This two volume set presents over 50 of the most groundbreaking contributions of Menahem M Schiffer. All of the reprints of Schiffer's works herein have extensive annotation and invited commentaries, giving new clarity and insight into the impact and legacy of Schiffer's work. A complete bibliography and brief biography make this a rounded and invaluable reference.

This monograph presents a collection of results, observations, and examples related to dynamical systems described by linear and nonlinear ordinary differential and difference equations. In particular, dynamical systems that are susceptible to analysis by the Liapunov approach are considered. The naive observation that certain "diagonal-type" Liapunov functions are ubiquitous in the literature attracted the attention of the authors and led to some natural questions. Why does this happen so often? What are the spe cial virtues of these functions in this context? Do they occur so frequently merely because they belong to the simplest class of Liapunov functions and are thus more convenient, or are there any more specific reasons? This monograph constitutes the authors' synthesis of the work on this subject that has been jointly developed by them, among others, producing and compiling results, properties, and examples for many years, aiming to answer these questions and also to formalize some of the folklore or "cul ture" that has grown around diagonal stability and diagonal-type Liapunov functions. A natural answer to these questions would be that the use of diagonal type Liapunov functions is frequent because of their simplicity within the class of all possible Liapunov functions. This monograph shows that, although this obvious interpretation is often adequate, there are many in stances in which the Liapunov approach is best taken advantage of using diagonal-type Liapunov functions. In fact, they yield necessary and suffi cient stability conditions for some classes of nonlinear dynamical systems."

The book constructs explicitly the fundamental solution of the sub-Laplacian operator for a family of model domains in Cn+1. This type of domain is a good point-wise model for a Cauchy-Rieman (CR) manifold with diagonalizable Levi form. Qualitative results for such operators have been studied extensively, but exact formulas are difficult to derive. Exact formulas are closely related to the underlying geometry and lead to equations of classical types such as hypergeometric equations and Whittaker's equations.

Simple Ordinary Differential Equations may have solutions in terms of power series whose coefficients grow at such a rate that the series has a radius of convergence equal to zero. In fact, every linear meromorphic system has a formal solution of a certain form, which can be relatively easily computed, but which generally involves such power series diverging everywhere. In this book the author presents the classical theory of meromorphic systems of ODE in the new light shed upon it by the recent achievements in the theory of summability of formal power series.

Beginning with the works of N.N.Krasovskii [81, 82, 83], which clari fied the functional nature of systems with delays, the functional approach provides a foundation for a complete theory of differential equations with delays. Based on the functional approach, different aspects of time-delay system theory have been developed with almost the same completeness as the corresponding field of ODE (ordinary differential equations) the ory. The term functional differential equations (FDE) is used as a syn onym for systems with delays 1. The systematic presentation of these re sults and further references can be found in a number of excellent books [2, 15, 22, 32, 34, 38, 41, 45, 50, 52, 77, 78, 81, 93, 102, 128]. In this monograph we present basic facts of i-smooth calculus ~ a new differential calculus of nonlinear functionals, based on the notion of the invariant derivative, and some of its applications to the qualitative theory of functional differential equations. Utilization of the new calculus is the main distinction of this book from other books devoted to FDE theory. Two other distinguishing features of the volume are the following: - the central concept that we use is the separation of finite dimensional and infinite dimensional components in the structures of FDE and functionals; - we use the conditional representation of functional differential equa tions, which is convenient for application of methods and constructions of i~smooth calculus to FDE theory.

The main theme is the integration of the theory of linear PDE and the theory of finite difference and finite element methods. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. The chapters on elliptic equations are preceded by a chapter on the two-point boundary value problem for ordinary differential equations. Similarly, the chapters on time-dependent problems are preceded by a chapter on the initial-value problem for ordinary differential equations. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. The required background on linear functional analysis and Sobolev spaces is reviewed in an appendix. The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering.

JeanVaillant L'oeuvre de Jean Leray est originale et profonde; ses theoremes et ses theories sont au coeur des recherches mathematiques actuelles: la beaute de chacun de ses travaux ne se divise pas. Son cours de Princeton, sous forme de notes en anglais (et d'une traduction en russe) en est une belle illustration: ce cours presente les equations aux derivees partielles a partir de la transformation de Laplace et du theoreme de Cauchy-Kowaleska et contient l'essentiel de nombreusesrecherchesmodernes. Lerayavaitpourbutderesoudreunprobleme, souvent d'origine mecanique ou physique - qui se pose, et non qu'on se pose -, de demontrer un theoreme; il construit alors son oeuvre de facon complete et essentiellement intrinseque. En fait, Leray construit une theorie dont l'extension tient a son origine naturelle, l'acuite, la perfection, la profondeur d'esprit de son auteur;enmemetempsildominelescalculs,qu'ilmeneavecplaisiretelegance: "Il n'y a pas de mathematiques sans calculs" disait-il. La science etait au centre de la vie de Jean Leray. Il s'inquietait de sa sauvegarde. Rappelons quelques phrases de ses textes de 1974: "D'ailleurs la science ne s'apprend pas: elle se comprend. Elle n'est pas lettre morte et les livres n'assurent pas sa perennite; elle est une pensee vivante. Pour la maitriser notre esprit doit, habilement guide, la redecouvrir de meme que notre corps a du revivre dans le sein mat- nel, toute l'evolution qui crea notre espece. Aussi n'y a-t-il qu'une facon ef?cace d'enseigner les sciences et les techniques: transmettre l'esprit de recherche.

This book contains a systematic study of ecological communities of two or three interacting populations. Starting from the Lotka-Volterra system, various regulating factors are considered, such as rates of birth and death, predation and competition. The different factors can have a stabilizing or a destabilizing effect on the community, and their interplay leads to increasingly complicated behavior. Studying and understanding this path to greater dynamical complexity of ecological systems constitutes the backbone of this book. On the mathematical side, the tool of choice is the qualitative theory of dynamical systems - most importantly bifurcation theory, which describes the dependence of a system on the parameters. This approach allows one to find general patterns of behavior that are expected to be observed in ecological models. Of special interest is the reaction of a given model to disturbances of its present state, as well as to changes in the external conditions. This leads to the general idea of "dangerous boundaries" in the state and parameter space of an ecological system. The study of these boundaries allows one to analyze and predict qualitative and often sudden changes of the dynamics - a much-needed tool, given the increasing antropogenic load on the biosphere.As a spin-off from this approach, the book can be used as a guided tour of bifurcation theory from the viewpoint of application. The interested reader will find a wealth of intriguing examples of how known bifurcations occur in applications. The book can in fact be seen as bridging the gap between mathematical biology and bifurcation theory.

This text emphasizes the physical interpretation of mathematical solutions and introduces applied mathematics while presenting differential equations. Coverage includes Fourier series, orthogonal functions, boundary value problems, Green's functions, and transform methods. This text is ideal for students in science, engineering, and applied mathematics.

Ne as' book "Direct Methods in the Theory of Elliptic Equations," published 1967 in French, has become a standard reference for the mathematical theory of linear elliptic equations and systems. This English edition, translated by G. Tronel and A. Kufner, presents Ne as' work essentially in the form it was published in 1967. It gives a timeless and in some sense definitive treatment of a number issues in variational methods for elliptic systems and higher order equations. The text is recommended to graduate students of partial differential equations, postdoctoral associates in Analysis, and scientists working with linear elliptic systems. In fact, any researcher using the theory of elliptic systems will benefit from having the book in his library.

The volume gives a self-contained presentation of the elliptic theory based on the "direct method," also known as the variational method. Due to its universality and close connections to numerical approximations, the variational method has become one of the most important approaches to the elliptic theory. The method does not rely on the maximum principle or other special properties of the scalar second order elliptic equations, and it is ideally suited for handling systems of equations of arbitrary order. The prototypical examples of equations covered by the theory are, in addition to the standard Laplace equation, Lame's system of linear elasticity and the biharmonic equation (both with variable coefficients, of course). General ellipticity conditions are discussed and most of the natural boundary condition is covered. The necessary foundations of the function space theory are explained along the way, in an arguably optimal manner. The standard boundary regularity requirement on the domains is the Lipschitz continuity of the boundary, which "when going beyond the scalar equations of second order" turns out to be a very natural class. These choices reflect the author's opinion that the Lame system and the biharmonic equations are just as important as the Laplace equation, and that the class of the domains with the Lipschitz continuous boundary (as opposed to smooth domains) is the most natural class of domains to consider in connection with these equations and their applications."

This book presents the basic algorithms, the main theoretical results, and some applications of spectral methods. Particular attention is paid to the applications of spectral methods to nonlinear problems arising in fluid dynamics, quantum mechanics, weather prediction, heat conduction and other fields.The book consists of three parts. The first part deals with orthogonal approximations in Sobolev spaces and the stability and convergence of approximations for nonlinear problems, as the mathematical foundation of spectral methods. In the second part, various spectral methods are described, with some applications. It includes Fourier spectral method, Legendre spectral method, Chebyshev spectral method, spectral penalty method, spectral vanishing viscosity method, spectral approximation of isolated solutions, multi-dimensional spectral method, spectral method for high-order equations, spectral-domain decomposition method and spectral multigrid method. The third part is devoted to some recent developments of spectral methods, such as mixed spectral methods, combined spectral methods and spectral methods on the surface.

This comprehensive book presents a rigorous and state-of-the-art treatment of variational inequalities and complementarity problems in finite dimensions. This class of mathematical programming problems provides a powerful framework for the unified analysis and development of efficient solution algorithms for a wide range of equilibrium problems in economics, engineering, finance, and applied sciences. New research material and recent results, not otherwise easily accessible, are presented in a self-contained and consistent manner. The book is published in two volumes, with the first volume concentrating on the basic theory and the second on iterative algorithms. Both volumes contain abundant exercises and feature extensive bibliographies. Written with a wide range of readers in mind, including graduate students and researchers in applied mathematics, optimization, and operations research as well as computational economists and engineers, this book will be an enduring reference on the subject and provide the foundation for its sustained growth.

The Complex Variable Boundary Element Method (CVBEM) has an important role to play in a number of technical engineering situations and can be a tremendous help to scholars and practitioners preoccupied with solving problems in areas such as heat transport, structural mechanics and river hydraulics. As well as describing the extremely useful applications of this method, the authors explain the mathematical background to the CVBEM, which is vital to understanding the subject as a whole. Advances in the Complex Variable Boundary Element Method is the most comprehensive of books on this subject, bringing together ten years of work and boasting the latest news in CVBEM technology. It will be of particular interest to those concerned with solving technical engineering problems - scientists, graduate students, computer programmers and those working in industry may all find the book helpful.

The Centre de recherches mathCmatiques (CRM) was created in 1968 by the Universite de Montreal to promote research in the mathematical sci- ences. It is now a national institute that hosts several groups, holds special theme years, summer schools, workshops, postdoctoral program. The focus of its scientific activities ranges from pure to applied mathematics, and includes satistics, theoretical computer science, mathematical methods in biology and life sciences, and mathematical and theoretical physics. The CRM also promotes collaboration between mathematicians and industry. It is subsidized by the Natural Sciences and Engineering Research Council of Canada, the Fonds FCAR od the Province of Quebec, the Canadian Institute for Advanced Research and has private endowments. Current ac- tivities, fellowships, and annual reports can be found on the CRM web page at http://www . CRM. UMontreal. CAl. The CRM Series in Mathematical Physics will publish monographs, lec- ture notes, and proceedings base on research pursued and events held at the Centre de recherches mathematiques. Yvan Saint-Aubin Montreal Preface The subject of this three-week school was the explicit integration, that is, analytical as opposed to numerical, of all kinds of nonlinear differential equations (ordinary differential, partial differential, finite difference). The result of such integration is ideally the "general solution," but there are numerous physical systems for which only a particular solution is accessible, for instance the solitary wave of the equation of Kuramoto and Sivashinsky in turbulence.

This text explains how advances in wavelet analysis provide new means for multiresolution analysis and describes its wide array of powerful tools. The book covers such topics as: the variations of the windowed Fourier transform; constructions of special waveforms suitable for specific tasks; the use of redundant representations in reconstruction and enhancement; applications of efficient numerical compression as a tool for fast numerical analysis; and approximation properties of various waveforms in different contexts.

to the English Translation This is a concise guide to basic sections of modern functional analysis. Included are such topics as the principles of Banach and Hilbert spaces, the theory of multinormed and uniform spaces, the Riesz-Dunford holomorphic functional calculus, the Fredholm index theory, convex analysis and duality theory for locally convex spaces. With standard provisos the presentation is self-contained, exposing about a h- dred famous "named" theorems furnished with complete proofs and culminating in the Gelfand-Nalmark-Segal construction for C*-algebras. The first Russian edition was printed by the Siberian Division of "Nauka" P- lishers in 1983. Since then the monograph has served as the standard textbook on functional analysis at the University of Novosibirsk. This volume is translated from the second Russian edition printed by the Sobolev Institute of Mathematics of the Siberian Division of the Russian Academy of Sciences. in 1995. It incorporates new sections on Radon measures, the Schwartz spaces of distributions, and a supplementary list of theoretical exercises and problems. This edition was typeset using AMS-'lEX, the American Mathematical Society's 'lEX system. To clear my conscience completely, I also confess that: = stands for the definor, the assignment operator, signifies the end of the proof."

The present book is the second of the two volume Proceedings of the Mark Krein International Conference on Operator Theory and Applications. This conference, which was dedicated to the 90th Anniversary of the prominent mathematician Mark Krein, was held in Odessa, Ukraine from 18-22 August, 1997. The conference focused on the main ideas, methods, results, and achievements of M. G. Krein. This second volume is devoted to operator theory and related topics. It opens with the bibliography of M. G. Krein and a number of survey papers about his work. The main part of the book consists of original research papers presenting the state of the art in operator theory and its applications. The first volume of these proceedings, entitled Differential Operators and related Topics, concerns the other aspects of the conference. The two volumes will be of interest to a wide-range of readership in pure and applied mathematics, physics and engineering sciences. Table of Contents Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Bibliography of Mark Grigorevich Krein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Review papers: M. G. Krein's Contributions to Prediction Theory H. Dym M. G. Krein's Contribution to the Moment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 AA Nudelman Research Papers: Solution of the Truncated Matrix Hamburger Moment Problem according to M. G. Krein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Y. M. Adamyan and I. M. Tkachenko Extreme Points of a Positive Operator Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 T. Ando M-accretive Extensions of Sectorial Operators and Krein Spaces . . . . . . . . . 67 Y. M. Arlinskii A Simple Proof of the Continuous Commutant Lifting Theorem . . . . . . . . . . 83 R. Bruzual and M.

This volume consists of eight papers on new advances in interpolation theory for matrix functions and completion theory for matrices and operators. Much emphasis is placed on different interpolation and completion problems when the interpolant is estimated in two different norms. The book also focusses on the study of the spectra of different completions of 2 x 2 block matrices when originally all entries are specified except the lower left corner. A third theme concerns two-sided tangential interpolation problems for real rational matrix functions, and also for the time varying case. A tangential moment problem is also analyzed. All papers deal with related problems of modern matrix analysis, operator theory, complex analysis and system theory and will appeal to a wide group of mathematicians and engineers. The material can be used for advance courses and seminars. Contents: Editorial Introduction ? D. Alpay/P. Loubaton: The tangential trigonometric moment problem on an interval and related topics ? M. Bakonyi/V.G. Kaftal/G. Weiss/H.J. Woerdeman: Maximum entropy and joint norm bounds for operator extensions ? J.A. Ball/I. Gohberg/M.A. Kaashoek: Bitangential interpolation for input-output operators of time varying systems: the discrete time case ? J.A. Ball/I. Gohberg/L. Rodman: Two-sided tangential interpolation of real rational matrix functions ? H. Du/C. Gu: On the spectra of operator completion problems ? C. Foias/A.E. Frazho/W.S. Li: The exact H2 estimate for the central H interpolant ? A.E. Frazho/s.M. Kherat: On mixed H2 - H tangential interpolation ? I. Gohberg/C.Gu: On a completion problem for matrices

This book contains a collection of research articles and surveys on recent developments on operator theory as well as its applications covered in the IWOTA 2011 conference held at Sevilla University in the summer of 2011. The topics include spectral theory, differential operators, integral operators, composition operators, Toeplitz operators, and more. The book also presents a large number of techniques in operator theory.

This volume of papers presented at the conference in honor of Calixto P. Calderon by his friends, colleagues, and students is intended to make the mathematical community aware of his important scholarly and research contributions in contemporary Harmonic Analysis and Mathematical Models applied to Biology and Medicine, and to stimulate further research in the future in this area of pure and applied mathematics.

The past decade has witnessed the rapid development of a new mathematical tool, called wavlet analysis, for analyzing complex signals. It has begin to play a serious role in applications ranging from communications to geophysics, and from simulations to image processing. Like Fourier analysis (of which it is a generalization), or musical notation, wavelet analysis provides a method for representing a set of complex phenomena in a simpler, more compact, and thus more efficient manner. This text introduces the ideas and methods of wavelet analysis, relates them to previously known methods in mathematics and engineering, and shows how to apply wavelet analysis to digital signal processing. It begins by describing the multiscale (sometimes called "fractal") nature of information in many aspects of thereal world; it then turns to the algebra and analysis of wavelet matrices, scaling and wavelet functions, and the corresponding analysis of square-integrable functins on a space. The discussion then turns from the continuous to the discrete and shows how a properly selected set of wavelets can be used to represent -- and even differentiate -- a wide range of signls efficiently and effectively. The last part of the book presents a wide variety of applications of wavelets to probllems in data compression and telecommunications.

This monograph, addressing researchers as well as engineers, is devoted to nonclassical thermoelastic modelling of the nonlinear dynamics of shells. Differential equations of different dimensionality and different type have to be combined and nonlinearities of different geometrical, physical or elasto-plastic categories are addressed. Special emphasis is given to the Bubnov--Galerkin method. It can be applied to many problems in the theory of plates and shells, even those with very complex geometries, holes and various boundary conditions. The authors made every effort to keep the text intelligible for both practitioners and graduate students, although they offer a rigorous treatment of both purely mathematical and numerical approaches presented so that the reader can understand, analyse and track the nonlinear dynamics of spatial systems (shells) with thermomechanical behaviours.

This book combining wavelets and the world of the spectrum focuses on recent developments in wavelet theory, emphasizing fundamental and relatively timeless techniques that have a geometric and spectral-theoretic flavor. The exposition is clearly motivated and unfolds systematically, aided by numerous graphics.This self-contained book deals with important applications to signal processing, communications engineering, computer graphics algorithms, qubit algorithms and chaos theory, and is aimed at a broad readership of graduate students, practitioners, and researchers in applied mathematics and engineering. The book is also useful for other mathematicians with an interest in the interface between mathematics and communication theory.

Thisvolumeofthe Operator Theory: Advances and Applications series (OTAA) isthe ?rst volume of a new subseries. This subseries is dedicated to connections between the theory of linear operators and the mathematical theory of linear systems and is named Linear Operators and Linear Systems (LOLS).Asthe- isting subseries Advances in Partial Di?erential Equations (ADPE), the new s- series will continue the traditions of the OTAA series and keep the high quality of the volumes. The editors of the new subseries are: Daniel Alpay (Beer-Sheva, - rael), Joseph Ball (Blacksburg, Virginia, USA) and Andr e Ran (Amsterdam, The Netherlands). In the last 25-30 years, Mathematical System Theory developed in an ess- tial way. A large part of this development was connected with the use of the state space method. Let us mention for instance the "theory of H control". The state ? space method allowed to introduce in system theory the modern tools of matrix and operator theory. On the other hand the state space approach had an imp- tant impact on Algebra, Analysis and Operator Theory. In particular it allowed to solve explicitly some problems from interpolation theory, theory of convolution equations, inverse problems for canonical di?erential equations and their discrete analogs. All these directions are planned to be present in the subseries LOLS. The editors and the publisher are inviting authors to submit their manuscripts for publication in this subseries.

This book concentrates on the topic of evaluation of Jacobians in some specific linear as well as nonlinear matrix transformations, in the real and complex cases, which are widely applied in the statistical, physical, engineering, biological and social sciences. It aims to develop some techniques systematically so that anyone with a little exposure to multivariable calculus can easily follow the steps and understand the various methods by which the Jacobians in complicated matrix transformations are evaluated. The material is developed slowly, with lots of worked examples, aimed at self-study. Some exercises are also given, at the end of each section.The book is a valuable reference for statisticians, engineers, physicists, econometricians, applied mathematicians and people working in many other areas. It can be used for a one-semester graduate level course on Jacobians and functions of matrix argument.