Algebraic Structures In Integrability
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Algebraic Structures In Integrability
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Author : Vladimir Sokolov
language : en
Publisher:
Release Date : 2020-05-26
Algebraic Structures In Integrability written by Vladimir Sokolov and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-05-26 with Science categories.
Relationships of the theory of integrable systems with various branches of mathematics are extremely deep and diverse. On the other hand, the most fundamental exactly integrable systems often have applications in theoretical physics. Therefore, many mathematicians and physicists are interested in integrable models. The book is intelligible to graduate and PhD students and can serve as an introduction to separate sections of the theory of classical integrable systems for scientists with algebraic inclinations. For the young, the book can serve as a starting point in the study of various aspects of integrability, while professional algebraists will be able to use some examples of algebraic structures, which appear in the theory of integrable systems, for wide-ranging generalizations. The statements are formulated in the simplest possible form. However, some ways of generalization are indicated. In the proofs, only essential points are mentioned, while for technical details, references are provided. The focus is on carefully selected examples. In addition, the book proposes many unsolved problems of various levels of complexity. A deeper understanding of every chapter of the book may require the study of more rigorous and specialized literature.
Algebraic Structures In Integrability Foreword By Victor Kac
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Author : Vladimir V Sokolov
language : en
Publisher: World Scientific
Release Date : 2020-06-05
Algebraic Structures In Integrability Foreword By Victor Kac written by Vladimir V Sokolov and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-06-05 with Science categories.
Relationships of the theory of integrable systems with various branches of mathematics are extremely deep and diverse. On the other hand, the most fundamental exactly integrable systems often have applications in theoretical physics. Therefore, many mathematicians and physicists are interested in integrable models.The book is intelligible to graduate and PhD students and can serve as an introduction to separate sections of the theory of classical integrable systems for scientists with algebraic inclinations. For the young, the book can serve as a starting point in the study of various aspects of integrability, while professional algebraists will be able to use some examples of algebraic structures, which appear in the theory of integrable systems, for wide-ranging generalizations.The statements are formulated in the simplest possible form. However, some ways of generalization are indicated. In the proofs, only essential points are mentioned, while for technical details, references are provided. The focus is on carefully selected examples. In addition, the book proposes many unsolved problems of various levels of complexity. A deeper understanding of every chapter of the book may require the study of more rigorous and specialized literature.
Algebraic Aspects Of Integrable Systems
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Author : A.S. Fokas
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Algebraic Aspects Of Integrable Systems written by A.S. Fokas 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 Mathematics categories.
A collection of articles in memory of Irene Dorfman and her research in mathematical physics. Among the topics covered are: the Hamiltonian and bi-Hamiltonian nature of continuous and discrete integrable equations; the t-function construction; the r-matrix formulation of integrable systems; pseudo-differential operators and modular forms; master symmetries and the Bocher theorem; asymptotic integrability; the integrability of the equations of associativity; invariance under Laplace-darboux transformations; trace formulae of the Dirac and Schrodinger periodic operators; and certain canonical 1-forms.
Geometric And Algebraic Structures In Differential Equations
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Author : P.H. Kersten
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Geometric And Algebraic Structures In Differential Equations written by P.H. Kersten 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 Mathematics categories.
The geometrical theory of nonlinear differential equations originates from classical works by S. Lie and A. Bäcklund. It obtained a new impulse in the sixties when the complete integrability of the Korteweg-de Vries equation was found and it became clear that some basic and quite general geometrical and algebraic structures govern this property of integrability. Nowadays the geometrical and algebraic approach to partial differential equations constitutes a special branch of modern mathematics. In 1993, a workshop on algebra and geometry of differential equations took place at the University of Twente (The Netherlands), where the state-of-the-art of the main problems was fixed. This book contains a collection of invited lectures presented at this workshop. The material presented is of interest to those who work in pure and applied mathematics and especially in mathematical physics.
Integrable Systems In The Realm Of Algebraic Geometry
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Author : Pol Vanhaecke
language : en
Publisher: Springer
Release Date : 2013-11-11
Integrable Systems In The Realm Of Algebraic Geometry written by Pol Vanhaecke and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-11 with Mathematics categories.
Integrable systems are related to algebraic geometry in many different ways. This book deals with some aspects of this relation, the main focus being on the algebraic geometry of the level manifolds of integrable systems and the construction of integrable systems, starting from algebraic geometric data. For a rigorous account of these matters, integrable systems are defined on affine algebraic varieties rather than on smooth manifolds. The exposition is self-contained and is accessible at the graduate level; in particular, prior knowledge of integrable systems is not assumed.
Integral Closure
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Author : Wolmer Vasconcelos
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-05-23
Integral Closure written by Wolmer Vasconcelos 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 2005-05-23 with Mathematics categories.
This book gives an account of theoretical and algorithmic developments on the integral closure of algebraic structures. It gives a comprehensive treatment of Rees algebras and multiplicity theory while pointing to applications in many other problem areas. Its main goal is to provide complexity estimates by tracking numerically invariants of the structures that may occur.
Symmetries And Singularity Structures
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Author : Muthuswamy Lakshmanan
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Symmetries And Singularity Structures written by Muthuswamy Lakshmanan 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 Mathematics categories.
Proceedings of the Workshop, Bharathidasan University, Tiruchirapalli, India, November 29 - December 2, 1989
Lie Algebraic Methods In Integrable Systems
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Author : Amit K. Roy-Chowdhury
language : en
Publisher: CRC Press
Release Date : 2021-01-04
Lie Algebraic Methods In Integrable Systems written by Amit K. Roy-Chowdhury and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-01-04 with Mathematics categories.
Over the last thirty years, the subject of nonlinear integrable systems has grown into a full-fledged research topic. In the last decade, Lie algebraic methods have grown in importance to various fields of theoretical research and worked to establish close relations between apparently unrelated systems. The various ideas associated with Lie algebra and Lie groups can be used to form a particularly elegant approach to the properties of nonlinear systems. In this volume, the author exposes the basic techniques of using Lie algebraic concepts to explore the domain of nonlinear integrable systems. His emphasis is not on developing a rigorous mathematical basis, but on using Lie algebraic methods as an effective tool. The book begins by establishing a practical basis in Lie algebra, including discussions of structure Lie, loop, and Virasor groups, quantum tori and Kac-Moody algebras, and gradation. It then offers a detailed discussion of prolongation structure and its representation theory, the orbit approach-for both finite and infinite dimension Lie algebra. The author also presents the modern approach to symmetries of integrable systems, including important new ideas in symmetry analysis, such as gauge transformations, and the "soldering" approach. He then moves to Hamiltonian structure, where he presents the Drinfeld-Sokolov approach, the Lie algebraic approach, Kupershmidt's approach, Hamiltonian reductions and the Gelfand Dikii formula. He concludes his treatment of Lie algebraic methods with a discussion of the classical r-matrix, its use, and its relations to double Lie algebra and the KP equation.
Integrability
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Author : Alexander Mikhailov
language : en
Publisher: Springer
Release Date : 2008-11-05
Integrability written by Alexander Mikhailov and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-11-05 with Science categories.
The principal aim of the book is to give a comprehensive account of the variety of approaches to such an important and complex concept as Integrability. Dev- oping mathematical models, physicists often raise the following questions: whether the model obtained is integrable or close in some sense to an integrable one and whether it can be studied in depth analytically. In this book we have tried to c- ate a mathematical framework to address these issues, and we give descriptions of methods and review results. In the Introduction we give a historical account of the birth and development of the theory of integrable equations, focusing on the main issue of the book – the concept of integrability itself. A universal de nition of Integrability is proving to be elusive despite more than 40 years of its development. Often such notions as “- act solvability” or “regular behaviour” of solutions are associated with integrable systems. Unfortunately these notions do not lead to any rigorous mathematical d- inition. A constructive approach could be based upon the study of hidden and rich algebraic or analytic structures associated with integrable equations. The requi- ment of existence of elements of these structures could, in principle, be taken as a de nition for integrability. It is astonishing that the nal result is not sensitive to the choice of the structure taken; eventually we arrive at the same pattern of eq- tions.
Algebraic Integrability Painlev Geometry And Lie Algebras
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Author : Mark Adler
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-14
Algebraic Integrability Painlev Geometry And Lie Algebras written by Mark Adler 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 2013-03-14 with Mathematics categories.
This Ergebnisse volume is aimed at a wide readership of mathematicians and physicists, graduate students and professionals. The main thrust of the book is to show how algebraic geometry, Lie theory and Painlevé analysis can be used to explicitly solve integrable differential equations and construct the algebraic tori on which they linearize; at the same time, it is, for the student, a playing ground to applying algebraic geometry and Lie theory. The book is meant to be reasonably self-contained and presents numerous examples. The latter appear throughout the text to illustrate the ideas, and make up the core of the last part of the book. The first part of the book contains the basic tools from Lie groups, algebraic and differential geometry to understand the main topic.