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This series is devoted to thorough, yet reasonably concise treatments of topics in any branch of mathematics. Typically, a Tract takes up a single thread in a wide subject, and follows its ramifications, thus throwing light on various of its aspects. Tracts are expected to be rigorous, definitive and of lasting value to mathematicians working in the relevant disciplines. Exercises can be included to illustrate techniques, summarize past work and enhance the book's value as a seminar text.

Random Variables and Probability Distributions

H. Cramer

H. Cramer

Cambridge Tracts in Mathematics (No. 36)

**Print Publication Year:** 1970

**Print ISBN:** 9780521076852

**Online Publication Date:** September 2009

**Online ISBN:** 9780511470936

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511470936

This tract develops the purely mathematical side of the theory of probability, without reference to any applications. When originally published, it was one of the earliest works in the field built on the axiomatic foundations introduced by A. Kolmogoroff in his book Grundbegriffe der Wahrscheinlichkeitsrechnung, thus treating the subject as a branch of the theory of completely additive set functions. The author restricts himself to a consideration of probability distributions in spaces of a finite number of dimensions, and to problems connected with the Central Limit Theorem and some of its generalizations and modifications. In this edition the chapter on Liapounoff's theorem has been partly rewritten, and now includes a proof of the important inequality due to Berry and Esseen. The terminology has been modernized, and several minor changes have been made.

The Lebesgue Integral

J. C. Burkill

J. C. Burkill

Cambridge Tracts in Mathematics (No. 40)

**Print Publication Year:** 1951

**Print ISBN:** 9780521043823

**Online Publication Date:** December 2009

**Online ISBN:** 9780511566127

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511566127

Dr Burkill gives a straightforward introduction to Lebesgue's theory of integration. His approach is the classical one, making use of the concept of measure, and deriving the principal results required for applications of the theory.

Ideal Theory

D. G. Northcott

D. G. Northcott

Cambridge Tracts in Mathematics (No. 42)

**Print Publication Year:** 1953

**Print ISBN:** 9780521058407

**Online Publication Date:** October 2009

**Online ISBN:** 9780511565908

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511565908

Ideal theory is important not only for the intrinsic interest and purity of its logical structure but because it is a necessary tool in many branches of mathematics. In this introduction to the modern theory of ideals, Professor Northcott assumes a sound background of mathematical theory but no previous knowledge of modern algebra. After a discussion of elementary ring theory, he deals with 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.

An Introduction to Homotopy Theory

P. J. Hilton

P. J. Hilton

Cambridge Tracts in Mathematics (No. 43)

**Print Publication Year:** 1953

**Print ISBN:** 9780521052658

**Online Publication Date:** February 2015

**Online ISBN:** 9780511666278

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511666278

Since the introduction of homotopy groups by Hurewicz in 1935, homotopy theory has occupied a prominent place in the development of algebraic topology. This monograph provides an account of the subject which bridges the gap between the fundamental concepts of topology and the more complex treatment to be found in original papers. The first six chapters describe the essential ideas of homotopy theory: homotopy groups, the classical theorems, the exact homotopy sequence, fibre-spaces, the Hopf invariant, and the Freudenthal suspension. The final chapters discuss J. H. C. Whitehead's cell-complexes and their application to homotopy groups of complexes.

Convexity

H. G. Eggleston

H. G. Eggleston

Cambridge Tracts in Mathematics (No. 47)

**Print Publication Year:** 1958

**Print ISBN:** 9780521077347

**Online Publication Date:** March 2010

**Online ISBN:** 9780511566172

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511566172

This account of convexity includes the basic properties of convex sets in Euclidean space and their applications, the theory of convex functions and an outline of the results of transformations and combinations of convex sets. It will be useful for those concerned with the many applications of convexity in economics, the theory of games, the theory of functions, topology, geometry and the theory of numbers.

Asymptotic Expansions

E. T. Copson

E. T. Copson

Cambridge Tracts in Mathematics (No. 55)

**Print Publication Year:** 1965

**Print ISBN:** 9780521047210

**Online Publication Date:** September 2009

**Online ISBN:** 9780511526121

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511526121

Certain functions, capable of expansion only as a divergent series, may nevertheless be calculated with great accuracy by taking the sum of a suitable number of terms. The theory of such asymptotic expansions is of great importance in many branches of pure and applied mathematics and in theoretical physics. Solutions of ordinary differential equations are frequently obtained in the form of a definite integral or contour integral, and this tract is concerned with the asymptotic representation of a function of a real or complex variable defined in this way. After a preliminary account of the properties of asymptotic series, the standard methods of deriving the asymptotic expansion of an integral are explained in detail and illustrated by the expansions of various special functions. These methods include integration by parts, Laplace's approximation, Watson's lemma on Laplace transforms, the method of steepest descents, and the saddle-point method. The last two chapters deal with Airy's integral and uniform asymptotic expansions.

The Theory of Cluster Sets

E. F. Collingwood, A. J. Lohwater

E. F. Collingwood, A. J. Lohwater

Cambridge Tracts in Mathematics (No. 56)

**Print Publication Year:** 1966

**Print ISBN:** 9780521046954

**Online Publication Date:** November 2009

**Online ISBN:** 9780511566134

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511566134

The book provides an introduction to the theory of cluster sets, a branch of topological analysis which has made great strides in recent years. The cluster set of a function at a particular point is the set of limit values of the function at that point which may be either a boundary point or (in the case of a non-analytic function) an interior point of the function's domain. In topological analysis, its main application is to problems arising in the theory of functions of a complex variable, with particular reference to boundary behaviour such as the theory of prime ends under conformal mapping. An important and novel feature of the book is the discussion of more general applications to non-analytic functions, including arbitrary functions. The authors assume a general familiarity with classical function theory but include the more specialised material required for the development of the theory of cluster sets, so making the treatment accessible to graduate students.

Metric Spaces

E. T. Copson

E. T. Copson

Cambridge Tracts in Mathematics (No. 57)

**Print Publication Year:** 1968

**Print ISBN:** 9780521047227

**Online Publication Date:** January 2010

**Online ISBN:** 9780511566141

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511566141

Metric space topology, as the generalization to abstract spaces of the theory of sets of points on a line or in a plane, unifies many branches of classical analysis and is necessary introduction to functional analysis. Professor Copson's book, which is based on lectures given to third-year undergraduates at the University of St Andrews, provides a more leisurely treatment of metric spaces than is found in books on functional analysis, which are usually written at graduate student level. His presentation is aimed at the applications of the theory to classical algebra and analysis; in particular, the chapter on contraction mappings shows how it provides proof of many of the existence theorems in classical analysis.

Proximity Spaces

S. A. Naimpally

S. A. Naimpally

Cambridge Tracts in Mathematics (No. 59)

**Print Publication Year:** 1971

**Print ISBN:** 9780521079358

**Online Publication Date:** May 2010

**Online ISBN:** 9780511569364

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511569364

This tract provides a compact introduction to the theory of proximity spaces and their generalisations, making the subject accessible to readers having a basic knowledge of topological and uniform spaces, such as can be found in standard textbooks. Two chapters are devoted to fundamentals, the main result being the proof of the existence of the Smirnov compactification using clusters. Chapter 3 discusses the interrelationships between proximity spaces and uniform spaces and contains some of the most interesting results in the theory of proximity spaces. The final chapter introduces the reader to several generalised forms of proximity structures and studies one of them in detail. The bibliography contains over 130 references to the scattered research literature on proximity spaces, in addition to general references.

Contiguity of Probability Measures

**Some Applications in Statistics**

George G. Roussas

George G. Roussas

Cambridge Tracts in Mathematics (No. 63)

**Print Publication Year:** 1972

**Print ISBN:** 9780521083546

**Online Publication Date:** February 2011

**Online ISBN:** 9780511804373

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511804373

This Tract presents an elaboration of the notion of 'contiguity', which is a concept of 'nearness' of sequences of probability measures. It provides a powerful mathematical tool for establishing certain theoretical results with applications in statistics, particularly in large sample theory problems, where it simplifies derivations and points the way to important results. The potential of this concept has so far only been touched upon in the existing literature, and this book provides the first systematic discussion of it. Alternative characterizations of contiguity are first described and related to more familiar mathematical ideas of a similar nature. A number of general theorems are formulated and proved. These results, which provide the means of obtaining asymptotic expansions and distributions of likelihood functions, are essential to the applications which follow.

Gibbs States on Countable Sets

Christopher J. Preston

Christopher J. Preston

Cambridge Tracts in Mathematics (No. 68)

**Print Publication Year:** 1974

**Print ISBN:** 9780521203753

**Online Publication Date:** October 2011

**Online ISBN:** 9780511897122

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511897122

The book is an introduction to some of the 1967–1974 results and techniques in classical lattice statistical mechanics. It is written in the language of probability theory rather than that of physics, and is thus aimed primarily at mathematicians who might have little or no background in physics. This area of statistical mechanics is presently enjoying a rapid growth and the book should allow a graduate student or research mathematician to find out what is happening in it. The book is self-contained except for some basic concepts of probability theory, and can be read by any undergraduate student in mathematics who has a reasonable background in probability.

Simple Noetherian Rings

John Cozzens, CArl Faith

John Cozzens, CArl Faith

Cambridge Tracts in Mathematics (No. 69)

**Print Publication Year:** 1975

**Print ISBN:** 9780521207348

**Online Publication Date:** October 2009

**Online ISBN:** 9780511565700

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511565700

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. The text is as elementary and self-contained as practicable, and the little background required in homological and categorical algebra is given in a short appendix. Full definitions are given and short, complete, elementary proofs are provided for such key theorems as the Morita theorem, the Correspondence theorem, the Wedderburn–Artin theorem, the Goldie–Lesieur–Croisot theorem, and many others. Complex mathematical machinery has been eliminated wherever possible or its introduction into the text delayed as long as possible. (Even tensor products are not required until Chapter 3.)

Finite Free Resolutions

D. G. Northcott

D. G. Northcott

Cambridge Tracts in Mathematics (No. 71)

**Print Publication Year:** 1976

**Print ISBN:** 9780521211550

**Online Publication Date:** December 2009

**Online ISBN:** 9780511565892

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511565892

An important part of homological algebra deals with modules possessing projective resolutions of finite length. This goes back to Hilbert's famous theorem on syzygies through, in the earlier theory, free modules with finite bases were used rather than projective modules. The introduction of a wider class of resolutions led to a theory rich in results, but in the process certain special properties of finite free resolutions were overlooked. D. A. Buchsbaum and D. Eisenbud have shown that finite free resolutions have a fascinating structure theory. This has revived interest in the simpler kind of resolution and caused the subject to develop rapidly. This Cambridge Tract attempts to give a genuinely self-contained and elementary presentation of the basic theory, and to provide a sound foundation for further study. The text contains a substantial number of exercises. These enable the reader to test his understanding and they allow the subject to be developed more rapidly. Each chapter ends with the solutions to the exercises contained in it.

Completeness and Basis Properties of Sets of Special Functions

J. R. Higgins

J. R. Higgins

Cambridge Tracts in Mathematics (No. 72)

**Print Publication Year:** 1977

**Print ISBN:** 9780521213769

**Online Publication Date:** December 2009

**Online ISBN:** 9780511566189

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511566189

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 first chapter contains the theoretical background to the subject, largely in a general Hilbert space setting, and theorems in which the structure of Hilbert space is revealed by properties of its bases are dealt with. Later parts of the book deal with methods: for example, the Vitali criterion, together with its generalisations and applications, is discussed in some detail, and there is an introduction to the theory of stability of bases. The last chapter deals with complete sets as eigenfunctions of differential and a table of a wide variety of bases and complete sets of special functions. Dr Higgins' account will be useful to graduate students of mathematics and professional mathematicians, especially Banach 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.

Approaches to the Theory of Optimization

J. P. Ponstein

J. P. Ponstein

Cambridge Tracts in Mathematics (No. 77)

**Print Publication Year:** 1980

**Print ISBN:** 9780521231558

**Online Publication Date:** October 2009

**Online ISBN:** 9780511526527

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511526527

Optimization is concerned with finding the best (optimal) solution to mathematical problems that may arise in economics, engineering, the social sciences and the mathematical sciences. As is suggested by its title, this book surveys various ways of penetrating the subject. The author begins with a selection of the type of problem to which optimization can be applied and the remainder of the book develops the theory, mainly from the viewpoint of mathematical programming. To prevent the treatment becoming too abstract, subjects which may be considered 'unpractical' are not touched upon. The author gives plausible reasons, without forsaking rigor, to show how the subject develops 'naturally'. Professor Ponstein has provided a concise account of optimization which should be readily accessible to anyone with a basic understanding of topology and functional analysis. Advanced students and professionals concerned with operations research, optimal control and mathematical programming will welcome this useful and interesting book.

Chain Conditions in Topology

W. W. Comfort, S. Negrepontis

W. W. Comfort, S. Negrepontis

Cambridge Tracts in Mathematics (No. 79)

**Print Publication Year:** 1982

**Print ISBN:** 9780521234870

**Online Publication Date:** November 2011

**Online ISBN:** 9780511897337

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511897337

A chain condition is a property, typically involving considerations of cardinality, of the family of open subsets of a topological space. (Sample questions: (a) How large a fmily of pairwise disjoint open sets does the space admit? (b) From an uncountable family of open sets, can one always extract an uncountable subfamily with the finite intersection property. This monograph, which is partly fresh research and partly expository (in the sense that the authors co-ordinate and unify disparate results obtained in several different countries over a period of several decades) is devoted to the systematic use of infinitary combinatorial methods in topology to obtain results concerning chain conditions. The combinatorial tools developed by P. Erdös and the Hungarian school, by Erdös and Rado in the 1960s and by the Soviet mathematician Shanin in the 1940s, are adequate to handle many natural questions concerning chain conditions in product spaces.

Module Categories of Analytic Groups

Andy R. Magid

Andy R. Magid

Cambridge Tracts in Mathematics (No. 81)

**Print Publication Year:** 1982

**Print ISBN:** 9780521242004

**Online Publication Date:** October 2011

**Online ISBN:** 9780511897177

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511897177

Knowledge of an analytic group implies knowledge of its module category. However complete knowledge of the category does not determine the group. Professor Magid shows here that the category determines another, larger group and an algebra of functions in this new group. The new group and its function algebra are completely described; this description thus tells everything that is known when the module category, as a category, is given. This categorical view brings together and highlights the significance of earlier work in this area by several authors, as well as yielding new results. 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.

Polycyclic Groups

Daniel Segal

Daniel Segal

Cambridge Tracts in Mathematics (No. 82)

**Print Publication Year:** 1983

**Print ISBN:** 9780521241465

**Online Publication Date:** December 2009

**Online ISBN:** 9780511565953

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511565953

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. As well as providing a connected and self-contained account of the group-theoretical background, it explains in detail how deep methods of number theory and algebraic group theory have been used to achieve some very recent and rather spectacular advances in the subject. Up to now, most of this material has only been available in scattered research journals, and some of it is new. This book is the only unified account of these developments, and will be of interest to mathematicians doing research in algebra, and to postgraduate students studying that subject.

General Irreducible Markov Chains and Non-Negative Operators

**For the Economic & Social Sciences**

Esa Nummelin

Esa Nummelin

Cambridge Tracts in Mathematics (No. 83)

**Print Publication Year:** 1984

**Print ISBN:** 9780521250054

**Online Publication Date:** September 2009

**Online ISBN:** 9780511526237

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511526237

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. The technique of embedded renewal processes, common in the study of discrete Markov chains, plays a particularly important role. The examples discussed indicate applications to such topics as queueing theory, storage theory, autoregressive processes and renewal theory. The book will therefore be useful to researchers in the theory and applications of Markov chains. It could also be used as a graduate-level textbook for courses on Markov chains or aspects of operator theory.

Consequences of Martin's Axiom

D. H. Fremlin

D. H. Fremlin

Cambridge Tracts in Mathematics (No. 84)

**Print Publication Year:** 1984

**Print ISBN:** 9780521250917

**Online Publication Date:** October 2011

**Online ISBN:** 9780511896972

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511896972

'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 1applications m set theory, infinitary combinatorics, general topology, measure theory, functional analysis and group theory. In this book Dr Fremlin has sought to collect together as many of these applications as possible into one rational scheme, with proofs of the principal results. His aim is to show how straightforward and beautiful arguments can be used to derive a great many consistency results from the consistency of Martin's axiom.

The Geometry of Fractal Sets

K. J. Falconer

K. J. Falconer

Cambridge Tracts in Mathematics (No. 85)

**Print Publication Year:** 1985

**Print ISBN:** 9780521256940

**Online Publication Date:** January 2010

**Online ISBN:** 9780511623738

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623738

This book contains a rigorous mathematical treatment of the geometrical aspects of sets of both integral and fractional Hausdorff dimension. Questions of local density and the existence of tangents of such sets are studied, as well as the dimensional properties of their projections in various directions. In the case of sets of integral dimension the dramatic differences between regular 'curve-like' sets and irregular 'dust like' sets are exhibited. The theory is related by duality to Kayeka sets (sets of zero area containing lines in every direction). The final chapter includes diverse examples of sets to which the general theory is applicable: discussions of curves of fractional dimension, self-similar sets, strange attractors, and examples from number theory, convexity and so on. There is an emphasis on the basic tools of the subject such as the Vitali covering lemma, net measures and Fourier transform methods.

Fredholm Theory in Banach Spaces

Anthony Francis Ruston

Anthony Francis Ruston

Cambridge Tracts in Mathematics (No. 86)

**Print Publication Year:** 1986

**Print ISBN:** 9780521248464

**Online Publication Date:** October 2009

**Online ISBN:** 9780511569180

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511569180

In this tract, Dr Ruston presents analogues for operators on Banach spaces of Fredholm's solution of integral equations of the second kind. Much of the presentation is based on research carried out over the last twenty-five years and has never appeared in book form before. Dr Ruston begins with the construction for operators of finite rank, using Fredholm's original method as a guide. He then considers formulae that have structure similar to those obtained by Fredholm, using, and developing further, the relationship with Riesz theory. In particular, he obtains bases for the finite-dimensional subspaces figuring in the Riesz theory. Finally he returns to the study of specific constructions for various classes of operators. Dr Ruston has made every effort to keep the presentation as elementary as possible, using arguments that do not require a very advanced background. Thus the book can be read with profit by graduate students as well as specialists working in the general area of functional analysis and its applications.

Exponential Diophantine Equations

T. N. Shorey, R. Tijdeman

T. N. Shorey, R. Tijdeman

Cambridge Tracts in Mathematics (No. 87)

**Print Publication Year:** 1986

**Print ISBN:** 9780521268264

**Online Publication Date:** May 2010

**Online ISBN:** 9780511566042

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511566042

This is a integrated presentation of the theory of exponential diophantine equations. The authors present, in a clear and unified fashion, applications to exponential diophantine equations and linear recurrence sequences of the Gelfond-Baker theory of linear forms in logarithms of algebraic numbers. Topics covered include the Thue equations, the generalised hyperelliptic equation, and the Fermat and Catalan equations. The necessary preliminaries are given in the first three chapters. Each chapter ends with a section giving details of related results.

Multiple Forcing

T. Jech

T. Jech

Cambridge Tracts in Mathematics (No. 88)

**Print Publication Year:** 1987

**Print ISBN:** 9780521266598

**Online Publication Date:** May 2010

**Online ISBN:** 9780511721168

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511721168

In this 1987 text Professor Jech gives a unified treatment of the various forcing methods used in set theory, and presents their important applications. Product forcing, iterated forcing and proper forcing have proved powerful tools when studying the foundations of mathematics, for instance in consistency proofs. The book is based on graduate courses though some results are also included, making the book attractive to set theorists and logicians.

Irregularities of Distribution

Jozsef Beck, William W. L. Chen

Jozsef Beck, William W. L. Chen

Cambridge Tracts in Mathematics (No. 89)

**Print Publication Year:** 1987

**Print ISBN:** 9780521307925

**Online Publication Date:** January 2010

**Online ISBN:** 9780511565984

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511565984

This book is an authoritative description of the various approaches to and methods in the theory of irregularities of distribution. The subject is primarily concerned with number theory, but also borders on combinatorics and probability theory. The work is in three parts. The first is concerned with the classical problem, complemented where appropriate with more recent results. In the second part, the authors study generalizations of the classical problem, pioneered by Schmidt. Here, they include chapters on the integral equation method of Schmidt and the more recent Fourier transform technique. The final part is devoted to Roth's '1/4-theorem'.

Divisors

Richard R. Hall, Gérald Tenenbaum

Richard R. Hall, Gérald Tenenbaum

Cambridge Tracts in Mathematics (No. 90)

**Print Publication Year:** 1988

**Print ISBN:** 9780521340564

**Online Publication Date:** March 2010

**Online ISBN:** 9780511566004

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511566004

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. Hardy and Ramanujan initiated this area of research and it was developed by Erdös in the thirties. His work led to some deep and basic conjectures of wide application which have now essentially been settled. This book contains detailed proofs, some of which have never appeared in print before, of those conjectures that are concerned with the propinquity of divisors. Consequently it will be essential reading for all researchers in analytic number theory.

Fibrewise Topology

I. M. James

I. M. James

Cambridge Tracts in Mathematics (No. 91)

**Print Publication Year:** 1989

**Print ISBN:** 9780521360906

**Online Publication Date:** November 2011

**Online ISBN:** 9780511896835

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511896835

The aim of this book is to promote a fibrewise perspective, particularly in topology, which is central to modern mathematics. 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. There are strong links with equivariant topology, a topic which has latterly been subject to great research activity. It is to be hoped that this book will provide a solid and invigorating foundation for the increasing research interest in fibrewise topology

Heat Kernels and Spectral Theory

E. B. Davies

E. B. Davies

Cambridge Tracts in Mathematics (No. 92)

**Print Publication Year:** 1989

**Print ISBN:** 9780521361361

**Online Publication Date:** October 2009

**Online ISBN:** 9780511566158

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511566158

An advanced monograph on a central topic in the theory of differential equations, Heat Kernels and Spectral Theory investigates the theory of second-order elliptic operators. While the study of the heat equation is a classical subject, this book analyses the improvements in our quantitative understanding of heat kernels. The author considers variable coefficient operators on regions in Euclidean space and Laplace-Beltrami operators on complete Riemannian manifolds. He also includes results pertaining to the heat kernels of Schrödinger operators; such results will be of particular interest to mathematical physicists, and relevant too to those concerned with properties of Brownian motion and other diffusion processes.

On L1-Approximation

Allan M. Pinkus

Allan M. Pinkus

Cambridge Tracts in Mathematics (No. 93)

**Print Publication Year:** 1989

**Print ISBN:** 9780521366502

**Online Publication Date:** January 2010

**Online ISBN:** 9780511526497

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511526497

This monograph discusses the qualitative linear theory of best L^T1-approximation from finite-dimensional subspaces. It presents a survey of recent research that extends "classical" results concerned with best-uniform approximation to the more general case. The work is organized to serve as a self-study guide or as a text for advanced courses. It begins with a basic introduction to the concepts of approximation theory before addressing 1- or 2-sided best approximations from finite-dimensional subspaces and approaches to the computation of these. At the end of each chapter is a series of exercises that give the reader an opportunity to test understanding and also contain some theoretical digressions and extensions of the text.

The Volume of Convex Bodies and Banach Space Geometry

Gilles Pisier

Gilles Pisier

Cambridge Tracts in Mathematics (No. 94)

**Print Publication Year:** 1989

**Print ISBN:** 9780521364652

**Online Publication Date:** May 2010

**Online ISBN:** 9780511662454

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511662454

Now in paperback, this popular book gives a self-contained presentation of a number of recent results, which relate the volume of convex bodies in n-dimensional Euclidean space and the geometry of the corresponding finite-dimensional normed spaces. The methods employ classical ideas from the theory of convex sets, probability theory, approximation theory, and the local theory of Banach spaces. The first part of the book presents self-contained proofs of the quotient of the subspace theorem, the inverse Santalo inequality and the inverse Brunn-Minkowski inequality. In the second part Pisier gives a detailed exposition of the recently introduced classes of Banach spaces of weak cotype 2 or weak type 2, and the intersection of the classes (weak Hilbert space). This text will be a superb choice for courses in analysis and probability theory.

Nonlinear Superposition Operators

Jürgen Appell, Petr P. Zabrejko

Jürgen Appell, Petr P. Zabrejko

Cambridge Tracts in Mathematics (No. 95)

**Print Publication Year:** 1990

**Print ISBN:** 9780521361026

**Online Publication Date:** February 2012

**Online ISBN:** 9780511897450

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511897450

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. The purpose of this book is to collect the basic facts about the superposition operator, to present the main ideas which are useful in studying its properties and to provide a comparison of its behaviour in different function spaces. Some applications are also considered, for example to control theory and optimization. Much of the work here has only appeared before in research literature, which itself is catalogued in detail here.

Algebraic Curves over Finite Fields

Carlos Moreno

Carlos Moreno

Cambridge Tracts in Mathematics (No. 97)

**Print Publication Year:** 1991

**Print ISBN:** 9780521342520

**Online Publication Date:** August 2012

**Online ISBN:** 9780511608766

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511608766

In this tract, Professor Moreno develops the theory of algebraic curves over finite fields, their zeta and L-functions, and, for the first time, the theory of algebraic geometric Goppa codes on algebraic curves. Among the applications considered are: the problem of counting the number of solutions of equations over finite fields; Bombieri's proof of the Reimann hypothesis for function fields, with consequences for the estimation of exponential sums in one variable; Goppa's theory of error-correcting codes constructed from linear systems on algebraic curves; there is also a new proof of the TsfasmanSHVladutSHZink theorem. The prerequisites needed to follow this book are few, and it can be used for graduate courses for mathematics students. Electrical engineers who need to understand the modern developments in the theory of error-correcting codes will also benefit from studying this work.

Entropy, Compactness and the Approximation of Operators

Bernd Carl, Irmtraud Stephani

Bernd Carl, Irmtraud Stephani

Cambridge Tracts in Mathematics (No. 98)

**Print Publication Year:** 1990

**Print ISBN:** 9780521330114

**Online Publication Date:** February 2012

**Online ISBN:** 9780511897467

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511897467

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, e.g. approximability and eigenvalue sequences, of such operators. The authors present many generalized results, some of which have not appeared in the literature before. In the final chapter, the authors demonstrate that, to a certain extent, the geometry of Banach spaces can also be developed on the basis of operator theory. All mathematicians working in functional analysis and operator theory will welcome this work as a reference or for advanced graduate courses.

Some Applications of Modular Forms

Peter Sarnak

Peter Sarnak

Cambridge Tracts in Mathematics (No. 99)

**Print Publication Year:** 1990

**Print ISBN:** 9780521402453

**Online Publication Date:** February 2012

**Online ISBN:** 9780511895593

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511895593

The theory of modular forms and especially the so-called Ramanujan Conjectures have recently been applied to resolve problems in combinatorics, computer science, analysis, and number theory. Professor Sarnak begins by developing the necessary background material in modular forms. He then considers in detail the solution of three problems: the Rusiewisz problem concerning finitely additive rotationally invariant measures on the sphere; the explicit construction of highly connected but sparse graphs, e.g. expander graphs and Ramanujan graphs; and the Linnik problem concerning the distribution of integers that represent a given large integer as a sum of three squares.

Analysis and Geometry on Groups

Nicholas T. Varopoulos, L. Saloff-Coste, T. Coulhon

Nicholas T. Varopoulos, L. Saloff-Coste, T. Coulhon

Cambridge Tracts in Mathematics (No. 100)

**Print Publication Year:** 1993

**Print ISBN:** 9780521353823

**Online Publication Date:** April 2010

**Online ISBN:** 9780511662485

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511662485

The geometry and analysis that is discussed in this book extends to classical results for general discrete or Lie groups, and the methods used are analytical, but are not concerned with what is described these days as real analysis. Most of the results described in this book have a dual formulation: they have a "discrete version" related to a finitely generated discrete group and a continuous version related to a Lie group. The authors chose to center this book around Lie groups, but could easily have pushed it in several other directions as it interacts with the theory of second order partial differential operators, and probability theory, as well as with group theory.

Infinite Electrical Networks

Armen H. Zemanian

Armen H. Zemanian

Cambridge Tracts in Mathematics (No. 101)

**Print Publication Year:** 1991

**Print ISBN:** 9780521401531

**Online Publication Date:** February 2012

**Online ISBN:** 9780511895432

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511895432

Over the past two decades a general mathematical theory of infinite electrical networks has been developed. This is the first book to present the salient features of this theory in a coherent exposition. Using the basic tools of functional analysis and graph theory, the author presents the fundamental developments of the past two decades and discusses applications to other areas of mathematics. The first half of the book presents existence and uniqueness theorems for both infinite-power and finite-power voltage-current regimes, and the second half discusses methods for solving problems in infinite cascades and grids. A notable feature is the recent invention of transfinite networks, roughly analogous to Cantor's extension of the natural numbers to the transfinite ordinals. The last chapter is a survey of applications to exterior problems of partial differential equations, random walks on infinite graphs, and networks of operators on Hilbert spaces. The jump in complexity from finite electrical networks to infinite ones is comparable to the jump in complexity from finite-dimensional to infinite-dimensional spaces. Many of the questions that are conventionally asked about finite networks are presently unanswerable for infinite networks, while questions that are meaningless for finite networks crop up for infinite ones and lead to surprising results, such as the occasional collapse of Kirchoff's laws in infinite regimes. Some central concepts have no counterpart in the finite case, as for example the extremities of an infinite network, the perceptibility of infinity, and the connections at infinity.

Designs and their Codes

E. F. Assmus, J. D. Key

E. F. Assmus, J. D. Key

Cambridge Tracts in Mathematics (No. 103)

**Print Publication Year:** 1992

**Print ISBN:** 9780521413619

**Online Publication Date:** October 2015

**Online ISBN:** 9781316529836

**Book DOI:** http://dx.doi.org/10.1017/CBO9781316529836

Algebraic coding theory has in recent years been increasingly applied to the study of combinatorial designs. This book gives an account of many of those applications together with a thorough general introduction to both design theory and coding theory developing the relationship between the two areas. The first half of the book contains background material in design theory, including symmetric designs and designs from affine and projective geometry, and in coding theory, coverage of most of the important classes of linear codes. In particular, the authors provide a new treatment of the Reed-Muller and generalized Reed-Muller codes. The last three chapters treat the applications of coding theory to some important classes of designs, namely finite planes, Hadamard designs and Steiner systems, in particular the Witt systems.

Sporadic Groups

Michael Aschbacher

Michael Aschbacher

Cambridge Tracts in Mathematics (No. 104)

**Print Publication Year:** 1994

**Print ISBN:** 9780521420495

**Online Publication Date:** August 2010

**Online ISBN:** 9780511665585

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511665585

Sporadic Groups provides for the first time a self-contained treatment of the foundations of the theory of sporadic groups accessible to mathematicians with a basic background in finite groups, such as in the author's text Finite Group Theory. Introductory material useful for studying the sporadics, such as a discussion of large extraspecial 2-subgroups and Tits' coset geometries, opens the book. A construction of the Mathieu groups as the automorphism groups of Steiner systems follows. The Golay and Todd modules and the 2-local geometry for M24 are discussed. This is followed by the standard construction of Conway of the Leech lattice and the Conway group. The Monster is constructed as the automorphism group of the Griess algebra using some of the best features of the approaches of Griess, Conway, and Tits plus a few new wrinkles. The existence treatment finishes with an application of the theory of large extraspecial subgroups to produce the twenty sporadics involved in the Monster. The Aschbacher-Segev approach addresses the uniqueness of the sporadics via coverings of graphs and simplicial complexes. The basics of this approach are developed and used to establish the uniqueness of five of the sporadics.

Fourier Integrals in Classical Analysis

Christopher D. Sogge

Christopher D. Sogge

Cambridge Tracts in Mathematics (No. 105)

**Print Publication Year:** 1993

**Print ISBN:** 9780521434645

**Online Publication Date:** September 2009

**Online ISBN:** 9780511530029

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511530029

Fourier Integrals in Classical Analysis is an advanced treatment of central problems in harmonic analysis. The main theme of the book is the interplay between ideas used to study the propagation of singularities for the wave equation and their counterparts in classical analysis. Using microlocal analysis, the author in particular studies problems involving maximal functions and Riesz means using the so-called half-wave operator. This self-contained book starts with a rapid review of important topics in Fourier analysis. The author then presents the necessary tools from microlocal analysis, and goes on to give a proof of the sharp Weyl formula which he then modifies to give sharp estimates for the size of eigenfunctions on compact manifolds. Finally, the tools that have been developed are used to study the regularity properties of Fourier integral operators, culminating in the proof of local smoothing estimates and their applications to singular maximal theorems in two and more dimensions.

Arithmetic of Quadratic Forms

Yoshiyuki Kitaoka

Yoshiyuki Kitaoka

Cambridge Tracts in Mathematics (No. 106)

**Print Publication Year:** 1993

**Print ISBN:** 9780521404754

**Online Publication Date:** March 2010

**Online ISBN:** 9780511666155

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511666155

This book provides an introduction to quadratic forms, building from basics to the most recent results. Professor Kitaoka is well known for his work in this area, and in this book he covers many aspects of the subject, including lattice theory, Siegel's formula, and some results involving tensor products of positive definite quadratic forms. The reader should have a knowledge of algebraic number fields, making this book ideal for graduate students and researchers wishing for an insight into quadratic forms.

Duality and Perturbation Methods in Critical Point Theory

N. Ghoussoub

N. Ghoussoub

Cambridge Tracts in Mathematics (No. 107)

**Print Publication Year:** 1993

**Print ISBN:** 9780521440257

**Online Publication Date:** December 2009

**Online ISBN:** 9780511551703

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511551703

Building on min-max methods, Professor Ghoussoub systematically develops a general theory that can be applied in a variety of situations. In so doing he also presents a whole new array of duality and perturbation methods. The prerequisites for following this book are relatively few; an appendix sketching certain methods in analysis makes the book self-contained.

Multivalent Functions

**Second edition**

W. K. Hayman

W. K. Hayman

Cambridge Tracts in Mathematics (No. 110)

**Print Publication Year:** 1994

**Print ISBN:** 9780521460262

**Online Publication Date:** November 2009

**Online ISBN:** 9780511526268

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511526268

The class of multivalent functions is an important one in complex analysis. They occur for example in the proof of De Branges' Theorem, which in 1985 settled the long-standing Bieberbach conjecture. The second edition of Professor Hayman's celebrated book contains a full and self-contained proof of this result, with a new chapter devoted to it. Another new chapter deals with coefficient differences. The text has been updated in several other ways, with recent theorems of Baernstein and Pommerenke on univalent functions of restricted growth, and an account of the theory of mean p-valent functions. In addition, many of the original proofs have been simplified. Each chapter contains examples and exercises of varying degrees of difficulty designed both to test understanding and illustrate the material.

Schur Algebras and Representation Theory

Stuart Martin

Stuart Martin

Cambridge Tracts in Mathematics (No. 112)

**Print Publication Year:** 1994

**Print ISBN:** 9780521415910

**Online Publication Date:** September 2009

**Online ISBN:** 9780511470899

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511470899

Schur algebras are an algebraic system that provide a link between the representation theory of the symmetric and general linear groups. Dr. Martin gives a self-contained account of this algebra and those links, covering the basic ideas and their quantum analogues. He discusses not only the usual representation-theoretic topics (such as constructions of irreducible modules, the structure of blocks containing them, decomposition numbers and so on) but also the intrinsic properties of Schur algebras, leading to a discussion of their cohomology theory. He also investigates the relationship between Schur algebras and other algebraic structures. Throughout, the approach uses combinatorial language where possible, thereby making the presentation accessible to graduate students. Some topics require results from algebraic group theory, which are contained in an appendix.

Spectral Decomposition and Eisenstein Series

**A Paraphrase of the Scriptures**

C. Moeglin, J. L. Waldspurger, Translated by Leila Schneps

C. Moeglin, J. L. Waldspurger, Translated by Leila Schneps

Cambridge Tracts in Mathematics (No. 113)

**Print Publication Year:** 1995

**Print ISBN:** 9780521418935

**Online Publication Date:** September 2009

**Online ISBN:** 9780511470905

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511470905

The decomposition of the space L2 (G(Q)\G(A)), where G is a reductive group defined over Q and A is the ring of adeles of Q, is a deep problem at the intersection of number and group theory. Langlands reduced this decomposition to that of the (smaller) spaces of cuspidal automorphic forms for certain subgroups of G. This book describes this proof in detail. The starting point is the theory of automorphic forms, which can also serve as a first step toward understanding the Arthur-Selberg trace formula. To make the book reasonably self-contained, the authors also provide essential background in subjects such as: automorphic forms; Eisenstein series; Eisenstein pseudo-series, and their properties. It is thus also an introduction, suitable for graduate students, to the theory of automorphic forms, the first written using contemporary terminology.

Introduction to Hp Spaces

**Second edition**

Paul Koosis

Paul Koosis

Cambridge Tracts in Mathematics (No. 115)

**Print Publication Year:** 1999

**Print ISBN:** 9780521455213

**Online Publication Date:** September 2009

**Online ISBN:** 9780511470950

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511470950

The first edition of this well-known book was noted for its clear and accessible exposition of the basic theory of Hardy spaces from the concrete point of view (in the unit circle and the half plane). This second edition retains many of the features found in the first--detailed computation, an emphasis on methods--but greatly extends its coverage. The discussions of conformal mapping now include Lindelöf's second theorem and the one due to Kellogg. A simple derivation of the atomic decomposition for RH1 is given, and then used to provide an alternative proof of Fefferman's duality theorem. Two appendices by V.P. Havin have also been added: on Peter Jones' interpolation formula for RH1 and on Havin's own proof of the weak sequential completeness of L1/H1(0). Numerous other additions, emendations and corrections have been made throughout.

Matrices of Sign-Solvable Linear Systems

Richard A. Brualdi, Bryan L. Shader

Richard A. Brualdi, Bryan L. Shader

Cambridge Tracts in Mathematics (No. 116)

**Print Publication Year:** 1995

**Print ISBN:** 9780521482967

**Online Publication Date:** February 2010

**Online ISBN:** 9780511574733

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511574733

In a sign-solvable linear system, the signs of the coefficients determine the signs of some entries in the solution. This type of system is part of a larger study that helps researchers understand if properties of a matrix can be determined from combinatorial arrangements of its elements. In this book, the authors present the diffuse body of literature on sign-solvability as a coherent whole for the first time, giving many new results and proofs and establishing many new connections. Brualdi and Shader describe and comment on algorithms implicit in many of the proofs and their complexity. The book is self-contained, assuming familiarity only with elementary linear algebra and graph theory. Intended primarily for researchers in combinatorics and linear algebra, it should also be of interest to computer scientists, economists, physicists, chemists, and engineers.

Generalized Topological Degree and Semilinear Equations

Wolodymyr V. Petryshyn

Wolodymyr V. Petryshyn

Cambridge Tracts in Mathematics (No. 117)

**Print Publication Year:** 1995

**Print ISBN:** 9780521444743

**Online Publication Date:** May 2010

**Online ISBN:** 9780511574832

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511574832

This book describes many new results and extensions of the theory of generalized topological degree for densely defined A-proper operators and presents important applications, particularly to boundary value problems of nonlinear ordinary and partial differential equations that are intractable under any other existing theory. A-proper mappings arise naturally in the solution to an equation in infinite dimensional space via the finite dimensional approximation. The theory subsumes classical theory involving compact vector fields as well as the more recent theories of condensing vector-fields, strongly monotone, and strongly accretive maps. Researchers and graduate students in mathematics, applied mathematics, and physics who make use of nonlinear analysis will find this an important resource for new techniques.

Sets of Multiples

Richard R. Hall

Richard R. Hall

Cambridge Tracts in Mathematics (No. 118)

**Print Publication Year:** 1996

**Print ISBN:** 9780521404242

**Online Publication Date:** January 2010

**Online ISBN:** 9780511566011

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511566011

The theory of sets of multiples, a subject that lies at the intersection of analytic and probabilistic number theory, has seen much development since the publication of "Sequences" by Halberstam and Roth nearly thirty years ago. The area is rich in problems, many of them still unsolved or arising from current work. In this book, the author gives a coherent, self-contained account of the existing theory, bringing the reader to the frontiers of research. One of the fascinations of the theory is the variety of methods applicable to it, which include Fourier analysis, group theory, high and ultra-low moments, probability and elementary inequalities, and several branches of number theory.

Continuum Percolation

Ronald Meester, Rahul Roy

Ronald Meester, Rahul Roy

Cambridge Tracts in Mathematics (No. 119)

**Print Publication Year:** 1996

**Print ISBN:** 9780521475044

**Online Publication Date:** November 2011

**Online ISBN:** 9780511895357

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511895357

Many phenomena in physics, chemistry, and biology can be modeled by spatial random processes. One such process is continuum percolation, which is used when the phenomenon being modeled is made up of individual events that overlap e.g., individual raindrops that eventually make the ground evenly wet. This is a systematic, rigorous account of continuum percolation. The authors treat two models, the Boolean model and the random connection model, in detail, and they discuss related continuum models. Meester and Roy explain all important techniques and methods and apply them to obtain results on the existence of phase transitions, equality and continuity of critical densities, compressions, rarefaction, and other aspects of continuum models.

Function Spaces, Entropy Numbers, Differential Operators

D. E. Edmunds, H. Triebel

D. E. Edmunds, H. Triebel

Cambridge Tracts in Mathematics (No. 120)

**Print Publication Year:** 1996

**Print ISBN:** 9780521560368

**Online Publication Date:** July 2010

**Online ISBN:** 9780511662201

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511662201

The distribution of the eigenvalues of differential operators has long fascinated mathematicians. Recent advances have shed new light on classical problems in this area, and this book presents a fresh approach, largely based on the results of the authors. The emphasis here is on a topic of central importance in analysis, namely the relationship between i) function spaces on Euclidean n-space and on domains; ii) entropy numbers in quasi-Banach spaces; and iii) the distribution of the eigenvalues of degenerate elliptic (pseudo) differential operators. The treatment is largely self-contained and accessible to nonspecialists.

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