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# MathFest 2005 Student Paper Winners

MathFest
2005

1. CUR^{1}
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 EM^{2} 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. SIAM^{3 }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. SIAM^{3 }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.

^{1}Council on
Undergraduate Research, http://www.cur.org

^{2}Special Interest Group of the MAA on Environmental
Mathematics, http://intellihawk.org/sigmaaem/index.html

^{3}Society for Industrial and Applied Mathematics, http://www.siam.org

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