PHD, Kent State University, 2021, College of Arts and Sciences / Department of Mathematical Sciences

In this thesis we develop efficient numerical methods for the approximation of matrix functionals of the form F(A):=w^Tf(A)v, where A is a large symmetric or nonsymmetric matrix, w,v are given vectors, and f is a function. Golub and Meurant describe a technique for computing upper and lower bounds for matrix functionals F(A) based on the connection between the Lanczos process, orthogonal polynomials, and Gauss-type quadrature rules. Their technique considers the expression F(A) as a Stieltjes integral. If the derivatives of the integrand f do not change sign on the convex hull of the support of the measure, then Gauss-type quadrature rules can be applied to compute upper and lower bounds for F(A). However, when A is symmetric and derivatives of the integrand f change sign in the convex hull of the spectrum of A, or when the matrix A is nonsymmetric, then this approach is not guaranteed to yield upper and lower bounds. We consider several extensions of the technique by Golub and Meurant for approximating matrix functions of the form F(A). Let A be a large symmetric matrix. Our first extension is based on the use of pairs of Gauss, and suitable generalized Gauss--Radau or generalized Gauss--Lobatto rules that yield upper and lower bounds for F(A) when some of the derivatives of f change sign on the convex hull of the support of the measure. We also describe new methods to evaluate these quadrature rules. Our other extensions are concerned with the situation when the function f cannot be approximated accurately by a polynomial of small to moderate degree. Then Gauss-type rules may yield poor approximations of the functional F(A). This situation occurs, for instance, when the function f has one or several singularities close to the support of the measure. This difficulty can be remedied by using rational Gauss rules. We discuss two approaches related to this case. First, we develop a technique to approximate matrix functionals of the form F(A) with A a large (open full item for complete abstract)

Committee: Lothar Reichel (Advisor); Miroslav Pranic' (Advisor); Jing Li (Committee Member); Jun Li (Committee Member); Mikhail Nesterenko (Committee Member); Arvind Bansal (Committee Member) Subjects: Applied Mathematics

Doctor of Philosophy (PhD), Ohio University, 2018, Mathematics (Arts and Sciences)

This dissertation entails four different topics related to coding theory. While much effort has gone recently into studying codes over ring alphabets (rather than the traditional field alphabets), we change this perspective slightly and focus not necessarily on the algebraic structure of the alphabet but in the ancillary structure of the ambient (the algebraically enhanced environment from which codes are drawn.) Doing so, we determine, for example, that when the ambient is a Frobenius algebra a version of the MacWilliams identities holds between the weight distribution of the appropriate codes and their properly defined duals. We further the consideration of rational power series and sequential codes, which had mostly been studied over field alphabets, to the case when the alphabet is a commutative local ring with nilpotent radical. Perhaps the most striking feature of our study is the introduction of a modified division algorithm that is based not on the degree of polynomials but on their codegree (degree of the lowest degree non-zero term.) As was the case in the initial papers on the subject, we explore criteria to recognize the Kronecker Criterion as well as multivariable rational functions among arbitrary multivariable power series with coefficients in a commutative local ring with nilpotent radical. The duals of sequential codes are the so-called polycyclic codes. We have studied intrinsic notions of duality based on the codegree of polynomials with sights to have a fully intrinsic perspective while studying codes in such ambients. We continue the ongoing explorations to recognize commutative local finite rings. Our study considers the feasibility of extending results in the literature for rings with 16 elements to rings with p4 elements (p an arbitrary prime.)

Committee: Sergio Lopez-Permouth (Advisor); Alexei Davydov (Committee Member); Adam Fuller (Committee Member); Jeffery Dill (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2010, EDU Teaching and Learning

The purpose of this research was to investigate student conceptions of the topic of asymptotes of rational functions and to understand the connections that students developed between the closely related notions of asymptotes, continuity, and limits. The participants of the study were university students taking Calculus 2 and were mostly freshmen. The study of rational functions and asymptotes follows the study of functions in College Algebra. The function concept is a fundamental topic in the field of mathematics and physical sciences. The concept of asymptotes is closely related to the concepts of limits, continuity, and indeterminate forms in Calculus 1. Therefore, investigating student beliefs of asymptotes and the connections with related topics could possibly shed light onto effective ways of instructing of these concepts. Qualitative methodology was used to conduct this investigation. The investigation was conducted through two problem-solving interviews and several teaching episodes. The goal of each problem-solving interview was to gain an indepth understanding of students' thinking processes while solving problems. Nineteen Calculus 2 students participated in the first problem solving interview that investigated student concept images of asymptotes of rational functions and the connections students have developed among these concepts. The interview was about two hours long. Based on the results of this interview, eight students were selected, and seven students completed teaching episodes. The teaching episodes lasted for one hour and thirty minutes and were conducted twice a week for four weeks. The participants of the teaching episodes were divided into two groups; one group consisted of 4 students, and the other group consisted of 3 students. Thus, adjustments could me made in the teaching episodes of the second group based on the observations of the first group. The purpose of these teaching episodes was to create a model of student thinking while th (open full item for complete abstract)

Committee: Douglas Owens (Advisor); Azita Manouchehri (Committee Member); Robert Brown (Committee Member) Subjects: Mathematics; Mathematics Education