Georgia Institute of Technology College of Sciences

We start studying open book foliations in this series of seminars. We will go through the theory and see how it is used in applications to contact topology.

The famous Doignon-Bell-Scarf theorem is a Helly-type result about the existence of integer solutions on systems linear inequalities. The purpose of this paper is to present the following ``weighted'' generalization: Given an integer k, we prove that there exists a constant c(k,n), depending only on the dimension n and k, such that if a polyhedron {x : Ax <= b} contains exactly k integer solutions, then there exists a subset of the rows of cardinality no more than c(k,n), defining a polyhedron that contains exactly the same k integer solutions. We work on both upper and lower bounds for this constant. This is joint work with Quentin Louveaux, Iskander Aliev and Robert Bassett.

In this work, we numerically studied the effect of the vorticity on the enhancement of heat transfer in a channel flow. Based on the model we proposed, we find that the flow exhibits different properties depending on the value of four dimensionless parameters. In particularly, we can classify the flows into two types, active and passive vibration, based on the sign of the incoming vortices. The temperature profiles according to the flow just described also show different characteristics corresponding to the active and passive vibration cases. In active vibration cases, we find that the heat transfer performance is directly related to the strength of the incoming vortices and the speed of the background flow. In passive vibration cases, the corresponding heat transfer process is complicated and varies dramatically as the flow changes its properties. Compared to the fluid parameters, we also find that the thermal parameters have much less effect on the heat transfer enhancement. Finally, we propose a more realistic optimization problem which is to minimize the maximum temperature of the solids with a given input energy. We find that the best heat transfer performance is obtained in the active vibration case with zero background flow.

The work in this dissertation is mainly focused on three subjects which are essentially related to linear systems on metric graphs and its application: (1) rank-determining sets of metric graphs, which can be employed to actually compute the rank function of arbitrary divisors on an arbitrary metric graph, (2) a tropical convexity theory for linear systems on metric graphs, and (3) smoothing of limit linear series of rank one on refined metrized complex (an intermediate object between metric graphs and algebraic curves),

This is a continuation of the previous talk.