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Author: J. Hietarinta
Publisher: Cambridge University Press
ISBN: 1107042720
Category : Mathematics
Languages : en
Pages : 461
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Book Description
A first introduction to the theory of discrete integrable systems at a level suitable for students and non-experts.
Author: J. Hietarinta
Publisher: Cambridge University Press
ISBN: 1107042720
Category : Mathematics
Languages : en
Pages : 461
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Book Description
A first introduction to the theory of discrete integrable systems at a level suitable for students and non-experts.
Author: Yuri B. Suris
Publisher: Birkhäuser
ISBN: 3034880162
Category : Mathematics
Languages : en
Pages : 1078
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Book Description
An exploration of the theory of discrete integrable systems, with an emphasis on the following general problem: how to discretize one or several of independent variables in a given integrable system of differential equations, maintaining the integrability property? This question (related in spirit to such a modern branch of numerical analysis as geometric integration) is treated in the book as an immanent part of the theory of integrable systems, also commonly termed as the theory of solitons. Most of the results are only available from recent journal publications, many of them are new. Thus, the book is a kind of encyclopedia on discrete integrable systems. It unifies the features of a research monograph and a handbook. It is supplied with an extensive bibliography and detailed bibliographic remarks at the end of each chapter. Largely self-contained, it will be accessible to graduate and post-graduate students as well as to researchers in the area of integrable dynamical systems.
Author: Alexander I. Bobenko
Publisher: American Mathematical Society
ISBN: 1470474565
Category : Mathematics
Languages : en
Pages : 432
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Book Description
An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry. For a given smooth geometry one can suggest many different discretizations. Which one is the best? This book answers this question by providing fundamental discretization principles and applying them to numerous concrete problems. It turns out that intelligent theoretical discretizations are distinguished also by their good performance in applications. The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results. The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question “How do we discretize differential geometry?” arising in their specific field. Prerequisites for reading this book include standard undergraduate background (calculus and linear algebra). No knowledge of differential geometry is expected, although some familiarity with curves and surfaces can be helpful.
Author: Basil Grammaticos
Publisher:
ISBN: 9783662144602
Category :
Languages : en
Pages : 460
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Book Description
Author: N.J. Hitchin
Publisher: Oxford University Press, USA
ISBN: 0199676771
Category : Mathematics
Languages : en
Pages : 148
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Book Description
Designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors, this book has its origins in a lecture series given by the internationally renowned authors. Written in an accessible, informal style, it fills a gap in the existing literature.
Author: J.J. Duistermaat
Publisher: Springer
ISBN: 9781441971166
Category : Mathematics
Languages : en
Pages : 627
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Book Description
This book is devoted to Quisped, Roberts, and Thompson (QRT) maps, considered as automorphisms of rational elliptic surfaces. The theory of QRT maps arose from problems in mathematical physics, involving difference equations. The application of QRT maps to these and other problems in the literature, including Poncelet mapping and the elliptic billiard, is examined in detail. The link between elliptic fibrations and completely integrable Hamiltonian systems is also discussed. The book begins with a comprehensive overview of the subject, including QRT maps, singularity confinement, automorphisms of rational elliptic surfaces, action on homology classes, and periodic QRT maps. Later chapters cover these topics and more in detail. While QRT maps will be familiar to specialists in algebraic geometry, the present volume makes the subject accessible to mathematicians and graduate students in a classroom setting or for self-study.
Author: Kenji Iohara
Publisher: Springer Science & Business Media
ISBN: 1447148630
Category : Mathematics
Languages : en
Pages : 633
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Book Description
This volume is the result of two international workshops; Infinite Analysis 11 – Frontier of Integrability – held at University of Tokyo, Japan in July 25th to 29th, 2011, and Symmetries, Integrable Systems and Representations held at Université Claude Bernard Lyon 1, France in December 13th to 16th, 2011. Included are research articles based on the talks presented at the workshops, latest results obtained thereafter, and some review articles. The subjects discussed range across diverse areas such as algebraic geometry, combinatorics, differential equations, integrable systems, representation theory, solvable lattice models and special functions. Through these topics, the reader will find some recent developments in the field of mathematical physics and their interactions with several other domains.
Author: Chaohao Gu
Publisher: Springer Science & Business Media
ISBN: 1402030886
Category : Science
Languages : en
Pages : 317
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Book Description
The Darboux transformation approach is one of the most effective methods for constructing explicit solutions of partial differential equations which are called integrable systems and play important roles in mechanics, physics and differential geometry. This book presents the Darboux transformations in matrix form and provides purely algebraic algorithms for constructing the explicit solutions. A basis for using symbolic computations to obtain the explicit exact solutions for many integrable systems is established. Moreover, the behavior of simple and multi-solutions, even in multi-dimensional cases, can be elucidated clearly. The method covers a series of important equations such as various kinds of AKNS systems in R1+n, harmonic maps from 2-dimensional manifolds, self-dual Yang-Mills fields and the generalizations to higher dimensional case, theory of line congruences in three dimensions or higher dimensional space etc. All these cases are explained in detail. This book contains many results that were obtained by the authors in the past few years. Audience: The book has been written for specialists, teachers and graduate students (or undergraduate students of higher grade) in mathematics and physics.
Author: Nalini Joshi
Publisher: American Mathematical Soc.
ISBN: 1470450380
Category : Mathematics
Languages : en
Pages : 154
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Book Description
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.
Author: Oleg Kupervasser
Publisher: Elsevier
ISBN: 012813318X
Category : Science
Languages : en
Pages : 308
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Book Description
Application of New Cybernetics in Physics describes the application of new cybernetics to physical problems and the resolution of basic physical paradoxes by considering external observer influence. This aids the reader in solving problems that were solved incorrectly or have not been solved. Three groups of problems of the new cybernetics are considered in the book: (a) Systems that can be calculated based on known physics of subsystems. This includes the external observer influence calculated from basic physical laws (ideal dynamics) and dynamics of a physical system influenced even by low noise (observable dynamics). (b) Emergent systems. This includes external noise from the observer by using the black box model (complex dynamics), external noise from the observer by using the observer's intuition (unpredictable dynamics), defining boundaries of application of scientific methods for system behavior prediction, and the role of the observer's intuition for unpredictable systems. (c) Methods for solution of basic physical paradoxes by using methods of the new cybernetics: the entropy increase paradox, Schrödinger's cat paradox (wave package reduction in quantum mechanics), the black holes information paradox, and the time wormholes grandfather paradox. All of the above paradoxes have the same resolution based on the principles of new cybernetics. Indeed, even a small interaction of an observer with an observed system results in their time arrows' alignment (synchronization) and results in the paradox resolution and appearance of the universal time arrow. - Provides solutions to the basic physical paradoxes and demonstrates their practical actuality for modern physics - Describes a wide class of molecular physics and kinetic problems to present semi-analytical and semi-qualitative calculations of solvation, flame propagation, and high-molecular formation - Demonstrates the effectiveness in application to complex molecular systems and other many-component objects - Includes numerous illustrations to support the text
