Expected publication date is January 9, 2004
Description
Ramsey theory is the study of the structure
of mathematical
objects that is preserved under partitions.
In its full
generality, Ramsey theory is quite powerful,
but can quickly
become complicated. By limiting the focus
of this book to Ramsey
theory applied to the set of integers, the
authors have produced
a gentle, but meaningful, introduction to
an important and
enticing branch of modern mathematics. Ramsey
Theory on the
Integers offers students something quite
rare for a book at this
level: a glimpse into the world of mathematical
research and the
opportunity for them to begin pondering unsolved
problems.
In addition to being the first truly accessible
book on Ramsey
theory, this innovative book also provides
the first cohesive
study of Ramsey theory on the integers. It
contains perhaps the
most substantial account of solved and unsolved
problems in this
blossoming subarea of Ramsey theory. The
result is a breakthrough
book that will engage students, teachers,
and researchers alike.
Contents
Preliminaries
Van der Waerden's theorem
Supersets of AP
Subsets of AP
Other generalizations of w(k;r)
Arithmetic progressions (mod m)
Other variations on van der Waerden's theorem
Schur's theorem
Rado's theorem
Other topics
Notation
Biobliography
Index
Details:
Series: Student Mathematical Library, Volume:
24
Publication Year: 2003
ISBN: 0-8218-3199-2
Paging: 317 pp.
Binding: Softcover
Expected publication date is December 23,
2003
Description
Singular integral operators play a central
role in modern
harmonic analysis. Simplest examples of singular
kernels are
given by Calderon-Zygmund kernels. Many important
properties of
singular integrals have been thoroughly studied
for Calderon-Zygmund
operators.
In the 1980's and early 1990's, Coifman,
Weiss, and Christ
noticed that the theory of Calderon-Zygmund
operators can be
generalized from Euclidean spaces to spaces
of homogeneous type.
The purpose of this book is to make the reader
believe that
homogeneity (previously considered as a cornerstone
of the theory)
is not needed. This claim is illustrated
by presenting two
harmonic analysis problems famous for their
difficulty.
The first problem treats semiadditivity of
analytic and Lipschitz
harmonic capacities. The volume presents
the first self-contained
and unified proof of the semiadditivity of
these capacities. The
book details Tolsa's solution of Painleve's
and Vitushkin's
problems and explains why these are problems
of the theory of
Calderon-Zygmund operators on nonhomogeneous
spaces. The
exposition is not dimension-specific, which
allows the author to
treat Lipschitz harmonic capacity and analytic
capacity at the
same time.
The second problem considered in the volume
is a two-weight
estimate for the Hilbert transform. This
problem recently found
important applications in operator theory,
where it is intimately
related to spectral theory of small perturbations
of unitary
operators.
The book presents a technique that can be
helpful in overcoming
rather bad degeneracies (i.e., exponential
growth or decay) of
underlying measure (volume) on the space
where the singular
integral operator is considered. These situations
occur, for
example, in boundary value problems for elliptic
PDE's in domains
with extremely singular boundaries. Another
example involves
harmonic analysis on the boundaries of pseudoconvex
domains that
goes beyond the scope of Carnot-Caratheodory
spaces.
The book is suitable for graduate students
and research
mathematicians interested in harmonic analysis.
Contents
Introduction
Preliminaries on capacities
Localization of Newton and Riesz potentials
From distribution to measure. Carleson property
Potential neighborhood that has properties
(3.13)-(3.14)
The tree of the proof
The first reduction to nonhomogeneous Tb
theorem
The second reduction
The third reduction
The fourth reduction
The proof of nonhomogeneous Cotlar's lemma.
Arbitrary measure
Starting the proof of nonhomogeneous nonaccretive
Tb theorem
Next step in theorem 10.6. Good and bad functions
Estimate of the diagonal sum. Remainder in
theorem 3.3
Two-weight estimate for the Hilbert transform.
Preliminaries
Necessity in the main theorem
Two-weight Hilbert transform. Towards the
main theorem
Long range interaction
The rest of the long range interaction
The short range interaction
Difficult terms and several paraproducts
Two-weight Hilbert transform and maximal
operator
Bibliography
Details:
Series: CBMS Regional Conference Series in
Mathematics, Number:
100
Publication Year: 2003
ISBN: 0-8218-3252-2
Paging: 167 pp.
Binding: Softcover
Expected publication date is January 8, 2004
Description
Science and engineering have been great sources
of problems and
inspiration for generations of mathematicians.
This is probably
true now more than ever as numerous challenges
in science and
technology are met by mathematicians. One
of these challenges is
understanding propagation of waves of different
nature in systems
of complex structure.
This book contains the proceedings of the
research conference,
"Waves in Periodic and Random Media".
Papers are
devoted to a number of related themes, including
spectral theory
of periodic differential operators, Anderson
localization and
spectral theory of random operators, photonic
crystals, waveguide
theory, mesoscopic systems, and designer
random surfaces.
Contributions are written by prominent experts
and are of
interest to researchers and graduate students
in mathematical
physics.
Contents
J. M. Combes, B. Gralak, and A. Tip -- Spectral
properties of
absorptive photonic crystals
J.-M. Combes, P. D. Hislop, and F. Klopp
-- Regularity properties
for the density of states of random Schrodinger
operators
P. Exner -- Spectral properties of Schrodinger
operators with a
strongly attractive delta interaction supported
by a surface
L. Friedlander -- Absolute continuity of
the spectra of periodic
waveguides
F. Germinet and A. Klein -- The Anderson
metal-insulator
transport transition
S. E. Golowich and M. I. Weinstein -- Theory
and computation of
scattering resonances of photonic microstructures
Yu. Karpeshina -- Asymptotic formulas for
eigenvalues and
eigenfunctions of periodic magnetic Schrodinger
operators
P. Kuchment and B. Ong -- On guided waves
in photonic crystal
waveguides
T. A. Leskova, A. A. Maradudin, E. R. Mendez,
and J. Munoz-Lopez
-- The design and photofabrication of random
achromatic optical
diffusers for uniform illumination
B. Pavlov and K. Robert -- Resonance optical
switch: Calculation
of resonance eigenvalues
A. V. Sobolev -- High energy asymptotics
of the density of states
for certain periodic pseudo-differential
operators in dimension
one
T. A. Suslina -- On discrete spectrum in
the gaps of a two-dimensional
periodic elliptic operator perturbed by a
decaying potential
V. V. Yurinsky -- On the principal eigenvalue
of the Schrodinger
operator with a scaled random potential
Details:
Series: Contemporary Mathematics, Volume:
339
Publication Year: 2003
ISBN: 0-8218-3286-7
Paging: approximately 232 pp.
Binding: Softcover
Expected publication date is January 15,
2004
Description
The book surveys some recent developments
in the arithmetic of
modular elliptic curves. It places a special
emphasis on the
construction of rational points on elliptic
curves, the Birch and
Swinnerton-Dyer conjecture, and the crucial
role played by
modularity in shedding light on these two
closely related issues.
The main theme of the book is the theory
of complex
multiplication, Heegner points, and some
conjectural variants.
The first three chapters introduce the background
and
prerequisites: elliptic curves, modular forms
and the Shimura-Taniyama-Weil
conjecture, complex multiplication and the
Heegner point
construction. The next three chapters introduce
variants of
modular parametrizations in which modular
curves are replaced by
Shimura curves attached to certain indefinite
quaternion algebras.
The main new contributions are found in Chapters
7-9, which
survey the author's attempts to extend the
theory of Heegner
points and complex multiplication to situations
where the base
field is not a CM field. Chapter 10 explains
the proof of
Kolyvagin's theorem, which relates Heegner
points to the
arithmetic of elliptic curves and leads to
the best evidence so
far for the Birch and Swinnerton-Dyer conjecture.
Contents
Elliptic curves
Modular forms
Heegner points on X_0(N)
Heegner points on Shimura curves
Rigid analytic modular forms
Rigid analytic modular parametrisations
Totally real fields
ATR points
Integration on mathcal{H}_ptimesmathcal{H}
Kolyvagin's theorem
Bibliography
Details:
Series: CBMS Regional Conference Series in
Mathematics,Number:
101
Publication Year: 2003
ISBN: 0-8218-2868-1
Paging: 129 pp.
Binding: Softcover
The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series
Expected publication date is January 14,
2004
Description
Modular forms appear in many ways in number
theory. They play a
central role in the theory of quadratic forms,
in particular, as
generating functions for the number of representations
of
integers by positive definite quadratic forms.
They are also key
players in the recent spectacular proof of
Fermat's Last Theorem.
Modular forms are at the center of an immense
amount of current
research activity. Also detailed in this
volume are other roles
that modular forms and q-series play in number
theory, such as
applications and connections to basic hypergeometric
functions,
Gaussian hypergeometric functions, super-congruences,
Weierstrass
points on modular curves, singular moduli,
class numbers, L-values,
and elliptic curves.
The first three chapters provide some basic
facts and results on
modular forms, which set the stage for the
advanced areas that
are treated in the remainder of the book.
Ono gives ample
motivation on topics where modular forms
play a role. Rather than
cataloging all of the known results, he highlights
those that
give their flavor. At the end of most chapters,
he gives open
problems and questions.
The book is an excellent resource for advanced
graduate students
and researchers interested in number theory.
Contents
Basic facts
Integer weight modular forms
Half-integral weight modular forms
Product expansions of modular forms on mathrm{SL}_2(mathbb{Z})
Partitions
Weierstrass points on modular curves
Traces of singular moduli and class equations
Class numbers of quadratic fields
Central values of modular L-functions and
applications
Basic hypergeometric generating functions
for L-values
Gaussian hypergeometric functions
Bibliography
Index
Details:
Series: CBMS Regional Conference Series in
Mathematics, Number:
102
Publication Year: 2003
ISBN: 0-8218-3368-5
Paging: 216 pp.
Binding: Softcover