These two volumes contain eighteen invited papers by distinguished mathematicians in honor of the eightieth birthday of Israel M. Gelfand, one of the most remarkable mathematicians of our time. Gelfand has played a crucial role in the development of functional analysis during the last half-century. His work and his philosophy have in fact helped shape our understanding of the term 'functional analysis'. The papers in these volumes largely concern areas in which Gelfand has a very strong interest today, including geometric quantum field theory, representation theory, combinatorial structures underlying various 'continuous' constructions, quantum groups and geometry.

As far as the number of new results and quoted papers is concerned the present book may be considered a monograph. However, it also has some features of a textbook. Firstly, it proceeds from concrete problems to abstract ones, and secondly, all considerations and procedures are presented in much detail when met for the first time (such very elementary expositions can be found especially at the beginning of Chapters III and V). Finally, the authors focus their attention on elementary problems which can be dealt with by relatively simple methods. The authors hope that all this will make it possible also for an applied or technical research worker with some mathematical training to read this book. Naturally, the reader is supposed to be familiar with some basic notions from mathematical analysis, functional analysis and theory of partial differential equations. Also, the arguments and procedures which are repeated in the book are presented more briefly when met again, the reader being expected to become gradually more thoroughly acquainted with them. The authors have tried to provide a complete bibliography of all relevant publications (their number reaches about 500) from the theory of time-periodic solutions to non-linear partial and abstract differential equations whose origin may be put in the early thirties of this century.

This edited monograph provides a comprehensive and in-depth analysis of sliding mode control, focusing on event-triggered implementation. The technique allows to prefix the steady-state bounds of the system, and this is independent of any boundary disturbances. The idea of event-triggered SMC is developed for both single input / single output and multi-input / multi-output linear systems. Moreover, the reader learns how to apply this method to nonlinear systems. The book primarily addresses research experts in the field of sliding mode control, but the book may also be beneficial for graduate students.

In 1979, I edited Volume 18 in this series: Solution Methods for Integral Equations: Theory and Applications. Since that time, there has been an explosive growth in all aspects of the numerical solution of integral equations. By my estimate over 2000 papers on this subject have been published in the last decade, and more than 60 books on theory and applications have appeared. In particular, as can be seen in many of the chapters in this book, integral equation techniques are playing an increas ingly important role in the solution of many scientific and engineering problems. For instance, the boundary element method discussed by Atkinson in Chapter 1 is becoming an equal partner with finite element and finite difference techniques for solving many types of partial differential equations. Obviously, in one volume it would be impossible to present a complete picture of what has taken place in this area during the past ten years. Consequently, we have chosen a number of subjects in which significant advances have been made that we feel have not been covered in depth in other books. For instance, ten years ago the theory of the numerical solution of Cauchy singular equations was in its infancy. Today, as shown by Golberg and Elliott in Chapters 5 and 6, the theory of polynomial approximations is essentially complete, although many details of practical implementation remain to be worked out."

The present edition differs from the original German one mainly in the following addi tional material: weighted norm inequalities for maximal functions and singular opera tors ( 12, Chap. XI), polysingular integral operators and pseudo-differential operators ( 7, 8, Chap. XII), and spline approximation methods for solving singular integral equations ( 4, Chap. XVII). Furthermore, we added two subsections on polynomial approximation methods for singular integral equations over an interval or with dis continuous coefficients (Nos. 3.6 and 3.7, Chap. XVII). In many places we incorporated new results which, in the vast majority, are from the last five years after publishing the German edition (note that the references are enlarged by about 150 new titles). S. G. Mikhlin wrote 7, 8, Chap. XII, and the other additions were drawn up by S. Prossdorf. We wish to express our deepest gratitude to Dr. A. Bottcher and Dr. R. Lehmann who together translated the text into English carefully and with remarkable expertise."

In the study of integrable systems, two different approaches in particular have attracted considerable attention during the past twenty years. (1) The inverse scattering transform (IST), using complex function theory, which has been employed to solve many physically significant equations, the soliton' equations. (2) Twistor theory, using differential geometry, which has been used to solve the self-dual Yang--Mills (SDYM) equations, a four-dimensional system having important applications in mathematical physics. Both soliton and the SDYM equations have rich algebraic structures which have been extensively studied. Recently, it has been conjectured that, in some sense, all soliton equations arise as special cases of the SDYM equations; subsequently many have been discovered as either exact or asymptotic reductions of the SDYM equations. Consequently what seems to be emerging is that a natural, physically significant system such as the SDYM equations provides the basis for a unifying framework underlying this class of integrable systems, i.e. soliton' systems. This book contains several articles on the reduction of the SDYM equations to soliton equations and the relationship between the IST and twistor methods. The majority of nonlinear evolution equations are nonintegrable, and so asymptotic, numerical perturbation and reduction techniques are often used to study such equations. This book also contains articles on perturbed soliton equations. PainlevA(c) analysis of partial differential equations, studies of the PainlevA(c) equations and symmetry reductions of nonlinear partial differential equations. (ABSTRACT) In the study of integrable systems, two different approaches in particularhave attracted considerable attention during the past twenty years; the inverse scattering transform (IST), for soliton' equations and twistor theory, for the self-dual Yang--Mills (SDYM) equations. This book contains several articles on the reduction of the SDYM equations to soliton equations and the relationship between the IST and twistor methods. Additionally, it contains articles on perturbed soliton equations, Painleve analysis of partial differential equations, studies of the Painleve equations and symmetry reductions of nonlinear partial differential equations.

This volume is a collection of up-to-date research and expository papers on different aspects of complex analysis, including relations to operator theory and hypercomplex analysis. The articles cover many important and essential subjects, such as the SchrAdinger equation, subelliptic operators, Lie algebras and superalgebras, Toeplitz and Hankel operators, reproducing kernels and Qp spaces, among others. Most of the papers were presented at the International Symposium on Complex Analysis and Related Topics held in Cuernavaca (Morelos), Mexico, in November 1996, which was attended by approximately 50 experts in the field. The book can be used as a reference work on recent research in the subjects covered. It is one of the few books stressing the relation between operator theory and complex and hypercomplex analyses. The book is addressed to researchers and postgraduate students in the fields named here and in related ones.

This book offers the reader an overview of recent developments of multivariable dynamic calculus on time scales, taking readers beyond the traditional calculus texts. Covering topics from parameter-dependent integrals to partial differentiation on time scales, the book's nine pedagogically oriented chapters provide a pathway to this active area of research that will appeal to students and researchers in mathematics and the physical sciences. The authors present a clear and well-organized treatment of the concept behind the mathematics and solution techniques, including many practical examples and exercises.

This book is intended as a fairly complete presentation of what..'We call the discretization approach to functional integrals, i.e. path integrals defined as limits of discretized axpressions. In its main parts it is based 0n the original work of the authors. We hope to have provided the readers with a rather complete and up-to-date bibliography, and we apologize to authors whose work has not been cited through ignorance ori our part. Our main concern has been to present a for malism that is practical and which can be adapted to make computations in the numerous areas where path integrals are being increasingly used. For these reasons applications, illustrative examples, and detailed calculations are included. The book is partially based on lectures given by one of us (E.T.) at the Institut de Physique Theorique of the u.c.L. (Louvain-la-Neuve). We thank Dr. M.E. Brachet (University of Paris) for his help in the redaction of chapter 8. We are indebted to many of our colleagues and especially to the members of the Instituut voor Theoretische Fysica, K.U. Leuven for their interest and encouragement. We also thank Professor Claudio Anguita, Dean of the Faculty of Physics and Mathematics of .the University of Chile, for his constant support. Special thanks are due to Christine Detroije and Lutgarde Dubois for their very fine and hard work in typing the manuscript."

A development of some of the principal applications of function theory in several complex variables to Banach algebras. The authors do not presuppose any knowledge of several complex variables on the part of the reader, and all relevant material is developed within the text. Furthermore, the book deals with problems of uniform approximation on compact subsets of the space of n complex variables. This third edition contains new material on maximum modulus algebras and subharmonicity, the hull of a smooth curve, integral kernels, perturbations of the Stone-Weierstrass Theorem, boundaries of analytic varieties, polynomial hulls of sets over the circle, areas, and the topology of hulls. The authors have also included a new chapter commenting on history and recent developments, as well as an updated and expanded reading list.

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 Backlund 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.

This book contains eleven refereed research papers on deformation quantization by leading experts in the respective fields. These contributions are based on talks presented on the occasion of the meeting between mathematicians and theoretical physicists held in Strasbourg in May 2001. Topics covered are: star-products over Poisson manifolds, quantization of Hopf algebras, index theorems, globalization and cohomological problems. Both the mathematical and the physical approach ranging from asymptotic quantum electrodynamics to operads and prop theory will be presented. Historical remarks and surveys set the results presented in perspective. Directed at research mathematicians and theoretical physicists as well as graduate students, the volume will give an overview of a field of research that has seen enourmous acticity in the last years, with new ties to many other areas of mathematics and physics.

The Plancherel formula says that the "L" DEGREES2 norm of the function is equal to the "L" DEGREES2 norm of its Fourier transform. This implies that at least on average, the Fourier transform of an "L" DEGREES2 function decays at infinity. This book is dedicated to the study of the rate of this decay under various assumptions and circumstances, far beyond the original "L" DEGREES2 setting. Analytic and geometric properties of the underlying functions interact in a seamless symbiosis which underlines the wide range influences and applications of the concepts under consideration.

Many physical problems are meaningfully formulated in a cylindrical domain. When the size of the cylinder goes to infinity, the solutions, under certain symmetry conditions, are expected to be identical in every cross-section of the domain. The proof of this, however, is sometimes difficult and almost never given in the literature. The present book partially fills this gap by providing proofs of the asymptotic behaviour of solutions to various important cases of linear and nonlinear problems in the theory of elliptic and parabolic partial differential equations.
The book is a valuable resource for graduates and researchers in applied mathematics and for engineers. Many results presented here are original and have not been published elsewhere. They will motivate and enable the reader to apply the theory to other problems in partial differential equations.

This volume provides a comprehensive review of the developments which have taken place during the last thirty years concerning the asymptotic properties of solutions of nonautonomous ordinary differential equations. The conditions of oscillation of solutions are established, and some general theorems on the classification of equations according to their oscillatory properties are proved. In addition, the conditions are found under which nonlinear equations do not have singular, proper, oscillatory and monotone solutions. The book has five chapters: Chapter I deals with linear differential equations; Chapter II with quasilinear equations; Chapter III with general nonlinear differential equations; and Chapter IV and V deal, respectively, with higher-order and second-order differential equations of the Emden-Fowler type. Each section contains problems, including some which presently remain unsolved. The volume concludes with an extensive list of references. For researchers and graduate students interested in the qualitative theory of differential equations.

For many years Serge Lang has given talks to undergraduates on selected items in mathematics which could be extracted at a level understandable by students who have had calculus. Written in a conversational tone, Lang now presents a collection of those talks as a book. The talks could be given by faculty, but even better, they may be given by students in seminars run by the students themselves. Undergraduates, and even some high school students, will enjoy the talks which cover prime numbers, the abc conjecture, approximation theorems of analysis, Bruhat-Tits spaces, harmonic and symmetric polynomials, and more in a lively and informal style.

This book provides analytical solutions to a number of classical problems in transport processes, i.e. in fluid mechanics, heat and mass transfer. Expanding computing power and more efficient numerical methods have increased the importance of computational tools. However, the interpretation of these results is often difficult and the computational results need to be tested against the analytical results, making analytical solutions a valuable commodity. Furthermore, analytical solutions for transport processes provide a much deeper understanding of the physical phenomena involved in a given process than do corresponding numerical solutions. Though this book primarily addresses the needs of researchers and practitioners, it may also be beneficial for graduate students just entering the field.

The aim of this book is a detailed study of topological effects related to continuity of the dependence of solutions on initial values and parameters. This allows us to develop cheaply a theory which deals easily with equations having singularities and with equations with multivalued right hand sides (differential inclusions). An explicit description of corresponding topological structures expands the theory in the case of equations with continuous right hand sides also. In reality, this is a new science where Ordinary Differential Equations, General Topology, Integration theory and Functional Analysis meet. In what concerns equations with discontinuities and differential inclu sions, we do not restrict the consideration to the Cauchy problem, but we show how to develop an advanced theory whose volume is commensurable with the volume of the existing theory of Ordinary Differential Equations. The level of the account rises in the book step by step from second year student to working scientist."

This book deals with the constructive Weierstrassian approach to the theory of function spaces and various applications. The first chapter is devoted to a detailed study of quarkonial (subatomic) decompositions of functions and distributions on euclidean spaces, domains, manifolds and fractals. This approach combines the advantages of atomic and wavelet representations. It paves the way to sharp inequalities and embeddings in function spaces, spectral theory of fractal elliptic operators, and a regularity theory of some semi-linear equations. The book is self-contained, although some parts may be considered as a continuation of the author's book "Fractals and Spectra" (MMA 91). It is directed to mathematicians and (theoretical) physicists interested in the topics indicated and, in particular, how they are interrelated.

This comprehensive monograph details polynomially convex sets. It presents the general properties of polynomially convex sets with particular attention to the theory of the hulls of one-dimensional sets. Coverage examines in considerable detail questions of uniform approximation for the most part on compact sets but with some attention to questions of global approximation on noncompact sets. The book also discusses important applications and motivates the reader with numerous examples and counterexamples, which serve to illustrate the general theory and to delineate its boundaries.

In view of the eminent importance of spectral theory of linear operators in many fields of mathematics and physics, it is not surprising that various attempts have been made to define and study spectra also for nonlinear operators. This book provides a comprehensive and self-contained treatment of the theory, methods, and applications of nonlinear spectral theory. The first chapter briefly recalls the definition and properties of the spectrum and several subspectra for bounded linear operators. Then some numerical characteristics for nonlinear operators are introduced which are useful for describing those classes of operators for which there exists a spectral theory. Since spectral values are closely related to solvability results for operator equations, various conditions for the local or global invertibility of a nonlinear operator are collected in the third chapter. The following two chapters are concerned with spectra for certain classes of continuous, Lipschitz continuous, or differentiable operators. These spectra, however, simply adapt the corresponding definitions from the linear theory which somehow restricts their applicability. Other spectra which are defined in a completely different way, but seem to have useful applications, are defined and studied in the following four chapters. The remaining three chapters are more application-oriented and deal with nonlinear eigenvalue problems, numerical ranges, and selected applications to nonlinear problems. The only prerequisite for understanding this book is a modest background in functional analysis and operator theory. It is addressed to non-specialists who want to get an idea of the development of spectral theory for nonlinear operators in the last 30 years, as well as a glimpse of the diversity of the directions in which current research is moving.

This book contains the extended abstracts presented at the 12th International Conference on Power Series and Algebraic Combinatorics (FPSAC '00) that took place at Moscow State University, June 26-30, 2000. These proceedings cover the most recent trends in algebraic and bijective combinatorics, including classical combinatorics, combinatorial computer algebra, combinatorial identities, combinatorics of classical groups, Lie algebra and quantum groups, enumeration, symmetric functions, young tableaux etc...

The Heisenberg group plays an important role in several branches of mathematics, such as representation theory, partial differential equations, number theory, several complex variables and quantum mechanics. This monograph deals with various aspects of harmonic analysis on the Heisenberg group, which is the most commutative among the non-commutative Lie groups, and hence gives the greatest opportunity for generalizing the remarkable results of Euclidean harmonic analysis. The aim of this text is to demonstrate how the standard results of abelian harmonic analysis take shape in the non-abelian setup of the Heisenberg group. Thangavelu 's exposition is clear and well developed, and leads to several problems worthy of further consideration. Any reader who is interested in pursuing research on the Heisenberg group will find this unique and self-contained text invaluable.

The study of quantum disorder has generated considerable research activity in mathematics and physics over past 40 years. While single-particle models have been extensively studied at a rigorous mathematical level, little was known about systems of several interacting particles, let alone systems with positive spatial particle density. Creating a consistent theory of disorder in multi-particle quantum systems is an important and challenging problem that largely remains open. Multi-scale Analysis for Random Quantum Systems with Interaction presents the progress that had been recently achieved in this area. The main focus of the book is on a rigorous derivation of the multi-particle localization in a strong random external potential field. To make the presentation accessible to a wider audience, the authors restrict attention to a relatively simple tight-binding Anderson model on a cubic lattice Zd. This book includes the following cutting-edge features: an introduction to the state-of-the-art single-particle localization theory an extensive discussion of relevant technical aspects of the localization theory a thorough comparison of the multi-particle model with its single-particle counterpart a self-contained rigorous derivation of both spectral and dynamical localization in the multi-particle tight-binding Anderson model. Required mathematical background for the book includes a knowledge of functional calculus, spectral theory (essentially reduced to the case of finite matrices) and basic probability theory. This is an excellent text for a year-long graduate course or seminar in mathematical physics. It also can serve as a standard reference for specialists.

With the first edition out of print, we decided to arrange for republi cation of Denumerrible Markov Ohains with additional bibliographic material. The new edition contains a section Additional Notes that indicates some of the developments in Markov chain theory over the last ten years. As in the first edition and for the same reasons, we have resisted the temptation to follow the theory in directions that deal with uncountable state spaces or continuous time. A section entitled Additional References complements the Additional Notes. J. W. Pitman pointed out an error in Theorem 9-53 of the first edition, which we have corrected. More detail about the correction appears in the Additional Notes. Aside from this change, we have left intact the text of the first eleven chapters. The second edition contains a twelfth chapter, written by David Griffeath, on Markov random fields. We are grateful to Ted Cox for his help in preparing this material. Notes for the chapter appear in the section Additional Notes. J.G.K., J.L.S., A.W.K."