Orthogonal Polynomials And Painlev Equations
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Orthogonal Polynomials And Painlev Equations
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Author : Walter Van Assche
language : en
Publisher: Cambridge University Press
Release Date : 2018
Orthogonal Polynomials And Painlev Equations written by Walter Van Assche and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with Mathematics categories.
There are a number of intriguing connections between Painlev equations and orthogonal polynomials, and this book is one of the first to provide an introduction to these. Researchers in integrable systems and non-linear equations will find the many explicit examples where Painlev equations appear in mathematical analysis very useful. Those interested in the asymptotic behavior of orthogonal polynomials will also find the description of Painlev transcendants and their use for local analysis near certain critical points helpful to their work. Rational solutions and special function solutions of Painlev equations are worked out in detail, with a survey of recent results and an outline of their close relationship with orthogonal polynomials. Exercises throughout the book help the reader to get to grips with the material. The author is a leading authority on orthogonal polynomials, giving this work a unique perspective on Painlev equations.
Classical And Quantum Orthogonal Polynomials In One Variable
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Author : Mourad Ismail
language : en
Publisher: Cambridge University Press
Release Date : 2005-11-21
Classical And Quantum Orthogonal Polynomials In One Variable written by Mourad Ismail and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-11-21 with Mathematics categories.
The first modern treatment of orthogonal polynomials from the viewpoint of special functions is now available in paperback.
Difference Equations Special Functions And Orthogonal Polynomials
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Author : Saber Elaydi
language : en
Publisher: World Scientific
Release Date : 2007
Difference Equations Special Functions And Orthogonal Polynomials written by Saber Elaydi and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Science categories.
This volume contains talks given at a joint meeting of three communities working in the fields of difference equations, special functions and applications (ISDE, OPSFA, and SIDE). The articles reflect the diversity of the topics in the meeting but have difference equations as common thread. Articles cover topics in difference equations, discrete dynamical systems, special functions, orthogonal polynomials, symmetries, and integrable difference equations.
Log Gases And Random Matrices Lms 34
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Author : Peter J. Forrester
language : en
Publisher: Princeton University Press
Release Date : 2010-07-01
Log Gases And Random Matrices Lms 34 written by Peter J. Forrester and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-07-01 with Mathematics categories.
Random matrix theory, both as an application and as a theory, has evolved rapidly over the past fifteen years. Log-Gases and Random Matrices gives a comprehensive account of these developments, emphasizing log-gases as a physical picture and heuristic, as well as covering topics such as beta ensembles and Jack polynomials. Peter Forrester presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems. Forrester develops not only the application and theory of Gaussian and circular ensembles of classical random matrix theory, but also of the Laguerre and Jacobi ensembles, and their beta extensions. Prominence is given to the computation of a multitude of Jacobians; determinantal point processes and orthogonal polynomials of one variable; the Selberg integral, Jack polynomials, and generalized hypergeometric functions; Painlevé transcendents; macroscopic electrostatistics and asymptotic formulas; nonintersecting paths and models in statistical mechanics; and applications of random matrix theory. This is the first textbook development of both nonsymmetric and symmetric Jack polynomial theory, as well as the connection between Selberg integral theory and beta ensembles. The author provides hundreds of guided exercises and linked topics, making Log-Gases and Random Matrices an indispensable reference work, as well as a learning resource for all students and researchers in the field.
Riemann Hilbert Problems Their Numerical Solution And The Computation Of Nonlinear Special Functions
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Author : Thomas Trogdon
language : en
Publisher: SIAM
Release Date : 2015-12-22
Riemann Hilbert Problems Their Numerical Solution And The Computation Of Nonlinear Special Functions written by Thomas Trogdon and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-12-22 with Mathematics categories.
Riemann?Hilbert problems are fundamental objects of study within complex analysis. Many problems in differential equations and integrable systems, probability and random matrix theory, and asymptotic analysis can be solved by reformulation as a Riemann?Hilbert problem.This book, the most comprehensive one to date on the applied and computational theory of Riemann?Hilbert problems, includes an introduction to computational complex analysis, an introduction to the applied theory of Riemann?Hilbert problems from an analytical and numerical perspective, and a discussion of applications to integrable systems, differential equations, and special function theory. It also includes six fundamental examples and five more sophisticated examples of the analytical and numerical Riemann?Hilbert method, each of mathematical or physical significance or both.?
Pad Methods For Painlev Equations
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Author : Hidehito Nagao
language : en
Publisher: Springer Nature
Release Date : 2021-09-01
Pad Methods For Painlev Equations written by Hidehito Nagao and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-09-01 with Science categories.
The isomonodromic deformation equations such as the Painlevé and Garnier systems are an important class of nonlinear differential equations in mathematics and mathematical physics. For discrete analogs of these equations in particular, much progress has been made in recent decades. Various approaches to such isomonodromic equations are known: the Painlevé test/Painlevé property, reduction of integrable hierarchy, the Lax formulation, algebro-geometric methods, and others. Among them, the Padé method explained in this book provides a simple approach to those equations in both continuous and discrete cases. For a given function f(x), the Padé approximation/interpolation supplies the rational functions P(x), Q(x) as approximants such as f(x)~P(x)/Q(x). The basic idea of the Padé method is to consider the linear differential (or difference) equations satisfied by P(x) and f(x)Q(x). In choosing the suitable approximation problem, the linear differential equations give the Lax pair for some isomonodromic equations. Although this relation between the isomonodromic equations and Padé approximations has been known classically, a systematic study including discrete cases has been conducted only recently. By this simple and easy procedure, one can simultaneously obtain various results such as the nonlinear evolution equation, its Lax pair, and their special solutions. In this way, the method is a convenient means of approaching the isomonodromic deformation equations.
Discrete Painlev Equations
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Author : Nalini Joshi
language : en
Publisher: American Mathematical Soc.
Release Date : 2019-05-30
Discrete Painlev Equations written by Nalini Joshi and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-05-30 with Mathematics categories.
Discrete Painlevé equations are nonlinear difference equations, which arise from translations on crystallographic lattices. The deceptive simplicity of this statement hides immensely rich mathematical properties, connecting dynamical systems, algebraic geometry, Coxeter groups, topology, special functions theory, and mathematical physics. This book necessarily starts with introductory material to give the reader an accessible entry point to this vast subject matter. It is based on lectures that the author presented as principal lecturer at a Conference Board of Mathematical Sciences and National Science Foundation conference in Texas in 2016. Instead of technical theorems or complete proofs, the book relies on providing essential points of many arguments through explicit examples, with the hope that they will be useful for applied mathematicians and physicists.
The Painlev Property
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Author : Robert Conte
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
The Painlev Property written by Robert Conte and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Science categories.
The subject this volume is explicit integration, that is, the analytical as opposed to the numerical solution, of all kinds of nonlinear differential equations (ordinary differential, partial differential, finite difference). Such equations describe many physical phenomena, their analytic solutions (particular solutions, first integral, and so forth) are in many cases preferable to numerical computation, which may be long, costly and, worst, subject to numerical errors. In addition, the analytic approach can provide a global knowledge of the solution, while the numerical approach is always local. Explicit integration is based on the powerful methods based on an in-depth study of singularities, that were first used by Poincar and subsequently developed by Painlev in his famous Leons de Stockholm of 1895. The recent interest in the subject and in the equations investigated by Painlev dates back about thirty years ago, arising from three, apparently disjoint, fields: the Ising model of statistical physics and field theory, propagation of solitons, and dynamical systems. The chapters in this volume, based on courses given at Cargse 1998, alternate mathematics and physics; they are intended to bring researchers entering the field to the level of present research.
Orthogonal Polynomials And Special Functions
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Author : Francisco Marcellàn
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-06-19
Orthogonal Polynomials And Special Functions written by Francisco Marcellàn and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-06-19 with Mathematics categories.
Special functions and orthogonal polynomials in particular have been around for centuries. Can you imagine mathematics without trigonometric functions, the exponential function or polynomials? In the twentieth century the emphasis was on special functions satisfying linear differential equations, but this has now been extended to difference equations, partial differential equations and non-linear differential equations. The present set of lecture notes containes seven chapters about the current state of orthogonal polynomials and special functions and gives a view on open problems and future directions. The topics are: computational methods and software for quadrature and approximation, equilibrium problems in logarithmic potential theory, discrete orthogonal polynomials and convergence of Krylov subspace methods in numerical linear algebra, orthogonal rational functions and matrix orthogonal rational functions, orthogonal polynomials in several variables (Jack polynomials) and separation of variables, a classification of finite families of orthogonal polynomials in Askey’s scheme using Leonard pairs, and non-linear special functions associated with the Painlevé equations.
Recent Progress In Special Functions
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Author : Galina Filipuk
language : en
Publisher: American Mathematical Society
Release Date : 2024-11-02
Recent Progress In Special Functions written by Galina Filipuk and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-11-02 with Mathematics categories.
This volume contains a collection of papers that focus on recent research in the broad field of special functions. The articles cover topics related to differential equations, dynamic systems, integrable systems, billiards, and random matrix theory. Linear classical special functions, such as hypergeometric functions, Heun functions, and various orthogonal polynomials and nonlinear special functions (e.g., the Painlev‚ transcendents and their generalizations), are studied from different perspectives. This volume serves as a useful reference for a large audience of mathematicians and mathematical physicists interested in modern theory of special functions. It is suitable for both graduate students and specialists in the field.