## Kvant Selecta: Combinatorics, I

Expected publication date is November 29, 2001

Description
There is a tradition in Russia that holds that mathematics can be both challenging and fun. One fine outgrowth of that tradition is the magazine, Kvant, which has been enjoyed by many of the best students since its founding in 1970. The articles in Kvant assume only a minimal background, that of a good high school student, yet are capable of entertaining mathematicians of almost any level. Sometimes the articles require careful thought or a moment's work with a pencil and paper. However, the industrious reader will be generously rewarded by the elegance and beauty of the subjects.

This book is the third collection of articles from Kvant to be published by the AMS. The volume is devoted mainly to combinatorics and discrete mathematics. Several of the topics are well known: nonrepeating sequences, detecting a counterfeit coin, and linear inequalities in economics, but they are discussed here with the entertaining and engaging style typical of the magazine. The two previous collections treat aspects of algebra and analysis, including connections to number theory and other topics. They were published as Volumes 14 and 15 in the Mathematical World series.

The articles are written so as to present genuine mathematics in a conceptual, entertaining, and accessible way. The books are designed to be used by students and teachers who love mathematics and want to study its various aspects, deepening and expanding upon the school curriculum.

Contents

I. M. Yaglom -- Two games with matchsticks
A. B. Katok -- Economics and linear inequalities
A. B. Katok -- Economics and linear inequalities (Continuation)
R. V. Freivald -- Switching networks
G. M. Adel'son-Vel'skii, I. N. Bernshtein, and M. L. Gerver -- Who will go to Rio?
A. L. Toom -- From the life of units
G. A. Gurevich -- Nonrepeating sequences
A. M. Stepin and A. T. Tagi-Zade -- Words with restrictions
S. Ovchinnikov -- Planar switching circuits
P. Bleher and M. Kel'bert -- Classification algorithms
G. Shestopal -- How to detect a counterfeit coin
M. Mamikon -- The generalized problem of counterfeit coins
P. Bleher -- Truthtellers, liars, and deceivers
V. A. Uspenskii and A. L. Semenov -- Solvable and unsolvable algorithmic problems
P. A. Pevzner -- Best bet for simpletons

Details:

Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Mathematical World, Volume: 17
Publication Year: 2002
ISBN: 0-8218-2171-7
Paging: 131 pp.
Binding: Softcover

## Function Theory of One Complex Variable: Second Edition

Expected publication date is December 28, 2001

From a review of the First Edition:

"The book is carefully and precisely written in a lively and soft style. It is extremely clear ... and very detailed. Moreover, it is stimulating and very suitable for self-study ... Certainly, the book reflects the authors' experience in teaching. The other features include the fruitful connection with real analysis ... the authors have produced a modern, quality work that could serve as an excellent model for writing and teaching graduate texts ... it will occupy a distinguished place in the extensive literature on the subject ... I read this book with great pleasure and I warmly recommend it for all those who are interested in complex analysis of one variable."

-- Mathematical Reviews

Description
Complex analysis is one of the most beautiful subjects that we learn as graduate students. Part of the joy comes from being able to arrive quickly at some "real theorems". The fundamental techniques of complex variables are also used to solve real problems in neighboring subjects, such as number theory or PDEs.

This book is a text for a first-year graduate course in complex analysis. It is an engaging and modern introduction to the subject, reflecting the authors' expertise both as mathematicians and as expositors.

All the material usually treated in such a course is covered here, but following somewhat different principles. To begin with, the authors emphasize how this subject is a natural outgrowth of multivariable real analysis. Complex function theory has long been a flourishing independent field. However, an efficient path into the subject is to observe how its rudiments arise directly from familiar ideas in calculus. The authors pursue this point of view by comparing and contrasting complex analysis with its real variable counterpart.

Explanations of certain topics in complex analysis can sometimes become complicated by the intermingling of the analysis and the topology. Here, the authors have collected the primary topological issues in a separate chapter, leaving the way open for a more direct and less ambiguous approach to the analytic material.

The book concludes with several chapters on special topics, including full treatments of special functions, the prime number theorem, and the Bergman kernel. The authors also treat $H^p$ spaces and Painleve's theorem on smoothness to the boundary for conformal maps.

A large number of exercises are included. Some are simply drills to hone the students' skills, but many others are further developments of the ideas in the main text. The exercises are also used to explore the striking interconnectedness of the topics that constitute complex analysis.

Contents

Fundamental concepts
Complex line integrals
Applications of the Cauchy integral
Meromorphic functions and residues
The zeros of a holomorphic function
Holomorphic functions as geometric mappings
Harmonic functions
Infinite series and products
Applications of infinite sums and products
Analytic continuation
Topology
Rational approximation theory
Special classes of holomorphic functions
Hilbert spaces of holomorphic functions, the Bergman kernel, and biholomorphic mappings
Special functions
The prime number theorem
Real analysis
The statement and proof of Goursat's theorem
References
Index

Details:

Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Graduate Studies in Mathematics, Volume: 40
Publication Year: 2002
ISBN: 0-8218-2905-X
Paging: approximately 561 pp.
Binding: Hardcover

## Attractors for Equations of Mathematical Physics

Expected publication date is December 1, 2001

Description
One of the major problems in the study of evolution equations of mathematical physics is the investigation of the behavior of the solutions to these equations when time is large or tends to infinity. The related important questions concern the stability of solutions or the character of the instability if a solution is unstable. In the last few decades, considerable progress in this area has been achieved in the study of autonomous evolution partial differential equations. For a number of basic evolution equations of mathematical physics, it was shown that the long time behavior of their solutions can be characterized by a very important notion of a global attractor of the equation.

In this book, the authors study new problems related to the theory of infinite-dimensional dynamical systems that were intensively developed during the last 20 years. They construct the attractors and study their properties for various non-autonomous equations of mathematical physics: the 2D and 3D Navier-Stokes systems, reaction-diffusion systems, dissipative wave equations, the complex Ginzburg-Landau equation, and others. Since, as it is shown, the attractors usually have infinite dimension, the research is focused on the Kolmogorov $\varepsilon$-entropy of attractors. Upper estimates for the $\varepsilon$-entropy of uniform attractors of non-autonomous equations in terms of $\varepsilon$-entropy of time-dependent coefficients are proved.

Also, the authors construct attractors for those equations of mathematical physics for which the solution of the corresponding Cauchy problem is not unique or the uniqueness is not proved. The theory of the trajectory attractors for these equations is developed, which is later used to construct global attractors for equations without uniqueness. The method of trajectory attractors is applied to the study of finite-dimensional approximations of attractors. The perturbation theory for trajectory and global attractors is developed and used in the study of the attractors of equations with terms rapidly oscillating with respect to spatial and time variables. It is shown that the attractors of these equations are contained in a thin neighborhood of the attractor of the averaged equation.

The book gives systematic treatment to the theory of attractors of autonomous and non-autonomous evolution equations of mathematical physics. It can be used both by specialists and by those who want to get acquainted with this rapidly growing and important area of mathematics.

Contents

Introduction
Attractors of autonomous equations
Attractors of autonomous ordinary differential equations
Attractors of autonomous partial differential equations
Dimension of attractors
Attractors of non-autonomous equations
Processes and attractors
Translation compact functions
Attractors of non-autonomous partial differential equations
Semiprocesses and attractors
Kernels of processes
Kolmogorov $\varepsilon$-entropy of attractors
Trajectory attractors
Trajectory attractors of autonomous ordinary differential equations
Attractors in Hausdorff spaces
Trajectory attractors of autonomous equations
Trajectory attractors of autonomous partial differential equations
Trajectory attractors of non-autonomous equations
Trajectory attractors of non-autonomous partial differential equations
Approximation of trajectory attractors
Perturbation of trajectory attractors
Averaging of attractors of evolution equations with rapidly oscillating terms
Proofs of Theorems II.1.4 and II.1.5
Lattices and coverings
Bibliography
Index

Details:

Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Colloquium Publications, Volume: 49
Publication Year: 2002
ISBN: 0-8218-2950-5
Paging: 363 pp.
Binding: Hardcover

### Yuji Shimizu and Kenji Ueno, Kyoto University, Japan

Expected publication date is December 7, 2001

Description
The word "moduli" in the sense of this book first appeared in the epoch-making paper of B. Riemann, Theorie der Abel'schen Funktionen, published in 1857. Riemann defined a Riemann surface of an algebraic function field as a branched covering of a one-dimensional complex projective space, and found out that Riemann surfaces have parameters. This work gave birth to the theory of moduli.

However, the viewpoint regarding a Riemann surface as an algebraic curve became the mainstream, and the moduli meant the parameters for the figures (graphs) defined by equations.

In 1913, H. Weyl defined a Riemann surface as a complex manifold of dimension one. Moreover, Teichmuller's theory of quasiconformal mappings and Teichmuller spaces made a start for new development of the theory of moduli, making possible a complex analytic approach toward the theory of moduli of Riemann surfaces. This theory was then investigated and made complete by Ahlfors, Bers, Rauch, and others. However, the theory of Teichmuller spaces utilized the special nature of complex dimension one, and it was difficult to generalize it to an arbitrary dimension in a direct way.

It was Kodaira-Spencer's deformation theory of complex manifolds that allowed one to study arbitrary dimensional complex manifolds. Initial motivation in Kodaira-Spencer's discussion was the need to clarify what one should mean by number of moduli. Their results, together with further work by Kuranishi, provided this notion with intrinsic meaning.

This book begins by presenting the Kodaira-Spencer theory in its original naive form in Chapter 1 and introduces readers to moduli theory from the viewpoint of complex analytic geometry. Chapter 2 briefly outlines the theory of period mapping and Jacobian variety for compact Riemann surfaces, with the Torelli theorem as a goal. The theory of period mappings for compact Riemann surfaces can be generalized to the theory of period mappings in terms of Hodge structures for compact Kahler manifolds. In Chapter 3, the authors state the theory of Hodge structures, focusing briefly on period mappings. Chapter 4 explains conformal field theory as an application of moduli theory.

This is the English translation of a book originally published in Japanese. Other books by Kenji Ueno published in this AMS series, Translations of Mathematical Monographs, include An Introduction to Algebraic Geometry, Volume 166, Algebraic Geometry 1: From Algebraic Varieties to Schemes, Volume 185, and Algebraic Geometry 2: Sheaves and Cohomology, Volume 197.

Contents

Kodaira-Spencer mapping
Torelli's theorem
Period mappings and Hodge theory
Conformal field theory
Prospects and remaining problems
Bibliography
Solutions to problems
Index

Details:

Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Translations of Mathematical Monographs, Volume: 206
Subseries: Iwanami Series in Modern Mathematics
Publication Year: 2002
ISBN: 0-8218-2156-3
Paging: 300 pp.
Binding: Softcover