Functional integration successfully entered physics as path integrals in the 1942 PhD dissertation of Richard P. Feynman, but it made no sense at all as a mathematical definition. Cartier and DeWitt-Morette have created, in this book, a fresh approach to functional integration. The book is self-contained: mathematical ideas are introduced, developed, generalised and applied. In the authors' hands, functional integration is shown to be a robust, user-friendly and multi-purpose tool that can be applied to a great variety of situations, for example: systems of indistinguishable particles; Aharonov-Bohm systems; supersymmetry; non-gaussian integrals. Problems in quantum field theory are also considered. In the final part the authors outline topics that can be profitably pursued using material already presented.

Kinetic theory is the link between the non--equilibrium statistical mechanics of many particle systems and macroscopic or phenomenological physics. Therefore much attention is paid in this book both to the derivation of kinetic equations with their limitations and generalizations on the one hand, and to the use of kinetic theory for the description of physical phenomena and the calculation of transport coefficients on the other hand. The book is meant for researchers in the field, graduate students and advanced undergraduate students. At the end of each chapter a section of exercises is added not only for the purpose of providing the reader with the opportunity to test his understanding of the theory and his ability to apply it, but also to complete the chapter with relevant additions and examples that otherwise would have overburdened the main text of the preceding sections. The author is indebted to the physicists who taught him Statistical Mechanics, Kinetic Theory, Plasma Physics and Fluid Mechanics. I gratefully acknowledge the fact that much of the inspiration without which this book would not have been possible, originated from what I learned from several outstanding teachers. In particular I want to mention the late Prof. dr. H. C. Brinkman, who directed my first steps in the field of theoretical plasma physics, my thesis advisor Prof. dr. N. G. Van Kampen and Prof. dr. A. N. Kaufman, whose course on Non-Equilibrium Statistical Mechanics in Berkeley I remember with delight.

The description of emerging collective phenomena and self-organization in systems composed of large numbers of individuals has gained increasing interest from various research communities in biology, ecology, robotics and control theory, as well as sociology and economics. Applied mathematics is concerned with the construction, analysis and interpretation of mathematical models that can shed light on significant problems of the natural sciences as well as our daily lives. To this set of problems belongs the description of the collective behaviours of complex systems composed by a large enough number of individuals. Examples of such systems are interacting agents in a financial market, potential voters during political elections, or groups of animals with a tendency to flock or herd. Among other possible approaches, this book provides a step-by-step introduction to the mathematical modelling based on a mesoscopic description and the construction of efficient simulation algorithms by Monte Carlo methods. The arguments of the book cover various applications, from the analysis of wealth distributions, the formation of opinions and choices, the price dynamics in a financial market, to the description of cell mutations and the swarming of birds and fishes. By means of methods inspired by the kinetic theory of rarefied gases, a robust approach to mathematical modelling and numerical simulation of multi-agent systems is presented in detail. The content is a useful reference text for applied mathematicians, physicists, biologists and economists who want to learn about modelling and approximation of such challenging phenomena.

The contents of this book correspond to Sessions VII and VIII of the International Workshop on Instabilities and Nonequilibrium Structures which took place in Vifia del Mar, Chile, in December 1997 and December 1999, respectively. We were not able to publish this book before and we apologize for this fact to the authors and participants of the meeting. We have made an effort to actualize the courses and articles which have been reviewed by the authors. Both Workshops were organized by Facultad de Ciencias Fisicas y Matematicas, Universidad de Chile, Instituto de Fisica of Universidad Cat61ica de Valparaiso and Centro de Fisica No Lineal y Sistemas Complejos de Santiago. We are glad to acknowledge here the support of the Facultad de Ingenieria of Universidad de los Andes of Santiago which also be from now on one of the organizing Institutions of future Workshops. Enrique Tirapegui PREFACE This book is divided in two parts. In Part I we have collected the courses given in Sessions VII and VIII of the Workshop and in Part II we include a selection of the invited Conferences and Seminars presented at both meetings.

This book focuses mainly on fractional Brownian fields and their extensions. It has been used to teach graduate students at Grenoble and Toulouse's Universities. It is as self-contained as possible and contains numerous exercises, with solutions in an appendix. After a foreword by Stephane Jaffard, a long first chapter is devoted to classical results from stochastic fields and fractal analysis. A central notion throughout this book is self-similarity, which is dealt with in a second chapter with a particular emphasis on the celebrated Gaussian self-similar fields, called fractional Brownian fields after Mandelbrot and Van Ness's seminal paper. Fundamental properties of fractional Brownian fields are then stated and proved. The second central notion of this book is the so-called local asymptotic self-similarity (in short lass), which is a local version of self-similarity, defined in the third chapter. A lengthy study is devoted to lass fields with finite variance. Among these lass fields, we find both Gaussian fields and non-Gaussian fields, called Levy fields. The Levy fields can be viewed as bridges between fractional Brownian fields and stable self-similar fields. A further key issue concerns the identification of fractional parameters. This is the raison d'etre of the statistics chapter, where generalized quadratic variations methods are mainly used for estimating fractional parameters. Last but not least, the simulation is addressed in the last chapter. Unlike the previous issues, the simulation of fractional fields is still an area of ongoing research. The algorithms presented in this chapter are efficient but do not claim to close the debate.

One service mathematics has rendered the 'Et BIOi. .... si j'avait su comment en revenir. human race. It has put common sense back je n'y serais point aile.' Jules Verne where it belongs. on the topmost shelf next to the dusty canister labelled 'discarded non The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Math@matics is a tool for thought. A highly necessary tool in a world where both feedback and non Iinearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series."

Advances in the dynamics of stellar systems have been made recently by applying mathematical methods of ergodic theory and chaotic dynamics, by numerous computer simulations, and by observations with the most powerful telescopes. This has led to a considerable change of our view on stellar systems. These systems appear much more chaotic than was previously thought and subject to various instabilities leading to new paths of evolution than previously thought. The implications are fundamental for our views on the evolution of the galaxies and the universe. Such questions are addressed in this book, especially in the 8 review papers by leading experts on various aspects of the N-body problem, explaining at the graduate/postgraduate level the concepts, methods, techniques and results.

Critical phenomena is one of the most exciting areas of modern physics. This 2007 book provides a thorough but economic introduction into the principles and techniques of the theory of critical phenomena and the renormalization group, from the perspective of modern condensed matter physics. Assuming basic knowledge of quantum and statistical mechanics, the book discusses phase transitions in magnets, superfluids, superconductors, and gauge field theories. Particular attention is given to topics such as gauge field fluctuations in superconductors, the Kosterlitz-Thouless transition, duality transformations, and quantum phase transitions - all of which are at the forefront of physics research. This book contains numerous problems of varying degrees of difficulty, with solutions. These problems provide readers with a wealth of material to test their understanding of the subject. It is ideal for graduate students and more experienced researchers in the fields of condensed matter physics, statistical physics, and many-body physics.

The aim of this Book is to give an overview, based on the results of nearly three decades of intensive research, of transient chaos. One belief that motivates us to write this book is that, transient chaos may not have been appreciated even within the nonlinear-science community, let alone other scientific disciplines.

The field of neural information processing has two main objects: investigation into the functioning of biological neural networks and use of artificial neural networks to sol ve real world problems. Even before the reincarnation of the field of artificial neural networks in mid nineteen eighties, researchers have attempted to explore the engineering of human brain function. After the reincarnation, we have seen an emergence of a large number of neural network models and their successful applications to solve real world problems. This volume presents a collection of recent research and developments in the field of neural information processing. The book is organized in three Parts, i.e., (1) architectures, (2) learning algorithms, and (3) applications. Artificial neural networks consist of simple processing elements called neurons, which are connected by weights. The number of neurons and how they are connected to each other defines the architecture of a particular neural network. Part 1 of the book has nine chapters, demonstrating some of recent neural network architectures derived either to mimic aspects of human brain function or applied in some real world problems. Muresan provides a simple neural network model, based on spiking neurons that make use of shunting inhibition, which is capable of resisting small scale changes of stimulus. Hoshino and Zheng simulate a neural network of the auditory cortex to investigate neural basis for encoding and perception of vowel sounds.

Computational Architectures Integrating Neural and Symbolic Processes: A Perspective on the State of the Art focuses on a currently emerging body of research. With the reemergence of neural networks in the 1980s with their emphasis on overcoming some of the limitations of symbolic AI, there is clearly a need to support some form of high-level symbolic processing in connectionist networks. As argued by many researchers, on both the symbolic AI and connectionist sides, many cognitive tasks, e.g. language understanding and common sense reasoning, seem to require high-level symbolic capabilities. How these capabilities are realized in connectionist networks is a difficult question and it constitutes the focus of this book. Computational Architectures Integrating Neural and Symbolic Processes addresses the underlying architectural aspects of the integration of neural and symbolic processes. In order to provide a basis for a deeper understanding of existing divergent approaches and provide insight for further developments in this field, this book presents: (1) an examination of specific architectures (grouped together according to their approaches), their strengths and weaknesses, why they work, and what they predict, and (2) a critique/comparison of these approaches. Computational Architectures Integrating Neural and Symbolic Processes is of interest to researchers, graduate students, and interested laymen, in areas such as cognitive science, artificial intelligence, computer science, cognitive psychology, and neurocomputing, in keeping up-to-date with the newest research trends. It is a comprehensive, in-depth introduction to this new emerging field.

Mathematicians often face the question to which extent mathematical models describe processes of the real world. These models are derived from experimental data, hence they describe real phenomena only approximately. Thus a mathematical approach must begin with choosing properties which are not very sensitive to small changes in the model, and so may be viewed as properties of the real process. In particular, this concerns real processes which can be described by means of ordinary differential equations. By this reason different notions of stability played an important role in the qualitative theory of ordinary differential equations commonly known nowdays as the theory of dynamical systems. Since physical processes are usually affected by an enormous number of small external fluctuations whose resulting action would be natural to consider as random, the stability of dynamical systems with respect to random perturbations comes into the picture. There are differences between the study of stability properties of single trajectories, i. e. , the Lyapunov stability, and the global stability of dynamical systems. The stochastic Lyapunov stability was dealt with in Hasminskii [Has]. In this book we are concerned mainly with questions of global stability in the presence of noise which can be described as recovering parameters of dynamical systems from the study of their random perturbations. The parameters which is possible to obtain in this way can be considered as stable under random perturbations, and so having physical sense. -1- Our set up is the following.

This book can be looked upon in more ways than one. On the one hand, it describes strikingly interesting and lucid hydrodynamic experiments done in the style of the "good old days" when the physicist needed little more than a piece of string and some sealing wax. On the other hand, it demonstrates how a profound physical analogy can help to get a synoptic view on a broad range of nonlinear phenomena involving self-organization of vortical structures in planetary atmo spheres and oceans, in galaxies and in plasmas. In particular, this approach has elucidated the nature and the mechanism of such grand phenomena as the Great of galaxies. A number of our Red Spot vortex on Jupiter and the spiral arms predictions concerning the dynamics of spiral galaxies are now being confirmed by astronomical observations stimulated by our experiments. This book is based on the material most of which was accumulated during 1981-88 in close cooperation with our colleagues, experimenters from the Plasma Physics Department of the Kurchatov Atomic Energy Institute (S. V. Antipov, A. S. Trubnikov, AYu. Rylov, AV. Khutoretsky) and astrophysics theoreticians from the Astronomical Council of the USSR Academy of Sciences (AM. Frid man) and from the Volgograd State University (AG. Morozov). To all of them we wish to express our gratitude. Whenever we speak of "our experiments," the participation of the entire team is implied."

Modern astronomy has been characterized by an enormous growth in data acquisition - from new technologies in telescopes, detectors, and computation. One can now compile catalogs of tens or hundreds of millions of stars or galaxies and databases from satellite-based observations are reaching terabit proportions. This wealth of data gives rise to statistical challenges not previously encountered in astronomy. This book is the result of a workshop held at Pennsylvania State University in August 1991 that brought together leading astronomers and statisticians to consider statistical challenges encountered in modern astronomical research. The chapters have all been thoroughly revised in the light of the discussions at the conference, and some of the lively discussion is recorded here as well.

This classic text provides an excellent introduction to a new and rapidly developing field of research. Now well established as a textbook in this rapidly developing field of research, the new edition is much enlarged and covers a host of new results.

Nonlinear dynamics is now recognized as playing a crucial role in a wide variety of disciplines. But what is only just beginning is the important process of cross fertilization and transfer of knowledge and expertise from one area to another. This book is intended to promote this process which will undoubtedly contribute greatly to furthering our understanding of complex systems. Contributions are provided by leading experts from the areas of sociology, cognitive science, chemistry, physiology, ecology, economics, neural networks and physics.

Over recent years, a considerable amount of effort has been devoted, both in industry and academia, towards the performance modelling, evaluation and prediction of Asynchronous Transfer Mode (ATM) networks. This book describes recent advances in ATM networks reflecting the state-of-the-art technology and research achievements worldwide. In addition, it provides a fundamental source of reference in the ATM field. Research topics discussed in detail include: * Traffic Modelling and Characterisation; * Routing; * Switch and Multiplexer Models; * Call Admission Control (CAC); * Congestion Control; * Resource Allocation; * Quality of Service (QoS); * Tools and Techniques. This volume contains recently extended refereed papers of the 5th International Workshop on Performance Modelling and Evaluation of ATM Networks, which was sponsored by the International Federation for Information Processing (IFIP) and held in Ilkley, UK in July 1997.Performance Analysis of ATM Networks continues the tradition established by the first three IFIP volumes on the subject, and it is ideal for personnel in computer/communication industries as well as academic and research staff in computer science and electrical engineering.

Computational neuroscience is best defined by its focus on understanding the nervous systems as a computational device rather than by a particular experimental technique. Accordinlgy, while the majority of the papers in this book describe analysis and modeling efforts, other papers describe the results of new biological experiments explicitly placed in the context of computational issues. The distribution of subjects in Computation and Neural Systems reflects the current state of the field. In addition to the scientific results presented here, numerous papers also describe the ongoing technical developments that are critical for the continued growth of computational neuroscience. Computation and Neural Systems includes papers presented at the First Annual Computation and Neural Systems meeting held in San Francisco, CA, July 26--29, 1992.

The idea of devoting a complete book to this topic was born at one of the Workshops on Nonlinear and Turbulent Processes in Physics taking place reg ularly in Kiev. With the exception of E. D. Siggia and N. Ercolani, all authors of this volume were participants at the third of these workshops. All of them were acquainted with each other and with each other's work. Yet it seemed to be somewhat of a discovery that all of them were and are trying to understand the same problem - the problem of integrability of dynamical systems, primarily Hamiltonian ones with an infinite number of degrees of freedom. No doubt that they (or to be more exact, we) were led to this by the logical process of scientific evolution which often leads to independent, almost simultaneous discoveries. Integrable, or, more accurately, exactly solvable equations are essential to theoretical and mathematical physics. One could say that they constitute the "mathematical nucleus" of theoretical physics whose goal is to describe real clas sical or quantum systems. For example, the kinetic gas theory may be considered to be a theory of a system which is trivially integrable: the system of classical noninteracting particles. One of the main tasks of quantum electrodynamics is the development of a theory of an integrable perturbed quantum system, namely, noninteracting electromagnetic and electron-positron fields."

Thermodynamic methods of analysis have in recent years found ever-growing extensions in diverse regions of modern tech nology. The object of the present book is to apply these methods to the description of materials of varying physical properties. I hope the book will illustrate the wide variety and usefulness of thermodynamics which was well described by Albert Einstein: "A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended is its area of applicability. Therefore the deep impression which classical thermodynamics made upon me." The work of the American thermodynamic school is well known in the Soviet Union, and thus it is a great pleasure to offer this book to American readers. V. V. Sychev v Preface At the present time, when a number of new areas of tech nology are rapidly evolving, it is difficult to present a modern course in technical thermodynamics without developing such sub jects as the thermodynamics of insulators, magnets, and super conductors, or without treating the features of thermodynamic systems located in a gravitational field and in conditions of weight lessness, etc. In fact the limited coverage of technical thermody namics in the usual textbooks and school equipment as a rule prevents the authors from giving any detailed discussion of these important problems. I therefore resolved to treat these problems in a separate text. I discussed the concept of this book with my teachers V. A."

This book describes significant tractable models used in solid mechanics - classical models used in modern mechanics as well as new ones. The models are selected to illustrate the main ideas which allow scientists to describe complicated effects in a simple manner and to clarify basic notations of solid mechanics. A model is considered to be tractable if it is based on clear physical assumptions which allow the selection of significant effects and relatively simple mathematical formulations. The first part of the book briefly reviews classical tractable models for a simple description of complex effects developed from the 18th to the 20th century and widely used in modern mechanics. The second part describes systematically the new tractable models used today for the treatment of increasingly complex mechanical objects - from systems with two degrees of freedom to three-dimensional continuous objects.

Et mai . ..., si j'avait su comment en revenir. One service mathematics has rendered the human race. It has put common sense back je n'y serais point aIIe.' Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent: therefore we may be sense' . able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series."

Thermodynamicsandstatisticalphysicsstudythephysicalproperties(mec- nical, thermal, magnetic, optical, electrical, etc.) of the macroscopic system. The tasks and objects of study in thermodynamics and statistical physics are identical. However, the methods of investigationinto macroscopicsystems are di?erent. Thermodynamics is a phenomenological theory. It studies the properties of bodies, without going into the mechanism of phenomena, i.e., not taking into consideration the relation between the internal structure of substance and phenomena, it generalizes experimental results. As a result of such a g- eralization, postulates and laws of thermodynamics made their appearance. These laws make it possible to ?nd general relations between the di?erent properties of macroscopic systems and the physical events occurring in them. Statisticalphysicsisa microscopic theory.Onthebasisoftheknowledgeof the type of particles a system consists of, the nature of their interaction, and thelawsofmotionoftheseparticlesissuingfromtheconstructionofsubstance, it explains the properties being observedon experiment, and predicts the new properties of systems. Using the laws of classical or quantum mechanics, and alsothe theoryofprobability, itestablishesqualitativelynewstatistical app- priatenesses of the physical properties of macroscopic systems, substantiates the laws of thermodynamics, determines the limits of their applicability, gives the statistical interpretation of thermodynamic parameters, and also works out methods of calculations of their means. The Gibbs method is based on statisticalphysics.Thismethodis themostcanonical.Therefore, inthis book, the exposition of the Gibbs method takes an important pla

Neural Networks and Fuzzy Systems: Theory and Applications discusses theories that have proven useful in applying neural networks and fuzzy systems to real world problems. The book includes performance comparison of neural networks and fuzzy systems using data gathered from real systems. Topics covered include the Hopfield network for combinatorial optimization problems, multilayered neural networks for pattern classification and function approximation, fuzzy systems that have the same functions as multilayered networks, and composite systems that have been successfully applied to real world problems. The author also includes representative neural network models such as the Kohonen network and radial basis function network. New fuzzy systems with learning capabilities are also covered. The advantages and disadvantages of neural networks and fuzzy systems are examined. The performance of these two systems in license plate recognition, a water purification plant, blood cell classification, and other real world problems is compared.

Arturo Carsetti According to molecular Biology, true invariance (life) can exist only within the framework of ongoing autonomous morphogenesis and vice versa. With respect to this secret dialectics, life and cognition appear as indissolubly interlinked. In this sense, for instance, the inner articulation of conceptual spaces appears to be linked to an inner functional development based on a continuous activity of selection and "anchorage" realised on semantic grounds. It is the work of "invention" and g- eration (in invariance), linked with the "rooting" of meaning, which determines the evolution, the leaps and punctuated equilibria, the conditions related to the unfo- ing of new modalities of invariance, an invariance which is never simple repetition and which springs on each occasion through deep-level processes of renewal and recovery. The selection perpetrated by meaning reveals its autonomy aboveall in its underpinning, in an objective way, the ongoing choice of these new modalities. As such it is not, then, concerned only with the game of "possibles," offering itself as a simple channel for pure chance, but with providing a channel for the articulation of the " le" in the humus of a semantic (and embodied) net in order to prepare the necessary conditionsfor a continuousrenewal and recoveryof original creativity. In effect, it is this autonomy in inventing new possible modules of incompressibility whichdeterminestheactualemergenceofnew(andtrue)creativity, whichalsotakes place through the "narration" of the effected construction.