This book has been written in a frankly partisian spirit-we believe that singularity theory offers an extremely useful approach to bifurcation prob lems and we hope to convert the reader to this view. In this preface we will discuss what we feel are the strengths of the singularity theory approach. This discussion then Ieads naturally into a discussion of the contents of the book and the prerequisites for reading it. Let us emphasize that our principal contribution in this area has been to apply pre-existing techniques from singularity theory, especially unfolding theory and classification theory, to bifurcation problems. Many ofthe ideas in this part of singularity theory were originally proposed by Rene Thom; the subject was then developed rigorously by John Matherand extended by V. I. Arnold. In applying this material to bifurcation problems, we were greatly encouraged by how weil the mathematical ideas of singularity theory meshed with the questions addressed by bifurcation theory. Concerning our title, Singularities and Groups in Bifurcation Theory, it should be mentioned that the present text is the first volume in a two-volume sequence. In this volume our emphasis is on singularity theory, with group theory playing a subordinate role. In Volume II the emphasis will be more balanced. Having made these remarks, Iet us set the context for the discussion of the strengths of the singularity theory approach to bifurcation. As we use the term, bifurcation theory is the study of equations with multiple solutions."

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).

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.

Building on the author's previous edition on the subject (Introduction to Linear Algebra, Jones & Bartlett, 1996), this book offers a refreshingly concise text suitable for a standard course in linear algebra, presenting a carefully selected array of essential topics that can be thoroughly covered in a single semester. Although the exposition generally falls in line with the material recommended by the Linear Algebra Curriculum Study Group, it notably deviates in providing an early emphasis on the geometric foundations of linear algebra. This gives students a more intuitive understanding of the subject and enables an easier grasp of more abstract concepts covered later in the course. The focus throughout is rooted in the mathematical fundamentals, but the text also investigates a number of interesting applications, including a section on computer graphics, a chapter on numerical methods, and many exercises and examples using MATLAB. Meanwhile, many visuals and problems (a complete solutions manual is available to instructors) are included to enhance and reinforce understanding throughout the book. Brief yet precise and rigorous, this work is an ideal choice for a one-semester course in linear algebra targeted primarily at math or physics majors. It is a valuable tool for any professor who teaches the subject.

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.

'Et moi, .... si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point alit.' human race. It has put common sense back Jules Verne where it belongs. on the topmost shelf next to the dusty canister labelled 'discarded non The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bcll o. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nOD linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series."

The purpose of the book is to take stock of the situation concerning Algebra via Category Theory in the last fifteen years, where the new and synthetic notions of Mal'cev, protomodular, homological and semi-abelian categories emerged. These notions force attention on the fibration of points and allow a unified treatment of the main algebraic: homological lemmas, Noether isomorphisms, commutator theory.
The book gives full importance to examples and makes strong connections with Universal Algebra. One of its aims is to allow appreciating how productive the essential categorical constraint is: knowing an object, not from inside via its elements, but from outside via its relations with its environment.
The book is intended to be a powerful tool in the hands of researchers in category theory, homology theory and universal algebra, as well as a textbook for graduate courses on these topics.

This book gives a comprehensive account of Mori¿s Program, that is an approach to the following problem: classify all the projective varieties X in P^n over C up to isomorphism. Mori¿s Program is a fusion of the so-called Minimal Model Program and the Iitaka Program toward the biregular and/or birational classification of higher dimensional algebraic varieties. The author presents this theory in an easy and understandable way with lots of background motivation. It is the first book in this extremely important and active area of research and will become a key resource for graduate students.

Mathematical summary for Digital Signal Processing Applications with Matlab consists of Mathematics which is not usually dealt in the DSP core subject, but used in DSP applications. Matlab programs with illustrations are given for the selective topics such as generation of Multivariate Gaussian distributed sample outcomes, Bacterial foraging algorithm, Newton's iteration, Steepest descent algorithm, etc. are given exclusively in the separate chapter. Also Mathematical summary for Digital Signal Processing Applications with Matlab is written in such a way that it is suitable for Non-Mathematical readers and is very much suitable for the beginners who are doing research in Digital Signal Processing.

This text is an introduction to the representation theory of the symmetric group from three different points of view: via general representation theory, via combinatorial algorithms, and via symmetric functions. It is the only book to deal with all three aspects of this subject at once. The style of presentation is relaxed yet rigorous and the prerequisites have been kept to a minimum¿undergraduate courses in linear algebra and group theory will suffice.

These books grew out of the perception that a number of important conceptual and theoretical advances in research on small group behavior had developed in recent years, but were scattered in rather fragmentary fashion across a diverse literature. Thus, it seemed useful to encourage the formulation of summary accounts. A conference was held in Hamburg with the aim of not only encouraging such developments, but also encouraging the integration of theoretical approaches where possible. These two volumes are the result. Current research on small groups falls roughly into two moderately broad categories, and this classification is reflected in the two books. Volume I addresses theoretical problems associated with the consensual action of task-oriented small groups, whereas Volume II focuses on interpersonal relations and social processes within such groups. The two volumes differ somewhat in that the conceptual work of Volume I tends to address rather strictly defined problems of consensual action, some approaches tending to the axiomatic, whereas the conceptual work described in Volume II is generally less formal and rather general in focus. However, both volumes represent current conceptual work in small group research and can claim to have achieved the original purpose of up-to-date conceptual summaries of progress on new theoretical work.

Introduction to Large Truncated Toeplitz Matrices is a text on the application of functional analysis and operator theory to some concrete asymptotic problems of linear algebra. The book contains results on the stability of projection methods, deals with asymptotic inverses and Moore-Penrose inversion of large Toeplitz matrices, and embarks on the asymptotic behavoir of the norms of inverses, the pseudospectra, the singular values, and the eigenvalues of large Toeplitz matrices. The approach is heavily based on Banach algebra techniques and nicely demonstrates the usefulness of C*-algebras and local principles in numerical analysis. The book includes classical topics as well as results obtained and methods developed only in the last few years. Though employing modern tools, the exposition is elementary and aims at pointing out the mathematical background behind some interesting phenomena one encounters when working with large Toeplitz matrices. The text is accessible to readers with basic knowledge in functional analysis. It is addressed to graduate students, teachers, and researchers with some inclination to concrete operator theory and should be of interest to everyone who has to deal with infinite matrices (Toeplitz or not) and their large truncations.

On the occasion of the 150th anniversary of Sophus Lie, an International Work shop "Modern Group Analysis: advanced analytical and computational methods in mathematical physics" has been organized in Acireale (Catania, Sicily, October 27 31, 1992). The Workshop was aimed to enlighten the present state ofthis rapidly expanding branch of applied mathematics. Main topics of the Conference were: * classical Lie groups applied for constructing invariant solutions and conservation laws; * conditional (partial) symmetries; * Backlund transformations; * approximate symmetries; * group analysis of finite-difference equations; * problems of group classification; * software packages in group analysis. The success of the Workshop was due to the participation of many experts in Group Analysis from different countries. This book consists of selected papers presented at the Workshop. We would like to thank the Scientific Committee for the generous support of recommending invited lectures and selecting the papers for this volume, as well as the members of the Organizing Committee for their help. The Workshop was made possible by the financial support of several sponsors that are listed below. It is also a pleasure to thank our colleague Enrico Gregorio for his invaluable help of this volume.

The theory of finite fields is of central importance in engineering and computer science, because of its applications to error-correcting codes, cryptography, spread-spectrum communications, and digital signal processing. Though not inherently difficult, this subject is almost never taught in depth in mathematics courses, (and even when it is the emphasis is rarely on the practical aspect). Indeed, most students get a brief and superficial survey which is crammed into a course on error-correcting codes. It is the object of this text to remedy this situation by presenting a thorough introduction to the subject which is completely sound mathematically, yet emphasizes those aspects of the subject which have proved to be the most important for applications. This book is unique in several respects. Throughout, the emphasis is on fields of characteristic 2, the fields on which almost all applications are based. The importance of Euclid's algorithm is stressed early and often. Berlekamp's polynomial factoring algorithm is given a complete explanation. The book contains the first treatment of Berlekamp's 1982 bit-serial multiplication circuits, and concludes with a thorough discussion of the theory of m-sequences, which are widely used in communications systems of many kinds.

This book provides a comprehensive treatment of Gr bner bases theory embedded in an introduction to commutative algebra from a computational point of view. The centerpiece of Gr bner bases theory is the Buchberger algorithm, which provides a common generalization of the Euclidean algorithm and the Gaussian elimination algorithm to multivariate polynomial rings. The book explains how the Buchberger algorithm and the theory surrounding it are eminently important both for the mathematical theory and for computational applications. A number of results such as optimized version of the Buchberger algorithm are presented in textbook format for the first time. This book requires no prerequisites other than the mathematical maturity of an advanced undergraduate and is therefore well suited for use as a textbook. At the same time, the comprehensive treatment makes it a valuable source of reference on Gr bner bases theory for mathematicians, computer scientists, and others. Placing a strong emphasis on algorithms and their verification, while making no sacrifices in mathematical rigor, the book spans a bridge between mathematics and computer science.

This volume is an outgrowth of the 1995 Summer School on Theoretical Physics of the Canadian Association of Physicists (CAP), held in Banff, Alberta, in the Canadian Rockies, from July 30 to August 12,1995. The chapters, based on lectures given at the School, are designed to be tutorial in nature, and many include exercises to assist the learning process. Most lecturers gave three or four fifty-minute lectures aimed at relative novices in the field. More emphasis is therefore placed on pedagogy and establishing comprehension than on erudition and superior scholarship. Of course, new and exciting results are presented in applications of Clifford algebras, but in a coherent and user-friendly way to the nonspecialist. The subject area of the volume is Clifford algebra and its applications. Through the geometric language of the Clifford-algebra approach, many concepts in physics are clarified, united, and extended in new and sometimes surprising directions. In particular, the approach eliminates the formal gaps that traditionally separate clas sical, quantum, and relativistic physics. It thereby makes the study of physics more efficient and the research more penetrating, and it suggests resolutions to a major physics problem of the twentieth century, namely how to unite quantum theory and gravity. The term "geometric algebra" was used by Clifford himself, and David Hestenes has suggested its use in order to emphasize its wide applicability, and b& cause the developments by Clifford were themselves based heavily on previous work by Grassmann, Hamilton, Rodrigues, Gauss, and others."

The theory of automorphisms and derivations of associative rings is a direct descendant of the development of classical Galois theory and the theory of invariants. This volume presents a comprehensive overview of the methods and results of that theory, which has been greatly enriched during the last twenty years. Some of the material included appears for the first time. Among the problems discussed in this book are the following: construction of a Galois theory for prime and semiprime rings and its application to domains and free algebras; investigation of the problems of the algebraic dependence of automorphisms and derivations; studies of the fixed rings for finite groups and rings of constants for differential Lie algebras acting on the rings; non-commutative invariants of linear groups; theorems of finite groups acting on modular lattices; actions of Hopf algebras. The monograph is meant for specialists in algebra, but it can also be useful for a wider range of mathematicians. The inclusions in the book of the latest achievements on the structural theory of rings with generalized identities makes it desirable reading for graduate students as well.

Techniques of physics find wide application in biology, medicine, engineering and technology generally. This series is devoted to techniques which have found and are finding application. The aim is to clarify the principles of each technique, to emphasize and illustrate the applications and to draw attention to new fields of possible employment.

Hans Duistermaat, an influential geometer-analyst, made substantial contributions to the theory of ordinary and partial differential equations, symplectic, differential, and algebraic geometry, minimal surfaces, semisimple Lie groups, mechanics, mathematical physics, and related fields. Written in his honor, the invited and refereed articles in this volume contain important new results as well as surveys in some of these areas, clearly demonstrating the impact of Duistermaat's research and, in addition, exhibiting interrelationships among many of the topics.

A smart city utilizes ICT technologies to improve the working effectiveness, share various data with the citizens, and enhance political assistance and societal wellbeing. The fundamental needs of a smart and sustainable city are utilizing smart technology for enhancing municipal activities, expanding monetary development, and improving citizens' standards of living. Data-Driven Mathematical Modeling in Smart Cities discusses new mathematical models in smart and sustainable cities using big data, visualization tools in mathematical modeling, machine learning-based mathematical modeling, and more. It further delves into privacy and ethics in data analysis. Covering topics such as deep learning, optimization-based data science, and smart city automation, this premier reference source is an excellent resource for mathematicians, statisticians, computer scientists, civil engineers, government officials, students and educators of higher education, librarians, researchers, and academicians.

The first interactive course covering first and second year algebra. Starting from such fundamental topics as integers and divisions, modular arithmetic and polynomials the content extends to rings, fields and permutation groups. The hypertext is written in Java-enhanced HTML, and Java applets illustrate the theory while also contributing interactive calculators for computing with integers, polynomials and permutations. The computer algebra system GAP is integrated throughout, allowing the calculation and manipulation of mathematical objects. In addition, collections for Mathematica notebooks and Maple worksheets review the algorithms presented. Multiple choice exercises provide users with instant feedback, while facilities for monitoring students and a bulletin board complete this digital course.

http://www.worldscientific.com/worldscibooks/10.1142/0131

http://www.worldscientific.com/worldscibooks/10.1142/0131

A collection of matrices is said to be triangularizable if there is an invertible matrix S such that S1 AS is upper triangular for every A in the collection. This generalization of commutativity is the subject of many classical theorems due to Engel, Kolchin, Kaplansky, McCoy and others. The concept has been extended to collections of bounded linear operators on Banach spaces: such a collection is defined to be triangularizable if there is a maximal chain of subspaces of the Banach space, each of which is invariant under every member of the collection. Most of the classical results have been generalized to compact operators, and there are also recent theorems in the finite-dimensional case. This book is the first comprehensive treatment of triangularizability in both the finite and infinite-dimensional cases. It contains numerous very recent results and new proofs of many of the classical theorems. It provides a thorough background for research in both the linear-algebraic and operator-theoretic aspects of triangularizability and related areas. More generally, the book will be useful to anyone interested in matrices or operators, as many of the results are linked to other topics such as spectral mapping theorems, properties of spectral radii and traces, and the structure of semigroups and algebras of operators. It is essentially self-contained modulo solid courses in linear algebra (for the first half) and functional analysis (for the second half), and is therefore suitable as a text or reference for a graduate course.