http://hdl.handle.net/1903/2261 2013-05-25T17:04:19Z http://hdl.handle.net/1903/13853 Title: Commutative Endomorphism Rings of Simple Abelian Varieties over Finite Fields Authors: Bradford, Jeremy Scott Abstract: In this thesis we look at simple abelian varieties defined over a finite field $k =\mathbb{F}_{p^n}$ with $\End_k(A)$ commutative. We derive a formula that connects the $p$-rank $r(A)$ with the splitting behavior of $p$ in $E = \mathbb{Q}(\pi)$, where $\pi$ is a root of the characteristic polynomial of the Frobenius endomorphism. We show how this formula can be used to explicitly list all possible splitting behaviors of $p$ in $\mathcal{O}_E$, and we do so for abelian varieties of dimension less than or equal to four defined over $\mathbb{F}_p$. We then look for when $p$ divides $[\mathcal{O}_E : \mathbb{Z}[\pi, \bar{\pi}]]$. This allows us to prove that the endomorphism ring of an absolutely simple abelian surface is maximal at $p$ when $p \geq 3$. We also derive a condition that guarantees that $p$ divides $[\mathcal{O}_E: \mathbb{Z}[\pi, \bar{\pi}]]$. Last, we explicitly describe the structure of some intermediate subrings of $p$-power index between $\mathbb{Z}[\pi, \bar{\pi}]$ and $\mathcal{O}_E$ when $A$ is an abelian 3-fold with $r(A) = 1$. 2012-01-01T00:00:00Z http://hdl.handle.net/1903/13656 Title: Small fluctuations in epitaxial growth via conservative noise Authors: Patrone, Paul Nathan Abstract: We study the combined effect of growth (material deposition from above) and nearest-neighbor entropic and force-dipole interactions in a stochastically perturbed system of N line defects (steps) on a vicinal crystal surface in 1+1 dimensions. First, we formulate a general model of conservative white noise, and we derive sim- plified formulas for the terrace width distribution (TWD) and pair correlations, particularly the covariance matrix of terrace widths, in the limit N → ∞ for small step fluctuations. Second, we apply our formalism to two specific noise models which stem, respectively, from: (i) the fluctuation-dissipation theorem for diffusion of adsorbed atoms; and (ii) the phenomenological consideration of deposition-flux- induced asymmetric attachment and detachment of atoms at step edges. We discuss implications of our analysis, particularly the narrowing of the TWD with the de- position flux, connection of noise structure to terrace width correlations, behavior of these correlations in the macroscopic limit, and comparison of our perturbation results to a known mean field approach. 2012-01-01T00:00:00Z http://hdl.handle.net/1903/13621 Title: Three Dimensional Edge Detection Using Wavelet and Shearlet Analysis Authors: Schug, David Albert Abstract: Edge detection determines the boundary of objects in an image. A sequence of images records a 2D representation of a scene changing over time, giving 3D data. New 3D edge detectors, particularly ones we developed using shearlets and hybrid shearlet-Canny algorithms, identify edges of complicated objects much more reliably than standard approaches, especially under high noise conditions. We also use edge information to track the position and velocity of objects using an optimization algorithm. 2012-01-01T00:00:00Z http://hdl.handle.net/1903/13620 Title: Modeling and Solving Arc Routing Problems in Street Sweeping and Snow Plowing Authors: Dussault, Ben Abstract: In arc routing problems, the goal is to determine an optimal path, or set of paths, that traverse a required subset of arcs on a graph with respect to a set of constraints and objective function. The Chinese Postman Problem (CPP) forms the basis for many arc routing problems. Let graph G =(V,A), where V is a set of vertices and A = {(i,j) | i,j in V} is a set of arcs that each connect exactly two vertices, each with its own cost of traversal c_ij. The objective of the CPP is to construct a least cost path that traverses each arc in A at least once. There are many practical applications for variants of the CPP, including winter street maintenance, and street sweeping that incorporate: [Rural Instances] Rural Postman Problems (RPP) stipulate that only a subset $A_R \subset A$ require traversal, allowing for non-servicing traversal on the rest of the graph. In the context of street sweeping, a street sweeper isn't responsible for sweeping all the streets. [Windy Graphs] In the CPP, the cost of traversal of an arc is the same, regardless of the direction of traversal. In the Windy Postman Problem (WPP), the cost of traversal is asymmetric. That is, it is possible for c_ij not equal c_ji. In the context of snow plowing, it is harder to plow uphill than downhill. [Multi-Vehicle] Instead of a single vehicle with a single tour, multiple tours are found for multiple vehicles. This is often accompanied with an objective function that seeks to minimize the cost of the largest cost route. This is motivated by practical applications, which seek to balance the cost of each route. In the case where route cost is measured in time, route balancing minimizes, for example, paid overtime. [Turn Penalties] UPS reported that it saved three million gallons of gasoline annually by avoiding unnecessary left-hand turns, which take longer to perform than going straight or turning right. Instances with turn penalties incorporate costs of turning, in addition to costs of traversal. The Windy Postman Problem (WPP) incorporates windy graphs and the Rural Postman Problem (RPP) incorporates rural instances. The RPP can be extended to include turn penalties (RPPTP). The Windy Rural Postman Problem (WRPP) incorporates instances that are both windy and rural. The WRPP can be extended to the MM k-WRPP which adds k plows. In this dissertation, we extend these variants to new problems with new problem attributes that are practically motivated. Our new attributes are listed below. [Multi-Period] The CPP solves for a single route, which can be interpreted to be traversed in a single day. It is possible that the set of required arcs is too long to service in a single day and therefore must be split among multiple days. In this case, we need to decide which day to assign service to each arc, before routing can take place. [Downhill Instances] In street snow plowing, it is faster to deadhead (traverse without servicing) a street rather than plowing it. In this case, there are different costs for deadheading and plowing a street. Moreover, it takes longer to plow uphill, resulting in four costs: plowing uphill, plowing downhill, deadheading uphill, and deadheading downhill. [Precedence] When considering downhill instances, the snow may be so deep that it is impossible for a snowplow to deadhead a street before the street is plowed. In this dissertation we present a variety of heuristics to solve these problems, all adaptations of the concept of cycle permutation based on Euclidean cycle decomposition. To our knowledge, the use of moving or permuting sub-cycles as a way to change and improve a Eulerian cycle is novel and we show that it is very robust at improving solutions. 2012-01-01T00:00:00Z