Mathematics Commons^™

Integer Partitions Under Certain Finiteness Conditions, Tim Wagner

Dissertations, Master's Theses and Master's Reports

This dissertation focuses on problems related to integer partitions under various finiteness restrictions. Much of our work involves the collection of partitions fitting inside a fixed partition $\lambda$, and the associated generating function $G_{\lambda}$.

In Chapter 2, we discuss the flawlessness of such generating functions, as proved by Pouzet using the Multicolor Theorem. We give novel applications of the Multicolor Theorem to re-prove flawlessness of pure $O$-sequences, and show original flawlessness results for other combinatorial sequences. We also present a linear-algebraic generalization of the Multicolor Theorem that may have far-reaching applications.

In Chapter 3, we extend a technique due to …

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

Graduate Theses, Dissertations, and Problem Reports (ETD)

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 …

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

Graduate Theses, Dissertations, and Problem Reports (ETD)

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 …

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

Graduate Theses, Dissertations, and Problem Reports (ETD)

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 (ETD)

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 …

Heterogeneous Generalizations Of Vertex Coloring, Lucia Ciletti Mazza

Graduate Theses, Dissertations, and Problem Reports (ETD)

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 …

Calabi-Yau Products: Graded Quivers For General Toric Calabi-Yaus, Sebastián Franco, Azeem Hasan

Publications and Research

The open string sector of the topological B-model on CY (m + 2)-folds is described by m-graded quivers with superpotentials. This correspondence generalizes the connection between CY (m + 2)-folds and gauge theories on the worldvolume of D(5 − 2m)-branes for m = 0, . . . , 3 to arbitrary m. In this paper we introduce the Calabi-Yau product, a new algorithm that starting from the known quiver theories for a pair of toric CY_m+2 and CY_n+2 produces the quiver theory for a related CY_{m+n+3 …}

Minimizing Reaction Systems, Matthew R. Thomas

UNF Graduate Theses and Dissertations

The theoretical model for reaction systems is a relatively new framework originally proposed as a mathematical model for biochemical processes which take place in living cells. Growing interest in this research area has lead to the abstraction of the model for non-biological purpose as well. Reaction systems, with a well understood behavior, have become important for studying transition systems. As with any mathematical model, we want to simplify a given implementation of the model as much as possible while maintaining functional equivalence. This paper discusses the formal model for reaction systems, how we can simplify them with minimization techniques, some …

Analysis Of A Time-Delayed Hiv/Aids Epidemic Model With Education Campaigns, Sedar Ngoma, Dawit Denu, Rachidi B. Salako

Mathematics

We consider a time-delayed HIV/AIDS epidemic model with education dissemination and study the asymptotic dynamics of solutions as well as the asymptotic behavior of the endemic equilibrium with respect to the amount of information disseminated about the disease. Under appropriate assumptions on the infection rates, we show that if the basic reproduction number is less than or equal to one, then the disease will be eradicated in the long run and any solution to the Cauchy problem converges to the unique disease-free equilibrium of the model. On the other hand, when the basic reproduction number is greater than one, we …

A Consistent Estimator Of Nontrivial Stationary Solutions Of Dynamic Neural Fields, Eddy Kwessi

Mathematics Faculty Research

Dynamics of neural fields are tools used in neurosciences to understand the activities generated by large ensembles of neurons. They are also used in networks analysis and neuroinformatics in particular to model a continuum of neural networks. They are mathematical models that describe the average behavior of these congregations of neurons, which are often in large numbers, even in small cortexes of the brain. Therefore, change of average activity (potential, connectivity, firing rate, etc.) are described using systems of partial different equations. In their continuous or discrete forms, these systems have a rich array of properties, among which is the …

A Continuous-Time Mathematical Model And Discrete Approximations For The Aggregation Of Β-Amyloid, A. S. Ackleh, Saber Elaydi, G. Livadiotis, A. Veprauskas

Mathematics Faculty Research

Alzheimer's disease is a degenerative disorder characterized by the loss of synapses and neurons from the brain, as well as the accumulation of amyloid-based neuritic plaques. While it remains a matter of contention whether β-amyloid causes the neurodegeneration, β-amyloid aggregation is associated with the disease progression. Therefore, gaining a clearer understanding of this aggregation may help to better understand the disease. We develop a continuous-time model for β-amyloid aggregation using concepts from chemical kinetics and population dynamics. We show the model conserves mass and establish conditions for the existence and stability of equilibria. We also develop two …

Pancyclicity And Vertex Pancyclicity For Some Products Of Graphs, Artchariya Muaengwaeng

Chulalongkorn University Theses and Dissertations (Chula ETD)

No abstract provided.

Global Analysis Of A Discrete Sirs Epidemic Model With Nonlinear Incidence Rate, Butsayapat Chaihao

Chulalongkorn University Theses and Dissertations (Chula ETD)

In this thesis, we prove some behaviors of a discrete SIRS epidemic model with a nonlinear incidence rate and a distributed time-delay. This model is constructed from the discretization of the corresponding continuous model by using a nonstandard finite difference method. The basic properties including the positivity and the boundedness of the solutions are established. We derive the existence of the disease-free equilibrium and the endemic equilibrium of the model. In addition, by applying Lyapunov function techniques, we prove that the disease-free equilibrium is globally attractive. Moreover, we give a sufficient condition for the permanence of the model. In order …

Hamiltonian Decompositions Of Hypergraphs, Chutima Saengchampa

Chulalongkorn University Theses and Dissertations (Chula ETD)

In this dissertation, we first discuss four versions of Hamiltonicity in hypergraphs. We mainly study the existence problem of Hamiltonian decompositions of uniform hypergraphs based on two versions of Hamiltonian cycles, so called ``KK-definition" and ``WJ-definition". For KK-definition, we create a recursive construction of KK-Hamiltonian decomposition of complete 3-uniform hypergraphs. Our construction method uses a KK-Hamiltonian decomposition of the complete 3-uniform hypergraph, Kt(3), and some well-known graph decompositions to obtain a KK-Hamiltonian decomposition of the complete t-partite 3-uniform hypergraph, Kt(n)(3), when t=4,8 (mod12), n>=2, as well as a KK-Hamiltonian decomposition of K2t(3). Therefore, together with the current results in …

Algebraic Degree Of Spectra Of Cayley Hypergraphs, Naparat Sripaisan

Chulalongkorn University Theses and Dissertations (Chula ETD)

No abstract provided.

Local Limit Theorems For Sums Of Independent Integer-Valued Lattice Random Variables, Punyapat Kammoo

Chulalongkorn University Theses and Dissertations (Chula ETD)

No abstract provided.

Hamiltonian Cycles Of Equational Graphs Over Finite Fields, Wachirawit Chaifongsri

Chulalongkorn University Theses and Dissertations (Chula ETD)

No abstract provided.

Closed-Form Formulas For Conditional Moments Of Generalized Cox-Ingersoll-Ross Processes, Phiraphat Sutthimat

Chulalongkorn University Theses and Dissertations (Chula ETD)

Cox-Ingersoll-Ross (CIR) process, introduced in 1985, is a one factor model used to describe the evolution of interest rate and pricing the financial derivatives. It was later extended to have time-dependent parameters called the extended CIR (ECIR) process, which is more widely studied and used in a variety of applications. The generalized versions of CIR process are also studied and investigated for more applications in finance. However, most of these applications rely on the knowledge and properties of conditional expectations and moments, which most of them are not yet fully developed into closed form. In this work, we propose closed-form …

Enhanced Heterogeneous Network Model With Ensemble Similarities For Identifying Protein Targets Of Drugs, Piyanut Tangmanussukum

Chulalongkorn University Theses and Dissertations (Chula ETD)

Currently, computational identification of drug target proteins is widely used to help saving cost and time for drug discovery and development. One of the most efficient approaches is a prediction of drug-target interactions based on similarity scores between drugs and target proteins. Despite various data about drugs and targets extensively available, only chemical structures and protein sequences are mostly used to compute drug-drug and target-target similarity scores, respectively. In this thesis, the Forward similarity integration (FSI) Framework is proposed for systematically integrating multiple similarity measures to construct a heterogeneous network propagation model with a suitable similarity integration. Seven drug-similarity measures, …

Newton Iterative Algorithm For Polynomial Modular Inversion Modulo Xn±1 For Some Patterns Of N, Samakorn Sripatthanakul

Chulalongkorn University Theses and Dissertations (Chula ETD)

This thesis presents an algorithm for computing the modular inverse of a polynomial in a ring of polynomials over a finite field $\mathbb{F}_q$ with a characteristic $p$. Given a polynomial $f$ and a natural number $r$, by applying the idea of the Newton iteration algorithm, the fast division algorithm used to find the inverse of $f$ under modulo $x^{p^r}-1$, $x^{p^r}+1$, $x^{2p^r}-1$ and $x^n-1$ where $n=2^r d$ for some $r,d\in\mathbb{N}$, is established. The cost analysis for these cases show that the algorithm has the computational complexity of $\mathcal{O}(n \log n)$ which is more efficient than the Half-GCD algorithm in terms of …