Equivariant cohomology on smooth manifolds is the subject of this book which is part of a collection of volumes edited by J. Bruning and V.W. Guillemin. The point of departure are two relatively short but very remarkable papers be Henry Cartan, published in 1950 in the Proceedings of the "Colloque de Topologie." These papers are reproduced here, together with a modern introduction to the subject, written by two of the leading experts in the field. This "introduction" comes as a textbook of its own, though, presenting the first full treatment of equivariant cohomology in the de Rahm setting. The well known topological approach is linked with the differential form aspect through the equivariant de Rahm theorem. The systematic use of supersymmetry simplifies considerably the ensuing development of the basic technical tools which are then applied to a variety of subjects, leading up to the localization theorems and other very recent results."

The Influence Line Approach to the Analysis of Rigid Frames offers a simple method of analysis of indeterminate structures. It is original and independent of other methods. The author derived these equations by applying an algebraic rather than an arithmetical method of distribution of fixed-end moments. His method is fully explained and illustrated by worked examples.

The equations listed in the Tables in The Influence Line Approach to the Analysis of Rigid Frames offer a simple approach to the analysis of rigid frames, including building frames, rendering them statically determinate for any system of loading, static or moving and including the self weight of a structure.

Particularly useful aspects to the reader are:
-The equations are of an elementary nature consisting only of distribution factors and the co-efficient of a span length and to which values from zero to unity are given.
-The equations can be used to analyze frames the members of which can be either of constant or variable cross-section, and in both cases distributions of fixed-end moments are not required.
-In addition, the evaluation of fixed-end moments is not required when the frame consists of members of constant cross-section.
-The equations are independent of other methods of analysis requiring neither the use of model analysis nor the application of linear equations.
-The equations offer a good indication of structural behavior.
-The Tables lend themselves to expansion catering for different degrees of end fixation. The Influence Line Approach to the Analysis of Rigid Frames can be taught not only to university undergraduate students, but also to those pursuing middle-levelcourses in Civil Engineering, Structural Engineering and Building. In addition, practicing assistant structural designers will find it a useful reference work.

Field Arithmetic explores Diophantine fields through their absolute Galois groups. This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. We use Haar measure on the absolute Galois group to replace counting arguments. New Chebotarev density variants interpret diophantine properties. Here we have the only complete treatment of Galois stratifications, used by Denef and Loeser, et al, to study Chow motives of Diophantine statements.

Progress from the first edition starts by characterizing the finite-field like P(seudo)A(lgebraically)C(losed) fields. We once believed PAC fields were rare. Now we know they include valuable Galois extensions of the rationals that present its absolute Galois group through known groups. PAC fields have projective absolute Galois group. Those that are Hilbertian are characterized by this group being pro-free. These last decade results are tools for studying fields by their relation to those with projective absolute group. There are still mysterious problems to guide a new generation: Is the solvable closure of the rationals PAC; and do projective Hilbertian fields have pro-free absolute Galois group (includes Shafarevich's conjecture)?

The third edition improves the second edition in two ways: First it removes many typos and mathematical inaccuracies that occur in the second edition (in particular in the references). Secondly, the third edition reports on five open problems (out of thirtyfour open problems of the second edition) that have been partially or fully solved since that edition appeared in 2005.

The fusion of algebra, analysis and geometry, and their application to real world problems, have been dominant themes underlying mathematics for over a century. Geometric algebras, introduced and classified by Clifford in the late 19th century, have played a prominent role in this effort, as seen in the mathematical work of Cartan, Brauer, Weyl, Chevelley, Atiyah, and Bott, and in applications to physics in the work of Pauli, Dirac and others. One of the most important applications of geometric algebras to geometry is to the representation of groups of Euclidean and Minkowski rotations. This aspect and its direct relation to robotics and vision will be discussed in several chapters of this multi-authored textbook, which resulted from the ASI meeting.

Moreover, group theory, beginning with the work of Burnside, Frobenius and Schur, has been influenced by even more general problems. As a result, general group actions have provided the setting for powerful methods within group theory and for the use of groups in applications to physics, chemistry, molecular biology, and signal processing. These aspects, too, will be covered in detail.

With the rapidly growing importance of, and ever expanding conceptual and computational demands on signal and image processing in remote sensing, computer vision, medical image processing, and biological signal processing, and on neural and quantum computing, geometric algebras, and computational group harmonic analysis, the topics of the book have emerged as key tools. The list of authors includes many of the world's leading experts in the development of new algebraic modeling and signal representation methodologies, novel Fourier-based andgeometric transforms, and computational algorithms required for realizing the potential of these new application fields.

The volume covers wide-ranging topics from Theory: structure of finite fields, normal bases, polynomials, function fields, APN functions. Computation: algorithms and complexity, polynomial factorization, decomposition and irreducibility testing, sequences and functions. Applications: algebraic coding theory, cryptography, algebraic geometry over finite fields, finite incidence geometry, designs, combinatorics, quantum information science.

Some Historical Background This book deals with the cohomology of groups, particularly finite ones. Historically, the subject has been one of significant interaction between algebra and topology and has directly led to the creation of such important areas of mathematics as homo logical algebra and algebraic K-theory. It arose primarily in the 1920's and 1930's independently in number theory and topology. In topology the main focus was on the work ofH. Hopf, but B. Eckmann, S. Eilenberg, and S. MacLane (among others) made significant contributions. The main thrust of the early work here was to try to understand the meanings of the low dimensional homology groups of a space X. For example, if the universal cover of X was three connected, it was known that H2(X; A. ) depends only on the fundamental group of X. Group cohomology initially appeared to explain this dependence. In number theory, group cohomology arose as a natural device for describing the main theorems of class field theory and, in particular, for describing and analyzing the Brauer group of a field. It also arose naturally in the study of group extensions, N"

In semigroup theory there are certain kinds of band decompositions, which are very useful in the study of the structure semigroups. There are a number of special semigroup classes in which these decompositions can be used very successfully. The book focuses attention on such classes of semigroups. Some of them are partially discussed in earlier books, but in the last thirty years new semigroup classes have appeared and a fairly large body of material has been published on them. The book provides a systematic review on this subject. The first chapter is an introduction. The remaining chapters are devoted to special semigroup classes. These are Putcha semigroups, commutative semigroups, weakly commutative semigroups, R-Commutative semigroups, conditionally commutative semigroups, RC-commutative semigroups, quasi commutative semigroups, medial semigroups, right commutative semigroups, externally commutative semigroups, E-m semigroups, WE-m semigroups, weakly exponential semigroups, (m, n)-commutative semigroups and n(2)-permutable semigroups. Audience: Students and researchers working in algebra and computer science.

With applications in quantum field theory, elementary particle physics and general relativity, this two-volume work studies invariance of differential operators under Lie algebras, quantum groups, superalgebras including infinite-dimensional cases, Schroedinger algebras, applications to holography. This first volume covers the general aspects of Lie algebras and group theory supplemented by many concrete examples for a great variety of noncompact semisimple Lie algebras and groups. Contents: Introduction Lie Algebras and Groups Real Semisimple Lie Algebras Invariant Differential Operators Case of the Anti-de Sitter Group Conformal Case in 4D Kazhdan-Lusztig Polynomials, Subsingular Vectors, and Conditionally Invariant Equations Invariant Differential Operators for Noncompact Lie Algebras Parabolically Related to Conformal Lie Algebras Multilinear Invariant Differential Operators from New Generalized Verma Modules Bibliography Author Index Subject Index

This volume begins with a description of Alladi Ramakrishnan's remarkable scientific career and his grand vision that led to the creation of The Institute of Mathematical Sciences (MATSCIENCE), in Madras (now Chennai), India, in 1962. The lists of his research publications, his PhD students, and other relevant facts relating to his eventful career are included. The inclusion of both research and survey articles by leading mathematicians, statisticians, and physicists who got to know Alladi Ramakrishnan over the years and admired his significant contributions to research and to the scientific profession, have been written and dedicated in this volume to Ramakrishnan's memory.

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.

This self-contained work, focusing on the theory of state spaces of C*-algebras and von Neumann algebras, explains how the oriented state space geometrically determines the algebra. The theory of orientation of C*-algebra state spaces is presented with a new approach that does not depend on Jordan algebras, and the theory of orientation of normal state spaces of von Neumann algebras is presented with complete proofs for the first time. The theory of operator algebras was initially motivated by applications to physics, but has recently found unexpected new applications to fields of pure mathematics as diverse as foliations and knot theory. Key features include: * first and only work devoted to state spaces of operator algebras--contains much material not available in existing books * prerequisites are standard graduate courses in real and complex variables, measure theory, and functional analysis * complete proofs of basic results on operator algebras presented so that no previous knowledge in the field is needed * detailed introduction develops basic tools used throughout the text * numerous chapter remarks on advanced topics of independent interest with references to the literature, or discussion of applications to physics "State Spaces of Operator Algebras" is intended for specialists in operator algebras, as well as graduate students and mathematicians seeking an overview of the field. The introduction to C*-algebras and von Neumann algebras may also be of interest in it own right for those wanting a quick introduction to basic concepts in those fields.

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.

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 invaluable book provides a concise and systematic introduction to the theory of compact connected Lie groups and their representations, as well as a complete presentation of the structure and classification theory. It uses a non-traditional approach and organization. There is a proper balance between, and a natural combination of, the algebraic and geometric aspects of Lie theory, not only in technical proofs but also in conceptual viewpoints. For example, the orbital geometry of adjoint action, is regarded as the geometric organization of the totality of non-commutativity of a given compact connected Lie group, while the maximal tori theorem of E. Cartan and the Weyl reduction of the adjoint action on G to the Weyl group action on a chosen maximal torus are presented as the key results that provide a clear-cut understanding of the orbital geometry.

This invaluable book provides a concise and systematic introduction to the theory of compact connected Lie groups and their representations, as well as a complete presentation of the structure and classification theory. It uses a non-traditional approach and organization. There is a proper balance between, and a natural combination of, the algebraic and geometric aspects of Lie theory, not only in technical proofs but also in conceptual viewpoints. For example, the orbital geometry of adjoint action, is regarded as the geometric organization of the totality of non-commutativity of a given compact connected Lie group, while the maximal tori theorem of E. Cartan and the Weyl reduction of the adjoint action on G to the Weyl group action on a chosen maximal torus are presented as the key results that provide a clear-cut understanding of the orbital geometry.

Many results, both from semi group theory itself and from the applied sciences, are phrased in discipline-specific languages and hence are hardly known to a broader community. This volume contains a selection of lectures presented at a conference that was organised as a forum for all mathematicians using semi group theory to learn what is happening outside their own field of research. The collection will help to establish a number of new links between various sub-disciplines of semigroup theory, stochastic processes, differential equations and the applied fields. The theory of semigroups of operators is a well-developed branch of functional analysis. Its foundations were laid at the beginning of the 20th century, while the fundamental generation theorem of Hille and Yosida dates back to the forties. The theory was, from the very beginning, designed as a universal language for partial differential equations and stochastic processes, but at the same time it started to live as an independent branch of operator theory. Nowadays, it still has the same distinctive flavour: it develops rapidly by posing new 'internal' questions and in answering them, discovering new methods that can be used in applications. On the other hand, it is influenced by questions from PDEs and stochastic processes as well as from applied sciences such as mathematical biology and optimal control, and thus it continually gathers a new momentum. Researchers and postgraduate students working in operator theory, partial differential equations, probability and stochastic processes, analytical methods in biology and other natural sciences, optimization and optimal control will find this volume useful.

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

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.

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

This book, in some sense, began to be written by the first author in 1983, when optional lectures on Abelian groups were held at the Fac ulty of Mathematics and Computer Science, 'Babes-Bolyai' University in Cluj-Napoca, Romania. From 1992, these lectures were extended to a twosemester electivecourse on abelian groups for undergraduate stu dents, followed by a twosemester course on the same topic for graduate students in Algebra. All the other authors attended these two years of lectures and are now Assistants to the Chair of Algebra of this Fac ulty. The first draft of this collection, including only exercises solved by students as home works, the last ten years, had 160pages. We felt that there is a need for a book such as this one, because it would provide a nice bridge between introductory Abelian Group Theory and more advanced research problems. The book InfiniteAbelianGroups, published by LaszloFuchsin two volumes 1970 and 1973 willwithout doubt last as the most important guide for abelian group theorists. Many exercises are selected from this source but there are plenty of other bibliographical items (see the Bibliography) which were used in order to make up this collection. For some of the problems stated, recent developments are also given. Nevertheless, there are plenty of elementary results (the so called 'folklore') in Abelian Group Theory whichdo not appear in any written material. It is also one purpose of this book to complete this gap."