Bøger i Modern Birkhauser Classics serien

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.

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.

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.

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.

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)

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.

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.

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 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.

The discoveries of the last decades have opened new perspectives for the old field of Hamiltonian systems and led to the creation of a new field: symplectic topology. Surprising rigidity phenomena demonstrate that the nature of symplectic mappings is very different from that of volume preserving mappings. This raises new questions, many of them still unanswered. On the other hand, analysis of an old variational principle in classical mechanics has established global periodic phenomena in Hamiltonian systems. As it turns out, these seemingly different phenomena are mysteriously related. One of the links is a class of symplectic invariants, called symplectic capacities. These invariants are the main theme of this book, which includes such topics as basic symplectic geometry, symplectic capacities and rigidity, periodic orbits for Hamiltonian systems and the action principle, a bi-invariant metric on the symplectic diffeomorphism group and its geometry, symplectic fixed point theory, the Arnold conjectures and first order elliptic systems, and finally a survey on Floer homology and symplectic homology. The exposition is self-contained and addressed to researchers and students from the graduate level onwards.

- Acta Scientiarum MathematicarumAvramov lecture: "... - Zentralblatt MATHValla lecture: "... since he is an acknowledged expert on Hilbert functions and since his interest has been so broad, he has done a superb job in giving the readers a lively picture of the theory."

This work explains early results of the theory of continuous maps of an interval to itself to mathematicians and theoretical physicists, and aims to inspire further inquiry into these phenomena of beautiful regularity, which often appear near chaotic systems.

The book deals with the two scales Bsp,q and Fsp,q of spaces of distributions, where -

This book is a well-informed and detailed analysis of the problems and development of algebraic topology, from Poincare and Brouwer to Serre, Adams, and Thom. This book demonstrates in the case of topology how these obstacles can be overcome, with enlightening results....

The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. As in the first edition, the book includes a self-contained treatment of the prerequisites from algebraic geometry and commutative algebra, as well as basic results on reductive groups.

The name of Bernard Riemann is well known to mathematicians and physicists around the world. His name is indelibly stamped on the literature of mathematics and physics. This remarkable work, rich in insight and scholarship, is addressed to mathematicians, physicists, and philosophers interested in mathematics.

With its connections to algebra, algebraic geometry, topology, and number theory, it has implications for a wide variety of researchers and students in mathematics. This new edition includes an appendix on algebraic geometry that contains required definitions and results needed to understand the core of the book.

Over the past several decades, we have witnessed a renaissance of theoretical work on the macroscopic behavior of microscopically heterogeneous materials. This work has implications for many physical problems involving optimal design, composite materials, and coherent phase transitions.

"An introduction to the ideas of algebraic geometry in the motivated context of system theory." This describes this two volume work which has been specifically written to serve the needs of researchers and students of systems, control, and applied mathematics. Without sacrificing mathematical rigor, the author makes the basic ideas of algebraic geometry accessible to engineers and applied scientists. The emphasis is on constructive methods and clarity rather than on abstraction. While familiarity with Part I is helpful, it is not essential, since a considerable amount of relevant material is included here.Part I, Scalar Linear Systems and Affine Algebraic Geometry, contains a clear presentation, with an applied flavor , of the core ideas in the algebra-geometric treatment of scalar linear system theory. Part II extends the theory to multivariable systems. After delineating limitations of the scalar theory through carefully chosen examples, the author introduces seven representations of a multivariable linear system and establishes the major results of the underlying theory. Of key importance is a clear, detailed analysis of the structure of the space of linear systems including the full set of equations defining the space. Key topics also covered are the Geometric Quotient Theorem and a highly geometric analysis of both state and output feedback. Prerequisites are the basics of linear algebra, some simple topological notions, the elementary properties of groups, rings, and fields, and a basic course in linear systems. Exercises, which are an integral part of the exposition throughout, combined with an index and extensive bibliography of related literature make this a valuable classroom tool or good self-study resource. The present, softcover reprint is designed to make this classic textbook available to a wider audience."The exposition is extremely clear. In order to motivate the general theory, the author presents a number of examples of two or three input-, two-output systems in detail. I highly recommend this excellent book to all those interested in the interplay between control theory and algebraic geometry." ¿Publicationes Mathematicae, Debrecen"This book is the multivariable counterpart of Methods of Algebraic Geometry in Control Theory, Part I¿. In the first volume the simpler single-input¿single-output time-invariant linear systems were considered and the corresponding simpler affine algebraic geometry was used as the required prerequisite. Obviously, multivariable systems are more difficult and consequently the algebraic results are deeper and less transparent, but essential in the understanding of linear control theory¿. Each chapter contains illustrative examples throughout and terminates with some exercises for further study." ¿Mathematical Reviews

First published in 2003, this is an accessible and thorough treatment of implicit and inverse function theorems and their applications. The book documents and places in context a substantial body of concepts that have played an role in modern mathematics.

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.

This volume is designed as a textbook for an introductory course on wavelet analysis and time-frequency analysis aimed at graduate students or advanced undergraduates in science and engineering.

The author introduces the four representations of a scalar linear system and establishes the major results of a similar theory for multivariable systems appearing in a succeeding volume (Part II: Multivariable Linear Systems and Projective Algebraic Geometry).

This textbook presents the basic concepts and methods of fluid mechanics, including Lagrangian and Eulerian descriptions, tensors of stresses and strains, continuity, momentum, energy, thermodynamics laws, and similarity theory.  The models and their solutions are presented within a context of the mechanics of multiphase media.  The treatment fully utilizes the computer algebra and software system Mathematica® to both develop concepts and help the reader to master modern methods of solving problems in fluid mechanics.Topics and features:Glossary of over thirty Mathematica® computer programsExtensive, self-contained appendix of Mathematica® functions and their useChapter coverage of mechanics of multiphase heterogeneous mediaDetailed coverage of theory of shock waves in gas dynamicsThorough discussion of aerohydrodynamics of ideal and viscous fluidsand gasesComplete worked examples with detailed solutionsProblem-solving approachFoundations of Fluid Mechanics with Applications is a complete and accessible text or reference for graduates and professionals in mechanics, applied mathematics, physical sciences, materials science, and engineering.  It is an essential resource for the study and use of modern solution methods for problems in fluid mechanics and the underlying mathematical models.  The present, softcover reprint is designed to make this classic textbook available to a wider audience.

The conjugate operator method is a powerful recently developed technique for studying spectral properties of self-adjoint operators. One of the purposes of this volume is to present a refinement of the original method due to Mourre leading to essentially optimal results in situations as varied as ordinary differential operators, pseudo-differential operators and N-body Schrodinger hamiltonians. Another topic is a new algebraic framework for the N-body problem allowing a simple and systematic treatment of large classes of many-channel hamiltonians. The monograph will be of interest to research mathematicians and mathematical physicists. The authors have made efforts to produce an essentially self-contained text, which makes it accessible to advanced students. Thus about one third of the book is devoted to the development of tools from functional analysis, in particular real interpolation theory for Banach spaces and functional calculus and Besov spaces associated with multi-parameter C0-groups. Certainly this monograph (containing a bibliography of 170 items) is a well-written contribution to this field which is suitable to stimulate further evolution of the theory. (Mathematical Reviews)

In a book written for mathematicians, teachers of mathematics, and highly motivated students, Harold Edwards has taken a bold and unusual approach to the presentation of advanced calculus.

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.

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)

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)

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.