Georgia Institute of Technology College of Sciences

In this talk, we demonstrate the application of Neural Networks with Locally Converging Inputs (NNLCI) to simulate the scattering of electromagnetic waves around two-dimensional perfect electric conductors (PEC). The NNLCIs are designed to output high-fidelity numerical solutions from local patches of two coarse grid numerical solutions obtained by a convergent numerical scheme. Once trained, the NNLCIs can play the role of a computational cost-saving tool for repetitive computations with varying parameters. To generate the inputs to our NNLCI, we design on uniform rectangular grids a second-order accurate finite difference scheme that can handle curved PEC boundaries systematically. More specifically, our numerical scheme is based on the Back and Forth Error Compensation and Correction method together with the construction of ghost points via a level set framework, PDE-based extension technique, and what we term guest values. We illustrate the performance of our NNLCI subject to variations in incident waves as well as PEC boundary geometries.

We will finish our proof of the key lemma for the probabilistic unique continuation principle used in Ding-Smart. We will also briefly recall enough of the theory of martingales to clarify a use of Azuma's inequality, and the basic definitions of \epsilon-nets and \epsilon-packings required to formulate the basic volumetric bound for these in e.g. the unit sphere, before using these to complete the proof.

Mathapalooza! is back at this year's Atlanta Science Festival! Come join us on Saturday, March 18, for an afternoon of mathematical fun beginning at 1:00pm at the Paideia School.  There will be interactive puzzles and games, artwork, music, stage acts, and mathematics in motion.

https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

G. W. Hill made major contributions to Celestial Mechanics. One of them is to develop his lunar theory as an alternative approach for the study of the motion of the Moon around the Earth, which is the classical Lunar Hill problem. The mathematical model we study is one of the extensions of the classical Hill approximation of the restricted three-body problem. Considering a restricted four body problem, with a hierarchy between the bodies: two larger bodies, a smaller one and a fourth infinitesimal body, we encounter the shapes of the three heavy bodies via oblateness. We first find that the triangular central configurations of the three heavy bodies is a scalene triangle. Through the application of the Hill approximation, we obtain the limiting Hamiltonian that describes the dynamics of the infinitesimal body in a neighborhood of the smaller body. As a motivating example, we identify the three heavy bodies with the Sun, Jupiter and the Jupiter’s Trojan asteroid Hektor.