Undergraduate Research in Mathematics

Senior Projects 2011

Deborah Schuda - The Axiom of Choice and Its Ramifications, Advisor Dr. Danny Otero

The Axiom of Choice in its general description is not very surprising and, in fact, seems obvious. That is why many mathematicians have used it without any hesitation. It is only within the last century or so that the Axiom of Choice was given its name and given any special attention. While at first glance the axiom seems obvious, some of its ramifications can be quite surprising. The Banach-Tarski Paradox is one such surprise. We will be presenting some statements equivalent to the Axiom of Choice although their intuitive comprehension can vary widely. We will also be presenting a few surprising and not so surprising results that directly hinge on the Axiom of Choice.

Matt Dulle - Modification of the Colley Matrix Ranking Method, Advisor Dr. Max Buot

For the past thirteen seasons of college football, the BCS has been responsible for picking the top two teams to play for the National Championship in Division 1-A college football. Due to heavy criticism, the way the BCS rankings are calculated has been changed over the years. Currently it is a combination of human polls, and computer polls. One of the computer polls is the Colley Matrix. The Colley Matrix is based on the expected value of a beta distribution, and uses a system of linear equations to incorporate a team's strength of schedule. In particular, we solve the matrix equation Ax = b, where the A matrix is determined by the game schedule, and the b vector is based on a team's overall record. In this project, we modify the b vector to incorporate information available from game box-score data.

Kevin Johns - Frieze Groups, Advisor Dr. Dena Morton

In the field of abstract algebra, one of the more widely studied subjects is that of symmetry groups (groups formed by the set of symmetries of an object). Frieze groups are a specific type of symmetry groups created by the symmetries of a frieze pattern, a periodic pattern that exists on an infinitely wide strip of the plane. The goal of this presentation will be to provide an introduction to the seven known frieze patterns and their corresponding symmetry groups and also provide a brief sketch of the proof that only these seven frieze patterns exist

Andrew Kroger - Backward Elimination Multivariate Stochastic Modeling, Advisor Dr. Ganesh Malla

In statistics, stochastic modeling is a widely used technique in a variety of fields. Backward elimination is a particular technique of stochastic modeling that can be used in a multivariate setting. This project looks at the systematic development of the theory and methods behind this modeling, and includes a case study. Properties of multi-collinearity and homoscedasticity have been tested for the model developed. The interactive stochastic model to determine 'the amount of arsenic present in drinking water' by using its 15 covariates will be discussed.

Will Meyer - Lights Out, Advisor Dr. Esmeralda Nastase

The Lights Out game is played on an n x n rectangular grid with an on/off light at each vertex. When a light is toggled, the lights adjacent non-diagonal neighbors and itself changes to its opposite state. An NP-complete problem arises from this game. Given a starting configuration (with m lights in the on state) on an n x n grid, G. By using a sequence of toggle operations, can G be transformed to a grid with at least k switches in the off position. We propose a new algorithm, called Wave Cancellation, which seemingly reduces the search space for this problem to significantly smaller exponentially sized set of states to be examined and leads into more research opportunities.

Patrick Schmidt - The Koch Curve and Its Properties, Advisor Dr. Danny Otero

Reading Helge Von Koch's paper, On a Continuous Curve without Tangents Constructible from Elementary Geometry, made me aware of some glaring abnormalities about the Koch curve. The area underneath the curve has a finite value; however the length of the perimeter is infinite. Also, the curve is continuous everywhere, but has no points which are differentiable. My research has gone in depth to discover how these properties are possible.

Samuel Volpenheim - Chaos, Advisor Dr. Danny Otero

Chaos is a relatively new branch of science that describes irregularities that occur within systems. These systems can be identified in a variety of places including the physical world, population models and economic data. An outline of the historical development of the topic and a discussion focusing on the landmark paper that resulted in the naming of this new science serves as a foundation for understanding its characteristics and relevance.