Series: Cambridge Series in Statistical and Probabilistic Mathematics (No. 22)
Hardback (ISBN-13: 9780521872508)
Modern statistical methods use complex, sophisticated models that can lead to intractable computations. Saddlepoint approximations can be the answer. Written from the user’s point of view, this book explains in clear language how such approximate probability computations are made, taking readers from the very beginnings to current applications. The core material is presented in chapters 1-6 at an elementary mathematical level. Chapters 7-9 then give a highly readable account of higher-order asymptotic inference. Later chapters address areas where saddlepoint methods have had substantial impact: multivariate testing, stochastic systems and applied probability, bootstrap implementation in the transform domain, and Bayesian computation and inference. No previous background in the area is required. Data examples from real applications demonstrate the practical value of the methods. Ideal for graduate students and researchers in statistics, biostatistics, electrical engineering, econometrics, and applied mathematics, this is both an entry-level text and a valuable reference.
* An accessible, readable introduction that equips the reader to use the
methods for real applications
* Abundant examples, both numerical and theoretical, build and reinforce
skills and understanding
Author is a major contributor to the field: this is, and will remain,
the book on saddlepoint approximation
Contents
Preface; 1. Fundamental approximations; 2. Properties and derivatives; 3. Multivariate densities; 4. Conditional densities and distribution functions; 5. Exponential families and tilted distributions; 6. Further exponential family examples and theory; 7. Probability computation with p*; 8. Probabilities with r*-type approximations; 9. Nuisance parameters; 10. Sequential saddlepoint applications; 11. Applications to multivariate testing; 12. Ratios and roots of estimating equations; 13. First passage and time to event distributions; 14. Bootstrapping in the transform domain; 15. Bayesian applications; 16. Non-normal bases; References; Index.
Series: Encyclopedia of Mathematics and its Applications (No. 111)
Hardback (ISBN-13: 9780521857215)
Some of the most beautiful mathematical objects found in the last forty years are the sporadic simple groups. But gaining familiarity with these groups presents problems for two reasons. Firstly, they were discovered in many different ways, so to understand their constructions in depth one needs to study lots of different techniques. Secondly, since each of them is in a sense recording some exceptional symmetry in spaces of certain dimensions, they are by their nature highly complicated objects with a rich underlying combinatorial structure. Motivated by initial results which showed that the Mathieu groups can be generated by highly symmetrical sets of elements, which themselves have a natural geometric definition, the author develops from scratch the notion of symmetric generation. He exploits this technique by using it to define and construct many of the sporadic simple groups including all the Janko groups and the Higman-Sims group. For researchers and postgraduates.
*The technique of symmetric generation and its applications is developed from scratch by the author, who is the leading researcher in this field
*Discusses in detail how symmetric generation can be exploited to provide concise and elementary definitions of many sporadic simple groups
*Will be of great interest to researchers and graduate students in combinatorial or computational group theory
Contents
Preface; Acknowledgements; Part I. Motivation: 1. The Mathieu group M12; 2. The Mathieu group M24; Part II. Involutory Symmetric Generators: 3. The progenitor; 4. Classical examples; 5. Sporadic simple groups; Part III. Non-involutory Symmetric Generators: 6. The progenitor; 7. Images of these progenitors.
Hardback (ISBN-13: 9780521876247)
Paperback (ISBN-13: 9780521699730)
Contains a wealth of inequalities used in linear analysis, and explains in detail how they are used. The book begins with Cauchy's inequality and ends with Grothendieck's inequality, in between one finds the Loomis-Whitney inequality, maximal inequalities, inequalities of Hardy and of Hilbert, hypercontractive and logarithmic Sobolev inequalities, Beckner's inequality, and many, many more. The inequalities are used to obtain properties of function spaces, linear operators between them, and of special classes of operators such as absolutely summing operators. This textbook complements and fills out standard treatments, providing many diverse applications: for example, the Lebesgue decomposition theorem and the Lebesgue density theorem, the Hilbert transform and other singular integral operators, the martingale convergence theorem, eigenvalue distributions, Lidskii's trace formula, Mercer's theorem and Littlewood's 4/3 theorem. It will broaden the knowledge of postgraduate and research students, and should also appeal to their teachers, and all who work in linear analysis.
*Establishes the fundamental inequalities of linear analysis
*Explains in detail how these important inequalities are used
*Provides breadth to courses on linear analysis
Contents
Introduction; 1. Measure and integral; 2. The Cauchy-Schwarz inequality; 3. The AM-GM inequality; 4. Convexity, and Jensen's inequality; 5. The Lp spaces; 6. Banach function spaces; 7. Rearrangements; 8. Maximal inequalities; 9. Complex interpolation; 10. Real interpolation; 11. The Hilbert transform, and Hilbert's inequalities; 12. Khintchine's inequality; 13. Hypercontractive and logarithmic Sobolev inequalities; 14. Hadamard's inequality; 15. Hilbert space operator inequalities; 16. Summing operators; 17. Approximation numbers and eigenvalues; 18. Grothendieck's inequality, type and cotype.
Hardback (ISBN-13: 9780521873413)
While many scientists are familiar with fractals, fewer are familiar with scale-invariance and universality which underly the ubiquity of their shapes. These properties may emerge from the collective behaviour of simple fundamental constituents, and are studied using statistical field theories. Initial chapters connect the particulate perspective developed in the companion volume, to the coarse grained statistical fields studied here. Based on lectures taught by Professor Kardar at MIT, this textbook demonstrates how such theories are formulated and studied. Perturbation theory, exact solutions, renormalization groups, and other tools are employed to demonstrate the emergence of scale invariance and universality, and the non-equilibrium dynamics of interfaces and directed paths in random media are discussed. Ideal for advanced graduate courses in statistical physics, it contains an integrated set of problems, with solutions to selected problems at the end of the book and a complete set available to lecturers at www.cambridge.org/9780521873413.
*Based on lecture notes from a course on Statistical Mechanics taught by the author at MIT
*Contains 65 exercises, with solutions to selected problems
*Features a thorough introduction to the methods of Statistical Field theory
*Ideal for graduate courses in Statistical Physics
Contents
1. Collective behaviour, from particles to fields; 2. Statistical fields; 3. Fluctuations; 4. The scaling hypothesis; 5. Perturbative renormalization group; 6. Lattice systems; 7. Series expansions; 8. Beyond spin waves; 9. Dissipative dynamics; 10. Directed paths in random media; Solutions to selected problems; Index.
Hardback (ISBN-13: 9780521660785)
en physicists first began to use statistical mechanics to study granular media. They are prototypical of complex systems, manifesting metastability, hysteresis and bistability, and a range of other fascinating phenomena. This book is a wide-ranging account of developments in granular physics, and lays out the foundations of the statics and dynamics of granular physics. It covers a wide range of subfields, ranging from fluidization to jamming, and these are modeled through a range of computer simulation and theoretical approaches. Written with an eye to pedagogy and completeness, this book will be a valuable asset for any researcher in this field. It includes the most recent ideas and contains discussions at the end of each chapter. The book also contains contributions from Professor Sir Sam Edwards, with Dr Raphael Blumenfeld; Professor Isaac Goldhirsch; and Professor Philippe Claudin.
*Written by Anita Mehta, a pioneer in granular physics
*Contains contributions from Professor Sir Sam Edwards (with Dr Raphael Blumenfeld), Professor Isaac Goldhirsch, and Professor Philippe Claudin - all distinguished researchers in this field
*A pedagogical approach covering a wide range of topics
Contents
1. Introduction; 2. Computer simulation approaches - an overview; 3. Structure of vibrated powders - numerical results; 4. Collective structures in sand - the phenomenon of bridging; 5. On angles of repose: bistability and collapse; 6. Compaction of disordered grains in the jamming limit: sand on random graphs; 7. Shaking a box of sand I - a simple lattice model; 8. Shaking a box of sand II - at the jamming limit, when shape matters!; 9. Avalanches with reorganising grains; 10. From earthquakes to sandpiles - stick-slip motion; 11. Coupled continuum equations: the dynamics of sand-pile surfaces; 12. Theory of rapid granular flows; 13. The thermodynamics of granular materials; 14. Static properties of granular materials; Author index; Subject index; Bibliography.
Hardback (ISBN-13: 9780521868341)
15 line diagrams 5 half-tones 52 exercises
Page extent: 284 pages
Size: 247 x 174 mm
This is the first introductory textbook on quantum field theory in gravitational backgrounds intended for undergraduate and beginning graduate students in the fields of theoretical astrophysics, cosmology, particle physics, and string theory. The book covers the basic (but essential) material of quantization of fields in an expanding universe and quantum fluctuations in inflationary spacetime. It also contains a detailed explanation of the Casimir, Unruh, and Hawking effects, and introduces the method of effective action used for calculating the back-reaction of quantum systems on a classical external gravitational field. The broad scope of the material covered will provide the reader with a thorough perspective of the subject. Every major result is derived from first principles and thoroughly explained. The book is self-contained and assumes only a basic knowledge of general relativity. Exercises with detailed solutions are provided throughout the book.
*The first introductory textbook on quantum field theory in gravitational backgrounds for undergraduate and beginning graduate students
*Contains exercises with detailed solutions
*Self-contained, covering a broad scope of material
Contents
Preface; Part I. Canonical Quantization and Particle Production: 1. Overview: a taste of quantum fields; 2. Reminder: Classical and quantum theory; 3. Driven harmonic oscillator; 4. From harmonic oscillators to fields; 5. Reminder: Classical fields; 6. Quantum fields in expanding universe; 7. Quantum fields in the de Sitter universe; 8. Unruh effect; 9. Hawking effect. Thermodynamics of black holes; 10. The Casimir effect; Part II. Path Integrals and Vacuum Polarization: 11. Path integrals; 12. Effective action; 13. Calculation of heat kernel; 14. Results from effective action; Appendices; Index.
Series: Cambridge Mathematical Library
Paperback (ISBN-13: 9780521705127)
40 figures
Page extent: 600 pages
Size: 228 x 152 mm
This classic work continues to offer a comprehensive treatment of the theory of univariate and tensor-product splines. It will be of interest to researchers and students working in applied analysis, numerical analysis, computer science, and engineering. The material covered provides the reader with the necessary tools for understanding the many applications of splines in such diverse areas as approximation theory, computer-aided geometric design, curve and surface design and fitting, image processing, numerical solution of differential equations, and increasingly in business and the biosciences. This new edition includes a supplement outlining some of the major advances in the theory since 1981, and some 250 new references. It can be used as the main or supplementary text for courses in splines, approximation theory or numerical analysis.
*Comprehensive reference, with preparatory material on polynomials, Tchebycheff systems etc, plus historical notes and comments, and comprehensive list of references
*Includes efficent algorithms for evaluating B-splines, treats generalized splines, gives full account of approximation properties of splines
*New supplement helps keep the book up to date
Contents
1. Introduction; 2. Preliminaries; 3. Polynomials; 4. Polynomial splines; 5. Computational methods; 6. Approximation power of splines; 7. Approximation power of splines (free knots); 8. Other spaces of polynomial splines; 9. Tchebycheffian splines; 10. L-Splines; 11. Generalized splines; 12. Tensor-product splines; 13. Some multidimensional tools; Supplement; References; New references; Index.