Atlanta Metropolitan College

Title: Project SMART III (Science and Mathematics are Right Together)

Director(s): John J. Morrell, Natural Science and Math Division

Dates of Program: June 6 - July 15, 2005

Summary: Atlanta Metropolitan College will host a six-week REU on the campus from June 6 to July 15. Project SMART (Science and Mathematics Are Right Together) will be comprised of six freshmen/sophomores formed into three teams of two each. Under the guidance of the PI, each team will research one of the following:

Initial Questions: Area of Investigation I: Modelingâ?¦numeric (computers): Participants are interested in finding the ’bestâ? lineup for a particular major league team. They have the statistics for the team for a year. There has been much public discussion about the qualifications needed for a player to be considered a ’good’ or even ’natural’ candidate for the lead-off or the clean-up position. A selected lineup will be tested on a computer simulation program that will be written by the participants. The participants will find the necessity to move to numeric modeling, simulation, and inductive reasoning in order to develop better lineups. At each stage in the modeling process, a measure of ’goodness of fit’ will be computed by the sum of the differences of the projected scores of the actual lineups used during the season with the actual scores. The result of this investigation will be a computerized best-fit program simulation of the line up. As a result, a most-productive lineup will be produced. During the research, participants will examine existent literature and findings. The simulations created by the students will be compared to the theoretic models as well as to the actual performances of the players and team.

Area of Investigation II: Modelingâ?¦analytic (Markov chains): a second team of participants will investigate a modeling. A standard situation to be considered in an elementary Linear Algebra course is to look at a transition matrix that describes the probability of movement from one state to another and to find an equilibrium state. Markov chains, which include such a stochastic transition matrix and the associated probability vectors have wide applications and have been extensively studied. In Project SMART, the students will investigate a situation where the entries of the matrix are cyclic (with the same period) and not simply constants. As an example, suppose an initial 2 X 2 constant matrix represented a fixed percentage of the population that moved among the states ’ill’ and ’healthy’. Finding the steady state, or equilibrium vector, would represent a standard problem in a course. Suppose, however, a flu season was introduced that affected the movement probabilities as the season progressed each year, peaking at mid-season. In addition, other factors could be introduced such as allowing probabilities to be affected by publicity, the amount of which is determined by the magnitude of category ’ill’. Ramifications of the effect of this type of varying transition matrix on the population probability vector will be examined. In particular, the possibility of developing a ’continuous’ transition function will be examined.

Area of Investigation III: Metrics, norms, distances involving sequences: A third team of participants will investigate and extend the concept of Fibonacci numbers where a general term is formed by the addition of the preceding two terms. In addition to the varied examples in art, architecture, biology, and other areas, the ratio of two succeeding Fibonacci numbers approaches a constant that is variously labeled the ’golden section’, ’golden mean’, ’golden ratio’, etc. and a specific equation can be developed to generate this value as a solution.

The participants will examine sequences where a general term is formed by the sum of the preceding k terms as well as the ratio of succeeding terms for each sequence. The ratios of succeeding terms of these sequences also have limits. The sequence formed by these sequences approaches the sequence { 2^n } and the sequence of the limits of the corresponding ratios, of course, approaches 2. The participants will examine whether the techniques that have been used to study the Fibonacci sequence can be usefully applied to these other sequences and have analogous results mathematically and whether there is analogous geometric interpretations of these sequences corresponding to those of the Fibonacci sequence. The participants will also examine the sequence of such sequences with an examination of various metrics applied to the set of sequences to determine if useful results can be obtained from such a study.

Student Researchers: