This book offers an original introduction to the representation theory of algebras, suitable for beginning researchers in algebra. It includes many results and techniques not usually covered in introductory books, some of which appear here for the first time in book form. The exposition employs methods from linear algebra (spectral methods and quadratic forms), as well as categorical and homological methods (module categories, Galois coverings, Hochschild cohomology) to present classical aspects of ring theory under new light. This includes topics such as rings with several objects, the Harada-Sai lemma, chain conditions, and Auslander-Reiten theory. Noteworthy and significant results covered in the book include the Brauer-Thrall conjectures, Drozd's theorem, and criteria to distinguish tame from wild algebras. This text may serve as the basis for a second graduate course in algebra or as an introduction to research in the field of representation theory of algebras. The originality of the exposition and the wealth of topics covered also make it a valuable resource for more established researchers.

This work presents invited contributions from the second "International Conference on Mathematics and Statistics" jointly organized by the AUS (American University of Sharjah) and the AMS (American Mathematical Society). Addressing several research fields across the mathematical sciences, all of the papers were prepared by faculty members at universities in the Gulf region or prominent international researchers. The current volume is the first of its kind in the UAE and is intended to set new standards of excellence for collaboration and scholarship in the region.

This edition has been revised and expanded, particularly the material on rings and fields, to provide a comprehensive first course in abstract algebra. The text is written for the student encountering this subject for the first time - the treatment is clear and patient, but also provides enough depth to demonstrate the insights that abstract algebra offers. The book includes many worked examples and each chapter contains a set of graded exercises, with partial solutions.

For more than five decades Bertram Kostant has been one of the major architects of modern Lie theory. Virtually all his papers are pioneering with deep consequences, many giving rise to whole new fields of activities. His interests span a tremendous range of Lie theory, from differential geometry to representation theory, abstract algebra, and mathematical physics. It is striking to note that Lie theory (and symmetry in general) now occupies an ever increasing larger role in mathematics than it did in the fifties. Now in the sixth decade of his career, he continues to produce results of astonishing beauty and significance for which he is invited to lecture all over the world. This is the fifth volume (1995-2005) of a five-volume set of Bertram Kostant's collected papers. A distinguished feature of this fifth volume is Kostant's commentaries and summaries of his papers in his own words.

The chapters in this contributed volume explore new results and existing problems in algebra, analysis, and related topics. This broad coverage will help generate new ideas to solve various challenges that face researchers in pure mathematics. Specific topics covered include maximal rotational hypersurfaces, k-Horadam sequences, quantum dynamical semigroups, and more. Additionally, several applications of algebraic number theory and analysis are presented. Algebra, Analysis, and Associated Topics will appeal to researchers, graduate students, and engineers interested in learning more about the impact pure mathematics has on various fields.

The modular representation theory of Iwahori-Hecke algebras and this theory's connection to groups of Lie type is an area of rapidly expanding interest; it is one that has also seen a number of breakthroughs in recent years. In classifying the irreducible representations of Iwahori-Hecke algebras at roots of unity, this book is a particularly valuable addition to current research in this field. Using the framework provided by the Kazhdan-Lusztig theory of cells, the authors develop an analogue of James' (1970) "characteristic-free'' approach to the representation theory of Iwahori-Hecke algebras in general. Presenting a systematic and unified treatment of representations of Hecke algebras at roots of unity, this book is unique in its approach and includes new results that have not yet been published in book form. It also serves as background reading to further active areas of current research such as the theory of affine Hecke algebras and Cherednik algebras. The main results of this book are obtained by an interaction of several branches of mathematics, namely the theory of Fock spaces for quantum affine Lie algebras and Ariki's theorem, the combinatorics of crystal bases, the theory of Kazhdan-Lusztig bases and cells, and computational methods. This book will be of use to researchers and graduate students in representation theory as well as any researchers outside of the field with an interest in Hecke algebras.

This book gives a comprehensive introduction to Universal Algebraic Logic. The three main themes are (i) universal logic and the question of what logic is, (ii) duality theories between the world of logics and the world of algebra, and (iii) Tarskian algebraic logic proper including algebras of relations of various ranks, cylindric algebras, relation algebras, polyadic algebras and other kinds of algebras of logic. One of the strengths of our approach is that it is directly applicable to a wide range of logics including not only propositional logics but also e.g. classical first order logic and other quantifier logics. Following the Tarskian tradition, besides the connections between logic and algebra, related logical connections with geometry and eventually spacetime geometry leading up to relativity are also part of the perspective of the book. Besides Tarskian algebraizations of logics, category theoretical perspectives are also touched upon. This book, apart from being a monograph containing state of the art results in algebraic logic, can be used as the basis for a number of different courses intended for both novices and more experienced students of logic, mathematics, or philosophy. For instance, the first two chapters can be used in their own right as a crash course in Universal Algebra.

Provides a thorough discussion of the orderability of a group. The book details the major developments in the theory of lattice-ordered groups, delineating standard approaches to structural and permutation representations. A radically new presentation of the theory of varieties of lattice-ordered groups is offered.;This work is intended for pure and applied mathematicians and algebraists interested in topics such as group, order, number and lattice theory, universal algebra, and representation theory; and upper-level undergraduate and graduate students in these disciplines.;College or university bookstores may order five or more copies at a special student price which is available from Marcel Dekker Inc, upon request.

The main theme of this book is the theory of heights as they appear in various guises. This includes a large body of results on Mahlers measure of the height of a polynomial. The authors'approach is very down to earth as they cover the rationals, assuming no prior knowledge of elliptic curves. The chapters include examples and particular computations, with all special calculation included so as to be self-contained. The authors devote space to discussing Mahlers measure and to giving some convincing and original examples to explain this phenomenon. XXXXXXX NEUER TEXT The main theme of this book is the theory of heights as it appears in various guises. To this End.txt.Int.:, it examines the results of Mahlers measure of the height of a polynomial, which have never before appeared in book form. The authors take a down-to-earth approach that includes convincing and original examples. The book uncovers new and interesting connections between number theory and dynamics and will be interesting to researchers in both number theory and nonlinear dynamics."

Some mathematical disciplines can be presented and developed in the context of other disciplines, for instance Boolean algebras, that Stone has converted in a branch of ring theory, projective geome- tries, characterized by Birkhoff as lattices of a special type, projec- tive, descriptive and spherical geometries, represented by Prenowitz, as multigroups, linear geometries and convex sets presented by Jan- tosciak and Prenowitz as join spaces. As Prenowitz and Jantosciak did for geometries, in this book we present and study several ma- thematical disciplines that use the Hyperstructure Theory. Since the beginning, the Hyperstructure Theory and particu- larly the Hypergroup Theory, had applications to several domains. Marty, who introduced hypergroups in 1934, applied them to groups, algebraic functions and rational fractions. New applications to groups were also found among others by Eaton, Ore, Krasner, Utumi, Drbohlav, Harrison, Roth, Mockor, Sureau and Haddad. Connections with other subjects of classical pure Mathematics have been determined and studied: * Fields by Krasner, Stratigopoulos and Massouros Ch. * Lattices by Mittas, Comer, Konstantinidou, Serafimidis, Leoreanu and Calugareanu * Rings by Nakano, Kemprasit, Yuwaree * Quasigroups and Groupoids by Koskas, Corsini, Kepka, Drbohlav, Nemec * Semigroups by Kepka, Drbohlav, Nemec, Yuwaree, Kempra- sit, Punkla, Leoreanu * Ordered Structures by Prenowitz, Corsini, Chvalina IX x * Combinatorics by Comer, Tallini, Migliorato, De Salvo, Scafati, Gionfriddo, Scorzoni * Vector Spaces by Mittas * Topology by Mittas , Konstantinidou * Ternary Algebras by Bandelt and Hedlikova.

The three volumes of Interest Rate Modeling present a comprehensive and up-to-date treatment of techniques and models used in the pricing and risk management of fixed income securities. Written by two leading practitioners and seasoned industry veterans, this unique series combines finance theory, numerical methods, and approximation techniques to provide the reader with an integrated approach to the process of designing and implementing industrial-strength models for fixed income security valuation and hedging. Aiming to bridge the gap between advanced theoretical models and real-life trading applications, the pragmatic, yet rigorous, approach taken in this book will appeal to students, academics, and professionals working in quantitative finance. The first half of Volume III contains a detailed study of several classes of fixed income securities, ranging from simple vanilla options to highly exotic cancelable and path-dependent derivatives. The analysis is done in product-specific fashion covering, among other subjects, risk characterization, calibration strategies, and valuation methods. In its second half, Volume III studies the general topic of derivative portfolio risk management, with a particular emphasis on the challenging problem of computing smooth price sensitivities to market input perturbations.

This comprehensive reference begins with a review of the basics followed by a presentation of flag varieties and finite- and infinite-dimensional representations in classical types and subvarieties of flag varieties and their singularities. Associated varieties and characteristic cycles are covered as well and Kazhdan-Lusztig polynomials are treated. The coverage concludes with a discussion of pattern avoidance and singularities and some recent results on Springer fibers.

A comprehensive presentation of abstract algebra and an in-depth treatment of the applications of algebraic techniques and the relationship of algebra to other disciplines, such as number theory, combinatorics, geometry, topology, differential equations, and Markov chains.

In his studies of cyclotomic fields, in view of establishing his monumental theorem about Fermat's last theorem, Kummer introduced "local" methods. They are concerned with divisibility of "ideal numbers" of cyclotomic fields by lambda = 1 - psi where psi is a primitive "p"-th root of 1 (p any odd prime). Henssel developed Kummer's ideas, constructed the field of "p"-adic numbers and proved the fundamental theorem known today. Kurschak formally introduced the concept of a valuation of a field, as being real valued functions on the set of non-zero elements of the field satisfying certain properties, like the "p"-adic valuations. Ostrowski, Hasse, Schmidt and others developed this theory and collectively, these topics form the primary focus of this book.

This modern translation of Sophus Lie's and Friedrich Engel's "Theorie der Transformationsgruppen I" will allow readers to discover the striking conceptual clarity and remarkably systematic organizational thought of the original German text. Volume I presents a comprehensive introduction to the theory and is mainly directed towards the generalization of ideas drawn from the study of examples. The major part of the present volume offers an extremely clear translation of the lucid original. The first four chapters provide not only a translation, but also a contemporary approach, which will help present day readers to familiarize themselves with the concepts at the heart of the subject. The editor's main objective was to encourage a renewed interest in the detailed classification of Lie algebras in dimensions 1, 2 and 3, and to offer access to Sophus Lie's monumental Galois theory of continuous transformation groups, established at the end of the 19th Century. Lie groups are widespread in mathematics, playing a role in representation theory, algebraic geometry, Galois theory, the theory of partial differential equations and also in physics, for example in general relativity. This volume is of interest to researchers in Lie theory and exterior differential systems and also to historians of mathematics. The prerequisites are a basic knowledge of differential calculus, ordinary differential equations and differential geometry.

This book collects important results concerning the classification and properties of nilpotent orbits in a Lie algebra. It develops the Dynkin-Kostant and Bala-Carter classifications of complex nilpotent orbits and derives the Lusztig-Spaltenstein theory of induction of nilpotent orbits.

For more than five decades Bertram Kostant has been one of the major architects of modern Lie theory. Virtually all his papers are pioneering with deep consequences, many giving rise to whole new fields of activities. His interests span a tremendous range of Lie theory, from differential geometry to representation theory, abstract algebra, and mathematical physics. It is striking to note that Lie theory (and symmetry in general) now occupies an ever increasing larger role in mathematics than it did in the fifties. Now in the sixth decade of his career, he continues to produce results of astonishing beauty and significance for which he is invited to lecture all over the world. This is the second volume (1965-1975) of a five-volume set of Bertram Kostant's collected papers. A distinguished feature of this second volume is Kostant's commentaries and summaries of his papers in his own words.

The book is devoted to the perturbation analysis of matrix equations. The importance of perturbation analysis is that it gives a way to estimate the influence of measurement and/or parametric errors in mathematical models together with the rounding errors done in the computational process. The perturbation bounds may further be incorporated in accuracy estimates for the solution computed in finite arithmetic. This is necessary for the development of reliable computational methods, algorithms and software from the viewpoint of modern numerical analysis.

In this book a general perturbation theory for matrix algebraic equations is presented. Local and non-local perturbation bounds are derived for general types of matrix equations as well as for the most important equations arising in linear algebra and control theory. A large number of examples, tables and figures is included in order to illustrate the perturbation techniques and bounds.

Key features:

The first book in this field
Can be used by a variety of specialists
Material is self-contained
Results can be used in the development of reliable computational algorithms
A large number of examples and graphical illustrations are given
Written by prominent specialists in the field
"

This book covers the material of an introductory course in linear algebra: sets and maps, vector spaces, bases, linear maps, matrices, determinants, systems of linear equations, Euclidean spaces, eigenvalues and eigenvectors, diagonalization of self-adjoint operators, and classification of matrices. The book is written for beginners. Its didactic features (the "book within a book" and multiple choice tests with commented answers) make it especially suitable for self-study.

This book offers an up-to-date, comprehensive account of determinantal rings and varieties, presenting a multitude of methods used in their study, with tools from combinatorics, algebra, representation theory and geometry. After a concise introduction to Groebner and Sagbi bases, determinantal ideals are studied via the standard monomial theory and the straightening law. This opens the door for representation theoretic methods, such as the Robinson-Schensted-Knuth correspondence, which provide a description of the Groebner bases of determinantal ideals, yielding homological and enumerative theorems on determinantal rings. Sagbi bases then lead to the introduction of toric methods. In positive characteristic, the Frobenius functor is used to study properties of singularities, such as F-regularity and F-rationality. Castelnuovo-Mumford regularity, an important complexity measure in commutative algebra and algebraic geometry, is introduced in the general setting of a Noetherian base ring and then applied to powers and products of ideals. The remainder of the book focuses on algebraic geometry, where general vanishing results for the cohomology of line bundles on flag varieties are presented and used to obtain asymptotic values of the regularity of symbolic powers of determinantal ideals. In characteristic zero, the Borel-Weil-Bott theorem provides sharper results for GL-invariant ideals. The book concludes with a computation of cohomology with support in determinantal ideals and a survey of their free resolutions. Determinants, Groebner Bases and Cohomology provides a unique reference for the theory of determinantal ideals and varieties, as well as an introduction to the beautiful mathematics developed in their study. Accessible to graduate students with basic grounding in commutative algebra and algebraic geometry, it can be used alongside general texts to illustrate the theory with a particularly interesting and important class of varieties.

A matroid is an abstract mathematical structure that captures combinatorial properties of matrices. This book offers a unique introduction to matroid theory, emphasizing motivations from matrix theory and applications to systems analysis.This book serves also as a comprehensive presentation of the theory and application of mixed matrices, developed primarily by the present author in the last decade. A mixed matrix is a convenient mathematical tool for systems analysis, compatible with the physical observation that "fixed constants" and "system parameters" are to be distinguished in the description of engineering systems.This book will be extremely useful to graduate students and researchers in engineering, mathematics and computer science.

Factorization algebras are local-to-global objects that play a role in classical and quantum field theory that is similar to the role of sheaves in geometry: they conveniently organize complicated information. Their local structure encompasses examples like associative and vertex algebras; in these examples, their global structure encompasses Hochschild homology and conformal blocks. In this second volume, the authors show how factorization algebras arise from interacting field theories, both classical and quantum, and how they encode essential information such as operator product expansions, Noether currents, and anomalies. Along with a systematic reworking of the Batalin-Vilkovisky formalism via derived geometry and factorization algebras, this book offers concrete examples from physics, ranging from angular momentum and Virasoro symmetries to a five-dimensional gauge theory.

The aim of the present work is two-fold. Firstly it aims at a giving an account of many existing algorithms for calculating with finite-dimensional Lie algebras. Secondly, the book provides an introduction into the theory of finite-dimensional Lie algebras. These two subject areas are intimately related. First of all, the algorithmic perspective often invites a different approach to the theoretical material than the one taken in various other monographs (e.g., 42], 48], 77], 86]). Indeed, on various occasions the knowledge of certain algorithms allows us to obtain a straightforward proof of theoretical results (we mention the proof of the Poincare-Birkhoff-Witt theorem and the proof of Iwasawa's theorem as examples). Also proofs that contain algorithmic constructions are explicitly formulated as algorithms (an example is the isomorphism theorem for semisimple Lie algebras that constructs an isomorphism in case it exists). Secondly, the algorithms can be used to arrive at a better understanding of the theory. Performing the algorithms in concrete examples, calculating with the concepts involved, really brings the theory of life.

This book addresses Birkhoff and Mal'cev's problem of describing subquasivariety lattices. The text begins by developing the basics of atomic theories and implicational theories in languages that may, or may not, contain equality. Subquasivariety lattices are represented as lattices of closed algebraic subsets of a lattice with operators, which yields new restrictions on the equaclosure operator. As an application of this new approach, it is shown that completely distributive lattices with a dually compact least element are subquasivariety lattices. The book contains many examples to illustrate these principles, as well as open problems. Ultimately this new approach gives readers a set of tools to investigate classes of lattices that can be represented as subquasivariety lattices.