MathFest 2005 Student Paper Winners

1. CUR¹ Award to Alexander Zupan of Gustavus Adolphus College for Numbers and Patterns in Segments in the Hausdorff Metric Geomtery.

The Hausdorff metric defines a geometry on the space of all non-empty compact subsets of n-dimensional Euclidean space. We will describe some Fibonacci-type patterns that are known in this geometry and elaborate on our search for new types of patterns. We will also discuss the surprising result that certain numbers do not appear at all in this geometry.

2. Carl Erickson of Stanford University for Class Number Divisibility of Global Fields part II.

Previous results have given sufficient conditions for the prime divisibility of the class number for quadratic number fields. In addition, necessary and sufficient conditions have been given for a quadratic number field’s class number to be divisible by 3. We explore analogues of these results in the function field case.

3. Alan Covert of Arizona State University for Dispersal and Connectivity in a Stochastic Multi-City Epidemic Model.

We consider the behavior of moderately infectious and lethal diseases such as SARS over a system of networked cities and SLIR dynamics. We include a disease-induced death rate and stochastic effects on intercity dispersal rates. We compute R0 for a theoretical system, and study system response to dispersal volumes and number of connections between cities.

4. Daniel Walton of Harvey Mudd College for Diophantine Approximations of Real Curves in the Plane.

Here we will discuss generalizations of Diophantine approximation in the Cartesian plane. We move beyond approximating real numbers by rationals to approximating curves in the plane with points having rational coordinates. The objective is to classify curves into categories such as ’badly approximativeâ? in a manner analogous to the classification of real numbers.

5. Thomas Kindred of Williams College for Surfaces Bounded by Alternating Knots.

An alternating knot is a knot with minimal crossing projection in which the crossings alternate between undercrossings and overcrossings. This talk will concern surfaces, orientable or not, bounded by alternating knots. By studying the surfaces of knot bounds, we can learn a great deal about the structure of a knot itself.

6. Nicholas Yates of Williams College for Irrational Numbers and the Notion of Equivalence.

When are two irrational numbers equivalent? When is a quadratic irrational equivalent to its conjugate? In this talk, we examine some classical and recent results relating equivalence to numbers’ continued fraction expansions. Then we begin a discussion of our work over the summer of 2005 toward extending those results.

7. Samuel Kolins of Bowdoin College for Spans of the Derivatives of Polynomials.

The span of a polynomial function with all real zeroes is the difference between its least and greatest root. We will examine the relationship between the placement of the roots of a polynomial with fixed span and the spans of the derivatives of the polynomial. In particular, we will share results on which arrangements of zeros result in the extreme cases for the spans for the spans of the derivatives.

8. Joesph Kolenick of Youngstown State University for his Solution to American Mathematical Monthly Problem #11103.

In this talk, a solution to problem 11103 (Proposed by Gregory Galperin and Hillel Gauchman, Eastern Illinois University, Charleston IL) from the American Mathematical Monthly will be presented. Only elementary methods will be used.
Problem: Prove that for every positive integer n,

9. Sarah Fritsch of Sam Houston State University for her study on The Life and Work of Georg Cantor.

In this talk, the life of Georg Cantor will be discussed. An overview of his ideas and research on infinite sets and the continuum hypothesis will be given.

10. Diana Davis of Williams College for her talk on Curvature in the Gauss Plane and Minimizing Curves.

We consider constant-curvature curves in the Euclidean plane with Gaussian density.

11. SIGMAA EM² award to Nicole Casacchia of Youngstown State University for her Statistical Analysis of Downed Trees in a Riparian Valley.

It is hypothesized that downed trees in a protected river valley fall with a random orientation. Data collected for the downed course woody debris (CWD) in Zoar Valley, New York included volume, orientation from the North, decay class, and tree species. Various tests, including the Kolmogorov-Smirnov Test, were conducted to evaluate orientation uniformity and to test the hypothesis.

12. SIAM³ award to Andrew Harrell of Texas A&M University for his development of Error Analysis in Moore-Penrose Interpolation Methods.

This presentation will demonstrate the power of the Moore-Penrose matrix pseudoinverse as a tool for interpolating data to a wide range of possible functions. Discussion will also include limitations on the method and possible errors in data analysis.

13. SIAM³ award to John Gemmer of Millersville University of Pennsylvania for his talk on The General Brachistochrone Problem

Consider a frictionless surface in a gravitational field that need not be uniform. Given two points, A and B, on the surface, what curve is traced out by a particle that starts at A and reaches B in the shortest time? This project studies this problem for simple surfaces such as surfaces of revolution. We solve this more general problem using the Euler-Lagrange equation and conservation of mechanical energy.

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¹Council on Undergraduate Research, http://www.cur.org
²Special Interest Group of the MAA on Environmental Mathematics, http://intellihawk.org/sigmaaem/index.html
³Society for Industrial and Applied Mathematics, http://www.siam.org