Monday, May 14, 2012
Six mathematics majors presented their capstone research at poster and oral presentation sessions that were recently held at Westminster College.
Poster Presentation Session: Monday, April 30, at 4:30 p.m., Witherspoon Maple Room.
Student participants: Emily Dolsak, Coty Hainsey, Richard Ligo, Robert Rhodes, and Chelsea Schrock
Oral Presentation Session: Thursday, May 10, 2010, 2:00, Hoyt 166.
Mathematics major Chelsea Wesp participated in this event.
Mathematics major Rachel Legg also completed her capstone research but was unable to present at either of the above sessions.
The following is a summary of the capstone students' titles, faculty sponsors, and abstracts:
Faculty Sponsor: Dr. Carolyn Cuff
Title: A Statistical Analysis of the Free Throw
Abstract: In this talk, we discuss the statistical analysis of free throws shot by a NCAA Division III women's basketball team in order to determine the variables that contribute to a successful free throw. Analysis was performed on both the team as a whole, and each individual shooter. Results indicate that knee bend, follow through, and time (a proxy for speed) contribute to a make in team analysis with very small explanation of variance. However, a combination of these and others appear to contribute in shooter analysis with higher explained variance.
Faculty Sponsor: Dr. David Offner
Title: A Look at Benford's Law
Abstract: Benford's Law governs the first digit of many sets and sequences of numbers we encounter all around us. Examples include heights of buildings, areas of countries, populations, and many other natural occurrences. Surprisingly, instead of each digit having the same probability of occurring as a first digit, lower numbers have a much greater chance than higher numbers. Benford's Law predicts what the probabilities are for each digit, d, from 1 to 9. The probability that d is the first digit of a number is . Since Frank Benford published this phenomenon in 1938, many mathematicians have studied Benford's Law to try to gain a better understanding of why this occurs and where it can be applied. One piece of my research examines what data sets do and do not follow Benford's Law and justifying why they do follow this law. I will present some well-known data sets that grow exponentially and do follow Benford's Law and show that they are scale-invariant and base-invariant under Benford's Law. I also investigated sequences of the form . These sequences do not follow Benford's Law, but they do follow a different law and have a surprising connection to Benford's Law. As p grows, the probabilities for the first digits of approach the probabilities for Benford's Law. Another research direction is an application to fraud detection. Benford's Law is not regarded as proof of fraud, but it can be a helpful indicator that there is something unusual going on. Benford's Law is already used for certain types of accounting and insurance fraud. Tax returns also follow Benford's Law. One controversial topic is the use of Benford's Law in election fraud. I will present data that suggests Benford's Law may also be useful in detecting election fraud.
Faculty Sponsor: Dr. Natacha Fontes-Merz
Title: Can You Think Out of The Box?
Abstract: Think-Tac-Toe is a single player game, similar to Minesweeper, in which the player uses given numberic clues to determine the location of X's and O's on a board. Although the game is normally played on a square board, in this talk we have adapted it to a triangular board. Our goal is to determine which boards have unique solutions when given a set of clues.
Faculty Sponsor: Dr. Jeffrey Boerner
Title: Diagrammatic Representations for a Specific Class of Knots
Abstract: Every knot can be represented diagrammatically. One method for representing knot is through a grid diagram. Every knot has some corresponding grid diagram, and every grid diagram represents exactly one knot. We define a special type of grid diagram called a diagonal diagram. From this definition, we prove that specific classes of diagonal diagrams represent the unknot, a torus knot, or multiple linked unknots. We also determine specific characteristics about the knot represented by a diagonal diagram.
Faculty Sponsors: Dr. David Offner and Dr. Jeffrey Boerner
Title: A Markov Chain Model for Predicting Wins in Baseball
Abstract: For over a century, baseball statistics such as batting average have been used to give players, owners, and fans the means to compare players and measure player production. In the past few decades, the growing popularity of sabermetrics (the study of baseball statistics) has given rise to new statistics that give people a better base of comparison and player production. One of the most famous sabermetritians, Bill James, discovered a formula that has since been refined, known as the Pythagorean expectation, in the 1980's. James' formula can be used to calculate a team's expected win percentage based on a team's number of runs scored and runs against. This calculation gives us a team's ``first order wins." A team's first order win percentage can be a better prediction of future win percentage then the teams actual win percentage, and it can also be used to assess whether a team is overachieving or underachieving. There have been numerous run estimators developed by sabermetritians. These run estimators are formulas that estimate a team's number of runs scored based on various offensive stats. When we use these results in the Pythagorean expectation, we calculate a teams` second order wins." This is the number of wins a team deserves based on that teams offensive production. This can be an even better predictor of future win percentage than first order wins. In my research I created a Markov chain simulator that estimates runs. The Markov chain model is very useful since a Markov chain models a process that advances from one state to another. In baseball, the number and location of runners on the bases and the number of outs provide us with the different states. When a batter steps up to the plate, he gets a hit or walks to first a certain percentage of the time. The simulator takes a team's hitting statistics and uses random number generators to simulate a batter's outcome and the runner's advancement. Thus, the simulator, with a given set of offensive statistics, can simulate many games or seasons to give us an estimate of the number of runs a team should score on average. Using this, we can calculate a second order win percentage for all teams in the Major League. Using data for the 2011 season, we will compare our results with other methods of calculating second order wins. In addition to calculating second order wins, there are a number of questions we can answer using simulation. For example, which type of hitter is more productive, a power hitter or a hitter that consistently gets on base? Finally, were the St. Louis Cardinals lucky to have beaten the Texas Rangers in the 2011 World Series?
Faculty Sponsor: Dr. David Offner
Title: How To Hang With the High Rollers
Abstract: My research focused on using the Monte Carlo Simulation to analyze games. I recreated a betting system made by Steven Skiena for the game Jai Alai using Java programming. I recreated a betting system made by Steven Skiena for the game Jai Alai using Java programming. This was done by using the Monte Carlo Simulation and the results closely resembled that of Steven Skiena in which the first two players of Jai Alai matches have the highest probability of winning. Once the creation of this betting system was understood, I made my own system for the game Knockout. The results confirmed that the last player in any Knockout game would have the highest probability of winning.
Faculty Sponsor: Mr. James Anthony
Title: Determining the Winning Player's Strategy in any Game of Juniper Green
Abstract: Primarily used in an educational setting, the game Juniper Green teaches children about multiplication and division. The idea of the game is simple but it does not have an obvious winning strategy. The purpose of this project was to find a method for determining the winning player's strategy in all possible games and to see if player one always has a winning strategy. A single method could not be found, but multiple methods for determining a players winning strategy were found. One method, The Three Medium Prime Strategy, can be used to find player one's winning strategy for an infinite number of games. The second method, The Two Medium Prime Strategy, can be used to find player one's winning strategy in a specific number of games. The last method, Using a Reduced Graph and Matching, allows us to find player one or player two's winning strategy when the first two strategies do not apply. Our results have shown that player one has a winning strategy for Juniper Green in all but a finite number of cases.