Leading researchers in the area of the origin and evolution of life in the universe contributed to Chemical Evolution: Physics of the Origin and Evolution of Life. This volume provides a review of this interdisciplinary field. In 35 chapters many aspects of the origin of life are discussed by 90 authors, with particular emphasis on the early paleontological record: physical, chemical, biological, and informational aspects of life's origin, instrumentation in exobiology and system exploration; the search for habitable planets and extraterrestrial intelligent radio signals. This book contains the proceedings of the Fourth Trieste Conference on Chemical Evolution that took place in September 1995, in which scientists from a wide geographical distribution joined in a Memorial to Cyril Ponnamperuma, who was a pioneer in the field of chemical evolution, the origin of life, and exobiology, and also initiated the Trieste Conferences on Chemical Evolution and the Origin of Life. This fourth Conference was therefore dedicated to his memory. Audience: Graduate students and researchers in the many areas of basic, earth, and life sciences that contribute to the study of chemical evolution and the origin of life.

Forty years ago, in 1957, the Principle of Maximum Entropy was first intro duced by Jaynes into the field of statistical mechanics. Since that seminal publication, this principle has been adopted in many areas of science and technology beyond its initial application. It is now found in spectral analysis, image restoration and a number of branches ofmathematics and physics, and has become better known as the Maximum Entropy Method (MEM). Today MEM is a powerful means to deal with ill-posed problems, and much research work is devoted to it. My own research in the area ofMEM started in 1980, when I was a grad uate student in the Department of Electrical Engineering at the University of Sydney, Australia. This research work was the basis of my Ph.D. the sis, The Maximum Entropy Method and Its Application in Radio Astronomy, completed in 1985. As well as continuing my research in MEM after graduation, I taught a course of the same name at the Graduate School, Chinese Academy of Sciences, Beijingfrom 1987to 1990. Delivering the course was theimpetus for developing a structured approach to the understanding of MEM and writing hundreds of pages of lecture notes."

Simple random walks - or equivalently, sums of independent random vari ables - have long been a standard topic of probability theory and mathemat ical physics. In the 1950s, non-Markovian random-walk models, such as the self-avoiding walk, were introduced into theoretical polymer physics, and gradu ally came to serve as a paradigm for the general theory of critical phenomena. In the past decade, random-walk expansions have evolved into an important tool for the rigorous analysis of critical phenomena in classical spin systems and of the continuum limit in quantum field theory. Among the results obtained by random-walk methods are the proof of triviality of the cp4 quantum field theo ryin space-time dimension d (:::: ) 4, and the proof of mean-field critical behavior for cp4 and Ising models in space dimension d (:::: ) 4. The principal goal of the present monograph is to present a detailed review of these developments. It is supplemented by a brief excursion to the theory of random surfaces and various applications thereof. This book has grown out of research carried out by the authors mainly from 1982 until the middle of 1985. Our original intention was to write a research paper. However, the writing of such a paper turned out to be a very slow process, partly because of our geographical separation, partly because each of us was involved in other projects that may have appeared more urgent.

A modern introduction to methods of statistical mechanics in turbulence, this volume explains the methodology of non-equilibrium statistical mechanics and how it plays an increasingly important role in modern turbulence research. The range of relevant tools and methods is so wide and developing so fast, that until now there has not been a single book covering the subject. This much-needed book is comprised of three harmonized lecture courses by world class experts in statistical physics and turbulence: John Cardy introduces Field Theory and Non-Equilibrium Statistical Mechanics; Gregory Falkovich discusses Turbulence Theory as part of Statistical Physics; and Krzysztof Gawedzki examines Soluble Models of Turbulent Transport. To encourage readers to deepen their understanding of the theoretical material, each chapter contains exercises with solutions. Essential reading for students and researchers in the field of theoretical turbulence, this volume will also interest any scientist or engineer who applies knowledge of turbulence and non-equilibrium physics to their work.

It is increasingly being recognized that the experimental and theoretical study of the complex system brain requires the cooperation of many disciplines, in cluding biology, medicine, physics, chemistry, mathematics, computer science, linguistics, and others. In this way brain research has become a truly interdis ciplinary endeavor. Indeed, the most important progress is quite often made when different disciplines cooperate. Thus it becomes necessary for scientists to look across the fence surrounding their disciplines. The present book is written precisely in this spirit. It addresses graduate students, professors and scientists in a variety of fields, such as biology, medicine and physics. Be yond its mathematical representation the book gives ample space to verbal and pictorial descriptions of the main and, as I believe, fundamental new insights, so that it will be of interest to a general readership, too. I use this opportunity to thank my former students, some of whom are my present co-workers, for their cooperation over many years. Among them I wish to mention in particular M. Bestehorn, L. Borland, H. Bunz, A. Daf fertshofer, T. Ditzinger, E. Fischer, A. Fuchs, R. Haas, R. Honlinger, V. Jirsa, M. Neufeld, M. Ossig, D. Reimann, M. Schanz, G. Schoner, P. Tass, C. Uhl. My particular thanks go to R. Friedrich and A. Wunderlin for their constant help in many respects. Stimulating discussions with a number of colleagues from a variety of fields are also highly appreciated.

We present here the lectures and a selection of the seminars given at the Ninth International Workshop on Instabilities and Nonequilibrium Structures which took place in Vifiadel Mar, Chile, in December 2001. The Workshop was organized by Facultad de Ciencias Fisicas y Matematicas, Universidad de Chile, Instituto de Fisica of Universidad Cat6lica de Valparaiso, Centro de Fisica No Lineal y Sistemas Complejos de Santiago and Facultad de Ingenieria, Universidad de los Andes, which starting from this year joins the other institutions in the coorganization ofthe Workshop. The organizers would like to express their gratitude to the following sponsors: Facultad de Ciencias Fisicas y Matematicas de la Universidad de Chile, Instituto de Fisica de la Universidad Cat6lica de Valparaiso, Facultad de Ingenieria de la Universidad de los Andes, Centro de Fisica No Lineal y Sistemas Complejos de Santiago, Academia Chilena de Ciencias, Ministere Francais des Affaires Etrangeres, CONICYT (Comisi6n Nacional de Investigaci6n Cientifica y Tecno16gicade Chile) and Departamento Tecnico de Investigaci6n y de Relaciones Internacionales de la Universidad de Chile. Enrique Tirapegui PREFACE This book consists of two parts, the first one has three lectures written by Professors H. R. Brand, M. Moreau and L. S. Tuckerman. H. R. Brand gives an overview about reorientation and undulation instabilities in liquid crystals, M. Moreau presents recent results on biased tracer diffusion in lattice gases, finally, L. S. Tuckerman summarizes some numerical methods used in bifurcation problems.

The thread of self-organization which is now recognized as permeating many dynamical transformations in diverse systems around us seems set to unleash a revolution as influential as that of Darwin in the last century. Darwin removed the 'originator' of a species; self-organization now seeks to remove the 'organizer' from an organism. Methods of nonlinear dynamics have played a crucial role in opening up this field and if these methods have a progenitor it is Henri Poi~care (1854 - 1912) whose first substantial compilation amongst his prolific productio~ was Les Methodes Nouvelles de la Mecanique Celeste, Vol 1. Thi~ work'aplleared in Paris in 1892, a century ago, thus offering us the opportunity to celebrate a centenary of this extraordinary revolutionary of science: M~:of. ~he ex;tensive contributions of Poincare helped to lay the foundation of this field of nonlinear dynamics, for _e. xam*lI~e mai:hematic~l theory of . dillJ. ensions~ . ( qualita. tively) global aspects of ph~ ~~c~ d~~~ics, topological analysis, . fiXed. pbiilt theo- rems, bifurcation COIiC~l?ts :{~o ?e u~d later for example for Andronov-Poincare bifurcation), diffe~!:. Iice' equation mappings in phase spa

This volume of research papers comprises the proceedings of the first International Conference on Mathematics of Neural Networks and Applications (MANNA), which was held at Lady Margaret Hall, Oxford from July 3rd to 7th, 1995 and attended by 116 people. The meeting was strongly supported and, in addition to a stimulating academic programme, it featured a delightful venue, excellent food and accommo dation, a full social programme and fine weather - all of which made for a very enjoyable week. This was the first meeting with this title and it was run under the auspices of the Universities of Huddersfield and Brighton, with sponsorship from the US Air Force (European Office of Aerospace Research and Development) and the London Math ematical Society. This enabled a very interesting and wide-ranging conference pro gramme to be offered. We sincerely thank all these organisations, USAF-EOARD, LMS, and Universities of Huddersfield and Brighton for their invaluable support. The conference organisers were John Mason (Huddersfield) and Steve Ellacott (Brighton), supported by a programme committee consisting of Nigel Allinson (UMIST), Norman Biggs (London School of Economics), Chris Bishop (Aston), David Lowe (Aston), Patrick Parks (Oxford), John Taylor (King's College, Lon don) and Kevin Warwick (Reading). The local organiser from Huddersfield was Ros Hawkins, who took responsibility for much of the administration with great efficiency and energy. The Lady Margaret Hall organisation was led by their bursar, Jeanette Griffiths, who ensured that the week was very smoothly run."

Nonlinear Modeling: Advanced Black-Box Techniques discusses methods on Neural nets and related model structures for nonlinear system identification; Enhanced multi-stream Kalman filter training for recurrent networks; The support vector method of function estimation; Parametric density estimation for the classification of acoustic feature vectors in speech recognition; Wavelet-based modeling of nonlinear systems; Nonlinear identification based on fuzzy models; Statistical learning in control and matrix theory; Nonlinear time-series analysis. It also contains the results of the K.U. Leuven time series prediction competition, held within the framework of an international workshop at the K.U. Leuven, Belgium in July 1998.

Modern physics rests on two fundamental building blocks: general relativity and quantum theory. General relativity is a geometric interpretation of gravity while quantum theory governs the microscopic behaviour of matter. Since matter is described by quantum theory which in turn couples to geometry, we need a quantum theory of gravity. In order to construct quantum gravity one must reformulate quantum theory on a background independent way. Modern Canonical Quantum General Relativity provides a complete treatise of the canonical quantisation of general relativity. The focus is on detailing the conceptual and mathematical framework, on describing physical applications and on summarising the status of this programme in its most popular incarnation, called loop quantum gravity. Mathematical concepts and their relevance to physics are provided within this book, which therefore can be read by graduate students with basic knowledge of quantum field theory or general relativity.

`Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications' builds from basic principles to advanced techniques, and covers the major phenomena, methods, and results of time-dependent systems. It is a pedagogic introduction, a comprehensive reference manual, and an original research monograph. Uniquely, the book treats time-dependent systems by close analogy with their static counterparts, with most of the familiar results of equilibrium thermodynamics and statistical mechanics being generalized and applied to the non-equilibrium case. The book is notable for its unified treatment of thermodynamics, hydrodynamics, stochastic processes, and statistical mechanics, for its self-contained, coherent derivation of a variety of non-equilibrium theorems, and for its quantitative tests against experimental measurements and computer simulations. Systems that evolve in time are more common than static systems, and yet until recently they lacked any over-arching theory. 'Non-equilibrium Thermodynamics and Statistical Mechanics' is unique in its unified presentation of the theory of non-equilibrium systems, which has now reached the stage of quantitative experimental and computational verification. The novel perspective and deep understanding that this book brings offers the opportunity for new direction and growth in the study of time-dependent phenomena. 'Non-equilibrium Thermodynamics and Statistical Mechanics' is an invaluable reference manual for experts already working in the field. Research scientists from different disciplines will find the overview of time-dependent systems stimulating and thought-provoking. Lecturers in physics and chemistry will be excited by many fresh ideas and topics, insightful explanations, and new approaches. Graduate students will benefit from its lucid reasoning and its coherent approach, as well as from the chem12physof mathematical techniques, derivations, and computer algorithms.

In the past decade, a number of different research communities within the computational sciences have studied learning in networks, starting from a number of different points of view. There has been substantial progress in these different communities and surprising convergence has developed between the formalisms. The awareness of this convergence and the growing interest of researchers in understanding the essential unity of the subject underlies the current volume. Two research communities which have used graphical or network formalisms to particular advantage are the belief network community and the neural network community. Belief networks arose within computer science and statistics and were developed with an emphasis on prior knowledge and exact probabilistic calculations. Neural networks arose within electrical engineering, physics and neuroscience and have emphasised pattern recognition and systems modelling problems. This volume draws together researchers from these two communities and presents both kinds of networks as instances of a general unified graphical formalism. The book focuses on probabilistic methods for learning and inference in graphical models, algorithm analysis and design, theory and applications. Exact methods, sampling methods and variational methods are discussed in detail. Audience: A wide cross-section of computationally oriented researchers, including computer scientists, statisticians, electrical engineers, physicists and neuroscientists.

The availability of large data sets has allowed researchers to uncover complex properties such as large-scale fluctuations and heterogeneities in many networks, leading to the breakdown of standard theoretical frameworks and models. Until recently these systems were considered as haphazard sets of points and connections. Recent advances have generated a vigorous research effort in understanding the effect of complex connectivity patterns on dynamical phenomena. This book presents a comprehensive account of these effects. A vast number of systems, from the brain to ecosystems, power grids and the internet, can be represented as large complex networks. This book will interest graduate students and researchers in many disciplines, from physics and statistical mechanics to mathematical biology and information science. Its modular approach allows readers to readily access the sections of most interest to them, and complicated maths is avoided so the text can be easily followed by non-experts in the subject.

We have classified the articles presented here in two Sections according to their general content. In Part I we have included papers which deal with statistical mechanics, math ematical aspects of dynamical systems and sthochastic effects in nonequilibrium systems. Part II is devoted mainly to instabilities and self-organization in extended nonequilibrium systems. The study of partial differential equations by numerical and analytic methods plays a great role here and many works are related to this subject. Most recent developments in this fascinating and rapidly growing area are discussed. PART I STATISTICAL MECHANICS AND RELATED TOPICS NONEQUILIBRIUM POTENTIALS FOR PERIOD DOUBLING R. Graham and A. Hamm Fachbereich Physik, Universitiit Gesamthochschule Essen D4300 Essen 1 Germany ABSTRACT. In this lecture we consider the influence of weak stochastic perturbations on period doubling using nonequilibrium potentials, a concept which is explained in section 1 and formulated for the case of maps in section 2. In section 3 nonequilibrium potentials are considered for the family of quadratic maps (a) at the Feigenbaum 'attractor' with Gaussian noise, (b) for more general non Gaussian noise, and (c) for the case of a strange repeller. Our discussion will be informal. A more detailed account of this and related material can be found in our papers [1-3] and in the reviews [4, 5], where further references to related work are also given. 1.

In recent years there has been tremendous activity in computational neuroscience resulting from two parallel developments. On the one hand, our knowledge of real nervous systems has increased dramatically over the years; on the other, there is now enough computing power available to perform realistic simulations of actual neural circuits. This is leading to a revolution in quantitative neuroscience, which is attracting a growing number of scientists from non-biological disciplines. These scientists bring with them expertise in signal processing, information theory, and dynamical systems theory that has helped transform our ways of approaching neural systems. New developments in experimental techniques have enabled biologists to gather the data necessary to test these new theories. While we do not yet understand how the brain sees, hears or smells, we do have testable models of specific components of visual, auditory, and olfactory processing. Some of these models have been applied to help construct artificial vision and hearing systems. Similarly, our understanding of motor control has grown to the point where it has become a useful guide in the development of artificial robots. Many neuroscientists believe that we have only scratched the surface, and that a more complete understanding of biological information processing is likely to lead to technologies whose impact will propel another industrial revolution. Neural Systems: Analysis and Modeling contains the collected papers of the 1991 Conference on Analysis and Modeling of Neural Systems (AMNS), and the papers presented at the satellite symposium on compartmental modeling, held July 23-26, 1992, in San Francisco, California. The papers included, present an update of the most recent developments in quantitative analysis and modeling techniques for the study of neural systems.

This volume contains the proceedings of a NATO Advanced study Institute held at Geilo, Norway between 2 - 12 april 1991. This institute was the eleventh in a series held biannually at Geilo on the subject of phase transitions. It was intended to capture the latest ideas on selforgan ized patterns and criticality. The Institute brought together many lecturers, students and active re searchers in the field from a wide range of NATO and non-NATO countries. The main financial support came from the NATO scientific Affairs Divi sion, but additional support was obtained from the Norwegian Research Council for Science and the Humanities (NAVF) and Institutt for energi teknikk. The organizers would like to thank all these contributors for their help in promoting an exciting and rewarding meeting, and in doing so are confident that they echo the appreciation of all the parti cipants. In cooperative, equilibrium systems, physical states are described by spatio-temporal correlation functions. The intimate connection between space and time correlations is especially apparent at the critical point, the second order phase transition, where the spatial range and the decay time of the correlation function both become infinite. The salient features of critical phenomena and the history of the devel opment of this field of science are treated in the first chapter of this book.

The motion of a particle in a random potential in two or more dimensions is chaotic, and the trajectories in deterministically chaotic systems are effectively random. It is therefore no surprise that there are links between the quantum properties of disordered systems and those of simple chaotic systems. The question is, how deep do the connec tions go? And to what extent do the mathematical techniques designed to understand one problem lead to new insights into the other? The canonical problem in the theory of disordered mesoscopic systems is that of a particle moving in a random array of scatterers. The aim is to calculate the statistical properties of, for example, the quantum energy levels, wavefunctions, and conductance fluctuations by averaging over different arrays; that is, by averaging over an ensemble of different realizations of the random potential. In some regimes, corresponding to energy scales that are large compared to the mean level spacing, this can be done using diagrammatic perturbation theory. In others, where the discreteness of the quantum spectrum becomes important, such an approach fails. A more powerful method, devel oped by Efetov, involves representing correlation functions in terms of a supersymmetric nonlinear sigma-model. This applies over a wider range of energy scales, covering both the perturbative and non-perturbative regimes. It was proved using this method that energy level correlations in disordered systems coincide with those of random matrix theory when the dimensionless conductance tends to infinity.

arise automatically as a result of the recursive structure of the task and the continuous nature of the SRN's state space. Elman also introduces a new graphical technique for study ing network behavior based on principal components analysis. He shows that sentences with multiple levels of embedding produce state space trajectories with an intriguing self similar structure. The development and shape of a recurrent network's state space is the subject of Pollack's paper, the most provocative in this collection. Pollack looks more closely at a connectionist network as a continuous dynamical system. He describes a new type of machine learning phenomenon: induction by phase transition. He then shows that under certain conditions, the state space created by these machines can have a fractal or chaotic structure, with a potentially infinite number of states. This is graphically illustrated using a higher-order recurrent network trained to recognize various regular languages over binary strings. Finally, Pollack suggests that it might be possible to exploit the fractal dynamics of these systems to achieve a generative capacity beyond that of finite-state machines."

analyzing the experimental data and constructing math.ematical models of the processes under study, one has to rely rather on the physical intuition than on the strict calculations. Now let us go one step higher and explain the main title of the book. The concepts of "laminary" and "turbulent" motions were first introduced in hydrodynamics. Since the old days these concepts have considerably broadened; now the laminar and the turbulent motions have been discovered and investigated at all levels of description of nonequilibrium processes in the open systems, from kinetics to reaction diffusion. In any case, one of the principal characteristics of the turbulent motion is the existence of a large number of well-developed macroscopic degrees of freedom. For this reason the turbulent motion is extremely complicated and to a large extent unpredictable. As the laminar and the turbulent flows play an important role in the processes of evolution in the open systems, and in particular, in the processes of self-organization, the need arises for assessing the relative degree of order of laminar and turbulent motions, and also for comparing the degree of order of various turbulent motions. Without being able to make such estimates it will be impossible to determine whether the evolution is going towards higher or towards lower organization when one turbulent state is replaced by another.

Over the past five de-:: ades researchers have sought to develop a new framework that would resolve the anomalies attributable to a patchwork formulation of relativistic quantum mechanics. This book chronicles the development of a new paradigm for describing relativistic quantum phenomena. What makes the new paradigm unique is its inclusion of a physically measurable, invariant evolution parameter. The resulting theory has been sufficiently well developed in the refereed literature that it is now possible to present a synthesis of its ideas and techniques. My synthesis is intended to encourage and enhance future research, and is presented in six parts. The environment within which the conventional paradigm exists is described in the Introduction. Part I eases the mainstream reader into the ideas of the new paradigm by providing the reader with a discussion that should look very familiar, but contains subtle nuances. Indeed, I try to provide the mainstream reader with familiar "landmarks" throughout the text. This is possible because the new paradigm contains the conventional paradigm as a subset. The foundation of the new paradigm is presented in Part II, fol owed by numerous applications in the remaining three parts. The reader should notice that the new paradigm handles not only the broad class of problems typically dealt with in conventional relativistic quantum theory, but also contains fertile research areas for both experimentalists and theorists. To avoid developing a theoretical framework without physical validity, numerous comparisons between theory and experiment are provided, and several predictions are made.

One of the first books in this area, this text focuses on important aspects of the system operation, analysis and performance evaluation of selected chaos-based digital communications systems a hot topic in communications and signal processing. "

In Chapter One we review the foundations of statistieal physies and frac tal functions. Our purpose is to demonstrate the limitations of Hamilton's equations of motion for providing a dynamical basis for the statistics of complex phenomena. The fractal functions are intended as possible models of certain complex phenomena; physical.systems that have long-time mem ory and/or long-range spatial interactions. Since fractal functions are non differentiable, those phenomena described by such functions do not have dif ferential equations of motion, but may have fractional-differential equations of motion. We argue that the traditional justification of statistieal mechan ics relies on aseparation between microscopic and macroscopie time scales. When this separation exists traditional statistieal physics results. When the microscopic time scales diverge and overlap with the macroscopie time scales, classieal statistieal mechanics is not applicable to the phenomenon described. In fact, it is shown that rather than the stochastic differential equations of Langevin describing such things as Brownian motion, we ob tain fractional differential equations driven by stochastic processes."

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.

This book consists of two parts, the first dealing with dissipative structures and the second with the structure and physics of chaos. The first part was written by Y. Kuramoto and the second part by H. Mori. Throughout the book, emphasis is laid on fundamental concepts and methods rather than applications, which are too numerous to be treated here. Typical physical examples, however, including nonlinear forced oscilla tors, chemical reactions with diffusion, and Benard convection in horizontal fluid layers, are discussed explicitly. Our consideration of dissipative structures is based on a phenomenolog ical reduction theory in which universal aspects of the phenomena under consideration are emphasized, while the theory of chaos is developed to treat transport phenomena, such as the mixing and diffusion of chaotic orbits, from the viewpoint of the geometrical phase space structure of chaos. The title of the original, Japanese version of the book is Sanitsu Kozo to Kaosu (Dissipative Structures and Chaos). It is part of the Iwanami Koza Gendai no Butsurigaku (Iwanami Series on Modern Physics). The first Japanese edition was published in March 1994 and the second in August 1997. We are pleased that this book has been translated into English and that it can now have an audience outside of Japan. We would like to express our gratitude to Glenn Paquette for his English translation, which has made this book more understandable than the original in many respects."

This monograph covers phenomena of deformation and machining of granular media: macroscopic particles of different size, shape, and surface properties which typically exhibit behavior similar to fluids, as well as the behavior of solids under deformation. The book analyses the behavior of granular media in soils, rocks and stones, metals and various synthetic materials, presenting a theoretical description, applications and understanding of basic phenomena in granular matter.