Digital Commons Network^™

Topological Data Analysis Using The Mapper Algorithm, Jessica Girard

Electronic Theses and Dissertations, 2020-

Topological data analysis is an expanding field that attempts to obtain qualitative information from a data set using topological ideas. There are two common methods of topological data analysis: persistent homology and the Mapper algorithm; the focus of this thesis is on the latter. In this thesis, we will be discussing the key ideas behind the Mapper algorithm, following the flow from Morse Theory to Reeb graphs to the topological version of the algorithm and finally to the statistical version. Lastly, we will present an application of Mapper to the USAIR97 data set using the RTDAmapper package.

Clustering Of Diverse Multiplex Networks, Yaxuan Wang

Electronic Theses and Dissertations, 2020-

This dissertation introduces the DIverse MultiPLEx Generalized Dot Product Graph (DIMPLE-GDPG) network model where all layers of the network have the same collection of nodes and follow the Generalized Dot Product Graph (GDPG) model. In addition, all layers can be partitioned into groups such that the layers in the same group are embedded in the same ambient subspace but otherwise all matrices of connection probabilities can be different. In common particular cases, where layers of the network follow the Stochastic Block Model (SBM) and Degree Corrected Block Model (DCBM), this setting implies that the groups of layers have common community …

Predation And Harvesting In Spatial Population Models, Connor R. Shrader

Honors Undergraduate Theses

Predation and harvesting play critical roles in maintaining biodiversity in ecological communities. Too much harvesting may drive a species to extinction, while too little harvesting may allow a population to drive out competing species. The spatial features of a habitat can also significantly affect population dynamics within these communities. Here, we formulate and analyze three ordinary differential equation models for the population density of a single species. Each model differs in its assumptions about how the species is harvested. We then extend each of these models to analogous partial differential equation models that more explicitly describe the spatial habitat and …

Weierstrass Vertices On Finite Graphs, Abrianna L. Gill

Honors Undergraduate Theses

The intent of this thesis is to explore whether any patterns emerge among families or through graph operations regarding the appearance of Weierstrass vertices on graphs. Currently, patterns have been identified and proven on cycles, complete graphs, complete bipartite graphs, and the house and house-x graphs. A Python program developed as part of this thesis to perform the algorithms used in this analysis confirms these findings. This program also revealed a pattern: if v is a Weierstrass vertex, then the vertex v* added to the graph as a pendant vertex to v is also a Weierstrass vertex. The converse is …

Asymptotic Regularity Estimates For Diffusion Processes, David Hernandez

Honors Undergraduate Theses

A fundamental result in the theory of elliptic PDEs shows that the hessian of solutions of uniformly elliptic PDEs belong to the Sobolev space ��^2,ε. New results show that for the right choice of c, the optimal hessain integrability exponent ε* is given by

ε* = ������ ����(1−������) / ����(1−��), �� ∈ (0,1)

Through the techniques of asymptotic analysis, the behavior and properties of this function are better understood to establish improved quantitative estimates for the optimal integrability exponent in the ��^2,ε-regularity theory.

Various Dynamical Regimes, And Transitions From Homogeneous To Inhomogeneous Steady States In Nonlinear Systems With Delays And Diverse Couplings, Ryan Roopnarain

Electronic Theses and Dissertations, 2020-

This dissertation focuses on the effects of distributed delays modeled by 'weak generic kernels' on the collective behavior of coupled nonlinear systems. These distributed delays are introduced into several well-known periodic oscillators such as coupled Landau-Stuart and Van der Pol systems, as well as coupled chaotic Van der Pol-Rayleigh and Sprott systems, for a variety of couplings including diffusive, cyclic, or dynamic ones. The resulting system is then closed via the 'linear chain trick' and the linear stability analysis of the system and conditions for Hopf bifurcations that initiate oscillations are investigated. A variety of dynamical regimes and transitions between …

The Effects Of Viscous Damping On Rogue Wave Formation And Permanent Downshift In The Nonlinear Schrödinger Equation, Evelyn Smith

Honors Undergraduate Theses

This thesis investigates the effect of viscous damping on rogue wave formation and permanent downshift using the higher-order nonlinear Schrödinger equation (HONLS). The strength of viscous damping is varied and compared to experiments with only linear damped HONLS.

Stability analysis of the linear damped HONLS equation shows that instability stabilizes over time. This analysis also provides an instability criterion in the case of HONLS with viscous damping.

Numerical experiments are conducted in the two unstable mode regime using perturbations of the Stokes wave as initial data. With only linear damping permanent downshift is not observed and rogue wave formation is …

Extensions Of The General Solution To The Inverse Problem Of The Calculus Of Variations, And Variational, Perturbative And Reversible Systems Approaches To Regular And Embedded Solitary Waves, Ranses Alfonso Rodriguez

Electronic Theses and Dissertations, 2020-

In the first part of this Dissertation, hierarchies of Lagrangians of degree two, three or four, each only partly determined by the choice of leading terms and with some coefficients remaining free, are derived. These have significantly greater freedom than the most general differential geometric criterion currently known for the existence of a Lagrangian and variational formulation since our existence conditions are for individual coefficients in the Lagrangian. For different choices of leading coefficients, the resulting variational equations could also represent traveling waves of various nonlinear evolution equations. Families of regular and embedded solitary waves are derived for some of …

Function Approximation Guarantees For A Shallow Neural Network Trained By Gradient Flow, Russell Gentile

Electronic Theses and Dissertations, 2020-

This work features an original result linking approximation and optimization theory for deep learning. Several examples from recent literature show that, given the same number of learnable parameters, deep neural networks can approximate richer classes of functions, with better accuracy than classical methods. The bulk of approximation theory results though, are only concerned with the infimum error for all possible parameterizations of a given network size. Their proofs often rely on hand-crafted networks, where the weights and biases are carefully selected. Optimization theory indicates that such models would be difficult or impossible to realize with standard gradient-based training methods. The …

Population Persistence And Disease Invasion In Heterogenous Networks, Poroshat Yazdanbakhshghahyazi

Electronic Theses and Dissertations, 2020-

The problem of understanding how biological species and infectious diseases can persist and spread in heterogeneous networks has brought a wide attention, recently highlighted due to the COVID-19 pandemic. This dissertation investigates the connection between the structures of heterogeneous networks and population persistence/disease invasion. To do so, we propose a new index for network heterogeneity by employing the Laplacian matrix of population dispersal and its corresponding group inverse. The network growth rate and reproduction number can be evaluated using the network average and the network heterogeneity index as the first and second order approximation, respectively. We also illustrate the impact …

Optimal Impulse Controls With Changing Running Cost And Applications In Mortgage Refinance, Yuchen Cao

Electronic Theses and Dissertations, 2020-

Almost all home buyers have mortgages and it is quite common to have mortgage refinanced. There are two main reasons that make people decide to refinance the mortgage: (i) need some cash for urgent purposes, and (ii) lower the monthly payment. In this dissertation, we are not going to discuss (i), and we are investigating problems related to (ii). To begin with, let us intuitively make the following observations: If the interest rate remains the same as the current mortgage interest rate, then the monthly payment will automatically lower if you start a new mortgage with the same term, say, …

Mathematical Model For Giraffe Population Dynamics, Huntir Bass

Electronic Theses and Dissertations, 2020-

Since the 1980s the overall giraffe population has dropped at least 40% causing some researchers to label this rapid decline as the "Silent Extinction." Due to this plummet, understanding the behaviors of the giraffe population is absolutely necessary before they are on the brink of extinction. Through the usage of mathematical modeling methodologies, a general model is created to illustrate the relationship between juvenile and adult female giraffes through numerous interaction parameters. Variations on specific variables generate different simulations, which allows more biological accuracy. With each variation having an established coexistence equilibrium between the juvenile and adult female populations, the …

The Parker Problem In Hall Magnetohydrodynamics Analytical And Numerical Solutions, Chad Malott

Electronic Theses and Dissertations, 2020-

In this thesis we follow on the mathematical aspects of the previous work of Shivamoggi (2009) on the Parker problem in Hall magnetohydrodynamics (MHD). We will present an analysis involving detailed analytical and numerical solutions to the Parker problem in Hall MHD. We give an analytical formulation for the Parker problem in Hall MHD, involving an initial value problem (IVP) associated with a first order Riccati equation (RE). We present Mathematica software exact solutions directly with special functions and more straightforward solutions that use the change of variables and power series methods without special functions. We give an asymptotic formulation …

Cholera Transmission Dynamic Model With Environmental Impacts Of Plankton Reservoirs, Sweety Sarker

Electronic Theses and Dissertations, 2020-

Cholera is an acute disease that is a global threat to the world and can kill people within a few hours if left untreated. In the last 200 years, seven pandemics occurred, and, in some countries, it remains endemic. The World Health Organization (WHO) declared a global initiative to prevent cholera by 2030. Cholera dynamics are contributed by several environmental factors such as salinity level of water, water temperature, presence of plankton especially zooplankton such as cladocerans, rotifers, copepods, etc. Vibrio cholerae (V. cholerae) bacterium is the main reason behind the cholera disease and the growth of V. cholerae depends …

Multicolor Ramsey And List Ramsey Numbers For Double Stars, Jake Ruotolo

Honors Undergraduate Theses

The core idea of Ramsey theory is that complete disorder is impossible. Given a large structure, no matter how complex it is, we can always find a smaller substructure that has some sort of order. For a graph H, the k-color Ramsey number r(H; k) of H is the smallest integer n such that every k-edge-coloring of K_n contains a monochromatic copy of H. Despite active research for decades, very little is known about Ramsey numbers of graphs. This is especially true for r(H; k) when k is at least 3, also known as the multicolor Ramsey number of …

Asymptotic Properties Of The Potentials For Greedy Energy Sequences On The Unit Circle, Ryan Edward Mccleary

Honors Undergraduate Theses

In this work, we analyze the asymptotic behavior of the minimum values of Riesz s-potentials generated by greedy s-energy sequences on the unit circle. The analysis is broken into the cases 0 < s < 1, s = 1, and s > 1, since the behavior of the minimum values of the Riesz s-potential undergoes a sharp transition at s = 1. For 0 < s < 1, the first-order behavior is already known. We obtain first-order asymptotic results for 0 < s < 1. We also prove first-order and second-order asymptotic formulas for s = 1 and investigate the first-order behavior for s > 1.

Regularized Estimations In Some Statistical Problems, Feng Yu

Electronic Theses and Dissertations, 2020-

In this dissertation, we consider estimations with regularization in three statistical problems relating to linear time system, stochastic block model and matrix variate regression. In the first part of the dissertation, we specifically investigate Mixture Multilayer Stochastic Block Model, where layers can be partitioned into groups of similar networks, and networks in each group are equipped with a distinct Stochastic Block Model. The goal is to partition the multilayer network into clusters of similar layers, and to identify communities in those layers. The Mixture Multilayer Stochastic Block Model was introduced by Bing-yi Jing [2020] and a clustering methodology, TWIST, is …

Computation Of Effective Properties Of Smart Composite Materials With Generalized Periodicity Using The Two-Scales Asymptotic Homogenization Method, David Guinovart Sanjuan

Electronic Theses and Dissertations, 2020-

In this work, a general mathematical models for flexoelectric heterogeneous equilibrium boundary value problems are considered. A methodology to find the local problems and the effective properties of flexoelectric composites with generalized periodicity is presented, using a two-scales asymptotic homogenization method. The model of the homogenized boundary values problem is presented. A procedure to solve the local problems of stratified multilayered composites with wavy geometry with perfect contact at the interface is proposed. Further, a study of a multilayered piezoelectric composite with imperfect contact at the interface and the influence of the flexoelectric constituents in the behavior of heterogeneous structures …

A Mathematical Model For Predicting Animal Population Persistence On Fragmented Landscapes, Allyson Jones

Electronic Theses and Dissertations, 2020-

The effects of roads, buildings, and cities on animal populations are widespread and, often times, disastrous. These structures fragment animals' homes, inhibiting their ability to obtain essential resources and to reproduce. The question arises then: Under what circumstances can an animal population persist in a fragmented landscape? To attempt to answer this question, we present a spatially explicit reaction-diffusion model with varying growth and diffusion rates that incorporates animal behavior at points where habitats are fragmented for four different habitats. The outcome of extinction or persistence of the animal population is determined by examining the effects of changing parameters on …

Algebraic And Combinatorial Approaches For Counting Cycles Arising In Population Biology, Brian Chau

Honors Undergraduate Theses

Within population biology, models are often analyzed for the net reproduction number or other generalized target reproduction numbers, which describe the growth or decline of the population based on specific mechanisms. This is useful in determining the strength and efficiency of control measures for inhibiting or enhancing population growth. The literature contains many algebraic and combinatorial approaches for deriving the net reproduction number and generalized target reproduction numbers from digraphs and associated matrices. Finding, categorizing, and counting the permutations of disjoint cycles, or cycles unions is a requirement of the Cycle Union approach by Lewis et al. (2019). These cycles …

Representations Of Cuntz Algebras Associated To Random Walks, Nicholas Christoffersen

Honors Undergraduate Theses

In the present thesis, we investigate representations of Cuntz algebras coming from dilations of row co-isometries. First, we give some general results about such representations. Next, we show that by labeling a random walk, a row co-isometry appears naturally. We give an explicit form for representations that come from such random walks. Then, we give some conditions relating to the reducibility of these representations, exploring how properties of a random walk relate to the Cuntz algebra representation that comes from it

Estimation And Clustering In Block Models, Majid Noroozi

Electronic Theses and Dissertations, 2020-

Networks with community structure arise in many fields such as social science, biological science, and computer science. Stochastic block models are popular tools to describe such networks. For this reason, in this dissertation which is composed of two parts we explore some stochastic block models and the relationship between them. In the first part of the dissertation, we study the Popularity Adjusted Block Model (PABM) and introduce its sparse case, the Sparse Popularity Adjusted Block Model (SPABM). The SPABM is the only existing block model which allows to set some probabilities of connections to zero. For both the PABM and …

Estimation And Clustering In Network And Indirect Data, Ramchandra Rimal

Electronic Theses and Dissertations, 2020-

The first part of the dissertation studies a density deconvolution problem with small Berkson errors. In this setting, the data is not available directly but rather in the form of convolution and one needs to estimate the convolution of the unknown density with Berkson errors. While it is known that the Berkson errors improve the precision of the reconstruction, it does not necessarily happen when Berkson errors are small. Furthermore, the choice of bandwidth in density estimation has been an open problem so far. In this dissertation, we provide an in-depth study of the choice of the bandwidth which leads …

Mean Field Optimal Control And Related Problems, Wei Yan

Electronic Theses and Dissertations, 2020-

It has been decades since the first paper that mean field problems were studied. More and more problems are considered or solved as new methods and new concepts have been developed. In this dissertation, we will present a series of results on (recursive) mean field stochastic optimal control problems. Comparing our results with those in the classical stochastic optimal control theory, there are following significant differences. First, the value function of a mean field optimal control problem is not Markovian any more, even when coefficient functions in the problem are deterministic. Second, the cost functional we considered is induced by …

Distributed Algorithms And Inverse Graph Filtering, Nazar Emirov

Electronic Theses and Dissertations, 2020-

Graph signal processing provides an innovative framework to handle data residing on distributed networks, smart grids, neural networks, social networks and many other irregular domains. By leveraging applied harmonic analysis and graph spectral theory, graph signal processing has been extensively exploited, and many important concepts in classical signal processing have been extended to the graph setting such as graph Fourier transform, graph wavelets and graph filter banks. Similarly, many optimization problems in machine learning, sensor networks, power systems, control theory and signal processing can be modeled using underlying network structure. In modern applications, the size of a network is large, …

Transfunctions And Other Topics In Measure Theory, Jason Bentley

Electronic Theses and Dissertations, 2020-

Measures are versatile objects which can represent how populations or supplies are distributed within a given space by assigning sizes to subregions (or subsets) of that space. To model how populations or supplies are shifted from one configuration to another, it is natural to use functions between measures, called transfunctions. Any measurable function can be identified with its push-forward transfunction. Other transfunctions exist such as convolution operators. In this manner, transfunctions are treated as generalized functions. This dissertation serves to build the theory of transfunctions and their connections to other mathematical fields. Transfunctions that identify with continuous or measurable push-forward …

Lattice-Valued T-Filters And Induced Structures, Frederick Reid

Electronic Theses and Dissertations

A complete lattice is called a frame provided meets distribute over arbitrary joins. The implication operation in this context plays a central role. Intuitively, it measures the degree to which one element is less than or equal to another. In this setting, a category is defined by equipping each set with a T-convergence structure which is defined in terms of T-filters. This category is shown to be topological, strongly Cartesian closed, and extensional. It is well known that the category of topological spaces and continuous maps is neither Cartesian closed nor extensional. Subcategories of compact and of complete spaces are …

Spectral Properties Of The Finite Hilbert Transform On Two Adjacent Intervals Via The Method Of Riemann-Hilbert Problem, Elliot Blackstone

Electronic Theses and Dissertations

In this dissertation, we study a self-adjoint integral operator $\hat{K}$ which is defined in terms of finite Hilbert transforms on two adjacent intervals. These types of transforms arise when one studies the interior problem of tomography. The operator $\hat{K}$ possesses a so-called "integrable kernel'' and it is known that the spectral properties of $\hat{K}$ are intimately related to a $2\times2$ matrix function $\Gamma(z;\lambda)$ which is the solution to a particular Riemann-Hilbert problem (in the $z$ plane). We express $\Gamma(z;\lambda)$ explicitly in terms of hypergeometric functions and find the small $\lambda$ asymptotics of $\Gamma(z;\lambda)$. This asymptotic analysis is necessary for the …

Semi-Analytical Solutions Of Non-Linear Differential Equations Arising In Science And Engineering, Mangalagama Dewasurendra

Electronic Theses and Dissertations

Systems of coupled non-linear differential equations arise in science and engineering are inherently nonlinear and difficult to find exact solutions. However, in the late nineties, Liao introduced Optimal Homotopy Analysis Method (OHAM), and it allows us to construct accurate approximations to the systems of coupled nonlinear differential equations. The drawback of OHAM is, we must first choose the proper auxiliary linear operator and then solve the linear higher-order deformation equation by spending lots of CPU time. However, in the latest innovation of Liao's "Method of Directly Defining inverse Mapping (MDDiM)" which he introduced to solve a single nonlinear ordinary differential …

Variational Inclusions With General Over-Relaxed Proximal Point And Variational-Like Inequalities With Densely Pseudomonotonicity, George Nguyen

Electronic Theses and Dissertations

This dissertation focuses on the existence and uniqueness of the solutions of variational inclusion and variational inequality problems and then attempts to develop efficient algorithms to estimate numerical solutions for the problems. The dissertation consists a total of five chapters. Chapter 1 is an introduction to variational inequality problems, variational inclusion problems, monotone operators, and some basic definitions and preliminaries from convex analysis. Chapter 2 is a study of a general class of nonlinear implicit inclusion problems. The objective of this study is to explore how to omit the Lipschitz continuity condition by using an alternating approach to the proximal …