Interdisciplinary Applied Mathematics – serie

Itisaspecialpleasureformetowritethisforewordforaremarkablebookbya remarkableauthor.MarcoPettiniisadeepthinker,whohasspentmanyyears probing the foundations of Hamiltonian chaos and statistical mechanics, in particular phase transitions, from the point of view of geometry and topology. Itisinparticularthequalityofmindoftheauthorandhisdeepphysical,as well as mathematical insights which make this book so special and inspiring. It is a “must” for those who want to venture into a new approach to old problems or want to use new tools for new problems. Although topology has penetrated a number of ?elds of physics, a broad participationoftopologyintheclari?cationandprogressoffundamentalpr- lems in the above-mentioned ?elds has been lacking. The new perspectives topology gives to the above-mentioned problems are bound to help in their clari?cation and to spread to other ?elds of science. The sparsity of geometric thinking and of its use to solve fundamental problems, when compared with purely analytical methods in physics, could be relieved and made highly productive using the material discussed in this book. It is unavoidable that the physicist reader may have then to learn some new mathematics and be challenged to a new way of thinking, but with the author as a guide, he is assured of the best help in achieving this that is presently available.

Disorder plays a fundamental role in many natural and man-made systems that are of industrial and scienti?c importance. Of all the disordered systems, hete- geneous materials are perhaps the most heavily utilized in all aspects of our daily lives, and hence have been studied for a long time. With the advent of new - perimental techniques, it is now possible to study the morphology of disordered materials and gain a much deeper understanding of their properties. Novel te- niqueshavealsoallowedustodesignmaterialsofmorphologieswiththeproperties that are suitable for intended applications. With the development of a class of powerful theoretical methods, we now have the ability for interpreting the experimental data and predicting many properties of disordered materials at many length scales. Included in this class are ren- malization group theory, various versions of effective-medium approximation, percolation theory, variational principles that lead to rigorous bounds to the - fective properties, and Green function formulations and perturbation expansions. Thetheoreticaldevelopmentshavebeenaccompaniedbyatremendousincreasein the computational power and the emergence of massively parallel computational strategies. Hence, we are now able to model many materials at molecular scales and predict many of their properties based on ?rst-principle computations.

4 Derivation of Elastic Networks from Continuum Elasticity 8. 6 Rigidity Percolation 8. 5 Mapping between rigidity percolation and resistor networks 8. 6 Nature of phase transition in rigidity percolation 8. 8 The Critical Path Method 8. 11 Elastic Percolation Networks with Bond-Bending Forces 8.

Je tiens impossible de connaître les parties sans connaître le tout, non plus que de connaître le tout sans connaître particulierement les parties -Pascal The eterna[ mystery of the world is its comprehensibility -Einstein This book deals with the application of mathematical tools to the study of physiological systems. It is directed toward an audience of physiologists, physicians, physicists, kinesiologists, psychologists, engineers, mathemati­ cians, and others interested in finding out more about the complexities and subtleties of rhythmic physiological processes from a theoretical per­ spective. We have attempted to give a broad view of the underlying notions behind the dynamics of physiological rhythms, sometimes from a theoretical perspective and sometimes from the perspective of the experimentalist. This book can be used in a variety of ways, ranging from a more tra­ ditional approach such as a textbook in a biomathematics course (at either the advanced undergraduate or graduatelevel) to a research re­ source in which someone interested in a particular problem might look at the corresponding discussion here to guide their own thinking. We hope that researchers at all levels will find inspiration from the way we have dealt with particular research problems to tackle completely new areas of investigation, or even approach these in totally new ways.

This book is intended to give students at the advanced undergraduate or introduc­ tory graduate level, and researchers in computer vision, robotics and computer graphics, a self-contained introduction to the geometry of three-dimensional (3- D) vision. This is the study of the reconstruction of 3-D models of objects from a collection of 2-D images. An essential prerequisite for this book is a course in linear algebra at the advanced undergraduate level. Background knowledge in rigid-body motion, estimation and optimization will certainly improve the reader's appreciation of the material but is not critical since the first few chapters and the appendices provide a review and summary of basic notions and results on these topics. Our motivation Research monographs and books on geometric approaches to computer vision have been published recently in two batches: The first was in the mid 1990s with books on the geometry of two views, see e. g. [Faugeras, 1993, Kanatani, 1993b, Maybank, 1993, Weng et aI. , 1993b]. The second was more recent with books fo­ cusing on the geometry of multiple views, see e. g. [Hartley and Zisserman, 2000] and [Faugeras and Luong, 2001] as well as a more comprehensive book on computer vision [Forsyth and Ponce, 2002]. We felt that the time was ripe for synthesizing the material in a unified framework so as to provide a self-contained exposition of this subject, which can be used both for pedagogical purposes and by practitioners interested in this field.

Population genetics occupies a central role in a number of important biological and social undertakings. It is fundamental to our understanding of evolutionary processes, of plant and animal breeding programs, and of various diseases of particular importance to mankind.This is the first of a planned two-volume work discussing the mathematical aspects of population genetics, with an emphasis on the evolutionary theory. This first volume draws heavily from the author's classic 1979 edition, which appeared originally in Springer's Biomathematics series. It has been revised and expanded to include recent topics which follow naturally from the treatment in the earlier edition, e.g., the theory of molecular population genetics.This book will appeal to graduate students and researchers in mathematical biology and other mathematically-trained scientists looking to enter the field of population genetics.

In the last few years there has been significant progress in the development of microfluidics and nanofluidics at the application as well as at the fundamental and simulation levels. This book provides a comprehensive summary of these changes describing fluid flow in micro and nano configurations. Where as in their previous book entitled Microflows: Fundamentals and Simulation the authors covered scales from one hundred nanometers to microns (and beyond), in this new book they discuss length scales from angstroms to microns (and beyond). While still maintaining the emphasis on fundamental concepts with a mix of semianalytical, experimental, and numerical results, this book outlines their relevance to modeling and analyzing functional devices. The text has been divided into three main subject categories: gas flows; liquid flows; and simulation techniques. The majority of the completely new developments in this book are in liquid flows and simulation techniques chapters withmodified information throughout the rest of the book. This book can be used in a two-semester graduate course. Also, selected chapters can be used for a short course or an undergraduate-level course. The book is suitable for graduate students and researchers in fluid mechanics, physics, and in electrical, mechanical and chemical engineering.Review of earlier volume: Applied Mechanics, 2002: "Among recent books that addressed the physics of micro devices, the present one ... is perhaps the best of the bunch. ... Microflows: Fundamentals and Simulation has a lot to offer and is certainly recommended as a good place to start for MEMS students interested in flow physics."

Remarkable progress has recently been made in the application of quantumtrajectories as the computational tool for solving quantum mechanical problems. This is the first book to present these developments in the broader context of the hydrodynamical formulation of quantum dynamics. In addition to a thorough discussion of the quantum trajectory equations of motion, there is considerable material that deals with phase space dynamics, adaptive moving grids, electronic energy transfer, and trajectories for stationary states.On the pedagogical side, a number of sections of this book will be accessible to students who have had an introductory quantum mechanics course. There is also considerable material for advanced researchers, and chapters in the book cover both methodology and applications. The book will be useful to students and researchers in physics, chemistry, applied math, and computational dynamics.

Systems biology and computational biology have recently become prominent areas of research in the biomedical community, especially in the area of cell biology. Given that much information on genes and their protein products has become available, the big question is how the individual components interact and work together, and how this determines the functioning of cells, organs, and organisms. Long before the popularity of systems biology in biomedicine, however, such approaches have been used successfully in a di?erent area of biology: population ecology. Research in the area of population dynamics - vestigated complex interactions between di?erent populations of organisms, such as the dynamics of competition and predation, food webs, community structure, as well as the epidemiology of infectious diseases. In this ?eld, t- oretical biology and mathematical modeling have become an integral part of research. Mathematical models allowed people to obtain interesting and counter-intuitive insights into how complex interactions among di?erent p- ulations can play out. Such mathematical studies not only gave rise to - teresting theoretical ideas, but also provided the basis for the design of new experimental work and de?ned major questions and directions of research. Around 1990, such population dynamic concepts, and the use of mathema- cal/computational approaches, started to be applied to the in vivo dynamics between viruses and the immune system. These interactions have many s- ilarities to ecological, epidemiological, and evolutionary principles. Consider theepidemiologicalspreadofapathogen(suchasthecommoncold)througha population of hosts.

The purpose of this book is to provide an accessible introduction to a new set of methods for the analysis of Lagrangian motion in geophysical ?ows. These methods were originally developed in the abstract mathem- ical setting of dynamical systems theory, through a geometric approach to di?erential equations that ultimately owes much to the insights of Poincar´ e (1892). In the 1980s and 1990s, researchers in applied mathematics and ?uid dynamics recognized the potential of this approach for the analysis of ?uid motion. Despite these developments and the existence of a substantial body of work on geophysical ?uid problems in the dynamical systems and geophysicalliterature,nointroductorytexthasbeenavailablethatpresents these methods in the context of geophysical ?uid ?ow. The text is meant to be accessible to geophysical ?uid scientists and students familiar with the mathematics of ordinary (mostly) and partial (sometimes) di?erential equations. It assumes little or no prior knowledge of dynamical systems theory. An e?ort is made to explain concepts from a physical point of view, and to avoid the theorem and proof constructions that appear in dynamical systems texts. We hope that this book will prove usefultograduatestudents,researchscientists,andeducatorsinanybranch of geophysical ?uid science in which the motion and transport of ?uid, and ofmaterialscarriedbythe?uid,isofinterest.Wehopethatitwillalsoprove interesting and useful to applied mathematicians who seek an introduction to an intriguing and rapidly developing area of geophysical ?uid dynamics.

Cancer is a complex and devastating disease, responsible for millions of deaths worldwide each year. While traditional oncology focuses on diagnosis and treatment through medical, surgical, radiation and clinical techniques, mathematical oncology has emerged as a powerful discipline that applies mathematical modelling to understand cancer growth, spread, and response to treatment.This volume provides a comprehensive exploration of mathematical approaches in oncology, offering a deep dive into differential equation models and biomechanical models. From reaction-diffusion equations that capture tumour growth and spread, to mechanical models that examine cellular interactions within tissues and vessels, this book presents both analytical techniques and computational modelling approaches that enhance our understanding of cancer dynamics.This book serves as both a reference for researchers and a foundation for integrating mathematical oncology into university curricula. By bridging the gap between mathematics, biology, and clinical research, it highlights the crucial role of mathematical modelling in advancing cancer treatment strategies and improving patient outcomes.

This richly illustrated book introduces the reader to a newly developed theory in Computer Vision yielding elementary techniques to analyze digital images. These techniques are inspired by, and are a mathematical formalization of the Gestalt theory. The authors maintain a publicly available software program called MegaWave, containing implementations of most of the image analysis techniques developed in the book. The volume is intended for a multidisciplinary audience of researchers and engineers. It is mathematically self-contained and requires only the basic understanding of probability and calculus. The text contains more than 130 illustrations. The authors offer numerous examples based on specific images on which the theory is tested. Detailed exercises at the end of each chapter help the reader develop a firm understanding of the concepts imparted.

There has been a long history of interaction  between mathematics and physiology. This book looks in detail at a wide selection of mathematical models in physiology, showing how physiological problems can be formulated and studied mathematically, and how such models give rise to interesting and challenging mathematical questions. With its coverage of many recent models it gives an overview of the field, while many older models are also discussed, to put the modern work in context. In this second edition the coverage of basic principles has been expanded to include such topics as stochastic differential equations, Markov models and Gibbs free energy, and the selection of models has also been expanded to include some of the basic models of fluid transport, respiration/perfusion, blood diseases, molecular motors, smooth muscle, neuroendrocine cells, the baroreceptor loop, turboglomerular oscillations, blood clotting and the retina. Owing to this extensive coverage,  the seond edition is published in two volumes. This first volume deals with the fundamental principles of cell physiology and the second with the physiology of systems. The book includes detailed illustrations and  numerous excercises with selected solutions. The emphasis throughout is on the applications; because of this interdisciplinary  approach, this book  will be of  interest to students and researchers, not only in mathematics, but also in bioengineering, physics, chemistry, biology, statistics and medicine.James Keener is a Distinguished Professor of Mathematics at the University of Utah. James Sneyd is the Professor of Applied Mathematics at the University of Auckland, New Zealand. He is best known for his work on the dynamics of intracellular calcium.Reviews of the first edition:...probably the best book ever written on the interdisciplinary field of mathematical physiology. Mathematical Reviews, 2000In addition to being good reading, excellent pedagogy, and appealing science, the exposition is lucid and clear, and there are many good problem sets to choose from... Highly recommended. Mathematical Biosciences, 1999Both authors are seasoned experts in the field of mathematical physiology and particularly in the field of excitability, calcium dynamics and spiral waves. It directs students to become not merely skilled technicians in biological research but masters of the science. SIAM, 2004The first edition was the winner of the prize for The Best Mathematics book of 1998 from the American Association of Publishers.

This book applies methods from nonlinear dynamics to problems in neuroscience. It uses modern mathematical approaches to understand patterns of neuronal activity seen in experiments and models of neuronal behavior. The intended audience is researchers interested in applying mathematics to important problems in neuroscience, and neuroscientists who would like to understand how to create models, as well as the mathematical and computational methods for analyzing them. The authors take a very broad approach and use many different methods to solve and understand complex models of neurons and circuits. They explain and combine numerical, analytical, dynamical systems and perturbation methods to produce a modern approach to the types of model equations that arise in neuroscience. There are extensive chapters on the role of noise, multiple time scales and spatial interactions in generating complex activity patterns found in experiments. The early chapters require little more than basic calculus and some elementary differential equations and can form the core of a computational neuroscience course. Later chapters can be used as a basis for a graduate class and as a source for current research in mathematical neuroscience. The book contains a large number of illustrations, chapter summaries and hundreds of exercises which are motivated by issues that arise in biology, and involve both computation and analysis. Bard Ermentrout is Professor of Computational Biology and Professor of Mathematics at the University of Pittsburgh. David Terman is Professor of Mathematics at the Ohio State University.

This monograph presents a general mathematical theory for biological growth. It provides both a conceptual and a technical foundation for the understanding and analysis of problems arising in biology and physiology. The theory and methods are illustrated on a wide range of examples and applications.A process of extreme complexity, growth plays a fundamental role in many biological processes and is considered to be the hallmark of life itself. Its description has been one of the fundamental problems of life sciences, but until recently, it has not attracted much attention from mathematicians, physicists, and engineers. The author herein presents the first major technical monograph on the problem of growth since D’Arcy Wentworth Thompson’s 1917 book On Growth and Form.The emphasis of the book is on the proper mathematical formulation of growth kinematics and mechanics. Accordingly, the discussion proceeds in order of complexity and the book is divided into five parts. First, a general introduction on the problem of growth from a historical perspective is given.  Then, basic concepts are introduced within the context of growth in filamentary structures. These ideas are then generalized to surfaces and membranes and eventually to the general case of volumetric growth. The book concludes with a discussion of open problems and outstanding challenges.  Thoughtfully written and richly illustrated to be accessible to readers of varying interests and background, the text will appeal to life scientists, biophysicists, biomedical engineers, and applied mathematicians alike.

This book provides a comprehensive introduction to the latest advances in the mathematical theory and computational tools for modeling high-dimensional data drawn from one or multiple low-dimensional subspaces (or manifolds) and potentially corrupted by noise, gross errors, or outliers. This challenging task requires the development of new algebraic, geometric, statistical, and computational methods for efficient and robust estimation and segmentation of one or multiple subspaces. The book also presents interesting real-world applications of these new methods in image processing, image and video segmentation, face recognition and clustering, and hybrid system identification etc. This book is intended to serve as a textbook for graduate students and beginning researchers in data science, machine learning, computer vision, image and signal processing, and systems theory. It contains ample illustrations, examples, and exercises and is made largely self-contained with three Appendices which survey basic concepts and principles from statistics, optimization, and algebraic-geometry used in this book.René Vidal is a Professor of Biomedical Engineering and Director of the Vision Dynamics and Learning Lab at The Johns Hopkins University. Yi Ma is Executive Dean and Professor at the School of Information Science and Technology at ShanghaiTech University. S. Shankar Sastry is Dean of the College of Engineering, Professor of Electrical Engineering and Computer Science and Professor of Bioengineering at the University of California, Berkeley.

For several decades developments in porous media have taken place in almost independent areas. In civilengineering, many papers were publisheddealing with the foundations offlow and transport through porous media. The method used in most cases is called averaging, and the notion ofa representative elementary vol- ume(REV)playsanimportantrole. Inchemicalengineering,papersonconceptual models were written on the theory ofmixtures. Intheoretical physics and stochas- tic analysis, percolation theory has emerged, providing probabilistic models for systems where theconnectedness propertiesofsomecomponentdominatethebe- havior. In mathematics, atheoryhasbeendevelopedcalled homogenizationwhich deals with partial differential equations having rapidly oscillating coefficients. Early work in these and related areas was - among others - done by the fol- lowing scientists: Maxwell [Max81] and Rayleigh [Ray92] studied the effective conductivity of media with small concentrations of randomly and periodically, respectively, arranged inclusions. Einstein [Ein06] investigated the effective vis- cosityofsuspensions with hard spherical particles in compressible viscous fluids.Marchenko and Khrouslov [MK64] looked at the asymptotic nature of homog- enization; they introduced a general approach of averaging based on asymptotic tools which can handle a variety ofdifferent physical problems. Unfortunately, up to now, little efforthas been made to bridge the gap between these different fields of research. Consequently, many results were and are dis- covered independently, and scientists are almost unable to understand each other because the respective languages have been developing in different directions.

In the last 40 years geophysicists have found that it is possible to construct images and even determine important physical characteristics of rocks that can yield information about oil and gas bearing structures in the earth. To make these images and extract this information requires the application of an advanced understanding of the mathematical physics of wave propagation. The oil and gas industry labels a major collection of the necessary seismic data processing methods by the name seismic "migration". This text ist the first to treat many kinds of migration in a unified mahtematical way. The audience is mathematically oriented geophysicists or applied mathematicians working in the field of "inverse scattering imaging". The text can serve as a bridge between the applied math and geophysics community by presenting geophysicists with a practical introduction to advanced engineering mathematics, while presenting mathematicians with a window into the world of the mathematically sophistiated geophysicist.

The interdisciplinary subject of random heterogeneous materials has experienced remarkable growth since the publication of the well-known monograph Statistical Con­ tinuum Theories by Beran ( 1968). Many of these advances, especially those concerning the statistical characterization of the microstructure and its effect on the physical prop­ erties of the material, have not been treated fully in any book. One of the intents of the present book is to fill this gap. This book also distinguishes itself in that it provides a unified rigorous framework to characterize the microstructures and macroscopic properties of the widely diverse types of heterogeneous materials found in nature and synthetic products. Emphasis is placed on providing foundational theoretical methods that can simultaneously yield results of practical utility. This book treats a wide breadth of topics, but the choice of subjects naturally reflects my own interests. The sheer enormity of the field has prevented me from covering many important topics. I apologize to those colleagues, known and unknown, who may not find enough of their own work cited in the ensuing pages.

It has been over a decade since the release of the now classic original edition of Murray's Mathematical Biology. Since then mathematical biology has grown at an astonishing rate and is well established as a distinct discipline. Mathematical modeling is now being applied in every major discipline in the biomedical sciences. Though the field has become increasingly large and specialized, 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 modeling can address. Due to the tremendous development in the field this book is being published in two volumes. This first volume is an introduction to the field, the mathematics mainly involves ordinary differential equations that are suitable for undergraduate and graduate courses at different levels. For this new edition Murray is covering certain items in depth, giving new applications such as modeling marital interactions andtemperature dependence sex determination.SIAM, 2004: "Murray's Mathematical Biology is a classic that belongs on the shelf of any serious student or researcher in the field. Together the two volumes contain well over 1000 references, a rich source of material, together with an excellent index to help readers quickly find key words. ... I recommend the new and expanded third edition to any serious young student interested in mathematical biology who already has a solid basis in applied mathematics."

It has been over a decade since the release first edition of the now classic original edition of Murray's Mathematical Biology. Since then mathematical biology and medicine has grown at an astonishing rate and has established itself as a distinct discipline. Mathematical modelling is now being applied in every major discipline in the biomedical sciences. Though the field has become increasingly large and specialized, this book remains important as a text that introduces some of the exciting problems which arise in the biomedical sciences and gives some indication of the wide spectrum of questions that modelling can address. Due to the tremendous development in recent years, this new edition is being published in two volumes. This second volume covers spatial models and biomedical applications. For this new edition, Murray covers certain items in depth, introducing new applications such as modelling growth and control of brain tumours, bacterial patterns, wound healing and wolf territoriality. In other areas, he discusses basic modelling concepts and provides further references as needed. He also provides even closer links between models and experimental data throughout the text. Graduate students and researchers will find this book invaluable as it gives an excellent background from which to begin genuinely practical interdisciplinary research in the biomedical sciences.

The subject of this monograph lies in the joint areas of applied mathematics and hydrogeology. The goals are to introduce various mathematical techniques and ideas to applied scientists while at the same time to reveal to applied math­ ematicians an exciting catalog of interesting equations and examples, some of which have not undergone the rigors of mathematical analysis. Of course, there is a danger in a dual endeavor-the applied scientist may feel the mathematical models lack physical depth and the mathematician may think the mathematics is trivial. However, mathematical modeling has established itself firmly as a tool that can not only lead to greater understanding of the science, but can also be a catalyst for the advancement of science. I hope the presentation, written in the spirit of mathematical modeling, has a balance that bridges these two areas and spawns some cross-fertilization. Notwithstanding, the reader should fully understand the idea of a mathe­ matical model. In the world of reality we are often faced with describing and predicting the results of experiments. A mathematical model is a set of equa­ tions that encapsulates reality; it is a caricature of the real physical system that aids in our understanding of real phenomena. A good model extracts the essen­ tial features of the problem and lays out, in a simple manner, those processes and interactions that are important. By design, mathematical models should have predictive capability.

This text is an introduction to dynamical modeling in cell biology. It is not meant as a complete overview of modeling or of particular models in cell biology. Rather, we use selected biological examples to motivate the concepts and techniques used in computational cell biology. This is done through a progression of increasingly more complex cellular functions modeled with increasingly complex mathematical and c- putational techniques. There are other excellent sources for material on mathematical cell biology, and so the focus here truly is computer modeling. This does not mean that there are no mathematical techniques introduced, because some of them are absolutely vital, but it does mean that much of the mathematics is explained in a more intuitive fashion, while we allow the computer to do most of the work. The target audience for this text is mathematically sophisticated cell biology or neuroscience students or mathematics students who wish to learn about modeling in cell biology. The ideal class would comprise both biology and applied math students, who might be encouraged to collaborate on exercises or class projects. We assume as little mathematical and biological background as we feel we can get away with, and we proceed fairly slowly. The techniques and approaches covered in the ?rst half of the book will form a basis for some elementary modeling or as a lead in to more advanced topics covered in the second half of the book.

Describes the chaos apparent in simple mechanical systems with the goal of elucidating the connections between classical and quantum mechanics. It develops the relevant ideas of the last two decades via geometric intuition rather than algebraic manipulation. The historical and cultural background against which these scientific developments have occurred is depicted, and realistic examples are discussed in detail. This book enables entry-level graduate students to tackle fresh problems in this rich field.

This book goes back a long way. There is a tradition of research and teaching in inelasticity at Stanford that goes back at least to Wilhelm Flugge ¨ and Erastus Lee. I joined the faculty in 1980, and shortly thereafter the Chairman of the Applied Mechanics Division, George Herrmann, asked me to present a course in plasticity. I decided to develop a new two-quarter sequence entitled “Theoretical and C- putational Plasticity” which combined the basic theory I had learned as a graduate student at the University of California at Berkeley from David Bogy, James Kelly, Jacob Lubliner, and Paul Naghdi with new computational techniques from the ?nite-element literature and my personal research. I taught the course a couple of times and developed a set of notes that I passed on to Juan Simo when he joined thefacultyin1985. IwasChairmanatthattimeandIaskedJuantofurtherdevelop the course into a full year covering inelasticity from a more comprehensive p- spective. Juan embarked on this path creating what was to become his signature course. He eventually renamed it “Computational and Theoretical Inelasticity” and it covered much of the material that was the basis of his research in material modeling and simulation for which he achieved international recognition. At the outset we decided to write a book that would cover the material in the course.

Provides a new and more realistic framework for describing the dynamics of non-linear systems. A number of issues arising in applied dynamical systems from the viewpoint of problems of phase space transport are raised in this monograph. Illustrating phase space transport problems arising in a variety of applications that can be modeled as time-periodic perturbations of planar Hamiltonian systems, the book begins with the study of transport in the associated two-dimensional Poincaré Map. This serves as a starting point for the further motivation of the transport issues through the development of ideas in a non-perturbative framework with generalizations to higher dimensions as well as more general time dependence. A timely and important contribution to those concerned with the applications of mathematics.

There has been a great deal of excitement in the last ten years over the emer­ gence of new mathematical techniques for the analysis and control of nonlinear systems: Witness the emergence of a set of simplified tools for the analysis of bifurcations, chaos, and other complicated dynamical behavior and the develop­ ment of a comprehensive theory of geometric nonlinear control. Coupled with this set of analytic advances has been the vast increase in computational power available for both the simulation and visualization of nonlinear systems as well as for the implementation in real time of sophisticated, real-time nonlinear control laws. Thus, technological advances havebolstered the impact of analytic advances and produced a tremendous variety of new problems and applications that are nonlinear in an essential way. Nonlinear controllaws have been implemented for sophisticated flight control systems on board helicopters, and vertical take offand landing aircraft; adaptive, nonlinearcontrollaws havebeen implementedfor robot manipulators operating either singly, or in cooperation on a multi-fingered robot hand; adaptive control laws have been implemented forjetengines andautomotive fuel injection systems, as well as for automated highway systems and air traffic management systems, to mention a few examples. Bifurcation theory has been used to explain and understand the onset of fiutterin the dynamics of aircraft wing structures, the onset of oscillations in nonlinear circuits, surge and stall in aircraft engines, voltage collapse in a power transmission network.

The story of this edition is a testament to an almost legendary gure in theoretical ecology and to the in uence his work and charisma has had on the eld. It is also a story that can only be told by a trip back in time, to the genesis of the First Edition and before. Akira kubo and I were students together, but never knew it at the time. He was a graduate student at The ohns Hopkins niversity, where I was an undergraduate in mathematics. e both studied modern physics, taught by Dino Franco asetti, and we decided years later that we must have been in the same class. Akira was then a chemical oceanographer, but ship time and his stomach did not agree. Sohe turned to theory,and the rest is history. His impact has been phenomenal, and the First Edition of this book was his most in uential work. Building on his famous work with dye-di¤usion e periments, he turned his attention to organisms and created a unique melding of ideas from physics and biology.

Geometry of Biological Time deals with dynamics of processes that repeat themselves regularly. Such rhythmic return through a cycle of change is an ubiquitous principle of organization in living systems. In this revised and updated edition the author plans to extend the thread from 1980 to the present concentrating on areas which he personally feels have been interesting and where he feels there will be much activity in the future. This involves going through spatial biochemical, electrophysiological, and organismic dynamical systems and patterns that were discovered by pursuing the theme of phase singularities that the original book introduced. In particular the work on excitability in cell membranes has been thoroughly updated as have the references throughout the book.