Da Prato, Giuseppe

An Introduction to Infinite-Dimensional Analysis

Series: Universitext
2006, 370 p., Softcover
ISBN: 3-540-29020-6
Due: March 2006

About this textbook

In this revised and extended version of his course notes from a 1-year course at Scuola Normale Superiore, Pisa, the author provides an introduction ? for an audience knowing basic functional analysis and measure theory but not necessarily probability theory ? to analysis in a separable Hilbert space of infinite dimension. Moreover, some details have been added as well as some new material on dynamical systems with dissipative nonlinearities and asymptotic behavior for gradient systems.

Table of contents

Gaussian Measures in Hilbert Spaces.- The Cameron-Martin Formula.- Brownian Motion.- Stochastic Perturbations of a Dynamical System.- Invariant Measures for Markov Semigroups.- Weak Convergence of Measures.- Existence and Uniqueness of Invariant Measures.- Examples of Markov Semigroups.- L2 Spaces with respect to a Gaussian Measure.- Sobolev Spaces.- Gradient Systems.- Linear Semigroups Theory.- Bibliography.

Mansuy, Roger, Yor, Marc

Random Times and Enlargements of Filtrations in a Brownian Setting

Series: Lecture Notes in Mathematics, Vol. 1873
2006, Approx. 120 p., Softcover
ISBN: 3-540-29407-4
Due: December 2, 2005

About this book

In November 2004, M. Yor and R. Mansuy jointly gave six lectures at Columbia University, New York. These notes follow the contents of that course, covering expansion of filtration formulae; BDG inequalities up to any random time; martingales that vanish on the zero set of Brownian motion; the Azema-Emery martingales and chaos representation; the filtration of truncated Brownian motion; attempts to characterize the Brownian filtration.

The book accordingly sets out to acquaint its readers with the theory and main examples of enlargements of filtrations, of either the initial or the progressive kind. It is accessible to researchers and graduate students working in stochastic calculus and excursion theory, and more broadly to mathematicians acquainted with the basics of Brownian motion

Table of contents

Preliminaries.- Enlargements of Filtrations.- Stopping and Non-Stopping Times.- On the Martingales which Vanish on the Set of Brownian Zeroes.- PRP and CRP for some Remarkable Martingales.- Unveiling the Brownian Path (or History) as the Level Rises.-Weak and Strong Brownian Filtrations.- Appendix: Sketches of Solutions for the Exercises.- References.- Index.

Singer, Ivan

Duality for Nonconvex Approximation and Optimization

Series: CMS Books in Mathematics
2006, Approx. 320 p. 17 illus., Hardcover
ISBN: 0-387-28394-3
Due: December 2005

About this book

The theory of convex optimization has been constantly developing over the past 30 years. Most recently, many researchers have been studying more complicated classes of problems that still can be studied by means of convex analysis, so-called "anticonvex" and "convex-anticonvex" optimizaton problems. This manuscript contains an exhaustive presentation of the duality for these classes of problems and some of its generalization in the framework of abstract convexity. This manuscript will be of great interest for experts in this and related fields.

Table of contents

Preliminaries.- Worst approximation.- Duality for quasi-convex supremization.- Optimal solutions for quasi-convex maximization.- Reverse convex best approximation.- Unperturbational duality for reverse convex infimization.- Optimal solutions for reverse convex infimization.- Duality for d. c. optimization problems.- Duality for optimization in the framework of abstract convexity.- Notes and remarks.

Lucchetti, Roberto

Convexity and Well-Posed Problems

Series: CMS Books in Mathematics
2006, Approx. 320 p. 46 illus., Hardcover
ISBN: 0-387-28719-1
Due: January 2006

About this textbook

This book deals with the study of convex functions and of their behavior from the point of view of stability with respect to perturbations. Convex functions are considered from the modern point of view that underlines the geometrical aspect: thus a function is defined as convex whenever its graph is a convex set.

A primary goal of this book is to study the problems of stability and well-posedness, in the convex case. Stability means that the basic parameters of a minimum problem do not vary much if we slightly change the initial data. On the other hand, well-posedness means that points with values close to the value of the problem must be close to actual solutions. In studying this, one is naturally led to consider perturbations of functions and of sets. This approach fits perfectly with the idea of regarding functions as sets. Thus the second part of the book starts with a short, yet rather complete, overview of the so-called hypertopologies, i.e. topologies in the closed subsets of a metric space.

While there exist numerous classic texts on the issue of stability, there only exists one book on hypertopologies [Beer 1993]. The current book differs from Beerfs in that it contains a much more condensed explication of hypertopologies and is intended to help those not familiar with hypertopologies learn how to use them in the context of optimization problems.

Table of contents

Preface.- Convex Sets and Convex Functions: the fundamentals.- Continuity and Gamma (X).- The Derivatives and the Subdifferential.- Minima and Quasi Minima.- The Fenchel Conjugate.- Duality.- Linar Programming and Game Theory.- Hypertopologies, Hyperconvergences.- Continuity of Some Operations Between Functions.- Well-Posed Problems.- Generic Well-Posedness.- More Exercises.- Appendix A: Functional Analysis.- Appendix B: Topology.- Appendix C: More Game Theory.- Appendix D: Symbols, Notations, Definitions and Important Theorems.- References, Index.

Greenspan, Donald

Numerical Solution of Ordinary Differential Equations
for Classical, Relativistic and Nano Systems

1. Edition - November 2005
2005. X, 206 Pages, Softcover
- Textbook -
ISBN 3-527-40610-7

Short description
This work meets the need for an affordable textbook on numerical solutions of ODE. Carefully structured by an experienced textbook author, it provides a survey of ODE for various applications, both classical and modern.

From the contents

I Euler's Method
II Runge-Kutta Methods
III The Method of Taylor Expansions
IV Large Second Order Systems with Application to Nano Systems
V Completely Conservative, Covariant Numerical Methodology
VI Instability
VII Numerical Solution of Tridiagonal Linear Algebraic Systems and Related Nonlinear Systems
VIII Approximate Solution of Boundary Value Problems
IX Special Relativistic Motion
X Special Topics
Appendix - Basic Matrix Operations
Bibliography