Part of Cambridge Texts in Applied Mathematics
Publication planned for: July 2015
format: Hardback
isbn: 9781107075399
format: Paperback
isbn: 9781107428201
The mathematical theory of stochastic dynamics has become an important tool in the modeling of uncertainty in many complex biological, physical, and chemical systems and in engineering applications - for example, gene regulation systems, neuronal networks, geophysical flows, climate dynamics, chemical reaction systems, nanocomposites, and communication systems. It is now understood that these systems are often subject to random influences, which can significantly impact their evolution. This book serves as a concise introductory text on stochastic dynamics for applied mathematicians and scientists. Starting from the knowledge base typical for beginning graduate students in applied mathematics, it introduces the basic tools from probability and analysis and then develops for stochastic systems the properties traditionally calculated for deterministic systems. The book's final chapter opens the door to modeling in non-Gaussian situations, typical of many real-world applications. Rich with examples, illustrations, and exercises with solutions, this book is also ideal for self-study.
1. Introduction
2. Background in analysis and probability
3. Noise
4. A crash course in stochastic differential equations
5. Deterministic quantities for stochastic dynamics
6. Invariant structures for stochastic dynamics
7. Dynamical systems driven by non-Gaussian Levy motions.
Publication planned for: June 2015
format: Hardback
isbn: 9781107075092
Catalan numbers are probably the most ubiquitous sequence of numbers in mathematics. This book gives for the first time a comprehensive collection of their properties and applications to combinatorics, algebra, analysis, number theory, probability theory, geometry, topology, and other areas. Following an introduction to the basic properties of Catalan numbers, the book presents 214 different kinds of objects counted by them in the form of exercises with solutions. The reader can try solving the exercises or simply browse through them. Some 68 additional exercises with prescribed difficulty levels present various properties of Catalan numbers and related numbers, such as Fuss-Catalan numbers, Motzkin numbers, Schroder numbers, Narayana numbers, super Catalan numbers, q-Catalan numbers and (q,t)-Catalan numbers. The book ends with a history of Catalan numbers by Igor Pak and a glossary of key terms. Whether your interest in mathematics is recreation or research, you will find plenty of fascinating and stimulating facts here.
1. Basic properties
2. Bijective exercises
3. Bijective solutions
4. Additional problems
5. Solutions to additional problems
Part of London Mathematical Society Lecture Note Series
Publication planned for: April 2015
format: Paperback
isbn: 9781107492967
There are still many arithmetic mysteries surrounding the values of the Riemann zeta function at the odd positive integers greater than one. For example, the matter of their irrationality, let alone transcendence, remains largely unknown. However, by extending ideas of Garland, Borel proved that these values are related to the higher K-theory of the ring of integers. Shortly afterwards, Bloch and Kato proposed a Tamagawa number-type conjecture for these values, and showed that it would follow from a result in motivic cohomology which was unknown at the time. This vital result from motivic cohomology was subsequently proven by Huber, Kings, and Wildeshaus. Bringing together key results from K-theory, motivic cohomology, and Iwasawa theory, this book is the first to give a complete proof, accessible to graduate students, of the Bloch?Kato conjecture for odd positive integers. It includes a new account of the results from motivic cohomology by Huber and Kings.
List of contributors
Preface A. Raghuram
1. Special values of the Riemann zeta function: some results and conjectures A. Raghuram
2. K-theoretic background R. Sujatha
3. Values of the Riemann zeta function at the odd positive integers and Iwasawa theory John Coates
4. Explicit reciprocity law of Bloch?Kato and exponential maps Anupam Saikia
5. The norm residue theorem and the Quillen?Lichtenbaum conjecture Manfred Kolster
6. Regulators and zeta functions Stephen Lichtenbaum
7. Soule's theorem Stephen Lichtenbaum
8. Soule's regulator map Ralph Greenberg
9. On the determinantal approach to the Tamagawa number conjecture T. Nguyen Quang Do
10. Motivic polylogarithm and related classes Don Blasius
11. The comparison theorem for the Soule?Deligne classes Annette Huber
12. Eisenstein classes, elliptic Soule elements and the ?-adic elliptic polylogarithm Guido Kings
13. Postscript R. Sujatha.
Part of Cambridge Series in Statistical and Probabilistic Mathematics
Publication planned for: May 2015
format: Hardback
isbn: 9781107065178
High-dimensional data appear in many fields, and their analysis has become increasingly important in modern statistics. However, it has long been observed that several well-known methods in multivariate analysis become inefficient, or even misleading, when the data dimension p is larger than, say, several tens. A seminal example is the well-known inefficiency of Hotelling's T2-test in such cases. This example shows that classical large sample limits may no longer hold for high-dimensional data; statisticians must seek new limiting theorems in these instances. Thus, the theory of random matrices (RMT) serves as a much-needed and welcome alternative framework. Based on the authors' own research, this book provides a firsthand introduction to new high-dimensional statistical methods derived from RMT. The book begins with a detailed introduction to useful tools from RMT, and then presents a series of high-dimensional problems with solutions provided by RMT methods.
1. Introduction
2. Limiting spectral distributions
3. CLT for linear spectral statistics
4. The generalised variance and multiple correlation coefficient
5. The T2-statistic
6. Classification of data
7. Testing the general linear hypothesis
8. Testing independence of sets of variates
9. Testing hypotheses of equality of covariance matrices
10. Estimation of the population spectral distribution
11. Large-dimensional spiked population models
12. Efficient optimisation of a large financial portfolio.
Part of London Mathematical Society Lecture Note Series
Publication planned for: July 2015
format: Paperback
isbn: 9781107477391
Reaction-diffusion theory is a topic which has developed rapidly over the last thirty years, particularly with regards to applications in chemistry and life sciences. Of particular importance is the analysis of semi-linear parabolic PDEs. This monograph provides a general approach to the study of semi-linear parabolic equations when the nonlinearity, while failing to be Lipschitz continuous, is Holder and/or upper Lipschitz continuous, a scenario that is not well studied, despite occurring often in models. The text presents new existence, uniqueness and continuous dependence results, leading to global and uniformly global well-posedness results (in the sense of Hadamard). Extensions of classical maximum/minimum principles, comparison theorems and derivative (Schauder-type) estimates are developed and employed. Detailed specific applications are presented in the later stages of the monograph. Requiring only a solid background in real analysis, this book is suitable for researchers in all areas of study involving semi-linear parabolic PDEs.
1. Introduction
2. The bounded reaction-diffusion Cauchy problem
3. Maximum principles
4. Diffusion theory
5. Convolution functions, function spaces, integral equations and equivalence lemmas
6. The bounded reaction-diffusion Cauchy problem with f e L
7. The bounded reaction-diffusion Cauchy problem with f e Lu
8. The bounded reaction-diffusion Cauchy problem with f e La
9. Application to specific problems
10. Concluding remarks.