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  • Global Fujita-Kato solution of 3-D inhomogeneous incompressible Navier-Stokes system
    Adv. Math. (IF 1.435) Pub Date : 2020-01-27
    Ping Zhang

    In this paper, we shall prove the global existence of weak solutions to 3D inhomogeneous incompressible Navier-Stokes system (INS) with initial density in the bounded function space and having a positive lower bound and with initial velocity being sufficiently small in the critical Besov space, B˙2,112. This result corresponds to the Fujita-Kato solutions of the classical Navier-Stokes system. The same idea can be used to prove the global existence of weak solutions in the critical functional framework to (INS) with one component of the initial velocity being large and can also be applied to provide a lower bound for the lifespan of smooth enough solutions of (INS).

    更新日期:2020-01-27
  • Semi-Calabi–Yau orbifolds and mirror pairs
    Adv. Math. (IF 1.435) Pub Date : 2020-01-24
    Alessandro Chiodo; Elana Kalashnikov; Davide Cesare Veniani

    We generalize the cohomological mirror duality of Borcea and Voisin in any dimension and for any number of factors. Our proof applies to all examples which can be constructed through Berglund–Hübsch duality. Our method is a variant of the so-called Landau–Ginzburg/Calabi–Yau correspondence of Calabi–Yau orbifolds with an involution that does not preserve the volume form. We deduce a version of mirror duality for the fixed loci of the involution, which are beyond the Calabi–Yau category and feature hypersurfaces of general type.

    更新日期:2020-01-24
  • On the non-vanishing of theta liftings of tempered representations of U(p,q)
    Adv. Math. (IF 1.435) Pub Date : 2020-01-21
    Hiraku Atobe

    In this paper, we give an explicit determination of the non-vanishing of the theta liftings of tempered representations for unitary dual pairs (U(p,q),U(r,s)) for arbitrary non-negative integers p,q,r,s. For discrete series representations, in terms of Harish-Chandra parameters, we give a complete criterion when the theta lifts are nonzero. For tempered representations, we determine the non-vanishing in terms of the local Langlands correspondence assuming the local Gan–Gross–Prasad conjecture.

    更新日期:2020-01-22
  • Weak Vopěnka's Principle does not imply Vopěnka's Principle
    Adv. Math. (IF 1.435) Pub Date : 2020-01-21
    Trevor M. Wilson

    Vopěnka's Principle says that the category of graphs has no large discrete full subcategory, or equivalently that the category of ordinals cannot be fully embedded into it. Weak Vopěnka's Principle is the dual statement, which says that the opposite category of ordinals cannot be fully embedded into the category of graphs. It was introduced in 1988 by Adámek, Rosický, and Trnková, who showed that it follows from Vopěnka's Principle and asked whether the two statements are equivalent. We show that they are not. However, we show that Weak Vopěnka's Principle is equivalent to the generalization of itself known as Semi-Weak Vopěnka's Principle, introduced by Adámek and Rosický in 1993.

    更新日期:2020-01-22
  • Class field theory, Diophantine analysis and the asymptotic Fermat's Last Theorem
    Adv. Math. (IF 1.435) Pub Date : 2020-01-21
    Nuno Freitas; Alain Kraus; Samir Siksek

    Recent results of Freitas, Kraus, Şengün and Siksek, give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over a specific number field. Those works in turn build on many deep theorems in arithmetic geometry. In this paper we combine the aforementioned results with techniques from class field theory, the theory of p-groups and p-extensions, Diophantine approximation and linear forms in logarithms, to establish the asymptotic Fermat's Last Theorem for many infinite families of number fields, and for thousands of number fields of small degree. For example, we prove the effective asymptotic Fermat's Last Theorem for the infinite family of fields Q(ζ2r)+ where r≥2.

    更新日期:2020-01-22
  • Positive specializations of symmetric Grothendieck polynomials
    Adv. Math. (IF 1.435) Pub Date : 2020-01-21
    Damir Yeliussizov

    It is a classical fundamental result that Schur-positive specializations of the ring of symmetric functions are characterized via totally positive functions whose parametrization describes the Edrei–Thoma theorem. In this paper, we study positive specializations of symmetric Grothendieck polynomials, K-theoretic deformations of Schur polynomials.

    更新日期:2020-01-22
  • Set relations and set systems induced by some families of integral domains
    Adv. Math. (IF 1.435) Pub Date : 2020-01-21
    G. Chiaselotti; F. Infusino; P.A. Oliverio

    In this paper, given an integral domain U, we investigate the main properties of a relation ←mod which is based on the interrelation between subdomains of U and finitely generated unitary submodules of U. We shall characterize it in terms of a second relation ≺⋄ between n-tuples (u1,…,un) of elements of U and subdomains D of U defined by the vanishing in (u1,…,un) of some polynomial p(Z1,…,Zn) belonging to a specific subset of the polynomial ring in several variables D[Z1,…,Zn]. Such an equivalence shall be used in order to introduce three specific collections of subdomains XU, BU and PU, whose algebraic properties present a close connection with geometrical and combinatorial properties induced by ←mod. On the other hand, the characterization of the subdomains of XU leads to the more general problem of finding a map Ψ associating with a subdomain D of U a collection Ψ(D) of subdomains of KU such that the intersection of some or of any member of Ψ(D) gives D. In this perspective, in the present paper we shall study two further collections of subdomains of U, denoted respectively by EU and LU, whose main properties are related to those of the families PU and BU. Finally, our investigation of all the aforementioned subdomain families shall be also related to the study of pairs (e,ξ), where e∈U∖{0} and ξ is an idempotent ring endomorphism of U whose kernel agrees with the ideal of U generated by e. We shall exhibit several results concerning the membership of ξ(U) and of ξ(U)[e] to the above subdomain families.

    更新日期:2020-01-22
  • C1,α isometric embeddings of polar caps
    Adv. Math. (IF 1.435) Pub Date : 2020-01-21
    Camillo De Lellis; Dominik Inauen

    We study isometric embeddings of C2 Riemannian manifolds in the Euclidean space and we establish that the Hölder space C1,12 is critical in a suitable sense: in particular we prove that for α>12 the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, whereas for any α<12 we construct C1,α isometric embeddings of portions of the standard 2-dimensional sphere for which such property fails.

    更新日期:2020-01-22
  • On the derivative of the Hausdorff dimension of the Julia sets for z2 + c, c∈R at parabolic parameters with two petals
    Adv. Math. (IF 1.435) Pub Date : 2020-01-21
    Ludwik Jaksztas; Michel Zinsmeister

    Let d(c) denote the Hausdorff dimension of the Julia set Jc of the polynomial fc(z)=z2+c. Let c0∈R be such that fc0 has a parabolic cycle with two petals: we investigate in this paper how the bifurcation that occurs as c∈R crosses c0 reflects on the variations of d(c).

    更新日期:2020-01-22
  • A boundedness criterion for singular integral operators of convolution type on the Fock space
    Adv. Math. (IF 1.435) Pub Date : 2020-01-20
    Guangfu Cao; Ji Li; Minxing Shen; Brett D. Wick; Lixin Yan

    We show that for an entire function φ belonging to the Fock space F2(Cn) on the complex Euclidean space Cn, the integral operatorSφF(z)=∫CnF(w)ez⋅w¯φ(z−w¯)dλ(w),z∈Cn, is bounded on F2(Cn) if and only if there exists a function m∈L∞(Rn) such thatφ(z)=∫Rnm(x)e−2(x−i2z)2dx,z∈Cn. Here dλ(w)=π−ne−|w|2dw is the Gaussian measure on Cn. With this characterization we are able to obtain some fundamental results of the operator Sφ, including the normality, the C⁎ algebraic properties, the spectrum and its compactness. Moreover, we obtain the reducing subspaces of Sφ. In particular, in the case n=1, we give a complete solution to an open problem proposed by K. Zhu for the Fock space F2(C) on the complex plane C (Zhu (2015) [30]).

    更新日期:2020-01-21
  • Augmented quasigroups and character algebras
    Adv. Math. (IF 1.435) Pub Date : 2020-01-16
    Jonathan D.H. Smith

    The conjugacy classes of groups and quasigroups form association schemes, in which the relation products are defined by collapsing group or quasigroup multiplications. In previous work, sharp transitivity was used to identify association schemes, such as certain Johnson schemes, which cannot appear as quasigroup schemes. Thus quasigroup schemes only constitute a fragment of the full set of all association schemes. Nevertheless, the current paper shows that every association scheme is in fact obtained by collapsing a quasigroup multiplication. In a second application of a similar technique, character quasigroups are constructed for each finite group, as analogues of the character groups of abelian groups, to encode the multiplicative structure of group characters. As infrastructure for these and related results, three key unifying concepts in compact closed categories are established: augmented comagmas, augmented magmas, and augmented quasigroups, the latter serving to capture such diverse structures as groups and Heyting algebras.

    更新日期:2020-01-17
  • Topological generation and matrix models for quantum reflection groups
    Adv. Math. (IF 1.435) Pub Date : 2020-01-17
    Michael Brannan; Alexandru Chirvasitu; Amaury Freslon

    We prove that the von Neumann algebras of quantum permutation groups and quantum reflection groups have the Connes embedding property. We do this by establishing several new topological generation results for the quantum permutation groups SN+ and the quantum reflection groups HNs+. We use these results to show that these quantum groups admit sufficiently many “matrix models”. In particular, all of these quantum groups have residually finite discrete duals (and are, in particular, hyperlinear), and certain “flat” matrix models for SN+ are inner faithful.

    更新日期:2020-01-17
  • Trigonometric Lie algebras, affine Lie algebras, and vertex algebras
    Adv. Math. (IF 1.435) Pub Date : 2020-01-15
    Haisheng Li; Shaobin Tan; Qing Wang

    In this paper, we explore natural connections among trigonometric Lie algebras, (general) affine Lie algebras, and vertex algebras. Among the main results, we obtain a realization of trigonometric Lie algebras as what were called the covariant algebras of the affine Lie algebra Aˆ of Lie algebra A=gl∞⊕gl∞ with respect to certain automorphism groups. We then prove that restricted modules of level ℓ for trigonometric Lie algebras naturally correspond to equivariant quasi modules for the affine vertex algebras VAˆ(ℓ,0) (or VAˆ(2ℓ,0)). Furthermore, we determine irreducible modules and equivariant quasi modules for the simple vertex algebra LAˆ(ℓ,0) with ℓ a positive integer. In particular, we prove that every quasi-finite unitary highest weight (irreducible) module of level ℓ for type A trigonometric Lie algebra gives rise to an irreducible equivariant quasi LAˆ(ℓ,0)-module.

    更新日期:2020-01-15
  • The Gromov-Hausdorff hyperspace of a Euclidean space
    Adv. Math. (IF 1.435) Pub Date : 2020-01-15
    Sergey A. Antonyan

    We investigate the hyperspace GH(Rn) of the isometry classes of all non-empty compact subsets of a Euclidean space in the Gromov-Hausdorff metric. It is proved that for any n≥1, GH(Rn) is homeomorphic to the orbit space 2Rn/E(n) of the hyperspace 2Rn of all non-empty compact subsets of a Euclidean space Rn equipped with the Hausdorff metric and the natural action of the Euclidean group E(n). This is further applied to prove that 2Rn/E(n) is homeomorphic to the open cone OCone(Ch(Bn)/O(n)), where Ch(Bn) stands for the set of all A∈2Rn for which the closed Euclidean unit ball Bn is the least circumscribed ball (the Chebyshev ball). These results lead to determine the complete topological structure of GH(Rn) for n≤2, namely, we prove that GH(Rn) is homeomorphic to the Hilbert cube with a removed point. We also prove that for n≤2, GH(Bn) is homeomorphic to the Hilbert cube.

    更新日期:2020-01-15
  • Asymptotic codimensions of Mk(E)
    Adv. Math. (IF 1.435) Pub Date : 2020-01-14
    Allan Berele; Amitai Regev

    We show that the codimension sequence of the algebra of k×k matrices over the Grassmann algebra, cn(Mk(E)), is asymptotic to αn1−k22(2k2)n, where α is an undetermined constant.

    更新日期:2020-01-15
  • Notes on Plücker's relations in geometric algebra
    Adv. Math. (IF 1.435) Pub Date : 2020-01-14
    Garret Sobczyk

    Grassmannians are of fundamental importance in projective geometry, algebraic geometry, and representation theory. A vast literature has grown up using many different languages of higher mathematics, such as multilinear and tensor algebra, matroid theory, and Lie groups and Lie algebras. Here we explore the Plücker relations in Clifford's geometric algebra. We discover that the Plücker relations can be fully characterized in terms of the geometric product, without the need for a confusing hodgepodge of many different formalisms and mathematical traditions found in the literature.

    更新日期:2020-01-15
  • On uniform measures in the Heisenberg group
    Adv. Math. (IF 1.435) Pub Date : 2020-01-14
    Vasilis Chousionis; Valentino Magnani; Jeremy T. Tyson

    We study uniform measures in the first Heisenberg group H equipped with the Korányi metric dH. We prove that 1-uniform measures are proportional to the spherical 1-Hausdorff measure restricted to an affine horizontal line, while 2-uniform measures are proportional to spherical 2-Hausdorff measure restricted to an affine vertical line. We also show that each 3-uniform measure which is supported on a vertically ruled surface is proportional to the restriction of spherical 3-Hausdorff measure to an affine vertical plane, and that no quadratic x3-graph can be the support of a 3-uniform measure. According to a result of Merlo, every 3-uniform measure is supported on a quadratic variety; in conjunction with our results, this shows that all 3-uniform measures are proportional to spherical 3-Hausdorff measure restricted to an affine vertical plane. We establish our conclusions by deriving asymptotic formulas for the measures of small extrinsic balls in (H,dH) intersected with smooth submanifolds. The coefficients in our power series expansions involve intrinsic notions of curvature associated to smooth curves and surfaces in H.

    更新日期:2020-01-15
  • Cluster variables, ancestral triangles and Alexander polynomials
    Adv. Math. (IF 1.435) Pub Date : 2020-01-13
    Wataru Nagai; Yuji Terashima

    In this paper, we show that Alexander polynomials for any 2-bridge knots are specializations of cluster variables. A key tool is an ancestral triangle which appeared in both quantum topology and hyperbolic geometry in different ways.

    更新日期:2020-01-14
  • Beyond cohomological assignments
    Adv. Math. (IF 1.435) Pub Date : 2020-01-10
    Victor Guillemin; Susan Tolman; Catalin Zara

    Let a torus T act in a Hamiltonian fashion on a compact symplectic manifold (M,ω). The assignment ring AT(M) is an extension of the equivariant cohomology ring HT(M); it is modeled on the GKM description of the equivariant cohomology of a GKM space. We show that AT(M) is a finitely generated S(t⁎)-module, and give a criterion guaranteeing that a given set of assignments generates (alternatively, is a basis for) this module. We define two new types of assignments, delta classes and bridge classes, and show that if the torus T is 2-dimensional, then all assignments of sufficiently high degree are generated by cohomological, delta, and bridge classes. In particular, if M is 6-dimensional, then we can find a basis of such classes.

    更新日期:2020-01-11
  • Linear inviscid damping and enhanced dissipation for the Kolmogorov flow
    Adv. Math. (IF 1.435) Pub Date : 2020-01-08
    Dongyi Wei; Zhifei Zhang; Weiren Zhao

    In this paper, we prove the linear inviscid damping and vorticity depletion phenomena for the linearized Euler equations around the Kolmogorov flow. These results confirm Bouchet and Morita's predictions based on numerical analysis. By using the wave operator method introduced by Li, Wei and Zhang, we solve Beck and Wayne's conjecture on the enhanced dissipation rate for the 2-D linearized Navier-Stokes equations around the bar state called Kolmogorov flow. The same dissipation rate is proved for the Navier-Stokes equations if the initial velocity is included in a basin of attraction of the Kolmogorov flow with the size of ν23+, here ν is the viscosity coefficient.

    更新日期:2020-01-08
  • Higher order convergence rates in theory of homogenization II: Oscillatory initial data
    Adv. Math. (IF 1.435) Pub Date : 2020-01-08
    Sunghan Kim; Ki-Ahm Lee

    We establish higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic Cauchy problems accompanied with rapidly oscillating initial data. Such result is new even for linear problems. Here we construct higher order initial layer and interior correctors, which describe the oscillatory behavior near the initial and interior time zone of the domain. To construct higher order correctors, we develop a regularity theory in macroscopic scales, and prove an exponential decay estimate for initial layer correctors. The higher order expansion requires an iteration process: successively correcting the initial layer, then the interior. This leads to a more complicated asymptotic expansion, as compared to the non-oscillating data case, and this complexity is present even in the linear case. A notable observation for fully nonlinear operators is that even if the given operator is space-time periodic, the interior correctors become aperiodic in the time variable as we proceed with the iteration process. Moreover, each interior corrector of higher order is paired with a space-time periodic version, and the difference between the two decays exponentially fast with time.

    更新日期:2020-01-08
  • On the complete separation of asymptotic structures in Banach spaces
    Adv. Math. (IF 1.435) Pub Date : 2020-01-08
    Spiros A. Argyros; Pavlos Motakis

    Let (ei)i denote the unit vector basis of ℓp, 1≤p<∞, or c0. We construct a reflexive Banach space with an unconditional basis that admits (ei)i as a uniformly unique spreading model while it has no subspace with a unique asymptotic model, and hence it has no asymptotic-ℓp or c0 subspace. This solves a problem of E. Odell. We also construct a space with a unique ℓ1 spreading model and no subspace with a uniformly unique ℓ1 spreading model. These results are achieved with the utilization of a new version of the method of saturation under constraints that uses sequences of functionals with increasing weights.

    更新日期:2020-01-08
  • Minimal surfaces from infinitesimal deformations of circle packings
    Adv. Math. (IF 1.435) Pub Date : 2019-12-30
    Wai Yeung Lam

    We study circle packings with the combinatorics of a triangulated disk in the plane and parametrize deformations of circle packings in terms of vertex rotation and cross ratios. We show that there is a Weierstrass representation formula relating infinitesimal deformations of circle packings to discrete minimal surfaces of Koebe type. Furthermore, every minimal surface of Koebe type can be extended naturally to a discrete minimal surface of general type. In this way, discrete minimal surfaces via Steiner's formula are unified.

    更新日期:2020-01-04
  • Orbits in (Pr)n and equivariant quantum cohomology
    Adv. Math. (IF 1.435) Pub Date : 2019-12-27
    Mitchell Lee; Anand Patel; Hunter Spink; Dennis Tseng

    We compute the GLr+1-equivariant Chow class of the GLr+1-orbit closure of any point (x1,…,xn)∈(Pr)n in terms of the rank polytope of the matroid represented by x1,…,xn∈Pr. Using these classes and generalizations involving point configurations in higher dimensional projective spaces, we define for each d×n matrix M an n-ary operation [M]ħ on the small equivariant quantum cohomology ring of Pr, which is the n-ary quantum product when M is an invertible matrix. We prove that M↦[M]ħ is a valuative matroid polytope association. Like the quantum product, these operations satisfy recursive properties encoding solutions to enumerative problems involving point configurations of given moduli in a relative setting. As an application, we compute the number of line sections with given moduli of a general degree 2r+1 hypersurface in Pr, generalizing the known case of quintic plane curves.

    更新日期:2020-01-04
  • Asymptotic behavior of supercuspidal representations and Sato-Tate equidistribution for families
    Adv. Math. (IF 1.435) Pub Date : 2019-12-27
    Ju-Lee Kim; Sug Woo Shin; Nicolas Templier

    We establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes. For the tame supercuspidals constructed by J.-K. Yu we prove the limit multiplicity property with error terms. Thereby we obtain a Sato-Tate equidistribution for the Hecke eigenvalues of these families. The main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive p-adic group tend to zero uniformly for every noncentral semisimple element.

    更新日期:2020-01-04
  • Horizontal non-vanishing of Heegner points and toric periods
    Adv. Math. (IF 1.435) Pub Date : 2019-12-27
    Ashay A. Burungale; Ye Tian

    Let F be a totally real number field and A a modular GL2-type abelian variety over F. Let K/F be a CM quadratic extension. Let χ be a class group character over K such that the Rankin-Selberg convolution L(s,A,χ) is self-dual with root number −1. We show that the number of class group characters χ with bounded ramification such that L′(1,A,χ)≠0 increases with the absolute value of the discriminant of K. We also consider a rather general rank zero situation. Let π be a cuspidal cohomological automorphic representation over GL2(AF). Let χ be a Hecke character over K such that the Rankin–Selberg convolution L(s,π,χ) is self-dual with root number 1. We show that the number of Hecke characters χ with fixed ∞-type and bounded ramification such that L(1/2,π,χ)≠0 increases with the absolute value of the discriminant of K. The Gross–Zagier formula and the Waldspurger formula relate the question to horizontal non-vanishing of Heegner points and toric periods, respectively. For both situations, the strategy is geometric relying on the Zariski density of CM points on self-products of a quaternionic Shimura variety. The recent result [26], [31], [1] on the André–Oort conjecture is accordingly fundamental to the approach.

    更新日期:2020-01-04
  • Equilibrium states and entropy theory for Nica-Pimsner algebras
    Adv. Math. (IF 1.435) Pub Date : 2019-12-27
    Evgenios T.A. Kakariadis

    We study the equilibrium simplex of Nica-Pimsner algebras arising from product systems of finite rank on the free abelian semigroup. First we show that every equilibrium state has a convex decomposition into parts parametrized by ideals on the unit hypercube. Secondly we associate every gauge-invariant part to a sub-simplex of tracial states of the diagonal algebra. We show how this parametrization lifts to the full equilibrium simplices of non-infinite type. The finite rank entails an entropy theory for identifying the two critical inverse temperatures: (a) the least upper bound for existence of non finite-type equilibrium states, and (b) the least positive inverse temperature below which there are no equilibrium states at all. We show that the first one can be at most the strong entropy of the product system whereas the second is the infimum of the tracial entropies (modulo negative values). Thus phase transitions can happen only in-between these two critical points and possibly at zero temperature.

    更新日期:2020-01-04
  • Bergman spaces of natural G-manifolds.
    Adv. Math. (IF 1.435) Pub Date : 2013-11-14
    Giuseppe Della Sala,Joe J Perez

    Let G be a unimodular Lie group, X a compact manifold with boundary, and M the total space of a principal bundle [Formula: see text] so that M is also a strongly pseudoconvex complex manifold. In this work, we show that if there exists a point [Formula: see text] such that [Formula: see text] is contained in the complex tangent space [Formula: see text] of bM at p, then the Bergman space of M is large. Natural examples include the gauged G-complexifications of Heinzner, Huckleberry, and Kutzschebauch.

    更新日期:2019-11-01
  • Fractal tiles associated with shift radix systems.
    Adv. Math. (IF 1.435) Pub Date : 2011-01-15
    Valérie Berthé,Anne Siegel,Wolfgang Steiner,Paul Surer,Jörg M Thuswaldner

    Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings. In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials. We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the d-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine).

    更新日期:2019-11-01
  • Orlov spectra as a filtered cohomology theory.
    Adv. Math. (IF 1.435) Pub Date : 2013-08-24
    Ludmil Katzarkov,Gabriel Kerr

    This paper presents a new approach to the dimension theory of triangulated categories by considering invariants that arise in the pretriangulated setting.

    更新日期:2019-11-01
  • The Steiner formula for Minkowski valuations.
    Adv. Math. (IF 1.435) Pub Date : 2013-03-09
    Lukas Parapatits,Franz E Schuster

    A Steiner type formula for continuous translation invariant Minkowski valuations is established. In combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations, this new Steiner type formula is used to obtain a family of Brunn-Minkowski type inequalities for rigid motion intertwining Minkowski valuations.

    更新日期:2019-11-01
  • The Andrews-Sellers family of partition congruences.
    Adv. Math. (IF 1.435) Pub Date : 2013-03-09
    Peter Paule,Cristian-Silviu Radu

    In 1994, James Sellers conjectured an infinite family of Ramanujan type congruences for 2-colored Frobenius partitions introduced by George E. Andrews. These congruences arise modulo powers of 5. In 2002 Dennis Eichhorn and Sellers were able to settle the conjecture for powers up to 4. In this article, we prove Sellers' conjecture for all powers of 5. In addition, we discuss why the Andrews-Sellers family is significantly different from classical congruences modulo powers of primes.

    更新日期:2019-11-01
  • Cycle decompositions: From graphs to continua.
    Adv. Math. (IF 1.435) Pub Date : 2012-02-03
    Agelos Georgakopoulos

    We generalise a fundamental graph-theoretical fact, stating that every element of the cycle space of a graph is a sum of edge-disjoint cycles, to arbitrary continua. To achieve this we replace graph cycles by topological circles, and replace the cycle space of a graph by a new homology group for continua which is a quotient of the first singular homology group [Formula: see text]. This homology seems to be particularly apt for studying spaces with infinitely generated [Formula: see text], e.g. infinite graphs or fractals.

    更新日期:2019-11-01
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