Abstract of Projects

Fall, 2006

Courtney Gilmore

“Moving Out: Modeling Population Migration Using Linear Algebra”

Abstract: A population migration model using linear algebra and Markov chains was examined. The model assumes that the total population of the system remains fixed, the migration rate for any two states is constant, and the eigenvectors of the population matrices are linearly independent. Changes were made to the model to incorporate relaxed assumptions. Limitations of the models were determined using proofs and an analysis of United States-Mexico population migration data.

Michael Henninger

“There’s Still Time to Change the Road You’re On”

Abstract: This is a statistical investigation of students changing their majors at Westminster College. Similar studies at other universities examine relationships between major-changing and graduation time. This study looks to determine is students who alter their course of study are more likely to finish there time here in one major versus another as determined by the major in which they began their career.

Amy Perkins

“Of Mice and Mean: A Statistical Analysis of Ephedrine’s Effect on Mouse Respiration”

Abstract: The diet craze in America led to the development, potentially health-hazardous use, and subsequent ban on ephedra-based diet pills, but ephedra has been used as Ma Huang in Traditional Chinese Medicine (TCM) for over 5,000 years to relieve asthma attacks and nasal congestion. Ephedra’s primary chemical alkaloid, ephedrine, can be extracted and combined with anhydrous caffeine to provide weight loss benefits. The proposed experiment investigated whether administration of ephedrine affected the respiration rate in mice. The experimental data was statistically analyzed utilizing analysis of variance and yielded no significant differences in the oxygen consumption among the control and ephedrine treatments on mice.

Allison Rook

“It’s Goal Time: Fitting NCAA Division I Women’s Soccer Data”

Abstract: In this study, the goals and goal times of championship games in the 2003 Division I women’s soccer tournament are fit to both the Poisson and Exponential Distributions. This study is modeled after Professor Chu-Chun-Ling of the University of Singapore in which he fit the data for the 1998 Men’s World Cup of France to both statistical distributions. Furthermore, the data from the NCAA tournament also fits the Poisson Process. However, this doesn’t hold true when consecutive years of data are used together.

Sarah Spardy

“Elliptic Curve Cryptography: Why are we making the switch?”

Abstract: In my research, I have attempted to discover why the National Security Agency (NSA) has decided to endorse Elliptic Curve Cryptography (ECC) rather than RSA, which was the preferred method for many years. In doing so, I will learn more about the mathematical background of these two methods, and further my knowledge about Abstract Algebra (more specifically groups and fields) and modular arithmetic to gain an understanding of how ECC works. Also, I will explore the discrete logarithm problem and its applications in ECC and determine its significance in Cryptography.

Tracy Wolf

“Linear Programming and Game Theory”

Abstract: I am going to show how linear programming can be used to solve game theory problems. I am going to demonstrate this using a common game called Rock, Paper, Scissors and also by using two variations of this game. I am also going to extend this information in order to determine home linear programming can be used to solve the game theory problem called Undercut.

Yenyo, Curtis

“The Dimension of the Coast of Westminster’s Lake Brittain: A Look at Fractal Dimensions”

Abstract: Fractal dimensions of objects, which are non-integer dimensions, differ from the standard Euclidean whole number dimensions. The fractal dimensions can be found in many different ways: The Hausdorff method, compass setting method, and others. So, what kind of a fractal dimension does Lake Brittain of Westminster College have? This is explored through looking at other fractals and fractal dimension methods, and the use of GPS and mapping software.

Fall, 2003

Higby, Alan “Measuring the Difficulty of a Maze”

Currently no metric exists (at least publicly) that measures the difficulty of a maze. To meet this need, a metric was developed. A maze generation program was used to compile statistics on mazes to aid in designing this metric. The metric developed depends on the number of branches off of a maze’s solution path, the number of turns in the solution path, and the percentage of time that a greedy algorithm is successful while traversing the maze. The metric’s validity was defended logically and through a small set of test mazes.

Klink, Heather “SET and All Its Glory”

By altering the original deck of cards for the game SET®, an investigation into the mathematics of the new deck evolves. The cards in the new deck still have four attributes, but now the attributes vary in number of values. The investigation includes the number of cards in the deck, the number of sets given a premise at different times and the cap for the game. The game can be played by finding sets of three cards or sets of four cards. Since the new deck has attributes that vary in number, different combinations for each set must also be considered.

Medjesky, Chris Markov Chains and the SORRY! Board Game

Investigations into board games using Markov Chains have been used in the past to provide information and strategies on how to win. Here the game SORRY! was the subject of investigation, this time focusing on the expected number of moves needed to be taken in order to move from the START position to HOME. Due to the restrictions of the game, a “one-player” scenario was created to allow Markov Chains to be properly applied to the situation. Each card was analyzed to determine the probability of movement from one space to another and these probabilities were placed into a 60X60 matrix. With all other probabilities of movement fixed, the probability of movement using the 10 card was varied in order to determine the minimal number of turns needed to complete the “one” player game.

Pazul, Andy “An Investigation of the Perpetual Calendar Algorithm”

The purpose of my paper was to work with the perpetual calendar algorithm and find out where all of the key numbers came from, and find a handful of different numbers that may work. The majority of my paper does in fact explain and show 10 different sets of numbers, each being their own separate algorithm, that work to give us the day of the week for a specific date in time; past, present, future. I determined the different key numbers by noticing a pattern in the month numbers. From there I worked through the algorithm to determine the century numbers. I found out that the what ever the new month key values differ from the original set of key numbers, that is how many the century numbers differ by, keeping the rules of addition mod7 in mind.

Potocnak, Nikki “The 8 -iamonds: What are they and where do they come from?”

This research centers on the idea of polyiamonds. One must first look at the studies of polyominos and the idea of how to correctly and easily construct a polyomino. This research furthers those studies to cover the topic of polyiamonds and, following closely to the polyomino, the methods one can use to create polyiamonds. The sixty-six 8-iamonds are discovered and analyzed with the method of tiling. Of the sixty-six possible, only five of the 8-iamonds did not tile. Finally, all of the –iamond configurations were taken, from the 4-iamonds to the 8-iamonds, and put in a recursion scheme. The goal of this scheme was to find a pattern that could be followed in order to complete future research of the 9-iamonds and beyond.

Smith, Jessalyn “Investigations of Wavelets with an Application in Heart Rate Analysis”

Wavelets are used to numerically analyze signals that vary in both space and frequency. Wavelets are more versatile than other numerical methods because there are several different bases to choose from. Any desired base may be used to fit any application. All calculations computed are done so in a time efficient manner. This paper investigates two of the more popular wavelet bases, Haar and Daubechies. It also makes connections between wavelet methods and Fourier methods. Concluding the investigation is a practical application, which demonstrates the use of the Daubechies wavelet in heart rate analysis.

Zielinski, Danielle “Cryptography”

This paper examines the uses of cryptology. Different types of encryption methods that have been considered, including the Julius Caesar cipher, the simple substitution cipher, the Vigenère cipher, the jigsaw cipher, and the two-letter cipher. In addition, the public key cryptography RSA, was investigated. Examples of deciphering each of these encryption schemes were included.

Spring, 2003

Caplinger, Joshua, “Activity Networks” Presented at the MAA regional Conference in DuBois, Pennsylvania, April 4. Activity networks are directed graphs used to coordinate a set of activities in some type of project. A path is defined to be a group of activities that establish a connection between the start and finish nodes. The critical path of an activity network is the most important path because a delay in any one of these activities will cause a delay in the overall project. In this study, each activity in a constructed activity network was given is the form of a triangular distribution. The expected values and the variances of each activity were calculated in order to determine which of the paths was the critical path. It was observed, using Tchebysheff’s theorem, that each path could possibly be a critical path.

Culp, Kevin, “Strategy for NCAA College Football Overtime Games” Presented at the MAA regional Conference in DuBois, Pennsylvania, April 4. Results from past NCAA division IA seasons are analyzed and hypothesis testing and regression analysis techniques are used to determine which independent variables are most important to success and which factors of overtime games contribute most to a team’s success. A strategy is determined and tested using other overtime games from past years in NCAA division IAA.

Deah, Emily, “Cutting Polyominos” Presented at the MAA regional Conference in DuBois, Pennsylvania, April 4. Two articles from the American Mathematical Monthly are examined. Iwan Praton’s “Cutting Polyominos into Equal-Area Triangles” discusses labeling, coloring and shrinking of polyominos. Sherman Stein’s “Cutting a Polyomino into Triangles of Equal Area” describes cutting polyominos into equal area triangles. The ideas from these two articles are combined and illustrated.

Klipa, Daniel, “Game Theory” This project was a study of introductory game theory, including von Neumann’s Minimax Theorem and Nash’s Theorem. Two zero-sum games were examined using von Neumann’s theorem, both modeling poker strategy in order to show the profitability of bluffing. It was predicted to be shown that bluffing is indeed a good mixed strategy, which one of the results did show, but the result of the other game was inconclusive. An application of Nash’s Theorem to a theoretical study of the Cuban Missile Crises is also explained.

Plimpton, Sarah, “Getting to the Core of Packing” Presented at the MAA regional Conference in DuBois, Pennsylvania, April 4. This paper is an investigation of packing techniques for unit apples. The paper begins by observing various packing techniques in two dimensions. It finds that hexagonal packing of circles, and alternating packing for trapezoids are the densest known packing strategies for the two shapes. Since sphere packing is not the same as apple packing, the paper only briefly touches on some sphere packing techniques. A few different packing techniques are defined and investigated for the unit apple. The densest packing strategy found is a combination of the hexagonal packing of the circles and the alternating packing or the trapezoids with a maximum packing density of 82%.

Sullivan, Brian, “An Ising Model of a Ferromagnet” Presented at the MAA regional conference in DuBois, Pennsylvania, April 4. Ferromagnetic materials are marked by the tendency of dipoles in the crystal lattice to spontaneously align their magnetic moments parallel to the moments of their neighbors. This behavior is strongly dependent on temperature, and magnetization only occurs below a certain temperature, known as the Curie temperature. The Ising model is a simple numerical simulation the structure and behavior of a ferromagnetic material. The model predicts certain critical exponents which relate various thermodynamic properties of the lattice to the temperature. In this project, an Ising has been coded in MATLAB. Data from the model are analyzed and compared with theory.

Abstract of Projects 2001

Clohessy, Erin.  "A Variation on the Game of Hex." Presented to the mathematics and computer science faculty at the end of the semester meeting.  Hex is a two-player game played on a N x N rhombus shaped board.  This paper examines the four player game played on an octagonal shaped board.  Strategies for winning are examined.  The winner of the 4-player Hex type game is determined in the first few moves of the game.

Kanaan, Simon.  "Dots and Boxes." Presented at the undergraduate research symposium held at Westminster College on April 28.  Strategies for the three player dots and boxes game are developed.  Known strategies for the two person game are converted to the three player game.  A computer simulation of the three person game was created.  A new strategy is developed.

McCaskey, Meredith.  Presented at the MAA regional conference in Altoona, April 6. "Seeding Teams in the Men's 2001 NCAA Basketball Tournament."    Two seeding methods for the NCAA tournament were examined, traditional, in which the best team faces the worst in in first round play and cohort randomized method in which several teams are grouped and then randomly assigned to play against each other.  The mathematics behind the seeding methods were examined.  Application of both methods were applied in a simulation of the tournament using Las Vegas odds.

Orr, Gabrielle.  "Class Scheduling in the Mathematics Department at Westminster College." Presented at the MAA regional conference in Altoona, April 6.  Prerequisites for and constraints on the mathematical courses for math majors were examined.  An integer program was developed to determine alternate course sequencing for the major based on starting semester in the program and courses offered alternate years.  Output from this was used for a second integer program determining teaching schedules for professors.

Quallich, Nicole.  Presented at the undergraduate research symposium held at Westminster College on April 28.  "The 3-player Cooperative Game of Couples."  Coalitions are formed, binding agreements made, and payoffs are determined in cooperative games.  The game of couples considered 2-person coalitions in which each person simultaneously chooses exactly one partner and the partners choose each other.  The normal form and characteristic function form were examined using different pay-off matrices.

Stamp, Aaron.  "Dots-and-Hexagons." Presented to the mathematics and computer science faculty at the end of the semester meeting.  The child's game, dots and boxes is converted to play on a hexagonal grid.  The known strategies for winning on the rectangular board are examined on the hexagonal board.  A new strategy for winning on the hexagonal board is developed.

Stefanis, Lee.  "Cuisenaire Rods and Partitions." Presented to the mathematics and computer science faculty at the end of the semester meeting.  Ordered and unordered partitions of an integer are examined.  The recursive relationship for unordered partitions is demonstrated using Cuisenaire Rods.

Vaccari, Amy.  "Routing of School Buses by Computer." Presented at the MAA regional conference in Altoona, April 6.  Data of current school bus stops was obtained from a local school district.  The three-opt branch exchange procedure was implemented and a traveling salesman tour was found.  The tour was divided into a set of routes feasible with respect ot the capacity of the bus and travel time of the students.  Sensitivity analysis of the bus capacity and fleet size was examined.