This is some of what I did while I was at Xavier. Old news now, though.
I have been working on this project for about a year now. The 3-Dimensional Matching problem has been proven to be NP-Complete. My research has been investigating both exact and approximation algorithms for this problem. I have designed two different exact algorithms and analyzed their time complexity. I have also designed an approximation algorithm and am currently finishing up analyzing its time complexity and closeness of approximation. The research is an example of how to approach NP-Complete problems, which do not have any known efficient algorithms for solving them. The results of the research will be presented at the 2002 National Conference of Undergraduate Research.
I am currently working on understanding the fundamentals of Quantum
Computing. This includes understanding the underlying phycis principles as
well as how these principles are used to create algorithms. Interesting
algorithms that I am currently studying include: Shor's algorithm for fast
factorization, Grover's search algorithm, and Quantum cryptography. After
achieving understanding of these algorithms and the fundamentals of Quantum
Computing, I will attempt to design a Quantum algorithm. This attempt will
help to uncover the possible difficulties in designing algorithms for
quantum computers. The results of my research will be presented to Xavier's
Math/CS dept. in late April or early May of 2002.
See web page for results of my efforts.
I am currently investigating the famous Waring's problem from number theory. The question arose out of the 4 squares problem: can every positive integer be written as the sum of finitely many squares? The answer is that every positive integer can be written as the sum of 4 squares. Waring claimed that the general assertion is also true: every positive integer can be written as finitely many n'th powers, where n can be any positive integer. Hilbert proved that this is in fact true. There is current research for trying to find the numbers g(n) and G(n). g(n) for a positive integer n is the smallest positive integer such that every positive integer can be written as that number of n'th powers. G(n) for a positive integer n is the smallest positive integer such that all but finitely many positive integers can be written as that number of n'th powers. I am currently investigating both the 4-squares theorem and Hilbert's existen proof. I will present the results of my research sometime in the spring of 2002 to the Math/CS dept. at Xavier.
This is a program that I put together for
Mike Matera that allows a
person to keep a log of their files that they can search through.
Unfortunately, I didn't actually finish it until after it would have been
useful. Oh Well. You could probably figure out how it works just by
fiddle farting around with it.
Download mp3organize.ZIP