First, we examine the DC response of an electrochemical thin film, such as the separator in a micro-battery, over a wide range of applied current densities. The model system consists of a binary electrolyte between parallel-plate electrodes, each possessing a compact Stern layer which mediates Faradaic reactions with nonlinear Butler-Volmer kinetics. Our analysis, which is based on the Poisson-Nernst-Planck equations subject to boundary conditions appropriate for an electrolytic/galvanic cell, leads to the general conclusion that both the system size and surface physics play a critical role in the response of the electrochemical cell. For instance, reaction rate constants and the Stern-layer capacitance have a strong influence on cell behavior. We also explore the structure of the cell as a function of the applied current density. At high currents, two significant features appear at the cathode end of the cell: (i) a nested boundary layer structure at the classical diffusion-limited current and (ii) an extended space charge region above the limiting current.

Next, we study the response of a metal colloid sphere in an electrolyte solution in a range of applied electric fields. This problem, which is applicable to novel electrokinetically driven microfluidic devices, has traditionally been analyzed using circuit models which neglect bulk concentration variations that arise due to double layer charging. Our analysis is based on the Nernst-Planck equations and involve the derivation of general effective boundary conditions for surface transport processes. For the steady problem, we find that bulk concentrations gradients become significant at high applied fields and affect both bulk and double layer transport processes. In addition, surface transport becomes important for strong applied fields as a result of enhanced absorption ions by the double layer. For the unsteady problem at applied fields that are not too strong, diffusion processes, which are necessary for the system to relax to steady-state, are suppressed at leading-order but appear as higher-order corrections. Unfortunately, the dynamic response of the system to large applied fields seems to introduce several complications that make the analysis (both mathematical and numerical) quite challenging; the resolution of these challenges is left for future work.

In this thesis, we study polyhedral and combinatorial properties of solutions to the Traveling Salesman Path Problem. Using the approach of linear programming, we study properties of the polyhedron corresponding to a linear programming relaxation of the traveling salesman path problem. Our results relate the structure of the underlying graph of the problem instance with polyhedral properties of the corresponding fractional path polytope.

We first characterize traveling salesman path perfect graphs, graphs for which the convex hull of incidence vectors of traveling salesman paths can be described by linear inequalities. For these graphs, the convex hull of solutions to the traveling salesman path problem has a known complete description by linear inequalities. We show these graphs have a simple description by way of forbidden minors and give a complete characterization of the set of such graphs. We extend these results to relate the structure of the underlying graph to the integrality gap of the corresponding fractional path polytope. We present graph operations which preserve integrality gap; these operations allow us to find the integrality gap of graphs built from smaller bricks, whose integrality gaps can be found by computational or other methods.

A p-adic Local Langlands correspondence would assign an L-packet consisting of finitely many admissible representations of a p-adic group, to each Langlands parameter. To identify particular representations, data extending a Langlands parameter is needed to make ``completed Langlands parameters.''