k u l ) = E R N • E M R (u * i v * j v k u l ) = E R N u * i E M R (v * j v k )u l = δ j,k δ i,l . Thus, {v j u i : i ∈ I and j ∈ J} is a unitary orthonormal basis for M/N . ✷ Two-sided basis versus unitary orthonormal basis. Some preliminary observations suggest that the above questions of Jones (Question 3.1) and Popa (Question 3.17) may be intimately interrelated in the case of integer index (extremal) subfactors. Below, we illustrate some such connections. The following fact is implicit in [6]. Proof. This proof is extracted verbatim from [6]. Suppose {λ i : 1 ≤ i ≤ n} is a unitary orthonormal basis for M/N . Thus, i λ i e 1 λ * i = 1. Now, put v k = n−1 i=0 ω ki λ i e 1 λ * i , 0 ≤ k ≤ n − 1, where ω is an n-th root of unity. In [6,Proposition 3.24], it has been shown that {v k } is a unitary orthonormal basis for M 1 /M . Clearly this is two-sided. Next, recall that N ⊂ M is extremal if and only if M ⊂ M 1 is extremal -see, for instance, [20]. Since M 1 /M has a two-sided basis, it is easily seen (see [2]) that it is extremal. ✷ Remark 3.28. Note that even if it can be shown that a finite index subfactor with a two-sided basis also admits a unitary orthonormal basis, then in view of [2], it will follow that Conjecture 3.20 holds true. Regular subfactors and weak Kac algebras As recalled in the introduction, a finite index irreducible regular inclusion of II 1 -factors is always of the form N ⊂ N ⋊ G with respect to an outer action of a finite group G. It is then natural to ask what happens if we drop the irreducibility condition. If his assertion is true, then it will automatically force N ⊂ M to be irreducible, whereas he has claimed to have characterized regular subfactors sans irreducibility. In fact, Ceccherini-Silberstein's oversight stems from an incomplete proof of an assertion made in [6,Theorem 4.5], as explained below: In the proof of [6,Theorem 4.6], in view of [6,Theorem 4.5], a unitary orthonormal basis {λ i } is chosen for M/N and then it is deduced that N ′ ∩ M 1 = Alg{λ i e 1 λ * i : i ∈ I}. Note that, the family {λ i e 1 λ * i } consists of mutually orthogonal projections with i λ i e 1 λ * i = 1. Thus, if N ⊂ M is regular with finite index, then according to [6,Theorem 4.6], N ′ ∩ M 1 is always commutative. However, this is known to be untrue. For instance, taking an irreducible regular subfactor K ⊂ L and putting N = C ⊗ K and M = M n (C) ⊗ L, it can be seen that N ⊂ M is regular (see Lemma 4.2) with integer index and N ′ ∩ M ∼ = M n (C); so that N ′ ∩ M 1 is not commutative. So, the question of characterizing (finite index) regular subfactors of type II 1 is still unresolved. In particular, we have i,j w i,j e 1 w * i,j = 1. Now, note that for any unitary u ∈ N we have w * i,j uw i,j = v i,j for some unitary v i,j ∈ N . Thus, u(w i,j e 1 w * i,j )u * = w i,j v i,j e 1 v * i,j w * i,j = w i,j e 1 w * i,j . This implies that w i,j e 1 w * i,j ∈ N ′ ∩ M 1 for all (i, j) ∈ I × J. Further, we readily see that (w i,j e 1 w * i,j )e 2 (w i,j e 1 w * i,j ) = τ w i,j e 1 w * i,j ∀ (i, j) ∈ I × J. Also, u ⊗ x i WOT −→ u ⊗ z. Hence, u ⊗ z ∈ ( * -alg N Mn⊗M (C ⊗ N )) ′′ . ✷ Hence, w i,j e 1 w * i,j ∈ (N ′ ∩ M 1 )e 2 (N ′ ∩ M 1 ) for every (i, j) ∈ I × J. Now, since 1 = i,j w i,j e 1 w * i,j and that ( Few remarks are in order which tell that the converse of the above result need not be true. N ′ ∩ M 1 )e 2 (N ′ ∩ M 1 ) is an ideal in N ′ ∩ M 2 , it follows that that (N ′ ∩ M 1 )e 2 (N ′ ∩ M 1 ) = N ′ ∩ M 2 . Remark 4.5. (1) A depth 2 subfactor having commutative first relative commutant need not be regular. For example, consider a finite dimensional Hopf C * -algebra (that is, a Kac algebra) K, which is not a group algebra, acting minimally on a type II 1 factor N . Then, N ⊂ N ⋊ K is a depth 2 subfactor. Being irreducible, this subfactor is not regular. (2) Notice that a depth 2 regular subfactor N ⊂ M may have a non-commutative first relative commutant. As an example, one may look at the subfactor illustrated in Remark 4.1. (1) If a finite group G acts innerly on a II 1 factor N in such a way that M = N ⋊ G is a II 1 factor, then the inclusion N ⊂ M is regular and N ′ ∩ M is nontrivial. 3 (2) Suppose N is a type II 1 factor. Then, the depth 1 subfactor C ⊗ N ⊂ M n (C) ⊗ N is an example of a regular subfactor with simple first relative commutant ( ∼ = M n (C)). (3) Let P be a II 1 -factor and α ∈ Aut(P ). Consider the diagonal inclusion N := x 0 0 α(x) : x ∈ P ⊂ M := P ⊗ M 2 (C), which is well known to be a subfactor of type II 1 with [M : N ] = 4. If α is an outer automorphism, then it is well known and can be easily seen that N ′ ∩ M = {diag(λ, µ) : λ, µ ∈ C}. Next, recall the Connes' outer conjugacy invariants p 0 (α) ∈ N and γ(α) ∈ C given by {n ∈ Z : α n ∈ Inn(P )} = p 0 (α)Z and α(u) = γ(α)u for some u ∈ U(P ) with α p0(α) = Ad u . Note that, if p 0 (α) = 2 (in particular, α is outer) and γ(α) = 1, then it is known that N ⊂ M has depth 2 -see [13, §7]. We show that N ⊂ M is regular as well, i.e., x ∈ N M (N ) ⊂ Q; so, by the maximality of D in M , it follows that Q = M (a factor), which contradicts the assumption that Q ′ ∩ Q ∼ = C ⊕ C. Hence, it just remains to prove the maximality of D in M . Let y ∈ M \ D. Without loss of generality, we can assume that y = 0 w z 0 with (w, z) = (0, 0). Futher, we can assume that w = 0. Since P is a II 1 -factor, it is algebraically simple; so, we have P wP = P . Thus, for each 0 = a ∈ P , we have a = i x i wy i for a finite collection {x i , y i : 1 ≤ i ≤ n} in P . Thus, Theorem 3.21 Let N ⊂ M be a finite index regular inclusion of factors of type II 1 . If N ′ ∩ M is either commutative or simple, then M admits a unitary orthonormal basis over N . Theorem 3.14 Let N ⊂ M be a finite index inclusion of type II 1 -factors of depth 2 with simple relative commutant N ′ ∩ M . Then, M 2 admits a two-sided Pimsner-Popa basis over M 1 . -that if N ⊂ M is of depth 2, then there exists a Kac algebra K and a minimal action of K on M 1 such that M 2 ∼ = M 1 ⋊ K and M = M H 1 . More generally, Nikshych and Vainerman obtained the following characterization: Theorem 2.9. [13, 5] A finite index inclusion N ⊂ M of II 1 -factors is of depth 2 if and only if there exists a biconnected weak Hopf C * -algebra H and a minimal action of H on M 1 such that M 2 ∼ = M 1 ⋊ H and M = M H 1 . Question 3. 1 . 1(Vaughan Jones) Let N be a II 1 -factor and N ⊂ M be an extremal subfactor of finite index. Then, does there always exist a two-sided Pimsner-Popa basis for M over N ? Example 3.2. Given a finite group G and a subgroup H, by Hall's Marriage Theorem, we can obtain a set of coset representatives which acts simultaneously as representatives of left and right cosets 2 of H in G. Therefore, if G acts outerly on a II 1 -factor N , then N ⋊ G always possesses a two-sided unitary orthonormal basis over N ⋊ H.This observation, therefore, allows us to think about the existence of a two-sided basis as a subfactor analogue of Hall's Marriage Theorem. Definition 3. 3 . 3An inclusion Q ⊂ P of von Neumann algebras is said to be regular if its group of normalizers N P (Q) := {u ∈ U(P) : uQu * = Q} generates P as von Neumann algebra, i.e., N P (Q) ′′ = P. Remark 3 . 4 . 34To the best of our knowledge, till date, too little progress has been made in answering Question 3.1. , it follows that [M : N ] = dim C (N ′ ∩ M ). Note that, if Λ denotes the inclusion matrix of the inclusion C ⊂ N ′ ∩ M , then Λ 2 = dim C (N ′ ∩ M ). In particular, Λ 2 = [M : N ], which then implies that C ⊂ N ′ ∩ M ⊂ N ′ ∩ M 1 is an instance of basic construction -see [8, Theorem 4.6.3 (vii)]. Thus, N ′ ∩ M ∼ = M n (C) for some n ≥ 2; so that [M : N ] = n 2 . This contradicts the hypothesis that [M : N ] is a prime number. Hence, N ⊂ M must be irreducible. The asserted structure of N ⊂ M is then well-known. ✷ Further, employing appropriate two-sided bases for the inclusions N ⊂ N ∨ (N ′ ∩ M ) and N ∨(N ′ ∩M ) ⊂ M , the following useful observation was proved explicitly in the first two paragraphs of the proof of [2, Theorem 3.12]. We will be using it crucially in the proof of Theorem 4.6 and shall not repeat the details here. [ 2 ] 2Let N ⊂ M be a finite index regular inclusion of II 1 -factors. Then, the Watatani index of the restriction of tr M to N ′ ∩ M is a scalar. Question 3. 8 . 8Let N ⊂ M be a depth 2 subfactor of type II 1 of finite index. Then, does M/N always have a two-sided basis? First, we need (a mild generalization of) a useful result of Popa [20, § 1.1.5]. Popa had proved it for any (left or right) orthonormal basis and it is easy to see that it holds for any (left or right) Pimsner-Popa basis as well. Recall that a commuting square (D, C, B, A) of von Neumann algebras is said to be non-degenerate if span[CB] S.O.T. = A = span[BC] S.O.T. . Lemma 3.9. (Popa) Let M be a finite von Neumann algebra with a faithful normal tracial state and (N , K, L, M) be a non-degenerate commuting square of von Neumann subalgebras of M. Then, any right basis for K/N is also a right basis for M/L. Lemma 3. 10 . 10Let M be a finite von Neumann algebra with a faithful normal tracial state and (N , P, Q, M) be a commuting square consisting of von Neumann subalgebras of M. If, either (1) Q ⊂ M is an inclusion of II 1 -factors with finite index and N ⊂ P is a connected inclusion of finite dimensional C * -algebras with [M : Q] = Λ 2 , where Λ denotes the inclusion matrix of N ⊂ P; or (2) both N ⊂ P and Q ⊂ M are connected inclusions of finite dimensional C * -algebras with Λ 2 = Γ 2 , where Λ and Γ denote the respective inclusion matrices, then (N , P, Q, M) is non-degenerate. Proof. A proof can be obtained on similar lines as that of [3, Lemma 18] based on the characterization of a basis illustrated in [1, Theorem 2.2]. ✷The next useful observation is a straight forward adaptation of [2, Proposition 3.3], which uses the notion of path algebras associated to inclusions of finite dimensional C * -algebras by Sunder and Ocneanu. We skip the details. are isomorphic. In particular, if Λ i denotes the inclusion matrix for the inclusion (N ′ ∩M i ⊂ N ′ ∩M i+1 ), then Λ 0 = Γ and Λ 1 = Ω; so, by [8, Theorem 4.6.3], we obtain Ω 2 = Λ 1 2 = [M : N ] = [N −1 : N −2 ] and Γ 2 = Λ 0 2 < [M : N ] = [N −1 : N −2 ]. For the basic construction part, it suffices to show that M 1 ⊂ M 2 has depth 2. The shift operator ψ : N ′ ∩ M 2 → M ′ 1 ∩ M 4 does the job as above. ✷ Corollary 3.13. Let N ⊂ M be a finite index depth 2 subfactor of type II 1 . Then, M k ⊂ M k+1 also has depth 2 for all k ≥ 0. In particular, N −k ⊂ N −k+1 has depth 2 for all k ≥ 1, for any tunnel construction M Theorem 3. 14 . 14Let N ⊂ M be a finite index inclusion of type II 1 -factors of depth 2 with simple relative commutant N ′ ∩ M . Then, M 2 admits a two-sided orthonormal basis over M 1 .Furthermore, M also admits a two-sided orthonormal basis over N . as was recalled in Proposition 3.12, we have [M 1 : M ] = [M : N ] = Λ 2 . Therefore, by Lemma 3.10, the quadruple G 1 is a non-degenerate commuting square. Thus, its extension (as defined in [20, §1.1.6]) is given by the quadruple by the proposition in §1.1.6 of [20], G 2 is a non-degenerate commuting square as well. On the other hand, note that if Γ denotes the inclusion matrix for M ′ ∩ M 1 ⊂ M ′ ∩ M 2 , then since M ⊂ M 1 is also of depth 2 (see Corollary 3.13), we have Γ 2 = [M 1 : M ] = [M : N ] = Λ T 2 ; so, the commuting square Corollary 3 . 15 . 315Every finite index irreducible inclusion of II 1 factors of depth 2 admits a twosided orthonormal basis. Proof. ( 1 ) 1This follows easily from [8, Proposition 2.7.2]. : 1 ≤ i ≤ n} be a unitary orthonormal basis for A over C. Consider the matrix U = [u ij ] ∈ M n whose entries are given by u ij = 1 √ n z (j) Lemma 3 . 24 . 324Let N ⊂ M be a finite index regular subfactor of type II 1 . Then, tr M \| N ′ ∩M is the Markov trace for the inclusion C ⊂ N ′ ∩ M . Proof. As pointed out in Proposition 3.7, it can be extracted from the proof of [2, Theorem 3.12] that the Watatani index ([25]) Ind(tr M ) of tr M is a scalar. Thus, it follows from [25, Corollary 2.4.3] and [11, Proposition 3.2.3] that tr M \| N ′ ∩M is indeed the Markov trace for the inclusion C ⊂ N ′ ∩ M . ✷ Proof of Theorem 3.21: Consider the intermediate von Neumann subalgebra R := N ∨ (N ′ ∩ M ). Then, as in the proof of [2, Lemma 3.4], we see that (C, N ′ ∩ M, N, R) is a non-degenerate commuting square. By Lemma 3.24, tr M \| N ′ ∩M is the Markov trace for C ⊂ N ′ ∩ M ; so, by Proposition 3.23, there exists a unitary orthonormal basis, say, {u i : i ∈ I} for N ′ ∩ M over C. Then, by Lemma 3.9, {u i : i ∈ I} is a unitary orthonormal basis for R/N as well. Lemma 3 . 25 . [ 6 ] 3256Let N ⊂ M be a subfactor of finite index. If M/N has a unitary orthonormal basis, then M 1 /M admits a two-sided unitary orthonormal basis.In particular, N ⊂ M is extremal. Proposition 3 . 26 . 326Let N ⊂ M be a finite index hyperfinite subfactor of type II 1 with finite depth. If M/N has a unitary orthonormal basis, then it also has a two-sided unitary orthonormal basis. Proof. By Lemma 3.25, it follows that M 1 /M , and hence, M 2 /M 1 has a two-sided unitary orthonormal basis. It is known that the standard invariants of the extremal subfactors N ⊂ M and M 1 ⊂ M 2 are isomorphic. Thus, by Popa's classification result (see[20]), N ⊂ M and M 1 ⊂ M 2 , both being hyperfinite, are isomorphic. Hence, N ⊂ M has a two-sided basis. This completes the proof. ✷ It will be good to know an answer of the following natural question. Question 3 . 27 . 327If N ⊂ M is a finite depth integer index subfactor of type II 1 , then is it true that M/N has a unitary orthonormal basis if and only if M/N has a two-sided Pimsner-Popa basis? Remark 4. 1 . 1Employing Szymański's characterization of depth 2 (irreducible) subfactors, Ceccherini-Silberstein (in[6, Theorem 4.6]) asserted that every finite index regular subfactor N ⊂ M of type II 1 is of the form N ⊂ N ⋊H with respect to an outer action of a finite dimensional Hopf * -algebra H on N . However, it had the following obvious oversight: Lemma 4 . 2 . 42Let N ⊂ M be a regular inclusion of von Neumann algebras. Then, C⊗N ⊂ M n ⊗M is also regular.Proof. Note that {u ⊗ v : u ∈ U (n), v ∈ N M (N )} ⊆ N Mn⊗M (C ⊗ N ). Thus, [ * -alg U (n)] ⊗ [ * -alg N M (N )] ⊆ * -alg N Mn⊗M (C ⊗ N ).It is enough to show that{u ⊗ z : u ∈ U (n), z ∈ M } ⊂ N Mn⊗M (C ⊗ N ) ′′ .Let z ∈ M and u ∈ U (n). Then, there exists a net {x i } in * -alg N M (N ) such that x i WOT −→ z. Thus, {u ⊗ x i } is a net in [ * -AC U (n)] ⊗ [ * -alg N M (N )], which is a * -subalgebra of * -alg N Mn⊗M (C ⊗ N ). Theorem 4. 3 . 3Let N ⊂ M be a finite index regular inclusion of type II 1 -factors such that N ′ ∩M is either simple or commutative. Then, N ⊂ M is of depth at most 2. Proof. Note that, by Theorem 3.21, M admits a unitary orthonormal basis over N . More precisely, taking R := N ∨ (N ′ ∩ M ), we saw that R/N admits a unitary orthonormal basis, say, {u i : i ∈ I} ⊂ U(N ′ ∩ M ) ⊂ N M (N ); M/R admits a unitary orthonormal basis {v j : j ∈ J} ⊂ N M (N ); and then, taking w i,j = v j u i we saw that {w i,j : i ∈ I, j ∈ J} is a unitary orthonormal basis for M/N and {w i,j : (i, j) ∈ I × J} ⊂ N M (N ). Theorem 4. 6 .. 6Let N ⊂ M be a finite index regular inclusion of II 1 -factors with commutative relative commutant N ′ ∩ M . Then, there exists a biconnected weak Kac algebra K and a minimal action ofK on N such that N ⊂ M is isomorphic to N ⊂ N ⋊ K.Proof. By Theorem 4.3 and Corollary 4.4, we observe that N ⊂ M has depth 2. Choose a 2-step downward basic constructionN −2 ⊂ N −1 ⊂ N ⊂ M . Then, N −2 ⊂ N −1 is also of depth twosee Corollary 3.13. Let K := N ′ −1 ∩ M . From[13], it will follow that K admits a biconnected weak Kac algebra structure (with an appropriate action on N ) provided the Watatani index oftr N \| N ′ −1 ∩N is a scalar. By Proposition 3.7, we know that Ind(tr M \| N ′ ∩M ) is a scalar. Let J : L 2 (N ) → L 2 (N )denote the modular conjugation operator. Since N −1 ⊂ N ⊂ M is an instance of basic construction, the map B(L 2 (N )) ∋ x → JxJ ∈ B(L 2 (N )) is an anti-isomorphism that maps N ′ −1 onto M and tr M = tr N ′ −1 • [J(·)J]; so that tr M \| N ′ ∩M = tr N ′ −1 • [J(·)J] \| N ′ ∩M . Also, J(N ′ ∩ M ) = N ′ −1 ∩ N ; so that, N ′ −1 ∩ N is commutative and (C ⊂ N ′ ∩ M ) ∼ = (C ⊂ N ′ −1 ∩ N ). Further, since N −1 ⊂ N is extremal (being of depth 2)Thus, tr M \| N ′ ∩M and tr N \| N ′ −1 ∩N have same trace vectors and hence Ind(tr N \| N ′ −1 ∩N ) = Ind(tr M \| N ′ ∩M ), which is a scalar. Thus, by [13, Corollary 4.7 and Theorem 4.17], K admits a weak Kac algebra structure, which is also biconnected, by [13, Remark 5.8 (ii)]. Further, by [13, Propositions 6.1 and 6.3, and Remark 6.4 (i)], K acts minimally on N such that N ⊂ M is isomorphic to N ⊂ N ⋊ K. This completes the proof. ✷We end our discussion with a few well-known classes of reducible regular subfactors. N M (N ) ′′ = M . Let Q := N M (N )′′ and fix a 1 = u ∈ U(P ) such that α 2 = Ad u . Since{diag(1, c) : c ∈ T \ {1}} ⊆ N M (N ) \ N,it is clear that Q = N . In view of the fact that N ⊂ M is a maximal subfactor (see, for instance, [24, Theorem 5.4]), it is sufficient to show that Q is a factor. To this end, we show that Q ′ ∩ Q = {diag(λ, µ) : λ, µ ∈ C}; this will then prove that Q is a factor asQ ′ ∩ Q ⊆ N ′ ∩ M ∼ = C ⊕ C.Suppose, on the contrary, that Q ′ ∩ Q = {diag(λ, µ) : λ, µ ∈ C}. Then, we see that the projection p :to Q ′ ∩ Q and hence) to Q; thus,{diag(x, y) : x, y ∈ P } = pM p ⊕ (1 − p)M (1 − p) = pN p ⊕ (1 − p)N (1 − p) ⊆ Q.We assert that the diagonal subalgebra D := {diag(x, y) : x, y ∈ P } is a maximal von Neumann subalgebra of M , i.e., {D, a} ′′ = M for any a ∈ M \ D. Assuming this assertion, if we consider x := 0 1 u 0 ∈ M , then x / ∈ D and a simple calculation shows that , y} ′′ . Hence, 0 a b 0 ∈ {D, y} ′′ for all a, b ∈ P , which then implies that {D, y} ′′ = M , i.e., D is maximal in M . 6, Theorem 4.5]) that if N ⊂ M is a regular subfactor with finite index, then M/N has a unitary orthonormal basis. However, his proof depends on a technique of Popa ([19, Theorem 2.3]) which holds for Cartan subalgebras. Since Popa's proof depends crucially on maximal abelian-ness of the subalgebra, it is not clear whether it holds, more generally, for regular subalgebras or not. So, the proof of [6, Theorem 4.5] seems to be incomplete; although, the statement may still be true, which we rephrase in Conjecture 3.20. (2) Furthermore, Ceccherini-Silberstein (in [6, Theorem 4.7]) had also asserted that if M/N has a unitary orthonormal basis then the subfactor M 1 ⊂ M 2 is of the form M 1 ⊂ M 1 ⋊ H Conjecture 3.20. Let N ⊂ M be a finite index regular inclusion of factors of type II 1 . Then, M/N has a unitary orthonormal basis.and hence has depth 2 (see for instance [13]), which then implies that N ⊂ M is also of depth 2 -see Proposition 3.12. However, this is well known to be incorrect as every group-subgroup subfactor (R × H) ⊂ (R ⋊ G) always has a unitary orthonormal basis (see Example 3.18) whereas it is not necessarily of depth 2. (3) Though not directly related to the present discussion, the characterization of index 3 sub- factors provided in Corollary 3.19 of [6] is known to be incorrect. Theorem 3.21. Let N ⊂ M be a finite index regular inclusion of factors of type II 1 . If N ′ ∩ M is either commutative or simple, then M admits a unitary orthonormal basis over N . Thus, in view of[8, Theorem 4.6.3], N ⊂ M has depth at most 2. ✷ Corollary 4.4. If N ⊂ M is a finite index regular inclusion of type II 1 factors such that N ′ ∩ M is commutative, then it has depth 2. For instance, during the second talk by M. Izumi in the workshop organized in honour of V S Sunder's 60th birthday at IMSc, Chennai during March-April 2012. 2 https://mathoverflow.net/questions/6647/do-subgroups-have-two-sided-bases. https://mathoverflow.net/questions/364547/action-of-a-finite-group-on-a-finite-factor Acknowledgements. The authors would like to thank Yongle Jiang for bringing Example 4.7(3) to our notice; and, Vijay Kodiyalam, Sebastien Palcoux and Leonid Vainerman for many fruitful exchanges. The authors would also like to mark their note of appreciation to T. 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[ "Characterizing volume via cone duality", "Characterizing volume via cone duality" ]	[ "Jian Xiao " ]	[]	[]	For divisors over smooth projective varieties, we show that the volume can be characterized by the duality between pseudo-effective cone of divisors and movable cone of curves. Inspired by this result, we give and study a natural intersection-theoretic volume functional for 1-cycles over compact Kähler manifolds. In particular, for numerical equivalence classes of curves over projective varieties, it is closely related to the mobility functional.	10.1007/s00208-016-1501-3	[ "https://arxiv.org/pdf/1502.06450v1.pdf" ]	117,912,637	1502.06450	0c01d9c65ba015550ab3f875460642d1dc30e5de	Characterizing volume via cone duality 23 Feb 2015 Jian Xiao Characterizing volume via cone duality 23 Feb 2015 For divisors over smooth projective varieties, we show that the volume can be characterized by the duality between pseudo-effective cone of divisors and movable cone of curves. Inspired by this result, we give and study a natural intersection-theoretic volume functional for 1-cycles over compact Kähler manifolds. In particular, for numerical equivalence classes of curves over projective varieties, it is closely related to the mobility functional. Introduction In this paper, all projective varieties are defined over C. The volume of a divisor on projective variety is a non-negative number measuring the positivity of the divisor. Let X be an ndimensional smooth projective variety, and let D be a divisor on X. By definition, the volume of D is defined to be vol(D) := lim sup m→∞ h 0 (X, mD) m n /n! . Thus vol(D) measures the asymptotic growth of the dimensions of the section space of multiplied divisors mD. We call D a big divisor if h 0 (X, mD) has growth of order m n as m tends to infinity, that is, D is big if and only if vol(D) > 0. The pseudo-effective cone of divisors (denoted by E N S ) is the closure of the cone generated by numerical classes of big divisors. It contains the cone of ample divisors as a subcone. It is well known that the volume vol depends only on the numerical class of the divisor, and vol 1/n is homogeneous of degree one, concave on the pseudo-effective cone and extends to a continuous function on the whole real Néron-Severi space which is strictly positive exactly on big classes. In the analytical context, from the work [Bou02a,Bou02b], we know that the volume can be characterized by Monge-Ampère mass, and from the work [Dem10], it can even be characterized by Morse type integrals. In this paper, the starting point is to give a new characterization of the volume of divisors by using cone duality. From the seminal work of Boucksom-Demailly-Paun-Peternell (see [BDPP13]), we know the duality of the pseudo-effective cone of divisors and the cone generated by movable curves, that is, E ∨ N S = M N S . Using this cone duality and an invariant of movable curve, we give the following new volume characterization of divisors by the infimum of intersection numbers between the pairings of E N S and M N S . Theorem 1.1. Let X be an n-dimensional smooth projective variety and let α ∈ N 1 (X, R) be a numerical class of divisor. Then the volume of α can be characterized as following: vol(α) = inf γ∈M NS,1 max( α, γ , 0) n where M N S,1 is a subset of the movable cone M N S (see Definition 2.3). Conversely, this volume characterization implies the cone duality E ∨ N S = M N S . Furthermore, we can also replace the movable cone M N S by the Gauduchon cone G or balanced cone B which is generated by special hermitian metrics. Remark 1.1. Under the conjecture on weak transcendental holomorphic Morse inequalities (see [BDPP13]), the above result also holds true for any Bott-Chern (1, 1)-class over compact Kähler manifolds. In particular, even without this assumption, for any α ∈ H 1,1 BC (X, R) over a hyper-Kähler manifold X, we have vol(α) = inf γ∈M 1 max( α, γ , 0) n . Inspired by the above volume characterization for divisors, using cone dualities, we introduce a volume functional for 1-cycles over compact Kähler manifolds. For smooth projective variety, by Kleiman's criterion, we have the cone duality Amp ∨ = N E where Amp is the ample cone generated by ample divisors and N E is the cone generated by irreducible curves. For ndimensional compact Kähler manifold, by Demailly-Paun's numerical characterization of Kähler cone (see [DP04]), we have the cone duality K ∨ = N where K is the Kähler cone generated by Kähler classes and N is the cone generated by d-closed positive (n − 1, n − 1)-currents. Definition 1.1. (1) Let X be an n-dimensional smooth projective variety, and let γ ∈ N 1 (X, R) be a numerical equivalence class of curve. Let Amp 1 be the set containing all numerical classes of ample divisors of volume one. Then the volume of γ is defined to be vol N E (γ) = inf β∈Amp 1 max( β, γ , 0) n n−1 . (2) Let X be an n-dimensional compact Kähler manifold, and let γ ∈ H n−1,n−1 BC (X, R) be a Bott-Chern (n − 1, n − 1)-class. Let K 1 be the set containing all Kähler classes of volume one. Then the volume of γ is defined to be vol N (γ) = inf γ∈K 1 max( β, γ , 0) n n−1 . From its definition, it is clear vol N E (resp. vol N ) has concave property. It also has other nice properties. Theorem 1.2. Let X be an n-dimensional smooth projective variety (resp. compact Kähler manifold). Then vol N E (resp. vol N ) is a continuous function on the whole vector space N 1 (X, R) (resp. H n−1,n−1 BC (X, R)). Furthermore, γ ∈ N E • (resp. N • ) if and only if vol N E (γ) > 0 (resp. vol N (γ) > 0). The functional vol N E is closely related to the mobility functional recently introduced by Lehmann (see [Leh13]). Mobility functional for cycles was suggested in [DELV11] as an analogue of the volume function for divisors. The motivation is that one can interpret the volume of a divisor D as an asymptotic measurement of the number of general points contained in members of \|mD\| as m tends to infinity. Let γ be a numerical equivalence class of k-cycles over an ndimensional integral projective variety X, following [DELV11], Lehmann defined the mobility of γ as following: where mc(mγ) is the mobility count of the cycle class mγ, which is the maximal non-negative integer b such that any b general points of X are contained in a cycle of class mγ. In particular, we can define the mobility for numerical classes of curves. Lehmann proved that the mobility functional also distinguishes interior points and boundary points. Thus, in the situation of curves, combining with Theorem 1.2, we have two functionals with this property. It is interesting to compare mob and vol N E over N E. The optimistical expectation is that there are two positive constants c 1 , c 2 depending only on the dimension of the underlying manifold such that c 1 vol N E (γ) ≤ mob(γ) ≤ c 2 vol N E (γ) for any γ ∈ N E. Moreover, we expect vol N E (γ) = mob(γ). In this paper, we obtain the positive constant c 2 by using Lehamnn's estimates of mobility count functional mc. In a subsequent joint work [LX15] with Lehmann, besides other results, we will obtain the positive constant c 1 . Indeed, for any fixed ample divisor A and boundary point γ ∈ ∂N E, it is not hard to obtain the asymptotic behaviour of the quotient mob(γ + εA n−1 )/ vol N E (γ + εA n−1 ) as ε tends to zero. Theorem 1.3. Let X be an n-dimensional smooth projective variety, and let N E be the closure of the cone generated by effective 1-cycles. Then for any γ ∈ N E, we have mob(γ) ≤ n!2 4n+1 vol N E (γ). And for any fixed ample divisor A and boundary point γ ∈ ∂N E, there is a positive constant c(A, γ) such that mob(γ + εA n−1 ) ≥ c(A, γ)ε vol N E (γ + εA n−1 ). In particular, we have lim inf ε→0 mob(γ + εA n−1 ) ε vol N E (γ + εA n−1 ) ≥ c(A, γ). With respect to our volume functional vol N , we want to study Fujita type approximation results for 1-cycles over compact Kähler manifolds. In this paper, following Boucksom's analytical version of divisorial Zariski decomposition (see [Bou04], [Bou02b]) (for the algebraic approach, see [Nak04]), we study Zariski decomposition for 1-cycles. In divisorial Zariski decomposition, the negative part is an effective divisor of Kodaira dimension zero, and indeed it contains only one positive (1, 1)-current. In our setting, we can prove this fact also holds for big 1-cycles. Comparing with other definitions of Zariski decomposition for 1-cycles (see e.g. [FL13]), effectiveness of the negative part is one of its advantage. Using his characterization of volume by Monge-Ampère mass, Boucksom showed that the Zariski projection preserves volume. It is also expected that in our setting the Zariski projection preserves vol N . Indeed, this follows from our another kind of Zariski decomposition for 1-cycles developed in [LX15], which is more closely related to vol N . Theorem 1.4. Let X be an n-dimensional compact Kähler manifold and let γ ∈ N • be an interior point. Let γ = Z(γ) + {N (γ)} be the Zariski decomposition in the sense of Boucksom, then N (γ) is an effective curve and it is the unique positive current contained in the negative part {N (γ)}. As a consequence, this implies vol N ({N (γ)}) = 0. Moreover, we have vol N (γ) = vol N (Z(γ)). 2 Characterizing volume for divisors 2.1 Technical preliminaries Smoothing movable classes Besides the well known cone duality E ∨ N S = M N S , we also have cone dualities between the cone defined by positive currents and the cone defined by positive forms. They provide a method to smooth movable classes, which will be useful in volume characterization by using special metrics. Let X be an n-dimensional compact complex manifold, then we have Bott-Chern cohomology groups H •,• BC (X, K) and Aeppli cohomology groups H •,• A (X, K) with K = R or C. Recall that we have canonical duality between H •,• BC (X, K) and H n−•,n−• A (X, K) (see [AT13]). Definition 2.1. Let X be an n-dimensional compact complex manifold. (1) The cone E is defined to be the convex cone in H 1,1 BC (X, R) generated by d-closed positive (1, 1)-currents; (2) The cone E A is defined to be the convex cone in H 1,1 A (X, R) generated by dd c -closed positive (1, 1)-currents; (3) The balanced cone B is defined to be the convex cone in H n−1,n−1 BC (X, R) generated by dclosed strictly positive (n − 1, n − 1)-forms; (4) The Gauduchon cone G is defined to be the convex cone in H n−1,n−1 A (X, R) generated by dd c -closed strictly positive (n − 1, n − 1)-forms. Under the duality of H 1,1 BC (X, R) and H n−1,n−1 A (X, R) and the duality of H 1,1 A (X, R) and H n−1,n−1 BC (X, R), we have the following cone dualities between the above positive cones. Proposition 2.1. Let X be an n-dimensional compact complex manifold, then we have E ∨ = G and E ∨ A = B Proof. Indeed, the above cone dualities are consequences of geometric Hahn-Banach theorem, for example, one can see [Sul76], [Lam99] or [Tom10]. For reader's convenience, let us sketch its proof. Firstly, we prove E ∨ = G. Let α be a real smooth (1, 1)-form. Applying Lamari's characterization of positive (1, 1)-currents, we know that there exists a distribution ψ such that α + dd c ψ is a positive (1, 1)-current if and only if Recall that the cone of movable curves M N S is generated by numerical equivalence classes of curves of the form µ * (Ã 1 ∧ ... ∧Ã n−1 ), where µ :X → X ranges among all modifications with X smooth projective andÃ 1 , ...,Ã n−1 range among all ample divisors overX. And its transcendental version is the movable cone M ⊆ H n−1,n−1 BC (X, R) over a compact Kähler manifold X. M is the cone generated by all the Bott-Chern classes of the form [µ * (ω 1 ∧ ... ∧ω n−1 )] BC , where µ :X → X ranges among all modifications withX Kähler andω 1 , ...,ω n−1 range among all Kähler metrics overX. Our first observation is that any current µ * (ω 1 ∧ ... ∧ω n−1 ) can be smoothed to be a Gaudu- chon metric G such that [µ * (ω 1 ∧ ... ∧ω n−1 )] A = [G] A . Proposition 2.2. Let µ :X → X be a modification between compact complex manifold, and letG be a Gauduchon metric onX. Then µ * G can be smoothed to be a Gauduchon metric G such that [µ * G ] A = [G] A . Proof. From the cone duality E ∨ = G, in order to prove [µ * G ] A ∈ G, we only need to verify that [µ * G ] A is an interior point of E ∨ (= G). For any α ∈ E \ {[0] BC }, since the pull-back µ * α is also pseudo-effective, we have [µ * G ] A , α = [G] A , µ * α ≥ 0. Take a positive currentT ∈ µ * α, then we have [G] A , µ * α = G ∧T . By the strictly positivity ofG, G ∧T = 0 if and only ifT = 0, and this contradicts to our assumption α = [µ * T ] BC ∈ E \ {[0] BC }. Thus [µ * G ] A ,] A = [G] A . Indeed, the current µ * (ω 1 ∧ ... ∧ω n−1 ) can not only be smoothed to be a Gauduchon class, it can also smoothed to be a balanced metric B such that [µ * (ω 1 ∧...∧ω n−1 )] BC = [B] BC . From the proof of Proposition 2.2, we see that a key ingredient is that the pull-back of cohomology class in E contains positive currents. Analogue to this fact, due to a result of [AB95], one can also always pull back Aeppli class in E A and get dd c -closed positive (1, 1)-currents on the manifold upstairs. Lemma 2.1. (see [AB95]) Let µ :X → X be a modification between compact complex manifold, and let T be a dd c -closed positive (1, 1)-current on X. Then there exists an unique dd c -closed positive (1, 1)-currentT ∈ µ * [T ] A such that µ * T = T . We remark that the above fact is already used by Toma (see [Tom10]), and the following proposition is essentially due to Toma. Proof. Similar to the proof of Proposition 2.2, we only need to show [µ * B ] BC , α > 0 for any α ∈ E A \ {[0] A }. Now, by using Lemma 2.1, for any α = [T ] A ∈ E A \ {[0] A }, one can find a non-zero dd c -closed positive (1, 1)-currentT ∈ µ * α, so we have [µ * B ] BC , α = [B] BC , µ * α = B ∧T > 0. And as a consequence, there exists a balanced metric B such that [µ * B ] BC = [B] BC . An invariant of movable classes In this subsection, we introduce an (universal) invariant M for movable, balanced or Gauduchon classes. This invariant is defined by cone duality and intersection numbers. We will see that they coincide with the volume of Kähler classes if the cohomology classes are given by the (n − 1)-power of Kähler classes. Definition 2.2. Let X be an n-dimensional compact Kähler manifold, and let γ be a movable (or balanced, or Gauduchon) class. Let E 1 be the set of pseudo-effective classes of volume one. Then the invariant M(γ) is defined as following: M(γ) := inf β∈E 1 β, γ n n−1 . Remark 2.1. In the case when X is a smooth projective variety, we can also define M(γ) for γ ∈ M N S . In this situation, the parings β, γ are the parings of numerical equivalence classes of divisors and curves. Remark 2.2. Recall that we have the cone dualities E ∨ = G and E ∨ A = B. Indeed, under the assumption of the conjectured transcendental cone duality E ∨ = M (see [BDPP13]), the movable cone M, the balanced cone B and the Gauduchon cone G should be the same, that is, [FX14]). This is why we call M is an universal invariant associated to movable, balanced and Gauduchon classes over compact Kähler manifolds. E ∨ = M = B = G (see e.g. It is clear that, from its definition, we have M(γ 1 + γ 2 ) n−1 n ≥ M(γ 1 ) n−1 n + M(γ 2 ) n−1 n . Proposition 2.4. Let X be an n-dimensional compact Kähler manifold, and let γ = ω n−1 for some Kähler class ω, then we have M(γ) = vol(ω). Proof. Firstly, let β = ω vol(ω) 1 n , it is clear that β, ω n−1 = vol(ω) n−1 n , which implies M(γ) ≤ vol(ω). On the other hand, we claim that, for any β ∈ E with vol(β) = 1, we have β, γ n n−1 ≥ vol(ω). This is just the Khovanskii-Teissier inequality which follows from the singular version of Calabi-Yau theorem (see [Bou02a]): there exists a positive (1, 1)-current T ∈ β such that T n ac = Φ almost everywhere, where Φ = ω n /vol(ω) and T ac is the absolutely continuous part of T with respect to Lebesgue measure. Here we use the same symbol ω to denote a Kähler metric in the Kähler class ω. Then we have β, γ = T ∧ ω n−1 ≥ T ac ∧ ω n−1 ≥ ( T n ac Φ ) 1 n ( ω n Φ ) n−1 n Φ = vol(ω) n−1 n . This implies the claim, thus finishing our proof. An easy corollary is the strictly positivity of M. Corollary 2.1. Let X be an n-dimensional compact Kähler manifold, and let γ ∈ G (resp. M or B) be an interior point, then we have M(γ) > 0. Next let µ :X → X be a modification between compact Kähler manifolds, we want to study the behaviour of M under µ. Firstly, we need the following elementary fact on the transform of the volume of pseudo-effective (1, 1)-classes under bimeromorphic maps. Lemma 2.2. Let µ :X → X be a modification between n-dimensional compact Kähler manifolds. Assume β ∈ E is a pseudo-effective class on X, then vol(β) = vol(µ * β); Assumeβ ∈ E is a pseudo-effective class onX, then vol(β) ≤ vol(µ * β ). Proof. Recall that the volume of β is defined to be the supremum of Monge-Ampère mass, that is, vol(β) = sup T T n ac , where T ranges among all positive (1, 1)-currents in the class β. For any positive current T ∈ β, we obtain a positive current µ * T ∈ µ * β. By the definition of the absolutely part with respect to Lebesgue measure, we have (µ * T ) ac = µ * T ac . And T ac is a (1, 1)-form with L 1 loc coefficients. In particular, analytic subset is of zero measure with respect to the measure T n ac , which yields (µ * T ) n ac = T n ac . This implies vol(µ * β) ≥ vol(β). On the other hand, for any positive currentT ∈ µ * β, we get a positive current µ * T ∈ β. By (µ * T ) ac = µ * Tac , we obtain vol(µ * β) ≤ vol(β). All in all we have vol(β) = vol(µ * β). Similarly, it is also easy to see vol(β) ≤ vol(µ * β ) for anyβ ∈ E. Now we can show that M has the same property as vol under bimeromorphic maps. We only state the result for Kähler manifolds. It is clear that M admits an extension to the closure of G (resp. M or B). Proposition 2.5. Let µ :X → X be a modification between n-dimensional compact Kähler manifolds. Assume γ ∈ G (resp. M or B ), then M(γ) = M(µ * γ); Assumeγ ∈ G (resp. M or B), then M(γ) ≤ M(µ * γ ). Proof. We firstly consider the pull-back case. By Lemma 2.2, for any fixedβ ∈Ẽ with vol(β) = 1, we have β , µ * γ = µ * β , γ ≥ µ * β vol(µ * β ) 1/n , γ ≥ M(γ) n−1 n . This clearly implies M(µ * γ) ≥ M(γ). For the other direction, for any fixed β ∈ E with vol(β) = 1, using Lemma 2.2 again, we have β, γ = µ * (µ * β), γ = µ * β, µ * γ ≥ M(µ * γ) n−1 n . Thus, M(µ * γ) ≤ M(γ), and as a consequence, we finish the proof of M(µ * γ) = M(γ). For the push-forward case, the proof of M(γ) ≤ M(µ * γ ) is the same. We remark that the inequality M(γ) ≤ M(µ * γ ) is important in the characterization of the volume of divisors in the following section. Volume characterization In this section, using the invariant M introduced in the previous section, we show the volume of divisors can be characterized by cone duality. Definition 2.3. Let X be an n-dimensional smooth projective variety, and let M N S be the cone of movable curves. Then M N S,1 is defined to be the subset containing all γ ∈ M N S with M(γ) = 1. Similarly, for compact Kähler manifolds, we can define G 1 , M 1 and B 1 in the same way. Theorem 2.1. Let X be an n-dimensional smooth projective variety and let α ∈ N 1 (X, R) be a numerical equivalence class of divisor. Then the volume of α can be characterized as following: (⋆) vol(α) 1 n = inf γ∈M NS,1 max( α, γ , 0). Conversely, this volume characterization implies the cone duality E ∨ N S = M N S . Moreover, we can also replace the cone of movable curves by Gauduchon or balanced cone, that is, vol(α) 1 n = inf γ∈G 1 max( α, γ , 0) = inf γ∈B 1 max( α, γ , 0). Proof. We first consider the case when α is not pseudo-effective, by the definition of volume of divisors, it is clear vol(α) = 0. On the other hand, the cone duality E ∨ N S = M N S implies there exists some interior point γ ∈ M N S such that α, γ < 0. Furthermore, using Corollary 2.1, we can even normalize γ such that M(γ) = 1. Thus inf γ∈M NS,1 max( α, γ , 0) = 0 = vol(α) 1 n . Next consider the case when α is given by a big divisor. By the very definition of M, for any γ ∈ M N S , it is clear that α vol(α) 1/n , γ ≥ M(γ) n−1 n , or equivalently, α, γ ≥ vol(α) 1 n M(γ) n−1 n . In particular, for any γ ∈ M N S,1 , this yields α, γ ≥ vol(α) 1 n . Thus we have vol(α) 1 n ≤ inf γ∈M NS,1 α, γ . In order to prove the equality, we need to show that, for any ε > 0, there exists a movable class γ ε ∈ M N S,1 such that α, γ ε ≤ vol(α) 1 n + ε. This mainly depends on approximating Zariski decomposition of Kähler currents and orthogonality estimates of the decomposition (see [BDPP13]). Since α is given by a big divisor, for any δ > 0, there exists a modification µ δ : X δ → X such that µ * δ α = β δ + [E δ ] with β δ given by an ample divisor and E δ given by an effective divisor. Moreover, we also have vol(α) − δ ≤ vol(β δ ) ≤ vol(α)(1) and [E δ ], β n−1 δ ≤ c(vol(α) − vol(β δ )) 1/2 (2) where c is a positive constant depending only on the class α and dimension n. Applying (1) and (2) to α, µ δ * β n−1 δ , we get α, µ δ * (β n−1 δ ) = µ * δ α, β n−1 δ (3) = vol(β δ ) + [E δ ], β n−1 δ (4) ≤ vol(α) + O(δ 1/2 ).(5) Next by Proposition 2.4 and Proposition 2.5, we know that M(µ δ * (β n−1 δ )) ≥ M(β n−1 δ ) = vol(β δ ).(6) We claim that γ δ := µ δ * (β n−1 δ )/M(µ δ * (β n−1 δ )) n−1 n is our desired movable class. Firstly, by the definition of M, it is obvious that M(γ δ ) = 1. Secondly, by using (1) and (6), we can estimate α, γ δ as following: α, γ δ ≤ µ * δ α, β n−1 δ vol(β δ ) n−1/n (7) ≤ vol(α) 1/n + [ vol(α) (vol(α) − δ) n−1 n − vol(α) 1/n ] + O(δ 1/2 )(8) Thus, for any ε > 0, we can choose some δ(ε) > 0, such that γ δ(ε) is our desired movable class. In summary, we have finished the proof of the equality vol(α) 1 n = inf γ∈M NS,1 α, γ for big class α. In the case when α lies on the boundary of E N S , for any ε > 0 and ample divisor A, apply the above proved equality for α + εA, we have vol(α + εA) 1 n = inf γ∈M NS,1 α + εA, γ . Take inf on both sides with respect to ε > 0, we get the equality for boundary class. For the volume characterization by Gauduchon or balanced cone, from the proof for movable cone, one can see that if we can show γ δ(ε) can be smoothed to be a Gauduchon or balanced class, then we have the desired equality. And this just follows from the results of Proposition 2.2 and Proposition 2.3. Now we show that (⋆) implies E ∨ N S = M N S . It is obvious E N S ⊆ M Remark 2.3. Let X be an n-dimensional compact Kähler manifold. Under the assumption of the conjectured weak transcendental holomorphic Morse inequality, that is, vol(α − β) ≥ α n − nα n−1 · β for any nef classes α, β (for recent progress of this problem, one can see [Xia13], [Pop14]), then we will also have orthogonality estimates for interior points of E (see [BDPP13]). By the arguments above, we will have volume characterization for any Bott-Chern (1, 1)-class α, that is, vol(α) 1 n = inf γ∈M 1 max( α, γ , 0). Moreover, this implies the cone duality E ∨ = M. Thus it is natural to ask whether one can prove this volume characterization without using orthogonality estimates of approximation Zariski decomposition. And this also provides new perspectives to prove the conjectured cone duality E ∨ = M. 3 Volume functional for 1-cycles Definition and properties Inspired by Theorem 2.1, using cone dualities, we introduce a volume functional for the numerical equivalence class of curves over smooth projective varieties and a volume functional for Bott-Chern (n − 1, n − 1)-classes over compact Kähler manifolds. For smooth projective variety, we have the ample cone Amp generated by ample divisors and the cone N E generated by irreducible curves. Then we have the cone duality Amp ∨ = N E which is just Kleiman's criterion. For n-dimensional compact Kähler manifold, we have Kähler cone K generated by Kähler classes and the cone N generated by d-closed positive (n − 1, n − 1)currents. Then we have the cone duality K ∨ = N which follows from Demailly-Paun's numerical characterization of Kähler cone (see [DP04]). Now we can give the following definition. (2) Let X be an n-dimensional compact Kähler manifold, and let γ ∈ H n−1,n−1 BC (X, R) be a Bott-Chern (n − 1, n − 1)-class. Let K 1 be the set containing all Kähler classes of volume one. Then the volume of γ is defined to be vol N (γ) = inf Unlike vol giving an uniform volume functional on E and K, we do not know whether they would coincide on the movable cone. In general, the nef cone K can be strictly contained in E, it seems possible that M may be smaller than vol N . However, if X is a projective or compact Kähler surface, both our volume functional vol N (or vol N E ) and M coincide with the usual volume for pseudo-effective classes. Example 3.1. To illustrate the definition of volume functional for 1-cycles, we propose to do some concrete calculations on an example similar to the one due to Cutkosky [Cut86] (it is also contained in [Bou04]). Let Y be a smooth projective surface, and let D, H be two very ample divisors over Y . Let X = P(O(D) ⊕ O(−H)) with its canonical projection π : X → Y . Denote by L = O X (1) the tautological bundle of X, then the nef cone K X of X is generated by π * K Y and π * H + L. In Cutkosky's example, Y is an Abelian surface (or more generally, a projective surface with K Y = E Y ). For simplicity, we consider the very simple case Y = P 2 with D = O(d), H = O(1), then we have L 3 = (d − 1) 2 + d, π * H 2 · L = 1, π * H · L 2 = d − 1, π * H 3 = 0. Let α = aπ * H + b(π * H + L) with a, b ∈ R + be a nef class, then the volume of α is as following vol(α) = b 3 ((d − 1) 2 ) + d) + 3b 2 (a + b)(d − 1) + 3(a + b) 2 b. Consider the 1-cycle γ(x, y) = xπ * H 2 + yπ * H · L with x, y ≥ 0, then we have α, γ(x, y) = (a + b)y + bx + by(d − 1). From the above expressions, we have an explicit formula of vol N E . In particular, if we take d = 1, then vol N E (γ(x, y)) = inf b 3 +3(a+b) 2 b=1 a,b≥0 (by + (a + b)x) 3 2 . The volume functionals vol N and vol N E have many nice properties. For simplicity, we only state the result for vol N . The argument for vol N E is similar. Proof. Property (1) just follows from the definition of vol N . Now let us first prove property (3). Let γ ∈ N • be an interior point, we want to show that vol N (γ) > 0. γ ∈ N • means that there exists some Kähler class ω such that γ − ω n−1 ∈ N , this implies vol N (γ) ≥ vol N (ω n−1 ). We claim that vol N (ω n−1 ) = vol(ω), The arguments are similar with the estimation of ρ ε , but with little modification. Once again, using the fact γ ∈ ∂N \ {[0] BC }, there exists some θ ∈ K \ {[0] BC } such that θ, γ = 0. We consider the following intersection number ρ δ,ε := θ + δω vol(θ + δω) 1/n , γ + εω n−1 (10) with δ positive to be determined. Using θ, γ = 0 and θ, ω n−1 > 0 again, it is easy to see that ρ δ,ε ≤ O(δ 1 n + δ 1 n ε + δ − n−1 n ε).(11) Take δ = ε, we get ρ δ,ε ≤ O(ε 1/n ), which implies vol N (γ + εω n−1 ) ≤ O(ε 1 n−1 ),(12) thus finishing the proof of continuity. We give a new interpretation of our volume functional as the infinimum of a family of geometric norms. We only work for vol N , and the arguments go through mutatis mutandis for the volume functional vol N E . Lemma 3.1. (see also Corollary 2.8 of [FL13]) Let X be an n-dimensional compact Kähler manifold. Then any Kähler class α gives a norm \|\| · \|\| α over H n−1,n−1 BC (X, R). Moreover, for γ ∈ N , we have \|\|γ\|\| α = α, γ . Proof. For any fixed Kähler class α, there exist d = h 1,1 Kähler classes α 1 , ..., α d such that α 1 , ..., α d constitute a basis of the real vector space H 1,1 BC (X, R), and α = 1≤i≤d α i . Then for any η ∈ H n−1,n−1 BC (X, R), we define \|\|η\|\| α as following: \|\|η\|\| α = 1≤i≤d \| α i , η \|. It is clear that the above \|\| · \|\| α is a norm, since it is just the sum of absolute values of the coordinates with respect to the basis α 1 , ..., α d . Now, for γ ∈ N , we have α i , γ ≥ 0. And this implies \|\|γ\|\| α = 1≤i≤d α i , γ = α, γ . Now by the definition of vol N , we have the following proposition. Relation with mobility In this section, we focus on comparing vol N E and Lehmann's mobility functional mob for 1cycles over smooth projective variety. Firstly, let us recall the definition of mobility of numerical equivalence classes of curves. Let γ be a 1-cycle class over X of dimension n, the mobility of γ is defined as following: Conjecture 3.1. Let X be a smooth projective variety, then mob = vol N E , or at least there exist two positive constants c 1 and c 2 depending only on the dimension of X such that c 1 vol N E ≤ mob ≤ c 2 vol N E . We observe that the constant c 2 is provided by the upper bound estimation of mobility count. For any fixed ample divisor A and boundary point γ ∈ ∂N E, it is clear that if we can find a positive constant c(A, γ) such that lim inf ε→0 mob(γ + εA n−1 ) vol N E (γ + εA n−1 ) ≥ c(A, γ), then we can obtain the desired uniform constant c 1 . In this direction, we can get a weaker asymptotic behaviour as ε tends to zero. Theorem 3.2. Let X be an n-dimensional smooth projective variety. Then for any γ ∈ N E, we have mob(γ) ≤ n!2 4n+1 vol N E (γ). And for any fixed ample divisor A and boundary point γ ∈ ∂N E, there is a positive constant c(A, γ) such that mob(γ + εA n−1 ) ≥ c(A, γ)ε vol N E (γ + εA n−1 ). In particular, we have lim inf ε→0 mob(γ + εA n−1 ) ε vol N E (γ + εA n−1 ) ≥ c(A, γ). Proof. The upper bound c 2 relies on the estimations of mobility counts. By homogeneity and continuity, we only need to consider the case when γ is given by a 1-cycle with Z-coefficients. Indeed, by inspection of the proof of Theorem 6.24 of [Leh13], any real number s ≥ 1 is sufficient for the above estimation of mc(γ). Fix a Q-ample divisor α, then there exists a positive integer m α such that m α α is very ample. And for this very ample divisor m α α, there exists a positive integer k α such that m α α, kγ vol(m α α) ≥ 1(13) for all positive integer k ≥ k α Applying Lehmann's mobility count estimation to kγ when A = m α α and s = mαα,kγ vol(mαα) , we get mc(kγ) ≤ 2 4n+1 m α α, kγ vol(m α α) n n−1 vol(m α α)(14) = 2 4n+1 α vol(α) 1/n , kγ Since any point of the ample cone can be approximated by Q-ample divisors, we obtain our desired mob(γ) ≤ n!2 4n+1 vol N E (γ).(17) Now let us consider the lower bound. In the proof of Theorem 3.1 (see (9)-(12)), we obtain the estimation of vol N (γ + εω n−1 ). Using similar argument, we can get the same estimation of vol N E (γ + εA n−1 ) with γ ∈ ∂N E and A ample, that is, Remark 3.2. In order to obtain such an uniform lower bound c 1 , what we expect is a better estimation of mob(γ + εA n−1 ), that is, vol N E (γ + εA n−1 ) ≤ O(ε 1 n−1 ).(18)mob(γ + εA n−1 ) ≥ O(ε 1 n−1 ), as ε tends to zero. To obtain this, we need deeper understanding of mob. Remark 3.3. Just from its definition, the mobility functional mob seems very hard to compute. For example, even in the case of complete intersection of ample divisor (see Question 7.1 of [Leh13]), we do not know how to calculate its mobility. However, using our volume functional, we have seen that vol N E (A n−1 ) = vol(A) for any ample divisor A. For the concavity of mob, it is conjectured (see Conjecture 6.20 of [Leh13]) that mob(γ 1 + γ 2 ) n−1 n ≥ mob(γ 1 ) n−1 n + mob(γ 2 ) n−1 n . For our vol N E , concavity just follows from its definition (see Theorem 3.1). Thus concavity of mob will follow if we can prove mob = vol N E . Towards Fujita approximation for 1-cycles In the work of [FL13], Fulger and Lehmann proved the existence of Zariski decomposition for big cycles with respect to mobility functional. Moreover, they also proved a Fujita type approximation for numerical class of curves. Our goal is to give such a Fujita type approximation for Bott-Chern classes of d-closed positive (n−1, n−1)-currents over compact Kähler manifolds with respect to our volume functional vol N , thus also give a Fujita type approximation for numerical class of curves over projective variety with respect to vol N E . Analogue to Fujita approximation for Kähler currents (see inequality (1)), one may conjecture the following: Let X be an n-dimensional compact Kähler manifold and let γ ∈ N • . Then for any ε > 0, there exists a proper modification µ : X → X with X Kähler such that µ * γ = β ε + [C ε ] and vol N (γ) − ε ≤ vol N (β ε ) ≤ vol N (γ), where β ε is an interior point of movable cone M (or balanced cone B) and C ε is an effective curve. Indeed, if we have the decomposition µ * γ = β ε + [C ε ], then we have γ − µ * β ε ∈ N . This implies vol N (µ * β ε ) ≤ vol N (γ). Now similar to Proposition 2.5, it is easy to see vol N (β ε ) ≤ vol N (µ * β ε ). Thus the above expected decomposition automatically implies vol N (β ε ) ≤ vol N (γ). Unfortunately, the pull-back µ * γ need not to be a pseudo-effective class in general. Note that µ * γ is pseudo-effective over X if and only if µ * γ,α ≥ 0 for any Kähler classα, which is equivalent to γ, µ * α ≥ 0. In general, µ * α is not a nef class on X. By the cone duality K ∨ = N , we have γ, µ * α < 0 if µ * α / ∈ K. Anyhow, if γ ∈ M is movable, then its pull-back µ * γ is also movable (thus pseudo-effective). For movable classes, it is possible to obtain the conjectured decomposition µ * γ = β ε + [C ε ] with desired properties. To prove Fujita approximation for γ with respect to our volume functional, the first step of our strategy is to decompose γ over the underlying manifold X into some "good" part with its volume near the volume of γ. We also call it the positive part, and call the difference the negative part. Here "good" means we can find a positive current in the class with less singularities, then we may get a movable or balanced class from its pull-back on some Kähler manifold X such that its volume is as near vol N (γ) as possible (this will be developed in our subsequent work [LX15]). Besides the desired positive part, we also want to obtain some effective curve from such a decomposition. For the Zariski decomposition of Fulger and Lehmann, in general the negative part is not the class of an effective curve (see Example 5.18 of [FL13]). In the work [Bou04], Boucksom defined a beautiful divisorial Zariski decomposition for any pseudo-effective (1, 1)class over compact complex manifolds. Boucksom's definition is totally analytic which depends on Siu decomposition of positive currents (see [Siu74]). And it can be seen as a cohomology version of Siu decomposition. As Siu decomposition holds for d-closed positive currents of any bidegree, the method of Boucksom provides a possible Zariski decomposition for pseudoeffective (n − 1, n − 1)-classes. However, unlike the (1, 1)-classes, we do not have an analogue of Demailly's regularization theorem (see [Dem92]) for d-closed positive (n − 1, n − 1)-currents. We know little about the singularities of such currents. Thus we can not expect too much about such decompositions. Following Boucksom's method of divisorial Zariski decomposition, we give such a decomposition for pseudo-effective (n − 1, n − 1)-classes. It shares many nice properties with divisorial Zariski decomposition. Firstly, we give the definition of minimal multiplicity. Definition 3.2. Let X be an n-dimensional compact Kähler manifold with a Kähler metric ω, and let γ ∈ N be a pseudo-effective (n − 1, n − 1)-class. (1) The minimal multiplicity of γ at the point x is defined to be ν(γ, x) := sup ε>0 inf Tε ν(T ε , x), where T ε ∈ γ ranges among all currents such that T ε ≥ −εω n−1 (we also denote this set by γ[−εω n−1 ]) and ν(T ε , x) is the Lelong number of T ε at x. (2) For any irreducible curve C, the minimal multiplicity of γ along C is defined to be ν(γ, C) := inf x∈C ν(γ, x). Remark 3.4. It is easy to see that ν(γ, x) is finite. And ν(γ, C) = ν(γ, x) for a generic point x ∈ C, here generic means outside at most countable union of analytic subsets. Intuitively, the positive part Z(γ) should share almost all positivity of γ and the negative part should have very little positivity. Indeed, in the divisorial Zariski decomposition case, using his volume characterization by Monge-Ampère mass, Boucksom showed that vol(α) = vol(Z(α)) for any α ∈ E over compact Kähler manifolds. In our setting, one way to compare the positivity of Z(γ) and γ is to compare their respective volumes vol N (Z(γ)) and vol N (γ). For the negative part {N (γ)}, like the one in divisorial Zariski decomposition, we find N (γ) is an effective curve which is very rigidly embedded in X if we assume γ is an interior point. This is an advantage compared with the other decompositions (e.g. the decompositions in [FL13] and [LX15]). Remark 3.5. From Theorem 3.1, it is clear that vol N (γ) = vol N (Z(γ)) = 0 if γ ∈ ∂N . And by the concavity of vol N , the equality vol N (γ) = vol N (Z(γ)) will imply vol N ({N (γ)}) = 0. Proof. We first prove the first part of the above theorem. Indeed, as the Zariski decomposition here is an (n − 1, n − 1)-analogue of Boucksom's divisorial Zariski decomposition, the statement concerning N (γ) can be proved using almost the same arguments as in [Bou04]. In [Bou04], some arguments use Demailly's regularization theorem. As we do not have such a regularization theorem for (n − 1, n − 1)-currents, for reader's convenience, we present the details here. The assumption γ ∈ N • will play the role as Demailly's regularization theorem in the divisorial Zariski decomposition situation. We Proof. To prove this, we only need to verify ν(γ, x) = inf 0≤T ∈γ ν(T, x) for any point x, then we will have ν(γ, C) = inf x∈C ν(γ, x) = inf x∈C inf 0≤T ∈γ ν(T, x) = inf 0≤T ∈γ ν(T, C). From the definition of ν(γ, x), we only need to prove ν(γ, x) ≥ inf 0≤T ∈γ ν(T, x).(20) As γ ∈ N • , there exists a positive current T ∈ γ such that T ≥ β n−1 for some Kähler metric β. Fix ε > 0, for any δ > 0 there exists a current T ε,δ ∈ γ[−εβ n−1 ] such that ν(T ε,δ , x) − δ < inf Tε ν(T ε , x),(21) where T ε ranges among γ[−εβ n−1 ]. Since T ≥ β n−1 , we have (1 − ε)T ε,δ + εT ≥ ε 2 β n−1 which is a positive current in γ, thus inf 0≤T ∈γ ν(T, x) ≤ ν((1 − ε)T ε,δ + εT, x) (22) ≤ (1 − ε)inf Tε ν(T ε , x) + (1 − ε)δ + εν(T, x).(23) Now let δ → 0 and then let ε → 0, we get the desired inequality ν(γ, x) ≥ inf Proof. Once again, γ ∈ N • implies there exists a positive current T ∈ γ such that T ≥ β n−1 for some Kähler metric β. Apply Siu decomposition to the d-closed positive current T − β n−1 : T − β n−1 = R + ν(T − β n−1 , C)[C] = R + ν(T, C)[C] for some residue positive current R. Then the definition of N (γ) implies T − β n−1 − N (γ) ≥ 0, which yields T − N (γ) ≥ β n−1 . This implies Z(γ) = {T − N (γ)} ∈ N • . Indeed, by the above arguments, Siu decomposition also shows that any positive current in Z(γ) are of the form T − N (γ) for some positive current T ∈ γ. With Lemma 3.2 and this fact, we get ν(Z(γ), C) = inf 0≤Γ∈Z(γ) ν(Γ, C) (24) = inf 0≤T ∈γ ν(T − N (γ), C) (25) = ν(γ, C) − ν(γ, C) = 0. (26) Lemma 3.4. Let γ ∈ N • , then {N (γ)} = {N ({N (γ)})}. Proof. By the definition of Z(·), it is easy to see that Z(γ 1 + γ 2 )− Z(γ 1 )− Z(γ 2 ) ∈ N for any two γ 1 , γ 2 ∈ N . In particular, we have Z Next we show N (γ) is an effective curve, that is, it is a finite sum of irreducible curves. Indeed, we will show N (γ) is a sum of at most ρ = dim R N 1 (X, R) irreducible curves. This follows from the following lemma. Finally let us prove Zariski projection preserves vol N . By the decomposition developed in [LX15], we know γ ∈ N • can be uniquely decomposed as following: γ = B n−1 γ + ζ γ(27) with B γ nef big, B γ , ζ γ = 0, vol N (B n−1 γ ) = vol N (γ) and ζ γ ∈ ∂N . Denote by the same symbol B γ a smooth (1, 1)-form in the class B γ . Since B γ is nef, for any ε > 0 there exists a smooth function ψ ε such that B γ + εω + i∂∂ψ ε > 0. From this, it is easy to see for any ε > 0 there exists a smooth (n − 2, n − 2)-form Ψ ε such that Ω ε := B n−1 γ + i∂∂Ψ ε ≥ −εω n−1 . Denote by T γ a positive (n − 1, n − 1)-current in the class ζ γ , then Ω ε + T γ ∈ γ[−εω n−1 ]. And by the definition of minimal multiplicity (see Definition 3.2), we get ν(γ, x) = sup ε>0 inf Tε ν(T ε , x) (28) ≤ sup ε>0 ν(Ω ε + T γ , x)(29) = ν(T γ , x). The last line follows because Ω ε is smooth. By Siu decomposition, the above inequality implies Combining with vol N (γ) ≥ vol N (Z(γ)) and vol N (γ) = vol N (B n−1 γ ), we finish the proof of the equality vol N (γ) = vol N (Z(γ)). Remark 3.6. It will be interesting to know whether the statement for N (γ) in Theorem 3.3 is still true for γ ∈ ∂N . Our above arguments show that the assumption γ ∈ N • is important in Lemma 3.2. And we need Lemma 3.2 to prove the other lemmas. Remark 3.7. One may expect that Z(γ) could be represented by some positive smooth (n − 1, n − 1)-form, more precisely, one may expect Z(γ) ∈ M. Thus, by Proposition 2.2 and Proposition 2.3, there exists a smooth positive (n − 1, n − 1)-form in the class Z(γ) if Z(γ) is an interior point of M. However, in general, Z(γ) could not be a movable class. Let π : X → P 3 be the blow-up along a point, and let E = P 2 be the exceptional divisor. Let ω F S be the Fubini-Study metric of P 3 and let P 1 ⊆ E be a line of E, then we claim that Z({π * (ω 2 F S ) + [P 1 ]}) = {π * (ω 2 F S ) + [P 1 ]} ∈ N • \ M. Firstly, it is easy to see {π * (ω 2 F S )+[P 1 ]} ∈ N • which of course implies Z({π * (ω 2 F S )+[P 1 ]}) ∈ N • . For any point x, we can always choose an integration current in the class [P 1 ] but with its support avoiding x. Then we have ν({π * (ω 2 F S ) + [P 1 ]}, x) = 0, which yields the equality Z({π * (ω 2 F S ) + [P 1 ]}) = {π * (ω 2 F S ) + [P 1 ]}. Since we have {π * (ω 2 F S ) + [P 1 ]} · E = −1, the class {π * (ω 2 F S ) + [P 1 ]} can not be movable. Comparing with the Zariski decompositions developed in [FL13] and [LX15], Z(γ) not always being movable is its disadvantage in many applications. Anyhow, if γ ∈ N • , then Lemma 3.2 and Lemma 3.3 show that we can always choose a positive current in the class Z(γ) with its Lelong number along any curve being arbitrarily small. In some sense, this means that Z(γ) is less singular than γ. Indeed, Z(γ) ∈ K if X is a Kähler surface. At the end of this section, we show that Zariski decomposition for 1-cycles is trivial for compact Kähler manifold with nef tangent bundle. Proposition 3.2. Let X be a compact Kähler manifold with nef tangent bundle, then γ = Z(γ) for any γ ∈ N . Indeed, we will have γ = Z(γ) ∈ M. Proof. This follows from Demailly's regularization theorem of positive (1, 1)-currents and our previous work on transcendental holomorphic Morse inequality. If T X is nef, then K = E (see Corollary 1.5 of [Dem92]). Now let α, β ∈ K be two nef classes such that α n − nα n−1 · β > 0, then α − β must be an interior point of E (see [Xia13], [Pop14]). By K = E, α − β must be a Kähler class. In particular, α − tβ ∈ K for t ∈ [0, 1]. Consider the difference vol(α − β) − vol(α), we have vol(α − β) − vol(α) = 1 0 d dt vol(α − tβ)dt = 1 0 −n(α − tβ) n−1 · βdt ≥ −nα n−1 · β, thus vol(α − β) ≥ α n − nα n−1 · β. Using the same arguments as [BDPP13], this of course implies the cone duality E ∨ = M. And this yields E ∨ = M = B (see e.g. [FX14]). Using K = E again, K ∨ = N implies N = B. Since γ ∈ N = B, for any ε > 0 there exists a smooth (n−1, n−1)-form Ω ε ∈ γ such that Ω ε ≥ −εω n−1 . Now by the definition of minimal multiplicity (see Definition 3.2), we get ν(γ, x) = 0 for every point, yielding N (γ) = 0. This implies γ = Z(γ). Further discussions 4.1 Another invariant of movable class As remarked in the previous section, under the assumption of the conjecture on transcendental holomorphic Morse inequality, we will have M = G = B. And invariant of Gauduchon or balanced classes would be invariant of movable class. Inspired by our previous work [FX14], we introduce another invariant M CY of Gauduchon class by using form-type Calabi-Yau equations (or complex Monge-Ampère equations for (n − 1)-plurisubharmonic functions) (see [FWW10], [TW13a], [TW13b]). Definition 4.1. Let X be an n-dimensional compact Kähler manifold, and let γ be a Gauduchon class. Then we define M CY (γ) as following: M CY (γ) := sup Φ,ω {c Φ,ω } where c Φ,ω is a positive constant satisfying ω n = c Φ,ω Φ such that Φ is a smooth volume form with Φ = 1 and ω n−1 ∈ γ is a Gauduchon metric. Assume γ = α n−1 for some α ∈ K, we prove that M CY (γ) = vol(α). Proposition 4.1. Let X be an n-dimensional compact Kähler manifold, and let γ = α n−1 for some Kähler class α. Then we have M CY (γ) = vol(α). Proof. Firstly, since α is a Kähler class, by Calabi-Yau theorem (see [Yau78]), there exists an unique Kähler metric α u ∈ α such that α n u = vol(α)Φ. In particular, c Φ,αu = vol(α), thus vol(α) ≤ M CY (γ). We claim that, for any Φ, ω in the definition of M CY (γ), we have c Φ,ω ≤ vol(α). For any fixed such Φ, ω, we first apply Calabi-Yau theorem to find a Kähler metric α ψ such that α n ψ = vol(α) c Φ,ω ω n . Using the following pointwise inequality ω n−1 ∧ α ψ ≥ ( α n ψ ω n ) 1 n ω n and ω n−1 ∈ γ = α n−1 being Gauduchon, we estimate vol(α) as following: vol(α) = γ ∧ α = ω n−1 ∧ α ψ ≥ ( α n ψ ω n ) 1 n ω n = vol(α) Note that M CY is an analytical invariant by solving non-linear PDEs, and M is an intersectiontheoretic invariant. It will be very interesting to compare M CY and M, and we have the following proposition. Proposition 4.2. Let X be an n-dimensional compact Kähler manifold, and let γ be a Gauduchon class. Then we always have M CY (γ) ≤ M(γ). Moreover, they coincide over Kähler classes, that is, M CY (α n−1 ) = M(α n−1 ) for any Kähler class α. Proof. For any smooth volume form Φ with Φ = 1 and any β ∈ E with vol(β) = 1, by the singular version of Calabi-Yau theorem (see [Bou02a]), there exists a positive (1, 1)-current T ∈ β such that T n ac = Φ almost everywhere. Now for any Gauduchon metric ω n−1 ∈ γ in the definition of c Φ,ω , we get β, γ = T ∧ ω n−1 ≥ T ac ∧ ω n−1 ≥ ( T n ac Φ ) 1 n ( ω n Φ ) n−1 n Φ = c n−1 n Φ,ω . Since β, ω n−1 and Φ are (conditionally) arbitrary, we get M CY (γ) ≤ M(γ). By Proposition 2.4 and Proposition 4.1, we have M CY (α n−1 ) = M(α n−1 ) for any Kähler class α. Remark 4.1. The above proposition also implies that M CY (γ) is always well defined over compact Kähler manifolds, that is, M CY (γ) < ∞, which is not explicit from its definition. Remark 4.2. Let X be an n-dimensional compact Kähler manifold, we do not know whether M CY (γ) = M(γ) for any γ ∈ G. As we always have M CY (γ) ≤ M(γ), we only need to show M CY (γ) ≥ M(γ). We also want to know the behaviour of M CY under bimeromorphic maps (compare with Proposition 2.5). In particular, we do not know whether we have M CY (µ * γ ) ≥ M CY (γ). If this would be true, then we can use this invariant in Theorem 2.1. It will also be very interesting to study the concavity of M CY . To study these problems, we need know more about the family of constants c Φ,ω in the definition of M CY . A general approach This section comes from a suggestion of Mattias Jonsson. Let C ⊆ V be a proper convex cone of a real vector space. Let u : C → R + be a continuous function. Let p > 1 be a constant. Let C ∨ ⊆ V * be the dual of C. In general, we can define the dual of u in the following way: u(x * ) := inf y∈C 1 x * , y q , where C 1 = {y ∈ C\| u(y) = 1} and 1 p + 1 q = 1. This is similar to some kind of Legendre-Fenchel transform. It is easy to see u 1 q is concave and homogeneous of degree one over C ∨ . If we assume u 1/p is concave and homogeneous of degree one. Then we have u(x * ) := inf y∈C ≥1 x * , y q , where C ≥1 = {y ∈ C\| u(y) ≥ 1}. Since u 1/p is concave, C ≥1 is a convex closed subset of C. In our definition of vol N for 1-cycles over compact Kähler manifold, we have C = K, u = vol and p = 1/n. For 1 < k < n − 1, let N k ∈ H k,k BC (X, R) be the cone generated by d-closed positive (k, k)-currents. It will be interesting if one can generalize this kind of construction of volume to N k , thus define a volume functional for general k-cycles. Principally, we first need to define a function u on some kind of smooth positive (n − k, n − k)-forms. However, unlike the case for the cone N , the structure of the dual of N k is not clear (and indeed this problem is still widely open). As a starting point, it will be very interesting to carry out the above general approach over toric varieties. preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Smoothing movable classes . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 An invariant of movable classes . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Volume characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Volume functional for 1-cycles 11 3.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Relation with mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Towards Fujita approximation for 1-cycles . . . . . . . . . . . . . . . . . . . . . . 17 4 Further discussions 22 4.1 Another invariant of movable class . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 A general approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 dd c -closed strictly positive (n − 1, n − 1)-form G (thus G = ω n−1 for some Gauduchon metric ω). Under the natural duality of H 1,1 BC (X, R) and H n−1,n−1 A (X, R), it is clear this implies the cone duality E ∨ = G. Using the same technique (Hahn-Banach theorem), one can also give a characterization of dd c -closed positive (1, 1)-currents. More precisely, there exists a (0, 1)-current θ such that α + ∂θ + ∂θ is a positive (1, 1)-current if and only if α ∧ B ≥ 0 for any d-closed strictly positive (n − 1, n − 1)-form B (thus B = ω n−1 for some balanced metric ω). Under the natural duality of H 1,1 A (X, R) and H n−1,n−1 BC (X, R), this implies the cone duality E ∨ A = B. Proposition 2 . 3 . 23Let µ :X → X be a modification between compact balanced manifold, and letB be a balanced metric onX. Then µ * B can be smoothed to be a balanced metric B such that [µ * B ] BC = [B] BC . ∨ N S . In order to prove the converse inclusion, we only need to show: if α is an interior point of M ∨ N S , then α is also an interior point of E N S (or equivalently, vol(α) > 0). Fix an ample divisor A. Since α is an interior point of M ∨ N S , for ε > 0 small, α − εA also lies in the interior of M ∨ N S . In particular, we have α, γ > εA, γ for any γ ∈ M N S \ [0]. Then (⋆) implies vol(α) 1 n = inf γ∈M NS,1 max( α, γ , 0) ≥ εvol(A) 1 n > 0. Definition 3. 1 . ( 1 ) 11Let X be an n-dimensional smooth projective variety, and let γ ∈ N 1 (X, R) be a numerical equivalence class of curve. Let Amp 1 be the set containing all numerical classes of ample divisors of volume one. Then the volume of γ is defined to be vol N E (γ) = inf β∈Amp 1 max( β, γ , 0) n n−1 . In the volume characterization of divisors, using the cone duality E ∨ N S = M N S (or the conjectured E ∨ = M), we introduce an invariant M for movable classes (see Definition 2.2). Now vol N gives another invariant of movable classes when it restricts on M. From their definitions, it is clear we have M(γ) ≤ vol N (γ) for any γ ∈ M. Theorem 3. 1 . 1Let X be an n-dimensional compact Kähler manifold. Then vol N has the following properties: and homogeneous of degree one.(2) vol N is continuous on the whole vector space H n−1,n−1 BC (X, R).(3) γ ∈ N • if and only if vol N (γ) > 0. which yields vol N (γ) ≥ vol(ω) > 0. The proof of this claim is the same with Proposition 2.4, so we omit it. Conversely, we need to show that if vol N (γ) > 0 then γ ∈ N • . Otherwise, γ ∈ ∂N \ {[0] BC }. And the cone duality K ∨ = N implies there exists some θ ∈ K \ {[0] BC } such that θ, γ = 0. Fix a Kähler class ω. For any ε > 0, we consider the Kähler class θ + εω and the following intersection numberρ ε := θ + εω vol(θ + εω) 1/n , γ .Since θ ∈ K \ {[0] BC }, the class θ contains at least one non-zero positive current, then we have θ, ω n−1 > 0. And we have vol(θ + εω) , γ = 0, we get ρ ε ≤ O(ε 1/n ). Thus, vol N (γ) = 0. In conclusion, we have proved that γ ∈ N • if and only if vol N (γ) > 0.Next we consider the continuity of vol N , thus proving property (2). Since concave function defined in a convex set is continuous in the interior. In order to show the continuity of vol N , we need to verify limε→0 vol N (γ + εω n−1 ) = 0 for any γ ∈ ∂N \ {[0] BC } and any Kähler class ω. Indeed, for γ ∈ ∂N \ {[0] BC }, we will prove vol N (γ + εω n−1 ) ≤ O(ε 1 n−1 ). Proposition 3. 1 . 1Let X be an n-dimensional compact Kähler manifold, then for γ ∈ N we have vol mc(mγ) is the mobility count of the 1-cycle class mγ defined as the maximal non-negative integer b such that any b general points of X are contained in a 1-cycle of class mγ. From Theorem 3.1 and Theorem A in[Leh13], both functionals take positive values exactly in N E • and are continuous over N E. Moreover, both of them are homogeneous over N E, it is natural to propose the following question. We need Lehmann's upper bound estimation (see Theorem 6.24 of [Leh13]): let A be a very ample divisor and let s be a positive integer such that A, γ ≤ svol(A), then mc(γ) ≤ 2 4n+1 s n n−1 vol(A). By the basic property of mobility functional (see Lemma 6.17 of[Leh13]), we havemob(γ + εA n−1 ) ≥ mob(εA n−1 ) = get mob(γ + εA n−1 ) ≥ c(A, γ)ε vol N E (γ + εA n−1 )for some positive constant c(A, γ). In particular, we have lim inf ε→0 mob(γ + εA n−1 ) ε vol N E (γ + εA n−1 ) ≥ c(A, γ). Definition 3 . 3 . 33Let γ ∈ N be a pseudo-effective (n − 1, n − 1)-class, the negative part N (γ) of γ is defined to be N (γ) := ν(γ, C)[C], where C ranges among all irreducible curves on X. And the positive part Z(γ) of γ is defined to be Z(γ) := γ − {N (γ)}. And we call γ = Z(γ) + {N (γ)} the Zariski decomposition of γ. Theorem 3. 3 . 3Let X be an n-dimensional compact Kähler manifold and let γ ∈ N • be an interior point. Let γ = Z(γ) + {N (γ)} be the Zariski decomposition in the sense of Boucksom, then N (γ) is an effective curve and it is the unique positive current contained in the negative part {N (γ)}. As a consequence, this implies vol N ({N (γ)}) = 0. Moreover, we have vol N (γ) = vol N (Z(γ)). first show the claim ( * ): N (γ) = ν(γ, C)[C] is the unique positive current in the class {N (γ)} if γ ∈ N • . We remark that claim ( * ) implies vol N ({N (γ)}) = 0 (or equivalently, {N (γ)} ∈ ∂N ). Otherwise, {N (γ)} ∈ N • . Fix a Kähler class ω, then there exists a positive constant δ > 0 such that {N (γ)} − δω n−1 ∈ N • . In particular, there exists a positive current Θ ∈ {N (γ)} such that Θ ≥ δω n−1 . Here we use the same symbol ω to represent a Kähler metric in the Kähler class ω. Note that H n−1,n−1 A (X, R) = {[0] A } over compact Kähler manifolds, thus there exists some smooth (n − 2, n − 2)-form ψ such that i∂∂ψ = 0. For ε > 0 small enough, Θ ε := Θ + εi∂∂ψ ∈ {N (γ)} is a positive current and Θ ε = Θ, contradicting our claim ( * ). Now let us begin the proof of the claim ( * ). The proof is divided into several steps. Lemma 3.2. Let γ ∈ N • , then ν(γ, C) = inf 0≤T ∈γ ν(T, C) for any irreducible curve C. Lemma 3. 3 . 3(compare with Proposition 3.8 of[Bou04]) Let γ ∈ N • , then Z(γ) ∈ N • and ν(Z(γ), C) = 0. (γ) − Z(Z(γ)) − Z({N (γ)}) ∈ N . Now Lemma 3.3 implies N (Z(γ)) = ν(Z(γ), C)[C] = 0, so we have Z(Z(γ)) = Z(γ) − {N (Z(γ))} = Z(γ). And this yields Z({N (γ)}) = 0, which is equivalent to the equality {N (γ)} = {N ({N (γ)})}. Now we can finish the proof of claim ( * ). Firstly, by the definition of N ({N (γ)}) and Siu decomposition, we have N (γ) ≥ N ({N (γ)}). As Lemma 3.4 shows they lie in the same Bott-Chern class, we must have N (γ) = N ({N (γ)}). For any positive current T ∈ {N (γ)}, using Siu decomposition and the definition of N ({N (γ)}) again, we have T ≥ ν(T, C)[C] ≥ N ({N (γ)}) = N (γ). Thus T = N (γ), and N (γ) is the unique positive current in the class {N (γ)}. Lemma 3.5. (compare with Proposition 3.11 of[Bou04]) Let γ ∈ N • , and let S the set of irreducible curves C satisfying ν(γ, C) > 0, then #S ≤ ρ.Proof. Take finite curves C 1 , ..., C k ∈ S, and let Γ= k i=1 a i [C i ] with a i ∈ R.We claim that if the class {Γ} = 0 then all a i = 0. This of course yields #S ≤ ρ. Write Γ = Γ + − Γ − such that both Γ + and Γ − are positive. Since we have assumed {Γ} = 0, we have N ({Γ + }) = N ({Γ − }). By the definition of N (γ), we can take a positive constant c large enough such that {cN (γ) − Γ + } ∈ N . By Lemma 3.4 we know Z({cN (γ)}) = cZ({N (γ)}) = 0, which implies Z({Γ + }) = 0. So we have {Γ + } = {N ({Γ + })}, and this implies Γ + = N ({Γ + }). This also holds for Γ − . Combining with N ({Γ + }) = N ({Γ − }), we get Γ = Γ + − Γ − = 0, which proves our claim. ζ γ − {N (γ)} ∈ N . Thus Z(γ) − B n−1 γ = ζ γ − {N (γ)} ∈ N , which yields vol N (Z(γ)) ≥ vol N (B This of course implies M CY (γ) ≤ vol(α). Combining with vol(α) ≤ M CY (γ), we get the desired equality M CY (γ) = vol(α). 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[ "Possible treatment of the Ghost states in the Lee-Wick Standard Model", "Possible treatment of the Ghost states in the Lee-Wick Standard Model" ]	[ "Abouzeid M Shalaby \nPhysics Department\nFaculty of Science\nMansoura University\nEgypt\n" ]	[ "Physics Department\nFaculty of Science\nMansoura University\nEgypt" ]	[]	Very recently, the Lee-Wick standard model has been introduced as a non-SUSY extension of the Standard model which solves the Hierarchy problem. In this model, each field kinetic term attains a higher derivative term. Like any Lee-Wick theory, this model suffers from existence of Ghost states. In this work, we consider a prototype scalar field theory with its kinetic term has a higher derivative term, which mimics the scalar sector in the Lee-Wick Standard model. We introduced an imaginary auxiliary field to have an equivalent non-Hermitian two-field scalar field theory. We were able to calculate the positive definite metric operator η in quantum mechanical and quantum field versions of the theory in a closed form. While the Hamiltonian is non-Hermitian in a Hilbert space with the Dirac sense inner product, it is Hermitian in a Hilbert space endowed by the inner product n\|η\|m as well as having a correct-sign propagator (no Lee-Wick fields). Besides, the obtained metric operator also diagonalizes the Hamiltonian in the two fields ( no mixing). Moreover, the Hermiticity of η constrained the two Higgs masses to be related as M > 2m, which has been obtained in another work using a very different regime and thus supports our calculation. Also, an equivalent Hermitian (in the Dirac sense) Hamiltonian is obtained which has no Ghost states at all, which is a forward step to make the Lee-Wick theories more popular among the Physicists.	10.1103/physrevd.80.025006	[ "https://arxiv.org/pdf/0812.3419v2.pdf" ]	118,550,572	0812.3419	d0e85953133a00b41c121c0400cd1c8650fe4048	Possible treatment of the Ghost states in the Lee-Wick Standard Model 20 Jan 2009 Abouzeid M Shalaby Physics Department Faculty of Science Mansoura University Egypt Possible treatment of the Ghost states in the Lee-Wick Standard Model 20 Jan 2009numbers: 1290+b1260Cn1130Er Keywords: non-Hermitian modelsPT -Symmetric theoriesgauge HierarchyLee-Wick standard model * Electronic address: amshalab@ mansedueg Very recently, the Lee-Wick standard model has been introduced as a non-SUSY extension of the Standard model which solves the Hierarchy problem. In this model, each field kinetic term attains a higher derivative term. Like any Lee-Wick theory, this model suffers from existence of Ghost states. In this work, we consider a prototype scalar field theory with its kinetic term has a higher derivative term, which mimics the scalar sector in the Lee-Wick Standard model. We introduced an imaginary auxiliary field to have an equivalent non-Hermitian two-field scalar field theory. We were able to calculate the positive definite metric operator η in quantum mechanical and quantum field versions of the theory in a closed form. While the Hamiltonian is non-Hermitian in a Hilbert space with the Dirac sense inner product, it is Hermitian in a Hilbert space endowed by the inner product n\|η\|m as well as having a correct-sign propagator (no Lee-Wick fields). Besides, the obtained metric operator also diagonalizes the Hamiltonian in the two fields ( no mixing). Moreover, the Hermiticity of η constrained the two Higgs masses to be related as M > 2m, which has been obtained in another work using a very different regime and thus supports our calculation. Also, an equivalent Hermitian (in the Dirac sense) Hamiltonian is obtained which has no Ghost states at all, which is a forward step to make the Lee-Wick theories more popular among the Physicists. One of the greatest puzzles in Particle Physics is the Hierarchy problem [1]. SUSY was invented to solve this problem [2]. However, very recently and on the guidance of a previous work of Lee and Wick [3,4], a Lee-Wick (LW) extension of the standard model has been introduced and investigated [5,6,7,8]. While the LW QED is a finite theory, the non-Abealian LW gauge theory is not finite. Although it is not finite, it has been shown that it solves the Hierarchy problem too. The main idea of any Lee-Wick model is that the regulator in Pauli-Villars corresponds to a physical degree of freedom. However, a QED with a Photon propagator with the regulator term is a theory with higher derivative. A great puzzle that makes such trends in the theory of particle physics not popular is that they include exotic fields called the Lee-Wick fields. For instance, in the LW extension of standard model every field of the conventional standard model has a higher derivative kinetic term and it has been shown that the theory can be converted into an equivalent one with more fields but some of them has a propagator with wrong sign (exotic). In a very different kind of studies, Carl Bender and Philip D Mannheim have shown that a quantum mechanical theory with higher derivatives which apparently suffers from negative norm problem can be converted into an equivalent one with the ghost states are disappeared [9]. In showing that, they stressed a higher derivative Pais-Uhlenbeck model. In fact, the regime of PT -symmetric theories has been used successfully in some other works [10,11,12]. In this letter, we show that the ideas can successfully applied to the different sectors in the LW Standard model introduced very recently. For that, we shall stress a type of scalar field theory very similar to that employed in the Lee-Wick standard model. We show that the theory is free from ghost states which then leads to the enhancement of the popularity of the Lee-Wick standard model as a possible theory free from ghosts as well as does not suffer from the Hierarchy puzzle. A prototype scalar field Lagrangian introduced in the Lee-Wick standard model is [5]; L = 1 2 ∂ µ φ∂ µ φ − 1 2M 2 ∂ 2 φ 2 − 1 2 m 2 φ 2 .(1) Fellowing the work in Ref. [5], one can introduce an auxiliary field φ 2 to get rid of the higher derivative in the theory such that; L = 1 2 ∂ µ φ∂ µ φ − 1 2 m 2 φ 2 − φ 2 ∂ 2 φ + 1 2 M 2 φ 2 2 ,(2) From the equation of motion of φ 2 we get; ∂L ∂ (∂ µ φ 2 ) = 0, ∂L ∂φ 2 = φ 2 M 2 − ∂ 2 φ. Then, the auxiliary field φ 2 is given by the relation φ 2 = 1 M 2 ∂ 2 φ. Let us define φ = φ 1 − φ 2 , Then; L = 1 2 ∂ µ (φ 1 − φ 2 ) ∂ µ (φ 1 − φ 2 ) − 1 2 m 2 (φ 1 − φ 2 ) 2 − φ 2 ∂ 2 (φ 1 − φ 2 ) + 1 2 M 2 φ 2 2 , = 1 2 ∂ µ φ 1 ∂ µ φ 1 + 1 2 ∂ µ φ 2 ∂ µ φ 2 − ∂ µ φ 2 ∂ µ φ 1 − φ 2 ∂ 2 φ 1 + φ 2 ∂ 2 φ 2 (3) − 1 2 m 2 (φ 1 − φ 2 ) 2 + 1 2 M 2 φ 2 2 , = 1 2 ∂ µ φ 1 ∂ µ φ 1 − 1 2 ∂ µ φ 2 ∂ µ φ 2 − 1 2 m 2 φ 2 1 + 1 2 M 2 − m 2 φ 2 2 + m 2 φ 2 φ 1 . The Hamiltonian corresponding to the Lagrangian in Eq.(3) can be obtained as; H = π 2 1 2 + 1 2 (∇φ 1 ) 2 + 1 2 m 2 φ 2 1 − π 2 2 2 − 1 2 (∇φ 2 ) 2 − 1 2 M 2 − m 2 φ 2 2 − m 2 φ 1 φ 2(4) Now, let us apply the canonical transformation φ 2 → iφ 2 , π 2 → −iπ 2 which preserve the commutation relation [9]; [φ 2 (x) , π 2 (y)] = [iφ 2 (x) , −iπ 2 (y)] = iδ 3 (x − y) .(5) Then the transformed Hamiltonian will take the form; H = π 2 1 2 + 1 2 (∇φ 1 ) 2 + 1 2 m 2 φ 2 1 + π 2 2 2 + 1 2 (∇φ 2 ) 2 + 1 2 M 2 − m 2 φ 2 2 − im 2 φ 1 φ 2(6) In other words, the negative norm manifested in the work of Ref. [5] by a negative kinetic term of the LW field (φ 2 ) is manifested here by the non-Hermiticity of the theory represented by the Lagrangian in Eq. (3). By assuming that φ 2 is a pseudo scalar, the Hamiltonian obtained from Eq. (3) is PT -symmetric too. Indeed, non-Hermitian PT -symmetric theories are suffering from the existence of ghost states however there exists known algorithms to recover such problems [9,13,13]. In fact, though the Hamiltonian in Eq. (6) is non-Hermitian in the Dirac sense, it is not only Hermitian in a Hilbert space endowed by the inner product n\|η\|m [13,14], where η is a positive definite metric operator, but also the kinetic terms have the correct form. To start the algorithm of curing the ghost states problem in the theory, for simplicity, let us investigate, first, the theory in 0 + 1 dimensions (Quantum mechanics). Since the Hamiltonian in Eq. (6) is pseudo-Hermitian, one can seek a positive definite metric operator of the form; η = exp (2 (ω 1 π 1 φ 2 + ω 2 π 2 φ 1 )) , where ω 1 and ω 2 are two real parameters to be obtained later in terms of the mass parameters m and M. Note that η is Hermitian and has the property [13,14] ηHη −1 = H † . Also, ρ = √ η has the property ρHρ −1 = h,(8) where h is a Hermitian (in the Dirac sense) as well as positive normed Hamiltonian equivalent to H. To determine the parameters ω 1 and ω 2 , we consider the transformations of the different fields in the Hamiltonian under the effect of ρ as follows; ρφ 1 ρ −1 = φ 1 − iω 1 φ 2 , ρπ 1 ρ −1 = π 1 + iω 2 π 2 , ρφ 2 ρ −1 = φ 2 − iω 2 φ 1 , ρπ 2 ρ −1 = π 2 + iω 1 π 1 . Accordingly; h = (π 1 + iω 2 π 2 ) 2 2 + 1 2 m 2 (φ 1 − iω 1 φ 2 ) 2 + (π 2 + iω 1 π 1 ) 2 2 + 1 2 M 2 − m 2 (φ 2 − iω 2 φ 1 ) 2 − im 2 (φ 1 − iω 1 φ 2 ) (φ 2 − iω 2 φ 1 ) , or h = 1 2 π 2 1 + iπ 1 ω 2 π 2 − 1 2 ω 2 2 π 2 2 + 1 2 m 2 φ 2 1 − 1 2 m 2 ω 2 1 φ 2 2 + 1 2 π 2 2 + iπ 2 ω 1 π 1 − 1 2 ω 2 1 π 2 1 1 2 M 2 − m 2 φ 2 2 − ω 2 2 φ 2 1 − m 2 ω 2 φ 2 1 − m 2 ω 1 φ 2 2 − im 2 φ 1 φ 2(9)+ im 2 ω 1 φ 2 ω 2 φ 1 − iφ 2 ω 2 φ 1 M 2 + iφ 2 ω 2 φ 1 m 2 − im 2 φ 1 ω 1 φ 2 . For h to be Hermitian, one has to put the constraints iω 2 + iω 1 = 0, −m 2 + m 2 ω 1 ω 2 − ω 2 M 2 + ω 2 m 2 − m 2 ω 1 = 0,(10) on the introduced parameters ω 1 and ω 2 . Equivalently, we have the relations ω 1 = −ω 2 , −m 2 − m 2 ω 2 1 + ω 1 M 2 − 2m 2 ω 1 = 0.(11) In terms of the mass parameters, ω 1 can be obtained as ω 1 = 1 2m 2 M 2 − 2m 2 ± √ M 4 − 4M 2 m 2 .(12) Also, due to the reality of ω 1 , the two Higgs masses are related by; M 2 4m 2 , which agrees with the results in Ref. [5] Then the Hermitian Hamiltonian h has the form; h = 1 2 π 2 1 1 − ω 2 1 + 1 2 m 2 φ 2 1 + 1 2 1 − ω 2 1 π 2 2 − 1 2 m 2 ω 2 1 φ 2 2 + 1 2 M 2 − m 2 φ 2 2 − ω 2 1 φ 2 1 + m 2 ω 1 φ 2 1 − m 2 ω 1 φ 2 2 , = 1 2 π 2 1 1 − ω 2 1 + 1 2 m 2 φ 2 1 + 1 2 1 − ω 2 1 π 2 2 + 1 2 m 2 ω 2 1 + m 2 ω 1 − 1 2 M 2 ω 2 1 φ 2 1 + − 1 2 m 2 ω 2 1 + 1 2 M 2 − m 2 ω 1 − 1 2 m 2 φ 2 2 . To make sure that the negative norm problem has been lifted, we plotted the propagatorsign governing factors of the form µ 2 0 = (1 − ω 2 1 ), µ 2 1 = 1 2 m 2 ω 2 1 + m 2 ω 1 − 1 2 M 2 ω 2 1 and µ 2 2 = − 1 2 m 2 ω 2 1 + 1 2 M 2 − m 2 ω 1 − 1 2 m 2 as a function of M for m = 1, in Fig.1., Fig.2. and Fig.3, respectively. In these plots, we have taken the root ω 1 = 1 2m 2 M 2 − 2m 2 − √ M 4 − 4M 2 m 2 , while the other root represents a theory of indefinite norm. One can realize that all these factors are positive for the available range of M which assures the remedy of the wrong sign in the propagator of the LW field. In higher dimensions (Quantum field theory), one needs to deal with operator densities and thus the metric operator will take the from; η = d 3 z exp (2 (ω 1 π 1 (z) φ 2 (z) + ω 2 π 2 (z) φ 1 (z))) . Accordingly, we have the relations ρφ 1 (x) ρ −1 = φ 1 (x) − iω 1 d 3 zφ 2 (z) δ 3 (x − z), ρπ 1 ρ −1 = π 1 + iω 2 d 3 zπ 2 (z) δ 3 (x − z), ρφ −1 2 ρ −1 = φ 2 − iω 2 d 3 zφ 1 (z) δ 3 (x − z), ρπ −1 2 ρ −1 = π 2 + iω 1 d 3 zπ 1 (z) δ 3 (x − z). And thus ρφ −1 1 ρ −1 = φ 1 − iω 1 φ 2 , ρπ −1 1 ρ −1 = π 1 + iω 2 π 2 , ρφ −1 2 ρ −1 = φ 2 − iω 2 φ 1 , ρπ −1 2 ρ −1 = π 2 + iω 1 π 1 . Also, note that ρ 1 2 (∇φ 1 (x)) 2 ρ −1 = 1 2 (∇φ 1 (x)) 2 − iω 1 ∇ x φ 1 (x)∇ x φ 2 (x) − ω 2 1 2 (∇ x φ 2 (x)) 2 , ρ 1 2 (∇φ 2 (x)) 2 ρ −1 = 1 2 (∇φ 2 (x)) 2 − iω 2 ∇ x φ 1 (x)∇ x φ 2 (x) − ω 2 2 2 (∇ x φ 1 (x)) 2 .(13) Again, with the choice ω 1 = −ω 2 = ω, one gets; ρ 1 2 (∇φ 1 (x)) 2 + 1 2 (∇φ 2 (x)) 2 ρ −1 = 1 2 1 − ω 2 (∇φ 1 (x)) 2 + (∇φ 2 (x)) 2 ,(14) and the quantum field Hermitian Hamiltonian takes the form; h = 1 2 π 2 1 1 − ω 2 + 1 2 1 − ω 2 (∇φ 1 (x)) 2 + 1 2 m 2 φ 2 1 + 1 2 1 − ω 2 π 2 2 + 1 2 1 − ω 2 (∇φ 2 (x)) 2 + 1 2 M 2 φ 2 2 + 1 2 m 2 ω 2 + m 2 ω − 1 2 M 2 ω 2 1 φ 2 1 (15) + − 1 2 m 2 ω 2 1 − m 2 ω 1 − 1 2 m 2 φ 2 2 . One can easily realize that the governing factors are all positive for the available range of the mass parameter M relative to the mass parameter m. Accordingly, the problem of wrong sign propagator has been recovered. Another benefit of the transformation mapping H → h, is that there exists no mixing terms in h (h is diagonal in the fields φ 1 and φ 2 ) . To make sure that the Hermitian equivalent Hamiltonian in Eq.(15) still bears the feature of quadratic divergence cancellation we rewrite it in the form; h = h 1 + h 2 , h 1 = 1 2 π 2 1 + (∇φ 1 ) 2 + 1 2 m 2 (1 + ω 1 ) 2 φ 2 1 + 1 2 −ω 2 1 π 2 2 + (∇φ 2 ) 2 + 1 2 −m 2 (1 + ω 1 ) 2 φ 2 2 , h 2 = 1 2 π 2 2 + (∇φ 2 ) 2 + 1 2 M 2 φ 2 2 + 1 2 −ω 2 1 π 2 1 + (∇φ 1 ) 2 + 1 2 −M 2 ω 2 1 φ 2 1 , which shows that although h is Hermitian and positive normed it can be decomposed into two terms each of which has the from of a normal and a Lee-Wick fields. Conclusions We considered a higher derivative scalar field theory of the form used in the Lee-Wick standard model. We were able to obtain a non-Hermitian but PT -symmetric two-field equivalent Hamiltonian. Using the tools applied to pseudo Hermitian Hamiltonians, we were able to obtain the positive definite metric operator both on the quantum mechanical and quantum field versions of the theory in a closed form. The so obtained equivalent Hermitian Hamiltonian has propagators of correct sign which mean that the Ghost problem has been cured. Moreover, the Hermitian Hamiltonian is diagonal in the fields. Note that, we discarded the potential term as it is used to break the symmetry and has no effect of the negative norm of the auxiliary field and thus one can add it at the end to the equivalent Hermitian Hamiltonian we obtained. We assert that the work presented here is fully new as it is the first time to obtain the exact metric operator for a realistic quantum field theory. Also, the idea here can be applied to all the sectors in the Lee-wick standard model and thus obtain a theory which is non-SUSY, has no Ghosts, as well as solves the Hierarchy problem. A note to be mentioned is that the mass of the auxiliary field is greater than the normal Higgs which means that it is out of any experimental tests carried out. We aim that in proving the non-existence of Ghosts in a Lee-Wick theory we make those theories attract the attentions of researchers as they introduce finite Quantum Electrodynamics theory as well as showing up a standard model free from the Hierarchy problem. FIG. 1 : 1The factor µ 2 0 = (1 − ω 2 ) plotted against the mass parameter M for m = 1. One can realize that the factor is positive for the available range of M . FIG. 2 : 2A contribution to the mass parameter squared of the field φ 1 of the form µ the Hermitian Hamiltonian h, plotted against the mass parameter M for m = 1. Since the other contribution is m 2 and from the plot µ 2 1 is always positive the mass squared as a whole is positive. FIG. 3 : 3The mass parameter squared of the field φ 2 given by µ the Hermitian Hamiltonian h, plotted against the mass parameter M for m = 1, which is again a positive quantity. AcknowledgmentsWe would like to thank Dr. Shabaan Khalil for his help. Djouadi Abdelhak, arXiv:hep-ph/0503172The Anatomy of Electro-Weak Symmetry Breaking. Abdelhak DJOUADI, The Anatomy of Electro-Weak Symmetry Breaking, arXiv:hep-ph/0503172. . J Wess, B Zumino, Nucl. Phys. B. 7039J. Wess and B. Zumino, Nucl. Phys. B 70 (1974) 39. . T D Lee, G C Wick, Nucl. Phys. B. 9209T. D. Lee and G. C. Wick, Nucl. Phys. B 9, 209 (1969). . T D Lee, G C Wick, Phys. Rev. D. 21033T. D. Lee and G. C. Wick, Phys. Rev. D 2, 1033 (1970). . Benjamín Grinstein, Mark B Connell, Wise, Phys.Rev. 7725012Benjamín Grinstein, Donal O'Connell, and Mark B. Wise, Phys.Rev.D77:025012 (2008). . C D Carone, R F Lebed, Phys. Lett. B. 668221C. D. Carone and R. F. Lebed, Phys. Lett. B 668, 221 (2008). . Thomas G Rizzo, 070670Thomas G. Rizzo, JHEP0706:070 (2007). . Benjamín José Ramón Espinosa, Grinstein, O' Donal, Mark B Connell, Wise, Phys.Rev. 7785002José Ramón Espinosa, Benjamín Grinstein, Donal O'Connell and Mark B. Wise, Phys.Rev.D77:085002 (2008). . M Carl, Philip D Bender, Mannheim, Phys.Rev.Lett. 100110402Carl M. Bender and Philip D. Mannheim, Phys.Rev.Lett.100:110402(2008). . Carl Bender, Stefan Boettcher, Phys.Rev.Lett. 80Carl Bender and Stefan Boettcher, Phys.Rev.Lett.80:5243-5246 (1998). Abouzeid shalaby. Phys.Rev. 7641702Abouzeid shalaby, Phys.Rev.D76:041702 (2007 ). . Abouzeid Shalaby, Eur.Phys.J. 50Abouzeid shalaby, Eur.Phys.J.C50:999-1006 (2007). . A Mostafazadeh, J. Math. Phys. 433944A. Mostafazadeh, J. Math. Phys., 43, 3944 (2002). . A Mostafazadeh, J. Math. Phys. 43205A. Mostafazadeh, J. Math. Phys. 43, 205 (2002).	[]
[ "Energy spectrum of Buoyancy-driven Flows", "Energy spectrum of Buoyancy-driven Flows" ]	[ "Abhishek Kumar \nDepartment of Physics\nIndian Institute of Technology -Kanpur\n208016India\n", "Anando G Chatterjee \nDepartment of Physics\nIndian Institute of Technology -Kanpur\n208016India\n", "Mahendra K Verma \nDepartment of Physics\nIndian Institute of Technology -Kanpur\n208016India\n" ]	[ "Department of Physics\nIndian Institute of Technology -Kanpur\n208016India", "Department of Physics\nIndian Institute of Technology -Kanpur\n208016India", "Department of Physics\nIndian Institute of Technology -Kanpur\n208016India" ]	[]	Using high-resolution direct numerical simulation and arguments based on the kinetic energy flux Πu, we demonstrate that for stably stratified flows, the kinetic energy spectrum Eu(k) ∼ k −11/5 , the entropy spectrum E θ (k) ∼ k −7/5 , and Πu(k) ∼ k −4/5 (Bolgiano-Obukhov scaling). This scaling is due to the depletion of kinetic energy because of buoyancy. For weaker buoyancy in stratified flows, Eu(k) follows Kolmgorov's spectrum with a constant energy flux. We also argue that for Rayleigh Bénard convection, the Bolgiano-Obukhov scaling will not hold for the bulk flow due to the positive energy supply by buoyancy and non-decreasing Πu(k).	null	[ "https://arxiv.org/pdf/1404.2148v1.pdf" ]	118,559,974	1404.2148	a3045014b746c8849d45e4cc0ada1784f5aefc3e	Energy spectrum of Buoyancy-driven Flows Abhishek Kumar Department of Physics Indian Institute of Technology -Kanpur 208016India Anando G Chatterjee Department of Physics Indian Institute of Technology -Kanpur 208016India Mahendra K Verma Department of Physics Indian Institute of Technology -Kanpur 208016India Energy spectrum of Buoyancy-driven Flows Using high-resolution direct numerical simulation and arguments based on the kinetic energy flux Πu, we demonstrate that for stably stratified flows, the kinetic energy spectrum Eu(k) ∼ k −11/5 , the entropy spectrum E θ (k) ∼ k −7/5 , and Πu(k) ∼ k −4/5 (Bolgiano-Obukhov scaling). This scaling is due to the depletion of kinetic energy because of buoyancy. For weaker buoyancy in stratified flows, Eu(k) follows Kolmgorov's spectrum with a constant energy flux. We also argue that for Rayleigh Bénard convection, the Bolgiano-Obukhov scaling will not hold for the bulk flow due to the positive energy supply by buoyancy and non-decreasing Πu(k). Buoyancy or density gradient drives flows in the atmosphere and interiors of planets and stars, as well as in electronic devices and industrial applications like heat exchangers, boilers, etc. Accordingly, scientists (including geo-, astro-, atmospheric-and solar physicists) and engineers have been working on understanding buoyancy driven flows for more than a century. An important unsolved problem in this field is how to quantify the spectra and fluxes of kinetic energy (KE) and entropy (u 2 /2 and θ 2 /2 respectively, where u and θ are the velocity and temperature fluctuations) of buoyancy driven flows [1,2]. In this letter, we will study these quantities and respective nonlinear fluxes using direct numerical simulations, and show that the spectrum differs from Kolmogorov's theory when buoyancy is strong. Flows driven by buoyancy can be classified in two categories: (a) convective flows in which hotter and lighter fluid at the bottom rises, while colder and heavier fluid at the top comes down. These flows are unstable; (b) Stably stratified flows in which lighter fluid rests above heavier fluid. Stably stratified flows are stable, hence their fluctuations vanish over time. Therefore, they need to be driven by an external force to obtain a steady turbulent state. Even though both types of flows are driven by density gradients, the properties of such flows are quite different, which we decipher using quantitative analysis of energy flux and energy supply rate by buoyancy. For stably stratified flows, Bolgiano [3] and Obukhov [4] first proposed a phenomenology, according to which the KE flux Π u of a stably stratified flow is depleted at different length scales due to conversion of kinetic energy to "potential energy" via buoyancy (u z θ). As a result, Π u (k) decreases with wavenumber (see Fig. 1(a)), and the energy spectrum is steeper than that prediced by Kolmogorov theory (E(k) ∼ k −5/3 , where k is the wavenumber). According to the phenomenology proposed by Bolgiano and Obukbhov (referred to as BO), for k < k B (k B = Bolgiano wavenumber [3]), the KE spectrum E u (k), entropy spectrum E θ (k), Π u , and entropy flux Π θ are: E u (k) = c 1 (α 2 g 2 θ ) 2/5 k −11/5 ,(1)E θ (k) = c 2 (αg) −2/5 4/5 θ k −7/5 ,(2)Π θ (k) = θ = constant,(3)Π u (k) = c 3 (α 2 g 2 θ ) 3/5 k −4/5 ,(4) where α, g, and θ are the thermal expansion coefficient, acceleration due to gravity, and the entropy dissipation rate respectively, and c i 's are constants. Random Force (a) In a stably stratified flow, Πu(k) decreases with k due to a negative energy supply rate uz(k)θ * (k) . (b) In a thermally driven flow (e.g., Rayleigh Bénard convection), Πu(k) is a non-decreasing function of k due to positive uz(k)θ * (k) . According to the BO theory, the decrease in Π u (k) occurs due to a negative energy supply rate F (k) = u z (k)θ * (k) , where stands for the real part of the argument. For the wavenumbers in the range k B < k < k d , Π u ≈ u , and E u (k), E θ (k) ∼ k −5/3 (see Fig. 1(a)). Here u is the KE dissipation rate, and k d is the wavenumber after which dissipation starts. In this letter we numerically compute F (k) and Π u (k) for stratified turbulence, and show a consistency with the BO scaling. We remark that many researchers describe the stably stratified flows in terms of density fluctuation ρ , which leads to an equivalent description since θ 2 /2 is proportional to ρ 2 /2 (usually referred to as "potential energy" [5]). Procaccia and Zeitak [6], L'vov [2], L'vov and Falkovich [8], and Rubinstein [9] proposed that a similar scaling is applicable to Rayleigh-Bénard convection (RBC). Their arguments hinges on an assumption that F (k) = u z (k)θ * (k) < 0 even for RBC. Note that for the inertial range regime, under a steady state, the variation of energy flux is given by d dk Π(k) = F (k) − D(k),(5) where D(k) is the viscous dissipation [1-3, 8, 12]. In this letter, we demonstrate using numerical simulations that F (k) > 0 for RBC [see Fig. 1(b)], unlike stably stratified flows where F (k) < 0. Consequently Π(k) would increase with k, and E(k) would be either Kolmogorov-like (E(k) ∼ k −5/3 ) or shallower (E(k) ∼ k −a with a < 5/3). These observations of non-decreasing Π(k) contradict the earlier predictions on RBC [2,6,8], but they are in agreement with the numerical results ofŠkandera et al. [13]. In the past there have been several attempts to verify BO scaling for stably stratified flows. Kimura and Herring [14] reported BO scaling for a narrow band of wavenumbers in their 128 3 decaying spectral simulation; the Richardson number of their simulations was greater than unity. Later, Kimura et al. [15], Lindborg [16,17], and Vallgren et al. [18] focussed on anisotropic energy spectrum, and attempted to explain k −3 KE spectrum observed for the synoptic scale of terrestrial atmosphere. For Rayleigh Bénard convection (RBC), which is a class of thermally-driven convection, the numerical and experimental results are largely inconclusive. Based on simulations with periodic boundary conditions, Borue and Orszag [19] andŠkandera et al. [13] reported KO scaling.Škandera et al. [13] reported a constant KE flux, somewhat consistent with the aforementioned argument [Eq. (5)]. Calzavarini et al. [20] reported BO scaling in the boundary layer, and KO scaling in the bulk. Mishra and Verma [4] reported KO scaling for zero-and low Prandtl number flows since F (k) = u z (k)θ * (k) is active only at low wavenumbers for such flows. Camussi and Verzicco [22,23] however reported BO scaling. The experimental results [24] are more divergent with some reporting BO scaling, and some reporting KO scaling. In this letter, we focus on Boussinesq stably stratified flows, whose equations in a non-dimensionalised form are ∂u ∂t + (u · ∇)u = −∇σ + GrPr 2 θẑ + Pr∇ 2 u + f u ,(6)∂θ ∂t + (u · ∇)θ = −u z + ∇ 2 θ,(7)∇ · u = 0,(8) where Pr = ν/κ is the Prandtl number, and Gr = αg dT dz d 4 /ν 2 is the Grashof number, which is a ratio of the buoyancy and dissipation terms. Another important non-dimensional number is Richardson number Ri = αg dT dz d 2 /u 2 rms , which is a ratio of the buoyancy and the nonlinear term (u · ∇)u. We demonstrate that BO scaling is observed when Ri = O(1), but Kolmogorov scaling E(k) ∼ k −5/3 [referred to as Kolmogorov-Obukhov (KO) scaling] is observed when Ri 1, or when buoyancy is negligible. To test whether BO scaling is valid or not for the stably stratified flows, we perform direct numerical simulation of Eqs. (6-8) using pseudospectral code Tarang [25] in a three-dimensional box of size (2π) 3 . We employ periodic boundary condition on all sides [15]. We use fourth-order Runge-Kutta (RK4) method for time stepping, Courant-Freidricks-Lewey (CFL) condition for computing time step ∆t, and 3/2 rule for dealiasing. To obtain a steady turbulent flow, we apply a random force to the flow in the band 2 ≤ k ≤ 4 using the scheme of Kimura and Herring [15]. We perform large-resolution simulations for Pr = 1 (close to that of air) and Richardson numbers Ri = 4 × 10 −7 , 0.01, and 0.5. Simulations for Ri = 0.01 have been performed on 1024 3 grid, while that for 4 × 10 −7 and 0.5 have been performed on 512 3 grid. The parameters of our runs are listed in Table I. All our simulations are fully resolved since k max η > 1, where k max is the maximum wavenumber of the run, and η is the Kolmogorov length scale. We compute the KE and entropy spectra and fluxes for the steady-state data of Pr = 1 and Ri = 0.01 run. In Fig. 2(a) we plot the normalized KE spectra, E u (k)k 11/5 (BO scaling) and E u (k)k 5/3 (KO scaling). In Fig. 2(b) we plot the normalized entropy spectra, E θ (k)k 7/5 (BO scaling) and E θ (k)k 5/3 (KO scaling). The figures indicate that for Ri = 0.01, BO scaling fits with the numerical data better than KO scaling. We cross check our spectrum results with those on KE and entropy fluxes, which are plotted in Fig. 3. Clearly, the KE flux Π u (k) is positive, and it decreases with k. However Π u (k)k 4/5 is almost flat, thus Π u (k) ∝ k −4/5 , consistent with the BO predictions [Eq. (4)]. This is consistent with the observed negative F (k) = u z (k)θ * (k) for this case (see Fig. 4 and [12]). We also observe that Π θ is a constant in the inertial range [Eq. (3)]. These results show that the BO scaling is valid for stably stratified flows for Ri = O(1). We also compute the Bolgiano wavenumber k B [3] using the numerical data of Eq. (4), and find that k B ≈ 8.5. Our plots on spectra and fluxes show that k B ≈ 8.5 is only 3 to 4 times smaller than k d , wavenumber where the dissipation range starts. Therefore a clear-cut crossover from k −11/5 to k −5/3 is not observed in our simulations. We are in the process of performing simulations on even higher resolution to probe the crossover region. We also performed 512 3 grid simulations for Ri = 0.5 and 4 × 10 −7 with Pr = 1. The normalized KE spectra for these two cases are exhibited in Figs. 5(a) and 5(b) respectively. Our results show that BO scaling is valid for Ri = 0.5, but KO scaling (with a constant Π u (k)) is valid for Ri = 4 × 10 −7 , which is as expected since buoyancy is significant only for moderate and large Ri's. The energy supply rate by buoyancy, F (k), is significant for Ri = 0.5, but insignificant for Ri = 4 × 10 −7 , consistent with the above observations [12]. Π u (k), Π θ (k) Π u (k) Π u (k)k 4/5 Π θ (k) To contrast the energy supply rate by buoyancy in stratified flows with that in Rayleigh Bénard convection (RBC), we numerically solve the nondimensionalized RBC equations for Pr = 1.0 and Rayleigh number Ra = 5 × 10 6 on 256 3 grid [4]. For this run, we plot F (k) in Fig. 4 that demonstrates that F (k) > 0, consistent with our aforementioned arguments, but differs from those of L'vov and Falkovich [8]. The ongoing work on the flux and spectrum for RBC will be reported in a future work. Thus, the behaviour of F (k) and Π u (k) for the stably stratified flow and RBC are quite different. We employ periodic boundary condition for the stably stratified flows in the vertical direction, thus eliminating the effects of boundary walls. In Fig. 6 we plot the planeaveraged mean temperature profileT (z) = T (x, y, z) xy . SinceT (z) is linear, a constant temperature gradient dT /dz (hence buoyancy) acts in the whole box. Therefore, BO scaling is expected everywhere. It is important to contrast the above profile with that for Rayleigh-Bénard convection in which most of the temperature drop takes place in a narrow thermal boundary layer [23,26], while the bulk flow has dT /dz ≈ 0. Thus we expect BO scaling in the boundary layers, and KO scaling in the bulk, as reported by Calzavarini et al. [20]. In summary, our numerical simulations demonstrate an existence of BO scaling in stably stratified flows. A major novelty in our approach is the quantitative analysis of the KE and entropy fluxes, as well as the energy supply rate by buoyancy (F (k)). We show that F (k) < 0 for stably stratified flows, but F (k) > 0 for Rayleigh Bénard convection. Consequently, for stably stratified flows with moderate Richardson numbers, the energy flux Π u (k) ∝ k −4/5 and E u (k) ∼ k −11/5 , as proposed in the BO phenomenology. However, F (k) is somewhat insignificant for small Richardson number, and we observe KO scaling. For RBC flows, Π u (k) is a non-decreasing function of k (since F (k) > 0), and the energy spectrum of KE cannot be steeper than k −5/3 in the bulk. Thus, energy flux and energy supply rate due to buoyancy provide valuable insights into the physics of stably stratified flows and RBC. Our numerical simulations were performed at Centre for Development of Advanced Computing (CDAC) and IBM Blue Gene P "Shaheen" at KAUST supercomputing laboratory, Saudi Arabia. This work was supported by a research grant SERB/F/3279/2013-14 from Science and SUPPLEMENTAL MATERIAL In Fourier space, the equation for the kinetic energy (KE) can be derived from Eq. (1) of the main paper as [1][2][3] ∂E u (k) ∂t = T u (k) + F (k) − D(k)(9) where T u (k) is energy transfer rate to a wavenumber shell of radius k, and it is related to the energy flux of KE as Π u (k) = − k 0 T u (k) dk.(10) The energy supply rate F (k) of Eq. (9) is given by F (k) = GrPr 2 \|k\|=k u z (k)θ * (k) + u(k) · f * (k) ,(11) where the first term is due to buoyancy, while the second term is due to the external random forcing, which is active only for 2 ≤ k ≤ 4. The viscous dissipation is given by D(k) = \|k\|=k 2Prk 2 E u (k)(12) From the above equation, we deduce that d dk Π(k) = −T (k) = − ∂E ∂t + F (k) − D(k).(13) Under a steady state (∂E/∂t = 0), we obtain d dk Π(k) = F (k) − D(k).(14) In our simulations of stably stratified flows, the external force f * (k) is active for the band 2 ≤ k ≤ 4. Therefore, we focus on wavenumbers k > 4 where only buoyancy force is active. We compute F (k) and D(k) using the numerical data computed for stably stratified flows with Pr = 1, and Ri = 0.01, 0.5, 4 × 10 −7 . These quantities are shown in Figs. 7, 8, and 9 respectively. In the inertial range, F (k) is negative for all the three cases, consistent with the predictions of Bolgiano-Obukhov (BO) phenomenology. We observe that for Ri = 0.01 and 0.5, F (k) is significant for small wavenumbers. However for Ri = 4 × 10 −7 , buoyancy is weak, and \|F (k)\| D(k). These results are consistent with the flux and spectra results presented in the main paper. We also find that \|F (k)\| drops sharply with k for Ri = 0.01, 0.5. Since Π(k) ∼ k −4/5 , we observe F (k) ∼ dΠ/dk ∼ −k −9/5 for a narrow band in the small wavenumber regime. In contrast, for Ri = 4 × 10 −7 , F (k) is much smaller than the corresponding F (k) for Ri = 0.01, 0.5, consistent with dΠ(k)/dk ≈ −D(k). We exhibit negative F (k) (lower k) with thick red curve, and positive F (k) (higher k) using thick dotted red curve. We also exhibit D(k) (thick green curve), [−F (k) + D(k)] (thick dotted blue curve), and − d dk Πu(k) (thin magenta curve). k −9/5 scaling is shown using black line. We contrast the above results with those for Rayleigh Bénard convection (RBC). We numerically solve the nondimensionalized RBC equations under the Boussinesq approximation for Pr = 1.0 and Rayleigh number Ra = 5 × 10 6 on 256 3 grid [4]. For this run, we plot F (k) and D(k) in Fig. 10 that demonstrates that F (k) > 0, consistent with our arguments. Note however that for this case, F (k) D(k) and dΠ u (k)/dk = −D(k) < 0, or Π u (k) is decreasing with k due to dominance of D(k) over F (k). We need to perform very large-resolution simulation for much higher Ra that would provide significant inertial range where Π u (k) could be a non-decreasing function of k. FIG. 1 . 1Schematic diagrams of energy flux Πu(k): FIG. 2 . 2For Pr = 1 and Ri = 0.01, plots of (a) normalized KE spectra and (b) normalized entropy spectra for Bolgiano-Obukhov (BO) and Kolmogorov-Obukhov (KO) scaling. BO scaling fits with the data better than KO scaling. FIG. 3 . 3For Pr = 1 and Ri = 0.01, plots of KE flux Πu(k), normalized KE flux Πu(k)k 4/5 , and entropy flux Π θ (k). Π θ (k) is multiplied by 10 5 to fit in the same plot. FIG. 4 . 4Plots of F (k) stably stratified flow (Pr = 1, Ri = 0.01 on 1024 3 grid), and for RBC (Pr = 1, Ra = 5 × 10 6 on 256 3 grid). F (k) < 0 for stratified flows, but F (k) > 0 for RBC. F (k) for RBC is multiplied by 10 4 to fit in the same plot. FIG. 5 . 5Plots of normalized KE spectra for Bolgiano-Obukhov (BO) scaling and Kolmogorov-Obukhov (KO) scaling for: (a) Ri = 0.5 and (b) Ri = 4 × 10 −7 FIG. 6 . 6The vertical variation of horizontally averaged mean temperatureT (z) = T (x, y, z) xy for Pr = 1 and Ri = 0.01 run. FIG. 7 . 7For Pr = 1 and Ri = 0.01, plots of −F (k) (thick red curve), D(k) (thick green curve), [−F (k)+D(k)] (dotted blue curve) , and − d dk Πu(k) (thin magenta curve). k −9/5 scaling is shown using black line. FIG. 8 . 8For Pr = 1 and Ri = 0.5, F (k) has both signs. FIG. 9 . 9For Pr = 1 and Ri = 4 × 10 −7 , plots of −F (k) (thick red curve), D(k) (thick green curve), [−F (k) + D(k)] (dotted blue curve) , and − d dk Πu(k) (thin magenta curve). F (k) is multiplied by 10 5 to fit in the same plot. 10. For Pr = 1 and Ra = 5 × 10 6 , plots of F (k) (thick red curve), D(k) (thick green curve), and − d dk Πu(k) (thin magenta curve) on 256 3 grid. F (k) is multiplied by 10 5 to fit in the same plot. TABLE I . IParameters of our numerical simulations: Richardson number Ri, Grashof number Gr, Prandtl number Pr, Grid size, kinetic energy dissipation rate u, thermal dissipation rate θ , Reynolds number Re, Reynolds number Re λ based on Taylor micro-scale, kmaxη where η is the Kolmogorov length, Bolgiano wavenumber kB, and averaged ∆t.Ri Gr Pr Grid u θ Re Re λ kmaxη kB ∆t 0.5 1 × 10 5 1 512 3 1.4 × 10 7 60.7 467 220 4.2 6.0 2.5 × 10 −5 0.01 5 × 10 3 1 1024 3 4.0 × 10 7 150 649 260 6.4 8.5 3.5 × 10 −6 4 × 10 −7 0.1 1 512 3 2.1 × 10 7 141 510 220 3.8 < 1 2.6 × 10 −6 10 0 10 1 10 2 k We thank Ambrish Pandey, Anindya Chatterjee, Pankaj Mishra, and Mani Chandra for valuable suggestions. * [email protected]. Engineering Research Board, IndiaEngineering Research Board, India. We thank Ambrish Pandey, Anindya Chatterjee, Pankaj Mishra, and Mani Chandra for valuable suggestions. * [email protected] . E D Siggia, Ann. Rev. Fluid Mech. 26137E. 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[ "Atomic monolayer deposition on the surface of nanotube mechanical resonators", "Atomic monolayer deposition on the surface of nanotube mechanical resonators" ]	[ "A Tavernarakis \nICFO -Institut de Ciencies Fotoniques\nMediterranean Technology Park08860Castelldefels, BarcelonaSpain\n", "J Chaste \nInstitut Català de Nanotecnologia\nCampus de la UABE-08193BellaterraSpain\n", "A Eichler \nICFO -Institut de Ciencies Fotoniques\nMediterranean Technology Park08860Castelldefels, BarcelonaSpain\n\nInstitut Català de Nanotecnologia\nCampus de la UABE-08193BellaterraSpain\n", "G Ceballos \nInstitut Català de Nanotecnologia\nCampus de la UABE-08193BellaterraSpain\n", "M C Gordillo \nDepartamento de Sistemas Físicos\nQuímicos\n", "J Boronat \nDepartament de Física i Enginyeria Nuclear\nUniversitat Politècnica de Catalunya\nCampus NordB4-B5, 08034BarcelonaSpain\n", "A Bachtold \nICFO -Institut de Ciencies Fotoniques\nMediterranean Technology Park08860Castelldefels, BarcelonaSpain\n\nInstitut Català de Nanotecnologia\nCampus de la UABE-08193BellaterraSpain\n", "\nUniversidad Pablo de Olavide\nNaturales\n", "\nCarretera de Utrera\nkm 1E-41013SevillaSpain\n" ]	[ "ICFO -Institut de Ciencies Fotoniques\nMediterranean Technology Park08860Castelldefels, BarcelonaSpain", "Institut Català de Nanotecnologia\nCampus de la UABE-08193BellaterraSpain", "ICFO -Institut de Ciencies Fotoniques\nMediterranean Technology Park08860Castelldefels, BarcelonaSpain", "Institut Català de Nanotecnologia\nCampus de la UABE-08193BellaterraSpain", "Institut Català de Nanotecnologia\nCampus de la UABE-08193BellaterraSpain", "Departamento de Sistemas Físicos\nQuímicos", "Departament de Física i Enginyeria Nuclear\nUniversitat Politècnica de Catalunya\nCampus NordB4-B5, 08034BarcelonaSpain", "ICFO -Institut de Ciencies Fotoniques\nMediterranean Technology Park08860Castelldefels, BarcelonaSpain", "Institut Català de Nanotecnologia\nCampus de la UABE-08193BellaterraSpain", "Universidad Pablo de Olavide\nNaturales", "Carretera de Utrera\nkm 1E-41013SevillaSpain" ]	[]	We studied monolayers of noble gas atoms (Xe, Kr, Ar, and Ne) deposited on individual ultraclean suspended nanotubes. For this, we recorded the resonance frequency of the mechanical motion of the nanotube, since it provides a direct measure of the coverage. The latter is the number of adsorbed atoms divided by the number of the carbon atoms of the suspended nanotube. Monolayers formed when the temperature was lowered in a constant pressure of noble gas atoms. The coverage of Xe monolayers remained constant at 1/6 over a large temperature range. This finding reveals that Xe monolayers are solid phases with a triangular atomic arrangement, and are commensurate with the underlying carbon nanotube. By comparing our measurements to theoretical calculations, we identify the phases of Ar and Ne monolayers as fluids, and we tentatively describe Kr monolayers as solid phases. These results underscore that mechanical resonators made from single nanotubes are excellent probes for surface science.	10.1103/physrevlett.112.196103	[ "https://arxiv.org/pdf/1404.5833v2.pdf" ]	18,595,485	1404.5833	b7ba7fd42b48c4980d354429610edd88e778f0cc	Atomic monolayer deposition on the surface of nanotube mechanical resonators 24 Apr 2014 A Tavernarakis ICFO -Institut de Ciencies Fotoniques Mediterranean Technology Park08860Castelldefels, BarcelonaSpain J Chaste Institut Català de Nanotecnologia Campus de la UABE-08193BellaterraSpain A Eichler ICFO -Institut de Ciencies Fotoniques Mediterranean Technology Park08860Castelldefels, BarcelonaSpain Institut Català de Nanotecnologia Campus de la UABE-08193BellaterraSpain G Ceballos Institut Català de Nanotecnologia Campus de la UABE-08193BellaterraSpain M C Gordillo Departamento de Sistemas Físicos Químicos J Boronat Departament de Física i Enginyeria Nuclear Universitat Politècnica de Catalunya Campus NordB4-B5, 08034BarcelonaSpain A Bachtold ICFO -Institut de Ciencies Fotoniques Mediterranean Technology Park08860Castelldefels, BarcelonaSpain Institut Català de Nanotecnologia Campus de la UABE-08193BellaterraSpain Universidad Pablo de Olavide Naturales Carretera de Utrera km 1E-41013SevillaSpain Atomic monolayer deposition on the surface of nanotube mechanical resonators 24 Apr 2014arXiv:1404.5833v2 [cond-mat.mes-hall]numbers: 6843-h6225Jk8107De Keywords: We studied monolayers of noble gas atoms (Xe, Kr, Ar, and Ne) deposited on individual ultraclean suspended nanotubes. For this, we recorded the resonance frequency of the mechanical motion of the nanotube, since it provides a direct measure of the coverage. The latter is the number of adsorbed atoms divided by the number of the carbon atoms of the suspended nanotube. Monolayers formed when the temperature was lowered in a constant pressure of noble gas atoms. The coverage of Xe monolayers remained constant at 1/6 over a large temperature range. This finding reveals that Xe monolayers are solid phases with a triangular atomic arrangement, and are commensurate with the underlying carbon nanotube. By comparing our measurements to theoretical calculations, we identify the phases of Ar and Ne monolayers as fluids, and we tentatively describe Kr monolayers as solid phases. These results underscore that mechanical resonators made from single nanotubes are excellent probes for surface science. Carbon nanotubes have motivated a considerable research effort for the study of gas adsorption onto substrates that approach the one-dimensional limit [1][2][3][4][5][6][7][8][9]. Many studies have been carried out on mats and films of nanotube bundles, but the interpretation of those measurements is complicated by the fact that the binding energy of the gas atoms on the substrate is not homogeneous. That is, the binding energy depends on whether the atom is located on an individual nanotube, at the junction between two crossing nanotubes, or along the interstitial channel formed between two parallel nanotubes. Recently, this homogeneity problem was solved by studying gas adsorption on individual nanotubes, a technical feat considering the tiny amount of adsorbed atoms [7][8][9]. For this, nanotubes were employed both as substrates for adsorption and as detectors. Namely, the nanotubes were operated as mechanical resonators, the resonance frequency being exquisitely sensitive to the number of adsorbed atoms [9][10][11][12]. Atoms adsorbed on graphitic surfaces can form a rich variety of different phases, such as vapor, liquid, supercritical fluids, and solids [13]. The solid phase can be either commensurate or incommensurate with the graphene surface ( Fig. 1(a,b)). The commensurate solid phase is robust, since the crystal formed by the adsorbed atoms is strongly pinned to the underlying carbon surface. Commensurate monolayers on graphite feature a well defined ratio between the number of adsorbed atoms and the number of carbon atoms at the surface. This ratio, called coverage, is often 1/6 for noble gas atoms, which corresponds to a registered √ 3 × √ 3 lattice (table 6.1 in Ref. [13]). This particular coverage value arises because, in this solid phase, noble gas atoms form a two-dimensional triangular arrangement in which atoms occupy the center of carbon hexagons, leaving an empty one in the center of the triangle (see Fig. 1b). Nanotubes are also expected to host commensurate solids; however, due to cylindrical boundary conditions, these solids exist only for some specific nanotube chiralities (n, m), namely when (n − m)/3 is an integer [7]. Remarkably, this is also the condition for nanotubes to be metallic. Solid and fluid monolayers made of noble gas atoms, such as Xe, Kr, and Ar, were measured on graphite surfaces only when the coverage was comparable to or larger than 1/6 (chapter 6 in Ref. [13]). When adsorbed on a nanotube surface, the coverage of incommensurate solids and fluids is expected to become larger than that measured on graphite due to the curvature of the nanotube [8]. This is because (i) adsorbed atoms form cylindrical monolayers with a surface larger than that of the carbon nanotube, and (ii) the two-dimensional density of noble gas atoms is, to a good approximation, independent of the curvature of the monolayer. Recently, monolayers of Kr and Ar were obtained on individual nanotubes by increasing the pressure of Kr or Ar gas surrounding the nanotube [7]. These monolayers were identified as solids, but these phases were fragile, since the number of atoms in the monolayer was very sensitive to temperature. Here, we report on the formation of monolayers of Xe, Kr, Ar, and Ne on individual ultraclean nanotubes upon decreasing temperature. The pressure was kept constant, typically in the 10 −7 mbar range. We prepared the nanotube by thoroughly current annealing it in order to remove contamination from the surface. The monolayer of Xe was found to be the most robust phase. Its coverage remained constant at 1/6 over a large temperature range, indicating the formation of a √ 3× √ 3 commensurate solid. The coverages of the monolayers made from Kr, Ar, and Ne were less stable against temperature variations. We compare our experimental findings to theoretical calculations in order to establish the nature of these phases. In order to demonstrate commensurate solid phases, we fabricated resonators based on ultra-clean nanotubes that are metallic. For this, we used the fabrication process that we described in Ref. [14]. As shown in Fig. 1(c), the nanotube is contacted by two electrodes and is suspended over a trench with a gate electrode at the bottom. The nanotube was grown by chemical vapor deposition in the last step of the fabrication process in order to reduce contamination [15] (Supplemental Material, Sec. I). The measurement of the electrical conductance as a function of the voltage applied to the gate electrode allowed us to select nanotubes that are metallic with a small energy gap (Supplemental Material, Sec. III). The mechanical motion was driven and detected using the frequency-modulation mixing technique [16] (Supplemental Material, Sec. III). We carried out the experiment in a homebuilt ultra-high vacuum cryostat that reaches a base pressure of ∼ 3 · 10 −11 mbar. The nanotube was cleaned by current annealing. Noble gas atoms were dosed from a roomtemperature supply with a pinhole microdoser. We studied 3 nanotubes yielding similar results. We discuss in the following the data for one device. Data for a second device are shown in Supplementary Material, Sec. VIII. Monolayers of noble gas atoms formed on the nanotube when the temperature (T ) was lowered while keeping a constant pressure of noble gas in the cryostat chamber. The for-mation was monitored by measuring the resonance frequency f 0 (that is, by continuously recording the response of the nanotube resonator to the driving frequency). Figure 1(d) shows prominent jumps of f 0 to lower frequencies upon lowering T (see arrows), indicating the sudden adsorption of a large quantity of atoms onto the nanotube. For comparison, when we did not dose atoms, the temperature dependence of f 0 is weak and monotonic (grey curve labeled "pristine" in Fig. 1(d)). This weak dependence is attributed to the thermal expansion of the electrodes which modifies the spring constant of the nanotube resonator [17]. The coverage at T is extracted using ϕ(T ) = N ads (T ) N C = m C m ads A · f 0 prist (T ) f 0 ads (T ) 2 − 1 ,(1) where N C is the number of C atoms of the suspended nanotube, N ads is the number of adsorbed atoms on the nanotube, and m C and m ads are the atomic masses of carbon and adsorbed atoms, respectively. Here, f 0 ads is the resonance frequency when dosing atoms for adsorption, and f 0 prist is the resonance frequency when not dosing atoms and keeping the nanotube pristine. The constant A is introduced to account for variations in the spring constant between the measurement of f 0 ads (T ) and that of f 0 prist (T ); indeed, the spring constant can be different, if for instance the gate voltage applied in the measurement of f 0 ads (T ) differs from that of f 0 prist (T ) (Supplemental Material, Sec. IV). The constant A is fixed so that ϕ = 0 at high T . In Eq. 1, we assume that the spring tension is insensitive to the tension induced by the interaction between noble gas atoms, which is two orders of magnitude weaker than that of covalent C-C bonds [7]. Above a characteristic temperature T c ≃ 48 K, the coverage remains at zero. On lowering temperature, the coverage jumps to ϕ ≃ 1/6 and remains close to this value until T ≃ 26 K. This behavior can be accounted for by the balance of atoms impinging on and departing from the nanotube. For T > T c , an impinging atom departs very rapidly from the nanotube, so that the number of adsorbed atoms remains close to zero ( Fig. 1(e)). For T < T c , it is energetically favourable for the atoms to stay on the nanotube (Fig. 1(f)) -the atoms forming a layer with ϕ ≃ 1/6. This layer is likely a monolayer, because the coverage ϕ ≃ 1/6 of Kr on graphite corresponds to a monolayer (chapter 6 in Ref. [13]). Upon further lowering temperature so that T << T c , the coverage gets larger than ϕ = 1/6, indicating that Kr atoms start to form the second layer. The coverage grows in a monotonic way without any additional plateaus even when the coverage gets larger than one (Supplementary Material, Sec. VII). The absence of additional plateaus above ϕ = 1/6 further supports the interpretation of the coverage ϕ ≃ 1/6 as being related to the monolayer. Key to this work is annealing the nanotube by passing a large current through it. After the measurements shown in Fig. 2(a), we exposed the nanotube to ambient air. We then baked the cryostat and the nanotube at 110 • C under vacuum for two days to reach a base pressure of ∼ 3 · 10 −11 mbar. We again measured the coverage upon lowering T while dosing Kr atoms. Fig. 2(b) shows that T c is much lower than before, and the coverage at T T c is significantly lower than 1/6. We had to anneal the nanotube with a current of ∼ 10 µA in order to recover the same measurement as in Fig. 2(a). These measurements suggest that the growth of monolayers is extremely sensitive to contamination, since a simple exposure to air prevents the formation of homogeneous monolayers. Another advantage of current annealing is that it brings the nanotube back to its pristine state after the adsorption of noble gas atoms on its surface. We grew different monolayers on the nanotube by dosing Xe, Kr, Ar, and Ne (Fig. 3). The nanotube surface was cleaned by current annealing before each growth. Upon decreasing temperature, the coverage increases rapidly from 0 to a plateau with ϕ ≃ 1/6, indicating the growth of the monolayer. The characteristic temperature of the monolayer growth depends on the atomic species; T c is higher when the atomic mass is larger (Fig. 3). We attribute the origin of the variation of T c to the polarizability of the atomic species and the van der Waals interaction between the atom and the nanotube; the polarizability and the interaction both increasing with the atomic radius. We also carried out experiments where we evaporated the monolayers from the nanotube by continuously increasing the temperature of the cryostat from 4 to ∼100 K. The coverage jumped from ≃ 1/6 to 0 at a temperature that is up to ∼10 K higher than T c (Supplemental Material, Sec. V). We measured the time of the growth of monolayers from ϕ = 0 to ϕ = 1/6 while keeping the temperature constant. This time gets longer for lower pressure (Supplementary Material, sec. VI). Xenon monolayers are particularly robust against temperature changes. Figure 4(a) shows coverage-temperature measurements recorded at different pressures and different temperature ramping rates. T c varies from one measurement to the next. However, the plateau in coverage at 1/6 is clearly reproducible. This shows that Xe monolayers are energetically sta-ble with the number of atoms being insensitive to temperature over a large parameter space. In contrast, measurements with Ne feature a plateau whose coverage depends significantly on T (Fig. 4(b)). As for Kr, the measured temperature dependence of the coverage is similar to that of Xe, supporting the scenario that Kr monolayers are commensurate solid phases. This result would be in agreement with experimental signatures of stability of a commensurate Kr layer on graphite up to quite high temperatures, T ∼ 130 K [18]. However, the coverage of Kr slightly depends on temperature in the plateau region (Fig. 3) [19,20]. Only by increasing in an empirical way the anisotropic part of the pair interaction the commensurate phase becomes stable. Our present simulations on Kr adsorbed on nanotubes show the same trend. Therefore, more work is needed to establish the phase of Kr monolayers on nanotubes. To conclude, we studied the formation of noble gas atom monolayers on individual nanotubes. We found that Xe atoms form robust commensurate solids, whereas Ar and Ne atoms organize themselves in fluids. These monolayers consist of ∼ 10 5 atoms, which is a tiny amount of material difficult to detect with most experimental techniques used in surface science. The study of these monolayers was here possible, because nanotube mechanical resonators are extremely sensitive probes. The second important aspect of our experiments is that the nanotube surface was ultra-clean; this was achieved by thoroughly current annealing the nanotube in ultra-high vacuum. These resonators made from ultra-clean nanotubes are promising for various future adsorption experiments, such as the measurement of new phase transitions emerging in the one-dimensional limit with narrower nanotubes, the investigation of quantum effects of He monolayers adsorbed on nanotubes [21], the study of the diffusion of adsorbed atoms over the resonator surface which is a topic of increasing interest [22], and the interplay between the strong mechanical nonlinearities of nanotubes [14,[23][24][25] and the diffusion of atoms [26][27][28]. We 3 · 10 −7 , and 3 · 10 −7 mbar, and the T ramping rate is 0.016, 0.033, and 0.008 K/s for the blue, green, and red lines, respectively. (b) Coverage upon lowering temperature while dosing Ne atoms. The pressure is 3 · 10 −7 mbar for all three measurements, and the T ramping rate is 0.016 K/s for the blue and the green lines and 0.008 K/s for the red line. Figure 2 ( 2a) shows the temperature dependence of the coverage while dosing Kr atoms. Fig. S4 . S4Monolayers of Ar and Ne are less stable, since the measured coverage depends significantly on temperature for T T c . Our calculations reveal that in the limit of zero temperature Ar and Ne monolayers are incommensurate solids with coverages 0.265 and 0.403, respectively.The measured coverages at T T c are much lower than these predicted values, suggesting that the monolayers observed experimentally are not in the solid phase. Moreover, our calculations show that incommensurate solids melt at temperatures as low as 5 K when the coverage is set to the values we typically measure at T T c . This further indicates that the monolayers of Ar and Ne observed experimentally at 25 − 35 K are in the liquid phase. FIG. 1 : 1thank A. Isacsson and J. Moser for discussions. We acknowledge support from the European Union through the Graphene Flagship (604391), the ERC-carbonNEMS project, and a Marie Curie grant (271938), the Spanish state (MAT2012-31338), and the Catalan government (AGAUR, SGR). C. G. and J. B. acknowledge partial financial support from the Junta de de Andalucía Group PAI-205, Grant No. FQM-5987, MICINN (Spain) Grants No.FIS2010-18356 and FIS2011-25275, and Generalitat de Catalunya Grant 2009SGR-1003. * Present address: CNRS, Laboratoire de Photonique et de Nanostructures, UPR20, route de Nozay, 91460 Marcoussis, France † Present address: Department of Physics, ETH Zurich, Schafmattstrasse 16, 8093 Zurich, (a) Growth of an atomic monolayer on a nanotube. (b) Schematics of monolayers in the solid phase that are commensurate (top) and incommensurate (bottom) with the carbon substrate. The adsorbed atoms are represented by red spheres, whereas the carbon surface is depicted by the honeycomb lattice. (c) Layout of the nanotube resonator. (d) Resonance frequency upon lowering temperature while dosing Xe and Ne using a pinhole micromanipulator. The curve labeled "pristine" corresponds to the T dependance of f 0 when we do not dose atoms. The pressure is 3 · 10 −7 mbar for the Xe and the Ne measurements and 3 · 10 −11 mbar for the pristine measurement.(e,f) Schematics showing the balance of atoms impinging on and departing from the nanotube above and below the characteristic temperature T c . upon dosing Kr atoms while lowering the temperature T with a ramping rate 0.016 K/s. (b) Same measurement recorded before having current annealed the nanotube with a T ramping rate 0.033 K/s. The pressure is 3 · 10 −7 mbar for both measurements. FIG. 3 : 3Coverage upon lowering temperature while dosing Xe, Kr, Ar, and Ne atoms. The pressure is 3 · 10 −7 mbar for all measurements. The T ramping rate is 0.008 K/s for the Xe measurement and 0.016 K/s for the other measurements. The black line corresponds to ϕ = 1/6. FIG. 4 : 4(a) Coverage upon lowering temperature while dosing Xe atoms. The pressure is 7 · 10 −8 , large temperature range. This robustness suggests that Xe atoms are strongly bound to the underlying carbon surface, as it is the case for commensurate solids. Our experimental results are accounted for by our theoretical calculations, which predict that the solid is aWe now discuss the nature of the monolayers of Xe, Kr, Ar, and Ne. For this, we carried out theoretical calculations to predict whether the solid phases are commensurate or incommensurate in the limit of zero temperature. In addition, we estimated the melting temperature of the different solid phases. 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[ "Distribution of the Number of Generations in Flux Compactifications", "Distribution of the Number of Generations in Flux Compactifications" ]	[ "Andreas P Braun \nDepartment of Mathematics\nKing's College\nWC2R 2LSLondonUK\n", "Taizan Watari \nKavli Institute for the Physics and Mathematics of the Universe\nthe University of Tokyo\n277-8583TokyoJapan\n" ]	[ "Department of Mathematics\nKing's College\nWC2R 2LSLondonUK", "Kavli Institute for the Physics and Mathematics of the Universe\nthe University of Tokyo\n277-8583TokyoJapan" ]	[]	Flux compactification of string theory generates an ensemble with a large number of vacua called the landscape. By using the statistics of various properties of low-energy effective theories in the string landscape, one can therefore hope to provide a scientific foundation to the notion of naturalness. This article discusses how to answer such questions of practical interest by using flux compactification of F-theory. It is found that the distribution is approximately in a factorized form given by the distribution of the choice of 7-brane gauge group, that of the number of generations Ngen and that of effective coupling constants. The distribution of Ngen is approximately Gaussian for the range \|Ngen\| < ∼ 10. The statistical cost of higher-rank gauge groups is also discussed.	10.1103/physrevd.90.121901	[ "https://arxiv.org/pdf/1408.6156v2.pdf" ]	118,728,463	1408.6156	30a013a80dfd41150f361efd75755803fecba447	Distribution of the Number of Generations in Flux Compactifications 26 Aug 2014 Andreas P Braun Department of Mathematics King's College WC2R 2LSLondonUK Taizan Watari Kavli Institute for the Physics and Mathematics of the Universe the University of Tokyo 277-8583TokyoJapan Distribution of the Number of Generations in Flux Compactifications 26 Aug 2014 Flux compactification of string theory generates an ensemble with a large number of vacua called the landscape. By using the statistics of various properties of low-energy effective theories in the string landscape, one can therefore hope to provide a scientific foundation to the notion of naturalness. This article discusses how to answer such questions of practical interest by using flux compactification of F-theory. It is found that the distribution is approximately in a factorized form given by the distribution of the choice of 7-brane gauge group, that of the number of generations Ngen and that of effective coupling constants. The distribution of Ngen is approximately Gaussian for the range \|Ngen\| < ∼ 10. The statistical cost of higher-rank gauge groups is also discussed. Introduction: String theory with compactified extra dimensions gives rise to a large number of vacua. The diversity of vacua originates from the choice of topology of compact internal space and flux configurations on it [1]. String theory as understood in this way today therefore does not predict a unique low-energy effective theory. Despite this lack of prediction, there are still many ways in which we can take advantage of such an ensemble of string vacua for a better understanding of particle physics in the real world. Such an ensemble containing a large number of vacua provided by a fundamental theory is referred to as a landscape of vacua, or landscape for short. At least two ideas have been proposed so far in how to take advantage of such landscape of string theory. One is in the context of understanding the non-vanishing (yet extremely small) dark energy. It has been realized that the very small value was predicted to be "natural" under the combination of three ansatzes, i) that the universe is occupied with many distinct areas facilitating effective theories with different values of the cosmological constant [2], ii) that the statistical distribution of dark energy is taken as a principle of arguing naturalness, and iii) that the observational factor (anthropics) is taken into account in the statistical distribution [3]. The string landscape indeed give rise to such an ensemble of low-energy effective theories with different values of the cosmological constant [7], and eternal inflation has this ensemble of string vacua realized in the universe [9]. Thus the string landscape provides a theoretical foundation for the attempt of understanding dark energy along the lines of i) ii) and iii). The string landscape can also provide a scientific foundation for a notion of naturalness for various kinds of parameters of the standard model, not just for the value of dark energy. Naturalness has been exploited for decades as a guiding principle in the quest of models beyond the standard model. Arguments relying on naturalness, however, tend to depend on the class of vacua (an ensemble) one has in mind. Since string theory is able to provide a well-motivated ensemble of vacua on which naturalness arguments can be built, this is hence another place where string theory can contribute to progress of theoretical particle physics. This article aims at making progress in the second direction above. It is known that flux compactification of Type IIB string theory / F-theory stabilizes not only complex structure moduli but also the brane configuration. This means that both gauge groups and coupling constants of the effective theory are determined once a topological flux configuration is given. Exploiting all of the theoretically possible topological flux configurations, an ensemble of low-energy effective theories with various gauge groups and coupling constants is generated in this framework, in principle. To get this done in practice, however, a clever approach is necessary. Low-energy effective theories are usually classified in terms of their algebraic information (such as gauge groups, matter representations and presence/absence of certain types of interactions) first, and then in terms of topological information (such as the number of generations of matter fields in a given representation). Effective theories with the same algebraic and topological information are then specified by the values of coupling constants. The string landscape will be of some use only when the statistical distribution of low-energy observables are presented and studied in compatibility with such a hierarchical classification of effective theories. This can be done along the line described in [12], which is built on top of pioneering works [8,10]. The study of [12], however, used K3 × K3 compactification of Ftheory, where analysis is a little easier, but we cannot even hope to obtain a semi-realistic model of low-energy physics. This article applies the method to more general Calabi-Yau fourfolds for F-theory compactification, derives answers to questions of practical interest, and exemplifies the potential power of the method. Section 2 is devoted to a review of the method in [8,10,12] along with new observations in [13]. The method is then applied to a class of compactifications that lead to semi-realistic supersymmetric grand unification (GUT) models in section 3. We derive distribution of the number of generations in flux vacua of SU(5) GUT models, and also study how the number of flux vacua scales when we require SO(10), SU (5) or no unification group on 7-branes, respectively. We find that the distribution is in a factorized form to a good approximation, independently of topological choice of geometry. Details and explanations omitted in this article are found in [13]. The Method: A family of geometries over a restricted moduli space, π : Y −→ M * , is a useful concept when one wants to focus on flux vacua with a given algebraic information. A restricted family and its moduli space, π : Y −→ M A4 * (resp. M D5 * ), is specified for a topological choice of (B 3 , [S]), where [S] is a divisor class in B 3 . Each mem- ber of the family, π −1 (p) = Y p for p ∈ M A4 * (resp. M D5 * ) , is a smooth elliptically fibred Calabi-Yau fourfold π Yp : Y p −→ B 3 with a section, and the discriminant locus of the fibration π Yp contains an irreducible component in [S] and the generic fibre over it is I 5 (resp. I * 1 ) in the Kodaira classification. Any one of such fourfolds in the family over M A4 * (resp. M D5 * ) can be used for F-theory compactification that results in a vacuum with an R = SU(5) (resp. R = SO(10)) unification group on 7-branes. M A4 * or M D5 * parametrizes the complex structure of such geometries. 1 Higher rank 7-brane gauge group implies a larger number of independent divisors [Ĉ i ] ∈ [H 2 (Y p ; Z) ∩ H 1,1 (Y p ; R)], i = 1, · · · , rank(R), (1) for a generic geometry π −1 (p) = Y p in p ∈ M * . An ensemble of F-theory flux vacua with a given algebraic and topological information is specified by a pair (H scan , G (4) fix ), where H scan ⊂ [H 4 (Y ; Z)] ker , G (4) fix ∈ [H 4 (Y ; 'Z')] prim ; (2) the subscript "prim" implies J ∧ G = 0 ∈ H 6 (Y ; R) (the D-term condition), and "ker" both J ∧ G = 0 and i * Ci (G) = 0 ∈ H 4 (Ĉ i ; Q). The last condition is to make sure that fluxes in H scan do not introduce gauge symmetry breaking (cf [13]). An ensemble of 4-form flux G (4) tot = G (4) scan + G (4) fix \| G (4) scan ∈ H scan(3) determines an ensemble of vacua of complex structure through the superpotential W ∝ Yp Ω Yp ∧ G tot . Statistics of such an ensemble of vacua can be presented as a distribution over the restricted moduli space M * . The distribution was worked out analytically in a 1 As in [12] very robust way [8] for the vacuum index density dµ I , d 2m z G (4) scan δ 2m (DW, DW ) det D 2 WDDW DDW D 2 W ,(4) to which each flux vacuum on M * contributes by a deltafunction with coefficient ±1. Here, m := dim C M * = h 3,1 (Y ), and the dz's are local holomorphic coordinates on M * . Derivatives in DW , DW etc. are with respect to the fields corresponding to the complex structure moduli tangent to M * . Making a continuous approximation [8] of the sum over flux configurations G (4) scan in (4), the vacuum index density is cast into the following form dµ I = (2πL * ) K/2 (K/2)! ρ I , K := dim R (H scan ⊗ R) . (5) Here, L * is the maximal D3-brane charge available. Although ρ I depends on the choice of H scan , it is given by c m (T M * ⊗ L) = det − R 2πi + ω 2π 1 m×m(6) for the Kähler form ω on M * whenever (H scan ⊗ R) contains the real primary horizontal subspace H 4 H * (Y ; R) ⊂ H 4 (Y ; R) [8, 10, 12]. H 4 H * (Y ; R) is the real part of the primary horizontal subspace in [4], Span C Ω Yp , DΩ Yp , D 2 Ω Yp , · · · ⊂ H 4 (Y ; C) ,(7) which does not depend on the choice of p ∈ M * . In order to see how H scan should be chosen to achieve the goal we have set in this article, note that the vector space H 4 (Y ; R) is decomposed as follows: We choose H scan ⊗ R to be H 4 H * (Y ; R) in this article. As discussed in more detail in [13], H 4 H * (Y ; R) is contained in [H 4 (Y ; R)] ker . Thus, for this choice of H scan , all the vacua share the same symmetry group from 7-branes in the effective theories below the Kaluza-Klein scale. This argument does not exclude the option to take H scan ⊗ R larger that H 4 H * (Y ; R), but we should not take it to be as large as [H 2,2 V * (Y ; R)] ⊥ . It is not hard to find families π : Y −→ M * with h 2,0 (Y p ) = 0 where there are algebraic four-forms in H 2,2 RM * (Y ; R) ∩ H 4 (Y ; Z) that break the symmetry of the 7-brane gauge group [13]. H 4 H * (Y ; R) ⊕ H 2,2 RM * (Y ; R) ⊕ H 2,2 V * (Y ; R) ;(8) A similar phenomenon has also been observed in [12], where Y = K3 × K3, and the four-form in H 2,2 RM * (Y ; R) is not dual to an algebraic cycle. The Results: Let us apply the method described in the previous section to derive statistical distributions of observables of practical interest. The study in [12] used a family of K3 × K3 for compactification of F-theory, where all the 7-branes are parallel, and there is no light matter fields except those in the adjoint representation of the 7-brane gauge groups. Ensembles of flux vacua in such a set-up do not include low-energy effective theories that look close to the (supersymmetric extensions of the) Standard Model. In this article, we therefore use a few other families of elliptically fibred Calabi-Yau fourfolds, for which the lowenergy effective theories are at least semi-realistic. These effective theories have SU(5) (resp. SO(10)) unification, and N gen generations of matter fields in the 10 +5 (resp. 16) representations. The task is to determine the value of L * and K for such families and to study how those values depend on the choice of the unification group or the number of generations N gen . 3.A Number of Generations We focus on a few choices of (B 3 , [S]) for which the restricted moduli spaces and families for SU(5) unification are constructed in the way stated at the beginning of the previous section as examples. We choose B 3 = P [O P 2 ⊕ O P 2 (n)] , −3 ≤ n ≤ 3,(9) which is a P 1 -fibration over P 2 , and let [S] be the "northpole section" of the P 1 -fibration corresponding to the zero of a section of O P 2 . The range of n is set so that the 7brane unification group at the S can be as small as SU (5), while the gauge group at the hidden sector (corresponding to the zero of a section of O P 2 (n)) can be completely Higgsed away. We set (H scan ⊗ R) = H 4 H * (Y ; R), and choose G (4) fix to be the F-theory dual of the chirality-generating bundle twist in [5], parametrized by λ F MW ∈ 1/2 + Z. This flux gives rise to the net chirality of the matter fields in the 10 vs 10 (also5 vs 5) representation of the SU(5) unification group; N gen = −(18 − n)(3 − n)λ F MW [6]. Since the vanishing cycles for the chiral matter belong to H 2,2 V * (Y ; Q), any flux vacua in the ensemble (3) have the same N gen [13]. In this way, we obtain an ensemble of F-theory flux vacua that share the same algebraic and topological (N gen ) information. The F-theory dual description of G (4) fix has been determined in [11]. L * = χ(Y )/24 − (G (4) fix ) 2 /2 is the upper bound on the net D3-brane charges from G (4) scan . It depends on N gen through G (4) fix [12] and a straightforward computation reveals that [11,13] L * = 2163 4 + 125 8 n (n + 7) − 5 N 2 gen 2 (18 − n)(3 − n) .(10) The maximal values of L max * range within ∼ 300 to 800 for the families with −3 ≤ n ≤ 3 ( Table I); details of the calculation are found in [13]. L * is always an upper convex quadratic function of N gen in F-theory, not just for the choice in (9). The other number we need to use in (5) is K = dim R H 4 H * (Y ; R) . This task boils down to the determination of the dimension of the horizontal component H 2,2 H * (Y p ; R), since K = 2 + 2m + dim R [H 2,2 H * (Y p ; R)]. A general recipe is to use mirror symmetry and determine the dimension of the vertical component of the mirror manifold of Y . The authors derived in [13] the formula for h 2,2 V , h 2,2 H and h 2,2 RM that is valid for any Calabi-Yau hypersurface of a toric 5-dimensional ambient space. We carried out the computation of h 2,2 V * (Y ), h 2,2 H * (Y ) and K; for the families for SU(5) unification with B 3 in (9), it turns out that h 2,2 RM = 0. Details of the computation are found in [13], and only the results are recorded in Table I With this preparation, we can derive the distribution of the number of generations N gen almost immediately. We have constructed ensembles of flux vacua that are labelled by λ F MW ∈ 1/2 + Z. Each one of those ensembles consists of vacua that lead to effective theories with common algebraic information (SU(5) GUT with chiral matter in the 10+5 representation), but their topological information N gen ∝ λ F MW varies from one ensemble to another. To compare the number of vacua that the individual ensembles contain, one simply needs to integrate ρ I over M * , which is to ignore the difference in the value of the effective coupling constants of the vacua in a given ensemble. Since the integral of ρ I over M * usually yields a number of order unity (some region of M * may have to be excluded; cf the discussion of D-limits in [8]), we simply make an approximation M * ρ I ≈ 1. Only the prefactor (2πL * ) K/2 /[(K/2)!] in (5) is then used as an estimate of the number of vacua in a given ensemble. We can use the value of K in Table I, and the N gen -dependence of L * has already been discussed in this article. The number of vacua depends on N gen in a way the volume of a K-dimensional sphere changes as the radius-square L * decreases quadratically in N gen . There is an absolute upper limit on N gen for a given family π : Y −→ M A4 * due to the D3-tadpole constraint L * ≥ 0. In the examples of (B 3 , [S]) we have chosen at the beginning of this section, however, we have L * ≪ K (see Table I). The continuous approximation [8] is not particularly good for such families, and Ref. [8] suggests that the estimate of the number of flux quanta (the prefactor) √ πL * K /[(K/2)!] be replaced by e √ 2πKL * , in caes with K ≫ L * rather than L * ≫ K. In fact, it is commonplace to find L * ≪ K, not just in the familes cosidered in this section, but in a broader class of familes π : Y −→ M R * of interest. For B 3 with h 1,1 (B 3 ) ≈ O(1), where c is the coefficient of N 2 gen in (10), which remains of order unity. The expansion in N 2 gen in the exponent is valid for N gen < ∼ 10, since χ(Y )/24 is often around 100-1000. The first factor of (11) depends on the choice of the 7-brane symmetry R (and on (B 3 , [S])), while the second factor is a Gaussian distribution on N gen for robust choice of (B 3 , [S]). Algebraic and topological data of effective theories have a factorized distribution. One may further bring the distribution ρ I of the effective coupling constants back to (11), without integrating it over M R * . 3.B Cost of Higher-Rank Gauge Groups The distribution (5,6,11) can be used to derive the statistical cost of requiring a higher rank gauge group on 7-branes. This idea was pursued already in [12], using Y = K3 × K3 for F-theory compactification; more examples are obtained in this article to estimate the systematics. The choices of (B 3 , [S]) here are also more realistic than that of K3 x K3. The SO(10) version of the family, π : Y −→ M D5 * , can also be constructed for the choice of (B 3 , [S]) in (9) using toric geometry. Various topological data for the families over M D5 * , M A4 * and M (where no 7-brane gauge symmetry is required) can be computed and the results are recorded in Table II for the n = 0 case. For more details of computations, see [13]. One can see from (11) that the difference in the value of K for different choices of 7-brane symmetry R determines the relative number of the corresponding flux quanta (vacua). Ensembles with higher rank 7-brane gauge group have smaller dimension K (Table II), confirming the same observation in [12] based on the family Y = K3 × K3. This leads to the observation that the rank-4 gauge group on 7-branes is not as statistically "natural" as vacua without a gauge group on 7-branes. In the choice of (B 3 , [S]) in (9) with n = 0, for example, vacua with the rank-4 SU(5) unification constitutes only the fraction e −∆K/2 ≈ e −3000 of the entire flux vacua (smaller than the fraction 10 −120 for the cosmological constant). The authors do not provide their interpretations for this inconvenient prediction; a popular attitude will be to hint at poor understanding of string theory, to count on cosmological factors that we did not study here, and/or to resort to anthropics. Outlook: This article only deals with the easiest applications of the method explained in section 2. Various ideas of using (11) and ρ I to address questions of practical interest are described in detail in [13]. the vertical component H 2,2 V * (Y ; R)is the subspace generated by the wedge products of integral (1, 1)-forms on Y p , which defines a subspace of H 4 (Y ; R) independent of p ∈ M * . The remaining component H 2,2 RM * (Y ; R), thus, should not depend on p ∈ M * . . Certainly the results on L max * and K in this table are only for a limited number of choices of B 3 and do not tell us whether they are typical among the results for all other choices of B 3 from 3-dimensional Fano varieties, or how much the values of L max * and K can vary for different B 3 . But the table at least provides the first example of such calculations. and literatures therein, we only consider flux vacua for (B 3 , [S]) of a given topology and for a given choice of Kähler form J on Yp that is in H 2 (Yp; Q)∩H 1,1 (Yp; R) modulo multiplication of R × . The (restricted) moduli space M or M * therefore refers only to that of complex structure. It is beyond the scope of this article or [13] to include the scanning over (B 3 , [S])'s of different topology or their Kähler moduli spaces. TABLE TABLE II : IIData for families over M, M A4 * and M D5 * . h 1,1 h 3,1 h 2,2 V * h 2,2 H * χ(Y ) K smooth CY 3 3277 4 13160 19728 19716 SU(5) model 7 2148 9 8655 12978 12953 SO(10) model 8 2138 10 8618 12924 12896 for example, h 1,1 (Y ) still remains O(1), while h 3,1 is much larger, due to the degrees of freedom for the complex structure of fibration. It is then a quite natural consequence that h 2,2 V and h 2,2 RM are not as large as h 2,2 H . 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[]	[ "KEKTakashi Kaneko \nINFN-LNF\n\n", "Barbara Sciascia \nINFN-LNF\n\n" ]	[ "INFN-LNF\n", "INFN-LNF\n" ]	[]	The actual limit of the Unitarity condition of the first row of the CKM matrix \|V ud \| 2 + \|V us \| 2 + \|V ub \| 2 = 1+ ∆ CKM is ∆ CKM = −0.0001(6). In 2010 the same was ∆ CKM,2010 = +0.0001(6). Despite the only difference of a sign, and with an absolute change of the value of one third of the accuracy, a substantial amount of work has been done in the last two years to improve the knowledge of all the contributions to this stringent limit to CKM unitarity, and more is expected in the next years. In this paper we present an organized summary of all the important contributions presented during the WG1 sessions, referring as much as possible to the contribution papers prepared by the individual authors.	null	[ "https://arxiv.org/pdf/1408.6374v1.pdf" ]	119,196,207	1408.6374	8508424bd945cb9a7fa0c47a74ca24b7166c2d21	27 Aug 2014 September -2 October 2012 KEKTakashi Kaneko INFN-LNF Barbara Sciascia INFN-LNF 27 Aug 2014 September -2 October 2012Precise determination of V ud and V us Summary of the WG1 contributions, written for the Proceedings of CKM 2012, the 7th International Workshop on the CKM Unitarity Triangle, University of Cincinnati, USA, 28 The actual limit of the Unitarity condition of the first row of the CKM matrix \|V ud \| 2 + \|V us \| 2 + \|V ub \| 2 = 1+ ∆ CKM is ∆ CKM = −0.0001(6). In 2010 the same was ∆ CKM,2010 = +0.0001(6). Despite the only difference of a sign, and with an absolute change of the value of one third of the accuracy, a substantial amount of work has been done in the last two years to improve the knowledge of all the contributions to this stringent limit to CKM unitarity, and more is expected in the next years. In this paper we present an organized summary of all the important contributions presented during the WG1 sessions, referring as much as possible to the contribution papers prepared by the individual authors. New bounds on violations of CKM unitarity translate into significant constraints on various new physics scenarios. Such tests may eventually turn up evidence of new physics. If the couplings of the W to quarks and leptons are indeed specified by a single gauge coupling, then for universality to be observed as the equivalence of the Fermi constant G F as measured in muon and hadron decays, the CKM matrix must be unitary. Currently, the most stringent test of CKM unitarity is obtained from the first-row relation \|V ud \| 2 + \|V us \| 2 + \|V ub \| 2 = 1 + ∆ CKM ; here we shortly discuss the ingredients that contribute to its accuracy and its possible new physics implications. We start with Vincenzo Cirigliano [1] that in his talk emphasized a model-independent EFT approach to β decays and Cabibbo universality tests. Given the hierarchy \|V ud \| 2 ≫ \|V us \| 2 , let us focus on effects of physics beyond the Standard Model (BSM) to \|V ud \|. Assuming that right-handed neutrinos do not appear as low-energy degrees of freedom, new physics introduces five operators ǫ Γ lγ µ (1 − γ 5 )ν l · uΓ µ d (Γ = V, A), ǫ Γ l(1 − γ 5 )ν l · uΓd (Γ = S, P, T ) into the Lagrangian of the d → ulν l transitions. The (axial-)vector and scalar couplings ǫ {V,A,S} are constrained at the level of 10 −3 from the superallowed nuclear β decays. The current precise knowledge of \|V ud \| from the nuclear decays together with a future competitive determination from neutron decays could constrain ǫ {S,T } at 0.02% level. The pseudo-scalar coupling ǫ P is strongly constrained from the ratio Γ(π → eν e )/Γ(π → µν µ ). The outlook is therefore quite positive: the effective couplings of all the BSM charged-current operators are currently probed or will be soon probed at the level of 10 −3 or better. This corresponds to probing maximal BSM physics scales Λ ranging from 7 TeV (for scalar and tensor interactions) to 11 TeV (for vector interactions). New physics can also modify the Lagrangian of the muon decays. Its effects would be encoded in the Fermi constant G F , which is best determined by the measurement of the positive muon lifetime τ µ . Tim Gorringe [2] reported results from the MuLan measurement of the positive muon lifetime, conducted at the Paul Scherrer Institute. The result is characterized by a part-per-million accuracy, largely dominated by the statistical contribution, and rock-solid systematic effects study. The MuLan measurement translates into G F = 1.1663787(6) × 10 −5 GeV −2 . The 0.5 ppm error is dominated by the 1.0 ppm uncertainty of the lifetime measurement, with contributions of 0.08 ppm from the muon mass measurement and 0.14 ppm from the theoretical corrections. τ µ and G F at 0.1 ppm is now the open challenge for the next years. \|V ud \| is best determined using 0 + → 0 + super-allowed nuclear β decays. The \|V ud \| state of the art has been presented by Dan Melconian [3]. Given the limited number of new experimental contributions, no new survey on \|V ud \| has been done and its world average is unchanged from the 2010 CKM edition [4]. The 0 + → 0 + transitions benefit from the conservation of the vector current and from small isospinbreaking corrections. The experimental inputs are combined in a quantity that should be nucleus-independent to first order. This is true only applying also the isospinbreaking corrections, δ C . These are a dominant contribution to the \|V ud \| accuracy and have been studied in recent years using a variety of theoretical methods. A new approach, proposed by Towner and Hardy [5], allow one to experimentally test the SU(2) correction. They succeeded in measuring the δ C for the elements with the largest isospin-breaking correction, 32 Ar and 32 Cl, and found it in agreement with the theoretical calculations. Other groups are beginning to develop complementary models of SU(2) in superallowed decays that will be important checks and may lead to smaller systematic error on this \|V ud \| determination. The alternative ways to determine \|V ud \|, from neutron lifetime, from mirror decays, and from pion β decay, are still limited by the experimental accuracy. It is nevertheless worth improving their accuracy because the involved processes probe different BSM operators. As a probe free from nuclear structure corrections, the decay of the free neutron has the potential to provide the most accurate value of \|V ud \|. However, the experimental sensitivity still needs to be further improved to become competitive with superallowed nuclear β decays. The current status of the neutron decay studies relevant for the determination of \|V ud \| has been presented by Oliver Zimmer [6]. In the standard V-A theoretical description of neutron decay both the vector and the axial-vector currents contribute, with g V and g A coupling constants respectively, so that two observables are needed to access \|V ud \|: the neutron lifetime τ n and the g A /g V ratio. For τ n , the accuracy needed to compete with 0 + → 0 + transitions in the \|V ud \| determination is ∼0.3 s on τ n (a factor three from the present accuracy). In this respect, a novelty is the solution of the 5.4 σ tension between different determinations, mainly due to a single 2005 measurement. A wide effort in the field and a new measurement in 2010, pushed the authors of the 2005 result to scrutinize their procedures and to recently publish a corrected value. With these changes included, the tension is reduced to 1.4 σ even if the the 1.8 scale factor applied by the PDG indicates that systematic uncertainties are not properly taken into account in all experiments. The highest experimental sensitivity on the g A /g V ratio has been achieved measuring the β asymmetry coefficient. The accuracy goal here is ∼0.0003 on g A /g V , about one order of magnitude below the present determination. For both τ n and g A /g V many projects are in the pipeline using ultra-cold neutrons or magnetic trapping. Some are in a very advanced state and have the potential to reach the accuracy goal in the next years. Nuclear mirror transitions occur between isobaric analogue states within an isospin doublet, where initial and final states have the same spin and parity. The determination of \|V ud \| from nuclear mirror transitions and the current experimental efforts aimed at improving its precision have been presented by Oscar Naviliat-Cuncic [7]. In addition, the mirror transitions are driven by a mixing of vector and axial-vector interactions, and the present \|V ud \| accuracy is dominated by the experimental error in the determination of the mixing ratio. As in the neutron lifetime case, two experimental inputs are needed. These are the half-life of the decay, and one of the following: the β − ν angular correlation, the β asymmetry, or the ν asymmetry. Recently a substantial activity has been initiated on the experimental and theoretical sides to improve all the relevant inputs: many experiments are on going, applying different techniques, to measure the β − ν angular correlation, and there are plans to measure also relative β asymmetries. These efforts should enable significant improvements in the precision on \|V ud \| from mirror transitions, which is currently a factor ∼8 less precise than the value extracted from Fermi transitions. Moving to \|V us \|, its best determination arises from K ℓ3 and K ℓ2 decays. A very comprehensive review of the \|V us \| determination from kaons and of its effects on the unitarity test of the first row has been presented by Matthew Moulson [8]. In the field, there have been a few significant new measurements and some important theoretical developments. The experimental inputs for the determination of \|V us \| from K ℓ3 decays are the rates and form factors for the decays of both charged and neutral kaons. There have been no new branching ratio measurements since the 2010 review. On the other hand, both the KLOE and KTeV collaborations have new measurements of the K S lifetime. Finally, the NA48/2 experiment has recently released preliminary results for the form factors for charged kaon decays. This is important because it helps to resolve a controversy: the older measurements of the K µ3 form factors for K L decays from NA48 are in such strong disagreement with the other existing measurements that they have been excluded from the FlaviaNet averages. The new NA48/2 measurements, on the other hand, are in good agreement with other measurements. For all of the above efforts, however, the value of and uncertainty on \|V us f + (0)\| are essentially unchanged. This is because the new results are nicely consistent with the older averages, and neither the K S lifetime nor the phase space integrals were significant contributors to the overall experimental uncertainty. The latter is dominated by the lifetime accuracy for the K L and by the branching ratio for the K S and K ± determinations. In the near future there are no kaon experiments planning new branching ratio or lifetime measurements. For the K ± , also the uncertainty on the theoretical isospin-breaking correction gives the largest contributions to the \|V us f + (0)\| uncertainty. Besides the latter, advances in algorithmic sophistication and computing power are leading to more and better lattice QCD estimates of the hadronic constants f + (0) and f K /f π , which enter into the determination of \|V us \| from K l3 and K µ2 decays, respectively. Due to their non-perturbative nature the only systematically improvable way to compute them are simulations of lattice QCD. Since the ultimate goal is a test of the SM, any model-dependence should be avoided and this is where progress in lattice simulations is currently being made. In addition, two groups working on the classification and averaging of results from lattice QCD have joined their efforts, constituting the newly formed Flavor Lattice Average Group (FLAG-2) [9] to provide recommended values of these constants. The talk by Andreas Jüttner [10] reported on the status and ongoing improvements of determinations of f + (0). A key observation that allows a precise extraction of f + (0) is the conservation of the vector current at zero momentum transfer in the SU(3) limit: the normalization is fixed in this limit and corrections start at the second order in SU(3) breaking effects. Because of this fortunate situation, recent lattice computations determine f + (0) at the level of 0.5 -1.0 % and show an excellent agreement among them. The uncertainty of f + (0) in the state-of-the-art calculations is dominated by the statistical error and the error due to the extrapolation to the physical pion mass. The latter is about to be removed by simulating very close to or at the physical mass. In his talk Jack Laiho [11] presented the lattice progress and the future prospects for f K /f π . This is a key input in the determination of \|V us \|/\|V ud \| via K l2 decays, which probe different BSM operators than the K l3 decays. While the normalization in the SU(3) limit is fixed, only the axial-vector current contributes to the decay constants and f K /f π can receives unsuppressed chiral corrections. However, recent simulations on realistic lattices, particularly those at small or even physical pion mass, have calculated this ratio to sub-percent precision leading to the world average with a 0.4% accuracy. A further reduction of the discretization error as well as effects of finite lattice volumes is needed to improve the precision, say, to a level of 0.2 %. We expect more simulations with different lattice formulations at the physical pion mass in the future for better understanding and control of these systematic uncertainties. At the level of precision now reached in the lattice determinations of f + (0) and f K /f π , the uncertainties of the electromagnetic and isospin corrections are becoming non-negligible. Traditionally, these corrections have been estimated by means of chiral perturbation theory (ChPT), in which an extension to higher orders is generally not easy due to rapidly increasing numbers of relevant diagrams and unknown effective couplings. An interesting possibility is to include these corrections into the lattice determinations in a fully non-perturbative way. In his talk Nazario Tantalo [12] presented a first principle lattice calculation of the QCD isospin corrections, namely those coming from the small difference of the up and down quark masses ∆m ud = (m u − m d )/2 in the absence of the electromagnetic interactions. Isospin is an approximate symmetry of QCD and most of the theoretical predictions on phenomenologically relevant hadronic observables have been derived by assuming the exact validity of isospin symmetry. This is also the case for most of the non-perturbative theoretical predictions on hadronic matrix elements obtained over the years by performing lattice QCD simulations, like the hadronic constants f + (0) and f K /f π . The RM123 collaboration has recently performed a lattice calculation of the QCD isospin corrections from numerical simulations of isosymmetric QCD combined with a systematic expansion of the partition function with respect to the small parameter ∆m ud . Their new method, therefore, does not need time-consuming simulations of the isospin-broken theory. They obtained encouraging results for the isospin corrections, which are of the same order of magnitude, though higher, of the ChPT estimate. It should be noted that the separation of QED from QCD isospinbreaking effects is prescription-dependent. It is therefore important to calculate the full (QCD+QED) correction on the lattice, which could be an interesting alternative to the conventional ChPT approach. The hadronic τ decays provide an alternative way to measure \|V us \| and to probe the relation of the first row of the CKM matrix. Measurements of \|V us \| from τ decays are complementary to those from kaon decays because new physics scenarios that couple primarily to the third generation could cause deviation between measurements of \|V us \| in the kaon and τ systems. In her talk Elvira Gámiz [13] gave an overview of the theoretical issues related to HFAG-Tau Winter 2012 Figure 1: An update of \|V us \| from the hadronic τ decays (see Ref. [14] and references therein). The three upper values are from the FlaviaNet Working Group's analysis of the K l3 and K l2 decays in 2010 and from the unitarity constraint and \|V ud \|. the \|V us \| extractions from inclusive and exclusive hadronic τ decays. Some exclusive decay channels, such as τ → Kν and τ → Kπν, can be used to extract \|V us \| in a similar manner to K l2 and K l3 decays. This method is therefore sensitive to the same lattice QCD uncertainties. The most precise measurement could be offered from the SU(3)-breaking effect in the inclusive rate δR = R τ,V +A \|V ud \| 2 − R τ,S \|V us \| 2 ,(2) where R τ,S = Γ(τ → X s ν τ )/Γ(τ → eν e ν τ ) is the Cabibbo-suppressed hadronic rate into strange particles (X s ) and R τ,V +A = Γ(τ → X non−s ν τ )/Γ(τ → eν e ν τ ) is the Cabibboallowed hadronic rate. The SU(3)-breaking effect δR can be estimated from finite energy sum rules (FESR). This technique makes the hadronic τ decays an ideal system to study low-energy QCD under rather clean conditions, allowing the determination of the strong coupling with the same precision achieved by lattice determinations. To profit from δR, we need to measure the inclusive strange and non-strange spectral density functions, which are constructed from the sum of invariant mass distributions for each of the strange and non-strange decay modes and normalized to the corresponding branching fractions. Since there are no solid predictions for the branching fractions of hadronic individual τ decays, all possible modes must be measured or upper bounds have to have placed on them. This technique is completely Figure 2: \|V ud \| from 0 + → 0 + β decays, \|V us \| from K l3 decays, and \|V us /V ud \| from K l2 decays in the plane of (\|V ud \|, \|V us \|) [8]. The yellow ellipse indicates the 1σ confidence interval for the fit to fix \|V ud \| and \|V us \|. The unitarity constraint is shown by the solid line. independent of the kaon measurements, and if all of the branching fractions and spectral functions were updated with the whole data sets of BELLE and BABAR experiments, this method would be expected to provide the most precise measurement of \|V us \|. Ian Nugent [14] presented an overview of the current status of the experiments. Currently, both the exclusive and the inclusive decays determine \|V us \| with an accuracy of about 1.0 %, which is slightly larger than 0.6 % achieved in the determinations from K l2 and K l3 decays. More importantly, the updated experimental results lead to only a slight change from the previous workshop [4] in \|V us \| from the inclusive decays. This result is about 3 σ below the K l2 and K l3 determinations as shown in Fig. 1. Since the uncertainty is limited by the experimental precision, further experimental data are needed before drawing any significant conclusion. While the reliability of the FESR analysis has been studied, a more thorough study, for instance about the stability against the choice of the weight, is also welcome. Presently, the most precise value of \|V ud \|, \|V ud \| = 0.97425(22),(3) is obtained from 0 + → 0 + nuclear decays and is unchanged from the previous CKM workshop. This together with the updated values of \|V us \| = 0.2254(13) from K l3 decays and \|V us /V ud \| = 0.2317(11) from K l2 can be combined in a single fit to determine the CKM elements and ∆ CKM [8]. As plotted in Fig. 2, the fit does not change the input value of \|V ud \| and yields \|V us \| = 0.2256 (8), ∆ CKM = +0.0001(6)(4) in perfect agreement with unitarity. The uncertainty of ∆ CKM is equally shared by \|V ud \| and \|V us \|. The precision of \|V ud \| is determined from the uncertainty of the transition independent radiative correction [15], which has been stable over the last several years. The uncertainty of \|V us \| is currently dominated by uncertainties in the lattice results for f + (0) and f K /f π , which are at the level of 0.5%. Thus, at the moment, the lattice offers the most certain prospects for further improvement. Although the accuracy of ∆ CKM is essentially unchanged from the previous workshop, the current precision already allow us to put significant constraints on new physics as summarized in the beginning of this paper. An interesting question is whether the low-energy observables from the nuclear, neutron, kaon and τ decays have a higher sensitivity than the rest of observables, for instance, from energyfrontier experiments. In his talk Martín González-Alonso [16] answered this question by comparing new physics bounds from the low-energy observables with those obtained by the CMS Collaboration analyzing 5 fb −1 of data recorded at √ s = 7 TeV in the pp → e + MET (Missing Transverse Energy) channel. Assuming that the heavy mediators that generate the BSM interactions in Eq. (1) are too massive to be produced at the LHC, we can again employ an EFT approach and put bounds on the ǫ i couplings from the LHC search. For the (axial) vector and pseudo-scalar ones low-energy probes are much more powerful. There is an interesting competition between low-and high-energy searches for the scalar and tensor couplings. In addition to Eq. (1), we can also introduce BSM interactions involving right-hand neutrinos, for which the LHC will dominate the search. Even if the LHC sensitivity will get better as more data is collected, future lowenergy experiments, such as with (ultra)cold neutrons, will improve in finding the bounds on new physics. Currently, the precise determination of \|V ud \| and \|V us \| provides one of the most stringent tests of the Standard Model and will play an important role, complementary to the collider searches, in probing new physics. A General Introduction to the First Row and Implications Beyond the SM" and references therein. V Cirigliano, These proceedingsV. Cirigliano, "A General Introduction to the First Row and Implications Beyond the SM" and references therein. These proceedings. . T Gorringe, ArXiv: 1301.0504τ µ from the MuLan Experiment" and references therein. These proceedingsT. Gorringe, "τ µ from the MuLan Experiment" and references therein. These proceedings. ArXiv: 1301.0504 The State of the Art for V ud Extraction from Nuclear β Decay" and references therein. D Melconian, These proceedingsD. Melconian, "The State of the Art for V ud Extraction from Nuclear β Decay" and references therein. These proceedings. \|V ud \| and \|V us \|: working group I summary. T Spadaro, R Young, ArXiv: 1112.0238T. Spadaro and R. Young, "\|V ud \| and \|V us \|: working group I summary", ArXiv: 1112.0238. Comparative tests of isospin-symmetry-breaking corrections to superallowed 0 + → 0 + nuclear β decay. I S Towner, J C Hardy, Phys. Rev. C. 8265501I. S. Towner and J. C. Hardy, "Comparative tests of isospin-symmetry-breaking corrections to superallowed 0 + → 0 + nuclear β decay", Phys. Rev. C 82 (2010) 065501. V ud from Cold and Ultra-cold Neutrons" and references therein. These proceedings. O Zimmer, ArXiv: 1301.1854O. Zimmer, "V ud from Cold and Ultra-cold Neutrons" and references therein. These proceedings. ArXiv: 1301.1854. O Naviliat-Cuncic, ArXiv: 1301:4153V ud from Nuclear Mirror Transitions" and references therein. These proceedings. O. Naviliat-Cuncic, "V ud from Nuclear Mirror Transitions" and references therein. These proceedings. ArXiv: 1301:4153. V us From Kaon Decays" and references therein. These proceedings. M Moulson, ArXiv: 1301.3046M. Moulson, "V us From Kaon Decays" and references therein. These proceedings. ArXiv: 1301.3046. S Aoki, arXiv:1310.8555Review of lattice results concerning low energy particle physics. S. Aoki et al., "Review of lattice results concerning low energy particle physics", arXiv:1310.8555. Lattice Prospects and Future Prospects on f + (0)" and references therein. A Jüttner, ArXiv: 1301.0212These proceedings. A. Jüttner, "Lattice Prospects and Future Prospects on f + (0)" and references therein. These proceedings. ArXiv: 1301.0212 Lattice Progress and Future Prospects on f K /f π " and references therein. J Laiho, These proceedingsJ. Laiho, "Lattice Progress and Future Prospects on f K /f π " and references therein". These proceedings. Lattice Calculation of Isospin Corrections to Kℓ2 and Kℓ3 Decays" and references therein. N Tantalo, ArXiv: 1301.2881These proceedings. N. Tantalo, "Lattice Calculation of Isospin Corrections to Kℓ2 and Kℓ3 Decays" and references therein. These proceedings. ArXiv: 1301.2881 V us Determination From τ Decays (Theoretical Review)" and references therein. E Gámiz, ArXiv: 1301.2206These proceedings. E. Gámiz, "V us Determination From τ Decays (Theoretical Review)" and refer- ences therein. These proceedings. ArXiv: 1301.2206 V us Determination From τ Decays (Experimental Review)"and references therein. I M Nugent, ArXiv: 1301.0637These proceedings. I. M. Nugent, "V us Determination From τ Decays (Experimental Review)"and references therein. These proceedings. ArXiv: 1301.0637 Improved calculation of electroweak radiative corrections and the value of \|V ud \|. W J Marciano, A Sirlin, Phys. Rev. Lett. 9632002W. J. Marciano and A. Sirlin, "Improved calculation of electroweak radiative corrections and the value of \|V ud \|", Phys. Rev. Lett. 96 (2006) 032002. Probing Novel Scalar and Tensor Interactions from (Ultra) Cold Neutrons to the LHC" and references therein. M González-Alonso, ArXiv: 1301.0287These proceedings. M. González-Alonso, "Probing Novel Scalar and Tensor Interactions from (Ultra) Cold Neutrons to the LHC" and references therein. These proceedings. ArXiv: 1301.0287	[]
[ "On the number of collisions in beta(2, b)-coalescents", "On the number of collisions in beta(2, b)-coalescents" ]	[ "Alex Iksanov \nFaculty of Cybernetics\nNational T. Shevchenko University\n01033KievUkraine\n", "Alex Marynych **[email protected] \nFaculty of Cybernetics\nNational T. Shevchenko University\n01033KievUkraine\n", "Martin Möhle \nMathematical Institute\nUniversity of Düsseldorf\nUniversitätsstraße 140225Düsseldorf, Ger\n" ]	[ "Faculty of Cybernetics\nNational T. Shevchenko University\n01033KievUkraine", "Faculty of Cybernetics\nNational T. Shevchenko University\n01033KievUkraine", "Mathematical Institute\nUniversity of Düsseldorf\nUniversitätsstraße 140225Düsseldorf, Ger" ]	[ "Bernoulli" ]	Expansions are provided for the moments of the number of collisions Xn in the β(2, b)-coalescent restricted to the set {1, . . . , n}. We verify that Xn/EXn converges almost surely to one and that Xn, properly normalized, weakly converges to the standard normal law. These results complement previously known facts concerning the number of collisions in β(a, b)-coalescents with a ∈ (0, 2) and b = 1, and a > 2 and b > 0. The case a = 2 is a kind of 'border situation' which seems not to be amenable to approaches used for a = 2.	10.3150/09-bej192	[ "https://arxiv.org/pdf/0909.0870v1.pdf" ]	402,214	0909.0870	e5a0e32d07dfc0db5ddbc87b897f91609d498a23	On the number of collisions in beta(2, b)-coalescents 2009 Alex Iksanov Faculty of Cybernetics National T. Shevchenko University 01033KievUkraine Alex Marynych **[email protected] Faculty of Cybernetics National T. Shevchenko University 01033KievUkraine Martin Möhle Mathematical Institute University of Düsseldorf Universitätsstraße 140225Düsseldorf, Ger On the number of collisions in beta(2, b)-coalescents Bernoulli 153200910.3150/09-BEJ192asymptotics of momentsbeta-coalescentnumber of collisionsrandom regenerative compositionrecursion with random indices Expansions are provided for the moments of the number of collisions Xn in the β(2, b)-coalescent restricted to the set {1, . . . , n}. We verify that Xn/EXn converges almost surely to one and that Xn, properly normalized, weakly converges to the standard normal law. These results complement previously known facts concerning the number of collisions in β(a, b)-coalescents with a ∈ (0, 2) and b = 1, and a > 2 and b > 0. The case a = 2 is a kind of 'border situation' which seems not to be amenable to approaches used for a = 2. Introduction and main results Let E denote the set of all equivalence relations (partitions) on N. For n ∈ N, let ̺ n : E → E n denote the natural restriction to the set E n of all equivalence relations on {1, . . . , n}. For ξ ∈ E n let \|ξ\| denote the number of blocks (equivalence classes) of ξ. Pitman [15] and Sagitov [17] independently introduced coalescent processes with multiple collisions. These Markovian processes with state space E are characterized by a finite measure Λ on [0, 1] and hence are also called Λ-coalescent processes. For a Λ-coalescent {Π t : t ≥ 0}, it is known that g nk := lim tց0 P{\|̺ n Π t \| = k} t = n k − 1 [0,1] x n−k−1 (1 − x) k−1 Λ(dx)(1) for k, n ∈ N with k < n. Let denote the total rates. Recently, there appeared several papers [2,3,4,6,8,9,10] dealing with certain functionals of the restricted coalescent process {̺ n Π t : t ≥ 0} (for some particular choices of Λ). Functionals under consideration in these papers are (i) the number X n of collision events (jumps) that take place until there is just a single block, and (ii) the total branch length L n , that is, the sum of the length of all branches of the restricted coalescent tree. Such functionals are important for biological and statistical applications because they are closely related to the number of mutations on the restricted coalescent tree, if it is assumed that mutations occur independently of the underlying genealogical tree (neutrality) on each branch of the tree according to some homogeneous Poisson process with parameter r > 0 (coalescent with mutation). In particular, the weak asymptotic behavior of the number of collisions X n is known for β(a, b)-coalescents with a ∈ (0, 2) and b = 1, and a > 2 and b > 0. We briefly recall the corresponding weak convergence results because they provide insight into the role of the parameter a of the beta distribution Λ = β(a, b) in this model. If 0 < a < 1 and b = 1, then (see [10]) X n − n(α − 1) (α − 1)n 1/α d → X, where α := 2 − a and X is an α-stable random variable with characteristic function Ee itX = exp(\|t\| α (cos(πα/2) + i sin(πα/2) sgn(t))), t ∈ R. Gnedin and Yakubovich ( [8], Theorem 9) used analytic methods to generalize this result to Λ-coalescents satisfying Λ([0, x]) = Ax a + O(x a+ζ ) as x ↓ 0, where a ∈ (0, 1), A > 0 and ζ > max{(2 − a) 2 /(5 − 5a + a 2 ), 1 − a}. If a = b = 1 (Bolthausen-Sznitman coalescent), then (see [4,9,10]) (log n) 2 n X n − log(n log n) d → X, where X is a 1-stable random variable with characteristic function Ee itX = exp(it log \|t\| − π 2 \|t\|), t ∈ R. If 1 < a < 2 and b = 1, then (see [10]) X n Γ(2 − α)n α d → ∞ 0 e −Ut dt, where α := 2 − a and {U t : t ≥ 0} is a drift-free subordinator with Lévy measure ν(dt) = e −t/α /((1 − e −t/α ) α+1 ) dt, t > 0. If a > 2 and b > 0, then (see [6]) X n − µ −1 1 log n (µ 2 µ −3 1 log n) 1/2 d → X, where X is a random variable with the standard normal law, µ 1 := Ψ(a − 2 + b) − Ψ(b), µ 2 := Ψ ′ (b) − Ψ ′ (a − 2 + b) and Ψ(z) := (d/dz) log Γ(z) denotes the logarithmic derivative of the gamma function. There is also very precise information available concerning the asymptotics of the moments of X n for β(a, 1)-coalescents with a ∈ (0, 1]. For more details, we refer to [10] and [14]. The convergence results above indicate, in particular, that the two special parameter values a = 1 and a = 2 play a kind of threshold role when studying the limiting behavior of X n . This paper focuses on the asymptotics of X n for β(a, b)-coalescents with parameter a = 2 (and arbitrary b > 0). To the best of our knowledge, no convergence results have yet been provided for these particular beta coalescents. From the structure of the coalescent process, it follows that {X n : n ∈ N} satisfies the recursion X 1 := 0 and X n d = X n−In + 1, n ∈ {2, 3, . . .},(2) where I n is a random variable independent of X 2 , . . . , X n−1 with distribution P{I n = n − k} = g nk /g n , k ∈ {1, . . . , n − 1}. The random variable n − I n is the (random) state of the process {\|̺ n Π t \| : t ≥ 0} after its first jump. As already mentioned above, our aim is to investigate the asymptotic behavior of X n for β(2, b)-coalescents with b > 0. In this case, I n has distribution P{I n = k} = Γ(n − k + b − 1)Γ(n + 1) (k + 1)Γ(n − k)Γ(n + b)H(n, b) , k ∈ {1, . . . , n − 1},(3) where H(n, b) := b b + n − 1 + Ψ(b + n − 1) − Ψ(b) − 1, n ∈ N, b > 0. Note that Ψ(b + n − 1) = log n + O(1/n), n → ∞ (see (6.3.18) in [1]) and therefore H(n, b) = log n − Ψ(b) − 1 + O 1 n , n → ∞.(4) In the proofs, we will need the asymptotics of the total rates g n = H(n, b) B(2, b) ∼ log n B(2, b) , n → ∞,(5) where B(x, y) : = 1 0 u x−1 (1 − u) y−1 du, x, y > 0, denotes the beta function. Moreover, we will use the Lévy measure µ b on (0, ∞) defined via µ b (dt) := e −bt 1 − e −t dt, t > 0, b > 0.(6) Note that µ b has moments m (b) r := (0,∞) t r µ b (dt) = (0,1) (− log(1 − x)) r (1 − x) b−1 x dx (7) = Γ(r + 1)ζ(r + 1, b), r > 0, which follows from a Hurwitz identity (see, for example, (23.2.7) in [1]). Here, ζ(z, b) = ∞ i=0 (i + b) −z , Re(z) > 1, is the Hurwitz zeta function. Our first result presents the asymptotic expansions of the moments of X n . For convenience, we use the notation log k n := (log(n)) k , k, n ∈ N. Theorem 1.1 (Expansion of moments). As n → ∞, for k ∈ N, (7)), and c := −Ψ(b) − 1. In particular, the variance DX n has the asymptotic expansion EX k n = 1 (2m 1 ) k log 2k n + 2k((2k + 1)m 2 + 6cm 1 ) 3(2m 1 ) k+1 log 2k−1 n + O(log 2k−2 n), where m 1 := m (b) 1 = ζ(2, b) and m 2 := m (b) 2 = 2ζ(3, b) (seeDX n = m 2 3m 3 1 log 3 n + O(log 2 n) = 2ζ(3, b) 3ζ 3 (2, b) log 3 n + O(log 2 n). Remark 1.2. Let {S t : t ≥ 0} be a drift-free subordinator with Lévy measure (6). For n ∈ N, let Y n (Z n ) be the number of parts (with more than one point) of a regenerative composition arising from throwing n independent (random) points, which are independent of {S t : t ≥ 0} and all uniformly distributed on [0, 1], on the closed range of the multiplicative subordinator {1 − e −St : t ≥ 0}. According to (19) and (22) in [7], EY n and EY 2 n admit almost the same asymptotic expansions as EX n and EX 2 n , the only difference being that our c equals −Ψ(b) − 1 and their c equals −Ψ(b). According to (19) and Theorem 14 in [7], EZ n admits exactly the same asymptotic expansion as EX n . According to (24) in [7], DY n has the same asymptotic expansion as DX n . These observations strongly suggest that X n and Y n may have a similar asymptotic behavior. Remark 1.3. For t ≥ 0, let {f i (t) : i ∈ N} be the sequence (in some order) of the asymptotic frequencies of the random exchangeable partition Π t . Note that [0,1] x −1 Λ(dx) < ∞ for Λ = β(2, b), b > 0. Therefore, by Proposition 26 in [15], { S t := − log(1 − ∞ i=1 f i (t)) : t ≥ 0} is a version of {S t : t ≥ 0} . We will come back to this remark later in the proofs. Corollary 1.4 (Strong law of large numbers). As n → ∞, X n / log 2 n → 1/(2m 1 ) almost surely, with m 1 defined as in Theorem 1.1. Our last main result is a central limit theorem for {X n : n ∈ N}. Theorem 1.5 (Central limit theorem). As n → ∞, the sequence X n − (1/(2m 1 )) log 2 n (m 2 /(3m 3 1 )) log 3 n weakly converges to the standard normal law, where m 1 and m 2 are defined as in Theorem 1.1. Remark 1.6. The proof of Theorem 1.5 presented in Section 3 draws heavily from coalescent theory and results on random exchangeable partitions. We leave open the question of whether it is possible to deduce the asymptotic normality of X n from the recursion (2) alone, that is, without using pathwise results available in the coalescent setting. Proofs of Theorem 1.1 and Corollary 1.4 Proof of Theorem 1.1. For k ∈ N, set a (k) n := EX k n . By induction on k, we will prove the asymptotic expansion a (k) n = α k log 2k n + r k log 2k−1 n + O(log 2k−2 n), k ∈ N,(8) where α := (2m 1 ) −1 and r k := 2 3 kα k+1 ((2k + 1)m 2 + 6cm 1 ).(9) Recall that m 1 = m (7)) and c := −Ψ(b) − 1. For k = 1, write a n instead of a (1) n , for simplicity. In view of (2), we have (b) 1 = ζ(2, b), m 2 = m (b) 2 = 2ζ(3, b) (seea 1 = 0, a n = 1 + n−1 i=1 a n−i P{I n = i}, n ∈ {2, 3, . . .}.(10) Put b n := a n − α log 2 n, n ∈ N. From (10), it follows that b 1 = 0 and b n = 1 + α n−1 i=1 (log 2 (n − i) − log 2 n)P{I n = i} + n−1 i=1 b n−i P{I n = i}(11)=: c n + n−1 i=1 b n−i P{I n = i}, n ∈ {2, 3, . . .}. Using Lemma A.1 (with k = 1 and k = 2), we get c n = 1 + α n−1 i=1 (log 2 (1 − i/n) + 2 log n log(1 − i/n))P{I n = i} = 1 + α H(n, b) m 2 + O log 2 n n b∧1 + 2 log n −m 1 + O log n n b∧1 = 1 − log n H(n, b) + m 2 2m 1 H(n, b) + O log n n b∧1 and, by (4), c n = 1 − H(n, b) + Ψ(b) + 1 + O(1/n) H(n, b) + m 2 2m 1 H(n, b) + O log n n b∧1 = m 2 /(2m 1 ) − Ψ(b) − 1 H(n, b) + O log n n b∧1 =: C 1 H(n, b) + O log n n b∧1 . Substituting this relation into (11) yields b n = C 1 H(n, b) + O log n n b∧1 + n−1 i=1 b n−i P{I n = i}. Set d n := b n − (C 1 /m 1 ) log n, n ∈ N. Then, d 1 = 0 and d n = C 1 H(n, b) + C 1 m 1 n−1 i=1 log(1 − i/n)P{I n = i} + O log n n b∧1 + n−1 i=1 d n−i P{I n = i}, n ∈ {2, 3, . . .}. Another application of Lemma A.1 leads to d n = C 1 H(n, b) + C 1 m 1 H(n, b) −m 1 + O log n n b∧1 + O log n n b∧1 + n−1 i=1 d n−i P{I n = i} = O log n n b∧1 + n−1 i=1 d n−i P{I n = i}. By Lemma A.2, d n = O(1). Therefore, a n = α log 2 n + r 1 log n + O(1), and we have already proven (8) for k = 1. The induction step from {1, . . . , k} to k + 1 works as follows. Using (2) and dropping terms of lower orders (which is possible due to the assumption of induction), we get a n = c (k+1) n + n−1 j=1 b (k+1) n−j P{I n = j}, n ∈ {2, 3, . . .},(12) where c (k+1) n := α k+1 n−1 j=1 (log 2k+2 (n − j) − log 2k+2 n)P{I n = j} + (k + 1)α k log 2k n + (k + 1)r k log 2k−1 n + O(log 2k−2 n). Binomial expansion of log 2k+2 (n − j) = (log(1 − j/n) + log n) 2k+2 leads to c (k+1) n = (k + 1)α k log 2k n + (k + 1)r k log 2k−1 n + O(log 2k−2 n) + α k+1 n−1 j=1 P{I n = j} 2k+1 i=0 2k + 2 i log 2k+2−i (1 − j/n) log i n = (k + 1)α k log 2k n + (k + 1)r k log 2k−1 n + O(log 2k−2 n) + α k+1 2k+1 i=0 2k + 2 i log i n n−1 j=1 P{I n = j} log 2k+2−i (1 − j/n). Using Lemma A.1 gives c (k+1) n = (k + 1)α k log 2k n + (k + 1)r k log 2k−1 n + O(log 2k−2 n) + α k+1 H(n, b) 2k+1 i=0 2k + 2 i log i n (−1) i m (b) 2k+2−i + O log 2k+2−i n n b∧1 = (k + 1)α k log 2k n + (k + 1)r k log 2k−1 n + O(log 2k−2 n) + α k+1 H(n, b) −m 1 2k + 2 2k + 1 log 2k+1 n + m 2 2k + 2 2k log 2k n = (k + 1)α k log 2k n 1 − log n H(n, b) + (k + 1)r k + α k+1 (2k + 1)(k + 1)m 2 log n H(n, b) log 2k−1 n + O(log 2k−2 n) = (k + 1)(r k + (2k + 1)α k+1 m 2 − (Ψ(b) + 1)α k ) log 2k−1 n + O(log 2k−2 n) =: c k log 2k−1 n + O(log 2k−2 n). Plugging the last expression into (12) + C k n−1 i=1 (log 2k+1 (n − i) − log 2k+1 n)P{I n = i} + n−1 j=1 e (k+1) n−j P{I n = j} = c k log 2k−1 n + O(log 2k−2 n) + C k n−1 i=1 P{I n = i} 2k j=0 2k + 1 j log j n log 2k+1−j (1 − i/n) + n−1 j=1 e (k+1) n−j P{I n = j}. The sequence {r k : k ∈ N} satisfies the recursion r k+1 = k + 1 (2k + 1)m 1 (r k + (2k + 1)α k+1 m 2 − (Ψ(b) + 1)α k ) with initial condition r 1 = m 2 /(2m 1 ) − Ψ(b) − 1 m 1 = ζ(3, b)/ζ(2, b) − Ψ(b) − 1 ζ(2, b) . The unique solution of this recursion is given by (9). The proof of Theorem 1.1 is thus complete. Proof of Corollary 1.4. For n ∈ N and ε > 0, set A n (ε) := {\|X n − EX n \| ≥ εEX n }. By Chebyshev's inequality, P{A n (ε)} ≤ DX n /(εEX n ) 2 . From Theorem 1.1, it follows that DX n (EX n ) 2 = 4m 2 3m 1 1 log n + O 1 log 2 n . Therefore, replacing n by n k := ⌊exp(k 2 )⌋, it follows that ∞ k=1 P{A n k (ε)} < ∞ and hence X n k /EX n k → 1 almost surely as k → ∞, by the Borel-Cantelli lemma. Thus, we have already verified the result along the subsequence {n k : k ∈ N}. For each integer n ≥ n 1 , there exists a unique index k = k(n) ∈ N such that n k ≤ n < n k+1 . By its definition, the sequence {X n : n ∈ N} is almost surely non-decreasing. Therefore, the corollary follows from the standard sandwich argument X n k EX n k EX n k EX n k+1 ≤ X n EX n ≤ X n k+1 EX n k+1 EX n k+1 EX n k almost surely and from EX n k /EX n k+1 ∼ log 2 n k / log 2 n k+1 ∼ k 4 /(k + 1) 4 → 1. Proof of Theorem 1.5 We will use Theorem 2.1 of Neininger and Rüschendorf [13], which is written down below in a modified form suggested by Gnedin, Pitman and Yor [7], Theorem 10. In the following, for random variables X, we use the notation X 3 := (E(\|X\| 3 )) 1/3 . for some n 0 ∈ N, where (J n , V n ) is independent of {U n : n ≥ n 0 }, J n takes values in {0, 1, . . ., n} and P{J n = n} < 1 for each integer n ≥ n 0 . Suppose, further, that U n 3 < ∞ and that for some constants C > 0 and α > 0, the following three conditions hold: (i) lim sup n→∞ E log( Jn∨1 n ) < 0 and sup n≥2 log( Jn∨1 n ) 3 < ∞; (ii) for some λ ∈ [0, 2α) and some κ > 0, as n → ∞, V n − µ n + µ Jn 3 = O(log κ n), DU n = C log 2α n + O(log λ n), where µ n := EU n ; (iii) α > 1/3 + max(κ, λ/2). Then, as n → ∞, the sequence (U n − µ n )/( √ C log α n) weakly converges to the standard normal law. (4), E log J n n = n−1 i=1 log 1 − i n P{I n = i} ∼ − m (b) 1 log n . Therefore, lim n→∞ E log(J n /n) = 0. In particular, the first part of condition (i) in Proposition 3.1 is not satisfied. Hence, Proposition 3.1 is not applicable to the recursion (2). Fix any T > 0. The total number X n of collisions is the sum of the numbers of collisions occurring during the time intervals [0, T ) (denote this by X n (T )) and [T, ∞) (denote this by X n (T )). Since the coalescent is a Markov process, X n (T ) d = X ′ \|̺nΠT \| , where (J n , V n ) := (\|̺ n Π T \|, X n (T )) is independent of {X ′ n : n ∈ N}, an independent copy of {X n : n ∈ N}. Thus, we have proven that {X n : n ∈ N} satisfies another recursion of the form (13), namely X n d = X \|̺nΠT \| + X n (T ).(14) Proof of Theorem 1.5. Let us prove that the recursion (14) satisfies all of the conditions of Proposition 3.1. Since X n ≤ n − 1 almost surely, X n 3 < ∞. Recall that X n (T ) is the number of jumps of the process {̺ n Π t : t ∈ [0, T )}. If Λ has no atom at the origin, then any Λ-coalescent can be constructed from a Poisson point process (see page 1872 in [15]). From this construction, it follows that with probability one, X n (T ) is bounded from above by a random variable with Poisson distribution with parameter T g n . By (5), T g n ∼ (T /B(2, b)) log n. Therefore, From the paintbox construction [12] of a random exchangeable partition, it follows that X n (T ) 3 = O(log n), n → ∞.(15)\|̺ n Π T \| d = ζ n (T ) + η n (T ). Arguing in the same way as on page 592 in [7], we conclude that as n → ∞, η n (T )/n → e − ST almost surely, which easily implies that lim n→∞ − log η n (T ) ∨ 1 n = S T(16) almost surely and that for each k ∈ N, lim n→∞ E log η n (T ) ∨ 1 n k = E S k T .(17) Note that, in view of (7), the right-hand side is finite for each k ∈ N. Interpreting the intervals as "boxes" and the points as "balls", the θ n (T ) is just the number of occupied boxes in the classical multinomial occupancy scheme. From the results on page 152 in [5], it follows that lim n→∞ n −1 E(θ n (T )\| f i (T ) : i ∈ N) = 0 almost surely. This fact, together with Proposition 2 of the same reference (see also Theorem 8 in [11]), leads to lim n→∞ θ n (T )/n = 0 almost surely conditionally on { f i (T ) : i ∈ N} and, hence, unconditionally. The latter implies that lim n→∞ \|̺ n Π T \|/n = e − ST almost surely and, hence, lim n→∞ − log \|̺ n Π T \| n = S T (18) almost surely. Since − log \|̺ n Π T \| n ≤ − log η n (T ) ∨ 1 n almost surely, (16)-(18) together imply that for each k ∈ N, lim n→∞ E log \|̺ n Π T \| n k = E S k T ,(19) by a variant of Fatou's lemma sometimes called Pratt's lemma (see [16]). Condition (i) of Proposition 3.1 follows from (19). The estimate µ n − µ Jn 3 = O(log n) follows from Theorem 1. 1 and (19). In view of this observation, (15) and Theorem 1.1, (ii) holds with κ = 1, α = 3/2 and λ = 2. Therefore, (iii) also holds. Proof. We first prove that J n (b, k) := n−1 i=1 1 − i n b−1 1 i − log 1 − i n k − m (b) k = O log k n n b∧1(21) and that L n (b, k) := n−1 i=1 1 − i n b−1 1 i + 1 − log 1 − i n k − m (b) k = O log k n n b∧1 .(22)f b i n − 1 δ f b (x) dx = n−1 i=[nδ]+1 (i+1)/n i/n f b i n − f b (x) dx − ([nδ]+1)/n δ f b (x) dx ≤ n−1 i=[nδ]+1 (i+1)/n i/n f b i n − f b i + 1 n dx + ([nδ]+1)/n δ f b (x) dx = O 1 n . It is easily checked that f b is continuously differentiable on (0, δ) with sup 0<x<δ \|f ′ b (x)\| < ∞. Therefore, exploiting the mean value theorem for differentiable functions, we have 1 n [nδ] i=1 f b i n − δ 0 f b (x) dx = O 1 n . Combining these two pieces and using the equality m (b) k = 1 0 f b (x) dx, we get J n (b, k) = O(1/n), which is more than we need. Assuming that b ∈ (0, 1], an application of the previous result to the function f b+1 , which satisfies f b+1 (x) = (1 − x) b−1 (− log(1 − x)) k x − (1 − x) b−1 (− log(1 − x)) k for x ∈ (0, 1), yields n−1 i=1 (1 − i/n) b−1 (− log(1 − i/n)) k i − (1 − i/n) b−1 (− log(1 − i/n)) k n (23) − 1 0 f b+1 (x) dx = O 1 n . Note that 1 0 f b+1 (x) dx = m (b) k − k!/b k+1 . For all n ∈ N with b log n ≥ 1, we now use the inequalities 1 n n−1 i=1 i n b−1 − log i n k ≥ 1 1/n x b−1 (− log x) k dx = k! b k+1 1 − n −b k i=0 (b log n) i i! ≥ k! b k+1 − k! log k n bn b k i=0 1 i! ≥ k! b k+1 − k!e log k n bn b to conclude that, as n → ∞, 1 n n−1 i=1 1 − i n b−1 − log 1 − i n k − k! b k+1 = O log k n n b . Combining this estimate with (23) yields (21). Let us now prove (22). If k ∈ N \ {1}, then 0 ≤ M n (b, k) := n−1 i=1 (1 − i/n) b−1 (− log(1 − i/n)) k i − n−1 i=1 (1 − i/n) b−1 (− log(1 − i/n)) k i + 1 ≤ n−1 i=1 (1 − i/n) b−1 (− log(1 − i/n)) k i 2 ∼ 1 n 1 0 (1 − x) b−1 (− log(1 − x)) k x 2 dx and the last integral is finite. Therefore, M n (b, k) = O(1/n), which, together with (21), proves (22) under the current assumptions. If k = 1, then 0 ≤ M n (b, k) ≤ n (1−b)∨0 n−1 i=1 − log(1 − i/n) i 2 = n (1−b)∨0 n−1 i=1 1 i 2 ∞ j=1 (i/n) j j ≤ n (1−b)∨0 n−1 i=1 1 i 2 ∞ j=1 i n j = n (1−b)∨0 n−1 i=1 1 i 2 i/n 1 − i/n = n (1−b)∨0 n−1 i=1 1 i(n − i) = n (1−b)∨0 1 n n−1 i=1 1 i + 1 n − i ∼ 2 log n n b∧1 . This relation, together with (21), proves (22). For b = 1, the left-hand side of (22) coincides with that of (20). Thus, we only have to check (20) for b = 1. To this end, keeping in mind (21) and (22), it suffices to show that n−1 i=1 Γ(n − i + b − 1)Γ(n + 1) Γ(n − i)Γ(n + b) − 1 − i n b−1 1 i + 1 − log 1 − i n k(24) = O log k n n b∧1 . First, we will prove that for any b > 0, there exists a constant M > 0 such that for all n ∈ N and all j ∈ {1, . . . , n − 1}, Γ(n − j + b − 1)Γ(n + 1) Γ(n − j)Γ(n + b) − 1 − j n b−1 ≤ M n 1 − j n b−2(25) or, equivalently, Γ(j + b − 1)Γ(n + 1) Γ(j)Γ(n + b) − j n b−1 ≤ M n j n b−2 .(26) The subsequent argument relies on the following inequality (see (6.1.47) in [1]). For b > 1, the function x → x −1 (1 − x) b−2 log k (1 − x) is integrable on [0, 1], which implies that the latter sum is bounded and the right-hand side in (24) is O(1/n). If b ∈ (0, 1), then noting that the function x → x −1 (− log(1 − x)) k is non-decreasing on (0, 1), we conclude that for n ∈ {2, 3, . . .}, = Γ(j + b − 1) Γ(j) − j b−1 Γ(n + 1) Γ(n + b) + Γ(n + 1) Γ(n + b) j b−1 − j n b−1 ≤ Γ(n + 1) Γ(n + b) Γ(j + b − 1) Γ(j) − j b−1 + j b−1 Γ(n + 1) Γ(n + b) − n 1−b ≤ Γ(n + 1) Γ(n + b) M b−1,0 j b−2 + j b−1 M 1,b n −b ≤ Γ(n + 1) Γ(n + b) − n 1−b M b−1,0 j b−2 + n 1−b M b−1,0 j b−2 + j b−1 M 1,b n −b ≤ M 1,b M b−1,Q n (b, k) = M n b n−1 i=1 (n − i) b−2 1 (i + 1)/n − log 1 − i n k ≤ M n b n−1 i=1 (n − i) b−2 1 i/n − log 1 − i n k ≤ 2M log k n n b n−1 i=1 (n − i) b−2 = O log k n n b . Thus, (24) is established and the proof is complete. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2009, Vol. 15, No. 3, 829-845. This reprint differs from the original in pagination and typographic detail. α k log 2k n + (k + 1)r k log 2k−1 n + O(log 2k−2 n) + j P{I n = j}, n ∈ {2, 3, . . .}. k+1 log 2k+2 n, n ∈ N. We then have b − C k log 2k+1 n, n ∈ N, where C k := c k /((2k + 1)m 1 ). The sequence thus defined is given by the recursione (k+1) n = c k log 2k−1 n + O(log 2k−2 n) P{I n = j}, by the choice of C k . For sufficiently large n, we can choose M k > 0 such that the O(log 2k−2 n) term is dominated by M k (kα k−1 log 2k−2 n + kr k−1 log 2k−3 n + O(log 2k−4 n)). 2k n). Setting r k+1 := C k = c k /((2k + 1)m 1 ), we obtain a (k+1) n = α k+1 log 2k+2 n + r k+1 log 2k+1 n + O(log 2k n). Proposition 3. 1 . 1Assume that a random sequence {U n : n ∈ N} of scalar random variables satisfies the recursion U n d = U Jn + V n , n ∈ {n 0 , n 0 + 1, . . .}, Remark 3 . 2 . 32The recursion (2) is of the form (13) with random indices J n := n − I n , where I n has distribution (3). By Lemma A.1 and Let Q(T ) := { f i (T ) : i ∈ N} be the decreasing rearrangement of the asymptotic frequencies of the random exchangeable partition Π T . According to Remark 1.3, 1 −∞ i=1 f i (T ) = e − ST . The elements of the set Q(T ) ∪ {1 − ∞ i=1 f i (T )}are the lengths of the intervals (from left to right) comprising the partition of [0, 1]. Let U 1 , . . . , U n be independent random variables (points), uniformly distributed on [0, 1] and independent of the lengths of the intervals. Let W n,i (T ) be the number of points falling in the interval of length f i (T ). Set η n (T ) := \|{i ∈ {1, . . . , n} : U i > 1 − e − ST }\|, ζ n (T ) := \|{i ≥ 1 : W n,i (T ) > 0}\|, θ n (T ) := ζ n (T ) + 1 {ηn(T )>0} . Fix k ∈ N . NFor b > 1, introduce the continuous non-negative function f b : [0, 1] → R via f b (x) := x −1 (1 − x) b−1 (− log(1 − x)) k for x ∈ (0, 1), f b (0) := 1 {k=1} and f b (1) := 0. Pick some δ ∈ (0, 1) such that f b is non-increasing on [δ, 1] For c, d > − 1 , 1there exists M c,d > 0 such that for all n ∈ N,Γ(n + c) Γ(n + d) − n c−d ≤ M c,d n c−d−1 . ( 26 ) 26now follows from the chain of inequalities Γ(j + b − 1)Γ(n + 1) Γ(j)Γ(n + b) − j n b−1 M := M 1,b M b−1,0 + M b−1,0 + M 1,b .Plugging (25) into the left-hand side of (Q n (b, k). AcknowledgementsThe first author was supported by the German Research Foundation DFG, Project 436UKR 113/93/0-1. The authors would like to thank the referees for their careful reading and for their suggestions leading to a significant improvement of the style of the manuscript.AppendixThe proof of Theorem 1.1 relies on the two following technical results.Lemma A.1. For all k ∈ N and b > 0, as n → ∞,where H(n, b) is the function defined after (3) and mis the kth moment (see(7)) of the Lévy measure(6).Lemma A.2. Fix k ∈ N and b > 0, and suppose that {a n : n ∈ N} is some sequence satisfying a n = O(n −b log k n). If the sequence {v n : n ∈ N} is defined recursively byProof. Since EI n ∼ n/(b log n), there exists an M > 0 such that for all n ∈ {2, 3, . . .It suffices to prove the following. 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[ "Dealing with elementary paths in the Kidney Exchange Problem", "Dealing with elementary paths in the Kidney Exchange Problem" ]	[ "Lucie Pansart \nUniv.Grenoble Alpes\nCNRS\n38000GrenobleG-SCOPFrance\n", "Hadrien Cambazard \nUniv.Grenoble Alpes\nCNRS\n38000GrenobleG-SCOPFrance\n", "Nicolas Catusse \nUniv.Grenoble Alpes\nCNRS\n38000GrenobleG-SCOPFrance\n" ]	[ "Univ.Grenoble Alpes\nCNRS\n38000GrenobleG-SCOPFrance", "Univ.Grenoble Alpes\nCNRS\n38000GrenobleG-SCOPFrance", "Univ.Grenoble Alpes\nCNRS\n38000GrenobleG-SCOPFrance" ]	[]	We study an elementary path problem which appears in the pricing step of a column generation scheme solving the kidney exchange problem. The latter aims at finding exchanges of donations in a pool of patients and donors of kidney transplantations. Informally, the problem is to determine a set of cycles and chains of limited length maximizing a medical benefit in a directed graph. The cycle formulation, a large-scale model of the problem restricted to cycles of donation, is efficiently solved via branch-and-price. When including chains of donation however, the pricing subproblem becomes NP-hard. This article proposes a new complete column generation scheme that takes into account these chains initiated by altruistic donors. The development of non-exact dynamic approaches for the pricing problem, the NG-route relaxation and the color coding heuristic, leads to an efficient column generation process.	null	[ "https://arxiv.org/pdf/2201.08446v1.pdf" ]	246,210,180	2201.08446	3620c57c1ac33dcad035a9027130a8fcba15dc10	Dealing with elementary paths in the Kidney Exchange Problem Lucie Pansart Univ.Grenoble Alpes CNRS 38000GrenobleG-SCOPFrance Hadrien Cambazard Univ.Grenoble Alpes CNRS 38000GrenobleG-SCOPFrance Nicolas Catusse Univ.Grenoble Alpes CNRS 38000GrenobleG-SCOPFrance Dealing with elementary paths in the Kidney Exchange Problem OR in health serviceskidney exchange problemelementary pathscolumn generation We study an elementary path problem which appears in the pricing step of a column generation scheme solving the kidney exchange problem. The latter aims at finding exchanges of donations in a pool of patients and donors of kidney transplantations. Informally, the problem is to determine a set of cycles and chains of limited length maximizing a medical benefit in a directed graph. The cycle formulation, a large-scale model of the problem restricted to cycles of donation, is efficiently solved via branch-and-price. When including chains of donation however, the pricing subproblem becomes NP-hard. This article proposes a new complete column generation scheme that takes into account these chains initiated by altruistic donors. The development of non-exact dynamic approaches for the pricing problem, the NG-route relaxation and the color coding heuristic, leads to an efficient column generation process. Introduction The kidney exchange problem models the barter market of kidney exchange programs, which were created to match patients waiting for a kidney transplant to living donors with the objective to find the best possible transplants to perform. Each participating patient is paired with a willing, but incompatible, donor. This donor accepts to give one kidney only if its associated patient receives one, creating cycles of donation (see Figure 1a). It is imperative that a patient is transplanted if its associated donor gives a kidney, so no donors should give before another and all the surgeries of a cycle must be done simultaneously. As a cycle requires twice its size operating teams and rooms, kidney exchange programs impose a limit K on the size of a cycle. In a lot of countries, programs also include altruistic donors. These donors are not expecting any transplant to happen in return, creating chains of donation (or domino chains, see Figure 1b). In this case, the simultaneity may not be required, as no donor would give his kidney before its patient receives another one. Yet, the failure of a transplant causes the failure of every remaining transplant in the chain. Thus, it is preferable to have several shorter chains than a big one and a lot of programs impose a limit L (usually L > K) on the length of a chain. An exchange in a kidney exchange program is therefore either a cycle or a chain of donation which implies at most K (resp. L) transplants. The kidney exchange problem (KEP) aims at finding the best set of exchanges in order to maximize the medical benefit of the performed transplants. The idea of kidney exchange was first mentioned by Rapaport in 1986 [32] and quickly set up in South Korea in 1991 [23,27]. This idea was promising for this country where the public opinion is hostile on deceased transplantation. The Switzerland was the first European country to perform a kidney exchange in 1999, but the first national kidney exchange program (KPD) in Europe was created by the Netherlands in 2004 [9]. Since then, a dozen states in Europe have created their own KPD [5]. In the rest of the world, and to the best of our knowledge, such programs exist only in Canada, Australia and the USA. We refer the reader to the survey of Ellison [12] for more details on the development of KPDs. It is worth to note that kidney exchange programs involve more and more participants as the usage is spreading in hospitals, but also due to transnational exchanges. The bigger the pool, the higher the chance to match patients, but the harder the kidney exchange problem. In 2019, the largest program in Europe involves 250 British patients [6], but considering that more than half a million Europeans [22] and even more USA citizens [35,36] are treated for end stage kidney disease, new efficient techniques must be developed to handle many more candidates for kidney exchange programs. Different approaches exist to solve the kidney exchange problem. When it contains only cycles of length 2, the KEP can be solved polynomially as a matching problem via Edmonds' algorithm, but as soon as K > 2, the problem is proved to be NP-complete [1,7]. Consequently, the KEP is often tackled with integer programs and the major ones are surveyed by Mak-Hau [24]. We focus on the cycle formulation. The original version of this formulation does not take into account altruistic donors and requires to compute every possible cycles. Abraham et al. developed a column generation approach to solve this integer program [1], which is still to this day the best way to solve the KEP without chains of donation. The cycle formulation can be equivalently applied when including these chains, and we will refer to it as the exchange formulation. In Chen et al., all exchanges are computed beforehand [8], but this is not a viable method when the patients pool grows. On the basis of Abraham et al. work, branch-and-price algorithms were developed [16,17,20,30] claiming to accommodate well altruistic donors via chains of donation. However some of these algorithms (in [16,17,30]) were proven wrong by Plaut et al. [31] and Klimentova et al. did not test their algorithm with altruistic donors [20]. Actually, Plaut et al. proved in 2016 that the pricing algorithm becomes NPcomplete in this case. In order to handle large-scale instances of the KEP, our objective is to address this NP-hard problem and establish efficient pricing strategies. We study its optimization version, that we refer to as the elementary minimum path problem with length constraint (EMPPLC). Based on a review of similar problems, we propose to generate lower and upper bounds using dynamic programming. This approach allows an efficient column generation process and thus to compute the linear relaxation of the exchange formulation. This upper bound on the optimal value can be used to assess the quality of the feasible solution that our algorithm constructs by solving the exchange formulation on generated columns. Our method turns out to be very efficient as the gap between lower and upper bounds is always smaller than 0.5%, even on instances with more than 800 vertices. The number of patients of these instances is larger than in the current literature or in the field, but is likely to be prevalent in a near future. In Section 2 we model the kidney exchange problem with the exchange formulation. Section 3 defines the elementary minimum path problem with length constraint and reviews the different approaches used to solve it, in particular the key idea of our contributions. We detail in Sections 4 and 5 the improvements we developed on the NG-route relaxation and the color coding. Their performance are compared in Section 6. Finally Section 7 shows how these algorithms are used to solve the kidney exchange problem. Note that the path problem studied in this article may be found in other applications and the contributions presented here used in these other cases as well. Using an exponential formulation for the KEP We model a kidney exchange program as a directed graph by creating one vertex for each participant and one arc for each possible transplant. Formally, the set P contains one vertex for each patient-donor pairs and the set N one vertex for each altruistic donor. To construct the compatibility graph D = (V = P ∪ N, A), we add an arc a = (uv) between u ∈ V and v ∈ P if the kidney of donor u can be transplanted to patient v. A weight function w : A → R + represents the medical benefit of each possible transplant. Note that determining the weight function is an upstream work and that w is an input in our case. This graph is generally quite sparse as it is rare for a patient and a donor to be compatible. Figure 2a shows an example of compatibility graph and differentiates altruistic donors (orange diamonds) from pairs (red circles). An exchange is a subgraph of D which represents either a cycle of donation between pairs or a domino chain initiated by altruistic donors. In the compatibility graph, exchanges are elementary cycles of length at most K, called valid cycles and elementary paths starting by a vertex of N and having at most L vertices, valid paths. A valid cycle could have several symmetrical representations but they are eliminated by restricting the first vertex of the cycle vector to have the lowest identifier. Thus, a valid cycle c is represented by a unique vector of vertices (v 1 , ..., v \|c\| ) such that v 1 < v j ∀j ∈ {2, ..., \|c\|}. C is the set of all valid cycles, P the set of all valid paths and E = C ∪ P the set of all possible exchanges. We refer to the set of vertices (resp. edges) of an exchange e as V (e) (resp. A(e)). The weight of an exchange e ∈ E is w(e) := a∈A(e) w a . In Figure 2a Formally, the kidney exchange problem is a maximum-weight set packing problem, where the considered sets are the exchanges. As each agent can give or receive at most one kidney, the exchanges must indeed be pairwise disjoint. Figure 2 shows the optimal solution of the KEP in our example. In the exchange formulation (EF), each exchange is associated with one binary variable indicating if it is chosen or not in the solution and a unique set of constraints (2) is required to model the disjonction of exchanges: ∀e ∈ E, x e = 1 if exchange e is chosen 0 otherwise z * = max e∈E w e x e (1) e∈E: i∈V (e) x e ≤ 1 ∀i ∈ V (2) x e ∈ {0, 1} ∀e ∈ E(3) The number of variables of EF grows exponentially with K and L, so even computing its linear relaxation z * LP may need an excessive amount of time or be impossible due to memory issues. However, the quality of its linear relaxation makes EF very promising. It is actually, up to now, the tightest formulation for the KEP, including compact and other large-scale formulations [11,24]. Moreover, these other integer programs are more complex and also suffer from scalability issues. To overcome the exponential growth of EF, its linear relaxation EF can be solved by a column generation approach. It constructs iteratively a (small) set of variables guaranteeing that an optimal solution uses only these variables with the following two steps: 1. Solve the restricted master problem (RMP): EFL restricted on a subset E ⊆ E. 2. Solve the pricing problem: find an "interesting" exchange to add in E (go to 1.) or prove none exists (end). When the restricted master problem is solved, it computes for each vertex v the dual values α v associated with constraints (2). The pricing problem of the exchange formulation aims at finding a new exchange with a positive reduced cost or proving that none exists. The reduced cost of an exchange e is given by rc e = w e − v∈V (e) α v . For a non-basis variable, it estimates the improvement of the objective function if a solution includes e, i.e., x e > 0. It is important to note that these reduced costs are in R and thus can be positive or negative. As there are two kinds of exchanges, we can decompose this pricing problem into two subproblems: • The cycle pricing problem: find a cycle of length at most K of positive reduced cost, or prove none exists. • The path pricing problem: find an elementary path of length at most L starting by an altruistic donor and with a positive reduced cost, or prove none exists. The cycle pricing problem can be solved in polynomial time with a Bellman-Ford algorithm. On the contrary, the path pricing problem is NP-complete and the proof, based on a reduction from the directed Hamiltonian path problem, was recently given by Plaut et al. [31]. We handle this decision problem with algorithms solving the associated optimization problem: the elementary minimum path problem with length constraint (EMPPLC). This article is dedicated to this problem and how to solve it. Finding elementary paths The elementary minimum path problem with length constraint belongs to the well-known family of paths problems. It is a special case of the elementary shortest path problem with resource constraints. However its specificity-the length constraint-can be exploited to strengthen existing algorithms. Description of the problem Let G = (V, A) be a digraph such that the set of vertices V includes a source s. Each arc (ij) ∈ A has a cost c ij . Let L be the limit on the length (number of arcs) of a path. An (l, i)-path p = (s, i 1 , ..., i l = i) is an elementary path of length l starting from the source s and ending in i. We denote by A(p) (resp. V (p)) the set of arcs (resp. vertices) of p and by c p = (ij)∈A(p) c ij its cost. The objective of the elementary minimum path problem with length constraint (EMPPLC) is to find p * an elementary (l, i)-path of minimum cost c * such that l ≤ L. EMPPLC in the KEP To cast the pricing problem of the exchange formulation as a minimization problem, we consider a new weight function c associating with each arc the opposite of its estimated reduced cost: ∀(uv) ∈ A, c uv = −w uv + α v . We also construct a new directed graph containing an artificial source s linked to each α v − w e = −rc e . As c e = −rc e , the pricing problem aims at finding paths of negative weight or prove none exists. Thus, solving EMPPLC on D provides an answer for both cases. altruistic donor D = (V ∪ {s}, A ) where A = A ∪ {(su) ∀u ∈ N }. Standard approaches Interest for elementary shortest path problems mainly arose from vehicle routing applications solved by column generation. The elementarity constraint is often relaxed to get a simpler problem, which can be relevant in many applications such as vehicle routing problems where a vehicle can visit the same place twice. In the standard case, one just wants to solve the shortest path problem (SPP), which can be done with a Bellman-Ford algorithm. In the presence of resource constraints, such as time windows or vehicle capacities, the problem becomes harder. In 1988, Desrochers [10] proposed an extension of the Bellman-Ford algorithm for the shortest path problem with resource constraints (SPPRC). However, Feillet et al. [13] argued that relaxing the elementarity constraint can lead to bounds of poor quality, thus proposed to extend Desrochers' algorithm in order to solve the NP-hard elementary shortest path problem with resource constraints (ESPPRC). In the ESPPRC, paths have limited resources (instead of having a maximum length as in EMPPLC). Let R be the number of resource types and g r ij ≥ 0 the consumption of resource r along the arc (ij). Each vertex i ∈ V constrains the path to reach it with a resource consumption belonging to [a r i , b r i ] for each resource r. The objective is to find a path of minimum cost c such that every resource constraint is satisfied. By considering a single resource with a unit consumption and by setting the bound on this resource consumption to the length limit, ESPPRC describes EMPPLC. Formally let R = 1 and, ∀i ∈ V : a i = 0 and b i = L. In addition we set g ij = 1, ∀(ij) ∈ A and observe that EMPPLC is a special case of ESPPRC. When dealing with the EMPPLC, like for any problem, the objective is generally to find the optimal solution. Exact algorithms are designed for this, but as EMPPLC is NP-hard, they might take a long time to get it. On the other hand, in a column generation framework it is not important to find optimal solutions, but to find a solution with the same sign as the optimal solution. Indeed, assume the pricing problem is to find a path with a negative cost. Then any feasible solution of EMPPLC is a new column to add to the restricted master problem. Similarly, if a lower bound has a positive (or zero) cost, the optimality proof of the linear relaxation is done. Lower bounds and feasible solutions can be computed with relaxations and heuristics respectively. Numerous approaches solving paths problems, including Desrochers' and Feillet', are based on dynamic programing and labeling algorithms following Held and Karp results [18]. This dynamic program solving the EMPPLC has a space complexity of O(\|V \|2 \|V \| ), a time complexity of O(\|A\|2 \|V \| ) and does not scale up to large instances. A common idea to overcome scaling issues is to relax the problem in order to deal with a smaller search space. Relaxing a minimization problem aims at quickly providing lower bounds. If we relax the constraint of elementarity, the search space is strongly reduced since the visited vertices are not remembered anymore. We can also relax the resource constraints to solve the shorstest path problem. More complex dynamic programs relax only partially these constraints. This is the case of the NG-route relaxation proposed by Baldacci et al. [4] which constructs partially elementary paths. We present their algorithm and how we reinforce it for our problem in Section 5. Note that in the kidney exchange context, the elementarity constraint cannot be relaxed to get a feasible solution, but it can be relaxed in the RMP, "temporarily", in order to speed up the column generation. Actually, compatibility graphs are generally sparse, unlike graphs of vehicle routing problems which are usually complete, and relaxed solutions may be elementary anyway. Another, and opposite, idea to reduce the search space in dynamic programming is to restrict the problem. More constrained problems will provide feasible solutions and thus upper bounds on the EMPPLC optimal solution. We introduce the color coding algorithm proposed by Alon et al. [3] and present our contributions in Section 4. i1 ir−1 ir it it+1 il (a) Initial path i1 ir−1 ir it it+1 il j1 jm (b) Insertion i1 ir−1 ir it it+1 il (c) Suppression i1 ir−1 ir it it+1 il j1 jm (d) Exchange These two dynamic programs, the color coding and the NG-route, are the core of our study of the EMPPLC. They provide a different type of solution and to assess their interest, we compare them to two alternative algorithms that serve as baseline methods. Firstly, we adapt an integer linear programming formulation of the traveling salesman problem called the time-stage formulation (TSF), which was proposed by Fox, Gavish and Graves [14] for the traveling salesman problem. In this model, a decision variable y l ij states if arc (ij) is taken at the l th position of the path. The integer program TSF is used to compute the optimal solutions of the different EMPPLC instances. It is also solved within a time limit of 1 second, giving the best lower and upper bounds TSF-lb and TSF-ub Secondly, we develop a quick heuristic based on local search: the algorithm moves from the current solution p to a better solution p in its neighborhood. This solution p is constructed by inserting, removing or exchanging up to 3 vertices in p (see figure 3) and kept if c p < c p . The solution with the minimum cost is returned after one second (solution LM), as well as the first path of negative cost (solution LF). If no path of negative cost is found in the time limit, then LF is equal to LM. Exploit the length constraint As we look for a path of length at most L, we can use this information to reinforce shortest path algorithms. We compute with Floyd-Warshall algorithm the distance function d : V × V → N ∪ {+∞} where d(i, j) is the shortest path between i and j, with respect to the number of arcs. We define the extended neighborhood γ(i) of vertex i as the set of vertices that may appear in any path starting at the source of length at most L includ- ing i: γ(i) := {j ∈ V : d(s, i) + d(i, j) ≤ L or d(s, j) + d(j, i) ≤ L}. Note that if i ∈ γ(j) then j ∈ γ(i). We define the extended predecessors γ − (i) of vertex i as the sets of vertices that can reach i in a path of length at most L: γ − (i) := {j ∈ V : d(s, j) + d(j, i) ≤ L}. The distance function is used first to perform a preprocessing on the graph G by removing every arc (and vertex) that is too far from the source to be contained in a path of length L. The sets of extended neighbors and predecessors also take part in the improvements of the next sections. The color coding restriction The randomized algorithm referred to as the color coding was introduced by Alon et al. [3] to solve the subgraph isomorphism problem. It can be applied in particular to solve the elementary minimum path problem with length constraint. Its principle is to randomly assign a color to each vertex, then to solve a dynamic program finding the path of minimum weight whose vertices all have different colors. Such a path is said to be colorful and searching for it is computationally more efficient than the search for an elementary path as there are C distinct vertex identifiers instead of \|V \|. However, this colorful path is not optimal in general. Indeed, for an optimal solution to be found with color coding, the random coloring must, by chance, assign different colors to its vertices. Thus, the two steps of the algorithm are repeated so that each solving of the dynamic program might return a different path, increasing the probability that one of them is optimal. In all its applications and theoretical studies, the coloring step is performed with a discrete uniform distribution. Only Kneis et al. [21] proposed another probability distribution to color the graph, but for a derandomization purpose. We introduce in this section new strategies for this step of the algorithm and focus on how this strategies are helpful to efficiently solve the EMPPLC for our pricing step. We refer the reader to Pansart et al. [25] for the theoretical study of these new strategies in the general context of subgraph isomorphism problem, but also for an extensive literature review on color coding and proofs of properties mentioned in the following. Description of the algorithm To solve the elementary minimum path problem with length constraint in a weighted graph G = (V, A), the color coding uses two steps: 1. Coloring: randomly assign a color c ∈ {1, ..., C} to each vertex v ∈ V \ {s} 2. Dynamic programming: finds the colorful path of minimum weight, of length at most L, whose vertices all have distinct colors. f * (C, i) is the minimal cost of a path from s to i, using \|C\| − 1 arcs and visiting vertices of each color of C: f * (C, i) = min j∈N − (i) cj ∈C\{ci} {f * (C \ {c i }, j) + c ji }(4) These two steps are repeated several times in order to guarantee a probability high enough that one of these trials finds an optimal path. In the standard version of the color coding, the coloring step applies a discrete uniform distribution to assign the colors and C = L. In this case, the number of trials required to ensure that a specific path (e.g., an optimal path) of length L is colorful with probability ρ is t = ln(1−ρ) ln(1− C! C C ) . Using more colors can however increase the chance for a path to be colorful, thus reduce the number of trials t, but it also increases the complexity of the dynamic programming step. Hüffner et al. [19] analyzed the best trade-off between these two parameters: the number of trials and the runtime of one trial. Another lever on the color coding efficiency is the probability distribution used to color the graph. Surprisingly though, the coloring strategy always follows the discrete uniform distribution, except in Kneis et al. [21] for a derandomization purpose. With this strategy, assigning a color to a vertex does not depend on other vertices colors. Yet, using a different probability distribution can significantly increase the probability that an optimal path is colorful, and, contrarily to increasing C, will not affect the computational performance of the dynamic program. The new method we introduce in the next section is based on this idea. New randomized strategies We propose to use a coloring technique that aims at spreading the colors in extended neighborhoods. To do so, it tries to make extended neighborhoods colorful by relying on three main ideas. First, a preprocessing step applies a local search to create an ordering of vertices x 1 , ...x n (x i ∈ V ) in which extended neighbors are gathered. Secondly, the vertices are colored by intervals of size C such that each interval is colorful. The color assigned to a vertex therefore depends on the color of other vertices in this interval. Finally, the ordering is shifted so that the intervals are made up of different extended neighbors each iteration. The preprocessing step constructs a coloring sequence with the objective to gather in this sequence vertices that might belong to a solution. Such vertices are identified thanks to the extended neighborhoods. Let x i be the position of vertex i in the sequence and δ be the sum of differences between the positions of two extended neighbors: δ = i∈V j∈γ(i) \|x i − x j \|. Our preprocessing constructs a coloring sequence such that δ is minimized, i.e., the average distance between the positions of two extended neighbors is minimized. This problem is known as the (minimum) linear arrangement problem [2] that was proven to be NPcomplete [15]. We handle it with a local search and this method is referred to as la ordering. We denote by (x 1 , ..., x n ) the coloring sequence of vertices of V \{s} provided by this preprocessing step. Our coloring strategy spread splits the coloring sequence into intervals of size C and colors vertices such that these intervals are colorful. Concretely, the colors vector of vertices in an interval I is a permutation of {1, ..., C} drawn with a uniform distribution over all the permutations. The color of a vertex thus depends on the colors taken by other vertices of the same interval, but on no other vertex. Thus, two vertices in the same interval cannot have the same color and two vertices in different intervals are assigned the same color with probability 1 C (proof in [25]). Once the dynamic program is solved, the coloring sequence is offset by 1 to the left. So after the first trial, the coloring sequence is (x 2 , ..., x n , x 1 ). This shifted-spread strategy means that, after C trials of the color coding, every subsequence of size C has been colorful once. Thus, if an optimal path has been gathered enough in the coloring sequence by the preprocessing, it will be colorful and found in C trials only. In general we cannot know when this happens, but it can be guaranteed by a parameter of the coloring sequence: the maximum distance between the positions of two extended neighbors ∆ = max i∈V j∈γ(i) \|x i − x j \|. When ∆ ≤ C, every optimal path is contained in a subsequence of size at most C and our algorithm guarantees to find them with a probability of 1 with at most C trials of the color coding, so only C calls to the costly dynamic program. The worst case for this coloring strategy is to have the positions of all the vertices of every optimal path in different intervals. In this case, which should not happen often in practice thanks to the la ordering, the probability that a path is colorful is the same than when using the discrete uniform distribution (proof in [25]). Thus, the method we propose always improve the chance to find an optimal solution over the original color coding. To sum up, our method lass (la ordering + shifted-spread) contains a preprocessing phase and a color coding phase with three steps: 1. Apply a spread coloring strategy 2. Solve the dynamic program Shift the coloring sequence When ∆ ≤ C these three steps are repeated only C times and the solution is optimal, otherwise we limit the running time and get the best solution after one second (solution CM). We also return the first negative solution (solution CF), if applicable. This time limit is small as the elementary minimum path problem with length constraint must be solved a lot of times in a column generation scheme. On the contrary, the la ordering is a preprocessing applied only once by instance of the master problem, so we let the local search runs for 300 seconds. When applied in the pricing step to solve the kidney exchange problem, this method often returns the optimal solution, even when ∆ > C. Indeed, graphs are sparse so the extended neighborhoods are small enough to make the preprocessing powerful. This performance allows a quick and effective solving of the pricing problem. Yet, if the color coding does not find a path of negative cost, it does not prove that the column generation has finished since this solution is only an upper bound. Thus, another method is still required to make this proof: this is the role of the NG-route relaxation. The ng-route relaxation In 2011, Baldacci et al. [4] proposed a new relaxation of the elementary shortest path problem with resource constraints called NG-route. The memory of a path is relaxed so that the search space of the dynamic program is reduced. In practice, a path constructed in the NG-route relaxation, called an NG-path, can forget that it went through some vertices and may visit them several times and be non elementary. However, if it turns out that the NG-path is elementary, then it is an optimal solution of the ESPPRC. We describe the adaptation of this algorithm for the EMPPLC special case. Note that solving the EMPPLC with the NG-route relaxation in a column generation scheme can lead to the introduction of non elementary columns in the RMP. In this case, the column generation does not solve the linear relaxation of the master problem, but a relaxation of this linear problem. In the KEP context, the solution obtained with non-elementary paths is thus an upper bound on z * LP , but it can be used similarly, for example to assess the quality of a feasible solution. Of course if only elementary paths were added by the NG-route relaxation, this upper bound is actually z * LP . As compatibility graphs are rather sparse, the elementarity will be often satisfied by NG-paths. Description of the algorithm In this relaxation, each vertex i has a "memory", also named NG-set, denoted by η i ⊆ V and such that i ∈ η i . If an NG-path goes through i, it can remember only vertices of η i . As it is true for each vertex, an NG-path only remembers the vertices appearing in every NG-set. An NG-path of minimum cost with respect to these NG-sets can be constructed by a dynamic program, either in a forward or backward scheme. We detail below the forward dynamic program and refer the reader to the paper of Baldacci et al. for the backward version. Each path p = (s, i 1 , ..., i l ) is associated with a set of remembered vertices Π(p) = i r : i r ∈ l t=r+1 η it , r = 1, ..., l − 1 {i l }. A forward NG-path (Π, l, i), is a (non necessarily elementary) path p = (s, i 1 , ..., i l = i) starting from s, ending in i, using l arcs and such that Π = Π(p). Π represents the memory of p, since no vertex of Π can be used to extend p. p is constructed by adding i to a smaller NG-path that belongs to the set Ψ − (Π, l, i): Ψ − (Π, l, i) = { (Π , l − 1, j) ng-paths s.t. : j ∈ N − (i), Π = (Π ∩ η i ) ∪ {i}, Π ⊆ η j , j ∈ Π , i / ∈ Π } f * (Π, l, i) is the minimal cost of an NG-path (Π, l, i) and can be computed with the following recursive formula: f * (Π, l, i) = min (Π ,l−1,j)∈Ψ − (Π,l,i) {f * (Π , l − 1, j) + c ji }(5) Decremental State-Space Relaxation Pecin et al. [29] proposed to use the Decremental State-Space Relaxation (DSSR) technique of Righini and Salani's [33], an iterative algorithm in which the search space is even more relaxed than in the pure NG-route relaxation. At each iteration k, each vertex i is associated with a set µ k i that takes the role of the NG-set η i in the dynamic program. At the first iteration the subsets µ 0 i are empty sets. When the solution p k of iteration k is not elementary and does not respect the chosen criterion, vertices are added to the sets µ k i and a new iteration begins. There are three main possible criteria in the DSSR NG-route algorithm. • predefined: original NG-sets η i are computed and the DSSR continues until p k is either elementary or a feasible NG-path with respect to these sets. Vertices are added to sets µ k i only if they belong to η i . • limited: the DSSR continues until p k is elementary or the sizes of sets µ k i exceed a given limit. • unlimited: the DSSR continues until p k is elementary. Without descent or with a predefined one, NG-sets are the heart of the algorithm since they determine the quality of the solution as well as the computation efficiency. When η i is empty for every vertex, there is absolutely no constraint on the elementarity of the path and the NG-route relaxation solves the SPPRC. When η i = V for every vertex, the NG-route is not a relaxation anymore and the solution is necessarily elementary. Except for the unlimited DSSR-NG-route, the NG-sets have a limited size of Λ and, in general, they are constructed randomly. We propose to exploit the length constraint of the problem and to take into account the extended neighborhoods in the NG-sets construction. Indeed, it is sufficient for a vertex to "remember" in η i only its extended predecessors since they are the only vertices that can appear in a path reaching i. Moreoever, when the EMPPLC is embedded in a column generation framework, vertices are associated with a dual value which usually represents the interest for the vertex to appear in the solution. The memory of a vertex can therefore be composed by its Λ extended neighbors with the highest dual value. We also apply the filtering proposed by Pecin et al. [29] based on the alternation between forward and backward computations of the NG-path in a DSSR scheme. Assume we have U B p an upper bound on the EMPPLC optimal value. At each iteration, the costs computed in the previous iteration can be used as completion bounds. Thus, for each state of the dynamic program, a lower bound on the cost of a NG-path constructed from this state can be calculated. If this bound is greater than U B p , then the state is pruned as it never be part of an optimal solution. A natural upper bound in our case is 0 as we are looking for a path of a negative cost. The solution of this relaxation can be a non elementary path. However in the context of the kidney exchange problem, graph are sparse so the elementarity is often achieved even with a relaxed memory. Moreover, the NG-route algorithm is not aimed at finding columns to add in the column generation, but rather at proving the end of the column generation. Ideally, it will be called only once at the end of the column generation and in practice the number of calls is indeed low (4.2 in our experiments) due to the efficiency of our color coding method. Experiments on EMPPLC algorithms We first detail the instances and the protocol, then we analyze the performance of the different algorithms studied in this article to solve the elementary minimum path problem with length constraint. Instances and protocol Experiments are conducted on several EMPPLC instances generated from the pricing step of KEP instances. Pools of patients and donors are created using an online 1 Saidman-based generator [34] with realistic parameters, leading to sparse graphs. In this benchmark called KBR, there are 27 different classes of instances. In particular, the number of patients varies between 50 and 250, K = 3 and L ∈ {4, 7, 13}. The exchange formulation is solved by column generation on one instance of each class and EMPPLC instances are extracted from the first, last and middle iterations of the pricing problem. Note that the 54 instances generated from the first and middle iterations (instances E-KBR − ) contain a solution of negative weight while the optimal value for the 27 last iterations (instances E-KBR 0 ) is zero 2 . The purpose of generating instances with such a procedure is to get dual values at different stages of the column generation, so different weight functions c. In a column generation, methods providing feasible solutions are designed to quickly find new columns to add, i.e., find a path with a negative reduced cost. For this reason, heuristics either stop when the first solution of negative cost is found, or run within a small time limit. On the contrary, relaxations must find good solutions to allow filtering and computation of good bounds. Therefore, the NG-route relaxation is not limited in time while the color coding and local search algorithms are set to return the first solution of negative cost and the best solution after 1 second of running time. Note that the color coding actually ends after at least one trial was completely executed, so the effective running time may exceed this time limit. The integer program is used to compute the optimal solution, but also best upper and lower bounds within 1 second. All in all, seven solution types are reported: • 5 upper bounds -CF: first solution of color coding -CM: best solution of color coding in 1 second -LF: first solution of local search -LM: best solution of local search in 1 second -TSF-ub: best feasible solution of the integer program in 1 second • 2 lower bounds -NG: best NG-route of the instance -TSF-lb: best lower bound of the integer program in 1 second The quality of a solution of cost c is evaluated against the optimal value c * with three performance indicators: its gap to optimal ( \|c−c * \| c * ), its optimality (c = c * ) and its sign compared to optimal solution. Since instances of E-KBR 0 have an optimal value of 0, solutions CF and LF are not reported for these instances and the gap is not computed either. All the experiments were performed on an Intel Xeon E5-2440 v2 @ 1.9 GHz processor and 32 GB of RAM. Color coding We conducted experiments to determine the best number of colors and the best probability distribution to solve the EMPPLC with color coding. We observe that our method lass outperforms the standard color coding algorithm which uses a unif distribution. Figures 4 show the gap for the E-KBR − benchmark. The 54 instances are grouped in three sets of 18 instances, according to the parameter L. The first negative solution CF for the lass strategy is almost always better than for the standard color coding unif. Its quality increases with the number of colors and this is due to the fact that a trial of color coding is more efficient with more colors, both for lass and unif configurations. However, an important condition for the color coding to return good solutions is to make many trials. As increasing C also increases the running time of one trial, it leads to poorer solutions in the same time limit, hence the growing gap for CM. When C is too big, the color coding can exceed the time limit (see Figure 5) because an iteration runs during more than one second. In this case, the color coding actually stops after a single iteration and CF = CM. From this analysis, the best compromise seems to run the color coding with a lass configuration and a small number of colors (we take C = L + 1). Five versions of the NG-route are implemented and tested, voluntarily omitting the unlimited DSSR as it provides no control on the computation time and memory space. Table 1 sums up the different versions and how they construct the NG-set and which DSSR is applied, if any. We tested different size limits for the NG-set (5 to 13), but it appears that they make no difference on the solution quality. On the other hand, increasing this size limit deteriorates the computation time, in particular for configurations without descent. Thus, only results for a size of 5 are kept. Figure 6a shows the number of instances optimally solved by the NG-route relaxation, i.e., the number of instances for which the NG-route returns an elementary NG-path, so the optimal solution. Figure 6b shows the number of instances for which the lower bound returned by the relaxation has the same sign as the optimal solution, a success criterion for the column generation algorithm. Both figures demonstrate the good performance of the limited DSSR compared to other configurations, even though all of them are quite efficient. Still, the limited DSSR is the only configuration that always returns a solution of the same sign as the optimal one and finds this optimal solution almost every time. The fact that the solution of the NG-route is often the optimal solution explains the fact that increasing the NG-set size is not interesting, as even when they are small the solution is elementary. NG-route NG-set creation Descent UN Uniform None NN Neighborhood None L - Limited UP Uniform Predefined NP Neighborhood Predefined Comparisons of four EMPPLC algorithms The four algorithms were implemented and tested on the two benchmarks E-KBR − and E-KBR 0 . Recall that for E-KBR 0 , the gap cannot be computed and that CF and LF solutions do not make sense as no feasible solutions of negative weight can be found for these instances. Thus, for this benchmark, Table 2 shows only the number of instances for which: • the solution has the same sign as the optimal one • the solution is the optimal one for solutions CM, LM, TSF-ub, NG and TSF-lb. Table 3 for E-KBL − is more complete as it also displays the gap with the optimal solution for instances having a solution of the same sign as the optimal one. This gap is therefore not computed on the same number of instances for all the solutions. These results illustrate the dominance of the dynamic programs to compute both upper and lower bounds. The NG-route relaxation always finds the optimal solution, except once. No time limit was given to this algorithm, but we observe in Table 4 optimal solution for 67 instances out of 81. Even when it does not succeed to find the optimum, the gap is the smallest among every feasible solutions. On the contrary, the local search sometimes (for 13 instances) fails to find a negative solution when there is one. Most importantly, the color coding and the NG-route always return a solution which has the same sign as the optimal solution in a small amount of time. Thus, these algorithms are really suitable for the pricing step of a column generation scheme. CM * * LM * * TSF * NG Average Table 4: Average and maximum running times to get solutions * Time limit of 1 second * * Soft time limit of 1 second (checked only between iterations) Solving the kidney exchange problem In the previous sections, algorithms and results are provided for the elementary minimum path problem with length constraint. Our initial goal is however to solve the kidney exchange problem. In this section, we present experimental results comparing the performance of the previous algorithms in a column generation scheme. The complete framework of our implementation called CG-dyn is shown in Figure 7. It solves only the path pricing problem while cycles of donation are all computed beforehand and form the initial set of variables E . This is more efficient when K = 3 than solving the complete pricing problem as preliminary experiments revealed. The path pricing problem is solved with the two dynamic programs (color coding and NG-route) studied in this article. The color coding is limited to 1 second at each iteration of the column generation. Then, if the color coding fails to find an improving path within this time limit, the NG-route relaxation is called. Recall that the KEP is a maximization problem so a path e should have a positive reduced cost rc e to be added in the column generation. When the NG-route relaxation does not find an improving path, i.e., every reduced cost is negative or zero, then the column generation stops. If the solution of this last NG-route relaxation was elementary, thus optimal, then the linear relaxation z * LP is returned. Otherwise, the last value of RMPz is only an upper bound UB on this value. If a given time limit was reached before finishing the column generation, then we cannot conclude on the link between z * LP and the current value of RMPz. However, we can still compute an upper bound UB with Lagrangian relaxation. This upper bound is computed by relaxing in the objective function the redundant constraint stating that there are at most \|V \|/2 exchanges in the compatibility graph (see [26] for more details). After the column generation, the final integer solution x is computed with the integer program exchange formulation restricted on E as it runs very quickly, especially compared to the column generation running time. This step is not required if the last computed solutionx is already integer. The NG-route relaxation is also used to filter suboptimal arcs and vertices of the kidney exchange problem instance. These results also demonstrate that adding all the subpaths of the EMP-PLC solution speeds up the column generation. Concretely, when the path (s, v 1 , ..., v l ) is added to E , we also add the paths (s, v 1 , ..., v l−1 ), (s, v 1 , ..., v l−2 ), and so on until (s, v 1 , v 2 ). Performance of the pricing step The efficiency of dynamic programming approaches to solve the elementary minimum path problem with length constraint was demonstrated in Section 6. To find out if this is still the case when they are embedded in a column generation scheme solving the KEP, we compared our algorithm CG-dyn with another column generation called CG-tsf, which handles the pricing step with the local search heuristic and the time-stage formulation. Every other parameter of the column generation framework is the same for both algorithms, in particular they generate all cycles in the master problem, add every subpath in E , and solve a restricted exchange formulation at the end instead of solving a complete branch-and-price. They are applied on 135 instances (5 in each class of KBR) 3 , each within a time limit of 2000 seconds. CG-dyn CG-tsf # z * LP computed 135 120 Average gap between UB and LB 0.13 % 3.02 % # z * computed (gap = 0) 93 84 Table 5 shows that our algorithm always finds the linear relaxation EFL while CG-tsf reaches the time limit for 15 instances (those with \|P \| = 250 and L = 13). Consequently, the CG-dyn algorithm finds the optimal solution of the integer program exchange formulation for more instances. For both algorithms, this integer solution is mostly found because the linear relaxation is integer (and valid), and in this case there is no need to call the integer program after the column generation. Moreover, considering the 120 instances for which both methods find the linear relaxation in 2000 seconds, the running time is significantly smaller for CG-dyn than CG-tsf (see Table 6). These results support the efficiency of the dynamic programs to solve the path pricing problem in a column generation for the KEP. Scaling up As kidney exchange programs are growing, we experiment CG-dyn on larger instances, generated as before but with different parameters. In particular, it seems reasonable to consider that most of the altruistic donors are already included in kidney exchange programs, unlike patients for whom such programs are very different from the standard procedure. Thus, the proportion of altruistic donors would probably be low in future large programs. In the end, we apply CG-dyn on 20 instances 4 , divided in 4 classes (L ∈ {4, 7}, \|P \| ∈ {500, 750}, p \|N \| = 1%). Results, summarized in Table 7, show that the quality of the solution is still very high. Every instance was solved in less than half an hour and in average rather quickly. We also tried to solve instances with L = 13 or \|P \| = 1000, but encountered memory issues. However, our implementation does not profit from the fact that instances are quite sparse, so a more efficient implementation should overcome these memory errors. Moreover, while solving the final integer program corresponds in average to less than 1% of the running time on realistic instances, for these large instances it represents more than 5%, sometimes almost 15%. It is likely that this proportion will increase with the size of instances, making necessary the development of new algorithms to find feasible solutions. Average gap between UB and LB 0.05% 0.18% 0.06% 0.23% Average running time (seconds) 5.6 123.6 23.1 1823.5 Table 7: Results of CG-dyn on 40 large instances Conclusion In this article, we designed a complete column generation framework to solve the kidney exchange problem including altruistic donors. Due to the hardness of the pricing problem in this case, an extension of previous column generation schemes containing only cycles was not possible. We therefore proposed a new framework showing excellent results on realistic instances and promising for larger instances. We believe that the memory issues encountered for instances with \|P \| = 1000 can be avoided with an implementation of the column generation using algorithms and data structures adapted to large and sparse instances. For example, the preprocessing step computing the distances between each pair of vertices could be performed with a Johnson's algorithm instead of Floyd-Warshall. This column generation relies on the efficiency of the dynamic approaches to handle the pricing problem, referred to as the elementary minimum path problem with length constraint. This problem has to be solved many times in the column generation framework. We adapted algorithms from the literature of shortest paths problem to better fit the specificity of our problem and designed an experimental protocol to assess their quality. The development of a new method for the color coding algorithm, including a preprocessing phase ordering the vertices, a new procedure to color the graph and a shifting technique, is promising for many applications. Indeed, it guarantees to find the optimal solution in only C calls to the dynamic program in some particular cases and in any case outperforms the original discrete uniform strategy. Our complete study of this method can be found in [25]. Other improvements can be considered to get even better results. In particular, an adaptation of the different techniques proposed by Pecin et al. [28], including memory cuts, would probably strengthen our implementation of the NG-route relaxation. We also intend to study the application of the color coding for other pricing problems where it is surprisingly absent, for example for vehicle routing problems. Figure 1 : 1Standard exchanges in a kidney exchange program. Chains and cycles may include more pairs. for example, by taking K = 3 and L = 4, there exists 8 exchanges: two cycles (e 1 = 5 − 7 − 6; e 2 = 4 − 6) and six paths (e 3 = 1 − 3; e 4 = 1 − 3 − 5; e 5 = 1 − 3 − 5 − 7; e 6 = 2 − 3; e 7 = 2 − 3 − 5; e 8 = 2 − 3 − 5 − 7). Figure 2 : 2Example of a compatibility graph of a kidney exchange program The function c is extended to these new arcs: ∀u ∈ N, c sv = α v . A valid path contains L + 1 vertices (including the source) and L arcs in D . For an exchange e, c e = (uv)∈A (e) c uv = v∈V (e) Figure 3 : 3Movements of the local search. Dashed arrows represent subpaths, plain arrows are edges and crossed arrows are removed in the movement. Figure 4 : 4Gap between CF or CM and OPT depending on the configuration and the number of colors, for the E-KBR − benchmark Figure 5 : 5Figures 6 illustrate the quality of the NG-route solution for the E-KBR Average and maximum color coding running times depending on the number of colors and the configuration on E-KBR − instances benchmark. The 81 instances are grouped into three sets of 27 instances, according to the parameter L. Figure 6 : 6Quality of the NG-route solution depending on the configuration on E-KBR instances Figure 7 : 7Our algorithm solving the kidney exchange problem: CG-dyn L = 4 L = 7 L = 4 L = 7 Table 1 : 1The five versions of the NG-route relaxation that it actually runs very quickly. TSF provides poorer results with the same average running time, which, besides, is bounded by the time limit. Similarly, the color coding is very powerful as it finds theUpper bounds Lower bounds CM LM TSF-ub NG TSF-lb # instances with good sign 27 27 24 27 19 # instances with optimal solution 27 26 24 27 19 Table 2 : 2Quality of solutions for the 27 E-KBR 0 instancesUpper bounds Lower bounds CF CM LF LM TSF-ub NG TSF-lb # instances with good sign 54 54 41 41 51 54 40 Average gap on such instances (%) 17.4 7.4 75.8 25.2 15.2 0.01 0 # instances with optimal solution 18 40 1 17 40 53 40 Table 3 : 3Quality of solutions for the 54 E-KBR − instances (b) Number of instances, out of 27, for which NG and OPT have the same signLength limit 4 7 13 18 25 27 20 25 27 26 27 27 17 25 27 21 25 27 NN NP L UN UP Configuration (a) Number of instances, out of 27, for which NG = OPT 25 25 27 25 25 27 27 27 27 24 25 27 25 25 27 NN NP L UN UP Configuration 7.1. Complete algorithm IP formulation EF Solve RMP on E Filtering Get path e from NG-route relaxation rc e > 0 z + 1 > UB e relaxation or time limit reached add e and subpaths to Erc e ≤ 0 rc e > 0 rc e ≤ 0 e optimal Get path e from color coding z, x, UB pricing problem master problem path Compute every cycles x * LP integer z * LP , x * LP MIP on E x * LP or x fractional x integer z, x x * Table 5 : 5Results for the two different CG schemes on 135 instances after 2000 seconds of running timeL CG-dyn CG-tsf 4 1.6 8.9 7 10 34.3 13 15.6 673.1 Table 6 : 6Average running time (in seconds), depending on L, for the 120 instances solved in the time limit available at http://www.dcs.gla.ac.uk/~jamest/kidney-webapp/#/generator 2 all the 81 instances are available at https://pagesperso.g-scop.grenoble-inp.fr/ pansartl/data/instances-EKBR.zip available at https://pagesperso.g-scop.grenoble-inp.fr/~pansartl/data/instances-KBR.zip available at https://pagesperso.g-scop.grenoble-inp.fr/~pansartl/data/instances-KBL.zip Clearing algorithms for barter exchange markets: Enabling nationwide kidney exchanges. 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[ "De th evelop heir Ap A B of Hyb tions fr", "De th evelop heir Ap A B of Hyb tions fr" ]	[ "Comp Hame \ntment of Min r University\n\n" ]	[ "tment of Min r University\n" ]	[]	Sep as been pre 1 Brief Repo brid In rom En plex Sy ed Owlade ning &Meta y of Technol Tehran, Ir p 2006ǦMay epared for 1 ort on: ntellige nginee ystem eghaffari allurgical E logy (Tehra ran y 2008 10th Young	null	[ "https://arxiv.org/pdf/0806.2356v1.pdf" ]	8,025,163	0806.2356	20ece45c7736895a24bee3ab876770b15a49899d	De th evelop heir Ap A B of Hyb tions fr Comp Hame tment of Min r University De th evelop heir Ap A B of Hyb tions fr (This pment pplicat Depart Amirkabir s version ha Sep as been pre 1 Brief Repo brid In rom En plex Sy ed Owlade ning &Meta y of Technol Tehran, Ir p 2006ǦMay epared for 1 ort on: ntellige nginee ystem eghaffari allurgical E logy (Tehra ran y 2008 10th Young Introduction Complex systems are often coincided with uncertainty and order-disorder transitions. Apart of uncertainty, fluctuations forces due to competition of between constructive particles of system drive the system towards order and disorder. There are numerous examples which their behaviors show such anomalies in their evolution, i.e., physical systems, biological, financial systems [1]. In other view, in monitoring of most complex systems, there are some generic challenges for example sparse essence, conflicts in different levels, inaccuracy and limitation of measurements ,which in beyond of inherent feature of such interacted systems are real obstacle in their analysis and predicating of behaviors. There are many methods to analyzing of systems include many particles that are acting on each other, for example statistical methods [2], Vicsek model [3]. Other solution is finding out of "main nominations of each distinct behavior which may has overlapping, in part, to others". This advance is to bate of some mentioned difficulties that can be concluded in the "information granules" proposed by Zadeh [4]. In fact, more complex systems in their natural shape can be described in the sense of networks, which are made of connections among the units. These units are several facets of information granules as well as clusters, groups, communities, modules [5]. Let us consider a more real feature: dynamic natural particles in their inherent properties have (had have-will have) several appearances of "natural" attributes as in individually or in group forms. On the other hand, in society, interacting of main such characteristics (or may extra-natural forces: metaphysic) in facing of predictable or unpredictable events, determines destination of the supposed society. Based upon the above, hierarchical nature of complex systems [6], developed (developing) several branches of natural computing (and related limbs) [7], collaborations [13], conflicts [11], emotions and other features of real complex systems, we propose a general framework of the known computing methods in the connected (or complex hybrid) shape, so that the aim is to inferring of the substantial behaviors of intricate and entangled large societies. Obviously, connections between units of computing cores (intelligent particles) can introduce part (or may full) of the comportments (demeanors-deportments...). Complexity of this system, called MAny Connected Intelligent Particles Systems (MACIPS), add to reactions of particles against information flow, and can open new horizons in studying of this big query: is there a unified theory for the ways in which elements of a system(or aggregation of systems) organize themselves to produce a behavior? [8]. With expanding of a few MACIPS ( Fig.1.) within a network (or complex network), we may construe events of our world within small world. Considering of growing, evolution, cliquing, competition and collaboration among supposed networks can instill a concomitant strategy on the insatiable problems of our world (Fig.2.). In this study, we select a few limited parts of MACIPS, as well as hybrid intelligent systems, and investigate several levels of responses in facing of real information. We show how relatively such our simple methods that can produce (mimic) complicated behavior such governmentnation interactions. Mutual relations between algorithms layers identify order-disorder transferring of such systems. So, we found our proposed methods have good ability in prediction and controlling of some engineering systems as well as those are emerged in Dam engineering, Geoseicince, Mineral processing. So, based on the mentioned system, we have developed a general intelligent rock engineering called: INtelligent Rock Engineering System (INRES). Developing of such intelligent hierarchical networks, investigations of their performances on the noisy information and exploration of possible relate between phase transition steps of the MACIPS and flow of in formations are new interesting fields, as well in various fields of science and economy. The proposed Algorithms 2-1-A general methodology in designing: INtelligent Rock Engineering System (INRES): See Details of the algorithms in the References. Figure 3. A general methodology for Rock Engineering Design using Information granulation theory &extended Modeling instruments ([16], [17], [18], [21] & [22]) Figure 4. Bridging between Knowledge Discovery Methods & Distinct Element Methods)-[23], [24], [26], [27] & [28] Figure 5. A strategy in "Modeling Instrument" to Permeability Analysis "-[25] 2-2-Developed Algorithms: Developed algorithms in Fig (6-10) use four basic axioms upon the balancing of the successive granules assumption: x Step (1): dividing the monitored data into groups of training and testing data x Step (2): first granulation (crisp) by SOM or other crisp granulation methods Step (2-1): selecting the level of granularity randomly or depend on the obtained error from the NFIS or RST (regular neuron growth) Step : construction of the granules (crisp). x Step (3): second granulation (fuzzy or rough granules) by NFIS or RST Step : crisp granules as a new data. Step : selecting the level of granularity; (Error level, number of rules, strength threshold...) Step : checking the suitability. (Close-open iteration: referring to the real data and reinspect closed world) Step (3)(4): construction of fuzzy/rough granules. x Step (4): extraction of knowledge rules Selection of initial crisp granules can be supposed as "Close World Assumption (CWA)" .But in many applications, the assumption of complete information is not feasible, and only cannot be used. In such cases, an "Open World Assumption (OWA)', where information not known by an agent is assumed to be unknown, is often accepted [13]. Step3Ͳfuzzy CͲmean algorithm (or other): creation C clusters (number of clusters) for each > @ N ii . SOM1 SOM2 SOM3 SOMn RST1 RST2 RST3 RSTn Rule strength= ... ij c M J V ª º « » « » « » « » « » « » ¬ ¼ . . .... indu plastic dis r V ª º « » « » « » « » « » ¬ ¼ Hard Step 4ͲVCC algorithm Step 4Ͳ1Ͳ set > @ , i j E , randomly. Step4Ͳ2Ͳfinding prototypes for i=1,…p and extraction of rules( linear format). Step4Ͳ3Ͳ determine i H ' (by testing data). Step4Ͳ4Ͳ i H [ ' d , if yesͲͲͲͲͲͲAEend, else go to 5 Step5ͲAnt colony system, set * t (repeat * t t d ) 5Ͳ1Ͳset 1 t t i i p t i i H H H ' ' ' ¦ 5Ͳ2Ͳ 1 t ij t t i j W H H ' ' ' 5Ͳ3Ͳ ( 1) (1 ) t t ij ij ij t W UW W ' ij E OW ================================================== Else (if * t t t ) ( Figure 1 . 1A schematic view of MAny Connected Intelligent Particles Systems (MACIPS) Figure 2. A schematic Small world perspective: interactions of some MACIPS within network Figure 6 .Figure 7 .Figure 8 .Figure 9 . 6789Self Organizing Neuro-Fuzzy Inference System (SONFIS) -([14] -[22]) Evolutionary Self Organizing Neuro-Fuzzy Inference System (E-SONFIS)-[29] Self Organizing Rough Set Theory-Random neuron growth & adaptive strength factor (SORST-R)-[15], [17], [19],[20] Bridging of hard computations and Self Organizing Rough Set Theory-Random neuron growth & adaptive strength factor (SORST-R)-[17], [22] Figure 10 . 10Self Organizing Rough Set Theory-Adaptive Scaling (SORST-AS)-[14] Figure 11. a combining between vertical collaborative clustering (VCC) [13] &balancing of granules using ant colony system in the granular worlds [30] .... N N ii ; 1,..., i p ; With different 2ͲDstructure (RandomlyͲby training data set). 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L N D Castro, Physics of Life Reviews. 4Castro,L.N.D.: Fundamentals of Natural Computing: An Overview. Physics of Life Reviews.4, pp1Ͳ 36(2006) The Bigger Picture. T Vicsek, Nature. 418131Vicsek, T.: The Bigger Picture. Nature.418, 131(2002) SelfͲOrganization and Associate Memory. T Kohonen, Springer -VerlagBerlin2nd editKohonen, T.: SelfͲOrganization and Associate Memory. 2nd edit. Springer -Verlag, Berlin (1983) . J Jang, S R Sun, C T Mizutani, E , Prentice HallNewjersyJang, J., S. R., Sun, C. T., Mizutani, E.: NeuroͲFuzzy and Soft Computing", Newjersy, Prentice Hall (1997) Rough Sets: Theoretical Aspects Reasoning about Data. Kluwer academic. Z Pawlak, BostonPawlak, Z.: Rough Sets: Theoretical Aspects Reasoning about Data. Kluwer academic, Boston (1991) Using Contextually Closed Queries for Local ClosedͲWorld Reasoning in Rough Knowledge Data Base. P Doherty, J Kachniarz, A Szatas, S K Pal, L Polkowski, A Skowron, roughͲneural computing techniques for comparing with words. Doherty, P., Kachniarz, J., Szatas, A.: Using Contextually Closed Queries for Local ClosedͲWorld Reasoning in Rough Knowledge Data Base. In roughͲneural computing techniques for comparing with words, eds. Pal, S. K., Polkowski, L., Skowron, A. pp.219-250(2004). Collaborative fuzzy clustering. W Pedrycz, Pattern Recognition Letters. 23Publications on the Proposal. Marked Documents are Available atPedrycz, W.: Collaborative fuzzy clustering. Pattern Recognition Letters 23;1675-1686, (2002) Publications on the Proposal: (Marked Documents are Available at: http://arxiv.org/find/all/1/all:+Owladeghaffari/ Bulletin of Engineering Geology and the Environment, in revising Ͳ An Application of Soft Granulation Theory to Permeability Analysis. H Owladeghaffari, E Bakhtavar&, K Shahriar, ; H Owladeghaffari, M Sharifzadeh, K Shahriar, & W Pedrycz, Ͳ Interactions of NationsͲGovernments Using. Journal of Systems Science and Complexity. Intelligent SystemsAn Intelligent Algorithm to Permeability Analysis, H.Owladeghaffari, E.Bakhtavar& K.Shahriar, Bulletin of Engineering Geology and the Environment, in revising Ͳ An Application of Soft Granulation Theory to Permeability Analysis , H. Owladeghaffari, M.Sharifzadeh ,K.Shahriar & W. Pedrycz, Int. Journal of Rock Mechanics and Mining Sciences, In revising Ͳ Modeling of Social phase Transitions Using Intelligent Systems, H.Owladeghaffari & K.Shahriar submitted to Journal of Systems Science and Complexity, May 2008 Ͳ Interactions of NationsͲGovernments Using Intelligent Systems; . H Owladeghaffari, ;K Shahriar, W Pedrycz, Applied Soft Computing. submitted toH.Owladeghaffari ;K.Shahriar and W. Pedrycz ; submitted to Applied Soft Computing, May 2008 Order to Disorder Transitions in Hybrid Intelligent Systems: a Hatch to the Interactions of Nations ͲGovernments. Order to Disorder Transitions in Hybrid Intelligent Systems: a Hatch to the Interactions of Nations ͲGovernments; H Owladeghaffari, The 2008 IEEE International Conference on Granular Computing. ChinaH.Owladeghaffari; The 2008 IEEE International Conference on Granular Computing (GrC 2008), China, Aug 2008 . Phase Transition in SONFIS&SORST. Phase Transition in SONFIS&SORST; H Owladeghaffari, ; W Pedrycz, The Sixth International Conference on Rough Sets and Current Trends in Computing. H.Owladeghaffari; W. Pedrycz; The Sixth International Conference on Rough Sets and Current Trends in Computing; . Ohio Akron, Usa, 16Akron , Ohio, USA,2008 16 . Graphical Estimation of Permeability Using RST&NFIS. Graphical Estimation of Permeability Using RST&NFIS ; H Owladeghaffari ; K.Shahriar, &w Pedrycz, The 27th Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS'08) Ͳ. New York, NY, USAH.Owladeghaffari ; K.Shahriar &W. Pedrycz; The 27th Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS'08) Ͳ New York, NY, USA, May 19Ͳ22, 2008 Rock Mechanics Modeling Based on Soft Granulation Theory. H.OwladeghaffariRock Mechanics Modeling Based on Soft Granulation Theory; H.Owladeghaffari, M Sharifzadeh, K Shahriar, & E Bakhtavar; 42nd, U S , Rock Mechanics Symposium 2nd U.S.Ͳ Canada Rock Mechanics Symposium. in pressM.Sharifzadeh ,K.Shahriar & E.Bakhtavar; 42nd U.S. Rock Mechanics Symposium 2nd U.S.Ͳ Canada Rock Mechanics Symposium, 2008(in press) Permeability Analysis Based on Information Granulation Theory. Permeability Analysis Based on Information Granulation Theory; . M Sharifzadeh, M.Sharifzadeh, H Owladeghaffari, E Bakhtavar, K Shahriar, The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG). H.Owladeghaffari , E.Bakhtavar ,K.Shahriar; The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG)1Ͳ6 ; . October, October, 2008 Analysis of Hydrocyclone Performance Based on Information Granulation Theory. Analysis of Hydrocyclone Performance Based on Information Granulation Theory; Mehdi Irannajad; 8th. World Congress on Computational Mechanics (WCCM8); 5th. O Hamed, Majid Ghaffari, Ejtemaei, European Congress on Computational Methods in Applied Sciences and Engineering. Hamed O.Ghaffari, Majid Ejtemaei & Mehdi Irannajad; 8th. World Congress on Computational Mechanics (WCCM8); 5th. European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2008); Knowledge Discovery of Hydrocyclone's Circuit Based on SONFIS&SORST. 20June 30 -July 5, 2008ͲVenice, Italy 20. Knowledge Discovery of Hydrocyclone's Circuit Based on SONFIS&SORST; Majid Ejtemaei & Mehdi Irannajad;11th International Mineral Processing SymposiumͲ21Ͳ23 OctoberͲAntalyaͲ Turkey 21. Assessment of Effective Parameters on Dilution Using Approximate Reasoning Methods in Longwall Mining Method. O Hamed, Ghaffari, Iran Coal MinesHamed O.Ghaffari , Majid Ejtemaei & Mehdi Irannajad;11th International Mineral Processing SymposiumͲ21Ͳ23 OctoberͲAntalyaͲ Turkey 21. Assessment of Effective Parameters on Dilution Using Approximate Reasoning Methods in Longwall Mining Method ; Iran Coal Mines; . H O Ghaffari, K Shahriar & Gh, Saeedi, H.O.Ghaffari, K.Shahriar & Gh.Saeedi; . The 21st World Mining Congress &EXPO. 711The 21st World Mining Congress &EXPO 2008; 7Ͳ11 ;September 2008,; Back Analysis Based on SOMͲRST System. Poland Krakow, 22, Krakow, Poland 22. Back Analysis Based on SOMͲRST System; . H Owladeghaffari, H.Owladeghaffari; ZhognͲKui Li 23. Toward Fuzzy Block Theory. H Aghababaei ; Ͳ Zuͳyu Chen, Jianͳmin Zhang, Ken Ho, Faͳquan Wu, Proceedings of the 10th Internatinoal Symposium on Landslides and Engineered Slopes. the 10th Internatinoal Symposium on Landslides and Engineered SlopesXi'an, Chinah.owladeghaffari, h.salariͲrad ;PROCEEDINGS OF THE 5THH.Aghababaei ; Proceedings of the 10th Internatinoal Symposium on Landslides and Engineered Slopes, 30 June -4 July 2008, Xi'an, China ;Editor(s) Ͳ ZuͲyu Chen, JianͲMin Zhang, Ken Ho, FaͲQuan Wu, ZhognͲKui Li 23. Toward Fuzzy Block Theory ; h.owladeghaffari, h.salariͲrad ;PROCEEDINGS OF THE 5TH Contact State Analysis Using NFIS &SOM; h.owladeghaffari; COMPUTATIONAL MECHANICS; ISCM2007. International, Workshop, Applications, Computational, In, Engineering Guimarães / Portugal / 1ͳ4, CDROM. 25. Analysis of Permeability Using BPF. Beijing,China; ANFIS & SOMINTERNATIONAL WORKSHOP ON APPLICATIONS OF COMPUTATIONAL MECHANICS IN GEOTECHNICAL ENGINEERING GUIMARÃES / PORTUGAL / 1Ͳ4 APRIL 2007ͲCDROM 24. *Contact State Analysis Using NFIS &SOM; h.owladeghaffari; COMPUTATIONAL MECHANICS; ISCM2007, July 30Ͳ August 1, 2007, Beijing,China ;2007Ͳ CDROM. 25. Analysis of Permeability Using BPF, ANFIS & SOM; . K Shahriar&, H Owladeghaffari, 1K. Shahriar& H. Owladeghaffari; 1st Editor(s) : Erik Eberhardt ,Doug Stead &Tom Morrison pp. . S Canadaͳu, Rock, Mechanics Symposium ͲRock Mechanics: Meeting Society's Challenges and Demands. Vancouver, Canada, 27Ͳ31 May303CanadaͲU.S. Rock Mechanics Symposium ͲRock Mechanics: Meeting Society's Challenges and Demands, Vancouver, Canada, 27Ͳ31 May 2007 ; Editor(s) : Erik Eberhardt ,Doug Stead &Tom Morrison pp.303Ͳ7, 2007 . Analysis of Key Blocks Stability Using FIS&SOM. 26C) NonͲrefereed PapersC) NonͲrefereed Papers 26. Analysis of Key Blocks Stability Using FIS&SOM; Parallelization of DDA Using Soft Computing Approaches. H Owladeghaffari, H. Owladeghaffari; fall 2006 27. Parallelization of DDA Using Soft Computing Approaches; towards Fuzzy Analysis on Contacts in Block System. H Owladeghaffari, H. Owladeghaffari; winter 2006 28. towards Fuzzy Analysis on Contacts in Block System; . H Owladeghaffari, 2007H. Owladeghaffari; winter 2007 A self organizing Vertical Collaborative Clustering. SONFIS &SORST based on Genetic. 30SONFIS &SORST based on Genetic algorithm ;Winter 2008 30. A self organizing Vertical Collaborative Clustering; . W Pedrycz & H.Owladeghaffari Ͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳͳ Hamed, Owladeghaffari, Tehran PolytechnicDepartment of Mining and Metallurgical Engineering; Amirkabir University of TechnologyGraduate education center-Email: [email protected]. Pedrycz & H.Owladeghaffari ͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲ Hamed Owladeghaffari: Department of Mining and Metallurgical Engineering; Amirkabir University of Technology (Tehran Polytechnic)-Graduate education center-Email: [email protected]	[]
[ "SOLUTION OF THE CAUCHY PROBLEM FOR THE NAVIER -STOKES AND EULER EQUATIONS", "SOLUTION OF THE CAUCHY PROBLEM FOR THE NAVIER -STOKES AND EULER EQUATIONS" ]	[ "A Tsionskiy ", "M Tsionskiy " ]	[]	[]	Some known results regarding the Euler and Navier-Stokes equations were obtained by different authors. Existence and smoothness of the Navier-Stokes solutions in two dimensions have been known for a long time. Leray [4] showed that the Navier-Stokes equations in three space dimensions have a weak solution. Scheffer [5], [6] and Shnirelman [7] obtained weak solution of the Euler equations with compact support in spacetime. Caffarelli-Kohn-Nirenberg [8] improved Scheffer's results , and F.-H. Lin [9] simplified the proof of the results of J. Leray. Many problems and conjectures about the behavior of solutions of the Euler and Navier-Stokes equations are described in the book of Bertozzi and Majda [1] or Constantin [2].Solutions of the Navier-Stokes and Euler equations with initial conditions (Cauchy problem) for two and three dimensions are obtained in the convergence series form by the iterative method using the Fourier and Laplace transforms in this paper. For several combinations of problem parameters numerical results were obtained and presented as graphs. * 2000 Mathematics Subject Classification. Primary 35Q30, Secondary 76D05.	null	[ "https://arxiv.org/pdf/1009.2198v3.pdf" ]	118,084,796	1009.2198	b8a97286e76615b3e85fe9e2fdbcc560f3c7873f	SOLUTION OF THE CAUCHY PROBLEM FOR THE NAVIER -STOKES AND EULER EQUATIONS 26 Sep 2010 A Tsionskiy M Tsionskiy SOLUTION OF THE CAUCHY PROBLEM FOR THE NAVIER -STOKES AND EULER EQUATIONS 26 Sep 2010 Some known results regarding the Euler and Navier-Stokes equations were obtained by different authors. Existence and smoothness of the Navier-Stokes solutions in two dimensions have been known for a long time. Leray [4] showed that the Navier-Stokes equations in three space dimensions have a weak solution. Scheffer [5], [6] and Shnirelman [7] obtained weak solution of the Euler equations with compact support in spacetime. Caffarelli-Kohn-Nirenberg [8] improved Scheffer's results , and F.-H. Lin [9] simplified the proof of the results of J. Leray. Many problems and conjectures about the behavior of solutions of the Euler and Navier-Stokes equations are described in the book of Bertozzi and Majda [1] or Constantin [2].Solutions of the Navier-Stokes and Euler equations with initial conditions (Cauchy problem) for two and three dimensions are obtained in the convergence series form by the iterative method using the Fourier and Laplace transforms in this paper. For several combinations of problem parameters numerical results were obtained and presented as graphs. * 2000 Mathematics Subject Classification. Primary 35Q30, Secondary 76D05. The mathematical setup. The Navier-Stokes equations describe the motion of a fluid in R N (N = 2 or 3). We look for a viscous incompressible fluid filling all of R N here. The Navier-Stokes equations are then given by ∂u k ∂t + N n=1 u n ∂u k ∂x n = ν∆u k − ∂p ∂x k + f k (x, t) (x ∈ R N , t ≥ 0, 1 ≤ k ≤ N ) (1.1) div u = N n=1 ∂u n ∂x n = 0 (x ∈ R N , t ≥ 0) (1.2) with initial conditions u(x, 0) = u 0 (x) (x ∈ R N ) (1.3) Here u(x, t) = (u k (x, t)) ∈ R N , (1 ≤ k ≤ N ) − is an unknown velocity vector (N = 2 or 3), p (x, t) − is an unknown pressure, u 0 (x) is a given, C ∞ divergence-free vector field , f k (x, t) are components of a given, externally applied force f (x, t), ν is a positive coefficient of the viscosity (if ν = 0 then (1.1) - (1.3) are the Euler equations), and ∆ = N n=1 ∂ 2 ∂x 2 n is the Laplacian in the space variables. Equation (1.1) is Newton's law for a fluid element subject. Equation (1.2) says that the fluid is incompressible. For physically reasonable solutions, we accept u k (x, t) → 0 , ∂u k ∂x n → 0 as \| x \| → ∞ (1 ≤ k ≤ N, 1 ≤ n ≤ N) (1.4) Hence, we will restrict attention to initial conditions u 0 and force f that satisfy \| ∂ α x u 0 (x) \| ≤ C αK (1+ \| x \|) −K on R N for any α and K > 0. (1.5) and \| ∂ α x ∂ β t f (x, t) \| ≤ C αβK (1+ \| x \| +t) −K on R N × [0, ∞) for any α, β and K > 0. (1.6) We add (− N n=1 u n ∂u k ∂xn ) to both sides of the equations (1.1). Then we have: ∂u k ∂t = ν ∆ u k − ∂p ∂x k + f k (x, t) − N n=1 u n ∂u k ∂x n (x ∈ R N , t ≥ 0, 1 ≤ k ≤ N ) (1.7) div u = N n=1 ∂u n ∂x n = 0 (x ∈ R N , t ≥ 0) (1.8) u(x, 0) = u 0 (x) (x ∈ R N ) (1.9) u k (x, t) → 0 , ∂u k ∂x n → 0 as \| x \| → ∞ (1 ≤ k ≤ N, 1 ≤ n ≤ N) (1.10) \| ∂ α x u 0 (x) \| ≤ C αK (1+ \| x \|) −K on R N for any α and K > 0. (1.11) \| ∂ α x ∂ β t f (x, t) \| ≤ C αβK (1+ \| x \| +t) −K on R N × [0, ∞) for any α, β and K > 0. (1.12) We shall solve the system of equations (1.7) -(1.12) by the iterative method. To do so we write this system of equations in the following form: ∂u jk ∂t = ν ∆ u jk − ∂p j ∂x k + f jk (x, t) (x ∈ R N , t ≥ 0, 1 ≤ k ≤ N ) (1.13) div u j = N n=1 ∂u jn ∂x n = 0 (x ∈ R N , t ≥ 0) (1.14) u j (x, 0) = u 0 (x) (x ∈ R N ) (1.15) u jk (x, t) → 0 , ∂u jk ∂x n → 0 as \| x \| → ∞ (1 ≤ k ≤ N, 1 ≤ n ≤ N) (1.16) \| ∂ α x u 0 (x) \| ≤ C αK (1+ \| x \|) −K on R N for any α , K > 0 and C αK > 0. (1.17) \| ∂ α x ∂ β t f (x, t) \| ≤ C αβK (1+ \| x \| +t) −K on R N × [0, ∞) for any α, β , K > 0 and C αβK > 0. (1.18) Here j is the number of the iterative process step (j = 1,2,3,...). (1.19) or the vector form f jk (x, t) = f k (x, t) − N n=1 u j−1,n ∂u j−1,k ∂x n (1 ≤ k ≤ N )f j (x, t) = f (x, t) − ( u j−1 · ∇ ) u j−1 (1.20) For the first step of the iterative process (j = 1) we have: U 0 k (γ 1 , γ 2 ) = F [u 0 k (x 1 , x 2 )] P j (γ 1 , γ 2 , t) = F [p j (x 1 , x 2 , t)] F jk (γ 1 , γ 2 , t) = F [f jk (x 1 , x 2 , t)] k, s = 1, 2 and then: ∂U j1 (γ 1 , γ 2 , t) ∂t = −ν(γ 2 1 + γ 2 2 )U j1 (γ 1 , γ 2 , t) + iγ 1 P j (γ 1 , γ 2 , t) + F j1 (γ 1 , γ 2 , t) (2.1) ∂U j2 (γ 1 , γ 2 , t) ∂t = −ν(γ 2 1 + γ 2 2 )U j2 (γ 1 , γ 2 , t) + iγ 2 P j (γ 1 , γ 2 , t) + F j2 (γ 1 , γ 2 , t) (2.2) γ 1 U j1 (γ 1 , γ 2 , t) + γ 2 U j2 (γ 1 , γ 2 , t) = 0 (2.3) U j1 (γ 1 , γ 2 , 0) = U 0 1 (γ 1 , γ 2 ) (2.4) U j2 (γ 1 , γ 2 , 0) = U 0 2 (γ 1 , γ 2 ) (2.5) Hence eliminate P j (γ 1 , γ 2 , t) from equations (2.1), (2.2) and find: ∂ ∂t [ U j2 (γ 1 , γ 2 , t) − γ 2 γ 1 U j1 (γ 1 , γ 2 , t)] = −ν(γ 2 1 + γ 2 2 )[ U j2 (γ 1 , γ 2 , t) − γ 2 γ 1 U j1 (γ 1 , γ 2 , t)] + [ F j2 (γ 1 , γ 2 , t) − γ 2 γ 1 F j1 (γ 1 , γ 2 , t)] (2.6) γ 1 U j1 (γ 1 , γ 2 , t) + γ 2 U j2 (γ 1 , γ 2 , t) = 0 (2.7) U j1 (γ 1 , γ 2 , 0) = U 0 1 (γ 1 , γ 2 ) (2.8) U j2 (γ 1 , γ 2 , 0) = U 0 2 (γ 1 , γ 2 ) (2.9) We use Laplace transform (A.4), (A.5) for equations (2.6), (2.7) and have: U ⊗ jk (γ 1 , γ 2 , η) = L[ U jk (γ 1 , γ 2 , t) ] k = 1, 2 η [ U ⊗ j2 (γ 1 , γ 2 , η) − γ 2 γ 1 U ⊗ j1 (γ 1 , γ 2 , η)] − [ U j2 (γ 1 , γ 2 , 0) − γ 2 γ 1 U j1 (γ 1 , γ 2 , 0)] = −ν (γ 2 1 + γ 2 2 )[ U ⊗ j2 (γ 1 , γ 2 , η) − γ 2 γ 1 U ⊗ j1 (γ 1 , γ 2 , η)] + [ F ⊗ j2 (γ 1 , γ 2 , η) − γ 2 γ 1 F ⊗ j1 (γ 1 , γ 2 , η)] (2.10) γ 1 U ⊗ j1 (γ 1 , γ 2 , η) + γ 2 U ⊗ j2 (γ 1 , γ 2 , η) = 0 (2.11) U j1 (γ 1 , γ 2 , 0) = U 0 1 (γ 1 , γ 2 ) (2.12) U j2 (γ 1 , γ 2 , 0) = U 0 2 (γ 1 , γ 2 ) (2.13) The solution of the system of equations (2.10) − (2.13) is: U ⊗ j1 (γ 1 , γ 2 , η) = [γ 2 2 F ⊗ j1 (γ 1 , γ 2 , η) − γ 1 γ 2 F ⊗ j2 (γ 1 , γ 2 , η) + γ 2 2 U 0 1 (γ 1 , γ 2 ) − γ 1 γ 2 U 0 2 (γ 1 , γ 2 )] (γ 2 1 + γ 2 2 )[η + ν(γ 2 1 + γ 2 2 )] (2.14) U ⊗ j2 (γ 1 , γ 2 , η) = [γ 2 1 F ⊗ j2 (γ 1 , γ 2 , η) − γ 1 γ 2 F ⊗ j1 (γ 1 , γ 2 , η) + γ 2 1 U 0 2 (γ 1 , γ 2 ) − γ 1 γ 2 U 0 1 (γ 1 , γ 2 )] (γ 2 1 + γ 2 2 )[η + ν(γ 2 1 + γ 2 2 )] (2.15) Then we use the convolution formula (A.6) and integral (A.7) for (2.14) , (2.15) and obtain: U j1 (γ 1 , γ 2 , t) = t 0 e −ν(γ 2 1 +γ 2 2 )(t−τ ) [γ 2 2 F j1 (γ 1 , γ 2 , τ ) − γ 1 γ 2 F j2 (γ 1 , γ 2 , τ )] (γ 2 1 + γ 2 2 ) dτ + + e −ν(γ 2 1 +γ 2 2 )t U 0 1 (γ 1 , γ 2 ) (2.16) U j2 (γ 1 , γ 2 , t) = t 0 e −ν(γ 2 1 +γ 2 2 )(t−τ ) [γ 2 1 F j2 (γ 1 , γ 2 , τ ) − γ 1 γ 2 F j1 (γ 1 , γ 2 , τ )] (γ 2 1 + γ 2 2 ) dτ + + e −ν(γ 2 1 +γ 2 2 )t U 0 2 (γ 1 , γ 2 ) (2.17) P j (γ 1 , γ 2 , t) is obtained from (2.1) or (2.2) : P j (γ 1 , γ 2 , t) = i [γ 1 F j1 (γ 1 , γ 2 , t) + γ 2 F j2 (γ 1 , γ 2 , t)] (γ 2 1 + γ 2 2 ) (2.18) Use of the Fourier inversion formula (A.2) and find: u j1 (x 1 , x 2 , t) = 1 2π ∞ −∞ ∞ −∞ t 0 e −ν(γ 2 1 +γ 2 2 )(t−τ ) [γ 2 2 F j1 (γ 1 , γ 2 , τ ) − γ 1 γ 2 F j2 (γ 1 , γ 2 , τ )] (γ 2 1 + γ 2 2 ) dτ + + e −ν(γ 2 1 +γ 2 2 )t U 0 1 (γ 1 , γ 2 ) e −i(x1γ1+x2γ2) dγ 1 dγ 2 = = 1 4π 2 ∞ −∞ ∞ −∞ γ 2 2 (γ 2 1 + γ 2 2 ) t 0 e −ν(γ 2 1 +γ 2 2 )(t−τ ) ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) f j1 (x 1 ,x 2 , τ ) dx 1 dx 2 dτ · · e −i(x1γ1+x2γ2) dγ 1 dγ 2 − − 1 4π 2 ∞ −∞ ∞ −∞ γ 1 γ 2 (γ 2 1 + γ 2 2 ) t 0 e −ν(γ 2 1 +γ 2 2 )(t−τ ) ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) f j2 (x 1 ,x 2 , τ ) dx 1 dx 2 dτ · · e −i(x1γ1+x2γ2) dγ 1 dγ 2 + + 1 4π 2 ∞ −∞ ∞ −∞ e −ν(γ 2 1 +γ 2 2 )t ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) u 0 1 (x 1 ,x 2 ) dx 1 dx 2 e −i(x1γ1+x2γ2) dγ 1 dγ 2 = = S 11 (f j1 ) + S 12 (f j2 ) + B(u 0 1 ) (2.19) u j2 (x 1 , x 2 , t) = 1 2π ∞ −∞ ∞ −∞ t 0 e −ν(γ 2 1 +γ 2 2 )(t−τ ) [γ 2 1 F j2 (γ 1 , γ 2 , τ ) − γ 1 γ 2 F j1 (γ 1 , γ 2 , τ )] (γ 2 1 + γ 2 2 ) dτ + + e −ν(γ 2 1 +γ 2 2 )t U 0 2 (γ 1 , γ 2 ) e −i(x1γ1+x2γ2) dγ 1 dγ 2 = = − 1 4π 2 ∞ −∞ ∞ −∞ γ 1 γ 2 (γ 2 1 + γ 2 2 ) t 0 e −ν(γ 2 1 +γ 2 2 )(t−τ ) ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) f j1 (x 1 ,x 2 , τ ) dx 1 dx 2 dτ · · e −i(x1γ1+x2γ2) dγ 1 dγ 2 + + 1 4π 2 ∞ −∞ ∞ −∞ γ 2 1 (γ 2 1 + γ 2 2 ) t 0 e −ν(γ 2 1 +γ 2 2 )(t−τ ) ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) f j2 (x 1 ,x 2 , τ ) dx 1 dx 2 dτ · · e −i(x1γ1+x2γ2) dγ 1 dγ 2 + + 1 4π 2 ∞ −∞ ∞ −∞ e −ν(γ 2 1 +γ 2 2 )t ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) u 0 2 (x 1 ,x 2 ) dx 1 dx 2 e −i(x1γ1+x2γ2) dγ 1 dγ 2 = = S 21 (f j1 ) + S 22 (f j2 ) + B(u 0 2 ) (2.20) p j (x 1 , x 2 , t) = i 2π ∞ −∞ ∞ −∞ [γ 1 F j1 (γ 1 , γ 2 , t) + γ 2 F j2 (γ 1 , γ 2 , t)] (γ 2 1 + γ 2 2 ) e −i(x1γ1+x2γ2) dγ 1 dγ 2 = = − i 4π 2 ∞ −∞ ∞ −∞ γ 1 (γ 2 1 + γ 2 2 ) ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) f j1 (x 1 ,x 2 , t) dx 1 dx 2 e −i(x1γ1+x2γ2) dγ 1 dγ 2 + + i 4π 2 ∞ −∞ ∞ −∞ γ 2 (γ 2 1 + γ 2 2 ) ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) f j2 (x 1 ,x 2 , t) dx 1 dx 2 e −i(x1γ1+x2γ2) dγ 1 dγ 2 = =S 1 (f j1 ) +S 2 (f j2 ) (2.u j =S · f j + B( u 0 ) , (2.22) whereS is the matrix -operator: S 11 S 12 S 21 S 22 We put f j from equation (1.20) into equation (2.22) and have: u j =S · ( f − ( u j−1 · ∇) u j−1 ) + B( u 0 ) = =S · f −S · ( u j−1 · ∇ ) u j−1 + B( u 0 ) = = u 1 −S · ( u j−1 · ∇) u j−1 (2.23) Here u 1 is the solution of the system of equations (1.13) − (1.20) with condition: 2 n=1 u n ∂u k ∂x n = 0 k = 1, 2 For j = 1 formula (2.22) can be written as follows: u 1 =S · f 1 + B( u 0 ) , f 1 (x, t) = f (x, t) (2.24) If t → 0 then u 1 → u 0 (look at integral-operatorsS, B() -integrals (2.19) , (2.20)). For j = 2 we define from equation (1.20): f 2 (x, t) = f 1 (x, t) − ( u 1 · ∇ ) u 1 (2.25) We denote: f * 2 = ( u 1 · ∇) u 1 (2.26) and then we have: f 2 (x, t) = f 1 (x, t) − f * 2 (2.27) Then we get u 2 from (2.22), (2.24): u 2 =S · f 2 + B( u 0 ) =S · ( f 1 − f * 2 ) + B( u 0 ) = u 1 − u * 2 (2.28) Here we have: u * 2 =S · f * 2 (2.29) If t → 0 then u * 2 → 0 (look at integral-operatorS -integrals (2.19) , (2.20)). Continue for j = 3. We define from equation (1.20): f 3 (x, t) = f 1 (x, t) − ( u 2 · ∇ ) u 2 (2.30) Here we have: ( u 2 · ∇) u 2 = (( u 1 − u * 2 ) · ∇ ) ( u 1 − u * 2 ) = f * 2 + f * 3 (2.31) We denote in (2.31): f * 3 = − ( u 1 · ∇) u * 2 − ( u * 2 · ∇) u 1 + ( u * 2 · ∇) u * 2 and then we have: f 3 (x, t) = f 1 (x, t) − f * 2 − f * 3 (2.32) Then we get u 3 from (2.22), (2.24), (2.29): u 3 =S · f 3 + B( u 0 ) =S · ( f 1 − f * 2 − f * 3 ) + B( u 0 ) = u 1 − u * 2 − u * 3 (2.33) Here we denote: u * 3 =S · f * 3 (2.34) If t → 0 then u * 3 → 0 (look at integral-operatorS -integrals (2.19) , (2.20)). For j = 4. We define from equation (1.20): f 4 (x, t) = f 1 (x, t) − ( u 3 · ∇ ) u 3 (2.35) Here we have: ( u 3 · ∇) u 3 = (( u 2 − u * 3 ) · ∇ ) ( u 2 − u * 3 ) = f * 2 + f * 3 + f * 4 (2.36) We denote in (2.36): f * 4 = − ( u 2 · ∇) u * 3 − ( u * 3 · ∇) u 2 + ( u * 3 · ∇) u * 3 and then we have: f 4 (x, t) = f 1 (x, t) − f * 2 − f * 3 − f * 4 (2.37) Then we get u 4 from (2.22), (2.24), (2.29), (2.34): u 4 =S · f 4 + B( u 0 ) =S · ( f 1 − f * 2 − f * 3 − f * 4 ) + B( u 0 ) = u 1 − u * 2 − u * 3 − u * 4 (2.38) Here we denote: u * 4 =S · f * 4 (2.39) If t → 0 then u * 4 → 0 (look at integral-operatorS -integrals (2.19) , (2.20)). For arbitrary number j (j ≥ 2). We define from equation (1.20): f j (x, t) = f 1 (x, t) − ( u j−1 · ∇ ) u j−1 (2.40) Here we have: ( u j−1 · ∇) u j−1 = j l=2 f * l (2.41) and it follows: f j = f 1 − j l=2 f * l (2.42) Then we get u j from (2.22), (2.24): u j =S · f j + B( u 0 ) =S · ( f 1 − j l=2 f * l ) + B( u 0 ) = u 1 − j l=2 u * l (2.43) Here we denote: u * l =S · f * l (2 ≤ l ≤ j) (2.44) If t → 0 then u * l → 0 (look at integral-operatorS -integrals (2.19) , (2.20)). We consider the equations (2.24) -(2.44) and see that the series (2.43) converge for j → ∞ with the conditions for the first step (j = 1) of the iterative process: 2 n=1 u 0n ∂u 0k ∂x n = 0 k = 1, 2 and conditions C αK ≤ 1 2 , C αβK ≤ 1 2 (2.45) Here C αK and C αβK are received from (1.17) , (1.18). Hence, we receive from equation (2.23) when j → ∞: u ∞ = u 1 −S · ( u ∞ · ∇) u ∞ (2.46) Equation (2.46) describes the converging iterative process. Then we have from formula (2.21) : p ∞ =S 1 (f ∞1 ) +S 2 (f ∞2 ) (2.47) Here f ∞ = (f ∞1 , f ∞2 ) is received from formula (2.42) . On the other hand we can transform the original system of differential equations (1.7) − (1.9) to the equivalent system of integral equations by the scheme of iterative process (2.22) , (2.23) for vector u: u = u 1 −S · ( u · ∇) u, (2.48) where u 1 is from formula (2.24). We compare the equations (2.46) and (2.48) and see that the iterative process (2.46) converge to the solution of the system (2.48) and hence to the solution of the differential equations (1.7) − (1.9) with conditions (2.45). In other words there exist smooth functions p ∞ (x, t), u ∞i (x, t) (i = 1, 2) on R 2 × [0, ∞) that satisfy (1.1), (1.2), (1.3) and p ∞ , u ∞i ∈ C ∞ (R 2 × [0, ∞)), (2.49) R 2 \| u ∞ (x, t)\| 2 dx < C (2.50) for all t ≥ 0. Solution. Case N = 3. We use Fourier transform (A.3) for equations (1.13) − (1.20) and get: U jk (γ 1 , γ 2 , γ 3 , t) = F [u jk (x 1 , x 2 , x 3 , t)] F [ ∂ 2 u jk (x 1 , x 2 , x 3 , t) ∂x 2 s ] = −γ 2 s U jk (γ 1 , γ 2 , γ 3 , t) [use(1.16)] U 0 k (γ 1 , γ 2 , γ 3 ) = F [u 0 k (x 1 , x 2 , x 3 )] P j (γ 1 , γ 2 , γ 3 , t) = F [p j (x 1 , x 2 , x 3 , t)] F jk (γ 1 , γ 2 , γ 3 , t) = F [f jk (x 1 , x 2 , x 3 , t)] k, s = 1, 2, 3 and then: dU j1 (γ 1 , γ 2 , γ 3 , t) dt = −ν(γ 2 1 + γ 2 2 + γ 2 3 )U j1 (γ 1 , γ 2 , γ 3 , t) + iγ 1 P j (γ 1 , γ 2 , γ 3 , t) + F j1 (γ 1 , γ 2 , γ 3 , t) (3.1) dU j2 (γ 1 , γ 2 , γ 3 , t) dt = −ν(γ 2 1 + γ 2 2 + γ 2 3 )U j2 (γ 1 , γ 2 , γ 3 , t) + iγ 2 P j (γ 1 , γ 2 , γ 3 , t) + F j2 (γ 1 , γ 2 , γ 3 , t) (3.2) dU j3 (γ 1 , γ 2 , γ 3 , t) dt = −ν(γ 2 1 + γ 2 2 + γ 2 3 )U j3 (γ 1 , γ 2 , γ 3 , t) + iγ 3 P j (γ 1 , γ 2 , γ 3 , t) + F j3 (γ 1 , γ 2 , γ 3 , t) (3.3) γ 1 U j1 (γ 1 , γ 2 , γ 3 , t) + γ 2 U j2 (γ 1 , γ 2 , γ 3 , t) + γ 3 U j3 (γ 1 , γ 2 , γ 3 , t) = 0 (3.4) U j1 (γ 1 , γ 2 , γ 3 , 0) = U 0 1 (γ 1 , γ 2 , γ 3 ) (3.5) U j2 (γ 1 , γ 2 , γ 3 , 0) = U 0 2 (γ 1 , γ 2 , γ 3 ) (3.6) U j3 (γ 1 , γ 2 , γ 3 , 0) = U 0 3 (γ 1 , γ 2 , γ 3 ) (3.7) Hence eliminate P j (γ 1 , γ 2 , γ 3 , t) from equations (3.1) − (3.3) and find: d dt [ U j2 (γ 1 , γ 2 , γ 3 , t) − γ 2 γ 1 U j1 (γ 1 , γ 2 , γ 3 , t)] = −ν(γ 2 1 + γ 2 2 + γ 2 3 )[ U j2 (γ 1 , γ 2 , γ 3 , t) − − γ 2 γ 1 U j1 (γ 1 , γ 2 , γ 3 , t)] + [ F j2 (γ 1 , γ 2 , γ 3 , t) − γ 2 γ 1 F j1 (γ 1 , γ 2 , γ 3 , t)] (3.8) d dt [ U j3 (γ 1 , γ 2 , γ 3 , t) − γ 3 γ 1 U j1 (γ 1 , γ 2 , γ 3 , t)] = −ν(γ 2 1 + γ 2 2 + γ 2 3 )[ U j3 (γ 1 , γ 2 , γ 3 , t) − − γ 3 γ 1 U j1 (γ 1 , γ 2 , γ 3 , t)] + [ F j3 (γ 1 , γ 2 , γ 3 , t) − γ 3 γ 1 F j1 (γ 1 , γ 2 , γ 3 , t)] (3.9) γ 1 U j1 (γ 1 , γ 2 , γ 3 , t) + γ 2 U j2 (γ 1 , γ 2 , γ 3 , t) + γ 3 U j3 (γ 1 , γ 2 , γ 3 , t) = 0 (3.10) U j1 (γ 1 , γ 2 , γ 3 , 0) = U 0 1 (γ 1 , γ 2 , γ 3 ) (3.11) U j2 (γ 1 , γ 2 , γ 3 , 0) = U 0 2 (γ 1 , γ 2 , γ 3 ) (3.12) U j3 (γ 1 , γ 2 , γ 3 , 0) = U 0 3 (γ 1 , γ 2 , γ 3 ) (3.13) We use Laplace transform (A.4), (A.5) for equations (3.8) − (3.10) and have: U ⊗ jk (γ 1 , γ 2 , γ 3 , η) = L[U jk (γ 1 , γ 2 , γ 3 , t)] k = 1, 2, 3 η[ U ⊗ j2 (γ 1 , γ 2 , γ 3 , η) − γ 2 γ 1 U ⊗ j1 (γ 1 , γ 2 , γ 3 , η)] − [ U j2 (γ 1 , γ 2 , γ 3 , 0) − γ 2 γ 1 U j1 (γ 1 , γ 2 , γ 3 , 0)] = −ν(γ 2 1 + γ 2 2 + γ 2 3 )[ U ⊗ j2 (γ 1 , γ 2 , γ 3 , η) − γ 2 γ 1 U ⊗ j1 (γ 1 , γ 2 , γ 3 , η)] + + [ F ⊗ j2 (γ 1 , γ 2 , γ 3 , η) − γ 2 γ 1 F ⊗ j1 (γ 1 , γ 2 , γ 3 , η)] (3.14) η[ U ⊗ j3 (γ 1 , γ 2 , γ 3 , η) − γ 3 γ 1 U ⊗ j1 (γ 1 , γ 2 , γ 3 , η)] − [ U j3 (γ 1 , γ 2 , γ 3 , 0) − γ 3 γ 1 U j1 (γ 1 , γ 2 , γ 3 , 0)] = −ν(γ 2 1 + γ 2 2 + γ 2 3 )[ U ⊗ j3 (γ 1 , γ 2 , γ 3 , η) − γ 3 γ 1 U ⊗ j1 (γ 1 , γ 2 , γ 3 , η)] + + [ F ⊗ j3 (γ 1 , γ 2 , γ 3 , η) − γ 3 γ 1 F ⊗ j1 (γ 1 , γ 2 , γ 3 , η)] (3.15) γ 1 U ⊗ j1 (γ 1 , γ 2 , γ 3 , η) + γ 2 U ⊗ j2 (γ 1 , γ 2 , γ 3 , η) + γ 3 U ⊗ j3 (γ 1 , γ 2 , γ 3 , η) = 0 (3.16) U j1 (γ 1 , γ 2 , γ 3 , 0) = U 0 1 (γ 1 , γ 2 , γ 3 ) (3.17) U j2 (γ 1 , γ 2 , γ 3 , 0) = U 0 2 (γ 1 , γ 2 , γ 3 ) (3.18) U j3 (γ 1 , γ 2 , γ 3 , 0) = U 0 3 (γ 1 , γ 2 , γ 3 ) (3.19) In the usual way the solution of the system of equations (3.14) − (3.16) with formulas (3.17) − (3.19) can be rewritten in the following form: U ⊗ j1 (γ 1 , γ 2 , γ 3 , η) = [(γ 2 2 + γ 2 3 )F ⊗ j1 (γ 1 , γ 2 , γ 3 , η) − γ 1 γ 2 F ⊗ j2 (γ 1 , γ 2 , γ 3 , η) − γ 1 γ 3 F ⊗ j3 (γ 1 , γ 2 , γ 3 , η)] (γ 2 1 + γ 2 2 + γ 2 3 )[η + ν(γ 2 1 + γ 2 2 + γ 2 3 )] + + U 0 1 (γ 1 , γ 2 , γ 3 ) [η + ν(γ 2 1 + γ 2 2 + γ 2 3 )] (3.20) U ⊗ j2 (γ 1 , γ 2 , γ 3 , η) = [(γ 2 3 + γ 2 1 )F ⊗ j2 (γ 1 , γ 2 , γ 3 , η) − γ 2 γ 3 F ⊗ j3 (γ 1 , γ 2 , γ 3 , η) − γ 2 γ 1 F ⊗ j1 (γ 1 , γ 2 , γ 3 , η)] (γ 2 1 + γ 2 2 + γ 2 3 )[η + ν(γ 2 1 + γ 2 2 + γ 2 3 )] + + U 0 2 (γ 1 , γ 2 , γ 3 ) [η + ν(γ 2 1 + γ 2 2 + γ 2 3 )] (3.21) U ⊗ j3 (γ 1 , γ 2 , γ 3 , η) = [(γ 2 1 + γ 2 2 )F ⊗ j3 (γ 1 , γ 2 , γ 3 , η) − γ 3 γ 1 F ⊗ j1 (γ 1 , γ 2 , γ 3 , η) − γ 3 γ 2 F ⊗ j2 (γ 1 , γ 2 , γ 3 , η)] (γ 2 1 + γ 2 2 + γ 2 3 )[η + ν(γ 2 1 + γ 2 2 + γ 2 3 )] + + U 0 3 (γ 1 , γ 2 , γ 3 ) [η + ν(γ 2 1 + γ 2 2 + γ 2 3 )]U j1 (γ 1 , γ 2 , γ 3 , t) = t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) [(γ 2 2 + γ 2 3 )F j1 (γ 1 , γ 2 , γ 3 , τ ) − γ 1 γ 2 F j2 (γ 1 , γ 2 , γ 3 , τ ) − γ 1 γ 3 F j3 (γ 1 , γ 2 , γ 3 , τ )] (γ 2 1 + γ 2 2 + γ 2 3 ) dτ + + e −ν(γ 2 1 +γ 2 2 +γ 2 3 )t U 0 1 (γ 1 , γ 2 , γ 3 ) (3.23) U j2 (γ 1 , γ 2 , γ 3 , t) = t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) [(γ 2 3 + γ 2 1 )F j2 (γ 1 , γ 2 , γ 3 , τ ) − γ 2 γ 3 F j3 (γ 1 , γ 2 , γ 3 , τ ) − γ 2 γ 1 F j1 (γ 1 , γ 2 , γ 3 , τ )] (γ 2 1 + γ 2 2 + γ 2 3 ) dτ + + e −ν(γ 2 1 +γ 2 2 +γ 2 3 )t U 0 2 (γ 1 , γ 2 , γ 3 ) (3.24) U j3 (γ 1 , γ 2 , γ 3 , t) = t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) [(γ 2 1 + γ 2 2 )F j3 (γ 1 , γ 2 , γ 3 , τ ) − γ 3 γ 1 F j1 (γ 1 , γ 2 , γ 3 , τ ) − γ 3 γ 2 F j2 (γ 1 , γ 2 , γ 3 , τ )] (γ 2 1 + γ 2 2 + γ 2 3 ) dτ + + e −ν(γ 2 1 +γ 2 2 +γ 2 3 )t U 0 3 (γ 1 , γ 2 , γ 3 ) (3.25) P j (γ 1 , γ 2 , γ 3 , t) is obtained from (3.1) [(3.2) or (3.3)] : P j (γ 1 , γ 2 , γ 3 , t) = i [γ 1 F j1 (γ 1 , γ 2 , γ 3 , t) + γ 2 F j2 (γ 1 , γ 2 , γ 3 , t) + γ 3 F j3 (γ 1 , γ 2 , γ 3 , t)] (γ 2 1 + γ 2 2 + γ 2 3 ) (3.26) Use of the Fourier inversion formula (A.3) and find: u j1 (x 1 , x 2 , x 3 , t) = 1 (2π) 3/2 ∞ −∞ ∞ −∞ ∞ −∞ t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) [(γ 2 2 + γ 2 3 )F j1 (γ 1 , γ 2 , γ 3 , τ )] (γ 2 1 + γ 2 2 + γ 2 3 ) dτ − − t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) [γ 1 γ 2 F j2 (γ 1 , γ 2 , γ 3 , τ ) + γ 1 γ 3 F j3 (γ 1 , γ 2 , γ 3 , τ )] (γ 2 1 + γ 2 2 + γ 2 3 ) dτ + + e −ν(γ 2 1 +γ 2 2 +γ 2 3 )t U 0 1 (γ 1 , γ 2 , γ 3 ) e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 = = 1 8π 3 ∞ −∞ ∞ −∞ ∞ −∞ (γ 2 2 + γ 2 3 ) (γ 2 1 + γ 2 2 + γ 2 3 ) t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) ∞ −∞ ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2+x3γ3) · ·f j1 (x 1 ,x 2 ,x 3 , τ ) dx 1 dx 2 dx 3 dτ e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 − − 1 8π 3 ∞ −∞ ∞ −∞ ∞ −∞ γ 1 γ 2 (γ 2 1 + γ 2 2 + γ 2 3 ) t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) ∞ −∞ ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2+x3γ3) · ·f j2 (x 1 ,x 2 ,x 3 , τ ) dx 1 dx 2 dx 3 dτ e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 − − 1 8π 3 ∞ −∞ ∞ −∞ ∞ −∞ γ 1 γ 3 (γ 2 1 + γ 2 2 + γ 2 3 ) t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) ∞ −∞ ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2+x3γ3) · ·f j3 (x 1 ,x 2 ,x 3 , τ ) dx 1 dx 2 dx 3 dτ e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 + + 1 8π 3 ∞ −∞ ∞ −∞ ∞ −∞ e −ν(γ 2 1 +γ 2 2 +γ 2 3 )t ∞ −∞ ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2+x3γ3) · · u 0 1 (x 1 ,x 2 ,x 3 ) dx 1 dx 2 dx 3 e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 = = S 11 (f j1 ) + S 12 (f j2 ) + S 13 (f j3 ) + B(u 0 1 ) (3.27) u j2 (x 1 , x 2 , x 3 , t) = 1 (2π) 3/2 ∞ −∞ ∞ −∞ ∞ −∞ t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) [(γ 2 3 + γ 2 1 )F j2 (γ 1 , γ 2 , γ 3 , τ )] (γ 2 1 + γ 2 2 + γ 2 3 ) dτ − − t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) [γ 2 γ 3 F j3 (γ 1 , γ 2 , γ 3 , τ ) + γ 2 γ 1 F j1 (γ 1 , γ 2 , γ 3 , τ )] (γ 2 1 + γ 2 2 + γ 2 3 ) dτ + + e −ν(γ 2 1 +γ 2 2 +γ 2 3 )t U 0 2 (γ 1 , γ 2 , γ 3 ) e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 = = − 1 8π 3 ∞ −∞ ∞ −∞ ∞ −∞ γ 2 γ 1 (γ 2 1 + γ 2 2 + γ 2 3 ) t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) ∞ −∞ ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2+x3γ3) · ·f j1 (x 1 ,x 2 ,x 3 , τ ) dx 1 dx 2 dx 3 dτ e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 + + 1 8π 3 ∞ −∞ ∞ −∞ ∞ −∞ (γ 2 3 + γ 2 1 ) (γ 2 1 + γ 2 2 + γ 2 3 ) t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) ∞ −∞ ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2+x3γ3) · ·f j2 (x 1 ,x 2 ,x 3 , τ ) dx 1 dx 2 dx 3 dτ e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 − − 1 8π 3 ∞ −∞ ∞ −∞ ∞ −∞ γ 2 γ 3 (γ 2 1 + γ 2 2 + γ 2 3 ) t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) ∞ −∞ ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2+x3γ3) · ·f j3 (x 1 ,x 2 ,x 3 , τ ) dx 1 dx 2 dx 3 dτ e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 + + 1 8π 3 ∞ −∞ ∞ −∞ ∞ −∞ e −ν(γ 2 1 +γ 2 2 +γ 2 3 )t ∞ −∞ ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2+x3γ3) · · u 0 2 (x 1 ,x 2 ,x 3 ) dx 1 dx 2 dx 3 e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 = = S 21 (f j1 ) + S 22 (f j2 ) + S 23 (f j3 ) + B(u 0 2 ) (3.28) u j3 (x 1 , x 2 , x 3 , t) = 1 (2π) 3/2 ∞ −∞ ∞ −∞ ∞ −∞ t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) [(γ 2 1 + γ 2 2 )F j3 (γ 1 , γ 2 , γ 3 , τ )] (γ 2 1 + γ 2 2 + γ 2 3 ) dτ − − t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) [γ 3 γ 1 F j1 (γ 1 , γ 2 , γ 3 , τ ) + γ 3 γ 2 F j2 (γ 1 , γ 2 , γ 3 , τ )] (γ 2 1 + γ 2 2 + γ 2 3 ) dτ + + e −ν(γ 2 1 +γ 2 2 +γ 2 3 )t U 0 3 (γ 1 , γ 2 , γ 3 ) e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 = = − 1 8π 3 ∞ −∞ ∞ −∞ ∞ −∞ γ 3 γ 1 (γ 2 1 + γ 2 2 + γ 2 3 ) t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) ∞ −∞ ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2+x3γ3) · ·f j1 (x 1 ,x 2 ,x 3 , τ ) dx 1 dx 2 dx 3 dτ e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 − − 1 8π 3 ∞ −∞ ∞ −∞ ∞ −∞ γ 3 γ 2 (γ 2 1 + γ 2 2 + γ 2 3 ) t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) ∞ −∞ ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2+x3γ3) · ·f j2 (x 1 ,x 2 ,x 3 , τ ) dx 1 dx 2 dx 3 dτ e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 + + 1 8π 3 ∞ −∞ ∞ −∞ ∞ −∞ (γ 2 1 + γ 2 2 ) (γ 2 1 + γ 2 2 + γ 2 3 ) t 0 e −ν(γ 2 1 +γ 2 2 +γ 2 3 )(t−τ ) ∞ −∞ ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2+x3γ3) · ·f j3 (x 1 ,x 2 ,x 3 , τ ) dx 1 dx 2 dx 3 dτ e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 + + 1 8π 3 ∞ −∞ ∞ −∞ ∞ −∞ e −ν(γ 2 1 +γ 2 2 +γ 2 3 )t ∞ −∞ ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2+x3γ3) · · u 0 3 (x 1 ,x 2 ,x 3 ) dx 1 dx 2 dx 3 e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 = = S 31 (f j1 ) + S 32 (f j2 ) + S 33 (f j3 ) + B(u 0 3 ) (3.29) p j (x 1 , x 2 , x 3 , t) = i (2π) 3/2 ∞ −∞ ∞ −∞ ∞ −∞ [γ 1 F j1 (γ 1 , γ 2 , γ 3 , t) + γ 2 F j2 (γ 1 , γ 2 , γ 3 , t)] (γ 2 1 + γ 2 2 + γ 2 3 ) + + γ 3 F j3 (γ 1 , γ 2 , γ 3 , t) (γ 2 1 + γ 2 2 + γ 2 3 ) e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 = = i 8π 3 ∞ −∞ ∞ −∞ ∞ −∞ γ 1 (γ 2 1 + γ 2 2 + γ 2 3 ) ∞ −∞ ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2+x3γ3) · ·f j1 (x 1 ,x 2 ,x 3 , t) dx 1 dx 2 dx 3 e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 + + i 8π 3 ∞ −∞ ∞ −∞ ∞ −∞ γ 2 (γ 2 1 + γ 2 2 + γ 2 3 ) ∞ −∞ ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2+x3γ3) · ·f j2 (x 1 ,x 2 ,x 3 , t) dx 1 dx 2 dx 3 e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 + + i 8π 3 ∞ −∞ ∞ −∞ ∞ −∞ γ 3 (γ 2 1 + γ 2 2 + γ 2 3 ) ∞ −∞ ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2+x3γ3) · ·f j3 (x 1 ,x 2 ,x 3 , t) dx 1 dx 2 dx 3 e −i(x1γ1+x2γ2+x3γ3) dγ 1 dγ 2 dγ 3 = =S 1 (f j1 ) +S 2 (f j2 ) +S 3 (f j3 ) (3.30) So,u j =S · f j + B( u 0 ) , (3.31) whereS is the matrix -operator:   S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33   We put f j from equation (1.20) into equation (3.31) and have: u j =S · ( f − ( u j−1 · ∇) u j−1 ) + B( u 0 ) = =S · f −S · ( u j−1 · ∇ ) u j−1 + B( u 0 ) = = u 1 −S · ( u j−1 · ∇) u j−1 (3.32) Here u 1 is the solution of the system of equations (1.13) − (1.20) with condition: 3 n=1 u n ∂u k ∂x n = 0 k = 1, 2, 3 For j = 1 formula (3.31) can be written as follows: u 1 =S · f 1 + B( u 0 ) , f 1 (x, t) = f (x, t) (3.33) If t → 0 then u 1 → u 0 (look at integral-operatorsS, B() -integrals (3.27) − (3.29)). For j = 2 we define from equation (1.20): f 2 (x, t) = f 1 (x, t) − ( u 1 · ∇ ) u 1 (3.34) We denote: f * 2 = ( u 1 · ∇) u 1 (3.35) and then we have: f 2 (x, t) = f 1 (x, t) − f * 2 (3.36) Then we get u 2 from (3.31), (3.33): u 2 =S · f 2 + B( u 0 ) =S · ( f 1 − f * 2 ) + B( u 0 ) = u 1 − u * 2 (3.37) Here we have: u * 2 =S · f * 2 (3.38) If t → 0 then u * 2 → 0 (look at integral-operatorS -integrals (3.27) − (3.29)). Continue for j = 3. We define from equation (1.20): f 3 (x, t) = f 1 (x, t) − ( u 2 · ∇ ) u 2 (3.39) Here we have: ( u 2 · ∇) u 2 = (( u 1 − u * 2 ) · ∇ ) ( u 1 − u * 2 ) = f * 2 + f * 3 (3.40) We denote in (3.40): f * 3 = − ( u 1 · ∇) u * 2 − ( u * 2 · ∇) u 1 + ( u * 2 · ∇) u * 2 and then we have: f 3 (x, t) = f 1 (x, t) − f * 2 − f * 3 (3.41) Then we get u 3 from (3.31), (3.33), (3.38): u 3 =S · f 3 + B( u 0 ) =S · ( f 1 − f * 2 − f * 3 ) + B( u 0 ) = u 1 − u * 2 − u * 3 (3.42) Here we denote: .29)). For j = 4. We define from equation (1.20): u * 3 =S · f * 3 (3.43) If t → 0 then u * 3 → 0 (look at integral-operatorS -integrals (3.27) − (3f 4 (x, t) = f 1 (x, t) − ( u 3 · ∇ ) u 3 (3.44) Here we have: ( u 3 · ∇) u 3 = (( u 2 − u * 3 ) · ∇ ) ( u 2 − u * 3 ) = f * 2 + f * 3 + f * 4 (3.45) We denote in (3.45): f * 4 = − ( u 2 · ∇) u * 3 − ( u * 3 · ∇) u 2 + ( u * 3 · ∇) u * 3 and then we have: f 4 (x, t) = f 1 (x, t) − f * 2 − f * 3 − f * 4 (3.46) Then we get u 4 from (3.31), (3.33), (3.38), (3.43): u 4 =S · ( f 1 − f * 2 − f * 3 − f * 4 ) + B( u 0 ) = u 1 − u * 2 − u * 3 − u * 4 (3.47) Here we denote: u * 4 =S · f * 4 (3.48) If t → 0 then u * 4 → 0 (look at integral-operatorS -integrals (3.27) − (3.29)). For arbitrary number j (j ≥ 2). We define from equation (1.20): f j (x, t) = f 1 (x, t) − ( u j−1 · ∇ ) u j−1 (3.49) Here we have: ( u j−1 · ∇) u j−1 = j l=2 f * l (3.50) and it follows: f j = f 1 − j l=2 f * l (3.51) Then we get u j from (3.31), (3.33) u j =S · f j + B( u 0 ) =S · ( f 1 − j l=2 f * l ) + B( u 0 ) = u 1 − j l=2 u * l (3.52) Here we denote: u * l =S · f * l (2 ≤ l ≤ j) (3.53) If t → 0 then u * l → 0 (look at integral-operatorS -integrals (3.27) , (3.29)) . We consider the equations (3.33) -(3.53) and see that the series (3.52) converge for j → ∞ with the conditions for the first step (j = 1) of the iterative process: 3 n=1 u 0n ∂u 0k ∂x n = 0 k = 1, 2, 3 and conditions C αK ≤ 1 2 , C αβK ≤ 1 2 . (3.54) Here C αK and C αβK are received from (1.17) , (1.18). Hence, we receive from equation (3.32) when j → ∞: u ∞ = u 1 −S · ( u ∞ · ∇) u ∞ (3.55) Equation (3.55) describes the converging iterative process. Then we have from formula (3.30) : p ∞ =S 1 (f ∞1 ) +S 2 (f ∞2 ) +S 3 (f ∞3 ) (3.56) Here f ∞ = (f ∞1 , f ∞2 , f ∞3 ) is received from formula (3.51) . On the other hand we can transform the original system of differential equations (1.7) − (1.9) to the equivalent system of integral equations by the scheme of iterative process (3.31) , (3.32) for vector u: u = u 1 −S · ( u · ∇) u, (3.57) where u 1 is from formula (3.33). We compare the equations (3.55) and (3.57) and see that the iterative process (3.55) converge to the solution of the system (3.57) and hence to the solution of the differential equations (1.7) − (1.9) with conditions (3.54). In other words there exist smooth functions p ∞ (x, t), u ∞i (x, t) (i = 1, 2, 3) on R 3 × [0, ∞) that satisfy (1.1), (1.2), (1.3) and p ∞ , u ∞i ∈ C ∞ (R 3 × [0, ∞)), (3.58) R 3 \| u ∞ (x, t)\| 2 dx < C (3.59) for all t ≥ 0. In the following chapters 4 and 5 we describe in further details examples of the solutions for the Navier-Stocks and Euler problems with various applied forces and different values of the viscosity coefficient ν. 4. Example of the solution of the Cauchy problem for the Euler equations by the described iterative method with a particular applied force (N = 2). We will consider an example of the solution of the Cauchy problem for the Euler equations (ν = 0) for N = 2 and with initial conditions: u(x, 0) = u 0 (x) = 0 (x ∈ R 2 ) (4.1) Hence, and from formulas (2.19), (2.20) for arbitrary step j of the iterative process, it follows: u j1 (x 1 , x 2 , t) = 1 4π 2 ∞ −∞ ∞ −∞ γ 2 2 (γ 2 1 + γ 2 2 ) t 0 ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) f j1 (x 1 ,x 2 , τ ) dx 1 dx 2 dτ · · e −i(x1γ1+x2γ2) dγ 1 dγ 2 − − 1 4π 2 ∞ −∞ ∞ −∞ γ 1 γ 2 (γ 2 1 + γ 2 2 ) t 0 ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) f j2 (x 1 ,x 2 , τ ) dx 1 dx 2 dτ · · e −i(x1γ1+x2γ2) dγ 1 dγ 2 (4.2) u j2 (x 1 , x 2 , t) = − 1 4π 2 ∞ −∞ ∞ −∞ γ 1 γ 2 (γ 2 1 + γ 2 2 ) t 0 ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) f j1 (x 1 ,x 2 , τ ) dx 1 dx 2 dτ · · e −i(x1γ1+x2γ2) dγ 1 dγ 2 + + 1 4π 2 ∞ −∞ ∞ −∞ γ 2 1 (γ 2 1 + γ 2 2 ) t 0 ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) f j2 (x 1 ,x 2 , τ ) dx 1 dx 2 dτ · · e −i(x1γ1+x2γ2) dγ 1 dγ 2 (4.3) We convert the Cartesian coordinates to the polar coordinates by formulas: x 1 = r · cosϕ ; x 2 = r · sinϕ ; γ 1 = ρ · cosψ ; γ 2 = ρ · sinψ ;x 1 =r · cosφ ;x 2 =r · sinφ; and obtain from formulas (4.2), (4.3): u j1 (r, ϕ, t) = 1 4π 2 ∞ 0 2π 0 sin 2 ψ t 0 ∞ 0 2π 0 e irρcos(φ−ψ) f j1 (r,φ, τ )rdrdφdτ · · e −irρcos(ψ−ϕ) ρdρdψ − − 1 4π 2 ∞ 0 2π 0 sinψcosψ t 0 ∞ 0 2π 0 e irρcos(φ−ψ) f j2 (r,φ, τ )rdrdφdτ · · e −irρcos(ψ−ϕ) ρdρdψ (4.4) u j2 (r, ϕ, t) = − 1 4π 2 ∞ 0 2π 0 sinψcosψ t 0 ∞ 0 2π 0 e irρcos(φ−ψ) f j1 (r,φ, τ )rdrdφdτ · · e −irρcos(ψ−ϕ) ρdρdψ + + 1 4π 2 ∞ 0 2π 0 cos 2 ψ t 0 ∞ 0 2π 0 e irρcos(φ−ψ) f j2 (r,φ, τ )rdrdφdτ · · e −irρcos(ψ−ϕ) ρdρdψ (4.5) We have the applied force f j for arbitrary step j of the iterative process: f jr (r,φ, τ ) = f jr (r)e injφ f jτ (τ ) , f jφ (r,φ, τ ) ≡ 0 (4.6) or f jr (r,φ, τ ) ≡ 0 , f jφ (r,φ, τ ) = f jφ (r)e injφ f jτ (τ ) (4.7) where f jr (r,φ, τ ) , f jφ (r,φ, τ ) − radial and tangential components of the applied force. n j -separate circumferential mode, n j = 0,1,2,3,... We take the radial and tangential components of the applied force (4.6), (4.7) with condition (1.18) . For the radial component of the applied force we use De Moivre's formulas (A.8) and have: f j1 (r,φ, τ ) = f jr (r)e injφ cosφf jτ (τ ) = 1 2 f jr (r) e i(nj −1)φ + e i(nj +1)φ f jτ (τ ) f j2 (r,φ, τ ) = f jr (r)e injφ sinφf jτ (τ ) = i 2 f jr (r) e i(nj −1)φ − e i(nj +1)φ f jτ (τ ) (4.8) We put the applied force components (4.8) in formulas (4.4), (4.5) , change the order of integration and find: u jr1 (r, ϕ, t) = 1 8π 2 ∞ 0 2π 0 sin 2 ψ ∞ 0 f jr (r) 2π 0 e irρcos(φ−ψ) (e i(nj −1)φ + e i(nj +1)φ )dφrdr · · e −irρcos(ψ−ϕ) ρdρdψ − − i ∞ 0 2π 0 sinψcosψ ∞ 0 f jr (r) 2π 0 e irρcos(φ−ψ) (e i(nj −1)φ − e i(nj +1)φ )dφrdr · · e −irρcos(ψ−ϕ) ρdρdψ t 0 f jτ (τ )dτ (4.9) u jr2 (r, ϕ, t) = 1 8π 2 − ∞ 0 2π 0 sinψcosψ ∞ 0 f jr (r) 2π 0 e irρcos(φ−ψ) (e i(nj −1)φ + e i(nj +1)φ )dφrdr · · e −irρcos(ψ−ϕ) ρdρdψ + + i ∞ 0 2π 0 cos 2 ψ ∞ 0 f jr (r) 2π 0 e irρcos(φ−ψ) (e i(nj −1)φ − e i(nj +1)φ )dφrdr · · e −irρcos(ψ−ϕ) ρdρdψ t 0 f jτ (τ )dτ (4.10) We denote: u jt (t) = t 0 f jτ (τ )dτ (4.11) and from formulas (4.9), (4.10) it follows: u jr1 (r, ϕ, t) = u jr1 (r, ϕ)u jt (t) u jr2 (r, ϕ, t) = u jr2 (r, ϕ)u jt (t) (4.12) where u jr1 (r, ϕ) = 1 8π 2 ∞ 0 2π 0 sin 2 ψ ∞ 0 f jr (r) 2π 0 e irρcos(φ−ψ) (e i(nj −1)φ + e i(nj +1)φ )dφrdr · · e −irρcos(ψ−ϕ) ρdρdψ − − i ∞ 0 2π 0 sinψcosψ ∞ 0 f jr (r) 2π 0 e irρcos(φ−ψ) (e i(nj −1)φ − e i(nj +1)φ )dφrdr · · e −irρcos(ψ−ϕ) ρdρdψ (4.13) u jr2 (r, ϕ) = 1 8π 2 − ∞ 0 2π 0 sinψcosψ ∞ 0 f jr (r) 2π 0 e irρcos(φ−ψ) (e i(nj −1)φ + e i(nj +1)φ )dφrdr · · e −irρcos(ψ−ϕ) ρdρdψ + + i ∞ 0 2π 0 cos 2 ψ ∞ 0 f jr (r) 2π 0 e irρcos(φ−ψ) (e i(nj −1)φ − e i(nj +1)φ )dφrdr · · e −irρcos(ψ−ϕ) ρdρdψ (4.14) Let us denote internal integrals in (4.13), (4.14) as I + (r, ρ, ψ) : I + (r, ρ, ψ) = 2π 0 e irρcos(φ−ψ) (e i(nj −1)φ + e i(nj +1)φ )dφ (4.15) We have two integrals here. Plus (+) is for the first part of each integral (4.13), (4.14) and minus (-) is for the second part. We substituteθ forφ:θ =φ -ψ , dθ = dφ and receive: Put I + (r, ρ, ψ) from (4.17) in formulas (4.13), (4.14) , change order of integration and obtain: I + (r, ρ, ψ) = e i(nj −1)ψ 2π−ψ −ψ e irρcosθ+i(nj −1)θ dθ + e i(nj +1)ψ 2π−ψ −ψ e irρcosθ+i(nj +1)θ dθu jr1 (r, ϕ) = 1 8π 2 ∞ 0 ∞ 0 f jr (r) 2π 0 sin 2 ψI + (r, ρ, ψ)−i sinψcosψI − (r, ρ, ψ) e −irρcos(ψ−ϕ) dψrdrρdρ (4.18) u jr2 (r, ϕ) = 1 8π 2 ∞ 0 ∞ 0 f jr (r) 2π 0 −sinψcosψI + (r, ρ, ψ)+i cos 2 ψI − (r, ρ, ψ) e −irρcos(ψ−ϕ) dψrdrρdρu jr1 (r, ϕ) = − n j i nj 2π ∞ 0 ∞ 0 f jr (r) 2π 0 sinψ e −irρcos(ψ−ϕ)+inj ψ dψ J nj (rρ) drdρu jr1 (r, ϕ) = n j 2 e inj ϕ ∞ 0 e iϕ J nj+1 (rρ) + e −iϕ J nj−1 (rρ) ∞ 0 f jr (r) J nj (rρ) drdρ (4.22) u jr2 (r, ϕ) = i n j 2 e inj ϕ ∞ 0 e iϕ J nj+1 (rρ) − e −iϕ J nj−1 (rρ) ∞ 0 f jr (r) J nj (rρ) drdρ (4.23) Let us denote: Then if n j = 0 it follows from (4.24), (4.25), (4.26), (4.27) that u jr1 (r, ϕ) = u jr2 (r, ϕ) = 0 and hence u 1 = u 2 = 0. R j,nj −1,r (r) = ∞ 0 ∞ 0 f jr (r) J nj (rρ)J nj −1 (rρ) drdρ (4.24) R j,nj +1,r (r) = ∞ 0 ∞ 0 f jr (r) J nj (rρ)J nj +1 (rρ) drdρ In the equations bellow we will consider n j ≥ 1. We change the order of integration in formulas (4.24), (4.25) and obtain: R j,nj −1,r (r) = ∞ 0 f jr (r) ∞ 0 J nj (rρ)J nj −1 (rρ) dρ dr (4.28) R j,nj +1,r (r) = ∞ 0 f jr (r) ∞ 0 J nj (rρ)J nj +1 (rρ) dρ drf j1 (r,φ, τ ) = −f jφ (r)e injφ sinφf jτ (τ ) = − i 2 f jφ (r) e i(nj −1)φ − e i(nj +1)φ f jτ (τ ) f j2 (r,φ, τ ) = f jφ (r)e injφ cosφf jτ (τ ) = 1 2 f jφ (r) e i(nj −1)φ + e i(nj +1)φ f jτ (τ ) (4.32) Hence formulas (4.32) are the components f j1 and f j2 from the tangential component of the applied force (4.7), while formulas (4.8) are the components f j1 and f j2 from the radial component of the applied force (4.6). Let us put (4.32) in formulas (4.4), (4.5) and do the operations as we did in (4.9) − (4.27) ( n j ≥ 1). We consider that f jφ (r) · f jτ (τ ) is restricted by condition (1.18) and get: R j,nj −1,ϕ (r) = − ∞ 0 ∞ 0 f jφ (r)r ′ r J nj (rρ)J nj −1 (rρ) drdρ (4.33) R j,nj +1,ϕ (r) = − ∞ 0 ∞ 0 f jφ (r)r ′ r J nj (rρ)J nj +1 (rρ) drdρ ,(4.u jϕ1 (r, ϕ) = − i 2 R j,nj −1,ϕ (r) e i(nj −1)ϕ + R j,nj +1,ϕ (r) e i(nj +1)ϕ (4.35) u jϕ2 (r, ϕ) = 1 2 R j,nj −1,ϕ (r) e i(nj −1)ϕ − R j,nj +1,ϕ (r) e i(nj +1)ϕ (4.36) We change the order of integration in formulas (4.33), (4.34) and obtain: R j,nj −1,ϕ (r) = − ∞ 0 f jφ (r)r ′ r ∞ 0 J nj (rρ)J nj −1 (rρ) dρdr (4.37) R j,nj +1,ϕ (r) = − ∞ 0 f jφ (r)r ′ r ∞ 0 J nj (rρ)J nj +1 (rρ) dρdr (4.38) Internal integrals in formulas (4.37), (4.38) are established by the discontinuous integral of Weber and Schafheitlin (A.10) [12]. Then we have from (4.37), (4.38): R j,nj −1,ϕ (r) = − r nj −1 ∞ r f jφ (r)r ′ r dr r nj (4.39) R j,nj +1,ϕ (r) = − 1 r nj +1 r 0 f jφ (r)r ′ rr nj dr (4.40) We have obtain formulas (4.8) -(4.40) for an arbitrary step j of the iterative process and the applied forces (4.6) or (4.7). Now we investigate the first step (j = 1 , n 1 = n = 1, 2, 3, ...) of the iterative process with the particular radial component of the applied force f 1 (x, t) [ look at (2.24)]: f 1r (r,φ, τ ) = f 1r (r)e inφ f 1τ (τ ) , f 1φ (r,φ, τ ) ≡ 0 f 1r (r) = F nr n+1 e −µnr , f 1τ (τ ) = e −σnτ (4.41) F n , µ n , σ n -constants. 0 < F n < ∞ , 1 < µ n < ∞ , 1 < σ n < ∞. Let us put the applied force (4.41) in formulas (4.30), (4.31), integrate and then we have: R 1,n−1,r (r) = r n−1 ∞ r F nr n+1 e −µnru 1r1 (r, ϕ, t) = u 1r1 (r, ϕ) u 1t (t) u 1r2 (r, ϕ, t) = u 1r2 (r, ϕ) u 1t (t) (4.47) We obtain the following equations by performing appropriate transformations: u 1r (r, ϕ, t) = n 2 R 1,n−1,r (r) + R 1,n+1,r (r) e inϕ u 1t (t) (4.48) u 1ϕ (r, ϕ, t) = i n 2 R 1,n−1,r (r) − R 1,n+1,r (r) e inϕ u 1t (t) (4.49) u 1r (r, ϕ, t), u 1ϕ (r, ϕ, t) are the radial and tangential components of the velocity u 1 . We have from formula (4.46): lim u 1t (t) = 0 t → 0 (4.50) Hence, and from formulas (4.47) − (4.49): lim u 1r1 (r, ϕ, t) = 0; lim u 1r2 (r, ϕ, t) = 0; t → 0; t → 0; lim u 1r (r, ϕ, t) = 0; lim u 1ϕ (r, ϕ, t) = 0; t → 0; t → 0; (4.51) In other words the velocity u 1 satisfies the initial conditions (4.1). We use the asymptotic properties of the incomplete gamma functions Γ(α, x), γ(α, x) and from formulas (4.42) − (4.45) , (4.48), (4.49) we have the velocity u 1 satisfies conditions (1.16) (for r → ∞). Let us continue investigation for the second step (j = 2) of the iterative process. Find f * 2 (r, ϕ, t) = {f * 21 , f * 22 } -the first correction of the particular radial applied force f 1 (x, t) (4.41). We have for f * 2 from formula (2.26): f * 21 = u 1r1 ∂u 1r1 ∂x 1 + u 1r2 ∂u 1r1 ∂x 2 (4.52) f * 22 = u 1r1 ∂u 1r2 ∂x 1 + u 1r2 ∂u 1r2 ∂x 2 (4.53) where u 1r1 , u 1r2 are the components of u 1 and were taken from formulas (4.47). We have here: Hence, we use formulas (4.44) − (4.47) for u 1r1 (r, ϕ, t), u 1r2 (r, ϕ, t) and have from (4.54): ∂u 1r1 (r, ϕ, t) ∂x 1 = ∂u 1r1 (r,∂u 1r1 (r, ϕ, t) ∂x 1 = n 2 R ′ 1,n−1,r (r) e i(n−1)ϕ + R ′ 1,n+1,r (r) e i(n+1)ϕ cosϕ + + i (n − 1)R 1,n−1,r (r) e i(n−1)ϕ + (n + 1)R 1,n+1,r (r) e i(n+1)ϕ − sinϕ r u 1t (t) ∂u 1r1 (r, ϕ, t) ∂x 2 = n 2 R ′ 1,n−1,r (r) e i(n−1)ϕ + R ′ 1,n+1,r (r) e i(n+1)ϕ sinϕ + + i (n − 1)R 1,n−1,r (r) e i(n−1)ϕ + (n + 1)R 1,n+1,r (r) e i(n+1)ϕ cosϕ r u 1t (t) ∂u 1r2 (r, ϕ, t) ∂x 1 = in 2 R ′ 1,n−1,r (r) e i(n−1)ϕ − R ′ 1,n+1,r (r) e i(n+1)ϕ cosϕ + + i (n − 1)R 1,n−1,r (r) e i(n−1)ϕ − (n + 1)R 1,n+1,r (r) e i(n+1)ϕ − sinϕ r u 1t (t) ∂u 1r2 (r, ϕ, t) ∂x 2 = in 2 R ′ 1,n−1,r (r) e i(n−1)ϕ − R ′ 1,n+1,r (r) e i(n+1)ϕ sinϕ + + i (n − 1)R 1,n−1,r (r) e i(n−1)ϕ − (n + 1)R 1,n+1,r (r) e i(n+1)ϕ cosϕ r u 1t (t) (4.55) where R ′ 1,n−1,r (r) = dR 1,n−1,r (r) dr = F n (n − 1)r n−2 µ 2 n Γ(2, µ n r) − r n e −µnr R ′ 1,n+1,r (r) = dR 1,n+1,r (r) dr = F n − (n + 1) r n+2 µ 2n+2 n γ(2n + 2, µ n r) + r n e −µnr (4.56) Let us put u 1r1 , u 1r2 , ∂u1r1 ∂x1 , ∂u1r1 ∂x2 , ∂u1r2 ∂x1 , ∂u1r2 ∂x2 from formulas (4.47), (4.55) in formulas (4.52), (4.53) for f * 21 , f * 22 . After completing appropriate operations we have: f * 21 (r, ϕ, t) = n 2 2 2 T 2,2n−1,r (r) e i(2n−1)ϕ + T 2,2n+1,r (r) e i(2n+1)ϕ T 2n (t) (4.57) f * 22 (r, ϕ, t) = in 2 2 2 T 2,2n−1,r (r) e i(2n−1)ϕ − T 2,2n+1,r (r) e i(2n+1)ϕ T 2n (t) (4.58) where T 2,2n−1,r (r) = R 1,n−1,r (r) + R 1,n+1,r (r) R ′ 1,n−1,r (r) − (n − 1)R 1,n−1,r (r) r R 1,n−1,r (r) − R 1,n+1,r (r) T 2,2n+1,r (r) = R 1,n−1,r (r) + R 1,n+1,r (r) R ′ 1,n+1,r (r) − (n + 1)R 1,n+1,r (r) r R 1,n−1,r (r) − R 1,n+1,r (r) (4.59) T 2n (t) = u 2 1t (t) (4.60) We use formulas (4.42), (4.43) for R 1,n−1,r (r) , R 1,n+1,r (r) and (4.56) for R ′ 1,n−1,r (r) , R ′ 1,n+1,r (r) then do appropriate operations for T 2,2n−1,r (r) , T 2,2n+1,r (r) and get: For radial f * 2r and tangential f * 2ϕ components of the first correction f * 2 (r, ϕ, t) of the particular radial applied force we have: We compare the particular radial applied force f 1 from (4.41) with the first correction f * 2 from ((4.62) − (4.65)) of this particular radial applied force, and we have: f * 2r (r, ϕ, t) = n 2 2 2 T 2,2n−1,r (r) + T 2,2n+1,r (r) e i 2nϕ T 2n (t) = n 2 2 2 T 2,2n,r (r) e i 2nϕ T 2n (t) (4.62) f * 2ϕ (r, ϕ, t) = in 2 2 2 T 2,2n−1,r (r) − T 2,2n+1,r (r) e i 2nϕ T 2n (t) = in 2 2 2 T 2,2n,ϕ (r) e i 2nϕ T 2n (t)\| f * 2 \| << \| f 1 \| (4.66) with condition F n ≤ 1 n (4.67) After the first step of the iterative process (j = 1) we obtained the velocity u 1 [ see (4.47)]. Now we will calculate u * 2 -the first correction of the velocity u 1 . Solution of this problem has two stages. On the first stage we find the part of the first correction u * 2r , corresponding to the radial component of the first correction of applied force f * 2r from (4.62): f * 2r (r, ϕ, t) = n 2 2 2 T 2,2n,r (r) e i 2nϕ T 2n (t) , f * 2ϕ (r, ϕ, t) ≡ 0 (4.68) On the second stage we calculate the other part of the first correction u * 2ϕ , corresponding to the tangential component of the first correction of applied force f * 2ϕ from (4.63): f * 2r (r, ϕ, t) ≡ 0 , f * 2ϕ (r, ϕ, t) = in 2 2 2 T 2,2n,ϕ (r) e i 2nϕ T 2n (t) (4.69) In other words u * 2 = u * 2r + u * 2ϕ , u * 2r = {u * 2r1 , u * 2r2 }, u * 2ϕ = {u * 2ϕ1 , u * 2ϕ2 }. (4.70) First stage: we use formulas (4.26) , (4.27) for components u jr1 (r, ϕ), u jr2 (r, ϕ) and formulas (4.30) , (4.31) for R j,nj −1,r (r), R j,nj +1,r (r) for j = 2 and then formulas (4.62) , (4.64) for f * 2r (r, ϕ, t) , T 2,2n,r (r). We do appropriate operations and have: u * 2r1 (r, ϕ) = n R 2,2n−1,r (r) e i(2n−1)ϕ + R 2,2n+1,r (r) e i(2n+1)ϕ (4.71) u * 2r2 (r, ϕ) = i n R 2,2n−1,r (r) e i(2n−1)ϕ − R 2,2n+1,r (r) e i(2n+1)ϕ (4.72) Here R 2,2n−1,r (r) = n 2 r 2n−1 2 2 ∞ r T 2,2n,r (r) r 2n dr (4.73) R 2,2n+1,r (r) = n 2 2 2 r 2n+1 r 0r 2n T 2,2n,r (r)dr (4.74) Second stage: we use formulas (4.35) , (4.36) for components u jϕ1 (r, ϕ), u jϕ2 (r, ϕ) and formulas (4.39) , (4.40) for R j,nj −1,ϕ (r), R j,nj +1,ϕ (r) for j = 2 and then formulas (4.63) , (4.65) for f * 2ϕ (r, ϕ, t) , T 2,2n,ϕ (r). We do appropriate operations and have: Here: u * 2ϕ1 (r, ϕ) = − i 2 R 2,R 2,2n−1,ϕ (r) = − in 2 r 2n−1 2 2 ∞ r (T 2,2n,ϕ (r) ·r) ′ r r 2n dr (4.77) R 2,2n+1,ϕ (r) = − in 2 2 2 r 2n+1 r 0r 2n (T 2,2n,ϕ (r) ·r) ′ r dr (4.78) Then we have: u * 21 (r, ϕ) = u * 2r1 (r, ϕ) + u * 2ϕ1 (r, ϕ) = = nR 2,2n−1,r (r) − i 2 R 2,2n−1,ϕ (r) e i(2n−1)ϕ + nR 2,2n+1,r (r) − i 2 R 2,2n+1,ϕ (r) e i(2n+1)ϕ (4.79) u * 22 (r, ϕ) = u * 2r2 (r, ϕ) + u * 2ϕ2 (r, ϕ) = = i nR 2,2n−1,r (r) − i 2 R 2,2n−1,ϕ (r) e i(2n−1)ϕ − i nR 2,2n+1,r (r) − i 2 R 2,2n+1,ϕ (r) e i(2n+1)ϕ (4.80) From formula (4.11) for j = 2 and formula (4.60) we have: u 2t (t) = t 0 T 2n (τ )dτ = 1 σ 2 n t − 2 σ n γ(1, σ n t) + 1 2σ n γ(1, 2σ n t) (4.81) Hence, and from equation (4.12) it follows: u * 21 (r, ϕ, t) = u * 21 (r, ϕ) u 2t (t) u * 22 (r, ϕ, t) = u * 22 (r, ϕ) u 2t (t) (4.82) After completing appropriate operations we have: u * 2r (r, ϕ, t) = nR 2,2n−1,r (r) − i 2 R 2,2n−1,ϕ (r) + nR 2,2n+1,r (r) − i 2 R 2,2n+1,ϕ (r) e i2nϕ u 2t (t) (4.83) u * 2ϕ (r, ϕ, t) = i nR 2,2n−1,r (r) − i 2 R 2,2n−1,ϕ (r) − nR 2,2n+1,r (r) − i 2 R 2,2n+1,ϕ (r) e i2nϕ u 2t (t) (4.84) Here u * 2r (r, ϕ, t), u * 2ϕ (r, ϕ, t) are the radial and tangential components of the first correction u * 2 of the velocity u 1 and nR 2,2n−1,r (r) − i 2 R 2,2n−1,ϕ (r) = F 2 n n 2 r 2n−1 2 2 − Γ(1, 2µ n r) 2µ 2 n − Γ(2, 2µ n r) 2 2 µ 2 n + + n ∞ l=0 Γ(l + 2, 2µ n r) (2n + 2) l+1 2 l+1 µ 2 n − ∞ l=0 Γ(l + 3, 2µ n r) (2n + 2) l+1 2 l+3 µ 2 n (4.85) nR 2,2n+1,r (r) − i 2 R 2,2n+1,ϕ (r) = F 2 n n 2 2 2 r 2n+1 − n(2n + 1) 2 4n−2 µ 4n+2 n γ(4n, 2µ n r) + + 2n(2n + 1) µ 2n+2 n γ(2n, µ n r) r 2n e −µnr (4.86) From formulas (4.82) or (4.83), (4.84) with properties of u 2t (t) -(4.81) it follows: lim u * 21 (r, ϕ, t) = 0; lim u * 22 (r, ϕ, t) = 0; t → 0; t → 0; (4.87) lim u * 2r (r, ϕ, t) = 0; lim u * 2ϕ (r, ϕ, t) = 0; t → 0; t → 0; (4.88) and we have the velocity u 2 = u 1 − u * 2 [ look at (2.28)] satisfying the initial conditions (4.1). We use the asymptotic properties of the incomplete gamma functions Γ(α, x), γ(α, x) and from formulas (4.85), (4.86) we have: the first correction u * 2 and therefore the velocity u 2 satisfies conditions (1.16) (for r → ∞). Let us compare the solution (4.47) or (4.48), (4.49) for u 1 of the first step of iterative process with the first correction (4.82) or (4.83), (4.84) for u * 2 , which is received on the second step of iterative process. We see that \| u * 2 \| << \| u 1 \| (4.89) with conditions F n ≤ 1 n t ≤ σ n (4.90) By continuing this iterative process we can obtain next parts u * 3 , u * 4 , ... of the converging series for u. For arbitrary step j of the iterative process we have by using formula (2.43): u j = u 1 − j l=2 u * l (4.91) and then: lim u j = u j → ∞ (4.92) where u is the solution of the problem (1.1) − (1.6) for ν = 0. Below we provide numerical analysis of these results for the following values of problem's parameters: Circumferential modes n = 1, 2, 3, 4, 5. σ n = 10. 0 ≤ t ≤ 10. Results were obtained for functions u 1 − (4.47) or (4.48), (4.49) ; u * 2 − (4.82) or (4.83), (4.84) with calculations of the incomplete gamma functions [11]. u 2 = u 1 − u * 2 and is shown in FIG. 4.1 -4.5. The vector field u 2 at distances r = 1, 2, 3, 5, 7 is represented by the dotted curves in left diagrams. The comparison of \| u 1 \| (dashed plots) and \| u * 2 \| (solid plots) in plane ϕ = [0, π], at distances 0 ≤ r ≤ 50 is represented in right diagrams. This comparison shows \| u * 2 \| << \| u 1 \| and is corresponding to the conclusion (4.89). 5. Example of the solution of the Cauchy problem for the Navier -Stokes equations by the described iterative method with a particular applied force (N = 2). We will consider an example of the solution of the Cauchy problem for the Navier -Stokes equations for N = 2 and with initial conditions: u(x, 0) = u 0 (x) = 0 (x ∈ R 2 ) (5.1) Hence, and from formulas (2.19), (2.20) for arbitrary step j of the iterative process, it follows: u j1 (x 1 , x 2 , t) = 1 4π 2 ∞ −∞ ∞ −∞ γ 2 2 (γ 2 1 + γ 2 2 ) t 0 e −ν(γ 2 1 +γ 2 2 )(t−τ ) ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) f j1 (x 1 ,x 2 , τ ) dx 1 dx 2 dτ · · e −i(x1γ1+x2γ2) dγ 1 dγ 2 − − 1 4π 2 ∞ −∞ ∞ −∞ γ 1 γ 2 (γ 2 1 + γ 2 2 ) t 0 e −ν(γ 2 1 +γ 2 2 )(t−τ ) ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) f j2 (x 1 ,x 2 , τ ) dx 1 dx 2 dτ · · e −i(x1γ1+x2γ2) dγ 1 dγ 2 (5.2) u j2 (x 1 , x 2 , t) = − 1 4π 2 ∞ −∞ ∞ −∞ γ 1 γ 2 (γ 2 1 + γ 2 2 ) t 0 e −ν(γ 2 1 +γ 2 2 )(t−τ ) ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) f j1 (x 1 ,x 2 , τ ) dx 1 dx 2 dτ · · e −i(x1γ1+x2γ2) dγ 1 dγ 2 + + 1 4π 2 ∞ −∞ ∞ −∞ γ 2 1 (γ 2 1 + γ 2 2 ) t 0 e −ν(γ 2 1 +γ 2 2 )(t−τ ) ∞ −∞ ∞ −∞ e i(x1γ1+x2γ2) f j2 (x 1 ,x 2 , τ ) dx 1 dx 2 dτ · · e −i(x1γ1+x2γ2) dγ 1 dγ 2 (5.3) We convert the Cartesian coordinates to the polar coordinates by formulas: x 1 = r · cosϕ ; x 2 = r · sinϕ ; γ 1 = ρ · cosψ ; γ 2 = ρ · sinψ ;x 1 =r · cosφ ;x 2 =r · sinφ; and obtain from formulas (5.2), (5.3): u j1 (r, ϕ, t) = 1 4π 2 ∞ 0 2π 0 sin 2 ψ t 0 e −νρ 2 (t−τ ) ∞ 0 2π 0 e irρcos(φ−ψ) f j1 (r,φ, τ )rdrdφdτ · · e −irρcos(ψ−ϕ) ρdρdψ − − 1 4π 2 ∞ 0 2π 0 sinψcosψ t 0 e −νρ 2 (t−τ ) ∞ 0 2π 0 e irρcos(φ−ψ) f j2 (r,φ, τ )rdrdφdτ · · e −irρcos(ψ−ϕ) ρdρdψ (5.4) u j2 (r, ϕ, t) = − 1 4π 2 ∞ 0 2π 0 sinψcosψ t 0 e −νρ 2 (t−τ ) ∞ 0 2π 0 e irρcos(φ−ψ) f j1 (r,φ, τ )rdrdφdτ · · e −irρcos(ψ−ϕ) ρdρdψ + + 1 4π 2 ∞ 0 2π 0 cos 2 ψ t 0 e −νρ 2 (t−τ ) ∞ 0 2π 0 e irρcos(φ−ψ) f j2 (r,φ, τ )rdrdφdτ · · e −irρcos(ψ−ϕ) ρdρdψ (5.5) We have the applied force f j for arbitrary step j of the iterative process: f jr (r,φ, τ ) = f jr (r, τ )e injφ , f jφ (r,φ, τ ) ≡ 0 (5.6) or f jr (r,φ, τ ) ≡ 0 , f jφ (r,φ, τ ) = f jφ (r, τ )e injφ (5.7) where f jr (r,φ, τ ) , f jφ (r,φ, τ ) − radial and tangential components of the applied force. n j -separate circumferential mode, n j = 0,1,2,3,... We take the radial and tangential components of the applied force (5.6), (5.7) with condition (1.18) . For the radial component of the applied force we use De Moivre's formulas (A.8) and have: f j1 (r,φ, τ ) = f jr (r, τ )e injφ cosφ = 1 2 f jr (r, τ ) e i(nj −1)φ + e i(nj +1)φ f j2 (r,φ, τ ) = f jr (r, τ )e injφ sinφ = i 2 f jr (r, τ ) e i(nj −1)φ − e i(nj +1)φ (5.8) We put the applied force components (5.8) in formulas (5.4), (5.5) and find: u jr1 (r, ϕ, t) = 1 8π 2 ∞ 0 2π 0 sin 2 ψ t 0 e −νρ 2 (t−τ ) ∞ 0 f jr (r, τ ) · · 2π 0 e irρcos(φ−ψ) (e i(nj −1)φ + e i(nj +1)φ )dφrdrdτ e −irρcos(ψ−ϕ) ρdρdψ − − i 8π 2 ∞ 0 2π 0 sinψcosψ t 0 e −νρ 2 (t−τ ) ∞ 0 f jr (r, τ ) · · 2π 0 e irρcos(φ−ψ) (e i(nj −1)φ − e i(nj +1)φ )dφrdrdτ e −irρcos(ψ−ϕ) ρdρdψ (5.9) u jr2 (r, ϕ, t) = − 1 8π 2 ∞ 0 2π 0 sinψcosψ t 0 e −νρ 2 (t−τ ) ∞ 0 f jr (r, τ ) · · 2π 0 e irρcos(φ−ψ) (e i(nj −1)φ + e i(nj +1)φ )dφrdrdτ e −irρcos(ψ−ϕ) ρdρdψ + + i 8π 2 ∞ 0 2π 0 cos 2 ψ t 0 e −νρ 2 (t−τ ) ∞ 0 f jr (r, τ ) · · 2π 0 e irρcos(φ−ψ) (e i(nj −1)φ − e i(nj +1)φ )dφrdrdτ e −irρcos(ψ−ϕ) ρdρdψ (5.10) Let us denote internal integrals in (5.9), (5.10) as I + (r, ρ, ψ) : I + (r, ρ, ψ) = 2π 0 e irρcos(φ−ψ) (e i(nj −1)φ + e i(nj +1)φ )dφ (5.11) We have two integrals here. Plus (+) is for the first part of each integral (5.9), (5.10) and minus (-) is for the second part. We substituteθ forφ:θ =φ -ψ , dθ = dφ and receive: I + (r, ρ, ψ) = e i(nj −1)ψ 2π−ψ −ψ e irρcosθ+i(nj −1)θ dθ + e i(nj +1)ψ 2π−ψ −ψ e irρcosθ+i(nj +1)θ dθ (5.12) Then we use the Bessel function's integral representation (A.9) and have: I + (r, ρ, ψ) = 2πi (nj−1) e i(nj −1)ψ J nj −1 (rρ) + 2πi (nj+1) e i(nj +1)ψ J nj +1 (rρ) (5.13) Let us put I + (r, ρ, ψ) from (5.13) in formulas (5.9), (5.10) , change order of integration and obtain: u jr1 (r, ϕ, t) = 1 8π 2 ∞ 0 t 0 e −νρ 2 (t−τ ) ∞ 0 f jr (r, τ ) · · 2π 0 sin 2 ψI + (r, ρ, ψ) − i sinψcosψI − (r, ρ, ψ) e −irρcos(ψ−ϕ) dψrdrdτ ρdρ (5.14) u jr2 (r, ϕ, t) = 1 8π 2 ∞ 0 t 0 e −νρ 2 (t−τ ) ∞ 0 f jr (r, τ ) · · 2π 0 −sinψcosψI + (r, ρ, ψ) + i cos 2 ψI − (r, ρ, ψ) e −irρcos(ψ−ϕ) dψrdrdτ ρdρ (5.15) Then we group parts in brackets of formulas (5.14), (5.15) , use De Moivre's formulas (A.8) and the Bessel function's properties. And we get: u jr1 (r, ϕ, t) = − n j i nj 2π ∞ 0 t 0 e −νρ 2 (t−τ ) ∞ 0 f jr (r, τ ) 2π 0 sinψ e −irρcos(ψ−ϕ)+inj ψ dψ J nj (rρ) drdτ dρ (5.16) u jr2 (r, ϕ, t) = n j i nj 2π ∞ 0 t 0 e −νρ 2 (t−τ ) ∞ 0 f jr (r, τ )u jr1 (r, ϕ, t) = n j 2 e inj ϕ ∞ 0 t 0 e −νρ 2 (t−τ ) e iϕ J nj+1 (rρ) + e −iϕ J nj−1 (rρ) ∞ 0 f jr (r, τ ) J nj (rρ) drdτ dρ (5.18) u jr2 (r, ϕ, t) = i n j 2 e inj ϕ ∞ 0 t 0 e −νρ 2 (t−τ ) e iϕ J nj+1 (rρ) − e −iϕ J nj−1 (rρ) ∞ 0 f jr (r, τ ) J nj (rρ) drdτ dρ (5.19) Let us denote: In the equations bellow we will consider n j ≥ 1. Now we integrate the solution (5.4), (5.5) by the tangential component of the applied force (5.7) ( for n j ≥ 1). Then we use De Moivre's formulas (A.8) and have: R j,nj −1,r (r, t) = ∞ 0 t 0 e −νρ 2 (t−τ ) J nj −1 (rρ) ∞ 0 f jr (r, τ ) J nj (rρ) drdτ dρ (5.20) R j,nj +1,r (r, t) = ∞ 0 t 0 e −νρ 2 (t−τ ) J nj +1 (rρ) ∞ 0 f jr (r, τ ) J nj (rρ) drdτ dρf j1 (r,φ, τ ) = −f jφ (r, τ )e injφ sinφ = − i 2 f jφ (r, τ ) e i(nj −1)φ − e i(nj +1)φ f j2 (r,φ, τ ) = f jφ (r, τ )e injφ cosφ = 1 2 f jφ (r, τ ) e i(nj −1)φ + e i(nj +1)φ (5.24) Hence formulas (5.24) are the components f j1 and f j2 from the tangential applied force (5.7), while formulas (5.8) are the components f j1 and f j2 from the radial applied force (5.6). Let us put (5.24) in formulas (5.4), (5.5) and do the operations as we did in (5.9) − (5.23) ( n j ≥ 1). We consider that f jφ (r, τ ) is restricted by condition (1.18) and get: R j,nj −1,ϕ (r, t) = − ∞ 0 t 0 e −νρ 2 (t−τ ) J nj −1 (rρ) ∞ 0 f jφ (r, τ ) ·r ′ r J nj (rρ) drdτ dρ (5.25) R j,nj +1,ϕ (r, t) = − ∞ 0 t 0 e −νρ 2 (t−τ ) J nj +1 (rρ) ∞ 0 f jφ (r, τ ) ·r ′ r J nj (rρ) drdτ dρ , (5.26) Here ′ r ≡ ∂ ∂r . Hence we have: u jϕ1 (r, ϕ, t) = − i 2 R j,nj −1,ϕ (r, t) e i(nj −1)ϕ + R j,nj +1,ϕ (r, t) e i(nj +1)ϕ (5.27) u jϕ2 (r, ϕ, t) = 1 2 R j,nj −1,ϕ (r, t) e i(nj −1)ϕ − R j,nj +1,ϕ (r, t) e i(nj +1)ϕ (5.28) We have obtain formulas (5.8) -(5.28) for an arbitrary step j of the iterative process and the applied forces (5.6) or (5.7). Now we investigate the first step (j = 1 , n 1 = n = 1, 2, 3, ...) of the iterative process with the particular radial applied force f 1 (x, t) [look at (2.24)]: f 1r (r,φ, τ ) = f 1r (r, τ )e inφ , f 1φ (r,φ, τ ) ≡ 0 f 1r (r, τ ) = F nr n+1 e −µ 2 nr 2 f 1τ (τ ) (5.29) F n , µ n -constants. , 0 < F n < ∞ , 1 < µ n < ∞. Let us put the particular radial applied force (5.29) in formulas (5.20) , (5.21) and for the internal integral we have by using formula (A.11), [11]: I(ρ, τ ) = ∞ 0 f 1r (r, τ ) J n (rρ) dr = F n f 1τ (τ ) ∞ 0r n+1 e −µ 2 nr 2 J n (rρ) dr = F n f 1τ (τ )ρ n (2µ 2 n ) n+1 e − ρ 2 4µ 2 n (5.30) Now we put I(ρ, τ ) from formula (5.30) in formulas (5.20) , (5.21) , change the order of integration and have by using formula (A.12), [11]: R 1,n−1,r (r, t) = F n t 0 f 1τ (τ ) ∞ 0 e − ν(t−τ )+ 1 4µ 2 n ρ 2 ρ n (2µ 2 n ) n+1 J n−1 (rρ)dρdτ = = F n r n−1 2µ 2 n t 0 f 1τ (τ )Φ n + 2, n + 2; −µ 2 n r 2 4µ 2 n ν(t−τ )+1 [4µ 2 n ν(t − τ ) + 1] n dτ (5.31) R 1,n+1,r (r, t) = F n t 0 f 1τ (τ ) ∞ 0 e − ν(t−τ )+ 1 4µ 2 n ρ 2 ρ n (2µ 2 n ) n+1 J n+1 (rρ)dρdτ = = F n r n+1 2(n + 1) t 0 f 1τ (τ )Φ n + 1, n + 2; −µ 2 n r 2 4µ 2 n ν(t−τ )+1 [4µ 2 n ν(t − τ ) + 1] n+1 dτ (5.32) Here Φ(a, c; x) is a confluent hypergeometric function [10]. We substitute y for τ : y = 1 [4µ 2 n ν(t−τ )+1] , dy = 4µ 2 n ν [4µ 2 n ν(t−τ )+1] 2 dτ and receive: R 1,n−1,r (r, t) = F n r n−1 8µ 4 n ν 1 1 [4µ 2 n νt+1] f 1τ (y) · y n−2 · Φ n + 2, n + 2; −µ 2 n r 2 y dy (5.33) R 1,n+1,r (r, t) = F n r n+1 8µ 2 n ν(n + 1) 1 1 [4µ 2 n νt+1] f 1τ (y) · y n−1 · Φ n + 1, n + 2; −µ 2 n r 2 y dy (5.34) Let us denote f 1τ (y) = y 2 and get: R 1,n−1,r (r, t) = F n r n−1 8µ 4 n ν 1 1 [4µ 2 n νt+1] y n · Φ n + 2, n + 2; −µ 2 n r 2 y dy (5.35) R 1,n+1,r (r, t) = F n r n+1 8µ 2 n ν(n + 1) 1 1 [4µ 2 n νt+1] y n+1 · Φ n + 1, n + 2; −µ 2 n r 2 y dy (5.36) We use formula (A.13) for integrand in the integral (5.35) and formula (A.14) for integrand in the integral (5.36) [10], integrate and then we have: R 1,n−1,r (r, t) = F n r n−1 8µ 4 n ν(n + 1) Φ n + 1, n + 2; −µ 2 n r 2 − Φ n + 1, n + 2; − µ 2 n r 2 (4µ 2 n νt+1) (4µ 2 n νt + 1) n+1 (5.37) Let us continue investigation for the second step (j = 2) of the iterative process. Find f * 2 (r, ϕ, t) = {f * 21 , f * 22 } -the first correction of the particular radial applied force f 1 (x, t) (5.29). We have for f * 2 from formula (2.26): R 1,n+1,r (r, t) = F n r n+1f * 21 = u 1r1 ∂u 1r1 ∂x 1 + u 1r2 ∂u 1r1 ∂x 2 (5.44) f * 22 = u 1r1 ∂u 1r2 ∂x 1 + u 1r2 ∂u 1r2 ∂x 2 (5.45) where u 1r1 , u 1r2 are the components of u 1 and were taken from formulas (5.39), (5.40). We have here: where R ′ 1,n−1,r (r, t) = ∂R 1,n−1,r (r, t) ∂r = F n (n − 1)r n−2 8µ 4 n ν(n + 1) ∂u 1r1 (r, ϕ, t) ∂x 1 = ∂u 1r1 (r, ϕ,∂u 1r1 (r, ϕ, t) ∂x 1 = n 2 R ′ 1,n−1,r (r, t) e i(n−1)ϕ + R ′ 1,n+1,r (r, t) e i(n+1)ϕ cosϕ + + i (n − 1)R 1,n−1,r (r, t) e i(n−1)ϕ + (n + 1)R 1,n+1,r (r, t) e i(n+1)ϕ − sinϕ r ∂u 1r1 (r, ϕ, t) ∂x 2 = n 2 R ′ 1,n−1,r (r, t) e i(n−1)ϕ + R ′ 1,n+1,r (r, t) e i(n+1)ϕ sinϕ + + i (n − 1)R 1,n−1,r (r, t) e i(n−1)ϕ + (n + 1)R 1,n+1,r (r, t) e i(n+1)ϕ cosϕ r ∂u 1r2 (r, ϕ, t) ∂x 1 = in 2 R ′ 1,n−1,r (r, t) e i(n−1)ϕ − R ′ 1,n+1,r (r, t) e i(n+1)ϕ cosϕ + + i (n − 1)R 1,n−1,r (r, t) e i(n−1)ϕ − (n + 1)R 1,n+1,r (r, t) e i(n+1)ϕ − sinϕ r ∂u 1r2 (r, ϕ, t) ∂x 2 = in 2 RΦ n + 1, n + 2; −µ 2 n r 2 − Φ n + 1, n + 2; − µ 2 n r 2 (4µ 2 n νt+1) (4µ 2 n νt + 1) n+1 − − F n r n 4µ 2 n ν(n + 2) Φ n + 2, n + 3; −µ 2 n r 2 − Φ n + 2, n + 3; − µ 2 n r 2 (4µ 2 n νt+1) (4µ 2 n νt + 1) n+2 (5.48) R ′ 1,n+1,r (r, t) = ∂R 1,n+1,r (r, t) ∂r = F n r n 8µ 2 n ν(n + 2) Φ n + 1, n + 3; −µ 2 n r 2 − Φ n + 1, n + 3; − µ 2 n r 2 (4µ 2 n νt+1) (4µ 2 n νt + 1) n+2 − − F n r n+2 4ν(n + 2)(n + 3) Φ n + 2, n + 4; −µ 2 n r 2 − Φ n + 2, n + 4; − After compliting appropriate operations we have: f * 21 (r, ϕ, t) = n 2 2 2 T 2,2n−1,r (r, t) e i(2n−1)ϕ + T 2,2n+1,r (r, t) e i(2n+1)ϕ (5.50) f * 22 (r, ϕ, t) = in 2 2 2 T 2,2n−1,r (r, t) e i(2n−1)ϕ − T 2,2n+1,r (r, t) e i(2n+1)ϕ (5.51) where T 2,2n−1,r (r, t) = R 1,n−1,r (r, t) + R 1,n+1,r (r, t) R ′ 1,n−1,r (r, t) − − (n − 1) r R 1,n−1,r (r, t) R 1,n−1,r (r, t) − R 1,n+1,r (r, t) T 2,2n+1,r (r, t) = R 1,n−1,r (r, t) + R 1,n+1,r (r, t) R ′ 1,n+1,r (r, t) − − (n + 1) r R 1,n+1,r (r, t) R 1,n−1,r (r, t) − R 1,n+1,r (r, t) (5.52) For radial f * 2r and tangential f * 2ϕ components of the first correction f * 2 (r, ϕ, t) of the particular radial applied force we have: f * 2r (r, ϕ, t) = n 2 2 2 T 2,2n−1,r (r, t) + T 2,2n+1,r (r, t) e i 2nϕ = n 2 2 2 T 2,2n,r (r, t) e i 2nϕ (5.53) f * 2ϕ (r, ϕ, t) = in 2 2 2 T 2,2n−1,r (r, t) − T 2,2n+1,r (r, t) e i 2nϕ = in 2 2 2 T 2,2n,ϕ (r, t) e i 2nϕ (5.54) where ( see (5.52)) T 2,2n,r (r, t) = R 1,n−1,r (r, t) + R 1,n+1,r (r, t) R ′ 1,n−1,r (r, t) + R ′ 1,n+1,r (r, t) − − 1 r (n − 1)R 1,n−1,r (r, t) + (n + 1)R 1,n+1,r (r, t) R 1,n−1,r (r, t) − R 1,n+1,r (r, t) T 2,2n,ϕ (r, t) = R 1,n−1,r (r, t) + R 1,n+1,r (r, t) R ′ 1,n−1,r (r, t) − R ′ 1,n+1,r (r, t) − − 1 r (n − 1)R 1,n−1,r (r, t) − (n + 1)R 1,n+1,r (r, t) R 1,n−1,r (r, t) − R 1,n+1,r (r, t) (5.55) We use formulas (5.37), (5.38) for R 1,n−1,r (r, t) , R 1,n+1,r (r, t) and (5.48), (5.49) for R ′ 1,n−1,r (r, t) , R ′ 1,n+1,r (r, t) then do the appropriate operations for T 2,2n−1,r (r, t) , T 2,2n+1,r (r, t), using formula (A.15), and get: T 2,2n,r (r, t) = −F 2 n · r 2n−1 16µ 6 n ν 2 (n + 1) 2 (n + 2) Φ n + 1, n + 2; −µ 2 n r 2 − Φ n + 1, n + 2; − µ 2 n r 2 (4µ 2 n νt+1) (4µ 2 n νt + 1) n+1 · · Φ n + 1, n + 3; −µ 2 n r 2 − Φ n + 1, n + 3; − µ 2 n r 2 (4µ 2 n νt+1) (4µ 2 n νt + 1) n+2 (5.56) T 2,2n,ϕ (r, t) = −F 2 n · r 2n−1 16µ 6 n ν 2 (n + 1)(n + 2) Φ n, n + 2; −µ 2 n r 2 − Φ n, n + 2; − µ 2 n r 2 (4µ 2 n νt+1) (4µ 2 n νt + 1) n+1 · · Φ n + 2, n + 3; −µ 2 n r 2 − Φ n + 2, n + 3; − µ 2 n r 2 (4µ 2 n νt+1) (4µ 2 n νt + 1) n+2 + + F 2 n · n · r 2n−1 16µ 6 n ν 2 (n + 1) 2 (n + 2) Φ n + 1, n + 2; −µ 2 n r 2 − Φ n + 1, n + 2; − µ 2 n r 2 (4µ 2 n νt+1) (4µ 2 n νt + 1) n+1 · · Φ n + 1, n + 3; −µ 2 n r 2 − Φ n + 1, n + 3; − µ 2 n r 2 (4µ 2 n νt+1) (4µ 2 n νt + 1) n+2 (5.57) By comparing particular radial applied force f 1 from (5.29) with the first correction f * 2 from ((5.53) − (5.57)) of this particular radial applied force we have: \| f * 2 \| << \| f 1 \| (5.58) with condition F n ≤ 1 n (5.59) After the first step of the iterative process (j = 1) we had the velocity u 1 -formulas (5.41) , (5.42). Now we will calculate u * 2 -the first correction of the velocity u 1 . Solution of this problem has two stages. On the first stage we find the part of the first correction u * 2r , corresponding to the first correction f * 2r from formula (5.53) of the applied force: f * 2r (r, ϕ, t) = n 2 2 2 T 2,2n,r (r, t) e i 2nϕ , f * 2ϕ (r, ϕ, t) ≡ 0 (5.60) On the second stage we calculate the other part of the first correction u * 2ϕ , corresponding to the first correction f * 2ϕ from formula (5.54) of the applied force: f * 2r (r, ϕ, t) ≡ 0 , f * 2ϕ (r, ϕ, t) = in 2 2 2 T 2,2n,ϕ (r, t) e i 2nϕ (5.61) In other words u * 2 = u * 2r + u * 2ϕ , u * 2r = {u * 2r1 , u * 2r2 }, u * 2ϕ = {u * 2ϕ1 , u * 2ϕ2 }. (5.62) First stage: After completing appropriate operations we have from formula (5.56): T 2,2n,r (r, t) = −F 2 n · r 2n−1 16µ 6 n ν 2 (n + 1) 2 (n + 2) Φ n + 1, n + 2; −µ 2 n r 2 · Φ n + 1, n + 3; −µ 2 n r 2 − − 1 (4µ 2 n νt + 1) n+2 · Φ n + 1, n + 2; −µ 2 n r 2 · Φ n + 1, n + 3; − µ 2 n r 2 (4µ 2 n νt + 1) − − 1 (4µ 2 n νt + 1) n+1 · Φ n + 1, n + 2; − µ 2 n r 2 (4µ 2 n νt + 1) · Φ n + 1, n + 3; −µ 2 n r 2 + + 1 (4µ 2 n νt + 1) 2n+3 · Φ n + 1, n + 2; − µ 2 n r 2 (4µ 2 n νt + 1) · Φ n + 1, n + 3; − µ 2 n r 2 (4µ 2 n νt + 1) (5.63) We take formulas (5.22) , (5.23) for components u jr1 (r, ϕ, t), u jr2 (r, ϕ, t) and formulas (5.20), (5.21) for R j,nj −1,r (r, t), R j,nj +1,r (r, t) for j = 2 and then formulas (5.60), (5.63) for f * 2r (r, ϕ, t) , T 2,2n,r (r, t) . We do appropriate operations and have: u * 2r1 (r, ϕ, t) = n R 2,2n−1,r (r, t) e i(2n−1)ϕ + R 2,2n+1,r (r, t) e i(2n+1)ϕ (5.64) u * 2r2 (r, ϕ, t) = i n R 2,2n−1,r (r, t) e i(2n−1)ϕ − R 2,2n+1,r (r, t) e i(2n+1)ϕ (5.65) After changing the order of integration we receive: R 2,2n−1,r (r, t) = n 2 2 2 t 0 ∞ 0 T 2,2n,r (r, τ ) ∞ 0 e −νρ 2 (t−τ ) J 2n−1 (rρ) J 2n (rρ) dρdrdτ (5.66) R 2,2n+1,r (r, t) = n 2 2 2 t 0 ∞ 0 T 2,2n,r (r, τ ) ∞ 0 e −νρ 2 (t−τ ) J 2n+1 (rρ) J 2n (rρ) dρdrdτ (5.67) Second stage: After operations with T 2,2n,ϕ (r, t) from formula (5.57) and using formula (A.15), we obtain: T 2,2n,ϕ (r, t) · r ′ r = ∂ T 2,2n,ϕ (r, t) · r ∂r = T 2,2n,ϕ,ϕ (r, t) + T 2,2n,ϕ,r (r, t) (5.68) We denote here T 2,2n,ϕ,ϕ (r, t) = −F 2 n · n · r 2n+1 8µ 4 n ν 2 (n + 1)(n + 2) 2 Φ n + 1, n + 3; −µ 2 n r 2 − Φ n + 1, n + 3; − µ 2 n r 2 (4µ 2 n νt+1) (4µ 2 n νt + 1) n+2 · · Φ n + 2, n + 3; −µ 2 n r 2 − Φ n + 2, n + 3; − µ 2 n r 2 (4µ 2 n νt+1) (4µ 2 n νt + 1) n+2 + + F 2 n · r 2n+1 8µ 4 n ν 2 (n + 1)(n + 3) Φ n, n + 2; −µ 2 n r 2 − Φ n, n + 2; − µ 2 n r 2 (4µ 2 n νt+1) (4µ 2 n νt + 1) n+1 · · Φ n + 3, n + 4; −µ 2 n r 2 − Φ n + 3, n + 4; − µ 2 n r 2 (4µ 2 n νt+1) (4µ 2 n νt + 1) n+3 (5.69) and T 2,2n,ϕ,r (r, t) = −F 2 n · n · r 2n−1 8µ 6 n ν 2 (n + 1) 2 (n + 2) Φ n + 1, n + 2; −µ 2 n r 2 − Φ n + 1, n + 2; − µ 2 n r 2 (4µ 2 n νt+1) T 2,2n,ϕ,ϕ (r, t) = −F 2 n · n · r 2n+1 8µ 4 n ν 2 (n + 1)(n + 2) 2 Φ n + 1, n + 3; −µ 2 n r 2 · Φ n + 2, n + 3; −µ 2 n r 2 − − 1 (4µ 2 n νt + 1) n+2 · Φ n + 1, n + 3; −µ 2 n r 2 · Φ n + 2, n + 3; − µ 2 n r 2 (4µ 2 n νt + 1) − − 1 (4µ 2 n νt + 1) n+2 · Φ n + 1, n + 3; − µ 2 n r 2 (4µ 2 n νt + 1) (4µ 2 n νt + 1) n+1 · · Φ n + 1, n + 3; −µ 2 n r 2 − Φ n + 1, n + 3; − µ 2 n r 2 (4µ 2 n νt+1) (4µ 2 n νt + 1) n+2 = 2n · T 2, · Φ n + 2, n + 3; −µ 2 n r 2 + + 1 (4µ 2 n νt + 1) 2n+4 · Φ n + 1, n + 3; − µ 2 n r 2 (4µ 2 n νt + 1) · Φ n + 2, n + 3; − µ 2 n r 2 (4µ 2 n νt + 1) + + F 2 n · r 2n+1 8µ 4 n ν 2 (n + 1)(n + 3) Φ n, n + 2; −µ 2 n r 2 · Φ n + 3, n + 4; −µ 2 n r 2 − − 1 (4µ 2 n νt + 1) n+3 · Φ n, n + 2; −µ 2 n r 2 · Φ n + 3, n + 4; − µ 2 n r 2 (4µ 2 n νt + 1) − − 1 (4µ 2 n νt + 1) n+1 · Φ n, n + 2; − µ 2 n r 2 (4µ 2 n νt + 1) · Φ n + 3, n + 4; −µ 2 n r 2 + + 1 (4µ 2 n νt + 1) 2n+4 · Φ n, n + 2; − µ 2 n r 2 (4µ 2 n νt + 1) · Φ n + 3, n + 4; − µ 2 n r 2 (4µ 2 n νt + 1) (5.71) We transform formulas (5.27), (5.28) for components u jϕ1 (r, ϕ), u jϕ2 (r, ϕ) and formulas (5.25), (5.26) for R j,nj −1,ϕ (r), R j,nj +1,ϕ (r) for j = 2 and then use formula (5.61) for f * 2ϕ (r, ϕ, t). We do several operations and have: u * 21 (r, ϕ, t) = u * 2r1 (r, ϕ, t) + u * 2ϕ1 (r, ϕ, t) = = nR 2,2n−1,r (r, t) − i 2 R 2,2n−1,ϕ (r, t) e i(2n−1)ϕ + nR 2,2n+1,r (r, t) − i 2 R 2,2n+1,ϕ (r, t) e i(2n+1)ϕ (5.80) u * 22 (r, ϕ, t) = u * 2r2 (r, ϕ, t) + u * 2ϕ2 (r, ϕ, t) = = i nR 2,2n−1,r (r, t) − i 2 R 2,2n−1,ϕ (r, t) e i(2n−1)ϕ − i nR 2,2n+1,r (r, t) − i 2 R 2,2n+1,ϕ (r, t) e i(2n+1)ϕ (5.81) We obtain by performing appropriate transformations: u * 2r (r, ϕ, t) = nR 2,2n−1,r (r, t) − i 2 R 2,2n−1,ϕ (r, t) + nR 2,2n+1,r (r, t) − i 2 R 2,2n+1,ϕ (r, t) e i2nϕ (5.82) u * 2ϕ (r, ϕ, t) = i nR 2,2n−1,r (r, t) − i 2 R 2,2n−1,ϕ (r, t) − nR 2,2n+1,r (r, t) − i 2 R 2,2n+1,ϕ (r, t) e i2nϕ (5.83) Here u * 2r (r, ϕ, t), u * 2ϕ (r, ϕ, t) are the radial and tangential components of the first correction u * 2 of the velocity u 1 . R 2,2n−1,r (r, t), R 2,2n+1,r (r, t) are taken from formulas (5.66), (5.67). R 2,2n−1,ϕ (r, t) = R 2,2n−1,ϕ,ϕ (r, t) − 2niR 2,2n−1,r (r, t) R 2,2n+1,ϕ (r, t) = R 2,2n+1,ϕ,ϕ (r, t) − 2niR 2,2n+1,r (r, t) From formulas (5.80), (5.81) with properties of R 2,2n−1,r (r, t), R 2,2n+1,r (r, t), R 2,2n−1,ϕ (r, t), R 2,2n+1,ϕ (r, t) it follows: lim u * 21 (r, ϕ, t) = 0; lim u * 22 (r, ϕ, t) = 0; t → 0; t → 0; (5.87) lim u * 2r (r, ϕ, t) = 0; lim u * 2ϕ (r, ϕ, t) = 0; t → 0; t → 0; (5.88) In other words the velocity u 2 = u 1 − u * 2 [look at (2.28)] satisfies the initial conditions (5.1). We use the asymptotic properties of the confluent hypergeometric function Φ(a, c; x) and have from formulas (5.66), (5.67), (5.78), (5.79): the first correction u * 2 and therefore the velocity u 2 satisfies conditions (1.16) ( for r → ∞). By comparing the solution u 1 from (5.39), (5.40) or (5.41), (5.42) of the first step of iterative process with the first correction u * 2 from (5.80), (5.81) or (5.85), (5.86), which is received on the second step of iterative process, we see that By continuing this iterative process we can obtain next parts u * 3 , u * 4 ..., of the converging series for u. For arbitrary step j of the iterative process we have by using formula (2.43): Below we provide numerical analysis of these results for the following values of problem's parameters: u j = u 1 − Circumferential modes n = 1, 2, 3, 4, 5. For condition t < τ the integrand is diminishing fast enough. It is easy to find upper limit of integration, so we can substitute integral (5.93) for I + (r,r, t, τ ) = A1 0 e −νρ 2 (t−τ ) J 2n+1 (rρ) · J 2n (rρ) dρ, (5.94) where (0 < A 1 = 200 < ∞), and hence we are integrating over the finite interval. For additional check let us increase A 1 in 1.5 times and change the number of steps of integration n 1 ( from 4001 to 6001 ). We see that the difference in result values of integral (5.94) is within the range of required precision ǫ 1 (10 −14 ). For condition t = τ the integral (5.93) is in fact an integral of Weber and Schafheitlin and it is possible to calculate it analytically (A.10) [12]. Let us now consider the calculation of middle integrals I + (r, t, τ ) = ∞ 0 T 2,2n,ϕ,ϕ (r, τ )I + (r,r, t, τ )dr (5.95) ( u 0 s 0· ∇ ) u 0 = 0 and f 1 (x, t) = f (x, t) 2. Solution. Case N = 2.We use Fourier transform (A.2) for equations (1.13) − (1.20) and get:U jk (γ 1 , γ 2 , t) = F [u jk (x 1 , x 2 , t)] F [ ∂ 2 u jk (x 1 , x 2 U jk (γ 1 , γ 2 , t) [use(1.16)] the integrals (2.19) − (2.21) exist by the restrictions (1.17) , (1.18) . Here S 11 (), S 12 (), S 21 (), S 22 (), B(),S 1 (),S 2 () are the integral -operators. S 12 () = S 21 () We have for the vector u j from the equations (2.19) − (2.20) : use the convolution formula (A.6) and integral (A.7) for (3.20) − (3.22) and obtain: the integrals (3.27) − (3.30) exist by the restrictions (1.17) , (1.18) . Here S 11 (), S 12 (), S 13 (), S 21 (), S 22 (), S 23 (), S 31 (), S 32 (), S 33 (), B(),S 1 (),S 2 (),S 3 () are the integral -operators. S 12 () = S 21 () S 13 () = S 31 () S 23 () = S 32 () We have for the vector u j from the equations (3.27) − (3.29) : use the Bessel function's integral representation (A.9) and have: I + (r, ρ, ψ) = 2πi (nj−1) e i(nj −1)ψ J nj −1 (rρ) + 2πi (nj+1) e i(nj +1)ψ J nj +1 (rρ) (4.17) (4. 19 ) 19Then we group parts in brackets of formulas (4.18), (4.19) , use De Moivre's formulas (A.8) and the Bessel function's properties. And we get: jr2 (r, ϕ) = n j i nj 2π e −irρcos(ψ−ϕ)+inj ψ dψ J nj (rρ) drdρ (4. 21 ) 21We substitute θ for ψ: θ = ψ -ϕ , dθ = dψ in the internal integrals of formulas (4.20), (4.21) , use De Moivre's formulas (A.8) and the Bessel function's integral representation (A.9) and have from formulas (4.20), (4.21): have from formulas (4.22), (4.23): u jr1 (r, ϕ) = n j 2 R j,nj −1,r (r) e i(nj −1)ϕ + R j,nj +1,r (r) e i(nj +1)ϕ (4.26) u jr2 (r, ϕ) = i n j 2 R j,nj −1,r (r) e i(nj −1)ϕ − R j,nj +1,r (r) e i(nj +1)ϕ (4.27) in formulas (4.28),(4.29) are established by the discontinuous integral of Weber and Schafheitlin (A.10)[12]. Then we have from (4.28), (4.29):R j,nj −1,r (r) = r nj −1 ∞ r f jr (r)r nj dr (4.30) R j,nj +1,r (rintegrate the solution (4.4), (4.5) by the tangential component of the applied force (4.7) for (n j ≥ 1). Then we use De Moivre's formulas (A.8) and have: α, x), γ(α, x) are the incomplete gamma functions[11]. Hence, and from formulas (4.26), (4.27) it follows by j = 1: u 1r1 (r, ϕ) = n 2 R 1,n−1,r (r) e i(n−1)ϕ + R 1,n+1,r (r) e i(r (r) e i(n−1)ϕ − R 1,n+1,r (r) e i(n+1)ϕ e −σnτ dτ = 1 σ n γ(1, σ n t) (4.46) And we have the velocity u 1 [look at (2.24)] from formulas (4.12): µnr γ(2n + 2, µ n r) 2n−1,ϕ (r) e i(2n−1)ϕ + R 2,2n+1,ϕ (r) e i(ϕ (r) e i(2n−1)ϕ − R 2,2n+1,ϕ (r) e i(2n+1)ϕ (4.76) FIG. 4 FIG 4.1. n = 1, F 1 = 1, µ 1 = 1 FIG.4.2. n = 2, F 2 = 0.5, µ 2 = 1 −irρcos(ψ−ϕ)+inj ψ dψ J nj (rρ) drdτ dρ(5.17) We substitute θ for ψ: θ = ψ -ϕ , dθ = dψ in the internal integrals of formulas (5.16), (5.17) , use De Moivre's formulas (A.8) and the Bessel function's integral representation (A.9) and have from formulas (5.16), (5.17): R j,nj −1,r (r, t) e i(nj −1)ϕ + R j,nj +1,r (r, t) e i(nj +1)ϕ(5.22) u jr2 (r, ϕ, t) = i n j 2 R j,nj −1,r (r, t) e i(nj −1)ϕ − R j,nj +1,r (r, t) e i(nj +1)ϕ (5.23)Then if n j = 0 it follows from (5.20), (5.21), (5.22), (5.23) that u jr1 (r, ϕ, t) = u jr2 (r, ϕ, t) = 0 and hence u 1 = u 2 = 0. and from formulas (5.22), (5.23) it follows for j = 1: u 1r1 (r, ϕ, t) = n 2 R 1,n−1,r (r, t) e i(n−1)ϕ + R 1,n+1,r (r, t) e i(n+1)ϕ(5.39) u 1r2 (r, ϕ, t) = i n 2 R 1,n−1,r (r, t) e i(n−1)ϕ − R 1,n+1,r (r, t) r (r, t) − R 1,n+1,r (r, t) e inϕ (5.42) u 1r (r, ϕ, t), u 1ϕ (r, ϕ, t)are the radial and tangential components of the velocity u 1 .We use the properties of the confluent hypergeometric function Φ(a, c; x) and have from formulas (5.37)− (5.42):lim u 1r1 (r, ϕ, t) = 0; lim u 1r2 (r, ϕ, t) = 0; t → 0; t → 0; lim u 1r (r, ϕ, t) = 0; lim u 1ϕ (r, ϕ, t) have velocity u 1 satisfies the initial conditions (5.1). Then we use the asymptotic properties of the confluent hypergeometric function Φ(a, c; x)[10] and from formulas (5.37) − (5.42) we have velocity u 1 satisfies conditions (1.16) ( for r → ∞). r (r, t) e i(n−1)ϕ − R ′ 1,n+1,r (r, t) e i(n+1)ϕ sinϕ + + i (n − 1)R 1,n−1,r (r, t) e i(n−1)ϕ − (n + 1)R 1,n+1,r (r, t) e i(n+1)ϕ cosϕ r (5.47) eeTeeee ϕ (r, t) e i(2n−1)ϕ + R 2,2n+1,ϕ (r, t) e i(ϕ (r, t) e i(2n−1)ϕ − R 2,2n+1,ϕ (r, t) e i(−νρ 2 (t−τ ) J 2n−1 (rρ) J 2n (rρ) dρdrdτ −νρ 2 (t−τ ) J 2n+1 (rρ) J 2n (rρ) dρdrdτ (5.75)Then we take T 2,2n,ϕ (r, τ ) ·r ′ r from formula (5.68) and with use of formula (5.70) put it in formulas (5.74), (5.75) and have:R 2,2n−1,ϕ (r, t) = − 2,2n,ϕ,ϕ (r, τ ) ∞ 0 e −νρ 2 (t−τ ) J 2n−1 (rρ) · J 2n (rρ) −νρ 2 (t−τ ) J 2n+1 (rρ) · J 2n (rρ) −νρ 2 (t−τ ) J 2n+1 (rρ) · J 2n (rρ) dρdrdτ = R 2,2n+1,ϕ,ϕ (r, t) − 2niR 2,2n+1,r (r, t) −νρ 2 (t−τ ) J 2n−1 (rρ) · J 2n (rρ) −νρ 2 (t−τ ) J 2n+1 (rρ) · J 2n (rρ) dρdrdτ (5.79)and R 2,2n−1,r (r, t) , R 2,2n+1,r (r, t) we take from formulas (5.66), (5.67). Then we use formulas (5.62), (5.64), (5.65), (5.72), (5.73) and have: ϕ,ϕ (r, t), R 2,2n+1,ϕ,ϕ (r, t) are taken from formulas (5.78), (5.79). Then we do appropriate operations and have from formulas (5.82), (5.83):u * 2r (r, ϕ, t) = − i 2 R 2,2n−1,ϕ,ϕ (r, t) + R 2,2n+1,ϕ,ϕ (r, t) ϕ,ϕ (r, t) − R 2,2n+1,ϕ,ϕ (r, t) e i2nϕ (5.86) \| u * 2 \| 2<< \| u 1 \| is the solution of the problem (1.1) − (1.6). were obtained for functions u 1 − (5.39), (5.40) or (5.41), (5.42) with calculations of the confluent hypergeometric functions [10]; u * 2 − (5.80), (5.81) or (5.85), (5.86) by using numerical integration of the triple integrals (5.78), (5.79). Each of those integrals is computed as an iterated integral. Let us consider first the calculation of the inner integrals from (5.78), (5.79): I + (r,r, t, τ ) = ∞ 0 e −νρ 2 (t−τ ) J 2n+1 (rρ) · J 2n (rρ) dρ (5.93) FIG.5.1.1. n = 1, F 1 = 1, ν = 1.5 We use asymptotical properties of confluent hypergeometric functions Φ(a, c; x)[10]and we have:Hence, we substitute integral (5.95) for I + (r, t, τ ) = A2 0 T 2,2n,ϕ,ϕ (r, τ )I + (r,r, t, τ )dr(5.97)where (0 < A 2 = 20 < ∞) and integration is really over the finite interval. For additional check let us increase value A 2 in 1.5 times and change the number of integration steps n 2 (from 201 to 301). We see that the difference in result values of integral (5.97) is within the range of required precision ǫ 2 (10 −11 ).Confluent hypergeometric functions Φ(a, c; x) were computed with precision ǫ(10 −15 ). The outer integrals in (5.78), (5.79) are the integrals over finite interval (0, t = 10). These integrals are computed with precision ǫ 3 (10 −5 ) and the number of steps of integration n 3 = 101. For additional check let us change the number of integration steps n 3 (from 101 to 201), and we see that the difference in result integral values is within the required precision ǫ 3 (10 −5 ). All integrals were computed by Simpson's method and ǫ 1 (10 −14 ) < ǫ 2 (10 −11 ) < ǫ 3 (10 −5 ).The vector field u 2 at distances r = 1, 2, 3, 5, 7 is represented by the dotted curves in top diagrams. The comparison of \| u 1 \| (dashed plots) and \| u * 2 \| (solid plots) in plane ϕ = [0, π], at distances 0 ≤ r ≤ 50 is represented in bottom diagrams. This comparison shows \| u * 2 \| << \| u 1 \| and is corresponding to the conclusion (4.89).Appendix A.The Fourier integral can be stated in the forms:The Laplace integral is usually stated in the following form:The convolution theorem A.1.If integralsDe Moivre's formulas: The discontinuous integral of Weber and Schafheitlin:Re γ 2 > 0, Re(µ + ρ) > 0. (A.12) d dy[y a · Φ(a, c; −βy)] = a · y a−1 · Φ(a + 1, c; −βy) (A.13) d dy[y c−1 · Φ(a, c; −βy)] = (c − 1) · y c−2 · Φ(a, c − 1; −βy) (A.14)Formula describing connection between the contiguous confluent hypergeometric functions: A Bertozzi, A Majda, Vorticity and Incompressible Flows. CambridgeCambridge U. PressA. Bertozzi, A. Majda, Vorticity and Incompressible Flows, Cambridge U. Press, Cambridge, 2002. P Constantin, Some open problems and research directions in the mathematical study of fluid dynamics, in Mathematics Unlimited-2001 and Beyond. BerlinSpringer VerlagP. Constantin, Some open problems and research directions in the mathematical study of fluid dynamics, in Mathematics Unlimited-2001 and Beyond, Springer Verlag, Berlin, 2001, 353-360. O Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flows. Gordon and Breach2nd editionO. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flows, (2nd edition), Gordon and Breach, 1969. Sur le Mouvement d'un Liquide Visquex Emplissent l'Espace. J Leray, Acta Math. J. 63J. Leray, Sur le Mouvement d'un Liquide Visquex Emplissent l'Espace. Acta Math. J. 63 (1934), 193-248. Turbulence and Hausdorff dimension. V Scheffer, Turbulence and the Navier-Stokes Equations. Springer Verlag565V. Scheffer, Turbulence and Hausdorff dimension, in Turbulence and the Navier-Stokes Equations. Lecture Notes in Math. No. 565, Springer Verlag, 1976, pp. 94-112. An inviscid flow with compact support in spacetime. V Scheffer, J. Geom. Analysis. 34V. Scheffer, An inviscid flow with compact support in spacetime. J. Geom. Analysis 3 No. 4 (1993), 343-401. On the nonuniqueness of weak solutions of the Euler equation. A Shnirelman, Communications on Pure and Applied Math. 50A. Shnirelman, On the nonuniqueness of weak solutions of the Euler equation. Communications on Pure and Applied Math. 50 (1997), 1260-1286. Partial regularity of suitable weak solutions of the Navier-Stokes equations. L Caffarelli, R Kohn, L Nirenberg, Communications on Pure and Applied Math. 35L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations. Communications on Pure and Applied Math. 35 (1982), 771-831. A new proof of the Caffarelli-Kohn-Nirenberg theorem. F.-H Lin, Communications on Pure and Applied Math. 51F.-H. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem. Communications on Pure and Applied Math. 51 (1998), 241-257. Higher Transcendental functions. H Bateman, A Erdelyi, Mc Graw-Hill Book Company, Inc1New York, Toronto, LondonH. Bateman, A. Erdelyi, Higher Transcendental functions. Volume 1. New York, Toronto, London. Mc Graw-Hill Book Company, Inc. 1953. Higher Transcendental functions. H Bateman, A Erdelyi, Mc Graw-Hill Book Company, Inc2New York, Toronto, LondonH. Bateman, A. Erdelyi, Higher Transcendental functions. Volume 2. New York, Toronto, London. Mc Graw-Hill Book Company, Inc. 1953. G N Watson, A treatise on the theory of Bessel functions. Cambridge Univercity Press2nd editionG.N. Watson, A treatise on the theory of Bessel functions. (2nd edition), Cambridge Univercity Press, 1944.	[]
[ "Data-driven Inferences of Agency-level Risk and Response Communication on COVID-19 through Social Media based Interactions", "Data-driven Inferences of Agency-level Risk and Response Communication on COVID-19 through Social Media based Interactions" ]	[ "Md Ashraf \nDepartment of Civil and Environmental Engineering College of Engineering and Computing\nMoss Department of Construction Management College of Engineering and Computing\nFlorida International University\n10555 West Flagler Street2900, 33174MiamiEC, FL\n", "Ph.DAhmed Student \nDepartment of Civil and Environmental Engineering College of Engineering and Computing\nMoss Department of Construction Management College of Engineering and Computing\nFlorida International University\n10555 West Flagler Street2900, 33174MiamiEC, FL\n", "Ph.DArif Mohaimin Sadri [email protected] \nSchool of Computing and Information Sciences College of Engineering and Computing\nFlorida International University\n10555 West Flagler Street2934, 33174MiamiEC, FL\n", "Ph.DM Hadi Amini \nFlorida International University\n11200 SW 8th Street, CASE 35433199MiamiFL\n", "ProfessorAssistant \nFlorida International University\n11200 SW 8th Street, CASE 35433199MiamiFL\n" ]	[ "Department of Civil and Environmental Engineering College of Engineering and Computing\nMoss Department of Construction Management College of Engineering and Computing\nFlorida International University\n10555 West Flagler Street2900, 33174MiamiEC, FL", "Department of Civil and Environmental Engineering College of Engineering and Computing\nMoss Department of Construction Management College of Engineering and Computing\nFlorida International University\n10555 West Flagler Street2900, 33174MiamiEC, FL", "School of Computing and Information Sciences College of Engineering and Computing\nFlorida International University\n10555 West Flagler Street2934, 33174MiamiEC, FL", "Florida International University\n11200 SW 8th Street, CASE 35433199MiamiFL", "Florida International University\n11200 SW 8th Street, CASE 35433199MiamiFL" ]	[]	Risk and response communication of public agencies through social media played a significant role in the emergence and spread of novel Coronavirus and such interactions were echoed in other information outlets. This study collected time-sensitive online social media data and analyzed such communication patterns from public health (WHO, CDC), emergency (FEMA), and transportation (FDOT) agencies using data-driven methods. The scope of the work includes a detailed understanding of how agencies communicate risk information through social media during a pandemic and influence community response (i.e. timing of lockdown, timing of reopening) and disease outbreak indicators (i.e. number of confirmed cases, number of deaths). The data includes Twitter interactions from different agencies (2.15K tweets per agency on average) and crowdsourced data (i.e. Worldometer) on COVID-19 cases and deaths were observed between February 21, 2020 and June 06, 2020. Several machine learning techniques such as (i.e. topic mining and sentiment ratings over time) are applied here to identify the dynamics of emergent topics during this unprecedented time. Temporal infographics of the results captured the agencylevels variations over time in circulating information about the importance of face covering, home quarantine, social distancing and contact tracing. In addition, agencies showed differences in their discussions about community transmission, lack of personal protective equipment, testing and medical supplies, use of tobacco, vaccine, mental health issues, hospitalization, hurricane season, airports, construction work among others. Findings could support more efficient transfer of risk and response information as communities shift to new normal as well as in future pandemics.	10.5055/jem.0589	[ "https://arxiv.org/pdf/2008.03866v1.pdf" ]	221,090,784	2008.03866	d87618a08f9b0ea02baf480bb93b854bb8b313d0	Data-driven Inferences of Agency-level Risk and Response Communication on COVID-19 through Social Media based Interactions Md Ashraf Department of Civil and Environmental Engineering College of Engineering and Computing Moss Department of Construction Management College of Engineering and Computing Florida International University 10555 West Flagler Street2900, 33174MiamiEC, FL Ph.DAhmed Student Department of Civil and Environmental Engineering College of Engineering and Computing Moss Department of Construction Management College of Engineering and Computing Florida International University 10555 West Flagler Street2900, 33174MiamiEC, FL Ph.DArif Mohaimin Sadri [email protected] School of Computing and Information Sciences College of Engineering and Computing Florida International University 10555 West Flagler Street2934, 33174MiamiEC, FL Ph.DM Hadi Amini Florida International University 11200 SW 8th Street, CASE 35433199MiamiFL ProfessorAssistant Florida International University 11200 SW 8th Street, CASE 35433199MiamiFL Data-driven Inferences of Agency-level Risk and Response Communication on COVID-19 through Social Media based Interactions (Corresponding Author) Word Count: 01 tables + 5 figures + 6,200 words = 6,450 word equivalents 2COVID-19TwitterAgencyInteractionsRiskResponseSocial Media Risk and response communication of public agencies through social media played a significant role in the emergence and spread of novel Coronavirus and such interactions were echoed in other information outlets. This study collected time-sensitive online social media data and analyzed such communication patterns from public health (WHO, CDC), emergency (FEMA), and transportation (FDOT) agencies using data-driven methods. The scope of the work includes a detailed understanding of how agencies communicate risk information through social media during a pandemic and influence community response (i.e. timing of lockdown, timing of reopening) and disease outbreak indicators (i.e. number of confirmed cases, number of deaths). The data includes Twitter interactions from different agencies (2.15K tweets per agency on average) and crowdsourced data (i.e. Worldometer) on COVID-19 cases and deaths were observed between February 21, 2020 and June 06, 2020. Several machine learning techniques such as (i.e. topic mining and sentiment ratings over time) are applied here to identify the dynamics of emergent topics during this unprecedented time. Temporal infographics of the results captured the agencylevels variations over time in circulating information about the importance of face covering, home quarantine, social distancing and contact tracing. In addition, agencies showed differences in their discussions about community transmission, lack of personal protective equipment, testing and medical supplies, use of tobacco, vaccine, mental health issues, hospitalization, hurricane season, airports, construction work among others. Findings could support more efficient transfer of risk and response information as communities shift to new normal as well as in future pandemics. INTRODUCTION AND MOTIVATION No person or place is safe from natural or man-made hazards or losses resulting from an extreme event. Infectious disease can significantly add to the far-reaching adverse consequences and sufferings of countries and communities. One way to reduce such impacts can be done by investing in communication networks similar to the resilience observed in civil infrastructure systems (1). However, advancing such resilience capacities is not that simple since often times it is quite challenging to identify low cost alternatives. Communicating risks, connecting community networks (2), and promoting resilience culture can aid local communities in making progress to increase their resilience. As such, community coalitions of elected representatives from the public and private sectors, with ties and funding from federal and state governments, and with input from local residents, are becoming very relevant. Such coalitions may be reinforced to determine the exposure and vulnerability of the community to hazard, to inform and communicate risk, and to analyze and extend the capacity of the community to manage such risks. A truly robust coalition will have a clear leadership and governance system at the core, and people with sufficient time, expertise, and commitment required to establish and sustain relationships among all the community stakeholders (3). Public agencies engagement in risk communication can lead to more effective decisionmaking and enhanced public feedback to the regulatory process. Risk and response communication of public agencies through social media played a significant role in the emergence and spread of novel Coronavirus (COVID- 19) and such interactions were echoed in other information outlets. The primary goal of this study is to mine and analyze time-sensitive agency-level social media data (rich spatio-temporal data) as well as crowd-sourced data (i.e. Worldometer). The scope of the work includes a detailed understanding of how agencies communicate risk information through social media during a pandemic and influence community response (i.e. timing of lockdown, timing of reopening) and disease outbreak indicators (i.e. number of confirmed cases, number of deaths). The specific aims are twofold: (1) to document how public agencies interact and communicate health risk information through their online social networks during a major disease outbreak; (2) to examine how online social networks influence the current COVID-19 pandemic situation in terms of the change in daily number of cases and deaths in United States. To achieve the goal and aims, the study have utilized social media interactions of four major public agencies e.g. World Health Organization (WHO), Center for Disease Control and Prevention (CDC), Federal Emergency Management Agency (FEMA) and Florida Department of Transportation (FDOT) with crowd-sourced information on COVID-19. The data includes Twitter interactions from these agencies (2.15K tweets per agency on average) and crowdsourced information about COVID-19 cases and deaths between February 21, 2020 and June 06, 2020. Several machine learning techniques such as (i.e. topic mining and sentiment ratings over time) are applied here to identify the dynamics of emergent topics during this unprecedented time. The contribution to the knowledge gaps based on empirical literature are listed below:  It has developed methods to advance our understanding of how public agencies influence online, while communicating health risks and interacting in their respective communities as the disease continues to spread;  It informed the literature on how information is exchanged among people who are socially connected online and exposed to health risk in such outbreaks of disease;  This study used novel machine learning techniques to identify the dynamics of risk communication strategies during this unprecedented time ;  It generated temporal infographics of the results captured the agency-levels variations over time in circulating information LITERATURE REVIEW The transmutability of the latest coronavirus disease in 2019 was tracked in Zhejiang, China accounting for transmissions from imported cases. While Zhejiang is one of the worst 21 affected provinces, an interruption in the transmission of the disease (i.e. an instant reproduction number < 1) was observed in early / mid-February following an early social-distancing response to the outbreak (4). Zhang et al. developed data-driven, susceptible-exposed-infectious-quarantinerecovered (SEIQR) models to simulate the outbreak of Coronavirus with measures of social distancing and epicenter lockdown. Population migration data combined with officially recorded data were used to estimate model parameters and then to measure the daily exported infected individuals by estimating the daily infected ratio and the daily susceptible population size (5). In 2003, public health interventions were crucial in the prevention of the SARS epidemic. Community containment requires steps ranging from increasing social distancing to city-wide quarantine. Whether these steps (isolation, quarantine, social distancing) will be appropriate to monitor 2019-nCoV depends on resolving some unanswered issues (6). Many countries only seek to achieve social distancing and hygiene measures when widespread transmission is apparent. It gives the virus a number of weeks to propagate with a higher basic reproductive number than if it had been in place before transmission was observed or widespread. Hence, preventive, low cost, enhanced hygiene and social distance in the sense of imminent population transmission of the novel coronavirus COVID-19 should be considered (7). Social media is considered as an emerging platform for efficient crisis communication in recent literature (8; 9). Traditional media is primarily intended for one-way communication whereas social media allows two-way communication, hence social media platforms are exclusively different (10). Sadri et al. explained the critical role of social media during crisis by facilitating communication and information dissemination to both evacuee and non-evacuee during hurricane Sandy (11). Zhang et al. envisages intelligent public disaster information and alert based on social media, which has three functions: (1) effectively and efficiently collecting disaster situational awareness information; (2) promoting self-organized assistance activities; and (3) enabling emergency management agencies to hear from the public. The results of this analysis highlight the importance of such research fields; (1) a fine-grained social media catastrophe ontology with semantic interoperability, (2) trend knowledge network pattern and emerging prominent users, (3) fine-grained societal impact assessment due to infrastructure failures, and (4) best practices for social media use during disasters (12). Austin et al. examines how audiences seek social and traditional media information, and what factors influence media usage during crises. Using the model of social-mediated crisis communication (SMCC), a review of crisis information and sources reveals that audiences use social media for insider information and check-in with family / friends during crises, and use traditional media for educational purposes. Convenience, interaction, and personal feedback encourage the use of conventional and social media; usage of both discourages overload. Humor and beliefs towards social media uses discourage the use of social media while legitimacy promotes the use of conventional media. The findings stressed the importance of the influence of third parties in crisis communication and the need to use traditional and social media in crisis response (13). Freimuth et al. describes the design, implementation, and assessment of a risk communication simulation during the first hours of a pandemic. The simulation design was focused on the communication of crisis and emergency risks principles upheld by the Centers for Disease Control and Prevention (CDC), as well as the author's collective experience. Several local health district risk communicators in Georgia responded to a scenario where after returning from an international conference, every community in the state had teenagers infected with avian flu. The evaluation revealed that, under the time pressures of a realistic and stressful event, local risk communicators had much greater difficulty following the principles of risk communication than they did in a tabletop workout. Strengths and weaknesses of local risk communicators' performance are identified in addition to the lessons learned on designing and implementing a simulation for risk management (14). Palen et al. (15) studied the rapid growth of social media in a number of disaster situations, exploring issues such as citizen engagement, community-oriented computing, distributed problem solving, and digital volunteerism as modes of socio-technical innovation, as well as issues of situational knowledge and truthfulness as opportunities and challenges emerging from the social media data deluge. The chapter also discusses the study that looks at integrating social media technologies and data into current emergency response work. Reflecting on the decade-old area of science, the authors warned of the danger that all "crisis" encounters can fail unintentionally without differentiation, which appears to happen because social media networks cross-cut all emergency situations. Through an effort to isolate what social media adds new, there is a tendency to fail to recognize how non-technological influences on cultural socio-behavioral scales greatly affect the usage of social media itself. Misinformation spreading in social media are also becoming an evolving concern. Monahan et al. mentioned that mass media play an important but often misunderstood role in the events of a catastrophe. Research has repeatedly shown that disaster-related media reporting appears to be riddled with disinformation and promotes misconceptions about race, social status, aggression and crime. Studies have found that powerful media campaigns can support prevention efforts, strengthen early warning systems, facilitate orderly and prompt evacuation procedures and help bring communities together in times of upheaval. Authors review research on the relationship between media and disaster to highlight the many ways media can positively impact disaster preparation and recovery, while also highlighting the many issues associated with disaster reporting. Future directions for media-disaster research are being discussed along with ways in which media staff and emergency response professionals can handle the media-disaster relationship more efficiently before, during and after emergency incidents (16). Battur et al. detected twitter bot, which is a software that sends fake tweets automatically to users. Detecting bots is necessary to identify the fake users and to protect the genuine users from misinformation and malicious intents. The study proposes an approach to detect the twitter bots using several machine learning algorithms; such as Decision Tree, Multinomial Naïve Bayes, Random Forest and Bag of Words. The algorithm with highest accuracy (Bag of Words) is used to test real time data (17). Asr et al. stated that misinformation detection at the level of full news articles is a text classification problem and reliably labeled data in this domain is rare. Previous work relied on news headlines, microblogs, tweets and articles collected from so-called "reputable" and "suspicious" websites and labeled accordingly. Authors leveraged fact-checking websites to collect individually labeled news articles with regard to the veracity of their content (18). Huang et al. proposed a systematic meta-analysis (SMA) looked at 38 studies involving real responses to hurricane warnings and 11 studies with expected responses to hypothetical hurricane performed since 1991 (19). METHODOLOGY The focus of this study is to examine how public agencies risk assessments, risk averting behaviors, and crisis communication patterns in online social network influence the pandemic situation. To reveal the interaction patterns, several machine learning algorithms have used in this study. At first, sentiment analysis over time for major public organizations have performed. Then Dynamic Topic Models (based on Topic Model theory) was applied to identify the topics over the same timeline which causes the change of sentiments. Sentiment analysis is a method of Natural Language Processing (NLP) task at many levels of granularity. Starting from a document level classification task, it has been handled at the sentence level and more recently at the phrase level (20). It is well recognized that twitter usergenerated content with rich sentiment information should be utilized for many applications such as search engines and other information systems. While tweet level sentiment analysis results indeed provide very useful information, the overall or general sentiment tendency towards topics are more appealing in some scenarios. For example, people are curious about how others feel about Apple's new product, "iPhone11," and it will offer great convenience for them if major opinions are collected from massive tweets (21). A topic model is a statistical model to explore the abstract "topics" that happen in an assembly of information in machine learning and natural language processing. This was first explored by David Blei according to the most common topic model named Latent Dirichlet Allocation (LDA). The instinct behind LDA is that the set of texts reveal numerous topics. In topic models, first, the algorithm chose a topic, then sample a set of words from the given topic. Clusters of similar terms are the themes or topics created by topic modelling technology. A topic model is a recurrently performed text-mining tool for the discovery of hidden semantic structures in a text body. Topic models can help us to organize and provide insights into understanding large collections of unstructured text bodies (22). The dynamic topic model (DTM) includes a group of probabilistic time series model, which is used to observe the time evolution of topics in huge document collections. This group of models was proposed by David Blei and John Lafferty and is an extension to Latent Dirichlet Allocation (LDA) that can handle chronological documents. In LDA, both the order and the words appear in a document, whereas words are still assumed to be interchangeable, but in a dynamic topic model, the order of the documents plays a key role. The method is to use state-space models to represent the topics on the natural parameters of the multinomial distributions. To perform approximate posterior inference over the latent topics, variation in approximations based on Kalman filters and nonparametric wavelet regression are developed. In addition to providing sequential, quantitative, and predictive models, DTM provides a qualitative window into the contents of a large document collection. It is assumed that the data is divided by time slice in a dynamic topic model, for example, by month. It is modeled as each slice's documents with a K component subject model, where slice t-related topics evolve from slice t-1-related topics. Let βt denote the V -vector of natural parameters for topic k in slice t for a K-component model with V terms. A multinomial distribution is usually represented by its mean parameterization. If we denote the mean parameter of a Vdimensional multinomial by π, the mapping βi = log (πi/πV) of the ith element of the natural parameter is given. Dirichlet distributions are used in typical language modeling applications to model uncertainty over word distributions. The Dirichlet, however, is not conducive to sequential modeling. Alternatively, we chain the natural parameters of each βt subject into a state-space model that evolves with Gaussian noise; equation (2) shows the simplest version of such a model. \| −1~( −1 , 2 )(1) In LDA, the document-specific topic proportions θ are drawn from a Dirichlet distribution. In the dynamic topic model, a logistic normal is used with mean α to express uncertainty over proportions. The sequential structure between models is again captured with a simple dynamic model is expressed by equation (3). \| −1~( −1 , 2 )(2) For simplicity, it did not model the dynamics of topic correlation, as it was done for static models by Blei and Lafferty (23). By chaining together topics and topic proportion distributions, it has sequentially tied a collection of topic models. DATA SOURCE AND DATA COLLECTION Traditional datasets have limited capacity to adequately capture user risk communication strategies and analyze user concerns with such details and coverage. As such, social media datasets, enriched with user activity information, will be useful to capture user sentiments and concerns in real time and help early detection of the people exposed to health-risks in the vulnerable communities. Twitter, in particular, provides unique features to release their data through their Application Programming Interface (API) and make it publicly available which could then be combined with other complementary information (e.g. crowdsourced data) specially over the timeline of COVID-19 crisis. For this study, Twitter Search APIs is used to collect and store public agencies crisis interactions through social media outlets. Natural language processing and machine learning techniques are adopted here to extract user concerns, response, and needs over time. Besides, Worldometer data (crowdsource data) is used for extracting the daily number of cases and deaths due to COVID-19 for United States. The goal of the study is to reveal different agencies perspective during COVID-19 that exists in social media interactions. To achieve this goal, the first key data source we considered is twitter data. We were particularly interested in the tweets generated from major public health agencies (WHO and CDC), disaster management (FEMA) and transportation agencies (FDOT). Hence, we collected historical tweets during the COVID-19 pandemic of these four major agencies to perform the analysis. The summary of the collected twitter data is given below, which shows that WHO was the most active on Twitter than any other organization. All the twitter data were collected from 02.21.2020 to 06.06.2020 which is around three and half months. ANALYSES AND RESULTS To reveal the interaction patterns of the public organizations, several machine learning algorithms; e.g. sentiment analysis and topic frequency over time are applied here. The graphical representations and the result interpretations of the tweets from four public organizations are listed below. WHO Interactions To understand the dynamics of communication pattern, 15 optimum number of topics were identified from static topic model analysis of WHO tweets. In Figure 1a, the topic frequencies are plotted along with the time and in Figure 1b CDC Interactions From CDC tweets, 8 optimum number of topics are plotted over the three and half month timeline in Figure 2a. CDC put importance on following their guideline consistently, increase in number of cases in whole March, hospitalization rates at the end of April which also found frequent after one month, pandemic stress in May rather than from the beginning, prevention of spread during mid-April, contact tracing at the beginning of April and risk of older people in March and May. After comparing Figure 2a and Figure 2b, positive spikes in tweet sentiment have found when CDC put emphasize on following their guideline from the beginning and mostly on May, importance of contact tracing at the beginning of April, reducing the spread of the virus as well as the pandemic stress. Besides, negative sentiments are observed when elderly people's risk, increased number of cases and hospitalization rates became more frequent in twitter. By interpreting Figure 3a and 3b together, the extreme negative sentiments are observed when the need of medical supplies, lack of food and the emergence of upcoming hurricane season along with the increased spreading of the virus are discussed. Then, positive sentiments have found while importance of COVID-19 testing, availability of responses, fund and spreading of critical information got more importance. FDOT Interactions FDOT tweets showed 8 optimum topics which are envisaged over time in Figure 4a. FDOT emphasized on working together to be successful over COVID-19 from May, travel and COVID-19 related information in March, use of airports in the beginning of April, roadway construction work consistently, importance of social distancing from April and the spreading of coronavirus in Floridians consistently. By comparing Figure 4a and 4b, spikes in positive sentiments have found when the importance of social distancing, effectiveness of working together throughout the pandemic and restricted roadway construction work have discussed. In the other hand, extreme negative sentiments have observed while alarming spread of COVID-19, increase in number of cases among Floridians and use of airport topics became more frequent in Twitter. DISCUSSION OF FINDINGS To identify the specific interaction patters along with the sentiments which caused the changes (increase and decrease) in the number of cases and deaths in US, the topic dynamics and sentiments over time are represented along with the number of cases and deaths of COVID-19 from Worldometer data (24) in Figure 5a and Figure 5b as following. From both Figure 5a and 5b, it is obvious that the number of cases and deaths started to increase rapidly from 20 March and hit the first pick (around 40,000 cases and 2,500 deaths per day) around 10 April. In this time duration, WHO discussed more about the lack of social measures, responses, requirement of adequate number of nurses, importance of wearing face masks; CDC emphasized on need of contact tracing, failure of preventing the spread and suggesting to follow their guidelines; FEMA mentioned about shortage of medical supplies, responses and need of critical information; and FDOT stated about the use of airports, continuous roadway construction works and lack of social distancing. All of these topics contribute to the increase in number of cases and deaths. If the entities put importance on these topics at the beginning of the pandemic (January 2020), then the number of cases and deaths could be much lower. Then, the number of cases and deaths remained nearly constant from 11 April to 10 May. In this time, importance of social distancing, increase in funding, response and support from government, essence of pandemic stress management, new measures to prevent the spread and the requirement of PPE have started becoming into the discussion focus. The result of emphasizing on these issues is obvious from the last portion (11 May to 6 June) of the graphs (Figure 5a and 5b) where both the number of cases and deaths decreased. In this timeline, increased response from people and government, awareness about the lowering hospitalization rates, following health guidance, understanding the risk of elderly people, increased medical supply, and working together also helped to reduce the number of cases and deaths which results in flattening the curve. The text-based infographics showed in previous sections have revealed different interaction patterns along with the sentiments from the four major public agencies during the pandemic. These interactions were actually generated from most frequently appeared words in different topics form these entities which are listed below in different timeline. The entire time is divided into four sub-period based on the lockdown (13 March) This study identified specific topics in Twitter which influenced the change in number of cases and deaths across US. Findings could help policy makers to take better preparation for the next steps of the ongoing pandemic as well as in future pandemic situations. CONCLUSIONS ANF FUTURE RECOMMENDATIONS Social media is considered as an effective information dissemination platform and showed prevalence in recent times. The influence of information received through social media on public behavior and decision-making is remarkable. However, spread of misinformation and information overload may impede public risk perception and personal protective actions. Exploring the origins of such information in times pandemic are inevitable, as such, Twitter interactions from agencies such as WHO, CDC, FEMA and FDOT were observed in this study in the emergence and outbreak of the novel coronavirus (COVID-19). A total of 8,600 tweets were analyzed from these four major public organizations. Several machine learning techniques such as (i.e. topic mining and sentiment ratings over time) are applied here to identify the dynamics of emergent topics during this unprecedented time. WHO was found to be the most active in social media during the first three and half month of COVID-19 pandemic as compared to the other agencies. Results also indicate that most agencies were less vocal on the importance of face covering except for WHO in April which could have brought into attention earlier. Lack of healthcare professional, medical supplies, contact tracing, and social distancing may have contributed to the sudden increase in both number of cases and deaths in US. However, the importance of virus testing was emphasized more by FEMA along with the concerns about the upcoming hurricane season. CDC focused more on pandemic stress management and monitoring the hospitalization rates. FDOT discussed more about COVID situation in Florida, travel information, use of airports and construction during the pandemic. Finally, higher levels of social media interactions from agencies on social distancing and vulnerability of older people seemed to have contributed in reducing number of cases and deaths in US. The findings of this research can support public health, emergency management, transportation and other agencies more efficient transfer of risk and response information as communities shift to new normal as well as in future pandemics. This can be done by identifying more effective information dissemination strategies for diverse user groups based on their social network characteristics, activities, and interactions in response to similar public health hazards. This study identified specific Twitter interaction topics from major agencies which may have influenced the community response and disease outbreak indicators of the pandemic. The methodologies, and implications of this research can be transferred in designing effective intervention policies to other natural and man-made disaster contexts in which public health risks become major concerns. The study considered up to 3,200 tweets from the four agencies due to the limitation set by Twitter on historical tweets, future studies should consider more agencies and more recent timeline.  World Health Organization (WHO) Tweets-4,434 tweets  Centers for Disease Control and Prevention (CDC) Tweets-868 tweets  Federal Emergency Management Agency (FEMA) Tweets-1,996 tweets  Florida Department of Transportation (FDOT) Tweets-1,262 tweets , average sentiment score of tweets are plotted over the same timestamp. WHO emphasized on community transmission and the need of vaccine in March; lack of nurses was discussed(Figure 1a) at the beginning of April; shortage of PPE, need of staying home and importance of wearing masks became prominent from late April and in May; all of these should be emphasized from January (beginning of the pandemic). Requirement of Government support got notified in the beginning of May. These late responses show the clear deficiency in preparedness to handle the pandemic situation. Also, the number of cases (COVID-19 in countries topic) found increasing after 1.5 months' time interval. Several other topics such as importance of health concerns, responses in different countries, social measures, and use of tobacco got attention also from WHO in twitter.By comparingFigure 1aandFigure 1b(rolling mean or 1-day running average), it can be said that the tweet sentiment went to most negative while WHO emphasized on alarming number of cases of COVID-19, lack of peoples' responses, shortage of PPE and negative impact of using tobacco. Then, positive sentiments have found when WHO discussed about need of government support, responses from people, development of vaccine, importance of home quarantine and use of tobacco to develop the vaccine. Hence, the use of tobacco considered as negative topic in the beginning, but later it turned into positive topic. Figure 1a . 1aTopic Frequency over time from WHO Tweets Figure 1b. Sentiment Analysis over time from WHO Tweets Figure 2a. Topic Frequency over time from CDC Tweets Figure 2b. Sentiment Analysis over time from CDC Tweets FEMA Interactions From around 2000 tweets of FEMA, 10 optimum number of topics frequency have visualized in Figure 3a. FEMA discussed about the importance of critical information in April, need of COVID-19 responses in mid-March and also after one month (mid-April), spreading of the virus at the end of April and need of supplies and food in April and May. Hurricane season also got attention of FEMA during the pandemic. Besides, FEMA emphasized on the importance of COVID-19 testing from April to May which is not mentioned by WHO and CDC. Also, lack of medical supply got notified from end of the March, April and May which shows after each month medical supply in US is needed. Figure 3a . 3aTopic Frequency over time from FEMA Tweets Figure 3b. Sentiment Analysis over time from FEMA Tweets Figure 4a . 4aTopic Frequency over time from FDOT Tweets Figure 4b. Sentiment Analysis over time from FDOT Tweets and reopening (beginning of May) time in US-After analyzing all the results and infographics, the noteworthy findings are listed in below- All the organizations discussed about the increased number of COVID-19 cases, need of responses from both people and government and spreading of the virus.  The unique topics WHO discussed are the importance of PPE and wearing masks, home quarantine, need of nurses and vaccine, emergence of community transmission and the both negative and positive use of tobacco.  CDC specifically mentioned about the importance of contact tracing, pandemic stress management and the negative impact of increased hospitalization rates.  FEMA is the only organization who emphasized on the importance of COVID-19 testing, lack of medical supplies and the emergence of upcoming hurricane season along with the pandemic.  FDOT put importance on social distancing, negative impact of using airports and roadway construction works. Figure 5a .Figure 5b . 5a5bOrganizations' Topic Dynamics with daily new Cases and Deaths in USA Organizations' Sentiment with daily new Cases and Deaths in USA Table 1 . 1Most Common Words from Organizations in specific timelines  Only WHO discussed about the importance of wearing face mask on April and FEMA emphasized on COVID-19 testing on May which should be focused earlier and contributed to the initial increase of the number of cases and deaths in US.  Before the lockdown time (Table 1), organizations discussed about community transmission, spreading of the virus, emergency management and health concerns along with the need of doctors.  Just after the lockdown (20 March to 10 April), organizations emphasized on shortage of nurses, medical supplies, social distancing and contact tracing, which contributed to the Sudden jump in cases and deaths.  During the lockdown (11 April-10 May), public agencies highlighted about the importance of social distancing, pandemic stress management, PPE requirement and the need of Government support to manage the pandemic.  In the reopening phase (11 May to 06 June), organizations focused on managing hospitalization rate, risk of older people, adequate medical supplies and following the helath guidance, which eventually helped to reduce the number of cases and deaths.Organizations 21 Feb-19 March 20 March-10 April 11 April-10 May 11 May-06 June Before the lockdown Beginning of the lockdown During the lockdown Re-opening phase WHO global, media, support, minister, personal, covid, increase, protect, health, safe, measures, countries, solidarity, community, transmission, cases, stop, now, spread, safehands, response, prevent coronavirus, protective, medical, equipment, nurses, midwives, violence, economic, social, support, stay, home, confirmed, masks world, international, response, fight, prime, emergency, advice, committee, tools, europe, physical, active, information, virus, infection health, outbreak, ebola, human, public, tobacco, use, healthyathome, vaccines, solidarity, pandemic CDC health, US cases, doctor, people, risk, hands, coronavirus, CDC, help facebook, coca, spread, older, travel, home, learn, CDC covid, age, protect, home, help, gas, spread, states, report cases, hospitalization rates, covidview, pandemic, stress, slow, guidance, learn, help, protect FEMA information, critical, need, emergency, firefighters, support, disaster, emotional, risk, emergency management, prevent, state, local, offcials supplies, need, equipment, social, spread, medical, help, federal, guard, hospital, states, national people, working, right, team, care, response, slodiers, response, medical, supplies, covid, million, funding local, state, million, grants, facts, distancing, hurricane, season, tropical, storm, testing, health, food, national, members FDOT project, order, florida, road, covid, public, staff social, distancing, executive, order, issued, travel, follow, visit, emergency, service, home, floridians information, win, together, survey, stay, construction, work, traffic, driving, florida, essential, safe, practice checkpoints, coordination, traveler, airports, enforcement, app, download, state, transportation, searching ACKNOWLEDGMENTSThe authors are grateful to National Science Foundation for the Rapid Response Research grant IIS-2027360 to support the research presented in this paper. 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[ "Singularities of Gaussian Random Maps into the Plane", "Singularities of Gaussian Random Maps into the Plane" ]	[ "Mishal Assif " ]	[]	[]	We compute the expected value of various quantities related to the biparametric singularities of a pair of smooth centered Gaussian random fields on an n-dimensional compact manifold, such as the lengths of the critical curves and contours of a fixed index and the number of cusps. We obtain certain expressions under no particular assumptions other than smoothness of the two fields, but more explicit formulae are derived under varying levels of additional constraints such as the two random fields being i.i.d, stationary, isotropic etc.	null	[ "https://arxiv.org/pdf/2202.08242v1.pdf" ]	246,866,977	2202.08242	6c091c7c00109a7cd68869c3c56516c3ea9394f8	Singularities of Gaussian Random Maps into the Plane February 17, 2022 Mishal Assif Singularities of Gaussian Random Maps into the Plane February 17, 2022 We compute the expected value of various quantities related to the biparametric singularities of a pair of smooth centered Gaussian random fields on an n-dimensional compact manifold, such as the lengths of the critical curves and contours of a fixed index and the number of cusps. We obtain certain expressions under no particular assumptions other than smoothness of the two fields, but more explicit formulae are derived under varying levels of additional constraints such as the two random fields being i.i.d, stationary, isotropic etc. Introduction Let N be an n-dimensional compact Riemannian manifold (n 2). Given a smooth function N p → h(p) = (f(p), g(p)) ∈ R 2 , a point p ∈ N is called a critical point if the derivative Dh(p) : T x N → R 2 at p is not surjective and the set of all critical points is called the critical curve of h. The critical point is an example of a singularity of the smooth function h, and the objective of this paper is to study the expected value of various quantities of interest associated with such singularities when the components of h are Gaussian random fields (GRFs). The expected number of critical points of a single GRF has been the subject of many papers e.g. [CS18, AAC13, AA13, ATW10, BBKS86,LH60], having applications in a wide variety of domains. The singularities of a pair of functions, being a two dimensional analogue of such one dimensional singularities, naturally warrant study. However, our main motivation to study these quantities come from biparametric persistent homology of smooth functions. Persistent homology (PH) is a topological data analysis technique used to extract robust topological features from data. The key idea in single parameter PH is that if X is a topological space and f : X → R is a nice enough function on X, one can encode the change in homologies of the sublevel sets {f a} as the single parameter a varies along the real line in the form of a simple planar diagram called the persistence diagram of f. If X is a smooth manifold and f is a Morse function, it is well known from Morse theory that the critical points of f are precisely where the homology of its sublevel sets change, and hence the behavior of critical points of f determine that of the persistence diagram of f. In biparametric persistence, one has a pair of functions h = (f, g) : X → R 2 and one tries to track the change in homologies of the sublevel sets {f a, g b} as the two parameters (a, b) vary in the plane. When X is a smooth manifold and the function h is smooth, biparametric persistence can be understood from the perspective of Whitney theory, analogous to the Morse theoretic perspective of single parameter PH, and there is a growing amount of literature regarding this [CEF19,BK21,BC21,APKB21]. We give a brief description of this Whitney theoretic perspective on biparametric persistence here, the details of which can be found in [APKB21]. For a generic function h, the critical curve is a 1-dimensional embedded submanifold of N, or a disjoint finite union of smooth circles. The image of the critical curve under the map h is called the visible contour. The visible contour will also be 1 arXiv:2202.08242v1 [math.PR] 16 Feb 2022 a finite union of closed curves in R 2 , although these curves may intersect each other and will be smooth only outside a finite number of points called cusps. The preimage of a cusp point can be characterized as a second order singularity of h, that is, a point of N where the derivatives of h upto order two satisfy certain conditions. In comparison, critical points are first order singularities of h since their description only involves conditions on derivatives of h upto order one. Figure 1b shows an example of a visible contour where the critical curve consists of a single circle. The visible contour is thus a single closed loop in R 2 , which has one point of self intersection and two cusp points where it loses smoothness. At the image of the critical points of the component functions f and g, the tangents to the visible contour are vertical and horizontal respectively. These points thus split the visible contour into segments with positive or negative slopes. The segments with negative slope are called Pareto segments of the visible contour. The curves indicated in black in Figure 1c are the Pareto segments of the visible contour shown in Figure 1b. At the image of critical points of f and g, we attach vertical and horizontal rays extending upward and rightward respectively, and call them the extension rays. The extension rays are the curves marked in blue in Figure 1c. The union of the Pareto segments of the visible contour and the extension rays is called the Pareto grid of h. The grid formed by the black and blue curves in Figure 1c form the Pareto grid of the visible contour in Figure 1b. The Pareto grid has certain additional points of non-smoothness where a Pareto segment attaches to an extension ray in a non-smooth manner and these corner points are called pseudocusps. One can see that there are four non-smooth points on the Pareto grid in Figure 1c indicated in red, and two of these are cusps of the visible contour, while the other two are pseudocusps. The cusps and pseudocusps split the Pareto grid into multiple smooth pieces. These smooth pieces of the Pareto grid are the biparametric analogues of critical values in single parameter persistent homology. Homology generators are born or killed as the parameter value (a, b) ∈ R 2 crosses these smooth pieces. One can define an index for each of these pieces determining the dimension of the cell attached at a crossing as well. The objective of this article is to study some statistical properties of these biparametric singularities of a GRF h. Given the Whitney theoretic description of biparametric persistence, understanding the properties of these singularities give us an idea about the complexity of the biparametric persistence of GRFs into the plane. We will mainly focus on computing the expected lengths of the critical curve and visible contour of each index. Unlike critical points, the cusp points are second order singularities which means their characterization involves derivatives upto order two. If we were to use the Kac-Rice formula [AT07, Theorem 12.1.1] to compute the expected number of cusps, the computations will involve derivatives of h upto order three making them really cumbersome. Hence, these computations are left to a later paper. However, the expected number of pseudocusps can be computed using standard techniques as they are critical points of single variable GRFs and these computations are done in the article. We derive expressions for these expectations for general pairs of GRFs, assuming only that they are smooth and centered (mean zero) and some additional mild technical assumptions. The expressions yield neater formulae as more assumptions such as the pair of GRFs being identical and independent, stationary or isotropic are imposed. However, all the general expressions we find here are written as expectations of functions of certain Gaussian random vectors and Gaussian random matrices. The additional assumptions on the GRFs make the distributions of these random vectors and matrices nicer, such as being independent, rotationally invariant etc. yielding better closed form solutions. The paper is structured as follows. The remaining part of section 1 introduces the notations and assumptions used in the article. In section 2, we derive expressions for the expected length of the critical curve, visible contour and Pareto segments, each of a fixed index. This section doesn't assume much about the GRF h beyond begin smooth and centered. In section 3, we simplify the expressions obtained in the previous section under the additional assumption that the component functions f and g are independent and identically distributed GRFs. We also show concrete examples of these computations in two settings, random bandlimited functions into the plane and random planar projections of the standard embedded torus in R 3 . In section 4, we obtain neater formulae under the assumptions that the manifold N is the n-sphere § n and the component GRFs f and g are isotropic and stationary. Finally, in section 5, we derive the expected number of pseudocusps of a fixed index. Notation We denote by N an n-dimensional compact orientable Riemannian manifold endowed with a Riemannian metric G. The corresponding volume form on N will be denoted by d N V. We will deal with certain one dimensional compact submanifolds on N, and the volume form endowed by the induced Riemannian metric on them will be denoted by d N l. S 1 will denote the unit circle endowed with the standard Riemannian metric. We will also need to look at certain one dimensional compact submanifolds on the product space S 1 × N endowed with the product Riemannian metric and the volume form endowed by the induced Riemannian metric on them will be denoted by d S 1 ×N l. Some of the computations in the article will be done in coordinate charts on N and if v ∈ R n is the coordinate representation of a tangent vector on N, v G will denote its norm in the Riemannian metric while v will denote its usual Euclidean norm. J r (N, R 2 ) is the r-jet space of N to R 2 and J r p (N, R 2 ) is the r-jet space at p ∈ N, which are both Euclidean spaces since the codomain R 2 is Euclidean. We denote by S k the corank k submanifold of J 1 (N, R 2 ) consisting of those jets with rank 2 − k. Given a smooth function h : N → R 2 , we denote by j r h : N → J r (N, R 2 ) its r-jet function (see [GG12] for details). N × Ω (p, ω) → h(p, ω) = (f(p, ω), g(p, ω)) ∈ R 2 will denote a Gaussian random field (GRF) on N. To make notations less cumbersome, we will avoid including ω and refer to h(p) freely as a GRF. The support of h is defined as supp (h) = {h ∈ C ∞ (N, R 2 ) such that P (h ∈ U) > 0 for all neighborhoods U ofh}. (see [S + 20] for more details). If h is smooth almost surely, its derivatives are also GRFs and so is j r h. If W is a submanifold of J r (N, R 2 ), then j r h W will denote that the function j r h intersects W transversally. Assumptions There are three standing assumptions throughout the article. Assumption 1.1. The GRF h is centered, that is, E h(p) = 0 for all p ∈ N. Our techniques and proofs work identically in the non-centered case as well, but the final formulae obtained are not very clean and exact computation is not possible when the mean is not zero. Assumption 1.2. The GRF h is smooth on N almost surely. This is not too strict an assumption, as there are conditions on the GRF that ensures this happens, such as [AT07, Theorem 11.3.4]. Assumption 1.3. The support of the 2-jet supp j 2 h(p) = J 2 p (N, R 2 ) for all p ∈ N, that is, the jointly Gaussian random vector f(p), g(p), ∇f(p), ∇g(p), ∇ 2 f(p), ∇ 2 g(p) is non degenerate. Smoothness of h isn't the only regularity condition we need for our computations; we will require the r-jet of h satisfy certain non-degeneracy conditions. We will see that the above assumption on a GRF, along with the following lemma, will be required to ensure this happens almost surely. Lemma 1.4 ( [S + 20], Theorem 23). Let h : N → R 2 be a smooth GRF and r ∈ N. Assume that for every p ∈ N we have supp (j r h(p)) = J r p (N, R 2 ). Then for any submanifold W ⊂ J r (N, R 2 ), we have P (j r h W) = 1. Length computations on general centered GRFs Expected length of critical curves In this section, we derive expressions for the average length of the critical curve of h. Recall that a point p ∈ N is called a critical point of h if the derivative Dh(p) : T x N → R 2 at p is not surjective. This is equivalent to saying that the gradient vectors ∇f(p), ∇g(p) are not linearly independent vectors lying in the tangent space T p N. The set of critical points of h is called the critical set of h. For a generic function h 1 the critical set will be a 1-dimensional embedded submanifold of N, or a collection of disjoint embedded circles in N, justifying the term critical curve. Proof. The critical curve Σ c = j 1 h −1 (S 1 ∪ S 2 ) . This means that the critical curve is closed, as S 1 ∪ S 2 is a closed subset of J 1 (N, R 2 ). As a consequence of Lemma 1.4, Assumption 1.3 guarantees that P j 1 h W = 1 for any submanifold W ⊂ J 1 (N, R 2 ). This implies j 1 h S r for r = 1, 2 almost surely. Since codim(S 2 ) = 2n > dim(N), j 1 h −1 S 2 is empty. Therefore, the critical curve Σ c = j 1 h −1 (S 1 ) . Since j 1 h intersects S 1 transversally, Σ c is a submanifold of N. In addition, codim (S 1 ) = n − 1 =⇒ codim(Σ c ) = n − 1 =⇒ dim(Σ c ) = 1. We are now in a position where the length of the critical curve makes sense almost surely, and will derive expressions for the average length in local coordinates first and then show that the expressions are coordinate invariant. Let x = (x 1 , x 2 , ..., x n ) be local coordinates on a coordinate neighborhood U of N, that is, R n (x 1 , ..., x n ) = x −→ φ(x) = p ∈ U is a diffeomorphism. Let K ⊂ U be the coordinate image of a compact set in R n . To avoid notational clutter, we will refer to the local representation of h (and f, g) as h itself. We can characterize the critical points in these coordinates as the projection onto N of the zeros of the R n valued function U × S 1 (x, θ) −→ V(x, θ) = cos(θ)∇f(x) + sin(θ)∇g(x) ∈ R n . (1) 1 generic refers to a set of functions that is open and dense in an appropriate metric on the space of smooth functions. We also define the infinitesimal length vector LV(x, θ) as LV(x, θ) = adj (∇ x V(x, θ)) ∇ θ V(x, θ). The following local result will be our first step. Proposition 2.2. If the GRF h satisfies assumptions 1.1-1.3 and K is a compact set contained in a coordinate chart U, then the expected value of the length of the critical curve in K is E len(Σ c ∩ K) = K [0,π] E LV(x, θ) G V(x, θ) = 0 (2π) n det(G(x)Var V(x, θ) ) d N Vdθ.(2) We will prove 2.2 after defining a few more objects and establishing a sequence of lemmas. Define the S 1 indexed family of real valued GRFs h θ as N p −→ h θ (p) := cos(θ)f(p) + sin(θ)g(p) ∈ R. Lemma 2.3. If h is a GRF satisfying assumptions 1.1-1.3, then h θ is a Morse function outside finitely many pairs (p, θ), almost surely. The function N × S 1 (p, θ) −→ h R (θ, p) := h θ (p) ∈ R is also a Morse function almost surely. Proof. Let W be the submanifold of J 2 (N × S 1 , R) defined as W = j 2 f(p, θ) f : N × S 1 −→ R, D 1 f(p, θ) = 0, rk(D 2 1,1 f(p, θ)) = n − 1 which satisfies codim(W) = n + 1. Assumption 1.3 and Lemma 1.4 tells us that j 2 h R W almost surely. Therefore, Z = (j 2 h R ) −1 (W) is a codimension n + 1 submanifold of N × S 1 , i.e. a finite set of points. h θ is not Morse at (p, θ) iff (p, θ) ∈ Z, which proves the first part of the lemma. We use the fact that h R is a Morse function only if j 1 h R S 1 ⊂ J 1 (N × S 1 , R), where S 1 denotes the corank-1 submanifold of J 1 (N × S 1 , R) [GG12, Proposition 6.4]. Assumption 1.3 guarantees that the GRF h R : N × S 1 −→ R satisfies supp j 1 h R (p, θ) = J 1 (p,θ) (N × S 1 , R) . Lemma 1.4 then says that j 1 h R S 1 almost surely, which means h R is a Morse function almost surely. Note that V(x, θ) is just the derivative of h θ in local coordinates. The fact that h θ is a Morse function ensures that ∇ x V(x, θ) = cos(θ)∇ 2 f(x) + sin(θ)∇ 2 g(x).(3) is non-degenerate at points where V(x, θ) = 0, excluding finitely many points on the critical curve. Since the length of the critical curve is not affected by removing finitely many points, we can ignore these points and proceed. This means that V(x, θ) is a submersion on V −1 (0), which further implies that V −1 (0) is a one dimensional submanifold of U×S 1 . If π : N×S 1 → N denotes projection to the first factor, then π(V −1 (0)) = Σ c ∩ U. We denote by dl the volume form on V −1 (0) corresponding to the metric induced on it by the Euclidean metric on R n × S 1 . Similarly, d N l denotes the volume form on Σ c corresponding to the metric induced by the Riemannian metric on N. Lemma 2.4. If h is a GRF satisfying assumptions 1.1-1.3, then the following equation holds almost surely. len(Σ c ∩ K) = 1 2 V −1 (0)∩(K×S 1 ) LV(x, θ) G det(∇ x V(x, θ)) 2 + LV(x, θ) 2 dl. Proof. The fact that ∇ x V(x, θ) is non-degenerate implies that V −1 (0) can be locally parametrized by θ, as per the implicit function theorem. We will denote this parametrization by x(θ). The derivative of this function is then dx dθ = (∇ x V(x, θ)) −1 (∇ θ V(x, θ)) = adj (∇ x V(x, θ)) ∇ θ V(x, θ) det (∇ x V(x, θ)) . Since π : V −1 (0) → Σ c ∩ U is a double cover of the critical curve, Σ c ∩K d N l = 1 2 V −1 (0)∩(K×S 1 ) π * d N l. We know that ( dx dθ , 1) lies tangent to V −1 (0). Then, π * d N l dx dθ , 1 = d N l dx dθ = dx dθ G , dl dx dθ , 1 = 1 + dx dθ 2 , =⇒ π * d N l = dx dθ G 1 + dx dθ 2 dl = adj (∇ x V(x, θ)) ∇ θ V(x, θ) G det(∇ x V(x, θ)) 2 + adj (∇ x V(x, θ)) ∇ θ V(x, θ) 2 dl, which proves the result. Proof of proposition 2.2. In lemma 2.4, we expressed the length of the critical curve as a weighted integral over the level set of the real valued GRF V(x, θ) on U × S 1 . To find the expectation of this weighted integral, we apply the generalized Rice formula ( [AW09, Theorem 6.10]). Assumption 1.3 along with lemma 2.1 ensure that the conditions required ((i)-(iv) in [AW09, Theorem 6.8]) to justify this are satisfied. If we denote the total derivative of V by ∇V(x, θ) = ∇ x V(x, θ) ∇ θ V(x, θ) , then E len(Σ c ∩ K) = 1 2 K S 1 E det ∇V.∇V 1 2 adj(∇ x V)∇ θ V G det(∇ x V) 2 + adj(∇ x V)∇ θ V 2 V = 0 p V (0)dxdθ, where we have dropped the obvious (x, θ) dependence to avoid clutter. Observe that det ∇V.∇V = det ∇ x V.∇ x V + ∇ θ V.∇ θ V = det(∇ x V) 2 det I + ∇ x V −1 ∇ θ V ∇ x V −1 ∇ θ V = det(∇ x V) 2 1 + ∇ x V −1 ∇ θ V 2 = det(∇ x V) 2 + adj(∇ x V)∇ θ V 2 . In addition, p V(x,θ) (0) is the density of V(x, θ) evaluated at 0. Since V(x, θ) is an n-dimensional mean zero Gaussian random vector, p V(x,θ) (0) = (2π) −n 2 (det(Var V(x, θ) ) −1 2 . In total, we get E len(Σ c ∩ K) = 1 2(2π) n 2 K S 1 E adj(∇ x V(x, θ))∇ θ V(x, θ) G V(x, θ) = 0 det(Var V(x, θ) ) dxdθ. The Riemannian volume form on N is related to the Euclidean volume form dx as d N V = det(G(x))dx. In addition, observe that ∇ x V(x, θ + π) = −∇ x V(x, θ), ∇ θ V(x, θ + π) = −∇ θ V(x, θ). Therefore, we can say that E len(Σ c ∩ K) = K [0,π] E adj(∇ x V(x, θ))∇ θ V(x, θ) G V(x, θ) = 0 (2π) n det(G(x)Var V(x, θ) ) d N Vdθ. We now show that the integrand in (2) is coordinate invariant. Let R n (y 1 , ..., y n ) = y −→ ψ(y) ∈ U be another coordinate chart on U. Let J be the Jacobin of the coordinate change map from (y 1 , ..., y n ) → (x 1 , ..., x n ). Then V(x, θ) = J V(y, θ), ∇ θ V(x, θ) = J V(y, θ), ∇ x V(x, θ) = J ∇ y V(y, θ)J. The Riemannian metric tensor transforms as G(x) = J G(y)J. This means adj(∇ x V(x, θ))∇ θ V(x, θ) = det(J) 2 J −1 adj(∇ y V(y, θ)J − J ∇ θ V(y, θ) = det(J) 2 J −1 ∇ y V(y, θ)∇ θ V(y, θ), det(G(x)Var V(x, θ) ) = det(J) 2 det(G(y)Var V(y, θ) . We can write adj(∇ x V(x, θ))∇ θ V(x, θ) G = ∇ θ V(x, θ) adj(∇ x V(x, θ)) G(x) adj(∇ x V(x, θ))∇ θ V(x, θ) = (det(J)) 2 ∇ θ V(y, θ) adj(∇ y V(y, θ)) J − G(x)J −1 adj(∇ y V(y, θ))∇ θ V(y, θ) = (det(J)) 2 ∇ θ V(y, θ) adj(∇ y V(y, θ)) G(y) adj(∇ y V(y, θ))∇ θ V(y, θ) = (det(J)) 2 adj(∇ y V(y, θ))∇ θ V(y, θ) G which immediately gives coordinate invariance of the integrand. We can now give each term in the integrand the following coordinate invariant characterization: V(x, θ) −→ ∇h θ (p), ∇ x V(x, θ) −→ ∇ 2 h θ (p), ∇ θ V(x, θ) −→ ∇h θ+ π 2 (p). If we define Var ∇h θ (p) i,j := E (∇h θ (p).v i )(∇h θ (p).v j ) where {v i } n i=1 is an orthonormal basis of T p N, then det(Var ∇h θ (p) ) = det(G(x)Var V(x, θ) ). We can now extend proposition 2.2 globally to get the main result of this section. Theorem 2.5. If the GRF h satisfies assumptions 1.1-1.3, then the expected value of the length of its critical curve is E len(Σ c ) = N [0,π] E adj ∇ 2 h θ (p) ∇h θ+ π 2 (p) G ∇h θ (p) = 0 (2π) n det Var ∇h θ (p) dVdθ.(4) Proof. Cover the manifold N by a finite number of compact coordinate disks K i , i = 1, ..., k. We already know E len(Σ c ) ∩ K i = K i [0,π] E adj ∇ 2 h θ (p) ∇h θ+ π 2 (p) G ∇h θ (p) = 0 (2π) n det Var ∇h θ (p) dVdθ. By the inclusion exclusion principle, E len Σ c = k l=1 i 1 <i 2 <...<i l (−1) l+1 E len Σ c ∩ K i 1 ∩ K i 2 ... ∩ K i l . But the integral of any function f : N → R can be written as N fd N V = k l=1 i 1 <i 2 <...<i l (−1) l+1 ∩ l j=1 K i j fd N V, from which the result follows directly. Remark 2.6. Notice that (∇ 2 h θ , ∇h θ+ π 2 , ∇h θ ) are jointly Gaussian random vectors and so the conditional expectation in (4) is just the expectation of a function of a Gaussian random vector. Expected length of the visible contour A point v ∈ R 2 is called a critical value of h if the preimage h −1 {v} contains a critical point, that is, γ c = h(Σ c ). The subset of R 2 consisting of all critical values is called the visible contour γ c of h. In this section, we will compute the average length of the visible contour of h in this section. The computation of the expected length of the visible contour follows the exact same procedure as the one we saw in the previous section. We have the following analogues of proposition 2.2, lemma 2.4. Proposition 2.7. If the GRF h satisfies assumptions 1.1-1.3 and K is a compact set contained in a coordinate chart U, then the expected value of the length of its critical curve in K is E len(h(Σ c ∩ K)) = K [0,π] E ∇ θ V(x, θ) LV(x, θ) V(x, θ) = 0 (2π) n det(G(x)Var V(x, θ) ) d N Vdθ.(5)len(Σ c ∩ K) = 1 2 V −1 (0)∩(K×S 1 ) ∇ θ V(x, θ) LV(x, θ) det(∇ x V(x, θ)) 2 + LV(x, θ) 2 dl. Proof. Here we have the sequence of maps V −1 (0) π −→ Σ c ∩ U h −→ γ c . If we denote by d R 2 l the volume form on γ c outside a finite set of cusps) induced from the Euclidean metric on R 2 , we can say that h(Σ c ∩K) d R 2 l = 1 2 V −1 (0)∩(K×S 1 ) (h • π) * d R 2 l. We know that ( dx dθ , 1) lies tangent to V −1 (0). Then, (h • π) * d R 2 l dx dθ , 1 = d R 2 l Dh dx dθ = Dh dx dθ , dl dx dθ , 1 = 1 + dx dθ 2 . Since the Euclidean norm on R 2 is invariant under rotation by an angle θ, we can say Dh dx dθ = (cos(θ)∇f(x) + sin(θ)∇g(x)) (− sin(θ)∇f(x) + cos(θ)∇g(x)) dx dθ .(6) However, since cos(θ)∇f(x) + sin(θ)∇g(x) = 0, we can further rewrite this term as − sin(θ)∇f(x) + cos(θ)∇g(x), dx dθ = ∇ θ V(x, θ) (∇ x V(x, θ)) −1 ∇ θ V(x, θ) .(7) Therefore, (h • π) * d R 2 l = Dh dx dθ 1 + dx dθ 2 = \|∇ θ V(x, θ) adj(∇ x V(x, θ)∇ θ V(x, θ)\| det(∇ x V(x, θ)) 2 + adj(∇ x V(x, θ)∇ θ V(x, θ) 2 dl. from which the result follows through the exact same steps as in the proof of lemma 2.4. The rest of the computations and justifications in the proof of 2.7 also follow the exact same pattern as in the proof of 2.2, and won't be repeated. The main result of this section also follows from 2.7 as Theorem 2.9. If the GRF h satisfies assumptions 1.1-1.3, then the expected value of the length of its visible contour is E len(γ c ) = N [0,π] E ∇h θ+ π 2 adj ∇ 2 h θ (p) ∇h θ+ π 2 (p) ∇h θ (p) = 0 (2π) n det Var ∇h θ (p) dVdθ.(8) Expected length of segments of fixed index The ind (x, θ) = ind (∇ x V(x, θ)) − 1 ∇ θ V(x, θ) (∇ x V(x, θ)) −1 ∇ θ V(x, θ) < 0 . Proof. As mentioned before, the biparametric index is just the usual index of f restricted to the level set {g = g(p)}, which can be computed using the method of Lagrange multipliers as follows. Define the Lagrangian L(x, λ) := f(x) + λ(g(x) − g(p)). If p is a critical point and ∇g(p) = 0, then there exists some multiplier λ * such that ∇f(p) + λ * ∇g(p) = 0. This means that (p, λ * ) is a critical point of L. The restricted index of p is then just ind ∇ 2 L(p, λ * ) − 1 where ∇ 2 L(p, λ * ) = ∇ 2 f(p) ∇g(p) ∇g(p) 0 . In order to symmetrize our computations, we can use the fact that the index can also be computed using the index of the Lagrangian L(x, λ) = (cos(θ)f(x) + sin(θ)g(x)) + λ (− sin(θ)f(x) + cos(θ)g(x)) so that ∇ 2 L(p, λ * ) = cos(θ)∇ 2 f(p) + sin(θ)∇ 2 g(p) − sin(θ)∇f(p) + cos(θ)∇g(p) − sin(θ)∇f(p) + cos(θ)∇g(p) 0 = ∇ x V(x, θ) ∇ θ V(x, θ) ∇ θ V(x, θ) 0 . To see why this is true, imagine what happens to the sublevel set when crossing the visible contour at a direction normal to it as opposed to the horizontal direction; the dimension of the cell attached must be the same in both cases. The above matrix is conjugate to ∇ x V(x, θ) 0 0 −∇ θ V(x, θ) (∇ x V(x, θ)) −1 ∇ θ V(x, θ) and since index is invariant under change of basis, the index of p is just ind (∇ x V(x, θ)) + 1 ∇ θ V(x, θ) (∇ x V(x, θ)) −1 ∇ θ V(x, θ) > 0 − 1 = ind (∇ x V(x, θ)) − 1 ∇ θ V(x, θ) (∇ x V(x, θ)) −1 ∇ θ V(x, θ) < 0 . The index at a point again is just a function of (∇ θ V, ∇ x V), which we denote as ind(x, θ). We denote the segments of the critical curve and contour of index k by Σ k c and γ k c respectively. We then have the obvious analogues of lemma 2.4, 2.8, which we state without proving. Lemma 2.13. If h is a GRF satisfying assumptions 1.1-1.3, then the following equations hold almost surely. len(Σ k c ∩ K) = 1 2 V −1 (0)∩(K×S 1 ) LV(x, θ) G 1 (ind(x, θ) = k) det(∇ x V(x, θ)) 2 + LV(x, θ) 2 dl, len(h(Σ k c ∩ K)) = 1 2 V −1 (0)∩(K×S 1 ) ∇ θ V(x, θ) LV(x, θ) 1 (ind(x, θ) = k) det(∇ x V(x, θ)) 2 + LV(x, θ) 2 dl. Proposition 2.14. If the GRF h satisfies assumptions 1.1-1.3 and K is a compact set contained in a coordinate chart U, then the expected value of the length of the critical curve of index k in K is E len(Σ k c ∩ K) = K [0,π] E LV(x, θ) G 1 (ind(x, θ) = k) V(x, θ) = 0 (2π) n det(G(x)Var V(x, θ) ) d N Vdθ,(9) and that of the visible contour of index k in K is E len(h(Σ k c ∩ K)) = K [0,π] E ∇ θ V(x, θ) LV(x, θ) 1 (ind(x, θ) = k) V(x, θ) = 0 (2π) n det(G(x)Var V(x, θ) ) d N Vdθ.(10) Proof. The sets O > k := {(M, v) ∈ Sym n (R) × R n M invertible, ind(M) = k, v M −1 v > 0}, O < k := {(M, v) ∈ Sym n (R) × R n M invertible, ind(M) = k, v M −1 v < 0} are open in Sym n (R) × R n . Observe that 1 (ind(x, θ) = k) = 1 (∇ x V(x, θ), ∇ θ V(x, θ)) ∈ O > k ∪ O < k+1 . The indicator function of any open set can be approximated pointwise by a sequence of bounded continuous functions, which means there exists continuous bounded functions Sym n (R) × R n (M, v) −→ 1 k (M, v) ∈ [0, 1] such that 1 k (M, v) ↑ 1 (M, v) ∈ O > k ∪ O < k+1 everywhere, as ↓ 0. If we now define 1 (ind(x, θ) = k) = 1 k (∇ x V(x, θ), ∇ θ V(x, θ)), and len (Σ k c ∩ K) = 1 2 V −1 (0)∩(K×S 1 ) adj(∇ x V(x, θ)∇ θ V(x, θ) G 1 (ind(x, θ) = k) det(∇ x V(x, θ)) 2 + adj(∇ x V(x, θ)∇ θ V(x, θ) 2 dl, then by the monotone convergence theorem, len (Σ k c ∩ K) ↑ len(Σ k c ∩ K) almost surely as → 0. We need this continuous bounded approximation for the application of [AW09, Theorem 6.10] in the proof of proposition 2.2. The same steps as in the proof of proposition 2.2 now give E len (Σ k c ∩ K) = K [0,π] E adj(∇ x V(x, θ))(∇ θ V(x, θ) G 1 (ind(x, θ) = k) V(x, θ) = 0 (2π) n det(G(x)Var V(x, θ) ) d N Vdθ. Applying the monotone convergence theorem on both sides and to the conditional expectation in the integrand, we get the required result. The proof for the visible contour follows exactly the same way. We can now globalize the above result using the exact same arguments as in the proof of 2.5 to get, h satisfies assumptions 1.1-1.3, then the expected value of the length of its critical curve and contour of index k are E ∇h θ+ π 2 adj ∇ 2 h θ (p) ∇h θ+ π 2 (p) 1 (ind(p, θ) = k) ∇h θ (p) = 0 (2π) n det Var ∇h θ (p) Theorem 2.15. If the GRF E len(Σ k c ) = N [0,π] E adj ∇ 2 h θ (p) ∇h θ+ π 2 (p) G 1 (ind(p, θ) = k) ∇h θ (p) = 0 (2π) n det Var ∇h θ (p) dVdθ,(11) dVdθ. (12) Length computations on independent and identical GRFs All the expressions derived for the expectation of average length depend on the joint distribution of (∇ θ V(x, θ), ∇ x V(x, θ)) conditioned on V(x, θ) = 0. In this section, we see that if we assume f and g are identical and independent random processes, the conditional distribution does not depend on θ allowing us to get rid of the θ integral in our formulae. We will also see two concrete examples of computations assuming i.i.d pairs in this section. Theorem 3.1. If the GRF h satisfies assumptions 1.1-1.3 and its components f and g are independent and identically distributed GRFs, then the expected value of the length of its critical curve and contour of index k are E len Σ k c = N πE adj ∇ 2 f(p) − E ∇ 2 f(p) ∇f(p) ∇f(p) G 1 (ind(p) = k) (2π) n Var ∇f(p) ) dV,(13) and E len γ k c = N πE ∇f(p) adj ∇ 2 f(p) − E ∇ 2 f(p) ∇f(p) ∇f(p) 1 (ind(p) = k) (2π) n Var ∇f(p) ) dV, (14) where ind(p) = ind ∇ 2 f(p) − 1 ∇f(p) ∇ 2 f(p) −1 ∇f(p) < 0 . Proof. Observe that if the pair (f, g) is i.i.d, Var V(x, θ) = Var cos(θ)∇f(x) + sin(θ)∇g(x) = Var ∇f(x) . In addition, see that ∇ θ V(x, θ) = − sin(θ)∇f(x) + cos(θ)∇g(x), ∇ x V(x, θ) = cos(θ)∇ 2 f(x) + sin(θ)∇ 2 g(x), which means Var ∇ θ V(x, θ) = Var ∇f(x) , Var ∇ x V(x, θ) = Var ∇ 2 f(x) , and Cov ∇ θ V(x, θ), V(x, θ) = 0, Cov ∇ θ V(x, θ), ∇ x V(x, θ) = 0. Putting all this together, we can say that (∇ θ V(x, θ), ∇ x V(x, θ)) are conditionally independent given V(x, θ) = 0, and that Var ∇ θ V(x, θ) V(x, θ) = 0 = Var ∇f(x) , Var ∇ x V(x, θ) V(x, θ) = 0 = Var ∇ 2 f(x)\|∇f(x) = 0 . This is just the joint distribution of ∇f(x), ∇ 2 f(x) − E ∇ 2 f(x) ∇f(x) . So the expected length of the critical curve can be rewritten as E len(Σ c ) = N πE adj ∇ 2 f(p) − E ∇ 2 f(p) ∇f(p) ∇f(p) G (2π) n Var ∇f(p) ) dV, and that of the contour as E len(γ c ) = N πE ∇f(p) adj ∇ 2 f(p) − E ∇ 2 f(p) ∇f(p) ∇f(p) (2π) n Var ∇f(p) ) dV. The index of a critical point can also be simplified as ind(x) = ind ∇ 2 f(x) − 1 ∇f(x) ∇ 2 f(x) −1 ∇f(x) < 0 giving the length of index k segments as E len Σ k c = N πE adj ∇ 2 f(p) − E ∇ 2 f(p) ∇f(p) ∇f(p) G 1 (ind(p) = k) (2π) n Var ∇f(p) ) dV and E len γ k c = N πE ∇f(p) adj ∇ 2 f(p) − E ∇ 2 f(p) ∇f(p) ∇f(p) 1 (ind(p) = k) (2π) n Var ∇f(p) ) dV. Examples We now see some concrete computations of these expectations. We don't show the index computations here either, as the i.i.d assumption is still not enough to get adequate structure on the Hessian of f for index computations. We will however do this in the next section while assuming isotropy. with the assumptions 1. F j (m,n) = F j (−m,−n) , so that (f, g) is real. 2. Re F j (m,n) and Im F j (m,n) are i.i.d with E \|F j (m,n) \| 2 = 1, 3. {F j (m,n) } (m 0,j=1,2) are pairwise independent. The above conditions ensure that f and g are identical and independent processes. We now compute ∂f(x, y) ∂x = (m,n)∈[−K,K] 2 2πimF 1 (m,n) e 2πimx e 2πiny , ∂f(x, y) ∂y = Observe that E F j (m,n) 2 = 0 due to assumption (2) above, and so we can compute the different variances as Var ∂f(x, y) ∂x = (m,n)∈[−K,K] 2 4π 2 m 2 = 16 3 π 2 K 4 1 + o 1 √ K := v 1 (K), Cov ∂f(x, y) ∂x , ∂f(x, y) ∂y = (m,n)∈[−K,K] 2 4π 2 mn = 0, Var ∂ 2 f(x, y) ∂ 2 x = (m,n)∈[−K,K] 2 16π 4 m 4 = 64 5 π 4 K 6 1 + o 1 √ K := v 2 (K), Var ∂ 2 f(x, y) ∂x∂y = (m,n)∈[−K,K] 2 16π 4 m 2 n 2 = 64 9 π 4 K 6 1 + o 1 √ K := v 3 (K), Cov ∂ 2 f(x, y) ∂ 2 x , ∂ 2 f(x, y) ∂ 2 y = (m,n)∈[−K,K] 2 16π 4 m 2 n 2 = 64 9 π 4 K 6 1 + o 1 √ K , Cov ∂ 2 f(x, y) ∂x∂y , ∂ 2 f(x, y) ∂ 2 y = 0, Cov ∇ 2 f, ∇f = 0. Assumptions 1.1-1.3 are clearly satisfied here. Note that f is a stationary process as well here. So the formula (13) reduces to E len(Σ c ) = E adj ∇ 2 f − E ∇ 2 f\|∇f ∇f 2 Var ∇f(x) ,(15) since n = 2 and N dV = 1 in this situation. Observe that ∇f is v 1 (K) times a standard normal 2-vector, ∇ 2 f is a random Gaussian symmetric matrix independent of ∇f. Also note that 1 (2K) 3 π 2 ∇ 2 f is a random Gaussian symmetric matrix M with Var M ii = 1 5 + o 1 √ K , Var M ij = Cov M ij , M ii = 1 9 + o 1 √ K . Finally, we compute det Var ∇f = 16 3 π 2 K 4 1 + o 1 √ K = v 1 (K), to apply (15) and say that the average length of the critical curve is 1 2v 1 (K) v 1 (K)(2K) 3 π 2 l ≈ √ 3πlK,(16) where l is a constant equal to E adj MZ where M is random Gaussian symmetric random matrix and Z is an independent standard normal 2-vector with We can also verify this linear relationship numerically in Mathematica. For each value of K in {2, 3, ..., 10}, we choose 20 sets of Fourier coefficients drawn randomly according to the assumptions given in the beginning of this example. We then computed the sample average length of the critical curve for each K and attach the plot in Figure 3. The red points in the plot show the sample average critical curve lengths and the blue line is the best linear fit, which has almost zero y-intercept and is consistent with the observation in (16). The slope of the best fit line is 3.33. We computed the constant l approximately using a sample average with a large number of samples and found it is approximately 0.607, which tells us that √ 3πl ≈ 3.3. This is very close to the slope of the best fit line in Figure 3. Var M ii = 1 5 , Var M ij = Cov M ij , M ii = 1 9 . We can compute the average length of the visible contour in a similar fashion as E len(γ c ) = E ∇f adj ∇ 2 f − E ∇ 2 f ∇f ∇f 2 Var ∇f(x) ,(17)which becomes 1 2v 1 (K) v 1 (K)(2K) 3 π 2 c ≈ 4π 2 cK 3 ,(18) where c = E Z MZ . Random linear projections of a thin doughnut In this example we consider N to be the 2-Torus embedded in R 3 in the shape of a hollow doughnut of radius R and cross sectional radius r, and the metric to be the metric induced from the Euclidean metric on R 3 . We denote by T the ratio R r . The embedding can be written in φ, ρ coordinates as [0, 2π] × [0, 2π] (φ, ρ) → E(φ, ρ) =   cos(φ) (R + r sin(ρ)) sin(φ) (R + r sin(ρ)) r cos(ρ))   ∈ R 3 . The induced metric tensor can then be written in (φ, ρ) coordinates as G(φ, ρ) = (R + r sin(ρ)) 0 0 r = r(T + sin(ρ)) 0 0 r . We consider random linear projections of this embedding onto R 2 . That is, we choose A ∈ R 2×3 such that each entry is drawn from a standard Gaussian distribution independently of each other, and then define h(φ, ρ) := A.E(φ, ρ) ∈ R 2 . The two components of h are clearly identical and independent since the two rows of A are i.i.d. and so equations (13) and (14) apply here. We can see that f(φ, ρ) = v E(φ, ρ) where v is drawn from a standard 3D Gaussian distribution. The computations in this example are a bit tedious and are done in Mathematica. We can show that Var ∇f(φ, ρ) = (R + r sin(ρ)) 2 0 0 r 2 = r 2 (T + sin(ρ)) 2 0 0 r 2 , and Var     f φ,φ f φ,ρ f ρ,ρ   ∇f = 0   (φ, ρ) = 1 4 r 2 (cos(4φ)+3) sin 2 (ρ)(T +sin(ρ)) 2 1 4 r 2 sin(4φ) sin 2 (ρ) cos(ρ)(T +sin(ρ)) 1 4 r 2 (cos(4φ)+3) sin 3 (ρ)(T +sin(ρ)) 1 4 r 2 sin(4φ) sin 2 (ρ) cos(ρ)(T +sin(ρ)) 1 4 r 2 (cos(4φ)+3) sin 3 (ρ)(T +sin(ρ)) 1 8 r 2 sin 2 (2φ) sin 2 (2ρ) 1 4 r 2 sin(4φ) sin 3 (ρ) cos(ρ) 1 4 r 2 sin(4φ) sin 3 (ρ) cos(ρ) 1 32 r 2 ((cos(4φ)+7)(cos(4ρ)+3)+8 sin 2 (2φ) cos(2ρ)) . We compute the length of the visible contour when T 1. To avoid cumbersome notation, from this point on in this example we will denote ∇ 2 f − E ∇ 2 f\|∇f as just ∇ 2 f. We remind that the variance of this random symmetric matrix is given in the previous equation, and it is independent of ∇f. Observe that adj ∇ 2 f = f ρ,ρ −f φ,ρ −f φ,ρ f φ,φ and ∇f (adj ∇ 2 f)∇f = f 2 φ f ρ,ρ + f 2 ρ f φ,φ − 2f φ f ρ f φ,ρ . In addition, det Var ∇f (φ, ρ) = r 2 (T + sin(ρ)). This random field is not stationary, so we will need to integrate over the torus unlike the previous example. Equation (14) gives us the average length of the visible contour as [0,2π] [0,2π] E f 2 φ f ρ,ρ + f 2 ρ f φ,φ − 2f φ f ρ f φ,ρ 2r 2 (T + sin(ρ)) dφdρ.(19) Observe that E \|f 2 φ f ρ,ρ \| = E \|f 2 φ E \|f ρ,ρ \| = r 2 (T + sin(ρ)) 2 2 π Var f ρ,ρ , E \|f 2 ρ f φ,φ \| = E \|f 2 ρ E \|f φ,φ \| = r 2 2 π Var f φ,φ = r 2 (T + sin(ρ)) 2 π Var f φ,φ (T + sin(ρ) 2 , E \|f φ f ρ f ρ,φ \| = E \|f φ \| E \|f ρ \| E \|f φ,ρ \| = 2 π r 2 (T + sin(ρ)) 2 π Var f φ,ρ , where the expectations split as a product because ∇ 2 f and ∇f are independent, and we have used the fact that E \|X\| = 2 π Var X for a Gaussian random variable. Clearly, as T → ∞ the E \|f 2 φ f ρ,ρ \| term dominates and we can say that The constant c can be computed numerically as ≈ 31.6 and so E len(γ c ) ≈ 12.6R, when R r 1. E len(γ c ) ≈ [0,2π] [0,2π] E f 2 φ f ρ,ρ 2r 2 (T + sin(ρ)) dφdρ = [0,2π] [0,2π] (T + sin(ρ)) Var f ρ,ρ √ 2π dφdρ = T r √ 2π [0,2π] [0,2π] (1 + sin(ρ) T ) Var f ρ,ρ r dφdρ, ≈ Rc √ 2π where A similar computation will show that the length of the critical curve also grows linearly in R when R r 1. Isotropic GRFs on Spheres All the formulae we computed in the previous section depends only on the joint distribution of (∇f(x), ∇ 2 f(x)). When the space N is S n , and the GRF f is isotropic and stationary, we will see that this joint distribution is particularly well structured. Under these conditions, ∇f and ∇ 2 f are independent, ∇f is a standard Gaussian random vector, and ∇ 2 f is distributed as a Gaussian Orthogonally Invariant (GOI) ensemble. We will see in the following section some of the properties of GOI ensembles that will lead to more reduced formulae for the various computations we did in earlier sections. Most of the results about GOI ensembles mentioned here are a review of what can be found in [CS18]. Gaussian Orthogonally Invariant Ensembles An n × n random matrix H ij is said to have Gaussian Orthogonal Ensemble (GOE) distribution if it is symmetric and all entries are centered Gaussian random variables with E H ij H kl = 1 2 (δ ik δ jl + δ il δ jk ). It is well known that the GOE ensemble is orthogonally invariant i.e. the distribution of H is the same as that of QHQ for any orthogonal matrix Q. Moreover, the entries of H are independent. However, we will need a slightly more general distribution to capture the structure of the Hessian of isotropic GRFs. An n × n random matrix M ij is said to have Gaussian Orthogonally Invariant distribution with covariance parameter c (GOI(c)) if it is symmetric and all entries are centered Gaussian random variables with E M ij M kl = 1 2 (δ ik δ jl + δ il δ jk + cδ ij δ kl ). The GOI distribution is also orthogonally invariant. In fact, upto a scaling constant any orthogonally invariant symmetric Gaussian random matrix has to have GOI(c) distribution. The only constraint on the covariance parameter is that c −1/N ( [CS18, Lemma 2.1]). We will see that the Hessian of isotropic GRFs are GOI ensembles. The orthogonal invariance of GOI random matrices will prove a very useful property in our computations. In addition, the density of the ordered eigenvalues of GOI(c) matrices can be written as f c (λ 1 , ..., λ n ) = 1 K n √ 1 + nc exp    − 1 2 n i=1 λ 2 i + c 2(1 + nc) n i=1 λ i 2    × 1 i<j n \|λ i − λ j \|1 ({λ 1 ... λ n })(21) where K n is the normalization constant K n = 2 n/2 n i=1 Γ i 2 . For any measurable function g, we will denote by E n GOI(c) g(λ 1 , ..., λ n ) := R n g(λ 1 , ..., λ n )f c (λ 1 , ..., λ n )dλ 1 ...dλ n the expectation under GOI(c) density. Length computations on Isotropic GRFs on spheres Let S n = (z 1 , .., z n+1 ) ∈ R n+1 n+1 i=1 z 2 i = 1 be the unit n-sphere embedded in R n+1 endowed with the induced Riemannian metric, and f be a centered, unit-variance, smooth isotropic GRF on S n . Due to isotropy, we can write the covariance function R of f as R(x, y) = C( x, y ) for some C : [−1, 1] → R. Define C = C (1), C = C (1), η = √ C / √ C , κ = C / √ C . Since an isotropic GRF is also stationary, we only need to compute the integrands in equations (13)-(14) at one point on the sphere. We will choose this point to be the north pole N = (0, ..., 0, 1), and use the fact that (z 1 , ..., z n ) forms a coordinate chart on the sphere in a neighborhood around this point. In this coordinate system, the metric tensor G(N) is simply the identity matrix I n . In addition, we have the following lemma giving us the distribution of the derivatives of f. where the derivatives are computed in the (z 1 , ..., z n ) coordinates at N. We then have the following result giving a nicer formula for the expected length of the visible contour, Theorem 4.2. If N = S n and the components of the GRF f and g are independent and identically distributed as centered, unit-variance, smooth isotropic GRFs with C = 0, C = 0, then the expected value of the length of its visible contour of index k is E len γ k c = √ 2π 3 n Γ ( n+1 2 ) κ η n E n−1 GOI 1+η 2 2 n−1 i=1 \|λ i \|1 (λ k < 0 < λ k+1 ) .(22) Proof. Assumption 1.3 is satisfied here, since (f, ∇f) are independent, f has unit-variance and lemma 4.1 implies ∇f is non degenerate. Equation (14) gives E len γ c = Vol (S n ). πC ( √ 2C ) n−1 ( √ 2πC ) n E v adj(M)v = 2 √ π n+1 Γ ( n+1 2 ) π √ 2 √ π n ( √ C ) n−1 ( √ C ) n−2 E v adj(M)v = √ 2π 3 Γ ( n+1 2 ) κ η n E v adj(M)v , where v is a standard unit Gaussian random vector independent of the GOI( 1+η 2 2 ) matrix M. Since Q adj(M)Q = adj(Q MQ) for any orthogonal matrix Q, E v adj(M)v = E v 2 (e 1 ) adj(R MR)e 1 = nE \|adj(M) 11 \| = nE n−1 GOI 1+η 2 2 n−1 i=1 \|λ i \| , which gives E len γ c = √ 2π 3 n Γ ( n+1 2 ) κ η n E n−1 GOI 1+η 2 2 n−1 i=1 \|λ i \| . We can similarly compute the expected length of the critical curve of index k using (14) as If the index of M is k, then the index ofM can either be k or k − 1. However, (−1) k det(M) > 0 implies that the index ofM is k. Similarly, if the index of M is k + 1, then the index ofM has to be either k + 1 or k, but (−1) k+1 det(M) < 0 implies that the the index ofM is k. This allows us to further reduce the above expectation as \|λ i \|1 (λ k < 0 < λ k+1 ) . E len γ k c = √ 2π 3 Γ ( n+1 2 ) κ η n E v adj(M)v 1 ind(M) − 1 v M −1 v < 0 = k = √ 2π 3 Γ ( n+1 2 ) κ η n nE e 1 adj(Q MQ)e 1 1 ind(Q MQ) − 1 e 1 (Q MQ) −1 e 1 < 0 = k = √ 2π 3 n Γ ( n+1 2 ) κ η n E Expected number of pseudocusps Pseudocusps can be split into two obvious types, vertical and horizontal, depending on whether they're the image of critical points of f or g respectively. Recall the notation from section 2; if a point p on N is a critical point of f, one can locally parametrize the critical curve in its neighborhood as a function x(θ) of θ such that x(0) = p. Recall that if V(x, θ) = cos(θ)∇f(x) + sin(θ)∇g(x) = 0 then the tangent line to the visible contour at h(x) lies along (− sin(θ), cos(θ)). This means the slope of the visible contour at the image of x(θ) is just θ − π 2 . For the image of a critical point p to be a vertical pseudocusp, the direction of the tangent vector to the visible contour along the direction of increasing slope must point upward, which means dh(x(θ)) dθ θ=0 , 0 1 > 0 =⇒ −∇g(p) ∇ 2 f(p) −1 ∇g(p) > 0. The index of the extension ray attached to a vertical pseudocusp, i.e. the dimension of the cell attached to the sublevel set when the parameter value crosses a point on the extension ray, is just the index of ∇ 2 f(p). We call this the index of a vertical pseudocusp. The same definition with g replacing f holds for horizontal pseudocusps. Therefore, a vertical pseudocusp of index k is characterized by the conditions ∇f(p) = 0, ∇g(p) ∇ 2 f(p) −1 ∇g(p) < 0, ind ∇ 2 f(p) = k. while a horizontal pseudocusp of index k is characterized by ∇g(p) = 0, ∇f(p) ∇ 2 g(p) −1 ∇f(p) < 0, ind ∇ 2 g(p) = k. If N k vpc and N k hpc denote the number of vertical and horizontal pseudocusps of index k, the next theorem gives a formula for the expected number of these points. Proof. This is a consequence of the characterizations (25), (26) and the Kac-Rice formula [AT07, Theorem 12.1.1]. Conclusion We have computed the expected length of the critical curve and visible contour of fixed index of a smooth centered Gaussian random map into the plane in this article. We derived more explicit expressions in the case where the components are identical and independent and a closed form expression under the additional assumption of isotropy. We also computed the expected number of pseudocusps of such a Gaussian random map. The one remaining singularity appearing in the description of biparametric persistence is the cusp point; these are the points where the visible contour loses smoothness. We have not treated these singularities in this article. The cusps points can be characterized as points where the 2-jet j 2 h intersects a certain submanifold S 1,1 ⊂ J 2 (N, R 2 ) of codimension n. If the support of j 2 h(p) is full for all p ∈ N, these intersections will be transverse at all p ∈ N almost surely. We can then compute the expected number of these cusps as the number of transverse intersections of the function j 2 h with the submanifold S 1,1 using a generalized Kac-Rice formula [Ste21]. However, these computations are a bit cumbersome and we will pursue these in a future work. Figure 1 : 1Figure (a)shows a bean sculpture, the surface of which is a two dimensional sphere. If h is the projection of the surface onto a plane behind the bean the corresponding visible contour is shown inFigure (b). The Pareto grid, along with cusps, pseudocusps and indices of various segments of the grid are shown in Figure (c). Lemma 2. 1 . 1If h satisfies assumptions 1.1-1.3 then the critical curve of h is a one dimensional compact submanifold of N on which the rank of Dh is 1 almost surely. Remark 2. 10 . 10The length of the visible contour does not depend on the choice of metric on N. Indeed, the formula given in equation (8) is invariant under a change of metric, since dV √ det G does not depend on the choice of metric. Remark 2.11. The Pareto segments of the visible contour are the parts of the contour where its slope is negative. The segments of the contour play a special role in biparametric persistent homology and the length of only these parts can also be easily computed. The only observation needed to do this is that h(x) lies on the Pareto segment only if θ lies in [0, π 2 ]. So one simply needs to replace the domain of evaluation of the inner θ integral in (8) with [0, π 2 ]. index of a critical point (and the corresponding critical value) of a single function f refers to the index of the Hessian of f at the critical point. The significance of the index is that if f(p) is a critical value of index k, then the sublevel set {f f(p) + } is obtained by attaching a k-cell to {f f(p) − } as long as (f(p) − , f(p) + ) contains no other critical values. In the single persistent homology setting, this translates to the fact that an index k critical value can either lead to a birth in k-th homology or a death in k + 1-th homology. The Pareto segments of the visible contour plays a similar role in bi-biparametric persistence. If p is a critical point of h and the corresponding value v = (f(p), g(p)) is a point on a Pareto segment of the visible contour of index k (to be defined soon), then the sublevel set {f f(p)+ , g g(p)+ } is obtained by attaching a k-cell to {f f(p) − , g g(p) − }. If ∇g(p) = 0, then p will be a critical point of the function f restricted to the submanifold {g = g(p)}, and the appropriate definition of the index of p is just the index of the Hessian of f restricted to {g = g(p)} at p. We have the following characterization of the biparametric index. Proposition 2.12. Suppose ∇g(p) = 0, and p is a critical point of the function f restricted to the submanifold {g = g(p)}. Then the index of this critical point is given by and E len(γ k c ) = N [0,π] Bandlimited functions on the flat torusWe choose N to be the 2-Torus identified as the quotient space of the unit square equipped with the standard flat metric on the unit square. The computations here will be done in the usual Euclidean coordinates of the unit square. The Riemannian metric tensor G(x) in this coordinate system is just the 2x2 identity matrix. SeeFigure 2for a few examples of critical curves of such bandlimited functions.Consider bandlimited functions with random Fourier coefficients n) e 2πimx e 2πiny . ,n)∈[−K,K] 2 −4π 2 mnF 1 (m,n) e 2πimx e 2πiny , ∂ 2 f(x, y) ∂ 2 y = (m,n)∈[−K,K] 2 −4π 2 n 2 F 1 (m,n) e 2πimx e 2πiny . Figure 2 : 2Critical curves of bandlimited functions on the flat torus Figure 3 : 3Average lengths of critical curves of bandlimited functions (4φ) + 7)(cos(4ρ) + 3) + 8 sin 2 (2φ) cos( e 1 adj(M)e 1 1 ind(M) − 1 e 1 (M) −1 e 1 < 0 = k .If we denote byM the first n − 1 × n − 1 principal minor of M, we can reduce the above expectation asE det(M) 1 ind(M) = k, (−1) k det(M) > 0 + 1 ind(M) = k + 1, (−1) k+1 det(M) < 0 . E det(M) 1 ind(M) = k, ind(M) = k + 1 ind(M) = k + 1, ind(M) = k = E det(M) 1 ind(M) = k . Theorem 5. 1 .E 1If the GRF h satisfies assumptions 1.1-1.3, then the expected number of pseudocusps of index k areE N k vpc = N E det ∇ 2 f(p) 1 ind ∇ 2 f(p) = k, ∇g(p) ∇ 2 f(p) −1 ∇g(p) < 0 ∇f(p) = 0 (2π) n det Var ∇f(p) det ∇ 2 g(p) 1 ind ∇ 2 g(p) = k, ∇f(p) ∇ 2 g(p) −1 ∇f(p) < 0 ∇g(p) = 0 (2π) n det Var ∇g(p) dV, Lemma 2.8. If h is a GRF satisfying assumptions 1.1-1.3, then the following equation holds almost surely. Lemma 4.1.[CS18, Lemma 4.1, 4.3] Let f be a centered, unit-variance, smooth isotropic GRF on S n . Then ∇f and ∇ 2 f are independent,1. ∇f is √ C times a standard Gaussian random vector, 2. ∇ 2 f is √ 2C times a GOI((1 + η 2 )/2) matrix, 3. Complexity of random smooth functions on the high-dimensional sphere. The Annals of Probability. Antonio Auffinger, Gerard Ben Arous, 41Antonio Auffinger and Gerard Ben Arous. Complexity of random smooth functions on the high-dimensional sphere. The Annals of Probability, 41(6):4214-4247, 2013. 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[ "Non-equilibrium theory of tunneling into localized state in superconductor", "Non-equilibrium theory of tunneling into localized state in superconductor" ]	[ "Ivar Martin \nMaterials Science Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA\n", "Dmitry Mozyrsky \nTheoretical Division\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n" ]	[ "Materials Science Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA", "Theoretical Division\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA" ]	[]	A single static magnetic impurity in a fully-gapped superconductor leads to formation of an intragap quasiparticle bound state. At temperatures much below the superconducting transition, the energy relaxation and spin dephasing of the state are expected to be exponentially suppressed. The presence of such a state can be detected in electron tunneling experiments as a pair of conductance peaks at positive and negative biases. Here we show, that for an arbitrarily weak tunneling strength, the peaks have to be symmetric with respect to the applied bias. This is in contrast to the standard result that the tunneling conductance is proportional to the local (in general particle-hole asymmetric) density of states. The asymmetry can be recovered is one allows for either a finite density of impurity states, or that impurities are coupled to another, non-superconducting, equilibrium bath.	10.1103/physrevb.90.100508	[ "https://arxiv.org/pdf/1312.2602v1.pdf" ]	96,453,799	1312.2602	c3a5355ee114cf4d9f5980c46e43021ab66d9de7	Non-equilibrium theory of tunneling into localized state in superconductor Ivar Martin Materials Science Division Argonne National Laboratory 60439ArgonneIllinoisUSA Dmitry Mozyrsky Theoretical Division Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA Non-equilibrium theory of tunneling into localized state in superconductor (Dated: December 11, 2013) A single static magnetic impurity in a fully-gapped superconductor leads to formation of an intragap quasiparticle bound state. At temperatures much below the superconducting transition, the energy relaxation and spin dephasing of the state are expected to be exponentially suppressed. The presence of such a state can be detected in electron tunneling experiments as a pair of conductance peaks at positive and negative biases. Here we show, that for an arbitrarily weak tunneling strength, the peaks have to be symmetric with respect to the applied bias. This is in contrast to the standard result that the tunneling conductance is proportional to the local (in general particle-hole asymmetric) density of states. The asymmetry can be recovered is one allows for either a finite density of impurity states, or that impurities are coupled to another, non-superconducting, equilibrium bath. Introduction. Conventional s-wave superconductors are remarkably robust with respect to nonmagnetic disorder [1]: potential scattering of electrons affects neither the superconducting gap, nor the transition temperature significantly. On the other hand, even weak magnetic impurities have been found to be strongly Cooper pairbreaking, leading to a rapid suppression of superconductivity [2]. An exact treatment of a quantum magnetic impurity in a superconductor is a complex problem, which has only been solved numerically so far [3]. However, in the case when the magnetic moment can be treated as static, (approximately the case for atoms with large spin, S, or when conduction electrons only couple to one of the components of the spin), within the BCS approximation, the problem is easily solvable. The key result is the appearance of a localized, so called Yu-Shiba-Rusinov (YSR) quasiparticle state [4][5][6]. For finite density of impurities, these states fill the superconducting gap, eventually destroying superconductivity. The presence of YSR-like states in superconductors has been confirmed by tunneling experiments [7,8] (see Fig. 1a). The metal-insulator-superconductor junction experiment of Ref. [7] on Mn doped Pb revealed a σ(V ) = dI/dV that is symmetric with respect to reversal of applied bias (particle-hole symmetry), with a clearly visible intra-gap peak whose energy remained approximately constant but the intensity grew with the increasing Mn concentration. Remarkably, the normal-tip STM experiment of Ref. [8], which allowed to look at individual magnetic ions of Mn or Gd on the surface of superconducting Nb, showed particle-hole asymmetric σ(V ). The asymmetry was attributed to the asymmetry in the particle and the hole content of the Bogoliubov quasiparticle associated with the YSR state. This however, raises a question why no such asymmetry had been observed in the earlier tunnel junction experiment [7]. It is interesting to note that individual YSR states bear strong resemblance to the localized impurity, e.g. donor, states in semiconductors. Each donor or acceptor state in a semiconductor can be populated by at most two electrons (incuding spin). Consequently, if one were to perform a tunneling experiment in a semiconductor, as long as the bias is insufficient to inject carries into conduction or valence band, the dc current will remain zero: after the tunneling electrons populate the initially unoccupied localized states, the current has to stop. What makes the YSR states different? Just as in a semiconductor, the individual YSR states are infinitely sharp resonances, since there are no continuum states that they could hybridize with. Therefore, it would seem that continuous tunneling into YSR states should be impossible, in conflict wight the experimental observations. That YSR assumes classical impurity cannot be the issue, since even for a quantum impurity, the spectrum has only one bound quasiparticle state associated with every impurity [3]. As we will show here, the reason that the intra-gap tunneling through the localized states in a superconductor is possible lies in the ability of superconductor to violate particle conservation law: While it is impossible to introduce a single electron with subgap energy into bulk of superconductor, two injected electrons with zero total energy can be absorbed by the condensate [9]. This problem can be analyzed by means of nonequilibrium Green function formalism for superconductors [10]. Here we will follow however a more physically transparent approach, valid in the case of singlet superconductors: By applying a partial particle-hole transformation, we convert the problem of tunneling from metallic tip to YSR state into the problem of tunneling between two non-superconducting spineless reservoirs through a single resonant level (Fig. 1b). Each transfer of a spineless particle between the reservoirs in the equivalent model corresponds to the transfer of a pair of electrons between the metallic tip and the superconductor. The mapping allows to see immediately that for a single impurity σ(V ) has to be symmetric, regardless of the local particle-hole content of YSR state. The origin of this surprising result is that since in the absence of coupling to the tip YSR state has zero energy width, any arbitrarily weak perturbation can drive it out of equilibrium. The height of the peaks of σ(V ) is of the order of conductance quantum, G 0 = 2e 2 /h. In contrast, the standard approach to calculating the tunneling conductance assumes that the YSR remains in equilibrium with the superconductor, leading to the erroneous conclusion that for single magnetic impurity the tunneling conductance is simply proportional to the tunneling density of states [11]. Why do some experiments show symmetric tunneling density of states [7], and others don't [8]? The reason most likely lies in the broadening of the resonant level due to the presence of other nearby magnetic impurities, which allows electrons to tunnel into multiple YSR states simultaneously, or due to an additional relaxation channel for YSR states. The latter can be modeled as a metallic reservoir that remains in equilibrium with the superconductor and thus can easily absorb quasiparticles injected into the YSR state. We will explicitly consider here this possibility. Model. The Hamiltonian for an s-wave superconductor with a magnetic impurity is [5] H = H BCS + H imp ,(1)H BCS = dr α ψ α (r) † − ∇ 2 2m − µ ψ α (r) +∆ 0 ψ ↑ (r) † ψ ↓ (r) † + ∆ 0 ψ ↓ (r)ψ ↑ (r) ,(2)H imp = JS ψ ↑ (0) † ψ ↑ (0) − ψ ↓ (0) † ψ ↓ (0) .(3) Here, ψ α (r) is the annihilation operator for electron with spin α at location r, m is the mass of electron, ∆ 0 is the unperturbed value of the superconducting order parameter (assumed real and positive for concreteness). For the impurity we assume a classical moment of size S polarized in the positive z-direction (in the continuum limit value of the coupling constant J is related to the atomic value by the factor of the unit cell volume, a 3 ). This Hamiltonian can be diagonalized by the Bogoliubov quasiparticles [12], γ n , which satisfy [H, γ † n ] = E n γ † n and can be expressed in terms of the electronic operators as γ n = dr[u n (r)ψ ↑ (r) + v n (r)ψ † ↓ (r)].(4) The solution of the Bogoliubov equations for u(r) and v(r) reveals that static magnetic impurity leads to formation of localized state inside the superconducting gap [6], with the energy E 0 = −∆ 0 sign(J) 1 − (πN 0 J) 2 1 + (πN 0 J) 2 ,(5) and (u, v) that oscillate with the Fermi wavevector and decay is space as exp(−r/ξ)/r. The exponential decay is governed by the lengthξ = v F / ∆ 2 0 − E 2 0 . Here v F is the Fermi velocity and N 0 is the normal state density of states in the superconductor. In general, u(r) = v(r). In addition to the localized states, there is a continuum of Bogoliubov's quasiparticles both for E n > ∆ 0 and E n < −∆ 0 . The Fermion operators can be expanded in terms of all Bogoliubov quasiparticles as ψ ↑ (r) = n u n (r)γ n and ψ † ↓ (r) = n v n (r)γ n . Hence, the local density of electronic states is N ↑ (ω) = n u 2 n (r)δ(ω−E n ) and N ↓ (ω) = n v 2 n (r)δ(ω + E n ). Note, that single YSR level contributes two delta-functions at energies ±E 0 with weights u 2 0 and v 2 0 that correspond to spin-up and spin-down states, respectively. According to the standard theory of electron tunneling from a metallic contact [11], at zero temperature the differential tunneling conductance σ(V ) is proportional to the density of states in the sample at E = V , which in the case of YSR states would correspond to, in general, asymmetric delta function peaks. However, as we discussed above, such treatment neglects the possibility of having non-equilibrium distribution function, which in fact, leads to a qualitatively different result. The tunneling between atomically sharp tip and the sample can be described by the tunneling Hamiltonian, H = H tip + t[d † σ (r 0 )ψ σ (r 0 ) + ψ † σ (r 0 )d σ (r 0 )],(6) where r 0 corresponds to the location where the tip and sample wavefunctions overlap, with the matrix element t, and H tip = kσ ( t k − µ t )d † kσ d kσ is the Hamiltonian of the tip, with modes d k . The tunneling part of the Hamiltonian can be conveniently expressed in terms of the Bogoliubov quasiparticles. Since we are interested in the subgap conductance due to the YSR state, out of the full expansion we only need to keep terms related to it, ψ ↑ (r 0 ) → u 0 (r 0 )γ 0 and ψ † ↓ (r 0 ) → v 0 (r 0 )γ 0 . In the spin-down channel this leads to terms of the form d † ↓ γ † 0 , which do not conserve the number of particles. A significant simplification occurs if one performs a particle-hole transformation of spin-down electrons in the tip,d ↓ = d † ↓ . For the spin down holes, t k → − t k , µ t → −µ t (relative to the chemical potential of the superconductor), and the state occupation numbers n k → 1 − n k . In the new basis, the tunneling Hamiltonian becomes, tu(r 0 )d † ↑ (r 0 )γ 0 − tv(r 0 )d † ↓ (r 0 )γ 0 + H.c. The full transformed Hamiltonian, which includes the superconductor, the tip, and the tunneling between them, now conveniently conserves the number of particles. It corresponds to the problem of tunneling of spinless particles between two reservoirs through a resonant level. The couplings to the two reservoirs are in general different due to the factors u(r 0 ), v(r 0 ). Schematically, the equivalent representation is illustrated in Figure 1b. The right reservoir correspond to spin-up electrons, and the left reservoir to spin-down holes. Notice that the process in which a particle is transferred from right reservoir to the left one, in terms of the original electrons corresponds to transferring two electrons (with spin up and spin down) into the superconductor, with the help of the YSR state. The initial and final energy of the spinless particle is the same; in the original language this corresponds to selecting two electrons with total energy equal to zero (relative to the superconductor's µ). The problem of tunneling through a resonant level is very well known [13]. The key quantities that enter are the tunneling rates between the level and the reservoirs, Γ 1 = πN t u 2 0 (r 0 )t 2 and Γ 2 = πN t v 2 0 (r 0 )t 2 . The sum of these two rates determines the resonant level broadening. Interestingly, even when Γ 1 = Γ 2 , the particle current through the resonant level does not depend on the direction of bias, reaching the maximum value of (2e/ ) × 2Γ 1 Γ 2 /(Γ 1 + Γ 2 ) for large bias. The ratio of the current to the level width, measured in the voltage units, gives, up to a constant, the differential conductance. Since the magnitude of the current does not depend on the direction of bias, subgap σ(V ) is symmetric with respect to the sign of V . With the numerical prefactors inluded, we find σ(±E 0 ) = 2e 2 h 4Γ 1 Γ 2 (Γ 1 + Γ 2 ) 2 = G 0 4u 2 0 v 2 0 (u 2 0 + v 2 0 ) 2 .(7) Thus the maximum value of conductance, which is achieved at the spatial locations r where u 0 (r) = v 0 (r) is equal to one quantum of conductance, and the spatial map of σ(±E 0 ) can be used to determine the spatial dependence of the quasiparticle particle-hole content, u 0 (r)/v 0 (r). Extra bath. We now turn to the case when magnetic impurity is not fully isolated within superconductor. To allow for additional relaxation, we introduce a gapless metallic bath, whose chemical potential is pinned to the chemical potential of superconductor, into which YSR state can decay with rate Γ 0 . If this rate is much faster than Γ 1,2 , the YSR state will remain in equilibrium with superconductor, and we expect to recover the "standard" result where σ(V ) is proportional to the density of states in superconductor. We study this problem within the normal-state nonequilibrium Green function formalism. The current through the system is fully determined by the resonant level Green function [14], which in this case is G > (ω) = −2i i=0,1,2 Γ i [1 − n i (ω)] (ω − E 0 ) 2 + (Γ 0 + Γ 1 + Γ 2 ) 2 ,(8)G < (ω) = 2i i=0,1,2 Γ i n i (ω) (ω − E 0 ) 2 + (Γ 0 + Γ 1 + Γ 2 ) 2 ,(9) with n 1(2) (ω) being the Fermi distribution functions for the reservoirs of spin up electrons and spin down holes, e.g. Fig. 1(b), n 1(2) (ω) = {1 + exp [(ω ± V )/T ]} −1 and n 0 is the distribution function for the bulk of the superconductor, n 0 (ω) = [1 + exp (ω/T )] −1 . The retarded (advanced) components are G R(A) = [ω − E 0 ± i(Γ 1 + Γ 2 + Γ 0 )] −1 . The current through the YSR level is given by I(V ) = ie dω 2π {(Γ 1 − Γ 2 )G < (ω) +[Γ 1 n 1 (ω) − Γ 2 n 2 (ω)][G R (ω) − G A (ω)]},(10) which is twice that of the case of a conventional resonant level [13]. The corresponding differential conductance σ(V ) = dI/dV at zero temperature has a simple two-Lorentzian form, σ = 2G 0 2Γ 1 Γ 2 + Γ 0 Γ 1 (V − E 0 ) 2 + Γ 2 T + 2Γ 1 Γ 2 + Γ 0 Γ 2 (V + E 0 ) 2 + Γ 2 T ,(11) with Γ T = Γ 0 + Γ 1 + Γ 2 . If Γ 0 Γ 1,2 , the heights of the Lorentzian peaks at ±E 0 are proportional to u 2 and v 2 , respectively, wich is the standard density of states result (see Fig 1c, solid line). Only when Γ 0 Γ 1,2 that the symmetric σ(V ) is recovered, e.g., Eq. (7). Finite temperature does not change this conclusion. In view of this result, we conclude that in the STM experiment of Ref. [8], the impurity states cannot be considered to be isolated, i.e., their (unrelated to coupling to STM) linewidth was larger than the electron tunneling rate. On the other hand, the planar tunnel junction experiment of Ref. [7] showed symmetric σ(V ), indicating that the magnetic impurities were sufficiently diluted and decoupled form any extrinsic relaxation baths, so that the tunneling current could drive them out of equilibrium. We note here that since the the crossover from asymmetric to symmetric σ(V ) occurs when Γ 0 ∼ Γ 1,2 , varying Γ 1,2 in STM experiments by means of changing the tunneling distance and lateral tip location, can be used to determine the broadening Γ 0 . Measurement of impurity spin. Spin-polarized tunneling into the YSR state allows to measure the impurity spin orientation. Upon impurity spin reversal, the Bogoliubov quasiparticles transform as E n → −E n , and (u n , v n ) → (v n , −u n ). Spin-polarized STM tip can be modeled by assuming different densities of states for up and down electrons, N t ↑ = N t ↓ . If impurity spin is up, then Γ 1↑ = πt 2 u 2 0 (r 0 )N t ↑ and Γ 2↑ = πt 2 v 2 0 (r 0 )N t ↓ ; for impurity spin down, Γ 1↓ = πt 2 v 2 0 (r 0 )N t ↑ and Γ 2↓ = πt 2 u 2 0 (r 0 )N t ↓ . Since Γ i↑ = Γ i↓ for \|u(r)\| = \|v(r)\|, the value of the current for the two impurity states will be different, and hence can be used to determine the spin orientation. Thus, the presence of YSR state enables the measurement of the local moment orientation. However, as we will now show, it also leads to dephasing of the local moment. From the Hamiltonian (1), the effective magnetic field acting on the local moment is h z = J[ψ † ↑ (0)ψ ↑ (0) − ψ † ↓ (0)ψ ↓ (0)].(12) with the main contribution to the fluctuation of h z deriving from YSR state; the delocalized Bogoliubov quasiparticles can be neglected at low temperatures, as we will show below. That leaves h z = J u 2 0 (0) 2 + v 2 0 (0) γ † 0 γ 0 − Jv 2 0 (0).(13) [Notably, within YSR approximation, the transverse field components are zero since they involve operator combinations γ 2 0 = (γ † 0 ) 2 ]. The spin dephasing time T 2 is related to the fluctuations of this field as 1 T 2 ∼ S 2 ∞ −∞ dt (h z (t) − h z )(h z (0) − h z ) , i.e., its determination reduces to evaluation of the zerofrequency correlation function of the YSR level occupation number. The zero-frequency fluctuations of occupancy reach maximum in the sequential tunneling regime. These fluctuations can be easily determined from the classical rate equations to be Γ 1 Γ 2 /(Γ 1 + Γ 2 ) 3 , which for the dephasing rate yields 1 T 2 seq ∼ J 2 S 2 Γ a ξ 0 6 . (we assumed here that Γ 1 ∼ Γ 2 ≡ Γ). For instance, in the case of Nb the ratio of the coherence length to the lattice constant ξ 0 /a ∼ 100. Taking J ∼ 1eV, and tunneling rate Γ ∼ 10 10 s −1 , which corresponds to the tunnel current of about 0.1 nA, the dephasing time is 10 −8 s. In the low-bias regime, such that \|E 0 \| (T, V ) Γ, the fluctuations can be found using the same Green function formalism as we used to determine current. In this regime, 1 T 2 l.b. ∼ Γ 3 max(T, V ) E 4 0 1 T 2 seq , which, for the same tunneling rate and E 0 of the order of ∆ 0 ∼ 1meV, gives T 2l.b. ∼ 10 −4 s. In this regime, the dephasing rate is proportional to Γ 2 . That the contribution of the delocalized states in the superconductor to spin dephasing can be neglected, can be seen from the following qualitative argument. Let us consider each delocalized state in the same way as we did the YSR state. Since these states are delocalized, their broadening will scale as u 2 (0), v 2 (0) ∼ 1/V . The number of these states is proportional to the sample volume V , and hence their overall contribution will scale as 1/V , vanishing for nonmicroscopic samples. Moving the tip away from the sample one can recover the dephasing and relaxation rates that are governed by thermal excitations, whose density is ∼ e −∆0/T . This long dephasing rate makes localized spin states in superconductors an appealing framework for various quantum computing applications, including those based on Majorana fermions [15,16]. The results obtained here apply not only to YSR states, but to any other localized intragap states in superconductors, e.g. states in the vortex cores [17], or to the case of normal-quantum dot-superconductor junctions [10,18]. In the case of quantum dots, the single particle states in the dot may provide the effective equilibrium reservoir that allows YSR level relaxation that we discussed above [19]. Experimentally it has been found that using a superconducting tip provides a way to sharpen the features associated with tunneling though the YSR state [20]. Theoretically, this problem can also be mapped onto tunneling of spinless particles between two reservoirs with energy dependent densities of states. Unlike in the normal tip case, however, the peaks that appear due to YSR states at ±(\|∆ tip \| + \|E 0 \|) are no longer symmetric [21] even in the absence of additional bath, consistent with experimental findings [20]. FIG . 1. (a) Schematic representation of the problem: Electrons from an STM tip tunnel into a superconductor containing a single YSR state; (b) Effective representation after the particle-hole transformation on the spin-down tip electrons is performed; (c) Differential conductance of the system. The punctured line is the conductance of an "ideal"system, i.e., when the broadening is caused by the couplig to the STM tip only. 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[ "Bib2vec: Embedding-based Search System for Bibliographic Information", "Bib2vec: Embedding-based Search System for Bibliographic Information" ]	[ "Takuma Yoneda \nDepartment of Advanced Science and Technology\nToyota Technological Institute\n2-12-1 Hisakata, Tempaku-kuNagoyaJapan\n", "Koki Mori \nDepartment of Advanced Science and Technology\nToyota Technological Institute\n2-12-1 Hisakata, Tempaku-kuNagoyaJapan\n", "Makoto Miwa [email protected] \nDepartment of Advanced Science and Technology\nToyota Technological Institute\n2-12-1 Hisakata, Tempaku-kuNagoyaJapan\n", "Yutaka Sasaki [email protected] \nDepartment of Advanced Science and Technology\nToyota Technological Institute\n2-12-1 Hisakata, Tempaku-kuNagoyaJapan\n" ]	[ "Department of Advanced Science and Technology\nToyota Technological Institute\n2-12-1 Hisakata, Tempaku-kuNagoyaJapan", "Department of Advanced Science and Technology\nToyota Technological Institute\n2-12-1 Hisakata, Tempaku-kuNagoyaJapan", "Department of Advanced Science and Technology\nToyota Technological Institute\n2-12-1 Hisakata, Tempaku-kuNagoyaJapan", "Department of Advanced Science and Technology\nToyota Technological Institute\n2-12-1 Hisakata, Tempaku-kuNagoyaJapan" ]	[ "Proceedings of the EACL 2017 Software Demonstrations" ]	We propose a novel embedding model that represents relationships among several elements in bibliographic information with high representation ability and flexibility. Based on this model, we present a novel search system that shows the relationships among the elements in the ACL Anthology Reference Corpus. The evaluation results show that our model can achieve a high prediction ability and produce reasonable search results.	10.18653/v1/e17-3028	[ "https://www.aclweb.org/anthology/E17-3028.pdf" ]	1,423,705	1706.05122	4b3082e1f9a6d8df689f7ca24db9fc79166190d5	Bib2vec: Embedding-based Search System for Bibliographic Information April 3-7 2017 Takuma Yoneda Department of Advanced Science and Technology Toyota Technological Institute 2-12-1 Hisakata, Tempaku-kuNagoyaJapan Koki Mori Department of Advanced Science and Technology Toyota Technological Institute 2-12-1 Hisakata, Tempaku-kuNagoyaJapan Makoto Miwa [email protected] Department of Advanced Science and Technology Toyota Technological Institute 2-12-1 Hisakata, Tempaku-kuNagoyaJapan Yutaka Sasaki [email protected] Department of Advanced Science and Technology Toyota Technological Institute 2-12-1 Hisakata, Tempaku-kuNagoyaJapan Bib2vec: Embedding-based Search System for Bibliographic Information Proceedings of the EACL 2017 Software Demonstrations the EACL 2017 Software DemonstrationsValencia, SpainApril 3-7 2017 We propose a novel embedding model that represents relationships among several elements in bibliographic information with high representation ability and flexibility. Based on this model, we present a novel search system that shows the relationships among the elements in the ACL Anthology Reference Corpus. The evaluation results show that our model can achieve a high prediction ability and produce reasonable search results. Introduction Modeling relationships among several types of information, such as nodes in information network, has attracted great interests in natural language processing (NLP) and data mining (DM), since their modeling can uncover hidden information in data. Topic models such as authortopic model (Rosen-Zvi et al., 2004) have been widely studied to represent relationships among these types of information. These models, however, need a considerable effort to incorporate new types and do not scale well in increasing the number of types since they explicitly model the relationships between types in the generating process. Word representation models, such as skip-gram and continuous bag-of-word (CBOW) (Mikolov et al., 2013), have made a great success in NLP. They have been widely used to represent texts, but recent studies started to apply these methods to represent other types of information, e.g., authors or papers in citation networks (Tang et al., 2015). We propose a novel embedding model that represents relationships among several elements in bibliographic information, which is useful to discover hidden relationships such as authors' interests and similar authors. We built a novel search system that enables to search for authors and words related to other authors based on the model using the ACL Anthology Reference Corpus (Bird et al., 2008). Based on skip-gram and CBOW, our model embeds vectors to not only words but also other elements of bibliographic information such as authors and references and provides a great representation ability and flexibility. The vectors can be used to calculate distances among the elements using similarity measures such as cosine distance and inner products. For example, the distances can be used to find words or authors related to a specific author. Our model can easily incorporate new types without changing the model structure and scale well in the number of types. Related works Most previous work on modeling several elements in bibliographic information is based on topic models such as author-topic model (Rosen-Zvi et al., 2004). Although the models work fairly well, they have comparably low flexibility and scalability since they explicitly model the generation process. Our model employs word representationbased models instead of topic models. Some previous work embedded vectors to the elements. Among them, large-scale information network embedding (LINE) (Tang et al., 2015) embedded a vector to each node in information network. LINE handles single type of information and prepares a network for each element separately. By contrast, our model simultaneously handles all the types of information. Method We propose a novel method to represent bibliographic information by embedding vectors to elements based on skip-gram and CBOW. Task definition We assume the bibliographic data set has the following structure. The data set is composed of bib-liographic information of papers. Each paper consists of several categories. Categories are divided into two groups: a textual category Ψ (e.g., titles and abstracts 1 ) and non-textual categories Φ (e.g., authors and references). Figure 1 illustrates an example structure of bibliographic information of a paper. Each category has one or more elements; the textual category usually has many elements while a non-textual category has a few elements (e.g., authors are not many for a paper). Proposed model Our model focuses on a target element, and predicts a context element from the target element. We use only the elements in non-textual categories as contexts to reduce the computational cost. Figure 1 shows the case when we use an element in a non-textual category as a target. For the blackpainted target element in category Φ 2 , the shaded elements in the same paper are used as its contexts. When we use elements in the textual category as a target, instead of treating each element as a target, we consider that the textual category has only one element that represents all the elements in the category like CBOW. Figure 1 exemplifies the case that we consider the averaged vector of the vectors of all the elements in the textual category as a target. We describe our probabilistic model to predict a context element e j O from a target e i I in a certain paper. We define two d-dimensional vectors υ i t and ω i t to represent an element e i t as a target and context, respectively. Similarly to the skip-gram model, the probability to predict element e j O in the context from input e i I is defined as follows: p(e j O \|e i I ) = exp(ω j O ·υ i I + β j O ) (ω j s ,β j s )∈S j exp(ω j s ·υ i I + β j s ) , e j O ∈ Φ, e i I ∈ Ψ ∪ Φ,(1) where β j s denotes a bias corresponds to ω j s , and S j denotes pairs of ω j s and β j s that belong to a category Φ j . As we mentioned, our model considers that the textual category Ψ has only one averaged vector. The vector υ j rep can be described as: υ j rep = 1 n n q=1 υ j q , e j ∈ Ψ(2) 1 Note that we have only one textual category since the categories for texts are usually not distinguished in most word representation models. Figure 1: Example of the bibliographic information of a paper when the target is the element in the non-textual category. The black element is a target and the shaded elements are contexts. Paper Average Element T ! " # $ Non-textual Category where D denotes a set of all the correct pairs of the elements in the data set. To reduce the cost of the summation in Eq. (1), we applied the noisecontrastive estimation (NCE) to minimize the loss (Gutmann and Hyvärinen, 2010). Predicting related elements We predict the top k elements related to a query element by calculating their similarities to the query element. We calculate the similarities using one of three similarity measures: the linear function in Eq. (1), dot product, and cosine distance. Experiments Evaluation settings We built our data set from the ACL Anthology Reference Corpus version 20160301 (Bird et al., 2008). The statistics of the data set and our model settings are summarized in Table 1. As pre-processing, we deleted commas and periods that sticked to the tails of words and removed non-alphabetical words such as numbers Table 1: Summary of our data set and model and brackets from abstracts and titles. We then lowercased the words, and made phrases using the word2phrase tool 2 . We prepared 5 categories: author, paper-id, reference, year and text. author consists of the list of authors without distinguishing the order of the authors. paper-id is an unique identifier assigned to each paper, and this mimics the paragraph vector model (Le and Mikolov, 2014). reference includes the paper ids of reference papers in this data set. Although ids in paper-id and reference are shared, we did not assign the same vectors to the ids since they are different categories. year is the publication year of the paper. text includes words and phrases in both abstracts and titles, and it belongs to the textual category Ψ, while each other category is treated as a non-textual category Φ i . We regard elements as unknown elements when they appear less than minimum frequencies in Table 1. We split the data set into training and test. We prepared 17,475 papers for training and the remaining 2,000 papers for evaluation. For the test set, we regarded the elements that do not appear in the training set as unknown elements. We set the dimension d of vectors to 300 and show the results with the linear function. Evaluation We automatically built multiple choice questions and evaluate the accuracy of our model. We also compared some results of our model with those of author-topic model. Our method models elements in several categories and allows us to estimate relationships among the elements with high flexibility, but this makes the evaluation complex. Since it is tough to evaluate all the possible combinations of inputs and targets, we focused on relationships between authors and other categories. We prepared an evaluation data set that requires to estimate an author from other elements. We removed an (not unknown) author from each paper in the evaluation set to ask the system to predict the removed author considering all the other elements in the paper. To choose a correct author from all the authors can be insanely difficult, so we prepared 10 selection candidates. In order to evaluate the effectiveness of our model, we compared the accuracy on this data set with that by logistic regression. As a result, when we use our model, we got 74.3% (1,486 / 2,000) in accuracy, which was comparable to 74.1% (1,482 / 2,000) by logistic regression. Table 2 shows the examples of the search results using our model. The leftmost column shows the authors we input to our model. In the rightmost two columns, we manually picked up words and authors belonging to a certain topic described in Sim et al. (2015) that can be considered to correspond to the input author. This table shows that our model can predict relative words or similar authors favorably well although the words are inconsistent with those by the author topic model. Figure 3 shows the screenshot of our system. The lefthand box shows words in the word cloud related to the query and the righthand box shows the close authors. We can input a query by putting it in the textbox or click one of the authors in the righthand box and select a similarity measure by selecting a radio button. Discussion When we train the model, we did not use elements in category Ψ as context. This reduced the computational costs, but this might disturbed the accuracy of the embeddings. Furthermore, we used the averaged vector for the textual category Ψ, so we do not consider the importance of each word. Our model might ignore the inter-dependency among elements since we applied skip-grams. To resolve these problems, we plan to incorporate attentions (Ling et al., 2015) so that the model can pay more attentions to certain elements that are important to predict other elements. We also found that some elements have several aspects. For example, words related to an author spread over several different tasks in NLP. We may be able to model this by embedding multiple vectors (Neelakantan et al., 2014). Conclusions This paper proposed a novel embedding method that represents several elements in bibliographic information with high representation ability and flexibility, and presented a system that can search for relationships among the elements in the bibliographic information. Experimental results in Table 2 show that our model can predict relative words or similar authors favorably well. We plan to extend our model by other modifications such as incorporating attention and embedding multiple vectors to an element. Since this model has high flexibility and scalability, it can be applied to not only papers but also a variety of bibliographic information in broad fields. Figure 2 : 2Example when the target is the elements in the textual category Our target loss can be defined as: − (ea,e b )∈D log p(e b \|e a ), Figure 3 : 3Screen shot of the system with the search results for the query "Ryan McDonald". Table 2 : 2Working examples of our model and author topic-model https://github.com/tmikolov/word2vec AcknowledgmentsWe would like to thank the anonymous reviewer for helpful comments and suggestions. Steven Bird, Robert Dale, Bonnie J Dorr, Bryan R Gibson, Mark Thomas Joseph, Min-Yen Kan, Dongwon Lee, Brett Powley, R Dragomir, Yee Fan Radev, Tan, The ACL anthology reference corpus: A reference dataset for bibliographic research in computational linguistics. LRECSteven Bird, Robert Dale, Bonnie J. Dorr, Bryan R. Gibson, Mark Thomas Joseph, Min-Yen Kan, Dong- won Lee, Brett Powley, Dragomir R. Radev, and Yee Fan Tan. 2008. The ACL anthology refer- ence corpus: A reference dataset for bibliographic research in computational linguistics. In LREC. Noisecontrastive estimation: A new estimation principle for unnormalized statistical models. Michael Gutmann, Aapo Hyvärinen, AISTATS. Michael Gutmann and Aapo Hyvärinen. 2010. Noise- contrastive estimation: A new estimation principle for unnormalized statistical models. In AISTATS, pages 297-304. Distributed representations of sentences and documents. V Quoc, Tomas Le, Mikolov, ICML. Quoc V. Le and Tomas Mikolov. 2014. Distributed representations of sentences and documents. In ICML, pages 1188-1196. Not all contexts are created equal: Better word representations with variable attention. Wang Ling, Yulia Tsvetkov, Silvio Amir, Ramon Fermandez, Chris Dyer, Alan W Black, Isabel Trancoso, Chu-Cheng Lin, EMNLP. Wang Ling, Yulia Tsvetkov, Silvio Amir, Ramon Fer- mandez, Chris Dyer, Alan W Black, Isabel Tran- coso, and Chu-Cheng Lin. 2015. Not all contexts are created equal: Better word representations with variable attention. In EMNLP, pages 1367-1372. Distributed representations of words and phrases and their compositionality. Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, Jeff Dean, NIPS. Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S. Cor- rado, and Jeff Dean. 2013. Distributed representa- tions of words and phrases and their compositional- ity. In NIPS, pages 3111-3119. Efficient nonparametric estimation of multiple embeddings per word in vector space. Arvind Neelakantan, Jeevan Shankar, Alexandre Passos, Andrew Mccallum, EMNLP. Arvind Neelakantan, Jeevan Shankar, Alexandre Pas- sos, and Andrew McCallum. 2014. Efficient non- parametric estimation of multiple embeddings per word in vector space. In EMNLP, pages 1059-1069. The authortopic model for authors and documents. Michal Rosen-Zvi, Thomas L Griffiths, Mark Steyvers, Padhraic Smyth, UAI. Michal Rosen-Zvi, Thomas L. Griffiths, Mark Steyvers, and Padhraic Smyth. 2004. The author- topic model for authors and documents. In UAI, pages 487-494. A utility model of authors in the scientific community. Yanchuan Sim, Bryan R Routledge, Noah A Smith, EMNLP. Yanchuan Sim, Bryan R. Routledge, and Noah A. Smith. 2015. A utility model of authors in the sci- entific community. In EMNLP, pages 1510-1519. LINE: large-scale information network embedding. Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, Qiaozhu Mei, WWW. Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu Mei. 2015. LINE: large-scale information network embedding. In WWW, pages 1067-1077.	[ "https://github.com/tmikolov/word2vec" ]
[ "ON THE DIMENSION OF CHOWLA-MILNOR SPACE", "ON THE DIMENSION OF CHOWLA-MILNOR SPACE" ]	[ "Tapas Chatterjee " ]	[]	[]	In a recent work, Gun, Murty and Rath defined the Chowla-Milnor space and proved a non-trivial lower bound for these spaces. They also obtained a conditional improvement of this lower bound and noted that an unconditional improvement of their lower bound will lead to irrationality of ζ(k)/π k for odd positive integers k > 1. In this paper, we give an alternate proof of their theorem about the conditional lower bound.	10.1007/s12044-012-0088-1	[ "https://arxiv.org/pdf/1308.6445v1.pdf" ]	119,283,766	1308.6445	46d853719103d9a3d1307ed9a1f1b60cae67490b	ON THE DIMENSION OF CHOWLA-MILNOR SPACE 29 Aug 2013 Tapas Chatterjee ON THE DIMENSION OF CHOWLA-MILNOR SPACE 29 Aug 2013arXiv:1308.6445v1 [math.NT] In a recent work, Gun, Murty and Rath defined the Chowla-Milnor space and proved a non-trivial lower bound for these spaces. They also obtained a conditional improvement of this lower bound and noted that an unconditional improvement of their lower bound will lead to irrationality of ζ(k)/π k for odd positive integers k > 1. In this paper, we give an alternate proof of their theorem about the conditional lower bound. Introduction For any complex number s ∈ C, with ℜ(s) > 1, one defines the Riemann zeta function as The Riemann zeta function defines an analytic function in the region ℜ(s) > 1 and can be extended meromorphically to the whole complex plane with a simple pole at s = 1 having residue 1. Hurwitz generalized the Riemann zeta function by ζ(s, x) which is defined as ζ(s, x) = ∞ n=0 1 (n + x) s where 0 < x ≤ 1 and s ∈ C with ℜ(s) > 1. He proved that ζ(s, x) can be extended meromorphically to the entire complex plane with a pole at s = 1. Note that for x = 1, ζ(s, 1) is the classical Riemann zeta function. Definition. For integers k > 1, q ≥ 2, define the Chowla-Milnor space V k (q) by V k (q) := Q − span of {ζ(k, a/q) : 1 ≤ a < q, (a, q) = 1}. As described in [1], the conjecture of Chowla and Milnor is the assertion that the dimension of V k (q) is equal to ϕ(q), where ϕ is the Euler's phifunction. Gun, Murty and Rath [1] show that the dimension of the above spaces is at least ϕ(q)/2. They also derived the following theorem. Theorem. Let k > 1 be an odd integer and q, r > 2 be two co-prime integers. Then either dim Q V k (q) ≥ ϕ(q) 2 + 1 or dim Q V k (r) ≥ ϕ(r) 2 + 1. The proof in [1] uses the expansion of Bernoulli polynomials. In this note, we give an alternate proof of the theorem by an explicit evaluation of co-tangent derivatives. Proof of the Theorem The following lemma 1 due to Okada [2] about the linear independence of co-tangent values at rational arguments plays a significant role in proving the theorem. Lemma 1. Let k and q be positive integers with k ≥ 1 and q > 2. Let T be a set of ϕ(q)/2 representations mod q such that the union T ∪ (−T ) constitutes a complete set of co-prime residue classes mod q. Then the set of real numbers d k−1 dz k−1 cot(πz)\| z=a/q , a ∈ T are linearly independent over Q. We first have the following lemma. Lemma 2. For an integer k ≥ 1, D k−1 (π cot πz) = π k × Z linear combination of (csc πz) 2l (cot πz) k−2l , for some non-negative integer l. Here D k−1 = d k−1 dz k−1 . Proof. We will prove this by induction on k. For k = 1, we have D k−1 (π cot(πz)) = π cot(πz). Assume that the statement is true for k − 1, i.e. D k−2 (π cot(πz)) = π k−1 a i (csc πz) 2l i (cot πz) (k−1)−2l i where a i 's are integers. Differentiating both sides with respect to z we get, D k−1 (π cot πz) = π k b i (csc πz) 2l i (cot πz) k−2l i + c i (csc πz) 2l i +2 (cot πz) k−(2l i +2) , where b i , c i 's are integers. This completes the proof of lemma 2. Lemma 3. For an integer k ≥ 2, ζ(k, a/q) + (−1) k ζ(k, 1 − a/q) = (−1) k−1 (k − 1)! D k−1 (π cot πz)\| z=a/q . Proof. L.H.S. = ζ(k, a/q) + (−1) k ζ(k, 1 − a/q) = ∞ n≥0 1 (n + a/q) k + (−1) k ∞ n≥0 1 (n + 1 − a/q) k = ∞ n≥0 1 (n + a/q) k + (−1) k ∞ n=1 1 (n − a/q) k = ∞ n≥0 1 (n + a/q) k + (−1) 2k ∞ n=1 1 (−n + a/q) k = n∈Z 1 (n + a/q) k . Again we know that for z / ∈ Z, π cot πz = n∈Z 1 z + n . This implies that D k−1 (π cot πz) = (−1) k−1 (k − 1)! n∈Z 1 (z + n) k . So, (−1) k−1 (k − 1)! D k−1 (π cot πz)\| z=a/q = n∈Z 1 (n + a/q) k , which completes the proof of lemma 3. Finally, we have lemma 4, whose proof is standard. Lemma 4. Let P be the set of primes. We have ζ(k) p∈P, p\|q (1 − p −k ) = q −k q−1 a=1 (a,q)=1 ζ(k, a/q). Proof of the Theorem. First note that the space V k (q) is also spanned by the following sets of real numbers: {ζ(k, a/q) + ζ(k, 1 − a/q)\|(a, q) = 1, 1 ≤ a < q/2}, {ζ(k, a/q) − ζ(k, 1 − a/q)\|(a, q) = 1, 1 ≤ a < q/2}. Now from the lemma 3, we have the following ζ(k, a/q) + (−1) k ζ(k, 1 − a/q) = (−1) k−1 (k − 1)! D k−1 (π cot πz)\| z=a/q . Applying the above lemma 1, we see that dim Q V k (q) ≥ ϕ(q) 2 . Now from lemma 2 and lemma 3 for an odd integer k, we have ζ(k, a/q) − ζ(k, 1 − a/q) (2πi) k = i 2 k × Q linear combinations of (csc πa/q) 2l (cot πa/q) k−2l . We note that i cot(πa/q) = 1 + ζ a q 1 − ζ a q belongs to Q(ζ q ) and hence so do the numbers csc(πa/q) 2l and cot(πa/q) 2l . Since k is odd, we have ζ(k, a/q) − ζ(k, 1 − a/q) (2πi) k ∈ Q(ζ q )(1) Now we go back to the main part of the proof. Let q and r be two co-prime integers. Suppose that dim Q V k (q) = ϕ(q) 2 . Then the numbers ζ(k, a/q) − ζ(k, 1 − a/q), where (a, q) = 1, 1 ≤ a < q/2 generate V k (q). Now from lemma 4, we get ζ(k) p\|q (1 − p −k ) = q −k q−1 a=1, (a,q)=1 ζ(k, a/q) ∈ V k (q). and hence ζ(k) = (a,q)=1 1≤a<q/2 λ a [ζ(k, a/q) − ζ(k, 1 − a/q)], λ a ∈ Q so that ζ(k) (2πi) k = (a,q)=1 1≤a<q/2 λ a [ζ(k, a/q) − ζ(k, 1 − a/q)] (2πi) k Thus by (1) ζ(k) iπ k ∈ Q(ζ q ). Similarly , if dim Q V k (r) = ϕ(r) 2 , then ζ(k) iπ k ∈ Q(ζ r ) and hence ζ(k) iπ k ∈ Q(ζ q ) ∩ Q(ζ r ). Since any non-trivial finite extension of Q is ramified, if Q(ζ q ) ∩ Q(ζ r ) = Q then there exists a prime which is ramified in Q(ζ q ) ∩ Q(ζ r ), hence both in Q(ζ q ) and Q(ζ r ). Note a prime which ramifies in this intersection must necessarily divide both q and r. This is impossible because (q, r) = 1. So Q(ζ q ) ∩ Q(ζ r ) = Q. Hence we arrive at a contradiction as ζ(k) π k is a real number. Thus dim Q V k (q) ≥ ϕ(q) 2 + 1 or dim Q V k (r) ≥ ϕ(r) 2 + 1. This completes the proof of the theorem. p −s ) −1 . Acknowledgement. I would like to thank Sanoli Gun for helpful discussions. On a conjecture of Chowla and Milnor, Canadian. S Gun, M Ram Murty, P Rath, J. Math. 636S. Gun, M. Ram Murty and P. Rath, On a conjecture of Chowla and Milnor, Cana- dian J. Math. 63(6),2011, 1328-1344. On an extension of a theorem of S. Chowla, Acta Arith. T Okada, 38T. Okada, On an extension of a theorem of S. Chowla, Acta Arith. 38 (1980/81), no. 4, 341-345. Tapas Chatterjee) The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600 113 India E-mail address. Tapas Chatterjee: [email protected]. (Tapas Chatterjee) The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600 113 India E-mail address, Tapas Chatterjee: [email protected]	[]
[ "Camera-based vehicle velocity estimation from monocular video", "Camera-based vehicle velocity estimation from monocular video" ]	[ "Moritz Kampelmühler [email protected] \nInstitute of Electrical Measurement and Measurement Signal Processing\n\n", "Michael G Müller [email protected] \nInstitute of Theoretical Computer Science\nGraz University of Technology\n\n", "Christoph Feichtenhofer [email protected] \nInstitute of Electrical Measurement and Measurement Signal Processing\n\n" ]	[ "Institute of Electrical Measurement and Measurement Signal Processing\n", "Institute of Theoretical Computer Science\nGraz University of Technology\n", "Institute of Electrical Measurement and Measurement Signal Processing\n" ]	[]	This paper documents the winning entry at the CVPR2017 vehicle velocity estimation challenge. Velocity estimation is an emerging task in autonomous driving which has not yet been thoroughly explored. The goal is to estimate the relative velocity of a specific vehicle from a sequence of images. In this paper, we present a light-weight approach for directly regressing vehicle velocities from their trajectories using a multilayer perceptron. Another contribution is an explorative study of features for monocular vehicle velocity estimation. We find that lightweight trajectory based features outperform depth and motion cues extracted from deep ConvNets, especially for far-distance predictions where current disparity and optical flow estimators are challenged significantly. Our light-weight approach is real-time capable on a single CPU and outperforms all competing entries in the velocity estimation challenge. On the test set, we report an average error of 1.12 m/s which is comparable to a (ground-truth) system that combines LiDAR and radar techniques to achieve an error of around 0.71 m/s.	null	[ "https://arxiv.org/pdf/1802.07094v1.pdf" ]	3,409,072	1802.07094	fe6409e8e09d47758d4e71981ad951423bdce212	Camera-based vehicle velocity estimation from monocular video Moritz Kampelmühler [email protected] Institute of Electrical Measurement and Measurement Signal Processing Michael G Müller [email protected] Institute of Theoretical Computer Science Graz University of Technology Christoph Feichtenhofer [email protected] Institute of Electrical Measurement and Measurement Signal Processing Camera-based vehicle velocity estimation from monocular video This paper documents the winning entry at the CVPR2017 vehicle velocity estimation challenge. Velocity estimation is an emerging task in autonomous driving which has not yet been thoroughly explored. The goal is to estimate the relative velocity of a specific vehicle from a sequence of images. In this paper, we present a light-weight approach for directly regressing vehicle velocities from their trajectories using a multilayer perceptron. Another contribution is an explorative study of features for monocular vehicle velocity estimation. We find that lightweight trajectory based features outperform depth and motion cues extracted from deep ConvNets, especially for far-distance predictions where current disparity and optical flow estimators are challenged significantly. Our light-weight approach is real-time capable on a single CPU and outperforms all competing entries in the velocity estimation challenge. On the test set, we report an average error of 1.12 m/s which is comparable to a (ground-truth) system that combines LiDAR and radar techniques to achieve an error of around 0.71 m/s. Introduction Camera sensors provide an inexpensive yet powerful alternative to range sensors based on LiDAR or radar. While LiDAR systems can provide very accurate measurements, they may also malfunction under adverse environmental conditions such as fog, snow, rain or even exhaust gas fumes [30,14]. Arguably vision based sensing is more closely related to how humans engage in driving situations and it should thus be possible to solve any task in autonomous driving based on visual input. This work addresses monocular vehicle velocity estimation, an emerging task in autonomous driving which has not yet been thoroughly explored. The specific task, which forms the base for this work, was introduced as the Autonomous Driving Velocity Estimation Challenge 1 at CVPR2017. The goal is to estimate the relative velocity of a specific vehicle from a sequence of monocular RGB images to aid autonomous driving algorithms such as for example collision avoidance [1] or adaptive cruise control [20]. Figure 1 shows an example image from the data 1 . Vehicle velocity estimation as such is not a new subject of interest, since it is extensively studied in the context of traffic surveillance [17,4], where , however, a stationary camera is employed. Under the restriction of a fixed camera pose the problem becomes significantly less complex, since with a calibrated camera system angular measurements can be obtained and from these measurements velocity estimates can readily be established. In contrast in our case the observer resides on a moving platform and inferring velocity in a similar fashion would re-quire additional information such as camera pose, ego-motion and foreground-background segmentation. Very recent research [41] shows that estimating ego-motion as well as disparity maps from monocular camera images by means of structure from motion is indeed possible, but still limited. Semantic segmentation of scenes, which is a fundamental problem in computer vision, has also more recently been tackled using deep neural networks [6,24]. In a more general sense the given task can be seen as a lightweight version of object scene flow as for example provided in the KITTI benchmark [11,28]. Object scene flow aims at estimating dense 3D motion fields, which in their temporal evolution carry highly valuable information about the geometric constellation of a given scene. Recent approaches [36,35] yield impressive results, but they rely on the availability of stereo image data. Furthermore, they come at the price of very high computational cost, such that the estimation for a temporal frame pair might take 5-10 minutes on a single CPU core. In autonomous driving scenarios computational resources are in general highly limited [13], which makes object scene flow currently not practically feasible. In this work we adopt recent deep learning architectures [18,12] for depth and motion estimation to leverage a mapping of the video input into a beneficial feature space for learning from the few training samples provided. Our approach employs a twostage process for monocular velocity estimation. In a first step we extract vehicle tracks as well as dense depth and optical flow information, followed by locally aggregating these depth and motion cues at the tracked vehicle locations and concatenating over the temporal dimension. After this feature extraction procedure we use the spatiotemporal depth, flow and location features to train a fully connected regression network for velocity estimation of the respective vehicles. Further on we conduct an extensive ablation study to investigate the impact of the individual features and combinations thereof on the regression performance as well as on the runtime of the estimation. We show that a light weight implementation can achieve excellent results, and that leveraging deep motion and depth cues does not necessarily improve performance for this task on the given data. Related Work Tracking. Object tracking is one of the fundamental problems in computer vision and has been extensively studied [40] and applied in many different tasks. Median Flow [21] is a method building on top of the Lucas-Kanade [25] method, which is an early optical flow algorithm operating on local intensity changes. This method is extended by a Forward-Backward error, which denotes the deviation between the trajectories obtained by tracking from I t−1 → I t and I t → I t−1 . Using Forward-Backward error, robust predictions can be identified. The Multiple Instance Learning tracker [2] first transforms the image into an appropriate feature space, and uses a classifier as well as a motion model to determine the presence of an object in a frame, which is referred to as tracking by detection. More recent methods like [15] employ convolutional neural networks to learn motion and appearance of objects. The feature maps of higher convolutional layers provide robust and accurate appearance representations, but lack spatial resolution. Lower layers on the other hand provide higher spatial resolution and less refined appearance representations. This hierarchical structure is used in [26] by inferring responses of correlation filters on each corresponding layer pair. Monocular Depth Estimation. Estimating the depth information of a scene seen from a given angle using only a single camera is not a well-posed problem. Some of the methods tackling this problem [31,39] apply supervised learning regimes requiring ground truth depth data. Since acquiring such ground truth data with sufficient accuracy requires immense effort, several methods are being developed that require little to no supervision. In [23] a semi-supervised approach is described that provides a fusion between using sparse ground truth data from a LiDAR sensor and stereo view synthesis, i.e. estimating one image in a stereo pair from the other, to infer dense depth maps. Others [12,38] in turn rely solely on stereo as a supervision signal, which comes with the benefit of easily available or obtainable data. Some recent work [41,10] shows the capability of inferring single image depth from monocular video only. This is achieved by leveraging the (small) temporal motion of the camera and its thus changing pose to learn from multiple views of the scene. Via novel view synthesis the future camera frames can be used as a self-supervision signal. The mapping thus learned implicitly carries information about the 3D scene geometry. Optical Flow Estimation. Optical flow is widely used in computer vision, e.g. for video object detection [42], to quantify pixel-wise motion in between frames of a moving scene. Traditional methods [16,37] employ variational motion estimation approaches, that regard the estimation of pixel level correspondences between frames as an optimization problem. With the growing interest in deep learning also optical flow estimation is now often successfully treated as a supervised learning problem [7]. This is achieved by employing convolutional neural networks for feature extraction and aggregation, followed by an 'upconvolutional' network, which concatenates the feature maps from the corresponding convolutional layers and jointly applies transposed convolution to increase spatial resolution. Further improvements on this approach have since been made [18] that provide improved performance and robustness as well as scalability. Vehicle velocity estimation We present an explorative study over features for vehicle velocity estimation from monocular camera videos and discuss effectiveness as well as significance of the methods employed. The task is to estimate the relative velocity as well as position of given vehicles seen in short dashcam video snippets. Our overall approach, shown in Figure 2, consists of a two stage process: First, features subsidiary to the task are extracted to subserve the second stage, which is a light-weight Multilayer Perceptron (MLP) architecture working on these features to regress velocity and positions of vehicle instances. Feature Extraction For the estimation of vehicle velocity, the raw RGB video data is first transformed into a beneficial feature space. We use three feature types of complementary nature: Vehicle tracks (i.e. trajectories of the 2D object outline over time), depth (i.e. disparity estimates from monocular imagery) and motion (i.e. optical flow estimates between consecutive frames). The remainder of this section describes the specific algorithmic instances used for extracting these cues. Tracks. For a given vehicle defined by a bounding box in a single frame, tracking over the temporal extent of the input serves for all further processing steps. A variety of well functioning tracking algorithms are readily available in literature. Since we aim for a lightweight tracker that should precisely localize the object outline, we employ fast trackers that operate directly at the pixel level, the Median Flow [21] and MIL [2] trackers, both implemented in the OpenCV library [3]. The Median Flow tracker comes with the benefit of being able to adapt bounding boxes over the trajectories, and provides a tight bounding box that can be a very useful feature when estimating the relative velocity of the objects. However, this tracker is unstable for occlusions since it employs forward-backward tracking. Whenever Median Flow detects a tracking failure the missing bounding boxes are substituted with MIL tracks. Depth. For dense depth map prediction we employ a recently described deep architecture [12], that learns monocular depth map prediction via novel view synthesis in a stereo environment. This is achieved by synthesizing e.g. the left camera image from the right, where the left camera image is used as supervision signal. The warping operation that is thus learned implicitly carries information on the disparities in the source image. We use a model pretrained on KITTI [11] and Cityscapes [5] stereo images for predicting the dense disparity maps (note that this model is trained without disparity groundtruth). Owing to architectural constraints and limited computational capacity the RGB input images are resized to 512x256p. Motion. Finally, we extract motion information by extracting dense optical flow maps using a state-ofthe-art neural network architecture, FlowNet2 [18]. FlowNet2 treats optical flow estimation as a supervised learning problem, where a convolutional neural network is trained on a volume of two stacked input frames with ground truth optical flow as a supervision signal. In our case we use a FlowNet2 architecture pre-trained on the synthetic Flying Chairs [8] dataset to calculate 39 dense u, v flow maps from 512x256p input images. A sample of the extracted feature maps is shown in Figure 3. In both feature maps the vehicle can be segmented from the background, but the capabilities are limited to close ranges. Transformation into feature space. The depth and motion cues are computed globally and further processed to serve as lightweight input features for a regression model. This is achieved by locally aggregating the dense predictions within the bounding boxes established by the tracking stage. The local aggregation is achieved by calculating the mean over the estimates within each tracked bounding box, after shrinking the box by 10% in width and height, which reduces the variance since flow and depth cues tend to be inaccurate at the object boundaries. Sub- sequently, the aggregated feature vectors are temporally smoothed using a Gaussian kernel of width 5, which is chosen in correspondence to the frame skip of 5 in the learning stage. For optical flow, this procedure is carried out individually for each the horizontal and vertical component u, v. The temporal smoothing provides robustness to short-term deviations of the camera orientation (e.g. caused by bumps when driving on a highway). Neural Network Model These pre-computed features allow us to use relatively shallow fully-connected neural networks. This is especially advantageous for the task at hand, in which the number of given training examples is relatively small and learning can take advantage of highly abstract features. We employ a comparatively simple, rather small and thus efficient 4-layer MLP architecture to regress from feature space to vehicle velocities. Data Split. The relationship of the pre-computed features to the learning output is highly nonlinear. Consider for instance the size of bounding boxes around other vehicles: they rapidly decrease when the vehicle is close to the camera, while remaining more or less constant if the vehicle is far away regardless of the velocity. To facilitate learning in these distinct regimes, we split the data set into three disjoint parts, based on the distance from the observer (near/mid/far), and train separate models for each. Figure 4 shows the bounding box size as a function of the distance from the observer for the training set, as well as our chosen splits. As indicator for the relative distance of the target vehicle we use the bounding box area of the last frame that the vehicle is shown. [32] as hidden unit activations; thus, all layer output sizes are twice the input size. The output layer activation is linear. We use an L2 loss to train the network by using the groundtruth velocity values, obtained from a combination of LiDAR and radar sensors. Training. Each individual network is trained on the MSE between network output and targets for 2000 epochs using minibatches of 50 samples. For regularization, weight decay of 10 −5 and Dropout [34] of 0.2 were used. We train using the ADAM optimizer [22]. To make use of all available training data, we use a partitioning scheme similar to k-fold cross-validation and train multiple networks per distance set. Each of them is split up into five partitions. Four of these are used as training set, the fifth is used for validation. After training for 2000 epochs on each possible combination, the model with the lowest validation error is saved. This results in 3 × 5 models for the entire dataset. Note that the number of examples per neural network is quite small, so overfitting may occur easily as the validation score can no longer be regarded as an accurate estimate of the error on the test set in most cases. The training data not only contains targets for the target vehicle velocity, but also for the relative vehicle position. In other machine learning domains, it is well-known that auxilliary targets can increase the learning performance (see e.g. for reinforcement learning [29,19,9]). We also make use of all targets during training and simultaneously regress for vehicle velocities and distance. Early stopping is performed using only the performance on the actual targets. This leads to slight increases in performance. Model Averaging. Above, we described how 3 × 5 partitions are trained. When evaluating the test set, we split the data according to the computed bounding box areas using the same procedure as for the training set. Then, the average over all five models for the respective distance is computed. This gives the final estimation for the relative vehicle velocity. Experiments Dataset The provided dataset includes 1074 driving sequences in freeway traffic, recorded by a single HD (1280x720p) camera, each 40 frames long captured at 20fps, as well as camera calibration matrices. For each sequence specific vehicles are annotated with a bounding box as well as position and velocity in both x and y coordinates in the last frame only, resulting in a total of 1442 annotated vehicles. For evaluation the individual vehicles are classified into three clusters according to their ground truth relative distance d in the last frame. d < 20 m is considered near range ( 12% of samples), 20 m ≥ d > 45 m medium range ( 65% of samples) and d ≥ 45 m far range ( 23% of samples). For each of those ranges different difficulties arise in the estimation. While in near range examples the perspective on vehicles can shift drastically in between instances and over time for individual instances, for far range samples the estimation is limited by the pixel resolution of the data. From the provided data a method should be developed, which is able to infer velocity as well as position of vehicles specified by a given bounding box in the last frame. For evaluation a test set consisting of 269 clips, or 375 vehicle tracks is provided, which is structurally identical to the training data, with the only difference being the absence of ground truth position and velocity data. Ablation study To investigate the impact of the features used in our initial approach on velocity estimation accuracy, we have conducted an ablation study to test combinations of all features and activation functions. Validation split. In order to be able to evaluate the performance of our approach we had to first split the available training data into a training and validation set. Earlier we used 80/20 random splits, but for more robust evaluation we decided to take into account the distribution of near, medium and far range samples (12/65/23%) and also the similarity of the test sequences. Since the training samples are sequences from a fixed set of drives, we decided to split training and validation splits such that the validation samples stem from unseen drives. To achieve this we compute the 4096 dimensional VGGNet fc2 feature vector [33], reduce it to 2D space using t-sne [27] and then cluster the resulting features into 7 clusters using k-means. We have then chosen a cluster consisting of 10% of the total training samples with a (14/63/21%) near, medium, far range distribution as validation set. The remaining data is used for training after defining a fixed 5-fold split for cross validation. Training. For each of the 5 train/test splits, 10 models with fixed 3 × 40 topologies are trained according to the same paradigm as described above. Here, training was stopped early if the velocity mean squared error (1) did not improve in the course of 500 iterations. The ablation was performed both training models on the whole data as well as training individual models on near, medium and far data splits. Results. The results of the ablation study described above are shown in Table 1. It can be seen that depth and motion cues can indeed help improve results of vehicle velocity estimation, but their impact is limited by range. Both depth and optical flow estimates show significantly degrading performances for distances larger than 20m (above near range). Within near range on the other hand, where rapid changes in perspective can occur, they are superior in performance to using only tracking. For medium range, where vehicles are predominantly viewed from their rear only, tracking cues only yield superior performance. For far range the pixel level appearance of vehicles changes so slightly at different velocities, that a robust estimation of their velocity can not be achieved by the method used. Hardware GPU GPU CPU GPU Timing 344 ms 69 ms 7 ms 3 ms Table 2: Runtime of the individual stages of our method for a single frame on Intel®Core TM i7-5930K, 32GB, NVIDIA Titan X. For depth and motion estimation inference times only are indicated. Motion Depth Tracking MLP Using tracking cues only is also the most efficient, as shown in Table 2. Inference of our model is generally very fast and thus the feature extraction is the biggest performance bottleneck of our approach. While using motion, depth and tracking cues provides a theoretical estimation rate of 2 frames per second, with tracking cues 100 frames can be estimated in one second. Challenge results On the leaderboard of the CVPR2017 Autonomous Driving velocity estimation challenge, the individual entries are ranked by overall average velocity mean squared error. For each given set of samples C, in this case the ranges near, medium and far, the error is evaluated according to (1) and then averaged (2). E V,C = 1 \|C\| c∈C \|\|V c −V c \|\| 2 (1) E V = E V,near + E V,med + E V,f ar 3 . ( In correspondence to the given evaluation metric, the results achieved on the test set are depicted in Table 3. The entry Ours full is the winning approach described above, using individual models trained on all three ranges separately, while incorporating tracking, flow and depth features. Ours tracking are the best performing models using only tracking cues obtained in the ablation study, which are again separate models for each range, but trained on the whole data. Notably, the ground-truth accuracy is at around 0.71 m/s which is obtained from a combination of LiDAR and radar sensors (this number was communicated by the challenge organizers). Therefore, our average estimation 1.12 m/s error (corresponding to 1.25 m 2 /s 2 MSE) is relatively close to a LiDAR and radar solu- Figure 5: Qualitative results on the validation data, near medium and far range (f.l.t.r.). In each example one frame of a sequence from the validation split is shown, annotated with the estimated velocity in m/s for (x, y) coordinates. Blue is ground truth, green the estimated velocity. tion, but uses only videos recorded from a monocular dash cam. E V E V,near E V,med E V, Conclusion This paper documents the winning entry at the CVPR2017 vehicle velocity estimation challenge. We have proposed a light-weight approach for directly regressing vehicle velocities from their tracks in monocular video sequences. By comparing complementary features for vehicle velocity estimation, we find that light-weight trajectory based features outperform depth and motion cues extracted from deep ConvNets. Our approach is real-time capable on a single CPU and outperforms all competing entries in the velocity estimation challenge. Future work shall address an end-to-end system for joint tracking and estimation. Figure 1 : 1A sample image from a training sequence. Velocity and position ground truth are provided for the vehicles surrounded by the green bounding boxes. Figure 2 : 2Overview of our overall architecture (see Section 3 for details). Our proposed light-weight architecture only uses the trajectory features estimated by the tracker. Figure 3 : 3Optical flow (left, Middlebury flow encoding), bounding box and depth map (right, darker values represent larger distances) of a close range sample vehicle overlayed over the RGB image. Figure 4 : 4Split of distance categories for neural network. For the labeled training set, the true distance is known. The empirically chosen boundaries are shown in green. Table 3 : 3Challenge leaderbord top 5. Ours full is the winning approach. Ours tracking denotes the best performing tracking only model. *submitted post dead-line Acknowledgments. We are grateful for discussions with Axel Pinz. The GPUs used for this research were donated by NVIDIA. Perception for collision avoidance and autonomous driving. 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[ "Real Fermionic Symmetry in Type II Supergravity", "Real Fermionic Symmetry in Type II Supergravity" ]	[ "Hadi Godazgar ", "Malcolm J Perry \nPerimeter Institute for Theoretical Physics\n31 Caroline St. NN2L 2Y5WaterlooOntarioCanada\n", "† Damtp ", "\nCentre for Mathematical Science\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeEngland\n" ]	[ "Perimeter Institute for Theoretical Physics\n31 Caroline St. NN2L 2Y5WaterlooOntarioCanada", "Centre for Mathematical Science\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeEngland" ]	[]	It is known that the transformations of fermionic T-duality, derived from the worldsheet theory, generically transform real supergravity backgrounds to complex supergravity backgrounds. We consider the low-energy target space theory and show that the type II supergravity equations admit a symmetry that transforms the Ramond-Ramond fields and the dilaton. The transformations given by this symmetry involve Killing spinors and include the transformations of Berkovits andMaldacena. However, we show that they also allow real transformations.	10.1007/jhep01(2011)032	[ "https://arxiv.org/pdf/1008.3128v2.pdf" ]	96,471,348	1008.3128	25e99999e2d219b8cdbbef30cf1bab3f44e0164f	Real Fermionic Symmetry in Type II Supergravity 13 Jan 2011 January 14, 2011 Hadi Godazgar Malcolm J Perry Perimeter Institute for Theoretical Physics 31 Caroline St. NN2L 2Y5WaterlooOntarioCanada † Damtp Centre for Mathematical Science University of Cambridge Wilberforce RoadCB3 0WACambridgeEngland Real Fermionic Symmetry in Type II Supergravity 13 Jan 2011 January 14, 2011 It is known that the transformations of fermionic T-duality, derived from the worldsheet theory, generically transform real supergravity backgrounds to complex supergravity backgrounds. We consider the low-energy target space theory and show that the type II supergravity equations admit a symmetry that transforms the Ramond-Ramond fields and the dilaton. The transformations given by this symmetry involve Killing spinors and include the transformations of Berkovits andMaldacena. However, we show that they also allow real transformations. Introduction In the past two decades, T-duality [1,2] has been highly successful in helping increase our understanding of string theory. In the 1990s, S, T and U-dualities were used to relate the five different string theories. The existence of these dualities was crucial in the conjecture that the five string theories are different limits of a theory that is the strong coupling limit of type IIA string theory, M-theory [3,4]. T-duality has also provided insights into the study of D-branes, which would perhaps be inaccessible otherwise. For example, Myers [5] used consistency of the world-volume action for the D-brane with T-duality to find the correct coupling of background Ramond-Ramond fields to D-branes, which was used to discover the Myers effect. Furthermore, another important application of T-duality has been to use it to generate solutions in supergravity [6]. In Buscher's formulation of T-duality [7], a shift symmetry of a target space coordinate, which corresponds to an isometry in the target space of the sigma-model, is used to make a field redefinition in the sigma model. The new sigma model is classically of the same form as the original sigma model except for the sigma model couplings, i.e. the metric and the 2-form field, which are different. The two sigma models are equivalent quantum-mechanically if the dilaton also transforms. This shows that the string theories described by the two sigma models with different couplings, which correspond to different backgrounds for the string theories, are equivalent. The transformed background fields are related to the original fields by the killing vector that corresponds to the isometry. Recently, this idea has been generalised to the case where the target space has a fermionic isometry, or supersymmetry, as opposed to an isometry, to find a duality of tree-level type II string theory, fermionic T-duality [8,9]. Under this duality the background Ramond-Ramond fields and the dilaton transform and the metric and the NSNS 2-form field are invariant. Analogously to T-duality, the transformation of the background supergravity fields are given by the Killing spinors corresponding to the supersymmetry in superspace. Given the success of T-duality, we expect that fermionic T-duality will also make important contributions to our understanding of string theory. In fact, fermionic T-duality was introduced to explain the dual superconformal symmetry of planar scattering amplitudes in N = 4 super Yang-Mills theory [8,9], which has no obvious origin in the weak coupling computations of these amplitudes in which this symmetry was found [10]. There have also been other studies of fermionic T-duality [11][12][13][14][15]. However, in contrast to T-duality, fermionic T-duality generically transforms a real supergravity background into a complex supergravity background. An ordinary T-duality along a time-like direction can then be applied to get back a real background. This means that the application of fermionic T-duality as a solution generating mechanism, one of the key applications of T-duality, is limited to supergravity solutions with a timelike Killing vector. In this paper, we consider fermionic T-duality from the spacetime viewpoint, rather than the worldsheet perspective in which it was found. We consider a general ansatz for the transformation of the Ramond-Ramond fields and the dilaton involving Killing spinors in both type IIA and type IIB supergravity. We then systematically impose that the supergravity equations are invariant under this transformation, i.e. we impose that the transformed fields are solutions of the supergravity equations. We find that the symmetry includes fermionic T-duality, as it must, but it also admits real transformations of the supergravity fields. The structure of the paper is as follows. In section 2, we review Büscher's T-duality and show how a symmetry in the target space allows a field redefinition in the worldsheet theory that gives rise to a duality. We also review the transformation rules of fermionic T-duality [8] in this section. Then, in section 3, we set our conventions by stating the type IIA supergravity Lagrangian and equations, and we construct the symmetry for type IIA supergravity. In section 4, we construct an analogous symmetry for type IIB supergravity. Finally, in section 5, we make some comments and outline future work. Review of T-duality and fermionic T-duality Büscher [7] showed that T-duality in curved backgrounds arises as a symmetry of the sigma-model. Consider the bosonic string sigma-model S = 1 4πα ′ d 2 σ √ hh αβ g ab ∂ α X a ∂ β X b + ǫ αβ B ab ∂ α X a ∂ β X b + α ′ √ hR (2) φ(X) . The field X a is the position of the point (σ 1 , σ 2 ) on the worldsheet in spacetime; g ab is the metric on the target space; B ab is the antisymmetric gauge potential; φ is the dilaton and R (2) is the curvature of the worldsheet metric h. Imposing conformal invariance in the quantum theory gives the equations of motion for the background fields [16]. In 26 dimensions, the equations of motion for the metric, two-form field and dilaton are R ab − 1 4 H cd a H bcd + 2∇ a ∇ b φ = 0, ∇ c H c ab − 2 (∂ c φ) H c ab = 0, 4 (∂φ) 2 − 4 φ − R + 1 12 H 2 = 0,(1) respectively. The tensor R ab is the Ricci tensor associated to the metric on the target space and H abc = 3∂ [a B bc] . If there is a Killing vector in the target space, k, then we can choose a coordinate system-we will let X a be such a coordinate system-in which k = ∂/∂X 0 . In this coordinate system, the metric, two-form field and the dilaton are independent of the X 0 coordinate. We can then write ∂ α X 0 = V α , in the action, but we must impose the constraint that V α is exact. For Euclidean worldsheets of spherical topology we can impose this constraint using a Lagrange multiplier term ǫ αβX 0 ∂ α V β . So we can write the action as S = 1 4πα ′ d 2 σ √ hh αβ g 00 V α V β + 2g 0i V α ∂ β X i + g ij ∂ α X i ∂ β X j +ǫ αβ 2B 0i V α ∂ β X i + B ij ∂ α X i ∂ β X j + 2ǫ αβX 0 ∂ α V β + α ′ √ hR (2) φ(X) , where a = (0, i). The equation of motion forX 0 gives that V is closed, which for a spherical worldsheet implies that V is exact, so we get back the original theory. The V equation of motion is V α = − 1 g 00 g 0i ∂ α X i + ǫ β α √ h B 0i ∂ β X i + ∂ βX 0 . Integrating the action over V we get the dual action that has the same form as the original action except that the metric and the two-form field are now g 00 = 1 g 00 , g 0i = B 0i g 00 ,B 0i = g 0i g 00 , g ij = g ij − g 0i g 0j − B 0i B 0j g 00 , B ij = B ij − g 0i B 0j − B 0i g 0j g 00 .(2) We would like to impose the condition that the T-dual theory is also conformally invariant. This can be imposed at one-loop using either the results of reference [16], i.e. equations (1), or by considering the change in the measure of the path-integral. Either method suggests that the dilaton is shifted [7] toφ = φ − 1 2 log g 00 .(3) Note that if we take the special case of toroidal compactification on a flat background, then we get the well-known result that the radius of the compactification circle is inverted and the string coupling constant, which is the exponential of the expectation value of the dilaton, is modified by a factor of √ α ′ /R. The argument given here is valid only for spherical worldsheets, hence the duality has only been proved to first order in string perturbation theory. By gauging the isometry the duality can be extended to higher genus worldsheets, but in this case the isometry orbits must be compact, or in other words the shift symmetry has to be along a compact coordinate [17]. Recently, Berkovits and Maldacena [8] have generalised Büscher's formulation of T-duality to the case where the worldsheet action is invariant under constant shifts of spacetime fermionic coordinates θ J , J = 1, . . . n. They show that under this duality the metric and the NS-NS 2-form potential do not change, and they give the transformation of the Ramond-Ramond fields in terms of the bispinor field strength. In type IIA string theory the bispinor field strength is F = 1 2 F (2) a 1 a 2 γ a 1 a 2 + 1 4! F (4) a 1 ...a 4 γ a 1 ...a 4 γ 11 , where the 2-form F (2) and 4-form F (4) are the RR field strengths. In our notation, γ−matrices are 32 by 32 matrix representations of the ten-dimensional Clifford algebra; two Majorana-Weyl spinors describing N = 2A supersymmetry are combined into a single Majorana-Dirac spinor. In type IIB string theory the Ramond-Ramond field strengths are the 1-form F (1) , 3-form F (3) and the self-dual 5-form F (5) , and we define the R-R bispinor field strength by F = F (1) a γ a σ 1 + 1 3! F (3) a 1 ...a 3 γ a 1 ...a 3 iσ 2 + 1 2.5! F (5) a 1 ...a 5 γ a 1 ...a 5 σ 1 . In type IIB theory, two Majorana-Weyl spinors, ε andε, with the same chirality are combined into an SO(2) vector ǫ = ε ε , which are rotated amongst each other by acting with e iσ 2 θ , and on which Pauli matrices act on in the obvious way. Fermionic T-duality transforms the bispinor field strength in the following way: e φ ′ F ′ = e φ F ± 32 N I,J=1 (ε I ⊗ε J ) M IJ ,(4) where we take + in type IIB theory and − in type IIA string theory. We will always combine two type IIA Weyl spinors into a Dirac spinor and type IIB spinors into an SO(2) vector. However, in equation (4), and only there, the spinors are Weyl Killing spinors. Furthermore, under fermionic T-duality, the dilaton is transformed to φ ′ = φ + 1 2 n I=1 log 1 2 M −1 II ,(5) where M −1 satisfies ∂ a M −1 IJ = 2ǭ I γ a γ 11 ǫ J for type IIA theory, and ∂ a M −1 IJ = 2ǭ I γ a σ 3 ǫ J for type IIB theory-of course ǫ has a different meaning for each theory, as described above. The spinors, ǫ J , are Killing spinors corresponding to the constant shift symmetry of the fermionic coordinate θ J in superspace θ I → θ I + ρ I , where ρ I is a Grassmann-valued constant. Since this symmetry is abelian, {ǫ I Q I , ǫ J Q J } = 0, where there is no summation over I and J. However, from the supersymmetry algebra {ǫ I Q I , ǫ J Q J } =ǭ I γ a ǫ J P a , where P is the generator for translations. Therefore, ǫ I γ a ǫ J = 0,(6) for all I, J = 1, . . . , n ∈ Z + . This condition can only be satisfied non-trivially for complex ǫ, so, in general, the transformation sends real supergravity backgrounds into complex backgrounds. Fermionic T-duality preserves the number of supersymmetries, which is necessary in order for it to be a duality of string theory. Explicitly, the Killing spinors in the T-dual theory are ǫ ′ I = M IJ ǫ J .(7) 3 Type IIA supergravity symmetry In our conventions, summarised in appendix A, the type IIA supergravity action is S = 1 2κ 2 d 10 x √ g e −2φ R + 4(∂φ) 2 − 1 12 H 2 − 1 2 1 2 F (2) 2 + 1 4! F (4) 2 − 1 144 1 √ g ǫ ∂C (3) ∂C (3) B . (8) The first square bracket is the action for the NSNS fields, the metric g, the 2-form field B, and the dilaton. The second set of terms constitute the action for the RR fields, the 1-form potential C (1) , and the 3-form potential C (3) . The last term is the Chern-Simons term. The field strengths, H, F (2) , F (4) are defined by H = dB, F (2) = dC (1) , F (4) = dC (3) − H ∧ C (1) . The Bianchi identities for the field strengths are dH = 0,(9)dF (2) = 0,(10)dF (4) − H ∧ F (2) = 0.(11) The equations of motion are d e −2φ ⋆ H + F (2) ∧ ⋆F (4) − 1 2 F (4) ∧ F (4) = 0,(12)d ⋆ F (2) + H ∧ ⋆F (4) = 0,(13)d ⋆ F (4) − H ∧ F (4) = 0.(14) The Einstein equation is R ab = − 1 4 g ab φ + 1 2 g ab (∂φ) 2 − 2∇ a ∇ b φ + 1 4 H acd H cd b − 1 12 g ab H 2 + 1 2 e 2φ F (2) ac F (2) c b − 1 16 g ab F (2) 2 + 1 12 e 2φ F (4) acde F (4) cde b − 3 32 g ab F (4) 2 ,(15) and the dilaton equation of motion is R + 4 φ − 4 (∂φ) 2 − 1 12 H 2 = 0.(16) We can check the consistency of these equations by showing that the contracted Bianchi identity holds. Indeed, using the Bianchi identities and equations of motion for the field strengths and the Einstein equation, ∇ a (R ab − 1 2 g ab R) = −∂ b φ R + 4 φ − 4 (∂φ) 2 − 1 12 H 2 , which vanishes by the dilaton equation of motion. The Killing spinor equations from the variations of the gravitino and dilatino are ∇ a ǫ − 1 8 H abc γ bc γ 11 ǫ − 1 16 e φ F (2) bc γ bc γ a γ 11 ǫ + 1 192 e φ F (4) bcde γ bcde γ a ǫ = 0,(17)γ a ∂ a φ − 1 12 H abc γ abc γ 11 − 3 8 e φ F (2) ab γ ab γ 11 + 1 96 e φ F (4) abcd γ abcd ǫ = 0,(18) respectively. We will consider a transformation in the RR fields only. We also allow the dilaton to transform because at the quantum level this restores the conformal invariance of the string sigma model. For a string theory to be quantum mechanically consistent, it can be shown that the background fields must satisfy the supergravity equations of motion by imposing the vanishing of the beta-function [16,18], or imposing κ−invariance of the Green-Schwarz action [19,20]. The other NSNS fields, the metric and the 3-form field strength H, are invariant under the transformation. We consider the most general ansatz for the transformation of the fields: e φ F (2) ab → e φ ′ F ′(2) ab = e φ F (2) ab +ǭ I γ ab (S 1 + S 2 γ 11 )η J M IJ , e φ F (4) abcd → e φ ′ F ′(4) abcd = e φ F (4) abcd +ǭ I γ abcd (S 3 + S 4 γ 11 )η J M IJ ,(19) where φ ′ , M IJ , S 1 , . . . , S 4 are arbitrary functions, and spinors ǫ I , η I , I = 1, . . . , n, satisfy the gravitino and dilatino Killing spinor equations. Both of the RR field strengths transform with the same spinors and functions, for if they transformed with different spinors and functions, then, for example, upon requiring that the Bianchi identity for the transformed 4-form field strength, equation (11), holds, they would be identified. We will identify them from the onset in the interests of clarity and terseness. Furthermore, from equation (12), φ ′ can be identified as the transformed dilaton, which we will write as φ ′ = φ + X, where X is an unknown function. Let us consider each Bianchi identity and equation of motion in turn. First, consider the transformed Bianchi identity for the RR 2-form, equation (10). Using the gravitino Killing spinor equation ∇ [a F ′(2) bc] = ∇ [a e −X F (2) bc] + e −(φ+X)ǭ I γ bc] (S 1 + S 2 γ 11 ) η J M IJ = −e −X F(2)[bc + e −φǭ I γ [bc (S 1 + S 2 γ 11 ) η J M IJ ∂ a] X − e −(φ+X)ǭ I γ [bc (S 1 + S 2 γ 11 ) η J M IJ ∂ a] φ + e −X ǫ α I η β J 1 4 e −φ H de[a S 1 γ bc] γ de γ 11 (αβ) + S 2 γ bc] γ de [αβ] + 1 8 F (2) de S 1 γ [bc γ de γ a] γ 11 (αβ) − S 2 γ [bc γ de γ a] [αβ] − 1 96 F (4) def g S 1 γ [bc γ def g γ a] γ 11 (αβ) − S 2 γ [bc γ def g γ a] [αβ] M IJ + e −(φ+X)ǭ I γ [bc ∂ a] [(S 1 + S 2 γ 11 ) M IJ ] η J .(20) The Greek indices, α, β = 1, . . . , 32, are spinor indices. Now we can use the dilatino Killing spinor equation to express the term involving the derivative of the dilaton in the expression above in terms of the field strengths. As ǫ I and η J satisfy the dilatino Killing spinor equation, (18), ǫ α I γ abc γ d ∂ d φ − 1 12 H def γ def γ 11 − 3 8 e φ F (2) de γ de γ 11 + 1 96 e φ F (4) def g γ def g (αβ) η β J = 0.(21)Using 3 γ a 1 ...am γ b 1 ...bn = min(m,n) k=0 c k mn γ [b 1 ...b n−k [a 1 ...a m−k δ b n−k+1 a m−k+1 . . . δ bn] am] ,(22) where c k mn = (−1) kn+ 1 2 k(k+1) m!n! k!(m − k)!(n − k)! , and (γ a 1 ...an ) αβ = (−1) 1 2 n(n+1)+1 (γ a 1 ...an ) βα , (γ a 1 ...an γ 11 ) αβ = (−1) 1 2 n(n+3) (γ a 1 ...an γ 11 ) βα ,(23) equation (21) implies that ǫ I γ [bc η J ∂ a] φ − 1 12 H de[aǭI 3γ de bc] − 2δ d b δ e c] γ 11 η J − 1 8 e φ F (2) deǭ I γ de abc − 6γ [a δ d b δ e c] γ 11 η J + 1 24 e φ F (4) def gǭ I γ def [ab δ g c] − 2γ d δ e a δ f b δ g c η J = 0. Similarly, ǫ I γ [bc γ 11 η J ∂ a] φ − 1 12 H de[aǭI 3γ de bc] − 2δ d b δ e c] η J − 3 4 e φ F (2) deǭ I γ d [bc δ e a] η J − 1 288 e φ F (4) def gǭ I γ def g abc − 36γ de [b δ f c δ g a] γ 11 η J = 0. Substituting the two equations above in equation (20) and using the gamma matrix identities (22) and (23), equation (20) becomes ∇ [a F ′(2) bc] = − e −X F(2)[bc + e −φǭ I γ [bc (S 1 + S 2 γ 11 ) η J M IJ ∂ a] X + e −(φ+X)ǭ I γ [bc ∂ a] [(S 1 + S 2 γ 11 ) M IJ ] η J − 1 3 e −(φ+X) H abcǭI (S 1 γ 11 + S 2 ) η J M IJ + 1 2 e −X F (2) deǭ I 2S 1 γ [a δ d b δ e c] γ 11 − S 2 γ d [bc δ e a] η J M IJ + 1 144 e −X F (4) def gǭ I 48S 1 γ d δ e a δ f b δ g c + S 2 γ def g abc + 36γ de [a δ f b δ g c] η J M IJ . The expression above must vanish for the transformed Bianchi identity to be satisfied. Since we are considering generic supergravity solutions, by looking at the terms proportional to the RR 2-form field strength we conclude that ∂ a X = S 1ǭI γ a γ 11 η J M IJ and S 2ǭI γ abc η J M IJ = 0.(24) From the terms proportional to the NSNS field strength we get that ǫ I (S 1 γ 11 + S 2 ) η J M IJ = 0,(25) and from the terms involving the RR 4-form field strength we get S 1ǭI γ a η J M IJ = 0 and S 2ǭI γ abc γ 11 η J M IJ = 0,(26) using S 2ǭI γ abc η J M IJ = 0, from equation (24). Finally from the remaining terms we have that ǫ I γ [bc ∂ a] X (S 1 + S 2 γ 11 ) M IJ − ∂ a] (S 1 + S 2 γ 11 ) M IJ η J = 0.(27) Using similar techniques to those used above, we can also show that ∇ [a F ′(4) bcde] − 2H [abc F ′(2) de] = −e −X F(4)[bcde + e −φǭ I γ [bcde (S 3 + S 4 γ 11 ) η J M IJ ∂ a] X − 2e −(φ+X) H [abcǭI γ de] ((S 2 + S 3 ) γ 11 + (S 1 + S 4 )) η J M IJ + 1 20 e −X F (2) f gǭ I S 3 γ f g abcde − 20γ [abc δ f d δ g e] γ 11 η J M IJ + 1 120 e −X F (4) f ghiǭ I 10S 3 γ f gh [bcde δ i a] + 12γ f [ab δ g c δ h d δ i e] +S 4 γ f ghi abcde − 120γ [a δ f b δ g c δ h d δ i e] γ 11 η J M IJ + e −(φ+X)ǭ I γ [bcde ∂ a] [(S 3 + S 4 γ 11 ) M IJ ] η J ,(28)∇ b F ′(2) ba − 1 6 H bcd F ′(4) abcd = − e −X F (2) ba + e −φǭ I γ ba (S 1 + S 2 γ 11 ) η J M IJ ∂ b X + e −(φ+X)ǭ I γ ba ∂ b [(S 1 + S 2 γ 11 ) M IJ ] η J − 1 6 e −(φ+X) H bcdǭI γ bcd a ((S 1 + S 4 )γ 11 + (S 2 + S 3 )) η J M IJ + 1 4 e −X F (2) bcǭ I 4S 1 γ b δ c a γ 11 + S 2 γ bc a η J M IJ − 1 6 e −X F (4) bcdaǭ I S 2 γ bcd η J M IJ(29) and ∇ d F ′(4) dabc − 1 144 ǫ abcd 1 ...d 7 H d 1 d 2 d 3 F ′(4)d 4 ...d 7 = − e −X F (4) dabc + e −φǭ I γ dabc (S 3 + S 4 γ 11 ) η J M IJ ∂ d X + e −(φ+X)ǭ I γ dabc ∂ d [(S 3 + S 4 γ 11 ) M IJ ] η J + 3 2 e −X F (2) d[cǭ I S 3 γ d ab] γ 11 − 2S 4 γ a δ d b] η J M IJ − 1 48 e −X F (4) def gǭ I S 3 γ def g abc + 36γ de [a δ f b δ g c] + 48S 4 γ d δ e a δ f b δ g c γ 11 η J M IJ .(30) For the transformed equation of motion for the RR 4-form field to hold, from equation (30), the following must be satisfied ∂ a X = −S 4ǭI γ a γ 11 η J M IJ ,(31)S 4ǭI γ a η J M IJ = 0,(32)S 3ǭI γ abc η J M IJ = 0, S 3ǭI γ abc γ 11 η J M IJ = 0,(33)ǫ I γ abcd ∂ d X (S 3 + S 4 γ 11 ) M IJ − ∂ d (S 3 + S 4 γ 11 ) M IJ η J = 0.(34) Note that if, for example,ǭ I γ abc γ 11 η J M IJ = 0, thenǭ I γ a 1 ...a 7 η J M IJ = 0, for γ a 1 ...am = − (−1) 1 2 (10−m)(10−m+1) (10 − m)! ǫ a 1 ...amb 1 ...b 10−m γ b 1 ...b 10−m γ 11 ,(35) proved in Appendix B. Now, if equations (24), (31-33) hold, then the expressions in equations (28) and (29) vanish only if (S 1 + S 4 )ǭ I γ ab η J M IJ + (S 2 + S 3 )ǭ I γ ab γ 11 η J M IJ = 0,(36)ǫ I γ [bcde ∂ a] X (S 3 + S 4 γ 11 ) M IJ − ∂ a] (S 3 + S 4 γ 11 ) M IJ η J = 0(37) and (S 1 + S 4 )ǭ I γ abcd γ 11 η J M IJ + (S 2 + S 3 )ǭ I γ abcd η J M IJ = 0,(38)ǫ I γ ab ∂ b X (S 1 + S 2 γ 11 ) M IJ − ∂ b (S 1 + S 2 γ 11 ) M IJ η J = 0,(39) respectively. In summary, the Killing spinors and functions that describe the transformation must satisfy ∂ a X = S 1ǭI γ a γ 11 η J M IJ , (S 1 + S 4 )ǭ I γ a γ 11 η J M IJ = 0, ǫ I (S 1 γ 11 + S 2 ) η J M IJ = 0, S 1ǭI γ a η J M IJ = 0, S 4ǭI γ a η J M IJ = 0, S 2ǭI γ abc η J M IJ = 0, S 2ǭI γ abc γ 11 η J M IJ = 0, S 3ǭI γ abc η J M IJ = 0, S 3ǭI γ abc γ 11 η J M IJ = 0, (S 1 + S 4 )ǭ I γ ab η J M IJ + (S 2 + S 3 )ǭ I γ ab γ 11 η J M IJ = 0, (S 1 + S 4 )ǭ I γ abcd γ 11 η J M IJ + (S 2 + S 3 )ǭ I γ abcd η J M IJ = 0, ǫ I γ [bc ∂ a] X (S 1 + S 2 γ 11 ) M IJ − ∂ a] (S 1 + S 2 γ 11 ) M IJ η J = 0, ǫ I γ ab ∂ b X (S 1 + S 2 γ 11 ) M IJ − ∂ b (S 1 + S 2 γ 11 ) M IJ η J = 0, ǫ I γ [bcde ∂ a] X (S 3 + S 4 γ 11 ) M IJ − ∂ a] (S 3 + S 4 γ 11 ) M IJ η J = 0 ǫ I γ abcd ∂ d X (S 3 + S 4 γ 11 ) M IJ − ∂ d (S 3 + S 4 γ 11 ) M IJ η J = 0 (40) in order for the Bianchi identities and equations of motion for the transformed RR fields to be satisfied. Let us consider the NSNS 3-form equations. The Bianchi identity for the NSNS 3-form field is invariant under the transformation, so we do not need to consider it. However, using the Killing spinor equations and equations (40), the equation of motion for the NSNS 3-form, (12), reduces to e −(φ+2X) F cdǭI (3S 3 − S 4 γ 11 ) δ c a δ d b − S 3 γ cd ab η J M IJ − e −(φ+2X) S 3 F abcdǭI γ cd γ 11 η J M IJ − 4e −2(φ+X)ǭ I γ [a η J ∂ b] (S 4 M IJ ) + 1 2 e −2(φ+X) (ǭ I γ abcd (S 3 + S 4 γ 11 ) η J ) ǭ K γ cd (S 1 + S 2 γ 11 ) η L M IJ M KL − 1 48 e −2(φ+X) (ǭ I γ cdef (S 3 + S 4 γ 11 ) η J ) (ǭ K γ cdef ab (S 3 γ 11 + S 4 ) η L )M IJ M KL = 0. (41) The supergravity fields we are considering are generic, so, in particular from the term proportional to the RR 2-form field, we must have that S 3ǭI γ abcd η J M IJ = 0, which implies that S 3 = 0, for this is precisely the combination that enters in the transformation of the 4-form RR field strength. Furthermore, since S 3 = 0, from (S 1 + S 4 )ǭ I γ ab η J M IJ + (S 2 + S 3 )ǭ I γ ab γ 11 η J M IJ = 0 we get that S 2ǭI γ ab γ 11 η J M IJ ∝ǭ I γ ab η J M IJ , hence without loss of generality we can set S 2 = 0. Since S 2 and S 3 vanish, we must have that at least one of ǫ I γ ab η J M IJ ,ǭ I γ abcd γ 11 η J M IJ are non-zero in order for the transformation to be non-trivial. Therefore, using equations (S 1 + S 4 )ǭ I γ ab η J M IJ = 0, (S 1 + S 4 )ǭ I γ abcd γ 11 η J M IJ = 0, from the set of equations (40) with S 2 = S 3 = 0, we deduce that S 4 = −S 1 . Without loss of generality we can let S 1 = 1. Furthermore, from the first term in equation (41), the spinors must satisfȳ ǫ I γ 11 η J M IJ = 0.(42) The last two terms in equation (41) are quartic in spinors and they can be simplified using Fierz identities. The Fierz identity for commuting spinors λ, χ, ψ, ϕ in d−dimensions is λ M χ ψ N ϕ = 2 −[d/2] I λ M O I N ϕ ψ O I χ , where M, N are arbitrary combination of gamma matrices and {O I } = {I, γ a ,{O I } = {I, γ a , iγ ab , iγ abc , γ abcd , . . . } is the dual basis. Using Fierz identities, equation (41) with S 1 = −S 4 = 1, S 2 = S 3 = 0, ǫ I γ 11 η J M IJ = 0 becomes 4ǭ I γ [a η J ∂ b] M IJ − 16 ǭ I γ [a γ 11 η J ǭ K γ b] η L − (ǭ I γ ab η J ) (ǭ K γ 11 η L ) + (ǭ I γ ab γ 11 η J ) (ǭ K η L ) + 1 2 (ǭ I γ abcd η J ) ǭ K γ cd γ 11 η L + 1 48 (ǭ I γ abcdef γ 11 η J ) ǭ K γ cdef η L M IJ M KL = 0. We can use equation (42) again to simplify the above equation to 4ǭ I γ [a η J ∂ b] M IJ − 16 ǭ I γ [a γ 11 η J ǭ K γ b] η L + (ǭ I γ ab γ 11 η J ) (ǭ K η L ) + 1 2 (ǭ I γ abcd η J ) ǭ K γ cd γ 11 η L + 1 48 (ǭ I γ abcdef γ 11 η J ) ǭ K γ cdef η L M IJ M KL = 0. (43) So far, having only the dilaton and Einstein equation to consider, we have the following conditions on the Killing spinors and functions in the transformation of the fields: S 1 = −S 4 = 1,(44)S 3 = S 2 = 0,(45)∂ a X =ǭ I γ a γ 11 η J M IJ ,(46)ǫ I γ 11 η J M IJ = 0,(47)ǫ I γ a η J M IJ = 0,(48)ǫ I γ [bc η J ∂ a] XM IJ − ∂ a] M IJ = 0,(49)ǫ I γ ab η J ∂ b XM IJ − ∂ b M IJ = 0,(50)ǫ I γ [bcde γ 11 η J ∂ a] XM IJ − ∂ a] M IJ = 0,(51)ǫ I γ abcd γ 11 η J ∂ d XM IJ − ∂ d M IJ = 0(52) and equation (43). The Dilaton equation for the transformed fields is R + 4 φ ′ − 4 ∂φ ′ 2 − 1 12 H 2 = 0, which using the dilaton equation for the original fields implies that X = 2∂ a φ∂ a X + ∂ a X∂ a X.(53) Using equation (46) and the Killing spinor equation from the variation of the gravitino, X = ∇ a (ǭ I γ a γ 11 η J M IJ ) = − 3 4 e φ F bcǭI γ bc η J M IJ + 1 48 e φ F bcdeǭI γ bcde γ 11 η J M IJ +ǭ I γ a γ 11 η J ∂ a M IJ . However, since ǫ I and η I , for all I = 1 . . . , n, satisfy the dilatino Killing spinor equation, (18), ǫ α I γ 11 γ a ∂ a φ − 1 12 H abc γ abc γ 11 − 3 8 e φ F (2) ab γ ab γ 11 + 1 96 e φ F (4) abcd γ abcd (αβ) η β J = 0, hence, using equation (23), − 3 4 e φ F bcǭI γ bc η J M IJ + 1 48 e φ F bcdeǭI γ bcde γ 11 η J M IJ = 2ǭ I γ a γ 11 η J ∂ a φM IJ . Therefore, X = 2ǭ I γ a γ 11 η J ∂ a φM IJ +ǭ I γ a γ 11 η J ∂ a M IJ . So, from equation (53), the transformed dilaton equation is satisfied if ǫ I γ a γ 11 η J (ǭ K γ a γ 11 η L M IJ M KL − ∂ a M IJ ) = 0.(54) Finally, we have to find conditions on the spinors and functions in the transformation in order for the Einstein equation to be satisfied for the transformed fields. Using equations (44), (45) and (53) and the Einstein equation for the original fields, Einstein's equation becomes 1 4 g ab X − 2∇ a ∇ b X + e φ F (2) c (aǭ I γ b)c η J − 1 16 g ab F (2) cdǭ I γ cd η J M IJ − 1 6 e φ F (4) cde (aǭ I γ b)cde γ 11 η J − 3 32 g ab F (4) cdefǭ I γ cdef γ 11 η J M IJ + 1 2 (ǭ I γ ac η J ) (ǭ K γ c b η L ) − 1 16 g ab (ǭ I γ cd η J ) ǭ K γ cd η L M IJ M KL + 1 12 (ǭ I γ acde γ 11 η J ) ǭ K γ cde b γ 11 η L − 3 32 g ab (ǭ I γ cdef γ 11 η J ) ǭ K γ cdef γ 11 η L M IJ M KL = 0.(55) Now, consider ǫ I γ (a\| γ 11 ∇ \|b) − 1 8 H \|b)cd γ cd γ 11 − 1 16 e φ F (2) cd γ cd γ \|b) γ 11 + 1 192 e φ F(4) cdef γ cdef γ \|b) η J = 0. Adding this to the same expression, but with ǫ I and η J interchanged we get ǫ I γ (a γ 11 ∇ b) η J +η J γ (a γ 11 ∇ b) ǫ I + 1 8 e φ F (2) cdǭ I 4γ c (a δ d b) + g ab γ cd η J − 1 96 e φ F(4) cdefǭ I 8γ cde (a δ f b) + g ab γ cdef γ 11 η J = 0, using equations (22) and (23). The above equation and the equation obtained by contracting the two free indices in the above equation can be used to reduce equation (55) to 1 4 g ab X − 2∇ a ∇ b X + 2∇ (a ǭ I γ b) γ 11 η J M IJ − 1 4 g ab ∇ c (ǭ I γ c γ 11 η J ) M IJ + 1 2 (ǭ I γ ac η J ) (ǭ K γ c b η L ) − 1 16 g ab (ǭ I γ cd η J ) ǭ K γ cd η L M IJ M KL + 1 12 (ǭ I γ acde γ 11 η J ) ǭ K γ cde b γ 11 η L − 3 32 g ab (ǭ I γ cdef γ 11 η J ) ǭ K γ cdef γ 11 η L M IJ M KL = 0.(56) Using equation (46) to simplify the terms on the first line, Fierz identities and equations (47) and (48), it can be shown that equation (56) is − 2ǭ I γ (a γ 11 η J ∂ b) M IJ + 2ǭ K γ b) γ 11 η L M IL M KJ + 1 4 g abǭI γ c γ 11 η J (∂ c M IJ + 2ǭ K γ c γ 11 η L M IL M KJ ) + 4 (ǭ I γ a η L ) (ǭ K γ b η J ) + 1 2 (ǭ I γ c a γ 11 η J ) (ǭ K γ bc γ 11 η L ) + 1 12 ǭ I γ cde a η J (ǭ K γ bcde η L ) + 1 16 g ab (ǭ I η J ) (ǭ K η L ) − 1 2 g ab (ǭ I γ c η L ) (ǭ K γ c η J ) − 1 32 g ab (ǭ I γ cd γ 11 η J ) ǭ K γ cd γ 11 η L − 1 128 g ab (ǭ I γ cdef η J ) ǭ K γ cdef η L M IJ M KL = 0.(57) This is a quartic condition on the spinors. Moreover, from equations (43), (49-52) and equation (54) we also have that the spinors must satisfy We have shown that the type IIA supergravity equations admit a symmetry described by the following transformations of the dilaton and RR field strengths 4ǭ I γ [a η J ∂ b] M IJ − 4ǭ K γ b] γ 11 η L M IL M KJ − (ǭ I γ ab γ 11 η J ) (ǭ K η L ) + 1 2 (ǭ I γ abcd η J ) ǭ K γ cd γ 11 η L + 1 48 (ǭ I γ abcdef γ 11 η J ) ǭ K γ cdef η L M IJ M KL = 0, (58) ǫ I γ [ab η J ∂ c] M IJ + 2ǭ K γ c] γ 11 η L M IL M KJ + 1 2 ǭ I γ d [ab η J ǭ K γ c]d γ 11 η L + 1 6 (ǭ I γ abc γ 11 η J ) (ǭ K η L ) − 1 4 ǭ I γ de [ab η J ǭ K γ c]de γ 11 η L + 1 3 (ǭ I γ abcd γ 11 η J ) ǭ K γ d η L + 2 3 (ǭ I γ abcd γ 11 η L ) ǭ K γ d η J − 1 36 (ǭ I γ abcdef γ 11 η J ) ǭ K γ def η L M IJ M KL = 0, (59) ǫ I γ ab η J ∂ b M IJ + 2ǭ K γ b γ 11 η L M IL M KJ + (ǭ I γ a η J ) (ǭ K γ 11 η L ) + 2 (ǭ I γ a η L ) (ǭ K γ 11 η J ) − 3 4 (ǭ I γ abc η J ) ǭ K γ bc γ 11 η L − 1 12 (ǭ I γ abcd η J ) ǭ K γ bcd γ 11 η L M IJ M KL = 0, (60) ǭ I γ [abcd γ 11 η J ∂ e] M IJ + 2ǭ K γ e] γ 11 η L M IL M KJ + ǭ I γ [abc γ 11 η J ǭ K γ de] γ 11 η L + ǭ I γ f [abc η J ǭ K γ de]f η L + 1 5 (ǭ I γ abcdef η J ) ǭ K γ f η L + 2 5 (ǭ I γ abcdef η L ) ǭ K γ f η J − 1 4 ǭ I γ f g [abcd γ 11 η J ǭ K γ e]f g γ 11 η L + 1 20 (ǭ I γ abcdef g γ 11 η J ) ǭ K γ f g γ 11 η L M IJ M KL = 0,(61)ǫ I γ abcd γ 11 η J ∂ d M IJ + 2ǭ K γ d γ 11 η L M IL M KJ + 3 ǭ I γ [ab η J ǭ K γ c] η L + 6 ǭ I γ [ab η L ǭ K γ c] η J + 1 2 (ǭ I γ abc η J ) (ǭ K η L ) + 3 2 ǭ I γ d [ab γ 11 η J ǭ K γ c]d γ 11 η L − 3 4 ǭ I γ de [ab η J ǭ K γ c]de η L − 1 12 (ǭ I γ abcdef γ 11 η J ) ǭ K γ def γ 11 η L M IJ M KL = 0, (62) ǫ I γ a γ 11 η J (∂ a M IJ + 2ǭ K γ a γ 11 η L M IL M KJ ) + (ǭ I γ a η J ) (ǭ K γ a η L ) + 2 (ǭ I γ a η L ) (ǭ K γ a η J ) − 1 12 ǭ I γ abc η J (ǭ K γ abc η L ) − 1 12 ǭ I γ abc γ 11 η J (ǭ K γ abc γ 11 η L ) M IJ M KL = 0,(63)φ → φ ′ = φ + X, e φ F (2) ab → e φ ′ F ′(2) ab = e φ F(2)ab +ǭ I γ ab η J M IJ , e φ F (4) abcd → e φ ′ F ′(4) abcd = e φ F (4) abcd −ǭ I γ abcd γ 11 η J M IJ ,(64) where the Killing spinors must satisfyǭ I γ a η J = 0,(65)ǫ I γ 11 η J M IJ = 0,(66)ǫ I η J M IJ = 0,ǭ I γ ab γ 11 η J M IJ = 0,(67) ǫ I γ abc η J M IJ = 0,ǭ I γ abc γ 11 η J M IJ = 0,ǭ I γ cdef η J M IJ = 0, ∂ a X =ǭ I γ a γ 11 η J M IJ ,(68)∂ a M IJ = −2ǭ K γ a γ 11 η L M IL M KJ .(69) Equation (70) is equivalent to ∂ a (M −1 ) IJ = 2ǭ J γ a γ 11 η I ,(71) and equation (69) can be solved to find X up to a constant of integration: X = 1 2 n I=1 log M −1 II .(72) The integrability conditions arising from equations (69) and (71) are trivial because ∇ [a ∇ b] X = 1 2 H abcǭI γ c η J M IJ and ∇ [a ∇ b] (M −1 ) IJ = H abcǭJ γ c η I , which vanish by equation (65). In the transformations given by Berkovits and Maldacena the spinors ǫ I and η I are identified. This is sufficient forǭ I η J M IJ = 0,ǭ I γ ab γ 11 η J M IJ = 0, ǫ I γ abc η J M IJ = 0,ǭ I γ abc γ 11 η J M IJ = 0,ǭ I γ cdef η J M IJ = 0. When ǫ I and η I are identified, only the symmetric part of M IJ contributes in the transformations of the fields, so without loss of generality we can let M IJ be symmetric in I and J, as a consequence of which the above equations are satisfied. If we identity ǫ I and η I then we recover the transformations of Berkovits and Maldacena, but with an extra condition on the spinors, namely that ǫ I γ 11 ǫ J M IJ = 0. When n = 1, we can explicitly show that the solution tō ǫη = 0,ǭγ ab γ 11 η = 0, ǫγ abc η = 0,ǭγ abc γ 11 η = 0,ǭγ cdef η = 0 is ǫ ∝ η. However, when n > 1, these conditions do not reduce to the transformation rules of fermionic T-duality. Type IIB supergravity symmetry The type IIB supergravity action is (2) . (73) In type IIB supergravity the RR fields are the scalar C (0) , the 2-form C (2) and the 4-form C (4) . In terms of potentials B, C (0) , C (2) and C (4) , the field strengths are defined to be (2) . S = 1 2κ 2 d 10 x √ g e −2φ R + 4(∂φ) 2 − 1 12 H 2 − 1 2 F (1) 2 + 1 3! F (3) 2 + 1 2.5! F (5) 2 − 1 192 1 √ g ǫ C (4) ∂B∂CH = dB, F (1) = dC (0) , F (3) = dC (2) − HC (0) , F (5) = dC (4) − 1 2 C (2) ∧ H + 1 2 B ∧ dC The 5-form field strength is constrained to be self-dual. The Bianchi identities for the field strengths are dH = 0,(74)dF (1) = 0,(75)dF (3) − H ∧ F (1) = 0,(76)dF (5) − H ∧ F (3) = 0.(77) The equations of motion are d e −2φ ⋆ H − F (1) ∧ ⋆F (3) − F (3) ∧ F (5) = 0,(78)d ⋆ F (1) + H ∧ ⋆F (3) = 0,(79)d ⋆ F (3) + H ∧ F (5) = 0.(80) The equation of motion for the 5-form field strength, F (5) , is equivalent to the Bianchi identity for the 5-form, equation (77), as it is self-dual. Moreover, the Einstein equation is R ab = − 1 4 g ab φ + 1 2 g ab (∂φ) 2 − 2∇ a ∇ b φ + 1 4 H acd H cd b − 1 12 g ab H 2 + 1 2 e 2φ F (1) a F (1) b + 1 4 e 2φ F (3) acd F (3) cd b − 1 12 g ab F (3) 2 + 1 96 e 2φ F (5) acdef F (5) cdef b ,(81) noting that F (5) 2 vanishes because the 5-form field is self-dual. Finally, the dilaton equation of motion is the same as the type IIA supergravity dilaton equation of motion, equation (16). Also, the twicecontracted Bianchi identity is again satisfied using the equations of motion for the fields. The Killing spinor equations from the variation of the gravitino and dilatino are ∇ a ǫ − 1 8 H abc γ bc σ 3 ǫ − 1 8 e φ F (1) b γ b γ a iσ 2 ǫ + 1 3! F (3) bcd γ bcd γ a σ 1 ǫ + 1 2.5! F (5) bcdef γ bcdef γ a iσ 2 ǫ = 0,(82)γ a ∂ a φ − 1 12 H abc γ abc σ 3 + e φ F (1) a γ a iσ 2 + 1 12 e φ F (3) abc γ abc σ 1 ǫ = 0,(83) respectively. We now consider the most general transformation of the RR field strengths and we will also allow the dilaton to transform: e φ F (1) a → e φ ′ F ′(1) a = e φ F (1) a +ǭ I γ a S (1) η J M IJ , e φ F (3) abc → e φ ′ F ′(3) abc = e φ F (3) abc +ǭ I γ abc S (2) η J M IJ , e φ F (5) abcde → e φ ′ F ′(5) abcde = e φ F (5) abcde +ǭ I γ abcde S (3) η J M IJ , where M IJ is an arbitrary function; the spinors ǫ I , η I satisfy the gravitino and dilatino Killing spinor equations; S (1,2,3) = µ S (1,2,3) µ σ ⋆µ , σ ⋆µ = I, σ 1 , iσ 2 , σ 3 µ . The field φ ′ is some arbitrary field, which is identified with the transformed dilaton upon considering the NSNS 3-form equation of motion with transformed fields. We let φ ′ = φ + X, where X is some arbitrary function. Note that the RR fields need not a priori transform with the same spinors and coefficients. However, as in section 3, if we let them transform with different spinors and coefficients, then we will find from equation (76), for example, that the spinors and functions have to be identified. As in section 3, we let the NSNS fields g and H be invariant under the transformation. It is important that the 5-form field strength remains self-dual after the transformation. The Hodge dual of δF (5) is ⋆ ǭ I γ a 1 ...a 5 S (3) η J M IJ dx a 1 ∧ · · · ∧ dx a 5 =ǭ I 1 5! ǫ a 1 ...a 5 b 1 ...b 5 γ b 1 ...b 5 S (3) η J M IJ dx a 1 ∧ · · · ∧ dx a 5 =ǭ I (γ a 1 ...a 5 γ 11 ) S (3) η J M IJ dx a 1 ∧ · · · ∧ dx a 5 , by identity (35). Hence if we let γ 11 η J = η J then the transformed 5-form field strength is self-dual. Recall that in type IIB supergravity all the Killing spinors have the same chirality, hence γ 11 ǫ I = ǫ I . We will now find the constraints that the various functions and the Killing spinors must satisfy so that the transformed fields satisfy the Bianchi identities and the equations of motion. First, let us consider the Bianchi identities. Using the gravitino Killing spinor equation, the Bianchi identity for the transformed RR 1-form field strength is ∇ [a F ′(1) b] = ∇ [a e −X F (1) b] + e −(φ+X)ǭ I γ b] S (1) η J M IJ = −e −X ∂ [a X F b] + e −φǭ I γ b] S (1) η J M IJ − e −(φ+X) ∂ [a φǭ I γ b] S (1) η J M IJ + e −X M IJ ǫ α I 1 8 e −φ H cd[a γ cd b] αβ S (1) σ 3 + γ cd b] βα σ 3 S (1) + 1 8 F (1) c γ [b γ c γ a] αβ iS (1) σ 2 − γ [b γ c γ a] βα iσ 2 S (1) + 1 8.3! F (3) cde γ [b γ cde γ a] αβ S (1) σ 1 + γ [b γ cde γ a] βα σ 1 S (1) + 1 16.5! F (5) cdef g γ [b γ cdef g γ a] αβ iS (1) σ 2 − γ [b γ cdef g γ a] βα iσ 2 S (1) η β J + e −(φ+X)ǭ I γ [b ∂ a] S (1) η J M IJ + e −(φ+X)ǭ I γ [b S (1) η J ∂ a] M IJ = 0. Now, from the dilatino Killing spinor equation ǫ I γ ba S (1) γ c ∂ c φ − 1 12 H cde γ cde σ 3 + e φ F (1) c γ c (iσ 2 ) + 1 12 e φ F(3) cde γ cde σ 1 η J = 0. Adding the above equation tō η J γ ba S (1)t γ c ∂ c φ − 1 12 H cde γ cde σ 3 + e φ F (1) c γ c (iσ 2 ) + 1 12 e φ F (3) cde γ cde σ 1 ǫ I = 0, where S t is the transpose of S, and using the first identity in the set of equations (23) we can show that ǫ I ∂ [a φγ b] S (1) η J = ǫ α I 1 48 H cde γ ba γ cde αβ S (1) σ 3 + γ ba γ cde βα σ 3 S (1) − 1 4 e φ F (1) c (γ ba γ c ) αβ S (1) iσ 2 − (γ ba γ c ) βα iσ 2 S (1) − 1 48 e φ F (3) cde γ ba γ cde αβ S (1) σ 1 + γ ba γ cde βα σ 1 S (1) η J . Therefore, using the above equation and performing some gamma matrix manipulations, using equations (22) and (23), we can show that ∇ [a F ′(1) b] = e −(φ+X)ǭ I γ [b ∂ a] S (1) M IJ − ∂ a] XS (1) M IJ η J − e −X ∂ [a XF (1) b] − 1 24 e −(φ+X) H cdeǭI γ cde ba + γ c δ d b δ e a S(1)0 σ 3 + S (1) 3 I η J M IJ + 1 4 e −X F (1) cǭI γ c ba S (1) 0 iσ 2 − S (1) 2 I − 4γ [b δ c a] S(1)1 σ 3 − S (1) 3 σ 1 η J M IJ + 1 4 e −X F (3) cd[aǭ I γ cd b] S (1) 2 σ 3 + S (1) 3 iσ 2 − 2γ c δ d b] S (1) 0 σ 1 + S (1) 1 I η J M IJ − 1 24 e −X F (5) bacdeǭ I γ cde S (1) 0 iσ 2 − S (1) 2 I η J M IJ . This expression must vanish if the transformed RR 1-form field strength is to satisfy the Bianchi identity. We are considering generic supergravity fields, so the expression vanishes only if ∂ a X =ǭ I γ a S(1)1 σ 3 − S(1)3 σ 1 η J M IJ ,(84)ǫ I γ abcde S(1)0 σ 3 + S(1)3 I η J M IJ = 0,(85)ǫ I γ a S(1)0 σ 3 + S(1)3 I η J M IJ = 0,(86)ǫ I γ abc S(1)0 iσ 2 − S(1)2 I η J M IJ = 0,(87)ǫ I γ abc S (1) 2 σ 3 + S (1) 3 iσ 2 η J M IJ = 0,(88)ǫ I γ a S(1)0 σ 1 + S (1) 1 I η J M IJ = 0,(89)ǫ I γ [b ∂ a] S (1) M IJ − ∂ a] XS (1) M IJ η J = 0.(90) We can also show that ∇ [a F ′(3) bcd] + F ′(1) [a H bcd] =e −(φ+X)ǭ I γ [bcd ∂ a] S (2) M IJ − ∂ a] XS (2) M IJ η J − e −X ∂ [a XF (3) bcd] − 1 48 e −(φ+X) H ef gǭI γ ef g bcda + 36γ e [bc δ f d δ g a] S(2)0 σ 3 + S(2)3 I + 48γ [b δ e c δ f d δ g a] S(1)0 I + S (1) 1 − S (2) 2 σ 1 + S (1) 2 − S (2) 1 iσ 2 + S (1) 3 σ 3 η J M IJ + 1 2 e −X F(1)[aǭ I γ bcd] S(2)3 σ 1 − S (2) 1 σ 3 η J M IJ − 1 48 e −X F (3) ef gǭ I γ ef g bcda + 36γ e [bc δ f d δ g a] S(2)0 σ 1 + S (2) 1 I + 48γ [b δ e c δ f d δ g a] S(2)2 σ 3 + S (2) 3 iσ 2 η J M IJ − 1 4 e −X F (5) ef [bcdǭ I γ e δ f a] S (2) 0 iσ 2 − S (2) 2 I + γ ef a] S(2)1 σ 3 − S (2) 3 σ 1 η J M IJ , ∇ [a F ′(5) bcdef ] + 10 3 F ′(3) [abc H def ] =e −(φ+X)ǭ I γ [bcdef ∂ a] S (3) M IJ − ∂ a] XS (3) M IJ η J − e −X ∂ [a XF (5) bcdef ] − 1 72 e −(φ+X) H ghiǭI γ ghi bcdef a + 90γ g [bcde δ h f δ i a] S(3)0 σ 3 + S(3)3 I + 240γ [bcd δ g e δ h f δ i a] S(2)0 I + S (2) 1 − S (3) 2 σ 1 + S (2) 2 − S (3) 1 iσ 2 + S (2) 3 σ 3 η J M IJ − 1 12 e −X F (1) gǭ I γ g bcdef a S (3) 0 iσ 2 − S (3) 2 I η J M IJ − 1 72 e −X F (3) ghiǭ I γ ghi bcdef a S (3) 0 σ 1 + S (3) 1 I + 9γ gh [bcdef δ i a] + 60γ [bcd δ g e δ h f δ i a] S(3)2 σ 3 + S(3)3 iσ 2 η J M IJ + 1 4 e −X F (5) g[def aǭ I 5γ g bc] S(3)0 iσ 2 − S(3)2 I − 4γ b δ g c] S(3)1 σ 3 − S(3)3 σ 1 η J M IJ . Both of the above expressions must vanish for the Bianchi identities for F ′(3) and F ′(5) , respectively, to hold. So we have that ∂ a X =ǭ I γ a S (2) 2 σ 3 + S (2) 3 iσ 2 η J M IJ ,(91)ǫ I γ abc S (2) 0 σ 3 + S(2)3 I η J M IJ = 0,(92)ǫ I γ a S(1)0 I + S (1) 1 − S (2) 2 σ 1 + S (1) 2 − S (2) 1 iσ 2 + S (1) 3 σ 3 η J M IJ = 0,(93)ǫ I γ abc S (2) 3 σ 1 − S (2) 1 σ 3 η J M IJ = 0,(94)ǫ I γ abc S(2)0 σ 1 + S(2)1 I η J M IJ = 0,(95)ǫ I γ a S(2)0 iσ 2 − S (2) 2 I η J M IJ = 0,(96)ǫ I γ [bcd ∂ a] S (2) M IJ − ∂ a] XS (2) M IJ η J = 0,(97) and ∂ a X =ǭ I γ a S(3)1 σ 3 − S (3) 3 σ 1 η J M IJ ,(98)ǫ I γ a S(3)0 σ 3 + S(3)3 I η J M IJ = 0,(99)ǫ I γ abcde S (3) 0 σ 3 + S(3)3 I η J M IJ = 0,(100) ǫγ abc S 0 I + S (2) 1 − S (3) 2 σ 1 + S (2) 2 − S (3) 1 iσ 2 + S (2) 3 σ 3 η J M IJ = 0,(2)ǫ I γ abc S (3) 0 iσ 2 − S (3) 2 I η J M IJ = 0,(101)ǫ I γ a S(102)0 σ 1 + S(3)1 I η J M IJ = 0,(3)ǫ I γ abc S(103)2 σ 3 + S(3)3 iσ 2 η J M IJ = 0,(3)ǫ I γ abc S (3) 0 iσ 2 − S (3) 2 I η J M IJ = 0,(104)ǫ I γ [bcdef ∂ a] S (3) M IJ − ∂ a] XS (3) M IJ η J = 0,(105) respectively. The NSNS 3-form field strength does not change, so the Bianchi identity for the 3-form field is the same as before. Now, we assume that equations (84-106) hold and consider the equations of motion. As before, using the Killing spinor equations for ǫ I and η J , the equation of motion for the transformed RR 1-form field strength can be simplified to ∇ a F ′(1) a + 1 6 H abc F ′(3)abc =e −(φ+X)ǭ I γ a ∂ a S (1) M IJ − ∂ a XS (1) M IJ η J + 1 6 e −(φ+X) H abcǭI γ abc S(2)0 I + S(2)1 − S(1)σ 1 + S(2)2 − S (1) 1 iσ 2 + S(2) 3 σ 3 η J M IJ = 0. So, we get the following conditions: ǫ I γ abc S (2) 0 I + S (2) 1 − S (1) 2 σ 1 + S (2) 2 − S (1) 1 iσ 2 + S (2) 3 σ 3 η J M IJ = 0,(107)ǫ I γ a ∂ a S (1) M IJ − ∂ a XS (1) M IJ η J = 0.(108) Similarly, the equation of motion for the transformed 3-form field strength becomes ∇ a F ′(3) abc + 1 6 H def F ′(5) bcdef =e −(φ+X)ǭ I γ abc ∂ a S (2) M IJ − ∂ a XS (2) M IJ η J + 1 6 e −(φ+X) H defǭ I γ bcdef S(3)0 I + S (3) 1 − S (2) 2 σ 1 + S (3) 2 − S (2) 1 iσ 2 + S(3)3 σ 3 η J M IJ = 0. hence we need to imposē ǫ I γ abcde S(3)0 I + S (3) 1 − S (2) 2 σ 1 + S (3) 2 − S (2) 1 iσ 2 + S (3) 3 σ 3 η J M IJ = 0,(109)ǫ I γ abc ∂ a S (2) M IJ − ∂ a XS (2) M IJ η J = 0.(110) We also need to show that the transformed fields satisfy the equation of motion for the NSNS 3-form: ∇ a e −2φ ′ H abc − F ′(1)a F ′(3) abc − 1 6 F ′(3)def F ′(5) bcdef = − 2e −2(φ+X) ∂ a XH abc − e −(φ+2X) M IJ F (1)aǭ I γ abc S (2) η J + F (3) abcǭ I γ a S (1) η J + 1 6 F (3)defǭ I γ bcdef S (3) η J + 1 6 F (5) bcdefǭ I γ def S (2) η J − e −2(φ+X) ǭ I γ a S (1) η J ǭ K γ abc S (2) η L M IJ M KL − 1 6 e −2(φ+X) ǭ I γ def S (2) η J ǭ K γ bcdef S (3) η L M IJ M KL = 0,(111) where the NSNS 3-form equation of motion with the original supergravity fields has been used in the first equality. Using the gravitino Killing spinor equation and the self-duality of the 5-form RR field strength we can show that F (1)aǭ I γ abc S (2) 2 iσ 2 η J M IJ =4∇ [b ǭ I γ c] η J S (2) 2 M IJ − 2H abcǭI γ a S(2)2 σ 3 η J M IJ − 1 6 e φ F(3)defǭ I γ def bc + γ d δ e b δ f c S (2) 2 σ 1 η J M IJ − 1 6 e φ F (5) bcdefǭ I γ def S (2) 2 iσ 2 η J M IJ ,(112) and using the dilatino Killing spinor equation and equations (92-94) we get that F (1)aǭ I γ abc S(2)0 I + S(2)1 σ 1 + S (2) 3 σ 3 η J M IJ = − 2∂ [b φǭ I γ c] S(2)0 iσ 2 η J M IJ + 1 12 H defǭI γ def bc − 6γ d δ e b δ f c S(2) 3 iσ 2 η J M IJ + 1 12 e φ F(3)defǭ I γ def bc − 6γ d δ e b δ f c S(2)1 iσ 2 η J M IJ = 0.(113) Substituting the above equations into equation (111), and using equations (91), (93) and (109), the NSNS 3-form equation of motion becomes − 4∇ [b ǭ I γ c] η J S (2) 2 M IJ + 2∂ [b φǭ I γ c] S (2) 0 iσ 2 η J M IJ − 1 12 H defǭI γ def bc + 18γ d δ e b δ f c S (2) 3 iσ 2 η J M IJ − 1 4 e φ F (3) defǭ I γ def bc + 2γ d δ e b δ f c S (2) 1 iσ 2 η J M IJ − 1 6 e φ F(5) bcdefǭ I γ def S 0 I + S(2)1 σ 1 + S (2) 3 σ 3 η J M IJ − ǭ I γ a S (1) η J ǭ K γ abc S (2) η L M IJ M KL − 1 6 ǭ I γ def S (2) η J ǭ K γ bcdef S (3) η L M IJ M KL = 0.(2) The supergravity fields are generic so the terms proportional to each supergravity field in the above expression must vanish. In particular, if we consider the expression proportional to the RR 5-form field strength, then as this expression is exactly the expression that enters in the transformation of the RR 3-form field strength S ∂ a X =ǭ I γ a σ 3 η J M IJ ,(115)ǫ I γ a η J M IJ = 0,(116)ǫ I γ [b σ 1 η J ∂ a] M IJ − ∂ a] XM IJ = 0,(117)ǫ I γ [bcd iσ 2 η J ∂ a] M IJ − ∂ a] XM IJ = 0,(118)ǫ I γ [bcdef σ 1 η J ∂ a] M IJ − ∂ a] XM IJ = 0,(119)ǫ I γ a σ 1 η J (∂ a M IJ − ∂ a XM IJ ) = 0,(120)ǫ I γ abc iσ 2 η J (∂ a M IJ − ∂ a XM IJ ) = 0.(121) The NSNS 3-form field strength equation of motion, equation (114), becomes 4∇ [b ǭ I γ c] η J M IJ + ǭ I γ a σ 1 η J ǭ K γ abc iσ 2 η L M IJ M KL + 1 6 ǭ I γ def iσ 2 η J ǭ K γ bcdef σ 1 η L M IJ M KL = 0, and using equation (116) this reduces to 4ǭ I γ [b η J ∂ c] M IJ + ǭ I γ a σ 1 η J ǭ K γ abc iσ 2 η L + 1 6 ǭ I γ def iσ 2 η J ǭ K γ bcdef σ 1 η L M IJ M KL = 0.(122) Fierz identities can be used to simplify the terms that are quartic in spinors. Just as the tensor product of a combination of gamma matrices, M and N, can be expanded in the basis {O I } = {I, γ a , iγ ab , iγ abc , γ abcd , . . . }, M α β N γ δ = 2 −[d/2] I M O I N α δ O γ I β , we can expand the tensor product of 2 × 2 matrices, Σ and Ξ, in the basis σ µ = (I, σ 1 , σ 2 , σ 3 ), Σ AB Ξ CD = 2 −1 µ (Σσ µ Ξ) AD σ µ CB , where uppercase Latin letters are SO(2) vector indices. Hence, for type IIB theory spinors, the Fierz identity is λ M Σχ ψ N Ξϕ = 1 64 I,µλ M O I N (Σσ µ Ξ) ϕψ (O I σ µ ) χ. Using the Fierz identity multiple times, we can show that ǭ I γ a σ 1 η J ǭ K γ abc iσ 2 η L + 1 6 ǭ I γ a 1 ...a 3 iσ 2 η J ǭ K γ a 1 ...a 3 bc σ 1 η L = − 16 ǭ I γ [b σ 3 η L ǭ K γ c] η J + ǭ I γ a iσ 2 η J ǭ K γ abc σ 1 η L + 1 6 ǭ I γ a 1 ...a 3 σ 1 η J ǭ K γ a 1 ...a 3 bc iσ 2 η L . So from equation (122), the NSNS 3-form equation of motion is satisfied if 4ǭ I γ [b η J ∂ c] M IJ + 4ǭ K γ c] σ 3 η L M KJ M IL + ǭ I γ a iσ 2 η J ǭ K γ abc σ 1 η L M IJ M KL + 1 6 ǭ I γ a 1 ...a 3 σ 1 η J ǭ K γ a 1 ...a 3 bc iσ 2 η L M IJ M KL = 0. (123) The dilaton equation, (16), for the transformed fields simply reduces to X − 2∂ a φ∂ a X − (∂X) 2 = 0.(124) Using ∂ a X =ǭ I γ a σ 3 η J M IJ , and the Killing spinor equations, the above equation reduces tō ǫ I γ a σ 3 η J ∂ a M IJ −ǭ K γ a σ 3 η L M IJ M KL = 0, which using Fierz identities becomes ǫ I γ a σ 3 η J ∂ a M IJ + 2ǭ K γ a σ 3 η L M IL M KJ + (ǭ I γ a η J ) (ǭ K γ a η L ) + 2 (ǭ I γ a η L ) (ǭ K γ a η J ) − 1 12 (ǭ I γ abc η J ) ǭ K γ abc η L − 1 12 ǭ I γ abc σ 3 η J ǭ K γ abc σ 3 η L M IJ M KL = 0.(125)2∇ (a ǭ I γ b) σ 3 η J M IJ − ∂ b) X − 1 4 g ab ∇ c ǭ I γ c σ 3 η J M IJ − ∂ c X − 2ǭ I γ (a σ 3 η J ∂ b) M IJ + 1 4 g abǭI γ c σ 3 η J ∂ c M IJ + 1 96 48 ǭ I γ a σ 1 η J ǭ K γ b σ 1 η L + 24 ǭ I γ cd a iσ 2 η J ǭ K γ bcd iσ 2 η L − 2g ab ǭ I γ cde iσ 2 η J ǭ K γ cde η L iσ 2 + ǭ I γ cdef a σ 1 η J ǭ K γ bcdef σ 1 η L M IJ M KL = 0. The first two terms vanish because of equation (115), and we can use Fierz identities to rewrite the terms that are quartic in spinors. Upon doing so, Einstein's equation becomes Therefore, the type IIB supergravity symmetry is described by the following transformations of the RR fields and dilaton − 2ǭ I γ (a σ 3 ǫ J ∂ b) M IJ + 2ǭ K γ b) σ 3 ǫ L M IL M KJ + 1 4 g abǭI γ c σ 3 ǫ J ∂ c M IJ + 2ǭ K γ c σ 3 ǫ L M IL M KJ + 1 96 384 (ǭ I γ a η L ) (ǭ K γ b η J ) + 48 ǭ I γ a iσ 2 η J ǭ K γ b iσ 2 η L + 24 ǭ I γ cd a σ 1 η J ǭ K γ bcd σ 1 η L + ǭ I γ cdef a iσ 2 η J ǭ K γ bcdef iσ 2 η L − 48g ab (ǭ I γ c η J ) (ǭ K γ c η L ) − 2g ab ǭ I γ cde σ 1 η J ǭ K γ cde σ 1 η L M IJ M KL = 0.(126)ǫ I γ [a σ 1 ǫ J ∂ b] M IJ + 2ǭ K γ b] σ 3 ǫ L M IL M KJ − 1 4 (ǭ I γ abc η J ) ǭ K γ c iσ 2 η L + 1 2 ǭ I γ abc iσ 2 η J (ǭ K γ c η L ) + ǭ I γ abc iσ 2 η L (ǭ K γ c η J ) − 1 4 ǭ I γ cd [a σ 3 η J ǭ K γ b]cd σ 1 η L − 1 24 ǭ I γ abcde iσ 2 η J ǭ K γ cde η L M IJ M KL = 0,(127)ǫ I γ [abc iσ 2 ǫ J ∂ d] M IJ + 2ǭ K γ d] σ 3 ǫ L M IL M KJ − 3 4 ǭ I γ e [ab σ 1 η J ǭ K γ cd]e η L + 1 2 ǭ I γ [abc σ 3 η J ǭ K γ d] iσ 2 η L + 1 4 ǭ I γ abcde σ 1 η J (ǭ K γ e η L ) + 1 2 ǭ I γ abcde σ 1 η L (ǭ K γ e η J ) + 1 4 ǭ I γ ef [abc iσ 2 η J ǭ K γ d]ef σ 3 η L + 1 48 (ǭ I γ abcdef g η J ) ǭ K γ ef g σ 1 η L M IJ M KL = 0,(128)ǭ I γ [abcde σ 1 ǫ J ∂ f ] M IJ + 2ǭ K γ f ] σ 3 ǫ L M IL M KJ − 5 3 ǭ I γ [abc σ 3 η J ǭ K γ def ] σ 1 η L + 5 4 ǭ I γ g [abcd iσ 2 η J ǭ K γ ef ]g η L − 1 12 (ǭ I γ abcdef g η J ) ǭ K γ g iσ 2 η L + 1 4 ǭ I γ gh [abcde σ 1 η J ǭ K γ f ]gh σ 3 η L + 1 6 ǭ I γ abcdef g iσ 2 η J (ǭ K γ g η L ) + 1 3 ǭ I γ abcdef g iσ 2 η L (ǭ K γ g η J ) M IJ M KL = 0,(129)ǫ I γ a σ 1 ǫ J ∂ a M IJ + 2ǭ K γ a σ 3 ǫ L M IL M KJ − 1 12 ǭ I γ abc σ 1 η J ǭ K γ abc σ 3 η L M IJ M KL = 0,(130)ǫ I γ abc iσ 2 ǫ J ∂ c M IJ + 2ǭ K γ c σ 3 ǫ L M IL M KJ + 2 ǭ I γ [a η J ǭ K γ b] σ 1 η L + 4 ǭ I γ [a η L ǭ K γ b] σ 1 η J − 1 2 ǭ I γ cd [a η J ǭ K γ b]cd σ 1 η L + 1 2 ǭ I γ abc σ 3 η J ǭ K γ c iσ 2 η L − 1 12 ǭ I γ abcde iσ 2 η J ǭ K γ cde σ 3 η L M IJ M KL = 0,(131)φ → φ ′ = φ + X, e φ F (1) a → e φ ′ F ′(1) a = e φ F (1) a +ǭ I γ a σ 1 η J M IJ , e φ F (3) abc → e φ ′ F ′(3) abc = e φ F (3) abc +ǭ I γ abc iσ 2 η J M IJ , e φ F (5) abcde → e φ ′ F ′(5) abcde = e φ F (5) abcde +ǭ I γ abcde σ 1 η J M IJ ,(132) where γ 11 ǫ I = ǫ I , γ 11 η I = η I , ǫ I γ a η J = 0,(133) ǫ I γ a iσ 2 η J M IJ = 0,ǭ I γ abc η J M IJ = 0,ǭ I γ abc σ 1 η J M IJ = 0, ǫ I γ abc σ 3 η J M IJ = 0,ǭ I γ abcde iσ 2 η J M IJ = 0,(135)∂ a X =ǭ I γ a σ 3 η J M IJ ,(136)∂ a M IJ = −2ǭ K γ a σ 3 η L M IL M KJ .(137) Equation (138) is equivalent to ∂ a (M −1 ) IJ = 2ǭ J γ a σ 3 η I ,(139) and up to a constant of integration X = 1 2 n I=1 log M −1 II .(140) The integrability conditions for equations (137) and (138) are satisfied by equation (134). If ǫ I = η I then the equations in the lines labelled by (135) and (136) are satisfied, and the transformations are precisely the transformations found by Berkovits and Maldacena, equations (4) and (5) in section 2. Furthermore, when n = 1 these equations can be explicitly solved to show that ǫ ∝ η. When n > 1 this is no longer true, and the conditions can be satisfied without identifying ǫ I and η I . Note that, since in our transformations it is not necessary to identify ǫ I and η I , we can solvē ǫ I γ a η J = 0 for real spinors. Comments In both type IIA and type IIB supergravity we have found a larger class of transformations that include the transformations of Berkovits and Maldacena [8]. In both cases, when n = 1, these transformations are precisely the transformations found by Berkovits and Maldacena. However, for n > 1 ǫ I ∝ η I is sufficient but no longer necessary for the conditions given by equations (67, 68) and (135, 136) in the analysis for type IIA and type IIB supergravity, respectively, to be satisfied. Indeed, in both cases, we have found spinors ǫ I = η I , where I = 1, 2, for which M IJ is antisymmetric in its I, J indices and the above-mentioned conditions hold. In the transformations of fermionic T-duality, the spinors were complexified in order to find non-trivial solutions toǭ I γ a ǫ J = 0. Note that, in the transformations that we have constructed η I does not necessarily have to be identified with ǫ I when n > 1. Therefore,ǭ I γ a η J = 0 can be solved for real spinors, keeping the transformation real. Furthermore, when the two set of Killing spinors ǫ I and η I are identified the supersymmetry of the transformed supergravity solution is the same as the original solution. In fact, the Killing spinors in the new background can be written explicitly in terms of the Killing spinors of the original background [8], equation (7). This must be true because fermionic T-duality is a duality of string theory, so the transformation must preserve supersymmetry. However, for our transformation it is not clear whether supersymmetry is preserved. If this is case, then the transformation could be a useful tool for generating backgrounds with lower supersymmetry. In general, however, the conditions given in equations (65-68) and (134-136) are difficult to solve explicitly. If this symmetry, and indeed fermionic T-duality, is to be a more practical solution-generating mechanism then a new technique must be found to solve these constraints. The original motivation for fermionic T-duality was to understand the dual superconformal invariance found in maximally supersymmetric Yang-Mills theory. Similarly, it is hoped that there will an understanding of the dual superconformal symmetry of ABJM [21,22] using fermionic T-duality in type IIA theory. The string theory dual to ABJM [23] theory is type IIA string theory on AdS 4 × CP 3 , and there has been work on trying to understand the self-duality of the AdS 4 × CP 3 background under a combination of T-duality and fermionic T-duality [11,24,25]. In [25], fermionic T-duality transformations on the partially κ-gauge fixed Green-Schwarz action is considered and found to be singular. However, the partially κ-gauge fixed action for the AdS 4 × CP 3 sigma model is not well-defined for all string configurations. It is not clear in [25] whether the singularity arises for this reason or not. The transformation rules for the type IIA supergravity fields derived in this paper can be used to perform the transformation from the target space point of view. Indeed this has recently been done by Bakhmatov [26]. The results of this paper are consistent with the singularity found in [25]. In [26] the transformation is done solely in supergravity, and hence the work suggests that the singularity found in [25] does not have its source in the sigma model. It is an intriguing problem to find out the origin of this singularity at the supergravity level. Finally, in the transformation rules for type IIA supergravity besides the conditions which have analogues in the type IIB supergravity transformations we also found that ǫ I γ 11 η J M IJ = 0 must hold. This condition may be physically interpreted as maintaining a zero Romans mass [27], for the Romans mass can be thought of as a constant 0-form field strength [28]. This suggests that the fermionic symmetry that we have constructed for type IIA supergravity can be extended to massive type IIA supergravity. We will report on this problem in a future paper. Another problem that we would like to address in the future is the complexity of the fermionic Tduality transformations, for in string theory the transformations cannot be made real. Understanding the physical interpretation of the complexity may reveal important, hitherto unknown, aspects of string theory. The metric signature is (− + · · · +). The permutation symbol is totally antisymmetric and its sign is defined by ǫ 01...9 = 1. respectively, where Fierz identities have been used to rewrite equations (49-52) and equation (54). The only set of quadratic constraints on the spinors that we have found that solve equations (57-63) is ǫ I γ a η J = 0,ǭ I η J M IJ = 0,ǭ I γ ab γ 11 η J M IJ = 0 ǫ I γ abc η J M IJ = 0,ǭ I γ abc γ 11 η J M IJ = 0,ǭ I γ cdef η J M IJ = 0, ∂ a M IJ = −2ǭ K γ a γ 11 η L M IL M KJ . a bc S (2) η J , we can show that S (1) = σ 1 and S (3) = σ 1 . Letting S (1) = S (3) = σ 1 and S (2) = iσ 2 , conditions (84-110) become to equations (123), (125-131) is ǫ I γ a η J = 0,ǭ I γ a iσ 2 η J M IJ = 0, ǫ I γ abc η J M IJ = 0,ǭ I γ abc σ 1 η J M IJ = 0, ǫ I γ abc σ 3 η J M IJ = 0,ǭ I γ abcde iσ 2 η J M IJ = 0, ∂ a M IJ = −2ǭ K γ a σ 3 η L M IL M KJ . iγ ab , iγ abc , γ abcd , . . . } forms a basis for 2 [d/2] × 2 [d/2] matrices and 2 This must hold in order for the dilaton equation to be satisfied for the transformed fields. Now, let us consider the Einstein equation. We can use the gravitino Killing spinor equation and the constraint from the dilaton equation of motion, equation (124), to show that the Einstein equation reduces to So, the transformed fields satisfy the type IIB supergravity equations if equations (115-121), (123), (125) and (126) are satisfied. Using Fierz identities, equations (117-121) are equivalent tō This identity can be proved by using induction on m with n = 1 and then by induction on n. 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[ "Evaporation of Schwarzschild Black Holes in Matrix Theory", "Evaporation of Schwarzschild Black Holes in Matrix Theory" ]	[ "T Banks [email protected] \nDepartment of Physics\nAstronomy Rutgers University\n08855-0849PiscatawayNJ\n", "W Fischler [email protected] \nDepartment of Physics\nTheory Group\nUniversity of Texas\n78712AustinTX\n", "I R Klebanov [email protected] \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJ\n" ]	[ "Department of Physics\nAstronomy Rutgers University\n08855-0849PiscatawayNJ", "Department of Physics\nTheory Group\nUniversity of Texas\n78712AustinTX", "Joseph Henry Laboratories\nPrinceton University\n08544PrincetonNJ" ]	[]	Recently, in collaboration with Susskind, we proposed a model of Schwarzschild black holes in Matrix theory. A large Schwarzschild black hole is described by a metastable bound state of a large number of D0-branes which are held together by a background, whose structure has so far been understood only in 8 and 11 dimensions. The Hawking radiation proceeds by emission of small clusters of D0-branes. We estimate the Hawking rate in the Matrix theory model of Schwarzschild black holes and find agreement with the semiclassical rate up to an undetermined numerical coefficient of order 1.	10.1016/s0370-2693(98)00118-x	[ "https://arxiv.org/pdf/hep-th/9712236v1.pdf" ]	18,092,606	hep-th/9712236	90225e252bcc668d54f575783fb08d5e2f115b65	Evaporation of Schwarzschild Black Holes in Matrix Theory arXiv:hep-th/9712236v1 26 Dec 1997 December 1997 T Banks [email protected] Department of Physics Astronomy Rutgers University 08855-0849PiscatawayNJ W Fischler [email protected] Department of Physics Theory Group University of Texas 78712AustinTX I R Klebanov [email protected] Joseph Henry Laboratories Princeton University 08544PrincetonNJ Evaporation of Schwarzschild Black Holes in Matrix Theory arXiv:hep-th/9712236v1 26 Dec 1997 December 1997 Recently, in collaboration with Susskind, we proposed a model of Schwarzschild black holes in Matrix theory. A large Schwarzschild black hole is described by a metastable bound state of a large number of D0-branes which are held together by a background, whose structure has so far been understood only in 8 and 11 dimensions. The Hawking radiation proceeds by emission of small clusters of D0-branes. We estimate the Hawking rate in the Matrix theory model of Schwarzschild black holes and find agreement with the semiclassical rate up to an undetermined numerical coefficient of order 1. Introduction In a recent paper [1] Susskind and the authors presented a model of Schwarzschild black holes in Matrix Theory [2]. The key feature of the model was the notion of a Boltzmann gas of D0-branes (which we will review briefly below). 1 We showed that the Matrix model for M theory in 11 noncompact dimensions, and also for its toroidal compactification to 8 noncompact dimensions, contained a set of states which obeyed (up to a numerical coefficient which we could not calculate) the Bekenstein-Hawking relation between entropy and transverse area for a Schwarzschild black hole. The relation between energy and entropy for these states also obeyed the black hole formula. 2 We were also able to compute the long distance Newtonian gravitational force between equal mass black holes, with an answer in agreement with classical gravity. The purpose of the present note is to calculate the rate of Hawking radiation from our model black holes. In [1] we showed that the individual bound D0-branes in the model, had the kinematic properties of Hawking radiation in the boosted frame in which we examine our black hole. The D0-branes are "tethered" to a classical background by harmonic forces. In this note we argue that the probability for the classical variables which produce these forces on an individual D0-brane to fluctuate to zero is independent of the mass of the black hole in the large mass limit. We show that when this estimate is combined with the proper phase space integral, it gives a decay rate for the boosted black hole which is just the Lorentz transform of the rest frame Hawking evaporation rate. To briefly summarize our black hole model: we consider the Hilbert space of the matrix theory which represents Discrete Light Cone Quantized M theory compactified on some manifold Y [10] in the sector with N units of longitudinal momentum (DLCQ N ). The radius of the light-like circle will be denoted by R. We will also choose the noncompact transverse spacetime dimension D to be greater than or equal to 6. The model contains a set of variables which includes matrices X i representing the transverse positions of N D0branes in a weakly coupled Type IIA string theory called "the analog model." We emphasize that these are not the Boltzmann D0-branes of which our black hole is constructed. The matrices also describe creation and annihilation operators for strings stretching between the D0-branes. We consider a semiclassical configuration of the variables of the model, which includes a semiclassical background X i cl for the transverse position matrices. This background configuration must satisfy a number of constraints, which were described in [1] . Boltzmann D0-brane positions are defined as perturbations of the background X i cl → X i cl + N n=1 r i n δ n (1.1) where the δ n are a set of independent commuting matrices chosen to minimize the quantity \|T r[X i cl , δ n ] 2 \|. This is a term in the matrix model Hamiltonian which gives rise to a harmonic potential binding these Boltzmann D0-branes to the classical configuration. In [1] we argued that for D ≥ 6 we could choose our classical configuration so that this harmonic potential did not interfere with the scaling argument we review in the next paragraph. In matrix theory, the Bose or Fermi statistics of particles arises as a residual gauge symmetry. Since the entire gauge symmetry of the model is broken by the classical background, this means that the variables r i n should be treated as the coordinates of distinguishable, or Boltzmann, particles. The approximately degenerate configurations of these particles then have an entropy of order N . In [1] we argued that the effective Hamiltonian of these degrees of freedom gave rise to bound states whose transverse area is of order G D N and whose light cone energy is of order N R(G D N ) −2/(D−2) . When translated into invariant mass, this gives precisely the mass/entropy/area relations of a Schwarzschild black hole. We were unable to calculate the numerical coefficients in these relations because we did not have the full effective Hamiltonian of the Boltzmann gas 3 and because the mean field approximation which we used was very crude. According to the above discussion, the light cone energy per particle of the Boltzmann [7]. Energetic considerations ensured that only strings attaching a given D0-brane to o(1) nearest neighbors on this lattice were excited. We will assume that similar pictures work for all D ≥ 6. gas is ∼ R(G D N ) −2/(D−2) , where G D is As a consequence, it seems reasonable to assume that the amplitude to "liberate a D0-brane" from the black hole is independent of N as N → ∞. It is the value of the wave function of the system on a submanifold of codimension which is o(1). Calculation of the Hawking Evaporation Rate Given the estimate of the D0-brane liberation amplitude in the previous section, we can proceed to calculate the Hawking rate. The quantum fluctuation of the background described in the previous section gives rise to a single D0-brane wave function which is, by the estimates of [1], a smooth function A(y) of rapid decrease in the variable y = \|p ⊥ \| (G D N ) −1/(D−2) ,(2.1) where p ⊥ is the transverse momentum and G D is the D-dimensional Newton constant. Thus, the amplitude to produce a D0-brane with momentum much larger than the Hawking momentum is highly suppressed. If we assume that the fluctuations which liberate any of the N D0-branes are independent and incoherent, then the probability per unit time to emit a Hawking particle is given by dN dx + ∼ N 1 R n>0 ∞ 0 dp + d D−2 p ⊥ δ n R − p 2 ⊥ p + \|A(n, y)\| 2 (2.2) where y is the variable defined above. In this equation we have generalized our considerations to include processes in which a cluster of n D0-branes, with n finite and independent of N , is emitted. Such systems also have the kinematic properties of Hawking radiation. However, since the cluster of D0-branes is connected to the classical background by o(n) degrees of freedom, we should expect the matrix element to fall off exponentially in n. The Theory is that the S matrix computed by the theory is indeed Lorentz invariant. We cannot prove this claim at present, nor can we show that our approximate evaluation of the matrix element is accurate. However, assuming that these claims are valid, the correct rate is obtained by integrating our matrix element against relativistic phase space. Our measure thus contains an extra factor of light cone energy, p + , compared to the nonrelativistic phase space of the D0-brane quantum mechanics. This factor, which endows the measure with the correct boost transformation properties, has been absorbed into the longitudinal momentum delta-function in (2.2) . Finally, we need to estimate the square of the matrix element for n D0-branes to be liberated, \|A(n, y)\| 2 . As we have explained, this quantity is appreciable only if n and y are of order 1. \|A(n, y)\| 2 has dimensions of length D−2 . The only dimensionful quantities at our disposal are R, l 11 and the radii of the compactification torus, L i . Since the measure in (2.2) transforms properly under boosts, any dependence of \|A(n, y)\| 2 on R would violate Lorentz invariance. Furthermore, we will assume that the dependence on l 11 and the radii is through the D-dimensional Newton constant only. 4 Thus, we are led to \|A(n, y)\| 2 ∼ G D , for n and y of order 1. Estimating the Hawking rate (2.2) with this assumption, we find dN dx + ∼ R(G D N ) −2/(D−2) . (2.3) By way of comparison, we now compute the Hawking radiation rate according to the conventional semiclassical formulae. First, let us write down the Hawking rate in the usual equal-time quantization. The answer can be written as dN dx 0 ∼ ∞ 0 dp + ∞ 0 dp − d D−2 p ⊥ δ(p + p − − p 2 ⊥ )e −p 0 /T H Ap 0 ,(2.4) where A = 4G D S is the horizon area. We have included the thermal factor appropriate for the Boltzmann statistics. The Hawking temperature is related to the Schwarzschild radius R S by T H ∼ 1 R S ,(2.5) and R S ∼ (SG D ) 1/(D−2) ∼ (N G D ) 1/(D−2) . (2.6) An explicit factor of p 0 is needed in (2.4) because the measure ∞ 0 dp + ∞ 0 dp − d D−2 p ⊥ δ(p + p − − p 2 ⊥ ) (2.7) is Lorentz invariant. The left-hand side, however, contains a derivative with respect to x 0 , hence transforms in the same way as p 0 . Now we perform a parallel computation in the light-cone frame. Here, the number of particles radiated per unit light-cone time is dN dx + ∼ ∞ 0 dp − ∞ 0 dp + d D−2 p ⊥ δ(p + p − − p 2 ⊥ )e −p + /T + e −p − /T − Ap + . (2.8) The factor of p + is needed for correct boost invariance, since the left-hand side contains a derivative with respect to x + . In the rest frame of the black hole, we have T rest + = T rest − ∼ T H ∼ 1/R S . (2.9) If we carry out a boost x − = R R S x − rest , x + = R S R x + rest ,(2.10) then the new temperatures are T − = T rest − R S R ∼ 1/R , T + = T rest + R R S ∼ R/R 2 S . (2.11) Doing the integrals, we find dN dx + ∼ A(T + T − ) (D−2)/2 T + ∼ T + ∼ R(G D S) −2/(D−2) . (2.12) If we compactify x − on a circle of radius R, and work with S ∼ N , then the integral over p − is replaced by sum, dN dx + ∼ 1 R n>0 ∞ 0 dp + d D−2 p ⊥ δ n R − p 2 ⊥ p + e −p + /T + e −n/(RT − ) G D N . The Rate of Mass Loss Using arguments analogous to those that led to (2.2), we may obtain from the matrix model a formula for the rate of light cone energy loss per unit light cone time, dE dx + ∼ N 1 R n>0 ∞ 0 dp + d D−2 p ⊥ δ n R − p 2 ⊥ p + p + \|A(n, y)\| 2 . (3.1) Estimating the integral using the previously stated assumptions, we find dE dx + ∼ T 2 + ∼ R 2 R 4 S . (3.2) Now using The Matrix theory result for the rate of mass loss in the rest frame, (3.5), is consistent with the standard semiclassical result. M 2 = 2E N R ,(3. What we have shown in this note is that plausible assumptions about the D0-brane emission process from the metastable bound state describing the black hole lead to the radiation rate consistent with the semiclassical calculations, up to a constant of proportionality of order 1. Just as in the semiclassical analysis, suppression of the rate for large entropy N comes from the smallness of the one-particle phase space available at an energy comparable to the Hawking temperature (T H scales as N −1/(D−2) ). Our analysis should be regarded as a plausibility argument. In particular, we need a microscopic argument for why at low momenta the square of the matrix element for liberating a D0-brane is of order G D . Nevertheless, we believe that we have described the correct mechanism for evaporation of Schwarzschild black holes in Matrix theory. the Newton constant in D non-compact dimensions. The transverse momentum per particle is ∼ (G D N ) −1/(D−2) and the longitudinal momentum is just 1/R. We showed that these were precisely the kinematical properties of the Hawking particles when boosted into a frame where the black hole has longitudinal momentum P − = N/R. This suggests the following attractive and simple picture of the Hawking evaporation process: the classical background provides a harmonic potential which binds the Boltzmann D0-branes to the black hole. The background itself should be represented as a coherent quantum state centered around a periodic solution of the classical equations of motion of the matrix model. According to the wave function of this state, there is a certain amplitude for the part of the classical configuration which interacts with a given D0-brane, to fluctuate to zero. From the point of view of a basis in which the particular D0-brane of interest (say δ 1 ) occupies the lowest right hand corner of the matrix, the relevant part of the classical solution is the part which occupies the last row and column. There are thus o(N ) possible degrees of freedom which might be excited. However, in the explicit classical backgrounds which we constructed in D = 11, 8 [1], only o(1) of these possible background degrees of freedom were actually utilized. In D = 11, this is a consequence of approximate locality of the action on the world volume of the classical membrane. In D = 8 the classical background for a typical black hole could be viewed as a lattice of D0-branes connected by strings on a 3-torus, with the lattice spacing of order N −1/3 overall factor of N in equation(2.2) represents the incoherent sum over processes in which a particular bound D0-brane is liberated. Finally, we have used relativistic phase space to integrate over the final states of the outgoing D0-brane. Although our calculation is done in a particular frame, chosen so that the geometrical structure of the black hole just fits inside the DLCQ N quantization volume, we expect that the matrix element which we have estimated is in fact, for large N , approximately the invariant S matrix element of a Lorentz covariant system. Of course, the crucial, as yet unproven, assumption of Matrix changes the normalization, the result (2.12) still holds. Thus, our formula (2.3) for the rate of emission of D0-branes from one of our matrix theory black holes coincides (up to uncalculated numerical factors of order one) with the semiclassical Hawking evaporation rate in the light cone frame, (2.12). For related work, see[3],[4],[5],[6] 2 In the supersymmetric Yang-Mills formulation on the dual torus the corresponding results were obtained in[7],[8],[9]. Recently, Liu and Tseytlin[11] have proposed to describe the interactions of the Boltzmann D0-branes by a Hamiltonian containing terms of all orders in the velocity[12]. Perhaps this can be used to make some progress on the numerical coefficients. This assumption is plausible because G D is the only quantity that appears in the D-brane interaction Hamiltonian, but its better justification is clearly necessary. ACKNOWLEDGEMENTSWe are grateful to Lenny Susskind for important discussions and to the Physics De- . T Banks, W Fischler, I R Klebanov, L Susskind, hep-th/9711005T. Banks, W. Fischler, I.R. Klebanov, L. Susskind, hep-th/9711005. . T Banks, W Fischler, S Shenker, L Susskind, hep-th/9610043Phys. Rev. 55T. Banks, W. Fischler, S. Shenker, L. Susskind, Phys. Rev. D55 (1997) 112, hep-th/9610043. . I Volovich, hep-th/9608137I.Volovich, hep-th/9608137. . G Horowitz, E Martinec, hep-th/9710217G. Horowitz and E. Martinec, hep-th/9710217. . M Li, hep-th/9710226M. Li, hep-th/9710226. . D Minic, hep-th/9712202D. Minic, hep-th/9712202. . T Banks, W Fischler, I R Klebanov, L Susskind, hep-th/9709091T. Banks, W. Fischler, I.R. Klebanov, L. Susskind, hep-th/9709091. . I R Klebanov, L Susskind, hep-th/9709108I.R. Klebanov, L. Susskind, hep-th/9709108. . S Das, S Mathur, S K Rama, P Ramadevi, hep-th/9711003S. Das, S. Mathur, S.K. Rama and P. Ramadevi, hep-th/9711003. hep-th/9704080; N. Seiberg. L Susskind, hep-th/9709220Phys. Rev. Lett. 79hepth/9710009; A. SenL. Susskind, hep-th/9704080; N. Seiberg, Phys. Rev. Lett. 79 (1997) 3577, hep- th/9710009; A. Sen, hep-th/9709220. . H Liu, A Tseytlin, hep-th/9712063H. Liu and A. Tseytlin, hep-th/9712063. . K Becker, M Becker, J Polchinski, A Tseytlin, hep-th/9706072Phys. Rev. 56K. Becker, M. Becker, J. Polchinski and A. Tseytlin, Phys. Rev. D56 (1997) 3174, hep-th/9706072.	[]
[ "BISHOP-PHELPS-BOLLOBÁS MODULI OF A BANACH SPACE", "BISHOP-PHELPS-BOLLOBÁS MODULI OF A BANACH SPACE" ]	[ "Mario Chica ", "Vladimir Kadets ", "Miguel Martín ", "ANDSoledad Moreno-Pulido ", "Fernando Rambla-Barreno " ]	[]	[]	We introduce two Bishop-Phelps-Bollobás moduli of a Banach space which measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollobás theorem in this space. We show that there is a common upper bound for these moduli for all Banach spaces and we present an example showing that this bound is sharp. We prove the continuity of these moduli and an inequality with respect to duality. We calculate the two moduli for Hilbert spaces and also present many examples for which the moduli have the maximum possible value (among them, there are C(K) spaces and L 1 (µ) spaces). Finally, we show that if a Banach space has the maximum possible value of any of the moduli, then it contains almost isometric copies of the real space(2) ∞ and present an example showing that this condition is not sufficient.	10.1016/j.jmaa.2013.10.083	[ "https://arxiv.org/pdf/1304.0376v1.pdf" ]	55,664,978	1304.0376	09f02d9c576eda586511b323256f306f065d127e	BISHOP-PHELPS-BOLLOBÁS MODULI OF A BANACH SPACE Mario Chica Vladimir Kadets Miguel Martín ANDSoledad Moreno-Pulido Fernando Rambla-Barreno BISHOP-PHELPS-BOLLOBÁS MODULI OF A BANACH SPACE Dedicated to the memory of Robert R. Phelps (1926-2013) We introduce two Bishop-Phelps-Bollobás moduli of a Banach space which measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollobás theorem in this space. We show that there is a common upper bound for these moduli for all Banach spaces and we present an example showing that this bound is sharp. We prove the continuity of these moduli and an inequality with respect to duality. We calculate the two moduli for Hilbert spaces and also present many examples for which the moduli have the maximum possible value (among them, there are C(K) spaces and L 1 (µ) spaces). Finally, we show that if a Banach space has the maximum possible value of any of the moduli, then it contains almost isometric copies of the real space(2) ∞ and present an example showing that this condition is not sufficient. Introduction The classical Bishop-Phelps theorem of 1961 [4] states that the set of norm attaining functionals on a Banach space is norm dense in the dual space. Few years later, B. Bollobás [5] gave a sharper version of this theorem allowing to approximate at the same time a functional and a vector in which it almost attains the norm (see the result bellow). The main aim of this paper is to study the best possible approximation of this kind that one may have in each Banach space, measuring it by using two moduli which we define. Before going further, we first present the original result by Bollobás which nowadays is known as the Bishop-Phelps-Bollobás theorem. We need to fix some notation. Given a (real or complex) Banach space X, we write B X and S X to denote the closed unit ball and the unit sphere of the space, and X * denotes the (topological) dual of X. We will also use the notation Π(X) := (x, x * ) ∈ X × X * : x = x * = x * (x) = 1 . Theorem 1.1 (Bishop-Phelps-Bollobás theorem [5]). Let X be a Banach space. Suppose x ∈ S X and x * ∈ S X * satisfy \|1 − x * (x)\| ε 2 /2 (0 < ε < 1/2). Then there exists (y, y * ) ∈ Π(X) such that x − y < ε + ε 2 and x * − y * ε. So the idea is that given (x, x * ) ∈ S X × S X * such that x * (x) ∼ 1, there exist y ∈ S X close to x and y * ∈ S X * close to x * for which y * (y) = 1. This result has many applications, especially for the theory of numerical ranges, see [5,6]. Our objective is to introduce two moduli which measures, for a given Banach space, what is the best possible Bollobás theorem in this space, that is, how close can be y to x and y * to x * in the result above depending on how close is x * (x) to 1. In the first modulus, we allow the vector and the functional to Remark 1.4. Let X be a Banach space. Then, for every δ ∈ (0, 2), one has Φ X (δ) := inf ε > 0 : ∀(x, x * ) ∈ B X × B X * with Re x * (x) > 1 − δ, ∃(y, y * ) ∈ Π(X) with d ∞ ((x, x * ), (y, y * )) < ε = inf ε > 0 : ∀(x, x * ) ∈ B X × B X * with Re x * (x) 1 − δ, ∃(y, y * ) ∈ Π(X) with d ∞ ((x, x * ), (y, y * )) < ε = inf ε > 0 : ∀(x, x * ) ∈ B X × B X * with Re x * (x) > 1 − δ, ∃(y, y * ) ∈ Π(X) with d ∞ ((x, x * ), (y, y * )) ε = inf ε > 0 : ∀(x, x * ) ∈ B X × B X * with Re x * (x) 1 − δ, ∃(y, y * ) ∈ Π(X) with d ∞ ((x, x * ), (y, y * )) ε , and Φ S X (δ) := inf ε > 0 : ∀(x, x * ) ∈ S X × S X * with Re x * (x) > 1 − δ, ∃(y, y * ) ∈ Π(X) with d ∞ ((x, x * ), (y, y * )) < ε = inf ε > 0 : ∀(x, x * ) ∈ S X × S X * with Re x * (x) 1 − δ, ∃(y, y * ) ∈ Π(X) with d ∞ ((x, x * ), (y, y * )) < ε = inf ε > 0 : ∀(x, x * ) ∈ S X × S X * with Re x * (x) > 1 − δ, ∃(y, y * ) ∈ Π(X) with d ∞ ((x, x * ), (y, y * )) ε = inf ε > 0 : ∀(x, x * ) ∈ S X × S X * with Re x * (x) 1 − δ, ∃(y, y * ) ∈ Π(X) with d ∞ ((x, x * ), (y, y * )) ε . Observe that the smaller are the functions Φ X (·) and Φ S X (·), the better is the approximation on the space. It can be deduced from the Bishop-Phelps-Bollobás theorem that there is a common upper bound for Φ X (·) and Φ S X (·) for all Banach spaces X. Our first result in the next section will be to present the best possible upper bound, namely we will show that (1) Φ S X (δ) Φ X (δ) √ 2δ 0 < δ < 2, X Banach space . This will follow from a result by R. Phelps [13]. A version for Φ S X (δ) for small δ's can be also deduced from the Brøndsted-Rockafellar variational principle [14,Theorem 3.17], as claimed in [7]. The sharpness of (1) can be verified by considering the real space X = (2) ∞ . This is the content of section 2. Next, we prove in section 3 that for every Banach space X, the moduli Φ X (δ) and Φ S X (δ) are continuous in δ. We prove that Φ X (δ) Φ X * (δ) and Φ S X (δ) Φ S X * (δ). Finally, we show that Φ X (δ) = √ 2δ if and only if Φ S X (δ) = √ 2δ. Examples of spaces for which the two moduli are computed are presented in section 4. Among other results, the moduli of R and of every real or complex Hilbert space of (real)-dimension greater than one are calculated, and there are presented a number of spaces for which the value of both moduli are √ 2δ (i.e. the maximal possible value) for small δ's: namely c 0 , 1 and, more in general, L 1 (µ), C 0 (L), unital C * -algebras with non-trivial centralizer. . . The main result of section 5 states that if a Banach space X satisfies Φ X (δ 0 ) = √ 2δ 0 (equivalently, Φ S X (δ 0 ) = √ 2δ 0 ) for some δ 0 ∈ (0, 1/2), then X contains almost isometric copies of the real space (2) ∞ . We finish presenting, for every δ ∈ (0, 1/2), an example of a three dimensional real space Z containing an isometric copy of (2) ∞ for which Φ Z (δ) < √ 2δ. This is the content of section 6. The upper bound of the moduli Our first result is the promised best upper bound of the Bishop-Phelps-Bollobás moduli. Theorem 2.1. For every Banach space X and every δ ∈ (0, 2), Φ X (δ) √ 2δ and so, Φ S X (δ) √ 2δ We deduce the above result from [13, Corollary 2.2], which was stated for general bounded convex sets on real Banach spaces. Particularizing the result to the case of the unit ball of a Banach space, using a routine argument to change non-strict inequalities to strict inequalities, and taking into account that the dual of a complex Banach space is isometric (taking real parts) to the dual of the real subjacent space, we get the following result. Proposition 2.2 (Particular case of [13,Corollary 2.2]). Let X be Banach space. Suppose that z * ∈ S X * , z ∈ B X and η > 0 are given such that Re z * (z) > 1 − η. Then, for any k ∈ (0, 1) there existỹ * ∈ X * andỹ ∈ S X such that ỹ * =ỹ * (ỹ), z −ỹ < η k , z * −ỹ * < k. Proof of Theorem 2.1. We have to show that given (x, x * ) ∈ B X × B X * with Re x * (x) > 1 − δ, there exists (y, y * ) ∈ Π(X) such that x − y < √ 2δ and x * − y * < √ 2δ. Let us first prove the result for the more interesting case of δ ∈ (0, 1). In this case, 0 < 1 − δ < x * 1, so, if we write η = x * − 1 + δ x * > 0, z * = x * / x * and z = x, one has Re z * (z) > 1 − η. Next, we consider k = η/ √ 2δ and claim that 0 < k < 1. Indeed, as the function (2) ϕ(t) = t − 1 + δ √ 2δ t (t ∈ R + ) is strictly increasing, k = ϕ( x * ) and 1 − δ < x * 1, we have that 0 = ϕ(1 − δ) < k ϕ(1) = √ δ √ 2 < 1, as desired. Therefore, we may apply Proposition 2.2 with z * ∈ S X * , z ∈ B X , η > 0 and 0 < k < 1 to obtainỹ * ∈ X * andỹ ∈ S X satisfying ỹ * =ỹ * (ỹ), z −ỹ < η k = √ 2δ, x * x * −ỹ * < k = x * − 1 + δ x * √ 2δ . As k < 1, we getỹ * = 0 and we may write y * =ỹ * ỹ * , y =ỹ, to get that (y, y * ) ∈ Π(X). We already have that x − y < √ 2δ. On the other hand, we have x * − y * = x * −ỹ * ỹ * x * − x * ỹ * + x * ỹ * −ỹ * ỹ * x * x * x * −ỹ * + x * ỹ * − 1 x * x * x * −ỹ * + x * ỹ * − x * + 1 − x * x * x * x * −ỹ * + ỹ * − 1 + 1 − x * 2 x * x * x * −ỹ * + 1 − x * < 2 √ 2δ x * − 1 + δ + 1 − x * . Now, as the function γ(t) = 2 √ 2δ t − 1 + δ + 1 − t t ∈ [0, 1] is strictly increasing (for this we only need 0 < δ < 2), we get γ( x * ) γ(1) = 2δ √ 2δ = √ 2δ. It follows that x * − y * < √ 2δ, as desired. Let us now prove the case when δ ∈ [1, 2). Here, it can be routinely verified that δ − 1 √ 2δ − 1 < √ 2δ − 1 so, writing ψ(δ) = 1 2 δ − 1 √ 2δ − 1 + √ 2δ − 1 we get (3) δ − 1 √ 2δ − 1 < ψ(δ) < √ 2δ − 1 δ ∈ [1, 2) . Now, we have to distinguish two situations. Let first suppose that x * ψ(δ). Then, we take any y ∈ S X such that x − y 1 and take y * ∈ S X * such that y * (y) = 1. Then, (y, y * ) ∈ Π(X), x − y 1 < √ 2δ and x * − y * 1 + x * 1 + ψ(δ) < √ 2δ by (3). Otherwise, suppose x * > ψ(δ). We then write η = x * − 1 + δ x * > 0 and k = η/ √ 2δ as in the previous case, and we have to show that k < 1. This is trivial for the case δ = 1 and for δ > 1, we use that the function ϕ defined in (2) is now strictly decreasing to get that k = ϕ( x * ) < ϕ ψ(δ) < ϕ δ − 1 √ 2δ − 1 = 1. Then, the rest of the proof follows the same lines of the case when δ ∈ (0, 1) since this hypothesis is not longer used. Let us comment that the above proof is much simpler if we restrict to x * ∈ S X * (in particular, to the spherical modulus Φ S X (δ)), but the result for non-unital functionals is stronger. Actually, the following stronger version can be deduced by conveniently modifying the election of k in the proof of Theorem 2.1. Remark 2.3. For every 0 < θ < 1 and every 0 < δ < 2, there is ρ = ρ(δ, θ) > 0 such that for every Banach space X, if x * ∈ B X * with x * θ, x ∈ B X satisfy that Re x * (x) > 1 − δ, then there is a pair (y, y * ) ∈ Π(X) satisfying x − y < √ 2δ − ρ and x * − y * < √ 2δ − ρ. Let us observe that, given 0 < θ < 1, the hypothesis above is not empty only when 1 − θ < δ. On the other hand, in the proof it is sufficient to consider only the case of δ < 1 + θ, because, otherwise, the evident inequality Re x * (x) > −θ = 1 − (1 + θ) implies that there is a pair (y, y * ) ∈ Π(X) satisfying x − y < 2(1 + θ) and x * − y * < 2(1 + θ), so the statement of our remark holds true with ρ := √ 2δ − 2(1 + θ). Next, we rewrite Theorem 2.1 in two equivalent ways. Corollary 2.4. Let X be a Banach space. (a) Let 0 < ε < 2 and suppose that x ∈ B X and x * ∈ B X * satisfy Re x * (x) > 1 − ε 2 /2. Then, there exists (y, y * ) ∈ Π(X) such that x − y < ε and x * − y * < ε. (b) Let 0 < δ < 2 and suppose that x ∈ B X and x * ∈ B X * satisfy Re x * (x) > 1 − δ. Then, there exists (y, y * ) ∈ Π(X) such that x − y < √ 2δ and x * − y * < √ 2δ. As the last result of this section, we present an example of a Banach space for which the estimate in Theorem 2.1 is sharp. Example 2.5. Let X be the real space (2) ∞ . Then, Φ S X (δ) = Φ X (δ) = √ 2δ for all δ ∈ (0, 2). Proof. Fix 0 < δ < 2. We consider z = (1 − √ 2δ, 1) ∈ S X and z * = √ 2δ 2 , 1 − √ 2δ 2 ∈ S X * , and observe that z * (z) = 1 − δ. Now, suppose we may find (y, y * ) ∈ Π(X) such that z − y < √ 2δ and z * − y * < √ 2δ. By the shape of B X , we only have two possibilities: either y is an extreme point of B X or y * is an extreme point of B X * (this is actually true for all two-dimensional real spaces). Suppose first that y is an extreme point of B X , which has the form y = (a, b) with a, b ∈ {−1, 1}. As z − y = max{\|1 − √ 2δ − a\|, \|1 − b\|} < √ 2δ, we are forced to have b = 1 and a = −1. Now, we have y * = (−t, 1 − t) for some 0 t 1 and z * − y * = √ 2δ 2 + t + t − √ 2δ 2 = max √ 2δ, 2t √ 2δ, a contradiction. On the other hand, if y * is an extreme point of B X * , then either y * = (a, 0) or y * = (0, b) for suitable a, b ∈ {−1, 1}. In the first case, as z * − y * = √ 2δ 2 − a + 1 − √ 2δ 2 < √ 2δ, we are forced to have a = 1 and so, y = (1, s) for suitable s ∈ [−1, 1]. But then z − y √ 2δ, which is impossible. In case y * = (0, b) with b = ±1, we have z * − y * = √ 2δ 2 + 1 − √ 2δ 2 − b < √ 2δ, so b = −1 and therefore, y = (s, −1) for suitable s ∈ [−1, 1], giving z − y 2, a contradiction. Basic properties of the moduli Our first result is the continuity of the Bishop-Phelps-Bollobás moduli. Proposition 3.1. Let X be a Banach space. Then, the functions δ −→ Φ X (δ) and δ −→ Φ S X (δ) are continuous in (0, 2). We need the following three lemmata which could be of independent interest. Lemma 3.2. For every pair (x 0 , x * 0 ) ∈ B X × B X * there is a pair (y, y * ) ∈ Π(X) with Re y * (x 0 ) + x * 0 (y) 0. Moreover, if actually Re x * 0 (x 0 ) > 0 then (y, y * ) ∈ Π(X) can be selected to satisfy Re y * (x 0 ) + x * 0 (y) 2 Re x * 0 (x 0 ) . Proof. 1. Take y 0 ∈ S X ∩ ker x * 0 and let y * 0 be a supporting functional at y 0 . Then Re y * 0 (x 0 ) + x * 0 (y 0 ) = Re y * 0 (x 0 ) If the right hand side is positive we can take y = y 0 , y * = y * 0 , in the opposite case take y = −y 0 , y * = −y * 0 . 2. Take y = x0 x0 and let y * be a supporting functional at y. Then, since for a fixed a > 0 the minimum of f (t) := t + a t for t > 0 equals 2 √ a, we get Re y * (x 0 ) + x * 0 (y) = x 0 + 1 x 0 Re x * 0 (x 0 ) 2 Re x * 0 (x 0 ). The above lemma allows us to prove the following result which we will use to show the continuity of the Bishop-Phelps-Bollobás modulus. Lemma 3.3. Let X be a Banach space. Suppose (x 0 , x * 0 ) ∈ A X (δ 0 ) with 0 < δ < δ 0 < 2. Then: Case 1: If δ, δ 0 ∈]0, 1] then dist (x 0 , x * 0 ), A X (δ) 2 √ 1 − δ − √ 1 − δ 0 1 − √ 1 − δ 0 . Case 2: If δ, δ 0 ∈ [1, 2) then dist (x 0 , x * 0 ), A X (δ) 2 2 − δ 0 δ 0 · δ 0 − δ δ 0 − 1 + √ 1 − 2δ + δδ 0 . Proof. Denote t = Re x * 0 (x 0 ). Let (y, y * ) ∈ Π(X) be from the previous lemma (in case 1 we use part 2 of the lemma, in case 2 we use part 1). For every λ ∈ [0, 1] we define x λ = (1 − λ)x 0 + λy and x * λ = (1 − λ)x * 0 + λy * . Both x λ and x * λ belong to corresponding balls, and dist ∞ ((x 0 , x * 0 ), (x λ , x * λ )) 2λ. We have: (4) Re x * λ (x λ ) = (1 − λ) 2 t + λ(1 − λ) Re y * (x 0 ) + x * 0 (y) + λ 2 , so in case 1 Re x * λ (x λ ) (1 − λ) 2 t + 2λ(1 − λ) √ t + λ 2 = (1 − λ) √ t + λ 2 . Now we are looking for a possibly small value of λ, for which ( x λ , x * λ ) ∈ A X (δ). If δ 1 − t, the value λ = 0 is already ok and dist ∞ ((x 0 , x * 0 ), A X (δ)) = 0. If 0 < δ < 1 − t then the positive solution in λ of the equation (1 − λ) √ t + λ 2 = 1 − δ is λ t = √ 1 − δ − √ t 1 − √ t . Evidently, λ t ∈ [0, 1], so (x λt , x * λt ) ∈ A X (δ). Since λ t decreases in t, dist ∞ ((x 0 , x * 0 ), A X (δ)) 2λ t 2λ 1−δ0 = 2 √ 1 − δ − √ 1 − δ 0 1 − √ 1 − δ 0 . This completes the proof of case 1. In the case 2 we may assume t 1 − δ (otherwise the corresponding distance is 0 and the job is done), so t 0. By part 1 of the previous lemma and (4) Re x * λ (x λ ) (1 − λ) 2 t + λ 2 , so we are solving in λ the equation (1 − λ) 2 t + λ 2 − 1 + δ = 0, i.e. (1 + t)λ 2 − 2tλ + (t − 1 + δ) = 0. The discriminant of this equation is D = −tδ − δ + 1. Remark that D −(1 − δ)δ − δ + 1 = (1 − δ) 2 0 and t − 1 + δ 0, so there is a positive solution of our equation given by λ t = 1 1 + t (t + √ D) = 1 1 + t (t + √ 1 − tδ − δ). This λ t decreases in t, so λ t λ 1−δ0 = 1 δ 0 (1 − δ 0 + 1 − 2δ + δδ 0 ) = 2 + δ 0 δ 0 · δ 0 − δ δ 0 − 1 + √ 1 − 2δ + δδ 0 . For the continuity of the spherical modulus, we need the following result. Lemma 3.4. Let X be a Banach space. Suppose (x 0 , x * 0 ) ∈ A S X (δ 0 ) with 0 < δ < δ 0 < 2. Then: Case 1: If δ < 1 then dist ∞ (x 0 , x * 0 ), A S X (δ) 4(δ 0 − δ) δ 0 . Case 2: If δ ∈ [1, 2) and 2 − √ 2 − δ 0 < δ < δ 0 , then dist ∞ (x 0 , x * 0 ), A S X (δ) 2(δ 0 − δ) 2 − δ . Proof. Let us start with case 1. Fix ξ ∈ (0, δ). As x * 0 = 1, we may find y ξ ∈ S X satisfying x * 0 (y ξ ) > 1−ξ. For every λ ∈ [0, 1] we define x(λ, ξ) = λx 0 + (1 − λ)y ξ . Consider λ ξ = δ−ξ δ0−ξ ∈ [0, 1] and write x ξ = x(λ ξ , ξ ). An straightforward verification shows that Re x * 0 (x ξ ) > 1 − δ and so, as 1 − δ 0, we have that x ξ = 0 and also that Re x * 0 x ξ x ξ > 1 − δ. Therefore, x ξ x ξ , x * 0 ∈ A S X (δ). It remains to estimate x 0 − x ξ x ξ as follows: x 0 − x ξ x ξ x 0 − x ξ + x ξ − x ξ x ξ 2 δ 0 − δ δ 0 − ξ + \| x ξ − 1\| 2 δ 0 − δ δ 0 − ξ + \| x ξ − x 0 \| 2 δ 0 − δ δ 0 − ξ + x ξ − x 0 4 δ 0 − δ δ 0 − ξ . We get the result by just letting ξ −→ 0. Let us prove case 2. We have to distinguish the values of Re x * 0 (x 0 ). If Re x * 0 (x 0 ) > 1 − δ, then the proof is done. Suppose otherwise that 1 − δ Re x * 0 (x 0 ) > 1 − δ 0 . Fix ξ ∈ 0, min{2 − δ 0 , 4δ−2−δ0−δ 2 δ−1 } (observe that 4δ−2−δ0−δ 2 δ−1 > 0 by the conditions on δ). As x * 0 = 1, we may find y ξ ∈ S X satisfying x * 0 (y ξ ) > 1 − ξ. Now, we consider λ ξ = δ 0 − δ 2 − δ − ξ and x ξ = x 0 + λ ξ y ξ . Notice that λ ξ ∈ (0, 1) (since δ < δ 0 and ξ < 2 − δ 0 ) and x ξ x 0 − λ y ξ = 1 − λ ξ > 0. Also, observe that Re x * 0 (x ξ ) 1 − δ + λ ξ = (1 − δ)(2 − δ − ξ) + δ 0 − δ 2 − δ − ξ so, Re x * 0 (x ξ ) 0 since ξ 4δ−2−δ0−δ 2 δ−1 . Now, Re x * 0 x ξ x ξ Re x * 0 x ξ 1 − λ ξ > 1 − δ 0 + λ ξ (1 − ξ) 1 − λ ξ = 1 − δ. Therefore, x ξ x ξ , x * 0 ∈ A S X (δ). It remains to estimate x 0 − x ξ x ξ as follows: x 0 − x ξ x ξ x 0 − x ξ + x ξ − x ξ x ξ δ 0 − δ 2 − δ − ξ + x ξ − 1 δ 0 − δ 2 − δ − ξ + x ξ − x 0 δ 0 − δ 2 − δ − ξ + x ξ − x 0 2 δ 0 − δ 2 − δ − ξ . Consequently, letting ξ −→ 0, we get dist ∞ (x 0 , x * 0 ), A S X (δ) 2(δ 0 − δ) 2 − δ . Proof of Proposition 3.1. Let us give the proof for Φ X (δ). Observe that for δ 1 , δ 2 ∈ (0, 2) with δ 1 < δ 2 , one has 0 < Φ X (δ 2 ) − Φ X (δ 1 ) = d H (A X (δ 2 ), Π(X)) − d H (A X (δ 1 ), Π(X)) d H (A X (δ 2 ), A X (δ 1 )) . Now, the continuity follows routinely from Lemma 3.3. An analogous argument allows to prove the continuity of Φ S X (δ) from Lemma 3.4. The following lemma will be used to show that the approximation in the space is not worse than the approximation in the dual. It is actually an easy application of the Principle of Local Reflexivity. Lemma 3.5. For ε > 0, let (x, x * ) ∈ B X × B X * and let (ỹ * ,ỹ * * ) ∈ Π(Y * ) such that x * −ỹ * < ε and x −ỹ * * < ε. Then there is a pair (y, y * ) ∈ Π(X) such that x − y < ε and x * − y * < ε. Proof. First chose ε < ε such that still x * −ỹ * < ε and x −ỹ * * < ε . Now, we consider ξ > 0 such that (1 + ξ)ε + ξ + 2ξ 1 + ξ < ε, and use the Principle of Local Reflexivity (see [1,Theorem 11.2.4], for instance) to get an operator T : Lin {x,ỹ * * } −→ X satisfying T , T −1 1 + ξ, T (x) = x,ỹ * (T (ỹ * * )) = y * * (ỹ * ) = 1. Next, we considerx = T (ỹ * * ) T (ỹ * * ) ∈ S X andx * =ỹ * ∈ S X * , observe that Rex * (x) > 1 1 + ξ = 1 − ξ 1 + ξ , and we use Corollary 2.4 to get (y, y * ) ∈ Π(X) satisfying that x − y < 2ξ 1 + ξ and x * − y * < 2ξ 1 + ξ . Let us show that (y, y * ) ∈ Π(X) fulfill our requirements: x − y T (x) − T (ỹ * * ) + T (ỹ * * ) −x + x − y < (1 + ξ)ε + ξ + 2ξ 1 + ξ < ε and, analogously, x * − y * x * −ỹ * + ỹ * − y * < ε + 2ξ 1 + ξ < ε. Proposition 3.6. Let X be a Banach space. Then Φ X (δ) Φ X * (δ) and Φ S X (δ) Φ S X * (δ) for every δ ∈ (0, 2). Proof. The proof is the same for both moduli, so we are only giving the case of Φ X (δ). Fix δ ∈ (0, 2). We consider any ε > 0 such that Φ X * (δ) < ε and for a given (x, x * ) ∈ A X (δ) consider (x * , x) ∈ A X * (δ) ( we identify X as a subspace of X * * ) and so we may find (ỹ * ,ỹ * * ) ∈ Π(Y * ) such that x * −ỹ * < ε and x −ỹ * * < ε. Now, an application of the previous lemma gives us a (y, y * ) ∈ Π(X) such that x − y < ε and x * − y * < ε. This means that Φ X (δ) ε and, therefore, Φ X (δ) Φ X * (δ), as desired. We do not know whether the inequalities in Proposition 3.6 can be strict. Of course, this can not be the case when the space is reflexive. Corollary 3.7. For every reflexive Banach space X, one has Φ X (δ) = Φ X * (δ) and Φ S X (δ) = Φ S X * (δ) for every 0 < δ < 2. Our last result in this section states that when the Bishop-Phelps-Bollobás modulus is the worst possible, then the spherical Bishop-Phelps-Bollobás modulus is also the worst possible. Proposition 3.8. Let X be a Banach space. For every δ ∈ (0, 2), the condition Φ X (δ) = √ 2δ is equivalent to the condition Φ S X (δ) = √ 2δ. Proof. Since Φ S X (δ) Φ X (δ) √ 2δ, the implication Φ S X (δ) = √ 2δ ⇒ Φ X (δ) = √ 2δ is evident. Let us prove the inverse implication. Let Φ X (δ) = √ 2δ. Then there is a sequence of pairs (x n , x * n ) ∈ B X × B X * such that Re x * n (x n ) > 1 − δ but for every (y, y * ) ∈ Π(X) we have x n − y √ 2δ − 1 n or x * n − y * √ 2δ − 1 n . An application of Remark 2.3 gives us that x * n −→ 1 as n → ∞. As the duality argument given in Lemma 3.5 implies the dual version of Remark 2.3, we also have x n −→ 1 as n → ∞. Denotex n = xn xn , x * n = x * n x * n . In the case when δ ∈ (0, 1], we have Rex * n (x n ) > 1 − δ but for every (y, y * ) ∈ Π(X) x n − y √ 2δ − 1 n − x n −x n or x * n − y * √ 2δ − 1 n − x * n − x * n . Since the right-hand sides of the above inequalities go to √ 2δ, we get the condition Φ S X (δ) = √ 2δ. In the case of δ ∈ (1, 2), we no longer know that Rex * n (x n ) > 1 − δ, but what we do know is that lim inf Rex * n (x n ) 1 − δ, and that gives us the desired condition Φ S X (δ) = √ 2δ thanks to the continuity of the spherical modulus (Proposition 3.1). Examples We start with the simplest example of X = R. Example 4.1. Φ R (δ) = δ if 0 < δ 1 √ δ − 1 + 1 if 1 < δ < 2 , Φ S R (δ) = 0 for every δ ∈ (0, 2). Proof. We first fix δ ∈ (0, 1]. First observe that taking x = 1 − δ, x * = 1, it is evident that Φ R (δ) δ. For the other inequality, we fix x, x * ∈ [−1, 1] with x * x > 1 − δ. Then, x and x * have the same sign and we have that \|x\| > 1 − δ and \|x * \| > 1 − δ. Indeed, if \|x\| < 1 − δ, as \|x * \| 1, one has x * x = \|x * x\| < 1 − δ, a contradiction; the other inequality follows in the same manner. Finally, one deduces that \|x−sign(x)\| < δ and \|x * − sign(x * )\| < δ, as desired. Second, fix δ ∈ (1, 2). On the one hand, taking x = √ δ − 1, x * = − √ δ − 1, one has x * x = 1 − δ. As \|x + 1\| = √ δ − 1 + 1 and \|x * − 1\| = √ δ − 1 + 1, it follows that Φ R (δ) √ δ − 1 + 1. For the other inequality, we fix x, x * ∈ [−1, 1] with x * x > 1 − δ. If x and x * have the same sign, which we may and do suppose positive, then \|x − 1\| 1 < δ and \|x * − 1\| 1 < δ and the same is true if one of them is null. Therefore, to prove the last case we may and do suppose that x > 0 and x * < 0. Now, if we suppose, for the sake of contradiction, that \|x − (−1)\| √ δ − 1 + 1 and \|x * − 1\| √ δ − 1 + 1, we get x √ δ − 1 and −x * √ δ − 1, so −x * x δ − 1 or, equivalently, x * x 1 − δ, a contradiction. Therefore, either \|x − (−1)\| < √ δ − 1 + 1 and \|x * − (−1)\| < 1 < √ δ − 1 + 1 or \|x * − 1\| < √ δ − 1 + 1 and \|x − 1\| < 1 < √ δ − 1 + 1. The result for Φ S R is an obvious consequence of the fact that S R = {−1, 1}. Let us observe that the above proof gives actually a lower bound for Φ X (δ) for every Banach space X when δ ∈ (0, 1]. Remark 4.2. Let X be a Banach space. Then Φ X (δ) δ for every δ ∈ (0, 1]. Indeed, consider x 0 ∈ S X and x * 0 ∈ S X * with x * 0 (x 0 ) = 1 and write x = (1 − δ)x 0 and x * = x * 0 . Then Re x * (x) = 1 − δ and dist (x, S X ) = δ. We do not know a result giving a lower bound for Φ X (δ) when δ > 1, outside of the trivial one Φ X (δ) 1. Also, we do not know if the lower bound for the behavior of Φ X (δ) in a neighborhood of 0 given in the remark above can be improved for Banach spaces of dimension greater than or equal to two. We next calculate the moduli of a Hilbert space of (real) dimension greater than one. Example 4.3. Let H be a Hilbert space of dimension over R greater than or equal to two. Then: (a) Φ S H (δ) = 2 − √ 4 − 2δ for every δ ∈ (0, 2). (b) For δ ∈ (0, 1], Φ H (δ) = max δ, 2 − √ 4 − 2δ . For δ ∈ (1, 2), Φ H (δ) = √ δ. Proof. As we commented in the introduction, both Φ H and Φ S H only depend on the real structure of the space, so we may and do suppose that H is a real Hilbert space of dimension greater than or equal to 2. Let us also recall that H * identifies with H and that the action of a vector y ∈ H on a vector x ∈ H is nothing but their inner product denoted by x, y . In particular, Π(H) = (z, z) ∈ S H × S H . Therefore, for every δ ∈ (0, 2), Φ H (δ) (resp. Φ S H (δ)) is the infimum of those ε > 0 such that whenever x, y ∈ B H (resp. x, y ∈ S H ) satisfies x, y 1 − δ, there is z ∈ S H such that x − z ε and y − z ε. We will use the following (easy) claim in both the proofs of (a) and (b). Claim: Given x, y ∈ S H with x + y = 0, write z = x+y x+y to denote the normalized midpoint. Then x − z = y − z = 2 − 2 + 2 x, y . Indeed, we have x − z 2 = 2 − 2 x, z and 2 x, z = 2 x, x + y x + y = 2 + 2 x, y 2 + 2 x, y , giving x − z = 2 − 2 + 2 x, y , being the other equality true by symmetry. (a). Let first prove that Φ S H (δ) 2 − √ 4 − 2δ. Take x, y ∈ S H with x, y 1 − δ (so x + y = 0), consider z = x+y x+y ∈ S H and use the claim to get that x − z = y − z = 2 − 2 + 2 x, y 2 − √ 4 − 2δ. To get the other inequality, we fix an ortonormal basis {e 1 , e 2 , . . .} of H, consider x = 1 − δ/2 e 1 + δ/2e 2 ∈ S H and y = 1 − δ/2 e 1 − δ/2e 2 ∈ S H and observe that x, y = 1 − δ. Now, given z ∈ S H , we write z 1 = z, e 1 , z 2 = z, e 2 , and observe that max{ z − x 2 , z − y 2 } = max ± \|z 1 − 1 − δ/2\| 2 + \|z 2 ± δ/2\| 2 + 1 − z 2 1 − z 2 2 = z 2 1 + 1 − δ/2 − 2z 1 1 − δ/2 + max ± \|z 2 ± δ/2\| 2 + 1 − z 2 1 − z 2 2 = 2 − 2z 1 1 − δ/2 + 2\|z 2 \| δ/2 2 − 2 1 − δ/2. It follows that Φ S H (δ) 2 − √ 4 − 2δ, as desired. (b). We first fix δ ∈ (0, 1) and write ε 0 = max δ, 2 − √ 4 − 2δ . The inequality Φ H (δ) ε 0 follows by Remark 4.2, the fact that Φ H (δ) Φ S H (δ) and the result in item (a). To get the other inequality, we first observe that (5) Φ H (δ) Φ Lin {x,y} (δ) ∀x, y ∈ B H with x, y = 1 − δ. This follows from the obvious fact that Φ · (δ) increases when we restrict to subspaces. So, we are done if we restrict to the two-dimensional case and consider two points P = ( P , 0), Q = (q 1 , q 2 ) with q 2 0 and P Q , satisfying P, Q 1 − δ, and we find z ∈ S H such that P − z ε 0 and Q − z ε 0 . Now, it is straightforward to check that we have P ∈ √ 1 − δ, 1 , and q 1 = 1−δ P ∈ 1 − δ, √ 1 − δ . Figure 1 helps to the better understanding of the rest of the proof. Consider M = 1−δ+ P 2 P , P −(1−δ) 2 P , which is the normalized midpoint between A = (1, 0) and 2 and write ∆ to denote the arc of the unit sphere of H between A and M . We claim that Q ∈ z∈∆ B(z, ε 0 ) and P ∈ z∈∆ B(z, ε 0 ). Observe that this gives that there is z ∈ ∆ ⊂ S H whose distance to P and Q is less than or equal to ε 0 , finishing the proof. Let us prove the claim. First, we show that Q = (q 1 , q 2 ) ∈ z∈∆ B(z, ε 0 ). If q 2 B = 1−δ P , 1 − ( 1−δ P )P −(1−δ) 2 P , the ball of radius ε 0 centered in the point of ∆ with second coordinate equal to q 2 contains the point Q since ε 0 dist ((q 1 , 0), A) dist (Q, ∆). For greater values of q 2 , write first C = q 1 , P −(1−δ) 2 P , which belongs to B(M, ε 0 ) by the previous argument. Also, as M is the normalized mid point between A and B, we have by the claim at the beginning of this proof that Figure 1. Calculating Φ H (δ) for δ ∈ (0, 1) so, also, M − D ε 0 . Therefore, both the points C and C belong to B(M, ε 0 ), so also the whole segment [C, D] is contained there, and this proves the first part of the claim. To show the second part of the claim, that P ∈ z∈∆ B(z, ε 0 ), we consider the function M − B = 2 − 2 + 2 A, B = 2 − 2 + 2 1 − δ P 2 − √ 4 − 2δ ε 0 A 1 − δ √ 1 − δ P 1−δ P B M q1 D Cf (p) := 1 + p 2 − 2p(p + 1 − δ) p ∈ [ √ 1 − δ, 1] and observe that it is a convex function, so f (p) max{f (1), f ( √ 1 − δ} ε 2 0 . It follows that P − M = 1 + P 2 − 2 P ( P + 1 − δ) ε 0 , hence M ∈ B(P, ε 0 ). As also A ∈ B(P, ε 0 ), it follows that the whole circular arc ∆ is contained in B(P, ε 0 ) or, equivalently, that P ∈ z∈∆ B(z, ε 0 ). Let now fix δ ∈ (1, 2). Analogously to what we did before in equation (5), to show that Φ H (δ) √ δ, it is enough to consider the two-dimensional case and that, given p = ( p , 0) ∈ B H , q = (q 1 , q 2 ) ∈ B H with q 2 0, to find z ∈ S H such that z − P , z − Q √ δ. Routine computations show that z = p + q 1 2 , 1 − p + q 1 2 2 ∈ S H does the job. For the other inequality, we fix an ortonormal basis {e 1 , e 2 , . . .} of H, consider P = √ δ − 1 e 1 ∈ B H , Q = − √ δ − 1 e 1 ∈ B H and observe that P, Q = 1 − δ. For any z ∈ S H , we write z 1 = z, e 1 and we compute max{ z − P 2 , z − Q 2 } = max \|z 1 − √ δ − 1\| 2 + 1 − \|z 1 \| 2 , \|z 1 + √ δ − 1\| 2 + 1 − \|z 1 \| 2 = max ± \|z 1 ± √ δ − 1\| 2 + 1 − \|z 1 \| 2 = (\|z 1 \| + √ δ − 1) 2 + 1 − \|z 1 \| 2 = δ + 2 √ δ − 1\|z 1 \| δ. It follows that Φ H (δ) √ δ, as desired. In the next section we will show that the latter is a necessary condition that it is not actually sufficient. The first result is about Banach spaces admitting an L-descomposition. As a consequence we will calculate the moduli of L 1 (µ) spaces. Proposition 4.4. Let X be a Banach space. Suppose that there are two (non-trivial) subspaces Y and Z such that X = Y ⊕ 1 Z. Then Φ X (δ) = Φ S X (δ) = √ 2δ for every δ ∈ (0, 1/2]. Proof. Fix δ ∈ (0, 1/2] and consider (y 0 , y * 0 ) ∈ Π(Y ) and (z 0 , z * 0 ) ∈ Π(Z) and write x 0 = √ 2δ 2 y 0 , 1 − √ 2δ 2 z 0 ∈ S X x * 0 = 1 − √ 2δ y * 0 , z * 0 ∈ S X * . It is clear that Re x * 0 (x 0 ) = 1 − δ. Now, suppose that we may choose (x, x * ) ∈ Π(X) such that x 0 − x < √ 2δ and x * 0 − x * < √ 2δ. Write x = (y, z) ∈ Y ⊕ 1 Z, x * = (y * , z * ) ∈ Y * ⊕ ∞ Z * and observe that 1 = Re x * (x) = Re y * (y) + Re z * (z) y * y + z * z y + z = 1, therefore, we have (6) Re y * (y) = y * y . Now, we have 1 − √ 2δ − y * 1 − √ 2δ y * 0 − y * < √ 2δ from which follows that y * < 1 and so, y = 0 by (6), giving z = x = 1. But then, x 0 − x = √ 2δ 2 y 0 + 1 − √ 2δ 2 z 0 − z √ 2δ 2 + 1 − √ 2δ 2 − z = √ 2δ, a contradiction. We have proved that Φ X (δ) √ 2δ, being the other inequality always true. The result above produces the following example. It is immediate that with a dual argument than the one given in Proposition 4.4 it is possible to deduce the same for a Banach space which decomposes as an ∞ -sum. Actually, in this case we will get a better result using ideals instead of subspaces. Proposition 4.6. Let X be a Banach space. Suppose that X * = Y ⊕ 1 Z where Y and Z are (non- trivial) subspaces of X * such that Y w * = X * and Z w * = X * (w * is the weak * -topology σ(X * , X)). Then Φ X (δ) = Φ S X (δ) = √ 2δ for every δ ∈ (0, 1/2]. Proof. We claim that there are y 0 , z 0 ∈ S X and y * 0 ∈ S Y and z * 0 ∈ S Z such that Re y * 0 (y 0 ) = 1, Re z * 0 (z 0 ) = 1, y * (z 0 ) = 0 ∀y * ∈ Y, z * (y 0 ) = 0 ∀z * ∈ Z. Indeed, we define y 0 and y * 0 , being z 0 and z * 0 analogous. By assumption there is y 0 ∈ S X such that z * (y 0 ) = 0 for every z * ∈ Z and we may choose x * ∈ S X * such that Re x * (y 0 ) = 1 and we only have to prove that x * ∈ Y and then write y * 0 = x * . But we have x * = y * + z * with y * ∈ Y , z * ∈ Z and 1 = Re x * (y 0 ) = Re y * (y 0 ) y * y * + z * = 1, so z * = 0 and x * ∈ Y . We now define x * 0 = √ 2δ 2 y * 0 , 1 − √ 2δ 2 z * 0 ∈ S X * x 0 = 1 − √ 2δ y 0 + z 0 ∈ X and first observe that x 0 1; indeed, for every x * = y * + z * ∈ S X * one has \|x * (x 0 )\| = 1 − √ 2δ y * (y 0 ) + z * (z 0 ) 1 − √ 2δ y * + z * y * + z * = 1. It is clear that Re x * 0 (x 0 ) = 1 − δ. Now, suppose that we may choose (x, x * ) ∈ Π(X) such that x 0 − x < √ 2δ and x * 0 − x * < √ 2δ. We consider the semi-norm · Y defined on X by x Y := sup{\|y * (x)\| : y * ∈ S Y } which is smaller than or equal to the original norm, write x * = y * + z * with y * ∈ Y and z * ∈ Z, and observe that 1 = Re x * (x) = Re y * (x) + Re z * (x) y * x Y + z * x y * + z * = 1. Therefore, we have, in particular, that (7) Re y * (x) = y * x Y . Now, we have 1 − √ 2δ − x Y = 1 − √ 2δ y 0 Y − x Y 1 − √ 2δ y 0 − x Y < √ 2δ from which follows that x Y < 1 and so, y * = 0 by (7) and z * = x * = 1. But then, x * 0 − x * = √ 2δ 2 y * 0 + 1 − √ 2δ 2 z * 0 − z * √ 2δ 2 + 1 − √ 2δ 2 − z * = √ 2δ, a contradiction. Again, we have proved that Φ X (δ) √ 2δ, being the other inequality always true. Of course, the first consequence of the above result is to Banach spaces which decompose as ∞ -sum of two subspaces. Indeed, if X = Y ⊕ ∞ Z for two (non-trivial) subspaces Y and Z, then X * = Y ⊥ ⊕ 1 Z ⊥ and Y ⊥ and Z ⊥ are w * -closed, so far away of being dense. Therefore, Proposition 4.6 applies. We have proved the following result. Corollary 4.7. Let X be a Banach space. Suppose that there are two (non-trivial) subspaces Y and Z such that X = Y ⊕ ∞ Z. Then Φ X (δ) = Φ S X (δ) = √ 2δ for every δ ∈ (0, 1/2]. As a consequence, we obtain the following examples, analogous to the ones presented in Example 4.5. Examples 4.8. (a) Let (Ω, Σ, µ) a measure space such that L ∞ (Ω) has dimension greater than one and let E be any non-zero Banach space. Then, Φ L∞(µ,E) = Φ S L∞(µ,E) (δ) = √ 2δ δ ∈ (0, 1/2] . (b) Let Γ be a set with more than one point and let E be any non-zero Banach space. Then, Φ c0(Γ,E) = Φ S c0(Γ,E) (δ) = √ 2δ and Φ c(Γ,E) = Φ S c(Γ,E) (δ) = √ 2δ δ ∈ (0, 1/2] . Our next aim is to deduce from Proposition 4.6 that also arbitrary C(K) spaces have the maximum moduli and for this we have to deal with the concept of M -ideal. Given a subspace J of a Banach space X, J is called M -ideal if J ⊥ is a L-summand on X * (use [10] for background). In this case, X * = J ⊥ ⊕ 1 J where J = {x * ∈ X * : x * = x * \| J } ≡ J * . Now, if X contain a non-trivial M -ideal J, one has X * = J ⊥ ⊕ 1 J and to apply Proposition 4.6 we need that J to be not σ(X * , X)-dense. Actually, J is not dense in X * if and only if there is x 0 ∈ X \ {0} such that x 0 + y = max{ x 0 , y } for every y ∈ J (this is easy to verify and a proof can be found in [3]). Let us enunciate what we have shown. Corollary 4.9. Let X be a Banach space. Suppose that there is a non-trivial M -ideal J of X and a point x 0 ∈ X \ {0} such that x 0 + y = max{ x 0 , y } for every y ∈ J. Then, Φ X (δ) = Φ S X (δ) = √ 2δ for every δ ∈ (0, 1/2]. With the above corollary we are able to prove that the moduli of any non-trivial C 0 (L) space are maximum. J = f ∈ C 0 (L, E) : f \| U = 0}, which is an M -ideal of C 0 (L, E) by [10, Corollary VI.3.4] (use the simpler [10, Example I.1.4.a] for the scalar-valued case) and it is non-zero since L \ U has non-empty interior. As U is open and nonempty, we may find a non-null function x 0 ∈ C 0 (L, E) whose support is contained in U . It follows that x 0 + y = max{ x 0 , y } for every y ∈ J by disjointness of the supports. A sufficient condition to be in the hypotheses of Corollary 4.9 is that a Banach space X contains two non-trivial M -ideals J 1 and J 2 such that J 1 ∩ J 2 = {0} since, in this case, J 1 and J 2 are complementary M -summands in J 1 + J 2 [10, Proposition I.1.17]. Let us comment that this is actually what happens in C(K) when K has more than one point. A sufficient condition for a Banach space to have two non-intersecting M -ideals is that its centralizer is non-trivial (i.e. has dimension at least two). We are not going into details, but roughly speaking, the centralizer Z(X) of a Banach space X is a closed subalgebra of L(X) isometrically isomorphic to C(K X ) where K X is a Hausdorff topological space, and it is possible to see X as a C(K X )-submodule of k∈K X X k for suitable X k 's. We refer to [2, §3.B] and [10, §I.3] for details. It happens that every M -ideal of C(K X ) produces an M -ideal of X in a suitable way (see [2, §4.A]) and if Z(X) contains more than one point, then two non-intersecting M -ideals appear in X, so our corollary above applies. Corollary 4.12. Let X a Banach space. If Z(X) has dimension greater than one, then Φ X (δ) = Φ S X (δ) = √ 2δ for every δ ∈ (0, 2]. To give some new examples coming from this corollary, we recall that the centralizer of a unital (complex) C * -algebra identifies with its center (see [10,Theorem V.4.7] or [2, Example 3 in page 63]). It would be interesting to see whether the algebra L(H) for a finite-or infinite-dimensional Hilbert space H has the maximum Bishop-Phelps-Bollobás moduli. None of the results of this section applies to it since its center is trivial and, despite it contains K(H) as an M -ideal, there is no element x 0 ∈ L(H) satisfying the requirements of Corollary 4.9 (see [3, page 538]). Let us also comment that the bidual of L(H) is a C * -algebra with non-trivial centralizer, so Φ L(H) * * (δ) = Φ S L(H) * * (δ) = √ 2δ for every δ ∈ (0, 1/2]. If there is δ ∈ (0, 1/2] such that Φ L(H) (δ) < √ 2δ, then this would be an example when the inequality in Proposition 3.6 is strict. We finish this section with two pictures: one with the Bishop-Phelps-Bollobás moduli of R, C and Banach spaces with the greatest possible modulus Our goal in this section is to show that Banach spaces with the greatest possible moduli contain almost isometric copies of the real 2 ∞ . Let us first recall the following definition. Definition 5.1. Let X, E be Banach spaces. X is said to contain almost isometric copies of E if, for every ε > 0 there is a subspace E ε ⊂ X and there is a bijective linear operator T : E −→ E ε with T < 1 + ε and T −1 < 1 + ε. The next result is well-known and has a straightforward proof. Lemma 5.2. A real Banach space E contains an isometric copy of (2) ∞ if and only if there are elements u, v ∈ S E such that u − v = u + v = 2. E contains almost isometric copies of (2) ∞ if and only if there are elements u n , v n ∈ S E , n ∈ N such that u n − v n −→ 2 and u n + v n −→ 2 as n → ∞. The class of spaces X that do not contain almost isometric copies of (2) ∞ was deeply studied by James [11] (see also the exposition in Van Dulst's book [9]), who gave to such spaces the name "uniformly non-square". He proved in particular, that every uniformly non-square space must be reflexive, that this property is stable under passing to subspaces, quotient spaces and duals. In fact, a general result is true [12]: for every 2-dimensional space E if a real Banach space X does not contain almost isometric copies of E then X is reflexive. The aim of this section is to prove that if a real Banach space X satisfies that its Bishop-Phelps-Bollobás modulus is √ 2δ in at least one point δ ∈ (0, 1/2), then X (and, equivalently, the dual space) contains almost isometric copies of (2) ∞ . Actually, as shown in Remark 3.8 , Φ X (δ) = √ 2δ if and only if Φ S X (δ) = √ 2δ. Therefore, we may use the formally stronger hypothesis of Φ S X (δ) = √ 2δ. We will use some lemmas and ideas of Bishop and Phelps [4], but for the reader's convenience we will refer to the corresponding lemmas in the already classical Diestel's book [8]. From now on, X will denote a real Banach space. For t > 1 and x * ∈ S X * , we denote K(t, x * ) := {x ∈ X : x t x * (x)}. Observe that K(t, x * ) is a convex cone with non-empty interior. . For every z ∈ B X , every x * ∈ S X * and every t > 1, there is x 0 ∈ S X such that x 0 − z ∈ K(t, x * ) and [K(t, x * ) + x 0 ] ∩ B X = {x 0 }. Lemma 5.4 ([8, Chapter 1, Lemma 2] with a little modification that follows from the proof there). Let x * , y * ∈ S X * and suppose that x * ker y * ∩ S X ⊂ (−∞, ε/2]. Then dist x * , Lin y * ε/2 and min{ x * − y * , x * + y * } ε. Lemma 5.5. Let z ∈ B X , x * ∈ S X * , t > 1, and let x 0 ∈ S X be from Lemma 5.3. Denote y * ∈ S X * a functional that separates x 0 + K(t, x * ) from B X , so y * (x 0 ) = 1 and y * K(t, x * ) ⊂ [0, ∞). Then x * ker y * ∩ S X ⊂ (−∞, 1/t] and so, dist x * , Lin y * 1/t and min{ x * − y * , x * + y * } 2/t. Proof. This also can be extracted from [8, Chapter 1], but it is better to give a proof. For every w ∈ ker y * ∩ S X we have that w does not belong to the interior of K(t, x * ), so 1 = w t x * (w), i.e. x * (ker y * ∩ S X ) ⊂ (−∞, 1/t]. An application of Lemma 5.4 completes the proof. Now we are passing to our results. At first, for the sake of simplicity, we consider the easier finitedimensional case. Lemma 5.6. Let X be a finite-dimensional real space. Fix ε ∈ (0, 1). Suppose that (x, x * ) ∈ S X × S X * satisfies that x * (x) = 1 − ε 2 2 and that max{ y − x , y * − x * } ε for every pair (y, y * ) ∈ Π(X). Then for t = 2 ε , there exists y 0 ∈ [x + K(t, x * )] ∩ S X such that x * (y 0 ) = 1. Proof. Consider a sequence t n > t, n ∈ N, with lim n t n = t. Using Lemma 5.3, we get y n ∈ S X such that (8) y n − x ∈ K(t n , x * ) and (K(t n , x * ) + y n ) ∩ B X = {y n }. Let y * n ∈ X * be a functional that separates K(t n , x * ) + y n from B X , i.e. y * n (y n ) = 1 and y * n (K(t n , x * )) ⊂ [0, ∞). Then, according to Lemma 5.5, (9) min{ x * − y * n , x * + y * n } 2/t n < ε. But x * + y * n (x * + y * n )(y n ) = 1 + x * (y n ) = 1 + x * (x) + x * (y n − x) = 2 − ε 2 2 + x * (y n − x). Since (y n − x) ∈ K(t n , x * ), we have x * (y n − x) (y n − x) /t n 0 so x * + y * n 2 − ε 2 2 > ε (we have used here that 0 < ε < 1). Comparing with (9), we get x * − y * n < ε, so the condition of our lemma says that x − y n ε. Without loss of generality (passing to a subsequence if necessary) we can assume that y n tend to some y 0 . Then ε lim n y n − x lim n t n x * (y n − x) = t(x * (y 0 ) − x * (x)) 2 ε (x * (y 0 ) − 1 + ε 2 2 ) 2 ε (1 − 1 + ε 2 2 ) = ε. This means that all the inequalities in the above chain are in fact equalities. In particular, x * (y 0 ) = 1 and y 0 − x = lim n y n − x = t x * (y 0 ) − x * (x) , i.e. y 0 ∈ [x + K(t, x * )] ∩ S X . Lemma 5.7. Under the conditions of Lemma 5.6, there are y * ∈ S X * and α 1 − ε 2 with (10) x * − αy * ε 2 and x * − y * ε, and there is v ∈ S X such that (11) x * (v) = y * (v) = 1. Proof. Let y 0 be from the previous lemma. Fix a strictly increasing sequence of t n > 1 with lim n t n = t and let us consider two cases. Case 1 : Suppose there exists m 0 ∈ N with int K(t m0 , x * ) + x ∩ B X = ∅. Then, using the fact that for every closed convex set with non-empty interior, the closure of the interior is the whole set, we get y 0 ∈ x + K(t, x * ) ∩ B X = int x + K(t, x * ) ∩ B X = n m0 int x + K(t n , x * ) ∩ B X . So, we can pick (12) z n ∈ x + K(t n , x * ) ∩ B X such that z n −→ y 0 . In particular, x * (z n ) −→ 1. Let us apply Lemma 5.3: there are v n ∈ S X such that (13) v n − z n ∈ K(t n , x * ) and K(t n , x * ) + v n ∩ B X = {v n }. (12) implies that z n − x ∈ K(t n , x * ) which, together with (13), mean that v n − x ∈ K(t n , x * ). Consequently, Then x * (v n − z n ) 0, i.e. 1 x * (v n ) x * (z n ) −→ 1, so x * (v n ) −→ 1. Conditionv n − x t n x * (v n − x) t n ε 2 2 < ε. If we denote y * n ∈ S X * to the functional that separates v n + K(t n , x * ) from B X , then (v n , y * n ) ∈ Π(X). Since we are working under the conditions of Lemma 5.6, it follows that y * n − x * ε. Also, by Lemma 5.5, dist (x * , Lin y * n ) 1/t n , so there are α n ∈ R such that x * − α n y * n 1/t n . Again, without loss of generality, we may assume that the sequences (α n ), (v n ) and (y * n ) have limits. Let us denote α := lim n α n , y * := lim n y * n , and v := lim n v n . Then v = 1, y * = 1, x * (v) = lim n x * (v n ) = 1, and y * (v) = lim n y * n (v n ) = 1. This proves (11). Also, x * − αy * = lim n x * − α n y * n 1 t = ε 2 . Consequently, ε 2 x * − αy * (x * − αy * )(v) = 1 − α, so, α 1 − ε 2 .(14) Case 2 : Assume that for every n ∈ N we have int K(t n , x * ) + x ∩ B X = ∅. Let us separate x + int (K(t n , x * )) from B X by a norm-one functional y * n , that is, y * n x + int K(t n , x * ) > 1, so, in particular, y * n (x) 1. Again, passing to a subsequence, we can assume that there exists y * = lim n y * n which satisfies y * = 1, 1 y * (x) lim n y * n (x) 1. So, y * (x) = 1, i.e. (x, y * ) ∈ Π(X). By the conditions of our lemma, this implies that y * − x * = max{ x − x , y * − x * } ε. Since y 0 ∈ x + K(t, x * ) = n∈N int x + K(t n , x * ) , we can select z n ∈ int x + K(t n , x * ) in such a way that z n −→ y 0 . Then y * (y 0 ) = lim n y * n (z n ) 1, hence, y * (y 0 ) = 1. This means that condition (11) works for v := y 0 . The remaining conditions can be deduced from Lemma 5.5 the same way as in the case 1. We are now able to state and prove the main result of the section in the finite-dimensional case. Theorem 5.8. Let X be a finite-dimensional real Banach space. Suppose that there is a δ ∈ (0, 1/2) such that Φ X (δ) = √ 2δ (or, equivalently, Φ S X (δ) = √ 2δ). Then X * contains an isometric copy of (2) ∞ (hence, X also contains an isometric copy of (2) ∞ ). Proof. Denote ε := √ 2δ ∈ (0, 1). There is a sequence of pairs (x n , x * n ) ∈ S X × S X * such that x * n (x n ) > 1 − δ = 1 − ε 2 2 and max{ y − x n , y * − x * n } ε − 1 n for every pair (y, y * ) ∈ Π(X). Since the space is finite-dimensional, we can find a subsequence of (x n , x * n ) that converges to a pair (x, x * ) ∈ S X × S X * . This pair satisfies that x * (x) 1 − δ and for every (y, y * ) ∈ Π(X), max{ y − x , y * − x * } max{ y − x n , y * − x * n } − max{ x − x n , x * − x * n } ε − 1 n − max{ x − x n , x * − x * n } −→ ε. Since by Theorem 2.1, x * (x) cannot be strictly smaller than 1 − δ, we have x * (x) = 1 − δ. Therefore, we may apply Lemma 5.7 to find y * ∈ S X * and α 1 − ε 2 for which conditions (10) and (11) are fulfilled. Now we claim that in fact there is only one number γ ∈ R for which (15) x * − γy * ε 2 and this γ equals 1 − ε 2 . So α = 1 − ε 2 and, we also claim that (16) x * − αy * = ε 2 and x * − y * = ε. Indeed, when we were proving equation (14), we proved that every γ ∈ R that satisfies (15) must satisfy γ 1 − ε 2 . On the other hand, the function γ −→ x * − γy * is convex, so the set G of those γ ∈ R satisfying (15) also must be convex; but 1 / ∈ G, so γ < 1. Finally, according to (10), ε 2 1 − γ = y * − γy * x * − y * − x * − γy * ε 2 . This means that all the inequalities above must be equalities, so γ 1 − ε 2 , and also (16) must be true. The claim is proved. and lim n x * Observe that we have shown that the values of the functional x * n on ker y * n ∩ S X do not exceed 1 t−rn + 2r n . Therefore, by Lemma 5.4, dist (x * n , Lin y * n ) 1 t − r n + 2r n −→ 1 t and so there are α n ∈ R such that lim n x * n − α n y * n 1 t . The remaining conditions in (21) and (22) can be deduced the same way as in the case 1. Finally, (21) and (22) imply that lim n x * n − y * n = lim n x * n − y * n = 2: the proof does not differ much from the corresponding part of the Theorem 5.8 demonstration. Corollary 5.10. Let X be a uniformly non-square Banach space. Then, Φ S X (δ) Φ X (δ) < √ 2δ for every δ ∈ (0, 1/2). Consequently, every superreflexive Banach space can be equivalently renormed in such a way that, in the new norm, Φ S X (δ) Φ X (δ) < √ 2δ for all δ ∈ (0, 1/2). It would be interesting to obtain a quantitative version of the above corollary. 6. A three dimensional space E containing (2) ∞ with Φ E (δ) < √ 2δ In the last section we decided to decorate our paper with two diamonds: The first of them represents the unit ball of the space D ε that we construct below, and on the second picture one can see the unit ball of D * ε . Like in the previous section, for every δ ∈ (0, 1/2) we denote ε = √ 2δ, so 0 < ε < 1. We denote B 3 ε ⊂ R 3 the absolute convex hull of the following 11 points A k , k = 1, . . . , 11 (or, what is the same, the convex hull of 22 points ±A k , k = 1, . . . , 11): A 1 = (0, 0,3 4 ) , A 2 = (1 − ε, 1, ε 2 ), A 3 = (1 − ε, −1, ε 2 ), A 4 = (ε − 1, 1, ε 2 ), A 5 = (ε − 1, −1,ε 2 ) , A 6 = (1, 1 − ε, ε 2 ), A 7 = (−1, 1 − ε, ε 2 ), A 8 = (1, ε − 1, ε 2 ), A 9 = (−1, ε − 1,ε 2 ) , A 10 = (1, 1, 0), A 11 = (1, −1, 0). Denote D ε ("D" from "Diamond") the normed space R 3 , · , for which B 3 ε is its unit ball. Then D * ε can be viewed as R 3 with the polar of B 3 ε as the unit ball, and the action of x * ∈ D * ε on x ∈ D ε is just the standard inner product in R 3 . Let us list, without proof, some properties of D ε whose verification is straightforward: Our next aim is to present a number of examples for which the values of the Bishop-Phelps-Bollobás moduli are the maximum possible, namely Φ S X(δ) = Φ X (δ) = √ 2δ for small δ's. As we always have Φ S X (δ) Φ X (δ) √2δ, it is enough if we prove the formally stronger result that Φ S X (δ) = √ 2δ for small δ's (actually, the two facts are equivalent, see Proposition 3.8), and this is what we will show. It happens that all of the examples have in common that they contains an isometric copy of the real space Example 4. 5 . 5Let (Ω, Σ, µ) be a measure space such that L 1 (µ) has dimension greater than one and let E be any non-zero Banach space. Then,Φ L1(µ,E) (δ) = Φ S L1(µ,E) (δ) = √2δ for every δ ∈ (0, 1/2]. Indeed, we may find two measurable sets A, B ⊂ Ω with empty intersection such that Ω = A ∪ B. Then Y = L 1 (µ\| A , E) and Z = L 1 (µ\| B , E) are non-null, L 1 (µ, E) = Y ⊕ 1 Z and so the results follows from Proposition 4.4. Particular case of the above example are 1 and L 1 [0, 1]. Example 4 . 10 . 410Let L be a locally compact Hausdorff topological space with at least two points and let E be any non-zero Banach space. Then Φ C0(L,E) (δ) = Φ S C0(L,E) (δ) = √ 2δ for every δ ∈ (0, 1/2]. Indeed, we may find a non-empty non-dense open subset U of L and consider the subspace Corollary 4 . 11 . 411Let X be a Banach space. Suppose there are two non-trivial M -ideals J 1 and J 2 such that J 1 ∩ J 2 = {0}. Then Φ X (δ) = Φ S X (δ) = √ 2δ for every δ ∈ (0, 1/2]. Example 4 . 13 . 413Let A be a unital C * -algebra with non-trivial center. Then, Φ A (δ) = Φ S A (δ) = √ 2δ for every δ ∈ (0, 1/2]. one with the corresponding values of the spherical Bishop-Phelps-Bollobás moduli. Figure 2 .Figure 3 . 23The value of Φ X (δ) for R, C and The value of Φ S X (δ) for R, C and(2)∞ Figure 4 . 2 Figure 5 . 425The unit ball of D 1 The unit ball of D * 1 2 Now, let us defineand let us show that functionals u * and y * span a subspace of X * isometric to(2)∞ . According to Lemma 5.2, it is sufficient to show that u * − y * = u * + y * = 2. Let us do this. At first,At second,Let us comment that for complex Banach spaces, we cannot expect that Theorem 5.8 provides a complex copy of(2)∞ in the dual of the space. Namely, the two-dimensional complex space X = (2) 1 satisfies Φ X (δ) = √ 2δ for δ ∈ (0, 1/2) but it does not contain the complex space(2)∞ (of course, it contains the real space(2)∞ as a subspace since(2) 1and(2)∞ are isometric in the real case). We do not know whether it is true a result saying that if a complex space X satisfies Φ X (δ) = √ 2δ for some δ ∈ (0, 1/2), then X contains a copy of the complex space(2)1 or a copy of the complex space(2)∞ . Let us extend the result of Theorem 5.8 to the infinite-dimensional case. Roughly speaking, we proceed as in the proof of such theorem, but instead of selecting convergent subsequences, we select subsequences such that their numerical characteristics (like norms of elements, pairwise distances, or values of some important functionals) have limits.. Then X * (and hence also X) contains almost isometric copies offor every pair (y, y * ) ∈ Π(X). Since we have x * n (x n ) 1 − (ε − 1 n ) 2 /2 by Theorem 2.1, we deduce that lim n x * n (x n ) = 1 − δ. Denote t = 2 ε . Now, we are going to proceed like in Lemma 5.6 in order to show that there is a sequence (y n ) of elements in S X such that (18) lim n y n − x n t lim n x * n (y n − x n ) and lim n x * n (y n ) = 1.Pick a sequence (t n ) with t n > t, n ∈ N and lim n t n = t. Using Lemma 5.3, for every n ∈ N we get y n ∈ S X such that (19) y n − x n ∈ K(t n , x * n ) and (K(t n , x * n ) + y n ) ∩ B X = {y n }. For given n ∈ N, let u * n ∈ S X * be a functional that separates K(t n , x * n ) + y n from B X , that is, satisfying u * n (y n ) = 1 and u * n (K(t n , x * n )) ⊂ [0, ∞). Then, according to Lemma 5.5, we have min{ x * n − u * n , x * n + u * n } 2/t n < ε. As we havewe get x * n − u * n < ε, so (17) says that x n − y n ε − 1 n . Without loss of generality, passing to a subsequence if necessary, we can assume that the following limits exist: lim n x n − y n , lim n x * n (y n − x n ) n (y n ). Then ε limThis means that all the inequalities in the above chain are in fact equalities. In particular, lim n x * n (y n ) = 1, andso the analogue of Lemma 5.6 is proved. Now, we proceed with analogue of Lemma 5.7: we need to show that there are y * n ∈ S X * and α n 0,and there is a sequence of v n ∈ S X such thatCase 1 : Assume that there exist r > 0 and n ∈ N such that, for all m > n, Observe first that λ m is smaller than every value of λ for whichOn the one hand, if y m − x m − tx * m (y m − x m ) 0, then λ = 0 belongs to the set in question, and the job is done. On the other hand, ifis positive and belongs to the set in question. This means thatbut the limit of the right-hand side equals 0 thanks to (20). So condition (23) is proved. This means that y m,λm ∈ x m + K(t, x * m ) and y m,λm − y m 2λ m −→ 0. Let us pick a little bit biggerλ m > λ m in such a way that we still have y m,λm − y m −→ 0, but for somet n < t witht n −→ t, we haveThen, in particular, lim n x * n (y n,λn ) = lim n x * n (y n ) = 1. Let us apply Lemma 5.3. There are v n ∈ S X such that (25) v n − y n,λn ∈ K(t n , x * n ) andThen x * n (v n − y n,λn ) 0, i.e. 1 x * n (v n ) x * n (y n,λn ) −→ 1, so x * n (v n ) −→ 1. This proves the first part of (22). Condition (24) imply that y n,λn − x n ∈ K(t n , x * n ) which, together with (25), mean that v n − x n ∈ K(t n , x * n ). Consequently,If we denote by y * n ∈ S X * the functional that separates v n + K(t n , x * ) from B X , then (v n , y * n ) ∈ Π(X) (this proves the second part of (22) even in a stronger form) so, thanks to (17),Also, by Lemma 5.5, dist (x * n , Lin y * n ) 1/t n , so there are α n ∈ R such that x * − α n y * n 1/t n .Again, without loss of generality, we may assume that the sequences (α n ) and x * n − α n y * n converge. Then,so lim n α n 1 − ε 2 . Starting at this point, (21) can be deduced in the same way as it was done for (16). Case 2 : Assume that there is a sequence of r n > 0, r n −→ 0 and that there is a subsequence of (x m , x * m ) (that we will again denote (x m , x * m )) such thatThen alsoLet us separatefrom B X by a norm-one functional y * n , that is, (26) y * n K(t − r m , x * m ) + x m > 1 − r m so, in particular, y * m (x m ) 1 − r m and lim m y * m (x m ) = 1. By the Bishop-Phelps-Bollobás theorem, there is a sequence (x n ,ỹ * n ) ∈ Π(X), such that max{ x n − x n , ỹ * n − y * n } −→ 0 as n → ∞. Again, passing to a subsequence, we can assume that all the numerical characteristics that appear here have the corresponding limits. According to (17), for n big enough, we haveWe can select z n ∈ x n + K(t − r n , x * n ) in such a way that z n − y n −→ 0. Then 1 lim n y * n (y n ) = lim n y * n (z n ) lim n (1 − r n ) = 1.This means that condition (22) works for v n := y n . Now consider an arbitrary w ∈ ker y * n ∩ S X . Taking a convex combination with an element h of the unit sphere where y * n (h) almost equals −1, we can construct an elementw ∈ B X such that w − w 2r n and y * n (w) = −r n . Then, by (26),w / ∈ int K(t − r n , x * n ) , so w (t − r n )x * n (w). Consequently,x * n (w) x * n (w) + 2r n 1 t − r n + 2r n .• The subspace of D ε formed by vectors of the form (x 1 , x 2 , 0) is canonically isometric to(2)∞ . • There are no other isometric copies of(2)∞ in D ε .• The subspace of D * ε formed by vectors of the form (x 1 , x 2 , 0) is canonically isometric to(2)1 (and so, is isometric to• There are no other isometric copies of(2)∞ in D * ε . • The following operators act as isometries both on D ε and D * ε : (. In other words, changing the sign of one coordinate or rearranging the first two coordinates do not change the norm of an element.The following theorem shows that the existence of an(2)∞ -subspace does not imply that Φ X (δ) = √ 2δ, even in dimension 3.Proof. Assume contrary that Φ X (δ) = √ 2δ. Like in the proof of Theorem 5.8, this implies the existence of a pair (x, x * ) ∈ S X × S X * with the following properties: x * (x) = 1 − δ and(27)max{ z − x , z * − x * } ε for every pair (z, z * ) ∈ Π(X).Also, repeating the proof of Theorem 5.8 for this x * ∈ S X * , we can find u * , y * ∈ S X * such that the pair (u * , y * ) is 1-equivalent to the canonical basis of(2)1 andThis means that x * = ε 2 u * + (1 − ε 2 )y * . What can be this (u * , y * ) if we take into account that there is only one isometric copy of(2)1 in X * ? It can be either u * = (1, 0, 0), y * = (0, 1, 0), or a pair of vectors that can be obtained from this one by application of isometries, i.e. just 8 possibilities. Consequently, x * either equals to the vector (ε/2, 1 − ε/2, 0), or to a vector that can be obtained from this one by application of isometries, again just 8 possibilities.By duality argument, there are u, y ∈ S X such that the pair (u, y) is 1-equivalent to the canonical basis of(2)1 andSince the only (up to isometries) pair u, y ∈ S X of this kind is u = (1, 1, 0), y = (1, −1, 0), we get x = (1, 1 − ε, 0), or can be obtained from this one by application of isometries. So there are 8 × 8 = 64 possibilities for the pair (x, x * ). Taking into account that x * (x) = 1 − δ we reduce this number to 8 possibilities: x = (1 − ε, 1, 0), x * = (ε/2, 1 − ε/2, 0) and images of this pair under remaining 7 reflections and rotations of the underlying R 2 . If we show that this choice of (x, x * ) do not satisfy condition (27) then, by symmetry, the remaining choices would not satisfy (27) neither, and this would give us the desired contradiction.Indeed, the pair (z, z * ) ∈ Π(X) that do not satisfy (27) for x = (1 − ε, 1, 0), x * = (ε/2, 1 − ε/2, 0) is the following one: z = (1 − ε, 1, ε/2), z * = (ε/2, 1 − ε/2, ε). Let us check the required properties. At first, z = A 2 ∈ S X . Then, z * (z) = 1. The last property means, that z * 1, so in order to check that z * = 1 it remains to show that \|z * (A k )\| 1 for all k. This is true for ε < 1. Finally, z − x = (0, 0, ε/2) = ε 2 4 3 A 1 = 2 3 ε < ε, and z * − x * = (0, 0, ε) = (0, 0, ε), A 1 = 3 4 ε < ε. F Albiac, N Kalton, Graduate Texts in Mathematics 233. New YorkSpringerTopics in Banach Space TheoryAlbiac F., Kalton N., Topics in Banach Space Theory, Graduate Texts in Mathematics 233, Springer, New York, 2006. . E Behrends, M -Structure, Banach-Stone The, Theorem, Lecture Notes in Math. 736Springer-VerlagE. Behrends, M -structure and the Banach-Stone Theorem, Lecture Notes in Math. 736, Springer-Verlag, Berlin, 1979. E Behrends, M -complements of M -ideals. 29E. Behrends, M -complements of M -ideals, Rev. Roumaine. Math. Pures Appl. 29 (1984), 537-541. A proof that every Banach space is subreflexive. E Bishop, R R Phelps, Bull. Amer. Math. Soc. 67E. Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc 67 (1961), 97-98. An extension to the theorem of Bishop and Phelps. B Bollobás, Bull. London Math. Soc. 2B. Bollobás, An extension to the theorem of Bishop and Phelps, Bull. London Math. Soc. 2 (1970), 181-182. Numerical Ranges II. F F Bonsall, J Duncan, London Math. Soc. Lecture Note Series. 10F. F. Bonsall and J. Duncan, Numerical Ranges II, London Math. Soc. Lecture Note Series 10, Cambridge 1973. A Bishop-Phelps-Bollobás type theorem for uniform algebras. B Cascales, V Kadets, A J Guirao, preprintB. Cascales, V. Kadets, and A. J. Guirao, A Bishop-Phelps-Bollobás type theorem for uniform algebras, preprint. Geometry of Banach spaces Lecture notes in Math. J Diestel, Springer-Verlag485BerlinJ. Diestel, Geometry of Banach spaces Lecture notes in Math. 485, Springer-Verlag, Berlin, 1975. Reflexive and superreflexive Banach spaces. D Van Dulst, Mathematical Centre Tracts. 102ppAmsterdam: Mathematisch CentrumD. Van Dulst, Reflexive and superreflexive Banach spaces. Mathematical Centre Tracts. 102. Amsterdam: Mathe- matisch Centrum. V, 1978, 273 pp. M -ideals in Banach spaces and Banach algebras. P Harmand, D Werner, D Werner, Lecture Notes in Math. 1547P. Harmand, D. Werner, and D. Werner, M -ideals in Banach spaces and Banach algebras, Lecture Notes in Math. 1547, 1993. Uniformly non-square Banach spaces. R C James, Ann. Math. 2R.C. James, Uniformly non-square Banach spaces. Ann. Math. (2) 80, (1964) 542-550 On two-dimensional universal Banach spaces. V M Kadets, C. R. Acad. Bulg. Sci. 35RussianKadets, V.M. On two-dimensional universal Banach spaces. (Russian) C. R. Acad. Bulg. Sci. 35(1982), 1331-1332. Support Cones in Banach Spaces and Their Applications. R R Phelps, Adv. Math. 13R. R. Phelps, Support Cones in Banach Spaces and Their Applications, Adv. Math. 13 (1974), 1-19. R R Phelps, Convex functions, monotone operators and differentiability. BerlinSpringer-Verlag1364second editionR. R. Phelps, Convex functions, monotone operators and differentiability (second edition), Lecture Notes in Math. 1364, Springer-Verlag, Berlin, 1993. . Spain Granada, Granada, Spain	[]
[ "A CNN Approach for 5G mmWave Positioning Using Beamformed CSI Measurements", "A CNN Approach for 5G mmWave Positioning Using Beamformed CSI Measurements" ]	[ "Ghazaleh Kia [email protected] \nDepartment of Computer Science\nDepartment of Computer Science\nUniversity of Helsinki Helsinki\nFinland\n", "Laura Ruotsalainen [email protected] \nUnit of Electrical Engineering\nUniversity of Helsinki Helsinki\nFinland\n", "Jukka Talvitie [email protected] \nTampere University Tampere\nFinland\n" ]	[ "Department of Computer Science\nDepartment of Computer Science\nUniversity of Helsinki Helsinki\nFinland", "Unit of Electrical Engineering\nUniversity of Helsinki Helsinki\nFinland", "Tampere University Tampere\nFinland" ]	[]	The advent of Artificial Intelligence (AI) has impacted all aspects of human life. One of the concrete examples of AI impact is visible in radio positioning. In this article, for the first time we utilize the power of AI by training a Convolutional Neural Network (CNN) using 5G New Radio (NR) fingerprints consisting of beamformed Channel State Information (CSI). By observing CSI, it is possible to characterize the multipath channel between the transmitter and the receiver, and thus provide a good source of spatiotemporal data to find the position of a User Equipment (UE). We collect ray-tracing-based 5G NR CSI from an urban area. The CSI data of the signals from one Base Station (BS) is collected at the reference points with known positions to train a CNN. We evaluate our work by testing: a) the robustness of the trained network for estimating the positions for the new measurements on the same reference points and b) the accuracy of the CNN-based position estimation while the UE is on points other than the reference points. The results prove that our trained network for a specific urban environment can estimate the UE position with a minimum mean error of 0.98 m.	10.48550/arxiv.2205.03236	[ "https://arxiv.org/pdf/2205.03236v1.pdf" ]	248,562,906	2205.03236	b92463fea60e389db55d4cec34481cff5f4c18c4	A CNN Approach for 5G mmWave Positioning Using Beamformed CSI Measurements 30 Apr 2022 Ghazaleh Kia [email protected] Department of Computer Science Department of Computer Science University of Helsinki Helsinki Finland Laura Ruotsalainen [email protected] Unit of Electrical Engineering University of Helsinki Helsinki Finland Jukka Talvitie [email protected] Tampere University Tampere Finland A CNN Approach for 5G mmWave Positioning Using Beamformed CSI Measurements 30 Apr 2022Index Terms-5G New Radio (NR)Artificial Intelligence (AI)Channel State Information (CSI)Convolutional Neural Network (CNN)FingerprintingMachine Learning (ML)Outdoor Posi- tioning The advent of Artificial Intelligence (AI) has impacted all aspects of human life. One of the concrete examples of AI impact is visible in radio positioning. In this article, for the first time we utilize the power of AI by training a Convolutional Neural Network (CNN) using 5G New Radio (NR) fingerprints consisting of beamformed Channel State Information (CSI). By observing CSI, it is possible to characterize the multipath channel between the transmitter and the receiver, and thus provide a good source of spatiotemporal data to find the position of a User Equipment (UE). We collect ray-tracing-based 5G NR CSI from an urban area. The CSI data of the signals from one Base Station (BS) is collected at the reference points with known positions to train a CNN. We evaluate our work by testing: a) the robustness of the trained network for estimating the positions for the new measurements on the same reference points and b) the accuracy of the CNN-based position estimation while the UE is on points other than the reference points. The results prove that our trained network for a specific urban environment can estimate the UE position with a minimum mean error of 0.98 m. I. INTRODUCTION In urban areas, where Global Navigation Satellite Systems (GNSS) signals are only partially available and degraded by multipath, it is difficult to reach a satisfactory positioning performance for many important use cases. To address the difficulty and achieve an accurate position solution, expensive and professional sensors such as Inertial Measurement Units (IMUs), cameras or other Radio Frequency (RF) signals are required to assist the positioning solution. Radio signals enable the position estimation by different methods, such as fingerprinting and signal features [1], [2], range measurement and multilateration [3], and the combination of the ranging and Angle of Arrival (AoA) [4]. In addition, known or estimated channel properties of signals which are known as Channel State Information (CSI) are able to characterize the environment by describing how a signal propagates from a transmitter to a receiver. Consequently, in dense urban areas the multi-path effect resulted from the environment, is embedded in the CSI data. Thus, CSI is a good candidate for positioning purposes, where one of the use cases is fingerprinting [5], [6]. Fingerprinting is mainly made of two phases: offline and online. In the offline phase, a large data set of fingerprints for each known position is collected and fed to the system. Later in the online phase, the test data is compared with the data set and maps the collected data to the corresponding position [7]. Compared to traditional ranging and angular based positioning methods, where Non-Line-of-Sight (NLOS) propagation can considerably decrease the accuracy [8], fingerprinting can exploit the richer features in NLOS scenarios to obtain a better positioning solution. CSI fingerprints of Wi-Fi systems have been frequently used for indoor positioning [6], [9], [10]. Nevertheless, in outdoor urban areas Wi-Fi signals are not always available, or they are of poor quality. On the other hand, in the emerging 5G networks, efficient communications require continuous exchange of CSI data between the BS and the User Equipment (UE) [11], which makes 5G New Radio (NR) CSI a convenient fingerprint in the urban area. The new millimeter Wave (mmWave) frequency band, introduced in the 5G NR, can be used to reach sub-meter level positioning accuracy [12]. However, due to the high frequencies, the mmWave signals suffer from high penetration and path loss as well as the rain fading, which typically limits the number of available BSs for positioning. On the other hand, the short wavelength in millimeter level allows implementation of larger antenna arrays compared to the previous technologies [12]. Partly resulting from the increased array sizes, another supplement provided by 5G NR technology is the beam-based operation, which enables highly directional beamforming via Massive Multiple Input Multiple Output (MIMO) technology [13]. Therefore, in this work, the utilized CSI data is beamformed and the CSI observations are performed per beam. These beam-wise CSI measurements work as an input to the proposed Convolutional Neural Network (CNN) based positioning engine. The main contributions of this paper are as follows: 1) We introduce a novel utilization of 5G mmWave in position estimation. To the best of our knowledge, 5G NR CSI-based fingerprinting utilizing beamforming technology has not been reported in the literature. 2) For obtaining the results, we utilize a realistic raytracing-based CSI data, considering a real-life city area with rich multipath propagation while following the guidelines of 5G NR specifications [16]. 3) We propose and analyze a neural network-based fingerprinting with CNN-architecture for positioning in urban area. This method requires low-power and low-memory. 4) We consider a single BS positioning scenario to reduce the system complexity, and avoid the demand of observing multiple BSs using mmWave signals under high path loss scenarios. In addition, wrapping up the neural network-based processing in a single BS enables efficient scalability of the proposed methods for various use cases, including cases with varying BS deployment. The remainder of the article is as follows. The related works are discussed in Section II, the preliminaries are provided in Section III. Our proposed CNN-based fingerprinting utilizing 5G NR CSI is presented in Section IV, the experiment and analysis are provided in Section V, followed by conclusion and future work in Section VI. II. RELATED WORKS Fingerprints of radio signals for positioning have been frequently used in the literature. Wu et al. in [5], have utilized Wi-Fi-based CSI data collected by an Atheros CSI tool in an indoor environment. They have proposed a supervised Deep Neural Network (DNN) to estimate the position of the UE. Authors proved that their proposed network architecture outperforms its counterpart in terms of being fast in the online phase and saving memory for the storage of the weights and biases of the network. In [14], DeepNar was proposed as a neural network-based fingerprinting method. Authors have used Wi-Fi Round Trip Time (RTT) fingerprints from 7 BSs to localize a UE in an indoor scenario. There, the RTT measurements are first normalized to stay in the range between [0,1] and then fed to the neural network. By utilizing DeepNar, the authors have achieved a minimum mean positioning error of 91 cm with the test samples. This error is the minimum calculated average error in estimation of test points positions using the test CSI samples. In [6], a deep residual sharing learning based system was presented utilizing Wi-Fi CSI data collected in an indoor environment with the Intel 5300 NIC tool. To prepare the input data for the neural network, authors have constructed CSI tensors utilizing CSI amplitude and AoA measurements. Considering the two types of measurements, the authors feed the data to two different channels. Thus, the trained weights and biases are different in the output of the channels and they will be shared at the end of a residual block. Authors have tested their method in the indoor environment of a laboratory and a corridor, and they were able to achieve the minimum mean error of 1.16 m, which is the least achieved average error among the estimations for the test samples. In urban environments, Wi-Fi signals are not always available. In [1], authors take advantage of the fingerprints including beamformed radio signal strength in a 5G NR network. They have also fused the position estimations with the available GNSS position data to achieve higher accuracy. The considered signals are transmitted by seven beamforming-capable BSs over 32 beams for each. The results in their work show that 3.4 m mean positioning error is achieved by the proposed fingerprinting method. This error decreased to 1.75 m mean by fusing the radio signal strength based estimates with GNSS position data utilizing a neural network. In [2], authors exploit deep learning-based fingerprinting to localize a UE in an urban area. They have used the Reference Signal Received Power (RSRP) of 5G beams and the beams ID as the input data to their proposed neural network. For simulating the data, the authors have used WinProp ray-tracing simulation tool. They have utilized 8 5G radio nodes (gNBs) and achieved the minimum average error of 1.4 m. Authors in [15] have utilized 5G CSI fingerprints and Siamese CNN to find the position of a UE in an indoor area. They have achieved a minimum mean error of 1.17 m using 4 BSs. In this work, despite the previous works, we are interested in beamformed CSI fingerprints. Furthermore, regarding the lack of Wi-Fi signals in outdoor environment, we take advantage of the beamforming technology of 5G signals to improve the results for an outdoor scenario. Besides, we utilize only one BS to evaluate the performance of the proposed method in an outdoor positioning scenario. A summary of methods considered in the literature, which utilize neural networks is presented in Table I. Besides the literature references, the environment type, radio signal fingerprint, and the minimum mean error achieved by the method are provided in the table. The minimum mean error is the lowest error among the averaged errors achieved in position estimation by feeding the test CSI samples to the trained network. III. PRELIMINARIES WITH DATA DESCRIPTION In this work, we take advantage of the 5G mmWave CSI fingerprints of a MIMO Orthogonal Frequency Devision Multiplexing (OFDM) system to train a CNN. The trained network will be then used for fingerprinting. A. Neural network based fingerprinting In traditional fingerprinting methods, a large amount of data is required during the real-time process to estimate the position of the UE. However, utilizing neural networks enables the system to eliminate the required memory and expense to carry the heavy database in the online phase. There are mainly two phases in the neural network-based fingerprinting method: 1) Offline phase: a large amount of CSI data are given as input to the system. The data are labeled with the position of each training point. The weights and biases of the neural network will be optimized based on the given labeled data. Finally, the trained network will be used in the online phase. 2) Online Phase: There will be no requirement of memorizing the training data set. The only necessity will be the weights and biases of the trained network. The test data which is not previously seen by the network will be provided as the input to the network. The network will then estimate the position of the UE. B. Channel State Information (CSI) While the radio signals propagate through the air, they experience different types of channel effects, such as diffractions, specular reflections, scattering, and path loss. The radio channel can be represented by CSI which is essentially a complexvalued frequency response of the channel in OFDM-based transmission. Assuming a single antenna UE, the received signal at the m th subcarrier of the b th beam can be represented as r m,b = H m,b F b s m,b + n m,b ,(1) where s m,b is the transmitted signal, H m,b ∈ R 1×NTX is the channel matrix between the transmitter and the receiver including the effect of all multipaths, F b ∈ R NTX×1 is the beamformer vector, and n m,b ∼ N (0, σ 2 ) is Additive White Gaussian Noise (AWGN). From (1), the effective beamformed CSI for the m th subcarrier and b th beam can be observed as H CSI m,b = H m,b F b . In practice, the CSI is estimated from the received signal assuming a known transmitted reference signal [5]. To prepare the input data for the neural network, the available CSI data are received by M number of subcarriers and B number of beams. Considering that the CSI data involve complex numbers, to prepare the CSI data in a way that they are applicable to CNN, we have added the real and imaginary parts of the CSI data separately in two consecutive columns. Using this method prevents any probable data loss during the CNN classification [11]. Thus, finally, we have the data of M × 2B matrices as the input to the network. Assuming a fixed and static environment, separate CSI measurements collected at the same location differ only by measurement noise. In addition, the noise on the CSI data can be considered to vary according to changes in the environment, such as the movement of vehicles [17]. Thus, for each considered measurement location simulated in the raytracing tool, a large amount of data can be generated by obtaining different random noise realizations. The quality of the beam-wise CSI measurements is dependent on the signal SNR, which is defined as SN R b = 10 log 10 m \|r m,b \| 2 M σ 2 [dB],(2) where σ 2 is the noise power. C. CNN and data splitting CNNs have the ability to extract the features of the given input and detect the important features in comparison with other traditional Machine Learning (ML) methods including random forests. The features extraction is performed by an element-wise product between the inputs and a small array of numbers, called a kernel. The kernel is applied to the input by sliding over all the locations to extract the features. Notice that the convolutional layer is linear. Thus, a nonlinear function such as a Rectifying Linear Unit (ReLU) is required to enable the back-propagation. Before feeding the data set as the input to the neural network, the data set is first split into the following categories: 1) Training data set: Some measurement points on the simulation area are considered as the reference points. A large amount of CSI data are collected at each reference point. Notice that the labels of the CSI data are the positions of each reference point. 2) Validation data set: This data set is generated for each reference point. However, the validation data has been never seen previously by the network as it has different noise from the training data set. This data set is used to evaluate the robustness of the neural network. 3) Test data set: In this data set, new points other than reference points are considered to evaluate the network performance and positioning accuracy. For this data set, there will not be any specific labels. Instead of finding only one label, the network will find the probabilities of the labels. The position will then be estimated based on the predicted probabilities. These data sets are then fed to the proposed CNN when considering the numerical evaluations shown in Section V. IV. PROPOSED CNN-BASED FINGERPRINTING UTILIZING 5G NR CSI The proposed system architecture is illustrated in Fig. 1, where the CNN is developed to extract the features in CSI data. The hyperparameter of number of layers is optimized by observing the generalization error while increasing the number of layers. The optimum number of layers is selected based on the lowest validation error while avoiding overfitting. We have utilized 5 convolutional2D layers with a ReLU function at each layer to define non-linearity. As the downsampling strategy, 2D max-pooling is used in the first 4 layers [18]. Then, in the 5th layer, flattening is done to prepare the input for the output layer of the network, which is the classifier. A. Hyperparameters To achieve the highest accuracy, the choice of hyperparameter of a network plays an important role. Furthermore, having optimal or close to optimal hyperparameters results in saving time, energy, computation, and money [19]. Hyperparameters consist of two types of variables: a) the variables which determine the network structure like the number of hidden layers, and b) the variables which affect the training of the network, such as the learning rate, weight decay, and batch size. Moreover, the tuned hyperparameters for the developed CNN are listed in Table II. Regularization is one of the most essential hyperparameters. In this work, the techniques utilized for regularization are L2 norm and Batch Normalization (BatchNorm). L2 norm is applied with the weight decay and BatchNorm normalizes the input of the layer. It subtracts the input by mean value of the mini-batch and finally divides it by the mini-batch standard deviation. This technique solves the problem of internal covariate shift. This problem occurs due to the varying distribution of the input weights to the neurons at each epoch. Neurons must adapt to this change, which is solved by the BatchNorm technique [20]. Another issue that DNNs might experience, is the vanishing or exploding gradient. Vanishing gradients occur due to the gradients getting smaller while B. Loss Function The parameters of the neural network are optimized using AdamW optimizer explained in detail in [21]. AdamW is an adaptive optimizer which provides the optimization of weight decay and learning rate separately. The loss function is used to calculate the error in predictions while training the network. We have used two different error calculations during the training and the test phase. 1) Training and the robustness of test performance: For training the network, we aim to minimize the Negative Log Likelihood (NLL) loss function given as loss = − N n=1 y n logŷ n ,(3) where N is the number of labels (number of reference points in our work), y n is the true label of the n th training data and y n represent the probabilities computed by the softmax layer asŷ n = Sof tmax(z j ) = exp z j K k=1 exp z k ,(4) where z j represents the neurons values in output layer. The values are calculated by the input to the neurons, weights and biases. We utilize NLL loss for both training and validation. The validation is done by CSI data which have never been seen by the network previously. However, this CSI data have the same labels with the reference points. 2) New Test Points: The final test is done for the CSI data collected at the points other than reference points. The loss/error in the predictions which are estimated positions for the test points is the Euclidean distance to the ground truth. The predicted coordinates (x pred ,y pred ) for the corresponding CSI data are estimated using the softmax layer probabilities as x pred = R i=1 p i R j=1 p j x i and y pred = R i=1 p i R j=1 p j y i ,(5) where (x i , y i ) represents the position of the training points, p i is the probability calculated by the softmax layer, and R is the number of selected training points, defined as R = 4 in this work. The R selected training points have the largest calculated probabilities p i among the others. The estimator is sensitive to the value of R. R = 4 is selected considering the consistency it provides in the accuracy of calculated positions. Higher values of R result in lower accuracy and lower values of R provides inconsistent solutions. V. NUMERICAL EXPERIMENT AND ANALYSIS The proposed method is tested by simulating CSI data in an urban environment in Helsinki metropolitan area called Punavuori. In order to obtain the CSI data, we utilize a proprietary ray-tracing software, following the map-based channel modeling principles proposed by the 3GPP in [16]. In total of 30 number of training point locations and 10 number of test point locations are considered in the environment, as shown in Fig. 2. The carrier frequency is defined as 30 GHz. At each location, the CSI is measured from a single BS based on a downlink transmitted Synchronization Signal Blocks (SSBs) consisting of 240 subcarriers with subcarrier spacing of 60 kHz (bandwidth of 14.4 MHz). Furthermore, the SSBs are beamformed over B = 32 beams using a uniform rectangular array with dimensions of 16 azimuth and 8 elevation elements. For each reference point, 1000 noisy CSI samples are obtained for training and validation, as discussed in Section III.B. The radio link budget is designed so that in Line-of-Sight (LOS) conditions, the SNR for the best beam is approximately 10 dB at 100 m distance from the BS. From the generated data 60% is used for training and 40% for validation using random splitting. Similarly, for testing purposes, 100 CSI samples are correspondingly generated for the test point locations. The training and validation accuracy are presented in Fig. 3. Considering the training accuracy, we can realize that the network has been appropriately trained. It shows that the proposed CNN is able to extract the features out of the CSI tensors. Furthermore, the validation accuracy, extracted from the validation set, has reached maximum 97.77%. The high validation accuracy indicates that first the network is wellregularized due to regularization methods, and second, the network is robust against the unseen data. The training and validation loss are presented in Fig. 4. It is shown that the training loss values are decreasing to a point of stability and the optimized model fits well to the training data. Moreover, considering that also the validation loss is decreasing to a good extent, the model seems to perform adequately. However, it can be seen that the validation loss begins to slightly increase around the 80 th epoch. This can be a result of the data shuffling at each epoch. It could be the case that the random sampler has taken a more complex data at these epochs in comparison with the previous epochs. However, at this point we expect the regularization method to modify the learning algorithm so that the model generalizes in the optimum way. The mean absolute positioning error for the test point locations is presented in Fig. 5. The time required for our system to estimate the position of a test point by feeding one CSI sample to the network is 8 ms. It should be yet emphasized that the test data is collected at different locations compared to the training data, and thus, by enhancing the generalization of the network, it should be possible to reduce the positioning error. Although the network is able to consistently reduce the positioning error when increasing the number of epochs, it is seen that similarly as with the validation loss, the network struggles around the 80 th epoch. It seems that it took a few epochs for the regularization method to modify the learning algorithm and reduce the generalization error and achieve the consistent downward trend in test error. In order to analyze the positioning errors at different map locations, the error distribution at each test point is illustrated in Fig. 6. It is shown that in most of the points, the average error is below one meter. Considering that points number 1532 and 1560 are in NLOS conditions and the rest are in LOS conditions, one noticeable observation is that the proposed method works accurately also in NLOS conditions. From the CSI perspective this is justified by the fact that the CSI is more unique per location in NLOS conditions compared to LOSdominated scenarios, where nearby locations share similar multipath characteristics with only a slight power difference. Thus, on contrary to traditional positioning methods, machine learning based methods seem to naturally manage positioning in both LOS and NLOS conditions in the considered urban environment. VI. CONCLUSION AND FUTURE WORK In this work, we have presented a deep learning-based position estimation method using 5G CSI fingerprints and beamforming technology for the first time in the literature. We have taken advantage of the neural networks to find the static single BS-based position of a UE in a specific outdoor area. The CSI fingerprints of 5G NR mmWave have been collected based on ray-tracing simulations. The results show that by using machine learning-based CSI fingerprinting with CNNs, it is possible to achieve a sub-meter positioning error regardless of the LOS condition of the channel. In the next steps of this research, higher number of reference points in a wider area will be considered. Furthermore, a mobile user positioning will be investigated by utilizing the time series and recurrent neural networks with measurements from several BSs. Fig. 1 . 1System architecture of the proposed positioning method. Fig. 2 . 2Reference and test points in the simulated environment. The green circles are the reference points, and the red circles with unique label numbering are the test points. The BS location is shown with a magenta marker accompanied with a pointer vector to illustrate the antenna direction. Fig. 4 . 4The loss of training and validation. Fig. 5 . 5The mean absolute error of positioning for the test points. Fig. 6 . 6The distribution of positioning errors at each test point. TABLE I NEURAL INETWORK-BASED FINGERPRINTING METHODSReference Environment Radio Signal Fingerprint Minimum Mean Error [5] indoor Wi-Fi CSI 1.8 m [14] indoor Wi-Fi RTT 0.91 m [6] indoor Wi-Fi CSI 1.16 m [1] outdoor 5G and GNSS radio signal strength 1.75 m [2] outdoor 5G RSRP and beam IDs 1.4 m [15] indoor 5G CSI from 4 BSs 1.17 m Proposed Work outdoor 5G CSI from 1 BSs with beamforming technology 0.98 m TABLE II HYPERPARAMETERS IIFOR TRAINING THE NETWORKHyperparameter Value Activation Function ReLU Regularization BatchNorm and Weight Decay Decay Rate 0.001 Learning Rate 0.000001 Epochs Size 150 Batch Size 20 the network is being trained. On the other hand, exploding gradients occur when big errors accumulate and represent large values to the network. This issue prevents the network to be trained and results in an unstable network. 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[ "Controllability Robustness under Actuator Failures: Complexities and Algorithms", "Controllability Robustness under Actuator Failures: Complexities and Algorithms" ]	[ "Yuan Zhang " ]	[]	[]	The problem of determining the minimal number of inputs (i.e., actuators) whose removal destroy controllability of a linear time invariant (LTI) system is addressed. This problem is inverse to the recently well-studied minimal controllability problems, and significant in measuring controllability robustness of an LTI system under denial of service attacks on actuators. It is first proven that this problem is generally NP-hard. Then, a pseudo-polynomial time algorithm is suggested for solving this problem on systems with bounded maximum geometric multiplicities. Moreover, this algorithm is extended to the case where each actuator has a non-negative cost to be removed.	null	[ "https://arxiv.org/pdf/1812.07745v1.pdf" ]	119,664,863	1812.07745	dd952fb41836570848bc3782b996bbd7e9279f56	Controllability Robustness under Actuator Failures: Complexities and Algorithms 19 Dec 2018 Yuan Zhang Controllability Robustness under Actuator Failures: Complexities and Algorithms 19 Dec 20181Index Terms Controllability robustnessactuator removalssecurity analysiscomplexity The problem of determining the minimal number of inputs (i.e., actuators) whose removal destroy controllability of a linear time invariant (LTI) system is addressed. This problem is inverse to the recently well-studied minimal controllability problems, and significant in measuring controllability robustness of an LTI system under denial of service attacks on actuators. It is first proven that this problem is generally NP-hard. Then, a pseudo-polynomial time algorithm is suggested for solving this problem on systems with bounded maximum geometric multiplicities. Moreover, this algorithm is extended to the case where each actuator has a non-negative cost to be removed. I. INTRODUCTION Input/output selections under the objectives to meet/optimize certain system performances have long been active but challenging issues in control community [8]. In a recent paper [5], it is first shown that, given an autonomous system as (1), it is NP-hard to determine the minimal number of state variables that need to be actuated by an input to ensure system controllability, x(t) = Ax(t),(1) where x(t) is the state vector, and A is state transition matrix. A more general proposition is that, given a collection of possible input matrix columns, it is NP-hard to choose the minimal number of columns to form an input matrix so that the resulting system can be controllable [5], [7]. In this paper, we consider an inverse problem to that. That is, given a state transition matrix and its associated input matrix, determining the minimal number of inputs whose removal destroy controllability of the resulting system. This problem is well-encountered when one needs to evaluate whether controllability of a system is robust against malicious attacks on its actuators, for example, the denial of service attacks on actuators. Main Contributions: It is proven that the above problem is generally NP-hard for the first time. Nevertheless, a pseudopolynomial time algorithm is provided for solving that problem on systems with bounded maximum geometric multiplicities. This algorithm uses traversals over a recursive tree built from the left eigenspaces of the system state transition matrix and the input matrix. Moreover, this algorithm is extended to the case where each actuator has a non-negative cost. In addition, the cardinality-constraint submodular function minimization structure of the involved problems are revealed. Notations: R, C, Z and N denote the set of real, complex, integral and non-negative integral numbers, respectively. For a matrix M , M ij denotes its (i, j)-th entry if no confusion is made. II. PROBLEM FORMULATION Consider the following linear time invariant (LTI) systeṁ x(t) = Ax(t) + Bu(t),(2) where x(t) ∈ R n is the state vector, u(t) ∈ R m is the input vector, A ∈ R n×n and B ∈ R n×m are respectively the state transition matrix and input matrix. Without loss of generality, assume that every column of B is a non-zero vector. Moreover, let V = {1, ..., m}, and B S denotes the submatrix of B consisting of columns of B indexed by S ⊆ V . System (2) is said to be controllable, if for any two states x 0 , x 1 ∈ R n , there exists an input u(t) that can drive the system states from x 0 to x 1 in finite time. We just simply say (A, B) is controllable if System (2) is controllable. For practical cyber-physical systems, we may want to know whether a given system like (2) can preserve its controllability under denial of service attacks on a cardinality-constrained set of actuators. Notice that the denial of service attack on an actuator means that the attacked actuator cannot inject signals into the state variables, which we just call 'actuator failure' or 'actuator removal'. We say System (2) is p-robust against actuator failures, p ∈ N, if under the failures of an arbitrary set of p actuators, the resulting system is still controllable. To measure such controllability robustness, the following optimization problem is considered in this paper Problem 1: Given (A, B) in (2) min S⊆V \|S\| (A, B V \S ) is uncontrollable Denote the cardinality of the optimal solution to Problem 1 by p * . Then, it is easy to see that, System (2) is p-robust against actuator failures for any nonzero integers p ≤ p * − 1. Furthermore, a variant of Problem 1 is to determine the minimum dimension of controllable subspaces of a given system under cardinality-constrained actuator failures. This problem asks for the worst damage in controllable subspaces that the failures of a set of actuators with cardinality upper bound can cause. The following well-know PBH test gives a necessary and sufficient condition for System (2) to be controllable. Lemma 1: System (2) is controllable, if and only if for each eigenvalue λ of A, there exists no x ∈ C n , x = 0, such that x ⊺ A = λx ⊺ and x ⊺ B = 0. Moreover, assume that A has p ≤ n distinct eigenvalues, and denote the ith eigenvalue by λ i , for i = 1, ..., p. Let k i be the dimension of the left null space of λ i I − A; that is, k i is the geometric multiplicity of λ i . In addition, let X i = [x i1 , ..., x iki ] be a left eigenbasis of A associated with λ i ; that is, X i is stacked by k i vectors which are linearly independent spanning the left null space of λ i I − A. With these definitions, from Lemma 1, it is easy to obtain the following corollary. Corollary 1: System (2) is controllable, if and only if X ⊺ i B is of full row rank for i = 1, ..., p. III. COMPLEXITY ANALYSIS A natural question is whether the above two problems are solvable in polynomial time. In this section we give a negative answer to that question. Theorem 1: Problem 1 and Problem 2 are both NP-hard. Proof: If Problem 2 is solvable in polynomial time, let l, which is the cardinality upper bound of S in Problem 2, increase from 0 to n. It is easy to see that, the first l that makes the optimal value of Problem 2 less than n, is exactly the optimal value of Problem 1. Hence, to show the NP-hardness of Problem 2, it suffices to show the NP-hardness of Problem 1. To this end, in the following we give a reduction from the Linear Degeneracy Problem [1]. The linear degeneracy problem is to determine whether a given d × p rational matrix F = [f 1 , ..., f p ] contains a degenerate submatrix of order d, i.e., det[f i1 , ..., f i d ] = 0, for some i 1 , ..., i d ∈ {1, ..., p}. In [1], it is proven that this problem is NPcomplete, and there are infinitely many integral matrices associated with which the linear degeneracy problem is NP-complete. Now, let X be an arbitrary k × n integral matrix (k < n) with full row rank, and α 1 = max{\|X ij \|}. Notice that the full row rank constraint does not alter the NP-completeness of the linear degeneracy problem associated X. Let X ⊥ be the basis matrix of the null space of X, i.e., X ⊥ is an n × (n − k) matrix spanning the null space of X. X ⊥ can be constructed via the Gaussian elimination method in polynomial time O(n 3 ). Moreover, let α 2 = max{\|X ⊥ ij \|}, and α max = max{α 1 , α 2 }. Notice that the entries of X ⊥ are rational, and X ⊥ multiplied by any nonzero scalars is still a basis matrix of the null space of X. Hence, the encoding length of α 2 (i.e., log 2 α 2 ) can be polynomially bonded by k and n; so is α max . Next, define H(η) as H(η) = X (X ⊥ ) T +η1 (n−k)×n , where 1 (n−k)×n denotes the (n − k) × n matrix whose entries are all one. Then, clearly det H(0) = 0. Select one rational number η * that satisfies η * > α 2 and det H(η * ) = 0. We will show that such η * can be found in polynomial time. First, det H(η) is a polynomial of η with degree at most one by noting that the coefficient matrix of η in H(η) is rank one, and that det H(0) = 0. Hence, in arbitrary set consisting of 2 distinct rational numbers which are bigger than α 2 , there must exist one η * , such that det H(0)(1 − η * ) = det H(1) * η * , leading to det H(η * ) = 0. Second, it holds that det H(0) ≤ α n max n n by noting that det H(0) consists of the summations of n! ≤ n n signed products of precisely one entry per row and column of H(0). Similarly, det H(1) ≤ (α max + 1) n n n . Hence, both det H(0) and det H(1) have encoding lengths polynomially bounded by n (i.e., the encoding lengths are n log 2 α max + n log 2 n and n log 2 (α max + 1) + n log 2 n respectively). After determining an η * satisfying the above requirements, let matrices P = H(η * ), Γ = diag{I k , 2, 3, ..., n − k + 1}, construct the system (A, B) as A = P −1 ΓP, B = I n . Since the encoding lengths of entries of P are polynomially bounded by n, its inversion P −1 = adj(P )/ det P can be computed in polynomial time and has polynomially bounded encoding lengths too, where adj(P ) denotes the adjugate matrix of P . Hence, A can be computed in polynomial time. We claim that the optimal value of Problem 1 associated with (A, B) is no more than n − k, if and only if there exists an k × k submatrix of X which has zero determinant. Indeed, from the construction of A, the (k + i)th row of P , denoted by P [k+i] , is the left eigenvector of A associated with the eigenvalue i + 1, i = 1, ..., n − k. Notice that, all entries of P [k+i] are nonzero. Hence, all n columns of B need to be removed, such that the resulting P [k+i] B V \S fail to be of full row rank, where V = {1, ..., n}. From Corollary 1, the optimal value of Problem 1 associated with (A, B) then equals the minimal number of columns whose removal from X makes the resulting X fail to be of full row rank. If such value is no more than n − k, there must exist an k × k + submatrix of X which is not of full row rank for some k + ≥ k. Then clearly, the linear degeneracy problem associated with X is yes. Conversely, suppose that there is an k × k submatrix of X with zero determinant, and denote it by XS,S ⊆ V . Then, clearly one just needs to remove the columns indexed by V \S from X, such that the resulting X fail to be of full row rank. Hence, the optimal value of Problem 1 is no more than \|V \S\| = n − k. Combining the fact that the linear degeneracy problem associated with X is NP-complete, this proves the NP-hardness of Problem 1. We present some corollaries of Theorem 1. First, notice that in the proof of Theorem 1, B = I n which means that each input actuates only one state variable. Hence, removing one input corresponds to that exactly one state variable loses its direct input signals. The following corollary is immediate. Corollary 2: Given a system (A, B), it is NP-hard to determine the minimal number of state variables that need to be blocked from their direct inputs, such that the resulting system becomes uncontrollable. Next, suppose that System (2) is measured by the following equation: y(t) = Cx(t),(3) where y(t) is the output vector, C is the output matrix. By duality between controllability and observability, it is easy to obtain the following corollary on observability robustness under output failures. Corollary 3: For System (2)-(3), it is NP-hard to determine the minimal number of outputs whose failures make the resulting system unobservable. Taking the input and output into consideration together, we have the following conclusion. Corollary 4: For System (2)-(3), it is NP-hard to determine the minimal total number of inputs and outputs whose failures make the resulting system neither controllable nor observable. Proof : Construct (A, B) as suggested in the proof of Theorem 1, i.e., A = P −1 ΓP , B = I n . Construct C such that C ∈ R k×n and (A, C) is observable. Since the eigenbases P of A are available, the matrix C can be constructed in polynomial time as suggested in [3]. Notice that k is the maximum geometric multiplicity of eigenvalues of A. According to [3], the minimal number of outputs that ensure observability of the associated system equals the maximum geometric multiplicity of A. Hence, removing any one of these k outputs can make the system unobservable. As a result, the minimal total number of inputs and outputs whose failures cause uncontrollability and unobservability equals p * + 1, where p * is the minimal number of inputs that need to be removed to make the associated system uncontrollable. The latter problem is shown to be NP-hard in Theorem 1. The required result follows. IV. PSEUDO-POLYNOMIAL TIME ALGORITHMS In this section, we give a pseudo-polynomial time algorithm for Problem 1 for systems with bounded eigenvalue geometric multiplicities. This algorithm is based on the traversals over a recursive tree, constructed from the eigenspaces of the system state transition matrix and input matrix. Then, we extend this algorithm to the case where each input has a non-negative cost, and the purpose is to find the minimal cost of input set whose removal causes uncontrollability. Finally, we point out that Problem 2 has the structure of cardinality-constrained submodular function minimization. We shall assume that a collection of left eigenbases of A are computationally available, and denote them by X i \| p i=1 . We will use r min to denote the optimal value of Problem 1 associated with (A, B). Recall that p is the number of distinct eigenvalues of A, and k i is the geometric multiplicity of the ith eigenvalue. Moreover, rewrite the input matrix as B = [b 1 , ..., b m ]. We first consider a simple case where A has no repeated eigenvalues (i.e., A is simple), and then the general case. A. Simple Dynamics Case Assume that matrix A has no repeated eigenvalues. Then, {X i \| n i=1 } becomes a collection of left eigenvectors of A. For the ith eigenvalue of A, define F i as the collection of columns of B that are not orthogonal to X i ; that is F i = {j : X ⊺ i b j = 0, j ∈ {1, ..., m}}. Since X i is a vector, X ⊺ i b j becomes a scalar. Then, from Corollary 1, it's obvious that r min = min i∈{1,...,n} \|F i \|. Obviously, finding r min can be done in polynomial time. Indeed the above analysis is nothing but trivial. Can we extend it to the general case where the geometric multiplicities of eigenvalues of A can be greater than one? B. General Case Now we assume that the geometric multiplicities of eigenvalues of A are bounded by some constant k ∈ N. That is, k i ≤ k, i = 1, ..., p, as n and m increase. For most practical systems, this assumption is reasonable. Indeed, it is found that random square matrices generically have no repeated eigenvalues [2]. Let us focus on an individual eigenvalue λ i , i ∈ {1, ..., p}. Let r i be the minimum number of columns whose removals from Y (i) . = [X ⊺ i b 1 , ..., X ⊺ i b m ] make the remaining matrix fail to be full of row rank. To determine r i , a pure combinatorial method needs to compute the ranks of at most m 1 + m 2 + · · · + m m − ki → O(2 m ) submatrices, which increases exponentially with m even when k i is bounded. Hence, the direct combinatorial method is not computationally efficient. In what follows, a pseudo-polynomial time algorithm based on traversals over a recursive tree is presented. The pseudo code of this algorithm is given in Algorithm 1. The intuition behind Algorithm 1 is that, for the κth eigenvalue of A, instead of directly determining r κ , we try to determine m − r κ , which is the maximum number of columns of Y (κ) that fail to have full row rank. Algorithm 1 first builds a recursive tree and then searches the maximum return value among the leafs of this recursive tree, i.e., T [κ] max , which equals exactly m − r κ . An illustrative example of this recursive tree is given in Fig. 1. To build the recursive tree, in the τ th iteration for the κth eigenvalue, all the matrices Y , ..., T (τ +1) c (τ ) i−1 +\|Ω (τ ) i \| }, such that each matrix Y (κ) T (τ +1) i has rank τ + 1, i = c (τ ) i−1 + 1, ..., c (τ ) i−1 + \|Ω (τ ) i \|. Theorem 2: Given a system (A, B) with geometric multiplicities of eigenvalues of A bounded byk, A ∈ R n×n , B ∈ R n×m , Algorithm 1 can determine the optimal solution to Problem 1 in time complexity at most O(k 2 mk +1 n + mn 3 ). Proof: As argued above, the main procedure of Algorithm 1 is to determine the maximum number of columns of Y (κ) that fail to have full row rank for each eigenvalue λ κ , κ = 1, ..., p. To justify the recursive procedure, observe that for any subset S ⊆ V , Initialize τ =0, T (0) 1 = φ, c (−1) 0 = 1. 4: while τ < k κ do 5: 7: for i = 1 to c (τ −1) end do 6:T (τ ) i = T (τ ) i ∪ j : span(Y (κ) j ) ∈ span(Y (κ) T (τ ) i ), j ∈ V \T (τ ) i .Ω (τ ) i = {j : span(Y (κ) j ) / ∈ span(T (τ ) i ), j ∈ V \T (τ ) i }. Let c (τ ) i = i l=1 \|Ω (τ ) l \|, c (τ ) 0 =0. 8: Rewrite if S ′ ⊆ V \S and span(Y Ω (τ ) i . = {j 1 , ..., j \|Ω (τ ) i \| } and let T (τ +1) c (τ ) i−1 +q =T (τ ) i {j q } for q = 1, ..., \|Ω (τ ) i \|.T [κ] max = arg max {T (kκ −1) i } \|T (kκ−1) 1 \|, · · · , \|T (kκ−1) c (kκ−2) end \| , F κ = V \T(κ) S ′ ) ⊆ span(Y (κ) S ), then rankY (κ) S S ′ = Y (κ) S ; if span(Y (κ) j ) / ∈ span(Y (κ) S ), then rankY (κ) S {j} = rankY (κ) S + 1. Hence, for the κth eigenvalue, the recursive tree has depth exactly k κ . Moreover, {T (kκ−1) 1 , · · · ,T (kκ−1) c (kκ−2) end } contain all the submatrices formed by columns of Y (κ) with the property that: it has rank k κ − 1, and adding any rest columns from Y (κ) can make it have full row rank. The optimality of the solution returned from Algorithm 1 then follows immediately. In the recursive tree of the κth eigenvalue, each parent node has at most m child nodes. Thus, the τ th layer has at most m τ nodes, τ = 0, ..., k κ − 1. Hence, the total nodes in that recursive tree is at most kκ−1 τ =0 m τ → m kκ . For each node, the rank update procedure incurs O(k 2 κ m) using the singular value decomposition (Line 6 to Line 7 of Algorithm 1). To obtain Y (κ) \| p κ=1 , it incurs computational complexity O(mn 3 ). Note that k κ ≤k for κ = 1, ..., p, and p ≤ n. To sum up, the total computational complexity of Algorithm 1 is at most O(k 2 mk +1 n + mn 3 ). Remark 1: In computational complexity theory, an algorithm runs in pseudo-polynomial time if its running time is a polynomial in the numeric value of the input (the largest integer present in the input). A NP-hard problem is said to be weakly NP-hard, if there is a pseudo-polynomial time algorithm for it; otherwise it is strongly NP-hard. The above results indicate that, provided that the left eigenbases of A are computationally available, Problem 1 is weakly NP-hard in a broader sense (with respect to the maximum geometric multiplicities of A). In other words, the computational intractability of Problem 1 is essentially caused by the geometric multiplicities of eigenvalues of A, rather than the dimensions of states and inputs of the systems. By contrast, from [5], it is known that the minimal controllability problems discussed therein are strongly NP-hard. 1 This distinction is somehow surprising to the authors. C. Minimal Cost Actuator Failures In actual systems, different actuators may incur different costs to be removed. Here cost can be used to measure the budget of an actuator to be removed, or the difficulty/fragility of an actuator to be attacked. In such case, an attacker/protector may be more interested in the minimal cost of a set of actuators whose removal causes uncontrollability. Let c(i) ≥ 0 denote the cost of the ith actuator, i ∈ V . Given S ⊆ V , let c(S) = i∈S c(i). Then, Problem 1 can be generalized to the following Problem 3 Problem 3: Given (A, B) in (2) min S⊆V c(S) (A, B V \S ) is uncontrollable The following theorem reveals that Problem 3 is pseudo-polynomial time solvable with bounded maximum geometric multiplicities of A. Theorem 3: Given a system (A, B) with geometric multiplicities of eigenvalues of A bounded byk, A ∈ R n×n , B ∈ R n×m , Algorithm 2 can determine the optimal solution to Problem 3 with complexity at most O(k 2 mk +1 n + mn 3 ). Run the same procedure from Line 3 to Line 11 in Algorithm 1. Proof: Note that the distinction between Algorithm 2 and Algorithm 1 lies in Line 4 of Algorithm 2. To be specific, in Algorithm 2, the cardinality of a set of actuators is replaced by its total cost. With this observation, the proof of Theorem 3 follows similar arguments to that of Theorem 2. The details are omitted. D. Structure of Problem 2 Given (A, B) in (2), define a function f (S) : 2 V → N as f (S) = rankC(A, B S ). It is known that f (S) is submodular on S ⊆ V . From the property of submodularity, f (V \S) is also submodular on S⊆ V . This means that Problem 2 can be formalized as a cardinality-constrained submodular function minimization problem. Although a submodular function minimization problem is strongly polynomial time solvable [6], any cardinality-constraint on it is NP-hard [4]. Currently, there is no known algorithm that can return a constant factor approximation for a general cardinality-constrained submodular function minimization problem [4]. Nevertheless, exploiting some special inherent structure of Problem 2 and finding efficient approximation algorithms remain our further work. V. SOME DISCUSSIONS The pseudo-polynomial time Algorithm 1 is based on the eigenspace decomposition of the system state transition matrix A. However, even when the exact value of A is known, its (left) eigenbases might cannot be precisely obtained in its numeric value within a polynomial bounded number of operations. A natural question arises that, can we find r min by simply querying rankC(A, B V \S )? In other words, can we solve Problem 1 by regarding the system (A, B) as a block-box system and querying rankC(A, B V \S ) within polynomial bounded times for S ⊆ V ? The results of [9] show that many random networks can be controllable by only one actuator. A more interesting extension of Problem 2 is to use some frequently-used quantitative controllability metrics [7] instead of the dimension of controllable subspaces, such as trace of the inverse of controllability Gramian, the minimal eigenvalue of controllability Gramian, etc. VI. CONCLUSIONS An inverse problem of the minimal controllability problems is addressed in this paper, that is, determining the minimal number of inputs whose removal destroy controllability of an LTI system. It is shown that this problem is weakly NP-hard with bounded maximum geometric multiplicities of the given state transition matrices for the first time. Some variants, as well as some further directions, are also discussed. Although most techniques/algorithms in this paper are designed for controllability, they may be applied to some other fields, like the minimal cost critical columns in the linear degeneracy problem [1]. Denote the controllability matrix of a pair (A, B) by C(A, B), i.e., C(A, B) = [B, AB, ..., A n−1 B]. Then, this problem can be formulated as Problem 2: Given (A, B) in (2) and l ∈ N min S⊆V,\|S\|≤l rankC(A, B V \S ) denotes the last element of the sequence {c by adding the maximal columns of Y (κ) to Y while its rank is preserved. Next, each setT child nodes in the recursive tree, namely {T ⊆ {1, ..., m τ } to denote a sequence of sorted integers with updated cardinality depending on τ , where c (τ ) end denotes the last element of this sequence. end for 14: Return F min = arg min {Fi} {\|F 1 \|, ..., \|F p \|} and r min = \|F min \|. Fig. 1 . 1An illustrative example of a recursive tree in Algorithm 1. Note that this recursive tree, depending on the considered system parameters, is not necessarily a binary tree. Return F min = arg min {Fi} {\|c(F 1 )\|, ..., \|c(F p )\|} and r min = c(F min ). Algorithm 2 : A pseudo-polynomial time algorithm for Problem 3 Input: System parameters (A, B) Output: The optimal solution to Problem 1 1: Calculate the left eigenbases {X i \| p i=1 } of A. 2: for κ = 1 to p do3: It is proven in[5] that the minimal controllability problems considered therein cannot be approximated within a constant multiplicative factor even when the involved state transition matrix has no repeated eigenvalues. On the complexity of approximating extremal determinants in matrices. L Khachiyan, Journal of Complexity. 111L. Khachiyan, On the complexity of approximating extremal determinants in matrices, Journal of Complexity, vol. 11, no. 1, pp. 138-153, 1995. T Tao, V Vu, arXiv:1412.1438Random matrices have simple spectrum. preprintT. Tao and V. Vu, Random matrices have simple spectrum, preprint, arXiv:1412.1438, 2014. Minimal inputs/outputs for a networked System. T Zhou, IEEE Control Systems Letters. 12T. Zhou, Minimal inputs/outputs for a networked System, IEEE Control Systems Letters, vol. 1, no. 2, pp. 298-303, 2017. Submodular approximation: Sampling-based algorithms and lower bounds. Z Svitkina, L Fleischer, SIAM Journal on Computing. 406Z. Svitkina and L. Fleischer, Submodular approximation: Sampling-based algorithms and lower bounds, SIAM Journal on Computing, vol. 40, no. 6, pp. 1715-1737, 2011. Minimal controllability problems. A Olshevsky, IEEE Transactions on Control of Network Systems. 13A. Olshevsky, Minimal controllability problems, IEEE Transactions on Control of Network Systems, vol. 1, no. 3, pp. 249-258, 2014. A combinatorial strongly polynomial algorithm for minimizing submodular functions. S Iwata, L Fleischer, S Fujishige, Journal of the ACM (JACM). 484S. Iwata, L. Fleischer, and S. Fujishige, A combinatorial strongly polynomial algorithm for minimizing submodular functions, Journal of the ACM (JACM), vol. 48, no. 4, pp. 761-777, 2001. On submodularity and controllability in complex dynamical networks. T H Summers, F L Cortesi, J Lygeros, IEEE Transactions on Control of Network Systems. 31T. H. Summers, F. L. Cortesi and J. Lygeros, On submodularity and controllability in complex dynamical networks, IEEE Transactions on Control of Network Systems, vol. 3, no. 1, pp. 91-101, 2016. A review of methods for input/output selection. V D , B D Jager, Automatica. 37V. D. Wal and B. D. Jager, A review of methods for input/output selection, Automatica, vol. 37, pp. 487-410, 2001. On a conjecture of Godsil concerning controllable random graphs. S O'rourke, B Touri, SIAM Journal on Control and Optimization. 546S. O'Rourke and B. Touri, On a conjecture of Godsil concerning controllable random graphs, SIAM Journal on Control and Optimization, vol. 54, no. 6, pp. 3347-3378, 2016.	[]
[ "Coefficient of performance at maximum figure of merit and its bounds for low-dissipation Carnot-like refrigerators", "Coefficient of performance at maximum figure of merit and its bounds for low-dissipation Carnot-like refrigerators" ]	[ "Yang Wang ", "Mingxing Li ", "Z C Tu ", "A Calvo Hernández ", "J M M Roco ", "\nDepartment of Physics\nDepartamento de Física Aplicada, and Instituto Universitario de Física Fundamental y Matemáticas (IUFFyM)\nBeijing Normal University\n100875BeijingChina\n", "\nUniversidad de Salamanca\nE-37008SalamancaSpain\n" ]	[ "Department of Physics\nDepartamento de Física Aplicada, and Instituto Universitario de Física Fundamental y Matemáticas (IUFFyM)\nBeijing Normal University\n100875BeijingChina", "Universidad de Salamanca\nE-37008SalamancaSpain" ]	[]	The figure of merit for refrigerators performing finite-time Carnot-like cycles between two reservoirs at temperature T h and Tc (< T h ) is optimized. It is found that the coefficient of performance at maximum figure of merit is bounded between 0 and ( √ 9 + 8εc − 3)/2 for the low-dissipation refrigerators, where εc = Tc/(T h − Tc) is the Carnot coefficient of performance for reversible refrigerators. These bounds can be reached for extremely asymmetric low-dissipation cases when the ratio between the dissipation constants of the processes in contact with the cold and hot reservoirs approaches to zero or infinity, respectively. The observed coefficients of performance for real refrigerators are located in the region between the lower and upper bounds, which is in good agreement with our theoretical estimation.	10.1103/physreve.86.011127	[ "https://arxiv.org/pdf/1205.2258v2.pdf" ]	23,530,178	1205.2258	2664fb6fca9f1dcfeaf393b228451bd52d933cd9	Coefficient of performance at maximum figure of merit and its bounds for low-dissipation Carnot-like refrigerators 5 Jun 2012 (Dated: May 1, 2014) Yang Wang Mingxing Li Z C Tu A Calvo Hernández J M M Roco Department of Physics Departamento de Física Aplicada, and Instituto Universitario de Física Fundamental y Matemáticas (IUFFyM) Beijing Normal University 100875BeijingChina Universidad de Salamanca E-37008SalamancaSpain Coefficient of performance at maximum figure of merit and its bounds for low-dissipation Carnot-like refrigerators 5 Jun 2012 (Dated: May 1, 2014) The figure of merit for refrigerators performing finite-time Carnot-like cycles between two reservoirs at temperature T h and Tc (< T h ) is optimized. It is found that the coefficient of performance at maximum figure of merit is bounded between 0 and ( √ 9 + 8εc − 3)/2 for the low-dissipation refrigerators, where εc = Tc/(T h − Tc) is the Carnot coefficient of performance for reversible refrigerators. These bounds can be reached for extremely asymmetric low-dissipation cases when the ratio between the dissipation constants of the processes in contact with the cold and hot reservoirs approaches to zero or infinity, respectively. The observed coefficients of performance for real refrigerators are located in the region between the lower and upper bounds, which is in good agreement with our theoretical estimation. Introduction.-The issue of efficiency at maximum power output has attracted much attention since the seminal achievements made by Yvon [1], Novikov [2], Chambadal [3], Curzon and Ahlborn [4], which gives rise to finite-time thermodynamics, a new branch of nonequilibrium thermodynamics, and opens open new avenues to the perspective of establishing more realistic theoretical bounds for real heat engines as well as refrigerators [5][6][7][8]. Previous reported works on this subject show that different model systems exhibit various kinds of behaviors at large relative temperature difference between two thermal reservoirs at temperatures T h and T c (< T h ), in spite that they show certain universal behavior at small relative temperature difference [9][10][11][12][13][14][15][16][17] leading to recent discussions on the bounds of efficiency at maximum power output for Carnot-like heat engines [18][19][20][21][22][23]. In particular, Esposito et al. investigated low-dissipation Carnotlike engines by assuming that the irreversible entropy production in each isothermal process is inversely proportional to the time required for completing that process [19]. Furthermore, they obtained that the efficiency at maximum power output for low-dissipation engines is bounded between η − ≡ η C /2 and η + ≡ η C /(2 − η C ) [19], where η C = 1 − T c /T h is the Carnot efficiency of reversible heat engines. Besides, Ma [24] proposed the perunit-time efficiency to be another criterion, which can be viewed as a compromise between the efficiency and the speed of the whole cycle. Two of present authors and their coworkers [25] proved that the efficiency of endoreversible heat engines performing at maximum per-unittime efficiency is bounded between η C /2 and 1− √ 1 − η C . However, it is relatively difficult to define an optimal * Corresponding author. Email: [email protected] criterion and obtain its corresponding coefficient of performance (COP) for refrigerators [26][27][28][29][30][31][32][33][34][35] in the way as we address the issue of efficiency at maximum power for heat engines provided that minimum power input is not an appropriate figure of merit in Carnot-like refrigerators. Velasco et al. [26] adopted the per-unit-time COP as an target function and proved ε CA ≡ √ ε c + 1 − 1 to be the upper bound of COP for endoreversible refrigerators operating at the maximum per-unit-time COP, being ε c = T c /(T h − T c ) the Carnot COP for reversible refrigerators. Allahverdyan et al. [27] investigated a quantum model which consists of two n-level systems interacting via a pulsed external field and took εQ c as the target function, where ε and Q c are the COP of refrigerators and the heat absorbed from the cold reservoir, respectively. They also proved that the COP of this model is bounded between ε CA and ε c at the small relative temperature difference. Chen and Yan [28] suggested to take χ = εQ c /t cycle as the target function, where t cycle is the time for completing the whole Carnot-like cycle. Recently, de Tomás and two of present authors [29] optimized χ for symmetric low-dissipation refrigerators and derived the COP at maximum χ to be ε CA = √ 1 + ε c −1. The above results give rise to two straightforward questions: (i) What target function could be appropriate as the figure of merit for refrigerators? (ii) Can we derive the bounds of COP at maximum figure of merit for general low-dissipation refrigerators as a counterpart to the bounds of efficiency at maximum power output for heat engines? We will address these problems in this work. We select χ = εQ c /t cycle as the figure of merit and derive that the COP at maximum figure of merit is bounded between 0 and ( √ 9 + 8ε c −3)/2 for low-dissipation refrigerators. Our theoretical prediction is in good agreement with the observed data from real refrigerators, which suggests χ = εQ c /t cycle is appropriate as the figure of merit for refrigerators. Model.-The refrigerator that we consider performs a Carnot-like cycle consisting of two isothermal processes and two adiabatic steps as follows. It must be noted that the word "isothermal" in this work also merely indicates that the working fluid is in contact with a reservoir at constant temperature. Here we do not introduce the effective temperature of working fluid because the effective temperature might not be well-defined in many cases [23]. (S1) Isothermal expansion. The working substance is in contact with a cold reservoir at temperature T c and the constraint on the system is loosened according to the external controlled parameter λ c (τ ) during the time interval 0 < τ < t c , where τ is a time variable. It is in the sense of loosening the constraint that this step is called an expansion process. A certain amount of heat Q c is absorbed from the cold reservoir. Then the variation of entropy in this process can be expressed as ∆S c = Q c /T c + ∆S ir c ,(1) where ∆S ir c ≥ 0 is the irreversible entropy production. We adopt the convention that the heat absorbed by the refrigerator is positive, so ∆S ir c ≤ ∆S c . (S2) Adiabatic compression. This step is idealized as the working substance suddenly decouples from the cold reservoir and then comes into contact with the hot reservoir instantaneously at time t c . During this transition, the controlled parameter is switched from λ c (t c ) to λ h (t h ) [> λ c (t c )] , that is, the constraint on the system is enhanced. It is in the sense of enhancing the constraint that this step is called a compression process. There is no heat exchange in this transition, i.e. Q 2 = 0. The distribution function of molecules of working substance is unchanged. Thus there is no entropy production in this transition, i.e. ∆S 2 = 0. (S3) Isothermal compression. The working substance is in contact with a hot reservoir at temperature T h and the constraint on the system is further enhanced according to the external controlled parameter λ h (τ ) during the time interval t c < τ < t c + t h . A certain amount of heat Q h is released to the hot reservoir T h . Thus the total variation of entropy in this process is ∆S h = −Q h /T h + ∆S ir h ,(2) where ∆S ir h ≥ 0 is the irreversible entropy production. (S4) Adiabatic expansion. Similar to the adiabatic compression process, the working substance suddenly decouples from the hot reservoir and then comes into contact with the cold reservoir instantaneously at time t c + t h . During this transition, the controlled parame- ter is switched from λ h (t c + t h ) to λ c (0) [< λ h (t c + t h )], that is, the constraint on the system is loosened. In this transition, both the heat exchange and the entropy production are vanishing, i.e. Q 4 = 0 and ∆S 4 = 0. Optimizing the figure of merit.-Having undergone a whole cycle, the system recovers its initial state. Thus the change of entropy is vanishing for the whole cycle, from which we can easily derive that the variations of entropy in two "isothermal" processes satisfy ∆S c = −∆S h ≡ ∆S > 0. Similarly, the total energy also remains unchanged for the whole cycle, thus the work input in the cycle can be expressed as W = Q h − Q c , and then the COP of refrigerators is reduced to ε = Q c /(Q h − Q c ).(3) Considering Eqs. (1)-(3) and t cycle = t c + t h , the figure of merit χ ≡ εQ c /t cycle is transformed into χ = T 2 c (∆S − ∆S ir c ) 2 [(T h − T c )∆S + T c ∆S ir c + T h ∆S ir h ](t h + t c ) . The variation of entropy ∆S is a state variable only depending on the initial and final states of the isothermal processes while ∆S ir c and ∆S ir h are process variables relying on the detailed protocols λ(τ ). In addition, ∆S ir c < ∆S according to Eq. (1). Thus Eq. (4) implies that the maximum of the figure of merit corresponds to minimizing irreversible entropy production ∆S ir c and ∆S ir h with respect to the protocols for given time intervals t c and t h , which is equivalent to that obtained for Carnot-like heat engines working at maximum power output. To continue our analysis, we denote the minimum irreversible entropy production with the optimized protocols as min{∆S ir c } ≡ L c (t c ) and min{∆S ir h } ≡ L h (t h ). Intuitively, L c (t c ) and L h (t h ) are the monotonous decreasing functions of t c and t h , respectively, because the larger time for completing the isothermal steps, the closer these steps are to quasistatic processes so that the irreversible entropy production ∆S ir c and ∆S ir h become much smaller. In particular, ∆S ir c and ∆S ir h should vanish in the long-time limit t c → ∞ and t h → ∞. For convenience, we can make a variable transformation x c = 1/t c and x h = 1/t h . If we consider Eqs. (1) and (2), the heat Q c and Q h can be expressed as Q c = T c [∆S − L c (x c )],(5) and Q h = T h [∆S + L h (x h )].(6) Substituting Eqs. (5) and (6) into (3), we derive the COP of refrigerators to be ε = Q c Q h − Q c = T c (∆S − L c ) (T h − T c )∆S + T c L c + T h L h .(7) Considering t cycle = t c + t h = 1/x c + 1/x h and the above equations (5)-(7), we optimize the figure of merit χ = εQ c /t cycle with respect to x h and x c and derive the following two equations: (Q h − Q c )x h = (2Q h /Q c − 1)T c L ′ c x c (x h + x c ), (8) (Q h − Q c )x c = T h L ′ h x h (x h + x c ),(9) where L ′ c ≡ dL c /dx c and L ′ h ≡ dL h /dx h . Considering Eqs. (5)-(7) and then dividing Eq. (8) by Eq. (9), we can derive that the COP at maximum figure of merit satisfies ε * T h L ′ h x 2 h = (ε * + 2)T c L ′ c x 2 c .(10) Similarly, adding Eq. (8) and Eq. (9), we can derive 1 ε * = 1 ε c + 1 N ε * + (2ε c − ε * )M/(1 + ε c )(11) with reducing parameters Bounds of COP at maximum figure of merit.-Now we turn to the low-dissipation refrigerators by assuming that L ′ c = Σ c and L ′ h = Σ h are two dissipation constants as Esposito et al. [19] proposed for low-dissipation heat engines. In this case, N = 1 and M = Σ c x c /(Σ c x c +Σ h x h ). Particularly, Σ c = Σ h = Σ for the symmetric lowdissipation cases investigated by de Tomás et al. [29], it is not hard for us to recover its COP at maximum maximum figure of merit to be ε CA = √ 1 + ε c − 1 from Eqs. (10) and (11). However, for the asymmetric lowdissipation cases where Σ c = Σ h , it is more difficult to obtain a concise analytic expression of ε * than the symmetric case. But we can still estimate its bounds from Eq. (11). According to this equation, we have N = (L ′ c x c + L ′ h x h )/(L c + L h ), M = L ′ c x c /(L c + L h ) and ε c = T c /(T h − T c ).ε * = ε c [ 1 + 8(1 + ε c )/M − 3] 2[(1 + ε c )/M − 1] ,(12) which is the key equation in the present work. Because ε c > 0 and 0 ≤ M ≤ 1, it is easy to prove that ε * is a monotonous increasing function of M . As a main result, from Eq. (12) we obtain the wished bounds as: 0 ≤ ε * ≤ ( √ 9 + 8ε c − 3)/2.(13) It is noted that M is also constrained by Eq. (10), which pushes us to further discuss the accessibility of the lower bound ε − ≡ 0 and the upper bound ε + ≡ ( √ 9 + 8ε c − 3)/2. Eliminating x c /x h from Eqs. (10) and (11), we have 2ε c y 2 − 3y − 1 = α(1 + 2y) 1/2 ,(14) where y = 1/ε * and α = T h Σ h /T c Σ c . In Fig. 1 . Noting that y = 1/ε * , we arrive at 0 ≤ ε * ≤ 1/y 0 = ( √ 9 + 8ε c −3)/2 which is exactly the same as inequality (13). Simultaneously, we obtain the condition for reaching the lower and upper bounds: ε * → 0 when Σ c /Σ h → 0 and ε * → ( √ 9 + 8ε c − 3)/2 when Σ c /Σ h → ∞. That is, the lower and upper bounds of COP at maximum figure of merit can be reached for extremely asymmetric low-dissipation refrigerators. Although the lower and upper bounds of efficiency at maximum power output can also be reached for extremely asymmetric low-dissipation heat engines [19], the subtle difference is that the lower bound can be reached when Σ c /Σ h → ∞ while the upper one can be reached when Σ c /Σ h → 0, which is in the inverse situation with respect to the refrigerators. However, this difference is quite reasonable because refrigerators need the input work to pump heat from the cold reservoir while heat engines utilize heat from the hot source to generate work. The numerical solutions to Eq. (14) can also be calculated by setting different values of ratio Σ c /Σ h . The corresponding values of ε * = 1/y are shown in Fig. 2, from which we find that the COP at maximum figure of merit indeed reaches the upper bound ε + = ( √ 9 + 8ε c − 3)/2 when the ratio Σ c /Σ h is relatively large while it approaches the lower bound ε − ≡ 0 when the ratio Σ c /Σ h is small enough. In addition, the curve with parameter Σ c /Σ h = 1 corresponds to ε CA = √ 1 + ε c − 1, which is also located in the region bounded between ε − and ε + . Now we compare our prediction with the observed COPs of some real refrigerators. The circles and squares in Fig. 3 respectively show the relationship between the observed COPs of two different kinds of real refrigerators working in different temperature regions and the corresponding Carnot COPs calculated according to the working temperatures. The raw data are adapted from Tables 6.1 and 10.1 in Ref. [36]. We stress from this figure that all data are located between the optimized COPs at Σ c /Σ h → ∞ (the solid line) and Σ c /Σ h = 0.01 (the dotted line), which reveals the capability of the lowdissipation assumption and the bounds of the optimized COP in order to reasonably estimate the experimental results for real refrigerators. Additionally, we also plot ε CA = √ 1 + ε c − 1 as the dashed line in Fig. 3, from which we see that ε CA is neither the upper bound nor lower bound of observed COPs. This result suggests that χ = εQ c /t cycle is indeed a very valuable figure of merit in comparing with experimental refrigerators data. Conclusion.-The issue of COP at maximum figure of merit for Carnot-like refrigerators is addressed. We obtain the universal lower and upper bounds of COP at maximum figure of merit for low-dissipation Carnot-like refrigerators. These bounds can be reached for extremely asymmetric dissipation cases. We compare our prediction with the observed COPs of real refrigerators and find that all measured COPs are located in between the prediction model. From a theoretical point of view, these results for low-dissipation refrigerators can be regarded as a counterpart of the bounds of efficiency at maximum power output obtained by Esposito et al. [19] for lowdissipation heat engines. In the future work, we will extend our discussions to the refrigerators working out of the low-dissipation regime based on the key equation (12) and our previous investigation on heat engines [37]. FIG. 1 . 1(Color online) Schematic diagrams of function. The dashed line is the diagram of f (y) = 2εcy 2 − 3y − 1 while the solid lines correspond to diagram of g(y) = α(1 + 2y) 1/2 with different values of α. The points of intersection represent the solutions to Eq. (14) for different values of α. y0 is the solution to f (y) = 2εcy 2 − 3y − 1 = 0. , we schematically plot the function f (y) = 2ε c y 2 − 3y − 1 (dashed line) and g(y) = α(1 + 2y) 1/2 for different values of α (solid lines). The points of intersection between the dashed line and solid lines correspond to the solutions to Eq. (14) for different values of α. It follows that the solutions to Eq. (14) increase with the increasing value of α. On the other hand, the values of α can be taken from 0 to ∞. Therefore we infer that the solutions to Eq. (14) are between y 0 = ( √ 9 + 8ε c + 3)/4ε c [solution to Eq. (14) for α = 0, i.e. Σ c /Σ h → ∞] and ∞ [solution to Eq. (14) for α → ∞, i.e. Σ c /Σ h → 0] FIG. 2 . 2(Color online) Numerical solutions to Eq.(14). The used values of parameter Σc/Σ h are marked nearby each curves. FIG. 3 . 3(Color online) Comparison between theoretical prediction and the observed COPs of refrigerators. 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[ "arXiv:1207.5544v1 [quant-ph] Security of distributed-phase-reference quantum key distribution", "arXiv:1207.5544v1 [quant-ph] Security of distributed-phase-reference quantum key distribution" ]	[ "Tobias Moroder \nNaturwissenschaftlich-Technische Fakultät\nUniversität Siegen\nWalter-Flex-Str. 3D-57068SiegenGermany\n", "Marcos Curty \nDept. of Signal Theory and Communications\nEI Telecomunicación\nUniversity of Vigo\nE-36310VigoSpain\n", "Charles Ci ", "Wen Lim \nGroup of Applied Physics\nUniversity of Geneva\nCH-1211GenevaSwitzerland\n", "Le Phuc Thinh \nCentre for Quantum Technologies\nNational University of Singapore\n3 Science Drive 2117543Singapore, Singapore\n", "Hugo Zbinden \nGroup of Applied Physics\nUniversity of Geneva\nCH-1211GenevaSwitzerland\n", "Nicolas Gisin \nGroup of Applied Physics\nUniversity of Geneva\nCH-1211GenevaSwitzerland\n" ]	[ "Naturwissenschaftlich-Technische Fakultät\nUniversität Siegen\nWalter-Flex-Str. 3D-57068SiegenGermany", "Dept. of Signal Theory and Communications\nEI Telecomunicación\nUniversity of Vigo\nE-36310VigoSpain", "Group of Applied Physics\nUniversity of Geneva\nCH-1211GenevaSwitzerland", "Centre for Quantum Technologies\nNational University of Singapore\n3 Science Drive 2117543Singapore, Singapore", "Group of Applied Physics\nUniversity of Geneva\nCH-1211GenevaSwitzerland", "Group of Applied Physics\nUniversity of Geneva\nCH-1211GenevaSwitzerland" ]	[]	Distributed-phase-reference quantum key distribution stands out for its easy implementation with present day technology. Since many years, a full security proof of these schemes in a realistic setting has been elusive. For the first time, we solve this long standing problem and present a generic method to prove the security of such protocols against general attacks. To illustrate our result we provide lower bounds on the key generation rate of a variant of the coherent-one-way quantum key distribution protocol. In contrast to standard predictions, it appears to scale quadratically with the system transmittance.	10.1103/physrevlett.109.260501	[ "https://arxiv.org/pdf/1207.5544v1.pdf" ]	8,873,107	1207.5544	9c06c587c4f0d855d59dea84fbfcdb2a549a9723	arXiv:1207.5544v1 [quant-ph] Security of distributed-phase-reference quantum key distribution 23 Jul 2012 (Dated: May 2, 2014) Tobias Moroder Naturwissenschaftlich-Technische Fakultät Universität Siegen Walter-Flex-Str. 3D-57068SiegenGermany Marcos Curty Dept. of Signal Theory and Communications EI Telecomunicación University of Vigo E-36310VigoSpain Charles Ci Wen Lim Group of Applied Physics University of Geneva CH-1211GenevaSwitzerland Le Phuc Thinh Centre for Quantum Technologies National University of Singapore 3 Science Drive 2117543Singapore, Singapore Hugo Zbinden Group of Applied Physics University of Geneva CH-1211GenevaSwitzerland Nicolas Gisin Group of Applied Physics University of Geneva CH-1211GenevaSwitzerland arXiv:1207.5544v1 [quant-ph] Security of distributed-phase-reference quantum key distribution 23 Jul 2012 (Dated: May 2, 2014)numbers: 0367Dd0367Hk0367Mn Distributed-phase-reference quantum key distribution stands out for its easy implementation with present day technology. Since many years, a full security proof of these schemes in a realistic setting has been elusive. For the first time, we solve this long standing problem and present a generic method to prove the security of such protocols against general attacks. To illustrate our result we provide lower bounds on the key generation rate of a variant of the coherent-one-way quantum key distribution protocol. In contrast to standard predictions, it appears to scale quadratically with the system transmittance. Introduction.-Quantum key distribution (QKD) is on the verge to become a standard tool for secure communications [1]. In its original proposal QKD is based on the transmission of single photons. However, since true single photon sources are not available yet most experimental prototypes and all current commercial products of QKD use weak laser pulses. A main drawback of these systems is that some signals contain more than one photon prepared in the same quantum state. This fact severely limits the distances that can be achieved by these techniques due to the photon number splitting attack [2]. To enhance the performance of practical QKD systems, several approaches have been proposed. One solution is to send a strong reference pulse together with the quantum signals [3]. A second approach is based on the decoy state method, where the transmitter sends states with different intensities [4]. Both schemes provide a secret key generation rate that scales linearly with the system transmittance [3,4]. A third alternative is to use distributed-phase-reference (DPR) QKD protocols. They differ from standard QKD schemes in that the receiver now performs joint measurements onto subsequent signals, often given in the form of coherence measurements [5,6]. This approach includes the differentialphase-shift [5] and the coherent-one-way (COW) [6] protocols. In the former, the sender prepares coherent states of equal intensity but modulates their phases; in the COW protocol all pulses share a common phase but their intensities vary. A complete security proof of DPR-QKD in a realistic setting has been missing since many years. Security has only been proven so far against restricted types of attacks [7][8][9], or assuming the use of ideal single photon sources [10]. In this Letter we present a generic method to prove security of practical DPR-QKD against general attacks. This solves a long standing open question in the field of quantum communications [1]. We illustrate our result by providing non-trivial lower bounds for a variant of the original COW protocol [11], which maintains all the practical advantages. Our analysis suggests that practical DPR-QKD might not be as robust against imperfections as initially foreseen, i.e., its key rate appears to scale quadratically with the system transmittance. Security discussion.-The challenge in DPR-QKD is to prove security against general, also termed coherent attacks. Usually such attacks are known to be of no advantage to the eavesdropper (Eve) in comparison to collective attacks by virtue of the de Finetti theorem [12]. This theorem applies, for instance, when the underlying quantum state shared by the legitimate users (Alice and Bob) is permutationally invariant. In standard QKD this is typically ensured by performing simultaneous random permutations on the classical measurement results. DPR-QKD defines however a fixed ordering of the signals by its coherence measurement and, therefore, it is not possible to permute the classical outcomes without destroying vital information [13]. However, such a predicament can be circumvented by grouping the entire signal stream into blocks. More specifically, consider that Alice and Bob group their signals into subsequent blocks of size m, where the length m is optimized for the expected behaviour. When permuting these blocks one preserves the coherence information within them while the information between the blocks is destroyed. Still this is enough to apply the de Finetti argument on the level of blocks. As a result, the state shared by Alice and Bob after distributing a large number mN of signals satisfies ρ mN AB ≈ ρ m ⊗N AB and security against collective attacks on these signal blocks implies security against coherent attacks in the original setting. Suppose that the state shared by Alice, Bob and Eve after transmitting an m block signal is ρ m ABE . Let us first consider the effect of public announcements by Alice and Bob based on their classical measurement results. This announcement, labelled as v, allows both parties to distinguish between conclusive events that contribute to the sifted key and inconclusive ones that are discarded. On the level of quantum states this is described by suitable maps Λ A v ⊗ Λ B v . Given an announcement v, that happens with probability p(v), the three parties share the state σ m ABE,v determined by Λ A v ⊗ Λ B v (ρ m ABE ) = p(v)σ m ABE,v . For each announcement v one can use a one-way classical post-processing key rate formula [14]. If systemĀ denotes a qubit and Alice's raw key is obtained by projecting this system onto the orthogonal states \|0 Ā , \|1 Ā , then a lower bound on the secret key rate is given by 1 − h 2 (e v ) − h 2 (δ v ). Here h 2 represents the binary entropy, e v is the symmetrized bit error between the key measurements of Alice and Bob, and δ v denotes the corresponding error, typically called phase error, when Alice performs a measurement in a mutually unbiased basis and Bob in his other different setting. This last parameter is used to upper bound Eve's knowledge on the sifted key generated by Alice. Note that δ v does not need to be measured directly, it only needs to be estimated. When Bob's measurements are similar qubit measurements like the ones of Alice then the expression above represents the Shor-Preskill key rate formula [15]. To consider that the output systemĀ is a qubit implies that Alice can, at best, distill one secret bit per block. Nevertheless this restriction should not have a significant impact on the key rate in a long distance regime, since Bob observes, if any, most often only one single conclusive event per m arriving signals due to the high losses in the channel (given that m is not too big). Instead of estimating separate phase errors δ v , it is often easier to combine all conclusive announcements v ∈ V c into an averaged version. Let G = v∈Vc p(v) ≤ 1 denote the total sifted key gain. Then, we have that the secret key rate per block can be bounded by R m ≥ inf ρ m ABE v∈Vc p(v) [1 − h 2 (e v ) − h 2 (δ v )] (1) ≥ inf ρ m ABE G 1 − h 2 (ē c ) − h 2 (δ/G) ≥ G 1 − h 2 (ē c ) − h 2 (δ max /G) .(2) Here one uses concavity of h 2 to lower bound R m by the averaged (conditional) error ratesē c = v∈Vc p(v)e v /G andδ = v∈Vc p(v)δ v . The last step takes into account thatē c and G are observed quantities and that the optimization is attained at the largest phase errorδ max compatible with the obtained data since h 2 increases in [0, 1 2 ]. Phase error estimation.-The main difficulty to compute Eq. (2) is to upper bound the average phase errorδ. This parameter can be expressed as an expectation value on the original bipartite state ρ m AB = tr E (ρ m ABE ) using adjoint maps δ = v∈Vc p(v) tr(σ m AB,v F δv ) = v∈Vc tr[Λ A v ⊗ Λ B v (ρ m AB )F δv ] = tr[ρ m AB v∈Vc Λ A † v ⊗ Λ B † v (F δv )] = tr(ρ m AB Fδ).(3) Here F δv denote the corresponding phase error operators on the state σ m AB,v . Partial knowledge of Alice and Bob about the state ρ m AB can be parsed as known expectation values k i = tr(ρ m AB K i ) for certain operators K i . This means that the search for the maximum phase errorδ max can be cast into the form of a semidefinite program [16], max tr(ρ m AB Fδ)(4)s.t. ρ m AB 0, tr(ρ m AB K i ) = k i ∀i. Such special convex optimization problems can be solved efficiently using standard tools to obtain the exact optimum, even for large dimensions. Available information and its description.-Let us be more precise about which expectation values k i are known in a prepare and measure scheme, where Alice sends potentially mixed states ρ m i with a priori probability p(i). This state preparation can be formulated in an entanglement based version as follows [17]: Alice first creates a source state \|Ψ m A b AsB = i p(i)\|i A b \|ρ m i AsB , where \|ρ m i AsB denote purifications of the signal states ρ m i to a shield system A s [18]. Afterwards, she measures her bit system A b in the standard basis, thereby producing the correct signal state at site B which is sent to Bob. Eve transforms the overall source state to the final tripartite state ρ m ABE with A = A b A s . On the receiving side, Bob performs a measurement modelled by B k . As a result, both Alice and Bob observe the expectation values of \|i A b i\| ⊗ ½ As ⊗ B k . Moreover, since Eve is restricted to interact only with Bob's system, the reduced density matrix ρ m A = tr BE (ρ m ABE ) is fixed and directly given by the source state. This information can be added by including expectation values of T k ⊗ ½ B , where T k denotes a tomographic complete operator set on A. Both sets of observables constitute the previously denoted set K i . The signal states and performed measurements in practical DPR-QKD are described by operators on an infinite dimensional Fock space of several modes. In order to apply the de Finetti argument [12], and to numerically obtain an upper bound on the phase error using Eq. (4), it is necessary to formulate this problem in a manageable, finite dimensional form. Clearly, system A b is finite. For Bob's measurements one can employ the squash model argument [19]. Here the real measurement is notionally decomposed into a two step procedure by first applying a map that transforms any incoming signal to a finite dimensional output state on which a specified target measurement B k is performed afterwards. Since this map can be even given to Eve, its output state only lowers the key generation capabilities of Alice and Bob, and one readily works in finite dimensions. For our simulations we assume that Bob has at his disposal inefficient photon number resolving detectors with state independent dark counts. Also, we consider that only the single photon events within the whole block are finally considered as conclusive. In this case the map outputs either a single signal states: [11] with an active measurement choice. Bob reads the raw key in detector D d . Moreover, he uses an optical switch to send some pairs of consecutive pulses to a monitoring line that examines the coherence between even and odd pulses sent by Alice. i-1 i-2 i-3 i+1 i+3 i i+2 photon, measured with the perfect detection scheme, or an auxiliary state that triggers all inconclusive events. For the shield system A s one uses only partial information of the reduced state. In the case of phase randomized signal blocks, an example that we consider later, a purification is given by storing the total photon number of the block in the shield system \|n As . Using only tomography on the subspace spanned by all n = 1, . . . , n cut , together with an ancilla state \|N As for all other cases, the shield system can effectively be described in finite dimensions. Description of the protocol.-To illustrate our results we analyze the security of a variant of the COW protocol [11]. The basic setup is shown in Fig. 1. Alice uses a laser, followed by an intensity modulator (IM), to prepare a sequence of coherent states \|0 \|α and \|α \|0 . On the receiving side, Bob employs an optical switch to distribute each pair of incoming pulses into the data or the monitoring line [20]. The data line measures the arrival time of the pulses in detector D d and creates the raw key. Whenever Bob sees a "click" in this detector in say time instance i, he decides at random whether to publicly announce a detection event in time instances i and i + 2 or i and i − 2. The first case is associated with a bit value "0", while the second one corresponds to a bit value "1". If the state sent by Alice in these time instances is \|0 \|α (\|α \|0 ) then she assigns to it a bit value "0" ("1") and tells Bob to keep his result. Otherwise, the result is discarded and does not contribute to the sifted key. Let us illustrate this procedure with a simple example drawn in Fig. 1, and assume that Bob observes a click in D d in time i. If he announces the pair i and i + 2, then this result is discarded. Note that in this case Alice sent \|α \|α and hence she cannot infer in which time slot Bob saw a "click". On the contrary, if Bob reports i and i − 2, then both parties assign to it a bit value "1". The monitoring line checks for eavesdropping by measuring the coherence between subsequent even and odd pulses sent by Alice. This is done by interfering adjacent pairs of pulses in a 50 : 50 beamsplitter and measuring the output states in detectors D + and D − . In the security analysis we assume that Alice and Bob discard coherence information between consecutive signal blocks. Moreover we consider that the sifted key is created only from signals within the same block. To guarantee this, one could discard those detection events where Bob declares time instances that belong to different blocks. Alternatively, one could change Bob's public announcement slightly. For example, one can reorder the 2m possible detection time slots of a given block to form a closed chain with the first and last time instances connected. Now, if Bob observes a "click" in the data line in say the first time slot he announces a detection event in time instances one and three or one and 2m−1 with equal probability, and similar for the other cases. This strategy preserves the original symmetry in Bob's announcement and we use it in our simulations. Simulation.-For simulation purposes, we consider that Bob's detectors are identical and have a dark count rate of 10 −7 . The channel model includes an intrinsic error rate of 1% in the data line together with an additional misalignment in the monitoring line that reduces the interferometric visibility to 99%. More details on this channel model and on the adapted security discussion to the COW protocol are given in the appendix. We study two different scenarios: (a) the case where all different m-signals blocks share the same phase, and (b) the scenario where each block is phase randomized. The resulting lower bounds on the secret key rate per pulse, i.e., R m /(2m), are illustrated in Fig. 2. For comparison, this figure includes as well a lower bound on the secret key rate for a coherent-state version of the standard BB84 protocol [21] with and without phase randomization [22,23]. For a given total system loss, i.e., including the losses in the channel and in Bob's detection apparatus, we optimize the lower bound over the respective signal strength α of Alice's source which is of order 0.1. As expected, we find that case (b) performs better than that where all blocks share a common phase, since the signal states are less distinguishable for an eavesdropper without a global phase. We obtain that the tolerable system loss for the COW protocol is, respectively, ≈ 19.5 dB (a), and ≈ 22.6 dB (b). The bit error and visibility at these cutoff points are, respectively, ≈ 3% and ≈ 96% (a), and ≈ 5.3% and ≈ 93.3% (b). Let us remark that the lower bound with m = 2 even holds for threshold detectors [24]. Our simulations reveal that a main limiting factor in DPR-QKD seems to be the dark count rate of Bob's detectors. For given experimental parameters, there is an optimal finite block size that allows a maximum tolerable total system loss. If one increases the block size further this does not translate into an improved lower bound or distance. This is due to the fact that, in the high loss regime, large sized blocks suffer from a higher dark count probability per block than smaller sized blocks, and this reduces the achievable secret key rate. A similar ef- (2) per pulse on a logarithmic scale (base 10) vs. the total system loss in dB for the COW protocol illustrated in Fig. 1 using signal blocks carrying m bits of information (i.e., 2m optical pulses) in the security proof. The upper figure corresponds to the case where all blocks of signals share a common phase, while the lower figure represents the situation where each block is phase randomized. For comparison, we include a lower bound on the secret key rate for a coherent-state version of the standard BB84 protocol [21] with and without phase randomization [22,23]. We consider three main error contributions: an intrinsic error rate of 1% in the data line, an additional misalignment in the monitoring line reducing the visibility to 99%, and a dark count rate in the detectors of 10 −7 . Moreover, in the lower figure we assume ncut = 2. fect was already observed in the security analysis for the differential-phase-shift protocol with true single photon sources [10]. For a dark count rate per pulse of 10 −7 the optimal block size in the COW scheme turns out to be m = 3, i.e., 6 optical pulses. Also, this figure shows that a coherent-state version of the BB84 protocol without decoy states can deliver notably higher key rates per signal than the analyzed COW protocol assuming the same channel model. The reason for this might be threefold: (1) the small optimal block size in the COW scheme; (2) considering blocks, it can be shown that certain multiphoton pulses are completely insecure; (3) most impor-tantly, while in the BB84 the phase error is measured directly, in the COW protocol it has to be estimated. Possible improvements.-To further improve the lower bounds shown in Fig. 2 there are several alternatives. Since a main limitation seems to come from dark counts, one may consider security in the fully calibrated device scenario where these errors are not attributed to Eve. As a quantitative bound on the performance of this scenario we investigated the case of a zero dark count rate, in which all key rate bounds shown in Fig. 2 shift by about 3 dB, though the difference between the COW and the BB84 protocol remains. Additionally, one can evaluate different public announcements in a similar spirit like the SARG protocol [25]. We considered different declarations, but unfortunately none of them enhanced the resulting key rate [26]. Another possibility is to include, for instance, an extra monitoring line on Bob's side to additionally check the coherence between subsequent pulses. The state distribution part of this protocol is then very similar to the one of the original COW scheme [6] with an additional decoy signal composed by two vacuum pulses as proposed in Ref. [9]. This hardware change improves the maximum tolerable system loss by about 1 dB. Another hardware change might be to include additional phase differences in the signal stream, such that the signals states get closer to the one used in a BB84 protocol. Finally, one may ask whether different security techniques might provide better lower bounds. For instance, one could consider more valid detection events per block. This needs however much larger block sizes such that one obtains at all a reasonable fraction of two or more click events in the long distance limit. Another alternative would be to bound the rate by the individual phase errors, i.e., directly using Eq. (1). This could give a benefit if, for example, bits at the boundary are much easier to infer by Eve than bit values originating from events well inside the block. Moreover, it might be of advantage if Eve's information is estimated by using different, possibly not mutually unbiased basis measurements. Here the more general key rate formula of Ref. [14] could be used. Clearly another option would be to abandon the block idea. However even in this case Eve could always attack the signals block-wise. Though a coherence measurement across blocks would then reveal the eavesdropper, any coherence measurement within them would be still fine. Hence when considering only an average visibility this effect will become less and less important. All these alternatives definitely deserve further investigations, but we do not expect a dramatic improvement. Conclusion.-We have presented a generic method to prove security of practical DPR-QKD against general attacks. With the explicit example of a variant of the COW protocol, we have shown that these schemes are indeed secure for certain distances at given rates. Its performance, however, seems to be less robust against practical imperfections than originally expected. We would like to thank H.-K. Lo, N. Lütkenhaus, V. Scarani, L. Sheridan and N. Walenta for stimulating discussions about the topic and technicalities, and L. M. Eriksson for comments on the presentation of the paper. T.M., M.C., and L.P.T. especially thank the Group of Applied Physics, University of Geneva, for hospitality and support during their stay at this institution, where parts of this research have been conducted. This work has been supported by the EU (Marie Curie CIG 293993/ENFOQI), the BMBF (Chist-Era Project QUASAR), the National Research Foundation and the Ministry of Education, Singapore, the National Centre of Competence in Research QSIT, the Swiss Nan-oTera project QCRYPT and the FP7 Marie-Curie IAAP QCERT project. accessible to the eavesdropper. Moreover let us point out that though a single purification \|ρ m i AsB is unique up to local unitary, here one requires that all signals ρ m i are purified to the same shield system As, which is not unique anymore. While certain collective purifications are clearly better than others, any choice is valid. [19] N. J. Beaudry, T. Moroder, and N. Lütkenhaus, Phys. Rev. Lett. 101, 093601 (2008) , it can be shown that, for 2-bit blocks, Bob's measurement apparatus can be described with a squash model and the security analysis applies directly. The resulting secret key rate, however, is almost identical to the case where he uses photon number resolving detectors, only slightly higher in the low loss regime where the probability to obtain a valid detection event is also higher. [25] V. Scarani et al., Phys. Rev. Lett. 92, 057901 (2004). [26] In particular, we have examined two further cases. In the first one, whenever Bob sees a "click" in detector D d in time instance i ∈ {1, 2} (i ∈ {2m, 2m − 1}) he always declares a detection event in time instances i and i + 2 (i and i − 2). Detection events not situated in the border of the blocks are treated as described in the main text. In the second strategy, detection events produced in the border of the blocks are only announced by Bob with probability 1/2. Both methods deliver lower key rates than the one described in the main text. APPENDIX In this appendix we apply the described security method to the explained version of the COW protocol [11]. In particular, we consider signal blocks carrying m bits of information. Since a single bit comprises two modes, one has 2m different temporal modes described by their creation and annihilation operators a † s and a s , respectively, with s = 1, . . . , 2m. We assume that the l-th bit relates to the modes with s = 2l − 1, 2l. Real and assumed measurement description.-At first let us concentrate on the real measurement model M real k and the way how we describe it in the security part, denoted as B k in the main text. For the real measurement setup we assume inefficient photon number resolving detectors that suffer from state-independent dark counts. The inefficiency of M real k is modelled by a global beamsplitter (BS) of transmittance η det located in front of a perfectly efficient scheme, labelled as M k , that still suffers from dark counts. This is schematically drawn in the first line of Fig. 3. In a second step, one models the efficient scheme M k as a map Λ s , sometimes called squashing or filter operation [19], in front of the assumed description B k . Let us emphasize that the security simulation is valid for any true measurement scheme that can be modelled as a physical map Λ followed by the measurement B k as shown in the third line of the figure. There are three different types of outcomes for the so far abstract outcome label "k". For a data line measurement we use d, with d = 1, . . . , 2m, to denote a single photon detection in temporal mode d only. The corresponding measurement operator M d is given by M d = ǫ(1 − ǫ) 2m−1 \|vac vac\| + (1 − ǫ) 2m \|d d\|,(5) with ǫ representing the dark count probability of Bob's detectors and \|d = a † d \|vac . In addition to a data line measurement Bob can also perform coherence measurements on subsequent bits employing the monitoring line. For instance, whenever he tests the coherence between bits l and l+1 he effectively mixes the modes 2l−1, 2l+1 and, at the same time, 2l, 2l + 2. For each pair of modes there are two single photon events, denoted as ±, that can be distinguished, depending on whether the single excitation is registered in the bright (D + ) or in the dark (D − ) detector. As an outcome label for the coherence measurements we use k = (c, ±), where c = 1, . . . , 2m − 2 denotes the first of the two interfering modes. In this case the measurement operators are given by M c,± = ǫ(1 − ǫ) 2m−1 \|vac vac\| + (1 − ǫ) 2m \|χ ± c χ ± c \|,(6) with \|χ ± c = (\|c ± \|c + 2 )/ √ 2. Let us emphasize that in these coherence measurements it is still necessary to check that all other modes are empty. Finally, note that each measurement setting has also other possible outcomes, e.g., "no click" or more than a single photon detection event. All these cases are grouped (via classical post-processing) into a single inconclusive outcome described by M inc . As the modelled measurement operators B k we use B d = \|d d\|, B c,± = \|χ ± c χ ± c \|, B inc = \|a a\|,(7) where \|a is the auxiliary state that describes the inconclusive outcome. These measurement operators B k act on a 2m + 1 dimensional Hilbert space. Both measurement sets can be made equivalent by an appropriate map Λ s such that tr(ρM k ) = tr[Λ s (ρ)B k ] holds for all possible states ρ and measurement outcomes "k" as schematically shown in Fig. 3. This map Λ s is given as follows. First one measures the total number of photons n within an arriving block. Whenever one finds n ≥ 2 one outputs the auxiliary state \|a . If n = 1 then with probability (1 − ǫ) 2m the single photon state stays untouched, otherwise the auxiliary state is thrown again. Finally, for n = 0 the map creates the completely mixed single photon state k \|k k\|/2m with probability 2mǫ(1 − ǫ) 2m−1 and \|a otherwise. This map is physical because we explicitly describe it in terms of measurements and conditional signal state preparations. Source state and reduced density matrix.-The following discussion provides the source states for both cases of pure or phase randomized COW block signals. These states determine the reduced density matrix ρ m A which belongs to the available information. Let us consider first the case of pure signal states. In the COW protocol analyzed Alice sends to Bob either the sequence \|α, 0 or \|0, α , with α ∈ Ê, depending on whether her raw key bit value is "0" or "1". Let us start with the scenario where Alice sends to Bob only one bit value, occurring with equal a priori probability. This corresponds to a block size m = 1. Then the source state is given by \|Ψ m=1 AB = 1 √ 2 \|0 A \|α, 0 B + \|1 A \|0, α B ,(8) and its reduced density matrix becomes ρ m=1 A = 1 2 1 e −α 2 e −α 2 1 .(9) Suppose now that Alice sends to Bob m bits according to this scheme. If i = (i 1 , i 2 , . . . , i m ) denotes the m-bit string being sent and \|φ i B refers to the corresponding signal state, then one obtains \|Ψ m AB = 2 − m 2 i∈{0,1} m \|i A \|φ i B = \|Ψ m A1...AmB = \|Ψ m=1 ⊗m AB .(10) In particular, from the last expression one finds that the reduced density matrix ρ m A is given by ρ m A = (ρ m=1 A ) ⊗m .(11) Next, let us turn to the case of phase randomized blocks. Since randomizing the phase of a block is equivalent to measuring the total number of photons contained in it, the true signals states are of the form ρ m i = ∞ n=0 Π n \|φ i B φ i \|Π n = ∞ n=0 p λ (n)\|ψ i n B ψ i n \|.(12) Here Π n stands for the projector onto the n-photon subspace of the 2m different modes. The outcome of such a photon number measurement follows a Poisson distribution p λ (n) = e −λ λ n /n! with mean λ = mα 2 . The projected n-photon signal states \|ψ i n B can be expressed as \|ψ i n B = m − n 2 √ n! n1,...nm m l=1 (a † 2l+i l −1 ) n l n l ! \|vac B ,(13) where the summation runs over all natural numbers n 1 , . . . , n m that satisfy m l=1 n l = n. These states fulfill the relation ψ i n \|ψ j n = δ nn m − ∆ ij m n ,(14) with ∆ ij being the Hamming distance between the bit strings i and j, i.e., the number of places they differ. Using the framework of mixed signal states as explained in the main text one must now choose an overall purification of all signal states \|ψ i n B . For our simulation we select \|ρ m i AsB = ∞ n=0 p λ (n)\|n As \|ψ i n B ,(15) which can be seen as a coherent storage of the total photon number n in the shield system A s . Let us remark that this choice satisfies ρ m i \|ρ m j = F (ρ m i , ρ m j ), with F being the fidelity of mixed states, which is also the maximal possible overlap between two signal states [27]. We find, therefore, that the source state in this scenario is given by \|Ψ m A b AsB = 2 − m 2 i∈{0,1} m \|i A b \|ρ m i AsB ,(16) with A b = A 1 . . . A m . This means that the reduced density matrix ρ m A , with A = A b A s , can be expressed as ρ m A = ∞ n=0 p λ (n)ρ n A b ⊗ \|n As n\|,(17) with ρ n A b given by ρ n A b = 2 −m i,j m − ∆ ij m n \|i A b j\|.(18) In our simulation we only use partial information of the reduced density matrix ρ m A . In particular, we transform A s toĀ s by making a shield measurement that distinguishes the different photon number cases mentioned in the main text such that one obtains ρ m A bĀs = ncut n=1 p λ (n)ρ n A b ⊗ \|n Ā s n\| + n ∈{1,...,ncut} p λ (n)ρ n A b ⊗ \|N Ā s N \|,(19) where \|N Ā s denotes an auxiliary system for all higher photon numbers. Let us point out that considering the reduced state given by Eq. (19) can be understood as "tagging" the n = 1, . . . , n cut signal states [22]. Announcement maps and phase operator.-The specific announcements v of the COW protocol can be phrased in terms of appropriate maps Λ v on the quantum state. Together with a chosen "phase setting" measurement this provides a concrete expression for the averaged phase error operator Fδ used in Eq. (3). As explained in the protocol description, Bob announces two consecutive even or odd time slots where he registered his single photon event. Suppose, for instance, that he announces v = (2l − 1, 2l + 1). These are the first arrival times of the modes associated with bits i l and i l+1 sent by Alice. In such cases, Alice and Bob agree to call the outcome in the first time instance "0" while the later event is "1". This announcement can be modelled as a filter operation Λ B v (ρ) = F B v ρF B † v given by F B v = 1 √ 2 (\|0 B B 2l − 1\| + \|1 B B 2l + 1\|) .(20) If Bob measures systemB in the standard basis \|0 B , \|1 B he obtains the real outcome he has observed. The prefactor 1/ √ 2 which appears in Eq. (20) takes into account that whenever Bob sees a single photon click in either 2l − 1 or 2l + 1 he announces this particular v with just 50% probability, i.e., F B † v F B v = (B 2l−1 + B 2l+1 )/2. Suppose Bob has actually declared v = (2l − 1, 2l + 1). Then, Alice has to look on her bit string to determine whether she can conclusively infer Bob's bit value. For that, only her bits i l and i l+1 matter. As shown in Fig. 4, if these two bits are equal it means that she had sent to Bob either two full or two empty pulses. In this scenario, she cannot infer Bob's bit value and they discard this result. However, if these bits differ then she knows Bob's sifted bit value precisely (in the error free case) and she tells Bob to keep it. Such a conclusive announcement by Alice can similarly be modelled as a filter operation Λ A v acting on her qubits l and l + 1, i.e., Λ A v (ρ m A1...Am ) = F A v ρ m A l A l+1 F A † v with F A v = \|0 Ā A l A l+1 01\| + \|1 Ā A l A l+1 10\|.(21) Again a measurement in the standard basis \|0 Ā , \|1 Ā provides Alice with her real outcomes. In order to determine the phase error δ v we assume that both parties perform measurements in the X-basis, i.e., they project the output signals from their filter operations onto the states \|± = (\|0 ± \|1 )/ √ 2. Then, the symmetrized phase error δ v = p(+, −) + p(−, +) can be expressed as p(v)δ v = p(v) tr[ 1 2 (½ ⊗ ½ − σ x ⊗ σ x )σ m AB,v ] = 1 2 p(v) − 1 2 tr(σ x ⊗ σ x p(v)σ m AB,v ) = 1 2 p(v) − tr(X ′ A ⊗ X ′ B ρ m AB ),(22) with σ x denoting the Pauli matrix σ x = \|0 1\| + \|1 0\|. In the last line of Eq. (22) we have defined the operators X ′ A = ½ A1...A l−1 ⊗ X A ⊗ ½ A l+2 ...Am , X ′ B = 1 2 F B † v σ x F B v = 1 4 (\|2l − 1 2l + 1\| + \|2l + 1 2l − 1\|) ,(23)with X A = F A † v σ x F A v = \|01 10\| + \|10 01\|. Similar arguments apply to the cases where Bob announces subsequent even outcome pairs or the special instances at the borders of the blocks. We find that the averaged phase errorδ = v∈Vc p(v)δ v can be written as δ = 1 2 v∈Vc p(v) − tr(Xδρ m AB ),(24) with an operator Xδ = m l=1 X A;l ⊗ X B;l . Here X A;l denotes the operator composed by the previously defined X A acting on qubits l and l + 1 and the identity operator acting on the remaining qubits (l = m means the first and last qubit). On Bob's side the operators X B;l are given by X B;l = 1 4 (\|2l − 1 2l + 1\| + \|2l + 1 2l − 1\| +\|2l 2l + 2\| + \|2l + 2 2l\|), with addition being carried out modulo 2m. Channel model.-In this section we present the employed channel model of the COW experiment used in our numerical simulations. Note, however, that the results presented in this article can be applied as well to any other quantum channel, as they only depend on the observed detection probabilities in both the data and monitoring lines. In particular, we characterize the losses in the channel with a BS of transmittance η channel . This parameter can be related with a transmission distance d measured in km for the given QKD scheme as η channel = 10 − αd 10 ,(26) where α represents the loss coefficient of the channel (e.g., an optical fiber) measured in dB/km. Together with the efficiency of the detectors the overall system transmittance is given by η sys = η channel η det .(27) The total system loss in dB is used as the x-axis in the secret key rate figures, i.e., −10 log 10 η sys . The channel misalignment is parametrized with an error probability e d that a signal hits Bob's detectors in the wrong time slot within the same bit. For simplicity, we assume that e d is a constant independent of the distance and we use e d = 1% for simulation purposes. This effect is modelled by a completely positive tracepreserving map Φ that incoherently flips the signal states within the same bit slot as \|0, √ η sys α → \| √ η sys α, 0 and \| √ η sys α, 0 → \|0, √ η sys α with probability e d . Here we consider that the input signals have been already affected by system losses. We have, therefore, that whenever Alice sends to Bob a corresponding COW signal state with coherent state \|α in temporal mode d, the probability that Bob observes a single photon detection event in this mode only (within the whole signal block) is given by p correct d = tr{Λ s [Φ ⊗m (ρ m loss )]M d } = ǫ(1 − ǫ) 2m−1 e −ηsysλ +(1 − ǫ) 2m (1 − e d )η sys µe −ηsysλ ,(28) where ρ m loss represents the output signal of the BS characterizing the total system loss, µ = α 2 , and λ = mµ. Similarly, when Alice sends to Bob a vacuum state in temporal mode d Bob can observe a single photon detection event in this mode only with probability p error d = ǫ(1 − ǫ) 2m−1 e −ηsysλ +(1 − ǫ) 2m e d η sys µe −ηsysλ . [27] M. A. Nielsen, and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press) (2000). FIG. 3 . 3Decomposition of Bob's measuring device. FIG. 4 . 4Announcement choices for Alice given that Bob has declared a detection event in time slots 2l − 1 and 2l + 1. FIG. 2. Lower bound on the secret key rate given by Eq.2 -1 1 5 10 15 20 25 30 Key generation rate Total dB loss Coherent states BB84 m=2 m=3 m=4 m=5 -8 -7 -6 -5 -4 -3 -2 -1 1 5 10 15 20 25 30 Key generation rate Total dB loss Phase randomized states BB84 m=2 m=3 m=4 Gottesman et al., Quant. Inf. Comp. 5, pp. 325-360 (2004). [23] H.-K. Lo, and J. Preskill, Quant. Inf. Comput. 8, 431 (2007) . [24] In the easiest setting Bob may want to use threshold detectors, i.e., simple click/no-click detectors, instead of photon number resolving detectors. Using similar arguments like in Ref. [19]; T. Tsurumaru, and K. Tamaki, Phys. Rev. A 78, 032302 (2008); [20] In our simulation we consider, for simplicity, an active measurement choice by Bob. In practice, this can be re- placed by a passive optical coupler to select between data and monitoring line, together with an unbalanced Mach- Zehnder interferometer for the coherence measurement. [21] C. H. Bennett and G. Brassard, in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India (IEEE Press, New York, 1984), pp. 175-179. [22] D. The total probability that Bob observes an inconclusive detection event in the data line is then given byIn the monitoring line we include an additional misalignment effect that reduces further the interferometric visibility. In particular, we assume that whenever two equal coherent states interfere at a 50 : 50 BS then the outcome signal can exit the BS through the wrong output port with error probability e m . In our simulations we use e m = 0.5%. Here we distinguish two possible scenarios, depending on whether the signals which interfere at the BS were prepared by Alice in the same quantum state or not. Let us assume that the first signal corresponds to bit i l while the later to bit i l+1 . That is, Bob interferes modes 2l − 1, 2l + 1 and, at the same time, 2l, 2l + 2.Let us consider first the situation where both signals were generated in the same state \|0, α . In this scenario, we find that Bob observes a single photon detection event in temporal mode 2l − 1 only (and no click in the remaining modes of the block) with probabilitywhere the superscript ± indicates whether the single excitation is registered in the bright (D + ) or in the dark (D − ) detector of the monitoring line. Similarly, we have that the probability that Bob sees a single photon detection in temporal mode 2l only is given byThe case where both signals were generated in the same state \|α, 0 is completely analogous. 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[ "Wasserstein Image Local Analysis: Histogram of Orientations, Smoothing and Edge Detection", "Wasserstein Image Local Analysis: Histogram of Orientations, Smoothing and Edge Detection" ]	[ "Jiening Zhu [email protected] \nStony Brook University\nNYUnited States\n", "Harini Veeraraghavan \nMemorial Sloan Kettering Cancer Center\nNYUnited States\n", "Larry Norton [email protected] \nMemorial Sloan Kettering Cancer Center\nNYUnited States\n", "Joseph O Deasy [email protected] \nMemorial Sloan Kettering Cancer Center\nNYUnited States\n", "Allen Tannenbaum [email protected] \nStony Brook University\nNYUnited States\n" ]	[ "Stony Brook University\nNYUnited States", "Memorial Sloan Kettering Cancer Center\nNYUnited States", "Memorial Sloan Kettering Cancer Center\nNYUnited States", "Memorial Sloan Kettering Cancer Center\nNYUnited States", "Stony Brook University\nNYUnited States" ]	[]	The Histogram of Oriented Gradient (HOG) is a widely used image feature, which describes local image directionality based on numerical differentiation. Due to its ill-posed nature, small noise in the image may lead to large errors. Thus conventional HOG may fail to produce meaningful directionality results in the presence of noise, which is common in medical radiographic imaging. We approach the directionality problem from a novel perspective by the use of the optimal transport map of a local image patch to a uni-color patch of its mean. Concretely, we decompose the transport map into sub-work costs each transporting in different directions. To test our approach, we evaluated the ability of the optimal transport to quantify tumor heterogeneity from MRI brain images of patients diagnosed with glioblastoma multiforme available from the TCIA. By considering the entropy difference of the extracted local directionality within tumor regions, we found that patients with higher entropy in their images, had statistically significant worse overall survival (p = 0.008), which indicates that tumors that have images exhibiting flows in many directions may be more malignant, perhaps reflecting high tumor histologic grade, a reflection of histologic disorganization. We also explored the possibility of solving classical image processing problems such as smoothing and edge detection via optimal transport. By looking for a 2-color patch with minimum transport distance to a local patch, we derive a nonlinear shock filter, which preserves edges. Moreover, we found that the color difference of the computed 2-color patch indicates whether there is a large change in color, i.e., an edge in the given patch. In summary, we expand the usefulness of optimal transport as an image local analysis tool, to extract robust measures of imaging tumor heterogeneity for outcomes prediction as well as image pre-processing required in several image analysis applications. Because of its robust nature, we find it offers several advantages over the classical approaches.	null	[ "https://arxiv.org/pdf/2205.05606v1.pdf" ]	248,693,628	2205.05606	5de8390ad263c4201efe010f9af513b61e292116	Wasserstein Image Local Analysis: Histogram of Orientations, Smoothing and Edge Detection Jiening Zhu [email protected] Stony Brook University NYUnited States Harini Veeraraghavan Memorial Sloan Kettering Cancer Center NYUnited States Larry Norton [email protected] Memorial Sloan Kettering Cancer Center NYUnited States Joseph O Deasy [email protected] Memorial Sloan Kettering Cancer Center NYUnited States Allen Tannenbaum [email protected] Stony Brook University NYUnited States Wasserstein Image Local Analysis: Histogram of Orientations, Smoothing and Edge Detection Image Processing · Optimal Transport · Tumor Analysis The Histogram of Oriented Gradient (HOG) is a widely used image feature, which describes local image directionality based on numerical differentiation. Due to its ill-posed nature, small noise in the image may lead to large errors. Thus conventional HOG may fail to produce meaningful directionality results in the presence of noise, which is common in medical radiographic imaging. We approach the directionality problem from a novel perspective by the use of the optimal transport map of a local image patch to a uni-color patch of its mean. Concretely, we decompose the transport map into sub-work costs each transporting in different directions. To test our approach, we evaluated the ability of the optimal transport to quantify tumor heterogeneity from MRI brain images of patients diagnosed with glioblastoma multiforme available from the TCIA. By considering the entropy difference of the extracted local directionality within tumor regions, we found that patients with higher entropy in their images, had statistically significant worse overall survival (p = 0.008), which indicates that tumors that have images exhibiting flows in many directions may be more malignant, perhaps reflecting high tumor histologic grade, a reflection of histologic disorganization. We also explored the possibility of solving classical image processing problems such as smoothing and edge detection via optimal transport. By looking for a 2-color patch with minimum transport distance to a local patch, we derive a nonlinear shock filter, which preserves edges. Moreover, we found that the color difference of the computed 2-color patch indicates whether there is a large change in color, i.e., an edge in the given patch. In summary, we expand the usefulness of optimal transport as an image local analysis tool, to extract robust measures of imaging tumor heterogeneity for outcomes prediction as well as image pre-processing required in several image analysis applications. Because of its robust nature, we find it offers several advantages over the classical approaches. Introduction Image feature extraction is an classical topic ever since image data was first digitalized. Over the past several decades, multiple different image feature de-arXiv:2205.05606v1 [eess.IV] 11 May 2022 scribers have been proposed. Steerable filters are synthesized filters of arbitrary orientations from linear combinations of basis filters [6]. Scale-Invariant Feature Transform (SIFT) and Histogram of Orientated Gradient (HOG) are two feature describers based on histogram of gradients [4,9]. Image feature describers are widely used for object detection, object matching, and network pretraining. A similar concept is used in the fields of medical imaging and radiomics. Cooccurrence of Local Anisotropic Gradient Orientations (CoLlAGe) uses a similar histogram of gradient method to capture subtle differences between benign and pathologic phenotypes on anatomic imaging [13]. Wasserstein distance is a powerful metric for distributions, known to be robust compared to other metrics (or divergences). It is defined naturally by the optimal cost of transporting one distribution to another, which was first motivated by the civil engineering problem of relocating a pile of soil to an excavation site by Gaspard Monge in 1781 [5,14,16,17]. A relaxation was proposed by the Russian mathematician Leonid Kantorovich [8] in 1942, and so the optimal transport problem is many times called the Monge-Kantorovich problem. It gives a distance of the space of probability distribtions, called the Wasserstein distance. In this note, we will only use the L 1 version, namely, the W 1 metric. Optimal mass transport methods are widely used in signal processing, machine learning, computer vision, meteorology, statistical physics, quantum mechanics, and network theory [1,2,7,10,11,15]. In the medical field, a number of works incorporate the Wasserstein distance and mass transport for various purposes. Some examples include [12,18] who use the distance to study multiomics networks for cancer subtype clustering, [3] that used regularized optimal transport to visualize fluid flows in the glymphatic system, as well as [19], which employed the unbalanced Wasserstein distance to identify high-risk normal tissue regions associated with worse mortality from spillover radiation in radiation treatments. In this work, we make use of the robust property of the Wasserstein distance for image local analysis in order to extract directional information from the optimal transport map instead of gradient. The optimal transport based computation increases robustness to noise, making it preferable for a variety of medical image analysis applications. Furthermore, we show the applicability of this methodology for other classical image processing tasks, namely, smoothing and edge detection. In the following sections, we sketch some of the background on optimal transport and and then detail our proposed methodology. Quantitative comparison and illustrative examples of our method are included in the experiments and results section. We conclude this note with some discussion about further applications to pathology data. Background and Methods Background: optimal transport The original Monge's formulation of optimal transport may be given a modern expression in terms of measure theory as follows [16,17]: W M (µ 0 , µ 1 ) = inf T { S c(x, T (x))dµ 0 (x) \| T # µ 0 = µ 1 },(1) where S denotes a subdomain of R n , µ 0 , µ 1 are two measures on S, T : S → S is the transport map, and c(·, ·) : S × S → R ≥0 is a convex cost function. In the present note, c(·, ·) will always be taken to be the distance function. Here T # denotes the push-forward of T (µ 1 (E) = µ 0 (T −1 (E)), ∀E ⊂ S). Kantorovich relaxed the model by replacing transport maps T in (1) by couplings π: W K (µ 0 , µ 1 ) = inf π∈Π(µ0,µ1) S c(x, y)π(dx, dy),(2) where Π(µ 0 , µ 1 ) denotes the set of all the couplings between µ 0 and µ 1 ( measures on S × S whose two marginals are µ 0 and µ 1 : π(· × S) = µ 0 (·), π(S × ·) = µ 1 (·)). Despite the relaxation, one may show that Kantorovich and Monge formulations are equivalent in a number of cases under certain continuity constraints; see [16,17] and the references therein. There are two major benefits of using Kantorovich form. First, it expands the possible transport maps. Mass can be split up, which Monge's form cannot handle. This relaxation guarantees a solution while Monge's form admits no solution in some cases. Second, in terms of discrete probability densities, Kantorovich form can be written as a standard linear programming problem, which may be solved very efficiently. The discrete form of Kantorovich optimal transport can be expressed as follows: d K (ρ 0 , ρ 1 ) = min Π∈R M ×N M i=1 N j=1 C(i, j) · Π(i, j) (3a) subject to: Π 1 N = ρ 0 ,(3b)Π T 1 M = ρ 1 ,(3c) where ρ 0 ∈ R M + , ρ 1 ∈ R N + are density of M/N desecrate locations of initial/target distribution and their total mass needs to be preserved as a prerequisite ( M i=1 ρ 0 (i) = N j=1 ρ 1 (j)). C is a point to point cost matrix and 1 N ( 1 M ) is an all-1 vector of length N (M ). Π specifies all the possible couplings with given marginals. The mass initially located at location i ∈ {1, 2, ..., M } in initial distribution may be transported to any one or more of the locations j ∈ {1, 2, ..., N } in target distribution. Π(i, j) specifies that amount of sub-transport mass and w(i, j) = C(i, j) · Π(i, j) is the workload of that sub-transport route. The overall work is the sum of all the work from all the sub-transport routes. So the sub-transport work matrix W = C. * Π reveals much of local information, which is very useful. Transport to 1-color patch: directionality Usually, the Wasserstein distance is computed globally, analyzing the difference between a pair of images. Here, also a key contribution, we employ optimal transport locally as a nonlinear filter, analyzing the image local features. For each local n × n image patch, we take ρ 0 ∈ R n 2 + to be the image intensity on that patch. An uni-color patch with the mean value of ρ 0 as intensity of all pixels on that patch is used as ρ 1 ∈ R n 2 + . Further, the cost matrix C ∈ R n 2 × R n 2 is taken as the standard pairwise distance matrix. With these definitions, the sub-transport work matrix W ∈ R n 2 × R n 2 can be solved via (3). Local image directions are extracted from the matrix W by regrouping its entries into bins (Fig. 1). Each entry of W has a corresponding orientation given by the transport starting point and end point locations. For the (i, j) entry of W (i = j), i'th location in the density vector corresponding to a coordinate (u 1 , v 1 ) in original n × n grid and j'th location in the density vector corresponding to another coordinate (u 2 , v 2 ) in original n × n grid. The directionality of that sub-transport route is given as: θ = arctan( v 1 − v 2 u 1 − u 2 ).(4) We evenly divide [0, π] into n b bins and add up all the w ij values in their corresponding bins. So each bin contains all the sub-work of the optimal transport in its corresponding direction. The n b -vector is a representation of local directionality. The sum of n b values in the feature vector coincides with the sum of all w ij , which is the Wasserstein distance. On the other hand, we decompose the distance into components in the n b directions. This gives a similar feature vector that characterizes local directionality as in HOG but more robust. We call our method Wasserstein HOG or WHOG, since its direction extraction is based on the Wasserstein distance. We performed detailed comparisons in the experiments and results section below. Transport to 2-color patches: smoothing, edge detection In addition to treating the transport between each local patch to a pure color patch, we also explore the transport from a local patch to 2-color patches. We seek the 2-color patch that has the minimum Wassertein distance to a local patch. The optimization problem may be expressed as follows: min Π∈R n 2 ×n 2 ,z∈{0,1} n 2 ,a,b∈R n 2 i=1 n 2 j=1 C(i, j) · Π(i, j),(5a) subjectto : Π 1 n 2 = ρ 0 , (5b) Π T 1 n 2 = a * z + b * (1 − z),(5c) where ρ 0 is the intensity vector of a local patch. ρ 1 = a * z + b * (1 − z) gives a 2-color patch that is closest to the original patch in term of the Wasserstein distance. Since ρ 1 has only two colors, it is smoothed with edge preservation at the same time. Moreover, the intensity difference \|a − b\| of that patch measures whether there is a large change in color, i.e., an edge in the given patch. In terms of the sub-transport work matrix W . We look at its projection onto the initial and target patches, which gives a an estimate pf pixelwise noise. Ideally, most of the mass stays at the same place and the work is just zero. Pixels that are considered to be noise are those that need extra work to move additional mass away or get mass from other pixels. Experiments and Results Compare with conventional HOG As a quick review of conventional HOG: The Sobel filters 1 0 −1 ,   1 0 −1   give x-and y-derivatives (g x , g y ) for each pixel. The directionality of that pixel is given by the direction of the gradient vector: θ = arctan( g y g x ).(6) Then all the pixels are put into corresponding bins in terms of θ. The sums of gradient vectors' norms g = g 2 x + g 2 y of all bins are combined together forming a feature vector of that patch. Feature vectors of neighboring patches are normalized together to alleviate artifacts caused by light variations. Notice that the way we get directionality (4) is very different from HOG (6). The direction of HOG is based on a gradient, which is sensitive to noise. WHOG, on the other hand, considers all the sub-transport routes and the direction is determined by grid points, making WHOG more robust. The examples in Fig.2 show that both methods work well when there is no noise. However, when more noise is added (Gaussian noise with mean=0, σ = {0, 0.01, 0.01, 0.1, 0.15}), WHOG almost doesn't change while HOG gradually lose directionality until the histogram is evenly distributed to all the bins. Bellow is an example of a medical image (Fig. 3). The rose plots illustrate the local orientations extracted by two methods. We observe that WHOG extracts the shapes of soft tissues, and we find that our proposed feature detector is better at extracting clear directionality, especially in the tissue area. Brain Glioma Biological evidence indicates that some of the flows within tumor regions are connected tumor severity. Those tumors with trajectories which are more spread out, seem to have greater disorganization and are malignant. We examined brain MRI images of glioblastoma multiforme patients from TCIA. WHOG local directionality feature vectors were computed within the tumor regions for each sample. By randomly taking the same amount of patches for different patients (tumor sizes differ), we computed the entropy of those distributions of directionality. Interestingly, we found significant survival differences (log-rank test p-value = 0.008 vs the p-value of HOG = 0.34) between the high entropy group and low entropy group (cut at the median value of entropy). The high entropy group which has more flows in many directions is more malignant, while the low entropy group with more steady flows are more benign. We compared a few slices from the largest and lowest entropy patients for both WHOG and HOG methods. We find that some patients are in the high entropy group from the HOG method is due to noise. We think the reason that the WHOG gets significant survival difference is the result of its unique ability to handle noise which helps to extract truly biological information of the tumor local fluid flows. To test whether our WHOG feature is related to scanner manufacturers and the magnet strengths. We computed the Wilcoxon rank sum test between GE vs non-GE; 1.5T vs 3T. p-values are: 0.6, 0.8 respectively. Thus for these data, the WHOG feature does not seem to correlate with GE or the MRI strength. Smoothing and Edge Detection We now illustrate our method via a smoothing and edge detection example. Referring to Figure 5, we see that the smoothed image reduces much of the background noise while all the edges are preserved. Most of the edges are extracted with good quality. Discussion and Conclusion The present work makes use of optimal transport in image local analysis to extract information such as directionality and edge information. Because of its robustness, we find our method performs well even in the presence of noise. As a general local analysis model, we considered the optimal transport between local patches and 1-color patches or 2-color patches to get interpretable results, which is useful in multiple classical image processing tasks. We got similar conclusion about image directionality and tumor heterogeneity in [13]. They did classifications based on their HOG based feature vetors. In principle, we can replace their HOG feature by our WHOG feature. We hope by doing this, classification accuracy can be further improved. We plan to further evaluate ability of our method to study pathology images. The microscopic flows in these high resolution images contain information for which our proposed method may better capture directionality, and thus distinguish different biological processes that are going on in different tissues. The method can be easily modified to 3D case by looking at the transport map between a local cube to a pure color cube. We also want to test on the 3D extension so that the directionality is within the transverse plane, frontal, and sagittal directions. Generally speaking, we can consider the transport between a local patch and any other distribution on that patch. Notice that the process of computing the optimal transport map acts like a local nonlinear filter, but unlike commonly used linear filters which may reduce the information in the original data, all the information is preserved, and may be recovered by projection. By considering the transport map, the results are in the product space with the pairwise information. This may be incorporated in a neural network as an upsampling layer. Fig. 1 . 1An illustration of the proposed WHOG method. Fig. 2 . 2HOG vs WHOG under different noise level: for both methods, we use 8 by 8 patch and 9 bins. Fig. 3 . 3An example of a head and neck slice from PDDCA dataset. Left: original slice, Middle: Rose plot from HOG, Right: Rose plot from WHOG. Fig. 4 . 4Kaplan-Meier plots of survival difference between the high entropy group and the low entropy group each with a characterized slice from a patient in each group from WHOG (left) and HOG (right). Fig. 5 . 5An example of optimal transport based smoothing and edge detection. Left: original image. Middle: smoothed image. Right: detected edges. Wasserstein GAN. 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[ "A Survey on Self-supervised Pre-training for Sequential Transfer Learning in Neural Networks", "A Survey on Self-supervised Pre-training for Sequential Transfer Learning in Neural Networks" ]	[ "Huanru Henry Mao [email protected] \nComputer Science and Engineering\nUniversity of California\nSan Diego\n" ]	[ "Computer Science and Engineering\nUniversity of California\nSan Diego" ]	[]	Deep neural networks are typically trained under a supervised learning framework where a model learns a single task using labeled data. Instead of relying solely on labeled data, practitioners can harness unlabeled or related data to improve model performance, which is often more accessible and ubiquitous. Self-supervised pre-training for transfer learning is becoming an increasingly popular technique to improve state-of-the-art results using unlabeled data. It involves first pre-training a model on a large amount of unlabeled data, then adapting the model to target tasks of interest. In this review, we survey self-supervised learning methods and their applications within the sequential transfer learning framework. We provide an overview of the taxonomy for self-supervised learning and transfer learning, and highlight some prominent methods for designing pre-training tasks across different domains. Finally, we discuss recent trends and suggest areas for future investigation.	null	[ "https://arxiv.org/pdf/2007.00800v1.pdf" ]	220,302,492	2007.00800	43bfcc4333920138613d6f9d76798f0c99452b92	A Survey on Self-supervised Pre-training for Sequential Transfer Learning in Neural Networks Huanru Henry Mao [email protected] Computer Science and Engineering University of California San Diego A Survey on Self-supervised Pre-training for Sequential Transfer Learning in Neural Networks Deep neural networks are typically trained under a supervised learning framework where a model learns a single task using labeled data. Instead of relying solely on labeled data, practitioners can harness unlabeled or related data to improve model performance, which is often more accessible and ubiquitous. Self-supervised pre-training for transfer learning is becoming an increasingly popular technique to improve state-of-the-art results using unlabeled data. It involves first pre-training a model on a large amount of unlabeled data, then adapting the model to target tasks of interest. In this review, we survey self-supervised learning methods and their applications within the sequential transfer learning framework. We provide an overview of the taxonomy for self-supervised learning and transfer learning, and highlight some prominent methods for designing pre-training tasks across different domains. Finally, we discuss recent trends and suggest areas for future investigation. Introduction Deep learning has led to significant improvements in state-of-the-art performance across many domains (LeCun et al., 2015) and has become the dominant approach in building intelligent systems over the last decade. Traditionally, deep networks are trained under a supervised learning framework where a model is trained tabula rasa (from scratch) to optimize the performance on a single task with the hopes of generalizing to unseen test examples. A task is typically provided as a set of labeled data with the assumption that the training and test set are drawn from the same underlying distribution. While effective when labeled data is abundant, the paradigm of learning a single task in isolation is limited when human-annotated data is lacking for tasks of interest, leading to poor model generalization (Pan and Yang, 2010). In contrast to the supervised learning framework, humans are able to learn priors about our environment without labels and adapt our knowledge to new tasks with only a few examples (Dubey et al., 2018). For instance, learning how to play piano can help us learn music fundamentals, which, subsequently, makes learning how to play violin easier. When an infant learns how to recognize faces, they can apply this knowledge to recognize other objects (Wallis and Bülthoff, 1999). Ideally, a similar approach could be applied to machine learning. Instead of relying solely on labeled data, practitioners can leverage unlabeled or related data, which is often more accessible and ubiquitous. Knowledge from a large corpus of unlabeled data can be extracted and transferred to improve performance on a target task where labeled data is either limited or unavailable. There is a large amount of literature on unsupervised and transfer learning. In this paper, we focus on surveying self-supervised learning methods for sequential transfer learn-ing. Self-supervised learning is a type of unsupervised learning where a model is trained on labels that are automatically derived from the data itself without human annotation (Erhan et al., 2010;. Self-supervised learning methods enable a model to learn useful knowledge about an unlabeled dataset by learning useful representations and parameters. Transfer learning focuses on how to transfer or adapt this learned knowledge from a source task to a target task (Pan and Yang, 2010). Specifically, we focus on a specific type of transfer learning called sequential transfer learning (Ruder, 2019) which adopts a "pre-train then fine-tune" paradigm. Self-supervised learning and transfer learning are two complementary research areas that, together, enable us to harness a source task with a large amount of unlabeled examples and transfer the learned knowledge to a target task of interest. These methods have grown in popularity due to their success and scalability in improving state-of-the-art results across domains. Finding useful self-supervised learning algorithms and transfer learning methods are areas of active investigation. Compared to other surveys that focus primarily on either computer vision (Schmarje et al., 2020;Jing and Tian, 2019) or natural language processing (NLP) (Ruder, 2019), we provide a broad review of self-supervised learning across domains in computer vision, natural language and audio/speech. This can, hopefully, provide a birds eye view of self-supervised research in deep learning and highlight areas for further investigation. We first provide a background overview of self-supervised pre-training and transfer learning in section 2 and 3. We then review self-supervised learning methods organized under the following categories: bottleneck-based methods (sec. 4) and prediction-based methods (sec. 5). Bottleneck-based methods drive learning by imposing an information bottleneck through a model's architecture. Prediction-based methods learn by asking a model to predict or generate relevant data with respect to the input. Finally, we provide a discussion of research trends and frontiers for future work in section 6. Transfer Learning We provide a more formal definition of transfer learning, following the definitions from Pan and Yang (2010); Ruder (2019). Transfer learning is a collection of techniques that focus on adapting knowledge between tasks and involves two concepts: a domain and a task. A domain D = {X , P (X)} has a feature space X and a marginal distribution P (X) over the feature space where X = {x 1 , . . . , x n } ∈ X . For image classification, X is the space of all images, x i corresponds to some image and X is a sample of images used for training. A task T is defined with respect to some domain D and consists of a label space Y, a prior distribution P (Y ) and a learned conditional probability distribution P (Y \|X). P (Y \|X) is typically learned from a training set of {x i , y i } where x i ∈ X, y i ∈ Y. For image classification, Y is the set of possible image classes. The aim of transfer learning is to learn the target task T t using knowledge learned from a source task T s . Specifically, we want to learn the target conditional probability distribution P t (Y t \|X t ) in D t from information learned from T s and D s where D s = D t and/or T s = T t . Transfer Learning Scenarios Following the taxonomy from Pan and Yang (2010), we can categorize transfer learning into three broad settings depending on the tasks in the source and target domain ( 1). When source and target tasks are the same T s = T t with labels only available in the source domain, we call it transductive transfer learning. A specific example of transductive transfer learning is domain adaptation (Wang and Deng, 2018), where a model could be trained on the source task of predicting sentiment on Amazon reviews and needs to be adapted to predict sentiment in news. When the tasks are different and labeled data is provided in the target domain T s = T t , we refer to this as inductive transfer learning. An example of inductive transfer learning is training a model on (optionally labeled) data to classify many images of natural scenery, then adapting the model to classify images of cats. When no labels are provided in either case, we refer to this setting as unsupervised transfer learning. Ruder (2019) further refines inductive transfer learning into two subcategories: multitask learning (Caruana, 1997) and sequential transfer learning. In multi-task learning, tasks T s and T t are learned simultaneously, typically through the joint optimization of multiple objective functions. In sequential transfer learning, T s is first learned, then the downstream task T t is learned. The first stage is often called pre-training while the second stage of learning is often called fine-tuning in the context of neural networks. The primary difference between these two types of transfer learning is when the target task is learned. More generalized schemes can define a multi-tasking schedule that interpolates between learning the source and target task (Liang and Shu, 2017) or contain multiple source tasks . Sequential transfer learning is our primary focus in this paper. Sequential transfer learning is more popular in practice as it is simple to set up a two-phase training pipeline and easy to distribute pre-trained models without needing to disclose the pre-training dataset. Most of the self-supervised learning techniques we review can be categorized under sequential transfer learning. Related Areas There are other related research areas to transfer learning that are beyond the scope of this paper, that we briefly mention here. Lifelong learning (Parisi et al., 2019) can be seen as form of sequential transfer learning of many tasks, with the additional goal of learning without forgetting previous tasks (e.g., catastrophic forgetting). Few shot learning (Wang et al., 2019) focuses on the general problem of learning with few labels and is achievable in certain extents with transfer learning. Meta learning (Vilalta and Drissi, 2002) focuses on algorithms that enable us to learn how to learn and can be considered a form of transfer learning where meta-knowledge is transferred to task-specific knowledge. Why Does Sequential Transfer Learning Work? In order to understand the success of sequential transfer learning it is useful to consider some theoretical arguments as to why it works. Multi-tasking Perspective We briefly summarize an analysis from Ruder (2017) on why multi-task learning is beneficial since it is highly related to sequential transfer learning. Multi-task learning (Caruana, 1997) has been shown to serve as a form of regularization as it reduces the Rademacher complexity of the model (the ability to fit random noise) (Søgaard and Goldberg, 2016). It biases the model to prefer representations that other tasks would likely prefer (Baxter, 2000) and allows the model to learn a task better through hints from another task (Abu- Mostafa, 1990). While multi-task and sequential transfer learning are not strictly the same, it is useful to consider these related effects especially when hybrid sequential and multi-task transfer learning approaches are used. Regularization To understand why sequential transfer learning works, we summarize an early work that provides useful insights in unsupervised pre-training. Erhan et al. (2010) analyzed a special case of unsupervised pre-training applied to deep belief networks , but the arguments presented there are more broadly applicable to sequential transfer learning. Erhan et al. (2010) hypothesizes that pre-training serves as a form of implicit regularization through parameter initialization by constraining the minima that the supervised objective can optimize to. Pre-training restricts learning to a subset of the parameter space bound by a basin of attraction achievable through fine-tuning the supervised target task. This hypothesis is supported experimentally by observing the training dynamics of MNIST filters (Erhan et al., 2010). Recent work has also shown some evidence to suggest that fine-tuning pre-trained language models does not deviate from the pre-trained weights significantly (Sanh et al., 2020). In other words, the final weights are mostly predetermined by pre-training, especially if the pre-training task dominates the total training time of the sequential transfer learning process. Inducing Priors Treating sequential transfer learning as simply a form of regularization underestimates its benefits. Similar to multi-task learning, sequential transfer learning also induces a prior on the model. Practitioners who use similar source and target tasks encode a prior on what knowledge is likely useful, thus the effects are akin to selecting good neural architectures and better hyperparameters. Implicit Meta Learning Another perspective we can consider is that pre-training, when given an appropriate and sufficiently large source task, can perform implicit meta learning (Brown et al., 2020a). This provides a similar effect as meta-learning algorithms such as MAML (Finn et al., 2017) that explicitly aim to learn an initialization that easily adapts to various problems. Self-supervised Learning Unsupervised learning is a family of approaches that learn from data without any supervision. A particular form of unsupervised learning of growing interest is self-supervised learning. The terms unsupervised and self-supervised have been, historically, used interchangeably in the literature, but recent work has preferred the term self-supervised learning for its specificity. In this review, we refer to self-supervised learning as any unsupervised learning approach that can be easily reduced into a supervised problem by generating labels. Thus, self-supervised learning can reap the advancements and breakthroughs from supervised learning. Self-supervised learning still requires labels, but it is unsupervised in the sense that these labels are derived from the data itself rather than annotated by humans. Early work in self-supervised pre-training for deep neural networks aimed to effectively train stacked auto-encoders and deep belief networks (DBN) without labels. These techniques train deep networks one layer at a time in a greedy fashion in order to circumvent poor local minima that prevented successful endto-end gradient descent (Bengio and Lecun, 2007). Once trained, the neural network is fine-tuned, where the model with pre-trained weights switches from unsupervised learning to supervised learning objective of the target task. This can lead to improved performance on the target task as opposed to simply learning the target task from scratch. In the last decade, greedy layer-wise unsupervised learning has fallen out of fashion in favor of end-toend learning where an entire deep network is trained in one operation. This shift is partly due to the architectural innovations (He et al., 2015), normalization (Ioffe and Szegedy, 2015) and better activation functions (Nair and Hinton, 2010) that enable training of very deep networks (Bachlechner et al., 2020) while avoiding local minima. In contrast to classic work on greedy self-supervised learning (Ackley et al., 1985;De-Mers and Cottrell, 1993), modern approaches focus on end-to-end learning. Self-supervised learning constructs a pre-training or "pretext" task that is used to extract knowledge from unlabeled data (Jing and Tian, 2019). After training a model on the pretext task, it can then be adapted to the target task through transfer learning. Pre-training tasks come in many forms. They usually involve transforming or imputing the input data with the goal of forcing the model to predict missing parts of the data or through introducing some information bottleneck, which we will review in later sections. Downstream Tasks Self-supervised learning has been used to transfer knowledge into a variety of target tasks. In this review, we do not focus on any specific downstream task since we are primarily concerned with pre-training methods. Instead, we briefly highlight some common tasks used to benchmark self-supervised learning algorithms. For computer vision, image classification is typically the downstream task of interest for self-supervised learning of still images and action recognition benchmarks are used to evaluate video self-supervised learning methods (Srivastava et al., 2015). For natural language, a popular benchmark, GLUE , has been used to test selfsupervised learning approaches on a bag of tasks including natural language inference, sentiment analysis and paraphrase identification. For speech, automatic speech recognition, phoneme identification and speaker identification are downstream tasks of interest (Chi et al., 2020). Information Content in Learning It is useful to consider the amount of information content that can be derived from different learning frameworks (Ruder, 2019). Yann LeCun has referred to the hierarchy of information given to learning algorithms metaphorically as a "cake": Reinforcement learning gets the cherry on top (a single scalar value per episode, (Williams, 1992)); supervised learning the frosting (10-10k bits per sample); and unsupervised and self-supervised learning is the foundation of the cake (millions of bits per example depending on the domain). Hence, in many cases, self-supervised learning can provide significantly more information per example for learning. Generative verses Discriminative Learning A critical decision when designing pre-training schemes is to consider whether we want to perform generative or discriminative learning. In this section we outline the differences between the two approaches. These approaches are somewhat orthogonal to the choice of pre-training tasks and either option can be used for a given task. Generative Approaches Generative approaches for self-supervised learning involve the process of producing all or parts of the training data as part of the model's output (Jing and Tian, 2019). For instance, we can take a frame in a video and ask the model to generate future frames (Srivastava et al., 2015). The labels, in this case, are typically in the feature space of the training data. Generative approaches have the advantage that the output is qualitatively interpretable as we can inspect samples from the model. In addition, generative models have other applications beyond self-supervised learning (Goodfellow et al., 2014). The drawback of generative learning is that it requires learning how to produce every single detail in the input feature space, which could be a substantial amount of dedicated computation and modeling resources. For example, generating an image requires predicting every single pixel in the output space of the model and the process of decoding an image is not necessarily helpful for transfer learning to downstream tasks. For continuous domain applications such as images or raw audio, generation is challenging when there are multiple "correct" answers (e.g., predicting the future audio frames spoken), sometimes leading to the model predicting the mean of all futures (which qualitatively results in blurry predictions). To avoid generating the average prediction, researchers have adopted alternative generative techniques using adversarial learning (GANs) (Goodfellow et al., 2014), which can lead to sharper generations. For a detailed survey on GANs, we refer the reader to Jabbar et al. (2020). Discriminative Approaches On a high level, discriminative approaches for self-supervised learning involve the process of determining positive samples from negative samples. When labels are provided, as in supervised learning, this is simply called classification. Discriminative approaches eschew the challenge of generation by asking the model to simply differentiate between pairs of input samples. In self-supervised learning, a common interpretation of discriminative learning without labels is mutual information maximization (Hjelm et al., 2018). The mutual information (MI) (Bell and Sejnowski, 1995) of two random variables X, Y measures the reduction in uncertainty of one variable when the other is observed. For instance, knowing that the background of an image contains grass x can make us less uncertain about the location y in which the image was photographed. For the purpose of self-supervised learning, it may be desirable to maximize the mutual information between certain features of the data (Hjelm et al., 2018). More formally, mutual information is defined as: I(X, Y ) = E p(X,Y ) log p(x, y) p(x)p(y)(1) It is intractable to compute I and sample-based estimators that maximize lower bounds on MI are used in practice. The most commonly used lower bound that has been shown to work well is Information Noise Contrastive Estimation (InfoNCE) (van den Oord et al., 2018). InfoNCE is a probabilistic contrastive loss (Chopra et al., 2005) that tries to separate positive examples from negative examples. Following the formulation and notation in Kong et al. (2019), the InfoNCE lower bound is defined as: I(A, B) ≥ E p(A,B)   f θ (a, b) − E q(B)   log b ∈B exp f θ (a,b)     + log \|B\|,(2) where a and b are the positive example pairs,B is a set of samples drawn from some proposal distribution q(B), and f θ ∈ R is a learned comparison function with parameters θ.B contains positive samples b and \|B\| − 1 negative samples. There are many ways to construct f θ . For instance, we can construct it as the dot product of features produced by two identical encoders, commonly known as Siamese Networks (Hadsell et al., 2006). In practice, training f θ involves sampling a pair of positive samples and \|B\| − 1 negative samples, then minimizing the cross entropy loss of the positive example over all samples. This is equivalent, in expectation, to maximizing Eq. 2. Contrastive learning can be used in self-supervised learning by trying to predict certain samples from negative samples, such as predicting future audio frames against random frames or image patches within the same images against random patches (details in section 5). This works well for various continuous domain tasks as shown in van den Oord et al. (2018). A challenge with contrastive learning is choosing proposal distribution q(B), which determines how negative samples are selected. Having a large number of negative samples can be helpful in certain domains (He et al., 2020). For discrete domain tasks such as natural language, Kong et al. (2019) show that language modeling and generation tasks that maximize cross entropy loss also maximizes In-foNCE. Indeed, cross entropy loss is a special case of InfoNCE whenB = B. For instance, language modeling predicts the next token by comparing against all possible tokens in the model's vocabulary. This is equivalent to performing a "negative sampling" scheme where all possible outputs are sampled at all times. Mutual information maximization alone is insufficient for learning good representations as suggested in Hjelm et al. (2018) and demonstrated empirically in . Instead, good representations also depend on the choice of architecture, task and parametrization of the MI estimators. Architectural Bottleneck Methods We categorize self-supervised learning approaches that primarily rely on an information bottleneck induced through a model's architecture as bottleneck-based methods. Bottleneckbased methods attempt to learn a low dimensional or constrained representation of the data typically by learning to reconstruct the input data (DeMers and Cottrell, 1993). Bottleneckbased methods are sometimes categorized in the literature as unsupervised rather than self-supervised learning. By learning a constrained representation, a model must discard irrelevant information and retain useful information. A direct application of bottleneck-based methods is in their learned representations, which can be used as feature extractors for downstream target tasks Cottrell and Metcalfe, 1991). Alternatively, the model's weights (usually the encoder) can also be transferred to target tasks via fine-tuning or learned jointly in a multi-task setting. We present a summary of bottleneck-based methods in this section. Dimensionality Reduction We first briefly review classical approaches to dimensionality reduction. The most well known technique for dimensionality reduction in machine learning is principal component analysis (PCA) (Wold et al., 1987). Given a dataset of d-dimensional vectors represented as a matrix X, PCA aims to find a low dimensional representation of the data by eliminating correlations between variables. In practice, PCA can be solved by using singular value decomposition. The low dimensional feature can then be used as input to other machine learning algorithms. Latent Semantic Analysis In natural language, inputs are often sequences of discrete tokens, which can be represented as one-hot vectors over some vocabulary. It is useful to extract low-dimensional representation of words, known as word embeddings, because one-hot vectors are large in dimensionality and do not contain semantic meaning of the words they represent. Latent semantic analysis (LSA) (Deerwester et al., 1990) is a classic technique used to extract low dimensional distributed representations of words based on the co-occurrence of words within a document context. First, a word-document matrix A is constructed by counting the occurrence of words that appear in each document. Then, we apply dimensionality reduction on A using singular value decomposition to factorize it into the product of three matrices. A = U ΣV T(3) Word embeddings E of dimension d can be extracted by truncating matrices U and Σ by retaining only the top d rows, resulting in U d and Σ d . Then, word embeddings can be computed by the product: E = U d Σ d(4) LSA can be considered as PCA applied to matrix A. Deep Autoencoders Approaches such as PCA and LSA extract low dimensional representations of data, but are linear approaches. Deep autoencoders (DeMers and Cottrell, 1993;Hinton and Salakhutdinov, 2006) with non-linearity are more expressive approaches that can extract better low-dimensional features from data. In the most high level definition, deep autoencoders can be formulated as two neural networks that contain an encoder enc and decoder dec. The encoder produces a latent representation from input x, h = enc(x).(5) While the decoder reconstructs the input x from the latent representation h, x = dec(h).(6) Autoencoders are trained to minimize the reconstruction error between x andx. Thus, autoencoders are considered as generative approaches. There are a variety of methods to impose an information bottleneck in the autoencoder and we summarize some prominent approaches in the follow subsections. Once an autoencoder has been trained, depending on the task, the decoder may be discarded and the encoder can be transferred to downstream tasks. We also highlight how some of these techniques have been applied for fine-tuning. Compression-based Autoencoders In order to learn a non-trivial mapping, the dimensionality of h is typically constrained to be less than the dimensionality of x, thus the model must learn what information to keep. Early work (Cottrell et al., 1987) demonstrated that this bottleneck, after quantization, can learn effective image compressors. Compression-based autoencoders have been successfully applied to natural language for self-supervised learning. In Dai and Le (2015), a sequence-to-sequence autoencoder is trained to take an input sequence x, produce a single latent vector h, which is then used to generate the original sequence x. The authors demonstrated improvements on sentiment analysis tasks through pre-training. Sparse Autoencoders Alternatively, h can be overcomplete (dimensionality of h is greater than dimensionality of x) but constrained by other means (Bengio et al., 2013). One popular constraint is by imposing a sparsity prior on the latent representation typically by minimizing the L1 loss of h. Sparsity prior has been motivated biologically by the human visual cortex (Olshausen and Field, 1997). Ranzato et al. (2008) suggests that this acts as a soft way of restricting "the volume of the input space over which the energy surface can take a low value". Makhzani and Frey (2013) introduced k-sparse autoencoders, where the latent representation is constrained to only have top k largest activations active and the rest are set to zero. They demonstrated experimentally that k-sparse autoencoders can be used as a pre-training step, then fine-tuned on image classification tasks for improved performance. One drawback of this technique is that it could lead to dead hidden units, which can be addressed by scheduling the sparsity level. Variational Autoencoders Another method to impose a constraint on the latent variables is by treating the latent variable as a stochastic variable, as introduced in variational autoencoders (VAE) (Kingma and Welling, 2013). VAEs impose a regularization term in addition to the reconstruction loss to minimize the KL divergence between the latent representation and a prior distribution, typically a multivariate Gaussian. One issue with VAE approaches in sequence models is the posterior collapse problem, where the latent variable is completely ignored when more powerful sequence decoders are used (Roberts et al., 2018). This can be mitigated by using hierarchical decoders. We have found that, generally, Gaussian VAEs have been less explored for the purpose of transfer learning. The prior distribution of VAEs could also be categorical. Vector quantized variational autoencoders (VQ-VAE) (van den Oord et al., 2017) and probabilistic variants (Sønderby et al., 2017) are a type of autoencoder with a discrete bottleneck. The learned discrete bottleneck provides several advantages such as enabling latent discrete modeling. Discrete latent autoencoders have been used in speech for unsupervised phoneme discovery. For example, Eloff et al. (2019) trained autoencoders to quantize speech and found that the learned discrete codes can be used for speech synthesis. Quantizing speech is a reasonable prior, since spoken words in raw wave forms often have corresponding discrete phonemes that represent them. Discrete latent models have also been combined with prediction-based methods for self-supervised speech recognition (Baevski et al., 2019b). In these scenarios, the extracted discrete latent code can be further processed using NLP pretraining techniques, such as BERT (Devlin et al., 2018), to learn even better representations. Dhariwal et al. (2020) extended this line of work to learn discrete latent codes for music generation. Other Approaches Autoencoders are not the only way to impose bottleneck learning. Wu et al. (2018) demonstrate that one can perform unsupervised learning by simply treating each data point as its own class, and to maximally scatter all data points onto a 128 embedding space using contrastive learning. By trying to compress the entire dataset into a low dimensional space similar inputs must cluster together. They show that this method can lead to competitive results on ImageNet when compared to other self-supervised techniques. This is similar to the bottleneck-based approaches in autoencoders except learning is performed using a contrastive loss without a decoder. In the authors train a bidirectional GAN (BiGAN) for selfsupervised representation learning. GANs typically employ a generator and a discriminator, which learns a latent to data space mapping. For self-supervised learning, we ideally want a data to latent mapping for downstream tasks (e.g. image classification). Thus, the authors propose a bidirectional GAN learning framework where an encoder is also learned jointly with a generator and discriminator. This model is not explicitly an autoencoder, but the adversarial constraint forces the encoder to invert the generator. The authors show that BiGAN is closely related to autoencoders with an l 0 loss. Limitations As mentioned in previous sections, bottleneck-based methods have shown success in various domains, especially when realized as autoencoders. However, bottleneck approaches have generally been found to be inferior to prediction-based methods (Zhang et al., 2017) and current state-of-the-art techniques are mostly prediction-based methods. This may stem from the fact that bottleneck approaches need to trade off between information content and representational capacity. Critics claim that autoencoders are an unsupervised learning approach, which, by definition, cannot be tailored to downstream tasks without additional priors (Rasmus et al., 2015). That is not to say that bottleneck approaches cannot be combined with prediction-based methods for improved performance and, indeed, many prediction-based methods have built upon bottleneck-based approaches. Prediction-Based Methods Prediction-based methods aim to learn useful representations of data through learning a relevant predictive task such as asking the model to predict the missing parts of an input given its related context. These techniques range from asking a model to predict the future given the present, predict missing patches of an image or missing words in a sentence. Intuitively, prediction forces a model to learn relationships between the global and local parts of the data. In this section, we review methods for continuous and discrete domains separately since they tend to have different methods. Pre-training for Continuous Domains In this section, we focus on self-supervised learning methods for continuous domain tasks such as vision and speech. An overarching theme among these approaches is to create selfsupervised tasks that learn high level features while discarding low level information and noise. Here, we summarize various commonly used pre-training tasks. Spatial Prediction Spatial prediction aims to learn representations by removing patches of an image and predicting the masked patches. When posed as a generative task, this technique is also known as image in-painting. In Context Encoders (Pathak et al., 2016), the authors train a convolutional neural network (CNN) autoencoder by blanking-out the center patch of an input image and ask the model to generate contents within the missing square. This is similar to a denoising autoencoder (Vincent et al., 2008), but differs in that the input mask is a contiguous block instead of random noise, and only the masked segments are predicted. An issue raised by the paper is that pixel-level prediction creates blurry in-paintings, since L2 loss encourages learning the average of all possible completions. Using an adversarial loss can mitigate this issue. Alternative spatial masking approaches perform self-supervised learning by using a discriminative loss where the ground truth patch must be correctly identified from negative samples. Hjelm et al. (2018) proposes to maximize mutual information between local and global features of an image. This is done by encoding an image into feature vectors for each patch, forming low-level features. A separate network summarizes the low-level features into high-level features. These low and high level features are grouped together but some high-level features are grouped with low-level features from another random image. A discriminator is trained to assign correct groupings with a higher score than random groupings. van den Oord et al. (2018) segments an image into overlapping patches, imposes a topleft to bottom-right ordering of all patches, then uses an auto-regressive model to predict "future" patches of the image using InfoNCE loss. "Future" is defined as the next patch in the imposed ordering. Follow up work demonstrated that adding more model capacity and increasing the task difficulty (e.g., predicting several steps into the future) improves performance (Hénaff et al., 2019). Trinh et al. (2019) proposes a similar approach but avoids imposing an ordering of patches by randomly masking input image patches and training the model to predict the masked patches. Channel Prediction Color or channel prediction methods perform self-supervised learning by removing channel information from an image and asking the model to predict the missing channel. Several work (Zhang et al., 2016;Larsson et al., 2017Larsson et al., , 2016 has shown that using colorization as a pre-training task can lead to improvements on ImageNet classification without labels. In Split-Brain Autoencoders (Zhang et al., 2017), the authors split a traditional autoencoder into two disjoint sub-networks with each sub-network receiving a subset of the input channels. The disjoint autoencoders are then trained to predict the missing channels of the other encoder. Temporal Prediction Temporal prediction focuses on exploiting temporal information to learn representations. Many work in this area are based on ideas from early work on slow feature analysis (SFA) (Wiskott and Sejnowski, 2002), which suggests that a good prior for feature extraction is to learn features that vary slowly with time. Learning to extract information that move slowly with time can naturally lead to higher-level representations and discard low-level noise. A modern realization of this idea in deep learning is found in Jayaraman and Grauman (2016), where temporally close representations are encouraged to exhibit small differences. Temporal prediction for computer vision focus on learning how to predict different perspectives of an image by leveraging either the camera movement or the motion of objects in the image. A motivation for this type of learning is that motion in video helps identify objects, since pixels of the same moving object will likely move together (Wertheimer, 1938). In Srivastava et al. (2015), the authors predict future frames in a video using an LSTM. Alternatively, a contrastive loss can be used to avoid modeling low level information (Han et al., 2019). Instead of learning directly to predict future frames, the motion of objects can be extracted as synthetic labels for training static images (Pathak et al., 2017). Temporal prediction has also been applied for self-supervised learning for speech. Schneider et al. (2019) trains a self-supervised model from raw audio waves to predict future speech features against negative samples from the same audio clip, similar to van den Oord et al. (2018). Chi et al. (2020) proposes to mask random speech frames (represented a spectrograms) and to predict those masked frames. These approaches have many commonalities with those in section 5.1.1. Order Prediction Order prediction approaches aim to train a model to predict the position of image patches. In Doersch et al. (2015), random pairs of image patches are sampled from one of 8 positions in the image. The model is asked to predict the relative position of one patch to another. In Noroozi and Favaro (2016), image patches are randomly shuffled and the model has to predict the permutation of the shuffle as a classification task. A follow up work increased the difficulty (Kim et al., 2018) of the task by randomly deleting an image patch and asking the model to also predict the color of the image. Misra et al. (2016) applies this principle of order prediction to videos to predict the ordering of frames given shuffled frames. Hybrid Approaches When choosing a self-supervision task, it is not necessary for us to choose only a single predictive learning task. Recent work has shown that a combination of different self-supervision tasks can yield much better results, rivaling the results of purely supervised learning for image classification. Pre-training for Discrete Domains In this section, we survey approaches that enable self-supervised learning in the discrete domain such as natural language processing (NLP). Natural language treats text as a sequences of discrete symbols (also called tokens). Although we primarily focus on self-supervised learning applied to NLP, techniques presented here are likely applicable to other forms of discrete sequences or non-natural languages (e.g., modeling music (Donahue et al., 2019) or programming languages (Lachaux et al., 2020)). Word Embeddings Skip-gram and Continuous Bag of Words (CBOW) (Mikolov et al., 2013a,b) are popular approaches developed in 2013 for learning high quality word embeddings. Skip-gram learns word embeddings by forcing words to predict nearby surrounding words within a given context. Given a context of word embeddings S = (s t−c , ..., s t , ..., s t+c ) with context length c, Skip-gram predicts s t+i , i ∈ [−c, c], i = t from s t . CBOW involves a similar idea to Skip-gram, but instead learns to predict s t using the sum of its surrounding embeddings,ŝ t = i∈[−c,c],i =t s i .(7) Once these embeddings are learned they can be used as input or fine-tuned as lower layers of other models. Contextual Embeddings Word embeddings are scalable and fast to train, but are limited in their representative power since they are usually learned using a linear model. Furthermore, words in isolation provide limited information for which features can be extracted. A natural extension to word embeddings is to learn deeper networks with contextual embeddings. Early work explored learning contextual representations by predicting contiguous sentences of an input using a recurrent neural network (Kiros et al., 2015). Contextual Word Vectors (McCann et al., 2017) provided embeddings based on a word and its entire sentence by leveraging the attention learned from machine translation. These models have shown some success in text classification tasks and question answering. Language Models Core to the recent surge in transfer learning in NLP arises from the success of self-supervised learning from language modeling tasks and their variants. Language modeling, in this context, is a pre-training task that learn to predict the probability of the next word or token given a historical context for an input sequence X = {x 1 , ..., x n }. p(x i \|x 1 , ..., x i−1 )(8) A seminal work that demonstrates the general transferability of language modeling is the paper Embeddings from Language Models (ELMo) (Sun et al., 2019a). ELMo learns a bidirectional LSTM (Hochreiter and Schmidhuber, 1997) language model and demonstrated strong improvements to a variety of downstream GLUE tasks with less labeled data. Transformer Language Models Since ELMo, researchers have transitioned to focus on training self-attention models instead of recurrent neural networks. Transformers (Vaswani et al., 2017) are a type of deep neural network that contain stacked layers of self-attention and feed-forward layers. When compared to recurrent neural networks, Transformers are more efficient to train and enable gradients signals to easily propagate to all positions of the input. The General Pre-trained Transformer (GPT) (Radford et al., 2018) is the first successful attempt at pre-training a Transformer and achieving strong target task performance for a variety of tasks. GPT learns a unidirectional language model on a large corpus of text. Follow up work (Brown et al., 2020b) scaled GPT to larger models and bigger datasets, observing strong generative capabilities and zero-shot performance on a variety of natural language tasks. Masked Language Modeling A major limitation with GPT is that it learns a unidirectional language model in which every token can only attend to the tokens left of it. Bidirectional Encoder Representations for Transformers (BERT) (Devlin et al., 2018;Baevski et al., 2019a) proposes to learn bidirectional Transformers using a masked language modeling (MLM) pre-training task. MLM randomly removes input tokens to the model and trains the model to predict the removed tokens. At every iteration, BERT masks 15% of its input tokens. The downside of BERT is the pre-training procedure is expensive (only 15% of positions are trained per iteration) and it does not explicitly learn conditional generation akin to language models. Several extensions of BERT have been proposed, such as SpanBERT (Joshi et al., 2020), a training procedure that masks out contiguous spans instead of individual tokens, and ERNIE (Sun et al., 2019b), which masks out full entities or phrase-level units. These strategies propose smarter masking strategies for better performance. Permutation Language Models XLNet (Yang et al., 2019) harnesses the benefits of language model conditioning with bidirectional training by introducing a permutation lan-guage modeling objective. However, BERT, with more training and better hyperparameters, can outperform XLNet (Yang et al., 2019). It is later shown that permutation language modeling can be seen as a masked language model with stochastic attention masks (Kong et al., 2019). Sequence to Sequence Pre-training BERT has shown a lot of success in natural language inference tasks, but it is less well suited for sequence to sequence tasks. Pre-training for sequence to sequence learning is explored T5 (Raffel et al., 2019), BART (Lewis et al., 2019) and MASS (Song et al., 2019). Raffel et al. (2019) provides an extensive analysis of various sequence to sequence pretraining tasks including prefix language modeling, masking and deshuffling. They found that masking input spans and asking the model to generate these masked spans leads to the best performance. Interestingly, learning how to deshuffle an input sequence performs the worse, which contradicts some of the success of order prediction techniques found in vision. Discriminative Pre-training Tasks An alternative to the popular approach of learning a generative language model is to consider discriminative pre-training tasks. Indeed, in the original BERT (Devlin et al., 2018) implementation the authors proposed to jointly perform masked language modeling and next sentence prediction. Next sentence prediction is a task where segments of text (specifically, sentences) are randomly swapped 50% of the time and the model must predict whether or not the swap occurred. This task has later been found to be not useful given masked language modeling (Liu et al., 2019b). Electra (Clark et al., 2020) proposes to pre-train a model by classifying whether or not a token in the input sequence was randomly replaced by a small BERT model. This focuses the model to learn how to differentiate real sequences from plausible alternatives. The authors demonstrated that learning a discriminator yields strong results on downstream tasks with much better sample efficiency, since every single position is trained per iteration. Discussion Throughout this review, we have seen a variety of approaches to enable self-supervised learning. The following are some general tips for self-supervision based on our observations of previous work. Pre-training should be challenging Choosing a pre-training task that is sufficiently difficult is desired and the difficulty should scale as models becomes larger. It is also critical to prevent models from exploiting shortcuts and cheating (Minderer et al., 2020) or leak statistical information from normalization . Combining pre-training tasks can be much better than using any single pre-training task alone . Ideally, pre-training tasks should be similar to the target task or subsume it. For example, language modeling have shown to implicitly perform few shot learning when the dataset and model is sufficiently large (Brown et al., 2020b), likely because the patterns that appear in a text corpus naturally contain relevant tasks. Designing better and more universal 1 pre-training tasks should be an active area for future investigation. More data and larger models are better Unsurprisingly, having more data and larger models lead to better results (Kolesnikov et al., 2019). This is even more important in self-supervised learning where a lot of information needs to be absorbed for fine-tuning. Furthermore, as seen in the trend of moving from word embeddings to contextual embeddings in NLP, the more parameters of a model that are pre-trained the better. Even under computational constraints, training a larger model with more parameters for fewer iterations on a sufficiently large dataset is better than training a small model . These large models can be subsequently pruned if fast inference is required (Frankle and Carbin, 2018). Choose flexible architectures Choosing model architectures that have more flexibility (trading off priors and bias) can be advantageous in the context of self-supervised learning. More flexible models enable a form of soft architectural search (Elsken et al., 2018). We see this example in NLP where Transformers have the advantage of having no positional bias as opposed to the receny bias of recurrent neural networks (Ravfogel et al., 2019). This lack of positional bias likely provides more opportunities for gradient descent to mold its learning, which explains Transformer's tendency to be more data hungry and appropriate for large scale self-supervised training. In computer vision, most self-supervised work has focused on ResNet (He et al., 2015) and it would be interesting to see if this trend holds across domains. Future Work There are many future directions to further explore self-supervised learning. Simply scaling existing approaches to larger models and datasets have diminishing returns (Kolesnikov et al., 2019) and even 175 billion parameter language models cannot learn commonsense physics and lack world knowledge (Brown et al., 2020b). Most self-supervised learning approaches have been focused on a single domain and it would be interesting to extend these techniques to multi-modal scenarios. After all, humans are multi-modal learners. Several work (Chen et al., 2019;Arandjelovic and Zisserman, 2017) have shown promising results in this direction by performing contrastive learning of audio and visual information or masked "language" modeling between images and text. Another area to explore is better ways to extract information from these pre-trained models. In this survey, we primarily focused on the popular fine-tuning approach, but other knowledge adaptation techniques exist (Ruder, 2019). For example, ; Raffel et al. (2019) explored learning multiple tasks in a multi-task learning framework while fine-tuning pre-trained language models, leading to better downstream performances. One interesting approach for pre-trained language models adopted by Brown et al. (2020b) is few-shot probing. This technique involves using natural language itself to specify the desired downstream task along with a few examples and requires no fine-tuning. It would be interesting to see if this type of probing works for other domains such as vision and speech. Conclusion Supervised learning's primary bottleneck is the availability of labeled data. Self-supervised learning is a powerful technique to extract knowledge from a large unlabelled corpus of data. After a model is trained in a self-supervised manner, it can attain significantly improved performance on tasks that have few labels and even on tasks that have plenty of labels. The value of self-supervision comes from its scalability with virtually unlimited data in certain domains and its ability to be fine-tuned to a variety of tasks. In the long term, self-supervision approaches are likely to outperform more task-specific approaches as computational resources become more ubiquitous (Sutton, 2019;LeCun et al., 2015). Figure 1 : 1Transfer learning scenarios based on Ruder (2019). . Universal can be defined as beneficial to all conceivable tasks that humans care about, which does not contradict the "No Free Lunch Theorem"(Wolpert, 2012). AcknowledgmentsThanks to Professor Garrison W. Cottrell for providing comments, advice and editorial assistance. Thanks to Bodhisattwa Prasad Majumder for providing proofreading assistance. Learning from hints in neural networks. 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[ "Λ-RINGS AND THE FIELD WITH ONE ELEMENT", "Λ-RINGS AND THE FIELD WITH ONE ELEMENT" ]	[ "James Borger " ]	[]	[]	The theory of Λ-rings, in the sense of Grothendieck's Riemann-Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce Λ-algebraic geometry. We show that Λ-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than Z and that it has many properties predicted for algebraic geometry over the mythical field with one element. Moreover, it does this is a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry.	null	[ "https://arxiv.org/pdf/0906.3146v1.pdf" ]	7,040,380	0906.3146	0649c93d77c257bc009d7b0615349a83a2dc271f	Λ-RINGS AND THE FIELD WITH ONE ELEMENT 17 Jun 2009 James Borger Λ-RINGS AND THE FIELD WITH ONE ELEMENT 17 Jun 2009 The theory of Λ-rings, in the sense of Grothendieck's Riemann-Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce Λ-algebraic geometry. We show that Λ-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than Z and that it has many properties predicted for algebraic geometry over the mythical field with one element. Moreover, it does this is a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry. Introduction Many writers have mused about algebraic geometry over deeper bases than the ring Z of integers. Although there are several, possibly unrelated reasons for this, here I will mention just two. The first is that the combinatorial nature of enumeration formulas in linear algebra over finite fields F q as q tends to 1 suggests that, just as one can work over all finite fields simultaneously by using algebraic geometry over Z, perhaps one could bring in the combinatorics of finite sets by working over an even deeper base, one which somehow allows q = 1. It is common, following Tits [60], to call this mythical base F 1 , the field with one element. (See also Steinberg [58], p. 279.) The second purpose is to prove the Riemann hypothesis. With the analogy between integers and polynomials in mind, we might hope that Spec Z would be a kind of curve over Spec F 1 , that Spec Z ⊗ F1 Z would not only make sense but be a surface bearing some kind of intersection theory, and that we could then mimic over Z Weil's proof [64] of the Riemann hypothesis over function fields. 1 Of course, since Z is the initial object in the category of rings, any theory of algebraic geometry over a deeper base would have to leave the usual world of rings and schemes. The most obvious way of doing this is to consider weaker algebraic structures than rings (commutative, as always), such as commutative monoids, and to try using them as the affine building blocks for a more rigid theory of algebraic geometry. This has been pursued in a number of papers, which I will cite below. Another natural approach is motived by the following question, first articulated by Soulé [57]: Which rings over Z can be defined over F 1 ? Less set-theoretically, on a ring over Z, what should descent data to F 1 be? The main goal of this paper is to show that a reasonable answer to this question is a Λ-ring structure, in the sense of Grothendieck's Riemann-Roch theory [31]. More precisely, we show that a Λ-ring structure on a ring can be thought of as descent data to a deeper base in the precise sense that it gives rise to a map from the bigétale topos of Spec Z to a Λ-equivariant version of the bigétale topos of Spec Z, and that this deeper base has many properties expected of the field with one element. Not only does the resulting algebraic geometry fit into the supple formalism of topos theory, it is also arithmetically rich-unlike the category of sets, say, which is the deepest topos of all. For instance, it is closely related to global class field theory, complex multiplication, and crystalline cohomology. So let us define an F 1 -algebra to be a Λ-ring. (The language of F 1 will quickly feel silly, but for most of this paper, it will be useful as an expository device.) More generally, define an F 1 -scheme to be a scheme equipped with a Λ-structure. The theory of Λ-structures on schemes was introduced in Borger [6] [7]. (Although see Grothendieck [34], p. 506.) Defining a Λ-structure on a general scheme X takes some time, as it does on a general ring. But when X is flat over Z, there is a simple equivalent definition: it is a commuting family of endomorphisms ψ p : X → X, indexed by the set of prime numbers p, such that each ψ p agrees with the p-th power Frobenius map on the special fiber X × Spec Z Spec F p . It is also true that any reduced Λ-scheme is flat over Z. Keeping these two facts in mind, it is possible to read most of this paper without knowing the definition in general. If we take this F 1 -to-Λ dictionary seriously, then the functor that removes the Λ-structure from a Λ-scheme should be thought of as removing the descent data, and hence as the base-change functor from F 1 to Z. So for instance, F 1 itself should be defined to be Z with its unique Λ-structure, where each ψ p is the identity map. It is the initial object in the category of Λ-rings. As indicated above, the definition of a Λ-structure extends not just to the category of schemes but to the entire ambient topos. By this, I mean the bigétale topos over Z, by definition the category of sheaves of sets on the category of affine schemes equipped with theétale topology. It is therefore natural to define the bigétale topos over F 1 to be the category of such sheaves with Λ-structure. The base-change functor v * that strips off the Λ-structure then induces a well-defined map v : Spec Z −→ Spec F 1 , not on the level of schemes, which would of course be meaningless, but on the level of bigétale toposes. The functor v * has not only a right adjoint v * , as required, but also a left adjoint v ! . If we think of v * as the base-change functor, we should think of v * as the Weil restriction of scalars functor and v ! as the base-forgetting functor. In terms of definitions rather than interpretations, v * sends a space to its arithmetic jet space, which is a multi-prime, algebraic version of Buium's p-jet space [9], which is in turn a formal p-adic lift of the Greenberg transform [29] [30]. On the other hand, v ! sends a space to its space of big Witt vectors. Both of these are given their natural Λ-structures. These constructions are exotic by the standards of algebraic geometry over fields, but even in the arithmetically local case, where we consider padic schemes with only one Frobenius lift ψ p , they have proven very useful. See, for example, Buium [9] for applications of the first and Illusie [37] for the second. (In fact, Buium has had similar ideas regarding Frobenius lifts and the field with one element. See [10], the preface of [9], or Buium-Simanca [11]. Manin [47] has also recently interpreted Witt vectors as being related to the field with one element.) For applications to usual, non-Λ arithmetic algebraic geometry, the most interesting spaces over F 1 are those obtained from schemes over Z. Whether we produce them by applying v ! or v * , the result is almost never a scheme of finite type over F 1 . So from the perspective of algebraic geometry over Z, the spaces of principal interest over F 1 are not those of finite type. In fact, as has been expected, there appear to be very few schemes of finite type over Z that descend to F 1 at all. But their infrequency is so extreme that they become interesting on their own terms. For example, it might be possible to describe the category of algebraic spaces of finite type over F 1 in purely combinatorial terms, without any mention of algebraic geometry or Λ-structures. This question is probably within reach, and the second purpose of this paper is to take some steps in that direction. For example, we prove the following theorem, stated here under restricted hypotheses: 0.1. Theorem. Let X be a smooth proper scheme over Z[1/M ], for some integer M 1. If X descends to F 1 (i.e., admits a Λ-structure), then is has the following properties: (a) The action of Gal(Q/Q) on any p-adicétale cohomology group H ń et (XQ, Q p ) factors through its abelianization. (b) There is an integer N , all of whose prime divisors divide M p, such that the restriction of the representation H ń et (XQ, Q p ) to Gal(Q/Q(ζ N )) is isomorphic to a sum of powers of the cyclotomic character. (c) The (i, j) Hodge number of the mixed Hodge structure on the singular cohomology H n (X an , C) is zero when i = j. Another way to express this theorem is that only abelian Artin-Tate motives can be defined over F 1 . (Compare Soulé [57], 6.4, question 4.) While this means there are no motivically interesting F 1 -schemes of finite type, the argument that shows this is rather interesting. In the zero-dimensional case, it is an elementary consequence of the Kronecker-Weber theorem and Chebotarev's density theorem. (See Borger-de Smit [4].) In the higher-dimensional case, the argument also uses the Lefschetz theorem and the proper base change theorem forétale cohomology, on the one hand, and on the other, p-adic Hodge theory, including the potentially semi-stable theorem for arbitrary varieties proved by Kisin [41], following the work of many others. The theorem above attests to the predicted combinatorial nature of schemes of finite type over F 1 . I emphasize once again that the combinatorial nature is not built into the foundations of the theory-it is a consequence of hard arithmetic results in the presence of finiteness conditions. Indeed, I expect that the cohomological theory of infinite-dimensional spaces over F 1 contains, via v ! and de Rham-Witt theory, the full theory of motives, and even in a visible way. It is hard to imagine a theory defined in combinatorial terms having this property. As strong as the cohomological restrictions above are, they are far from being sharp. In a future version of this paper, I hope to show that all examples of finite type come from toric varieties, in a certain precise sense. Here is a partial result in this direction whose proof does appear in the present version. It is probably even a necessary step in the proof of the full classification theorem. 0.2. Theorem. Let X be a Λ-scheme which is of finite type over Z. Then X has a point with coordinates in a cyclotomic field. When X is proper, there is even a Λ-morphism Spec Z → X. In other words, every F 1 -scheme of finite type has a cyclotomic point and, if proper, has an F 1 -point. I believe that this result had not been predicted. Observe that in the affine case, the theorem is a new result about Λ-rings, which can be stated independently of the theory. It says that every Λ-ring which is finitely generated as a ring admits a ring map to some cyclotomic field. Even in this case, the proof uses all the deep arithmetic results above. If we apply theorem 0.1 and the Lefschetz fixed-point formula at various primes p, we obtain the following result, which was essentially predicted (implicitly by the followers of Tits [60], and by definition in Kurokawa [44] 1 with q ≡ 1 mod N , the number of F q -valued points of X is P (q), where P is the Hodge polynomial of X: P (t) = i,n (−1) n r i,n t i , where r i,n is the (i, i) Hodge number of H n c (X an , C) . In particular, if X is smooth and proper over Z, the number of such points is given by the polynomial P (q) for all prime powers q > 1. Let us now consider F 1 -points. It is not difficult to show, when X is separated and of finite type over F 1 , that X(F 1 ) is finite. So it is tempting to hope that the number of F 1 -valued points would be P (1), the Euler characteristic of X. This is typically false, but if we restrict to complemented points-those whose complement is an open sub-F 1 -scheme of X-then it is true in many situations. For instance, it is true for toric varieties, when given a certain natural, "toric" Λ-structure. But it is not always true. It fails for the toric varieties A 1 and P 1 if they are given the Chebychev Λ-structure (6.7). On the other hand, if all Λ-varieties can indeed be built from toric varieties with the toric Λ-structure, it might be possible to salvage the formula X(F 1 ) = P (1) = χ(X) by cleverly reinterpreting the definitions. Our final purpose is to consider variations on the definition of Λ-structure, such as function-field analogues. For example, instead of working over Spec Z, we can work over a smooth curve S over a finite field k. Then our family of commuting Frobenius lifts would be indexed by the closed points of S, and the same procedure as over Z gives the notion of a Λ S -structure, a topos of F S 1 -spaces, and a topos map v S : S −→ Spec F S 1 . Having invented an absolute algebraic geometry relative to S, we can ask how it relates to the usual absolute algebraic geometry relative to S, that is, algebraic geometry over k. The answer is that the structure map s : S → Spec k factors naturally as a composition of topos maps: S vS E Spec F S 1 f y y s s s s s Spec k. The map f is not an equivalence, but it is not far from being one. For example, f * embeds the category of reduced algebraic spaces over k fully faithfully into the category of spaces over F S 1 , but there can be schemes defined over F S 1 (given by certain rank-one Drinfeld modules) that do not descend to k. As a check to see if this could lead to a proof of the Riemann hypothesis for Z, one might examine the translation of Weil's proof of the Riemann hypothesis for S from algebraic geometry over k to that over F S 1 . But since the current version of our theory says nothing about the archimedean place of Q, it is hard to imagine this succeeding without further ideas. Even so, it should be done. Let us now list the contents of this paper. In section 1, we recall the foundations of the theory of spaces over F 1 (that is, Λ-spaces). In section 2, we give examples of F 1 -schemes. In section 3, we discuss sub-F 1 -spaces and in particular F 1 -valued points. In section 4, we discuss function spaces over F 1 and especially GL n . In section 5, we show that abelian motives are Artin-Tate. This is a result in usual, non-Λ number theory needed in the following section. In section 6, we discuss the p-adicétale cohomology of F 1 -schemes of finite type and implications for point counting. And in section 7, we consider variations on our approach to F 1 for function fields and number fields larger than Q. Other work In the early days of this project, I was greatly inspired by Manin's exposition [48], Soulé's paper [57], and Deninger's program, for example [25]. In a strict mathematical sense, this paper does not owe them much, but their spiritual effect has been deep. There are, of course, many approaches to absolute algebraic geometry which I did not mention above. Here are some I know about. One is that an F 1 -algebra should be some kind of algebraic structure that is set-theoretically weaker than a commutative ring, for example a commutative monoid. From this point a view an F 1 -vector space is often taken to be a set, perhaps with some additional weak structure. One could investigate the K-theory that comes out of this, and even aspire to see the place at infinity by incorporating archimedean information in these structures. Another approach has been to pursue notions of ζ-functions over F 1 , perhaps independently of any formal definition of F 1 . A final approach is to find and prove analogues over number fields of basic geometric results over function fields. For these approaches see the following references: Baez [ [59], and Toën-Vaquié [61]. Several of these writers have other papers on the subject, but I believe these are representative of their approaches. Acknowledgments My focused work on Λ-algebraic geometry began on October 22, 2003, when I read an email from Ivan Fesenko asking if there were relations between my paper [5] with Ben Wieland and the field with one element. That question instantly gave direction to some scattered thoughts about Λ-algebraic geometry and the p-adic absolute point (7.10). My interests at the time were arithmetically local, and as obvious as the global connection is in retrospect, it had not occurred to me before that day. So I thank him greatly. I would also like to thank Mark Kisin for many conversations about this project over several years. Most of his influence on this project is on forthcoming work, but even with this paper, if he had not insisted that I begin writing up what I knew in the case of Λ-varieties of finite type over Z, I never would have uncovered much of what is here. And since the foundational papers [6] [7] owe their existence to the present paper, perhaps they also owe some of it to him. I did some of this work in 2004-2005 at the Institut des HautesÉtudes Scientifiques and the Max-Planck-Institut für Mathematik. It is a pleasure to thank them for their generous support and nearly ideal working environments. I did more work on this topic in January 2006, as a visitor at the University of Chicago. I thank Alexander Beilinson and Vladimir Drinfeld for making that possible. 1. Λ-spaces 1.1. General Λ-spaces. Let us quickly review [7]. Let W * = W * ∞ denote the infinite-length big Witt vector functor from the category of spaces to itself. (The category of spaces is, by definition, the category of sheaves of sets on the category of affine schemes under theétale topology.) Then W * carries a monad structure, and a Λ-structure on a space X is by definition an action of W * on X. Note that if X is an algebraic space, then the space W * (X) is ind-algebraic but typically not algebraic. This is just the familiar fact that the ring of infinite-length Witt vectors is naturally a projective limit of rings. Here are some important examples. If A is a ring, then we have W * (Spec A) = colim n Spec W n (A), where W n denotes the functor of (big) Witt vectors of length n. Therefore a Λ-structure on the space Spec A is the same as a Λ-ring structure on A in the usual sense. If X is a flat algebraic space over Z, then Λ-structure on X is the same as a commuting family of endomorphisms ψ p : X → X, one for each prime p, such that ψ p agrees with the p-th power Frobenius map on X × Spec Z Spec F p . If a reduced algebraic space admits a Λ-structure, then it must be flat over Z. Finally, if X is a Λ-algebraic space, then X red is a closed Λ-algebraic subspace. (Because a Λ-structure is given by a monad action, a subspace of a Λ-space can have at most one compatible Λ-structure.) Non-reduced Λ-algebraic spaces appear only sporadically in this paper. The functor W * has a right adjoint W * = W ∞ * , the arithmetic jet space functor, which is a generalized version of Buium's p-jet space functor [9]. If A is a ring, then W * (Spec A) = Spec Λ ⊙ A, where Λ ⊙ A denotes the Λ-ring freely generated by A. For example, W * (A 1 Z ) = Spec Λ, where Λ is the free Λ-ring on one generator, the ring of symmetric functions in infinitely many variables. Let S = Spec Z, and let Sp S = Sp Z denote the category of spaces. Let Sp S/Λ denote the category of Λ-spaces (with Λ-equivariant, or rather W * -equivariant, morphisms). This can also be described using W * . By adjunction, W * inherits a comonad structure from the monad structure on W * . The category of Λ-spaces is the same as the category of spaces equipped with an action of the comonad W * . It is therefore a topos, like Sp S . Categorical structure. Let v * : Sp S/Λ → Sp S denote the functor that simply strips off the Λ-structure. The point of this paper is that v * can also be thought of as the functor that strips off the descent data from Z to F 1 , and hence that it can be thought of as the base-change functor from F 1 to Z. Therefore, we have the following equation: Sp S v * ṽ! Sp Z Weil restrictf orget base = Sp S/Λ v * O O Sp F1 . S×F 1 − O O This means that the structure on the right is defined to be that on the left, or that we think of the precisely defined left-hand side using the geometric language of the right-hand side. Each functor is the left adjoint of the one to its right. The left adjoint v ! of v * sends X to its Witt space W * (X) with the natural Λ-structure, and the right adjoint v * sends X to its arithmetic jet space W * (X), again with the natural Λ-structure. As always the left adjoint of a base-change functor is called base-forgetting, and the right adjoint is called Weil restriction of scalars. In particular, if one accepts the premise of this paper, the space Spec Z× Spec F1 Spec Z must be defined to be the Witt space W * (Spec Z). These three adjoint functors form, by definition, an essential topos map v : Sp Z → Sp F1 . This is what one would would hope to have with any algebraic geometry over a deeper base than Spec Z. Similarly, the base-forgetting functor v ! is faithful but not full. (See [6], 12.2.) Yet another way of expressing the point of this paper, in the playful tradition of the field with one element, is the nonsense formula "Λ = Z[Gal(Z/F 1 )]". The meaning of this is that if descent from Z to F 1 were controlled by a finite group, one would call it Gal(Z/F 1 ). In that case, descent for rings would alternatively be controlled by the plethory Z[Gal(Z/F 1 )] whose underlying ring would be the polynomial ring freely generated by the set Gal(Z/F 1 ). (See Borger-Wieland [5].) But the plethory that actually controls this is Λ. So while the group Gal(Z/F 1 ) does not exist, the polynomial algebra it would generate if it did exist does. Note that everything above extends to the case where S is the spectrum of the ring of integers of any number field or any smooth curve over a finite field. Thus for any such S, there is a topos Sp F S 1 = Sp S/ΛS , the topos of spaces over the S-variant of the field with one element. These toposes are all related as S varies, and as one would expect, S = Spec Z, as above, gives the deepest one. We will return to this in section 7. 1.3. Λ-modules. An important topic we will not discuss in this paper is the Λanalogue of a quasi-coherent sheaf. The non-linear nature of Λ-structures makes module theory slightly subtler than it is in equivariant algebraic geometry under actions of monoids or Lie algebras, or more generally in the context of Toën-Vaquié [61]. The reason for this, in the language of Borger-Wieland [5], is that the additive bialgebra of the plethory Λ does not agree with the cotangent algebra of Λ. Therefore one cannot properly speak about modules without specifying certain extra information. In the case of Λ, these are essentially the slopes of the Frobenius operators. I mention this here only because F 1 -modules are a frequent concern in papers on the field with one element and I will not address them. Examples 2.1. The point. The ring Z has a unique Λ-structure-each ψ p is the identity. Under this structure, it is the initial object in the category of Λ-rings. It is therefore reasonable to denote it F 1 and call Spec Z, viewed as a Λ-space, the absolute point. Monoid algebras. If M is any commutative monoid, the monoid algebra Z[M ] has a natural Λ-action induced by ψ p : m → m p for any m ∈ M , and so Spec Z[M ] descends naturally to F 1 . I will call this Λ-action the toric Λ-action. For example, given a choice of coordinates, A r × G s m equals Spec Z[N r × Z s ], which descends naturally to F 1 . In fact, we have even more: the monoid scheme structure on Spec Z[M ] also descends to F 1 , as does the group structure if M is a group. Indeed, the coalgebra structure on Z[M ] given by m → m ⊗ m for all m ∈ M is a map of Λ-rings. The same is true of the counit and, if M is a group, the antipode m → m −1 . For example, all split tori can be thought of as group schemes over F 1 . The group scheme µ n = Spec Z[x]/(x n − 1) = Spec Z[Z/nZ] also descends to F 1 . Following Kapranov-Smirnov [39], Soulé calls this the base change to Z of F 1 n , the field with 1 n elements [57]. Limits and colimits. The category of Λ-rings has products and coproducts, and their underlying rings agree with the same constructions taken in the category of rings. In fact, this is true for all limits and colimits-in particular pull-backs, push-outs, direct limits, and inverse limits. This is because the forgetful functor Ring Λ → Ring Z has a left and a right adjoint. Since it is a topos, Sp S/Λ also has all limits and colimits. And since v * has a left and a right adjoint, the space underlying any limit, or colimit, of Λ-spaces agrees with the limit, or colimit, or the underlying spaces. Further, since algebraization is a left adjoint, it commutes with W * . Therefore the algebraization of a Λ-space is an algebraic Λ-space. In particular, given a system of algebraic spaces with Λ-actions, the colimit taken in the category of algebraic spaces has a unique compatible Λaction. (The analogous fact for limits is true simply because the subcategory of Sp S consisting of algebraic spaces is closed under limits.) For the same reason, the affinization of an algebraic Λ-space is an affine Λ-space. Toric varieties. A toric variety is a colimit of spectra of monoid rings, where the maps in the system are open immersions induced by maps of the monoids. Therefore the colimit in Sp S is an algebraic space (and even a scheme). By 2.2 and 2.3, they carry natural Λ-actions. Thus toric varieties descend to F 1 . (Compare Soulé [57].) In particular, projective spaces P n do, once we choose homogeneous coordinates. Then the Frobenius lifts are given by ψ p : [x 0 , . . . , x n ] → [x p 0 , . . . , x p n ]. 2.5. The Chebychev line. Clauwens [12] has used Ritt's work [51][52] to argue that, up to isomorphism, the affine line Spec Z[x] has exactly one Λ-structure besides the toric one of 2.2. It can be described as follows. The ring Z[t ±1 ], endowed with the toric Λ-action, has a Λ-involution t → t −1 . By 2.3, the fixed subring is naturally a Λ-ring. It is freely generated as a ring by x = t + t −1 , and this gives the other Λaction on Z[x]. It also has a simple K-theoretic interpretation as the representation ring of the algebraic group SL 2 , but in this paper we are regarding the connection between Λ-rings and K-theory as a curiosity. The polynomials ψ p (x) are Chebychev polynomials: ψ 2 (x) = x 2 − 2, ψ 3 (x) = x 3 − 3x, ψ 5 (x) = x 5 − 5x 3 + 5x, . . . . More generally, any subgroup of GL n (Z) acts on the toric Λ-ring Z[x ±1 1 , . . . , x ±1 n ] , and the invariant subrings give more examples of Λ-rings. For example, if we take the permutation representation S n → GL n (Z) of the n-th symmetric group, the invariant subring is isomorphic to Z[λ 1 , . . . , λ n−1 , λ ±1 n ] and thus gives a non-toric Λ-action on A n−1 × G m . It would be interesting to generalize Clauwens' results. For example, are there only finitely many isomorphism classes of Λ-structures on A 2 ? 2.6. Singular lines. We can divide out, not just by group actions, as in 2.5, but also by any Λ-equivalence relation, whether we take the quotient in the category of affine Λ-spaces, algebraic Λ-spaces, or all Λ-spaces. For instance, on G m with the toric Λ-action, we can identify 1 and −1 to make a nodal line A = {f (z) ∈ Z[t ±1 ] \| f (−1) = f (1)}, or we can identify 1 with itself to order two to make a cuspidal line A ′ = {f (z) ∈ Z[t ±1 ] \| f ′ (1) = 0}. It is easy to check that these subrings of Z[t ±1 ] are sub-Λ-rings. Observe that we can identify the points 1, q ∈ G m only when q = ±1-otherwise, the ψ p operators would fail to descend to the quotient. But if we use Λ-algebraic parameter spaces, there are non-discrete moduli. For instance, in the family G m × G m pr 1 −→ G m , we can identify the diagonal section ∆ and the identity section id Gm × 1, to get the family of nodal lines G m /(1 ∼ q) parameterized by q ∈ G m . The result is a perfectly legitimate Λ-algebraic family of nodal Λ-quotients of G m . The reason why before q could only lie in a finite set is that there we insisted that the endomorphisms ψ p act trivially on the parameter space. We will see below that an algebraic Λ-space of finite type over Z has only finitely many points (with coordinates in C, say) fixed by the ψ p operators (3.2). Therefore in any Λ-algebraic family, only finitely many of the fibers will be stable under the ψ p . In particular this will be true for any universal family, assuming the Λ-moduli space is of finite type. So the finiteness phenomenon above is rather general. We can also contract any Λ-invariant modulus on G m . If f (x) is a product of polynomials the form x n − 1, then the two maps Z[x ±1 ] ⇉ Z[x ±1 ]/(f (x)), one given by x → x and the other by x → 1, are Λ-ring maps. Therefore their equalizer is a Λ-ring. Its spectrum is G m with the zero locus of f (x) contracted to a point. Last, these constructions can be used to make Λ-schemes that cannot be covered by open affine Λ-schemes. In particular, it is inaccurate to say that Λ-schemes are formed by gluing Λ-rings together in the Zariski topology, though it is generally the right idea. The following example is due to Ben Wieland. Consider P 1 with the toric Λ-structure, and let X be the quotient by the Λ-equivalence relation 0 ∼ ∞. Then there is no open immersion j : U → X which has the following properties: U is an affine Λ-scheme, j is a Λ-map, and the nodal point 0 = ∞ is in the image. Indeed, if there were such a neighborhood U , then since it would be affine, the set U (C) would be the complement in X(C) of a finite nonempty subset T of G m (C). But U ∩ G m would also be a sub-Λ-space of G m . Since ψ n on G m is the n-th power map, T would have to be closed under the extraction of n-th roots, which is not the case for any finite nonempty subset of C * . Zero-dimensional varieties. As shown in Borger-de Smit [4], it follows from class field theory that every Λ-ring which is both finite over Z and reduced is contained in a product of cyclotomic fields. It is in fact isomorphic to a sub-Λ-ring of a product of toric Λ-rings of the form Z[x]/(x n − 1). This can be viewed as an integral version of the Kronecker-Weber theorem. Thus even zero-dimensional Λalgebraic geometry is somewhat interesting. In fact, the proofs of several theorems about higher-dimensional Λ-varieties use the zero-dimensional theory in key ways. 2.8. Non-example: flag varieties. It follows from a theorem of Paranjape and Srinivas [50] that no flag varieties besides projective spaces P n admit even one Frobenius lift ψ p . Therefore, besides P n , flag varieties are not defined over F 1 . In 6.2 below, we show that there are strong motivic conditions on a variety for it to descend to F 1 . But in the case of flag varieties, the obstruction is not in the motive. Indeed, flag varieties are paved by affine spaces and are therefore indistinguishable from them from the point of view of motives. It would be interesting to know whether flag varieties admit a weakened version of a Λ-structure but which is still stronger than being paved by Λ-varieties. 2.9. Non-example: curves of genus g 1. Let C Q be a connected smooth proper curve over Q of genus g 1. Choose in integer M 1 such that C Q has a a connected smooth proper model C over Z[1/M ]. Then C has no Λ-structure. One can see this as follows. If g 2, let p be a prime number such that p ∤ M . Then since g 2, the map ψ p cannot be a constant map on the fiber over Q because it is not on the fiber over F p . Therefore it must be an automorphism on the fiber over Q. Further, there must exist an integer n 1 such that ψ •n p is the identity on the fiber over Q. But as C Q is dense in C, we see that ψ •n p is the identity. This contradicts the fact that ψ p is the Frobenius map on C Fp , which is a nonempty curve. If g 1, then for any prime p ∤ M , there is a finite extension K of Q p and point e ∈ C(K) = C(O K ), where O K denote the integral closure of Z p in K. Let us now take the group law on C OK to be the one for which e is the identity, and let E denote the endomorphism ring End(CQ p ). By enlarging K, we may assume E = End(C K ). Note that E is an integral domain of rank 1 or 2 over Z. Then there is an element ϕ ∈ E such that ψ p (x) = ψ p (e) + ϕ(x), for all points x. Therefore on each fiber of C OK over O K , the degree of ψ p is ϕφ ∈ Z. But on the fiber over the residue field of O K , the map ψ p agrees with the base-change of the p-th power Frobenius map, which has degree p. Therefore, we have ϕφ = p. This rules out E = Z. To rule out the other case, observe that the same equation implies p is not inert in E. But since p was allowed to be any sufficiently large prime, this is impossible. Sub-Λ-spaces 3.1. Periodic primes. Let X be a Λ-space. Let us say that a prime number p is periodic if there exists an integer m 1 such that the endomorphism ψ •m p of X is the identity. We also say that m a period of p. (Also see Davydov [16].) Proposition . Let X be a separated algebraic Λ-space of finite type over Z with infinitely many periodic primes. Then X is affine and quasi-finite over Z. Proof. Let us consider quasi-finiteness first. It suffices to assume X is reduced and, hence, flat over Z. (See [7].) Therefore it is enough to show that X × Spec Z Spec Q is finite over Spec Q. Let p be one of the given primes, and let X p denote the fiber of X over p. Then ψ p is periodic, and therefore the p-th power Frobenius map on X p is periodic. By 3.3 below, the fiber X p is finite over F p . Because there are infinitely many such p, X is finite over Z at a dense set of scheme-theoretic points. And because X is of finite type, its fiber over Q is finite. (See EGA IV (9.2.6.2) [33].) Affineness follows from quasi-finiteness. Since X is separated, of finite type, and quasi-finite over Z, Zariski's Main Theorem [32], III 4.4.3, implies it is an open subscheme of a scheme which is finite over Z, and any such scheme is affine. 3.3. Lemma. Let X be an algebraic space of finite type over F p , and let Fr X denote the p-th power Frobenius map on X. If Fr •n X is the identity map, then X is a finite disjoint union of spaces of the form Spec F , where F is a field of degree at most n over F p . Proof. Let U = Spec B be an affineétale cover of X. Then Fr •n U is the identity map on U . In particular, B is reduced and is hence a subring of a finite product of fields. Because the p-th power map on B has period n, the image in each field is a field of degree at most n over F p . Since U is a finite disjoint union of spectra of finite fields, and since U covers X, X is also such a space. 3.4. Proposition. Let X and Y be separated algebraic Λ-spaces of finite type over Z. Assume that X is reduced and has infinitely many periodic primes. Then Hom Λ (X, Y ) is finite. Proof. Since X is reduced, we can assume Y is reduced. We can also assume that Y satisfies the same periodicity conditions as X. Indeed, let p be a periodic prime of X, and let m p denote its minimal period. Then the equalizer Y p of ψ •mp p and the identity map is a closed sub-Λ-space of Y . Therefore, so is the reduced subspace of Y ′ = ∩ p Y p , where p runs over all periodic primes of X. But any map X → Y factors through Y ′ , which as a closed subspace of Y is of finite type over Z. Therefore it is enough to assume Y = Y ′ , which is to say that Y satisfies the same periodicity conditions as X. By 3.2, there are generically finite rings A and B such that X = Spec A and Y = Spec B. Because A and B are reduced Λ-rings, they are torsion free, and thus Hom Λ (X, Y ) = Hom(B, A) ⊆ Hom(Q ⊗ Z B, Q ⊗ Z A). By Galois theory, there are only finitely many ring maps between two finiteétale algebras over a field; so Hom Λ (X, Y ) is finite. 3.5. Corollary. Let X be a separated algebraic space of finite type over F 1 . Then there are only finitely many Λ-maps µ n → X, where µ n is defined in (2.2). In particular, X has only finitely many F 1 -valued points. 3.6. Primitive Λ-spaces and complemented sub-Λ-spaces. Let f : Y → X be a map of Λ-algebraic spaces which is a closed (resp. open) immersion. Then Y (or better, f ) is said to be complemented if the complementary open (resp. reduced closed) algebraic subspace T admits a Λ-structure such that the map T → X is a Λ-map. (Compare SGA 4 IV 9.1.13c [1].) Note that because T → X is a monomorphism, the Λ-structure on T , when it exists, must be unique. We also say that a Λ-space X is primitive if it is nonempty and its only nonempty complemented closed sub-Λ-space is itself. Observe that if X ′ → X is a Λ-morphism and Y is a complemented closed (resp. open) algebraic subspace of X, then its preimage X ′ × X Y is a complemented closed (resp. open) algebraic subspace of X ′ . Also it is clear that finite intersections of complemented closed (resp. open) Λ-algebraic subspaces are again complemented. Therefore the same is true for finite unions. For example, consider A 1 with the toric Λ-structure. The Z-valued Λ-points of Proof. Let Z be a nonempty complemented closed sub-Λ-space of G d m . Since Z red is flat over Z ( [7]) and nonempty, the space Z has a C-valued point z. Write z = (e w1 , . . . , e w d ) for some numbers w 1 , . . . , w d ∈ C. For each integer s 1, the point z s = (e w1/s , . . . , e w d /s ) satisfies ψ s (z s ) = z ∈ Z. Because Z is complemented, we then have z s ∈ Z. But in the analytic topology we have lim s→∞ z s = (1, . . . , 1). Since Z is closed (in the Zariski and hence analytic topology), it contains (1, . . . , 1), and because it is complemented, it must contain ψ −1 n (1, . . . , 1) = µ d n for any integer n 1. On the other hand, ∪ n µ n is Zariski dense in G m . Therefore ∪ n µ d n is Zariski dense in G d m , and so we have Z = G d m . Proposition. Let X be a toric variety. Then a closed sub-Λ-space is complemented if and only if it is a union of closures of torus orbits. For background on toric varieties, see Fulton's notes [28], especially sections 2.1 and 3.2. Proof. Let Z be a closed sub-Λ-space of X. Suppose Z is a union of closures Z i of torus orbits. Then each Z i is the toric subvariety corresponding to a fan, and is therefore complemented. Since there are only finitely many torus orbits, the union Z of the Z i is complemented. Now assume instead that Z is complemented. Then its intersection with any torus orbit Y is either Y or ∅, by 3.7. Therefore Z is a union of torus orbits. Since Z is closed, it is also a union of their closures. 3.9. Corollary. The complemented F 1 -points of a toric variety X are the fixed points of the torus action. In particular, the number of complemented F 1 -points is the Euler characteristic. Function spaces In this section, we discuss GL n over F 1 . The group scheme GL n does not descend to F 1 . (See Buium [8].) But we can realize GL n as the automorphism group of something that does descend to F 1 . This might be surprising, because formation of function spaces commutes with base change. Indeed, that is one important way in which the topos map v : Sp Z → Sp F1 is not like a true map of spaces. For discussion of GL n in other approaches to F 1 , see for example Connes-Consani [13] and Toën-Vaquié [61]. This section was directly inspired by some questions Kirsten Wickelgren asked me. 4.1. Let X and Y be objects of Sp F1 . Write Hom F1 (X, Y ) for the set of maps from X to Y , and write Hom Z (X, Y ) for the set of maps v * X → v * Y between the underlying objects of Sp Z . Let Hom F1 (X, Y ) denote the usual Sp F1 -object of maps from X to Y . It is defined by Hom F1 (X, Y ) : T → Hom T (X × T, Y × T ), for any space T ∈ Sp F1 . We define Hom Z (X, Y ) similarly. Then we have a map (4.1.1) Hom F1 (X, Y ) −→ v * (Hom Z (X, Y )), which sends an F 1 -map a : X × T → Y × T to its underlying Z-map v * (a). This map is clearly injective. Adjunction then gives another map (4.1.2) v * : v * (Hom F1 (X, Y )) −→ Hom Z (X, Y ). Note that this map is generally neither a monomorphism nor an epimorphism. This is one way in which the topos map v : Sp Z → Sp F1 is different from one induced by a true map of spaces. For example, if X = W (Spec Q), then (4.1.2) is identified with the map Y Z → Y N , which is rarely a monomorphism. On the other hand, if X = Y = A 1 F1 , then the map is not an epimorphism. Recall that an endomorphism A n → A n is a linear transformation if and only if it is equivariant under the action of G m on A n given by scalar multiplication: Z[x 1 , . . . , x n ] xi →xi⊗z −→ Z[x 1 , . . . , x n ] ⊗ Z Z[z, z −1 ]. Now observe that if we give G m and A n their toric Λ-structures, then this action morphism is Λ-equivariant. Therefore it is reasonable to define Z = Spec Z[b 1 , . . . , b n ]/(b i b j : i = j). Further the operators on Z induced by ψ p take each coordinate b i to b p i . Proof. We just calculate the B-valued points of v * Hom Gm/F1 (A n , A 1 ) for any ring B. We have (v * Hom Gm/F1 (A n , A 1 ))(Spec B) = (Hom Gm/F1 (A n , A 1 ))(v ! (Spec B)). But since Hom Gm/F1 (A n , A 1 ) sits inside Hom Gm (A n , A 1 ), which is affine, any map v ! (Spec B) → Hom Gm/F1 (A n , A 1 ) factors through the affinization Spec W (B) of v ! (Spec B). Therefore we have (v * Hom Gm/F1 (A n , A 1 ))(Spec B) = (Hom Gm/F1 (A n , A 1 ))(Spec W (B)). If we think of points of A n as n-dimensional column vectors, then it is natural to think of the points of Hom Gm (A n , A 1 ) as 1 × n matrices. Given such a matrix (a 1 , . . . , a n ) ∈ W (B) n , the corresponding map in Hom Gm (A n , A 1 )(Spec W (B)) is an F 1 -map if and only if the ring map (4.2.1) ϕ : Z[t] −→ Z[x 1 , . . . , x n ] ⊗ W (B) determined by t → j x j ⊗ a j is a Λ-ring map, where Z[x 1 , . . . , x n ] has the toric Λ-structure. Thus to finish the proof, it suffices to show that necessary and sufficient conditions for ϕ to be a Λ-ring map are, first, that each element a j is the Teichmüller lift of some element b j ∈ B and, second, that b i b j = 0 for all i = j. Let us first show the necessity. So assume ϕ is a Λ-ring map. Consider, for each i, the map π i : Z[x 1 , . . . , x n ] ⊗ W (B) −→ W (B) given by x j → δ ij and by the identity on W (B). Then π i • ϕ is a Λ-ring map under which the image of t is a i . It follows that a i must be a Teichmüller lift. Indeed, giving a Λ-map Z[t] → W (B) is by adjunction the same as giving a ring map Z[t] → B. This in turn is the same as an element of b i ∈ B. Tracing through these identifications shows that a i = [b i ], and thus the necessity of the first condition above. To see the necessity of the second condition, use the operator λ 2 ∈ Λ: 0 = ϕ(0) = ϕ(λ 2 (t)) = λ 2 (ϕ(t)) = λ 2 j x j ⊗ [b j ] = i<j x i x j ⊗ [b i ][b j ]. Thus for i = j, we have [b i b j ] = [b i ][b j ] = 0 and hence b i b j = 0. Sufficiency is similar. Suppose ϕ is defined by ϕ(t) = j x j ⊗ [b j ] with b i b j = 0 for all i = j. To show ϕ is a Λ-ring map, it is enough to show it commutes with λ n for n 2. But we have that λ n (x j ⊗ [b j ]) = 0 for all n 2. Therefore λ n ( j x j ⊗[b j ]) is the n-th elementary symmetric function in the x j ⊗[b j ]. Because b i b j = 0 for i = j, each of the elementary monomials is zero when n 2. Therefore we have λ n (ϕ(t)) = λ n j x j ⊗ [b j ] = λ n (0) = ϕ(0) = ϕ(λ n (t)), for n 2. Thus ϕ is a map of Λ-rings. Corollary. We have the following equalities of subspaces of M n : M n/F1 = Z n and GL n/F1 = S n ⋉ G n m . Proof. The first equality follows from the universal property of products and 4.2: Hom Gm/F1 (A n , A n ) = Hom Gm/F1 (A n , A 1 ) n = Z n . For the second equality, recall that a morphism of Λ-spaces is an isomorphism if and only if it becomes one after applying v * . Therefore we have GL n/F1 = GL n ∩ M n/F1 = S n ⋉ G n m . Corollary. (a) M n/F1 (F 1 ) is the set of n × n matrices with the property that every entry is either 0 or 1 and every row has at most one 1. (b) GL n/F1 (F 1 ) = S n . 4.5. Remarks. Note that none of the F 1 -valued points of GL n/F1 are complemented. Also note that the determinant map GL n/F1 → G m is not a Λ-map, because it fails to commute with ψ 2 . It does commute with the other ψ p though. Recall that we can define cohomology with compact support because, by Nagata's theorem, any separated morphism of finite type between noetherian schemes is compactifiable. (See Conrad [15], theorem 4.1. Actually, according to a forthcoming paper of Conrad-Lieblich-Olsson, the scheme-theoretic hypotheses can be removed.) Proof. The sheaf R n f ! (Z/pZ) is constructible (Deligne SGA 4 1/2 [21] [Arcata] IV (6.2)) and hence lisse away from a finite set T of primes. For m 0, if R n f ! (Z/p m Z) is lisse, then all subsheaves and quotient sheaves of R n f ! (Z/p m Z) are lisse away from T . On the other hand, by the long exact sequence of cohomology, R n f ! (Z/p m+1 Z) is an extension of a subsheaf of R n f ! (Z/pZ) by a quotient sheaf of R n f ! (Z/p m Z) and is therefore also lisse away from T . By induction R n f ! (Z/p m Z) is lisse away from T for all m. Therefore R n f ! (Z p ) and hence R n f ! (Q p ) are lisse away from T . The final statement follows from Deligne, SGA 4 1/2 [21] [Arcata] V (3.1). Cyclotomic representations. Let E be a finite extension of Q p . Let K be a finite extension of Q or Q p , and let V be a d-dimensional continuous E-linear representation of Gal(K/K). Let us say that V is cyclotomic if there exist integers j 1 , . . . , j d such that V is isomorphic to E(j 1 ) ⊕ · · · ⊕ E(j d ). Let us say that V is potentially cyclotomic if there exists a finite extension L of K such that V is cyclotomic as a representation of Gal(K/L). Observe that these concepts are independent of E in the sense that for any finite extension E ′ of E, we have Proof. By the main theorem of p-adic Hodge theory, H ń et,c (XQ, Q p ) is a potentially semi-stable representation of Gal(Q p /Q p ). (This is the work of many people. See [41] for the final form of the theorem.) By 5.1, the set of ramified primes is finite, and if X has a smooth proper model over Z[1/M ], it contains only primes dividing M p. The theorem is then an immediate consequence of 5.4. (5.2.1) E ′ ⊗ E V Theorem. Let E be a finite extension of Q p . Let T be a finite set of prime numbers containing p, and let V be a finite-dimensional representation of G Q satisfying the following properties: (a) G Q acts on V through its abelianization, (b) V is unramified away from T , (c) V is potentially semi-stable at p. Then there is an integer N divisible only by the primes in T such that the restriction of V to Gal(Q/Q(ζ N )) is cyclotomic. Proof. By the Kronecker-Weber theorem, there is an integer n 1, divisible only by primes in T , such that the action of G Q,S on V factors through the cyclotomic character G Q,S → (Z/nZ) * × Z * p . Let H be the torsion subgroup of (Z/nZ) * × Z * p . Let us show that it is enough to consider the case where H acts trivially on V . Let G denote the quotient (Z/nZ) * × Z * p /H. Since G ∼ = Z p , we have (Z/nZ) * × Z * p ∼ = G × H. By (5.2.1), we can assume every irreducible E-linear representation of H is onedimensional. For each character ρ : H → E * , let V ρ denote the summand of V on which H acts via ρ. Since G × H is abelian, V ρ is stable under G × H. Therefore we have a decomposition of Galois representations V = ρ V ρ . It is enough to show each V ρ is potentially cyclotomic. On the other hand, because each V ρ satisfies (a)-(c), it is enough to assume V = V ρ for some ρ. Now observe that ρ, viewed as a Galois representation by the composition G × H −→ H ρ −→ E * , is potentially cyclotomic, since it is potentially trivial. Therefore it is enough to show V ⊗ ρ −1 is potentially cyclotomic. Similarly, ρ −1 satisfies (a)-(c) above, and hence so does V ⊗ ρ −1 . But H acts trivially on V ⊗ ρ −1 . Therefore it is indeed enough to consider the case where H acts trivially on V . Let g be a pro-generator of G/H. By (5.2.1), it is sufficient to assume that E contains all the eigenvalues of g acting on V . Then V has a g-stable filtration 0 = V 0 ⊆ V 1 ⊆ · · · ⊆ V d = V by sub-E-vector spaces such that dim E (V i /V i−1 ) = 1, for i = 1, . . . , d. Because g is a pro-generator of G/H, the filtration is G/H-stable. Because V is potentially semi-stable, so is each subquotient V i /V i−1 . For each i, the lemma 5.5 implies there is finite extension K i of Q p in Q p (ζ p ∞ ) such that the action of Gal(Q p (ζ p ∞ )/K i ) on V i /V i−1 is an integral power of the cyclotomic character. Take an integer r 0 such that Q p (ζ p r ) contains K 1 , . . . , K d . Then the action of Gal(Q(ζ p ∞ )/Q p (ζ p r )) on V i /V i−1 is an integral power of the cyclotomic character. In other words, V is an iterated extension of Gal(Q(ζ p ∞ )/Q p (ζ p r ))representations of the form E(j). But by 5.6, it must then be a direct sum of such representations. Thus it is cyclotomic as a representation of Gal(Q(ζ p ∞ )/Q p (ζ p r )). But because the natural map Gal(Q p (ζ p ∞ )/Q p (ζ p r )) −→ Gal(Q(ζ p ∞ )/Q(ζ p r )) is an isomorphism, V is cyclotomic as a representation of G Q(ζ p r ) . Lemma. Let E be a finite extension of Q p , and let V be a one-dimensional E-vector space with a continuous E-linear action of Gal(Q p (ζ p ∞ )/Q p ). Assume further that V is potentially semi-stable (as a Q p -representation of G Qp ). Then there is a finite extension K of Q p such that for some integer i, we have V ∼ = E(i) as representations of Gal(K(ζ p ∞ )/K). Proof. Let d = [E : Q p ]. Let M be the weakly admissible module associated to V . Then M inherits from V an action of E. That is, every e ∈ E acts as a morphism of weakly admissible modules. Therefore the Hodge filtration on M is a filtration by sub-E-modules and the Frobenius and monodromy operators are E-linear. Because dim E (M ) = dim Qp (M )/d = 1, there is an integer j such that F j M = M and F j+1 M = 0, and there is an element α ∈ E such that the endomorphism ϕ of M is multiplication by α. Therefore the the Hodge number of det Qp (M ) is dj and the slope is dv Qp (α). Because M is weakly admissible, these two must be equal, and so we have v Qp (α) = j. Replacing M by M (−j), it suffices to assume j = 0. We now want to show that M becomes trivial after a finite extension. Let k denote the residue field of K. Let M ′ = M ⊗ W (k) W (k). Then M ′ is a one-dimensional potentially semi-stable weakly admissible E-module over W (k) with j = 0. Let us show that M ′ is isomorphic to the trivial weakly admissible module. Let us now show that we can change basis of M ′ so that α = 1. (This is a consequence of Manin's theorem when E = Q p .) The ring E ⊗ W (k) has an endomorphism id ⊗ σ. The ratio of this with the identity map gives a group endomorphism f of (E ⊗ W (k)) * sending a ⊗ x → a ⊗ σ(x)/x. We need to show this is a surjection. Define a filtration A i of (E ⊗ W (k)) * by setting A i (A ⊗ W (k)) * −→ (E/m i ⊗ W (k)) * , where m is the maximal ideal of E. Because the ring E ⊗ W (k) is m-adically complete, the group (E ⊗ W (k)) * is complete with respect to this filtration. Therefore it is enough to show f is surjective on the associated graded abelian group. We have gr 0 A = (E/m ⊗k) * and for i 1, we have gr i A ≡ (m i /m i+1 ⊗k) by the map x → x − 1. The map f becomes x → x p−1 on gr 0 A, and it becomes a ⊗ x → a ⊗ (x p − x) on gr i A for i 1. Becausek is algebraically closed, both these maps are surjective. Let us now show that the monodromy operator N is zero. But this holds because ϕ and N are two E-linear endomorphisms of a one-dimensional vector space satisfying ϕN = pN ϕ and ϕ = 0. And so the weakly admissible module associated to the G Q⊗W (k) -representation V is trivial. Therefore there is a finite extension L of Q ⊗ W (k) such that G L acts trivially on V . Therefore there is a finite extension K of Q p such that the inertia group I K of G K acts trivially on V . Since the action of G Qp on V factors through Gal(Q p (ζ p ∞ )/Q p ), the action of G K . 5.6. Lemma. Let E and K be finite extensions of Q p , and let V be a finitedimensional continuous E-representation of Gal(K(ζ p ∞ )/K(ζ p b )). If V is Hodge-Tate and has a semi-simplification which is isomorphic to a sum of cyclotomic representations, then V itself is isomorphic to a sum of cyclotomic representations. Proof. Let G denote Gal(K(ζ p ∞ )/K(ζ p b )). Giving a representation of G is the same as giving a representation of its Lie algebra, which is one-dimensional; and so this is the same as giving a matrix θ. We have assumed that θ is upper-triangular with only integers on the diagonal. One can change basis to make θ a blockdiagonal matrix, where each block is upper-triangular and has a single integer on the diagonal. Therefore it is enough to assume θ has is upper triangular and has a single integer on the diagonal. Twisting by a cyclotomic character, we can assume θ is nilpotent. Thus it suffices to assume that V has a trivial semi-simplification. By induction on dim E (V ), we need only prove that any Hodge-Tate extension W of the trivial representation by itself is split. Therefore the representation is given by a group map G → E into the upper-right corner of the matrix. Since the Hodge-Tate weights of W are both 0, it must in fact be C p -admissible. Therefore by Sen's theorem, it factors through finite quotient of G. But the only group map G → E with this property is the trivial map. Therefore V is a split extension. p-adicétale cohomology Let X be a separated Λ-scheme of finite type over Z. Let r m,n denote the Hodge number h m,m of H n c (X an , C), where X an is the complex-analytic space underlying X. (See Deligne [23], (2.3.7), for the definition of the Hodge numbers of a mixed Hodge structure and Deligne [22] for the fact that H n c (X an , C) carries a natural mixed Hodge structure.) Let us write P (t) = m,n (−1) n r m,n t m ∈ Z[t]. Finally, let us fix an algebraic closureQ of Q. Proof. By Deligne [21], [Arcata] IV (6.2), the sheaf R n f ! (Z/sZ) is constructible. Therefore, there exists a finite Galois extension K/Q and an integer N > 0 such that the restriction of R n f ! (Z/sZ) to Spec O K [1/N ] is the constant sheaf associated to an abelian group V . Here, O K denotes the ring of integers of K. By functoriality V has an action of G = Gal(K/Q). Let us also assume that N is a multiple of the discriminant of K. By the base-change theorem, we have an isomorphism H ń et,c (XQ, Z/sZ) ∼ = V of representation of Gal(Q/Q) (which acts on V via the map to G). Therefore it is enough to show the action of G on V is through its abelianization. Let p be a prime not dividing N . Let p be a prime of K over p, let k p denote O K /p, and let D denote the decomposition subgroup of G corresponding to p. Because p ∤ N , the map D → Gal(k p /F p ) is an isomorphism. By the proper base-change theorem, we have an isomorphism V ∼ = H ń et,c (XF p , Z/sZ) of representations of Gal(F p /F p ) . Further, the endomorphism ψ * p of V (induced by the endomorphism ψ p of X) corresponds to the endomorphism Fr * p of the righthand side. By generalities about the Frobenius map, Fr * p acts on H ń et,c (XF p , Z/sZ) as the inverse of the residual arithmetic Frobenius element Frob p ∈ D/I, the automorphism defined by Frob p (x) = x p . Therefore ψ * p acts on V in the same way as Frob −1 D where Frob D is the element of D mapping to Frob p . On the other hand, the endomorphisms ψ p of X commute with each other as p varies. Therefore the endomorphisms ψ * p of H ń et,c (XQ, Z/sZ) commute with each other. By the above, the action of any Frobenius elements Frob D for p ∤ N , commute with each other. By Chebotarev's theorem (see Neukirch [49], V (6.4)), every element of G is such a Frobenius element. Therefore G acts on V through its abelianization. Corollary. For any prime number p, there is an integer N 1 such that there is an isomorphism of representations of Gal(Q/Q(ζ N )): (6.2.1) H ń et,c (XQ, Q p ) ∼ = m Q p (−m) rm,n . If X is smooth and proper over Z[1/M ], for some integer M > 0, then N can be taken such that all its prime divisors are divisors of M p. Let us call an integer N > 0 satisfying the conclusion of 6.2 a conductor of X. Proof. By 6.1 and 5.3, there exists an integer N 1 such that H ń et,c (XQ, Q p ) is a cyclotomic representation of Gal(Q/Q(ζ N )). The fact that the multiplicities are given by Hodge numbers as shown is because H ń et,c (XQ, Q p ) is potentially semistable. See Kisin [41] (3.2). Last, when X is smooth and proper over Z[1/M ], the Galois representation H ń et,c (XQ, Q p ) is unramified at primes not dividing M p, by 5.1. Then 5.3 implies that N can be taken as asserted. 6.3. Remark. Note that the cohomology of the nodal curve of 2.6 when q = −1 is non-pure mixed Tate, but the extension class is q = −1 ∈ Q * , which being torsion vanishes when coefficients are taken in Q p . Therefore the Q p -cohomology is pure mixed Tate, and there is no contradiction. On the other hand, the previous theorem would be false with cohomology with coefficients in Z 2 . It would be interesting to see which other mixed Tate motives with torsion classes can be realized in Λ-algebraic geometry. 6.4. Corollary. Let N be a conductor for X. Then there is a finite set T of prime numbers such that for any finite field k whose cardinality q is relatively prime to every element of T and satisfies q ≡ 1 mod N , the number of k-valued points of X is P (q). More precisely, the set T can be taken such that it contains only prime numbers p with the property that for every prime ℓ = p the sheaf R n f ! (Q ℓ ) is not lisse at p, where f denotes the map X × Spec Z[1/ℓ] → Spec Z. Proof. Fix a prime number ℓ = p. Let T denote the set of prime numbers at which R n f ! (Q ℓ ) is not lisse. By 5.1, T is finite. Because of the restrictions on q, we have a factorization Spec k a −→ Spec Z[ζ N , T −1 ] b −→ Spec Z. Let D a denote a decomposition group in Gal(Q/Q(ζ N )) at the point a, and let k denote the corresponding algebraic closure of k. Then since b * R n f ! (Q ℓ ) is lisse, the proper base change theorem implies H n (Xk, Q ℓ ) ∼ = H n (XQ, Q ℓ ) as representations of D a , where D a acts on the left side through Gal(k/k). In particular, the trace of the geometric Frobenius element F ∈ Gal(k/k) on H n (Xk, Q ℓ ) agrees with the trace of the inverse of an arithmetic Frobenius element Frob a of Gal(Q/Q(ζ N )) on H n (XQ, Q ℓ ). Therefore by the Lefschetz fixed-point formula (Houzel, SGA 5, exp. XV [2]) and 6.2, the number of F q -valued points of X is Proof. The representation H ń et,c (XQ, Q p ) of Gal(Q/Q) is unramified away from M , and so there is an integer N satisfying the property of 6.2 and such that it has the same prime divisors as M . Now let q > 1 be a prime power with q ≡ 1 mod N . Let p be the prime number dividing q, let ℓ = p be another prime number. Observe that p ∤ M ℓ. n (−1) n tr(F \| H ń et,c (XF q , Q ℓ )) = n (−1) n tr(Frob −1 a \| H ń et, The map f : X × Spec Z[1/ℓ] → Spec Z[1/M ℓ] is smooth and proper, and so by Deligne (SGA 4 1/2, [Arcata] V (3.1) [21]), the sheaf R n f * (Q ℓ ) is lisse. In particular, it is lisse at p. Therefore by 6.4, there is a set of primes T not containing p such that the conclusion of 6.4 holds. Since q ≡ 1 mod N , the number of F q -valued points of X is P (q). 6.6. Corollary. Let X be a smooth proper scheme over F 1 . Then for all prime powers q > 1, the number of F q -valued points of X is P (q). 6.7. Remark. The number of complemented F 1 -valued points of X equals P (1) when X = A n or X = P n , with their toric Λ-structures. On the other hand, if X is the Chebychev line (2.5), it has no complemented F 1 -valued points, but P (1) = 1. 6.9. Proposition. Let X be a nonempty separated Λ-scheme of finite type over Z. Let P (t) and N be as in 6.1. Then for every prime p ≫ 0 and every integer r 1 with p r ≡ 1 mod N , there is an F q -valued point of X. Proof. By 6.4, for every prime p not in set T supplied and for every integer r with p r ≡ 1 mod N , the number of F p r -valued points of X is P (p r ). On the other hand, it follows from general facts about Hodge numbers of varieties proved by Deligne [24], (8.2.4), that P (t) → ∞ as t → ∞. Indeed, since X C is nonempty, its dimension d is non-negative, the degree of P (t) is 2d, and the coefficient of t 2d is the number of connected components of X C , which is positive. Thus for sufficiently large p and all r as above, we see that the number of F p rvalued points is P (p r ) and that P (p r ) is positive. 6.10. Theorem. Let X be a nonempty separated Λ-scheme of finite type over Z. Then there is a nonempty closed Λ-subscheme Z of X which isétale over Z. Proof. Let us first reduce to the case where X is reduced and quasi-finite over Z. Define a sequence X 0 ⊇ X 1 ⊇ · · · of nonempty closed sub-Λ-schemes of X recursively as follows: Let X 0 = X. For n 0, assume X n has already been defined. Then let p = p n+1 be a prime number distinct from p 1 , . . . , p n such that X n has an F p -valued point x. This exists by 6.9. (For definiteness, we can take p to be the smallest such prime, say.) Let X n+1 be the fixed locus of X n under ψ p , which is to say the equalizer in the category of Λ-spaces of ψ p : X n → X n and the identity map. Because X n is separated, X n+1 is a closed sub-Λ-scheme of X n . Because x is an F p -valued point of X n , it is fixed by the Frobenius map on the special fiber of X n over p. Therefore it is fixed by ψ p , and so x is also a point of X n+1 . In particular, X n+1 is nonempty. Let Z = n 0 X n . Because X is of finite type over Z, it is noetherian. Therefore there is an integer n 0 such that Z = X n . It follows that Z is nonempty and, by 3.2, it is also affine and quasi-finite over Z. Therefore we can assume X = Z = Spec B, where B is quasi-finite over Z. Since the reduced subscheme of any nonzero Λ-ring of finite type over Z is the same, we can also assume B is reduced. Now let us show that B has a quotient Λ-ring which isétale over Z. Suppose B is notétale over Z at some prime p. For each integer m 1, let I m denote the kernel of ψ m : B → B. The Λ-ideals I m are ordered by divisibility on m, and the ordering is cofinal. Let I denote the Λ-ideal ∪ m I m , and let C denote the quotient of B/I by the ideal of torsion elements. Then C is a Λ-ring quotient of C and, hence, of B. Note that 1 ∈ I, so B/I is a nonzero Λ-ring. But 1 is not a torsion element in any nonzero Λ-ring. Therefore C is nonzero and is flat, quasi-finite, and of finite type over Z. Let us finally show that C is actuallyétale over Z. For each integer m 1, the endomorphism ψ m of C is injective. Indeed, if b is a lift to B of any element of the kernel, then nψ m (b) = 0 for some integer n 1. Therefore ψ m (nb) = 0 and hence nb = 0 and hence b is torsion. Therefore the image of b in C is 0. Now let p be a prime. Since ψ p is an injective endomorphism of C, it induces an injective endomorphism of Q⊗ Z C. Since Q⊗ Z C is finite over Q, this endomorphism is in fact an automorphism of finite order. Since C is flat over Z, we have C ⊆ Q⊗ Z C, and so ψ p is an automorphism of C. Therefore the Frobenius endomorphism of C/pC is an automorphism. Thus C/pC is reduced, and so C isétale at p. 6.11. Corollary. Let X be a nonempty proper scheme over F 1 . Then X(F 1 ) = ∅. Proof. The scheme Z supplied by 6.10 is proper andétale over Z. Therefore each of its finitely many connected components must be Spec Z, by Minkowski's theorem. But the only Λ-structure on such a space is the disjoint-union Λ-structure. Indeed, it is flat over Z, so it suffices to show that every Frobenius lift ψ p is the identity. But the Frobenius map on each special fiber is the identity. Since Z is a disjoint union of copies of Spec Z, each ψ p must be the identity. Therefore Z, as a nonempty disjoint union of copies of Spec F 1 , has an F 1 -valued point, and hence so does X. 6.12. Corollary. Let U a nonempty open Λ-subscheme of a Λ-scheme X which is proper over Z. Then U has a Q-valued Λ-point. Proof. By the theorem above, there is a nonempty closed Λ-subscheme Z of U which isétale over Z. Let Y denote the closure of Z in X with the reduced subscheme structure. Then Y is a closed Λ-subscheme of X. (Basic property of Λ-ideals.) Because Y is reduced, it is flat. Because it is the closure of Z it is generically finite over Z. A closed subscheme of X, it is therefore finite over Z. By 6.11, it has an Z-valued Λ-point, and hence a Q-valued Λ-point. Since we have Spec Q × Spec Z Y = Spec Q × Spec Z Z ⊆ Spec Q × Spec Z U, we see that U has a Q-valued Λ-point. 6.13. Remark. The condition that such a compactification X exists cannot be dropped. For example, for any integer n > 0, we can make Z[ζ n , 1/n] a Λ-ring by taking ψ p to be anything for p \| n, and to be the unique choice ζ n → ζ p n for p ∤ n. But this ring has no maps to Q. 6.14. For any F 1 -valued point x of X, let Z x denote the closure of the pre-images of x under the maps ψ p , viewed as a reduced closed subscheme. Then Z x is a complemented closed Λ-subspace of X. Therefore the assignment x → Z x defines a function 6.15. Corollary. Let X be a proper Λ-scheme over Z. Assume that X is irreducible as a scheme. Then there is a Z-valued Λ-point x such that Z x = X. Proof. Since the set X(F 1 ) is finite and complementary closed Λ-subschemes are stable under finite union, the union Z = x∈X(F1) Z x , with its reduced scheme structure is a complementary closed Λ-subscheme. By construction, there are no Q-valued Λ-points of X − Z. Therefore by 6.12, this is only possible if X = Z. Finally since Z is irreducible, we must have Z x = X for some x ∈ X(F 1 ). Variants One of the motivations behind work on the field with one element has been to imitate over number fields the theory of function fields over a finite field k, where we can work over the absolute point Spec k. But Λ-algebraic geometry works perfectly well over function fields, too. So we can compare k-algebras to the function-field analogues of F 1 -algebras. 7.1. Λ S,E -spaces. Let S be a scheme of finite type over Z, and let E be a set of regular closed points of codimension 1. For each point s ∈ E, let q s denote the cardinality of s. Let X be a flat algebraic space over S. We can then define a Λ S,Eaction on X just as we define Λ-actions, but now we use commuting endomorphisms ψ s : X → X, one for each point s ∈ E, such that ψ s agrees with the q s -power Frobenius operator on the fiber X s . (See [7].) If E is the set of all regular closed points of codimension 1, then we write Λ S = Λ S,E . For example, if S = Spec Z, then a Λ S -space is just a Λ-space. The general theory works just as well when S is arbitrary. Thus we get a topos Sp F S,E Since S × k T is flat over S, giving a Λ S -action on S × k T , is the same as giving a commuting family of Frobenius lifts. For any maximal ideal m of O S , define ψ m : S × k T → S × k T by ψ m = id S × Fr m T , where Fr m T denotes the endomorphism of T which on O T acts as x → x qm , where q m denotes the cardinality of the residue field O S /m. It is clear the ψ m commute with each other and lift the appropriate Frobenius maps. The functor Aff S → Sp F S 1 just defined preserves covering families. Indeed, a map U → V in Sp F S 1 is an epimorphism if and only if the induced map v * U → v * V is. But v * f * preserves covering families. Therefore f * does. For a similar reason, f * sends products to products. Therefore it extends uniquely to a topos map f : Sp F S 1 → Sp Spec k , which yields the commutative diagram (7.2.1). In fact, f is essential, meaning that f * has a left adjoint, denoted f ! . We let f ! (U ) be the colimit of the coequalizers of the diagrams U Fr m U / / ψm / / U over all m. In other words, f ! (U ) is the largest quotient U ′ of U on which each ψ m acts as Fr m U ′ . This is clearly the left adjoint of f * . f is not an isomorphism of toposes. For example, let T be an affine space over S, the unit of the adjunction f ! ⊣ f * at v ! (T ) is (7.3.1) v ! (T ) −→ f * f ! v ! (T ) = f * s ! (T ) which, by definition, is a map (7.3.2) W S (T ) −→ S × k T, where W S := W S,E is the E-typical Witt vector functor over S. (See [7].) The composition N (E) T γ −→ W S (T ) −→ S × k T. with the ghost map γ is the map that, on the component n ∈ N (E) , is simply T (pr,Fr n T ) / / S × k T where pr denotes the structure map T → S, and Fr n T is the Frobenius map defined on functions by x → x q deg(n) , where deg(n) denotes m n m [O S /m : k], the degree of the effective divisor corresponding to n. In particular, if T = S, then the image of this map is the union in S × k S of the graphs of all powers of the Frobenius map on S. These components are not disjoint. For instance, let x and y be two distinct closed points of S with the same residue field. Then the the components of W S (S) of indices x and y are distinct but have the same image in S × k S. So (7.3.1) is not a monomorphism. Therefore f ! is not faithful, and hence f is not an isomorphism of toposes. One can also show that (7.3.1) is not an epimorphism. But the image of (7.3.2), or equivalently (7.3.1), does see much of the geometry of S × k T . Another way of expressing the point of this paper is that it is reasonable to think of it as being almost an isomorphism. For example, we have the following result. Proposition. The map W S (T ) −→ S × k T of (7.3.2) has Zariski dense image. Proof. Let m be a maximal ideal of O S with residue cardinality r. It suffices to show the image of the composition N × T −→ N (E) × T −→ W S (T ) −→ S × k T is dense. The map satisfies (n, t) → h(t), Fr r n (t) . We may assume S and T are affine. Indeed, it suffices to show density locally. Therefore we can replace S with an affine open subscheme S ′ and replace T with T ′ = S ′ × S T . We can then replace T ′ with an affine scheme T ′′ mapping to T ′ by anétale map. Let us then write S = Spec R and T = Spec B, where R is a Dedekind domain and B is an R-algebra. In terms of rings, the map in question is (7.4.1) R ⊗ k B −→ B × B × . . . and is defined by a ⊗ b → ab, ab r , ab r 2 , . . . . We need to show that any element of its kernel is nilpotent. So let d j=1 a j ⊗b j be an element of its kernel. Assume without loss of generality that the elements a j are linearly independent over k. Then for every m ∈ N, we have Since the family a 1 , . . . , a d is linearly independent, the matrix a r i j has nonzero determinant. (This is the Moore matrix from the theory of function fields. The vanishing of its determinant is equivalent to the linear dependence of the a j . The proof is by a degree argument, just as for the familiar, analogous result about Vandermonde matrices.) Therefore we have b r d j = 0 for all j. It follows that the element d j=1 a j ⊗ b j r d is zero. So every element of the kernel of (7.4.1) is nilpotent. 7.5. Corollary. Let X and Y be separated reduced algebraic spaces over k. Then any map f ! f * X → Y factors uniquely through the map f ! f * X → X. Proof. Consider the composite map S × X −→ f ! f * X −→ Y, and let Γ = (S × X) × Y (S × X) denote the induced equivalence relation on S × X. Recall that f ! f * X is defined to be the quotient of S × X by the equivalence relation Γ ′ generated by the image of W (S × X) in S × S × X under the map (7.3.2) when T = S × k X. Therefore Γ contains Γ ′ . On the other hand, by 7.4, Γ ′ is dense in Γ ′′ = S × S × X = (S × X) × X (S × X). Therefore Γ agrees with Γ ′′ , and so the map S × X → Y factors uniquely through the quotient of S × X by Γ ′′ , which is just X. 7.6. Corollary. The functor f * : Sp k → Sp F S 1 is faithful, and it embeds the full subcategory of separated reduced algebraic spaces over k fully faithfully in the category Sp F S 1 . Some restriction on nilpotent elements is necessary. For example, suppose S = Spec R. Let A be a non-reduced k-algebra. Choose a square-zero element a ∈ A and an element r in R but not in k. Then the element r ⊗ a of R ⊗ k A does not lie in the subring A. On the other hand, we have (r ⊗ a) qm = 0 = r ⊗ a qm = ψ m (r ⊗ a) for any maximal ideal m of R. Therefore the map f ! f * Spec A → Spec A is not an isomorphism, because r ⊗ a is a function on the affinization of f ! f * Spec A that does not come from A. 7.7. Analogy. It is rare for the pull-back functor for a map of spaces to be fully faithful. So let us consider a similar, but more familiar situation where this happens. Let S be a complex algebraic space and let Γ be an equivalence relation on S which is Zariski dense in S × S. For instance, Γ could be given by the action of a discrete group with a dense orbit or, if S is connected, by the formal neighborhood of the diagonal. Then algebraic spaces (perhaps under some mild conditions) form a full subcategory of the category of Γ-equivariant spaces over S. The condition for a Γ-equivariant algebraic space X over S to descend to the point is then a property on X, rather than a structure. We might then say the Γ-action is uniform, or constant. Therefore, following the previous corollary, it is natural to interpret a descent datum on a reduced algebraic space X over S to the point Spec k as being a descent datum to F S 1 with a similar algebraic uniformity property. So, objects of F S 1 are generalized-but not weakened-versions of separated reduced algebraic spaces over the point Spec k. Of course this makes essential use of equal characteristic. The corresponding interpretation of Λ-spaces in the usual sense, over Z, would be that while it is possible to say what it means to descend an algebraic space to F 1 -that is, to give it a Λ-action-we do not know if there is a uniformity property, which is what we would need to create a true arithmetic analogue of the base point Spec k. (Buium has come to similar ideas independently. He proposed in conversation that it might be reasonable to consider a Λ-structure on a scheme as being an isotrivialization relative to F 1 .) 7.8. Drinfeld modules. Using the theory of Drinfeld modules, we can give examples of objects of Sp F S 1 that do not descend to Sp k . Let C be a connected smooth projective curve over F p , let ∞ ∈ C be a closed point, and let A = Γ(C − {∞}, O C ). Let S be a smooth F p -curve over which there is a Drinfeld A-module ϕ : A → End S (G a ) of rank 1 and of generic characteristic. (See Laumon [46] section (1.2), say.) Then for any closed point s of S, the fiber of ϕ over s gives a Drinfeld R-module ϕ s over s of rank 1. Because of the assumption that ϕ has generic characteristic, the characteristic of ϕ s is s. A basic result of Drinfeld's ( [46] (2.2.2)(ii)) then implies that there is a unique element Π s ∈ A such that ϕ s (Π s ) is the q s -th power Frobenius endomorphism of G a over s, where q s denotes the residue cardinality of s. If we set ψ s = ϕ(Π s ), then the various ψ s are commuting endomorphisms of G a over S, each agreeing with the q s -power Frobenius map on the fiber over s. This gives a Λ S -structure on G a (which also respects the group structure). For example, the Carlitz module is defined when A = F p [t] and S = Spec A by ρ(t) = t + τ , where τ is the Frobenius map of G a . Then for each maximal ideal m of k[t], the operator ψ m is ρ(f (t)), where f (t) denotes the monic generator of m. Observe that none of these "Drinfeld Λ S -structures" on A 1 S descends from Sp F S 1 to Sp k . Indeed, for every object in the image of f * , the operators ψ s act as zero on the conormal sheaf of the identity section S ⊂ G a . But ϕ was assumed to have generic characteristic. Therefore every Π s acts faithfully on the conormal sheaf, and hence so does every ψ s . Another important use of the construction S × X is in the study of shtukas, also due to Drinfeld. Indeed, it is possible to mimic this using W S (X) instead of S × X. And this concept can be translated to number fields. This paper is not, however, the place to discuss this in any detail. 7.9. Complex multiplication by number fields. Let R be a Dedekind domain whose field of fractions is a number field. Assume that there is an abelian scheme X over R of dimension d having the property that Q ⊗ End R (X) contains a field F of degree 2d over Q. As above, X has a natural Λ R -structure. For each maximal ideal m of R, there is a unique element π m ∈ F ∩ End R (X) such that π m induces the q m -th power Frobenius map on the fiber of X over m. (See Serre-Tate [54].) Because each π m lies in F , they all commute. Therefore putting ψ m = π m is a Λ R -structure on X. Observe that we can modify X to make Λ R -varieties that are not CM varieties in the usual sense. For instance, let G is a finite subgroup of Aut R (X). Because every automorphism commutes with the complex multiplications, G acts Λ R -equivariantly on X. Therefore the quotient X/G is also a Λ R -space. (Because G is finite, the quotient is an algebraic space, by Artin's theorem.) For instance, if X is an elliptic curve and G = Aut(X), then X/G is a projective line. In particular, it seems likely that explicit class field theory for imaginary quadratic fields could be expressed in terms of Λ R -structures on P 1 , just like in the case of Q and function fields. Suppose instead that R is a Dedekind domain whose field of fractions is a real quadratic number field. In light of Ritt's work [51] [52], it seems unlikely that there are Λ R -actions on P 1 which do not come from Λ Z -actions. It might, however, be possible to find such Λ R -actions on surfaces or higher-dimensional varieties, and any example would without a doubt lead to another example of an explicit class field theory. Of course, it would only be interesting if it could see more than the maximal cyclotomic extension of R. On the other hand, it would also be interesting to prove that no such examples exist. Here are some precise questions. Is there an algebraic Λ R -space X of finite type over R with the property that for any abelianétale R-algebra R ′ , the Λ R -space Spec R ′ × Spec R X is not isomorphic as a Λ R -space to one of the form Spec R ′ × Spec Z Y , for any algebraic Λ Z -space Y ? (Abelian here means that Q ⊗ Z R ′ is a product of abelian extensions of Q ⊗ Z R.) Are there any algebraic Λ R -spaces X of finite type over R whose generic fiber is geometrically connected and which do not descend to Λ Z -spaces? I do not even know if many particular varieties can be ruled out: are there any Λ R -structures on X = P 2 R with this property? 7.10. Local number fields. Let S = Spec Z, let p be a prime number, and let E = {p}. Let us write Λ S,E = Λ p . The corresponding Witt functor W * n = W * S,E,n is, up to re-indexing, the p-typical Witt vector functor defined by Witt [65] in 1936. Of course, Λ p -rings and p-typical Witt vectors are now ubiquitous in work on p-adic cohomology. (See [17] or [37], say.) Now let A be a complete discrete valuation ring with perfect residue field k of characteristic p. If A is of equal characteristic, then the map A → k has a unique section. Now suppose A has mixed characteristic. Of course A → k cannot have a section defined over Z, but remarkably, it does have a unique section defined over F S,E 1 , the p-typical field with one element, and hence over F 1 . By definition, this means that the map W (A) → W (k) of Λ p -rings has a unique section. Indeed, because k is perfect, there is a unique ring map W (k) → A compatible with the projections to k. (See [53], say.) By adjointness, this then lifts to a unique map W (k) → W (A) of Λ p -rings, which is easily seen to be a section of the map in question. A 1 are in bijection with Hom Λ (Z[x], Z), which agrees with {x ∈ Z \| x p = x for all primes p} = {0, 1}.So 0 and 1 are the only F 1 -valued points of the toric A 1 . The point 0 is complemented, because its complement is the toric G m . But the point 1 is not complemented, because the preimage of 1 under ψ 2 , say, is µ 2 , which is not contained in {1}. More generally, the set of F 1 -valued points of A d is {0, 1} d , but the only complemented point is the origin. 3. 7 . 7Proposition. The Λ-space G d m is primitive, for any integer d 0. For example, the only complemented F 1 -valued point of the toric A n is the origin. The toric P n has exactly n+1 complemented points with values in F 1 . They are [1, 0, . . . , 0], [0, 1, 0, . . . , 0], and [0, . . . , 0, 1]. These facts have been predicted in earlier speculations on the field with one element [60], p. 285. n/F1 = Hom Gm/F1 (A n , A n ),where for general F 1 -spaces X, Y with G m -actions, we define Hom Gm/F1 (X, Y ) to be the sub-F 1 -space of Hom F1 (X, Y ) consisting of the G m -equivariant maps. Let us also set(4.1.4) GL n/F1 = Aut Gm/F1 (A n ),to be locus of M n/F1 consisting of invertible maps. Of course, M n/F1 is a monoid object in Sp F1 and GL n/F1 is a group object. In 4.3 below, we will describe them concretely. Note that because v * is faithful, we can view v * Hom Gm/F1 (X, Y ) as a subspace of Hom Gm (v * X, v * Y ). Consider this when X = A n and Y = A 1 , both with the toric Λ-structures. Then for any ring B, a B-valued point of Hom Gm (A n , A 1 ) is just a B-module homomorphism B n → B, or simply a 1 × n matrix (b 1 , . . . , b n ) with entries in B. This furnishes an identification(4.1.5) Hom Gm (A n , A 1 ) = Spec Z[b 1 , . . . , b n ] = A n 4.2. Proposition. Under the identification (4.1.5), the subspace of A n corresponding to the subspace Hom Gm/F1 (A n , A 1 ) of Hom Gm (A n , A 1 ) is the union of the axes 5 . 5Aside: abelian motives are potentially cyclotomic Let p be a prime number. The purpose of this section is to establish the result (5.3) in usual, non-Λ algebraic geometry that abelian p-adic Galois representations of geometric origin are Artin-Tate. The proof is an result in p-adic Hodge theory. It is well within the scope of established techniques, but to my knowledge, it is not actually in the literature. For results of a similar flavor, see Wang [63], Kisin-Lehrer [40], and van dan Bogaart-Edixhoven [62]. 2 5.1. Proposition. Let X be a separated scheme of finite type over Z[1/p]. Let f : X → Spec Z[1/p] denote the structure map. Then R n f ! (Q p ) is lisse away from a finite set of primes. If X is smooth and proper over Z[1/M ], then the sheaf is lisse at all primes not dividing M . 6. 1 . 1Theorem. For any integer s > 0, the action of Gal(Q/Q) on H ń et,c (XQ, Z/sZ) factors through Gal(Q/Q) ab . ,n q m = P (q).6.5. Corollary. Let X be a smooth proper Λ-scheme over Z[1/M ]. Then there is an integer N divisible only by the primes dividing M such that for all prime powers q > 1 with q ≡ 1 mod N , the number of F q -valued points of X is P (q). 6. 8 . 8Remark. Some conditions on q of the kind in 6.4 are necessary. For example, let X = Spec Z[ζ N , 1/N M ]. For primes p dividing N M , let ψ p = id; and for all other p, let ψ p be the unique Frobenius lift. Then r 0,0 = φ(N ) and all other r m,n are zero. Therefore P (t) = φ(N ). On the other hand, there are φ(N ) points in X(F q ) if and only if both N M ∈ F * q and q ≡ 1 mod N . Furthermore the sheaf R n f ! (Q ℓ ) is lisse exactly at the prime numbers that do not divide N M . {F 1 1-points of X} −→ {complemented reduced closed Λ-subspaces of X}. various powers of Fr r , we have the system of equations d j=1 a r i j b r d j = 0 where i = 0, . . . , d − 1. This can be expressed as the matrix equation a r i j ij · b r d j j = 0. ): 0.3. Corollary. Let X be a smooth proper scheme over Z[1/M ], for some integer M 1. If X descends to F 1 , then there is an integer N 1 divisible only by the primes dividing M such that for all prime powers q > is (potentially) cyclotomic if and only if V is. 5.3. Theorem. Let X be a separated scheme of finite type over Q. Suppose that Gal(Q/Q) acts on H ń et,c (XQ, Q p ) through its abelianization. Then there is an integer N 1 such that the restriction of H ń et,c (XQ, Q p ) to Gal(Q/Q(ζ N )) is cyclotomic. If X has a smooth and proper model over Z[1/M ], then N can be taken such that all its prime divisors are divisors of M p. The style of this section is somewhat clumsy. I hope to improve it in a future version. Proof. This is 6.5 in the case M = 1. 1, whose objects we call spaces over the generalized field with one element F S,E 1 . If S ′ and E ′ are another instance of this data, and a : S → S ′ is a map such that a(E) ⊆ E ′ , we have a diagram of toposesrendered commutative by a certain invertible 2-morphism. Here b is as in[7].In particular, Sp F1 is the deepest.Function fields.Let q denote the cardinality of k, and let S be a smooth geometrically connected curve over k. We will now construct a factorization of topos mapsLet us first define f * (T ) for affine (or algebraic) T . As a space, set f * (T ) = S × k T. Séminaire de Géométrie Algébrique du Bois-Marie. M Artin, A Grothendieck, J L Verdier, Théorie des topos et cohomologieétale des schémas. Tome 1: Théorie des topos. BerlinSpringer-Verlag269Lecture Notes in MathematicsThéorie des topos et cohomologieétale des schémas. 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[ "Concurrent Training of a Control Policy and a State Estimator for Dynamic and Robust Legged Locomotion", "Concurrent Training of a Control Policy and a State Estimator for Dynamic and Robust Legged Locomotion" ]	[ "Gwanghyeon Ji \nIEEE Copyright Notice\n\n", "Juhyeok Mun \nIEEE Copyright Notice\n\n", "Hyeongjun Kim \nIEEE Copyright Notice\n\n", "Jemin Hwangbo \nIEEE Copyright Notice\n\n" ]	[ "IEEE Copyright Notice\n", "IEEE Copyright Notice\n", "IEEE Copyright Notice\n", "IEEE Copyright Notice\n" ]	[]	In this paper, we propose a locomotion training framework where a control policy and a state estimator are trained concurrently. The framework consists of a policy network which outputs the desired joint positions and a state estimation network which outputs estimates of the robot's states such as the base linear velocity, foot height, and contact probability. We exploit a fast simulation environment to train the networks and the trained networks are transferred to the real robot. The trained policy and state estimator are capable of traversing diverse terrains such as a hill, slippery plate, and bumpy road. We also demonstrate that the learned policy can run at up to 3.75 m/s on normal flat ground and 3.54 m/s on a slippery plate with the coefficient of friction of 0.22.	10.1109/lra.2022.3151396	[ "https://arxiv.org/pdf/2202.05481v2.pdf" ]	246,822,591	2202.05481	5d5812c5236c5fc576ba308dd9ccbd573f5dd7ef	Concurrent Training of a Control Policy and a State Estimator for Dynamic and Robust Legged Locomotion 2 Mar 2022 Gwanghyeon Ji IEEE Copyright Notice Juhyeok Mun IEEE Copyright Notice Hyeongjun Kim IEEE Copyright Notice Jemin Hwangbo IEEE Copyright Notice Concurrent Training of a Control Policy and a State Estimator for Dynamic and Robust Legged Locomotion 2 Mar 202210.1109/LRA.2022.3151396This paper has been accepted for publication in IEEE Robotics And Automation Letters (RA-L). In this paper, we propose a locomotion training framework where a control policy and a state estimator are trained concurrently. The framework consists of a policy network which outputs the desired joint positions and a state estimation network which outputs estimates of the robot's states such as the base linear velocity, foot height, and contact probability. We exploit a fast simulation environment to train the networks and the trained networks are transferred to the real robot. The trained policy and state estimator are capable of traversing diverse terrains such as a hill, slippery plate, and bumpy road. We also demonstrate that the learned policy can run at up to 3.75 m/s on normal flat ground and 3.54 m/s on a slippery plate with the coefficient of friction of 0.22. I. INTRODUCTION In recent years, reinforcement learning (RL) has become one of the most popular control approaches for legged robots. For quadrupedal robots, there have been remarkable improvements in learning dynamic locomotion skills. Hwangbo et al. [1] trained control policies for the ANYmal robot [2] for robust and high-speed locomotion while keeping the balance under large disturbances. In the later works [3] and [4], RL-trained policies made a quadrupedal robot traverse over various challenging terrains such as slippery ground, vegetation, and rocky terrain. They trained an encoder which compresses environmental information and enabled effective environmentaware locomotion. Moreover, Peng et al. [5] reproduced agile motions of animals by imitating recorded motion trajectory data. They made the Laikago robot walk and turn at moderate speed. More complicated trained behaviors, such as dynamic recovery from a fall [1], [6], have been reported in the literature. These complex behaviors can be composed of a single framework using pre-trained expert networks and a gate neural network, and manifest agile and effective motions [7] on the real robot. Furthermore, RL can be utilized for bipedal robots to climb up stairs [8] or to display diverse locomotion patterns such as standing, walking, and running [9]. The existing control approaches for quadrupedal locomotion rely on accurately estimated state input [1], [3], [4], [7], [10]- [13]. However, we observed that existing state estimation algorithms become unreliable [14], [15] on challenging terrains, such as ice and sand. Moreover, many state estimation algorithms, such as the one built in to the Mini Cheetah robot, require gait patterns a priori. Most neural network control policies often do not provide such information because the patterns are learned as well. Alternatively, contact states can be estimated either from dedicated contact sensors or from the model [16], but the former is computationally costly and the latter is prone to permanent damages during foot landing. Therefore, it is desirable to develop a control framework that does not rely on contact information. In addition, to walk and run on challenging terrains blind, information about the terrain must be estimated. An analytical method can be employed [10] to estimate part of the information. Alternatively, it can be estimated implicitly using a trained neural network and proprioceptive state history [3], [4]. The proprioceptive state history is useful for estimating both intrinsic robot states and extrinsic environment variables. However, this approach has two different major drawbacks. First, because the latent vectors are not interpretable, they cannot be used in conjunction with other modules that require state information. Second, the encoder training causes a significant computational overhead. An alternative to this approach is to directly estimate observable state variables such as the terrain angle. In our proposed approach, this information is indirectly estimated as a distance from the terrain to the foot. To address the aforementioned shortcomings of the existing methods, we present a learning-based state estimation network, which is concurrently trained with the policy network. The efficacy of our method is demonstrated using the Mini Cheetah robot [17], which is a lightweight and highly dynamic quadrupedal robot. Kim et al. [11] reports an MPC-based controller that could make Mini Cheetah run at up to 3.7 m/s on a treadmill and [17] achieves 2.45 m/s in outdoor environments. We hereby report highly dynamic locomotion at 3.74 m/s in various indoor and outdoor environments and robust and reliable locomotion behaviors on a slippery plate, bumpy asphalt road, and hills. Our main contributions are as follows: • We propose a simple end-to-end locomotion learning framework that concurrently trains a control policy and a state estimator. • Using the trained networks, we demonstrate dynamic locomotion on slippery terrains and slopes. • We share the training details, such as the dynamic randomization and curriculum, so that our work can be reproduced by other researchers. II. METHOD Our goal is to develop an RL-based control framework that can follow the given velocity command, which consists of desired base linear velocities in the forward and lateral directions, and the desired yaw rate. We assume that the robot is equipped with an Inertial Measurement Unit (IMU) and joint encoders. An overview of our control framework is illustrated in Fig. 2. The framework consists of three different neural networks: the estimator, critic, and actor. The estimator network estimates multiple relevant state variables for control using the onboard sensors and feeds them to the actor network which outputs actuator commands. The critic network helps reduce variance in the policy gradient estimate from the RL algorithms. All neural networks are trained in simulation using RaiSim [18]. We use Proximal Policy Optimization (PPO) [19] for training the actor and critic, and supervised learning for training the estimator network. After a collection of a batch of trajectories in RaiSim, we update all three networks using their corresponding loss function. This process repeats like in the vanilla PPO until the performance metric converges. This concurrent learning of the two networks ensures that the policy network can adapt to the performance characteristics of the estimator network. For example, when the estimation is unreliable due to slippery foot contacts, the policy network will be trained with unreliable state estimates and manifest conservative behaviors. The Mini Cheetah robot [17] is the robotic platform used in this work. Its compact size and powerful actuators enable us to tackle difficult tasks such as high-speed locomotion on slippery terrain. In addition, the source code for robot operation is available online 1 , which includes a high-performance locomotion controller [11]. Furthermore, its IMU sensor, 3DMGX5-AHRS made by Lord Corporation, provides not only the linear acceleration and angular velocity but also the estimated orientation based on the extended Kalman filter. Our version of the Mini Cheetah is 1.8 kg heavier and 1 cm longer compared to the original version presented in [17]. Our implementation code for the real robot can be found online 2 . A. Training In Simulation The policy is trained on flat terrain in 800 different environments to efficiently collect samples. In each environment, the robot is initialized with highly random initial states as shown in the table I. This helps the robot to recover from unexpected external disturbances such as interactions with humans or sudden changes in terrain parameters. With the probability of 25 %, the robot is initialized with the final state of the previous episode, whereby the robot can learn to overcome sudden changes in the velocity command in the real world. For further improvements, we also train the model in environments where an uneven flat terrain or slopes up to ±10 • are randomly generated. The uneven terrain was created using Perlin noise of the following parameters (fractal octaves = 5, fractal lacunarity = 3.0, fractal gain = 0.45, z-scale = mi n(0.21, 0.21·t −1 ) where t is the number of iteration). To train a policy more effectively, we set up a curriculum where the velocity command in the x-direction (i.e., forward/backward direction) gradually increases over each PPO iteration. At the early stage of training, the linear velocity command in the x-direction is uniformly sampled from U 1 (−0.5, 1.0) m/s. This range gradually enlarges up to U 1 (−1.75, 3.5) m/s, with the maximum forward command given according to V x,max = 1 + 2.5 1 + exp(−0.002 · (t − 1000)) ,(1) where t is the number of training iterations. Ten percent of the trajectories are then selected to have a zero velocity command for learning a standing still behavior. We define reward functions and their coefficients as shown in the table II. The reward function is designed for two objectives: to follow the given command and to run in an efficient and natural way. The linear and angular velocity rewards are related to the former objective, and the other rewards are for the latter one. Most of the reward functions are shaped by referring to [1]. Among them, the foot clearance reward is important for a successful simto-real transfer because the relative foot positions to the ground and terrain geometry might be uncertain in some situations. The square-root function in the foot clearance reward is to increase its influence on the policy when the commanded velocity is too low. Airtime reward is designed for controlling swing-stance timing and generating standing still motions. r v = k v exp(−\|\|cmd vx y − V x y \|\| 2 ) Angular velocity r ω = k ω exp(−1.5(cmd ωz − ω z ) 2 ) Airtime r ai r,i =      if stance cmd: k a cl i p(T s,i − T a,i ,−0.3,0.3) else: if T max,i < 0.25 : k a mi n(max(T s,i ,T a,i ),0.2) else: 0 Foot slip r sl i p,i = k sl i p C f ,i \|\|V f ,x y,i \|\| 2 Foot clearance r cl ,i = k cl ( w p f ,z,i − w p d e s f ,z ) 2 \|\|V f ,x y,i \|\| 0.5 Orientation r or i = k or i (ang l e(φ bod y,z ,φ w or l d ,z )) 2 Joint torque r τ = k τ \|\|τ\|\| 2 Joint position r q = k q \|\|q t − q nomi nal \|\| 2 Joint speed rq = kq \|\|q t \|\| 2 Joint acceleration rq = kq \|\|q t −q t −1 \|\| 2 Action smoothness 1 r s1 = k s1 \|\|q d e s t − q d e s t −1 \|\| 2 Action smoothness 2 r s2 = k s2 \|\|q d e s t − 2q d e s t −1 + q d e s t −2 \|\| 2 Base motion r base = k base (0.8V 2 z + 0.2\|ω x \| + 0.2\|ω y \|) Reward Coefficients k v 3.0 k cl -15.0 kq -6e-4 k ω 3.0 k or i -3.0 kq -0.02 k a 0.3 k τ -6e-4 k s1 ,k s2 -2.5, -1.2 k sl i p -0.08 k q -0.75 k base -1.5 In the table II, cmd is an abbreviation of command and i is an index of the foot. T a,i and T s,i represent the time since last takeoff and touchdown, respectively, while being initialized to zero whenever a transition happens. C f ,i in the foot slip reward is the contact state of each foot. In the foot clearance reward, w p d es f ,z is the desired foot height and it is set to 0.09 m. We define a positive reward sum as r pos = r v +r ω + 3 i=0 r air,i and a negative reward sum as r neg = 3 i=0 (r sl i p,i + r cl ,i ) + r or i + r τ + r q + rq + rq + r s1 + r s2 + r base . The total reward is defined as r t ot = r pos · exp(0.2r neg )(2) This form of a reward function ensures that the resulting reward is always positive and discourages the policy to choose an early termination. Whenever the body of the robot except the knees and feet contacts the environment, the episode terminates and the robot is punished with a reward of -10. Therefore, the policy is trained toward reducing unnecessary collisions. B. Network Architecture Our neural network structure consists of 3 components: an actor, a critic, and an estimator. All of them are designed as a Multi-Layer Perceptron (MLP) network, with the actor and critic having a [512×256×64] structure, and the estimator having a [256×128] structure. An MLP is the simplest neural network structure and computationally more efficient than other memory-based networks such as RNNs. We trained policies in a form of an LSTM but could not find meaningful differences in performance. The actor maps an observation to an action and the critic [20] estimates the value of the current state. The estimator network is to estimate states of the robot such as the base linear velocity. Those values are estimated by taking an observation o t as an input, and fed to the actor. The estimator network is trained using supervised learning with data from the simulation. Both the policy and the estimator are running synchronously at 100 Hz. The network structure is shown in Fig. 2. Our system takes sensor data as an input, and outputs desired joint positions for each actuator. Our framework still uses an analytical estimate of the gravity vector expressed in the body frame because it is computed by the IMU sensor. Furthermore, the estimation algorithms for the orientation are simple and reliable. Joint velocities are computed on the motor controllers by applying the finite difference method on joint positions. The observation tuple is defined as o t = (φ, ω, q,q, q d es t −1 , q d es t −2 ,Q hi st ,Q hi st , b p f , cmd) (3) where φ and ω are the base orientation and angular velocity, q andq are the joint positions and velocities, q d es t −1 and q d es t −2 are the desired joint position targets for two previous time steps, Q hi st andQ hi st are the joint position error history and joint velocity history, b p f is the Cartesian positions of the feet relative to the center of mass expressed in the body frame, and cmd is the given velocity command. The Cartesian foot positions are for observing where the feet are located, and it is known to be helpful for training a policy for complicated systems [21]. For our study, joint state history is selected at t − 0.02s, t −0.04s, and t −0.06s. For stable learning and control, all observation variables are normalized to have a mean of 0.0 and a standard deviation of 1.0. For the same reason, policy outputs are multiplied by a nominal value and then added to nominal joint positions to obtain the desired joint target distributions. This relationship is expressed as q d es t = q nominal + σ a a t , where q nominal is the nominal joint positions, which is the same as the standing up configuration, σ a = 0.1 is a predefined action scaling factor, and a t is an output of the policy network. The desired joint positions are computed at 100 Hz and converted to joint torques by a PD controller module with K p =17 N·m·rad −1 and K d =0.4 N·m·s·rad −1 , at 40 kHz on the real robot. The estimator network is designed to predict the state of the robot without utilizing a dedicated estimation algorithm. In this paper, the linear velocity, foot height, and contact probability are estimated. The linear velocity estimate is essential in the following velocity command. By removing the necessity of sophisticated state estimation algorithms, the implementation on the robot becomes much simpler. It also has an advantage that the controllers become robust against inevitable errors of the state estimator. As illustrated in [15], estimation of the linear velocity under highly erroneous environments is vulnerable to foot slips. Our end-to-end neural network structure avoids such a challenge in two ways. First, the estimator network is trained in environments where the feet slip often. Therefore, it can still produce a reasonable estimate of the linear velocity using other sensor information and previous observations. Second, the policy network is trained with imperfect information such that it is aware of possible slippages. The idea behind learning foot height and contact probability is to achieve the sufficient foot clearance. Due to the wide range of velocity command and the clearance reward that penalizes high speed, the foot clearances become smaller at low speeds. Foot clearance plays an important role in a sim-to-real transfer because insufficient foot clearance might lead to an early foot landing or tripping while running. We discovered that the reward term alone is not sufficient to learn to maintain sufficient foot clearances. Our solution to this problem is to estimate the foot clearances and feed them directly to the policy network. We note that a learning-based estimator is capable of approximately estimating the foot clearance from the observation. First, foot contact states are obtainable from joint position errors. Second, a terrain slope becomes observable from the foot contact states, orientation, and joint positions. Therefore, as the slope is observable, the estimator network can compute the foot height under the assumption that the terrain is even. C. Dynamics Randomization Dynamics randomization is important for a successful sim-to-real transfer of policies trained in simulation [22]. In our case, the robot controller learned without dynamics randomization exhibits shaky motions when deployed on the real robot. It comes from the fact that kinematic and dynamic parameters such as leg length, actuator positions, and center of mass, are not exactly the same as those used in the simulation. Consequently, this reality gap often makes the robot unable to reach sufficient performance. To eliminate this gap, we randomize several components as follows: • observation noise • motor frictions • PD controller gains • foot positions and collision geometry • ground friction These parameters are randomized at the start of each episode or iteration. The observation noise is added during the training phase in the simulation. On the real robot, joint velocity measurement comes from numerical differentiation of the joint positions, which causes errors in the joint velocity observation. Moreover, fluctuation in the logging frequency might lead to a failure in updating the velocity values for a single time step. Such an event corresponds to a delay of 2 milliseconds. Therefore, the joint position and velocity measurements in simulation are randomized to reflect the true distribution of the measurements; they are sampled from U 12 (-0.05, 0.05) rad and U 12 (-0.5, 0.5) rad/s, respectively. For the same reason, a uniformly distributed noise U 4 (-0.03, 0.03), U 4 (-0.03, 0.03) m, and U 3 (-0.1, 0.1) rad/s are added to the base orientation, foot position, and base angular velocity, respectively. Motor friction is randomized to reflect the differences between actuator units. We chose a conservative range of U 1 (0, 0.3) N·m for the hip abduction/adduction (HAA) and hip flexion/extension (HFE), and U 1 (0.1, 0.7) N·m for the knee flexion/extension (KFE). Their measured friction values on the real robot are 0.2, 0.2, and 0.5 N·m for HAA, HFE, and KFE, respectively. The KFE joints have higher friction because they have an extra belt transmission. The PD controller gains are randomized to mitigate the effects of motor friction and damping. We added a uniform noise of U 1 (-2, 2) N·m·rad −1 , and U 1 (-0.1, 0.1) N·m·s·rad −1 for the position and velocity gains, respectively. The foot position and collision geometry are randomized to reduce both effects of measurement errors and the deformation of the rubber feet. The foot position observations are disturbed with a uniform noise of U 1 (-10, 10) mm in the longitudinal direction, U 1 (-5, 5) mm in the lateral direction, and U 1 (-20, 20) mm in the leg length direction. These noises are added to the measured foot positions. The foot sphere radii are randomized to U 1 (6, 10) mm. Finally, the ground friction was randomized to U 1 (0.4, 1.0). Owing to this randomization, the robot can run not only on very slippery ground but also on the ground with very high friction like asphalt. III. RESULTS Part of the results in this section can also be found in the accompanying video. A. Controller Descriptions To evaluate the performance of the proposed network structures and analyze how each component affects different performance metrics, we test the following settings: • Implicit: As a baseline, the explicit state estimator in our proposed framework is substituted with an implicit estimator as in [3], [4]. • Sequential: In phase 1, a policy is trained with the ground truth robot states. In phase 2, a state estimator was trained with the final policy of phase 1. For implementation, the estimator replaces the ground truth input. • Built-in MPC: The MPC controller described in [17] • RL-LKF: An RL policy in a single MLP form trained with the linear velocity data from the simulator and uses an LKF state estimator on the real robot. • Concurrent: Our proposed control framework trained on relatively flat ground • Concurrent+Slope: Our proposed control framework trained on randomly generated slopes. In addition to the above models, we also created four other network models by excluding one of the three estimated states or all of them (i.e., w /o Linvel Estimator, w/o FootHeight Estimator, w/o Contact Estimator, w/o Estimator) for ablation studies in simulation. For comparison of the explicit and implicit estimators, we trained the Implicit model. The implicit estimator follows the framework in [4]. During the phase 1, the encoder takes the observation o t \(Q hi st ,Q hi st ), linear velocity, foot height, and foot contact as an input. For the adaptation module, the history length of 20 is used and the network consists of 3 layers of 1D CNN. The output dimension of the encoder and adaptation module is 11. All the other settings are the same as our framework. The total training time is 7 hours for phases 1 and 2 altogether. Fig. 3. All the models were trained until they converged to a stable expected return. After 2500 iterations, which consumed about 800 million samples and 4 hours of training in real-time, they all converged to a stable value. The rewards of the trained models are summarized in the table III. B. Evaluation of the Performance in Simulation As shown in Fig. 3 and the table III, the Concurrent model converges to the highest rewards while the w/o Estimator and Implicit models converge to the lowest. Out of the three estimated states, linear velocity estimation plays the most important role in improving the policy. Omitting the linear velocity estimation leads to a significant drop in metrics: the total, linear velocity, and foot clearance rewards. This result proves that the linear velocity is crucial for learning the locomotion of legged robots. Another noticeable improvement comes from the foot height estimation. Explicitly estimating the foot height improves the foot clearance of the robot, resulting in a higher foot clearance reward. The effectiveness of the higher foot clearance will be discussed in the Locomotion on rough terrains section. Foot contact probability estimation makes the least impact on the final performance, but it stabilizes and accelerates learning processes as shown in the total reward graph in Fig. 3. 2) Tracking and Estimation Error: For further investigation on the impact of the estimator network, we tested the Concurrent, w/o Estimator, and Implicit models in simulation. All models were given the same random commands every 20 seconds and for 10 minutes in total while the friction coefficient of the flat ground is kept at 0.6. The performance was only measured for the steadystate errors. The velocity commands are sampled from U 1 (-1.75, 3.5) m/s, U 1 (-1, 1) m/s, and U 1 (-2, 2) rad/s, for V x ,V y , and ω, respectively. The result is shown in the table IV. The Concurrent model has the smallest RMS errors for following the given linear velocity. It means that the estimated states help the robot to stabilize itself. Furthermore, the tracking errors of the Implicit model are on a similar level to that of w/o Estimator. From this fact, we suggest that tracking performance does not benefit a lot from utilizing the implicit estimator. Models with the implicit estimator having a latent vector of sizes 8 and 20 showed worse performance than the presented one. Interestingly, our concurrently trained model performs better than a sequentially trained model. From the ta- ble IV, the Sequential model has slight performance degradation. We hypothesize that this is because the policy trained with a state estimator tends to avoid states where the state estimator becomes unreliable. This problem can be easily solved by training them concurrently as we proposed. Note also that the concurrent training requires only one training dataset, which is more efficient than the sequential training. 3) Locomotion on rough terrains: We investigated the effectiveness of the foot clearance of the learned models. The foot clearance should be sufficiently high for traversing over the rough terrains. Therefore, in this experiment, we compare the average time to fall on the rough terrains with z-scale of 0.525. Commands are sampled from U 1 (-1, 1) m/s, and U 1 (-1, 1) rad/s where the foot clearance is relatively small. From the table IV, the Concurrent model shows an overwhelming performance compared to the other models. It is robust against a fall due to increased foot clearance. On the other hand, an implicit estimator and a single policy do not have sufficient foot clearance. Eventually, they are easy to fall and have worse locomotion performance. In conclusion, we suggest that the explicit estimation of the foot height significantly contributes to locomotion performance. 4) Computational Cost: The trained estimator has computational benefits over analytical state estimators. Using a single core of Ryzen9 5950x, the estimator network takes 7 µs for a forward pass, while the linear Kalman filter on the Mini Cheetah consumes 34 µs. Also, the implicit estimator with 20 history inputs takes 20 µs which is about three times longer than the simple explicit estimator. Learning curves of the total reward, linear velocity reward, and foot clearance reward are shown. The linear velocity, foot height, and contact probability are estimated. The Concurrent model has the highest performance and learning stability, while the model without an estimator (w/o Estimator) has the lowest performance. C. Evaluation of the Performance on the Real Robot We evaluated the performance of controllers on the real robot in terms of command following, state estimation, the maximum running speed, the maximum traversable slope angle, and foot clearance. The Concurrent+Slope model requires 5000 iterations for convergence due to the challenging terrains. The summary of the performance is shown in the table V. 1) Network Implementation on the Real Robot: For the experiments in the real environments, we compared the five aforementioned control models: Built-in MPC, RL-LKF, Concurrent, and Concurrent+Slope. As described in the Controller Description section, the RL-LKF model requires an LKF state estimator. However, we could not use the built-in contact estimator as it assumes a periodic gait schedule, while our learned policy inherently changes gait patterns over speeds. Therefore, when estimating the contact state of the RL-LKF model for the LKF, we used the proprioceptive touchdown detection method described in [23]. When the difference between the KFE joint position and its desired position is over the threshold of -0.4 rad, we assume that the leg is in contact. 2) Command Following and State Estimation: In the real environments, each controller was tested on normal and slippery ground with a step velocity command of 1.0 m/s, 0.8 m/s, and 2.0 rad/s for linear velocities in x and y directions, and a yaw rate, respectively. The commands continued for 1 second and the robot started from zero commands. The slippery plate covered with boric acid powder has a friction coefficient of 0.22, which is much lower than that of the training environments. The results are shown in the table V. In some cases, the Built-in MPC controller fell and those instances are marked with '-' in the table. On the other hand, RL policies performed robustly against all sudden step inputs in this experiment. The Concurrent model has the best tracking performance, while the Concurrent+Slope model has a slightly higher error possibly due to the fact that the Concurrent model is overfitted to simpler terrains. We also recorded the estimation data from the LKF while the Concurrent and Concurrent+Slope models are running. The errors from the estimator networks are written in the parentheses. We note that estimation error does not necessarily lead to a deterioration in tracking performance. We suppose that other observation inputs, such as joint state history, mitigate the effects of the estimation errors so that the concurrent learning framework becomes robust against these errors. 3) High-Speed Locomotion: The Concurrent+Slope model has the highest maximum speed. We tested each controller repeatedly until the robot fell. The maximum outdoor speed of our model, 3.75 m/s, is comparable to the one reported by Kim et al. [11], who report outdoor speed of over 1.7 m/s and treadmill speed of 3.7 m/s. Our Concurrent+Slope controller is capable of running at 3.54 m/s on a slippery plate with µ = 0.22. When large foot slippages occurred, the robot recovered quickly, even when the robot is running near the maximum speed. If the command is suddenly set to zero while running, the robot makes a stable pose to stop quickly. We assume that this high performance is achieved owing to two factors: the policy trained on low friction terrains and its independence on a state estimation algorithm. Because our proposed framework is trained to be aware of possible foot slippages, it can be robust against estimation errors. On both normal and slippery ground, the Concurrent model exhibits better performance than the Built-in MPC and RL-LKF models. The Built-in MPC model could not reach speeds over 1.7 m/s on the normal ground and was easy to fall on the slippery ground at speed under 1.3 m/s. RL-LKF model also runs at rather conservative speeds lower than 2.2 m/s. On the other hand, all the RL controllers are able to run consistently on all tested terrains as they are trained on terrains with the various friction coefficients. 4) Locomotion on Hills: Training a policy on slopes with random angles and friction coefficients significantly improves climbing performance. The Concurrent+Slope model is capable of climbing a normal hill up to 19.1 • , which is steeper than slope angles of the training environments, ±10 • . Also, we note that Concurrent+Slope model can walk up a slippery hill up to 9.0 • . Although the feet of the robot are slipping on it, the robot managed to climb up the slope by pushing the ground with adequate force. This behavior is partially learned in simulation, but the robot adapts its motion for the much more slippery slope of the friction coefficient of 0.22. For outdoor environments, we demonstrate the performance of our policy on a bumpy and hilly asphalt road. Please refer to the supplementary video for the demonstration. The other controllers could not climb up a normal hill with angles steeper than 12.4 • . Because the RL controllers except for the Concurrent+Slope model are trained only on nearly flat terrains, they have unsatisfactory climbing performance. Although the other controllers' maximum traversable slope angles are on a similar level, they display different locomotion characteristics. The Built-in MPC model has higher foot clearances, but its non-stopping gait makes the robot unstable. The other RL models show relatively lower foot clearances, while their standing-still behavior significantly improves the stability of the robot. 5) Foot Clearance: We evaluated the foot clearance of each controller while the robot is running at 1.0 m/s. The maximum foot clearance is shown in the table V. We could recognize that foot clearance of the Concurrent and Concurrent+Slope models on the real robot is higher than the models without the estimator network, RL-LKF and w/o Estimator. As shown in the simulation test, lower foot clearance hinders stable locomotion on highly uneven terrains. This issue remains equally problematic on the real robot. We discovered that the RL-LKF and w/o Estimator models exhibit lower foot clearance at low speeds. Therefore, we suggest that explicit estimation of the foot clearance is effective for improving the performance. 6) Contact Estimation: Our proposed estimator network outputs contact probability for each foot. In Fig. 4, the contact state probability of the front right foot is drawn with the real contact state, KFE joint position error and joint velocity. The estimated contacts are shifted by 0.04 seconds from the actual contacts, which corresponds to 3 control steps excluding 0.01 seconds of communication delay. This is because there is a reality gap due to the compliance of the chain and the rubber feet of the real robot. In addition, the joint history inputs are sparsely sampled with 0.02-second intervals and it introduces further delay in detection. The estimator network detects a contact when the joint abruptly stops by impacts with the ground. The diagram also justifies the threshold of -0.4 rad of joint position error for the contact estimation used for LKF. IV. CONCLUSION We presented a framework for concurrent training of a control policy and a state estimator. This framework requires neither an advanced control algorithm nor an accurate state estimation algorithm. Therefore, it requires significantly less effort for implementation on the real-legged system. Furthermore, our concurrent training method outperforms implicit estimation methods and a sequential training method in many aspects such as command tracking, robustness on rough terrain, and training time. To the best of our knowledge, our record is the fastest reported legged locomotion using reinforcement learning. The robot is also able to stably run on a slippery plate even under foot slippages. Although foot slippages often compromise the quality of the state estimation, the concurrently trained policy is robust against such issues. The proposed learning-based state estimation can be useful without the control policy in many applications. It can provide reliable state estimates for motion analysis. Furthermore, we expect that the interpretable state outputs from our proposed network can be useful in conjunction with other controllers, such as MPC-based ones. This work was supported by Samsung Research Funding & Incubation Center for Future Technology at Samsung Electronics under Project Number SRFC-IT2002-02. The Mini Cheetah robot was provided by MIT Biomimetic Robotics Lab and Naver Labs Corporation. * All authors are with Robotics and Artificial Intelligence Lab in the department of Mechanical Engineering, KAIST, Daejeon 34141, Republic of Korea. [email protected] Fig. 1 . 1Dynamic and robust locomotion on a slippery plate of the friction coefficient of 0.22. The Mini Cheetah is traversing over it at the average speed of 3.54 m/s. Fig. 2 . 2An overall control diagram and a proposed training framework for concurrent training of a control policy and a state estimator are shown. The estimator network takes sensor data o t as an input and outputs state variables, which are the base linear velocity, foot height, and contact probabilities. These estimated states are fed into the policy network together with the observation o t . The estimator network is trained with supervised learning to reduce MSE between the estimated robot states and their corresponding true values obtained from simulation. An actor network is a policy which produces desired joint positions based on the state estimates. Both the critic and actor are trained using the PPO algorithm. Fig. 3 . 3Fig. 3. Learning curves of the total reward, linear velocity reward, and foot clearance reward are shown. The linear velocity, foot height, and contact probability are estimated. The Concurrent model has the highest performance and learning stability, while the model without an estimator (w/o Estimator) has the lowest performance. Fig. 4 . 4A contact estimation diagram for the front right foot is shown. KFE joint position error, joint velocity, and contact state are drawn. Contact starts with an abrupt change of the joint velocity by impact with the ground and ends with upward joint motion. TABLE I IINITIAL STATE NOISEState Noise Quaternion U 4 (−0.2,0.2), then normalized Joint positions U 12 (−0.2,0.2) rad Joint velocities U 12 (−2.5,2.5) rad/s X linear velocity U 1 (−1,1) m/s YZ linear velocities U 2 (−0.5,0.5) m/s Angular velocities U 3 (−0.7,0.7) rad/s TABLE II REWARD FUNCTIONS IIFUNCTIONSReward Expression Linear velocity 1 ) 1Learning Performance: First, we compared the performance of the learned controllers (i.e., Concurrent, w/o Linvel Estimator, w/o FootHeight Estimator, w/o Contact Estimator, and w/o Estimator) in simulation. Each network structure is trained four times and their average learning curves are shown in TABLE III ABLATION IIISTUDY FOR THE ESTIMATOR: REWARD OF THE LEARNED MODELSModel Reward Total Linear Velocity Foot Clearance Concurrent 4.7212 2.7595 -0.1338 w/o LinVel Estimator 4.5555 2.6643 -0.1432 w/o FootHeight Estimator 4.7293 2.7625 -0.1504 w/o Contact Estimator 4.7125 2.7543 -0.1382 w/o Estimator 4.4577 2.6284 -0.1565 Implicit 4.439 2.611 -0.1447 TABLE IV TRACKING IVAND ESTIMATION ERROR & LOCOMOTION ON ROUGH TERRAINSTask Model V x [m/s] V y [m/s] ω z [rad/s] Command Following Concurrent 0.1112 0.0708 0.1222 w/o Estimator 0.1725 0.0959 0.1137 Implicit 0.1679 0.1233 0.1379 Sequential 0.1358 0.0996 0.1201 Estimation Error Concurrent 0.0243 0.0199 - Sequential 0.06 0.036 - Average time to fall [sec] Rough Terrain Concurrent 85.7 w/o Estimator 25.0 Implicit 30.0 Sequential 75.0 TABLE V VPERFORMANCE OF CONTROLLERS ON THE REAL ROBOTTask Terrain (friction) Command Model Built-in MPC RL-LKF w/o Estimator Concurrent Concurrent+Slope Command Following : Tracking Error Normal (µ = 0.6-0.88) V x [m/s] 0.3041 0.2744 0.2635 0.2387 0.2488 V y [m/s] - 0.2031 0.2334 0.1722 0.1817 ω z [rad/s] 0.462 0.2857 0.2415 0.2427 0.2395 Slippery (µ = 0.22) V x [m/s] - 0.3255 0.2888 0.2874 0.2532 V y [m/s] - 0.285 0.2617 0.2183 0.2446 ω z [rad/s] 0.525 0.3229 0.2709 0.2504 0.2654 State Estimation : Estimation Error Normal (µ = 0.6-0.88) V x [m/s] 0.1869 0.0808 0.0653 0.0783 (0.1069) 0.0852 (0.0946) V y [m/s] - 0.1115 0.0666 0.1227 (0.0793) 0.0557 (0.078) Slippery (µ = 0.22) V x [m/s] - 0.1575 0.0741 0.0533 (0.1209) 0.1168 (0.1029) V y [m/s] - 0.0623 0.0612 0.0555 (0.0848) 0.0833 (0.1061) Maximum Average Speed Normal (µ = 0.6-0.88) [m/s] 1.72 2.18 3.20 3.33 3.75 Slippery (µ = 0.22) [m/s] 1.34 2.19 3.14 3.25 3.54 Maximum Slope Normal (µ = 0.6-0.88) / Slippery (µ = 0.22) [ • ] 12.4 12.4 9.6 12.4 19.1 / 9.0 Maximum Foot Height Normal [cm] 5 2 2 3 3 https://github.com/mit-biomimetics/Cheetah-Software https://github.com/karlji1021/Cheetah-Software Learning agile and dynamic motor skills for legged robots. 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[ "DETECTION OF HERMITIAN CONNECTIONS IN WAVE EQUATIONS WITH CUBIC NON-LINEARITY", "DETECTION OF HERMITIAN CONNECTIONS IN WAVE EQUATIONS WITH CUBIC NON-LINEARITY" ]	[ "X I Chen ", "Matti Lassas ", "ANDLauri Oksanen ", "Gabriel P Paternain " ]	[]	[]	We consider the geometric non-linear inverse problem of recovering a Hermitian connection A from the source-to-solution map of the cubic wave equation A φ + κ\|φ\| 2 φ = f , where κ = 0 and A is the connection wave operator in Minkowski space R 1+3 . The equation arises naturally when considering the Yang-Mills-Higgs equations with Mexican hat type potentials. Our proof exploits the microlocal analysis of nonlinear wave interactions, but instead of employing information contained in the geometry of the wave front sets as in previous literature, we study the principal symbols of waves generated by suitable interactions. Moreover, our approach relies on inversion of a novel non-abelian broken light ray transform.	10.4171/jems/1136	[ "https://arxiv.org/pdf/1902.05711v1.pdf" ]	119,327,550	1902.05711	007851a95c5b30620a91eca2d008815b5a7a5945	DETECTION OF HERMITIAN CONNECTIONS IN WAVE EQUATIONS WITH CUBIC NON-LINEARITY X I Chen Matti Lassas ANDLauri Oksanen Gabriel P Paternain DETECTION OF HERMITIAN CONNECTIONS IN WAVE EQUATIONS WITH CUBIC NON-LINEARITY To the memory of our friend and colleague Slava Kurylev We consider the geometric non-linear inverse problem of recovering a Hermitian connection A from the source-to-solution map of the cubic wave equation A φ + κ\|φ\| 2 φ = f , where κ = 0 and A is the connection wave operator in Minkowski space R 1+3 . The equation arises naturally when considering the Yang-Mills-Higgs equations with Mexican hat type potentials. Our proof exploits the microlocal analysis of nonlinear wave interactions, but instead of employing information contained in the geometry of the wave front sets as in previous literature, we study the principal symbols of waves generated by suitable interactions. Moreover, our approach relies on inversion of a novel non-abelian broken light ray transform. This paper considers an inverse problem for a non-linear wave equation motivated by theoretical physics and differential geometry. The main problem we wish to address is the following: can the geometric structures governing the wave propagation be globally determined from local information, or more physically, can an observer do local measurements to determine the geometric structures in the maximal region where the waves can propagate and return back? There has been recent progress on this question when the geometric structure is space-time itself and the relevant PDEs are the Einstein equations [20]. Here we propose the study of a natural non-linear wave equation when the Lorentzian background is fixed and the goal is the reconstruction of a Hermitian connection. The main difference between the inverse problems for the Einstein equations and the equation considered here is that, in the former case, the geometric structure (the metric) to be reconstructed appears in the leading order terms, and in the latter case, it (the connection) appears in the lower order terms. This difference poses novel challenges, since a perturbation in the leading order affects the wave front sets of solutions whereas lower order perturbations do not. The leading order terms can frequently be reconstructed via study of distances (or time separations/earliest arrival times), whereas lower order terms often require reductions to light ray transform questions. Nevertheless, our approach exploits the recent philosophy that non-linear interaction of waves creates new singularities and enriches the dynamics [20,23,28]. As we shall see this interaction leads to a broken non-abelian light ray transform on lightlike geodesics that has not been previously studied. Our main long term goal is the study of inverse problems for the Yang-Mills-Higgs equation. The present paper is the first stepping stone in this direction and our objective here is to start exposing the main features that this problem will have by considering a simplified, but non-trivial model case. Since the bundle M × g is trivial, A is a connection, and it is called the Yang-Mills potential; Φ is the Higgs field. The Yang-Mills-Higgs equations are D * A F A + [Φ, D A Φ] = 0; (1) D * A D A Φ + V (\|Φ\| 2 )Φ = 0,(2) where F A := dA + A ∧ A is the curvature of A, D A Φ := dΦ + [A, Φ] is the associated covariant derivative, and V is the derivative of a smooth function V : [0, ∞) → R. The adoint D * A is taken with respect to g and hence A := D * A D A is the wave operator associated with g and A. An extensively studied case is the Yang-Mills-Higgs equations with the Mexican hat type potential, V (\|Φ\| 2 ) = 1 2 κ(\|Φ\| 2 − b) 2 ,(3) where κ, b ∈ R, see e.g. [8,Eq. (10.5)] where the Lagrangian formulation of the problem is used. We will consider the potential (3) with κ = 0, and to simplify the notations, with b = 0. The case b = 0 is not substantially different. Our choice can be viewed as the simplest potential introducing a non-linearity. We refer also to [36] where Yang-Mills-Higgs equations, with the potential (3), are discussed in a purely mathematical context, (M, g) being a Riemannian manifold there. As it is well known, equations (1)- (2) are invariant under the group of gauge transformations which in this case coincides with the set of maps u ∈ C ∞ (M ; G) and the action on pairs is (A, Φ) → (u −1 du + u −1 Au, u −1 Φu). When Φ = 0 we obtain the pure Yang-Mills equation D * A F A = 0. 1.2. Formulation of the inverse problem in the model case. Dealing with the equations (1)-(2) from the outset might be too ambitious, so here we propose a simplified model. We shall suppose that we have a trivial bundle E = M × C n and a Hermitian connection A on E giving rise to a covariant derivative d + A. In this case, the gauge group is U (n). We take V to be the Mexican hat type potential (3) with b = 0, discard equation (1) completely and focus on the analogue of equation (2), with M × g replaced by E. That is, we consider the equation (4) A φ + κ\|φ\| 2 φ = 0, where φ is a section of E, A = (d + A) * (d + A) and \|φ\| is the norm with respect to the standard Hermitian inner product of C n . We shall further simplify matters by assuming that M is R 1+3 and that g is the Minkowski metric. Let us consider Cartesian coordinates (t = x 0 , x 1 , x 2 , x 3 ) on R 1+3 . Let 0 > 0 and define B( 0 ) = {y ∈ R 3 : \|y\| < 0 } and = (0, 1) × B( 0 ). (5) We will formulate a source-to-solution map, associated to (4), that corresponds physically to measurements gathered in . We can think that the measurements are performed by an observer travelling along the path µ : [0, 1] → R 1+3 , µ(t) = (t, 0). (6) In what follows, we consider only finite time intervals, and write M = (−1, 2) × R 3 . and let C be a small neighbourhood of the zero section in C 4 0 ( ; E). Then the source-to-solution map L A f := φ\| , f ∈ C,(7) is well-defined, where φ is the solution of A φ + κ\|φ\| 2 φ = f, in M,(8) φ\| t<0 = 0. We discuss the existence of L A in more detail in Section 2 below. The goal of the observer is to determine the Yang-Mills potential A up to the natural obstructions, given the source-to-solution map L A . The causal structure of (M, g) encodes the finite speed of propagation for the wave equation (4). Given x, y ∈ M we say that x ≤ y if x = y or x can be joined to y by a future pointing causal curve, and denote the causal future of x ∈ M by J + (x) = {y ∈ M : x ≤ y}. The causal future J + (x) is the largest set that waves generated at x can reach. The causal past of a point z ∈ M is denoted by J − (z) = {y ∈ M : y ≤ z}. If waves generated at x are recorded at z, the finite speed of propagation dictates that no information on the potential A outside the causal diamond J + (x) ∩ J − (z) is obtained. The model problem is to determine A given L A in the largest domain possible, that is, in D = x,z∈ J + (x) ∩ J − (z) ,(9) up to the natural gauge, A → u −1 du + u −1 Au,(10) where u ∈ C ∞ (D; U (n)) and u\| = id. The sets and D are visualized in Figure 2. Observe that if we have two connections A and B on M such that there exists a smooth map u : M → U (n) with the property that B = u −1 du + u −1 Au and u\| = id, then B = u −1 A u and \|uφ\| = \|φ\|. Moreover, as f has compact support in it holds that uf = f . Therefore φ solves (8) for B if and only if uφ solves (8) for A, and it follows that L A = L B . This shows that the gauge (10) is indeed natural. Our main theorem asserts that the model problem has a unique solution, or in more physical terms, the measurements performed on , as encoded by L A , determine the gauge equivalence class of the Yang-Mills potential A, in the largest possible causal diamond D. As D is strictly larger than , we can view the determination of the equivalence class of A as a form of remote sensing. We emphasize that the gauge equivalence classes of Yang-Mills potentials, not the potentials themselves, correspond to physically distinct configurations. Theorem 1. Let A and B be two connections in R 1+3 such that L A = L B where the source-to-solution map L A is defined as above, and L B is defined analogously, with A replaced by B in (8). Suppose that κ = 0 in (8). Then there exists a smooth u : D → U (n) such that u\| = id and B = u −1 du + u −1 Au. It is straightforward to see that L A = L B implies A = B on . The non-trivial content of the theorem is the gauge equivalence away from . To see that A and B coincide on , we fix y ∈ and choose φ ∈ C ∞ 0 ( ; E) such that φ(y) = 0. Then for small > 0 it holds that f := ( A φ + κ\|φ\| 2 φ) ∈ C. Since L A = L B we see that at y: −A 0 ∂ t φ + 3 j=1 A j ∂ x j φ = −B 0 ∂ t φ + 3 j=1 B j ∂ x j φ, cf. (14) below, and since dφ at y is arbitrary, there holds A = B at y. 1.3. Comparison with previous literature. The previous results on inverse problems for non-linear wave equations, such as [20,23,28], are based on analysis of four singular, interacting waves. A new feature in the present paper is that we consider interactions of three waves only. This leads to a more economic proof, and is particularly well suited for the cubic non-linearity in (8). A more detailed comparison of interaction three versus four waves is given in the beginning of Section 3.1. Let us briefly explain what we mean by the interactions of three waves. The idea is to choose a source of the form f = 1 f 1 + 2 f 2 + 3 f 3 where j > 0 are small and f j are conormal distributions. Then the cross-derivative ∂ 1 ∂ 2 ∂ 3 φ\| =0 satisfies a linear wave equation with a right-hand side that corresponds to a certain product of ∂ j φ\| =0 , j = 1, 2, 3. Here = ( 1 , 2 , 3 ). As also the functions ∂ j φ\| =0 satisfy the linear wave equation, we can view the cross-derivative as a result of their interaction. The wave front set of the above cross-derivative was studied in the case of the 1 + 2-dimensional Minkowski space by Rauch and Reed [32], see also [6,30] for later results of similar nature. What is new in the present paper, is that, contrary to [32] and the previous results on inverse problems for non-linear wave equations, e.g. [20,23,28], we employ more precise information on the singular structure of the cross derivative than just its wave front set. Namely, in a suitable microlocal sense, the cross-derivative has a principal symbol, and the proof of Theorem 1 uses information contained in the principal symbol in order to recover a novel broken non-abelian light ray transform of the connection A along lightlike geodesics. The proof of Theorem 1 is completed by solving the subsidiary geometric inverse problem of inverting this transform in the Minkowski space, see Proposition 2 below, a result which has independent interest. In more physical terms, we can say that the interaction of the three waves ∂ j φ\| =0 , j = 1, 2, 3, produce an artificial source, that can be viewed either as two moving point sources or as a filament in spacetime, and that emits a wave encoded by the crossderivative ∂ 1 ∂ 2 ∂ 3 φ\| =0 . We show that, when the sources f j , j = 1, 2, 3, are chosen carefully, the singular wave front emitted by the artificial source returns to . This wave front is visualized in Figure 1. Stretching the physical analogy further, we can say the leading amplitude of this singular wave front is the information used in the proof. Paradoxically, Theorem 1 is open for the linear case, κ = 0, but a positive solution is known if A and B are supposed to be time-independent [24]. In the time-dependent case there are results [34] available only in the abelian case of a line bundle, n = 1, and it is an open problem if recovery of A in the optimal causal diamond D is possible in this case. Let us also mention that the linear, abelian, time-independent case has been studied extensively, see e.g. [2,3,17], but these results do not carry over to the time-dependent case. The reason for this is that they (and also above mentioned [24]) are based on Tataru's unique continuation principle [35], that again is known to fail for equations with time-dependent coefficients [1]. We emphasize once more that the focus of the current paper is on the recovery of the lower order terms in a non-linear wave equation. This makes definite progress towards Open Problem 5 in [25], and is different from the previous results where only the determination of the leading order terms is considered. See for instance [23,28] where the determination of the metric tensor (or its conformal class) is studied for scalar-valued non-linear equations, or [20,27] where the determination of the metric tensor is studied for the Einstein equations coupled with different matter field equations. The difference between recovery of leading and lower order terms All the three pieces intersect in the two black points, moving along the vertical axis over the point where the black lines intersect. The points act as artificial sources that produce a new propagating wave, the red surface. The line segment traced by the two points can be viewed also as the projection of a one dimensional filament acting as an artifical source. The filament curves in spacetime since the two points move with a non-constant speed. Bottom right. As time progresses, the red propagating wave front grows. Eventually it will reach the points where the pieces of the spherical waves originate from. is reflected in the key novelty of our approach, namely, in the study of principal symbols instead of wave front sets. Such difference is apparent also in the existing theory of inverse problems for linear wave equations. The case of a linear wave equation with time independent coefficients, and with sources and observations in disjoint sets, illustrates this. In this case the theory is still under active development, and the best results available are very different for leading and lower order terms: the recovery of the metric [26] is based on distance functions, whereas the recovery of the lower-order terms [19] is based on focussing of waves. The latter also requires additional convexity assumptions, that are not present in the former case. 1.4. A conjecture on higher order non-linearities. One outcome of the current paper is the following emergent principle for dealing with inverse problems for waves with polynomial non-linearities using an approach similar to ours. Assume for simplicity that we are in the line bundle case (i.e. n = 1) and consider an equation of the form A φ + κφ N = f,(11) where κ = 0. Then to recover A (up to gauge) from a source-to-solution map it is necessary to consider the J-fold linearization of (11), where J ≥ max{3, N }.(12) The necessity is discussed further in Remarks 2 and 4 below. We conjecture that (12) is also a sufficient condition, but the present paper establishes this only in the case N = 3. 1.5. Outline of the paper. This paper is organized as follows. Section 2 contains preliminaries mostly having to do with the direct problem (8). Section 3 contains the microlocal analysis for the interaction of three waves and shows that we can recover the broken non-abelian light ray transform along lightlike geodesics from the knowledge of L A . Section 4 solves the geometric inverse problem of determining A up to gauge from the broken non-abelian light ray transform and completes the proof of Theorem 1. Appendix A recalls the theory of conormal and Intersecting Pair of Lagrangian (IPL) distributions; Appendix B contains certain technical details concerning symplectic transformations to a model pair of intersecting Lagrangians, and Maslov bundles; and Appendix C gives full description of the wave front set of the cross-derivative ∂ 1 ∂ 2 ∂ 3 φ\| =0 , that is, the red surface in Figure 1. We would like to dedicate this paper to the memory of our friend and colleague Slava Kurylev who was instrumental in initiating the present line of research on inverse problems for the Yang-Mills-Higgs equations. Acknowledgements. LO thanks Allan Greenleaf, Alexander Strohmaier and Gunther Uhlmann for discussions on microlocal analysis. ML was supported by Academy of Finland grants 320113 and 312119. LO was supported by EPSRC grants EP/P01593X/1 and EP/R002207/1 and XC and GPP were supported by EPSRC grant EP/R001898/1. GPP thanks the University of Washington for hospitality while this work was in progress and the Leverhulme trust for financial support. Preliminaries In this section, to accommodate further work, we let (M, g) be an arbitrary, globally hyperbolic Lorentzian manifold of dimension 1 + m. Also E can be taken as an arbitrary Hermitian vector bundle over M . Recall that a Lorentzian manifold (M, g) is globally hyperbolic if there are no closed causal paths in M , and the causal diamond J + (x) ∩ J − (z) is compact for any pair of points x, z ∈ M , see [5]. A globally hyperbolic manifold (M, g) is isometric to a product manifold R × M 0 with the Lorentzian metric given by (13) g = −c(t, x )dt 2 + g 0 (t, x ), (t, x ) ∈ R × M 0 , where c : R × M 0 → R + is smooth and g 0 is a Riemannian metric on M 0 depending smoothly on t, see [4]. Moreover, the vector field ∂ t gives time-orientation on M . To simplify the discussion, we make the further assumption that all the geodesics of (M, g) are defined on the whole R. 2.1. Direct problem. We write occasionally ∇ = d + A for the covariant derivative associated to the connection A, and view it as a map ∇ : C ∞ (M ; E) −→ C ∞ (M ; T * M ⊗ E). Writing g = g ij dx i dx j in coordinates, we denote by \|g\| and g ij the determinant and inverse of g ij , respectively. Moreover, A = A j dx j is a 1-form, and each A j is a skew-Hermitian matrix. Let us now write the wave operator A = ∇ * ∇ in coordinates. Consider compactly supported sections φ and ψ = ψ j dx j of E and T * M ⊗ E, respectively. Then ∇φ, ψ L 2 (M ;T * M ⊗E) = M g ij (∂ x i φ + A i φ), ψ j E dV g , where ·, · E is the inner product on E, and dV g denotes the volume form on (M, g). Integrating by parts and using the fact that A is skew-Hermitian, we have dφ, ψ L 2 (M ;T * M ⊗E) = − M φ, \|g\| −1/2 ∂ x i \|g\| 1/2 g ij ψ j E dV g , Aφ, ψ L 2 (M ;T * M ⊗E) = − M φ, g ij A i ψ j E dV g . Consequently, A φ takes the form −\|g\| −1/2 ∂ x i \|g\| 1/2 g ij ∂ x j φ − 2g ij A i ∂ x j φ − \|g\| −1/2 ∂ x i \|g\| 1/2 g ij A j φ − g ij A i A j φ. Remark 1. To prove Theorem 1, we will need the operator A exclusively in Minkowski space where we can explicitly write A φ = φ − 2 −A 0 ∂ t φ + 3 j=1 A j ∂ x j φ + divA + A 2 0 − 3 j=1 A 2 j φ,(14) where and div are the usual wave operator and divergence, φ = ∂ 2 t φ − 3 j=1 ∂ 2 x j φ, divA = ∂ t A 0 − 3 j=1 ∂ x j A j . Let T > 0 and let us consider the following nonlinear Cauchy problem (15) A φ(t, x) = H(t, x, φ(t, x)) + f (t, x), on (−∞, T ) × M 0 , φ = 0, for t < 0, where H : (R × M 0 ) × E → E is a smooth map operating section-wise such that H(t, x, 0) = 0, and f is a section of E. We will now give sufficient conditions on f in order for (15) to have a unique solution. As the leading term of A is simply the canonical wave operator on (M, g), acting on each component of φ, we can use the standard results for quasilinear hyperbolic equations to show existence and uniqueness of solutions to this Cauchy problem. See, for example, Theorem 6 of [18] and its proof (with the notations explained in detail in Appendix C in [21]) or Theorems I-III and the proof of Lemma 2.7 in [15]. By these results, we have that for an integer r > m/2+2 and any compact set K ⊂ (0, T )×M 0 , there is 0 > 0 such that for any f ∈ C r 0 (K) satisfying f C r (K) < 0 , the initial value problem (15) has a unique solution. Recall that m is the dimension of the underlying space M 0 . In particular, in the case of the Minkowski space R 1+3 , we may take r = 4, and see that source-to-solution map L A is well-defined by (7). 2.2. Notations for microlocal analysis. For a conic Lagrangian submanifold Λ 0 ⊂ T * M \ 0 and a vector bundle E over M , we denote by I p (M ; Λ 0 ; E) the space of Lagrangian distributions of order p ∈ R associated to Λ 0 , and taking values in E. If Λ 1 ⊂ T * M \ 0 is another conic Lagrangian submanifold intersecting Λ 0 cleanly, we denote by I p (M ; Λ 0 , Λ 1 ; E) the space of Intersecting Pair of Lagrangian (IPL) distributions of order p ∈ R associated to (Λ 0 , Λ 1 ), and taking values in E. We use occasionally the notation ·, · for the duality pairing between covectors and vectors, and I(M ; Λ 0 ; E) = p∈R I p (M ; Λ 0 ; E), I(M ; Λ 0 , Λ 1 ; E) = p∈R I p (M ; Λ 0 , Λ 1 ; E). If Λ 0 coincides with the conormal bundle N * K = {(x, ξ) ∈ T * M : x ∈ K, ξ, v = 0 for all v ∈ T x K} of a submanifold K ⊂ M in the sense that Λ 0 = N * K \ 0, then the distributions in I(M ; Λ 0 ; E) are called conormal distributions. Although removing the zero section from N * K, when considering it as a conic Lagrangian manifold, is somewhat awkward notationally, it is natural to consider N * K as a submanifold of T * M , since then the fibres N * x K ⊂ T * x M , x ∈ K, are linear subspaces. We recall the basic properties of conormal and IPL distributions in Appendix A below. The wave front set of a distribution u ∈ D (M ) is denoted by WF(u), see [11,Def. 2.5.2]. It is a subset of T * M \0, and its projection on M is called the singular support singsupp(u) of u. The wave front set WF(u) is conical and closed in T * M \ 0, and it is occasionally convenient to use the notation ccl B = {(x, λξ) ∈ T * M \ 0 : (x, ξ) ∈ B, λ > 0} for the conical closure of a set B ⊂ T * M \ 0. For u ∈ I(M ; Λ 0 ; E) it holds that WF(u) ⊂ Λ 0 . If K is the Schwartz kernel of a pseudodifferential operator χ on M , then the projection of WF(K ) ⊂ (T * M \ 0) 2 on the first factor T * M \ 0 is called the essential support of χ. (As WF(K ) is contained in the conormal bundle of the diagonal {(x, y) ∈ M 2 : x = y}, the choice between the first and second factor makes no difference.) Following [11, p. 124] we write WF(χ) for this set. We denote by Ω 1/2 the half-density bundle over M . When Λ 0 and Λ 1 \ Λ 0 coincide with conormal bundles, and E = E ⊗ Ω 1/2 , there is a coordinate invariant way to define the principal symbol σ[u] of u ∈ I(M ; Λ 0 ; E), respectively u ∈ I(M ; Λ 0 , Λ 1 ; E), as an equivalence class of sections of E ⊗ Ω 1/2 over Λ 0 , respectively Λ 1 \ Λ 0 . We will not emphasize the difference between the equivalence class σ[u] and a representative of it, and we will also use the same notation for the half-density bundles over M and Λ j , j = 0, 1. Let us remark that there is typically no natural way to relate these bundles. For example, while it is natural to use \|g\| 1/4 to trivialize Ω 1/2 over M , the Lorentzian metric g on M typically does not induce a natural trivialization of Ω 1/2 over Λ j . For IPL distributions in I(M ; Λ 0 , Λ 1 ; E), there is also a refined notion of principal symbol, with components on both Λ 0 and Λ 1 . We will use the refined principal symbol only in Appendix A. The notation σ[χ] is used also for the principal symbol of a pseudodifferential operator χ on M . In this case, σ[χ] is represented by a section of T * M \ 0. 2.3. Microlocal analysis of the wave operator. It is convenient to rescale (8), and consider the following non-linear wave operator Q 0 (φ) = 1 2 ( A φ + κ\|φ\| 2 φ), where \|·\| = \|·\| E is the norm with respect to the inner product ·, · = ·, · E . In order to make use of the microlocal machinery developed in [9], we conjugate the operator Q 0 with the half density \|g\| 1/4 and consider the operator Q(u) := \|g\| 1/4 Q 0 (\|g\| −1/4 u) acting on the sections of E ⊗ Ω 1/2 . Writing P u = \|g\| 1/4 A (\|g\| −1/4 u)/2, the operator Q reads Q(u) = P u + κ 2 \|g\| −1/4 u 2 u.(16) For the sake of convenience, we will slightly abuse the notation, and write uv = \|g\| 1/4 ((\|g\| −1/4 u)(\|g\| −1/4 v)) for products of half-densities as functions. Then Q(u) = P u + κ\|u\| 2 u/2. Writing ı = √ −1, the full symbol of the operator P reads P (x, ξ) = g ij ξ i ξ j /2 + ı −1 (∂ x i g ij /2 + g ij A i )ξ j +σ[P ](x, ξ) = g ij ξ i ξ j /2, σ sub [P ](x, ξ) = ı −1 g ij A i ξ j , (x, ξ) ∈ T * M.(17) We write also σ[P ] = ξ, ξ g /2 where ξ, ξ g denotes the inner product with respect to g. Let us remark that the subprincipal symbol transforming as a connection is discussed in [16] in the more general context of pseudodifferential operators on vector bundles. We denote by H P the Hamiltonian vector field associated to σ[P ], and by Σ(P ) the characteristic set of P . That is, H P = g ij ξ j ∂ x i − 1 2 (∂ x i g jk )ξ j ξ k ∂ ξ i ,(18)Σ(P ) = {(x, ξ) ∈ T * M \ 0 : ξ, ξ g = 0}. The covectors ξ satisfying ξ, ξ g = 0 are called lightlike. We denote by Φ s , s ∈ R, the flow of H P , and define for a set B ⊂ Σ(P ) the future flowout of B by {(y, η) ∈ Σ(P ); (y, η) = Φ s (x, ξ), s ∈ R, (x, ξ) ∈ B, y ≥ x}.(19) Let us recall the parametrix construction for the linear wave equation P u = f, in R × M 0 , u\| t<0 = 0,(20) that originates from [9]. We will follow the purely symbolic construction from [31], the only difference being that u is vector valued in our case. For the convenience of the reader, we give a proof of the below theorem in Appendix A. Theorem 2. Let f ∈ I k (M ; Λ 0 ; E ⊗ Ω 1/2 ) for a conic Lagrangian Λ 0 ⊂ T * M \ 0. Suppose that H P is nowhere tangent to Λ 0 , write B = Λ 0 ∩Σ(P ), and denote by Λ 1 the future flowout of B. Then the solution u of (20) is in I k−2+1/2 (M ; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ). Moreover, (L H P + ıσ sub [P ])σ[u] = 0, on Λ 1 \ Λ 0 ,(21)σ[u] = R((σ[P ]) −1 σ[f ]), on B,(22) where L H P is the Lie derivative with respect to H P , σ[u] and σ[f ] are the principal symbols of u and f on Λ 1 and Λ 0 , respectively, and R is a map, defined by (72) in Appendix A below, R : S k−1/2+n/4 (Λ 0 \ ∂Λ 1 ; Ω 1/2 ) −→ S k+n/4 (Λ 1 ; Ω 1/2 )\| ∂Λ 1 , that acts as a multiplication by a scalar on E. Here it is assumed that Λ 0 and Λ 1 \ Λ 0 coincide with conormal bundles. 2.4. Flowout from a point in the Minkowski space. The following case will be of particular importance for us. We have also included a detailed discussion of Theorem 2 in the context of this example case in Appendix B. Example 1. Let (M, g) be the 1 + 3-dimensional Minkowski space, and consider a lightlike vector ξ 0 ∈ T 0 R 1+3 \ 0 of the form ξ 0 = (1, θ 0 ) where θ 0 is in the unit sphere S 2 = {θ ∈ R 3 : \|θ\| = 1}. Let c ∈ E 0 \ 0, that is, c is a non-zero vector in the fibre of E over the origin, and let χ be a pseudodifferential operator such that σ[χ] = 0 near ccl{(0, ξ 0 )}. We define f ∈ I(M ; Λ 0 ; E ⊗ Ω 1/2 ), f = c\|g\| 1/4 χδ, where Λ 0 = T 0 R 1+3 \ 0 = N * {0} \ 0 and δ is the Dirac delta distribution at the origin. The corresponding future flowout Λ 1 satisfies Λ 1 \ Λ 0 ⊂ N * K \ 0 where K = {(t, tθ) ∈ R 1+3 : t > 0, θ ∈ S 2 }(23) is the future light-cone in the spacetime R 1+3 emanating from the origin. Letting u be the solution of (20), its restriction on R 1+3 \ 0 is a conormal distribution in I(R 1+3 \ 0; N * K \ 0; E ⊗ Ω 1/2 ). As σ[f ](0, ξ 0 ) = 0, Theorem 2 implies that for all s > 0 it holds that σ[u](γ(s), ξ 0 ) = 0 where γ(s) = (s, sθ 0 ) , and ξ 0 is viewed also as an element of T γ(s) R 1+3 . The smaller the essential support WF(χ) is chosen around ccl(0, ±ξ 0 ) := ccl{(0, ξ 0 ), (0, −ξ 0 )}, the smaller is singsupp(u) ⊂ K = K ∪ {0} around γ(R + ) := {γ(s) : s ≥ 0}. The pseudodifferential operator χ can be chosen for example as follows. Choose functions χ 1 ∈ C ∞ (S 2 ), χ 2 ∈ C ∞ (R) and χ 3 ∈ C ∞ (R 1+3 ) such that χ 1 (θ 0 ) = 1, χ 2 (1) = 1 and χ 3 (0) = 1. Let also q ∈ R. Then, writing ξ = (ξ 0 , ξ ) and c(ξ) = ( √ 2\|ξ\|) −1 , with \|ξ\| the Euclidean norm of ξ, we define the function χ 0 (x, ξ) = χ 3 (x)χ 2 (c(ξ)ξ 0 )χ 1 (c(ξ)ξ )\|ξ\| q .(24) Now χ 0 is positively homogeneous of degree q. Choose, furthermore, χ 4 ∈ C ∞ 0 (R 1+4 ) such that χ 4 = 1 near the origin. Then (1 − χ 4 (ξ))χ 0 (x, ξ) is smooth also near ξ = 0, and it is a symbol in the sense of [11, Def. 1.1.1]. Now we define a pseudodifferential operator by χu = R 1+4 e ıξx (1 − χ 4 (ξ))χ 0 (x, ξ)û(ξ)dξ. Ignoring 2π factors, the full symbol of χ is simply (1−χ 4 (ξ))χ 0 (x, ξ), and the principal symbol σ[χ] is the corresponding equivalence class modulo symbols of one degree lower order. Choose now anotherχ 4 ∈ C ∞ 0 (R 1+4 ) such thatχ 4 = 1 near the origin. Then (1 − χ 4 (ξ))χ 0 (x, ξ) − (1 −χ 4 (ξ))χ 0 (x, ξ) = (χ 4 (ξ) − χ 4 (ξ))χ 0 (x, ξ) is smooth near ξ = 0 sinceχ 4 − χ 4 = 0 there. Moreover, it is a symbol of order −∞ as both χ 4 andχ 4 are compactly supported. Therefore also ( 1 −χ 4 (ξ))χ 0 (x, ξ) is a representative of σ[χ] . As the support of χ 4 can be chosen arbitrarily small it makes sense to say that σ[χ] = χ 0 although this does not quite hold for any representative. Also for q = 0 it holds, in this sense, that σ[χ] = 1 in ccl{(0, ξ 0 )}. Observe that WF(χ) becomes small around ccl{(0, ξ 0 )} when the supports of χ j , j = 1, 2, 3, are small around θ 0 , 1, and 0 respectively. Also χu(x) = 0 for any x outside the support of χ 3 , and for any k ∈ N we can choose q < 0 negative enough, so that χδ ∈ H k (R 1+3 ). Let us check that indeed Λ 1 \ Λ 0 = N * K \ 0. The lightlike vectors in T 0 R 1+3 \ 0 are given by (λ, λθ) with λ ∈ R \ 0 and θ ∈ S 2 . In the Minkowski space, the tangentcotangent isomorphism corresponds to changing the sign of the first component. Therefore, Λ 1 \ Λ 0 = {(sλ, sλθ; −λ, λθ) ∈ R 1+3 × R 1+3 : λ ∈ R \ 0, θ ∈ S 2 , sλ > 0}. We can also reparametrize Λ 1 \ Λ 0 = {(t, tθ; −λ, λθ) ∈ R 1+3 × R 1+3 : λ ∈ R \ 0, θ ∈ S 2 , t > 0}. Clearly the projection of Λ 1 \ Λ 0 on the base space R 1+3 is K. For the convenience of the reader, we will still compute explicitly the conormal bundle of K. Toward that end, we choose local coordinates R 2 ⊃ B a → Θ(a) ∈ S 2 on S 2 and see that the tangent space of K at (t, θ 0 ) is given by the range of 1 0 θ 0 t dΘ da .(25) Writing θ ⊥ 0 = {v ∈ R 3 : v · θ 0 = 0} where · denotes the Euclidean inner product, we have that θ ⊥ 0 is the range of dΘ/da. Writing (ξ 0 , ξ ) ∈ N * (t,θ 0 ) K, this fibre is characterized by ξ 0 δt + ξ θ 0 δt + ξ δθ = 0, δt ∈ R, δθ ∈ θ ⊥ 0 . Taking first δt = 1 and δθ = 0 we have ξ 0 = −ξ θ 0 . Letting then δθ vary we see that ξ ⊥ θ ⊥ 0 , that is, ξ = λθ 0 where λ ∈ R. Here we can view θ 0 ∈ R 3 as a covector since the tangent-cotangent isomorphism in R 3 is the identity. Hence N * (t,θ 0 ) K \ 0 = {(−λ, λθ 0 ) : λ ∈ R \ 0}, and indeed Λ 1 \ Λ 0 = N * K \ 0. Observe that Λ 1 \ Λ 0 is embedded in the following smooth submanifold of T * M \ 0, that is the flowout to both past and futurê Λ 1 = {(t, tθ; −λ, λθ) ∈ R 1+3 × R 1+3 : λ ∈ R \ 0, θ ∈ S 2 , t ∈ R}.(26) Note that while K is singular at t = 0, we see thatΛ 1 is not by considering the derivative analogous to (25),     1 0 0 θ 0 t dΘ da 0 0 0 −1 0 λ dΘ da θ 0     . This matrix is injective since λ = 0. We will next write the transport equation (21) for the principal symbol σ[u] as a parallel transport equation with respect to the covariant derivative ∇, and we begin by discussing Ω 1/2 over the flowout Λ 1 . 2.5. Trivialization of the half-density bundle over the flowout. We want to trivialize Ω 1/2 in a way that preserves homogeneity properties, as possessed for example by χ 0 in (24). Let us point out that, even in the context of Example 1, there appears to be no canonical choice of a non-vanishing section of Ω 1/2 over the conormal bundle Λ 1 \ Λ 0 = N * K \ 0. For example, the Sasaki metric on T * R 1+3 , associated with the Minkowski metric, is degenerate when restricted on N * K \ 0. The submanifold K in Example 1 is of codimension one in R 1+3 . This holds in general in the sense that, if Λ 0 and Λ 1 in Theorem 2 satisfy Λ 1 \ Λ 0 = N * K \ 0 (27) for a submanifold K ⊂ M , then K is of codimension one. This can be seen as follows. Observe first that Λ 1 ⊂ Σ(P ) simply because Λ 1 is the future flowout from Λ 0 ∩ Σ(P ). Therefore, for any x ∈ K, the fibre N * x K can contain only lightlike vectors with respect to g. On the other hand, if ξ 1 , ξ 2 ∈ T * x M are lightlike and linearly independent, then their linear span satisfies, see e.g. [33, Cor. 1.1.5], {ξ ∈ span(ξ 1 , ξ 2 ) : ξ is lightlike} = span(ξ 1 ) ∪ span(ξ 2 ).(28) In particular span(ξ 1 , ξ 2 ) contains vectors that are not lightlike, and therefore at most one of ξ j , j = 1, 2, can belong to N * x K. This shows that N * x K is of dimension one, or equivalently, K is of codimension one in M . We will trivialize Ω 1/2 over Λ 1 by choosing a strictly positive half-density ω in C ∞ (Λ 1 ; Ω 1/2 ) that is positively homogeneous of degree 1/2. We begin by recalling the definition of positive homogeneity following [13, p. 13]. Let λ ∈ R \ 0 and define m λ : T * M \ 0 → T * M \ 0, m λ (x, ξ) = (x, λξ). Then m λ restricts as a map on Λ 1 in a natural way, and we denote the restriction still by m λ . The half-density ω is said to be positively homogeneous of degree q ∈ R if m * λ ω = λ q ω for all λ > 0. If in local coordinates (29) K = {(x 0 , x ) ∈ R 1+m : x 0 = 0}, then N * K = {(x 0 , x ; ξ 0 , ξ ) ∈ R 1+m × R 1+m : x 0 = 0, ξ = 0}. We emphasize that, as the conormal bundle of K coincides with the flowout Λ 1 in the sense of (27), the local coordinate x 0 is not the time coordinate t in (13). Here (x, ξ) = (x 0 , x ; ξ 0 , ξ ) are the induced coordinates on T * M . Considering the restriction m λ : N * K \ 0 → N * K \ 0, the pullback m * λ ω is given by m * λ ω(x , ξ 0 ) = dm λ d(x , ξ 0 ) 1/2 ω(x , λξ 0 ) = λ 1/2 ω(x , λξ 0 ). Thus ω being positively homogeneous of degree 1/2 means that ω(x , λξ 0 ) = ω(x , ξ 0 ), moreover, ω being strictly positive means that ω(x , ξ 0 ) > 0. We will, in fact, choose a half-density ω that is also symmetric in the sense that ω(x , λξ 0 ) = ω(x , ξ 0 ), λ ∈ R \ 0,(30) a coordinate invariant formulation of which reads m * λ ω = \|λ\| 1/2 ω for all λ ∈ R \ 0. In general, a strictly positive half-density ω ∈ C ∞ (Λ 1 ; Ω 1/2 ) satisfying (30) can be constructed by choosing an auxiliary Riemannian metric on M , restricting the associated Sasaki metric h on T * M \ 0 to Λ 1 , and taking ω = \|h\| 1/4 . In particular, when (M, g) is the Minkowski space, it feels natural to choose the Euclidean metric on M . 2.6. Parallel transport equation for the principal symbol. Let us fix a strictly positive half-density ω over N * K satisfying (30). Here the closure is taken in T * M , and we remark that, in view of (27) , Λ 1 ⊂ N * K. We write σ[u] = aω where a is a section in C ∞ (N * K \ 0; E) . A computation in coordinates shows that L H P (aω) = (H P a)ω + aL H P ω. Introducing the notation div ω H P = ω −1 L H P ω, we write ω −1 L H P (aω) = H P a + adiv ω H P . We want to further rewrite this as a conjugated differentiation along bicharacteristics, that is, along the flow curves of H P . Recall that Φ s , s ∈ R, denotes the flow of H P . Writing β(s) = Φ s (x 0 , ξ 0 ) for the bicharacteristic through (x 0 , ξ 0 ) ∈ N * K \ 0, we have (H P a) • β(s) = ∂ r a(Φ r (β(s)))\| r=0 = ∂ s (a • β)(s). We further write α = a • β, and define ρ(s) = s 0 ρ (s )ds , ρ = div ω H P • β.(31) Then ω −1 L H P (aω) • β = ∂ s α + ρ α = e −ρ ∂ s (e ρ α) . We denote by γ the projection of the bicharacteristic β to the base manifold M , and byγ * the tangent vector of γ as a covector, that is,γ * i = g ijγ j . It follows from (18) that γ is a geodesic of (M, g), and that β(s) = (γ(s),γ * (s)). As β(s) ∈ Σ(P ), the geodesic γ is lightlike. Moreover, using (17), ı(σ sub [P ] • β)(s) = g ij A iγ * j (s) = A iγ i (s) = A,γ(s) , where ·, · is again the duality pairing between covectors and vectors. The covariant derivative on the bundle E along the geodesic γ(s) is given by ∇γ = ∂ s + A,γ(s) . Therefore (21) along β is equivalent with e −ρ ∇γ(e ρ α) = 0. If we define the following symbol along β for conormal distributions in I(M ; N * K \ 0; E ⊗ Ω 1/2 ), σ ω,β [·] = e ρ (ω −1 σ[·]) • β, then we can rewrite (21) along β as follows (32) ∇γσ ω,β [u] = 0. This is the parallel transport equation along γ with respect to the connection A. If x = γ(s 0 ) and y = γ(s 1 ) for some s 0 , s 1 ∈ R, then we write P A y←x : E x → E y for the parallel transport map from x to y along γ. That is, P A y←x v = w(s 1 ) where w is the solution of (33) ẇ + A,γ w = 0, on (s 0 , s 1 ), w(s 0 ) = v. In general, the map P A y←x depends on the geodesic γ joining x and y, but not on the parametrization of γ. We do not emphasize the dependency on γ in our notation, since we are mainly interested in the Minkowski case, and in this case P A y←x depends only on the points x and y. To summarize, writing ξ =γ * (s 0 ) and η =γ * (s 1 ), it follows from (32) that e ρ(s 1 ) (ω −1 σ[u])(y, η) = e ρ(s 0 ) P A y←x ((ω −1 σ[u])(x, ξ)).(34) 2.7. Positively homogeneous symbols. We will now consider how (34) changes under rescaling of ξ ∈ N * x K \ 0, assuming that σ[u] is positively homogeneous of degree q + 1/2 ∈ R at (x, ξ), that is, (ω −1 σ[u])(x, λξ) = λ q (ω −1 σ[u])(x, ξ), λ > 0.(35) Proposition 1. Let Λ j , j = 0, 1, and u be as in Theorem 2, and suppose that (27) holds. Let (x, ξ) ∈ N * K \ 0 and (y, η) = Φ s 1 (x, ξ) for some s 1 ∈ R. Suppose that (35) holds at (x, ±ξ) and that (y, η) ∈ N * K \ 0. Then e ρ(s 1 ) (ω −1 σ[u])(y, ±λη) = λ q P A y←x ((ω −1 σ[u])(x, ±ξ)), λ > 0.(36) Recall that B = Λ 0 ∩ Σ(P ) = Λ 0 ∩ Λ 1 . As the symbol σ[u] on Λ 1 \ Λ 0 is smooth up to B, equation (36) holds also when (x, ±ξ) ∈ B. Then (y, η) ∈ N * K \ 0 implies x < y, that is, the causal relation x ≤ y holds and x = y. Writing again γ(s) for the projection of β(s) = Φ s (x, ξ) to M , we have γ(0) = x < y = γ(s 1 ). Hence, if ξ is future pointing then s 1 > 0, and if ξ is past pointing then s 1 < 0. We emphasize that also past pointing singularities are propagated forward in time by the wave equation (20). Proof. For λ ∈ R \ 0, the bicharacteristic through (x, λξ) is given by β λ (s) = (γ(λs), λγ * (λs)), where γ denotes still the projection of β = β 1 to M . Analogously to (31), we write ρ λ = div ω H P • β λ and ρ λ (s) = s 0 ρ λ (s )ds , with the usual convention that s 0 ρ λ (s )ds = − 0 s ρ λ (s )ds for s < 0. By applying (34) to β λ , we obtain e ρ λ (s 1 /λ) (ω −1 σ[u])(y, λη) = P A y←x ((ω −1 σ[u])(x, λξ)) = \|λ\| q P A y←x ((ω −1 σ[u])(x, ±ξ)) , where the sign is the one in λ = ±\|λ\|. It remains to show that e ρ λ (s 1 /λ) = e ρ(s 1 ) . We denote the restriction of the flow Φ s on N * K \ 0 still by Φ s . In coordinates satisfying (29), the Lie derivative L H P ω is of the form L H P ω(x , ξ 0 ) = ∂ s Φ * s ω(x , ξ 0 )\| s=0 = ∂ s dΦ s d(x , ξ 0 ) 1/2 ω(Φ s (x , ξ 0 )) s=0 = ∂ s dΦ s d(x , ξ 0 ) 1/2 s=0 ω(x , ξ 0 ) + ∂ s ω(Φ s (x , ξ 0 )) s=0 , where we used the fact that Φ 0 = id, and therefore \|dΦ 0 /d(x , ξ 0 )\| 1/2 = 1. We write H P (x , ξ 0 ) = H 0 (x , ξ 0 )∂ ξ 0 + n i=1 H i (x , ξ 0 )∂ x i for the Hamiltonian vector field H P on N * K \ 0. Then, see e.g. p. 418 of [29], ∂ s dΦ s d(x , ξ 0 ) 1/2 s=0 = 1 2 ∂ s dΦ s d(x , ξ 0 ) s=0 = 1 2 ∂ ξ 0 H 0 + n i=1 ∂ x i H i . Equation (18), together with the fact that ξ = 0 on N * K, implies that ∂ ξ 0 H 0 + n j=1 ∂ x i H i = −(∂ x 0 g 00 )ξ 0 + n i=1 (∂ x i g 0i )ξ 0 . In particular, (∂ ξ 0 H 0 )(x , λξ 0 ) + n j=1 (∂ x i H i )(x , λξ 0 ) = λ(∂ ξ 0 H 0 )(x , ξ 0 ) + λ n j=1 (∂ x i H i )(x , ξ 0 ). Moreover, as ω satisfies (30), ∂ s ω(Φ s (x , λξ 0 ))\| s=0 = ∂ s ω(γ(λs), λγ * (λs))\| s=0 = ∂ s ω(γ(λs),γ * (λs))\| s=0 = λ∂ s ω(γ(s),γ * (s))\| s=0 = λ∂ s ω(Φ s (x , ξ 0 ))\| s=0 . The above six equations imply that L H P ω(x , λξ 0 ) = λL H P ω(x , ξ 0 ). This again implies that div ω H P (x , λξ 0 ) = ω −1 (x , λξ 0 )L H P ω(x , λξ 0 ) = λdiv ω H P (x , ξ 0 ), and therefore the change of variables s = λs gives ρ λ (s 1 /λ) = s 1 /λ 0 div ω H P (γ(λs), λγ * (λs))ds = s 1 0 div ω H P (γ(s ),γ * (s ))ds = ρ(s 1 ). Microlocal analysis of the interaction of three waves The core idea of the proof of Theorem 1 is to choose the source f in (8) as the weighted superposition of three singular sources, f ( ) = 3 j=1 j f j , = ( 1 , 2 , 3 ), j ∈ R,(37) where f j , j = 1, 2, 3, are conormal distributions supported in satisfying supp(f j ) ∩ J + (supp(f k )) = ∅, for j = k.(38) Recall that is the set where the measurements are gathered, see the definition (7) of the source-to-solution map L A . Each f j will be similar to the source f in Example 1. We denote by φ = φ( ) the solution of (8) with the source f = f ( ), and write φ in = ∂ 1 φ\| =0 , φ out = ∂ 1 ∂ 2 ∂ 3 φ\| =0 .(39) We will choose f 1 so that φ in is singular along a lightlike geodesic γ in intersecting . Then we choose f 2 and f 3 to be such perturbations of f 1 that φ out is singular along another lightlike geodesic γ out that intersects both γ in and . Let us fix the parametrization of γ in and γ out so that γ in (0) = x ∈ , γ in (s in ) = γ out (0) = y, γ out (s out ) = z ∈(40) for some s in , s out ∈ R. Taking into account the fact that the wave equation (8) propagates singularities forward in time, we consider only the case x < y < z, see Figure 2, left. (5), is the blue cylinder, and set D, see (9), is the diamond like region drawn with dashed curves. Left. Points x, z ∈ and y ∈ D as in (40). The geodesic segment from x to y is γ in and the segment from y to z is γ out . We write also γ in = γ y←x and γ out = γ z←y . Right. Points x j , j = 1, 2, 3, see (50), together with the geodesic segments γ y←x j . Here x 1 = x. We will use the following shorthand notations related to (40), L = {(x, y) ∈ M 2 : there is a lightlike geodesic joining x and y}, S + ( ) = {(x, y, z) ∈ M 3 : (x, y), (y, z) ∈ L, x < y < z, x, z ∈ , y / ∈ }. Assume from now on that (M, g) is the 1 + 3-dimensional Minkowski space, and denote the geodesic joining a pair (x, y) ∈ L by γ y←x . What follows is independent from the parametrization of γ y←x , but let us fix it by requiring that γ y←x (0) = x andγ y←x (0) = (1, θ) where θ ∈ S 2 . We define also the broken non-abelian light ray transform of the connection A by S A z←y←x = P A z←y P A y←x , (x, y, z) ∈ S + ( ).(42) Now we can summarize the above discussion as follows: for each (x, y, z) ∈ S + ( ) we want to choose f j , j = 1, 2, 3, so that φ in is singular along γ y←x and that φ out is singular along γ z←y . Moreover, we will choose f j , j = 1, 2, 3, in such a way that, in a suitable microlocal sense, φ out is a conormal distribution. Then we will show using Proposition 1 that S A z←y←x is determined by the principal symbol of φ out \| when f j , j = 1, 2, 3, are varied. As each f j is supported in , this will show that the source-to-solution map L A determines S A z←y←x for all (x, y, z) ∈ S + ( ). 3.1. The linear span of three lightlike vectors. We will begin with a lemma in linear algebra that will be a key component when proving that singularities propagating along any γ z←y , with the pair (y, z) in S out ( ) = {(y, z) : (x, y, z) ∈ S + ( ) for some x ∈ }, can be generated by choosing suitable f j , j = 1, 2, 3, supported in . We believe that this lemma can be also used to simplify the proofs of previous results, such as [22,23,28]. These results are based on using a superposition of four singular sources, which leads to more complicated computations. To highlight the difference, we recall that all the previous results consider nonlinear wave equations on 1 + 3-dimensional Lorentzian manifolds, and use the fact that if N * K j \ 0, j = 1, 2, 3, 4, are flowouts analogous to N * K \ 0 in Example 1, then generically 4 j=1 K j is discrete. It follows that N * y ( 4 j=1 K j ) = T * y M for a point y ∈ 4 j=1 K j . Then, using notation analogous with the above, it is shown that φ out = ∂ 1 ∂ 2 ∂ 3 ∂ 4 φ\| =0 can be made singular along any γ z←y with (y, z) ∈ S out ( ). Asγ z←y (0) ∈ N * y ( 4 j=1 K j ) trivially, there is no geometric obstruction for φ out being singular onγ z←y in view of the product calculus for conormal distributions, see Lemma 2 below. On the other hand, we will use only three flowouts, and the intersection 3 j=1 K j will be of dimension one. Lemma 1 implies, however, that for any fixed y ∈ K 1 and any fixed lightlike η ∈ T * y M we can guarantee that (y, η) ∈ N * ( 3 j=1 K j ) by choosing K 2 and K 3 carefully. Remark 2. Consider the solution φ( ) of (11) with vanishing initial conditions and the source f ( ) = 1 f 1 + 2 f 2 , with f j a conormal distribution supported in satisfying (38). Regardless of the degree of non-linearity N ≥ 2, it is not possible to make ∂ 1 ∂ 2 φ\| =0 singular along arbitrary γ z←y with (y, z) ∈ S out ( ). Indeed, using (28) it can be shown that singsupp(∂ 1 ∂ 2 φ\| =0 ) ⊂ 2 j=1 singsupp(∂ j φ\| =0 ). This explains the lower bound J ≥ 3 in (12). The following lemma is formulated for the Minkowski space but, being of pointwise nature, it generalizes to an arbitrary Lorentzian manifold. Lemma 1. Let y be a point in R 1+3 , and let ξ 1 , η ∈ T y R 1+3 \ 0 be lightlike. In any neighbourhood of ξ 1 in T y R 1+3 , there exist two lightlike vectors ξ 2 , ξ 3 such that η is in span(ξ 1 , ξ 2 , ξ 3 ). Proof. The statement is invariant with respect to non-vanishing rescaling of ξ 1 and η, and we assume without loss of generality that ξ 1 = (1, ξ 1 ) and η = (1, η ). Let ν ∈ S 2 be such that ξ 1 , η ⊥ ν. After a rotation in R 3 , we may assume that ν = (0, 0, 1). Then ξ 1 = (θ 1 , 0) and η = (θ 0 , 0) with θ 1 , θ 0 ∈ S 1 . After a rotation in R 2 , we may assume that θ 1 = (1, 0). We write a(r) = √ 1 − r 2 for r ∈ [−1, 1]. Then there is r 0 ∈ [−1, 1], and a choice of sign, such that θ 0 = (±a(r 0 ), r 0 ). To summarize, we may assume without loss of generality that ξ 1 = (1, 1, 0, 0), η = (1, ±a(r 0 ), r 0 , 0).(43) It is enough to show that in any neighbourhood U 0 ⊂ S 1 of θ 1 there are θ 2 , θ 3 ∈ U 0 such that (1, θ j ), j = 1, 2, 3, span R 3 . We set θ 2 = (a(r), r) and θ 3 = (a(r), −r) where r > 0 is chosen to be small enough so that θ j ∈ U 0 , j = 2, 3. Now (1, θ j ), j = 1, 2, 3, are linearly independent since 1 1 1 1 a(r) a(r) 0 r −r = 2r(1 − a(r)) = 0. To summarize, we may choose ξ 2 = (1, a(r), r, 0), ξ 3 = (1, a(r), −r, 0).(44) We will need also an explicit version of the above lemma. As in the proof, we assume without loss of generality that (43) holds, and choose ξ 2 and ξ 3 according to (44). Then r 2 η = (−2b + O(r))ξ 1 + (b + O(r))ξ 2 + (b + O(r))ξ 3 ,(45) where b = 1 ∓ a(r 0 ) = 1 ∓ 1 − r 2 0 . Remark 3. In the context of (40), we will take ξ 1 =γ in (s in ) and η =γ out (0). Assuming that y / ∈ , the condition γ out (s out ) ∈ implies that the second component of η in (43), that is, ±a(r 0 ), must have negative sign. Then b ≥ 1, and in particular, η is not in span(ξ j , ξ k ) for any j, k ∈ {1, 2, 3}. This fact will be important in what follows, and the set was chosen to be of the particular form (5) to avoid technicalities related, for example, to geodesics that return to after exiting it. To simplify the notation, we assume that κ = −1 in (4). The general case is analogous. We introduce u j 1 ,...,j l = ∂ j 1 · · · ∂ j l u, v j 1 ,··· ,j l = u j 1 ,··· ,j l \| =0 , and write Re for the real part of a complex number. With these notations, we differentiate (46), and obtain P u j − Re( u j , u )u − 1 2 \|u\| 2 u j = \|g\| 1/4 f j . Together with the fact that u\| =0 = 0, this implies P v j = \|g\| 1/4 f j . Differentiating the equation (46) twice yields (47) P u jk − Re( u jk , u + u j , u k )u − Re( u j , u )u k − Re( u k , u )u j − 1 2 \|u\| 2 u jk = 0, and v jk = 0. Finally, one more differentiation gives the desired linear equation, that we call the three-fold linearization, (48) P v 123 = Re( v 1 , v 2 )v 3 + Re( v 1 , v 3 )v 2 + Re( v 2 , v 3 )v 1 . The right-hand side of (48) is the sum of products of v 1 , v 2 , v 3 , and we call it the three-fold interaction of these three solutions to the linear wave equation (20). Remark 4. Consider the solution φ( ) of (11) with vanishing initial conditions and the source f ( ) = N j=1 j f j , with f j a distribution supported in . Analogously to (47), we see that ∂ α φ\| =0 = 0 for any multi-index α ∈ N 1+3 satisfying \|α\| < N where N is the degree of non-linearity in the equation (11). This explains the lower bound J ≥ N in (12). 3.3. Wave interactions as products of conormal distributions. Consider a triple (x, y, z) ∈ S + ( ) and let us construct f j , j = 1, 2, 3, such that φ out , defined by (39), is singular on γ z←y . We write η =γ * z←y (0), ξ 1 =γ * y←x (s in ),(49) where s in > 0 satisfies γ y←x (s in ) = y. The geodesic γ with the initial condition (x, ξ) is denoted by γ(·; x, ξ). Let V ⊂ T * y M be a small enough neighbourhood of ξ 1 so that for all ξ ∈ V it holds that γ(−s in ; y, ξ) ∈ and that ξ is future pointing. Let ξ 2 , ξ 3 ∈ V be two lightlike vectors as in Lemma 1 and write x j = γ(−s in ; y, ξ j ), j = 1, 2, 3. (50) With a slight abuse of notation, we write also ξ j =γ(−s in ; y, ξ j ), see Figure 2, right, for the geometric setup. Analogously to Example 1, we let c j ∈ E x j \ 0, and let χ j be a pseudodifferential operator with the following properties: (χ1) σ[χ j ] is positively homogeneous of degree q, symmetric with respect to 0 ∈ T * M , and real valued, (χ2) σ[χ j ] = 0 near (x j , ξ j ), (χ3) WF(χ j ) is contained in a small neighbourhood of ccl(x j , ±ξ j ). The degree q of χ j is chosen to be small enough so that f j ∈ C 4 (M ) where f j ∈ I(M ; N * {x j } \ 0; E), f j = c j χ j δ x j .(51) Here δ x j is the Dirac delta distribution at x j . Moreover, we choose χ j so that supp(χ j δ j ) ⊂ and that the support condition (38) is satisfied. Recalling that C ⊂ C 4 0 ( ; E) is the domain of the source-to-solution map L A , see (7), we have then that the linear combination 1 f 1 + 2 f 2 + 3 f 3 is in C for small enough j ≥ 0. Recall that v j denotes the solution of P v j = \|g\| 1/4 f j , v j \| t<0 = 0. (52) Moreover, we denote the restriction of v j on R 1+3 \ {x j } still by v j , and write K j = x j + K where K is as in (23). That is, writing x j = (t j , x j ), K j = {(t j + s, x j + sθ) ∈ R 1+3 : s > 0, θ ∈ S 2 }. (53) Then v j ∈ I(R 1+3 \ x j ; N * K j \ 0; E ⊗ Ω 1/2 ). As in Example 1, property (χ3) implies that singsupp(v j ) is contained in a small neighbourhood S j ⊂ K j of γ y←x j (R + ). Clearly y ∈ 3 j=1 γ y←x j (R + ) ⊂ 3 j=1 K j . It follows from Remark 3 that the covectors ξ j , j = 1, 2, 3, are linearly independent. This again implies that N * y (K 1 ∩ K 2 ∩ K 3 ) = span(ξ 1 , ξ 2 , ξ 3 ). When WF(χ j ), j = 1, 2, 3, are small enough, this also guarantees for distinct j, k, l that K j and K k , as well as K j and K k ∩ K l , are transversal in S := S 1 ∪ S 2 ∪ S 3 . Let us now consider products of the form Re( v j , v k )v l , with distinct j, k, l, that appear on the right-hand side of (48). As s in > 0, it holds that y = x j for each j = 1, 2, 3, and v j are conormal distributions near y. Thus their products can be analysed using the product calculus for conormal distributions. We recall this calculus in the next lemma, that is a variant of [10, Lemma 1.1]. With obvious modifications it holds also when the conormal distributions u j , j = 1, 2, take values on E ⊗ Ω 1/2 and the product is defined in terms of ·, · E , and also when u 1 takes values on Ω 1/2 and u 2 on E ⊗ Ω 1/2 and the product is defined in terms of the scalar-vector product on E. Lemma 2. Let K j ⊂ M , j = 1, 2, be transversal submanifolds, and let u j be a conormal distribution in I(M ; N * K j \ 0; Ω 1/2 ). For a fixed nowhere vanishing halfdensity µ ∈ C ∞ (M ; Ω 1/2 ), we define the product of u 1 and u 2 by u 1 u 2 = µ((µ −1 u 1 )(µ −1 u 2 )). Let χ be a pseudodifferential operator with WF(χ) disjoint from both N * K j , j = 1, 2. It follows that χ(u 1 u 2 ) ∈ I(M ; N * (K 1 ∩ K 2 ) \ 0; Ω 1/2 ), with the principal symbol, ignoring the 2π and ı factors, σ[χ(u 1 u 2 )](x, ξ) = µ −1 (x)σ[χ](x, ξ)σ[u 1 ](x, ξ (1) )σ[u 2 ](x, ξ (2) ). where ξ = ξ (1) + ξ (2) with ξ (j) ∈ N * x K j \ 0 and x ∈ K 1 ∩ K 2 . Let us remark that, due to the transversality of K 1 and K 2 , it holds that N * x (K 1 ∩ K 2 ) = N * x K 1 ⊕ N * x K 2 , x ∈ K 1 ∩ K 2 . Therefore the decomposition ξ = ξ (1) + ξ (2) , ξ (j) ∈ N * x K j , is unique for any covector ξ ∈ N * x (K 1 ∩ K 2 ) . The condition that both ξ (1) and ξ (2) are non-zero is equivalent with (x, ξ) ∈ N * (K 1 ∩ K 2 ) \ (N * K 1 ∪ N * K 2 ). We return to our study of v j , j = 1, 2, 3. Due to Remark 3, we can choose a pseudodifferential operator χ such that χ = 1 near ccl(y, η), and that WF(χ) is contained in a small conical neighbourhood of ccl(y, η). Then applying Lemma 2 twice, we obtain χ(Re( v j , v k )v l ) ∈ I(R 1+3 ; Λ 0 ; E ⊗ Ω 1/2 ),(54) where Λ 0 = N * (K 1 ∩K 2 ∩K 3 )\0 and j, k, l ∈ {1, 2, 3} are distinct. For the convenience of the reader, we give a detailed proof of this. Proof of (54). To simplify the notation, we will consider the product Re( v 1 , v 2 )v 3 , the other cases being analogous. Write η = η (1) + η (2) + η (3) , η (j) ∈ N * y K j = span(ξ j ),(55) and let χ 0 and χ 1 be pseudodiffential operators such that (2) ) and χ 1 = 1 near ccl(y, η). Recall that η is not in span(ξ j , ξ k ) for any j, k ∈ {1, 2, 3}, see Remark 3, and this guarantees that χ 0 and χ 1 with the above properties exist. Furthermore, as s in > 0, we can choose χ 0 so that WF(χ 0 ) is also disjoint from {x 1 , x 2 , x 3 }. Now Lemma 2 implies that WF(χ 0 ) ∩ (N * K 1 ∪ N * K 2 ) = ∅, WF(χ 1 ) ∩ (N * (K 1 ∩ K 2 ) ∪ N * K 3 ) = ∅, χ 0 = 1 near ccl(y, η (1) + ηχ 0 (Re v 1 , v 2 ) ∈ I(R 1+3 ; N * (K 1 ∩ K 2 ) \ 0; Ω 1/2 ), χ 1 (χ 0 (Re v 1 , v 2 )v 3 ) ∈ I(R 1+3 ; Λ 0 ; E ⊗ Ω 1/2 ).(56) We write w = (1 − χ 0 )(Re v 1 , v 2 )v 3 and will show that (y, η) / ∈ WF(w). As WF(w) is conical and closed, it then follows that WF(χ 1 ) ∩ WF(w) = ∅ whenever WF(χ 1 ) is contained in a small enough conical neighbourhood of ccl(y, η). In this case χ 1 w is smooth, and (56), together with χ 1 (Re( v 1 , v 2 )v 3 ) = χ 1 (χ 0 (Re v 1 , v 2 )v 3 ) + χ 1 w, implies that χ 1 (Re( v 1 , v 2 )v 3 ) ∈ I(R 1+3 ; Λ 0 ; E ⊗ Ω 1/2 ). To simplify the notation, we write WF y (w) = WF(w) ∩ T * y R 1+3 . It remains to show that η / ∈ WF y (w). We have WF y (v j ) ⊂ N * y K j = span(ξ j ). As the vectors ξ 1 and ξ 2 are linearly independent, [14,Th. 8.2.10] implies that ξ 2 ). Moreover, as χ 0 = 1 near ccl(y, η (1) + η (2) ), (2) ). WF y ( v 1 , v 2 ) ⊂ N * y (K 1 ∩ K 2 ) = span(ξ 1 ,WF y ((1 − χ 0 )(Re v 1 , v 2 )) ⊂ span(ξ 1 , ξ 2 ) \ span(η (1) + η Using again [14,Th. 8.2.10], we have WF y (w) ⊂ {ζ 1 + ζ 2 : ζ 1 ∈ (span(ξ 1 , ξ 2 ) \ span(η (1) + η (2) )) ∪ {0}, ζ 2 ∈ span(ξ 3 )}. As the direct sum decomposition (55) is unique, and as η (1) + η (2) = 0 by Remark 3, it holds that η / ∈ WF y (w). Let us denote the right-hand side of (48) by f out , that is, f out = Re( v 1 , v 2 )v 3 + Re( v 1 , v 3 )v 2 + Re( v 2 , v 3 )v 1 . Furthermore, we write v out and w for the solutions of the following two wave equations P v out = χf out , v out \| t<0 = 0, P w = (1 − χ)f out , w\| t<0 = 0,(57) where χ is as in (54). Then the solution v 123 of (48) satisfies v 123 = v out + w, and due to (54), it holds that χf out ∈ I(R 1+3 ; Λ 0 ; E ⊗ Ω 1/2 ). We will treat w as a remainder term. Recall that y = γ z←y (0) and η =γ * z←y (0). As χ = 1 near ccl(y, η), it holds that (y, η) / ∈ WF((1 − χ)f out ). As singsupp(f out ) ⊂ S and as S is a small neighbourhood of 3 j=1 γ y←x j (R + ) in 3 j=1 K j , we see that the bicharacteristic β out (s) = (γ z←y (s), γ * z←y (s)) (58) does not intersect WF((1 − χ)f out ). Writing (z, ζ) = β out (s out ) for some s out > 0 and ζ ∈ T * z M \ 0, it follows from [12, Th. 23.2.9] that (z, ζ) / ∈ WF(w). Let us now consider the future flowout Λ 1 from Λ 0 ∩ Σ(P ), with Λ 0 as in (54). We will show in Appendix C that Λ 1 \ Λ 0 coincides with a conormal bundle, and thus we can apply Theorem 2 to the equation for v out in (57). Let us point out that this study of the structure of Λ 1 is not essential for the proof. We could alternatively treat v out as a Lagrangian distribution, but then its symbol should be viewed as a section of E ⊗ Ω 1/2 ⊗ M , with M the Maslov bundle over Λ 1 . However, we prefer to avoid technicalities related to general Lagrangian distributions in the present paper. Theorem 2 implies that v out ∈ I(M ; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ), and as (z, ζ) / ∈ WF(w), we may choose a pseudodifferential operatorχ acting on D ( ) such that σ[χ] = 1 near (z, ζ) and thatχ(v 123 \| ) ∈ I( ; Λ 1 ; E ⊗ Ω 1/2 ). Then σ[χ(v 123 \| )](z, ζ) = σ[v out ](z, ζ) and in terms of the source-to-solution map this reads as σ[χ\|g\| 1/4 ∂ 1 ∂ 2 ∂ 3 L A ( 1 f 1 + 2 f 2 + 3 f 3 )\| =0 ](z, ζ) = σ[v out ](z, ζ).(59) This shows that L A determines σ[v out ](z, ζ). We will recover S A z←y←x by considering σ[v out ] at (z, ζ) ∈ Λ 1 . From the source-to-solution map to the broken light ray transform. Having understood the propagation and interaction of the linear waves, we now prove that the source-to-solution map determines the broken non-abelian light ray transform. Theorem 3. Let A and B be two connections in R 1+3 such that L A = L B , as in Theorem 1. Then S A z←y←x = S B z←y←x for all (x, y, z) ∈ S + ( ). We give a constructive proof of Theorem 3 in the form of a method that recovers S A z←y←x for any (x, y, z) ∈ S + ( ) given L A . Therefore we continue considering only a single connection A. Let (x, y, z) ∈ S + ( ) and let η, ξ 1 ∈ T * y M be defined by (49). Without loss of generality we may assume that η and ξ 1 are of the form (43), and letting again ξ 2 , ξ 3 ∈ T * y M be as in Lemma 1, we have (45). Recall that K j is defined by (53). Decomposing η as in (55), it holds for small r > 0 that η (1) = −r −2 λ 1 ξ 1 , η (j) = r −2 λ j ξ j , j = 2, 3,(60) where λ 1 = 2b + O(r), λ j = b + O(r), j = 2, 3. We recall also that b ≥ 1, see Remark 3. In particular, λ j > 0, j = 1, 2, 3, for small r > 0. Let v j be again the solution of (52) with f j as in (51), and writê v j = σ[v j ](y, η (j) ), j = 1, 2, 3. The product calculus in Lemma 2 implies that χf out in (57) satisfies \|g(y)\| 1/2 σ[χf out ](y, η) = Re( v 1 ,v 2 )v 3 + Re( v 1 ,v 3 )v 2 + Re( v 2 ,v 3 )v 1 ,(61) where we used χ(y, η) = 1. We will apply Proposition 1 to (52). Toward that end, we choose a strictly positive ω j ∈ C ∞ (N * K j ; Ω 1/2 ) satisfying (30), and denote by ρ j the factor (31) along the bicharacteristic β j (s) = (γ y←x j (s),γ * y←x j (s)). It follows from (51) and (χ1) that in any coordinates there holds, ignoring 2π factors, σ[\|g\| 1/4 f j ](x j , ±λξ j ) = c j \|g(x j )\| 1/4 σ[χ j ](x j , ±λξ j ) = λ q c j α j , λ > 0,(62) where α j = \|g(x j )\| 1/4 σ[χ j ](x j , ξ j ). Of course, \|g(x j )\| 1/4 = 1 in the Cartesian coordinates, but α j transforms as half-density. Observe, however, that α j > 0 in any coordinate system. Let us translate the origin to x 1 and pass to the coordinates (x 0 ,x ,ξ 0 ,ξ ) in (81) in Appendix B below. In these coordinates (x 1 , −λξ 1 ) = (0; 0, −λξ 1 ) for λ > 0. Moreover, (82) and (62) give σ[v 1 ](0, −λξ 1 ) = −ι 0 λ q−1 c 1 α 1 , λ > 0, where ι 0 ∈ C \ 0 is a constant that contains all the 2π and ı factors. In particular, the positive homogeneity (35) holds for v 1 at (x 1 , −ξ 1 ). The coordinates (81) can be used in a neighbourhood of the bicharacteristic segment β − 1 (s) = (γ y←x 1 (s), −γ * y←x 1 (s)), s ∈ [0, s in ], and in these coordinates (y, −ξ 1 ) = (s in 0; 0, −ξ 1 ). It follows from Proposition 1 and (60) that e ρ 1 (s in ) (ω −1 1 σ[v 1 ])(s in , −r −2 λ 1 ξ 1 ) = (r −2 λ 1 ) q−1 P A y←x 1 ((ω −1 1 σ[v 1 ])(0, −ξ 1 )), and therefore in the coordinates (81), v 1 = −(r −2 λ 1 ) q−1 ι 0α1 P A y←x 1 c 1 ,(63) whereα 1 := e −ρ 1 (s in ) ω 1 (s in , ξ 1 )ω −1 1 (0, ξ 1 )α 1 contains all the volume factors. Analogously, we may use coordinates of the form (79) in a neighbourhood of the bicharacteristic segment β j (s), s ∈ [0, s in ], j = 2, 3, to obtain v j = (r −2 λ j ) q−1 ι 0αj P A y←x j c j , j = 2, 3,(64)whereα j = e −ρ j (s in ) ω j (s in , ξ j )ω −1 j (0, ξ j )α j . We have given the volume factorsα j , j = 1, 2, 3, in different coordinate systems, however, the way they transform under changes of coordinates is inconsequential. For our present purposes it is enough to observe that they are independent from the connection A and that they satisfyα j > 0 in any coordinate system. Let us also emphasize thatv j , as the value of a section of E ⊗ Ω 1/2 over K j at the point (y, η (j) ), is a coordinate invariant quantity. Combining (61), (63) and (64) yields \|g(y)\| 1/2 σ[χf out ](y, η) = α in c in where c in = Re( c 1,in , c 2,in )c 3,in + Re( c 1,in , c 3,in )c 2,in + Re( c 2,in , c 3,in )c 1,in ), α in = −r −6(q−1) (λ 1 λ 2 λ 3 ) q−1 \|ι 0 \| 2 ι 0α1α2α3 , c j,in = P A y←x j c j . Similarly with the above, we may apply Proposition 1 to (57) and the bicharacteristic β out defined by (58). This gives, ignoring ı and 2π factors, σ[v out ](z, ζ) = α out α in P A z←y c in , for some volume factor α out > 0 independent from the connection A. Recall that the source-to-solution map L A determines σ[v out ](z, ζ) via (59), and observe that both the factors α in and α out are independent from A. Therefore L A determines the parallel transport P A z←y c in . Letting x 2 , x 3 → x 1 = x and c 2 , c 3 → c 1 , we have P A z←y c in → 3\|P A y←x c 1 \| 2 P A z←y P A y←x c 1 = 3\|c 1 \| 2 S A z←y←x c 1 , where we used the fact that P A y←x is unitary. Thus L A determines S A z←y←x after varying c 1 ∈ E x \ 0, and we have shown Theorem 3. Inversion of the broken light ray transform From now on we assume that E is the trivial bundle M × C n . Recall the definition (5) of . More generally, we write ( ) = (0, 1) × B( ) where B( ) is the open ball of radius > 0, centred at the origin of R 3 . We use also the shorthand notation D( ( )) = {y ∈ M : there is (x, y, z) ∈ S + ( ( ))}, 0 < < 0 , where S + ( ( )) is defined by (41). We recall that S A z←y←x is defined by (42), that is, S A z←y←x = P A z←y P A y←x , (x, y, z) ∈ S + ( ), and that the parallel transport map P A y←x is the fundamental solution to the ordinary differential equation (33). In this section we will prove the following: Proposition 2. Let A and B be two connections in R 1+3 such that for all 0 < < 0 there holds (65) S A z←y←x = S B z←y←x , for all (x, y, z) ∈ S + ( ( )). Then there exists a smooth u : D(Ω( 0 )) → U (n) such that u\| ( 0 ) = id and B = u −1 du + u −1 Au. Observe that the causal diamond, defined by (9), satisfies D = D( ( 0 )), and therefore Theorem 1 follows immediately by combining Theorem 3 and Proposition 2. Similarly to S out ( ) we define S in ( ) = {(x, y) : (x, y, z) ∈ S + ( ) for some z ∈ }. Lemma 3. Let A and B be two connections in R 1+3 . We define u(y, x) = P A y←x P B x←y for (x, y) ∈ S in ( ). If (65) holds then u(y, x 1 ) = u(y, x 2 ) for all (x j , y) ∈ S in ( ), j = 1, 2. Proof. Note that P A y←x is a linear isomorphism and (P A y←x ) −1 = P A x←y . In particular, (S A z←y←x ) −1 = S A x←y←z , and (65) implies that S A x←y←z = S B x←y←z for all (x, y, z) ∈ S + ( ). Consider now y, x 1 and x 2 as in the claim. Then there is z ∈ such that (x j , y, z) ∈ S + ( ) for j = 1, 2. We have S A x 2 ←y←z S A z←y←x 1 = P A x 2 ←y P A y←z P A z←y P A y←x 1 = P A x 2 ←y P A y←x 1 . But S A x 2 ←y←z S A z←y←x 1 = S B x 2 ←y←z S B z←y←x 1 , and therefore P A x 2 ←y P A y←x 1 = P B x 2 ←y P B y←x 1 . We apply P A y←x 2 on left and P B x 1 ←y on right, and obtain P A y←x 1 P B x 1 ←y = P A y←x 2 P B x 2 ←y . Proof of Proposition 2. For y ∈ D( ) there are x, z ∈ such that (x, y, z) ∈ S + ( ) and we define u(y, x) as in Lemma 3. It follows from Lemma 3 that u(y, x) = u(y) and u can be viewed as a function of y ∈ D( ). The parallel transport map takes values in U (n) and therefore u is a section of U (n) over D( ). Observe that ∩ D( ) = ∅ by the definition of S + ( ), see (41). For this reason we shrink , that is, we define u as above but replace with ( ), 0 < < 0 . This allows us to define u on ( 0 ) \ µ([0, 1]), see (6) for the definition of the path µ. By the continuity of the parallel transport map, we can define u on the whole ( 0 ). Using the continuity again, we let x → y ∈ (0, 1) × ∂B( ) in u(y, x), and see that u(y) = id for any y ∈ (0, 1) × ∂B( ) and any 0 < < 0 . Using the continuity once more, we see that u = id in the whole = ( 0 ). We writeÃ = u −1 du + u −1 Au. Let (x, y) ∈ S in ( ( )), 0 < < 0 , and consider the geodesic γ y←x from x to y. Let s in > 0 satisfy y = γ y←x (s in ). By using the choice u(y) = u(y, x) we will show that (66) Ã ,γ y←x = B,γ y←x . We write γ = γ y←x and consider the fundamental matrix solution U A : R 2 → U (n) of (33). That is, for fixed s ∈ R, the function U A (t, s) in t is the solution of (67) ∂ t U A (t, s) + A,γ(t) U A (t, s) = 0, on R, U A (s, s) = id . Clearly P A γ = U A (s in , 0). We define U B analogously. By (67) it holds that ∂ t U A (s, s) = − A,γ(s) U A (s, s) = − A,γ(s) . Moreover, differentiating (67) in s, gives ∂ t ∂ s U A (t, s) + A,γ(t) ∂ s U A (t, s) = 0, ∂ t U A (s, s) + ∂ s U A (s, s) = 0. Therefore W(t, s) = ∂ s U A (t, s) satisfies ∂ t W(t, s) + A,γ(t) W(t, s) = 0, W(s, s) = A,γ(s) , and writing W in terms of the fundamental solution U A gives (68) ∂ s U A (t, s) = U A (t, s) A,γ(s) . We have u(γ(t), y) = U A (t, 0)U B (0, t), and using (67) for A and (68) for B, du,γ(t) = ∂ t (U A (t, 0)U B (0, t)) = − A,γ(t) U A (t, 0)U B (0, t) + U A (0, t)U B (0, t) B,γ(t) = − Au,γ(t) + uB,γ(t) . We obtain (66) after rearranging and multiplying both sides with u −1 , u −1 du + u −1 Au,γ(t) = B,γ(t) . We have shown that (66) holds for any (x, y) ∈ S in ( ( )). As ( ) is open, it follows from Lemma 1, that the vectorsγ y←x span T y M as x varies in the set {x ∈ : (x, y) ∈ S in ( ( ))}. HenceÃ = B at y for each y ∈ D( ( )) and each 0 < < 0 . Appendix A. Conormal distributions and IPL distributions We formulate the framework of E ⊗ Ω 1/2 valued conormal and IPL distributions for the application to the connection wave equation. The contents of the appendix is modelled on the pioneering works of Hörmander [11], Duistermaat-Hörmander [9] and Melrose-Uhlmann [31]. Additionally, we refer the reader to Hörmander's books [14,12] for the basics of distributions and half densities. Let us begin with the notion of conormal distribution. Definition 1. Let X be an n-dimensional manifold and Y an (n − N )-dimensional closed submanifold. We say a section-valued distribution u ∈ D (X, E ⊗ Ω 1/2 ) is a member of the conormal distributions I m (X; N * Y ; E ⊗ Ω 1/2 ), if in local coordinates (x , x ) ∈ R N +(n−N ) on X, such that Y is defined by x = 0, the distribution u takes the following form u = (2π) −(n+2N )/4 e −ıπN/4 R N e iθx a(x, θ) dθ,(69) where a ∈ S m+n/4 (R n × (R N \ 0); E ⊗ Ω 1/2 ). From the viewpoint of Lagrangian distributions, u is associated with the conormal bundle N * Y \0 and WF(u) ⊂ N * Y \0. If ξ = (ξ , ξ ) denotes the induced coordinates on the cotangent space of X, N * Y is defined by {x = 0, ξ = 0}. The principal symbol σ[u] ∈ S m+n/4 (N * Y \ 0; E ⊗ Ω 1/2 ) is defined as σ[u](x , ξ ) = a(0, x , ξ ). We have the following exact sequence 0 → I m−1 (X; N * Y \ 0; E ⊗ Ω 1/2 ) → I m (X; N * Y \ 0; E ⊗ Ω 1/2 ) σ − → S m+n/4 (N * Y \ 0; E ⊗ Ω 1/2 ) → 0. When the source term for a linear wave equation is a conormal distribution, the solution is not in the same class, and the wider class of IPL distributions was introduced in [31] to tackle this problem. We begin with the model case whereΛ 0 = T * 0 R n \ 0 and Λ 1 = {(x, ξ) ∈ T * R n \ 0 : ξ 1 = 0, x 1 ≥ 0, x 2 = · · · = x n = 0}. We write x = (x 1 , x ) and ξ = (ξ 1 , ξ ). Definition 2. We say that a compactly supported distribution u ∈ E (R n ; E ⊗ Ω 1/2 ) belongs to the space I m (R n ;Λ 0 ,Λ 1 ; E ⊗ Ω 1/2 ), if modulo compactly supported smooth functions, it can be expressed as u = ∞ 0 R n exp(ıξ 1 (x 1 − s) + ξ x ) a(s, x, ξ) dξds, where the amplitude a ∈ S m+1/2−n/4 (R n × (R n \ 0); E ⊗ Ω 1/2 ) is supported on the conical closure of a compact set. Observe thatΛ 0 = N * {0}\0, and thatΛ 1 \∂Λ 1 ⊂ N * Y \0 where Y = {x = 0}. The wavefront set of A is contained in the unionΛ 0 ∪Λ 1 and away from the intersection ∂Λ 1 =Λ 0 ∩Λ 1 , the IPL distribution A is a conormal distribution in the following sense, see [31, pp. Proposition 3. Suppose u ∈ I m (R n ;Λ 0 ,Λ 1 ; E ⊗ Ω 1/2 ) and χ is a zero-th order pseudodifferential operator. We have χu ∈ I m (R n ; N * Y \ 0; E ⊗ Ω 1/2 ) when WF(χ) ∩Λ 0 = ∅; χu ∈ I m−1/2 (R n ; N * {0} \ 0; E ⊗ Ω 1/2 ) when WF(χ) ∩Λ 1 = ∅. This microlocalization leads to the following local principal symbol maps, ignoring 2π and ı factors related to the normalization in (69), σ (1) [u] = a(x 1 , (x 1 , 0), (0, ξ )) onΛ 1 \ ∂Λ 1 ; σ (0) [u] = −ıa(0, 0, ξ)/ξ 1 onΛ 0 \ ∂Λ 1 ; ξ 1 σ (0) [u] = −ıσ (1) [u] on ∂Λ 1 .(70) IPL distributions can be defined on any pair of Lagrangians (Λ 0 , Λ 1 ), Λ j ⊂ T * X \0, with a clean intersection. By a clean intersection of two Lagrangians, we mean T λ (Λ 0 ) ∩ T λ (Λ 1 ) = T λ (∂Λ 1 ) for any λ ∈ ∂Λ 1 , given two Lagrangians Λ 0 and Λ 1 with Λ 0 ∩ Λ 1 = ∂Λ 1 . However, we will omit discussion of Lagrangian distributions, and make the additional assumption that Λ j \ ∂Λ 1 ⊂ N * Y j \ 0, j = 0, 1,(71) for some submanifolds Y j ⊂ X. Definition 3. Suppose (Λ 0 , Λ 1 ), a pair of Lagrangians over a smooth n-manifold X, intersect cleanly at Λ 1 . The space I m (X; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ) consists of distributions of the form u 0 + u 1 + j F j v j , where u 0 ∈ I m−1/2 (X; Λ 0 ; E ⊗ Ω 1/2 ); u 1 ∈ I m (X; Λ 1 \ ∂Λ 1 ; E ⊗ Ω 1/2 ); {F j } is a family of zero-th order Fourier integral operators associated with the inverse of the homogeneous symplectic transformation from V j to T * R n , where {V j } is a locally finite, countable covering of ∂Λ 1 ; and v j ∈ I m (R n ;Λ 0 ,Λ 1 ; E ⊗ Ω 1/2 ). To symbolically construct the parametrix of the wave operator, the key thing is to understand the principal symbols of IPL distributions. As in the model case, away from the intersection of Λ 0 and Λ 1 , the corresponding IPL distributions are conormal distributions assuming (71). Proposition 4. Suppose that (71) holds. Let u ∈ I m (X; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ) and let χ be a properly supported zero-th order pseudodifferential operator. We have χu ∈ I m (X; N * Y 1 \ 0; E ⊗ Ω 1/2 ) when WF(χ) ∩ Λ 0 = ∅; χu ∈ I m−1/2 (X; N * Y 0 \ 0; E ⊗ Ω 1/2 ) when WF(χ) ∩ Λ 1 = ∅. Choosing χ so that σ[χ](x, ξ) = 0 at (x, ξ) ∈ (N * Y 0 \ 0 ∪ N * Y 1 \ 0) \ ∂Λ 1 , we can define the principal symbols of u away from the intersection ∂Λ 1 , σ (1) [u] = σ[χu]/σ[χ] ∈ S m+n/4 (N * Y 1 \ 0; E ⊗ Ω 1/2 ); σ (0) [u] = σ[χu]/σ[χ] ∈ S m−1/2+n/4 (N * Y 0 \ 0; E ⊗ Ω 1/2 ). However, the map from the IPL distributions I m (X; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ) to the symbols S m+n/4 (Λ 1 \ ∂Λ 1 ; E ⊗ Ω 1/2 ) × S m−1/2+n/4 (Λ 0 \ ∂Λ 1 ; E ⊗ Ω 1/2 ) can not be defined as the principal symbol map, since it is not surjective. Indeed, Proposition 3 implies that σ (1) [u] extends to a smooth section of E ⊗ Ω 1/2 up to ∂Λ 1 , whilst hσ (0) [u] extends to a smooth section over Λ 0 if h is a smooth function vanishing on ∂Λ 1 . Hence the principal symbol space should be a proper subspace of the product space of symbols. Moreover, the fact that the Lagrangians Λ j , j = 0, 1, may not be conormal bundles near the intersection ∂Λ 1 causes additional complications. In the case of applications that we are interested in, Λ 1 fails to be a conormal bundle near ∂Λ 1 . The principal symbol of a general Lagrangian distribution is not a section of E ⊗ Ω 1/2 but a section of E ⊗ Ω 1/2 ⊗ L where L is the Maslov bundle. To avoid discussion of Maslov bundles over Λ j , j = 0, 1, we make the assumption that they are trivial. In all the cases that we are interested in, Λ 0 is a conormal bundle and Λ 1 is the future flowout of Λ 0 ∩ Σ(P ), with P the wave operator in the Minkowski space. We will show that the Maslov bundles over the flowouts of interest are trivial in Section B.1 below. To understand the principal symbol space, assuming triviality of the Maslov bundles, we now elucidate the relationship between σ (1) [u] and σ (0) [u]. Following [31], we define a map R : S m−1/2+n/4 (Λ 0 \ ∂Λ 1 ; E ⊗ Ω 1/2 ) → S m+n/4 (Λ 1 ; E ⊗ Ω 1/2 )\| ∂Λ 1 , that encodes the relation (70). Then the sections,        (a (1) , a (0) ) a (0) ∈ S m−1/2+n/4 (Λ 0 \ ∂Λ 1 ; E ⊗ Ω 1/2 ), a (1) ∈ S m+n/4 (Λ 1 ; E ⊗ Ω 1/2 ), a (1) \| ∂Λ 1 = Ra (0) , ha (0) is smooth on ∂Λ 1 if h vanishes on ∂Λ 1 .        , where R = e ıπ/4 (2π) 1/4 R,(72) will be the desired principal symbol space, that we will call S m (Λ 1 , Λ 0 ; E ⊗ Ω 1/2 ), and we have the exact sequence 0 → I m−1/2 (X; Λ 0 ; E ⊗ Ω 1/2 ) + I m−1 (X; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ) → I m (X; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ) σ − → S m+n/4 (Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ) → 0.(73) We remark that the constant e ıπ/4 (2π) 1/4 in (72) results from the fact that Λ 0 is parametrized with n + 1 phase variables whilst Λ 1 is parametrized with n phase variables, cf. the normalization in (69). Apart from the Hermitian bundle factor, the map R was constructed by Melrose-Uhlmann [31, p.491-493]. But that factor is harmless. Indeed, after passing to the model case, we can simply define R by (70), that is, Ra (0) = ı(ξ 1 a (0) )\| ∂Λ 1 . This definition entails, of course, that R does not depend on the choice of a homogeneous symplectic transformation that maps (Λ 0 , Λ 1 ) locally to the model case (Λ 0 ,Λ 1 ). Such coordinate invariance was shown in [31]. Having the principal symbol map (73), we are ready to give a proof of Theorem 2, following [31]. Proof of Theorem 2. We will construct a parametrix for (20), and the claim will then follow from solving (20) with a smooth function f on the right-hand side. The first step of the construction is to find u (0) ∈ I k−2+1/2 (M ; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ) such that (74) P u (0) = f + f (1) + e (1) , where f (1) ∈ I k−1 (M ; Λ 0 ; E ⊗ Ω 1/2 ) and e (1) ∈ I k−3/2 (M ; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ). To do so, we use the symbol calculus. First of all, since P is elliptic on Λ 0 \ ∂Λ 1 , we choose σ[u (0) ] = p −1 σ[f ] on Λ 0 \ ∂Λ 1 , where we used the shorthand notation p = σ[P ] for the principal symbol of P . As an element of I k−2+1/2 (M ; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ), u (0) has the following principal symbol on ∂Λ 1 , (75) σ[u (0) ]\| ∂Λ 1 = R(p −1 σ[f ]). This can be viewed as the initial condition of the bicharacteristic flow on Λ 1 emanating from Λ 0 . On the other hand, the symbol calculus on Λ 1 obeys the transport equation, σ[P u] = (ı −1 L H P + c)σ[u] on Λ 1 , where c = σ sub [P ] is the subprincipal symbol of P . Noting that WF(f ) does not intersect Λ 1 \ ∂Λ 1 , we have (L H P + ıc)σ[u (0) ] = 0, on Λ 1 \ ∂Λ 1 .(76) Combining (75) and (76), we have solved (74). Next, we iteratively solve the following equations (77) P u (j−1) = f (j−1) + e (j−1) + f (j) + e (j) , where f (j) ∈ I k−j (M ; Λ 0 ; E ⊗ Ω 1/2 ) and e (j) ∈ I k−j−1/2 (M ; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ). This can be done by choosing u (j) obey σ[u (j) ] = p −1 σ[f (j) ], on Λ 0 ; (L H P + ıc)σ[u (j) ] = ıσ[e (j) ], on Λ 1 \ ∂Λ 1 ; σ[u (j) ] = R(p −1 σ[f (j) ]), on ∂Λ 1 . We complete the proof by adding up the equations (76) and (77) for j = 1, · · · , N , and letting N → ∞. Appendix B. Theorem 2 in the context of Example 1 Let Λ j , j = 0, 1, be as in Example 1. That is, Λ 0 = N * {0} \ 0 and, taking into account the microlocal cutoff χ, Λ 1 = {(t, tθ; −λ, λθ) ∈ R 1+3 × R 1+3 : λ ∈ R \ 0, θ ∈ V, t ≥ 0},(78) where V is a small neighbourhood of θ 0 in S 2 . The smaller WF(χ) is, the smaller we can choose V. Observe that Λ 1 has two components Λ ± 1 = {(t, tθ; −λ, λθ) ∈ R 1+3 × R 1+3 : ±λ > 0, θ ∈ V, t ≥ 0}. Let us give an explicit choice of two homogeneous symplectic transformations F ± taking (Λ 0 , Λ ± 1 ) to the model case (Λ 0 ,Λ 1 ). We define in a neighbourhood of Λ + 1 in T * R 1+3 \ 0 the map F + (x 0 , x ; ξ 0 , ξ ) = (x 0 , x − x 0 ξ /\|ξ \|; ξ 0 + \|ξ \|, ξ ). Then F + takes N * {0} \ 0 to itself and Λ + 1 toΛ 1 . Indeed, F + (0, 0; ξ 0 , ξ ) = (0, 0; ξ 0 + \|ξ \|, ξ ), and for λ > 0, F + (t, tθ; −λ, λθ) = (t, tθ − tθ; −λ + λ, λθ) = (t, 0; 0, λθ). Analogously, we define F − (x 0 , x ; ξ 0 , ξ ) = (x 0 , x + x 0 ξ /\|ξ \|; ξ 0 − \|ξ \|, ξ ) taking N * {0} \ 0 to itself and Λ − 1 toΛ 1 . We will show that F − is symplectic, the proof for F + being analogous. It holds that dF − d(x 0 , x ; ξ 0 , ξ ) =     1 0 0 0 ξ /\|ξ \| id 0 B 0 0 1 −(ξ ) T /\|ξ \| 0 0 0 id     , where B is a symmetric matrix, the precise form of which is inconsequential. Writing shortly dF − for the above derivative and J for the symplectic form on T * R 1+3 as a matrix, we assert that (dF − ) T JdF − = J. We write η = ξ /\|ξ \| and Let us now write the initial condition (22) in this context. We denote by (x,ξ) = (x 0 ,x ;ξ 0 ,ξ ) the local coordinates on T * R 1+3 given by F + , that is, (x 0 ,x ;ξ 0 ,ξ ) = F + (x 0 , x ; ξ 0 , ξ ). Recall that p = σ[P ] is a section of T * R 1+3 \ 0, and that 2p(x, ξ) = −ξ 2 0 + \|ξ \| 2 in the Cartesian coordinates (x, ξ) = (x 0 , x ; ξ 0 , ξ ). Therefore, p(x,ξ) = −(ξ 0 − \|ξ \|) 2 /2 + \|ξ \| 2 /2 =ξ 0 (\|ξ \| −ξ 0 /2), and (22) can be written with the same ι 0 . B.1. Trivialization of the Maslov bundle over the flowout. It is well-known [11,Th. 3.3.4] that the Maslov bundle over a conormal bundle N * K \ 0 is trivial. However, even in the case the flowout Λ 1 in Example 1, more care is needed, since Λ 1 fails to coincide with a conormal bundle near B = Λ 1 ∩ Λ 0 . We will discuss the Maslov bundle in detail only in the context of Example 1, this being the most important case for the purposes of the present paper. When the essential support WF(χ) is small around ccl(0, ±ξ 0 ), the future flowout Λ 1 is embedded in the following flowout to both past and future, a localized version of (26),Λ Appendix C. Flowout from the triple intersection We consider the intersection the three cones, defined by (53), that is, K j = {(t j + t, x j + tθ) ∈ R 1+3 : t > 0, θ ∈ S 2 }, j = 1, 2, 3, where, for small r > 0, (t j , x j ) = x j = γ(−s in ; y, ξ j ), ξ j = (1, 1 − r 2 j , r j , 0), r 1 = 0, r 2 = r, r 3 = −r, see (43), (44) and (50). After a translation, we may assume without loss of generality that y = 0. Then t j = −s in for all j = 1, 2, 3, and x j = (−s in 1 − r 2 j , −s in r j , 0), j = 1, 2, 3. The points (t, x ) ∈ R 1+3 in K 1 ∩ K 2 ∩ K 3 satisfy \|x − x j \| 2 = \|t + s in \| 2 for each j = 1, 2, 3, or equivalently, writing x = (x 1 , x 2 , x 3 ), \|x 1 + s in 1 − r 2 j \| 2 + \|x 2 + s in r j \| 2 + \|x 3 \| 2 = \|t + s in \| 2 , j = 1, 2, 3. To simplify the notation, we write z = x 3 . Taking x 1 = x 2 = 0, the above three equations simplify to the single equation z 2 = t 2 + 2s in t. This equation defines the filament in spacetime that acts as an artificial source, as discussed in the introduction. Solving for z gives an equation for the two moving point sources in Figure 1. In the figure, we have taken s in = 2 and r = 0.8. For the study of the flowout from the filament it is more convenient to solve for t rather than for z. Near y = 0 this yields t = T (z) = −s in + s 2 in + z 2 . As K 1 ∩ K 2 ∩ K 3 is a smooth, one dimensional manifold near y, it holds locally that K 1 ∩ K 2 ∩ K 3 = {(T (z), 0, 0, z) : z ∈ R}. For a covector (ζ 0 , . . . , ζ 4 ) ∈ R 4 in the fibre of Λ 0 = N * (K 1 ∩ K 2 ∩ K 3 ) \ 0 at (T (z), 0, 0, z) it holds that ζ 0 T (z) + ζ 4 = 0 where T is the derivative of T . Therefore Λ 0 = {(T (z), 0, 0, z; ζ 0 , ζ 1 , ζ 2 , −ζ 0 T (z)) : z ∈ R, (ζ 0 , ζ 1 , ζ 2 ) ∈ R 3 \ 0}. We will proceed to compute the future flowout Λ 1 of Λ 0 ∩ Σ(P ). To simplify the notation, we write ζ 0 = −λ. Then (−λ, ζ 1 , ζ 2 , λT (z)) is lightlike if and only if λ 2 = ζ 2 1 + ζ 2 2 + λ 2 \|T (z)\| 2 . We write (ζ 1 , ζ 2 ) = µθ with µ ≥ 0 and θ ∈ S 1 . Then µ 2 = (1 − \|T (z)\| 2 )λ 2 , and Λ 1 \ Λ 0 = {(T (z) + sλ, sλθ, z + sλε; −λ, λθ, λε) :θ = √ 1 − ε 2 θ, ε = T (z), θ ∈ S 1 , λ ∈ R \ 0, z ∈ R, sλ > 0}. Observe that ε = T (z) = z/ s 2 in + z 2 , and this is small when we localize near y = 0 as the notation suggests. Let us show that Λ 1 \ Λ 0 coincides with a conormal bundle. It is enough to verify that its projection to the base space R 1+3 is a smooth manifold. For our purposes it is enough to consider Λ 1 \ Λ 0 only over the compact diamond D and therefore it is enough verify that the map F (t, θ, z) = (T (z) + t, tθ, z + tε) is injective and has injective differential for t > 0, θ ∈ S 1 and small \|z\|. Suppose that F (t 1 , θ 1 , z 1 ) = F (t 2 , θ 2 , z 2 ). As t > 0 and ε is small, the second component of (83) implies θ 1 = θ 2 and t 1 1 − \|T (z 1 )\| 2 = t 2 1 − \|T (z 2 )\| 2 . Writing Z j = s 2 in + z 2 j , and using 1 − \|T (z j )\| 2 = s 2 in + z 2 j − z 2 j s 2 in + z 2 j = s 2 in Z 2 j , the latter equation reduces to t 1 /Z 1 = t 2 /Z 2 . On the other hand, the last component of (83) gives z 1 (1 + t 1 /Z 1 ) = z 2 (1 + t 2 /Z 2 ). As 1 + t 1 /Z 1 = 1 + t 2 /Z 2 > 0, we get z 1 = z 2 . This together with t 1 /Z 1 = t 2 /Z 2 implies that also t 1 = t 2 . We have shown that F is injective. Observe that, writing Z = Z j when z 1 = z, d dz = T (z) = 1 Z − z 2 Z 3 = 1 − ε 2 Z . Therefore, letting R ⊃ B a → Θ(a) ∈ S 1 be local coordinates on S 1 , dF =   1 0 ε θ t √ 1 − ε 2 Θ −t ε √ 1−ε 2 1−ε 2 Z θ ε 0 1 + t 1−ε 2 Z   . When z = 0, this reduces to dF =   1 0 0 θ tΘ 0 0 0 1 + t Z   , which is invertible since t > 0, Z > 0 and Θ = 0. The same holds when \|z\| is small enough, and we have shown that Λ 1 \ Λ 0 coincides with a conormal bundle. Let us also remark that an argument similar to that in Section B.1 shows that the Maslov bundle over Λ 1 is trivial. Microlocal analysis of the wave operator 12 2.4. Flowout from a point in the Minkowski space 13 2.5. Trivialization of the half-density bundle over the flowout 16 2.6. Parallel transport equation for the principal symbol 17 2.7. Positively homogeneous symbols 18 3. Microlocal analysis of the interaction of three waves 20 3.1. The linear span of three lightlike vectors 22 Date: Compiled February 18, 2019. 1. 1 . 1The Yang-Mills-Higgs equations. Let (M, g) be a Lorentzian manifold of dimension 1 + 3 and consider a compact Lie group G with Lie algebra g. We choose a positive definite inner product on g invariant under the adjoint action. To simplify the exposition we discuss the case of the trivial bundle M × g over M . The Yang-Mills-Higgs equations are PDEs on a pair (A, Φ), where (A, Φ) ∈ C ∞ (M ; T * M ⊗ g) ⊕ C ∞ (M ; g). Figure 1 . 1The interaction of three pieces of spherical waves, the blue surfaces. Top left. The pieces propagate along the black lines, and do not intersect yet. Top right. The pieces intersect along the blue curves. Pairwise intersections do not produce new propagating wave fronts. Bottom left. Figure 2 . 2Set , see 3. 2 . 2Three-fold linearization of cubic connection wave equations. Let us consider a source f ( ) of the form (37), and the half-density u( ) = \|g\| 1/4 φ( ) where φ( ) denotes again the solution of (8) with the source f = f ( ). Recalling the definition of Q, see (16), we observe that u = u( ) satisfies the wave equation Q(u) = \|g\| 1/4 f. 0 ∈ C \ 0 is a constant containing the 2π and ı factors. Analogously, setting(x 0 ,x ;ξ 0 ,ξ ) = F − (x 0 , x ; ξ 0 , ξ ),(81)we obtain p(x,ξ) = −(ξ 0 + \|ξ \|) 2 /2 + \|ξ \| 2 /2 = −ξ 0 (\|ξ \| +ξ 0 /2), and (22) reads σ[u](0,ξ ) = −ι 0 \|ξ \| −1 σ[f ](0; 0,ξ ), X. CHEN, M. LASSAS, L. OKSANEN, AND G.P. PATERNAIN = {(t, tθ; −λ, λθ) ∈ R 1+3 × R 1+3 : λ ∈ R \ 0, θ ∈ V, t ∈ R},where V ⊂ S 2 is as in (78). We suppose that V is chosen so that there is a diffeomorphism Θ : B → V where B is an open ball in R 2 , with the centre at the origin, and Θ(0) = θ 0 .Analogously to Λ 1 , alsoΛ 1 has two componentŝ Λ ± 1 = {(t, tθ; −λ, λθ) ∈ R 1+3 × R 1+3 : ±λ > 0, θ ∈ V, t ∈ R}.We will show thatΛ + 1 is contractible. The same holds forΛ − 1 with an analogous proof. It then follows that any vector bundle overΛ 1 =Λ + 1 ∪Λ − 1 is trivial[7]. In particular, the Maslov bundle over Λ 1 is trivial. 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[ "Anomalous multicritical phenomena and frustration induced by synthetic magnetic fields", "Anomalous multicritical phenomena and frustration induced by synthetic magnetic fields" ]	[ "Jinchen Zhao \nDivision of Natural and Applied Sciences\nDuke Kunshan University\n215300Kunshan, JiangsuChina\n", "Myung-Joong Hwang \nDivision of Natural and Applied Sciences\nDuke Kunshan University\n215300Kunshan, JiangsuChina\n\nZu Chongzhi Center for Mathematics and Computational Science\nDuke Kunshan University\n215300Kunshan, JiangsuChina\n" ]	[ "Division of Natural and Applied Sciences\nDuke Kunshan University\n215300Kunshan, JiangsuChina", "Division of Natural and Applied Sciences\nDuke Kunshan University\n215300Kunshan, JiangsuChina", "Zu Chongzhi Center for Mathematics and Computational Science\nDuke Kunshan University\n215300Kunshan, JiangsuChina" ]	[]	We consider a one-dimensional Dicke lattice with complex photon hopping amplitudes and investigate the influence of time-reversal symmetry breaking due to synthetic magnetic fields. We show that, by tuning the total flux threading the loop, the universality class of superradiant phase transition (SPT) changes from that of the mean-field fully-connected systems to one that features anomalous critical phenomena. The anomalous SPT exhibits a closing of the energy gap with different critical exponents on both sides of transition and a discontinuity of correlations and fluctuation despite it being a continuous phase transition. In the anomalous normal phase, we find that a non-mean-field critical exponent for the closing energy gap and non-divergent fluctuations and correlations appear, which we attribute to the asymmetric dispersion relation. Moreover, we show that the nearest neighborhood complex hopping induces effective long-range interactions for position quadratures of the cavity fields, whose competition leads to a series of first order phase transitions among superradiant phases with varying degrees of frustration. The resulting multicritical points also show anomalous features such as two coexisting critical scalings on both sides of the transition. Our work shows that the interplay between the broken time-reversal symmetry and frustration on bosonic lattice systems can give rise to anomalous critical phenomena that have no counter part in fermionic or spin systems or time-reversal symmetric quantum optical systems.	null	[ "https://export.arxiv.org/pdf/2208.02268v1.pdf" ]	251,320,625	2208.02268	d70d78f76024c4cf1c7fa69cb53f45293d1150de	Anomalous multicritical phenomena and frustration induced by synthetic magnetic fields Jinchen Zhao Division of Natural and Applied Sciences Duke Kunshan University 215300Kunshan, JiangsuChina Myung-Joong Hwang Division of Natural and Applied Sciences Duke Kunshan University 215300Kunshan, JiangsuChina Zu Chongzhi Center for Mathematics and Computational Science Duke Kunshan University 215300Kunshan, JiangsuChina Anomalous multicritical phenomena and frustration induced by synthetic magnetic fields We consider a one-dimensional Dicke lattice with complex photon hopping amplitudes and investigate the influence of time-reversal symmetry breaking due to synthetic magnetic fields. We show that, by tuning the total flux threading the loop, the universality class of superradiant phase transition (SPT) changes from that of the mean-field fully-connected systems to one that features anomalous critical phenomena. The anomalous SPT exhibits a closing of the energy gap with different critical exponents on both sides of transition and a discontinuity of correlations and fluctuation despite it being a continuous phase transition. In the anomalous normal phase, we find that a non-mean-field critical exponent for the closing energy gap and non-divergent fluctuations and correlations appear, which we attribute to the asymmetric dispersion relation. Moreover, we show that the nearest neighborhood complex hopping induces effective long-range interactions for position quadratures of the cavity fields, whose competition leads to a series of first order phase transitions among superradiant phases with varying degrees of frustration. The resulting multicritical points also show anomalous features such as two coexisting critical scalings on both sides of the transition. Our work shows that the interplay between the broken time-reversal symmetry and frustration on bosonic lattice systems can give rise to anomalous critical phenomena that have no counter part in fermionic or spin systems or time-reversal symmetric quantum optical systems. Introduction.-Inspired by the discovery of remarkable phenomena for charged particles moving in magnetic fields such as integer and fractional quantum Hall effects [1,2], there have been intense theoretical and experimental efforts to realize synthetic magnetic fields for uncharged particles such as photons [3][4][5][6][7], phonons [8,9] and neutral atoms [10,11]. In a lattice of photonic resonators, for example, the synthetic magnetic fields have been realized to observe unique topological photonic properties and robust edge states [12][13][14]. Moreover, the chiral photon current due to the time-reversal symmetry breaking induced by synthetic magnetic fields has also been observed [5]. The light-matter interaction between such chiral photons and quantum emitters may give rise to novel quantum optical phenomena [15][16][17][18][19] A bosonic mode coupled to two-level systems, described by the Dicke model, exhibits a superradiant phase transition (SPT) [20][21][22][23][24][25][26][27] when the spin-boson coupling strength exceeds a threshold. The spins in the Dicke model effectively realize an infinite-range interaction through their interaction with a common bosonic mode; as such, the SPT of the Dicke model belongs to the universality class of fully-connected systems characterized by mean-field exponents [3,23,[28][29][30]. Finding SPTs that exhibit critical phenomena going beyond the mean-field type phase transition may lead to a discovery of novel phases of coupled light and matter; recently discovered examples include spin glass phases induced by the multimode cavity fields [32][33][34][35][36][37][38][39] and a frustrated SPT in the Dicke lattice [40]. Recently, a tricritical SPT in the Rabi lattice in the synthetic magnetic field has been discovered [41,42]. In this Letter, we consider a 1D Dicke lattice model with complex photon hopping amplitudes whose phase determines the magnetic flux threading the loop and show that the interplay between the synthetic magnetic fields and the local spinboson interaction in fact leads to a novel universality class of SPTs with anomalous multicritical phenomena. We find a multicritical flux point, above which a standard mean-field SPT occurs and below which an anomalous SPT occurs. The anomalous SPT exhibits several exotic properties. First, critical exponents for the energy gap on both sides of transitions are not only different from the mean-field exponent, but they are different from each other; this adds an experimentally accessible counter-example to the common expectation that critical exponents on both sides of the critical point are same due to the identical renormalization properties [43]. Second, the fluctuation and correlation, represented by the photon number and entanglement in our model, do not diverge in the anomalous normal phase. To the best of our knowledge, all SPTs discovered so far show a divergent photon number and entanglement [3,27,44]. Moreover, since photon number and entanglement do diverge in the superradiant phase, they become discontinuous on both sides of a continuous secondorder phase transition. We find that the anomalous SPT occurs when an elementary excitation that becomes critical carries a finite momentum and the asymmetry in the dispersion relation due to the broken-time reversal symmetry gives rise to the non-mean-field exponent and the non-diverging correlation. Moreover, by integrating out the momentum quadratures of the cavity fields, we construct an effective semiclassical model for the position quadratures with long-range effective photon hopping interactions. Our effective model shows that the first order phase transition from a mean-field to anomalous SPT occurs when the effective nearest and next-nearest photon hopping interaction cannot be simultaneously minimized for odd lattice sites, namely, when frustration occurs analogous to the J 1 − J 2 Ising model. The resulting anomalous multicritical flux point exhibits two modes that become critical in both phases with distinct critical exponents. Our model further predicts that there are first order phase transitions within the frustrated SP, whenever the configuration for the frustrated mean-field solutions that minimize the energy changes. Model.-We consider a Dicke lattice model where each lattice site realizes the Dicke model and neighboring lattices are connected by the photon hopping interaction with complex amplitudes. The model Hamiltonian reads ANP NP (k = − 2π 3 ) (k = 0) ∝ \|g − g c \| γ θ ∈ (0, θ k1,k0 c ) γ = 1 γ = 3/2 (θ = θ k1,k0 c ) γ = 1/2 γ = 1 g = g c θ ∈ (θ k1,k0 c , π) γ = 1/2 0.00 0.25 0.50 0.75 1.00 θ/π −0.1 0.0 0.1 J eff m θ k1,k0 c m = 1 (a) (b) 0.9 1.0 1.1 g FSP FSP SP ANP ANP NP (k = − 4π 5 ) (k = − 2π 5 ) (k = 0) J 1 J 2 0.45 0.50 0.55 θ/π −0.01 0.00 0.01 J eff m θ k2,k1 c θ k1,k0 c m = 1 m = 2 ∝ \|g − g c \| γ (0, θ k2,k1 c ), (θ k2,k1 c , θ k1,k0 c ) γ = 1 γ = 5/2 (θ = θ k2,k1 c ) γ = 1 g = g c (θ = θ k1,k0 c ) γ = 1/2 γ = 1 (c) (d)H N = N n=1 H n + J e iθ a † n a n+1 + h.c. H n = ωa † n a n + Ω 2 Na j=1 σ z n,j + λ √ N a a n + a † n Na j=1 σ x n,j(1) with a periodic boundary condition a N +1 = a 1 . The phase θ ∈ (0, π) represents the total flux of synthetic magnetic fields threading the loop. At nth lattice site, the oscillator of frequency ω is described by an annihilation operator a n and there are N a two-level systems of frequency Ω described by σ x,z j . The local spin-boson coupling is characterized by λ. The Hamiltonian H N commutes with the parity operator Π = exp iπ N n=1 (a † n a n + 1 2 Na j=1 σ z n + Na 2 ) and thus respects a global Z 2 symmetry in addition to the translational symmetry. However, the time reversal symmetry is broken due to the synthetic magnetic field. We consider the thermodynamic limit of N a → ∞ with a finite number of lattice sites N and find a rich phase diagram as a function of g = 2λ/ √ ωΩ and θ and anomalous critical properties as shown in Fig. 1 for N = 3 and N = 5. Anomalous normal phase with bounded correlation.-Let us begin by investigating the normal phase. In the thermodynamic limit, we introduce the Holstein-Primakoff transformation J + n √ N a b † n and J z n = N a /2− b † n b n with b n , b † n = 1 and perform a Fourier transform, a † n = k e ikn a † k / √ N , b † n = k e ikn b † k / √ N with k = 0, ±2π/N, · · · , ±(N − 1)π/N , to derive the effective Hamil- tonian H np = k ω k a † k a k + Ωb † k b k − λ(a k + a † −k )(b −k + b † k ) ,(2) where ω k = ω +2J cos(θ −k). Note that only the modes with the same magnitude of momentum, a ±k and b ±k , are coupled with each other; thus, the Hamiltonian can be diagonalized for each k. Four excitation energies are ε (1),(2) k = 2 j=1 A (1),(2) j,k + ∆ k ,(3) where ∆ k = (ω k − ω −k )/4. The expression for A (1),(2) j,k is given in the supplementary material [45]. For simplicity, let us denote the excitation energy that becomes critical as ε k at g c (k) = 2ω k ω −k ω(ω k + ω −k ) .(4) For a fixed θ, the true critical point where the normal phase becomes unstable is thus determined by g c (θ) = min{g c (k j )\|j = 0, 1, · · · , (N − 1)/2} where k j = −2πj/N . We define flux critical points θ ki,kj c , the boundary between two regions where k i and k j mode become critical, respectively. They can be determined by solving g c (k i ) = g c (k j ) for θ. We find that 0 < θ Fig. 1 (b) and (d)]. We call the latter an anomalous NP, which spans g < g c (θ < θ k1,k0 c ). k (N −1)/2 ,k (N −3)/2 c < θ k (N −3)/2 ,k (N −5)/2 c [ To understand the emergence of the anomalous NP, we note that ε k consists of a sum of square root terms and a constant shift ∆ k = (ω k − ω −k )/4 = J sin θ sin k. The latter is the difference in frequencies of lattice photons with opposite momentums and it is non-zero only for k = 0 when the time-reversal symmetry is broken (θ = 0, π). For k = 0, ∆ k=0 = 0; therefore, ε 0 closes the gap with the square root ε 0 ∝ \|g − g c \| 1/2 , a typical mean-field behavior. For k = 0, however, ε k becomes zero before the square root term becomes singular due to the cancelation with ∆ k < 0 for k < 0. In this case, the energy gap closes when an analytical function simply crosses the zero and the exponent becomes 1, i.e. ε 0 ∝ \|g − g c \| 1 . Furthermore, at the boundary between the normal phase and anomalous normal phase, namely, at g c (θ k1,k0 c ), we find that both the k 0 and k 1 mode simultaneously become critical, whose scaling exponents are 1/2 and 1. In addition, at other flux critical points between the non-zero momentum modes, i.e., θ ki+1,ki c with i > 0, the two critical excitations with an identical exponent γ = 1 appear. See Figs. 1(b) and 1(d). In the anomalous NP, the local photon number a † a n and the bipartite entanglement S n between the nth site and the rest of chain remains finite at the critical point [ Fig. 2(a)]. This is a striking observation because the fluctuation and correlation are typically expected to diverge at the critical point [44,46,47]. To gain further insight, we adiabatically eliminate the atomic degrees of freedom in the infinitely frequency ratio limit (Ω/ω → ∞) [3], and derive the analytical expression [45] for the excitation energy as ε k = √ A k + 2∆ k and the photon number as a † n a n np = 1 N k ω k + ω −k 2 √ A k − 2 √ A k ω k + ω −k − 2 . (5) From this, we see that a † n a n np → ∞ only if √ A k → 0 and for the anomalous NP with ∆ k < 0, the photon number is bounded. This in turn leads to the bounded entanglement among cavity fields as the entanglement is generated by multimode squeezing with a bounded photon number. On the other hand, for θ > θ k1,k0 c , both a † a n and S n diverges with meanfield exponent [45] as the photon number is proportional to the inverse of the square root term √ A k in ε k . Multicriticality and frustration in the broken symmetry phase.-When g > g c (θ), a second-order continuous phase transition occurs giving rise to spontaneous coherence i.e. a n = x n + iy n = 0. We first replace the operators in Eq. (1) with their mean values to derive the mean-field energy. By minimizing the mean-field energy over the atomic degree of freedom [45], we havē In the right inset of (a) and (b), the divergence of the 3rd site is given separately as its peak is much narrower than the rest. Here, we show that the origin of first order transitions is the frustration of cavity fields, which leads to the recently discovered frustrated superradiant phase [40]. To this end, we derive an effective mean-field energy for the position quadrature x n of the cavity fields only by eliminating the momentum quadrature y n at the global minimum of Eq. (6) [45], which leads tō E N = N n=1 x 2 n +ȳ 2 n − 1 2 1 + 4g 2x2 n + 2J cos θ(x nxn+1 +ȳ nȳn+1 ) + 2J sin θ(x n+1ȳn −x nȳn+1 ) ,(6)whereJ = J/ω,Ē = E/N a Ω,x n = ω 0 /N a Ω x n , y n = ω 0 /NE GS N = N n=1  x 2 n − 1 2 1 + 4g 2x2 n + (N −1)/2 m=0J eff mxnxn+m   .(7) Eq. (7) shows that the nearest-neighborhood complex photon hopping effectively realizes long-range interactions amongx n mediated byȳ n . In particular, we find that the dominant terms are the nearest and next-nearest neighborhood interaction,J eff m>2 J eff 1,2 . As the flux modulates signs and magnitudes ofJ eff 2 andJ eff 1 , frustrated sign configurations forx n may occur, analogous to the J 1 and J 2 Ising model. We illustrate this point using N odd lattices. For N = 3, Eq. (7) becomes identical with the mean-field energy of the Dicke lattice model with a real photon hopping [40]. As shown in Fig. 1 (a),J eff 1 changes the sign at the critical flux point θ = θ k1,k0 c . Therefore, the broken symmetry phase undergoes a first order phase transition between the non-frustrated SP forJ eff 1 < 0 and the frustrated SP for J eff 1 > 0 with the ground-state degeneracy D = 6. We note that a similar phase diagram has been found in the Rabi triangle model [41], which can also be understood from our effective description. For N = 5, when θ < θ k2,k1 c , one hasJ eff 1 >J eff 2 > 0 and the nearest neighbors should be anti-aligned to minimize the energy, which is incompatible with a 1D chain with odd N . Therefore, a frustrated configuration emerges where a single pair of neighboring sites are aligned, called a ferromagnetic pair, with D = 10. For θ > θ k2,k1 c , one hasJ eff 2 >J eff 1 > 0 that favors the next-nearest neighbors x n and x n+2 to be antialigned, which leads to a frustrated configuration with three ferromagnetic pair (D = 10). This configuration persists even whenJ eff 1 becomes negative; however, when the negativeJ eff 1 becomes the dominant energy scale and all x n have the same sign, leading to a non-frustrated SP with the degeneracy D = 2. Moreover, the first order phase transition line can be independent of g as in the case of θ = θ k1,k0 Fig. 1(c)], or it can be dependent on g like the one stemming from the tricritical point θ k1,k0 c for N = 5. We note that both cases may generally appear for larger N . This can be understood by considering a large glimit, \|g − g c \| 1, where the − 1 2 1 + 4g 2x2 n term in Eq. (7) dominates so that \|x n \| ∼x. For the three configurations shown in Fig. 1(c), the interaction energy (from left to right) becomes c of N = 3 or θ = θ k2,k1 c of N = 5 [(−3J eff 1 +J eff 2 )x 2 , (J eff 1 −3J eff 2 )x 2 , (5J eff 1 +5J eff 2 ) x 2 , respectively. By comparing the first two configurations we see that they are equal atJ eff 1 =J eff 2 , which happens at θ = θ k2,k1 c . Therefore, the first order PT line is independent of g. By comparing the latter two, we find that the crossing happens atJ eff 1 = −2J eff 2 , where θ is slightly less than the tricritical flux point θ = θ k1,k0 c . These two limiting values are smoothly connected leading to the g-dependent first order PT line. Our analysis can be straightforwardly extended to a larger lattice size for both odd and even N [45]. For even N , if J 2 > 0 is dominant over J 1 , there could be frustration for odd N/2, but no frustration for even N/2. Excitation and fluctuation in the superradiant phase.-Let us discuss the excitation and fluctuation in various SPs. For detailed derivation, we refer to Ref. [45]. For θ ≥ θ k1,k0 c , since all mean valuesx n are identical with no frustration, the resulting effective Hamiltonian preserves the translational symmetry and we find that the k 0 momentum mode becomes critical, with the mean-field exponent γ = 1/2 [45]. For 0 < θ < θ k1,k0 c , the translational symmetry of the system is broken due to the frustration. Therefore, we numerically calculate the excitation spectra for N = 3, N = 5 [See Fig. 1(b,d)]. For N = 3, through asymptotic expansions, we analytically derive that the excitation energy gap closes with an exponent 3/2 [45], which agrees with the numerical result. For N = 5, we find the exponent to be 5/2. Therefore, we have ε ∝ (g c − g) γ− (for g < g c ) ε ∝ (g − g c ) γ+(N ) (for g > g c )(8) with γ − = 1 and γ + (N ) = N/2 for N = 3, 5. We note that the possibility of having different critical exponents on both sides of phase transition has been recently discussed in Ref. [43] and the anomalous SPT in the synthetic magnetic fields exhibits such unique properties. We also calculate the photon number and bipartite entanglement S n in the SP. Unlike the anomalous NP where both are non-divergent, we find that they do diverge at the critical point as shown in Fig. 2 for N = 3. Therefore, there is a discontinuity of both quantities at the critical point of a continuous phase transition. We discovered that the SPT for 0 < θ < θ k1,k0 c exhibits highly unusual anomalous critical properties summarized above, and hence we call it an anomalous SPT. Anomalous multicritical points.-Finally, we discuss the properties at the multicritical points. We have found that there are two types of multicritical points: i) one is g c θ k1,k0 c , where the boundary between the NP and anomalous NP and the boundary between non-frustrated and frustrated SP meet. At this point, two critical scalings for the closing energy gap with γ = 1 and γ = 1/2 coexist. ii) Others are g c θ ki+1,ki c with 1 ≤ i ≤ (N − 3)/2 where the momentum of the critical mode changes from k i to k i+1 in the normal phase and the sign configuration for the frustrated SP changes. At this point, there are two critical modes on both sides of the multicritical point, but their exponents are both γ = 1. While it is generally expected that the critical exponents at the multicritical point are different from that of the continuous phase transition, two coexisting critical scalings are unique properties of multicritical points of the anomalous SPT. Discussions.-We have demonstrated that the light-matter interaction for photons experiencing synthetic magnetic fields leads to a novel universality class of the SPT with anomalous critical properties that are commonly found neither in the time-reversal symmetric light-matter systems nor condensed matter systems. Our work shows that the breaking of the timereversal symmetry offers a unique mechanism for the normal phase of lattice bosons to become unstable with bounded fluctuation and that the complex nearest-neighborhood hopping amplitudes effectively mediate long-range interactions which may lead to exotic frustrated quantum phases of coupled light and matter. Note added.-Upon completion of this work, we became aware of a recent work on the Rabi lattice models in synthetic magnetic fields [42]. Anomalous multicritical phenomena and frustration induced by synthetic gauge fields -Supplemental Material - EFFECTIVE HAMILTONIAN In this section, we derive the effective Hamiltonian for the Dicke lattice model in the thermodynamic limit N a → ∞ [S1]. Let us define the mean values of the collective spin operators J x,y,z n = 1 2 Na j=1 σ x,y,z n,j and the cavity fields as a n = α n = x n + iy n , ( J x n , J y n , J z n ) = N a 2 (sin θ n cos φ n , sin θ n sin φ n , cos θ n ). One can separate the mean-field contribution by applying a unitary transformation U = N n=1 exp −iφ n J z n − iθ n J y n + (α n a † n − α * n a n ) to Eq. (1) in the main text, to get H = U † HU = N n=1 ω(a † n + α * n )(a n + α n ) + Ω (− sin θ n J x n + cos θ n J z n ) + 2λ √ N a a n + a † n + 2x n × (cos θ n cos φ n J x n + sin θ n cos φ n J z n − sin φ n J y n ) + J e iθ (a † n + α * n )(a n+1 + α n+1 ) + h.c. . (S2) We choose parameters so that linear terms in J x,y n , a n , and a † n to be zero, which leads to − Ω sin θ n + 4λ √ N a x n cos θ n cos φ n = 0, 4λ √ N a x n sin φ n = 0. and ωα n + λ N a sin θ n cos φ n + J e iθ α n+1 + e −iθ α n−1 = 0. Therefore, we can rewrite Eq. (S2) as H = N n=1 ωa † n a n +Ω n J z n + 2λ n √ N a a n + a † n J x n + J(e iθ a † n a n+1 + h.c.) + E 0 ,(S5) whereΩ n = Ω cos θ n + 4λx n sin θ n cos φ n / √ N a ,λ n = λ cos θ n cos φ n and E 0 is a constant. In the limit N a → ∞, we apply the Holstein-Primakoff transformation , i.e. J x n = √ Na 2 (b n + b † n ), J z n = Na 2 − b † n b n to rewrite the displaced Hamiltonian asH DL = H DL q + E DL GS , where H q is quadratic in relevant operators and E GS is the semiclassical ground state energy H q = N n=1 ωa † n a n −Ω n b † n b n +λ n a n + a † n (b n + b † n ) + J(e iθ a † n a n+1 + h.c.) ,(S6a)E GS = N n=1 ωα * n α n + N a 2Ω n + J e iθ α * n α n+1 + e −iθ α n α * n+1 .(S6b) We will use Eq. (S6b) to determine the semiclassical ground state by minimizing it and use Eq.(S6a) to study the fluctuations around the mean values. NORMAL PHASE In this section, we present the diagonalization of the normal phase effective Hamiltonian, where α n = 0, which reads H DL np = 3 n=1 ωa † n a n + Ωb † n b n − λ a n + a † n b n + b † n + J(e iθ a † n a n+1 + h.c.) . By introducing the Fourier transformation a † n = k e ikn a † k / √ 3, b † n = k e ikn b † k / √ 3 with k = 0, ±2π/3, we have H DL np = k [ω + 2J cos(θ − k)]a † k a k + Ωb † k b k − λ(a k + a † −k )(b −k + b † k ) ,(S8) The above Hamiltonian can be written as H k = 1 2 ψ † M ψ, where ψ = (a k , b k , a −k , b −k , a † k , b † k , a † −k , b † −k ) and M = M 1 M 2 M 2 M 1 , where M 1 =     ω k λ 0 0 λ Ω 0 0 0 0 ω −k λ 0 0 λ Ω     , M 2 =     0 0 0 λ 0 0 λ 0 0 λ 0 0 λ 0 0 0     .(S9) The excitation energies are the eigenvalues of the matrix D = I − M , where I − = I 4 0 0 −I 4 (S10) and I 4 is the 4 × 4 identity matrix. The characteristic polynomial of D can be written as P = P + P − , where P ± = (ε ± ω k ) (ε ∓ ω −k ) ε 2 − Ω 2 − 1 2 g 2 (ω k + ω −k ) ωΩ 2 .(S11) The critical point g c can be derived by letting ε = 0 in the above equation, which gives Eq. (4) in the main text. To solve for ε, we use the general expression for the roots of a quartic polynomial [S2]. We first define a cubic equation t 3,k v 3 k + t 2,k v 2 k + t 1,k v k + t 0,k = 0, whose coefficients are given as t 3,k = (ω k − ω −k ) 3 − 4(ω k − ω −k )(−Ω 2 − ω k ω −k ) − 8(ω k − ω −k )Ω 2 , t 2,k = (ω k − ω −k ) 2 (−Ω 2 − ω k ω −k ) − 4(−Ω 2 − ω k ω −k ) 2 − 2(ω k − ω −k ) 2 Ω 2 + 16[ω k ω −k − 1 2 g 2 ω (ω k + ω −k )]Ω 2 , t 1,k = − (ω k − ω −k ) 3 Ω 2 + 4(−Ω 2 − ω k ω −k )(ω k − ω −k )Ω 2 + 8(ω k − ω −k )[ω k ω −k − 1 2 g 2 ω (ω k + ω −k )]Ω 2 , t 0,k = − (ω k − ω −k ) 2 Ω 4 + (ω k − ω −k ) 2 [ω k ω −k − 1 2 g 2 ω (ω k + ω −k )]Ω 2 . (S12) We choose v k to be the smallest root of the above cubic equation, and use it to express the excitation energies as ε (1) k = √ 2 Y k + X k (ω k − ω −k + 4v k ) + X k + (ω k − ω −k ) 4 , ε (2) k = √ 2 Y k − X k (ω k − ω −k + 4v k ) − X k + (ω k − ω −k ) 4 , (S13) where X k = (ω k − ω −k ) 3 − 8Ω 2 (ω k − ω −k ) − 4(ω k − ω −k )(−Ω 2 − ω k ω −k ) ω k − ω −k + 4v k , Y k = (ω k − ω −k ) 3 + 4 2 Ω 2 (ω k − ω −k ) − 2(ω k − ω −k )(−Ω 2 − ω k ω −k ) + 6(ω k − ω −k ) 2 v k − 16(−Ω 2 − ω k ω −k )v k ω k − ω −k + 4v k . (S14) Therefore, we find that the excitation energy of Eq. (S13) is the sum of a constant term ∆ k = (ω k − ω −k )/4 and two square root terms A (1),(2) 1,k = ±X k /4, A (1),(2) 2,k = √ 2 Y k ± X k (ω k − ω −k + 4v)/4,(S15) enabling us to write the excitation energy as Eq. (3) in the main text. In Fig. S1, we plot the two square root terms of the critical excitation energy ε −k1 . One can find that the square root term A (2) 1,−k1 becomes non analytic after the critical point g c , showing that the instability of the normal phase does not come from the square root singularity unlike the mean-field PT. The analytical expressions for the excitation energy for the Dicke lattice are too complicated to gain further insights. By taking the infinite frequency limit Ω/ω → ∞ for N a = 1 [S3], the number of degrees of freedom reduces from six to three and it allows us to find the explicit relation between the excitation and fluctuation. This becomes the Rabi lattice model and our analysis here also shows that the anomalous NP exists for the Rabi lattice model and its universality class is different from that of the single Dicke model contrary to the claim in Ref. [S4]. We first adiabatically eliminate the atomic degrees of freedom by considering the infinite frequency limit N a = 1, Ω/ω → ∞ [S3].By applying the Schrieffer-Wolff transformation U S = N n=1 exp (λ n /Ω n ) a + a † (σ + − σ − ) to remove the coupling between spin states and project the Hamiltonian to the spin subspace H ↓ [S3], we find the effective Hamiltonian in the normal phase, H RL np = 3 n=1 ωa † n a n − λ 2 ωΩ a n + a † n 2 + J(e iθ a † n a n+1 + h.c.) . (S16) By introducing the Fourier transformation a † n = k e ikn a † k / √ 3 with k = 0, ±2π/3, we have H RL np = k 1 − g 2 2 ω + 2J cos(θ − k) a † k a k − ωg 2 4 a k a −k + a † k a † −k . (S17) For each momentum k, the above quadratic Hamiltonian can be diagonalized by a two-mode squeezed operator, S = exp k r k a † k a † −k − a k a −k with r k = 1 8 ln ω k +ω −k ω k +ω −k −2ωg 2 . We find the excitation spectra to be ε RL k = A k + 2∆ k (S18) where A k = (ω k + ω −k ) 2 /4 − (ω k + ω −k )ωg 2 /2, ∆ k = (ω k − ω −k )/4 (S19) Note that there is a single square root term instead of two and the constant shift is twice as that of the Dicke lattice model. The photon number expectation value for the ground state, \|ψ GS = S k \|0 k , for each momentum k is a † k a k np = k 0\|S † k a † k a k S k \|0 k = ω k + ω −k 2 √ A k + 2 √ A k ω k + ω −k − 2. (S20) the local photon number is therefore given by a † n a n np = 1 N k a † k a k = 1 N k ω k + ω −k 2 √ A k + 2 √ A k ω k + ω −k − 2 ,(S21) which is Eq. (5) in the main text. Therefore, we clearly see that the divergence of the photon number is determined by the singularity of the square root term in the excitation energy. Therefore, the non-divergent fluctuation in the anomalous NP originates from the asymmetry in the dispersion relation (∆ k = 0) that prevents the square-root singularity of the excitation energy to occur. MEAN-FIELD SOLUTION Derivation of the effective mean-field energy with long-range interaction From Eq. (S3), we find that the mean-field ground-state energy, Eqs. (S6b), are minimized when cos φ n = −x n /\|x n \|, cos θ n = −Ω/ Ω 2 + 16λ 2 x 2 n . We insert these relations to the GS energy and rescale byᾱ n = ω 0 /N a Ω α n , g = 2λ/ √ ω 0 Ω,Ē GS = E GS /(N a Ω),J = J/ω 0 . We then have, E N = N n=1 x 2 n +ȳ 2 n − 1 2 1 + 4g 2x2 n + 2J cos θ(x nxn+1 +ȳ nȳn+1 ) + 2J sin θ(x n+1ȳn −x nȳn+1 ) ,(S23) which is Eq. (6) in the main text. The rescaled energyĒ N is minimized when ∂Ē ∂ȳ n = 2ȳ n + 2J [cos θ(ȳ n−1 +ȳ n+1 ) + sin θ(x n+1 −x n−1 )] = 0,(S24) which is equivalent to the imaginary part of Eq. (S4). We take advantage of the translational symmetry of the system to integrate out the momentum quadratures y n of the cavity fields in the mean-field energy. As we shall see below and we have discussed in the main text, this allows us to understand the nature of first order transitions in the broken symmetry phase. Let y = (y 1 , · · · , y n ) and x = (x 1 , · · · , x n ) , then Eq. (S24) becomes M 1 y = M 2 x (S25) where M 1 and M 2 are cyclic matrices, i.e. M 1 = circ{1, J cos θ, 0, · · · , 0, J cos θ}, M 2 = circ{0, −J sin θ, 0, · · · , 0, +J sin θ}. Therefore, they can both be diagonalized by M 1 = QD 1 Q † , M 2 = QD 2 Q † (S26) where D 1 and D 2 are diagonal matrices and Q is a discrete Fourier transformation matrix D 1 = diag[1 + 2J cos θ cos(2πj/N )], j = 0, 1, · · · , N − 1 D 2 = diag[2iJ sin θ sin(2πj/N )], j = 0, 1, · · · , N − 1 Q ij = ω (i−1)(j−1) N , i, j = 1, 2, · · · , N(S27) where ω N = e −i2π/N is a primitive N th root of unity. Consider the bilinear terms of y n inĒ GS , which can be expressed as E y = [y 2 n + 2J cos θy n y n+1 + 2J sin θy n (x n+1 − x n−1 )] = y M 1 y − 2y M 2 x = −y M 2 x = −x QD 2 D −1 1 D 2 Q † x.(S28) This shows that the nearest neighborhood interaction for the momentum quadratures can be written as long-range interactions for the position quadratures. In the absence of the hopping terms for theȳ n , the Eq. (6) in the main text can be trivially minimized overȳ n and becomes E GS = N n=1 x 2 n − 1 2 1 + 4g 2x2 n + 2J cos θx nxn+1 + x QD yx Q † x D yx = D 2 D −1 1 D 2 = −diag [2J sin θ sin(2πj/N )] 2 /[1 + 2J cos θ cos(2πj/N )] j = 0, 1, · · · , N − 1 (S29) The above equation can be written as This is Eq. (7) in the main text. E GS N = N n=1  x 2 n − 1 2 1 + 4g 2x2 n + (N −1)/2 m=0J eff mxnxn+m   .(S30) Mean-field solution of the effective mean-field energy with J1-J2 interaction The mediated interaction amongx n byȳ n is on the order ofJ 2 and we haveJ eff 2 J eff m>2 . See Fig. S2 for N = 5 and N = 7 as an example. This stems from the fact that a longer range interaction requires a higher order process to mediate. WhileJ eff 1 , which is on the order ofJ, is the dominant interaction, it is modulated byJ cos θ so thatJ eff 1 andJ eff 2 can be comparable in magnitude for θ ∼ π/2 and their competition could drive a series of first order transition. If \|J eff 1 \| \|J eff 2 \|, the next-nearest neighborhood interaction can be neglected and the mean-field energy in Eq. (7) of the main text becomes identical with that of the Dicke lattice model with real hopping [S1]. Therefore, forJ eff 1 > 0 the frustrated superradiant phase where all neighboring sites have an opposite sign forx n , except a single pair which has the same sign. ForJ eff 1 < 0, on the other hand, allx n has the same sign, leading to the non-frustrated superradiant phase. If \|J eff 1 \| becomes comparable withJ eff 2 , then the relative magnitudes and signs compete to favor one frustrated solution over the other. As one varies θ from 0 to π, the number of ferromagnetic pairs (neighbors that have the same sign forx n ) increases from 1, 3,..., and N − 1 for odd N . For even N , as shown in Ref. [S1], there is no frustration whenJ eff 1 is the dominant energy scale. However, ifJ eff 2 > 0 becomes larger thanJ eff 1 , the next-nearest neighborhood tends to be anti-aligned, which is incompatible with a lattice where N/2 is an odd number; on the other hand, if N/2 is an even number, there would be no frustration. This point is illustrated in Fig. S3. Therefore, our analysis for odd N can be straightforwardly generalized to even N , particularly to N = 6, 10.. where geometric frustration is expected. Non-frustrated solution When the superradiant solution is not frustrated, we can show that the superradiance take the same value to minimize the mean field energy [S1], i.e. x =x n . Moreover, we haveȳ n = 0 by Eq. (S24). The rescaled mean field energy therefore reduces toĒ N = N (1 + 2J cos θ)x 2 − 1 2 1 + 4g 2 x 2 (S32) The above equation is minimized at x = 1 2g [g/g c (0)] 4 − 1 (S33) where g c (0) = √ 1 + 2J cos θ by Eq. (4) in the main text. Asymptotic solution for N = 3 Following the above analysis, the mean-field energy of the Dicke triangle can be reduced tō E GS 3 = 3 n=1 x 2 n − 1 2 1 + 4g 2x2 n + 1 m=0J eff mxnxn+m = 3 n=1 (1 +J eff 0 )x 2 n − 1 2 1 + 4g 2x2 n +J eff 1xnxn+1 .(S34) which is of the same form as the mean-field energy derived in Ref. [S1] where the photon hopping is considered to be real. The minimum of this function cannot be computed analytically, but we can find an asymptotic solution near the critical point g + c = g c (2π/3):x 3 − 2\|g − g + c \| 1/2 √ 3(g + c ) 3/2 − \|g − g + c \| 3/2 6 √ 3(g + c ) 5/2 ,(S35a)x 1 =x 2 \|g − g + c \| 1/2 √ 3(g + c ) 3/2 + (8 − 7J)\|g − g + c \| 3/2 12 √ 3J(g + c ) 5/2 .(S35b) The asymptotic solutions are useful in Sec. where we derive the novel scaling of the excitation energy in FSP. It also indicates a second-order phase transition becausex n is continuous. We have verified that the order parameter is continuous in all phases during the transition at g = g c and the ground-state energy is discontinuous in the second derivative, as shown in Fig. S4 for the frustrated superradiant phase. Ground state degeneracies at the first-order phase transition lines We note that the configurations on both sides of the first-order transition can coexist on the first-order boundary. Therefore, the degeneracy on the first-order line is the sum of the degeneracy of two sides. Moreover, for N = 3, it is interesting to note thatJ eff 1 = 0 at θ = θ k1,k0 c . This indicates that the magnetic flux effectively decouples the three cavities in the mean-field energy. The mean field solution of each cavity is therefore independent and can have both plus and minus signs, leading to a degeneracy of D(N = 3, θ = θ k1,k0 c ) = 2 3 = 8. The degeneracy here can also be interpreted as a sum of the degeneracy of both sides, i.e. 6 + 2 = 8. EXCITATION AND FLUCTUATION IN THE SUPERRADIANT PHASES We have α n = 0 in SP and the system preserves the translational symmetry only if x n are the same. When there is translational symmetry, we use the same strategy as NP to analyze the momentum of the critical mode. Otherwise, we use the symplectic transform to diagonalize the effective Hamiltonian. Non-frustrated Superradiant Phase Since all x n have the same value, we letλ = λ n ,Ω = Ω n and implement the same Fourier transform as NP to have H sp = k [ω + 2J cos(θ − k)]a † k a k +Ωb † k b k −λ(a k + a † −k )(b −k + b † k ) ,(S36) whereg = 2λ/ ωΩ. The Hamiltonian can be diagonalized in a similar way as in NP. One can find that ∆ sp k = 0 for the zero momentum mode, so the excitation energy follows the typical mean-field behavior ε 0 ∝ \|g − g c \| 1/2 . The result for the photon number and entanglement across the mean-field SPT, calculated using the method outlined below, is given in Fig. . The scaling behaviors are identical to that of the single Dicke model. Frustrated Superradiant Phase When the translational symmetry is spontaeneouly broken due to frustration, we introduce q n = (a n + a † n )/ √ 2, p n = (a † n − a n )/ √ 2, Q n = (b n + b † n )/ √ 2, P n = (b † n − b n )/ √ 2 to write the effective Hamiltonian Eqs. (S6a) as H sp = N n=1 ω 0 2 q 2 n + p 2 n −Ω n Q 2 n + P 2 n +λ n Q n q n + J cos θ(q n q n+1 + p n p n+1 ) + J sin θ(q n+1 p n − q n p n+1 ) . (S37) The above Hamiltonian is bilinear in operators H sp = 1 2 r H sp r with r = (q 1 , p 1 , Q 1 , P 1 , . . . , q n , p n , Q n , P n ), and can be diagonalized by symplectic transform [S5], so that SH sp S = diag(ε 1 , ε 1 , . . . , ε 2n , ε 2n ) (S38) The covariance matrix C of the ground state can be calculated as C = 1 2 S S (S39) We compute the covariance matrix numerically, then obtain the photon number and bipartite entanglement by a † n a n = 1 2 (∆q n ) 2 + (∆p n ) 2 − 1 (S40a) S n = 1 2 ln det 2C n (S40b) where C n denotes the nth 4 × 4 block diagonal matrix of C. The fluctuations in the normal phase and the non-frustrated superradiant phase are also computed by the covariance matrix in a similar way. In addition to the N = 3 example in the main text, we show here the result for N = 5. Just like the N = 3 case, we find non-divergent correlation in the normal phase, as shown in Fig. (S6). On the other hand, they are divergent in the frustrated superradiant phase as for N = 3, but with different scaling. For the photon number we have γ = 3/2 while for entanglement we have ν = 3/4. This indicates that the critical exponents for the frustrated SP depend on the size of the lattice, consistent with the findings in Ref. [S1]. Novel critical scaling for N = 3 in FSP Although we cannot derive an analytic expression for the critical excitation for N = 3, we derive the critical exponent by using asymptotic expansion near the critical point [S1]. We first investigate the characteristic polynomial of the matrix iΩ 0 H sp in the symplectic transform P c (ε) = 6 s=0 C 2s ε 2s = 0. We note that if we expand P c (ε) in terms of \|g − g c \|, then every order of \|g − g c \| should vanishes in order to satisfy Eq. (S41). Let us assume that the energy gap closes with an unknown exponent γ, i.e., ε ∝ \|g − g c \| γ . Except the constant term C 0 , the lowest order term in \|g − g c \| of P c (ε) is \|g − g c \| 2γ , since C 2s \| g=g + c = 0 if s ≥ 2. On the other hand, we find that C 0 ∝ \|g − g c \| 3 ,(S42) by inserting the asymptotic solutions of Eq. (S35) and expanding C 0 around g = g c . In order to make the term in the order of \|g − g c \| 2γ vanish in Eq. (S41), \|g − g c \| 2γ must cancel with C 0 ∝ \|g − g c \| 3 , and this allows us to identify that γ = 3/2 (S43) FIG. 1 . 1Phase diagrams and excitation energies for (a)(b) N = 3 and (c)(d) N = 5. (a) and (c) Top panel: Phase diagram in the g − θ space. NP and SP stand for the normal and superradiant phase, respectively; ANP and FSP stand for anomalous NP and frustrated SP, respectively. k is the momentum of the critical mode in NP. Arrows denote sign configurations of the mean values xn = Re( an ). Up arrows denote xn > 0 and down arrows denote xn < 0. The lines connecting lattice sites correspond to the effective hopping shown in the bottom panel. (a) and (c) Bottom panel: The effective photon hoppingJ eff m in the mean-field energy as a function of θ.J eff 1 is the nearest neighbor hopping whileJ eff 2 is the next nearest neighbor hopping. (b) and (d) Critical excitation energies as a function of g for different values of θ. γ denotes the scaling of the corresponding excitation. FIG. 2 . 2(a) Bipartite entanglement between nth site with the two remaining sites and (b) photon number for nth cavity as a function of g/gc for N = 3 and θ < θc across the anomalous SPT. Lines are analytical and shapes are numerical results. Both quantities are bounded in the anomalous NP and divergent in the frustrated SP. The insets on the left side show scaling of the two quantities in the frustrated SP, both of which are γ = 1/2. a Ω y n . We numerically minimize Eq. (6) and draw the phase diagram for N = 3 and N = 5 as shown in Figs. 1(a) and 1(c). For odd N , there exists (N − 1)/2 first order transition lines that meet with the continuous transition line g c (θ) at each flux critical point θ ki,ki+1 c , making them tricritical points. FIG . S1. Two square root terms of the critical excitation energy in Eq. (3) of the main text as a function of g/gc at θ = π/4. The vertical dashed line denotes the critical point. It shows that one of the square root terms becomes zero at g > gc and the other remains finite.The origin of the non-divergent fluctuation in the anomalous normal phase FIG . S2. Bottom panel: Effective hopping energyJ eff m as a function of θ for (a) N = 7 and (b) N = 9. Top panel: Configurations of mean-field solutions xn in corresponding regions. Up arrows denote xn > 0 and down arrows denote xn < 0. [ FIG. S3. Bottom panel: Effective hopping energyJ eff m as a function of θ for (a) N = 4 and (b) N = 6. Top panel: Favored configurations of mean-field solutions xn in corresponding regions. In (b), the red question mark denotes the occurrence of frustration as the positive next nearest neighbor coupling is dominant. For even N , the frustration occurs only when N/2 is odd and whenJ eff 2 > 0 is the dominant energy scale.with effective photon hopping amplitudes, 2J sin θ sin(2πj/N )]2 1 + 2J cos θ cos(2πj/N ) cos 2π(j − m + 1) N , m = 0, 2, · · · , N − 1 .(S31) FIG . S4. (a) Ground state energy (black) and its second derivative (red) as a function of g/gc for θ = π/4. (b) One of the six frustrated superradiance solutions as a function of g/gc. x1 and x2 have the same value while x3 has a different sign and larger magnitude. The two figures indicate a continuous phase transition. . S5. (a) Bipartite entanglement and (b) photon number as a function of g/gc in the non-frustrated superradiant phase (θ = 3π/4). Both quantities satisfy the typical mean-field scaling γ = 1/2. S6. 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[ "Long distance co-propagation of quantum key distribution and terabit classical optical data channels", "Long distance co-propagation of quantum key distribution and terabit classical optical data channels" ]	[ "Liu-Jun Wang \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n", "Kai-Heng Zou \nState Key Laboratory of Advanced Optical Communication Systems and Networks\nPeking University\n100871BeijingChina\n", "Wei Sun \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n", "Yingqiu Mao \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n", "Yi-Xiao Zhu \nState Key Laboratory of Advanced Optical Communication Systems and Networks\nPeking University\n100871BeijingChina\n", "Hua-Lei Yin \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n", "Qing Chen \nQuantumCTek Co., Ltd\n230088HefeiAnhuiChina\n", "Yong Zhao \nQuantumCTek Co., Ltd\n230088HefeiAnhuiChina\n", "Fan Zhang \nState Key Laboratory of Advanced Optical Communication Systems and Networks\nPeking University\n100871BeijingChina\n", "Teng-Yun Chen \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n", "Jian-Wei Pan \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n" ]	[ "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "State Key Laboratory of Advanced Optical Communication Systems and Networks\nPeking University\n100871BeijingChina", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "State Key Laboratory of Advanced Optical Communication Systems and Networks\nPeking University\n100871BeijingChina", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "QuantumCTek Co., Ltd\n230088HefeiAnhuiChina", "QuantumCTek Co., Ltd\n230088HefeiAnhuiChina", "State Key Laboratory of Advanced Optical Communication Systems and Networks\nPeking University\n100871BeijingChina", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina" ]	[]	Quantum key distribution (QKD) generates symmetric keys between two remote parties, and guarantees the keys not accessible to any third party. Wavelength division multiplexing (WDM) between QKD and classical optical communications by sharing the existing fibre optics infrastructure is highly desired in order to reduce the cost of QKD applications. However, quantum signals are extremely weak and thus easily affected by the spontaneous Raman scattering effect from intensive classical light. Here, by means of wavelength selecting and spectral and temporal filtering, we realize the multiplexing and long distance co-propagation of QKD and Terabit classical coherent optical communication system up to 80km. The data capacity is two orders of magnitude larger than the previous results. Our demonstration verifies the feasibility of QKD and classical communication to share the resources of backbone fibre links, and thus taking the utility of QKD a great step forward.	10.1103/physreva.95.012301	[ "https://arxiv.org/pdf/1610.04475v1.pdf" ]	115,011,138	1610.04475	6d6679e7a4ae29f88bbd2504316e53314473c7a3	Long distance co-propagation of quantum key distribution and terabit classical optical data channels 14 Oct 2016 Liu-Jun Wang Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics University of Science and Technology of China 230026HefeiAnhuiChina Kai-Heng Zou State Key Laboratory of Advanced Optical Communication Systems and Networks Peking University 100871BeijingChina Wei Sun Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics University of Science and Technology of China 230026HefeiAnhuiChina Yingqiu Mao Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics University of Science and Technology of China 230026HefeiAnhuiChina Yi-Xiao Zhu State Key Laboratory of Advanced Optical Communication Systems and Networks Peking University 100871BeijingChina Hua-Lei Yin Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics University of Science and Technology of China 230026HefeiAnhuiChina Qing Chen QuantumCTek Co., Ltd 230088HefeiAnhuiChina Yong Zhao QuantumCTek Co., Ltd 230088HefeiAnhuiChina Fan Zhang State Key Laboratory of Advanced Optical Communication Systems and Networks Peking University 100871BeijingChina Teng-Yun Chen Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics University of Science and Technology of China 230026HefeiAnhuiChina Jian-Wei Pan Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics University of Science and Technology of China 230026HefeiAnhuiChina Long distance co-propagation of quantum key distribution and terabit classical optical data channels 14 Oct 2016 Quantum key distribution (QKD) generates symmetric keys between two remote parties, and guarantees the keys not accessible to any third party. Wavelength division multiplexing (WDM) between QKD and classical optical communications by sharing the existing fibre optics infrastructure is highly desired in order to reduce the cost of QKD applications. However, quantum signals are extremely weak and thus easily affected by the spontaneous Raman scattering effect from intensive classical light. Here, by means of wavelength selecting and spectral and temporal filtering, we realize the multiplexing and long distance co-propagation of QKD and Terabit classical coherent optical communication system up to 80km. The data capacity is two orders of magnitude larger than the previous results. Our demonstration verifies the feasibility of QKD and classical communication to share the resources of backbone fibre links, and thus taking the utility of QKD a great step forward. to simultaneously transmit QKD is 40 Gigabits per second (Gbps) [26,27]. In fact, from the simulation results of Patel et al. [26], when quantum signals wavelength is located at C-band (1530 -1565 nm), the maximum bandwidth of data channels achievable was predicted to be 140 Gbps. This is because that as the bandwidth increases, the classical light launch power also increases, resulting in stronger spontaneous Raman scattering noise and linear crosstalk induced by the classical light, which are the main obstacles in WDM integration of QKD and classical optical data channels. Besides, in previous experiments the classical communication generally used on-off keying (OOK) modulation schemes, which is intensity modulated and detected directly by photodiodes. With this kind of modulation, a bit rate of 1 Gbps typically corresponds to a launch power of 0 dBm (1 mW). As the bit rate is basically proportional to the launch power, 1 Tbps OOK data communication would require 30-dBm classical light, which will result in the unacceptably severe Raman scattering noise. Fortunately, the Tbps classical data channels are currently implemented by coherent optical communication combined with M-ary quadrature amplitude modulation (QAM) formats. By using 16-QAM and 64-QAM in our experiment, the classical optical power is about 10 dBm at Tbps level, which provides the possibility of QKD multiplexing. We note that high-order QAMs require higher optical signal to noise ratios (OSNR) than OOK modulations. The low launch power will lead to a worse OSNR while the high launch power will result in severe fibre nonlinear distortions that deteriorate the signal quality. Therefore, for a specific transmission distance, there exists an optimum launch power as a trade-off to balance the influence of noise and nonlinear interference. There are some points of consideration when multiplexing QKD with such a high capacity classical optical communication. Firstly, we need to suppress the in-band noise that has the same wavelength as our quantum signals, which comes from the background fluorescence of the classical light source and the amplified spontaneous emission noise generated from erbium-doped fibre amplifiers (EDFA). Secondly, we require a high degree of isolation to reduce the out-of-band noise, which corresponds to the probability of classical light being detected by the single-photon detectors at the QKD receiver site. These two kinds of noise are proportional to the incident light power and generally referred to as the linear crosstalk. In fact, the main challenge of WDM comes from the spontaneous Raman scattering effect from the intensive classical light [28][29][30][31]. Here, we show that through wavelength selection and sharp optical filtering, the multiplexing between QKD and Terabit classical data channels can be successfully achieved. Such coexistence leverages the existing backbone fibre cables, realizing large cost savings potentials over deploying dedicated quantum links. II. RAMAN NOISE AND SECURE KEY RATES AT 1550.12 nm AND 1310 nm In order to quantify the impact of Raman scattering on QKD, we first need to determine the classical signal wavelength λ c and quantum signal wavelength λ q . The commercial dense WDM (DWDM) technology usually uses the C-band with relatively low fibre loss. Note that ED-FAs are generally necessary to compensate for the fibre link attenuation. In contrast, the quantum signals cannot be amplified in principle because of the no-cloning theorem [32,33]. Therefore, in previous point-to-point WDM experiments, λ c was usually chosen in the C-band, while λ q was also located at C-band because of its low fibre loss [34][35][36][37][38], or at the O-band (1260 -1360 nm) because of its low Raman noise [17,18,39,40]. Hence, we need to consider the two factors together to determine the appropriate quantum signal wavelength in different classical optical communication environments. We measure the forward Raman scattering noises, which transmit in the same direction as the incident light, at both 1550.12 nm and 1310 nm using an InGaAs avalanche photodiode (APD) based single-photon detector, operating at 1.25 GHz with a 180-ps full width at half maximum (FWHM) gate width. Figure 1 shows the count rate of Raman noise generated from a continuous wave laser source tuned from 1530 nm to 1570 nm and launched with a power level of 6 dBm. We note that in the λ q =1550.12 nm configuration, the QKD receiver used a 20-GHz fibre Bragg gating (FBG) to filter the Raman noise, which induces an extra loss of 3.2 dB. While in the λ q =1310 nm configuration, a bandpass filter with center wavelength of 1310.0 nm, passband width of 100 GHz, and insertion loss of 0.5 dB was used. Consequently, the Raman noise at 1550.12 nm strongly depends on the incident light wavelength, which has a count rate of 440.4 kilo counts per second (kcps) on average between 1550.12 ± 3 nm, and two times more counts beyond 1550.12 ± 10 nm. Moreover, Fig. 1 shows the intensity of the anti-Stokes scattering slightly weaker than that of the Stokes scattering. Meanwhile, the averaged noise count rate at 1310 nm is 6.2 kcps, and decreases slightly with the increasing classical signal wavelength. We can see that although the received bandwidth of 1550.12 nm is 1/5 of that of 1310 nm, the Raman noise at 1550.12 nm is approximately two orders of magnitude higher than that of 1310 nm. Nevertheless, typical fibre attenuation at 1310 nm is 0.33 dB/km, which is larger than the loss of 0.2 dB/km at 1550.12 nm. In order to compare the secure key rates of the two quantum signal wavelengths, we consider a scenario of QKD co-propagating with classical channels in a 50 km fibre, and simulate the key rate as a function of classical launch power as shown in Fig. 2. The Raman scattering coefficient we used is obtained from the measured data of Raman noise in Fig. 1, and the QKD key rate simulation follows the decoy method [41]. One can see that the key rate corresponding to λ q =1550.12 nm is higher than that of 1310 nm when the classical launch power is less than -0.76 dBm, because the Raman noise for both cases is small at low classical launch power, while 1550.12 nm has the advantage of low fibre loss. As the optical launch power increases, the disturbance from Raman noise becomes evident that it deteriorates quantum signals of 1550.12 nm much more severely than that of 1310 nm, thus resulting in its rapid key rate decline. When the launch power reaches -0.76 dBm, the key rates of both wavelengths are the same, and afterwards 1310 nm quantum signals display more advantage from the low level of Raman noise at 1310 nm. In addition, as we can see, when the power is greater than 2.0 dBm, 1550.12 nm could not generate any secure keys while 1310 nm could still performs well up to a power level of about 10 dBm, which corresponds to the Tbps Secure key rate comparison between 1550.12 nm and 1310 nm. The secure key rates were calculated at 50 km fibre length as a function of classical launch power. The blue region indicates that 1550.12 nm offers a higher key rate than 1310 nm, while the green area indicates vice versa and that 1310 nm is more suitable for the quantum signal wavelength in a Tbps environment, where the classical launch power is around 10 dBm. level of classical communication. Similar results could be obtained for the QKD counter-propagation scenario. Consequently, we choose 1310 nm as the wavelength of quantum signals in our experiment, which not only allows one to achieve higher degrees of isolation in suppressing the linear crosstalk through low-cost coarse wavelength division multiplexer (CWDM), but also avoid nonlinear four-wave mixing (FWM) effects when multiple C-band classical channels are used, which may produce additional noise to a 1550.12-nm quantum channel [20,35]. III. CO-PROPAGATION OF QKD AND FOUR 64-QAM CLASSICAL CHANNELS Figure 3 shows the experimental setup. Classical communication includes multiple DWDM channels within the C-band with wavelengths λ 1 , λ 2 , . . . , λ 2n−1 , λ 2n . Meanwhile, our QKD system employs polarization encoding based BB84 protocol [1] and the decoy state method against photon-number-splitting (PNS) attacks [41][42][43][44]. The clock synchronization between the QKD transmitter and receiver (referred to as Alice and Bob) is achieved with 100 kHz optical pulses at wavelength of 1570 nm. The classical, the quantum, and the synchronization channels are multiplexed and de-multiplexed using CWDMs to transmit over a single standard singlemode fibre. The CWDMs provide about 83-dB suppression of the in-band noise in the multiplexing and >180 dB isolation between the classical and the quantum channels in the de-multiplexing, which is sufficient to reduce the linear crosstalk to a negligible level. Before the detection of quantum signals, we use a custom-made 1310-nm bandpass filter with a bandwidth of 100 GHz, to diminish the Raman noise down to about 1/24 of that passes through the de-multiplexing CWDMs. The single-photon detectors can also effectively reduce the Raman noise in time domain through narrow gate widths. In the first set of experiments, the classical optical communication system consists of 4 channels modulated with 64-QAM format. The channel spacing is 50 GHz with wavelengths ranging from 1549.1 to 1550.3 nm (the optical spectrum is shown as Supplementary Fig. 1). The bit rate of each channel is 336 Gbps, and thus the total gross data capacity is 1.344 Tbps. The co-and counter-propagating WDM layouts each induce a total loss of about 1.6 dB to classical channels (see Supplementary Fig. 2 and Fig. 3). Figure 4 shows the measured classical bit error rate (BER) and Raman noise as functions of classical launch power after 50 km standard single-mode fibre (SSMF) transmission, with QKD copropagating with classical channels in a WDM way. One can see that the classical BER is slightly higher with QKD than that of without, due to the additional attenuation induced by QKD multiplexing. The BER has a minimum value at 4-dBm launch power. As the power increases, the nonlinear distortions will degrade the signal quality and thus increase the BER. In addition, as shown in Fig. 4, the amount of Raman noise generated from the classical signal at 1310 nm is proportional to the incident light power, indicating that the spontaneous Raman scattering is a linear effect. Figure 5(a) shows the Raman noise and the classical BER measured at different fibre distances in the WDM environment. Both the forward and the backward Raman noises are measured at 4-dBm launch power. As the transmission distance increases, the forward Raman noise first increases and then decreases, while the backward Raman noise increases gradually until saturation. Also, the backward noise count is much higher than the forward noise, which is consistent with theoretical calculations. It should be noted that, for classical communication forward error correction (FEC) is usually adopted, which can correct a pre-FEC BER of 0.45% or 2.4% to a level of 10 −15 or less by adding hard-or soft-decision FEC with 7% or 20% overhead, respectively [45,46]. Since Tbps communication is generally deployed in optical trunk links, we demonstrate the co-propagation of QKD and the four classical data channels at moderately longer distances. Figure 5(b) shows the QKD secure key rate and quantum bit error rate (QBER) in this scenario. The QKD secure key rate after 50 km transmission is 18.7 kbps, and the classical launch power is kept at 4 dBm from 50 km to 70 km with the BER below 2.4% (see Supplementary Fig. 4). The maximum distance we achieved is 80 km with a fibre loss of 27.1 dB at 1310 nm, and the secure key rate is 1.2 kbps and QBER is 3.1%. For 80 km co-propagation, we have to increase the optical power of the classical channels to 8 dBm in order to ensure its BER to be below 2.4% (2.14% in the experimental measurement, see Supplementary Fig. 5). Considering the soft-decision FEC with 20% redundancy and frame overhead, the net bit rate of the classical communi- cation is actually 1.07 Tbps. In the counter-propagating case, QKD suffers from much stronger backward Raman scattering. From Fig. 5(a) one can see that the backward Raman noise count at 50 km is 3.2 times more than its forward noise, resulting in QBER of 1.98%, and key rate of 17.7 kbps. The maximum distance we achieved in the counter-propagating case is 70 km with QBER at 2.62% and key rate of 3.7 kbps, where we have increased the classical launch power to 5 dBm with a measured BER of 2.18% ( <2.4%), and the net bit rate of the classical channels is still 1.07 Tbps. IV. CO-PROPAGATION OF QKD AND 32 16-QAM CLASSICAL CHANNELS In the second set of experiments, we build up a classical optical communication system consisting of 32 channels modulated with a 16-QAM format. The channel spacing is 100 GHz with wavelengths ranging from 1535.7 to 1559.7 nm, and the optical spectrum is shown as Fig. 6. The bit rate of each channel is 224 Gbps, thus the total gross bandwidth amounts to 7.168 Tbps. The WDM layouts introduce about 2-dB loss to the classical channels (see Supplementary Fig. 6 and Fig. 7). We successfully implement the WDM of QKD and classical communication at different fibre distances for both co-and counterpropagating cases. In Table I we list the measured results of 50 km and the maximum distance achievable. We have measured that the optimal launch power for 50 km transmission is around 11 dBm (see Supplementary Fig. 8). We obtain the classical BER to be below 0.45% when fibre distance is less than or equal to 70 km (see Supplementary Fig. 9), so we can perform error correction by adding 7% overhead, therefore the effective throughput of classical channels reaches 6.38 Tbps, improving two orders of magnitude compared with previous results [26,27]. We achieve maximum transmission distances of 80 km and 60 km in the co-and counterpropagating cases, respectively. V. CONCLUTION In our experiments, the WDM optical arrangements follow the principle of guaranteeing sufficient isolation of linear crosstalk, while using as few filters as possible, so as to reduce optical loss and cost. WDM filters generally have three ports: a common-port, a pass-port, and a reflect-port. We find that the pass-ports have much higher isolation than the reflect-ports. For instance, the pass-ports of 1550-nm (1310-nm) CWDMs have about 83-dB (90-dB) isolation to the light with wavelength 1310 nm (1550 nm), while the reflect-ports have an isolation of only about 20 dB for the filter center wavelength. Therefore, we use one 1550-nm CWDM to suppress the inband noise, and two cascaded 1310-nm CWDMs to suppress the out-band noise. In addition, the second set of experiments have similar arrangements except that the 1550-nm CWDMs are replaced by 1550-nm filter-based wavelength division multiplexers (FWDMs), which have wider passband to accommodate all 32 channels. For wavelength division multiplexing the main challenges of suppressing linear crosstalk and reducing Raman noise are irrelevant to the implemented QKD protocol and encoding format. Therefore, although we adopt BB84 protocol with polarization encoding in our experiment, the wavelength division multiplexing principle and methods we propose are adaptive to other QKD protocols, like differential phase shift QKD and measurement device independent QKD, and other encoding formats, like phase and time-bin encoding. Furthermore, our methods may also be used by continuous variable QKD or other kinds of quantum communications when they co-propagate with classical data channels over opti- FIG. 6. The spectrum of 32 classical data channels and QKD clock synchronous channel, which is measured back-to-back (BTB) and after transmission over 80 km standard single mode fibre. One can see the obvious amplified spontaneous emission (ASE) generated from the erbium-doped fibre amplifier (EDFA) which should be suppress in advance to reduce the crosstalk. cal fibre. The secure key rate and transmission distance are two important parameters of QKD, and are related to the performance of single-photon detectors and parameter estimation process. In our experiment, we use semiconductor APD based detectors, but currently the superconducting nanowire single-photon detectors (SNSPDs) have better performances with detection efficiency of >70% and dark count rate of <100 counts per second. Therefore, if the detector of our QKD system upgrades to SNSPD, the secure key rates and transmission distances of QKD will improve drastically. In addition, the finite key length we use to estimate parameters is 1 × 10 6 . By increasing the statistical length we can obtain tighter parameter estimation, which would result in higher secure key rate and longer transmission distance. In conclusion, we analyze the suitable wavelength for QKD transmission when multiplexed with C-band classical optical communication, and find that compared with 1550.12 nm, 1310 nm quantum signals are more adaptable in a Tbps classical data transmission environment with about 10-dBm launch power. Under this wavelength allocation, we have achieved more sufficient crosstalk isolation against classical channels using low-cost CWDMs. In addition, we have reduced the Raman noise through 100-GHz passband filters and single photon detectors with 180 ps gate width. Consequently, we demonstrate the wavelength division multiplexing of QKD with 16-QAM/64-QAM coherent optical communication, with a maximum throughput of 6.38 Tbps and a maximum transmission distance of 80 km, which is the typical span distance in classical communications. We note that although the secure key rate at 80 km is relatively low, the key rate at 50 km is still enough for voice and text encryption using one-time pad, and through using SNSPDs or trusted relays we can realize farther key distributions. Due to the high capacity of coherent optical communication, it will be a mainstream in future and may be applied in metropolitan and access networks, and thus QKD can be deployed in more classical optical communication environments and provide high security applications at low costs. ACKNOWLEDGMENTS This work has been supported by the Science and Technological Fund of Anhui Province for Outstanding Youth (No. 1508085J02), the National Natural Science Foundation of China (No. 61475004) and the Chinese Academy of Sciences (No. XDA04030213). We thank Dan Wang for discussions. Appendix A: Classical communication subsystem In our experiments, the classical communication subsystem conveys multichannel WDM optical signals with digital Nyquist pulse shaping. The high order modulation formats such as 16-/64-QAM are adopted. We build two sets of transmitters and carry out transmission experiments of Terabit Nyquist polarization division multiplexing (PDM) 16/64-QAM signals (see Supplementary Methods). In our experiments, the arbitrary waveform generators (Keysight M8195A) operating at 56 GSample/s with 2-point DAC up-sampling generate baseband signals of 28 Gbaud. The digital RRC filters with a rolloff factor of 0.1 are chosen for Nyquist pulse shaping. We digitized and recorded the received data with a real-time oscilloscope (Keysight DSA-X 96204Q) for offline digital signal processing and signal quality evaluation. In the data frame of PDM Nyquist pulse shaping signal, the preamble of each polarization consist of synchronization and training sequences with a total length of 4696 symbols, which is followed by 102400 data symbols. Two pilot symbols are inserted in every 512 data symbols for carrier phase recovery. The data frame and the DSP diagrams of the transmitter and the receiver are detailed in Supplementary Fig. 11 and Fig. 12. The bit error rate is used to signal quality evaluation. For each measurement, we record a total of ∼ 10 6 data symbols, that is, we evaluate ∼ 4 × 10 6 received bits for 16-QAM signal and ∼ 6 × 10 6 received bits for 64-QAM signal. To determine the BER, we perform the error counting by comparing the decoded symbols with the known bit sequence. In our experiment, two raw BER criterions of 4.5×10 −3 and 2.4 × 10 −2 are adopted, which are the respective thresholds for error-free transmission when secondgeneration hard-decision FEC with 7% overhead [45] or soft-decision FEC with 20% overhead are used [46]. If the FEC works properly, the corrected output BER is less than 1 × 10 −15 , which can be considered as error-free transmission. Appendix B: Quantum key distribution subsystem Our QKD system operates at 625 MHz using polarization encoding. Alice encode the information using weak coherent laser sources and Bob detects the signals with InGaAs APD based single-photon detectors. The detectors work at gated mode with detection efficiency of 10% and dark count rate of 1 × 10 −6 per clock cycle. To reduce the probability of afterpulsing, we set the dead time of detectors to 200 ns. In addition, we implement the decoy-state method to protect the system from potential photon-number-splitting attacks. Alice launches the signal states, decoy states, and vacuum states with a probability ratio of 6:1:1, and the average photon numbers of signal and decoy states are 0.6 and 0.2, respectively. We note that the vacuum state is also a kind of decoy state. In the simulation, we estimate the single-photon parameters and calculate the QBER and secure key rate following the decoy state approach [41,47]. The quantum bit error rate (QBER) is given by E µ = 1 Q µ 1 2 Y 0 + e opt (1 − Y 0 )(1 − e −ηµ )(B1) where Q µ and Y 0 are the probabilities of an detection event given Alice emits a signal state and a vacuum state, respectively, e opt is the probability that a photon hit the erroneous detector due to finite polarization contrast, which is about 0.5% for our system, and is the overall transmittance, including fibre loss, 3-dB loss of Bob's optical components, and the 10% detection efficiency of single-photon detectors. In our experiment, the Y 0 contains two kinds of noise, including the dark count and afterpulsing of detectors, and the spontaneous Raman scattering from classical light. Meanwhile, the secure key rate per clock cycle is given by R = q{−Q µ f H 2 (E µ ) + Q 1 [1 − H 2 (e 1 )] + Q 0 } (B2) where q is the probability of Alice emits signal states and Alice and Bob choose the same bases, f is the inefficiency of error correction which is about 1.25, e 1 is the estimated error rate of single-photon states, and Q 1 and Q 0 are the fractions of detection events by Bob that is due to the single-photon and vacuum ingredients of signal states, respectively. H 2 (x) = −xlog 2 (x) − (1 − x)log 2 (1 − x) is the binary entropy function. The data block size we used to estimate parameters is 1 × 10 6 , and we consider the statistical fluctuations of 5 standard derivations. In the experiment, we implement the entire QKD postprocessing [48] based on hardware, including message authentication with pre-shared symmetric keys, error correction with a cascade algorithm [49], error verification with a cyclic redundancy check (CRC), and privacy amplification with a Toeplitz matrix [50]. FIG. 1 . 1Raman noise at 1550.12 nm (black dots) and 1310 nm (red dots). The forward Raman noises are measured in kilo counts per second (kcps) as a function of classical light wavelength, in 13.6 km standard single-mode fibre at room temperature. The classical launch power is 6 dBm. FIG. 2. Secure key rate comparison between 1550.12 nm and 1310 nm. The secure key rates were calculated at 50 km fibre length as a function of classical launch power. The blue region indicates that 1550.12 nm offers a higher key rate than 1310 nm, while the green area indicates vice versa and that 1310 nm is more suitable for the quantum signal wavelength in a Tbps environment, where the classical launch power is around 10 dBm. FIG. 3 . 3Multiplexing schematic of QKD and Tbps data channels. (a) Classical transmitter. To simulate a real communication environment, we build up two sets of transmitters. The odd channels are combined by beam splitter (BS) to enter modulator 1, and the even channels are combined to enter modulator 2. Then the odd and even channels are combined in an interleaving way. The two in-phase and quadrature modulators (IQM) are driven by electrical signals with different data sequences, ensuring the adjacent channels carrying independent data. After an emulator of polarization division multiplexing, we use an EDFA to amplify and control the power of the classical light which enters the fibre link. (b) Classical receiver. At the receiver site, an EDFA amplifies the classical signals first, then a tunable bandpass filter at C-band (BPF1) is used to select the channel to be detected. In a polarization and phase diversity coherent receiver, the signal light and a local oscillator (LO) laser with approximately the same frequency are passed onto a polarization splitter and mixed in an optical hybrid, from which we operate balanced detection (BD) using paired photodiodes to accurately extract the signal amplitude and phase information. (c) Quantum transmitter. Four non-orthogonal states are generated through two polarizing beam splitters (PBS) and a polarization controller (PC), and the incident power of quantum states is adjusted by a variable optical attenuator (VOA). (d) Quantum receiver. We use a 100-GHz bandpass filter (BPF2) at 1310 nm to effectively suppress the Raman noise. The quantum signals are detected in two conjugate bases using InGaAs avalanche photodiode (APD) based single-photon detectors, and the states of polarization are controlled with automatic feedback. FIG. 4 . 4The classical bit error rate (BER) and forward Raman noise (measured at 1310 nm) as functions of classical launch power at 50 km. FIG. 5 . 5Classical BER and QKD performance with WDM. (a) Measured (symbols) and simulated (solid lines) forward and backward Raman noise as a function of fibre length, and the measured classical BER (red dots) with WDM. (b) Measured and simulated QKD secure key rate (green dots and line) and QBER (blue squares and line), with quantum signals copropagating with 4 classical channels. TABLE I . 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[ "EXPONENTIAL SPLINES AND PSEUDO-SPLINES: GENERATION VERSUS REPRODUCTION OF EXPONENTIAL POLYNOMIALS", "EXPONENTIAL SPLINES AND PSEUDO-SPLINES: GENERATION VERSUS REPRODUCTION OF EXPONENTIAL POLYNOMIALS" ]	[ "Costanza Conti ", "ANDLuca Gemignani ", "Lucia Romani " ]	[]	[]	Subdivision schemes are iterative methods for the design of smooth curves and surfaces. Any linear subdivision scheme can be identified by a sequence of Laurent polynomials, also called subdivision symbols, which describe the linear rules determining successive refinements of coarse initial meshes. One important property of subdivision schemes is their capability of exactly reproducing in the limit specific types of functions from which the data is sampled. Indeed, this property is linked to the approximation order of the scheme and to its regularity. When the capability of reproducing polynomials is required, it is possible to define a family of subdivision schemes that allows to meet various demands for balancing approximation order, regularity and support size. The members of this family are known in the literature with the name of pseudo-splines. In case reproduction of exponential polynomials instead of polynomials is requested, the resulting family turns out to be the non-stationary counterpart of the one of pseudo-splines, that we here call the family of exponential pseudo-splines. The goal of this work is to derive the explicit expressions of the subdivision symbols of exponential pseudo-splines and to study their symmetry properties as well as their convergence and regularity.	null	[ "https://arxiv.org/pdf/1404.6624v2.pdf" ]	119,328,514	1404.6624	43f8215d24e3932e2a034f4cabcd51b2ad54e568	EXPONENTIAL SPLINES AND PSEUDO-SPLINES: GENERATION VERSUS REPRODUCTION OF EXPONENTIAL POLYNOMIALS 13 Nov 2014 Costanza Conti ANDLuca Gemignani Lucia Romani EXPONENTIAL SPLINES AND PSEUDO-SPLINES: GENERATION VERSUS REPRODUCTION OF EXPONENTIAL POLYNOMIALS 13 Nov 2014 Subdivision schemes are iterative methods for the design of smooth curves and surfaces. Any linear subdivision scheme can be identified by a sequence of Laurent polynomials, also called subdivision symbols, which describe the linear rules determining successive refinements of coarse initial meshes. One important property of subdivision schemes is their capability of exactly reproducing in the limit specific types of functions from which the data is sampled. Indeed, this property is linked to the approximation order of the scheme and to its regularity. When the capability of reproducing polynomials is required, it is possible to define a family of subdivision schemes that allows to meet various demands for balancing approximation order, regularity and support size. The members of this family are known in the literature with the name of pseudo-splines. In case reproduction of exponential polynomials instead of polynomials is requested, the resulting family turns out to be the non-stationary counterpart of the one of pseudo-splines, that we here call the family of exponential pseudo-splines. The goal of this work is to derive the explicit expressions of the subdivision symbols of exponential pseudo-splines and to study their symmetry properties as well as their convergence and regularity. Introduction Subdivision schemes are efficient tools for the design of smooth curves and surfaces in many applicative areas such as computer-aided geometric design, curve and surface reconstruction, signal/image processing. Since in all these areas the capability of representing shapes described by polynomial, trigonometric or hyperbolic functions is fundamental (see Figure 1), interpolating and approximating subdivision schemes based on exponential B-splines and inheriting their generation properties, have been recently introduced [1,2,4,6,9,10,12,13,22,28]. The property of reproduction of exponential polynomials is also important since strictly connected to the approximation order of subdivision schemes and to their regularity [14]. In fact, the higher is the number of exponential polynomials reproduced, the higher is the approximation order and the possible regularity of the scheme. Indeed, in application, we aim at subdivision schemes with exponential polynomial reproduction properties, that allow to meet various demands for balancing approximation order, regularity and support size. Such kind of schemes turn out to constitute the family of exponential pseudo-splines, the non-stationary counterpart of polynomial pseudo-splines. The latter family neatly fills the gap between B-splines and 2n-point interpolatory subdivision schemes -both extreme cases of pseudo-splines: while B-splines stand out due to their high smoothness and short support, they provide a rather poor approximation order; in contrast, the limit functions of 2npoint interpolatory subdivision schemes have optimal approximation order but low smoothness and large support. Figure 1. Examples of reproduction of shapes described by exponential polynomials using non-stationary subdivision schemes: initial control points and corresponding limit shapes for the subdivision schemes in [13]. Binary, primal and dual (polynomial) pseudo-splines -the first originally presented by Dong and Shen [19], the latter successively discovered by Dyn et al. [20] and generalized to any arity and to arbitrary parametrizations by Conti and Hormann [11]-are both obtained by means of stationary subdivision schemes whose symbols can be read as a suitable polynomial "correction" of the order-N polynomial B-spline symbol B N (z), since of the form a M,N (z) = B N (z)c M (z). The polynomial correction c M (z) is such that the subdivision schemes with symbols a M,N (z) are the ones of minimal support that, besides generating polynomials of degree N − 1, satisfy the conditions for reproduction of polynomials of degree M − 1, with M ≤ N . We recall that while with generation we mean the subdivision capability to provide specific type of limit functions, with reproduction we mean the capability of a subdivision scheme to reproduce in the limit exactly the same function from which the data is sampled. Similarly to the stationary case, we here define exponential pseudo-splines by means of k-level subdivision symbols which are a suitable "correction" of the k-level subdivision symbols B M,Γ (z). Here Γ identifies the particular space of exponential polynomials EP Γ we deal with, while N and M are related to the number of exponential polynomials that are being generated and reproduced, respectively. Again, c The main contribution of this paper consists in showing how the symbols of exponential pseudo-splines can be explicitly derived. Indeed, we provide the expressions of the inverse matrices of the linear systems arising by imposing the algebraic conditions for exponential polynomial reproduction which were firstly given in [13] and successively extended to any arbitrary arity in [6]. We also show that, under the symmetry assumption on Γ (or on its subset), the symbol a N,Γ (z). To prove the latter we also discover remarkable algebraic properties, never highlighted so far, of symmetric non-stationary subdivision symbols. As a minor contribution, we show how the k-level normalization factor of the exponential B-spline symbol can be selected in accordance with the shift parameter in order to ensure that the exponential B-spline is correctly normalized, namely, besides generating the space EP Γ , it reproduces a specific pair of exponential polynomials {e θ ℓ x , e −θ ℓ x } ∈ EP Γ . Finally, we additionally provide a convergence and regularity analysis of the non-stationary subdivision schemes corresponding to the exponential pseudo-spline symbols here derived. This is possible by first showing that exponential pseudo-splines are asymptotically similar to polynomial pseudo-splines, and then combining recent advances on convergence and regularity of non-stationary subdivision schemes presented in [5] and in [14]. The remainder of the paper is organized as follows. In Section 2 we recall basic notions on non-stationary subdivision schemes reproducing exponential polynomials. Then, in Section 3 we discuss new important results concerning symmetry properties of such subdivision schemes. Symmetric exponential B-splines are recalled in Section 4 where accordance between their parameter shift and their normalization factor is also considered with respect to their reproduction capabilities. The derivation of the symbols of exponential pseudo-splines is provided in Section 5 where the symmetry properties of such symbols are also discussed. Convergence and regularity of the new family of (non-stationary) exponential pseudo-spline subdivision schemes are then investigated in Section 6. As an example of application of the presented theoretical results, the expression of the subdivision symbols of a new family of exponential pseudo-splines is also explicitly derived in Section 7, where pictures of basic limit functions of the corresponding subdivision schemes are also given. The closing Section 8 is to draw conclusions. Non-stationary subdivision schemes and exponential polynomials This paper deals with non-stationary subdivision schemes and reproduction of exponential polynomials. The interest in non-stationary subdivision schemes arose in the last ten years after it was pointed out that they are able to reproduce conic sections, spirals or widely used trigonometric/hyperbolic curves and surfaces, as well as they are featured by tension parameters that allow, on the one side, to obtain considerable variations of shape and, on the other side, to get close to the initial mesh as much as desired (see [12,28,29]). Since numerical methods based on subdivision schemes are relatively simple to implement and highly intuitive in use, they are currently widely exploited in modeling freeform curves and surfaces in computer games and animated movies. The potential of subdivision schemes has recently become apparent also in the context of Isogeometric Analysis (IgA), a modern computational approach that offers the possibility of integrating finite element analysis (FEA) into conventional CAD systems (see, e.g., [3,7,8]). However, the employment of IgA in conjunction with subdivision schemes is nowadays only restricted to the class of stationary methods. This is due to the fact that nonstationary subdivision schemes still require the development of further theoretical results that turn out to be fundamental to support their practical use. Following the notation in [21], for any k ≥ 0 we denote by a (k) := {a (k) j , j ∈ Z} the finite set of real coefficients corresponding to the so called k-level mask of a non-stationary subdivision scheme and we define by a (k) (z) := j∈Z , a (k) j z j , z ∈ C\{0} , the Laurent polynomial whose coefficients are exactly the entries of a (k) . The previous polynomial is commonly known as the k-level symbol of the nonstationary subdivision scheme. With any mask a (k) comes a linear subdivision operator identifying a refinement process, that is the process which transforms a set of real data at level k, f (k) = {f (k) i ∈ R, i ∈ Z}, into the denser set f (k+1) given by (2.1) f (k+1) := S a (k) f (k) , where (S a (k) f (k) ) i := j∈Z a (k) i−2j f (k) j , ∀ k ≥ 0. The subdivision scheme consists in the repeated application of the subdivision operators starting from any initial "data" sequence f (0) ≡ f := {f i ∈ R, i ∈ Z}, and therefore is shortly denoted by {S a (k) , k ≥ 0}. Since the subdivision process generates denser and denser sequences of data, attaching the data f (k) i generated at the k-th step to the parameter values t (k) i with t (k) i < t (k) i+1 and t (k) i+1 − t (k) i = 2 −k , k ≥ 0, a notion of convergence can be established by taking into account the piecewise linear function F (k) that interpolates the data (namely F (k) (t (k) i ) = f (k) i , F (k) \| [t (k) i ,t (k) i+1 ] ∈ Π 1 , i ∈ Z, k ≥ 0) . If the sequence of continuous functions {F (k) , k ≥ 0} converges uniformly, we denote its limit by g f := lim k→+∞ S a (k) S a (k−1) · · · S a (0) f = lim k→+∞ F (k) , and say that g f is the limit function of the non-stationary subdivision scheme based on the rules in (2.1) for the data f . If the non-stationary subdivision scheme is convergent, and g f ≡ 0 if and only if f ≡ 0, then the subdivision scheme is termed non-singular. In the forthcoming discussion we restrict ourselves to non-singular schemes only. As it will be better clarified later, with respect to the subdivision capability of reproducing specific classes of functions, the standard parametrization (corresponding to the choice t (k) i := i 2 k , i ∈ Z) is not always the optimal one. Indeed, the choice (2.2) t (k) i := i + p 2 k , i ∈ Z, p ∈ R, k ≥ 0 , with p suitably set, turns out to be a better selection. In particular, when p ∈ Z the parametrization is termed primal, whereas if p ∈ Z 2 dual. For a complete discussion concerning the choice of the parametrization in the analysis of the polynomial reproduction properties of stationary subdivision schemes, we refer the reader to the papers [4,11,20]. In consideration of the fact that the main goal of this work is the construction of a special class of non-stationary subdivision symbols capable of generating as limit functions exponential polynomials, we continue by recalling the following definitions (see, e.g, [6,13,28]). Definition 1 (Exponential polynomials). Let Γ := {(θ 1 , τ 1 ), . . . , (θ n , τ n )} with θ i ∈ R ∪ iR, θ i = θ j if i = j and τ i ∈ N, i = 1, · · · , n. We define the space of exponential polynomials EP Γ as EP Γ := span{ x rj e θjx , r j = 0, · · · , τ j − 1, j = 1, · · · , n} . For a fixed Γ, and for the corresponding space EP Γ , we recall the following definition. Definition 2 (E-Generation and E-Reproduction). Let {a (k) (z), k ≥ 0} be a sequence of subdivision symbols. The subdivision scheme associated with the symbols {a (k) (z), k ≥ 0} is said to be EP Γ -generating if it is convergent and for f ∈ EP Γ there exists an initial sequence f (0) := {f (t (0) i ), i ∈ Z},f ∈ EP Γ such that lim k→+∞ S a (k) S a (k−1) · · · S a (0) f (0) = f . Moreover, it is said to be EP Γ -reproducing if it is convergent and for f ∈ EP Γ and for the initial sequence f (0) := {f (t (0) i ), i ∈ Z}, lim k→+∞ S a (k) S a (k−1) · · · S a (0) f (0) = f . In the following theorem we recall the algebraic conditions on the k-level symbol that fully identify the generation and reproduction properties of a non-singular, univariate, binary, non-stationary subdivision scheme. A more general version of these conditions, holding for non-stationary subdivision schemes of arbitrary arity, has recently appeared in [6]. Theorem 2.1. [13, Theorem 1] Let Γ := {(θ 1 , τ 1 ), . . . , (θ n , τ n )} with θ ℓ ∈ R ∪ iR, θ ℓ = θ j if ℓ = j and τ ℓ ∈ N, ℓ = 1, · · · , n. Let also z (k) ℓ := e −θ ℓ 2 k+1 , ℓ = 1, . . . , n. A non-singular, non-stationary subdivision scheme associated with the symbols {a (k) (z), k ≥ 0} generates the space of exponential polynomials EP Γ if and only if, for each k ≥ 0, the following conditions are satisfied (2.3) d r a (k) (−z (k) ℓ ) dz r = 0, ℓ = 1, . . . , n, r = 0, . . . , τ ℓ − 1. Furthermore, it reproduces the space of exponential polynomials EP Γ if and only if, for each k ≥ 0, in addition to (2.3) the following conditions are satisfied (2.4) d r a (k) (z (k) ℓ ) dz r = 2 z (k) ℓ p−r r−1 i=0 (p − i), ℓ = 1, . . . , n, r = 0, . . . , τ ℓ − 1, where an empty product is understood to be equal to 1, and p ∈ R is the shift parameter identifying the parametrization in (2.2). Symmetric subdivision symbols reproducing exponential polynomials In this section we analyze in detail the case of EP Γ -reproducing subdivision schemes featured by symmetric symbols, since they are considered of remarkable interest in several applications. To this purpose, we first introduce the definition of k-level symmetric symbol, then we point out the symmetric structure required on the set Γ identifying the space of exponential polynomials EP Γ reproduced by a symmetric subdivision scheme. Definition 3 (Symmetric k-level symbol). A k-level subdivision symbol a (k) (z) is called odd-symmetric if a (k) (z) = a (k) (z −1 ) and even-symmetric if z a (k) (z) = a (k) (z −1 ). In terms of k-level masks the odd/even symmetry translates into the condition a i−1 , i ∈ Z, respectively. Remark 1. It is worth mentioning that a subdivision scheme has to be considered symmetric even if its k-level symbol satisfies the above condition after a suitable shift, i.e. after multiplication by z s , s ∈ Z. Note that, as shown in [13], the shift s does affect the value of the parameter p in a well-known way: the parameter p s , characterizing the parametrization of the shifted scheme, is simply p s = p + s. A symmetric set Γ is characterized as in the following definition. Definition 4 (Symmetric set Γ). Let Γ = {γ 1 , ..., γ N } be the set of cardinality N in Definition 1, containing all θ values counted with their multiplicities. The set Γ is said to be symmetric if (3.1) Γ :=    {(θ ℓ , τ ℓ ), (−θ ℓ , τ ℓ )} ℓ=1,..., N 2 , when N is even, {(θ ℓ , τ ℓ ), (−θ ℓ , τ ℓ )} ℓ=1,..., N −1 2 ∪ {0}, when N is odd with θ ℓ ∈ R + ∪ i[0, π), θ ℓ = θ j if ℓ = j, and R + := {x ∈ R : x > 0}. The space of exponential polynomials EP Γ associated to a symmetric set Γ is also said to be symmetric. In the remainder of the paper we focus our attention on EP Γ -reproducing symmetric subdivision schemes. We thus always assume the set Γ to be featured by the symmetric structure specified in Definition 4. The next proposition proves two very important properties of EP Γ -reproducing symmetric subdivision schemes. Proposition 3.1. A non-singular, non-stationary subdivision scheme associated with odd-symmetric or even-symmetric symbols {a (k) (z), k ≥ 0} reproduces the pair of exponential polynomials {e θ ℓ x , e −θ ℓ x }, θ ℓ ∈ R + ∪ i(0, π), only if p = 0 or p = − 1 2 , respectively. Moreover, in case θ ℓ = 0, the subdivision scheme reproduces {1, x} only if p = 0 or p = − 1 2 , respectively. Proof. Let z (k) ℓ := e −θ ℓ 2 k+1 , θ ℓ ∈ R + ∪ i(0, π). We know from conditions (2.4) that the reproduction of the pair {e θ ℓ x , e −θ ℓ x } is equivalent to the existence of a shift parameter p such that a (k) (z (k) ℓ ) = 2(z (k) ℓ ) p and a (k) ((z (k) ℓ ) −1 ) = 2(z (k) ℓ ) −p . Thus, if the k-level symbols are odd-symmetric, we can write 2(z (k) ℓ ) p = 2(z (k) ℓ ) −p , and the latter equation is satisfied only if the shift parameter p = 0 is chosen. Otherwise, if the k-level symbols are even-symmetric, we can write 2(z (k) ℓ ) p+1 = 2(z (k) ℓ ) −p , and the latter equation is fulfilled only if the shift parameter p = − 1 2 is fixed. To conclude the proof we observe that, when θ ℓ = 0, the reproduction of the pair {1, x} is obtained by setting p = 0 if the k-level symbol is odd-symmetric and p = − 1 2 if it is even-symmetric, as shown in [11]. We continue by analyzing useful algebraic properties fulfilled by symmetric subdivision symbols. Proposition 3.2. Let Γ = {(θ ℓ , d), (−θ ℓ , d)} with θ ℓ ∈ R + ∪ i[0, π) be a set of cardinality 2d, d ∈ N. For z (k) ℓ = e −θ ℓ 2 k+1 the even-symmetric subdivision symbols {a (k) (z), k ≥ 0} satisfy d r a (k) (z (k) ℓ ) dz r = 2 z (k) ℓ − 1 2 −r r−1 i=0 − 1 2 − i , r = 0, . . . , d − 1, if and only if the associated odd-symmetric subdivision symbols {b (k) (z), k ≥ 0} with b (k) (z) = z a (k) (z 2 ) − 2 satisfy d r b (k) ((z (k) ℓ ) 1 2 ) dz r = 0, r = 0, . . . , d − 1. In the above equation, an empty product is understood to be equal to 1. Proof. We start showing that the k-level symbol a (k) (z) is even-symmetric if and only if b (k) (z) = z a (k) (z 2 ) − 2 is odd-symmetric. Indeed z a (k) (z) = a (k) (z −1 ) if and only if z 2 a (k) (z 2 ) = a (k) (z −2 ) if and only if b (k) (z) + 2 = b (k) (z −1 ) + 2 if and only if b (k) (z) = b (k) (z −1 ). The rest of the proof is inductive on r. The case r = 0 is easy to check. Therefore we consider the case r > 0 and use the Leibniz formula and the induction for r = 0 to write the derivatives of a (k) (z) = z − 1 2 (b (k) (z 1 2 )+ 2) evaluated at z (k) ℓ . Recall that z − 1 2 = e − 1 2 log(z) can be defined as a single-valued function, analytic on C \ (−∞, 0]. Thus we have d r a (k) (z) dz r z=z (k) ℓ = r s=0 r s d s (b (k) (z 1 2 )+2) dz s d r−s (z − 1 2 ) dz r−s z=z (k) ℓ = d r (z − 1 2 ) dz r (b (k) (z 1 2 ) + 2) z=z (k) ℓ + r s=1 r s d s (b (k) (z 1 2 )+2) dz s d r−s (z − 1 2 ) dz r−s z=z (k) ℓ = 2 r−1 i=0 − 1 2 − i (z (k) ℓ ) − 1 2 −r + r s=1 r s d s (b (k) (z 1 2 )) dz s d r−s (z − 1 2 ) dz r−s z=z (k) ℓ . We continue by using the Faà di Bruno's formula (see [25] or [26]) to write d s (b (k) (z 1 2 )) dz s z=z (k) ℓ = s j=1 d j b (k) (y) dy j y=(z (k) ℓ ) 1 2 A j,s (z) z=z (k) ℓ , with A j,s (z) given functions whose value is important to know only for j = s. In particular, we have A s,s (z) = 1 2 z − 1 2 s . In fact, for j = 1, · · · , s, s < r, using the induction assumption we know that d j b (k) (y) dy j y=(z (k) ℓ ) 1 2 = 0 and therefore the above sum reduces to the last term only, that is to d s b (k) (y) dy s y=(z (k) ℓ ) 1 2 1 2 (z (k) ℓ ) − 1 2 s . Hence, using the fact that d r a (k) (z) dz r z=z (k) ℓ = 2 z (k) ℓ − 1 2 −r r−1 i=0 − 1 2 − i , we arrive at 2 z (k) ℓ − 1 2 −r r−1 i=0 − 1 2 − i = 2 r−1 i=0 − 1 2 − i − 1 2 (z (k) ℓ ) − 1 2 −r + r s=1 r s d s b (k) (y) dy s y=(z (k) ℓ ) 1 2 1 2 (z (k) ℓ ) − 1 2 s r−s−1 i=0 − 1 2 − i − 1 2 (z (k) ℓ ) − 1 2 −(r−s) . Now, using again the inductive hypothesis that d s b (k) (y) dy s y=(z (k) ℓ ) 1 2 = 0 for all s = 1, ..., r − 1 we obtain 0 = d r b (k) (y) dy r y=(z (k) ℓ ) 1 2 1 2 (z (k) ℓ ) − 1 2 r (z (k) ℓ ) − 1 2 , which is the required value of the r-th derivative of b (k) (z) at (z (k) ℓ ) 1 2 , i.e. d r b (k) (y) dy r y=(z (k) ℓ ) 1 2 = 0 . This concludes the induction step and therefore the proof. Remark 2. As a by-product of the proof of Proposition 3. 2 from b (k) (z) = b (k) (z −1 ) we obtain that, for z (k) ℓ = 1, the condition d r b (k) ((z (k) ℓ ) 1 2 ) dz r = 0 for r = 0, . . . , 2j, with 0 ≤ 2j ≤ τ ℓ − 1, implies that d 2j+1 b (k) ((z (k) ℓ ) 1 2 ) dz 2j+1 = 0 and, therefore, the corresponding condition on d 2j+1 a (k) (z (k) ℓ ) dz 2j+1 . The next proposition shows that conditions (2.4) are compatible with symmetry properties of subdivision symbols. Indeed we prove that symmetric subdivision symbols are such that, if conditions (2.4) are satisfied at a given z (k) ℓ , they are also satisfied at (z (k) ℓ ) −1 . Proposition 3.3. Let {a (k) (z), k ≥ 0} be the odd-symmetric (even-symmetric) symbols of a non-singular, non-stationary subdivision scheme associated with the shift parameter p = 0 (p = − 1 2 ). For z (k) ℓ := e −θ ℓ 2 k+1 with θ ℓ ∈ R + ∪ i[0, π) we have that d r a (k) (z (k) ℓ ) dz r = 2 z (k) ℓ p−r r−1 i=0 (p − i), r = 0, . . . , d − 1, d ∈ N if and only if d r a (k) ((z (k) ℓ ) −1 ) dz r = 2 (z (k) ℓ ) −1 p−r r−1 i=0 (p − i) r = 0, . . . , d − 1, d ∈ N , where an empty product is understood to be equal to 1. Proof. We show the claim by induction on r. The case r = 0 has been already considered in Proposition 3.1. For r > 0 we first consider the odd-symmetric case and we start proving one of the two implications. Computing the r-th derivative of the equation a (k) (z) = a (k) (z −1 ) via the Faà di Bruno's formula (see [25] or [26]) and evaluating it at z (k) ℓ , we obtain d r a (k) (z) dz r z=z (k) ℓ = r j=1 d j a (k) (y) dy j y=(z (k) ℓ ) −1 A j,r (z (k) ℓ ) with A j,r (z) = q∈M j , \|q\|=r r! q! (−1) r z −r−j r i=1 N (q, i)! , where M j = {q = (q 1 , q 2 , ..., q j ) ∈ N j , q 1 ≥ q 2 ≥ ... ≥ q j ≥ 1}, \|q\| = q 1 + ... + q j , and N (q, i) denoting the number of times the positive integer i appears in the j- tuple q ∈ N j . Now, by the inductive hypothesis we know that d r a (k) (z) dz r z=z (k) ℓ = 0 for r = 1, ..., d − 2 implies d r a (k) (z) dz r z=(z (k) ℓ ) −1 = 0 for r = 1, ..., d − 2. Hence, d d−1 a (k) (z) dz d−1 z=z (k) ℓ = d d−1 a (k) (y) dy d−1 y=(z (k) ℓ ) −1 A d−1,d−1 (z (k) ℓ ), and since A d−1,d−1 (z (k) ℓ ) = (−1) d−1 (z (k) ℓ ) −2(d−1) = 0, we easily get that d d−1 a (k) (z) dz d−1 z=z (k) ℓ = 0 ⇒ d d−1 a (k) (y) dy d−1 y=(z (k) ℓ ) −1 = 0 , which concludes one direction of the proof in the odd-symmetric case. The proof of the converse implication can be repeated analogously. For the even-symmetric case, in view of Proposition 3.2, we use the same argument as above for the odd-symmetric k-level symbol b (k) (z) = z a (k) (z 2 ) − 2 and for the roots (z (k) ℓ ) 1 2 , (z (k) ℓ ) − 1 2 , so completing the proof. Remark 3. The above proposition proves the equivalence between the conditions for exponential polynomial reproduction given in [13] and [23,24] when p = 0 or p = − 1 2 . Symmetric exponential B-splines and their normalization factors For a symmetric set Γ as in Definition 4, we introduce the following notation where, for a given L ∈ R, ⌊L⌋ := max{M ∈ Z : M ≤ L}. For ℓ = 1, · · · , ⌊ N 2 ⌋, in case N is even we denote by Γ ℓ,e := Γ \ {θ ℓ , −θ ℓ }, and in case N is odd we define Γ ℓ,o := Γ \ {θ ℓ , −θ ℓ , 0}. In the following, for a given L ∈ R, we also use the notation ⌈L⌉ := min{M ∈ Z : M ≥ L}. A symmetric (not-normalized) exponential B-spline is based on the sequence of symbols (4.1)B (k) N,Γ (z) := z −⌈ N 2 ⌉ N i=1 e γ i 2 k+1 z + 1 , γ i ∈ Γ, k ≥ 0 . By definition of Γ ℓ,e we easily see thatB (k) N,Γ (z) satisfies a "recursion" formula since (4.2) B (k) N,Γ (z) = z −1 e θ ℓ 2 k+1 z + 1 e −θ ℓ 2 k+1 z + 1 B (k) N −2,Γ ℓ,e (z) , ℓ = 1, . . . , N 2 . For later use we also observe that for any θ i = \|θ ℓ \| , if N is odd. Moreover, the symbols in (4.1) satisfy the necessary and sufficient conditions for EP Γ -generation (4.4)                                     B (k) N,Γ −e ±θ j 2 k+1 = 0, d rB (k) N,Γ −e ±θ j 2 k+1 dz r = 0, r = 1, . . . , τ j − 1, j = 1, . . . , N 2 , if N is even; B (k) N,Γ (−1) = 0,B (k) N,Γ −e ±θ j 2 k+1 = 0, d rB (k) N,Γ −e ±θ j 2 k+1 dz r = 0, r = 1, . . . , τ j − 1, j = 1, . . . , N −1 2 , if N is odd. For reproduction purposes it may be convenient to consider normalized exponential B-splines. Their symbols are defined by multiplyingB (k) N,Γ (z) in (4.1) with an extra factor K (k) ℓ ∈ R, namely by (4.5) B (k) N,Γ (z) := K (k) ℓB (k) N,Γ (z) = K (k) ℓ z −⌈ N 2 ⌉ N i=1 e γ i 2 k+1 z + 1 , γ i ∈ Γ, k ≥ 0 , where the k-level coefficient K (k) ℓ can be selected in accordance with the parameter p in order to ensure that the normalized exponential B-spline, besides generating EP Γ , reproduces the pair of exponential polynomials {e θ ℓ x , e −θ ℓ x } ∈ EP Γ . This fact is discussed in the next proposition. (ii) p = − 1 2 and (K (k) ℓ ) −1 = e −θ ℓ 2 k+2 + e θ ℓ 2 k+2 e −θ ℓ 2 k+1 + e θ ℓ 2 k+1 B (k) N −3,Γ ℓ,o (e θ ℓ 2 k+1 ), for N odd. Proof. Let us start analyzing the case N even. Introducing the abbreviation z B (k) N,Γ (z (k) ℓ ) = 2(z (k) ℓ ) p , B (k) N,Γ (z (k) ℓ ) −1 = 2(z (k) ℓ ) −p , that is K (k) ℓB (k) N,Γ z (k) ℓ = 2(z (k) ℓ ) p , K (k) ℓB (k) N,Γ (z (k) ℓ ) −1 = 2 z (k) ℓ −p . Exploiting the recurrence relation in (4.2) and recalling thatB (k) N −2,Γ ℓ,e z (k) ℓ = B (k) N −2,Γ ℓ,e (z (k) ℓ ) −1 , we have K (k) ℓ (z (k) ℓ ) −2 + 1 B (k) N −2,Γ ℓ,e (z (k) ℓ ) −1 = (z (k) ℓ ) p−1 , K (k) ℓ (z (k) ℓ ) −2 + 1 B (k) N −2,Γ ℓ,e (z (k) ℓ ) −1 = (z (k) ℓ ) −p−1 . The solution of this system in the unknowns p and K (k) ℓ is therefore given by p = 0 and (K (k) ℓ ) −1 = z (k) ℓ + (z (k) ℓ ) −1 B (k) N −2,Γ ℓ,e (z (k) ℓ ) −1 , which concludes the proof of subcase (i). We continue studying the case N odd. Again, using the recurrence relation (4.2), the two conditions to be satisfied for the reproduction of {e θ ℓ x , e −θ ℓ x } can be written as 2K (k) ℓ z (k) ℓ −2 (z (k) ℓ ) 2 + 1 z (k) ℓ + 1 B (k) N −3,Γ ℓ,o z (k) ℓ = 2(z (k) ℓ ) p , 2K (k) ℓ z (k) ℓ 2 (z (k) ℓ ) −2 + 1 (z (k) ℓ ) −1 + 1 B (k) N −3,Γ ℓ,o (z (k) ℓ ) −1 = 2(z (k) ℓ ) −p . Now, since N −3 is even andB (k) N −3,Γ ℓ,o (z (k) ℓ ) −1 =B (k) N −3,Γ ℓ,o z (k) ℓ , we can write the simplified expressions K (k) ℓ z (k) ℓ + 1 (z (k) ℓ ) −1 + z (k) ℓ B (k) N −3,Γ ℓ,o (z (k) ℓ ) −1 = (z (k) ℓ ) p+1 , K (k) ℓ z (k) ℓ + 1 (z (k) ℓ ) −1 + z (k) ℓ B (k) N −3,Γ ℓ,o (z (k) ℓ ) −1 = (z (k) ℓ ) −p . The solution of this system in the unknowns p and K (k) ℓ is given by p = − 1 2 and (K (k) ℓ ) −1 = (z (k) ℓ ) 1 2 + (z (k) ℓ ) − 1 2 z (k) ℓ + (z (k) ℓ ) −1 B (k) N −3,Γ ℓ,o (z (k) ℓ ) −1 , which concludes the proof of subcase (ii). Note that similar results concerning the normalization of exponential B-splines are also given in [23]. Two special situations are considered in the next result. Corollary 4.2. For N ≥ 2 the exponential B-splines B (k) N,Γ (z) reproduce {1, x} if 0 = θ ℓ ∈ Γ with τ ℓ = 2, (K (k) ℓ ) −1 = 2B (k) N −2,Γ\{θ ℓ ,−θ ℓ } (1), for N even; Deriving the symbols of exponential pseudo-splines and investigating their symmetry properties For any p ∈ R and N, M ∈ N the binary pseudo-spline subdivision scheme is defined to be the stationary scheme with minimal support that generates polynomials of degree N − 1 and whose symbol, a M,N (z), satisfies the conditions [11,17,18]). The main contribution of this paper consists in generalizing the family of binary pseudo-splines to the non-stationary setting. The resulting family is called the family of binary exponential pseudo-splines. For Γ as in (3.1) and p = 0 if N is even, p = − 1 2 if N is odd, the family of symmetric binary exponential pseudo-splines is defined to be the family of symmetric subdivision schemes with minimal support that generates the space of exponential polynomials EP Γ and whose k-level symbol satisfies the conditions in Using the Leibniz rule we can write the set of conditions in (5.1) in the equivalent form (5.2) s i=0 s i d i c (k) M,Γ (z (k) ℓ ) dz i d s−i B (k) N,Γ (z (k) ℓ ) dz s−i = v ℓ,s , ℓ = 1, . . . , m, m ≤ n s = 0, . . . , τ ℓ − 1, where v ℓ,s := 2 z (p − i), and an empty product is understood to be equal to 1. We start by considering the case m = n (which means M = N ). In the latter case equations (5.2) can be rewritten as the linear system (5.3) Aw = v, A ∈ R N ×N , w ∈ R N , v ∈ R N , N = n j=1 τ j , where w := d i c (k) N,Γ (z (k) ℓ ) dz i , i = 0, · · · , τ ℓ − 1, ℓ = 1, · · · , n T , v is defined as v := (v 1,0 , . . . , v 1,τ1−1 , v 2,0 , . . . , v 2,τ2−1 , . . .) T , and A is the block diagonal lower triangular matrix given by A :=    A 1 . . . A n    , A j ∈ C τj ×τj , where, for 1 ≤ j ≤ n, A j :=         B (k) N,Γ (z (k) j ) 1 0 d 1 B (k) N,Γ (z (k) j ) dz 1 B (k) N,Γ (z (k) j ) . . . . . . τj −1 0 d τ j −1 B (k) N,Γ (z (k) j ) dz τ j −1 τj−1 1 d τ j −2 B (k) N,Γ (z (k) j ) dz τ j −2 . . . B (k) N,Γ (z (k) j )         . The structure of A −1 j follows from the next result. Lemma 5.1. For a given z (k) j such that B (k) N,Γ (z (k) j ) = 0 the matrix A j is invertible and A −1 j = G j :=         G (k) N,Γ (z (k) j ) 1 0 d 1 G (k) N,Γ (z (k) j ) dz 1 G (k) N,Γ (z (k) j ) . . . . . . τj −1 0 d τ j −1 G (k) N,Γ (z (k) j ) dz τ j −1 τj −1 1 d τ j −2 G (k) N,Γ (z (k) j ) dz τ j −2 . . . G (k) N,Γ (z (k) j )         with Gd i G (k) N,Γ (z (k) j ) dz i d s−i B (k) N,Γ (z (k) j ) dz s−i = 0, which means that the subdiagonal entries in the first column of S j are zero. For the remaining entries observe that (S j ) ℓ,r = ℓ i=r ℓ − 1 i − 1 i − 1 r − 1 d ℓ−i B (k) N,Γ (z (k) j ) dz ℓ−i d i−r G (k) N,Γ (z (k) j ) dz i−r , and, hence, by settingĩ := i − r andl := ℓ − r (S j ) ℓ,r = ℓ − 1 r − 1 l ĩ =0 l i dl −ĩ B (k) N,Γ (z (k) j ) dzl −ĩ dĩ G (k) N,Γ (z (k) j ) dzĩ = δ ℓ,r , where δ ℓ,r denotes the Kronecker symbol. Assuming that Γ is defined as in (3.1), we have that z (k) j = −z (k) ℓ , 1 ≤ j, ℓ ≤ N and thus Lemma 5.1 yields the following. 2 k+1 , ℓ = 1, · · · , n, we have (5.4) d s c (k) N,Γ (z (k) ℓ ) dz s = s i=0 s i v ℓ,i d s−i G (k) N,Γ (z (k) ℓ ) dz s−i , ℓ = 1, . . . , n, s = 0, . . . , τ ℓ − 1, where v ℓ,i := 2 z (k) ℓ p−i i−1 r=0 (p − r) and G (k) N,Γ (z) := 1 B (k) N,Γ (z) . It is worth pointing out that, once we have specified the support of c γ η = {z ∈ C : \| n ℓ=1 (z − z (k) ℓ ) τ ℓ \| = η N }. It is worth noticing that n ℓ=1 (z − z (k) ℓ ) τ ℓ = ξ k (−z) ⌈ N 2 ⌉ B (k) N,Γ (−z), for a suitable ξ k ∈ C. Let us introduce the infinite sequence {β i } i∈N obtained by cyclically repeating each β i , i.e., β i = β mod(i,N ) , i ≥ 1. For m 1 ≤ m 2 let f [{β i } m2 i=m1 ] be the divided difference of the meromorphic function f (z) := z ℓ a (k) N,N,Γ (z)/B (k) N,Γ (z) on the set of pointsβ i , m 1 ≤ i ≤ m 2 , where ℓ ∈ Z is a given fixed integer. Then the relation f (z) = +∞ i=1 f [{β s } i s=1 ] i−1 j=1 (z −β j ) holds in the following sense: the partial sums of the Newton series converge uniformly to f (z) in any closed set lying in the interior of γ η for any η > 0 such that f is analytic in γ η . Since +∞ i=1 f [{β s } i s=1 ] i−1 j=1 (z −β j ) = N i=1 f [{β s } i s=1 ] i−1 j=1 (z −β j ) + B (k) N,Γ (−z)R k (z), one deduces that a (k) N,N,Γ (z) = z −ℓ B (k) N,Γ (z) N i=1 f [{β s } i s=1 ] i−1 j=1 (z −β j ) + z −ℓ B (k) N,Γ (z)B (k) N,Γ (−z)R k (z), and, therefore, in view of (4.4), we can set c (k) N,Γ (z) = z −ℓ N i=1 f [{β s } i s=1 ] i−1 j=1 (z −β j ), which is the shifted Newton form of the Hermite interpolant of . By using the symmetry of both the symbol and the distribution of nodes and from the uniqueness of the Laurent polynomial we may conclude that this latter polynomial is symmetric. The case where N is even is a bit more involved. In principle, using the same arguments as above we find that the interpolating Laurent polynomial is supported in [− N 2 , N 2 − 1] or [− N 2 + 1, N 2 ]. However, by expressing the polynomial in the new variable t := z + z −1 we find that there exists a uniquely determined symmetric interpolating Laurent polynomial supported in [− N 2 + 1, N 2 − 1]. The precise statement is given below. ℓ ) −1 and viceversa. Hence, for N even we have to impose N/2 independent conditions, while for N odd we have τ ℓ = 2j + 1 conditions at z (k) ℓ = 1 plus (N − 1)/2 − j independent conditions. Since from Remark 2 the conditions at z (k) ℓ = 1 yield j + 1 independent conditions, we obtain (N + 1)/2 = ⌈ N 2 ⌉ conditions also for the odd case. Defining t := z + z −1 let us introduce the functions p j (t) = z j + z −j , j ≥ 0. Such functions are monic Chebyshev-like polynomials of degree j which, starting from p 0 (t) = 2, p 1 (t) = t, satisfy the three-term recurrence relation p j+1 (t) = tp j (t) − p j−1 (t), j ≥ 1. Hence, by writing c (k) N,Γ (z) = c 0 2 p 0 (t) + ⌈ N 2 ⌉−1 j=1 c j p j (t) = ψ(t), the proof follows from the existence and uniqueness of the interpolating polynomial ψ(t) on the considered set of nodes. The results given so far immediately generalize to the case where we consider a subsetΓ of Γ. . For the effective construction of the polynomial c (k) M,Γ (z) we can proceed as follows. By setting (5.5) c (k) M,Γ (z) = ψ(z + z −1 ), using the Faà di Bruno's Formula we get that, for s = 1, . . . , τ ℓ − 1, (5.6) d s c (k) M,Γ (z (k) ℓ ) dz s = s r=1 d r ψ z (k) ℓ + (z (k) ℓ ) −1 dz r A r,s (z (k) ℓ ). Since A s,s (z (k) ℓ ) =   1 − 1 (z (k) ℓ ) 2   s , we obtain that for z (k) ℓ = ±1 the triangular system is invertible and, therefore, it enables the computation of the highest derivative of ψ(z + z −1 ) to be performed. In this way ψ(z + z −1 ) and then c Proof. The proof exploits the fact that for p = 0 and N even from a subdivision symbol satisfying (2.3) we are able to construct a subdivision symbol satisfying (2.4), and viceversa. In particular, in the case p = 0 and N even, from conditions (5.4) we find that a (k) N,N,Γ (z)−2 = B (k) N,Γ (−z) q (k) (z) for a certain Laurent polynomial q (k) (z). Hence, a (k) N,N,Γ (z) = 2 + B (k) N,Γ (−z) q (k) (z) = B (k) N,Γ (z) c (k) N,Γ (z). Since the relation holds for any z we also have 2 + B (k) N,Γ (z) q (k) (−z) = B (k) N,Γ (−z) c Convergence and regularity of exponential pseudo-spline subdivision schemes The aim of this section is two-fold. First, we show that the strategy proposed in the previous section allows us to construct polynomial pseudo-splines in the stationary case. Second, we prove convergence and regularity of exponential pseudospline subdivision schemes. To achieve the first goal we introduce a new set of z-functions, that are meant to be shifted B-spline symbols, of the form B N (z) := (z + 1) N 2 N −1 z − N 2 = (z+z −1 +2) ρ 2 2ρ−1 , if N = 2ρ (z+z −1 +2) ρ+ 1 2 2 2ρ , if N = 2ρ + 1. Obviously, in case N is evenB N (z) = B N (z), whereas for N oddB N (z) = z 1 2 B N (z). We also need the following result whose proof is obtained by induction, following the lines of the proof of Proposition 3.2. Lemma 6.1. The even-symmetric symbol a(z) satisfies d r a(1) dz r = 2 r−1 i=0 − 1 2 − i , r ≥ 0, if and only if the associated odd-symmetric z-function b(z) = z 1 2 a(z) satisfies d r b(1) dz r = 2δ r,0 , r ≥ 0. By means of these preliminary results, we can prove the following proposition. [20,Section 6], that is a M,N (z) =                2 σ ρ (z) ⌊ M −1 2 ⌋ s=0 ρ + s − 1 s δ s (z), if N = 2ρ, z+1 z σ ρ (z) ⌊ M −1 2 ⌋ s=0 ρ − 1 2 + s s δ s (z), if N = 2ρ + 1,G N (z) = 1 B N (z) = 2 2ρ−2p−1 (z + z −1 + 2) −ρ+p , p ∈ 0, − 1 2 . Thus, using the Faà di Bruno's formula (see [25] or [26]) we can write d rḠ N (z) dz r = 2 2ρ−2p−1 r j=1 (−1) j ρ − p + j − 1 j j!(z + z −1 + 2) −ρ+p−j A j,r (z), where A j,r (z) = q∈M j , \|q\|=r r! q! j i=1 δ qi,1 + (−1) qi q i !z −(qi+1) r i=1 N (q, i)! with M j = {q = (q 1 , q 2 , ..., q j ) ∈ N j , q 1 ≥ q 2 ≥ ... ≥ q j ≥ 1}, \|q\| = q 1 + ... + q j and N (q, i) denoting the number of times the integer i ∈ N appears in the j-tuple q ∈ N j . Evaluating at z = 1 we obtain d rḠ N (1) dz r = r j=1 (−1) j ρ − p + j − 1 j j!2 −2j−1 A j,r (1) so that, recalling (5.4), for s = 0, . . . , M − 1 we find d s c M (1) dz s = s i=0 s i v i s−i j=1 (−1) j ρ − p + j − 1 j j!2 −2j−1 A j,s−i (1). Hence, c M (1) = 1 (6.1) d s c M (1) dz s = s j=1 (−1) j ρ − p + j − 1 j j!2 −2j A j,s (1), s = 1, . . . , M − 1. On the other hand, recalling (5.5)-(5.6) we can write c M (z) = ψ(z + z −1 ), d s c M (z) dz s = s j=1 d j ψ(z + z −1 ) dz j A j,s (z), s = 1, . . . , M − 1, so that when evaluating at z = 1 we obtain c M (1) = ψ(2), (6.2) d s c M (1) dz s = s j=1 d j ψ(2) dz j A j,s (1), s = 1, . . . , M − 1. In this way, by comparison of (6.1) with (6.2), from all even s we can find the values attained by all derivatives of ψ at 2, i.e. ψ(2) = 1, d j ψ(2) dz j = (−1) j ρ − p + j − 1 j j!2 −2j , j = 1, . . . , M − 1 2 , and, exploiting the customary Hermite interpolation formula we can thus get the analytic expression of c M (z) c M (z) ≡ ψ(z+z −1 ) = ⌊ M −1 2 ⌋ j=0 (z + z −1 − 2) j j! d j ψ(2) dz j = ⌊ M −1 2 ⌋ j=0 ρ − p + j − 1 j δ j (z), with δ(z) = − (1−z) 2 4z . Making distinction between N even and N odd, and rewriting B N (z) in terms of σ(z) = (1+z) 2 4z , the claim is proven. Now, in order to study the asymptotical behaviour of a (k) i = β (k) mod(i,M) , i ≥ 1, we get f [{β (k) s } i s=1 ] = χ (k) [{β (k) s } i s=1 ] = 1 2πi C χ (k) (z)dz i s=1 (z −β (k) i ) , where C is a simple closed curve in the complex plane enclosing a simply connected region which contains the points {β (k) s } i s=1 . Therefore, lim k→+∞ χ (k) [{β (k) s } i s=1 ] = 1 (i − 1)! ∂ i−1 ∂z i−1 χ (+∞) (z) z=1 , which means that as k goes to infinity c Collecting all previous results we finally arrive at an important asymptotical result. Corollary 6.5. The exponential pseudo-spline subdivision masks are asymptotically similar to the polynomial pseudo-spline subdivision masks, i.e. , Before proceeding, we recall that in [19] and in [16] the authors prove convergence and regularity of the subdivision schemes associated with the symbol a M,N (z) with N even and odd, respectively. In the following we continue analyzing the values attained by {a (k) M,N,Γ (z), k ≥ 0} and its derivatives at z = −1 and, in turn, we study the convergence and regularity of the associated subdivision schemes. Proof. The proof of (6.5) is based on the recent results proven in [14,Theorem 10] and is a direct consequence of the exponential polynomial generation properties of {a An application example This section contains an interesting example of a family of purely non-stationary symmetric exponential pseudo-splines. As far as we know this is the first derivation of purely non-stationary exponential pseudo-spline symbols to appear. In fact, the recently published paper [27] merely discusses the interpolatory subcase of a family of exponential pseudo-splines which reproduces function spaces spanned by an arbitrary number of polynomials and just a pair of exponential polynomials. Let θ ∈ R + ∪ i[0, π) and for all k ≥ 0 define v (k) = 1 2 (e θ 2 k+1 + e − θ 2 k+1 ). We consider the exponential B-spline with k-level symbol B (k) N,Γ (z) = (z + z −1 + 2v (k) ) ρ 2 2ρ−1 (v (k) ) ρ , with ρ ∈ N, Γ = {(θ, ρ), (−θ, ρ)}, and N = ♯Γ = 2ρ, which is obtained from the general formulation with n = 2, z N,Γ (z) = 2 2ρ−1 (v (k) ) ρ (z + z −1 + 2v (k) ) −ρ and, using the Faà di Bruno's formula we compute d r G (k) N,Γ (z) dz r = 2 2ρ−1 (v (k) ) ρ r j=1 (−1) j ρ + j − 1 j j!(z + z −1 + 2v (k) ) −ρ−j A j,r (z) where A j,r (z) is the same as the one appearing in the proof of Proposition 6.2. Hence, for ℓ = 1, 2 G (k) N,Γ (z (k) ℓ ) = 1 2 and d r G (k) N,Γ (z (k) ℓ ) dz r = r j=1 (−1) j ρ + j − 1 j j!2 −2j−1 (v (k) ) −j A j,r (z (k) ℓ ), r = 1, . . . , ρ−1. Therefore, from (5.4) we can write for ℓ = 1, 2 (7.2) d s c (k) N,Γ (z (k) ℓ ) dz s = s i=0 s i v ℓ,i s−i j=1 (−1) j ρ + j − 1 j j!2 −2j−1 (v (k) ) −j A j,s−i (z (k) ℓ ), with s = 0, ..., ρ − 1. Now, taking into account that when p = 0 then v ℓ,i = 2δ i,0 , i = 0, . . . , s, equation (7.2) can be rewritten for ℓ = 1, 2 as c (k) N,Γ (z (k) ℓ ) = 1, (7.3) d s c (k) N,Γ (z (k) ℓ ) dz s = s j=1 (−1) j ρ + j − 1 j j!2 −2j (v (k) ) −j A j,s (z (k) ℓ ), s = 1, . . . , ρ − 1. (7.4) On the other hand, from (5.5)-(5.6) we have that ℓ ) −1 = 2v (k) for ℓ = 1, 2. At this point, comparing (7.3) with (7.5) and (7.4) with (7.6), we respectively obtain ψ(2v (k) ) = 1, d j ψ(2v (k) ) dz j = (−1) j ρ + j − 1 j j!2 −2j (v (k) ) −j , j = 1, ..., ρ − 1. c (k) N,Γ (z) = ψ(z + z −1 ) d s c (k) N,Γ (z) dz s = s j=1 d j ψ(z + z −1 ) dz j A j, Using the customary Hermite interpolation formula we can thus compute (7.7) ψ(z + z −1 ) = ρ−1 j=0 (z + z −1 − 2v (k) ) j j! d j ψ(2v (k) ) dz j = ρ−1 j=0 (z + z −1 − 2v (k) ) j (−1) j ρ + j − 1 j 2 −2j (v (k) ) −j , which provides the expression of the polynomial correction c N,Γ (z) into the interpolatory scheme with the same generation properties (see [2,9,10]). N,Γ (z) = c 2ρ (z), then the derived family of exponential pseudo-splines with klevel symbol a (k) N,N,Γ (z) can be considered a non-stationary extension of the family of interpolatory (2ρ)-point Dubuc-Deslauriers schemes [15]. A general construction for families of interpolatory schemes reproducing exponential polynomials was originally proposed in [22]. However, to the best of our knowledge, an algebraic approach for deriving the subdivision symbols of exponential pseudo-splines (including as a special case all non-stationary variants of Dubuc-Deslauriers schemes) was never investigated before. For instance, note that when ρ = 2, equation (7.7) yields c (k) N,Γ (z) = − 1 2v (k) z + 2 − 1 2v (k) z −1 and the resulting exponential pseudo spline is an interpolatory 4-point scheme with k-level mask (7.8) − 1 16(v (k) ) 3 , 0, 3(4(v (k) ) 2 − 1) 16(v (k) ) 3 , 1, 3(4(v (k) ) 2 − 1) 16(v (k) ) 3 , 0, − 1 16(v (k) ) 3 , while, when ρ = 3, from equation (7.7) we obtain c (k) N,Γ (z) = 3 8(v (k) ) 2 z 2 − 9 4v (k) z + 3+16(v (k) ) 2 4(v (k) ) 2 − 9 4v (k) z −1 + 3 8(v (k) ) 2 z −2 and thus the resulting exponential pseudo spline is an interpolatory 6-point scheme with k-level mask (7.9) 3 256(v (k) ) 5 , 0, − 5(8(v (k) ) 2 − 3) 256(v (k) ) 5 , 0, 15(8(v (k) ) 4 − 4(v (k) ) 2 + 1) 128(v (k) ) 5 , 1, 15(8(v (k) ) 4 − 4(v (k) ) 2 + 1) 128(v (k) ) 5 , 0, − 5(8(v (k) ) 2 − 3) 256(v (k) ) 5 , 0, 3 256(v (k) ) 5 . Figure 2 shows the graph of the basic limit functions for the interpolatory 4-point and 6-point schemes with k-level mask in (7.8) and (7.9), respectively. Such interpolatory 4-and 6-point schemes are new non-stationary variants of the well-known Dubuc-Deslauriers schemes in [15]. They differ from the ones previously proposed in [1,28] for the space of exponential polynomials they reproduce. Figure 2. Basic limit functions for the interpolatory 4-and 6point schemes with k-level mask in (7.8) and (7.9) with θ = i (lower function), θ = 3 2 (middle function) and θ = 2 (upper function). Conclusions In this work we have proposed an algebraic strategy to derive the subdivision symbols of exponential pseudo-splines from the subdivision symbols of exponential B-splines. The presented strategy is featured by the following key properties: • it allows the user to pass from subdivision schemes generating a space of exponential polynomials to subdivision schemes reproducing the same space, or any desired of its subspaces; • it provides the subdivision symbols of minimal support that fulfill the set of conditions ensuring reproduction of the desired space of exponential polynomials; • it preserves the symmetry properties of the given exponential B-spline symbols; • it contains the stationary case of polynomial pseudo-splines as a special subcase. Moreover, we have proved convergence and regularity of the non-stationary subdivision schemes obtained from the repeated application of exponential pseudo-spline symbols exploiting the property of asymptotical similarity to the stationary symbols of the well-known polynomial pseudo-splines. N,Γ (z) are of minimal support and satisfy the conditions for reproduction of the space of exponential polynomials EP Γ generated by the exponential B-splines with symbols B (k) N,Γ (z), or a subset of it. Proposition 4. 1 . 1Let Γ be given and θ ℓ ∈ Γ. The symbols in (4.5) satisfy the {e θ ℓ x , e −θ ℓ x }-reproduction condition if (i) p = 0 and (K ), for N even, , in view of Theorem 2.1 the reproduction of {e θ ℓ x , e −θ ℓ x } requires the fulfillment of the conditions N − 3 , 3Γ\{θ ℓ ,−θ ℓ ,0} (1), for N odd; and p = 0, for N even; − 1 2 , for N odd. Moreover, when Γ = {(θ 1 , τ 1 ) = (0, N )}, the symbol of B Γ (z) does not depend on k any longer and becomes the stationary symbol of the shifted order-N (polynomial) B-spline that we simply denote by B N (z) = z −⌈ N 2 ⌉ (1+z) N 2 N −1 . p − i), r = 0, . . . , M − 1 for reproduction of polynomials up to degree M −1. Its actual degree of polynomial reproduction is thus min{N −1, M −1} (see of elements in EPΓ, whereΓ denotes a symmetric subset of Γ of cardinality M ≤ N , with M and N of the same parity. The k-level symbol of an exponential pseudo-spline is therefore of the form Γ (z) is the k-level symbol of the normalized exponential B-spline in (Γ (z) is the k-level Laurent polynomial of lowest possible degree such that a 1, . . . , m, m ≤ n; s = 1, . . . , τ ℓ and (θ ℓ , τ ℓ ) ∈Γ ⊂ Γ for ℓ = 1, ..., m, with M := m j=1 τ j . Obviously, (2.3) are satisfied by construction. Proof. Let S j = A j G j be the product matrix. The claim follows from the relation B conditions (5.4) enable the computation of its coefficients by means of an interpolation process. In particular, the construction of c(k) N,Γ (z)can rely upon the following functional approach. Let {β i } N i=1 , N = i τ i , denote a finite sequence of nodes generated from the distinct points z (k) ℓ , 1 ≤ ℓ ≤ n, each of them repeated τ ℓ times. Moreover, for a given η > 0 let γ η be the lemniscata defined by ℓ , ℓ = 1, · · · , n. In this way for any value of ℓ we may determine a Laurent polynomial satisfying(5.4) whose support lies in [−ℓ, −ℓ + N − 1]. If N is odd, then by choosing ℓ = N −1 2 we obtain the unique Laurent polynomial supported in [ 1−N 2 , N −1 2 ] Γ (z) is the unique symmetric Laurent polynomial supported in [−⌈ N 2 ⌉+1, ⌈ N 2 ⌉−1] which satisfies the conditions (5.4) with p, N such that p = 0 if N even and p = − 1 2 if N odd. Proof. We are looking for a polynomial of the form Theorem 5. 4 . 4Let Γ andΓ ⊂ Γ be symmetric sets of cardinality N and M , respectively, with M and N of the same parity. Moreover let EP Γ , EPΓ be the corresponding sets of exponential polynomials, and assume p = 0 in case N and M are both even, p = − 1 2 in case N and M are both odd. Then there exists a unique symmetric Laurent polynomial c , (θ ℓ , τ ℓ ) ∈Γ, ℓ = 1, ..., m, the generalized interpolation conditions d s c dz s−i , ℓ = 1, . . . , m, m ≤ n, s = 0, . . . , τ ℓ − 1, . Γ (z) of even order give information about the derivatives of ψ(z + z −1 ). The case z In the case p = 0 and M = N even, the subdivision symbol of the symmetric exponential pseudo-spline a (k) N,N,Γ (z) derived from Theorem 5.4 is always interpolatory. N,Γ (z) + a (k) N,N,Γ (−z) = 2. Proposition 6. 2 . 2For the order-N B-spline symbol B N (z) = 2 −(N −1) z −⌈ N 2 ⌉ (z + 1) N , let c M (z) be the polynomial correction such that − i), s = 0, . . . , M − 1, with p = 0 if N = 2ρ and p = − 1 2 if N = 2ρ + 1. Then, a M,N (z) = B N (z) c M (z) is the subdivision symbol of the polynomial pseudo-spline given in We start by observing that, since c M (c M (1) dz i d s−iB N (1) dz s−i = 2v s , s = 0, . . . , M − 1 withB N (z) = 2 −(N −1) z − N 2 (z + 1) N and v s = 2δ s,0 . We continue by takinḡ ρℓ N,Γ (z) when k approaches infinity, we introduce the following definition.Definition 6.3. The sequence of subdivision masks {a (k) , k ≥ 0} and {a} are called asymptotically similar if lim k→+∞ a (k) − a ∞ equivalently in terms of symbols, if lim k→+∞ a (k) (z) = a(z).The following proposition proves the asymptotical similarity between the polynomial corrections c(k) M,Γ (z) and c M (z). Proposition 6.4. As k → +∞ the exponential B-spline B (k) N,Γ (z) converges to the polynomial B-spline B N (z), and the polynomial correction c + s − 1 s δ s (z), if p = 0 ( i.e. N = 2ρ), s (z), if p = − 1 2 ( i.e. N = 2ρ + 1), with δ(z) := − (1−z) 2 4z . Proof. We start by the simple observation that lim k→+∞ B (k) N,Γ (z) = B N (z). Next, we first consider the case p = 0 and N even. From equation (5.1) we have that the Hermite interpolant of f (z) , ℓ = 1, · · · , m coincides with the Hermite interpolant of χ (k) ℓ , a finite sequence of nodes generated from the distinct points z (k) ℓ , 1 ≤ ℓ ≤ m, each of them repeated τ ℓ times, and bỹ β ⌉ (z+1) N . The remaining case p = −1/2 and N odd reduces to the previous analysis by observing that c N,Γ (z) = a M,N (z)with a M,N (z) denoting the well-known polynomial pseudo-spline symbol. Proposition 6 . 6 . 66Let M > 1. The subdivision symbols {a N,Γ (−1) = O(2 −k(N −s) ), s = 0, . . . , N − 1, k → +∞. Moreover, the non-stationary subdivision scheme with symbols {a (k) M,N,Γ (z), k ≥ 0} converges and has the same regularity as the stationary one with symbol a M,N (z). N,Γ (z), k ≥ 0} and the asymptotical similarity of {a(k) M,N,Γ , k ≥ 0} to {a M,N },previously shown in Proposition 6.4. Then, for the convergence and regularity result we can rely on[5]. and τ 1 = τ 2 = ρ. The subdivision scheme with symbol B θx , e −θx , xe θx , xe −θx , · · · , x ρ−1 e θx , x ρ−1 e −θx } and reproduces span{e θx , e −θx } with respect to the parameter shift p = 0. In order to apply Theorem 5.4 with m = n, namely with M = N (the only possibility we have here), we define G (k) s (z), s = 1, ..., ρ − 1, Γ (z) which reproduces the space of exponential polynomials (7.1) with respect to the parameter shift p = 0. We can thus refer to c(k) N,Γ (z) as to the polynomial correction transforming the approximating scheme having symbol B (k) Remark 4 . 4Since, as previously observed, lim k→+∞ B (k) N,Γ = B 2ρ (z) and lim k→+∞ c (k) A non-stationary uniform tension controlled interpolating 4-point scheme reproducing conics. C Beccari, G Casciola, L Romani, Comput. Aided Geom. Design. 24Beccari, C., Casciola, G., Romani, L.: A non-stationary uniform tension controlled interpolat- ing 4-point scheme reproducing conics. Comput. Aided Geom. 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F Cirak, M J Scott, E K Antonsson, M Ortiz, P Schröder, Computer-Aided Design. 34Cirak, F., Scott, M.J., Antonsson, E.K., Ortiz, M., Schröder, P.: Integrated modeling, finite- element analysis, and engineering design for thin-shell structures using subdivision. Computer- Aided Design 34, 137-148 (2002) From symmetric subdivision masks of Hurwitz type to interpolatory subdivision masks. C Conti, L Gemignani, L Romani, Linear Algebra Appl. 431Conti, C., Gemignani, L., Romani, L.: From symmetric subdivision masks of Hurwitz type to interpolatory subdivision masks. Linear Algebra Appl. 431, 1971-1987 (2009) From approximating to interpolatory non-stationary subdivision schemes with the same generation properties. C Conti, L Gemignani, L Romani, Adv. Comput. Math. 352-4Conti, C., Gemignani, L., Romani, L.: From approximating to interpolatory non-stationary subdivision schemes with the same generation properties. Adv. Comput. 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Math. 681Mortini, R.: The Faà di Bruno's formula revisited. Elem. Math. 68(1), 33-38 (2013) Building blocks for designing arbitrarily smooth subdivision schemes with conic precision. P Novara, L Romani, J. Comput. Appl. Math. accepted for publicationNovara, P., Romani, L.: Building blocks for designing arbitrarily smooth subdivision schemes with conic precision. J. Comput. Appl. Math., accepted for publication. From approximating subdivision schemes for exponential splines to highperformance interpolating algorithms. L Romani, J. Comput. Appl. Math. 2241Romani, L.: From approximating subdivision schemes for exponential splines to high- performance interpolating algorithms. J. Comput. Appl. Math. 224(1), 383-396 (2009) Subdivision methods for geometric design -A constructive approach. J Warren, H Weimer, Morgan-KaufmannWarren, J., Weimer, H.: Subdivision methods for geometric design -A constructive approach, Morgan-Kaufmann (2002) . Dipartimento Di Ingegneria Industriale, Viale Morgagni. 4050134Università di FirenzeDipartimento di Ingegneria Industriale, Università di Firenze, Viale Morgagni 40/44, 50134 . Firenze, Italy E-mail address: [email protected], Italy E-mail address: [email protected] . Dipartimento Di Informatica, 356127Largo Bruno PontecorvoUniversità di PisaDipartimento di Informatica, Università di Pisa, Largo Bruno Pontecorvo, 3 56127 Italy E-mail address: [email protected]. Pisa, Italy E-mail address: [email protected] . Dipartimento Di Matematica E Applicazioni, Via R. Cozzi. 55Università di Milano-BicoccaDipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via R. Cozzi 55, 20125 Milano, Italy E-mail address: [email protected]	[]
[ "Entropy of non-local gravity", "Entropy of non-local gravity" ]	[ "Ali Teimouri \nConsortium for Fundamental Physics\nLancaster University\nLA1 4YBLancasterUK\n" ]	[ "Consortium for Fundamental Physics\nLancaster University\nLA1 4YBLancasterUK" ]	[]	For higher derivative theories of gravity, it is possible to write the action in terms of auxiliary fields. In such case, one can show that the equations of motion for both actions are equivalent and hence the actions themselves. In this paper we show that one can obtain the Wald's entropy from the equivalent action. We use this useful approach to localise a non-local gravitational action and calculate its associated entropy.	null	[ "https://arxiv.org/pdf/1705.11164v1.pdf" ]	119,431,279	1705.11164	d47d662bef24af56e662bc8f9e436020148a7220	Entropy of non-local gravity May 2017 Ali Teimouri Consortium for Fundamental Physics Lancaster University LA1 4YBLancasterUK Entropy of non-local gravity May 2017 For higher derivative theories of gravity, it is possible to write the action in terms of auxiliary fields. In such case, one can show that the equations of motion for both actions are equivalent and hence the actions themselves. In this paper we show that one can obtain the Wald's entropy from the equivalent action. We use this useful approach to localise a non-local gravitational action and calculate its associated entropy. I. INTRODUCTION Einstein's theory of general relativity can be modified in number of ways to address different aspects of cosmology. One approach to account for modified gravity in the infrared (IR) regime is the non-local gravity. Non-local gravity is constructed by inversed d'Alembertian operators that are accountable in the IR regime. Such modification to the theory of general relativity arises naturally as quantum loop effect [2] and used initially by [1] to explain the cosmic acceleration. Non-local gravity further used to explain dark energy [3]. Since such gravity is associated with large distances, it is also possible to use it as an alternative to understand the cosmological constant [4]. Additionally, non-local corrections arise, in the leading order, in the context of bosonic string [5]. It is argued in [6] that, non-locality may have a positive rule in understanding the black hole information problem. Recently, the non-local effect was studied in the context of Schwarzschild black hole [7,8]. In similar manner, the entropy of some non-local models were studied in [9]. In the light of these developments, in this paper, we are going to obtain the entropy for a non-local gravitational action. In this theory the non-local operators (i.e. inversed d'Alembertian operators) can be of any order, namely finite or infinite. The entropy is being obtained by using the Wald's description and getting use of an equivalent action, where we introduced auxiliary fields to localise the theory. II. HIGHER DERIVATIVE AND NON-LOCAL GRAVITY In this section we shall introduce two types of higher derivative actions and show how they can be written equivalently in terms of auxiliary fields. * [email protected], [email protected] A. Higher Derivative Gravity General relativity is a diffeomorphism invariant theory, thus one can add covariant higher derivatives to the action. Such modifications to the gravity can be found in [10]. The higher derivative action can be formulated as: S 0 + S 1 = 1 16πG d 4 x √ −g R + RF (¯ )R , with: F (¯ ) = m n=0 f n¯ n ,(1) where G is the Newton's gravitational constant, R is scalar curvature, = ∇ µ ∇ µ is the d'Alembertian operator and¯ ≡ /M 2 , this is due to the fact that has dimension mass squared and we wish to have dimensionless F (¯ ), we shall note that f n 's are dimensionless coefficients of the series expansion. In the above action we denoted the Einstein Hilbert (EH) term as S 0 . Finally, m is some finite positive integer which also can be taken to infinity as in [10]. The above action can be written as [19], S 0 +S 1 = 1 16πG d 4 x √ −g R + m n=0 Rf n η n + Rχ n (η n −¯ n R) ,(2) where we introduced two auxiliary fields χ n and η n . By solving the equations of motion for χ n , we obtain: η n = n R, and hence the original action given in Eq. (1 ) can be recovered. B. Non-Local Gravity It is possible to formulate a non-local action to address the IR aspects of gravity [11]. The non-local action can be written as, S 0 + S 2 = 1 16πG d 4 x √ −g R + RG(¯ )R , with: G(¯ ) = m n=0 c n¯ −n .(3) In this case the inversed d'Alembertian operators are acting on the scalar curvature. The equivalent action can be written as, S 0 +S 2 = 1 16πG d 4 x √ −g R + m n=0 Rc n ψ n + Rξ n (¯ n ψ n − R) . (4) Again, we introduced two auxiliary fields ξ n and ψ n . Solving the equations of motion for ξ n , results in having:¯ n ψ n = R or ψ n =¯ −n R. Thus, the original action given in Eq. (3) can be recovered. III. ENTROPY By varying the action, it is possible to find the Nöether current. When the current is being conserved (i.e. when the equations of motion are satisfied to be zero) one can define an associated potential. Now if we have a 4dimensional space with 3-dimensional hyper surface, one can define the associated Nöether charge by an integral over the 2-dimensional space-like boundary of the hypersurface. Wald, [12], used this method and proved that the first law of the black hole thermodynamics can be satisfied when the entropy is defined in terms of a specific Nöether charge. Given we have a static spherically symmetric metric, ds 2 = −f (r)dt 2 + f (r) −1 dr 2 + r 2 dΩ 2 2 .(6) The Wald's entropy can be obtained as: S = −8π δL δR rtrt r 2 dΩ 2 2 = −8πA H δL δR rtrt .(7) Where A H = 4πr 2 is the area of the horizon in 4dimension. A. Case I: Higher Derivative Gravity The entropy for S 0 + S 1 as appeared in Eq. (1) is given by [13]: S 0 + S 1 = A H 4G 1 + 2F (¯ )R .(8) Now let us calculate the entropy forS 1 , S 1 = − A H 2G × m n=0 (− 1 2 f n η n − 1 2 χ n η n + χ n¯ n R) = − A H 2G × m n=0 (− 1 2 f n η n − 1 2 χ n η n + χ n η n ) = A H 4G × m n=0 (2f n η n ) = A H 4G (2F (¯ )R).(9) Where we fixed the lagrange multiplier as χ n = −f n . It is clear that both S 1 andS 1 are giving the same result for the entropy as they should (see also [13]). This is to verify that it is always possible to use the equivalent action and find the correct entropy. This method is very advantageous in the case of non-local gravity, where we have inversed operators. B. Case II: Non-Local Gravity In this case calculating the Wald's entropy can be a challenging task, this is due to the fact that for an action of the form Eq. (3), the functional differentiation contains inversed operators acting on the scalar curvature. However, by introducing the equivalent action as given in Eq. (4), one can obtain the entropy as it had been done in the previous case. We know that the contribution of the EH term to the entropy is S 0 = A H /4G. Thus we shall consider the entropy ofS 2 : S 2 = − A H 2G × m n=0 (− 1 2 c n ψ n − 1 2 ξ n¯ n ψ n + ξ n R) = − A H 2G × m n=0 (− 1 2 c n ψ n − 1 2 ξ n¯ n ψ n + ξ n¯ n ψ n ) = A H 4G × m n=0 (c n ψ n + c n¯ n ψ n ) = A H 4G × m n=0 (c n (¯ −n R) + c n¯ n (¯ −n R)) = A H 4G × m n=0 (c n¯ −n R + c n R),(10) where we took ξ n = −c n . Furthermore, we used the fact that¯ n (¯ −n R) = R, [11]. Clearly, for n = 0, we get: S 20 = A H 4G × (2c 0 R).(11) This is an expected result, since Eq. (3), for n = 0, reduces to, S 0 + S 20 = 1 16πG d 4 x √ −g R + c 0 R 2 .(12) By fixing c 0 = (6M 2 ) −1 , one recovers the Starobinsky's model of gravity [18]. de Sitter non-local gravity Non-local gravity was studied in the de Sitter (dS) universe in [14]. For dS case, the action takes the form of: S dS = 1 16πG d 4 x √ −g R − 2Λ + RG(¯ )R ,(13) where Λ = 3/l 2 is the cosmological constant in 4dimension, R = 12/l 2 , and l denotes the cosmological horizon. Thus, the entropy would be of the form: S dS = A dS H 4G 1 + 2c 0 ( 12 l 2 ) = A dS H 4G 1 + 24c 0 l 2 . (14) The area of the horizon in dS space is given by: A dS H = 4πl 2 in 4-dimensions. This result is the same as for the infinite derivative gravity, [13]. The matching of the results is due the fact that, in dS background, we have constant curvatures, and thus the operators are not accountable. IV. SUMMARY We have shown that for a non-local action of the form Eq. (3), the Wald's entropy takes the following form: S nonloc = A H 4G 1 + m n=0 c n R + G(¯ )R .(15) It can be seen that the correction to the entropy of nonlocal gravity consists of two terms, one local and one non-local. We have shown the entropy for a generic background, of the form given in Eq. 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[ "A Linear Time Algorithm for Solving #2SAT on Cactus Formulas", "A Linear Time Algorithm for Solving #2SAT on Cactus Formulas" ]	[ "M A López \nUniversidad Autónoma del Estado de México\nMéxico\n", "J R Marcial [email protected] \nG. De Ita, Benemérita\nUniversidad Autónoma del Estado de México\nMéxico\n", "H A Montes-Venegas \nUniversidad Autónoma de Puebla\nMéxico\n", "M A López \nUniversidad Autónoma del Estado de México\nMéxico\n", "J R Marcial \nUniversidad Autónoma del Estado de México\nMéxico\n", "G De Ita [email protected] \nUniversidad Autónoma del Estado de México\nMéxico\n", "H A Montes-Venegas \nUniversidad Autónoma del Estado de México\nMéxico\n", "R Alejo \nUniversidad Autónoma del Estado de México\nMéxico\n" ]	[ "Universidad Autónoma del Estado de México\nMéxico", "G. De Ita, Benemérita\nUniversidad Autónoma del Estado de México\nMéxico", "Universidad Autónoma de Puebla\nMéxico", "Universidad Autónoma del Estado de México\nMéxico", "Universidad Autónoma del Estado de México\nMéxico", "Universidad Autónoma del Estado de México\nMéxico", "Universidad Autónoma del Estado de México\nMéxico", "Universidad Autónoma del Estado de México\nMéxico" ]	[]	An+-time algorithm is presented for counting the number of models of a two Conjunctive Normal Form Formula F that represents a Cactus graph, where is the number of variables and is the number of clauses of F. Although, it was already known that this class of formulas could be computed in polynomial time, we compare our proposal algorithm with two state of the art implementations for the same problem, sharpSAT and countAntom. The results of the comparison show that our algorithm outperforms both implementations, and it can be considered as a base case for general counting of two Conjunctive Normal Formulas.	null	[ "https://arxiv.org/pdf/1702.08581v1.pdf" ]	6,936,223	1702.08581	e31fd43c401272cd99e16bddcf5a5d231ec9db2a	A Linear Time Algorithm for Solving #2SAT on Cactus Formulas M A López Universidad Autónoma del Estado de México México J R Marcial [email protected] G. De Ita, Benemérita Universidad Autónoma del Estado de México México H A Montes-Venegas Universidad Autónoma de Puebla México M A López Universidad Autónoma del Estado de México México J R Marcial Universidad Autónoma del Estado de México México G De Ita [email protected] Universidad Autónoma del Estado de México México H A Montes-Venegas Universidad Autónoma del Estado de México México R Alejo Universidad Autónoma del Estado de México México A Linear Time Algorithm for Solving #2SAT on Cactus Formulas 1 An+-time algorithm is presented for counting the number of models of a two Conjunctive Normal Form Formula F that represents a Cactus graph, where is the number of variables and is the number of clauses of F. Although, it was already known that this class of formulas could be computed in polynomial time, we compare our proposal algorithm with two state of the art implementations for the same problem, sharpSAT and countAntom. The results of the comparison show that our algorithm outperforms both implementations, and it can be considered as a base case for general counting of two Conjunctive Normal Formulas. I. INTRODUCCIÓN L PROBLEMA ( ), donde es una fórmula Booleana, consiste en decidir si tiene un modelo, es decir, una asignación a las variables de tal que, al ser evaluada con respecto a la lógica proposicional, devuelve un valor verdadero. Si esta en dos Forma Normal Conjuntiva (2-FNC), entonces ( ) se puede resolver en tiempo polinomial. Sin embargo, si esta en -FNC, > 2, entonces ( ) es un problema NP-Completo. Por otro lado, existe el problema de conteo denotado como # ( ) que consiste en contar el número de modelos de . A diferencia de ( ) que se puede considerar como un problema de decisión, # ( ) es un problema de conteo. # ( ) pertenece a la clase # -Completo aún cuando este en 2-FNC, este último denotado como #2 [1]. Aunque el problema #2 es # -Completo, existen instancias que se pueden resolver en tiempo polinomial [2]. Por ejemplo, si la gráfica que representa la fórmula es acíclica, entonces #2 puede resolverse en tiempo polinomial. #2 es considerado un problema fundamental en el establecimiento de la frontera entre problemas de conteo intratables y aquellos que pueden resolverse eficientemente. # ( ) puede reducirse a diferentes problemas en el área de razonamiento aproximado. Por ejemplo, cuando se quiere estimar el grado de creencia, en la generación de explicaciones para consultas en bases de datos lógicas, en la inferencia Bayesiana y en el mantenimiento de sistemas de M [3][4] [5]. Los anteriores problemas provienen de aplicaciones de la Inteligencia Artificial, tales como planeación, sistemas expertos, razonamiento automático, etc. Actualmente, los algoritmos utilizados para # ( ) para cualquier fórmula en 2-FNC, descomponen en subfórmulas hasta obtener casos base en los que se puede contar modelos de forma eficiente. El algoritmo de menor orden de complejidad conocido hasta el momento fue desarrollado por Wahlström [3], éste es de (1.2377 ! ), donde representa el número de variables de la fórmula. El algoritmo de Wahlström utiliza como criterio de elección de una variable, el número de veces que aparece en la fórmula (sea la variable o su negación). Los dos criterios de paro del algoritmo son cuando = ∅ o cuando ∅ ∈ . Por otro lado, están las implementaciones para # en donde el objetivo es buscar estrategias que permitan resolver el problema de forma eficiente para instancias en donde el número de variables es considerable. Las herramientas que a la fecha se consideran las más eficientes son relsat [6] y sharpSAT [7], ambas bajo el paradigma secuencial. También se han implementado métodos paralelos, como: countAntom [8]. relsat esta basado en el algoritmo DPLL (Davis-Putnam algorithm) cuya ventaja es poder procesar rápidamente fórmulas disjuntas, es decir, cuyos conjuntos de variables no se intersectan. La herramienta sharpSAT es una mejora de relsat en donde se elige una literal heurísticamente. Asimismo, utiliza descomposición y cache de componentes para agilizar el cálculo. La herramienta countAntom es una implementación paralela basada en sharpSAT cuyos resultados muestran que se mejora el tiempo en las instancias de prueba con respecto a sharpSAT. En este artículo se presenta un algoritmo de compeljidad lineal en tiempo para el conteo de modelos de fórmulas en 2-FNC y cuya gráfica de restricciones sea tipo cactus. Los experimentos presentados muestran que nuestra propuesta mejora sustancialmente el tiempo para realizar el conteo de modelos con respecto a las herramientas de software actuales; relsat, sharpSAT y countAntom, por lo que nuestro algoritmo puede utilizarse como caso base en los algoritmos de conteo de modelos basados en descomposición de fórmulas en FNC. → {0,1}. Una asignación puede también considerarse como un conjunto de pares de literales no complementario. Si ! ∈ ∈ {0,1}, siendo una asignación, entonces convierte a ! en verdadero y a !!! en falso. Considerando una cláusula y una asignación como un conjunto de literales, se dice que se satisface por sí y solo si ∩ ≠ ∅ y si para toda ! ∈ , si !!! ∈ , entonces falsifíca a . II. PRELIMINARES Sea = { ! , … , ! } Sea una fórmula Booleana en FNC, se dice que se satisface por la asignación si cada cláusula en se satisface por . Por otro lado, se dice que se contradice por si al menos una cláusula de se falsifíca por . Un modelo de es una asignación para ( ) tal que satisface . Dada una fórmula en FNC, SAT consiste en determinar si tiene un modelo, mientras que #SAT consiste en contar el número de modelos que tiene sobre ( ). Por otro lado, #2SAT denota #SAT para fórmulas en 2-FNC. II.I. La gráfica restringida de una 2-FNC. Existen algunas representaciones gráficas de una Forma Normal Conjuntiva, en este caso, se utilizará la gráfica primal signada (gráfica restringida) [9]. Sea una 2-FNC, su gráfica restringida se denota por ! = ( , ( )) donde los vértices de la gráfica son las variables = , y las cláusulas las aristas = { , ∶ ∨ ∈ }, esto es, para cada cláusula en existe una arista , ∈ ( ). Para ∈ ( ), denota su grado, es decir, el número de aristas incidentes en x. Cada arista = { , ( )} ∈ ( ) se asocia con un par ( ! , ! ) de signos, que se asignan como etiquetas de la arista que conecta las variables de la cláusula en la gráfica. Los signos ! y ! pertenecen a las variables x y y respectivamente. Por ejemplo, la cláusula ( ! ∨ ! ) determina la arista con etiqueta " !! ", que es equivalente a la arista " !! ". Sea = +, − un conjunto de signos. Una gráfica con aristas etiquetadas en es el par , , dónde = , es una gráfica restringida, y es una función con dominio y rango . se denomina a la etiqueta de la arista ∈ . Sea = , , una gráfica restringida con aristas etiquetadas en × y y vértices en , si = , es una arista y = , ! , entonces ' es el signo adyacente a . Note que la gráfica restringida ! = , , de una 2-FC F puede contener aristas paralelas. Nosotros consideraremos gráficas simples (sin aristas paralelas), ya que éstas últimas pueden ser pre-procesadas, tal y como se presenta en [4], sin que este pre-procesamiento modifique la complejidad en tiempo de nuestra propuesta algorítmica. Sea una gráfica conectada de vértices, un árbol de expansión de es un subconjunto de − 1 aristas tal que forman un árbol de . Se denomina coárbol al subconjunto de aristas que son el complemento de un árbol, cada una de estas aristas forman un ciclo en la gráfica. Una gráfica cactus es una gráfica = ( ! , ! ) dónde: -Cada arista de ! pertenece a lo más a un ciclo. -Cualquier par de ciclos comparten a lo más un vértice. En este artículo, para encontrar el árbol de expansión y el coárbol se utiliza el método de búsqueda primero en profundidad [10] que permite construir la tupla (árbol, coárbol) de una gráfica. III. ALGORITMO En esta sección se presenta el algoritmo utilizado para el conteo de modelos en gráficas cactus en tiempo lineal. El método principal consiste de 3 pasos: construcción de una tabla en donde se almacenan las cláusulas, construcción de un árbol de expansión en donde se marcan los ciclos de la gráfica y finalmente, se realiza el conteo de modelos sobre el árbol de expansión. El algoritmo I muestra las entradas y salidas de cada paso. Ya que la lectura de cláusulas se realiza desde un archivo en formato DIMACS (http://logic.pdmi.ras.ru/~basolver/dimacs.html), se utilizan dos tablas dinámicas para almacenar cada cláusula leída. El índice del renglón h de cada tabla representa a la variable ! . Por cada cláusula ( ! , ! ), se agrega la entrada ! al renglón i en la tabla uno, y la entrada ! al renglón j en la tabla dos. La duplicidad de cláusulas permite agilizar las búsquedas durante la construcción del árbol de expansión. Adicionalmente, las cláusulas se insertan considerando el índice menor de sus dos variables. El algoritmo II muestra la construcción de la tabla. III.II. Creación del árbol. El par (árbol, coárbol) se crea a partir de las tablas que almacenan cláusulas. El algoritmo utilizado para la construcción del árbol es primero en profundida (depth first search, por sus siglas en inglés) [10]. Cada vez que se lee una cláusula, se revisa si ambos vértices están ya en el árbol, si es el caso, se marca el camino entre ambos vértices para denotar que se tiene una arista del coarbol, es decir, se encontró un ciclo. El algoritmo III muestra la forma en que se recorre la tabla para insertar sus elementos en el árbol, mientras que el algoritmo IV presenta la forma de insertar una variable en el árbol considerando si ésta aparece de forma positiva o negativa en la cláusula de la que proviene. ALGORITMO II CONSTRUCCIÓN DE LA TABLA DE CLÁUSULAS _____________________________________________ Como se puede apreciar en el algoritmo IV, se tiene la condición para conocer si la gráfica de entrada es cactus, lo anterior con la finalidad de hacer una comparación equitativa con las herramientas sharpSAT y countAntom, ya que en ellas no se conoce de antemano si la gráfica de entrada es cactus. ALGORITMO Una vez que se tiene el árbol de la gráfica cactus con los caminos de los ciclos marcados (coárbol), se realiza el conteo de los modelos mediante un recorrido en postorden del árbol. A cada nodo ! del árbol se le asigna un par ( ! , ! ) donde ! denota el número de modelos en donde la variable ! toma un valor verdadero y ! el número de modelos en donde ! toma un valor falso. Los modelos contabilizados por cada par serán relativos a la subfórmula hasta ese momento calculada. Si el nodo del árbol esta marcado como de apertura de un ciclo, se genera un segundo par ! ! , ! ! que estará activo hasta que el ciclo se cierre. El número de modelos sobre el nodo de cierre de ciclo, se calcula como: ! , ! − ( ! ! , ! ! ), donde la resta se realiza a pares. ____________________________________________ Inicialmente, a los nodos hoja se les asigna el par (1,1) y si el nodo hoja es a la vez la apertura de un ciclo, se le asigna como segundo par (0,1). Durante el recorrido, el par de cada nodo interior se calcula utilizando la recurrencia (1) que se aplica de acuerdo a los signos de las variables del nodo actual y del nodo hijo (es decir, de la cláusula representada por la arista hijo-padre en el árbol). ! , ! = !!! , !!! + !!! ! , ! = (−, −) !!! + !!! , !!! ! , ! = (−, +) !!! , !!! + !!! ! , ! = (+, −) !!! + !!! , !!! ! , ! = (+, +)(1) Si un nodo interior es la apertura de un ciclo, su segundo par se calcula dependiendo de su signo, la recurrencia (2) establece el valor para el segundo par ! ! , ! ! = 0, ! ! !! si = + ! ! !! , 0 si = − (2) Finalmente, si el nodo que se esta evaluando tiene mas de un hijo, el número de modelos se calcula mediante el producto de los pares de cada uno de ellos. El algoritmo V presenta el conteo de los modelos en el árbol. Más detalles del método de conteo se puede consultar en [11], en donde se muestra cómo contar el número de modelos para árboles y ciclos por separado. En este mismo artículo se puede ver la demostración de la validez del método. Finalmente, el total de modelos se obtiene cuando se llega al nodo raíz del árbol, y se deriva de la suma de sus dos componentes: y . ALGORITMO V CONTEO DE MODELOS EN EL ÁRBOL QUE REPRESENTA LA GRÁFICA CACTUS. _____________________________________________ IV. RESULTADOS En esta sección se muestran los resultados de comparar la propuesta aquí presentada contra sharpSAT y countAntom, las dos herramientas más eficientes hasta el momento reportadas en la literarura para resolver #SAT. Se realizaron dos tipos de pruebas, la primera consistió en generar de manera aleatoria 22 gráficas cactus que tienen entre 9,000 y 240,000 vértices y entre 12,500 y 320,000 aristas. La segunda prueba consisitió en generar gráficas cactus con el número máximo de ciclos que puede tener este tipo de gráficas, este es el peor de los casos en donde por cada tres vértices se generó un ciclo. Ya que sharpSAT y countAntom utilizan la estructura de la fórmula para realizar el conteo, éste segundo tipo de prueba generan los mejores casos para estas herramientas y para este tipo de gráficas. Todas las pruebas se realizaron en una Computadora con procesador de dos núcleos a una velocidad de 2.4 Mhz con 8 GB en RAM y sistema operativo Ubuntu, ya que éste es el sistema operativo en donde se provee el código fuente para poder compilar y ejecutar ambas herramientas. La Tabla 1 muestra los resultados de realizar el primer tipo de pruebas. Como se puede observar, nuestra propuesta obtiene mejores tiempos de ejecución en los 22 casos considerados, con respecto a las dos herramientas antes mencionadas. El tiempo de ejecución máximo para el conteo de modelos fue de 25 minutos. También se puede observar que countAntom no fue capaz de producir una respuesta para 17 de los 22 casos. La Gráfica 1 muestra el crecimiento en el tiempo de ejecución, tanto para sharpSAT como para nuestra propuesta. En la gráfica no se incluyen los tiempos de countAntom, ya que a partir de la sexta entrada, esta herramienta tardó mas de 25 minutos en dar una respuesta, por lo que se considera que sus tiempos de respuesta no son competitivos con respecto a los otros dos programas. Como se puede observar en la Gráfica 1, nuestra propuesta obtiene mejores tiempos de ejecución en todos los casos que los obtenidos con sharpSAT. GRÁFICA I CRECIMIENTO DEL TIEMPO DE EJECUCIÓN PARA 22 GRÁFICAS CACTUS. EL TIEMPO REPORTADO ESTA EN SEGUNDOS Para realizar pruebas considerando los peores casos para nuestra propuesta, se generaron gráficas cactus con el máximo número de ciclos posibles, es decir, se formó un ciclo por cada tres vértices. La diferencia principal con este tipo de gráficas es que el árbol generado tiene un tercio de nodos de inicio y un tercio de nodos de cierre de ciclos lo cual significa que se tienen dos variables de conteo en todo el recorrido del árbol. La Tabla II muestra los resultados del tiempo de ejecución para el conteo de modelos sobre este tipo de gráficas. Como se puede observar en los 7 casos, nuestra propuesta obtiene menores tiempos de ejecución que el de las otras dos herramientas. Similar al primer caso, counAntom dio un resultado para los primero 5 casos, sin embargo, para los dos últimos no fue capaz de producir un resultado en los 5 minutos de ejecución que se dieron como máximo em esta segunda prueba. La Gráfica II muestra el crecimiento de los tiempos de ejecución del conteo de modelos para sharpSAT y para nuestra propuesta. Al igual que en la Gráfica I, no se muestran los resultados para countAntom, ya que sus tiempos reportados no son competitivos con respecto a los reportados para los otros dos programas. Similar a los resultados reportados para la primera prueba, nuestra propuesta mejora significativamente el tiempo de ejecución con respecto al de sharpSAT. IV. COMPLEJIDAD EN TIEMPO DEL ALGORITMO El algoritmo de esta propuesta consta de tres partes, crear tabla, crear árbol y conteo de modelos en el árbol. Para la creación de la tabla es suficiente con recorrer las cláusulas de la fórmula de entrada por lo que la complejidad de este paso es de orden ( ). Mientras que para creación del árbol, se recorre cada entrada en la tabla, lo que requiere también del orden operaciones básicas. Finalmente, el recorrido del árbol es en postorden, y mientras se visitan los nodos del árbol, se va también visitando cada una de las aristas (cláusulas) de la gráfica de restricciones, al mismo tiempo que se van aplicando las recurrencias (1) y (2). Este último proceso nos genera del orden de ( + ) operaciones básicas. Así, la complejidad en tiempo de nuestra propuesta algorítmica es lineal y de orden ( + ). En este artículo se presentó un algoritmo para el conteo de modelos de fórmulas Booleanas en 2-FNC y cuya gráfica de restricciones es tipo cactus. El algoritmo presentado tiene una complejidad en tiempo de orden ( + ), donde es el número de cláusulas y el número de variables de la fórmula de entrada. Nuestra propuesta detecta en tiempo lineal y al inicio de su procesamiento, si la fórmula de entrada es representada efectivamente por una gráfica de restricciones tipo cactus simple. Las pruebas realizadas sobre dos diferentes clases de fórmulas muestran que la propuesta aquí presentada obtiene mejores tiempos de ejecución con respecto a las dos herramientas que se reportan en la literatura como las más eficientes; sharpSAT y countAtompara resolver este tipo de problemas de conteo. Por tanto, se considera que nuestro algoritmo puede incluirse como un caso base en el conteo de modelos de sharpSAT o de otras herramientas que realicen descomposición de la fórmula de entrada, típico de los métodos de ramificación y corte, hasta llegar a los casos bases. Para el caso general #SAT, se podrían realizar descomposiciones sobre la fórmula de entrada, hasta generar gráficas cactus y entonces, procesar éstas últimas subfórmulas usando nuestro algoritmo, lo que creemos que puede impactar en la complejidad en tiempo de estos algoritmos de conteeo. Marco Antonio López Medina, En el 2016 obtiene el título de Ingeniero en Computación en la Universidad Autónoma del Estado de México. Actualmente estudia la Maestría en Ciencias de la Ingeniería con línea de acentuación en Computación en la Universidad Autónoma del Estado de México. José Raymundo Marcial-Romero, Actualmente profesor investigador de tiempo completo en la Facultad de Ingeniería de la Universidad Autónoma del Estado de México. Miembro del Sistema Nacional de Investigadores del Consejo Nacional de Ciencia y Tecnología, Nivel 1. En el año 2000 obtuvo el título de Licenciatura en Ciencias de la Computación y en el año 2007 el grado de Doctor en Ciencias de la Computación por The University of Birmingham UK en The School of Computer Science. Guillermo De Ita Luna, Obtuvo su doctorado en Ingeniería Eléctrica por el CINVESTAV-IPN, México. Ha trabajado como desarrollador y consultor en sistemas de bases de datos y sistemas de información geográfica para diferentes empresas en México. Ha realizado estancias de investigación en la Universidad de Chicago, Texas A&M, INAOEP Puebla, en el instituto INRIA en Lille-1 y en la Fac. de Ing. de la UAEMEX. Actualmente, es profesor investigador de la Facultad de Ciencias de la Computación, BUAP, Puebla, México. Héctor Alejandro Montes-Venegas, Es actualmente profesor investigador de tiempo completo en la Facultad de Ingeniería de la Universidad Autónoma del Estado de México. Es también ingeniero en Sistemas Computacionales por el Instituto Tecnológico de León y Maestro en Ciencias Computacionales por el Instituto Tecnológico y de Estudios Superiores de Monterrey. Roberto Alejo Eleuterio Doctor en Ciencias Computacionales, adscrito al Instituto de Estudio Superiores de Jocotitlán, Miembro del Sistema Nacional de Investigadores del CONACYT. Los intereses en investigación se centran en la aplicación de inteligencia artificial a la solución de problemas reales. .I. Construcción de la Tabla de Cláusulas. ALGORITMO IV AGREGAR VÉRTICES AL ÁRBOL Y DETERMINAR SI LAGRÁFICA ES CACTUS. ---------- GRÁFICA II CRECIMIENTO DEL DEL TIEMPO DE EJECUCIÓN PARA 7 GRÁFICAS CACTUS QUE REPRESENTAN EL PEOR CASO PARA NUESTRA PROPUESTA. TIEMPO REPORTADO EN SEGUNDOS V. CONCLUSIONES . A. López, Universidad Autónoma del Estado de México, México,[email protected] J. R. Marcial, Universidad Autónoma del Estado de México, México, [email protected] G. De Ita, Benemérita Universidad Autónoma de Puebla, México, [email protected] H. A. Montes-Venegas, Universidad Autónoma del Estado de México, México, [email protected] razonamiento un conjunto de variables Booleanas.Una literal es una variable ! ( ! ! ) o la variable negada ¬ ! ( ! ! ). Una cláusula es una disyunción de literales distintas. Una fórmula Booleana en forma normal conjuntiva (FNC) es una conjunción de cláusulas. Sea ( ) el conjunto de variables involucradas en el E A Linear Time Algorithm for Solving #2SAT on Cactus Formulas M. A. López, J. R. Marcial, G. De Ita, H. A. Montes-Venegas and R. Alejo objeto , dónde puede ser una literal, una cláusula o una fórmula Booleana. Por ejemplo, para la cláusula = ! ¬ ! , = { ! , ! }. Una asignación para es una función Booleana : TABLA I TIEMPO DE EJECUCIÓN DEL CONTEO DE MODELOS PARA 22 GRÁFICAS CACTUS. TIEMPO REPORTADO EN SEGUNDOSVértices Aristas Ésta Propuesta sharpSAT countAntom 1 9382 12508 0.076 0.213 11.297 2 19999 26664 0.056 0.490 851.425 3 10000 13332 0.060 0.231 427.430 4 16000 21332 0.065 0.380 197.042 5 25999 34664 0.071 0.700 1334.007 6 30001 40000 0.078 0. TABLA II TIEMPO DE EJECUCIÓN DEL CONTEO DE MODELOS EN 7 GRÁFICAS CACTUS CON EL MÁXIMO NÚMERO DE CICLOS.TIEMPO REPORTADO EN SEGUNDOS Vértices Aristas Ésta Propuesta sharpSAT countAntom 1 9999 14997 0.028 0.240 1.637 2 15999 23997 0.05 0.399 8.562 3 19999 29997 0.062 0.515 8.960 4 39999 59997 0.115 1.312 50.023 5 79999 119997 0.225 3.572 239.328 . G Brifhtwell, Winkler Peter, Counting linear extensions. Order. 8Brifhtwell G., Winkler Peter. (1991). Counting linear extensions. Order, Vol. 8, issue e, pp. 225-242. Model counting for CNF formulas ofbounded modular treewidth. D Paulusma, F Slivovsky, S Szeider, STACS. LIPIcs. Portier, N., Wilke, T.20Schloss Dagstuhl -Leibniz-Zentrum fuer InformatikPaulusma, D., Slivovsky, F., Szeider, S. (2013). Model counting for CNF formulas ofbounded modular treewidth. In: Portier, N., Wilke, T. (eds.) STACS. LIPIcs,vol. 20, pp. 55-66. Schloss Dagstuhl -Leibniz- Zentrum fuer Informatik. A tighter bound for counting max-weight solutions to 2SAT instances. Magnus Wahlström, Proc. 3rd Int. Workshop on Parametrical and Exact Computation(IWPEC). 3rd Int. Workshop on Parametrical and Exact Computation(IWPEC)Magnus Wahlström. (2008). A tighter bound for counting max-weight solutions to 2SAT instances. In Proc. 3rd Int. Workshop on Parametrical and Exact Computation(IWPEC), pp. 202-213. Polynomial Classes of Boolean Formulas for Computing the Degree of Belief. De Ita, G , Lectures Notes in Artificial Intelligence. 3315De Ita G. (2004). Polynomial Classes of Boolean Formulas for Computing the Degree of Belief, Lectures Notes in Artificial Intelligence Vol. 3315, pp. 430-440. On the hardness of approximate reasoning. D Roth, Artificial Intelligence. 82Roth D., (1996). On the hardness of approximate reasoning, Artificial Intelligence, Vol. 82, pp. 273-302 Using CSP look-back techniques to solve real-world SAT instances. R Bayardo, R Schrag, Proceedings of the Fourteenth National Conference on Artificial Intelligence (AAAI-97). the Fourteenth National Conference on Artificial Intelligence (AAAI-97)New Providence, RIBayardo R. and R. Schrag. (1997). Using CSP look-back techniques to solve real-world SAT instances. In: Proceedings of the Fourteenth National Conference on Artificial Intelligence (AAAI-97). New Providence, RI, pp. 203-208. sharpSAT -Counting models with Advanced Component Caching and Implicit BCP. M Thurley, Proceedings of the 9th International Conference on Theory and Applications of Satisfiability Testing. the 9th International Conference on Theory and Applications of Satisfiability TestingThurley M. (2006). sharpSAT -Counting models with Advanced Component Caching and Implicit BCP. Proceedings of the 9th International Conference on Theory and Applications of Satisfiability Testing (SAT 2006). pp. 424-429. Laissez-Faire Caching for Parallel #SAT Solving. Jan Burchard, Tobias Schubert, Bernd Becker, Jan Burchard, Tobias Schubert, Bernd Becker. (2015). Laissez-Faire Caching for Parallel #SAT Solving. SAT 2015. pp. 46-61. On fixed-parameter tractable parameterizations of SAT. S Szeider, SAT 2003. Giunchiglia, E., Taccella, A.HeidelbergSpringer2919Szeider S. (2004). On fixed-parameter tractable parameterizations of SAT. In: Giunchiglia, E., Taccella, A. (eds.) SAT 2003. LNCS, vol 2919, pp. 188-202. Springer, Heidelberg. . J A Bondy, U S R Murty, Theory, Graduate Texts in Mathematics. Springer VerlagBondy J. A. and Murty U.S.R. Graph Theory. Springer Verlag, Graduate Texts in Mathematics, 2010. Parametric Polynomial Deterministic Algorithm for #2SAT. J R Marcial-Romero, G De Ita, J A Hernández, R M A Valdovinos, Lecture Notes in Computer Science. 9413Marcial-Romero J. R., De Ita G., Hernández, J. A., Valdovinos, R. M. A. (2015). Parametric Polynomial Deterministic Algorithm for #2SAT, Lecture Notes in Computer Science, Vol. 9413, pp. 202-213.	[]
[ "Dust productivity and impact collision of the asteroid (596) Scheila", "Dust productivity and impact collision of the asteroid (596) Scheila" ]	[ "L Neslusan \nAstronomical Institute\nSlovak Academy of Sciences\n05960Tatranská LomnicaSlovakia\n", "O Ivanova \nMain Astronomical Observatory of NAS of Ukraine\nAkademika Zabolotnoho 2703680KyivUkraine\n", "M Husarik \nAstronomical Institute\nSlovak Academy of Sciences\n05960Tatranská LomnicaSlovakia\n", "J Svoren \nAstronomical Institute\nSlovak Academy of Sciences\n05960Tatranská LomnicaSlovakia\n", "Z Seman Krisandova \nAstronomical Institute\nSlovak Academy of Sciences\n05960Tatranská LomnicaSlovakia\n" ]	[ "Astronomical Institute\nSlovak Academy of Sciences\n05960Tatranská LomnicaSlovakia", "Main Astronomical Observatory of NAS of Ukraine\nAkademika Zabolotnoho 2703680KyivUkraine", "Astronomical Institute\nSlovak Academy of Sciences\n05960Tatranská LomnicaSlovakia", "Astronomical Institute\nSlovak Academy of Sciences\n05960Tatranská LomnicaSlovakia", "Astronomical Institute\nSlovak Academy of Sciences\n05960Tatranská LomnicaSlovakia" ]	[]	a b s t r a c tPhotometric observations of asteroid (596) Scheila were obtained during and after its 2010 outburst. The estimated radius of the body (spherical approximation of the asteroidal body) was 51.273.0 km and 50.6 73.0 km for different methods. The ejected dust mass from the asteroid ranged from 2:5 Â 10 7 to 3:4 Â 10 7 kg for different methods. An impact mechanism for triggering Scheila's activity is discussed. A few days before the impact, Scheila passed through the corridors of two potential cometary streams.	10.1016/j.pss.2016.01.017	[ "https://arxiv.org/pdf/2012.08434v1.pdf" ]	123,433,789	2012.08434	f736a5edfc6b12485b0bfc5e743fa3fd3e91af3b	Dust productivity and impact collision of the asteroid (596) Scheila L Neslusan Astronomical Institute Slovak Academy of Sciences 05960Tatranská LomnicaSlovakia O Ivanova Main Astronomical Observatory of NAS of Ukraine Akademika Zabolotnoho 2703680KyivUkraine M Husarik Astronomical Institute Slovak Academy of Sciences 05960Tatranská LomnicaSlovakia J Svoren Astronomical Institute Slovak Academy of Sciences 05960Tatranská LomnicaSlovakia Z Seman Krisandova Astronomical Institute Slovak Academy of Sciences 05960Tatranská LomnicaSlovakia Dust productivity and impact collision of the asteroid (596) Scheila a b s t r a c tPhotometric observations of asteroid (596) Scheila were obtained during and after its 2010 outburst. The estimated radius of the body (spherical approximation of the asteroidal body) was 51.273.0 km and 50.6 73.0 km for different methods. The ejected dust mass from the asteroid ranged from 2:5 Â 10 7 to 3:4 Â 10 7 kg for different methods. An impact mechanism for triggering Scheila's activity is discussed. A few days before the impact, Scheila passed through the corridors of two potential cometary streams. Introduction The main-belt asteroid (596) Scheila (1906 UA) was discovered by A. Kopff in Heidelberg in 1906. It is classified as the Tholen PCD class or SMASSII T class asteroid (Bus and Binzel, 2002). Licandro et al. (2011) showed that its surface was homogeneous and corresponded to a dark primitive D-type asteroid. It is a middle-sized body with the diameter of 113 km and visual geometric albedo of 0.038 (Tedesco and Desert, 2002). The asteroid moves in the orbit with semi-major axis 2.938 AU, eccentricity 0.165, and inclination 14.7°. Its orbit is situated in the outer asteroid belt (Fig. 1), where several main-belt comets (Hsieh and Jewitt, 2006) have been found. Similarly, Scheila is a typical asteroid according to the value of its Tisserand parameter with respect to Jupiter, which equals T J ¼ 3:21. The observations taken with the 0.68-m Catalina Schmidt telescope on December 11.44-11.47 (UT), 2010, showed a cometary activity (Larson et al., 2010). The archival Catalina observations showed that the activity was triggered before December 3, when Scheila appeared as a slightly diffuse object (V ¼13.2 mag). With a main-belt orbit (T J 4 3) and a comet-like morphology, Scheila obeys the definition of a main-belt comet or active asteroid (Hsieh and Jewitt, 2006). At the time of outburst, Scheila was the seventh known example of this specific class of the objects in the Solar System. The currently known main-belt comets (MBCs, hereinafter) Active asteroids are now recognized as a new class of objects in the Solar System. These objects are remarkable since they have both the orbital characteristics of asteroids and the physical characteristics of comets. It means that they look like comets because they show comae and tails, but they have the whole orbits inside the Jupiter's orbit. In the literature, sublimation, impact, electrostatics and rotational bursting, thermal effects, and radiation pressure sweeping have been proposed as the mechanisms of their activity (Hsieh and Jewitt, 2006;Jewitt, 2012;Jewitt et al., 2016). Because of their activity, the MBCs seem to be related to the objects in the well-established comet reservoirs, Oort cloud and Kuiper belt. However, there is no dynamical pathway between two distant reservoirs and the MBCs in the Solar System. In the period of activity of Scheila, its spectra were obtained and emissions were searched for. Bodewits et al. (2011) showed that no gases were detected by Swift UV-optical observations, suggesting that the outburst was not triggered by the ice sublimation. Other physical evidence suggests (Yang and Hsieh, 2011;Jewitt et al., 2011;Ishiguro et al., 2011) that Scheila likely collided, recently, with another, smaller asteroid, and that its activity is unlikely produced by sublimation. Since Scheila is the largest known active asteroid, or MBC, it was a suitable candidate also for observations performed at the Skalnaté Pleso Observatory during the period of its activity. In 2010 and 2011, Husárik (2012) performed a relative photometry to compare and determine whether and, if so, how much the rotational period was changed by an outburst with respect to the data published by Warner in 2006. The comparison shows very similar lightcurves and, hence, no significant change in the rotational period. In our article, we analyze the photometric data of the Scheila in the period of its activity and after. We estimate the dust productivity in the outburst phase and radius of the asteroid after this period, when object was not active. Furthermore, we expanded our work to find possible sources of the impact, which triggered the Scheila's sudden activity. In principle, there are three source regions of the impactor: (i) an individually moving small asteroid, fragment of comet, or sporadic meteoroid, (ii) a meteoroid from a cometary or asteroidal stream, and (iii) a component of a known binary asteroid. In the first case, we cannot identify the previously unknown impactor and, thus, say anything more about the circumstances of the collision. However, another research is possible, if the impactor is a member of a meteoroid stream or satellite of known minor body still existing in the Solar System. Because of this possibility, we look for an appropriate, cometary and asteroidal, potential parent body of meteoroid stream that could be crossed by Scheila in time of the outburst. As well, we calculate the approaches of all small bodies, with well-known orbits, to Scheila in the most probable time of the collision. We also predict the dates of future passages of Scheila near the former collisional point of its orbit. If the impactor was a meteoroid from a stream, a certain, at least indicative, activity could again appear at the crossing of the stream corridor. Observation and reduction The observations of asteroid Scheila were obtained between December 15, 2010, and May 22, 2011, with the 0.61-m f/4.3 reflector at the Skalnaté Pleso Observatory. The photometric data of Scheila were obtained through the V and R broadband filters. The CCD detector SBIG ST-10XME with 3 Â 3 binning and resolution of 1.6 arcsec/px was used. The reduction of the raw data, which included bias subtraction, dark and flat field correction, and removal of cosmic ray tracks, was made. The morning sky was exposed to provide flat field corrections for the non-uniform sensitivity of the CCD chip. We applied the standard calibration frames with IRAF tools. The observational circumstance for the asteroid in each observational night are listed in Table 1 and Fig. 2. Morphology of (596) Scheila during outburst An impact has been proposed as the cause of the activity of (596) Scheila, in the literature. The modeling of the outburst of this asteroid implied an estimate of the impact occurrence on November 27, with an estimated accuracy of 73 days (Moreno et al., 2011). Assuming the same density for the impactor and asteroid (ρ ¼1500 kg m À 3 ), the authors estimated the projectile mean radius in the range of 30-90 m. Ishiguro et al. (2011), analyzing the changes of shape of the light curve, estimated the mass of ejecta to Fig. 1. The distribution of the known main-belt comets (big black circles), asteroids (grey dots), and comets (bigger grey dots) in the semi-major axis vs. eccentricity plane. The dashed lines show the semi-major axes of the orbits of Mars, Jupiter, and the location of the 2:1 mean-motion resonance with Jupiter. Table 1 List of our observations of (596) Scheila. r and Δ are the heliocentric and geocentric distances, respectively, texp: is the duration of exposure, and n is the number of exposures in a given sequence. be ð1:5-4:9Þ Â 10 8 kg, suggesting that equivalent mass of 500-800 m crater was excavated at the event. The value of impactor size (20-50 m) obtained by the authors is consistent with the impactor size derived by Jewitt et al. (2011) and Bodewits et al. (2011). We observed Scheila during its activity from December 15, 2010 to January 9, 2011. Our observations of the object showed the outburst with a structure in dust environment. To recognize the weakly contrasted structures within the images of the dust coma, we used the special software Astroart. 1 It can be applied for a number of digital filters: the Larson-Sekanina filter (Larson and Sekanina, 1984), the unsharp masking, and the Gaussian blur. The detailed information on the digital filters can be found in the paper by Ivanova et al. (2012). Such the technique of recognizing the structures in the coma was successfully used by Manzini et al. (2007) for comet C/2002 C1 (Ikeya-Zhang), Korsun et al. (2008Korsun et al. ( , 2010 for distant comets, and by Ivanova et al. (2012) for comet 29P/Schwassmann-Wachmann 1. In Fig. 3 we present the results of processing of images. Photometry of the asteroid We used the observations of Scheila obtained through the R broadband filter to calculate the magnitude in the period of activity and through the V broadband filter after this period. A radius of the nucleus and the dust production were also obtained. In the period of the outburst, Scheila was more similar to a comet than an asteroid, therefore a cometary description is more relevant. The cometary magnitude is determined by m a ðλÞ ¼ À2:5 log 10 I a ðλÞ I s ðλÞ þ m st À 2:5 log 10 ½PðλÞΔM; where m st is the magnitude of the standard star, I s and I a are the measured fluxes of the star and the asteroid in counts, respectively, P is wavelength dependent sky transparency, ΔM is the difference between the comet and star airmasses. As we used the field stars for calibration, the sky transparency is not considered. We estimate the absolute magnitude of the object, m a ð1; 1; 0Þ, i.e. the magnitude being corrected to r ¼ Δ ¼ 1 AU and phase angle α¼0°, by using the H, G1, G2 model by Penttilä et al. (2016) (see also Muinonen et al., 2010). According to this model, the value of the magnitude, at phase angle α between the Sun and the observer as seen from the object, reduced to unit distance is m a ð1; 1; αÞ ¼ m a ð1; 1; 0ÞÀ2:5 log 10 ½G 1 Φ 1 ðαÞþG 2 Φ 2 ðαÞ þð1 À G 1 À G 2 ÞΦ 3 ðαÞ:ð2Þ At the same time, the reduced magnitude m a ð1; 1; αÞ is related to the observed magnitude of object, which is in the heliocentric and geocentric distances r and Δ (in AU), respectively, and seen under the space angle α, as m a ð1; 1; αÞ ¼ m a ðr; Δ; αÞÀ5 log 10 ðrΔÞ; The basic functions Φ 1 , Φ 2 , and Φ 3 in relation (2) (2) can be gained using the calculator provided by these authors). The results for Scheila are given in Table 2. Radius of the asteroid We estimated the radius of the asteroid, using our observations obtained through V broadband filter in the post-activity period (March 8.9, 2011), when the object had not longer any detectable dust "coma", and looked like a typical asteroid. On March 2, 2011, Ishiguro et al. (2011) detected a faint straight tail connected to Scheila. This structure was not, however, seen in our images. Specifically, we calculate the radius by using two formulas. To estimate the effective radius considering the cometary appearance of our object, we use relation by Meech et al. (2009): p v R 2 a ¼ 2:24 Â 10 22 10 0:4½m À mað1;1;0Þ : Here, p v ¼ 0:038 7 0:040 is the albedo of asteroid (Tedesco and Desert, 2002); m is the magnitude of the Sun (the latter being $ À27:094 for R band and $ À26:74 for V band, (Holmberg et al., 2006). We also estimated the diameter using the formula for an asteroid-like object, which describes the relationship between the flux and size and albedo of asteroid. This relationship is given by Fig. 3. R band images of (596) Scheila taken in six nights and processed with digital filters. The structure of outburst is changed during the time of observation. After January 9, 2011, we could not detect the dust environment. We observed Scheila as a stellar object. (Harris and Lagerros, 2002) D ¼ 1329 ffiffiffiffiffi p v p 10 0:2 mað1;1;0Þ ; where D is the diameter of asteroids in kilometers. Using formulas (4) and (5) Dust production of the asteroid We also used our observations of the asteroid obtained in period of "activity" (see Table 1) to estimate the dust production of the object. We used different methods for calculation of this parameter. We calculated the effective scattering cross section of the coma from equation (Jewitt et al., 2011) C C ¼ C n 10 Δm À 1 ;ð6Þ where C n ¼ πR 2 N ¼ 0:92 Â 10 4 km 2 is the geometric cross section of the nucleus (we use aperture radius 1.6 arcsec), Δm ¼ m n À m, where m n is absolute R magnitude of the nucleus and m is total R-band magnitude measured within a 66.72 arcsec radius aperture. We further estimated the mass of dust particles in the coma from the scattering cross section using the relation M d ¼ ρaC C ;ð7Þ where ρ is the particle density, taken to be ρ¼2000 kg m À 3 , and a ¼ 1 μm is the average particle radius in the coma. We used these parameters to compare our value of radius with results obtained in article by Jewitt et al. (2011). In addition, we calculated the dust production rate by using the methods described by Newburn et al. (1981) and Newburn and Spinrad (1985) as well as Weiler et al. (2003), taking into account the discussion by Fink and Rubin (2012). We used a density from 1000 to 2000 kg m À 3 (Moreno et al., 2011;Jewitt et al., 2011). For our estimation, we used the dust size distribution function in the form of f ðaÞ $ a À n for two variants of parameter n. Specifically, we considered Table 3. Table 3 The measured magnitude, effective scattering cross-section of coma, and mass of ejected dust from our R-band photometry of asteroid (596) The determination of the mass of dust escaping from Scheila to the interplanetary space by various authors differs up to almost two orders of magnitude. Our estimate, yielding the ejected mass from 2:5 Â10 7 to 3:4 Â 10 7 kg, roughly agrees with the values of 4:4 Â 10 7 and 3:0 Â 10 7 kg found by Jewitt et al. (2011) and 3:0 Â 10 7 kg by Hsieh et al. (2012). Both these teams however used the observations performed in V-filter, in contrast to our R-filter observations. Ishiguro et al. (2011) as well as Bodewits et al. (2011) obtained an order of magnitude larger mass of the dust escaped from Scheila. Some discrepancy can be caused from the different parameters used in the calculations. For example, the assumed value of albedo varied from 0.038 to 0.1, densities of the particles from 1670 to 2000 kg m À 3 , dust size from 0.1 μm to 10 cm, and assumed ejection velocity from 55 to 100 m s À 1 . When comparing the results, it is necessary to distinguish between the total "excavated" material and that "ejected away", into the interplanetary space. For example, Moreno et al. (2011) estimated the excavated dust mass of 2 Â 10 10 kg, which is much larger than the amount observed in the object's coma and, obviously, escaping into the interplanetary space. (The latter values are given in Table 3.) Bodewits et al. (2014) estimated that only about 1-10% of the excavated material escapes into the interplanetary space (and is photometrically observed). The rest of material falls back on the surface changing its optical properties. Potential meteoroid streams crossed Cometary streams Considering all known periodic comets from the Catalogue of Cometary Orbits (Marsden and Green, 2005), we calculated their approaches to Scheila's orbit. If there is such an approach within 0.15 AU (the maximum distance of the orbit of the cometary parent body of well-known meteor shower from the Earth's orbit; Neslušan et al., 1998) and comet produces a meteoroid stream, then we can suppose that Scheila passes through the corridor of this stream and can collide with its meteoroids. Taking into account the result by Larson et al. (2010) that Scheila was impacted on December 3, 2010, roughly, or perhaps a short time earlier, we regard December 3.5, 2010, as the latest possible date of the impact and are interested in the events 10 days before this critical time. In this context, we found two approaches of periodic comets, with the orbital period up to 1000 years, to Scheila's orbital arc corresponding to the time interval from ten days before until the latest supposed time of collision. Specifically, comets 127P and P/2005 K3 passed within minimum distances of 0.130 and 0.034 AU, respectively, to Scheila's orbit.(Within ten days after the critical time, comet P/2003 O2 passed in the minimum distance of 0.147 AU.) Asteroidal streams Although the asteroidal streams do not seem to be so numerous than cometary, some asteroids still produce their streams. This circumstance motivated us to search also for the potential asteroidal streams, the corridors of which could be crossed by Scheila. To perform this search, we used the MPCORB database 2 downloaded on June 11, 2015. It contained 686 093 orbits in total and 499 763 orbits determined from the observations covering the period of at least three oppositions and with the perihelion and aphelion of nominal orbit enabling the minimum distance to the Scheila's orbit less than 0.02 AU. We consider this limit in the case of asteroids, because the asteroidal streams are expected to be less dispersed than their cometary counterparts. The value of 0.02 AU is the minimum distance between the orbit of the Geminid parent, (3200) Phaethon, and the Earth's orbit. We found 108 approaches, within 0.02 AU, to the Scheila's orbital arc corresponding to its position from ten days before until the supposed time of collision. If we consider a shorter orbital arc of Scheila, which corresponds to its position from two days before until the time of collision, then there were 25 such approaches within 0.02 AU. Specifically, the orbits of minor planets Nos. 37257, 63520, 67241, 82009, 83469, 91906, B5596, H1994, P9337, R7003, S0122, Y1562, Y4649, Y7882, Z2557, b3473, c7996, d1643, d3266, d6361, f1223, K01U57W, K08R77R, K08SS4W, and K14D91K approached the above-mentioned Scheila's orbital arc. The number of 108 is a quite large. However, the asteroids (3200) Phaethon and 196 256 (2003 EH1), which are the parent bodies of two known asteroidal showers, Geminids and Quadrantids, respectively, move in the orbits with the perihelion distance lower than 0.2 AU (Phaethon) or the perihelion distance periodically changes and decreases below this limit during a certain evolutionary period (2003 EH1) (Neslušan et al., 2013). The proximity of the perihelion near the Sun causes a stress of the material in asteroidal surface layer. Breaking of the material due to the stress seems to be the mechanism leading to an occurrence of a meteoroid stream. Among the objects participating in the found 108 approaches to the Scheila's orbit, none has the perihelion distance shorter than 0.2 AU. A collision of Scheila with a meteoroid of asteroidal origin does not seem to be very probable. Probable boulder stream of Příbram and Neuschwanstein Two meteorites with the known orbit, Příbram falling on April 7, 1959(Ceplecha, 1961, and Neuschwanstein falling on April 6, 2002 (Oberst et al., 2004), moved in very similar orbits before colliding with our planet (Spurný et al., 2003). Most probably, these meteorites were the members of stream of larger boulders orbiting the Sun in a common corridor (Kornoš et al., 2008). Since the boulder stream is supposed to contain a less number of larger bodies, these do not collide with the Earth every year as the particles of a common meteoroid stream. Nevertheless, the case of Příbram and Neuschwanstein indicates that a collision is possible, whereby we can expect that the boulder stream of these two meteorites is not sole boulder stream in the interplanetary space. It is only the single known such stream at the present. We mean, the possibility of collision of Scheila with an object from a boulder stream should also be considered in our analysis. We calculated the approach of Scheila to the orbit of Příbram (Neuschwanstein). It occurred that Scheila actually approached this orbit to the relatively small minimum distance of 0.083 AU (0.149 AU). However, the approach became in a point far from the supposed site of collision. The boulder stream of Příbram cannot account for the Scheila's outburst. Close approaches of small bodies Likely, the most probable impactor on Scheila was a minor asteroidal object from the asteroid belt. And, likely, the impactor was not discovered and recorded to a database before the event. Nevertheless, to surely exclude this possibility or the possibility that it could be a satellite of a known binary asteroid, we calculate the minimum distance of known small objects from Scheila in the period from 10 days before to 10 days after the supposed time of collision on December 3.5, 2010. No extremely close approach of any periodic-comet (up to 1000 years) to Scheila in the supposed time of collision within 0.15 AU was found. Concerning the known, three-opposition asteroids, 109 (72, 33, 10, and 3) approaches within the distance of 0.15 AU (0.125, 0.10, 0.075, and 0.05 AU, respectively) were found. The closest approach was found for minor planet W8706. However, the minimum distance between this object and Scheila was 0.022 AU, too large the W8706 or even its potential satellite could be the candidate for the impactor. Future passages through the collisional point If the impactor was a boulder released in a not very distant past from a periodic comet or minor planet, which produces a meteoroid stream around its orbit, or was a member of boulder stream, then Scheila can obviously be impacted by another stream member when it will cross the corridor of the stream again. Of course, the probability of another impact of a larger object is extremely small. However, there is expected a large number of tiny particles in a typical stream, which could still impact the Scheila's surface and blast into the surrounding space a small, but possibly detectable amount of dust. Because of this possibility, we integrated Scheila's motion, considering the perturbations of all big planets, to the future and calculated the closest approaches to the supposed point of collision for three next orbital revolutions. These approaches will occur on (1) December 6.7, 2015 (JDT ¼2 457 363.2), (2) December 9.9, 2020 (TJD ¼2 459 193.4), and (3) December 16.0, 2025 (TJD ¼2 461 025.5). Fortunately, the observational conditions will be good in all three periods around these dates, especially from the northern hemisphere of the Earth. Results We found that: (1) The morphology of outburst of the asteroid changed during the period of our observation. (2) The radius of the asteroid is estimated to be 51.2 73.0 and 50.6 73.0 km. (3) The mass of dust ejecta is estimated to be ð2:5 À3:4Þ Â 10 7 kg. This estimate was obtained considering the different parameters of the size, density, and velocity of dust grains. (4) In the 10-day time interval before the latest estimated time of collision between the Scheila and an impactor, Scheila passed closely (within 0.15 AU) around the orbits of two periodic comets, 127P and P/2005 K3. If these comets produce their meteoroid streams, the impactor could be a member of one of these streams. (5) In the 10-day time interval before the collision, Scheila also passed closely (within 0.2 AU in this case) around the orbits of 108 other main-belt asteroids. However, no perihelion distance of these objects was short and, thus, any thermal stress of surface material leading to a production of potential meteoroid stream could be expected. Most probably, the impactor did not originate from an asteroidal stream. (6) In near future, Scheila will again pass near the collision point on December 6.7, 2015 (JDT¼2 457 363.2), December 9.9, 2020 (TJD¼2 459 193.4), and December 16.0, 2025 (TJD¼2 461 025.5). are 133P/Elst-Pizarro, 176P/LINEAR, 238P/Read, 259P/Garradd, P/2010 A2 (LINEAR), 324P/La Sagra, 2006 VW139, P/2012 F5 (Gibbs), P/2012 T1 (PANSTARRS), 311P/ PANSTARRS, and P/2013 R3 (Catalina-PANSTARRS). Fig. 2 . 2The Scheila's orbit projected in the ecliptic plane. The asterisks correspond to our observational dates from December 15, 2010 to January 9, 2011 and empty squares correspond to the dates from January 26, 2011 to May 22, 2011. P indicates the position of perihelion of the Scheila's orbit. The orbits of the terrestrial planets and Jupiter are also plotted for a comparison. , we obtained the asteroid effective radii 51.27 3.0 and 50.6 73.0 km, respectively. These values are close to estimations presented in the literature(Tedesco and Desert, 2002;Bauer et al., 2012;Licandro et al., 2011). n¼ 3.0(Moreno et al., 2011) and n ¼3.5 (Ishiguro et al., 2011). In calculation, we considered the visual geometric albedo of asteroid (Tedesco and Desert(2002)) and used the distribution with the lower and upper limits of dust grain radii taken at 0.8 μm and 5 cm(Moreno et al., 2011;Ishiguro et al., 2011), respectively. The outflow dust velocity ranged from 50 to 80 m s À 1(Moreno et al., 2011;Ishiguro et al., 2011). Our obtained results and results from the literature are presented in were given in the tabulated form by Penttilä et al. (2016) (or, the parameters necessary to evaluate relation Table 2 2The absolute magnitudes of asteroid (596) Scheila determined from its photometry. The used symbols are explained in the text.Observation time (UT) r (AU) Δ (AU) Phase angle (deg) m a mað1; 1; 0Þ Filters 11.12″ 66.72″ 11.12″ 66.72″ 2010 12 15.0 3.104 2.496 16.0 13.747 0.01 13.357 0.02 8.727 0.01 8.337 0.02 R 2010 12 16.1 3.102 2.482 15.8 13.687 0.01 13.32 7 0.01 8.677 0.01 8.317 0.01 R 2010 12 17.1 3.101 2.469 15.7 13.717 0.01 13.39 7 0.01 8.71 70.01 8.39 7 0.01 R 2010 12 27.9 3.085 2.338 13.7 13.34 70.02 13.067 0.07 8.44 70.01 8.167 0.07 R 2011 01 04.0 3.074 2.263 12.1 13.277 0.01 13.01 70.02 8.417 0.01 8.167 0.02 R 2011 01 06.0 3.071 2.244 11.6 13.25 70.01 13.08 7 0.01 8.417 0.01 8.25 7 0.01 R 2011 01 09.9 3.065 2.209 10.7 13.28 70.01 13.157 0.02 8.46 7 0.01 8.337 0.02 R 2011 01 26.8 3.040 2.105 7.0 13.377 0.01 - 8.57 70.01 - V 2011 01 28.1 3.037 2.097 6.7 13.337 0.01 - 8.54 7 0.01 - V 2011 02 01.1 3.029 2.082 6.2 13.277 0.01 - 8.487 0.01 - V 2011 03 08.9 2.982 2.133 11.6 13.80 70.01 - 9.147 0.01 - V 2011 03 22.7 2.952 2.272 16.1 13.99 70.01 - 9.29 7 0.01 - V 2011 03 23.7 2.943 2.318 17.1 13.88 70.01 - 9.167 0.01 - V 2011 04 02.9 2.933 2.377 18.0 14.017 0.01 - 9.25 7 0.01 - V 2011 04 22.8 2.900 2.594 20.0 14.34 70.01 - 9.45 7 0.01 - V 2011 05 06.8 2.877 2.753 20.4 14.36 70.01 - 9.36 7 0.01 - V 2011 05 08.8 2.873 2.776 20.4 14.43 70.01 - 9.40 7 0.01 - V 2011 05 22.8 2.850 2.932 20.0 14.31 7 0.01 - 9.197 0.01 - V Scheila. The result of other authors is also given for a comparison. The symbols are explained in the text.Observation time (UT) r (AU) Δ (AU) Phase angle (deg) Δm 10 4 C C (km 2 ) 10 7 M d (kg) 10 7 M d (kg) Filter Source n¼3.0 n¼ 3.5 2010 12 15.0 3.104 2.496 16.0 1.13 1.69 3.29 2.51 2.48 R This work 2010 12 16.1 3.102 2.482 15.8 1.09 1.60 3.21 2.49 2.44 R This work 2010 12 17.1 3.101 2.469 15.7 1.11 1.64 3.29 2.50 2.45 R This work 2010 12 27.9 3.085 2.338 13.7 1.07 1.55 3.11 2.52 2.51 R This work 2011 01 04.0 3.074 2.263 12.1 1.01 1.42 2.84 2.50 2.50 R This work 2011 01 06.0 3.071 2.244 11.6 0.94 1.27 2.55 2.50 2.50 R This work 2011 01 09.9 3.065 2.209 10.7 0.95 1.29 2.59 2.49 2.47 R This work 2010 12 27.9 3.085 2.338 13.7 1.26 2.2 4.4 - - V Jewitt et al. (2011) 2011 01 04.9 3.073 2.254 11.9 1.00 1.5 3.0 - - V 2010 12 12-2011 02 02 3.107-2.984 2.526-2.130 16.3-11.4 - - - - 15-49 Rc Ishiguro et al. (2011) 2010 12 14-2010 12 15 3.104 2.496 16.0 - 200 - - 60 V Bodewits et al. 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[ "DECOMPOSITION OF THE REGULAR REPRESENTATION FOR DIHEDRAL QUANDLES", "DECOMPOSITION OF THE REGULAR REPRESENTATION FOR DIHEDRAL QUANDLES" ]	[ "Mohamed Elhamdadi ", "Prasad Senesi ", "Emanuele Zappala " ]	[]	[]	We decompose the regular quandle representation of a dihedral quandle Rn into irreducible quandle subrepresentations.	null	[ "https://arxiv.org/pdf/2206.06311v1.pdf" ]	249,626,300	2206.06311	0482c7e8f7c225848cfe409452544ce4b15e9956	DECOMPOSITION OF THE REGULAR REPRESENTATION FOR DIHEDRAL QUANDLES 13 Jun 2022 Mohamed Elhamdadi Prasad Senesi Emanuele Zappala DECOMPOSITION OF THE REGULAR REPRESENTATION FOR DIHEDRAL QUANDLES 13 Jun 2022 We decompose the regular quandle representation of a dihedral quandle Rn into irreducible quandle subrepresentations. Introduction Let R n be the dihedral quandle of order n, and let CR n be the C-vector space of all C-valued functions on R n . Then CR n has the structure of a quandle representation of R n , which we call the regular representation of R n . In this note we provide, for all positive integers n, the decomposition of this regular representation into irreducible quandle subrepresentations. This problem was first addressed using the notion of a quandle ring in [2] in 2019, although the results given there are incomplete and do not address dihedral quandles of odd order. Our main result here was recently reported by one of the authors at the Knots in Washington Meeting on April 23, 2022, and will be included in a more comprehensive forthcoming publication on the representation theory of quandles. The authors would also like to note the recent arxiv submission [7], where these results are obtained with a different approach for the dihedral quandles of even order. Review of racks and quandles We give a quick review of racks and quandles in this section. Definition 2.1. [3,5,6] A rack is a set X provided with a binary operation ⊲ : X × X −→ X (x, y) −→ x ⊲ y such that (i) for all x, y ∈ X, there is a unique z ∈ X such that y = z ⊲ x; (ii) (right distributivity) for all x, y, z ∈ X, we have (x ⊲ y) ⊲ z = (x ⊲ z) ⊲ (y ⊲ z). For x ∈ X, we denote by R x : X → X the map given by R x (y) = y ⊲ x. Observe that property (i) also reads that for any fixed element x ∈ X, the map R x is a bijection. Also, notice that the right distributivity condition is equivalent to the relation R x (y ⊲ z) = R x (y) ⊲ R x (z) for all y, z ∈ X; that is, R x is an automorphism of the quandle X. Unless otherwise stated, we will always assume our racks (or quandles) to be finite racks (or quandles). Definition 2.2. A quandle is a rack such that x ⊲ x = x, ∀x ∈ X. As each R x permutes the elements of X, we can consider the subgroup R x x∈X in S X (the symmetric group on X) generated by the R x , which we denote by Inn(X) (the group of inner automorphisms of X). This group acts on the set X, and so the group action partitions X into distinct orbits. We say the quandle X is connected if Inn(X) acts transitively on X (and hence there is only one orbit). If (X, ⊲) and (Y, ⊲) are two quandles, a map f : X → Y is a quandle homomorphism if f (x ⊲ y) = f (x) ⊲ f (y) for all x, y ∈ X. Example 2.3. Any group G with the operation x ⊲ y = yxy −1 is a quandle, called conjugation quandle and denoted by Conj(G). Example 2.4. Any group G with the operation x ⊲ y = yx −1 y is a quandle, called the Core quandle of G and denoted Core(G). Example 2.5. Let G be a group and f ∈ Aut(G). Then one can define a quandle structure on G by x ⊲ y = f (xy −1 )y. It is called a generalized Alexander quandle. If G is abelian, the operation becomes x ⊲ y = f (x) + (Id − f )(y), where Id stands for the identity map. This quandle is called an Alexander quandle. Example 2.6. The set R n = {1, 2, . . . , n}, with quandle operation x ⊲ y = 2y − x (mod n) is a quandle, called a dihedral quandle. This is an Alexander quandle on the the group Z n where f is multiplication by −1. For the remainder of this manuscript, R n will denote the dihedral quandle of order n. Representation theory As mentioned in the introduction, the results here will be included in a later manuscript providing more results for the (complex) representation theory of (finite) quandles. The terminology used here to describe this theory is adapted from group representation theory, a topic for which there is a wealth of literature. There are many accessible introductions to this topic; see, for example, Chapters 4 and 5 of [4]. All vector spaces henceforth are defined over C. For any positive integer k, we set ω k = e 2πi/k . A representation of a finite quandle (X, * ) on a finite dimensional complex vector space V is a quandle homomorphism ρ : X → Conj(GL(V )). In other words, for all x, y ∈ X, we have ρ(x * y) = ρ(y)ρ(x)ρ(y) −1 . To simplify the notation, we will denote ρ(x) by just ρ x . Let V and W be two representations of a quandle X. If ρ V : X → Conj(GL(V )) and ρ W : X → Conj(GL(W )) are two representations of X, then a linear map φ : V → W for which the following square commutes V φ / / ρ V x W ρ W x V φ / / W. for all x is called an intertwiner. If, furthermore, φ is an isomorphism then we say that the two representations are isomorphic as representations of X, or equivalent. If φ : V ֒→ W is the inclusion of the linear subspace V into W , then we say that V is a subrepresentation of W . Equivalently, a subspace U of W is a subrepresentation of W if U is invariant under ρ W (X). A representation V of a quandle X is called irreducible if its only quandle subrepresentations are {0} and V . A representation ia called indecompasable if it cannot be written as direct sum of nontrivial subrepresentations. If φ : Inn(X) → Aut(V ) is a group homomorphism, then the induced map φ : X → Conj(Aut(V )), φ(x) := φ(R x ), is a quandle homomorphism. Therefore any group representation of Inn(X) induces a quandle representation of X. The regular representation of a quandle For any finite quandle X, we denote by CX the C-vector space of C-valued functions on X; equivalently it is the C-vector space generated by basis vectors {e x } x∈X , whose elements are formal sums f = x∈X a x e x , a x ∈ C. The regular representation of X is λ : X → Conj(GL(CX)), where λ t (f )(x) := f (R −1 t (x) ). This action of X is equivalent to the right X-action on CX given by the linear extension of e x · t = e Rt(x) = e x * t . We note that a subspace W of CX is a quandle subrepresentation of CX if and only if W is a subrepresentation of CX as a group representation of Inn(X). If a distinction is necessary, we will refer to a vector space as either a quandle representation or a group representation. If two vector spaces U, V are isomorphic as quandle representations of X, we will write U ∼ = X V ; if they are isomorphic as group representations of some group G, we will write U ∼ = G V . When \|X\| ≥ 2, The regular representation always has two nontrivial subrepresentations, one of which is always irreducible: the one-dimensional subspace C1, where 1 is the constant function 1(x) = 1, for all x ∈ X. Its vector space complement, (C1) ⊥ , is also a subrepresentation, since (C1) ⊥ = x∈X a x e x \| a x = 0 , and action by any t ∈ X only permutes the coefficients a x . So CX decomposes into a direct sum of subrepresentations CX = C1 ⊕ (C1) ⊥ . For x, y ∈ X let v xy = e x − e y . If we enumerate X = {x 1 , . . . , x n }, then a basis for (C1) ⊥ is v x 1 x 2 , v x 2 x 3 , . . . , v x n−1 xn . The regular representation of a dihedral quandle The rest of this paper is devoted to the regular quandle representation of a dihedral quandle R n . We first explicitly describe some quandle representations of R n . Since quandle subrepresentations of CR n are in correspondence with group subrepresentations of CR n (under the action of Inn(R n )), we obtain the full decomposition of CR n as a quandle representation of R n by finding the decomposition as a group representation of Inn(R n ). We note that this correspondence between group and quandle representations does not hold in general, but it is characteristic of the regular representation. In general, there is no reason to expect any correspondence or similarity between quandle representations of a quandle X and group representations of Inn(X). By virtue of the aforementioned correspondence, and the fact that the group Inn(R n ) is a dihedral group, we will consider in detail the representation theory of dihedral groups. We will let D m be the dihedral group of order 2m. Then it is shown in [1] that Inn(R n ) ∼ = D n/2 , n even, D n , n odd. We describe the (isomorphism classes of) finite-dimensional irreducible group representations of the dihedral group D m here, to facilitate statement of results below. The dihedral group D m has a presentation given by D m = α, β \| \|α\| = \|β\| = 2, \|αβ\| = m . We fix generators α, β for this presentation. The classification of the finite-dimensional irreducible group representations of D m falls into two cases: m even and m odd. If λ, µ ∈ C, we will denote by C(λ, µ) the one-dimensional group representation of D m on which α and β act by scalar multiplication by λ and µ, respectively. And for nonzero λ ∈ C, we will denote by W (λ) the two-dimensional group representation of D m for which the matrix representations of α and β are α → 0 1 1 0 , β → 0 λ λ −1 0 . With this notation fixed, we provide the classification of the finite-dimensional irreducible group representations of D m . All such group representations are either 1-or 2-dimensional, and are listed here: D m m even m odd 1-dimensional C(1, 1), C(1, −1), group representations C(−1, 1), C(−1, −1) C(1, 1), C(−1, −1) 2-dimensional W (ω s m ) , W (ω s m ) , group representations 1 ≤ s ≤ m/2 − 1 1 ≤ s ≤ (m − 1)/2 We will denote by Γ(D m ) the set of (isomorphism classes of) finite-dimensional irreducible group representations of D m . For example, D 8 has 4 representations of dimension 1 and 4 irreducible representations of dimension 2, and they are Γ(D 8 ) = C(±1, ±1), C(±1, ∓1), W (ω 8 ), W (ω 2 8 ), W (ω 3 8 ) , while D 9 has 2 representations of dimension 1 and 3 irreducible representations of dimension 2, and they are Γ(D 9 ) = C(±1, ±1), W (ω 9 ), W (ω 2 9 ), W (ω 3 9 ), W (ω 4 9 ) . Since Inn(R n ) is a dihedral group generated by R 1 and R 2 , these group representations also provide us with examples of 1-dimensional and 2-dimensional irreducible quandle representations of R n . We will make this precise as follows: let C(λ, µ) be the 1-dimensional representation of Inn(R n ) for which R 1 and R 2 act by scalar multiplication by λ and µ, respectively, and let W (λ) be the two-dimensional representation of Inn(R n ) for which R 1 and R 2 have matrix representations R 1 → 0 1 1 0 , R 2 → 0 λ λ −1 0 . Now we consider the regular quandle representation CR n of R n and the subrepresentation (C1) ⊥ , in cases n even or n odd. In either case, (C1) ⊥ is spanned by {v ij } 1≤i,j≤n . If n = 2r is even, R n has two orbits: the even orbit {2, 4, . . . , 2r}, and the odd orbit We call Φ n,0 and Φ n,1 the even and odd subspaces, respectively, of (C1) ⊥ . For i = 0, 1, ∆ n,i is a basis for Φ n,i . If n is odd, R n has a single orbit. In this case, we set Φ n = sp {v ij } i =j , ∆ n = {v 12 , v 23 , . . . , v n−1 n } , and ∆ n is a basis for Φ n . Because the orbits of R n are as described above, all of the subspaces Φ n,i , Φ n are quandle subrepresentations of CR n . Theorem 5.1. If n is even, then the decomposition of the regular representation CR n of R n into irreducible quandle subrepresentations is, for n ≡ 0 mod 4, CR n ∼ = Rn C(1, 1) ⊕2 ⊕ C(±1, ∓1) n/4−1 s=1 W (ω s n/2 ) ⊕2 , and for n ≡ 2 mod 4, CR n ∼ = Rn C(1, 1) ⊕2 (n−2)/4 s=1 W (ω s n/2 ) ⊕2 . If n is odd, then the decomposition of CR n into irreducible quandle subrepresentations of R n is CR n ∼ = Rn C(1, 1) (n−1)/2 s=1 W (ω 2s n ). Proof. For arbitrary n, let1 = n i=1 (−1) i e i ∈ (C1) ⊥ . Then, as already pointed out before, CR n decomposes as CR n = C1 ⊕ (C1) ⊥ , and C1 ∼ = Rn C1 ∼ = Rn C(1, 1). We proceed distinguishing the cases n even or n odd. Case 1: n is even. Let n = 2r. In this case, each Φ n,i (for i = 0, 1) is a subrepresentation of CR n of dimension r − 1, and (C1) ⊥ decomposes into R n -subrepresentations as (C1) ⊥ = C1 ⊕ Φ n,0 ⊕ Φ n,1 . Now set [a 1 , . . . , a r−1 ] 0 = r−1 i=1 a i v 2i,2i+2 (the coordinate vector in the basis ∆ n,0 ). The right multiplication operators R 1 and R 2 together generate all of Inn(R n ) ∼ = D n/2 , and with respect to the basis ∆ n,0 , the matrix representations of these operators on Φ n,0 are [R 1 ] ∆ n,0 =        0 0 · · · 0 −1 0 0 · · · −1 0 . . . . . . . . . . . . . . . 0 −1 · · · 0 0 −1 0 · · · 0 0        , [R 2 ] ∆ n,0 =        1 0 · · · 0 0 1 0 · · · 0 −1 . . . . . . . . . . . . . . . 1 0 · · · 0 0 1 −1 · · · 0 0        .(1) For example, for n = 12, we have R 1 ([a 1 , a 2 , a 3 , a 4 , a 5 ] 0 ) = [−a 5 , −a 4 , −a 3 , −a 2 , −a 1 ] 0 , R 2 ([a 1 , a 2 , a 3 , a 4 , a 5 ] 0 ) = [a 1 , a 1 − a 5 , a 1 − a 4 , a 1 − a 3 , a 1 − a 2 ] 0 . For 1 ≤ s ≤ r − 1, let u s := 1 − ω s r , 1 − (ω s r ) 2 , · · · , 1 − (ω s r ) r−1 0 = r−1 i=1 1 − (ω s r ) i v 2i,2i+2 , and v s := R 1 (u s ). Then v s = r−1 i=1 (ω s r ) r−i − 1 v 2i,2i+2 , and R 2 (u s ) = r−1 i=1 (ω s r ) r+1−i − ω s r v 2i,2i+2 = ω s r r−1 i=1 (ω s r ) r−i − 1 v 2i,2i+2 = ω s r v s . Since all R i are involutions, we also have R 2 (v s ) = (ω −1 r ) s u s . Let U s,0 be the subspace of Φ n,0 spanned by u s and v s . The above identities show that each U s,0 is a quandle subrepresentation of Φ n,0 . If u s and v s are linearly dependent, then U s,0 is a 1-dimensional quandle subrepresentation. And if u s and v s are linearly independent, the matrix representations of R 1 and R 2 with respect to the basis {u s , v s } are [R 1 ] {us,vs} = 0 1 1 0 , [R 2 ] {us,vs} = 0 ω s r ω −s r 0 , which gives us U s,0 ∼ = W (ω s r ) (both an R n -quandle and an Inn(R n )-group isomorphism). From the identity (ω r−s r ) ℓ = (ω s r ) −ℓ (for any integers ℓ, s), we obtain u s = −v r−s , for 1 ≤ s ≤ r − 1. Therefore U s,0 = U r−s,0 , 1 ≤ s ≤ r − 1. And since each W (ω s r ) is an irreducible group representation for Inn(R n ) for 1 ≤ s ≤ r − 1, it follows that U s,0 ∼ = Rn U v,0 if and only if s = v, for 1 ≤ s, v ≤ ⌊r/2⌋, and U s,0 ∩ U v,0 = {0} for 1 ≤ s = v ≤ ⌊r/2⌋. We summarize these results here: For r even, U ℓ,0 ∩ U s,0 = {0} for 1 ≤ ℓ = s ≤ r/2, and dim(U s,0 ) = 2, 1 ≤ s < r/2 1, s = r/2 . For r odd, U ℓ,0 ∩ U s,0 = {0} for 1 ≤ ℓ = s ≤ (r − 1)/2, and dim(U s,0 ) = 2 for all s. Hence we obtain a decomposition of Φ n,0 into irreducible quandle subrepresentations: Φ n,0 ∼ = Rn ⌊r/2⌋ s=1 U s,0 ∼ = Rn    W (ω r ) ⊕ · · · ⊕ W ω r/2−1 r ⊕ C(−1, 1), r even, W (ω r ) ⊕ · · · ⊕ W ω (r−1)/2 r , r odd. The decomposition of Φ n,1 is achieved in an similar fashion -with a few small changes. We first replace R 1 , R 2 with R n/2 and R 1 , respectively. The matrix representations of these transformations, with respect to the basis ∆ n,1 , are given by R n/2 ∆ n,1 =        0 0 · · · 0 −1 0 0 · · · −1 0 . . . . . . . . . . . . . . . 0 −1 · · · 0 0 −1 0 · · · 0 0        , [R 1 ] ∆ n,1 =        1 0 · · · 0 0 1 0 · · · 0 −1 . . . . . . . . . . . . . . . 1 0 · · · 0 0 1 −1 · · · 0 0        .(2) Then we define u s := 1 − ω s r , 1 − (ω s r ) 2 , · · · , 1 − (ω s r ) r−1 1 = r−1 i=1 1 − (ω s r ) i v 2i−1,2i+1 , and v s := ω −s r R n/2 (u s ). Then we can show that R 1 (u s ) = v s , and since For r odd (n ≡ 2 mod 4), the one-dimensional quandle subrepresentations of CR n are: R 2 = R n/2+2 = R n/2 ⊲1 = R 1 R n/2 R −1 1 = R 1 R n/2 R 1 , we have R 2 (u s ) = R 1 R n/2 R 1 (u s ) = ω s r v s . For 1 ≤ s ≤ r − 1, C1 ∼ = C(1, 1) and C1 ∼ = C(1, 1), and the two-dimensional irreducible quandle subrepresentations are U s,0 ∼ = U s,1 ∼ = W (ω s r ), 1 ≤ s ≤ (r − 1) / 2. Case 2: n odd. Let n = 2r + 1. With respect to the basis ∆ n = {v 12 , v 23 , . . . , v n−1,n }, the elements R 1 , R 2 ∈ Inn(R n ) have matrix representations [R 1 ] ∆n =        0 · · · 0 −1 1 0 · · · −1 0 1 . . . . . . . . . . . . . . . −1 0 · · · 0 1 0 0 · · · 0 1        , [R 2 ] ∆n =        1 0 · · · 0 0 1 0 · · · 0 −1 . . . . . . . . . . . . . . . 1 0 · · · 0 0 1 −1 · · · 0 0        . For 1 ≤ s ≤ n − 1, let u s = 1 − ω s n , 1 − (ω s n ) 2 , · · · , 1 − (ω s n ) n−1 , and v s = R 1 (u s ). We can then show that v s = ω −s n ω −s n − 1, (ω −s n ) 2 − 1, . . . , (ω −s n ) n−1 − 1 . Let U s be the subspace of Φ n spanned by u s and v s . We can show that u s = v n−s , hence U s = U n−s . Furthermore, we have R 2 (u s ) = R 2 (1 − ω s n , 1 − (ω s n ) 2 , . . . , 1 − (ω s n ) n−1 ) = (1 − ω s n , (ω s n ) n−1 − ω s n , (ω s n ) n−2 − ω s n , . . . , (ω s n ) 2 − ω s n ) = ω 2s n ω −s n (ω −s n − 1, (ω −s n ) 2 − 1, . . . , (ω −s n ) n−1 − 1) = ω 2s n v s . Therefore, if u s and v s are linearly independent, the matrix representations of R 1 and R 2 with respect to the basis {u s , v s } of U s are again [R 1 ] {us,vs} = 0 1 1 0 , [R 2 ] {us,vs} = 0 ω 2s n ω −2s n 0 , We can show linear independence of u s and v s for 1 ≤ s ≤ r, which gives us the decomposition (C1) ⊥ = r s=1 U s , and we obtain the desired decomposition into irreducible representations CR n ∼ = C(1, 1) r s=1 W (ω 2s n ). Examples. The decomposition of CR n given in Theorem 5.1 falls into 3 cases: n ≡ 0 mod 4, n ≡ 2 mod 4, and n odd. We provide three examples below, one for each of these cases, wherein we provide the decomposition and all irreducible subrepresentations of CR 10 , CR 11 , and CR 12 : Regular quandle representation of R 10 : CR 10 = C1 ⊕ (C1) ⊥ C1 ⊕ U 1,0 ⊕ U 2,0 Φ 10,0 ⊕ U 1,1 ⊕ U 2,1 Φ 10,1 subrep of CR 10 irrep of D 5 generated by dimension C1 ∼ = C(1, 1) 1 1 C1 ∼ = C(1, 1)1 1 U 1,0 ∼ = W (ω 5 ) 4 i=1 1 − ω i 5 v 2i,2i+2 2 U 2,0 ∼ = W (ω 2 5 ) 4 i=1 1 − ω 2i 5 v 2i,2i+2 2 U 1,1 ∼ = W (ω 5 ) 4 i=1 1 − ω i 5 v 2i−1,2i+1 2 U 2,1 ∼ = W (ω 2 5 ) 4 i=1 1 − ω 2i 5 v 2i−1,2i+1 2 Regular quandle representation of R 11 : CR 11 = C1 ⊕ (C1) ⊥ U 1 ⊕ U 2 ⊕ U 3 ⊕ U 4 ⊕ U 5 subrep of CR 11 irrep of D 11 generated by dimension C1 ∼ = C(1, 1) 1 1 U 1 ∼ = W (ω 2 11 ) 8 i=1 1 − ω 2i 11 e i 2 U 2 ∼ = W (ω 4 11 ) 8 i=1 1 − ω 4i 11 e i 2 U 3 ∼ = W (ω 6 11 ) 8 i=1 1 − ω 6i 11 e i 2 U 4 ∼ = W (ω 8 11 ) 8 i=1 1 − ω 8i 11 e i 2 U 5 ∼ = W (ω 10 11 ) 8 i=1 1 − ω 10i 11 e i 2 Regular quandle representation of R 12 : CR 12 = C1 ⊕ (C1) ⊥ C1 ⊕ U 1,0 ⊕ U 2,0 ⊕ U 3,0 Φ 12,0 ⊕ U 1,1 ⊕ U 2,1 ⊕ U 3,1 Φ 12,1 subrep of CR 12 irrep of D 6 generated by dimension C1 ∼ = C(1, 1) 1 1 C1 ∼ = C(1, 1)1 1 U 1,0 ∼ = W (ω 6 ) 5 i=1 1 − ω i 6 v 2i,2i+2 2 U 2,0 ∼ = W (ω 2 6 ) {1, 3, . . . , 2r − 1}. Let Φ n,0 = sp {v ij } i =j even , ∆ n,0 = {v 24 , v 46 , . . . , v 2r−2 2r } , and Φ n,1 = sp {v ij } i =j odd , ∆ n,1 = {v 13 , v 35 , . . . , v 2r−3 2r−1 } . we define U s,1 to be the subspace of Φ n,1 spanned by u s and v s . Then, as above, we have dim(U s,1 ) = 1, s = r/2, r even 2, s = r/2, and for s = r/2, the matrix representations of R 1 and R 2 are [R 1 ] {us,even (n ≡ 0 mod 4), the one-dimensional quandle subrepresentations of CR n are:C1 ∼ = C(1, 1), C1 ∼ = C(1, 1), U r/2,0 ∼ = C(−1, 1), U r/2,1 ∼ = C(1, −1), and the two-dimensional irreducible quandle subrepresentations are U s,0 ∼ = U s,1 ∼ = W (ω s r ), 1 ≤ s ≤ r/2 − 1. Automorphism groups of quandles. Mohamed Elhamdadi, Jennifer Macquarrie, Ricardo Restrepo, 10.1142/S0219498812500089.MR2900878J. Algebra Appl. 1119Mohamed Elhamdadi, Jennifer Macquarrie, and Ricardo Restrepo, Automorphism groups of quandles, J. Algebra Appl. 11 (2012), no. 1, 1250008, 9, DOI 10.1142/S0219498812500089. MR2900878 Ring theoretic aspects of quandles. Mohamed Elhamdadi, Neranga Fernando, Boris Tsvelikhovskiy, J. Algebra. 526153915329Mohamed Elhamdadi, Neranga Fernando, and Boris Tsvelikhovskiy, Ring theoretic aspects of quandles, J. Algebra 526 (2019), no. 15, 166-187. MR3915329 Quandles-an introduction to the algebra of knots. Mohamed Elhamdadi, Sam Nelson, Student Mathematical Library. 743379534American Mathematical SocietyMohamed Elhamdadi and Sam Nelson, Quandles-an introduction to the algebra of knots, Student Mathematical Library, vol. 74, American Mathematical Society, Providence, RI, 2015. MR3379534 Introduction to representation theory. Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina, Student Mathematical Library. Slava Gerovitch. MR280816059American Mathematical SocietyPavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina, Introduction to representation theory, Student Mathematical Library, vol. 59, American Mathematical Society, Providence, RI, 2011. With historical interludes by Slava Gerovitch. MR2808160 A classifying invariant of knots, the knot quandle. D Joyce, DOI10.1016/0022-4049(82)90077-9.MR638121J. Pure Appl. Algebra. 231D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), no. 1, 37-65, DOI 10.1016/0022-4049(82)90077-9. MR638121 Distributive groupoids in knot theory. S V Matveev, Mat. Sb. (N.S.). 119161672410RussianS. V. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.) 119(161) (1982), no. 1, 78-88, 160 (Russian). MR672410 Dilpreet Kaur, Pushpendra Singh, arXiv:2206.03162Decomposition of quandle rings of dihedral quandles. Dilpreet Kaur and Pushpendra Singh, Decomposition of quandle rings of dihedral quandles, arXiv:2206.03162, 2022.	[]
[ "Electric-circuit realization of non-Hermitian higher-order topological systems", "Electric-circuit realization of non-Hermitian higher-order topological systems" ]	[ "Motohiko Ezawa \nDepartment of Applied Physics\nUniversity of Tokyo\nHongo 7-3-1113-8656Japan\n" ]	[ "Department of Applied Physics\nUniversity of Tokyo\nHongo 7-3-1113-8656Japan" ]	[]	Topological phases are characterized by the bulk topological number, and signaled by the emergence of zeroenergy boundary modes. LC electric circuits appropriately designed are known to have topological phases together with topological phase transitions. We show that LCR circuits realize them in non-Hermitian topological systems. In particular, higher-order topological phases are obtained in the anisotropic honeycomb and diamond circuits. Topological phase transitions, being induced by tuning variable capacitors, are clearly observable by measuring the impedance resonance due to zero-admittance corner modes. Remarkably, only the topological resonance peaks remain prominent in non-Hermitian systems, since all other resonances are suppressed by resistors. arXiv:1810.04527v1 [cond-mat.mes-hall]	10.1103/physrevb.99.121411	[ "https://arxiv.org/pdf/1810.04527v1.pdf" ]	53,125,589	1810.04527	040d1db11fb880db3bffccd9c5397b9eb47ddd40	Electric-circuit realization of non-Hermitian higher-order topological systems Motohiko Ezawa Department of Applied Physics University of Tokyo Hongo 7-3-1113-8656Japan Electric-circuit realization of non-Hermitian higher-order topological systems Topological phases are characterized by the bulk topological number, and signaled by the emergence of zeroenergy boundary modes. LC electric circuits appropriately designed are known to have topological phases together with topological phase transitions. We show that LCR circuits realize them in non-Hermitian topological systems. In particular, higher-order topological phases are obtained in the anisotropic honeycomb and diamond circuits. Topological phase transitions, being induced by tuning variable capacitors, are clearly observable by measuring the impedance resonance due to zero-admittance corner modes. Remarkably, only the topological resonance peaks remain prominent in non-Hermitian systems, since all other resonances are suppressed by resistors. arXiv:1810.04527v1 [cond-mat.mes-hall] I. INTRODUCTION Topological physics is one of the most important concepts in contemporary physics, among which topological insulators and its generalization to higher-order topological insulators [1][2][3][4][5][6][7][8][9][10][11][12] are fascinating. They are characterized by the bulk topological numbers, where the bulk-boundary correspondence and its generalization play a key role. In particular, topological zero-energy corner modes emerge for the secondorder topological insulators in two dimensions and for the third-order topological insulators in three dimensions. They are robust against impurities. They have so far been studied mainly in fermionic systems in materials. However, since topological features are based solely on the homotopy of the linear algebra inherent to the system, various systems such as photonic [13][14][15] , phononic [16][17][18][19][20] and microwave 21,22 systems share the same topological properties. LC electric circuits have also topological phases [23][24][25][26][27][28] . As great merits, electric circuits are easily designed to yield topological phases, and furthermore, topological phase transitions are induced simply by controlling variable capacitors. Topological zero-energy boundary modes are observed as zero-admittance boundary modes. Recently, non-Hermitian topological systems attract increasing attentions. Although quantum mechanics should be Hermitian, the dissipation effects can be simulated by introducing non-Hermitian terms into the system. They are realized in photonic systems [29][30][31][32] , microwave resonators 33 , wave guides 34 , quantum walks 35,36 and cavity systems 37 . In the parity-time-reversal (P T ) symmetric non-Hermitian systems [38][39][40][41] , the bulk energy remains to be real. On the other hand, in the chiral-symmetric non-Hermitian systems [42][43][44][45][46][47] , the bulk energy becomes complex in general. The non-Hermitian Su-Scrieffer-Heeger (SSH) model is the most studied example 30,32,33,48,49 . The Hermitian SSH model has been generalized to higher dimensions 2,[4][5][6]9,50 . Especially, They are realized in the anisotropic honeycomb and diamond lattice models 50 . These models realize higher-order topological phases, which are characterized by the emergence of topological zero-energy corner modes. A natural question is whether there is a non-Hermitian counter part of higher-order topological phases and how to realize them in electric circuits. In this work, we show that LCR electric circuits present a concrete playground to investigate non-Hermitian topological physics, where resistors naturally lead to non-Hermitian terms by way of the Joule heating energy dissipation. We focus on the chiral symmetric topological electric circuits. We find that zero-admittance modes in the absence of resistors remain as they are even when non-Hermitian terms are introduced by resistors although the bulk admittance becomes complex. Furthermore, in the presence of resistors, all impedance resonances are drastically suppressed except for the topological zero-admittance modes. It will be easy to perform experimental observations of these predictions electrically since we have variable resistors and capacitors. We explicitly investigate topological properties in the SSH, anisotropic honeycomb and diamond LCR circuits as non-Hermitian generalizations, where the ordinary, second-order and third-order topological systems are realized. II. RESULTS A. Kirchhoff's law and non-Hermitian systems We investigate a class of electric circuits, where each node a is connected to the ground via the inductance L. See examples in Fig.1. Let I a be the current between node a and the ground via the inductance, V a be the voltage at node a, C ab and R ab be the capacitance and the resistance between nodes a and b, respectively. When we apply an AC voltage V (t) = V (0) e iωt , the Kirchhoff's current law reads 23,24 I a (ω) = b J ab (ω) V b (ω) ,(1) where the sum is taken over all adjacent nodes b, and J ab (ω) is the circuit Laplacian, J ab (ω) = iωδ ab   − 1 ω 2 L + c =a 1 1/C ac + iωR ac   − iωH ab (ω) ,(2) where H ab (ω) = 1 1/C ab + iωR ab(3) for a = b and H aa (ω) = 0. An important observation is that we are able to regard H ab (ω) as the tight-binding Hamiltonian of a lattice system in condensed-matter physics, where the transfer integral between adjacent sites a and b is given by t ab = C ab / (1 + iωC ab R ab ) .(4) Accordingly we may examine various topological concepts by using electric circuits. We are particularly interested in the topological boundary modes associated with the bulkboundary correspondence in non-Hermitian topological systems. Note that the Hamiltonian H ab (ω) is naturally non-Hermitian due to the Joule heating loss by resistors. We analyze an electric circuit forming a lattice structure. A lattice is constructed by translating a unit cell repeatedly, which allows us to define the crystal momentum together with the Brilloin zone even in electric circuits. Bipertite circuits: To make the idea as concrete as possible, we focus on a bipertite electric circuit, where a unit cell contains only two nodes. As we illustrate in Fig.1, a unit electric circuit consists of two nodes A and B, two capacitors C A and C B , and two resistors R A and R B . In addition, we require the chiral symmetry σ z satisfying {H (k) , σ z } = 0, which assures the symmetric spectrum E ↔ −E. Topological numbers: The generic non-Hermitian bipertite Hamiltonian with chiral symmetry is given by H (k) = 0 h 1 (k) h 2 (k) 0 .(5) It allows us to define two winding numbers 51 , Hence the system is characterized 45,52 by Z ⊕ Z. They define the bulk topological numbers. In our cases, we find W 2 = −W 1 and the topological number is given by one wind- W α = π −π dk 2πi ∂ k log h α .(6)ing number W = (W 2 − W 1 ) /2 = W 2 . Admittance spectrum: The admittance spectrum consists of the eigenvalues of the circuit Laplacian [23][24][25][26][27][28] . It corresponds to the band structure in condensed-matter physics: See an instance in Fig.2. The emergence of the zero-admittance modes is the signal of the bulk topology owing to the bulk-boundary correspondence. Impedance: A measurable quantity of electric circuits is the two-point impedance, which is given by 23,24 Z ab = V a − V b I ab = G aa + G bb − G ab − G ba ,(7) where G is the green function defined by the inverse of the Laplacian G ≡ J −1 . It diverges at the frequency satisfying J = 0. Especially, it is possible to detect the topological zero-admittance modes by measuring the divergence of the impedance. After the diagonalization, the circuit Laplacian has the form J n (ω) = iω   − 1 ω 2 L + α=A,B n α C α 1 + iωC α R α   − iωε n (ω) ,(8) where n α is the number of the nodes adjacent to node α, and ε n is the eigen-modes of the circuit Laplacian. When R α = 0, the impedance diverges at the resonance frequencies ω R (ε n ) = −ε n + α n α C α L ,(9) which is the solution of J n (ω) = 0. Especially, we find a resonance at the zero-admittance mode (ε 0 = 0) which is a topological impedance resonance. Indeed, the emergence of the zero-admittance modes is a requisite of the topological phase. On the other hand, all other resonances are trivial in the sense that they have no topological origin. However, it is nontrivial to differentiate these resonances unless we know the value of the resonance frequency. We shall soon see that only the topological resonances survive in the presence of resistors (R α = 0), where the imaginary part of ε n introduced by resistors suppresses trivial resonances. Actually, a resonance does not always occur at all resonance frequencies given by (9). For instance, although both solid and dotted curves are solutions of (9) in Fig.3(a), actual resonances appear only on solid curves in Fig.3(b). It is due to the exact cancellation between the contributions from G aa + G bb and G ab + G ba in (7). B. Non-Hermitian SSH circuits with chiral symmetry The simplest model of the non-Hermitian topological system is the SSH model 30,32,33,48,49 . The P T symmetric SSH model is especially well studied, where the gain and loss are balanced 30,44 . However, it is nontrivial to construct this model in passive electric circuits consisting only capacitors, inductors and resistors due to the absence of gains. On the other hand, the chiral symmetric SSH model is adequate for electric circuits. We consider a circuit shown in Fig.1(a), where capacitors and resistors are connected directly, and all nodes are grounded by inductors. The circuit Laplacian is written in the form J ab (ω) = iωδ ab   − 1 ω 2 L + α=A,B C α 1 + iωC α R α   − iωH ab (ω) ,(10) with the Hamiltonian H = 0 t A + t B e −ik t A + t B e ik 0 .(11) Here, t A = C A 1 + iωC A R A , t B = C B 1 + iωC B R B(12) are complex numbers. It is reduced to the normal SSH model when t A and t B are real, i.e., R A = R B = 0. Otherwise, the Hamiltonian is non-Hermitian. By evaluating the winding number (6), we find that the system is topological (W = 1) for \|t A /t B \| < 1 and trivial (W = 0) for \|t A /t B \| > 1. We show the admittance spectrum as a function of t A /t B in Fig.2. First we calculate the spectrum without resistors, where the Hamiltonian is Hermitian. There appear zero-admittance modes for \|t A /t B \| < 1, indicating that the system is in the topological phase. When resistors are included, the admittance of the bulk becomes complex but the zero-admittance modes remains as they are. Next we study a finite chain containing L unit cell made of the SSH circuit. We show the two-point impedance between the two outermost edge nodes in Fig.3. First, we note that the frequency formula (9) explains very well the numerical results of resonances: See Fig.3(a) and (b). Next, there appear many resonant peaks without resistors (R A = R B = 0) both in the trivial and topological phases, as in Fig.3(c) and (e). Especially, there is a topological resonance at ω 0 = 1/ L (C A + C B ). Nevertheless, it is actually difficult to distinguish these two phases by the emergence of large resonant peaks. However, once resistors are included, all resonance peaks are washed away except for the resonance peak in the topological phase, as in Fig.3 (d) and (f). Topological stability against randomness: We study the effects of the randomness of the capacitors and resistors. For this purpose, we make substitution C α → C α (1 + η α ) and R α → R α (1 + ξ α ), where η α and ξ α are uniformly distributed random variables ranging from −δ to δ. We have calculated the admittance spectrum by choosing δ = 0.2. We find the zero-admittance modes survive even in the presence of the randomness whether resistors exist or not: See Fig.3(g) and (h). This is because there is always zero-admittance solution for a finite chain due to its edges even in the presence of the randomness. See Methods B. C. Non-Hermitian honeycomb circuits Next we generalize our investigation to electric circuits corresponding to two-dimensional lattices. A typical example is the anisotropic honeycomb circuit consisting of two types of the capacitors and resistors, as in Fig.1(b). The anisotropic honeycomb lattice model is known to become a second-order topological insulator, where topological corner modes emerge 50 . The circuit Laplacian is written in the form J ab (ω) = iωδ ab − 1 ω 2 L + 2C A 1 + iωC A R A + C B 1 + iωC B R B − iωH ab (ω) ,(13) with the Hamiltonian H = 0 h 1 h 2 0 ,(14) and Second-order topological circuits: It has been shown that 50 the system is topological for \|t A /t B \| < 1/2 in the absence of resistors. It can be easily shown that the same result is derived even in the presence of resistors. We calculate the impedance of the rhombus geometry of the anisotropic honeycomb lattice with \|t A /t B \| < 1/2, where the second-order topological phase is realized and topological corner modes emerge at two corners [See Fig.1(c)]. We fix one node a in the vicinity of the center of the rhombus, and measure the impedance Z ab between node a and another node b. By moving b over all nodes, we obtain a space distribution of the two-point impedance. We show the results in the topological phase in Fig.1(c) and (d). When one node is fixed at node A or B, a prominent peak emerges at corner B or A. This is a peculiar property of the bipertite circuit structure that the zero-admittance eigenmodes are localized only at one subcircuit. They are the result of the topological corner modes corresponding to the ones in second-order topological insulators. h 1 = 2t A cos √ 3k x 2 + t B e −i 3ky 2 ,(15)h 2 = 2t A cos √ 3k x 2 + t B e i 3ky 2 .(16) We show the corner impedance in Fig.4. The topological resonance emerges at ω 0 = 1/ √ 2C A + C B in the absence of resistors. However, there are also many other resonances. All these trivial ones are suppressed and only the topological one is clearly seen in the presence of resistors, as in Fig.4(b). Wannier centers: It has been argued that the topological numbers are given by the Wannier center in the Hermitian anisotropic honeycomb lattice model 50 . We show that the definition is not modified in the non-Hermitian model. The Wannier center is the set of the polarization (p x , p y ), which is defined in non-Hermitian models by where µ = x, y; V is the volume of the Brillouin zone and p µ = − 1 V BZ d j kA µ ,(17)A µ = −i ψ R ∂ kµ ψ L(18) is the non-Hermitian Berry connection 53,54 , ψ L is the left eigen-function satisfying H ψ L = ε L ψ L ,(19) and ψ R is the right eigen-function satisfying H † ψ R = ε R ψ R .(20) We have already defined the winding numbers in the bipertite system by (6). There is a relation p α = W α /2 as shown in Method A. Then the non-Hermitian polarization is half quantized, p µ = 0, 1/2, which we may as the topological numbers as in the case of Hermitian higher-order topological insulators. The Wannier center is (0, 1/2) for the topological phase and is (0, 0) for the trivial phase. D. Non-Hermitian diamond circuits Finally, we study the anisotropic diamond lattice 50 , which has a third-order topological insulator phase for \|t A /t B \| < 1/3. For the anisotropic diamond lattice, we take h 1 = t A e ik·X2 + e ik·X3 + e ik·X4 + t B e ik·X1 ,(21)h 1 = t A e −ik·X2 + e −ik·X3 + e −ik·X4 + t B e −ik·X1(22) with the four lattice vectors pointing the tetrahedron directions X 1 = (1, 1, 1) , X 2 = (1, −1, −1) , X 3 = (−1, 1, −1) and X 4 = (−1, −1, 1). We show the corner impedance in Fig.5. When we fix one terminal at A (B) in the vicinity of the rhombus center, an enhanced topological resonance emerges at the corner B (A). III. METHODS A. Topological numbers The chiral symmetric 2×2 non-Hermitian Hamiltonian systems is H = 0 h 1 h 2 0 .(23) There are several way of defining topological numbers. Here we prove their equivalence. i) We have adopted the definition (6), or W α,µ = π −π dk µ 2πi ∂ kµ log h α .(24) For the non-Hermitian SSH model, we obtain W 1,x = −1 for \|t A /t B \| < 1 0 for \|t A /t B \| > 1 ,(25)W 2,x = 1 for \|t A /t B \| < 1 0 for \|t A /t B \| > 1 .(26) We find the relation W 2 = −W 1 , and hence there is only one winding number, W = (W 2 − W 1 ) /2, in the present system. ii) Another definition of the non-Hermitian winding number 44 is W µ = 1 2πi ψ R ∂ kµ ψ L dk µ ,(27) where ψ L is the left eigen-function satisfying H ψ L = ε L ψ L ,(28) and ψ R is the right eigen-function satisfying H † ψ R = ε R ψ R .(29) It is identical to the non-Hermitian Wannier center defined by (17) when A µ depends only on k µ , which is the case in gapped phases since the integration over the codimension of k µ is constant and cancels with V . In our model, the relation \|h 1 \| = \|h 2 \| holds and they are given by ψ L = 1 √ 2 − h 1 h 2 , 1 , ψ R = 1 √ 2 − h * 2 h * 1 , 1 ,(30) and hence p = 1 2πi ∂ k log h 1 − ∂ k log h 2 4 dk = W 1 − W 2 4 .(31) We conclude that the non-Hermitian polarization is half quantized for the topological phase since there is a relation p = −W/2. iii) In the chiral symmetric model there exists the non-Hermitian chiral index, γ = π −π dk 2πi Tr σ z H −1 ∂ k H = W 2 − W 1 . This index is equivalent to the topological number since γ = 2W . B. Zero-admittance edge modes of SSH circuits We construct an analytic form of the eigen-functions at the zero-energy state in the SSH model (11). We label the eigenfunction at the outer most node as ψ 1 , and that of the node next to it as ψ 2 , and as so on. The eigen-function is ψ = {ψ 1 , ψ 2 , · · · , ψ N } if there are N nodes across the chain. The Hamiltonian is explicitly written as H =      0 t A 0 0 · · · t A 0 t B 0 · · · 0 t B 0 t A · · · 0 0 t A 0 · · · · · · · · · · · · · · · · · ·      . The eigenvalue problem Hψ = 0 is explicitly given by t A ψ 2 = 0, t A ψ 1 + t B ψ 3 = 0, · · · t A ψ 2n + t B ψ 2n+2 = 0, t A ψ 2n+1 + t B ψ 2n+3 = 0.(34) By solving (34) recursively from the outer most cite, we obtain the relation ψ 2n+1 = [t A /t B ] n ψ 1 and the analytic form of the eigen mode for odd cite n, ψ 2n+1 = 1 − t A t B 2 t A t B n .(35) On the other hand, the wave function is zero for even cite ψ 2n = 0. In order for the edge modes to exist, the eigen-function must be normalizable, whose condition is t A t B < 1.(36) It is consistent with the topological phase. For the random case, the Hamiltonian is altered to be H =      0 t A,1 0 0 · · · t A,1 0 t B,1 0 · · · 0 t B,1 0 t A,2 · · · 0 0 t A,2 0 · · · · · · · · · · · · · · · · · ·      . Then the zero-admittance solution is given by ψ 2n+1 = t A,n t B,n 1 + t A,n t B,n 2 ,(38) and ψ 2n = 0. FIG. 1 : 1(a) Illustration of the SSH circuit and its unit circuit. (b) Those in the anisotropic honeycomb circuit. A lattice is constructed by translating a unit circuit repeatedly. Unit circuits are indicated by dotted boxes. (c) Spatial distribution of two-point impedance in the topological phase of the anisotropic honeycomb circuit, where one node is fixed at A (B) in the vicinity of the rhombus center. Then, an enhanced topological peak emerges only at the corner B (A). FIG. 2 : 2Admittance spectrum in the chiral non-Hermitian SSH model in the absence and presence of resistors. (a-b) Bird's eye's view of the spectrum. The x, y and z axes are tA/tB, Im[ε] and Re[ε]. Admittance becomes complex in the presence of resistance. (c-d) Real part of the spectrum Re[ε]. (e-f) Imaginary part of the spectrum Im[ε]. (g-h) Bird's eye's view of the spectrum in the presence of 20% randomness. The zero-admittance modes marked in red remain as they are even in the presence of strong randomness. (a,c,e,g) for R = 0, and (b,d,f,h) for R = 0. FIG. 3 : 3Impedance spectrum in the chiral non-Hermitian SSH model in the absence and presence of resistors. (a-b) Resonance frequency in the chiral non-Hermitian SSH model. We have taken Ra = R b = 0. The horizontal axis is tA/tB, and the vertical axis is ω. Curves are given by(9). (c-d) Edge impedance Z is given in the (tA/tB)-ω plane. (e-f) Edge impedance Z as a function of ω/ω0 at fixed (tA/tB). (g-h) Edge impedance Z in the presence of 20% randomness. The vertical axis Z is in the unit of Ω. (c,e,g) for RA = RB = 0, and (d,f,h) for RA = RB = 0.1Ω. We have taken CA/CB = 0.5 with CA = 1µF and L = 10 in (e,f,g,h). There are many impedance peaks when R = 0, but all of them are washed away except for the topological impedance peaks for R = 0. FIG. 4 : 4Corner impedance of the anisotropic honeycomb circuit in rhombus geometry in the absence and presence of resistors. (a-b) Corner impedance Z is given in the (tA/tB)-ω plane. (c-d) Corner impedance Z as a function of ω/ω0 at fixed (tA/tB). We have set CA/CB = 0.25 with CA = 1µF and RA = RB = 0.01Ω. There are many impedance peaks when R = 0, but all of them are washed away except for the topological impedance peaks for R = 0. FIG. 5 : 5Corner impedance of the anisotropic diamond circuit in rhombohedron geometry. We have represented the spatial distribution of two-point impedance by the size of a ball in the topological phase. When one node is fixed at B (A) in the vicinity of the rhombus center, a huge ball is found at the corner A (B). AcknowledgementsThe author is very much grateful to N. Nagaosa for helpful discussions on the subject. 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[ "Commutative Families of the Elliptic Macdonald Operator", "Commutative Families of the Elliptic Macdonald Operator" ]	[ "Yosuke Saito \nMathematical Institute\nTohoku University\nSendaiJapan\n" ]	[ "Mathematical Institute\nTohoku University\nSendaiJapan" ]	[]	In the paper [1], using the Ding-Iohara algebra and the trigonometric Feigin-Odesskii algebra, Feigin, Hashizume, Hoshino, Shiraishi, and Yanagida constructed two families of commuting operators which contain the Macdonald operator (commutative families of the Macdonald operator). In the previous paper [3], the author constructed the elliptic Ding-Iohara algebra and the free field realization of the elliptic Macdonald operator. In this paper, we show that by using the elliptic Ding-Iohara algebra and the elliptic Feigin-Odesskii algebra, we can construct commutative families of the elliptic Macdonald operator.In Appendix, we will show a relation between the elliptic Macdonald operator and its kernel function by the free field realization.	10.3842/sigma.2014.021	[ "https://arxiv.org/pdf/1305.7097v6.pdf" ]	15,417,657	1305.7097	d9d5a2ad229946e9009a9b1c1c120c79e982e141	Commutative Families of the Elliptic Macdonald Operator 25 Jul 2013 July 26, 2013 Yosuke Saito Mathematical Institute Tohoku University SendaiJapan Commutative Families of the Elliptic Macdonald Operator 25 Jul 2013 July 26, 2013arXiv:1305.7097v3 [math.QA] In the paper [1], using the Ding-Iohara algebra and the trigonometric Feigin-Odesskii algebra, Feigin, Hashizume, Hoshino, Shiraishi, and Yanagida constructed two families of commuting operators which contain the Macdonald operator (commutative families of the Macdonald operator). In the previous paper [3], the author constructed the elliptic Ding-Iohara algebra and the free field realization of the elliptic Macdonald operator. In this paper, we show that by using the elliptic Ding-Iohara algebra and the elliptic Feigin-Odesskii algebra, we can construct commutative families of the elliptic Macdonald operator.In Appendix, we will show a relation between the elliptic Macdonald operator and its kernel function by the free field realization. Notations. In this paper, we use the following symbols. Z : The set of integers, Z ≥0 := {0, 1, 2, · · · }, Z >0 := {1, 2, · · · }, Q : The set of rational numbers, C : The set of complex numbers, C × := C \ {0}, C[[z, z −1 ]] : The set of formal power series of z, z −1 over C. If a sequence λ = (λ 1 , · · · , λ N ) ∈ (Z ≥0 ) N (N ∈ Z >0 ) satisfies the condition λ i ≥ λ i+1 (1 ≤ i ≤ N), λ is called a partition. We denote the set of partitions by P. For a partition λ, ℓ(λ) := ♯{i : λ i = 0} denotes the length of λ and \|λ\| := ℓ(λ) i=1 λ i denotes the size of λ. Let q, p ∈ C be complex parameters satisfying \|q\| < 1, \|p\| < 1. We define the q-infinite product as (x; q) ∞ := n≥0 (1 − xq n ) and the theta function as Θ p (x) := (p; p) ∞ (x; p) ∞ (px −1 ; p) ∞ . We set the double infinite product as (x; q, p) ∞ := m,n≥0 (1 − xq m p n ) and the elliptic gamma function as Γ q,p (x) := (qpx −1 ; q, p) ∞ (x; q, p) ∞ . Organization of this paper. In section 1, we review the trigonometric case treated in the paper [1]. In section 2, first we recall related materials of the elliptic Ding-Iohara algebra and the free field realization of the elliptic Macdonald operator. Second we show that using the elliptic Ding-Iohara algebra and the elliptic Feigin-Odesskii algebra, we can obtain commutative families of the elliptic Macdonald operator. In Appendix, by the free field realization of the elliptic Macdonald operator, we show a functional equation of the elliptic kernel function. Trigonometric case In this section, we review the construction of the commutative families of the Macdonald operator by Feigin, Hashizume, Hoshino, Shiraishi, Yanagida [1]. Ding-Iohara algebra U (q, t) The Ding-Iohara algebra is a quantum group obtained from the free field realization of the Macdonald operator [1]. Here we define the Ding-Iohara algebra and collect some basic facts. Definition 1.1 (Ding-Iohara algebra U(q, t)). Let us define the structure function g(x) as g(x) := (1 − qx)(1 − t −1 x)(1 − q −1 tx) (1 − q −1 x)(1 − tx)(1 − qt −1 x) . (1.1) Let γ be a central, invertible element and x ± (z) := n∈Z x ± n z −n , ψ ± (z) := n∈Z ψ ± n z −n be currents satisfying the relations : [ψ ± (z), ψ ± (w)] = 0, ψ + (z)ψ − (w) = g(γz/w) g(γ −1 z/w) ψ − (w)ψ + (z), ψ ± (z)x + (w) = g γ ± 1 2 z w x + (w)ψ ± (z), ψ ± (z)x − (w) = g γ ∓ 1 2 z w −1 x − (w)ψ ± (z), x ± (z)x ± (w) = g z w ±1 x ± (w)x ± (z), [x + (z), x − (w)] = (1 − q)(1 − t −1 ) 1 − qt −1 δ γ w z ψ + γ 1/2 w − δ γ −1 w z ψ − γ −1/2 w . (1.2) Here we set the delta function by δ(x) := n∈Z x n . We define the Ding-Iohara algebra U(q, t) to be an associative C-algebra generated by {x ± n } n∈Z , {ψ ± n } n∈Z , and γ. The free field realization of the Ding-Iohara algebra is stated as follows. In the following, let q, t ∈ C be parameters and we assume \|q\| < 1. First we define the algebra B of boson to be generated by {a n } n∈Z\{0} and the relation : [a m , a n ] = m 1 − q \|m\| 1 − t \|m\| δ m+n,0 . (1.3) We set the normal ordering : • : as : a m a n := a m a n (m < n), a n a m (m ≥ n). Let \|0 be the vacuum vector which satisfies a n \|0 = 0 (n > 0). For a partition λ, we set a −λ := a −λ 1 · · · a −λ ℓ(λ) and define the boson Fock space F as a left B module : F := span{a −λ \|0 : λ ∈ P}. Proposition 1.2 (Free field realization of the Ding-Iohara algebra U(q, t)). Set γ := (qt −1 ) −1/2 and define operators η(z), ξ(z), ϕ ± (z) : F → F ⊗ C[[z, z −1 ]] as follows : η(z) :=: exp − n =0 (1 − t n )a n z −n n :, ξ(z) :=: exp n =0 (1 − t n )γ \|n\| a n z −n n :, ϕ + (z) :=: η(γ 1/2 z)ξ(γ −1/2 z) :, ϕ − (z) :=: η(γ −1/2 z)ξ(γ 1/2 z) : . (1.4) Then the map x + (z) → η(z), x − (z) → ξ(z), ψ ± (z) → ϕ ± (z) gives a representation of the Ding-Iohara algebra U(q, t). Set the q-shift operator as T q,x f (x) := f (qx). We define the Macdonald operator H N (q, t) (N ∈ Z >0 ) as H N (q, t) := N i=1 j =i tx i − x j x i − x j T q,x i . (1.5) In the following [f (z)] 1 denotes the constant term of f (z) in z. 1 − t n 1 − q n a −n z n n . (1.6) We use the symbol φ N (x) := N j=1 φ(x j ). (1) The operator η(z) reproduces the Macdonald operator H N (q, t) as follows. [η(z)] 1 φ N (x)\|0 = t −N {(t − 1)H N (q, t) + 1}φ N (x)\|0 . (1.7) (2) The operator ξ(z) reproduces the Macdonald operator H N (q −1 , t −1 ) as follows. [ξ(z)] 1 φ N (x)\|0 = t N {(t −1 − 1)H N (q −1 , t −1 ) + 1}φ N (x)\|0 . (1.8) We also have the dual version of the proposition 1.3. Let 0\| be the dual vacuum vector which satisfies the condition 0\|a n = 0 (n < 0) and define the dual boson Fock space F * as a right B module : F * := span{ 0\|a λ : λ ∈ P} (a λ := a λ 1 · · · a ℓ(λ) ). Proposition 1.4 (Dual version of the proposition 1.3). Let us define an operator φ * (z) : F * → F * ⊗ C[[z, z −1 ]] as φ * (z) := exp n>0 1 − t n 1 − q n a n z n n . (1.9) We use the symbol φ * N (x) := N j=1 φ * (x j ). (1) The operator η(z) reproduces the Macdonald operator H N (q, t) as follows. 0\|φ * N (x)[η(z)] 1 = t −N {(t − 1)H N (q, t) + 1} 0\|φ * N (x). (1.10) (2) The operator ξ(z) reproduces the Macdonald operator H N (q −1 , t −1 ) as follows. 0\|φ * N (x)[ξ(z)] 1 = t N {(t −1 − 1)H N (q −1 , t −1 ) + 1} 0\|φ * N (x). (1.11) Remark 1.5. The kernel function of the Macdonald operator Π(q, t)({x i } M i=1 , {y j } N j=1 ) (M, N ∈ Z >0 ) is defined by Π(q, t)({x i } M i=1 , {y j } N j=1 ) := 1≤i≤M 1≤j≤N (tx i y j ; q) ∞ (x i y j ; q) ∞ . Then the kernel function Π(q, t)( {x i } M i=1 , {y j } N j=1 ) is reproduced from the operators φ * M (x), φ N (y) as 0\|φ * M (x)φ N (y)\|0 = Π(q, t)({x i } M i=1 , {y j } N j=1 ). Trigonometric Feigin-Odesskii algebra A In this subsection, we review basic facts of the trigonometric Feigin-Odesskii algebra [1]. Definition 1.6 (Trigonometric Feigin-Odesskii algebra A). Let ε n (q; x) (n ∈ Z >0 ) be a function defined as ε n (q; x) := 1≤a<b≤n (x a − qx b )(x a − q −1 x b ) (x a − x b ) 2 . (1.12) We also define ω(x, y) as ω(x, y) := (x − q −1 y)(x − ty)(x − qt −1 y) (x − y) 3 . (1.13) For a N-variable function f (x 1 , · · · , x N ), we define the action of the N-th symmetric group S N on f (x 1 , · · · , x N ) by σ · (f (x 1 , · · · , x N )) := f (x σ(1) , · · · , x σ(N ) ) (σ ∈ S N ). We define the symmetrizer as Sym[f (x 1 , · · · , x N )] := 1 N! σ∈S N σ · (f (x 1 , · · · , x N )). (1.14) For a m-variable function f (x 1 , · · · , x m ) and a n-variable function g(x 1 , · · · , x n ), we define the star product * as follows. (f * g)(x 1 , · · · , x m+n ) := Sym f (x 1 , · · · , x m )g(x m+1 , · · · , x m+n ) 1≤α≤m m+1≤β≤m+n ω(x α , x β ) . (1.15) For a partition λ, we define ε λ (q; x) as ε λ (q; x) := ε λ 1 (q; x) * · · · * ε λ ℓ(λ) (q; x). (1.16) Set A 0 := C, A n := span{ε λ (q; x) : \|λ\| = n} (n ≥ 1). We define the trigonometric Feigin-Odesskii algebra A := n≥0 A n whose algebraic structure is given by the star product * . Remark 1.7. The definition of the trigonometric Feigin-Odesskii algebra A above is a reduced version of the paper [1]. For instance, there would be a question why the function ε n (q; x) appears. For more detail of the trigonometric Feigin-Odesskii algebra A, see [1]. In the paper [1], the following fact is shown. O(f ) := f (z 1 , · · · , z n ) 1≤i<j≤n ω(z i , z j ) −1 η(z 1 ) · · · η(z n ) 1 (f ∈ A n ). (1.17) Here [f (z 1 , · · · , z n )] 1 denotes the constant term of f (z 1 , · · · , z n ) in z 1 , · · · , z n , and we extend the map O linearly. Proof . To prove the proposition, for f ∈ A m and g ∈ A n , we show O(f * g) = O(f )O(g). First we have O(f * g)(x 1 , · · · , x m+n ) = Sym f (x 1 , · · · , x m )g(x m+1 , · · · , x m+n ) 1≤α≤m m+1≤β≤m+n ω(x α , x β ) × 1≤i<j≤m+n ω(x i , x j ) −1 η(x 1 ) · · · η(x m+n ) 1 . Then from the relation η(z)η(w) = g z w η(w)η(z), we have the following : 1 ω(z, w) η(z)η(w) = 1 ω(w, z) η(w)η(z). (1.18) The relation shows that the operator-valued function 1≤i<j≤N ω(x i , x j ) −1 η(x 1 ) · · · η(x N ) (1.19) is symmetric in x 1 , · · · , x N . Further we have [Sym(F (x 1 , · · · , x N ))] 1 = [F (x 1 , · · · , x N )] 1 . This follows from the fact that the constant term is invariant under the action of the symmetric group. In addition for a symmetric function f (x 1 , · · · , x N ), we have σ(f (x 1 , · · · , x N )g(x 1 , · · · , x N )) = f (x 1 , · · · , x N )σ(g(x 1 , · · · , x N )) (σ ∈ S N ). Hence we have Sym(f (x 1 , · · · , x N )g(x 1 , · · · , x N )) = f (x 1 , · · · , x N )Sym(g(x 1 , · · · , x N )). From them we have the following. O(f * g) = Sym f (x 1 , · · · , x m )g(x m+1 , · · · , x m+n ) 1≤α≤m m+1≤β≤m+n ω(x α , x β ) × 1≤i<j≤m+n ω(x i , x j ) −1 η(x 1 ) · · · η(x m+n ) 1 = f (x 1 , · · · , x m )g(x m+1 , · · · , x m+n ) 1≤α≤m m+1≤β≤m+n ω(x α , x β ) × 1≤i<j≤m+n ω(x i , x j ) −1 η(x 1 ) · · · η(x m+n ) 1 = f (x 1 , · · · , x m ) 1≤i<j≤m ω(x i , x j ) −1 η(x 1 ) · · · η(x m ) × g(x m+1 , · · · , x m+n ) m+1≤i<j≤m+n ω(x i , x j ) −1 η(x m+1 ) · · · η(x m+n ) 1 = O(f )O(g). The trigonometric Feigin-Odesskii algebra A is commutative by means of the star product * , therefore we have the following corollary. Let V be a C-vector space and T : V → V be a C-linear operator. Then for a subset W ⊂ V , the symbol T \| W denotes the restriction of T on W . For a subset M ⊂ End C (V ), we use the symbol M\| W := {T \| W : T ∈ M} (W ⊂ V ). (2) The space M\| Cφ N (x)\|0 consists of commuting q-difference operators which contains the Macdonald operator H N (q, t) (commutative family of the Macdonald operator H N (q, t)). Proof . (1) This statement follows from the commutativity of A and the compatibility of the star product * and the map O. We have O(f * g) = O(f )O(g) and also have O(f * g) = O(g * f ) = O(g)O(f ). This shows [O(f ), O(g)] = 0. (2) Due to the free field realization of the Macdonald operator H N (q, t), the operator O(ε r (q; z)) (r ∈ Z >0 ) acts on φ N (x)\|0 (N ∈ Z >0 ) as a r-th order q-difference operator. By the fact that M = O(A) is generated by {O(ε r (q; z))} r∈Z >0 and the relation O(ε 1 (q; z))φ N (x)\|0 = [η(z)] 1 φ N (x)\|0 = t −N {(t − 1)H N (q, t) + 1}φ N (x)\|0 and (1) in the corollary 1.11, the restriction M\| Cφ N (x)\|0 is a space of commuting qdifference operators which contains the Macdonald operator H N (q, t). The Macdonald operator H N (q −1 , t −1 ) is reproduced from the operator ξ(z). By this fact, we can construct another commutative family of the Macdonald operator. Definition 1.12 (Map O ′ ) . Set a function ω ′ (x, y) as follows. ω ′ (x, y) := (x − qy)(x − t −1 y)(x − q −1 ty) (x − y) 3 . (1.20) Define a linear map O ′ : A → End(F ) as O ′ (f ) := f (z 1 , · · · , z n ) 1≤i<j≤n ω ′ (z i , z j ) −1 ξ(z 1 ) · · · ξ(z n ) 1 (f ∈ A n ). (1.21) We extend the map O ′ linearly. Lemma 1.13. Define another star product * ′ as follows : (f * ′ g)(x 1 , · · · , x m+n ) := Sym f (x 1 , · · · , x m )g(x m+1 , · · · , x m+n ) 1≤α≤m m+1≤β≤m+n ω ′ (x α , x β ) . In the trigonometric Feigin-Odesskii algebra A, we have * ′ = * . Proof . From the relation ω(x, y) = ω ′ (y, x), for f ∈ A m , g ∈ A n we have the following. (f * ′ g)(x 1 , · · · , x m+n ) = Sym f (x 1 , · · · , x m )g(x m+1 , · · · , x m+n ) 1≤α≤m m+1≤β≤m+n ω ′ (x α , x β ) = Sym f (x 1 , · · · , x m )g(x m+1 , · · · , x m+n ) 1≤α≤m m+1≤β≤m+n ω(x β , x α ) = Sym g(x 1 , · · · , x n )f (x n+1 , · · · , x n+m ) 1≤α≤n n+1≤β≤n+m ω(x α , x β ) = (g * f )(x 1 , · · · , x m+n ) = (f * g)(x 1 , · · · , x m+n ) (∵ A is commutative by means of * ). Since the relation holds for any f ∈ A m , g ∈ A n , we have * ′ = * . We can check the map O ′ and the star product * ′ are compatible in the similar way of the proof of the proposition 1.10. Furthermore by the lemma 1.13 as * ′ = * , we have the following corollary. (2) The space M ′ \| Cφ N (x)\|0 consists of commuting q-difference operators which contains the Macdonald operator H N (q −1 , t −1 ) (commutative family of the Macdonald operator H N (q −1 , t −1 )). Proof . This proposition follows from the existence of the Macdonald symmetric functions. That is, elements of the commutative families are simultaneously diagonalized by the Macdonald symmetric functions. From the proposition 1.15, commutative families M\| Cφ N (x)\|0 , M ′ \| Cφ N (x)\|0 also com- mute with each other : [M\| Cφ N (x)\|0 , M ′ \| Cφ N (x)\|0 ] = 0. Elliptic case In this section, we are going to construct a commutative family of the elliptic Macdonald operator by using the elliptic Ding-Iohara algebra and the elliptic Feigin-Odesskii algebra. In the following, for complex parameters q, p ∈ C we assume \|q\| < 1, \|p\| < 1. Elliptic Ding-Iohara algebra U (q, t, p) The elliptic Ding-Iohara algebra is an elliptic analog of the Ding-Iohara algebra introduced by the author [3]. First we recall the definition of the elliptic Ding-Iohara algebra and its free field realization. Definition 2.1 (Elliptic Ding-Iohara algebra U(q, t, p)). Set the structure function g p (x) as g p (x) := Θ p (qx)Θ p (t −1 x)Θ p (q −1 tx) Θ p (q −1 x)Θ p (tx)Θ p (qt −1 x) . Let x ± (p; z) := n∈Z x ± n (p)z −n , ψ ± (p; z) := n∈Z ψ ± n (p)z −n be currents and γ be central, invertible element satisfying the following relations : [ψ ± (p; z), ψ ± (p; w)] = 0, ψ + (p; z)ψ − (p; w) = g p (γz/w) g p (γ −1 z/w) ψ − (p; w)ψ + (p; z), ψ ± (p; z)x + (p; w) = g p γ ± 1 2 z w x + (p; w)ψ ± (p; z), ψ ± (p; z)x − (p; w) = g p γ ∓ 1 2 z w −1 x − (p; w)ψ ± (p; z), x ± (p; z)x ± (p; w) = g p z w ±1 x ± (p; w)x ± (p; z), [x + (p; z), x − (p; w)] = Θ p (q)Θ p (t −1 ) (p; p) 3 ∞ Θ p (qt −1 ) δ γ w z ψ + (p; γ 1/2 w) − δ γ −1 w z ψ − (p; γ −1/2 w) . (2.1) Here we define the delta function as δ(x) := n∈Z x n . We define the elliptic Ding-Iohara algebra U(q, t, p) to be an associative C-algebra generated by {x ± n (p)} n∈Z , {ψ ± n (p)} n∈Z and γ. Theorem 2.2 (Free field realization of the elliptic Ding-Iohara algebra U(q, t, p)). Set an algebra of boson B a,b generated by {a n } n∈Z\{0} , {b n } n∈Z\{0} and the following relations : [a m , a n ] = m(1 − p \|m\| ) 1 − q \|m\| 1 − t \|m\| δ m+n,0 , [b m , b n ] = m 1 − p \|m\| (qt −1 p) \|m\| 1 − q \|m\| 1 − t \|m\| δ m+n,0 , [a m , b n ] = 0. Let \|0 be the vacuum vector which satisfies the condition a n \|0 = b n \|0 = 0 (n > 0) and set the boson Fock space F as a left B a,b module. F = span{a −λ b −µ \|0 : λ, µ ∈ P}. We also define the normal ordering : • : as usual : : a m a n := a m a n (m < n), a n a m (m ≥ n), : b m b n := b m b n (m < n), b n b m (m ≥ n). Set γ := (qt −1 ) −1/2 and operators η(p; z), ξ(p; z), ϕ ± (p; z) : F → F ⊗ C[[z, z −1 ]] as η(p; z) :=: exp − n =0 1 − t −n 1 − p \|n\| p \|n\| b n z n n exp − n =0 1 − t n 1 − p \|n\| a n z −n n :, Then the map ξ(p; z) :=: exp n =0 1 − t −n 1 − p \|n\| γ −\|n\| p \|n\| b n z n n exp n =0 1 − t n 1 − p \|n\| γ \|n\| a n z −x + (p; z) → η(p; z), x − (p; z) → ξ(p; z), ψ ± (p; z) → ϕ ± (p; z) gives a representation of the elliptic Ding-Iohara algebra U(q, t, p). The elliptic Macdonald operator H N (q, t, p) (N ∈ Z >0 ) is defined as follows. H N (q, t, p) := N i=1 j =i Θ p (tx i /x j ) Θ p (x i /x j ) T q,x i . (2.2) By the operators η(p; z), ξ(p; z) which are in the theorem 2.2, we can reproduce the elliptic Macdonald operator as follows [3]. (1 − t n )(qt −1 p) n (1 − q n )(1 − p n ) b −n z −n n exp n>0 1 − t n (1 − q n )(1 − p n ) a −n z n n . (2.3) We use the symbol φ N (p; x) := N j=1 φ(p; x j ). (1) The elliptic Macdonald operator H N (q, t, p) is reproduced by the operator η(p; z) as follows. [η(p; z) − t −N (η(p; z)) − (η(p; p −1 z)) + ] 1 φ N (p; x)\|0 = t −N +1 Θ p (t −1 ) (p; p) 3 ∞ H N (q, t, p)φ N (p; x)\|0 . (2.4) Here we use the notation (η(p; z)) ± as (η(p; z)) ± := exp − ±n>0 1 − t −n 1 − p \|n\| p \|n\| b n z n n exp − ±n>0 1 − t n 1 − p \|n\| a n z −n n . (2) The elliptic Macdonald operator H N (q −1 , t −1 , p) is reproduced by the operator ξ(p; z) as follows. [ξ(p; z) − t N (ξ(p; z)) − (ξ(p; p −1 z)) + ] 1 φ N (p; x)\|0 = t N −1 Θ p (t) (p; p) 3 ∞ H N (q −1 , t −1 , p)φ N (p; x)\|0 . (2.5) Here we use the notation (ξ(p; z)) ± as (ξ(p; z)) ± := exp ±n>0 1 − t −n 1 − p \|n\| γ −\|n\| p \|n\| b n z n n exp ±n>0 1 − t n 1 − p \|n\| γ \|n\| a n z −n n . The theorem 2.3 is also stated as follows. First we set zero mode generators a 0 , Q which satisfy the relation : (2.6) We also set the condition a 0 \|0 = 0. For a complex number α ∈ C we define \|α := e αQ \|0 . Then we can check a 0 \|α = α\|α . For α ∈ C, we set F α := span{a −λ b −µ \|α : λ, µ ∈ P}. Theorem 2.4. Set η(p; z) := (η(p; z)) − (η(p; p −1 z)) + , ξ(p; z) := (ξ(p; z)) − (ξ(p; p −1 z)) + . Using these symbols we define operators E(p; z), F (p; z) as follows : E(p; z) := η(p; z) − η(p; z)t −a 0 , F (p; z) := ξ(p; z) − ξ(p; z)t a 0 . (2.7) Then the elliptic Macdonald operators H N (q, t, p), H N (q −1 , t −1 , p) are reproduced by the operators E(p; z), F (p; z) as follows. [E(p; z)] 1 φ N (p; x)\|N = t −N +1 Θ p (t −1 ) (p; p) 3 ∞ H N (q, t, p)φ N (p; x)\|N , (2.8) [F (p; z)] 1 φ N (p; x)\|N = t N −1 Θ p (t) (p; p) 3 ∞ H N (q −1 , t −1 , p)φ N (p; x)\|N . (2.9) The dual versions of the theorem 2.3, 2.4 are also available. Let 0\| be the dual vacuum vector which satisfies the condition 0\|a n = 0\|b n = 0 (n < 0) and 0\|a 0 = 0. We define the dual boson Fock space as a right B a,b module : F * := span{ 0\|a λ b µ : λ, µ ∈ P}. For a complex number α ∈ C, set α\| := 0\|e −αQ . Then we have α\|a 0 = α α\|. For α ∈ C, we set F * α := span{ α\|a λ b µ : λ, µ ∈ P}. (1 − t n )(qt −1 p) n (1 − q n )(1 − p n ) b n z −n n exp n>0 1 − t n (1 − q n )(1 − p n ) a n z n n . (2.10) We use the symbol φ * N (p; x) := N j=1 φ * (p; x j ). (1) The operators η(p; z), ξ(p; z) reproduce the elliptic Macdonald operators H N (q, t, p), H N (q −1 , t −1 , p) as follows. 0\|φ * N (p; x)[η(p; z) − t −N (η(p; z)) − (η(p; p −1 z)) + ] 1 = t −N +1 Θ p (t −1 ) (p; p) 3 ∞ H N (q, t, p) 0\|φ * N (p; x), (2.11) 0\|φ * N (p; x)[ξ(p; z) − t N (ξ(p; z)) − (ξ(p; p −1 z)) + ] 1 = t N −1 Θ p (t) (p; p) 3 ∞ H N (q −1 , t −1 , p) 0\|φ * N (p; x). (2.12) (2) The operators E(p; z), F (p; z) reproduce the elliptic Macdonald operators H N (q, t, p), H N (q −1 , t −1 , p) as follows. N\|φ * N (p; x)[E(p; z)] 1 = t −N +1 Θ p (t −1 ) (p; p) 3 ∞ H N (q, t, p) N\|φ * N (p; x), (2.13) N\|φ * N (p; x)[F (p; z)] 1 = t N −1 Θ p (t) (p; p) 3 ∞ H N (q −1 , t −1 , p) N\|φ * N (p; x). (2.14) Remark 2.6. Let Π(q, t, p)({x i } M i=1 , {y j } N j=1 ) (M, N ∈ Z >0 ) be the kernel function of the elliptic Macdonald operator defined as Π(q, t, p)({x i } M i=1 , {y j } N j=1 ) := 1≤i≤M 1≤j≤N Γ q,p (x i y j ) Γ q,p (tx i y j ) . Then the kernel function Π(q, t, p) ({x i } M i=1 , {y j } N j=1 ) is reproduced from the operators φ * M (p; x), φ N (p; y) as 0\|φ * M (p; x)φ N (p; y)\|0 = Π(q, t, p)({x i } M i=1 , {y j } N j=1 ). Elliptic Feigin-Odesskii algebra A(p) The elliptic Feigin-Odesskii algebra is defined in the similar way of the trigonometric case except the emergence of elliptic functions [1]. Definition 2.7 (Elliptic Feigin-Odesskii algebra A(p)). Define a n-variable function ε n (q, p; x) (n ∈ Z >0 ) as follows. ε n (q, p; x) := 1≤a<b≤n Θ p (qx a /x b )Θ p (q −1 x a /x b ) Θ p (x a /x b ) 2 . (2.15) Set a function ω p (x, y) as ω p (x, y) := Θ p (q −1 y/x)Θ p (ty/x)Θ p (qt −1 y/x) Θ p (y/x) 3 . (2.16) Define the star product * as (f * g)(x 1 , · · · , x m+n ) := Sym f (x 1 , · · · , x m )g(x m+1 , · · · , x m+n ) 1≤α≤m m+1≤β≤m+n ω p (x α , x β ) . (2.17) For a partition λ, we set ε λ (q, p; x) as ε λ (q, p; x) := ε λ 1 (q, p; x) * · · · * ε λ ℓ(λ) (q, p; x). (2.18) Set A 0 (p) := C, A n (p) := span{ε λ (q, p; x) : \|λ\| = n} (n ≥ 1). We define the elliptic Feigin-Odesskii algebra as A(p) := n≥0 A n (p) whose algebraic structure is given by the star product * . Similar to the trigonometric case, the following is shown [1]. Proposition 2.8. The elliptic Feigin-Odesskii algebra (A(p), * ) is unital, associative, and commutative algebra. Commutative families M(p), M ′ (p) For the operators E(p; z), F (p; z) which are used in the theorem 2.4, we have the following proposition [3]. Proposition 2.9. (1) Operators E(p; z), F (p; z) satisfy the relation as E(p; z)E(p; w) = g p z w E(p; w)E(p; z), (2.19) F (p; z)F (p; w) = g p z w −1 F (p; w)F (p; z). (2.20) Due to the relations operator-valued functions as 1≤i<j≤N ω p (x i , x j ) −1 E(p; x 1 ) · · · E(p; x N ), 1≤i<j≤N ω ′ p (x i , x j ) −1 F (p; x 1 ) · · · F (p; x N ) are symmetric in x 1 , · · · , x N . (2) Commutator of E(p; z) and F (p; z) takes the following form. O p (f ) := f (z 1 , · · · , z n ) 1≤i<j≤n ω p (z i , z j ) −1 E(p; z 1 ) · · · E(p; z n ) 1 (f ∈ A n (p)). (2.22) Here [f (z 1 , · · · , z n )] 1 denotes the constant term of f (z 1 , · · · , z n ) in z 1 , · · · , z n . We extend the map linearly. [E(p; z), F (p; w)] = Θ p (q)Θ p (t −1 ) (p; p) 3 ∞ Θ p (qt −1 ) δ γ w z {ϕ + (p; γ 1/2 w) − ϕ + (p; γ 1/2 p −1 w)}. In the similar way of the trigonometric case, we can check the following. Proposition 2.11. The map O p and the star product * are compatible : for f, g ∈ A(p), A(p)). The space is a commutative algebra of boson operators. we have O p (f * g) = O p (f )O p (g). (2) The space M(p)\| Cφ N (p;x)\|N consists of commuting elliptic q-difference operators which contains the elliptic Macdonald operator H N (q, t, p) (commutative family of the elliptic Macdonald operator H N (q, t, p)). A commutative family of the elliptic Macdonald operator H N (q −1 , t −1 , p) is also constructed as follows. Definition 2.13 (Map O ′ p ). Set a function ω ′ p (x, y) as ω ′ p (x, y) := Θ p (qy/x)Θ p (t −1 y/x)Θ p (q −1 ty/x) Θ p (y/x) 3 . (2.23) We define a linear map O ′ p : A(p) → End(F α ) (α ∈ C) as follows. O ′ p (f ) := f (z 1 , · · · , z n ) 1≤i<j≤n ω ′ p (z i , z j ) −1 F (p; z 1 ) · · · F (p; z n ) 1 (f ∈ A n (p)). (2.24) We extend the map linearly. In the same way of the trigonometric case, we have the following lemma. Lemma 2.14. Set another star product * ′ as (f * ′ g)(x 1 , · · · , x m+n ) := Sym f (x 1 , · · · , x m )g(x m+1 , · · · , x m+n ) 1≤α≤m m+1≤β≤m+n ω ′ p (x α , x β ) . (2.25) In the elliptic Feigin-Odesskii algebra A(p), we have * ′ = * . The theorem 2.16 is the elliptic analog of the proposition 1.15. But we can't prove the theorem 2.16 in the similar way of the proof of the proposition 1.15, because we don't have an elliptic analog of the Macdonald symmetric functions. Hence we will show the theorem 2.16 in a direct way. For the proof we prepare the following lemma. Lemma 2.17. Assume that a r-variable function A(x 1 , · · · , x r ) and a s-variable function B(x 1 , · · · , x s ) have a period p, i.e. T p,x i A(x 1 , · · · , x r ) = A(x 1 , · · · , x r ) (1 ≤ i ≤ r), T p,x i B(x 1 , · · · , x s ) = B(x 1 , · · · , x s ) (1 ≤ i ≤ s). Then we have [[A(z 1 , · · · , z r )E(p; z 1 ) · · · E(p; z r )] 1 , [B(w 1 , · · · , w s )F (p; w 1 ) · · · F (p; w s )] 1 ] = 0. (2.26) Proof . First we have [A 1 · · · A r , B 1 · · · B s ] = r i=1 s j=1 A 1 · · · A i−1 B 1 · · · B j−1 [A i , B j ]B j+1 · · · B s A i+1 · · · A r . (2.27) Set c(q, t, p) := Θ p (q)Θ p (t −1 )/(p; p) 3 ∞ Θ p (qt −1 ) and we denote the p-difference of f (z) by ∆ p f (z) := f (pz) − f (z) . By the equation (2.27) and the proposition 2.9 (2) (2.21), we have the following. [A(z 1 , · · · , z r )E(p; z 1 ) · · · E(p; z r ), B(w 1 , · · · , w s )F (p; w 1 ) · · · F (p; w s )] = r i=1 s j=1 A(z 1 , · · · , z r )B(w 1 , · · · , w s )E(p; z 1 ) · · · E(p; z i−1 )F (p; w 1 ) · · · F (p; w j−1 ) × [E(p; z i ), F (p; w j )]F (p; w j+1 ) · · · F (p; w s )E(p; z i+1 ) · · · E(p; z r ) = c(q, t, p) r i=1 s j=1 E(p; z 1 ) · · · E(p; z i−1 )F (p; w 1 ) · · · F (p; w j−1 ) × A(z 1 , · · · , i-th γw j , · · · , z r )B(w 1 , · · · , j-th w j , · · · , w s )δ γ w j z i ∆ p ϕ + (p; γ 1/2 p −1 w j ) × F (p; w j+1 ) · · · F (p; w s )E(p; z i+1 ) · · · E(p; z r ). (2.28) By picking up the constant term of z i , w j dependence part of (2.28), we have A(z 1 , · · · , i-th γw j , · · · , z r )B(w 1 , · · · , j-th w j , · · · , w s )δ γ w j z i ∆ p ϕ + (p; γ 1/2 p −1 w j ) z i ,w j ,1 = A(z 1 , · · · , i-th γw j , · · · , z r )B(w 1 , · · · , j-th w j , · · · , w s )∆ p ϕ + (p; γ 1/2 p −1 w j ) w j ,1 = A(z 1 , · · · , i-th γw j , · · · , z r )B(w 1 , · · · , j-th w j , · · · , w s )ϕ + (p; γ 1/2 w j ) w j ,1 − A(z 1 , · · · , i-th γw j , · · · , z r )B(w 1 , · · · , j-th w j , · · · , w s )ϕ + (p; γ 1/2 p −1 w j ) w j ,1 . (2.29) We recall [f (z)] 1 = [f (az)] 1 (a ∈ C) and the functions A(z 1 , · · · , z r ) and B(w 1 , · · · , w s ) have a period p. Hence we have A(z 1 , · · · , i-th γw j , · · · , z r )B(w 1 , · · · , j-th w j , · · · , w s )δ γ w j z i ∆ p ϕ + (p; γ 1/2 p −1 w j ) z i ,w j ,1 = 0. The equation holds for any i, j, hence we have the lemma 2.17. Proof of the theorem 2.16. For the proof what we have to show is [O p (ε r (q, p; z)), O ′ p (ε s (q, p; w))] = 0 (r, s ∈ Z >0 ). By the definition of O p , O ′ p , operators O p (ε r (q, p; z)), O ′ p (ε s (q, p; w)) are the constant terms of the following operators. ε r (q, p; z) 1≤i<j≤r ω p (z i , z j ) −1 E(p; z 1 ) · · · E(p; z r ), (2.30) ε s (q, p; w) 1≤i<j≤s ω ′ p (w i , w j ) −1 F (p; w 1 ) · · · F (p; w s ). (2.31) Then their functional parts take the following forms. (Functional part of (2.30)) = ε r (q, p; z) 1≤i<j≤r ω p (z i , z j ) −1 = 1≤i<j≤r Θ p (z i /z j )Θ p (q −1 z i /z j ) Θ p (t −1 z i /z j )Θ p (q −1 tz i /z j ) , (2.32) (Functional part of (2.31)) = ε s (q, p; w) 1≤i<j≤s ω ′ p (w i , w j ) −1 = 1≤i<j≤s Θ p (w i /w j )Θ p (qw i /w j ) Θ p (tw i /w j )Θ p (qt −1 w i /w j ) . H N (q, t) := N i=1 j =i tx i − x j x i − x j T q,x i (T q,x f (x) := f (qx)) (3.1) and its kernel function Π(q, t)({x i } M i=1 , {y j } N j=1 ) (M, N ∈ Z >0 ) as Π(q, t)({x i } M i=1 , {y j } N j=1 ) := 1≤i≤M 1≤j≤N (tx i y j ; q) ∞ (x i y j ; q) ∞ . (3.2) Then we have the following functional equation. {H M (q, t) x − t M −N H N (q, t) y }Π(q, t)({x i } M i=1 , {y j } N j=1 ) = 1 − t M −N 1 − t Π(q, t)({x i } M i=1 , {y j } N j=1 ). (3.3) Here we denote the Macdonald operator which acts on functions of x 1 , · · · , x M by H M (q, t) x . In the following, we will show the elliptic analog of the theorem 3.1 by the free field realization of the elliptic Macdonald operator. Recollection : free field realization of the elliptic Macdonald operator The elliptic Macdonald operator H N (q, t, p) (N ∈ Z >0 ) is defined as H N (q, t, p) := N i=1 j =i Θ p (tx i /x j ) Θ p (x i /x j ) T q,x i . (3.4) First we review the free field realization of the elliptic Macdonald operator. In the following we use the notations in section 2.1. (1 − t n )(qt −1 p) n (1 − q n )(1 − p n ) b −n z −n n exp n>0 1 − t n (1 − q n )(1 − p n ) a −n z n n . (3.5) We use the symbol φ N (p; x) := N j=1 φ(p; x j ). The elliptic Macdonald operator H N (q, t, p) is reproduced by the operator η(p; z) as follows. (1 − t n )(qt −1 p) n (1 − q n )(1 − p n ) b n z −n n exp n>0 1 − t n (1 − q n )(1 − p n ) a n z n n . (3.7) [η(p; z) − t −N (η(p; z)) − (η(p; p −1 z)) + ] 1 φ N (p; x)\|0 = t −N +1 Θ p (t −1 ) (p; p) 3 ∞ H N (q, t, p)φ N (p; x)\|0 . We use the symbol φ * N (p; x) := N j=1 φ * (p; x j ). The elliptic Macdonald operator H N (q, t, p) is reproduced by the operator η(p; z) as follows. 0\|φ * N (p; x)[η(p; z) − t −N (η(p; z)) − (η(p; p −1 z)) + ] 1 = t −N +1 Θ p (t −1 ) (p; p) 3 ∞ H N (q, t, p) 0\|φ * N (p; x). (3.8) Elliptic kernel function and its functional equation For a partition λ, set n λ (a) := ♯{i : λ i = a}, z λ := a≥1 a n λ (a) n λ (a)! and define z λ (q, t, p), z λ (q, t, p) by z λ (q, t, p) := z λ ℓ(λ) i=1 (1 − p λ i ) 1 − q λ i 1 − t λ i , z λ (q, t, p) := z λ ℓ(λ) i=1 1 − p λ i (qt −1 p) λ i 1 − q λ i 1 − t λ i . We define a bilinear form •\|• : F * × F → C by the following conditions. (1) 0\|0 = 1, (2) 0\|a λ 1 b λ 2 a −µ 1 b −µ 2 \|0 = δ λ 1 µ 1 δ λ 2 µ 2 z λ 1 (q, t, p)z λ 2 (q, t, p). By the free field realization, we can show the following theorem. Γ q,p (x i y j ) Γ q,p (tx i y j ) . (3.9) We also define C M N (p; x, y) as C M N (p; x, y) := 0\|φ * M (p; x)[(η(p; z)) − (η(p; p −1 z)) + ] 1 φ N (p; y)\|0 Π(q, t, p)( {x i } M i=1 , {y j } N j=1 ) = M i=1 Θ p (t −1 x i z) Θ p (x i z) N j=1 Θ p (z/y j ) Θ p (t −1 z/y j ) 1 . (3.10) For the elliptic Macdonald operator and the function Π(q, t, p)({x i } M i=1 , {y j } N j=1 ), we have the following functional equation. {H M (q, t, p) x − t M −N H N (q, t, p) y }Π(q, t, p)({x i } M i=1 , {y j } N j=1 ) = (1 − t M −N )(p; p) 3 ∞ Θ p (t) C M N (p; x, y)Π(q, t, p)({x i } M i=1 , {y j } N j=1 ). (3.11) Proof . The proof is straightforward. What we have to do is to calculate the matrix element 0\|φ * M (p; x)[η(p; z)] 1 φ N (p; y)\|0 by the theorem 3.2, 3.3 in different two ways as follows : 0\|φ * M (p; x)[η(p; z)] 1 φ N (p; y)\|0 = t −M +1 Θ p (t −1 ) (p; p) 3 ∞ H M (q, t, p) x Π(q, t, p)({x i } M i=1 , {y j } N j=1 ) + t −M C M N (p; x, y)Π(q, t, p)({x i } M i=1 , {y j } N j=1 ) = t −N +1 Θ p (t −1 ) (p; p) 3 ∞ H N (q, t, p) y Π(q, t, p)({x i } M i=1 , {y j } N j=1 ) + t −N C M N (p; x, y)Π(q, t, p)({x i } M i=1 , {y j } N j=1 ). Therefore we obtain the theorem 3.4. Remark 3.5. We can check the following : C M N (p; x, y) = M i=1 Θ p (t −1 x i z) Θ p (x i z) N j=1 Θ p (z/y j ) Θ p (t −1 z/y j ) 1 −−→ p→0 M i=1 1 − t −1 x i z 1 − x i z N j=1 1 − z/y j 1 − t −1 z/y j 1 = 1. Hence by taking the limit p → 0 the equation (3.11) reduces to the equation (3.3). Proposition 1 . 8 . 18The trigonometric Feigin-Odesskii algebra (A, * ) is unital, associative, and commutative. give an overview of the construction of the commutative families of the Macdonald operator by using the Ding-Iohara algebra and the trigonometric Feigin-Odesskii algebra.Definition 1.9 (Map O). Define a linear map O : A → End(F ) as Proposition 1 . 10 . 110The map O and the star product * are compatible : for f, g ∈ A, we have O(f * g) = O(f )O(g). Corollary 1 . 111 (Commutative family M). (1) Set M := O(A). The space M consists of operators commuting with each other : [O(f ), O(g)] = 0 (f, g ∈ A). Corollary 1 . 114 (Commutative family M ′ ). (1) Set M ′ := O ′ (A). The space M ′ consists of operators commuting with each other. From the relation [[η(z)] 1 , [ξ(w)] 1 ] = 0, we have the following proposition. Proposition 1 . 15 . 115The commutative families M, M ′ satisfy [M, M ′ ] = 0 : For any a ∈ M and a ′ ∈ M ′ , we have [a, a ′ ] = 0. ϕ + (p; z) :=: η(p; γ 1/2 z)ξ(p; γ −1/2 z) :, ϕ − (p; z) :=: η(p; γ −1/2 z)ξ(p; γ 1/2 z) : . Theorem 2. 3 ( 3Free field realization of the elliptic Macdonald operator). Let φ(p; z) : F → F ⊗ C[[z, z −1 ]] be an operator defined as follows. φ(p; z) := expn>0 [a 0 0, Q] = 1, [a n , a 0 ] = [b n , a 0 ] = 0, [a n , Q] = [b n , Q] = 0 (n ∈ Z \ {0}). Theorem 2. 5 ( 5Dual versions of the theorem 2.3, 2.4). Let us define an operator φ * (p; z) : F * → F * ⊗ C[[z, z −1 ]] as follows. φ * (p; z) := expn>0 relation (2.21) we have [[E(p; z)] 1 , [F (p; w)] 1 ] = 0. This corresponds to the commutativity of the elliptic Macdonald operators [H N (q, t, p), H N (q −1 , t −1 , p)] = 0. Definition 2. 10 ( 10Map O p ). We define a linear map O p : A(p) → End(F α ) (α ∈ C) as follows. Theorem 2 . 15 ( 215Commutative family M ′ (p)). (1) Set M ′ (p) := O ′ p (A(p)). The space is a commutative algebra of boson operators. (2) The space M ′ (p)\| Cφ N (p;x)\|N consists of commuting elliptic q-difference operators which contains the elliptic Macdonald operator H N (q −1 , t −1 , p) (commutative family of the elliptic Macdonald operator H N (q −1 , t −1 , p)). Similar to the proposition 1.15, we can show that the commutative families M(p), M ′ (p) commute with each other. Theorem 2.16. For the commutative families M(p), M ′ (p), we have the commutativity [M(p), M ′ (p)] = 0. check (2.32), (2.33) have a period p. By the lemma 2.17, we have the theorem 2.16. By the theorem 2.16, commutative families M(p)\| Cφ N (p;x)\|N , M ′ (p)\| Cφ N (p;x)\|N also commute with each other : [M(p)\| Cφ N (p;x)\|N , M ′ (p)\| Cφ N (p;x)\|N ] = 0.Acknowledgement. The author would like to thank Koji Hasegawa and Gen Kuroki for helpful discussions and comments. kernel function and its functional equation By Komori, Noumi, and Shiraishi[2], the following theorem is shown. Theorem 3.1 ([2]). Define the Macdonald operator H N (q, t) (N ∈ Z >0 ) as Theorem 3. 2 ( 2Free field realization of the elliptic Macdonald operator). Let φ(p; z) : F → F ⊗ C[[z, z −1 ]] be an operator defined as follows. φ(p; z) := expn>0 . 3 ( 3Dual version of the theorem 3.2). Let φ * (p; z) : F * → F * ⊗C[[z, z −1 ]] be an operator defined as follows. φ * (p; z) := expn>0 A Commutative Algebra on Degenerate CP 1 and Macdonald Polynomials. B Feigin, K Hashizume, A Hoshino, J Shiraishi, S Yanagida, arXiv:0904.2291J. Math. Phys. 50B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, S. Yanagida. A Commutative Algebra on Degenerate CP 1 and Macdonald Polynomials. J. Math. Phys. 50 (2009) arXiv:0904.2291. Kernel Functions for Difference Operators of Ruijsenaars Type and Their Applications. Y Komori, M Noumi, J Shiraishi, arXiv:0812.0279v4SIGMA. Symmetry, Integrability and Geometry : Methods and Applications. 5Y. Komori, M. Noumi, J. Shiraishi. Kernel Functions for Difference Operators of Ruijsenaars Type and Their Applications. SIGMA. Symmetry, Integrability and Ge- ometry : Methods and Applications. Volume 5 (2009) arXiv:0812.0279v4. Elliptic Ding-Iohara Algebra and the Free Field Realization of the Elliptic Macdonald Operator. Yosuke Saito, arXiv:1301.4912Yosuke Saito. Elliptic Ding-Iohara Algebra and the Free Field Realization of the El- liptic Macdonald Operator. (2013) arXiv:1301.4912.	[]
[ "Optimal probabilistic storage and retrieval of unitary channels", "Optimal probabilistic storage and retrieval of unitary channels" ]	[ "Michal Sedlák \nInstitute of Physics\nRCQI\nSlovak Academy of Sciences\nDúbravská cesta 984511BratislavaSlovakia\n\nFaculty of Informatics\nMasaryk University\nBotanická 68a60200BrnoCzech Republic\n", "Alessandro Bisio \nQUIT group\nDipartimento di Fisica\nINFN Sezione di Pavia\nvia Bassi 627100PaviaItaly\n", "Mário Ziman \nInstitute of Physics\nRCQI\nSlovak Academy of Sciences\nDúbravská cesta 984511BratislavaSlovakia\n\nFaculty of Informatics\nMasaryk University\nBotanická 68a60200BrnoCzech Republic\n", "\n⊗U )\n\n" ]	[ "Institute of Physics\nRCQI\nSlovak Academy of Sciences\nDúbravská cesta 984511BratislavaSlovakia", "Faculty of Informatics\nMasaryk University\nBotanická 68a60200BrnoCzech Republic", "QUIT group\nDipartimento di Fisica\nINFN Sezione di Pavia\nvia Bassi 627100PaviaItaly", "Institute of Physics\nRCQI\nSlovak Academy of Sciences\nDúbravská cesta 984511BratislavaSlovakia", "Faculty of Informatics\nMasaryk University\nBotanická 68a60200BrnoCzech Republic", "⊗U )\n" ]	[]	U J ⊗ I m (j) J , (S.6) which induces the Hilbert space decomposition H j ⊗H = J∈Irr(U * j ⊗U ) H J ⊗ H m (j) J . Let us denote by j JK the set of values ofUsing Eqs. (S.21) and (S.23) we can assume [27] that R sJ \|. Given this the left hand side of Eq. (S.22) reads ψ\|R s \|ψ = J λ J \|I I\| + ν J I − 1 d \|I I\| , (S.7) U J ⊗ I m (j) J , (S.23) which induces the Hilbert space decompositionFirst, we notice that the multiplicity spaces H m (j) J and H m (j) K are one dimensional and therefore I m (j) J are rank one. From the Schur-Weyl duality, any irreducible representation U j of SU (d) is in correspondence with a young diagram Y j . The defining representation U is represented by a single box . One can verify that there cannot be two equivalent Young diagrams in the decomposition Y j × = K Y K . For a more detailed treatment we refer to [4]. Then we have thatj and H m JJ = span({\|I m (j) J }, j ∈ j JJ ). From Eq. (S.25) the commutation relation of Eq. (S.21) becomes [R s , JK U J ⊗ V K ⊗ I m JK ] = 0, which, thanks to the Schur's lemma, gives R s = J,K I J ⊗ I K ⊗ s (JK) , (S.26) where s (JK) ∈ L(H m JK ), s (JK) ≥ 0. Due to \|I I\| being a rank one operator and R s being the sum of the positive operators from Eq. (S.26) we have that Eq. (S.22) holds if and only if ψ\|I J ⊗ I K ⊗ s (JK) \|ψ = λ JK \|I I\| ∀J, K. (S.27) From the identity I j ⊗ I = J∈Irr(U * j ⊗U ) I J ⊗ I m (j) J (we remind that I m (j) J has rank one), we obtain \|ψ \|I =	10.1103/physrevlett.122.170502	[ "https://arxiv.org/pdf/1809.04552v1.pdf" ]	119,479,836	1809.04552	e372d9a76f14ac6a698cdfac1bc65e5c59b40071	Optimal probabilistic storage and retrieval of unitary channels Michal Sedlák Institute of Physics RCQI Slovak Academy of Sciences Dúbravská cesta 984511BratislavaSlovakia Faculty of Informatics Masaryk University Botanická 68a60200BrnoCzech Republic Alessandro Bisio QUIT group Dipartimento di Fisica INFN Sezione di Pavia via Bassi 627100PaviaItaly Mário Ziman Institute of Physics RCQI Slovak Academy of Sciences Dúbravská cesta 984511BratislavaSlovakia Faculty of Informatics Masaryk University Botanická 68a60200BrnoCzech Republic ⊗U ) Optimal probabilistic storage and retrieval of unitary channels U J ⊗ I m (j) J , (S.6) which induces the Hilbert space decomposition H j ⊗H = J∈Irr(U * j ⊗U ) H J ⊗ H m (j) J . Let us denote by j JK the set of values ofUsing Eqs. (S.21) and (S.23) we can assume [27] that R sJ \|. Given this the left hand side of Eq. (S.22) reads ψ\|R s \|ψ = J λ J \|I I\| + ν J I − 1 d \|I I\| , (S.7) U J ⊗ I m (j) J , (S.23) which induces the Hilbert space decompositionFirst, we notice that the multiplicity spaces H m (j) J and H m (j) K are one dimensional and therefore I m (j) J are rank one. From the Schur-Weyl duality, any irreducible representation U j of SU (d) is in correspondence with a young diagram Y j . The defining representation U is represented by a single box . One can verify that there cannot be two equivalent Young diagrams in the decomposition Y j × = K Y K . For a more detailed treatment we refer to [4]. Then we have thatj and H m JJ = span({\|I m (j) J }, j ∈ j JJ ). From Eq. (S.25) the commutation relation of Eq. (S.21) becomes [R s , JK U J ⊗ V K ⊗ I m JK ] = 0, which, thanks to the Schur's lemma, gives R s = J,K I J ⊗ I K ⊗ s (JK) , (S.26) where s (JK) ∈ L(H m JK ), s (JK) ≥ 0. Due to \|I I\| being a rank one operator and R s being the sum of the positive operators from Eq. (S.26) we have that Eq. (S.22) holds if and only if ψ\|I J ⊗ I K ⊗ s (JK) \|ψ = λ JK \|I I\| ∀J, K. (S.27) From the identity I j ⊗ I = J∈Irr(U * j ⊗U ) I J ⊗ I m (j) J (we remind that I m (j) J has rank one), we obtain \|ψ \|I = We address the question of a quantum memory storage of quantum dynamics. In particular, we design an optimal protocol for N → 1 probabilistic storage-and-retrieval of unitary channels on d-dimensional quantum systems. If we may access the unknown unitary gate only N -times, the optimal success probability of perfect retrieval of its single use is N/(N − 1 + d 2 ). The derived size of the memory system exponentially improves the known upper bound on the size of the program register needed for probabilistic programmable quantum processors. Our results are closely related to probabilistic perfect alignment of reference frames and probabilistic port-based teleportation. Introduction. Since the discovery of the first quantum algorithms [1,2] and protocols [3,4] the information processing with quantum systems has challenged basic paradigms and existing limitations of computer science. In the last few decades we have discovered that quantum information cannot be cloned [5], its "logical value" cannot be inverted [6], quantum processors cannot be universally programmed [7], and universal multimeters do not exist [8,9]. No doubt, any of these programmable devices would represent a very useful piece of quantum technology, thus, their approximate realisations are of foundational interest [8][9][10][11][12]. The no-go restrictions imposed by quantum theory are treated in two ways. Either we ask for an approximate performance, or we allow that the perfect performance happens with some probability of failure. Studies of optimal approximate cloners initiated by Hillery and Bužek [10] demonstrated that such non-ideal devices are of practical relevance and this motivated the study of other universal devices. In particular, it was shown that quantum theory limits the fidelity of 1 → N clones of qubits to (2N + 1)/3N [13]. For quantum processors Nielsen and Chuang [7] proved that perfect (error free) implementation of k distinct unitary transformations requires at least k dimensional program register. Recently, the cloning was considered also for quantum transformations [14,15]. This unveiled an unexpected feature called super-replication [16,17]. In this protocol, starting with N copies of a qubit unitary transformation U one deterministically generates up to N 2 copies of U with an exponentially small error rate. While studying cloning of unitaries it was realized there is a closely related task of storage-and-retrieval (SAR), which only differs in the causal order of available resources. While in the cloning the cloned device is available after the input states are at the disposal, one can consider also a task where this order is reversed, thus, the device is available only before the input states. In such case, we need to learn [18] and somehow store the action of the device and retrieve it once the input states are available. Problem formulation. The devices transforming states of d-dimensional quantum systems associated with Hilbert space H are formalized as quantum channels, i.e. completely positive trace-preserving linear maps on the space L(H) of linear operators on H. Suppose an unknown channel U is provided for experiments and we may access it N times. However, we are asked to apply U on an unknown state ξ only after we lost the access to this channel. Therefore, our aim is to find an optimal strategy that stores U in a state of a quantum memory (associated with Hilbert space H M ) and allows us to retrieve its action when needed. In the approximative settings this task (for unitary channels) was studied in Ref. [19]. Our goal is to investigate the probabilistic version of the SAR problem, in particular, we aim to find the optimal N → 1 probabilistic storage and retrieval procedure (PSAR). Moreover, we require the retrieved channel to be implemented perfectly and with the same probability of success ("covariance" property) for all considered channels. We will design the strategy maximizing the probability for the set of unitary channels, i.e. U(ξ) = U ξU † for some unitary operator U . Due to no-programming theorem [7], the retrieving part of any PSAR strategy cannot be deterministic. Thus, the successful retrieval is described by a trace-non-increasing completely positive linear map (quantum operation) T U : L(H) → L(H) proportional to the unknown unitary channel, T U = λ U U. Consequently, the success probability is λ U = tr[T U (ξ)] and the condition of covariance implies λ U = λ for all U . One-to-one probabilistic storage-and-retrieval. L(H in ⊗ H M ) → L(H out ) with H = H in = H out . The retrieving quantum instrument plays the role of a probabilistic programmable processor and the state \|ψ U programs a unitary transformation U to be performed on a state ξ. Using the Choi isomorphism [20] we have that R s (ξ ⊗ \|ψ U ψ U \|) = tr in,M [(I ⊗ ξ T ⊗ \|ψ U ψ U \| T )R s ] = λtr in [(I ⊗ ξ T )\|U U \|] = λU ξU † , where R s ∈ L(H out ⊗ H in ⊗ H M ) and \|U = √ d(U ⊗ I)\|ψ + with \|ψ + = d −1/2 j \|j ⊗ \|j (vectors {\|j } form an orthonormal basis of H = H in = H out ). Since the above identity must hold for any ξ and \|ψ U ψ U \| T = \|ψ * U ψ * U \| (both the transposition and the conjugation are defined with respect to the same basis of H M ) we obtain the following perfect retrieval condition ψ * U \|R s \|ψ * U = λ\|U U \| ∀U ∈ SU (d) . (S.1) Already this simple case shows that the maximization of probability of success λ involves the simultaneous optimization of the storing phase (choice of \|ψ ) and the retrieving phase (choice of quantum instrument R). It turns out that the optimal performance is achieved by the (incomplete) quantum teleportation protocol [4] that is a known example of a universal probabilistic quantum processor [21]. Let us note that this is similar to quantum gate teleportation invented by Gottesman and Nielsen [22], yet it is different, because PSAR must work perfectly for any unitary transformation. In particular, for the storing phase we set \|ψ = \|ψ + . Then the optimal retrieval is achieved by a quantum teleportation of state ξ using the stored state \|ψ U = d −1/2 \|U (see Fig. S.1). The generalized Bell measurement performed on ξ and one part of \|ψ U results in an outcome k with probability 1/d 2 . In such case we are left with the second part of \|ψ U in the state U σ k ξσ k U † , where σ k are generalized Pauli operators. In case of σ k = I (associated with the Bell measurement projection onto \|ψ + ) the stored unitary channel is successfully retrieved. For all the other outcomes, the unwanted σ k rotation can not be undone, because the unitary U is unknown. In conclusion, the teleportation-based PSAR succeeds with probability 1/d 2 . Its optimality follows from our subsequent discussion of the optimal N → 1 PSAR. N-to-one probabilistic storage-and-retrieval. The general PSAR strategy with N uses of a channel in the storing phase involves all combinations of their parallel, successive and adaptive processing and corresponds to a quantum circuit with open slots, where the N uses of a channel can be inserted. Such framework is described within the theory of quantum networks [1][2][3]24] and any quantum circuit with open slots is represented by a positive operator (see [27] for a short introduction). The where λ gives the success probability. Let us stress that the probability of success, i.e. the value of λ, is required to be the same for all U ∈ SU (d). Thanks to this assumption we can without loss of generality apply the methods of [19] to conclude that the optimal storing phase is parallel as illustrated in Fig. S.2b. Consider the decomposition U ⊗N = j∈Irr(U ⊗N ) U j ⊗ I mj into irreducible representations (IRRs), where U j is a unitary operator on H j and I mj denotes the identity operator on the multiplicity space. This corresponds to the following decomposition of the Hilbert space H A := j∈Irr(U ⊗N ) H j ⊗ H mj , and we set d j = dim(H j ). The result of [19] implies that the memory state \|ψ can be taken of the following form \|ψ := j p j d j \|I j ∈ H M p j ≥ 0, j p j = 1 , (S.3) where I j denotes the identity operator on H j and H M := j∈Irr(U ⊗N ) H j ⊗ H j ⊆ H A ⊗ H A . The state \|ψ undergoes the action of the unitary channels and becomes \|ψ U := j pj dj \|U j . Clearly, \|ψ U ∈ H M for any U . Let us now focus on the retrieving quantum instrument R from L(H in ⊗H M ) to L(H out ), where in/out labels the system on which the retrieved channel is applied. The perfect retrieval condition is again given by Eq. (S.1) with \|ψ * U = j pj dj \|U * j . As a consequence of Eq. (S.2) the optimal Choi operator R s can be chosen to satisfy the commutation relation Theorem 1. For optimal PSAR the success probability λ is given by the following linear programming problem: [R s , U * V ⊗ U in ⊗ V * out ] = 0, (S.4) where U := j U j ⊗ I j , V := j I j ⊗ V j .maximize µ J ,pj λ = J∈C d 3 J µ J , (S.8) subject to 0 ≤ d J µ J ≤ p j d 2 j ∀j ∈ j JJ ∀J ∈ C p j ≥ 0 j∈Irr(U ⊗N ) p j = 1 , where C = {J ∈ Irr(U ⊗N ⊗ U * )\|dd J = j∈j JJ d j }. Proof. We will sketch only the key steps. The complete proof is in [27]. First, one shows that J / ∈ C implies s (J) = 0. Then, (for any J ∈ C) ν J = 0 and s (J) ≥ 0 im- Case study: N → 1 PSAR for qubit channels. In case [28] and U j are the IRRs of spin j with dimension d j = 2j + 1. For convenience we work with even N (for odd N see [27]), so j = 0, 1, . . . , N/2. ply that √ p j p j s (J) jj = µ J d 3 j d 3 j for some µ J ≥ 0. Thus, λ = J∈C j,j ∈j JJ d J µ J d 2 d j d j = J∈C d 3 J µ J .of qubit (d = 2) the decomposition of U ⊗N into IRRs of SU (2) reads U ⊗N = N/2 j=(N mod 2)/2 U j ⊗ I mj , where m j = 2j+1 N/2+j+1 N N/2+jFor SU (2) the complex conjugate representation U * j is equivalent to IRR U j . Thus, in Eq. (S.23) we get either J = j +1/2 or J = j −1/2. Altogether, J can have values J ∈ C = {1/2, . . . , (N − 1)/2} or J = (N + 1)/2 / ∈ C, because j∈j JJ d j = d J−1/2 + d J+1/2 = dd J and d N/2 = 2d (N +1)/2 . The constraints in Eq. (S.51) imply for any j but j = 0, N/2 the following two inequalities µ j+1/2 d 2 j d j+1/2 ≤ p j , (S.9) µ j−1/2 d 2 j d j−1/2 ≤ p j . (S.10) For j = 0, N/2 only one of them exists. Let us define f j ∈ [0, 1] for j = 0, . . . , N 2 as f j = 1 2 2j 2j+1 2j+2 N + 1 . Since f 0 = 0 and f N/2 = 1 we can multiply Eq. (S.9) by 1 − f j and Eq. (S.10) by f j , and take the sum for all j. A straightforward calculation gives the upper bound: N + 3 N N −1 2 J= 1 2 d 3 J µ J ≤ 1 ⇔ λ ≤ N N + 3 . (S.11) Finally, by choosing p j = (2j+1) 2 /L, µ j+1/2 = 1/(L(2j+ 2)) (where L = (N + 1)(N + 2)(N + 3)/6), one proves that that conditions in Eq. (S.51) are satisfied and the upper bound (S.11) is achieved. The knowledge of µ J and p j completely specifies the state \|ψ and the retrieving operation R s which can be explicitly expressed (see Fig. S.2b). Let \|j, j z ∈ H j with j z ∈ {−j, . . . , j} be an orthonormal basis of the spin j IRR. By definition \|I j = j jz=−j \|j, j z ⊗ \|j, j z . Consequently, from Eq. (S.59), the dimension of the quantum memory is dim H M = N/2 j=0 d 2 j = L and the optimal input state for storage is \|ψ = N/2 j=0 2j+1 L \|I j . Optimal PSAR for qudit unitary transformations. The optimization of N → 1 PSAR of qudit channels follows similar steps as for the qubit case and it exploits a combinatorial identity (Proposition 3 in [6]) which was discovered and proved as a byproduct of this analysis. The proof is given in [27]. Clearly, as N goes to infin- ity λ ∼ 1 − d 2 −1 N , and λ ≈ 1 2 implies N ≈ d 2 . Reminding that a d-dimensional unitary transformation has d 2 parameters, we see that roughly one use per unknown parameter is needed for reliable storage and retrieval of the transformation. Let us note that the storage state in Theorem 2 is optimal also for the estimation of a group transformation in the maximum likelihood approach [30]. Further, it is worth to stress that the optimal PSAR protocol is achieved by a coherent retrieval, hence, the quantum memory is essential. In contrast, optimal approximate SAR [19] is equivalent to quantum estimation in the maximum fidelity approach and classical memory is sufficient as an output of the storing phase. Use of the optimal storage state in the design of an approximate SAR leads to fidelity that scales as 1 − O(N −1 ), however, for the optimal approximate SAR the fidelity scales as 1 − O(N −2 ) [19]. This O(N ) difference is the price to pay for the perfect retrieval in case of PSAR. Alignment of reference frames [8]. (ARF) Let us note that the correction of alignment errors can be modeled as a PSAR protocol in which N uses of an unknown U are stored and the aim is to retrieve the inverse transformation U † . For SU (2), we can show that, given N uses of U, the inverse transformation U −1 can be perfectly retrieved with the same optimal probability of success λ (see Probabilistic port-based teleportation. (PPBT) As the first step of PPBT [32] Alice and Bob share N suitably entangled pairs of quantum systems. Their goal is to teleport an unknown state ξ to Bob in a way that this state appears in one of his systems (called ports [33,34]). In order to achieve this goal (see also Fig. S.4) Alice performs a specific measurement resulting in n ∈ {0, 1, . . . , N } (0 labels the failure of the protocol), communicates this information to Bob who selects the system from the nth port to accomplish the teleportation. If Bob applies a channel U on each of his ports (storing phase) and Alice starts the teleportation (retrieving phase) of ξ afterwards, the nth port will output U(ξ). Strictly speaking, we swap nth port into a fixed quantum system and effectively we achieve N → 1 PSAR. Let us stress that while any PPBT protocol can be turned into a PSAR protocol, the converse does not hold. In a sense, PPBT scheme provides a structurally simple realization of an optimal PSAR protocol. Our results show that the optimal probability of PPBT [35] coincides with the optimal success probability of PSAR. However, the memory dimension dim H M of the optimal PSAR is exponentially smaller (see the following paragraph) in comparison with 2N qudits used in PPBT construction. Implications for covariant probabilistic programmable processors. Up to now the best bound on the size of the program register for universal covariant probabilistic processors was provided by family of PPBT processors for which dim H M ≈ (d 2(d 2 −1) ) 1/f , where f = 1 − λ is the failure probability. In contrast, the retrieving phase of optimal N → 1 PSAR defines a class of processors for which the program register size reads dim H M = j∈Irr(U ⊗N ) d 2 j = N +d 2 −1 N , where we used Schur's result [36]. In terms of the failure probability it reads dim H M ∝ (1/f ) (d 2 −1) , which is exponentially smaller (for fixed d and f → 0) in comparison with PPBT-based processors. This result can be viewed as a quantification of achievable tradeoffs imposed by the no-programming theorem [7] on universal covariant probabilistic processors. Although PSAR provides only an upper bound on the size of the program register, we conjecture that the lower bound will have the same scaling. However, this question remains open. Summary. We showed that optimal probabilistic storage-and-retrieval of unknown unitary channels on ddimensional quantum systems can be designed with success probability λ = N/(N −1+d 2 ), where N is the number of uses of the channel in the storing phase. This probability coincides with the success probability for probabilistic port-based teleportation [35], and, for the SU (2) case, with the probability of success for probabilistic alignment of reference frames. Optimal PPBT can be rephrased as an optimal protocol for PSAR, but for the PSAR protocol designed here the storing memory system is exponentially smaller and optimal in this parameter. On the other hand, N → 1 PPBT-based PSAR implements all quantum channels (not only unitary ones), thus, its performance is universal. The question of potential reduction of memory system while keeping the universality for all channels remains open. I. SUPPLEMENTAL MATERIAL This Supplemental Material provides a short introduction to theory of quantum networks, detailed proofs of Theorems 1,2 and more precise clarification of the relation of the presented work to the alignment of reference frames. II. QUANTUM NETWORKS AND GENERALIZED INSTRUMENTS The mathematical formalization of the perfect learning of a unitary channel can be easily given within the framework of quantum networks. In this section we provide a small review of the subject and we refer to the literature [1][2][3] for a complete presentation. We will start by introducing some notation. If H and K are finite-dimensional Hilbert spaces, then we denote with L(H) the set of linear operator on H and with L(H, K) the set of linear operator from H to K. We will use the one-to-one correspondence between linear operators A ∈ L(H, K) and vectors \|A ∈ K ⊗ H and given by \|A = dim(K)O * O = Tr K [(O ⊗ I 456 )(I 123 ⊗ O T K )] (S.16) where O T K denotes the partial transposition of O on the Hilbert space K and I ijk denotes the identity operator on H i ⊗H j ⊗H k . We can interpret Eq. (S.15) as an instance of Eq. (S.16). A quantum network R consists in a sequence of multipartite quantum operations {O i , i = 1, . . . N } where some output of a O i is connected to some input of the following quantum operation O i+1 as we illustrate in the following diagram: 0 O 1 1 2 O 2 3 2N −2 O N 2N −1 · · · , (S.17) where the folating wires correspond to the input and output systems of the quantum network. R is called a deterministic quantum network if all the quantum operations in Eq. (S.17) are trace preserving, and it is called a probabilistic quantum network otherwise. A quantum network can be represented by a Choi operator (commonly called quantum comb) which is given by the link product of all the component quantum operations. The Choi operator R of a deterministic quantum network R obeys the following constraints Tr 2k−1 [R (k) ] = I 2k−2 ⊗ R (k−1) k = 1, . . . , N (S.18) where, referring to the diagram in Eq. (S.17), the Hilbert space of the wire labelled by j is H j , R (N ) = R, R (0) = 1, R (k) ∈ L(H odd k ⊗ H even k ) with H even k = k−1 j=0 H 2j and H odd k = k−1 j=0 H 2j+1 . R (k) is the Choi operator of the reduced network R (k) obtained by discarding the last N − k teeth. The set of of positive operators satisfying Eq. (S.18) and the set of deterministic quantum networks are in one to one correspondence. On the other hand, a given deterministic quantum network R can be realized as a composition of quantum channels in many different ways. In the probabilistic case, the Choi operator of a probabilistic quantum network T , must satisfy 0 ≤ T ≤ R (S.19) where R is the Choi operator of a deterministic quantum network. A given probabilitic quantum network T can be realised as a composition of quantum operations in many different ways. In particular, any probabilitic quantum network T can be realised by a composition of channels {C} and a final quantum operation O as follows: T = 0 C 1 1 2 C 2 3 2N −2 O 2N −1 · · · . (S.20) A set of probabilistic quantum networks {R i }, with the same input and output wires, is called a generalised quantum instrument if the sum of their Choi operators i R i =: R is the Choi operator of a deterministic quantum networks. As in the analogous case of quantum instruments, the index i which labels the elements of a generalised quantum instrument represents the classical outcome which is available after the quantum network has been provided with some input. If the outcome i is obtained then it means that the probabilistic quantum network R i happened. Any generalised quantum instrument can always be realised by a a composition of channels followed by a final quantum intrument. We notice that for any probabilistic quantum network there exists a generalised quantum instrument which it belongs to. III. RELEVANT SUB-BLOCKS OF RETRIEVING OPERATION Rs As we stated in the main text, Choi operator R s of the retrieving operation can be chosen to satisfy the commutation relation [R s , U * V ⊗ U in ⊗ V * out ] = 0, (S.21) where U := j U j ⊗ I j , V := j I j ⊗ V j . We remind also the perfect retrieving condition ψ\|R s \|ψ = λ\|I I\|. (S.22) For convenience we placed here also the decomposition U * j ⊗ U = J∈Irr(U * j ⊗U ) p j d j \|I m (j)ψ\|R (J) s \|ψ = = ψ\|(U U * ⊗ U * ⊗ U )R (J) s (U U * ⊗ U * ⊗ U ) † \|ψ = (U * ⊗ U ) ψ\|R (J) s \|ψ (U * ⊗ U ) † ∀U (S.33) Thanks to the Schur's lemma Eq. (S. 33) gives ψ\|R (J) s \|ψ = λ J \|I I\| + ν J I − 1 d \|I I\| . (S.34) By taking the trace of Eq. (S.34) we have Tr[ ψ\|R (J) s \|ψ ] = ψ\| Tr out in [R (J) s ]\|ψ = ψ\| j∈j JJ I j ⊗ I j q (J) j \|ψ = j∈j JJ p j q (J) j = λ J d + ν J (d 2 − 1) (S.35) q (J) j := d 2 J d 2 j s (J) jj . If we insert Eq. (S.29) into Eq. (S.30) we have λ J = d J d 2 j,j ∈j JJ p j p j d j d j s (J)ν J = 0 ⇐⇒ j,j ∈j JJ δ j,j d d J p j d 2 j s (J) jj − p j p j d j d j s (J) jj = 0, (S.37) which is the most explicit form of the perfect retrieving condition that constraints the relation between the state \|ψ parametrized by probabilities p j and the structure of the retrieving operation parameterized by s (J) jk . V. N → 1 PSAR AS A LINEAR PROGRAMMING PROBLEM In this section we provide complete proof of Theorem 1 from the main text. First, we prove the following technical lemma, which will be needed. Lemma 1. Suppose a matrix X = j,j X jj \|j j \| ≥ 0 obeys j,j X jj = j 1 cj X jj , where c j > 0 and j c j = 1. This implies X ∝ \|χ χ\|, where \|χ = j c j \|j . Let us restate Theorem 1 from the main text. Theorem 3. For optimal PSAR the success probability λ is given by the following linear programming problem: maximize µ J ,pj λ = J∈C d 3 J µ J , (S.51) subject to 0 ≤ d J µ J ≤ p j d 2 j ∀j ∈ j JJ ∀J ∈ C p j ≥ 0 j∈Irr(U ⊗N ) p j = 1 , where C = {J ∈ Irr(U ⊗N ⊗ U * )\|dd J = j∈j JJ d j }. Proof. We first need to examine relations between IRR's that appear in the decomposition of U ⊗N and those that appear in j∈Irr(U ⊗N ) U (j) ⊗ U * . We remind that from the Schur-Weyl duality, any irreducible representation U j of SU (d) is in correspondence with a young diagram Y j . The defining representation U is represented by a single box and IRR defined via U * is represented by a column of d − 1 boxes. Decomposition of U ⊗N into IRRs can be obtained by collecting the decompositions of the tensor products U k ⊗U of all Young diagrams k appearing with multiplicity m k in the decomposition of U ⊗N −1 and putting together equivalent IRRs (those with the same Young diagram). This can be mathematically stated as follows. Let Irr(U ⊗N ) denote the set of Young diagrams that appear in the decomposition of U ⊗N into IRRs of SU (d). We have that K ∈ Irr(U ⊗N ) if and only if ∃k ∈ Irr(U ⊗N −1 ) such that K ∈ Irr(U k ⊗ U ) and m K = k∈k K m k , where k K denotes the set of values of k such that U K is in the decomposition of U k ⊗ U . On the other hand, thanks to Schur-Weyl duality the multiplicity m K = d K (m k = d k ) is given by the dimension of the IRRs of the symmetric group S(N ) (S(N − 1)) with the Young diagram K (k), respectively. Hence, we obtained a known identity [4] d K = k∈k K d k , (S.52) where k K can be equivalently specified as those Young diagrams k, which by addition of a single box become K. Next, we consider decomposition of U j ⊗ U * (or more conveniently U * ⊗ U j ), where j ∈ Irr(U ⊗N ). We denote Young diagram Y j with r rows and n i boxes in the i-th row as (n 1 , n 2 , . . . , n r ). A valid Young diagram of SU (d) IRR has r ≤ d, n r > 0 and n i ≥ n i+1 ∀i (we set n r+1 = 0). Rows i in which n i > n i+1 we call corners of (n 1 , n 2 , . . . , n r ) and we denote the number of corners by s and we write i ∈ Cor j . Suppose Y j ↔ (n 1 , n 2 , . . . , n r ) has r ≤ d − 1. Then the decomposition of U * ⊗U j contains s+1 Young diagrams each with multiplicity one. One of them is given as Young diagram (n 1 + 1, n 2 + 1, . . . , n r + 1, 1, . . . , 1) with d − 1 rows, which we denote Y \|j and for each i ∈ Cor j we have Young diagram Y j\ i ↔ (n 1 , . . . , n i − 1, . . . , n r ). The above statement follows from the Littlewood-Richardson rules [4] if one realizes, that either one of the corner boxes completes the first column into d boxes (the remaining boxes can be only attached to the right in the original order) or the whole Young diagram is attached from the right to the column of d − 1 boxes. If Y j ↔ (n 1 , n 2 , . . . , n r ) has r = d the situation is the same except for the diagram Y \|j not appearing in the decomposition, because it would not be a valid Young diagram. Let us note that Young diagram Y \|j can emerge in our setting only from diagram Y l , where l = j. We can also easily verify that dd \|j = d j , which can be seen from the formula for the dimension of SU (d) IRRs [5] by calculating the fraction d \|j /d j for a general j. Therefore, we conclude that for J = \|j Eq. (S.37) can be satisfied only if s (\|j) jj = 0, which in turn thanks to Eq. (S.30) implies λ \|j = 0. Thus, Young diagrams C = {Y \|j , j ∈ Irr(U ⊗N )} correspond to those J that do not belong to the set C defined in the theorem. On the other hand, consistently with the notation for C, we define C = {Y j\ i , j ∈ Irr(U ⊗N ), i ∈ Cor j }. Let us remind that Irr(U ⊗N ) is exactly constituted by all Young diagrams consisting of N boxes and having at most d rows. This implies C = Irr(U ⊗N −1 ), because by removing in any possible way a single box from Young diagrams in Irr(U ⊗N ) we get all possible Young diagrams in Irr(U ⊗N −1 ). More operationally, for any Young diagram J ∈ C we can add a box to the first row and get some element j ∈ Irr(U ⊗N ), which can be reversed to prove the claim. Moreover, for every subset C j = {Y j\ i , i ∈ Cor j } of C we have that d j = J∈Cj d J , (S.53) which is just a reformulation of Eq. (S.52), because Young diagrams J ∈ C j have N − 1 boxes and an addition of a single box changes them to Young diagram j consisting of N boxes. Let us pick any element J ∈ C. Let us now specify all the Young diagrams Y j , j ∈ Irr(U ⊗N ), which contain J in the decomposition of U j ⊗ U * . We denote such set j J and it coincides with j JJ defined below Eq. (S.25). These are such Young diagrams j in which by removing one corner box we get Y J . This is the same as saying that j J is the set of Young diagrams of SU (d) group that can be obtained from J by addition of a single box, because Irr(U ⊗N ) contains all possibly emerging Young diagrams. This implies that dd J = j∈j J d j , because this corresponds to the decomposition of an operator U J ⊗ U , which acts on dd J dimensional space. Thus, we proved that the set C can be equivalently defined as C = {J ∈ Irr(U ⊗N ⊗ U * )\|dd J = j∈j JJ d j } = {Y j\ i , j ∈ Irr(U ⊗N ), i ∈ Cor j } = Irr(U ⊗N −1 ) (S.54) Furthermore, we showed that for J / ∈ C s J = 0 and consequently λ J = 0. In order to proceed we apply Lemma 1 for every J ∈ C. Expression pj p j dj d j s (J) jj plays the role of X jj , c j = dj dd J and the remaining assumption is guaranteed by Eq. (S.37). As a consequence, we get that the condition (S.22) of perfect retrieving and s (J) ≥ 0 is equivalent to s (J) jj = µ J d 3 j d 3 j p j p j µ J ≥ 0 ∀J ∈ C (S.55) Thus, fulfillment of Eq. (S.55) guarantees the perfect retrieving of unitary transformations and we can rewrite the probability of success as jj . This implies λ = J∈C j,j ∈j JJ d J d 2 µ J d j d j = J∈C d 3 J µ J ,(Tr D [R s ] ≤ I ⇔ d J d j s (J) jj ≤ 1 ∀J, ∀j ∈ j JJ . (S.57) Let us express the above condition via coefficients µ J using Eq. (S.55) µ J d 2 j ≤ p j d J ∀J, ∀j ∈ j JJ . (S.58) Let us remind the definition of state \|ψ from the main text. All the steps are completely analogical to the derivation valid for even N presented in the main text. The main difference is that the IRR's with minimum and maximum spin (J = 0 and J = N +1 \|ψ := j p j d j \|I j ∈H p j ≥ 0, j p j = 1 , 2 ) have only multiplicity one. For odd N (identically as for even N ) the investigation of the conditions of perfect learning reveals that s N +1 2 has to be zero. On the other hand, J = 0 can be involved in the perfect storing and retrieving. Other expressions remain identical, but now J is an integer. In particular, we choose f J according to the same formula as in the main text f j = 1 2 2j 2j + 1 2j + 2 N + 1 (S.60) and the whole proof goes on analogically to the case of even N . VII. N → 1 PSAR FOR QUDIT CHANNELS - THE GENERAL CASE OF SU (d) The goal of this section is to prove Theorem 4 from the main text. For the proof of the main theorem we need a new theorem from combinatorics [6] and a technical lemma. Theorem 5. For any Young diagram J consisting of N − 1 boxes it holds that j∈J (C j − R j ) 2 d j = N (N − 1) d J , (S.63) where the sum runs through all Young diagrams j that can be obtained from J by addition of a single box, • d J , d j are dimensions of IRRs of the symmetric group S(N − 1), S(N ), respectively • C j is the number of the column of the added box, • R j is the number of the row of the added box that leads from diagram J to the diagram j. Lemma 2. For any Young diagram J ∈ Irr(U ⊗N −1 ) the following identity for dimensions d J , d j of IRRs of SU (d) group and for dimensions of d J , d j of the symmetric group, holds j∈j JJ d 2 j d j = N − 1 + d 2 N d 2 J d J ∀J ∈ Irr(U ⊗N −1 ), (S.64) where j JJ = {j ∈ Irr(U ⊗N ) \| J ∈ Irr(U j ⊗ U * )}. Proof. Let us remind expressions for the dimensions of IRRs that are involved (for detailed explanation see [4]): d j = l j h j d J = l J h J d j = N ! h j d J = (N − 1)! h J , (S.65) where h j , h J denote the hook lengths factors and l j is Thus, proving Lemma 2 is equivalent to proving that Eq. (S.66) holds. We start by direct evaluation of the left hand side. We obtain: box∈Yj (d − R i + C i ) (here R i (C i )j∈j JJ l j 2 l J 2 h J h j = j∈j JJ (d − R j + C j ) 2 h J h j (S.67) where R j is the row number and C j the column number of the additional box in Young diagram Y j with respect to Y J . At this point it is useful to realize that for Young diagrams J ∈ Irr(U ⊗N −1 ) with d-rows, there is a difference between the set j JJ = j J and the set J of all Young diagrams that can be obtained from J by addition of a single box. The difference is exactly one Young diagram, which is obtained from J by adding the box into the d + 1-th row, in the first column. Luckily, the bracket (d − R j + C j ) for this diagram evaluates to zero (d − (d + 1) + 1 = 0), so we can sum also through this term in Eq. (S.67) without changing its value. This is useful especially for d < N , because later on we want to apply Theorem 5, where the summation runs through the set J . Thus, left hand side of Eq. (S.66) can be equivalently rewritten as j∈j JJ l j 2 l J 2 h J h j = j∈J (d − R j + C j ) 2 h J h j = F + G + H, (S.68) where we expanded the square and we defined F = d 2 j∈J h J h j G = j∈J (C j − R j ) 2 h J h j (S.69) H = j∈J 2d(C j − R j ) h J h j . (S.(d − R j + C j ) h J h j = j∈J l j l J h J h j = j∈j JJ d j d J = d, (S.73) where we used Eq. (S.54) and the fact that d j = 0 if j has more than d rows. On the other hand j∈J (d − R j + C j ) h J h j = d j∈J h J h j + j∈J (C j − R j ) h J h j (S.74) = d + 1 2d H (S.75) and then H = 0. Combining the above considerations equation (S.68) reads j∈j JJ l j 2 l J 2 h J h j = d 2 + j∈J (C j − R j ) 2 h J h j . (S.76) Comparing Eq. (S.76) with Eq. (S.66) we conclude we still need to prove j∈J (C j − R j ) 2 h J h j = N − 1 . (S.77) Luckily, the above equation is exactly the claim of Theorem 5 written using Eq. (S.65). Thus, relaying on Theorem 5 we conclude the proof. Let us continue with the proof of Theorem 4. We multiply inequality (S.58) for every J ∈ Irr(U ⊗N −1 ) and every j ∈ j JJ by f (j, J). We sum these inequalities and thanks to Eqs. (S.62), (S.59) we get J∈Irr(U ⊗N −1 ) j∈j JJ f (j, J) d 2 j d J µ J ≤ j∈Irr(U ⊗N ) J∈Cj f (j, J)p j ≤ j∈Irr(U ⊗N ) p j = 1 . (S.78) Let us define z J ≡ d J j∈j JJ f (j, J) d 2 j ∀J ∈ Irr(U ⊗N −1 ), = d J d J j∈j JJ d 2 j d j = N − 1 + d 2 N d 3 J (S.79) where we used Eq. (S.61) and Lemma 2. Using the definition (S.79) we can rewrite inequality (S.78) as J∈Irr(U ⊗N −1 ) z J µ J ≤ 1. (S.80) We remind that Irr(U ⊗N −1 ) = C. Taking this into account inequality (S.80) directly implies where we used Lemma 2, exchanged the order of the sums and used the Eq. (S.62). Thanks to knowledge of µ J and p j we can completely specify the state \|ψ and the retrieving operation R s . Thus, we can build valid storing and retrieving strategy, which succeeds with probability N/(N − 1 + d 2 ) saturating the upper bound (S.81) and concluding the proof. N − 1 + d 2 N J∈C d 3 J µ J ≤ 1 ⇔ λ = J∈C λ J ≤ N N − 1 + d 2 . VIII. ALIGNMENT OF REFERENCE FRAMES We now review the quantum protocol for the alignment of reference frames in a quantum communication scenario, as it was considered in Ref. [8]. Let us consider the scenario in which one party, called Alice, wants to send a qubit to another distant party, denoted as Bob. If the qubit is encoded into a spin-1/2 particle Bob can recover the quantum state \|ϕ if he and Alice share a reference frame for orientation. Otherwise, the lack of a shared frame amounts to having a noisy channel and Bob receives a decohered state ρ = SU (2) U \|ϕ ϕ\|U † dU . This problem can be circumvented if Alice, along with the quantum message \|ϕ , sends a state \|ψ as a token of her reference frame. Then Bob receives the state ρ ψ = SU (2) \|ψ U ψ U \| ⊗ U \|ϕ ϕ\|U † dU , from which he tries to retrieve the message \|ϕ . In the perfect retrieving scenario, Bob wants to maximize the probability for recovering \|ϕ without any error. This scenario is equivalent to a storage and retrieval protocol: • the token \|ψ plays the role of the storage state. \|ψ is a multipartite state \|ψ ∈ H ⊗N • The effect of the misalignment can be thought of as the storing phase in which the state \|ψ U := U ⊗N \|ψ is created. • In the retrieving phase, Bob exploits the state \|ψ U to retrieve the inverted channel U † which is applied to the qubit U \|ϕ . There are two differences between this protocol and the SAR we consider in our work. The first difference is that N uses of U are given, but we are required to retrieve U † . However, for U ∈ SU (2), we can show that our optimal PSAR protocol which stores U ⊗N and retrieves U , can be turned into a PSAR protocol which retrieves U † with the same probability of success. If we had \|ψ U † , then the retrieval phase of our optimal PSAR protocol would recover U † with the optimal probability of success λ (which is the same for any U ∈ SU (2)). In particular, for U ∈ SU (2), the storage state \|ψ U † can be created by exploiting N uses of U as follows \|ψ U † = U †⊗N ⊗ I\|ψ = I ⊗ U * ⊗N \|ψ = = I ⊗ (σ y U σ y ) ⊗N \|ψ (S.84) where \|ψ is the optimal state for storage. The second difference between SAR and the alignment protocol is that we are not allowed to use an external reference system, i.e. the ancillary system H A in our protocol, since it would correspond to a partially shared reference frame. Since our protocol is less constraint than the alignment protocol, the probability of success λ of PSAR is an upper bound for the probability of success of perfect alignment. However both the strategy of Ref. [8] and the optimal PSAR protocol achieve the same O(N −1 ) scaling, which is then optimal. PACS numbers: 03.67.-a, 03.67.Ac, 03.65.Fd . S.1: Optimal 1 → 1 PSAR of unitary channels. FIG . S.2: Illustration of N → 1 PSAR Top: PSAR with the most general strategy. Bottom: PSAR with parallel use of unitary channels. storing network is described by an operator S. It accepts N channels as its input and it outputs a memory state \|ψ U ∈ H M (see Fig. S.2a). As in 1 → 1 case the retrieving phase is described by a two-valued instrument R = {R s , R f }. The overall action of PSAR is a composition of S and R determining a generalized quantum instrument L = {L s , L f }. In the Choi picture the input of PSAR corresponds to \|U U \| ⊗N ∈ L(H A ⊗ H B ) and L s ∈ L(H A ⊗H B ⊗H out ⊗H in ), where H A = H B = H ⊗N . The perfect retrieval condition (similarly to Eq. (S.1)) is U * \| ⊗N L s \|U * ⊗N = λ\|U U \| ∀U ∈ SU (d), (S.2) Thanks to Eq. (S.21), U \|ψ = \|ψ U and \|ψ * I = \|ψ the perfect retrieval condition becomes ψ\|R s \|ψ = λ\|I I\| (S.5) and the success probability reads λ = 1 d 2 I\| ψ\|R s \|ψ \|I . Let us now consider the decomposition where ν J are specified in [27], λ J = d J d 2 φ J \|s (J) \|φ J and \|φ J = j∈j JJ pj dj \|I m (j) J . Since R s ≥ 0, the perfect learning condition of Eq. (S.22) holds only if ν J = 0 for all J. Then, the success probability is λ = J λ J . The following result translates the optimisation of λ from an operator optimisation problem into a linear program. Theorem 2 . 2The optimal probability of success of N → 1 probabilistic storage and retrieval of a unitary channel U(.) = U.U † , U ∈ SU (d) equals λ = N/(N −1+d 2 ). The optimal state for storage is \|ψ := j dj L \|I j ∈ H M where L := j d 2 j and j ∈ Irr(U ⊗N ). Storing Retrieving Success FIG. S.3: A modified optimal 1 → 1 PSAR in which U is stored and the inverse transformation U † is retrieved (SU (2) case). The generalisation to the N → 1 is straightforward. Fig. S.3 and [27]). It follows that the success probability of the probabilistic ARF protocol[8] achieves the optimal scaling O(N −1 ) (see[27]). FIG. S. 4 : 4Use of port-based teleportation scheme for PSAR. are two fixed orthonormal bases for K and H, respectively. For A, B and C operators on H one can verify the identityA ⊗ B\|C = \|ABC T (S.13)where X T denotes the transpose of X with respect to the orthonormal basis \|n . A quantum operation O from L(H) to L(K) is a completely positive trace non increasing map which can be represented by its Choi operator O ∈ L(K ⊗ H). The operator O must satisfyO ≥ 0, Tr K [O] ≤ I H (S.14)where Tr K denotes the partial trace on K and I H the identity operator on H. The two constraints in Eq. (S.14) correspond to the complete positivity and trace non increasing of the quantum operation O. By making use of the notation in Eq. (S.12), the Choi operator for a unitary channel U can be written as the rank one projector \|U U \|.The action of the quantum operation O on a quantum state ρ ∈ L(H) can be described in terms of the Choi operator O as follows O(ρ) = Tr K [O(I K ⊗ ρ T )] =: O * ρ (S.15) where we introduce the link product between the operators O and ρ. The composition of two quantum operations can be represented in terms of their Choi operators too. Let us consider two quantum operations O, Owith multipartite input and output, i.e. O goes from L(H 1 ⊗ H 2 ) to L(H 3 ⊗ K) and O goes from L(H 4 ⊗ K) to L(H 5 ⊗ H 6 ). We can connect the output of O on L(K) with the input of O on L(K) obtaining a new quantum operation from L(H 1 ⊗ H 2 ⊗ H 4 ) to L(H 3 ⊗ H 5 ⊗ H 6 ). The Choi operator of the resulting quantum operation is given by the link product of the two quantum operations, as follows: . (S.26), (S.28) into λ = 1 d 2 I\| ψ\|R s \|ψ \|I we obtain λ = J λ JJ λ JJ = d J d 2 φ J \|s (JJ) \|φ J (S.30)where the λ JK 's were defined in Eq. (S.27). It is now easy to show that we can assumeR s = J I J ⊗ I J ⊗ s (J) , (S.31)where s (J) := j,j ∈j JJ s Indeed, let R s = JK I J ⊗ I K ⊗ s (JK) be the optimal quantum operation and let us define the operatorsR s = J I J ⊗ I J ⊗ s (J) where s (J) = s (JJ) and R s = J =K I J ⊗ I K ⊗ s (JK) . Since both R s and R s are positive and R s + R s = R s , we have that Tr D [R s ] ≤ I implies Tr D [R s ] ≤ I i.e. R s is a quantum operation. Finally, from Eq. (S.28) we have that ψ\|R s \|ψ = ψ\|R s \|ψ , thus proving that also {R s , \|ψ } is an optimal solution of our optimization problem.IV. EXPLICIT FORM OF THE RETRIEVED CHANNELDue to commutation relation (S.21) and the form of R s given by Eq. (S.31) the retrieved channel has the following Choi operator ψ\|R s \|ψ = J λ J \|I I\| + ν J I − 1 d \|I I\| . (S.32) As we stated in the main text the perfect retrieving condition is satisfied if and only if ν J = 0 for all J. This happens because positive-semidefiniteness of R (J) s := I J ⊗I J ⊗s (J) implies ν J ≥ 0 and the requirement J ν J = 0 implies that all the terms must vanish. Let us study separately every operator R (J) s , which by definition satisfies the commutation relation of Eq. (S.21). We have that From Eq. (S.35) and Eq. (S.36) we have j,j X jj = j 1 cj X jj can be written as v\|A − X\|v = 0. (S.41) Matrix H ij is positive semidefinite if and only if H ii ≥ 0 ∀i and \|H ij \| ≤ H ii H jj ∀i = j. Using this criterion and j c j = 1 one can easily show that both A − X and A − \|ρ ρ\| are positive semidefinite matrices. Moreover, using (X jj ) ≤ \|X jj \| ≤ X jj X j j one can easily prove the inequality v\|(A − X)\|v ≥ v\|(A − \|ρ ρ\|)\|v , (S.42) which also gives v\|(A − X)\|v = 0 =⇒ v\|(A − \|ρ ρ\|)\|v = 0, (S.43) due to A−\|ρ ρ\| ≥ 0. Moreover, let us rewrite expression v\|A\|v as v\|A\|v = ρ\|B\|ρ (S.44) consequence we have v\|(A − \|ρ ρ\|)\|v = ρ\|(B − \|v v\|)\|ρ . (S.46) Thanks to c j > 0 ∀j we have that B − \|v v\| is a positive matrix, which has either trivial or one dimensional kernel. This together with Eqs. (S.43),(S.46) allows us to write a necessary condition for matrix X ρ\|(B − \|v v\|)\|ρ = 0 (S.47) =⇒ B\|ρ − \|v v\|ρ = 0 (S.48) Explicitly solving the above equation we get the only possible solution 1 c j X jj = 1 c j X j j =⇒ X jj = µc 2 j , (S.49) which is unique up to a constant µ, as we expected due to the rank one deficiency of B − \|v v\|. Once the diagonal elements X jj respect Eq. (S.49) we have v\|(A − \|ρ ρ\|)\|v = 0, but to fulfill LHS of Eq. (S.43) we need also the saturation of the bound (S.42). This happens if and only if X jj = X jj X j j , (S.50) which together with Eq. (S.49) proves the claim of the lemma. S.56) where we used Eqs. (S.29),(S.30) and the defining property of the set C. The constraint that R s is a quantum operation translates into its Choi operator as Tr D [R s ] ≤ I. Since R s satisfies Eq. (S.21), we obtain [Tr D [R s ], U V ⊗ U * C ] = 0, which implies Tr D [R s ] = J j∈j JJ I J ⊗I .(S.56), (S.55),(S.58) and (S.59) we see that the optimization of probabilistic storage and retrieval is reduced to a linear program stated in the Theorem 3. VI. N → 1 PSAR FOR QUBIT CHANNELS -THE CASE OF ODD N Theorem 4 . 4The optimal probability of success of N → 1 probabilistic storage and retrieval of a unitary channel U(.) = U.U † , U ∈ SU (d) equals λ = N/(N −1+d 2 ). The optimal state for storage is \|ψ := j dj L \|I j ∈ H M where L := j d 2 j and j ∈ Irr(U ⊗N ). Proof. The idea of the proof is analogical to the case of qubit unitary transformations. However, in the qudit case the relations between IRRs are more complicated and we will need some of the facts derived in the proof of Theorem 3 and a new combinatorial identities, which were derived in[6] by some of us.Let us define positive functionf (j, J) = d J d j , (S.61)for all j ∈ Irr(U ⊗N ), J ∈ C j or equivalently for all J ∈ Irr(U ⊗N −1 ), j ∈ j JJ = j J . Let us note that thanks to Eq. (S.53) we haveJ∈Cj f (j, J) = 1 ∀j ∈ Irr(U ⊗N ). (S.62) is the row (column) of the current box from the Young diagram j). Using Eq. (S.65) we can write Eq. (S. we finish the proof of Theorem 4 by showing that the upper bound (S.81) can be saturated. One can choose them into Eqs. (S.51). It is easy to see that requirements on p j are satisfied and inequalities between p j and µ J are actually all saturated. Let us now evaluate λ. Inserting Eq. (S.82) into Eq. (S.(U ⊗N ) d 2 k j∈Irr(U ⊗N ) d 2 j = N N − 1 + d 2 , (S.83) case the unknown unitary U is applied on a suitably chosen state \|ψ (in general bipartite and entangled), which yields state \|ψ U ∈ H M and concludes the storing phase. Afterwards, once we want to apply unitary U on some state ξ, we employ a retrieving quantum instrument R = {R s , R f }, which acts on ξ ⊗ \|ψ U ψ U \| and in case of success outputs an sub-normalized state λU ξU † , i.e. R s :In such arXiv:1809.04552v1 [quant-ph] 12 Sep 2018 The constraint that R s is a quantum operation gives tr out [R s ] ≤ I. Eq. (S.21) implies [tr out [R s ], U V ⊗ U * in ] = 0 and tr out [R s ] = J j∈j JJ I J ⊗I j pj ≤ 1 must hold for all J and j ∈ j JJ . Conditions on p j are from Eq. (S.59).d J dj s (J) jj . Thus, d J µ J d 2 j Using the identity (S.72) we have that F = d 2 . Moreover, we have70) It is known [7] that j∈J d j = N d J ∀J ∈ Irr(U ⊗N −1 ), (S.71) which can be using Eq. 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[ "Thermal Transport in a Noncommutative Hydrodynamics", "Thermal Transport in a Noncommutative Hydrodynamics" ]	[ "Michael Geracie \nKadanoff Center for Theoretical Physics\nUniversity of Chicago\n60637Illinois, ChicagoUSA\n", "Dam Thanh Son \nKadanoff Center for Theoretical Physics\nUniversity of Chicago\n60637Illinois, ChicagoUSA\n" ]	[ "Kadanoff Center for Theoretical Physics\nUniversity of Chicago\n60637Illinois, ChicagoUSA", "Kadanoff Center for Theoretical Physics\nUniversity of Chicago\n60637Illinois, ChicagoUSA" ]	[]	We find the hydrodynamic equations of a system of particles constrained to be in the lowest Landau level. We interpret the hydrodynamic theory as a Hamiltonian system with the Poisson brackets between the hydrodynamic variables determined from the noncommutativity of space. We argue that the most general hydrodynamic theory can be obtained from this Hamiltonian system by allowing the Righi-Leduc coefficient to be an arbitrary function of thermodynamic variables. We compute the Righi-Leduc coefficients at high temperatures and show that it satisfies the requirements of particle-hole symmetry, which we outline.	10.1134/s1063776115030061	[ "https://arxiv.org/pdf/1407.4460v2.pdf" ]	118,561,179	1407.4460	f8aef61d4eabd672c4496a0d318989a0fcb86e06	Thermal Transport in a Noncommutative Hydrodynamics 16 Jul 2014 Michael Geracie Kadanoff Center for Theoretical Physics University of Chicago 60637Illinois, ChicagoUSA Dam Thanh Son Kadanoff Center for Theoretical Physics University of Chicago 60637Illinois, ChicagoUSA Thermal Transport in a Noncommutative Hydrodynamics 16 Jul 2014 We find the hydrodynamic equations of a system of particles constrained to be in the lowest Landau level. We interpret the hydrodynamic theory as a Hamiltonian system with the Poisson brackets between the hydrodynamic variables determined from the noncommutativity of space. We argue that the most general hydrodynamic theory can be obtained from this Hamiltonian system by allowing the Righi-Leduc coefficient to be an arbitrary function of thermodynamic variables. We compute the Righi-Leduc coefficients at high temperatures and show that it satisfies the requirements of particle-hole symmetry, which we outline. Introduction.-Interacting electrons in very high magnetic fields show extremely rich behaviors, the most wellknown of which is the fractional quantum Hall (FQH) effect [1,2]. In the most interesting limit, all the physics occurs in the lowest Landau level (LLL) and originates from the interactions. In this paper, we study the finite-temperature dynamics of electrons in a magnetic field so high that all particles are constrained to be on the LLL. This problem is the finite-temperature counterpart of the FQH problem. While many quantum phenomena are smeared out by the temperature, the hydrodynamic theory, which takes hold at distances and time scales much larger than the mean free path/time, is expected to be universal. We assume the system is clean, without impurities, and the only relaxation mechanism is the interactions between particles. This regime is particularly relevant for the proposed realizations of the FQH regime in cold atomic gases [3][4][5] length/time. The main outcome of our investigation is the set of hydrodynamic equations [Eqs. (8), (18), and (29)] which describes the long-wavelength dynamics of the system and the identification of the kinetic coefficients. Previous studies of transport in high magnetic field include Ref. [6][7][8][9]. high In particular, in Ref. [9] a general approach based on conservation laws is developed for the hydrodynamics of a system in a quantizing magnetic field. This is the approach that we will follow in this paper. We concentrate here, however, on the LLL limit (the zero mass limit), which should be a regular limit when the particle carries a magnetic moment corresponding to the gyromagnetic factor g = 2. This allows us to consider the response of the system to variations of the magnetic field, as well as to discuss the particle-hole symmetry of the hydrodynamic equations. An important concept in our discussion is that a particle in the lowest Landau level effectively lives on a noncommutative space [10], with its two coordinates x 1 , x 2 satisfying the commutation law [x 1 , x 2 ] = −iℓ 2 B . This idea has attracted some attention in the context of the quantum Hall effect; see, e.g. Refs. [11,12]. We use this noncommutativity to argue for a particular Poisson bracket algebra between hydrodynamic variables, and proceed to derive the hydrodynamic equations from the Poisson brackets with the Hamiltonian. This approach is inspired by the Hamiltonian formulation of classical hydrodynamics [13]. The Hamiltonian equations that follow from the formalism form a self-consistent hydrodynamic theory, but we will argue that they need a slight modification to become the most general set of equations consistent with conservation laws and the second law of thermodynamics. This modification is related to the Righi-Leduc (thermal Hall) effect [14,15]. Thermodynamics.-Let us recall the basic thermodynamic functions of a system in an external magnetic field [9]. The grand potential is an extensive thermodynamic variable which depends on the temperature, chemical potential, and magnetic field: Ω = −V P (T, µ, B). The partial derivatives of P are the entropy density, particle number density and magnetization: dP = sdT + ndµ + M dB. The hydrodynamic pressure is not P but its Legendre transform with respect to B: p = P − M B, and hence dp = sdT + ndµ − BdM . The energy density is ε = T s + µn − P . The system.-We consider a system of nonrelativistic particles of mass m and gyromagnetic factor g = 2 moving in a background magnetic field B, and will be interested in the regime where all higher Landau levels can be neglected. We will study the respond of the system to arbitrary fluctuations of both electric and magnetic fields, assuming that the B does not vanish at any place in space and time so that the separation between the lowest and the higher Landau levels is always maintained. The LLL limit corresponds to taking m → 0 and all the physics should be finite in this limit for g = 2. The Hamiltonian for our system is H = dx \|D i ψ\| 2 2m − A 0 + B 2m ψ † ψ + interactions,(1) where D i = ∂ i − iA i (we use the = c = 1 unit system and absorb the electron charge e into the gauge potential A µ ). We can also think about our system as that of particles with zero magnetic moment (g = 0), subjected to an external field in which the scalar potential is tuned to deviate from B/2m by an amount which remains finite when m → 0. The conservation laws are the conventional ones, with the replacement A 0 → A 0 + B/2m, ∂n ∂t + ∂ i j i = 0,(2)∂ ∂t (mj i ) + ∂ k Π ik = n E i + ∂ i B 2m + ǫ ik j k B, (3) ∂ε ∂t + ∂ i ε i = j i E i + ∂ i B 2m .(4) We now extract the part divergent at m → 0 from the conserved currents and the stress tensor in the following manner, j i = j i + ǫ ij 2m ∂ j n,(5)Π ik = Π ik + 1 2 (ǫ ij ∂ jj k + ǫ kj ∂ jj i ) − nB 2m δ ik ,(6)ε = ε − nB 2m ,ε i = ε i − 1 2m (Bj i + ǫ ij E j n). (7) For the number current (5), this procedure of extracting the 1/m part was done in Ref. [16]. The conservation laws are regular in the m → 0 limit in terms of the newly defined quantities, ∂n ∂t + ∂ ij i = 0, (8a) ∂ ∂t (mj i ) + ∂ kΠ ik = nE i + ǫ ikj k B,(8b)∂ε ∂t + ∂ iε i =j i E i .(8c) These equations can also be obtained within the Newton-Cartan formalism [18]. Moreover, in the limit m → 0 the first term in the left-hand side of Eq. (8b) can be dropped, and it becomes a force-balance condition. From now on we will drop the tildes in the finite currents. To close the equations we need to express j i , ε i and Π ik through the derivatives of the local temperature and chemical potential. To first order in derivatives in the equations, we can limit ourselves to the leading order contribution to the stress tensor: Π ik = pδ ik . Hamiltonian model of a noncommutative fluid.-We start with a simple model of particles moving the lowest Landau level. We number the particles by the index A = 1 . . . N , and by i the spatial coordinates. The coordinates of a particle do not commute with each other, but commute with those of other particles, {x i A , x j B } = δ AB ǫ ij B(x A ) .(9) The particle number density, n(x) = A δ(x − x A ),(10) then has the following Poisson bracket, {n(x), n(y)} = −ǫ ij ∂ i n B ∂ j δ(x − y).(11) We now need to understand the Possion brackets involving the entropy density. Recall that in ideal hydrodynamics the entropy per particle s/n is conserved along fluid worldlines. We can assume that each particle A carries an entropy s A for all time, s(x) = A s A δ(x − x A ).(12) as s ′ is in the continuum picture. We find {s(x), n(y)} = −ǫ ij ∂ i s B ∂ j δ(x − y),(13){s(x), s(y)} = −ǫ ij ∂ i c ∂ j δ(x − y),(14) where c = A s 2 A B(x A ) δ(x − x A ).(15) In order to close the Poisson algebra, we should express c in terms of s and n. In the "mean field" approximation we may expect c = s 2 /nB. We shall for now assume the most general c compatible with the Jacobi identity, which can be shown to be c = n B f s n .(16) Now the hydrodynamic equations can be obtained by computing Poisson brackets with the Hamiltonian H = dx ε(s(x), n(x), B(x)) − A 0 (x)n(x) .(17) Note that both the total particle number and the total entropy are Casimirs of the Poisson algebra, so they are automatically conserved. We find, for example, ∂ t n = −∂ i j i where the particle number current j i is j i = ǫ ij B [n(E j − ∂ j µ) − s∂ j T ] + ǫ ij ∂ j α.(18) where α cannot be determined from charge conservation alone. This can be done using the force balance equation (8b), into which we substitute Π ik = pδ ik , ∂ i p = nE i + ǫ ik j k B,(19) which, by using dp = sdT + ndµ − BdM completely determines j i , and the result corresponds to α = M . The first term on the right-hand side of Eq. (18) corresponds to the "transport current," while the second part is the "magnetization current." Computing the Poisson bracket of s with the Hamiltonian, we can find the conservation law for the entropy, ∂ t s + ∂ i s i = 0, s i = ǫ ij s B (E j − ∂ j µ) − c∂ j T .(20) For energy density, we can use ∂ t ε = T ∂ t s + µ∂ t n and derive from Eq. (8c) the energy current ε i = ǫ ij ε + p B (E j −∂ j µ)−M ∂ j µ− µs B +cT ∂ j T +∂ j M E . (21) We have introduced the "energy magnetization" M E whose contribution to the energy current is divergencefree. Středa formulas.-We note here in passing that the Středa formula can be derived from our equation for the current. Expanding the current in terms of derivatives of thermodynamic variables, including the derivative of B, j i = ǫ ij (σ H E j + σ µ H ∂ j µ + σ T H ∂ j T + σ B H ∂ j B),(22) we can then read out, for example σ H = n B , σ µ H = − n B + ∂M ∂µ T B .(23) The naive Einstein relation σ µ H = −σ H does not hold due the nonvanishing magnetization current in thermal equilibrium. Note that in a zero-temperature incompressible phase n/B is constant and σ µ H = 0, consistent with the expectation that small spatial variations of µ should not have any physical effect in such a phase. In thermal equilibrium the chemical potential traces the electric field, µ = A 0 , and so the only current flowing in the system is the magnetization current, equal to j i = σ eq H ǫ ij E j where σ eq H = σ H + σ µ H = ∂M ∂µ T,B = ∂n ∂B µ,T ,(24) where we have used a Maxwell's relation. This is the Středa formula [17]. The thermopower can be read out from our expression for the transport current: it is equal to entropy per particle s/n, a known result [9]. Note also that a gradient of the magnetic field only leads to a magnetization (but not transport) current. The noncommutative model above gives a complete expression for the energy current in terms of the function c appearing in the Poisson algebra and the energy magnetization M E . Namely, if we write, in zero electric field, ε i = ǫ ij (κ µ H ∂ j µ + κ T H ∂ j T + κ B H ∂ j B), then κ µ H = − T s + µn B + ∂M E ∂µ T,B ,(25)κ T H = − µs B − cT + ∂M E ∂T µ,B ,(26)κ B H = ∂M E ∂B T,µ .(27) the More general equations.-The Poisson bracket formalism, while giving a self-consistent set of equations, rely on certain unjustified assumptions. For example, the effect of dissipative heat conduction cannot be taken into account in this formalism. Fortunately, we can show that beside this effect, the most general hydrodynamic equations have the same forms as the equations derived above; the only modification is that there is now no restriction on the form of c, which to this point has been required to be of the form (16). To write down the most general system of hydrodynamic equations, first we notice that the particle number current cannot be modified due to the force balance condition. Thus the only place where modifications can be made is in the constitutive relation for the energy current (21). The dissipative part has the familiar form of longitudinal heat conduction and shall not be discussed here. The most general additional transverse terms one can add to the energy current is ǫ ij Σ a ∂ j X a where X a , a = 1, 2, 3, are three independent thermodynamic variables (which can be chosen to be, e.g., µ, T , and B, but any other choice is equally valid) and Σ a = Σ a (X) are the corresponding three kinetic coefficients. It is convenient to introduce the one-form Σ ≡ Σ a dX a in the space of thermodynamic variables. The constraint on Σ a is that one can modify the entropy current by adding to Eq. (20) a contribution of the form ǫ ij ζ a ∂ j X a and still preserve the entropy production rate, which should receive no contributions from these new kinetic terms. By direct calculation using the thermodynamic relation ds = T −1 (dε − µdn) and the conservation of energy and particle number, one can find the divergence of the new entropy current ∂ t s + ∂ i s i = 1 2 ǫ ij dζ − 1 T dΣ ab ∂ i X a ∂ j X b .(28) Therefore, for entropy conservation we need to have dζ = T −1 dΣ. The most general solution to this equation is ζ = db 0 + c 0 dT , Σ = dσ 0 + T c 0 dT , where b 0 , c 0 and σ 0 are scalar functions (zero-forms) of thermodynamic variables. In the energy current, σ 0 can be absorbed into the magnetization current, and c 0 into c to make the latter a unconstrained function of three thermodynamic variables. The full energy current therefore is ε i = ǫ ij ε + p B (E j − ∂ j µ) − M ∂ j µ + ∂ j M E − c RL T ∂ j T ,(29) where c RL = c + µs/BT , corresponding to the Righi-Leduc effect with thermal Hall conductivityK H = T c RL . This means that in a gapped quantum Hall phase at low temperature c RL = π 6 (c R − c L ), where c R and c L are the numbers of right and left moving modes, respectively [14]. We emphasize here that the fact that the energy current is parametrized in terms of two functions c and M E implies one relationship between the coefficient κ µ H , κ T H , and κ B H . The response to the Luttinger potential coupled to the energy density [19] can also be expressed in terms of these two functions [20]. Righi-Leduc coefficient at high temperature.-At low temperature as µ changes c RL is expected to vary in a complicated fashion as the system scans through many quantum Hall plateaux. When the temperature is large compared to the interaction energy, the system is weakly interacting and the Righi-Leduc coefficient c RL can be computed reliably. One can follow the method of Ref. [6], but one can also employ the following short-cut. In thermal equilibrium, all states in the lowest Landau level has occupation number ν = (e −βµ +1) −1 , which depends only on µ/T but not µ and T separately. The current and energy current out of equilibrium, where µ and T vary in space, thus depend only on µ/T . But these quantities have nonzero dimension, and hence they have to vanish at high temperature. The grand partition function in this regime is P = BT 2π ln(1 + e µ/T ),(30) from which all other thermodynamic potentials can be computed. In particular, ε = p = 0, M = P/B. The condition of vanishing energy current reads M ∂ j µ = ∂ j M E − c RL T ∂ j T , which means ∂M E ∂µ = M, ∂M E ∂T − T c RL = 0.(31) The solution to these equations is M E = − 1 2π T 2 Li 2 (−e µ/T ),(32)c RL = − 1 2π µ T ln(1 + e µ/T ) + 2Li 2 (−e µ/T ) = − 1 2π ln ν 1 − ν ln 1 1 − ν + 2Li 2 − ν 1 − ν .(33) The Righi-Leduc coefficient approaches 0 as ν → 0, and π/6 as ν → 1. The latter value matches exactly with that expected for the ν = 1 integer quantum Hall state with a single chiral edge mode [14]. In the limit of low filling fraction ν ≪ 1 the formulas simplify. For example, c RL ≈ − 1 2π (2 − µ/T )e µ/T . It is interesting to note that, to leading order in ln(1/ν), c = c RL − µs BT ≈ e µ/T 2π µ 2 T 2 = s 2 nB ,(34) which is of the form (16) and moreover coincides with our initial "mean-field" guess for c. Particle-hole symmetry.-When the interaction between fermions is two-body, the system has particle-hole symmetry (for simplicity, here we assume the magnetic field is uniform and constant). This should also be a symmetry of the hydrodynamic equations. If one normalizes the one-body potential so that µ = 0 corresponds to a half-filled Landau level (ν = 1/2), then particle-hole symmetry is the symmetry under µ → −µ. It is easy to check that the hydrodynamic equations are particle-hole symmetric if P , M E and c RL satisfy P (T, µ) = P (T, µ) − Bµ 2π ,(35)M E (T, −µ) = 1 4π µ 2 + µ 2 3 T 2 − M E (T, µ),(36)c RL (T, −µ) = π 6 − c RL (T, µ).(37) In particular, at half filling c RL = π/12 if particle-hole symmetry is not spontaneously broken, which should be the case at least at sufficiently high temperature. Note that these properties are satisfied by Eqs. (32) and (33). Conclusion.-We have shown that the full finitetemperature hydrodynamics of a system of particles confined to the lowest Landau level can be written down based on general principles. We have assumed that the interaction between the electrons are short-ranged. In the case of long-ranged Coulomb interaction, we have to add a Poisson equation for the scalar potential, which can be done in a straightforward manner. The Righi-Leduc coefficient c RL should be viewed as a fundamental property of a system on the LLL at any filling fraction and temperature. In principle, the coefficient can be measured; but the task may be complicated by the edge transport, as well as from the contributions from the other terms in the energy current (29). We defer a more detailed study to future work. To the order that we are working on, we are not sensitive to the first-order corrections to the stress tensor, including the dissipative shear and bulk viscosities and the dissipationless Hall viscosity. These can be introduced; half and it would be interesting to investigate their behaviors under particle-hole symmetry. We thank Sean Hartnoll for discussions. This work is supported, in part, by the US DOE grant No. DE-FG02-13ER41958, and the ARO-MURI 63834-PH-MUR grant, and a Simon Investigator grant from the Simons Foundation. . D C Tsui, H L Stormer, A C Gossard, Phys. Rev. Lett. 481559D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982). . R B Laughlin, Phys. Rev. Lett. 501395R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). . N R Copper, Adv. Phys. 57539N. R. Copper, Adv. Phys. 57, 539 (2008). . N R Cooper, J Dalibard, Phys. Rev. Lett. 110185301N. R. Cooper and J. Dalibard, Phys. Rev. Lett. 110, 185301 (2013). . N Y Yao, A V Gorshkov, C R Laumann, A M Luchli, J Ye, M D Lukin, Phys. Rev. Lett. 110185302N. Y. Yao, A. V. Gorshkov, C. R. Laumann, A. M. Luchli, J. Ye, and M. D. Lukin, Phys. Rev. Lett. 110, 185302 (2013). . P S Zyryanov, Phys. Stat. Sol. 6401P. S. Zyryanov, Phys. Stat. Sol. 6, 401 (1964). . Yu N Obraztsov, Sov. Phys. Solid State. 7455Yu. N. Obraztsov, Sov. Phys. Solid State 7, 455 (1965). . L Smrčka, P Středa, J. Phys. C. 102153L. Smrčka and P. Středa, J. Phys. C 10, 2153 (1977). . N R Cooper, B I Halperin, I M Ruzin, Phys. Rev. B. 552344N. R. Cooper, B. I. Halperin, and I. M. Ruzin, Phys. Rev. B 55, 2344 (1997). . S M Girvin, A H Macdonald, P M Platzman, Phys. Rev. B. 332481S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Phys. Rev. B 33, 2481 (1986). . L Susskind, L. Susskind, . E Fradkin, V Jejjala, R G Leigh, Nucl. Phys. B. 642483E. Fradkin, V. Jejjala, and R. G. Leigh, Nucl. Phys. B 642, 483 (2002). . L D Landau, J. Phys. USSR. 571L. D. Landau, J. Phys. USSR 5, 71 (1941). . C L Kane, M P A Fisher, Phys. Rev. B. 5515832C. L. Kane and M. P. A. Fisher, Phys. Rev. B 55, 15832 (1997). . M Stone, Phys. Rev. B. 85184503M. Stone, Phys. Rev. B 85, 184503 (2012). . S H Simon, A Stern, B I Halperin, Phys. Rev. B. 5411114S H. Simon, A. Stern, and B. I. Halperin, Phys. Rev. B 54, 11114(R) (1996). . P Středa, J. Phys. C. 15717P. Středa, J. Phys. C 15, L717 (1982). . M Geracie, D T Son, S.-F Wu, C Wu, to be publishedM. Geracie, D. T. Son, S.-F. Wu, and C. Wu, to be pub- lished. . J M Luttinger, Phys. Rev. 1351505J. M. Luttinger, Phys. Rev. 135, A1505 (1964). . M Geracie, D T Son, M. Geracie and D. T. 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[ "FMOS near-IR spectroscopy of Herschel selected galaxies: star formation rates, metallicity and dust attenuation at z ∼ 1", "FMOS near-IR spectroscopy of Herschel selected galaxies: star formation rates, metallicity and dust attenuation at z ∼ 1" ]	[ "I G Roseboom \nInstitute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK\n\nAstronomy Centre\nDept. of Physics & Astronomy\nUniversity of Sussex\nBN1 9QHBrightonUK\n", "2⋆ A Bunker \nAstrophysics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK\n", "M Sumiyoshi \nDepartment of Astronomy\nFaculty of Science\nKyoto University\n606-8502KyotoJapan\n", "L Wang \nAstronomy Centre\nDept. of Physics & Astronomy\nUniversity of Sussex\nBN1 9QHBrightonUK\n", "G Dalton \nAstrophysics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK\n\nRALSpace\nOX11 0QXRutherford Appleton Laboratory, Chilton, DidcotOxfordshireUK\n", "M Akiyama \nAstronomical Institute\nTohoku University\nAoba-ku980-8578SendaiJapan\n", "J Bock \nCalifornia Institute of Technology\n1200 E. California Blvd91125PasadenaCAUSA\n\nJet Propulsion Laboratory\n4800 Oak Grove Drive91109PasadenaCAUSA\n", "D Bonfield \nCentre for Astrophysics Research\nUniversity of Hertfordshire\nCollege LaneAL10 9ABHatfieldHertfordshireUK\n", "V Buat \nLaboratoire d'Astrophysique de Marseille\nOAMP\nUniversité Aix-marseille\nCNRS\n38 rue Frédéric Joliot-Curie13388Marseille cedex 13France\n", "C Casey \nInstitute for Astronomy\nUniversity of Hawaii\n96822ManoaHIUSA ; Canada, France\n\nHawaii Telescope Corp\n96743KamuelaHIUSA\n", "E Chapin \nDepartment of Physics & Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadV6T 1Z1VancouverBCCanada\n", "D L Clements \nAstrophysics Group\nBlackett Laboratory\nImperial College London\nPrince Consort RoadSW7 2AZLondonUK\n", "A Conley \nCenter for Astrophysics and Space Astronomy\n593 UCB, 80309-0593BoulderCoUSA\n", "E Curtis-Lake \nInstitute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK\n", "A Cooray \nCalifornia Institute of Technology\n1200 E. California Blvd91125PasadenaCAUSA\n\nDept. of Physics & Astronomy\nUniversity of California\n92697IrvineCAUSA\n", "J S Dunlop \nInstitute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK\n", "D Farrah \nAstronomy Centre\nDept. of Physics & Astronomy\nUniversity of Sussex\nBN1 9QHBrightonUK\n", "S J Ham \nAstrophysics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK\n", "E Ibar \nUK Astronomy Technology Centre\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK\n", "F Iwamuro \nDepartment of Astronomy\nFaculty of Science\nKyoto University\n606-8502KyotoJapan\n", "M Kimura \nSubaru Telescope\nNAOJ\n650 North Aohoku Place96720HiloHIUSA\n", "I Lewis \nAstrophysics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK\n", "E Macaulay \nAstrophysics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK\n", "G Magdis \nAstrophysics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK\n", "T Maihara \nDepartment of Astronomy\nFaculty of Science\nKyoto University\n606-8502KyotoJapan\n", "G Marsden \nDepartment of Physics & Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadV6T 1Z1VancouverBCCanada\n", "T Mauch \nAstrophysics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK\n\nCentre for Astrophysics Research\nUniversity of Hertfordshire\nCollege LaneAL10 9ABHatfieldHertfordshireUK\n", "Y Moritani \nDepartment of Astronomy\nFaculty of Science\nKyoto University\n606-8502KyotoJapan\n", "K Ohta \nDepartment of Astronomy\nFaculty of Science\nKyoto University\n606-8502KyotoJapan\n", "S J Oliver \nAstronomy Centre\nDept. of Physics & Astronomy\nUniversity of Sussex\nBN1 9QHBrightonUK\n", "M J Page \nMullard Space Science Laboratory\nUniversity College London\nHolmbury St. Mary\nRH5 6NTDorkingSurreyUK\n", "B Schulz \nJet Propulsion Laboratory\n4800 Oak Grove Drive91109PasadenaCAUSA\n\nInfrared Processing and Analysis Center\nCalifornia Institute of Technology\nJPL\n100-22, 91125PasadenaMS, CAUSA\n", "Douglas Scott \nDepartment of Physics & Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadV6T 1Z1VancouverBCCanada\n", "M Symeonidis \nMullard Space Science Laboratory\nUniversity College London\nHolmbury St. Mary\nRH5 6NTDorkingSurreyUK\n", "N Takato \nSubaru Telescope\nNAOJ\n650 North Aohoku Place96720HiloHIUSA\n", "N Tamura \nSubaru Telescope\nNAOJ\n650 North Aohoku Place96720HiloHIUSA\n", "T Totani \nDepartment of Astronomy\nFaculty of Science\nKyoto University\n606-8502KyotoJapan\n", "K Yabe \nDepartment of Astronomy\nFaculty of Science\nKyoto University\n606-8502KyotoJapan\n", "M Zemcov \nCalifornia Institute of Technology\n1200 E. California Blvd91125PasadenaCAUSA\n\nJet Propulsion Laboratory\n4800 Oak Grove Drive91109PasadenaCAUSA\n" ]	[ "Institute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK", "Astronomy Centre\nDept. of Physics & Astronomy\nUniversity of Sussex\nBN1 9QHBrightonUK", "Astrophysics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK", "Department of Astronomy\nFaculty of Science\nKyoto University\n606-8502KyotoJapan", "Astronomy Centre\nDept. of Physics & Astronomy\nUniversity of Sussex\nBN1 9QHBrightonUK", "Astrophysics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK", "RALSpace\nOX11 0QXRutherford Appleton Laboratory, Chilton, DidcotOxfordshireUK", "Astronomical Institute\nTohoku University\nAoba-ku980-8578SendaiJapan", "California Institute of Technology\n1200 E. California Blvd91125PasadenaCAUSA", "Jet Propulsion Laboratory\n4800 Oak Grove Drive91109PasadenaCAUSA", "Centre for Astrophysics Research\nUniversity of Hertfordshire\nCollege LaneAL10 9ABHatfieldHertfordshireUK", "Laboratoire d'Astrophysique de Marseille\nOAMP\nUniversité Aix-marseille\nCNRS\n38 rue Frédéric Joliot-Curie13388Marseille cedex 13France", "Institute for Astronomy\nUniversity of Hawaii\n96822ManoaHIUSA ; Canada, France", "Hawaii Telescope Corp\n96743KamuelaHIUSA", "Department of Physics & Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadV6T 1Z1VancouverBCCanada", "Astrophysics Group\nBlackett Laboratory\nImperial College London\nPrince Consort RoadSW7 2AZLondonUK", "Center for Astrophysics and Space Astronomy\n593 UCB, 80309-0593BoulderCoUSA", "Institute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK", "California Institute of Technology\n1200 E. California Blvd91125PasadenaCAUSA", "Dept. of Physics & Astronomy\nUniversity of California\n92697IrvineCAUSA", "Institute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK", "Astronomy Centre\nDept. of Physics & Astronomy\nUniversity of Sussex\nBN1 9QHBrightonUK", "Astrophysics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK", "UK Astronomy Technology Centre\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK", "Department of Astronomy\nFaculty of Science\nKyoto University\n606-8502KyotoJapan", "Subaru Telescope\nNAOJ\n650 North Aohoku Place96720HiloHIUSA", "Astrophysics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK", "Astrophysics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK", "Astrophysics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK", "Department of Astronomy\nFaculty of Science\nKyoto University\n606-8502KyotoJapan", "Department of Physics & Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadV6T 1Z1VancouverBCCanada", "Astrophysics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK", "Centre for Astrophysics Research\nUniversity of Hertfordshire\nCollege LaneAL10 9ABHatfieldHertfordshireUK", "Department of Astronomy\nFaculty of Science\nKyoto University\n606-8502KyotoJapan", "Department of Astronomy\nFaculty of Science\nKyoto University\n606-8502KyotoJapan", "Astronomy Centre\nDept. of Physics & Astronomy\nUniversity of Sussex\nBN1 9QHBrightonUK", "Mullard Space Science Laboratory\nUniversity College London\nHolmbury St. Mary\nRH5 6NTDorkingSurreyUK", "Jet Propulsion Laboratory\n4800 Oak Grove Drive91109PasadenaCAUSA", "Infrared Processing and Analysis Center\nCalifornia Institute of Technology\nJPL\n100-22, 91125PasadenaMS, CAUSA", "Department of Physics & Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadV6T 1Z1VancouverBCCanada", "Mullard Space Science Laboratory\nUniversity College London\nHolmbury St. Mary\nRH5 6NTDorkingSurreyUK", "Subaru Telescope\nNAOJ\n650 North Aohoku Place96720HiloHIUSA", "Subaru Telescope\nNAOJ\n650 North Aohoku Place96720HiloHIUSA", "Department of Astronomy\nFaculty of Science\nKyoto University\n606-8502KyotoJapan", "Department of Astronomy\nFaculty of Science\nKyoto University\n606-8502KyotoJapan", "California Institute of Technology\n1200 E. California Blvd91125PasadenaCAUSA", "Jet Propulsion Laboratory\n4800 Oak Grove Drive91109PasadenaCAUSA" ]	[ "Mon. Not. R. Astron. Soc" ]	We investigate the properties (e.g. star formation rate, dust attentuation, stellar mass and metallicity) of a sample of infrared luminous galaxies at z ∼ 1 via near-IR spectroscopy with Subaru-FMOS. Our sample consists of Herschel SPIRE and Spitzer MIPS selected sources in the COSMOS field with photometric redshifts in the range 0.7 < z phot < 1.8, which have been targeted in 2 pointings (0.5 sq. deg.) with FMOS. We find a modest success rate for emission line detections, with candidate Hα emission lines detected for 57 of 168 SPIRE sources (34 per cent). By stacking the near-IR spectra we directly measure the mean Balmer decrement for the Hα and Hβ lines, finding a value of E(B −V ) = 0.51±0.27 for L IR = 10 12 L ⊙ sources at z = 1.36. By comparing star formation rates estimated from the IR and from the dust uncorrected Hα line we find a strong relationship between dust attenuation and star formation rate. This relation is broadly consistent with that previously seen in star-forming galaxies at z ∼ 0.1. Finally, we investigate the metallicity via the N2 ratio, finding that z ∼ 1 IR-selected sources are indistinguishable from the local mass-metallicity relation. We also find a strong correlation between dust attentuation and metallicity, with the most metal-rich IR-sources experiencing the largest levels of dust attenuation.	10.1111/j.1365-2966.2012.21777.x	[ "https://arxiv.org/pdf/1207.5564v1.pdf" ]	8,381,530	1207.5564	61de690f923ea10978a53631dd290d3012d3760b	FMOS near-IR spectroscopy of Herschel selected galaxies: star formation rates, metallicity and dust attenuation at z ∼ 1 2011 I G Roseboom Institute for Astronomy University of Edinburgh Royal Observatory Blackford HillEH9 3HJEdinburghUK Astronomy Centre Dept. of Physics & Astronomy University of Sussex BN1 9QHBrightonUK 2⋆ A Bunker Astrophysics University of Oxford Keble RoadOX1 3RHOxfordUK M Sumiyoshi Department of Astronomy Faculty of Science Kyoto University 606-8502KyotoJapan L Wang Astronomy Centre Dept. of Physics & Astronomy University of Sussex BN1 9QHBrightonUK G Dalton Astrophysics University of Oxford Keble RoadOX1 3RHOxfordUK RALSpace OX11 0QXRutherford Appleton Laboratory, Chilton, DidcotOxfordshireUK M Akiyama Astronomical Institute Tohoku University Aoba-ku980-8578SendaiJapan J Bock California Institute of Technology 1200 E. California Blvd91125PasadenaCAUSA Jet Propulsion Laboratory 4800 Oak Grove Drive91109PasadenaCAUSA D Bonfield Centre for Astrophysics Research University of Hertfordshire College LaneAL10 9ABHatfieldHertfordshireUK V Buat Laboratoire d'Astrophysique de Marseille OAMP Université Aix-marseille CNRS 38 rue Frédéric Joliot-Curie13388Marseille cedex 13France C Casey Institute for Astronomy University of Hawaii 96822ManoaHIUSA ; Canada, France Hawaii Telescope Corp 96743KamuelaHIUSA E Chapin Department of Physics & Astronomy University of British Columbia 6224 Agricultural RoadV6T 1Z1VancouverBCCanada D L Clements Astrophysics Group Blackett Laboratory Imperial College London Prince Consort RoadSW7 2AZLondonUK A Conley Center for Astrophysics and Space Astronomy 593 UCB, 80309-0593BoulderCoUSA E Curtis-Lake Institute for Astronomy University of Edinburgh Royal Observatory Blackford HillEH9 3HJEdinburghUK A Cooray California Institute of Technology 1200 E. California Blvd91125PasadenaCAUSA Dept. of Physics & Astronomy University of California 92697IrvineCAUSA J S Dunlop Institute for Astronomy University of Edinburgh Royal Observatory Blackford HillEH9 3HJEdinburghUK D Farrah Astronomy Centre Dept. of Physics & Astronomy University of Sussex BN1 9QHBrightonUK S J Ham Astrophysics University of Oxford Keble RoadOX1 3RHOxfordUK E Ibar UK Astronomy Technology Centre Royal Observatory Blackford HillEH9 3HJEdinburghUK F Iwamuro Department of Astronomy Faculty of Science Kyoto University 606-8502KyotoJapan M Kimura Subaru Telescope NAOJ 650 North Aohoku Place96720HiloHIUSA I Lewis Astrophysics University of Oxford Keble RoadOX1 3RHOxfordUK E Macaulay Astrophysics University of Oxford Keble RoadOX1 3RHOxfordUK G Magdis Astrophysics University of Oxford Keble RoadOX1 3RHOxfordUK T Maihara Department of Astronomy Faculty of Science Kyoto University 606-8502KyotoJapan G Marsden Department of Physics & Astronomy University of British Columbia 6224 Agricultural RoadV6T 1Z1VancouverBCCanada T Mauch Astrophysics University of Oxford Keble RoadOX1 3RHOxfordUK Centre for Astrophysics Research University of Hertfordshire College LaneAL10 9ABHatfieldHertfordshireUK Y Moritani Department of Astronomy Faculty of Science Kyoto University 606-8502KyotoJapan K Ohta Department of Astronomy Faculty of Science Kyoto University 606-8502KyotoJapan S J Oliver Astronomy Centre Dept. of Physics & Astronomy University of Sussex BN1 9QHBrightonUK M J Page Mullard Space Science Laboratory University College London Holmbury St. Mary RH5 6NTDorkingSurreyUK B Schulz Jet Propulsion Laboratory 4800 Oak Grove Drive91109PasadenaCAUSA Infrared Processing and Analysis Center California Institute of Technology JPL 100-22, 91125PasadenaMS, CAUSA Douglas Scott Department of Physics & Astronomy University of British Columbia 6224 Agricultural RoadV6T 1Z1VancouverBCCanada M Symeonidis Mullard Space Science Laboratory University College London Holmbury St. Mary RH5 6NTDorkingSurreyUK N Takato Subaru Telescope NAOJ 650 North Aohoku Place96720HiloHIUSA N Tamura Subaru Telescope NAOJ 650 North Aohoku Place96720HiloHIUSA T Totani Department of Astronomy Faculty of Science Kyoto University 606-8502KyotoJapan K Yabe Department of Astronomy Faculty of Science Kyoto University 606-8502KyotoJapan M Zemcov California Institute of Technology 1200 E. California Blvd91125PasadenaCAUSA Jet Propulsion Laboratory 4800 Oak Grove Drive91109PasadenaCAUSA FMOS near-IR spectroscopy of Herschel selected galaxies: star formation rates, metallicity and dust attenuation at z ∼ 1 Mon. Not. R. Astron. Soc 0002011arXiv:1207.5564v1 [astro-ph.CO] Printed 2 (MN L A T E X style file v2.2)galaxies: evolution, submillimetre: galaxies We investigate the properties (e.g. star formation rate, dust attentuation, stellar mass and metallicity) of a sample of infrared luminous galaxies at z ∼ 1 via near-IR spectroscopy with Subaru-FMOS. Our sample consists of Herschel SPIRE and Spitzer MIPS selected sources in the COSMOS field with photometric redshifts in the range 0.7 < z phot < 1.8, which have been targeted in 2 pointings (0.5 sq. deg.) with FMOS. We find a modest success rate for emission line detections, with candidate Hα emission lines detected for 57 of 168 SPIRE sources (34 per cent). By stacking the near-IR spectra we directly measure the mean Balmer decrement for the Hα and Hβ lines, finding a value of E(B −V ) = 0.51±0.27 for L IR = 10 12 L ⊙ sources at z = 1.36. By comparing star formation rates estimated from the IR and from the dust uncorrected Hα line we find a strong relationship between dust attenuation and star formation rate. This relation is broadly consistent with that previously seen in star-forming galaxies at z ∼ 0.1. Finally, we investigate the metallicity via the N2 ratio, finding that z ∼ 1 IR-selected sources are indistinguishable from the local mass-metallicity relation. We also find a strong correlation between dust attentuation and metallicity, with the most metal-rich IR-sources experiencing the largest levels of dust attenuation. INTRODUCTION Accurate measurements of the characteristic properties of galaxies; star formation rate (SFR), metallicity and stellar mass, are central to our understanding of their evolution. Significant progess in our ability to measure these properties in distant (z > 1) galaxies has been made in the last two decades. Deep surveys with the Hubble space telescope have enabled the star formation rates of large numbers of galaxies up to z ∼ 7 to be measured (e.g. Madau et al. 1996;Bunker et al. 2004;Bouwens et al. 2006;Bouwens et al. 2009., McLure et al. 2010. Large scale optical and near-IR spectroscopic surveys have targetted key emission lines allowing the study of gas metallicity to z < 4 (e.g. Tremonti et al. 2004;Erb et al. 2006;Mannucci et al. 2009;Zahid et al. 2011;Cresci et al. 2012). Finally, deep optical and near-IR photometric surveys have allowed an accurate assessment of the stellar mass contained in galaxies, and its build-up with redshift out to z ∼ 5 (e.g. Fontana et al. 2006;Ilbert et al. 2010;Caputi et al. 2011). While these advances have re-shaped our understanding of galaxy formation and evolution, they typically rely observations in a single wavelength window i.e. UV, optical/near-IR, far-IR, etc. Meanwhile at low and intermediate redshifts it is becoming clear that large scale, multi-wavelength, studies of galaxies are needed to determine unbiased estimates of their properties. In the case of SFR estimates, where the corrections for dust attenuation tend to be large, comparisons of UV and IR SFR estimates at z ∼ 0 (Hao et al. 2011), z ∼ 1 (Buat et al. 2010) and z ∼ 2 (Reddy et al. 2011) show that widely used in band (i.e. in the same waveband as the SFR estimate) dust attenuation estimators (e.g. the UV continuum slope; Meurer et al. 1999) have large errors (δ log 10 SF R ∼ 0.3 dex) and can have significant systemic biases for certain populations of galaxies (low SFR spirals, ULIRGs 10 12 L⊙ and young starbursts). By comparison SFR estimators based on combinations of the far-IR, Hα line or radio have much smaller errors (δ log 10 SF R ∼ 0.1 dex; Kennicutt et al. 2009;Hao et al. 2011) and are universally valid. Measurements of other galaxy properties also benefit from a multi-wavelength approach. Metallicity estimates from single tracers e.g. the N 2 or R32 methods (Pettini & Pagel 2004) can disagree by up to ∆[log 10 (O/H)] = 0.7 dex (Kewley & Ellison 2008). Stellar mass estimates obtained via the fitting the stellar population models require multi-band observations in the optical and near-IR to be reliable; omitting near-IR observations introduces an error of ∼ 0.1dex to stellar mass estimates at z ∼ 1 (Pozzetti et al. 2007;Ilbert et al. 2010). In order to put galaxy evolution at high−z on a firm footing multi-wavelength observations at the same restframe wavelengths as our low-z benchmarks (e.g. SDSS, Spitzer, IRAS) for a large number of high-z galaxies is needed. This necessitates wide-field imaging and spectrocopy in the IR. Here we investigate the rest-frame optical-to-far IR properties of a sample of Herschel 1 (Pilbratt et al. 2010) sources which were targetted for near-IR spectroscopy with FMOS (Kimura et al. 2010). The key goal of this work is to determine the key galaxy properties (SFR, dust attenuation, stellar mass and metallicity) between a sample of high−z (0.8 < z < 1.7), IR luminous (> 10 11 L⊙) sources using the same tracers commonly used for low-z samples. In this way we can be sure that our results are fully consistent (in terms of both calibration and selection effects) with those at low-z. The datasets used in this work are described in §2, §3.1 presents the detection rate of Hα, §3.2 the aggregate near-IR spectral properties and §3.3 a comparison of the star formation rates from the IR and Hα line. In §3.4 we investigate the stellar mass and metallicity of our sample and, finally, §4 summarises our conclusions. Throughout we assume a ΛCDM cosmology with ΩΛ = 0.7, Ωm = 0.3 and H0 = 70 km s −1 Mpc −1 . DATA Pre-existing COSMOS data The starting point for this work is the SPIRE observations of the COSMOS field (Scoville et al. 2007) taken as part of the Herschel Multi-tiered Extragalactic Survey (HerMES; Oliver et al. 2012). The SPIRE instrument, its in-orbit performance and its scientific capabilities are described by Griffin et al. (2010); its calibration methods and accuracy are outlined in Swinyard et al. (2010). Here we make use of SPIRE maps as described in Levenson et al. (2010). At the time of writing HerMES observations of COSMOS cover ∼ 4.8 sq. deg. to a 1σ instrumental noise of ∼2 mJy/Beam at the three SPIRE wavelengths of 250, 350 and 500 µm. As the SPIRE data offer an instrumental noise significantly lower than the confusion noise (∼ 6 mJy; Nyugen et al. 2010) we make use of prior source positions from higher angular resolution data to extract SPIRE photometry. The MIPS 24 µm channel is the most obvious prior for SPIRE data as it offers a significant improvement in angular resolution (6 arcsec FWHM for MIPS 24 µm vs. 18.6 arcsec FWHM for SPIRE 250 µm) while also being able to account for > 80 per cent of the flux at SPIRE wavelengths at the 24 µm depths now available in a large fraction of HerMES fields (Bethermin et al. 2012;Oliver et al. 2012). To construct our prior catalogue for SPIRE photometry we begin with the MIPS 24 µm imaging from the Spitzer COSMOS survey (Le Floc'h et al. 2009). Here we make use of the publicly available imaging, performing source extraction via the starfinder IDL package (Diolaiti et al. 2000). The resulting catalogue covers ∼ 2.1 sq. deg. and has a typical 1σ sensitivity of σ = 15 µJy. In order to provide the most accurate positional information, for both our SPIRE photometry and FMOS fibre positioning, we cross-match our 24 µm catalogue to the publically available HST ACS IF814W-band catalogue of Leauthaud et al. (2007). This catalogue covers 1.64 sq. deg. to a limiting magnitude of IF814W < 26.5. In addition to improving the positional accuracy this matching helps eliminate spurious 24 µm sources produced by artifacts in the image, in particular those located close to bright (> 1 mJy) sources. Of the 35,914 24 µm sources located well within the ACS i-band coverage, 33,071 (92 per cent) have IF814W-band counterparts within 2 arcsec. SPIRE photometry is performed using the IF814W-band positions of the 24 µm sources as a prior, following the algorithms described in Roseboom et al. (2010) and Roseboom et al. (2012). All > 3σ (∼ 60 µJy) 24 µm sources are considered as potential SPIRE counterparts. Using the residual map statistics we estimate that our prior-driven SPIRE catalogue reaches a typical point source sensitivity of σtot =2.7, 3.5, and 3.2 mJy at 250, 350 and 500 µm, including the contribution from source confusion. To complete our multi-wavelength COSMOS dataset we add multi-band optical/near-IR data and photometric redshifts from the catalogue of Ilbert et al. (2009). This catalogue is limited to i + AB < 25, and hence our HerMES-COSMOS sample is similarly restricted. FMOS observations and emission line measurements IR-selected sources were targeted in two pointings (0.5 sq. deg.) located within the COSMOS field with FMOS as part of the GTO program. The FMOS instrument (Kimura et al. 2008) consists of 400 1.2-arcsec diameter fibres which can be placed within a 30-arcmin diameter field of view. We used the low-resolution mode (R ∼ 600), allowing instantaneous coverage of both the J and H band (0.9 < λ < 1.8 µm), with cross-beam switching i.e. two fibres for each target; one placed on the sky and one on the target, with the target/sky "switched" between them at regular intervals. Potential targets for FMOS fibre allocation were selected from our HerMES-COSMOS parent catologue by requiring a photometric redshift in the range 0.65 < z phot < 1.75 from the catalogue of Ilbert et al. (2009). This restriction was introduced to ensure that the Hα line was likely within the FMOS wavelength coverage. Fibre allocation preference was given to sources detected at both 24 µm and 250 µm (> 3σtot), followed by 24 µm only sources. As well as science targets, a number (typically 2-4) of 2MASS (Skrutskie et al. 2006) selected stars were included in the observations for flux calibration purposes. The first of our FMOS pointings was dedicated to solely HerMES-COSMOS targets, while for the second pointing (2010 November 24 and 25) we shared fibres with the evolSMURF project (Bunker et al., in prep). While the split between the samples was roughly 50-50, this was aided by the overlap between the samples (42 sources). Table 1 details the exposure times, together with the number of 24 and 250 µm detected (henceforth refered to as 24 ∩ 250 µm), and 24 µm only sources in each pointing. In total 241 fibres were allocated to IR-selected sources, with four sources appearing in both configurations, resulting in 237 unique targets (168 unique 24∩250 µm targets). All data were reduced using the standard FMOS pipeline (Iwamuro et al., 2011). Emission lines were identified in the 2D reduced frames, after flux calibration, via a semi-automated procedure. At each pixel the line profile was fitted to the surrounding 9×9 pixels. The pixel scale was 5Å in the spectral direction and 0.13 arcsec in the fibre direction. We only considered pixels in the wavelength ranges 1.1-1.36 µm and 1.42-1.7 µm. Pixels within 5Å of an OH line were excluded from consideration. The line profile was assumed to be Gaussian with FWHM= λ/600Å in the spectral direction, and 6.9 pixels in the fibre direction. The noise was estimated by taking the variance of all illuminated pixels at that wavelength on the detector. Regions where the noise is exceptionally high (> 10 µJy pixel −1 ) were excluded. The local continuum was estimated by taking the median pixel value in a window of 20 pixels (200Å), excluding the closest 7 pixels. For each fit the line signal-to-noise ratio (SNR), peak SNR and correlation coefficient between the line profile and 2D spectrum was measured. The line SNR (σ line ) was calculated via Eqn. 1, σ line = i (di − ci)Pi/σ 2 i i (P 2 i /σ 2 i ) −1/2 ,(1) where di is the pixel intensity at position i, ci is the continuum at pixel i, Pi the line profile at position i, and σi the noise estimate at position i. The peak SNR is defined as the ratio of the peak flux density, taken to be the mean flux density in a 3×3 pixel window less the local continuum, to the standard deviation of the surrounding pixels in the spectrum. Finally the correlation coefficient (ρ line ) is calculated Eqn. 2, ρ line = i (di −d)(Pi −P ) σP σ d(2) where σP and σ d are the standard deviation of the line profile and data values, respectively. Line fits which have σ line > 4, σ peak > 2.5 and ρ line > 20 were considered as candidate emission lines. All candidate emission lines that have a wavelength within the range (1 + z phot − 0.16) × 6563.4Å< λ < (1 + z phot + 0.16) × 6563.4Å were considered to be Hα. The window of δz = 0.16 equates to 4σ phot−z for the typical photo-z error quoted by Ilbert et al. (2009) at z ∼ 1. A total of 85 candidate Hα emission lines were found from the sample of 237 unique 24 µm targets. We assessed the reliability of our line identification technique in two ways. Firstly the line identification was repeated, but with the proposed redshifts (and hence wavelength search window) shifted. To ensure that the mock search windows are sufficiently far away from real lines, but still within the wavelength coverage of FMOS, sources at z phot < 1.1 were given z mock = z phot + 0.16 + δ, while those at z phot > 1.1 were set to z mock = z phot − 0.15 − δ, where δ is a random number between 0 and 0.15. As a result of this process seven lines were identified, giving an estimate of the false positive line detection rate of 7/85 or 8±3 per cent. No false lines were returned with σ line > 8. We compare the redshifts, as determined by the wavelength of our candidate Hα lines, and the known spectroscopic redshifts. From our sample of 85 Hα line emitters, 28 are found to have reliable (z qual > 3) spectroscopic redshifts in the zCOSMOS bright (i < 22.5) sample of Lilly Lilly et al. (2007). While this result is encouraging, it is likely that the reliability of our line identification is a function of brightness, and the zCOSMOS bright sample is limited to i + AB < 22.5; roughly 54 per cent (46/85) of our candidate Hα line emitters have 22.5 < i + AB < 25. Via a similar process we can estimate the completeness of our line identification process. Assuming the random redshifts, z mock , described above, we inject mock emission lines into our data at a wavelength corresponding to 1 + z mock ) 6564.3Å. We estimate the completeness for the two pointings independantly. Table 2 details the completeness (i.e. the ratio of sources detected to those injected) as a function of flux. No aperture correction is assumed, injected line fluxes are considered to be those contained within the 1.2 arcsec diameter fibre of FMOS. The completeness never reaches 100 per cent as lines at certain wavelengths will always be undetectable due to the gap in wavelength coverage from 1.36-1.42 µm, as well as the masking of OH sky lines. In total 125 OH lines are suppressed which, combined with the gap due to atmospheric absorption, remove 24 per cent of the potential wavelength coverage. It can be seen that our identification process reaches this maximum level above a line flux of fHα > ∼ 4 × 10 −16 ergs cm −2 s −1 , while we are > ∼ 50 per cent complete above a line flux of fHα > ∼ 1.5 × 10 −16 ergs cm −2 s −1 . The completeness estimates for the second pointing are marginally higher at fHα < ∼ 3×10 −16 ergs cm −2 s −1 due to the increased exposure time (12600 sec for the second pointing vs. 7200 sec for the first). Note that these line fluxes consider only the flux contained within the 1.2 arcsec diameter fibres used on FMOS; no aperture correction has been applied yet. For each Hα line emitter we attempt to measure the flux of the neighbouring [NII] 6584 line. Line fluxes are estimated in the same manner as Hα, but with the central wavelength fixed at (1 + z) 6584Å. Of the 85 Hα line emitters; 33 also have [NII] at SN R > 3. Finally, both Hα and [NII] line fluxes are corrected for the limited aperture and unknown stellar continuum. The aperture correction is determined via the ratio of IF814W flux within the FMOS fibre (1.2 arcsec diameter) to that within the Kron radius. The typical aperture correction is ∼ 2-3. For the continuum correction, the near-IR spectra are not deep enough to detect the continuum emission near the Hα line. Thus, continuum emission at the wavelength of the line is estimated, and removed, using the available broadband optical and near-IR imaging from the COSMOS survey (Capak et al. 2007, McCracken et al. 2010. For sources where the line lies at λ < 1.4 µm we use the J band magnitude, and the [z + −J] colour to estimate the continuum flux. For those lines where λ > 1.4 µm the K band magnitude and [J − K] colour are used. In all cases we assume that the Hα line is coincident with stellar absorption of equivalent width EW = 4.4 (Moustakas & Kennicutt 2006). Appendix A contains the line fluxes and photometric properties for our 85 Hα line emitters. While we list the full catalogue of Hα line detections resulting from our observations, in the following analysis we consider only the 57 Hα line emitters which have robust detections at 250 µm. This restriction is implemented as 24 µm alone is not a good tracer of total IR luminosity at these redshifts (Elbaz et al. 2010). Enforcing 250 µm detections also helps to minimise contamination by AGN activity (Hatziminaoglou et al. 2010). SDSS comparison sample To establish a low redshift baseline for our z ∼ 1 measurements we identify a sample of Spitzer 160 µm selected sources from the SWIRE survey (Lonsdale et al. 2003) in the Lockman Hole field with SDSS DR6 (Adelman- McCarthy et al. 2008) spectroscopy. Sources are required to be brighter than 50 mJy at 160 µm and to lie at z < 0.4. The 160 µm photometry is complemented with SWIRE MIPS 24, 70 µm data, and SPIRE photometry at 250, 350 and 500 µm from HerMES. Hα line fluxes are calculated using the GAN-DALF package (Sarzi et al. 2006), with aperture corrections based on the ratio of the SDSS Petrosian-to-fibre rmagnitudes. We retain sources with a peak line flux density to noise ratio of greater than three. The typical limiting Hα line flux (3σ) is 5.8 × 10 16 erg cm −2 s −1 . Our final low-z sample consists of 156 160 µm-detected sources with reliable Hα line flux measurements, at a mean redshift of z = 0.1. RESULTS Hα detection rate for IR-selected sources The raw Hα detection rate for IR-sources (24∩250 µm) is 57/168 (34±4 per cent). In Fig. 1 we present the Hα detection rate as a function of 250 µm and 24 µm flux densities. While the Hα detection rate appears to be insensitive to 250 µm flux density, a modest increase in the detection rate is seen for sources with S24 µm > 200 µJy. Fig. 2 shows the redshift distribution of IR-sources targeted with FMOS, and those with Hα line detections. The contrast between the two distributions highlights the visibility of Hα with FMOS as a function of redshift. Prior to FMOS observations, only photometric redshifts were available for the vast majority of targets. Thus while targets were photo−z selected to be in the redshift range where Hα is visible, some fraction of sources were expected to have Hα fall at wavelengths outside the FMOS wavelength range. Despite this the observed detection rate compares well with what would be predicted given the known completeness ( §2.2). For each detected source we calculate the completeness at that line flux, c l , by interpolating the values in Table 2. If the Hα line fluxes of the undetected sources are distributed similarly to the detected ones (i.e. the bulk of the incompleteness is due to OH sky lines) then the sum l 1/c l should be equal to the number of targetted objects. For our 24∩250 µm sample l 1/c l = 177; close to the actual number of sources targetted (168). We can compare our detection rate for IR-selected sources with FMOS to optically selected sources with VLT VIMOS. For the VVDS-DEEP survey (Le Févre et al. 2005), the detection rate for i-selected sources with similar magnitudes (median for our sample is i + AB ∼ 23.5) and redshifts (z ∼ 1.2) to our sample is ∼ 70 per cent (for 4.5 hr exposures; Ilbert et al. 2005). While our FMOS detection rate is almost one-half of this, IR-sources have quite a low areal density and cannot make the most of the large multiplex of VIMOS. For our parent sample of 24 & 250 µm detected, i + AB < 25 and 0.65 < z phot < 1.75 sources the areal density is ∼ 1500 per sq. deg. Using these numbers as a guide, the expected number of redshifts recovered in a single 2 hr FMOS pointing (0.36×200 fibres equals 72 sources), is comparable to the number expected from a single 4.5 hr VIMOS pointing (0.7×91 targets equals 64). Thus for Herschel selected targets FMOS is a competitive facility for redshift recovery in the range 0.7 < z < 1.8. Composite spectrum for Hα detected sources While our near-IR spectra are of sufficient quality to robustly measure fluxes for bright emission lines, very little additional information can be extracted from the individual spectra. This is partly due to the low SNR, but also because of the large number of OH sky lines in the near-IR, which significantly limit the wavelength coverage. However, using the spectral coverage unaffected by OH sky lines from each of our Hα-detected sources we can build a composite spectrum across a reasonably wide and continuous range of rest-frame wavelengths. Fig. 3 shows the composite rest-frame spectrum in the region of Hα and Hβ produced from our sample of 57 sources. The composite spectrum is produced by first adopting a grid in rest-frame wavelength and calculating the contribution of each observed near-IR spectrum via a Gaussian kernel. The FWHM of the Gaussian kernel is taken to be the spectral resolution, λ/600Å. Pixels within 10Å of an OHsuppressed line are excluded, as are those which lie outside the wavelength range 1.1-1.36 µm or 1.42-1.7 µm. The variance in the composite spectrum is estimated via jack-knife resampling of the sources contributing to each wavelength bin. The absolute flux calibration of the composite is corrected by comparing the continuum level in the composite to the mean broadband photometry in the J and K bands. Several well-known spectral lines can be seen in addition to Hα: [NII] (Baldwin, Phillips & Terlevich 1981;BPT). Measuring the strengths of these lines from Fig. 3 we determine f [NII] 6584 /fHα = 0.24 ± 0.07, while f [OIII] 5008 /f Hβ = 1.3 ± 0.6. Comparing these ratios with the BPT diagnostic plots suggests that, on average, the primary origin of emission lines, and by proxy the IR-luminosity, in our sample is star formation. The standard approach to estimating the dust attenuation in star-forming galaxies is to use the flux ratios of Balmer lines. Under typical conditions (i.e. Te ∼ 10 4 K), and in the absence of dust, the line ratio fHα/f Hβ = 2.86 (Osterbrock 1989 known as the Balmer decrement, can be attributed to differential dust attenuation at the rest-frame wavelengths of the Balmer lines. Using our composite spectrum we estimate the aggregate value of the Balmer decrement for SPIRE sources at z > ∼ 1. The Hβ line is visible in the FMOS wavelength coverage at 1.26 < z < 2.5, excluding sky lines. To account for this we build a second composite spectrum, using only those sources which have 'clean' (i.e. no overlapping sky lines) FMOS wavelength coverage at the wavelength of both Hβ and Hα; only 24 sources satisfy this criterion. The composite spectrum from these sources in the region of Hα and Hβ is shown in Fig. 3. The Hβ observable sources have a mean redshift of z = 1.36 and a mean IR luminosity of LIR = 10 45.5 ergs s −1 (10 12 L⊙). We measure line fluxes of fHα =6.8±0.6 × 10 −16 ergs cm −2 s −1 and f Hβ =1.3±0.4 × 10 −16 ergs cm −2 s −1 , after correcting both lines for stellar absorption (EWHα = 4.4 and EW Hβ =2.8Å; Moustakas & Kennicutt 2006) and applying an aperture correction of 2.8 (the mean value for these 24 sources; see §3.3). This gives a Balmer decrement of R = fHα/f Hβ = 5.2 ± 1.6, resulting in a dust attenuation of E(B − V ) = log 10 (R/2.86)/0.4[k(λHα) − k(λ Hβ )] = 0.51 ± 0.26, where k(λHα) = 4.596 and k(λ Hβ ) = 3.325 (Calzetti et al. 2000). This is equivalent to Av = 2.1 mags, similar to that found for local IR-luminous galaxies (e.g. Hopkins et al. 2001, Wijesinghe et al. 2011). Relationship between Hα and IR star formation rate estimates In the absence of AGN activity or strongly non-solar metallicity, differences between the Hα estimated SFR (SFRHα), and the best estimate of the total SFR (SFRtot) can be attributed to the effect of dust attenuation. Thus the ratio SFRtot/SFRHα can be used as an estimator of the level of dust attenuation (e.g. Hopkins et al. 2001;Kewley et al. 2002). In order to calculate SFRtot and SFRHα the following steps were taken. IR luminosities are calculated by fitting template SEDs to the 24 µm and SPIRE data, then integrating the best fit template in the range 8-1000 µm. IR template SEDs are taken from Rieke et al. (2009). Both IR and Hα luminosities are converted into SFR via the relations presented in Kennicutt (1998), assuming a Chabrier (2003) IMF, i.e; SFRIR = 2.61 × 10 −44 LIR (erg s −1 ), and SFRHα = 4.61 × 10 −42 LHα (erg s −1 ) Finally, SFRtot is calculated by adding together the IR and Hα star formation rates, i.e. SFRtot = SFRIR + SFRHα. While other combined Hα+IR estimators of SFR exist (e.g. Kennicutt et al. 2009), these may not give good results for the class of IR-luminous galaxies (nor Herschel derived IR luminosities). We can confirm our assumption that SFRtot/SFRHα is a good tracer of dust attenuation by comparing to estimates of the attenuation from the Balmer decrement for our SDSS sample. All 156 of our SDSS comparison sample have reliable (> 3σ) Hβ line flux estimates. We produce estimates of the dust attenuation independant of the far-IR measurements using the Balmer decrement. Fig. 4 compares the dust attenuation estimated from the Balmer decrement (E(B − V )BD) to the infered value assuming SFRtot/SFRHα is a good estimate of AHα. Alternative estimates of the mean (and standard deviation) of dust attenuation measured from the Balmer decrement of individual SDSS spectra, and the aggregate FMOS value from our composite spectrum ( §3.2) are also shown in Fig. 6. Encouragingly the direct estimates of the mean dust attenuation are in good agreement with that inferred from SFRIR/SFRHα, and both the dust attenuation-SFR relations. In Fig. 6 are also shown unlensed 850 µm selected sources (SMGs) at z > ∼ 2, with Hα line measurements from Swinbank et al. (2004). Here we make use of the SFR(Hα) and L(FIR) quantities given in Table 2 attenuation -SFR relation, suggesting that SMGs experience enhanced attenuation. However it is worth noting that the L(FIR) estimates for the SMGs come from pre-Herschel submm-radio estimates and hence may be overestimated (see Magnelli et al. 2012). Future studies with FMOS in this SFR range, as well as a re-assessment of the SMG population with Herschel photometry will allow this trend to be confirmed. The modest Hα detection limits achievable with FMOS mean we will not recover IR-sources which have very large SFRtot/SFRHα. To quantify this we calculate the typical maximum observable limit of SFRtot/SFRHα as a function of SFRtot for both the SDSS and FMOS samples. For the SDSS sample we assume a detection limit of fHα = 5.8 × 10 16 erg cm −2 s −1 , while for the FMOS sample we assume a detection limit of fHα = 1 × 10 16 erg cm −2 s −1 . In both cases these limits include a correction for the mean loss due to the limited aperture of the fibres (1.2 arcsec diameter for FMOS, 3 arcsec diameter for SDSS). While the maximum limit for the SDSS sample is significantly higher (∼ 0.2 dex) than the observed values of SFRtot/SFRHα, the limits for the FMOS dataset appear quite close to the observed data points. Given ∼ 65 per cent of our parent sample is undetected in Hα a potential explanation for this large incompleteness is a significant population of sources with SFRtot/SFRHα above these selection limits. Hence the observed consistency with the Hopkins et al. (2001), and our Eqn. 5 may be a result of a bias towards low SFRtot/SFRHα. While we cannot rule out the existence of large SFRtot/SFRHα sources (as we cannot detect them), we can estimate the observed completeness for our parent sample assuming our best-fit to the E(B − V )-SFRtot relation is a good description for the whole population. For each source in our parent sample of 168 24 ∩ 250 µm sources we first estimate LIR, assuming the photo-z from Ilbert et al. (2009) and the SED fitting process described above. The Hα line flux is then predicted from LIR (assuming SFRIR =SFRtot) using the best fit log-linear relation from Fig. 6 (with 0.5 dex of intrinsic scatter) and the mean loss due to the fibre aperture (2.8). Applying the completeness curves from Table 2 we would expect a completeness of 35 per cent, in good agreement with the observed completeness of 34±4 per cent, and consistent with a similar assessment of the completeness presented in §3.1. If a large fraction of our parent IR-selected sample had dust attenuation levels (i.e. SFRtot/SFRHα) significantly above that predicted by Eqn. 5 we would expect much lower completeness in our FMOS observations than achieved. Given the good agreement between the observed and expected completeness we conclude that Eqn. 5 must hold for the bulk of IR-luminous sources at z ∼ 1. The existence of a relationship between stellar mass and metallicity has been confirmed across a wide range in redshift (0 < z < 3; Lequeux et al. 1979;Tremonti et al. 2004;Erb et al. 2006;Maiolino et al. 2008;Zahid et al. 2011), with a steady trend towards lower metallicity for a given stellar mass with increasing redshift. To investigate the massmetallicity relation for our z ∼ 1 IR-sources we combine our metallicity estimates with stellar masses as derived by Wang et al. (2012). Stellar mass estimates from Wang et al. (2012) are calculated by finding the best-fit stellar population model to the observed multi-wavelength photometry using the Le Phare software (Arnouts et al. 2002;Ilbert et al. 2006), combined with stellar population synthesis models from Bruzual & Charlot (2003) and assuming a Chabrier (2003) IMF. The mass-metallicity relation for IR-galaxies The left panel of Fig. 7 compares the stellar mass to the metallicity for both our sample of z ∼ 1 Herschel-FMOS and comparison sample of z ∼ 0.1 SDSS sources. A tentative trend of metallicity with stellar mass can be seen in both samples, although with significant scatter. No discernable evolution is seen in the metallicity between the z ∼ 1 and z ∼ 0.1 IR-selected samples. Also shown in the left panel of Fig. 7 are studies using stacking of near-IR spectra by Erb et al. (2006) Yabe et al. (2012) study the difference in the metallcities is surprising, although likely due to a combination of the different mass ranges probed, the stacking nature of the Yabe et al. (2012) result and our bias towards dusty galaxies. In the middle panel of Fig. 7 we show the relationship between specific SFR (SSFR;SFR/M⋆) and metallicity. Over the range in SSFR well-sampled by our Herschel-FMOS and SDSS samples we see no discernable trends, although there are hints of decreasing metallicity with increasing SSFR above SSFR= 0.5 Gyr −1 . It has been proposed that the metallicity is a function of both SFR and stellar mass, with a "fundamental mass relation" (FMR) linking the three parameters (Mannucci et al. 2010). In Fig. 7 we show the fit to the FMR from Mannucci et al. (2010) for values of SFR representative of our study. The metallcities of both our Herschel-FMOS and SDSS are consistent with the prediction from the FMR. Finally, we consider the relationship between dust attenuation and metallicity. In the right panel of Fig. 7 is shown the metallicity as a function of dust attenuation (as traced by the ratio SFRtot/SFRHα). A clear trend between dust attenuation and metallicity is seen in both the Herschel-FMOS and SDSS samples, with the most metalrich galaxies experiencing the largest levels of attenuation. The best fit log-linear relation between attenuation and metallicity is found to be: 12 + log 10 O/H = 0.19 log 10 SFRtot/SFRHα + 8.57. Converting to E(B − V ) via the Calzetti et al. (2000) attenuation curve we find Eqn. 6, 12 + log 10 O/H = 0.24 E(B − V ) + 8.57 This correlation is somewhat expected; the dust grains responsible for attenuating starlight (and nebular line emission) are synthesised from metals in the ISM. Similar studies, using the IR to UV luminosity ratios as a proxy for dust attentuation, have also found a correlation between metallicity and attenuation for nearby galaxies (Heckman et al. 1998) and z ∼ 2 (Reddy et al. 2010). Given the correlation between metallicity with attenuation the high metallicity of our Herschel-FMOS sample (when compared to Yabe et al. 2012 at z ∼ 1.4) is to be expected as our IR-selected sample must be biased towards the most obscured sources. DISCUSSION AND CONCLUSIONS We have investigated the properties of z ∼ 1 IR luminous galaxies by performing near-IR spectroscopy of a sample of Spitzer and Herschel selected sources with FMOS. Candidate emission lines were identified in the 2D reduced FMOS frames via a semi-automated procedure. Via comparison with known spectroscopic redshifts, and direct testing of the line detection algorithm, we estimate that our Hα line sample is > ∼ 90 per cent reliable. Our scientific conclusions are; • Robust detections of Hα were found for 57 of 168 24∩250 µm-sources, resulting in a detection rate of 34±4 per cent. This detection rate is consistent with the expected incompleteness measured by simulating the line detection process. For sparse targets such are Herschel-selected sources, (Kewley & Ellison (2008); KE08) and z ∼ 2.2 (Erb et al. 2006; E06) are shown as dotted and dot-dashed lines in the left panel, respectively. Results from a previous FMOS study for star-forming galaxies at z ∼ 1.4 (Yabe et al. 2012) FMOS is a competitive redshift recovery instrument with equivalent optical MOS instruments on other 8m class telescope (e.g. VIMOS on VLT). • The mean dust attenuation, estimated via the Balmer decrement for the Hα and Hβ emission lines for a composite spectrum of Hα detected sources, is E(B − V ) = 0.51 ± 0.27 for LIR = 10 12 L⊙ sources at z = 1.36. • Good agreement was found between the dust attenuation estimated from the Balmer decrement and that inferred from the ratio of Hα-estimated to best estimate of the total star formation rate. Using SFRtot/SFRHα as an indicator of attenuation in the Hα line we derive a relationship between dust attenuation and star formation rate of E(B − V ) = (0.135 ± 0.06) log 10 SFRtot + 0.35 ± 0.08. These results are broadly consistent with the relationship between star formation rate and dust attenuation seen both in low-z star forming galaxies (e.g. Hopkins et al. 2001). • The gas phase metallicity relation was investigated for the subset of Herschel-FMOS, and SDSS, sources with robust measurements of the [NII] 6584 line. For IR-selected sources with M⋆ ∼ 10 10.5 M⊙ the typical metallicity is found not to evolve between z ∼ 0.1 to z ∼ 1.2. No discernable trend with specfic SFR is seen, but a strong correlation between metallicity and dust attentuation is seen, described by a best-fit log-linear relation; 12 + log 10 O/H = 0.24 E(B − V ) + 8.57. University, the United States Naval Observatory, and the University of Washington. SPIRE has been developed by a consortium of institutes led by Cardiff Univ. (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UK ATC, Univ. Sussex (UK); Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC and UKSA (UK); and NASA (USA). FMOS was funded jointly by STFC and the Japanese Monbukagakusho, and we gratefully acknowledge the support of the staff at the Subaru Telescope throughout the instrument commissioning phase. The data presented in this paper will be released through the Herschel Database in Marseille HeDaM (hedam.oamp.fr/HerMES) Figure 1 . 1Hα line detection rate of FMOS targeted sources as a function of 24 µm and 250 µm flux density. The detection rate is relatively insensitive to 250 µm flux density, while a modest gain is seen for sources with S 24 µm > 200 µJy. Figure 2 . 2Redshift distribution of FMOS targeted sources and sources with Hα line detections. For FMOS targets we use photometric redshifts fromIlbert et al. (2009). For detected sources the redshift implied by the location of the Hα line is used. Figure 4 . 4Left panel: Comparison of dust attenuation estimates from the Balmer decrement (E(B − V ) BD ) and from the ratio of SFRtot/SFR Hα (E(B − V ) IR ) for SDSS galaxies. The dashed line represents E(B − V ) IR = E(B − V ) BD . The two estimates are excellent agreement, with a RMS difference of 0.17. Right panel: Comparison of dust attenuation estimates from the Balmer decrement (E(B − V ) BD ) and from the ratio of SFR K09 /SFR Hα , where SFR K09 is the combined Hα+IR SFR estimator presented in Kennicutt et al. 2009 (K09). Again, the dashed line represents E(B − V ) IR = E(B − V ) BD It can be seen that the K09 SFR calibration gives much poorer agreement with the Balmer decrement estimated attenuations (RMS= 0.27). Fig. 5 Figure 5 . 55shows the relationship between SFRHα and SFRtot for our sample of 57 sources detected at 24 µm, Comparison of Hα to total (Hα + Far-IR) SFRs, assuming the conversions ofKennicutt (1998) with a Chabrier (2003) IMF. FMOS detected sources (red triangles) and the lowz sample of 160 µm-selected SDSS galaxies (blue squares) are shown. The dashed line shows the best-fit log-linear relation to the data.250 µm and Hα. Fitting this observed correlation with a loglinear function results in the best fit; log 10 SFRHα = (0.82 ± 0.08) log 10 SFRIR + 0.47 ± 0.11 (M⊙ yr −1 )Using theCalzetti et al. 2000 model for the variation of dust attenuation with wavelength we can use this result to convert the ratio SFRtot/SFRHα to E(B − V ). Taking our best fit correlation between these values we derive the following relationship between SFR and dust attenuation; E(B−V )IR = (0.135±0.06) log 10 SFRtot+0.35±0.08 (M⊙ yr −1 )(5)Fig. 6presents dust uncorrected SFRtot/SFRHα vs. SFRtot. We also show inFig. 6our best fit relationship between E(B − V ) and SFRtot, and the empirical relationship for low-z IRAS galaxies as derived byHopkins et al. (2001). These two relations show reasonable agreement, with theHopkins et al. (2001) relation slightly below our simple loglinear fit. Figure 6 . 6of Swinbank et al. (2004), converting SFR(Hα) to the Chabrier (2003) IMF used here and calculating SFRIR from L(FIR) using the equation given above. SFR(Hα) as quoted by Swinbank et al. (2004) includes corrections for slit-loss and so should be compatible with the values we derive from FMOS and SDSS data. Interestingly the z > ∼ 2 SMGs appear slightly above both the Hopkins et al. (2001) and our best-Ratio of IR to dust uncorrected Hα-based SFR vs. IRbased star formation rate (SFR). The equivalent E(B − V ) is also given, assuming, a Calzetti et al. (2000) attenuation curve. FMOS detected sources (red triangles) and the low-z sample of 160 µmselected SDSS galaxies (blue squares) are shown. The dashed line shows the local relation between SFR and E(B − V ) from Hopkins et al.(2001). The E(B − V ) determined from the Balmer decrement of our FMOS composite spectrum(Fig. 3), and mean value for SDSS spectra are shown as cyan and orange circles, respectively. Also shown are the positions of z ∼ 2 SMGs with Hα line measurements fromSwinbank et al. (2004). Dotted lines show the effect of the Hα detection limit (5σ) at the mean redshift for sources with that SFRtot for the SDSS (blue) and FMOS (red) samples. A significant fraction (28/57) of our sample have robust (SNR> 3) measurements of the [NII] 6584 line. This allows us to investigate the gas phase metallicity (12+log 10 O/H) of our IR-sources via the ratio of the [NII] 6584 to the Hα line (N2 method; Kewley & Dopita 2002; Pettini & Pagel 2004). For each source with a robust [NII] measurement we estimate the metallicity via Eqn. 1 of Pettini & Pagel (2004; PP04); 12 + log 10 O/H = 8.9 + 0.57 × log 10 f [NII] /fHα Figure 7 . 7Relationship between stellar mass (left panel), specific SSFR (middle panel), dust attenuation (right panel) vs. gas phase metallicity as seen in our IR-selected sample. All sources with [NII] line flux measurements with SNR> 3 are shown, while limits are shown for those sources without robust [NII] line measures. Sources from our Herschel-FMOS sample at z ∼ 1.2 are shown as red dots, while the ∼ 0.1 Herschel-SDSS samples are blue squares. Previous estimates of the stellar mass -metallicity relation at z ∼ 0.1 are shown as orange stars. Where necessary, comparison samples have been converted to use the Pettini & Pagel (2004) calibration of the N2 metallicity tracer (KE08) and a Chabrier (2003) IMF. In the middle panel the position of the "fundamental metallicity relation" of Mannucci et al. (2010)is shown for a range of SFRs representative of our sample as the dotted lines. In the right panel the best-fit log-linear relation between metallicity and dust attenuation (as traced by the ratio SFRtot/SFR Hα ) is shown as a dashed line. Table 1 . 1Summary of FMOS observationsDate Texp N 24 µm N 24∩250 µm P1 2010 Nov. 22 8 × 900 s 136 102 P2 2010 Nov. 24 & 25 14 × 900 s 105 67 Total (Unique) 241 (237) 169 (168) Table 2 . 2Completeness of line identification process as a function of flux. Completeness is estimated via injection of mock emission lines into the 2D spectra at random wavelengths/redshifts. Note that this is for flux contained within the 1.2 arcsec diameter fibre of FMOS; no aperture effects are considered.Line flux Completeness P1 P2 10 −16 ergs cm −2 s −1 . per cent per cent 0.5 4.5 1.4 1. 23.2 25.3 1.5 52.9 57.8 2. 65.8 67.8 2.5 71.0 72.1 3. 74.2 74.7 3.5 77.4 77.6 4. 77.6 78.7 et al. (2007). Of these, 27 (96 per cent) are found to be within δz = 0.01 of our assumed Hα redshift. No incorrect redshifts are found amongst the 11 sources with candidate Hα lines at σ line > 8 and a spectroscopic redshift from 6584, although this is blended with Hα; a blend of the [SII] doublet at 6716 and 6731Å; and weak signatures of [OIII] 5008 and Hβ. The line ratio Hα/[NII] 6584 compared to the line ratio [OIII] 5008/Hβ is often used as diagnostic of AGN activity ). Observed differences in this ratio, also The dashed line is the composite spectrum for only those 24 sources which are visible at the rest frame wavelength of Hβ. The grey solid line represents the 1σ variance in the composite. Well-known spectral features are marked. Encouragingly, our composite spectrum recovers several other spectral lines, such as[NII] 6584, the blended doublet of [SII] 6716 and [SII] 6731, and weak signatures of [OIII] 5008 and Hβ.4800 4850 4900 4950 5000 5050 5100 Rest Wavelength (Å) 0.0 0.2 0.4 0.6 F λ (10 −17 ergs cm −2 s −1 Å −1 ) Hβ [OIII] [OIII] 6500 6550 6600 6650 6700 6750 Rest Wavelength (Å) 0.0 0.5 1.0 1.5 [NII] Hα [NII] [SII][SII] Figure 3. Composite rest-frame spectrum for the 57 24 ∩ 250 µm sources with robust line detections in the region of Hα (top) and Hβ (bottom). with Keck NIRSPEC at z ∼ 2.2 and Yabe et al. (2012) with FMOS at z ∼ 1.4 (We convert the Yabe et al. 2012 stellar masses to a Chabrier 2003 IMF for consistency). Given the similar redshift range between our Herschel-FMOS sample and the Table A1 : A1Details of Herschel 250 µm and Spitzer 24 µm-selected sources with reliable Hα line detections.RA a Dec. Redshift b I F814W S 24 µm S c 250 µm S c 350 µm S c 500 µm f d Hα f e [NII] deg. deg. AB mag mJy mJy mJy mJy 10 16 erg cm −2 s −1 10 16 erg cm −2 s −1 149.94205 2.32077 1.03 21.94±0.01 0.20±0.01 9.80 ±2.73 < 12.22 < 15.98 4.32±0.30 < 3.95 149.96689 2.44185 0.90 22.44±0.02 0.08±0.01 < 8.20 < 13.15 < 9.28 2.84±0.32 < 5.78 149.97292 2.49010 1.36 23.64±0.03 0.43±0.01 15.44 ±2.73 19.04 ±3.22 < 15.41 1.52±0.41 < 6.17 149.95179 2.48627 1.03 22.00±0.01 0.28±0.01 10.13 ±2.74 11.98 ±3.38 < 14.81 2.33±0.17 < 2.62 149.99931 2.45197 1.51 23.13±0.03 0.33±0.01 21.61 ±2.73 15.34 ±3.61 18.14 ±4.87 2.96±0.40 < 3.05 150.03484 2.26353 0.90 21.46±0.01 0.25±0.01 11.26 ±2.73 < 26.43 < 33.24 5.44±0.37 < 6.56 150.04760 2.62123 1.38 22.73±0.02 0.23±0.01 20.49 ±2.73 13.12 ±3.25 < 17.00 3.20±0.27 < 2.19 150.02476 2.35211 0.93 21.76±0.01 0.15±0.01 < 8.20 < 13.09 < 35.23 4.55±0.38 < 5.37 150.03650 2.31730 1.46 22.36±0.02 0.44±0.01 26.35 ±2.73 25.45 ±3.48 < 17.85 11.43±0.33 < 4.66 150.02440 2.44388 1.03 21.72±0.02 0.39±0.01 < 41.18 < 32.57 < 25.69 10.69±0.56 < 7.19 150.04820 2.46775 1.17 23.33±0.03 0.34±0.01 11.71 ±2.73 < 9.31 < 14.30 7.69±0.34 2.57±0.78 150.03637 2.49426 1.03 21.86±0.01 0.41±0.01 9.23 ±2.74 < 9.59 < 15.54 6.86±0.28 < 3.75 150.08336 2.53619 1.42 23.70±0.04 0.30±0.01 22.99 ±2.73 25.93 ±4.79 33.71 ±9.86 7.97±0.28 < 3.39 150.05928 2.51858 1.03 22.10±0.01 0.38±0.01 12.46 ±2.74 < 9.61 < 16.43 1.92±0.13 < 2.17 150.05386 2.58972 0.70 18.89±0.00 3.45±0.01 < 8.19 < 10.02 < 14.47 68.67±0.31 47.99±1.39 150.10975 2.60274 0.98 21.80±0.01 0.32±0.01 < 8.21 < 33.17 < 9.27 10.12±0.19 - 150.10019 2.48157 0.89 22.11±0.02 0.08±0.01 < 8.23 < 9.27 < 14.32 4.88±0.48 < 5.28 150.13340 2.26201 0.75 21.63±0.01 0.08±0.01 < 8.19 < 9.94 < 14.36 3.31±0.32 < 4.71 150.14563 2.29341 0.88 22.27±0.02 0.10±0.01 < 8.20 < 9.29 < 4.77 3.77±0.37 < 4.26 150.13445 2.61448 0.89 22.44±0.01 0.15±0.01 < 8.22 < 16.66 < 16.33 2.93±0.16 < 1.78 150.16162 2.69151 1.48 23.51±0.03 0.52±0.01 21.25 ±2.73 24.13 ±3.95 17.66 ±5.36 5.05±0.61 - 150.15563 2.67708 1.04 22.28±0.02 0.21±0.01 10.36 ±2.74 < 29.40 < 32.03 3.14±0.15 < 2.70 150.12472 2.66871 1.28 22.56±0.01 0.19±0.01 21.05 ±2.73 < 12.30 < 14.92 5.82±0.12 1.43±0.45 150.15229 2.21933 0.92 21.86±0.01 0.36±0.01 < 8.21 < 12.19 < 7.61 3.60±0.19 < 2.96 150.21805 2.52182 1.18 21.72±0.01 0.30±0.01 13.05 ±2.74 22.89 ±3.56 < 16.66 5.98±0.34 < 2.59 150.21025 2.56547 1.40 22.29±0.02 0.34±0.01 30.92 ±2.73 22.22 ±3.69 < 36.15 5.34±0.15 2.24±0.66 150.25980 2.29235 0.99 22.42±0.01 0.14±0.01 < 8.20 < 16.07 < 18.66 3.48±0.27 - 150.22858 2.31620 0.90 21.25±0.01 0.30±0.01 < 8.22 < 9.56 < 17.02 8.68±0.17 < 2.55 150.22203 2.62003 0.69 21.50±0.01 0.65±0.01 27.91 ±2.73 22.13 ±7.22 < 21.83 9.14±0.48 < 9.70 150.24813 2.39912 0.68 21.03±0.01 0.15±0.01 < 2.73 < 3.35 < 17.64 9.01±0.50 < 6.14 150.29832 2.46967 0.85 21.75±0.01 0.18±0.01 < 8.19 < 10.17 < 25.53 2.91±0.20 < 3.79 150.29408 2.51691 0.84 21.89±0.01 0.16±0.01 < 8.21 < 18.92 < 12.42 3.02±0.30 < 0.84 150.25672 2.48480 1.24 23.13±0.03 0.19±0.01 17.77 ±2.74 12.31 ±3.66 < 30.69 4.84±0.64 4.32±0.66 150.29414 2.57633 0.78 22.40±0.02 0.21±0.01 < 8.21 < 11.11 < 19.23 1.98±0.20 < 5.92 150.31964 2.61813 1.00 21.59±0.01 0.24±0.01 < 30.62 < 31.33 < 24.63 5.03±0.36 - 150.27952 2.59666 1.49 23.03±0.02 0.17±0.01 13.49 ±2.73 13.22 ±3.15 < 15.18 3.81±0.47 < 2.73 150.34435 2.73311 0.85 21.56±0.01 0.26±0.02 < 8.19 15.09 ±3.95 < 48.57 2.33±0.12 1.10±0.14 150.32742 2.68368 0.96 21.62±0.01 0.45±0.01 < 8.21 < 18.05 < 15.67 9.08±0.22 3.84±0.28 150.34866 2.45537 1.02 22.12±0.02 0.22±0.01 11.54 ±2.74 < 9.68 < 21.07 5.77±0.25 1.67±0.27 150.31731 2.50830 0.98 22.31±0.01 0.09±0.01 14.61 ±2.74 12.64 ±3.38 < 33.01 2.73±0.29 1.01±0.30 150.36745 2.55928 0.91 22.20±0.02 0.13±0.01 < 8.19 < 8.07 < 16.49 2.94±0.26 < 4.69 150.35415 2.51916 0.96 21.32±0.01 0.75±0.01 29.87 ±2.74 < 23.46 < 31.47 6.18±0.50 1.63±0.30 150.35350 2.55854 1.04 21.63±0.01 0.17±0.01 10.25 ±2.73 < 13.46 < 14.98 4.17±0.24 3.08±0.30 150.34722 2.57026 0.92 21.48±0.01 0.30±0.01 < 8.19 < 20.10 < 22.18 3.01±0.19 < 3.12 150.36110 2.60817 1.30 22.89±0.02 0.11±0.01 < 8.19 < 20.03 < 24.99 3.96±0.21 < 0.58 150.33580 2.64975 1.33 22.58±0.02 0.12±0.01 20.33 ±2.73 23.03 ±3.78 < 14.14 3.66±0.21 1.49±0.20 150.34937 2.36957 1.57 23.49±0.04 0.13±0.02 < 8.23 < 18.35 < 15.06 7.85±0.70 2.70±0.64 150.39351 2.44529 0.92 21.67±0.01 0.28±0.01 < 8.19 < 11.84 < 15.06 7.62±0.25 1.18±0.26 150.37070 2.49822 0.82 20.52±0.01 0.43±0.01 19.88 ±2.73 11.93 ±3.54 < 17.47 8.12±0.28 7.45±0.66 150.41783 2.55859 1.21 22.13±0.01 0.76±0.01 47.01 ±2.74 34.07 ±3.15 19.10 ±4.88 2.47±0.32 1.97±0.30 150.42288 2.58332 0.82 20.43±0.01 0.69±0.01 29.20 ±2.74 < 28.48 < 15.74 30.07±0.65 3.95±1.03 150.42188 2.32331 0.83 23.16±0.02 0.61±0.01 20.59 ±2.74 < 9.86 < 18.94 16.98±1.98 - 150.40063 2.33453 1.21 22.24±0.01 0.36±0.01 26.17 ±2.74 21.99 ±3.20 < 15.97 5.93±0.47 2.43±0.43 150.42068 2.62304 1.29 23.56±0.04 0.22±0.01 14.60 ±2.73 < 15.63 < 16.45 4.58±0.21 - 150.44572 2.76095 1.35 23.11±0.03 0.18±0.01 12.08 ±2.74 < 10.95 < 21.60 2.02±0.18 < 0.74 150.49513 2.37789 0.89 21.62±0.01 0.62±0.01 52.22 ±2.73 40.12 ±3.56 22.55 ±6.23 3.88±0.25 < 1.12 150.48572 2.71969 0.89 21.95±0.01 0.58±0.01 27.84 ±2.72 17.44 ±3.12 < 15.71 7.59±0.33 < 1.65 150.51527 2.56268 1.27 23.40±0.03 0.20±0.01 12.75 ±2.74 < 51.16 < 46.81 6.56±0.64 3.43±0.42 150.51142 2.57740 0.79 21.35±0.01 0.14±0.01 10.93 ±2.74 13.80 ±3.40 < 16.00 2.88±0.24 0.91±0.21 150.50345 2.65037 1.29 23.02±0.03 0.22±0.02 13.67 ±2.73 13.00 ±3.09 15.19 ±4.77 1.52±0.11 - 150.55036 2.73248 0.85 20.93±0.01 0.52±0.01 14.30 ±2.74 < 10.32 < 18.21 10.68±0.49 3.41±0.32 150.56193 2.47613 0.82 20.86±0.01 0.48±0.01 11.91 ±2.73 < 19.96 < 16.95 9.00±0.28 4.25±0.34 150.54374 2.49755 0.88 21.73±0.01 0.15±0.02 < 8.21 < 28.80 < 33.36 5.60±0.48 3.26±0.61 150.59733 2.61799 1.41 21.75±0.00 0.28±0.01 20.36 ±2.73 < 45.00 18.81 ±5.39 1.72±0.15 - 150.56845 2.64125 1.26 23.55±0.03 0.19±0.01 16.23 ±2.75 < 10.33 < 11.99 8.51±0.46 - 150.59931 2.38340 0.89 21.78±0.01 0.37±0.01 9.10 ±2.73 < 12.41 < 15.17 3.58±0.22 2.07±0.25 150.59031 2.53405 1.50 22.60±0.01 0.18±0.01 8.46 ±2.74 10.75 ±3.54 < 18.56 3.20±0.25 - 150.62185 2.55274 0.88 21.45±0.01 0.46±0.01 18.95 ±2.74 15.05 ±3.73 16.41 ±5.12 6.22±0.66 2.49±0.43 150.63049 2.55622 1.29 22.29±0.02 0.12±0.01 9.19 ±2.74 < 45.35 < 52.71 5.96±0.25 - 150.63901 2.61703 1.33 22.80±0.02 0.22±0.01 12.32 ±2.72 < 9.65 < 15.44 15.27±0.41 3.06±0.32 150.62021 2.62575 1.32 21.14±0.01 0.34±0.01 21.49 ±2.74 < 11.06 < 18.35 3.00±0.20 1.99±0.19 150.62429 2.72533 1.21 22.73±0.02 0.17±0.01 8.48 ±2.73 < 10.81 < 11.59 6.74±0.26 1.34±0.24 150.66639 2.44691 0.87 21.51±0.01 0.32±0.01 < 8.23 < 16.02 < 16.12 5.26±0.33 - 150.63611 2.74441 1.38 23.63±0.04 0.35±0.01 39.39 ±2.73 < 34.02 < 29.14 5.12±0.31 3.12±0.33 150.66376 2.49340 1.34 22.26±0.02 0.27±0.01 12.90 ±2.73 < 11.30 < 15.90 3.48±0.24 - 150.67631 2.52723 1.38 24.10±0.05 0.69±0.01 22.50 ±2.73 20.77 ±3.66 < 21.77 3.43±0.41 < 0.76 150.68868 2.61353 0.96 22.15±0.02 0.36±0.01 18.86 ±2.73 < 11.03 < 12.07 3.97±0.21 - 150.66590 2.64705 1.40 22.45±0.02 0.18±0.01 13.48 ±2.74 < 9.29 < 4.78 6.61±0.16 1.81±0.20 150.70877 2.52972 0.98 21.54±0.01 0.49±0.01 23.98 ±2.73 11.34 ±3.28 < 16.55 3.34±0.19 2.58±0.26 Table A1 : A1(continued) 16 erg cm −2 s −1 10 16 erg cm −2 s −1 Position from I F814W catalogue. b Redshift from detected Hα line. c Limits are 3σ, including confusion noise. d All values corrected for the limited aperture (1.2 arcsec diameter) of the FMOS fibre. e All values corrected for the limited aperture (1.2 arcsec diameter) of the FMOS fibre. Limits are 3σ, no value given in cases where the [NII] 6584 line falls on a OH sky line.RA a Dec. Redshift b I F814W S 24 µm S c 250 µm S c 350 µm S c 500 µm f d Hα f e [NII] deg. deg. AB mJy mJy mJy mJy 10 150.70577 2.56658 0.90 22.51±0.02 0.07±0.01 < 20.36 < 11.73 < 17.71 1.02±0.11 < 0.52 150.72242 2.61250 1.20 22.76±0.03 0.23±0.01 15.70 ±2.74 < 12.59 < 17.81 4.58±0.46 1.10±0.31 150.31065 2.31922 1.43 25.53±0.13 0.17±0.01 < 8.21 < 18.13 < 19.54 3.28±0.26 < 2.85 150.37685 2.45953 1.30 25.53±0.14 0.36±0.01 19.29 ±2.74 22.76 ±4.10 < 19.70 4.51±0.38 < 2.60 150.46438 2.63802 1.46 24.59±0.10 0.09±0.01 17.27 ±2.74 25.30 ±3.15 26.53 ±4.91 1.80±0.32 < 1.50 150.68825 2.45364 1.38 25.95±0.15 0.18±0.01 19.43 ±2.73 16.53 ±3.66 < 17.36 6.78±0.45 2.20±0.27 a Herschel is an ESA space observatory with science instruments provided by Principal Investigator consortia. It is open for proposals for observing time from the worldwide astronomical community. 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[ "A. Derdzinski, Geometry of elementary particles", "A. Derdzinski, Geometry of elementary particles" ]	[ "R E Greene \nDepartment of Mathematics\nThe Ohio State University\n43210Bȩdlewo, ColumbusOhioPoland\n", "PartS.-T Yau \nDepartment of Mathematics\nThe Ohio State University\n43210Bȩdlewo, ColumbusOhioPoland\n" ]	[ "Department of Mathematics\nThe Ohio State University\n43210Bȩdlewo, ColumbusOhioPoland", "Department of Mathematics\nThe Ohio State University\n43210Bȩdlewo, ColumbusOhioPoland" ]	[ "Geometry of the Standard Model of Elementary Particles, Texts and Monographs in Physics" ]	Geometry of the Standard ModelAndrzej Derdzinski 0. Introduction. The Standard Model of particles and interactions is the currently-accepted theory of elementary particles. It can be naturally divided into the classical part, a description of which is possible in the language of vector bundles over the spacetime and operations on them, and a field quantization procedure that transforms the classical part into a reasonable model of physical reality.This note covers only the classical part of the Standard Model. Similar but more detailed expositions of this topic can be found in the following texts:	null	[ "https://arxiv.org/pdf/0803.2668v1.pdf" ]	20,202,301	0803.2668	179b54d19f36232aa0029747bcec005f512af73f	A. Derdzinski, Geometry of elementary particles Springer-VerlagCopyright Springer-Verlag1992. 1993 R E Greene Department of Mathematics The Ohio State University 43210Bȩdlewo, ColumbusOhioPoland PartS.-T Yau Department of Mathematics The Ohio State University 43210Bȩdlewo, ColumbusOhioPoland A. Derdzinski, Geometry of elementary particles Geometry of the Standard Model of Elementary Particles, Texts and Monographs in Physics Berlin-Heidelberg-New YorkSpringer-Verlag541992. 1993Presented at Conference on Geometry Geometry of the Standard ModelAndrzej Derdzinski 0. Introduction. The Standard Model of particles and interactions is the currently-accepted theory of elementary particles. It can be naturally divided into the classical part, a description of which is possible in the language of vector bundles over the spacetime and operations on them, and a field quantization procedure that transforms the classical part into a reasonable model of physical reality.This note covers only the classical part of the Standard Model. Similar but more detailed expositions of this topic can be found in the following texts: 1. Interactions. Aside from gravity, the known kinds of particle interactions, ordered by decreasing strength, are the strong, electromagnetic, and weak forces. The latter two may be combined into the electroweak interaction ( §5). The strength of an interaction amounts to the probability of its occurrence in the given circumstances. Table 4.2). Baryons naturally form the disjoint classes of baryons proper and antibaryons, which consist of each other's antiparticles (Table 4.2). The same principle applies to leptons. Table 4.3. Interactions and gauge fields physics: interactions geometry: Yang-Mills fields a NATURAL VECTOR BUNDLE η → M of a FREE matter particle first order (the free-particle bundle of the given species); naturality amounts to direct observability an INTERACTION a NON-NATURAL vector bundle δ → M (the interacof some given kind tion bundle) with some geometry, mainly a G-structure CARRIERS of live in the AFFINE BUNDLE C(δ) whose secthe interaction tions are the compatible connections in δ an INTERACT- the INTERACTING-PARTICLE BUNDLE α = α(δ, η), ING matter functorial in both δ, η and "homogeneous linear" in particle the free-particle bundle η (basic example: α = δη) Usually, neither α nor C(δ) is natural. This contradicts the obvious requirement that carriers of interactions and interacting matter should be directly observable. One resolves this problem by "restoring" naturality of the bundles in question using bound states, or symmetry breaking, as described below. (Notations: N is the fibre dimension of the fixed interaction bundle δ; an integer k > 0 represents the product vector bundle M × C k .) BOUND STATES MORPHISMS of α 1 . . . α n onto NATURAL bundles, of n particles obtained by naturally "canceling" the δ-related factors SYMMETRY BREAKING selection of an ADDITIONAL STRUCTURE in δ, leading to reduction of G to a subgroup TRIVIALIZATION of δ, so δ = N and, e.g., δη = Nη FORMAL symmetry breaking (the interacting particle comes in N separate versions), (a thought experiment) while C(δ) = (dim G)T * , i.e., the carriers appear as dim G species of matter particles living in T * = T * M SPONTANEOUS symmetry breaking (in nature, for in-Example: the ELECTROWEAK MODEL ( §5). teractions of low strength) 5. The standard model. 1929 1961-1967 1964 The possibility of The model describes one gena unified descrip-eration of (anti)leptons ( §2) Hadrons appear tion of electromag-at a time. Choose, e.g., e, ν e : as composites netism for all par-the electron and electronic neu-of quarks and com-ticles expresses trino. Their free-particle bunantiquarks ments the fact that the dles are: σ for e and σ L for ν e , (abbreviation: electric charge is where σ denotes a fixed Dirac q,q), coming quantized, i.e., oc-spinor bundle, M is assum-in several flacurs in multiples ed orientable, and σ = σ L +σ R vors (species). of a fixed amount. (Weyl spinor bundles), σ R = σ L . free-par-a fixed Dirac spinor σ for quarks, ticle any η bundle σ for the whole σ for bundle generation e, ν e antiquarks inter-α = λ k η if partiα = ισ L + ι ∧2 σ R α = ρσ acting cle carries k units or, if neutrinos are massive, (for quarks) particle of electron charge even simpler: P n j=1 k j = 0 by , : ρρ → 1, α 1 . . . α n (electrically neu-Θ : ρ 3 → 1, or ↓ tral systems, none of interest Θ : ρ 3 →1, getting ζ e.g., atoms), qq pairs (mesons), (where both as λλ = 1 and q triples (baryζ and ↓ λ k 1 . . . λ k n = ons proper),q tri- G, δ G = U(1), δ = λ G = U(2), δ = ι G = SU(3), δ = ρ what δ is a complexα = ρσ bundle (with λ −k = λ k ) α = ισ(antiquarks)are natural) λ P j k j under , ples (antibaryons) λ = 1, α = η, ρ = 3, α = 3σ or formal C(λ) = T * . Mat-3σ: each q,q flavor symmetry ter: same as free. of no interest comes in 3 colors. breaking Carriers: just As C(ρ) = 8T * , the one species, carriers appear as the photon γ. 8 species of gluons. Choice of a section φ of ι with \|φ\| = constant > 0 reduces U(2) to U(1). Call λ = φ ⊥ the electromagneticinteraction bundle, so ι = 1+λ, 1 = Span φ, ι ∧2 = λ. spontaneous none: Thus, α = σ L +λσ describ-none: symmetry too strong es e, ν e with their correct much breaking charges, and the summands too strong of C(ι) = C(λ)+ λT * + T * represent the carriers: the photon γ, and the massive, matterlike weak-interaction carriers, W ± (charged), and Z (neutral). 6. Coupling constants and the Weinberg angle. An additional ingredient of the geometry of any interaction bundle δ is provided by a fixed natural fibre metric (, ) in C(δ), obtained in the obvious way from a biinvariant metric on G. Since G = U(2) is reducible, the freedom in choosing (, ) for the electroweak model involves not merely a scale factor (referred to as a coupling constant), but also an angular parameter (the Weinberg angle). The latter is physically meaningful, since the decomposition of C(ι) in Table 5.2 is (, )-orthogonal. In general, the coupling constant of (, ) also has a physical interpretation, namely in terms of the strength of the interaction described by δ. 2 . 2Taxonomy of particle species. (See also §3, §5.) 3. Definitions. Interaction carriers mediate interactions, matter particles do not. Leptons can't interact strongly, hadrons can. Mesons are bosons, baryons are fermions (seeknown particles z }\| { interaction carriers: matter particles 8 species of gluons, z }\| { γ (the photon), leptons hadrons W − , W + , Z 0 z }\| { z }\| { leptons proper, 3 generations: e, ν e µ, ν µ τ, ν τ antileptons, 3 generations: e + , ν e µ + , ν µ τ + , ν τ mesons (> 100) baryons z }\| { baryons proper (> 50) antibaryons (> 50) Table 4 . 1 . 41Particles and bundlesphysics geometry a PARTICLE species a BUNDLE ζ with some geometry over the spacetime (M, g); the particle is represented by (or lives in) ζ classical STATES of SECTIONS ψ of the bundle ζ the particle EVOLUTION of the states FIELD EQUATIONS imposed on ψ a MATTER particle a VECTOR bundle Table 4 . 42. Operations Table 4 . 4 . 44Bound states and symmetry breakingphysics geometry Table 5 . 1 . 51Geometry of interactionsinter- ELECTRO- ELECTROWEAK STRONG action MAGNETIC credits Weyl, Glashow, Salam, Weinberg, Gell-Mann, Zweig, Table 5 . 52. Bound states and symmetry breaking in the standard model inter- ELECTRO- ELECTROWEAK STRONG action MAGNETIC bound only if cancel ρ factors states:	[]
[ "Hypercharge Flux, Exotics, and Anomaly Cancellation in F-theory GUTs", "Hypercharge Flux, Exotics, and Anomaly Cancellation in F-theory GUTs" ]	[ "Joseph Marsano \nEnrico Fermi Institute\nUniversity of Chicago\n5640 S Ellis Ave60637ChicagoILUSA\n" ]	[ "Enrico Fermi Institute\nUniversity of Chicago\n5640 S Ellis Ave60637ChicagoILUSA" ]	[]	We sharpen constraints related to hypercharge flux in F-theory GUTs that possess U (1) symmetries and argue that they arise as a consequence of 4-dimensional anomaly cancellation. This gives a physical explanation for all restrictions that were observed in spectral cover models while demonstrating that the phenomenological implications for a well-motivated set of models are not tied to any particular formalism.	10.1103/physrevlett.106.081601	[ "https://arxiv.org/pdf/1011.2212v2.pdf" ]	20,378,835	1011.2212	3abcd551d21a69873f7e206e60e271fc85476e64	Hypercharge Flux, Exotics, and Anomaly Cancellation in F-theory GUTs 17 Mar 2011 Joseph Marsano Enrico Fermi Institute University of Chicago 5640 S Ellis Ave60637ChicagoILUSA Hypercharge Flux, Exotics, and Anomaly Cancellation in F-theory GUTs 17 Mar 2011 We sharpen constraints related to hypercharge flux in F-theory GUTs that possess U (1) symmetries and argue that they arise as a consequence of 4-dimensional anomaly cancellation. This gives a physical explanation for all restrictions that were observed in spectral cover models while demonstrating that the phenomenological implications for a well-motivated set of models are not tied to any particular formalism. I. INTRODUCTION The vastness of the string landscape presents a serious obstacle for studying particle physics in string theory. To make progress, it is often helpful to adopt a bottomup approach [1] that mirrors the successful techniques of effective field theory. Type II string theories provide a natural setting for this since the charged degrees of freedom can localize on branes that probe only a small part of the compactification geometry. The low energy physics associated to these branes is captured by a non-Abelian gauge theory whose bare coupling constants at the compactification scale are determined by local geometric data. This approach is particularly appealing for the construction of Grand Unified Theories (GUTs) [2][3][4] as the charged sector is engineered on a single stack of branes. The volume of the internal cycle wrapped by the branes introduces a new scale into the problem that can help to realize the small observed hierarchy between M GUT and M Planck . In this setting, the large top Yukawa coupling suggests an underlying exceptional group structure [5] that motivates the study of nonperturbative type II configurations described by M-theory or F-theory. The latter has received significant attention over the past few years in large part because powerful techniques of algebraic geometry are available to simplify the analysis. Most approaches to F-theory GUTs make crucial use of two important ingredients. The first is the presence of U (1) symmetries that can be used to protect against proton decay [5][6][7][8][9] or to motivate scenarios for how supersymmetry breaking is mediated to the Standard Model [10]. The second important ingredient is "hypercharge flux", which provides an elegant mechanism for breaking the GUT group while addressing the doublet-triplet splitting problem [3]. In explicit constructions based on spectral cover techniques [11], these two ingredients appear to be interrelated [6,8]; spectral cover models with a particular set of U (1) symmetries tend to exhibit tight constraints on how "hypercharge flux" can be distributed among the matter curves where charged fields localize [6]. This, in turn, has a striking impact on the 4-dimensional physics of all F-theory GUT models built to date. The goal of this letter is to understand the nature and source of these constraints. Because of the dramatic phenomenological implications [6], it is crucial to understand if the relationship between U (1) symmetries and "hypercharge flux" represents a limitation of our current modelbuilding toolbox or a more general lesson with an intrinsic physical origin. One indication of the latter can be found in a recent paper of Dudas and Palti [12], who noticed a simple pattern in the distribution of "hypercharge flux" in a set of spectral cover models. It is not hard to prove their relations for generic (suitably nondegenerate) spectral cover models and we do this in the upcoming paper [13]. More intriguing, however, is that we can rewrite the original Dudas-Palti observation in a simple way that does not make explicit reference to spectral covers at all ω Y = 5 matter curves, i q i Σ (i) 5 ω Y(1) Here, q a denotes the common U (1) charge of 10 or 5 fields that localize along curves Σ (x) in the compactification and ω Y is a "hypercharge flux" that is chosen to ensure that the U (1) Y gauge boson remains massless. A relation this simple should have a physical origin and, in this letter, we will demonstrate that it is a consequence of 4-dimensional anomaly cancellation. In addition to clarifying the physics of all known constraints of spectral cover models, this observation allows us to derive a generalization of (1) that must be satisfied by any F-theory GUT that combines U (1) symmetries and "hypercharge flux" regardless of how it is constructed. Among the many implications for phenomenology, our results imply that any U (1) symmetry in a model that combines "hypercharge flux" with the flavor scenario of [14] must be U (1) B−L , which cannot address µ or dimension 5 proton decay. Insisting on the existence of a U (1) P Q symmetry to deal with these necessarily introduces charged exotics into the spectrum. II. F-THEORY GUTS AND ANOMALY CANCELLATION A. Spectrum and "Hypercharge Flux" The charged sector of an F-theory GUT model is described by the 8-dimensional worldvolume theory that describes the physics of a stack of 7-branes. This theory, which we take to have gauge group SU (5) GUT , is compactified on a complex surface S GUT and can be UV completed by embedding that surface into a consistent F-theory compactification. Adjoint-valued fields propagate throughout the 8-dimensional worldvolume but the model contains additional degrees of freedom in the 10 and 5 representations (and their conjugates) that localize on holomorphic "matter curves" in S GUT . Determining the 4-dimensional spectrum requires a dimensional reduction in either case and can be influenced by introducing suitable fluxes into the model. While most of these fluxes descend from the bulk of the compactification, worldvolume flux plays an important role. An internal flux of the U (1) Y gauge field can break SU (5) GUT down to the MSSM gauge group and, when chosen correctly, remove unwanted degrees of freedom like Higgs triplets and leptoquarks [3]. In general, the net chirality of leptoquarks that descend from the SU (5) GUT adjoint is determined by an index theorem [3] n (3,2) −5/6 − n (3,2) +5/6 = SGUT c 1 (S GUT ) ∧ c 1 (L 5/6 Y ) where L Y is a line bundle that specifies the "hypercharge flux". The spectrum on a matter curve Σ, on the other hand, is computed as [3] n R −n R = Σ c 1 V Σ ⊗ L YR Y = Σ c 1 (V Σ ) + M Σ c 1 (L YR Y ) where V Σ is a bundle of rank M Σ that roughly encodes the "bulk" fluxes and Y R is the U (1) Y charge of fields in the representation R. The bundle V Σ and its rank M Σ are intrinsic properties of the matter curve Σ but the charges Y R can differ among the various MSSM multiplets contained in the SU (5) GUT multiplet that localizes there. In this way, a nontrivial "hypercharge flux" can be used to generate incomplete GUT-multiplets, which is very useful for obtaining Higgs doublets without their triplet partners. The ranks M Σ are all 1 for spectral cover models that are suitably nondegenerate but can be larger in more general constructions [4,15]. B. Constraints on "Hypercharge Flux" from MSSM Gauge Anomalies When building models, we need some freedom to distribute "hypercharge flux" among the matter curves that are present. This freedom must be limited, though, because "hypercharge flux" induces a chiral spectrum with respect to the MSSM gauge groups that generically leads to anomalies. The SU (3) 3 anomaly, for instance, is proportional to where [c 1 ] is the anti-canonical curve of S GUT . This relation is well-known [11,16] for constructions with M Σ (i) 10 = M Σ (a) 5 = 1 and has been derived using a "stringy" anomaly cancellation argument [11]. It is amusing to see, however, that it can be understood already as a consequence of anomaly cancellation in 4-dimensions. Cancellation of mixed gauge anomalies involving U (1) Y is not guaranteed for generic choices of L Y because, in most cases, the hypercharge gauge boson is lifted through an induced coupling to RR fields [3]. The conditions that L Y must satisfy in order to prevent this are known in F-theory and correspond to constructing L Y from a (1, 1)-form, ω Y ∼ c 1 (L Y ), that trivializes in the full compactification. Any "hypercharge flux" of this type will necessarily be constrained; at the very least, its distribution among the matter curves must guarantee that all MSSM gauge anomalies are cancelled. This leads to the conditions 0 = 10 matter curves, i M Σ (i) 10 Σ (i) 10 c 1 (L Y ) = 5 matter curves, a M Σ (a) 5 Σ (a) 5 c 1 (L Y )(2) that are easy to verify in generic F-theory GUT models [16] with a massless U (1) Y . C. Implications of Mixed Gauge Anomalies We would now like to ask if a "hypercharge flux" ω Y that doesn't lift U (1) Y exhibits any additional properties in a geometry that engineers bulk U (1) symmetries in addition to SU (5) GUT [17]. To address this, let us consider what happens when we turn on this flux and no other fluxes. Our flux will induce a nontrivial spectrum but, because all U (1)'s remain massless, it cannot give rise to any gauge anomalies [18]. Of particular interest to us are mixed anomalies with insertions of both MSSM and U (1) currents since these only get contributions from the chiral fields that localize on matter curves in S GUT . We will see that the Dudas-Palti relations (1) for spectral cover models simply express a set of nontrivial relations that the (1, 1)-form ω Y must satisfy in order for these 4-dimensional mixed gauge anomalies to cancel. To make things completely explicit, we use ω Y to define a line bundle L Y on the GUT 7-branes that defines a nontrivial U (1) Y background. We further normalize that background so that all charged fields on matter curves are sections of the integer quantized gauge bundles listed below SU (5) SU (3) × SU (2) × U (1) Y Bundle 10 (1, 1) +1 L 6 Y (3, 2) +1/6 L Y (3, 1) −2/3 L −4 Y 5 (3, 1) +1/3 L 2 Y (1, 2) −1/2 L −3 Y (3) We now determine the contributions to mixed gauge anomalies that arise from the chiral spectrum on a generic 10 or 5 matter curve. To obtain (1) and its generalization beyond spectral cover models, it will be sufficient to consider anomalies of the type G 2 SM × U (1), where G SM denotes a Standard Model gauge group. Consider first the contribution from fields that localize on a 10 curve, Σ the contributions to mixed G 2 SM × U (1) anomalies are Multiplet Chir SU (3) 2 U (1) SU (2) 2 U (1) U (1) 2 Y U (1) (1, 1) +1 6N a 0 0 6q a N a (3, 2) +1/6 N a 2q a N a 3q a N a q a N a /6 (3, 1) −2/3 −4N a −4q a N a 0 −16q a N a /3 Total −2q a N a 3q a N a 5q a N a /6 Note that a negative chirality means that we obtain zero modes of the conjugate multiplet, which carry an opposite U (1) charge. We now do the same thing for fields on a 5 (i) curve that carry U (1) charge q i . Letting N i denote the M Σ 5 (i) -weighted U (1) Y flux N i = M Σ 5 (i) Σ 5 (i) c 1 (L) = M Σ 5 (i) Σ 5 (i) ω Y(5) we find Multiplet Chir SU (3) 2 U (1) SU (2) 2 U (1) U (1) 2 Y U (1) (3, 1) +1/3 2N i 2q i N i 0 2q i N i /3 (1, 2) −1/2 −3N i 0 −3q i N i −3q i N i /2 Total 2q i N i −3q i N i −5q i N i /6 From this, we see that cancellation of all G 2 SM × U (1) anomalies implies that ω Y must satisfy (1). We refer to (6) as the generalized Dudas-Palti relations, which must hold for any ω Y that can be used to construct "hypercharge flux" in an SU (5) GUT F-theory GUT model with an extra U (1) symmetry. It is easy to see that other mixed anomalies, as well as the U (1) 3 anomaly, vanish without giving rise to any additional constraints. Though the story is less constrained than in 6-dimensions [19], it would be interesting to pursue a more general analysis of anomaly cancellation in 4-dimensional F-theory compactifications in the future. III. IMPLICATIONS OF THE GENERALIZED DUDAS-PALTI RELATIONS The first question to ask about (6) and (2) is whether they represent all of the nontrivial constraints on the distribution of "hypercharge flux" in F-theory GUTs. In the case of spectral cover models, we suspect that they do because it appears that one can use spectral covers to construct, at least in principle, all distributions of "hypercharge flux" that satisfy them [13]. Based on this, it is natural to conjecture that, even for more general classes of F-theory GUTs, (6) and (2) represent the only constraints. In light of this, we should correct some misstatements that were made in [6]. There, it was claimed that the presence of "hypercharge flux" on 5 matter curves automatically implied that "hypercharge flux" must thread some 10 matter curves as well. The DP relations (1) do not forbid a configuration in which "hypercharge flux" threads only 5 curves, though, and it is possible to construct spectral covers that do precisely this [13]. Finally, let us comment on implications of the generalized Dudas-Palti relations (6) for F-theory model building. While several approaches to flavor have been suggested in the past few years [20], the mechanism of wave function overlaps is particularly attractive [14]. This mechanism requires all three generations of the 10 to localize on one matter curve and similar for all three generations of the 5. The Higgs fields then lie on distinct matter curves, Σ = 1 and carry +1 and -1 units of "hypercharge flux", respecitvely, to lift the triplets [3]. Crucial to this scenario is that "hypercharge flux" not be allowed to thread any curve Σ other than Σ ; if it did, we would obtain massless matter fields on Σ that do not comprise a complete GUT multiplet. As one assumes that the Standard Model fields are engineered as complete GUT multiplets, the threading of "hypercharge flux" through such a Σ necessarily introduces new charged exotics into the spectrum [6]. If we wish to combine this scenario with a U (1) symmetry, the generalized Dudas-Palti relations (6) imply that the charges q Hu and q H d associated to the matter curves Σ 5 (Hu ) and Σ 5 (H d ) must satisfy q Hu − q H d = 0 The doublet H u comes from a 5 rather than a 5, though, so its charge is actually −q Hu . Writing (7) in terms of the actual H u and H d charges we get Q(H u ) + Q(H d ) = 0 (8) What type of U (1) symmetry can this be? Because all 10's (5's) are engineered on a single curve, all of them must carry a common charge. The only U (1) symmetry of this type that commutes with SU (5), satisfies (8), and preserves the MSSM superpotential is the famous U (1) χ , which is the linear combination of U (1) Y and U (1) B−L that enters naturally in SO(10) unification models. We see that P Q symmetries, broadly defined as U (1)'s for which (8) does not hold, cannot be combined with the desired distribution of hypercharge flux. If we insist on realizing all 3 generations of 10's (5's) on a single mat-ter curve, the presence of U (1) P Q implies the existence of additional charged matter fields that do not come in complete GUT multiplets [6]. 1 (S GUT ) ∧ c 1 (L Y ) which carry a U (1) charge q a . Denoting the M Σ (a) 10 -weighted U (1) Y flux there by N a N a = M Σ for M Σ (a) 10 = M Σ (i) 5 = 1, is nothing other than the Dudas-Palti relations Acknowledgements: I am grateful to N. Saulina and S. Schäfer-Nameki for valuable discussions during the course of this work and many enjoyable collaborations on the study of F-theory GUTs. I am also grateful to S. Sethi for encouragement and helpful discussion as well as S. Cecotti, C. Cordova, J. Heckman, and C. Vafa for explaining a crucial aspect of their work[15]. I thank the Physics Department at The Ohio State University and the organizers of the String Vacuum Project 2010 Fall meeting for their hospitality. This research is supported by DOE grant DE-FG02-90ER-40560 and NSF grant PHY-0855039. . G Aldazabal, L E Ibanez, F Quevedo, JHEP. 0008106H. Verlinde, M. WijnholtJHEPG. Aldazabal, L. E. Ibanez, F. Quevedo et al., JHEP 0008, 002 (2000), H. Verlinde, M. Wijnholt, JHEP 0701, 106 (2007). . R Donagi, M Wijnholt, ; C Beasley, J J Heckman, C Vafa, arXiv:0802.2969JHEP. 0158R. Donagi and M. Wijnholt, arXiv:0802.2969, C. Beasley, J. J. Heckman, and C. Vafa, JHEP 01 (2009) 058, . C Beasley, J J Heckman, C Vafa ; 059, R Donagi, M Wijnholt, arXiv:0808.2223JHEP. 01C. Beasley, J. J. Heckman, and C. Vafa, JHEP 01 (2009) 059, R. Donagi and M. Wijnholt, arXiv:0808.2223. . H Hayashi, R Tatar, Y Toda, T Watari, M Yamazaki, Nucl. Phys. 806H. Hayashi, R. Tatar, Y. Toda, T. Watari, and M. Yamazaki, Nucl. Phys. B806 (2009) 224-299, . 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While a nonzero G-flux typically induces an anomalous spectrum for our U (1), any additional contributions to the anomalies from turning on ωY must vanish because the 4d Green-Schwarz mechanism that cancels them is independent of ωY . More specifically, 4d anomalies are cancelled by the exchange of fieldsĈ0/Ĉ2 that satisfy * dĈ0 = dĈ2 and couple to flux as. A subtlety arises here because we are not always allowed to switch off the G-flux in an F-theory compactification due to the quantization condition of. Ĉ2 ∧ F +Ĉ0 ∧ FMSSM ∧ FMSSM + . . .). In F-theory, these fields descend from C4 and their. 4d couplings come from the bulk interaction C4 ∧ G ∧ G, to which ωY cannot contribute because it is globally trivialA subtlety arises here because we are not always allowed to switch off the G-flux in an F-theory compactification due to the quantization condition of [23]. 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[]	[ "Nicolas Kovensky \nDepartamento de Física\nFacultad de Ciencias Exactas\nInstituto de Física La Plata-UNLP-CONICET. Boulevard\n113 e 63 y 64, 1900) La PlataBuenos AiresArgentina\n\nUniversidad Nacional de La Plata\nCalle 49 y 115, C.C. 67, 1900) La PlataBuenos AiresArgentina\n\nInstitut de Physique Thèorique\nCEA\n91191Gif-sur-Yvette CedexSaclay, France\n", "Gustavo Michalski \nDepartamento de Física\nFacultad de Ciencias Exactas\nInstituto de Física La Plata-UNLP-CONICET. Boulevard\n113 e 63 y 64, 1900) La PlataBuenos AiresArgentina\n\nUniversidad Nacional de La Plata\nCalle 49 y 115, C.C. 67, 1900) La PlataBuenos AiresArgentina\n", "Martin Schvellinger \nDepartamento de Física\nFacultad de Ciencias Exactas\nInstituto de Física La Plata-UNLP-CONICET. Boulevard\n113 e 63 y 64, 1900) La PlataBuenos AiresArgentina\n\nUniversidad Nacional de La Plata\nCalle 49 y 115, C.C. 67, 1900) La PlataBuenos AiresArgentina\n" ]	[ "Departamento de Física\nFacultad de Ciencias Exactas\nInstituto de Física La Plata-UNLP-CONICET. Boulevard\n113 e 63 y 64, 1900) La PlataBuenos AiresArgentina", "Universidad Nacional de La Plata\nCalle 49 y 115, C.C. 67, 1900) La PlataBuenos AiresArgentina", "Institut de Physique Thèorique\nCEA\n91191Gif-sur-Yvette CedexSaclay, France", "Departamento de Física\nFacultad de Ciencias Exactas\nInstituto de Física La Plata-UNLP-CONICET. Boulevard\n113 e 63 y 64, 1900) La PlataBuenos AiresArgentina", "Universidad Nacional de La Plata\nCalle 49 y 115, C.C. 67, 1900) La PlataBuenos AiresArgentina", "Departamento de Física\nFacultad de Ciencias Exactas\nInstituto de Física La Plata-UNLP-CONICET. Boulevard\n113 e 63 y 64, 1900) La PlataBuenos AiresArgentina", "Universidad Nacional de La Plata\nCalle 49 y 115, C.C. 67, 1900) La PlataBuenos AiresArgentina" ]	[]	The structure functions F 1 and F 2 of the hadronic tensor of vector mesons are obtained at order 1/N and strong coupling using the gauge/gravity duality. We find that the large N limit and the high energy one do not commute. Thus, by considering the high energy limit first, our results of the first moments of F 1 for the rho meson agree well with those from lattice QCD, with an important improvement of the accuracy with respect to the holographic dual calculation in the planar limit.	10.1103/physrevd.99.046005	[ "https://arxiv.org/pdf/1809.10515v1.pdf" ]	118,914,849	1809.10515	1ce30faaa790505d24a33248a58b2fe04b85f744	27 Sep 2018 Nicolas Kovensky Departamento de Física Facultad de Ciencias Exactas Instituto de Física La Plata-UNLP-CONICET. Boulevard 113 e 63 y 64, 1900) La PlataBuenos AiresArgentina Universidad Nacional de La Plata Calle 49 y 115, C.C. 67, 1900) La PlataBuenos AiresArgentina Institut de Physique Thèorique CEA 91191Gif-sur-Yvette CedexSaclay, France Gustavo Michalski Departamento de Física Facultad de Ciencias Exactas Instituto de Física La Plata-UNLP-CONICET. Boulevard 113 e 63 y 64, 1900) La PlataBuenos AiresArgentina Universidad Nacional de La Plata Calle 49 y 115, C.C. 67, 1900) La PlataBuenos AiresArgentina Martin Schvellinger Departamento de Física Facultad de Ciencias Exactas Instituto de Física La Plata-UNLP-CONICET. Boulevard 113 e 63 y 64, 1900) La PlataBuenos AiresArgentina Universidad Nacional de La Plata Calle 49 y 115, C.C. 67, 1900) La PlataBuenos AiresArgentina 27 Sep 20181/N corrections to F 1 and F 2 structure functions of vector mesons from holography The structure functions F 1 and F 2 of the hadronic tensor of vector mesons are obtained at order 1/N and strong coupling using the gauge/gravity duality. We find that the large N limit and the high energy one do not commute. Thus, by considering the high energy limit first, our results of the first moments of F 1 for the rho meson agree well with those from lattice QCD, with an important improvement of the accuracy with respect to the holographic dual calculation in the planar limit. Introduction The idea of the present work is to investigate the leading 1/N corrections to the structure functions F 1 and F 2 of the hadronic tensor of unpolarized vector mesons at strong 't Hooft coupling λ, using the gauge/gravity duality. For this purpose we consider vector mesons from the D3D7-brane system in type IIB string theory [1]. We are interested in the electromagnetic deep inelastic scattering (DIS) of a charged lepton from a vector meson. The DIS cross section is given by the contraction of a leptonic tensor, l µν , with a hadronic one, W µν . The process involves an incoming charged lepton interacting with a hadron with momentum P through the exchange of a virtual photon with momentum q, with the condition q 2 >> −P 2 . We consider the definitions given in [2], however we use the mostly-plus signature. Thus, the DIS differential cross section is given by d 2 σ dx dy dφ = e 4 16π 2 q 4 y l µν W µν ,(1) where y is the lepton fractional energy loss and e denotes the electron charge. The hadronic tensor depends on the hadron structure, where there are important contributions from soft QCD processes. For this reason the gauge/string theory duality becomes a suitable tool for the calculation of this tensor, and therefore the structure functions. In this work we focus on the structure functions associated with unpolarized vector mesons 4 . The corresponding hadronic tensor has the form W µν = F 1 (x, q 2 ) η µν − F 2 (x, q 2 ) P · q P µ P ν . DIS is related to the forward Compton scattering (FCS) through the optical theorem, which is an special case of the Cutkosky rules, based on the fact that the S-matrix is unitary. It relates the imaginary part of the FCS amplitude to the DIS amplitude. Then, the tensor T µν is defined as T µν = i P, Q\| T (J µ (q) J ν (0))\|P, Q ,(3) where J µ and J ν are the electromagnetic current operators, Q denotes the charge of the hadron, while T represents time-ordered product. Tildes indicate the Fourier transform. In terms of the optical theorem we can write ImF j = 2π F j ,(4) beingF j the j-th structure function of the T µν tensor, while F j is the one corresponding to the W µν tensor. At this point it is convenient to define the Bjorken parameter x = −q 2 /(2P · q) for q 2 > 0, being its physical kinematic range 0 ≤ x ≤ 1. On the other hand, in the unphysical region for 1 ≪ x the product of the two electromagnetic currents in the hadron can be written as an operator product expansion (OPE), in terms of operators O n,k multiplied by powers of (Λ 2 /q 2 ) γ n,k /2 , where n is the spin of O n,k , while δ n,k , γ n,k , and ∆ n,k = δ n,k + γ n,k , represent the engineering, the anomalous and the total scaling dimensions of the operator, respectively [6]. Then, we can define the twist of each operator as τ n,k = ∆ n,k − n. The relation with the physical parametric region 0 ≤ x ≤ 1 is given by a contour argument, which allows to connect the OPE with the moments of the structure functions in the DIS process. Thus, the n-moment of the j-th structure function can be expressed as the sum of three contributions 5 M j n (q 2 ) ≈ 1 4 k C j n,k A n,k Λ 2 q 2 1 2 τ n,k −1 + 1 4 Qp=Q C j n,p A n,p Λ 2 q 2 τp−1 + 1 4 1 N Qp =Q C j n,p a n,p Λ 2 q 2 τp−1 ,(5) where the coefficients C j n,p are dimensionless, while A n,p and a n,p depend on the matrix elements of the operators < P, Q\|O n,k \|P, Q > for a hadron state with four-momentum P and charge Q. Let us briefly explain how different contributions behave in equation (5). We can study this equation for the photon virtuality q to be large, intermediate or small, in comparison with the confinement scale Λ. At weak coupling the Feynman's parton model gives a suitable description of hadrons, thus the leading contribution comes from the first term. This contribution only sums over terms associated with operators with the lowest twist τ n,k ≈ 2 at large q 2 . In this case perturbative methods of quantum field theory are suitable. On the other hand, the second sum dominates at strong coupling and in the planar limit, i.e. 1 ≪ λ ≪ N. In this case protected double-trace operators constructed from the protected single-trace ones have the smaller twist at strong coupling. Therefore, the calculations can be done by using the gauge/gravity duality, considering a forward Compton scattering with the exchange of a single on-shell particle between incoming and outgoing states. Within this regime exchange of more than one intermediate state is suppressed by 1/N powers. The third sum in equation (5) becomes the leading one when q 2 ≥ Λ 2 N 1/(τ Q −τc) . In this case τ Q is the minimum twist of protected operators with charge Q, while τ c is the minimum twist of all electrically charged protected operators. The 1/N suppression of the third sum is expected for mesons, while for glueballs there is a 1/N 2 suppression. In addition, we should comment on the different parametric regions in terms of x and the 't Hooft coupling. For 1/ √ λ ≪ x ≪ 1 only supergravity states contribute since in this region the ten-dimensional s-channel Mandelstam variable satisfiess ≪ 1/α ′ , where α ′ is the string constant. When exp (− √ λ) ≪ x ≪ 1/ √ λ excited strings are produced and their dynamics becomes important. The holographic dual calculation is derived from fourpoint superstring theory scattering amplitudes. Finally, for the exponentially small region the size of the excited string becomes comparable with the AdS radius. In this case dual Pomeron techniques are useful [7,8,9,10,11,12,13,14,15,16,17]. In previous papers we have calculated F 1 and F 2 by considering the FCS process with the exchange of a single intermediate state for scalar and vector mesons [3,4,5,18]. Then, we have also calculated these functions by considering the exchange of two intermediate states for glueballs [19] and for scalar mesons [20] in the D3D7-brane system of reference [1]. In both cases we found that the large N limit does not commute with the high energy one. By considering the high energy limit first, which corresponds to the physical situation, in the case of the pion we have obtained the first moments of the structure function F 2 and compared them with lattice QCD calculations [21,22,23], obtaining a substantial improvement of the accuracy, namely: from 10.8% for a single intermediate state [18] to 1.27% in the case of two intermediate states [20]. Then, a natural question is whether or not this effect also occurs in the case of vector mesons. The present work answers it positively as we shall explain in detail in the next sections. For finite values of N we can expand the structure functions of mesons as follows F j = f (0) j Λ 2 q 2 τ in −1 + 1 N f (1) j Λ 2 q 2 + 1 N 2 f (2) j Λ 2 q 2 + · · ·(6) where τ in is the twist of the incident dual vector meson state in type IIB supergravity, f (6) one can easily see that the high energy (q 2 ≫ Λ 2 ) and the large N limits do not commute. Moreover, by taking first the high energy limit, since the power of Λ 2 /q 2 in the first term is larger than for the rest, it vanishes, and then in the 1/N expansion the second term dominates. We would like to emphasize that 1/N corrections to the F 1 and F 2 structure functions for scalar mesons were studied in [20], but not for vector mesons. Therefore, we consider this calculation to be important for the investigation of such limits beyond scalar hadrons, since there are also lattice QCD results of the first moments of F 1 for the rho meson to compare with [21]. The work is organized as follows. In section 2 we discuss generalities of DIS in the context of the D3D7-brane system at large N. In section 3 we consider the 1/N expansion and obtain the relevant Feynman-Witten diagram in the bulk theory. Also in this section we develop the calculation of the F 1 and F 2 structure functions for vector mesons. In section 4 we carry out the analysis of our results. We focus on the calculation of the first moments of the structure function F 1 of the rho meson and compare them with the available results from lattice QCD. 2 DIS in the D3D7-brane system DIS processes of charged leptons from scalar and vector mesons in the D3D7-brane system have been studied in several papers [3,4,5], by considering the large N limit, which means that the final state has only a single hadron. A more realistic calculation for vector mesons must include 1/N corrections. This corresponds to final multi-particle states. In this work we consider 1/N corrections of DIS of charged leptons from unpolarized vector mesons. Firstly, we give a brief description of the D3D7-brane system. Let us consider N coincident D3-branes in type IIB superstring theory. The corresponding near-horizon geometry is the AdS 5 × S 5 spacetime, with the metric ds 2 = r 2 R 2 η µν dx µ dx ν + R 2 r 2 d Z · d Z ,(7) where Z are coordinates of the directions perpendicular to the D3-branes, being the radial coordinate r = \| Z\|. The radius of AdS 5 is R = (4πg s Nα ′2 ) 1/4 , where g s is the string coupling. Now, one can add a D7-brane in the probe approximation, at a distance L = \| Z\| in the (8,9) plane. The induced metric on the D7-brane is given by ds 2 = ρ 2 + L 2 R 2 η µν dx µ dx ν + R 2 ρ 2 + L 2 dρ 2 + R 2 ρ 2 ρ 2 + L 2 dΩ 2 3 ,(8) where ρ 2 = r 2 − L 2 and the angles contained in Ω 3 span a three-sphere. For L = 0 equation (8) gives the AdS 5 × S 3 metric, otherwise the metric is only asymptotically AdS 5 × S 3 . This is the situation where the conformal symmetry is preserved. For L > 0 the 3-7 quarks become massive, and meson type excitations are energetically favored. Mesons correspond to excitations of open strings ending on the D7-brane. The dynamics of these fluctuations is described by the action S D7 = −µ 7 d 8 ξ − det (P [g] ab + 2πα ′ F ab ) + (2πα ′ ) 2 2 µ 7 P C (4) ∧ F ∧ F ,(9) where µ 7 = [(2π) 7 g s α ′4 ] −1 is the D7-brane tension, ξ a denotes the D7-brane coordinates, g ab stands for the metric (8), and P is the pullback of the background fields on the D7-brane. The second term is the Wess-Zumino term. It is possible to induce excitations in the transverse directions to the D7-brane. These are two types of scalar excitations φ and χ, related to the Z 5 and Z 6 coordinates, respectively. On the other hand, it is also possible to perturb the gauge fields F ab = ∂ a B b − ∂ b B a on the Dirac-Born-Infeld (DBI) action. In this case, there are three types of solutions for the B a modes, related to the expansion the solutions in scalar or vector spherical harmonics on S 3 . The three classes of solutions are [1] type I : B µ = 0 , B ρ = 0 , B i = φ ± I (ρ) e ik·x Y l± i (Ω) ,(10) type II : B µ = ǫ µ φ II (ρ) e ik·x Y l (Ω) , k · ǫ = 0 , B ρ = 0 , B i = 0 ,(11) type III : B µ = 0 , B ρ = φ III (ρ) e ik·x Y l (Ω) , B i =φ III (ρ) e ik·x ∇ i Y l (Ω) . (12) Y l (Ω) and Y l i (Ω) are scalar and vector spherical harmonics, respectively. Some of their properties are described in the appendix and in references therein. Note that in this case type I and III are scalar modes, while type II modes represent vector fields from the (asymptotically) AdS perspective. The different modes of the scalar and vector perturbations are shown in table 1, together with their relevant quantum numbers. Table 1: Some features of D7-brane fluctuations on the AdS 5 × S 3 background relevant to this work. The integer l indicates the SO(4) ∼ SU (2) × SU (2) irreducible representation (irrep) and it defines the corresponding Kaluza-Klein mass. The relation between the scaling dimension of the associated operator ∆ and l is also presented. Field Type of field in 5D Built from ∆(l) SU(2) × SU(2) irrep φ, χ scalars φ, χ l + 3, l ≥ 0 l 2 , l 2 B µ vector B II µ l + 3, l ≥ 0 l 2 , l 2 φ − I scalar B I i l + 1, l ≥ 1 l+1 2 , l−1 2 φ + I scalar B I i l + 5, l ≥ 1 l−1 2 , l+1 2 φ III scalar B III i,z l + 3, l ≥ 1 l 2 , l 2 Beyond the quadratic order, the interaction Lagrangian for these modes has been derived in reference [20]. Up to this point we have described the D3D7-brane system presented in [1], where the solutions were computed in terms of hypergeometric functions. In the context of DIS from mesons, one identifies the parameter that controls the separation between the D7 and the D3branes in the (Z 5 , Z 6 ) plane with the IR scale Λ introduced as a cutoff in the radial direction to induce confinement [6]. Thus, we take L ∼ ΛR 2 . Therefore, the relevant interactions take place at values of ρ considerably larger than L, and in this region the solutions are well approximated by the typical AdS 5 expressions in terms of Bessel functions, which we write in section 3. The AdS masses can only take discrete values. The presence of a small but non-zero value of L is important for the vertices we will need to consider. 2.1 One-particle exchange: the N → ∞ limit For unpolarized vector mesons we shall study only the contributions to the F i structure functions. These functions can be written in terms of W µν and the vector v µ = 1 q (P µ + q µ 2x ) as F 1 (x, q 2 ) = 1 2 g µν − q µ q ν q 2 W µν + 2x 2 v µ v ν W µν ,(13)F 2 (x, q 2 ) = x g µν − q µ q ν q 2 W µν + 12 x 3 v µ v ν W µν .(14) The FCS amplitude can be derived by using the gauge/string theory duality, by studying a four-point interaction with vector modes on the D7-brane and gravi-photons related to current insertion on the boundary as external states. This gauge field arises from a particular decomposition of the graviton mode in ten dimensions: δg mj = A m (ρ, x) v j (Ω 3 ) ,(15) where v j are the Killing vectors on S 3 , and m = (µ, ρ). The structure functions have been calculated in this context by considering a single intermediate hadron state in [3,4], obtaining the following results at leading order F 1 (x, q 2 ) = A(x) 1 12x 3 (1 − x) ,(16)F 2 (x, q 2 ) = A(x) 1 6x 3 (1 − x) ,(17) with A(x) = A 0 Q 2 µ 2 7 (α ′ ) 4 Λ 8 Λ 2 q 2 l+1 x l+6 (1 − x) l ,(18) and A 0 = \|c i \| 2 \|c X \| 2 2 6+2l [Γ(3 + l)] 2 π 5 is a dimensionless constant. Also, c i and c X are the normalization constants of the incident and intermediate dual hadron states, respectively, while Q is the charge of the hadron, associated to a U(1) subgroup of the S 3 isometries. These results are valid for DIS from mesons considered in the context of the D3D7-brane system. However, it is important to keep in mind that [3,4] showed that completely analogous formulas hold in the context of different Dp-brane models, such as the D4D8D8-brane system [24] and the D4D6D6-brane system [25], both in type IIA superstring theory. These models are very different to each other, and each of them shares certain phenomenological features with large-N QCD. In consequence, it is reasonable to expect that the qualitative form of the structure functions we just described is universal, in the sense that it would hold in any holographic Dp-brane model for mesons at strong coupling and in the planar limit. Although in the present work we focus on the D3D7-model, we expect this universality to hold also for the leading one-loop correction, at least at the qualitative level. DIS in the 1/N expansion The non-planar 1/N corrections to F 1 and F 2 structure functions for scalar mesons were studied in [20], and also for the N = 4 SYM theory glueball in reference [19]. From the point of view of DIS, they correspond to processes with multi-particle final states. The first non-trivial contribution comes from considering a two-hadron final state in the hadronic tensor W µν 2 , which can be related to a FCS process with two intermediate on-shell states, denoted as T µν 2 . Writing this tensor in terms of the U(1) conserved current J µ we obtain [19] Im (T µν 2 ) = π X 1 ,X 2 P, Q\|J µ (q) \|X 1 , X 2 X 1 , X 2 \| J ν (0) \|P, Q = π M 2 ,M 3 d 3 q ′ 2E q ′ (2π) 3 d 3 p ′ 2E p ′ (2π) 3 P, Q\|J µ (q) \|X 1 , X 2 X 1 , X 2 \| J ν (0) \|P, Q = 4π 3 M 2 ,M 3 d 4 q ′ (2π) 4 δ M 2 2 − q ′2 δ M 2 3 − (P + q − q ′ ) 2 × P, Q\| J µ (0) \|X 1 , X 2 X 1 , X 2 \| J ν (0) \|P, Q ,(19) where X 1 and X 2 are the intermediate states with momenta p ′ and q ′ respectively, as shown in figure 1. The current J µ matrix element is related to its Fourier transform as P, Q\|J µ (q) \|X 1 , X 2 = (2π) 4 δ (4) (P + q − p ′ − q ′ ) P, Q\| J µ (0) \|X 1 , X 2 .(20) Using the AdS/CFT duality, this current can be related to an specific field in the bulk theory. In [19], considering 1/N corrections to DIS from a scalar meson we have shown that the leading contribution to the DIS process with two hadron final states is given by a specific Feynman diagram where one of these two outgoing hadrons has the lowest twist. In the next subsection we will explain the amplitude we need to calculate in the case of a spin-1 meson. Leading diagram for vector mesons at order 1/N We want to study the 1/N corrections to the DIS process from a vector meson (for instance a rho meson), associated to a type II vector mode B II µ on the D7-brane. Based on the results of [20], the leading diagram is the s-channel one, where the exchanged particle is the one with the lowest twist, τ = ∆ − n. This can be done by looking at table 1, which gives the relevant quantum numbers of the different solutions. Since the lowest τ is associated to the lowest ∆, the exchanged field should be the φ − I mode with τ = ∆ = 2. This is what has been done in [20]. However, the interaction term between B II µ and φ − I modes given in [20] L 3 = −µ 7 (2πα ′ ) 3 √ −g L ρ 2 + L 2 φ F aJ F aJ − F aµ F aµ ,(21) vanishes, due to the nature of the field solutions. It can be seen from equations (10) and (11), that a type I scalar has only angular components, while type II vectors have only µ components. Therefore, the interaction between these two modes vanishes. The vector mode with τ = 2 does not contribute to the DIS process either. This can be easily seen by analyzing the charge of this vector associated with the 3-sphere. For τ = 2 we need to consider ∆ = 3, but this implies that l = 0, meaning that the vector mode has no charge over the 3-sphere. Thus, there is no interaction with the A m photon. Another possible interaction could arise from the Wess-Zumino term in the low-energy action of the D7-brane. This is because the gauge field is actually a particular linear combination of the ten-dimensional graviton and RR 4-form perturbations. This is described in detail in [16,17], where this type of vertex has been used to study the antisymmetric contributions to the hadronic tensor for glueballs and spin-1/2 hadrons. However, it can be seen that the relevant angular integrals vanish in the present case. The next step is to consider the exchange of τ = 3 modes. There are two possibilities: 1. φ, χ scalars, with ∆ = 3. B µ vector, with ∆ = 4. In the former case the perturbations have l = 0, thus they are not charged with respect to the U(1) we are considering. Therefore, the only possibility is the exchange of a type II vector mode with l = 1. In the IR region, the relevant interaction includes two type II modes (one associated to the τ = 3 mode we just discussed and the other one with the incoming vector meson) and a scalar field, and from (21) we see that it is described by [1,20] S IR ≈ µ 7 (2πα ′ ) 3 L R 4 d 8 ξ √ g z 2 φ F µν F µν .(22) Note that we only keep the first term of the L ≪ ρ expansion, where ρ ≃ R 2 /z, being z the Poincaré radial coordinate of AdS 5 . For the UV vertex we have to consider the interaction between the A m gauge field and two B µ modes. The standard interaction is [4] S UV = −µ 7 (πα ′ ) 2 iQ d 5 x √ g A µ (B * ν F µν − B ν (F µν ) * ) ,(23) where we have already integrated over the The diagram we need to calculate is shown in figure 1, where there is an incoming photon with momentum q µ , which interacts with a massive B µ vector with momentum q ′ µ (q ′2 = −M 2 2 ) and τ min = 3. This vector interacts with an incident rho meson of momentum P µ , and with an outgoing scalar field with momentum p ′ m (p ′2 = −M 2 3 ) and conformal dimension τ ′ . In order to calculate the diagram we need the AdS 5 solutions of the fields. In the axial gauge the gravi-photon solution is A µ (x, z) = n µ e iq·x (qz)K 1 (qz) , A z (x, z) = 0 ,(24) while the fields on the probe D7-brane are given at small L by the following approximate expressions [4] B * µ (x, z) = 1 √ N c * B ζ µ e −iq ′ ·x Λ M 2 z J 2 (M 2 z) , ρ µ (x ′ , z ′ ) = 1 √ N c ρ ǫ µ e iP ·x ′ Λ M 1 zJ τ −1 (M 1 z ′ ) , φ * (x ′ , z ′ ) = 1 √ N c * φ e −ip ′ ·x ′ ΛM 3 zJ τ ′ −1 (M 3 z ′ ) ,(25) where c's are numerical constants, and we have also included the polarization vectors. We have only written the AdS 5 solution, the full ten-dimensional solution includes the 3-sphere contribution that only have the product of the scalar spherical harmonics Y l (Ω 3 ). On the other hand, the propagator of the type II vector mode is given by [28] G µν (x, x ′ , z, z ′ ) = − i N d 4 k (2π) 4 G (k) (z, z ′ )T µν e ik·(x−x ′ ) = − i N d 4 k (2π) 4 dω 2 2 (zz ′ )J τ −1 (ωz)J τ −1 (ωz ′ ) k 2 + ω 2 − iǫ T µν e ik·(x−x ′ ) ,(26) where T µν = η µν + kµkν ω 2 . Recall that in our case of interest the exchanged mode has conformal dimension τ min = 3. Finally, we redefine the D7-brane fields as Φ → Φ √ N , such that they are canonically normalized in terms of N. The 1/N-power counting shows that the interaction terms now scale as S U V → 1 N S U V , S IR → 1 N 3/2 S IR .(27) Structure functions for vector mesons We now derive the FCS amplitude related to the one-point function n µ X 1 , X 2 \| J µ (0) \|P, Q . Looking at the diagram of figure 1, the associated amplitude is A = −8i µ 2 7 (πα ′ ) 5 Q L R 4 (2π) 4 δ (4) (P + q − p ′ − q ′ ) I dzdz ′ √ g g ′ z ′2 ∂ ′[α ρ β] (z ′ )φ * (z ′ ) × B * ν (z) (∂ ′ α ∂ µ G νβ (z, z ′ ) − ∂ ′ α ∂ ν G µβ (z, z ′ )) − ∂ ′ α G νβ (z, z ′ ) ∂ µ B * ν (z) − ∂ ν B * µ (z) A µ (z) ,(28) where ∂ ′ α and ∂ ′ β are derivatives with respect to the primed coordinates. Also, we have already performed the integrals in the variables x and x ′ , which lead to the momentum conservation relations k µ = q ′ µ − q µ = P µ − p ′ µ .(29) In equation (28) I represents the integral of the scalar spherical harmonics over the 3-sphere, which is given in the appendix. Replacing the solutions (24), (25) and (26) in the amplitude, and using the relations (13), (14) and (19), the structure functions F i (i = 1, 2, L) can be written as F i (q 2 , x) = \|c\| 2 Q 2 µ 4 7 L 2 α ′10 R 8 M 2 ,M 3 d 3 q ′ 2E q ′ (2π) 3 d 3 p ′ 2E p ′ (2π) 3 δ (4) (P + q − q ′ − p ′ ) ×Λ 3 M 3 M 1 M 2 q 2 \|C t \| 2 F T i (x, P, q, q ′ ) ,(30) where \|c\| 2 = \|c φ \| 2 \|c B \| 2 \|c ρ \| 2 , C t contains the radial coordinate integrals, given by C t = dz dz ′ dω ω (q − q ′ ) 2 + ω 2 × z ′4 J τ −1 (M 1 z ′ )J τ ′ −1 (M 3 z ′ )J 2 (ωz ′ ) z 2 K 1 (qz)J 2 (M 2 z)J 2 (ωz) .(31) In order to obtain F T i , the factor in (30) which depends only of the tensor contractions, we need to calculate J T µ J T * ν , with J T µ = ζ * ν (k µ T νβ − k ν T µβ ) + T νβ q ′ µ ζ * ν − q ′ν ζ * µ k α P [α ǫ β] .(32) Recall that B * µ is an intermediate state, thus we need to sum over the outgoing vector polarizations ζ µ λ ζ µ ζ * ν = −q ′2 η µν + q ′ µ q ′ ν .(33) Since we are only interested in the unpolarized structure functions, we also average over the polarization vector of the incoming hadron ǫ µ ǫ * ν = 1 3 −P 2 η µν + P µ P ν .(34) By comparing J T µ J T * ν with equations (13) and (14) we obtain the following expressions F T 1 = 1 24q 2 x 2 P 2 128x 4 (q ′ ) 2 (P · q ′ ) 4 + 128x 3 (q ′ ) 2 q · q ′ (P · q ′ ) 3 + 8q 6 x (q ′ ) 2 P · q ′ +40q 4 x 2 (q ′ ) 2 (P · q ′ ) 2 − 8q 4 x 2 (q · q ′ ) (P · q ′ ) 2 + 32q 4 x (q ′ ) 4 P · q ′ −16q 4 x (q · q ′ ) 2 P · q ′ + 28q 4 x (q ′ ) 2 (q · q ′ ) (P · q ′ ) − 16q 2 x 4 (P · q ′ ) 4 +128q 2 x 3 (q ′ ) 2 (P · q ′ ) 3 − 48q 2 x 3 (q · q ′ ) (P · q ′ ) 3 + 32q 2 x 2 (q ′ ) 4 (P · q ′ ) 2 −40q 2 x 2 (q · q ′ ) 2 (P · q ′ ) 2 + 128q 2 x 2 (q ′ ) 2 (q · q ′ ) (P · q ′ ) 2 + 2q 8 (q ′ ) 2 +9q 6 (q ′ ) 4 − 2q 6 (q · q ′ ) 2 − 2q 6 (q ′ ) 2 q · q ′ (35) F T 2 = 1 12q 2 x P 2 384x 4 (q ′ ) 2 (P · q ′ ) 4 + 384x 3 (q ′ ) 2 (q · q ′ ) (P · q ′ ) 3 +64x 2 (q ′ ) 2 (q · q ′ ) 2 (P · q ′ ) 2 + 8q 6 x (q ′ ) 2 P · q ′ + 96q 4 x 2 (q ′ ) 2 (P · q ′ ) 2 −8q 4 x 2 (q · q ′ ) (P · q ′ ) 2 + 32q 4 x (q ′ ) 4 (P · q ′ ) − 16q 4 x (q · q ′ ) 2 (P · q ′ ) +76q 4 x (q ′ ) 2 (q · q ′ ) (P · q ′ ) − 48q 2 x 4 (P · q ′ ) 4 + 384q 2 x 3 (q ′ ) 2 (P · q ′ ) 3 −144q 2 x 3 (q · q ′ ) (P · q ′ ) 3 + 32q 2 x 2 (q ′ ) 4 (P · q ′ ) 2 − 136q 2 x 2 (q · q ′ ) 2 (P · q ′ ) 2 +384q 2 x 2 (q ′ ) 2 (q · q ′ ) (P · q ′ ) 2 − 32q 2 x (q · q ′ ) 3 (P · q ′ ) + 64q 2 x (q ′ ) 2 (q · q ′ ) 2 (P · q ′ ) +2q 8 (q ′ ) 2 + 9q 6 (q ′ ) 4 − 2q 6 (q · q ′ ) 2 − 2q 6 (q ′ ) 2 (q · q ′ ) + 8q 4 (q ′ ) 2 (q · q ′ ) 2 .(36) In order to calculate the integrals in (31) we need to use a few reasonable approximations, in a similar way as in [20,19]. The main assumption is that Λ and the masses of the hadrons are small in comparison with the momentum of the virtual photon. The IR integral selects the mass of the exchanged field as follows [19] Λ −1 0 dz ′ z ′4 J τ −1 (M 1 z ′ )J τ ′ −1 (M 3 z ′ )J 2 (ωz ′ ) ≈ (37) 1 Λ 3 1 √ M 1 M 3 [(−1) α δ(ω − (M 1 + M 3 )) + (−1) β δ(ω − (M 1 − M 3 ))] , for some integers α and β. The integral leads to ω = \|M 1 ± M 3 \|. Then, the UV integral can be obtained by expanding J 2 (ωz) ≈ ω 2 z 2 /8 for ω ≪ q, and taking the upper limit as infinity since K 1 decays quickly in the bulk. We obtain ∞ 0 dz ω 2 8 z 4 K 1 (qz)J 2 (M 2 z) ≈ 6M 2 2 q ω 2 (M 2 2 + q 2 ) 4 .(38) With these two equations we can obtain C t , after noticing that the leading term comes from ω = M 1 − M 3 [20], we obtain \|C t \| 2 = q 2 36M 4 2 (M 2 2 + q 2 ) 8 1 Λ 6 (M 1 − M 3 ) 6 M 1 M 3 1 ((q − q ′ ) 2 + (M 1 − M 3 ) 2 ) 2 .(39) The next step is to integrate over the on-shell momenta p ′ and q ′ , and sum over the corresponding masses 6 . The final results for the structure functions are F 1 (x, q 2 ) = 1 λN C M 1 Λ 6 Λ 2 q 2 1 2 x 3 (1 − x) 3 (40) F 2 (x, q 2 ) = 1 λN C M 1 Λ 6 Λ 2 q 2 x 3 (1 − x) 3 (4 − 3x) (41) F L (x, q 2 ) = 1 λN C M 1 Λ 6 Λ 2 q 2 4 x 3 (1 − x) 4 ,(42) where C is a numerical constant. We expect qualitatively similar results to hold in the context of different Dp-brane models. Discussion and conclusions We have obtained the 1/N corrections to the F 1 , F 2 and F L structure functions corresponding to vector mesons, using the gauge/gravity duality. Motivated by previous work for N = 4 SYM theory glueballs, and particularly for scalar mesons in the D3D7-brane system, the idea is to investigate how two very different limits behave, namely: the large N limit in comparison with the high energy limit (Λ 2 /q 2 → 0). Our first result is that they do not commute for the vector mesons. Then, since the physical way to consider these limits implies to take first the high energy one, followed by the large N limit, we find that in this situation the third term in the expansion of equation (5) dominates the moments of the structure functions. This is a very interesting result which says that at strong coupling this third term becomes the leading one for q 2 ≥ Λ 2 N 1/(τ Q −τc) , where τ Q and τ c are the minimum twist of protected operators with charge Q and the minimum twist of all electrically charged protected operators, respectively. This is similar to what happens for the glueball in the IR deformed version of N = 4 SYM [6,19], where the process is given in terms of closed string modes and at strong coupling one finds that the 1/N result dominates for q 2 ≥ Λ 2 N 2/(τ Q −τc) . However, note that in the present case, i.e. for mesons, the correction is proportional to the 1/N instead of 1/N 2 , as expected. Thus, at large N the critical value for the photon virtuality q where this happens is much smaller. The same occurs for the results presented in [20] for the case of scalar mesons. The physical implication of this result is that, at strong coupling, for the above energy range DIS is dominated by a two-hadron final state. From the viewpoint of the FCS process it corresponds, through the optical theorem, to a situation where there are two intermediate hadron states. The structure functions F 1 and F 2 behave as (1 − x) 3 as x approaches 1. We have also considered the longitudinal structure function F L which behaves as (1 − x) 4 as x approaches 1. We should notice that, although the states as well as the interactions for the vector and scalar mesons are different, all these structure functions have the same dependence on Λ 2 /q 2 , 1/N, 1/λ and M 1 /Λ as in the case of scalar mesons in the D3D7-brane system. It is important to consider the moments of the structure functions defined as M n [F i ] = 1 0 dx x n−1 F i (x, q 2 ) ,(43) where F i can be F 1 and F 2 in this case. Several moments of these structure functions have been calculated in reference [18] in the large N limit, i.e. by considering a single intermediate hadron state in the FCS process. This has been done for the first moments of the structure function F 2 in the case of the pion as well as for F 1 of the rho meson. In [18] we have compared these results with lattice QCD data from references [21,22,23] for the pion, associated with the lightest pseudoscalar mode. In addition, in the case of the rho meson associated to the l = 2 spin-1 mode of the type II solutions, the comparison has been made with respect to results from lattice QCD of [21]. The best fit for the case of the pion leads to results with an accuracy of 10.8% or better, while in the case of the rho meson the accuracy is of 18.5% or better 7 , for the D3D7-brane system. In [18] also the Sakai-Sugimoto model of the D4D8D8-brane system and the D4D6D6-brane system, both in type IIA string theory, have been considered for FCS with a single intermediate exchanged state. The next step has been done in [20] where we have considered the leading 1/N corrections to the structure functions. The accuracy is notoriously enhanced to 1.27% for the scalar mesons in the D3D7-brane system in this case. It leads to a natural question which is whether for vector mesons the accuracy of the fit can also be substantially improved by considering 1/N corrections. In order to investigate this point we have carried out the best fit of the structure F 1 including 1/N corrections in comparison with lattice QCD data from [21]. Recall that the results of the present work have been obtained in the type IIB supergravity regime, i.e. where 1/ √ λ ≪ x < 1, which means that for the calculation of the moments we have integrated our result for the functions between x = 0.1 and x = 1. On the other hand, we also need to carry out the integration over the range exp (− √ λ) ≪ x ≪ 1/ √ λ, where we assume that the behavior of the structure functions is similar to the behavior shown in [5] and used in [18], i.e. F small−x L ∝ x −1 . We support this assumption on the fact that, in the 1/ √ λ ≪ x < 1 range the difference between the large-N calculation (where there is only a single on-shell hadron state exchanged in the FCS process) and the 1/N calculation (where the leading Feynman-Witten diagram has two on-shell hadron states exchanged) is that in the former the dependence with the photon virtuality and the Bjorken parameter is given by τ in corresponding to the incident meson, while in the later the q 2 and x dependence is determined by τ min . This corresponds to one of the lowest conformal dimension from the supergravity excitations. However, for exp (− √ λ) ≪ x ≪ 1/ √ λ things are different, namely: the calculation from the two-open and two-closed strings scattering amplitudes is independent of τ in . Thus, we assume that in this low-x regime the genus-zero result from type IIB superstring theory should not be very different with respect to a much more complicated calculation on the torus. Then, for this string theory regime of the holographic dual calculation we use the expressions for the structure functions at tree level from [18]. For our numerical calculation at low-x we consider that the integration for the moments is performed between x = 0.0001 and x = 0.1 as before [18,20]. Then, we split each structure function in two parts, each of one having a dimensionless constant to be fixed by fitting with respect to lattice QCD data [21]. There is a constant C 1 multiplying the low-x F 1 function. In addition, there is a second constant C 2 on the large-x F 1 function. Results of the first moments of F 1 of the rho meson are presented in table 2. The values we obtain for the constants are C 1 = 0.0087 and C 2 = 32.1939. Note that they are of the same order as the ones found in our previous work [18] in the large N limit, for which the constants associated with the small-x F 1 and with the large-x F 1 of the rho meson are 0.012 [21] and in comparison with previous results presented in [18]. Uncertainties in the lattice QCD computations are omitted. Model / Moment M 2 (F 1 ) M 3 (F 1 ) M 4 (F 1 ) Lattice QCD 0. and 78.07, respectively. Figure 2 shows the structure function F 1 as a function of x. The blue bell-shaped curve indicates the 1/N calculation of this work. The black dashed bell-shaped line corresponds to the case obtained in reference [18] for the large N limit. For small-x values we use the result from [5], leading to the monotonically decreasing curves. The difference between the two curves at small values of the Bjorken parameter comes from the slightly different constants which correspond to the best fit developed in each situation. There is an important improvement with the inclusion of the leading 1/N correction. As it happened in the scalar case, the location of the maximum is shifted to the left with respect to the results obtained in the planar limit, matching better the phenomenological expectations 8 . From table 2 we can appreciate the enhancement of the accuracy of the moments of F 1 for the case of the rho meson which goes from 18.5% for one particle exchange in the FCS, down to 12.5% in the leading 1/N contribution, i.e. for the exchange of two intermediate on-shell states. This is very important because it confirms the trend found previously for glueballs in N = 4 SYM theory [19] and for the scalar mesons of the N = 2 SYM theory from the D3D7-brane system [20]. Thus, it indicates that in order to infer realistic conclusions for physical systems it is crucial to consider the 1/N expansion of the observables, and consider first the large momentum transfer limit. Possibly, this behavior can be identified in other physical process, and although we have restricted our investigation to strongly coupled gauge theories it would be very interesting to investigate the relations between the large N limit and the high energy one for gauge field theories at perturbative level. Appendix: Angular integrals Scalar spherical harmonics on the 3-sphere belong to the (l/2, l/2) representation of SU(2) × SU(2) ≡ SO(4), where l is a non-negative integer, while −l/2 ≤ m , n ≤ l/2. They form a basis of eigenfunctions of the Laplace operator on the sphere, and satisfy the orthogonality relation S 3 (Y m,n l ) * Y m ′ ,n ′ l ′ = δ ll ′ δ mm ′ δ nn ′ .(45) which is the value of I present in equation (28). the structure functions at order in 1/N n , with j = 1, 2 and n = 1, . . . . Notice that in the definitions of f (n) j 's powers of Λ 2 /q 2 have been factored out. The index n indicates the number of exchanged intermediate on-shell states of the FCS Feynman diagram. This corresponds to the number of hadrons in the final state of the DIS process. From expression S 3 , being Q the charge of the vector mode, given by the eigenvalue equation v i ∂ i B µ = iQB µ . Note that Q does not have to coincide with Q, the charge of the original hadron. In fact, one has to sum over all possible intermediate states, and in particular all possible values of Q (see appendix). Figure 1 : 1Feynman-Witten diagram associated to the DIS process with two intermediate hadron states. The momentum and the twist of each field are indicated, while the solutions are given in section 2. The incident hadron is represented by a double line, the intermediate spin-one modes are indicated by dashed lines, while the solid line denotes a scalar mode. The vertical dotted line represents the cut of the Cutkosky rule. Figure 2 : 2F 1 as a function of x. We display the leading results at low and moderate (0.1 < x ≤ 1)values of x. Dashed curves represent the F 1 computed for N → ∞, while the blue ones correspond to the 1/N corrections. The constants C 1 and C 2 are those which give the moments of F 1 shown in table 2. − 1 . 1For any of these values, equation (49) does no vanish only if l ′′ = 1, 2, 3. We want to sum over all possible exchanged and outgoing states, this is a sum over m ′ , n ′ = ±1/2, and over l ′′ . The final result is the same for the three values of m, Table 2 : 2Comparison of our results for the first moments of the structure function F 1 of the lightest vector meson for a suitable choice (best fitting) of the normalization constants with respect to the results of the lattice QCD simulations in The polarized structure functions b 1,2,3,4 and g 1,2[2] have been obtained at strong coupling and in the planar limit in[3,4] from supergravity and in[5] from superstring theory scattering amplitudes. 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[ "Emergent Newtonian dynamics and the geometric origin of mass", "Emergent Newtonian dynamics and the geometric origin of mass" ]	[ "Luca D'alessio \nDepartment of Physics\nThe Pennsylvania State University\n16802University ParkPAUSA\n\nPhysics Department\nBoston University\n02215BostonMAUSA\n", "Anatoli Polkovnikov \nPhysics Department\nBoston University\n02215BostonMAUSA\n" ]	[ "Department of Physics\nThe Pennsylvania State University\n16802University ParkPAUSA", "Physics Department\nBoston University\n02215BostonMAUSA", "Physics Department\nBoston University\n02215BostonMAUSA" ]	[]	We consider a set of macroscopic (classical) degrees of freedom coupled to an arbitrary many-particle Hamiltonian system, quantum or classical. These degrees of freedom can represent positions of objects in space, their angles, shape distortions, magnetization, currents and so on. Expanding their dynamics near the adiabatic limit we find the emergent Newton's second law (force is equal to the mass times acceleration) with an extra dissipative term. In systems with broken time reversal symmetry there is an additional Coriolis type force proportional to the Berry curvature. We give the microscopic definition of the mass tensor relating it to the non-equal time correlation functions in equilibrium or alternatively expressing it through dressing by virtual excitations in the system. In the classical (high-temperature) limit the mass tensor is given by the product of the inverse temperature and the Fubini-Study metric tensor determining the natural distance between the eigenstates of the Hamiltonian. For free particles this result reduces to the conventional definition of mass. This finding shows that any mass, at least in the classical limit, emerges from the distortions of the Hilbert space highlighting deep connections between any motion (not necessarily in space) and geometry. We illustrate our findings with four simple examples.	10.1016/j.aop.2014.03.009	[ "https://arxiv.org/pdf/1309.6354v2.pdf" ]	119,228,038	1309.6354	e87800ca80cf0f3ec2f56caaa69ebcd984831fc5	Emergent Newtonian dynamics and the geometric origin of mass May 22, 2014 24 Feb 2014 Luca D'alessio Department of Physics The Pennsylvania State University 16802University ParkPAUSA Physics Department Boston University 02215BostonMAUSA Anatoli Polkovnikov Physics Department Boston University 02215BostonMAUSA Emergent Newtonian dynamics and the geometric origin of mass May 22, 2014 24 Feb 2014Preprint submitted to ElsevierMany-body systemsOpen quantum systemsDriven dissipative systemsDynamics and Geometry We consider a set of macroscopic (classical) degrees of freedom coupled to an arbitrary many-particle Hamiltonian system, quantum or classical. These degrees of freedom can represent positions of objects in space, their angles, shape distortions, magnetization, currents and so on. Expanding their dynamics near the adiabatic limit we find the emergent Newton's second law (force is equal to the mass times acceleration) with an extra dissipative term. In systems with broken time reversal symmetry there is an additional Coriolis type force proportional to the Berry curvature. We give the microscopic definition of the mass tensor relating it to the non-equal time correlation functions in equilibrium or alternatively expressing it through dressing by virtual excitations in the system. In the classical (high-temperature) limit the mass tensor is given by the product of the inverse temperature and the Fubini-Study metric tensor determining the natural distance between the eigenstates of the Hamiltonian. For free particles this result reduces to the conventional definition of mass. This finding shows that any mass, at least in the classical limit, emerges from the distortions of the Hilbert space highlighting deep connections between any motion (not necessarily in space) and geometry. We illustrate our findings with four simple examples. Introduction Newton's second law, F = mẌ, applies to a wide range of phenomena and it is the cornerstone of the classical physics. This equation certainly describes the motion of particles in free space but it also predicts the dynamics of macroscopic objects like rotating bodies where X represents the angle and m is the moment of inertia or the behavior of the electrical current in a LC circuits where X is a current and m is a combination of capacitance and inductance. However, as we all know, when an object moves through a medium it dissipates energy into the surrounding environment increasing its entropy and, as a result, it slows down and eventually stops. In this case, to properly describe the object's dynamics, Newton's law need to be supplemented with a dissipative (drag) term: mẌ + ηẊ = F.(1) Drag is not specific to the motion in real space and it is present every time a physical quantity changes in time. For example, when the magnetic flux through a coil is increased in time, the environment (electrons in the coil) will react by producing a drag force which opposes this change (i.e. the Faradays law). Likewise coherent spin oscillations always decay in time due to dissipation in the spin environment and so on. A pragmatic and very successful approach consists in seeing the mass m and the dissipation η in Eq. (1) as emergent properties (i.e. phenomenological parameters) which need to be fitted to reproduce the correct dynamics of macroscopic degrees of freedom. This approach is conceptually unsatisfactory and, for this reason, a lot of research effort has been focused on trying to derive the mass and the dissipation from a more fundamental theory (see e.g. Refs. [1,2,3,4]). Normally one starts from the microscopic approach for both the object and the medium and using various approximations derives effective equations of motion for the macroscopic degree of freedom (d.o.f.) [5,6,7,8,9]. From this approach one finds both renormalization of the bare mass by dressing with elementary excitations and the drag force (see for example [10]). For example, in the Landau-Fermi Liquid theory [11,12] the electron-electron interaction lead to a renormalization of the bare electron mass (and other dynamical properties). In this work we take a different approach (see [2] and references therein), where we use statistical equilibrium of a complex system as a starting point. Namely we consider an arbitrary many-body system with possibly infinitely many degrees of freedom interacting with few macroscopic parameters λ(t), which are allowed to slowly change in time (see Fig.1). We assume that if λ is constant the system equilibrates at some temperature (as we will discuss later this assumption can be further relaxed). By extending the Kubo linear response theory [13] to such setups we provide a framework to compute the energy exchange between the many-body system and macroscopic d.o.f. in time derivatives of λ(t). In the first two orders (for systems with time-reversal invariance) the resulting expression takes the form of the Newton's second law with the additional drag term: κ νµλµ + η νµλµ = M ν ,(2) where κ νµ is the mass tensor, η νµ is the drag tensor, and M ν is the force (we choose this notation to reserve the letter F for the Berry curvature). In the absence of symmetries all three coefficients can explicitly depend on λ predicting nontrivial dynamics. We express κ and η through the non-equal time equilibrium correlation functions and show that in general the mass is related to virtual (off-shell) excitations while the drag coefficient is determined by real (on-shell) processes. The expansion in derivatives ofλ µ used to derive Eqs. (2) relies on the time scale separation between slow dynamics of the microscopic parameters λ and the fast dynamics of the system coupled to λ. In the situations where this time scale separation does not hold, e.g. near phase transitions or low dimensional gapless system the validity of the expansion in time derivatives of λ should be checked on the case by case basis (see Sec. 3 .2.2). Our result for the drag coefficient in Eq. (2) agrees with the recent classical result of Ref. [14] and, as shown there, can be used to define the dissipative metric. In addition we find that in the high-temperature or classical limit the mass tensor κ is equal to the product of the inverse temperature and the thermally averaged Fubini-Study metric tensor g µν . The latter characterizes deformations of the eigenstates of the systems to the change of λ [15,16,17]: κ νµ = βg νµ(3) This result allows one to interpret the (inertial) mass as an emergent property deriving from the response of the Hilbert space to the change of λ. This statement echoes the result in general relativity in which the mass is seen as an emergent property which stems from the curvature of space-time. Let us point that the metric tensor g µν is equal to the covariance matrix of the gauge potentials A µ = i∂ µ 1 responsible for the translations in the parameter 1 Except where explicitly mentioned we work in the units = 1. space [17]: 2g µν = A µ A ν c + µ ↔ ν ≡ A ν A µ − A ν A µ + µ ↔ ν, where . . . implies averaging with respect to the equilibrium density matrix. For translations in real space this gauge potential is nothing but the momentum operator A x = p x , for rotations in xy plain it is the angular momentum operator A φ = L z . In general these gauge potentials are complicated manybody operators, which however satisfy properties of locality [17]. For simple setups like a free particle in a vacuum our result m = κ xx = β p 2 x reduces to a conventional equipartition theorem p 2 x = mT (see Sec 5.1 for a detailed example). So this finding shows that even ordinary mass of a particle has a direct geometric interpretation and is related to the distortion of the Hilbert space surrounding the particle as it moves in space. At arbitrary temperatures instead of Eq. (3) we obtain 2κ νµ = β 0 dτ A H,ν (−iτ )A H,µ (0) + ν ↔ µ 0,c ,(4) where A H,µ (−iτ ) is the imaginary time Heisenberg representation of the gauge potential. In the high-temperature limit Eq. (4) obviously reduces to Eq. (3). This general expression can serve as the formal definition of the mass tensor for an arbitrary system. It also can be thought of as an extension of the equipartition theorem to quantum systems. In less trivial situations of interacting systems and motion in the parameter space rather than the real space our results suggest an experimentally feasible way to indirectly measure the Fubini-Study metric and to directly analyze the quantum geometry of the system including various singularities and geometric invariants [17]. We illustrate how this can be done by analyzing three simple examples (Sec. 5). Setup and main results We consider the following Hamiltonian: H tot ( λ) = H 0 ( λ) + H( λ),(5) where H 0 ( λ) is the Hamiltonian describing the bare motion of the macroscopic d.o.f. λ, which can be multi-component and H( λ) is the Hamiltonian of the interacting system of interest which depends on λ. The choice of splitting H tot between H 0 and H is somewhat arbitrary and we can well choose H 0 = 0 so that H tot = H, however, for an intuitive interpretation of the results, it is convenient to assume that H 0 ( λ) represents a massive degree of freedom in some external potential V ( λ): H 0 ( λ) = p 2 λ 2m + V ( λ). In the infinite mass limit (m → ∞), λ represents an external (control) parameter whose dynamics is specified a priori. When m is finite, λ is a dynamical variable and its dynamics needs to be determined self-consistently. One can identify two different sources of the energy change in the system. The first contribution is related to the dependence of the energy eigenstates of the system on λ. This contribution does not vanish in the adiabatic limit and it is reversible. The second contribution is related to the excitations created in the system and, as we shall see, it contains both reversible and irreversible terms. To shorten the notation we term the first contribution as the adiabatic work and the second one as the heat. Formally these two contributions can be defined as: ∂ t H (t) ≡ T r {∂ t (H(t)ρ(t))} =Ẇ (t) +Q(t) where ρ(t) is the density matrix and the adiabatic work rate,Ẇ (t), and the heat rate,Q(t), are defined as: W (t) =˙ λ n ∂ n ∂ λ ρ n (t) =˙ λ · ∇ λ E( λ) ≡ −˙ λ · M λ ,(6)Q(t) = nρ n (t) n ,(7) where E = n n ρ n is the total energy of the system, ρ n (t) is the probability of occupying the n-th instantaneous energy eigenstate, n is the instantaneous energy and M λ = −∇ λ E( λ) is the conventional generalized force which we introduced in Eq. (2) [18]. This force appears, for example, in the Born-Oppenheimer approximation schemes in which the energy of the electrons, calculated for fixed ions' positions, acts as a potential for the (classical) motion of the ions. In the strictly adiabatic limit, according to the quantum adiabatic theorem, there are no transitions between the instantaneous energy levels anḋ Q = 0. 2 Using time-dependent perturbation theory to second order inλ, we have computedρ(t), and obtained the leading non-adiabatic contributions to the heat production rate (see Appendix A): Q(t) =λ ν (t) t 0 dt β 0 dτλ µ (t − t ) M H,ν (t)M H,µ (t − t + iτ ) 0,c =λ ν (t) t 0 dt λ µ (t − t ) n =m ρ 0 n − ρ 0 m m − n e i( m− n)t m λ \|M ν \|n λ n λ \|M µ \|m λ(8) where we imply summation over repeated indices, \|n λ denote the (manybody) eigenstates of the Hamiltonian H( λ(t)) with energies n , ρ 0 n is the initial stationary occupation probability which, for the most of the paper, we assume to be thermal ρ n = exp[−β n ]/Z, c implies the connected part of the correlation function and M H,ν (t) = − exp[iHt] (∂ λν H) exp[−iHt](9) is the Heisenberg representation (for time-independent Hamiltonian) of the conjugate force coupled to λ ν . The equilibrium expectation value of this force M λ 0 = M λ = −∇ λ E( λ) is the conventional generalized force introduced earlier. Note that n and \|n are the many-body levels and many-body eigenstates of the full interacting system (including the bath if it is present). If the system is weakly coupled to the bath then Eq. (8) can be simplified using standard tools of the many-body perturbation theory [19]. For simplicity in Eq. (8) and (9) we assume that the energy spectrum n does not change with λ but in the Appendix A we show a complete derivation without such an assumption. Then essentially by energies n and the matrix elements one needs to understand instantaneous values taken at time t (see Appendix A). Eq. (8) applies to arbitrary times t and as such describes both transient and the long time regimes. It is clear that if t becomes longer than the relaxation time then the heat production rate becomes insensitive to the initial time t = 0 but, for small isolated systems and at low temperatures, transients can be important (see Sec. 5.3 for an example). This expression generalizes an earlier result by D. A. Sivak and G. E. Crooks [14] to quantum systems at arbitrary temperatures. In the high temperature (or classical) limit the integration over the imaginary time component τ reduces to a factor of the inverse temperature β and we recover Eq. (11) from Ref. [14]. If the rate˙ λ changes slowly in time on the scales determined by the relaxation rate in the system then we can expand Eq. (8) to the leading order in time derivatives of λ. For systems with the Hamiltonian H obeying time-reversal symmetry we get (terms proportional to higher time derivatives of λ are discussed in Appendix B) Q(t) =λ ν η νµλµ +λ ν κ νµλµ(10) The first term represents the usual drag or friction tensor and the second term (as we will see below) amounts to the mass renormalization of the external parameter. At zero temperature the first dissipative term generically vanishes as required by the fluctuation-dissipation relations while the second contribution remains finite. Both tensors η and κ are symmetric under ν ↔ µ and positive semi-definite (assuming positive temperature). Before describing the friction and mass tensors in details we discuss some consequences of our main result (10). 3. Implications of Eq.(10). Energy absorption in a driven system As the first implication of our results, we discuss qualitative features of energy absorption in systems driven by the external parameter λ(t). We will be interested in protocols longer than the relaxation time in the system such that susceptibilities η and κ become effectively time independent (at shorter times one expects transients, which can be analyzed using the short time expansion of Eq. (8)). To avoid possible singularities we assume that the protocol starts smoothly in time, i.e.˙ λ(0) = 0. Integrating Eq. (10) we find Q(t) ≈ η µν t 0 dt λ µ (t )λ ν (t ) + κ µν 2λ µ (t)λ ν (t) + . . .(11) From this expression we see that indeed η plays the role of dissipation while κ plays the role of the additional mass associated with the dressing of the parameter λ by excitations. It is clear that the first (dissipative) term gives a contribution proportional to the total duration of the process and it is dominant at long times t → ∞. However, at low temperatures or nearly isolated small systems the coefficient η can be very small and the second (mass) term can be dominant for long times. In particular, ifλ(t) approaches a constant value, after an initial transient, Q(t) will display a plateau for t t * = κ/η. Emergent equations of motion Next we consider λ to be a dynamic d.o.f. describing a "particle" coupled to a many-body system and study its dynamics. 3.2.1. Dynamics of a scalar particle. First let us assume that λ is a single component scalar object. The Hamiltonian is given by Eq. (5). Noting that the total energy of the particle and the system is conserved and using Eq. (6) and Eq. (10) we find d dt mλ 2 2 + V (λ) + E(λ) = 0 ⇒ mλλ +λ∂ λ V +Ẇ +Q = 0, ⇒ mλλ +λ∂ λ V −λM λ + ηλ 2 + κλλ = 0.(12) Dividing this equation byλ we find (m + κ)λ +λη = − ∂V ∂λ + M λ(13) i.e. we find that near the slow limit the dynamics of λ is given by the classical Newton's equations of motion with an additional dissipative term. Note that the Newtonian form of the equation of motion (13) was not postulated. It rather emerges from the leading non-adiabatic expansion of the density matrix describing the system coupled to λ. Thus the effect of the system on the dynamics of the macroscopic d.o.f. reduces to three contributions: mass renormalization (κ), drag coefficient (η), and an extra force (M λ ). As mentioned earlier, M λ is the force which appears in the Born-Oppenheimer approximation in which the only effect of the system is to lead to an effective (conservative) potential for the motion of the macroscopic d.o.f.. For example in this approximation schemes the electrons in a solid lead to an effective potential for the motion of the ions. By considering the non-adiabatic correction to the energy of the system, which we named heat (see Eq. (8)), we go beyond the Born-Oppenheimer approximation and find the emergent Newton's equation (13). Note that in general κ, η and M λ depend on λ and Eq. (13) can predict non-trivial dynamics. As we pointed earlier the Hamiltonian H 0 can be absorbed into the definition of H. In this case m = 0 and ∂ λ V = 0 and Eq. (13) reduces to the scalar version of Eq. (2). Moreover when H 0 = 0 the total mass and the total force are entirely determined by the interactions with the system. Let us note that when κ depends on λ it is more accurate to write the mass term in Eq. (13) as d dt (κλ). The difference between this term and κλ is of orderλ 2 and it is beyond the accuracy of our expansion. However a more careful analysis shows that this term is energy conserving, which implies that it comes from the full derivative of κλ 2 . Berry curvature and Coriolis force. The previous derivation of the equations of motion was based on using the conservation of the total energy conservation leading to Eq. (12) and dividing it byλ. For a multicomponent parameter energy conservation is not sufficient to fix the equations of motion, since, as it is well known in the case of rotations or magnetic field, the Coriolis or the Lorentz forces affect the dynamics but not the energy. To proceed within the same framework of non-adiabatic response we can evaluate the expectation value of the gener- alized force M ν (t) ≡ T r [ρ(t)M ν ]. We give the details of the derivation in Appendix A and here only quote the final result obtained using the same approximation as Eq. (8): M ν (t) = M ν − t 0 dt λ µ (t−t ) n =m ρ 0 n − ρ 0 m m − n e i( m− n)t m λ \|M ν \|n λ n λ \|M µ \|m λ .(14) We remind that the generalized force M ν is by definition the equilibrium expectation value of M ν . If we now do the expansion ofλ µ (t − t ) near t = 0 up to the second derivative we find [20] M ν (t) = M ν + F νµλµ − η νµλµ − κ νµλµ − F νµλ µ ,(15) where F νµ = i [A ν , A µ ] = i n ρ 0 n n λ \| ← − ∂ ν − → ∂ µ − ← − ∂ µ − → ∂ ν \|n λ(16) is the Berry curvature and F µν = iπ n =m ρ 0 n − ρ 0 m n − m m λ \|M ν \|n λ n λ \|M µ \|m λ δ ( n − m )(17) is another on-shell antisymmetric tensor (δ (x) stands for the derivative of the delta-function) and η and κ are the friction and the mass tensors discussed before. Without the acceleration terms Eq. (15) extends earlier results on the dynamical Hall response [21,22,23] to finite temperatures and possibly open systems without extra assumptions about the Lindbladian dynamics [4]. As in the familiar situation of the current response to the electric field (time dependent vector potential) the transverse Hall like response determined by the Berry curvature is non-dissipative (off-shell) while the longitudinal onshell response proportional to the drag η is directly related to dissipation [19]. We can now self-consistently combine Eq. (15) with the classical (Lagrangian) equations of motion for the parameter λ: m νµ dλ ν dt = p ν , dp ν dt = − ∂V ∂λ ν + M ν (t)(18) and get the multicomponent dissipative Newton's equations: (m νµ + κ νµ + F νµ )λ µ + (η νµ − F νµ )λ µ = − ∂V ∂λ ν + M ν .(19) which generalize Eq. (13). The first term in this equation represents the renormalized mass (as we discussed we can choose the bare mass to be zero by absorbing H 0 into H). The term η νµλµ is the dissipative term also appearing in the single component case. And finally there are two new antisymmetric terms, one proportional to the Berry curvature is clearly the analogous of the Coriolis force and the other antisymmetric on-shell contribution encoded in F is effectively antisymmetric mass term. This suggests that the Berry curvature can be indirectly measured via the Coriolis force acting upon a macroscopic d.o.f. [24]. In systems with time-reversal symmetry the two tensors F, F vanish [25] and therefore, by fitting the long time dynamics of λ(t) to Eq. (19) with the additional knowledge of equilibrium generalized force one can extract both the drag tensor, η νµ , and the mass tensor, κ νµ . We emphasize again that, at high temperatures, the mass tensor reduces to the Fubini-Study metric tensor which is a covariance matrix of the gauge potential, or the momentum operator, which translates the Hamiltonian eigenstates in the parameter space. Therefore the very notion of the mass is related to the distance between eigenstates of the Hamiltonian H( λ) induced by the change of the parameter λ. Let us point that Eq. (26) [see below] essentially guarantees UV convergence of the mass as long as the variance of the generalized force is finite. On the contrary in gapless systems or near singularities like continuous phase transitions or glassy systems where correlation functions slowly decay in time the mass can acquire infra-red divergences and the energy absorption becomes non-analytic function of the rate as discussed in Ref. [26]. Equation (19) has another interesting implication. At zero temperature both dissipative tensors (η and F ) vanish (unless the system is tuned to a critical point or if it has gapless low-dimensional excitations [26]). In this case Eq. (19) can be viewed as a Lagrangian equations of motion. If fact, it is easy to see that the Lagrangian: L = 1 2λ ν (m + κ) νµλµ +λ µ A µ ( λ) − V ( λ) − E 0 ( λ)(20) reproduces Eq. (19) where A µ ( λ) = 0 λ \|A µ \|0 λ and E 0 ( λ) = 0 λ \|H( λ)\|0 λ are the value of the Berry connection and Hamiltonian (see (5)) in the instantaneous ( λ-dependent) ground state and we have used (see Eq. (16)): ∂A µ ∂λ ν − ∂A ν ∂λ µ = i [∂ ν 0 λ \|∂ µ 0 λ − ∂ µ 0 λ \|∂ ν 0 λ ] = i 0 λ \| ← − ∂ ν − → ∂ µ − ← − ∂ µ − → ∂ ν \|0 λ = F νµ . From the Lagrangian (20) we can define the canonical momenta conjugate to the coordinates λ ν : p ν ≡ ∂L ∂λ ν = (m νµ + κ νµ )λ µ + A ν ( λ)(21) and the emergent Hamiltonian: H ≡λ ν p ν − L = 1 2 (p ν − A ν )(m + κ) −1 νµ (p µ − A µ ) + V ( λ) + E 0 ( λ). (22) Clearly the Berry connection term plays the role of the vector potential. Thus we see that the whole formalism of the Hamiltonian dynamics for arbitrary macroscopic degrees of freedom is actually emergent. Without mass renormalization this Hamiltonian was first derived in Ref. [27] in which it was also shown that when the slow d.o.f. is quantum, there is an addittional force proportional to the Fubini-Study metric tensor g νµ . Away from the ground state the dissipative tensors (η and F ) are, in general, non-zero and it is not possible to reformulate Eq. (19) via Hamiltonian dynamics. 3.2.3. Dynamics of a conserved degree of freedom. Emergent equilibrium from dynamics. It is straightforward to apply the results above to the setup where two systems are coupled by a single conserved degree of freedom, i.e. H = H 1 (λ 1 ) + H 2 (λ 2 ) with the additional constraint λ 1 + λ 2 = const. Then using the additivity of the mass κ and drag η, which are obvious from the microscopic expressions (see Eq. (24) and (26) below) and invariance of Eq. (10) under λ → −λ, we obtain the dissipative non-Markovian dynamics for λ 1 (t): (κ 1 + κ 2 )λ 1 + (η 1 + η 2 )λ 1 = M (1) λ 1 − M (2) λ 2(23) We note that the Markovian limit of Eq. (23) is obtained by setting κ 1 = κ 2 = 0. As expected from basic thermodynamics, the equilibrium between two systems (λ 1 = const) is only possible when the generalized forces between two systems are equal to each other. Unlike in statistical mechanics, where this statement follows from the maximum entropy principle, here we explicitly derive this condition from the dynamical equilibrium. Note that this condition for the dynamical equilibrium does not require the two systems to be ergodic. In our derivation we only relied on starting from a stationary state, thermal or not. Detailed description of friction and mass tensor We now discuss the friction and mass tensors in details. The friction tensor can be expressed as [28] η νµ = 1 2 t 0 dt β 0 dτ M H,ν (t − iτ )M H,µ (0) + ν ↔ µ c = n =m ρ 0 n − ρ 0 m m − n m λ \|M ν \|n λ n λ \|M µ \|m λ sin (( m − n )t) m − n(24) from which it is clear that at positive temperatures (or more generally for any passive density matrix such that (ρ 0 n − ρ 0 m )( m − n ) > 0) the friction tensor is positive semi-definite and can therefore be used to define a metric. In particular, the metric associated with the friction η was used in Ref. [29] for finding the paths of minimal dissipation in the parameter space. At long times, t → ∞, we have sin( m − n )t m − n → πδ( n − m ), ρ 0 n − ρ 0 m m − n → β ρ 0 m . Therefore the friction tensor is defined by the on-shell processes, which is expected since we are dealing with slow, zero frequency, driving η νµ = πβ n =m ρ 0 m m λ \|M ν \|n λ n λ \|M µ \|m λ δ( n − m ). So at t → ∞ the drag is always determined by the high temperature asymptote of Eq. (8) (and thus the result of Ref. [14] always holds). At finite times, the expression for η has different high temperature and low temperature asymptotes. Similarly we can analyze the mass tensor κ κ νµ = − 1 2 t 0 dt t β 0 dτ M H,ν (t − iτ )M H,µ (0) + ν ↔ µ c ,(25) which in the long time limit can be written as κ νµ = n =m ρ 0 n − ρ 0 m ( m − n ) 3 m λ \|M ν \|n λ n λ \|M µ \|m λ = n =m ρ 0 n − ρ 0 m m − n m λ \| ← − ∂ λν \|n λ n λ \| − → ∂ λµ \|m λ(26) where we have used the identity (see e.g. Ref. [30]) ( m − n ) n λ \| − → ∂ λµ \|m λ = n λ \|M µ \|m λ We emphasize again that we put aside the issue of convergence of the sum in Eq. (26) when we deal with macroscopic ergodic systems at finite temperatures. This convergence is essentially guaranteed if the non-equal time correlation functions entering Eq. (25) decay fast in time, which was our key assumption. Unlike for the friction tensor η, the mass tensor κ νµ has, in the infinite time limit, non-vanishing asymptotes both in the high and zero-temperature limits. At low temperatures, β → ∞, κ νµ ≈ m =0 0 λ \|M ν \|m λ m λ \|M µ \|0 λ + ν ↔ µ ( m − 0 ) 3 .(27) At high temperatures (or near the classical limit) we find κ νµ ≈ β 2 m ρ m m λ \| ← − ∂ λν − → ∂ λµ \|m λ c + ν ↔ µ = β g νµ(28) where g νµ is the finite temperature version Fubini-Study metric tensor characterizing statistical average of the distance between many-body eigenstates [17]. Let us mention that in traditional units the expressions for the mass Eqs. (26) - (28) should be multiplied by 2 , which follows from the correct definition of the Gauge potentials i∂ λµ → i ∂ λµ . We also point that the mass tensor κ can be written through the integrated connected imaginary time correlation function of the gauge potentials A ν and A µ : κ νµ = 1 2 β 0 dτ A H,ν (−iτ )A H,µ (0) + ν ↔ µ 0,c(29) At high temperatures the imaginary time integral reduces to a factor β and this result clearly reduces to Eq. (28). We point that the expressions for η in the second line of (24) and for κ in (26) apply to arbitrary stationary distributions {ρ 0 n }, not necessarily thermal. It is interesting to note that short time transient dynamics also bears the geometric information. Thus integrating Eq. (8) over short times (and again assuming high temperature or classical limit) gives Q(t) ≈ β 2 δλ ν (t) f νµ δλ µ (t), where δλ ν (t) ≡ λ ν (t) − λ ν (0) and f νµ = 1 2 M ν M µ + M µ M ν 0,c = ∂ 2 ln(Z) ∂λ µ ∂λ ν(30) is the thermodynamic metric tensor defined through the second derivative matrix of the partition function [32,31]. Examples Mass Renormalization of a particle interacting with identical quantum oscillators. We now show explicitly how the mass of a classical object coupled to a quantum environment is renormalized. Let us consider the Hamiltonian (5) where H 0 = P 2 λ 2µ + V (λ), H(λ) = N j=1 p 2 j 2m + k 2 (x j − λ) 2 + i>j U (x i −x j ) describes a macroscopic d.o.f. interacting with a collection of coupled quantum oscillators (QOs). We note that the coupling between the oscillators, U (x i −x j ), does not need to be harmonic so instead of oscillators we can deal with interacting particles. Here, for simplicity we consider the 1-dimensional case andx j andp j are quantum operators satisfying the usual commutation relations [x j ,p l ] = iδ j,l and we assume that the macroscopic d.o.f is subject to a constant external force, i.e. −∂ λ V (λ) = F 0 . The basic expectation is that, due to the interaction between the macroscopic d.o.f. and the QOs, the macroscopic d.o.f. will drag the QOs with itself as it moves and as a consequence its mass will be renormalized. This expectation can be confirmed by analyzing the evolution of the center of mass of the system which, by definition, is determined only by the external forces m ef fẌcm (t) = F 0 , where m ef f = µ + N m is the total mass. The position of the macroscopic d.o.f. is the vector sum of the position of the center of mass and the relative coordinate (which will in general have an oscillatory behavior) λ(t) = X cm (t) + X λ (t). We therefore conclude that, up to the oscillations of the relative coordinate, the macroscopic d.o.f. evolves with the renormalized mass m ef f = µ + N m which can be several times larger than its bare mass µ. If the coupling between oscillators is zero, U = 0, the dynamics of the macroscopic d.o.f. can be solved exactly and it can be verified that our argument is correct and that the oscillations of the relative coordinate have frequency Ω ∼ N 1/2 and amplitude B ∼ N −1/2 , which can be neglected for N 1. We now show how this behavior is reproduced using the emergent Newton's law Eq. (13). Let's first analyze the case of N independent Harmonic Oscillators (HO), i.e. U = 0. Then the drag term η is zero since the harmonic oscillators have a discrete spectrum and there are not gapless excitations [33]. Moreover the extra force term M λ ≡ n ρ n ∂ λ n is also zero since n = ω(n + 1 2 ) is independent on λ. Therefore we only need the mass tensor κ which can be computed via Eq. (26) where M λ ≡ −∂ λ H = k(x − λ). Expressing the position via the ladder operatorsx = 1 2mω (a + a † ) and using the standard properties a\|n = √ n\|n−1 and a † \|n = √ n + 1\|n+1 together with the thermal occupations ρ n = e −βωn 1 − e −βω it is straightforward to show that κ = m. Since there are N independent HOs each contributing equally to the renormalization of the mass Eq. (13) becomes: (µ + N m)λ = F 0(31) in perfect agreement with our discussion above. We can extend the above analysis to arbitrary interaction by using the equipartition theorem (see Eq. (26)): κ = β ( j p j )( l p l ) 0 = β p 2 tot 0 = N m where p tot = j p j is the total momentum of the oscillators. Therefore even when U = 0 we find that the renormalized mass is m ef f = µ + N m. Mass renormalization in a quantum piston. Let us consider a massless spring connected to the potential wall. In this section we will explicitly insert all factors of . We also imagine that a quantum particle of mass m is initially prepared in the ground state of the confining potential (see panel "a" in Fig. 2). As in the previous example we will compute how the mass of a classical object (the spring) coupled to a quantum environment (the potential well) is renormalized. According to Eq. (27) the mass renormalization is given by κ R = 2 2 n =0 \| n λ \|M\|0 λ \| 2 ( n − 0 ) 3 ,(32) where λ = X R is the position of the right potential wall. We approximate the confining potentail as a very deep square well potential. Then M ≡ −∂ λ H = −V δ(x − X R ) and we find κ R = 2 2 n =0 V 2 \|ψ 0 (X R )\| 2 \|ψ n (X R )\| 2 ( n − 0 ) 3 .(33) Using the well known result for a finite (and deep) square well potentail \|ψ n (X R )\| = 2 L n V where the factor of 2/L comes from the normalizazion of the wave-function in a square potential of lenght L, we obtain κ R = 2 2 2 L 2 n =0 0 n ( n − 0 ) 3 .(34) Substituting n = 2 k 2 n 2m , k n = n + 1 L π, ∀n ≥ 0 we arrive at κ R = m 16 π 2 n≥1 (n + 1) 2 [(n + 1) 2 − 1] 3 = m 2π 2 − 3 6π 2 ≈ 0.28m The result is identical if we connect the piston to the left wall, i.e. κ L = κ R . Now let us consider a slightly different setup where the spring connects to the whole potential well (see panel "b" in Fig. 2) so that λ now indicates the center of mass of the well. From the Galilean invariance we expect κ = m. In fact, since now both potentials walls are moving, our expression gives M = −∂ λ H = −V (δ(x − X R ) − δ(x − X L )), where X L and X R are the left and right positions of the walls. Thus using Eq. (32) we obtain κ + = 2 2 n =0 V 2 (ψ 0 (X L )ψ n (X L ) − ψ 0 (X R )ψ n (X R )) 2 ( n − 0 ) 3(35) Since in a symmetric potential well ψ n (X R ) = (−1) n ψ n (X L ) only the odd terms contribute in the equation above. Following the same line of reasoning as before we arrive at (note the extra factor of 4 with respect to Eq. (34)) κ + = 2 2 2 L 2 4 n=odd 0 n ( n − 0 ) 3 = m 64 π 2 n=odd (n + 1) 2 [(n + 1) 2 − 1] 3 = m So indeed we recover the expected result. This simple calculation illustrates that indeed we can understand the notion of the mass as a result of virtual excitations created due to the acceleration of the external coupling (position of the wall(s) in this case). If instead we analyze the setup where the two walls are connected to a spring and move towards each other so that λ is the (instantaneous) length of the potential well we find κ − = 2 2 n =0 V 2 (ψ 0 (X L )ψ n (X L ) + ψ 0 (X R )ψ n (X R )) 2 ( n − 0 ) 3 = m 64 π 2 n=even n =0 (n + 1) 2 [(n + 1) 2 − 1] 3 = m π 2 − 6 3π 2 ≈ 0.13m.(36) b) a) Figure 2: (Color on-line) Schematic of a quantum piston. a) The spring is connected to a wall of the potential in which a quantum particle of mass m is initially confined into the ground state. b) As in a) but now the spring is connected to the whole potential well which moves rigidly. The wave-lines in a) and b) represent the low energy wavefunctions of the quantum particle in the confining potential. Let us point another peculiar property of the mass. Clearly κ L + κ R ≈ 0.56m = κ + , κ − , i.e. the mass renormalization of the two walls is not the same as the sum of the mass renormalization of each wall measured separately. This is the result of quantum interference, which is apparent in Eqs. (35) and (36). Note that (κ + + κ − )/2 = κ L + κ R . Thus the mass behaves similarly to the intensity in the double pass interferometer, where the sum of intensities in the symmetric and antisymmetric channels is conserved. At a high temperature or in the classical limit the interference term will disappear and we will find κ L = κ R ≈ 0.5m. Energy absorption of a spin in a rotating magnetic field. As a next example we consider a system of independent spin-1 2 in a rotating magnetic field in the xz plane. Here we assume that the magnetic filed is an external parameter whose dynamics is given a priori. First we assume that these spins are isolated and later add a weak coupling to a bath. The Hamiltonian of each of the spins reads H (λ(t)) = −∆ (cos(λ) σ z + sin(λ) σ x ) + H SB ,(37) where H SB is the part of the Hamiltonian representing possible coupling to the bath and which does not depend on the angle λ. The instantaneous eigenstates change during the dynamics described by the protocol λ(t) while the energy levels remain unchanged. The eigenstates of this Hamiltonian (excluding the bath) are trivially \|gs λ = cos λ 2 sin λ 2 , \|ex λ = sin λ 2 − cos λ 2(38) corresponding to the energies E gs = −∆ and E ex = ∆ respectively. It is straightforward to compute the mass κ using Eq. (26): κ = tanh(β∆) 4∆(39) In the high temperature limit κ ≈ β/4, which is indeed a product of the inverse temperature and the single component metric tensor, also known as the fidelity susceptibility of the spin. Because this expression is nonsingular, a small coupling to the bath can at most introduce small corrections to κ. However, this is not the case with dissipative coefficient η, which clearly vanishes in the long time limit without the bath because there are no gapless excitations. For typical coupling to the bath the transverse spin-spin correlation functions entering Eq. (24) oscillate with frequency 2∆ and decay with the relaxation time τ c set by the bath [34,35] so η = ∆ tanh(β∆) ∞ 0 dt cos(2∆t ) exp − t τ c = tanh(β∆) ∆τ c 1 + (2∆τ c ) 2(40) Within the same approximation the mass κ is also modified: κ = ∆ tanh(β∆) ∞ 0 dt (−t ) cos(2∆t ) exp − t τ c = tanh(β∆) ∆τ 2 c (2∆τ c ) 2 − 1 ((2∆τ c ) 2 + 1) 2 (41) which reduces to Eq. (39) for τ c ∆ 1. In the high temperature limit β∆ 1 this expression for the mass is equal to the inverse temperature times the metric tensor but now of the full system including bath degrees of freedom. These expressions Eq.(40) and (41) do not apply for τ c < ∆ −1 since the simple approximation to the exponentially decaying correlation function breaks in this limit due to the Zeno effect. Therefore we only consider the situation ∆τ c 1. In Fig. 3 we show the spin energy obtained by numerically integrating the expressionQ(t) =λ(t) t 0 dt λ (t − t ) cos(2∆t ) exp[−t /τ c ] for the protocolλ(t) = v 0 tanh 2 (t/τ v ). Note thatλ(t) starts smoothly and approaches the constant value v 0 for t 2τ v . For t < ∆ −1 the heat Q(t) is well approximated by the short time expansion (see Eq. (30)) while for ∆ −1 < t t * = κ/η ≈ τ c we observe a plateau in agreement with the general discussion in 3.1. (11)). The horizontal (dotted red) line is the (constant) contribution from the mass renormalization. (Inset) At short time, t < ∆ −1 the heat is well approximated by Q(t) = ∆ tanh(β∆) (δλ(t)) 2 2 (see Eq. 30) (red dashed line). In both panels ∆ = 1, τ v = 10, τ c = 10 3 so that the condition τ c τ v > ∆ −1 is well satisfied and the the plateau is well visible. Central spin model. Extracting the Chern number from the Coriolis force. As a third example let us consider a macroscopic rotator (angular momentum) interacting with independent spin-1 2 particles. If instead of the rotator we use the quantum spin operator, this model is known as the central spin model. Because dynamics of a spin in any external field is always classical (the evolution of the Wigner function is exactly described by classical trajectories) there is no difference between quantum and classical dynamics in this case and we do not need to assume that the rotator is macroscopic. The Hamiltonian describing the system is (5) with H 0 = L 2 2I + V ( n), H = − n · N i=1 ∆ i σ i ,(42) where I is the momentum of inertia, V ( n) is a time-dependent external potential, L is the angular momentum and n is the three-dimensional unit vector which can be parameterized using spherical angles: n = (n x , n y , n z ) = (sin θ cos φ, sin θ sin φ, cos θ). This example is similar to the previous one except that the effective magnetic field is no longer confined to the xz plane and we no longer assume that it is given by an external protocol. The time evolution of this system needs to be found self-consistently. On the one hand, each spin evolves according to the von Neumann equation with the time-dependent Hamiltonian H( n(t)): i∂ t ρ = [H( n(t)), ρ](43) and, on the other hand, the rotator evolves according to the Hamilton equations of motion I˙ n = L × n,˙ L = n × M ext + − ∂H ∂ n = n × M ext + i ∆ i σ i (44) where M ext = − ∂V ( n) ∂ n is the external force and . . . indicates the quantum average over the density matrix ρ(t) (see Eq. (43)). We assume that initially n 0 = (0, 0, 1) and the spins are in thermal equilibrium with respect the Hamiltonian H( n 0 ), i.e. σ We now compare the exact dynamics with the emergent Newton's law. First, we note that the form of the equations (44) immediately implieṡ n · L = 0, n ·˙ L = 0 → n · L = consṫ n · n = 0 → \| n\| 2 = const,¨ n · n = −\|˙ n\| 2 (45) Next we need to compute the generalized force M = − ∂H ∂ n , and the tensors κ and F . The drag term η and the anti-symmetric mass F are zero since there are no gapless excitations. Therefore Eq. (15) reduces to: M = M 0 + F ν,µṅµ − κ ν,µnµ where ν, µ = x, y, z. The ground and excited states of each spin-1 2 are: \|gs i θ,φ = cos θ 2 e −iφ/2 sin θ 2 e +iφ/2 , \|ex i θ,φ = sin θ 2 e −iφ/2 − cos θ 2 e +iφ/2(46) with energy ±∆ i respectively from which it follows M 0 = 0, F = F 0   0 n z −n y −n z 0 n x n y −n x 0   , κ = κ 0   1 − n 2 x −n x n y −n x n z −n y n x 1 − n 2 y −n y n z −n z n x −n z n y 1 − n 2 z   . where F 0 ≡ 1 2 i tanh(β∆ i ) and κ 0 ≡ i tanh(β∆ i ) 4∆ i . Substituting these expressions in Eq. (44) we find I˙ n = L × n,˙ L = n × M ext + F 0˙ n − κ 0 ( n ×¨ n) where we have used standard properties of the vector triple product together with Eqs. (45). In the equations above we can substitute L → L ⊥ where by definition L ⊥ = L−( L· n) n. We now compute I¨ n =˙ L ⊥ × n+ L ⊥ ×˙ n and using standard properties of the vector triple product together with L ⊥ = I ( n ×˙ n) and Eqs. (45) we arrive at: I ef f¨ n = n × M ext × n + F 0 (˙ n × n) − I ef f \|˙ n\| 2 n(47) where we have defined the renormalized momentum of inertia is I ef f = I +κ 0 . From this equation we see that the moment of inertia of the rotator is renormalized by the interaction with the spin-1 2 particles. Moreover we see that, even when the external force is absent ( M ext = 0), the Berry curvature (F 0 ) causes the Coriolis type force tilting the rotation plane of the rotator. Indeed if we start with uniform rotations of the rotator in the xz plane, i.e. n,˙ n lie in the xz plane, we immediately see that the Berry curvature causes acceleration orthogonal to the rotation plane. The physics behind the Coriolis force is intuitively simple. At any finite rotator's velocity, the spins will not be able to follow adiabatically the rotator and thus will be somewhat behind. As a result there will be a finite angle between the instantaneous direction of the spins and the rotator so the spins will start precessing around the rotator. This precession will result in a finite tilt of the spin orientation with respect to the rotation plane proportional to the Berry curvature [23]. In turn this tilt will result in precession of the rotator around the spins and cause the tilt of the rotation plane. It is interesting that despite the motion of the rotator is completely classical the Coriolis force given by the Berry curvature is quantum in nature. In particular, at zero temperature the integral of the Berry curvature over the closed manifold (4π spherical angle is this case) is quantized. Therefore by measuring the Coriolis force and averaging it over the angles θ and φ one should be able to accurately see a quantized value: π 0 dθ 2π 0 dφ F θ,φ = 2πN.(48) It is interesting that this result remains robust against any small perturbations in the system, which do not close the gap in the spectrum. Similarly the origin of the extra mass (moment of inertia) κ in this simple example is purely quantum, i.e. despite this mass describes the classical Newtonian motion, it can not be computed within the classical framework. We now analyze the approximated Equation (47) in detail. We consider the situation in which M ext (t) is slowly turned on (off) at time t = 0 (t = t c ). When M ext (t) = 0 Eq. (47) describes a uniform circular motion whose solution can be written as: n ap (t) = 1 − A 2 apr + A ap [cos(ω ap t + φ)ĉ 1 + sin(ω ap t + φ)ĉ 2 ](49) where A ap is the amplitude, ω ap is the angular frequency,r is the vector orthogonal to the plane of motion (see Fig. 4) andĉ 1 ,ĉ 2 are two orthogonal unit vectors spanning the plane of motion. Substituting the ansatz (49) in Eq. (47) (with M ext = 0) we obtain ω ap = − F 0 I ef f 1 − A 2 ap . (50) The amplitude and the orientation of the plane of motion can not be computed from the initial conditions of Eq. (47) since this equation is only valid after a transient time. We therefore extract them from the exact numerical solution: 1 − A 2 apr = lim T →∞ 1 T tc+T tc dt n ex (t ) and compute ω ap via Eq. (50). In Fig. 4 we compare the exact trajectory n ex (t) obtained by numerically solving Eq. (43) and Eq. (44) for the rotator coupled to N = 20 spins with gaps ∆ i uniformly distributed between (1,2) to the approximated trajectory n ap (t) (49). We note that the frequency of the approximated motion (estimated via Eq.(50)) underestimates the exact frequency by 8% however if we had used the bare momentum of inertia in (50) with the chosen simulation parameters (I ef f /I ≈ 1.5) we would have overestimated the exact frequency by 28%. The accuracy will be higher if we increase the number of spins N . Conclusions In this paper we showed how macroscopic Newtonian dynamics for slow degrees of freedom coupled to an arbitrary system emerges in the leading order of expansion about the adiabatic limit. Our results apply to open and closed systems, quantum and classical. In particular, we found closed microscopic expressions for the friction tensor and for the mass tensor due to dressing of the macroscopic parameter with excitations. We showed that the mass tensor is directly related to the changing geometry of the Hilbert space of the system due to the adiabatic motion of the degree of freedom. In this sense its origin is similar in nature to the origin of the dynamical Casimir force [36,37]. At a high temperature (or in the classical) limit the mass term becomes equal to the product of the inverse temperature and the Fubini-Study metric tensor describing the system. While we focused on coupling to ergodic systems, which equilibrate at finite temperatures in the absence of motion, our results are more general. In particular, they will apply to non-ergodic, e.g. integrable or glassy systems, as long as they reach some steady state (or approximately steady state). In this case the mass and the friction will be different from the equilibrium values. Thus the mass of the object coupled to a weakly non-integrable system can change in time as the latter slowly equilibrates. Our results are non perturbative in the coupling between the macroscopic parameter and the (classical or quantum) system and can therefore be applied to situations in which mass renormalization of the macroscopic parameter is large [38]. This is in contrast with the standard many-body Green function approach [19] in which the real and imaginary part of the self energy (which are responsible of the mass renormalization and dissipation respectively) are perturbative in the coupling. Moreover our integral equations (8) and (14) contain non-Markovian effects and are valid even when the relaxation time in the system is long. Our results are based on the adiabatic perturbation theory which is based on a separation of time-scales between the dynamics of the (slow) macroscopic parameter and the (fast) dynamics of the system [2]. Therefore our approach is different and complementary to the usual Born-Markov scheme of open quantum systems [7,8,9]. Acknowledgments We gratefully acknowledge M. Kolodrubetz and E. Katz for many stimulating conversations and M. V. Berry for providing valuable feedback to this work and for pointing to us Ref. [27]. This work was partially supported by BSF 2010318, NSF DMR-0907039, AFOSR FA9550-10-1-0110 A. Appendix: Microscopic derivation of Eqs. (8) and (14) Let us show the details of the derivation of Eq. (8) and Eq. (14). We start from the generic Hamiltonian H(λ) with eigenstates \|m λ . Both the Hamiltonian and the eigenstates depend on time through the parameter λ(t) which can be multicomponents. For brevity we use the notation H(t) ≡ H(λ(t)) and \|m t ≡ \|m λ(t) . Our goal is to compute the evolution of the density matrix ρ(t) subject to the initial condition [ρ 0 , H(λ 0 )] = 0, i.e. the initial density matrix is stationary with respect the initial Hamiltonian. In the course of our derivation we will implement a sequence of two unitary transformations. First we make a unitary (time-dependent) transformation, R(λ), from the fixed (lab) frame to a new frame which is co-moving with the Hamiltonian. In the following we reserve the tilda sign to the quantities in the co-moving frame. In the co-moving frame the HamiltonianH(λ) ≡ R † (λ)H(λ)R(λ) assumes a diagonal form and its eigenstates \|m are λ-independent: n λ \|H(λ)\|m λ = ñ\|H(λ)\|m = n (λ) δ n,m . (A.1) where \|m λ ≡ R(λ)\|m . We note that \|m = \|m λ 0 where \|m λ 0 is the static basis, which diagonalizes the Hamiltonian at the initial value of λ = λ 0 . In the co-moving frame the wave-function is \|ψ = R † (λ)\|ψ and the density matrix is:ρ = R † (λ)ρR(λ). We assume that the dynamical process starts at t = 0 so that R(0) is the identity operator and thus ρ 0 andρ 0 coincide (this condition can be relaxed). The Hamiltonian which governs the time-evolution in the co-moving frame isH ef f :H ef f ≡H −λ νÃν (A.2) The first termH is diagonal and it is therefore only responsible for phase accumulation. The second "Galilean" type term (see e.g. Ref. [30]) originates from the fact that the basis transformation, R(λ) is time-dependent and plays the role of translation operator in the parameter space [17]. The operator A ν (λ) ≡ iR † (λ)∂ λν R(λ) (A.3) is a gauge potential. Clearly it can be also written as A ν = i∂ λν in a sense that i n λ \| − → ∂ λν \|m λ = ñ\|Ã ν (λ)\|m . (A.4) Gauge potentials are generally very complicated many-particle operators. For systems coupled to a bath they involve both the system's and bath's degrees of freedom as well as the system-bath interactions. In the co-moving basis this term has off-diagonal components and is responsible for the transition between the energy levels. Before proceeding let us illustrate this formalism with the example used in the main text (see Sec. 5.3). The Hamiltonian in the fixed (lab) frame is: H (λ) = −∆ (cos(λ) σ z + sin(λ) σ x ) with eigenstates \|gs λ = cos λ 2 sin λ 2 , \|ex λ = − sin λ 2 cos λ 2 The unitary transformation, which diagonalizes the instantaneous Hamiltonian, H(λ), is the rotation around the y-axis by the angle λ: R(λ) = exp[−i σ y 2 λ] = cos (λ/2) − sin (λ/2) sin (λ/2) cos (λ/2) Note that R(λ 0 = 0) = Identity. In the rotated (co-moving) frame we have: H ≡ R † (λ)H(λ)R(λ) = −∆σ z which is diagonal and (in this case) λ-independent. The eigenstates are trivially: \|gs = 1 0 , \|ẽx = 0 1 which are identical to the eigenstates at λ 0 = 0: \|gs = \|gs λ 0 , \|ẽx = \|ex λ 0 ) Finally the Gauge potential is: A ≡ iR † (λ) (∂ λ R(λ)) = σ y 2 , which in this example is also λ-independent. i dρ H dt = −λ ν Ã H,ν (t),ρ H , (A.5) which is equivalent to the integral equatioñ ρ H (t) =ρ H (0) + i t 0 dt λ ν (t ) Ã H,ν (t ),ρ H (t ) Note that if the moving HamiltonianH is time dependent (which is the case when the energy spectrum depends explicitly on λ, see Eq. (A.1)) the Heisenberg picture is different from the conventional one. But becauseH is always diagonal in the co-moving basis the difference appears only in the phase factor: ñ\|Õ H (λ(t))\|m = exp i t 0 dt ( n (t ) − m (t )) ñ\|Õ(λ(t))\|m = exp i t 0 dt ( n (t ) − m (t )) n t \|O(λ(t))\|m t (A.6) which follows from the definitioñ O H (t) ≡ e i t 0 dτH(τ ) R † (t)O(t)R(t) e −i t 0 dτH(τ ) We emphasize that this expression is not the same as the Heisenberg representation with respect to the original Hamiltonian H(λ(t)). The representation we use is perhaps more correctly termed as the adiabatic Heisenberg representation since it uses adiabatic energy levels n (t), while all the transitions (off-diagonal terms) are treated perturbatively. Clearly if n is time independent the expression (A.6) reduces to the conventional Heisenberg representation of the operator O(λ(t)) which can depend on time through the parameter λ(t). We can now solve Eq. (A.5) iteratively to the second order inλ: ∂ tρH (t) = iλ ν (t) Ã H,ν (t), ρ 0 −λ ν (t) t 0 dt λ µ (t ) Ã H,ν (t), Ã H,µ (t ), ρ 0 + O(\|λ\| 3 ), where we usedρ 0 H =ρ 0 = ρ 0 . To derive Eq. (8) we analyze the energy generation rate ∂ t H = ∂ t Tr ρ H (t)H(t) =Ẇ (t) +Q(t), whereẆ (t) = Tr ρ H (t) (∂ tH ) ,Q(t) = Tr (∂ tρH )H(t) The adiabatic work rate, related to the change of the spectrum in the Hamiltonian, in turn can be rewritten aṡ W (t) = −λ ν M ν + O \|λ\| 3 , where M ν = − n n 0 \|ρ 0 \|n 0 (∂ λν n (λ)) ≡ − ∂ λν H 0 ≡ M ν 0 is the generalized force with respect the initial (stationary) density matrix and we have defined M ν ≡ −∂ λν H. For an initially stationary density matrix, [ρ 0 , H(λ 0 )] = 0, it is easy to see that ρ 0 , ∂ tH = 0 (recall thatH(t) is diagonal for any time t) and there is no second order contribution inλ to the work rate. In general we find that for an initially stationary density matrix the work rate (heat rate) is odd (even) inλ. Then the leading contribution to the heat rateQ , related to the change of occupation of the instantaneous eigenstates of the Hamiltonian, is quadratic inλ and we finḋ Q(t) = −λ ν (t) t 0 dt λ µ (t ) × Tr H (t) Ã H,ν (t), Ã H,µ (t ), ρ 0 =λ ν (t) t 0 dt λ µ (t ) Ã H,ν (t),H(t) ,Ã H,µ (t ) 0 (A.7) Note that in this expression the HamiltonianH(t) depends on time only through the spectrum and it is diagonal for any time t. From the definition ofÃ ν ≡ iR † (∂ λν R) andH ≡ R † HR it follows that: Ã ν (t),H(t) = −iM ν − i∂ λνH (A.8) whereM ν ≡ R † M ν R ≡ −R † (∂ λν H) R is the generalized force in the co-moving basis and we have used the identity (∂ λν R) R † = −R ∂ λν R † . Eq. (A.8) naturally follows from the interpretation of the gauge potential A ν as the (negative) momentum operator along the direction λ ν : A ν = i∂ λν (see Eq. (A.4)). The second term in the RHS of Eq. (A.8) is diagonal in the co-moving basis and does not contribute to Eq. (A.7), which then simplifies tȯ Q = −iλ ν (t) t 0 dt λ µ (t ) M H,ν (t),Ã H,µ (t ) 0 (A.9) And finally, evaluating this expression in the co-moving basis (see Eq. (A.6)), and using the identity ñ\|Ã µ (λ)\|m = i n λ \|∂ λµ \|m λ = −i n λ \|M µ (λ)\|m λ m (λ) − n (λ) = −i ñ\|M µ (λ)\|m m (λ) − n (λ) (A.10) we arrive aṫ Q(t) =λ ν (t) t 0 dt λ µ (t ) × n =m ρ 0 n − ρ 0 m m (t ) − n (t ) e i t t dτ ( m(τ )− n(τ )) m t \|M ν (λ(t))\|n t n t \|M µ (λ(t ))\|m t =λ ν (t) t 0 dt λ µ (t ) × n =m ρ 0 n − ρ 0 m m (t ) − n (t ) e i t t dτ ( m(τ )− n(τ )) m\|M ν (λ(t))\|ñ ñ\|M µ (λ(t ))\|m =λ ν (t) t 0 dt λ µ (t ) n =m ρ 0 n − ρ 0 m m (t ) − n (t ) m\|M H,ν (t)\|ñ ñ\|M H,µ (t )\|m =λ ν (t) t 0 dt β 0 dτλ µ (t ) n =m ρ 0 m m\|M H,ν (t)\|ñ ñ\|M H,µ (t + iτ )\|m =λ ν (t) t 0 dt β 0 dτλ µ (t ) M H,ν (t)M H,µ (t + iτ ) 0 , (A.11) Eq. (A.11) gives the microscopic heat production rate in the most general form and in the last two lines we have used the fact that, for thermal distri- which gives the microscopic force in the most general form. In Eqs. (A.11) and (A.14) we now perform a change of dummy integration variable t → t − t and observe that the leading order in \|λ\| corresponds to evaluate the spectrum, the eigenstates and the force at the final time of the evolution, i.e. at t = 0. This is because the energies, the forces and the eigenstates depend on time only through the parameter λ(t) and can therefore be expanded in powers of \|λ\|. For example m (t − t ) ≡ m (λ(t − t )) = m (t) − t λ (t)∂ λ m + · · · ≈ m (t) We then obtain the leading contributions: Q(t) =λ ν (t) t 0 dt λ µ (t − t ) × n =m e i( m(t)− n(t))t ρ 0 n − ρ 0 m m (t) − n (t) m t \|M ν (λ(t))\|n t n t \|M µ (λ(t))\|m t (A.15) M ν (t) = M ν (t) − t 0 dt λ µ (t − t ) × n =m e i( m(t)− n(t))t ρ 0 n − ρ 0 m m (t) − n (t) m t \|M ν (λ(t))\|n t n t \|M µ (λ(t))\|m t (A. 16) which for time independent spectrum, m (λ(t)) = m , and time independent force, M µ (λ(t)) = M µ can be rewritten as the equations in the main text (see Eqs. (8) and (14)): Q(t) =λ ν (t)= iπ∂ ω G ν,µ \| ω=0 + P +∞ −∞ dω S ν,µ (ω) ω 3 = iπ 2 (∂ ω G ν,µ \| ω=0 − ∂ ω G µ,ν \| ω=0 ) + 1 2 P +∞ −∞ dω ω 3 (S ν,µ (ω) + S µ,ν (ω)) where P indicates the principal value, used the symmetry of S (see Eq. (B.6)) and we have defined the auxiliary function G ν,µ (ω) ≡ Sν,µ(ω) ω = G µ,ν (−ω). We note that: F ν,µ ≡ i ← − ∂ λν − → ∂ λµ − ← − ∂ λµ − → ∂ λν 0 and that when the occupations are thermal we can write: Finally we note that both χ 1 and χ 2 have a symmetric (under ν ↔ µ) and anti-symmetric terms. For χ 1 the symmetric part is on-shell and the anti-symmetric part is off-shell. The situation is opposite for χ 2 . In general, the on-shell terms describe dissipation and vanish for gapped system and zero temperature while the off-shell terms describe the renormalization of the systems' parameters. For system with time-reversal symmetry all antisymmetric contributions vanish [25] and, as a result, all odd susceptibilities are dissipative and on-shell while all the even susceptibilities are off-shell and describe renormalization of systems' parameters. These findings generalize the result of Berry and Robbins [2]. Finally we note that all susceptibilities change sign with the temperature and that for positive temperature (β > 0) the macroscopic degrees of freedom loses energy to the environment in all orders in the derivatives of λ(t). Figure 1 : 1(Color on-line) Schematic setup considered. A macroscopic d.o.f. is initially at equilibrium (light red ball) with a complex environment (indicated by the wavebackground). Then it starts moving with velocityλ through the environment, which has to rearrange, and generates both a drag force and a renormalization of the mass of the d.o.f.. The drag coefficient is related to real transitions between eigenstates of the environment (see Eq. (24)) while the mass is related to virtual transitions which describe dressing of the d.o.f. by the virtual exvitations of the environment (see Eq.(26)). Figure 3 : 3(Color on-line) Rescaled heat, ξ(t) ≡ tanh(β∆) , for the protocolλ(t) discussed in the main text. (Main graph) At long time, t τ v , ∆ −1 ,the heat is well approximated by the sum of the mass renormalization and dissipative contributions (dashed blue line) (see Eq. x i 0 0= σ y i 0 = 0 and σ z i 0 = tanh(β∆ i ). For the toy model proposed here these coupled equations can be easily solved numerically. In fact, according to the Ehrenfest theorem, the evolution of the expectation values follow the classical equation of motion and the von Neumann equation (43) can be replaced with the much simpler equatioṅ m i = 2∆ i m i × n where m i = σ i . Therefore the exact dynamics of the system consists of the vectors ( L, n, { m i }) precessing around each other. Figure 4 : 4(Color on-line) Dynamics of a rotator coupled to N = 20 spins-1 2 . (Left panel) The exact trajectory of the rotator obtained by numerically solving Eq. (43) and (44) (blue line) coincides with the approximated trajectory Eq. (49) which delimits the shaded red plane. (Right panel) Behavior of n z (t) for the exact (continuous blue) and approximated evolution (dashed red) versus time. The parameters of the exact simulations are: N = 20, β = 0.1, I = 1, ∆ i are randomly distributed in(1,2). For these parameters F 0 ≈ 1.5, κ 0 ≈ 0.5 and I ef f /I ≈ 1.5. The initial conditions are n 0 = (0, 0, 1) and L 0 = (0, 0, 0) and initially the spins are in thermal equilibrium (see main text). We chose the time-dependent external force to be M ext (t) = 250 t tc 1 − t tc 4x for 0 ≤ t ≤ t c = 3 and zero otherwise. of Eq.(14) is even simpler since it requires only going to the first order perturbation theory:ρ H (t) ≈ ρ 0 + i t 0 dt λ µ (t ) Ã H,µ (t ), ρ 0 . (A.13) Therefore M ν (t) = M ν (t) + i t 0 dt λ µ (t ) [M H,ν (t),Ã H,µ (t )] 0 ,where we recall that by definitions M ν (t) ≡ M ν (t) 0 . Evaluating this expression in the co-moving basis (see Eq. (A.6)) and using the identity (A.10) we arrive atM ν (t) = M ν (t) dτ ( m(τ )− n(τ )) m t \|M ν (λ(t))\|n t n t \|M µ (λ(t ))\|m t = M ν (t) dτ ( m(τ )− n(τ )) m\|M ν (λ(t))\|ñ ñ\|M µ (λ(m\|M H,ν (t)\|ñ ñ\|M H,µ (t + iτ )\|m = M ν (t) µ (t ) M H,ν (t)M µ (t + iτ ) 0 , (A.14) µµ( (t − t ) M H,ν (t )M H,µ (iτ ) (t − t ) M H,ν (t )M H,µ (iτ ) 0 (A.17) where we have used the fact that, for thermal distribution, ρ 0 m = Z −1 exp[−β m The expressions (A.17) are correct even if the spectrum and force are time dependent but vary slowly on the scale of the relaxation time in the system. e iωt {M H,ν (t), M H,µ (0)} 0 = n =m δ( n − m − ω)(ρ 0 n + ρ 0 m ) m\|M ν \|n n\|M µ \|m which satisfy the symmetry relation Φ ν,µ (ω) = Φ µ,ν (−ω).Finally using the identities (B.4) we obtain: ν \|n n\|M µ \|m δ( n − m ) n − m ) 2 m\|M ν \|n n\|M µ \|m = πG ν,µ (ω = 0) ν \|n n\|M µ \|m δ ( n − m − n ) 3 m\|M ν \|n n\|M µ \|m F m\|M ν \|n n\|M µ \|m δ( n − m ) = 2 m\|M ν \|n n\|M µ \|m − β ∂ ∂( n − m ) ( m\|M ν \|n n\|M µ \|m ) δ( n − m ). Then Eq. (A.4) is verified by direct inspection. Next, to derive Eq. (8) and Eq. (14), we solve the von Neumann's equation using standard time-dependent perturbation theory with the Galilean term being the perturbation. The easiest way to do so is to go to the interaction picture (the Heisenberg representation with respect toH), where the von Neumann's equation becomesi dρ dt = H −λ νÃν ,ρ For this paper we put aside the question of what happens when the system is macroscopic and ergodic so that the level spacings are exponentially small. Here we analyze the full Taylor expansion of Eq.(8)and and Eq.(14), which we rewrite in the Lehmann's representation (see Eq. (A.16)). In order for the following expansion to be valid both the velocity and the energy spectrum need to change slowly compared to the relaxation time scale on the system, i.e. the relaxation time must be the shortest time scale. For times t longer than the relaxation time we can set the upper limit of integration in Eq. 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However, one can show that these corrections are of the order of \|˙ λ\| 3. which is beyond the accuracy considered in this paperBecause we are taking the expectation value with the time- dependent density matrix ρ n (t) there will be corrections to the generalized force. However, one can show that these corrections are of the order of \|˙ λ\| 3 , which is beyond the accuracy considered in this paper. G Mahan, Many-Particle Physics. Springer3rd editionG. Mahan, Many-Particle Physics, 3rd edition (Springer, 2000). The minus signs are conventional. The sign of F is chosen to reproduce known expressions for the Berry curvature. The minus signs are conventional. The sign of F is chosen to reproduce known expressions for the Berry curvature. . N Bode, S Viola, R Kusminskiy, F Egger, Von Oppen, Phys. Rev. Lett. 10736804N. Bode, S. Viola Kusminskiy, R. Egger and F. von Oppen, Phys. Rev. Lett. 107, 036804 (2011) . M Thomas, T Karzig, S Viola, G Kusminskiy, F Zarand, Von Oppen, Phys. Rev. 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[ "The effect of gluon condensate on imaginary potential and thermal width from holography", "The effect of gluon condensate on imaginary potential and thermal width from holography" ]	[ "Yan-Qing Zhao \nCollege of Science\nThree Gorges University\n443002YichangChina, China\n", "Zhou-Run Zhu \nInstitute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOS)\nCentral China Normal University\n430079WuhanChina\n", "Xun Chen \nInstitute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOS)\nCentral China Normal University\n430079WuhanChina\n" ]	[ "College of Science\nThree Gorges University\n443002YichangChina, China", "Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOS)\nCentral China Normal University\n430079WuhanChina", "Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOS)\nCentral China Normal University\n430079WuhanChina" ]	[]	By the use of the gauge/gravity duality, we calculate the imaginary part of heavy quarkonium potential and thermal width with the effect of gluon condensate which is absent in AdS 5 background. Our results show that the dropping gluon condensate reduces the absolute value of imaginary potential and therefore decreases the thermal width both in "exact" and "approximate" approach implying that the heavy quarkonium has a weaker bound with the increase of gluon condensate. In addition, the thermal width will disappear at a critical condensate value, which indicates the dissociation of quarkonium. We conclude that increasing gluon condensate will lead to easier dissociation of heavy quarkonium for fixed temperature.PACS. PACS-key 11.25.Tq, 25.75.Nq	10.1140/epja/s10050-020-00072-5	[ "https://arxiv.org/pdf/1909.04994v3.pdf" ]	202,558,797	1909.04994	2ac0351b49ca2c519f42bb701bc5fa15b5a38e62	The effect of gluon condensate on imaginary potential and thermal width from holography arXiv:1909.04994v3 [hep-ph] 10 Mar 2020 Yan-Qing Zhao College of Science Three Gorges University 443002YichangChina, China Zhou-Run Zhu Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOS) Central China Normal University 430079WuhanChina Xun Chen Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOS) Central China Normal University 430079WuhanChina The effect of gluon condensate on imaginary potential and thermal width from holography arXiv:1909.04994v3 [hep-ph] 10 Mar 2020Received: date / Revised version: dateEPJ manuscript No. (will be inserted by the editor)PACS PACS-key 1125Tq, 2575Nq By the use of the gauge/gravity duality, we calculate the imaginary part of heavy quarkonium potential and thermal width with the effect of gluon condensate which is absent in AdS 5 background. Our results show that the dropping gluon condensate reduces the absolute value of imaginary potential and therefore decreases the thermal width both in "exact" and "approximate" approach implying that the heavy quarkonium has a weaker bound with the increase of gluon condensate. In addition, the thermal width will disappear at a critical condensate value, which indicates the dissociation of quarkonium. We conclude that increasing gluon condensate will lead to easier dissociation of heavy quarkonium for fixed temperature.PACS. PACS-key 11.25.Tq, 25.75.Nq Introduction It is well-known that the experiments, RHIC and LHC, have found a new state of matter which is called as quark gluon plasma(QGP) produced by the heavy ion collisions [1,2]. Heavy quark-antiquark pair can be regarded as one of probes in the process of quark-gluon plasma(QGP) formation because the dissolution of quarkonium implies the occurrence of deconfinement phase transition [3,4]. We usually use heavy quarkonium potential V QQ to describe the interaction energy between quark and anti-quark. It is found that the potential may be in possession of a imaginary part at non-zero temperature, which is closely related to the decouple of heavy quarkonium, and it is believed that the dissolution of quarkonium is not because the binding energy disappear but because the reduced binding energy becomes as big as the thermal width which can be calculated by imaginary potential [5][6][7][8][9][10][11][12]. So far, there are two main mechanisms for quarkonium dissociation or the appearance of imaginary potential, one is from the Landau damping of approximately static fields [5,13,14] and the other is its color singlet to color octet thermal break up [15]. In the past few years, a lot of works about the imaginary part of the heavy quarkonium potential have been done in a weakly coupled theory [16][17][18][19]. However, QCD theory is a strong coupling theory. Gauge/gravity duality [20][21][22][23], breaking the a email: [email protected] b The first two authors contribute equally c email: [email protected] d email: [email protected] conformal symmetry at low energy, provides a very important tool to research the properties of hadron physics in a strong coupling systems [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]. Many scholars used holographic approach to study imaginary potential by considering various background like taking into account of chemical potential [44], they observed that the presence of the chemical potential decreases the dissociation length. In addition, many researches show that the process of heavy ion collisions will produce strong magnetic field [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59], so a numbers of papers based on this have been published, such as Ref. [60], which shows that increasing magnetic field enhances the imaginary potential and decreases thermal width. In Ref. [61], the energy loss of heavy and light quarks has been calculated in holographic model. Moving case was also taken into account in [62,63] which drew a conclusion that slowly moving quarkonium are less stable than static case. When we consider a full quantum theory of QCD, a nonzero the trace of the energy-momentum tensor is manifested, since there is an anomaly, implying a nonzero gluon condensate. One can calculate the gluon condensate by trace anomaly [64][65][66] as follows ∆ G 2 (T ) = G 2 (T ) − G 2 (0) = −(ε(T ) − 3P(T)),(1) where G 2 (T ) denotes the gluon condensate of limited temperature, G 2 (0), being equal to the condensate value at deconfinement transition temperature, is the condensate vale of zero temperature, ε(T ) is the energy density, p(T ) is the pressure of QGP system. However, recent lattice calculations based on a QCD sum rule method have shown that the gluon condensate behaves a rapid change around T c [67,68]. If we ignore shift in the width of the quarknia, this change will result in a reducing heavy quarkonium mass around T c . Therefore, one can find that the gluon condensate is quite sensitive to the deconfinement phase transition of QCD and could be regarded as the signal to study phase transition. A lot of works on gluon dynamics have been investigated in holographic model [31,38,39,[69][70][71][72][73][74][75][76]. Inspired by this, we research the impact of gluon condensate on imaginary potential and thermal width in this work. The heavy quark anti-quark potential, at finite temperature, can be drawn from the vacuum expectation value in Wilson loop operator [77][78][79]. From the standpoint of AdS/CFT correspondence, the value of W (C) in the limit of large N c of strongly coupled 4-dimensional gauge theory is dual to gravity in the 5-dimensional bulk geometry. So we can get the stringy partition function. The stringy partition function in the classical gravity approximation is approximately equal to the exponent of the imaginary number unit i multiplied by the classical string action, which could be regarded as the Nambu-Goto action. Then, the imaginary part of potential and thermal width can be calculated. In this paper, we will take saddle point approximation approach to study the effect of the gluon condensate on imaginary part of heavy quark-antiquark potential, which leads to some restrictive conditions. Besides, the calculating results show that there is a critical condensate value, where the thermal width will disappear implying the dissociation of quarkonium. The rest of this paper is organized as follows. In section 2, we briefly review the holographic model with the effect of gluon condensate. In section 3, the results of the impact of gluon condensate on imaginary potential and thermal width are displayed. In section 4, conclusion and further discussion are given. The holographic model To begin with, we briefly review the 5-dimensional gravity background in Minkowski space with a dilaton [80] S = 1 2k 2 d 5 x √ g R + 12 L 2 − 1 2 ∂ µ φ ∂ µ φ ,(2) where k is the 5-dimensional gravitational coupling, L is the radius of the asymptotic AdS 5 spacetime. φ is a massless scalar coupled with the gluon operator G µν G µν . By solving the above action, we can get dilaton equation of motion(EOM) and the Einstein equation. The solutions can be solved from EOM with a suitable metric ansatz. One is the dilaton-wall solution [80,81], ds 2 = L 2 z 2 ( 1 − c 2 z 8 (dx 2 + dt 2 ) + dz 2 ),(3)φ (z) = 3 2 log 1 + cz 4 1 − cz 4 + φ 0 ,(4) where φ 0 is a constant and x = x 1 , x 2 , x 3 are orthogonal spatial boundary coordinates. z denotes the 5th dimension radial coordinate. We assume the asymptotically AdS 5 boundary for gravity dual is at z = 0. On top of this, c = 1/z 4 0 . The value of 1/z 0 can be determined by the mass of the lowest meson [82] or lightest glueball [81]. The other metric is the dilaton black hole solution [27,83]. ds 2 = L 2 z 2 A(z)dx 2 − B(z)dt 2 + dz 2 ,(5) where A(z) = (1 + f z 4 ) ( f +a)/2 f (1 − f z 4 ) ( f −a)/2 f ,(6)B(z) = (1 + f z 4 ) ( f −3a)/2 f (1 − f z 4 ) ( f +3a)/2 f ,(7)f 2 = a 2 + c 2 ,(8) and the corresponding dilaton profile is φ (z) = c f 3 2 log 1 + f z 4 1 − f z 4 + φ 0 .(9) A Taylor expansion is taken near the boundary z = 0 for the two dilatons above, one can get φ (z) = φ 0 + √ 6cz 4 + · · · .(10) we expect the constant piece to correspond to the source for the operator TrG 2 and coefficient of the z 4 to give the gluon condensate, see Ref. [31] for a detailed discussion. Note that the position of the singularity is determined by f = z −4 c , where z c is an IR cut-off. One can easily find that the dilaton black hole solution becomes the AdS black hole solution when c = 0, and the solution reduces to the dilaton wall solution with a = 0. In addition, a Hawking-Page transition occurs between the dilaton wall background and the dilaton black hole background at some critical value of a. Therefore, the dilaton wall solution corresponds to confined phase, and the dilaton black hole solution describes deconfined phase. More details can be seen from [69]. Next, we convert z coordinate to r coordinate by taking r = L 2 z , then the metric (5) is ds 2 = r 2 L 2 A(r)dx 2 − B(r)dt 2 + L 2 r 2 dr 2 ,(11) with φ (r) = c f 3 2 log 1 + f r −4 1 − f r −4 + φ 0 ,(12) where A(r) = (1 + f r −4 ) ( f +a)/2 f (1 − f r −4 ) ( f −a)/2 f ,(13)B(r) = (1 + f r −4 ) ( f −3a)/2 f (1 − f r −4 ) ( f +3a)/2 f ,(14)f 2 = a 2 + c 2 .(15) Parameter a is related to the temperature, a = (πT 4 )/4. The location of horizon r h = a 1 4 . Note that the presence of the cutoff, the range becomes r f < r < ∞. In addition, we set the value of gluon condensate 0 ≤ c ≤ 0.9GeV 4 [70,71]. We set L = 1 in this paper. The effect of gluon condensate on imaginary potential In this section, following the step in [8,9,84], we will calculate the imaginary potential and thermal width under the background metric (11). Now, we consider a rectangular Wilson loop, whose length of short side is L in the spatial direction and a long side is T along a time direction. Set a dipole located in the direction of short side and the quarks are located at x 1 = ± L 2 . Then, we use the string worldsheet coordinates t = τ, x 1 = σ , x 2 = x 3 = const, r = r(σ ).(16) The Nambu-Goto action of the string is S NG = − 1 2πα ′ dσ dτ √ −g,(17) where α ′ is related to 't Hooft coupling as 1 α ′ = √ λ . g is the determinant of the induced metric given by g 00 = r 2 B(r), g 01 = g 10 = 0, g 11 = r 2 A(r) + r ′2 r 2 .(18) In the calculation of imaginary potential, we will not consider the impact of dilaton. Then we can get the lagrangian density L = U(r) + V (r)r ′2 ,(19) where U(r) = r 4 A(r)B(r), V (r) = B(r), and the prime represents derivative with respect to σ . Since the Lagrangian in this paper has the same form as that in Refs. [8,9,84], we will straightforwardly give the formulas of separation distance, imaginary potential and thermal width according to Refs. [8,9,84], as follows L = 2 r 0 ∞ dr U(r 0 )V (r) U(r) (U(r) − U(r 0 )) ,(21)ImV QQ = − 1 2 √ 2α ′ V (r * ) U ′ (r * ) 2U ′′ (r * ) − U(r * ) U ′ (r * ) ,(22)Γ QQ T = − 4 (a 0 T ) 3 dω ω 2 e −2ω a 0 T ImV QQ (ω) T ,(23) where ω = LT . A upper and lower bound need be given for this integral. A upper limit can be easily obtained by the maximum of LT , while a lower bound can be derived by this condition σ ≥ 0 [8,9,84]. So the range of this integral is LT min ≤ ω ≤ LT max . With all the preparations in place, LT versus rh/r 0 for different values of c is shown in Fig. 1. One can find clearly that increasing c leads to decreasing LT max . Here, we define LT max as the dissolution length of heavy quarkonium. Namely, the increase of gluon condensate leads to a easier dissociation of heavy quarkonium. In Fig. 2, we draw ImV /( √ λ T ) against LT for c = 0.2GeV 4 , 0.5GeV 4 , 0.8GeV 4 . The range of each line of imaginary potential starts at a certain value LT min and ends at LT max , which is corresponding to r 0max and r 0min in Fig. 1, respectively. LT min is a point where the imaginary potential becomes negative and LT max is the maximum value during the whole valid range of rh/r 0 . In addition, we find that increasing gluon condensate causes the imaginary potential to start from smaller distance and have a larger absolute value, which indicates gluon condensate makes the suppression of heavy quarkonium stronger. In the following, There are two kinds of way to evaluate the thermal width [8,9,72]. One approach, named "exact" approach, is only to take the integral range of imaginary potential from LT min to LT max . Another approximate approach is to use a straight-line fitting for ImV and the integral range will be from LT min to infinite when evaluating the thermal width. As discussed in these papers, the conservative approach gives a lower bound for the thermal width while the approximate approach will considerablely overestimates the thermal width. It is clearly shown in Fig. 3. Another point we need to be cautious is the consistence of results calculated in two approaches. Since it is found the nontrivial behavior of thermal width for increasing rapidities, the reason maybe is that the thermal width is dominated by different processes, such as gluo-dissociation and Landau damping for weakly couple plasma [19]. More concretely, the thermal width calculated by conservative approach is decreasing while the thermal width calculated by approximate approach is increasing both for increasing rapidity in holographic calculation [85]. For comparison, we use both approaches to evaluate thermal width as shown in Fig. 4 and Fig. 5, respectively. We display Γ /T as a function of a 0 T for some different values of c. Instead of the non-trivial behavior in moving case, it is found that the large value of gluon condensate will lead to large width in both two approaches. Namely, increasing gluon condensate results in a weaker bound for heavy QQ pair. In Ref. [70], they investigated the gluon condensate on real potential. they found the heavy quark potential becomes deeper as c decreases, implying dissociation of heavy quarkonium is easy at large gluon condensate. This conclusion is consistent with our analysis of thermal width. In Fig. 6, by setting m Q = 4.7GeV , corresponding to a 0 = 0.621GeV −1 and the t'Hooft coupling λ = 9, we plot the thermal width of ϒ (1S) state as a function of the gluon condensate. It is found that there is a critical value c 0 below which the thermal width will disappear. It can be inferred the critical value of gluon condensate indicating the dissociation of heavy quarkonium. Since, the high temperature leads to a dramatically decreasing of gluon condensate, that is to say, small condensate means very high temperature. But the relationship between temperature and gluon condensate is not related in this metric. In this work, there are a lot of constraints but the most important constraint is small thermal fluctuations for the saddle point approximation, called as mathematical approximation. In addition, there is a constrain imposed by physics itself, named as physical approximation, existing a threshold c 0 for fixed temperature, in which the imaginary potential will disappear, equivalently, thermal width will disappear. In some sense, it may indicate the dissociation of heavy quarkonium. Conclusion and discuss In this paper, we have used the string worldsheet fluctuation approach around the deepest point of the string to study the effect of gluon condensate on imaginary potential of heavy quarkonium in strongly coupled quark gluon plasma by using the AdS/CFT correspondence. The imaginary potential and thermal width are calculated by using a general formula, as shown in (22) also been given, LT min ≤ ω ≤ LT max , where LT min is determined by σ c which is a real, and LT max corresponds to the maximum of LT . The thermal width is calculated by a ground-state wave function of a particle in a Coulomb-like potential and the Bohr radius is inversely proportional with one half mass of groundstate particle. One can find that the decrease of gluon condensate enlarges the inter-distances and makes the onset of the imaginary potential happen for larger LT which means that the suppression becomes weaker. In addition, the dropping gluon condensate reduces the absolute value of imaginary potential. Since two different approaches of evaluating thermal width give two opposite behavior in non-vanishing rapidity, for security, we compute the thermal width with these two approaches in non-vanishing gluon condensate and find the consistence of the two approaches. Thus, we can conclude that increasing the value of gluon condensate leads to increasing thermal width. Namely, the dropping gluon condensate makes heavy quarkonium have a stronger bound for fixed tempareture, which shows the heavy quarkonium is easier to dissociation. This conclusion is in agreement with Ref. [70], in which they investigate the effect of gluon condensate on real potential. We also find there is a critical value of gluon condensate below which the thermal width will no longer exist. We presume it relates to the dissociation of heavy quarkonium. The drawback of this model is that the value of gluon condensate don't have a direct connection with temperature. In some sense, the condensation is piecewise constant function, which allows us to consider the effect of gluon condensate and temperature on thermal width separately [69]. But a real situation requires us to examine the back reaction of gluon condensate and temperature in a more complicated holographic model [86]. This problem can be further discussed in future work. acknowledgement We would like to thank Zi-qiang Zhang for useful discussions of imaginary potential. Fig. 1 .Fig. 2 .Fig. 3 . 123LT as a function of rh/r 0 for some choices of c with a fixed temperature T = 0.2GeV . Solid line represents c = 0.2GeV 4 , dashed line denotes c = 0.5GeV 4 , dotted line is c = 0.8GeV 4 . ImV /( √ λ T ) versus LT for some choices of c with fixed temperature T = 0.2GeV . Solid line represents c = 0.2GeV 4 , dashed line denotes c = 0.5GeV 4 , dotted line is c = 0.8GeV 4 . Compare of exact and approximate approach at c = 0.2GeV 4 and T = 0.2GeV. Solid line represents conservative approach, dashed line represents approximate approach. Fig. 4 .Fig. 5 . 45Γ /T as a function of a 0 T calculated by approximate approach at fixed T = 0.2GeV . Solid line represents c = 0.2GeV 4 , dashed line denotes c = 0.5GeV 4 , dotted line is c = 0.8GeV 4 . 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[ "IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 1 HapTable: An Interactive Tabletop Providing Online Haptic Feedback for Touch Gestures", "IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 1 HapTable: An Interactive Tabletop Providing Online Haptic Feedback for Touch Gestures" ]	[ "Senem Ezgi Emgin ", "Amirreza Aghakhani ", "T Metin Sezgin ", "Cagatay Basdogan " ]	[]	[]	We present HapTable; a multi-modal interactive tabletop that allows users to interact with digital images and objects through natural touch gestures, and receive visual and haptic feedback accordingly. In our system, hand pose is registered by an infrared camera and hand gestures are classified using a Support Vector Machine (SVM) classifier. To display a rich set of haptic effects for both static and dynamic gestures, we integrated electromechanical and electrostatic actuation techniques effectively on tabletop surface of HapTable, which is a surface capacitive touch screen. We attached four piezo patches to the edges of tabletop to display vibrotactile feedback for static gestures. For this purpose, the vibration response of the touch screen, in the form of frequency response functions (FRFs), was obtained by a laser Doppler vibrometer for 84 grid points on its surface. Using these FRFs, we have developed a new technique to display localized vibrotactile feedback on the surface for static gestures. For dynamic gestures, we utilize electrostatic actuation technique to modulate the frictional forces between user's fingers and tabletop surface by applying voltage to the conductive layer of the touch screen. To our knowledge, this hybrid haptic technology is one of a kind and has not been implemented or tested on a tabletop. It opens up new avenues for gesture-based haptic interaction not only on tabletop surfaces but also on touch surfaces used in mobile devices with potential applications in data visualization, user interfaces, games, entertainment, and education. Here, we present two examples of such applications, one for static and one for dynamic gesture, along with detailed user studies. In the first one, user detects the direction of a flow, such as that of wind or water, by putting her/his hand on the surface and feels a vibrotactile stimulus traveling underneath it. In the second example, user rotates a virtual knob on the surface to select an item from a menu while feeling the knob's detents and resistance to rotation in the form of frictional haptic feedback.	10.1109/tvcg.2018.2855154	[ "https://arxiv.org/pdf/2103.16510v1.pdf" ]	51,625,585	2103.16510	c4ef185dfd9da4d5561b42c1a06d1044eea5edbf	IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 1 HapTable: An Interactive Tabletop Providing Online Haptic Feedback for Touch Gestures Senem Ezgi Emgin Amirreza Aghakhani T Metin Sezgin Cagatay Basdogan IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 1 HapTable: An Interactive Tabletop Providing Online Haptic Feedback for Touch Gestures Index Terms-Electrostatic actuationgesture recognitionhaptic interfaceshuman-computer interactionmultimodal systemsvibrotactile haptic feedback We present HapTable; a multi-modal interactive tabletop that allows users to interact with digital images and objects through natural touch gestures, and receive visual and haptic feedback accordingly. In our system, hand pose is registered by an infrared camera and hand gestures are classified using a Support Vector Machine (SVM) classifier. To display a rich set of haptic effects for both static and dynamic gestures, we integrated electromechanical and electrostatic actuation techniques effectively on tabletop surface of HapTable, which is a surface capacitive touch screen. We attached four piezo patches to the edges of tabletop to display vibrotactile feedback for static gestures. For this purpose, the vibration response of the touch screen, in the form of frequency response functions (FRFs), was obtained by a laser Doppler vibrometer for 84 grid points on its surface. Using these FRFs, we have developed a new technique to display localized vibrotactile feedback on the surface for static gestures. For dynamic gestures, we utilize electrostatic actuation technique to modulate the frictional forces between user's fingers and tabletop surface by applying voltage to the conductive layer of the touch screen. To our knowledge, this hybrid haptic technology is one of a kind and has not been implemented or tested on a tabletop. It opens up new avenues for gesture-based haptic interaction not only on tabletop surfaces but also on touch surfaces used in mobile devices with potential applications in data visualization, user interfaces, games, entertainment, and education. Here, we present two examples of such applications, one for static and one for dynamic gesture, along with detailed user studies. In the first one, user detects the direction of a flow, such as that of wind or water, by putting her/his hand on the surface and feels a vibrotactile stimulus traveling underneath it. In the second example, user rotates a virtual knob on the surface to select an item from a menu while feeling the knob's detents and resistance to rotation in the form of frictional haptic feedback. INTRODUCTION n contrast to p ersonal com p uters utilizing ind irect input d evices such as m ouse and keyboard , interactive tabletops allow users to d irectly m anipulate d igital content via touch gestures. They intuitively couple gesture input w ith d irect graphical outp ut, w hich requires m inim al learning and enables natural interaction. They also provid e a large horizontal su rface, allow ing m ultiple u sers to collaborate sim ultaneously and interact w ith each other [1]. H ow ever, they lack the physicality of an interaction as experienced w ith the input d evices, and consequ ently, require full visual attention of the user, w hich is tiring and results in d eterioration in task perform ance [2]. One of the key senses for interaction is haptics. H aptic feed back is know n to imp rove task performance (in term s of com pletion tim e and precision) and realism . It also helps to red uce cognitive load and enables representation and d igestion of complex d ata more easily [3]. There is an ongoing effort in research community for ad d ing haptic feed back to interactive tabletops and surface d isplays. One such effort is to d evelop shape-changing surfaces. For example, FEELEX is m ad e of an array of 36 linear actu ators, each m oving ind ivid ually in vertical d irection to project the surface contour of a d igital im age on a flexi-ble surface [4]. Lum en is an array of m ovable light guid es w hose height and color can be controlled ind ivid ually to create images, shapes and physical motions [5]. The motion of each light guid e is controlled by a string, m ad e of shape mem ory alloy (SMA), attached to the guid e. More recently, Follm er et al. [6] d eveloped inFORM, w hich enables d ynam ic afford ances, constraints, and actuation of passive objects. This system utilizes 900 motorized pins (30x30) to actu ate 150 board s m oving u p and d ow n to rend er d ynam ic shapes on the surface. H aptic feed back is d isplayed to the user by ad justing the stiffnesses of pins via a PID control. As stated by the authors, shape changing d isplays are currently not practical d ue to their large size and cost of manufactu ring. Another com mon line of effort for d isplaying haptic feed back through a touch surface is to utilize electrom echanical or electrostatic actuation. Poup yrev et al. attached four piezo actuators to the corners of a pen-based touch d isplay, in betw een LCD and protective glass panel, to convey vibrotactile haptic feed back to users by varying am plitud e and frequency of input signal [7]. The results of their user stud y show ed that subjects preferred haptic feed back w hen it was combined w ith an active gesture, such as w hile d ragging a slid er or highlighting a text using the pen interface. Jansen et al. d eveloped Mu d -Pad , a d evice that utilizes magnetorheological fluid combined w ith sm all electrom agnets placed und er the d isplay su rface [8]. The flu id 's physical properties are altered using electromagnets, thus the frictional properties of the surface are controlled to provid e active tactile feed back to u ser. Due to the electromagnets und er the surface, this system is not compact and requires visual projection from top. Bau et al. presented TeslaTouch, a capacitive touchscreen u tilizing electrostatic actu ation [9]. The d evice controls the attractive electrostatic force betw een user finger and d isplay surface by mod ulating the voltage applied to the cond uctive layer of the screen. Yamam oto et al. also used the same principle in a tactile telepresentation system to realize explorations of rem ote surface textures w ith real-time tactile feed back to user [10]. To d isplay a rich set of haptic effects for the gestures performed on a tabletop, w e integrated electrom echanical and electrostatic actuation techniqu es effectively on H ap-Table (Fig. 1). We attached four piezo patches to the ed ges of the table's interaction surface to control its out-of-plane vibrations and d isplay localized vibrotactile hap tic feedback to user for static gestures. For tabletop interactions using d ynamic gestures, w e convey hap tic feed back to user via electrostatic actuation technique introd uced in [9,10]. We mod ulate the frictional forces betw een user's finger(s) and our tabletop su rface (a large-size surface cap acitive tou ch screen, also referred to as touch screen in the text) in real time accord ing to the d ynamic gesture performed on the surface. Using this hybrid actuation approach, the type of haptic feed back that can be d isplayed through H apTable varies in com plexity from sim ple frictional effects to m ore com p lex localized vibrotactile flow effects. Our particular approach for creating localized vibrotactile effects on H ap- Table requires vibrational characterization of its touch surface and intense precomputations. To d em onstrate how haptic feed back can im prove user's interactions w ith H apTable, w e present tw o example applications, su pp orted by d etailed user stu d ies. In the first example, user d etects the d irection of a travelling vibrotactile flow , mim icking a flow of w ind or w ater, by placing her/ his hand on the surface. In the second example, user rotates a virtual haptic knob using tw o fingers to select an item from a menu w hile feeling the d etents of the knob and receiving frictional feed back accord ing to her/ his rotational m ovem ent. This paper is organized as follow s: Section 2 introd uces our table and its hard w are components. Section 3 introd uces ou r method s for recognizing static and dynamic hand gestures in real tim e. Section 4 d iscusses ou r hap tic rend ering m ethod s utilizing electrom echanical and electrostatic actuation techniques. Section 5 presents our user stud ies, investigating how haptic feed back may augment tabletop interactions triggered by static and d ynamic gestures. The results of the user stud ies are d iscussed in Section 6. The final section conclud es this paper and elaborates on ou r future w ork. DESIGN OF HAPTABLE HapTable system consists of three main modules: gesture detection, visual display, and haptic feedback (Fig. 1). Gesture d etection mod ule is responsible for registering and d etecting high resolution images of static and dynam ic hand gestures performed on H apTable surface. Althou gh there are tou ch su rfaces com m ercially available in the market for d etecting finger and / or hand gestu res, they may potentially interfere w ith our haptic feed back mod ule and may not capture hand contour in sufficient d etail for correct recognition of hand gestures. For examp le, the piezo actuators that are used to generate vibrotactile haptic effects in H apTable m ay interfere w ith the travelling sou nd w aves utilized in surface acou stic tou ch sensors to d etect finger poisiton. Sim ilarly, infrared touch frames have occlusion problems and are not good at d etecting hand contour. For these reasons, H ap Table uses rear d iffused illum ination (Rear DI) to register hand poses [1]. The tabletop surface is evenly illum inated w ith w id eangle infrared LEDs (50-Mod ule IR Kit, Environm ental Lights) in configuration of three row s: top row is placed parallel to the table surface to illu m inate its ed ges, w hereas the remaining tw o row s are perpend icular to the first row to illuminate the surface center (Fig. 1b). When a user touches the H apTable surface, light is reflected from contact points and captured by an infrared camera (Eye 3, PlayStation). This camera captures 60 frames per second w ith a resolution of 640 x 480 pixels. The visual display of digital images on the tabletop surface is achieved by a projector (B1M , A sus). We selected this projector specifically because it does not emit infrared light that may interfere w ith gesture detection module, and it has a short throw distance that allow s minimal table depth. The throw distance is extended by using an additional mirror (Fig. 1c), allow ing users to interact w ith H apTable even in a sitting position. The hap tic feed back m od u le integrates electromechanical and electrostatic actuation techniqu es to d isplay a w id e range of haptic effects w hile users interact w ith d igital im ages and objects through static and d ynamic gestures. These tw o actuation techniques complement each other. For static gestures, H apTable d isplays vibrotactile haptic feed back to user. Four piezoelectric p atches (PI-876.A 12, Physik Instru m ente, 61 x 35 x 0.5m m ) w ere attached beneath the touch surface to generate mechanical vibrations on the surface. The p rop agation of the vibrations from the touch surface to the table itself is prevented by rubber seals placed und er the tabletop surface (Fig. 1c). We utilize electrostatic hap tic feedback for d ynamic gestu res. This technology d oes not use any form of m echanical actuation but tactile sensations can be created by controlling frictional forces betw een the tabletop su rface and user's fingers. In ord er to generate friction on the surface based on electrostatic actuation, a large su rface capacitive screen (SCT-3250, 3M, 743.46 x 447.29 x 3.18m m ) is u sed as the tou ch su rface of the table. When a period ic voltage is applied to the cond uctive layer of the screen, normally used for sensing finger p osition, an attractive electrostatic force d evelops betw een finger skin and the screen su rface in the norm al d irection. This electrostatic force is sm all and cannot be sensed d irectly by finger w hile it is stationary on the surface. H owever, if finger slid es on the surface of the touch screen, a resistive frictional force is felt by user in tangential d irection. To control the voltage transmitted to the piezo patches for vibrotactile haptic feed back and also to the touch screen for electrovibration ind epend ently; a sound card , tw o high-voltage am p lifiers (E413.D2, Physik Instru mente, Gain: 50), and tw o solid state relay arrays (Yocto-MaxiCoupler, Yoctopuce) are used (Fig. 2). The haptic signals generated by the left and right output channels of the sound card of a personal computer (PC) are first transmitted to the high-voltage am plifiers. Each am plifier's positive output is connected to a multichannel solidstate relay array, controlled and pow ered by the USB ports of PC. This relay is fast, can sw itch voltages up to 350 Vp p for sm all load s of cu rrent (u p to 100 m A), and d oes not require any external pow er. The outputs of these relays are connected to each piezo patch and the touch screen, as show n in Fig. 2. This architecture enables us to excite any number of patches and the touch screen sim ultaneou sly. H ow ever, w e can only apply ind epend ent voltage signals to at most one piezo patch and the touch screen since the sound card has only tw o output channels. GESTURE RECOGNITION An imp ortant feature of H apTable is the real-tim e recognition of hand gestures performed on its touch surface. In general, hand gestures performed on touch surfaces can be classified as: (a) static and (b) d ynam ic. In static gestures (e.g. pressing a button, pointing an object), the hand has fixed position and orientation, while in d ynam ic gestures (e.g. d ragging a fold er, rotating a virtual knob), it has tim e-varying position and orientation. In [11], au thors d ifferentiate static gestures from d ynamic ones by examining the positional change of hand pose for a specific tim e w ind ow . If it d oes not change in time, then the gesture is classified as static. H ow ever, this algorithm recognizes the trajectory of hand gesture, rather than hand contour. Authors in [12] examine the hand shape and recognize a gesture in 1.5 second s using Self-Grow ing and Self-Organized N eu ral Gas (SGON G) algorithm . H ow ever, this d uration is long and not feasible for realtime haptic interactions on our table. Accord ing to [13], users expect a response in less than 20 m illisecond s in interactive systems. H ence, to provid e real-tim e haptic feed back for a gesture, ou r system has to recognize the gesture at its early stage of evolution. For this reason, it is cru cial for us to select sim p le but d iscrim inative features for gesture recognition. Since the user can make gestures anyw here on the table, the selected features should also be ind epend ent of translation, orientation, and hand size. We d eveloped a simple yet efficient algorithm that can recognize pre-selected five static (Fig. 3a) and five d ynam ic hand gestures (Fig. 3b). The first step in our algo- rithm is to d ecid e if a gesture is static or d ynam ic based on the change in position and orientation of the hand . Then, the camera images are sent to the relevant classifier accord ingly for further processing. If a hand gesture is static, it is rotated into a canonical orientation with respect to a reference edge and wrist is removed from the pose since users can approach the table from any side (Fig. 4). Aligning a hand pose with respect to the reference edge is achieved as follows: we first apply a high-pass filter to the raw image in order to highlight the parts that are in contact with the table (Fig. 4b). Then, the smallest circle enclosing the highlighted hand (i.e. bounding circle) is calculated (Fig. 4c). The arc intersecting the bounding circle is defined as the wrist (Fig. 4d). To rotate the hand and make it perpendicular to the reference edge, the angle θ between this reference edge (l1 in Fig. 4e) and the line connecting the midpoint of the wrist to the center of the bounding circle (l2 in Fig. 4e) is calculated. If the hand pose is rotated by an angle of θ degrees in counter-clockwise direction about the center point (c in Fig. 4e), it becomes perpendicular to the reference edge as shown in Fig. 4f. Then, the silhouette of the hand posture is recognized via Fourier descriptors [14] using Support Vector Machine [15]. Com pared to a static hand gesture, w e need to recognize a d ynamic hand gesture at its early stage of evolution to provid e the user w ith haptic feed back im m ed iately. For this reason, our algorithm utilizes the first four fram es of a d ynamic gesture for feature extraction. In ad d ition to Fourier d escriptors, the number of fingers in contact and the trajectory of contact points are also u sed as d iscriminating features to classify the d ynamic gestures. The gesture evaluation experiments w ere cond ucted w ith 5 subjects (2 males and 3 females). A sketch for each gesture w as presented to the subjects, and they w ere asked to repeat this gesture 40 tim es in d ifferent positions and orientations on touch surface. Hence, the total number of gestu res performed on the tabletop surface w as 2000 (5 su bjects x 10 gestu res x 40 rep etitions). We trained and tested our recognition algorithm using tw o-fold cross valid ation ap proach. Recognition rates of 98% and 91% w ere achieved for static and d ynamic gestures, resp ec-tively, w ithout com prom ising the responsiveness of the system. HAPTIC FEEDBACK In our table, hap tic feed back is d isplayed to user according to the type of gesture she/ he perform s and the d igital content she/ he is interacting w ith. Haptic Feedback for Static Gestures We d isplay localized vibrotactile haptic feed back for static gestures. For this purp ose, w e first construct a vibration m ap of the touch screen in ad vance and then d isplay hap tic feed back for the gesture accord ingly d uring realtim e interaction. To construct the vibration m ap of touch screen, w e d ivid ed its surface into 84 grid points (7 row s by 12 colu m ns, Fig. 5b). The size of each grid w as 6 x 6 cm . The ou t-of-plane vibrations at each grid point w ere measu red w hen each piezo patch w as excited ind ivid ually and w hen all patches w ere excited together. For this purp ose, a linear sine sw eep signal, varying in frequency from 0 to 625 H z, w as generated by a signal generator, am plified by one of the high-voltage am plifiers in ou r setu p (E413.D2, Physik Instrumente, Gain: 50), and then transmitted to the terminals of the piezo p atches. A Laser Doppler Vibrometer (LDV, PDV-100, Polytec) w as u sed to measu re the out-of-plane vibrations at each grid point (Fig. 5a). A signal analyzer (N etDB, 01d B-Metravib) w as used to record and analyze the signals com ing from LDV and the signal generator. H aving d efined the signal generator's output as the reference channel in the signal ana- lyzer, the experimental frequency response functions (FRFs) betw een the velocity output and piezoelectric voltage input w ere obtained. The same process w as repeated 3 times for the cases w hen each piezo patch w as active and w hen all piezo patches w ere active together in parallel configuration. The velocity FRF of each grid point w as estimated by averaging the data of three full sw eeps, and then converted to displacement FRFs (Fig. 6). The averaged FRFs for patches PA and PC, and PB and PD are similar due to their symmetrical configurations on H ap- Table su rface, as show n in Fig. 5b. In ord er to d isplay localized vibrotactile haptic feedback on the touch su rface, w e utilize the five FRF functions (one for each piezo patch and one for all together) of 84 grid p oints (referred to as "vibration m ap" in the text). For each grid point, there is an excitation frequency at w hich the am plitud e is m axim u m . If the su rface is excited at this frequency, a localized haptic effect can be generat- In ord er to extend the vibrotactile flow concept to all grid points on the surface efficiently (and hence to all points on the surface throu gh bilinear interpolation), w e have d evelop ed a sophisticated preprocessing ap p roach. We construct and store three lookup tables (Fig. 8), which are used to determine the excitation parameters during user interaction in real time. These tables store the maximum d ifference in vibration am p litud es of the grid points (Fig. 8a), the correspond ing excitation frequencies (Fig. 8b), and the actuator ID ( Fig. 8c; either the patch PA, PB, PC, PD, or PALL). For exam p le, if the points L and R, show n in Fig. 7a, are selected as active and passive points respectively (they correspond to the grid points 51 and 52 on the surface) and inputted to the tables, a maximu m vibration d ifference of 0.201 μm / Vp (Fig. 8a) at 465 H z (Fig. 8b) is retu rned . This d ifference in am plitud e is obtained w hen the surface is actuated by piezo patch PA (Fig. 8c). On the Fig. 8. Excitation lookup tables for HapTable: To create a vibrotactile flow from an "active" grid point to a "passive" one, our haptic rendering algorithm acquires maximum displacement difference in vibration amplitudes from table (a), the corresponding excitation frequency from table (b), and the actuator ID (i.e. which actuator to use to create that displacement difference at the corresponding excitation frequency) from table (c). other hand , if point R is active and point L is passive, the vibration d ifference now becom es 1.607 μm / Vp (Fig. 8a), w hich is obtained at a d ifferent actuation frequ ency of 428 Hz (Fig. 8b) and w hen the surface is actu ated by PALL (Fig. 8c). Using this inform ation, a haptic stim ulus is created for d isplaying a vibrotactile flow effect from left to right as show n in Fig. 9b. First part of the stim uli excites PA to create a substantially high vibration d isplacem ent at point L compared to that of at point R, w hereas the second p art excites PALL to accom p lish vice versa. The am plitud es on both parts of the stimulus are ad justed accord ing to the hum an sensitivity to vibrotactile excitation [16,17] to create an equ ivalent haptic effect in m agnitud e (Fig. 9b). Then, a linear am plitud e m od ulating envelope is applied to the beginning and end of the signals in each part of the stimulu s to make the transitions smoother d uring the activation and d eactivation period s of the piezo pathces ( Fig. 9b and 9c). We d emonstrate in Section 5 how this technique can be used to create a d irectional vibrotactile flow betw een the ind ex fingers of left and right hand s and also und erneath one hand placed on the tabletop surface. Haptic Feedback for Dynamic Gestures To d isplay haptic feed back for d ynam ic gestures, w e mod u late the frictional forces betw een user's fingers and the touch screen used as the tabletop surface of Hap Table. When an alternating current voltage is applied to the cond u ctive layer of a tou ch screen, electrostatic attraction force (f e ) d evelops betw een fingers and the su rface of touch screen (Eq. 1). The magnitude of this attractive force is governed by the applied voltage, V(t), contact area, A, perm ittivity of the vacuu m , insulating layer of the touch screen, and ou ter finger skin (ε 0 , ε i , and ε s respectively), and the insulator and outer skin thicknesses (ti, ts), as w ritten in below [18]: (1) The magnitud e of this attractive force is too small to be perceived by a stationary finger on the touch su rface. H ow ever, it results in a perceivable change in frictional force in tangential d irection w hen human finger moves on the su rface. ( By controlling the frequency, am plitu d e, and the w aveform of the applied voltage, it is possible to create d ifferent haptic effects on the surface [9]. In the upcoming section, w e d emonstrate the use of this technology in a case stud y involving a virtual knob, w hich is rotated by tw o fingers on the su rface. USER STUDIES We d emonstrate the functionality of H apTable via tw o example applications supported by d etailed user studies. In the first one, as an exemplar for static gestures, w e rend er localized d irectional vibrotactile flow betw een ind ex fingers of two hand s and also und erneath a hand p laced on the surface. We investigate if u sers can d ifferentiate the d irection of vibrotactile flow . As an exam ple for d ynam ic gestu res, w e hap tically rend er a virtual knob on the surface. The user receives frictional haptic feed back as she/ he rotates the knob using tw o fingers in ord er to select an item from a pull-d ow n menu. We investigate if haptic feed back imp roves task performance and the user's subjective sense of performing the task successfully. Vibrotactile Flow The vibrotactile flow , i.e. haptic illusion of apparent tactile movement on human skin, w as investigated by Sherrick and Rogers in 1960s [19]. They attached tw o vibration motors on user's thigh separated by a d istance and then ad justed the stimulu s d uration and the d elay betw een the actuation tim es to create an effect of a traveling haptic stimulus. They show ed that stimuli d u ration and the interstimulus offset interval (ISOI, i.e. the tem poral interval betw een the offset of one vibration to the onset of another one), are the key parameters that affect the subjects' haptic perception. Tan and Pentland [20] and Israr and Poupyrev [21] extend ed this concept to 2D surfaces by placing an array of vibration motors on the cushion of a chair to create d irectional tactile strokes. In a separate stud y, Israr and Poupyrev [22] investigated the control parameter space for p rod u cing reliable continu ous moving patterns on forearm and back. The results of their user stu d y show ed that ISOI sp ace for the forearm w as influenced by both the motion d irection and spacing of the actuators, w hereas ISOI sp ace for the back w as affected only by the d irection of actuation. Arasan et al. [23] applied the apparent tactile motion to a pen-based stylus that can be used w ith a tablet or a m obile d evice. Tw o vibration m otors w ere placed at the proxim al and d istal Vibrotactile Flow Experiments Experiment 1: Vibrotactile flow between two points We cond ucted an experiment w ith 5 subjects (2 female, 3 male) having an average age of 31.4 years (SD = 6.9) to investigate if they can d etect a d irectional vibrotactile flow betw een tw o grid points on the touch su rface. In ord er to create a d irectional vibrotactile flow , w e used an am plitud e-m od ulated voltage signal having tw o parts (see the p rofile in Fig. 9). Each part w as played by the appropriate piezo actuator. The frequency of the signal in each part and the actuator that plays the signal w ere carefully selected from the excitation lookup tables, as d iscussed in the previous section. Subjects p laced their left and right ind ex fingers on the d esignated locations (test points) of the tabletop su rface (Fig. 10a) and w ere asked the perceived d irection of vibrotactile flow : left to right hand , or right to left hand. All subjects w ere asked to w ear active noise-canceling head phones playing w hite noise to prevent them hearing aud itory cues caused by the vibrations. Only one subject w as left-hand ed . Prior to the experim entation, all subjects w ere informed about the nature of the experimental proced u re. Experiment started w ith a familiarization session, consisting of 10 trials (5 repetitions for each d irection in rand om ord er) perform ed w ith a single pair (p air F-F in Fig. 10a). Subjects w ere allow ed to replay eash stim ulus as many times as need ed d uring the familiarization session. Afterw ard s, the actu al experiment w as cond ucted w ith 4 pairs of test points, located at d ifferent regions on the touch screen. Subjects could replay each stim ulus only once in the actual experim ent. More information about the selected test pairs, the physical d istance betw een them, the actuators played the voltage signal in each part of stimulus, and the d ifference betw een their vibration amplitud es are reported in Fig. 10b. All vibration am plitud es in the experiment w ere above the absolute vibrotactile threshold of hum an finger for the excitation frequencies listed in Fig. 10b. The experim ent consisted of 40 trials (4 pairs x 2 d irections x 5 repetitions) d isplayed in rand om ord er. All subjects identified the direction of vibrotactile flow with a perfect accuracy of 100% for all pairs. The results of this experiment showed that subjects could easily differentiate the direction of vibrotactile flow with their index fingers even if the test points are diagonal to each other, as in the pairs of 1-1, 3-3, and 4-4. Vibrotactile flow under hand Based on the encou raging results of the first experiment, w e expand ed our stud y to investigate d irectional vibrotactile flow und er a hand placed on the touch surface. We assu med that human hand covers an area of 12 x 12 cm . We d ivid ed this area into nine equal squares (a), and each square is further d ivid ed into 15 x 15 subgrid points for finer resolution (FRFs for these points w ere calculated in ad vance using bilinear interpolation). Sim ilar to the concept of active and passive points introd uced in the first experiment, active and passive squares are d efined in this experiment. A square is assu med to be active if at least half of its subgrid points have a su fficiently high vibration (three JN D above the human vibrotactile threshold for the applied excitation frequency). The threshold and just noticeable difference values for d ifferent excitation frequ encies w ere obtained from the hum an vibrotactile sensitivity curve reported in [16,17]. Using these active and passive squares, w e aimed to generate a vibrotactile flow in horizontal and vertical d irections. As in the case of the first experiment, this required to select proper excitation frequ encies from the FRFs. In ord er to create a d irectional vibrotactile flow , the active and passive squares should be sym metric w ith 10. (a) The locations of the test pairs selected for the familiarization (pair: F-F) and the actual experiment (pairs: 1-1, 2-2, 3-3, 4-4) sessions. (b) Distance between the test points for each pair, actuation frequencies (f 1 , f 2 ), the piezo patches used for actuation (p 1 , p 2 ), and the displacement differences between the test points (Δd 1 , Δd 2 ) are given in the table on the right. respect to the horizontal (vertical) axis passing through the center point of the area representing hand (point C in Fig. 11a) for vertical (horizontal) flow . Fig. 11 illustrates example vibration maps that are acceptable (Fig. 11b, d ) and unacceptable (Fig. 11c, e) based on our algorithm . The second experiment w as cond ucted w ith eleven subjects (4 female, 7 male) having an average age of 29.6 years (SD: 6.0). Only tw o subjects w ere left-hand ed. The average hand w id th and length of the subjects w ere m easu red as 8.48 cm (SD: 0.52 cm ) and 18.42 cm (SD: 6.24 cm ), respectively. During the experim ent, all subjects stood in front of H apTable and placed their hand on the five d esignated regions (one for preliminary and fou r for actual experiment), rand om ly d istributed on the tabletop surface (Fig. 12a). They w ore active noise-cancelling head phones playing w hite noise to block any aud itory cu es. Experiment started w ith a preliminary session to help subjects fam iliarize w ith haptic stim uli and interface. It consisted of 40 trials (10 repetitions x 4 d irections) perform ed in Rprelim region (Fig. 12a). During this session, subjects could replay the stimulus as many times as they d esired , and ask questions to the experimenter about the experimentational procedure. In the actual exp erim ent, subjects com p leted 160 trials (4 regions x 4 d irections x 10 repetitions). For each region, the flow d irections w ere d isplayed in rand om ord er w hile the ord er w as sam e for each subject. In regions R 1 and R 2 , stim ulu s w as applied to the left hand of subjects; w hereas in other regions, R 3 and R 4 , it w as applied to their right hand . Subjects w ere allow ed to replay the stim ulus only once. At the end of each trial, they w ere asked to select the d irection of vibrotactile flow by pressing one of the four arrow buttons, representing the flow d irections of travelling u p , d ow n, left, and right, d isplayed on the screen (Fig. 12b). The recognition rates of the subjects for all d irections and regions are show n in Fig. 13. The average accuracy of the subjects for all d irections w as 90% (SD = 3.6%, Fig. 13a). Fig. 13b show s the regional recognition accu racy of all subjects (Mean = 90%, SD = 3.1%). A tw o-w ay repeated measures AN OVA w as used to investigate the statistically significant effects of region and d irection on recognition accu racy. Mau chly's test w as applied to check the violation of sp hericity assu mption. If need ed , the d egrees of freed om w ere corrected using Greenhouse-Geisser correction. Finally, Bonferroni corrected post-hoc analysis w as carried out to further investigate the statistical d ifferences betw een the groups. Tw o-w ay repeated measures AN OVA show ed a statistically significant interaction betw een region and d irection (p =0.006, η 2 partial = 0.317). Analysis of simple main effects for direction revealed that the direction had a significant for region R4 (p < 0.0005, η 2 partial = 0.682), but not for other regions. Pairwise comparisons for this region showed that the difference in the recognition accuracy between left to right and remaining (down to up, up to down, and right to left) directions were statistically significant (p = 0.001, p = 0.005, and p= 0.002 respectively). Mod e shape analysis revealed that d uring the first part of the stim ulus d isplayed in this region for vibrotactile flow of left to right, there is a vibration on both sid es of the hand , resulting in confusion about the d irection (Fig. 14b). Haptic Knob In contrast to physical controls such as buttons, slid ers, and knobs, virtual controls d isplayed on tabletops cannot be felt. As a result, task precision and performance d rop [2]. Moreover, lack of haptic feed back requ ires continuou s visual attention on the controller. For example, graphical knobs w ith visual d etents (notches) are frequently used in tabletop d isplays to rotate a virtual object in the scene or select an item from a p ull-d ow n menu using a rotation 12. (a) First, a preliminary experiment was conducted at the region R prelim and then the actual experiment was conducted at four different regions, R 1 , R 2 , R 3 , and R 4 . The size of each region was 12 by 12 cm. (b) Subjects were guided to align their hand position according to the hand image displayed on the screen. After the haptic stimulus was displayed, they were asked to determine the direction of vibrotactile flow by pressing one of the four arrow buttons on the screen. gesture. Lack of haptic feedback makes it difficult for the user to precisely rotate the object or quickly select the item from the menu. M oreover, she/ he cannot rest her/ his fingers on the knob and focus on the virtual object or the menu. A n alternative to a virtual control is a tangible control. In this approach, portable physical controls are directly placed on tabletop to augment visual interfaces w ith haptic feedback. These controls are detected w ith the help of touch sensing overlays or cameras. For example, Photo-Helix is a physical knob placed on a tabletop to interact w ith digital photo collection [27]. In this approach, one hand rotates the physical knob to control position on a helix-shaped calendar w hile the other hand inspects and modifies the digital photos. The translucent tangible knob in SLA P w idgets can be used in various modes to interact w ith digital content depending on the application [28]. For exam ple, the knob can be used in "jog w heel" mod e to find and m ark specific fram es in a vid eo or in "menu mod e" to navigate through hierarchical m enus. Weiss et al. used the knob in "jog mod e" and cond ucted a user stud y w ith 10 participants. SLAP knob outperformed a virtual knob in terms of task completion tim e and accuracy. While tangible controls provid e increased task performance, they have a fixed physical appearance unlike easily configurable virtual controls. Fu rtherm ore, they red uce the usable size of interaction area on tabletop su rface and also restrict user m ovements. Haptic Knob Experiment We used real-time dynamic gesture recognition ability of our table and electrostatic actuation technique to display a haptic knob on our tabletop surface, a large size touch screen. In our experiments, subjects rotated the knob to navigate through a menu. We modulated the frictional forces between their fingers rotating the knob and the touch screen to investigate if haptic feedback improved their task completion time, precision, and subjective sense of accomplishing the task. To modulate the frictional forces, we applied voltage signal to the conductive layer of the touch screen in various forms as discussed below. There were four sensory conditions (i.e. feedback types) in our study (Fig. 15): 1. Virtual (V): No artificial haptic feedback was displayed. Haptic Detent (HD): A pulse signal was transmitted to the touch screen to generate a "notch" (detent) effect while the subjects crossed a sector during rotation (Fig. 15b). The purpose of haptic detent was to provide users with confirmation for the sector crossings, similar to a volume knob in a car. Haptic Detent and Constant Friction (HD+CF): In addition to the pulse signals at sector crossings, a sinusoidal voltage signal with a constant amplitude (100 Vpp) and frequency (180 Hz, at which minimum electrovibration detection threshold for human finger was obtained for constant voltage by authors in [9]) was transmitted to the touch screen within the sector boundaries to display frictional haptic feed-back during rotation for better control and precision (Fig. 15c). Haptic Detent and Velocity-based Friction (HD+VF): The magnitude of the resistive frictional force was adjusted based on the subjects' angular velocity (the motivation for this type of haptic feedback stems from a rate-controlled joystick used in gaming applications, which simply displays more feedback force to faster movements). This was achieved by modulating the frequency of the input voltage applied to the screen between 60 and 180 Hz while keeping the amplitude constant at 100 Vpp. Sixteen subjects (2 female, 14 male) w ith an average age of 29 years (SD: 5.2) participated in this experim ent. Three subjects w ere left-hand ed , and none of the subjects had prior experience w ith electrostatic haptic feed back. They w ore an antistatic w ristband to their non-d ominant hand , to connect their bod ies d irectly to the ground , thus increasing the intensity of electrovibration. Subjects also put on active noise-cancelling head phones, playing w hite-noise, to block the environm ental noise. The exp eriment took approximately sixty minutes to complete. The experiment consisted of three consecutive sessions: (i) preliminary, (ii) testing, and (iii) subjective evaluation. Prior to the experim entation, all su bjects w ere informed about the experimental proced u re. The preliminary session helped subjects to get fam iliar w ith frictional haptic feed back d isplayed by electrovibration, rotation gesture, and the task itself (i.e. rotating the knob to navigate through a menu of items). A particular rotation gesture w as chosen to provid e comparable interaction experience across subjects. This gesture is performed w ith tw o fingers; thumb w as the pivot point w hile ind ex finger follow ed a circular arc. Du ring the experiment, su bjects w ere asked to rotate the knob to navigate from a start city to a target city (marked w ith red color) on the menu consisting of rand omized city names (Fig. 16a). As they navigated on the menu, a blue box highlighted the city that they w ere cu rrently on. We investigated the effects of sector size, angular d istance betw een start and target cities, and the senso- ry cond itions on the task performance in terms of task completion time and task accu racy. Three d ifferent sector sizes (8, 16, 32 sectors) and three d ifferent angular d istances betw een start and target cities (135, 270, 450 degrees) w ere used in the experiments. For a given angular d istance, the number of cities on the menu and the frequency of haptic d etents betw een start and target cities varied accord ingly (since each sector alw ays corresp onded to one item on the menu). Subjects com pleted a total of 216 trials (4 sensory cond itions x 3 sector sizes x 3 angular d istances x 6 repetitions). The trials w ere d isplayed in rand om ord er w hile the ord er w as same for all subjects. After the experiment, subjects w ere asked to fill a questionnaire, d isplayed d igitally on the table surface, containing a total of fourteen questions (7 categories x 2 rephrased qu estions for each category). The questions aimed to measure their subjective experience und er the four sensory cond itions (V, H D, H D+CF, H D+VF). For each question, 4 knobs (one for each sensory cond ition) w ere d isplayed on the screen at the same tim e (Fig. 16b), allow ing subjects to experience and com pare the sensory cond itions, and enter their experience for each cond ition using a 7-point Likert scale. As a remind er of the task performed in the actual experiment, w e also provid ed the subjects w ith the menu (list of cities) in each question. Results for the Haptic Knob Experiment Quantitative Results To investigate the effects of sector size, angular d istance betw een start and target cities, and the sensory cond itions, w e applied three-w ay repeated measures AN OVA on d epend ent variables of task com pletion tim e, overshoot rate, and recovery time. Mau chly's test of sphericity w as first performed to check w hether the d ifferences betw een the levels of the w ithin-subject factors have equal variance. If the sphericity assum ption w as violated , the d egrees of freed om w ere corrected using Greenhouse-Geisser correction. Finally, Bonferroni corrected post-hoc analysis w as carried out to investigate w here the statistically significant d ifferences betw een the levels of w ithinsubject factors lie. The results for each quantitative metric can be su m m arized as follow s: 1. Task completion time is the time it takes for a subject to navigate from start to target city in milliseconds. The results show ed that there w as no statistically significant three-w ay interaction (p=0.399). H ow ever, there w as a significant tw o-w ay interaction betw een sector size and angular distance (p=0.013). Observations from simple main effects of these tw o factors show ed that increasing either angular distance or number of sectors resulted in a statistically significant increase in task completion time. When angular distance w as fixed, although the physical distance that subjects' fingers travel remained unchanged, subjects opted to slow dow n since they observed a greater number of cities had to be crossed betw een start and target cities. Type of sensory feedback did not influence task completion time. 2. Overshoot rate is the total number of times that a subject missed target city. A lthough there w as no statistically significant three-w ay interaction, a statistically significant tw o-w ay interaction w as observed betw een sector size and angular distance again (p=0.001). Increasing sector size or decreasing angular distance increased overshoot rate. Type of sensory feedback did not influence overshoot rate. 3. Recovery time is the time that it takes for a subject to reach target city after the first miss. Results show ed that there w as no significant interaction betw een any pairs of independent variables. H owever, sector size and angular distance had statistically significant main effect on the dependent variable (p=0.05). Subjects spent more time to recover the target w hen either sector size or angular distance w as increased. Type of sensory feedback did not influence recovery time. Qualitative Results The response of the subjects to the questions (subjective scores) are given in Fig. 17. To evaluate these responses, w e used one-w ay repeated measures A N OVA . The results of A N OVA for each category in the questionnaire are summarized below : Dependability: Subjects felt greater confid ence w hile completing the task w ith the haptic knobs, since they perceived them to be more d epend able and supportive than the virtual one (p=0.013). DISCUSSION Vibrotactile Flow Experiments In the first experiment, subjects w ere asked to put their ind ex fingers of both hand s on d ifferent locations on the table and then d ifferentiate the d irection of vibrotactile flow (either from the left ind ex finger to the right one or vice versa). For a wide range of positional configurations, the subjects achieved perfect performance (100% correct identification) in this task that had a guess rate of fiftypercent. On the other hand , w hen they w ere asked to d ifferentiate the d irection of vibrotactile w ave (up, d ow n, left, right) travelling und erneath their hand in the second experiment, the success rate d ropped slightly. When rend ering a vibrotactile flow und erneath the subjects' hand , the rend ering approach utilized for the first experiment had to be m od ified since the contact w ith the surface involved an area (hand ) rather than a point (tip of an ind ex finger). We d ivid ed the area und er the user's hand into sm aller regions and consid ered the amplitud e of vibrations in those regions and their sym metry w ith respect to the horizontal and vertical axes d ivid ing the area into tw o equ al halves. Although this m od ification required extensive precomp u tations to id entify the symm etric regions having a significant d ifference in vibration am plitud e for all the excitation frequencies varied from 0 to 650 Hz, it w as d one only once. The results of the second experiment show ed that the subjects could d ifferentiate the d irection of haptic flow w ith an average accu racy of 90% (SD = 3.6%) across the fou r d irections. The slight d rop in recognition accu racy compared to the first experim ent is not suprising for several reasons. First of all, the sensitivity of ind ex finger to vibrotactile stim ulus is higher than p alm [16,17]. Second , the guess rate in the second experiment w as 25% compared to that of 50% in the first experim ent. Third , traveling vibrations can be better localized by ind ex fingers of tw o separate hand s rather than those traveling beneath one hand only. Finally, our cu rrent approach assumes a fixed -size area for human hand (w ithin 12 x 12 cm squ are) and im plem ents the hap tic stim uli accord ingly. H ence, a person w ith a hand sm aller or larger than that assumed area cou ld be in slight d isadvantage in ou r current approach. The success rate for the d irectional flow can be further im proved if the algorithm is auto-tuned w ith respect to the actual hand d imensions of the user. This can be accomplished using our gesture d etection system that alread y extracts the hand contour and orientation. Haptic Knob Experiment The quantitative results of the haptic knob experiment show ed that frictional haptic feed back d id not really improve the task performance in terms of completion time, num ber of overshoots, and recovery tim e after the first overshoot, w hen the same metrics w ere compared to that of no artificial haptic feed back. This resu lt m ay initially appear to be surprising since several earlier stud ies in other d om ains have show ed that haptic feed back improves task performance. H ow ever, it is important to emphasize that there is a major d ifference betw een ou r stud y and the earlier ones. In those stud ies, subjects w ho performed the task und er visual feed back cond ition d id not receive any haptic feed back at all. In our stud y, although no artificial friction w as d isplayed to the subjects und er visu al feed back cond ition, they still felt som e amount of friction w hen their fingers performed the rotation gesture. It appears that the ad d itional friction d isplayed by electrovibration d id not help them m uch in executing the task faster and w ith less error. This outcom e may be related to the type of rotation gesture used in ou r stu d y. Two-finger rotation gesture alread y provid es m ore control to a user d ue to constrained w rist motion and slow er rotational speed . Voelker et al. reported that the task accuracy w ith virtual rotary knob controlled w ith tw o fingers is comparable to those of tangible knobs, w hile the task com pletion is longer [32]. H ow ever, w e have chosen tw o-finger gesture because we observed in our initial experiments that the intensity of haptic feedback drops as the number of fingers increases. Finally, the outcome of our haptic knob stu d y may also be related to the w ay that frictional haptic feed back d isplayed to the subjects in our experiments. Although we have tried a wide range of alternatives, there are still many other options that can be explored in the future. For example, in H D cond ition, w e utilized a pulse signal at sector crossings to imitate the feeling of d etents. Alternatively, a d etent can be, for example, rend ered by leaving a gap betw een tw o subsequent p ulses. A similar argu ment can be extend ed to the other haptic cond itions. In our stud y, w e utilized a sinusoid al voltage to d isplay constant frictional haptic feed back u nd er H D+CF cond ition. H ow ever, a sinusoid al voltage signal that is am plitud e m od ulated to d isplay m ore friction close to the sector bound aries (in ord er to slow d ow n the rotational speed of user) cou ld m ake im provements in task performance. As obvious from the short d iscussion above, there are several alternative choices for the d esign of haptic knob, w hich need s to be further exp lored in the future. On the other hand , the subjective assessment follow ing the knob experiments (via 14 questions in 7 d ifferent categories) show ed that the subjects strongly preferred the haptic knobs over the virtu al one in alm ost all categories (Fig. 17). The results are encouraging and suggest that ad d ing haptic feed back to a virtual knob im p roves interaction qu ality, user experience, and also the confid ence of user. For example, feeling the d etents w hile rotating the knob d oes not perhaps help much in terms of task perform ance, bu t allows the user to receive a confirmation, w hich appears to improve her/ his personal interaction experience and confid ence. CONCLUSION This paper p resents the d esign of a novel multimod al tabletop that effectively combines visu al and haptic mod alities to provid e an interactive exp erience to a user. Users convey their intention of interaction via static and d ynam ic hand gestu res [29], and H ap Table recognizes these gestures in real tim e to d isplay hap tic feed back to the user accord ingly. We d emonstrated the haptic feedback capabilities of our table via tw o exam ple applications along w ith d etailed user stud ies; one for static and one for d ynamic gestu res. H ow ever, H ap Table is not restricted to only these gestures and can be potentially used to d isplay haptic feed back for a variety of other hand gestures, w ith applications in inform ation and d ata visualization, games, entertainment, and ed ucation. As an example for an interaction triggered by a static gesture, w e d isplayed d irectional vibrotactile haptic feedback to the ind ex fingers and hand s of the subjects. Using the piezo patches attached to the ed ges of the tabletop surface, w e su ccessfu lly created an illusion of travelling vibrotactile flow beneath their fingers and hand s in all four d irections. We are not aw are of any earlier stud y investigating vibrotactile flow on large-size tou ch screens such as ours. Using the method s presented in this paper, it is possible to d isplay localized and d irectional vibrotactile hap tic feed back on tabletop surfaces. Our futu re stu d ies w ill investigate the potential applications of this technology in d ata visualization, ed ucation, and gaming. For exam ple, in clim ate visualization, w e im agine that a user can put their hand (s) on the tabletop surface to feel the d irection of w ind forces, virtually overlaid on some other graphical clim ate d ata. Consid ering the fact that clim ate d ata is complex and multi-d im ensional, w hich overload s the visual channel, com m unicating som e clim ate information, such as the w ind forces, through haptic channel m ay alleviate the perceptual and cognitive load on the user, as suggested in [3]. Sim ilarly, in an ed ucational setting, a u ser could better appreciate granular materials such as sand, pebbles, beads, and seeds by shaking virtual cu p s containing them to feel the d ifferences in their vibrations, rather than just observing their m ovements visu ally. As an examp le for d ynam ic gestures, w e haptically rend ered a virtual knob on the table surface using the principles of electrostatic actuation. We investigated the potential benefits of frictional haptic feed back on task performance and user experience in selecting an item from a pull-d ow n menu by rotating the knob. We are not aw are of any earlier stud ies on electrostatic haptic rendering of a virtual knob on a touch surface. This required recognition of rotation gesture, tracking of ind ivid u al finger positions, and d isplaying frictional forces to the user accord ingly, all in real tim e. A knob is just one type of virtual control used in u ser interfaces and our futu re stud ies w ill investigate the haptic versions of the others such as slid er, sw itch, button, and keyboard . For example, a haptic slid er can be rend ered by m od ulating the friction betw een the user's finger and the surface as in the case of haptic knob, w hile a key press can be simulated by localized vibrotactile effects using piezo patches. Once these controls are tested through user stud ies and d esigned as haptic w id gets, they can be customized and integrated into various applications as a part of user interface. In the applications mentioned above and the other potential ones, the challenge is to find the most effective mapping betw een the user hand gestures and the haptic effects. In fact, surface haptics is such a new area of research and even the more fund amental relations betw een the voltage signals applied to the actuators and our haptic sensing and perception are not w ell know n yet. Without fully und erstand ing those relations, d eveloping an effective mapping betw een a gesture and haptic effect is highly challenging. For example, our recent w ork show s that electrovibration generated on a touch surface using a square voltage signal is perceived rougher than a sinusoid al one at low excitation frequencies [33]. For this reason, w e preferred square pulses to rend er the d etents of our haptic knobs over sinu soid al ones (Fig. 15) to make the sector crossings more d etectable by the subjects. Finally, in our current stud y, only one type of haptic m od ality (either vibrotactile or electrovibration) w as utilized to rend er haptic effects in each of the exemplar cases. H owever, the integration of tw o mod alities on the same application m ay lead to richer haptic effects. For exam ple, in our stud y, the d etents of the knob could be d isplayed by vibrotactile feed back w hile frictional forces are conveyed to the user via electrovibration d u ring the rotational movements. Fig. 1 . 1(a) User performing a gesture on the proposed table to interact with a digital scene while receiving suitable haptic feedback. The hardware components of the proposed table are shown in (b) rear and (c) front views. Fig. 3 . 3Selected gestures for HapTable: (a) static (from left to right: 1finger, 2-finger, L-shape, hand with closed and open fingers), and (b) dynamic (from left to right: dragging, rotation, spread/pile, wipe, zoom in/out) gestures. Fig. 2 . 2Schematic showing how voltage is transmitted to piezo patches for vibrotactile feedback, and to touch screen for electrostatic feedback. This design allows HapTable to send different stimulus to an individual or combination of piezo patches and to the electrostatic touch screen. Fig. 6 . 6Displacement FRFs for five excitation cases: patches PA, PB, PC, and PD are excited individually and together in parallel configuration (PALL). Fig. 4 . 4The steps of rotating a hand pose according to a reference edge: (a) image frame obtained by the infrared camera is subtracted from the background, (b) a high-pass filter is applied to reveal parts contacting the table surface, (c) smallest circle enclosing the hand posture is found, (d) the wrist (red arc) and the center point of the hand (point c) are determined, (e) the angle (θ) that the wrist makes with the reference line (l 1 ) is estimated, and (f) the adjusted hand pose perpendicular to the reference edge is obtained. Fig. 5 . 5(a) The setup for vibration measurements: (1) Laser doppler vibrometer (LDV), (2) signal generator, (3) signal analyzer, and (4) touch surface. (b) The tabletop surface was divided into 84 equallyspaced grid points for the measurements (7 rows by 12 columns. PA, PB, PC, and PD represent the piezo patches glued to the edges underneath the touch surface. ed at and around that point. Furthermore, it is even possible to generate d irectional vibrotactile flow betw een any tw o grid points on the screen by simply sw itching betw een the excitation frequ encies correspond ing to the m axim u m d ifference in their d isplacements. For non-grid points, FRFs can be estim ated by bilinear interpolation.For instance, consid er the tw o points illu strated inFig. 7a, contacted by the ind ex fingers of both left and right hand s. To generate a vibrotactile flow from point L to point R, tw o excitation frequencies are carefully chosen from the FRF plots(Fig. 7b)so that point L is the active point and has higher vibration amplitud e than that of point R in the first part of the stim ulus, and vice versa in the second part of the stimulus. The d ifference in FRF plots of points L and R are show n inFig. 7b. This plot show s that the vibration amplitud e of p oint L makes a m axim u m d ifference w ith that of point R (marked w ith orange circle) at 465 H z w hen the p iezo patch PA is active. Sim ilarly, the blue circle ind icates that the vibration am plitud e of point R is significantly larger than that of point L at 428 H z w hen all piezo p atches are excited simultaneously (PALL). H ence, PA is actuated first and then PALL. Fast solid -state relays show n inFig. 2are used to sw itch betw een the actuators PA and PALL d uring the d isplay of haptic stim ulus. Fig. 7 . 7To create a flow from point L to R on the touch screen (a), the FRF graph of point R is subtracted from that of point L for the five excitation cases: when piezo-patches are excited individually and all together (b). The maximum and minimum differences in vibration amplitude are marked by orange and blue circles, respectively. Fig. 9 . 9(a) To create a vibrotactile flow from left (L) to right (R), we first actuate the surface at a certain frequency such that the vibration amplitude of point L is significantly higher than that of the point R, and then actuate the surface at a different frequency such that the vibration amplitude of point R is significantly higher than that of the point L. Note that different piezo actuators could be used to play the voltage signals shown on the first (0 -1.5 seconds) and second (1.5 -3.0 seconds) parts in the figure. the stylus to create a tactile illusion of traveling w ave along its long axis (up to dow n or dow n to up). They demonstrated the potential applications of this stylus in computer games and d ata visualization.The illusion of tactile apparent m otion can also be generated by amplitud e or frequency m od u lation. Kim et al.[24] created the sensation of a traveling w ave betw een tw o vibration actuators embed d ed in a cell phone by ad justing the magnitud e and timing of the actuators. Lim et al.[25] used frequency mod u lation to create a vibrotactile flow betw een tw o hand s hold ing a tablet equip ped w ith vibration motors. Kang et al.[26] used piezo-p atches glued to the short ed ges of a tablet-size glass plate for creating a vibrotactile flow betw een them via frequency m od u lation. They m od ulated the frequency from zero to the first mod e of the plate in ord er to create an illusion of moving tactile stimulu s from one short ed ge of the plate to the other on the opposite sid e. Fig. 11 . 11(a) The area under hand is divided into nine equal squares, s i,j where i and j represent the column and row numbers. In the example shown above, the goal is to create a travelling vibrotactile flow from left to right. The active and passive squares shown in (b) and (e) are symmetric with respect to the vertical line, but (c) and (d) are not. Note that the line of symmetry is vertical (horizontal) for a horizontal (vertical) flow. Fig. Fig. 10. (a) The locations of the test pairs selected for the familiarization (pair: F-F) and the actual experiment (pairs: 1-1, 2-2, 3-3, 4-4) sessions. (b) Distance between the test points for each pair, actuation frequencies (f 1 , f 2 ), the piezo patches used for actuation (p 1 , p 2 ), and the displacement differences between the test points (Δd 1 , Δd 2 ) are given in the table on the right. Fig. 13 . 13Vibrotactile flow under hand: a) percentage of correct responses for each direction, b) percentage of correct responses for each region in the second experiment. The bars represent the mean values while deviations are the standard error of means. Fig. Fig. 12. (a) First, a preliminary experiment was conducted at the region R prelim and then the actual experiment was conducted at four different regions, R 1 , R 2 , R 3 , and R 4 . The size of each region was 12 by 12 cm. (b) Subjects were guided to align their hand position according to the hand image displayed on the screen. After the haptic stimulus was displayed, they were asked to determine the direction of vibrotactile flow by pressing one of the four arrow buttons on the screen. Fig. 14 . 14Evolution of vibration maps for vibrotactile flow under hand: a) left-to-right direction at R 3 recognized with 95.5% accuracy (SD: 6.6%). The first and second parts of the input voltage signal successfully creates localized vibrations on the left and right sides of the hand sequentially, resulting in high accuracy in the subjects' perception of flow direction. b) left-to-right direction at R 4 recognized with 66.4% accuracy (SD: 17.7%). The first part of the signal vibrates the bottom left and right sides of the hand region at the same time, causing confusion in subject's perception about the direction of flow. Fig. 15 . 15(a) A knob with eight sectors where each sector is mapped to an item on the menu. In our experiments, subjects navigate on the menu under four different sensory conditions: 1) virtual (no artificial haptic feedback), 2) haptic detent at sector crossings (b), 3) haptic detent and constant friction (c), and 4) haptic detent and velocity-based friction (d). • Effectiveness: Subjects reported that haptic knobs (H D, H D+CF, H D+VF) w ere significantly m ore effective than the virtual knob (V) (p=0.007). • Ease of use: Although the subjective scores suggest that the haptic knob w as easier to use com pared to the virtual one, the d ifference betw een them w ere not statistically significant. Fig. 16 . 16(a) A knob with eight sectors used in the actual experiments, and the menu is shown on the left, (b) user interface for the subjective evaluation phase.• Efficiency (accuracy): Subjects stated that they selected the target more accurately w hen haptic feedback w as present (p=0.023). This outcome d oes not agree w ith the quantitative results for overshoot count and recovery time metrics. • Efficiency (time): Although subjects personally stated that they completed the task fastest w ith knobs d isplaying H D+CF and H D+VF feed back, the subjective evaluation scores w ere not statistically significant. This outcom e agrees w ith the quantitative results for task completion time. • Interaction quality: Subjects perceived that the haptic knobs w ere more intuitive than the virtual knob d uring their interaction (p=0.026). • A ttractiveness: The results suggest that haptic knobs w ere m ore pleasant and attractive than the virtu al one (p = 0.030). • Fig. 17 . 17Means and standard errors of the subjective measures for each sensory condition (* The mean difference is significant at p=0.05 level). 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[ "ON THE STRUCTURE OF THE FUNDAMENTAL SERIES OF GENERALIZED HARISH-CHANDRA MODULES", "ON THE STRUCTURE OF THE FUNDAMENTAL SERIES OF GENERALIZED HARISH-CHANDRA MODULES" ]	[ "Ivan Penkov \nIVAN PENKOV AND GREGG ZUCKERMAN\n\n", "Gregg Zuckerman \nIVAN PENKOV AND GREGG ZUCKERMAN\n\n" ]	[ "IVAN PENKOV AND GREGG ZUCKERMAN\n", "IVAN PENKOV AND GREGG ZUCKERMAN\n" ]	[]	We continue the study of the fundamental series of generalized Harish-Chandra modules initiated in[PZ2]. Generalized Harish-Chandra modules are (g, k)-modules of finite type where g is a semisimple Lie algebra and k ⊂ g is a reductive in g subalgebra. A first result of the present paper is that a fundamental series module is a g-module of finite length. We then define the notions of strongly and weakly reconstructible simple (g, k)-modules M which reflect to what extent M can be determined via its appearance in the socle of a fundamental series module.In the second part of the paper we concentrate on the case k ≃ sl(2) and prove a sufficient condition for strong reconstructibility. This strengthens our main result from [PZ2] for the case k = sl(2). We also compute the sl(2)-characters of all simple strongly reconstructible (and some weakly reconstructible) (g, sl(2))-modules. We conclude the paper by discussing a functor between a generalization of the category O and a category of (g, sl(2))-modules, and we conjecture that this functor is an equivalence of categories.	10.4310/ajm.2012.v16.n3.a8	[ "https://arxiv.org/pdf/1109.1804v1.pdf" ]	9,312,223	1109.1804	489946598455745b02aee2848ddc1469c3d73141	ON THE STRUCTURE OF THE FUNDAMENTAL SERIES OF GENERALIZED HARISH-CHANDRA MODULES 8 Sep 2011 Ivan Penkov IVAN PENKOV AND GREGG ZUCKERMAN Gregg Zuckerman IVAN PENKOV AND GREGG ZUCKERMAN ON THE STRUCTURE OF THE FUNDAMENTAL SERIES OF GENERALIZED HARISH-CHANDRA MODULES 8 Sep 2011 We continue the study of the fundamental series of generalized Harish-Chandra modules initiated in[PZ2]. Generalized Harish-Chandra modules are (g, k)-modules of finite type where g is a semisimple Lie algebra and k ⊂ g is a reductive in g subalgebra. A first result of the present paper is that a fundamental series module is a g-module of finite length. We then define the notions of strongly and weakly reconstructible simple (g, k)-modules M which reflect to what extent M can be determined via its appearance in the socle of a fundamental series module.In the second part of the paper we concentrate on the case k ≃ sl(2) and prove a sufficient condition for strong reconstructibility. This strengthens our main result from [PZ2] for the case k = sl(2). We also compute the sl(2)-characters of all simple strongly reconstructible (and some weakly reconstructible) (g, sl(2))-modules. We conclude the paper by discussing a functor between a generalization of the category O and a category of (g, sl(2))-modules, and we conjecture that this functor is an equivalence of categories. Introduction 2 1. Notation and preliminary results 3 1.1. Conventions 3 1.2. Reductive subalgebras, compatible parabolics and generic k-types 3 1.3. The fundamental series of generalized Harish-Chandra modules 4 2. On the fundamental series of (g, k)-modules 5 3. On the n-cohomology of (g, k)-modules 8 4. Reconstruction of (g, k)-modules 9 5. Preliminary results on (g, sl (2))-modules 10 6. Strong reconstruction of (g, sl(2))-modules 12 7. k-characters and composition multiplicities of the fundamental series of (g, sl(2))-modules 14 8. Six examples 16 8.1. Background on the principal series of Harish-Chandra modules 17 8.2. g = sl(2) ⊕ sl(2), k is a diagonal sl(2)-subalgebra 17 8.3. g = sl(3), k is a root sl(2)-subalgebra 18 8.4. g = sl(3), k is a principal sl(2)-subalgebra 19 8.5. g = sp(4), k is a long root sl(2)-subalgebra 20 8.6. g = sp(4), k is a short root sl(2)-subalgebra 20 8.7. g = sp(4), k is a principal sl(2)-subalgebra 21 9. Towards an equivalence of categories 22 References 23 Introduction This paper is a continuation of our work [PZ2]. By g we denote a semisimple Lie algebra and by k an arbitrary reductive in g subalgebra. In [PZ2] we introduced the fundamental series of generalized Harish-Chandra modules (or equivalently, (g, k)-modules of finite type over k) and proved that any simple generalized Harish-Chandra module with generic minimal k-type arises as the socle of an appropriate fundamental series module. Using this result we were able to show that any simple generalized Harish-Chandra module with generic minimal k-type can be reconstructed from its n-cohomology. This led to a classification of generalized Harish-Chandra modules with generic minimal k-type. In the present paper we study the fundamental series further. After recalling the necessary preliminaries from [PZ2], we prove in Section 2 that any fundamental series generalized Harish-Chandra module has finite length. In Section 4 we introduce the concepts of a strongly reconstructible and a weakly reconstructible simple generalized Harish-Chandra module. Theorem 3 from [PZ2] implies that any simple Harish-Chandra module with generic minimal k-type is strongly reconstructible; however our aim is to study strong and weak reconstructibility of simple generalized Harish-Chandra modules which do not necessarily have a generic minimal k-type. From Section 5 on, we concentrate on the case when the subalgebra k of g is isomorphic to sl(2), i.e. we consider generalized Harish-Chandra (g, sl(2))-modules. Under this assumption we are able to considerably strengthen the results of [PZ2] and establish new results about strong and weak reconstructibility. In particular, we prove that if M is a simple (g, sl(2))-module with minimal k-type V(µ) satisfying µ ≥ 1 2 (λ 1 + λ 2 ) (note that µ ∈ Z ≥0 as k ≃ sl(2)), λ 1 , λ 2 being the maximum and submaximum eigenvalues in g of a Cartan subalgebra t of k = sl(2), then M is reconstructible by its n-cohomology. This yields a classification of simple (g, sl(2))-modules M with µ ≥ 1 2 (λ 1 + λ 2 ) and proves that all such simple (g, k)-modules have finite type over k. For the principal sl(2)-subalgebra the bound 1 2 (λ 1 + λ 2 ) is linear in rk g, while the bound established in [PZ2] is cubic in rk g. In addition, when k is a direct summand of a symmetric subalgebrak of g, we obtain new reconstruction results for Harish-Chandra modules. In Section 7 we prove that for µ ≥ 1 2 λ 1 , the socle of the fundamental series module is isomorphic to R 1 Γ k,t (L p (E)), where L p (E) is the simple lowest weight module associated to the data (p, E). The relative Kazhdan-Lusztig theory [CC] yields an explicit formula for the t-character of L p (E). In turn, the theory of the derived Zuckerman functors yields an explicit formula for the k-character of the strongly reconstructible module R 1 Γ k,t (L p (E)). Section 8 is devoted to examples. We consider six explicit pairs (g, sl(2)) with rk g = 2 and we compute the respective sharp bounds on µ which ensure that a simple (g, sl(2))-module with minimal sl(2)-type V(µ) is strongly reconstructible. For a principal sl(2)-subalgebra of sp(4) this sharp bound is the one established in the present paper, in the other five cases it turns out to be the bound from [PZ2]. In the final Section 9 we discuss the possibility that, for k ≃ sl(2) and a large enough n, the functor R 1 Γ k,t is an equivalence between a certain category ofp-finite modules Cp ,t,n+2 and a category of (g, k)-modules C k,n . Proving or disproving this statement is an open problem. We conjecture that if n ≥ 1 2 (λ 1 + λ 2 ), then R 1 Γ k,t is an equivalence of categories between Cp ,t,n+2 and C k,n . Acknowledgements. We thank Vera Serganova for pointing out to us Examples 1 and 2 in Subsection 8.7. Both authors acknowledge partial support through DFG Grant PE 980/3-1(SPP 1388). I. Penkov acknowledges the hospitality and partial support of Yale university, and G. Zuckerman acknowledges the hospitality of Jacobs University Bremen. Notation and preliminary results We start by recalling the setup of [PZ2]. 1.1. Conventions. The ground field is C, and if not explicitly stated otherwise, all vector spaces and Lie algebras are defined over C. The sign ⊗ denotes tensor product over C. The superscript * indicates dual space. The sign ⊂ + stands for semidirect sum of Lie algebras (if l = l ′ ⊂ + l ′′ , l ′ is an ideal in l and l ′′ l/l ′ ). H · (l, M) stands for the cohomology of a Lie algebra l with coefficients in an l-module M, and M l = H 0 (l, M) stands for space of l-invariants of M. By Z(l) we denote the center of l. Λ · ( ) and S · ( ) denote respectively the exterior and symmetric algebra. If l is a Lie algebra, then U(l) stands for the enveloping algebra of l and Z U(l) denotes the center of U(l). We identify l-modules with U(l)-modules. It is well known that if l is finite dimensional and M is a simple l-module (or equivalently a simple U(l)-module), Z U(l) acts on M via a Z U(l) -character, i.e. via an algebra homomorphism θ M : Z U(l) → C. We say that an l-module M is generated by a subspace M ′ ⊂ M if U(l) · M ′ = M, and we say that M is cogenerated by M ′ ⊂ M, if for any non-zero homomorphism ψ : M →M, M ′ ∩ ker ψ = {0}. By SocM we denote the socle (i.e. the unique maximal semisimple submodule) of an l-module M; by TopM we denote the unique maximal semisimple quotient of M, when M has finite length. If l is a Lie algebra, M is an l-module, and ω ∈ l * , we put M ω := {m ∈ M \| l · m = ω(l)m ∀l ∈ l}. By supp l M we denote the set {ω ∈ l * \| M ω 0}. A finite multiset is a function f from a finite set D into N. A submultiset of f is a multiset f ′ defined on the same domain D such that f ′ (d) ≤ f (d) for any d ∈ D. For any finite multiset f , defined on an additive monoid D, we can put ρ f := 1 2 d∈D f (d)d. If dim M < ∞ and M = ω∈l * M ω , then M determines the finite multiset ch l M which is the function ω → dim M ω defined on supp l M. 1.2. Reductive subalgebras, compatible parabolics and generic k-types. Let g be a finite-dimensional semisimple Lie algebra. By g-mod we denote the category of g-modules. Let k ⊂ g be an algebraic subalgebra which is reductive in g. We fix a Cartan subalgebra t of k and a Cartan subalgebra h of g such that t ⊂ h. By ∆ we denote the set of h-roots of g, i.e. ∆ = {supp h g} \ {0}. Note that, since k is reductive in g, g is a t-weight module, i.e. g = λ∈t * g λ . We set ∆ t := {supp t g} \ {0}. Note also that the R-span of the roots of h in g fixes a real structure on h * , whose projection onto t * is a well-defined real structure on t * . In what follows, we will denote by Reλ the real part of an element λ ∈ t * . We fix also a Borel subalgebra b k ⊂ k with b k ⊃ t. Then b k = t⊃ + n k , where n k is the nilradical of b k . We set ρ := ρ ch t n k . The quintet g, h, k, b k , t will be fixed throughout the paper. By W we denote the Weyl group of g, and by C(·) -centralizer in g. As usual, we will parametrize the characters of Z U(g) via the Harish-Chandra homomorphism. More precisely, if b is a given Borel subalgebra of g with b ⊃ h (b will be specified below), the Z U(g) -character corresponding to κ ∈ h * via the Harish-Chandra homomorphism defined by b will be denoted by θ κ (θ ρ ch h b is the trivial Z U(g) -character). By , we denote the unique g-invariant symmetric bilinear form on g * such that α, α = 2 for any long root of a simple component of g. The form , enables us to identify g with g * . Then h is identified with h * , and k is identified with k * . We will sometimes consider , as a form on g. The superscript ⊥ indicates orthogonal space. Note that there is a canonical k-module decomposition g = k ⊕ k ⊥ . We also set κ 2 := κ, κ for any κ ∈ h * . We say that an element λ ∈ t * is (g, k)-regular if Reλ, σ 0 for all σ ∈ ∆ t . To any λ ∈ t * we associate the following parabolic subalgebra p λ of g: p λ = h ⊕ ( α∈∆ λ g α ), where ∆ λ := {α ∈ ∆ \| Reλ, σ ≥ 0}. By m λ and n λ we denote respectively the reductive part of p (containing h) and the nilradical of p. In particular p λ = m λ ⊃ + n λ , and if t is b k -dominant, then p λ ∩ k = b λ . We call p λ a t-compatible parabolic subalgebra. A t-compatible parabolic subalgebra p = m⊃ + n (i.e. p = p λ for some λ ∈ t * ) is minimal if it does not properly contain another t-compatible parabolic subalgebra. It is an important observation that if p = m⊃ + n is minimal, then t ⊂ Z(m). In fact, a t-compatible parabolic subalgebra p is minimal if and only if m equals the centralizer C(t) of t in g, or equivalently if and only if p = p λ with λ (g, k)-regular. In this case n ∩ k = n k . Any t-compatible parabolic subalgebra p = p λ has a well-defined opposite parabolic subalgebrā p := p −λ ; clearly p is minimal if and only ifp is minimal. Lemma 1.1. If Reλ([k, k] ∩ t) 0, then p λ and k generate the Lie algebra g. Proof Since Reλ([k, k] ∩ t) 0, there exists an sl(2)-subalgebra k ′ ⊂ k such that Reλ(t ∩ k ′ ) 0. By definition, p λ contains all k ′ -singular vectors of g. Hence p λ generates g as a k ′ -module, i.e. p λ and k ′ generate g. A k-type is by definition a simple finite-dimensional k-module. By V(µ) we denote a k-type with b khighest weight µ (µ is then k-integral and b k -dominant). Let V(µ) be a k-type such that µ + 2ρ is (g, k)-regular, and let p = m⊃ + n be the minimal compatible parabolic subalgebra p µ+2ρ . Putρ n := ρ ch h n and ρ n := ρ ch t n . Clearly ρ n =ρ n \| t . We define V(µ) to be generic if the following two conditions hold: (1) Reµ + 2ρ − ρ n , α ≥ 0 ∀α ∈ supp t n k ; (2) Reµ + 2ρ − ρ S , ρ S > 0 for every submultiset S of ch t n. It is easy to show that there exists a positive constant C depending only on g, k and p such that Reµ + 2ρ, α > C for every α ∈ supp t n implies p µ+2ρ = p and that V(µ) is generic. In agreement with [PZ2], we define a g-module M to be a (g, k)-module if M is isomorphic as a k-module to a direct sum of isotypic components of k-types. If M is a (g, k)-module, we write M[µ] for the V(µ)-isotypic component of M, and we say that V(µ) is a k-type of M if M[µ] 0. We say that a (g, k)-module M is of finite type if dim M[µ] ∞ for every k-type V(µ). We will also refer to (g, k)-modules of finite type as generalized Harish-Chandra modules. Let Θ k be the discrete subgroup of Z(k) * generated by supp Z(k) g. By M we denote the class of (g, k)modules M for which there exists a finite subset S ⊂ Z(k) * such that supp Z(k) M ⊂ (S + Θ k ). If M is a module in M, a k-type V(µ) of M is minimal if the function µ ′ → Reµ ′ + 2ρ 2 defined on the set {µ ′ ∈ t * \| M[µ ′ ] 0} has a minimum at µ. Any non-zero (g, k)-module M in M has a minimal k-type. This follows from the fact that the squared length of a vector has a minimum on every shifted lattice in Euclidean space. 1.3. The fundamental series of generalized Harish-Chandra modules. Recall that the functor of k-locally finite vectors Γ k,t is a well-defined left exact functor on the category of (g, t)-modules with values in (g, k)modules, Γ k,t (M) = M ′ ⊂M,dim M ′ =1,dim U(k)·M ′ <∞ M ′ . By R · Γ k,t := i≥0 R i Γ k,t we denote as usual the total right derived functor of Γ k,t , see [PZ1] and the references therein. If M is a (g, k)-module of finite type, then Γ k,0 (M * ) is a well-defined (g, k)-module of finite type and Γ k,0 (· * ) is an involution on the category of (g, k)-modules of finite type. We put Γ k,0 (M * ) = M * k . There is an obvious g-invariant non-degenerate pairing M × M * k → C. Lemma 1.2. Let W be a finite-dimensional g-module and M be a finite length (g, k)-module of finite type over k. Then a) W ⊗ M is a (g, k)-module of finite type. b) W ⊗ M is a g-module of finite length. Proof a) Since k is finite dimensional and reductive in g, the class of (g, k)-modules is closed under tensor products. Let V(µ) be a k-type. Since W is finite dimensional, Hom k (V(µ), W ⊗ M) Hom k (V(µ) ⊗ W * , M), which is finite dimensional, since V(µ) ⊗ W * is finite dimensional and M has finite type over k. b) Since M has finite length, M is finitely generated over g. Note that M * k , the k-finite dual of M is a (g, k)-module of finite length and hence M * k is finitely generated over g and likewise W * ⊗ M * k is finitely generated. Hence, (W ⊗ M) * k is finitely generated, and satisfies the ascending chain condition. We have already seen that W ⊗ M is finitely generated; thus W ⊗ M satisfies the ascending chain condition. We conclude that W ⊗ M has finite length. We also introduce the following notation: if q is a subalgebra of g and J is a q-module, we set ind g q J := U(g) ⊗ U(q) J and pro g q J := Hom U(q) (U(g), J). For a finite-dimensional p-orp-module E we set N p (E) := Γ t,0 (pro g p (E ⊗ Λ dim n (n))), Np(E * ) := Γ t,0 (pro ḡ p (E * ⊗ Λ dim n (n * ))). Note that both N p (E) and Np(E * ) have simple socles, as long as E itself is simple. The fundamental series of (g, k)-modules of finite type F · (p, E) is defined as follows. Let p = m⊃ + n be a minimal compatible parabolic subalgebra, E be a simple finite dimensional p-module on which t acts via the weight ω ∈ t * , and µ : = ω + 2ρ ⊥ n where ρ ⊥ n := ρ n − ρ. Set F · (p, E) := R · Γ k,t (N p (E)). Then the following assertions hold under the assumptions that p = p µ+2ρ and that µ is b k -dominant, k-integral and yields a generic k-type V(µ) (Theorem 2 of [PZ2]). a) F · (p, E) is a (g, k)-module of finite type in the class M, and Z U(g) acts on F · (p, E) via the Z U(g) -character θ ν+ρ whereρ := ρ ch h b for some fixed Borel subalgebra b of g with b ⊃ h, b ⊂ p and b ∩ k = b k , and where ν is the b-highest weight of E (note that ν\| t = ω). b) F i (p, E) = 0 for i s := dim n k . c) There is a k-module isomorphism F s (p, E)[µ] C dim E ⊗ V(µ), and V(µ) is the unique minimal k-type of F s (p, E). d) LetF s (p, E) be the g-submodule of F s (p, E) generated by F s (p, E)[µ]. ThenF s (p, E) is the unique simple submodule of F s (p, E), and moreover, F s (p, E) is cogenerated by F s (p, E)[µ]. This implies that F s (p, E) * t is generated by F s (p, E) * t [w m (−µ)], where w m ∈ W k is the element of maximal length in the Weyl group W k of k. e) For any non-zero g-submodule M of F s (p, E) there is an isomorphism of m-modules H r (n, M) ω E. 2. On the fundamental series of (g, k)-modules In the rest of the paper, p is a minimal t-compatible parabolic subalgebra and E is a simple finitedimensional p-module. Then n · E = 0 and E is a simple m = C(t)-module. Fix a Borel subalgebra b m in m such that h ⊆ b m . Write b = b m ⊃ + n; then b is a Borel subalgebra of g. Setρ = ρ ch h b . Theorem 2.1. Assume that p = p µ+2ρ and that µ is generic. Assume in addition that N p (E) is a simple g-module. Then F s (p, E) is a simple (in particular, non-zero) g-module. Proof By the Duality Theorem from [EW], (1) (R i Γ k,t (X)) * k R 2s−i Γ k,t (X * t ) for any (g, t)-module X of finite type over t. Set X = N p (E) . Then X is a (g, t)-module of finite type over t (see for instance [Z]), and (1) yields for i = s F s (p, E) * k R s Γ k,t (N p (E) * t ). We have (ind g p (E * ⊗ Λ dim n (n) * )) * pro g p (E ⊗ Λ dim n (n)). Thus N p (E) = Γ k,0 (pro g p (E ⊗ Λ dim n (n))) (ind g p (E * ⊗ Λ dim n (n * ))) * t . Moreover, N p (E) has finite type over t. Hence, N p (E) * t ind g p (E * ⊗ Λ dim n (n * ) ). By Frobenius reciprocity, there is a canonical g-module homomorphism N p (E) * t ind g p (E * ⊗ Λ dim n (n * )) ϕ → Np(E * ) whose restriction to E * ⊗ Λ dim n (n * ) is the identity. As N p (E) * t is simple by our assumption, ϕ is injective. Moreover, ϕ must be surjective as the t-characters of N p (E) * t and Np(E * ) are equal. Therefore there is a commutative diagram of isomorphisms R s Γ k,t (N p (E) * t ) ∼ → R s Γ k,t (Np(E * )) ≀ ↓ ≀ ↓ γ : F s (p, E) * k ∼ → F s (p, E * ). The fact that V(µ) is generic for p implies immediately that V(µ) * is generic forp. Thus F s (p, E) is cogenerated by its minimal k-isotypic component F s (p, E) [µ], and F s (p, E * ) cogenerated by its minimal k-isotypic component. On the other hand, the isomorphism γ implies that F s (p, E * ) is also generated by its minimal k-isotypic component as F s (p, E) * k is generated by its minimal k-isotypic component. We conclude that F s (p, E * ) F s (p, E) * k is simple, which in turn shows that F s (p, E) (F s (p, E) * k ) * k is simple. Assume that the b-highest weight of E is ν ∈ h * . Set ω := ν\| t and µ := ω + 2ρ ⊥ n . Corollary 2.2. Let ν +ρ be b-dominant, i.e. Re ν +ρ, γ ≥ 0 for any root γ of h in b. Then, under the assumption that µ is generic and that p = p µ+2ρ , we have F i (p, E) = 0 for i s. Moreover F s (p, E) is simple. Thus, F i (p, E) has finite length for all i ≥ 0. Proof Under the hypothesis on ν, ind g p (E * ⊗ Λ dim n (n * )) N p (E) * t is simple, hence N p (E) is simple, and the statement follows from Theorem 2.1 In the rest of this section, p is an arbitrary minimal t-compatible parabolic subalgebra. Lemma 2.3. For any C ∈ Z ≥0 , there exists a b-dominant integral weight σ 0 of g such that σ 0 \| t , α > C for every weight α of t in n. Proof Since p is t-compatible, there exists κ ∈ t * such that 2 κ, γ\| t γ, γ ∈ Z ≥0 for every root γ of h in n. Regard t * as a subspace of h * via the Killing form of g restricted to h. Then, 2 κ, γ γ, γ ∈ Z ≥0 for every root γ of h in n and κ, γ = 0 for every root γ of h in m. Hence, κ is a dominant integral weight of h in g. Finally, choose a positive integer r such that rκ, α > C for every weight α of t in n. Then σ 0 = rκ is a b-dominant weight of g as required. Proposition 2.4. Suppose that ν +ρ is b-dominant. Then, F i (p, E) = 0 for i s, and F s (p, E) has finite length. Proof Fix a constant C ∈ Z >0 . Chose m ∈ Z >0 such that Reµ + 2ρ, α ≥ −mC for every α ∈ supp t n. Set σ 1 = (m + 1)σ 0 , where σ 0 is defined in Lemma 2.3. Then Reµ + σ 1 + 2ρ, α ≥ C for every α ∈ supp t n, and by possibly making C larger, we can assume that µ + σ 1 is generic. In particular, p µ+σ 1 +2ρ = p. Next, let ν 1 := ν + σ 1 , and E 1 be a simple finite-dimensional m-module with highest weight ν 1 . Set µ 1 := µ + σ 1 \| t . Then, by Corollary 2.2, F i (p, E 1 ) = 0 for i s, and F s (p, E 1 ) is a simple g-module. Furthermore, by Propositions 2.6 and 2.12 in [Z], F i (p, E) is a direct summand of V g (σ 1 ) * ⊗ F i (p, E 1 ) where V g (σ 1 ) stands for the finite-dimensional g-module with b-highest weight σ 1 . Lemma 1.2 implies the statement. Remark. By a more refined argument with translation functors [BG] one can show using the result of [PZ3] that F s (p, E) is simple and hence non-zero,while F i (p, E) = 0 if i s. Theorem 2.5. The (g, k)-module F i (p, E) has finite length for any simple p-module E and any i ∈ Z ≥0 . Proof We will assume at first that ν +ρ is a regular weight of h in g. Then, there exists a unique element w ∈ W such that w −1 (ν +ρ) is dominant for h in g. Denote by d(ν) the length l(w). We will argue by induction on d(ν). The theorem is true for d(ν) = 0 by Proposition 2.4. Suppose we assume the theorem for d(ν) = n ∈ Z >0 , n being fixed. If d(ν) = n + 1, we can choose a root γ of h in g such that d(s γ (ν)) = n. Let D have highest weight s γ (ν +ρ) −ρ. We will show that the finiteness of the length of F i (p, D) for all i implies the finiteness of the length of F i (p, E) for all i. Case I: ∈ Z ≥0 . Choose a translation functor Ψ so that Ψ(N p (D)) has a central character which is singular with respect to precisely γ. Let Φ be the translation functor adjoint to Ψ. By highest weight module theory we have a short exact sequence 0 → N p (D) → Φ • Ψ(N p (D)) → N p (E) → 0. This short exact sequence yields a long exact sequence ... → R i Γ k,t (N p (D)) → R i Γ k,t Φ • Ψ(N p (D)) → R i Γ k,t (N p (E)) → R i+1 Γ k,t (N p (D)) → ... . We can rewrite this long exact sequence as ... → Φ • Ψ(F i (p, D)) → F i (p, E) → F i+1 (p, D) → .... By assumption, F i (p, D) and F i+1 (p, D) have finite length. Hence, by Lemma 1.2 Φ • Ψ(F i (p, D)) has finite length. By the long exact sequence, F i (p, E) has finite length. Case II: D)), and by Lemma 1.2 the finiteness of the length of F i (p, D) implies the same for F i (p, E). Z ≥0 . Choose an integral weight σ ∈ h * such that if w(ν +ρ) is dominant, s γ w(ν − σ +ρ) is dominant. Let D be a finite-dimensional simple m-module such that the highest weight of D is ν − σ. There exists a translation functor Ψ such that Ψ(N p (E)) = N p (D). For the adjoint functor Φ, Φ(N p (D)) = N p (E). Then R i Γ k,t (Φ(N p (D))) = R i Γ k,t (N p (E)). Hence, F i (p, E) = Φ(F i (p,Corollary 2.6. Let A be a finite-dimensional (p, t)-module. Then R i Γ k,t (N p (A)) has finite length for all i. Proof Induction on the length of A as a p-module: if A has length 1, then n · A = 0, and we are back to Theorem 2.5. In what follows we denote by Cp ,t the full subcategory of g-modules consisting of finitely generated (g, t)-modules which are locallyp-finite. Lemma 2.7. Let N ∈ Cp ,t . Then N has finite length, each simple constituent of N is isomorphic to SocN p (D) for some simple finite-dimensional m-module D, and N has finite type over t. Proof Let {v 1 , ..., v n } generate N over U(g) and let B = (U(p))v 1 + (U(p))v 2 + ... + (U(p))v n . Then B is a finite-dimensional p-module. Moreover, N is a quotient of ind ḡ p B, for which the lemma is well known. Proposition 2.8. Let N ∈ Cp ,t . Then R i Γ k,t (N) has finite length for all i. Proof By Lemma 2.7, N * t ∈ C p,t . Any module in C p,t admits a resolution by modules of the form ind g p C., where each C k is a finite-dimensional (p, t)-module: ind g p C. → N * t → 0. By considering the t-finite dual of this resolution, we obtain a resolution of ( N * t ) * t N by modules of the form N p (A · ), where each A k is a finite-dimensional (p, t)-module. Write this resolution as N ֒→ N p (A · ). We have a convergent spectral sequence of g-modules with E a,b 2 = R a Γ k,t (N p (A b )), and which abuts to R · Γ k,t (N). If we fix i ∈ Z ≥0 , there are only finitely many terms E a,b 2 with a + b = i, since both a ≥ 0 and b ≥ 0. Hence, a+b=i E a,b ∞ has finite length. Finally, R i Γ k,t (N) has finite length By C k we denote the full subcategory of g-mod consisting of (g, k)-modules which have finite type over k and have finite length over g. Theorem 2.9. If N ∈ Cp ,t and i ≥ 0, then R i Γ k,t (N) ∈ C k . Proof The statement follows from Proposition 2.8 and from the "finiteness statement" of Theorem 2.4 c) in [Z]. If A is a full abelian subcategory of g-mod, let K 0 (A) be the Grothendieck group of A. Definition 2.1. If N ∈ Cp ,t , let Θ k,t (N) = (−1) i [R i Γ k,t (N)] in K 0 (C k ). The fact that Θ k,t (N) is well-defined follows from the vanishing statement of Theorem 2.4 b) in [Z]. Proposition 2.10. The map N → Θ k,t (N) yields a non-zero homomorphism Θ k,t : K 0 (Cp ,t ) → K 0 (C k ). Proof This is a well-known fact which follows from the long exact sequence of cohomology. Example. a) If E is a finite-dimensional simple m-module with highest weight ν such that ν +ρ is regular and g-dominant, then Θ k,t (N p (E)) = (−1) s [F s (p, E)]. If µ is b k -dominant and k-integral, then Θ k,t (N p (E)) 0 by the remark after Proposition 2.4. b) Θ k,t (C) = \|W k \|[C], where W k is the Weyl group of k. Indeed, in the proof of Theorem 2.4 in [Z] it is shown that Hom k (V, R i Γ k,t (C)) Ext i k,t (V, C) for any simple finite-dimensional k-module V. Since Ext i k,t (V, C) = 0 for V C, we conclude that Θ k,t (C) = i (−1) i dim Ext i k,t (C, C). Moreover, Ext i k,t (C, C) = H i (k, t, C), where H i (k, t, C) stands for the relative Lie algebra cohomology. It is well-known that H i (k, t, C) is the cohomology of the variety K 0 /T 0 , K 0 being a connected affine real algebraic group with Lie algebra k and T 0 being a torus in K 0 with Lie T 0 = t 0 . Moreover, K 0 /T 0 is homeomorphic to the flag variety of K and hence the Euler characteristic of K 0 /T 0 is \|W k \|, by the Bruhat decomposition of the flag variety. Thus, i (−1) i dim H i (k, t, C) = \|W k \|. 3. On the n-cohomology of (g, k)-modules We start by recalling [PZ2,Proposition 1] in the case when k is semisimple. In what follows, M will denote a (g, k)-module. Note that M is automatically in the class M. Proposition 3.1. In the category of t-weight modules, there exists a bounded (not necessarily first quadrant) cohomology spectral sequence which converges to H · (n, M), with E a,b 1 = H a+b−R(a) (n k , M) ⊗ V * a , where a runs over {0, . . . , y} for some y, R is a monotonic function on {0, . . . , y} with values in Z >0 such that R(a) ≤ a and R(y) = r, V a is a t-submodule of Λ R(a) (n ∩ k ⊥ ) for every a, and V y = Λ r (n ∩ k ⊥ ). We also have R(a)=p V a = Λ p (n ∩ k ⊥ ). Suppose we are interested in H j (n, M) for a fixed j. Write E j 1 := r p=0 H j−p (n k , M) ⊗ Λ p (n ∩ k ⊥ ) * . Then E j 1 = a+b=l E a,b 1 . Lemma 3.2. Fix κ ∈ t * and j, 0 ≤ j ≤ dim n = n. Assume that (E j−1 1 ) κ = (E j+1 1 ) κ = 0. Then (2) H j (n, M) κ ≃ (E j 1 ) κ = r p=0 H j−p (n k , M) ⊗ Λ p (n ∩ k ⊥ ) * κ . Proof This follows immediately from the definition of a convergent spectral sequence of vector spaces. As a special case we have the following lemma. Recall that s := dim(n ∩ k), r := dim(n ∩ k ⊥ ). Lemma 3.3. If H s (n k , M) κ ′ = 0 for κ ′ := κ + 2ρ ⊥ n , then H n (n, M) κ = 0. Proof The isomorphism (2) implies H n (n, M) κ ≃ H s (n k , M) κ ′ ⊗ Λ r ((n ∩ k ⊥ ) * ). As a consequence we have the following. Proposition 3.4. If κ ′ = κ + 2ρ ⊥ n is k-dominant integral, then H n (n, M) κ = 0. Proof Kostant's theorem [Ko] implies that if η is k-dominant integral, H s (n k , V(η)) has pure weight −w m (η) − 2ρ, where w m is the longest element of W k . We have as a consequence that H s (n k , V(η)) κ ′ = 0 if κ ′ is k-dominant integral. Since M is a direct sum of simple finite dimensional k-modules V(η) for various η, we conclude that H s (n k , M) κ ′ = 0. (We have used our assumption that rk k ss > 0). Then H n (n, M) κ = 0 by Lemma 3.3. We now recall that H · (n, M) is an (m, m ∩ k)-module. This is established in [V2,Ch. 5] in the case when k is a symmetric subalgebra but the argument extends to the case of a general reductive in g subalgebra k. Note that m ∩ k = t. The following statement is identical to [PZ2,Corollary 3]. Proposition 3.5. a) If M is a (g, k)-module of finite type, then H · (n, M) is an (m, t)-module of finite type. Moreover, if M is Z U(g) -finite (i.e. the action of Z U(g) on M factors through a finite-dimensional quotient of Z U(g) ) then H · (n, M) is Z U(m) -finite. b) If p is a minimal compatible parabolic subalgebra and M is a (g, k)-module of finite type which is in addition Z U(g) -finite, then H · (n, M) is finite dimensional. 4. Reconstruction of (g, k)-modules Suppose M is simple. Let V(µ) be a minimal k-type of M (a priori V(µ) is not unique). Suppose µ + 2ρ is (g, k)-regular and p = p µ+2ρ . Definition 4.1. The pair (M, µ) as above is strongly reconstructible if H r (n, M) µ−2ρ ⊥ n is a simple m-module and there is an isomorphism of g-modules (3) M ≃ SocF s (p, H r (n, M) µ−2ρ ⊥ n ). The isomorphism (3) implies via Theorem 2 of [PZ3] that V(µ) is the unique minimal k-type of M. Moreover, dim Hom k (V(µ), M) ≤ dim E. Therefore, if the pair (M, µ) is strongly reconstructible, µ is determined by M as the highest weight of the unique minimal k-type of M. This allows us to simply speak of strongly reconstructible simple (g, k)-modules rather than of strongly reconstructible pairs. The first reconstruction theorem of [PZ2] now implies the following. Theorem 4.1. If M is a simple (g, k)-module of finite type with a generic minimal k-type V(µ), then M is strongly reconstructible. Below we will see (in particular in Subsection 8.2) that the converse to the above theorem is false. We will also see (in Subsection 8.7) examples of simple finite-dimensional m-modules E such that F s (p, E) has a reducible socle. Theorem 4.2. Let M 1 , M 2 ∈ C k be simple with central characters θ λ 1 and θ λ 2 respectively. Assume λ 1 and λ 2 are dominant regular with respect to a Borel subalgebra b ⊂ g; assume further that λ 2 − λ 1 is dominant integral . Finally assume that Φ is a translation functor such that Φ(M 1 ) M 2 . Then, M 2 is weakly reconstructible if and only if M 1 is weakly reconstructible. Proof Assume that M 1 is weakly reconstructible. Then for some simple quotient E 1 of H r (n, M 1 ), we have an injection of g-modules α 1 : M 1 → F s (p, E 1 ). By assumption, Φ is an equivalence of categories. Hence we have an injection α 2 : M 2 → Φ(F s (p, E 1 )). By [Z], we have an isomorphism Φ(F s (p, E 1 )) F s (p, E 2 ) for a simple m-module E 2 . Moreover, we have a translation functor Φ m such that Φ m (E 1 ) E 2 . Thus, we have an injection α 1 2 : M 2 → F s (p, Φ m (E 1 )). Now let θ m χ i be the central character of E i for i = 1, 2. Let P m χ 1 and P m χ 2 be the respective projection functors. By assumption we have a surjection of m-modules β 1 : P m χ 1 (H r (n, M 1 )) → E 1 . Apply the translation functor Φ m to β 1 to obtain a surjection β 2 : Φ m P m χ 1 (H r (n, M 1 )) → Φ m (E 1 ) E 2 . By [KV,Ch. 7], Φ m P m χ 1 (H r (n, M 1 )) P m χ 2 (H r (n, M 1 )). Thus, E 2 is a quotient of H r (n, M 2 ). Finally, the injection α 1 2 yields an injection M 2 → F s (p, TopH r (n, M 2 )). Hence M 2 is weakly reconstructible. Preliminary results on (g, sl(2))-modules From now on we assume that k is isomorphic to sl(2, C). We fix a standard basis {e, h, f } for k; the eigenvalues of ad h in g will be integers. Let t = Ch be the Cartan subalgebra of k generated by h, and let p = p h * where h * ∈ t * , h * (h) = 1. The subalgebra p is automatically a minimal t-compatible parabolic subalgebra. For k ≃ sl(2), our Lemma 3.2 simplifies to the following. Lemma 5.1. Fix κ ∈ t * and j, 0 ≤ j ≤ r + 1. Write E j 1 = H 0 (n k , M) ⊗ Λ j (n ∩ k ⊥ ) * ⊕ H 1 (n k , M) ⊗ Λ j−1 (n ∩ k ⊥ ) * . Suppose that (E j−1 1 ) κ = (E j+1 1 ) κ = 0. Then there is an isomorphism of t-modules H j (n, M) κ ≃ (E j 1 ) κ (H 0 (n k , M) ⊗ Λ j (n ∩ k ⊥ ) * ⊕ H 1 (n k , M) ⊗ Λ j−1 (n ∩ k ⊥ ) * ) κ . In particular, let j = r and assume κ ′ = κ + 2ρ ⊥ n is dominant integral for k. Then (E r−1 1 ) κ = 0 implies (4) H r (n, M) κ ≃ H 0 (n k , M) κ ′ ⊕ H 1 (n k , M) ⊗ (n ∩ k ⊥ ) κ ′ . More generally, κ ′ dominant integral implies (5) dim H r (n, M) κ ≤ dim H 0 (n k , M) κ ′ + dim H 1 (n k , M) ⊗ (n ∩ k ⊥ ) κ ′ . From now on we identify integral weights κ of t with the corresponding integers, κ(h). Let λ 1 and λ 2 be the maximum and submaximum weights of t in n ∩ k ⊥ (we consider λ 1 and λ 2 as integers); if λ 1 has multiplicity at least two in n ∩ k ⊥ , then λ 2 = λ 1 . Proposition 5.2. Let µ be a nonnegative integer and let M be a (g, k)-module with the property that δ < µ implies M[δ] = 0. a) If µ ≥ 1 2 λ 1 , then dim H r (n, M) ω ≤ dim H 0 (n k , M) µ . b) If µ ≥ 1 2 (λ 1 + λ 2 ), then dim H r (n, M) ω = dim H 0 (n k , M) µ . Proof a) Our hypothesis on M implies that if M[δ] 0, then δ ≥ µ ≥ 1 2 λ 1 . Since H 1 (n k , M) has weights −δ − 2 with δ as above, we see that (2) implies the inequality in a). b) It suffices to show that (E r−1 1 ) ω = 0. Then the statement from (4), taking into account the vanishing of (H 1 (n k , M) ⊗ (n ∩ k ⊥ )) µ , implies (b). H 1 (n k , M) ⊗ (n ∩ k ⊥ ) µ = 0. Hence We now check that (E r−1 1 ) ω = 0. We have (E r−1 1 ) ω = (H 0 (n k , M) ⊗ Λ r−1 ((n ∩ k ⊥ ) * )) ω ⊕ (H 1 (n k , M) ⊗ Λ r−2 ((n ∩ k ⊥ ) * )) ω . Furthermore, Λ r−1 ((n ∩ k ⊥ ) * ) (n ∩ k ⊥ ) ⊗ Λ r ((n ∩ k ⊥ ) * ), Λ r−2 ((n ∩ k) * ) Λ 2 (n ∩ k ⊥ ) ⊗ Λ r ((n ∩ k ⊥ ) * ) implies (6) (E r−1 1 ) ω = (H 0 (n k , M) ⊗ (n ∩ k ⊥ )) µ ⊕ (H 1 (n k , M) ⊗ Λ 2 (n ∩ k ⊥ ) ) µ as the weight of Λ r (n ∩ k ⊥ ) equals 2ρ ⊥ n . The first term of (6) vanishes as the t-weights of m ∩ k ⊥ are strictly positive and the smallest t-weight of H 0 (n k , M) is µ. The maximal t-weight of the second term is −µ − 2 + λ 1 + λ 2 , hence the inequality µ ≥ λ 1 +λ 2 2 implies the vanishing of the second term of (6). Our next task is to state and prove a vanishing theorem for F 0 (p, E), where E is a simple finite dimensional m-module. Let ω ∈ t * be the weight of t in E. Proposition 5.3. Suppose µ = ω + 2ρ ⊥ n and µ ≥ 0. Then F 0 (p, E) = 0. Proof By definition F 0 (p, E) = Γ k,t (N p (E)). We have N p (E) * t ind g p (E * ⊗ Λ dim(n) (n * ) ) and the b-highest weight of ind g p (E * ⊗ Λ dim n (n * )) equals −ν − 2ρ n ∈ h * . On the other hand, ν ′ (h) ≥ 0 for any b-dominant weight. This follows from the fact that any bdominant weight is a non-negative linear combination of roots of b (see for instance [Kn], p. 686). The g-module N p (E) has a finite-dimensional submodule if and only if N p (E) * t has a finite-dimensional quotient. Note that Γ k,t (N p (E)) is an integrable g-module as N p (E) isp-locally finite and k andp generate g. Therefore, Γ k,t (N p (E)) = 0 whenever N p (E) has no finite-dimensional submodule, i.e. whenever −γ − 2ρ n is not b-dominant. The fact that (−γ − 2ρ n )(h) = −ω − 2ρ n = µ − 2ρ < 0, allows us to conclude that ν − 2ρ n is not b-dominant, i.e. that Γ k,t (N p (E)) = 0. Proposition 5.4. F 2 (p, E) = 0. Proof The statement is a direct corollary of Proposition 3 a) in [PZ2]. Note that this proof does not use genericity. Proposition 5.5. (cohomological Frobenius reciprocity) Let µ ≥ 0. If M is a (g, k)-module such that H · (n, M) is finite dimensional, then we have a natural isomorphism Hom g (M, F 1 (p, E)) Hom m (H r (n, M) ω , E). Proof This follows from the existence of a (not necessarily first quadrant) spectral sequence with E 2term E a,b 2 = Ext a m,t (H r−b (n, M), E) converging to Ext a+b g,k (M, F 1 (p, E)), see Proposition 6 of [PZ2]. By assumption H · (n, M) is finite dimensional. Choose b 0 to be the least possible integer with Ext · m,t (H r−b 0 (n, M), E) 0. By the same argument as in the proof of Theorem 2, b) in [PZ2], we conclude that Hom m (H r−b 0 (n, M), E) 0. Thus, E 0,b 0 2 0 and E a,b 2 = 0 for b < b 0 . Consequently, E 0,b 0 2 E 0,b 0 ∞ and we deduce that Ext b 0 g,k (M, F 1 (p, E)) 0. Hence, b 0 ≥ 0 and the spectral sequence is a first quadrant spectral sequence, with corner isomorphism Hom g (M, F 1 (p, E)) Hom m (H r (n, M), E). Corollary 5.6. Suppose µ ∈ Z ≥0 . a) Let X be any g-submodule of F 1 (p, E). Then E is a quotient of H r (n, X) ω . In particular, if X is simple, then X is weakly reconstructible. b) Let M be a simple (g, k)-module such that H · (n, M) is finite dimensional and E is isomorphic to a quotient of H r (n, M) ω . Then M is isomorphic to a submodule of SocF 1 (p, E). In particular M is weakly reconstructible and M has finite type over k. Corollary 5.7. Fix a central character θ. The set of isomorphism classes of simple (g, k)-modules M with central character θ such that dim H r (n, M) < ∞ and H r (n, M) κ 0 for some κ ∈ Z ≥−2 is finite. Proof A g-module M as in the corollary is isomorphic by Corollary 5.6 (b) to a g-submodule of F 1 (p, E ′ ), where E ′ runs over finitely many simple finite-dimensional p-modules. Since F 1 (p, E ′ ) has finite length for each E ′ by Theorem 2.5, the statement follows. Proof a) Since M is simple, Proposition 3.5 b) implies that H · (n, M) is finite dimensional. Note that the regularity of k in g implies that m is a Cartan subalgebra of g. Hence there exists a 1-dimensional simple m-quotient E of H r (n, M) ω . Proposition 5.5 implies now that any such E induces an injective homomorphism of M into F 1 (p, TopH r (n, M)). In particular, M is weakly reconstructible and is of finite type over k. b) Follows from a). 6. Strong reconstruction of (g, sl (2))-modules In the rest of the paper E, p, m, t, µ, ω are as in Section 1. We start with the following result on the of (g, k)-modules. Theorem 6.1. Let µ ≥ 1 2 λ 1 . Then SocF 1 (p, E) =F 1 (p, E) andF 1 (p, E) is simple. In particular, dim Hom k (V(µ), SocF 1 (p, E)) = dim E. Proof Let X be a non-zero submodule of F 1 (p, E). Since F 1 (p, E) is a (g, k)-module of finite type and is Z Ug -finite, Proposition 3.5 b) and Proposition 5.5 apply to X, yielding a surjective homomorphism of m-modules H r (n, X) ω → E. Hence dim E ≤ dim H r (n, X) ω . Next, by [PZ3], dim H 0 (n k , F 1 (p, E)) ω = dim E, therefore dim H 0 (n k , X) ω ≤ dim E by the left exactness of H 0 (n k , ·). Finally, by Proposition 5.2 a), dim H r (n, X) ω ≤ dim H 0 (n k , X) µ . Combining these inequalities we see that dim H r (n, X) ω = dim H 0 (n k , X) µ = dim E. Hence X[µ] = F 1 (p, E)[µ] , or equivalently X ⊇F 1 (p, E). Since in this wayF 1 (p, E) is contained in any non-zero submodule of F 1 (p, E),F 1 (p, E) is simple andF 1 (p, E) = SocF 1 (p, E). Corollary 6.2. Under the assumptions of Theorem 6.1, let X 0 be a g-submodule of F 1 (p, E). Then the minimal k-type of X is V(µ), dim Hom k (V(µ), X) = dim E, and there is an isomorphism of m-modules H r (n, X) ω E. Proof The statement was established in the proof of Theorem 6.1. Corollary 6.3. Let M be a simple (g, k)-module whose minimal k-type V(µ) satisfies µ ≥ 1 2 λ 1 . Then, if H · (n, M) is finite-dimensional and H r (n, M) ω 0, M is strongly reconstructible. Proof Let E ′ be a simple quotient of the m-module H r (n, M) ω . By Proposition 5.5, M is a simple submodule of F 1 (p, E ′ ), hence by Theorem 6.1 M SocF 1 (p, E ′ ). Corollary 6.4. Let M be a simple (g, k)-module of finite type such that its minimal k-type V(µ) satisfies µ ≥ 1 2 (λ 1 +λ 2 ). Then H · (n, M) is finite-dimensional, and H r (n, M) ω 0, hence M is strongly reconstructible by Corollary 6.3. Proof The statement follows from Corollary 6.4 via Proposition 5.2 b). Corollary 6.5. The correspondences M H r (n, M) ω E SocF 1 (p, E) induce mutually inverse bijections between the set of isomorphism classes of simple (g, k)-modules of finite type M whose minimal k-type V(µ) satisfies µ ≥ 1 2 (λ 1 +λ 2 ) and the set of isomorphism classes of finite dimensional m-modules on which t acts via ω = µ − 2ρ ⊥ n , where Z ≥0 ∋ µ ≥ 1 2 (λ 1 + λ 2 ). Corollary 6.6. Suppose k is regular in g (i.e. let t contain an element regular in g). Suppose M is a simple (g, k)module (not necessarily of finite type over k) with lowest k-type V(µ). If µ ≥ 1 2 (λ 1 + λ 2 ), then E = H r (n, M) ω is a 1-dimensional m-module and M F 1 (p, E). Thus M is strongly reconstructible. In particular, M has finite type over k. Proof We apply Theorem 5.8 a) and Proposition 5.2 b) to conclude that, for any 1-dimensional quotient E of H r (n, M) ω , we have an injection M → F 1 (p, E). Since µ ≥ 1 2 (λ 1 + λ 2 ), there are isomorphisms M F 1 (p, E) and H r (n, M) ω E (the latter is an isomorphism of m-modules). Example. Let g be a classical simple Lie algebra of rank n and k sl(2) be a principal subalgebra. Then the claim of Corollary 6.5 is proved in [PZ2] under the assumption that µ + 1 ≥ 2( i r i ), whereρ = 1 2 i r i α i , α i being the simple roots of b. It is well-known that 2( i r i ) grows cubically with the growth of n, while the value 1 2 (λ 1 + λ 2 ) has linear growth in n. Therefore, for large n, the result of Corollary 6.5 strengthens considerably Theorem 3 of [PZ2] for k being a principal sl(2)-subalgebra . On the other hand, we will see in Section 8 that for n = 2, there are cases where the bound 2( i r i ) − 1 is lower that 1 2 (λ 1 + λ 2 ). Set nowk := k ⊕ C(k) and note thatk is a reductive in g subalgebra. Recall that C(k) ss ⊂ m ss . Moreover, m ss ⊂ C(k) ⇔ m ss = C(k) ss . Proposition 6.7. If m ss = C(k) ss , then for any simple (g,k)-module M of finite type overk, H · (n, M) is finite dimensional. Proof By Proposition 3.5 a), H · (n, M) is an (m, m ∩k)-module of finite type as p isk-compatible (see also [V2,Corollary 5.2.4]). But m ∩k = (Z m ∩k) ⊕ m ss . Hence H · (n, M) is an integrable m-module. Finally, [PZ2, Corollary 3 a)] implies now that H · (n, M) is finite dimensional. We conclude this section with some applications to the case whenk is a symmetric subalgebra. Proposition 6.8. Assume that g is simple andk is symmetric. a) If g is classical with rank ≥ 4, the only case of a symmetric pair of the form (g,k) for which m ss is not equal to C(k) ss is the series (so(2n), so(3) ⊕ so(2n − 3)), where k = so(3). b) If g is exceptional, then m ss = C(k) ss . In fact,k is symmetric if and only if k is conjugate to the sl(2)-subalgebra of a highest root of g. Proof Follows from the classification of symmetric pairs. Corollary 6.9. If rkk = rk g andk is symmetric, then m ss = C(k) ss . Proof Follows from Proposition 6.8, but a more elegant proof is based on Borel -De Siebenthal [BdS]. Corollary 6.10. Assume that k is symmetric and m ss = C(k) ss . Let M be a simple (g,k)-module. a) If H r (n, M) ω 0, then M is strongly reconstructible as a (g, k)-module; in particular M has finite type over k. b) If µ ≥ 1 2 (λ 1 + λ 2 ), then M is strongly reconstructible as a (g, k)-module; in particular M has finite type over k. 7. k-characters and composition multiplicities of the fundamental series of (g, sl(2))-modules Assume µ ∈ Z. Set L p (E) = SocN p (E) and recall that L p (E) is simple. Also note that N p (E) and L p (E) are objects of Cp ,t . Denote by D a variable simple finite-dimensional p-module on which t acts via µ D − 2ρ ⊥ n . Non-negative integers m (E, D) are determined from the equality [N p (E)] = m (E, D)[L p (D)] in the Grothendieck group K 0 (Cp ,t ). We arrange the integers m(E, D) into a matrix (m(E, D)) with rows indexed by all possible E and columns indexed by all possible D; the rows and columns of (m(E, D)) are finitary, i.e. each row and each column have finitely many non-zero entries. The algorithm for computing the integers m(E, D) is discussed in [CC]. Lemma 7.1. Suppose D is not isomorphic to E. Then m(E, D) > 0 implies µ D ≥ µ. Proof We claim that the minimum t-weight of N p (E) is µ + 2. To see this it suffices to note that N p (E) * t ≃ ind g p (E ⊗ Λ dim n (n)) * U(n) ⊗ (E ⊗ Λ dim n (n)) * , as the maximum weight of N p (E) * t is −ω − 2ρ n = −ω − 2ρ ⊥ n + 2ρ ⊥ n − 2ρ n = −µ − 2ρ = −µ − 2 (note that ρ = 1). Thus, if L p (D) is a composition factor of N p (E) and D E, we have µ D + 2 > µ + 2, or equivalently, µ D > µ. Proposition 7.2. Let µ ≥ 0. Then [F 1 (p, E)] = D m(E, D)[R 1 Γ k,t (L p (D))]. Proof Taking into account that Θ is a homomorphism, it suffices to prove: a) F i (p, E) = R i Γ k,t (N p (E)) = 0 for i 1 and µ ≥ 0; b) R i Γ k,t (L p (D)) = 0 for i 1, m(E, D) > 0, µ ≥ 0. Part a) follows from Proposition 5.3 and Proposition 5.4. To prove part b), note that m(E, D) > 0 implies µ D ≥ µ ≥ 0 by Lemma 7.1. Then F 0 (p, D) = 0 implies R 0 Γ k,t (L p (D)) = 0 as R 0 Γ k,t (L p (D)) ⊆ F 0 (p, D). To see that R 2 Γ k,t (L p (D)) = 0, note that, by the Duality Theorem in [EW], (R 2 Γ k,t (L p (D))) * k Γ k,t (L p (D) * t ). The g-module L p (D) * t is p-locally finite, simple and infinite dimensional. As p and k generate g, Γ k,t (L p (D) * t ) 0 would imply dim L p (D) * t < ∞. As the latter is false, Γ k,t (L p (D) * t ) = 0. Part b) is proved. Proposition 7.3. Suppose µ ≥ 0. a) Then R 1 Γ k,t (L p (E)) 0, and the lowest k-type of R 1 Γ k,t (L p (E)) is V(µ) of multiplicity dim E. b) We have the following inclusions of (g, k)-modules: E) is proper and non-zero, and hence F 1 (p, E) is reducible. F 1 (p, E) ⊆ R 1 Γ k,t (L p (E)) ⊆ F 1 (p, E). c) If N p (E) is reducible then the submodule R 1 Γ k,t (L p (E)) of F 1 (p, Proof a) By Proposition 7.2, we have under our hypothesis (7) F 1 (p, E)[µ] = D R 1 Γ k,t (L p (D))[µ]. By Theorem 2 from [PZ3], We conclude that if m(E, D) > 0 and D E, R 1 Γ k,t (L p (D))[µ] = 0. So, from formula (7) above, we deduce that F 1 (p, E)[µ] (dim E)V(µ). Next, apply Proposition 7.2 to D. We see that [R 1 Γ k,t (L p (D))] is a summand of [F 1 (p, D)]. Thus, dim(R 1 Γ k,t (L p (D))[µ]) ≤ dim(R 1 Γ k,t (N p (D))[µ]).(8) F 1 (p, E)[µ] = R 1 Γ k,t (L p (E))[µ]. This proves part a). b) By the vanishing theorems a) and b) in the proof of Proposition 7.2, the inclusion of L p (E) into N p (E) yields an injection of R 1 Γ k,t (L p (E)) into F 1 (p, E); part b) follows immediately from (8) and the definition of D) 0. So, part c) follows from Proposition 7.2 and Proposition 7.3 a). F 1 (p, E). c) If N p (E) is reducible then for some D with µ D > µ ≥ 0, m(E, Conjecture 7.4. If µ ≥ 0, then R 1 Γ k,t (L p (E)) is a semisimple g-module. If true, this conjecture would imply that all simple constituents of R 1 Γ k,t (L p (E)) are weakly reconstructible for µ ≥ 0. This would follow from Corollary 5.6 a). See Subsection 8.7, Examples 1 and 2 for cases when R 1 Γ k,t (L p (E)) is reducible. Theorem 7.5. Assume µ ≥ λ 1 2 . Then R 1 Γ k,t (L p (E)) is a simple (in particular, non-zero) submodule of F 1 (p, E) and R 1 Γ k,t (L p (E)) =F 1 (p, E) = SocF 1 (p, E). Proof By Proposition 7.3, we haveF 1 (p, E) ⊆ R 1 Γ k,t (L p (E)). By the Duality Theorem, (R 1 Γ k,t (L p (E))) * k R 1 Γ k,t (L p (E) * t ). LetF 1 (p, E) ⊥ be the submodule of R 1 Γ k,t (L p (E) * t ) consisting of vectors which are orthogonal to R 1 Γ k,t (L p (E)) via the above duality. By Proposition 7.3, F 1 (p, E)[µ] = R 1 Γ k,t (L p (E))[µ]. Hence,F 1 (p, E) ⊥ [−µ] = 0. By construction,F 1 (p, E) ⊥ is a submodule of F 1 (p, E * ). It follows from the proof of Corollary 6.2 that F 1 (p, E) ⊥ = 0 and henceF 1 (p, E) = R 1 Γ k,t (L p (E)). The statement of Theorem 6.1 implies the remainder of the proof of Theorem 7.5. Corollary 7.6. If µ ≥ λ 1 2 , then: where (p(E,D)) is the matrix inverse to (m (E, D)). e) ch kF 1 (p, E) = p(E, D)ch k F 1 (p, D). (See [PZ1] for a formula for ch k F 1 (p, D).) f) H r (n, R 1 Γ k,t (L p (E))) ω E; in particular, the g-module R 1 Γ k,t (L p (E)) determines E up to isomorphism. a) [F 1 (p, E)] = m(E, D)[F 1 (p, D)]. b) If N p (E) is irreducible, thenF 1 (p, E) = R 1 Γ k,t (L p (E)) = F 1 (p, E), and F 1 (p, E) is irreducible. 2 c)F 1 (p, E)[µ] = F 1 (p, E)[µ]( C dim E V(µ)). d) [F 1 (p, E)] = p(E, D)[F 1 (p, D)], Proof a) Apply Proposition 7.2 and Theorem 7.5. b) If N p (E) is irreducible, then m(E, D) = 0 for D E and m(E, E) = 1. Now apply Corollary 7.6 a). c) Combine formula (8) with Theorem 7.5. d) Follows from a) and the definition of the matrix (p (E, D)). e) Follows from c). f) Apply Corollary 6.2 and Theorem 7.5. Let n ∈ Z and let Cp ,t,n be the full subcategory of Cp ,t consisting of (g, t)-modules N whose weight spaces N α satisfy α ∈ Z and α ≥ n + 2. Let C k,n be the full subcategory of C k consisting of (g, k)-modules M with minimal k-type V(µ) for µ ≥ n. Assume N is a non-zero object in Cp ,t,2 . Lemma 7.7. R i Γ k,t (N) = 0 for i = 0 and 2; R 1 Γ k,t (N) 0. Proof N has a finite composition series with simple subquotients L p (D) in Cp ,t,2 . We know that R i Γ k,t (L p (D)) = 0 for i = 0 and 2. Therefore our claim follows from the long exact sequence for right derived functors. Proposition 7.8. The restriction of R 1 Γ k,t (·) to the full subcategory Cp ,t,2 is a faithful exact functor. Proof The exactness follows from Lemma 7.7. Every map in Cp ,t,2 factors into a composition of surjection followed by an injection. Lemma 7.7 implies that R 1 Γ k,t (·) maps a nonzero surjection to a nonzero surjection and a nonzero injection to a nonzero injection. Proposition 7.9. Suppose M is a (g, k)-module. Then Ext i g,k (M, R 1 Γ k,t (N)) Ext i+1 g,t (M, N) for i ≥ 0. Proof Apply the Frobenius Reciprocity Spectral Sequence in Ch. 6 of [V2], then quote Lemma 7.7. Corollary 7.10. a) If M is a finite dimensional g-module, then Ext i g,k (M, R 1 Γ k,t (N)) Ext i+1 g,t (M, N). b) Suppose N 1 and N 2 are objects in Cp ,t,2 . Then, Hom g,k (R 1 Γ k,t (N 1 ), R 1 Γ k,t (N 2 )) Ext 1 g,t (R 1 Γ k,t (N 1 ), N 2 ) as finite-dimensional vector spaces.Thus, dim Hom g (N 1 , N 2 ) ≤ dim Ext 1 g,t (R 1 Γ k,t (N 1 ), N 2 ). Assume again thatk is symmetric. Let E be a simple finite dimensional m-module. Then R 1 Γ k,t (N p (E)) is a (g, k)-module of finite type, and hence a (g,k)-module of finite type overk, i. e. a Harish-Chandra module. By the Comparison Principle [PZ4,Proposition 2.6], we have a g-module isomorphism R 1 Γ k,t (N p (E)) R 1 Γ˜k ,t⊕C(k) (N p (E)). The (g, k)-module R 1 Γ˜k ,t⊕C(k) (N p (E)), denoted by A(p, E), has been studied extensively in the Harish-Chandra module literature. (See for example [KV].) Corollary 7.11. If µ ≥ 1 2 λ 1 , the Harish-Chandra module A(p, E) has a simple socle, and SocA(p, E) R 1 Γ k,t (L p (E)). Six examples In this section we consider six different pairs (g, k) such that rk g = 2 and k ≃ sl(2). 2 Corollary 7.6 b) is a strengthening of Theorem 2.1 under the assumption that k sl(2). Background on the principal series of Harish-Chandra modules. We start by recalling the construction of the algebraic principal series of (g, s)-modules for a symmetric subalgebra s ⊂ g, [D]. We use this construction in subsections 8.3 -8.6 below. Let s ⊂ g be a symmetric subalgebra of g. Denote by a I a maximal toral subalgebra of s ⊥ . If s is proper, a I is non-zero. Let h I be a Cartan subalgebra of g such that h I = (h I ∩s)⊕a I . Choose an element a ∈ a I such that the eigenvalues of a on g are real and C(a) = (C(a)∩s)⊕a I . Let p I,a = α(a)≥0 g α = m I ⊃ + n I . The following results are proved in [D]. Proposition 8.1. a) g = s + p I,a ; s ∩ p I,a = m I = C(a I ). b) If b I is a Borel subalgebra of m I such that h I ⊂ b I , then b I ⊃ + n I is a Borel subalgebra of g. Hence, p I,a is a parabolic subalgebra of g. c) If a ′ ∈ a I such that C(a ′ ) = C(a), then p I,a ′ is conjugate to p I,a under the connected algebraic subgroup S ⊂ Autg whose Lie algebra is s. We define an element a ∈ s ⊥ to be nondegenerate if C(a) ∩ s ⊥ is a toral subalgebra of s ⊥ . Moreover, an Iwasawa parabolic subalgebra for the pair (g, s) is any subalgebra of the form p I,a for some nondegenerate element a ∈ s ⊥ , such that ada has real eigenvalues in g. Fix an Iwasawa parabolic subalgebra p I ⊂ g. Let L be a finite-dimensional simple module over m I . Endow L with a p I -module structure by setting n I · L = 0. p I , L) is the (g, s)-module Y(q, L) = Γ s (Hom U(q) (U(g), L)), where q is a subalgebra containing p I and the p I -module structure of L extends to a q-module structure. 8.2. g = sl(2)⊕sl(2), k is a diagonal sl(2)-subalgebra. Let g = sl(2)⊕sl(2) and k be the diagonal sl(2)-subalgebra of g. The subalgebra k is regular in g. The pair (g, k) is symmetric and its Harish-Chandra modules have been studied for over half century, see [GN], [B] and [HC]. The parabolic subalgebra p is a Borel subalgebra, and λ 1 = λ 2 = 2. We have ρ n = 2, hence a minimal k-type V(µ) is generic if µ ≥ ρ n − 1 = 1 and there is a bijection between the 1-dimensional complex family {ν ∈ h * \| ν(h) = µ − 2} and the set of isomorphism classes of (g, k)-modules with minimal k-type V(µ). Hence any simple (g, k)-module M with minimal k-type µ ≥ 1 is strongly reconstructible by Theorem 4 in [PZ2]. On the other hand, Corollary 6.5 above, implies this fact under the stronger assumption µ ≥ 2. Note that for each µ, there exists a 1-dimensional complex family of simple (g, k)-modules with minimal k-type V(µ). These modules are multiplicity-free; a self-contained purely algebraic description of these modules in given in [PS]. We now consider the case µ = 0. Proposition 8.5. For any infinite-dimensional simple (g, k)-module M with minimal k-type C = V(0) (i. e. spherical simple (g, k)-module), there exists an h-module E such that M ≃ F 1 (p, E). Proof As a k-module, M is isomorphic to j∈Z ≥0 V(2j), and there is no finite-dimensional simple g-module with the same central character as M (see for instance [PS]). Now choose a 1-dimensional h-module E (in the case we consider, m = h) such that ω = −2 and F 1 (p, E) has the same central character as M. Then F 0 (p, E) = 0, since otherwise F 0 (p, E) would be a finitedimensional (g, k)-module with the same central character as M. By an application of the Euler characteristic principle [PZ1,Theorem 11], F 1 (p, E) is isomorphic as a k-module to j∈Z ≥0 V(2j). Therefore, F 1 (p, E) has the same central character and the same k-character as M, i.e. M F 1 (p, E). Note that modules M as in Proposition 8.5 are not strongly reconstructible. Indeed, if the central character of M is regular, it is not difficult to show that there are precisely two 1-dimensional modules E 1 and E 2 such that M ≃ F 1 (p, E i ) for i = 1, 2. In addition, in this case H 1 (n, M) ω=−2 E 1 ⊕ E 2 . In the case of a singular central character H 1 (n, M) ω=−2 is a non-trivial self-extension of the unique 1-dimensional m-module E such that M ≃ F 1 (p, E). Finally, it is true that any simple (g, k)-module for the pair considered is weakly reconstructible. We also remark that for any µ ≥ 0, F 1 (p, E) can be either irreducible or reducible. 8.3. g = sl(3), k is a root sl(2)-subalgebra. Let g = sl(3) and k be the sl(2)-subalgebra of g generated by the root spaces g ±(ε 1 −ε 2 ) . The subalgebra k is regular in g andk = k ⊕ C(k) is a symmetric subalgebra of g isomorphic to gl(2). The parabolic subalgebra p is a Borel subalgebra with roots ε 1 − ε 3 , ε 3 − ε 2 and ε 1 − ε 2 . Hencẽ ρ n = ε 1 − ε 2 , ρ n = 2, and any simple (g, k)-module M of finite type over k is strongly reconstructible for µ ≥ ρ n − 1 ≥ 1 by Theorem 4 in [PZ2]. On the other hand, λ 1 = 2, λ 2 = 1, hence Corollary 6.3 above implies the strong reconstructibility under the stronger assumption µ ≥ 3 2 . For completeness we note that for a k-type V(μ) ofk, a necessary, but not sufficient condition for V(μ) to be generic is thatμ(h) ≥ 1. Next, ρ ⊥ n = 1 2 ((ε 1 − ε 3 ) + (ε 3 − ε 2 ))(h) = 1 2 (ε 1 − ε 2 )(h), and hence 2ρ ⊥ n = 2. Fix µ ∈ Z ≥1 . As k is regular in g, by Theorem 4 of [PZ2] there exists a bijection between isomorphism classes of simple (g, k)-modules with lowest k-type µ and h-weights ν such that ν(h) = µ − 2. If k is a generator of C(k), observe that ν(k) is a free continuous parameter of ν. An Iwasawa parabolic subalgebra p I ⊂ g relative tok ⊂ g is a Borel subalgebra of g. Hence a finite-dimensional simple p I -module L is 1-dimensional. Write p I = h I ⊃ + n I , and L = L χ for χ ∈ h * I . Proposition 8.6. The principal series module X(p I , L χ ) has finite type over k and has lowest k-type C = V(0). Proof By Frobenius reciprocity for the principal series, Hom˜k(V(μ), X(p I , L χ )) Hom˜k ∩p I (V(μ), L χ ). The right hand side can be computed explicitly. Writet = t + Z(k),t being a Cartan subalgebra ofk with basis h and k. Let ζ ∈t * satisfy ζ(h) = 0, ζ(k) = 1. Then {ρ, ζ} is the basis oft * dual to the basis {h, k} of t. If V(μ) is ak-type we can now writeμ = aρ + bζ, with a ∈ Z ≥0 and b ∈ C. Next,k ∩ p I is a toral subalgebra oft and is spanned over C by h I := 3h + k. The eigenvalues of h I in V(μ) are 3a + b − 6j for j ∈ Z ≥0 , 0 ≤ j ≤ a, all of multiplicity one. The single eigenvalue of h I in L χ is χ(h I ). Hence, Hom Ch I (V(μ), L χ ) 0 precisely when there exists j ∈ N with 0 ≤ j ≤ a such that 3a + b − 6j = χ(h I ) ∈ C. Thus, by Frobenius reciprocity, V(μ) is ak-type of X(p I , L χ ) iff there exists j ∈ Z ≥0 with 0 ≤ j ≤ a such that b = χ(h I ) − 3a + 6j. As a consequence, if V(μ) is ã k-type of X(p I , L χ ), then χ(h I ) − 3a ≤ b ≤ χ(h I ) + 3a. If we restrict the action on X(p I , L χ ) fromk to k we see that the multiplicity of V(a) in X(p I , L χ ) is a + 1 = dim V(a). In Figure 1 we indicate the convex hull of thek-support of X(p I , L χ ). In general, χ(h I ) ∈ C, but in the figure we take χ(h I ) > 0. Proof The existence of an open dense subset U ′ ⊂ h * I such that X(p I , L χ ) is simple for χ ∈ U ′ is established in [Kra]. Moreover, this implies the claim as the set of weakly reconstructible modules depends on one complex and one integer parameters, while the set of irreducible principal series modules depends on two complex parameters. Corollary 8.8. The bound µ ≥ 1 is sharp relative to weak (and also strong) reconstruction for (g, k)-modules of finite type over k. For a classification of simple (g,k)-modules, see [Kra]. 8.4. g = sl(3), k is a principal sl(2)-subalgebra. Let g = sl(3) and k = so(3), the principal sl(2)-subalgebra of g. The subalgebra k is regular in g and it is a symmetric subalgebra of g. The parabolic subalgebra p is a Borel subalgebra and has positive roots ε 1 − ε 2 , ε 2 − ε 3 and ε 1 − ε 3 . Hence,ρ n = ε 1 − ε 3 . Moreover, it is easy to check that ρ n = 4, which shows that µ ∈ Z >0 is generic if µ ≥ ρ n − 1 = 3. On the other hand, λ 1 = 4, λ 2 = 2, so the condition µ ≥ 1 2 (λ 1 + λ 2 ) is equivalent to the same inequality: µ ≥ 1 2 (4 + 2) = 3. Furthermore, we have 2ρ ⊥ n = 4. Fix µ ∈ Z ≥3 . By Theorem 4 of [PZ2], there exists a bijection between isomorphism classes of simple (g, k)-modules with lowest k-type µ and h-weights ν such that ν(h) = µ − 4. If k is a generator of k ⊥ ∩ h, observe that ν(k) is a free continuous parameter of ν. Let p I ⊂ g be an Iwasawa parabolic subalgebra relative to k. The principal series X(p I , L χ ) has two free complex parameters. Let I χ be the sum in X(p I , L χ ) of the k-types V(0), V(2), V(4), .... Since g k ⊕ V(4), it is easy to see that I χ is a g-submodule of X(p I , L χ ) (see for instance [V2,Ch. 4]). A much deeper fact is that I χ splits as a direct summand of two submodules J χ and K χ , where the lowest k-type of J χ is 0 and the lowest k-type of K χ is 2. Furthermore, we have the following. Proposition 8.9. There exists an open dense subset U ⊂ h * I such that K χ , for χ ∈ U, is simple and not weakly reconstructible. Proof The existence of an open dense subset U ′ ⊂ h * I , such that the modules J χ and K χ are simple for χ ∈ U ′ , is established in [V2,Ch. 8]. This implies the claim as (similarly to the proof of Proposition 8.7) the set of weakly reconstructible modules depends on one complex and one integer parameters, while the set of irreducible principal series modules depends on two complex parameters. Corollary 8.10. The bound µ ≥ 3 is sharp relative to weak (and also strong) reconstruction for (g, k)-modules of finite type over k. 8.5. g = sp(4), k is a long root sl(2)-subalgebra. Let g = sp(4). The h-roots of g are ±2ε 1 , ±2ε 2 , ±(ε 1 −ε 2 ), ±(ε 1 + ε 2 ) ∈ h * . Let k be the sl(2)-subalgebra generated by g ±2ε 1 . The nilradical of p has roots ε 1 − ε 2 , ε 1 + ε 2 , 2ε 1 . Hence,ρ n = 2ε 1 , ρ n = 2, and a weight µ ≥ 0 is generic if µ ≥ ρ n − 1 = 1. On the other hand, λ 1 = 2, λ 2 = 1, so the condition µ ≥ 1 2 (2 + 1) ≥ 3 2 is stronger than the genericity condition. Note that 2ρ ⊥ n = 2. Finally, m = t ⊕ C(k), where C(k) is the sl(2)-subalgebra generated by g ±2ε 2 . Fix µ ∈ Z ≥1 . By Theorem 3 of [PZ2], we have a bijection between the following sets: a) isomorphism classes of simple (g, k)-modules M having finite type over k and lowest k-type µ; b) isomorphism classes of simple, finite-dimensional m-modules E such that the highest weight ν of E satisfies ν(h) = µ − 2. Moreover, if k is a generator of h∩C(k), ν(k) is a free but discrete parameter of ν; in this the fundamental series of (g, k)-modules depends on two discrete parameters. Let p I ⊂ g be an Iwasawa parabolic subalgebra relative to the symmetric subalgebrak = k ⊕ C(k). We can choose a Levi decomposition p I = m I ⊃ + n I such that m I ∩k is the diagonal sl(2)-subalgebra ink. Let L χ be a simple finite-dimensional m I -module with highest weight χ ∈ h * I , where h I is a Cartan subalgebra of m I . The weight χ has one discrete and one continuous parameter. Proposition 8.11. The principal series module X(p I , L χ ) has finite type over k and has lowest k-type 0. Proof It is completely analogous to the proof of Proposition 8.6, and Figure 2 is the analogue of Figure 1. Corollary 8.12. Every simple (g,k)-module has finite type over k. Proposition 8.13. If χ is nonintegral as a weight of g, then X(p I , L χ ) is simple and not weakly reconstructible. Proof See [Co], Theorem 2.3.1. Corollary 8.14. The bound µ ≥ 1 is sharp relative to weak (and also strong) reconstruction for (g, k)-modules of finite type over k. Proof The set of simple modules of the form X(p I , L χ ) is not countable while the set of simple weakly reconstructible (g, k)-modules is countable. This implies the claim. 8.6. g = sp(4), k is a short root sl(2)-subalgebra. Let g = sp(4) and k be generated by g ±(ε 1 −ε 2 ) . Then ε 1 (h) = 1, ε 2 (h) = −1. The nilradical of the parabolic subalgebra p has roots ε 1 − ε 2 , 2ε 1 and −2ε 2 . Hence, ρ n = 3 2 (ε 1 − ε 2 ), ρ n = 3, and µ ∈ Z ≥0 is generic if µ ≥ ρ n (h) − 1 = 2. On the other hand, λ 1 = 2, λ 2 = 2, so the condition µ ≥ 1 2 (λ 1 + λ 2 ) ≥ 2 is equivalent to being generic. Note that 2ρ ⊥ n = 4. Finally, m = t ⊕ C(k), where C(k) is the sl(2)-subalgebra generated by g ±(ε 1 +ε 2 ) . Figure 2 Fix µ ∈ Z ≥2 . Theorem 3 of [PZ2], or equivalently Corollary 6.5, implies that we have a bijection between the following sets: a) isomorphism classes of simple (g, k)-modules M having finite type over k and lowest k-type µ; b) isomorphism classes of simple finite-dimensional m-modules E such that the highest weight ν of E satisfies ν(h) = µ − 4. If h ′ is a generator of [g ε 1 +ε 2 , g −ε 1 −ε 2 ], observe that ν(h ′ ) is a free but discrete parameter for ν. We will exhibit a simple (g, k)-module M of finite type over k such that M has lowest k-type 1 but M is not weakly reconstructible. Letk = k ⊕ C(k) gl(2). Let p I be an Iwasawa parabolic subalgebra of g relative tok. This is a Borel subalgebra. Let q be a maximal parabolic subalgebra of g such that q ⊃ p I , q p I . (There are two choices for q.) Write q = l⊃ + u, where l is a reductive part of q and u is the nilradical of q. Note that we can choose l so that l ∩ k = 0 and l ∩k is a 1-dimensional toral subalgebra of l. Next, let L χ be a simple finite-dimensional l-module with h I -highest weight χ. Write Y(q, L χ ) for the degenerate principal series module Γ˜k(Hom U(q) (U(g), L χ )). Since g k ⊕ 2V(2) as a k-module, Y(q, L χ ) is a direct sum of two submodules, Y(q, L χ ) 0 and Y(q, L χ ) 1 corresponding to even highest weights of k and odd highest weights of k, respectively. Lemma 8.15. a) Y(q, L χ ) is a (g, k)-module of finite type over k. b) The lowest k-type of Y(q, L χ ) 0 is C = V(0); the lowest k-type of Y(q, L χ ) 1 is V(1). c) We can choose χ so that the central character of Y(q, L χ ) is not equal to the central character of a fundamental series module for (g, k). Proof Straightforward calculation. Proposition 8.16. The bound µ ≥ 2 is sharp relative to weak reconstruction for (g, k)-modules of finite type over k. Proof We take M to be a simple quotient of U(g) · (Y(q, L χ ) 0 [1]) and we chose χ so that M does not have the central character of a fundamental series module. 8.7. g = sp(4), k is a principal sl(2)-subalgebra. Let g = sp(4) and let k be a principal sl(2)-subalgebra. Here m = h. It is easy to check that ρ n = 7, and the results of [PZ2] (see formula (16) in [PZ2]) imply that any simple (g, k)-module with minimal k-type V(µ) for µ ≥ 6 is strongly reconstructible (and in particular is of finite type). The same follows from Corollary 6.6 under the weaker assumption that µ ≥ 5. Since ρ ⊥ n = 6, λ 1 = 6, λ 2 = 4, Corollary 6.6 implies that, for any µ ∈ Z ≥5 , we have a bijection between the set {ν ∈ h * \| ν(h) = µ − 12} and the set of isomorphism classes of (g, k)-modules with minimal k-type V(µ). Proposition 8.17. The bound µ ≥ 5 is sharp relative to the theorem on strong reconstruction for (g, k)-modules of finite type over k. Proof In [PS] a simple multiplicity-free (g, k)-module M 0 with k-character V(4) ⊕ V(10) ⊕ V(16) ⊕ ... and central character θ M 0 = θ 3 2 e 1 + 1 2 e 2 is exhibited: see equation 6.2 in [PS]. On the other hand, there are 8 fundamental series modules of the form F 1 (b, E) such that θ F 1 (b,E) = θ 3 2 ε 1 +ε 2 . A non-difficult computation shows that their respective minimal k-types are V(10), V(9), V(8), V(5), V(5), V(2), V(1) and V(0). This shows that M 0 is not strongly reconstructible. We do not know whether the bound µ ≥ 5 is sharp relative to weak reconstruction. The following examples demonstrate that Theorem 7.5 does not extend to the case 0 ≤ µ < λ 1 2 . Example 1. There is a unique 1-dimensional b-module E 0 such that θ F 1 (b,E 0 ) = θ 3 2 ε 1 +ε 2 and such that the minimal k-type of F 1 (b, E 0 ) is V(0). By direct computation, X 0 = R 1 Γ k,t (L b (E 0 )) is multiplicity free over k. By comparison with the simple multiplicity free modules discussed in [PS], we conclude that ch k X 0 is the sum of two simple characters. By the Duality Theorem and the fact that X 0 is multiplicity free, we conclude that X 0 is the direct sum of two simple submodules with lowest k-types V(0) and V(4) respectively. This decomposition is consistent with Conjecture 7.4. Moreover, the proper inclusions of (g, k)-modulesF 1 (b, E 0 ) ⊂ R 1 Γ k,t (L b (E 0 )) ⊂ F 1 (b, E 0 ) demonstrate that the inclusions discussed in Proposition 7.3 b) are generally proper. Example 2. There is a unique 1-dimensional b-module E 1 such that θ F 1 (b,E 1 ) = θ 3 2 ε 1 +ε 2 and the minimal k-type of F 1 (b, E 1 ) is V(1). As in Example 1, we find that R 1 Γ k,t (L b (E 1 )) is a direct sum of two simple multiplicity free (g, k)-modules with lowest k-types V(1) and V(3) respectively. Towards an equivalence of categories Recall the categories Cp ,t,n and C k,n introduced in Section 7. Proposition 7.2 and Proposition 7.8 imply that R 1 Γ k,t is a well-defined faithful and exact functor between Cp ,t,n+2 and C k,n for n ≥ 0. Conjecture 9.1. Let n ≥ 1 2 (λ 1 + λ 2 ). Then R 1 Γ k,t is an equivalence between the categories Cp ,t,n+2 and C k,n . Theorem 7.5 implies that if n ≥ λ 1 2 , the simple objects L(E) of the category Cp ,t,n+2 are being mapped by R 1 Γ k,t into simple objects of C k,n , and Corollary 6.4 ensures that, under the stronger condition n ≥ λ 1 +λ 2 2 , R 1 Γ k,t induces a bijection on the isomorphism classes of simple objects of Cp ,t,n+2 and C k,n . Conjecture 9.1 implies the existence of an isomorphism Ext 1 Cp ,t,n+2 (L(E 1 ), L(E 2 )) Ext 1 g (R 1 Γ k,t (L(E 1 )), R 1 Γ k,t (L(E 2 ))) for any simple objects L(E 1 ), L(E 2 ) of Cp ,t,n+2 where n ≥ 1 2 (λ 1 + λ 2 ). We have checked the existence of such an isomorphism by direct computations in the cases of subsections 8.2 and 8.4. In conclusion we note that it is easy to check that Conjecture 9.1 holds for the case when k = g sl(2). In this case 1 2 (λ 1 + λ 2 ) = 0 and R 1 Γ k,t is an equivalence of the categories Cp ,t,n+2 and C k,n for any n ≥ 0. Definition 4. 2 . 2A simple (g, k)-module M of finite type over k is weakly reconstructible if for some minimal k-compatible parabolic subalgebra p, there exists an injective homomorphism of g-modules M ֒→ F s (p, TopH r (n, M)). Theorem 5. 8 . 8Suppose k is regular in g. Let M be a simple (g, k)-module, not necessarily of finite type over k, with lowest k-type V(µ) for µ ≥ 0. a) If H r (n, M) ω 0, there exists a 1-dimensional simple quotient E of H r (n, M) ω . For any such E we have an injective homomorphism M → F 1 (p, E). Hence M is weakly reconstructible and M is of finite type over k. b) If M is of infinite type over k, then H r (n, M) ω = 0. 1 Assume now that m(E, D) > 0 and E D. Then µ D > µ by Lemma 7.1. Applying Theorem 2 from [PZ3] a second time, we have F 1 (p, D)[µ] = 0. Definition 8.1. a) The Iwasawa principal series module corresponding to the pair (p I , L) is the (g, s)-module X(p I , L) = Γ s (Hom U(p I ) (U(g), L)). b) A degenerate principal series module corresponding to the pair ( Lemma 8. 2 . 2There is an isomorphism of s-modules X(p I , L) Γ s (Hom U(m I ∩s) (U(s), L)).In particular, X(p I , L) is a (g, s)-module of finite type; if V is a simple finite-dimensional s-module, then Hom s (V, X(p I , L)) Hom m I ∩s (V, L), hence dim Hom s (V, X(p I , L)) ≤ dim V.A similar statement holds for Y(q, L).Theorem 8.3. (Harish-Chandra's subquotient theorem) Let M be a simple (g, s)-module. Then there exists a simple finite-dimensional m-module L such that M is a subquotient of X(p I , L). Corollary 8.4. For any simple (g, s)-module M and for any s-type V, dim Hom s (V, M) ≤ dim V. Figure 1 1Figure 1 Proposition 8. 7 . 7There exists an open dense subset U ⊂ h * I such that X(p I , L χ ) is simple and not weakly reconstructible for every χ ∈ U. Reν+ρ, γ γ, γ Reν+ρ, γ γ, γ See Theorem 9 of[PZ1]. Irreducible unitary representations of the Lorentz group. V Bargmann, Ann. of Math. 2V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math. (2) 48 (1947), 586-640. Les sous-groups fermés de rang maximum de groupes de Lie clos. A Borel, J De Siebenthal, Comm. Math. Helv. 231A. Borel, J. De Siebenthal, Les sous-groups fermés de rang maximum de groupes de Lie clos, Comm. Math. Helv. 23(1949), No. 1, 200-221. Tensor products of finite and infinite-dimensional representations of semisimple Lie algebras. J Bernstein, S Gelfand, Comp. Math. 41J. Bernstein, S. Gelfand, Tensor products of finite and infinite-dimensional representations of semisimple Lie algebras, Comp. Math. 41 (1980), 245-285. Representations of rank one Lie groups. D Collingwood, Research Notes in Mathematics. 137D. Collingwood, Representations of rank one Lie groups, Research Notes in Mathematics 137, Pitman, London, 1985. The Kazhdan-Lusztig Conjecture for Generalized Verma Modules. L Casian, D Collingwood, Math. Z. 195L. Casian, D. Collingwood, The Kazhdan-Lusztig Conjecture for Generalized Verma Modules, Math. Z. 195 (1987), 581-600. Enveloping Algebras. J Dixmier, American Math. SocJ. Dixmier, Enveloping Algebras, American Math. Soc., 1996. Notes on homological algebra and representations of Lie algebras. T J Enright, N R Wallach, Duke Math. J. 47T.J. Enright, N.R. Wallach, Notes on homological algebra and representations of Lie algebras, Duke Math. J. 47 (1980), 1-15. Unitary representations of the Lorentz group. I M Gelfand, M A Naimark, Izvestiya Akad. Nauk SSSR, Ser. Mat. 11I. M. Gelfand, M. A. Naimark, Unitary representations of the Lorentz group, Izvestiya Akad. Nauk SSSR, Ser. Mat. 11 (1947), 411-504. Infinite irreducible representations of the Lorentz group. Harish-Chandra, Proc. Roy. Soc. London, Ser. A. 189Harish-Chandra, Infinite irreducible representations of the Lorentz group, Proc. Roy. Soc. London, Ser. A 189 (1947), 372-401. A Knapp, Lie Groups: Beyond an Introduction. BostonBirkhauserA. Knapp, Lie Groups: Beyond an Introduction, Birkhauser, Boston, 2002. Lie algebra cohomology and the generalized Borel-Weil theorem. B Kostant, Ann. of Math. 74B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. 74 (1961), 329-387. H Kraljević, Representations of the universal covering group of the group SU(n,1). 28H. Kraljević, Representations of the universal covering group of the group SU(n,1), Glasnik Mat., Ser. III 28 (1973), 23-72. A Knapp, D Vogan, Cohomological Induction and Unitary Representations. Princeton University PressA. Knapp, D. Vogan, Cohomological Induction and Unitary Representations, Princeton Mathematical Series, Princeton University Press, 1995. Bounded simple (g; sl(2))-modules for rkg = 2. I Penkov, V Serganova, Journal of Lie Theory. 20I. Penkov, V. Serganova, Bounded simple (g; sl(2))-modules for rkg = 2, Journal of Lie Theory 20 (2010), 581-615. Generalized Harish-Chandra modules: a new direction of the structure theory of representations. I Penkov, G Zuckerman, Acta Applicandae Mathematicae. 81I. Penkov, G. Zuckerman, Generalized Harish-Chandra modules: a new direction of the structure theory of representations, Acta Applicandae Mathematicae 81(2004), 311-326. Generalized Harish-Chandra modules with generic minimal k-type. I Penkov, G Zuckerman, Asian J. of Math. 8I. Penkov, G. Zuckerman, Generalized Harish-Chandra modules with generic minimal k-type, Asian J. of Math. 8 (2004), 795-812. A construction of generalized Harish-Chandra modules with arbitrary minimal k-type. I Penkov, G Zuckerman, Canadian Mathematical Bulletin. 50I. Penkov, G. Zuckerman, A construction of generalized Harish-Chandra modules with arbitrary minimal k-type, Canadian Mathematical Bulletin 50(2007), 603-609. A construction of generalized Harish-Chandra modules for locally reductive Lie algebras. I Penkov, G Zuckerman, Transformation Groups. 13I. Penkov, G. Zuckerman, A construction of generalized Harish-Chandra modules for locally reductive Lie algebras, Trans- formation Groups 13 (2008), 799-817. The algebraic structure of the representations of semisimple Lie groups I. D Vogan, Ann. of Math. 109D. Vogan, The algebraic structure of the representations of semisimple Lie groups I, Ann. of Math. 109(1979), 1-60. Representations of Real Reductive Lie Groups. D Vogan, Progress in Math. 15BirkhauserD. Vogan, Representations of Real Reductive Lie Groups, Progress in Math. 15, Birkhauser, Boston,1981. Generalized Harish-Chandra Modules. 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[ "ON THE OPENNESS OF THE IDEMPOTENT BARYCENTER MAP", "ON THE OPENNESS OF THE IDEMPOTENT BARYCENTER MAP" ]	[ "Taras Radul " ]	[]	[]	We show that the openness of the idempotent barycenter map is equivalent to the openness of the map of Max-Plus convex combination. As corollary we obtain that the idempotent barycenter map is open for the spaces of idempotent measures. 2010 Mathematics Subject Classification. 52A30; 54C10; 28A33. Key words and phrases. open map; idempotent (Maslov) measure; idempotent barycenter map.	10.1016/j.topol.2019.07.003	[ "https://arxiv.org/pdf/1706.06823v3.pdf" ]	119,135,182	1706.06823	1780a115d4bf483251dec9958a582aba42b3ce82	ON THE OPENNESS OF THE IDEMPOTENT BARYCENTER MAP 31 Jul 2017 Taras Radul ON THE OPENNESS OF THE IDEMPOTENT BARYCENTER MAP 31 Jul 2017arXiv:1706.06823v2 [math.GN] We show that the openness of the idempotent barycenter map is equivalent to the openness of the map of Max-Plus convex combination. As corollary we obtain that the idempotent barycenter map is open for the spaces of idempotent measures. 2010 Mathematics Subject Classification. 52A30; 54C10; 28A33. Key words and phrases. open map; idempotent (Maslov) measure; idempotent barycenter map. Introduction The notion of idempotent (Maslov) measure finds important applications in different part of mathematics, mathematical physics and economics (see the survey article [6] and the bibliography therein). Topological and categorical properties of the functor of idempotent measures were studied in [12]. There are some parallels between the theory of probability measures and idempotent measures (see [10]). The problem of the openness of the barycentre map of probability measures was investigated in [3], [4], [2], [8] and [9]. In particular, it is proved in [8] that the barycentre map for a compact convex set in a locally convex space is open iff the map (x, y) → 1/2(x + y) is open. Zarichnyj defined in [12] the idempotent barycentre map for idempotent measures and asked the following two questions: Question 7.2. [12] Characterize the class of max-plus convex compact spaces for which the idempotent barycenter map is open. In particular, is the latter property equivalent to the openness of the map (x, y) → x ⊕ y? It is proved in [4] In this paper we characterize when the idempotent barycenter map is open. However we show that the openness of the idempotent barycenter map is not equivalent to the openness of the map (x, y) → x ⊕ y. We also give an affirmative answer to the second question. Idempotent measures: preliminaries In the sequel, all maps will be assumed to be continuous. Let X be a compact Hausdorff space. We shall denote the Banach space of continuous functions on X endowed with the sup-norm by C(X). For any c ∈ R we shall denote the constant function on X taking the value c by c X . Let R max = R ∪ {−∞} be the metric space endowed with the metric ̺ defined by ̺(x, y) = \|e x − e y \|. Following the notation of idempotent mathematics (see e.g., [7]) we use the notations ⊕ and ⊙ in R as alternatives for max and + respectively. The convention −∞ ⊙ x = −∞ allows us to extend ⊙ and ⊕ over R max . Max-Plus convex sets were introduced in [13]. Let τ be a cardinal number. Given x, y ∈ R τ and λ ∈ R max , we denote by y ⊕ x the coordinatewise maximum of x and y and by λ ⊙ x the vector obtained from x by adding λ to each of its coordinates. A subset A in R τ is said to be Max-Plus convex if α ⊙ a ⊕ b ∈ A for all a, b ∈ A and α ∈ R max with α ≤ 0. It is easy to check that A is Max-Plus convex iff ⊕ n i=1 λ i ⊙ δ xi ∈ A for all x 1 , . . . , x n ∈ A and λ 1 , . . . , λ n ∈ R max such that ⊕ n i=1 λ i = 0. In the following by Max-Plus convex compactum we mean a Max-Plus convex compact subset of R τ . We denote by ⊙ : R × C(X) → C(X) the map acting by (λ, ϕ) → λ X + ϕ, and by ⊕ : C(X) × C(X) → C(X) the map acting by (ψ, ϕ) → max{ψ, ϕ}. Definition 2.1. [12] A functional µ : C(X) → R is called an idempotent measure (a Maslov measure) if (1) µ(1 X ) = 1; (2) µ(λ ⊙ ϕ) = λ ⊙ µ(ϕ) for each λ ∈ R and ϕ ∈ C(X); (3) µ(ψ ⊕ ϕ) = µ(ψ) ⊕ µ(ϕ) for each ψ, ϕ ∈ C(X). Let IX denote the set of all idempotent measures on a compactum X. We consider IX as a subspace of R C(X) . It is shown in [12] that IX is a compact Max-Plus subset of R C(X) . The construction I is functorial what means that for each continuous map f : X → Y we can consider a continuous map If : IX → IY defined as follows If (µ)(ψ) = µ(ψ • f ) for µ ∈ IX and ψ ∈ C(Y ). By δ x we denote the Dirac measure supported by the point x ∈ X. We can consider a map δX : X → IX defined as δX(x) = δ x , x ∈ X. The map δX is continuous, moreover it is an embedding [12]. It is also shown in [12] that the set I ω X = {⊕ n i=1 λ i ⊙ δ xi \| λ i ∈ R max , i ∈ {1, . . . , n}, ⊕ n i=1 λ i = 0, x i ∈ X, n ∈ N}, (i.e., the set of idempotent probability measures of finite support) is dense in IX. Let A ⊂ R T be a compact max-plus convex subset. For each t ∈ T we put f t = pr t \| A : A → R where pr t : R T → R is the natural projection. Given µ ∈ A, the point β A (µ) ∈ R T is defined by the conditions pr t (β A (µ)) = µ(f t ) for each t ∈ T . It is shown in [12] that β A (µ) ∈ A for each µ ∈ I(A) and the map β A : I(A) → A is continuous. The map β A is called the idempotent barycenter map. It follows from results of [12] that for each compactum X we have β IX • I(δX) = id IX and for each map f : X → Y between compacta X and Y we have β IY • I 2 f = If • β IX . The openness of Max-Plus convex combination of idempotent measures Let X be a Max-Plus convex compactum. We consider a map s X : X × X × [−∞, 0] → X defined by the formula s X (x, y, t) = t ⊙ x ⊕ y. The main goal of this section is to prove that the map s IX : IX × IX × [−∞, 0] → IX is open for each compactum X. We start with a finite X. Proof. Let X = {1, . . . , n}. Since the functor I preserves the weight [12], the compactum IX is metrizable. Consider any (λ, β, t) ∈ IX × IX × [−∞, 0] and a sequence (α j ) in IX converging to t ⊙ λ ⊕ β. It is enough to find sequences (λ j ), (β j ) in IX and a sequence (t j ) in [−∞, 0] such that the sequence (λ j , β j , t j ) converges to (λ, β, t) and t j ⊙ λ j ⊕ β j = α j for each j ∈ N. We have λ = ⊕ n i=1 λ i ⊙ δ i , β = ⊕ n i=1 β i ⊙ δ i and α j = ⊕ n i=1 α j i ⊙ δ i where λ i , β i , α j i ∈ R max such that ⊕ n i=1 λ i = ⊕ n i=1 β i = ⊕ n i=1 α j i = 0. Then t ⊙ λ ⊕ β = ⊕ n i=1 (t ⊙ λ i ⊕ β i ) ⊙ δ i and we have that the sequence α j i converges to t ⊙ λ i ⊕ β i for each i ∈ {1, . . . , n}. We can assume (passing to a subsequence if necessary) that there exists i 0 ∈ {1, . . . , n} such that α j i0 = 0 for each j ∈ N. Consider the case t = 0. We can represent X = A ⊔ B ⊔ C where A = {i ∈ {1, . . . , n}\|λ i < β i }, B = {i ∈ {1, . . . , n}\|λ i > β i } and C = {i ∈ {1, . . . , n}\|λ i = β i }. We can assume that α j i > λi+βi 2 for each j ∈ A ∪ B. Consider the case i 0 ∈ C. Then λ i0 = β i0 = 0. Put λ j i = λ i , i ∈ A, α j i , i / ∈ A and β j i = β i , i ∈ B, α j i , i / ∈ B We have λ j i ≤ 0, β j i ≤ 0 and λ j i0 = β j i0 = 0. Put λ j = ⊕ n i=1 λ j i ⊙ δ i and β j = ⊕ n i=1 β j i ⊙ δ i . Then the sequence (λ j , β j , 0) converges to (λ, β, 0) and λ j ⊕ β j = α j for each j ∈ N. Consider the case i 0 ∈ A. (The proof is analogous for the case i 0 ∈ B.) Put c j = max{α j i \|i / ∈ A}. The sequence (c j ) converges to 0. Put λ j i = λ i , i ∈ A, α j i − c j , i / ∈ A and β j i = β i , i ∈ B, α j i , i / ∈ B We have λ j i ≤ 0, β j i ≤ 0 and β j i0 = 0. We also have λ j i = 0 for each i / ∈ A such that c j = α j i . Put λ j = ⊕ n i=1 λ j i ⊙ δ i and β j = ⊕ n i=1 β j i ⊙ δ i . Then the sequence (λ j , β j , c j ) converges to (λ, β, 0) and c j ⊙ λ j ⊕ β j = α j for each j ∈ N. Finally consider the case t < 0. We have X = A ⊔ B ⊔ C where A = {i ∈ {1, . . . , n}\|t ⊙ λ i < β i }, B = {i ∈ {1, . . . , n}\|t ⊙ λ i > β i } and C = {i ∈ {1, . . . , n}\|t ⊙ λ i = β i }. We can assume that α j i > t+λi+βi 2 for each j ∈ A ∪ B. We also have i 0 ∈ A and β i0 = 0. Put c j = max{α j i − t − λ i \|i / ∈ A} if there exists s ∈ A such that λ s = 0 and c j = max{α j i − t\|i / ∈ A} otherwise. The sequence (c j ) converges to 0. Put λ j i = λ i , i ∈ A, α j i − c j − t, i / ∈ A and β j i = β i , i ∈ B, α j i , i / ∈ B We have λ j i ≤ 0, β j i ≤ 0 and β j i0 = 0. If λ j s = 0 for each s ∈ A, we have λ j i = 0 for each i / ∈ A such that c j = α j i − t. Put λ j = ⊕ n i=1 λ j i ⊙ δ i and β j = ⊕ n i=1 β j i ⊙ δ i . Then the sequence (λ j , β j , t ⊙ c j ) converges to (λ, β, t) and (c j ⊙ t) ⊙ λ j ⊕ β j = α j for each j ∈ N. Let X 1 p − −−− → X 2   f1   f2 Y 1 q − −−− → Y 2 be a commutative diagram. The map χ : X 1 → X 2 × Y2 Y 1 = {(x, y) ∈ X 2 × Y 1 \| f 2 (x) = q(y)} defined by χ(x) = (p(x), f 1 (x)) is called a characteristic map of this diagram. The diagram is called bicommutative if the map χ is onto. Proof. Represent X as the limit of an inverse system C = {X α , p α β , A} consisting of finite compacta and epimorphisms. It is easy to check that s IX = lim{s I(Xα) }. By Proposition 2.10.9 [11] and Lemma 3.1 in order to prove that the map s IX is open, it is sufficient to prove that the diagram I(X α ) × I(X α ) × [−∞, 0] I(p α β )×I(p α β )×id [−∞,0] −−−−−−−−−−−−−−→ I(X β ) × I(X β ) × [−∞, 0]   s I(Xα )   s I(X β ) I(X α ) I(p α β ) − −−− → I(X β ) is bicommutative for each α ≥ β. Without loss of generality, one may assume that X α = {x 1 , . . . , x n+1 }, X β = {y 1 , . . . , y n } (all the points are assumed to be distinct) and the map p α β act as follows: p α β (x i ) = y i for each i ∈ {1, . . . , n} and p α β (x n+1 ) = y n . Thus, given (ν, (µ, α, t)) ∈ I(X α ) × I(X β ) I(X β ) × I(X β ) × [−∞, 0] one can write ν = ⊕ n+1 i=1 ν i ⊙ δ xi , µ = ⊕ n i=1 µ i ⊙ δ yi and α = ⊕ n i=1 α i ⊙ δ yi . Since I(p α β )(ν) = t ⊙ µ ⊕ α, we have ν i = t ⊙ µ i ⊕ α i , i ∈ {1, . . . , n − 1} and ν n ⊕ ν n+1 = t ⊙ µ n ⊕ α n . Put λ i = µ i , η i = α i , i ∈ {1, . . . , n − 1}, λ n = min{µ n , ν n − t}, λ n+1 = min{µ n , ν n+1 − t} and η n = min{α n , ν n }, η n+1 = min{α n , ν n+1 }. It is a routine checking that λ n ⊕ λ n+1 = µ n , η n ⊕ η n+1 = α n and ν n = t ⊙ λ n ⊕ η n , ν n+1 = t ⊙ λ n+1 ⊕ η n+1 . Hence we obtain s I(Xα) (λ, η, t) = ν and I(p α Proof. Choose a continuous onto map f : Y → X such that Y is a 0-dimensional compactum and there exists a continuous l : X → IY such that If • l = δX. Existence of such map was proved in [12]. (It is called β ) × I(p α β ) × id [−∞,0] (λ, η, t) = (µ, α, t) for λ = ⊕ n+1 i=1 λ i ⊙ δ xi and η = ⊕ n+1 i=1 η i ⊙ δ xi . The main results We characterize openness of the barycenter map in this section. Since the set I ω X is dense in IX, the following lemma can be obtained by direct checking for idempotent measures of finite support. β X (⊕ k i=1 λ i ⊙ µ i ) = ⊕ k i=1 λ i ⊙ β X (µ i ).∈ I ω X ∩ U such that β X (ν) = β X (µ). Proof. By d µ we denote the density of µ. Let U = {U 1 , . . . , U k } be a closed Max-Plus convex cover of X. For i ∈ {1, . . . , k} put s i = max{d µ (y) \| y ∈ U i } and define a function d i : X → [−∞, 0] by the formula d i (x) = d µ (x) − s i , x ∈ U i , −∞, x / ∈ U i . It is easy to check that d i is an upper semicontinuous function with max d i = 0. Denote by µ i the idempotent measure determined by d i and put x i = β X (µ i ). Define ν U ∈ I ω X by the formula ν U = ⊕ k i=1 s i ⊙ δ xi . By Corollary 4.2 we have β X (ν U ) = β X (⊕ k i=1 s i ⊙ µ i ). Since ⊕ k i=1 s i ⊙ µ i = µ, we obtain β X (ν U ) = β X (µ) . Now {ν U } forms a net where the set of all finite closed Max-Plus convex covers is ordered by refinement. Then ν U → µ. (β −1 X (V ) × β −1 X (U ) × O) is open in IX. Then β X • s IX (β −1 X (V ) × β −1 X (U ) × O) is open in X. Let us show that β X • s IX (β −1 X (V ) × β −1 X (U ) × O) = s X (V × U × O). Consider any y ∈ β X • s IX (β −1 X (V ) × β −1 X (U ) × O). Then there exists (µ, ν, p) ∈ β −1 X (V ) × β −1 X (U ) × O such that β X • s IX (µ, ν, p) = y. It follows from Lemma 4.1 that β X • s IX (µ, ν, p) = s X (β X (µ), β X (ν), p), hence y ∈ s X (V × U × O). Now take any z ∈ s X (V × U × O). Then there exists (r, q, p) ∈ V × U × O such that z = s X (r, q, p). By Lemma 4.1 we have z = β X • s IX (δ r , δ q , p). Hence z ∈ β X • s IX (β −1 X (V ) × β −1 X (U ) × O). 3.⇒ 1. Consider any ν = ⊕ k i=1 λ i ⊙ δ xi ∈ I ω X. We will prove that for each net {x α } converging to β X (ν) there exists a net {ν α } converging to ν such that β X (ν α ) = x α for each α. We use the induction by k. For k = 1 the statement is obvious. Let us assume that we have proved the statement for each k ≤ l ≥ 1. Consider k = l + 1. Then ν = ⊕ l+1 i=1 λ i ⊙ δ xi . We can assume that λ l+1 = 0. Put t = ⊕ l i=1 λ i and ν 1 = ⊕ l i=1 ((−t) ⊙ λ i ) ⊙ δ xi . We have t ⊙ ν 1 ⊕ δ x l+1 = ν. Hence t ⊙ β X (ν 1 ) ⊕ x l+1 = β X (ν). Consider any net {x α } in X converging to β X (ν). Since the map s X is open, there exists a net {(y α , x α l+1 , t α )} in X × X × [−∞, 0] converging to (β X (ν 1 ), x l+1 , t) such that t α ⊙ y α ⊕ x α l+1 = x α . By the induction assumption there exists a net {ν α 1 } converging to ν 1 such that β X (ν α 1 ) = y α . Then the net {t α ⊙ ν α 1 ⊕ δ x α l+1 } converges to ν and β X (t α ⊙ ν α 1 ⊕ δ x α l+1 ) = x α for each α. = {(µ, γ) ∈ ID × ID \| \|µ(ϕ)\| < 1/2} of (δ 0 , δ 1 ) in ID × ID. Consider any pair (α, β) ∈ ID ×ID such that α⊕β = ν −1/i for some i ∈ N. We have α = α 0 ⊙δ 0 ⊕α 1 ⊙δ 1 and β = β 0 ⊙δ 0 ⊕β 1 ⊙δ 1 for some α 0 , α 1 , β 0 , β 1 ∈ R max such that α 0 ⊕ α 1 = β 0 ⊕ β 1 = 0. Since α 0 ≤ −1/i, we have α 1 = 0 and α(ϕ) ≥ 1/2. Hence (α, β) / ∈ O and the map (x, y) → x ⊕ y is not open. Let {X α } α∈A be a family of Max-Plus convex compacta. Then the product X = α∈A X α has a natural structure of Max-Plus convexity with coordinatewise operation: Let X be a Max-Plus convex compactum. Following [5] we call a point x ∈ X an extremal point if for each two points y, z ∈ X and for each t ∈ [−∞, 0] the equality x = t ⊙ y ⊕ z implies x ∈ {y, z}. The set of extremal points of a Max-Plus convex compactum X we denote by ext(X). t ⊙ (x α ) ⊕ (y α ) = (t ⊙ x α ⊕ y α ) where (x α ), (y α ) ∈ X and t ∈ [−∞, 0].x γ α , y γ α , t γ )} converging to (x α , y α , t) in X α × X α × [−∞, 0] such that s Xα (x γ α , y γ α , t γ ) = z γ α . Theorem 4.7. Let X be a Max-Plus convex compactum such that the map β X is open. Then the set ext(X) is closed in X. Proof. Suppose the contrary. There exists a net {x α } in ext(X) converging to a point x / ∈ ext(X). Then there exist y, z ∈ X and t ∈ [−∞, 0] such that x = t ⊙ y ⊕ z and x / ∈ {y, z}. Evidently, y = z. There exist open neighborhoods V , U of y and z correspondingly such that x / ∈ Cl(V ∪ U ). We can suppose that x α / ∈ (V ∪ U ) for each α. Since the map s X is open, there exist α, y α ∈ V , z α ∈ U and p ∈ [−∞, 0] such that x α = p ⊙ y α ⊕ z α . Since x α ∈ ext(X), we have x α ∈ {y α , z α } ⊂ V ∪ U and we obtain a contradiction. that the product of barycentrically open compact convex sets (i.e. compact convex sets for which the barycentre map is open) is again barycentrically open. Question 7.3. [12]Is an analogous fact true for idempotent barycentrically open Max-Plus convex sets? Lemma 3 . 1 . 31The map s IX is open for each finite compactum X. Lemma 3. 2 . 2The map s IX is open for each 0-dimensional compactum X. The map s IX is open for each compactum X. an idempotent Milyutin map.) Define a map γ : IX → IY by the formula γ = β IY • Il. Then we have If• γ = If • β IY • Il = β IX • I 2 f • Il = β IX • I(If • l) = β IX • I(δX) = id IX .Since I preserves surjective maps, γ is an embedding and we can consider IX as a subset of IY . (We identify IX with γ(IX)).PutT = s −1 IY (IX). The map s IY \| T : T → IX is open. Then we have s IY \| T = s IX • (If × If × id [−∞,0] ) and s IX is open being a left divisor of the open map s IY \| T . Lemma 4 . 1 . 41The equality β X • s IX = s X • (β X × β X × id [−∞,0]) holds for each Max-Plus convex compactum X. Corollary 4 . 2 . 42Let X be a Max-Plus convex compactum, µ 1 , . . . , µ k ∈ IX and λ 1 , . . . , λ k ∈ [−∞, 0] be numbers such that max{λ 1 , . . . , λ k } = 0. Then we have The notion of density for an idempotent measure was introduced in[1]. Let µ ∈ IX. Then we can define a function d µ : X → [−∞, 0] by the formula d µ (x) = inf{µ(ϕ\|ϕ ∈ C(X) such that ϕ ≤ 0 and ϕ(x) = 0}, x ∈ X. The function d µ is upper semicontinuous and is called the density of µ. Conversely, each upper semicontinuous function f : X → [−∞, 0] with max f = 0 determines an idempotent measure ν f by the formula ν f (ϕ) = max{f (x)⊙ϕ(x)\|x ∈ X}, for ϕ ∈ C(X). Lemma 4 . 3 . 43Let X be a Max-Plus convex compactum, µ ∈ IX and U be an open neighborhood of µ. Then there exists ν Theorem 4 . 4 . 44Let X be a Max-Plus convex compactum. Then the following statements are equivalent:(1) the map β X \| Iω X : I ω X → X is open; (2) the map β X is open; (3) the map s X is open. Proof.The implication 1.⇒ 2. follows from Lemma 4.3. 2.⇒ 3. Consider any (x, y, t) ∈ X × X × [−∞, 0] and let W be an open neighborhood of (x, y, t). We can suppose that W = V × U × O where V , U and O are open neighborhoods of x, y and t correspondingly. Since the map s IX is open by Theorem 3.3, the set s IX Theorems 3.3 and 4.4 yield the following corollary. Corollary 4. 5 . 5The map β IX is open for each compactum X. Let us consider an example of a Max-Plus convex compactum K such that the map β K is open but the map (x, y) → x ⊕ y is not. This example gives a negative answer to the second part of Zarichnyi Question 7.2 and demonstrate some difference between the theory of probability measures and idempotent measures. Put K = ID where D = {0, 1} is a two-point discrete compactum. Then the map β K is open by Corollary 4.5. Put ν t = t⊙δ 0 ⊕δ 1 . Then the sequence {ν −1/i } converges to ν 0 = δ 0 ⊕δ 1 . Consider a function ϕ ∈ C(D) defined by the formula ϕ(i) = i, i ∈ {0, 1} and an open neighborhood O Theorem 4 . 6 . 46Let {X α } α∈A be a family of Max-Plus convex compacta such that all the maps β Xα are open. Then the map β X is open. Proof. We have by Theorem 4.4 that all the maps s Xα: X α × X α × [−∞, 0] → X α are open. Consider the map s X : X × X × [−∞, 0] → X. Take any point ((x α ), (y α ), t) ∈ X × X × [−∞, 0] and put z = (z α ) = s X ((x α ), (y α ), t).Let {z γ } be a net converging to z. Consider any α ∈ A. We have that the net {z γ α } converges to z α . Since the map s Xα is open, there exists a net {( Then the net {((x γ α ), (y γ α ), t γ )} converges to ((x α ), (y α ), t) in X × X × [−∞, 0] and s X ((x γ α ), (y γ α ), t γ ) = z γ . Hence the map s X is open. The map β X is open by Theorem 4.4. An example of a convex compactum with the closed set of extremal points and not open barycenter map was build in[8]. We construct an idempotent counterpart. Consider a subset Y ⊂ [−2, 0] 2 defined as followsY = A ∪ B ∪ C where A = {(x, y) ∈ [−2, 0] 2 \| x ∈ [−2, −1], y = −1}, B = {(x, y) ∈ [−2, 0] 2 \| x = −1 ∈ [−2, −1], y ∈ [−2, −1]} and C = {(x, y) ∈ [−1, 0] 2 \| x = y}.It is easy to see that Y is a Max-Plus convex compactum. Consider points a = (−2, −1), b = (−1, −2), c = (−1, −1) and a sequence (c i ) where c i = (−1 + 1 i , −1 + 1 i ) for i ∈ N. Evidently the sequence (c i ) converges to c. Put ν = δ a ⊕ δ b . We have b Y (ν) = c. It is easy to check that there is no sequence (ν i ) converging to ν and such that b Y (ν i ) = c i . Hence b y is not open but ext(X) = {a, b, (0, 0)}. Densities of idempotent measures and large deviations. M Akian, Trans. of Amer.Math.Soc. 35111M.Akian, Densities of idempotent measures and large deviations, Trans. of Amer.Math.Soc. 351 (1999), no. 11, 4515-4543. Openness of convex averaging. L Q Eifler, Glasnik Mat. Ser. III. 321L.Q. Eifler, Openness of convex averaging, Glasnik Mat. Ser. III, 32 (1977), no. 1, 67-72. On a barycentric map of probability measures. V V Fedorchuck, Vestn. Mosk. Univ, Ser. I. 1V.V. Fedorchuck, On a barycentric map of probability measures, Vestn. Mosk. Univ, Ser. I, No 1, (1992), 42-47. V V Fedorchuck, On barycentrically open bicompacta. 33V.V. Fedorchuck, On barycentrically open bicompacta, Siberian Mathematical Journal, 33 (1992), 1135-1139. Katz Max-plus convex geometry, Relations and Kleene algebra in computer science. S Gaubert, R , Lecture Notes in Comput. Sci. 4136SpringerS. Gaubert, R. Katz Max-plus convex geometry, Relations and Kleene algebra in computer science, 192206, Lecture Notes in Comput. Sci., 4136, Springer, Berlin, 2006. The Maslov dequantization, idempotent and tropical mathematics: a very brief introduction, Idempotent mathematics and mathematical physics. G L Litvinov, Contemp. Math. 117Amer. Math. SocG. L.Litvinov, The Maslov dequantization, idempotent and tropical mathematics: a very brief introduction, Idempotent mathematics and mathematical physics, 117, Contemp. Math., 377, Amer. Math. Soc., Providence, RI, 2005. V P Maslov, S N Samborskii, Idempotent Analysis. ProvidenceAmer. Math. Soc13V.P. Maslov, S.N. Samborskii, Idempotent Analysis, Adv. Soviet Math., vol. 13, Amer. Math. Soc., Providence, 1992. On the openness of the barycentre map. R C O'brien, Math. Ann. 223R.C. O'Brien, On the openness of the barycentre map, Math. Ann., 223 (1976), 207-212. On the geometry of stable compact convex sets. S Papadopoulou, Math. Ann. 229S. Papadopoulou, On the geometry of stable compact convex sets, Math. Ann., 229 (1977), 193-200. Absolute retracts and equiconnected monads. T , Topology Appl. 202T. Radul, Absolute retracts and equiconnected monads, Topology Appl. 202 (2016), 1-6. Categorical Topology of Compact Hausdorff Spaces. A Teleiko, M Zarichnyi, VNTL Publishers. LvivA. Teleiko, M. Zarichnyi, Categorical Topology of Compact Hausdorff Spaces, VNTL Publishers. Lviv, 1999. Spaces and mappings of idempotent measures. M Zarichnyi, Izv. Ross. Akad. Nauk Ser. Mat. 74M. Zarichnyi, Spaces and mappings of idempotent measures, Izv. Ross. Akad. Nauk Ser. Mat. 74 (2010), 45-64. A general separation theorem in extremal algebras. K Zimmermann, Ekon.-Mat. Obz. 13K. Zimmermann, A general separation theorem in extremal algebras, Ekon.-Mat. Obz. 13 (1977) 179-201.	[]
[ "Medium-modification of photon-tagged jets in AA collisions", "Medium-modification of photon-tagged jets in AA collisions" ]	[ "B G Zakharov \nCenter for Interdisciplinary Studies in Physics and Related Areas\nGuizhou University of Finance and Economics\n550025GuiyangChina\n\n) L.D. Landau Institute for Theoretical Physics\nKosygina Str. 2GSP-1, 117940, 117334MoscowRussia\n" ]	[ "Center for Interdisciplinary Studies in Physics and Related Areas\nGuizhou University of Finance and Economics\n550025GuiyangChina", ") L.D. Landau Institute for Theoretical Physics\nKosygina Str. 2GSP-1, 117940, 117334MoscowRussia" ]	[]	We study nuclear modification of the photon-tagged jets in AA collisions within the jet quenching scheme based on the light-cone path integral approach to the induced gluon emission. The calculations are performed for running coupling. Collisional energy loss is treated as a perturbation to the radiative mechanism. We obtain a reasonable agreement with the recent data from the STAR Collaboration on the mid-rapidity nuclear modification factor IAA for Au+Au collisions at √ s = 200 GeV for parametrization of running αs consistent with that necessary for description of the data on suppression of the high-pT spectra.PACS numbers:	10.1134/s1063776117100107	[ "https://arxiv.org/pdf/1706.03980v2.pdf" ]	119,189,628	1706.03980	a15a768e248f4aaabc2e7c196279e6a24ff43dcb	Medium-modification of photon-tagged jets in AA collisions 15 Jun 2017 B G Zakharov Center for Interdisciplinary Studies in Physics and Related Areas Guizhou University of Finance and Economics 550025GuiyangChina ) L.D. Landau Institute for Theoretical Physics Kosygina Str. 2GSP-1, 117940, 117334MoscowRussia Medium-modification of photon-tagged jets in AA collisions 15 Jun 2017(Dated: March 25, 2018)PACS numbers: We study nuclear modification of the photon-tagged jets in AA collisions within the jet quenching scheme based on the light-cone path integral approach to the induced gluon emission. The calculations are performed for running coupling. Collisional energy loss is treated as a perturbation to the radiative mechanism. We obtain a reasonable agreement with the recent data from the STAR Collaboration on the mid-rapidity nuclear modification factor IAA for Au+Au collisions at √ s = 200 GeV for parametrization of running αs consistent with that necessary for description of the data on suppression of the high-pT spectra.PACS numbers: I. INTRODUCTION Results from RHIC and LHC on heavy-ion collisions give strong evidence for production of a deconfined quark gluon plasma (QGP). One of the main signature of the QGP formation in AA collisions is the discovery at RHIC and LHC of the extremely strong suppression of the highp T hadron spectra. It is commonly believed that this suppression is a consequence of the jet modification (jet quenching) due to the final state interaction with the QGP produced in the initial stage of AA collisions. The jet quenching is caused by the radiative [1][2][3][4][5] and collisional [6] energy loss of fast partons in the QGP. The RHIC and LHC data on the nuclear modification factor R AA , characterizing the suppression of the high-p T spectra, can be reasonably described by the radiative and collisional parton energy loss in the QGP with dominant contribution from the radiative mechanism due to the induced gluon emission. A consistent analysis of the jet quenching phenomenon requires understanding multiple gluon emission. The available approaches to the induced gluon emission [1,2,4,5] deal with one gluon emission. At the one gluon level in the light-cone path integral (LCPI) [2] approach the spectrum of gluon emission by a quark may be expressed via the retarded Green function of a two dimensional Schrödinger equation, in which the longitudinal coordinates (along the initial quark momentum) plays the role of time, and the imaginary potential is proportional the cross section of interaction of the threebody qq-system with the medium constituent. The diagram technique developed in Refs. [2] allows one to go beyond the one gluon level. However, already for the double gluon emission, even in a crude oscillator approximation [7,8] (when the potential is approximated by a quadratic form), calculations become extremely complicated [9]. And up to now there are no phenomenological schemes for the jet quenching analyses that treat accurately the double gluon emission. Anyway the double gluon level is insufficient for analyses of the jet quenching data from RHIC and LHC. In the presently available analyses of the nuclear modification factor R AA the effect of multiple gluon emission is usually accounted for in the approximation of independent gluon emission [10][11][12][13], similar to the Landau method for multiple soft photon emission in QED. This approximation does not account for the effect of the gluon cascading that may be important for the medium-modified fragmentation functions (FFs) in the soft region z ≪ 1. Nevertheless, this approximations seems reasonable to calculate the nuclear modification factor R AA . Because it depends mostly on the form of the medium-modified FFs for parton→hadron transitions in the region of intermediate and large z, where the main effect of multiple gluon emission is the Sudakov suppression which should not be sensitive to the details of the in-medium parton cascading at z ≪ 1. Since at small z the approximation of independent gluon emission becomes questionable, it is of course highly desirable to perform comparison of the theoretical predictions obtained in the approximation of independent gluon emission [10][11][12][13] with experimental observables that are sensitive to the form of the mediummodified FFs in a broad range of z. Experimentally the information about the medium jet modification in a broad range of z can be obtained from measurement of the photon-tagged jet FFs in γ+jet events [14]. The medium effects in jet fragmentation in γ+jet events are characterized by the nuclear modification factor I AA , defined as the ratio I AA = D AA h (z) D pp h (z) ,(1) where D AA h (z) and D pp h (z) are the γ-triggered jet FFs for AA and pp collisions, respectively. The mid-rapidity factor I AA has been recently measured by the STAR Collaboration [15] at RHIC for central Au+Au collisions at √ s = 200 GeV in a broad range of z for the photon energies 12 < E γ T < 20 GeV. In the present paper we perform a comparison of the STAR data [15] with the theoretical predictions for I AA obtained within the model of jet quenching that we de-veloped in Ref. [13], and previously used in Refs. [16][17][18] for successful description of the data on the nuclear modification factor R AA . The scheme is based on the LCPI approach [2] to the induced gluon emission. The method allows one to treat accurately the Landau-Pomeranchuk-Migdal (LPM) effect and the finite-size effects. The calculations are performed for running coupling. We perform numerical calculations beyond the oscillator approximation when parton multiple scattering in the medium can be described in terms the well known transport coefficientq [1]. The model also includes the contribution of the collisional energy loss. The plan of the paper is as follows. In Sec. 2 we review our theoretical framework. In the first subsection we discuss our model for the in-medium FFs. In the second subsection we discuss the model of the QGP fireball used in our numerical calculations. In the last subsection we discuss calculations of the nuclear modification factor I AA . In Sec. 3 we present comparison of our numerical results with the STAR data [15] on the nuclear modification factor I AA . Sec. 4 summarizes our work. II. THE THEORETICAL FRAMEWORK In this section, we review the main aspects of our theoretical framework for calculation of the nuclear modification factor I AA that characterizes the jet medium modification in AA collisions. A. Medium-modified FFs As was said in Introduction, currently the first principle analysis of the jet medium modification in AA collisions is impossible. Our treatment of the medium modified FFs D m h/i for parton→ hadron transitions for AA collisions is similar to that developed in our analysis [13] of the nuclear modification factor R AA . It is based on the LCPI approach [2] to the induced gluon emission from fast partons in the QGP. For reader's convenience, and since some of the details have been omitted in our concise paper [13], in this subsection we discuss the important points of our model for calculation of the medium modified FFs. We assume that that the parton→hadron transition consists of the three stages: the DGLAP cascading, the induced gluon emission stage in the QGP, and parton hadronization outside the QGP. For a given jet trajectory in the fireball the FF D m h/i reads D m h/i (z, Q) = 1 z dz ′ z ′ D h/j (z/z ′ , Q 0 )D m j/i (z ′ , Q) ,(2) where D h/j describes parton hadronization outside the QGP, and D m j/i corresponds to transition of the initial hard parton i to the parton j escaping from the QGP. The partonic FF D m j/i includes the parton evolution in the DGLAP stage and medium modification in the QGP. We write it as a convolution D m j/i (z, Q) = 1 z dz ′ z ′ D in j/k (z/z ′ , E k ) ×D DGLAP k/i (z ′ , Q 0 , Q) ,(3) where E k = z ′ Q. The FF D DGLAP k/i (z, Q 0 , Q) describes the first DGLAP stage for parton→parton transition in the parton cascading from the initial parton virtuality Q to a small virtuality scale Q 0 , where the DGLAP cascade is stopped. The FF D in j/k (z, E k ) corresponds to the inmedium parton→parton transition in the QGP fireball. It depends on the energy of the parton k. In the absence of the medium the D in j/k in Eq. (3) reduces to the unit operator δ jk δ(z − 1), and D m j/i (z, Q) becomes equal to D DGLAP j/i (z, Q 0 , Q). In this case (2) reduces to its vacuum counterpart corresponding to FF for pp collisions D h/i (z, Q) = 1 z dz ′ z ′ D h/j (z/z ′ , Q 0 ) ×D DGLAP j/i (z ′ , Q 0 , Q) .(4) As in Ref. [13] we take Q 0 = 2 GeV for the FFs D h/j (z, Q 0 ) in Eq. (2), describing parton→hadron transition outside the QGP (and in Eq. (4) for pp collisions). For these FFs we use the KKP [19] parametrization. The DGLAP FFs D DGLAP k/i (z, Q 0 , Q) have been computed with the help of the PYTHIA event generator [20]. It was used to create a grid of values for D DGLAP k/i (z, Q 0 , Q) in the z − Q plane. Our method for calculation of the FFs for pp collisions in the form of the convolution of the KKP FF at Q 0 = 2 GeV and the DGLAP FFs guarantees that in the limit of the vanishing induced radiation the medium-modified FFs given by Eqs. (2), (3) exactly reduce to the pp FFs (4). We checked that quantitatively our formula (4) reproduces reasonably well the Qdependence of the KKP FFs [19]. Nevertheless, to avoid the effect of a possible difference between the KKP FFs and the FFs given by Eq. (4) on predictions for I AA , the use of the form (4) is clearly preferred for numerical calculations of the I AA . The form given by Eqs. (2), (3) assumes that the DGLAP and the induced gluon emission stages are approximately ordered in time. This picture seems to be reasonable for initial parton energies ∼ < 100 GeV, because in this region the typical formation time for emission of the first most energetic gluon in the DGLAP cascade turns out to be relatively small. The gluon formation length emitted by a fast quark in vacuum is approximately L f (x, k T ) ∼ 2E q x(1 − x)/(k 2 T + ǫ 2 ) ,(5) where x is the gluon fractional momentum, and ǫ 2 = m 2 q x 2 + m 2 g (1 − x) . From Eq. (5) using the vacuum spectrum of the gluon emission from a quark dN dk 2 T dx = C F α s (k 2 T ) πx 1 − x + x 2 /2) k 2 T (k 2 T + ǫ 2 ) 2(6) we obtained that for E ∼ < 100 GeV the typical gluon formation lengthL f ∼ 0.3 − 0.5 fm (we take m g = Q 0 since we are interested in the typical length for gluons with virtuality Q ∼ > Q 0 ). This says that the hardest gluon emission in the DGLAP cascade typically occurs before formation of the equilibrated QGP, which is expected at the proper time τ 0 ∼ 0.5 fm. The above estimate for the typical time of the DGLAP stage agrees with qualitative L-dependence of the fast parton virtuality Q(L) ∼ Q/L (which can be obtained from the uncertainty relation ∆E∆t ∼ 1). This says that for L ∼ τ 0 ∼ 0.5 fm for the initial partons with E ∼ 10 − 50 GeV we have Q(L) ∼ 2 − 4 GeV. For this reason the scale Q 0 ∼ 2 GeV, that is reasonable for the lower end of the DGLAP evolution, at the same time translates to the longitudinal scale that agrees qualitatively with the QGP production time where the induced gluon emission comes into play. Note however, that our numerical calculations show that the results are practically insensitive to the value of Q 0 . For this reason it does not make sense to try to find more appropriate value of Q 0 which better corresponds to the space-time picture of the QGP production and transition of the DGLAP stage to the stage of the induced gluon emission in the QGP. Anyway, in the DGLAP cascade the time of parton splitting can be only estimated very roughly. One remark about the arguments of the FFs in Eqs. (2), (3) is in order. For the DGLAP and the hadronization stages the FFs in Eqs. (2), (3) are written as functions of the parton virtualities. But the in-medium FF D in j/k is a function of the parton energy. It is because we evaluate the induced gluon spectrum within the old fashioned perturbation theory in the coordinate representation in which particles are not characterized by virtuality. In this formulation the virtuality may be estimated from the length scale L and the parton energy E with the help the uncertainty relation that gives Q ∼ E/L which, of course, matches the above formula for Q(L). We are fully aware that the picture with the time ordering of the DGLAP and the induced gluon emission stages may be questionable for the DGLAP gluon emission with sufficiently small transverse momenta (about the Debye mass of the QGP) when the vacuum gluon formation length becomes as large as the typical formation length for the induced gluon emission. Note that just because of the interference of the vacuum gluon emission with the induced one the induced gluon spectrum vanishes for zero medium size. In the form (2), (3) these interference effects are assigned to the in-medium FF D in j/k . In the absence of a consistent approach to the in-medium parton cascading it is difficult to estimate the theoretical uncertainties from the use of the representation given by Eqs. (2), (3). It is worth noting that a formal interchange in Eq. (3) of the DGLAP and in-medium FFs practically does not change the results. We calculate the in-medium FFs D in j/k (z, E k ) in the approximation of independent induced gluon emission [10] using for the one gluon emission distribution the induced gluon spectrum in the form obtained in Ref. [21] within the LCPI approach [2]. The form of Ref. [21] does not require calculation of the singular Green's function as in the original representation of the spectrum of Refs. [2]. The method of Ref. [21] reduces calculation of the gluon spectrum to solving a two-dimensional Schrödinger equation in backward time direction with a smooth boundary condition. It is convenient for accurate numerical calculations beyond the oscillator approximation. For the reader's convenience in Appendix A we give the necessary formulas. In the approximation of independent gluon emission the quark fractional energy loss distribution in ξ = ∆E/E can be written as [10] (hereafter, for notational simplicity, we omit argument E) W (ξ) = W 0 ∞ n=1 1 n! n i=1 1 0 dx i dP dx i δ ξ − n i=1 x i ,(7) where W 0 = exp − 1 0 dx dP dx (8) is the no gluon emission probability, dP/dx is the probability distribution for the q → gq transition with x = E g /E q . At ξ ≪ 1 the main effect of the multiple gluon emission is the Sudakov suppression. It is well seen from the approximate calculation of (7) at the level of twogluon emission for the regime when the relative energy loss ∆E/E = 1 0 dxxdP/dx is much smaller than unity. In this regime, similarly to the electron energy loss [22], from (7) one can obtain W (ξ) ≈ dP dξ exp   − 1 ξ dx dP dx    {1 − 1 2 ξ 0 dx 1 dP dx 1 + dP dx 2 − dP dx 1 dP dx 2 dP dξ −1    ,(9) where x 1 + x 2 = ξ. The exponential Sudakov suppression factor in (9) reflects a simple fact that emission of gluons with the fractional momentum bigger than ξ is forbidden. For accurate numerical calculation of the distribution W (ξ) given by Eq. (7) it is convenient to rewrite it as a series [11] W (ξ) = ∞ n=1 W n (ξ) ,(10) where W n are determined by the recurrence relations W n+1 (ξ) = 1 n + 1 ξ 0 dxW n (ξ − x) dP dx (11) with W 1 (ξ) = W 0 dP dξ .(12) In numerical calculations we set dP/dx = 0 at x < m g /E q and 1 − x < m q /E q . The expression (7) satisfies the relations ∞ 0 dξW (ξ) = 1 ,(13)∞ 0 dξξW (ξ) = 1 0 dxx dP (x) dx .(14) For any value of the ratio ∆E/E, the formula (7) leads to some leakage of the probability and the fractional momentum to the unphysical region of ξ > 1. The effect is small at ∆E/E ≪ 1, but for the conditions of the jet quenching in AA collisions when ∆E/E is not very small, say, ∼ 0.2 [23] for light quarks at E ∼ 10 − 20 GeV for RHIC conditions, the effect may be sizeable. To ensure the flavor conservation 1 0 dzD in q/q (z) = 1 ,(15) we define a renormalized distribution in the physical region ξ < 1 W R (ξ) = K qq W (ξ)(16) with K qq = ∞ 0 dξW (ξ) 1 0 dξW (ξ) .(17) Then this renormalized distribution is used to define the in-medium q → q FF D in q/q (z) = W R (1 − z)) .(18) We define the FF for q → g transition as D in g/q (z) = K gq dP/dz.(19) We determine the coefficient K gq from the momentum sum rule 1 0 dzz D in q/q (z) + D in g/q (z) = 1 .(20) In the limit ∆E/E → 0 K gq → 1. Indeed, at one gluon emission level the z-distribution for q → g transition is connected to that for q → q by interchange of the arguments z ↔ 1 − z. Then after setting the upper limit of ξ-integration in Eqs. (13), (14) to 1, we conclude that the momentum sum rule (20) is satisfied for K gq = 1. For the g → g transition we use the following procedure. In the first step we define D in g/g (z) in the region z > 0.5, where the Sudakov suppression is important, through the independent gluon emission distribution W (ξ) (with ξ = 1 − z) given by Eq. (7) using for dP/dx the x-distribution for g → gg induced transition. Note that for g → gg transition due to the x ↔ 1 − x symmetry of the function dP/dx we can use 0.5 for the upper limit in x-integrations in Eqs. (7), (8) (it means that we view the softest gluon with x < 0.5 as a radiated gluon). In the soft region z < 0.5, where one can expect a strong compensation of the multiple gluon emission and the Sudakov suppression, we use simply the one gluon distribution dP/dx (with x = z). This procedure can violate the momentum sum rule 1 0 dzzD in g/g (z) = 1 ,(21) which should be satisfied. To cure this drawback we multiply the FF obtained at the first step by a renormalization coefficient K gg defined from the momentum conservation (21). Note that for the jet path length in the QGP L ∼ 5 fm and the typical jet energy ∼ 15 GeV, that is relevant to conditions of the STAR experiment [15], the necessary values of the renormalization coefficients turn out to be not very far from unity (K qq ≈ 1.1, K gq ≈ 0.8 and K gg ≈ 0.7). This says that we are in a regime when the leakage of the probability to the unphysical region ξ > 1 and the violation of the momentum sum rule for the distribution (7) are relatively small. Note that in our calculations the process g → qq is included into the DGLAP FF in Eq. (3), but we neglect the induced gluon conversion into qq pairs in calculations of the in-medium FF D in j/k . Calculations within the LCPI formalism [2], using the formulas given in Appendix A, show that for light quarks for the QGP produced in AA collisions for RHIC and LHC conditions the probability of the induced g → qq transition turns out to be relatively small. For conditions of the STAR data [15], when the typical gluon energy E ∼ 15 GeV and the typical path length in the QGP L ∼ 5 fm, the probability of the gluon conversion into the qq states ∼ < 10%. From the point of view of the nuclear modification factors I AA and R AA the effect of this process should be negligible since the hadronization of the qq state should be similar to that of a gluon. The above formulas do not include the effect of the the collisional energy loss. Presently there is no a consistent approach for incorporating of the collisional energy loss in jet quenching calculations on an even footing with the radiative mechanism. In the present analysis, as in Refs. [13,[16][17][18], we treat the collisional energy loss (that is relatively small [23]) as a perturbation to the radiative mechanism, and incorporate it by a small renormalization of the initial QGP temperature for the radiative in-medium FFs D in i/k according to the change in the ∆E due to the collisional energy loss (see Ref. [13] for details). We evaluate the collisional energy loss using the Bjorken method [6], but with an accurate treatment of kinematics of the binary collisions (the details can be found in Ref. [23]). In calculations of the induced gluon emission distribution dP/dx we take for parton masses the quasiparticle masses in the QGP. We use the quasiparticle masses m q = 300 and m g = 400 MeV supported by the analysis of the lattice data [24]. Note that the results are practically insensitive to the quark mass. We use the Debye mass µ D in the QGP obtained in the lattice analysis [25], which gives µ D /T slowly decreasing with T (µ D /T ≈ 3 at T ∼ 1.5T c , µ D /T ≈ 2.4 at T ∼ 4T c ). However, the sensitivity of the results to the Debye mass is relatively weak. Because the spectrum dP/dx is mostly controlled by the behavior of the dipole cross section (see Eq. (A5)) in the region ρ ∼ < 1/µ D , where it depends on µ D only logarithmically. Both for radiative and collisional mechanisms we use the one-loop running α s frozen at low momenta at some value α f r s (see Appendix A). The use of the same parametrization of running α s for the radiative and collisional mechanisms is important for minimizing the theoretical uncertainties in the fraction of the collisional contribution. The results of the analyses of the low-x structure functions [26] and of the heavy quark energy loss in vacuum [27] show that for gluon emission in vacuum for this parametrization α f r s ≈ 0.7 − 0.8. However, the thermal effects can suppress the in-medium QCD coupling, and we treat α f r s as a free parameter. B. Model of the QGP fireball As in our previous analyses of the nuclear modification factor R AA [13,[16][17][18] we use the ideal gas model of the QGP with Bjorken's 1+1D expansion [28]. It gives T 3 0 τ 0 = T 3 τ , where τ 0 is the thermalization time of the matter. We take τ 0 = 0.5 fm. As in Refs. [13,[16][17][18], for simplicity, we neglect variation of T 0 with the transverse coordinates. We take the medium density ∝ τ for τ < τ 0 . This is just an ad hoc prescription to account for the fact that the QGP production is clearly not an instantaneous process. Note however, that the effect of the region τ < τ 0 is small. It is due to a strong finite-size suppression of the induced gluon emission in the regime when the parton path length is smaller than the gluon formation length [7,8]. We fix the initial temperature of the plasma fireball in AA collisions from the initial entropy density determined via the charged particle multiplicity pseudorapidity density, dN AA ch /dη, at mid-rapidity (η = 0) calculated from the two component Glauber wounded nucleon model [29] dN AA ch dη = (1 − α) 2 N part + αN coll dN pp ch dη ,(22) where dN pp ch /dη is the multiplicity density for pp collisions, N part and N coll for a given impact parameter b are given by the Glauber formulas N part (b) = 2 dρT A (ρ) {1 − exp[−T A (b − ρ)σ N N in ] ,(23)N coll (b) = σ N N in dρT A (ρ)T A (b − ρ)(24) with T A (b) = dzρ A (b, z) the nuclear profile function. The N part and N coll have been calculated with the Woods-Saxon nuclear distribution ρ A (r) = ρ 0 1 + exp[(r − R A )/a](25) with R A = (1.12A 1/3 − 0.86/A 1/3 ), and a = 0.54 fm [30]. We use dN pp ch /dη = 2.65 and σ in pp = 35 mb obtained by the UA1 Collaboration [31] for non-single-diffractive events for pp collisions at √ s = 200 GeV. We take for the fraction of the binary collisions α = 0.135 adjusted to reproduce the experimental STAR data [32] on centrality dependence of mid-rapidity dN AA ch /dη in Au+Au collisions at √ s = 200 GeV [33]. We determine the entropy density with the help of the Bjorken relation [28] s 0 = C τ 0 πS f dN AA ch dη .(26) Here C = dS/dy dN AA ch /dη ≈ 7.67 [34] is the entropy/multiplicity ratio, and S f is the transverse area of the QGP fireball. We define it as the overlapping area of the colliding nuclei. In calculating the S f we use the nuclear matter disk radius r = R A +ka with k = 1.5 used in our analysis [18] of the nuclear modification factor R AA . In the physically reasonable range k ∼ 1 − 2, the results for I AA have a weak dependence on k. This occurs because for the parton energy loss the change in the jet path length with variation of k is approximately compensated by the corresponding change in the fireball density. For Au+Au collisions at √ s = 200 GeV for 0 − 12% centrality bin as in Ref. [15] this procedure gives T 0 ≈ 327 MeV (we take N f = 2.5 to account for the mass suppression for the strange quarks in the QGP). For the above value of the initial temperature in 1 + 1D Bjorken's expansion the QGP reaches T ∼ T c (here T c ≈ 160 is the crossover temperature) at τ QGP ∼ 4 − 5 fm. We treat the crossover region as a mixed phase [28]. For the relevant values of the proper time τ ∼ < 8 − 10 fm (see below) the QGP fraction in the mixed phase is approximately ∝ 1/τ [28]. For this reason we can use in calculating jet quenching the 1/τ dependence of the number density of the scattering centers in the whole range of τ (but with the Debye mass defined for T ≈ T c at τ > τ QGP ). The Bjorken 1 + 1D model [28]), used in our analysis, neglects the transverse expansion of the matter that becomes very important at τ ∼ R A ∼ 6 fm. However, from the point of view of the parton energy loss its effect should not be large due to compensation between the enhancement of the energy loss caused by increase of the medium size and its suppression caused by reduction of the medium density. The fact that the transverse expansion of the fireball does not affect strongly jet quenching in AA collisions was demonstrated, for the first time, in Ref. [35] within the BDMPS approach [1]. C. Calculation of IAA In leading order (LO) pQCD the energy of the hard parton produced in the direction opposite to the tagged direct photon, E T , coincides with the photon energy E γ T . For E T = E γ T the medium modification factor for a given photon energy reads I AA (z, E γ T ) = D AA h (z, E γ T ) D pp h (z, E γ T ) ,(27) where the nominator and the denominator are the FFs for the photon-tagged jets in AA and pp collisions, respectively. For pp collisions D pp h (z, E γ T ) = i r pp i (E γ T )D h/i (z, E γ T ) ,(28) where D h/i is the FF for i → h process defined by Eq. (4) with Q = E γ T , and r pp i is the fraction of the γ + i parton state in the γ+jet events in pp collisions. The FF for AA collisions can be written as D AA h (z, E γ T ) = i r AA i (E γ T )D m h/i (z, E γ T ) ,(29) where D m h/i is the medium modified FF for i → h process defined by Eqs. (2), (3) with Q = E γ T , and r AA i is the fraction of the γ + i parton state in the γ+jet events for AA collisions, means averaging over the impact parameter of AA collision, the jet direction and the position of jet production. The distribution of the jet production points in the transverse plane for a given impact vector b is ∝ T A (ρ)T A (b − ρ). This averaging procedure leads to fluctuations of the fast parton path length L in the QGP. In Fig. 1 we show the distribution in L for Au+Au collision for the 0 − 12% centrality bin corresponding to the conditions of the STAR experiment [15] calculated for our model of the QGP fireball. For the STAR experiment [15] E γ T ∼ < 20 GeV. In this case the sum over all relevant types of partons on the right hand side of Eqs. (28), (29) is dominated by gluon and light quarks. Since the medium effects for all light quarks are very similar, we consider all light quarks as one effective light quark state q with r q = 1 − r g . We calculate the r q,g using the LO pQCD with the CTEQ6 [36] parton distribution functions. As in the PYTHIA event generator [20], to simulate the higher order effects in calculation of the partonic cross sections we take for the virtuality scale in α s the value cQ with c = 0.265. For AA collisions we account for the nuclear modification of the parton densities (which leads to a small deviation of I AA from unity even without parton energy loss) with the help of the EKS98 correction [37]. This calculation gives for E γ T ∼ 12 − 20 GeV r pp g /r pp q ∼ 0.27 − 0.33, and somewhat smaller values for AA collisions r AA g /r AA q ∼ 0.24 − 0.3. The LO pQCD formulas do not account for the effect of the intrinsic parton transverse momentum on the photontagged jet FFs. The intrinsic parton transverse momenta lead to fluctuation of the jet energy E T around the photon energy E γ T both for pp and AA collisions [14]. In the pQCD the intrinsic parton transverse momenta in PDFs emerge in NLO calculations due to the initial state radiation in jet production. The NLO calculations performed in Ref. [38] show that for AA collisions the smearing correction, ∆ sm , to the medium modification factor I AA (z) at E γ T ∼ 7 − 8 blows up at z ∼ > 0.8 − 0.9. In Appendix B we demonstrate that ∆ sm ≈ F (z, E γ T )dI AA /dz/E γ 2 T , where F (z, E γ T ) is a smooth function of E γ T . Using this formula with the help of the results of Ref. [38] we can obtain the smearing correction to I AA for the conditions of the STAR experiment [15]. The STAR data are obtained for z ∼ 0.15 − 0.8. The results of Ref. [38] show that at E γ ∼ 7 − 9 GeV, in the potentially dangerous region z ∼ 0.8 ∆ sm increases I AA by ∼ 30%. For the photon energy interval 12 < E γ T < 20 GeV studied in Ref. [15] from the formula for ∆ sm we conclude that the smearing correction to I AA should be ∼ < 8% at z ∼ 0.8 (in fact even at z = 0.9 it is relatively small), and for z ∼ < 0.6 it should be very small. For this reason for the photon energy region studied in Ref. [15] accuracy of the LO pQCD predictions for I AA should be quite good. III. NUMERICAL RESULTS AND DISCUSSION The data of Ref. [15] are obtained for the photon energy region 12 < E γ T < 20 GeV. We define the nuclear modification factor I AA for the photon energy window E 1 < E < E 2 (hereafter, for notational simplicity, we omit the T and γ indeces of the photon energy) as I AA (z, E 1 , E 2 ) = D AA h (z, E 1 , E 2 ) D pp h (z, E 1 , E 2 ) ,(30) where the FFs in the denominator and numerator are averaged over the photon energy with the help of the LO pQCD cross section for photon-jet events dσ/dydE defined as D i h (z, E 1 , E 1 ) = E2 E1 dED i h (z, E) dσ i dydE E2 E2 dE dσ i dydE (31) with superscript i = AA, pp. The cross section dσ AA /dydE is calculated with the EKS98 [37] correction to the nuclear PDFs. Before presenting our results for the medium modification factor I AA , it is instructive to first compare our results for the pp photon-tagged FF and to illustrate the z and L dependence of the medium effects in our model. In Fig. 2 we compare our results for the pp photon-tagged FF for charged hadrons for the STAR energy window between E 1 = 12 and E 2 = 20 GeV with that measured in Ref. [15]. From Fig. 2 one sees that the theoretical photon-tagged FF agrees reasonably with that from STAR [15]. In Fig. 2 in addition to the prediction for the photon-tagged jet FF calculated using FFs given by Eq. (4) we also plotted the curve obtained using the pure KKP FFs D h/i (z, Q). One can see that the results for both the methods agree reasonably well. However, in principle, from the point of view of the theoretical predictions for I AA the quality of description of the pp photon-tagged FF is not very important. To illustrate the relative contribution from gluon and quarks to the photon-tagged FFs in Fig. 3 we plot fraction of the gluon contribution to D pp,AA h . The results for D AA h are calculated for 0 − 12% central Au+Au collisions (as we noted in Sec. 2 it corresponds in our model of the QGP fireball to the initial QGP temperature T 0 ≈ 327 MeV at τ 0 = 0.5 fm). We used α f r s = 0.5 that gives a reasonable description of I AA (see below). To illustrate better the medium effects we present predictions for D AA h obtained with and without the EKS98 corrections to the nuclear PDFs. One can see that both for pp and AA collisions the relative gluon contribution decreases strongly with increasing z. The effect of the nuclear modification of the PDFs is relatively small. From the curves for the Au+Au collision, one can see that the suppression of the gluon fraction at large z is considerably stronger than for pp collisions. This is a consequence of a bigger energy loss in the QGP for gluons. In order to better demonstrate the difference between the strength of jet quenching for gluon and quark jets in (2) calculated for α f r s = 0.5 for the jet energy E T = 15 GeV (we included the argument L that has been omitted for clarity in (2)) L i (z) = dLLP (L)D m h/i (z, E T , L) dLP (L)D m h/i (z, E T , L) ,(32) where P (L) is the jet path length distribution for Au+Au collisions for 0 − 12% centrality bin shown in Fig. 1. One sees that the typical path lengths for gluon and quark jets, that contribute to hadron production at small z, are close to each other. But for large z the typical jet path length for gluon jets becomes by a factor of ∼ 1.5 − 2 smaller than that for quark jets. Both for quarks and gluons the value of L(z) decreases with increase of z. It occurs because hadron production at large z is biased to events with smaller parton energy loss, i.e. to events with smaller parton path length through the medium. The effect is more pronounced for gluons that have larger energy loss. To illustrate the L-dependence of the medium modification of the photon-tagged FF, in Fig. 5 we plot the medium modification factor for several path lengths in the QGP (we denote it by I m (z, L)). To separate the medium effect from influence of the nuclear corrections to the factors r q,g , we calculated I m without the EKS98 corrections, i.e. using for the nominator the mediummodified FFs (2) weighted by the factors r pp q,g D m h (z, E γ T ) = i r pp i (E γ T )D m h/i (z, E γ T ) ,(33) as for pp case (28). We perform averaging over energy as for the factor I AA given by Eqs. (30), (31). From Fig. 5 one sees that the L-dependence of the medium modification becomes rather weak at L ∼ 8 − 10 fm. One can see that, in agreement with the decrease of L q,g (z) with increase of z seen from Fig. 4, the photon-tagged FF at large z should be biased to the events with smaller jet path lengths through the QGP. Finally, after presenting illustrative results, in Fig. 6 we confront our results for the medium modification factor I AA obtained for α f r s = 0.4, 0.5 and 0.6 with the data from STAR [15]. Our previous analyses [18,39] of the RHIC data on the nuclear modification factor R AA in Au+Au collisions at √ s = 200 GeV at p T ∼ 10 − 20 GeV support α f r s ∼ 0.5. From Fig. 6 one sees that α f r s ∼ 0.5 gives also a reasonable agreement with the STAR data [15] on the nuclear modification factor I AA in the whole range of z studied in Ref. [15] from z ∼ 0.8 down to z ∼ 0.15 − 0.25. Note that at z ∼ < 0.4 our results depend weakly on the value of α f r s . In Fig. 6 we included the region of very small z down to z ∼ 0.03 that corresponds to hadrons with momentum ∼ 0.4 − 0.5 GeV. It is done just to illustrate the flow of the jet momentum into the soft region in our model, which satisfies the momentum sum rule. Of course, the production of such low energy hadrons cannot be treated in the jet fragmentation picture, because it should involve fragmentation of partons with energy comparable to the energy of the thermal partons in the produced QGP. As we noted in Introduction, the approximation of independent gluon emission, based on Landau's method [10], has no a rigorous theoretical justification at z ≪ 1, where cascading of the primary gluons radiated from fast partons may come into play. The typical energy for partons fragmenting to hadrons in the region z ∼ 0.15 − 0.25 is ∼ 3 − 6 GeV (we take the jet energy E ∼ 15 GeV that is approximately the mean energy for the STAR window E ∼ 12−20 GeV [15]). It corresponds to the parton fractional momentum x ∼ 0.2 − 0.4 The formation length for the induced radiation, L in f , of a primary gluon with the fractional momentum x from a fast quark can be esti- L in f ∼ 2Ex(1 − x)S LP M ǫ 2 ,(34) where as in Eq. (5) ǫ 2 = m 2 q x 2 + m 2 g (1 − x), S LP M is the LPM suppression factor. From the formula (34) one can obtain L in f ∼ 3 − 4 fm. We used S LP M ∼ 0.3 that corresponds to τ ∼ 3 fm that is interesting to us. The formation length for radiation of the secondary gluons with energy about half of that for the primary gluons is about ∼ 2 − 3 fm. It means that typically splitting of the primary gluons can occur at τ ∼ 5 − 6 fm. In this region the density of the QGP is at least by a factor of ∼ 10 smaller than that at the initial time τ 0 . For this reason the induced gluon splitting of the primary radiated gluons, that are neglected in our analysis, may be a relatively weak effect. The present analysis assumes that the medium effects are present only in AA collisions. It is possible that a small-size QGP is produced in pp collisions as well. The idea that the QGP may be produced in hadron collisions is very old [40]. Application of the Bjorken relation (26) to pp collisions shows that at RHIC energy √ s = 200 GeV the initial temperature of the mini-QGP fireball may be as large as ∼ 200 − 230 MeV [39] which is well above the deconfinement temperature. In the scenario with the mini-QGP production the theoretical nuclear modification factor R AA should be divided by the medium modification factor for pp collisions R pp [39] that accounts for jet quenching in the mini-QGP produced in pp collisions. For RHIC energy √ s = 200 GeV numerical calculations within the LCPI approach give R pp ∼ 0.7 − 0.8 at p T ∼ 10 − 20 GeV [39]. The results of Ref. [39] show that in the scenario with mini-QGP production some increase of the nuclear modification factor R AA due to the additional factor 1/R pp can be imitated by reduction of the α f r s . Since a direct measurements of R pp is impossible, it is practically impossible to distinguish the scenarios with and without the mini-QGP production in pp collisions using the data on the R AA . For the nuclear modification factor I AA the situation is similar. In the scenario with mini-QGP production in pp collisions the theoretical I AA should be divided by the medium modification factor I pp for pp collisions, which, similarly to R pp , is unobservable quantity. And its effect on the theoretical I AA can be imitated by some change in α s . However, contrary to the medium effects in pp collisions on the high-p T spectra, the photon-tagged jet FF, at least in principle, enables us [41] to study the effect of mini-QGP by measuring its variation with the multiplicity of soft off-jet particles (the so-called underlying events, see Ref. [42] for a review). But this requires measurements of the photon-tagged jet FF for very high underlying event multiplicities dN UE ch /dη ∼ 40 [41]. Such multiplicities are too high to be measured at RHIC energies in the γ+jet events. IV. SUMMARY We have calculated the nuclear modification factor I AA for the photon-tagged jets within the jet quenching scheme based on the LCPI approach [2] to the induced radiation with the collisional energy loss treated as a perturbation to the radiative mechanism. The calculations are performed for running α s frozen at low momenta. Our scheme for calculation of the mediummodified photon-tagged jet FF function in AA collisions preserves the flavor and the momentum conservation. We have compared the theoretical predictions with the recent data from STAR [15] for Au+Au collisions at √ s = 200 GeV for 12 < E γ T < 20 GeV. We obtained a reasonable agreement with the STAR data in the whole range of z from z ∼ 0.8 down to z ∼ 0.15 for running coupling constant frozen at the value α f r s ∼ 0.5. This value agrees well with that necessary for description of the RHIC data on the nuclear modification factor R AA . Appendix A. Formulas for the spectrum of the induced a → bc transition In this appendix for the reader convenience and completeness we give formulas for calculations of the probabilities of q → gq, g → gg, and g → qq transitions in the LCPI approach [2]. We use the representation of the induced spectrum obtained in Ref. [21] which is convenient for numerical calculations. In general in the LCPI approach the x-distribution for the induced a → bc partonic transition (hereafter x = E b /E a ) for parton a with momentum along the z-axis produced in a hard process at z = 0 in the matter of thickness L can be written as dP dx = L 0 dz n(z) dσ BH ef f (x, z) dx ,(A1) where n(z) is the medium number density, dσ BH ef f /dx is an effective Bethe-Heitler cross section. It accounts for both the LPM and finite-size effects and can be written as dσ BH ef f (x, z) dx = − P b a (x) πM Im z 0 dξα s (Q 2 (ξ)) × exp −i ξ L f ∂ ∂ρ F (ξ, ρ) √ ρ ρ=0 .(A2) Here P b a (x) is the usual pQCD splitting function for a → bc transition, M = E a x(1 − x) , L f = 2M/ǫ 2 with ǫ 2 = m 2 b (1 − x) + m 2 c x − m 2 a x(1 − x), Q 2 (ξ) = aM/ξ with a ≈ 1.85 (this value of the parameter a was fixed in Ref. [23] by comparison of the induced gluon spectrum calculated in the coordinate representation with that obtained in the momentum representation for the dominant N = 1 rescattering contribution), F is the solution to the radial Schrödinger equation for the azimuthal quantum number m = 1 i ∂F (ξ, ρ) ∂ξ = − 1 2M ∂ ∂ρ 2 + v(ρ, x, z − ξ) + 4m 2 − 1 8M ρ 2 F (ξ, ρ) (A3) with the potential v(ρ, x, z) = −i n(z)σ 3 (ρ, x, z) 2 . (A4) Note, that in terms of the original longitudinal variable z along the fast parton momentum we solve the Schrödinger equation backward in time/z. The point ξ = 0 corresponds to the last rescattering of the bcā system on a medium constituent located at z. This technical trick allows us to have a smooth boundary condition for F at ξ = 0: F (ξ = 0, ρ) = √ ρσ 3 (ρ, x, z)ǫK 1 (ǫρ) (K 1 is the Bessel function). The function σ 3 (ρ, x, z) is the cross section of interaction of the bcā system with a medium constituent located at z (the argument ρ is the transverse distance between b and c). In the transverse plane the partonā in the bcā system is located at the center of mass of the bc pair. In the Schrödinger equation (A3) M plays the role of the reduced "Schrödinger mass" for the bc pair (since the "masses" for the b and c partons are E a x and E a (1 − x), respectively). The three-body cross section σ 3 can be written in terms of the well known dipole cross section for the color singlet qq pair [43] that is given by σ qq (ρ, z) = C T C F dqα 2 s (q 2 ) [1 − exp(iqρ)] [q 2 + µ 2 D (z)] 2 . (A5) Here C F,T are the color Casimir for the quark and thermal parton (quark or gluon), and µ D (z) is the local Debye mass. In terms of the dipole cross section (A5) the three-body cross sections for q → gq, g → gg, and g → qq The LPM suppression and finite-size suppression for the in-medium a → bc process is characterized by the factor S(M, x) = dP/dx dP BH /dx , where the denominator is the Bethe-Heitler spectrum calculated using in Eq. (A1) the real Bethe-Heitler cross section for a → bc transition given by [44] dσ BH (x, z) dx = dρ\|Ψ bc a (ρ, x)\| 2 σ 3 (ρ, x, z) , where Ψ bc a (ρ, x) is the light-cone wave function for a → bc transition. Since \|Ψ bc a (ρ, x)\| 2 contains the splitting function P b a (x) 1 , the factor S(M, x) for a given value of M has a smooth dependence on x. It occurs because the x-dependence of the integrand of (A2), that comes only from the x-dependence of ǫ 2 and of the three-body cross section σ 3 , is relatively weak. This fact allows one to reduce considerably CPU time in numerical calculations by creating a grid of values of S(M, x) in the M −x plane with relatively small number of points in x. Then this grid can be used for calculation of the spectrum dP/dx by performing interpolation in x and using the Bethe-Heitler spectrum that does not require much computing power. In the above formulas for the effective Bethe-Heitler cross section (A2) and for the dipole cross section (A5) we use the following parametrization for α s (Q 2 ) α s (Q 2 ) =    α f r s if Q ≤ Q f r , 4π 9 log(Q 2 /Λ 2 QCD ) if Q > Q f r (A11) with Q f r = Λ QCD exp 2π/9α f r s , Λ QCD = 300 MeV. FIG. 1 : 1Probability distribution P (L) for the jet path length in the QGP in our model of the QGP fireball for 0 − 12% centrality bin for Au+Au collisions at √ s = 200 GeV. FIG. 2 : 2The photon-tagged jet FF for 12 < E γ T < 20 GeV for pp collisions at √ s = 200 GeV calculated using FFs given by Eq. (4) (solid) and obtained for pure KKP FFs D h/i (z, Q) (dotted). Data points are from Ref.[15]. Fig. 4 4we show the z-dependence of the average jet path length in the QGP defined via the medium modified FFs FIG. 3 :FIG. 4 : 34Fraction of the gluon contribution to the photontagged jet FF for 12 < E γ T < 20 GeV at √ s = 200 GeV for pp collisions and for Au+Au collisions for 0 − 12% centrality bin. Solid line: pp collisions; dashed and dotted line: Au+Au collisions, calculations for α Average jet path length defined by Eq. (32) for quark and gluon jets (top to bottom) vs z for jet energy ET = 15 GeV calculated for α f r s = 0.5 with the path length distribution P (L) for 0 − 12% centrality bin of Au+Au collisions at √ s = 200 GeV. FIG. 5 : 5Medium modification factor Im(z, L) for the photontagged jet FFs (28), (33) averaged over energy in the window 12 < E γ T < 20 GeV (see text for details) calculated with α f r s = 0.5 for L = 2, 4, 6, 8, and 10 fm (top to bottom at large z). FIG. 6 : 6Medium modification factor IAA for the photontagged jet FF at 12 < E γ T < 20 GeV for Au+Au collisions at √ s = 200 GeV for 0 − 12% centrality bin. The curves are for α f r s = 0.4, 0.5 and 0.6 (top to bottom at large z). Data points are from Ref. [15].mated as[2] qq (ρ, z) + σ qq ((1 − x)ρ, z)] − 1 8 σ qq (xρ, z) , (A6) σ 3 (ρ, x, z)\| g→gg = 9 8 [σ qq (ρ, z) + σ qq ((1 − x)ρ, z)+σ qq (xρ, z)] , (A7)σ 3 (ρ, x, z)\| g→qq = 9 8 [σ qq (xρ, z) + σ qq ((1 − x)ρ, z)] − 1 8 σ qq (ρ, z) . (A8) It is worth noting that the formula for the Bethe-Heitler cross section (A10) exactly corresponds to that for the effective Bethe-Heitler cross section given by Eq. (A2) with the infinite upper limit of the ξ-integration and the function F calculated for v = 0. AcknowledgmentsI would like to thank Wenchang Xiang for the invitation to visit Guizhou University of Finance and Economics, where this work was completed. This work is supported in part by the grant RFBR 15-02-00668-a.Appendix B. The smearing correction to IAA.In this appendix we discuss the energy dependence of the smearing correction to the LO predictions for the nuclear modification factor I AA . This correction potentially may be important at z close to unity. The question is where the regime of large smearing correction begins. To understand this we use as a plausible estimate of the smearing effect the results of the NLO model[38]. This analysis shows that for AA collisions the smearing correction to the medium modification factor I AA (z) blows up at z ∼ > 0.8−0.9 for E ∼ 8 GeV. The results of Ref.[38]can be easily rescaled to our conditions. Indeed, let us write the LO I AA aswhere D h is the FF for pp collisions, ∆z = z∆E/E is shift of z due to the radiative parton energy loss ∆E in the QGP (here, for simplicity, we omit averaging over ∆z which should be made, also for clarity we omit the argument E and superscript pp on the FF). The I AA in the presence of the smearing (we denote itĪ AA ) can be written aswhere δz = zq/E and q is the shift of the jet energy (E jet = E + q) (we assume that the smearing correction is not very large, and keep only the second order terms in δz). From (B2) we obtain an equation which does not contain ∆zĪThe second term in the square brackets in Eq. (B4) can be neglected (since I AA is a smooth function as compared to D h ). Then we obtainwhere F (z, E) is a smooth function of energy, and does not depend on the strength of the medium suppression at all. The fact that ∆ sm ∝ dI AA /dz is quite natural. For a flat I AA the smearing effect should vanish since the numerator and denominator are affected in the same way. Note that the ∆ sm ∝ dI AA /dz scaling agrees with the results for ∆ sm from Ref.[38]obtained for different magnitudes of the energy loss (shown inFig. 2of Ref.[38]). 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