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

• TITLE: Dictatorships Are Not the Only Option: An Exploration of Voting Theory

AUTHOR: Geneva Bahrke
ABSTRACT: The field of social choice theory, also known as voting theory, examines the methods by which the individual preferences of voters are compiled into a single social preference. These social preferences form the results of elections, ranging from choices as mundane as deciding which movie a group of friends should watch to the far more consequential elections of presidents and prime ministers. This paper explores multiple types of election methods and the unexpected problems that arise from each type of method. Including an analysis of Arrow's Theorem, one of the main results of social choice theory, from both a combinatorial and geometric perspective, this paper offers a brief introduction to multiple aspects of this relevant and important field.

• TITLE: Trigonometry in the Hyperbolic Plane

AUTHOR: Tiffani Traver
ABSTRACT: The primary objective of this paper is to discuss trigonometry in the context of hyperbolic geometry. This paper will be using the Poincaré model. In order to accomplish this, the paper is going to explore e the hyperbolic trigonometric functions and how they relate to the traditional circular trigonometric functions. In particular, the angle of parallelism in hyperbolic geometry will be introduced, which provides a direct link between the circular and hyperbolic functions. Using this connection, triangles, circles, and quadrilaterals in the hyperbolic plane will be explored. The paper is also going to look at the ways in which familiar formulas in Euclidean geometry correspond to analogous formulas in hyperbolic geometry. While hyperbolic geometry is the main focus, the paper will briefly discuss spherical geometry and will show how many of the formulas we consider from hyperbolic and Euclidean geometry also correspond to analogous formulas in the spherical plane.

• TITLE: An Introduction to Genetic Algorithms

AUTHOR: Jenna Carr
ABSTRACT: Genetic algorithms are a type of optimization algorithm, meaning they are used to find the maximum or minimum of a function. In this paper we introduce, illustrate, and discuss genetic algorithms for beginning users. We show what components make up genetic algorithms and how to write them. Using MATLAB, we program several examples, including a genetic algorithm that solves the classic Traveling Salesman Problem. We also discuss the history of genetic algorithms, current applications, and future developments.

• TITLE: Expression Parsing & Visualizing Complex Mappings

AUTHOR: James Edison
ABSTRACT: The purpose of this paper is to explore the theory behind expression parsing which will then be applied for use in function evaluation. This theory driven function evaluator will then be used to calculate results of complex-valued functions, specifically, complex mappings. After calculating the values of a complex mapping, the domain of the mapping can then be colored according to each point's image location in the complex-plane, which will be represented by a 12-part color wheel. The resulting coloring helps visualize the way in which a complex mapping reshapes the entire complex domain. A basic understanding of complex numbers and programming is recommended, but not required.

• TITLE: Squares in Arithmetic Progression (re-print available upon request to Russ Gordon)

AUTHOR: Sara Graham
ABSTRACT: In this paper we lay out and prove the foundational results necessary to show that there cannot be four positive integer squares in arithmetic progression. We then present and discuss two proofs that there cannot be four squares in arithmetic progression: one that has been commonly reproduced yet contains a logical flaw and anther that does not. We conclude the paper with a geometric result that is a simple consequence of there being no four squares in arithmetic progression.

• TITLE: Sample Size Determination for Clinical Trials

AUTHOR: Paivand Jalalian
ABSTRACT: An important component of clinical trials is determining the smallest sample size that provides accurate inferences. The Frequentist approach to determining sample size is most common; however there has been a recent shift towards using a Bayesian approach due to its flexibility. This paper will review both the Frequentist and Bayesian branches of statistics and their respective approaches to sample size determination. As well, the advantages and disadvantages of using each method will be considered. Finally, along with the Bayesian approach to sample size determination, we will also discuss a Bayesian adaptive design for clinical trials that allows for sample size adjustments during the trial.

• TITLE: Volumes of n-Dimensional Spheres and Ellipsoids

AUTHOR: Michael Jorgensen
ABSTRACT: This paper starts with an exploration of the volume of sphere of radius r in n dimensions. We then proceed to present generalized results for the volume of a sphere under different p-norms or metrics also in n dimensions. We use a linear transformation to find the volume of an n dimensional ellipse, and use the Fundamental Theorem of Calculus in a clever way to find the surface area of a generalized spheroid. This paper is accessible to those familiar with calculus and linear algebra, but select parts of the paper use pieces of real analysis.

• TITLE: A Holistic Statistical Test for Fairness in Video Poker

AUTHOR: Evan Kleiner
ABSTRACT: Electronic gambling has been around for decades and has gained popularity in recent years on the internet. In particular, video poker is a popular game where computers are used to simulate traditional poker. This paper addresses the possibility of rigging electronic games by artificially manipulating the probabilities associated with the game. Such modifications are unethical and illegal. This paper develops a statistical test capable of detecting this cheating by casinos. We begin with an introduction to video poker and the statistical methods used. The final part of the paper describes the test and details simulation results.

• TITLE: Rings on the Direct Product of Two Cyclic Groups (re-print available upon request to Pat Keef)

AUTHOR: Brett Leroux
ABSTRACT: The focus of this paper is a classification of rings whose additive group is the direct product of two cyclic groups. Such rings are represented by a quotient ring of the polynomials with integer coefficients. The paper begins with an overview of general ring theory including the Chinese Remainder theorem and the theory of local/irreducible rings. We then introduce Hensel's lemma which is later used as the main tool for classifying rings on the direct product of two cyclic groups. It is shown that two of these rings are isomorphic if and only if there is a solution to a particular quadratic equation in two variables mod n. We derive a new form of Hensel's lemma that applies directly to quadratic equations in two variables. It is used to systematically solve the quadratics in question and thus obtain a complete classification of rings on the direct product of two cyclic groups.

• TITLE: Applications of Expander Graphs in Cryptography

AUTHOR: Aleksander Maricq
ABSTRACT: Cryptographic hash algorithms form an important part of information security. Over time, advances in technology lead to better, more efficient attacks on these algorithms. Therefore, it is beneficial to explore either improvements on existing algorithms or novel methods of hash algorithm construction. In this paper, we explore the creation of cryptographic hash algorithms from families of expander graphs. We begin by introducing the requisite background information in the subjects of graph theory and cryptography. We then define what an expander graph is, and explore properties held by this type of graph. We conclude this paper by discussing the construction of hash functions from walks on expander graphs, giving examples of expander hash functions found in literature, and covering general cryptanalytic attacks on these algorithms.

• TITLE: Singular Value Decomposition (re-print available upon request to Doug Hundley)

AUTHOR: Carter Muenchau
ABSTRACT: When dealing with large amounts of data it is often difficult to find the patterns without some help. There are many patterns that we could look for. In particular, we will be looking at clustering. Clustering (or cluster analysis) is grouping a set of objects such that objects in the same group are more similar to each other than objects in other groups. These groups are generally referred to as clusters. Similar can be defined in a large number of ways. The most common way and the way we will be grouping our data is through distance. Generally, the closer two observations are to each other the more likely they'll be in the same cluster. We will be looking at two types of clustering methods; k-means clustering and spectral clustering. Throughout this paper we will give some reasoning as to why these methods get useful results then analyze the two methods against each other.

• TITLE: Exploring the Rate of Convergence of Approximations to the Riemann Integral

AUTHOR: Lukas Owens
ABSTRACT: There are many well known ways of approximating the value of the Riemann integral of a real-valued function. The endpoint rules, the midpoint rule, the trapezoid rule and Simpson's rule each produce sequences that converge to the value of the integral. Some methods are better than others and this paper seeks to quantify how quickly the error of each approximation converges to zero. After studying these classical results, we extend them in a variety of ways. Sums and products of functions with different tags and improper integrals are considered, as well as higher order expansion of the error term.

• TITLE: Automorphism Groups of Simple Graphs

AUTHOR: Luke Rodriguez
ABSTRACT: Group and graph theory both provide interesting and meaningful ways of examining relationships between elements of a given set. In this paper we investigate connections between the two. This investigation begins with automorphism groups of common graphs and an introduction of Frucht's Theorem, followed by an in-depth examination of the automorphism groups of generalized Petersen graphs and cubic Hamiltonian graphs in LCF notation. In conclusion, we examine how Frucht's Theorem applies to the specific case of cubic Hamiltonian graphs.

• TITLE: Survey Model-Assisted Estimation with the LASSO (re-print available upon request to Kelly McConville)

AUTHOR: Cooper Schumacher
ABSTRACT: In survey statistics, the design-based Horvitz-Thompson estimator is a commonly used method of estimation. However, large amounts of auxiliary data are often available and can augment the survey data. Model-assisted regression estimators use this auxiliary data to fit regression models, which can improve the estimation of population quantities. In this paper we focus on the lasso model, proposed by Tibshirani (1996), and present several lasso-based regression estimators. A logistic lasso regression estimator is used for estimating a population proportion. In order to study the behavior of the lasso under various constraints, we ran simulations which assess the efficiency of the lasso-based estimators compared to other survey estimators. We then applied the estimators to Colorado forestry data.