# MUMS 2014 Abstracts

Here are the abstracts for the student talks and poster presentations (submitted as of April 6, 2014). The abstracts are listed in alphabetical order according to the presenter’s last name. If you are a presenter and need to edit your abstract, email MUMS@simpson.edu.

**▶ Chad Bockholt** (Drake University)

*Comparisons of solutions to the Grunert equations of camera tracking*

Camera tracking is an important problem in virtual reality and robotics. A system of algebraic equations known as the Grunert equations has been long studied in connection with camera tracking. Though the equations can be solved analytically, first introducing a Grobner basis may reduce the computational time. Building on recent research, I will demonstrate a method of finding a Grobner basis using the Buchberger algorithm with several different monomial orderings to determine other bases. Comparisons will be made of the numerical solutions to the analytical solution. (POSTER)

**▶ Ben Castle** (University of Northern Iowa)

*Curve Shortening Flow for Polygons*

Curve shortening flows have been shown to provide a method of continuously shortening arbitrary smooth curves into circles. In this talk, I will examine a discrete version of a curve shortening flow and its applications to polygons. I will discuss results concerning certain special classes of polygons and progress toward characterizing the behavior of all polygons under this flow. (TALK)

**▶ Casey Croson, Louis Joslyn, Sara Reed** (Simpson College)

*Determining Critical Locations in a Road Network*

Critical locations in infrastructure are roads that, if damaged, would cause a large disruption in the ability of vehicles to navigate a city. In this paper, we consider critical locations in the road network of Indianola, Iowa. The presence of cut vertices and values of betweenness for a given road segment are used in determining the importance of that given road segment. We present a model that uses these critical factors to order the importance of separate road segments. Finally, we explore models that focus on betweenness to improve accuracy when discovering critical locations. (POSTER)

**▶ Emily Dornbusch** (Simpson College)

*The Relationship Between Fractal Dimensions and Arthropod Body Size Distributions*

Fractal dimensions can be used to explain natural occurrences such as the population density of arthropods in a given area. It is believed that as the fractal dimension of vegetation increases there will be a higher population density of smaller sized arthropods. This idea is based off of many biological assumptions such as how metabolism is related to body size. Many questions surround population density and body size because little research has been done about this relationship. My project studies this theory using the same assumptions once made by Morse, D.R. et al. I focused my research on the Simpson Campus as well as Oak Grove, which is a recreational area at Saylorville. This project includes an introduction to fractal geometry and dimensions with a focus on how equations for dimensions were developed. This talk will show you how to use a computer program called FrakOut! to determine the fractal dimension of objects. (TALK)

**▶ James Ethington** (Simpson College)

*Evolutionary Game Theory: The Continuous Prisoner’s Dilemma*

The focus of my research is to use the Continuous Prisoner’s Dilemma Problem to create a computer simulation that tracks the behavior of a population and how it changes over time. My program will work by creating a population of players with varying levels of cooperation. The cooperation value is how I measure the behavior of each player. When all of the players are created, I set matches between players using a payoff formula. The winner of a match is the player with the higher payoff. Once matches are complete, the reproductive cycle occurs. Players who have a higher payoff from their matches will be more likely to reproduce and pass on their traits. Other factors that I am adding to this simulation include a punishment factor that harms individuals with lower cooperation levels. Mutations are also be factored into the simulation. (POSTER)

**▶ Michael Frank **(Simpson College)

*Investigating Anthropogenic Mammoth Extinction with Mathematical Models*

One extinction theory of the Columbian mammoth (*Mammuthus columbi*), called overkill, hypothesizes that early humans overhunted the animal. We will employ three different approaches to test this theory mathematically: analyze the stability of the equilibria of a differential equations system, develop a differential equations model, and develop a discrete stochastic simulation. The system of ODEs is a modified predator-prey model that also includes immigration and emigration. The simulation is a stochastic temporospatial model based on a rectangular grid system designed to represent North America at the end of the last ice age. Using this simulation, we model the migration of humans into North America and the response in the mammoth population. These approaches show evidence that human-mammoth interaction would have affected the extinction of the Columbian mammoth during the late Pleistocene. [Joint work with Anneliese Slaton (Mary Baldwin College) and Teresa Tinta (University of Maryland Eastern Shore); advised by Alex Capaldi (Valparaiso University)]. (TALK)

**▶ Kyle Goranson, Ehren Gaebler** (Coe College)

*Minimal packing on a triangular grid*

We studied placements of diamond pieces on the triangular grid. We required these placements to be edge-disjoint, and attempted to find minimum density packing. We studied these packings on both an infinite plane and triangular bounded regions. We then generalized the number of diamonds required to fill such a region based on the side length of the triangle. (TALK)

**▶ Rob Heise** (Simpson College)

*Probability of Rocket Failure*

When new launch vehicles are being tested they have an initially high probability of failure which then decreases quickly. A Bivariate Approach can be used to calculate this initial probability. By using a Bivariate Approach the initial probability can be estimated to be much lower than other techniques. This presentation describes the Bivariate Approach, gives an example of it being applied and describes the future work that we plan to do with this technique. (POSTER)

**▶ Mike Henry** (Simpson College)

*Death of a Star*

We simulate spherical collapse using a Monte Carlo algorithm to determine if a nucleation event initiates the transition. Metastability before collapse would suggest nucleation as a possible transition mechanism. Nucleation could explain observed asymmetry in stellar collapse. (TALK)

**▶ Jessica Hernandez-Islas** (College of Saint Mary)

*CSM students and math anxiety*

An abbreviated mathematics anxiety rating scale questionnaire was given to most of students who took Intermediate Algebra, College Algebra, Elementary Statistics, Biostatistics, or Differentiation in Spring 13 – Fall 13. A multi-dimensional statistical method is used to analysis the collected data. The goal of this study is to find underlying factors of math anxiety and investigate the relationship between math anxiety and other factors. (TALK)

**▶ Jonathan Krein, Jesse Moeller** (University of Northern Iowa)

*Computing Curve Shortening Flows for Polygons*

The polygon shortening flow is a version of curve shortening where we move the vertices of a polygon in the direction of their Menger curvature. Menger curvature for polygons is the discrete analogue to curvature for smooth curves. In order to understand the polygon shortening flow, it is important to have visualizations. Using C/C++ programming we model our polygon shortening flow and gather experimental data to help us find theoretical results. (TALK)

**▶ Zack Lindeberg** (Simpson College)

*Star Wars galactic space battles*

Simple battles of attrition can be modeled using a system of equations called Lanchester’s equations. These equations can be expanded to show the use of specific units, like archers or cavalry. My model attempts to show a space battle between the Rebel Alliance and the Empire from the fictional universe Star Wars. This model includes fighters, bombers, and star ships along with their specific interactions with units from the opposing side. It also attempts to show how a hero can effect a battle. (POSTER)

**▶ Hannah Longstreet, Tony Saucedo, Lauren Tirado, Demetre Van Arsdale** (Simpson College)

*A Mathematical Model of Prairie Restoration*

The objective of this project is to create a modified SIR-type model to show the changes in land usage between prairie land, converted land (to agricultural area, settlement areas, and other), and restored prairie land. The model is analyzed by finding and classifying equilibrium solutions. The model is then applied to Midwestern prairies since European settlement in the nineteenth century. (TALK)

**▶ Hannah Longstreet, Tony Saucedo, Lauren Tirado, Demetre Van Arsdale** (Simpson College)

*A Statistical Analysis of a Nebraskan Restored Prairie*

Prairies are an important type of ecosystem that supply resources necessary for many endangered animals and plant species. However, since humans began populating the area, prairie land has decreased dramatically. In this project, we perform statistical analysis of different plot treatments (high diversity, low diversity, and monoculture) to see how different plant seedings can affect the invertebrate community. We use data collected from a restored prairie along the Platte River near Grand Island, Nebraska. Data was collected in June 2012. We compared measures of community structure (i.e. richness and diversity) for both the plant and ground-dwelling invertebrate communities. We preformed NMDS and ANOSIM tests using DECODA software. Plant diversity comparisons were found to have differences amongst the various plot treatments. However, invertebrate diversity among different plot treatments yielded insignificant results. (TALK)

**▶ Al Lucero** (Iowa State University)

*Cantor Set Sums*

The Cantor set can be described as all numbers between 0 and 1 whose digits in ternary expansion are not 1. The Cantor Set can be generalized to form Cantor type sets which exclude a different number in their base 3 representation. This project takes numbers from this the Cantor Set and a Cantor type set and sums them to form the real numbers between 0 and 1. We consider which of these real numbers can be written as two different sums of elements of the cantor sets. For some numbers this is impossible, while for others there may exist infinitely many different ways to write it as a sum of two elements from Cantor sets. Considering base 4 Cantor sets yields different interesting results. (TALK)

**▶ Heather Malbon **(Simpson College)

*Bargaining and Power in Networks*

Networks are the basis of a wide range of economic models, theories, and are interwoven into our lives in many regards. Networks are constantly evolving and adapting for various reasons, changes both within and outside of a network have the potential to change the network. An individual’s position within a network has a large impact on the power they hold within that network. We will look at what is required to have a balanced and stable network. (POSTER)

**▶ Kathryn Manternach** (Central College)

*Parameter Values for Oriented Graphs*

Combinatorial Matrix Theory is a branch of mathematics that uses linear algebra and graph theory concepts to analyze graphs. In this talk, I will present results from the Combinatorial Matrix Group of the 2013 Iowa State University Math REU. I will describe how to find the *Hierarchal Orientation* of a graph. Given this orientation, we can prove that the path cover number of this oriented graph is equal to the path cover number of the underlying simple graph. (TALK)

**▶ Nathan May** (Simpson College)

*Using Centrality Measures to Predict Simpson’s Homecoming Court*

Centrality measures are used in graph theory and network analysis to define the importance of a vertex as it relates to the rest of the graph. There are several different various centrality measures. This presentation will focus on betweenness and eigenvector centrality. Physical applications for centrality measures include how traveled an intersection is in a road map or what rooms are used most in a building. In this presentation we will focus on how centrality measures can be used to predict the homecoming court at Simpson College. (TALK)

**▶ Ethan Newman** (Simpson College)

*What Light Through Yonder Window Breaks: Analysis of Light and Color used in Theatre *

Lighting design in theatre works to affect the time, location, emotion, and mood of a play through light intensity, location, direction, and color. The color of light is a powerful medium to influence an audience and has an interesting base in mathematics. I analyzed how we are able to visualize color, how light can be quantified on the electromagnetic spectrum, and further how we can find the precise color, hue, and saturation of a light source on the chromaticity diagram. (POSTER)

**▶ Max Nguyen** (Simpson College)

*The Value of Flexibility in MLB Rosters*

One of the most important aspects in MLB is the depth and flexibility a team’s roster can provide. A team with a multitude of depth and flexibility gives itself the freedom to execute strategic decisions and overcome injuries in pursuit of maximizing its win total and the chance at a World Series title. Dr. Timothy C. Y. Chan from the University of Toronto and Dr. Douglas S. Fearing from the Harvard Business School wrote a paper analyzing the amount of flexibility in each MLB roster in 2012. Their results drew inspiration from the theory of production flexibility in manufacturing networks and they provided the first optimization-based analysis for positional flexibility factoring injury risk. We will use this analysis to compute the 2013 regular season flex vs. no-flex values for the American League teams and use it to investigate other trends and roster makeup.

**▶ Addison O’Conner** (Simpson College)

*Carrying Capacity: Will we be eating green crackers to survive?*

Today there are many concerns about food quantity, water availability, climate change, human population, and more. Is there a specific limit to the food or water? How much will the storms and sea level changes affect us? My research focused on these questions to find the carrying capacity of humans on Earth. Will we eventually run out of space? Could technology advancement raise that capacity limit? I investigated various types of population models and theories to determine a model that describes human carrying capacity. (POSTER)

**▶ Ruth Ann Roberts, Katie Westlund, Nick Yaeger **(Simpson College)

*The MLB All-Star Challenge*

We developed logistic models to determine which batters in each position should make the Major League Baseball All Star teams. We collected 36 different statistics on players from 2009 to 2012 and analyzed them using the statistical software JMP. Our models are based on a feature subset selection of only 3 of these statistics and they are effective at predicting the 2008 and 2013 All Star batters. Our results suggest that although players make the All Star team based on their on-field performance, other factors seem to determine which are the starters versus the reserves. (POSTER)

**▶ Mark Ronnenberg** (University of Northern Iowa)

*Discrete Modeling of Orbits on the Complex Unit Disk*

On the complex unit disk, the orbits of all points under an analytic function which are not automorphisms tend to a point in the closed disk, called the Denjoy-Wolff point. We wish to model the behavior of orbits on the disk in a discrete setting using graphs. Since Schwarz’s Lemma guarantees that analytic self-maps of the disk are contractions, the functions we study are contractions on the vertices of a graph in the graph metric. We will completely describe the fixed point set of such contractions for trees, proving that fixed point set is convex. (TALK)

**▶ Mike Rundle** (Simpson College)

*The Importance of Insuring Young Adults*

The Affordable Care Act (ACA) has changed the insurance landscape. The law was created by members of Congress and backed by the President to fix the fast-rising cost of healthcare. There are many provisions within the law that lead to insuring more people including, but not limited to: the age a person can remain on his/her parents’ insurance plan, insurance companies cannot deny coverage due to preexisting health conditions, the expansion of Medicaid, etc. With more people being insured, there is a chance that these people could be higher risks for insurance companies to insure. To stop the rising cost of premiums, insurance companies need to insure lower risk people. This group of people tends to belong to the young adult group. I will examine low risk groups and use premium and risk statistics to show why it is important for young adults to sign up for health insurance. (POSTER)

**▶ Corey Sterling** (Simpson College)

*The Colored Bridges Problem*

The Colored Bridges Problem originated in the city of Königsberg. The city was split up into four land regions separated by the Pregel River and had seven bridges connecting the regions. The citizens of Königsberg wondered if it was possible to walk through the entire town and pass over each bridge one time. This problem eventually was known as the Königsberg bridge problem. I will discuss the Colored Bridges Problem following the work of Gary Chartrand, Kyle Kolasinski, and Ping Zhang in *The Colored Bridges Problem: In Memory of Frank Harary (1921-2005)*. (TALK)

**▶ Kylie Van Houten** (Simpson College)

*Continuous Prisoner’s Dilemma: Cooperation in Evolution*

My research has been focused on an extension of the Prisoner’s Dilemma problem. Instead of an option of full cooperation or defection, each player can cooperate at any level on a continuous spectrum in between the two. I am creating a program that will simulate cooperation in individuals in a population. I will randomly generate the cooperation level for players that will be matched up with one another. Those with the best strategies will pass on their genes and the strategy will become dominant. The next step is to add punishment to those with too low of cooperation to see at what level cooperation will become a dominating strategy. (TALK)

**▶ Linsey Williams, Casey Croson** (Simpson College)

*Exploring Art through Robotics*

Art is traditionally seen as a human expression. As an alternative approach, we used mathematics and computer science to develop a non-human form of art. Using Lego Mindstorms NXT hardware and leJOS Java firmware, we implemented a motor-based design to create a drawing robot. We began by researching different ways to use the NXT to draw. These methods included a wheeled robot, a Cartesian system modeled after an etch-a-sketch, and a Polar system controlled by two motors. After building and testing these various methods, we determined the Polar system best fit our goals. Once we determined our hardware, we researched several methods of analyzing and representing an image without human aid. To achieve a final image, we experimented with scatter plots connected using a traveling salesman algorithm, grids with shaded pixels of various sizes using both Cartesian and Polar grids, and finally a Polar plotter to create a contour drawing. (POSTER)