Undergraduate Research in Mathematics

Senior Projects 2013

Tiara Anderson - What's in a Sound? Advisor Fr. Joseph Wagner, S.J.

Every sound wave can be understood as a composition of sine waves, each having different amplitudes, frequencies and phases. This phenomenon allows various instruments to produce a unique sound that distinguishes them from other instruments. Fourier analysis is used to mathematically describe periodic, piecewise continuous functions as the sum of a series whose terms represent the components that contribute to an overall sound. We will demonstrate Fourier analysis through various examples and show how it applies to the general wave equation for strings.

Alexander Antonelli - Algebraic Coding Theory Yesterday, Today, and Tomorrow, Advisor Dr. Dena Morton

Living in the technology driven society of today, I began to wonder what mathematics takes place behind the scenes, acting as the catalyst for the speed and security we are ever-seeking. What were we taking for granted? In this paper I explore different types of codes used in Algebraic Coding Theory. I begin with a brief overview of Coding theory, along with an overview of codes in general. I discuss the characteristics of a code that make that code desirable. The main focus of the paper is on error-detecting codes, specifically the International Standard Book Number (ISBN) code, and what makes this code so adequate and practical. Error-correcting codes, in short, are discussed as well. Finally, I address some effects Algebraic Coding Theory has had on the past and the present, and question what will be in store for the future.

Jillian Dolciato - Guaranteed to Win: Combinatorial Game Theory and the Game of Nim, Advisor Dr. Dena Morton

This paper will explore how Combinatorial Game Theory can be used to determine the winners and their strategies of multiple games. Through combinatorial game theory, we will be able to assign values to a game which will identify the winner. Throughout this paper we will look at the mathematical games like Subtraction, Hackenbush, and Domineering. These examples, along with new definitions and theorems, will lead to the final game, Nim. Lastly, this paper will uncover the winning strategy for any game of regular Nim.

Brandon Freeman - Incompleteness Theorem, Advisor Dr. Danny Otero

Do you think that mankind will ever reach the end of the universe? If so, how will we know if we get to the end? Will there be a sign saying you have reached the end of the universe, now turn around there is nothing else to know? Is it possible to even know everything about the universe? How about a very basic topic like arithmetic; you know, the thing everyone has been doing since second grade. Is it possible to know everything about arithmetic? Kurt GÖdel came up with a proof in 1931 that guaranteed that we cannot know everything in the theory of arithmetic. He worked through the theory of arithmetic and was able to find an undecidable statement that was true if and only if it was unprovable. Since this statement is undecidable it cannot be proven true or false, so the provability of the statement is unknown. This paper goes through the way that GÖdel came to his conclusion of his Incompleteness Theorem.

Brianne Lehnig - Actuarial Modeling, Advisor Dr. Max Buot

Modeling events is not only a very important aspect of Actuarial Science, but is used in economics, engineering, insurance, etc. There are various ways to model these events and most often they are too complicated to work by hand and must be modeled with some type of program. In this project WinBUGS and other add-in programs in the statistics program R are used to model and simulate the given data. This project explores how Bayesian Statistics, Markov Chain Monte Carlo and Gibbs sampling can be used to model actuarial data.

Andrew Leyden - Sabermetrics: The Search for Objective Knowledge about Baseball, Advisor Dr. Ganesh Malla

Baseball, America's pastime, has progressed into a generation where data reigns supreme. We will discover how these two are intertwined through an overview of Sabermetrics, the search for objective knowledge about America's oldest professional sport. We will do this by analyzing various statistic-based tools that are implemented in front offices around the MLB today. Lastly, we will explore where the relationship between the future of professional sports and data is headed.

Maeve Maloney - How Social Segregation Affects Undergraduate Educational Outcomes: An Application of the Perron-Frobenius Theory of Nonnegative Matrices, Advisor Dr. Minnie Catral

This research project examines the effect of social segregation on the educational outcomes of undergraduates. This research has two main parts. The first uses the Perron-Frobenius Theory of Nonnegative Matrices to fully understand the mechanism through which the Segregation Index designed by Echenique and Fryer [Fryer R, Echenique F. A Measure of Segregation Based on Social Interactions. Quarterly Journal of Economics, 2007] guarantees the axioms of monotonicity, homogeneity, and linearity. The second applies the Index to a set of original data. The Index provides the relative level of segregation of each sample member. These results will be regressed against the students' GPA, as a measurement of educational outcomes. The results of the regression are still in the process of being analyzed. If conclusive the project will demonstrate empirical evidence of the effect of social segregation on educational outcomes.

John McClellan - An Exploration of Goldbach's Conjecture in the Polynomial Field, Advisor Dr. Danny Otero

The Goldbach Conjecture has fascinated mathematicians for the past 3 centuries, eluding the best efforts for an acceptable proof. We will outline the conjecture's history, and the progress through a possible proof. Finally, we shall introduce (and prove) a theorem that could be considered a "Goldbach Theorem" for the polynomial field, and speak to the consequences of such a proof.

Jack Speth - A Survey of Graph Pebbling, Advisor Dr. Esmerald Nastase

A graph is a mathematical object that contains a set of vertices and edges, such that each edge connects any two vertices. With a graph we place some number of pebbles on the vertices of a graph. A pebbling move on a graph consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. In this paper we will explore traditional pebbling and the cover pebbling number of various graphs. In the traditional pebbling problem we try to reach a specified vertex of the graph through a sequence of pebbling moves. The cover pebbling number of a graph is the minimum number of pebbles such that no matter how we place them initially on the vertices we can end with at least a pebble on every vertex simultaneously. We find the pebbling and cover pebbling numbers of paths, complete graphs, trees, and some other graphs. We also consider a more general problem where we require a certain number of pebbles for specified vertices.

Michael Zelik - Fractals: Investigating Nature's Beauty, Advisor Dr. Bernd Rossa

This project is an investigation into fractal geometry. We will discuss what fractals are and how certain fractals are created. Specifically, we will investigate the fractal fern and the fractal tree. Resembling a real fern and tree found in nature, these objects are determined by a handful of numbers. Using a family of affine transformations and what is called an Iterated Function System, these natural objects can be created on programs such as Maple. We will take a look at these numbers and transformations and make sense of them. Ultimately, we will develop a method of deriving the transformations by hand. This will give us the ability to create fractals purely on our own.