Edited by Pavel M. Bleher, Alexander R. Its

Random Matrix Models and Their Applications

(Mathematical Science Research Insitute Publications,vol.40)

Description
Random matrices arise from, and have important applications to, number theory, probability, combinatorics, representation theory, quantum mechanics, solid state physics, quantum field theory, quantum gravity, and many other areas of physics and mathematics. This volume of surveys and research results, based largely on lectures given at the Spring 1999 MSRI program of the same name, covers broad areas such as topologic and combinatorial aspects of random matrix theory; scaling limits, universalities and phase transitions in matrix models; universalities for random polynomials; and applications to integrable systems. Its stress on the interaction between physics and mathematics will make it a welcome addition to the shelves of graduate students and researchers in both fields, as will its expository emphasis.

Chapter Contents
1. Symmetrized random permutations Jinho Baik and Eric M. Rains; 2. Hankel determinants as Fredholm determinants Estelle L. Basor, Yang Chen and Harold Widom; 3. Universality and scaling of zeros on symplectic manifolds Pavel Bleher, Bernard Shiffman and Steve Zelditch; 4. Z measures on partitions, Robinson-Schensted-Knuth correspondence, and random matrix ensembles Alexei Borodin and Grigori Olshanski; 5. Phase transitions and random matrices Giovanni M. Cicuta; 6. Matrix model combinatorics: applications to folding and coloring Philippe Di Francesco; 7. Inter-relationships between orthogonal, unitary and symplectic matrix ensembles Peter J. Forrester and Eric M. Rains; 8. A note on random matrices John Harnad; 9. Orthogonal polynomials and random matrix theory Mourad E. H. Ismail; 10. Random words, Toeplitz determinants and integrable systems I, Alexander R. Its, Craig A. Tracy and Harold Widom; 11. Random permutations and the discrete Bessel kernel Kurt Johansson; 12. Solvable matrix models Vladimir Kazakov; 13. Tau function for analytic Curves I. K. Kostov, I. Krichever, M. Mineev-Vainstein, P. B. Wiegmann and A. Zabrodin; 14. integration over angular variables for two coupled matrices G. Mahoux, M. L. Mehta and J.-M. Normand; 15. SL and Z-measures Andrei Okounkov; 16. Integrable lattices: random matrices and random permutations Pierre Van Moerbeke; 17. Some matrix integrals related to knots and links Paul Zinn-Justin.

Title Details
ISBN: 0-521-80209-1
Binding: Hardback
available from July 2001

Patrick Blackburn, Maarten de Rijke, Yde Venema

Modal Logic

(Cambridge Tracts in Theoretical Computer Science, vol.53)

Description
This is a modern, advanced textbook on modal logic, a field which caught the attention of computer scientists in the late 1970s. Researchers in areas ranging from economics to computational linguistics have since realized its worth. The book is intended both for novices and for more experienced readers, with two distinct tracks clearly signposted at the start of each chapter. The development is mathematical; prior acquaintance with first-order logic and its semantics is assumed, and familiarity with the basic mathematical notions of set theory is required. The authors focus on the use of modal languages as tools to analyze the properties of relational structures, including their algorithmic and algebraic aspects, and applications to issues in logic and computer science such as completeness, computability and complexity are considered. Three appendices supply basic background information and numerous exercises are provided. The work is ideal for anyone wanting to learn modern modal logic.

Chapter Contents
1. Basic concepts; 2. Models; 3. Frames; 4. Completeness; 5. Algebras and general frames; 6. Computability and complexity; 7. Extended modal logic.

Title Details
ISBN: 0-521-80200-8
Binding: Hardback
Bibliographic information:54 line diagrams
available from April 2001

Andreas Engel, Christian P. L. Van den Broeck

Statistical Mechanics of Learning

Description
Learning is one of the things that humans do naturally, and it has always been a challenge for us to understand the process. Nowadays this challenge has another dimension as we try to build machines that are able to learn and to undertake tasks such as datamining, image processing and pattern recognition. We can formulate a simple framework, artificial neural networks, in which learning from examples may be described and understood. The contribution to this subject made over the last decade by researchers applying the techniques of statistical mechanics is the subject of this book. The authors provide a coherent account of various important concepts and techniques that are currently only found scattered in papers, supplement this with background material in mathematics and physics and include many examples and exercises to make a book that can be used with courses, or for self-teaching, or as a handy reference.

Chapter Contents
1. Getting started; 2. Perceptron learning - basics; 3. A choice of learning rules; 4. Augmented statistical mechanics formulation; 5. Noisy teachers; 6. The storage problem; 7. Discontinuous learning; 8. Unsupervised learning; 9. On-line learning; 10. Making contact with statistics; 11. A bird痴 eye view: multifractals; 12. Multilayer networks; 13. On-line learning in multilayer networks; 14. What else?; Appendix A. Basic mathematics; Appendix B. The Gardner analysis; Appendix C. Convergence of the perceptron rule; Appendix D. Stability of the replica symmetric saddle point; Appendix E. 1-step replica symmetry breaking; Appendix F. The cavity approach; Appendix G. The VC-theorem.

Title Details
ISBN: 0-521-77479-9
Binding: Paperback
Bibliographic information:1 table 136 exercises 54 figures
available from May 2001

J. H. van Lint, R. M. Wilson

A Course in Combinatorics Second edition

Description
This is the second edition of a popular book on combinatorics, a subject dealing with ways of arranging and distributing objects, and which involves ideas from geometry, algebra and analysis. The breadth of the theory is matched by that of its applications, which include topics as diverse as codes, circuit design and algorithm complexity. It has thus become essential for workers in many scientific fields to have some familiarity with the subject. The authors have tried to be as comprehensive as possible, dealing in a unified manner with, for example, graph theory, extremal problems, designs, colorings and codes. The depth and breadth of the coverage make the book a unique guide to the whole of the subject. The book is ideal for courses on combinatorical mathematics at the advanced undergraduate or beginning graduate level. Working mathematicians and scientists will also find it a valuable introduction and reference

Chapter Contents
Preface; 1. Graphs; 2. Trees; 3. Colorings of graphs and Ramsey's theorem; 4. Turan's theorem and extremal graphs; 5. Systems of distinct representatives; 6. Dilworth's theorem and extremal set theory; 7. Flows in networks; 8. De Bruijn sequences; 9. The addressing problem for graphs; 10. The principle of inclusion and exclusion; inversion formulae; 11. Permanents; 12. The Van der Waerden conjecture; 13. Elementary counting; Stirling numbers; 14. Recursions and generating functions; 15. Partitions; 16. (0,1)-matrices; 17. Latin squares; 18. Hadamard matrices, Reed-Muller codes; 19. Designs; 20. Codes and designs; 21. Strongly regular graphs and partial geometries; 22. Orthogonal Latin squares; 23. Projective and combinatorial geometries; 24. Gaussian numbers and q-analogues; 25. Lattices and Mvbius inversion; 26. Combinatorial designs and projective geometries; 27. Difference sets and automorphisms; 28. Difference sets and the group ring; 29. Codes and symmetric designs; 30. Association schemes; 31. Algebraic graph theory: eigenvalue techniques; 32. Graphs: planarity and duality; 33. Graphs: colorings and embeddings; 34. Electrical networks and squared squares; 35. Pslya theory of counting; 36. Baranyai痴 theorem; Appendices; Name index; Subject index.

Title Details
ISBN: 0-521-80340-3
Binding: Hardback
ISBN: 0-521-00601-5
Binding: Paperback

Ravi P. Agarwal, Maria Meehan, Donal O'Regan

Fixed Point Theory and Applications

(Cambridge Tracts in Mathematics, vol.141)

Description
This book provides a clear exposition of the flourishing field of fixed point theory, an important tool in the fields of differential equations and functional equations, among others. Starting from the basics of Banach痴 contraction theorem, most of the main results and techniques are developed: fixed point results are established for several classes of maps and the three main approaches to establishing continuation principles are presented. The theory is applied to many areas of current interest in analysis, with topological considerations playing a crucial role, including a final chapter on the relationship with degree theory. Researchers and graduate students in applicable analysis will find this to be a useful survey of the fundamental principles of the subject. The very extensive bibliography and close to 100 exercises mean that it can be used both as a text and as a comprehensive reference work, currently the only one of its type.

Chapter Contents
Preface; 1. Contraction; 2. Nonexpansive maps; 3. Continuation methods for contractive nonexpansive maps; 4. The theorems of Brouwer, Schauder and Mvnch; 5. Nonlinear alternatives of Leray-Schauder type; 6. Continuation principles for condensing maps; 7. Fixed point theorems in conical shells; 8. Fixed point theory in Hausdorff locally convex linear topological spaces; 9. Contractive and nonexpansive multivalued mappings; 10. Multivalued maps with continuous selection; 11. Multivalued maps with closed graph; 12. Degree theory; 13. References; Index.

Title Details
ISBN: 0-521-80250-4
Binding: Hardback
Bibliographic information:95 exercises
available from March 2001