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.

Functional Analysis has become one of the main branches in Chinese mathematics. Many outstanding contributions and results have been achieved over the past sixty years. This authoritative collection is complementary to Western studies in this field, and seeks to summarise and introduce the historical progress of the development of Functional Analysis in China from the 1940s to the present. A broad range of topics is covered, such as nonlinear functional analysis, linear operator theory, theory of operator algebras, applications including the solvability of some partial differential equations, and special spaces that contain Banach spaces and topological vector spaces. Some of these papers have made a significant impact on the mathematical community worldwide. Audience: This volume will be of interest to mathematicians, physicists and engineers at postgraduate level.

Analysis on Symmetric Cones is the first book to provide a systematic and clear introduction to the theory of symmetric cones, a subject of growing importance in number theory and multivariate analysis. Beginning with an elementary description of the Jordan algebra approach to the geometric and algebraic foundations of the theory, the book goes on to discuss harmonic analysis and special functions associated with symmetric cones, tying these results together with the study of holomorphic functions on bounded symmetric domains of tube type. Written by algebraic geometers, the book contains a detailed exposition of the spherical polynomials, multivariate hypergeometric functions, and invariant differential operators. The approach is based on Jordan algebras; all that is needed from the theory of these is developed in the first few chapters. The book will be read by students and theoreticians in pure mathematics, non-commutative harmonic analysis, Jordan algebras, and multivariate statistics.

The renormalization group (RG) has nowadays achieved the status of a meta-theory, which is a theory about theories. The theory of the RG consists of a set of concepts and methods which can be used to understand phenomena in many different ?elds of physics, ranging from quantum ?eld theory over classical statistical mechanics to nonequilibrium phenomena. RG methods are particularly useful to understand phenomena where ?uctuations involving many different length or time scales lead to the emergence of new collective behavior in complex many-body systems. In view of the diversity of ?elds where RG methods have been successfully applied, it is not surprising that a variety of apparently different implementations of the RG idea have been proposed. Unfortunately, this makes it somewhat dif?cult for beginners to learn this technique. For example, the ?eld-theoretical formulation of the RG idea looks at the ?rst sight rather different from the RG approach pioneered by Wilson, the latter being based on the concept of the effective action which is ite- tively calculated by successive elimination of the high-energy degrees of freedom. Moreover, the Wilsonian RG idea has been implemented in many different ways, depending on the particular problem at hand, and there seems to be no canonical way of setting up the RG procedure for a given problem.

Different Faces of Geometry - edited by the world renowned geometers S. Donaldson, Ya. Eliashberg, and M. Gromov - presents the current state, new results, original ideas and open questions from the following important topics in modern geometry: Amoebas and Tropical Geometry Convex Geometry and Asymptotic Geometric Analysis Differential Topology of 4-Manifolds 3-Dimensional Contact Geometry Floer Homology and Low-Dimensional Topology Kahler Geometry Lagrangian and Special Lagrangian Submanifolds Refined Seiberg-Witten Invariants. These apparently diverse topics have a common feature in that they are all areas of exciting current activity. The Editors have attracted an impressive array of leading specialists to author chapters for this volume: G. Mikhalkin (USA-Canada-Russia), V.D. Milman (Israel) and A.A. Giannopoulos (Greece), C. LeBrun (USA), Ko Honda (USA), P. Ozsvath (USA) and Z. Szabo (USA), C. Simpson (France), D. Joyce (UK) and P. Seidel (USA), and S. Bauer (Germany). "One can distinguish various themes running through the different contributions. There is some emphasis on invariants defined by elliptic equations and their applications in low-dimensional topology, symplectic and contact geometry (Bauer, Seidel, Ozsvath and Szabo). These ideas enter, more tangentially, in the articles of Joyce, Honda and LeBrun. Here and elsewhere, as well as explaining the rapid advances that have been made, the articles convey a wonderful sense of the vast areas lying beyond our current understanding. Simpson's article emphasizes the need for interesting new constructions (in that case of Kahler and algebraic manifolds), a point which is also made by Bauer in the context of 4-manifolds and the "11/8 conjecture". LeBrun's article gives another perspective on 4-manifold theory, via Riemannian geometry, and the challenging open questions involving the geometry of even "well-known" 4-manifolds. There are also striking contrasts between the articles. The authors have taken different approaches: for example, the thoughtful essay of Simpson, the new research results of LeBrun and the thorough expositions with homework problems of Honda. One can also ponder the differences in the style of mathematics. In the articles of Honda, Giannopoulos and Milman, and Mikhalkin, the "geometry" is present in a very vivid and tangible way; combining respectively with topology, analysis and algebra. The papers of Bauer and Seidel, on the other hand, makes the point that algebraic and algebro-topological abstraction (triangulated categories, spectra) can play an important role in very unexpected ways in concrete geometric problems." - From the Preface by the Editors

This book was written expressly to serve as a textbook for a one- or two-semester introductory graduate course in functional analysis. Its (soon to be published) companion volume, Operators on Hilbert Space, is in tended to be used as a textbook for a subsequent course in operator theory. In writing these books we have naturally been concerned with the level of preparation of the potential reader, and, roughly speaking, we suppose him to be familiar with the approximate equivalent of a one-semester course in each of the following areas: linear algebra, general topology, complex analysis, and measure theory. Experience has taught us, however, that such a sequence of courses inevitably fails to treat certain topics that are important in the study of functional analysis and operator theory. For example, tensor products are frequently not discussed in a first course in linear algebra. Likewise for the topics of convergence of nets and the Baire category theorem in a course in topology, and the connections between measure and topology in a course in measure theory. For this reason we have chosen to devote the first ten chapters of this volume (entitled Part I) to topics of a preliminary nature. In other words, Part I summarizes in considerable detail what a student should (and eventually must) know in order to study functional analysis and operator theory successfully."

This volume contains solicited articles by speakers at the workshop ranging from expository surveys to original research papers, each of which carefully refereed. They all bear witness to the very rich mathematics that is connected with the study of elementary operators, may it be multivariable spectral theory, the invariant subspace problem or tensor products of C*-algebras.

In topological measure theory, Radon measures are the most important objects. In the context of locally compact spaces, there are two equivalent canonical definitions. As a set function, a Radon measure is an inner compact regular Borel measure, finite on compact sets. As a functional, it is simply a positive linear form, defined on the vector lattice of continuous real-valued functions with compact support. During the last few decades, in particular because of the developments of modem probability theory and mathematical physics, attention has been focussed on measures on general topological spaces which are no longer locally compact, e.g. spaces of continuous functions or Schwartz distributions. For a Radon measure on an arbitrary Hausdorff space, essentially three equivalent definitions have been proposed: As a set function, it was defined by L. Schwartz as an inner compact regular Borel measure which is locally bounded. G. Choquet considered it as a strongly additive right continuous content on the lattice of compact subsets. Following P.A. Meyer, N. Bourbaki defined a Radon measure as a locally uniformly bounded family of compatible positive linear forms, each defined on the vector lattice of continuous functions on some compact subset.

The current form of modern approximation theory is shaped by many new de velopments which are the subject of this series of conferences. The International Meetings on Approximation Theory attempt to keep track in particular of fun damental advances in the theory of function approximation, for example by (or thogonal) polynomials, (weighted) interpolation, multivariate quasi-interpolation, splines, radial basis functions and several others. This includes both approxima tion order and error estimates, as well as constructions of function systems for approximation of functions on Euclidean spaces and spheres. It is a piece of very good fortune that at all of the IDoMAT meetings, col leagues and friends from all over Europe, and indeed some count ries outside Europe and as far away as China, New Zealand, South Africa and U.S.A. came and dis cussed mathematics at IDoMAT conference facility in Witten-Bommerholz. The conference was, as always, held in a friendly and congenial atmosphere. After each meeting, the delegat es were invited to contribute to the proceed ing's volume, the previous one being published in the same Birkhauser series as this one. The editors were pleased about the quality of the contributions which could be solicited for the book. They are refereed and we should mention our gratitude to the referees and their work."

This second volume of the series Lecture Notes in Applied and Computational Mechanics is the second part of the compendium of reviewed articles presented at the 11th EUROMECH-MECAMAT conference entitled "Mechanics of microstructured solids: cellular materials, fibre reinforced solids and soft tissues", which took place in Torino (Italy) in March 10-14, 2008, at the Museo Regional delle Scienze. This EUROMECH-MECAMAT conference was jointly organized by the Dipartimento di Matematica dell'Universita di Torino, Italy and the INPL Institute (LEMTA, Nancy-Universite, France). Prof. Franco Pastrone and Prof. Jean-Francois Ganghoffer were the co-chairmen.

Most real-world spectrum analysis problems involve the computation of the real-data discrete Fourier transform (DFT), a unitary transform that maps elements N of the linear space of real-valued N-tuples, R , to elements of its complex-valued N counterpart, C , and when carried out in hardware it is conventionally achieved via a real-from-complex strategy using a complex-data version of the fast Fourier transform (FFT), the generic name given to the class of fast algorithms used for the ef?cient computation of the DFT. Such algorithms are typically derived by explo- ing the property of symmetry, whether it exists just in the transform kernel or, in certain circumstances, in the input data and/or output data as well. In order to make effective use of a complex-data FFT, however, via the chosen real-from-complex N strategy, the input data to the DFT must ?rst be converted from elements of R to N elements of C . The reason for choosing the computational domain of real-data problems such N N as this to be C , rather than R , is due in part to the fact that computing equ- ment manufacturers have invested so heavily in producing digital signal processing (DSP) devices built around the design of the complex-data fast multiplier and accumulator (MAC), an arithmetic unit ideally suited to the implementation of the complex-data radix-2 butter?y, the computational unit used by the familiar class of recursive radix-2 FFT algorithms.

This book presents a geometric theory of complex analytic integrals representing hypergeometric functions of several variables. Starting from an integrand which is a product of powers of polynomials, integrals are explained, in an open affine space, as a pair of twisted de Rham cohomology and its dual over the coefficients of local system. It is shown that hypergeometric integrals generally satisfy a holonomic system of linear differential equations with respect to the coefficients of polynomials and also satisfy a holonomic system of linear difference equations with respect to the exponents. These are deduced from Grothendieck-Deligne's rational de Rham cohomology on the one hand, and by multidimensional extension of Birkhoff's classical theory on analytic difference equations on the other.

The 7th International Workshop in Analysis and its Applications (IWAA) was held at the University of Maine, June 1-6, 1997 and featured approxi mately 60 mathematicians. The principal theme of the workshop shares the title of this volume and the latter is a direct outgrowth of the workshop. IWAA was founded in 1984 by Professor Caslav V. Stanojevic. The first meeting was held in the resort complex Kupuri, Yugoslavia, June 1-10, 1986, with two pilot meetings preceding. The Organization Committee to gether with the Advisory Committee (R. P. Boas, R. R. Goldberg, J. P. Kahne) set forward the format and content of future meetings. A certain number of papers were presented that later appeared individually in such journals as the Proceedings of the AMS, Bulletin of the AMS, Mathematis chen Annalen, and the Journal of Mathematical Analysis and its Applica tions. The second meeting took place June 1-10, 1987, at the same location. At the plenary session of this meeting it was decided that future meetings should have a principal theme. The theme for the third meeting (June 1- 10, 1989, Kupuri) was Karamata's Regular Variation. The principal theme for the fourth meeting (June 1-10, 1990, Kupuri) was Inner Product and Convexity Structures in Analysis, Mathematical Physics, and Economics. The fifth meeting was to have had the theme, Analysis and Foundations, organized in cooperation with Professor A. Blass (June 1-10, 1991, Kupuri)."

Non-Archimedean functional analysis, where alternative but equally valid number systems such as p-adic numbers are fundamental, is a fast-growing discipline widely used not just within pure mathematics, but also applied in other sciences, including physics, biology and chemistry. This book is the first to provide a comprehensive treatment of non-Archimedean locally convex spaces. The authors provide a clear exposition of the basic theory, together with complete proofs and new results from the latest research. A guide to the many illustrative examples provided, end-of-chapter notes and glossary of terms all make this book easily accessible to beginners at the graduate level, as well as specialists from a variety of disciplines.

This volume contains the proceedings of the Workshop on app1ications of linear operator theory to systems and networks, which was held at the Weizmann Institute of Science in the third week of June, 19S3,just be fore the MTNS Conference in Beersheva. For a 10ng time these subjects were studied indepen- dent1y by mathematica1 ana1ysts and e1ectrica1 engineers. Never- the1ess, in spite of the lack of communication, these two groups often deve10ped parallel theories, though in different languages, at different levels of genera1ity and typica11y quite different motivations. In the last severa1 years each side has become aware of the work of the other and there is a seeming1y ever- increasing invo1vement of the abstract theories of factorization, extension and interpolation of operators (and operator/matrix va1ued functions) to the design and analysis of systems and net- works. Moreover, the problems encountered in e1ectrica1 engineering have genera ted new mathematica1 problems, new approaches, and usefu1 new formu1ations. The papers contained in this volume constitute a more than representative se1ection of the presented talks and dis- cussion at the workshop, and hopefu11y will also serve to give a reasonably accurate picture of the problems which are under active study today and the techniques which are used to deal with them.

Et mai . ..., si j'avait su comment en revenir. One service mathematics has rendered the human race. It has put common sense back je n'y serais point aIIe.' Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent: therefore we may be sense' . able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series."

This is the first volume of a two volume set that provides a modern account of basic Banach algebra theory including all known results on general Banach *-algebras. This account emphasises the role of *-algebra structure and explores the algebraic results which underlie the theory of Banach algebras and *-algebras. This first volume is an independent, self-contained reference on Banach algebra theory. Each topic is treated in the maximum interesting generality within the framework of some class complex algebras rather than topological algebras. In both volumes proofs are presented in complete detail at a level accessible to graduate students. In addition, the books contain a wealth of historical comments, background material, examples, particularly in noncommutative harmonic analysis, and an extensive bibliography. Together these books will become the standard reference for the general theory of *-algebras.

The main goal of this book is to introduce readers to functional analysis methods, in particular, time dependent analysis, for reliability models. Understanding the concept of reliability is of key importance - schedule delays, inconvenience, customer dissatisfaction, and loss of prestige and even weakening of national security are common examples of results that are caused by unreliability of systems and individuals. The book begins with an introduction to C0-semigroup theory. Then, after a brief history of reliability theory, methods that study the well-posedness, the asymptotic behaviors of solutions and reliability indices for varied reliability models are presented. Finally, further research problems are explored. Functional Analysis Methods for Reliability Models is an excellent reference for graduate students and researchers in operations research, applied mathematics and systems engineering.

Over the last years, stochastic analysis has had an enormous progress with the impetus originating from different branches of mathematics: PDE's and the Malliavin calculus, quantum physics, path space analysis on curved manifolds via probabilistic methods, and more.

This volume contains selected contributions which were presented at the 8th Silivri Workshop on Stochastic Analysis and Related Topics, held in September 2000 in Gazimagusa, North Cyprus.

The topics include stochastic control theory, generalized functions in a nonlinear setting, tangent spaces of manifold-valued paths with quasi-invariant measures, and applications in game theory, theoretical biology and theoretical physics.

Contributors: A.E. Bashirov, A. Bensoussan and J. Frehse, U. Capar and H. Aktuglul, A.B. Cruzeiro and Kai-Nan Xiang, E. Hausenblas, Y. Ishikawa, N. Mahmudov, P. Malliavin and U. Taneri, N. Privault, A.S. Ustunel"

On May 20-24. 1968, a Conference on Functional Analysis and Related Fields was held at the Center for Continuing Education of the University cl Chicago in honor of ProfessoLMARSHALL HARVEY STONE on the occasion of his retirement from active service at the University. The Conference received support from the Air Force Office of Scientific Research under the Grant AFOSR 68-1497. The Organizing committee for this Conference consisted of ALBERTO P. CALDERON, SAUNDERS MACLANE, ROBERT G. POHRER, and FELIX E. BROWDER (Chairman). The present volume contains some of the papers presented at the Conference. nther talks which were presented at the Conference for which papers are noLinduded hereare: K. CHANDRASEKHARAN, "Zeta functions of quadratic fields"; J. L. DooB, "An application of prob ability theory to the Choquet boundary"; HALMOS, "Irreducible operators"; P. R. KADISON, "Strong continuity of operator functions"; L. NIRENBERG, "Intrinsic norms on complex manifolds"; D. SCOTT, "Some problems and recent results in Boolean algebras"; 1. M. SINGER, "A conjecture relating the Reidemeister torsion and the zeta function of the Laplacian." A banquet in honor of Professor STONE was held during the Con ference, with brief talks by S. S. CHERN, A. A. ALBERT, S. MACLANE, E. HEWITT, K. CHANDRASEKHARAN, and F. E. BROWDER (as Toast master), as weH as a response by Professor STONE."

This volume contains a selection of papers by the participants of the 6. International Conference on Probability in Banach Spaces, Sand bjerg, Denmark, June 16-D1, 1986. The conference was attended by 45 participants from several countries. One thing makes this conference completely different from the previous five ones, namely that it was ar ranged jointly in Probability in Banach spaces and Banach space theory with almost equal representation of scientists in the two fields. Though these fields are closely related it seems that direct collaboration between researchers in the two groups has been seldom. It is our feeling that the conference, where the participants were together for five days taking part in lectures and intense discussions of mutual problems, has contributed to a better understanding and closer collaboration in the two fields. The papers in the present volume do not cover all the material pre sented in the lectures; several results covered have been published else where. The sponsors of the conference are: The Carlsberg Foundation, The Danish Natural Science Research Council, The Danish Department of Education, The Department of Mathematics, Odense University, The Department of Mathematics, Aarhus University, The Knudsen Foundation, Odense, Odense University, The Research Foundation of Aarhus University, The Thborg Foundation. The participants and the organizers would like to thank these institu tions for their support. The Organizers. Contents A. de Acosta and M. Ledoux, On the identification of the limits in the law of the iterated logarithm in Banach spaces. . . . ."

The notion of stability of functional equations of several variables in the sense used here had its origins more than half a century ago when S. Ulam posed the fundamental problem and Donald H. Hyers gave the first significant partial solution in 1941. The subject has been revised and de veloped by an increasing number of mathematicians, particularly during the last two decades. Three survey articles have been written on the subject by D. H. Hyers (1983), D. H. Hyers and Th. M. Rassias (1992), and most recently by G. L. Forti (1995). None of these works included proofs of the results which were discussed. Furthermore, it should be mentioned that wider interest in this subject area has increased substantially over the last years, yet the pre sentation of research has been confined mainly to journal articles. The time seems ripe for a comprehensive introduction to this subject, which is the purpose of the present work. This book is the first to cover the classical results along with current research in the subject. An attempt has been made to present the material in an integrated and self-contained fashion. In addition to the main topic of the stability of certain functional equa tions, some other related problems are discussed, including the stability of the convex functional inequality and the stability of minimum points. A sad note. During the final stages of the manuscript our beloved co author and friend Professor Donald H. Hyers passed away."

In July of 1996, the conference Nonlinear Analysis and its Applications in Engineering and Economics took place on the Greek island of Samos, the birthplace of Pythagoras. During this conference, a special session was held on th the occasion of the 50 birthday of the well known mathematician and math ematical economist Professor Charalambos Aliprantis, who, by his numerous friends, is usually called Roko. The story behind this nickname is not quite clear yet; it will be investigated further and will be made public prior to his th 60 birthday. (At this moment we have already found out that it has nothing to do with the famous movie Rocco and his Brothers even though Roko does have two brothers. ) Roko was born on the Greek island of Cephalonia on May 12,1946, and his elementary and secondary school education took place there. At 18 he entered the Mathematics Department at the University of Athens. Upon graduation from the University of Athens he proceeded with his graduate studies at Cal tech, where in 1973 he completed his Ph. D. degree in Mathematics under the supervision of Professor W. A. J. Luxemburg. His research career can be divided into two periods. The first one, till 1981, was devoted entirely to pure mathematics. The other one, after 1981, has been subdivided between pure mathematics and mathematical economics. The main objects of Roko's work in pure mathematics are spaces with order structure (Riesz spaces) and operators acting on them."

One of the major concerns of theoretical computer science is the classifi cation of problems in terms of how hard they are. The natural measure of difficulty of a function is the amount of time needed to compute it (as a function of the length of the input). Other resources, such as space, have also been considered. In recursion theory, by contrast, a function is considered to be easy to compute if there exists some algorithm that computes it. We wish to classify functions that are hard, i.e., not computable, in a quantitative way. We cannot use time or space, since the functions are not even computable. We cannot use Turing degree, since this notion is not quantitative. Hence we need a new notion of complexity-much like time or spac that is quantitative and yet in some way captures the level of difficulty (such as the Turing degree) of a function."

Analytic and Geometric Inequalities and Applications is devoted to recent advances in a variety of inequalities of Mathematical Analysis and Geo metry. Subjects dealt with in this volume include: Fractional order inequalities of Hardy type, differential and integral inequalities with initial time differ ence, multi-dimensional integral inequalities, Opial type inequalities, Gruss' inequality, Furuta inequality, Laguerre-Samuelson inequality with extensions and applications in statistics and matrix theory, distortion inequalities for ana lytic and univalent functions associated with certain fractional calculus and other linear operators, problem of infimum in the positive cone, alpha-quasi convex functions defined by convolution with incomplete beta functions, Chebyshev polynomials with integer coefficients, extremal problems for poly nomials, Bernstein's inequality and Gauss-Lucas theorem, numerical radii of some companion matrices and bounds for the zeros of polynomials, degree of convergence for a class of linear operators, open problems on eigenvalues of the Laplacian, fourth order obstacle boundary value problems, bounds on entropy measures for mixed populations as well as controlling the velocity of Brownian motion by its terminal value. A wealth of applications of the above is also included. We wish to express our appreciation to the distinguished mathematicians who contributed to this volume. Finally, it is our pleasure to acknowledge the fine cooperation and assistance provided by the staff of Kluwer Academic Publishers. June 1999 Themistocles M. Rassias Hari M."