Stone Spaces

Author: Peter T. Johnstone

Publisher: Cambridge University Press

ISBN: 9780521337793

Category: Mathematics

Page: 370

View: 6598

A unified treatment of the corpus of mathematics that has developed out of M. H. Stone's representation theorem for Boolean algebras (1936) which has applications in almost every area of modern mathematics.
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Introduction to Model Spaces and their Operators

Author: Stephan Ramon Garcia,Javad Mashreghi,William T. Ross

Publisher: Cambridge University Press

ISBN: 1316390438

Category: Mathematics

Page: N.A

View: 1253

The study of model spaces, the closed invariant subspaces of the backward shift operator, is a vast area of research with connections to complex analysis, operator theory and functional analysis. This self-contained text is the ideal introduction for newcomers to the field. It sets out the basic ideas and quickly takes the reader through the history of the subject before ending up at the frontier of mathematical analysis. Open questions point to potential areas of future research, offering plenty of inspiration to graduate students wishing to advance further.
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Banach Spaces for Analysts

Author: P. Wojtaszczyk

Publisher: Cambridge University Press

ISBN: 9780521566759

Category: Mathematics

Page: 382

View: 5431

This book is intended to be used with graduate courses in Banach space theory.
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Martingales in Banach Spaces

Author: Gilles Pisier

Publisher: Cambridge University Press

ISBN: 1316679462

Category: Mathematics

Page: N.A

View: 2937

This book focuses on the major applications of martingales to the geometry of Banach spaces, and a substantial discussion of harmonic analysis in Banach space valued Hardy spaces is also presented. It covers exciting links between super-reflexivity and some metric spaces related to computer science, as well as an outline of the recently developed theory of non-commutative martingales, which has natural connections with quantum physics and quantum information theory. Requiring few prerequisites and providing fully detailed proofs for the main results, this self-contained study is accessible to graduate students with a basic knowledge of real and complex analysis and functional analysis. Chapters can be read independently, with each building from the introductory notes, and the diversity of topics included also means this book can serve as the basis for a variety of graduate courses.
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Wavelets and Operators:

Author: Yves Meyer

Publisher: Cambridge University Press

ISBN: 9780521458696

Category: Mathematics

Page: 244

View: 9244

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.
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Introduction to Banach Spaces: Analysis and Probability

Author: Daniel Li,Hervé Queffélec

Publisher: Cambridge University Press

ISBN: 1107162629

Category: Mathematics

Page: 412

View: 9962

This two-volume text provides a complete overview of the theory of Banach spaces, emphasising its interplay with classical and harmonic analysis (particularly Sidon sets) and probability. The authors give a full exposition of all results, as well as numerous exercises and comments to complement the text and aid graduate students in functional analysis. The book will also be an invaluable reference volume for researchers in analysis. Volume 1 covers the basics of Banach space theory, operatory theory in Banach spaces, harmonic analysis and probability. The authors also provide an annex devoted to compact Abelian groups. Volume 2 focuses on applications of the tools presented in the first volume, including Dvoretzky's theorem, spaces without the approximation property, Gaussian processes, and more. Four leading experts also provide surveys outlining major developments in the field since the publication of the original French edition.
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A User's Guide to Spectral Sequences

Author: John McCleary

Publisher: Cambridge University Press

ISBN: 9780521567596

Category: Mathematics

Page: 561

View: 3811

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.
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Algebraic Number Theory

Author: A. Fröhlich,M. J. Taylor,Martin J. Taylor

Publisher: Cambridge University Press

ISBN: 9780521438346

Category: Mathematics

Page: 355

View: 9795

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.
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Groups as Galois Groups

An Introduction

Author: Helmut Volklein

Publisher: Cambridge University Press

ISBN: 9780521562805

Category: Mathematics

Page: 248

View: 7440

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 realisation 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.
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Holomorphic Dynamics

Author: S. Morosawa

Publisher: Cambridge University Press

ISBN: 9780521662581

Category: Mathematics

Page: 338

View: 8592

This book, first published in 2000, is a graduate text on complex analytic dynamics - the mathematics behind things like the Mandelbrot set.
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Geometry of Sets and Measures in Euclidean Spaces

Fractals and Rectifiability

Author: Pertti Mattila

Publisher: Cambridge University Press

ISBN: 9780521655958

Category: Mathematics

Page: 343

View: 3012

This book studies the geometric properties of general sets and measures in euclidean space.
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An Introduction to Lie Groups and Lie Algebras

Author: Alexander Kirillov

Publisher: Cambridge University Press

ISBN: 0521889693

Category: Mathematics

Page: 222

View: 9330

This book is an introduction to semisimple Lie algebras; concise and informal, with numerous exercises and examples.
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An Introduction to the Theory of Reproducing Kernel Hilbert Spaces

Author: Vern I. Paulsen,Mrinal Raghupathi

Publisher: Cambridge University Press

ISBN: 1316558738

Category: Mathematics

Page: N.A

View: 9306

Reproducing kernel Hilbert spaces have developed into an important tool in many areas, especially statistics and machine learning, and they play a valuable role in complex analysis, probability, group representation theory, and the theory of integral operators. This unique text offers a unified overview of the topic, providing detailed examples of applications, as well as covering the fundamental underlying theory, including chapters on interpolation and approximation, Cholesky and Schur operations on kernels, and vector-valued spaces. Self-contained and accessibly written, with exercises at the end of each chapter, this unrivalled treatment of the topic serves as an ideal introduction for graduate students across mathematics, computer science, and engineering, as well as a useful reference for researchers working in functional analysis or its applications.
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Classical and Multilinear Harmonic Analysis

Author: Camil Muscalu,Wilhelm Schlag

Publisher: Cambridge University Press

ISBN: 0521882451

Category: Mathematics

Page: 387

View: 5937

"This two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained, and will be useful to graduate students and researchers in both pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. This first volume starts with classical one-dimensional topics: Fourier series; harmonic functions; Hilbert transform. Then the higher-dimensional Calderâon-Zygmund and Littlewood-Paley theories are developed. Probabilistic methods and their applications are discussed, as are applications of harmonic analysis to partial differential equations. The volume concludes with an introduction to the Weyl calculus. The second volume goes beyond the classical to the highly contemporary, and focuses on multilinear aspects of harmonic analysis: the bilinear Hilbert transform; Coifman-Meyer theory; Carleson's resolution of the Lusin conjecture; Calderâon's commutators and the Cauchy integral on Lipschitz curves. The material in this volume has not previously appeared together in book form"--
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Real Analysis and Probability

Author: R. M. Dudley

Publisher: CRC Press

ISBN: 1351093096

Category: Mathematics

Page: 450

View: 7490

Written by one of the best-known probabilists in the world this text offers a clear and modern presentation of modern probability theory and an exposition of the interplay between the properties of metric spaces and those of probability measures. This text is the first at this level to include discussions of the subadditive ergodic theorems, metrics for convergence in laws and the Borel isomorphism theory. The proofs for the theorems are consistently brief and clear and each chapter concludes with a set of historical notes and references. This book should be of interest to students taking degree courses in real analysis and/or probability theory.
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Riemannian Geometry

A Modern Introduction

Author: Isaac Chavel

Publisher: Cambridge University Press

ISBN: 1139452576

Category: Mathematics

Page: N.A

View: 6376

This book provides an introduction to Riemannian geometry, the geometry of curved spaces, for use in a graduate course. Requiring only an understanding of differentiable manifolds, the author covers the introductory ideas of Riemannian geometry followed by a selection of more specialized topics. Also featured are Notes and Exercises for each chapter, to develop and enrich the reader's appreciation of the subject. This second edition, first published in 2006, has a clearer treatment of many topics than the first edition, with new proofs of some theorems and a new chapter on the Riemannian geometry of surfaces. The main themes here are the effect of the curvature on the usual notions of classical Euclidean geometry, and the new notions and ideas motivated by curvature itself. Completely new themes created by curvature include the classical Rauch comparison theorem and its consequences in geometry and topology, and the interaction of microscopic behavior of the geometry with the macroscopic structure of the space.
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Geometry and Complexity Theory

Author: J. M. Landsberg

Publisher: Cambridge University Press

ISBN: 110819141X

Category: Computers

Page: N.A

View: 7494

Two central problems in computer science are P vs NP and the complexity of matrix multiplication. The first is also a leading candidate for the greatest unsolved problem in mathematics. The second is of enormous practical and theoretical importance. Algebraic geometry and representation theory provide fertile ground for advancing work on these problems and others in complexity. This introduction to algebraic complexity theory for graduate students and researchers in computer science and mathematics features concrete examples that demonstrate the application of geometric techniques to real world problems. Written by a noted expert in the field, it offers numerous open questions to motivate future research. Complexity theory has rejuvenated classical geometric questions and brought different areas of mathematics together in new ways. This book will show the beautiful, interesting, and important questions that have arisen as a result.
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Fourier Analysis and Hausdorff Dimension

Author: Pertti Mattila

Publisher: Cambridge University Press

ISBN: 1107107350

Category: Mathematics

Page: 452

View: 1967

Modern text examining the interplay between measure theory and Fourier analysis.
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Frames and Locales

Topology without points

Author: Jorge Picado,Aleš Pultr

Publisher: Springer Science & Business Media

ISBN: 3034801548

Category: Mathematics

Page: 398

View: 4919

Until the mid-twentieth century, topological studies were focused on the theory of suitable structures on sets of points. The concept of open set exploited since the twenties offered an expression of the geometric intuition of a "realistic" place (spot, grain) of non-trivial extent. Imitating the behaviour of open sets and their relations led to a new approach to topology flourishing since the end of the fifties.It has proved to be beneficial in many respects. Neglecting points, only little information was lost, while deeper insights have been gained; moreover, many results previously dependent on choice principles became constructive. The result is often a smoother, rather than a more entangled, theory. No monograph of this nature has appeared since Johnstone's celebrated Stone Spaces in 1983. The present book is intended as a bridge from that time to the present. Most of the material appears here in book form for the first time or is presented from new points of view. Two appendices provide an introduction to some requisite concepts from order and category theories.
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Algebraic Homotopy

Author: Hans J. Baues

Publisher: Cambridge University Press

ISBN: 9780521333764

Category: Mathematics

Page: 466

View: 4479

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.
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