טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentEyal Masad
SubjectSkew Products of Topological Flows and Applications to
Nonlinear Analysis
DepartmentDepartment of Mathematics
Supervisor Professor Emeritus Reich Simeon
Full Thesis textFull thesis text - English Version


Abstract

In the thesis we present various applications of topological skew products in linear and nonlinear analysis. We use a topological flow as a means of selecting nonlinear operators from a compact set of operators mapping a Banach space into itself. This method yields sequences (or paths) of operators with special topological properties.  We are specifically interested in the limits (and partial limits) of the products of these sequences, which enable us to solve some well known optimization problems.

Products of operators have long been used for solving optimization problems. Many real-world problems are solved using this method. However, most of these algorithms are confined to problems with finitely many constraints. The new approach which is presented in this thesis enables us to solve problems with infinitely many constraints. It also gives us a new way of constructing the algorithms.

First we look at a problem which occupied many mathematicians - generalizing von Neumann's alternating projections theorem. The theorem was originally formulated and proved for two subspaces of a Hilbert space. Later it was generalized to the case of finitely many subspaces W1,…,Wn, but not for infinitely many subspaces.

Through the use of skew-products we define almost periodic sequences of projections. We then prove some convergence results for infinite sets of subspaces. We also look at sequences of other types of operators in Hilbert and Banach spaces.

Next we examine the Perceptron model which is at the heart of machine learning algorithms. We prove that through the use of almost periodic sequences, perceptrons can find the linear separator of bounded (linearly separable) sets in infinite dimensional Hilbert spaces.

We continue with a generalization of an algorithm by Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, which solves a convex feasibility problem called MSCFP. Our generalization shows that the algorithm which was designed for finite dimensional spaces solves the same problem in infinite dimensional spaces as well.

In the last chapter we define a special class of skew products of an R+-semi-flow and the semigroup of nonexpansive mappings on a Banach space. These skew-products describe the solutions of a differential inclusion induced by a path of accretive operators selected by the skew product. In particular we show how these skew products can be used to find a common minimal point of an infinite set of convex functionals.