Books in the Cambridge Tracts in Mathematics series

"A self-contained text providing all the basics of fractional Sobolev spaces. The book provides the background and relevant theory with explanations, alternatives and comparisons, and includes discussion of the Hardy and Rellich inequalities, first in classical format and then in fractional versions. Ideal for researchers and graduate students"--

Entropy quantities are connected with the 'degree of compactness' of compact or precompact spaces, and so are appropriate tools for investigating linear and compact operators between Banach spaces. The main intention of this Tract is to study the relations between compactness and other analytical properties.

This 1999 book is concerned with Diophantine approximation on smooth manifolds embedded in Euclidean space. In particular, this book deals with Khintchine-type theorems and with the Hausdorff dimension of the associated null sets. All researchers with an interest in Diophantine approximation will welcome this book.

In this introduction to the modern theory of ideals, Professor Northcott first discusses the properties of Noetherian rings and the algebraic and analytical theories of local rings. In order to give some idea of deeper applications of this theory the author has woven into the connected algebraic theory those results which play outstanding roles in the geometric applications.

This work specifically surveys simple Noetherian rings. The authors present theorems on the structure of simple right Noetherian rings and, more generally, on simple rings containing a uniform right ideal U.

This tract presents an exposition of methods for testing sets of special functions for completeness and basis properties, mostly in L2 and L2 spaces. The emphasis on methods of testing and their applications will also interest scientists and engineers engaged in fields such as the sampling theory of signals in electrical engineering and boundary value problems in mathematical physics.

The purpose of this 1982 book is to present an introduction to developments which had taken place in finite group theory related to finite geometries. This book is practically self-contained and readers are assumed to have only an elementary knowledge of linear algebra.

Since the 1930s ergodic theory has been central to pure mathematics. This introduction provides sections on the classical ergodic theorems, topological dynamics, uniform distribution, Martingales, information theory and entropy. There is a chapter on mixing and one on special examples.

A chain condition is a property, typically involving considerations of cardinality, of the family of open subsets of a topological space.

The theory of polycyclic groups is a branch of infinite group theory which has a rather different flavour from the rest of that subject. This book is a comprehensive account of the present state of this theory.

By including many examples and computations Professor Magid has written a complete account of the subject that is accessible to a wide audience. Graduate students and professionals who have some knowledge of algebraic groups, Lie groups and Lie algebras will find this a useful and interesting text.

In this tract Dr Ruston presents analogues for operators on Banach spaces of Fredholm's solution of integral equations of the second kind. Every effort has been made to keep the presentation as elementary as possible, using arguments that do not require a very advanced background, for those working in the general area of functional analysis and its applications.

The purpose of this book is to present the theory of general irreducible Markov chains and to point out the connection between this and the Perron-Frobenius theory of nonnegative operators. The author begins by providing some basic material designed to make the book self-contained, yet his principal aim throughout is to emphasize recent developments.

'Martin's axiom' is one of the most fruitful axioms which have been devised to show that certain properties are insoluble in standard set theory. It has important applications m set theory, infinitary combinatorics, general topology, measure theory, functional analysis and group theory.

This is a integrated presentation of the theory of exponential diophantine equations.

This book is an authoritative description of the various approaches to and methods in the theory of irregularities of distribution.

Shmuel Weinberger describes here analogies between geometric topology, differential geometry, group theory, global analysis, and noncommutative geometry. He develops deep tools in a setting where they have immediate application. The connections between these fields enrich each and shed light on one another.

This is a systematic account of the multiplicative structure of integers, from the probabilistic point of view. The authors are especially concerned with the distribution of the divisors, which is as fundamental and important as the additive structure of the integers, and yet until now has hardly been discussed outside of the research literature.

The aim of this book is to promote a fibrewise perspective, particularly in topology. Already this view is standard in the theory of fibre bundles and therefore in such subjects as global analysis. It has a role to play also in general and equivariant topology.

This book is a self-contained account of knowledge of the theory of nonlinear superposition operators: a generalization of the notion of functions. The theory developed here is applicable to operators in a wide variety of function spaces, and it is here that the modern theory diverges from classical nonlinear analysis.

Heat Kernels and Spectral Theory investigates the theory of second-order elliptic operators.

This book provides an introduction to quadratic forms that builds from basics up to recent results covering many aspects including lattice theory, Siegel's formula and results involving tensor products of positive definite quadratic forms. It is ideal for graduate students and researchers wishing for an insight into quadratic forms.

Totally positive matrices constitute a particular class of matrices, the study of which was initiated by analysts because of its many applications in diverse areas. This account of the subject is comprehensive and thorough, with careful treatment of the central properties of totally positive matrices, full proofs and a complete bibliography. The history of the subject is also described: in particular, the book ends with a tribute to the four people who have made the most notable contributions to the history of total positivity: I. J. Schoenberg, M. G. Krein, F. R. Gantmacher and S. Karlin. This monograph will appeal to those with an interest in matrix theory, to those who use or have used total positivity, and to anyone who wishes to learn about this rich and interesting subject.

This elegant book is certain to become the standard introduction to synthetic differential geometry. It deals with some classical spaces in differential geometry, namely 'prolongation spaces' or neighbourhoods of the diagonal. The clear presentation makes this tract ideal for graduate students and researchers wishing to familiarize themselves with the field.

This book provides the first comprehensive introduction to the circle of ideas developed around Mori's program, the prerequisites being only a basic knowledge of the algebraic geometry. It will be of great interest to graduate students and researchers working in algebraic geometry and related fields.

This tract provides an introduction to four finite geometrical systems and to the theory of projective planes. Of the four geometries, one is based on a nine-element field and the other three can be constructed from the nine-element 'miniquaternion algebra', a simple system which has many though not all the properties of a field.

Poincare duality algebras originated in the work of topologists on the cohomology of closed manifolds, and Macaulay's dual systems in the study of irreducible ideals in polynomial algebras. These two ideas are tied together using basic commutative algebra involving Gorenstein algebras. Steenrod operations also originated in algebraic topology, but may best be viewed as a means of encoding the information often hidden behind the Frobenius map in characteristic p0. They provide a noncommutative tool to study commutative algebras over a Galois field. In this Tract the authors skilfully bring together these ideas and apply them to problems in invariant theory. A number of remarkable and unexpected interdisciplinary connections are revealed that will interest researchers in the areas of commutative algebra, invariant theory or algebraic topology.

Ridge functions are a rich class of simple multivariate functions which have found applications in a variety of areas. These include partial differential equations (where they are sometimes termed 'plane waves'), computerised tomography, projection pursuit in the analysis of large multivariate data sets, the MLP model in neural networks, Waring's problem over linear forms, and approximation theory. Ridge Functions is the first book devoted to studying them as entities in and of themselves. The author describes their central properties and provides a solid theoretical foundation for researchers working in areas such as approximation or data science. He also includes an extensive bibliography and discusses some of the unresolved questions that may set the course for future research in the field.

This monograph is concerned with the qualitative theory of best L1-approximation from finite-dimensional subspaces. It presents a survey of recent research that extends 'classical' results concerned with best uniform approximation to the L1 case. The work is organized in such a way as to be useful for self-study or as a text for advanced courses.

This book, first published in 2001, is a detailed exposition, in a single volume, of both the theory and applications of torsors to rational points. It is demonstrated that torsors provide a unified approach to several branches of the theory which were hitherto developing in parallel.