This unique book on the subject addresses fundamental problems and will be the standard reference for a long time to come. The authors have different scientific origins and combine these successfully, creating a text aimed at graduate students and researchers that can be used for courses and seminars.

The n-dimensionalmetaplectic groupSp(n,R) is the twofoldcoverof the sympl- n n tic group Sp(n,R), which is the group of linear transformations ofX = R xR that preserve the bilinear (alternate) form x y [( ), ( )] =? x, ? + y, ? . (0. 1) ? ? 2 n There is a unitary representation of Sp(n,R)intheHilbertspace L (R ), called the metaplectic representation,the image of which is the groupof transformations generated by the following ones: the linear changes of variables, the operators of multiplication by exponentials with pure imaginary quadratic forms in the ex- nent, and the Fourier transformation; some normalization factor enters the de?- tion of the operators of the ?rst and third species. The metaplectic representation was introduced in a great generality in [28] - special cases had been considered before, mostly in papers of mathematical physics - and it is of such fundamental importancethat the two concepts (the groupand the representation)havebecome virtually indistinguishable. This is not going to be our point of view: indeed, the main point of this work is to show that a certain ?nite covering of the symplectic group (generally of degree n) has another interesting representation, which enjoys analogues of most of the nicer properties of the metaplectic representation. We shall call it the anaplectic representation - other coinages that may come to your mind sound too medical - and shall consider ?rst the one-dimensional case, the main features of which can be described in quite elementary terms.

This book builds upon the earlier volume Problems in Analysis, more than doubling it with a new section of problems on complex analysis. The problems on real analysis from the earlier book have all been checked, and stylistic, typographical, and mathematical errors have been corrected. The problems in complex analysis cover most of the principal topics in the theory of functions of a complex variable. The problems in the book cover, in real analysis: set algebra, measure and topology, real- and complex-valued functions, and topological vector spaces; in complex analysis: polynomials and power series, functions holomorphic in a region, entire functions, analytic continuation, singularities, harmonic functions, families of functions, and convexity theorems.

Quadrature domains were singled out about 30 years ago by D. Aharonov and H.S. Shapiro in connection with an extremal problem in function theory. Since then, a series of coincidental discoveries put this class of planar domains at the center of crossroads of several quite independent mathematical theories, e.g., potential theory, Riemann surfaces, inverse problems, holomorphic partial differential equations, fluid mechanics, operator theory. The volume is devoted to recent advances in the theory of quadrature domains, illustrating well the multi-facet aspects of their nature. The book contains a large collection of open problems pertaining to the general theme of quadrature domains.

During the last twenty-five years, the development of the theory of Banach lattices has stimulated new directions of research in the theory of positive operators and the theory of semigroups of positive operators. In particular, the recent investigations in the structure of the lattice ordered (Banach) algebra of the order bounded operators of a Banach lattice have led to many important results in the spectral theory of positive operators. The contributions contained in this volume were presented as lectures at a conference organized by the Caribbean Mathematics Foundation, and provide an overview of the present state of development of various areas of the theory of positive operators and their spectral properties. This book will be of interest to analysts whose work involves positive matrices and positive operators.

This book presents a coherent framework for understanding the dynamics of piecewise-smooth and hybrid systems. An informal introduction expounds the ubiquity of such models via numerous. The results are presented in an informal style, and illustrated with many examples. The book is aimed at a wide audience of applied mathematicians, engineers and scientists at the beginning postgraduate level. Almost no mathematical background is assumed other than basic calculus and algebra.

In the Part at hand the authors undertake to give a presentation of the historical development of the theory of imbedding of function spaces, of the internal as well as the externals motives which have stimulated it, and of the current state of art in the field, in particular, what regards the methods employed today. The impossibility to cover all the enormous material connected with these questions inevitably forced on us the necessity to restrict ourselves to a limited circle of ideas which are both fundamental and of principal interest. Of course, such a choice had to some extent have a subjective character, being in the first place dictated by the personal interests of the authors. Thus, the Part does not constitute a survey of all contemporary questions in the theory of imbedding of function spaces. Therefore also the bibliographical references given do not pretend to be exhaustive; we only list works mentioned in the text, and a more complete bibliography can be found in appropriate other monographs. O.V. Besov, v.1. Burenkov, P.1. Lizorkin and V.G. Maz'ya have graciously read the Part in manuscript form. All their critical remarks, for which the authors hereby express their sincere thanks, were taken account of in the final editing of the manuscript.

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gu ik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; use Stein spaces. And in addition to this there are and prediction and electrical engineering can such new emerging subdisciplines as "experimental mathematics," "CFD," "completely integrable systems," "chaos, synergetics and large-scale order," which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

This is the second of a five-volume exposition of the main principles of nonlinear functional analysis and its applications to the natural sciences, economics, and numerical analysis. The presentation is self -contained and accessible to the nonspecialist. Part II concerns the theory of monotone operators. It is divided into two subvolumes, II/A and II/B, which form a unit. The present Part II/A is devoted to linear monotone operators. It serves as an elementary introduction to the modern functional analytic treatment of variational problems, integral equations, and partial differential equations of elliptic, parabolic and hyperbolic type. This book also represents an introduction to numerical functional analysis with applications to the Ritz method along with the method of finite elements, the Galerkin methods, and the difference method. Many exercises complement the text. The theory of monotone operators is closely related to Hilbert's rigorous justification of the Dirichlet principle, and to the 19th and 20th problems of Hilbert which he formulated in his famous Paris lecture in 1900, and which strongly influenced the development of analysis in the twentieth century.

This monographs presents new spherical mean value relations for classical boundary value problems of mathematical physics. The derived spherical mean value relations provide equivalent integral formulations of original boundary value problems. Direct and converse mean value theorems are proved for scalar elliptic equations (the Laplace, Helmholtz and diffusion equations), parabolic equations, high-order elliptic equations (biharmonic and metaharmonic equations), and systems of elliptic equations (the Lami equation, systems of diffusion and elasticity equations). In addition, applications to the random walk on spheres method are given.

The book offers a direct and up-to-date introduction to the theory of one-parameter semigroups of linear operators on Banach spaces. It contains the fundamental results of the theory such as the Hille-Yoshida generation theorem, the bounded perturbation theorem, and the Trotter-Kato approximation theorem. It also treats the spectral theory of semigroups and its consequences for the qualitative behavior. The book is intended for students and researchers who want to become acquainted with the concept of semigroups in order to work with it in fields like partial and functional differential equations. Exercises are provided at the end of the chapters.

First-year calculus presented roughly in the order in which it first was discovered. The first two chapters show how the ancient calculations of practical problems led to infinite series, differential and integral calculus and to differential equations, while the establishment of mathematical rigour for these subjects in the 19th century for one and several variables is treated in chapters III and IV. The text is complemented by a large number of examples, calculations and mathematical pictures and will provide stimulating and enjoyable reading for students, teachers, and researchers alike.

For this edition, a number of typographical errors and minor slip-ups have been corrected. In addition, following the persistent encouragement of Olga Oleinik, I have added a new chapter, Chapter 25, which I titled "Recent Results." This chapter is divided into four sections, and in these I have discussed what I consider to be some of the important developments which have come about since the writing of the first edition. Section I deals with reaction-diffusion equations, and in it are described both the work of C. Jones, on the stability of the travelling wave for the Fitz-Hugh-Nagumo equations, and symmetry-breaking bifurcations. Section II deals with some recent results in shock-wave theory. The main topics considered are L. Tartar's notion of compensated compactness, together with its application to pairs of conservation laws, and T.-P. Liu's work on the stability of viscous profiles for shock waves. In the next section, Conley's connection index and connection matrix are described; these general notions are useful in con structing travelling waves for systems of nonlinear equations. The final sec tion, Section IV, is devoted to the very recent results of C. Jones and R. Gardner, whereby they construct a general theory enabling them to locate the point spectrum of a wide class of linear operators which arise in stability problems for travelling waves. Their theory is general enough to be applica ble to many interesting reaction-diffusion systems."

Understanding Real Analysis, Second Edition offers substantial coverage of foundational material and expands on the ideas of elementary calculus to develop a better understanding of crucial mathematical ideas. The text meets students at their current level and helps them develop a foundation in real analysis. The author brings definitions, proofs, examples and other mathematical tools together to show how they work to create unified theory. These helps students grasp the linguistic conventions of mathematics early in the text. The text allows the instructor to pace the course for students of different mathematical backgrounds. Key Features: Meets and aligns with various student backgrounds Pays explicit attention to basic formalities and technical language Contains varied problems and exercises Drives the narrative through questions

¿The present book is a marvelous introduction in the modern theory of manifolds and differential forms. The undergraduate student can closely examine tangent spaces, basic concepts of differential forms, integration on manifolds, Stokes theorem, de Rham- cohomology theorem, differential forms on Riema-nnian manifolds, elements of the theory of differential equations on manifolds (Laplace-Beltrami operators). Every chapter contains useful exercises for the students.¿ ¿ ZENTRALBLATT MATH

Strongly coupled (or cross-diffusion) systems of parabolic and elliptic partial differential equations appear in many physical applications. This book presents a new approach to the solvability of general strongly coupled systems, a much more difficult problem in contrast to the scalar case, by unifying, elucidating and extending breakthrough results obtained by the author, and providing solutions to many open fundamental questions in the theory. Several examples in mathematical biology and ecology are also included. Contents Interpolation Gagliardo-Nirenberg inequalities The parabolic systems The elliptic systems Cross-diffusion systems of porous media type Nontrivial steady-state solutions The duality RBMO( )-H1( )| Some algebraic inequalities Partial regularity

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.

This volume contains the proceedings of the NATO Advanced Research Workshop on "Asymptotic-induced Numerical Methods for Partial Differ ential Equations, Critical Parameters, and Domain Decomposition," held at Beaune (France), May 25-28, 1992. The purpose of the workshop was to stimulate the integration of asymp totic analysis, domain decomposition methods, and symbolic manipulation tools for the numerical solution of partial differential equations (PDEs) with critical parameters. A workshop on the same topic was held at Argonne Na tional Laboratory in February 1990. (The proceedings were published under the title Asymptotic Analysis and the Numerical Solu.tion of Partial Differ ential Equations, Hans G. Kaper and Marc Garbey, eds., Lecture Notes in Pure and Applied Mathematics. Vol. 130, .Marcel Dekker, Inc., New York, 1991.) In a sense, the present proceedings represent a progress report on the topic area. Comparing the two sets of proceedings, we see an increase in the quantity as well as the quality of the contributions. 110re research is being done in the topic area, and the interest covers serious, nontrivial problems. We are pleased with this outcome and expect to see even more advances in the next few years as the field progresses."

This book presents problems and solutions in calculus with curvilinear coordinates. Vector analysis can be performed in different coordinate systems, an optimal system considers the symmetry of the problem in order to reduce calculatory difficulty. The book presents the material in arbitrary orthogonal coordinates, and includes the discussion of parametrization methods as well as topics such as potential theory and integral theorems. The target audience primarily comprises university teachers in engineering mathematics, but the book may also be beneficial for advanced undergraduate and graduate students alike.

Inverse scattering theory is an important area of applied mathematics due to its central role in such areas as medical imaging , nondestructive testing and geophysical exploration. Until recently all existing algorithms for solving inverse scattering problems were based on using either a weak scattering assumption or on the use of nonlinear optimization techniques. The limitations of these methods have led in recent years to an alternative approach to the inverse scattering problem which avoids the incorrect model assumptions inherent in the use of weak scattering approximations as well as the strong a priori information needed in order to implement nonlinear optimization techniques. These new methods come under the general title of qualitative methods in inverse scattering theory and seek to determine an approximation to the shape of the scattering object as well as estimates on its material properties without making any weak scattering assumption and using essentially no a priori information on the nature of the scattering object. This book is designed to be an introduction to this new approach in inverse scattering theory focusing on the use of sampling methods and transmission eigenvalues. In order to aid the reader coming from a discipline outside of mathematics we have included background material on functional analysis, Sobolev spaces, the theory of ill posed problems and certain topics in in the theory of entire functions of a complex variable. This book is an updated and expanded version of an earlier book by the authors published by Springer titled Qualitative Methods in Inverse Scattering Theory Review of Qualitative Methods in Inverse Scattering Theory All in all, the authors do exceptionally well in combining such a wide variety of mathematical material and in presenting it in a well-organized and easy-to-follow fashion. This text certainly complements the growing body of work in inverse scattering and should well suit both new researchers to the field as well as those who could benefit from such a nice codified collection of profitable results combined in one bound volume. SIAM Review, 2006

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."

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."

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.