Larry Smith / West Virginia University
Dagmar Meyer / Uniwersytet Wroclawski, Poland

Poincare Duality Algebras, Macaulay's Dual Systems, and Steenrod Operations

Series: Cambridge Tracts in Mathematics (No. 167)
Hardback ISBN 0521850649
Not yet published - available from May 2005 (Stock level updated: 17:58 GMT, 18 January 2005)

Poincare duality algebras originated in the work of topologists on the cohomology of closed manifolds, and Macaulay's dual systems in the study of irreducible ideals in polynomial algebras. These two ideas are tied together using basic commutative algebra involving Gorenstein algebras. Steenrod operations also originated in algebraic topology, but may best be viewed as a means of encoding the information often hidden behind the Frobenius map in characteristic p<>0. They provide a noncommutative tool to study commutative algebras over a Galois field. In this Tract the authors skilfully bring together these ideas and apply them to problems in invariant theory. A number of remarkable and unexpected interdisciplinary connections are revealed that will interest researchers in the areas of commutative algebra, invariant theory or algebraic topology.

Contents

Introduction; Part I. Poincare Duality Quotients; Part II. Macaulay’s Dual Systems and Frobenius Powers; Part III. Poincare Duality and the Steenrod Algebra; Part IV. Dickson, Symmetric, and other Coinvariants; Part V. The Hit Problem mod 2; Part VI. Macaulay’s Inverse Systems and Applications; References; Notation; Index.

Ke Chen
University of Liverpool

Matrix Preconditioning Techniques and Applications

Series: Cambridge Monographs on Applied and Computational Mathematics

Hardback ISBN 0521838282
Not yet published - available from July 2005

Preconditioning techniques have emerged as an essential part of successful and efficient iterative solutions of matrices. Ke Chen’s book offers a comprehensive introduction to these methods. A vast range of explicit and implicit sparse preconditioners are covered, including the conjugate gradient, multi-level and fast multi-pole methods, matrix and operator splitting, fast Fourier and wavelet transforms, incomplete LU and domain decomposition, Schur complements and approximate inverses. In addition, aspects of parallel realization using the MPI are discussed. Very much a users-guide, the book provides insight to the use of these techniques in areas such as acoustic wave scattering, image restoration and bifurcation problems in electrical power stations. Supporting MATLAB files are available from the Web to support and develop readers’ understanding, and provide stimulus for further study. Pitched at graduate level, the book is intended to serve as a useful guide and reference for students, computational practitioners, engineers and researchers alike.

Contents

1. Introduction; 2. Direct methods; 3. Iterative methods; 4. Matrix splitting preconditioners [t1]; 5. Approxi,ate inverse preconditioners [t2]; 6. Multilevel methods and preconditioners [t3]; 7. Multilevel recursive Schur complements preconditioners; 8. Wavelet preconditioners [t5] for ?A n x n and ?A -1 n x n; 9. Wavelet Schur preconditioners [t6]; 10. Implicit wavelet preconditioners [t7]; 11. Application I - acoustic scattering modelling; 12. Application II - coupled matrix problems; 13. Application III - image restoration and inverse problems; 14. Application IV-voltage stability in electrical power systems; 15. Parallel computing by examples.

Kyriakos Tamvakis
University of Ioannina, Greece

Problems and Solutions in Quantum Mechanics

Hardback ISBN 0521840872
PaperbackISBN 052160057X

Not yet published - available from September 2005

Textbook
Lecturers can request inspection copies of this title.
Courses: Quantum Mechanics Modern Physics Theoretical Physics
This collection of solved problems corresponds to the standard topics covered in established undergraduate and graduate courses in Quantum Mechanics. Completely up-to-date, problems are also included on topics of current interest which are absent in the existing literature. Solutions are presented in considerable detail, to enable students to follow each step. The emphasis is on stressing the principles and methods used, allowing students to master new ways of thinking and problem-solving techniques. The problems themselves are longer than those usually encountered in textbooks and consist of a number of questions based around a central theme, highlighting properties and concepts of interest. For undergraduate and graduate students, as well as those involved in teaching Quantum Mechanics, the book can be used as a supplementary text or as an independent self-study tool.

Contents

1. Wave functions; 2. The free particle; 3. Simple potentials; 4. The harmonic oscillator; 5. Angular momentum; 6. Quantum behaviour; 7. General motion; 8. Many particle systems; 9. Approximation methods; 10. Scattering.

Paul Cohn
University College London

Free Ideal Rings and Localization in General Rings

Series: Encyclopedia of Mathematics and its Applications
Hardback ISBN 0521853370

Not yet published - available from October 2005

Proving that a polynomial ring in one variable over a field is a principal ideal domain can be done by means of the Euclidean algorithm, but this does not extend to more variables. However, if the variables are not allowed to commute, giving a free associative algebra, then there is a generalization, the weak algorithm, which can be used to prove that all one-sided ideals are free. This book presents the theory of free ideal rings (firs) in detail. Particular emphasis is placed on rings with a weak algorithm, exemplified by free associative algebras. There is also a full account of localization which is treated for general rings but the features arising in firs are given special attention. Each section has a number of exercises, including some open problems, and each chapter ends in a historical note.

Contents

Preface; Note to the reader; Terminology, notations and conventions used; List of special notation; 0. Preliminaries in modules; 1. Principal ideal domains; 2. Firs, semifirs and the weak algorithm; 3. Factorization; 4. 2-firs with a distributive factor lattice; 5. Modules over firs and semifirs; 6. Centralizers and subalgebras; 7. Skew fields of fractions; Appendix; Bibliography and author index; Subject index.