(We would like to carry reviews of any of these books in future issues of UK Nonlinear News.)
Despite the great strides in nonlinear dynamics over the past 40 years, applying nonlinear dynamics to polymeric systems has not received much attention. This book addresses this absence by covering present theory, modeling, and experiments of nonlinear dynamics in polymeric systems. Oscillating chemical reactions, propagating fronts, far-from equilibrium pattern formation, Turing structures, and chaos are but some of the exotic phenomena discussed in this book.
The book is the result of the Symposium on Nonlinear Dynamics in Polymeric Systems sponsored by the Division of Polymer Chemistry and the Division of Physical Chemistry at the August 18-22, 2002 American Chemical Society National Meeting in Boston.
ACS Symposium Series, 869
368 pages; 146 line illus., 50 halftones & 8-page color insert; 6 x 9
In the last twenty years, the theory of holomorphic dynamical systems has had a resurgence of activity, particularly concerning the fine analysis of Julia sets associated with polynomials and rational maps in one complex variable. At the same time, closely related theories have had a similar rapid development, for example the qualitative theory of differential equations in the complex domain.
The meeting, "Etat de la recherche", held at Ecole Normale Supérieure de Lyon, presented the current state of the art in this area, emphasizing the unity linking the various sub-domains. This volume contains four survey articles corresponding to the talks presented at this meeting.
D. Cerveau describes the structure of polynomial differential equations in the complex plane, focusing on the local analysis in neighborhoods of singular points. E. Ghys surveys the theory of laminations by Riemann surfaces which occur in many dynamical or geometrical situations. N. Sibony describes the present state of the generalization of the Fatou-Julia theory for polynomial or rational maps in two or more complex dimensions. Lastly, the talk by J.-C. Yoccoz, written by M. Flexor, considers polynomials of degree 2 in one complex variable, and in particular, with the hyperbolic properties of these polynomials centered around the Jakobson theorem.
This is a general introduction that gives a basic history of holomorphic dynamical systems, demonstrating the numerous and fruitful interactions among the topics. In the spirit of the "Etat de la recherche de la SMF" meetings, the articles are written for a broad mathematical audience, especially students or mathematicians working in different fields. This book is translated from the French edition by Leslie Kay.
SMF/AMS Texts and Monographs, Volume 10
October 2003, 197 pages,
First posed by Hermann Weyl in 1910, the limit-point/limit-cycle problem has inspired, over the last century, several new developments in the asymptotic analysis of nonlinear differential equations. This self-contained monograph traces the evolution of this problem from its inception to its modern-day extensions to the study of deficiency indices and analogous properties for nonlinear equations. With over 120 references, many open problems, and illustratice examples, this work will be valuable to graduate students and researchers in differential equations, functional analysis, operator theory, and related fields.
2003, 140pp, 10 illustrations
This book presents a new method for studying the asymptotic behaviour of solutions to evolution partial differential equations; much of the text is dedicated to the application of this method to a wide class of nonlinear diffusion equations. The Stability Theorem, on whih the method hinges, is examined in the first chapter, followed by a review of basic results and methods - many original to the authors - for the solution of nonlinear diffusion equations. Further chapters provide a self-contained analysis of specific equations and their applications, along with carefully-constructed theorems, proofs, and references.
2003, 430 pages, 10 illustrations
Progress in Nonlinear Differential Equations and their Application
This introductory textbook includes linear and non-linear models of populations, markov models of molecular evolution, phylogenetic tree construction from DNA sequence data, genetics, and infectious disease models.
ISBN 0-521-81980-6 Hardback
ISBN 0-521-52586-1 Softback
The concept of entropy arose in the physical sciences during the nineteenth century, particularly in thermodynamics and statistical physics, as a measure of the equilibria and evolution of thermodynamic systems. Two main views developed: the macroscopic view formulated originally by Carnot, Clausius, Gibbs, Planck, and Caratheodory and the microscopic approach associated with Boltzmann and Maxwell. Since then both approaches have made possible deep insights into the nature and behavior of thermodynamic and other microscopically unpredictable processes. However, the mathematical tools used have later developed independently of their original physical background and have led to a plethora of methods and differing conventions.
The aim of this book is to identify the unifying threads by providing surveys of the uses and concepts of entropy in diverse areas of mathematics and the physical sciences. Two major threads, emphasized throughout the book, are variational principles and Ljapunov functionals. The book starts by providing basic concepts and terminology, illustrated by examples from both the macroscopic and microscopic lines of thought. In-depth surveys covering the macroscopic, microscopic and probabilistic approaches follow. Part I gives a basic introduction from the views of thermodynamics and probability theory. Part II collects surveys that look at the macroscopic approach of continuum mechanics and physics. Part III deals with the microscopic approach exposing the role of entropy as a concept in probability theory, namely in the analysis of the large time behavior of stochastic processes and in the study of qualitative properties of models in statistical physics. Finally in Part IV applications in dynamical systems, ergodic and information theory are presented.
The chapters were written to provide as cohesive an account as possible, making the book accessible to a wide range of graduate students and researchers. Any scientist dealing with systems that exhibit entropy will find the book an invaluable aid to their understanding.
Princeton Series in Applied Mathematics
Cloth, 2003, 448 pp.
ISBN: 0-691-11338-6 .
This book introduces mathematicians to real applications from physiology. using mathematics to analyze physiological systems, the authors focus on models reflecting current research in cardiovascular and pulmonary physiology. In particular, they present models describing blood flow in the heart and in the cardiovascular system, as well as the transport of oxygen and carbon dioxide through the respiratory system and a model for baroreceptor regulation.
Applied Mathematical Models in Human Physiology is the only book available that analyses up-to-date models of the physiological system at several levels of detail. Some are simple `real-time' models that can be directly used in larger systems, while others are more detailed `reference' models that show the underlying physiological mechanisms and provide parameters for and validation of simpler models. The book also covers two-dimensional modeling of the fluid dynamics in the heart and its ability to pump, and includes a discussion of modeling wave-propogation throughout the systemic arteries.
Approximately xiv+298 pages. Softcover.
This book gives a unified presentation in an abstract setting of the main theorems in bifurcation theory, as well as more recent and lesser known results. It covers both the local and global theory of one-parameter bifurcations for operators acting in infinite-dimensional Banach spaces, and shows how to apply the theory to problems involving partial differential equations. In addition to existence, qualitative properties such as stability and nodal structure of bifurcating solutions are treated in depth. This volume will serve as an important reference for mathematicians, physicists, and theoretically-inclined engineers working in bifurcation theory and its applications to partial differential equations.
2003, 336 pages, 38 illustrations, harcover.
Applied Mathematical Sciences Volume 156.
It has only been a couple of decades since Benoit Mandelbrot published his famous picture of what is now called the mandelbrot set. That picture, now seemingly graphically primitive, has changed out view of the mathematical and physical universe. The properties and circumstances of the discovery of the Mandelbrot Set continue to generate much interest in the research community and beyond. This book contains the hard-to-obtain original papers, many unpublished illustrations ating back to 1979 and extensive documented historical context hosinwg how Mandlebrot helped change our way of looking at the world.
2004, 290 pp, 100 illustrations
This book deals with the application of mathematics in modelling and understanding physiological systems, especially those involving rhythms. One novel feature of the book is the inclusion of classroom-tested computer exercises throughout, designed to form a bridge between the mathematical theory and physiological experiments. This volume will be interest to students and researchers in the natural and physical sciences wanting to learn about the complexities and subtleties of physiological systems from a mathematical perspective.
2003. 428 pages. 160 illustrations. Hardcover.
Interdisciplinary Applied Mathematics, Volume 20.