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.

This book describes and discusses the properties of heterogeneous materials. The properties considered include the conductivity (thermal, electrical, magnetic), elastic moduli, dielectrical constant, optical properties, mechanical fracture, and electrical and dielectrical breakdown properties. Both linear and nonlinear properties are considered. The nonlinear properties include those with constitutive non-linearities as well as threshold non-linearities, such as brittle fracture and dielectric breakdown. A main goal of this book is to compare two fundamental approaches to describing and predicting materials properties, namely, the continuum mechanics approach, and those based on the discrete models. The latter models include the lattice models and the atomistic approaches. The book provides comprehensive and up to date theoretical and computer simulation analysis of materials' properties. Typical experimental methods for measuring all of these properties are outlined, and comparison is made between the experimental data and the theoretical predictions.Volume I covers linear properties, while Volume II considers non-linear and fracture and breakdown properties, as well as atomistic modeling. This multidisciplinary book will appeal to applied physicists, materials scientists, chemical and mechanical engineers, chemists, and applied mathematicians.

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.

This book deals with the application of mathematics in modeling and understanding physiological systems, especially those involving rhythms. It is divided roughly into two sections. In the first part of the book, the authors introduce ideas and techniques from nonlinear dynamics that are relevant to the analysis of biological rhythms. The second part consists of five in-depth case studies in which the authors use the theoretical tools developed earlier to investigate a number of physiological processes: the dynamics of excitable nerve and cardiac tissue, resetting and entrainment of biological oscillators, the effects of noise and time delay on the pupil light reflex, pathologies associated with blood cell replication, and Parkinsonian tremor. 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 book will be of 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.The authors are members of the Centre for Nonlinear Dynamics in Physiology and Medicine. The material in this book was developed for use in courses and was presented in three Summer Schools run by the authors in Montreal.

This book gives senior undergraduate and beginning graduate students and researchers in computer vision, applied mathematics, computer graphics, and robotics a self-contained introduction to the geometry of 3D vision; that is the reconstruction of 3D models of objects from a collection of 2D images. Following a brief introduction, Part I provides background materials for the rest of the book. The two fundamental transformations, namely rigid body motion and perspective projection are introduced and image formation and feature extraction discussed. Part II covers the classic theory of two view geometry based on the so-called epipolar constraint. Part III shows that a more proper tool for studying the geometry of multiple views is the so- called rank considtion on the multiple view matrix. Part IV develops practical reconstruction algorithms step by step as well as discusses possible extensions of the theory. Exercises are provided at the end of each chapter. Software for examples and algorithms are available on the author's website.

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.

Fluid flow through small channels has become a popular research topic due to the emergence of biochemical lab-on-the-chip systems and microelectromechanical system fabrication technologies, which began in the late 1980's. There has been significant progress in the development of microfluidics and nanofluidics both at the application as well as at the fundamental and simulation levels in the last few years. This book provides a comprehensive summary of using computational tools to describe fluid flow in micro and nano configuration. This new version of the original book now covers length scales from Anstroms to microns (and beyond). The authors have maintained the emphasis on fundamental concepts with a mix of semianalytical, experimental and numerical results, and have outlined their relevance to modelling and analyzing functional devices. The material is devided into the three main categories of gas flows, liquid flows, and simulation techniques. The book contains many new developments and information has been updated and modified throughout.It is appropriate for researchers in fluid mechanics interested in this new flow field as well as for electrical or mechanical engineers or physicists who need to incorporate flow modeling into their work.Review of earlier book by Karniadakis/Beskok on Microflows:"For those who want to compute flows at the micro scale, this monograph is a must. It describes the state of the art and helps by providing coefficients, such as [are] needed in situations of slip. Those who wonder what new fluid dynamics there is in the microworld are served by the overview of theory and treasures of numerical methods."EUROPEAN JOURNAL OF MECHANICS B / FLUIDS

Remarkable progress has recently been made in the development and application of quantum trajectories as the computational tool for solving the time dependent Schrodinger equation. Analogous methods for stationary bound states are also being developed. The purpose of this book is to present recent developments and applications of quantum trajectory methods in the broader context of the hydrodynamical formulation of quantum dynamics. While many chapters of the book deal with Lagrangian quantum trajectories in which the velocity matches that of the probability fluid, other chapters deal with what will be termed post-Lagrangian trajectories. There are also many state-of-the-art topics covered that are unique to this book. On the pedagogical side, a number of sections will be accessible to students who have had at least one course in quantum dynamics. There is also considerable material for advanced researchers, and chapters in the book cover both methodology and applications.

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.

Written by a specialist in geophysical fluid dynamics and an applied mathematician, this book provides an accessible introduction to new methods for analysing Lagrangian motion in geophysical flows, and surveys research in geophysical fluid dynamics that makes use of them.

Preliminary: Over the last decade or so there has been renewed interest in the mathematical modeling of tumor growth. This book presents an overview of recent mathematical models developed to examine the many stages of cancer growth and development from a single mutant cell through to the metastatic spread of the disease. The mathematical models employed are ordinary and partial differential equation models, discrete models, and continuum mechanics models. Various analytical and numerical methods are used in the analysis of these models and at all times clinical implications of the model results are considered.

This book introduces a new theory in Computer Vision yielding elementary techniques to analyze digital images. These techniques are a mathematical formalization of the Gestalt theory. From the mathematical viewpoint the closest field to it is stochastic geometry, involving basic probability and statistics, in the context of image analysis. The book is mathematically self-contained, needing only basic understanding of probability and calculus. The text includes more than 130 illustrations, and 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.

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.

This book applies methods from nonlinear dynamics to problems in neuroscience. 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.

This monograph presents a general mathematical theory for biological growth. 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.

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 overall aim of this book is to provide a rigorous means to characterize the microstructure and properties of heterogeneous materials that can simultaneously yield results of practical utility. A unified treatment of both microstructure and properties is emphasized.The book is divided into two parts. Part I deals with the quantitative characterization of the microstructure of heterogeneous via theoretical, computer-simulation and imaging techniques. Emphasis is placed on theoretical methods. Part II treats a wide variety of effective properties of heterogeneous materials and how they are linked to the microstructure. This is accomplished using rigorous methods. Wherever possible, theoretical predictions for the effective properties are compared to available, experimental and computer-simulation data.This book will be of interest to graduate students and researchers in applied mathematics, physics, chemistry, materials science and engineering.

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.

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.

This book develops the basic ideas of transport models in hydrogeology, including diffusion-dispersion processes, advection, and adsorption or reaction. While balancing the mathematical and physical concepts, it is written in a scientific pedagogical style that is accessible to graduate students and researchers in applied mathematics, the geosciences, civil engineering, and other environmental sciences. The book is organized into six chapters that focus on the diffusion equation, the transport of contaminants and other solutes through porous media, the existence of traveling wave fronts in such systems, filtration theory, the kinematics and dynamics of ground water transport, and the flow and reactions in permeable rocks.Analytic methods for both elementary linear and nonlinear partial differential equations are discussed in detail. Appendices develop numerical methods for partial differential equations, including the method of lines and finite difference methods, and numerical methods for inverting Laplace transforms. Exercises are interspersed throughout the book and over one hundred and twenty references are cited.The book serves as an excellent text or supplementary reading in courses in applied mathematics, contaminant hydrology, ground water modeling, or hydrogeology.

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.

A description of the theoretical foundations of inelasticity, its numerical formulation and implementation, constituting a representative sample of state-of-the-art methodology currently used in inelastic calculations. Among the numerous topics covered are small deformation plasticity and viscoplasticity, convex optimisation theory, integration algorithms for the constitutive equation of plasticity and viscoplasticity, the variational setting of boundary value problems and discretization by finite element methods. Also addressed are the generalisation of the theory to non-smooth yield surface, mathematical numerical analysis issues of general return mapping algorithms, the generalisation to finite-strain inelasticity theory, objective integration algorithms for rate constitutive equations, the theory of hyperelastic-based plasticity models and small and large deformation viscoelasticity. Of great interest to researchers and graduate students in various branches of engineering, especially civil, aeronautical and mechanical, and applied mathematics.

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.

This book surveys a wide variety of mathematical models of diffusion in the ecological context. It is written with the primary intent of providing scientists, particularly physicists but also biologists, with some background of the mathematics and physics of diffusion and how they can be applied to ecological problems. The secondary intent is to provide a specialized text book for graduate students, who are interested in mathematical ecology. The reader is assumed to have a basic knowledge of probability and differential equations. Each chapter in this new edition is substantially updated by appopriate leading researchers in the field and contains much new material covering developments in the field in the last 20 years.

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 will be thoroughly updated as will references throughout the book.