For about a decade I have made an effort to study quadratic forms in infinite dimensional vector spaces over arbitrary division rings. Here we present in a systematic fashion half of the results found du ring this period, to wit, the results on denumerably infinite spaces (" ~O- forms") . Certain among the resul ts included here had of course been published at the time when they were found, others appear for the first time (the case, for example, in Chapters IX, X, XII where I in clude results contained in the Ph.D.theses by my students w. Allenspach, L. Brand, U. Schneider, M. Studer). If one wants to give an introduction to the geometric algebra of infinite dimensional quadratic spaces, a discussion of ~ -dimensional 0 spaces ideally serves the purpose. First, these spaces show a large nurober of phenomena typical of infinite dimensional spaces. Second, most proofs can be done by recursion which resembles the familiar pro cedure by induction in the finite dimensional Situation. Third, the student acquires a good feeling for the linear algebra in infinite di mensions because it is impossible to camouflage problems by topological expedients (in dimension ~O it is easy to see, in a given case, wheth er topological language is appropriate or not) .

The theory of Lyapunov exponents originated over a century ago in the study of the stability of solutions of differential equations. Written by one of the subject's leading authorities, this book is both an account of the classical theory, from a modern view, and an introduction to the significant developments relating the subject to dynamical systems, ergodic theory, mathematical physics and probability. It is based on the author's own graduate course and is reasonably self-contained with an extensive set of exercises provided at the end of each chapter. This book makes a welcome addition to the literature, serving as a graduate text and a valuable reference for researchers in the field.

Since the introduction of the functional classes HW (lI) and WT HW (lI) and their peri- odic analogs Hw (1I') and ~ (1I'), defined by a concave majorant w of functions and their rth derivatives, many researchers have contributed to the area of ex- tremal problems and approximation of these classes by algebraic or trigonometric polynomials, splines and other finite dimensional subspaces. In many extremal problems in the Sobolev class W~ (lI) and its periodic ana- log W~ (1I') an exceptional role belongs to the polynomial perfect splines of degree r, i.e. the functions whose rth derivative takes on the values -1 and 1 on the neighbor- ing intervals. For example, these functions turn out to be extremal in such problems of approximation theory as the best approximation of classes W~ (lI) and W~ (1I') by finite-dimensional subspaces and the problem of sharp Kolmogorov inequalities for intermediate derivatives of functions from W~. Therefore, no advance in the T exact and complete solution of problems in the nonperiodic classes W HW could be expected without finding analogs of polynomial perfect splines in WT HW .

This book contains thirty-six papers from among the forty-five papers presented at the Third International Conference on Fibonacci Numbers and Their Applications which was held in Pisa, Italy from July 25 to July 29, 1988 in honor of Leonardo de Pisa. These papers have been selected after a careful review by well known referees in the field, and they range from elementary number theory to probability and statistics. The Fibonacci numbers are their unifying bond. It is anticipated that this book, like its two predecessors, will be useful to research workers and graduate students interested in the Fibonacci numbers and their applications. August 1989 The Editors Gerald E. Bergum South Dakota State University Brookings, South Dakota, U. S. A. Andreas N. Philippou Ministry of Education Nicosia, Cyprus Alwyn F. Horadam University of New England Armidale N. S. W. , Australia xv THE ORGANIZING COMMITTEES LOCAL COMMITTEE INTERNATIONAL COMMITTEE Dvornicich, Roberto, Chairman Horadam, A. F. (Australia), Co-chairman Filipponi, Piero Philippou, A. N. (Cyprus), Co-chairman Perelli, Alberto Ando, S. (Japan) Viola, Carlo Bergum, G. E. (U. S. A. ) Zannier, Umberto Johnson, M. B. (U. S. A. ) Kiss, P. (Hungary) Tijdeman, Robert (The Netherlands) Tognetti, K. (Australia) XVII LIST OF CONTRIBUTORS TO THE CONFERENCE' ADLER, I. , RR 1, Box 532, North Bennington, VT 05257-9748. "Separating the Biological from the Mathematical Aspects of Phyllotaxis. " *AKRITAS, A. G. , (coauthor P. G. Bradford). "The Role of the Fibonacci Sequence in the Isolation of the Real Roots of Polynomial Equations.

Elasticity theory is a classical discipline. The mathematical theory of elasticity in mechanics, especially the linearized theory, is quite mature, and is one of the foundations of several engineering sciences. In the last twenty years, there has been significant progress in several areas closely related to this classical field, this applies in particular to the following two areas. First, progress has been made in numerical methods, especially the development of the finite element method. The finite element method, which was independently created and developed in different ways by sci entists both in China and in the West, is a kind of systematic and modern numerical method for solving partial differential equations, especially el liptic equations. Experience has shown that the finite element method is efficient enough to solve problems in an extremely wide range of applica tions of elastic mechanics. In particular, the finite element method is very suitable for highly complicated problems. One of the authors (Feng) of this book had the good fortune to participate in the work of creating and establishing the theoretical basis of the finite element method. He thought in the early sixties that the method could be used to solve computational problems of solid mechanics by computers. Later practice justified and still continues to justify this point of view. The authors believe that it is now time to include the finite element method as an important part of the content of a textbook of modern elastic mechanics."

The study of phase transitions is one of the fundamental problems of physics. The goal of this seminar was to understand better the spectacular progress made recently in constructing continuum models. Concentrating on a few examples such as the microstructure of crystals, defects in liquid crystals and liquid-vapor interfaces, several key points are described, for example the structure and evolution of the interfaces, regularization via interfacial energy, and equilibrium theories. The mathematical treatment of these questions involves large-oscillation theories (Young's measures, compensated compactness), spectral theory, admissibility of shock waves, long-time behavior of dynamical systems, high-order perturbations, group actions, solitons, and others.

This book provides a very readable description of a technique, developed by the author years ago but as current as ever, for proving that solutions to certain (non-elliptic) partial differential equations only have real analytic solutions when the data are real analytic (locally). The technique is completely elementary but relies on a construction, a kind of a non-commutative power series, to localize the analysis of high powers of derivatives in the so-called bad direction. It is hoped that this work will permit a far greater audience of researchers to come to a deep understanding of this technique and its power and flexibility.

This comprehensive two-volume work is devoted to the most general beginnings of mathematics. It goes back to Hausdorff's classic Set Theory (2nd ed., 1927), where set theory and the theory of functions were expounded as the fundamental parts of mathematics in such a way that there was no need for references to other sources. Along the lines of Hausdorff's initial work (1st ed., 1914), measure and integration theory is also included here as the third fundamental part of contemporary mathematics. The material about sets and numbers is placed in Volume 1 and the material about functions and measures is placed in Volume 2. Contents Historical foreword on the centenary after Felix Hausdorff's classic Set Theory Fundamentals of the theory of functions Fundamentals of the measure theory Historical notes on the Riesz - Radon - Frechet problem of characterization of Radon integrals as linear functionals

Advanced Calculus is intended as a text for courses that furnish the backbone of the student's undergraduate education in mathematical analysis. The goal is to rigorously present the fundamental concepts within the context of illuminating examples and stimulating exercises. This book is self-contained and starts with the creation of basic tools using the completeness axiom. The continuity, differentiability, integrability, and power series representation properties of functions of a single variable are established. The next few chapters describe the topological and metric properties of Euclidean space. These are the basis of a rigorous treatment of differential calculus (including the Implicit Function Theorem and Lagrange Multipliers) for mappings between Euclidean spaces and integration for functions of several real variables. Special attention has been paid to the motivation for proofs. Selected topics, such as the Picard Existence Theorem for differential equations, have been included in such a way that selections may be made while preserving a fluid presentation of the essential material. Supplemented with numerous exercises, ""Advanced Calculus"" is a perfect book for undergraduate students of analysis.

Pierre Grisvard, one of the most distinguished French mathematicians, died on April 22, 1994. A Conference was held in November 1994 out of which grew the invited articles contained in this volume. All of the papers are related to functional analysis applied to partial differential equations, which was Grisvard's specialty. Indeed his knowledge of this area was extremely broad. He began his career as one of the very first students of Jacques Louis Lions, and in 1965, he presented his "State Thesis" on interpolation spaces, using in particular, spectral theory for linear operators in Banach spaces. After 1970, he became a specialist in the study of optimal regularity for par tial differential equations with boundary conditions. He studied singulari ties coming from coefficients, boundary conditions, and mainly non-smooth domains, and left a legacy of precise results which have been published in journals and books. Pierre Grisvard spent most of his career as a full professor at the University of Nice, where he started in 1967. For shorter or longer periods, he visited several foreign countries, and collaborated with some of the most famous mathematicians in his field. He was also an excellent organizer and directed a large number of Ph.D. students. Finally, this volume contains a bibliography of Grisvard's works as well as one paper which he wrote and which has not been published before."

Automorphic forms and Galois representations have played a central role in the development of modern number theory, with the former coming to prominence via the celebrated Langlands program and Wiles' proof of Fermat's Last Theorem. This two-volume collection arose from the 94th LMS-EPSRC Durham Symposium on 'Automorphic Forms and Galois Representations' in July 2011, the aim of which was to explore recent developments in this area. The expository articles and research papers across the two volumes reflect recent interest in p-adic methods in number theory and representation theory, as well as recent progress on topics from anabelian geometry to p-adic Hodge theory and the Langlands program. The topics covered in volume two include curves and vector bundles in p-adic Hodge theory, associators, Shimura varieties, the birational section conjecture, and other topics of contemporary interest.

This selection of outstanding articles - an outgrowth of the QMath9 meeting for young scientists - covers new techniques and recent results on spectral theory, statistical mechanics, Bose-Einstein condensation, random operators, magnetic Schroedinger operators and more. The book's pedagogical style makes it a useful introduction to the research literature for postgraduate students. For more expert researchers it will serve as a concise source of modern reference.

Coding theory, system theory, and symbolic dynamics have much in common. A major new theme in this area of research is that of codes and systems based on graphical models. This volume contains survey and research articles from leading researchers at the interface of these subjects.

The present book contains three articles: "Systems of Linear Differential Equations," by V. P. Palamodov; "Fredholm Operators and Their Generalizations," by S. N. Krachkovskii and A. S. Di kanskii; and "Representations of Groups and Algebras in Spaces with an Indefinite Metric" by M. A. Naimark and R. S. Ismagilov. In the fi.rst article the accent is on those characteristics of systems of differential equations which distinguish the systems from the scalar case. Considerable space is devoted in particular to "nonquadratic systems," a topic that has very recently stimulated interest. The second article is devoted to the algebraic aspects of the theory of operators (determinant theory in particular) in Banach and linear topological spaces. The third article reflects the present state of the art in the given area of the theory of representations, which has been re ceiving considerable attention in connection with its applications in physics (particularly in quantum field theory) and in the theory of differential equations."

In the first half of the 19th century geometry changed radically, and withina century it helped to revolutionize both mathematics and physics. It also put the epistemology and the philosophy of science on a new footing. In this volume a sound overview of this development is given by leading mathematicians, physicists, philosophers, and historians of science. This interdisciplinary approach gives this collection a unique character. It can be used by scientists and students, but it also addresses a general readership.

The importance of mathematics in the study of problems arising from the real world, and the increasing success with which it has been used to model situations ranging from the purely deterministic to the stochastic, in all areas of today's Physical Sciences and Engineering, is well established. The purpose of the sets of volumes, the present one being the first in a planned series of sequential sets, is to make available authoritative, up to date, and self-contained accounts of some of the most important and useful of these analytical approaches and techniques. Each volume in each set will provide a detailed introduction to a specific subject area of current importance, and then goes beyond this by reviewing recent contributions, thereby serving as a valuable reference source. The progress in applicable mathematics has been brought about by the extension and development of many important analytical approaches and techniques, in areas both old and new, frequently aided by the use of computers without which the solution of realistic problems in modern Physical Sciences and Engineering would otherwise have been impossible.A case in point is the analytical technique of singular perturbation theory (Volume 3), which has a long history. In recent years it has been used in many different ways, and its importance has been enhanced by its having been used in various fields to derive sequences of asymptotic approximations, each with a higher order of accuracy than its predecessor. These approximations have, in turn, provided a better understanding of the subject and stimulated the development of new methods for the numerical solution of the higher order approximations. A typical example of this type is to be found in the general study of nonlinear wave propagation phenomena as typified by the study of water waves. Elsewhere, as with the identification and emergence of the study of inverse problems (volumes 1 and 2), new analytical approaches have stimulated the development of numerical techniques for the solution of this major class of problems.Such work divides naturally into two parts, the first being the identification and formulation of inverse problems, the theory of ill-posed problems and the class of one-dimensional inverse problems, and the second being the study and theory of multidimensional inverse problems. Volume 1: Inverse Problems 1 Volume 2: Inverse Problems 2 Alexander G. Ramm, Author These volumes present the theory of inverse spectral and scattering problems and of many other inverse problems for differential equations in an essentially self-contained way.Highlights of these volumes include novel presentation of the classical theories (Gel'fand-Levitan's and Marchenko's), analysis of the invertibility of the inversion steps in these theories, study of some new inverse problems in one-and multi-dimensional cases; I-function and applications to classical and new inverse scattering and spectral problems, study of inverse problems with incomplete data, study of some new inverse problems for parabolic and hyperbolic equations, discussion of some non-overdetermined inverse problems, a study of inverse problems arising in the theory of ground-penetrating radars, development of DSM (dynamical systems method) for solving ill-posed nonlinear operator equations, comparison of the Ramm's inversion method for solving fixed-energy inverse scattering

This book collects the Proceedings of a Congress held in Frascati (Rome) in the period July 1 -July 10, 1991, on the subject of harmonic analysis and discrete potential theory, and related topics. The Congress was made possible by the financial support of the Italian National Research Council ("Gruppo GNAFA"), the Ministry of University ("Gruppo Analisi Funzionale" of the University of Milano), the University of Rome "Tor Vergata", and was also patronized by the Centro "Vito Volterra" of the University of Rome "Tor Vergata". Financial support for publishing these Proceedings was provided by the University of Rome "Tor Vergata", and by a generous contribution of the Centro "Vito Volterra". I am happy of this opportunity to acknowledge the generous support of all these Institutions, and to express my gratitude, and that of all the participants. A number of distinguished mathematicians took part in the Congress. Here is the list of participants: M. Babillot, F. Choucroun, Th. Coulhon, L. Elie, F. Ledrappier, N. Th. Varopoulos (Paris); L. Gallardo (Brest); Ph. Bougerol, B. Roynette (Nancy); O. Gebuhrer (Strasbourg); G. Ahumada-Bustamante (Mulhouse); A. Valette (Neuchatel); P. Gerl (Salzburg); W. Hansen, H. Leptin (Bielefeld); M. Bozejko, A. Hulanicki, T. Pytlik (Wroclaw); C. Thomassen (Lyngby); P. Sjogren (Goteborg); V. Kaimanovich (Leningrad); A. Nevo (Jerusalem); T. Steger (Chicago); S. Sawyer, M. Taibleson, G. Weiss (St. Louis); J. Cohen, S. S ali ani (Maryland); D. Voiculescu (Berkeley); A. Zemanian (Stony Brook); S. Northshield (Plattsburgh); J. Taylor (Montreal); J.

In this volume selected papers delivered at the special session on "Spectral and scattering theory" are published. This session was organized by A. G. Ramm at the first international congress ofISAAC (International Society for Analysis, Applications and Computing) which was held at the University of Delaware, June 3-7, 1997. The papers in this volume deal with a wide va riety of problems including some nonlinear problems (Schechter, Trenogin), control theory (Shubov), fundamental problems of physics (Kitada), spectral and scattering theory in waveg uides and shallow ocean (Ramm and Makrakis), inverse scattering with incomplete data (Ramm), spectral theory for Sturm-Liouville operators with singular coefficients (Yurko) and with energy-dependent coefficients (Aktosun, Klaus, and van der Mee), spectral theory of SchrOdinger operators with periodic coefficients (Kuchment, Vainberg), resolvent estimates for SchrOdinger-type and Maxwell's operators (Ben-Artzi and Nemirovsky), SchrOdinger oper ators with von Neumann-Wignertype potentials (Rejto and Taboada), principal eigenvalues for indefinite-weight elliptic operators (pinchover), and symmetric solutions of Ginzburg-Landau equations (Gustafson). These papers will be of interest to a wide audience including mathematicians, physicists, and theoretically oriented engineers. A. G. Ramm Manhattan, KS v CONTENTS 1. Wave Scattering in 1-0 Nonconservative Media . . . . . . . . . . . . . . . . . . . Tuncay Aktosun, Martin Klaus, and Comelis van der Mee 2. Resolvent Estimates for SchrOdinger-type and Maxwell Equations with Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Matania Ben-Artzi and Jonathan Nemirovsky 3. Symmetric Solutions of Ginzburg-Landau Equations 33 S. Gustafson 4. Quantum Mechanics and Relativity: Their Unification by Local Time . . . . . . . 39 Hitoshi Kitada 5."

The theory of almost-periodic functions with complex values, created by H. Bohr [1] in his two classical papers published in Acta Mathematica in 1925 and 1926, has been developed by many authors and has had note- worthy applications: we recall the works of Weyl, De la Vallee Poussin, Bochner, Stepanov, Wiener, Besicovic, Favard, Delsarte, Maak, Bogoliu- bov, Levitan. This subject has been widely treated in the monographs by Bohr [2], Favard [1], Besicovic [1], Maak [1], Levitan [1], Cinquini [1], Corduneanu [1], [2]. An important class of almost-periodic functions was studied at the beginning of the century by Bohl and Esclangon. Bohr's theory has been extended by Muckenhoupt [1] in a particular case and, subsequently, by Bochner [1] and by Bochner and Von Neumann [1] to very general abstract spaces. The extension to Banach spaces is, in particular, of great interest, in view of the fundamental importance of these spaces in theory and application.

The purpose of this book is to give an introduction to the Laplace transform on the undergraduate level. The material is drawn from notes for a course taught by the author at the Milwaukee School of Engineering. Based on classroom experience, an attempt has been made to (1) keep the proofs short, (2) introduce applications as soon as possible, (3) concentrate on problems that are difficult to handle by the older classical methods, and (4) emphasize periodic phenomena. To make it possible to offer the course early in the curriculum (after differential equations), no knowledge of complex variable theory is assumed. However, since a thorough study of Laplace. transforms requires at least the rudiments of this theory, Chapter 3 includes a brief sketch of complex variables, with many of the details presented in Appendix A. This plan permits an introduction of the complex inversion formula, followed by additional applications. The author has found that a course taught three hours a week for a quarter can be based on the material in Chapters 1, 2, and 5 and the first three sections of Chapter 7. If additional time is available (e.g., four quarter-hours or three semester-hours), the whole book can be covered easily. The author is indebted to the students at the Milwaukee School of Engineering for their many helpful comments and criticisms.

Under the guidance and inspiration of Dr. Ajit Iqbal Singh, an International Conference on Harmonie Analysis took place at the Uni- versity of Delhi, India, from December 18 to 22, 1995. Twenty-one dis- tinguished mathematicians from around the world, as weIl as many from India, participated in this successful and stimulating conference. An underlying theme of the conference was hypergroups, the the- ory of wh ich has developed and been found useful in fields as diverse as special functions, differential equations, probability theory, representa- tion theory, measure theory, Hopf algebras and quantum groups. Some other areas of emphasis that emerged were harmonie analysis of analytic functions, ergo die theory and wavelets. This book includes most of the proceedings of this conference. I chaired the Editorial Board for this publication; the other members were J. M. Anderson (University College London), G. L. Litvinov (Centre for Optimization and Mathematical Modeling, Institute for New Technolo- gies, Moscow), Mrs. A. I. Singh (University ofDelhi, India), V. S. Sunder (Institute of Mathematical Sciences, C.LT., Madras, India), and N. J. Wildberger (University of New South Wales, Australia). I appreciate all the help provided by these editors as weIl as the help and cooperation of Our authors and referees of their papers. I especially appreciate techni- cial assistance and advice from Alan L. Schwartz (University of Missouri - St. Louis, USA) and Martin E. Walter (University of Colorado, USA). Finally, I thank Our editor, Ann Kostant, for her help and encouragement during this project.

The International Symposium on Generalized Functions and Their Applications was organized by the Department of Mathematics, Banaras Hindu University, and held December 23-26, 1991, on the occasion of the Platinum Jubilee Celebration of the university. More than a hundred mathematicians from ten countries participated in the deliberations of the symposium. Thirty lectures were delivered on a variety of topics within the area. The contributions to the proceedings of the symposium are, with a few exceptions, expanded versions of the lectures delivered by the invited speakers. The survey papers by Komatsu and Hoskins and Sousa Pinto provide an up-to-date account of the theory of hyperfunctions, ultradistributions and microfunctions, and the nonstandard theory of new generalized functions, respectively; those by Stankovic and Kanwal deal with structures and asymptotics. Choquet-Bruhat's work studies generalized functions on manifold and gives applications to shocks and discrete models. The other contributions relate to contemporary problems and achievements in theory and applications, especially in the theory of partial differential equations, differential geometry, mechanics, mathematical physics, and systems science. The proceedings give a very clear impression of the present state of the art in this field and contain many challenges, ideas, and open problems. The volume is very helpful for a broad spectrum of readers: graduate students to mathematical researchers.

This book contains survey papers based on the lectures presented at the 3rd International Winter School "Modern Problems of Mathematics and Mechanics" held in January 2010 at the Belarusian State University, Minsk. These lectures are devoted to different problems of modern analysis and its applications. An extended presentation of modern problems of applied analysis will enable the reader to get familiar with new approaches of mostly interdisciplinary character. The results discussed are application oriented and present new insight into applied problems of growing importance such as applications to composite materials, anomalous diffusion, and fluid dynamics.

"This book presents a functional calculus for "n"-tuples of not necessarily commuting linear operators. In particular, a functional calculus for quaternionic linear operators is developed. These calculi are based on a new theory of hyperholomorphicity for functions with values in a Clifford algebra: the so-called slice monogenic functions which are carefully described in the book. In the case of functions with values in the algebra of quaternions these functions are named slice regular functions."

Except for the appendix and the introduction all results are new and appear for the first time organized in a monograph. The material has been carefully prepared to be as self-contained as possible. The intended audience consists of researchers, graduate and postgraduate students interested in operator theory, spectral theory, hypercomplex analysis, and mathematical physics."

Despite the fact that Maple is one of the most popular computer algebra systems on the market, surprisingly few users realise its potential for scientific visualisation. This book equips readers with the graphics tools needed on the voyage into the complex and beautiful world of curves and surfaces. A comprehensive treatment of Maples graphics commands and structures is combined with an introduction to the main aspects of visual perception, with priority given to the use of light, colour, perspective, and geometric transformations. Numerous examples cover all aspects of Maple graphics, and these may be easily tailored to the individual needs of the reader. The approach is context-independent, and as such will appeal to students, educators, and researchers in a broad spectrum of scientific disciplines. For the general user at any level of experience, this book will serve as a comprehensive reference manual. For the beginner, it offers a user-friendly introduction to the subject, with mathematical requirements kept to a minimum, while, for those interested in advanced mathematical visualisation, it explains how to maximise Maples graphical capabilities.