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.