Series: Cambridge Studies in Advanced Mathematics (No. 124)
Hardback (ISBN-13: 9780521768078)
10 b/w illus. 220 exercises
Page extent: 450 pages
Size: 228 x 152 mm
The representation theory of finite groups has seen rapid growth in recent years with the development of efficient algorithms and computer algebra systems. This is the first book to provide an introduction to the ordinary and modular representation theory of finite groups with special emphasis on the computational aspects of the subject. Evolving from courses taught at Aachen University, this well-paced text is ideal for graduate-level study. The authors provide over 200 exercises, both theoretical and computational, and include worked examples using the computer algebra system GAP. These make the abstract theory tangible and engage students in real hands-on work. GAP is freely available from www.gap-system.org and readers can download source code and solutions to selected exercises from the book's web page.
* Gives hands-on experience with representation theory * Uses the computer
algebra systems GAP, which is freely available for download * Source code,
errata and solutions to selected exercises are available online
Preface; Frequently used symbols; 1. Representations and modules; 2. Characters; 3. Groups and subgroups; 4. Modular representations; List of notation; Bibliography; Index.
Series: Encyclopedia of Mathematics and its Applications (No. 43)
Paperback (ISBN-13: 9780521135047)
Page extent: 264 pages
In this treatise, the authors present the general theory of orthogonal polynomials on the complex plane and several of its applications. The assumptions on the measure of orthogonality are general, the only restriction is that it has compact support on the complex plane. In the development of the theory the main emphasis is on asymptotic behaviour and the distribution of zeros. In the following chapters, the author explores the exact upper and lower bounds are given for the orthonormal polynomials and for the location of their zeros; regular n-th root asymptotic behaviour; and applications of the theory, including exact rates for convergence of rational interpolants, best rational approximants and non-diagonal Pade approximants to Markov functions (Cauchy transforms of measures). The results are based on potential theoretic methods, so both the methods and the results can be extended to extremal polynomials in norms other than L2 norms. A sketch of the theory of logarithmic potentials is given in an appendix.
* The essential encyclopedic reference work on general orthogonal polynomials
* Intended for physicists as well as mathematicians
Introduction; 1. Upper and lower bounds; 2. Zero distribution of orthogonal polynomials; 3. Regular n-th root asymptotic behaviour of orthogonal polynomials; 4. Regularity criteria; 5. Localization; 6. Applications; Appendix; Notes and bibliographical references; Bibliography; List of symbols; Index.
Series: Encyclopedia of Mathematics and its Applications (No. 47)
Paperback (ISBN-13: 9780521135085)
Page extent: 442 pages
Size: 234 x 156 mm
The notion of ‘stopping times’ is a useful one in probability theory; it can be applied to both classical problems and fresh ones. This book presents this technique in the context of the directed set, stochastic processes indexed by directed sets, and many applications in probability, analysis and ergodic theory. Martingales and related processes are considered from several points of view. The book opens with a discussion of pointwise and stochastic convergence of processes, with concise proofs arising from the method of stochastic convergence. Later, the rewording of Vitali covering conditions in terms of stopping times clarifies connections with the theory of stochastic processes. Solutions are presented here for nearly all the open problems in the Krickeberg convergence theory for martingales and submartingales indexed by directed set. Another theme of the book is the unification of martingale and ergodic theorems.
* A unified treatment of multiparameter martingale and ergodic theory *
Martingale and ergodic theories are HOT topics * Applies the theory to
classical mathematical problems as well as fresh ones * Encyclopedic coverage
Introduction; 1. Stopping times; 2. Infinite measure and Orlicz spaces; 3. Inequalities; 4. Directed index set; 5. Banach-valued random variables; 6. Martingales; 7. Derivation; 8. Pointwise ergodic theorems; 9. Multiparameter processes; References; Index.
Series: Encyclopedia of Mathematics and its Applications (No. 48)
Paperback (ISBN-13: 9780521135078)
Page extent: 620 pages
Size: 234 x 156 mm
Research in computational group theory, an active subfield of computational algebra, has emphasised three areas: finite permutation groups, finite solvable groups, and finitely presented groups. This book deals with the third of these areas. The author emphasises the connections with fundamental algorithms from theoretical computer science, particularly the theory of automata and formal languages, computational number theory, and computational commutative algebra. The LLL lattice reduction algorithm and various algorithms for Hermite and Smith normal forms from computational number theory are used to study the abelian quotients of a finitely presented group. The work of Baumslag, Cannonito and Miller on computing nonabelian polycyclic quotients is described as a generalisation of Buchberger’s Grobner basis methods to right ideals in the integral group ring of a polycyclic group. Researchers in computational group theory, mathematicians interested in finitely presented groups and theoretical computer scientists will find this book useful.
* Comprehensive text presenting fundamental algorithmic ideas which have
been developed to compute with finitely presented groups * Emphasises connection
with fundamental algorithms from theoretical computer science * Comprehensive,
yet accessible to graduate students
1. Basic concepts; 2. Rewriting systems; 3. Automata and rational languages; 4. Subgroups of free products of cyclic groups; 5. Coset enumeration; 6. The Reidemeister-Schreier procedure; 7. Generalized automata; 8. Abelian groups; 9. Polycyclic groups; 10. Module bases; 11. Quotient groups
Series: Encyclopedia of Mathematics and its Applications (No. 59)
Paperback (ISBN-13: 9780521135092)
Page extent: 762 pages
Size: 234 x 156 mm
The first edition of this book was reviewed in 1982 as ‘the most extensive treatment of Pade approximants actually available’. This second edition has been thoroughly updated, with a substantial chapter on multiseries approximants. Applications to statistical mechanics and critical phenomena are extensively covered, and there are extended sections devoted to circuit design, matrix Pade approximation, and computational methods. This succinct and straightforward treatment will appeal to scientists, engineers, and mathematicians alike.
* Most comprehensive treatment available * Inclusion of numerical methods
* Applications to quantum mechanics and field theory * Fresh chapter on
multiseries approximants
1. Introduction and definitions; 2. Elementary developments; 3. Pade approximants
and numerical methods; 4. Connection with continued fractions; 5. Stieltjes
series and Polya series; 6. Convergence theory; 7. Extensions of Pade approximants;
8. Multiseries approximants; 9. Connection with integral equations and
quantum mechanics; 10. Connection with numerical analysis; 11. Connection
with quantum field theory; Bibliography; Appendix: a FORTRAN program.