(We would like to carry reviews of any of these books in future issues of UK Nonlinear News.)
This book presents the proceedings from the International Conference held in Halifax, NS in July 1997. Funded by The Fields Institute and Le Centre de Recherches Mathématiques, the conference was held in honour of the retirement of Professors Lynn Erbe and Herb I. Freedman (University of Alberta). Featured topics include ordinary, partial, functional, and stochastic differential equations and their applications to biology, epidemiology, neurobiology, physiology and other related areas.
The 41 papers included in this volume represent the recent work of leading researchers over a wide range of subjects, including bifurcation theory, chaos, stability theory, boundary value problems, persistence theory, neural networks, disease transmission, population dynamics, pattern formation and more. The text would be suitable for a graduate or advanced undergraduate course study in mathematical biology.
Fields Institute Communications, Volume 21
November 1998, approximately 528 pages, Hardcover
The research topic for this IAS/PCMS Summer Session was nonlinear wave phenomena. Mathematicians from the more theoretical areas of PDEs were brought together with those involved in applications. The goal was to share ideas, knowledge, and perspectives.
How waves, or "frequencies", interact in nonlinear phenomena has been a central issue in many of the recent developments in pure and applied analysis. It is believed that wavelet theory--with its simultaneous localisation in both physical and frequency space and its lacunarity--is and will be a fundamental new tool in the treatment of the phenomena.
Included in this volume are write-ups of the "general methods and tools" courses held by Jeff Rauch and Ingrid Daubechies. Rauch's article discusses geometric optics as an asymptotic limit of high-frequency phenomena. He shows how nonlinear effects are reflected in the asymptotic theory. In the article "Harmonic Analysis, Wavelets and Applications" by Daubechies and Gilbert the main structure of the wavelet theory is presented.
Also included are articles on the more "specialised" courses that were presented, such as "Nonlinear Schrödinger Equations" by Jean Bourgain and "Waves and Transport" by George Papanicolaou and Leonid Ryzhik. Susan Friedlander provides a written version of her lecture series "Stability and Instability of an Ideal Fluid", given at the Mentoring Program for Women in Mathematics, a preliminary program to the Summer Session.
This Summer Session brought together students, fellows, and established mathematicians from all over the globe to share ideas in a vibrant and exciting atmosphere. This book presents the compelling results.
IAS/Park City Mathematics Series, Volume 5.
November 1998, 466 pages, Hardcover, ISBN 0-8218-0592-4
"Presents an exceptionally complete overview of the latest developments in the field... There are numerous examples and the emphasis is on applications to almost all areas of science and engineering... This reviewer recommends [this book] strongly for individual students, practitioners, and libraries alike."
Applied Mechanics Review.
Hardcover ISBN 0-8176-3902-0
This introduction to topological dynamics and ergodic theory is divided into a number of relatively short chapters with the intention that each may be used as a component of a lecture course tailored to the particular audience. The authors provide a number of applications, principally to number theory and arithmetic progressions (through Van der Waerden's theorem and Szmerdi's theorem).
London Mathematical Society Student Texts 40
Beginning with fundamental concepts of motion, the book goes on to discuss linear and nonlinear waves, thermodynamics, and constitutive models for a variety of gases, liquids, and solids. Among the important areas of research and application are impact analysis, shock wave research, explosive detonation, nonlinear acoustics, and hypersonic aerodynamics.
This book covers developments in the field of the theory of oscillations from the diverse viewpoints, reflecting the field's multidisciplinary nature. Addressing researchers and students in mechanics, physics, applied mathematics, and engineering, it describes the state of the art in this area, including applications. For the first time a treatment of the asymptotic and homogenisation methods in the theory of oscillations in combination with Padé approximations is presented. Because of its wealth of interesting examples, this book will prove useful as an introduction to the field for novices and a reference for specialists.
1998, approximately 310 pp, hardcover
Springer Series in Synergetics
This is a book on nonlinear dynamical systems and their bifurcations under parameter variation. It provides a reader with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems. Special attention is given to efficient numerical implementations of the developed techniques. Several examples from recent research papers are used as illustrations. The book is designed for advanced undergraduate or graduate students in applied mathematics, as well as for PhD students and researchers in physics, biology, engineering, and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used. This new edition preserves the structure of the 1st edition while updating the context to incorporate recent theoretical developments, in particular new and improved numerical methods for bifurcation analysis.
1998, 616pp, hardcover
Applied Mathematical Sciences, Volume 112
This volume is concerned with various kinds of limit theorems for stochastic processes defined as a result of random perturbations of dynamical systems, especially with the long-term behaviour of the perturbed system. The authors main tools are the large deviation theory, the central limit theorem for stochastic processes, and the averaging principle --- all presented in great detail. The results allow for explicit calculations of the asymptotics of many interesting characteristics of the perturbed system. Most of the results are closely connected with PDE's and the authors approach presents a powerful method for studying the asymptotic behaviour of the solutions of initial boundary value problems for corresponding PDE's. The most essential additions and changes in this new edition concern the averaging principle. A new chapter on random perturbations of Hamiltonian systems has been added along with new results on fast oscillating perturbations of systems with conservation laws. New sections on wave front propagation in semilinear PDE's and on random perturbations of certain infinite-dimensional dynamical systems have been incorporated into the chapter on sharpenings and generalisations.
1998, approximately 448 pages
Hardcover, ISBN 0-387-98362-7