Part of Cambridge Tracts in Mathematics
Publication planned for: January 2017
availability: Not yet published - available from January 2017
format: Hardback
isbn: 9781107026698
Stochastic systems provide powerful abstract models for a variety of important real-life applications: for example, power supply, traffic flow, data transmission. They (and the real systems they model) are often subject to phase transitions, behaving in one way when a parameter is below a certain critical value, then switching behaviour as soon as that critical value is reached. In a real system, we do not necessarily have control over all the parameter values, so it is important to know how to find critical points and to understand system behaviour near these points. This book is a modern presentation of the 'semimartingale' or 'Lyapunov function' method applied to near-critical stochastic systems, exemplified by non-homogeneous random walks. Applications treat near-critical stochastic systems and range across modern probability theory from stochastic billiards models to interacting particle systems. Spatially non-homogeneous random walks are explored in depth, as they provide prototypical near-critical systems.
1. Introduction
2. Semimartingale approach and Markov chains
3. Lamperti's problem
4. Many-dimensional random walks
5. Heavy tails
6. Further applications
7. Markov chains in continuous time
Glossary of named assumptions
Bibliography
Index.
Publication planned for: March 2017
availability: Not yet published - available from March 2017
format: Hardback
isbn: 9781107159389
This thorough work presents the fundamental results of modular function theory as developed during the nineteenth and early-twentieth centuries. It features beautiful formulas and derives them using skillful and ingenious manipulations, especially classical methods often overlooked today. Starting with the work of Gauss, Abel, and Jacobi, the book then discusses the attempt by Dedekind to construct a theory of modular functions independent of elliptic functions. The latter part of the book explains how Hurwitz completed this task and includes one of Hurwitz's landmark papers, translated by the author, and delves into the work of Ramanujan, Mordell, and Hecke. For graduate students and experts in modular forms, this book demonstrates the relevance of these original sources and thereby provides the reader with new insights into contemporary work in this area.
1. The basic modular forms
2. Gauss's contributions to modular forms
3. Abel and Jacobi on elliptic functions
4. Eisenstein and Hurwitz
5. Hermite's transformation of theta functions
6. Complex variables and elliptic functions
7. Hypergeometric functions
8. Dedekind's paper on modular functions
9. The n function and Dedekind sums
10. Modular forms and invariant theory
11. The modular and multiplier equations
12. The theory of modular forms as reworked by Hurwitz
13. Ramanujan's Euler products and modular forms
14. Dirichlet series and modular forms
15. Sums of squares
16. The Hecke operators.
Part of London Mathematical Society Lecture Note Series
Publication planned for: April 2017
availability: Not yet published - available from April 2017
format: Paperback
isbn: 9781107674424
This introduction to random walks on infinite graphs gives particular emphasis to graphs with polynomial volume growth. It offers an overview of analytic methods, starting with the connection between random walks and electrical resistance, and then proceeding to study the use of isoperimetric and Poincare inequalities. The book presents rough isometries and looks at the properties of a graph that are stable under these transformations. Applications include the 'type problem': determining whether a graph is transient or recurrent. The final chapters show how geometric properties of the graph can be used to establish heat kernel bounds, that is, bounds on the transition probabilities of the random walk, and it is proved that Gaussian bounds hold for graphs that are roughly isometric to Euclidean space. Aimed at graduate students in mathematics, the book is also useful for researchers as a reference for results that are hard to find elsewhere.
Preface
1. Introduction
2. Random walks and electrical resistance
3. Isoperimetric inequalities and applications
4. Discrete time heat kernel
5. Continuous time random walks
6. Heat kernel bounds
7. Potential theory and Harnack inequalities
Appendix A
References
Index.
Part of Cambridge Series in Statistical and Probabilistic Mathematics
Publication planned for: June 2017
availability: Not yet published - available from June 2017
format: Hardback
isbn: 9780521878265
Explosive growth in computing power has made Bayesian methods for infinite-dimensional models - Bayesian nonparametrics - a nearly universal framework for inference, finding practical use in numerous subject areas. Written by leading researchers, this authoritative text draws on theoretical advances of the past twenty years to synthesize all aspects of Bayesian nonparametrics, from prior construction to computation and large sample behavior of posteriors. Because understanding the behavior of posteriors is critical to selecting priors that work, the large sample theory is developed systematically, illustrated by various examples of model and prior combinations. Precise sufficient conditions are given, with complete proofs, that ensure desirable posterior properties and behavior. Each chapter ends with historical notes and numerous exercises to deepen and consolidate the reader's understanding, making the book valuable for both graduate students and researchers in statistics and machine learning, as well as in application areas such as econometrics and biostatistics.
Preface
Glossary of symbols
1. Introduction
2. Priors on function spaces
3. Priors on spaces of probability measures
4. Dirichlet processes
5. Dirichlet process mixtures
6. Consistency: general theory
7. Consistency: examples
8. Contraction rates: general theory
9. Contraction rates: examples
10. Adaptation and model selection
11. Gaussian process priors
12. Infinite-dimensional Bernstein?von Mises theorem
13. Survival analysis
14. Discrete random structures
Appendices
References
Author index
Subject index.
Publication planned for: May 2017
availability: Not yet published - available from May 2017
format: Paperback
isbn: 9781316620243
A concise yet elementary introduction to measure and integration theory, which are vital in many areas of mathematics, including analysis, probability, mathematical physics and finance. In this highly successful textbook, core ideas of measure and integration are explored, and martingales are used to develop the theory further. Other topics are also covered such as Jacobi's transformation theorem, the Radon?Nikodym theorem, differentiation of measures and Hardy?Littlewood maximal functions. In this second edition, readers will find newly added chapters on Hausdorff measures, Fourier analysis, vague convergence and classical proofs of Radon?Nikodym and Riesz representation theorems. All proofs are carefully worked out to ensure full understanding of the material and its background. Requiring few prerequisites, this book is suitable for undergraduate lecture courses or self-study. Numerous illustrations and over 400 exercises help to consolidate and broaden knowledge. Full solutions to all exercises are available on the author's webpage at www.motapa.de.