(We would like to carry reviews of any of these books in future issues of UK Nonlinear News.)
This volume contains papers written by participants at the Conference on Functional Differential and Difference Equations held at the Instituo Superior Técnico in Lisbon, Portugal. The conference brought together mathematicians working in a wide range of topics, including qualitative properties of solutions, bifurcation and stability theory, oscillatory behaviour, control theory and feedback systems, biological models, state-dependent delay equations, Lyapunov methods, etc. Articles are written by leading experts in the field. A comprehensive overview is given of these active areas of research.
Fields Institute Communications, Volume 29; 2001;
378 pages;
Hardcover; ISBN 0-8218-2701-4.
The study of dynamic equations on a measure chain (time scale) is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on measure chains can build bridges between continuous and discrete mathematics. This work is an introduction to measure chain theory with particular emphasis on its usefulness in allowing for the simultaneous development of differential equations and difference equations. The study of measure chain theory has also led to several important applications, e.g., in the study of insect population models, neural networks, heat transfer, and epidemic models.
With numerous examples throughout, the book may be used in a special topics seminar at the senior undergraduate or beginning graduate levels. May also serve as a good reference to stimulate the development of new kinds of equations with potentially new applications.
Approximately 320 pages, hardcover
ISBN 0-8176-4225-0
Spatial Patterns offers a study of nonlinear higher order model equations that are central to the description and analysis of spatio-temporal pattern formation in the natural sciences. Through a unique combination of results obtained by rigorous mathematical analysis and computational studies, the text exhibits the principal families of solutions, such as kinks, pulses and periodic solutions, and their dependence on critical eigenvalue parameters, and points to a rich structure, much of which still awaits exploration.
The exposition unfolds systematically, first focusing on a single equation to achieve optimal transparency, and then branching out to wider classes of equations. The presentation is based on results from real analysis and the theory of ordinary differential equations.
The book is intended for mathematicians who wish to become acquainted with this new area of partial and ordinary differential equations, for mathematical physicists who wish to learn about the theory developed for a class of well-known higher order pattern-forming model equations, and for graduate students who are looking for an exciting and promising field of research.
Progress in Nonlinear Differential Equations and their Applications, Volume 45
June 2001, Approximately 350 pages, 288 illustrations
ISBN 0-8176-4110-6, Hardcover
As the population exceeds the six billion mark, questions of population explosion, of how many people the earth can support and under which conditions, become pressing. Some of the questions and challenges raised can be addressed through the use of mathematical models, but not all. The goal of this book is to search for a balance between simple and analyzable models and unsolvable models which are capable of addressing important questions such as these. Part I focuses on single species simple models including those which have been used to predict the growth of human and animal population in the past. Single population models are, in some sense, the building blocks of more realistic models - the subject of Part II. Their role is fundamental to the study of ecological and demographic processes including the role of population structure and spatial heterogeneity - the subject of Part III. This book, which will include both examples and exercises, will be useful to practitioners, graduate students, and scientists working in the field.
2001. 408pp. Hardcover
ISBN 0-387-98902-1
Texts in Applied Mathematics, Volume 40.
This is a research monograph on soliton solutions of elliptical partial differential equations arising in quantum field theory, solutions such as vortices, instantons, monopoles, dyons, and cosmic strings. The book presents in-depth descriptions of important problems of current interest from several major branches of field theory, Yang-Mills gauge theory, and cosmology. It forges a link between mathematical analysis and physics and seeks to stimulate further research in the area. Mathematically, the book involves Riemannian geometry, Lie groups and Lie algebras, algebraic topology (characteristic classes and homotopy) and emphasizes modern nonlinear functional analysis. While field theory has long been of interest to algebraists, geometers and topolgists, it has gradually begun to attract the attention of more analysts. Written for researchers and graduate students, this book will also appeal to mathematicians and theoretical physicists.
2001, approximately 576 pages, hardcover
ISBN 0-387-95242-X
Applied Mathematical Sciences, Volume 146
This book is the first one devoted to high-dimensional (or large-scale) diffusion stochastic processes (DSPs) with nonlinear coefficients. These processes are closely associated with nonlinear Ito's stochastic ordinary differential equations (ISODEs) and with the space-discretized versions of nonlinear Ito's stochastic partial integro-differential equations. The latter models include Ito's stochastic partial differential equations (ISPDEs).
The book presents the new analytical treatment which can serve as the basis of a combined, analytical-numerical approach to greater computational efficiency in engineering problems. A few examples discussed in the book include: the high-dimensional DSPs described with the ISODE systems for semiconductor circuits; the nonrandom model for stochastic resonance (and other noise-induced phenomena) in high-dimensional DSPs; the modification of the well-known stochastic-adaptive-interpolation method by means of bases of function spaces; ISPDEs as the tool to consistently model non-Markov phenomena; the ISPDE system for semiconductor devices; the corresponding classification of charge transport in macroscale, mesoscale and microscale semiconductor regions based on the wave-diffusion equation; the fully time-domain nonlinear-friction aware analytical model for the velocity covariance of particle of uniform fluid, simple or dispersed; the specific time-domain analytics for the long, non-exponential "tails" of the velocity in case of the hard-sphere fluid.
These examples demonstrate not only the capabilities of the developed techniques but also emphasize the usefulness of the complex-system-related approaches to solve some problems which have not been solved with the traditional, statistical-physics methods yet. From this viewpoint, the book can be regarded as a kind of complement to such books as "Introduction to the Physics of Complex Systems. The Mesoscopic Approach to Fluctuations, Nonlinearity and Self-Organization" by Serra, Andretta, Compiani and Zanarini, "Stochastic Dynamical Systems. Concepts, Numerical Methods, Data Analysis" and "Statistical Physics: An Advanced Approach with Applications" by Honerkamp which deal with physics of complex systems, some of the corresponding analysis methods and an innovative, stochastics-based vision of theoretical physics.
To facilitate the reading by nonmathematicians, the introductory chapter outlines the basic notions and results of theory of Markov and diffusion stochastic processes without involving the measure-theoretical approach. This presentation is based on probability densities commonly used in engineering and applied sciences.
Readership: Nonmathematicians (e.g., theoretical physicists, engineers in industry, specialists in models for finance or biology, computing scientists), mathematicians, undergraduate and postgraduate students of the corresponding specialties, managers in applied sciences and engineering dealing with the advancements in the related fields, any specialists who use diffusion stochastic processes to model high-dimensional (or large-scale) nonlinear stochastic systems.