
Summer
2003 Projects
Tumor Angiogenesis
Laura Lutz is a student at the University of Chicago
Erica Naves is a student at Florida A& M University,
Professor
Howard Levine is a Distinguished Professor and is a potential
mentor for 2004.
The students began by learning some cell biology and organic chemistry.
They modeled the onset of tumor angiogenesis with a system of partial differential
equations. The model focuses on the breakdown of fibronectin and movement
of endothelial cells along the capillary wall as critical steps in the
formation of new capillaries. It uses the principles of chemical
kinetics and random walks to describe those processes.
A highly vascularized tumor (taken from http://www.maths.dundee.ac.uk/~sanderso/tumour.htm)
The model appears to be more accurate than other models in certain
respects. Laura and Erica's paper was entitled A Rigorous Mathematical
Model for Tumor Angiogenesis and the Inhibition of Angiogenesis and is
being revised for submission. There are many aspects of this problem
for future REU students to study.
Elizabeth Blankenship is a student at Iowa State University
Jonathan Gandrud is a student at Iowa State University
Professor Dan Ashlock is
a member of the Bioinformatics program at ISU and is a potential
mentor for 2004.
This project involved computer simulation of evolution. Evolutionary
algorithms are computerbased problem solving systems that emulate features
of biological evolution to produce a sort of software evolution.
The students worked with virtual robots that build and paint (see
sample), and cannibal robots. The latter dealt with a population
of 200 virtual autonomous agents (kanibots) that live on a simple grid;
they evolved to perform the task of eating each other. The experiments
are related to the wellknown prisoner's dilemma, which has also been studied
by computer simulation. Their study serves as additional evidence
for a form of general adaptation in evolutionary computation systems using
an agentvsagent competitive fitness function. The students wrote
individual final REU papers (Elizabeth's was Experimentation with Evolutionary
Robots and Building Structures and Jonathan's was A Note on
General Adaptation in Populations of Cannibal Robots). They also
collaborated with Prof. Ashlock to produce a paper A
Note on General Adaptation in Populations of Painting Robots, which
has been accepted for presentation at the 2003 Congress on Evolutionary
Computation to be held in Australia and will appear in the proceedings
of the conference.
Nonnegative PMatrix Completion
Problem
Job Evers is a student at Massachusetts Institute of Technology
Steve Shaner is now (fall 2004) a graduate student at Iowa State University
Professor Leslie
Hogben is the leader of the ISU
Combinatorial Matrix Theory Research Group, codirector of the summer
2004 REUand is a potential mentor for 2005.
A partial matrix is a matrix in which some entries are specified and
others are not. A completion of a partial matrix is a matrix obtained
by choosing values for the unspecified entries. This project examined
the completion problem for the class of nonnegative Pmatrices. A
matrix is a Pmatrix if the determinant of every principal submatrix is
positive, and a matrix is nonnegative if all entries are nonnegative.
Together with Prof. Hogben they have submitted a paper entitled The
Positive and Nonnegative Pmatrix Completion Problems.
These results
were also presented at the 11th annual meeting of the International Society,
Coimbra, Portugal, July 2004.
Modular Pascal Triangle
Doug Doan is a student at Iowa State University
Tiya Sykes is a student at Florida A& M University
Professor
Jonathan Smith is the Director
of Graduate Studies for the Department of Mathematics and is a potential
mentor for 2004
Pascal's Triangle read to a prime modulus leads
to some standard fractal objects such as the Sierpinski gasket. Doug
and Tiya produced the following picture of the Pascal triangle mod 2 using
Matlab
for their paper entitled The Modular Pascal's
Triangle. In addition to exploring ideas in the literature, they investigated
Pascal's Triangle modulo a composite number, investigating the appropriate
tile size to reproduce patterns. There are many questions coming out of
this project, which would be fertile ground for projects in future years.
Eric Blabac is a 2003 graduate of ISU and is currently
a graduate student in mathematics at ISU.
Lonnie Colbert is a student at Jackson State University
This project was entitled Iterating
Analytic SelfMaps of Discs and Applications to Dynamical Systems.
It was based on a paper by Robert Burkel in the Monthly. The main
idea behind the paper was to use the notion of conjugacy of dynamical systems
to convert the not so transparent situation with Mobius maps on the
unit disc to a very simple system in the upper half plane. The Mobius maps
are classified as elliptic, parabolic, or hyperbolic, and the results in
each case are different. The students also wrote a program in Matlab
to model the results, and see if they agreed with the theory.
There
are several aspects of the project that were
not investigated, such as sensitivity near the fixed points, that would
be viable projects for summer 2004.
Dynamic and Statistical
Modeling of Data
Lindsay Hartman is a student at Central College
Anna Veit is a student at Grinnell College
Professor Wolfgang Kliemann is
both a professor of mathematics and an Associate Dean of the College of
Liberal Arts and Sciences and is a potential mentor
for 2004
The Mathematical Systems Theory group at Iowa State
University consists of 13 faculty members from the departments of Mathematics,
Statistics, Physics, and several engineering departments. The group collaborates
on various interdisciplinary research projects in the areas of control
theory, nanotechnology, national infrastructure, bioinformatics, and stochastic
systems. Lindsay and Anna worked with the group on the project 'Dynamic
and statistical modeling of data'. They first studied the mathematical
basics of dynamical systems, including the concepts of chaotic behavior,
attractors and reconstruction algorithms. They then selected two data sets
(stock market data and racing times of the Kentucky Derby) and analyzed
these data sets using a reconstruction method. For the statistical part,
they studied some basic concepts, linear regression, and time series modeling
via ARMA and AREMA approaches. They then analyzed the data sets using
these statistical models and compared prediction results obtained by using
the two methods. Anna and Lindsay's report that is currently being
reworked for submission.
Building Graphs from DNA Data to Reconstruct
Evolutionary History
Neha Kothari is a student at Iowa State University
Arian Sibila is a student at Iowa State University
This project required students to learn about the
DNA of various mammals and use Markov chains to see how the DNA mutated,
and ultimately defining an "evolutionary distance'' between mammals.
Neha's final report was entitled Method for Finding Evolutionary Distances
using PAM Matrices. Arian wrote a separate paper Inferring
phylogeny from Protein/DNA sequence.
Summer 04 REU
Mathematics
Department Homepage
Web page maintained by Leslie
Hogben
Last Update: August 22, 2004 