Activities in the Mathematical Sciences

There are many opportunities for undergraduate research in the mathematical sciences at the University of Louisville.  Below is a sample of possible projects.  This list includes projects offered by

## I. Professor Patricia B. Cerrito

1. An examination of the quality-of-life for patients with melanoma. Data were
collected at baseline, during, and after chemotherapy. It was found that during
treatment, many patients suffered both physical and emotional symptoms. After
treatment, the physical symptoms disappeared but the emotional symptoms
continued. The data were analyzed using traditional statistical techniques but also
data mining techniques of neural network analysis and genetic programming.
2. An examination of the occurrence of exercise-induced asthma in college athletes.
This project is ongoing. Athletes had to complete a screening survey as part of a
mandatory physical. Athletes who indicated some breathing difficulties during
practice were referred to a physician for follow-up and management of the
breathing problems. It was discovered early on that athletes in some sports were
reluctant to respond honestly to the questions because of a perceived stigma
against athletes with asthma. An education program was developed to reduce or
eliminate this stigma as a barrier to treatment. The data will also be compared to
geographic information using data mining techniques to determine the relationship
between asthma symptoms and environmental hazards.

3. An examination of the compliance of patients undergoing physical therapy for
shoulder injuries. All patients are given home exercises to perform regularly during
physical therapy. Each patient will be surveyed weekly to determine the level of
compliance. This will be related to the recovery of their shoulder injuries, and the
ability to avoid surgical repair. Categorical data analysis will be used to examine
the data

## II. Professor Kevin Clancey

Modeling Ohio River Barge Waves
The mathematical theory of shallow water waves dates back to 1849 and the
Scottish Engineer, John Scott Russell. It is known that shallow water waves satisfy a
certain nonlinear differential equation - the so called Korteveg - de Vries equation. It
develops that classical special functions called theta functions also satisfy such
equations. As a consequence, it is natural to try to use theta functions to
simulate wave motions. More specifically, computer software will be used with linear
sums of theta functions to mathematically model water waves behind barges on the
Ohio river. The coefficients in such sums and quasi-periods of theta functions will be
related to river and barge parameters.
Prerequisites: Some experience with mathematical software such as Maple and
elementary differential equations.

## III. Professors Ryan Gill and Kiseop Lee

Objective: To provide students with various tools of statistical data analysis. They
learn how to analyze real data sets and how to write a report in a scientific format.

Description: This project provides various methods of statistical data analysis,
especially with linear models. Students are asked to collect data sets from ‘real
world settings’ such as sports, medicine, the climate, population demographics,
stock markets, etc. They are required to complete a report with the goal of study,
methods of data collection, tools of data analysis, conclusions and discussion.

Plan:
1. Study the following topics :collection of data, sampling survey, t-test, F-test,paired t-test, two sample t-test, simple linear regression, multiple linearregression, regression diagnostics, experimental design, generalized linearmodel, categorical data analysis
2. Learn the statistical software: ex) R, SAS, S-plus, Microsoft Excel]
3. Collect data sets and analyze them.
4. Write a report.
It will be roughly a two semester project. Students do steps 1 and 2 in the first] semester and steps 3 and 4 in the second semester.  Students with a background] in statistics can complete a project as a summer research experience.]

## IV. Professor André Kézdy

Determinants and Rotations      SOLVED! (September, 2003)
Consider an n x n matrix  A=[aij], whose entries are n2 independent variables. For
an arbitrary selection r1,r2,...,rn from {0,...n-1} (repetition allowed), let A[
r1,r2,...,rn]
denote the matrix obtained from A by, for each i=1,...,n, rotating row i a total of ri
cells to the right with wrap around.  For example when n= 4, by definition the matrix
A is

 a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44

while the matrix A[1,2,0,1] is

 a14 a11 a12 a13 a23 a24 a21 a22 a31 a32 a33 a34 a44 a41 a42 a43

Let det(A[r1,r2,...,rn]) be the determinant of the matrix A[r1,r2,...,rn].  Observe that this
determinant is a multilinear function of the variables
aij.  We focus on the collection
of all of the determinants we can obtain from A this way:

S = { det(A[r1,r2,...,rn]) : for all r1,r2,...,rn from {0,...n-1} }.
Naturally S contains nn determinants, since there are n independent choices for
values of each of the
ri's.   Next consider the special subset of n! of these
determinants:

T  = {
det(A[r1,r2,...,rn]) : for all r1,r2,...,rn, where 0 ri n - i}.
OPEN PROBLEM: Prove that every element in S can be written as an integer
linear combination of the elements in T.

`National Center for Disease Control and Prevention has compiled on its web site the extensive national mortality tables by different mortality causes. The data is  organized by the county of residence. Preliminary analysis  indicated that Jefferson county Kentucky occupies (unfortunately) a very prominent place in these statistics with respect to several different causes of death including suicide auto-accidents and various cancers (most notably brain and lung). This project would be concerned with developing a precise mathematical model of the geographical mortality pattern for the parts of the states of Kentucky and Indiana neighboring Jefferson county which could possibly provide an answer to the question whether indeed the residents of Jeff county are at a higher risk of death for cancer and auto-accident, comparing to their neighbors.  The model would also compare Jeff county to similar  socio-economic areas in the different parts of the US.  Most of the data is available on the internet. Methodological papers, textbooks and other material on geo-spatial mathematical  models in epidemiology are available from the library and from myself. gr `