Investing in the stock market is much like a gambling game based on calculated risks. The Kelly Criterion method helps investors and gamblers calculate what percentage of their money they should allocate to each investment or bet. We analyze the volatility of the stock market from 1962-1991 and 1992-2021 to find the optimal amount to invest to maximize gains and the chaotic ruin point beyond which investors will inevitably lose everything.

This talk will serve as a summation of my studies in the Directed Reading Program I participated in at The University of Texas at Austin this summer. In this program, I studied a theory formulated by Wesley Pegden and a team of mathematicians who set out to model a method of detecting partisan bias in redistrictings. Here, we discuss the valuable research of these mathematicians and dissect their methods. Namely, their use of Markov Chains to model the redistricting process. Through their process, the districting in question serves as the initial step in the chain, which allows a computer program to make small random changes to the districts, and hence, creates more steps in the chain. Once the chain has been established to a particular length, statistical testing (Square Root of Epsilon Test) involving compactness measures, bias measures, and comparison tests ensues. It is through these results that we can determine if our data is statistically significant, and therefore, if the districting is significantly biased.

Pandemic vs. Epidemic : A Statistical Correlation Study Between COVID-19 and HIV In the last three years, significant public health efforts have gone towards fighting the COVID-19 pandemic in the world. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection is currently the third leading cause of death in the world today. In 2021 COVID-19 was the leading cause of death in the United States. Individuals with previous underlying health conditions are at a higher risk of experiencing severe, and even fatal, symptoms upon infection. Simultaneously, in many metropolitan areas, the 40-year-long HIV epidemic remains extremely prevalent. Thus, as different regions of the US took different approaches to combating the COVID-19 pandemic, our focus was on assessing state-level COVID-19 rates and socioeconomic factors as predictors of HIV rates using statistical models. Washington D.C. has one of the highest rates of HIV infections and has also struggled with high COVID-19 infection rates throughout the pandemic. This study aims to determine if there are any statistically significant correlations between COVID-19 infection, HIV transmission rates, and other social-economic factors such as marriage rates, race index, college degree rates in the Nation.

The classical Dedekind sum is well-studied both inside and outside of number theory due to its connections with the Dedekind eta function and its applications in topology and combinatorial geometry. There exists an algorithm to compute classical Dedekind sums in polynomial time, which has a close relation to the Euclidean algorithm. In 2020, Stucker, Vennos, and Young developed a generalized Dedekind sum defined on the congruence subgroup \(\Gamma_0(N)\), which requires exponential time to compute from the definition. We construct an algorithm that reduces the complexity for computing generalized Dedekind sums from exponential to polynomial time. We do so by using an efficient word rewriting process in group theory.

Koch curves were typical subjects of fractals. Instead of studying the dimensions, this work considers areas of the regions enclosed by Koch curves and their generalizations. The area of the region enclosed by the classical Koch curve is well known. We used five different generalization approaches to obtain five classes of curves and found a universal formula for areas of the regions from each class. Our generalized methods extend those in literatures and our formula for the areas provide interesting context for infinite series' that occur throughout calculus courses.

The notion of "factorization" extends far beyond the familiar realm of integers and polynomials. While efforts in Number Theory have shown that prime factorization is guaranteed in not only the domain of integers, but in lots of other structures, the computation of such factorization is highly nontrivial. This presentation opens with a result from Kulosman et al, on the irreducibility of binomials over zero-characteristic polynomial rings, and will cover the consequences of allowing finite fields and restrictions to exponents. The results from such work include a classification of primes satisfying the factorization criteria, as well as a computer-assisted approach on generating such primes, and investigating their theoretical density under Artin's primitive root conjecture.

Choanoflagellates, eukaryotes that are important predators on bacteria in aquatic ecosystems, are used as a model system to study the evolution of animals from protozoan ancestors. The choanoflagellate, Salpingoeca rosetta, has a complex life cycle that includes unicellular and multicellular stages, provides a model system to study the consequences of different cell morphologies, being free-swimming vs. sessile, or being a single cell vs. a multicellular colony. A unicellular S. rosetta has an ovoid cell body and a single flagellum surrounded by a collar of microvilli. The cell swims by waving its flagellum, which also creates a water current that brings bacteria to the collar of prey-capturing microvilli. One measure of the performance of a suspension-feeding organism is the volume of fluid that it can move into its collar during a beat cycle. The inward flux of fluid acts as a proxy for the rate of bacterial capture. While this is a good measure of uptake of dissolved nutrients, it is only an approximate measure of prey capture. Here we use a regularized Stokeslet framework to model the hydrodynamics of a unicellular choanoflagellate, the captured bacterial prey of non-zero volume, and their effect on swimming performance and clearance rate. We compare the model predictions with high-speed microvideography. Moreover, we will discuss current model assumptions, and future model improvements that, together with coordinated lab experiments, will help us probe this intriguing biophysical system.

In this talk, we lay the groundwork for a formal justification of the plane wave assumptions underlying Arthur Compton's scattering calculation. Using a recent relativistic formulation of Bohmian mechanics, we start by examining the behavior of a single free photon and a single free electron. We offer significant evidence that the electron wave function becomes asymptotically plane wave-like, meaning the particle's momentum and energy approach fixed values as assumed in Compton's calculations. We examine the effects the wave function's initial parameters have on the behavior of particle trajectories, and we explore different ways to visualize trajectories and this asymptotic behavior prior to interaction.

The Schroedinger equation of quantum mechanics is a linear partial differential equation that describes the spatiotemporal evolution of a quantum state. Solving it efficiently and accurately is essential in order to make predictions for physical phenomena and apply quantum mechanics in technology. In this project, I present a Java program that solves the energy eigenvalue problem numerically for a given potential. It uses the fourth order Runge-Kutta algorithm, instead of the commonly used Numerov algorithm. It also uses a root search algorithm called the secant method, in order to ensure differentiability of the wavefunction. The benchmark test performed on the code showed that the wavefunctions produced by the code matched the theoretically predicted wavefunctions. The Runge-Kutta method was found to be an efficient and viable algorithm to study the Schroedinger equation.

Fractional Differential Equations (FDEs) have demonstrated great adequacy in modeling biological and physical processes such as the transmission of infectious diseases, design of heat flux meters, and rheological properties of rocks, to name a few. The main objective of this research project is to establish highly accurate predictive models of processes modeled by FDEs and DEs arising in biological and physical sciences. In this research project, we also construct analytical and numerical solutions for FDEs through Wright-type integral transformations of their analogous integer-order differential equations. We establish numerical simulations using Monte Carlo integration and Physics-Informed Neural Networks implemented on MATLAB and TensorFlow environments.

To control outbreaks, epidemiologists rely heavily on accurate and detailed mathematical models predicting how the disease will spread. Classical models of disease spread involve systems of ordinary differential equations (ODEs) that assume homogeneous mixing and rely on delicate parameters that are difficult to estimate. In contrast, agent-based models allow individuals to have different characteristics and experiences, however, they can be prohibitively computationally expensive. This project uses stochastic processes on social networks to strike an optimal balance of individualization and computational efficiency. We present new mathematical models and Python-based simulations to extend methodology from a 2019 article1 that incorporates relationships (edges) between individuals (nodes) to create a network graph on which a disease outbreak is modeled as a stochastic process. This provides a more realistic and detailed model than classical ODE systems which can incorporate the same biological mechanisms while adding novel social and medical influences. Our results show that individual factors and model choices which ODEs models cannot capture play a significant role in controlling the transmission of diseases

The Voting Rights Act of 1965 required the creation of “minority opportunity” voting districts to provide a significant racial minority group the chance to elect a candidate of preference to represent their interests in state and federal government. Many legal cases (such as Thornburg v Gingles) have struggled to determine precise criteria to determine how many such districts are needed and whether a proposed district will truly function in practice as a minority opportunity district. Most recently, this issue played a critical role in the Fall 2021 redistricting cycle and connects to our group’s overarching goal to improve mathematical redistricting tools. Prior efforts to create quantitative criteria for minority opportunity districts relied either on legal tests that are difficult to apply or data sources that are inconsistent and underrepresent rural areas. To solve this problem, we created a detailed and robust dataset based on the 2020 Decennial Census, 10 years of the American Community Survey, and 10 years of precinct-level Texas election outcomes. We then built hierarchical multivariate regression models that predict voting participation rates and candidate preferences for different racial groups using high-quality publicly available data. These models have proven highly accurate, achieving vote predictions with 2% of the actual election results in U.S. presidential and senatorial elections 2016-2020."

The Poincare Ball is the model of three dimensional hyperbolic space that holds the sphere at infinity. Using a discrete approach, we find the minimal splitting surface of the regular hyperbolic octahedron inscribed in the ball. In particular, we find the area-minimizing hypersurface that divides the positive and negative vertices of our octahedron. This problem is equivalent to finding a minimal surface of genus-2 with curvature \(k\leq 1\) in a hyperbolic manifold due to the fact that the regular ideal octahedron tessellates the space for a group action. In this talk, we discuss our path to a solution: finding the discrete mean curvature flow (negative gradient flow for area) and applying this to a triangulation to continually minimize the area of our discrete hypersurface. We also discuss the computational challenges to this approach and implications to other discrete minimal area problems.

There is ever-growing research indicating that high spatial ability correlates with student and professional success in science, technology, engineering, and mathematics (STEM) courses and career fields. A few valid and reliable testing instruments have been developed to measure specific constructs of spatial thinking in sighted populations. However, due to a lack of accessibility, most of these testing instruments are unable to be utilized by blind or low-vision (BLV) populations. The project 3 team of the Research Experience for Undergraduates (REU) hosted by Utah State University's Engineering Education Department analyzed the Tactile Metal Cutting Test (TMCT)'s reliability in measuring spatial constructs of rotation, cutting plane, and proportion in BLV populations. Data was collected from BLV individuals attending National Federation of the Blind (NFB) conventions, learning centers for the blind, and STEM-oriented NFB summer camps for high school students. The analysis results indicate that the split form of the TMCT instrument is showing reliability in both subtests with the BLV population. This project reports on a National Science Foundation (NSF) funded effort to establish the instrument’s validity and reliability.

The Safety Net Problem, coined by myself and the members of my research team, asks the following question: If given an arbitrarily weighted grid graph, G, and a set of vertices, R, that we will further refer to as “required” vertices, what is the most efficient way to draw a low-cost path that connects every vertex but, if a random edge is deleted, would ensure that any required vertex would still have a path to get to every other required vertex? In our approach to answering this question, my team and I concluded that minimum spanning trees would be integral to creating an algorithm that could solve this question for any randomly generated grid graph. We then found two approaches to begin working off of: the Minimum Spanning Tree (MST) First approach and the Path-First approach. In the MST-First approach, the algorithm begins by finding the MST of the grid graph and then, using A* (a variation of Dijkstra's Algorithm), finding paths between all required vertices. We find a path from one vertex to the next until we end up back at the original vertex--much like a Hamiltonian path. Then, we would add these paths to the MST and then eliminate any unnecessary edges--edges which are connected to vertices that have a degree of 3 or more and are also not part of the path which connected the required vertices. This final MST with the addition of the paths which connect the required vertices, which we refer to as the Safety Net, is what we wish our algorithm to produce. In the Path-First approach, we create an algorithm very similar to the MST-First approach. The difference is that we find the paths that connect the required vertices before we find the MST. In this approach, we would still have a goal of the algorithm producing an MST combined with the Safety Net. The conclusion my research team and I came to, as our research was completed in a summer program and is still ongoing, is that there are flaws to both approaches, though the MST-First approach seems more promising as it may be more applicable to various grid graphs and to an increased amount of required vertices (such as if the grid graph was comprised of only required vertices). Moving forward, we hope to find solutions to any holes in the MST-First algorithm and, eventually, expand our problem to more than just grid graphs.

We explore a probabilistic approach to circuit theory. Given a resistor network and the properties of a resistor, we define the admittance of a state as the probability of an electron being in that state. We then explore the next possible states an electron can travel to by using a probabilistic matrix, but there is a crucial restriction that must be imposed for the resistor network to have valid resistor values.

Classical models of disease outbreaks rely on systems of nonlinear ordinary differential equations. Though ODE models have been widely successful for common diseases, they rely on assumptions that are often invalid or unverifiable for rare, novel, or localized diseases with limited reliable data. Traditional point-estimates of critical model parameters are highly susceptible to bias from data quality problems. To combat this problem, this project replaces point-estimates with Bayesian inference methods to create more robust distributional estimates for parameters. Bayesian methods provide two advantages: they naturally reflect the quality of available data via model certainty and they create an ensemble of forecasts for the future course of the outbreak (traditional point-estimates generate only a single forecast). This forecast ensemble gives the full range of courses the outbreak may take, including the most likely and best/worse-case scenarios.

Lotería is a popular Mexican-Latinx household game that is played similarly to Bingo. However, instead of announcing letter-number pairs, small-numbered cards with culturally relevant images are called, and players mark their Lotería cards, called tablas, if that image is present. We determine the cumulative probability distribution of achieving a Lotería for a single tabla, looking at only vertical, horizontal and diagonal Loterías. This probability distribution is determined by the number of squares covered by a combination of Loterías. We also determine the corresponding probability density function and calculate the expected value of the number of calls needed to obtain the first Lotería.

This summer research and teaching experience investigated the proportional reasoning of middle school students. Six preservice teachers and two graduate students developed and implemented curriculum for a one-week summer math camp emphasizing formative assessment. At the end of July, twenty-five middle school students participated in the one-week summer math camp facilitated by the preservice teachers and graduate students. This presentation will highlight the observations of the middle school students’ proportional reasoning.

Mapping tori are a topological construction that have a rich geometry. We define a class of mapping tori that we call linearly-mismatched. We prove that linearly mismatched mapping tori are asynchronously automatic. Asynchronously automatic groups are finitely generated groups whose elements can be characterized by finite state automata, and satisfy the asynchronous k-fellow traveler property. This tells us that these groups have nice algorithmic properties, including a solvable word problem.

A field is called \(O^*\) if each partial order can be extended to a total order. \(O^*\)-fields must be algebraic over the field of rational numbers. An \(O^*\)-field which is a number field cannot contain any imaginary embeddings. Examples of \(O^*\)-fields are presented.

In this talk, we examine the history of Pell's equations as well as their solutions, both from a comparative point of view. The "ancient" development in India up to the 12th century and the "modern" investigation from Fermat to Lagrange will be our twin foci; some additional remarks may be included.

We examine three data sets from experiments on heat transfer in an apparatus including a heating element wrapped around a cylinder containing steel shot spheres. We show the construction and purpose of the apparatus, as well as analysis of the data to determine temperature as a function of time. In this preliminary report, we focus on curve fitting to model the phases for each data set when the system is heating up and when it is cooling down.

This study investigates a special class of mathematical equations, known as partial differential equations (PDEs), used to model multivariable systems and real-world phenomena. Numerical approaches, including the employment of MATLAB’s pdepe function and the central-difference method, are utilized to compare results with known analytical formulations. Specifically, this research focuses on three types of PDEs. The first is the equation for one-dimensional heat conduction, which is fundamental to study related to heat transfer in engineering, while the second is the equation for a wave on a string, an elementary concept for understanding wave propagation in physics. Lastly, it is shown that the renowned Schrödinger Equation takes the form of a PDE and can be solved using numerical methods; this is a well-established law in the field of quantum mechanics, hence, its significance is the motivation for this research. Solutions to example PDEs exhibited in this study are depicted as graphs and surfaces in MATLAB, which can then be used for conceptual analysis.

DiseaseNet: a unified approach to disease detection Many diseases are correlated to detectable methylation data biomarkers. Methylation signatures of noninfectious diseases could reveal disease presence before symptoms occur, allowing for earlier disease surveillance and intervention. DiseaseNet seeks to provide a unified approach to disease detection using methylation by training a deep learning model on cancer, schizophrenia, asthma, arthritis, and normal samples. Using transfer learning, we trained an additional VAE classifier onto the CancerNet model to classify the original 33 cancer types, 3 new disease types, and one normal. Training this model resulted in an organized latent space and an average F-measure of approximately 89.2%. This reveals not only that general disease signatures exist in methylation data, but also that these diseases, cancer, and normal samples are distinguishable from one another. This model serves as a proof of concept that many human diseases may be encodable into the same model.

Supraventricular Tachycardia (SVT) is when the heart’s upper chambers beat either too quickly or out of rhythm with the heart’s lower chambers. This out-of-step heart beating is a leading cause of strokes, heart attacks, and heart failure. The most successful treatment for SVT is catheter ablation, a process where an electrophysiologist (EP) maps the heart to find areas with abnormal electrical activity. The EP then runs a catheter into the heart to burn the abnormal area, blocking the electrical signals. Much is not known about what triggers SVT and where to place scar tissue for optimal patient outcomes. We have produced a dynamic model of the right atrium accelerated on NVIDIA GPUs. An interface allows researchers to insert ectopic signals into the simulated atria and ablate sections of the atria allowing them to rapidly gain insight into what causes SVTs and how to terminate them.

Current mathematical models for the neural activity in the hippocampus do not accurately represent observations in neuroscience. Based on observations from rats with tetrodes, we have developed simulations that reproduce spikes in activity. Place cells contribute to the phenomenon of reverse replay which occurs within the first few seconds of a rat stopping along a track that it ran across. Our model seeks to replicate this by combining Tsodyks's differential equation model and implementing spike time dependent plasticity (STDP). The model displays activity bumps at the place field centers and can potentially display procession, which indicates that there may be reverse replay present. We explain some of the code in Matlab.

Onchocerciasis is a parasitic disease endemic in Sub-Saharan Africa and South America that spreads from black flies to humans. The disease causes skin nodules, itching, and in severe cases, permanent blindness; Contributing to its nickname, River Blindness. The World Health Organization’s current approach to Onchocerciasis is mass drug administration of Ivermectin. This treatment plan has two glaring issues. The first issue concerns the results of our previous work, prolonged use of Ivermectin may cause drug resistance which we’ve shown is likely present within the population at present. The second issue is that even without resistance eradication is still not possible and the only method of eliminating the parasite is in a joint treatment of Ivermectin and Doxycycline. It also should be said that this method isn’t perfect either since resistance is even more likely with the antibiotic Doxycycline. The goal of our project is to model the spread of Onchocerciasis with resistance, analyze the impact of possible Ivermectin resistance and figure out a treatment plan with doxycycline that can eliminate the disease without causing widespread resistance. After obtaining this goal we hope to expand the model to include Loiasis, another eye worm disease that may cause death when taking ivermectin.

Hamiltonian systems are special among dynamical systems, possessing many important and theoretically interesting properties. For this reason, it is advantageous to express physical theories in Hamiltonian form. However, this is not always possible for dissipative theories such as Landau collisions. One proposed extension to Hamiltonian dynamics is metriplectic dynamics, which adds another bracket to the existing Poisson structure on phase space. In many respects, metriplectic dynamics is a successful and powerful theory, preserving the essence of Hamiltonian dynamics while being applicable to a more general class of theories. However, despite its success, metriplectic dynamics is still problematic in several regards. In this talk, we review Hamiltonian dynamics in the geometrical context of Poisson manifolds. We then introduce metriplectic dynamics as an extension of the Lie algebra structure on observables. Publication permitting, we will then talk about a new generalization of metriplectic dynamics illuminating the elusive geometrical and algebraic structure of many dissipative theories while simultaneously resolving the problems that arise in metriplectic dynamics. "

My topic was modeling hanging chains using numerical methods. Typically, this problem is solved using mathematical minimization through the calculus of variations. This approach works well for graduate level students, but lacks the ability to easily simulate the movement and forced on the chain over time. The original approach we used was to attempt to predict the motion of each segment of the chain using finite difference method. We found this method to be inefficient and ill-suited. We then built our own approach, using object-oriented programming and basic equations of motion. We simulated the motion of a chain with two fixed endpoints over time.

Hispanic Americans are among the most underrepresented groups in STEM fields. Invisible barriers and lack of resources in Hispanic communities contribute to this underrepresentation. In this talk, we will explore what it means to be a mathematics student at a border college with small, geographically isolated campuses located hundreds of miles away from the main university campus.

Introductory STEM courses not only underrepresent female and racial minority students in their demographic environment, which promotes self-efficacy and performance gaps and strengthens negative stereotypes, but also commonly act as dropout points for these students from STEM majors. Yet, studies show that the percentage of females in undergraduate STEM courses predicts gender participation and performance gaps, as female-identifying students show better results in courses where they have a greater perception of belonging. We investigate how student identity (e.g. racial/ethnic identity, gender, first-generation status, deaf and hard-of-hearing support) and classroom demographic representation (e.g. female representation in gateway STEM courses) affect the short- and long-term success of undergraduate STEM majors. From longitudinal student data, we extract more than 200 Mathematics and Physics majors who took introductory Mathematics, Physics, and Computer Science courses at a medium-sized private university. We utilize Markov models to understand the short-term responses and logistic regression and odds ratio contribution to interpret the long-term effects. These dichotomous responses include yearly retention rates, 6-year graduation, and retention as math and physics majors. We find that female and non-male identifying Math and Physics majors show increased odds of graduating within 6 years, which is the average graduation time at the university, in addition to persisting with Math and Physics degrees. We also discover many positive correlations between greater female representation in gateway STEM courses and downstream responses. We find higher 6-year graduation rates for both men and women that take gateway Math and Physics courses with a higher percentage of female students, specifically over the median threshold of twenty-percent female composition found among the three STEM disciplines. Finally, we discover that increased female representation in introductory Math and Computer Science courses serves as a positive predictor of persistence of female and non-male identifying Math and Physics majors in STEM. Our results shed light on how these features play a significant role in advancing the success of underrepresented students to provide motivation for creating representative, diverse STEM classrooms. Future results point to expanding our analyses to include a greater sample size, including all STEM majors at the university. In addition to analyzing the impact of female representation in the three gateway disciplines of STEM, we plan to study the effect of racial and ethnic minority representation.

We use an upper bound on piercing numbers for axis-parallel boxes with the \((3,2)\)-property to improve the lower bound on minimum agreement proportion for \((2,3)\)-agreeable box societies. Additionally, we improve the lower bound on boxicity for such societies and show that it is sharp when the associated piercing number is less than six. We also present novel computational methods for determining the boxicity of intersection graphs and use these methods to explore whether our improved lower bound on minimum agreement proportion is sharp for \((2,3)\)-agreeable box societies with boxicity two.

Recently, Schneider and Schneider defined a new class of partitions called sequentially congruent partitions, in which each part is congruent to the next part modulo its index, and they proved two partition bijections involving these partitions. We introduce a new partition notation specific to sequentially congruent partitions which allows us to more easily study these bijections and their compositions, and we reinterpret them in terms of Young diagram transformations. We also define a generalization of sequentially congruent partitions, and we provide several new partition bijections for these generalized sequentially congruent partitions. Finally, we investigate a question of Schneider– Schneider regarding how sequentially congruent partitions fit into Andrews’ theory of partition ideals. We prove that the maximal partition ideal of sequentially congruent partitions has infinite order and is therefore not linked, and we identify its order 1 subideals.

In this talk, we introduce a brief history of the factorial, in particular a well-known asymptotic formula called Stirilng's formula, and its relationship to other factorial-like objects such as the q-factorial. Furthermore, we investigate a special case of a class of factorials pioneered by M. Bhargava in the early 2000s and prove these class of factorials admit a Stirling's-like formula. Lastly, we explore possible ways to generalize our methods to a broader class of factorials researched by M. Bhargava.