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Contents: Almost Periodic Schrödinger Operators (J Bellissard, R Lima, D Testard)Energy Forms and Diffusion Processes (M Fukushima)Block Spin Renormalization (K Gawedzki)Decomposition of Functions into Wavelets of Constant Shape, and Related Transforms (A Grossmann, J Morlet)Brownian Functionals and the Rotation Group (T Hida)Local Field Representations of the Conformal Group and their Applications (I T Todorov)Readership: Mathematicians and Physicists.Key Features: ICCE is the foremost international conference on coastal engineering Includes contributions from leading researchers across a broad range of topics Documents the latest advances in coastal engineering

Contents: The Inverse Method in Quantum Mechanics (H Grosse)An Invitation to Alain Connes'Cyclic Cohomology (D Kastler)Topological Methods in Field Theory (L A-Gaumé)Non-Standard Analysis: Applications to Probability Theory and Mathematical Physics (S Albeverio)Nonlinear Evolution Equation: Cauchy Problem and Scattering Theory (J Ginibre & G Velo)and other papersReadership: Mathematical and quantum physicists.Key Features: Provides an important source of new and not-yet-published results from the growing Asian and international geoscience community Presents a unique view of the rapid scientific progress made by Asian researchers in topics crucial to the future of the global environment Highlights a first-hand description of how the largest scientific population in the world is working together to manage the environmental problems which will determine the economic and social growth of the world itself

The book is a collection of research and review articles in several areas of modern mathematics and mathematical physics published in the span of three decades. The ICM Kyoto talk “Mathematics as Metaphor” summarises the author's view on mathematics as an outgrowth of natural language.

'Whatever regrets may be, we have done our best.'(Sir Ernest Shack­ 0 leton, turning back on 9 January 1909 at 88 23'South.) Brahms struggled for 20 years to write his first symphony. Compared to this, the 10 years we have been working on these two volumes may even appear short. This second volume treats stiff differential equations and differential algebraic equations. It contains three chapters: Chapter IV on one-step (Runge-Kutta) meth­ ods for stiff problems, Chapter V on multistep methods for stiff problems, and Chapter VI on singular perturbation and differential-algebraic equations. Each chapter is divided into sections. Usually the first sections of a chapter are of an introductory nature, explain numerical phenomena and exhibit numerical results. Investigations of a more theoretical nature are presented in the later sections of each chapter. As in Volume I, the formulas, theorems, tables and figures are numbered con­ secutively in each section and indicate, in addition, the section number. In cross references to other chapters the (latin) chapter number is put first. References to the bibliography are again by'author'plus'year'in parentheses. The bibliography again contains only those papers which are discussed in the text and is in no way meant to be complete.

Includes index.

A clear, practical and self-contained presentation of the methods of asymptotics and perturbation theory for obtaining approximate analytical solutions to differential and difference equations. Aimed at teaching the most useful insights in approaching new problems, the text avoids special methods and tricks that only work for particular problems. Intended for graduates and advanced undergraduates, it assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations, then develops local asymptotic methods for such equations, and explains perturbation and summation theory before concluding with an exposition of global asymptotic methods. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach readers how an applied mathematician tackles problems. There are 190 computer-generated plots and tables comparing approximate and exact solutions, over 600 problems of varying levels of difficulty, and an appendix summarizing the properties of special functions.

The scope ofthis book is to discuss recent numerical and algorithmic tools for the solution of certain flow problems arising in Computational Fluid Dynam­ ics (CFD). Here, we mainly restrict ourselves to the case ofthe incompressible Navier-Stokes equations, Ut - v~u + U. V'u+ V'p = f, V'·u = o. (1) These basic equations already play an important role in CFD, both for math­ ematicians as well as for more practical scientists: Physically important facts with'real life'character can be described by them, including also economical aspects in industrial applications. On the other hand, the equations in (1) provide the complete spectrum of numerical problems nowadays concerning the mathematical treatment of partial differential equations. Although this field of research may appear to be a small part only inside of CFD, it was and still is of great interest for mathematicians as well as engineers, physicists, computer scientists and many more: a fact which can be easily checked by counting the numerous publications. Nevertheless, our contribution has some unique characteristics since it contains a few ofthe lat­ est results for the numerical solution of (complex) flow problems on modern computer platforms. In this book, our particular emphasis lies on the solu­ tion process ofthe resulting high dimensional discrete systems ofequations which is often neglected in other works. Together with the included CDROM, which contains the'FEATFLOW 1.

This book provides an introduction to the ideas and methods of linear fu- tional analysis at a level appropriate to the?nal year of an undergraduate course at a British university. The prerequisites for reading it are a standard undergraduate knowledge of linear algebra and real analysis (including the t- ory of metric spaces). Part of the development of functional analysis can be traced to attempts to?nd a suitable framework in which to discuss di?erential and integral equations. Often, the appropriate setting turned out to be a vector space of real or complex-valued functions de?ned on some set. In general, such a v- tor space is in?nite-dimensional. This leads to di?culties in that, although many of the elementary properties of?nite-dimensional vector spaces hold in in?nite-dimensional vector spaces, many others do not. For example, in general in?nite-dimensionalvectorspacesthereisnoframeworkinwhichtomakesense of analytic concepts such as convergence and continuity. Nevertheless, on the spaces of most interest to us there is often a norm (which extends the idea of the length of a vector to a somewhat more abstract setting). Since a norm on a vector space gives rise to a metric on the space, it is now possible to do analysis in the space. As real or complex-valued functions are often called functionals, the term functional analysis came to be used for this topic. We now brie?y outline the contents of the book.

Extremality results proved in this Monograph for an abstract operator equation provide the theoretical framework for developing new methods that allow the treatment of a variety of discontinuous initial and boundary value problems for both ordinary and partial differential equations, in explicit and implicit forms. By means of these extremality res

In March 1997, we launched the Japan Association for Evolutionary Economics {JAFEE) to gather the academic minds that, out of dissatisfaction with established dynamic approaches, were separately searching for new approaches to economics. To our surprise and joy,as many as 500members, including graduate students,joined us. Later that year Prof. Horst Hanusch, then President of the International [oseph A. Schumpeter Society, remarked that such a start would take a couple of decades in Europe to prepare for. Since then we have been developing our activities incessantly not only in terms of the number of members, but also in terms of the intensity of international academic exchange. Originally the planning of this book came about as the successful outcome of our fourth annual conference organized as an international one, JAFEE 2000.Incorporat­ ing other international contributions related to our preceding conferences, this book has eventually turned out to be one of the most enterprising anthologies on evolu­ tionary economics ever published. Specifically, it contains excellent papers on such topics as streams of evolutionary economics, evolutionary nonlinear dynamics, experimental economics and evolution, multiagent systems and complexity, new frontiers for evolutionary economics, and economic heresies. In short, this book will provide a vivid and full-fledged picture of up-to-date evolutionary economics.

This book focuses treatable This class on exactly many'body problems. does not include most We are therefore reminded'of physical problems. the of the man home late at after an alcoholic who, story returning night the for his under he was a knew, evening, scanning ground key lamppost; be that he had it somewhere but under the to sure, dropped else, only Yet was there to conduct a searcW'. light lamppost enough proper we feel the interest for such models is nowadays sufficiently widespread because of their their mathematical relevance and their multi beauty, farious that need be made for no our apologies applicative potential choice. In whoever undertakes to read this book will know from any case, its title what she is in for! Yet this title a of it some may require explanations: gloss (including its extended inside front follows. version, see cover) and nonrelativistic'Classical'we mean nonquantal (although By consider the which indeed some are Ruijsenaars Schneider models, treated in this relativistic versions as known, nonre book, of, previously lativistic is focussed see our on models; below): presentation mainly of whose time evolution is determined many body point particles systems Newtonian of motion to by equations (acceleration proportional force).

The motto of connectivity and superconductivity is that the solutions of the Ginzburg–Landau equations are qualitatively in?uenced by the topology of the boundaries. Special attention is given to the “zero set”,the set of the positions (usually known as “quantum vortices”) where the order parameter vanishes. The paradigm of connectivity and superconductivity is the Little– Parks e?ect,discussed in most textbooks on superconductivity. This volume is intended to serve as a reference book for graduate students and researchers in physics or mathematics interested in superconductivity, or in the Schr¨ odinger equation as a limiting case of the Ginzburg–Landau equations. The e?ects considered here usually become important in the regime where the coherence length is of the order of the dimensions of the sample. While in the Little–Parks days a lot of ingenuity was required to achieve this regime, present microelectronic techniques have transformed it into a routine. Mo- over,measurement and visualization techniques are developing at a pace which makes it reasonable to expect veri?cation of distributions,and not only of global properties. Activity in the?eld has grown and diversi?ed substantially in recent years. We have therefore invited experts ranging from experimental and theoretical physicists to pure and applied mathematicians to contribute articles for this book. While the skeleton of the book deals with superconductivity,micron- works and generalizations of the Little–Parks situation,there are also articles which deal with applications of the Ginzburg–Landau formalism to several fundamental topics,such as quantum coherence,cosmology,and questions in materials science.

The present volume, published at the occasion of his 100th birthday anniversary, is a collection of articles that reviews the impact of Kolomogorov's work in the physical sciences and provides an introduction to the modern developments that have been triggered in this way to encompass recent applications in biology, chemistry, information sciences and finance. This book addresses scientists and postgraduate students in applied mathematics and theoretical physics.

For most mathematicians and many mathematical physicists the name Erich Kähler is strongly tied to important geometric notions such as Kähler metrics, Kähler manifolds and Kähler groups. They all go back to a paper of 14 pages written in 1932. This, however, is just a small part of Kähler's many outstanding achievements which cover an unusually wide area: From celestial mechanics he got into complex function theory, differential equations, analytic and complex geometry with differential forms, and then into his main topic, i.e. arithmetic geometry where he constructed a system of notions which is a precursor and, in large parts, equivalent to the now used system of Grothendieck and Dieudonné. His principal interest was in finding the unity in the variety of mathematical themes and establishing thus mathematics as a universal language. In this volume Kähler's mathematical papers are collected following a'Tribute to Herrn Erich Kähler'by S. S. Chern, an overview of Kähler's life data by A. Bohm and R. Berndt, and a Survey of his Mathematical Work by the editors. There are also comments and reports on the developments of the main topics of Kähler's work, starting by W. Neumann's paper on the topology of hypersurface singularities, J.-P. Bourguignon's report on Kähler geometry and, among others by Berndt, Bost, Deitmar, Ekeland, Kunz and Krieg, up to A. Nicolai's essay'Supersymmetry, Kähler geometry and Beyond'. As Kähler's interest went beyond the realm of mathematics and mathematical physics, any picture of his work would be incomplete without touching his work reaching into other regions. So a short appendix reproduces three of his articles concerning his vision of mathematics as a universal Theme together with an essay by K. Maurin giving an'Approach to the philosophy of Erich Kähler'.

The subject of local dynamical systems is concerned with the following two questions: 1. Given an n×n matrix A, describe the behavior, in a neighborhood of the origin, of the solutions of all systems of di?erential equations having a rest point at the origin with linear part Ax, that is, all systems of the form x? = Ax+···, n where x? R and the dots denote terms of quadratic and higher order. 2. Describethebehavior(neartheorigin)ofallsystemsclosetoasystem of the type just described. To answer these questions, the following steps are employed: 1. A normal form is obtained for the general system with linear part Ax. The normal form is intended to be the simplest form into which any system of the intended type can be transformed by changing the coordinates in a prescribed manner. 2. An unfolding of the normal form is obtained. This is intended to be the simplest form into which all systems close to the original s- tem can be transformed. It will contain parameters, called unfolding parameters, that are not present in the normal form found in step 1. vi Preface 3. The normal form, or its unfolding, is truncated at some degree k, and the behavior of the truncated system is studied.

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in - search and teaching, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathematical Sciences (AMS) series, whichwill focus on advanced textbooks and research-level monographs.

This book is about the mathematical theory of light propagation in media on general-relativistic spacetimes. The first part discusses the transition from Maxwell's equations to ray optics. The second part establishes a general mathematical framework for treating ray optics as a theory in its own right, making extensive use of the Hamiltonian formalism. This part also includes a detailed discussion of variational principles (i.e., various versions of Fermat's principle) for light rays in general-relativistic media. Some applications, e.g. to gravitational lensing, are worked out. The reader is assumed to have some basic knowledge of general relativity and some familiarity with differential geometry. Some of the results are published here for the first time, e.g. a general-relativistic version of Fermat's principle for light rays in a medium that has to satisfy some regularity condition only.

'Provides an excellent introduction for physicists interested in the statistical properties of financial markets... basic financial terms such as shorts, limit orders, puts, calls, and other terms are clearly defined... an excellent starting point for the interested physicist.'PHYSICS TODAYThis introductory treatment describes parallels between statistical physics and finance, both long established and new research results on capital markets. Forming the core of Voit's treatment are the concepts of random walks, scaling of data, and risk control. Voit discusses the underlying assumptions using empirical financial data and analogies to physical models such as fluid flows and turbulence. He formulates theories of derivative pricing and risk control, and shows how computer simulations of markets provide insights into price fluctuations and how crashes are modelled in ways analogous to phase transitions. This corrected edition has been updated with several new and significant developments, e.g. the dynamics of volatility smiles and implied volatility surfaces, path integral approaches to option pricing, a new simulation scheme for options, multifractals, the application of nonextensive statistical mechanics to financial markets, and the minority game.

The book is a collection of contributions concerning the theories, applications and perspectives of Variable Structure Systems (VSS). Variable Structure Systems have been a major control design methodology for many decades. The term Variable Structure Systems was introduced in the late 1950's, and the fundamental concepts were developed for its main branch Sliding Mode Control by Russian researchers Emelyanov and Utkin. The 20th Century has seen the formation and consolidation of VSS theory and its applications. It has also seen an emerging trend of cross-fertilization and integration of VSS with other control and non-control techniques such as feedback linearization,?atness, passivity based control, adaptive and learning? control, system identi?cation, pulse width modulation, H geometric and algebraic methods, arti?cial intelligence, modeling and optimization, neural networks, fuzzy logic, to name just a few. This trend will continue and?ourish in the new millennium. To re?ect these major developments in the 20th Century, this book - cludes 16 specially invited contributions from well-known experts in VSS theory and applications, covering a wide range of topics. The?rst chapter, “First Stage of VSS: People and Events” written by Vadim Utkin, the founder of VSS, oversees and documents the historical developments of VSS in the 20th Century, including many interesting events not known to the West until now. The second chapter, “An Integrated Learning Variable Structure Control Method” written by Jian-Xin Xu, addresses an important issue regarding control integration between variable structure control and learning control.

The notion of group is fundamental in our days, not only in mathematics, but also in classical mechanics, electromagnetism, theory of relativity, quantum mechanics, theory of elementary particles, etc. This notion has developed during a century and this development is connected with the names of great mathematicians as E. Galois, A. L. Cauchy, C. F. Gauss, W. R. Hamilton, C. Jordan, S. Lie, E. Cartan, H. Weyl, E. Wigner, and of many others. In mathematics, as in other sciences, the simple and fertile ideas make their way with difficulty and slowly; however, this long history would have been of a minor interest, had the notion of group remained connected only with rather restricted domains of mathematics, those in which it occurred at the beginning. But at present, groups have invaded almost all mathematical disciplines, mechanics, the largest part of physics, of chemistry, etc. We may say, without exaggeration, that this is the most important idea that occurred in mathematics since the invention of infinitesimal calculus; indeed, the notion of group expresses, in a precise and operational form, the vague and universal ideas of regularity and symmetry. The notion of group led to a profound understanding of the character of the laws which govern natural phenomena, permitting to formulate new laws, correcting certain inadequate formulations and providing unitary and non­ contradictory formulations for the investigated phenomena.

Almost 12 years have passed by since we wrote Chaos and Fractals. At the time we were hoping that our approach of writing a book which would be both accessible without mathematical sophistication and portray these exiting new fields in an authentic manner would find an audience. Now we know it did. We know from many reviews and personal letters that the book is used in a wide range of ways: researchers use it to acquaint themselves, teachers use it in college and university courses, students use it for background reading, and there is also a substantial audience of lay people who just want to know what chaos and fractals are about. Every book that is somewhat technical in nature is likely to have a number of misprints and errors in its first edition. Some of these were caught and brought to our attention by our readers. One of them, Hermann Flaschka, deserves to be thanked in particular for his suggestions and improvements. This second edition has several changes. We have taken out thetwo appendices from the firstedition. At the time of the first edition Yuval Fishers contribution, which we published as an appendix was probably the first complete expository account on fractal image compression. Meanwhile, Yuvals book Fractal Image Compression: Theory and Application appeared and is now the publication to refer to.

Partial differential equations are fundamental to the modeling of natural phenomena, arising in every field of science. Consequently, the desire to understand the solutions of these equations has always had a prominent place in the efforts of mathematicians; it has inspired such diverse fields as complex function theory, functional analysis and algebraic topology. Like algebra, topology, and rational mechanics, partial differential equations are a core area of mathematics. This book aims to provide the background necessary to initiate work on a Ph.D. thesis in PDEs for beginning graduate students. Prerequisites include a truly advanced calculus course and basic complex variables. Lebesgue integration is needed only in Chapter 10, and the necessary tools from functional analysis are developed within the course. The book can be used to teach a variety of different courses. This new edition features new problems throughout and the problems have been rearranged in each section from simplest to most difficult. New examples have also been added. The material on Sobolev spaces has been rearranged and expanded. A new section on nonlinear variational problems with'Young-measure'solutions appears. The reference section has also been expanded.

Phase transformations in solids typically lead to surprising mechanical behaviour with far reaching technological applications. The mathematical modeling of these transformations in the late 80s initiated a new field of research in applied mathematics, often referred to as mathematical materials science, with deep connections to the calculus of variations and the theory of partial differential equations. This volume gives a brief introduction to the essential physical background, in particular for shape memory alloys and a special class of polymers (nematic elastomers). Then the underlying mathematical concepts are presented with a strong emphasis on the importance of quasiconvex hulls of sets for experiments, analytical approaches, and numerical simulations.

The?nite element method is one of the major tools used in the numerical solution of partial di?erential equations. This book o?ers a fundamental and practical introduction to the method, its variants, and their applications. In presenting the material, I have attempted to introduce every concept in the simplest possible setting and to maintain a level of treatment that is as rigorous as possible without being unnecessarily abstract. The book is based on the material that I have used in a graduate course at Southern Methodist University for several years. Part of the material was alsousedformyseminarnotesatPurdueUniversity,UniversityofMinnesota, andTexasA&MUniversity.Furthermore,thisbookwasthebasisforsummer schools on the?nite element method and its applications held in China, Iran, Mexico, and Venezuela. This book covers six major topics and four applications. In Chap. 1, the 1 2 standard (H-and H -conforming)?nite element method is introduced. In Chaps.2and3,twocloselyrelated?niteelementmethods,thenonconforming and the mixed?nite element methods, are discussed. The discontinuous and characteristic?niteelementmethodsarestudiedinChaps.4and5;thesetwo methods have been recently developed. The adaptive?nite element method is considered in Chap. 6. The last four chapters are devoted to applications of these methods to solid mechanics (Chap. 7),?uid mechanics (Chap. 8),?uid?ow in porous media (Chap. 9), and semiconductor modeling (Chap. 10).

Mathematical Tools for Physicists is a unique collection of 18 carefully reviewed articles, each one written by a renowned expert working in the relevant field. The result is beneficial to both advanced students as well as scientists at work; the former will appreciate it as a comprehensive introduction, while the latter will use it as a ready reference. The contributions range from fundamental methods right up to the latest applications, including: - Algebraic/ analytic / geometric methods - Symmetries and conservation laws - Mathematical modeling - Quantum computation The emphasis throughout is ensuring quick access to the information sought, and each article features: - an abstract - a detailed table of contents - continuous cross-referencing - references to the most relevant publications in the field, and - suggestions for further reading, both introductory as well as highly specialized. In addition, a comprehensive index provides easy access to the vast number of key words extending beyond the range of the headlines.

This volume is the first of two volumes containing the revised and completed notes lectures given at the school'Quantum Independent Increment Processes: Structure and Applications to Physics'. This school was held at the Alfried-Krupp-Wissenschaftskolleg in Greifswald during the period March 9 – 22, 2003, and supported by the Volkswagen Foundation. The school gave an introduction to current research on quantum independent increment processes aimed at graduate students and non-specialists working in classical and quantum probability, operator algebras, and mathematical physics. The present first volume contains the following lectures:'Lévy Processes in Euclidean Spaces and Groups'by David Applebaum,'Locally Compact Quantum Groups'by Johan Kustermans,'Quantum Stochastic Analysis'by J. Martin Lindsay, and'Dilations, Cocycles and Product Systems'by B.V. Rajarama Bhat.

This is the second of two volumes containing the revised and completed notes of lectures given at the school'Quantum Independent Increment Processes: Structure and Applications to Physics'. This school was held at the Alfried-Krupp-Wissenschaftskolleg in Greifswald in March, 2003, and supported by the Volkswagen Foundation. The school gave an introduction to current research on quantum independent increment processes aimed at graduate students and non-specialists working in classical and quantum probability, operator algebras, and mathematical physics. The present second volume contains the following lectures:'Random Walks on Finite Quantum Groups'by Uwe Franz and Rolf Gohm,'Quantum Markov Processes and Applications in Physics'by Burkhard Kümmerer, Classical and Free Infinite Divisibility and Lévy Processes'by Ole E. Barndorff-Nielsen, Steen Thorbjornsen, and'Lévy Processes on Quantum Groups and Dual Groups'by Uwe Franz.

Stochastic geometry is a relatively new branch of mathematics. Although its predecessors such as geometric probability date back to the 18th century, the formal concept of a random set was developed in the beginning of the 1970s. Theory of Random Sets presents a state of the art treatment of the modern theory, but it does not neglect to recall and build on the foundations laid by Matheron and others, including the vast advances in stochastic geometry, probability theory, set-valued analysis, and statistical inference of the 1990s. The book is entirely self-contained, systematic and exhaustive, with the full proofs that are necessary to gain insight. It shows the various interdisciplinary relationships of random set theory within other parts of mathematics, and at the same time, fixes terminology and notation that are often varying in the current literature to establish it as a natural part of modern probability theory, and to provide a platform for future development.

Stochastic differential equations (SDEs) are a powerful tool in science, mathematics, economics and finance. This book will help the reader to master the basic theory and learn some applications of SDEs. In particular, the reader will be provided with the backward SDE technique for use in research when considering financial problems in the market, and with the reflecting SDE technique to enable study of optimal stochastic population control problems. These two techniques are powerful and efficient, and can also be applied to research in many other problems in nature, science and elsewhere.

Professor Noubari's recommendation:'Professor Starks book provides an effective entry into the field for engineering students who have little or no prior knowledge of this important subject. Avaibility of collection of computer codes and mfiles in combination with topics of the book, makes the book highly valuable to enhance student learning of the subject matter.'

Over two hundred novel and innovative computer algebra worksheets or'recipes'will enable readers in engineering, physics, and mathematics to easily and rapidly solve and explore most problems they encounter in their mathematical physics studies. While the aim of this text is to illustrate applications, a brief synopsis of the fundamentals for each topic is presented, the topics being organized to correlate with those found in traditional mathematical physics texts. The recipes are presented in the form of stories and anecdotes, a pedagogical approach that makes a mathematically challenging subject easier and more fun to learn. This is a self-contained and standalone text using MAPLE that may be used in the classroom, for self-study, as a reference, or as a text for an online course.

'Entropy and Energy- a Universal Competition'is a students textbook as well as a scientific monograph. The concepts of entropy and energy embody the effects of random walk in a body and of deterministic strife respectively, and are therefore often in competition. The book gives instructive examples from elementary thermodynamics and physico-chemistry and extrapolates the notion to non-standard thermodynamic subjects like shape memory, dissipation of the earth's atmosphere, and sociology. The works of the thermodynamic pioneers are presented, in particular Clausius, Carathéodory, Boltzmann, Gibbs, and Planck. The laws of thermodynamics and their limitations are discussed; also the pertinacious Gibbs paradox. The reader has numerous possibilities to influence the programs and thus develop an understanding for the thermodynamic principles.

My purpose in this monograph is to present an essentially self-contained account of the mathematical theory of Galerkin?nite element methods as appliedtoparabolicpartialdi?erentialequations. Theemphasesandselection of topics re?ects my own involvement in the?eld over the past 25 years, and my ambition has been to stress ideas and methods of analysis rather than to describe the most general and farreaching results possible. Since the formulation and analysis of Galerkin?nite element methods for parabolic problems are generally based on ideas and results from the corresponding theory for stationary elliptic problems, such material is often included in the presentation. The basis of this work is my earlier text entitled Galerkin Finite Element Methods for Parabolic Problems, Springer Lecture Notes in Mathematics, No. 1054, from 1984. This has been out of print for several years, and I have felt a need and been encouraged by colleagues and friends to publish an updated version. In doingso I have included most of the contents of the 14 chapters of the earlier work in an updated and revised form, and added four new chapters, on semigroup methods, on multistep schemes, on incomplete iterative solution of the linear algebraic systems at the time levels, and on semilinear equations. The old chapters on fully discrete methods have been reworked by?rst treating the time discretization of an abstract di?erential equation in a Hilbert space setting, and the chapter on the discontinuous Galerkin method has been completely rewritten.

Pierre-Francois Verhulst, with his seminal work using the logistic map to describe population growth and saturation, paved the way for the many applications of this tool in modern mathematics, physics, chemistry, biology, economics and sociology. Indeed nowadays the logistic map is considered a useful and paradigmatic showcase for the route leading to chaos. This volume gathers contributions from some of the leading specialists in the field to present a state-of-the art view of the many ramifications of the developments initiated by Verhulst over a century ago.

The well known transport laws of Navier-Stokes and Fourier fail for the simulation of processes on lengthscales in the order of the mean free path of a particle that is when the Knudsen number is not small enough. Thus, the proper simulation of flows in rarefied gases requires a more detailed description. This book discusses classical and modern methods to derive macroscopic transport equations for rarefied gases from the Boltzmann equation, for small and moderate Knudsen numbers, i.e. at and above the Navier-Stokes-Fourier level. The main methods discussed are the classical Chapman-Enskog and Grad approaches, as well as the new order of magnitude method, which avoids the short-comings of the classical methods, but retains their benefits. The relations between the various methods are carefully examined, and the resulting equations are compared and tested for a variety of standard problems. The book develops the topic starting from the basic description of an ideal gas, over the derivation of the Boltzmann equation, towards the various methods for deriving macroscopic transport equations, and the test problems which include stability of the equations, shock waves, and Couette flow.

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.

1) Phase Transitions, represented by generalizations of the classical Stefan problem. This is studied by Kenmochi and Rodrigues by means of variational techniques.2) Hysteresis Phenomena. Some alloys exhibit shape memory effects, corresponding to a stress-strain relation which strongly depends on temperature; mathematical physical aspects are treated in Müller's paper. In a general framework, hysteresis can be described by means of hysteresis operators in Banach spaces of time dependent functions; their properties are studied by Brokate.3) Numerical analysis. Several models of the phenomena above can be formulated in terms of nonlinear parabolic equations. Here Verdi deals with the most updated approximation techniques.

A tribute to the vision and legacy of Israel Moiseevich Gelfand, the invited papers in this volume reflect the unity of mathematics as a whole, with particular emphasis on the many connections among the fields of geometry, physics, and representation theory. Written by leading mathematicians, the text is broadly divided into two sections: the first is devoted to developments at the intersection of geometry and physics, and the second to representation theory and algebraic geometry. Topics include conformal field theory, K-theory, noncommutative geometry, gauge theory, representations of infinite-dimensional Lie algebras, and various aspects of the Langlands program. Graduate students and researchers will benefit from and find inspiration in this broad and unique work, which brings together fundamental results in a number of disciplines and highlights the rewards of an interdisciplinary approach to mathematics and physics. Contributors: M. Atiyah; A. Braverman; H. Brezis; T. Coates; A. Connes; S. Debacker; V. Drinfeld; L.D. Faddeev; M. Finkelberg; D. Gaitsgory; I.M. Gelfand; A. Givental; D. Kazhdan; M. Kontsevich; B. Kostant; C-H. Liu; K. Liu; G. Lusztig; D. McDuff; M. Movshev; N.A. Nekrasov; A. Okounkov; N. Reshetikhin; A. Schwarz; Y. Soibelman; C. Vafa; A.M. Vershik; N. Wallach; and S-T. Yau.

One of the key issues related to superfluidity is the existence of vortices. In very recent experiments on Bose–Einstein condensates, vortices have been observed by rotating the trap holding the atoms. In contrast to a classical fluid for which the equilibrium velocity corresponds to solid body rotation, a quantum fluid such as a Bose–Einstein condensate can rotate only through the nucleation of quantized vortices. This monograph is dedicated to the mathematical modelling of these phenomena. The mathematical tools employed are energy estimates, Gamma convergence, and homogenization techniques. The mathematical analysis is made in the framework of the Gross–Pitaevskii energy. Results are presented and open problems related to recent experiments are explained. The work can serve as a reference for mathematical researchers and theoretical physicists interested in superfluidity and quantum condensates, and can also complement a graduate seminar in elliptic PDEs or modelling of physical experiments.

Prior to the computer era, analytical methods in elasticity had already been developed and - proved up to impressive levels. Relevant mathematical techniques were extensively exploited, contributing signi?cantly to the understanding of physical phenomena. In recent decades, - merical computerized techniques have been re?ned and modernized, and have reached high levels of capabilities, standardization and automation. This trend, accompanied by convenient and high resolution graphical visualization capability, has made analytical methods less attr- tive, and the amount of effort devoted to them has become substantially smaller. Yet, with some tenacity, the tremendous advances in computerized tools have yielded various mature programs for symbolic manipulation. Such tools have revived many abandoned analytical methodologies by easing the tedious effort that was previously required, and by providing additional capab- ities to perform complex derivation processes that were once considered impractical. Generally speaking, it is well recognized that analytical solutions should be applied to re- tively simple problems, while numerical techniques may handle more complex cases. However, it is also agreed that analytical solutions provide better insight and improved understanding of the involved physical phenomena, and enable a clear representation of the role taken by each of the problem parameters. Nowadays, analytical and numerical methods are considered as c- plementary: that is, while analytical methods provide the required understanding,numerical solutions provide accuracy and the capability to deal with cases where the geometry and other characteristics impose relatively complex solutions.

As individual needs have arisen in the fields of physics, electrical engineering and computational science, each has created its own theories of information to serve as conceptual instruments for advancing developments. This book provides a coherent consolidation of information theories from these different fields. The author gives a survey of current theories and then introduces the underlying notion of symmetry, showing how information is related to the capacity of a system to distinguish itself. A formal methodology using group theory is employed and leads to the application of Burnside's Lemma to count distinguishable states. This provides a versatile tool for quantifying complexity and information capacity in any physical system. Written in an informal style, the book is accessible to all researchers in the fields of physics, chemistry, biology, computational science as well as many others.

Economic Systems exhibit complex dynamics evidenced by large-amplitude and aperiodic fluctuations in economic variables, such as foreign exchange rates and stock market prices, indicating that these systems are driven far from the equilibrium. Characterization of the complex behavior of economic cycles, by identifying regular and irregular patterns and regime switching in economic time series, is the key for pattern recognition and forecasting of economic cycles. Statistical analysis of stock markets and foreign exchange markets has demonstrated the intermittent nature of economic time series. A nonlinear model of business cycles is able to simulate intermittency arising from order-chaos and chaos-chaos transitions. This monograph introduces new concepts of unstable periodic orbits and chaotic saddles which are unstable structures embedded in a chaotic attractor, responsible for economic intermittency.

The monograph is devoted to the systematic presentation of the so called'dressing method'for solving differential equations (both linear and nonlinear) of mathematical physics. The essence of the dressing method consists in a generation of new non-trivial solutions of a given equation from (maybe trivial) solution of the same or related equation. The Moutard and Darboux transformations discovered in XIX century as applied to linear equations, the Bäcklund transformation in differential geometry of surfaces, the factorization method, the Riemann-Hilbert problem in the form proposed by Shabat and Zakharov for soliton equations and its extension in terms of the d-bar formalism comprise the main objects of the book. Throughout the text, a generally sufficient'linear experience'of readers is exploited, with a special attention to the algebraic aspects of the main mathematical constructions and to practical rules of obtaining new solutions. Various linear equations of classical and quantum mechanics are solved by the Darboux and factorization methods. An extension of the classical Darboux transformations to nonlinear equations in 1+1 and 2+1 dimensions, as well as its factorization are discussed in detail. The applicability of the local and non-local Riemann-Hilbert problem-based approach and its generalization in terms of the d-bar method are illustrated on various nonlinear equations.

The book introduces the general theory of relativity and includes applications to cosmology. The book contains a thorough introduction to tensor calculus and curved manifolds. After the necessary mathematical tools are introduced, we give a thorough presentation of the theory of relativity. Also, some advanced topics not previously covered by textbooks; e.g. Kaluza-Klein theory, Israel's formalism and branes. Anisotropic cosmological models are also included. The book contains a large number of new exercises and examples, each with separate headings. The reader will get an updated introduction to general relativity including the most recent developments in cosmology.

Fuchsian reduction is a method for representing solutions of nonlinear PDEs near singularities. The technique has multiple applications including soliton theory, Einstein's equations and cosmology, stellar models, laser collapse, conformal geometry and combustion. Developed in the 1990s for semilinear wave equations, Fuchsian reduction research has grown in response to those problems in pure and applied mathematics where numerical computations fail. This work unfolds systematically in four parts, interweaving theory and applications. The case studies examined in Part III illustrate the impact of reduction techniques, and may serve as prototypes for future new applications. In the same spirit, most chapters include a problem section. Background results and solutions to selected problems close the volume. This book can be used as a text in graduate courses in pure or applied analysis, or as a resource for researchers working with singularities in geometry and mathematical physics.

Itisaspecialpleasureformetowritethisforewordforaremarkablebookbya remarkableauthor.MarcoPettiniisadeepthinker,whohasspentmanyyears probing the foundations of Hamiltonian chaos and statistical mechanics, in particular phase transitions, from the point of view of geometry and topology. Itisinparticularthequalityofmindoftheauthorandhisdeepphysical,as well as mathematical insights which make this book so special and inspiring. It is a “must” for those who want to venture into a new approach to old problems or want to use new tools for new problems. Although topology has penetrated a number of?elds of physics, a broad participationoftopologyintheclari?cationandprogressoffundamentalpr- lems in the above-mentioned?elds has been lacking. The new perspectives topology gives to the above-mentioned problems are bound to help in their clari?cation and to spread to other?elds of science. The sparsity of geometric thinking and of its use to solve fundamental problems, when compared with purely analytical methods in physics, could be relieved and made highly productive using the material discussed in this book. It is unavoidable that the physicist reader may have then to learn some new mathematics and be challenged to a new way of thinking, but with the author as a guide, he is assured of the best help in achieving this that is presently available.

This book demonstrates scientific computing by presenting twelve computational projects in several disciplines including Fluid Mechanics, Thermal Science, Computer Aided Design, Signal Processing and more. Each follows typical steps of scientific computing, from physical and mathematical description, to numerical formulation and programming and critical discussion of results. The text teaches practical methods not usually available in basic textbooks: numerical checking of accuracy, choice of boundary conditions, effective solving of linear systems, comparison to exact solutions and more. The final section of each project contains the solutions to proposed exercises and guides the reader in using the MATLAB scripts available online.

This book is an elementary introduction to some ideas and techniques that have revolutionized enumerative geometry: stable maps and quantum cohomology. A striking demonstration of the potential of these techniques is provided by Kont- vich's famous formula, which solves a long-standing question: How many plane rational curves of degree d pass through 3d — 1 given points in general position? The formula expresses the number of curves for a given degree in terms of the numbers for lower degrees. A single initial datum is required for the recursion, namely, the case d = I, which simply amounts to the fact that through two points there is but one line. Assuming the existence of the Kontsevich spaces of stable maps and a few of their basic properties, we present a complete proof of the formula, and use the formula as a red thread in our Invitation to Quantum Cohomology. For more information about the mathematical content, see the Introduction. The canonical reference for this topic is the already classical Notes on Stable Maps and Quantum Cohomology by Fulton and Pandharipande [29], cited henceforth as FP-NOTES. We have traded greater generality for the sake of introducing some simplifications. We have also chosen not to include the technical details of the construction of the moduli space, favoring the exposition with many examples and heuristic discussions.