Senior Projects 2024

Nate Beard - Triple Systems and t-designs, Advisor Dr. Esmeralda Năstase

In this presentation, we investigate block designs, combinatorial systems made up of a finite set of varieties, V, and a set of nonempty subsets of V making up blocks called the block set, B. There are various types of designs, all having distinct properties. These properties allow us to make generalizations and have better structure to our designs. One such type of design is a triple system, more specifically a Steiner triple system. We will discuss the properties of these designs as well as how to construct a Steiner triple system. Another such design is a t-design which builds upon the structure of other designs. In this presentation, we explore examples of these designs, theorems that are known about them, as well as a graph theory application of a design.

 

Nicholas Ciancio - Stochastic Matrices and Markov Chains, Advisor Dr. Minnie Catral

A stochastic matrix is a matrix for which every entry is a nonnegative number in the closed interval between 0 and 1, and whose row sums are all equal to 1. This project explores the properties of stochastic matrices using the Perron-Frobenius Theorem for nonnegative matrices. As an application, we explore the theory behind finite Markov Chains. To illustrate the theory, we explore a weather probability problem as well as a problem that includes webpage clicking.

 

Valary Leland - Pedagogy of the Quadratic Formula and How to Teach it in Classrooms Today, Advisor Dr. Daniel Otero

The pedagogy of the quadratic formula will be explored dated all the way back to the Babylonian time. From there, we will explore different methods that will solve quadratic problems including geometry, trigonometry, completing the square, midpoint method, and of course algebra. Lastly, the best way to teach quadratics in the classroom will be shown that incorporates these methods as well as developing a new way of thinking about quadratic problems.

 

Taylor Luck - Coalescing Pairs of Cospectral Graphs and their Resulting Properties, Advisor Dr. Eric Bucher

Coalescing is the act of attaching a graph to a specific vertex or set of vertices of another graph. Two graphs coalesce cospectrally if they are cospectral for a matrix and have identifiable vertices upon which another graph can be coalesced whilst preserving their cospectral properties. Consider two graphs, G and H, which are coalescing cospectrally for the adjacency and distance matrices. Now consider another pair of graphs, E and F, that are also coalescing cospectrally to each other for the same matrices listed above. We look deeper into what happens when copies of graph E are coalesced onto graph G and when copies of graph F are coalesced onto graph H. The result of these coalescings are entirely new graphs, ˜G and ˜H. We discuss the properties of these resulting graphs and present examples of specific coalescing pairs.

 

Pascal Mosberger - Principal Component Analysis Theory and its Applications, Advisor Dr. Minnie Catral

This research explores multi-dimensional data and its analysis through the method of principal component analysis. We study statistical tools, linear algebra, and even some computer coding in python. In particular, we look at covariance matrices, spectral decomposition, singular value decomposition and eventually principal component analysis. This talk explores the theory behind representing data categorized by large dimensions in smaller subspaces. Our research finds that principal component analysis is extremely useful in factorizing, approximating, compressing and analyzing data. We conclude by showing that we can reduce variables within data significantly yet keep large amounts of variation.

 

Mac Peloquin and Sonia Vargas - Respiratory Illness Clinical Trial Modeling, Advisor Dr. Hem Raj Joshi

Respiratory illnesses significantly burden the lives of people today, and these infectious diseases claim many lives each year. The COVID-19 pandemic proved that such illnesses can become severe and even alter the way society functions. One significant change due to pandemics is that several large companies released their clinical trial data in the public domain. This helped us to understand the general procedure of clinical trials. To halt the spread and deaths caused by these diseases, efforts to design vaccines for these respiratory illnesses can rapidly begin, and clinical trials are needed to test those vaccines. We studied different phases of clinical trials and developed a mathematical model. Diseases affect both adults and children alike, but the process for approval for the different age groups is not the same. That is why we have developed a model representing two classes: adults and children. Our model could be applied to general respiratory diseases, in the hope that if a future pandemic were to arise, the model could be adjusted to predict the pathogenesis of the disease as well as the clinical trial that would follow. We ran several numerical simulations and present our results.

 

Sheny Perez - Post-quantum Cryptographic Schemes, Advisor Dr. Eric Bucher

With the growing threat of quantum computers, many of the current cryptographic systems will be broken. Thus, there is a need to find quantum-resistant, a.k.a. post-quantum, digital signature schemes. This project focuses on the Unbalanced Oil and Vinegar (UOV) signature scheme and its application to mercurial signatures.

 

Mick Rathbone - Regression Trees and their Sports Applications, Advisor Dr. Max Buot

As statistics become increasingly prevalent in making key decisions for sports organizations, there is a growing need for deeper analytical models to help analysts determine key performance factors for both players and the team. Data regression models and regression trees examine the relevance of specific parameters to an overall model. This project explores how these models and trees are built, as well as how algorithms use tree to create forests and build the best model possible.

 

Andrew Roden - Fundamental Properties of Block Designs, Advisor Dr. Esmeralda Năstase

Block designs are a subtopic of combinatorics. They were originally conceptualized in the 1840’s by Wesley Woolhouse. A block design includes a collection of blocks where each block is a set of varieties. Each design has distinct properties that make them useful. A design can be regular, uniform, and/or balanced. A design can also be represented as a matrix and can be used to model tournaments, lotteries, and statistical problems.

 

Lewis Wentler - Analyzing Stock Market Movements in the Ghana Stock Exchange Using Markov Chain Models , Advisor Dr. Minnie Catral

The goal of this project is to determine which stocks in the Ghana Stock Exchange from January 2017 to December 2020 are the best investments. A Markov Chain Model is designed to find a transition matrix representing the probability of each stock having upward, downward, or no change in price on a weekly basis. From the transition matrix, the steady state vector and mean recurrent times are determined for each of the stocks. The stocks are ranked using the steady state vector and mean recurrent time to determine which would be the best investments.