He'lein, F., Ecole Normale Supe'rieure de Cachan, Cachan Cedex, France

Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems

Lectures in Mathematics - ETH Zurich

2001. Approx. 124 pages. Softcover
ISBN 3-7643-6576-5
English

Due in June 2001

This book intends to give an introduction to harmonic maps between a surface and a symmetric manifold and constant mean curvature surfaces as completely integrable systems. The presentation is accessible to undergraduate and graduate students in mathematics but will also be useful to researchers. It is among the first textbooks about integrable systems, their interplay with harmonic maps and the use of loop groups, and it presents the theory, for the first time, from the point of view of a differential geometer. The most important results are exposed with complete proofs (except for the last two chapters, which require a minimal knowledge from the reader). Some proofs have been completely rewritten with the objective, in particular, to clarify the relation between finite mean curvature tori, Wente tori and the loop group approach - an aspect largely neglected in the literature. The book helps the reader to access the ideas of the theory and to acquire a unified perspective of the subject.

Karapetiants, N., Rostov State University, Russia,
Samko, S., University of Algarve, Faro, Portugal

Equations with Involutive Operators

2000. Approx. 400 pages. Hardcover
ISBN 3-7643-4157-2 English

Due in July 2001

This text demonstrates a fascinating interplay between abstract and concrete operator theory. The focus is on the investigation of a number of equations, which, while seemingly different, are all unified by the same idea: they are all realizations of some operator equations in Banach spaces. One permeating theme throughout involves the role of the Fredholm property of these equations.

This text demonstrates an important interplay between abstract and concrete operator theory. The focus is on the investigation of a number of equations, which, while seemingly different, are all unified by the same idea: they are all realizations of some operator equations in Banach spaces. One permeating theme in these equations involves the role of the Fredholm property.
The work is carefully written, is self-contained and covers a broad range of topics and results. Key ideas are developed in a step-by step approach, beginning with the required background material and culminating in the final chapters with state-of-the art topics. Experts in operator theory, integral equations and function theory as well as students in these areas will find open problems for further investigations. Good examples, bibliography and index make this text a valuable reference resource as well.

Table of Contents:

Introduction
Notation
Chapter I. On Fredholmness of Singular and Convolution Operators
Chapter II. On Fredholmness of Other Singular-Type Operators
Chapter III. Functional and Singular Integral Equations with Carleman Shifts in Case of Continuous Coefficients
Chapter IV. Two Term Equations $(A+Qb)\Varphi =F$ with an Involutive Operator $Q$: An Abstract Approach and Applications
Chapter V. Equations with Several Generalized Involutive Operators: A Matrix Abstract Approach and Applications
Chapter VI. Application of the Abstract Approach to Singular Equations on the Real Line with Fractional-Linear Shift
Index

Wallis, W.D., Southern Illinois University, Carbondale, USA

Magic Graphs

2001. Approx. 144 pages. Softcover
ISBN 3-7643-4252-8

Due in July 2001

This concise, self-contained book is unique in its focus on the theory of magic graphs/labeling and its applications to a number of new areas, e.g., networks, the construction of rulers, and pulse codes. It may serve as a graduate text for a special topics seminar in mathematics or computer science, or as a professional text for the researcher. Some key features: concise exposition from basic topics in graph theory to current research; theorems from graph theory and interesting counting arguments.

Magic squares, their origins lost in antiquity, are among the more popular mathematical recreations. Over the years a number of generalizations have been proposed, going back in the last century to Sedla'vcek (early 1960s) who asked whether "magic" ideas could be applied to graphs. Around the same time Kotzig and Rosa formulated the study of graph labelings, or valuations as they were first called.
In the last decade, there has been a resurgence of interest in "magic labelings" and other graph valuations, e.g., graceful labelings, due to a number of interesting results that have applications and are related to the problems of decomposing graphs into trees. Trees remain an elusive subject. From the pure mathematics viewpoint, no progress has been made in answering the question: Does every tree have an edge-magic total labeling? However, the corresponding problem for vertex-magic total labelings has been solved, and the details are examined in this volume. The book also contains a number of recent constructions of magic graphs and verifications that families of graphs are magic.

Table of Contents:

Preface
List of Figures
1. Preliminaries
1.1 Magic
1.2 Graphs
1.3 Labelings
1.4 Magic Labeling
1.5 Some Applications of Magic Labelings
2. Edge-Magic Total Labelings
2.1 Basic Ideas
2.2 Graphs with No EMTL
2.3 Cliques and Complete Graphs
2.4 Cycles
2.5 Complete Bipartite Graphs
2.6 Wheels
2.7 Trees
2.8 Disconnected Graphs
2.9 Strong Edge-Magic Total Labelings
2.10 Edge-Magic Injections
3. Vertex-Magic Total Labelings
3.1 Basic Ideas
3.2 Regular Graphs
3.3 Cycles and Paths
3.4 VMTLs of Wheels
3.5 VMTLs of Complete Bipartite Graphs
3.6 Graphs on Vertices of Degree One
3.7 The Complete Graphs
3.8 Disconnected Graphs
3.9 Vertex-Magic Injections
4. Totally Magic Labelings
4.1 Basic Ideas
4.2 Isolates and Stars
4.3 Forbidden Configurations
4.4 Unions of Triangles
4.5 Small Graphs
4.6 Totally Magic Injections
Bibliography
Solutions to Selected Exercises
Index

Freistu"hler, H., Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, Szepessy, A., Royal Institute of Technology, Stockholm, Sweden, (Eds.)

Recent Advances in the Theory of Shock Waves

Progress in Nonlinear Differential Equations vol 47.
2001. Approx. 480 pages. Hardcover
ISBN 3-7643-4187-4 English

Due in July 2001

This volume provides a comprehensive treatment of central themes in the modern mathematical theory of shock waves. Authored by leading scientists in the area, the five unified articles cover:
the Cauchy problem for hyperbolic systems of conservation laws (T.-P.Liu)
the stability problem for shock waves in the hyperbolic (inviscid) setting (G. Me'tivier)
shock wave solutions of the Einstein-Euler equations of General Relativity (J. Smoller and B. Temple)
fundamental properties of hyperbolic systems with relaxation (W.-A. Yong)
the stability problem for shock waves in the parabolic (viscous) setting (K. Zumbrun)
Recent Advances in the Theory of Shock Waves ((kursiv)) combines the rigor of mathematical analysis with attention to the physical origins of the field. The topics covered provide ideal starting points for seminars and courses for mathematicians, physicists, and theoretically motivated engineers.

From the contents:

Preface
"Well Posedness Theory for a System of Hyperbolic Conservation Laws "/ T.-P. LIU
"Stability of Multidimensional Weak Shocks" / G. METIVIER
"Shock Wave Solutions of the Einstein Equations: A General Theory with Examples" / J. SMOLLER and B. TEMPLE
"Basic Aspects of Hyperbolic Relaxation Systems" / W.-A. YONG
"Multidimensional Stability of Planar Viscous Shock Waves" / K. ZUMBRUN

Cruzeiro, A.B., Zambrini, J.-C., both University of Lisbon, Portugal, (Eds.)

Stochastic Analysis and Mathematical Physics

Progress in Probability vol. 50.

2001. Approx. 168 pages. Hardcover
ISBN 3-7643-4246-3

Due in June 2001

Nine survey articles in this volume extend concepts from classical probability and stochastic processes to a number of areas of mathematical physics.
Key topics covered: nonlinear stochastic wave equations, completely positive maps, Mehler-type semigroups on Hilbert spaces, entropic projections, martingale problem and Markov uniqueness of infinite-dimensional Nelson diffusions, analysis in geometric probability theory, measure-preserving shifts on the Wiener space, cohomology on loop spaces, and stochastic Volterra equations

The work, an outgrowth of a workshop on stochastic analysis held in Lisbon, serves as a good reference text for researchers and advanced students in the fields of probability, stochastic processes, analysis, geometry, mathematical physics, and physics.

Table of Contents:

Preface
1. Functorial Analysis in Geometric Probability Theory (H. Airault/ P. Malliavin)
2. Stochastic Volterra Equations with Singular Kernels (L. Coutin/ L. Decreusefond)
3. Stochastic Diffeology and Homotopy (R. Le'andre)
4. Some Results on Entropic Projections (C. Le'onard)
5. Mehler-type Semigroups on Hilbert Spaces and Their Generators (P. Lescot)
6. Singular Limiting Behavior in Nonlinear Stochastic Wave Equations (M. Oberguggenberger/ F. Russo)
7. Complete Positivity and Open Quantum Systems (R. Rebolledo)
8. Properties of Measure-preserving Shifts on the Wiener Space (A.S. Ustunel)
9. Martingale and Markov Uniqueness of Infinite Dimensional Nelson Diffusions (L. Wu)