On-line learning is one of the most powerful and commonly used techniques for training large layered networks and has been used successfully in many real-world applications. Traditional analytical methods have been recently complemented by ones from statistical physics and Bayesian statistics. This powerful combination of analytical methods provides more insight and deeper understanding of existing algorithms and leads to novel and principled proposals for their improvement. This book presents a coherent picture of the state-of-the-art in the theoretical analysis of on-line learning. An introduction relates the subject to other developments in neural networks and explains the overall picture. Surveys by leading experts in the field combine new and established material and enable non-experts to learn more about the techniques and methods used. This book, the first in the area, provides a comprehensive view of the subject and will be welcomed by mathematicians, scientists and engineers, whether in industry or academia.

The representation theory of reductive algebraic groups and related finite reductive groups is a subject of great topical interest and has many applications. The articles in this volume provide introductions to various aspects of the subject, including algebraic groups and Lie algebras, reflection groups, abelian and derived categories, the Deligne-Lusztig representation theory of finite reductive groups, Harish-Chandra theory and its generalisations, quantum groups, subgroup structure of algebraic groups, intersection cohomology, and Lusztig's conjectured character formula for irreducible representations in prime characteristic. The articles are carefully designed to reinforce one another, and are written by a team of distinguished authors: M. Broué, R. W. Carter, S. Donkin, M. Geck, J. C. Jantzen, B. Keller, M. W. Liebeck, G. Malle, J. C. Rickard and R. Rouquier. This volume as a whole should provide a very accessible introduction to an important, though technical, subject.

Certain nonlinear optimization problems arising in such disparate areas as the theory of computation, pure and applied probability and mathematical physics, can be solved by linear methods, provided one replaces the usual number system with one in which addition satisfies the idempotent law. This systematic study of the subject has emerged, triggered in part by a workshop organized by Hewlett-Packard's Basic Research Institute in the Mathematical Sciences (BRIMS), which brought together many leading researchers in the area. This volume is a record of that workshop, but it also includes other invited contributions, a broad Introduction to Idempotency, written specially for the book, and a bibliography of the subject. In sum, the articles cover both practical and more theoretical considerations, making it essential reading for all workers in the area.

Numerical Methods in Finance has emerged as a discipline at the intersection of probability theory, finance and numerical analysis. This book, based on lectures given at the Newton Institute as part of a broader programme, describes a wide variety of numerical methods used in financial analysis: computation of option prices, especially of American option prices, by finite difference and other methods; numerical solution of portfolio management strategies; statistical procedures; identification of models; Monte Carlo methods; and numerical implications of stochastic volatilities. Articles have been written in a pedagogic style and made reasonably self-contained, covering both mathematical matters and practical issues in numerical problems. Thus the book has something to offer economists, probabilists and applied mathematicians working in finance.

The aim of this volume is to present developments in semantics and logics of computation in a way that is accessible to graduate students. The book is based on a summer school at the Isaac Newton Institute and consists of a sequence of linked lecture courses by international authorities in the area. The whole set has been edited to form a coherent introduction to these topics, most of which have not been presented pedagogically before.

Infectious disease accounts for more death and disability globally than either non-infectious disease or injury. This book contains a breadth of different quantitative approaches to understanding the patterns of infectious diseases in populations, and the design of control strategies to lessen their effect. The contributors bring a great variety of mathematical expertise (including deterministic and stochastic modelling and statistical data analysis) and involvement in a wide range of applied fields across the spectrum of biological, medical and social sciences. The aim is to increase interaction between specialities by describing research on many of the infectious diseases that affect humans, including both viral diseases like measles and AIDS and tropical parasitic infections. The papers are divided into groups dealing with problems relating to transmissible diseases, vaccination strategies, the consequences of treatment interventions, the dynamics of immunity, heterogeneity of populations, and prediction.

This is a major synthesis of the theory and empirical knowledge about the ecology and epidemiology of infectious diseases in natural, unmanaged, animal and plant populations. Throughout the book a dialogue is developed between the patterns observed in empirical studies of disease in natural populations and the mathematical models used to dissect and examine the observed epidemiological patterns. The book arose from a symposium at the Newton Institute at Cambridge University. It is divided into a number of reviews by experts in various fields and four group reports: two of these synthesize important issues relating to the dynamics of microparasites and macroparasites, while the others discuss spatial patterns in disease dynamics and the evolutionary biology of parasites, pathogens and their hosts.

Many of the fundamental theories of modern physics can be considered as descriptions of dynamical systems subjected to constraints. The study of these constrained dynamical systems, in particular the problems encountered in formulating them as quantum systems, has many profound links with geometry. These links were explored in the Symposium on Geometry and Gravity, held at the Newton Institute in 1994. This book arose from a conference held during that symposium, and is a collection of papers devoted to problems such as Chern–Simons theory, sigma-models, gauge invariance and loop quantization, general relativity and the notion of time and quantum gravity. They present a lively, varied and topical perspective on this important branch of theoretical physics, from some of the leading authorities in the subject.

Dynamo theory, the study of the generation and maintenance of magnetic fields by fluid motions, is important in many areas of physics, ranging from stellar and galactic dynamics, through solar physics and geomagnetism, to reactor physics. This volume contains the lectures given by leading specialists, for an intensive course held at the Newton Institute, as part of a NATO Advanced Study Institute. The course was intended for beginning graduate students, so starts with background material, before leading on to describe areas closer to current research. Topics covered include planetary and solar dynamos, fast dynamos and the use of symmetry principles to derive evolution equations. Detailed bibliographies are provided. As the only modern introduction to the subject, this will be welcome reading for students in planetary and solar physics, plasma physics and astrophysics.

Dynamo theory is the study of how large-scale magnetic fields can arise in bodies such as the Earth, Sun and stars, and it brings together researchers in such diverse subjects as geophysics, astronomy and nuclear reactor technology. In this book are papers on the theory of dynamos, delivered at the Isaac Newton Institute for Mathematical Sciences in Cambridge in September 1992, and these have been carefully brought together to form a unified presentation. Topics covered include: modelling of solar magnetic field generation, theoretical studies of the galaxies, and papers on 'fast dynamos', study of magnetic field regeneration in fluid of very high electrical conductivity. This volume represents a comprehensive record of research into the theory of dynamos.