# Atlanta Metropolitan College

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

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

Email:
jmorrell@atlm.edu

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:

- Zahir Alam
- Dieter Carreon-Heras
- John Mardell
- Maxwell Nyamekye
- James Sibley
- Jaiteh Suwaiboh

Program Contacts:

Bill Hawkins

MAA SUMMA

bhawkins@maa.org

202-319-8473

Michael Pearson

MAA Programs & Services

pearson@maa.org

202-319-8470