Eigenvalues, Inequalities, and Ergodic Theory

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Springer Science & Business Media, Mar 30, 2006 - Mathematics - 228 pages
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A problem of broad interest – the estimation of the spectral gap for matrices or differential operators (Markov chains or diffusions) – is covered in this book. The area has a wide range of applications, and provides a tool to describe the phase transitions and the effectiveness of random algorithms. In particular, the book studies a subset of the general problem, taking some approaches that have, up till now, only appeared largely in the Chinese literature.

Eigenvalues, Inequalities and Ergodic Theory serves as an introduction to this developing field, and provides an overview of the methods used, in an accessible and concise manner. The author starts with an overview chapter, from which any of the following self-contained chapters can be read.

Each chapter starts with a summary and, in order to appeal to non-specialists, ideas are introduced through simple examples rather than technical proofs. In the latter chapters readers are introduced to problems and application areas, including stochastic models of economy.

Intended for researchers, graduates and postgraduates in probability theory, Markov processes, mathematical physics and spectrum theory, this book will be a welcome introduction to a growing area of research.

 

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Contents

An Overview of the Book
1
Optimal Markovian Couplings
17
New Variational Formulas for the First Eigenvalue
41
Ten Explicit Criteria in Dimension
89
PoincaréType Inequalities in Dimension
113
Functional Inequalities
131
A Diagram of Nine Types of Ergodicity
149
ReactionDiffusion Processes
163
Stochastic Models of Economic Optimization
181
Appendix A Some Elementary Lemmas
193
References 209
208
89
212
Author Index
223
98
224
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About the author (2006)

Mu-Fa Chen is Professor of Mathematics at Beijing Normal University, in the People’s Republic of China, and Member of the Chinese Academy of Sciences. In 1999 he received a Prize for Progress on Sciences and Technology from the Ministry of Education and a National Prize on Natural Sciences from the Ministry of Science and Technology in the People’s Republic of China.

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