Elizabet Dieguez and Jake Touchet
Dr. Weizhong Dai
Heat transfer is the study of how thermal energy moves between physical systems. The most widely studied form of heat transfer, conduction, is modeled by the heat equation, the solution to which can be numerically approximated by the Crank-Nicolson scheme. This scheme is stable and accurate, but it becomes exponentially more expensive for finer solutions. Additionally, complex boundary conditions can be difficult to implement. Physics-Informed Neural Networks (PINNs), by contrast, produce naturally continuous solutions and can be trained on complex boundaries with no special modifications. Our project compares the accuracy and cost of using the Crank-Nicolson scheme versus a PINN on the problem of cooking a medium-rare steak. By juxtaposing the two solutions, as well as comparing them to a real steak, we will demonstrate the strengths and weaknesses of these two methods on real-world heat-transfer problems.
Adelle Douglas
Dr. Stacey McAdams
Planning an efficient day at the Walt Disney World® Resort is the topic of countless articles and travel blogs. However, currently, an algorithmic approach has not been considered. The purpose of this paper is to explore the use of a greedy algorithm to plan a day at Disney World’s Magic Kingdom. Greedy algorithms such as Kruskal’s algorithm, Dijkstra’s algorithm, and Prim’s algorithm were researched. Using a similar greedy approach, we created a heuristic algorithm that takes into consideration popularity of attractions, average wait times of attractions, and physical distance between attractions. The algorithm uses a graph with weighted vertices and weighted edges as the input and then makes locally optimal choices to create a path for visiting the vertices, and thus, a plan for a day in Magic Kingdom is created. Future research would be to create an algorithm that could adapt to changes throughout a day in the park.
Billy Deroche
Dr. Nathan Green
The Riemann Zeta function has historically been of paramount importance to multiple
fields of mathematics. While it is in many ways elusive and has an important though
yet-unproven conjecture associated with it, there are many versions of it that exist
in a characteristic-p setting that we know a lot more about. A paper published in
1990 constructs the Anderson-Thakur polynomials in a mathematical setting called the
Carlitz module that is used to prove certain properties of Zeta values in relation
to values of the Carlitz Logarithm. It was conjectured that a similar family of polynomials
should also exist over Elliptic curves, and that is precisely what we have found.
To accomplish this, we used an equations in Carlitz and considered these equations
in an elliptic curve setting to reverse-engineer the processes used by Anderson and
Thakur to construct the original family of polynomials. We did manage this, but a
more general construction over any arbitrary curve has not yet been found. We have
concocted a few possible avenues to go down to accomplish this by framing the problem
in terms of a collection of transforms in a general Drinfeld Module setting.
Kathleen Edwards
Ms. Allyson Donald
The distribution of population within large United States metropolitan region exhibits
non-uniform spatial density patterns influenced by urban structure and geographic
constraints. In this project, the use of graph theory provides a discrete framework
for modeling spatial population distribution through weighted network representations.
Each metroplex is modeled as a weighted graph G = (V,E), where vertices represent
spatial subregions, census tracts or districts, and vertex weights encode population
density. Edges represent geographic adjacency or proximity to the other vertices.
A nested weighted structure is used to layer the cities into subregions that are easier
to associate within G. These metrics are used to quantify concentration, dispersion,
and clustering of population within each metropolitan system. Analysis is ongoing
to determine how these network properties characterize intra-metropolitan population
distribution patterns.
Jackson Slater
Dr. Nathan Ponder
This project was conducted to research and compare methods that can be applied to predict the prices of the trading card game Yu-Gi-Oh. Data on sales for a handful of cards have been gathered from the past few months. This data is then applied within different time series models to predict future prices of the same cards.
Alex Bazzelle
Dr. Jonathon Walters
A two-phase variant of the iterated Prisoner's Dilemma (IPD) is introduced. In this variation, each round of the classic IPD is split into an unscored, non-binding "Voice" phase preceding a scored "Choice" phase. A method of measuring and visualizing similarities between strategies is established using weighted network graphs. The Voice phase's simplistic communication maintains the programmatic ability to study the IPD while offering insights into the value of trust and deception.
Brandon Franks
Stan McCaa
In 2024, Walker, Louisiana completed fed mitigation in Walker, Louisiana, focusing on rainfall-induced infiltration across 28 lift stations. Following a city-wide upgrade of manhole infrastructure in 2024, regression models were utilized to quantify improvements in pump runtimes. The methodology incorporates direct rainfall data and the Antecedent Precipitation Index (API), with the decay constant k optimized to 0.6 via iterative regression to account for soil moisture. Structural stability of the regression models was assessed using the Chow Test to identify changes between pre-mitigation and post-mitigation periods. The analysis confirms a statistically significant break in the regression equations, demonstrating that the 2025 runtime slope against rainfall is significantly lower than in preceding years. These findings prove that the mitigation efforts effectively reduced groundwater infiltration into the sewer system, thereby decreasing wastewater treatment demand. The results provide a mathematical framework for validating infrastructure performance through regressive analysis of system runtimes.