Physical Sciences and Mathematics Commons^™

Methods Of Nonparametric Multivariate Ranking And Selection, Jeremy Entner

Mathematics - Dissertations

In a Ranking and Selection problem, a collection of k populations is given which follow some (partially) unknown probability distributions. The problem is to select the "best" of the k populations where "best" is well defined in terms of some unknown population parameter. In many univariate parametric and nonparamentric settings, solutions to these ranking and selection problems exist. In the multivariate case, only parametric solutions have been developed. We have developed several methods for solving nonparametric multivariate ranking and selection problems. The problems considered allow an experimenter to select the "best" populations based on nonparametric notions of dispersion, location, and …

Extension Properties Of Pluricomplex Green Functions, Sidika Zeynep Ozal

Mathematics - Dissertations

In 1985, Klimek introduced an extremal plurisubharmonic function on bounded domains in Cn that generalizes the Green's function of one variable. This function is called the pluricomplex Green function of Ω with logarithmic pole at a and is denoted by g_Ω(.,a). The aim of this thesis was to investigate the extension properties of g_Ω(.,a). Let Ω0 be a bounded domain of Cn and E be a compact subset of Ω0 such that Ω : = Ω0 E is connected. In general, g_Ω(.,a) cannot be extended as a pluricomplex Green function to any subdomain of …

Discrete Sparse Fourier Hermite Approximations In High Dimensions, Ashley Prater

Mathematics - Dissertations

In this dissertation, the discrete sparse Fourier Hermite approximation of a function in a specified Hilbert space of arbitrary dimension is defined, and theoretical error bounds of the numerically computed approximation are proven. Computing the Fourier Hermite approximation in high dimensions suffers from the well-known curse of dimensionality. In short, as the ambient dimension increases, the complexity of the problem grows until it is impossible to numerically compute a solution. To circumvent this difficulty, a sparse, hyperbolic cross shaped set, that takes advantage of the natural decaying nature of the Fourier Hermite coefficients, is used to serve as an index …

The Clar Structure Of Fullerenes, Elizabeth Jane Hartung

Mathematics - Dissertations

A fullerene is a 3-regular plane graph consisting only of pentagonal and hexagonal faces. Fullerenes are designed to model carbon molecules. The Clar number and Fries number are two parameters that are related to the stability of carbon molecules. We introduce chain decompositions, a new method to find lower bounds for the Clar and Fries numbers. In Chapter 3, we define the Clar structure for a fullerene, a less general decomposition designed to compute the Clar number for classes of fullerenes. We use these new decompositions to understand the structure of fullerenes and achieve several results. In Chapter 4, …

Excess Porteous, Coherent Porteous, And The Hyperelliptic Locus In M3, Thomas S. Bleier

Mathematics - Dissertations

In the moduli space of curves of genus 3, the locus of hyperelliptic curves forms a divisor, that is a closed subscheme of codimension 1. J. Harris and I. Morrison compute an expression for the class of this divisor, in the Chow ring of the moduli space, using a map of vector bundles and by applying the Thom-Porteous formula to determine an expression for a certain degeneracy locus of this map. One would like to extend their idea in order to compute an expression for the divisor associated to the closure of the hyperelliptic locus, in the Chow ring of …

Potential Theory On Compact Sets, Tony Perkins

Mathematics - Dissertations

The primary goal of this work is to extend the notions of potential theory to compact sets. There are several equivalent ways to define continuous harmonic functions H(K) on a compact set K in [the set of real numbers]ⁿ. One may let H(K) be the uniform closure of all functions in C(K) which are restrictions of harmonic functions on a neighborhood of K, or take H(K) as the subspace of C(K) consisting of functions which are finely harmonic on the fine interior …

Mixed Problems And Layer Potentials For Harmonic And Biharmonic Functions, Moises Venouziou

Mathematics - Dissertations

The mixed problem is to find a harmonic or biharmonic function having prescribed Dirichlet data on one part of the boundary and prescribed Neumann data on the remainder. One must make a choice as to the required boundary regularity of solutions. When only weak regularity conditions are imposed, the harmonic mixed problem has been solved on smooth domains in the plane by Wendland, Stephan, and Hsiao. Significant advances were later made on Lipschitz domains by Ott and Brown. The strain of requiring a square-integrable gradient on the boundary, however, forces a strong geometric restriction on the domain. Well-known counterexamples by …

Complexity Over Finite-Dimensional Algebras, Marju Purin

Mathematics - Dissertations

In this thesis we study two types of complexity of modules over finite-dimensional algebras.

In the first part, we examine the Ω-complexity of a family of self-injective k-algebras where k is an algebraically closed field and Ω is the syzygy operator. More precisely, let T be the trivial extension of an iterated tilted algebra of type H. We prove that modules over the trivial extension T all have complexities either 0, 1, 2 or infinity, depending on the representation type of the hereditary algebra H. As part of the proof, we show that a stable equivalence between self-injective algebras preserves …

Mathematical Knowledge For Teaching Teachers: The Case Of Multiplication And Division Of Fractions, Dana E. Olanoff

Mathematics - Dissertations

This study attempts to answer the question, What is the mathematical knowledge required by teachers of elementary mathematics content courses in the area of multiplication and division of fractions? Beginning in the mid-1980s, when Shulman (1986) introduced the idea of pedagogical content knowledge, researchers have been looking at the knowledge needed to teach in a variety of different content areas. One area that has garnered much of the research is that of mathematics. Researchers have developed frameworks for what they call mathematical knowledge for teaching, but there has been little work done looking at the knowledge requirements for teachers of …