Expected publication date is January 11, 2003
Description
This is the first introductory book on the theory of
prehomogeneous vector spaces, introduced in the 1970s by Mikio
Sato. The author was an early and important developer of the
theory and continues to be active in the field.
The subject combines elements of several areas of mathematics,
such as algebraic geometry, Lie groups, analysis, number theory,
and invariant theory. An important objective is to create
applications to number theory. For example, one of the key topics
is that of zeta functions attached to prehomogeneous vector
spaces; these are generalizations of the Riemann zeta function, a
cornerstone of analytic number theory. Prehomogeneous vector
spaces are also of use in representation theory, algebraic
geometry and invariant theory.
This book explains the basic concepts of prehomogeneous vector
spaces, the fundamental theorem, the zeta functions associated
with prehomogeneous vector spaces and a classification theory of
irreducible prehomogeneous vector spaces. It strives, and to a
large extent succeeds, in making this content, which is by its
nature fairly technical, self-contained and accessible. The first
section of the book, "Overview of the theory and contents of
this book," Is particularly noteworthy as an excellent
introduction to the subject.
Contents
Overview of the theory and contents of this book
Algebraic preliminaries
Relative invariants of prehomogeneous vector spaces
Analytic preliminaries
The fundamental theorem of prehomogeneous vector spaces
The zeta functions of prehomogeneous vector spaces
Convergence of zeta functions of prehomogeneous vector spaces
Classification of prehomogeneous vector spaces
Appendix: Table of irreducible reduced prehomogeneous vector
spaces
Bibliography
Index of symbols
Index
Details:
Series: Translations of Mathematical Monographs, Volume: 215
Publication Year: 2002
ISBN: 0-8218-2767-7
Paging: 288 pp.
Binding: Hardcover
Expected publication date is January 12, 2003
Description
Masaki Kashiwara is undoubtedly one of the masters of the theory
of D-modules, and he has created a good, accessible entry point
to the subject. The theory of D-modules is a very powerful point
of view, bringing ideas from algebra and algebraic geometry to
the analysis of systems of differential equations. It is often
used in conjunction with microlocal analysis, as some of the
important theorems are best stated or proved using these
techniques. The theory has been used very successfully in
applications to representation theory.
Here, there is an emphasis on b-functions. These show up in
various contexts: number theory, analysis, representation theory,
and the geometry and invariants of prehomogeneous vector spaces.
Some of the most important results on b-functions were obtained
by Kashiwara.
A hot topic from the mid '70s to mid '80s, it has now moved a bit
more into the mainstream. Graduate students and research
mathematicians will find that working on the subject in the two-decade
interval has given Kashiwara a very good perspective for
presenting the topic to the general mathematical public.
Contents
Basic properties of D-modules
Characteristic varieties
Construction of mathcal{D}
Functorial properties of mathcal{D}
Regular holonomic systems
b-functions
Ring of formal microdifferential operators
Microlocal analysis of holonomic systems
Microlocal calculus of b-functions
Appendix
Bibliography
Index
Index of notations
Other titles in this series
Details:
Series: Translations of Mathematical Monographs,Volume: 217
Publication Year: 2003
ISBN: 0-8218-2766-9
Paging: 254 pp.
Binding: Softcover
Description
This book contains the proceedings of the Special Session,
Interaction of Inverse Problems and Image Analysis, held at the
January 2001 meeting of the AMS in New Orleans, LA.
The common thread among inverse problems, signal analysis, and
image analysis is a canonical problem: recovering an object (function,
signal, picture) from partial or indirect information about the
object. Both inverse problems and imaging science have emerged in
recent years as interdisciplinary research fields with profound
applications in many areas of science, engineering, technology,
and medicine. Research in inverse problems and image processing
shows rich interaction with several areas of mathematics and
strong links to signal processing, variational problems, applied
harmonic analysis, and computational mathematics.
This volume contains carefully referred and edited original
research papers and high-level survey papers that provide
overview and perspective on the interaction of inverse problems,
image analysis, and medical imaging.
The book is suitable for graduate students and researchers
interested in signal and image processing and medical imaging.
Contents
R. B. Alexeev and A. B. Smirnova -- Regularization of nonlinear
unstable operator equations by secant methods with application to
gravitational sounding problem
J. J. Benedetto and S. Sumetkijakan -- A fractal set constructed
from a class of wavelet sets
M. Boutin -- Joint invariant signatures for curve recognition
T. F. Chan and J. Shen -- Inpainting based on nonlinear transport
and diffusion
U. Clarenz, M. Droske, and M. Rumpf -- Towards fast non-rigid
registration
C. De Mol and M. Defrise -- A note on wavelet-based inversion
algorithms
M. El-Gamel and A. I. Zayed -- A comparison between the wavelet-Galerkin
and the Sinc-Galerkin methods in solving nonhomogeneous heat
equations
B. Fischer and J. Modersitzki -- Fast diffusion registration
C. W. Groetsch and O. Scherzer -- Iterative stabilization and
edge detection
F. A. Grunbaum -- Backprojections in tomography, spherical
functions and addition formulas: A few challenges
B. A. Mair and J. A. Zahnen -- Mathematical models for 2d
positron emission tomography
O. Scherzer -- Explicit versus implicit relative error
regularization on the space of functions of bounded variation
F. Stenger, A. R. Naghsh-Nilchi, J. Niebsch, and R. Ramlau --
Sampling methods for approximate solution of pde
J. Weickert and T. Brox -- Diffusion and regularization of vector-
and matrix-valued images
I. Yamada, N. Ogura, and N. Shirakawa -- A numerically robust
hybrid steepest descent method for the convexly constrained
generalized inverse problems
Details:
Series: Contemporary Mathematics, Volume: 313
Publication Year: 2002
ISBN: 0-8218-2979-3
Paging: 305 pp.
Binding: Softcover
PMP - Progress in Mathematical Physics
2002. Approx. 200 pages. Hardcover
ISBN 0-8176-4282-X
English
Due in November 2002
Several well-established geometric and topological methods are
used in this work on a beautiful and important physical
phenomenon known as the 'geometric phase.' Going back to the
intense interest in this subject since the mid-1980s and the
seminal work of M. Berry and B. Simon, this book examines
geometric phases, bringing together different physical phenomena
under a unified mathematical scheme.
Key background material, beginning with the notion of manifolds
and differential forms, as well as basic mathematical tools -
fiber bundles, connections and holonomies - are presented in
Chapter 1. Topological invariants such as Chern classes and
homotopy theory are explained in simple, concrete language with
emphasis on physical applications.
The exposition then unfolds systematically. The adiabatic phases
of Berry, the Wilczek-Zee nonabelian factor, and a classical
counterpart called Hannay's angles focus on the physical side of
the geometric phase problem. Thereafter the geometry of quantum
evolution is treated. Here the reader learns about different
geometries (such as symplectic and metric structures) living on a
quantum phase space in connection with both abelian and
nonabelian geometric phases.
The concluding section on Examples and Applications paves the way
for a continuing study of the geometric PHASES. Throughout the
text, material is presented on a level suitable for graduate
students and researchers in applied mathematics and physics with
an understanding of classical and quantum mechanics.
Table of Contents
Introduction
0. PRELIMINARY MATHEMATICAL BACKGROUND
I. THE ADIABATIC PHASE
1. The Adiabatic Phase in Quantum Mechanics
2. The Adiabatic Phase in Classical Mechanics
3. Adiabatic phase and holonomy
II. DIFFERENTIAL GEOMETRY AND THE GEOMETRIC PHASE
4. Geometry of a sphere and the geometric phase
III. THE GEOMETRY OF QUANTUM EVOLUTION
5. The Aharonov-Anandan phase
6. Geometric phase for a non-cyclic evolution
IV. EXAMPLES AND APPLICATIONS
7.Harmonic oscillator
8. Geometric phase in optics
9. Geometric phase in molecular systems
10. Geometric phase and the motion in noninertial frames
Mathematical Appendices
References
Index
2002. Approx. 184 pages. Softcover
ISBN 0-8176-4274-9
English
Due in December 2002
This work examines derivatives and integrals of functions of
several real variables. Topics from advanced calculus are
covered, including: differentiability and its relation to partial
derivatives, directional derivatives and the gradient, surfaces,
inverse and implicit functions, integrability and properties of
integrals, and the theorems of Fubini, Stokes, and Gauss.
The order of topics reflects the order of development in calculus:
limits, continuity, derivatives, integrals - a sequencing that
allows generalizations from and analogies to elementary results,
such as the second-derivative test and the Chain Rule.
Derivatives and Integrals of Multivariable Functions has a
definition-theorem-proof format, together with a conversational
style, including historical comments, an abundance of questions,
and discussions about strategy, difficulties, and alternative
paths. It is aimed at advanced undergraduate pure mathematics
majors whose next course will be real analysis with measure
theory. Required background includes theoretical work in linear
algebra, one-variable calculus, properties of continuous
functions, and related topological material. The last two are
used in the context of Euclidean space, but a strong grounding in
the corresponding real-line topics will suffice.
Table of Contents
1. Differentiability Of Multivariable Functions
2. Derivatives Of Scalar Functions
3. Derivatives Of Vector Functions
4. Integrability Of Multivariable Functions
5. Integrals Of Scalar Functions
6. Vector Integrals And The Field Theorems
Bibliography
Index