Dynamical Systems Of Algebraic Origin
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Dynamical Systems Of Algebraic Origin
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Author : Klaus Schmidt
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-01-05
Dynamical Systems Of Algebraic Origin written by Klaus Schmidt 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 2012-01-05 with Mathematics categories.
Although much of classical ergodic theory is concerned with single transformations and one-parameter flows, the subject inherits from statistical mechanics not only its name, but also an obligation to analyze spatially extended systems with multidimensional symmetry groups. However, the wealth of concrete and natural examples which has contributed so much to the appeal and development of classical dynamics, is noticeably absent in this more general theory. The purpose of this book is to help remedy this scarcity of explicit examples by introducing a class of continuous Zd-actions diverse enough to exhibit many of the new phenomena encountered in the transition from Z to Zd, but which nevertheless lends itself to systematic study: the Zd-actions by automorphisms of compact, abelian groups. One aspect of these actions, not surprising in itself but quite striking in its extent and depth nonetheless, is the connection with commutative algebra and arithmetical algebraic geometry. The algebraic framework resulting from this connection allows the construction of examples with a variety of specified dynamical properties, and by combining algebraic and dynamical tools one obtains a quite detailed understanding of this class of Zd-actions.
Applied Algebraic Dynamics
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Author : Vladimir Anashin
language : en
Publisher: Walter de Gruyter
Release Date : 2009
Applied Algebraic Dynamics written by Vladimir Anashin and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Mathematics categories.
The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Cear , Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)
Integrability Of Dynamical Systems Algebra And Analysis
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Author : Xiang Zhang
language : en
Publisher: Springer
Release Date : 2017-03-30
Integrability Of Dynamical Systems Algebra And Analysis written by Xiang Zhang and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-30 with Mathematics categories.
This is the first book to systematically state the fundamental theory of integrability and its development of ordinary differential equations with emphasis on the Darboux theory of integrability and local integrability together with their applications. It summarizes the classical results of Darboux integrability and its modern development together with their related Darboux polynomials and their applications in the reduction of Liouville and elementary integrabilty and in the center—focus problem, the weakened Hilbert 16th problem on algebraic limit cycles and the global dynamical analysis of some realistic models in fields such as physics, mechanics and biology. Although it can be used as a textbook for graduate students in dynamical systems, it is intended as supplementary reading for graduate students from mathematics, physics, mechanics and engineering in courses related to the qualitative theory, bifurcation theory and the theory of integrability of dynamical systems.
Algebraic And Symbolic Computation Methods In Dynamical Systems
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Author : Alban Quadrat
language : en
Publisher: Springer Nature
Release Date : 2020-05-30
Algebraic And Symbolic Computation Methods In Dynamical Systems written by Alban Quadrat and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-05-30 with Science categories.
This book aims at reviewing recent progress in the direction of algebraic and symbolic computation methods for functional systems, e.g. ODE systems, differential time-delay equations, difference equations and integro-differential equations. In the nineties, modern algebraic theories were introduced in mathematical systems theory and in control theory. Combined with real algebraic geometry, which was previously introduced in control theory, the past years have seen a flourishing development of algebraic methods in control theory. One of the strengths of algebraic methods lies in their close connections to computations. The use of the above-mentioned algebraic theories in control theory has been an important source of motivation to develop effective versions of these theories (when possible). With the development of computer algebra and computer algebra systems, symbolic methods for control theory have been developed over the past years. The goal of this book is to propose a partial state of the art in this direction. To make recent results more easily accessible to a large audience, the chapters include materials which survey the main mathematical methods and results and which are illustrated with explicit examples.
Partial Dynamical Systems Fell Bundles And Applications
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Author : Ruy Exel
language : en
Publisher: American Mathematical Soc.
Release Date : 2017-09-20
Partial Dynamical Systems Fell Bundles And Applications written by Ruy Exel 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 2017-09-20 with Banach spaces categories.
Partial dynamical systems, originally developed as a tool to study algebras of operators in Hilbert spaces, has recently become an important branch of algebra. Its most powerful results allow for understanding structural properties of algebras, both in the purely algebraic and in the C*-contexts, in terms of the dynamical properties of certain systems which are often hiding behind algebraic structures. The first indication that the study of an algebra using partial dynamical systems may be helpful is the presence of a grading. While the usual theory of graded algebras often requires gradings to be saturated, the theory of partial dynamical systems is especially well suited to treat nonsaturated graded algebras which are in fact the source of the notion of “partiality”. One of the main results of the book states that every graded algebra satisfying suitable conditions may be reconstructed from a partial dynamical system via a process called the partial crossed product. Running in parallel with partial dynamical systems, partial representations of groups are also presented and studied in depth. In addition to presenting main theoretical results, several specific examples are analyzed, including Wiener–Hopf algebras and graph C*-algebras.
Integrable Systems In The Realm Of Algebraic Geometry
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Author : Pol Vanhaecke
language : en
Publisher: Springer
Release Date : 2003-07-01
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 2003-07-01 with Mathematics categories.
This book treats the general theory of Poisson structures and integrable systems on affine varieties in a systematic way. Special attention is drawn to algebraic completely integrable systems. Several integrable systems are constructed and studied in detail and a few applications of integrable systems to algebraic geometry are worked out. In the second edition some of the concepts in Poisson geometry are clarified by introducting Poisson cohomology; the Mumford systems are constructed from the algebra of pseudo-differential operators, which clarifies their origin; a new explanation of the multi Hamiltonian structure of the Mumford systems is given by using the loop algebra of sl(2); and finally Goedesic flow on SO(4) is added to illustrate the linearizatin algorith and to give another application of integrable systems to algebraic geometry.
Holomorphic Dynamical Systems
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Author : Nessim Sibony
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-07-31
Holomorphic Dynamical Systems written by Nessim Sibony 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 2010-07-31 with Mathematics categories.
The theory of holomorphic dynamical systems is a subject of increasing interest in mathematics, both for its challenging problems and for its connections with other branches of pure and applied mathematics. A holomorphic dynamical system is the datum of a complex variety and a holomorphic object (such as a self-map or a vector ?eld) acting on it. The study of a holomorphic dynamical system consists in describing the asymptotic behavior of the system, associating it with some invariant objects (easy to compute) which describe the dynamics and classify the possible holomorphic dynamical systems supported by a given manifold. The behavior of a holomorphic dynamical system is pretty much related to the geometry of the ambient manifold (for instance, - perbolic manifolds do no admit chaotic behavior, while projective manifolds have a variety of different chaotic pictures). The techniques used to tackle such pr- lems are of variouskinds: complexanalysis, methodsof real analysis, pluripotential theory, algebraic geometry, differential geometry, topology. To cover all the possible points of view of the subject in a unique occasion has become almost impossible, and the CIME session in Cetraro on Holomorphic Dynamical Systems was not an exception.
Chaos
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Author : Kathleen Alligood
language : en
Publisher: Springer
Release Date : 2012-12-06
Chaos written by Kathleen Alligood and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
BACKGROUND Sir Isaac Newton hrought to the world the idea of modeling the motion of physical systems with equations. It was necessary to invent calculus along the way, since fundamental equations of motion involve velocities and accelerations, of position. His greatest single success was his discovery that which are derivatives the motion of the planets and moons of the solar system resulted from a single fundamental source: the gravitational attraction of the hodies. He demonstrated that the ohserved motion of the planets could he explained hy assuming that there is a gravitational attraction he tween any two ohjects, a force that is proportional to the product of masses and inversely proportional to the square of the distance between them. The circular, elliptical, and parabolic orhits of astronomy were v INTRODUCTION no longer fundamental determinants of motion, but were approximations of laws specified with differential equations. His methods are now used in modeling motion and change in all areas of science. Subsequent generations of scientists extended the method of using differ ential equations to describe how physical systems evolve. But the method had a limitation. While the differential equations were sufficient to determine the behavior-in the sense that solutions of the equations did exist-it was frequently difficult to figure out what that behavior would be. It was often impossible to write down solutions in relatively simple algebraic expressions using a finite number of terms. Series solutions involving infinite sums often would not converge beyond some finite time.
Differential Equations Dynamical Systems And An Introduction To Chaos
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Author : Morris W. Hirsch
language : en
Publisher: Academic Press
Release Date : 2004
Differential Equations Dynamical Systems And An Introduction To Chaos written by Morris W. Hirsch and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Business & Economics categories.
Thirty years in the making, this revised text by three of the world's leading mathematicians covers the dynamical aspects of ordinary differential equations. it explores the relations between dynamical systems and certain fields outside pure mathematics, and has become the standard textbook for graduate courses in this area. The Second Edition now brings students to the brink of contemporary research, starting from a background that includes only calculus and elementary linear algebra. The authors are tops in the field of advanced mathematics, including Steve Smale who is a recipient of.
Algebraic Structure Of Dynamical Systems
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Author : James P. Talisse
language : en
Publisher:
Release Date : 2017
Algebraic Structure Of Dynamical Systems written by James P. Talisse and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with Dynamics categories.
A dynamical system is a mathematical object which describes the motion of a set of points over time. Dynamical systems can be used to study differential equations, cryptography, computer science, and even biology. Viewed as a purely mathematical object, one can ask questions about the behavior of the dynamical system based on the structure of algebraic objects associated with it. In this project we study two algebraic objects, centralizers and topological full groups, associated to symbolic dynamical systems. The centralizer group tells us about the symmetries a system possesses. Results relating to the centralizer historically have indicated that the more complex the dynamical system is, captured by the Topological Entropy, the more structure its centralizer has. Similarly, low complexity systems have been shown to have very simple centralizers. This seems to suggest that one can recover information about the dynamical system based upon its centralizer group. In particular, if a system is known to have a certain centralizer group, we might want to draw conclusions about the complexity of the system. In this project we present a class of high complexity systems which have a very rigid centralizer, which shows the relationship is more subtle than may have been originally thought. We also study the topological full group of a dynamical system. This group completely defines the system up to time reversal. We apply numerical estimates to draw conclusions about the algebraic properties of this group. In particular, we seek to know when the topological full group of a dynamical system is amenable. Amenability is an algebraic property that can be thought of as having a probability measure on G. This measure would answer the question: given a subset A of G, what is the probability that a random element of G is in A? We apply Grigorchuk’s amenability criterion to answer this question. Both these results provide us with information about the algebraic structure of dynamical systems. If we know certain information about the different groups associated with a dynamical system, we can make conclusions about the system itself. As such, questions about dynamical systems can now become questions about algebra, and vice versa. These results mostly reveal the structure of symbolic dynamical systems and address the fundamental question of mathematics about what is possible. However, our construction of a positive entropy system with trivial centralizer can be interpreted as the existence of an information channel with positive capacity that cannot be encrypted with substitution ciphers.