Peyre, E., Universite de Grenoble I et CNRS, Saint-Martin d'Heres, France, Tschinkel, Y., Princeton University, USA, (Eds.)
Rational Points on Algebraic Varieties
Progress in Mathematics, Vol. 199
2001. Approx. 464 pages. Hardcover
ISBN 3-7643-6612-5
English
This book is devoted to the study of rational and integral points on higher-dimensional algebraic varieties. It contains carefully selected research papers and gives a glimpse of the state of the art of this rapidly expanding domain in arithmetic geometry. It is a valuable resource for graduate students and researchers in analytic number theory, arithmetic geometry and algebraic geometry.
This book is devoted to the study of rational and integral points on higher-dimensional algebraic varieties. It contains carefully selected research papers addressing the arithmetic geometry of varieties which are not of general type, with an emphasis on how rational points are distributed with respect to the classical, Zariski and adelic topologies.
The present volume gives a glimpse of the state of the art of this rapidly expanding domain in arithmetic geometry. The techniques involve explicit geometric constructions, ideas from the minimal model program in algebraic geometry as well as analytic number theory and harmonic analysis on adelic groups.
Davis, J.H., Queen's University, Kingston, Canada
Foundations of Deterministic and Stochastic Control
2002. Approx. 400 pages. Hardcover
ISBN 3-7643-4257-9
English
Control theory has applications to a number of areas in engineering and communication theory. This introductory text on the subject is fairly self-contained and aimed primarily at advanced mathematics and engineering students in various disciplines.
The topics covered include realization problems, linear-quadratic optimal control, stability theory, stochastic modeling and recursive estimation algorithms in communications and control, and distributed system modeling. These topics have a wide range of applicability, and provide background for further study in the control and communications areas.
In the early chapters the basics of linear control systems as well as the fundamentals of stochastic control are presented in a unique way so that the methods generalize to a useful class of distributed parameter and nonlinear system models. The control of distributed parameter systems (systems governed by PDEs) is based on the framework of linear quadratic Gaussian optimization problems. The approach here utilizes methods based on Wiener-Hopf integral equations.
Additionally, the important notion of state space modeling of distributed systems is examined. Basic results due to Gohberg and Krein on convolution are given and many results are illustrated with some examples that carry throughout the text.
The standard linear regulator problem is studied in both the continuous and discrete time cases, followed by a discussion of the (dual) filtering problems. Later chapters treat the stationary regulator and filtering problems with a Wiener-Hopf approach. This leads to spectral factorization problems and useful iterative algorithms that follow naturally from the methods employed. The interplay between time and frequency domain approaches is emphasized.
Table of Contents:
1 Introduction
2 State Space Realizations
3 Least Squares Control
4 Random Variables and Processes
5 Gaussian Variables
6 Kalman Bucy Filters
7 Continuous Time Models
8 The Separation Theorem
9 Luenberger Observers
10 Nonlinear and Finite State Problems
11 Wiener Hopf Methods
12 Distributed System Regulators
13 Filters Without Riccati Equations
14 Newton's Method for Riccati Equations
15 Numerical Spectral Factorization
A Problems
B Hilbert Spaces
C The Projection Theorem
D Measure Theoretic Probability
References
Index
Rollet, L., Universite Nancy, Nancy Cedex, France, (Ed.)
Henri Poincare: Scientific Opportunism / L'Opportunisme scientifique : An Anthology
2002. 234 pages. Hardcover
ISBN 3-7643-6539-0
English / French
During his lifetime, Henri Poincare published three major philosophical books which achieved great success: La science et l'hypothese (1902), La valeur de la science (1905) and Science et methode (1908). After his death, a fourth volume of his philosophical works was published by his heirs as Dernieres pensees (1913). Around 1919, Gustave Le Bon wrote to Poincare's widow. As the director of the Bibliotheque de Philosophie Scientifique at Flammarion, he asked her permission to publish a second posthumous volume. L'Opportunisme scientifique was intended to be the fifth and final volume of Poincare's philosophical writings. Louis Rougier had elaborated the project, with the collaboration of Gustave Le Bon, and the approval of the philosopher Emile Boutroux and his son Pierre. Because of the reservations of the mathematician's heirs, this book was never published and Dernieres pensees remained his last philosophical book. The aim of this book is to restore Rougier's project while throwing some light on its history and the early posterity of Poincare's philosophical thinking.
Steinbach, J., Augsburg, Germany
A Variational Inequality Approach to Free Boundary Problems with Applications in Mould Filling
International Series of Numerical Mathematics, ISNM 136
2002. Approx. 304 pages. Hardcover
ISBN 3-7643-6582-X
English
This monograph is devoted to the study of an evolutionary variational inequality approach to a degenerate moving free boundary problem both with respect to analytical and numerical methods. It is primarily addressed to applied mathematicians working in the field of nonlinear partial differential equations and their applications, especially in numerical treatment. However, the book will also be useful for scientists from the application areas, in particular, applied scientists from engineering and physics.
The inequality approach of obstacle type results from the application of an integral transformation. It takes an intermediate position between elliptic and parabolic inequalities and comprises an elliptic differential operator, a memory term and time-dependent convex constraint sets. The study of such inequality problems is motivated by applications to injection and compression moulding, to electro-chemical machining and other quasi-stationary Stefan type problems.
The mathematical analysis of the problem covers existence, uniqueness, regularity and time evolution of the solution. This is carried out in the framework of the variational inequality theory. The numerical solution in two and three space dimensions is discussed using both finite element and finite volume approximations. Finally, a description of injection and compression moulding is presented in terms of different mathematical models, a generalized Hele-Shaw flow, a distance concept and Navier-Stokes flow.
Bouwknegt, P., University of Adelaide, Australia,
Wu, S., University of Colorado, Boulder, USA
Geometric Analysis and Applications to Quantum Field Theory
Progress in Mathematics, PM 205
2002. Approx. 208 pages. Hardcover
ISBN 3-7643-4287-0
English
In recent years, there has been tremendous progress on the interface of geometry and mathematical physics. This book reflects the expanded articles of several lectures in these areas delivered at the University of Adelaide, with an audience of primarily graduate students.
The aim of this volume is to provide surveys of recent progress without assuming too much prerequisite knowledge and with a comprehensive bibliography, so that researchers and graduate students in geometry and mathematical physics will benefit.
The contributors cover a number of areas in mathematical physics.
Chapter 1 offers a self-contained derivation of the partition function of Chern-Simons gauge theory in the semiclassical approximation.
Chapter 2 considers the algebraic and geometric aspects of the Knizhnik-Zamolodchikov equations in conformal field theory, including their relation to the braid group, quantum groups and infinite dimensional Lie algebras.
Chapter 3 surveys the application of the representation theory of loop groups to simple models in quantum field theory and to certain integrable systems.
Chapter 4 examines the variational methods in Hermitian geometry from the viewpoint of the critical points of action functionals together with physical backgrounds.
Chapter 5 is a review of monopoles in non-Abelian gauge theories and the various approaches to understanding them.
Chapter 6 covers much of the exciting recent developments in quantum cohomology, including relative Gromov-Witten invariant, birational geometry, naturality and mirror symmetry.
Chapter 7 explains the physics origin of the Seiberg-Witten equations in four-manifold theory and a number of important concepts in quantum field theory, such as vacuum, mass gap, (super)symmetry, anomalies and duality.
Contributors: D.H. Adam, P. Bouwknegt, A.L. Carey, A. Harris, E. Langmann, M.K. Murray, Y. Ruan, S. Wu
D. H. Adams: Semiclassical Approximation in Chern-Simons Gauge Theory
P. Bouwknegt: The Knizhnik-Zamolodchikov Equations
A. L. Carey and E. Langmann: Loop Groups and Quantum Fields
A. Harris: Some Applications of Variational Calculus in Hermitian Geometry
M. K. Murray: Monopoles
Y. Ruan: On Gromov-Witten Invariants and Quantum Cohomology
S. Wu The Geometry and Physics of the Seiberg-Witten Equations