Demuth, M., Technische/Universitat Clausthal, Clausthal-Zellerfeld, Germany,
Casteren, J.A. van, /University of Antwerp, Belgium
Stochastic Spectral Theory for Selfadjoint Feller Operators
A functional integration approach
2000. 480 pages. Hardcover
Probability and its Applications
ISBN 3-7643-5887-4
A beautiful interplay between probability theory (Markov processes, martingale theory) on the one hand and operator and spectral theory on the other yields a uniform treatment of several kinds of Hamiltonians such as the Laplace operator, relativistic Hamiltonian, Laplace-Beltrami operator, and generators of Ornstein-Uhlenbeck processes. For such operators regular and singular perturbations of order zero and their spectral properties are investigated.
A complete treatment of the Feynman-Kac formula is given. The theory is applied to such topics as
compactness or trace class properties of differences of Feynman-Kac semigroups, preservation of absolutely
continuous and/or essential spectra and completeness of scattering systems.
The unified approach provides a new viewpoint of and a deeper insight into the subject. The book is aimed at
advanced students and researchers in mathematical physics and mathematics with an interest in quantum physics, scattering theory, heat equation, operator theory, probability theory and spectral theory.
Balakrishnan, A.V., Prof., University of California, Los Angeles, USA
Semigroups of Operators: Theory and Applications
International Conference in Newport Beach, December 14 - 18, 1998
Progress in Non-Linear Differential Equations vol.42
2000. 380 pages. Hardcover
ISBN 3-7643-6310-X
English
This volume contains a collection of refereed papers by eminent experts originating from the
"International Conference on Semigroups of Operators: Theory and Control", held in December 1998 in Newport Beach, California.
They highlight recent advances in the theory of semigroups of operators which provide the framework for the
time-domain solutions of time-invariant boundary value and initial value problems of partial differential
equations. There is a firewall between the abstract theory and the applications, and one of the conference
aims, which is reflected in this collection, was to bring them together for the benefit of both communities.
Dehornoy, P., Universite de Caen
Braids and Self-Distributivity
Progress in Mathematics vol. 192
2000. 648 pages. Hardcover
ISBN 3-7643-6343-6
English
The aim of this book is to present recently discovered connections between Artin's braid groups and left
self-distributive systems, which are sets equipped with a binary operation satisfying the identity x(yz) = (xy)(xz). Order properties are crucial.
In the 1980s new examples of left self-distributive systems were discovered using unprovable axioms of set theory, and purely algebraic statements were deduced.
The quest for elementary proofs of these statements led to a general theory of self-distributivity centered on a certain group that captures the geometrical properties of this identity. This group happens to be
closely connected with Artin's braid groups, and new properties of the braids naturally arose as an application, in particular the existence of a left invariant linear order, which subsequently received alternative
topological constructions. The text proposes a first synthesis of this area of research. Three domains are
considered here, namely braids, self-distributive systems, and set theory. Although not a comprehensive course on these subjects, the exposition is self-contained, and a number of basic results are established. In particular, the first chapters include a rather complete algebraic study of Artin's braid groups.
Galdi, G. P., Profs., University of Pittsburgh, USA,
Heywood, J. G., Univ. of British Columbia, Vancouver; Canada,
Rannacher, R., Universite Heidelberg, Germany,
(Ed.)
Fundamental Directions in Mathematical Fluid Mechanics
Advances in Mathematical Fluid Mechanics
Approx. 304 pages. Hardcover
ISBN 3-7643-6414-9
English
Due in August 2000
This set of six papers, written by eminent experts in the field, is concerned with that part of fluid mechanics that seeks its foundation in the rigorous mathematical treatment of the Navier-Stokes equations.
In particular, an overview is given on state of research regarding the global existence of smooth solutions, for which uniqueness and continuous dependence on the data can be proven. Then, the book moves on to a
discussion of recent developments of the finite element Galerkin method, with an emphasis on a priori and a posteriori error estimation and adaptive mesh refinement. A further article elaborates on spectral Galerkin
methods and their extension to domains with complicated geometries by employing the techniques of domain decomposition. The rigorous explanation of bifurcation phenomena in fluids has long been a central
topic in the theory of Navier-Stokes equations. Here, bifurcation theory is introduced in a general setting that is particularly convenient for application to such problems. Finally, the extension of Navier-Stokes theory to compressible viscous flows, studied in two more papers, opens up a fascinating panorama of
theoretical and numerical problems. While some of the contributions are expository, others primarily present new results within a wider context and fuller exposition than is usual for research papers. The book is meant to introduce researchers and advanced students to the research level on some of the most important topics of the field.
Jungel, A., University of Konstanz, Germany
Quasi-hydrodynamic Semiconductor Equations
PNLDE 41
Progress in Non-Linear Differential Equations vol.41
Approx. 296 pages. Hardcover
ISBN 3-7643-6349-5
English
Due in August 2000
In this book a hierarchy of macroscopic models for semiconductor devices is presented. Three classes of models are studied in detail: isentropic drift-diffusion equations, energy-transport models, and quantum
hydrodynamic equations.
The derivation of each of the models is shown, including physical discussions. Furthermore, the corresponding
mathematical problems are analyzed, using modern techniques for nonlinear partial differential equations. The equations are discretized employing mixed finite-element methods. Also, numerical simulations for
modern semiconductor devices are performed, showing the particular features of the models.
Modern analytical techniques have been used and further developed, such as positive solution methods, local energy methods for free-boundary problems and entropy methods.
The book is aimed at applied mathematicians and physicists interested in mathematics, as well as graduate and postdoc students and researchers in these fields.