Series: De Gruyter Textbook
To be published: September 2012
ISBN: 978-3-11-028311-2
Standard work on numerics of partial differential equations
Elementary and understandable presentation of the mathematical theory
Covers efficiency of numerical algorithms and scientific computing
Incudes in-depth mathematical tools in the appendix
Numerical mathematics is a subtopic ofscientific computing. The focuslies on the efficiencyof algorithms, i.e. speed, reliability, and robustness.This leads to adaptive algorithms.The theoreticalderivation und analysesof algorithmsare kept as elementary as possible in this book; the neededsligtly advancedmathematical theory is summarized in the appendix.Numerous figures and illustrating examples explain the complex data,as non-trivial examples serve problems from nanotechnology, chirurgy, and physiology.
Series: De Gruyter Studies in Mathematics 46
To be published: December 2012
ISBN: 978-3-11-026255-1
This book is the first complete study and monograph dedicated to singular traces. The text mathematically formalises the study of traces in a self contained theory of functional analysis. Extensive notes will treat the historical development. The final section will contain the most complete and concise treatment known of the integration half of Connes' quantum calculus.
Singular traces are traces on ideals of compact operators that vanish on the subideal of finite rank operators. Singular traces feature in A. Connes' interpretation of noncommutative residues. Particularly the Dixmier trace,which generalises the restricted Adler-Manin-Wodzicki residue of pseudo-differential operators and plays the role of the residue for a new catalogue of 'geometric' spaces, including Connes-Chamseddine standard models, Yang-Mills action for quantum differential forms, fractals, isospectral deformations, foliations and noncommutative index theory.
The theory of singular traces has been studied after Connes' application to non-commutative geometry and physics by various authors. Recent work by Nigel Kalton and the authors has advanced the theory of singular traces.Singular traces can be equated to symmetric functionals of symmetric
sequence or function spaces, residues of zeta functions and heat kernel asymptotics, and characterised by Lidksii and Fredholm formulas. The traces and formulas used in noncommutative geometry are now completely understood in this theory, with surprising new mathematical and physical consequences.
For mathematical readers the text offers fundamental functional analysis
results and, due to Nigel Kalton's contribution, a now complete theory
of traces on compact operators. For mathematical physicists and other users
of Connes' noncommutative geometry the text offers a complete reference
to Dixmier traces and access to the deeper mathematical features of traces
on ideals associated to the harmonic sequence. These features, not known
and not discussed in general texts on noncommutative geometry, are undoubtably
physical and probe to the fascinating heart of classical limits and quantization.
Series: De Gruyter Textbook
To be published: January 2013
ISBN: 978-3-11-025024-4
Introduction forreaders who start to work in statistics of stochastic processes
Presentation of local asymptotics in statistics
Includes numerous examples and exercises
Also attractive for probabilists
Thistextbook is devoted to the general asymptotic theory of statistical experiments. Local asymptotics for statistical models in the sense of local asymptotic (mixed) normality or local asymptotic quadraticity make up the core of the book. Numerous examples deal with classical independent and identically distributed models and with stochastic processes.
The book is suitable for graduate students starting to work in statistics of stochastic processes, as well as for researchers interested in a precise introduction to this area.
IRMA Lectures in Mathematics and Theoretical Physics Vol. 18
ISBN 978-3-03719-105-7
DOI 10.4171/105
January 2012, 461 pages, softcover, x cm.
This book contains carefully revised and expanded versions of eight courses that were presented at the University of Strasbourg, during two geometry master classes, in 2008 and 2009. The aim of the master classes was to give to fifth-year students and PhD students in mathematics the opportunity to learn new topics that lead directly to the current research in geometry and topology. The courses were held by leading experts. The subjects treated include hyperbolic geometry, three-manifold topology, representation theory of fundamental groups of surfaces and of three-manifolds, dynamics on the hyperbolic plane with applications to number theory, Riemann surfaces, Teichmuller theory, Lie groups and asymptotic geometry.
The text is addressed to students and mathematicians who wish to learn the subject. It can also be used as a reference book and as a textbook for short courses on geometry.
EMS Monographs in Mathematics
ISBN 978-3-03719-106-4
DOI 10.4171/106
February 2012, 490 pages, hardcover, 16.5 x 23.5 cm.
Wave maps are the simplest wave equations taking their values in a Riemannian manifold (M,g) . Their Lagrangian is the same as for the scalar equation, the only difference being that lengths are measured with respect to the metric g . By Noether's theorem, symmetries of the Lagrangian imply conservation laws for wave maps, such as conservation of energy.
In coordinates, wave maps are given by a system of semillinear wave equations. Over the past 20 years important methods have emerged which address the problem of local and global wellposedness of this system. Due to weak dispersive effects, wave maps defined on Minkowski spaces of low dimensions, such as R 2+1 t,x , present particular technical difficulties. This class of wave maps has the additional important feature of being energy critial, which refers to the fact that the energy scales exactly like the equation.
Around 2000 Daniel Tataru and Terence Tao, building on earlier work of Klainerman?Machedon, proved that smooth data of small energy lead to global smooth solutions for wave maps from 2+1 dimensions into target manifolds satisfying some natural conditions. In contrast, for large data, singularities may occur in finite time for M=S 2 as target. This monograph establishes that for H as target the wave map evolution of any smooth data exists globally as a smooth function.
While we restrict ourselves to the hyperbolic plane as target the implementation of the concentration-compactness method, the most challenging piece of this exposition, yields more detailed information on the solution. This monograph will be of interest to experts in nonlinear dispersive equations, in particular to those working on geometric evolution equations.