Bøger i Modern Birkhauser Classics serien

The true history of physics can only be read in the life stories of those who made its progress possible. Matvei Bronstein was one of those for whom the vast territory of theoretical physics was as familiar as his own home: he worked in cosmology, nuclear physics, gravitation, semiconductors, atmospheric physics, quantum electrodynamics, astro- physics and the relativistic quantum theory. Everyone who knew him was struck by his wide knowledge, far beyond the limits of his trade. This partly explains why his life was closely intertwined with the social, historical and scientific context of his time. One might doubt that during his short life Bronstein could have made truly weighty contributions to science and have become, in a sense, a symbol of his time. Unlike mathematicians and poets, physicists reach the peak of their careers after the age of thirty. His thirty years of life, however, proved enough to secure him a place in the Greater Soviet Encyclopedia. In 1967, in describing the first generation of physicists educated after the 1917 revolution, Igor Tamm referred to Bronstein as "e;an exceptionally brilliant and promising"e; theoretician [268].

"The book is largely self-contained...There is a nice introduction to symplectic geometry and a charming exposition of equivariant K-theory. This is the only available introduction to geometric representation theory...it has already proved successful in introducing a new generation to the subject."

This book is devoted to one of the fastest developing fields in modern control theory - the so-called H-infinity optimal control theory. The book can be used for a second or third year graduate level course in the subject, and researchers working in the area will find the book useful as a standard reference.

This elegantly written text includes a wealth of exercises for students as it weaves classical probability theory into the quantum framework. It deepens our understanding of classical and quantum views on the dynamics of systems subject to the laws of chance.

This volume develops methods for proving the non-vanishing of certain L-functions at points in the critical strip. It begins at a very basic level and continues to develop, providing readers with a theoretical foundation that allows them to understand the latest discoveries in the field.

Many physical processes in fields such as mechanics, thermodynamics, electricity, magnetism or optics are described by means of partial differential equations. The aim of the present book is to demontstrate the basic methods for solving the classical linear problems in mathematical physics of elliptic, parabolic and hyperbolic type. In particular, the methods of conformal mappings, Fourier analysis and Green`s functions are considered, as well as the perturbation method and integral transformation method, among others. Every chapter contains concrete examples with a detailed analysis of their solution.The book is intended as a textbook for students in mathematical physics, but will also serve as a handbook for scientists and engineers.

In a coherent, exhaustive and progressive way, this book presents the tools for studying local bifurcations of limit cycles in families of planar vector fields. A systematic introduction is given to such methods as division of an analytic family of functions in its ideal of coefficients, and asymptotic expansion of non-differentiable return maps and desingularisation. The exposition moves from classical analytic geometric methods applied to regular limit periodic sets to more recent tools for singular limit sets.The methods can be applied to theoretical problems such as Hilbert's 16th problem, but also for the purpose of establishing bifurcation diagrams of specific families as well as explicit computations.- - -The book as a whole is a well-balanced exposition that can be recommended to all those who want to gain a thorough understanding and proficiency in the recently developed methods. The book, reflecting the current state of the art, can also be used for teaching special courses. (Mathematical Reviews)

Winner of the 1983 National Book Award, The Mathematical Experience conveyed the power and beauty of its topic to a broad audience. The study version added exercises and other classroom aids. This softcover edition includes new epilogues by the original authors.

A compact, moderately general book which encompasses many fluid models of current interest...The book is written very clearly and contains a large number of exercises and their solutions.

This book is devoted to the relation between two different concepts of integrability: the complete integrability of complex analytical Hamiltonian systems and the integrability of complex analytical linear differential equations. For linear differential equations, integrability is made precise within the framework of differential Galois theory. The connection of these two integrability notions is given by the variational equation (i.e. linearized equation) along a particular integral curve of the Hamiltonian system. The underlying heuristic idea, which motivated the main results presented in this monograph, is that a necessary condition for the integrability of a Hamiltonian system is the integrability of the variational equation along any of its particular integral curves. This idea led to the algebraic non-integrability criteria for Hamiltonian systems. These criteria can be considered as generalizations of classical non-integrability results by Poincare and Liapunov, as well as more recent results by Ziglin and Yoshida. Thus, by means of the differential Galois theory it is not only possible to understand all these approaches in a unified way but also to improve them. Several important applications are also included: homogeneous potentials, Bianchi IX cosmological model, three-body problem, Henon-Heiles system, etc.The book is based on the original joint research of the author with J.M. Peris, J.P. Ramis and C. Simo, but an effort was made to present these achievements in their logical order rather than their historical one. The necessary background on differential Galois theory and Hamiltonian systems is included, and several new problems and conjectures which open new lines of research are proposed.

Singularity theory is a far-reaching extension of maxima and minima investigations of differentiable functions, with implications for many different areas of mathematics, engineering (catastrophe theory and the theory of bifurcations), and science.

Presents the tools for studying local bifurcations of limit cycles in families of planar vector fields. This work offers an introduction to such methods as division of an analytic family of functions in its ideal of coefficients, and asymptotic expansion of non-differentiable return maps and desingularisation.

This three-volume work contains articles collected on the occasion of Alexander Grothendieck's sixtieth birthday and originally published in 1990. The articles were offered as a tribute to one of the world's greatest living mathematicians.

This book introduces the notions and methods of formal logic from a computer science standpoint, covering propositional logic, predicate logic, and foundations of logic programming. It presents applications and themes of computer science research such as resolution, automated deduction, and logic programming in a rigorous but readable way.

This text covers differentiable manifolds, global calculus, differential geometry, and related topics constituting a core of information for the first or second year graduate student preparing for advanced courses and seminars in differential topology and geometry.

Indiscrete Thoughts gives a glimpse into a world that has seldom been described - that of science and technology as seen through the eyes of a mathematician. Cherished myths are debunked along the way as Gian-Carlo Rota takes pleasure in portraying, warts and all, some of the great scientific personalities of the period.

This softcover book summarizes Lyapunov design techniques for nonlinear systems and raises important issues concerning large-signal robustness and performance. The researcher who wishes to enter the field of robust nonlinear control could use this book as a source of new research topics.

Automorphic forms on the upper half plane have been studied for a long time. He extended Hecke's relation between automorphic forms and Dirichlet series to real analytic automorphic forms.

n This book deals with several aspects of fractal geometry in ? which are closely connected with Fourier analysis, function spaces, and appropriate (pseudo)differ- tial operators. It emerged quite recently that some modern techniques in the theory of function spaces are intimately related to methods in fractal geometry. Special attention is paid to spectral properties of fractal (pseudo)differential operators; in particular we shall play the drum with a fractal layer. In some sense this book may be considered as the fractal twin of [ET96], where we developed adequate methods to handle spectral problems of degenerate n pseudodifferential operators in ? and in bounded domains. Besides a few special properties of function spaces we relied there on sharp estimates of entropy numbers of compact embeddings between these spaces and their relations to the distribution of eigenvalues. Some of the main assertions of the present book are based on just these techniques but now in a fractal setting. Since virtually nothing of these new methods is available in literature, a substantial part of what we have to say deals with recent developments in the theory of function spaces, also for their own sake. In this respect the book might also be considered as a continuation of [Tri83] and [Tri92].

Schroedinger Equations and Diffusion Theory addresses the question "What is the Schroedinger equation?" in terms of diffusion processes, and shows that the Schroedinger equation and diffusion equations in duality are equivalent.

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. It is directed to mathematicians and (theoretical) physicists interested in the topics indicated and, in particular, how they are interrelated. - - - The book under review can be regarded as a continuation of [his book on "e;Fractals and spectra"e;, 1997] (...) There are many sections named: comments, preparations, motivations, discussions and so on. These parts of the book seem to be very interesting and valuable. They help the reader to deal with the main course. (Mathematical Reviews)

This book gives an exposition of the fundamentals of the theory of linear representations of ?nite and compact groups, as well as elements of the t- ory of linear representations of Lie groups. As an application we derive the Laplace spherical functions. The book is based on lectures that I delivered in the framework of the experimental program at the Mathematics-Mechanics Faculty of Moscow State University and at the Faculty of Professional Skill Improvement. My aim has been to give as simple and detailed an account as possible of the problems considered. The book therefore makes no claim to completeness. Also, it can in no way give a representative picture of the modern state of the ?eld under study as does, for example, the monograph of A. A. Kirillov [3]. For a more complete acquaintance with the theory of representations of ?nite groups we recommend the book of C. W. Curtis and I. Reiner [2], and for the theory of representations of Lie groups, that of M. A. Naimark [6]. Introduction The theory of linear representations of groups is one of the most widely - pliedbranchesof algebra. Practically every timethatgroupsareencountered, their linear representations play an important role. In the theory of groups itself, linear representations are an irreplaceable source of examples and a tool for investigating groups. In the introduction we discuss some examples and en route we introduce a number of notions of representation theory. 0. Basic Notions 0. 1.

1. Historical Remarks Convex Integration theory, ?rst introduced by M. Gromov [17], is one of three general methods in immersion-theoretic topology for solving a broad range of problems in geometry and topology. The other methods are: (i) Removal of Singularities, introduced by M. Gromov and Y. Eliashberg [8]; (ii) the covering homotopy method which, following M. Gromov's thesis [16], is also referred to as the method of sheaves. The covering homotopy method is due originally to S. Smale [36] who proved a crucial covering homotopy result in order to solve the classi?cation problem for immersions of spheres in Euclidean space. These general methods are not linearly related in the sense that succ- sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Con- quently, each method has a range of applications to problems in topology that are best suited to its particular insight. For example, a distinguishing feature of ConvexIntegrationtheoryisthatitappliestosolveclosed relationsinjetspaces, including certain general classes of underdetermined non-linear systems of par- 1 tial di?erential equations. As a case of interest, the Nash-Kuiper C -isometric immersion theorem can be reformulated and proved using Convex Integration theory (cf. Gromov [18]). No such results on closed relations in jet spaces can be proved by means of the other two methods. On the other hand, many classical results in immersion-theoretic topology, such as the classi?cation of immersions, are provable by all three methods.

1 Aim and General Description of the Anthology The purpose of this anthology is to introduce the English speaking public to the wide spectrum of texts authored predominently by physicists portraying the ac tual and perceived role of physics in the Nazi state.

Generalized Polygons is the first book to cover, in a coherent manner, the theory of polygons from scratch. In particular, it fills elementary gaps in the literature and gives an up-to-date account of current research in this area, including most proofs, which are often unified and streamlined in comparison to the versions generally known. Generalized Polygons will be welcomed both by the student seeking an introduction to the subject as well as the researcher who will value the work as a reference. In particular, it will be of great value for specialists working in the field of generalized polygons (which are, incidentally, the rank 2 Tits-buildings) or in fields directly related to Tits-buildings, incidence geometry and finite geometry. The approach taken in the book is of geometric nature, but algebraic results are included and proven (in a geometric way!). A noteworthy feature is that the book unifies and generalizes notions, definitions and results that exist for quadrangles, hexagons, octagons - in the literature very often considered separately - to polygons. Many alternative viewpoints given in the book heighten the sense of beauty of the subject and help to provide further insight into the matter.

Although much of classical ergodic theory is concerned with single transformations and one-parameter flows, the subject inherits from statistical mechanics not only its name, but also an obligation to analyze spatially extended systems with multidimensional symmetry groups. However, the wealth of concrete and natural examples which has contributed so much to the appeal and development of classical dynamics, is noticeably absent in this more general theory. The purpose of this book is to help remedy this scarcity of explicit examples by introducing a class of continuous Zd-actions diverse enough to exhibit many of the new phenomena encountered in the transition from Z to Zd, but which nevertheless lends itself to systematic study: the Zd-actions by automorphisms of compact, abelian groups. One aspect of these actions, not surprising in itself but quite striking in its extent and depth nonetheless, is the connection with commutative algebra and arithmetical algebraic geometry. The algebraic framework resulting from this connection allows the construction of examples with a variety of specified dynamical properties, and by combining algebraic and dynamical tools one obtains a quite detailed understanding of this class of Zd-actions.

Combining algebraic groups and number theory, this volume gathers material from the representation theory of this group for the first time, doing so for both local (Archimedean and non-Archimedean) cases as well as for the global number field case.

In a detailed and comprehensive introduction to the theory of plane algebraic curves, the authors examine this classical area of mathematics that both figured prominently in ancient Greek studies and remains a source of inspiration and a topic of research to this day. Arising from notes for a course given at the University of Bonn in Germany, "e;Plane Algebraic Curves"e; reflects the authorsE concern for the student audience through its emphasis on motivation, development of imagination, and understanding of basic ideas. As classical objects, curves may be viewed from many angles. This text also provides a foundation for the comprehension and exploration of modern work on singularities. --- In the first chapter one finds many special curves with very attractive geometric presentations the wealth of illustrations is a distinctive characteristic of this book and an introduction to projective geometry (over the complex numbers). In the second chapter one finds a very simple proof of Bezout's theorem and a detailed discussion of cubics. The heart of this book and how else could it be with the first author is the chapter on the resolution of singularities (always over the complex numbers). (...) Especially remarkable is the outlook to further work on the topics discussed, with numerous references to the literature. Many examples round off this successful representation of a classical and yet still very much alive subject. (Mathematical Reviews)

This book deals with evolutionary systems whose equation of state can be formulated as a linear Volterra equation in a Banach space. The main feature of the kernels involved is that they consist of unbounded linear operators. The aim is a coherent presentation of the state of art of the theory including detailed proofs and its applications to problems from mathematical physics, such as viscoelasticity, heat conduction, and electrodynamics with memory. The importance of evolutionary integral equations which form a larger class than do evolution equations stems from such applications and therefore special emphasis is placed on these. A number of models are derived and, by means of the developed theory, discussed thoroughly. An annotated bibliography containing 450 entries increases the book's value as an incisive reference text. ---This excellent book presents a general approach to linear evolutionary systems, with an emphasis on infinite-dimensional systems with time delays, such as those occurring in linear viscoelasticity with or without thermal effects. It gives a very natural and mature extension of the usual semigroup approach to a more general class of infinite-dimensional evolutionary systems. This is the first appearance in the form of a monograph of this recently developed theory. A substantial part of the results are due to the author, or are even new. ( It is not a book that one reads in a few days. Rather, it should be considered as an investment with lasting value. (Zentralblatt MATH)In this book, the author, who has been at the forefront of research on these problems for the last decade, has collected, and in many places extended, the known theory for these equations. In addition, he has provided a framework that allows one to relate and evaluate diverse results in the literature. (Mathematical Reviews)This book constitutes a highly valuable addition to the existing literature on the theory of Volterra (evolutionary) integral equations and their applications in physics and engineering. ( and for the first time the stress is on the infinite-dimensional case. (SIAM Reviews)

Instead, A Probability Path is designed for those requiring a deep understanding of advanced probability for their research in statistics, applied probability, biology, operations research, mathematical finance and engineering.