Reflection groups and their invariant theory provide the main themes of this book and the first two parts focus on these topics. The first 13 chapters deal with reflection groups (Coxeter groups and Weyl groups) in Euclidean Space while the next thirteen chapters study the invariant theory of pseudo-reflection groups. The third part of the book studies conjugacy classes of the elements in reflection and pseudo-reflection groups. The book has evolved from various graduate courses given by the author over the past 10 years. It is intended to be a graduate text, accessible to students with a basic background in algebra.

The international conference on which the book is based brought together many of the world's leading experts, with particular effort on the interaction between established scientists and emerging young promising researchers, as well as on the interaction of pure and applied mathematics. All material has been rigorously refereed. The contributions contain much material developed after the conference, continuing research and incorporating additional new results and improvements. In addition, some up-to-date surveys are included.

Devoted to the theory of Lie algebras and algebraic groups, this book includes a large amount of commutative algebra and algebraic geometry so as to make it as self-contained as possible. The aim of the book is to assemble in a single volume the algebraic aspects of the theory, so as to present the foundations of the theory in characteristic zero. Detailed proofs are included, and some recent results are discussed in the final chapters.

A recurring theme in a traditional introductory graduate algebra course is the existence and consequences of relationships between different algebraic structures. This is also the theme of this book, an exposition of connections between representations of finite partially ordered sets and abelian groups. Emphasis is placed throughout on classification, a description of the objects up to isomorphism, and computation of representation type, a measure of when classification is feasible. David M. Arnold is the Ralph and Jean Storm Professor of Mathematics at Baylor University. He is the author of "Finite Rank Torsion Free Abelian Groups and Rings" published in the Springer-Verlag Lecture Notes in Mathematics series, a co-editor for two volumes of conference proceedings, and the author of numerous articles in mathematical research journals. His research interests are in abelian group theory and related topics, such as representations of partially ordered sets and modules over discrete valuation rings, subrings of algebraic number fields, and pullback rings. He received his Ph. D. from the University of Illinois, Urbana and was a member of the faculty at New Mexico State University for many years.

This text brings the reader to the frontiers of current research in topological rings. The exercises illustrate many results and theorems while a comprehensive bibliography is also included.

The book is aimed at those readers acquainted with some very basic point-set topology and algebra, as normally presented in semester courses at the beginning graduate level or even at the advanced undergraduate level. Familiarity with Hausdorff, metric, compact and locally compact spaces and basic properties of continuous functions, also with groups, rings, fields, vector spaces and modules, and with Zorn's Lemma, is also expected.

Introduction to Lie Groups and Lie Algebra, 51

This book is the first monograph wholly devoted to the investigation of differential and difference dimension theory. The differential dimension polynomial describes in exact terms the degree of freedom of a dynamic system as well as the number of arbitrary constants in the general solution of a system of algebraic differential equations.

Difference algebra arises from the study of algebraic difference equations and therefore bears a considerable resemblance to its differential counterpart. Difference algebra was developed in the same period as differential algebra and it has the same founder, J. Ritt. It grew to a mathematical area with its own ideas and methods mainly due to the work of R. Cohn, who raised difference algebra to the same level as differential algebra. The relatively new science of computer algebra has given strong impulses to the theory of dimension polynomials, now that packages such as MAPLE enable the solution of many problems which cannot be solved otherwise. Applications of differential and difference dimension theory can be found in many fields of mathematics, as well as in theoretical physics, system theory and other areas of science.

Audience: This book will be of interest to researchers and graduate students whose work involves differential and difference equations, algebra and number theory, partial differential equations, combinatorics and mathematical physics.

From the reviews:
.."., the book must be of great help for a researcher who already has some idea of Lie theory, wants to employ it in his everyday research and/or teaching, and needs a source for customary reference on the subject. From my viewpoint, the volume is perfectly fit to serve as such a source, ... On the whole, it is quite a pleasure, after making yourself comfortable in that favourite office armchair of yours, just to keep the volume gently in your hands and browse it slowly and thoughtfully; and after all, what more on Earth can one expect of any book?"
"The New Zealand Mathematical Society Newsletter"
..". Both parts are very nicely written and can be strongly recommended."
"European Mathematical Society"

This work presents new and old constructions of nearrings. Links between properties of the multiplicative of nearrings (as regularity conditions and identities) and the structure of nearrings are studied. Primality and minimality properties of ideals are collected. Some types of simpler' nearrings are examined. Some nearrings of maps on a group are reviewed and linked with group-theoretical and geometrical questions.
Audience: Researchers working in nearring theory, group theory, semigroup theory, designs, and translation planes. Some of the material will be accessible to graduate students.

In response to a growing interest in Total Least Squares (TLS) and Errors-In-Variables (EIV) modeling by researchers and practitioners, well-known experts from several disciplines were invited to prepare an overview paper and present it at the third international workshop on TLS and EIV modeling held in Leuven, Belgium, August 27-29, 2001. These invited papers, representing two-thirds of the book, together with a selection of other presented contributions yield a complete overview of the main scientific achievements since 1996 in TLS and Errors-In-Variables modeling. In this way, the book nicely completes two earlier books on TLS (SIAM 1991 and 1997). Not only computational issues, but also statistical, numerical, algebraic properties are described, as well as many new generalizations and applications. Being aware of the growing interest in these techniques, it is a strong belief that this book will aid and stimulate users to apply the new techniques and models correctly to their own practical problems.

While there are many excellent books available on fundamental and applied electromagnetics, most introduce operator concepts in an ad hoc manner, and few discuss the subject within the general framework of operator theory. This is in contrast to quantum theory, where the use of operators and concepts from functional analysis is common. However, casting electromagnetic problems in terms of operator theory produces useful insights into the mathematical properties and physical characteristics of solutions. For instance, the commonly used modal expansion of fields in waveguides are immediately justified upon identifying the differential operators as being of the appropriate Sturm-Liouville type. As another example, existence, uniqueness and solvability of integral formulations can often be settled by appealing to the theory of Fredholm operators. Many other examples that illustrate the value of abstracting problems to an operator level are provided. Although the book focuses on mathematical fundamentals, it is written from the perspective of engineers and applied scientists working in electromagnetics. The book begins with a review of electromagnetic theory, including a discussion of singular integral operators commonly encountered in applications. It then turns to a self-contained introduction to operator theory, including basic functional analysis, linear operators, Green¿s functions and Green¿s operators, spectral theory, and Sturm-Liouville operators. The discussion is at an introductory mathematical level, presenting definitions and theorems, as well as proofs of the theorems when these are particularly simple or enlightening. The tools developed in this first part of the book are then applied to problems in classical electromagnetic theory: boundary-value problems and potential theory, transmission lines, waves in layered media, scattering problems in waveguides, and electromagnetic cavities.

The objective of this monograph is to present a coherent picture of the almost mysterious role that analytic methods and, in particular, multidimensional residue have recently played in obtaining effective estimates for problems in commutative algebra. Bezout identities, i. e., f1g1 + ... + fmgm = 1, appear naturally in many problems, for example in commutative algebra in the Nullstellensatz, and in signal processing in the deconvolution problem. One way to solve them is by using explicit interpolation formulas in Cn, and these depend on the theory of multidimensional residues. The authors present this theory in detail, in a form developed by them, and illustrate its applications to the effective Nullstellensatz and to the Fundamental Principle for convolution equations.

This monograph presents a unified exposition of latin squares and mutually orthogonal sets of latin squares based on groups. Its focus is on orthomorphisms and complete mappings of finite groups, while also offering a complete proof of the Hall-Paige conjecture. The use of latin squares in constructions of nets, affine planes, projective planes, and transversal designs also motivates this inquiry. The text begins by introducing fundamental concepts, like the tests for determining whether a latin square is based on a group, as well as orthomorphisms and complete mappings. From there, it describes the existence problem for complete mappings of groups, building up to the proof of the Hall-Paige conjecture. The third part presents a comprehensive study of orthomorphism graphs of groups, while the last part provides a discussion of Cartesian projective planes, related combinatorial structures, and a list of open problems. Expanding the author's 1992 monograph, Orthomorphism Graphs of Groups, this book is an essential reference tool for mathematics researchers or graduate students tackling latin square problems in combinatorics. Its presentation draws on a basic understanding of finite group theory, finite field theory, linear algebra, and elementary number theory-more advanced theories are introduced in the text as needed.

From July 31 through August 3,1997, the Pennsylvania State University hosted the Topics in Number Theory Conference. The conference was organized by Ken Ono and myself. By writing the preface, I am afforded the opportunity to express my gratitude to Ken for beng the inspiring and driving force behind the whole conference. Without his energy, enthusiasm and skill the entire event would never have occurred. We are extremely grateful to the sponsors of the conference: The National Sci ence Foundation, The Penn State Conference Center and the Penn State Depart ment of Mathematics. The object in this conference was to provide a variety of presentations giving a current picture of recent, significant work in number theory. There were eight plenary lectures: H. Darmon (McGill University), "Non-vanishing of L-functions and their derivatives modulo p. " A. Granville (University of Georgia), "Mean values of multiplicative functions. " C. Pomerance (University of Georgia), "Recent results in primality testing. " C. Skinner (Princeton University), "Deformations of Galois representations. " R. Stanley (Massachusetts Institute of Technology), "Some interesting hyperplane arrangements. " F. Rodriguez Villegas (Princeton University), "Modular Mahler measures. " T. Wooley (University of Michigan), "Diophantine problems in many variables: The role of additive number theory. " D. Zeilberger (Temple University), "Reverse engineering in combinatorics and number theory. " The papers in this volume provide an accurate picture of many of the topics presented at the conference including contributions from four of the plenary lectures."

This volume brings together recent, original research and survey articles by leading experts in several fields that include singularity theory, algebraic geometry and commutative algebra. The motivation for this collection comes from the wide-ranging research of the distinguished mathematician, Antonio Campillo, in these and related fields. Besides his influence in the mathematical community stemming from his research, Campillo has also endeavored to promote mathematics and mathematicians' networking everywhere, especially in Spain, Latin America and Europe. Because of his impressive achievements throughout his career, we dedicate this book to Campillo in honor of his 65th birthday. Researchers and students from the world-wide, and in particular Latin American and European, communities in singularities, algebraic geometry, commutative algebra, coding theory, and other fields covered in the volume, will have interest in this book.

In the early 70's and 80's the field of integrable systems was in its prime youth: results and ideas were mushrooming all over the world. It was during the roaring 70's and 80's that a first version of the book was born, based on our research and on lectures which each of us had given. We owe many ideas to our colleagues Teruhisa Matsusaka and David Mumford, and to our inspiring graduate students (Constantin Bechlivanidis, Luc Haine, Ahmed Lesfari, Andrew McDaniel, Luis Piovan and Pol Vanhaecke). As it stood, our first version lacked rigor and precision, was rough, dis connected and incomplete. . . In the early 90's new problems appeared on the horizon and the project came to a complete standstill, ultimately con fined to a floppy. A few years ago, under the impulse of Pol Vanhaecke, the project was revived and gained real momentum due to his insight, vision and determination. The leap from the old to the new version is gigantic. The book is designed as a teaching textbook and is aimed at a wide read ership of mathematicians and physicists, graduate students and professionals."

The projectors are considered as simple but important type of matrices and operators. Their basic theory can be found in many books, among which Hal mas [177], [178] are of particular significance. The projectors or projections became an active research area in the last two decades due to ideas generated from linear algebra, statistics and various areas of algorithmic mathematics. There has also grown up a great and increasing number of projection meth ods for different purposes. The aim of this book is to give a unified survey on projectors and projection methods including the most recent results. The words projector, projection and idempotent are used as synonyms, although the word projection is more common. We assume that the reader is familiar with linear algebra and mathemati cal analysis at a bachelor level. The first chapter includes supplements from linear algebra and matrix analysis that are not incorporated in the standard courses. The second and the last chapter include the theory of projectors. Four chapters are devoted to projection methods for solving linear and non linear systems of algebraic equations and convex optimization problems.

Blending algebra, analysis, and topology, the study of compact Lie groups is one of the most beautiful areas of mathematics and a key stepping stone to the theory of general Lie groups. Assuming no prior knowledge of Lie groups, this book covers the structure and representation theory of compact Lie groups. It included the construction of the Spin groups, Schur Orthogonality, the Peter-Weyl Theorem, the Plancherel Theorem, the Maximal Torus Theorem, the Commutator Theorem, the Weyl Integration and Character Formulas, the Highest Weight Classification, and the Borel-Weil Theorem. The necessary Lie algebra theory is also developed in the text with a streamlined approach focusing on linear Lie groups. It's key features include: provides an approach that minimizes advanced prerequisites; self-contained and systematic exposition requiring no previous exposure to Lie theory; advances quickly to the Peter-Weyl Theorem and its corresponding Fourier theory; streamlined Lie algebra discussion reduces the differential geometry prerequisite and allows a more rapid transition to the classification and construction of representations; and exercises sprinkled throughout. related topics, assumes familiarity with elementary concepts from group theory, analysis, and manifold theory. Students, research mathematicians and physicists interested in Lie theory, will find this text very useful.

This text provides an introduction to group theory with an emphasis on clear examples. The authors present groups as naturally occurring structures arising from symmetry in geometrical figures and other mathematical objects. Written in a 'user-friendly' style, where new ideas are always motivated before being fully introduced, the text will help readers to gain confidence and skill in handling group theory notation before progressing on to applying it in complex situations. An ideal companion to any first or second year course on the topic.

This volume contains the proceedings of the NATO Advanced Study Institute "Symmetric Functions 2001: Surveys of Developments and Per- spectives", held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK, during the two weeks 25 June - 6 July 2001. The objective of the ASI was to survey recent developments and outline research perspectives in various fields, for which the fundamental questions can be stated in the language of symmetric functions (along the way emphasizing interdisciplinary connections). The instructional goals of the event determined its format: the ASI consisted of about a dozen mini-courses. Seven of them served as a basis for the papers comprising the current volume. The ASI lecturers were: Persi Diaconis, William Fulton, Mark Haiman, Phil Hanlon, Alexander Klyachko, Bernard Leclerc, Ian G. Macdonald, Masatoshi Noumi, Andrei Okounkov, Grigori Olshanski, Eric Opdam, Ana- toly Vershik, and Andrei Zelevinsky. The organizing committee consisted of Phil Hanlon, Ian Macdonald, Andrei 0 kounkov, G rigori 0 lshanski (co-director), and myself ( co-director). The original ASI co-director Sergei Kerov, who was instrumental in determining the format and scope of the event, selection of speakers, and drafting the initial grant proposal, died in July 2000. Kerov's mathemat- ical ideas strongly influenced the field, and were presented at length in a number of ASI lectures. A special afternoon session on Monday, July 2, was dedicated to his memory.

The theory of table algebras was introduced in 1991 by Z. Arad and H. Blau in order to treat, in a uniform way, products of conjugacy classes and irreducible characters of finite groups. Today, table algebra theory is a well-established branch of modern algebra with various applications, including the representation theory of finite groups, algebraic combinatorics and fusion rules algebras. This book presents the latest developments in this area. Its main goal is to give a classification of the Normalized Integral Table Algebras (Fusion Rings) generated by a faithful non-real element of degree 3. Divided into 4 parts, the first gives an outline of the classification approach, while remaining parts separately treat special cases that appear during classification. A particularly unique contribution to the field, can be found in part four, whereby a number of the algebras are linked to the polynomial irreducible representations of the group SL3(C). This book will be of interest to research mathematicians and PhD students working in table algebras, group representation theory, algebraic combinatorics and integral fusion rule algebras.

Signal processing applications have burgeoned in the past decade. During the same time, signal processing techniques have matured rapidly and now include tools from many areas of mathematics, computer science, physics, and engineering. This trend will continue as many new signal processing applications are opening up in consumer products and communications systems.
In particular, signal processing has been making increasingly sophisticated use of linear algebra on both theoretical and algorithmic fronts. This volume gives particular emphasis to exposing broader contexts of the signal processing problems so that the impact of algorithms and hardware can be better understood; it brings together the writings of signal processing engineers, computer engineers, and applied linear algebraists in an exchange of problems, theories, and techniques. This volume will be of interest to both applied mathematicians and engineers.

For every mathematician, ring theory and K-theory are intimately connected: al- braic K-theory is largely the K-theory of rings. At ?rst sight, polytopes, by their very nature, must appear alien to surveyors of this heartland of algebra. But in the presence of a discrete structure, polytopes de?ne a?ne monoids, and, in their turn, a?ne monoids give rise to monoid algebras. Teir spectra are the building blocks of toric varieties, an area that has developed rapidly in the last four decades. From a purely systematic viewpoint, "monoids" should therefore replace "po- topes" in the title of the book. However, such a change would conceal the geometric ?avor that we have tried to preserve through all chapters. Before delving into a description of the contents we would like to mention three general features of the book: (?) the exhibiting of interactions of convex geometry, ring theory, and K-theory is not the only goal; we present some of the central results in each of these ?elds; (?) the exposition is of constructive (i. e., algorithmic) nature at many places throughout the text-there is no doubt that one of the driving forces behind the current popularity of combinatorial geometry is the quest for visualization and computation; (? ) despite the large amount of information from various ?elds, we have strived to keep the polytopal perspective as the major organizational principle.

Probability theory on compact Lie groups deals with the interaction between chance and symmetry, a beautiful area of mathematics of great interest in its own sake but which is now also finding increasing applications in statistics and engineering (particularly with respect to signal processing). The author gives a comprehensive introduction to some of the principle areas of study, with an emphasis on applicability. The most important topics presented are: the study of measures via the non-commutative Fourier transform, existence and regularity of densities, properties of random walks and convolution semigroups of measures and the statistical problem of deconvolution. The emphasis on compact (rather than general) Lie groups helps readers to get acquainted with what is widely seen as a difficult field but which is also justified by the wealth of interesting results at this level and the importance of these groups for applications.

The book is primarily aimed at researchers working in probability, stochastic analysis and harmonic analysis on groups. It will also be of interest to mathematicians working in Lie theory and physicists, statisticians and engineers who are working on related applications. A background in first year graduate level measure theoretic probability and functional analysis is essential; a background in Lie groups and representation theory is certainly helpful but the first two chapters also offer orientation in these subjects."

The Bia owie a workshops on Geometric Methods in Physics, taking place in the unique environment of the Bia owie a natural forest in Poland, are among the important meetings in the field. Every year some 80 to 100 participants both from mathematics and physics join to discuss new developments and to interchange ideas. The current volume was produced on the occasion of the XXXI meeting in 2012. For the first time the workshop was followed by a School on Geometry and Physics, which consisted of advanced lectures for graduate students and young researchers. Selected speakers of the workshop were asked to contribute, and additional review articles were added. The selection shows that despite its now long tradition the workshop remains always at the cutting edge of ongoing research. The XXXI workshop had as a special topic the works of the late Boris Vasilievich Fedosov (1938 2011) who is best known for a simple and very natural construction of a deformation quantization for any symplectic manifold, and for his contributions to index theory.