Rings, Modules, Algebras, and Abelian Groups summarizes the proceedings of a recent algebraic conference held at Venice International University in Italy. Surveying the most influential developments in the field, this reference reviews the latest research on Abelian groups, algebras and their representations, module and ring theory, and topological algebraic structures, and provides more than 600 current references and 570 display equations for further exploration of the topic. It provides stimulating discussions from world-renowned names including Laszlo Fuchs, Robert Gilmer, Saharon Shelah, Daniel Simson, and Richard Swan to celebrate 40 years of study on cumulative rings. Describing emerging theories

Over the past 20 years, the emergence of clone theory, hyperequational theory, commutator theory and tame congruence theory has led to a growth of universal algebra both in richness and in applications, especially in computer science. Yet most of the classic books on the subject are long out of print and, to date, no other book has integrated these theories with the long-established work that supports them.

Universal Algebra and Applications in Theoretical Computer Science introduces the basic concepts of universal algebra and surveys some of the newer developments in the field. The first half of the book provides a solid grounding in the core material. A leisurely pace, careful exposition, numerous examples, and exercises combine to form an introduction to the subject ideal for beginning graduate students or researchers from other areas. The second half of the book focuses on applications in theoretical computer science and advanced topics, including Mal'cev conditions, tame congruence theory, clones, and commutators.

The impact of the advances in universal algebra on computer science is just beginning to be realized, and the field will undoubtedly continue to grow and mature. Universal Algebra and Applications in Theoretical Computer Science forms an outstanding text and offers a unique opportunity to build the foundation needed for further developments in its theory and in its computer science applications.

Today Lie group theoretical approach to differential equations has been extended to new situations and has become applicable to the majority of equations that frequently occur in applied sciences. Newly developed theoretical and computational methods are awaiting application. Students and applied scientists are expected to understand these methods. Volume 3 and the accompanying software allow readers to extend their knowledge of computational algebra. Written by the world's leading experts in the field, this up-to-date sourcebook covers topics such as Lie-B lund, conditional and non-classical symmetries, approximate symmetry groups for equations with a small parameter, group analysis of differential equations with distributions, integro-differential equations, recursions, and symbolic software packages. The text provides an ideal introduction to the modern group analysis and addresses issues to both beginners and experienced researchers in the application of Lie group methods.

Clear prose, tight organization, and a wealth of examples and computational techniques make Basic Matrix Algebra with Algorithms and Applications an outstanding introduction to linear algebra. The author designed this treatment specifically for freshman majors in mathematical subjects and upper-level students in natural resources, the social sciences, business, or any discipline that eventually requires an understanding of linear models. With extreme pedagogical clarity that avoids abstraction wherever possible, the author emphasizes minimal polynomials and their computation using a Krylov algorithm. The presentation is highly visual and relies heavily on work with a graphing calculator to allow readers to focus on concepts and techniques rather than on tedious arithmetic. Supporting materials, including test preparation Maple worksheets, are available for download from the Internet. This unassuming but insightful and remarkably original treatment is organized into bite-sized, clearly stated objectives. It goes well beyond the LACSG recommendations for a first course while still implementing their philosophy and core material. Classroom tested with great success, it prepares readers well for the more advanced studies their fields ultimately will require.

With a unique approach and presenting an array of new and intriguing topics, Mathematical Quantization offers a survey of operator algebras and related structures from the point of view that these objects are quantizations of classical mathematical structures. This approach makes possible, with minimal mathematical detail, a unified treatment of a variety of topics.

Detailed here for the first time, the fundamental idea of mathematical quantization is that sets are replaced by Hilbert spaces. Building on this idea, and most importantly on the fact that scalar-valued functions on a set correspond to operators on a Hilbert space, one can determine quantum analogs of a variety of classical structures. In particular, because topologies and measure classes on a set can be treated in terms of scalar-valued functions, we can transfer these constructions to the quantum realm, giving rise to C*- and von Neumann algebras.

In the first half of the book, the author quickly builds the operator algebra setting. He uses this as a unifying theme in the second half, in which he treats several active research topics, some for the first time in book form. These include the quantum plane and tori, operator spaces, Hilbert modules, Lipschitz algebras, and quantum groups.

For graduate students, Mathematical Quantization offers an ideal introduction to a research area of great current interest. For professionals in operator algebras and functional analysis, it provides a readable tour of the current state of the field.

This second edition of Compact Numerical Methods for Computers presents reliable yet compact algorithms for computational problems. As in the previous edition, the author considers specific mathematical problems of wide applicability, develops approaches to a solution and the consequent algorithm, and provides the program steps. He emphasizes useful applicable methods from various scientific research fields, ranging from mathematical physics to commodity production modeling. While the ubiquitous personal computer is the particular focus, the methods have been implemented on computers as small as a programmable pocket calculator and as large as a highly parallel supercomputer. New to the Second Edition Presents program steps as Turbo Pascal code Includes more algorithmic examples Contains an extended bibliography The accompanying software (available by coupon at no charge) includes not only the algorithm source codes, but also driver programs, example data, and several utility codes to help in the software engineering of end-user programs. The codes are designed for rapid implementation and reliable use in a wide variety of computing environments. Scientists, statisticians, engineers, and economists who prepare/modify programs for use in their work will find this resource invaluable. Moreover, since little previous training in numerical analysis is required, the book can also be used as a supplementary text for courses on numerical methods and mathematical software.

This popular textbook was thoughtfully and specifically tailored to introducing undergraduate students to linear algebra. The second edition has been carefully revised to improve upon its already successful format and approach. In particular, the author added a chapter on quadratic forms, making this one of the most comprehensive introductory texts on linear algebra.

The first and second editions of this successful textbook have been highly praised for their lucid and detailed coverage of abstract algebra. In this third edition, the author has carefully revised and extended his treatment, particularly the material on rings and fields, to provide an even more satisfying first course in abstract algebra.

"Presents the proceedings of the recently held Third International Conference on Commutative Ring Theory in Fez, Morocco. Details the latest developments in commutative algebra and related areas-featuring 26 original research articles and six survey articles on fundamental topics of current interest. Examines wide-ranging developments in commutative algebra, together with connections to algebraic number theory and algebraic geometry."

Written by pioneers in this exciting new field, Algebraic Statistics introduces the application of polynomial algebra to experimental design, discrete probability, and statistics.

It begins with an introduction to Gröbner bases and a thorough description of their applications to experimental design. A special chapter covers the binary case with new application to coherent systems in reliability and two level factorial designs. The work paves the way, in the last two chapters, for the application of computer algebra to discrete probability and statistical modelling through the important concept of an algebraic statistical model.

As the first book on the subject, Algebraic Statistics presents many opportunities for spin-off research and applications and should become a landmark work welcomed by both the statistical community and its relatives in mathematics and computer science.

Designed for use in a second course on linear algebra, Matrix Theory and Applications with MATLAB covers the basics of the subject-from a review of matrix algebra through vector spaces to matrix calculus and unitary similarity-in a presentation that stresses insight, understanding, and applications. Among its most outstanding features is the integration of MATLAB throughout the text. Each chapter includes a MATLAB subsection that discusses the various commands used to do the computations in that section and offers code for the graphics and some algorithms used in the text.

All of the material is presented from a matrix point of view with enough rigor for students to learn to compose arguments and proofs and adjust the material to cover other problems. The treatment includes optional subsections covering applications, and the final chapters move beyond basic matrix theory to discuss more advanced topics, such as decompositions, positive definite matrices, graphics, and topology.

Filled with illustrations, examples, and exercises that reinforce understanding, Matrix Theory and Applications with MATLAB allows readers to experiment and visualize results in a way that no other text does. Its rigor, use of MATLAB, and focus on applications better prepares them to use the material in their future work and research, to extend the material, and perhaps obtain new results of their own.

Model theory investigates mathematical structures by means of formal languages. These so-called first-order languages have proved particularly useful. The text introduces the reader to the model theory of first-order logic, avoiding syntactical issues that are not too relevant to model-theory. In this spirit, the compactness theorem is proved via the algebraically useful ultraproduct technique, rather than via the completeness theorem of first-order logic. This leads fairly quickly to algebraic applications, like Malcev's local theorems (of group theory) and, after a little more preparation, also to Hilbert's Nullstellensatz (of field theory). Steinitz' dimension theory for field extensions is obtained as a special case of a much more general model-theoretic treatment of strongly minimal sets. The final chapter is on the models of the first-order theory of the integers as an abelian group. This material appears here for the first time in a textbook of introductory level, and is used to give hints to further reading and to recent developments in the field, such as stability (or classification) theory. The latter itself is not touched upon. The undergraduate or graduate, is assumed t

A remarkable interplay exists between the fields of elliptic functions and orthogonal polynomials. In the first monograph to explore their connections, Elliptic Polynomials combines these two areas of study, leading to an interesting development of some basic aspects of each. It presents new material about various classes of polynomials and about the odd Jacobi elliptic functions and their inverses.

The term elliptic polynomials refers to the polynomials generated by odd elliptic integrals and elliptic functions. In studying these, the authors consider such things as orthogonality and the construction of weight functions and measures, finding structure constants and interesting inequalities, and deriving useful formulas and evaluations.

Although some of the material may be familiar, it establishes a new mathematical field that intersects with classical subjects at many points. Its wealth of information on important properties of polynomials and clear, accessible presentation make Elliptic Polynomials valuable to those in real and complex analysis, number theory, and combinatorics, and will undoubtedly generate further research.

Monomial Algebras, Second Edition presents algebraic, combinatorial, and computational methods for studying monomial algebras and their ideals, including Stanley-Reisner rings, monomial subrings, Ehrhart rings, and blowup algebras. It emphasizes square-free monomials and the corresponding graphs, clutters, or hypergraphs. New to the Second Edition Four new chapters that focus on the algebraic properties of blowup algebras in combinatorial optimization problems of clutters and hypergraphs Two new chapters that explore the algebraic and combinatorial properties of the edge ideal of clutters and hypergraphs Full revisions of existing chapters to provide an up-to-date account of the subject Bringing together several areas of pure and applied mathematics, this book shows how monomial algebras are related to polyhedral geometry, combinatorial optimization, and combinatorics of hypergraphs. It directly links the algebraic properties of monomial algebras to combinatorial structures (such as simplicial complexes, posets, digraphs, graphs, and clutters) and linear optimization problems.

This book describes, in a basic way, the most useful and effective iterative solvers and appropriate preconditioning techniques for some of the most important classes of large and sparse linear systems. The solution of large and sparse linear systems is the most time-consuming part for most of the scientific computing simulations. Indeed, mathematical models become more and more accurate by including a greater volume of data, but this requires the solution of larger and harder algebraic systems. In recent years, research has focused on the efficient solution of large sparse and/or structured systems generated by the discretization of numerical models by using iterative solvers.

About the book In honor of Edgar Enochs and his venerable contributions to a broad range of topics in Algebra, top researchers from around the world gathered at Auburn University to report on their latest work and exchange ideas on some of today's foremost research topics. This carefully edited volume presents the refereed papers of the participants of these talks along with contributions from other veteran researchers who were unable to attend. These papers reflect many of the current topics in Abelian Groups, Commutative Algebra, Commutative Rings, Group Theory, Homological Algebra, Lie Algebras, and Module Theory. Accessible even to beginning mathematicians, many of these articles suggest problems and programs for future study. This volume is an outstanding addition to the literature and a valuable handbook for beginning as well as seasoned researchers in Algebra. about the editors H. PAT GOETERS completed his undergraduate studies in mathematics and computer science at Southern Connecticut State University and received his Ph.D. in 1984 from the University of Connecticut under the supervision of William J. Wickless. After spending one year in a post-doctoral position in Wesleyan University under the tutelage of James D. Reid, Goeters was invited for a tenure track position in Auburn University by Ulrich F. Albrecht. Soon afterwards, William Ullery and Overtoun Jenda were hired, and so began a lively Algebra group. OVERTOUN M. G. JENDA received his bachelor's degree in Mathematics from Chancellor College, the University of Malawi. He moved to the U.S. 1977 to pursue graduate studies at University of Kentucky, earning his Ph.D. in 1981 under the supervision of Professor Edgar Enochs. He then returned to Chancellor College, where he was a lecturer (assistant professor) for three years. He moved to the University of Botswana for another three-year stint as a lecturer before moving back to the University of Kentucky as a visi

This volume collects presentations from the international workshop on local cohomology held in Guanajuato, Mexico, including expanded lecture notes of two minicourses on applications in equivariant topology and foundations of duality theory, and chapters on finiteness properties, D-modules, monomial ideals, combinatorial analysis, and related topics. Featuring selected papers from renowned experts around the world, Local Cohomology and Its Applications is a provocative reference for algebraists, topologists, and upper-level undergraduate and graduate students in these disciplines.

Over the last thirty years, the subject of nonlinear integrable systems has grown into a full-fledged research topic. In the last decade, Lie algebraic methods have grown in importance to various fields of theoretical research and worked to establish close relations between apparently unrelated systems. The various ideas associated with Lie algebra and Lie groups can be used to form a particularly elegant approach to the properties of nonlinear systems. In this volume, the author exposes the basic techniques of using Lie algebraic concepts to explore the domain of nonlinear integrable systems. His emphasis is not on developing a rigorous mathematical basis, but on using Lie algebraic methods as an effective tool. The book begins by establishing a practical basis in Lie algebra, including discussions of structure Lie, loop, and Virasor groups, quantum tori and Kac-Moody algebras, and gradation. It then offers a detailed discussion of prolongation structure and its representation theory, the orbit approach-for both finite and infinite dimension Lie algebra. The author also presents the modern approach to symmetries of integrable systems, including important new ideas in symmetry analysis, such as gauge transformations, and the "soldering" approach. He then moves to Hamiltonian structure, where he presents the Drinfeld-Sokolov approach, the Lie algebraic approach, Kupershmidt's approach, Hamiltonian reductions and the Gelfand Dikii formula. He concludes his treatment of Lie algebraic methods with a discussion of the classical r-matrix, its use, and its relations to double Lie algebra and the KP equation.

Renewed interest in vector spaces and linear algebras has spurred the search for large algebraic structures composed of mathematical objects with special properties. Bringing together research that was otherwise scattered throughout the literature, Lineability: The Search for Linearity in Mathematics collects the main results on the conditions for the existence of large algebraic substructures. It investigates lineability issues in a variety of areas, including real and complex analysis. After presenting basic concepts about the existence of linear structures, the book discusses lineability properties of families of functions defined on a subset of the real line as well as the lineability of special families of holomorphic (or analytic) functions defined on some domain of the complex plane. It next focuses on spaces of sequences and spaces of integrable functions before covering the phenomenon of universality from an algebraic point of view. The authors then describe the linear structure of the set of zeros of a polynomial defined on a real or complex Banach space and explore specialized topics, such as the lineability of various families of vectors. The book concludes with an account of general techniques for discovering lineability in its diverse degrees.

Elementary Linear Algebra is written for the first undergraduate course. The book focuses on the importance of linear algebra in many disciplines such as engineering, economics, statistics, and computer science. The text reinforces critical ideas and lessons of traditional topics. More importantly, the book is written in a manner that deeply ingrains computational methods.

This book is a collection of research papers and surveys on algebra that were presented at the Conference on Groups, Rings, and Group Rings held in Ubatuba, Brazil. This text familiarizes researchers with the latest topics, techniques, and methodologies in several branches of contemporary algebra. With extensive coverage, it examines broad themes from group theory and ring theory, exploring their relationship with other branches of algebra including actions of Hopf algebras, groups of units of group rings, combinatorics of Young diagrams, polynomial identities, growth of algebras, and more. Featuring international contributions, this book is ideal for mathematicians specializing in these areas.

The feedback control of nonlinear differential and algebraic equation systems (DAEs) is a relatively new subject. Developing steadily over the last few years, it has generated growing interest inspired by its engineering applications and by advances in the feedback control of nonlinear ordinary differential equations (ODEs). This book-the first of its kind-introduces the reader to the inherent characteristics of nonlinear DAE systems and the methods used to address their control, then discusses the significance of DAE systems to the modeling and control of chemical processes. Within a unified framework, Control of Nonlinear Differential Algebraic Equation Systems presents recent results on the stabilization, output tracking, and disturbance elimination for a large class of nonlinear DAE systems.

Written at a basic mathematical level-assuming some familiarity with analysis and control of nonlinear ODEs-the authors focus on continuous-time systems of differential and algebraic equations in semi-explicit form. Beginning with background material about DAE systems and their differences from ODE systems, the book discusses generic classes of chemical processes, feedback control of regular and non-regular DAE systems, control of systems with disturbance inputs, the connection of the DAE systems considered with singularly perturbed systems, and finally offers examples that illustrate the application of control methods and the advantages of using high-index DAE models as the basis for controller design.
Mathematicians and engineers will find that this book provides unique, timely results that also clearly documents the relevance of DAE systems to chemical processes.

A thorough understanding of statistical mechanics depends strongly on the insights and manipulative skills that are acquired through the solving of problems. Problems on Statistical Mechanics provides over 120 problems with model solutions, illustrating both basic principles and applications that range from solid-state physics to cosmology. An introductory chapter provides a summary of the basic concepts and results that are needed to tackle the problems, and also serves to establish the notation that is used throughout the book. The problems themselves occupy five chapters, progressing from the simpler aspects of thermodynamics and equilibrium statistical ensembles to the more challenging ideas associated with strongly interacting systems and nonequilibrium processes. Comprehensive solutions to all of the problems are designed to illustrate efficient and elegant problem-solving techniques. Where appropriate, the authors incorporate extended discussions of the points of principle that arise in the course of the solutions. The appendix provides useful mathematical formulae.

"Provides a thorough introduction to the algebraic theory of systems of differential equations, as developed by the Japanese school of M. Sato and his colleagues. Features a complete review of hyperfunction-microfunction theory and the theory of D-modules. Strikes the perfect balance between analytic and algebraic aspects."