(We would like to carry reviews of any of these books in future issues of UK Nonlinear News.)
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.
Topics treated in the book include:
The book is intended for graduate students who have a good command of basic measure theory and functional analysis and who would like to master the subject. It contains many detailed examples and many exercises, usually with indications of solutions. It can serve equally well as a textbook for graduate courses, for independent study, supplementary reading, or as a streamlined introduction for non-specialists who wish to learn about modern aspects of ergodic theory.
Mathematical Surveys and Monographs, Volume 101.
Hardcover, ISBN 0-8218-3372-3.
The study of epidemic models is one of the central topics of mathematical biology. This volume is the first to present in monograph form the rigorous mathematical theory developed to analyse the asymptotic behaviour of certain types of epidemic models.
The main model discussed is the so-called spatial deterministic epidemic in which infected individuals are not allowed to again become susceptible, and infection is spread by means of contact distributions. Results concern the existence of travelling wave solutions, the asymptotic speed of propagation, and the spatial final size. A central result for radially symmetric contact distributions is that the speed of propagation is the minimum wave speed. Further results are obtained using a saddle point method, suggesting that this result also holds for more general situations.
Methodology, used to extend the analysis from one-type to multi-type models, is likely to prove useful when analysing other multi-type systems in mathematical biology. This methodology is applied to two other areas in the monograph, namely epidemics with return to the susceptible state and contact branching processes.
This book presents an elegant theory that has been developed over the past quarter century. It will be useful to researchers and graduate students working in mathematical biology.
Mathematical Surveys and Monographs, Volume 102.
ISBN 0-8218-0499-5.
Presents concepts on chaos in discrete time dynamics that are accessible to anyone who has taken a first course in undergraduate calculus. Constitutes the first elementary presentation of a traditionally advanced subject.
Australian Mathematical Society Lecture Series 18
ISBN 0-521-53104-7.
It has been over a decade since the release of the now classic original edition of Murray's Mathematical Biology. Though the field has become increasingly large and specialised, this book remains important as a text that introduces some of the exciting problems that arise in biology and gives some indication of the wide spectrum of questions that modelling can address. Due to the tremendous development in the field, this book is being published in two volumes. This second volume covers new applications such as modelling the growth of cancer tumours, spatial patterns and wolf territorialty. Graduate students and researchers will find this book invaluable, as it gives an excellent background from which to begin genuine interdisciplinary research in the biomedical sciences.
2003. 808pp. 296 illustrations. Hardcover. ISBN 0-387-95228-4.
Interdisciplinary Applied Mathematics, Volume 18.
Applied mathematics is a central connecting link between scientific observations and their theoretical interpretation. Nonlinear analysis has contributed major developments which nowadays shape the face of applied mathematics. At the beginning of the millennium, all sciences are expanding at increased speed. Technological, ecological, economical and medical problem solving is a central issue of every modern society. Mathematical models help to expose fundamental structures hidden in these problems and serve as unifying tools to deepen our understanding. What are the new challenges applied mathematics has to face with the increased diversity of scientific problems? In which direction should the classical tools of nonlinear analysis be developed further? How do new available technologies influence the development of the field? How can problems be solved which have been beyond reach in former times? It is the aim of this book to explore new developments in the field by way of discussion of selected topics from nonlinear analysis.
Table of Contents:
A detailed Table of Contents can be found at
http://www.springer.de/books/toc/3540441980-c.pdf.
Keywords: Nonlinear analysis, dynamical systems, homogenisation, material sciences, mathematical biology, numerical mathematics, partial differential equations.
2003 XVI, 419 p. Hardcover
3-540-44198-0