This book introduces the concepts of linear algebra through the careful study of two and three-dimensional Euclidean geometry. This approach makes it possible to start with vectors, linear transformations, and matrices in the context of familiar plane geometry and to move directly to topics such as dot products, determinants, eigenvalues, and quadratic forms. The later chapters deal with n-dimensional Euclidean space and other finite-dimensional vector space.

This book summarizes the application of linear algebra-based controllers (LABC) for trajectory tracking for practitioners and students across a range of engineering disciplines. It clarifies the necessary steps to apply this straight-forward technique to a non-linear multivariable system, dealing with continuous or discrete time models, and outlines the steps to implement such controllers. In this book, the authors present an approach of the trajectory tracking problem in systems with dead time and in the presence of additive uncertainties and environmental disturbances. Examples of applications of LABC to systems in real operating conditions (mobile robots, marine vessels, quadrotor and pvtol aircraft, chemical reactors and First Order Plus Dead Time systems) illustrate the controller design in such a way that the reader attains an understanding of LABC.

This book is aimed at graduate students in physics who are studying group theory and its application to physics. It contains a short explanation of the fundamental knowledge and method, and the fundamental exercises for the method, as well as some important conclusions in group theory. The book has been designed as a supplement to the author's textbook Group Theory for Physicists, also published by World Scientific. Together these two books can be used in a course on group theory for first-year graduate students in physics, especially theoretical physics. They are also suitable for some graduate students in theoretical chemistry.

This book and the following second volume is an introduction into modern algebraic geometry. In the first volume the methods of homological algebra, theory of sheaves, and sheaf cohomology are developed. These methods are indispensable for modern algebraic geometry, but they are also fundamental for other branches of mathematics and of great interest in their own.
In the last chapter of volume I these concepts are applied to the theory of compact Riemann surfaces. In this chapter the author makes clear how influential the ideas of Abel, Riemann and Jacobi were and that many of the modern methods have been anticipated by them.
For this second edition the text was completely revised and corrected. The author also added a short section on moduli of elliptic curves with N-level structures. This new paragraph anticipates some of the techniques of volume II.

The book contains a comprehensive account of the structure and classification of Lie groups and finite-dimensional Lie algebras (including semisimple, solvable, and of general type). In particular, a modern approach to the description of automorphisms and gradings of semisimple Lie algebras is given. A special chapter is devoted to models of the exceptional Lie algebras. The book contains many tables and will serve as a reference. At the same time many results are accompanied by short proofs. Onishchik and Vinberg are internationally known specialists in their field and well-known for their monograph "Lie Groups and Algebraic Groups" (Springer-Verlag 1990). This Encyclopaedia volume will be immensely useful to graduate students in differential geometry, algebra and theoretical physics.

Geometric algebra has established itself as a powerful and valuable mathematical tool for solving problems in computer science, engineering, physics, and mathematics. The articles in this volume, written by experts in various fields, reflect an interdisciplinary approach to the subject, and highlight a range of techniques and applications. Relevant ideas are introduced in a self-contained manner and only a knowledge of linear algebra and calculus is assumed.

Features and Topics:

* The mathematical foundations of geometric algebra are explored

* Applications in computational geometry include models of reflection and ray-tracing and a new and concise characterization of the crystallographic groups

* Applications in engineering include robotics, image geometry, control-pose estimation, inverse kinematics and dynamics, control and visual navigation

* Applications in physics include rigid-body dynamics, elasticity, and electromagnetism

* Chapters dedicated to quantum information theory dealing with multi-particle entanglement, MRI, and relativistic generalizations

Practitioners, professionals, and researchers working in computer science, engineering, physics, and mathematics will find a wide range of useful applications in this state-of-the-art survey and reference book. Additionally, advanced graduate students interested in geometric algebra will find the most current applications and methods discussed.

An Introduction to Homological Algebra discusses the origins of algebraic topology. It also presents the study of homological algebra as a two-stage affair. First, one must learn the language of Ext and Tor and what it describes. Second, one must be able to compute these things, and often, this involves yet another language: spectral sequences. Homological algebra is an accessible subject to those who wish to learn it, and this book is the author's attempt to make it lovable. This book comprises 11 chapters, with an introductory chapter that focuses on line integrals and independence of path, categories and functors, tensor products, and singular homology. Succeeding chapters discuss Hom and ?; projectives, injectives, and flats; specific rings; extensions of groups; homology; Ext; Tor; son of specific rings; the return of cohomology of groups; and spectral sequences, such as bicomplexes, Kunneth Theorems, and Grothendieck Spectral Sequences. This book will be of interest to practitioners in the field of pure and applied mathematics.

For more than 30 years, the author has studied the model-theoretic aspects of the theory of valued fields and multi-valued fields. Many of the key results included in this book were obtained by the author whilst preparing the manuscript. Thus the unique overview of the theory, as developed in the book, has been previously unavailable. The book deals with the theory of valued fields and mutli-valued fields. The theory of PrA1/4fer rings is discussed from the geometric' point of view. The author shows that by introducing the Zariski topology on families of valuation rings, it is possible to distinguish two important subfamilies of PrA1/4fer rings that correspond to Boolean and near Boolean families of valuation rings. Also, algebraic and model-theoretic properties of multi-valued fields with near Boolean families of valuation rings satisfying the local-global principle are studied. It is important that this principle is elementary, i.e., it can be expressed in the language of predicate calculus. The most important results obtained in the book include a criterion for the elementarity of an embedding of a multi-valued field and a criterion for the elementary equivalence for multi-valued fields from the class defined by the additional natural elementary conditions (absolute unramification, maximality and almost continuity of local elementary properties). The book concludes with a brief chapter discussing the bibliographic references available on the material presented, and a short history of the major developments within the field.

This book gives an introduction to C*-algebras and their representations on Hilbert spaces. We have tried to present only what we believe are the most basic ideas, as simply and concretely as we could. So whenever it is convenient (and it usually is), Hilbert spaces become separable and C*-algebras become GCR. This practice probably creates an impression that nothing of value is known about other C*-algebras. Of course that is not true. But insofar as representations are con cerned, we can point to the empirical fact that to this day no one has given a concrete parametric description of even the irreducible representations of any C*-algebra which is not GCR. Indeed, there is metamathematical evidence which strongly suggests that no one ever will (see the discussion at the end of Section 3. 4). Occasionally, when the idea behind the proof of a general theorem is exposed very clearly in a special case, we prove only the special case and relegate generalizations to the exercises. In effect, we have systematically eschewed the Bourbaki tradition. We have also tried to take into account the interests of a variety of readers. For example, the multiplicity theory for normal operators is contained in Sections 2. 1 and 2. 2. (it would be desirable but not necessary to include Section 1. 1 as well), whereas someone interested in Borel structures could read Chapter 3 separately. Chapter I could be used as a bare-bones introduction to C*-algebras. Sections 2."

The conference on Ordered Algebraic Structures held in Curat;ao, from the 26th of June through the 30th of June, 1995, at the Avila Beach Hotel, marked the eighth year of ac tivities by the Caribbean Mathematics Foundation (abbr. CMF), which was the principal sponsor of this conference. CMF was inaugurated in 1988 with a conference on Ordered Algebraic Structures. During the years between these two conferences the field has changed sufficiently, both from my point of view and, I believe, that of my co-organizer, W. Charles Holland, to make one wonder about the label "Ordered Algebraic Structures" itself. We recognized this from the start, and right away this conference carried a subtitle, or, if one prefers, an agenda: we concentrated on the one hand, on traditional themes in the theory of ordered groups, including model-theoretic aspects, and, on the other hand, on matters in which topology (more precisely C(X)-style topology) and category theory would play a prominent role. Plainly, ordered algebra has many faces, and it is becoming increas ingly difficult to organize an intimate conference, such as the ones encouraged in the series sponsored by CMF, in this area on a broad set of themes. These proceedings reflect, accurately we think, the spirit of the conferees, but it is not a faithful record of the papers presented at the conference."

This book aims to provide a comprehensive study of the mathematical theory of the vortex method, from its origins in the 1930s, through the developments of the '70s when the use of computers made advanced research possible, to current work on this subject in China and elsewhere. The five chapters treat vortex methods for the Euler and Navier-Stokes equations; mathematical theory for incompressible flows; convergence of vortex methods for the Euler equations; convergence of viscosity splitting; and convergence of the random vortex method. Audience: This volume will be of interest to researchers and graduate students of applied mathematics, scientists in fluid dynamics, and aviation engineers.

A 30-article volume, introducing an active and attractive part of algebra that has gained much from its position at the crossroads of mathematics over the years. The papers stimulate the reader to consider and actively investigate the topics and problems they contain.

Geometric constructions have been a popular part of mathematics throughout history. The first chapter here is informal and starts from scratch, introducing all the geometric constructions from high school that have been forgotten or were never learned. The second chapter formalises Plato's game, and examines problems from antiquity such as the impossibility of trisecting an arbitrary angle. After that, variations on Plato's theme are explored: using only a ruler, a compass, toothpicks, a ruler and dividers, a marked rule, or a tomahawk, ending in a chapter on geometric constructions by paperfolding. The author writes in a charming style and nicely intersperses history and philosophy within the mathematics, teaching a little geometry and a little algebra along the way. This is as much an algebra book as it is a geometry book, yet since all the algebra and geometry needed is developed within the text, very little mathematical background is required. This text has been class tested for several semesters with a master's level class for secondary teachers.

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.

A NATO Advanced Study Institute entitled "Algebraic K-theory: Connections with Geometry and Topology" was held at the Chateau Lake Louise, Lake Louise, Alberta, Canada from December 7 to December 11 of 1987. This meeting was jointly supported by NATO and the Natural Sciences and Engineering Research Council of Canada, and was sponsored in part by the Canadian Mathematical Society. This book is the volume of proceedings for that meeting. Algebraic K-theory is essentially the study of homotopy invariants arising from rings and their associated matrix groups. More importantly perhaps, the subject has become central to the study of the relationship between Topology, Algebraic Geometry and Number Theory. It draws on all of these fields as a subject in its own right, but it serves as well as an effective translator for the application of concepts from one field in another. The papers in this volume are representative of the current state of the subject. They are, for the most part, research papers which are primarily of interest to researchers in the field and to those aspiring to be such. There is a section on problems in this volume which should be of particular interest to students; it contains a discussion of the problems from Gersten's well-known list of 1973, as well as a short list of new problems.

Elliptic operators arise naturally in several different mathematical settings, notably in the representation theory of Lie groups, the study of evolution equations, and the examination of Riemannian manifolds. This book develops the basic theory of elliptic operators on Lie groups and thereby extends the conventional theory of parabolic evolution equations to a natural non-commutative context. In order to achieve this goal, the author presents a synthesis of ideas from partial differential equations, harmonic analysis, functional analysis, and the theory of Lie groups. He begins by discussing the abstract theory of general operators with complex coefficients before concentrating on the central case of second-order operators with real coefficients. A full discussion of second-order subellilptic operators is also given. Prerequisites are a familiarity with basic semigroup theory, the elementary theory of Lie groups, and a firm grounding in functional analysis as might be gained from the first year of a graduate course.

An overview of the most successful algorithms and techniques for solving large, sparse systems of equations and some algorithms and strategies for solving optimization problems. The most important topics dealt with concern iterative methods, especially Krylov methods, ordering techniques, and some iterative optimization tools. The book is a compendium of theoretical and numerical methods for solving large algebraic systems, special emphasis being placed on convergence and numerical behaviour as affected by rounding errors, accuracy in computing solutions for ill-conditioned matrices, preconditioning effectiveness, ordering procedures, stability factors, hybrid procedures and stopping criteria. Recent advances in numerical matrix calculations are presented, especially methods to accelerate the solution of symmetric and unsymmetric linear systems. Convergence analysis of the multi-grid method using a posteriori error estimation in second order elliptic equations are presented. Some inverse problems are also included. Evolution based software is described, such as genetic algorithms and evolution strategies, relations and class hierarchising to improve the exploration of large search spaces and finding near-global optima. Recent developments in messy genetic algorithms are also described. The tutorial nature of the book makes it suitable for mathematicians, computer scientists, engineers and postgraduates.

The first book on commutative semigroups was Redei's The theory of .finitely generated commutative semigroups, published in Budapest in 1956. Subsequent years have brought much progress. By 1975 the structure of finite commutative semigroups was fairly well understood. Recent results have perfected this understanding and extended it to finitely generated semigroups. Today's coherent and powerful structure theory is the central subject of the present book. 1. Commutative semigroups are more important than is suggested by the stan- dard examples ofsemigroups, which consist ofvarious kinds oftransformations or arise from finite automata, and are usually quite noncommutative. Commutative of factoriza- semigroups provide a natural setting and a useful tool for the study tion in rings. Additive subsemigroups of N and Nn have close ties to algebraic geometry. Commutative rings are constructed from commutative semigroups as semigroup algebras or power series rings. These areas are all subjects of active research and together account for about half of all current papers on commutative semi groups. Commutative results also invite generalization to larger classes of semigroups. Archimedean decompositions, a comparatively small part oftoday's arsenal, have been generalized extensively, as shown for instance in the upcoming books by Nagy [2001] and Ciric [2002].

This book takes a theoretical perspective on the study of school algebra, in which both semiotics and history occur. The Methodological design allows for the interpretation of specific phenomena and the inclusion of evidence not addressed in more general treatments. The book gives priority to "meaning in use" over "formal meaning." These approaches and others of similar nature lead to a focus on competence rather than a user 's activity with mathematical language.

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceara, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

The main purpose of this book is to show how ideas from combinatorial group theory have spread to two other areas of mathematics: the theory of Lie algebras and affine algebraic geometry. Some of these ideas, in turn, came to combinatorial group theory from low-dimensional topology at the beginning of the 20th Century. This book is divided into three fairly independent parts. Part I provides a brief exposition of several classical techniques in combinatorial group theory, namely, methods of Nielsen, Whitehead, and Tietze. Part II contains the main focus of the book. Here the authors show how the aforementioned techniques of combinatorial group theory found their way into affine algebraic geometry, a fascinating area of mathematics that studies polynomials and polynomial mappings. Part III illustrates how ideas from combinatorial group theory contributed to the theory of free algebras. The focus here is on Schreier varieties of algebras (a variety of algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free in the same variety of algebras).

The study of systems of special partial differential operators that arise naturally from the use of Clifford algebra as a calculus tool lies in the heart of Clifford analysis. The focus is on the study of Dirac operators and related ones, together with applications in mathematics, physics and engineering. At the present time, the study of Clifford algebra and Clifford analysis has grown into a major research field. There are two sources of papers in this collection. One is from a satellite conference to the ICM 2002 in Beijing, held August 15-18 at the University of Macau; and the other stems from invited contributions by top-notch experts in the field.

One of the characteristics of modern algebra is the development of new tools and concepts for exploring classes of algebraic systems, whereas the research on individual algebraic systems (e. g. , groups, rings, Lie algebras, etc. ) continues along traditional lines. The early work on classes of alge bras was concerned with showing that one class X of algebraic systems is actually contained in another class F. Modern research into the theory of classes was initiated in the 1930's by Birkhoff's work [1] on general varieties of algebras, and Neumann's work [1] on varieties of groups. A. I. Mal'cev made fundamental contributions to this modern development. ln his re ports [1, 3] of 1963 and 1966 to The Fourth All-Union Mathematics Con ference and to another international mathematics congress, striking the ories of classes of algebraic systems were presented. These were later included in his book [5]. International interest in the theory of formations of finite groups was aroused, and rapidly heated up, during this time, thanks to the work of Gaschiitz [8] in 1963, and the work of Carter and Hawkes [1] in 1967. The major topics considered were saturated formations, Fitting classes, and Schunck classes. A class of groups is called a formation if it is closed with respect to homomorphic images and subdirect products. A formation is called saturated provided that G E F whenever Gjip(G) E F.

Two contributions on closely related subjects: the theory of linear algebraic groups and invariant theory, by well-known experts in the fields. The book will be very useful as a reference and research guide to graduate students and researchers in mathematics and theoretical physics.

This volume proposes and explores a new definition of logarithmic mappings as invertible selectors of multifunctions induced by linear operators with domains and ranges in an algebra over a field of characteristic zero. Several important previously published results are presented. Amongst the applications of logarithmic and antilogarithmic mappings are the solution of linear and nonlinear equations in algebras of square matrices. Some results may also provide numerical algorithms for the approximation of solutions. Audience: Research mathematicians and other scientists of other disciplines whose work involves the solution of equations.