Edited by: N. N. Uraltseva, St. Petersburg State University, Russia
Proceedings of the St. Petersburg Mathematical Society, Volume VIII
Expected publication date is May 9, 2002
Description
The articles in this collection present new results in partial differential equations, numerical analysis, probability theory, and geometry. The results, ideas, and methods given in the book will be of interest to a broad range of specialists.
Contents
A. N. Borodin -- Distribution of the supremum for linear combinations of local times of random processes
O. L. Vinogradov and V. V. Zhuk -- Sharp Jackson type inequalities for differentiable functions and minimization of the step of the moduli of continuity
Yu. K. Dem'yanovich -- Biorthogonal system of minimal splines on a nonuniform grid
V. V. Zhuk and G. I. Natanson -- S. N. Bernstein and direct and converse theorems of contructive function theory
I. A. Ibragimov -- On S. N. Bernstein's work in probability
N. M. Ivochkina -- Some trends in the development of the theory of fully nonlinear second order evolution equations
R. Lauter, B. A. Plamenevskii, and O. V. Sarafanov -- Coefficients in asymptotics of solutions to pseudodifferential equations on manifolds with conical points
G. A. Leonov -- The Brockett problem in the theory of nonstationary stabilization of linear differential equations
V. N. Malozemov and A. N. Sergeev -- Discrete nonperiodic splines on a uniform grid
A. I. Nazarov and Ya. Yu. Nikitin -- Some extremal problems for Gaussian and empirical random fields
F. V. Petrov and S. E. Rukshin -- Two theorems about convex polyhedra
Details:
Series: American Mathematical Society Translations--Series 2, Volume: 205
Publication Year: 2002
ISBN: 0-8218-2941-6
Paging: 208 pp.
Binding: Hardcover
Edited by: John Harnad, University of Montreal, PQ, Canada,
and Alexander Its, Indiana University - Purdue University, Indianapolis, IN
Isomonodromic Deformations and Applications in Physics
Description
The area of inverse scattering transform method or soliton theory has evolved over the past two decades in a vast variety of exciting new algebraic and analytic directions and has found numerous new applications. Methods and applications range from quantum group theory and exactly solvable statistical models to random matrices, random permutations, and number theory. The theory of isomonodromic deformations of systems of differential equations with rational coefficents, and most notably, the related apparatus of the Riemann-Hilbert problem, underlie the analytic side of this striking development.
The contributions in this volume are based on lectures given by leading experts at the CRM workshop (Montreal, Canada). Included are both survey articles and more detailed expositions relating to the theory of isomonodromic deformations, the Riemann-Hilbert problem, and modern applications.
The first part of the book represents the mathematical aspects of isomonodromic deformations; the second part deals mostly with the various appearances of isomonodromic deformations and Riemann-Hilbert methods in the theory of exactly solvable quantum field theory and statistical mechanical models, and related issues. The book elucidates for the first time in the current literature the important role that isomonodromic deformations play in the theory of integrable systems and their applications to physics.
Contents
Isomonodromic deformations
A. Bolibruch -- Inverse problems for linear differential equations with meromorphic coefficients
J. Harnad -- Virasoro generators and bilinear equations for isomonodromic tau functions
A. A. Kapaev -- Lax pairs for Painleve equations
D. A. Korotkin -- Isomonodromic deformations and Hurwitz spaces
Y. Ohyama -- Classical solutions of Schlesinger equations and twistor theory
M. A. Olshanetsky -- $W$-geometry and isomonodromic deformations
C. A. Tracy and H. Widom -- Airy kernel and Painleve II
Applications in physics and related topics
M. Bertola -- Jacobi groups, Jacobi forms and their applications
P. A. Clarkson and C. M. Cosgrove -- Symmetry, the Chazy equation and Chazy hierarchies
F. Gohmann -- Universal correlations of one-dimensional electrons at low density
F. Gohmann and V. E. Korepin -- A quantum version of the inverse scattering transformation
Y. Nakamura -- Continued fractions and integrable systems
A. Yu. Orlov and D. M. Scherbin -- Hypergeometric functions related to Schur functions and integrable systems
J. Palmer -- Ising model scaling functions at short distance
N. A. Slavnov -- The partition function of the six-vertex model as a Fredholm determinant
Details:
Series: CRM Proceedings & Lecture Notes, Volume: 31
Publication Year: 2002
ISBN: 0-8218-2804-5
Paging: 218 pp.
Binding: Softcover
Jose Bertin / Jean-Pierre Demailly / Luc Illusie / Chris Peters
Introduction to Hodge Theory
Expected publication date is May 1, 2002
Description
Hodge theory originated as an application of harmonic theory to the study of the geometry of compact complex manifolds. The ideas have proved to be quite powerful, leading to fundamentally important results throughout algebraic geometry. This book consists of expositions of various aspects of modern Hodge theory. Its purpose is to provide the nonexpert reader with a precise idea of the current status of the subject. The three chapters develop distinct but closely related subjects: $L^2$ Hodge theory and vanishing theorems; Frobenius and Hodge degeneration; variations of Hodge structures and mirror symmetry. The techniques employed cover a wide range of methods borrowed from the heart of mathematics: elliptic PDE theory, complex differential geometry, algebraic geometry in characteristic $p$, cohomological and sheaf-theoretic methods, deformation theory of complex varieties, Calabi-Yau manifolds, singularity theory, etc. A special effort has been made to approach the various themes from their most natural starting points. Each of the three chapters is supplemented with a detailed introduction and numerous references. The reader will find precise statements of quite a number of open problems that have been the subject of active research in recent years.
The reader should have some familiarity with differential and algebraic geometry, with other prerequisites varying by chapter. The book is suitable as an accompaniment to a second course in algebraic geometry.
Contents
J.-P. Demailly -- $L^2$ Hodge theory and vanishing theorems
L. Illusie -- Frobenius and Hodge degeneration
J. Bertin and C. Peters -- Variations of Hodge structure, Calabi-Yau manifolds and mirror symmetry
Details:
Series: SMF/AMS Texts and Monographs, Volume: 8
Publication Year: 2002
ISBN: 0-8218-2040-0
Paging: 232 pp.
Binding: Softcover
Victor M. Buchstaber and Taras E. Panov, Moscow State University, Russia
Torus Actions and Their Applications in Topology and Combinatorics
Expected publication date is May 16, 2002
Description
Here, the study of torus actions on topological spaces is presented as a bridge connecting combinatorial and convex geometry with commutative and homological algebra, algebraic geometry, and topology. This established link helps in understanding the geometry and topology of a space with torus action by studying the combinatorics of the space of orbits. Conversely, subtle properties of a combinatorial object can be realized by interpreting it as the orbit structure for a proper manifold or as a complex acted on by a torus. The latter can be a symplectic manifold with Hamiltonian torus action, a toric variety or manifold, a subspace arrangement complement, etc., while the combinatorial objects include simplicial and cubical complexes, polytopes, and arrangements. This approach also provides a natural topological interpretation in terms of torus actions of many constructions from commutative and homological algebra used in combinatorics.
The exposition centers around the theory of moment-angle complexes, providing an effective way to study invariants of triangulations by methods of equivariant topology. The book includes many new and well-known open problems and would be suitable as a textbook. It will be useful for specialists both in topology and in combinatorics and will help to establish even tighter connections between the subjects involved.
Contents
Introduction
Polytopes
Topology and combinatorics of simplicial complexes
Commutative and homological algebra of simplicial complexes
Cubical complexes
Toric and quasitoric manifolds
Moment-angle complexes
Cohomology of moment-angle complexes and combinatorics of triangulated manifolds
Cohomology rings of subspace arrangement complements
Bibliography
Index
Details:
Series: University Lecture Series, Volume: 24
Publication Year: 2002
ISBN: 0-8218-3186-0
Paging: 144 pp.
Binding: Softcover
Nikolai K. Nikolski, University of Bordeaux I, Talence, France
Operators, Functions, and Systems: An Easy Reading:
Volume 2: Model Operators and Systems
Expected publication date is May 11, 2002
Description
This unique work combines together in two volumes four formally distinct topics of modern analysis and its applications:
A. Hardy classes of holomorphic functions
B. Spectral theory of Hankel and Toeplitz operators
C. Function models for linear operators and free interpolations, and
D. Infinite-dimensional system theory and signal processing
Volume I, Hardy, Hankel, and Toeplitz, contains parts A and B; this volume, Volume II, contains Parts C and D.
Hardy classes of holomorphic functions: This topic is known to be the most powerful tool of complex analysis for a variety of applications, starting with Fourier series, through the Riemann $\zeta$-function, all the way to Wiener's theory of signal processing.
Spectral theory of Hankel and Toeplitz operators: These now become the supporting pillars for a large part of harmonic and complex analysis and for many of their applications. In this book, moment problems, Nevanlinna-Pick and Caratheodory interpolation, and the best rational approximations are considered to illustrate the power of Hankel and Toeplitz operators.
Function models for linear operators and free interpolations: This is a universal topic and, indeed, is the most influential operator theory technique in the post-spectral-theorem era. In this book, its capacity is tested by solving generalized Carleson-type interpolation problems.
Infinite-dimensional system theory and signal processing: This topic is the touchstone of the three previously developed techniques. The presence of this applied topic in a pure mathematic environment reflects important changes in the mathematical landscape of the last 20 years, in that the role of the main consumer and customer of harmonic, complex, and operator analysis has more and more passed from differential equations, scattering theory, and probability, to control theory and signal processing.
These volumes are geared toward a wide audience of readers, from graduate students to professional mathematicians. They develop an elementary approach while retaining an expert level that can be applied in advanced analysis and selected applications.
Contents
Model operators and free interpolation
Contents
Foreword to part C
The basic function model
Elements of spectral theory in the language of the characteristic function
Decompositions into invariant subspaces and free interpolation
Analytic problems in linear system theory
Contents
Foreword to part D
Basic theory
First optimizations: Multiplicity of the spectrum and the DISC
Eigenvector decompositions, vector valued exponentials, and squared optimization
A glance at bases of exponentials and of reproducing kernels
A brief introduction to $H^{\infty}$ control
Bibliography
Author index
Subject index
Symbol index
Errata to volume 1
Details:
Series: Mathematical Surveys and Monographs, Volume: 93
Publication Year: 2002
ISBN: 0-8218-2876-2
Paging: 439 pp.
Binding: Hardcover