This book consists of eighteen articles in the area of `Combinatorial Matrix Theory' and `Generalized Inverses of Matrices'. Original research and expository articles presented in this publication are written by leading Mathematicians and Statisticians working in these areas. The articles contained herein are on the following general topics: `matrices in graph theory', `generalized inverses of matrices', `matrix methods in statistics' and `magic squares'. In the area of matrices and graphs, speci_c topics addressed in this volume include energy of graphs, q-analog, immanants of matrices and graph realization of product of adjacency matrices. Topics in the book from `Matrix Methods in Statistics' are, for example, the analysis of BLUE via eigenvalues of covariance matrix, copulas, error orthogonal model, and orthogonal projectors in the linear regression models. Moore-Penrose inverse of perturbed operators, reverse order law in the case of inde_nite inner product space, approximation numbers, condition numbers, idempotent matrices, semiring of nonnegative matrices, regular matrices over incline and partial order of matrices are the topics addressed under the area of theory of generalized inverses. In addition to the above traditional topics and a report on CMTGIM 2012 as an appendix, we have an article on old magic squares from India.

This book is a collection of the various old and new results, centered around the following simple and beautiful observation of J.L. Walsh - If a function is analytic in a finite disc, and not in a larger disc, then the difference between the Lagrange interpolant of the function, at the roots of unity, and the partial sums of the Taylor series, about the origin, tends to zero in a larger disc than the radius of convergence of the Taylor series, while each of these operators converges only in the original disc.

This book will be particularly useful for researchers in approximation and interpolation theory.

Spinors are used extensively in physics. It is widely accepted that they are more fundamental than tensors, and the easy way to see this is through the results obtained in general relativity theory by using spinors -- results that could not have been obtained by using tensor methods only.

The foundation of the concept of spinors is groups; spinors appear as representations of groups. This textbook expounds the relationship between spinors and representations of groups. As is well known, spinors and representations are both widely used in the theory of elementary particles.

The authors present the origin of spinors from representation theory, but nevertheless apply the theory of spinors to general relativity theory, and part of the book is devoted to curved space-time applications.

Based on lectures given at Ben Gurion University, this textbook is intended for advanced undergraduate and graduate students in physics and mathematics, as well as being a reference for researchers.

Fifteen years ago, most mathematicians who worked in the intersection of function theory and operator theory thought that progress on the Bergman spaces was unlikely, yet today the situation has completely changed. For several years, research interest and activity have expanded in this area and there are now rich theories describing the Bergman spaces and their operators. This book is a timely treatment of the theory, written by three of the major players in the field.

The main theme in classical ring theory is the structure theory of rings of a particular kind. For example, no one text book in ring theory could miss the Wedderburn-Artin theorem, which says that a ring R is semisimple Artinian iffR is isomorphic to a finite direct sum of full matrix rings over skew fields. This is an example of a finiteness condition which, at least historically, has dominated in ring theory. Ifwe would like to consider a requirement of a lattice-theoretical type, other than being Artinian or Noetherian, the most natural is uni-seriality. Here a module M is called uni-serial if its lattice of submodules is a chain, and a ring R is uni-serial if both RR and RR are uni-serial modules. The class of uni-serial rings includes commutative valuation rings and closed under homomorphic images. But it is not closed under direct sums nor with respect to Morita equivalence: a matrix ring over a uni-serial ring is not uni-serial. There is a class of rings which is very close to uni-serial but closed under the constructions just mentioned: serial rings. A ring R is called serial if RR and RR is a direct sum (necessarily finite) of uni-serial modules. Amongst others this class includes triangular matrix rings over a skew field. Also if F is a finite field of characteristic p and G is a finite group with a cyclic normal p-Sylow subgroup, then the group ring FG is serial.

Algebra, as we know it today, consists of many different ideas, concepts and results. A reasonable estimate of the number of these different items would be somewhere between 50,000 and 200,000. Many of these have been named and many more could (and perhaps should) have a name or a convenient designation. Even the nonspecialist is likely to encounter most of these, either somewhere in the literature, disguised as a definition or a theorem or to hear about them and feel the need for more information. If this happens, one should be able to find enough information in this Handbook to judge if it is worthwhile to pursue the quest.

In addition to the primary information given in the Handbook, there are references to relevant articles, books or lecture notes to help the reader. An excellent index has been included which is extensive and not limited to definitions, theorems etc.

The Handbook of Algebra will publish articles as they are received and thus the reader will find in this third volume articles from twelve different sections. The advantages of this scheme are two-fold: accepted articles will be published quickly and the outline of the Handbook can be allowed to evolve as the various volumes are published.

A particularly important function of the Handbook is to provide professional mathematicians working in an area other than their own with sufficient information on the topic in question if and when it is needed.
- Thorough and practical source for information
- Provides in-depth coverage of new topics in algebra
- Includes references to relevant articles, books and lecture notes

This present volume is the Proceedings of the 14th International Conference on Near rings and Nearfields held in Hamburg at the Universitiit der Bundeswehr Hamburg, from July 30 to August 06, 1995. This Conference was attended by 70 mathematicians and many accompanying persons who represented 22 different countries from all five continents. Thus it was the largest conference devoted entirely to nearrings and nearfields. The first of these conferences took place in 1968 at the Mathematische For schungsinstitut Oberwolfach, Germany. This was also the site of the conferences in 1972, 1976, 1980 and 1989. The other eight conferences held before the Hamburg Conference took place in eight different countries. For details about this and, more over, for a general historical overview of the development of the subject, we refer to the article "On the beginnings and development of near-ring theory" by G. Betsch 3]. During the last forty years the theory of nearrings and related algebraic struc tures like nearfields, nearmodules, nearalgebras and seminearrings has developed into an extensive branch of algebra with its own features. In its position between group theory and ring theory, this relatively young branch of algebra has not only a close relationship to these two more well-known areas of algebra, but it also has, just as these two theories, very intensive connections to many further branches of mathematics."

This book is a revised and up-dated fourth edition of a textbook designed for upper division courses in linear algebra. It includes the basic results on vector spaces over fields, determinants, the theory of a single linear tranformation, and inner product spaces. While it does not presuppose an ealier course, many connections between linear algebrta and calculus are worked into the discussion, making it best suited for students who have completed the calulus sequence. A special feature of the book is the inclusion of sections devoted to applications of linear algebra, which can either be part of a course, or used for independent study. The topics covered in these secions are the geometric interpretation of systems of linear equations, the classification of finite symmetry groups in two and three dimensions, the exponential of a matrix and its application to solving systems of first order linear differential equations with constant coefficients, and Hurwitz's theorem on the composition of quadratic forms. This revised fourth edition contains a new section on analytic methods in matrix theory, with applications to Markov chains in probability theory. Proofs of all the main theorems are included, and are presented on an equal footing with methods for solving numerical problems. Worked examples are included in almost every section, to bring out the meaning of the theorems, and to illustrate techniques for solving problems. Many numerical exercises are included, which use all the ideas, and develop computational skills. There are also exercises of a theoretical nature, which provide opportunities for students to discover interesting theings for themselves.

Polynomial extremal problems (PEP) constitute one of the most important subclasses of nonlinear programming models. Their distinctive feature is that an objective function and constraints can be expressed by polynomial functions in one or several variables. Let: e = {: e 1, ...: en} be the vector in n-dimensional real linear space Rn; n PO(: e), PI (: e), ..., Pm (: e) are polynomial functions in R with real coefficients. In general, a PEP can be formulated in the following form: (0.1) find r = inf Po(: e) subject to constraints (0.2) Pi (: e) =0, i=l, ..., m (a constraint in the form of inequality can be written in the form of equality by introducing a new variable: for example, P( x) 0 is equivalent to P(: e) + y2 = 0). Boolean and mixed polynomial problems can be written in usual form by adding for each boolean variable z the equality: Z2 - Z = O. Let a = {al, ..., a } be integer vector with nonnegative entries {a;}f=l. n Denote by R a](: e) monomial in n variables of the form: n R a](: e) = IT: ef';;=1 d(a) = 2:7=1 ai is the total degree of monomial R a]. Each polynomial in n variables can be written as sum of monomials with nonzero coefficients: P(: e) = L caR a](: e), aEA{P) IX x Nondifferentiable optimization and polynomial problems where A(P) is the set of monomials contained in polynomial P

Computational intelligence (CI) lies at the interface between engineering and computer science; control engineering, where problems are solved using computer-assisted methods. Thus, it can be regarded as an indispensable basis for all artificial intelligence (AI) activities. This book collects surveys of most recent theoretical approaches focusing on fuzzy systems, neurocomputing, and nature inspired algorithms. It also presents surveys of up-to-date research and application with special focus on fuzzy systems as well as on applications in life sciences and neuronal computing.

The present book deals with factorization problems for matrix and operator functions. The problems originate from, or are motivated by, the theory of non-selfadjoint operators, the theory of matrix polynomials, mathematical systems and control theory, the theory of Riccati equations, inversion of convolution operators, theory of job scheduling in operations research. The book systematically employs a geometric principle of factorization which has its origins in the state space theory of linear input-output systems and in the theory of characteristic operator functions. This principle allows one to deal with different factorizations from one point of view. Covered are canonical factorization, minimal and non-minimal factorizations, pseudo-canonical factorization, and various types of degree one factorization.

Considerable attention is given to the matter of stability of factorization which in terms of the state space method involves stability of invariant subspaces.invariant subspaces.

"Categorical Perspectives" consists of introductory surveys as well as articles containing original research and complete proofs devoted mainly to the theoretical and foundational developments of category theory and its applications to other fields. A number of articles in the areas of topology, algebra and computer science reflect the varied interests of George Strecker to whom this work is dedicated. Notable also are an exposition of the contributions and importance of George Strecker's research and a survey chapter on general category theory. This work is an excellent reference text for researchers and graduate students in category theory and related areas.

Contributors: H.L. Bentley * G. Castellini * R. El Bashir * H. Herrlich * M. Husek * L. Janos * J. Koslowski * V.A. Lemin * A. Melton * G. Preua * Y.T. Rhineghost * B.S.W. Schroeder * L. Schr"der * G.E. Strecker * A. Zmrzlina"

This volume presents the lectures given during the second French-Uzbek Colloquium on Algebra and Operator Theory which took place in Tashkent in 1997, at the Mathematical Institute of the Uzbekistan Academy of Sciences. Among the algebraic topics discussed here are deformation of Lie algebras, cohomology theory, the algebraic variety of the laws of Lie algebras, Euler equations on Lie algebras, Leibniz algebras, and real K-theory. Some contributions have a geometrical aspect, such as supermanifolds. The papers on operator theory deal with the study of certain types of operator algebras. This volume also contains a detailed introduction to the theory of quantum groups. Audience: This book is intended for graduate students specialising in algebra, differential geometry, operator theory, and theoretical physics, and for researchers in mathematics and theoretical physics.

This work provides the current theory and observations behind the cosmological phenomenon of dark energy. The approach is comprehensive with rigorous mathematical theory and relevant astronomical observations discussed in context. The book treats the background and history starting with the new-found importance of Einstein's cosmological constant (proposed long ago) in dark energy formulation, as well as the frontiers of dark energy. The authors do not presuppose advanced knowledge of astronomy, and basic mathematical concepts used in modern cosmology are presented in a simple, but rigorous way. All this makes the book useful for both astronomers and physicists, and also for university students of physical sciences.

Semisimple Lie groups, and their algebraic analogues over fields other than the reals, are of fundamental importance in geometry, analysis, and mathematical physics. Three independent, self-contained volumes, under the general title Lie Theory, feature survey work and original results by well-established researchers in key areas of semisimple Lie theory.

Harmonic Analysis on Symmetric Spacesa "General Plancherel Theorems presents extensive surveys by E.P. van den Ban, H. Schlichtkrull, and P. Delorme of the spectacular progress over the past decade in deriving the Plancherel theorem on reductive symmetric spaces.

Van den Bana (TM)s introductory chapter explains the basic setup of a reductive symmetric space along with a careful study of the structure theory, particularly for the ring of invariant differential operators for the relevant class of parabolic subgroups. Advanced topics for the formulation and understanding of the proof are covered, including Eisenstein integrals, regularity theorems, Maassa "Selberg relations, and residue calculus for root systems. Schlichtkrull provides a cogent account of the basic ingredients in the harmonic analysis on a symmetric space through the explanation and definition of the Paleya "Wiener theorem. Approaching the Plancherel theorem through an alternative viewpoint, the Schwartz space, Delorme bases his discussion and proof on asymptotic expansions of eigenfunctions and the theory of intertwining integrals.

Well suited for both graduate students and researchers in semisimple Lie theory and neighboring fields, possibly even mathematical cosmology, Harmonic Analysis on Symmetric Spacesa "General Plancherel Theorems provides abroad, clearly focused examination of semisimple Lie groups and their integral importance and applications to research in many branches of mathematics and physics. Knowledge of basic representation theory of Lie groups as well as familiarity with semisimple Lie groups, symmetric spaces, and parabolic subgroups is required.

This text offers an introduction to error-correcting linear codes for researchers and graduate students in mathematics, computer science and engineering. The book differs from other standard texts in its emphasis on the classification of codes by means of isometry classes. The relevant algebraic are developed rigorously. Cyclic codes are discussed in great detail. In the last four chapters these isometry classes are enumerated, and representatives are constructed algorithmically.

The second volume of this work contains Parts 2 and 3 of the "Handbook of Coding Theory". Part 2, "Connections", is devoted to connections between coding theory and other branches of mathematics and computer science. Part 3, "Applications", deals with a variety of applications for coding.

This book contains two contributions: "Combinatorial and Asymptotic Methods in Algebra" by V.A. Ufnarovskij is a survey of various combinatorial methods in infinite-dimensional algebras, widely interpreted to contain homological algebra and vigorously developing computer algebra, and narrowly interpreted as the study of algebraic objects defined by generators and their relations. The author shows how objects like words, graphs and automata provide valuable information in asymptotic studies. The main methods emply the notions of Grobner bases, generating functions, growth and those of homological algebra. Treated are also problems of relationships between different series, such as Hilbert, Poincare and Poincare-Betti series. Hyperbolic and quantum groups are also discussed. The reader does not need much of background material for he can find definitions and simple properties of the defined notions introduced along the way. "Non-Associative Structures" by E.N.Kuz'min and I.P.Shestakov surveys the modern state of the theory of non-associative structures that are nearly associative. Jordan, alternative, Malcev, and quasigroup algebras are discussed as well as applications of these structures in various areas of mathematics and primarily their relationship with the associative algebras. Quasigroups and loops are treated too. The survey is self-contained and complete with references to proofs in the literature. The book will be of great interest to graduate students and researchers in mathematics, computer science and theoretical physics."

Orthogonal designs have proved fundamental to constructing code division multiple antenna systems for more efficient mobile communications. Starting with basic theory, this book develops the algebra and combinatorics to create new communications modes. Intended primarily for researchers, it is also useful for graduate students wanting to understand some of the current communications coding theories.

For courses in Prealgebra. Trusted author content. Thoughtful innovation. In this revision of the Bittinger Paperback Worktext Series, the Bittinger author team brings their extensive experience to developmental math courses, paired with thoughtful integration of technology and content. The Bittinger Series enables students to get the most out of their MyLab (TM) Math course through an updated learning path, new review videos, and engaging new exercises that offer the support they need, when they need it. Bittinger offers superior content written by author-educators, tightly integrated with MyLab Math - the #1 choice in digital learning. Bringing the authors' voices and their approach into the MyLab course encourages student motivation and engagement, while reinforcing their understanding of the skills and concepts they need to master algebra. Also available with MyLab Math By combining trusted author content with digital tools and a flexible platform, MyLab Math personalizes the learning experience and improves results for each student. Note: You are purchasing a standalone product; MyLab Math does not come packaged with this content. Students, if interested in purchasing this title with MyLab Math, ask your instructor to confirm the correct package ISBN and Course ID. Instructors, contact your Pearson representative for more information. If you would like to purchase both the physical text and MyLab Math, search for: 0135218284 / 9780135218280 Prealgebra Plus MyLab Math with Pearson eText - Access Card Package Package consists of: 0135182565 / 9780135182567 Prealgebra 0135184371 / 9780135184370 MyLab Math with Pearson eText - Standalone Access Card - for Prealgebra

This book, an outgrowth of the author¿s lectures at the University of California at Berkeley, is intended as a textbook for a one-semester course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semisimple rings, Jacobson¿s theory of the radical, representation theory of groups and algebras, prime and semiprime rings, local and semilocal rings, perfect and semiperfect rings, etc. By aiming the level of writing at the novice rather than the connoisseur and by stressing the role of examples and motivation, the author has produced a text that is suitable not only for use in a graduate course, but also for self-study in the subject by interested graduate students. More than 400 exercises testing the understanding of the general theory in the text are included in this new edition.

This book concentrates on the topic of evaluation of Jacobians in some specific linear as well as nonlinear matrix transformations, in the real and complex cases, which are widely applied in the statistical, physical, engineering, biological and social sciences. It aims to develop some techniques systematically so that anyone with a little exposure to multivariable calculus can easily follow the steps and understand the various methods by which the Jacobians in complicated matrix transformations are evaluated. The material is developed slowly, with lots of worked examples, aimed at self-study. Some exercises are also given, at the end of each section.The book is a valuable reference for statisticians, engineers, physicists, econometricians, applied mathematicians and people working in many other areas. It can be used for a one-semester graduate level course on Jacobians and functions of matrix argument.