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.

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.

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 .

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."

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.

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.

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.

The method of exponential sums is a general method enabling the solution of a wide range of problems in the theory of numbers and its applications. This volume presents an exposition of the fundamentals of the theory with the help of examples which show how exponential sums arise and how they are applied in problems of number theory and its applications. The material is divided into three chapters which embrace the classical results of Gauss, and the methods of Weyl, Mordell and Vinogradov; the traditional applications of exponential sums to the distribution of fractional parts, the estimation of the Riemann zeta function; and the theory of congruences and Diophantine equations. Some new applications of exponential sums are also included. It is assumed that the reader has a knowledge of the fundamentals of mathematical analysis and of elementary number theory.

The theory of operators stands at the intersection of the frontiers of modern analysis and its classical counterparts; of algebra and quantum mechanics; of spectral theory and partial differential equations; of the modern global approach to topology and geometry; of representation theory and harmonic analysis; and of dynamical systems and mathematical physics. The present collection of papers represents contributions to a conference, and they have been carefully selected with a view to bridging different but related areas of mathematics which have only recently displayed an unexpected network of interconnections, as well as new and exciting cross-fertilizations. Our unify ing theme is the algebraic view and approach to the study of operators and their applications. The complementarity between the diversity of topics on the one hand and the unity of ideas on the other has been stressed. Some of the longer contributions represent material from lectures (in expanded form and with proofs for the most part). However, the shorter papers, as well as the longer ones, are an integral part of the picture; they have all been carefully refereed and revised with a view to a unity of purpose, timeliness, readability, and broad appeal. Raul Curto and Paile E. T."

This book is designed as a text for a first-year graduate algebra course. The choice of topics is guided by the underlying theme of modules as a basic unifying concept in mathematics. Beginning with standard topics in groups and ring theory, the authors then develop basic module theory, culminating in the fundamental structure theorem for finitely generated modules over a principal ideal domain. They then treat canonical form theory in linear algebra as an application of this fundamental theorem. Module theory is also used in investigating bilinear, sesquilinear, and quadratic forms. The authors develop some multilinear algebra (Hom and tensor product) and the theory of semisimple rings and modules and apply these results in the final chapter to study group represetations by viewing a representation of a group G over a field F as an F(G)-module. The book emphasizes proofs with a maximum of insight and a minimum of computation in order to promote understanding. However, extensive material on computation (for example, computation of canonical forms) is provided.

With this translation, the classic monograph UEber die Klassenzahl abelscher Zahlkoerper by Helmut Hasse is now available in English for the first time. The book addresses three main topics: class number formulas for abelian number fields; expressions of the class number of real abelian number fields by the index of the subgroup generated by cyclotomic units; and the Hasse unit index of imaginary abelian number fields, the integrality of the relative class number formula, and the class number parity. Additionally, the book includes reprints of works by Ken-ichi Yoshino and Mikihito Hirabayashi, which extend the tables of Hasse unit indices and the relative class numbers to imaginary abelian number fields with conductor up to 100. The text provides systematic and practical methods for deriving class number formulas, determining the unit index and calculating the class number of abelian number fields. A wealth of illustrative examples, together with corrections and remarks on the original work, make this translation a valuable resource for today's students of and researchers in number theory.

During the last fifty years, Gopinath Kallianpur has made extensive and significant contributions to diverse areas of probability and statistics, including stochastic finance, Fisher consistent estimation, non-linear prediction and filtering problems, zero-one laws for Gaussian processes and reproducing kernel Hilbert space theory, and stochastic differential equations in infinite dimensions. To honor Kallianpur's pioneering work and scholarly achievements, a number of leading experts have written research articles highlighting progress and new directions of research in these and related areas. This commemorative volume, dedicated to Kallianpur on the occasion of his seventy-fifth birthday, will pay tribute to his multi-faceted achievements and to the deep insight and inspiration he has so graciously offered his students and colleagues throughout his career. Contributors to the volume: S. Aida, N. Asai, K. B. Athreya, R. N. Bhattacharya, A. Budhiraja, P. S. Chakraborty, P. Del Moral, R. Elliott, L. Gawarecki, D. Goswami, Y. Hu, J. Jacod, G. W. Johnson, L. Johnson, T. Koski, N. V. Krylov, I. Kubo, H.-H. Kuo, T. G. Kurtz, H. J. Kushner, V. Mandrekar, B. Margolius, R. Mikulevicius, I. Mitoma, H. Nagai, Y. Ogura, K. R. Parthasarathy, V. Perez-Abreu, E. Platen, B. V. Rao, B. Rozovskii, I. Shigekawa, K. B. Sinha, P. Sundar, M. Tomisaki, M. Tsuchiya, C. Tudor, W. A. Woycynski, J. Xiong

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics," "CFD," "completely integrable systems," "chaos, synergetics and large-scale order," which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

The theory of Kac lagebras and their duality, elaborated independently in the seventies by Kac and Vainermann and by the authors of this book, has nowreached a state of maturity which justifies the publication of a comprehensive and authoritative account in bookform. Further, the topic of "quantum groups" has recently become very fashionable and attracted the attention of more and more mathematicians and theoretical physicists. However a good characterization of quantum groups among Hopf algebras in analogy to the characterization of Lie groups among locally compact groups is still missing. It is thus very valuable to develop the generaltheory as does this book, with emphasis on the analytical aspects of the subject instead of the purely algebraic ones. While in the Pontrjagin duality theory of locally compact abelian groups a perfect symmetry exists between a group and its dual, this is no longer true in the various duality theorems of Tannaka, Krein, Stinespring and others dealing with non-abelian locally compact groups. Kac (1961) and Takesaki (1972) formulated the objective of finding a good category of Hopf algebras, containing the category of locally compact groups and fulfilling a perfect duality. The category of Kac algebras developed in this book fully answers the original duality problem, while not yet sufficiently non-unimodular to include quantum groups. This self-contained account of thetheory will be of interest to all researchers working in quantum groups, particularly those interested in the approach by Lie groups and Lie algebras or by non-commutative geometry, and more generally also to those working in C* algebras or theoretical physics.

It is because mathematics is often misunderstood, it is commonly believed it has nothing to say about politics. The high school experience with mathematics, for so many the lasting impression of the subject, suggests that mathematics is the study of numbers, operations, formulas, and manipulations of symbols. Those believing this is the extent of mathematics might conclude mathematics has no relevance to politics. This book counters this impression. The second edition of this popular book focuses on mathematical reasoning about politics. In the search for ideal ways to make certain kinds of decisions, a lot of wasted effort can be averted if mathematics can determine that finding such an ideal is actually impossible in the first place. In the first three parts of this book, we address the following three political questions: (1) Is there a good way to choose winners of elections? (2) Is there a good way to apportion congressional seats? (3) Is there a good way to make decisions in situations of conflict and uncertainty? In the fourth and final part of this book, we examine the Electoral College system that is used in the United States to select a president. There we bring together ideas that are introduced in each of the three earlier parts of the book.

Kaye Stacey' Helen Chick' and Margaret Kendal The University of Melbourne' Australia Abstract: This section reports on the organisation' procedures' and publications of the ICMI Study' The Future of the Teaching and Learning of Algebra. Key words: Study Conference' organisation' procedures' publications The International Commission on Mathematical Instruction (ICMI) has' since the 1980s' conducted a series of studies into topics of particular significance to the theory and practice of contemporary mathematics education. Each ICMI Study involves an international seminar' the "Study Conference"' and culminates in a published volume intended to promote and assist discussion and action at the international' national' regional' and institutional levels. The ICMI Study running from 2000 to 2004 was on The Future of the Teaching and Learning of Algebra' and its Study Conference was held at The University of Melbourne' Australia fromDecember to 2001. It was the first study held in the Southern Hemisphere. There are several reasons why the future of the teaching and learning of algebra was a timely focus at the beginning of the twenty first century. The strong research base developed over recent decades enabled us to take stock of what has been achieved and also to look forward to what should be done and what might be achieved in the future. In addition' trends evident over recent years have intensified. Those particularly affecting school mathematics are the "massification" of education-continuing in some countries whilst beginning in others-and the advance of technology.

In the summer of 1991 the Department of Mathematics and Statistics of the Universite de Montreal was fortunate to host the NATO Advanced Study Institute "Algebras and Orders" as its 30th Seminaire de mathematiques superieures (SMS), a summer school with a long tradition and well-established reputation. This book contains the contributions of the invited speakers. Universal algebra- which established itself only in the 1930's- grew from traditional algebra (e.g., groups, modules, rings and lattices) and logic (e.g., propositional calculus, model theory and the theory of relations). It started by extending results from these fields but by now it is a well-established and dynamic discipline in its own right. One of the objectives of the ASI was to cover a broad spectrum of topics in this field, and to put in evidence the natural links to, and interactions with, boolean algebra, lattice theory, topology, graphs, relations, automata, theoretical computer science and (partial) orders. The theory of orders is a relatively young and vigorous discipline sharing certain topics as well as many researchers and meetings with universal algebra and lattice theory. W. Taylor surveyed the abstract clone theory which formalizes the process of compos ing operations (i.e., the formation of term operations) of an algebra as a special category with countably many objects, and leading naturally to the interpretation and equivalence of varieties."

This book is a basic reference in the modern theory of holomorphic foliations, presenting the interplay between various aspects of the theory and utilizing methods from algebraic and complex geometry along with techniques from complex dynamics and several complex variables. The result is a solid introduction to the theory of foliations, covering basic concepts through modern results on the structure of foliations on complex projective spaces.

A modern approach to number theory through a blending of complementary algebraic and analytic perspectives, emphasising harmonic analysis on topological groups. The main goal is to cover John Tates visionary thesis, giving virtually all of the necessary analytic details and topological preliminaries -- technical prerequisites that are often foreign to the typical, more algebraically inclined number theorist. While most of the existing treatments of Tates thesis are somewhat terse and less than complete, the intent here is to be more leisurely, more comprehensive, and more comprehensible. While the choice of objects and methods is naturally guided by specific mathematical goals, the approach is by no means narrow. In fact, the subject matter at hand is germane not only to budding number theorists, but also to students of harmonic analysis or the representation theory of Lie groups. The text addresses students who have taken a year of graduate-level course in algebra, analysis, and topology. Moreover, the work will act as a good reference for working mathematicians interested in any of these fields.

This book is intended as an introductory text on the subject of Lie groups and algebras and their role in various fields of mathematics and physics. It is written by and for researchers who are primarily analysts or physicists, not algebraists or geometers. Not that we have eschewed the algebraic and geo metric developments. But we wanted to present them in a concrete way and to show how the subject interacted with physics, geometry, and mechanics. These interactions are, of course, manifold; we have discussed many of them here-in particular, Riemannian geometry, elementary particle physics, sym metries of differential equations, completely integrable Hamiltonian systems, and spontaneous symmetry breaking. Much ofthe material we have treated is standard and widely available; but we have tried to steer a course between the descriptive approach such as found in Gilmore and Wybourne, and the abstract mathematical approach of Helgason or Jacobson. Gilmore and Wybourne address themselves to the physics community whereas Helgason and Jacobson address themselves to the mathematical community. This book is an attempt to synthesize the two points of view and address both audiences simultaneously. We wanted to present the subject in a way which is at once intuitive, geometric, applications oriented, mathematically rigorous, and accessible to students and researchers without an extensive background in physics, algebra, or geometry."

Besides giving readers the techniques for solving polynomial equations and congruences, "An Introduction to Mathematical Thinking" provides preparation for understanding more advanced topics in Linear and Modern Algebra, as well as Calculus. This book introduces proofs and mathematical thinking while teaching basic algebraic skills involving number systems, including the integers and complex numbers. Ample questions at the end of each chapter provide opportunities for learning and practice; the "Exercises" are routine applications of the material in the chapter, while the "Problems" require more ingenuity, ranging from easy to nearly impossible. Topics covered in this comprehensive introduction range from logic and proofs, integers and diophantine equations, congruences, induction and binomial theorem, rational and real numbers, and functions and bijections to cryptography, complex numbers, and polynomial equations. With its comprehensive appendices, this book is an excellent desk reference for mathematicians and those involved in computer science.

Fraleigh and Beauregard's text is known for its clear presentation and writing style, mathematical appropriateness, and overall student usability. Its inclusion of calculus-related examples, true/false problems, section summaries, integrated applications, and coverage of Cn make it a superb text for the sophomore or junior-level linear algebra course. This Third Edition retains the features that have made it successful over the years, while addressing recent developments of how linear algebra is taught and learned. Key concepts are presented early on, with an emphasis on geometry.