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Abstracts for 1999–2000

Brownian Motion and Planar Regions: Constructing Boundaries from h-Functions

Otto Cortez, May, 2000.

Abstract (in PDF).

Examples of Volume-Preserving Great Circle Flows of S3

Ryan Haskett, May, 2000.

This summer Herman Gluck and Weiqing Gu proved the last step in a process that took conformal maps between two complex spaces and related them to Volume-Preserving Great Circle Fibrations of S3. These fibrations, which are non-intersecting flows, break down under certain conditions. We obtained the fibrations by applying the process to different conformal maps then calculated the angles where they intersect. This paper centers around the developments in the method for converting the conformal maps and finding the critical angles. Finally, the examples are included in their various stages of completeness.

Reconstruction of Convex Bodies in the Plane from Three Non-Collinear Point Source Directed X-Rays

Michael Lauzon, May, 2000.

When one takes an x-ray, one learns how much material is along the line between the x-ray source and the x-ray sensor. The goal of tomography is to learn what one can about an object, by knowing how much material is on a collection of lines or rays passing through that object. Mathematically, this is a collection of line integrals of a density function of the object. In this paper, we provide and prove reconstructions from a class of convex objects of uniform density using x-rays from three point sources.

Optimal Expected Values for Cribbage Hands

Philip Martin, May, 2000.

The game of Cribbage has a complex way of counting points in the hands that are dealt to each player. Each player has a choice of what cards to keep and what cards to throw into an extra hand, called the crib, that one of the players gets to count towards his score. Ideally, you could try to keep the most points possible in your hand and your crib, or conversely, the most points in your hand with the fewest points in your opponent's crib. To add to the fun, a final card is randomly chosen that all three hands share. This thesis deals with finding optimal expected values for each player's hand and the crib. Unfortunately, finding the exact optimal values is very difficult. However, I managed to get bounds on the optimal values.

Rates of Convergence to Self-Similar Solutions of Burgers' Equation

Joel Miller, May, 2000.

Abstract (in PDF).

N-Person Coalitional Games: The Power of More Than One

Helen Monroe, May, 1999.

This paper outlines the main features of Game Theory. The author has never attended a course in Game Theory, so it is assumed that the reader has no experience either.

The early sections deal with 2-Person Game Theory—its definitions, presentations, classifications, and so on. Later sections deal with N-person Game Theory and the consequences of allowing coalitions into the Game.

The final sections deal with the calculation of the Shapley value and some of its implications in the theory of voting. The author believes that there is real scope for some extensive research into the application of the Shapley value and voting procedures and counting.

In researching for this paper, the author found some interesting and intriguing uses of Game Theory, which are included. The scope of Game Theory is very wide, especially given that this area of mathematics was unknown until the 1940's.

This paper is aimed at undergraduate students of mathematics, and it is hoped that the reader will be as inspired as the author to find out more.

Combinatorial Proofs of Generalizations of Sperner's Lemma

Elisha Peterson, May, 2000.

In this thesis, we provide constructive proofs of several generalizations of Sperner's Lemma, a combinatorial result which is equivalent to the Brouwer Fixed Point Theorem. This lemma makes a statement about the number of a certain type of a certain type of simplices in a triangulation of a simplex with a special labeling. We prove generalizations for polytopes with simplical facets, for arbitrary 3-polytopes, and for polygons. We introduce a labeled graph which we call a nerve graph to prove these results. We also suggest a possible non-constructive proof for polytopal generalization.

Finite Invariance of Cayley Calibration Form

Yinan Song, May, 2000.

In the further development of the string theory, one needs to understand 3 or 4-dimensional volume minimizing subvarieties in 7 or 8-dimensional manifolds. As one example, one would like to understand 4-dimensional volume-minimizing cycles in a torus T8. The Cayley calibration form can be used to find all volume-minimizing cycles in each homology class of T8. In order to apply the Cayley form to 8-dimensional tori, we need to understand the finite symmetry of the Cayley form, which has a continuous symmetry group Spin(7). We have found one finite symmetry group of order eight generated by three elements. We have also studied the symmetry groups of tori based on the results of H. S. M. Coxeter, and have had a simple description of the four crystallographic groups in O(8). They can be used to classify all finite symmetry groups of the Cayley form.