Senior Projects 2023

Emily Armstrong - The Collector Problem, Advisor Dr. Eric Bucher

The Tax Collector Problem is a game that falls in the intersection between game theory and number theory. We will introduce the game, how a player wins, and some generalized theories applied to any size of the game.

 

Ronnie Cole - Jakob Bernoulli's Tractatus de Seriebus Infiniti, Advisor Dr. Eric Bucher

Jacob Bernoulli is a 17th-century mathematician who is known for their work, The Art of Conjecture. He provided a proof for the Law of Large Numbers and contributed heavily to the field of probability. Another lesser-known work of Jacob’s was published along with his Art of Conjecture. This work was a five-part treatise on infinite series called the Tractatus de Seriebus Infinitis. Dr. Otero has created an English translation of the Tractatus, allowing English readers to read Jacob’s work. Our goal was to provide supplementary material that would provide background information and modernize the mathematics within the Tractatus. To this end, a short Biography of Jacob was written, as well as a summary of the 3 parts of the Tractatus. The summary of the Tractatus expands on Bernoulli’s proofs and his older mathematical techniques were modernized to make it more comprehensible and complete. This research allowed for the content within the Tractatus to be studied by English readers. The content of which are unique methods of finding the sums of infinite series.

 

Jack Haskins - Change Point Detection, Advisor Dr. Grigory Sokolov

In this presentation, we will explore an area of Time Series called Change Point Detection. The goal of Change Point Detection is to detect a change, if any, in the density function of observations. More specifically, this presentation will focus on various stopping procedures based on log-likelihood ratio statistics, quantifying delay to detection, estimating average run length to a false alarm, and stopping procedure thresholds. Finally, performance results for stopping procedures will be compared using simulations in Julia.

 

Jake Heyser - Decompositions of Circulant Graphs, Advisor Dr. Esmeralda Năstase

While the origin of graph theory dates back a couple of centuries, circulant graphs are a relatively new family of graphs. Although many of the circulant graphs’ theoretic properties have been studied, there are still many structural properties that have not been researched. In this talk, we introduce circulant graphs, explore cycles within circulant graphs, and prove some results related to these. We also discuss decompositions of circulant graphs, and some related results.

 

Brett Holcomb - Public School Funding: Do Schools Need More?, Advisor Dr. Carla Gerberry and Dr. Michael Flic

“Public School Funding: Do Schools Need More?” explores the complex world of public education through a statistical lens. There is no doubt that many public schools are struggling, but what is the reason for this? One idea is funding, and we will test this theory by comparing the expenditure per-pupil of public schools in the state of Ohio to three performance metrics; graduation rate, honors diploma rate, and percent of students partaking in 3+ credit dual-enrollment courses to check for any correlations.

 

Kayla Reichert - National Teacher Shortage Crisis: The Decline in New Educators, Advisors Dr. Carla Gerberry & Dr. Michael Flick

Is there really a mass exodus of educators in the United States? In recent years, there has been a focus on a national teacher shortage, which indicates that teachers are leaving the profession in masses. There are several complicated factors involved in the shortage, and the following is a study designed to investigate one of those factors: why fewer young people are choosing to pursue a career in education. The smaller number of new teachers is drastically increasing the workload for veteran teachers, causing them to change careers or retire early. A one-page survey was distributed to undergraduate students in different colleges, programs, and classes. The survey was constructed to determine which of the factors cited in related literature were most prevalent to students. Overall, the average starting salary of a public-school teacher is the biggest deterrent for college students at Xavier University, but there were several differences in opinions across age, gender, and program.

 

Jessica Solon - The Group of Multiplicative Functions and Characterization of Prime Numbers, Advisor Dr. Jim Snodgrass

Arithmetic functions arise in number theory but can be studied from an algebraic point of view.  The goal of this presentation is to give some interesting examples of arithmetic functions, demonstrate the operation of convolution and show how the set of arithmetic functions forms a group under this operation.  We will also see that arithmetic functions can be used to determine whether a positive integer is prime.

 

Madeleine Touchette - Correlation Analysis with Missing Observations in Ecological Research Data, Advisor Dr. Max Buot

Scientific realms of academia attempt to answer questions about the natural world. In doing so, researchers set up experiments, collect measurements, and apply statistical methods to their data. In this project, I apply correlation analysis to an ecological data sets obtained from a previous research project. Specifically, for my Environmental Science senior thesis, I investigated how elevated stomatal conductance (i.e., a plant’s ability to rehydrate) mitigated plant temperatures during high stress living conditions. This project aims to mathematically support or debunk this relationship by determining Pearson, Spearman, and Kendall correlation statistics. Since the data set contained missing values, we examined the effect of an imputation algorithm based on Classification and Regression Trees (CART). The statistical analysis yielded richer insights into the environmental factors at play in my plant-water relation study.

 

Geoffrey White - A Naive Search for Perfect Numbers, Advisor Dr. Daniel Otero

A perfect number is one whose factors sum up to two times itself. In this presentation, we will go over the relationship between these numbers, powers of two, and prime numbers. Then, we will go over and witness in action various algorithms that can, in theory, help us find more of these perfect numbers. We will then compare the time complexity of these algorithms, as well as discuss avenues for optimization. Finally, we will go over Carmichael numbers, and discuss how they ruin an otherwise very promising route for finding perfect numbers.

 

Ashton Wine - Regression With Applications to Time Series and the VIX, Advisor Dr. Max Buot

Regression models have been used since as early as 1805 to predict the motion and movement of planets. However, depending on the situation, there are better methods to model data. In this presentation I will be examining regression models as a way to build up to a time series model known as Seasonal-Trend Decomposition using LOWESS (STL), where LOWESS stands for Locally Weighted Scatterplot Smoothing. I will explain the STL algorithm and apply it to the VIX index. VIX, a volatility statistic, provides the expected next 30-day risk neutral volatility of the S&P 500. This presentation contains the necessary background to understand the VIX. Lastly, I will use STL to determine anomalies in the VIX from 1990 to mid 2022.