Mathematics Commons^™

Advanced Techniques In Time Series Forecasting: From Deterministic Models To Deep Learning, Xue Bai

Graduate Theses, Dissertations, and Problem Reports

This dissertation discusses three instances of temporal prediction, applied to population dynamics and deep learning.

In population modeling, dynamic processes are frequently represented by systems of differential equations, allowing for the analysis of various phenomena. The first application explores modeling cloned hematopoiesis in chronic myeloid leukemia (CML) via a nonlinear system of differential equations. By tracking the evolution of different cell compartments, including cycling and quiescent stem cells, progenitor cells, differentiated cells, and terminally differentiated cells, the model captures the transition from normal hematopoiesis to the chronic and accelerated-acute phases of CML. Three distinct non-zero steady states are identified, representing …

On Eulerian Subgraphs And Hamiltonian Line Graphs, Yikang Xie

Graduate Theses, Dissertations, and Problem Reports

A graph {\color{black}$G$} is Hamilton-connected if for any pair of distinct vertices {\color{black}$u, v \in V(G)$}, {\color{black}$G$} has a spanning $(u,v)$-path; {\color{black}$G$} is 1-hamiltonian if for any vertex subset $S \subseteq {\color{black}V(G)}$ with $|S| \le 1$, $G - S$ has a spanning cycle. Let $\delta(G)$, $\alpha'(G)$ and $L(G)$ denote the minimum degree, the matching number and the line graph of a graph $G$, respectively. The following result is obtained. {\color{black} Let $G$ be a simple graph} with $|E(G)| \ge 3$. If $\delta(G) \geq \alpha'(G)$, then each of the following holds. \\ (i) $L(G)$ is Hamilton-connected if and only if $\kappa(L(G))\ge …

Finite Matroidal Spaces And Matrological Spaces, Ziyad M. Hamad

Graduate Theses, Dissertations, and Problem Reports

The purpose of this thesis is to present new different spaces as attempts to generalize the concept of topological vector spaces. A topological vector space, a well-known concept in mathematics, is a vector space over a field \mathbb{F} with a topology that makes the addition and scalar multiplication operations of the vector space continuous functions. The field \mathbb{F} is usually \mathbb{R} or \mathbb{C} with their standard topologies. Since every vector space is a finitary matroid, we define two spaces called finite matroidal spaces and matrological spaces by replacing the linear structure of the topological vector space with a finitary matroidal …

Studies On Depth And Torsion In Tensor Products Of Modules, Uyen Huyen Thao Le

Graduate Theses, Dissertations, and Problem Reports

This dissertation represents an in-depth exploration of two distinct yet interconnected research topics within commutative algebra: one centered around a conjecture of Huneke and R. Wiegand and the other concerns a depth inequality of Auslander. It consists of the following three papers as well as the author's work under the direction of Professor Olgur Celikbas:

• Remarks on a conjecture of Huneke and Wiegand and the vanishing of (co)homology, Journal of Mathematical Society of Japan Advance Publication. (joint work with Olgur Celikbas, Hiroki Matsui, and Arash Sadeghi).
• An extension of a depth inequality of Auslander, Taiwanese Journal of Mathematics, …

Feature Extraction Of Footwear Impression Images For Quality Assessment, Alexandra Hill

Graduate Theses, Dissertations, and Problem Reports

Forensic footwear impression analysis is a valuable tool in criminal investigations. Extracting useful features from images of footwear impressions is a critical step in this process. However, the quality of these images can vary widely, making feature extraction challenging. In order to give a quality assessment rating to a footwear impression image, the image should first be analyzed to extract features from the impression. In this paper, we present a method to extract features from a 2D grayscale footwear impression image. A Hierarchical Grid Model implementation has been adapted from use on a 3D dataset to assist in finding features, …

Longest Path And Cycle Transversal And Gallai Families, James A. Long Jr

Graduate Theses, Dissertations, and Problem Reports

A longest path transversal in a graph G is a set of vertices S of G such that every longest path in G has a vertex in S. The longest path transversal number of a graph G is the size of a smallest longest path transversal in G and is denoted lpt(G). Similarly, a longest cycle transversal is a set of vertices S in a graph G such that every longest cycle in G has a vertex in S. The longest cycle transversal number of a graph G is the size of a smallest longest cycle transversal in G and …

Structure-Dependent Characterizations Of Multistationarity In Mass-Action Reaction Networks, Galyna Voitiuk

Graduate Theses, Dissertations, and Problem Reports

This project explores a topic in Chemical Reaction Network Theory. We analyze networks with one dimensional stoichiometric subspace using mass-action kinetics. For these types of networks, we study how the capacity for multiple positive equilibria and multiple positive nondegenerate equilibria can be determined using Euclidian embedded graphs. Our work adds to the catalog of the class of reaction networks with one-dimensional stoichiometric subspace answering in the affirmative a conjecture posed by Joshi and Shiu: Conjecture 0.1 (Question 6.1 [26]). A reaction network with one-dimensional stoichiometric subspace and more than one source complex has the capacity for multistationarity if and only …

On Generalizations Of Supereulerian Graphs, Sulin Song

Graduate Theses, Dissertations, and Problem Reports

A graph is supereulerian if it has a spanning closed trail. Pulleyblank in 1979 showed that determining whether a graph is supereulerian, even when restricted to planar graphs, is NP-complete. Let $\kappa'(G)$ and $\delta(G)$ be the edge-connectivity and the minimum degree of a graph $G$, respectively. For integers $s \ge 0$ and $t \ge 0$, a graph $G$ is $(s,t)$-supereulerian if for any disjoint edge sets $X, Y \subseteq E(G)$ with $|X|\le s$ and $|Y|\le t$, $G$ has a spanning closed trail that contains $X$ and avoids $Y$. This dissertation is devoted to providing some results on $(s,t)$-supereulerian graphs and …

Cycle Decomposition For Integral Current Homology, Kristin Julia Duling

Graduate Theses, Dissertations, and Problem Reports

A standard graph theoretical result states that every element of the cycle space of a graph has a cycle decomposition. Georgakopoulos expands this result to a primitive decomposition and minimal representation of each element in a modified 1-dimensional singular homology. We modify the m-dimensional integral current homology in order to ensure a primitive decomposition for each element.

Polychromatic Colorings Of Certain Subgraphs Of Complete Graphs And Maximum Densities Of Substructures Of A Hypercube, Ryan Tyler Hansen

Graduate Theses, Dissertations, and Problem Reports

If G is a graph and H is a set of subgraphs of G, an edge-coloring of G is H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, poly_HG, is the largest number of colors in an H-polychromatic coloring. We determine poly_HG exactly when G is a complete graph on n vertices, q a fixed nonnegative integer, and H is the family of one of: all matchings spanning n-q vertices, all 2-regular graphs spanning at least n-q vertices, or all cycles of length precisely n-q. …

New Techniques In Celestial Mechanics, Ali Abdulrasool Abdulhussein

Graduate Theses, Dissertations, and Problem Reports

It is shown that for the classical system of the N body problem ( Newtonian Motion), if the motion of the N particles starts from a planar initial motion at t=t_{0}, then the motion of the N particles continues to be planar for every t\in[t_{0},t_{1}], assuming that no collisions occur between the N particles. Same argument is shown about the linear motion, namely, for the classical system of the N body problem, if the motion of the N particles starts from a linear initial motion at t=t_{0}, then the motion of the N particles continues to be linear for every …

On The Classification Of Generalized Pseudo-Orthogonal Lie Groups Via Curvature, Cohomology, And Algebraic Structure, Adam C. Fletcher

Graduate Theses, Dissertations, and Problem Reports

The study of Lie groups has yielded a rich catalogue of mathematical spaces that, in some sense, provide a theoretical and computational framework for describing the “world in which we live.” In particular, these topological groups that represent the rigid motions of a space, the behavior of subatomic particles, and the shape of the expanding universe consist of specialized matrices. In what follows, we define a new collection of matrices with a very specific transposition relation and attempt to classify this Lie group algebraically, geometrically, and topologically. We consider fields, $\Bbb{F},$ of characteristic zero and define the group of pseudo-orthogonal …

On The Lagrangian Description And Uniqueness For The One-Dimensional Pressureless Euler System, Mark David Suder

Graduate Theses, Dissertations, and Problem Reports

In this work we show that the one-dimensional pressureless Euler system admits a Lagragian characterization under fairly general initial conditions, extending recent results by Hynd [7]. Moreover, we show that if the initial velocity is right-continuous and bounded, then we have uniqueness of this Lagrangian solution (called Sticky Particles Flow, or SPF solution), which coincides with the Scalar Conservation Laws (or SCL) solution. An important tool we employed in order to prove existence is a result by Gangbo et al. [5], which establishes a canonical (i.e. the flow is given by the optimal maps pushing the Lebesgue measure restricted to …

Heterogeneous Generalizations Of Vertex Coloring, Lucian Ciletti Mazza

Graduate Theses, Dissertations, and Problem Reports

This dissertation proves a collection of results in some heterogeneous generalizations of vertex coloring, i.e. generalizations in which the relationship between the colors of two adjacent vertices may differ throughout the graph. For the most part, the results are from group coloring, group list coloring, and DP coloring. The main results are as follows: a group list coloring analogue of Brooks' Theorem for multigraphs, a result linking group structure (rather than only group size) with group coloring, some results involving coloring edge-disjoint unions, and an examination of the relationship between the group and DP coloring numbers of a multigraph and …

Analyzing Applied Calculus Student Understanding Of Definite Integrals In Real-Life Applications, Cody Hood

Graduate Theses, Dissertations, and Problem Reports

An individual’s knowledge of definite integrals can range from rote memorization to a strong foundational connection harkening back to its Riemann sum limit definition. In my research, I conducted seven task-based face-to-face interviews with Applied Calculus students. Through the use of real-life examples and guided reinvention, I analyzed ways in which these students, who all initially demonstrated rote memorization, could exhibit a Riemann sum based level of comprehension. This research was conducted in the confines of a student population with definite integral experience, but no formal instruction on limits. My results show that the lack of computational emphasis in class …

Algebraic, Analytic, And Combinatorial Properties Of Power Product Expansions In Two Independent Variables., Mohamed Ammar Elewoday

Graduate Theses, Dissertations, and Problem Reports

Let $F(x,y)=I+\hspace{-.3cm}\sum\limits_{\substack{p=1\\m+n=p}}^{\infty}\hspace{-.3cm}A_{m,n}x^my^n$ be a formal power series, where the coefficients $A_{m,n}$ are either all matrices or all scalars. We expand $F(x,y)$ into the formal products $\prod\limits_{\substack{p=1\\m+n=p}}^{\infty}\hspace{-.3cm}(I+G_{m,n}x^m y^n)$, $\prod\limits_{\substack{p=1\\m+n=p}}^{\infty}\hspace{-.3cm}(I-H_{m,n}x^m y^n)^{-1}$, namely the \textit{ power product expansion in two independent variables} and \textit{inverse power product expansion in two independent variables} respectively. By developing new machinery involving the majorizing infinite product, we provide estimates on the domain of absolute convergence of the infinite product via the Taylor series coefficients of $F(x,y)$. This machinery introduces a myriad of "mixed expansions", uncovers various algebraic connections between the $(A_{m,n})$ and the $(G_{m,n})$, and uncovers various algebraic …

Remediation Of Spatial Skills In First Semester Calculus Using Haptics-Based Applications, Kristen L. Murphy

Graduate Theses, Dissertations, and Problem Reports

Spatial reasoning is required for many topics in undergraduate mathematics as well as other STEM fields. In calculus specifically, there are few interventions that assess students’ spatial skills and provide remediation. Haptic technology is a novel approach to spatial skills remediation due to its unfamiliarity to students and its flexibility with regard to models. The purpose of this dissertation is two-fold: first, to determine what level of spatial abilities students possess upon entering an undergraduate calculus course, and second, to determine whether haptic feedback will enhance an intervention for improving spatial skills.

The dissertation research was a mixed-methods study using …

Cycle Double Covers And Integer Flows, Zhang Zhang

Graduate Theses, Dissertations, and Problem Reports

My research focuses on two famous problems in graph theory, namely the cycle double cover conjecture and the integer flows conjectures. This kind of problem is undoubtedly one of the major catalysts in the tremendous development of graph theory. It was observed by Tutte that the Four color problem can be formulated in terms of integer flows, as well as cycle covers. Since then, the topics of integer flows and cycle covers have always been in the main line of graph theory research. This dissertation provides several partial results on these two classes of problems.

Weighted Modulo Orientations Of Graphs, Jianbing Liu

Graduate Theses, Dissertations, and Problem Reports

This dissertation focuses on the subject of nowhere-zero flow problems on graphs. Tutte's 5-Flow Conjecture (1954) states that every bridgeless graph admits a nowhere-zero 5-flow, and Tutte's 3-Flow Conjecture (1972) states that every 4-edge-connected graph admits a nowhere-zero 3-flow. Extending Tutte's flows conjectures, Jaeger's Circular Flow Conjecture (1981) says every 4k-edge-connected graph admits a modulo (2k+1)-orientation, that is, an orientation such that the indegree is congruent to outdegree modulo (2k+1) at every vertex. Note that the k=1 case of Circular Flow Conjecture coincides with the 3-Flow Conjecture, and the case of k=2 implies the 5-Flow Conjecture. This work is devoted …

Circuits And Cycles In Graphs And Matroids, Yang Wu

Graduate Theses, Dissertations, and Problem Reports

This dissertation mainly focuses on characterizing cycles and circuits in graphs, line graphs and matroids. We obtain the following advances.

1. Results in graphs and line graphs. For a connected graph G not isomorphic to a path, a cycle or a K1,3, let pc(G) denote the smallest integer n such that the nth iterated line graph Ln(G) is panconnected. A path P is a divalent path of G if the internal vertices of P are of degree 2 in G. If every edge of P is a cut edge of G, then P is a bridge divalent path of G; …

Theory And Techniques Of Convergence Of Topological Transformation Group Actions, Murtadha Jaber Sarray

Graduate Theses, Dissertations, and Problem Reports

n this dissertation, we present new set functions called strongly limit and strongly prolongation limit sets. We show the new sets, especially strongly prolongation limit sets, characterize proper action under an arbitrary setting. That is, we characterize proper action for wider class of proper ܩ-spaces. Also, we show the new version of the sets could be derived from strongly exceptional sets which have been used as a good technique for the characterization of a proper maps. Moreover, we review properties of well-known limit sets and prolongations and properties for the new version of limit sets under an arbitrary setting on …

Global Existence And Asymptotic Behaviors For Some Nonlinear Partial Differential Equations., Ismahan Dhaw Binshati

Graduate Theses, Dissertations, and Problem Reports

We study global existence and asymptotic behavior of the solutions for two-fluid compressible isentropic Euler-Maxwell equations by the Fourier transform and energy method. We discuss the case when the pressure for two fluids is not identical and we also add the friction between two fluids. In addition, we discuss the rates of decay of $L^{p}-L^{q}$ norms for a linear system. Moreover, we use the result for $L^{p}-L^{q}$ estimates to prove the decay rates for the nonlinear systems. In addition, we prove existence of heteroclinic orbits for the nonlinear Vlasov and the one-dimensional Vlasov-Poisson systems. In the nonlinear Vlasov case with …

Infinite Matroids And Transfinite Sequences, Martin Andrew Storm

Graduate Theses, Dissertations, and Problem Reports

A matroid is a pair M = (E,I) where E is a set and I is a set of subsets of E that are called independent, echoing the notion of linear independence. One of the leading open problems in infinite matroid theory is the Matroid Intersection Conjecture by Nash-Williams which is a generalization of Hall’s Theorem. In [31] Jerzy Wojciechowski introduced µ- admissibility for pairs of matroids on the same ground set and showed that it is a necessary condition for the existence of a matching. A pair of matroids (M,W) with common ground set E is µ-admissible if a …

Edge Colorings Of Graphs On Surfaces And Star Edge Colorings Of Sparse Graphs, Katherine M. Horacek

Graduate Theses, Dissertations, and Problem Reports

In my dissertation, I present results on two types of edge coloring problems for graphs.

For each surface Σ, we define ∆(Σ) = max{∆(G)| G is a class two graph with maximum degree ∆(G) that can be embedded in Σ}. Hence Vizing’s Planar Graph Conjecture can be restated as ∆(Σ) = 5 if Σ is a sphere. For a surface Σ with characteristic χ(Σ) ≤ 0, it is known ∆(Σ) ≥ H(χ(Σ))−1, where H(χ(Σ)) is the Heawood number of the surface, and if the Euler char- acteristic χ(Σ) ∈ {−7, −6, . . . , −1, 0}, ∆(Σ) is already …

Analyzing Mathematicians' Concept Images Of Differentials, Timothy Shawn Mccarty

Graduate Theses, Dissertations, and Problem Reports

The differential is a symbol that is common in first- and second-year calculus. It is perhaps expected that a common mathematical symbol would be interpreted universally. However, recent literature that addresses student interpretations of differentials, usually in the context of definite integration, suggests that this is not the case, and that many interpretations are possible. Reviews of textbooks showed that there was not a lot of discussion about differentials, and what interpretations there were depended upon the context in which the differentials were presented. This dissertation explores some of these issues. Since students may not have the experience necessary to …

Stationary Automata, Anaam Mutar Bidhan

Graduate Theses, Dissertations, and Problem Reports

In this dissertation, we investigate new automata, we call it stationary automata or ST-automata. This concept is based on the definition of TF-automaton by Wojciechowski [Woj2]. What is new in our approach is that we incorporate stationary subsets of limit ordinals of uncountable cofinality. The first objective of the thesis is to motivate the new construction of automata. This concept of ST-automata allows us to make a connection with infinite graph theory. Aharoni, Nash-Williams, and Shelah [AhNaSh] formulated a condition that is necessary and sufficient for a bipartite graph to have a matching. For a bipartite …