A publication of the Societe Mathematique de France.
ISBN: 978-2-85629-335-5
Series: Asterisque, Number 341
Published: 15 February 2012; Copyright Year: 2012; Pages: 113; Softcover;
Subject Classification
Differential Equations
Analysis
Readership: Graduate students and research mathematicians interested in Birkhoff normal forms, quasi-linear
Hamiltonian equations, almost global existence, and Klein-Gordon equations.
Consider a nonlinear Klein-Gordon equation on the unit circle, with smooth data of size → 0. A
solution u which, for any κ ∈ N, may be extended as a smooth solution on a time-interval ] ? cκ ?κ , cκ ?κ [
for some cκ > 0 and for 0 < < κ , is called an almost global solution. It is known that when the nonlinearity
is a polynomial depending only on u , and vanishing at order at least 2 at the origin, any smooth
small Cauchy data generate, as soon as the mass parameter in the equation stays outside a subset of zero
measure of R?
+ , an almost global solution, whose Sobolev norms of higher order stay uniformly bounded.
The goal of this book is to extend this result to general Hamiltonian quasi-linear nonlinearities. These are
the only Hamiltonian nonlinearities that depend not only on u but also on its space derivative. To prove
the main theorem, the author develops a Birkhoff normal form method for quasi-linear equations.
A publication of the Societe Mathematique de France.
ISBN: 978-2-85629-334-8
Series, Volume: Asterisque, Number 342
Bibliographic Information: Published: 15 February 2012; Copyright Year: 2012; Pages: 127; Softcover
Subject Classification
Differential Equations
Analysis
Readership: Graduate students and research mathematicians interested in pure mathematics.
A class of pseudodifferential operators on the Heisenberg group is defined. As it should be, this
class is an algebra containing the class of differential operators. Furthermore, those pseudodifferential
operators act continuously on Sobolev spaces and the loss of derivatives may be controlled by the order
of the operator. Although a large number of works have been devoted in the past to the construction
and the study of algebras of variable-coefficient operators, including some very interesting works on the
Heisenberg group, the authors’ approach is different, and in particular puts into light microlocal directions
and completes, with the Littlewood-Paley theory initiated in 2000 by Bahouri, Gerard, and Xu: a microlocal
analysis of the Heisenberg group.
viii+176 pages, soft cover, ISBN 978-3-88538-232-4, EUR 28.00, 2011
Order is a theoretical model of preference which is used in everyday life and has applications in economical, sociological, technical and natural sciences, and in particular in mathematics. The formal theory dealing with ordered sets as a mathematical concept was treated by a number of authors in papers and several monographs.
There are ordered sets with particular properties that can be considered as algebras. The best known examples are semilattices and lattices. An important class of ordered sets which generalize semilattices is the class of up- (or down-) directed sets, i.e. ordered sets in which every pair of elements has a common upper (or lower) bound. Directoids, the main mathematical concept studied in this monograph, are an algebraic version of up- (or down-) directed sets. A common upper (or lower) bound is assigned to every pair x, y of elements in such a way that it coincides with max(x,y) (or min(x,y)) in case x, y are comparable.
Hence, directoids are algebras with one binary operation, which is not necessarily associative or commutative. However, the corresponding operation can be characterized by several simple identities and hence the class of directoids forms a variety.
Directoids can be enriched by complementation, pseudocomplementation or relative pseudocomplementation. Such algebras serve as an algebraic axiomatization of certain non-classical logics, in particular the logic of quantum mechanics. The basic properties of directoids, their variety and several applications are studied in this monograph.
1 Preliminaries 7
2 The concept of a directoid 17
3 Varieties of directoids 27
4 λ-lattices 43
5 Pseudocomplemented directoids 49
6 Relatively pseudocomplemented directoids 57
7 Bounded directoids with an antitone involution 73
8 Directoids with sectional involutions 89
9 A non-associative generalization of MV-algebras 97
10 Weak MV-algebras 119
11 A representation of effect algebras by means of commutative directoids 141
12 Relational systems 159
ISBN: 978-0-470-59669-2
Hardcover
404 pages
April 2012
Everyday decisions are made without the benefit of accurate information. Optimal Learning develops the needed principles for gathering information to make decisions, especially when collecting information is time-consuming and expensive. Designed for readers with an elementary background in probability and statistics, the book presents effective and practical policies illustrated in a wide range of applications, from energy, homeland security, and transportation to engineering, health, and business.
This book covers the fundamental dimensions of a learning problem and presents a simple method for testing and comparing policies for learning. Special attention is given to the knowledge gradient policy and its use with a wide range of belief models, including lookup table and parametric and for online and offline problems. Three sections develop ideas with increasing levels of sophistication:
* Fundamentals explores fundamental topics, including adaptive learning, ranking and selection, the knowledge gradient, and bandit problems
* Extensions and Applications features coverage of linear belief models, subset selection models, scalar function optimization, optimal bidding, and stopping problems
* Advanced Topics explores complex methods including simulation optimization, active learning in mathematical programming, and optimal continuous measurements
Each chapter identifies a specific learning problem, presents the related, practical algorithms for implementation, and concludes with numerous exercises. A related website features additional applications and downloadable software, including MATLAB and the Optimal Learning Calculator, a spreadsheet-based package that provides an introduc-tion to learning and a variety of policies for learning.
Oxford Mathematical Monographs
336 pages | 23 b/w illustrations | 234x156mm
978-0-19-965410-9 | Hardback | September 2012 (estimated)
Presents original content in the use of stochastic calculus in chaos theory and the introduction to relativistic diffusion
Elementary and self-contained access to to hyperbolic geometry (using special relativity), stochastic calculus, and chaotic dynamics
Interplay between several fields of mathematics
Clearly displayed key results and proofs
Hyperbolic Dynamics and Brownian Motion illustrates the interplay between distinct domains of mathematics. There is no assumption that the reader is a specialist in any of these domains: only basic knowledge of linear algebra, calculus and probability theory is required.
The content can be summarized in three ways:
Firstly, this book provides an introduction to hyperbolic geometry, based on the Lorentz group. The Lorentz group plays, in relativistic space-time, a role analogue to the rotations in Euclidean space. The hyperbolic geometry is the geometry of the unit pseudo-sphere. The boundary of the hyperbolic space is defined as the set of light rays. Special attention is given to the geodesic and horocyclic flows. Hyperbolic geometry is presented via special relativity to benefit from the physical intuition.
Secondly, this book introduces basic notions of stochastic analysis: the Wiener process, Ito's stochastic integral, and calculus. This introduction allows study in linear stochastic differential equations on groups of matrices. In this way the spherical and hyperbolic Brownian motions, diffusions on the stable leaves, and the relativistic diffusion are constructed.
Thirdly, quotients of the hyperbolic space under a discrete group of isometries are introduced. In this framework some elements of hyperbolic dynamics are presented, as the ergodicity of the geodesic and horocyclic flows. This book culminates with an analysis of the chaotic behaviour of the geodesic flow, performed using stochastic analysis methods. This main result is known as Sinai's central limit theorem.
Readership: Graduate students and researchers in a wide range of topics in mathematics and mathematical/theoretical physicists.
Introduction
Summary
1: The Lorentz-Mobius group PSO(1; d)
2: Hyperbolic Geometry
3: Operators and Measures
4: Kleinian groups
5: Measures and flows on ?\Fd
6: Basic Ito Calculus
7: Brownian motions on groups of matrices
8: Central Limit Theorem for geodesics
9: Appendix relating to geometry
10: Appendix relating to stochastic calculus
11: Index of notation, terms, and gures
References