Geometric Interpretation of the Two Dimensional Poisson Kernel And Its Applications.
Year of Dissertation:
2011
Hermann Schwarz, while studying complex analysis, introduced the geometric interpretation for the Poisson kernel in 1890. We shall see here that the geometric interpretation can be useful to develop a new approach to some old classical problems as well as to obtain several new results, mostly related to hyperbolic geometry.
The Geometry of Gauss' Composition Law
Year of Dissertation:
2010
Gauss' identification of a composition law for primitive integral binary quadratic forms of given discriminant D--which provides the set FD of SL2(Z) equivalence classes of such forms with a group structure--essentially amounts to the discovery of the class group of an order in a quadratic number field. We consider quadratic extensions of the field of rational functions k(u), where k is an algebraically closed field, and seek an analogue of Gauss composition in this context.
Sensor Strip Cover: Maximizing Network Lifetime on an Interval
Year of Dissertation:
2012
Suppose that n sensors are deployed on a one-dimensional region (a strip, or interval) that we wish to cover with a wireless sensor network. Each sensor is equipped with a finite battery, and has an adjustable sensing range, which we control. If each sensor's battery drains in inverse linear proportion to its sensing radius, which schedule will maximize the lifetime of the resulting network? We study this Sensor Strip Cover problem and several related variants. For the general Sensor Strip Cover problem, we analyze performance in both the worst-case and average-case for several algorithms, and show that the simplest algorithm, in which the sensors take turns covering the entire line, has a tight 3/2-approximation ratio. Moreover, we demonstrate a more sophisticated algorithm that achieves an expected lifetime of within 12% of the theoretical maximum against uniform random deployment of the sensors. We show that if the sensing radii can be set only once, then the resulting Set Once Strip Cover problem is NP-hard. However, if all sensors must be activated immediately, then we provide a polynomial time algorithm for the resulting Set Radius Strip Cover problem. Finally, we consider the imposition of a duty cycling restriction, which forces disjoint subsets of the sensors (called shifts) to act in concert to cover the entire interval. We provide a polynomial-time solution for the case in which each shift contains at most two sensors. For shifts of size k, we provide worst-case and average-case analysis for the performance of several algorithms.
Normal Families and Mondromies of Holomorphic Motions
Year of Dissertation:
2012
We explore some generalizations of results in holomorphic motions that result from Earle's infinite-dimensional generalization of Montel's Theorem. We then investigate topological obstructions to extending holomorphic motions. We finish with some miscellaneous facts.
Points of Canonical Height Zero on Projective Varieties
Year of Dissertation:
2010
Let k be an algebraically closed field of characteristic zero, C a smooth connected projective curve defined over k, K =k(C) the function field of C. Let Y be a projective K-variety, L a very ample line bundle on Y and α : Y &rarr Y a K-morphism such that α *<\super>L = L × d. We prove that a projective integral C-scheme Y is isotrivial when it is covered by a projective integral k-scheme X= X0<\sub> × C, where X0<\sub> is a k-scheme. This result provides a setup for a conjecture of L. Szpiro on parametrization of points of canonical height zero of the dynamical system (Y,L, α).
On the Arithmetic and Geometry of Quaternion Algebras: a spectral correspondence for Maass waveforms
Year of Dissertation:
2011
Late Points of Projections of Planar Symmetric Random Walks on the Lattice Torus
Year of Dissertation:
2012
We examine the cover time and set of late points of a symmetric random walk on Z2 projected onto the torus Z2K. This extends the work done for the simple random walk in [Late Points, DPRZ, 2006] to a large class of random walks. The approach uses comparisons between planar and toral hitting times and distributions on annuli, and uses only random walk methods. There are also generalizations of Green's functions, hitting times, and hitting distributions on Z2 and Z2K which are of independent interest.
Uniqueness Theorems for Some Nonlinear Parabolic Equations
Year of Dissertation:
2012
We study the uniqueness of solutions of the Cauchy problem of two nonlinear parabolic equations in this thesis.
Some Results on Large Cardinals and the Continuum Function
Year of Dissertation:
2012
I prove several new relative consistency results concerning large cardinals and the continuum function.
The Differentiability of Renormalized Triple Intersection Local Times
Year of Dissertation:
2013
The evolution of the theory of triple intersection times over the past, approximately, two decades has centered primarily on two dimensional Brownian Motion and planar symmetric stable processes. The one dimensional cases have gone largely unstudied. In this thesis, we examine the differentiability of renormalized triple intersection local times for the two aforementioned Markov processes in R1. In more detail, we prove that the single partial derivative with respect to each spatial variable exists and show that each partial derivative is, in fact, jointly continuous in both space and time variables. During the course of our analysis, we discover that these results hold for the class of symmetric stable process for which 3/2<β<2.