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Books in Mathematics

The Mathematics collection presents a range of foundational and advanced research content across applied and discrete mathematics, including fields such as Computational Mathematics; Differential Equations; Linear Algebra; Modelling & Simulation; Numerical Analysis; Probability & Statistics.

  • Operators and Representation Theory

    Canonical Models for Algebras of Operators Arising in Quantum Mechanics
    • 1st Edition
    • Volume 147
    • P.E.T. Jorgensen
    • English
    Historically, operator theory and representation theory both originated with the advent of quantum mechanics. The interplay between the subjects has been and still is active in a variety of areas.This volume focuses on representations of the universal enveloping algebra, covariant representations in general, and infinite-dimensional Lie algebras in particular. It also provides new applications of recent results on integrability of finite-dimensional Lie algebras. As a central theme, it is shown that a number of recent developments in operator algebras may be handled in a particularly elegant manner by the use of Lie algebras, extensions, and projective representations. In several cases, this Lie algebraic approach to questions in mathematical physics and C*-algebra theory is new; for example, the Lie algebraic treatment of the spectral theory of curved magnetic field Hamiltonians, the treatment of irrational rotation type algebras, and the Virasoro algebra.Also examined are C*-algebraic methods used (in non-traditional ways) in the study of representations of infinite-dimensional Lie algebras and their extensions, and the methods developed by A. Connes and M.A. Rieffel for the study of the Yang-Mills problem.Cutting across traditional separations between fields of specialization, the book addresses a broad audience of graduate students and researchers.

  • Generalized Solutions of Nonlinear Partial Differential Equations

    • 1st Edition
    • Volume 146
    • E.E. Rosinger
    • English
    During the last few years, several fairly systematic nonlinear theories of generalized solutions of rather arbitrary nonlinear partial differential equations have emerged. The aim of this volume is to offer the reader a sufficiently detailed introduction to two of these recent nonlinear theories which have so far contributed most to the study of generalized solutions of nonlinear partial differential equations, bringing the reader to the level of ongoing research.The essence of the two nonlinear theories presented in this volume is the observation that much of the mathematics concerning existence, uniqueness regularity, etc., of generalized solutions for nonlinear partial differential equations can be reduced to elementary calculus in Euclidean spaces, combined with elementary algebra in quotient rings of families of smooth functions on Euclidean spaces, all of that joined by certain asymptotic interpretations. In this way, one avoids the complexities and difficulties of the customary functional analytic methods which would involve sophisticated topologies on various function spaces. The result is a rather elementary yet powerful and far-reaching method which can, among others, give generalized solutions to linear and nonlinear partial differential equations previously unsolved or even unsolvable within distributions or hyperfunctions.Part 1 of the volume discusses the basic limitations of the linear theory of distributions when dealing with linear or nonlinear partial differential equations, particularly the impossibility and degeneracy results. Part 2 examines the way Colombeau constructs a nonlinear theory of generalized functions and then succeeds in proving quite impressive existence, uniqueness, regularity, etc., results concerning generalized solutions of large classes of linear and nonlinear partial differential equations. Finally, Part 3 is a short presentation of the nonlinear theory of Rosinger, showing its connections with Colombeau's theory, which it contains as a particular case.

  • Numerical Modelling: Applications to Marine Systems

    • 1st Edition
    • Volume 145
    • J. Noye
    • English
    The thirteen papers presented in this book are based on talks given at the workshop on Numerical Modelling of Marine Systems held at the University of Adelaide, South Australia in February 1986. Several of the articles are a direct outcome of two special sessions held on modelling of Open Boundary Conditions and on the Transport of Pollutants.Other articles in the book cover topics such as numerical modelling of wind-driven flow in shallow seas, sediment transport in estuaries, internal tides and comparison of numerical methods for solving tidal and pollutant transport problems.

  • Difference Schemes

    An Introduction to the Underlying Theory
    • 1st Edition
    • Volume 19
    • S.K. Godunov + 1 more
    • English
    Much applied and theoretical research in natural sciences leads to boundary-value problems stated in terms of differential equations. When solving these problems with computers, the differential problems are replaced approximately by difference schemes.This book is an introduction to the theory of difference schemes, and was written as a textbook for university mathematics and physics departments and for technical universities. Some sections of the book will be of interest to computations specialists.While stressing a mathematically rigorous treatment of model problems, the book also demonstrates the relation between theory and computer experiments, using difference schemes created for practical computations.

  • The Algebraic Structure of Crossed Products

    • 1st Edition
    • Volume 142
    • G. Karpilovsky
    • English
    In the past 15 years, the theory of crossed products has enjoyed a period of vigorous development. The foundations have been strengthened and reorganized from new points of view, especially from the viewpoint of graded rings.The purpose of this monograph is to give, in a self-contained manner, an up-to-date account of various aspects of this development, in an effort to convey a comprehensive picture of the current state of the subject. It is assumed that the reader has had the equivalent of a standard first-year graduate course, thus familiarity with basic ring-theoretic and group-theoretic concepts and an understanding of elementary properties of modules, tensor products and fields. A chapter on algebraic preliminaries is included, which briefly surveys topics required later in the book.

  • Geometry of Numbers

    • 2nd Edition
    • Volume 37
    • C.G. Lekkerkerker + 1 more
    • English
    This volume contains a fairly complete picture of the geometry of numbers, including relations to other branches of mathematics such as analytic number theory, diophantine approximation, coding and numerical analysis. It deals with convex or non-convex bodies and lattices in euclidean space, etc.This second edition was prepared jointly by P.M. Gruber and the author of the first edition. The authors have retained the existing text (with minor corrections) while adding to each chapter supplementary sections on the more recent developments. While this method may have drawbacks, it has the definite advantage of showing clearly where recent progress has taken place and in what areas interesting results may be expected in the future.

  • The Jacobson Radical of Group Algebras

    • 1st Edition
    • Volume 135
    • G. Karpilovsky
    • English
    Let G be a finite group and let F be a field. It is well known that linear representations of G over F can be interpreted as modules over the group algebra FG. Thus the investigation of ring-theoretic structure of the Jacobson radical J(FG) of FG is of fundamental importance. During the last two decades the subject has been pursued by a number of researchers and many interesting results have been obtained. This volume examines these results.The main body of the theory is presented, giving the central ideas, the basic results and the fundamental methods. It is assumed that the reader has had the equivalent of a standard first-year graduate algebra course, thus familiarity with basic ring-theoretic and group-theoretic concepts and an understanding of elementary properties of modules, tensor products and fields. A chapter on algebraic preliminaries is included, providing a survey of topics needed later in the book. There is a fairly large bibliography of works which are either directly relevant to the text or offer supplementary material of interest.

  • Foundations of Analysis over Surreal Number Fields

    • 1st Edition
    • Volume 141
    • N.L. Alling
    • English
    In this volume, a tower of surreal number fields is defined, each being a real-closed field having a canonical formal power series structure and many other higher order properties. Formal versions of such theorems as the Implicit Function Theorem hold over such fields. The Main Theorem states that every formal power series in a finite number of variables over a surreal field has a positive radius of hyper-convergence within which it may be evaluated. Analytic functions of several surreal and surcomplex variables can then be defined and studied. Some first results in the one variable case are derived. A primer on Conway's field of surreal numbers is also given.Throughout the manuscript, great efforts have been made to make the volume fairly self-contained. Much exposition is given. Many references are cited. While experts may want to turn quickly to new results, students should be able to find the explanation of many elementary points of interest. On the other hand, many new results are given, and much mathematics is brought to bear on the problems at hand.

  • Inverse Spectral Theory

    • 1st Edition
    • Volume 130
    • Jurgen Poschel
    • English

  • Barrelled Locally Convex Spaces

    • 1st Edition
    • Volume 131
    • P. Pérez Carreras + 1 more
    • English
    This book is a systematic treatment of barrelled spaces, and of structures in which barrelledness conditions are significant. It is a fairly self-contained study of the structural theory of those spaces, concentrating on the basic phenomena in the theory, and presenting a variety of functional-analytic techniques.Beginning with some basic and important results in different branches of Analysis, the volume deals with Baire spaces, presents a variety of techniques, and gives the necessary definitions, exploring conditions on discs to ensure that they are absorbed by the barrels of the space. The abstract theory of barrelled spaces is then presented, as well as local completeness and its applications to the inheritance of the Mackey topology to subspaces. Further discussed is the abstract study of bornological and ultrabornological spaces; B- and Br-completeness; inductive limits; strong barrelledness conditions; characterizations of barrelled, bornological and (DF)-spaces in the context of spaces of type C(X); the stability of barrelledness conditions of topological tensor products and the related questions of commutability of inductive limits and tensor products; and the holomorphically significant properties of locally convex spaces as developed by Nachbin and others.

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