Mathematical Theory of Entropy (Encyclopedia of Mathematics and its Applications, Series Number 12)

Originally published in 1981, this excellent treatment of the mathematical theory of entropy gives an accessible exposition of the ways in which this idea has been applied to information theory, ergodic theory, topological dynamics and statistical mechanics. Scientists who want a quick understanding of how entropy is applied in disciplines not their own, or simply desire a better understanding of the mathematical foundation of the entropy function will find this to be a valuable book. 2.2 The Information Function of an Experiment and Entropy2.3 An Example; 2.4 Conditional Information and Conditional Entropy; 2.5 Properties of Entropy and Conditional Entropy; 2.6 Entropy of Arbitrary Measurable Partitions and Limit Theorems; 2.7 Rate of Information Generation; 2.8 Entropy of Dynamical Systems; 2.9 Factor Automorphisms and Factor Systems; 2.10 Shannon's Theorem and the Equipartition Property; 2.11 Entropy as a Function of Distributions; 2.12 Examples; 2.12.1 Direct Products; 2.12.2 Skew Products; 2.12.3 Powers of Endomorphisms; 2.12 A Flows; 2.12.5 Induced Automorphisms 4.3 K-Systems and K-Automorphisms4.4 Spaces of Ordered Partitions, Weak Independence, and Weak Dependence; 4.5 Coding and Ornstein's Fundamental Lemma; 4.6 The Isomorphism Theorem for Bernoulli Systems; 4.7 Characterization of Bernoulli Systems; 4.8 Relative Isomorphism; 4.9 Special Flows and Equivalence Theory; CHAPTER 5 Topological Dynamics; 5.1 Introduction; 5.2 Definition and Basic Properties of Topological Entropy; 5.3 Connection between Topological and Measure Theoretic Entropy; 5.4 An Alternative Definition of Topological Entropy; CHAPTER 6 Statistical Mechanics; 6.1 Introduction 2.12.6 Periodic Automorphisms2.12.7 Rotations of the Circle; 2.12.8 Ergodic Automorphisms of Compact Abelian Groups; 2.12.9 Bernoulli Shifts; 2.12.10 Markov Shifts; 2.12.11 S-Automorphisms; 2.12.12 Unilateral Shifts; 2.12.13 Continued Fraction Transformations; 2.12.14 f-Transformations; 2.13 Sequence Entropy and r-Entropy; CHAPTER 3 Information Theory; 3.1 A Model of an Information System; 3.2 The Source; 3.3 Coding; 3.4 The Channel; 3.5 The Noisy Channel Coding Theorem; 3.6 Source Coding; CHAPTER 4 Ergodic Theory; 4.1 Introduction; 4.2 Unitary Operator of a System and Bernoulli Shifts Cover; Half Title; Series Page; Title; Copyright; Dedication; CONTENTS; Editor's Statement; Foreword; Preface; Special Symbols; CHAPTER 1 Topics from Probability Theory; 1.1 Probability Spaces; 1.2 Measurable Partitions and Lebesgue Spaces; 1.3 The Lattice of Measurable Partitions; 1.4 Random Variables; 1.5 Conditional Probability and Independence; 1.6 Conditional Expectation of Random Variables; 1.7 Stochastic Processes and Dynamical Systems; 1.8 The Ergodic Theorem and the Martingale Convergence Theorem; CHAPTER 2 Entropy and Information; 2.1 Information and Uncertainty of Events 6.2 Classical Continuous Systems6.3 Classical Lattice Systems; 6.4 Gibbs States for Lattice Systems; 6.5 Equilibrium States and the Concepts of Entropy and Pressure; BIBLIOGRAPHY; Index This excellent 1981 treatment of the mathematical theory of entropy gives an accessible exposition its application to other fields Nathaniel F.g. Martin, James W. England ; Foreword By James K. Brooks. Includes Index. Bibliography: P. 245-251.