Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

Neural networks have led to new and state of the art approaches for image recovery. They provide a contrast to standard image processing methods based on the ideas of sparsity and wavelets.  In this talk, we will study two different random neural networks.  One acts as a model for a learned neural network that is trained to sample from the distribution of natural images.  Another acts as an unlearned model which can be used to process natural images without any training data.  In both cases we will use high dimensional concentration estimates to establish theory for the performance of random neural networks in imaging problems.

I will consider the motion of a rigid body with an interior cavity that is completely filled with a viscous fluid. The equilibria of the system will be characterized and their stability properties are analyzed. It will be shown that the fluid exerts a stabilizing effect, driving the system towards a state where it is moving as a rigid body with constant angular velocity. In addition, I will characterize the critical spaces for the governing evolution equation, and I will show how parabolic regularization in time-weighted spaces affords great flexibility in establishing regularity and stability properties for the system. The approach is based on the theory of Lp-Lq maximal regularity. (Joint work with G. Mazzone and J. Prüss).

TBA

We present a topological mechanism of diffusion in a priori chaotic systems. The method leads to a proof of diffusion for an explicit range of perturbation parameters. The assumptions of our theorem can be verified using interval arithmetic numerics, leading to computer assisted proofs. As an example of application we prove diffusion in the Neptune-Triton planar elliptic restricted three body problem. Joint work with Marian Gidea.