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作者：Downarowicz, Tomasz; Maass, Alejandro
作者单位：Wroclaw University of Science & Technology; Universidad de Chile; Universidad de Chile
摘要：We prove that if X denotes the interval or the circle then every transformation T:X -> X of class C (r) , where r > 1 is not necessarily an integer, admits a symbolic extension, i.e., every such transformation is a topological factor of a subshift over a finite alphabet. This is done using the theory of entropy structure. For such transformations we control the entropy structure by providing an upper bound, in terms of Lyapunov exponents, of local entropy in the sense of Newhouse of an ergodic...
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作者：Chen, Xiuxiong
作者单位：University of Wisconsin System; University of Wisconsin Madison
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作者：Looijenga, Eduard
作者单位：Utrecht University
摘要：The period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20. We determine the image of this period map (thus confirming a conjecture of Hassett) and give at the same time a new proof of the theorem of Voisin that asserts that this period map is an open embedding. An algebraic version of our main result is an identification of the algebra of SL (6,a,)-invariant polynomials on the representation space Sym (3)(a,(6))(*) with a certain algebra ...
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作者：Aharoni, Ron; Berger, Eli
作者单位：Technion Israel Institute of Technology; University of Haifa
摘要：We prove that Menger's theorem is valid for infinite graphs, in the following strong version: let A and B be two sets of vertices in a possibly infinite digraph. Then there exist a set P of disjoint A-B paths, and a set S of vertices separating A from B, such that S consists of a choice of precisely one vertex from each path in P. This settles an old conjecture of Erdos.
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作者：Garofalo, Nicola; Petrosyan, Arshak
作者单位：Purdue University System; Purdue University
摘要：We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity and the second one is a family of Monneau type formulas suited for the study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity. This allows to prove a structural theorem for the singular...
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作者：Khare, Chandrashekhar; Wintenberger, Jean-Pierre
作者单位：Utah System of Higher Education; University of Utah; Universites de Strasbourg Etablissements Associes; Universite de Strasbourg
摘要：This paper is the first part of a work which proves Serre's modularity conjecture. We first prove the cases p not equal 2 and odd conductor, and p = 2 and weight 2, see Theorem 1.2, modulo Theorems 4.1 and 5.1. Theorems 4.1 and 5.1 are proven in the second part, see Khare and Wintenberger (Invent. Math., doi: 10.1007/s00222-009-0206-6, 2009). We then reduce the general case to a modularity statement for 2-adic lifts of modular mod 2 representations. This statement is now a theorem of Kisin (In...
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作者：Bae, Myoungjean; Chen, Gui-Qiang; Feldman, Mikhail
作者单位：University of Wisconsin System; University of Wisconsin Madison; Fudan University; Northwestern University
摘要：The shock reflection problem is one of the most important problems in mathematical fluid dynamics, since this problem not only arises in many important physical situations but also is fundamental for the mathematical theory of multidimensional conservation laws that is still largely incomplete. However, most of the fundamental issues for shock reflection have not been understood, including the regularity and transition of different patterns of shock reflection configurations. Therefore, it is ...
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作者：Brosnan, Patrick; Fang, Hao; Nie, Zhaohu; Pearlstein, Gregory
作者单位：University of British Columbia; University of Iowa; Pennsylvania Commonwealth System of Higher Education (PCSHE); Pennsylvania State University; Michigan State University
摘要：In a recent paper, M. Green and P. Griffiths used R. Thomas' work on nodal hypersurfaces to sketch a proof of the equivalence of the Hodge conjecture and the existence of certain singular admissible normal functions. Inspired by their work, we study normal functions using Morihiko Saito's mixed Hodge modules and prove that the existence of singularities of the type considered by Griffiths and Green is equivalent to the Hodge conjecture. Several of the intermediate results, including a relative...
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作者：Valko, Benedek; Virag, Balint
作者单位：University of Wisconsin System; University of Wisconsin Madison; University of Toronto
摘要：We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general beta siblings converge to Sine(beta), a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane. The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We sh...
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作者：Bian, Baojun; Guan, Pengfei
作者单位：McGill University; Tongji University
摘要：We study microscopic convexity property of fully nonlinear elliptic and parabolic partial differential equations. Under certain general structure condition, we establish that the rank of Hessian a double dagger (2) u is of constant rank for any convex solution u of equation F(a double dagger (2) u,a double dagger u,u,x)=0. The similar result is also proved for parabolic equations. Some of geometric applications are also discussed.
