This is an English translation of the now classic "Algèbre Locale - Multiplicités" originally published by Springer as LNM 11, in several editions since 1965. It gives a short account of the main theorems of commutative algebra, with emphasis on modules, homological methods and intersection multiplicities ("Tor-formula"). Many modifications to the original French text have been made by the author for this English edition: they make the text easier to read, without changing its intended informal character.

Structured matrices serve as a natural bridge between the areas of algebraic computations with polynomials and numerical matrix computations, allowing cross-fertilization of both fields. This book covers most fundamental numerical and algebraic computations with Toeplitz, Hankel, Vandermonde, Cauchy, and other popular structured matrices. Throughout the computations, the matrices are represented by their compressed images, called displacements, enabling both a unified treatment of various matrix structures and dramatic saving of computer time and memory. The resulting superfast algorithms allow further dramatic parallel acceleration using FFT and fast sine and cosine transforms. Included are specific applications to other fields, in particular, superfast solutions to: various fundamental problems of computer algebra; the tangential Nevanlinna--Pick and matrix Nehari problems The primary intended readership for this work includes researchers, algorithm designers, and advanced graduate students in the fields of computations with structured matrices, computer algebra, and numerical rational interpolation. The book goes beyond research frontiers and, apart from very recent research articles, includes yet unpublished results. To serve a wider audience, the presentation unfolds systematically and is written in a user-friendly engaging style. Only some preliminary knowledge of the fundamentals of linear algebra is required. This makes the material accessible to graduate students and new researchers who wish to study the rapidly exploding area of computations with structured matrices and polynomials. Examples, tables, figures, exercises, extensive bibliography, and index lend this text toclassroom use or self-study.

This monograph covers in a comprehensive manner the current state of classification theory with respect to infinite abelian groups. A wide variety of ways to characterise different classes of abelian groups by invariants, isomorphisms and duality principles are discussed.

A look at solving problems in three areas of classical elementary mathematics: equations and systems of equations of various kinds, algebraic inequalities, and elementary number theory, in particular divisibility and diophantine equations. In each topic, brief theoretical discussions are followed by carefully worked out examples of increasing difficulty, and by exercises which range from routine to rather more challenging problems. While it emphasizes some methods that are not usually covered in beginning university courses, the book nevertheless teaches techniques and skills which are useful beyond the specific topics covered here. With approximately 330 examples and 760 exercises.

This book presents tensors and tensor analysis as primary mathematical tools for engineering and engineering science students and researchers. The discussion is based on the concepts of vectors and vector analysis in three-dimensional Euclidean space, and although it takes the subject matter to an advanced level, the book starts with elementary geometrical vector algebra so that it is suitable as a first introduction to tensors and tensor analysis. Each chapter includes a number of problems for readers to solve, and solutions are provided in an Appendix at the end of the text. Chapter 1 introduces the necessary mathematical foundations for the chapters that follow, while Chapter 2 presents the equations of motions for bodies of continuous material. Chapter 3 offers a general definition of tensors and tensor fields in three-dimensional Euclidean space. Chapter 4 discusses a new family of tensors related to the deformation of continuous material. Chapter 5 then addresses constitutive equations for elastic materials and viscous fluids, which are presented as tensor equations relating the tensor concept of stress to the tensors describing deformation, rate of deformation and rotation. Chapter 6 investigates general coordinate systems in three-dimensional Euclidean space and Chapter 7 shows how the tensor equations discussed in chapters 4 and 5 are presented in general coordinates. Chapter 8 describes surface geometry in three-dimensional Euclidean space, Chapter 9 includes the most common integral theorems in two- and three-dimensional Euclidean space applied in continuum mechanics and mathematical physics.

This book features selected papers from The Seventh International Conference on Research and Education in Mathematics that was held in Kuala Lumpur, Malaysia from 25 - 27th August 2015. With chapters devoted to the most recent discoveries in mathematics and statistics and serve as a platform for knowledge and information exchange between experts from academic and industrial sectors, it covers a wide range of topics, including numerical analysis, fluid mechanics, operation research, optimization, statistics and game theory. It is a valuable resource for pure and applied mathematicians, statisticians, engineers and scientists, and provides an excellent overview of the latest research in mathematical sciences.

"Proofs and Fundamentals: A First Course in Abstract Mathematics" 2nd edition is designed as a "transition" course to introduce undergraduates to the writing of rigorous mathematical proofs, and to such fundamental mathematical ideas as sets, functions, relations, and cardinality. The text serves as a bridge between computational courses such as calculus, and more theoretical, proofs-oriented courses such as linear algebra, abstract algebra and real analysis. This 3-part work carefully balances Proofs, Fundamentals, and Extras. Part 1 presents logic and basic proof techniques; Part 2 thoroughly covers fundamental material such as sets, functions and relations; and Part 3 introduces a variety of extra topics such as groups, combinatorics and sequences. A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and in writing are never compromised. New to the second edition: 1) A new section about the foundations of set theory has been added at the end of the chapter about sets. This section includes a very informal discussion of the Zermelo- Fraenkel Axioms for set theory. We do not make use of these axioms subsequently in the text, but it is valuable for any mathematician to be aware that an axiomatic basis for set theory exists. Also included in this new section is a slightly expanded discussion of the Axiom of Choice, and new discussion of Zorn's Lemma, which is used later in the text. 2) The chapter about the cardinality of sets has been rearranged and expanded. There is a new section at the start of the chapter that summarizes various properties of the set of natural numbers; these properties play important roles subsequently in the chapter. The sections on induction and recursion have been slightly expanded, and have been relocated to an earlier place in the chapter (following the new section), both because they are more concrete than the material found in the other sections of the chapter, and because ideas from the sections on induction and recursion are used in the other sections. Next comes the section on the cardinality of sets (which was originally the first section of the chapter); this section gained proofs of the Schroeder-Bernstein theorem and the Trichotomy Law for Sets, and lost most of the material about finite and countable sets, which has now been moved to a new section devoted to those two types of sets. The chapter concludes with the section on the cardinality of the number systems. 3) The chapter on the construction of the natural numbers, integers and rational numbers from the Peano Postulates was removed entirely. That material was originally included to provide the needed background about the number systems, particularly for the discussion of the cardinality of sets, but it was always somewhat out of place given the level and scope of this text. The background material about the natural numbers needed for the cardinality of sets has now been summarized in a new section at the start of that chapter, making the chapter both self-contained and more accessible than it previously was. 4) The section on families of sets has been thoroughly revised, with the focus being on families of sets in general, not necessarily thought of as indexed. 5) A new section about the convergence of sequences has been added to the chapter on selected topics. This new section, which treats a topic from real analysis, adds some diversity to the chapter, which had hitherto contained selected topics of only an algebraic or combinatorial nature. 6) A new section called ``You Are the Professor'' has been added to the end of the last chapter. This new section, which includes a number of attempted proofs taken from actual homework exercises submitted by students, offers the reader the opportunity to solidify her facility for writing proofs by critiquing these submissions as if she were the instructor for the course. 7) All known errors have been corrected. 8) Many minor adjustments of wording have been made throughout the text, with the hope of improving the exposition.

The most difficult computational problems nowadays are those of higher dimensions. This research monograph offers an introduction to tensor numerical methods designed for the solution of the multidimensional problems in scientific computing. These methods are based on the rank-structured approximation of multivariate functions and operators by using the appropriate tensor formats. The old and new rank-structured tensor formats are investigated. We discuss in detail the novel quantized tensor approximation method (QTT) which provides function-operator calculus in higher dimensions in logarithmic complexity rendering super-fast convolution, FFT and wavelet transforms. This book suggests the constructive recipes and computational schemes for a number of real life problems described by the multidimensional partial differential equations. We present the theory and algorithms for the sinc-based separable approximation of the analytic radial basis functions including Green's and Helmholtz kernels. The efficient tensor-based techniques for computational problems in electronic structure calculations and for the grid-based evaluation of long-range interaction potentials in multi-particle systems are considered. We also discuss the QTT numerical approach in many-particle dynamics, tensor techniques for stochastic/parametric PDEs as well as for the solution and homogenization of the elliptic equations with highly-oscillating coefficients. Contents Theory on separable approximation of multivariate functions Multilinear algebra and nonlinear tensor approximation Superfast computations via quantized tensor approximation Tensor approach to multidimensional integrodifferential equations

This book explores the application of field theory (patterns of interaction) to Russian economic history, and how social and political fields mediate the influences of institutions, structures, discourses and ideologies in the creation and dissemination of economic thinking, theory and practice. Using focused cases on Russia's economy from the mid-nineteenth century to the present, Hass and co-authors expand the empirical basis of field studies to provide new material on Russian economic history. The cases are divided into two complementary halves: i) The role of fields of institutions, discourses, and structures in the development of Russian economic thought, especially economic theories and discourses; and ii) The role of fields in the real adoption and implementation of policies in Soviet and Russian economic history. With developed discussion of fields and field theory, this book moves beyond sociology to demonstrate to other disciplines the relation of fields and field theory to other frameworks and methodological considerations for field analysis, as well as providing new empirical insights and narratives not as well-known abroad.

The theme of this book is operator theory on C*-algebras. The main novel tool employed is the concept of local multipliers. Originally devised by Elliott and Pedersen in the 1970's in order to study derivations and automorphisms, local multipliers of C*-algebras were developed into a powerful device by the present authors in the 1990's. The book serves two purposes. The first part provides the reader - specialist and advanced graduate student alike - with a thorough introduction to the theory of local multipliers. Only a minimal knowledge of algebra and analysis is required, as the prerequisites in both non-commutative ring theory and basic C*-algebra theory are presented in the first chapter. In the second part, local multipliers are used to obtain a wealth of information on various classes of operators on C*-algebras, including (groups of) automorphisms, derivations, elementary operators, Lie isomorphisms and Lie derivations, as well as others. Many of the results appear in print for the first time. The authors have made an effort to avoid intricate technicalities thus some of the results are not pushed to their utmost generality. Several open problems are discussed, and hints for further developments are given.

From the reviews: "Serre's notes on groups acting on trees have appeared in various forms (all in French) over the past ten years and they have had a profound influence on the development of many areas, for example, the theory of ends of discrete groups. This fine translation is very welcome and I strongly recommend it as an introduction to an important subject. In Chapter I, which is self-contained, the pace is fairly gentle. The author proves the fundamental theorem for the special cases of free groups and tree products before dealing with the (rather difficult) proof of the general case."(A.W. Mason in Proceedings of the Edinburgh Mathematical Society 1982)

The mathematical theory of wavelets is less than 15 years old, yet already wavelets have become a fundamental tool in many areas of applied mathematics and engineering. This introduction to wavelets assumes a basic background in linear algebra (reviewed in Chapter 1) and real analysis at the undergraduate level. Fourier and wavelet analyses are first presented in the finite-dimensional context, using only linear algebra. Then Fourier series are introduced in order to develop wavelets in the infinite-dimensional, but discrete context. Finally, the text discusses Fourier transform and wavelet theory on the real line. The computation of the wavelet transform via filter banks is emphasized, and applications to signal compression and numerical differential equations are given. This text is ideal for a topics course for mathematics majors, because it exhibits and emerging mathematical theory with many applications. It also allows engineering students without graduate mathematics prerequisites to gain a practical knowledge of wavelets.

to Group Rings by Cesar Polcino Milies Instituto de Matematica e Estatistica, Universidade de sao Paulo, sao Paulo, Brasil and Sudarshan K. Sehgal Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton. Canada SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A c.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4020-0239-7 ISBN 978-94-010-0405-3 (eBook) DOI 10.1007/978-94-010-0405-3 Printed an acid-free paper AII Rights Reserved (c) 2002 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2002 Softcover reprint ofthe hardcover Ist edition 2002 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, inc1uding photocopying, recording Of by any information storage and retrieval system, without written permis sion from the copyright owner. Contents Preface ix 1 Groups 1 1.1 Basic Concepts . . . . . . . . . . . . 1 1.2 Homomorphisms and Factor Groups 10 1.3 Abelian Groups . 18 1.4 Group Actions, p-groups and Sylow Subgroups 21 1.5 Solvable and Nilpotent Groups 27 1.6 FC Groups .

This book presents interpolation theory from its classical roots beginning with Banach function spaces and equimeasurable rearrangements of functions, providing a thorough introduction to the theory of rearrangement-invariant Banach function spaces. At the same time, however, it clearly shows how the theory should be generalized in order to accommodate the more recent and powerful applications. Lebesgue, Lorentz, Zygmund, and Orlicz spaces receive detailed treatment, as do the classical interpolation theorems and their applications in harmonic analysis.
The text includes a wide range of techniques and applications, and will serve as an amenable introduction and useful reference to the modern theory of interpolation of operators.

Symmetry and Economic Invariance: An Introduction explores how symmetry and invariance of economic models can provide insights into their properties. While the professional economist is nowadays adept at many of the mathematical techniques used in static and dynamic optimization models, group theory is still not among his or her repertoire of tools. The authors aim to show that group theoretic methods form a natural extension of the techniques commonly used in economics and that they can be easily mastered.

History, Trauma and Shame provides an in-depth examination of the sustained dialogue about the past between children of Holocaust survivors and descendants of families whose parents were either directly or indirectly involved in Nazi crimes. Taking an autobiographical narrative perspective, the chapters in the book explore the intersection of history, trauma and shame, and how change and transformation unfolds over time. The analyses of the encounters described in the book provides a close examination of the process of dialogue among members of The Study Group on Intergenerational Consequences of the Holocaust (PAKH), exploring how Holocaust trauma lives in the 'everyday' lives of descendants of survivors. It goes to the heart of the issues at the forefront of contemporary transnational debates about building relationships of trust and reconciliation in societies with a history of genocide and mass political violence. This book will be great interest for academics, researchers and postgraduate students engaged in the study of social psychology, Holocaust or genocide studies, cultural studies, reconciliation studies, historical trauma and peacebuilding. It will also appeal to clinical psychologists, psychiatrists and psychoanalysts, as well as upper-level undergraduate students interested in the above areas.

Vertex algebra was introduced by Boreherds, and the slightly revised notion "vertex oper- ator algebra" was formulated by Frenkel, Lepowsky and Meurman, in order to solve the problem of the moonshine representation of the Monster group - the largest sporadie group. On the one hand, vertex operator algebras ean be viewed as extensions of eertain infinite-dimensional Lie algebras such as affine Lie algebras and the Virasoro algebra. On the other hand, they are natural one-variable generalizations of commutative associative algebras with an identity element. In a certain sense, Lie algebras and commutative asso- ciative algebras are reconciled in vertex operator algebras. Moreover, some other algebraie structures, such as integral linear lattiees, Jordan algebras and noncommutative associa- tive algebras, also appear as subalgebraic structures of vertex operator algebras. The axioms of vertex operator algebra have geometrie interpretations in terms of Riemman spheres with punctures. The trace functions of a certain component of vertex operators enjoy the modular invariant properties. Vertex operator algebras appeared in physies as the fundamental algebraic structures of eonformal field theory, whieh plays an important role in string theory and statistieal meehanies. Moreover,eonformalfieldtheoryreveals animportantmathematiealproperty,the so called "mirror symmetry" among Calabi-Yau manifolds. The general correspondence between vertex operator algebras and Calabi-Yau manifolds still remains mysterious. Ever since the first book on vertex operator algebras by Frenkel, Lepowsky and Meur- man was published in 1988, there has been a rapid development in vertex operator su- peralgebras, which are slight generalizations of vertex operator algebras.

This book presents the state-of-the-art research on the teaching and learning of linear algebra in the first year of university, in an international perspective. It provides university teachers in charge of linear algebra courses with a wide range of information from works including theoretical and experimental issues.

Accurate and efficient computer algorithms for factoring matrices, solving linear systems of equations, and extracting eigenvalues and eigenvectors. Regardless of the software system used, the book describes and gives examples of the use of modern computer software for numerical linear algebra. It begins with a discussion of the basics of numerical computations, and then describes the relevant properties of matrix inverses, factorisations, matrix and vector norms, and other topics in linear algebra. The book is essentially self- contained, with the topics addressed constituting the essential material for an introductory course in statistical computing. Numerous exercises allow the text to be used for a first course in statistical computing or as supplementary text for various courses that emphasise computations.

This proceedings volume, the fifth in a series from the Combinatorial and Additive Number Theory (CANT) conferences, is based on talks from the 19th annual workshop, held online due to the COVID-19 pandemic. Organized every year since 2003 by the New York Number Theory Seminar at the CUNY Graduate Center, the workshops survey state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. The CANT 2021 meeting featured over a hundred speakers from North and South America, Europe, Asia, Australia, and New Zealand, and was the largest CANT conference in terms of the number of both lectures and participants. These proceedings contain peer-reviewed and edited papers on current topics in number theory. Topics featured in this volume include sumsets, minimal bases, Sidon sets, analytic and prime number theory, combinatorial and discrete geometry, numerical semigroups, and a survey of expansion, divisibility, and parity. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory.

The topic of this book is finite group actions and their use in order to approach finite unlabeled structures by defining them as orbits of finite groups of sets. Well-known examples are graph, linear codes, chemical isomers, spin configurations, isomorphism classes of combinatorial designs etc.The second edition is an extended version and puts more emphasis on applications to the constructive theory of finite structures. Recent progress in this field, in particular in design and coding theory, is described.This book will be of great use to researchers and graduate students.

A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.

ClDo _ IIIIIIIoaIIIics bu _ die 'EI JDDi, sij'_ . . . . -. . . _. je _ . . . . . lIbupalaJllllllllll __ D'y_poa: wbae it beIoap. . . . die . . ." . ., . . _ DOD to dlecluly __ __ . 1110 _ is dioapaI; -. . e _ may be EricT. BeD IbIetodo--'_iL O. 1feaoriIide Mathematics is a tool for dloogIrt. A bighly necessary tool in a world where both feedback and noolineari ties abound. Similarly, all kinds of parts of IIIIIIhcmatiI: s serve as tools for odIcr parts and for ocher sci eoccs. Applying a simple rewriting rule to the quote on the right above one finds suc: h stalements as: 'One ser vice topology has rcncIerM mathematical physics . . . '; 'One service logic has rendered computer science . . '; 'One service category theory has rmdcn: d mathematics . . . '. All arguably true. And all statements obrainable this way form part of the raison d'etm of this series. This series, Mathmlatics tDIII Its Applications, saaned in 1977. Now that over one hundred volumcs have appeared it seems opportune to reexamine its scope. AI. the time I wrote "Growing spccialization and divenification have brought a host of monographs and textbooks on incJeasingly specialized topics. However, the 'tree' of knowledge of JJJatbcmatics and reIatcd ficIds docs not grow only by putting forth new bnDdIcs. It also happens, quite often in fact, that brancbes which were thought to be comp1etcly disparate am suddenly seen to be rdatcd."

Carl Friedrich Gauss's textbook, Disquisitiones arithmeticae, published in 1801 (Latin), remains to this day a true masterpiece of mathematical examination. .