Physical Sciences and Mathematics Commons^™

The Calculus War: The Ultimate Clash Of Genius, Walker Briles Bussey-Spencer

Chancellor’s Honors Program Projects

No abstract provided.

Construction And Classification Results For Commuting Squares Of Finite Dimensional *-Algebras, Chase Thomas Worley

Doctoral Dissertations

In this dissertation, we present new constructions of commuting squares, and we investigate finiteness and isolation results for these objects. We also give applications to the classification of complex Hadamard matrices and to Hopf algebras.

In the first part, we recall the notion of commuting squares which were introduced by Popa and arise naturally as invariants in Jones' theory of subfactors. We review some of the main known examples of commuting squares such as those constructed from finite groups and from complex Hadamard matrices. We also recall Nicoara's notion of defect which gives an upper bound for the number of …

Classification Results Of Hadamard Matrices, Gregory Allen Schmidt

Masters Theses

In 1893 Hadamard proved that for any n x n matrix A over the complex numbers, with all of its entries of absolute value less than or equal to 1, it necessarily follows that

|det(A)| ≤ n^n/2 [n raised to the power n divided by two],

with equality if and only if the rows of A are mutually orthogonal and the absolute value of each entry is equal to 1 (See [2], [3]). Such matrices are now appropriately identified as Hadamard matrices, which provides an active area of research in both theoretical and applied fields …

Approximation Of Invariant Subspaces, Faruk Yilmaz

Doctoral Dissertations

For a real number α [alpha] the Dirichlet-type spaces 𝔇_α [script D sub alpha] are the family of Hilbert spaces consisting of all analytic functions f(z) = ∑_n=0^∞[sum over n equals zero to infinity] ˆf(n) [f hat of n] zⁿ [z to the n] defined on the open unit disc 𝔻 [unit disc] such that

∑_n=0^∞ (n+1)^α | ˆf(n) |²

[sum over n equals 0 to infinity] [(n+1) to α] [ | f hat of n | to 2]

is finite.

For α < 0, the spaces 𝔇_α are known as weighted Bergman spaces. When …

Generalizations And Variations Of The Zero-Divisor Graph, Grace Elizabeth Mcclurkin

Doctoral Dissertations

We explore generalizations and variations of the zero-divisor graph on commutative rings with identity. A zero-divisor graph is a graph whose vertex set is the nonzero zero-divisors of a ring, wherein two distinct vertices are adjacent if their product is zero. Variations of the zero-divisor graph are created by changing the vertex set, the edge condition, or both. The annihilator graph and the extended zero-divisor graph are both variations that change the edge condition, whereas the compressed graph and ideal-based graph change the vertex set. By combining these concepts, we define and investigate graphs where both the vertex set and …

Localization Of Large Scale Structures, Ryan James Jensen

Doctoral Dissertations

We begin by giving the definition of coarse structures by John Roe, but quickly move to the equivalent concept of large scale geometry given by Jerzy Dydak. Next we present some basic but often used concepts and results in large scale geometry. We then state and prove the equivalence of various definitions of asymptotic dimension for arbitrary large scale spaces. Some of these are generalizations of asymptotic dimension for metric spaces, and many of the proofs are new. Particularly useful in proving the equivalences of the various definitions is the notion of partitions of unity, originally set forth by Jerzy …

Generalizations Of Coarse Properties In Large Scale Spaces, Kevin Michael Sinclair

Doctoral Dissertations

Many results in large scale geometry are proven for a metric space. However, there exists many large scale spaces that are not metrizable. We generalize several concepts to general large scale spaces and prove relationships between them. First we look into the concept of coarse amenability and other variations of amenability on large scale spaces. This leads into the definition of coarse sparsification and connections with coarse amenability. From there, we look into an equivalence of Sako's definition of property A on uniformly locally finite spaces and prove that finite coarse asymptotic definition implies it. As well, we define large …

Dependence Structures In Lévy-Type Markov Processes, Eddie Brendan Tu

Doctoral Dissertations

In this dissertation, we examine the positive and negative dependence of infinitely divisible distributions and Lévy-type Markov processes. Examples of infinitely divisible distributions include Poissonian distributions like compound Poisson and α-stable distributions. Examples of Lévy-type Markov processes include Lévy processes and Feller processes, which include a class of jump-diffusions, certain stochastic differential equations with Lévy noise, and subordinated Markov processes. Other examples of Lévy-type Markov processes are time-inhomogeneous Feller evolution systems (FES), which include additive processes. We will provide a tour of various forms of positive dependence, which include association, positive supermodular association (PSA), positive supermodular dependence (PSD), and positive …

Various Topics On Graphical Structures Placed On Commutative Rings, Darrin Weber

Doctoral Dissertations

In this dissertation, we look at two types of graphs that can be placed on a commutative ring: the zero-divisor graph and the ideal-based zero-divisor graph. A zero-divisor graph is a graph whose vertices are the nonzero zero-divisors of a ring and two vertices are connected by an edge if and only if their product is 0. We classify, up to isomorphism, all commutative rings without identity that have a zero-divisor graph on 14 or fewer vertices.

An ideal-based zero-divisor graph is a generalization of the zero-divisor graph where for a ring R and ideal I the vertices are { …

Statistical Computational Topology And Geometry For Understanding Data, Joshua Lee Mike

Doctoral Dissertations

Here we describe three projects involving data analysis which focus on engaging statistics with the geometry and/or topology of the data.

The first project involves the development and implementation of kernel density estimation for persistence diagrams. These kernel densities consider neighborhoods for every feature in the center diagram and gives to each feature an independent, orthogonal direction. The creation of kernel densities in this realm yields a (previously unavailable) full characterization of the (random) geometry of a dataspace or data distribution.

In the second project, cohomology is used to guide a search for kidney exchange cycles within a kidney paired …

Jet-Hadron Correlations Relative To The Event Plane Pb--Pb Collisions At The Lhc In Alice, Joel Anthony Mazer

Doctoral Dissertations

In relativistic heavy ion collisions at the Large Hadron Collider (LHC), a hot, dense and strongly interacting medium known as the Quark Gluon Plasma (QGP) is produced. Quarks and gluons from incoming nuclei collide to produce partons at high momenta early in the collisions. By fragmenting into collimated sprays of hadrons, these partons form 'jets'. Within the framework of perturbative Quantum Chromodynamics (pQCD), jet production is well understood in pp collisions. We can use jets measured in pp interactions as a baseline reference for comparing to heavy ion collision systems to detect and study jet quenching. The jet quenching mechanism …

The Loewner Equation And Weierstrass' Function, Gavin Ainsley Glenn

Chancellor’s Honors Program Projects

No abstract provided.

"Returning To The Root" Of The Problem: Improving The Social Condition Of African Americans Through Science And Mathematics Education, Vanessa R. Pitts Bannister, Julius Davis, Jomo Mutegi, Latasha Thompson, Deborah Lewis

Catalyst: A Social Justice Forum

The underachievement and underrepresentation of African Americans in STEM (Science, Technology, Engineering and Mathematics) disciplines have been well documented. Efforts to improve the STEM education of African Americans continue to focus on relationships between teaching and learning and factors such as culture, race, power, class, learning preferences, cultural styles and language. Although this body of literature is deemed valuable, it fails to help STEM teacher educators and teachers critically assess other important factors such as pedagogy and curriculum. In this article, the authors argue that both pedagogy and curriculum should be centered on the social condition of African Americans – …