The general theory of orthogonal polynomials was developed in the late 19th century from a study of continued fractions by P. L. Chebyshev, even though special cases were introduced earlier by Legendre, Hermite, Jacobi, Laguerre, and Chebyshev himself. It was further developed by A. A. Markov, T. J. Stieltjes, and many other mathematicians. The book by Szego, originally published in 1939, is the first monograph devoted to the theory of orthogonal polynomials and its applications in many areas, including analysis, differential equations, probability and mathematical physics. Even after all the years that have passed since the book first appeared, and with many other books on the subject published since then, this classic monograph by Szego remains an indispensable resource both as a textbook and as a reference book. It can be recommended to anyone who wants to be acquainted with this central topic of mathematical analysis.

Preface; E. Folke Bolinder. Lecture Series Clifford Numbers and Spinors given by Professor Marcel Riesz, October 1957--January 1958. Preface. I. Clifford Numbers. II. Rotations and Reflections. III. Canonical Representations of Isometrics. IV. Representation of Isometrics by Infinitesimal Transformation and Clifford Bivectors. Bibliography. Index. Errata to Clifford Numbers and Spinors. Marcel Riesz's Lecture Notes on Clifford Algebra dictated to E. Folke Bolinder, April 1959. Marcel Riesz's Work on Clifford Algebras; P. Lounesto.

see attached] This second edition of {it Representation Theory and Complex Geometry} provides an overview of significant advances in representation theory from a geometric standpoint. A geometrically-oriented treatment has long been desired, especially since the discovery of { l D}-modules in the early '80s and the quiver approach to quantum groups in the early '90s. The first half of the book fills the gap between the standard knowledge of a beginner in Lie theory and the much wider background needed by the working mathematician. Thus, Chapters 1-3 and 5-6 provide some basics in symplectic geometry, Borel--Moore homology, the geometry of semisimple groups, equivariant algebraic K-theory "from scratch," and the topology and algebraic geometry of flag varieties and conjugacy classes, respectively. The material covered by Chapters 5 and 6, as well as most of Chapter 3, has never been presented in book form. Chapters 3-4 and 7-8 present a uniform approach to representation theory of three quite different Weyl groups, Lie algebra sln, and the Iwahori--Hecke algebra. The results of Chapters 4 and 8, with complete proofs are not to be found elsewhere in the literature. This second edition contains substantial updates and revisions to include more standard classical results in chapters 2, 3, 5, and 6 as well as two new chapters. Chapter 9 treats the applications of { l D}-modules to Lie groups, and includes the study of * Differential operators on a semisimple group and on its flag manifold; * the famous Beilinson--Bernstein Localization Theorem reducing the study of {it g}-modules to that of { l D} modules; * the so-called Harish--Chandra holonomic system. Chapter 10 isdevoted to some very exciting developments connecting the representations of quantum groups to the geometry of "quiver varieties," introduced by Lusztig and Nakajima. The subject is closely related to many other important topics such as the McKay correspondence, semismall resolutions and Hilbert schemes. Overall, this chapter puts the representation theory of Kac--Moody algebras and quantum groups in this broader context. The exposition is practically self-contained with each chapter potentially serving as a basis for a graduate course or seminar. An excellent glossary of notation, comprehensive bibliography and extensive index round out this new edition. The techniques developed here play an essential role in the development of the Langlands program and can be successfully applied to representation theory, quantum groups and quantum field theory, affine Lie algebras, algebraic geometry, and mathematical physics.

Arnold's Problems contains mathematical problems brought up by Vladimir Arnold in his famous seminar at Moscow State University over several decades. In addition, there are problems published in his numerous papers and books. The invariable peculiarity of these problems was that Arnold did not consider mathematics a game with deductive reasoning and symbols, but a part of natural science (especially of physics), i.e. an experimental science. Many of these problems are still at the frontier of research today and are still open, and even those that are mainly solved keep stimulating new research, appearing every year in journals all over the world. The second part of the book is a collection of commentaries, mostly by Arnold's former students, on the current progress in the problems' solutions (featuring a bibliography inspired by them). This book will be of great interest to researchers and graduate students in mathematics and mathematical physics.

Vladimir Arnold is one of the greatest mathematical scientists of our time, as well as one of the finest, most prolific mathematical authors. This first volume of his Collected Works focuses on representations of functions, celestial mechanics and KAM theory.

The main goal of this work is to revisit the proof of the global stability of Minkowski space by D. Christodoulou and S. Klainerman, [Ch-KI]. We provide a new self-contained proof of the main part of that result, which concerns the full solution of the radiation problem in vacuum, for arbitrary asymptotically flat initial data sets. This can also be interpreted as a proof of the global stability of the external region of Schwarzschild spacetime. The proof, which is a significant modification of the arguments in [Ch-Kl], is based on a double null foliation of spacetime instead of the mixed null-maximal foliation used in [Ch-Kl]. This approach is more naturally adapted to the radiation features of the Einstein equations and leads to important technical simplifications. In the first chapter we review some basic notions of differential geometry that are sys tematically used in all the remaining chapters. We then introduce the Einstein equations and the initial data sets and discuss some of the basic features of the initial value problem in general relativity. We shall review, without proofs, well-established results concerning local and global existence and uniqueness and formulate our main result. The second chapter provides the technical motivation for the proof of our main theorem.

"Theory of Functionals: And of Integral and Integro-Differential Equations" is a foundational treatise in the field of mathematical analysis, representing the culmination of Vito Volterra's pioneering research into the calculus of variations and the generalization of functions. As one of the most significant figures in late 19th and early 20th-century mathematics, Volterra introduces a systematic approach to functionals-functions that depend on other functions-providing a rigorous framework for what is now known as functional analysis. The work explores the properties of integral and integro-differential equations, detailing their development and their profound implications for both pure mathematics and mathematical physics. Volterra examines the transition from finite-dimensional space to infinite-dimensional space, offering insights into the behavior of systems where the current state depends on past history. Edited and expanded from his influential lectures, this volume serves as an essential resource for understanding the historical development of modern analysis. It remains a vital reference for mathematicians and physicists interested in the evolution of linear operators and the mathematical modeling of complex physical phenomena. This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you may see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.