Sergey V. Matveev (Chelyabinsk State University, Russia):

Lectures on Algebraic Topology

EMS Series of Lectures in Mathematics
ISBN 3-03719-023-X
April 2006, 108 pages, softcover, 17.0 cm x 24.0 cm.

Algebraic topology is the study of the global properties of spaces by means of algebra. It is an important branch of modern mathematics with a wide degree of applicability to other fields, including geometric topology, differential geometry, functional analysis, differential equations, algebraic geometry, number theory, and theoretical physics.

This book provides an introduction to the basic concepts and methods of algebraic topology for the beginner. It presents elements of both homology theory and homotopy theory, and includes various applications.

The author's intention is to rely on the geometric approach by appealing to the reader's own intuition to help understanding. The numerous illustrations in the text also serve this purpose. Two features make the text different from the standard literature: first, special attention is given to providing explicit algorithms for calculating the homology groups and for manipulating the fundamental groups. Second, the book contains many exercises, all of which are supplied with hints or solutions. This makes the book suitable for both classroom use and for independent study.

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Sergei B. Kuksin (Heriot-Watt University, Edinburgh, UK):

Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions

Zurich Lectures in Advanced Mathematics
ISBN 3-03719-021-3
April 2006, 104 pages, softcover, 17.0 cm x 24.0 cm.

The book gives an account of recent achievements in the mathematical theory of two-dimensional turbulence, described by the 2D Navier-Stokes equation, perturbed by a random force. The main results presented here were obtained during the last five to ten years and, up to now, have been available only in papers in the primary literature. Their summary and synthesis here, beginning with some preliminaries on partial differential equations and stochastics, make the book a self-contained account that will appeal to readers with a general background in analysis.

After laying the groundwork, the author goes on to recent results on ergodicity of random dynamical systems, which the randomly forced Navier-Stokes equation defines in the function space of divergence-free vector fields, including a Central Limit Theorem. The physical meaning of these results is discussed as well as their relations with the theory of attractors. Next, the author studies the behaviour of solutions when the viscosity goes to zero. In the final section these dynamical methods are used to derive the so-called balance relations - the infinitely many algebraical relations satisfied by the solutions.

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Alberto Cattaneo - Bernhard Keller - Charles Torossian - Alain Bruguieres

Deformation, Quantification, Theorie de Lie

Panoramas et Syntheses 20 (2005), viii+186 pages

Resume :

En 1997, M. Kontsevich demontra que toute variete de Poisson admet une quantification formelle, canonique a equivalence pres, resolvant ainsi un probleme ancien de physique mathematique. Par sa demonstration, et l'interpretation qu'il fit d'une demonstration ulterieure due a Tamarkin, M. Kontsevich a ouvert des voies de recherche nouvelles en theorie de Lie, groupes quantiques, theorie des deformations, theorie des operades... et revele des liens fascinants entre ces sujets et la theorie des nombres, la theorie des n?uds et la theorie des motifs. Ce travail sur la quantification par deformation va continuer a influencer ces domaines dans les annees a venir. Dans les trois parties de ce volume, nous allons 1) presenter les resultats principaux de la prepublication de 1997 de Kontsevich et esquisser son interpretation de l'approche de Tamarkin, 2) montrer la pertinence du theoreme de Kontsevich pour la theorie de Lie et 3) expliquer l'idee provenant de la theorie des cordes topologiques qui a inspire l'approche de Kontsevich. Un appendice est consacre a la geometrie des espaces de configurations.

Mots clefs : Theorie des deformations, quantification par deformation, physique mathematique, variete de Poisson, cohomologie de Hochschild, algebre de Lie, isomorphisme de Duflo, formule de Campbell-Baker-Hausdorff, theorie des cordes, espace de configurations.

Abstract:

Deformation, Quantization, Lie theory
In 1997, M. Kontsevich proved that every Poisson manifold admits a formal quantization, canonical up to equivalence. In doing so he solved a longstanding problem in mathematical physics. Through his proof and his interpretation of a later proof given by Tamarkin, he also opened up new research avenues in Lie theory, quantum group theory, deformation theory and the study of operads... and uncovered fascinating links of these topics with number theory, knot theory and the theory of motives. Without doubt, his work on deformation quantization will continue to influence these fields for many years to come. In the three parts of this volume, we will 1) present the main results of Kontsevich's 1997 preprint and sketch his interpretation of Tamarkin's approach, 2) show the relevance of Kontsevich's theorem for Lie theory and 3) explain the idea from topological string theory which inspired Kontsevich's proof. An appendix is devoted to the geometry of configuration spaces.

Key words: Deformation theory, deformation quantization, mathematical physics, Poisson manifold, Hochschild cohomology, Lie algebra, Duflo isomorphism, Campbell-Baker-Hausdorff formula, string theory, configuration space.

Theodore G. Faticoni

The Mathematics of Infinity: A Guide to Great Ideas

ISBN: 0-471-79432-5
Hardcover
304 pages
June 2006

The Mathematics of Infinity addresses infinite cardinals and is appropriate for readers at any level. Inviting the reader to imagine constructing an infinite chain infinities, which are called cardinals, the author successfully prepares and motivates readers for topics covered within the book. The most unique feature of the book is that set theoretic depth is achieved without losing the target audience. Complementing existing popular books on infinity by actually doing the mathematics involved in addition to talking about the mathematics, the reader is gently led into the world of mathematical proofs.


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1. Elementary Set Theory.
2. Functions.
3. Counting Infinite Sets.
4. Infinite Cardinals.
5. Well Ordered Sets.
6. Inductions and Numbers.
7. Prime Numbers.
8. Thoughts on Logic Beyond Mathematics.

Douglas C. Montgomery, Elizabeth A. Peck, G. Geoffrey Vining

Introduction to Linear Regression Analysis, 4th Edition

ISBN: 0-471-75495-1
Hardcover
640 pages
August 2006

The Fourth Edition of Introduction to Linear Regression Analysis describes both the conventional and less common uses of linear regression in the practical context of today's mathematical and scientific research. This popular book blends both theory and application to equip the reader with an understanding of the basic principles necessary to apply regression model-building techniques in a wide variety of application environments. It assumes a working knowledge of basic statistics and a familiarity with hypothesis testing and confidence intervals, as well as the normal, t, x2, and F distributions.

Illustrating all of the major procedures employed by the contemporary software packages MINITABR, SASR, and S-PLUSR, the Fourth Edition begins with a general introduction to regression modeling, including typical applications. A host of technical tools are outlined, such as basic inference procedures, introductory aspects of model adequacy checking, and polynomial regression models and their variations. The book discusses how transformations and weighted least squares can be used to resolve problems of model inadequacy and also how to deal with influential observations. Subsequent chapters discuss:

Indicator variables and the connection between regression and analysis-of-variance models
Variable selection and model-building techniques and strategies
The multicollinearity problem?its sources, effects, diagnostics, and remedial measures
Robust regression techniques such as M-estimators, and properties of robust estimators
The basics of nonlinear regression
Generalized linear models
Using SASR for regression problems
This book is a robust resource that offers solid methodology for statistical practitioners and professionals in the fields of engineering, physical and chemical sciences, economics, management, life and biological sciences, and the social sciences. Both the accompanying FTP site, which contains data sets, extensive problem solutions, software hints, and PowerPointR slides, as well as the book's revised presentation of topics in increasing order of complexity, facilitate its use in a classroom setting.

With its new exercises and structure, this book is highly recommended for upper-undergraduate and beginning graduate students in mathematics, engineering, and natural sciences. Scientists and engineers will find the book to be an excellent choice for reference and self-study.

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Erich Zauderer

Partial Differential Equations of Applied Mathematics, 3rd Edition

ISBN: 0-471-69073-2
Hardcover
1010 pages
July 2006


This new edition features the latest tools for modeling, characterizing, and solving partial differential equations

The Third Edition of this classic text offers a comprehensive guide to modeling, characterizing, and solving partial differential equations (PDEs). The author provides all the theory and tools necessary to solve problems via exact, approximate, and numerical methods. The Third Edition retains all the hallmarks of its previous editions, including an emphasis on practical applications, clear writing style and logical organization, and extensive use of real-world examples.

Among the new and revised material, the book features:

A new section at the end of each original chapter, exhibiting the use of specially constructed Maple procedures that solve PDEs via many of the methods presented in the chapters. The results can be evaluated numerically or displayed graphically.
Two new chapters that present finite difference and finite element methods for the solution of PDEs. Newly constructed Maple procedures are provided and used to carry out each of these methods. All the numerical results can be displayed graphically.
A related FTP site that includes all the Maple code used in the text.
New exercises in each chapter, and answers to many of the exercises are provided via the FTP site. A supplementary Instructor's Solutions Manual is available.
The book begins with a demonstration of how the three basic types of equations?parabolic, hyperbolic, and elliptic?can be derived from random walk models. It then covers an exceptionally broad range of topics, including questions of stability, analysis of singularities, transform methods, Green's functions, and perturbation and asymptotic treatments. Approximation methods for simplifying complicated problems and solutions are described, and linear and nonlinear problems not easily solved by standard methods are examined in depth. Examples from the fields of engineering and physical sciences are used liberally throughout the text to help illustrate how theory and techniques are applied to actual problems.

With its extensive use of examples and exercises, this text is recommended for advanced undergraduates and graduate students in engineering, science, and applied mathematics, as well as professionals in any of these fields. It is possible to use the text, as in the past, without use of the new Maple material

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