This is the second in a series of three volumes dealing with important topics in algebra. Volume 2 is an introduction to linear algebra (including linear algebra over rings), Galois theory, representation theory, and the theory of group extensions. The section on linear algebra (chapters 1-5) does not require any background material from Algebra 1, except an understanding of set theory. Linear algebra is the most applicable branch of mathematics, and it is essential for students of science and engineering As such, the text can be used for one-semester courses for these students. The remaining part of the volume discusses Jordan and rational forms, general linear algebra (linear algebra over rings), Galois theory, representation theory (linear algebra over group algebras), and the theory of extension of groups follow linear algebra, and is suitable as a text for the second and third year students specializing in mathematics.

Commutative algebra is a rapidly growing subject that is developing in many different directions. This volume presents several of the most recent results from various areas related to both Noetherian and non-Noetherian commutative algebra. This volume contains a collection of invited survey articles by some of the leading experts in the field. The authors of these chapters have been carefully selected for their important contributions to an area of commutative-algebraic research. Some topics presented in the volume include: generalizations of cyclic modules, zero divisor graphs, class semigroups, forcing algebras, syzygy bundles, tight closure, Gorenstein dimensions, tensor products of algebras over fields, as well as many others. This book is intended for researchers and graduate students interested in studying the many topics related to commutative algebra.

First comprehensive treatment in book form of shape-preserving approximation by real or complex polynomials in one or several variables

Of interest to grad students and researchers in approximation theory, mathematical analysis, numerical analysis, Computer Aided Geometric Design, robotics, data fitting, chemistry, fluid mechanics, and engineering

Contains many open problems to spur future research

Rich and updated bibliography

This volume contains invited articles by top-notch experts who focus on such topics as: modular representations of algebraic groups, representations of quantum groups and crystal bases, representations of affine Lie algebras, representations of affine Hecke algebras, modular or ordinary representations of finite reductive groups, and representations of complex reflection groups and associated Hecke algebras.
"Representation Theory of Algebraic Groups and Quantum Groups" is intended for graduate students and researchers in representation theory, group theory, algebraic geometry, quantum theory and math physics.
Contributors:
H. H. Andersen, S. Ariki, C. Bonnaf, J. Chuang, J. Du, M. Finkelberg, Q. Fu, M. Geck, V. Ginzburg, A. Hida, L. Iancu, N. Jacon, T. Lam, G.I. Lehrer, G. Lusztig, H. Miyachi, S. Naito, H. Nakajima, T. Nakashima, D. Sagaki, Y. Saito, M. Shiota, J. Xiao, F. Xu, R. B. Zhang

This book, Algebraic Computability and Enumeration Models: Recursion Theory and Descriptive Complexity, presents new techniques with functorial models to address important areas on pure mathematics and computability theory from the algebraic viewpoint. The reader is first introduced to categories and functorial models, with Kleene algebra examples for languages. Functorial models for Peano arithmetic are described toward important computational complexity areas on a Hilbert program, leading to computability with initial models. Infinite language categories are also introduced to explain descriptive complexity with recursive computability with admissible sets and urelements. Algebraic and categorical realizability is staged on several levels, addressing new computability questions with omitting types realizably. Further applications to computing with ultrafilters on sets and Turing degree computability are examined. Functorial models computability is presented with algebraic trees realizing intuitionistic types of models. New homotopy techniques are applied to Marin Lof types of computations with model categories. Functorial computability, induction, and recursion are examined in view of the above, presenting new computability techniques with monad transformations and projective sets. This informative volume will give readers a complete new feel for models, computability, recursion sets, complexity, and realizability. This book pulls together functorial thoughts, models, computability, sets, recursion, arithmetic hierarchy, filters, with real tree computing areas, presented in a very intuitive manner for university teaching, with exercises for every chapter. The book will also prove valuable for faculty in computer science and mathematics.

Noncommutative Polynomial Algebras of Solvable Type and Their Modules is the first book to systematically introduce the basic constructive-computational theory and methods developed for investigating solvable polynomial algebras and their modules. In doing so, this book covers: A constructive introduction to solvable polynomial algebras and Groebner basis theory for left ideals of solvable polynomial algebras and submodules of free modules The new filtered-graded techniques combined with the determination of the existence of graded monomial orderings The elimination theory and methods (for left ideals and submodules of free modules) combining the Groebner basis techniques with the use of Gelfand-Kirillov dimension, and the construction of different kinds of elimination orderings The computational construction of finite free resolutions (including computation of syzygies, construction of different kinds of finite minimal free resolutions based on computation of different kinds of minimal generating sets), etc. This book is perfectly suited to researchers and postgraduates researching noncommutative computational algebra and would also be an ideal resource for teaching an advanced lecture course.

Combines discussion of a theoretical model with essays about social class and classism experience, offering a unique approach that helps students learn theory and apply it to different contexts. Accompanied by an Instructors' Guide which offers more in-depth exploration and commentary, as well as discussion questions, writing prompts, and activity suggestions, making this the ideal resource for lecturers teaching courses on social class. Addresses the gap in psychological literature in terms of considering social class in relation to psychology and shows how the SCWM-R model can be applied to real life scenarios.

This contributed volume highlights two areas of fundamental interest in high-performance computing: core algorithms for important kernels and computationally demanding applications. The first few chapters explore algorithms, numerical techniques, and their parallel formulations for a variety of kernels that arise in applications. The rest of the volume focuses on state-of-the-art applications from diverse domains. By structuring the volume around these two areas, it presents a comprehensive view of the application landscape for high-performance computing, while also enabling readers to develop new applications using the kernels. Readers will learn how to choose the most suitable parallel algorithms for any given application, ensuring that theory and practicality are clearly connected. Applications using these techniques are illustrated in detail, including: Computational materials science and engineering Computational cardiovascular analysis Multiscale analysis of wind turbines and turbomachinery Weather forecasting Machine learning techniques Parallel Algorithms in Computational Science and Engineering will be an ideal reference for applied mathematicians, engineers, computer scientists, and other researchers who utilize high-performance computing in their work.

Structures are defined by laws of composition, rules of generation, and relations. The objects on which these laws operate may be numbers, geometric objects like points and lines, or abstract symbols. Algebra is the study of mathematical laws, with a search for general principles that do not depend on what the objects are. Fundamental Structures of Algebra and Discrete Mathematics is an introduction to the twelve basic kinds of structures - sets, ordered sets, groups, rings, fields, vector spaces, graphs, lattices, matroids, topological spaces, universal algebras, and categories - that underlie algebra and discrete mathematics. Beginning with the most basic type of structure, sets, this unique reference provides a detailed look at the theoretical underpinning of each structure, shedding light on the significance of each structure as well as their interrelation. Using a self-contained approach that requires little previous knowledge of mathematical definitions, results, or methods, the book examines selected key aspects of these structures, including closure systems, generators, substructures, homomorphisms and congruences, equational axioms, connections with basic set theory, finiteness conditions, combinatorial properties, and discrete algorithmic procedures. Several classical results are proved, including the Abel-Ruffini theorem on unsolvability by radicals, Helly's theorem on intersecting convex sets, and a simplified version of the Godel-Herbrand completeness theorem. Many of the results are relevant to current research. Highly interactive in approach, the book features numerous exercises and examples woven throughout the text that allow the reader to become fully acquainted withthe character and function of each structure. Additional questions, designed to stretch the reader's analytical skills, appear at the end of each section. Similar to a grammar book that explains the fundamental structures essential to mastering a language, Fundamental Structures of Algebra and Discrete Mathematics is a systematic examination of the basic algebraic structures needed to understand and manipulate advanced mathematical concepts. A cornerstone reference that is both a clear primer and rigorous study guide, Fundamental Structures of Algebra and Discrete Mathematics is indispensable to the student and professional seeking to learn or use the methods of modern algebra.

Originally published in 1982, Time Resources, Society and Ecology examines and seeks to examine the time dimension in terms of the ecology, technology, social organization and spatial structure of the human habitat. Approaches to time resources - sociological time-budget studies, anthropological activity analysis, and economic analysis of money allocation - have been limited by their sectoral scope or their failure to relate effectively to the processes of social interaction, technological change and environmental structure. In this book, the book's articulation of time resources is developed in a general theoretical framework of action and interaction in time and space. The book examines constraints and possibilities facing preindustrial societies and throws light on the impact of technology on modern societies. Basic models of time allocation are presented, and, finally, a cross-cultural comparison is made of the mobilization of time resources in preindustrial societies. Geographers, social anthropologists and human ecologists should find this work directly relevant to their interest in understanding the interactions between man and environment.

The purpose of Numerical Linear Algebra in Signals, Systems and Control is to present an interdisciplinary book, blending linear and numerical linear algebra with three major areas of electrical engineering: Signal and Image Processing, and Control Systems and Circuit Theory. Numerical Linear Algebra in Signals, Systems and Control will contain articles, both the state-of-the-art surveys and technical papers, on theory, computations, and applications addressing significant new developments in these areas. The goal of the volume is to provide authoritative and accessible accounts of the fast-paced developments in computational mathematics, scientific computing, and computational engineering methods, applications, and algorithms. The state-of-the-art surveys will benefit, in particular, beginning researchers, graduate students, and those contemplating to start a new direction of research in these areas. A more general goal is to foster effective communications and exchange of information between various scientific and engineering communities with mutual interests in concepts, computations, and workable, reliable practices.

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.

Differential geometry is the study of the curvature and calculus of curves and surfaces. "A New Approach to Differential Geometry using Clifford's Geometric Algebra" simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space.

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Complete with chapter-by-chapter exercises, an overview of general relativity, and brief biographies of historical figures, this comprehensive textbook presents a valuable introduction to differential geometry. It will serve as a useful resource for upper-level undergraduates, beginning-level graduate students, and researchers in the algebra and physics communities.

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LINEAR ALGEBRA EXPLORE A COMPREHENSIVE INTRODUCTORY TEXT IN LINEAR ALGEBRA WITH COMPELLING SUPPLEMENTARY MATERIALS, INCLUDING A COMPANION WEBSITE AND SOLUTIONS MANUALS Linear Algebra delivers a fulsome exploration of the central concepts in linear algebra, including multidimensional spaces, linear transformations, matrices, matrix algebra, determinants, vector spaces, subspaces, linear independence, basis, inner products, and eigenvectors. While the text provides challenging problems that engage readers in the mathematical theory of linear algebra, it is written in an accessible and simple-to-grasp fashion appropriate for junior undergraduate students. An emphasis on logic, set theory, and functions exists throughout the book, and these topics are introduced early to provide students with a foundation from which to attack the rest of the material in the text. Linear Algebra includes accompanying material in the form of a companion website that features solutions manuals for students and instructors. Finally, the concluding chapter in the book includes discussions of advanced topics like generalized eigenvectors, Schur's Lemma, Jordan canonical form, and quadratic forms. Readers will also benefit from the inclusion of: A thorough introduction to logic and set theory, as well as descriptions of functions and linear transformations An exploration of Euclidean spaces and linear transformations between Euclidean spaces, including vectors, vector algebra, orthogonality, the standard matrix, Gauss-Jordan elimination, inverses, and determinants Discussions of abstract vector spaces, including subspaces, linear independence, dimension, and change of basis A treatment on defining geometries on vector spaces, including the Gram-Schmidt process Perfect for undergraduate students taking their first course in the subject matter, Linear Algebra will also earn a place in the libraries of researchers in computer science or statistics seeking an accessible and practical foundation in linear algebra.

Most students in abstract algebra classes have great difficulty making sense of what the instructor is saying. Moreover, this seems to remain true almost independently of the quality of the lecture. This book is based on the constructivist belief that, before students can make sense of any presentation of abstract mathematics, they need to be engaged in mental activities which will establish an experiential base for any future verbal explanation. No less, they need to have the opportunity to reflect on their activities. This approach is based on extensive theoretical and empirical studies as well as on the substantial experience of the authors in teaching astract algebra. The main source of activities in this course is computer constructions, specifically, small programs written in the mathlike programming language ISETL; the main tool for reflections is work in teams of 2-4 students, where the activities are discussed and debated. Because of the similarity of ISETL expressions to standard written mathematics, there is very little programming overhead: learning to program is inseparable from learning the mathematics. Each topic is first introduced through computer activities, which are then followed by a text section and exercises. This text section is written in an informed, discusive style, closely relating definitions and proofs to the constructions in the activities. Notions such as cosets and quotient groups become much more meaningful to the students than when they are preseted in a lecture.

The Center and Focus Problem: Algebraic Solutions and Hypotheses, M. N. Popa and V.V. Pricop, ISBN: 978-1-032-01725-9 (Hardback) This book focuses on an old problem of the qualitative theory of differential equations, called the Center and Focus Problem. It is intended for mathematicians, researchers, professors and Ph.D. students working in the field of differential equations, as well as other specialists who are interested in the theory of Lie algebras, commutative graded algebras, the theory of generating functions and Hilbert series. The book reflects the results obtained by the authors in the last decades. A rather essential result is obtained in solving Poincare's problem. Namely, there are given the upper estimations of the number of Poincare-Lyapunov quantities, which are algebraically independent and participate in solving the Center and Focus Problem that have not been known so far. These estimations are equal to Krull dimensions of Sibirsky graded algebras of comitants and invariants of systems of differential equations.

This is the first in a series of three volumes dealing with important topics in algebra. It offers an introduction to the foundations of mathematics together with the fundamental algebraic structures, namely groups, rings, fields, and arithmetic. Intended as a text for undergraduate and graduate students of mathematics, it discusses all major topics in algebra with numerous motivating illustrations and exercises to enable readers to acquire a good understanding of the basic algebraic structures, which they can then use to find the exact or the most realistic solutions to their problems.

This book presents the various algebraic techniques for solving partial differential equations to yield exact solutions, techniques developed by the author in recent years and with emphasis on physical equations such as: the Maxwell equations, the Dirac equations, the KdV equation, the KP equation, the nonlinear Schrodinger equation, the Davey and Stewartson equations, the Boussinesq equations in geophysics, the Navier-Stokes equations and the boundary layer problems. In order to solve them, I have employed the grading technique, matrix differential operators, stable-range of nonlinear terms, moving frames, asymmetric assumptions, symmetry transformations, linearization techniques and special functions. The book is self-contained and requires only a minimal understanding of calculus and linear algebra, making it accessible to a broad audience in the fields of mathematics, the sciences and engineering. Readers may find the exact solutions and mathematical skills needed in their own research.

* Solid research basis, drawing on findings from a 4-year research project with in-depth interviews with judges, attorneys, and seasoned forensic neuropsychologists and psychologists as well as further interviews with professionals in other fields such as engineering, physics and economics. * Provides focused attention on how experts interact with judges, attorneys, and juries * Challenges experts to avoid the traps of professional jargon and traditional manners of presenting information/knowledge/opinions. * Provides a step-by-step approach to orienting the new academic to expert witnessing

* Solid research basis, drawing on findings from a 4-year research project with in-depth interviews with judges, attorneys, and seasoned forensic neuropsychologists and psychologists as well as further interviews with professionals in other fields such as engineering, physics and economics. * Provides focused attention on how experts interact with judges, attorneys, and juries * Challenges experts to avoid the traps of professional jargon and traditional manners of presenting information/knowledge/opinions. * Provides a step-by-step approach to orienting the new academic to expert witnessing

Linear Algebra: Algorithms, Applications, and Techniques, Fourth Edition offers a modern and algorithmic approach to computation while providing clear and straightforward theoretical background information. The book guides readers through the major applications, with chapters on properties of real numbers, proof techniques, matrices, vector spaces, linear transformations, eigen values, and Euclidean inner products. Appendices on Jordan canonical forms and Markov chains are included for further study. This useful textbook presents broad and balanced views of theory, with key material highlighted and summarized in each chapter. To further support student practice, the book also includes ample exercises with answers and hints.

In this new examination of Babylonian cuneiform texts, Jens Hoyrup proposes a new interpretation, based on the fact that the tablets are almost entirely students¿ workbooks. The knowledge of mathematics expressed in these tablets is entirely ¿practical,¿ for use in surveying, accounting, and building, rather than theoretical. Hoyrup argues that the notion of algebraic manipulation, like other parts of a theoretical mathematics is indeed a later invention.

This unique text is an introduction to harmonic analysis on the simplest symmetric spaces, namely Euclidean space, the sphere, and the Poincare upper half plane. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. Written with an informal style, the book places an emphasis on motivation, concrete examples, history, and, above all, applications in mathematics, statistics, physics, and engineering.

Many corrections and updates have been incorporated in this new edition. Updates include discussions of P. Sarnak and others' work on quantum chaos, the work of T. Sunada, Marie-France Vigneras, Carolyn Gordon, and others on Mark Kac's question "Can you hear the shape of a drum?," A. Lubotzky, R. Phillips and P. Sarnak's examples of Ramanujan graphs, and, finally, the author's comparisons of continuous theory with the finite analogues.

Topics featured throughout the text include inversion formulas for Fourier transforms, central limit theorems, Poisson's summation formula and applications in crystallography and number theory, applications of spherical harmonic analysis to the hydrogen atom, the Radon transform, non-Euclidean geometry on the Poincare upper half plane H or unit disc and applications to microwave engineering, fundamental domains in H for discrete groups, tessellations of H from such discrete group actions, automorphic forms, and the Selberg trace formula and its applications in spectral theory as well as number theory."

The present review volume not only covers a wide range of topics pertinent to nuclear science and technology, but has attracted a distinguished international authorship, for which the editors are grateful. The opening review by Drs. Janet Tawn and Richard Wakeford addresses the difficult matter of questioning sci- tific hypotheses in a court of law. The United Kingdom experienced a substantial nuclear accident in the 1950s in the form of the Windscale Pile fire. This in itself had both good and bad consequences; the setting up of a licensing authority to ensure nuclear safety was one, the understandable public sentiment concerning nuclear power (despite the fire occurring in a weapons pile) the other. Windscale today is subsumed in the reprocessing plant at Sellafield operated by British Nuclear Fuels plc and it was inevitable perhaps that when an excess cluster of childhood leukaemia was observed in the nearby village of Seascale that public concern should be promoted by the media, leading to the hearing of a claim of compensation brought on behalf of two of the families of BNFLs workers who had suffered that loss. The review article demonstrates the complexity of und- standing such a claim against the statistical fluctuations inherent and shows how the courts were persuaded of the need to propose a biological mechanism if responsibility were to be held. The Company were undoubtedly relieved by the finding.

Algebra & Geometry: An Introduction to University Mathematics, Second Edition provides a bridge between high school and undergraduate mathematics courses on algebra and geometry. The author shows students how mathematics is more than a collection of methods by presenting important ideas and their historical origins throughout the text. He incorporates a hands-on approach to proofs and connects algebra and geometry to various applications. The text focuses on linear equations, polynomial equations, and quadratic forms. The first few chapters cover foundational topics, including the importance of proofs and a discussion of the properties commonly encountered when studying algebra. The remaining chapters form the mathematical core of the book. These chapters explain the solutions of different kinds of algebraic equations, the nature of the solutions, and the interplay between geometry and algebra. New to the second edition Several updated chapters, plus an all-new chapter discussing the construction of the real numbers by means of approximations by rational numbers Includes fifteen short 'essays' that are accessible to undergraduate readers, but which direct interested students to more advanced developments of the material Expanded references Contains chapter exercises with solutions provided online at www.routledge.com/9780367563035