Atlanta Metropolitan College

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

Director(s): John Morrell

Dates of Program: June 4 - July 14, 2007

Summary: Atlanta Metropolitan College will host a six-week REU on the campus from June 4 to July 14. 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: A team of participants will investigate a modeling problem that is amenable to analytic techniques. 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'. In particular, the possibility of developing a 'continuous' transition function will be examined. Participants will introduce a pattern to determine the intermediate transition matrices to be introduced between two extremes when the transitions are to be evaluated at evenly spaced time intervals. The Project SMART 2007 cohort will investigate whether an algorithm for computing the equilibrium states for such approximations to a continuously varying transition model will produce a 'bounded' situation. Ramifications of the effect of this type of variable transition matrix on the population probability vector will be examined.

Area of Investigation II: A second 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 such sequence. The ratios of succeeding terms of these sequences also have limits. These sequences approaches the sequence { 2^n } and the sequence of the limits of the corresponding ratios 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.

Area of Investigation III: A third team will investigate developing an algorithm that will, given a finite set of points in the plane {x_i: i= 1.n}, divide the plane into n regions {R_i} each of which will correspond to a point and whose characteristic is that all the points in region R_i lie closer to x_i than to any other given point. They will also examine the dual problem of determining, given a planar partition, the conditions that allow for the construction of such a set of points as well as a method to find such a set.

Student Researchers Supported by MAA: