Seminars, 2004
Lectures:
• 21.06.2004
Emil Saucan, Technion and ORT Braude College
Can one see the shape of a network?

Abstract
Geometric point view of information flow. We propose an information measure that naturally associates a weighted graph for every network. This measure endows the graph in question with an intrinsic metric, yielding it as a geodesic metric space. We show how features from differential geometry, in particular geodesic distance and Gaussian Curvature translate into this setting. Furthermore, the relations between the weights given by the internal logic of the network and the geometric structure presented herein are investigated. In particular, the Euler Characteristic is computed and compared with the one of the physical .

• 24.05.2004
Prof. Alexander G.Ramm, Kansas State University, U.S.A
DYNAMICAL SYSTEMS METHOD FOR SOLVING NONLINEAR AND LINEAR OPERATOR EQUATIONS

Abstract
Consider an operator equation F(u)=0 in a Hilbert space. Let us call the problem of solving this equation ill-posed if the operator F'(u) is not boundedly invertible, and well-posed otherwise. A general method, DSM (dynamical systems method), for solving linear and nonlinear ill-posed problems in a Hilbert space is presented. This method consists of the construction of a dynamical system, that is, a Cauchy problem, which has the following properties:

1. it has a global solution,
2. this solution tends to a limit as time tends to infinity,
3. the limit solves the original linear or non-linear problem.

DSM is justified for arbitrary linear solvable equations with bounded operator, for well-posed nonlinear equations with twice Fr\'echet differentiable operator F, for ill-posed nonlinear equations with monotone operators, and for ill-posed nonlinear equations with some non-monotone operators. In Newton-type schemes the main difficulty is to invert the derivative of the operator. A novel scheme, based on DSM, allows one to avoid this inversion. Global convergence theorem is obtained for the regularized continuous analog of Newton's method for monotone operators. Global convergence means that convergence is established for an arbitrary initial approximation, not necessarily the one which is sufficiently close to the solution. A general approach to constructing convergent iterative schemes for solving well-posed nonlinear operator equations is described and convergence theorems are obtained for such schemes. Stopping rules for stable solution of the ill-posed problems with noisy data are given.

• 03.05.2004
Andrei Lerner, Bar-Ilan University, Israel
On rearrangements of maximal operators

Abstract
We discuss a general approach to obtaining rearrangement inequalities of maximal operators. Being applied to specific operators it leads not only to reproving well-known relations in a simpler and shorter way but also to new profitable results.

• 19.04.2004
Victor Khatskevich, ORT Braude College, Israel
The Koenigs Embedding Problem: new results

Abstract
Let D be a domain in a Banach space and f a self-mapping of D. The following is the Koenigs Embedding Problem: if the semigroup of iterates of f can be embedded into a strongly continuous one parameter semigroup F(t) of self-mappings sush that F(1)=f ?. We consider this problem in the case when D is the open unit ball of L(X,Y), X, Y are Hilbert spaces, and f is a linear fractional transform. Relations to Abel-Schroeder type functional equations and to operator theory in Krein spaces are discussed.

• 04.04.2004
Prof. Giuseppe Zampieri , Podiva University, Italy
Symplectic methods for CR geometry and $\bar\partial$ problem

Abstract

• 23.03.2004
It is well-known that dimensions of Euclidean spaces add up, if one considers their product, ${\Bbb{R}}^d={\Bbb{R}}^m \times {\Bbb{R}}^n$, $d=m+n$. For Riemannian manifolds, the notion of dimension is more delicate, e.g. the topological dimension does not reflect their geometry at infinity. However, one may introduce {\it an isoperimetric dimension} through isoperimetric inequalities. The dimension introduced in this way is not a number but a family of functions indexed by a parameter $p$, $1 < p < \infty$. Our main result generalizes the addition of dimensions in the euclidean case using the notion of the isoperimetric dimension. (This is a joint work with T. Coulhon and A. Grigor'yan).