Combining control theory and modeling, this textbook introduces and builds on methods for simulating and tackling concrete problems in a variety of applied sciences. Emphasizing "learning by doing," the authors focus on examples and applications to real-world problems. An elementary presentation of advanced concepts, proofs to introduce new ideas, and carefully presented MATLAB(r) programs help foster an understanding of the basics, but also lead the way to new, independent research.

With minimal prerequisites and exercises in each chapter, this work serves as an excellent textbook and referencefor graduate and advanced undergraduatestudents, researchers, and practitioners in mathematics, physics, engineering, computer science, as well as biology, biotechnology, economics, and finance."

Pattern Formation in Morphogenesis is a rich source of interesting and challenging mathematical problems. The volume aims at showing how a combination of new discoveries in developmental biology and associated modelling and computational techniques has stimulated or may stimulate relevant advances in the field. Finally it aims at facilitating the process of unfolding a mutual recognition between Biologists and Mathematicians of their complementary skills, to the point where the resulting synergy generates new and novel discoveries. It offers an interdisciplinary interaction space between biologists from embryology, genetics and molecular biology who present their own work in the perspective of the advancement of their specific fields, and mathematicians who propose solutions based on the knowledge grasped from biologists.

The polymer industry raises a large number of relevant mathematical problems with respect to the quality of manufactured polymer parts. These include in particular questions about: - the production of the polymeric material from a monomer (based on the Ziegler-Natta catalytic process) - the crystallization kinetic of the polymer melt - the coupling of the crystallization process with the fluid dynamics of the manufacturing process such as extrusion, injection moulding of film blowing, etc.This book provides the first unified presentation of the mathematical modelling of polymerization, crystallization and extrusion of polymer melts, by means of advanced methods, presented in an accessible way for applied scientists and engineers. The present volume is the result of a long-term cooperation between different research teams in Europe within the ECMI Special Interest Group on "Polymers".

The Summer School has been dedicated to one of the proponents and ?rst Chairman of the Strategy Board of MACSI-net, the late Jacques Louis Lions (see the dedication by Roland Glowinski). MACSI-net is a European Network of Excellence, where both enterprises and university institutions co-operate to solve challenging problems to their mutual bene?t. In particular the network focuses on strategies to enhance interactions between industry and academia. The aim is to help industry (in particular SMEs) alert academia about industrial needs in terms of advanced mathematical and computational methods and tools. The network is mul- disciplinary oriented, combining the power of applied mathematics, scienti?c computing and engineering, for modeling and simulation. It was set up by a joint e?ort of ECCOMAS and ECMI European associations. Thisparticularevent,occurredduringMarch17-22,2003,wasajointe?ort ofthe TrainingCommittee (chairedby VC)andIndustrialRelationsComm- tee (chairedby JP)to alert both Academia and Industry about the increasing role of Multidisciplinary Methods and Tools for the design of complex pr- uctsinvariousareasofindustrialinterest.Thisincreasingcomplexityisdriven by societal constraints to be satis? ed in a simultaneous and a?ordable way. The mastering of complexity implies the sharing of di?erent tools by di?erent actors which require much higher level of communication between culturally di?erent people. The school o?ered to young researchers the opportunity to be exposed to the presentation of real industrial and societal problems and the relevant innovative methods used; the need of further contributions from mathematics to improve or provide better solutions had also been considered.

This textbook, now in its fourth edition, offers a rigorous and self-contained introduction to the theory of continuous-time stochastic processes, stochastic integrals, and stochastic differential equations. Expertly balancing theory and applications, it features concrete examples of modeling real-world problems from biology, medicine, finance, and insurance using stochastic methods. No previous knowledge of stochastic processes is required. Unlike other books on stochastic methods that specialize in a specific field of applications, this volume examines the ways in which similar stochastic methods can be applied across different fields. Beginning with the fundamentals of probability, the authors go on to introduce the theory of stochastic processes, the Ito Integral, and stochastic differential equations. The following chapters then explore stability, stationarity, and ergodicity. The second half of the book is dedicated to applications to a variety of fields, including finance, biology, and medicine. Some highlights of this fourth edition include a more rigorous introduction to Gaussian white noise, additional material on the stability of stochastic semigroups used in models of population dynamics and epidemic systems, and the expansion of methods of analysis of one-dimensional stochastic differential equations. An Introduction to Continuous-Time Stochastic Processes, Fourth Edition is intended for graduate students taking an introductory course on stochastic processes, applied probability, stochastic calculus, mathematical finance, or mathematical biology. Prerequisites include knowledge of calculus and some analysis; exposure to probability would be helpful but not required since the necessary fundamentals of measure and integration are provided. Researchers and practitioners in mathematical finance, biomathematics, biotechnology, and engineering will also find this volume to be of interest, particularly the applications explored in the second half of the book.

Pattern Formation in Morphogenesis is a rich source of interesting and challenging mathematical problems. The volume aims at showing how a combination of new discoveries in developmental biology and associated modelling and computational techniques has stimulated or may stimulate relevant advances in the field. Finally it aims at facilitating the process of unfolding a mutual recognition between Biologists and Mathematicians of their complementary skills, to the point where the resulting synergy generates new and novel discoveries. It offers an interdisciplinary interaction space between biologists from embryology, genetics and molecular biology who present their own work in the perspective of the advancement of their specific fields, and mathematicians who propose solutions based on the knowledge grasped from biologists.

Polymers are substances made of macromolecules formed by thousands of atoms organized in one (homopolymers) or more (copolymers) groups that repeat themselves to form linear or branched chains, or lattice structures. The concept of polymer traces back to the years 1920's and is one of the most significant ideas of last century. It has given great impulse to indus try but also to fundamental research, including life sciences. Macromolecules are made of sm all molecules known as monomers. The process that brings monomers into polymers is known as polymerization. A fundamental contri bution to the industrial production of polymers, particularly polypropylene and polyethylene, is due to the Nobel prize winners Giulio Natta and Karl Ziegler. The ideas of Ziegler and Natta date back to 1954, and the process has been improved continuously over the years, particularly concerning the design and shaping of the catalysts. Chapter 1 (due to A. Fasano ) is devoted to a review of some results concerning the modelling of the Ziegler- Natta polymerization. The specific ex am pie is the production of polypropilene. The process is extremely complex and all studies with relevant mathematical contents are fairly recent, and several problems are still open.

Realizing the need of interaction between universities and research groups in industry, the European Consortium for Mathematics in Industry (ECMI) was founded in 1986 by mathematicians from ten European universities. Since then it has been continuously extending and now it involves about all Euro pean countries. The aims of ECMI are * To promote the use of mathematical models in industry. * To educate industrial mathematicians to meet the growing demand for such experts. * To operate on a European Scale. Mathematics, as the language of the sciences, has always played an im portant role in technology, and now is applied also to a variety of problems in commerce and the environment. European industry is increasingly becoming dependent on high technology and the need for mathematical expertise in both research and development can only grow. These new demands on mathematics have stimulated academic interest in Industrial Mathematics and many mathematical groups world-wide are committed to interaction with industry as part of their research activities. ECMI was founded with the intention of offering its collective knowledge and expertise to European Industry. The experience of ECMI members is that similar technical problems are encountered by different companies in different countries. It is also true that the same mathematical expertise may often be used in differing industrial applications.

The Summer School has been dedicated to one of the proponents and ?rst Chairman of the Strategy Board of MACSI-net, the late Jacques Louis Lions (see the dedication by Roland Glowinski). MACSI-net is a European Network of Excellence, where both enterprises and university institutions co-operate to solve challenging problems to their mutual bene?t. In particular the network focuses on strategies to enhance interactions between industry and academia. The aim is to help industry (in particular SMEs) alert academia about industrial needs in terms of advanced mathematical and computational methods and tools. The network is mul- disciplinary oriented, combining the power of applied mathematics, scienti?c computing and engineering, for modeling and simulation. It was set up by a joint e?ort of ECCOMAS and ECMI European associations. Thisparticularevent,occurredduringMarch17-22,2003,wasajointe?ort ofthe TrainingCommittee (chairedby VC)andIndustrialRelationsComm- tee (chairedby JP)to alert both Academia and Industry about the increasing role of Multidisciplinary Methods and Tools for the design of complex pr- uctsinvariousareasofindustrialinterest.Thisincreasingcomplexityisdriven by societal constraints to be satis? ed in a simultaneous and a?ordable way. The mastering of complexity implies the sharing of di?erent tools by di?erent actors which require much higher level of communication between culturally di?erent people. The school o?ered to young researchers the opportunity to be exposed to the presentation of real industrial and societal problems and the relevant innovative methods used; the need of further contributions from mathematics to improve or provide better solutions had also been considered.

The theory of stochastic processes indexed by a partially ordered set has been the subject of much research over the past twenty years. The objective of this CIME International Summer School was to bring to a large audience of young probabilists the general theory of spatial processes, including the theory of set-indexed martingales and to present the different branches of applications of this theory, including stochastic geometry, spatial statistics, empirical processes, spatial estimators and survival analysis. This theory has a broad variety of applications in environmental sciences, social sciences, structure of material and image analysis. In this volume, the reader will find different approaches which foster the development of tools to modelling the spatial aspects of stochastic problems.

The 1990 CIME course on Mathematical Modelling of Industrial Processes set out to illustrate some advances in questions of industrial mathematics, i.e.of the applications of mathematics (with all its "academic" rigour) to real-life problems. The papers describe the genesis of the models and illustrate their relevant mathematical characteristics. Among the themesdealt with are: thermally controlled crystal growth, thermal behaviour of a high-pressure gas-discharge lamp, the sessile-drop problem, etching processes, the batch-coil- annealing process, inverse problems in classical dynamics, image representation and dynamical systems, scintillation in rear projections screens, identification of semiconductor properties, pattern recognition with neural networks. CONTENTS: H.K. Kuiken: Mathematical Modelling of Industrial Processes.- B. Forte: Inverse Problems in Mathematics for Industry.- S. Busenberg: Case Studies in Industrial Mathematics.

These Proceedings contain a large part of the papers presented at the International Conference "Mathematics in Biology and Medicine" organized by the Dipartimento di Matematica and the Istituto di Igiene at the University of Bari, Italy, July 18-22, 1983. The main objective of the Conference was to bring together scientists in pure and applied mathematics and scientists in biology and medicine. The purpose was to exchange ideas and discuss the common problems encoun~red in the formulation, ana- lysis and numerical treatment of mathematical models in the biomedical sciences. Si- mulation methods and problems of validation of models vs. experimental data were al- so treated. SUrveys on recent mathematical results motivated by biological questioffi were given. Altogether,we think that a balance between mathematical and biomedical aspects was maintained. The Scientific Committee consisted of E. Biondi (Milano), P. Colli-Franzone (Udine), I. Galligani (Bologna), M. Iannelli (Trento), G. Koch (Roma), E. Marubini (Milano), C. Matessi (Pavia), M. Primicerio (Firenze), L.M. Ricciardi (Napoli), R. Rinaldi (Milano), A. Zampieri (Roma), and the European Liaison Committee consisted of R.M. Anderson (United Kingdom), N.T.J. Bailey (Switzerland), O. Diekmann (The Netherlands), K. Dietz (F. R. Germany), K.P. Hadeler (F* .*. R. Germany), J.P. Kernevez (France).

The aim of this volume that presents lectures given at a joint CIME and Banach Center Summer School, is to offer a broad presentation of a class of updated methods providing a mathematical framework for the development of a hierarchy of models of complex systems in the natural sciences, with a special attention to biology and medicine. Mastering complexity implies sharing different tools requiring much higher level of communication between different mathematical and scientific schools, for solving classes of problems of the same nature. Today more than ever, one of the most important challenges derives from the need to bridge parts of a system evolving at different time and space scales, especially with respect to computational affordability. As a result the content has a rather general character; the main role is played by stochastic processes, positive semigroups, asymptotic analysis, kinetic theory, continuum theory, and game theory.

The dynamics of infectious diseases represents one of the oldest and ri- est areas of mathematical biology. From the classical work of Hamer (1906) and Ross (1911) to the spate of more modern developments associated with Anderson and May, Dietz, Hethcote, Castillo-Chavez and others, the subject has grown dramatically both in volume and in importance. Given the pace of development, the subject has become more and more di?use, and the need to provide a framework for organizing the diversity of mathematical approaches has become clear. Enzo Capasso, who has been a major contributor to the mathematical theory, has done that in the present volume, providing a system for organizing and analyzing a wide range of models, depending on the str- ture of the interaction matrix. The ?rst class, the quasi-monotone or positive feedback systems, can be analyzed e?ectively through the use of comparison theorems, that is the theory of order-preserving dynamical systems; the s- ond, the skew-symmetrizable systems, rely on Lyapunov methods. Capasso develops the general mathematical theory, and considers a broad range of - amples that can be treated within one or the other framework. In so doing, he has provided the ?rst steps towards the uni?cation of the subject, and made an invaluable contribution to the Lecture Notes in Biomathematics. Simon A. Levin Princeton, January 1993 Author's Preface to Second Printing In the Preface to the First Printing of this volume I wrote: \ . .