Series: Cambridge Tracts in Mathematics (No. 185)
ISBN: 9780521879095 Hardback
3 b/w illus.
Dimensions: 228 x 152 mm
available from June 2011
This self-contained monograph presents rigidity theory for a large class of dynamical systems, differentiable higher rank hyperbolic and partially hyperbolic actions. This first volume describes the subject in detail and develops the principal methods presently used in various aspects of the rigidity theory. Part I serves as an exposition and preparation, including a large collection of examples that are difficult to find in the existing literature. Part II focuses on cocycle rigidity, which serves as a model for rigidity phenomena as well as a useful tool for studying them. The book is an ideal reference for applied mathematicians and scientists working in dynamical systems and a useful introduction for graduate students interested in entering the field. Its wealth of examples also makes it excellent supplementary reading for any introductory course in dynamical systems.
* Builds a bridge between classical and emerging theory and emphasizes the new phenomena that appear
* Includes a chapter providing preparatory results in analysis
* Contains a large collection of examples of higher rank Anosov and partially hyperbolic actions
Introduction: an overview
Part I. Preliminaries from Dynamics and Analysis: 1. Properties of abelian group actions
2. Principal classes of algebraic actions
3. Preparatory results from analysis
Part II. Cocycles, Cohomology and Rigidity: 4. First cohomology and rigidity for vector-valued cocycles
5. Cocycles with non-abelian range
6. Higher order cohomology
Bibliography
Index.
Series: Cambridge Tracts in Mathematics (No. 187)
ISBN: 9781107007314 Hardback
8 b/w illus.
Dimensions: 228 x 152 mm
available from May 2011
Preface
1. Convex functions and sets
2. Orlicz spaces
3. Gauges and locally convex spaces
4. Separation theorems
5. Duality: dual topologies, bipolar sets, and Legendre transforms
6. Monotone and convex matrix functions
7. Loewner's theorem: a first proof
8. Extreme points and the Krein*Milman theorem
9. The strong Krein*Milman theorem
10. Choquet theory: existence
11. Choquet theory: uniqueness
12. Complex interpolation
13. The Brunn*Minkowski inequalities and log concave functions
14. Rearrangement inequalities: a) Brascamp*Lieb*Luttinger inequalities
15. Rearrangement inequalities: b) Majorization
16. The relative entropy
17. Notes
References
Author index
Subject index.
Series: London Mathematical Society Lecture Note Series (No. 385)
ISBN: 9780521111133 Paperback
8 b/w illus.
Dimensions: 228 x 152 mm
available from June 2011
Hidden Markov processes (HMPs) are important objects of study in many areas of pure and applied mathematics, including information theory, probability theory, dynamical systems and statistical physics, with applications in electrical engineering, computer science and molecular biology. This collection of research and survey papers presents important new results and open problems, serving as a unifying gateway for researchers in these areas. Based on talks given at the Banff International Research Station Workshop, 2007, this volume addresses a central problem of the subject: computation of the Shannon entropy rate of an HMP. This is a key quantity in statistical physics and information theory, characterising the fundamental limit on compression and closely related to channel capacity, the limit on reliable communication. Also discussed, from a symbolic dynamics and thermodynamical viewpoint, is when mappings between dynamical systems map Markov measures to Markov (or Gibbs) measures or allow for Markov lifts of Markov chains.
* Includes the latest results in an active area of research
* Connects approaches from various disciplines
* Discusses open problems such as the computation of the Shannon entropy rate of an HMP
1. Hidden Markov processes in the context of symbolic dynamics Mark Boyle and Karl Petersen
2. On the preservation of Gibbsianess under symbol amalgamation Jean-Rene Chazottes and E. Ugalde
3. A note on a complex Hilbert metric with application to domain of analyticity for entropy rate of hidden Markov processes Guangyue Han, Brian Marcus and Yuval Peres
4. Bounds on the entropy rate of binary hidden Markov processes Erik Ordentlich and Tsachy Weissman
5. Entropy rate for hidden Markov chains with rare transitions Yuval Peres and Anthony Quas
6. The capacity of finite-state channels in the high-noise regime Henry Pfister
7. Computing entropy rates for hidden Markov processes Mark Pollicott
8. Factors of Gibbs measures for full shifts Mark Pollicott and Thomas Kempton
9. Thermodynamics of hidden Markov processes Evgeny Verbitskiy.
Series: Cambridge Studies in Advanced Mathematics (No. 131)
ISBN: 9781107005969 Hardback
95 exercises
Dimensions: 228 x 152 mm
available from June 2011
Fusion systems are a recent development in finite group theory and sit at the intersection of algebra and topology. This book is the first to deal comprehensively with this new and expanding field, taking the reader from the basics of the theory right to the state of the art. Three motivational chapters, indicating the interaction of fusion and fusion systems in group theory, representation theory and topology are followed by six chapters that explore the theory of fusion systems themselves. Starting with the basic definitions, the topics covered include: weakly normal and normal subsystems; morphisms and quotients; saturation theorems; results about control of fusion; and the local theory of fusion systems. At the end there is also a discussion of exotic fusion systems. Designed for use as a text and reference work, this book is suitable for graduate students and experts alike.
* The first book to bring together all of the literature in the field, making it accessible to first-time learners
* Almost 100 exercises aid the reader's learning
* Includes new proofs, providing multiple ways of viewing theorems
Preface
Part I. Motivation: 1. Fusion in finite groups
2. Fusion in representation theory
3. Fusion in topology
Part II. The Theory: 4. Fusion systems
5. Weakly normal subsystems, quotients, and morphisms
6. Proving saturation
7. Control in fusion systems
8. Local theory of fusion systems
9. Exotic fusion systems
References
Index of notation
Index.
Series: Cambridge Series in Statistical and Probabilistic Mathematics
ISBN: 9781107009653 Hardback
98 b/w illus. 102 tables 77 exercises
Dimensions: 253 x 215 mm
available from August 2011
This book introduces basic and advanced concepts of categorical regression with a focus on the structuring constituents of regression, including regularization techniques to structure predictors. In addition to standard methods such as the logit and probit model and extensions to multivariate settings, the author presents more recent developments in flexible and high-dimensional regression, which allow weakening of assumptions on the structuring of the predictor and yield fits that are closer to the data. A generalized linear model is used as a unifying framework whenever possible in particular parametric models that are treated within this framework. Many topics not normally included in books on categorical data analysis are treated here, such as nonparametric regression; selection of predictors by regularized estimation procedures; ternative models like the hurdle model and zero-inflated regression models for count data; and non-standard tree-based ensemble methods, which provide excellent tools for prediction and the handling of both nominal and ordered categorical predictors. The book is accompanied an R package that contains data sets and code for all the examples.
* Covers modern topics such as high-dimensional regression and nonparametric models
* The book can be used as a text for courses on categorical data for students from different fields
* The book is written from the perspective of an applied statistician, the focus is on basic concepts and applications rather than formal mathematical theory
1. Introduction
2. Binary regression: the logit model
3. Generalized linear models
4. Modeling of binary data
5. Alternative binary regression models
6. Regularization and variable selection for parametric models
7. Regression analysis of count data
8. Multinomial response models
9. Ordinal response models
10. Semi- and nonparametric generalized regression
11. Tree-based methods
12. The analysis of contingency tables: log-linear and graphical models
13. Multivariate response models
14. Random effects models
15. Prediction and classification
Appendix A. Distributions
Appendix B. Some basic tools
Appendix C. Constrained estimation
Appendix D. Kullback-Leibler distance and information-based criteria of model fit
Appendix E. Numerical integration and tools for random effects modeling.
Series: Cambridge Texts in Applied Mathematics (No. 45)
ISBN: 9780521193191 hard cover
ISBN: 9780521149273 soft cover
80 b/w illus.
Dimensions: 228 x 152 mm
available from September 2011
Understanding the behaviour of particles suspended in a fluid has many important applications across a range of fields, including engineering and geophysics. Comprising two main parts, this book begins with the well-developed theory of particles in viscous fluids, i.e. microhydrodynamics, particularly for single- and pair-body dynamics. Part II considers many-body dynamics, covering shear flows and sedimentation, bulk flow properties and collective phenomena. An interlude between the two parts provides the basic statistical techniques needed to employ the results of the first (microscopic) in the second (macroscopic). The authors introduce theoretical, mathematical concepts through concrete examples, making the material accessible to non-mathematicians. They also include some of the many open questions in the field to encourage further study. Consequently, this is an ideal introduction for students and researchers from other disciplines who are approaching suspension dynamics for the first time.
* An introduction to the mathematical ideas of the area suitable for non-mathematicians
* Enables students and researchers from other disciplines to understand the relevance of mathematical theory
* Provides a multi-scale view of suspension mechanics
List of illustrations
Preface
Prologue
Part I. Microhydrodynamics: 1. Basic concepts in viscous *ow
2. One sphere in Stokes *ow
3. Sophisticated techniques
4. Particle pair interactions
iNTERLUDE: fROM THE MICROSCOPIC TO THe MACROSCOPIC
5. Statistical and stochastic concepts
PART II. Toward a Description of Macroscopic Phenomena in Suspensions: 6. Sedimentation
7. Rheology and microstructure
8. Beyond Stokes *ow: finite inertia
Epilogue
References
Index.