ABSTRACTS  for the 17th Annual Rose-Hulman Undergraduate Mathematics Conference
  - The abstracts for this year's conference will appear in late February or early March as the conference schedule is filled.

Abstracts of Student Talks

Abstracts are listed first by day and then time of presentation.
Click the title to go to the abstract.

Speakers:  Here are some  SPEAKER GUIDELINES for this conference.


Index

Friday Afternoon
Title Speakers Institution Room/Time
Large Primes Patrick Kessel & Westley Haggard Northern Kentucky University G219/3:00
Parallel Computation of the Polar Decomposition Johnnell Parrish  University of Maryland Eastern Shore  G222/3:00
Error Bounds Involving Almost-Binomial Approximations  Abu Jalal Wabash College G219/3:25
The Irrationality of Pi Amar Parmarr  Manchester College G222/3:25
Farmer Ted Goes 3D Shawn Alsburgh  Taylor University  G219/4:00
Patterns of Powers Kendell Dorsey College of Maint Saint Joseph  G222/4:00
Chaotic Behavior over the Naturals Steve Young  Rose-Hulman Institute of Technology  G219/4:25
Good Sampling Times for Sampled-Data Systems Mustafa Bashir  University of Nebraska-Lincoln  G222/4:25
Saturday Morning
Title Speakers Institution Room/Time
Energizer Fractions; They Keep Going, and Going, and ... Erin Bergman  Saint Norbert College  G219/10:10
 Rookies in the Math Modeling Contest Michael Ewing , Peter Nei & Tom Schneider  Rose-Hulman Institute of Technology G222/10:10
Roots, Roots and More Roots Michelle Budzban  Saint Norbert College  G219/10:35
NP Complete Radio Tower Placement Dennis Lin Rose-Hulman Institute of Technology G222/10:35
Bordered Klein Surfaces with Maximal Automorphism Groups Shaun McCance Purdue University - Calumet  G219/11:10
  G222/11:10
Cwatsets and Graph Isomorphisms Nancy-Elizabeth Bush  Wheaton College  G219/11:35
    G222/11:35

Abstracts

Friday Afternoon

LARGE PRIMES
by Patrick Kessel & Westley Haggard of Northern Kentucky University (G219 / 3:00 pm)

Abstract:
Public-key cryptography relies heavily on prime numbers. Anyone who knows the definition of a prime number can show that a number a is prime by showing that no positive integer divides a besides 1 and a.  But for a public-key cryptosystem to be secure, the number a must be hundreds of digits long.  One certainly cannot show that a is prime by showing that no positive integer divides it besides itself and 1.  This would take too much time. So how can one show that a number, which is hundred of digits long, is prime?


PARALLEL COMPUTATION OF THE POLAR DECOMPOSITION
by Johnnell Parrish of University of Maryland Eastern Shore (G222 / 3:00 pm)

Abstract:
A nonsingular complex matrix A as a polar decomposition such that A=UH, where U is a unitary matrix and H is a Hermitian positive definite matrix. In this discussion, parallel computation of the polar decomposition using a method by Higham and Papadimitriou is considered, and specific applications of the polar decomposition are identified. Furthermore, the efficiency of computing the polar decomposition on a distributed memory computer is shown, and the importance of utilizing the computer's resources effectively is pinpointed.


ERROR BOUNDS INVOLVING ALMOST-BINOMIAL APPROXIMATIONSOF PROBABILITIES INVOLVING SUMS OF INDEPENDENT HYPERGEOMETRIC RANDOM VARIABLES
by Abu Jalal of Wabash College  (G219 / 3:25 pm)

Abstract:
Problems involving sums of several independent hypergeometric random variables generally involve complex calculations. Straightforward approximations have been developed using the Poisson distribution, the binomial distribution, and the almost-binomial distribution, a generalization of the binomial distribution where n is allowed to take on noninteger values.  In this talk, we give error-bounds for the almost-binomial approximation and compare these with error- bounds for the other approximation methods.


THE IRRATIONALITY OF PI
by Amar Parmarr of Manchester College (G222 / 3:25 pm)

Abstract:
Everyone knows that Pi is irrational and yet proving this can be somewhat of a challenge. Mathematicians have spent centuries studying and estimating Pi. I will explain Ivan Niven's half-page elementary proof that Pi is irrational. I was interested in also obtaining elementary estimates for Pi.  Using geometric methods, I was able estimate Pi with a scientific calculator, but this method had many limitations. I then was able to obtain a good estimate of Pi using only techniques from Calculus.  I hope that my clumsy estimates give you an appreciation for estimates of Pi that are accurate to 200 million decimal digits.


FARMER TED GOES 3D
by Shawn Alsburgh of Taylor University (G219 / 4:00 pm)

Abstract:
Let s(n) be the minimum surface area of a three dimensional rectangular box with integer sides and a volume of n. Let F(n) be the ratio n/s(n).  Let A be the set of all integers n such that  for all integer  Because of the similarity with a property of cubes, we define the set of almost-cubes to be the set A.  We attempt to characterize almost-cubes.  We  note the relationship between almost-cubes and their two-dimensional counterpart almost-squares.  Three different classes of almost-cubes are described, including how to calculate which almost-cubes fall into each class.  This is a generalization of the work of Greg Martin, who characterized almost-squares in a 1999 paper.


PATTERNS OF POWERS
by Kendell L. Dorsey of College of Mount Saint Joesph(G222 / 4:00 pm)

Abstract:
In examining the sequence  with a calculator, one can see that the greatest integer less than or equal to each term is odd. In this talk we establish this result and consider which pairs (a,b) have the same pattern, i.e. the greatest integer less than or equal to  is odd for all n. We also contemplate other parity patterns for the greatest integer of .


CHAOTIC BEHAVIOR OVER THE NATURALS
by Steve Young of  Rose-Hulman Institute of Technology (G219 / 4:25 pm)

Abstract:
Typically when speaking of a chaotic data set, the set resides in the Reals.  This talk intends to show how a chaotic data set can be extended so that is can be thought of as residing in the Natural numbers and how a subset of the Naturals can be thought of as a chaotic data set by transforming it into the Reals.


GOOD SAMPLING TIMES FOR SAMPLED-DATA SYSTEMS
by Mustafa Bashir of  University of Nebraska-Lincoln (G222 / 4:25 pm)

Abstract:
We consider a continuous-time system modeled by a differential equation with a continuous-time control.  We examine its stability in terms of a continuous-time solution.  We apply a sampled-data control and compare the stability of the new solution with that of the continuous-time solution.  Our primary interest is in relating these solutions and understanding how the sampled-data control works on the system.  We finally examine the stabilities of several sample systems and form a conjecture about the stability of an infinite-dimensional system using its finite-dimensional truncation.



 
 
 
 

Saturday Morning


ENERGIZER FRACTION; THEY KEEPGOING, AND GOING, AND ...
by Erin M. Bergman of Saint Norbert College (G219 / 10:10 am)

Abstract:
This will be an introduction to continued fractions.  We will give a variety of examples and discuss some elementary properties.


ROOKIES IN THE MATH MODELING CONTEST
by Michael Ewing, Peter Nei, and Tom Schneider of Rose-Hulman Institute of Tecnology (G222 / 10:35 am)

Abstract:
We will describe our experiences in our rookie season in the math modeling contest.  Describing what we would have liked to know before hand and what we will would do differently if we could repeat our rookie season.


ROOTS, ROOTS, AND MORE ROOTS
by Michelle Budzban of Saint Norbert College (G219 / 10:35 am)

Abstract:
We will review the time-honored method of hand-calculating square roots, investigate the thery behind this method, and extend the method to other roots.



NP-COMPLETE RADIO TOWER PLACEMENT: HOW TO AVOID WORK BY CONVINCING PEOPLE THAT THE PROBLEM IS TOO HARD
by Dennis Lin of Rose-Hulman Institute of Tecnology (G222 / 10:35 am)

Abstract:
One of the problems in this year's Mathematical Contest in Modeling involved assigning frequencies radio towers arranged in a hexagonal pattern. The purpose was to minimize span of the frequency assigned while avoiding interference between towers. We showed that the general problem where an arbitrary number transmitters is placed on a hexagonal grid is NP-Complete. People have been unsuccessfully trying to solve NP-Complete problem for many years now.  Also, modern encryption depend on the fact that such problems are hard.  Thus, we conclude that we are unlikely to find an efficient solution to our problem.
 


BORDERED KLEIN SURFACES WITH MAXIMAL AUTOMORPHISM GROUPS
by Shaun McCance of Purdue University Calumet (G219 / 11:10 am)

Abstract:
In this talk, we consider tilings of the upper half plane by hyperbolic polygons. We use these tilings to construct Klein surfaces with nontrivial automorphism groups. It is known that if is a bordered Klein surface of algebraic genus p the order of its automorphism group must be less than or equal to 12(p --1).  We conclude the talk by examining Klein surfaces that possess this maximal number of automorphisms.


CWATSETS AND GRAPH ISOMORPHISM
by Nancy-Elizabeth Bush of Wheaton College (G219 / 11:35 am)

Abstract:
Cwatsets were first introduced at Rose-Hulman as a topic suitable for undergraduate research, and applications have been found for them in statistics and coding theory. Here we introduce another important application to the field of graph theory: we show how the equivalence class of a graph can be described using cwatsets. Studying invariance leads to a discovery of which of these cwatsets are groups