Tim Clem, Automatic Differentiation, Summer 05
Faculty Advisor: Dan Kalman
Automatic differentiation is a capability that can be built into a
computer language. With this capability, a program that computes the value
of a function f(x) will automatically also produce the value of f '(x) as
a biproduct, with no additional programming required. In this research project,
a couple of different implementations of this capability were compared in
terms of execution speed. Tim was familiar with programming in several languages,
and wrote test programs in C++ and Lisp. He used these programs to analyze
performance of the different algorithms. He also presented a paper describing
his work at the
Young Mathematicians Conference at Ohio State University.
Marieta Pehlivanova, Strange Things Do Happen, Summer 05
Faculty Advisor: Mary Gray
This project concerns events which happen in spite of the low
probability of their occurrence. The focus is on situations which have legal or medical
implications. Marieta has examined Bayesian approaches to statistical
evidence in various legal contexts, demonstrating how credibility,
reliability and understandability might be improved.
Stephen Wheatley, Calculations on Riemann Surfaces, Summer 05
Faculty Advisor: I-Lok Chang
This project was an investigation of the mathematics of Riemann Surfaces,
an advanced mathematics topic. Steve, who is a masters student, has a strong
background in math necessary for study at this level. His project was featured
in an
article
in the AU science magazine Catalyst.
Sarah Gourlie, Geometric Patterns in Complex Power Series, Summer 04
Faculty Advisor: Dan Kalman
One way to visualize the convergence of a complex power series is to
graph the successive terms vector fashion in the complex plane. So, for
the series
a0 + a1 + a2+ ...
draw a0 as a vector starting at the origin, then
draw a1 as a vector starting from the end of a0,
and so on. When this is done for the power series 1 + z + z2
+ ... , the terms all depend on the selected value of z. This can be viewed
on a computer screen, with z varying as a mouse is dragged across the complex
plane. The results are visually quite stunning. See some samples below.
The idea of this research project is to identify and classify various types
of geometric patterns that arise in this fashion, and to determine the values
of z that produce them. The original inspiration for the project was provided
by the following webpage:
http://www.math.ucla.edu/~sigmaa/
Sarah developed computer activities to explore this topic using software
called Mathwright. She identified and classified several different types
of patterns, and then determined which values of z correspond to which patterns.
Continuing her research in a capstone project for the honors program, Sarah
extended her investigation to other infinite series. Sarah presented the
results of her summer research at a regional meeting of the Mathematical
Association of America, where she won an award for an outstanding student
paper. For more information about the presentation and award, click
here.
Resnik Boone, Cocktail Party Conversation Separation, Summer 04
Faculty Advisor: I-Lok Chang
In this project, Resnik investigated the mathematics of signal processing
connected with the problem of separating two signals tangled together in
one data stream. For example, consider a microphone that picks up a conversation
that is masked by a loud musical performance in the background. To the device
that plays back the recording, there is simply one signal. How can that
signal be divided into one part that is the music and one part that is the
conversation? Interestingly, with two separate microphones recording the
same combination of sounds, it is possible to effectively separate the input
signals. Resnik worked on implementing the appropriate mathematical processing
to accomplish this goal.
Jason Richwine, Voting Power in the Electoral College, Summer 03
Faculty Advisor: Dan Kalman
This project concerned a mathematical/statistical analysis of the power
wielded by various states in the electoral college. Jason revisited a paper
from 1980 and updated the analysis to include all the presidential elections
that have since occurred. He presented his work at the
Young Mathematicians Conference at Ohio State University.
His project was also featured in an
article in the AU science magazine Catalyst.
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