Toeplitz Operators and Random Matrices

Toeplitz Operators and Random Matrices
Author :
Publisher : Springer Nature
Total Pages : 606
Release :
ISBN-10 : 9783031138515
ISBN-13 : 3031138511
Rating : 4/5 (15 Downloads)

Synopsis Toeplitz Operators and Random Matrices by : Estelle Basor

This volume is dedicated to the memory of Harold Widom (1932–2021), an outstanding mathematician who has enriched mathematics with his ideas and ground breaking work since the 1950s until the present time. It contains a biography of Harold Widom, personal notes written by his former students or colleagues, and also his last, previously unpublished paper on domain walls in a Heisenberg–Ising chain. Widom's most famous contributions were made to Toeplitz operators and random matrices. While his work on random matrices is part of almost all the present-day research activities in this field, his work in Toeplitz operators and matrices was done mainly before 2000 and is therefore described in a contribution devoted to his achievements in just this area. The volume contains 18 invited and refereed research and expository papers on Toeplitz operators and random matrices. These present new results or new perspectives on topics related to Widom's work.

Combinatorics and Random Matrix Theory

Combinatorics and Random Matrix Theory
Author :
Publisher : American Mathematical Soc.
Total Pages : 478
Release :
ISBN-10 : 9780821848418
ISBN-13 : 0821848410
Rating : 4/5 (18 Downloads)

Synopsis Combinatorics and Random Matrix Theory by : Jinho Baik

Over the last fifteen years a variety of problems in combinatorics have been solved in terms of random matrix theory. More precisely, the situation is as follows: the problems at hand are probabilistic in nature and, in an appropriate scaling limit, it turns out that certain key quantities associated with these problems behave statistically like the eigenvalues of a (large) random matrix. Said differently, random matrix theory provides a “stochastic special function theory” for a broad and growing class of problems in combinatorics. The goal of this book is to analyze in detail two key examples of this phenomenon, viz., Ulam's problem for increasing subsequences of random permutations and domino tilings of the Aztec diamond. Other examples are also described along the way, but in less detail. Techniques from many different areas in mathematics are needed to analyze these problems. These areas include combinatorics, probability theory, functional analysis, complex analysis, and the theory of integrable systems. The book is self-contained, and along the way we develop enough of the theory we need from each area that a general reader with, say, two or three years experience in graduate school can learn the subject directly from the text.

An Introduction to Random Matrices

An Introduction to Random Matrices
Author :
Publisher : Cambridge University Press
Total Pages : 507
Release :
ISBN-10 : 9780521194525
ISBN-13 : 0521194520
Rating : 4/5 (25 Downloads)

Synopsis An Introduction to Random Matrices by : Greg W. Anderson

A rigorous introduction to the basic theory of random matrices designed for graduate students with a background in probability theory.

Spectra and Pseudospectra

Spectra and Pseudospectra
Author :
Publisher : Princeton University Press
Total Pages : 634
Release :
ISBN-10 : 0691119465
ISBN-13 : 9780691119465
Rating : 4/5 (65 Downloads)

Synopsis Spectra and Pseudospectra by : Lloyd N. Trefethen

Pure and applied mathematicians, physicists, scientists, and engineers use matrices and operators and their eigenvalues in quantum mechanics, fluid mechanics, structural analysis, acoustics, ecology, numerical analysis, and many other areas. However, in some applications the usual analysis based on eigenvalues fails. For example, eigenvalues are often ineffective for analyzing dynamical systems such as fluid flow, Markov chains, ecological models, and matrix iterations. That's where this book comes in. This is the authoritative work on nonnormal matrices and operators, written by the authorities who made them famous. Each of the sixty sections is written as a self-contained essay. Each document is a lavishly illustrated introductory survey of its topic, complete with beautiful numerical experiments and all the right references. The breadth of included topics and the numerous applications that provide links between fields will make this an essential reference in mathematics and related sciences.

Toeplitz Matrices and Operators

Toeplitz Matrices and Operators
Author :
Publisher : Cambridge University Press
Total Pages : 453
Release :
ISBN-10 : 9781107198500
ISBN-13 : 110719850X
Rating : 4/5 (00 Downloads)

Synopsis Toeplitz Matrices and Operators by : Nikolaï Nikolski

A friendly introduction to Toeplitz theory and its applications throughout modern functional analysis.

The Random Matrix Theory of the Classical Compact Groups

The Random Matrix Theory of the Classical Compact Groups
Author :
Publisher : Cambridge University Press
Total Pages : 225
Release :
ISBN-10 : 9781108317993
ISBN-13 : 1108317995
Rating : 4/5 (93 Downloads)

Synopsis The Random Matrix Theory of the Classical Compact Groups by : Elizabeth S. Meckes

This is the first book to provide a comprehensive overview of foundational results and recent progress in the study of random matrices from the classical compact groups, drawing on the subject's deep connections to geometry, analysis, algebra, physics, and statistics. The book sets a foundation with an introduction to the groups themselves and six different constructions of Haar measure. Classical and recent results are then presented in a digested, accessible form, including the following: results on the joint distributions of the entries; an extensive treatment of eigenvalue distributions, including the Weyl integration formula, moment formulae, and limit theorems and large deviations for the spectral measures; concentration of measure with applications both within random matrix theory and in high dimensional geometry; and results on characteristic polynomials with connections to the Riemann zeta function. This book will be a useful reference for researchers and an accessible introduction for students in related fields.

Introduction to the Theory of Toeplitz Operators with Infinite Index

Introduction to the Theory of Toeplitz Operators with Infinite Index
Author :
Publisher : Birkhäuser
Total Pages : 306
Release :
ISBN-10 : 9783034882132
ISBN-13 : 3034882130
Rating : 4/5 (32 Downloads)

Synopsis Introduction to the Theory of Toeplitz Operators with Infinite Index by : Vladimir Dybin

This book is devoted to Toeplitz and singular integral operators with symbols that have discontinuities of the oscillating type. Criteria for the normal solvability of such operators are established and several methods for describing the kernel and image spaces of the operators are presented. The approach is based on the idea of modelling discontinuities with an "infinite index" by appropriate inner functions, especially by infinite Blaschke products. The corresponding techniques have been elaborated by the authors during the last two decades, and they are applicable to both symbols with slowly and rapidly increasing arguments. Moreover, the book reveals exciting connections between invariant subspaces of the shift operator, bases in Banach spaces, and various classes of entire and meromorphic functions. The book aims at making advanced topics accessible to a broad readership. It is addressed to graduate and postgraduate students and to mathematicians interested in functional analysis, the theory of functions of a complex variable, or mathematical physics.

Toeplitz Operators and Related Topics

Toeplitz Operators and Related Topics
Author :
Publisher : Birkhäuser
Total Pages : 208
Release :
ISBN-10 : 9783034885430
ISBN-13 : 3034885431
Rating : 4/5 (30 Downloads)

Synopsis Toeplitz Operators and Related Topics by : Estelle L. Basor

This volume is dedicated to Harold Widom, a distinguished mathematician and renowned expert in the area of Toeplitz, Wiener-Hopf and pseudodifferential operators, on the occasion of his sixtieth birthday. The book opens with biographical material and a list of the mathematician's publications, this being followed by two papers based on Toeplitz lectures which he delivered at Tel Aviv University in March, 1993. The rest of the book consists of a selection of papers containing some recent achievements in the following areas: Szegö-Widom asymptotic formulas for determinants of finite sections of Toeplitz matrices and their generalizations, the Fisher-Hartwig conjecture, random matrices, analysis of kernels of Toeplitz matrices, projectional methods and eigenvalue distribution for Toeplitz matrices, the Fredholm theory for convolution type operators, the Nehari interpolation problem with generalizations and applications, and Toeplitz-Hausdorff type theorems. The book will appeal to a wide audience of pure and applied mathematicians.

Toeplitz and Circulant Matrices

Toeplitz and Circulant Matrices
Author :
Publisher : Now Publishers Inc
Total Pages : 105
Release :
ISBN-10 : 9781933019239
ISBN-13 : 1933019239
Rating : 4/5 (39 Downloads)

Synopsis Toeplitz and Circulant Matrices by : Robert M. Gray

The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes. The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes.

Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics

Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics
Author :
Publisher : Birkhäuser
Total Pages : 757
Release :
ISBN-10 : 9783319491820
ISBN-13 : 3319491822
Rating : 4/5 (20 Downloads)

Synopsis Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics by : Dario A. Bini

This book presents a collection of expository and research papers on various topics in matrix and operator theory, contributed by several experts on the occasion of Albrecht Böttcher’s 60th birthday. Albrecht Böttcher himself has made substantial contributions to the subject in the past. The book also includes a biographical essay, a complete bibliography of Albrecht Böttcher’s work and brief informal notes on personal encounters with him. The book is of interest to graduate and advanced undergraduate students majoring in mathematics, researchers in matrix and operator theory as well as engineers and applied mathematicians.