| MAIN SPEAKERS
(One hour talks)
|
| P.E.T. JØRGENSEN (USA),
|
| (Department of Mathematics, University of
Iowa, IA - USA) |
 |
| Selected Problems In Nonlinear Dynamics.
(Joint work with Myung-Sin Song.) |
| Abstract:
Our work is motivated by, but not restricted to, the problem of separating variables in stochastic processes, say Xt from Geometric Analysis; processes that arise from statistical noise, for example from fractional Brownian motion. Since the initial inception in mathematical statistics, the operator algebraic contents of the arguments have crystallized as follows: Starting from the process Xt , for simplicity assume zero mean, i.e., E(Xt) = 0; create a correlation matrix C(s,t) = E(Xs Xt) . |
| A key analytic step in the Karhunen-Loeve method is to then apply the Spectral Theorem from operator theory to a corresponding selfadjoint operator, or to some operator naturally associated with C: Hence the name, the Karhunen-Loeve Decomposition (KLC). In favorable cases (discrete spectrum), an orthogonal family of functions in the time variable arise, and a corresponding family of eigenvalues. By integrating the basis functions against Xt, we get a sequence of random variables Yn. This sequence is independent, and if the initial random process Xt is Gaussian, then so are the random variables Yn. |
| In the talk, we note that the initial settings of the K-L method place rather stronger assumptions. We argue how modern nonlinear applications dictate the use of more general theorems, which we present. |
| We cover three things: (i) We extend the original Karhunen-Loeve idea to case of continuous spectrum; (ii) we give frame theoretic uses of the Karhunen-Loeve idea which arise in various wavelet contexts and which go beyond their initial uses; and finally (iii) give applications. |
| These applications are from image analysis: the problem of statistical recognition and detection; e.g., to nonlinear variance, for example due to illumination effects. Then the (KLD), also known as Principal Component Analysis (PCA) applies to the intensity images. This is traditional in statistical signal detection and in estimation theory. Adaptations to compression and recognition are of a more recent vintage. In brief outline, each intensity image is converted into vector form. (This is the simplest case of a purely intensity-based coding of the image, and it is not necessarily ideal for the application of KL-decompositions.) |
| The ensemble of vectors used in a particular conversion is assumed to have a multi-variate Gaussian distribution since human faces form a dense cluster in image space. The PCA method generates small set of basis vectors forming subspaces whose linear combination offer better (or perhaps ideal) approximation to the original vectors in the ensemble. In facial recognition, the new bases are said to span intra-face and inter-face variations, permitting Euclidean distance measurements to exclusively pick up changes in for example identity and expression. |
| Our presentation will start with various operator theoretic tools, and is for the union of four overlapping audiences, dynamics, information theory, wavelets, and image processing. |
| N. KAMRAN (Canada) |
| (Department of Mathematics and Statistics,
McGill University, Canada) |
 |
| The Penrose process and the wave equation in Kerr geometry. |
| Abstract:
We shall review the Penrose process for extracting mass and angular
momentum from the Kerr black hole solution of the Einstein equations.
We will show that Christodoulou's bound on the maximal energy gain by
the classical Penrose process can be realized by choosing suitable wave
packet initial data for the scalar wave equation in Kerr geometry,
thereby putting super-radiance on a rigorous mathematical footing. This is
joint work with Felix Finster, Joel Smoller and Shing-Tung Yau. |
| V. LYCHAGIN (Norway), |
| (Institute of Mathematics and
Statistics, University of Tromsø, Norway) |
 |
| Differential Invariants for Differential Equations. |
| Abstract:
Different pseudo group Lie actions give rise different symmetry and classification
problems for nonlinear differential equations.
We'll discuss various methods for computations of the corresponding differential invariants
algebras with applications for PDE and ODE classification. |
| A. PRÁSTARO (Italy) |
| (Department
of Methods and Mathematical Models for Applied Sciences (MEMOMAT), University of Rome
''La Sapienza'', Italy) |
 |
| On the Stability in the Geometric Theory of PDE's. A New MHD-PDE. |
| Abstract:
The stability of PDE's and solutions in PDE's are considered in a new unifying geometric theory, founded on the integral and quantum bordism groups of PDE's. This theory is applied to some important equations of the Mathematical Physics, like Navier-Stokes equations and anisotropic incompressible fluid plasmas equations (MHD-PDE's). New equations are proposed that allow to simulate these systems like stable extended crystals.
|
| Th.M. RASSIAS (Greece) |
| (Department of Mathematics, National Technical University of Athens, Greece) |
 |
| On the A.D. Aleksandrov Problem in Analysis, Geometry and Physics. |
| Abstract:
The problem of A.D.Aleksandrov for distance preserving mappings in the
spirit of Mathematical Analysis and Geometry will be considered.
Some new and old results and open problems will be discussed in a unified
manner. The connection of these results with certain problems of
Mathematical Physics will be analyzed.
|
