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

Saturday Morning

Abstracts

DIVISIBILITY OF DEDEKIND FINITE SETS
by David Blair of University of Michigan

Abstract:
A set X is said to be divisible by a natural number n if it can be partitioned into sets each of cardinality n. A set X is Dedekind-finite if there exists a one-to-one map from X onto a proper subset of X. Also, a set X is finite if there exists a one-to-one map of X onto the set {0,1,2,...,n-1} for some natural number n. In Set Theory with the Axiom of Choice (AC) Dedekind-finiteness is equivalent to regular finiteness. However, without AC one only has that if a set X is finite then it is Dedekind-finite. Our notion of divisibility is explored for Dedekind-finite sets. In particular, it is consistent in Set Theory without AC for a Dedekind-finite set to have as divisors any specified subset of N (containing 1 but not 0) and no other natural numbers. We show this by constructing finite support permutation models, in Set Theory with Atoms, in which AC does not hold and our result is true. Then, by Embedding Theorems, we obtain the desired result for Set Theory without AC.

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DETERMINING DIVISIBILITY RULES WITH PERMUTATIONS
By Gabe Shaughnessy of Southern Illinois University-Carbondale

Abstract:
We all know the divisibility rules for some primes like 2,3,5,9 or even 7 and 11. But what are the rules for any prime? Divisibility of an integer by a prime can be expressed in terms of the symmetric group (permutations). Further, the group properties of the divisibility specific permutations will also be shown along with certain symmetries inherent to these permutations.

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CROSS PRODUCT IN FINITE-DIMENSIONAL EUCLIDEAN SPACES
By Daniar Hussain of MIT

Abstract:
We present a thorough analysis of vector cross products in finite dimensional Euclidean vector spaces. The most common case in R³ is examined, and the cross product expression is derived separately for two and four dimensions. The most general task of defining a cross product in n-dimensions is tackled. Some applications are also presented.

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PYTHAGOREAN TRIPLES FROM GENERALIZED HARMONIC SEQUENCES
By Randy Tanner of Southern Illinois University

Abstract:
Many areas of mathematics have been correlated to Pythagorean Triples and their properties. In this talk, I plan to examine and display the relationship held between Pythagorean Triples and generalized harmonic sequences. It will be shown that corresponding to every Pythagorean Triple, there is a unique generalized harmonic sequence. And that, conversely, every generalized harmonic sequence is the generator of a set of Pythagorean Triples.

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IMPROVING SOLAR CAR RACE STRATEGY
By Brad Berron of Rose-Hulman Institute of Technology

Abstract:
We will look at modeling a few important aspects of the Rose-Hulman Solar Phantom VI solar car project, with the goal of improving race strategy for the upcoming 2001 American Solar Challenge. We have focused on two basic issues: calculating available power from sunlight and computing torque-power curves for various speeds and hill grades. The intensity of solar radiation changes over the course of a day, and depends on the current latitude, time of year, cloud conditions, and angle of the solar cell array on the car. The resulting power available versus time of day curve can then be used to help determine race strategy for that day (e.g., the maximum speed allowed by the available power for current road and weather conditions). To complement these calculations, we combined the efficiency curves provided by the engine manufacturer with the vast amounts of data compiled by past solar car runs to find torque-power curves for different constant speeds and hill grades. These models can use the GPS data supplied by the race coordinators, giving information like latitude and altitude along the racecourse, to help determine optimal race strategies.

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CRAYOLA MEETS MATROIDS
By Robin DeGracia of Dartmouth College

Abstract:
We will start by discussing the history of matroids and the search for a proof of the four-color theorem. Matroids were first talked about in 1935 by Hassler Whitney, a mathematician who greatly advanced the field of graph theory. Later Gian-Carlo Rota discussed the chromatic polynomial, P(), of a graph, G, and proved that ⁿP() is equivalent to Q() where n is the number of components in G and Q() is the characteristic polynomial of the bond lattice of G. The bond lattice of G has an interesting connection with the cycle matroid of G. We will discuss matroids and their characteristic polynomials along with graph theory and coloring. We shall tie the two together by showing that the characteristic polynomial of the cycle matroid of G is equal to the characteristic polynomial of the bond lattice of G.

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BOOLEAN SIMILARITY MEASURES--OR HOW TO HELP LAZY PROGRAMMERS
By Sylvain Hallé of Université Laval

A common problem in computer science consists in choosing, for a new given problem, between writing a whole new program or modifying some
others chosen in an existing collection. This talk will present a brief exploration of various boolean similarity measures to qualify the distance between two program specifications, and to provide a simple method allowing to refine any given expression formed of relational algebra operators.

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REDUCIBILITY OF CONFIGURATIONS IN THE PANCAKE PROBLEM
By William Cuckler of University of Illinois at Urbana-Champaign

The pancake problem is the problem of finding f(n), the worst-case number of prefix reversals required to sort a permutation of n integers
(a permutation is sorted if its elements are in increasing order).  This is an open combinatorial problem; the known bounds are 15n/14 <
f(n) < (5n+5)/3.  We introduce a concept of a-reducible configuration and show that if every permutation of [n] lacking at least r adjacencies contains an a-reducible configuration, then f(n) < a n+ c(r) for some constant  c(r). Here an adjacency is a contiguous appearance of successive values in
the permutation.  We present some configurations that are a-reducible for a strictly less than 5/3, seeking an asymptotic improvement in the upper bound.  Two blocks (strings of adjacent elements) are successive if the largest element of one block and the smallest of the other are successive in the size order; we show
that every configuration having eight successive blocks, two sets of seven successive blocks, or four sets of five successive blocks is strictly 5/3-reducible. We also put severe restrictions on permutations that have no strictly 5/3-reducible configuration.

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MAPLE SIMULATIONS OF A SET OF ODEs MODELING AN ANALYTICAL MECHANICS PROBLEM
By  Robert Clark  and  Hanna Wagner  of Ohio Northern University

The talk will explore a problem in analytical mechanics starting from a simpler to a more complex form. The system of ordinary differential equations that describes the problem rapidly evolves from a fairly analytically-solvable one (albeit through elliptic integrals), to a system for which only numerical solutions can be found. Computer simulations using MAPLE are used in combination with classical mechanics principles in order to explain the behavior of the chosen system.

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EXPLORATIONS OF EULER'S PHI FUNCTION
By David Aukerman of Taylor University

Euler's phi function has been a topic of interest to number theorists, particularly those interested in RSA cryptography.  Euler's phi function takes as input a positive integer n and returns the number of positive integers (less than n) that are relatively prime to n.  Iterations of the phi function necessarily "reduce" n to 1.  This talk will examine interesting patterns that emerge upon examining graphs of phi(n) and the function's iterations.  In particular, counting the number of iterations necessary to reduce n to 1 partitions the positive integers into distinct classes.  The smallest integer in each class has surprising properties, such as the fact that its prime factors function as the smallest integers in other similar classes.

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ELLIPTIC CURVES: THE BASICS
By Ben Harwood of Northern Kentucky University

When one tries to calculate the arc length of a piece of an ellipse, a complex integral must be evaluated. Buried away amongst the rather hairy integrand is an interesting equation. These equations are so interesting in fact, that they were given a special name and an entire area of mathematics is devoted to studying these “elliptic curves.” In this talk we will define what an elliptic curve is, prove a rather curious fact that elliptic curves form a group, and time permitting, we will examine one of the many applications elliptic curves have: Public-Key Cryptology.

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ON THE KLEIN BOTTLE
By Jessamy Ofcarcik of Greenville University

In this paper, we discuss the Klein bottle.  After an introduction to manifolds, we show how a Klein bottle can be formed from a flat torus.  Also, we give examples of two different types of Klein bottles.

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TOURNAMENT SEEDING: IS THERE A FAIR WAY?
By John E. Slewitzke of the College of Mount St. Joseph

The standard method for seeding an eight-team tournament, in most major sporting events, is for the 1st seed to play the 8th seed; the 4th plays the 5th, etc. This will sometimes lead to unfair favoritism for certain seeds. Another way to seed tournaments is by cohort randomized seeding, which groups similar strength teams together and randomly places them in the bracket. Randomized seeding minimizes the amount of favoritism that each team receives; thus a fairer way to seed tournaments can be accomplished.

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ALTERNATIVE  TO  FACTORING
By Stephen Young  of  Rose-Hulman Institute of Technology

Many of the current methods of breaking an RSA public key involve the factorization of the modulus.  This talk will focus on an  alternative method of determining the private key without factoring  the modulus of the public key.

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