Undergraduate Research in Mathematics

Senior Projects 2014

Josh Brackman - The Mathematical Study of American Roulette, Advisor Dr. Bernd Rossa

Roulette is one of the most popular casino games in America; most players believe they have a good chance at winning money by playing this game. This project will discuss the basic fundamentals of American Roulette, as well as show how it is nearly impossible for a player to win at this game in the long run. The different types of bets that a player can make, and certain betting strategies that many players have come up with in order to "guarantee" that they will win will be considered.

Robby Cagle - Continuous Transformations in Music and their Discrete Applications, Advisor Fr. Joseph Wagner, S.J.

Finnish composer Kaija Saariaho wrote of her atonal composition, Vers le blanc, "The harmony is a continuous stream and cannot be heard as a series of changing chords. One only notices from time to time that the harmonic situation has changed." Music theorist and mathematician Clifton Callender explored these harmonic situations in his paper, "Continuous Transformations." This talk includes a discussion of Callender's mathematical model for Vers le blanc and some interesting compositional applications I have derived from the ideas that Callender presented.

Sarah Cook - Space-Filling Curves, Advisor Dr. Dena Morton

This project has investigated space-filling curves, which are curves that are mapped in such a way as to pass through every point of a two-dimensional space. In other words, space-filling curves are a counter-intuitive method of taking a line and making it fill in a square with mathematical precision. Research focused mainly on the construction and applications of the Hilbert Curve, examining the creation of it and the special properties that come about due to its construction.

Sean Gravelle - Graph Theoretic Models of Interdependent Preferences in Referendum Elections, Advisor Dr. Esmeralda Nastase

In referendum elections, voters are frequently required to cast votes simultaneously on multiple questions or proposals. The separability problem occurs when a voter's preferences on the outcome of one or more proposals depend on the predicted outcomes of other proposals, often leading to unsatisfactory or even paradoxical election outcomes. We will discuss a new framework for modeling voter preferences using graph theory, which can help characterize and construct interdependent voter preferences. We apply this model to the complete bipartite graph and explore connections to prior research and open problems in voting theory.

Erin Gulick - Sabermetrics and the Major League Baseball Hall of Fame, Advisors Dr. Bernd Rossa and Dr. Richard Pulskamp

Induction into the Major League Baseball Hall of Fame is extremely subjective. In an effort to make this process less so, both traditional baseball statistics and sabermetrics were studied. Using logistic regression to analyze both members' and nonmembers' statistics, a sabermetric was constructed that predicts whether or not a player will be inducted into the Hall of Fame with 92% certainty. This still leaves some room for the intangibles (integrity, sportsmanship, character, etc.), but standardizes some of the statistical requirements.

Nate Han - Characteristics of Oncolytic Viruses Influencing Effectiveness as a Tumor Therapy, Advisor Dr. Hem Raj Joshi

Cancer is a very prominent disease in today's society with odds as high as one out of every seven people dying from the disease. Billions of dollars have been poured into research for a cure for cancer, but consistently effective therapies have yet to be obtained. One promising therapy points towards oncolytic viruses which specifically target tumorigenic cells via proteins expressed on the outside of infected cells. In this paper, we study a system of differential equations and use MATLAB to simulate results. We determine the optimal strategy to eliminate tumors using a variety of parameters (e.g. efficacy of viral infection, cytopathicity, viral growth, cancer growth) and find that weakly cytopathic replicating viruses and non replicating strongly cytopathic viruses are suitable choices to administer as effective anti-cancer treatments either as individual therapies or in conjunction with other therapies.

Tom Kelly - Credibility Theory in Insurance Practices, Advisor Dr. Max Buot

A fundamental aspect of life is that risk is almost always associated with it. Be it working every day or driving across town, the threat of losing money as a result of an accident is always present. Individuals buy insurance to offset this risk. Insurance companies, in essence, assume this risk from individuals. They collect premiums from their policyholders; to determine the amount each policyholder pays, credibility theory is utilized. Credibility theory uses statistical methods to create a premium blending individual and collective risk profiles. Credibility is a tool frequently utilized by many actuaries to effectively price insurance products.

Robbie Knetsche - Quantum Cryptography, Advisors Dr. Dena Morton, Dr. Heidrun Schmitzer, Dr. Gary Lewandowski, Mr. Dennis Tierney, and Dr. Steven Herbert

Quantum cryptography makes use of quantum mechanics in order to better protect sensitive data.While the field is still mostly at a theoretical level, there is still proof of concept work that can be done using our current technology. This talk explores the field of quantum cryptography in general, and demonstrates how to find the factors of very large numbers quickly by making use of Gaussian sums.

Dominic Masotti - Banach-Tarski Theorem, Advisor Dr. Danny Otero

This project studied the paradoxical Banach-Tarski Theorem which states that, given two bounded finite sets in space, one can partition one of them into a finite number of pieces and reassemble these to form the other set. The Banach-Tarski Theorem is often stated in the form: it is possible to partition a sphere into finitely many pieces which can be reassembled to form two spheres of the same size and volume of the original sphere. A proof involves working in the group of all possible rotations of the sphere, and sorts all such transformations into three distinct subsets G1 , G2 , and G3 , such that G1 = G2 = G3 = G1 U G2 (where = denotes congruence). This congruence result helps us partition the sphere into necessary subsets which we move through space to cover two distinct spheres.

Mitch McCord - Adaptive Designs in Clinical Trials, Advisor Dr. Max Buot

In medicine once clinical trials are set in motion, they are usually left untouched until they are finished. Adaptive designs allow the researcher to change the design of an ongoing study based on the results of the trial. If one treatment seems more effective, then a greater proportion of patients can be funneled into it. Using the randomized play the winner rule, we determine how many patients should be placed in each treatment in any clinical trial.

Vu Nguyen - Estimation and Modeling of the Hospital Charge Rate for Patients with Alcohol and Drug Abuse in the United States, Advisor Dr. Ganesh Malla

Data for our study came from a data set of about 800 million observations and 300 variables from the Healthcare Cost and Utilization Project (HCUP) - Nationwide Inpatient Sample (NIS), year 2011. Using various statistical techniques, we determine that the hospital regions, DRGs and admission type are statistically significant variables, while race, sex, hospital location, hospital teaching status, hospitals ownership, and median households income of the patients statistically insignificant for the Hospital Charge Rate determination. We employed the backward elimination regression method to model the Hospital Charge Rate for the same data. All significant variables (categorical and quantitative) of the demographics of the patients and various degrees of interactions of the quantitative variables are used to design interactive multivariate regression models.

Michael Reis - Transformation Geometry and Symmetry Groups, Advisor Dr. Minnie Catral

Transformation geometry is a modern approach to Euclidean geometry that beautifully brings together geometry and abstract algebra. This talk gives an introduction to transformation geometry and establishes classification theorems of symmetries in the plane. In particular, the ornamental groups (the rosette, frieze, and wallpaper groups) are discussed and applications of the frieze and wallpaper groups are presented.

Mackenzie Wall - A Survey of Projective Geometry, Advisor Dr. Danny Otero

We journey through the development of projective geometry through the lens of history. This unique topic arose as mathematicians developed theories from the ideas that artists came up with when faced with drawing in perspective. We will look at examples of projective space by giving Euclidean space projective definitions. The mathematics behind the projection of lines will set the stage for linear fractional functions to be introduced. I will show how the field axioms established for projective geometry rely heavily upon specific theorems. Proofs of these axioms display the ability to do arithmetic purely geometrically.