(We would like to carry reviews of any of these books in future issues of UK Nonlinear News.)
The geometrical study of differential equations has a long and distinguished history, dating back to the classical investigations of Sophus Lie, Gaston Darboux, and Elie Cartan. Currently, these ideas occupy a central position in several areas of pure and applied mathematics, including the theory of completely integrable evolution equations, the calculus of variations, and the study of conservation laws. In this book, the author gives an overview of a number of significant ideas and results developed over the past decade in the geometrical study of differential equations.
Topics covered in the book include symmetries of differential equations and variational problems, the variational bi-complex and conservation laws, geometric integrability for hyperbolic equations, transformations of submanifolds and systems of conservation laws, and an introduction to the characteristic cohomology of differential systems.
The exposition is sufficiently elementary so that non-experts can understand the main ideas and results by working independently. The book is also suitable for graduate students and researchers interested in the study of differential equations from a geometric perspective. It can serve nicely as a companion volume to The Geometrical Study of Differential Equations, Volume 285 in the AMS Contemporary Mathematics series.
CBMS Regional Conference Series in Mathematics, Number 96.
115 pages, Softcover
ISBN: 0-8218-2639-5
This book systematically explores two important areas of mathematical biology and biocomputing: cellular automata and pattern formation modelling. The first part of the book deals with general principles, theories, and models of pattern formation, the second part with cellular automata modelling, and the third part with applications. Besides applications in medicine and immunobiology, extensions to other fields are presented.
The book is aimed at professional researchers in applied mathematics, computational physics, computer science, and biology interested in a cellular automata approach to modelling. Graduates and advanced undergraduates in biomodelling and biocomputing will also find the book an accessible interdisciplinary approach and presentation of the topic.
Approximately 300 pages, 85 illustrations.
ISBN 0-8176-4281-1
This book concentrates on the branching solutions of nonlinear operator equations and the theory of degenerate operator-differential equations especially applicable to algorithmic analysis and nonlinear PDE's in mechanics and mathematical physics.
The authors expound the recent result on the generalised eigenvalue problem, the perturbation method, Schmidt's pseudo-inversion for regularisation of linear and nonlinear problems in the branching theory and group methods in bifurcation theory. The book covers regular iterative methods in a neighbourhood of branch points and the theory of differential-operator equations with a non-invertible operator in the main expression is constructed. Various recent results on theorems of existence are given including asymptotic, approximate and group methods.
The reduction of some mathematics, physics and mechanics problems (capillary-gravity surface wave theory, phase transitions theory, Andronov-Hopf bifurcation, boundary-value problems for the Vlasov-Maxwell system, filtration, magnetic insulation) to operator equations gives rich opportunities for creation and application of stated common methods for which existence theorems and the bifurcation of solutions for these applications are investigated.
Audience: The book will be of interest to mathematicians, mechanics, physicists and engineers interested in nonlinear equations and applications to nonlinear and singular systems as well as to researchers and students of these topics.
Kluwer Academic Publishers, Dordrecht
Hardbound, ISBN 1-4020-0941-0
November 2002, 566 pages.
EUR 174.00. USD 160.00. GBP 110.00
For more information see Nikolay Sidorov's homepage http://www.isu.ru/facs/math/kafedra/matan/prep/snaeng.html.
The new book "1089 and All That; A Journey into Mathematics" has just been published by Oxford University Press.
It aims to make a serious breakthrough in bringing some of the ideas and pleasures of mathematics to a wide public, in a light-hearted way, illustrated by several world-class cartoons. It includes some elementary material on nonlinear systems, together with computer animations. These can be downloaded from the website http://www.jesus.ox.ac.uk/~dacheson where further details of the book can be found.
Source: David Acheson.
Research on interior-point methods (IPMs) has dominated the field of mathematical programming for the last two decades. Two contrasting approaches in the analysis and implementation of IPMs are the so-called small-update and large-update methods, although, until now, there has been a notorious gap between the theory and practical performance of these two strategies. This book comes close to bridging that gap, presenting a new framework for the theory of primal-dual IPMs based on the notion of the self-regularity of a function.
Researchers and postgraduate students in all areas of linear and nonlinear optimisation will find this book an important and invaluable aid to their work.
Paper. April 2002. (US) $29.95. £ 19.95. ISBN: 0-691-09193-5
Cloth. April 2002. (US) $65.00. £ 45.00. ISBN: 0-691-09192-7
For further information, please visit http://www.pup.princeton.edu .
The modelling of stochastic dependence is fundamental for understanding random systems evolving in time. When measured through linear correlation, many of these systems exhibit a slow correlation decay - a phenomenon often referred to as long-memory or long-range dependence. An example of this is the absolute returns of equity data in finance. Selfsimilar stochastic processes (particularly fractional Brownian motion) have long been postulated as a means to model this behaviour, and the concept of selfsimilarity for a stochastic process is now proving to be extraordinarily useful. Selfsimilarity translates into the equality in distribution between the process under a linear time change and the same process properly scaled in space, a simple scaling property that yields a remarkably rich theory with far-flung applications.
Though the text uses the mathematical language of the theory of stochastic processes, researchers and end-users from such diverse fields as mathematics, physics, biology, telecommunications, finance, econometrics, and environmental science will find it an ideal entry point for studying the already extensive theory and applications of selfsimilarity.
Cloth. 2002. (US) $29.95. £19.95
ISBN: 0-691-09627-9
For further information, please visit http://www.pup.princeton.edu.
This volume celebrates J.E. Marsden's influence as a teacher, propagator of new ideas, and mentor of young talent. It presents both survey articles and research articles in his major areas of interest, including elasticity and analysis, fluid mechanics, dynamical systems theory, geometric mechanics, geometric control theory, and relativity and quantum mechanics. The common thread throughout is the use of geometric methods that serve to unify diverse disciplines and bring a wide variety of scientists and mathematicians together in a way that enhances dialogue and encourages cooperation.
2002. 584 pages. 68 illustrations.
ISBN 0-387-95518-6.
As systems evolve, they are subjected to random operating environments. In addition, random errors occur in measurements of their outputs and in their design and fabrication where tolerances are not precisely met. This book develops methods for describing random dynamical systems, and it illustrates how the methods can be used in a variety of applications. The first half of the book concentrates on finding approximations to random processes using the methodologies of probability theory. The second half of the book derives approximations to solutions of various problems in mechanics, electronic circuits, population biology, and genetics. In each example, the underlying physical or biological phenomenon is described in terms of nonrandom models taken from the literature, and the impact of random noise on the solutions is investigated. The mathematical problems in these applications involve random perturbations of gradient systems, Hamiltonian systems, toroidal flows, Markov chains, difference equations, filters, and nonlinear renewal equations. The models are analysed using the approximation methods described here and are visualised using MATLAB-based computer simulations. This book will appeal to those researchers and graduate students in science and engineering who require tools to investigate stochastic systems.
Keywords: Dynamical systems,Perturbation, Probability theory, Random dynamical systems, Random perturbation, Random perturbation method, Stochastic processes
Contents: Introduction.- Ergodic Theorems.- Convergence Properties of Stochastic Processes.- Averaging.- Normal Deviation.- Diffusion Approximation.- Stability.- Markov Chains with Random Transition Probabilities.- Randomly Perturbed Mechanical Systems.- Dynamical Systems on a Torus.- The Phase Locked Loop.- Models in Population Biology.- Genetics.- Appendices.- Index.
2002, 488 pages 31 illustrations. Hardcover 0-387-95427-9
Series: Applied Mathematical Sciences. Volume. 150
This invaluable book studies synchronisation of coupled chaotic circuits and systems, as well as its applications. It shows how one can use stability results in nonlinear control to derive synchronisation criteria for coupled chaotic circuits and systems. It also discusses the use of Lyapunov exponents in deriving synchronisation criteria. Both the case of two coupled systems and the case of arbitrarily coupled arrays of systems are considered. The book examines how synchronisation properties in arrays of coupled systems are dependent on graph-theoretical properties of the underlying coupling topology. Finally, it studies some applications of synchronised chaotic circuits and systems, including spread spectrum and secure communications, coupled map lattices and graph colouring.
Readership: Graduate students, researchers and academics in electrical engineering and nonlinear science.
World Scientific Series on Nonlinear Science, Series A - Volume 41.
188pp
ISBN 981-02-4713-3