Georgia Institute of Technology College of Sciences

In this paper, we consider selection of a sliding vector fieldof Filippov type on a discontinuity manifold $\Sigma$ of co-dimension 3(intersection of three co-dimension 1 manifolds). We propose an extension of the “moments vector field”to this case, and - under the assumption that $\Sigma$ is nodally attractive -we prove that our extension delivers a uniquely definedFilippov vector field. As it turns out, the justification of our proposed extension requiresestablishing invertibility of certain sign matrices. Finally,we also propose the extension of the moments vector field todiscontinuity manifolds of co-dimension 4 and higher.

By showing a duality relation between the Sobolev and Hardy-Littlewood-Sobolev inequalities, I discuss a proof of the sharp Sobolev inequality. The duality relation between these two inequalities is known since 1983 and has led to interesting recent work on the inequalities (which may be the topic of future talks).

We introduce a new model for cell phone signal problem, which is stochastic van der Pol oscillator with condition that ensures global boundedness in phase space and keeps unboundedness for frequency. Also we give a new definition for stochastic Poincare map and find a new approximation to return time and point. The new definition is based on the numerical observation. Also we develop a new approach by using dynamic tools, such as method of averaging and relaxation method, to estimate the return time and return point. Thus we can show that the return time is always not Gaussian and return point's distribution is not symmetric under certain section.

Fix k vertices in a graph G, say a_1,...,a_k, if there exists a cycle that visits these vertices with this specified order, we say such a cycle is (a_1,a_2,...,a_k)-ordered. It is shown by Thomas and Wollan that any 10k-connected graph is k-linked, therefore any 10k-connected graph has an (a_1,a_2,...,a_k)-ordered for any a_1,...,a_k. However, it is possible that we can improve this bound when k is small. It is shown by W. Goddard that any 4-connected maximal planar graph has an (a_1,...,a_4)-ordered cycle for any choice of 4 vertices. We will present a complete characterization of 4-ordered cycle in planar graphs. Namely, for any four vertices a,b,c,d in planar graph G, if there is no (a,b,c,d)-ordered cycle in G, then one of the follows holds: (1) there is a cut S separating {a,c} from {b,d} with |S|\leq 3; (2) roughly speaking, a,b,d,c "stay" in a face of G with this order.