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Westminster College Mathematics Majors Presented Capstone Projects

Wednesday, May 15, 2013

NEW WILMINGTON, Pa. - Seven Westminster College senior mathematics majors presented their capstone research posters April 29 in the Carlson Atrium of the McKelvey Campus Center.

Bethany Ekimoff presented "Checkerboard Surfaces on Knots," advised by Dr. Jeffrey Boerner, assistant professor of mathematics.

Ekimoff studied properties of knots and their orientability. Specifically focusing on whether a checkerboard surface of a knot is orientable or nonorientable in a minimal diagram. She examined the twist knot family and identified patterns in the use of a Cromwell translation to change a projection's orientability. She defined the o-excess and n-excess of a knot to describe the minimal number of crossings in an orientable and nonorientable projection. Using this property she concluded that some knots do not have projections with a minimal number of crossings an orientable checkerboard surface and that for every knot has a finite n-excess.

Ekimoff is a daughter of David Ekimoff of Webster, N.Y., and a graduate of Webster Schroeder High School.

Christina Erceg presented "Backgammon End Game Doubling Strategies in Money Play and Match Play," advised by Dr. David Offner, assistant professor of mathematics. 

Erceg studied end game situations and doubling strategy in Backgammon. Doubling is a key strategy in the game since it determines the worth of the game and is a key factor for determining a winner. The paper began with an overview of how to play Backgammon. Next, she determined the expected values for when to double and accept a double in money play, based on what point in time she was in the game. Then, she looked at doubling from a match play perspective. Overall, she compared doubling decisions for end game situations in money play to match play.

Erceg is a daughter of Eli and Mary Jo Erceg of Hermitage and a graduate of Hickory High School.

Lisa Kaylor presented "Matrices over Z (mod p) with Eigenvalues in the Same Field," advised by Offner.

Kaylor counted the number of 3x3 and 4x4 matrices with entries in Zp whose eigenvalues also lie in Zp. After providing a count of the invertible matrices with entries in Zp, we present a condensed version of the solution found in [6]. These ideas are then extended to both the 3 and 4 dimensional case. With the resulting counts from each case, we conjecture and prove a theorem stating that the number of matrices sharing this property for some n x n matrix has a limiting value if 1/n! as the prime p gets large.

Kaylor is a daughter of Karen Kaylor of Chicora and the late Terry Kaylor and a graduate of Karns City Area High School.

Andrew Kieffer presented "Using a Markov Chain Model to Compare the 2012 MLB Playoff System to the 2011 System," advised by Boerner and Offner. 

Kiefer looked at the playoff system in Major League Baseball (MLB). The current form in the 2012 playoffs has an extra wild card team in each league. He wanted to investigate how much that may change the outcome of the playoffs, i.e., what teams, if any, will benefit from, or get hurt by, this new format. To do this, he updated and added to a simulator that uses a Markov chain model to simulate baseball games, and then simulated the playoff scenarios. He compared the results from the new format to the possible outcomes if the same teams would play the playoffs using the previous system with only one wild card team in each league. When analyzing the simulator, he wanted to be able to check to see, along with how the playoff system affects who wins, which pitchers are the best to have in a starting rotation.

Kieffer, who is also majoring in economics, is a son of David and JoEllen Kieffer and a graduate of Boardman High School.

Domenic Neely presented "Quarantine for an Infectious Disease," advised by Dr. Carolyn Cuff, professor of mathematics.

Neely considered adapting the classic SIR model of an epidemic to include quarantines of the Susceptibles and/or the Infected. The model developed can be used to predict the best time to impose the quarantine. This will help to determine how to make an epidemic more manageable which will result in lower death rates.

Neely is a son of Yvonne Neely of Prospect and a graduate of Slippery Rock Area High School.

Daniel Rutkowski presented "An Exploration in Cover Pebbling," advised by James Anthony, visiting mathematics lecturer, and Dr. Natacha Fontes-Merz, associate professor of mathematics.

Graph pebbling is a recent development in graph theory in which pebbles are placed on the vertices of a graph G. Pebbles are moved from vertex to vertex by a process known as a pebbling move. Rutkowski explored a variation of graph pebbling known as cover pebbling. Specifically, he gave formulas for the cover pebbling numbers of certain families of graphs, investigated line graphs and their cover pebbling numbers, and explored cover pebbling and its connection to the maximum eccentricity of a graph.

Rutkowski is a son of Daniel and Lori Rutkowski of Beaver and a graduate of Beaver Area High School.

Morgan Swartz presented "Statistical Analysis of an Analytical Chemistry Experiment: Applications of Partial Least Squares Regression to Biodiesel Blends and Frackwater Samples," advised by Cuff.

Swartz is a daughter of Sheri Swartz and Kevin Swartz, Hubbard, Ohio, and a graduate of West Middlesex Area High School.

Swartz used a linear regression statistical model to analyze data collected from an analytical chemistry experiment using gas chromatography mass spectrometry and chemometrics to predict biodiesel blend percent com-position. This experiment will serve as a reference for a similar experiment to be conducted in the future dealing with Marcellus Shale fracturing, or fracking.

At Westminster, the final component of liberal studies is a senior study, or capstone, course.  The capstone is a four-semester-hour (minimum) course within the major designed to provide an opportunity for students to evaluate and assess the strengths and limitations of their major field.

The capstone experience permits opportunity for structured reflection on the value of education in and beyond the major and provides an opportunity to strengthen communication and problem-solving skills.

Contact Offner at (724) 946-7293 or email for additional information. 

About Westminster College...
Founded in 1852 and related to the Presbyterian Church (U.S.A.), Westminster College ranks first in the nation as "Best College for Women in Science, Technology, Engineering and Math," according to Westminster is a top-tier liberal arts college and a national leader in graduation rate performance, according to U.S. News Best Colleges guide. Westminster is also honored as one of "The Best 377 Colleges" by The Princeton Review, and is named to the President's Honor Roll for excellence in service learning.

Nearly 1,600 undergraduate and graduate students benefit from individualized attention from dedicated faculty while choosing from 41 majors and nearly 100 organizations on the New Wilmington, Pa., campus. Visit to view "Advantage: Westminster" A Strategic Plan 2010-2020.

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Bethany Ekimoff


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Christina Erceg


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Lisa Kaylor


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Andrew Kieffer


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Daniel Rutkowski


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Morgan Swartz