This is the tenth volume in the series "Mathematics in Industrial Prob lems. " The motivation for these volumes is to foster interaction between Industry and Mathematics at the "grass roots level;" that is, at the level of specific problems. These problems come from Industry: they arise from models developed by the industrial scientists in ventures directed at the manufacture of new or improved products. At the same time, these prob lems have the potential for mathematical challenge and novelty. To identify such problems, I have visited industries and had discussions with their scientists. Some of the scientists have subsequently presented their problems in the IMA Seminar on Industrial Problems. The book is based on the seminar presentations and on questions raised in subse quent discussions. Each chapter is devoted to one of the talks and is self contained. The chapters usually provide references to the mathematical literature and a list of open problems which are of interest to the industrial scientists. For some problems a partial solution is indicated briefly. The last chapter of the book contains a short description of solutions to some of the problems raised in the previous volume, as well as references to papers in which such solutions have been published. The speakers in the Seminar on Industrial Problems have given us at the IMA hours of delight and discovery."

This book presents mathematical models that arise in current photographic science. The book contains seventeen chapters, each dealing with one area of photographic science, and a final chapter containing exercises. Each chapter, except the two introductory chapters, begin with general background information at a level understandable by graduate and undergraduate students. It then proceeds to develop a mathematical model, using mathematical tools such as ordinary differential equations, partial differential equations, and stochastic processes. Next, some mathematical results are mentioned, often providing a partial solution to problems raised by the model. Finally, most chapters include open problems. The last chapter of the book contains "Modeling and Applied Mathematics" exercises based on the material presented in the earlier chapters. These exercises are intended primarily for graduate students and advanced undergraduates.

This is the eighth volume in the series "Mathematics in Industrial Prob lems." The motivation for these volumes is to foster interaction between Industry and Mathematics at the "grass roots level"; that is, at the level of specific problems. These problems come from Industry: they arise from models developed by the industrial scientists in ventures directed at the manufacture of new or improved products. At the same time, these prob lems have the potential for mathematical challenge and novelty. To identify such problems, I have visited industries and had discussions with their scientists. Some of the scientists have subsequently presented their problems in the IMA Seminar on Industrial Problems. The book is based on the seminar presentations and on questions raised in subsequent discussions. Each chapter is devoted to one of the talks and is self-contained. The chapters usually provide references to the mathematical literature and a list of open problems that are of interest to industrial scientists. For some problems, a partial solution is indicated briefly. The last chapter of the book contains a short description of solutions to some of the problems raised in the previous volume, as well as references to papers in which such solutions have been published."

This is the second volume in the series "Mathematics in Industrial Prob lems." The motivation for both volumes is to foster inter action between Industry and Mathematics at the "grass roots"; that is at the level of spe cific problems. These problems come from Industry: they arise from models developed by the industrial scientists in venture directed at the manufac ture of new or improved products. At the same time, these problems have the potential for mathematical challenge and novelty. To identify such problems, I have visited industries and had discussions with their scientists. Some of the scientists have subsequently presented their problems in the IMA seminar on Industrial Problems. The book is based on questions raised in the seminar and subsequent discussions. Each chapter is devoted to one of the talks and is self-contained. The chap ters usually provide references to the mathematical literat ure and a list of open problems which are of interest to the industrial scientists. For some problems partial solution is indicated briessy. The last chapter of the book contains a short description of solutions to some of the problems raised in the first volume, as weIl as references to papers in which such solutions have been published. The experience of the last two years demonstrates a growing fruitful interaction between Industry and Mathematics. This interaction benefits Industry by increasing the mathematical knowledge and ideas brought to bear upon its concern, and benefits Mathematics through the infusion of exciting new problems."

Developed from the cooperation between mathematicians and industrial scientists on the "grass roots" level of specific problems, this book is the most recent in a collection of self-contained volumes which present industrial problems to mathematicians. Topics include: imaging and visualization, diffusion in glassy and swelling polymers, composite materials, plastic flows, coating of fiber optics, communications, colloidal dispersion, stress in semiconductors, micromagnetics, photobleaching, and machine vision. Many chapters offer open problems and references, while the last chapter contains solutions to problems raised in previous volumes of Mathematics in Industrial Problems, Parts 2, 3, and 4, published in the "IMA" series as "Volumes " "24, 31, and 38" respectively.

Building a bridge between mathematicians and industry is both a chal lenging task and a valuable goal for the Institute for Mathematics and its Applications (IMA). The rationale for the existence of the IMA is to en courage interaction between mathematicians and scientists who use math ematics. Some of this interaction should evolve around industrial problems which mathematicians may be able to solve in "real time." Both Industry and Mathematics benefit: Industry, by increase of mathematical knowledge and ideas brought to bear upon their concerns, and Mathematics, through the infusion of exciting new problems. In the past ten months I have visited numerous industries and national laboratories, and met with several hundred scientists to discuss mathe matical questions which arise in specific industrial problems. Many of the problems have special features which existing mathematical theories do not encompass; such problems may open new directions for research. However, I have encountered a substantial number of problems to which mathemati cians should be able to contribute by providing either rigorous proofs or formal arguments. The majority of scientists with whom I met were engineers, physicists, chemists, applied mathematicians and computer scientists. I have found them eager to share their problems with the mathematical community. Often their only recourse with a problem is to "put it on the computer." However, further insight could be gained by mathematical analysis."

This IMA Volume in Mathematics and its Applications VARIATIONAL AND FREE BOUNDARY PROBLEMS is based on the proceedings of a workshop which was an integral part of the 1990- 91 IMA program on "Phase Transitions and Free Boundaries. " The aim of the workshop was to highlight new methods, directions and problems in variational and free boundary theory, with a concentration on novel applications of variational methods to applied problems. We thank R. Fosdick, M. E. Gurtin, W. -M. Ni and L. A. Peletier for organizing the year-long program and, especially, J. Sprock for co-organizing the meeting and co-editing these proceedings. We also take this opportunity to thank the National Science Foundation whose financial support made the workshop possible. Avner Friedman Willard Miller, Jr. PREFACE In a free boundary one seeks to find a solution u to a partial differential equation in a domain, a part r of its boundary of which is unknown. Thus both u and r must be determined. In addition to the standard boundary conditions on the un known domain, an additional condition must be prescribed on the free boundary. A classical example is the Stefan problem of melting of ice; here the temperature sat isfies the heat equation in the water region, and yet this region itself (or rather the ice-water interface) is unknown and must be determined together with the tempera ture within the water. Some free boundary problems lend themselves to variational formulation."

This is the third volume in the series "Mathematics in Industrial Prob lems." The motivation for these volumes is to foster interaction between Industry and Mathematics at the "grass roots"; that is, at the level of spe cific problems. These problems come from Industry: they arise from models developed by the industrial scientists in ventures directed at the manufac ture of new or improved products. At the same time, these problems have the potential for mathematical challenge and novelty. To identify such problems, I have visited industries and had discussions with their scientists. Some of the scientists have subsequently presented their problems in the IMA seminar on Industrial Problems. The book is based on questions raised in the seminar and subsequent discussions. Each chapter is devoted to one of the talks and is self-contained. The chap ters usually provide references to the mathematical literature and a list of open problems which are of interest to the industrial scientists. For some problems partial solution is indicated briefly. The last chapter of the book contains a short description of solutions to some of the problems raised in the second volume, as well as references to papers in which such solutions have been published."

This is the seventh volume in the series "Mathematics in Industrial Prob lems. " The motivation for these volumes is to foster interaction between Industry and Mathematics at the "grass roots level;" that is, at the level of specific problems. These problems come from Industry: they arise from models developed by the industrial scientists in ventures directed at the manufacture of new or improved products. At the same time, these prob lems have the potential for mathematical challenge and novelty. To identify such problems, I have visited industries and had discussions with their scientists. Some of the scientists have subsequently presented their problems in the IMA Seminar on Industrial Problems. The book is based on the seminar presentations and on questions raised in subse quent discussions. Each chapter is devoted to one of the talks and is self contained. The chapters usually provide references to the mathematical literature and a list of open problems which are of interest to the industrial scientists. For some problems a partial solution is indicated briefly. The last chapter of the book contains a short description of solutions to some of the problems raised in previous volumes, as well as references to papers in which such solutions have been published. The speakers in the Seminar on Industrial Problems have given us at the IMA hours of delight and discovery. My thanks to David K. Lambert (Gen eral Motors Research and Development), David S."

This is the sixth volume in the series "Mathematics in Industrial Prob lems. " The motivation for these volumes is to foster interaction between Industry and Mathematics at the "grass roots level"; that is, at the level of specific problems. These problems come from Industry: they arise from models developed by the industrial scientists in ventures directed at the manufacture of new or improved products. At the same time, these prob lems have the potential for mathematical challenge and novelty. To identify such problems, I have visited industries and had discussions with their scientists. Some of the scientists have subsequently presented their problems in the IMA Seminar on Industrial Problems. The book is based on the seminar presentations and on questions raised in subse quent discussions. Each chapter is devoted to one of the talks and is self contained. The chapters usually provide references to the mathematical literature and a list of open problems which are of interest to the industrial scientists. For some problems a partial solution is indicated briefly. The last chapter of the book contains a short description of solutions to some of the problems raised in previous volumes, as well as references to papers in which such solutions have been published. The speakers in the seminar on Industrial Problems have given us at the IMA hours of delight and discovery. My thanks to Thomas Hoffend (3M), John Spence (Eastman Kodak Company), Marius Orlowski (Mo torola, Inc. ), Robert J."

This book presents industrial problems to mathematicians, including the mathematical formulation of the problems. The first twenty chapters of the book include the industrial background, relevant mathematical literature, a list of open mathematical problems and, in some cases, reference to a solution or a partial solution of the problem. Most of the problems, however, are still open and they are addressed to mathematicians. The last chapter of the book contains reference to solutions of problems presented in the previous volume of Mathematics in Industrial Problems, Part 3 published in the IMA series, as volume 31. The topics of the book include semiconductor devices and processing; particles dynamics; polymer chains and electrophoresis; catalytic converter, robotics and CFD in the automobile industry, superconductivity, magnetic storage devices, signal processing, and experimental design. The book will be of interest to mathematicians seeking to work on mathematical problems which arise in industry.It will also be of interest to mathematicians and scientists who would like to learn about the interaction between mathematics and industry, what type of problems arise, how they are modeled, etc. Scientists working in industry may also be interested in the book as they discover that some of the topics dealt with are connected to their own work.

This is the 9th volume in Avner Friedman's collection of Mathematics in Industrial problems. This book aims to foster interaction between industry and mathematics at the "grass roots" level of specific problems. The problems presented in this book arise from models developed by industrial scientists engaged in research and development of new or improved products. The topics explored in this volume include diffusion in porous media and in rubber/glass transition, coating flows, solvation of molecules, semiconductor processing, optoelectronics, photographic images, density-functional theory, sphere packing, performance evaluation, causal networks, electrical well logging, general positioning system, sensor management, pursuit-evasion algorithms, and nonlinear viscoelasticity. Open problems and references are incorporated into most of the chapters. The final chapter contains some solutions to problems raised in earlier volumes.

This book offers an introduction to fast growing research areas in evolution of species, population genetics, ecological models, and population dynamics. It reviews the concept and methodologies of phylogenetic trees, introduces ecological models, examines a broad range of ongoing research in population dynamics, and deals with gene frequencies under the action of migration and selection. The book features computational schemes, illustrations, and mathematical theorems.

This volume introduces some basic mathematical models for cell cycle, proliferation, cancer, and cancer therapy. Chapter 1 gives an overview of the modeling of the cell division cycle. Chapter 2 describes how tumor secretes growth factors to form new blood vessels in its vicinity, which provide it with nutrients it needs in order to grow. Chapter 3 explores the process that enables the tumor to invade the neighboring tissue. Chapter 4 models the interaction between a tumor and the immune system. Chapter 5 is concerned with chemotherapy; it uses concepts from control theory to minimize obstacles arising from drug resistance and from cell cycle dynamics. Finally, Chapter 6 reviews mathematical results for various cancer models.

This volume introduces some basic theories on computational neuroscience. Chapter 1 is a brief introduction to neurons, tailored to the subsequent chapters. Chapter 2 is a self-contained introduction to dynamical systems and bifurcation theory, oriented towards neuronal dynamics. The theory is illustrated with a model of Parkinson's disease. Chapter 3 reviews the theory of coupled neural oscillators observed throughout the nervous systems at all levels; it describes how oscillations arise, what pattern they take, and how they depend on excitory or inhibitory synaptic connections. Chapter 4 specializes to one particular neuronal system, namely, the auditory system. It includes a self-contained introduction, from the anatomy and physiology of the inner ear to the neuronal network that connects the hair cells to the cortex, and describes various models of subsystems.

This book presents mathematical models that arise in current photographic science. The book contains seventeen chapters, each dealing with one area of photographic science, and a final chapter containing exercises. Each chapter, except the two introductory chapters, begin with general background information at a level understandable by graduate and undergraduate students. It then proceeds to develop a mathematical model, using mathematical tools such as ordinary differential equations, partial differential equations, and stochastic processes. Next, some mathematical results are mentioned, often providing a partial solution to problems raised by the model. Finally, most chapters include open problems. The last chapter of the book contains "Modeling and Applied Mathematics" exercises based on the material presented in the earlier chapters. These exercises are intended primarily for graduate students and advanced undergraduates.

This is the tenth volume in the series "Mathematics in Industrial Prob lems. " The motivation for these volumes is to foster interaction between Industry and Mathematics at the "grass roots level;" that is, at the level of specific problems. These problems come from Industry: they arise from models developed by the industrial scientists in ventures directed at the manufacture of new or improved products. At the same time, these prob lems have the potential for mathematical challenge and novelty. To identify such problems, I have visited industries and had discussions with their scientists. Some of the scientists have subsequently presented their problems in the IMA Seminar on Industrial Problems. The book is based on the seminar presentations and on questions raised in subse quent discussions. Each chapter is devoted to one of the talks and is self contained. The chapters usually provide references to the mathematical literature and a list of open problems which are of interest to the industrial scientists. For some problems a partial solution is indicated briefly. The last chapter of the book contains a short description of solutions to some of the problems raised in the previous volume, as well as references to papers in which such solutions have been published. The speakers in the Seminar on Industrial Problems have given us at the IMA hours of delight and discovery."

This is the 9th volume in Avner Friedman's collection of Mathematics in Industrial problems. This book aims to foster interaction between industry and mathematics at the "grass roots" level of specific problems. The problems presented in this book arise from models developed by industrial scientists engaged in research and development of new or improved products. The topics explored in this volume include diffusion in porous media and in rubber/glass transition, coating flows, solvation of molecules, semiconductor processing, optoelectronics, photographic images, density-functional theory, sphere packing, performance evaluation, causal networks, electrical well logging, general positioning system, sensor management, pursuit-evasion algorithms, and nonlinear viscoelasticity. Open problems and references are incorporated into most of the chapters. The final chapter contains some solutions to problems raised in earlier volumes.

This is the eighth volume in the series "Mathematics in Industrial Prob lems." The motivation for these volumes is to foster interaction between Industry and Mathematics at the "grass roots level"; that is, at the level of specific problems. These problems come from Industry: they arise from models developed by the industrial scientists in ventures directed at the manufacture of new or improved products. At the same time, these prob lems have the potential for mathematical challenge and novelty. To identify such problems, I have visited industries and had discussions with their scientists. Some of the scientists have subsequently presented their problems in the IMA Seminar on Industrial Problems. The book is based on the seminar presentations and on questions raised in subsequent discussions. Each chapter is devoted to one of the talks and is self-contained. The chapters usually provide references to the mathematical literature and a list of open problems that are of interest to industrial scientists. For some problems, a partial solution is indicated briefly. The last chapter of the book contains a short description of solutions to some of the problems raised in the previous volume, as well as references to papers in which such solutions have been published."

Mathematical biomedicine is a rapidly developing interdisciplinary field of research that connects the natural and exact sciences in an attempt to respond to the modeling and simulation challenges raised by biology and medicine. There exist a large number of mathematical methods and procedures that can be brought in to meet these challenges and this book presents a palette of such tools ranging from discrete cellular automata to cell population based models described by ordinary differential equations to nonlinear partial differential equations representing complex time- and space-dependent continuous processes. Both stochastic and deterministic methods are employed to analyze biological phenomena in various temporal and spatial settings. This book illustrates the breadth and depth of research opportunities that exist in the general field of mathematical biomedicine by highlighting some of the fascinating interactions that continue to develop between the mathematical and biomedical sciences. It consists of five parts that can be read independently, but are arranged to give the reader a broader picture of specific research topics and the mathematical tools that are being applied in its modeling and analysis. The main areas covered include immune system modeling, blood vessel dynamics, cancer modeling and treatment, and epidemiology. The chapters address topics that are at the forefront of current biomedical research such as cancer stem cells, immunodominance and viral epitopes, aggressive forms of brain cancer, or gene therapy. The presentations highlight how mathematical modeling can enhance biomedical understanding and will be of interest to both the mathematical and the biomedical communities including researchers already working in the field as well as those who might consider entering it. Much of the material is presented in a way that gives graduate students and young researchers a starting point for their own work.

This book on mathematical modeling of biological processes includes a wide selection of biological topics that demonstrate the power of mathematics and computational codes in setting up biological processes with a rigorous and predictive framework. Topics include: enzyme dynamics, spread of disease, harvesting bacteria, competition among live species, neuronal oscillations, transport of neurofilaments in axon, cancer and cancer therapy, and granulomas. Complete with a description of the biological background and biological question that requires the use of mathematics, this book is developed for graduate students and advanced undergraduate students with only basic knowledge of ordinary differential equations and partial differential equations; background in biology is not required. Students will gain knowledge on how to program with MATLAB without previous programming experience and how to use codes in order to test biological hypothesis."

Intended for students who have already completed a one-year course in elementary calculus, this rigorous two-part treatment advances from functions of one variable to those of several variables. Topics include differentiation, multiple integrals, and line and surface integrals. Complete solutions to all problems appear at the end. 1971 edition.

This self-contained treatment develops the theory of generalized functions and the theory of distributions, and it systematically applies them to solving a variety of problems in partial differential equations. A major portion of the text is based on material included in the books of L. Schwartz, who developed the theory of distributions, and in the books of Gelfand and Shilov, who deal with generalized functions of any class and their use in solving the Cauchy problem. In addition, the author provides applications developed through his own research. Geared toward upper-level undergraduates and graduate students, the text assumes a sound knowledge of both real and complex variables. Familiarity with the basic theory of functional analysis, especially normed spaces, is helpful but not necessary. An introductory chapter features helpful background on topological spaces. Applications to partial differential equations include a treatment of the Cauchy problem, the Goursat problem, fundamental solutions, existence and differentiality of solutions of equations with constants, coefficients, and related topics. Supplementary materials include end-of-chapter problems, bibliographical remarks, and a bibliography.