EDWARD OTT
University of Maryland

Chaos in Dynamical Systems

2nd Edition

Description: Over the past two decades scientists, mathematicians, and engineers have come to understand that a large variety of systems exhibit complicated evolution with time. This complicated behavior is known as chaos. In the new edition of this classic textbook Edward Ott has added much new material and has significantly increased the number of homework problems. The most important change is the addition of a completely new chapter on control and synchronization of chaos. Other changes include new material on riddled basins of attraction, phase locking of globally coupled oscillators, fractal aspects of fluid advection by Lagrangian chaotic flows, magnetic dynamos, and strange nonchaotic attractors. This new edition will be of interest to advanced undergraduates and graduate students in science, engineering, and mathematics taking courses in chaotic dynamics, as well as to researchers in the subject.

Contents: Preface; 1. Introduction and overview; 2. One-dimensional maps; 3. Strange attractors and fractal dimensions; 4. Dynamical properties of chaotic systems; 5. Nonattracting chaotic sets; 6. Quasiperiodicity; 7. Chaos in Hamiltonian systems; 8. Chaotic transitions; 9. Multifractals; 10. Control and synchronization of chaos; 11. Quantum chaos.

Essential Information
ISBN, Binding, Price: 0-521-01084-5 Paperback
Pages: 496
Approximate Publication Date: c.01/07/2002
Main Subject Category: Nonlinear science

JAN SMIT
University of Amsterdam

Introduction to Quantum Fields on a Lattice

Description: This book provides a concrete introduction to quantum fields on a lattice: a precise and non-perturbative definition of quantum field theory obtained by replacing continuous space-time by a discrete set of points on a lattice. The path integral on the lattice is explained in concrete examples using weak and strong coupling expansions. Fundamental concepts such as ‘triviality’ of Higgs fields and confinement of quarks and gluons into hadrons are described and illustrated with the results of numerical simulations. The book also provides an introduction to chiral symmetry and chiral gauge theory, as well as quantized non-abelian gauge fields, scaling and universality. Based on the lecture notes of a course given by the author, this book contains many explanatory examples and exercises, and is suitable as a textbook for advanced undergraduate and graduate courses.

Contents: Preface; 1. Introduction; 2. Path integral and lattice regularisation; 3. O(n) models; 4. Gauge field on the lattice; 5. U(1) and SU(n) gauge theory; 6. Fermions on the lattice; 7. Low mass hadrons in QCD; 8. Chiral symmetry; Appendix 1. SU(n); Appendix 2. Temporal gauge quantization in the continuum; Appendix 3. Fermionic coherent states; Appendix 4. Spinor fields.

Essential Information
ISBN, Binding, : 0-521-89051-9 Paperback
Pages: 232
Approximate Publication Date: c.01/07/2002
Main Subject Category: Theoretical, mathematical physics
Series: Cambridge Lecture Notes in Physics, No. 15

Market (Subject)
theoretical physics, quantum field theory, lattice fields, particle physics

Level
graduate students, undergraduate students

Bibliographic Details
74 line diagrams 1 table 34 exercises

Comparable titles: MONTVAY and MUNSTER/Quantum Fields on a Lattice/1994/0521 404320 CREUTZ/Quarks, Gluons and Lattices/1984/0521 244056

GRAEME MILTON
University of Utah

The Theory of Composites

Description: Some of the greatest scientists including Poisson, Faraday, Maxwell, Rayleigh, and Einstein have contributed to the theory of composite materials. Mathematically, it is the study of partial differential equations with rapid oscillations in their coefficients. Although extensively studied for more than a hundred years, an explosion of ideas in the last four decades (and particularly in the last two decades) has dramatically increased our understanding of the relationship between the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective (electrical, thermal, elastic) moduli which govern the macroscopic behavior. This renaissance has been fueled by the technological need for improving our knowledge base of composites, by the advance of the underlying mathematical theory of homogenization, by the discovery of new variational principles, by the recognition of how important the subject is to solving structural optimization problems, and by the realization of the connection with the mathematical problem of quasiconvexification. This book surveys these exciting developments at the frontier of mathematics and presents many new results.

Contents: 1. Introduction; 2. Equations of interest and numerical approaches; 3. Duality transformations; 4. Translations and equivalent media; 5. Microstructure independent exact relations; 6. Exact relations for coupled equations; 7. Assemblages of inclusions; 8. Tricks for exactly solvable microgeometries; 9. Laminate materials; 10. Approximations and asymptotic formulae; 11. Wave propagation in the quasistatic limit; 12. Reformulating the problem; 13. Variational principles and inequalities; 14. Series expansions; 15. Correlation functions and series expansions; 16. Other perturbation solutions; 17. The general theory of exact relations; 18. Analytic properties; 19. Y-tensors; 20. Y-tensors and effective tensors in circuits; 21. Bounds on the properties of composites; 22. Classical variational principle bounds; 23. Hashin-Shtrikman bounds; 24. Translation method bounds; 25. Choosing translations and finding geometries; 26. Bounds incorporating three-point statistics; 27. Bounds using the analytic method; 28. Fractional linear transformations for bounds; 29. The field equation recursion method; 30. G-closure properties and extremal composites; 31. Bounding and quasiconvexification.

Essential Information
ISBN, Binding, : 0-521-78125-6 Hardback
Pages: 748
Approximate Publication Date: 16/05/2002
Main Subject Category: Applied mathematics, mathematical physics
Series: Cambridge Monographs on Applied and Computational Mathematics, No. 6

Market (Subject)
applied mathematics, solid mathematics

Level
graduate students, academic researchers

Comparable titles: HULL and CLYNE /An Introduction to Composite Materials 2ed/1996/HB 0521 381908; PB 0521 388554 SHENOI and WELLICOME/Composite Materials in Maritime v1+2/1993/0521 451531 CLYNE and WITHERS/Introduction Metal Matrix Composites/1995/HB 0521 418089; PB 0521 483573 KOLLAR and SPRINGER/Mechanics of Composite Structures/2001/0521 801656

KLAUS BICHTELER
University of Texas, Austin

Stochastic Integration with Jumps

Description: Stochastic processes with jumps and random measures are gaining importance as drivers in applications like financial mathematics and signal processing. This book develops stochastic integration theory for both integrators (semimartingales) and random measures from a common point of view. Using some novel predictable controlling devices, the author furnishes the theory of stochastic differential equations driven by them, as well as their stability and numerical approximation theories. Highlights feature DCT and Egoroff’s Theorem, as well as comprehensive analogs to results from ordinary integration theory, for instance previsible envelopes and an algorithm computing stochastic integrals of cągląd integrands pathwise. Full proofs are given for all results, and motivation is stressed throughout. A large appendix contains most of the analysis that readers will need as a prerequisite. This will be an invaluable reference for graduate students and researchers in mathematics, physics, electrical engineering and finance who need to use stochastic differential equations.

Contents: 1. Introduction; 2. Integrators and martingales; 3. Extension of the integral; 4. Control of integral and integrator; 5. Stochastic differential equations; Appendix A. Complements to topology and measure theory; Appendix B. Answers to selected problems; References; Index.

Essential Information
ISBN, Binding,: 0-521-81129-5 Hardback
Pages: 516
Approximate Publication Date: c.01/07/2002
Main Subject Category: Mathematics - analysis, probability
Series: Encyclopedia of Mathematics and its Applications, No. 89

Market (Subject)
probability, analysis

Level
academic researchers, graduate students

Bibliographic Details
17 line diagrams 1 table 745 exercises

Comparable titles: SACHKOV/Probabilistic Methods in Combinatorial Analysis/1997/0521 45512X

ALEXANDER POLISHCHUK
Boston University

Abelian Varieties, Theta Functions and the Fourier Transform

Description: The aim of this book is to present a modern treatment of the theory of theta functions in the context of algebraic geometry. The novelty of its approach lies in the systematic use of the Fourier-Mukai transform. The author starts by discussing the classical theory of theta functions from the point of view of the representation theory of the Heisenberg group (in which the usual Fourier transform plays the prominent role). He then shows that in the algebraic approach to this theory (originally due to Mumford) the Fourier-Mukai transform can often be used to simplify the existing proofs or to provide completely new proofs of many important theorems. Graduate students and researchers with strong interest in algebraic geometry will find much of interest in this volume.

Essential Information
ISBN, Binding, : 0-521-80804-9 Hardback
Pages: 300
Approximate Publication Date: c.01/06/2002
Main Subject Category: Number theory
Series: Cambridge Tracts in Mathematics, No. 153

Market (Subject)
mathematics (number theory, algebraic geometry)

Level
graduate students, academic researchers