Physical Sciences and Mathematics Commons^™

On Crown-Free Set Families, Diffusion State Difference, And Non-Uniform Hypergraphs, Edward Lawrence Boehnlein

Theses and Dissertations

We present results in three different arenas of discrete mathematics. Let La(n, H) denote the cardinality of the largest family on the Boolean lattice that does not contain H as a subposet. Denote π(H) := limn→∞ La(n,H) (bn/ n2c) . A crown O2k for k ≥ 2 is a poset on 2 levels whose Hasse diagram is a cycle. Griggs and Lu (2009) showed π(O4k) = 1 for k ≥ 2. Lu (2014) proved π(O2k) = 1 for odd k ≥ 7. We prove that the maximum size of a O6 -free family, when restricted to the middle two levels …

Structure Of The Stable Marriage And Stable Roommate Problems And Applications, Joe Hidakatsu

Theses and Dissertations

The well-known Gale-Shapley algorithm is a solution to the stable marriage problem, but always results in the same stable marriage, regardless of how the algorithm is executed. Robert Irving and Paul Leather constructed the rotation poset, whose downward closed sets are in one-to-one correspondence with the set of stable marriage assignments. We discuss how to use the rotation poset to find the k-optimal matching, and prove that a k-optimal matching is the same as a minimum regret matching for high enough k. Finally, Dan Gusfield defines the rotation poset for the stable roommate problem, and uses it to efficiently enumerate …

Some Extremal And Structural Problems In Graph Theory, Taylor Mitchell Short

Theses and Dissertations

This work considers three main topics. In Chapter 2, we deal with König-Egerváry graphs. We will give two new characterizations of König-Egerváry graphs as well as prove a related lower bound for the independence number of a graph. In Chapter 3, we study joint degree vectors (JDV). A problem arising from statistics is to determine the maximum number of non-zero elements of a JDV. We provide reasonable lower and upper bounds for this maximum number. Lastly, in Chapter 4 we study a problem in chemical graph theory. In particular, we characterize extremal cases for the number of maximal matchings in …

Chebyshev Inversion Of The Radon Transform, Jared Cameron Szi

Theses and Dissertations

In its two-dimensional form, the Radon transform of an image (function) is a collection of projections of the image which are parameterized by a set of angles (from the positive x-axis) and distances from the origin. Computational methods of the Radon transform are important in many image processing and computer vision problems, such as pattern recognition and the reconstruction of medical images. However, computability requires the construction of a discrete analog to the Radon transform, along with discrete alternatives for its inversion. In this paper, we present discrete analogs using classical methods of Chebyshev polynomial reconstruction, along with a new …

On A Constant Associated With The Prouhet-Tarry-Escott Problem, Maria E. Markovich

Theses and Dissertations

For n a positive integer, the Prouhet-Tarry-Escott Problem asks for two different sets of n positive integers for which the sum of the kth powers of the elements of one set is equal to the sum of the kth powers of the elements of the second set for each positive integer k < n. For n > 12, it is not known whether such sets exist. I will give some background on this problem and then show how Newton polygons can be used to determine information on the size of the 2-adic value of a certain constant associated with the problem.

Binary Quartic Forms Over Fp, Daniel Thomas Kamenetsky

Theses and Dissertations

Let Vp denote the five dimensional vector space of binary quartic forms over the finite field Fp, with p a prime greater than 3. There is a natural action of the group GL1(Fp)×GL2(Fp) on Vp. This action partitions Vp into orbits, the number of which increases with p. In this thesis, we determine explicitly, for a given p, the number of orbits under the action of GL1(Fp) × GL2(Fp) on Vp. Moreover, we determine the size of each orbit and the general structure of the forms each orbit contains. We also introduce an application of understanding these orbits to the …

Modeling Of Structural Relaxation By A Variable-Order Fractional Differential Equation, Su Yang

Theses and Dissertations

In physical point of view, relaxation usually describes the return from a perturbed system into equilibrium and each process has its own characteristic relaxation time. In 1946, Tool first formulated the notion of fictive temperature to characterize the structure of a glass-forming melt. Since then, people used to simulate structural relaxation by first order model. Since fractional-based models have not widely applied in modeling the fictive temperature, I want to explore the the possibility of modeling structural relaxation by fractional differential equation.

In this thesis, I will first introduce the definitions of two different kinds of fractional derivatives: Riemann-Liouville fractional …