Edited by Katrin Tent
Tits Buildings and the Model Theory of Groups
London Mathematical Society Lecture Note Series, vol.291.
Description
This volume contains selected papers by leading researchers from the international conference entitled Tits Buildings and the Model Theory of Groups, held in WEzburg in 2000. The first part of the book provides a general introduction to many aspects of buildings and their geometries, based on short lecture courses given at the conference. The rest of the book comprises survey and research articles on model theoretic results and techniques, showing the vitality and richness of these branches of mathematics. Among the most fruitful techniques, amalgamation constructions Ela Hrushovski are explained and classified as they continue to play an important role both in model theory and geometry. The articles succeed in demonstrating the close connection between geometry, group theory and model theory. The book will be invaluable to graduate students as well as experienced researchers working in these areas of mathematics.
Chapter Contents
Preface; 1. Basics on buildings Theo Grundhφfer; 2. An Introduction to generalized polygons Hendrik Van Maldeghem; 3. Buildings and classical groups Linos Kramer; 4. Twin buildings Bernhard MElherr; 5. Twin trees and twin buildings Mark Ronan; 6. Simple groups of finite Morley rank of even type Tuna Altinel; 7. BN-pairs and groups of finite Morley rank Katrin Tent; 8. CM-trivial stable groups Andreas Baudisch; 9. Amalgames de Hruchovski Bruno Poizat; 10. Rank and homogeneous structures John T. Baldwin; 11. Constructions of semilinear towers of Steiner systems Keith Johnson; 12. Introduction to the Lascar group Martin Ziegler.
ISBN: 0-521-01063-2
Binding: Paperback
Pages: 308
Figures: 8 line diagrams 1 table
available from November 2001
Francis Bardou, Jean-Philippe Bouchaud, Alain Aspect, Claude Cohen-Tannoudji
Levy Statistics and Laser Cooling
Description
Laser cooling of atoms provides an ideal case study for the application of Levy statistics in a privileged situation where the statistical model can be derived from first principles. This book demonstrates how the most efficient laser cooling techniques can be simply and quantitatively understood in terms of non-ergodic random processes dominated by a few rare events. Levy statistics are now recognised as the proper tool for analysing many different problems for which standard Gaussian statistics are inadequate. Laser cooling provides a simple example of how Levy statistics can yield analytic predictions that can be compared to other theoretical approaches and experimental results. The authors of this book are world leaders in the fields of laser cooling and light-atom interactions, and are renowned for their clear presentation. This book will therefore hold much interest for graduate students and researchers in the fields of atomic physics, quantum optics, and statistical physics.
Chapter Contents
1. Introduction; 2. Subrecoil laser cooling and anomalous random walks; 3. Trapping and recyling. Statistical properties; 4. Broad distributions and Levy statistics: a brief overview; 5. Proportion of atoms trapped in quasi-dark states; 6. Momentum distribution; 7. Physical discussion; 8. Tests of the statistical approach; 9. Example of application: optimization of the peak of cooled atoms; 10. Conclusion; Appendix A. Correspondence of the parameters of the statistical models with atomic and laser parameters; Appendix B. The Doppler case; Appendix C. The special case mu = 1.
ISBN: 0-521-80821-9
Binding: Hardback
ISBN: 0-521-00422-5
Binding: Paperback
Pages: 208
Figures: 41 line diagrams 2 tables
available from December 2001
Robert Churchhouse
Codes and Ciphers
Description
The design of code and cipher systems has undergone major changes in modern times. Powerful personal computers have resulted in an explosion of e-banking, e-commerce and e-mail, and as a consequence the encryption of communications to ensure security has become a matter of public interest and importance. This book describes and analyses many cipher systems ranging from the earliest and elementary to the most recent and sophisticated, such as RSA and DES, as well as wartime machines such as the ENIGMA and Hagelin, and ciphers used by spies. Security issues and possible methods of attack are discussed and illustrated by examples. The design of many systems involves advanced mathematical concepts and this is explained in detail in a major appendix. This book will appeal to anyone interested in codes and ciphers as used by private individuals, spies, governments and industry throughout history and right up to the present day.
Chapter Contents
1. Introduction; 2 .From Julius Caesar to simple substitution; 3 .Polyalphabetic systems; 4. Jigsaw systems; 5.Two-letter ciphers; 6. Codes; 7. Ciphers for spies; 8. Producing random numbers and letters; 9. The ENIGMA cipher machine; 10. The Hagelin cipher machine; 11. Beyond the ENIGMA; 12. Public key cryptography; 13. Encipherment and the internet; 14. Appendix 1: References; 15. Appendix 2: Solutions to problems; 16. Appendix 3: Mathematical aspects.
ISBN: 0-521-81054-X
Binding: Hardback
ISBN: 0-521-00890-5
Binding: Paperback
Pages: 220
Figures: 7 colour plates
available from January 2002
Teo Mora
Systems of Polynomial Equations: Kronecker-Duval Philosophy
Encyclopedia of Mathematics and Its applications
Description
Polynomial equations have been long studied, both theoretically and with a view to solving them. Until recently, manual computation was the only solution method and the theory was developed to accommodate it. With the advent of computers, the situation changed dramatically. Many classical results can be more usefully recast within a different framework which in turn lends itself to further theoretical development tuned to computation. This first book in a trilogy is devoted to the new approach. It is a handbook covering the classical theory of finding roots of a univariate polynomial, emphasising computational aspects, especially the representation and manipulation of algebraic numbers, enlarged by more recent representations like the Duval Model and the Thom Codification. Mora aims to show that solving a polynomial equation really means finding algorithms that help one manipulate roots rather than simply computing them; to that end he also surveys algorithms for factorizing univariate polynomials.
Chapter Contents
Preface; Part I. The Kronecker-Duval Philosophy: 1. Euclid; 2. Intermezzo: Chinese remainder theorems; 3. Cardano; 4. Intermezzo: multiplicity of roots; 5. Kronecker I: Kronecker's philosophy; 6. Intermezzo: Sylvester; 7. Galois I: finite fields; 8. Kronecker II: Kronecker's model; 9. Steinitz; 10. Lagrange; 11. Duval; 12. Gauss; 13. Sturm; 14. Galois II; Part II. Factorization: 15. Ouverture; 16. Kronecker III: factorization; 17. Berlekamp; 18. Zassenhaus; 19. Fermeture; Bibliography; Index.
ISBN: 0-521-81154-6
Binding: Hardback
Pages: 450
available from March 2002
A. A. Ivanov, Sergei V. Shpectorov
Geometry of Sporadic Groups,Volume 2
Representations and Amalgams
Encyclopedia of Mathematics and its Applications, vol.91.
Description
This is the second volume in a two-volume set, which provides a complete self-contained proof of the classification of geometries associated with sporadic simple groups: Petersen and tilde geometries. The second volume contains a study of the representations of the geometries under consideration in GF(2)-vector spaces as well as in some non-abelian groups. The central part is the classification of the amalgam of maximal parabolics, associated with a flag transitive action on a Petersen or tilde geometry. The classification is based on the method of group amalgam, the most promising tool in modern finite group theory. Via their systematic treatment of group amalgams, the authors establish a deep and important mathematical result. This book will be of great interest to researchers in finite group theory, finite geometries and algebraic combinatorics.
Chapter Contents
1. Preliminaries; Part I. Representations: 2. General features; 3. Classical geometries; 4. Mathieu groups and Held group; 5. Conway groups; 6. Involution geometries; 7. Large sporadics; Part II. Amalgams: 8. Method of group amalgams; 9. Action on the derived graph; 10. Shapes of amalgams; 11. Amalgams for P-geometries; 12. Amalgams for T-geometries; Concluding remarks: 13. Further developments.
ISBN: 0-521-62349-9
Binding: Hardback
Pages: 255
Figures: 55 line diagrams 15 tables
available from March 2002