2024 COES Design and Research Conference

Congruencies of Sums of Fibonacci Numbers

Team Member: Tyler Warzynak

Advisor: Dr. Nathan Green

This paper serves as an extension/application of a method detailed by Chang et al. in two separate papers, which worked with sums of the form ∑ 𝑎𝑘 𝑟𝑝−1 𝑛=0 modulo p for combinatorial sequences 𝑎𝑘, positive integers r, and prime numbers p. Using a known formula, the Fibonacci numbers were converted into a sum of binomial coefficients to apply the aforementioned method. A hypothesis was formed using computational methods, where a pattern was observed to hold for the first 50,000 prime numbers. This claim was then proven following the methodology from Chang et al. using properties of Laurent polynomials, definitions and theorems related to the Fibonacci numbers, and elements of modular arithmetic, including the freshman’s dream congruency identity and quadratic residues. It was found that for 𝑟 = 1, the sum is congruent to -1 mod p if p is congruent to 2 or 3 mod 5, to 0 mod p if p is congruent to 1 or 4 mod 5, and to 2 mod p if p is 5. For 𝑟 = 2, the sum is congruent to 0 mod p if p is congruent to 2 or 3 mod 5, to 1 mod p if p is congruent to 1 or 4 mod 5, and to 3 mod p if p is 5.

Predicting State GDP Using Time Series Forecasting

Team Member: Zachary Kilpatrick

Advisor: Dr. Galen Turner

In this paper, we consider uniquely pancyclic graphs, described as graphs with n vertices and a unique cycle for each vertex 3 to n. We will be building upon Markstrom’s results (2008), which asserted that there are no new uniquely pancyclic graphs with less than 60 edges and no uniquely pancyclic graphs with 5 chords. The paper aims to narrow down what a UPC could look like to aid the search in finding new UPC’s. More specifically, we will focus on the different ways we can construct the 3 and 4 cycles for these graphs and if they can use more than 1 edge not from the Hamilton cycle

Factors that Contribute to Completing Gateway Math Courses at Southern University

Team Member: Emma Colvin

Advisor: Dr. Beth Hegab

At Southern University in Shreveport, there are several pathways a student can take to complete the institution’s required Gateway Math Course (GMC). The GMC requires students to complete Math 133, Math 135, and Math 136. Currently, there is a low passing rate for the GMC, potentially due to several factors that will be examined throughout this project. These factors include a student’s math ACT score, age (maturity level), whether or not the student took these courses online or in person, and completing the optional developmental math courses before entering the GMC or a combination of the three.

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2-Accessibility of the Lucas Numbers and Fibonacci-Like Sequences

Team Member: Cameron LeJuene

Advisor: Dr. Stacey McAdams

In this project, we will delve into the combinatorics side of mathematics, using a basic graph theory idea such as set coloring. It will continue a question formed by the works of Bruce M. Landman and Aaron Robertson, ”3-Accessibility of the Fibonacci Numbers.” They were able to prove the Fibonacci Numbers to be 2-accessible, and while they did not supply the induction proof, we could construct and provide proof. This project will use their lemmas and propositions to prove the 2-Accessibility of the Lucas Numbers. This will, in return, allow us to attempt 2-Accessibility of Fibonacci-Like Sequences as well.

Matroids Stemming from the Maximal Relaxation of Graphic Matroids

Team Member: Landen Nguyen

Advisor: Ms. Allyson Donal

This research explores the base perspective for maximal relaxation in matroids. We begin by covering the basics of matroid theory required to conduct this research, such as definitions of matroids, such as relaxation, hyperplanes, and uniform matroids. We first analyze the problem in matroids obtained from simple connected graphs and then attempt to explore what may or may not be different in general matroids.

Abstract Measures on Totally Ordered Sets

Team Member: Gregory Whitehurst

Advisor: Dr. Jonathan Walters

In analysis, no topic is captivating, such as integration. Integration is so critical of a topic that it is generalized to abstract measure spaces. Two specific notions of integration were the motivation of this paper: Riemann-Stieltjes and Lebesgue-Stieltjes integration. The question that arose from these notions was the idea of generalization. How can Stieltjes integration be generalized to an arbitrary measure? The buildup to this question requires a beautiful combination of measure theory and order theory to deliver a measure on totally ordered sets that provide structure for a potential generalization. This paper shows that the chains in an ordered set is a σ-algebra and defines cumulative cardinality measure on (X, Σ) to give insight into the structure of an ordered set X as a measurable space.

Comparison of Linear Control Techniques for the Underactuated Nonlinear Quadcopter System

Team Member: Ian Golsby

Advisor: Dr. Jonathan Walters

Uncrewed aerial vehicles (UAVs) are a prevalent technology in many fields. They must be lightweight, efficient, and stable in order to carry out their objectives or support a payload. The control system that maintains a UAV’s attitude directly contributes to the stability and efficiency of the UAV, and more efficient UAVs can be made more lightweight by reducing battery size. Because the UAV has only four degrees of control (one per motor) but requires twelve dimensions to describe its orientation and position over time, it is considered an under-actuated nonlinear complex system. In this study, we compare various linear control systems and implementations in a simulated environment with MATLAB to inform engineers of each approach’s relevant benefits and trade-offs. The control systems studied were the Proportional-Integral-Derivative controller (PID), the Linear Quadratic Gaussian controller (LQG), the Model-Predictive controller (MPC), and the Feedback Linearization Controller (FLC). The UAV kinematics were modeled in MATLAB. We benchmarked the code performance for each control algorithm and ran the most intensive controllers at lower update frequencies to provide a fair comparison, assuming comparable hardware was used. We primarily considered how long a controller took to reach its desired state initially (rise time), how much a controller overshot its target state (overshoot), and how long a controller took to stabilize at its target state (settling time).

Does the Freshman 15 Exist?

Team Member: Peyton Albritton

Advisor: Dr. Stacey McAdams

This experiment was performed to determine whether the environmental changes of adapting from high school to college impact a student’s weight. We collected data from 58 college students in two Psychology 102 classes, ages seventeen to twenty-two, using a Google form. The students were asked a series of questions regarding their age and gender, as well as their eating, sleeping, and activity habit changes. Data was analyzed using hypothesis testing and a t-test.

Can Compiled Player WAR Predict MLB Team Win Percentage

Team Member: Thomas Hertel

Advisor: Mr. Stan McCaa

The baseball statistic Wins Above Replacement (WAR) is a complex yet compelling metric for approximating a player’s contribution to his team by representing how many more wins his team ought to garner with a AAAA player in his stead. The intent, then, of this paper is to test whether the accumulation of individual player’s WAR can be extended to satisfy another discipline of SABRmetrics, i.e., predicting an entire team’s future performance through modeling the sum’s correspondence (or lack thereof) to wins. If WAR is shown to extend this way, franchise front offices could feasibly isolate it as a singular metric for team construction, rendering negligible the intangible elements of team chemistry, which could theoretically surface when mix-mashing individuals’ stats. Using numbers from the last 10 seasons of Major League Baseball, linear regressions, and multiple regressions for predictive analysis, among other methods, are employed to aid in the above determination as well as to answer several ancillary questions, including, for instance, concluding the relative importance of batting, pitching, and fielding WAR.

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Using Multiple Regression Analysis to Determine the Strength of Certain Factors on Student Absenteeism

Team Member: Emmitt Antwine

Advisor: Mr. Stan McCaa

During the 2015-16 academic year, approximately 16% of the student population—exceeding 7 million students—was absent from school for 15 days or more. The escalation in chronic absenteeism is influenced by various factors, including poor health conditions, nonstandard workschedules of parents, socioeconomic disadvantages, changes in household compositions, frequent residential relocation, and substantial family responsibilities. Previous research on student absenteeism has examined the negative impacts that chronic absenteeism has had on students in diverse communities, such as racial minorities, disabilities, and English Language Learners communities. We utilized information obtained from the U.S. Department of Education’s Civil Rights Data Collection to conduct multiple regression analysis, aiming to investigate the correlations between chronic absenteeism and factors such as poverty rates, healthcare accessibility, teacher salaries, and gross domestic product, including all U.S. states along with the District of Columbia. Our findings indicate that in certain states, there is a strong correlation between the factors used in this project.

Injuries on Artificial Turf vs. Natural Grass in the NFL

Team Member: Robert Emory

Advisor: Dr. Xiyuan Liu

The purpose of this research is to determine if artificial turf causes moreinjuries than natural grass. By referencing different statistics from a sample ofgames from the 2023 National Football League season. Out of the ten gamessampled, five were played on natural grass and five were played on artificial turf. We ran the data collected from these games through two different regression analysis models that output p-values to show what truly caused the injuries. Our model uses a player’s position, height, weight, age, and snap counts along with the field surface type to see how if an injury occurs. With over 900 lines of data collected from only ten games, studying the entire season may give a different outlook.

An Exploration of the Sums of Two Squares and Pentagonal Numbers

Team Member: Jacob Von Tress

Advisor: Dr. Stacey McAdams

In the field of number theory, square numbers are very significant, and finding the sums of square numbers is a topic of certain interest to mathematicians. The most immediate application for adding together two square numbers is to identify Pythagorean triples. However, apart from seeking sums of two squares that are squares themselves, interesting patterns emerge that have fascinated number theorists for decades. Particularly, the distribution of a number’s divisors can explicitly determine how many ways that number can be written as a sum of two squares. Furthermore, pentagonal numbers, similar to square numbers, can be visualized by drawing a pentagon with the same number of dots on each side. The purpose of this research is to explore how a theorem concerning the sums of two squares leads to a potential set of characteristics describing the pentagonal numbers.