A Comprehensive Analysis of the Sequence X(n+2) = imX(n+1) + X(n) for X(1) = X(2) = 1 + i — Jasmine Stefano

This research investigates the recursive sequence given by $X_{n+2}=im{X}_{n+1}+X_n$, where $X_1=X_2=1+i$, with $i=\sqrt{-1}$ and $m$ being a parameter that assumes real values. We analyze the recursive sequence as a second-order difference equation with constant coefficients. By solving this equation, we derive explicit bounds on the parameter $m$ and obtain general equations to describe the resulting geometric shapes. Our analysis reveals how said bounds on $m$ influence the behavior of the sequence. We plot the sequence in the complex plane and as $m$ varies, we can see interesting geometric shapes formed from conic sections. When $m$ lies in the interval (-2,2), the sequence exhibits bounded behavior with points in the complex plane tracing two ellipses. For $|m|>2$, the sequence becomes unbounded, leading to hyperbolic trajectories. This work explores the geometric nature of the solutions, examining how the sequence's behavior evolves as the parameter $m$ varies.

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A Hidden Pattern in the Co-Primes: A Journey Through an open Erdös Problem — Jonas Barfield

This presentation will delve into a fascinating open Erdös Problem, which poses the question: Is it true that there are no solutions to $$n!=x^k\pm y^k$$ with $x,y,n,k\in \mathbb{N}$, $xy>1$, and $k>2$? We will show that a solution does indeed exist. However, in exploring whether this solution is unique, we uncover an intriguing pattern among the co-primes. This presentation will feature beautiful plots and reveal unexpected phenomena that invite deeper investigation. Join us as we navigate through these mathematical curiosities and uncover the hidden structures within.

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A New Generalization of the Continuous Bernoulli Distribution — Garrett Nix

Many known distributions are useful in modeling data but often have limited shapes, such as only being right skewed, left skewed, or symmetric. These limitations make it difficult to use these distributions in broader applications, where data may not follow the shape of these models. By generalizing distributions, adding more parameters increases the flexibility of these distributions in most cases. This allows the new generalizations to have a wider range of applications than the original distribution. The aim of this research was to create a new generalization of the continuous Bernoulli (CB) distribution. In achieving this, a new family of unit-interval distributions could be created to model complex data. With this research, we propose a new generalization of the continuous Bernoulli distribution using the T-R{Y} framework, where we define random variables T, R, and Y that follow specified distributions. With this framework we introduce the T-CB{Cauchy} as well as the T-CB {logistic} families of distributions and investigate the properties of these families. We also investigate the properties of members within the families of these distributions, introducing the normal-CB{Cauchy} and the normal-CB {logistic} distributions. The flexibility of the distributions is observed by fitting the models to different data. Using the T-R{Y} framework, two families of generalized continuous Bernoulli distributions are defined. The properties are investigated, as well as the flexibility is observed in model fitting.

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A Universal Chord Theorem for Triangulable Spaces — Kalev Martinson

Where $A$ is a topological space, let $f:[0,1]\to A$. Define the *Horizontal Chord Set* $D_f:=\{\ell \in \mathbb{R}|\mathrm{\exists }x\in [0,1-\ell ],f(x)=f(x+\ell )\}$. Let $\mathcal{L}A$ be the *loop space* of $A$. It has been previously proven that $\bigcap _{f\in \mathcal{L}\mathbb{R}}{D}_f=\{\frac{1}{n}|n\in \mathbb{N}\}\cup \{0\}$. It has also previously been proven that the Lebesgue measure $\lambda (D_f)\ge \frac{1}{2}$ for $f\in \mathcal{L}\mathbb{R}$. For a topological space $A$, denote the constant $k_A=inf\{\lambda (D_f):f\in \mathcal{L}A\}$. In this paper we characterize $k_A$ for triangulable spaces, proving $k_A=1$ when $A$ is 0-dimensional, and $k_A=0$ when $A$ is more than 2-dimensional, or is 1-dimensional and has a cycle. When $A$ is 1-dimensional and has no cycles it is a tree, we prove that $k_A\le \frac{1}{n}$ where $n$ is the number of leaves, and conjecture that $k_A=\frac{1}{n}$. We show that the map $f\mapsto \lambda (D_f)$ is not continuous, making the proof of this conjecture difficult. We finally generalize Paul Levy's Universal Chord Theorem by showing that for any tree $A$ with a vertex of degree three or more, $\bigcap _{f\in \mathcal{L}\mathbb{A}}{D}_f=\{0,1\}$.

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Algebraic Methods for Exploring Phylogenetic Networks — Demmi Ramos

Phylogenetics is the study of evolutionary relationships between organisms. Our goal is to reconstruct the evolutionary history of collections of species by building phylogenetic trees (family trees) from biological data. While mathematically interesting, trees are often too simple to model the complex nature of real gene transfers. This leads to the study of phylogenetic networks, which incorporate hybridization, allowing separate species to come together. This talk explores assigning algebraic invariants to phylogenetic networks in order to find the network that best explains the data given. One approach to assigning these invariants involves using matrices of conditional probabilities to describe models of DNA sequence evolution.

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An Alternative Proof of a Sandwich Type Inequality — Alexander Alvarez

In the book, *Algebraic Inequalities: New Vistas*, Andreescu and Saul proved an inequality in one of the exercises: for fractions $\frac{a_1}{b_1},\frac{a_2}{b_2},\cdots ,\frac{a_n}{b_a}$ , if $m$ and $M$ are the smallest and largest of these fractions, we have $m\le \frac{a_1+\cdots +a_n}{b_1+\cdots +b_n}\le M$. Recently, while solving a problem in the journal, *MathAMATYC Educator*, Vol.15, No.3, Problem Section, we realized that the solution to this problem can be generalized to a proof of the inequality by Andreescu and Saul. In this talk, we will introduce the proof by Andreescu and Saul, and then we will present our new proof.

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Cohomogeneity two Bazaikin spaces — Rachel Flores

We study the sectional curvature of all of the cohomogeneity two Bazaikin spaces with respect to a Riemannian metric construction due to Wilking. We show that, in contrast to the cohomogeneity one and homogeneous case, for all of the cohomogeneity two examples, the set of points with strictly positive curvature does not have full measure.

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Comparing mathematical reasoning and logic across AI platforms — Abigail Hyatt

In this presentation, we discuss a series of experiments designed to test the logical and mathematical reasoning skills of various AI tools. We present our experiments and some initial results, which so far have been conducted across three well-known LLMs.

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Connectedness of Discretized Configuration Spaces — Justin Brentwood

The discretized configuration space of a finite one-dimensional cell complex $G$ can be connected in many ways, few ways, or not at all, depending on $G$. This project begins to take an algorithmic approach to describe sufficient or necessary conditions to determine if a configuration space is connected through primarily graph properties and secondarily algorithmic abilities.

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Dynamic Flexible Tile Modeling of DNA Self-Assembly — Sloane Kinley

Deoxyribonucleic Acid (DNA) is proven to be a valuable building block for constructing nanostructures capable of targeted drug delivery. DNA is also characterized as self-assembling; when a sticky end is introduced to complementary unbonded base pairs, DNA will hydrogen bond without any mechanical assistance. Graph theory can be implemented to model this phenomenon, and one such framework is referred to as the flexible tile model of DNA self-assembly. This model allows for symmetry and predicted bonding where it may not exist in the lab setting. We add to the flexible tile model in order to calculate the probability of self-assembly of certain graphs. If a specific graph type or family is desired, or even a single specific graph, the probability of this event occurring can be calculated. This theoretical experiment explores the possibility and probability that cohesive end types, in the flexible tile model, can bond to other end types when exposed to one another in a lab setting. This exploration specifically focused on small cycle graphs of order three and four.

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Error Analysis of a Symplectic Algorithm for a Hamiltonian System — Josiah Hays

In a Hamiltonian dynamical system, ‘area’ in the phase space of the system is conserved over time. Symplectic algorithms are numerical algorithms that solve for the time evolution of a Hamiltonian system and are explicitly designed to preserve areas in phase space. Due to their area-preserving properties, symplectic algorithms often demonstrate much higher accuracy over long time intervals than the equivalent non-symplectic numerical algorithms for Hamiltonian systems. However, symplectic algorithms are still susceptible to rounding error as they use finite-precision floating-point numbers in their computations. We describe a modified symplectic method due to Robert Skeel that takes rounding errors into account, and we mathematically verify the accuracy of this method for the case of the simple harmonic oscillator.

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Exploratory Analysis of the Top-Ranked Athletes' Performances in ATP Tour — Austin Hitt

Exploratory Analysis of the Top-Ranked Athletes' Performances in ATP Tour Austin Manning Hitt Faculty Advisors: Dr. Bo Li and Dr. Chao Gu In this talk, we investigate the relationship between the overall scores of the top-ranked tennis players and the metrics provided in the database “Official Site of Men’s Professional Tennis/ATP Tour” through an exploratory statistical analysis. To improve the overall scores, it is the aim to identify the underlying predictor variables for the statistical model, since not all the metrics play a role in explaining the overall scores. We perform the step-down model selection algorithm to identify the underlying predictor variables. The statistical exploratory analysis result shows that the best evaluation of the overall performances are the adjusted values of the return points won, 2nd serve points won, and the total number of Aces.

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Food Insecurities in a Blooming Rural Town: A Statistical Analysis — Christine Jator

Food is an essential part of daily life, providing vital nutrition to the mind and body. The lack of an essential resource can significantly hinder one’s quality of life. In the south, “The state of Tennessee measures at 11.9% food insecurity” (Durnell, 2023, p.1). This statistic reveals that several people cannot acquire healthy food, decreasing their quality of life. Clarksville, a growing rural town in Tennessee struggles with the same issue. In this study, we seek to understand what factors classify food-insecure populations in Clarksville, TN, and how they compare on a smaller and larger scale. Using data from the United Census Bureau, we will analyze trends and formulate conclusions from Clarksville, TN data, and compare these results with Murfreesboro, TN, and USA data. The data contains food stamp information counts and percentages for different populations. This research aspires to aid food pantries and food-insecure populations in growing rural areas similar to Clarksville, TN. This project sheds light on remembering the impoverished and those who struggle to obtain healthy food in a progressing society.

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Higher Dimension Analogues for Pythagorean Triples — Cara Admiraal

There is a known correspondence between rational points on the unit circle and Pythagorean triples. Using problem-posing strategies, such as those found in Brown and Walter's book *The Art of Problem Posing*, we will explore high-dimensional analogues of Pythagorean Triples such as by giving classifications or generalizing the generating formula. Beginning with a visual aid, in the form of a lattice, our generalization will proceed via the stereographic projection of rational points $S^n$ sphere.

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Mathematical Modeling of COVID-19 With a Focus on Asymptomatic Transmission — Lauren Beuerle

In 2020, the global COVID-19 pandemic erupted. Without knowledge to combat the disease, hospitals around the globe were overrun. Scientists and mathematicians, using past information on extremely infectious viruses, began investigating the effectiveness of social distancing, facial coverings, and eventually, vaccinations. Mathematical models can be used to explore the quantitative effectiveness of vaccinations, facial coverings, and create predictive models to aid the creation of policies in order to prevent future surges in cases. This project will utilize the ordinary differential equation SIR model to explore susceptible (S), infectious (I), recovered (R) populations to explore impact of asymptomatic individuals. The products of this model can be used as a reference for preventative measures for future epidemics that follow a similar pattern to COVID-19.

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Numerical Simulation of Jellyfish Swimming — Jillian Thomas

Jellyfish are considered the most energetically efficient swimmers to have ever existed, so their propulsion method can be researched to improve our own underwater vehicle designs. These unique animals need to be very efficient because of their untraditional bodily components, and because most consume limited food while they prey passively during swimming. Jellyfish accomplish their efficiency through vortex propulsion. The contraction of a jellyfish’s bell generates a vortex ring as it swims, which due to its axial symmetry we simplify into two dimensions using two point vortices. We model these vortices using a system of differential equations, for which parameters can be selected to adjust the strength, location, and direction of their rotation. To numerically solve the system, we use the fourth order Runge-Kutta method in MATLAB. The first goal of the project is to examine the jellyfish's propulsion and maneuvering mechanism, which involves creating simulations for various parameter values in the system of differential equations. Secondly, we study the material transport in the vicinity of jellyfish and its implication on food acquisition. Massless particles are inserted into the fluid flow to observe how the jellyfish and particles around it move through the water as affected by the vortices. To work towards these goals, we have implemented bell shapes to represent moon jellyfish, Pacific sea nettles, cannonball jellyfish, and lion’s mane jellyfish to provide a breadth of bell shapes and sizes. Each of these bell shapes serve as a barrier to material transport, so the jellyfish can capture particles from the surrounding environment. These results have significant implications for fields such as biomimetic engineering, including improvements to sub-aquatic vehicle efficiency and reliability, especially in cases when speed is not a priority.

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Orientable Quadrilateral Embeddings of Cartesian Products of Cycles — Matthew Farnsworth

In the spirit of Pisanski (1989) we consider orientable quadrilateral embeddings of Cartesian products of cycles on surfaces. We offer a constructive example of such an embedding of three low-order cycles. Then we show more generally that such embeddings exist for products of the form $C_2\times C_{2n}\times C_m$. We represent our graphs using rotation schemes to show this existence. Use of rotation schemes led to the ultimate characterization of our findings visually, providing conjectures for generalizations of products of three cycles.

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Quad Packing — Taiki Aiba

Quads or EvenQuads is a pattern-recognition game similar to Set. The goal of the game is to find "quads", which are sets of 4 cards that satisfy a particular pattern. In this talk, we will discuss our research into the maximum number of quads in a $k$ card layout. Specifically, we will demonstrate an upper bound for the number of quads in a $k$ card for all $k$ and prove this bound is tight for when $k$ is a power of two. Furthermore, we will share the results for $4\le k\le 6$.

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The Danceability Index — Lila Snodgrass

In this presentation, we will introduce a new knot invariant, the danceability index, which draws inspiration from the dynamic movement of dancers across a stage. We will examine the connections between this invariant and two other fundamental knot invariants: the braid index and the bridge index. Designed with an undergraduate audience in mind, this talk will offer a brief yet informative overview of the key concepts and foundational ideas within the field of knot theory.

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Theory to Identity: The Role of Mathematical Modeling in Shaping Graduate Students as Mathematicians — Christina Anderson

Students have a rare opportunity to combine fundamental concepts with practical applications in the mathematics classroom, even at the graduate level. In order to understand how their courses, interactions with peers, mentors, and multidisciplinary projects contribute to their developing sense of self as mathematicians, this study looks at the experiences of graduate students and instructors who are actively involved in advanced mathematical studies. In addition to addressing the difficulties students encounter in striking a balance between mathematical precision and real-world application, the study highlights the crucial role modeling plays in promoting competence, confidence, and professional purpose. These results highlight how important it can be to incorporate mathematical modeling as an instructional method and not just a course. This will foster identity formation and get students ready for meaningful employment in mathematics and related disciplines.

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Using Math to Win Fantasy Football — Gabe Boone

This presentation explores how mathematical strategies can give one an advantage in fantasy football. It covers key aspects of success, including drafting strategies, managing lineups, and making trades, while emphasizing the importance of adapting to changing player performance and injuries throughout the season. The presentation also discusses playoff preparation and analyzes the role of probability and projections in improving one’s chances of success. Through examples and statistical insights, it demonstrates how consistent, informed decision-making leads to better outcomes in fantasy football.

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“Using Multiplication and Addition to Manipulate Sequences Equating to 0 modulo n” — Ryan Legg

This poster was created from Mathematics Magazine Problem 2204. In any sequence, does there exist a way to manipulate how the terms of the sequence are added and multiplied to form a solution equivalent to 0 mod n, where n is the number of terms in the sequence? We look at whether this was possible by examining several sequences of numbers, differing in length and combination. Using techniques such as trial and error, as well as picking up on several different patterns, a method was formed to help prove that any sequence can be multiplied and added together in such a manner that the sequence could always come out to a remainder of zero. We prove that any sequence of numbers could be arranged in a certain way so that when multiplied and added together, there would always be a remainder of 0 modulo n.

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