National Research Experience for Undergraduates Program
at St. Mary's College of Maryland
Bugs, Curves and Games: Mathematics and Applications
May 30 - July 7, 2006
| The St. Mary's College of Maryland Department of Mathematics and Computer Science gratefully acknowledges the support of the Mathematical Association of America, the National Security Agency, the National Science Foundation, and the Moody's Foundation. | |
Please address all inquiries to Dr. Katherine Socha,
ksocha smcm.edu,
(240) 895-4353.
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| Project Director | Dr. Katherine Socha
(ksochasmcm.edu)
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| Schedule | See the Scheduled Activities |
| Student Participants | Michael Firrisa Helen King Heather Langdon Esrael Seyum Weitzel Scholar Dwayne Moxey |
| Faculty Mentors | Dr. Matthew Burke
(mmburkesmcm.edu) Dr. David Kung (dtkung smcm.edu)Dr. Katherine Socha |
| Seminar Leaders | Dr. Susan Goldstine
(sgoldstinesmcm.edu) Dr. Katherine Socha |
Research Projects | 1. The well-studied hanging chain problem has a solution that happens to be the curve traced by the focus of a parabola as the parabola is rolled along a straight line; the curve is a catenary. Another well-studied problem is to design a road along which a regular polygonal wheel can travel smoothly---the solution is also a series of catenaries. In this research project, the problem of approximating a catenary by rolling a polygonal approximation to a parabola along a straight line will be studied. The project will be guided by reading "A New Minimization Proof for the Brachistochrone," G. Lawlor, American Mathematical Monthly, 103 (1996), 242-249. The question to be studied is, given fixed endpoints of a polygonal approximation to a parabola (3 point approximation, 4 point approximation, and so on), where should the intermediate points be placed in order to generate the best approximation to the catenary formed by the full parabola? |
| 2. The game of Nim is played with one or more piles of beads. On a player's turn, she is allowed to choose one pile and remove as many beads from that pile as she likes. The last player able to remove beads is the winner. A winning strategy for Nim was discovered by Bouton in 1906. In the mid-1930s, Sprague and Grundy independently showed that the winning strategy for any impartial combinatorial game is related to Nim's strategy. It is not always a simple matter, however, to determine the relationship for some particular game. For this project, the student will explore a variant of Nim known as Hub-and-Spoke Nim. In this variant one pile is the hub and the others are arranged in rows radiating from the hub. A player may only remove beads from a pile that is at the end of a spoke. There is a similar variant known as Burning-the-Candle-at-Both-Ends which was solved by Albert and Knowakoski. Their work should provide guidance for attacking this problem. | |
| 3. The simplest of differential equations can be used to model the population growth of an infectious agent. More complicated models include influences such as limited resources, competition from other infectious agents, and disease resistance in recovered organisms. Following an introduction to differential equations and mathematical modeling, the students will study Smith and Moore's module "The SIR Model for Spread of Disease" in the Journal of Online Mathematics and its Applications. Computer modeling as well as theoretical work will complement each other. If time warrants, this work will be applied to a possible avian flu outbreak. | |