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Author : Marcelo Viana,Krerley Oliveira
Publisher : Cambridge University Press
Page : 547 pages
File Size : 55,8 Mb
Release : 2016-02-15
Category : Mathematics
ISBN : 9781107126961
Self-contained introductory textbook suitable for a variety of one- or two-semester courses. Rich with examples, applications and exercises.
Author : Karl E. Petersen,Karl Petersen
Publisher : Cambridge University Press
Page : 348 pages
File Size : 41,9 Mb
Release : 1989-11-23
Category : Mathematics
ISBN : 0521389976
The study of dynamical systems forms a vast and rapidly developing field even when one considers only activity whose methods derive mainly from measure theory and functional analysis. Karl Petersen has written a book which presents the fundamentals of the ergodic theory of point transformations and then several advanced topics which are currently undergoing intense research. By selecting one or more of these topics to focus on, the reader can quickly approach the specialized literature and indeed the frontier of the area of interest. Each of the four basic aspects of ergodic theory - examples, convergence theorems, recurrence properties, and entropy - receives first a basic and then a more advanced, particularized treatment. At the introductory level, the book provides clear and complete discussions of the standard examples, the mean and pointwise ergodic theorems, recurrence, ergodicity, weak mixing, strong mixing, and the fundamentals of entropy. Among the advanced topics are a thorough treatment of maximal functions and their usefulness in ergodic theory, analysis, and probability, an introduction to almost-periodic functions and topological dynamics, a proof of the Jewett-Krieger Theorem, an introduction to multiple recurrence and the Szemeredi-Furstenberg Theorem, and the Keane-Smorodinsky proof of Ornstein's Isomorphism Theorem for Bernoulli shifts. The author's easily-readable style combined with the profusion of exercises and references, summaries, historical remarks, and heuristic discussions make this book useful either as a text for graduate students or self-study, or as a reference work for the initiated.
Author : Daniel J. Rudolph
Publisher : Oxford University Press, USA
Page : 190 pages
File Size : 44,7 Mb
Release : 1990
Category : Mathematics
ISBN : UOM:39015019619942
This book is designed to provide graduate students and other researchers in dynamical systems theory with an introduction to the ergodic theory of Lebesgue spaces. The author's aim is to present a technically complete account which offers an in-depth understanding of the techniques of the field, both classical and modern. Thus, the basic structure theorems of Lebesgue spaces are given in detail as well as complete accounts of the ergodic theory of a single transformation, ergodic theorems, mixing properties and entropy. Subsequent chapters extend the earlier material to the areas of joinings and representation theorems, in particular the theorems of Ornstein and Krieger. Prerequisites are a working knowledge of Lebesgue measure and the topology of the real line as might be gained from the first year of a graduate course. Many exercises and examples are included to illustrate and to further cement the reader's understanding of the material. The result is a text which will furnish the reader with a sound technical background from the foundations of the subject to some of its most recent developments.
Author : R. Jancel
Publisher : Elsevier
Page : 440 pages
File Size : 42,9 Mb
Release : 2013-10-22
Category : Science
ISBN : 9781483186269
Foundations of Classical and Quantum Statistical Mechanics details the theoretical foundation the supports the concepts in classical and quantum statistical mechanics. The title discusses the various problems set by the theoretical justification of statistical mechanics methods. The text first covers the the ergodic theory in classical statistical mechanics, and then proceeds to tackling quantum mechanical ensembles. Next, the selection discusses the the ergodic theorem in quantum statistical mechanics and probability quantum ergodic theorems. The selection also details H-theorems and kinetic equations in classical and quantum statistical mechanics. The book will be of great interest to students, researchers, and practitioners of physics, chemistry, and engineering.
Author : Eli Glasner
Publisher : American Mathematical Soc.
Page : 384 pages
File Size : 44,9 Mb
Release : 2015-01-09
Category : Electronic
ISBN : 9781470419516
This book introduces modern ergodic theory. It emphasizes a new approach that relies on the technique of joining two (or more) dynamical systems. This approach has proved to be fruitful in many recent works, and this is the first time that the entire theory is presented from a joining perspective. Another new feature of the book is the presentation of basic definitions of ergodic theory in terms of the Koopman unitary representation associated with a dynamical system and the invariant mean on matrix coefficients, which exists for any acting groups, amenable or not. Accordingly, the first part of the book treats the ergodic theory for an action of an arbitrary countable group. The second part, which deals with entropy theory, is confined (for the sake of simplicity) to the classical case of a single measure-preserving transformation on a Lebesgue probability space.
Author : Olav Kallenberg
Publisher : Springer Science & Business Media
Page : 670 pages
File Size : 48,6 Mb
Release : 2002-01-08
Category : Mathematics
ISBN : 0387953132
The first edition of this single volume on the theory of probability has become a highly-praised standard reference for many areas of probability theory. Chapters from the first edition have been revised and corrected, and this edition contains four new chapters. New material covered includes multivariate and ratio ergodic theorems, shift coupling, Palm distributions, Harris recurrence, invariant measures, and strong and weak ergodicity.
Author : Aleksandr I?Akovlevich Khinchin
Publisher : Courier Corporation
Page : 130 pages
File Size : 49,6 Mb
Release : 1957-01-01
Category : Mathematics
ISBN : 9780486604343
First comprehensive introduction to information theory explores the work of Shannon, McMillan, Feinstein, and Khinchin. Topics include the entropy concept in probability theory, fundamental theorems, and other subjects. 1957 edition.
Author : I. P. Cornfeld,S. V. Fomin,Y. G. Sinai
Publisher : Springer Science & Business Media
Page : 487 pages
File Size : 48,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461569275
Ergodic theory is one of the few branches of mathematics which has changed radically during the last two decades. Before this period, with a small number of exceptions, ergodic theory dealt primarily with averaging problems and general qualitative questions, while now it is a powerful amalgam of methods used for the analysis of statistical properties of dyna mical systems. For this reason, the problems of ergodic theory now interest not only the mathematician, but also the research worker in physics, biology, chemistry, etc. The outline of this book became clear to us nearly ten years ago but, for various reasons, its writing demanded a long period of time. The main principle, which we adhered to from the beginning, was to develop the approaches and methods or ergodic theory in the study of numerous concrete examples. Because of this, Part I of the book contains the description of various classes of dynamical systems, and their elementary analysis on the basis of the fundamental notions of ergodicity, mixing, and spectra of dynamical systems. Here, as in many other cases, the adjective" elementary" i~ not synonymous with "simple. " Part II is devoted to "abstract ergodic theory. " It includes the construc tion of direct and skew products of dynamical systems, the Rohlin-Halmos lemma, and the theory of special representations of dynamical systems with continuous time. A considerable part deals with entropy.
Author : Steven Kalikow,Randall McCutcheon
Publisher : Cambridge University Press
Page : 183 pages
File Size : 48,9 Mb
Release : 2010-03-25
Category : Mathematics
ISBN : 9781139484251
This informal introduction provides a fresh perspective on isomorphism theory, which is the branch of ergodic theory that explores the conditions under which two measure preserving systems are essentially equivalent. It contains a primer in basic measure theory, proofs of fundamental ergodic theorems, and material on entropy, martingales, Bernoulli processes, and various varieties of mixing. Original proofs of classic theorems - including the Shannon–McMillan–Breiman theorem, the Krieger finite generator theorem, and the Ornstein isomorphism theorem - are presented by degrees, together with helpful hints that encourage the reader to develop the proofs on their own. Hundreds of exercises and open problems are also included, making this an ideal text for graduate courses. Professionals needing a quick review, or seeking a different perspective on the subject, will also value this book.
Author : M. M. Rao
Publisher : Courier Corporation
Page : 320 pages
File Size : 40,6 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9780486296531
This volume considers fundamental theories and contrasts the natural interplay between real and abstract methods. No prior knowledge of probability is assumed. Numerous problems, most with hints. 1981 edition.
Author : Aleksandr I?Akovlevich Khinchin
Publisher : Courier Corporation
Page : 212 pages
File Size : 42,5 Mb
Release : 1949-01-01
Category : Mathematics
ISBN : 0486601471
Phase space, ergodic problems, central limit theorem, dispersion and distribution of sum functions. Chapters include Geometry and Kinematics of the Phase Space; Ergodic Problem; Reduction to the Problem of the Theory of Probability; Application of the Central Limit Theorem; Ideal Monatomic Gas; The Foundation of Thermodynamics; and more.
Author : Manfred Einsiedler,Thomas Ward
Publisher : Springer Science & Business Media
Page : 481 pages
File Size : 52,9 Mb
Release : 2010-09-11
Category : Mathematics
ISBN : 9780857290212
This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence. Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits. Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.
Author : Robert M. Gray
Publisher : Springer Science & Business Media
Page : 309 pages
File Size : 40,8 Mb
Release : 2013-04-18
Category : Mathematics
ISBN : 9781475720242
This book has been written for several reasons, not all of which are academic. This material was for many years the first half of a book in progress on information and ergodic theory. The intent was and is to provide a reasonably self-contained advanced treatment of measure theory, prob ability theory, and the theory of discrete time random processes with an emphasis on general alphabets and on ergodic and stationary properties of random processes that might be neither ergodic nor stationary. The intended audience was mathematically inc1ined engineering graduate students and visiting scholars who had not had formal courses in measure theoretic probability . Much of the material is familiar stuff for mathematicians, but many of the topics and results have not previously appeared in books. The original project grew too large and the first part contained much that would likely bore mathematicians and dis courage them from the second part. Hence I finally followed the suggestion to separate the material and split the project in two. The original justification for the present manuscript was the pragmatic one that it would be a shame to waste all the effort thus far expended. A more idealistic motivation was that the presentation bad merit as filling a unique, albeit smaIl, hole in the literature.
Author : A. Ya. Khinchin
Publisher : Courier Corporation
Page : 130 pages
File Size : 47,6 Mb
Release : 2013-04-09
Category : Mathematics
ISBN : 9780486318448
First comprehensive introduction to information theory explores the work of Shannon, McMillan, Feinstein, and Khinchin. Topics include the entropy concept in probability theory, fundamental theorems, and other subjects. 1957 edition.
Author : Avrim Blum,John Hopcroft,Ravindran Kannan
Publisher : Cambridge University Press
Page : 433 pages
File Size : 50,9 Mb
Release : 2020-01-23
Category : Computers
ISBN : 9781108485067
Covers mathematical and algorithmic foundations of data science: machine learning, high-dimensional geometry, and analysis of large networks.