Arnold, D.N.; Bochev, P.B.; Lehoucq, R.B.; Nicolaides, R.A.; Shashkov, M. (Eds.)
Compatible Spatial Discretizations
Series: The IMA Volumes in Mathematics and its Applications, Vol. 142
2006, XIV, 224 p. 64 illus., Hardcover
ISBN: 0-387-30916-0
Due: May 2006
About this book
The IMA Hot Topics workshop on compatible spatialdiscretizations
was held May 11-15, 2004 at the University of Minnesota. The
purpose of the workshop was to bring together scientists at the
forefront of the research in the numerical solution of PDEs to
discuss recent advances and novel applications of geometrical and
homological approaches to discretization. This volume contains
original contributions based on the material presented at the
workshop. A unique feature of the collection is the inclusion of
work that is representative of the recent developments in
compatible discretizations across a wide spectrum of disciplines
in computational science.
Compatible spatial discretizations are those that inherit or
mimic fundamental properties of the PDE such as topology,
conservation, symmetries, and positivity structures and maximum
principles. The papers in the volume offer a snapshot of the
current trends and developments in compatible spatial
discretizations. The reader will find valuable insights on
spatial compatibility from several different perspectives and
important examples of applications compatible discretizations in
computational electromagnetics, geosciences, linear elasticity,
eigenvalue approximations and MHD. The contributions collected in
this volume will help to elucidate relations between different
methods and concepts and to generally advance our understanding
of compatible spatial discretizations for PDEs. Abstracts and
presentation slides from the workshop can be accessed at http://www.ima.umn.edu/talks/workshops/5-11-15.2004/.
Table of contents
Foreword.- Preface.- Numerical Convergence of the MPFA O-method
for General Quadrilateral Grids in Two and Three Dimensions.-
Differential Complexes and Stability of Finite Element Methods I.
The de Rham Complex.- Differential Complexes and Stability of
Finite Element Methods II. The Elasticity Complex.- On the Role
of Involutions in the Discontinuous Galerkin Discretization of
Maxwell and Magnetohydrodynamic Systems.- Principles of Mimetic
Discretizations of Differential Operators.- Compatible
Discretizations for Eigenvalue Problems.- Conjugated Bubnov-Galerkin
Infinite Element for Maxwell equations.- Covolume Discretization
of Differential Forms.- Mimetic Reconstruction of Vectors.- A
Cell-Centered Finite Difference Method on Quadrilaterals.-
Development and Application of Compatible Discretizations of
Maxwell's Equations.
Cerf, Raphael / Picard, Jean (Ed.)
The Wulff Crystal in Ising and Percolation Models
Ecole d'Ete de Probabilites de Saint-Flour XXXIV - 2004
Series: Lecture Notes in Mathematics, Vol. 1878
2006, Approx. 280 p. 32 illus., Softcover
ISBN: 3-540-30988-8
Due: May 3, 2006
About this book
This volume is a synopsis of recent works aiming at a
mathematically rigorous justification of the phase coexistence
phenomenon, starting from a microscopic model. It is intended to
be self-contained. Those proofs that can be found only in
research papers have been included, whereas results for which the
proofs can be found in classical textbooks are only quoted.
Written for:
Researchers and graduate students interested in optimization,
probability theory and stochastic processes
Table of contents
Phase coexistence and subadditivity.- Ising models.- Bernoulli
percolation. - FK or random cluster model.- The Wulff crystal.-
Large deviation theory.- Surface large deviation principles.-
Volume large deviations.- Coarse graining.- Decoupling.- Surface
tension.- Interface estimate.- Sets of finite perimeter.- Surface
energy.- The Wulff theorem.- LDP for the cluster shapes.-
Enhanced upper bound.- LDP for FK percolation.- LDP for ising.
Slade, Gordon / Picard, Jean (Ed.)
The Lace Expansion and its Applications
Ecole d'Ete de Probabilites de Saint-Flour XXXIV - 2004
Series: Lecture Notes in Mathematics, Vol. 1879
2006, Approx. 240 p. 38 illus., Softcover
ISBN: 3-540-31189-0
Due: May 3, 2006
About this book
The lace expansion is a powerful and flexible method for
understanding the critical scaling of several models of interest
in probability, statistical mechanics, and combinatorics, above
their upper critical dimensions. These models include the self-avoiding
walk, lattice trees and lattice animals, percolation, oriented
percolation, and the contact process. This volume provides a
unified and extensive overview of the lace expansion and its
applications to these models. Results include proofs of existence
of critical exponents and construction of scaling limits. Often,
the scaling limit is described in terms of super-Brownian motion.
Written for:
Researchers and graduate students interested in probability
theory, statistical mechanics and combinatorics
Table of contents
Preface.- 1 Simple Random Walk.- 2 The Self-Avoiding Walk.- 3 The
Lace Expansion for the Self-Avoiding Walk.- 4 Diagrammatic
Estimates for the Self-Avoiding Walk.-5 Convergence for the Self-Avoiding
Walk.- 6 Further Results for the Self-Avoiding Walk.- 7 Lattice
Trees.- 8 The Lace Expansion for Lattice Trees.- 9 Percolation.
Attal, Stephane, Joye, Alain, Pillet, Claude-Alain
Open Quantum Systems I
The Hamiltonian Approach
Series: Lecture Notes in Mathematics, Vol. 1880
2006, Approx. 325 p., Softcover
ISBN: 3-540-30991-8
Due: May 18, 2006
Open Quantum Systems II
The Markovian Approach
Series: Lecture Notes in Mathematics, Vol. 1881
2006, Approx. 230 p., Softcover
ISBN: 3-540-30992-6
Open Quantum Systems III
Recent Developments
Series: Lecture Notes in Mathematics, Vol. 1882
2006, Approx. 300 p., Softcover
ISBN: 3-540-30993
About this book
Understanding dissipative dynamics of open quantum systems
remains a challenge in mathematical physics. This problem is
relevant in various areas of fundamental and applied physics.
From a mathematical point of view, it involves a large body of
knowledge. Significant progresses in the understanding of such
systems have been made during the last decade. These books
present in a self-contained way the mathematical theories
involved in the modelization of such phenomena. They describe
physically relevant models, develop their mathematical analysis
and derive their physical implications.
In Volume I the Hamiltonian description of quantum open systems
is discussed. This includes an introduction to quantum
statistical mechanics and its operator algebraic formulation,
modular theory, spectral analysis and their applications to
quantum dynamical systems.
Volume II is dedicated to the Markovian formalism of classical
and quantum open systems. A complete exposition of noise theory,
Markov processes and stochastic differential equations, both in
the classical and the quantum context, is provided. These
mathematical tools are put into perspective with physical
motivations and applications.
Volume III is devoted to recent developments and applications.
The topics discussed include the non-equilibrium properties of
open quantum systems, the Fermi Golden Rule and weak coupling
limit, quantum irreversibility and decoherence, qualitative
behaviour of quantum Markov semigroups and continual quantum
measurements.
Written for:
Researchers and graduate students interested in both the
mathematical description of Quantum Open Systems and the analysis
of physically relevant models
Table of contents
I
Preface.- A. Joye: Introduction to the Theory of Linear Operators.-
A.Joye. Introduction to Quantum Statistical Mechanics.- A. Attal:
Elements of Operator Algebras and Modular Theory.- C.-A. Pillet:
Quantum Dynamical Systems.- M. Merckli: The Ideal Quantum Gas.- V.
Jak?ic: Topics in Spectral Theory.- Index.
II
Preface.- L. Rey-Bellet: Ergodic Properties of Markov Processes.-
L. Rey-Bellet: Open Classical Systems.- S. Attal: Quantum Noises.-
R. Rebolledo: Complete Positivity and the Markov Structure of
Open Quantum Systems.- F. Fagnola: Quantum Stochastic
Differential Equations and Dilation of Completely Prositive
Semigroups.- Index
III
Preface.- W. Aschbacher, V. Jakic, Y. Pautrat, C.-A. Pillet:
Topics in Non-Equilibrium Quantum Statistical Mechanics.- J.
Derezinski, R. Fruboes: Fermi Golden Rule and Open Quantum
Systems.- Ph. Blanchard, R. Olkiewicz: Decoherence as
Irreversible Dynamical Process in Open Quantum Systems.- F.
Fagnola, R. Rebolledo: Notes on the Qualitative Behaviour of
Quantum Markov Semigroups.- A. Barchielli: Continual Measurements
in Quantum Mechanics and Quantum Stochastic Calculus.- Index.
Coates, John, Sujatha, R.
Cyclotomic Fields and Zeta Values
Series: Springer Monographs in Mathematics
2006, Approx. 120 p., Hardcover
ISBN: 3-540-33068-2
Due: July 2006
About this book
Cyclotomic fields have always occupied a central place in number
theory, and the so called "main conjecture" on
cyclotomic fields is arguably the deepest and most beautiful
theorem known about them. It is also the simplest example of a
vast array of subsequent, unproven "main conjectures'' in
modern arithmetic geometry involving the arithmetic behaviour of
motives over p-adic Lie extensions of number fields. These main
conjectures are concerned with what one might loosely call the
exact formulae of number theory which conjecturally link the
special values of zeta and L-functions to purely arithmetic
expressions.
Written by two leading workers in the field, this short and
elegant book presents in full detail the simplest proof of the
"main conjecture'' for cyclotomic fields . Its motivation
stems not only from the inherent beauty of the result. The
masterly exposition is accessible to graduate students and
addressed also to non-experts in Iwasawa theory.
Table of contents
Cyclotomic Fields.- Local Units.- Iwasawa Algebras and p-adic
measures.- Cyclotomic Units and Iwasawa's Theorem.- Euler Systems.-
Main Conjecture.- Appendix.- References.