2003 XIII, 279 p. Hardcover
3-540-02234-1
In this text the authors consider the Korteweg-de
Vries (KdV)
equation (ut = - uxxx + 6uux) with periodic
boundary conditions.
Derived to describe long surface waves in
a narrow and shallow
channel, this equation in fact models waves
in homogeneous,
weakly nonlinear and weakly dispersive media
in general.
Viewing the KdV equation as an infinite dimensional,
and in fact
integrable Hamiltonian system, we first construct
action-angle
coordinates which turn out to be globally
defined. They make
evident that all solutions of the periodic
KdV equation are
periodic, quasi-periodic or almost-periodic
in time. Also, their
construction leads to some new results along
the way.
Subsequently, these coordinates allow us
to apply a general KAM
theorem for a class of integrable Hamiltonian
pde's, proving that
large families of periodic and quasi-periodic
solutions persist
under sufficiently small Hamiltonian perturbations.
The pertinent nondegeneracy conditions are
verified by
calculating the first few Birkhoff normal
form terms -- an
essentially elementary calculation.
Contents:
1. The Beginning. 2. Classical Background.
3. Birkhoff
Coordinates. 4. Perturbed KdV Equations.
5. The KAM Proof. 6.
Kuksin's Lemma. 7. Background Material. 8.
Psi-Functions and
Frequencies. 9. Birkhoff Normal Forms. 10.
Some Technicalities.
References. Index. Notation
Series: Ergebnisse der Mathematik und ihrer
Grenzgebiete. 3.
Folge / A Series of Modern Surveys in Mathematics.
Vol.. 45
2003 Approx. 330 p. Hardcover
0-387-00500-5
" This useful book, which grew out of
the author's lectures
at Berkeley, presents some 400 exercises
of varying degrees of
difficulty in classical ring theory, together
with complete
solutions, background information, historical
commentary,
bibliographic details, and indications of
possible improvements
or generalizations. The book should be especially
helpful to
graduate students as a model of the problem-solving
process and
an illustration of the applications of different
theorems in ring
theory. The author also discusses "the
folklore of the
subject: the 'tricks of the trade' in ring
theory, which are well
known to the experts in the field but may
not be familiar to
others, and for which there is usually no
good reference".
The problems are from the following areas:
the Wedderburn-Artin
theory of semisimple rings, the Jacobson
radical, representation
theory of groups and algebras, (semi)prime
rings, (semi)primitive
rings, division rings, ordered rings, (semi)local
rings, the
theory of idempotents, and (semi)perfect
rings. Problems in the
areas of module theory, category theory,
and rings of quotients
are not included, since they will appear
in a later book. "
(T. W. Hungerford, Mathematical Reviews)
Contents: * Preface * Notes to the Reader
* Wedderburn-Artin
Theory * Jacobson Radical Theory * Introduction
to Representation
Theory * Prime and Primitive Rings * Introduction
to Division
Rings * Ordered Structures in Rings * Local
Rings, Semilocal
Rings, and Idempotents, Perfect and Semiperfect
Rings * Name
Index * Subject Index
Series: Problem Books in Mathematics.
2003 Approx. 300 p. Hardcover
3-540-00820-9
Exploring the connections between arithmetic
and geometric
properties of algebraic varieties has been
the object of much
fruitful study for a long time, especially
in the case of curves.
The aim of the Summer School and Conference
on "Higher
Dimensional Varieties and Rational Points"
held in Budapest,
Hungary during September 2001 was to bring
together students and
experts from the arithmetic and geometric
sides of algebraic
geometry in order to get a better understanding
of the current
problems, interactions and advances in higher
dimension. The
lecture series and conference lectures assembled
in this volume
give a comprehensive introduction to students
and researchers in
algebraic geometry and in related fields
to the main ideas of
this rapidly developing area.
Contents:
C. Araujo and J. Kollar: Rational Curves
on Varieties.
J.-L. Colliot-Thelene: Points rationnels
sur les fibrations.
O. Debarre: Fano Varieties.
B. Hassett: Density of Rational Points on
K3 Surfaces and their
Symmetric Products.
J. Kollar: Rationally Connected Varieties
and Fundamental Groups.
S. J. Kovacs: Families of Varieties of General
Type: The
Shafarevich Conjecture and Related Problems.
Y. Tschinkel: Fujita's Program and Rational
Points.
Series: Bolyai Society Mathematical Studies.
Vol.. 12
2003 Approx. 630pp. Hardcover
3-540-44258-8
With breathtaking detail, Maria Georgiadou
sheds light on the
work and life of Constantin Caratheodory,
who until now has been
ignored by historians. In her thought-provoking
book, Georgiadou
maps out the mathematician?s oeuvre, life
and turbulent
historical surroundings. Descending from
the Greek elite of
Constantinople, Caratheodory graduated from
the military school
of Brussels, became engineer at the Assiout
dam in Egypt and
finally dedicated a lifetime to mathematics
and education. He
significantly contributed to: calculus of
variations, the theory
of point set measure, the theory of functions
of a real variable,
pdes, and complex function theory. An exciting
and well-written
biography, once started, difficult to put
down.
Contents: 1. Origin and the formative years.-
2. Academic Career
in Germany.- 3. The Asia Minor Project.-
4. A Scholar of World
Reputation.- 5. National Socialism and War.-
6. The Last years.
2003 XII, 178 p. Hardcover
0-387-00454-8
Sample data alone never suffice to draw conclusions
about
populations. Inference always requires assumptions
about the
population and sampling process. Statistical
theory has revealed
much about how strength of assumptions affects
the precision of
point estimates, but has had much less to
say about how it
affects the identification of population
parameters. Indeed, it
has been commonplace to think of identification
as a binary event
? a parameter is either identified or not
? and to view point
identification as a precondition for inference.
Yet there is
enormous scope for fruitful inference using
data and assumptions
that partially identify population parameters.
This book explains
why and shows how. The book presents in a
rigorous and thorough
manner the main elements of Charles Manski's
research on partial
identification of probability distributions.
One focus is
prediction with missing outcome or covariate
data. Another is
decomposition of finite mixtures, with application
to the
analysis of contaminated sampling and ecological
inference. A
third major focus is the analysis of treatment
response. Whatever
the particular subject under study, the presentation
follows a
common path. The author first specifies the
sampling process
generating the available data and asks what
may be learned about
population parameters using the empirical
evidence alone. He then
ask how the (typically) setvalued identification
regions for
these parameters shrink if various assumptions
are imposed. The
approach to inference that runs throughout
the book is
deliberately conservative and thoroughly
nonparametric.
Contents: Missing Outcomes * Instrumental
Variables * Conditional
Prediction with Missing Data * Contaminated
Outcomes *
Regressions, Short and Long * Response-Based
Sampling * Analysis
of Treatment Response * Monotone Treatment
Response * Monotone
Instrumental Variables * The Mixing Problem
Series: Springer Series in Statistics.