Students are encouraged to present research and results of special projects in poster format. The session is scheduled for Saturday morning, concurrent with contributed talks.
Poster presenters will be provided with a tripod, posterboard, and binder clips. Printed posters can be any size, but 48” x 36” is the most common. Appropriate content for a poster includes, but is not limited to, a new result, a new proof of a known result, a new mathematical model, an innovative solution to a Putnam problem, or a method of solution to an applied problem. Contact organizer Shalmali Bandyopadhyay (sbandyo5@utm.edu) if you need additional information.
A family of sets is union-closed if the union of any two sets in the family is also contained in the family. The Union-Closed Sets Conjecture, also known as Frankl’s Conjecture, posits that for any finite union-closed family of sets containing at least one nonempty set, there exists an element which belongs to at least half of the sets in the family. Although deceptively simple in its statement, the conjecture has defied a definitive proof, despite extensive study through various approaches and equivalent formulations, which have thus far yielded only partial results (Bruhn and Schaudt, 2015). We introduce a new formulation of the conjecture based on the Hamming distance (metric) between sets, which is defined as the cardinality of their symmetric difference. Using this framework, we derive multiple equivalent formulations of the conjecture and provide a complete characterization of the set of all elements that satisfy the conjecture. Furthermore, we demonstrate that our approach enhances the averaging method, a widely used approach that has led to numerous partial results. We also obtain a criterion that relaxes the requirement of the averaging method for verifying the validity of the conjecture. Our findings suggest that the Hamming metric provides a powerful perspective on the Union-Closed Sets Conjecture. By connecting structural properties of union-closed families to Hamming distances, this approach offers promising avenues for further exploration. Future research will focus on extending this framework to improve existing partial results and analyze larger families of sets, identifying additional equivalences, and hopefully progressing toward a complete proof of the conjecture.
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Lights Out is a game featuring a grid of light-up buttons that begins with some lights on and some off. The goal is to turn off all lights but pressing a button changes its state and the states of the cardinal neighboring buttons. In this paper, we explore a Lights Out game in which the board is placed on a cylinder and the lights have $k$ states with a specific starting configuration. We try to turn off all lights using a light chasing strategy in which we methodically turn off the lights row by row. We model this process using recursive equations. A connection to the Fibonacci sequence then allows us to determine the number of rows of buttons the board should have in order for us to turn off all lights using our light chasing strategy.
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Let $x_n = a_n + b_n i$, where $a_n , b_n \in \mathbb{R}$, and $ n \in \mathbb{N} $, and suppose that the sequence $\{x_n\}$ is governed by the recurrence relation $x_{n+2} = x_n + i\,m x_{n+1}$, where $m \in \mathbb{R}$ . We investigate the conditions under which two subsequences $x_{2k}$ and $x_{2k+1}$ lie on perpendicular lines in the complex plane. Specifically, we express $x_1$ and $x_2$ as $x_1 = a_1 + b_1 i $ and $ x_2 = a_2 + b_2 i$, satisfying the condition $ a_1 a_2 + b_1 b_2 = 0 $. We show that there exist two perpendicular lines $ l_1$ and $l_2$ passing through the origin in the complex plane, such that for all $k \in \mathbb{N} $, the sequence alternates between the lines: $ x_{2k} \in l_1 $ and $ x_{2k+1} \in l_2$. This provides a geometric interpretation of the recurrence relation. Furthermore, the ratios $ \frac{b_{2k}}{a_{2k}} = \frac{b_2}{a_2}$ and $\frac{b_{2k+1}}{a_{2k+1}} = \frac{b_1}{a_1} $remain constant for all $k \in \mathbb{N} $.
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We investigate classifying the stages of wounds from a dietetic wound imaging dataset with alternative machine learning algorithms to the standard approaches like Convolutional Neural Networks (CNN) and Support Vector Machines (SVM). Although these methods usually classify with high accuracy, we search for more explainable methods for clinician and patient interpretability. One approach previously studied in the literature is using a Bayesian Network model, specifically Naive Bayes Nearest Neighbor (NBNN). The goal of the authors was to make a computationally simple method while being competitive in image classification. We make some alterations to their method by trying alternative estimates of the probabilities in NBNN to improve the model's performance and equally important, its explainability. In particular, we consider K-Nearest Neighbor (KNN) for probability density estimation without using a kernel and then compare their classic NBNN to our approach using k-fold cross-validation. Ultimately, we address whether Bayesian Networks can be competitive in performance but also provide explainable wound imaging predictions.
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Pecans, known for their rich flavor and nutritional value, are a popular tree nut crop cultivated widely in the USA. The quality and yield of pecans are important factors for pecan farmers and growers, impacting both economic returns and consumer satisfaction. This research investigates the relationship between pecan size, weight, and quality, as well as the yield efficiency of three pecan trees located on Fort valley State University campus. The trees are not managed agriculturally and thus could be considered as wild pecan trees. Around the beginning of December 2024, we collected a total of 300 pecans, 100 from each tree, that were fallen naturally from the trees on the ground without shaking trees. Only intact pecans with no visible cracks were selected. Measurements of pecan length, diameter (using a caliper with 0.1 mm accuracy), and weight (using a scale with 0.1 g accuracy) were recorded. Each pecan was then cracked open to categorize its shelled quality as good (light brownish color and firm texture), blank, or bad (highly dried, mushy, or dark). Data were compiled in Excel for analysis. Both simple and multiple linear regression models were developed to explore relationships between pecan size (length and diameter) and weight. Statistical metrics, including minimum, maximum, average, standard deviation, and sample distribution, were calculated to characterize the yield of each tree. Yield efficiency was determined based on the percentage of good pecans produced by each tree. The initial analysis reveals that the average length, diameter, and weight of the pecans are 40.7 mm, 21.9 mm, and 7.6 grams, respectively. Correlation analysis showed relatively good relationships exist between length and weight (r = 0.54) and diameter and weight (r = 0.55). Simple linear regression showed linear trends between these variables. Furthermore, multiple linear regression using length and diameter as independent variables and weight as the dependent variable provided a better fit compared to simple linear regression models. The findings will provide valuable insights into tree management practices such as pruning, fertilizing, and pesticide application to enhance the quality and yield of pecan trees. This research highlights the importance of data-driven approaches in optimizing agricultural productivity.
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The ElonU Tangential's student group solved the AMM problem 12476 which states: Let $C$ be one arch of the elliptic cycloid generated by the ellipse $x^2+\frac{1}{4}(y-2)^2=1$. That is, let $C$ be the curve traced by the vertex at the origin as the ellipse rolls without slipping along the $x$-axis for one revolution. What is the area under $C$ and above the $x$-axis? Our solution uses a clever change of perspective letting a tangent line roll around the ellipse, instead of the ellipse roll along the $x$-axis. The desired area turns out to be a surprisingly simple number.
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Bootstrap percolation (BP) is a process on a graph which can be used to model the spread of an infection throughout a graph. First, an initially active set of vertices is given. Then, loosely, vertices are activated if they’re in contact with enough active vertices. More precisely, suppose G is a graph, k is the bootstrap parameter, and a set of active vertices at time 0 is given. In the next time step, any inactive vertex with at least k edges to active vertices becomes active; the process continues until no new active vertices are created. To see how this can be used to model the spread of disease, take people as vertices, interactions between people as edges, and make “active” mean “infected”.
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While exploring Brocard’s Equation $n! + 1 = m^2$, which is known to have 3 solutions, st. $m,n \in \mathbf{N} $, we develop an algorithm to effectively store factorials of large numbers. First, we begin by prime-decomposing the factorial with the help of Legendre’s Formula. Then, we convert the resulting product of prime powers into a sum by taking logarithms of convenient base. Finally, we implement the algorithm to search for potential solutions in $ \mathbf{N} $ for Brocard’s Equation up to $100 \,000!$.
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One powerful application of complex analysis is the visualization of streamlines and velocity fields of irrotational and incompressible fluids using equipotential curves. With this poster, we aim to leverage the complex potential function to provide valuable insights for both theoretical research and practical applications. Our methods utilize complex analysis to determine solutions to the differential equations describing fluid flow, focusing on circular objects. These techniques offer an unparalleled ability to accurately describe flow around circular objects. We analyze planar velocity fields using conformal mappings and employ the Joukowski transformation to model flow around airfoils. Additionally, we construct ideal fluid flows confined within a given domain, known as "streamlining." These methods offer a robust framework for understanding and optimizing fluid dynamics problems. The findings have wide-ranging applications, including pollution spread, blood flow, aerodynamics, and water distribution systems, contributing to more efficient and sustainable designs.
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Prior to the development of optical systems, it’s essential to understand their potential performance, often allowing for easier testing of ideas and a smoother development process. The performance of grazing incidence optical systems, used to collect x-rays, is generally less understood than that of most optical systems. Thus, this research focuses on understanding and modeling the energy performance of grazing incidence mirrors as a step towards developing a grazing incidence telescope design system to assist in the development of grazing incidence optics. Currently, this work will assist in the development of metrics for the Lynx X-Ray Observatory: a novel x-ray system with the capability of achieving 10 times the effective area of the Chandra X-Ray Observatory at an equivalent or better resolution. Multiple linear regression and beta regression were used to generate a model for reflectivity given grazing angles of 0$^\circ$ to 2.5$^\circ$ and energies of 0.5 to 10 keV based on data from the Center for X-Ray Optics – X-Ray Database. The chosen model was then applied to data on grazing incidence mirror shells to calculate their reflectivity and thus effective area. Effective area was then analyzed as a function of energy and radius. It was found that energies of 0.5 to 10 keV and radii of 10 to 150 cm yielded effective areas of 0.03 to 220 cm$^2$. In general, lower energies and higher radii resulted in a higher effective area. Specifically, it was observed that effective area decreases rapidly as energy increases and decreases more steadily as radii decreases. Following this work, a ray tracing program for Wolter-I type optics is to be developed as the next step toward developing a grazing incidence telescope design system. This work is supported by the NSF REU solar physics program at SAO, grant number AGS-2244112.
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In this project, we seek to understand how we can utilize machine learning (ML) to inform the design of biosensors to emit a strong fluorescence in the presence of dopamine. Dopamine is a neurotransmitter and plays a role in a variety of functions such as memory, learning, and reward systems. Detecting dopamine levels could help with diagnosing addiction, mental illness, and neurodegenerative disorders. We develop a framework for one-hot encoding nucleotide sequences for training ML models, create a data preprocessing pipeline to pad sequences and normalize output values, and construct neural net model architecture for training on both sequence and numerical data. Using a published toehold switch dataset along with a ribosensor dataset, we explore the accuracy of several different regression models in predicting biosensor effectiveness by training on both nucleotide sequence data and calculated thermodynamic parameter data. We find that a neural network model, specifically a multilayer perceptron, typically outperforms other regression models such as linear regression, random forests, and support vector machines in both datasets, and that training on sequence data appears to be more predictive than training on thermodynamic parameter data. We also suggest potential directions to pursue transfer learning between the two datasets.
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Current research supports the use of statistical methods in forensic science to analyze Fourier Transform Infrared Spectroscopy (FTIR) data for identifying polyvinyl chloride (PVC) electrical tape used in the construction of explosive devices. For this project FTIR data is collected for both the adhesive and backing sides of multiple rolls of black PVC electrical tape across the 3M brand’s product range. Principal components analysis is then applied to reduce the dimensionality of this FTIR data and cluster the rolls into similar groupings. This analysis is then used to classify other examples of similar PVC electrical tape, and the percentage of correct classifications is investigated.
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In response to the profound impact of the SARS-CoV-2 pandemic, the scientific community has focused considerable research efforts to understand the spread of the virus. Despite a tremendous volume of research in this area, the dynamics of the immune response to SARS-CoV-2 has been hampered due to limited analysis of the experimental or clinical information to date. Mathematical models that account for the interaction between SARS-CoV-2 and the human immune system will improve the scientific community’s ability to analyze the amount of data available.We develop a mathematical model for the immune response to SARS-CoV-2 to investigate the role of various pathways in successful viral clearance and the key mechanisms responsible for disease severity. These interactions are captured by a system of ordinary and delayed differential equations. We conduct parameter estimation based on experimental data and investigate model behaviors via computational simulations. Our model demonstrates key aspects of the immune response to SARS-CoV-2 which might be responsible for disease severity. Specifically, Natural Killer Cell dysfunction impairs its ability to eliminate infected cells effectively. This could be used to serve as a foundation for the development of therapeutic strategies.
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This poster presents a Diffusion-Limited Aggregation (DLA) model in the context of crystallization. The DLA model describes how particles randomly diffuse until they come in contact with the existing cluster. The underlying physics that creates DLA is Brownian motion, which is simulated stochastically through random walks. This model explores the formation of crystal structures through dendritic solidification. Dendritic solidification is a process where a liquid cools and forms a solid with tree-like patterns, known as dendrites. This principle is fundamental in forming naturally occurring patterns and structures, such as metal castings and ice crystals. By understanding how dendrites form, the microstructures of these materials can be optimized for improved mechanical properties such as strength and durability. With this model we demonstrate how changes in expansion rate and particle fusion probability affect the structural integrity of dendritic materials.
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The Discrete Element Method (DEM) is a numerical technique used to model interactions between discrete particles. DEM simulates the behavior of granular materials by calculating the forces and motions resulting from particle collisions, friction, and cohesion. Newton’s second law of motion, combined with force-displacement laws, maintains a crucial role in enabling accurate simulation of particle interactions in DEM. Applications of DEM encompass various fields including civil engineering for simulating soil mechanics, pharmaceuticals for optimizing powder mixing processes, and geophysics for studying natural phenomena like landslides and earthquakes. Building on these diverse applications, this research seeks to enhance the understanding of granular flow dynamics by leveraging DEM to analyze the interactions within granular materials. A series of simulations were conducted by using Python to explore how particle shape, size distribution, and contact properties influence flow behavior.
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We study nonlinear boundary value problem on discrete time-scale with continuous nonlinearity on the boundary and inside the domain. We define upper and lower solution for the problem and use Brower fixed point theory to prove existence of positive solution.
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We introduce a triangle of numbers, similar to the Fibonacci Triangle and the Pascal Triangle and discuss some of its properties. As an open question, this triangle was given as an OEIS Challenge (P419) from “The Playground” in the Math Horizons magazine. The triangle is constructed in the following way: The entries of the first diagonal from the left (or the first back slash diagonal) of the triangle are all equal to 2, and the second back slash diagonal is made up of the Fibonacci numbers. Any entry in a particular back slash diagonal after that is obtained using a recursive formula that involves adding up all previous entries in the respective diagonal along with a constant which depends on the position of the entry that is being computed. We provide a formal definition for the entries of the triangle using a recursive formula. We also discuss how an L-shaped pattern in the triangle gives rise to a formula for the entries of the third slash diagonal (or third diagonal from the right), involving entries of the previous slash diagonals and Triangular numbers. We finally provide a generalized proof of this formula using Mathematical Induction. Other patterns are also included.
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Heart disease remains the leading cause of death in the United States, with heart failure contributing significantly to these mortality rates. Defined by the CDC as the heart's inability to pump sufficient blood and oxygen to support other organs, heart failure affects over 6 million Americans, with a five-year post-diagnosis mortality rate of nearly 50%. This study leverages deep learning to predict mortality risk in heart failure patients using a dataset of 299 records with 17 clinical features. Deep neural networks (DNNs) and convolutional neural networks (CNNs) were developed and optimized through hyperparameter tuning to enhance prediction accuracy. Model interpretability was achieved using SHapley Additive exPlanations (SHAP), which identified key predictors such as follow-up duration, ejection fraction, serum creatinine, and diabetes. These findings align with prior research highlighting ejection fraction and serum creatinine as critical factors, while also emphasizing time, diabetes, and age as significant predictors. This work demonstrates the potential of combining explainable AI with deep learning to support clinical decision-making in heart failure management.
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In this study, we examine the predictive power of seven key bat tracking variables in determining Expected Weighted On-Base Average (xwOBA), a widely accepted measure of hitter success. xwOBA is considered a more reliable indicator of a player’s skill than traditional Weighted On-Base Average (wOBA), as it removes defensive factors from its calculation. Unlike wOBA, which depends on actual outcomes, xwOBA estimates a hitter's performance based on batted ball characteristics—exit velocity, launch angle, and other relevant factors—assigning probabilities for singles, doubles, triples, and home runs based on contact quality. To identify the most predictive variables for xwOBA, we utilize multiple linear regression (employing forward and full-subset selection) alongside decision tree models, including random forests and boosted trees. Model performance is evaluated via Root Mean Square Error (RMSE), and we identify the most influential metrics in predicting hitter success. These findings offer valuable insights for player evaluation and development.
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We investigate the properties of the eigenvalues of the fractal Laplacian. We begin by defining the fractal Laplacian operator in one dimension and formulate the corresponding Dirichlet eigenvalue problem. Analytical solutions are obtained for specific fractal parameters, and computational results illustrate the structure of eigenvalues and their associated eigenfunctions. We extend our analysis to two dimensions using separation of variables. Our findings contribute to a deeper understanding of how fractal geometry affects the spectral characteristics of differential operators.
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We solved Problem B-1337 from The Fibonacci Quarterly. In this problem, we show that the sum and difference of ratios of Fibonacci numbers is equivalent the product of two consecutive Fibonacci numbers. We prove a similar result for Lucas numbers. We use the Gelin-Cesaro Identity as well as a related identity to prove the equations of Problem B-1337.
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Norovirus, a highly contagious illness, affects the digestive system and is a common cause of morbidity globally. This study explores the spread of Norovirus using a hybrid SIR model integrated with cellular automata. Our simulation models the transmission and recovery processes in a sampled population under varying probabilities of susceptibility and infection. By employing cellular automata, the model captures localized interactions and provides a visual representation of spatial disease progression. Our model assumes a static population, deterministic state transitions, and fixed durations for infection and immunity. The simulation outcomes demonstrate how initial conditions, such as high susceptibility or infection rates, influence epidemic patterns over time. High susceptibility results in rapid disease spread due to limited initial immunity. Conversely, high initial infection rates lead to rapid saturation with immune individuals, followed by cyclical reinfections as immunity wanes. The findings highlight the interactions between susceptibility, infection, and temporary immunity, offering insights into effective epidemic control measures.
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In this project, we model the time advantage of speeding on a highway with traffic lights as a stochastic process. The object of study is the sequence of random variables $\{T_k\}$, representing the time benefit of the speeding car at the $k$ th traffic light. For simplicity, we assume identical traffic light cycles with independently distributed initial phases and constant distance between lights. With this simplified model, we investigate the distribution of $T_k$, proving results on the expected value of $T_k$ and deriving the conditional distribution $T_k\vert T_{k-1}$. The first result gives some intuition on the situation being modeled and the second result shows that $T_k\vert T_{k-1}$ and thus $T_{k}$ are examples of mixed distributions. Though the model serves more as a toy model, it provides an interesting example of a family of distributions having both a continuous and discrete part that arises from a simple mathematical model.
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