Other Mathematics Commons^™

Bridging Biological Systems With Social Behavior, Conservation, Decision Making, And Well-Being Through Hybrid Mathematical Modeling, Maggie Renee Sullens

Faculty Publications and Other Works -- Mathematics

This dissertation defense presentation highlights the power of hybrid mathematical modeling and addresses crucial issues such as:

1️. The Impact of Industry Collapse on Community Mental Health: A Complex Contagion ODE Model.

2️. Budget Allocation and Illegal Fishing: A Game Theoretic Model.

3️. Reactive Scope Model with an Energy Budget and Multiple Mediators: An ODE Model

The overarching theme of Hybrid Mathematical Modeling beautifully captures the essence of this work, demonstrating its potential to unravel ecological issues while addressing the intricate interactions between humans and the environment.

The G_2-Hitchin Component Of Triangle Groups: Dimension And Integer Points, Hannah E. Downs

Doctoral Dissertations

The image of $\PSL(2,\reals)$ under the irreducible representation into $\PSL(7,\reals)$ is contained in the split real form $G_{2}^{4,3}$ of the exceptional Lie group $G_{2}$. This irreducible representation therefore gives a representation $\rho$ of a hyperbolic triangle group $\Gamma(p,q,r)$ into $G_{2}^{4,3}$, and the \textit{Hitchin component} of the representation variety $\Hom(\Gamma(p,q,r),G_{2}^{4,3})$ is the component of $\Hom(\Gamma(p,q,r),G_{2}^{4,3})$ containing $\rho$.

This thesis is in two parts: (i) we give a simple, elementary proof of a formula for the dimension of this Hitchin component, this formula having been obtained earlier in [Alessandrini et al.], \citep{Alessandrini2023}, as part of a wider investigation using Higgs bundle techniques, …

Adaptive And Topological Deep Learning With Applications To Neuroscience, Edward Mitchell

Doctoral Dissertations

Deep Learning and neuroscience have developed a two way relationship with each informing the other. Neural networks, the main tools at the heart of Deep Learning, were originally inspired by connectivity in the brain and have now proven to be critical to state-of-the-art computational neuroscience methods. This dissertation explores this relationship, first, by developing an adaptive sampling method for a neural network-based partial different equation solver and then by developing a topological deep learning framework for neural spike decoding. We demonstrate that our adaptive scheme is convergent and more accurate than DGM -- as long as the residual mirrors the …

Preconditioned Nesterov’S Accelerated Gradient Descent Method And Its Applications To Nonlinear Pde, Jea Hyun Park

Doctoral Dissertations

We develop a theoretical foundation for the application of Nesterov’s accelerated gradient descent method (AGD) to the approximation of solutions of a wide class of partial differential equations (PDEs). This is achieved by proving the existence of an invariant set and exponential convergence rates when its preconditioned version (PAGD) is applied to minimize locally Lipschitz smooth, strongly convex objective functionals. We introduce a second-order ordinary differential equation (ODE) with a preconditioner built-in and show that PAGD is an explicit time-discretization of this ODE, which requires a natural time step restriction for energy stability. At the continuous time level, we show …

The Calculus War: The Ultimate Clash Of Genius, Walker Briles Bussey-Spencer

Chancellor’s Honors Program Projects

No abstract provided.

Dependence Structures In Lévy-Type Markov Processes, Eddie Brendan Tu

Doctoral Dissertations

In this dissertation, we examine the positive and negative dependence of infinitely divisible distributions and Lévy-type Markov processes. Examples of infinitely divisible distributions include Poissonian distributions like compound Poisson and α-stable distributions. Examples of Lévy-type Markov processes include Lévy processes and Feller processes, which include a class of jump-diffusions, certain stochastic differential equations with Lévy noise, and subordinated Markov processes. Other examples of Lévy-type Markov processes are time-inhomogeneous Feller evolution systems (FES), which include additive processes. We will provide a tour of various forms of positive dependence, which include association, positive supermodular association (PSA), positive supermodular dependence (PSD), and positive …

The Loewner Equation And Weierstrass' Function, Gavin Ainsley Glenn

Chancellor’s Honors Program Projects

No abstract provided.

Mathematical Approaches To Sustainability Assessment And Protocol Development For The Bioenergy Sustainability Target Assessment Resource (Bio-Star), Nathan Louis Pollesch

Doctoral Dissertations

Bioenergy is renewable energy made of materials derived from biological, non-fossil sources. In addition to the benefits of utilizing an energy source that is renewable, bioenergy is being researched for its potential positive impact on climate change mitigation, job creation, and regional energy security. It has also been studied to investigate possible challenges related to indirect and direct land-use change and food security. Bioenergy sustainability assessment provides a method to identify, quantify, and interpret indicators, or metrics, of bioenergy sustainability in order to study trade-offs between environmental, social, and economic aspects of bioenergy production and use. Assessment is crucial to …

Extension Theorems On Matrix Weighted Sobolev Spaces, Christopher Ryan Loga

Doctoral Dissertations

Let D a subset of Rⁿ [R n] be a domain with Lipschitz boundary and 1 ≤ p < ∞ [1 less than or equal to p less than infinity]. Suppose for each x in Rⁿ that W(x) is an m x m [m by m] positive definite matrix which satisfies the matrix A_p [A p] condition. For k = 0, 1, 2, 3;... define the matrix weighted, vector valued, Sobolev space [L p k of D,W] with

[the weighted L p k norm of vector valued f over D to the p power equals the sum over all alpha with order less than k of the integral over D of the the pth power …

A Survey On Hadamard Matrices, Adam J. Laclair

Chancellor’s Honors Program Projects

No abstract provided.

Loewner Space-Filling Curves, Hannah Marie Clark

Chancellor’s Honors Program Projects

No abstract provided.

Examining The Process Of Identification In The Mathematics Classroom And The Role Of Students’ Academic Communities, Richard J. Robinson

Doctoral Dissertations

The primary purpose of this research was to provide insight into the identities students develop as they interact in a high school mathematics classroom. A normative divide developed which eventually split the classroom into two distinct academic factions: those who resisted the emerging local definition of what it meant to do mathematics and those who did not resist (i.e. complied or identified). A secondary purpose of this research was to understand the role of students’ academic communities in mathematics identity development. Student narratives helped uncover mathematical spaces outside the classroom that each developed their own unique definition of what it …

Indefinite Knapsack Separable Quadratic Programming: Methods And Applications, Jaehwan Jeong

Doctoral Dissertations

Quadratic programming (QP) has received significant consideration due to an extensive list of applications. Although polynomial time algorithms for the convex case have been developed, the solution of large scale QPs is challenging due to the computer memory and speed limitations. Moreover, if the QP is nonconvex or includes integer variables, the problem is NP-hard. Therefore, no known algorithm can solve such QPs efficiently. Alternatively, row-aggregation and diagonalization techniques have been developed to solve QP by a sub-problem, knapsack separable QP (KSQP), which has a separable objective function and is constrained by a single knapsack linear constraint and box constraints. …

An Analysis Of The Patterns Of Crime And Socioeconomic Status Visualized Through Self-Organized Maps, Jason Carlin Kaufman

Masters Theses

This work is research to explore the association of spatial patterns between crime and socioeconomic status (SES) through the use of self-organized maps (SOM). It had been found that the spatial patterns of crime could be associated with those of socioeconomic, and this work sought to further these analyses in order to better understand how crime patterns and SES were related. To explore this association, patterns of crime and SES were examined in three cities: Nashville, TN; Portland, OR; and Tucson, AZ. Three SOMs were used in each city: one to analyze the patterns of crime, a second to analyze …

A Study Of Poisson And Related Processes With Applications, Phillip Mingola

Chancellor’s Honors Program Projects

No abstract provided.

Validation Of Weak Form Thermal Analysis Algorithms Supporting Thermal Signature Generation, Elton Lewis Freeman

Masters Theses

Extremization of a weak form for the continuum energy conservation principle differential equation naturally implements fluid convection and radiation as flux Robin boundary conditions associated with unsteady heat transfer. Combining a spatial semi-discretization via finite element trial space basis functions with time-accurate integration generates a totally node-based algebraic statement for computing. Closure for gray body radiation is a newly derived node-based radiosity formulation generating piecewise discontinuous solutions, while that for natural-forced-mixed convection heat transfer is extracted from the literature. Algorithm performance, mathematically predicted by asymptotic convergence theory, is subsequently validated with data obtained in 24 hour diurnal field experiments for …

On Decision Making: Bayesian And Stochastic Optimization Approaches, Yang Shen

Masters Theses

Decision analysis provides a framework for searching an optimal solution under uncertainties and potential risks. This thesis focuses on two problems arising in transportation engineering and computer sciences, respectively.

First, it is considered a centralized controller which imposes actions on a number of interacting subsystems. Employing an appropriate Markov Decision Process framework, we establish that the Pareto optimal solution of each subsystem will be optimal for the entire system. Synthetic data have been taken into account for verifying this claim.

Next, we focus on a supercomputing problem utilizing a hierarchical Bayesian model. We estimate an optimal solution in order to …

On A Quantum Form Of The Binomial Coefficient, Eric J. Jacob

Masters Theses

A unique form of the quantum binomial coefficient (n choose k) for k = 2 and 3 is presented in this thesis. An interesting double summation formula with floor function bounds is used for k = 3. The equations both show the discrete nature of the quantum form as the binomial coefficient is partitioned into specific quantum integers. The proof of these equations has been shown as well. The equations show that a general form of the quantum binomial coefficient with k summations appears to be feasible. This will be investigated in future work.

Spatiotemporal Dynamics In A Lower Montane Tropical Rainforest, Robert Michael Lawton

Doctoral Dissertations

Disturbance in a forest’s canopy, whether caused by treefall, limbfall, landslide, or fire determines not only the distribution of well-lit patches at any given time, but also the ways in which the forest changes over time. In this dissertation, I use a 25 year record of treefall gap formation find a novel and highly patterned process of forest disturbance and regeneration, providing a local mechanism by examining the factors that influence the likelihood of treefall. I then develop a stochastic cellular automaton for disturbance and regeneration based on the analysis of this long term data set and illustrate the potential …

The Origins Of Mathematical Societies And Journals, Eric S. Savage

Masters Theses

We investigate the origins of mathematical societies and journals. We argue that the origins of today’s professional societies and journals have their roots in the informal gatherings of mathematicians in 17th century Italy, France, and England. The small gatherings in these nations began as academies and after gaining government recognition and support, they became the ancestors of the professional societies that exist today. We provide a brief background on the influences of the Renaissance and Reformation before discussing the formation of mathematical academies in each country.

A Comparison Of The Deck Group And The Fundamental Group On Uniform Spaces Obtained By Gluing, Raymond David Phillippi

Doctoral Dissertations

We de…ne a uniformity on a glued space under uniformly continuous attachment maps. If the component spaces are uniform coverable then the resulting glued space is uniform coverable. We consider examples including the glued uniformity on a …nite dimensional CW complex which is shown to be uniformly coverable. For one dimensional CWcomplexes, the resulting deck group is equivalent to the fundamental group. Other properties of the deck group are explored.