2003 Approx. 247p. 35 illus. Softcover
3-540-44160-3
The book gives a streamlined introduction
to quantum mechanics,
while describing the basic mathematical structures
underpinning
this discipline. Starting with the description
of key physical
experiments illustrating the origin of the
physical foundations,
the book proceeds to a description of the
basic notions of
quantum mechanics and their mathematical
content.
It then makes its way to topics of current
interest, specifically
those in which mathematics plays an important
role. The topics
presented include spectral theory, many-body
theory, positive
temperatures, path integrals and quasiclassical
asymptotics, the
theory of resonances, an introduction to
quantum field theory and
the theory of radiation.
The book can serve as a text for an intermediate
course in
quantum mechanics, or a more advanced topics
course.
Contents:
Physical background.- Mathematical detour:
operator theory.-
Dynamics.- Mathematical detour: the Fourier
transform.-
Observables.- The uncertainty principle.-
Spectral theory.-
Scattering states.- Special cases.- Many-particle
systems.-
Density matrices.- The Feynman path integral.-
Mathematical
detour: the calculus of variations.- Mathematical
detours: the
stationary phase method and operator determinants.-
Quasi-classical
analysis.-Resonances.- Introduction to quantum
field theory.-
Quantum electrodynamics of non-relativistic
particles: the theory
of radiation.- Supplement: renormalization
group.-Comments on
missing topics, literature, and further reading.
Series: Universitext.
2003 V, 194 p. Softcover
3-540-00351-7
The numerical treatment of partial differential
equations with
meshfree discretization techniques has been
a very active
research area in recent years. Up to now,
however, meshfree
methods have been in an early experimental
stage and were not
competitive due to the lack of efficient
iterative solvers and
numerical quadrature. This volume now presents
an efficient
parallel implementation of a meshfree method,
namely the
partition of unity method (PUM). A general
numerical integration
scheme is presented for the efficient assembly
of the stiffness
matrix as well as an optimal multilevel solver
for the arising
linear system. Furthermore, detailed information
on the parallel
implementation of the method on distributed
memory computers is
provided and numerical results are presented
in two and three
space dimensions with linear, higher order
and augmented
approximation spaces with up to 42 million
degrees of freedom.
Series: Lecture Notes in Computational Science
and Engineering.
Volume. 29
2003 X, 187 p. Softcover
3-540-00402-5
In image processing, "motions by curvature"
provide an
efficient way to smooth curves representing
the boundaries of
objects. In such a motion, each point of
the curve moves, at any
instant, with a normal velocity equal to
a function of the
curvature at this point. This book is a rigorous
and self-contained
exposition of the techniques of "motion
by curvature".
The approach is axiomatic and formulated
in terms of geometric
invariance with respect to the position of
the observer. This is
translated into mathematical terms, and the
author develops the
approach of Olver, Sapiro and Tannenbaum,
which classifies all
curve evolution equations. He then draws
a complete parallel with
another axiomatic approach using level-set
methods: this leads to
generalized curvature motions. Finally, novel,
and very accurate,
numerical schemes are proposed allowing one
to compute the
solution of highly degenerate evolution equations
in a completely
invariant way. The convergence of this scheme
is also proved.
Contents:
Preface.- Part I. The curve smoothing problem:
1. Curve evolution
and image processing; 2. Rudimentary bases
of curve geometry.-
Part II. Theoretical curve evolution: 3.
Geometric curve
shortening flow; 4. Curve evolution and level
sets.- Part III.
Numerical curve evolution: 5. Classical numerical
methods for
curve evolution; 6. A geometrical scheme
for curve evolution.-
Conclusion and perspectives.- A. Proof of
Thm. 4.3.4.- References.-
Index.
Series: Lecture Notes in Mathematics. Volume.
1805
2003 XIII, 169 p. Softcover
3-540-00403-3
The authors consider applications of singularity
theory and
computer algebra to bifurcations of Hamiltonian
dynamical systems.
They restrict themselves to the case were
the following
simplification is possible. Near the equilibrium
or (quasi-)
periodic solution under consideration the
linear part allows
approximation by a normalized Hamiltonian
system with a torus
symmetry. It is assumed that reduction by
this symmetry leads to
a system with one degree of freedom. The
volume focuses on two
such reduction methods, the planar reduction
(or polar
coordinates) method and the reduction by
the energy momentum
mapping. The one-degree-of-freedom system
then is tackled by
singularity theory, where computer algebra,
in particular,
Grobner basis techniques, are applied. The
readership addressed
consists of advanced graduate students and
researchers in
dynamical systems.
Contents:
Introduction.- I. Applications: Methods I:
Planar reduction;
Method II: The energy-momentum map.- II.
Theory: Birkhoff
Normalization; Singularity Theory; Grobner
bases and Standard
bases; Computing normalizing transformations.-
Appendix A.1.
Classification of term orders; Appendix A.2.
Proof of Proposition
5.8.- References.- Index.
Series: Lecture Notes in Mathematics. Volume.
1806
Reprint of the 1st ed. New York 1979 2003
Approx. 255 p. 5
illus. Softcover
3-540-00373-8
From the reviews:
"Theory of Stein Spaces provides a rich
variety of methods,
results, and motivations - a book with masterful
mathematical
care and judgement. It is a pleasure to have
this fundamental
material now readily accessible to any serious
mathematician."
J. Eells in Bulletin of the London Mathematical
Society (1980)
"Written by two mathematicians who played
a crucial role in
the development of the modern theory of several
complex
variables, this is an important book."
J.B. Cooper in Internationale Mathematische
Nachrichten (1979)
Contents:
Introduction.- Sheaf Theory.- Cohomology
Theory.- Coherence
Theory for Finite Holomorphic Maps.- Differential
Forms and
Dolbeault Theory.- Theorems A and B for Compact
Blocks Cm.- Stein
Spaces.- Applications of Theorems A and B.-
The Finiteness
Theorem.- Compact Riemann Surfaces.- Bibliography.-
Subject Index.-
List of Symbols.
Series: Classics in Mathematics.
2003 Approx. 331 p. 80 illus. Hardcover
3-540-00256-1
In many fields of modern mathematics specialised
scientific
software becomes increasingly important.
Hence, tremendous effort
is taken by numerous groups all over the
world to develop
appropriate solutions.
This book contains surveys and research papers
on mathematical
software and algorithms. The common thread
is that the field of
mathematical applications lies on the border
between algebra and
geometry. Topics include polyhedral geometry,
elimination theory,
algebraic surfaces, Grobner bases, triangulations
of point sets
and the mutual relationship. This diversity
is accompanied by the
abundance of available software systems which
often handle only
special mathematical aspects. Therefore the
volume's other focus
is on solutions towards the integration of
mathematical software
systems. This includes low-level and XML
based high-level
communication channels as well as general
framework for modular
systems.
Keywords: mathematical software, software
engineering
2nd ed. 2003 XXIV, 540 p. Hardcover
3-540-00384-3
The theory of large cardinals is currently
a broad mainstream of
modern set theory, the main area of investigation
for the
analysis of the relative consistency of mathematical
propositions
and possible new axioms for mathematics.
The first of a projected
multi-volume series, this book provides a
comprehensive account
of the theory of large cardinals from its
beginnings and some of
the direct outgrowths leading to the frontiers
of contemporary
research. A "genetic" approach
is taken, presenting the
subject in the context of its historical
development. With
hindsight the consequential avenues are pursued
and the most
elegant or accessible expositions given.
With open questions and
speculations provided throughout the reader
should not only come
to appreciate the scope and coherence of
the overall enterprise
but also become prepared to pursue research
in several specific
areas by studying the relevant sections.
Keywords: infinitary combinatorics, large
cardinals, relative
consistency results, set theory, new axioms
for set theory
Series: Springer Monographs in Mathematics.