Titles are hyperlinked to pdf copies of the final project write-up. Course coordinator: Laura Schueller

• TITLE: Variational Methods in Optimization
AUTHOR: Henok Alazar
ABSTRACT: After a review of some well-known optimization problems, properties of vector spaces, and a close examination of functionals, a familiar approach to solving max and min problems is generalized from elementary calculus in order to find solutions to more difficult extremum problems. Using the Gateaux variation, a fundamental necessary condition for an extremum is established and applied. Optimization problems with one constraint are explored along with weak continuity of variations. Following a statement of the Euler-Lagrange Multiplier Theorem, more extremum problems are solved and then applications of the Euler-Lagrange Multiplier Theorem in the Calculus of Variations end the work.

• TITLE: Enumeration by Algebraic Combinatorics
AUTHOR: Carolyn Atwood
ABSTRACT: Polya's theorem can be used to enumerate objects under permutation groups. Using group theory, combinatorics, and many examples, Burnside's theorem and Polya's theorem are derived. The examples used are a hexagon, cube, and tetrahedron under their respective dihedral groups. Generalizations using more permutations and applications to graph theory and chemistry are looked at.

• TITLE: Piano Tuning and Continued Fractions
AUTHOR: Matthew Bartha
ABSTRACT: In this paper, we first establish algorithms for creating continued fractions representing rational numbers. From there, we prove that infinitely long fraction expressions represent irrational numbers, along with methods for rationally approximating these numbers. As we analyze the effectiveness of any given approximation, we provide examples for finding these numbers. Next, we use of these fractions to evaluate how pianos are tuned and why one cannot be tuned perfectly. We focus mainly on the most common way to tune pianos in Western music, but will briefly explore alternate scales, such as the one used in Chinese music. Finally, we conclude the paper with a discussion of theoretical alternative scales to the ones in place, and why the ones that are used are the most popular.

• TITLE: Centroidal Voronoi Tessellations
AUTHOR: Jared Burns
ABSTRACT: The Voronoi diagram and centroidal Voronoi tessellation (CVT) are defined and their properties explored. The Lloyd and MacQueen algorithms for determining a CVT from an ordinary Voronoi diagram are defined. In order to compare the efficiency of the two, a few stopping parameters including the energy functional are examined. Then, the data analysis of the computation time of Lloyd and MacQueen's algorithms given the same initial Voronoi diagram are presented and the differences discussed. Finally, a new hybrid method using the Lloyd and MacQueen algorithms as a template is constructed and shown to be more efficient then either method alone.

• TITLE: Newton's Method and Fractals
AUTHOR: Aaron Burton
ABSTRACT: In this paper Newton's method is derived, the general speed of convergence of the method is shown to be quadratic, the basins of attraction of Newton's method are described, and finally the method is generalized to the complex plane.

• TITLE: On Metric Topologies and the Refinable Chain Condition
AUTHOR: Dan Crytser
ABSTRACT: A fundamental result in finite Abelian group theory is the so-called structure theorem, which uniquely presents a finite Abelian group as the direct product of cyclic groups of prime-power order. Analogous to this is a result in the theory of vector spaces which reduces the question of vector isomorphism to the question of basis cardinality. These are in fact just two of a collection of theorems which serve to analyze and pick apart algebraic structures by certain size factors, or cardinal invariants. In this paper we consider questions of this type along with a much subtler class of problem which arises from the introduction of topological properties into Abelian groups and vector spaces.

• TITLE: Kepler's Laws
AUTHOR: Andy Erickson
ABSTRACT: In this paper, Kepler's three Laws of Planetary Motion are proven using Newton's Law of Universal Gravitation. In addition, several results pertaining to the orbital period of a satellite are derived. An equation for the velocity of a satellite, as well as the minimum and maximum velocities necessary for a satel- lite to stay in orbit are also derived. Finally, the anomalous orbit of Mercury is examined using Newton's Law of Universal Gravitation and Einstein's Theory of Relativity. This section assumes a basic familiarity with General Relativity though a knowledge of tensor calculus is not required to follow the analysis of Mercury's orbit.

• TITLE: The Chromatic Polynomial
AUTHOR: Cody Fouts
ABSTRACT: It is shown how to compute the Chromatic Polynomial of a simple graph utilizing bond lattices and the Mobius Inversion Theorem, which requires the establishment of a refinement ordering on the bond lattice and an exploration of the Incidence Algebra on a partially ordered set.

• TITLE: Functions of Bounded Variation
ABSTRACT: In this paper we explore functions of bounded variation. We discuss properties of functions of bounded variation and consider three related topics. The related topics are absolute continuity, arc length, and the Riemann-Stieltjes integral.

• TITLE: A Brief Introduction to Hilbert Space and Quantum Logic
AUTHOR: Joel Klipfel
ABSTRACT: We explore some of the fundamentals of Hilbert space theory from the perspective of a mathematician and use our insights gained to begin an investigation of one mathematical formulation of quantum mechanics called quantum logic.

• TITLE: Differential Geometry (voted outstanding senior project)

AUTHOR: Alex Masarie
ABSTRACT: A survey of Differential Geometry is presented with emphasis on surfaces in $R^3$. Differentiation on surfaces and a dual approach to normal, Gauss, and mean curvature involving the Shape Operator and fundamental forms are developed. Gauss's Theorem Egregium is proved and a supporting discussion of diffeomorphic and isomorphic surfaces is included. Minimal surfaces are examined and Aleksandrov's "Soap Bubble" Theorem regarding compact surfaces of constant mean curvature is proved.

• TITLE: Inverse Problem: An Exploration of Heat Flow
AUTHOR: Michael McKenzie
ABSTRACT: We discuss the notion of an inverse problem and then explore a particular inverse problem based on the heat equation. A relationship is established between the deformation location within a one-dimensional rod and the heat flux from the ends of the rod. A finite difference method serves as the tool to solve the one-dimensional heat equation that results from a moving heat source and deformation. We show that for certain simple deformations, it is possible to determine the location within the rod using only heat flux measurements at one end.

• TITLE: Optimizing Sizes of Codes
AUTHOR: Linda Mummy
ABSTRACT: This paper shows how to determine if codes are of optimal size. We discuss coding theory terms and techniques, including Hamming codes, perfect codes, cyclic codes, dual codes, and parity-check matrices.

• TITLE: Convexity and the Art Gallery Problem
AUTHOR: Bahiyyih Parish
ABSTRACT: Basic ideas from convex geometry in Euclidean space are developed. The finite and infinite versions of Helly's theorem are proved. Some applications of Helly's theorem are examined. Our study culminates in proof of Krasnosselsky's Art Gallery Theorem in $n$-dimensional space.

• TITLE: The Saturn-Janus-Epimetheus System
AUTHOR: Charli Sakari
ABSTRACT: The Saturn-Janus-Epimetheus system is a prime example of a circular restricted three body problem. In this paper we examine the difficulties behind the three-body problem, the assumptions that must be made in order to solve it, the limitations behind these assumptions, and possible methods of solving the three-body problem. Finally, we use the unique properties of the Saturn-Janus-Epimetheus system to show that Epimetheus' orbit traces out a horseshoe pattern in a rotating frame in which Saturn and Janus are fixed. The system is then numerically integrated and animated in MATLAB.