Traffic Distributions And Independence Permutation Invariant Random Matrices And The Three Notions Of Independence

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Traffic Distributions and Independence: Permutation Invariant Random Matrices and the Three Notions of Independence

Camille Male
Traffic Distributions and Independence Permutation Invariant Random Matrices and the Three Notions of Independence Book PDF

Author : Camille Male
Publisher : American Mathematical Society
Page : 88 pages
File Size : 41,5 Mb
Release : 2021-02-10
Category : Mathematics
ISBN : 9781470442989

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Traffic Distributions and Independence: Permutation Invariant Random Matrices and the Three Notions of Independence by Camille Male Pdf

Voiculescu's notion of asymptotic free independence is known for a large class of random matrices including independent unitary invariant matrices. This notion is extended for independent random matrices invariant in law by conjugation by permutation matrices. This fact leads naturally to an extension of free probability, formalized under the notions of traffic probability. The author first establishes this construction for random matrices and then defines the traffic distribution of random matrices, which is richer than the $^*$-distribution of free probability. The knowledge of the individual traffic distributions of independent permutation invariant families of matrices is sufficient to compute the limiting distribution of the join family. Under a factorization assumption, the author calls traffic independence the asymptotic rule that plays the role of independence with respect to traffic distributions. Wigner matrices, Haar unitary matrices and uniform permutation matrices converge in traffic distributions, a fact which yields new results on the limiting $^*$-distributions of several matrices the author can construct from them. Then the author defines the abstract traffic spaces as non commutative probability spaces with more structure. She proves that at an algebraic level, traffic independence in some sense unifies the three canonical notions of tensor, free and Boolean independence. A central limiting theorem is stated in this context, interpolating between the tensor, free and Boolean central limit theorems.

Random Matrices and Non-Commutative Probability

Arup Bose
Random Matrices and Non Commutative Probability Book PDF

Author : Arup Bose
Publisher : CRC Press
Page : 420 pages
File Size : 45,8 Mb
Release : 2021-10-26
Category : Mathematics
ISBN : 9781000458824

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Random Matrices and Non-Commutative Probability by Arup Bose Pdf

This is an introductory book on Non-Commutative Probability or Free Probability and Large Dimensional Random Matrices. Basic concepts of free probability are introduced by analogy with classical probability in a lucid and quick manner. It then develops the results on the convergence of large dimensional random matrices, with a special focus on the interesting connections to free probability. The book assumes almost no prerequisite for the most part. However, familiarity with the basic convergence concepts in probability and a bit of mathematical maturity will be helpful. Combinatorial properties of non-crossing partitions, including the Möbius function play a central role in introducing free probability. Free independence is defined via free cumulants in analogy with the way classical independence can be defined via classical cumulants. Free cumulants are introduced through the Möbius function. Free product probability spaces are constructed using free cumulants. Marginal and joint tracial convergence of large dimensional random matrices such as the Wigner, elliptic, sample covariance, cross-covariance, Toeplitz, Circulant and Hankel are discussed. Convergence of the empirical spectral distribution is discussed for symmetric matrices. Asymptotic freeness results for random matrices, including some recent ones, are discussed in detail. These clarify the structure of the limits for joint convergence of random matrices. Asymptotic freeness of independent sample covariance matrices is also demonstrated via embedding into Wigner matrices. Exercises, at advanced undergraduate and graduate level, are provided in each chapter.

Differential Function Spectra, the Differential Becker-Gottlieb Transfer, and Applications to Differential Algebraic K-Theory

Ulrich Bunke,David Gepner
Differential Function Spectra the Differential Becker Gottlieb Transfer and Applications to Differential Algebraic K Theory Book PDF

Author : Ulrich Bunke,David Gepner
Publisher : American Mathematical Soc.
Page : 177 pages
File Size : 45,8 Mb
Release : 2021-06-21
Category : Education
ISBN : 9781470446857

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Differential Function Spectra, the Differential Becker-Gottlieb Transfer, and Applications to Differential Algebraic K-Theory by Ulrich Bunke,David Gepner Pdf

We develop differential algebraic K-theory for rings of integers in number fields and we construct a cycle map from geometrized bundles of modules over such a ring to the differential algebraic K-theory. We also treat some of the foundational aspects of differential cohomology, including differential function spectra and the differential Becker-Gottlieb transfer. We then state a transfer index conjecture about the equality of the Becker-Gottlieb transfer and the analytic transfer defined by Lott. In support of this conjecture, we derive some non-trivial consequences which are provable by independent means.

Gromov-Witten Theory of Quotients of Fermat Calabi-Yau Varieties

Hiroshi Iritani,Todor Milanov,Yongbin Ruan, Yefeng Shen
Gromov Witten Theory of Quotients of Fermat Calabi Yau Varieties Book PDF

Author : Hiroshi Iritani,Todor Milanov,Yongbin Ruan, Yefeng Shen
Publisher : American Mathematical Soc.
Page : 92 pages
File Size : 53,6 Mb
Release : 2021-06-21
Category : Education
ISBN : 9781470443634

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Gromov-Witten Theory of Quotients of Fermat Calabi-Yau Varieties by Hiroshi Iritani,Todor Milanov,Yongbin Ruan, Yefeng Shen Pdf

Gromov-Witten theory started as an attempt to provide a rigorous mathematical foundation for the so-called A-model topological string theory of Calabi-Yau varieties. Even though it can be defined for all the Kähler/symplectic manifolds, the theory on Calabi-Yau varieties remains the most difficult one. In fact, a great deal of techniques were developed for non-Calabi-Yau varieties during the last twenty years. These techniques have only limited bearing on the Calabi-Yau cases. In a certain sense, Calabi-Yau cases are very special too. There are two outstanding problems for the Gromov-Witten theory of Calabi-Yau varieties and they are the focus of our investigation.

Resolvent, Heat Kernel, and Torsion under Degeneration to Fibered Cusps

Pierre Albin,Frédéric Rochon,David Sher
Resolvent Heat Kernel and Torsion under Degeneration to Fibered Cusps Book PDF

Author : Pierre Albin,Frédéric Rochon,David Sher
Publisher : American Mathematical Soc.
Page : 126 pages
File Size : 42,5 Mb
Release : 2021-06-21
Category : Education
ISBN : 9781470444228

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Resolvent, Heat Kernel, and Torsion under Degeneration to Fibered Cusps by Pierre Albin,Frédéric Rochon,David Sher Pdf

Manifolds with fibered cusps are a class of complete non-compact Riemannian manifolds including many examples of locally symmetric spaces of rank one. We study the spectrum of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold undergoing degeneration to a manifold with fibered cusps. We obtain precise asymptotics for the resolvent, the heat kernel, and the determinant of the Laplacian. Using these asymptotics we obtain a topological description of the analytic torsion on a manifold with fibered cusps in terms of the R-torsion of the underlying manifold with boundary.

Local Well-Posedness and Break-Down Criterion of the Incompressible Euler Equations with Free Boundary

Chao Wang
Local Well Posedness and Break Down Criterion of the Incompressible Euler Equations with Free Boundary Book PDF

Author : Chao Wang
Publisher : American Mathematical Soc.
Page : 119 pages
File Size : 51,7 Mb
Release : 2021-07-21
Category : Education
ISBN : 9781470446895

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Local Well-Posedness and Break-Down Criterion of the Incompressible Euler Equations with Free Boundary by Chao Wang Pdf

In this paper, we prove the local well-posedness of the free boundary problem for the incompressible Euler equations in low regularity Sobolev spaces, in which the velocity is a Lipschitz function and the free surface belongs to C 3 2 +ε. Moreover, we also present a Beale-Kato-Majda type break-down criterion of smooth solution in terms of the mean curvature of the free surface, the gradient of the velocity and Taylor sign condition.

Paley-Wiener Theorems for a p-Adic Spherical Variety

Patrick Delorme,Pascale Harinck,Yiannis Sakellaridis
Paley Wiener Theorems for a p Adic Spherical Variety Book PDF

Author : Patrick Delorme,Pascale Harinck,Yiannis Sakellaridis
Publisher : American Mathematical Soc.
Page : 102 pages
File Size : 49,9 Mb
Release : 2021-06-21
Category : Education
ISBN : 9781470444020

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Paley-Wiener Theorems for a p-Adic Spherical Variety by Patrick Delorme,Pascale Harinck,Yiannis Sakellaridis Pdf

Let SpXq be the Schwartz space of compactly supported smooth functions on the p-adic points of a spherical variety X, and let C pXq be the space of Harish-Chandra Schwartz functions. Under assumptions on the spherical variety, which are satisfied when it is symmetric, we prove Paley–Wiener theorems for the two spaces, characterizing them in terms of their spectral transforms. As a corollary, we get relative analogs of the smooth and tempered Bernstein centers — rings of multipliers for SpXq and C pXq.WhenX “ a reductive group, our theorem for C pXq specializes to the well-known theorem of Harish-Chandra, and our theorem for SpXq corresponds to a first step — enough to recover the structure of the Bern-stein center — towards the well-known theorems of Bernstein [Ber] and Heiermann [Hei01].

Bounded Littlewood Identities

Eric M. Rains,S. Ole Warnaar
Bounded Littlewood Identities Book PDF

Author : Eric M. Rains,S. Ole Warnaar
Publisher : American Mathematical Soc.
Page : 115 pages
File Size : 42,7 Mb
Release : 2021-07-21
Category : Education
ISBN : 9781470446901

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Bounded Littlewood Identities by Eric M. Rains,S. Ole Warnaar Pdf

We describe a method, based on the theory of Macdonald–Koornwinder polynomials, for proving bounded Littlewood identities. Our approach provides an alternative to Macdonald’s partial fraction technique and results in the first examples of bounded Littlewood identities for Macdonald polynomials. These identities, which take the form of decomposition formulas for Macdonald polynomials of type (R, S) in terms of ordinary Macdonald polynomials, are q, t-analogues of known branching formulas for characters of the symplectic, orthogonal and special orthogonal groups. In the classical limit, our method implies that MacMahon’s famous ex-conjecture for the generating function of symmetric plane partitions in a box follows from the identification of GL(n, R), O(n) as a Gelfand pair. As further applications, we obtain combinatorial formulas for characters of affine Lie algebras; Rogers–Ramanujan identities for affine Lie algebras, complementing recent results of Griffin et al.; and quadratic transformation formulas for Kaneko–Macdonald-type basic hypergeometric series.

Linear Dynamical Systems on Hilbert Spaces: Typical Properties and Explicit Examples

S. Grivaux,É. Matheron,Q. Menet
Linear Dynamical Systems on Hilbert Spaces Typical Properties and Explicit Examples Book PDF

Author : S. Grivaux,É. Matheron,Q. Menet
Publisher : American Mathematical Soc.
Page : 147 pages
File Size : 53,6 Mb
Release : 2021-06-21
Category : Education
ISBN : 9781470446635

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Linear Dynamical Systems on Hilbert Spaces: Typical Properties and Explicit Examples by S. Grivaux,É. Matheron,Q. Menet Pdf

We solve a number of questions pertaining to the dynamics of linear operators on Hilbert spaces, sometimes by using Baire category arguments and sometimes by constructing explicit examples. In particular, we prove the following results. (i) A typical hypercyclic operator is not topologically mixing, has no eigen-values and admits no non-trivial invariant measure, but is densely distri-butionally chaotic. (ii) A typical upper-triangular operator with coefficients of modulus 1 on the diagonal is ergodic in the Gaussian sense, whereas a typical operator of the form “diagonal with coefficients of modulus 1 on the diagonal plus backward unilateral weighted shift” is ergodic but has only countably many unimodular eigenvalues; in particular, it is ergodic but not ergodic in the Gaussian sense. (iii) There exist Hilbert space operators which are chaotic and U-frequently hypercyclic but not frequently hypercyclic, Hilbert space operators which are chaotic and frequently hypercyclic but not ergodic, and Hilbert space operators which are chaotic and topologically mixing but not U-frequently hypercyclic. We complement our results by investigating the descriptive complexity of some natural classes of operators defined by dynamical properties.

Weakly Modular Graphs and Nonpositive Curvature

Jérémie Chalopin,Victor Chepoi,Hiroshi Hirai,Damian Osajda
Weakly Modular Graphs and Nonpositive Curvature Book PDF

Author : Jérémie Chalopin,Victor Chepoi,Hiroshi Hirai,Damian Osajda
Publisher : American Mathematical Soc.
Page : 85 pages
File Size : 40,9 Mb
Release : 2021-06-18
Category : Education
ISBN : 9781470443627

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Weakly Modular Graphs and Nonpositive Curvature by Jérémie Chalopin,Victor Chepoi,Hiroshi Hirai,Damian Osajda Pdf

This article investigates structural, geometrical, and topological characteri-zations and properties of weakly modular graphs and of cell complexes derived from them. The unifying themes of our investigation are various “nonpositive cur-vature” and “local-to-global” properties and characterizations of weakly modular graphs and their subclasses. Weakly modular graphs have been introduced as a far-reaching common generalization of median graphs (and more generally, of mod-ular and orientable modular graphs), Helly graphs, bridged graphs, and dual polar graphs occurring under different disguises (1–skeletons, collinearity graphs, covering graphs, domains, etc.) in several seemingly-unrelated fields of mathematics: * Metric graph theory * Geometric group theory * Incidence geometries and buildings * Theoretical computer science and combinatorial optimization We give a local-to-global characterization of weakly modular graphs and their sub-classes in terms of simple connectedness of associated triangle-square complexes and specific local combinatorial conditions. In particular, we revisit characterizations of dual polar graphs by Cameron and by Brouwer-Cohen. We also show that (disk-)Helly graphs are precisely the clique-Helly graphs with simply connected clique complexes. With l1–embeddable weakly modular and sweakly modular graphs we associate high-dimensional cell complexes, having several strong topological and geometrical properties (contractibility and the CAT(0) property). Their cells have a specific structure: they are basis polyhedra of even 􀀁–matroids in the first case and orthoscheme complexes of gated dual polar subgraphs in the second case. We resolve some open problems concerning subclasses of weakly modular graphs: we prove a Brady-McCammond conjecture about CAT(0) metric on the orthoscheme.

The 2D Compressible Euler Equations in Bounded Impermeable Domains with Corners

Paul Godin
The 2D Compressible Euler Equations in Bounded Impermeable Domains with Corners Book PDF

Author : Paul Godin
Publisher : American Mathematical Soc.
Page : 72 pages
File Size : 41,9 Mb
Release : 2021-06-21
Category : Education
ISBN : 9781470444211

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The 2D Compressible Euler Equations in Bounded Impermeable Domains with Corners by Paul Godin Pdf

We study 2D compressible Euler flows in bounded impermeable domains whose boundary is smooth except for corners. We assume that the angles of the corners are small enough. Then we obtain local (in time) existence of solutions which keep the L2 Sobolev regularity of their Cauchy data, provided the external forces are sufficiently regular and suitable compatibility conditions are satisfied. Such a result is well known when there is no corner. Our proof relies on the study of associated linear problems. We also show that our results are rather sharp: we construct counterexamples in which the smallness condition on the angles is not fulfilled and which display a loss of L2 Sobolev regularity with respect to the Cauchy data and the external forces.

Infinity Operads And Monoidal Categories With Group Equivariance

Donald Yau
Infinity Operads And Monoidal Categories With Group Equivariance Book PDF

Author : Donald Yau
Publisher : World Scientific
Page : 486 pages
File Size : 49,9 Mb
Release : 2021-12-02
Category : Mathematics
ISBN : 9789811250941

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Infinity Operads And Monoidal Categories With Group Equivariance by Donald Yau Pdf

This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads, and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of the universal cover of the framed little 2-disc operad is an infinity ribbon operad.In Part 4 we define general monoidal categories equipped with an action operad equivariant structure and provide a unifying treatment of coherence and strictification for them. Examples of such monoidal categories include symmetric, braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups and of coboundary Hopf algebras and in the theory of crystals of finite dimensional complex reductive Lie algebras.

Local Dynamics of Non-Invertible Maps Near Normal Surface Singularities

William Gignac,Matteo Ruggiero
Local Dynamics of Non Invertible Maps Near Normal Surface Singularities Book PDF

Author : William Gignac,Matteo Ruggiero
Publisher : American Mathematical Society
Page : 100 pages
File Size : 51,7 Mb
Release : 2021-11-16
Category : Mathematics
ISBN : 9781470449582

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Local Dynamics of Non-Invertible Maps Near Normal Surface Singularities by William Gignac,Matteo Ruggiero Pdf

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Goodwillie Approximations to Higher Categories

Gijs Heuts
Goodwillie Approximations to Higher Categories Book PDF

Author : Gijs Heuts
Publisher : American Mathematical Society
Page : 108 pages
File Size : 47,7 Mb
Release : 2021-11-16
Category : Mathematics
ISBN : 9781470448936

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Goodwillie Approximations to Higher Categories by Gijs Heuts Pdf

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