Michael B. Marcus
Jay Rosen

Markov Processes, Gaussian Processes, and Local Times

Series: Cambridge Studies in Advanced Mathematics (No. 100)
Hardback (ISBN-10: 0521863007 | ISBN-13: 9780521863001)
Not yet published - available from September 2006 (Stock level updated: 08:00 GMT, 23 January 2006)

Written by two foremost researchers in the field, this book studies the local times of Markov processes by employing isomorphism theorems that relate them to certain associated Gaussian processes. It builds to this material through self-contained but harmonized ‘mini-courses’ on the relevant ingredients, which assume only knowledge of measure-theoretic probability. The streamlined selection of topics creates an easy entrance for students and for experts in related fields. The book starts by developing the fundamentals of Markov process theory and then of Gaussian process theory, including sample path properties. It then proceeds to more advanced results, bringing the reader to the heart of contemporary research. It presents the remarkable isomorphism theorems of Dynkin and Eisenbaum then shows how they can be applied to obtain new properties of Markov processes by using well-established techniques in Gaussian process theory. This original, readable book will appeal to both researchers and advanced graduate students.

* Starts with streamlined ‘mini-courses’ that give easy entry to important areas in stochastic process theory

* Material on the isomorphism theorems has never before appeared in book form

*Authors are the acknowledged experts in this area

Contents

1. Introduction; 2. Brownian motion and Ray-Knight theorems; 3. Markov processes and local times; 4. Constructing Markov processes; 5. Basic properties of Gaussian processes; 6. Continuity and boundedness; 7. Moduli of continuity; 8. Isomorphism theorems; 9. Sample path properties of local times; 10. p-Variation; 11. Most visited site; 12. Local times of diffusions; 13. Associated Gaussian processes; Appendices: A. Kolmogorov’s theorem for path continuity; B. Bessel processes; C. Analytic sets and the projection theorem; D. Hille-Yosida theorem; E. Stone-Weierstrass theorems; F. Independent random variables; G. Regularly varying functions; H. Some useful inequalities; I. Some linear algebra; References; Index.

Rob Sturman / Julio Ottino / Stephen Wiggins

The Mathematical Foundations of Mixing
The Linked Twist Map as a Paradigm in Applications: Micro to Macro, Fluids to Solids

Series: Cambridge Monographs on Applied and Computational Mathematics (No. 22)
Hardback (ISBN-10: 0521868130 | ISBN-13: 9780521868136)
Not yet published - available from September 2006 (Stock level updated: 08:00 GMT, 23 January 2006)


Mixing processes occur in many technological and natural applications, with length and time scales ranging from the very small to the very large. The diversity of problems can give rise to a diversity of approaches. Are there concepts that are central to all of them? Are there tools that allow for prediction and quantification?

The authors show how a variety of flows in very different settings possess the characteristic of streamline crossing. This notion can be placed on firm mathematical footing via Linked Twist Maps (LTMs), which is the central organizing principle of this book.

The authors discuss the definition and construction of LTMs, provide examples of specific mixers that can be analyzed in the LTM framework and introduce a number of mathematical techniques which are then brought to bear on the problem of fluid mixing. In a final chapter, they present a number of open problems and new directions.

* First book explaining new important ideas in our understanding of how fluids mix

* Authors are well-know book writers and leading researchers

* Starts with a broad, accessible and thought-provoking chapter, and ends with a collection of open problems

Contents

Preface; 1. Mixing: Physical Issues; 2. Linked Twist Maps; 3. The Ergodic Hierarchy; 4. Existence of a Horseshoe; 5. Hyperbolicity; 6. The Ergodic Partition for Toral LTMs; 7. Ergodicity and Bernoulli for TLTMs; 8. Linked twist maps on the plane; 9. Further Directions and Open Problems; Bibliography; Index

Richard A. Brualdi

Combinatorial Matrix Classes

Series: Encyclopedia of Mathematics and its Applications (No. 108)
Hardback (ISBN-10: 0521865654 | ISBN-13: 9780521865654)
Not yet published - available from September 2006 (Stock level updated: 08:08 GMT, 23 January 2006)

A natural sequel to the author’s previous book Combinatorial Matrix Theory written with H.J. Ryser, this is the first book devoted exclusively to existence questions, constructive algorithms, enumeration questions, and other properties concerning classes of matrices of combinatorial significance.

Several classes of matrices are thoroughly developed including the classes of matrices of 0's and 1's with a specified number of 1's in each row and column (equivalently, bipartite graphs with a specified degree sequence), symmetric matrices in such classes (equivalently, graphs with a specified degree sequence), tournament matrices with a specified number of 1's in each row (equivalently, tournaments with a specified score sequence), nonnegative matrices with specified row and column sums, and doubly stochastic matrices.

Most of this material is presented for the first time in book format and the chapter on doubly stochastic matrices provides the most complete development of the topic to date.

* Thorough and complete development of some important matrix classes of combinatorial importance

* Both existence questions and algorithms are given for most matrix classes treated

* Important combinatorial and linear-algebraic parameters associated with these classes are thoroughly investigated

Contents

1. Introduction; 2. Basic existence theorems for matrices with prescribed properties; 3. The class A(R; S) of (0,1)-matrices; 4. More on the class A(R; S) of (0,1)-matrices; 5. The class T(R) of tournament matrices; 6. Interchange graphs; 7. Classes of symmetric integral matrices; 8. Convex polytopes of matrices; 9. Doubly stochastic matrices.

Gerhard Michler

Theory of Finite Simple Groups

Series: New Mathematical Monographs (No. 8)
Hardback (ISBN-10: 0521866251 | ISBN-13: 9780521866255)
Not yet published - available from October 2006 (Stock level updated: 08:08 GMT, 23 January 2006)

This book provides the first representation theoretic and algorithmic approach to the theory of abstract finite simple groups. It presents self-contained proofs of classical and new group order formulas, and a new structure theorem for abstract finite simple groups. This, and the famous Brauer-Fowler theorem, provides the theoretical background for the author's algorithm which constructs all finite simple groups G having a 2-central involution z with a given centralizer CG(z)=H. The methods presented are designed for the construction of matrix representations, permutation representations and character tables of large finite groups. The author constructs all the simple satellites of the known simple groups that are not uniquely determined by a given centralizer H. Uniform existence and uniqueness proofs are given for the modern sporadic simple groups discovered by Janko, Higman and Sims, Harada, and Thompson. This latter result proves a long standing open problem in the theory. The experimental results (courtesy of M. Weller) for Chapter 12 are documented in the accompanying DVD.

* Provides a new theoretical and algorithmic approach to the study of finite simple groups

* Presents new algorithms for the construction of concrete character tables of arbitrary finite groups and shows how they can be used to provide uniform existence and uniqueness proofs

* The accompanying DVD contains a large amount of experimental data

Contents

Introduction; 1. Prerequisites from group theory; 2. Group representations and character theory; 3. Modular representation theory; 4. Group order formulas and structure theorem; 5. Permutation representations; 6. Concrete character tables of matrix groups; 7. Methods for constructing finite simple groups; 8. Finite simple groups with proper satellites; 9. Janko group J1; 10. Higman-Sims group HS; 11. Harada group Ha; 9. Thompson group Th; Bibliography; List of symbols; Index.