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The Cambridge Studies in Advanced Mathematics is a series of books each of which aims to introduce the reader to an active area of mathematical research. All topics in pure mathematics are covered, and treatments are suitable for graduate students, and experts from other branches of mathematics, seeking access to research topics.

A Course in Finite Group Representation Theory

Peter Webb

Peter Webb

Cambridge Studies in Advanced Mathematics

**Print Publication Year:** 2016

**Print ISBN:** 9781107162396

**Online Publication Date:** August 2016

**Online ISBN:** 9781316677216

**Book DOI:** http://dx.doi.org/10.1017/CBO9781316677216

This graduate-level text provides a thorough grounding in the representation theory of finite groups over fields and rings. The book provides a balanced and comprehensive account of the subject, detailing the methods needed to analyze representations that arise in many areas of mathematics. Key topics include the construction and use of character tables, the role of induction and restriction, projective and simple modules for group algebras, indecomposable representations, Brauer characters, and block theory. This classroom-tested text provides motivation through a large number of worked examples, with exercises at the end of each chapter that test the reader's knowledge, provide further examples and practice, and include results not proven in the text. Prerequisites include a graduate course in abstract algebra, and familiarity with the properties of groups, rings, field extensions, and linear algebra.

Period Mappings and Period Domains

**edition**

James Carlson, Stefan Müller-Stach, Chris Peters

James Carlson, Stefan Müller-Stach, Chris Peters

Cambridge Studies in Advanced Mathematics

**Print Publication Year:** 2017

**Print ISBN:** 9781108422628

**Online Publication Date:** August 2017

**Online ISBN:** 9781316995846

**Book DOI:** http://dx.doi.org/10.1017/9781316995846

This up-to-date introduction to Griffiths' theory of period maps and period domains focusses on algebraic, group-theoretic and differential geometric aspects. Starting with an explanation of Griffiths' basic theory, the authors go on to introduce spectral sequences and Koszul complexes that are used to derive results about cycles on higher-dimensional algebraic varieties such as the Noether–Lefschetz theorem and Nori's theorem. They explain differential geometric methods, leading up to proofs of Arakelov-type theorems, the theorem of the fixed part and the rigidity theorem. They also use Higgs bundles and harmonic maps to prove the striking result that not all compact quotients of period domains are Kähler. This thoroughly revised second edition includes a new third part covering important recent developments, in which the group-theoretic approach to Hodge structures is explained, leading to Mumford–Tate groups and their associated domains, the Mumford–Tate varieties and generalizations of Shimura varieties.

Algebraic Automata Theory

M. Holcombe

M. Holcombe

Cambridge Studies in Advanced Mathematics (No. 1)

**Print Publication Year:** 1982

**Print ISBN:** 9780521231961

**Online Publication Date:** September 2009

**Online ISBN:** 9780511525889

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511525889

This is a self-contained, modern treatment of the algebraic theory of machines. Dr Holcombe examines various applications of the idea of a machine in biology, biochemistry and computer science and gives also a rigorous treatment of the way in which these machines can be decomposed and simulated by simpler ones. This treatment is based on fundamental ideas from modern algebra. Motivation for many of the newer results is provided by way of applications so this account should be accessible and valuable for those studying applied algebra or theoretical computer science at advanced undergraduate or beginning postgraduate level, as well as for those undertaking research in those areas.

Ergodic Theory

Karl E. Petersen

Karl E. Petersen

Cambridge Studies in Advanced Mathematics (No. 2)

**Print Publication Year:** 1983

**Print ISBN:** 9780521236324

**Online Publication Date:** June 2012

**Online ISBN:** 9780511608728

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511608728

The author presents the fundamentals of the ergodic theory of point transformations and several advanced topics of intense research. The study of dynamical systems forms a vast and rapidly developing field even when considering only activity whose methods derive mainly from measure theory and functional analysis. Each of the basic aspects of ergodic theory--examples, convergence theorems, recurrence properties, and entropy--receives a basic and a specialized treatment. The author's accessible style and the profusion of exercises, references, summaries, and historical remarks make this a useful book for graduate students or self study.

Ultrametric Calculus

**An Introduction to p-Adic Analysis**

W. H. Schikhof

W. H. Schikhof

Cambridge Studies in Advanced Mathematics (No. 4)

**Print Publication Year:** 1985

**Print ISBN:** 9780521242349

**Online Publication Date:** January 2010

**Online ISBN:** 9780511623844

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623844

This is an introduction to p-adic analysis which is elementary yet complete and which displays the variety of applications of the subject. Dr Schikhof is able to point out and explain how p-adic and 'real' analysis differ. This approach guarantees the reader quickly becomes acquainted with this equally 'real' analysis and appreciates its relevance. The reader's understanding is enhanced and deepened by the large number of exercises included throughout; these both test the reader's grasp and extend the text in interesting directions. As a consequence, this book will become a standard reference for professionals (especially in p-adic analysis, number theory and algebraic geometry) and will be welcomed as a textbook for advanced students of mathematics familiar with algebra and analysis.

Commutative Ring Theory

H. Matsumura, Translated by Miles Reid

H. Matsumura, Translated by Miles Reid

Cambridge Studies in Advanced Mathematics (No. 8)

**Print Publication Year:** 1987

**Print ISBN:** 9780521259163

**Online Publication Date:** June 2012

**Online ISBN:** 9781139171762

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139171762

In addition to being an interesting and profound subject in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytical geometry. Matsumura covers the basic material, including dimension theory, depth, Cohen-Macaulay rings, Gorenstein rings, Krull rings and valuation rings. More advanced topics such as Ratliff's theorems on chains of prime ideals are also explored. The work is essentially self-contained, the only prerequisite being a sound knowledge of modern algebra, yet the reader is taken to the frontiers of the subject. Exercises are provided at the end of each section and solutions or hints to some of them are given at the end of the book.

Characteristic Classes and the Cohomology of Finite Groups

C. B. Thomas

C. B. Thomas

Cambridge Studies in Advanced Mathematics (No. 9)

**Print Publication Year:** 1987

**Print ISBN:** 9780521256612

**Online Publication Date:** November 2011

**Online ISBN:** 9780511897344

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511897344

The purpose of this book is to study the relation between the representation ring of a finite group and its integral cohomology by means of characteristic classes. In this way it is possible to extend the known calculations and prove some general results for the integral cohomology ring of a group G of prime power order. Among the groups considered are those of p-rank less than 3, extra-special p-groups, symmetric groups and linear groups over finite fields. An important tool is the Riemann - Roch formula which provides a relation between the characteristic classes of an induced representation, the classes of the underlying representation and those of the permutation representation of the infinite symmetric group. Dr Thomas also discusses the implications of his work for some arithmetic groups which will interest algebraic number theorists. Dr Thomas assumes the reader has taken basic courses in algebraic topology, group theory and homological algebra, but has included an appendix in which he gives a purely topological proof of the Riemann - Roch formula.

Finite Group Theory

**Second edition**

M. Aschbacher

M. Aschbacher

Cambridge Studies in Advanced Mathematics (No. 10)

**Print Publication Year:** 2000

**Print ISBN:** 9780521781459

**Online Publication Date:** June 2012

**Online ISBN:** 9781139175319

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139175319

This second edition develops the foundations of finite group theory. For students already exposed to a first course in algebra, it serves as a text for a course on finite groups. For the reader with some mathematical sophistication but limited knowledge of finite group theory, the book supplies the basic background necessary to begin to read journal articles in the field. It also provides the specialist in finite group theory with a reference on the foundations of the subject. Unifying themes include the Classification Theorem and the classical linear groups. Lie theory appears in chapters on Coxeter groups, root systems, buildings, and Tits systems. This second edition has been considerably improved with a completely rewritten Chapter 15 considering the 2-Signalizer Functor Theorem, and the addition of an appendix containing solutions to exercises.

Local Representation Theory

**Modular Representations as an Introduction to the Local Representation Theory of Finite Groups**

J. L. Alperin

J. L. Alperin

Cambridge Studies in Advanced Mathematics (No. 11)

**Print Publication Year:** 1986

**Print ISBN:** 9780521306607

**Online Publication Date:** January 2010

**Online ISBN:** 9780511623592

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623592

Representation theory has applications to number theory, combinatorics and many areas of algebra. The aim of this text is to present some of the key results in the representation theory of finite groups. Professor Alperin concentrates on local representation theory, emphasizing module theory throughout. In this way many deep results can be obtained rather quickly. After two introductory chapters, the basic results of Green are proved, which in turn lead in due course to Brauer's First Main Theorem. A proof of the module form of Brauer's Second Main Theorem is then presented, followed by a discussion of Feit's work connecting maps and the Green correspondence. The work concludes with a treatment, new in part, of the Brauer-Dade theory. Exercises are provided at the end of most sections; the results of some are used later in the text.

The Logarithmic Integral

**Volume 1**

Paul Koosis

Paul Koosis

Cambridge Studies in Advanced Mathematics (No. 12)

**Print Publication Year:** 1988

**Print ISBN:** 9780521309066

**Online Publication Date:** January 2010

**Online ISBN:** 9780511566196

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511566196

The theme of this unique work, the logarithmic integral, lies athwart much of twentieth century analysis. It is a thread connecting many apparently separate parts of the subject, and is a natural point at which to begin a serious study of real and complex analysis. Professor Koosis' aim is to show how, from simple ideas, one can build up an investigation that explains and clarifies many different, seemingly unrelated problems; to show, in effect, how mathematics grows. The presentation is straightforward, so this, the first of two volumes, is self-contained, but more importantly, by following the theme, Professor Koosis has produced a work that can be read as a whole. He has brought together here many results, some new and unpublished, making this a key reference for graduate students and researchers.

An Introduction to the Theory of the Riemann Zeta-Function

S. J. Patterson

S. J. Patterson

Cambridge Studies in Advanced Mathematics (No. 14)

**Print Publication Year:** 1988

**Print ISBN:** 9780521335355

**Online Publication Date:** August 2012

**Online ISBN:** 9780511623707

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623707

This is a modern introduction to the analytic techniques used in the investigation of zeta-function. Riemann introduced this function in connection with his study of prime numbers, and from this has developed the subject of analytic number theory. Since then, many other classes of "zeta-function" have been introduced and they are now some of the most intensively studied objects in number theory. Professor Patterson has emphasized central ideas of broad application, avoiding technical results and the customary function-theoretic approach.

Algebraic Homotopy

Hans Joachim Baues

Hans Joachim Baues

Cambridge Studies in Advanced Mathematics (No. 15)

**Print Publication Year:** 1989

**Print ISBN:** 9780521333764

**Online Publication Date:** March 2010

**Online ISBN:** 9780511662522

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511662522

This book gives a general outlook on homotopy theory; fundamental concepts, such as homotopy groups and spectral sequences, are developed from a few axioms and are thus available in a broad variety of contexts. Many examples and applications in topology and algebra are discussed, including an introduction to rational homotopy theory in terms of both differential Lie algebras and De Rham algebras. The author describes powerful tools for homotopy classification problems, particularly for the classification of homotopy types and for the computation of the group homotopy equivalences. Applications and examples of such computations are given, including when the fundamental group is non-trivial. Moreover, the deep connection between the homotopy classification problems and the cohomology theory of small categories is demonstrated. The prerequisites of the book are few: elementary topology and algebra. Consequently, this account will be valuable for non-specialists and experts alike. It is an important supplement to the standard presentations of algebraic topology, homotopy theory, category theory and homological algebra.

Cellular Structures in Topology

Rudolf Fritsch, Renzo Piccinini

Rudolf Fritsch, Renzo Piccinini

Cambridge Studies in Advanced Mathematics (No. 19)

**Print Publication Year:** 1990

**Print ISBN:** 9780521327848

**Online Publication Date:** November 2011

**Online ISBN:** 9780511983948

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511983948

This book describes the construction and the properties of CW-complexes. These spaces are important because firstly they are the correct framework for homotopy theory, and secondly most spaces that arise in pure mathematics are of this type. The authors discuss the foundations and also developments, for example, the theory of finite CW-complexes, CW-complexes in relation to the theory of fibrations, and Milnor's work on spaces of the type of CW-complexes. They establish very clearly the relationship between CW-complexes and the theory of simplicial complexes, which is developed in great detail. Exercises are provided throughout the book; some are straightforward, others extend the text in a non-trivial way. For the latter; further reference is given for their solution. Each chapter ends with a section sketching the historical development. An appendix gives basic results from topology, homology and homotopy theory. These features will aid graduate students, who can use the work as a course text. As a contemporary reference work it will be essential reading for the more specialized workers in algebraic topology and homotopy theory.

Introductory Lectures on Siegel Modular Forms

Helmut Klingen

Helmut Klingen

Cambridge Studies in Advanced Mathematics (No. 20)

**Print Publication Year:** 1990

**Print ISBN:** 9780521350525

**Online Publication Date:** December 2009

**Online ISBN:** 9780511619878

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511619878

This volume aims to present a straightforward and easily accessible survey of the analytic theory of quadratic forms. Written at an elementary level, the book provides a sound basis from which the reader can study advanced works and undertake original research. Roughly half a century ago C.L. Siegel discovered a new type of automorphic forms in several variables in connection with his famous work on the analytic theory of quadratic forms. Since then Siegel modular forms have been studied extensively because of their significance in both automorphic functions in several complex variables and number theory. The comprehensive theory of automorphic forms to subgroups of algebraic groups and the recent arithmetical theory of modular forms illustrate these two aspects in an illuminating manner. The text is based on the author's lectures given over a number of years and is intended for a one semester graduate course, although it can serve equally well for self study . The only prerequisites are a knowledge of algebra, number theory and complex analysis.

The Logarithmic Integral

**Volume 2**

Paul Koosis

Paul Koosis

Cambridge Studies in Advanced Mathematics (No. 21)

**Print Publication Year:** 1992

**Print ISBN:** 9780521309073

**Online Publication Date:** January 2010

**Online ISBN:** 9780511566202

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511566202

The theme of this unique work, the logarithmic integral, is found throughout much of twentieth century analysis. It is a thread connecting many apparently separate parts of the subject, and so is a natural point at which to begin a serious study of real and complex analysis. The author's aim is to show how, from simple ideas, one can build up an investigation that explains and clarifies many different, seemingly unrelated problems; to show, in effect, how mathematics grows.

Banach Spaces for Analysts

P. Wojtaszczyk

P. Wojtaszczyk

Cambridge Studies in Advanced Mathematics (No. 25)

**Print Publication Year:** 1991

**Print ISBN:** 9780521356183

**Online Publication Date:** May 2010

**Online ISBN:** 9780511608735

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511608735

This is an introduction to modern Banach space theory, in which applications to other areas such as harmonic analysis, function theory, orthogonal series, and approximation theory are also given prominence. The author begins with a discussion of weak topologies, weak compactness, and isomorphisms of Banach spaces before proceeding to the more detailed study of particular spaces. The book is intended to be used with graduate courses in Banach space theory, so the prerequisites are a background in functional, complex, and real analysis. As the only introduction to the modern theory of Banach spaces, it will be an essential companion for professional mathematicians working in the subject, or to those interested in applying it to other areas of analysis.

Clifford Algebras and Dirac Operators in Harmonic Analysis

J. Gilbert, M. Murray

J. Gilbert, M. Murray

Cambridge Studies in Advanced Mathematics (No. 26)

**Print Publication Year:** 1991

**Print ISBN:** 9780521346542

**Online Publication Date:** November 2009

**Online ISBN:** 9780511611582

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511611582

The aim of this book is to unite the seemingly disparate topics of Clifford algebras, analysis on manifolds, and harmonic analysis. The authors show how algebra, geometry, and differential equations play a more fundamental role in Euclidean Fourier analysis. They then link their presentation of the Euclidean theory naturally to the representation theory of semi-simple Lie groups.

Algebraic Number Theory

A. Fröhlich, M. J. Taylor

A. Fröhlich, M. J. Taylor

Cambridge Studies in Advanced Mathematics (No. 27)

**Print Publication Year:** 1991

**Print ISBN:** 9780521366649

**Online Publication Date:** June 2012

**Online ISBN:** 9781139172165

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139172165

This book provides a brisk, thorough treatment of the foundations of algebraic number theory on which it builds to introduce more advanced topics. Throughout, the authors emphasize the systematic development of techniques for the explicit calculation of the basic invariants such as rings of integers, class groups, and units, combining at each stage theory with explicit computations.

Topics in Metric Fixed Point Theory

Kazimierz Goebel, W. A. Kirk

Kazimierz Goebel, W. A. Kirk

Cambridge Studies in Advanced Mathematics (No. 28)

**Print Publication Year:** 1990

**Print ISBN:** 9780521382892

**Online Publication Date:** October 2009

**Online ISBN:** 9780511526152

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511526152

Metric fixed point theory has proved a flourishing area of research for the past twenty-five years. This book offers the mathematical community an accessible, self-contained document that can be used as an introduction to the subject and its development. It will be understandable to a wide audience, including nonspecialists and provides a source for examples, references and new approaches for those currently working in the subject.

Reflection Groups and Coxeter Groups

James E. Humphreys

James E. Humphreys

Cambridge Studies in Advanced Mathematics (No. 29)

**Print Publication Year:** 1990

**Print ISBN:** 9780521375108

**Online Publication Date:** June 2012

**Online ISBN:** 9780511623646

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623646

In this graduate textbook Professor Humphreys presents a concrete and up-to-date introduction to the theory of Coxeter groups. He assumes that the reader has a good knowledge of algebra, but otherwise the book is self contained. The first part is devoted to establishing concrete examples; the author begins by developing the most important facts about finite reflection groups and related geometry, and showing that such groups have a Coxeter representation. In the next chapter these groups are classified by Coxeter diagrams, and actual realizations of these groups are discussed. Chapter 3 discusses the polynomial invariants of finite reflection groups, and the first part ends with a description of the affine Weyl groups and the way they arise in Lie theory. The second part (which is logically independent of, but motivated by, the first) starts by developing the properties of the Coxeter groups. Chapter 6 shows how earlier examples and others fit into the general classification of Coxeter diagrams. Chapter 7 is based on the very important work of Kazhdan and Lusztig and the last chapter presents a number of miscellaneous topics of a combinatorial nature.

Representations and Cohomology

**Volume 1**
**, Basic Representation Theory of Finite Groups and Associative Algebras**

D. J. Benson

D. J. Benson

Cambridge Studies in Advanced Mathematics (No. 30)

**Print Publication Year:** 1991

**Print ISBN:** 9780521361347

**Online Publication Date:** January 2010

**Online ISBN:** 9780511623615

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623615

This is the first of two volumes providing an introduction to modern developments in the representation theory of finite groups and associative algebras, which have transformed the subject into a study of categories of modules. Thus, Dr. Benson's unique perspective in this book incorporates homological algebra and the theory of representations of finite-dimensional algebras. This volume is primarily concerned with the exposition of the necessary background material, and the heart of the discussion is a lengthy introduction to the (Auslander-Reiten) representation theory of finite dimensional algebras, in which the techniques of quivers with relations and almost-split sequences are discussed in some detail.

Representations and Cohomology

**Volume 2**
**, Cohomology of Groups and Modules**

D. J. Benson

D. J. Benson

Cambridge Studies in Advanced Mathematics (No. 31)

**Print Publication Year:** 1991

**Print ISBN:** 9780521361354

**Online Publication Date:** February 2010

**Online ISBN:** 9780511623622

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623622

The heart of the book is a lengthy introduction to the representation theory of finite dimensional algebras, in which the techniques of quivers with relations and almost split sequences are discussed in some detail.

Cohomological Methods in Transformation Groups

Christopher Allday, Volker Puppe

Christopher Allday, Volker Puppe

Cambridge Studies in Advanced Mathematics (No. 32)

**Print Publication Year:** 1993

**Print ISBN:** 9780521350228

**Online Publication Date:** December 2009

**Online ISBN:** 9780511526275

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511526275

In the large and thriving field of compact transformation groups an important role has long been played by cohomological methods. This book aims to give a contemporary account of such methods, in particular the applications of ordinary cohomology theory and rational homotopy theory with principal emphasis on actions of tori and elementary abelian p-groups on finite-dimensional spaces. For example, spectral sequences are not used in Chapter 1, where the approach is by means of cochain complexes; and much of the basic theory of cochain complexes needed for this chapter is outlined in an appendix. For simplicity, emphasis is put on G-CW-complexes; the refinements needed to treat more general finite-dimensional (or finitistic) G-spaces are often discussed separately. Subsequent chapters give systematic treatments of the Localization Theorem, applications of rational homotopy theory, equivariant Tate cohomology and actions on Poincaré duality spaces. Many shorter and more specialized topics are included also. Chapter 2 contains a summary of the main definitions and results from Sullivan's version of rational homotopy theory which are used in the book.

Lectures on Arakelov Geometry

C. Soulé, D. Abramovich, J. F. Burnol, J. K. Kramer

C. Soulé, D. Abramovich, J. F. Burnol, J. K. Kramer

Cambridge Studies in Advanced Mathematics (No. 33)

**Print Publication Year:** 1992

**Print ISBN:** 9780521416696

**Online Publication Date:** January 2010

**Online ISBN:** 9780511623950

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623950

Arakelov theory is a new geometric approach to diophantine equations. It combines algebraic geometry, in the sense of Grothendieck, with refined analytic tools such as currents on complex manifolds and the spectrum of Laplace operators. It has been used by Faltings and Vojta in their proofs of outstanding conjectures in diophantine geometry. This account presents the work of Gillet and Soulé, extending Arakelov geometry to higher dimensions. It includes a proof of Serre's conjecture on intersection multiplicities and an arithmetic Riemann-Roch theorem. To aid number theorists, background material on differential geometry is described, but techniques from algebra and analysis are covered as well. Several open problems and research themes are also mentioned.

Representation Theory of Artin Algebras

Maurice Auslander, Idun Reiten, Sverre O. Smalo

Maurice Auslander, Idun Reiten, Sverre O. Smalo

Cambridge Studies in Advanced Mathematics (No. 36)

**Print Publication Year:** 1995

**Print ISBN:** 9780521411349

**Online Publication Date:** May 2010

**Online ISBN:** 9780511623608

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623608

This book serves as a comprehensive introduction to the representation theory of Artin algebras, a branch of algebra. Written by three distinguished mathematicians, it illustrates how the theory of almost split sequences is utilized within representation theory. The authors develop several foundational aspects of the subject. For example, the representations of quivers with relations and their interpretation as modules over the factors of path algebras is discussed in detail. Thorough discussions yield concrete illustrations of some of the more abstract concepts and theorems. The book includes complete proofs of all theorems and numerous exercises. It is an invaluable resource for graduate students and researchers.

Wavelets and Operators

**Volume 1**

Yves Meyer, Translated by D. H. Salinger

Yves Meyer, Translated by D. H. Salinger

Cambridge Studies in Advanced Mathematics (No. 37)

**Print Publication Year:** 1993

**Print ISBN:** 9780521420006

**Online Publication Date:** December 2009

**Online ISBN:** 9780511623820

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623820

Over the last two years, wavelet methods have shown themselves to be of considerable use to harmonic analysts and, in particular, advances have been made concerning their applications. The strength of wavelet methods lies in their ability to describe local phenomena more accurately than a traditional expansion in sines and cosines can. Thus, wavelets are ideal in many fields where an approach to transient behaviour is needed, for example, in considering acoustic or seismic signals, or in image processing. Yves Meyer stands the theory of wavelets firmly upon solid ground by basing his book on the fundamental work of Calderón, Zygmund and their collaborators. For anyone who would like an introduction to wavelets, this book will prove to be a necessary purchase.

An Introduction to Homological Algebra

Charles A. Weibel

Charles A. Weibel

Cambridge Studies in Advanced Mathematics (No. 38)

**Print Publication Year:** 1994

**Print ISBN:** 9780521435000

**Online Publication Date:** March 2013

**Online ISBN:** 9781139644136

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139644136

The landscape of homological algebra has evolved over the past half-century into a fundamental tool for the working mathematician. This book provides a unified account of homological algebra as it exists today. The historical connection with topology, regular local rings, and semi-simple Lie algebras is also described. The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences. Homology of group and Lie algebras illustrate these topics. Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras. The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit: simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors.

Cohen-Macaulay Rings

**Second edition**

Winfried Bruns, H. Jürgen Herzog

Winfried Bruns, H. Jürgen Herzog

Cambridge Studies in Advanced Mathematics (No. 39)

**Print Publication Year:** 1998

**Print ISBN:** 9780521566742

**Online Publication Date:** December 2009

**Online ISBN:** 9780511608681

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511608681

In the past two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the subject. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. The general theory is applied to a number of examples and the connections with combinatorics are highlighted. Throughout each chapter, the authors have supplied many examples and exercises.

Explicit Brauer Induction

**With Applications to Algebra and Number Theory**

Victor P. Snaith

Victor P. Snaith

Cambridge Studies in Advanced Mathematics (No. 40)

**Print Publication Year:** 1994

**Print ISBN:** 9780521460156

**Online Publication Date:** January 2010

**Online ISBN:** 9780511600746

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511600746

Explicit Brauer Induction is a new and important technique in algebra, discovered by the author in 1986. It solves an old problem, giving a canonical formula for Brauer's induction theorem. In this book it is derived algebraically, following a method of R. Boltje--thereby making the technique, previously topological, accessible to algebraists. Once developed, the technique is used, by way of illustration, to reprove some important known results in new ways and to settle some outstanding problems. As with Brauer's original result, the canonical formula can be expected to have numerous applications and this book is designed to introduce research algebraists to its possibilities. For example, the technique gives an improved construction of the Oliver-Taylor group-ring logarithm, which enables the author to study more effectively algebraic and number-theoretic questions connected with class-groups of rings.

Cohomology of Drinfeld Modular Varieties

**Part 1**
**, Geometry, Counting of Points and Local Harmonic Analysis**

Gérard Laumon

Gérard Laumon

Cambridge Studies in Advanced Mathematics (No. 41)

**Print Publication Year:** 1995

**Print ISBN:** 9780521470605

**Online Publication Date:** May 2010

**Online ISBN:** 9780511666162

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511666162

Cohomology of Drinfeld Modular Varieties aims to provide an introduction to both the subject of the title and the Langlands correspondence for function fields. These varieties are the analogs for function fields of Shimura varieties over number fields. This present volume is devoted to the geometry of these varieties and to the local harmonic analysis needed to compute their cohomology. To keep the presentation as accessible as possible, the author considers the simpler case of function rather than number fields; nevertheless, many important features can still be illustrated. It will be welcomed by workers in number theory and representation theory.

Spectral Theory and Differential Operators

E. Brian Davies

E. Brian Davies

Cambridge Studies in Advanced Mathematics (No. 42)

**Print Publication Year:** 1995

**Print ISBN:** 9780521472500

**Online Publication Date:** January 2010

**Online ISBN:** 9780511623721

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623721

In this book, Davies introduces the reader to the theory of partial differential operators, up to the spectral theorem for bounded linear operators on Banach spaces. He also describes the theory of Fourier transforms and distributions as far as is needed to analyze the spectrum of any constant coefficient partial differential operator. He also presents a completely new proof of the spectral theorem for unbounded self-adjoint operators and demonstrates its application to a variety of second order elliptic differential operators. Finally, the book contains a detailed account of the application of variational methods to estimate the eigenvalues of operators with measurable coefficients defined by the use of quadratic form techniques. Illustrated with many examples, it is well-suited to graduate-level work.

Absolutely Summing Operators

Joe Diestel, Hans Jarchow, Andrew Tonge

Joe Diestel, Hans Jarchow, Andrew Tonge

Cambridge Studies in Advanced Mathematics (No. 43)

**Print Publication Year:** 1995

**Print ISBN:** 9780521431682

**Online Publication Date:** October 2009

**Online ISBN:** 9780511526138

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511526138

We can best understand many fundamental processes in analysis by studying and comparing the summability of series in various modes of convergence. This text provides the reader with basic knowledge of real and functional analysis, with an account of p-summing and related operators. The account is panoramic, with detailed expositions of the core results and highly relevant applications to harmonic analysis, probability and measure theory, and operator theory. This is the first time that the subject and its applications have been presented in such complete detail in book form. Graduate students and researchers in real, complex and functional analysis, and probability theory will benefit from this text.

Geometry of Sets and Measures in Euclidean Spaces

**Fractals and Rectifiability**

Pertti Mattila

Pertti Mattila

Cambridge Studies in Advanced Mathematics (No. 44)

**Print Publication Year:** 1995

**Print ISBN:** 9780521465762

**Online Publication Date:** August 2012

**Online ISBN:** 9780511623813

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623813

The focus of this book is geometric properties of general sets and measures in Euclidean spaces. Applications of this theory include fractal-type objects, such as strange attractors for dynamical systems, and those fractals used as models in the sciences. The author provides a firm and unified foundation for the subject and develops all the main tools used in its study, such as covering theorems, Hausdorff measures and their relations to Riesz capacities and Fourier transforms. The last third of the book is devoted to the Besicovitch-Federer theory of rectifiable sets, which form in a sense the largest class of subsets of Euclidean space possessing many of the properties of smooth surfaces.

Positive Harmonic Functions and Diffusion

Ross G. Pinsky

Ross G. Pinsky

Cambridge Studies in Advanced Mathematics (No. 45)

**Print Publication Year:** 1995

**Print ISBN:** 9780521470148

**Online Publication Date:** December 2009

**Online ISBN:** 9780511526244

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511526244

In this book, Professor Pinsky gives a self-contained account of the construction and basic properties of diffusion processes, including both analytic and probabilistic techniques. He starts with a rigorous treatment of the spectral theory of elliptic operators with nice coefficients on smooth, bounded domains, and then develops the theory of the generalized principal eigenvalue and the related criticality theory for elliptic operators on arbitrary domains. He considers Martin boundary theory and calculates the Martin boundary for several classes of operators. The book provides an array of criteria for determining whether a diffusion process is transient or recurrent. Also introduced are the theory of bounded harmonic functions, and Brownian motion on a manifold. Many results that form the folklore of the subject are given a rigorous exposition, making this book a useful reference for the specialist, and an excellent guide for the graduate student.

Enumerative Combinatorics

**Volume 1**
**, Second edition**

Richard P. Stanley

Richard P. Stanley

Cambridge Studies in Advanced Mathematics (No. 49)

**Print Publication Year:** 2011

**Print ISBN:** 9781107015425

**Online Publication Date:** June 2012

**Online ISBN:** 9781139058520

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139058520

Richard Stanley's two-volume basic introduction to enumerative combinatorics has become the standard guide to the topic for students and experts alike. This thoroughly revised second edition of Volume 1 includes ten new sections and more than 300 new exercises, most with solutions, reflecting numerous new developments since the publication of the first edition in 1986. The material in Volume 1 was chosen to cover those parts of enumerative combinatorics of greatest applicability and with the most important connections with other areas of mathematics. The four chapters are devoted to an introduction to enumeration (suitable for advanced undergraduates), sieve methods, partially ordered sets, and rational generating functions. Much of the material is related to generating functions, a fundamental tool in enumerative combinatorics. In this new edition, the author brings the coverage up to date and includes a wide variety of additional applications and examples, as well as updated and expanded chapter bibliographies. Many of the less difficult new exercises have no solutions so that they can more easily be assigned to students. The material on P-partitions has been rearranged and generalized; the treatment of permutation statistics has been greatly enlarged; and there are also new sections on q-analogues of permutations, hyperplane arrangements, the cd-index, promotion and evacuation, and differential posets.

Enumerative Combinatorics

**Volume 1**

Richard P. Stanley, Foreword by Gian-Carlo Rota

Richard P. Stanley, Foreword by Gian-Carlo Rota

Cambridge Studies in Advanced Mathematics (No. 49)

**Print Publication Year:** 1997

**Print ISBN:** 9780521553094

**Online Publication Date:** June 2012

**Online ISBN:** 9780511805967

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511805967

This book, the first of a two-volume basic introduction to enumerative combinatorics, concentrates on the theory and application of generating functions, a fundamental tool in enumerative combinatorics. Richard Stanley covers those parts of enumerative combinatorics with the greatest applications to other areas of mathematics. The four chapters are devoted to an accessible introduction to enumeration, sieve methods--including the Principle of Inclusion-Exclusion, partially ordered sets, and rational generating functions. A large number of exercises, almost all with solutions, augment the text and provide entry into many areas not covered directly. Graduate students and research mathematicians who wish to apply combinatorics to their work will find this an authoritative reference.

Clifford Algebras and the Classical Groups

Ian R. Porteous

Ian R. Porteous

Cambridge Studies in Advanced Mathematics (No. 50)

**Print Publication Year:** 1995

**Print ISBN:** 9780521551779

**Online Publication Date:** September 2009

**Online ISBN:** 9780511470912

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511470912

This book reflects the growing interest in the theory of Clifford algebras and their applications. The author has reworked his previous book on this subject, Topological Geometry, and has expanded and added material. As in the previous version, the author includes an exhaustive treatment of all the generalizations of the classical groups, as well as an excellent exposition of the classification of the conjugation anti-involution of the Clifford algebras and their complexifications. Toward the end of the book, the author introduces ideas from the theory of Lie groups and Lie algebras. This treatment of Clifford algebras will be welcomed by graduate students and researchers in algebra.

Geometric Control Theory

Velimir Jurdjevic

Velimir Jurdjevic

Cambridge Studies in Advanced Mathematics (No. 52)

**Print Publication Year:** 1996

**Print ISBN:** 9780521495028

**Online Publication Date:** October 2009

**Online ISBN:** 9780511530036

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511530036

This book describes the mathematical theory inspired by the irreversible nature of time-evolving events. The first part of the book deals with the ability to steer a system from any point of departure to any desired destination. The second part deals with optimal control--the problem of finding the best possible course. The author demonstrates an overlap with mathematical physics using the maximum principle, a fundamental concept of optimality arising from geometric control, which is applied to time-evolving systems governed by physics as well as to man-made systems governed by controls. He draws applications from geometry, mechanics, and control of dynamical systems. The geometric language in which the author expresses the results allows clear visual interpretations and makes the book accessible to physicists and engineers as well as to mathematicians.

Groups as Galois Groups

**An Introduction**

Helmut Volklein

Helmut Volklein

Cambridge Studies in Advanced Mathematics (No. 53)

**Print Publication Year:** 1996

**Print ISBN:** 9780521562805

**Online Publication Date:** September 2009

**Online ISBN:** 9780511471117

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511471117

This book describes various approaches to the Inverse Galois Problem, a classical unsolved problem of mathematics posed by Hilbert at the beginning of the century. It brings together ideas from group theory, algebraic geometry and number theory, topology, and analysis. Assuming only elementary algebra and complex analysis, the author develops the necessary background from topology, Riemann surface theory and number theory. The first part of the book is quite elementary, and leads up to the basic rigidity criteria for the realization of groups as Galois groups. The second part presents more advanced topics, such as braid group action and moduli spaces for covers of the Riemann sphere, GAR- and GAL- realizations, and patching over complete valued fields. Graduate students and mathematicians from other areas (especially group theory) will find this an excellent introduction to a fascinating field.

Automorphic Forms and Representations

Daniel Bump

Daniel Bump

Cambridge Studies in Advanced Mathematics (No. 55)

**Print Publication Year:** 1997

**Print ISBN:** 9780521550987

**Online Publication Date:** December 2009

**Online ISBN:** 9780511609572

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511609572

This book covers both the classical and representation theoretic views of automorphic forms in a style that is accessible to graduate students entering the field. The treatment is based on complete proofs, which reveal the uniqueness principles underlying the basic constructions. The book features extensive foundational material on the representation theory of GL(1) and GL(2) over local fields, the theory of automorphic representations, L-functions and advanced topics such as the Langlands conjectures, the Weil representation, the Rankin-Selberg method and the triple L-function, and examines this subject matter from many different and complementary viewpoints. Researchers as well as students in algebra and number theory will find this a valuable guide to a notoriously difficult subject.

Cohomology of Drinfeld Modular Varieties

**Part 2**
**, Automorphic Forms, Trace Formulas and Langlands Correspondence**

Gérard Laumon, Appendix by Jean Loup Waldspurger

Gérard Laumon, Appendix by Jean Loup Waldspurger

Cambridge Studies in Advanced Mathematics (No. 56)

**Print Publication Year:** 1997

**Print ISBN:** 9780521470612

**Online Publication Date:** March 2010

**Online ISBN:** 9780511661969

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511661969

Cohomology of Drinfeld Modular Varieties aims to provide an introduction to this subject and to the Langlands correspondence for function fields. These varieties are the analogues for function fields of the Shimura varieties over number fields. The Langlands correspondence is a conjectured link between automorphic forms and Galois representations over a global field. By analogy with the number-theoretic case, one expects to establish the conjecture for function fields by studying the cohomology of Drinfeld modular varieties, which has been done by Drinfeld himself for the rank two case. This second volume is concerned with the ArthurSHSelberg trace formula, and to the proof in some cases of the Ramanujan-Petersson conjecture and the global Langlands conjecture for function fields. The author uses techniques that are extensions of those used to study Shimura varieties. Though the author considers only the simpler case of function rather than number fields, many important features of the number field case can be illustrated. Several appendices on background material keep the work reasonably self-contained. This book will be of much interest to all researchers in algebraic number theory and representation theory.

A User's Guide to Spectral Sequences

**Second edition**

John McCleary

John McCleary

Cambridge Studies in Advanced Mathematics (No. 58)

**Print Publication Year:** 2000

**Print ISBN:** 9780521561419

**Online Publication Date:** January 2010

**Online ISBN:** 9780511626289

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511626289

Spectral sequences are among the most elegant and powerful methods of computation in mathematics. This book describes some of the most important examples of spectral sequences and some of their most spectacular applications. The first part treats the algebraic foundations for this sort of homological algebra, starting from informal calculations. The heart of the text is an exposition of the classical examples from homotopy theory, with chapters on the Leray-Serre spectral sequence, the Eilenberg-Moore spectral sequence, the Adams spectral sequence, and, in this new edition, the Bockstein spectral sequence. The last part of the book treats applications throughout mathematics, including the theory of knots and links, algebraic geometry, differential geometry and algebra. This is an excellent reference for students and researchers in geometry, topology, and algebra.

Local Cohomology

**An Algebraic Introduction with Geometric Applications**

M. P. Brodmann, R. Y. Sharp

M. P. Brodmann, R. Y. Sharp

Cambridge Studies in Advanced Mathematics (No. 60)

**Print Publication Year:** 1998

**Print ISBN:** 9780521372862

**Online Publication Date:** May 2010

**Online ISBN:** 9780511629204

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511629204

This book provides a careful and detailed algebraic introduction to Grothendieck's local cohomology theory, and illustrates many applications for the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Castelnuovo-Mumford regularity, the Fulton-Hansen connectedness theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. It is designed for graduate students who have some experience of basic commutative algebra and homological algebra, and also for experts in commutative algebra and algebraic geometry.

Analytic Pro-P Groups

**Second edition**

J. D. Dixon, M. P. F. Du Sautoy, A. Mann, D. Segal

J. D. Dixon, M. P. F. Du Sautoy, A. Mann, D. Segal

Cambridge Studies in Advanced Mathematics (No. 61)

**Print Publication Year:** 1999

**Print ISBN:** 9780521650113

**Online Publication Date:** December 2009

**Online ISBN:** 9780511470882

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511470882

The first edition of this book was the indispensable reference for researchers in the theory of pro-p groups. In this second edition the presentation has been improved and important new material has been added. The first part of the book is group-theoretic. It develops the theory of pro-p groups of finite rank, starting from first principles and using elementary methods. Part II introduces p-adic analytic groups: by taking advantage of the theory developed in Part I, it is possible to define these, and derive all the main results of p-adic Lie theory, without having to develop any sophisticated analytic machinery. Part III, consisting of new material, takes the theory further. Among those topics discussed are the theory of pro-p groups of finite coclass, the dimension subgroup series, and its associated graded Lie algebra. The final chapter sketches a theory of analytic groups over pro-p rings other than the p-adic integers.

Enumerative Combinatorics

**Volume 2**

Richard P. Stanley, Appendix by Sergey Fomin

Richard P. Stanley, Appendix by Sergey Fomin

Cambridge Studies in Advanced Mathematics (No. 62)

**Print Publication Year:** 1999

**Print ISBN:** 9780521560696

**Online Publication Date:** January 2010

**Online ISBN:** 9780511609589

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511609589

This second volume of a two-volume basic introduction to enumerative combinatorics covers the composition of generating functions, trees, algebraic generating functions, D-finite generating functions, noncommutative generating functions, and symmetric functions. The chapter on symmetric functions provides the only available treatment of this subject suitable for an introductory graduate course on combinatorics, and includes the important Robinson-Schensted-Knuth algorithm. Also covered are connections between symmetric functions and representation theory. An appendix by Sergey Fomin covers some deeper aspects of symmetric function theory, including jeu de taquin and the Littlewood-Richardson rule. As in Volume 1, the exercises play a vital role in developing the material. There are over 250 exercises, all with solutions or references to solutions, many of which concern previously unpublished results. Graduate students and research mathematicians who wish to apply combinatorics to their work will find this an authoritative reference.

Uniform Central Limit Theorems

R. M. Dudley

R. M. Dudley

Cambridge Studies in Advanced Mathematics (No. 63)

**Print Publication Year:** 1999

**Print ISBN:** 9780521461023

**Online Publication Date:** May 2010

**Online ISBN:** 9780511665622

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511665622

This book shows how, when samples become large, the probability laws of large numbers and related facts are guaranteed to hold over wide domains. The author, an acknowledged expert, gives a thorough treatment of the subject, including several topics not found in any previous book, such as the Fernique-Talagrand majorizing measure theorem for Gaussian processes, an extended treatment of Vapnik-Chervonenkis combinatorics, the Ossiander L2 bracketing central limit theorem, the Giné-Zinn bootstrap central limit theorem in probability, the Bronstein theorem on approximation of convex sets, and the Shor theorem on rates of convergence over lower layers. Other recent results of Talagrand and others are surveyed without proofs in separate sections. Problems are included at the end of each chapter so the book can be used as an advanced text. The book will interest mathematicians with an interest in probability, mathematical statisticians, and computer scientists working in computer learning theory.

Modular Forms and Galois Cohomology

Haruzo Hida

Haruzo Hida

Cambridge Studies in Advanced Mathematics (No. 69)

**Print Publication Year:** 2000

**Print ISBN:** 9780521770361

**Online Publication Date:** December 2009

**Online ISBN:** 9780511526046

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511526046

This book provides a comprehensive account of a key, perhaps the most important, theory that forms the basis of Taylor-Wiles proof of Fermat's last theorem. Hida begins with an overview of the theory of automorphic forms on linear algebraic groups and then covers the basic theory and recent results on elliptic modular forms, including a substantial simplification of the Taylor-Wiles proof by Fujiwara and Diamond. He offers a detailed exposition of the representation theory of profinite groups (including deformation theory), as well as the Euler characteristic formulas of Galois cohomology groups. The final chapter presents a proof of a non-abelian class number formula.

Fourier Analysis and Partial Differential Equations

Rafael José Iorio, Jr, Valéria de Magalhães Iorio

Rafael José Iorio, Jr, Valéria de Magalhães Iorio

Cambridge Studies in Advanced Mathematics (No. 70)

**Print Publication Year:** 2001

**Print ISBN:** 9780521621168

**Online Publication Date:** July 2014

**Online ISBN:** 9780511623745

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623745

This modern introduction to Fourier analysis and partial differential equations is intended to be used with courses for beginning graduate students. With minimal prerequisites the authors take the reader from fundamentals to research topics in the area of nonlinear evolution equations, including a fairly complete discussion of local and global well-posedness for the nonlinear Schrödinger and the Korteweg-de Vries equations; they turn their attention, in the two final chapters, to the nonperiodic setting, concentrating on problems that do not occur in the periodic case.

Analysis in Integer and Fractional Dimensions

Ron Blei

Ron Blei

Cambridge Studies in Advanced Mathematics (No. 71)

**Print Publication Year:** 2001

**Print ISBN:** 9780521650847

**Online Publication Date:** August 2009

**Online ISBN:** 9780511543012

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511543012

This book provides a thorough and self-contained study of interdependence and complexity in settings of functional analysis, harmonic analysis and stochastic analysis. It focuses on "dimension" as a basic counter of degrees of freedom, leading to precise relations between combinatorial measurements and various indices originating from the classical inequalities of Khintchin, Littlewood and Grothendieck. Topics include the (two-dimensional) Grothendieck inequality and its extensions to higher dimensions, stochastic models of Brownian motion, degrees of randomness and Fréchet measures in stochastic analysis. This book is primarily aimed at graduate students specializing in harmonic analysis, functional analysis or probability theory. It contains many exercises and is suitable as a textbook. It is also of interest to computer scientists, physicists, statisticians, biologists and economists.

Galois Theories

Francis Borceux, George Janelidze

Francis Borceux, George Janelidze

Cambridge Studies in Advanced Mathematics (No. 72)

**Print Publication Year:** 2001

**Print ISBN:** 9780521803090

**Online Publication Date:** January 2010

**Online ISBN:** 9780511619939

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511619939

Starting from the classical finite-dimensional Galois theory of fields, this book develops Galois theory in a much more general context. The authors first formalize the categorical context in which a general Galois theorem holds, and then give applications to Galois theory for commutative rings, central extensions of groups, the topological theory of covering maps and a Galois theorem for toposes. The book is designed to be accessible to a wide audience, the prerequisites are first courses in algebra and general topology, together with some familiarity with the categorical notions of limit and adjoint functors. For all algebraists and category theorists this book will be a rewarding read.

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