HRUMC 2015 Student Abstracts

Hudson River Undergraduate Mathematics Conference was held April 11, 2015 at Union College in Schenectady, New York. Below are titles and abstracts for students that presented.

Danny Driscoll - Clustering of NHL Goalies

Abstract: Predicting the career path of an NHL goalie has important consequences for an NHL franchise. In this project we use data on NHL goalies since the 2000 season, focusing particularly on goalie age, save percentage, and number of shots faced. Using save percentage and number of shots faced as performance metrics, we attempt to identify different clusters, and determine which cluster each goalie belongs to using not just their performance but their trajectory as they age.  To do this we apply a new approach for the clustering of multivariate longitudinal data developed by Komaraek and Komarkova (2013) to the career trajectories of NHL goalies.  In this talk we focus on the identifying the different clusters of NHL goalies.

Christina Gay - A New Spin on Trapezoidal Numbers

Trapezoidal numbers, which are numbers obtained by adding two or more consecutive positive integers, are well understood.  So we consider a related problem: what values can be obtained by adding consecutive integers of the form 3k+1, i.e. a block of numbers from the list ..., -5, -2, 1, 4, 7, ...?  We will show that one can determine whether or not a particular sum can be obtained by analyzing its prime factorization.  For example, we will prove that it is possible to obtain all of 2012 through 2015 in this manner, but impossible to get 2016.  We present our approach to the problem both analytically and visually.

Mitchell Joseph - Erdos-Faber-Lovasz Conjecture

Abstract: A conjecture of Erdos, Faber, and Lovasz states that if p complete graphs, each having exactly p vertices, have the property that every pair of complete graphs has at most one shared vertex, then the union of the graphs can be colored with p colors. In this presentation we will attempt to prove the conjecture true given the restraint that if two vertices have degree greater than 2(p-1) then they are not adjacent.

Alaina Larkin - Untangling Knot Theory

Abstract:  Aptly named, knot theory is the study of knots, which are simple closed curves in three-space. One way to tell knots apart is to find a knot invariant, in which a value is assigned to a knot that stays the same regardless of the knot diagram.  We have developed a knot invariant that detects chirality and is relatively simple to compute.  Our method involves the idea of ‘smoothing’ each crossing in a knot diagram to obtain a single loop that covers every arc of the knot diagram, then computing a sum over all such smoothings.  (This approach is based on the Kauffman Polynomial invariant.)  There are other knot invariants that also detect chirality, such as the Jones Polynomial and the Vassiliev Invariant; however, they are computationally intensive.  Our method cuts down on the time spent computing the invariant from exponential time to what we believe to be quadratic time.

 Brandon Lustig - Head-to-Head Comparison Models to Rank PGA Tour Players

Abstract: The current ranking systems for professional golfers, such as the Official World Golf rankings and the Fed Ex Cup standings, put emphasis on winning tournaments and finishing near the top instead of looking at every round played with equal importance. To provide alternatives that are not so heavily weighted on winning tournaments and give more emphasis on playing consistently well, we use models, such as Bradley-Terry, that rely on head-to-head comparisons for all pairs of players for every tournament round. We discuss the details of finding ratings using these models, estimating the probability of a player beating another in a round, and compare the results using data from the 2013-2014 PGA tour season.

Spencer Nelson - Sums of Reciprocals of Irreducible Polynomials Over Finite Fields

Abstract:  We will discuss some integral concepts and principles of finite fields as well as arithmetic involving polynomials over finite fields. After developing a sufficient background, we will examine a peculiar result involving the sums of reciprocals of monic polynomials of a given degree over a specified finite field. L. Carlitz proved in 1935 that for, where denotes the set of monic polynomials of degree n over a finite field of order q. Expanding on this result, we will discuss the equally curious case where instead of adding all monic polynomials of a given degree, we only consider those that are irreducible.

Nathaniel Shenton - The Double Pendulum A Case Study of Chaotic Behavior

The double pendulum is a dynamic system in which a second pendulum is connected to the first, allowing the second to swing freely. The motion of the resulting dynamical system will associated to the motion of the pendulum. This system both periodic and chaotic tendencies. We will use this example to show what it means for a system to be considered chaotic. To do this we will explore the known visual tools that can help detect chaos, including an actual built pendulum.

Yunsi Yang - ANOVA and Beyond: When is a difference really a difference?

Abstract: Analysis of variance (ANOVA) is a common statistical tool to look for difference between groups. The goal is to detect differences that really exist (power), but avoid calling two groups different that are really the same (Type 1 Error). Power and Type 1 Error rate depend on various factors, such as sample sizes, number of group, variability within groups, and variability between groups. We use simulations to explore some of these relationships and also look at alternatives to the traditional ANOVA test based on randomization procedures.