2003 IX, 586 p. Hardcover
3-540-00356-8
In the eighties, a group of theoretical physicists
introduced
several models for certain disordered systems,
called "spin
glasses". These models are simple and
rather canonical
random structures, that physicists studied
by non-rigorous
methods. They predicted spectacular behaviors,
previously unknown
in probability theory. They believe these
behaviors occur in many
models of considerable interest for several
branches of science (statistical
physics, neural networks and computer science).
This book introduces in a rigorous manner
this exciting new area
to the mathematically minded reader. It requires
no knowledge
whatsoever of any physics, and contains proofs
in complete detail
of much of what is rigorously known on spin
glasses at the time
of writing.
Keywords: Random Structures, Disordered Systems,
Sherrington
Kirkpatrick Model, Hopfield Model, Neural
Networks
Contents:
0. Introduction.- 1. A Toy Model, the REM.-
2. The Sherrington-Kirkpatrick
Model.- 3. The Capacity of the Perceptron:
The Ising Case.- 4.
Capacity of the Perceptron: The Gaussian
and the Spherical Case.-
5. The Hopfield Model.- 6. The p-Spin Interaction
Model at Low
Temperature.- 7. The Diluted SK Model and
the K-Sat Problems.- 8.
An Assignment Problem.- A. Appendix.- Elements
of Probability
Theory.- References.- Index.
Series: Ergebnisse der Mathematik und ihrer
Grenzgebiete. 3.
Folge / A Series of Modern Surveys in Mathematics.
Vol.. 46
2003 VIII, 296 p. Softcover
3-540-40335-3
In World Mathematical Year 2000 the traditional
St. Flour Summer
School was hosted jointly with the European
Mathematical Society.
Sergio Albeverio reviews the theory of Dirichlet
forms, and gives
applications including partial differential
equations, stochastic
dynamics of quantum systems, quantum fields
and the geometry of
loop spaces. The second text, by Walter Schachermayer,
is an
introduction to the basic concepts of mathematical
finance,
including the Bachelier and Black-Scholes
models. The fundamental
theorem of asset pricing is discussed in
detail. Finally Michel
Talagrand, gives an overview of the mean
field models for spin
glasses. This text is a major contribution
towards the proof of
certain results from physics, and includes
a discussion of the
Sherrington-Kirkpatrick and the p-spin interaction
models.
Contents:
Part I, S. Albeverio: Theory of Dirichlet
forms and applications:
0. Introduction.- 1. Functional analytic
background: semigroups,
generators, resolvents.- 2. Closed symmetric
coersive forms
associated with C0-contraction semigroups.-
3. Contraction
properties of forms, positivity preserving
and submarkovian
semigroups.- 4. Potential Theory and Markov
Processes associated
with Dirichlet Forms.- 5. Diffusions and
stochastic differential
equations associated with classical Dirichlet
Forms.- 6.
Applications.- References.- Index.- Part
II, W. Schachermayer:
Introduction to the Mathematics of Financial
Markets: 1.
Introduction: Bachelier's Thesis from 1900.-
2. Models of
Financial Markets on Finite Probability Spaces.-
3. The Binomial
Model, Bachelier's Model and the Blach-Scholes
Model.- 4. The No-Arbitrage
Theory for General Processes.- 5. Some Applications
of the
Fundamental Theorem of Asset Pricing.- References.-
Part III, M.
Talagrand: Mean field models for spin glasses:
a first course: 1.
Introduction.- 2. What this is all about:
the REM.- 3. The
Sherrington-Kirkpatrick model at high temperature.-
4. The p-spin
interaction model.- 5. External field and
the replica-symmetric
solution.- 6. Exponential inequalities.-
7. Central limit
theorems and the Almeida-Thoules line.- 8.
Emergence and
separation of the lumps in the p-spin interaction
model.-
Bibliography.
Series: Lecture Notes in Mathematics. Vol..
1816
2003 XI, 353 p. Softcover
3-540-40392-2
The C.I.M.E. session in Diophantine Approximation,
held in
Cetraro (Italy) June 28 - July 6, 2000 focused
on height theory,
linear independence and transcendence in
group varieties, Baker's
method, approximations to algebraic numbers
and applications to
polynomial-exponential diophantine equations
and to diophantine
theory of linear recurrences. Very fine lectures
by D. Masser, Y.
Nesterenko, H.-P. Schlickewei, W.M. Schmidt
and M. Walsschmidt
have resulted giving a good overview of these
topics, and
describing central results, both classical
and recent,
emphasizing the new methods and ideas of
the proofs rather than
the details. They are addressed to a wide
audience and do not
require any prior specific knowledge.
Contents:
Preface.- D. Masser: Heights, Transcendence,
and Linear
Independence on Commutative Group Varieties.-
Yu. Nesterenko:
Linear Forms in Logarithms of Rational Numbers.-
H. P.
Schlickewei: Approximation of Algebraic Numbers.-
W. M. Schmidt:
Linear Recurrence Sequences.- M. Waldschmidt:
Linear Independence
Measures for Logarithms of Algebraic Numbers.
Series: Lecture Notes in Mathematics. Vol..
1819
2004 Approx. 150 p. Hardcover
3-540-40469-4
Contributions by three authors treat aspects
of noncommutative
geometry that are related to cyclic homology.
The authors give
rather complete accounts of cyclic theory
from different points
of view. The connections between (bivariant)
K-theory and cyclic
theory via generalized Chern-characters are
discussed in detail.
Cyclic theory is the natural setting for
a variety of general
abstract index theorems. A survey of such
index theorems is given
and the concepts and ideas involved in these
theorems are
explained.
Keywords: cyclic homology, K-theory, non-commutative
geometry
Contents:
I. Cyclic Theory, Bivariant K-Theory and
the Bivariant Chern-Connes
Character by J. Cuntz: 1. Cyclic Theory;
2. Cyclic Theory for
Locally Convex Algebras; 3. Bivariant K-Theory;
4. Infinite-Dimensional
Cyclic Theories; A. Locally Convex Algebras;
B. Standard
Extensions.- II. Noncommutative Geometry,
the Transverse
Signature Operator, and Hopf Algebras (after
A. Connes and H.
Moscovici) by G. Skandalis: 1. Preliminaries;
2. The Local Index
Formula; 3. The Diff-Invariant Signature
Operator; 4. The "Transverse"
Hopf Algebra.- III. Cyclic Homology by B.
Tsygan: 1.
Introduction; 2. Hochschild and Cyclic Homology
of Algebras; 3.
The Cyclic Complex C^{\lambda}_{\bullet};
4. Non-Commutative
Differential Calculus; 5. Cyclic Objects;
6. Examples; 7. Index
Theorems; 8. Riemann-Roch Theorem for D-Modules.
Series: Encyclopaedia of Mathematical Sciences.
Vol.. 121
2003 Approx. 510 p. 66 illus. Hardcover
0-387-40319-1
Mathematical systems theory is a vibrant
research area in its own
right. The theory has an impact in numerous
applications areas
including aeronautics, biological systems,
chemical engineering,
communication systems, financial engineering
and robotics to name
just a few. This volume contains survey and
research articles by
some of the leading researchers in mathematical
systems theory.
Many authors have taken special care that
their articles are self-contained
and accessible also to non-specialists. The
articles contained in
this volume are from those presented as plenary
lectures, invited
one hour lectures and minisymposia at the
15th International
Symposium on the Mathematical Theory of Networks
and Systems held
at the University of Notre Dame, August 12-16,
2002.
Contents:
On cellular automaton approaches to modeling
biological cells.-
Crystalline stochastic systems and curvature
driven flows.-
Overdetermined multidimensional systems:
state space and
frequency domain methods.- Dissipative dynamics
in classical and
quantum conservative systems.- A state space
approach to control
of interconnected systems.- Multi-dimensional
capacity, pressure
and Hausdorff dimension.- Numerics versus
control.- Manipulating
matrix inequalities automatically.- A dynamical
system approach
to matrix eigenvalue algorithms.- Belief
propagation on partially
ordered sets.- Smoothing by Savitzky-Golay
and Legendre filters.-
On stochastic control in finance.- Control
and financial
engineering.- Passive and conservative infinite-dimensional
impedance and scattering systems (from a
personal point of view).-
Spectral theory for neural delay equations
with applications to
control and stabilization.- On factor graphs
and electrical
networks.- Dissipative distributed systems.
Series: The IMA Volumes in Mathematics and
its Applications. Vol..
134
2003 VI, 307 p. Softcover
3-540-40594-1
The authors develop thorough and complete
foundations for the
method of almost etale extensions, which
is at the basis of
Faltings' approach to p-adic Hodge theory.
The central notion is
that of an "almost ring". Almost
rings are the
commutative unitary monoids in a tensor category
obtained as a
quotient V-Mod/S of the category V-Mod of
modules over a fixed
ring V; the subcategory S consists of all
modules annihilated by
a fixed ideal m of V, satisfying certain
natural conditions.
The reader is assumed to be familiar with
general categorical
notions, some basic commutative algebra and
some advanced
homological algebra (derived categories,
simplicial methods).
Apart from these general prerequisites, the
text is as self-contained
as possible. One novel feature of the book
- compared with
Faltings' earlier treatment - is the systematic
exploitation of
the cotangent complex, especially for the
study of deformations
of almost algebras.
Keywords: Almost rings, deformations, valuations,
rigid geometry
Contents:
Introduction.- Homological Theory.- Almost
Ring Theory.- Fine
Study of Almost Projective Modules.- Henselian
Pairs.- Valuation
Theory.- Analytic Geometry.- Appendix.- References.-
Index.
Series: Lecture Notes in Mathematics. Vol..
1800