The Eastern Equine Encephalitis Virus (EEEV) is an erratic and deadly neurological disease that spans across the northeastern coast of the United States. To determine the rate at which the virus is spread between the Black-Tailed Mosquito (Culiseta melanura) and select avian species we began by analyzing the migration patterns of both the mosquito and the avian species. It was found that certain species of avians shared similar, or even identical, flight patterns with the Black-Tailed Mosquito. Through this research, we develop and analyze a system of Ordinary Differential Equations (ODEs) to gain insight into how and when transmission of the virus to avians is at its highest. We incorporate a host stage-structured model where the avian host group is split into two categories, adults and young-of-the-year birds (YOY). Using this we explored the extent to which fluctuations occurred in transmission rates according to host/vector abundances, mosquito biting rate, and type of host. We evaluate the hypothesis that YOY avians are more readily exposed to the mosquito vector as they lack a defense mechanism, unlike their adult counterpart using the compartmental model.

The process of successful skin healing from a wound involves different combinations of interactions. Moreover, by clearly understanding this process, we can provide and determine the appropriate amount of medicine to give to patients with varying types of wounds. Thus, this can improve the healing process of patients. In this research, we use the ADI method to solve a partial differential equation that models the wound healing process. Moreover, we try to explore the relationship between parameters in the model for different patients. Wound images are used as our dataset experiment. To segment the image's wound, we implement U-Net, a deep learning-based model, as our model for this segmentation problem. We believe the combination of ADI and Deep Learning helps us understand the process of wound healing.

Neglected tropical diseases are diseases that affect mainly developing countries and in turn are poorly funded and under researched. Onchocerciasis is a neglected tropical disease that exists primarily in Sub-Saharan Africa and South America. This is because the black flies that spread this disease are common to these regions and since these flies bite humans they can infect humans at a noticeable rate. When the black fly bites you it can pass a parasite called filarial nematode, or Onchocerca volvulus, which is transmitted solely by black flies during blood feeding. Some people do not experience symptoms while infected with Onchocerciasis,however as the larvae can migrate through the human body without provoking a response from the immune system. But many people do have symptoms, which include itchy skin rashes, nodules under the skin, and vision changes Onchocerciasis is currently being treated with ivermectin through mass drug administration. The goal of our research project is to find out the optimal way to distribute the treatment to eradicate the disease and to find if the current approach will eventually lead to drug resistance. Our focus was to model this situation and data fit our model to the country of Cameroon. Our research showed that the current treatment plan of Onchocerciasis may never completely eradicate the disease from the population due to resistance. We also found evidence that there may already be resistance in the population.

This project aims to generate a network graph representing the interactions among fungal species based on three levels of burn severity. Fungal data was gathered by St. Edward's University students and faculty after the Bastrop County Complex fire in 2011. 498 distinct OTUs (operational taxonomic units) were found at 30 sample sites across Bastrop County. To determine the level of species' interactions with each other, we use Pearson's correlation coefficient and attempt to find a sufficient cutoff to determine which values are significant and which can be disregarded. We slowly increase the initial cutoff according to Random Matrix Theory, which states that once the eigenvalues of the corresponding adjacency matrix follow a Poisson distribution, we have reached the appropriate cutoff. We then use this cutoff to generate a network graph that visualizes the species' interactions according to three distinct levels of burn severity. We also utilize principal component analysis (PCA) to visualize the similarity of fungal species composition among each of the sampling sites. We found that a cutoff between \(0.50\) and \(0.70\) is sufficient, and due to the large number of distinct OTUs, we are still working on generating the final network graph. We also found that sampling sites with no-burn had very similar fungal species composition, while the low-burn and high-burn sampling sites had much more variation.

Social determinants of health (SDoH) are environmental and social factors influencing health outcomes that are generally divided into five domains: economic stability; educational access and quality; health care access and quality; neighborhood and built environment; and social and community context. This presentation explores associations between SDoH and health care outcomes for emergency department and inpatient visits with the aid of machine learning algorithms. As one of the leading causes of adult disability and death in the U.S., complications arising from ischemic stroke are of particular interest among these outcomes. The analysis begins by applying dimensionality reduction techniques to the SDoH, including principal components analysis (PCA), nonlinear PCA with optimal scaling, and hierarchical clustering. We then present a predictive model of health care outcomes based on the transformed social determinants. Random forests are especially useful for this objective, as they provide variable importance plots aiding in model interpretation.

Supraventricular Tachycardia (SVT) is when the heart’s upper chambers, the atria, beat either too quickly, or out of rhythm with the heart’s lower chambers, the ventricles. This out-of-step beating of the heart is not in itself deadly but is indirectly the leading cause of strokes, heart attacks, and heart failure. The most successful treatment for most forms of SVT is catheter ablation, a process where an electrophysiologist (EP) makes a 3-dimensional mapping of the heart to find areas with abnormal electrical activity. The EP then runs a catheter up the femoral vein of the groin up to the heart and uses RF energy to burn the abnormal area of the heart to produce scar tissue that blocks the abnormal electrical signals. Though ablation and 3-dimensional imaging technology have made considerable advancements over the last 10 years there is still a great deal that is not known about what triggers SVT and where to place scar tissue for optimal patient outcomes. We will produce a computer-generated atrium that can beat in real-time. Sections of this computer-generated heart can be turned off to simulate an actual ablation. This will help doctors and researchers rapidly gain insight into new ablation techniques without the risks and time constraints of experimenting on living hearts. This project has the potential to save thousands of lives.

Erdos conjectured that the equation \(x_1 x_2 \cdots x_n = n (x_1 + x_2 + \cdots x_n)\) could be solved by at-least \(n^\theta\) collections of \(n\) natural numbers. We approach the problem by focusing on the sum \(s = x_1 + x_2 + \cdots x_n\) and length \(n\). Using this new formulation it is straightforward to show that \(s\) is always composite. We suggest every composite greater than \(6\) is such a sum and present a novel bound contingent on this conjecture.

The set of defective parking functions are vectors with positive integer entries which describe the preference of cars entering a one-way street with n parking spots. If d cars fail to park within these parking spots, then we call the vector a defective parking function of defect \(d\). We will create orbits of defective parking functions by grouping together permutations of the same defective vectors and set up a recursion for counting orbits of certain defects.

We provide a recursive formula for orbits of \(d=1\), which has a link to Catalan Numbers. We also conjecture a recursive formula for orbits of \(d \geq 2\), which has a link to Catalan Convolutions.

With 2020 Census data now available, political redistricting is in full swing. Markov Chain Monte Carlo Methods are powerful mathematical tools for producing large ensembles of districting plans. However, without guidance, the resulting maps typically fail the “eye-ball” test by producing districts with highly irregular boundaries that wantonly ignore political borders like county lines. We present techniques that encourage Recombination MCMC to respect these borders while still producing unbiased ensembles of legally-viable districting plans.

Parking functions are vectors that describe the parking preferences of \(n\) cars that enter a one-way street containing \(n\) parking spots numbered 1 through \(n\). A list of each car’s preferences is also compiled into vectors in which we denote as (\(a_{1}, \ldots , a_{n}\)), such that \(a_i\) is the parking preference for car \(i\). The classical parking rule allows cars to enter the street one at a time going to their preferred parking spot and parking, if that space is unoccupied. If it is occupied, they then proceed down the one-way street and park in the first available parking spot. If all cars can park, we say the vector (\(a_{1}, \ldots , a_{n}\)) is a parking function. In our research, we introduce new variants of parking function rules with backward movement called \(k\)-Zone, preferential, and inverse preferential functions. We study the relationship between \(k\)-Zone parking functions and \(k\)-Naples parking functions and count the number of parking functions under these new parking rules which allow cars that find their preferred spot occupied to back up a certain parameter. One of our main results establishes that the set of non-increasing preference vectors are \(k\)-Naples if and only if they are \(k\)-Zone. For one of our findings we provide a table of values enumerating these new combinatorial objects in which we discover a unique relationship to the order of the alternating group \(A_{n+1}\), numbers of Hamiltonian cycles on the complete graph, \(K_n\), and the number of necklaces with \(n\) distinct beads for \(n!\) bead permutations.

Traditional methods of using a neural network to create a mathematical model is to use regression to map the transformation of the input x to the output y along discrete steps. While this process creates accurate models, this is not an optimal method of mapping continuous, time series data. Neural Ordinary Differential Equations (NODEs) are a family of neural networks that utilize differential equations to model the continuous transformation of the input x to the output y. Augmented Neural Ordinary Differential Equations (ANODEs) improve upon this by reducing the complexity of the ODE, allowing for more expressive mathematical models with simpler flows. I explore the effectiveness of using both NODEs and ANODEs to generate mathematical models for different types of curves of varying complexity.

A prime labeling on a graph on \(n\) vertices is an assignment of \([n]\) to the vertices so that neighboring vertices are relatively prime. There are many results concerning the prime labeling of ladder graphs which utilize a consecutive cyclic labeling. In this talk we introduce an “extended” ladder graph \(P_n \times P_3\) , and address ways that prime labelings can be applied to these graphs. We introduce path consecutive labeling as an alternative to consecutive cyclic labeling, in an attempt to find systematic prime labelings for extended ladder graphs. Additionally, using an algorithm written in Python, we address a related problem in number theory that seeks to answer the question of whether or not every ladder graph has a prime labeling.

In the field of differential geometry, the condition on the Riemannian metric so that a manifold has positive scalar curvature (PSC) is important for a number of reasons. Many famous researchers have contributed gradually to this area of geometry, and in this project, we study more about PSC metrics on such manifolds. Specifically, we refine and provide some details to the proof of Gromov and Lawson that the connected sum of 2 n-dimensional manifolds will admit a PSC metric, provided each of the manifold has a metric with the same condition. We then derive some useful formulas related to the Riemann curvature tensor, the Ricci tensor, and the scalar curvature in many different scenarios. Specifically, we compute the said quantities for a manifold equipped with an orthonormal frame and its dual coframe. We also compute the relationships between the scalar curvature of a manifold \(X\) and its different “components” when dividing the tangent space into the direct sum of two different subspaces. Our ultimate goal with these computations is to eventually come up with some conditions on the structure of a Riemannian manifold and its corresponding metric in order for it to have PSC.

Our research observes the engagement of video game players with video games from the online gaming platform Steam, utilizing methods from Topological Data Analysis. We specifically use the Sliding Windows 1D Persistent Structures (SW1PerS) method, which breaks down a time series dataset (a signal function) into windows of time and observes the homology structure of the windows. In our case, this process observes whether players settle into a regular pattern of replayability or if the game’s popularity dissipates into nothing. This metric of persistence is measured by the persistence score. We then utilize a linear regression to see what factors of a video game correlate with a good persistence score (ie. a regular player base). Our research indicates that financial metrics, such as the game’s price, are the most significant factors regarding persistence scores, and suggest further work investigate the impact the financials behind a video game have on the game’s success, irregardless of genre or other playstyle details. Ultimately, this research can be used to further understand video game player psychology, and inform game creators, game marketers, and community builders of their players’ overall habits and trends.

This project assesses the impact of social media traction, particularly through Google searches and user signup on the internet community known as Reddit, onto the closing price of various “meme stocks”. Meme stocks are defined as stocks affected by the initial Gamestop short squeeze, which occurred in late January to roughly March of 2021. Using a General Additive Model, the data and regressions demonstrated that user activity in specific subcommunities, known as Subreddits, actually had a statistically significant detrimental effect on meme stock closing prices. In particular, this community is the Subreddit called r/wallstreetbets. r/wallstreetbets is a Subreddit notorious for its irrational and impulsive user base, which commonly invests disproportionate amounts of their savings into stocks which have already begun increasing in price, or out-of-money call options dangerously close to expiration. As nearly all other Subreddit activity demonstrated a statistically significant positive correlation to a stock, it can be argued that online retail investors that took the time to educate and broaden their investing scope proved beneficial to a meme stock’s performance, while retail investors who were interested in r/wallstreetbets did not. Also assessed were the Subreddit performances on log relative return and volume, which all demonstrated similar consistent patterns. A look into the anomaly of r/wallstreetbets demonstrates a clear example of retail investors acting irrationally, against expected market efficiency.

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. We also improve the lower bound on boxicity for such societies and show that it is sharp when the associated piercing number is less than six.

Reading fluency is a skill developed during childhood that is needed to understand the meaning of texts and literary pieces. It is therefore important to understand the statistical properties of and to accurately model reading fluency data. In this research, we consider one such dataset obtained from elementary school-aged children who were given passages of different lengths to read. We aim to find a suitable model that accurately captures their oral reading fluency (ORF) as measured by the number of words read incorrectly (WRI) scores. The dataset contains information about the length of the passages (number of words) and WRI scores obtained from recorded reading sessions. The idea is to find passage-level parameter estimates with good mean squared error (MSE) properties. By using penalized likelihood estimation, estimators are obtained that often have smaller MSE than maximum likelihood. Three statistical models, namely the binomial, zero-inflated binomial, and beta-binomial, are considered as candidates for WRI scores. Additionally, different choices of penalty functions are explored. Ultimately, cross-validation is used to choose an appropriate model and penalty function. It is established that the beta-binomial with moderate parameter shrinkage has the smallest cross-validation score.

A convex figure is of constant width if and only if the distance between two parallel supporting lines is the same. Convex figures of constant width have led to significant advancement in a multitude of areas such as mathematics, optimization, engineering, and physics. Applications of convex figures of constant width can be seen in architecture, mapmaking, and mechanical design. The goal of this presentation is to discuss some properties of convex figures of constant width in the Euclidean spaces. Animations of constructing infinitely many convex figures of constant width will be demonstrated.

This project was developed as a response to a question I received early in my investigation of the heat equation, "Why is a burner on a stove shaped the way it is? Could it be better?". This question was the guiding point and purpose of the continuation of this project. We wanted to know if the "orientation of heating", defined as the specific points on the plate that will have heat energy added, affected the time it took to reach a minimum temperature across the entire plate. We also were curious to see if we could find a heating orientation better than just heating the entire plate. We used Numerical Methods, namely the 2D Crank-Nicolson method, with homogeneous Neumann boundary questions. We tested numerous heating orientation and although the differences between them were substantial no orientation cam close to beating just applying the heat evenly over the entire plate. Looking forward, we want to test randomly heating percentages of specific materials, to see if our question even had a possibility of succeeding.