The Hermitian Two Matrix Model With An Even Quartic Potential

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The Hermitian Two Matrix Model with an Even Quartic Potential

Maurice Duits
The Hermitian Two Matrix Model with an Even Quartic Potential Book PDF

Author : Maurice Duits
Publisher : Unknown
Page : 105 pages
File Size : 42,6 Mb
Release : 2011
Category : Boundary value problems
ISBN : 0821887564

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The Hermitian Two Matrix Model with an Even Quartic Potential by Maurice Duits Pdf

We consider the two matrix model with an even quartic potential W(y)=y4/4+αy2/2 and an even polynomial potential V(x). The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices M1. The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a 4×4 matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of M1. Our results generalize earlier results for the case α=0, where the external field on the third measure was not present.

The Hermitian Two Matrix Model with an Even Quartic Potential

Maurice Duits,Arno B. J. Kuijlaars,Man Yue Mo
The Hermitian Two Matrix Model with an Even Quartic Potential Book PDF

Author : Maurice Duits,Arno B. J. Kuijlaars,Man Yue Mo
Publisher : American Mathematical Soc.
Page : 105 pages
File Size : 53,7 Mb
Release : 2012
Category : Boundary value problems
ISBN : 9780821869284

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The Hermitian Two Matrix Model with an Even Quartic Potential by Maurice Duits,Arno B. J. Kuijlaars,Man Yue Mo Pdf

The authors consider the two matrix model with an even quartic potential $W(y)=y^4/4+\alpha y^2/2$ and an even polynomial potential $V(x)$. The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices $M_1$. The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a $4\times4$ matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of $M_1$. The authors' results generalize earlier results for the case $\alpha=0$, where the external field on the third measure was not present.

Random Matrix Theory, Interacting Particle Systems and Integrable Systems

Percy Deift,Peter Forrester
Random Matrix Theory Interacting Particle Systems and Integrable Systems Book PDF

Author : Percy Deift,Peter Forrester
Publisher : Cambridge University Press
Page : 539 pages
File Size : 52,5 Mb
Release : 2014-12-15
Category : Language Arts & Disciplines
ISBN : 9781107079922

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Random Matrix Theory, Interacting Particle Systems and Integrable Systems by Percy Deift,Peter Forrester Pdf

This volume includes review articles and research contributions on long-standing questions on universalities of Wigner matrices and beta-ensembles.

Potential Wadge Classes

Dominique Lecomte
Potential Wadge Classes Book PDF

Author : Dominique Lecomte
Publisher : American Mathematical Soc.
Page : 83 pages
File Size : 40,9 Mb
Release : 2013-01-25
Category : Mathematics
ISBN : 9780821875575

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Potential Wadge Classes by Dominique Lecomte Pdf

Let $\bf\Gamma$ be a Borel class, or a Wadge class of Borel sets, and $2\!\leq\! d\!\leq\!\omega$ be a cardinal. A Borel subset $B$ of ${\mathbb R}^d$ is potentially in $\bf\Gamma$ if there is a finer Polish topology on $\mathbb R$ such that $B$ is in $\bf\Gamma$ when ${\mathbb R}^d$ is equipped with the new product topology. The author provides a way to recognize the sets potentially in $\bf\Gamma$ and applies this to the classes of graphs (oriented or not), quasi-orders and partial orders.

A Mutation-Selection Model with Recombination for General Genotypes

Steven Neil Evans,David Steinsaltz,Kenneth W. Wachter
A Mutation Selection Model with Recombination for General Genotypes Book PDF

Author : Steven Neil Evans,David Steinsaltz,Kenneth W. Wachter
Publisher : American Mathematical Soc.
Page : 128 pages
File Size : 55,9 Mb
Release : 2013-02-26
Category : Mathematics
ISBN : 9780821875698

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A Mutation-Selection Model with Recombination for General Genotypes by Steven Neil Evans,David Steinsaltz,Kenneth W. Wachter Pdf

The authors investigate a continuous time, probability measure-valued dynamical system that describes the process of mutation-selection balance in a context where the population is infinite, there may be infinitely many loci, and there are weak assumptions on selective costs. Their model arises when they incorporate very general recombination mechanisms into an earlier model of mutation and selection presented by Steinsaltz, Evans and Wachter in 2005 and take the relative strength of mutation and selection to be sufficiently small. The resulting dynamical system is a flow of measures on the space of loci. Each such measure is the intensity measure of a Poisson random measure on the space of loci: the points of a realization of the random measure record the set of loci at which the genotype of a uniformly chosen individual differs from a reference wild type due to an accumulation of ancestral mutations. The authors' motivation for working in such a general setting is to provide a basis for understanding mutation-driven changes in age-specific demographic schedules that arise from the complex interaction of many genes, and hence to develop a framework for understanding the evolution of aging.

On First and Second Order Planar Elliptic Equations with Degeneracies

Abdelhamid Meziani
On First and Second Order Planar Elliptic Equations with Degeneracies Book PDF

Author : Abdelhamid Meziani
Publisher : American Mathematical Soc.
Page : 77 pages
File Size : 52,8 Mb
Release : 2012
Category : Degenerate differential equations
ISBN : 9780821853122

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On First and Second Order Planar Elliptic Equations with Degeneracies by Abdelhamid Meziani Pdf

This paper deals with elliptic equations in the plane with degeneracies. The equations are generated by a complex vector field that is elliptic everywhere except along a simple closed curve. Kernels for these equations are constructed. Properties of solutions, in a neighborhood of the degeneracy curve, are obtained through integral and series representations. An application to a second order elliptic equation with a punctual singularity is given.

Zeta Functions for Two-Dimensional Shifts of Finite Type

Jung-Chao Ban,Wen-Guei Hu,Song-Sun Lin,Yin-Heng Lin
Zeta Functions for Two Dimensional Shifts of Finite Type Book PDF

Author : Jung-Chao Ban,Wen-Guei Hu,Song-Sun Lin,Yin-Heng Lin
Publisher : American Mathematical Soc.
Page : 60 pages
File Size : 49,9 Mb
Release : 2013-01-25
Category : Mathematics
ISBN : 9780821872901

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Zeta Functions for Two-Dimensional Shifts of Finite Type by Jung-Chao Ban,Wen-Guei Hu,Song-Sun Lin,Yin-Heng Lin Pdf

This work is concerned with zeta functions of two-dimensional shifts of finite type. A two-dimensional zeta function $\zeta^{0}(s)$, which generalizes the Artin-Mazur zeta function, was given by Lind for $\mathbb{Z}^{2}$-action $\phi$. In this paper, the $n$th-order zeta function $\zeta_{n}$ of $\phi$ on $\mathbb{Z}_{n\times \infty}$, $n\geq 1$, is studied first. The trace operator $\mathbf{T}_{n}$, which is the transition matrix for $x$-periodic patterns with period $n$ and height $2$, is rotationally symmetric. The rotational symmetry of $\mathbf{T}_{n}$ induces the reduced trace operator $\tau_{n}$ and $\zeta_{n}=\left(\det\left(I-s^{n}\tau_{n}\right)\right)^{-1}$. The zeta function $\zeta=\prod_{n=1}^{\infty} \left(\det\left(I-s^{n}\tau_{n}\right)\right)^{-1}$ in the $x$-direction is now a reciprocal of an infinite product of polynomials. The zeta function can be presented in the $y$-direction and in the coordinates of any unimodular transformation in $GL_{2}(\mathbb{Z})$. Therefore, there exists a family of zeta functions that are meromorphic extensions of the same analytic function $\zeta^{0}(s)$. The natural boundary of zeta functions is studied. The Taylor series for these zeta functions at the origin are equal with integer coefficients, yielding a family of identities, which are of interest in number theory. The method applies to thermodynamic zeta functions for the Ising model with finite range interactions.

Infinite-dimensional Representations of 2-groups

John C. Baez
Infinite dimensional Representations of 2 groups Book PDF

Author : John C. Baez
Publisher : American Mathematical Soc.
Page : 120 pages
File Size : 45,5 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821872840

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Infinite-dimensional Representations of 2-groups by John C. Baez Pdf

A “$2$-group'' is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, $2$-groups have representations on “$2$-vector spaces'', which are categories analogous to vector spaces. Unfortunately, Lie $2$-groups typically have few representations on the finite-dimensional $2$-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional $2$-vector spaces called ``measurable categories'' (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional representations of certain Lie $2$-groups. Here they continue this work.

They begin with a detailed study of measurable categories. Then they give a geometrical description of the measurable representations, intertwiners and $2$-intertwiners for any skeletal measurable $2$-group. They study tensor products and direct sums for representations, and various concepts of subrepresentation. They describe direct sums of intertwiners, and sub-intertwiners--features not seen in ordinary group representation theory and study irreducible and indecomposable representations and intertwiners. They also study “irretractable'' representations--another feature not seen in ordinary group representation theory. Finally, they argue that measurable categories equipped with some extra structure deserve to be considered “separable $2$-Hilbert spaces'', and compare this idea to a tentative definition of $2$-Hilbert spaces as representation categories of commutative von Neumann algebras.

Extended Graphical Calculus for Categorified Quantum Sl(2)

Mikhail Khovanov
Extended Graphical Calculus for Categorified Quantum Sl 2 Book PDF

Author : Mikhail Khovanov
Publisher : American Mathematical Soc.
Page : 87 pages
File Size : 45,7 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821889770

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Extended Graphical Calculus for Categorified Quantum Sl(2) by Mikhail Khovanov Pdf

A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in a paper (arXiv:0803.3652) by Aaron D. Lauda. Here the authors enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. They obtain explicit diagrammatical formulas for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms; the latter are in a bijection with the Lusztig canonical basis elements.

These formulas have integral coefficients and imply that one of the main results of Lauda's paper--identification of the Grothendieck ring of his 2-category with the idempotented quantum sl(2)--also holds when the 2-category is defined over the ring of integers rather than over a field. A new diagrammatic description of Schur functions is also given and it is shown that the the Jacobi-Trudy formulas for the decomposition of Schur functions into elementary or complete symmetric functions follows from the diagrammatic relations for categorified quantum sl(2).

General Relativistic Self-similar Waves that Induce an Anomalous Acceleration Into the Standard Model of Cosmology

Joel Smoller,Blake Temple
General Relativistic Self similar Waves that Induce an Anomalous Acceleration Into the Standard Model of Cosmology Book PDF

Author : Joel Smoller,Blake Temple
Publisher : American Mathematical Soc.
Page : 69 pages
File Size : 48,5 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821853580

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General Relativistic Self-similar Waves that Induce an Anomalous Acceleration Into the Standard Model of Cosmology by Joel Smoller,Blake Temple Pdf

The authors prove that the Einstein equations for a spherically symmetric spacetime in Standard Schwarzschild Coordinates (SSC) close to form a system of three ordinary differential equations for a family of self-similar expansion waves, and the critical ($k=0$) Friedmann universe associated with the pure radiation phase of the Standard Model of Cosmology is embedded as a single point in this family. Removing a scaling law and imposing regularity at the center, they prove that the family reduces to an implicitly defined one-parameter family of distinct spacetimes determined by the value of a new acceleration parameter $a$, such that $a=1$ corresponds to the Standard Model. The authors prove that all of the self-similar spacetimes in the family are distinct from the non-critical $k\neq0$ Friedmann spacetimes, thereby characterizing the critical $k=0$ Friedmann universe as the unique spacetime lying at the intersection of these two one-parameter families. They then present a mathematically rigorous analysis of solutions near the singular point at the center, deriving the expansion of solutions up to fourth order in the fractional distance to the Hubble Length. Finally, they use these rigorous estimates to calculate the exact leading order quadratic and cubic corrections to the redshift vs luminosity relation for an observer at the center.

Global Regularity for the Yang-Mills Equations on High Dimensional Minkowski Space

Joachim Krieger,Jacob Sterbenz
Global Regularity for the Yang Mills Equations on High Dimensional Minkowski Space Book PDF

Author : Joachim Krieger,Jacob Sterbenz
Publisher : American Mathematical Soc.
Page : 99 pages
File Size : 47,7 Mb
Release : 2013-04-22
Category : Mathematics
ISBN : 9780821844892

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Global Regularity for the Yang-Mills Equations on High Dimensional Minkowski Space by Joachim Krieger,Jacob Sterbenz Pdf

This monograph contains a study of the global Cauchy problem for the Yang-Mills equations on $(6+1)$ and higher dimensional Minkowski space, when the initial data sets are small in the critical gauge covariant Sobolev space $\dot{H}_A^{(n-4)/{2}}$. Regularity is obtained through a certain ``microlocal geometric renormalization'' of the equations which is implemented via a family of approximate null Cronstrom gauge transformations. The argument is then reduced to controlling some degenerate elliptic equations in high index and non-isotropic $L^p$ spaces, and also proving some bilinear estimates in specially constructed square-function spaces.

A Theory of Generalized Donaldson-Thomas Invariants

Dominic D. Joyce,Yinan Song
A Theory of Generalized Donaldson Thomas Invariants Book PDF

Author : Dominic D. Joyce,Yinan Song
Publisher : American Mathematical Soc.
Page : 199 pages
File Size : 54,6 Mb
Release : 2012
Category : Calabi-Yau manifolds
ISBN : 9780821852798

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A Theory of Generalized Donaldson-Thomas Invariants by Dominic D. Joyce,Yinan Song Pdf

This book studies generalized Donaldson-Thomas invariants $\bar{DT}{}^\alpha(\tau)$. They are rational numbers which `count' both $\tau$-stable and $\tau$-semistable coherent sheaves with Chern character $\alpha$ on $X$; strictly $\tau$-semistable sheaves must be counted with complicated rational weights. The $\bar{DT}{}^\alpha(\tau)$ are defined for all classes $\alpha$, and are equal to $DT^\alpha(\tau)$ when it is defined. They are unchanged under deformations of $X$, and transform by a wall-crossing formula under change of stability condition $\tau$. To prove all this, the authors study the local structure of the moduli stack $\mathfrak M$ of coherent sheaves on $X$. They show that an atlas for $\mathfrak M$ may be written locally as $\mathrm{Crit}(f)$ for $f:U\to{\mathbb C}$ holomorphic and $U$ smooth, and use this to deduce identities on the Behrend function $\nu_\mathfrak M$. They compute the invariants $\bar{DT}{}^\alpha(\tau)$ in examples, and make a conjecture about their integrality properties. They also extend the theory to abelian categories $\mathrm{mod}$-$\mathbb{C}Q\backslash I$ of representations of a quiver $Q$ with relations $I$ coming from a superpotential $W$ on $Q$.

Networking Seifert Surgeries on Knots

Arnaud Deruelle,Katura Miyazaki,Kimihiko Motegi
Networking Seifert Surgeries on Knots Book PDF

Author : Arnaud Deruelle,Katura Miyazaki,Kimihiko Motegi
Publisher : American Mathematical Soc.
Page : 130 pages
File Size : 47,8 Mb
Release : 2012
Category : Complex manifolds
ISBN : 9780821853337

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Networking Seifert Surgeries on Knots by Arnaud Deruelle,Katura Miyazaki,Kimihiko Motegi Pdf

The authors propose a new approach in studying Dehn surgeries on knots in the $3$-sphere $S^3$ yielding Seifert fiber spaces. The basic idea is finding relationships among such surgeries. To describe relationships and get a global picture of Seifert surgeries, they introduce ``seiferters'' and the Seifert Surgery Network, a $1$-dimensional complex whose vertices correspond to Seifert surgeries. A seiferter for a Seifert surgery on a knot $K$ is a trivial knot in $S^3$ disjoint from $K$ that becomes a fiber in the resulting Seifert fiber space. Twisting $K$ along its seiferter or an annulus cobounded by a pair of its seiferters yields another knot admitting a Seifert surgery. Edges of the network correspond to such twistings. A path in the network from one Seifert surgery to another explains how the former Seifert surgery is obtained from the latter after a sequence of twistings along seiferters and/or annuli cobounded by pairs of seiferters. The authors find explicit paths from various known Seifert surgeries to those on torus knots, the most basic Seifert surgeries. The authors classify seiferters and obtain some fundamental results on the structure of the Seifert Surgery Network. From the networking viewpoint, they find an infinite family of Seifert surgeries on hyperbolic knots which cannot be embedded in a genus two Heegaard surface of $S^3$.

Characterization and Topological Rigidity of Nobeling Manifolds

Andrzej Nagórko
Characterization and Topological Rigidity of Nobeling Manifolds Book PDF

Author : Andrzej Nagórko
Publisher : American Mathematical Soc.
Page : 92 pages
File Size : 49,8 Mb
Release : 2013-04-22
Category : Mathematics
ISBN : 9780821853665

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Characterization and Topological Rigidity of Nobeling Manifolds by Andrzej Nagórko Pdf

The author develops a theory of Nobeling manifolds similar to the theory of Hilbert space manifolds. He shows that it reflects the theory of Menger manifolds developed by M. Bestvina and is its counterpart in the realm of complete spaces. In particular the author proves the Nobeling manifold characterization conjecture.

The Poset of $k$-Shapes and Branching Rules for $k$-Schur Functions

Thomas Lam,Luc Lapointe,Jennifer Morse,Mark Shimozono
The Poset of k Shapes and Branching Rules for k Schur Functions Book PDF

Author : Thomas Lam,Luc Lapointe,Jennifer Morse,Mark Shimozono
Publisher : American Mathematical Soc.
Page : 101 pages
File Size : 50,6 Mb
Release : 2013-04-22
Category : Mathematics
ISBN : 9780821872949

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The Poset of $k$-Shapes and Branching Rules for $k$-Schur Functions by Thomas Lam,Luc Lapointe,Jennifer Morse,Mark Shimozono Pdf

The authors give a combinatorial expansion of a Schubert homology class in the affine Grassmannian $\mathrm{Gr}_{\mathrm{SL}_k}$ into Schubert homology classes in $\mathrm{Gr}_{\mathrm{SL}_{k+1}}$. This is achieved by studying the combinatorics of a new class of partitions called $k$-shapes, which interpolates between $k$-cores and $k+1$-cores. The authors define a symmetric function for each $k$-shape, and show that they expand positively in terms of dual $k$-Schur functions. They obtain an explicit combinatorial description of the expansion of an ungraded $k$-Schur function into $k+1$-Schur functions. As a corollary, they give a formula for the Schur expansion of an ungraded $k$-Schur function.