# MUMS 2013 Abstracts

Here are the abstracts for the student talks and poster presentations (submitted as of April 4, 2013). 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.

**▶ Andy Ardueser, Rachel Rice, Kelly Woodard ** (Simpson College)

*An Analysis of Beggar-My-Neighbor*

We will present our work completed in the summer of 2012 during the Dr. Albert H. and Greta A. Bryan Summer Research Program at Simpson College. We furthered the analysis of the card game Beggar-My-Neighbor specifically with the intent of discovering a deal that leads to an infinite game in a 52-card deck. We used combinatorics and programs written in Mathematica to examine and refine the large number of possible deals based on structures that lead to cyclic behavior. (POSTER)

**▶ Lane Bloome** (Millikin University)

*Sequences Generated by Appending Digits to Sierpiński Numbers*

In 1960, Sierpiński proved that there are infinitely many odd numbers *k* such that *k*2^{n} + 1 is composite. In 2011, Jones and White investigated the effects of appending sequences of digits to the left and to the right of positive integers. Using a technique pioneered by Paul Erdős, we construct arithmetic progressions of Sierpiński numbers that remain composite upon appending digits d ∈ {1,3,7,9} to the right. We also investigate cases in which Sierpiński numbers maintain their characteristic property upon repeated appending of digits. (TALK)

**▶ Wade Bloomquist** (University of Iowa)

*The Skein Algebra of the Punctured Torus*

Results have been found showing the structure of the Kauffman bracket skein algebra of a compact oriented surface at a root of unity. Central to this progress are some formulas for computing products of skeins in the skein algebra of a punctured torus. The proofs of these formulas depend on deep results from lattice gauge field theory. This talk will be focused on work that is attempting to give self-contained proofs of these formulas. (TALK)

**▶ Jayde Boyle ** (Simpson College)

*Tactile Approach to Solving One-Step Equations*

My research project consisted of a study done in two Indianola High School Algebra 1 classes. The study consisted of teaching two lessons over the same concept, but varying the technique used to teach the lessons. I found that using a hands-on approach can be beneficial to many students as well as when would be the most beneficial times to use hands-on approaches. (POSTER)

**▶ Chris Brown ** (Coe College)

*Spin-a-Block: A Grundy Analysis of all Two-Block Games*

Spin-a-Block is a two-player game in which players take turns spinning blocks in order to make all adjacent edges the same color. I studied a version of the game where only ninety degree clockwise turns are allowed. I construct a digraph for each two-block game that shows all possible moves that will lead to a win and then use an algorithm to calculate the Grundy values of all valid moves in each game. This algorithm comes from Fraenkel’s and Yesha’s paper, “Theory of Annihilation Games”. The Grundy values tell us the outcomes of each game. (POSTER)

**▶ Olya Cholewick **(Millikin University)

*Answer Key to Taxicab Geometry: An Adventure in Non-Euclidean Geometry by Eugene F. Krause*

This project focuses on Taxicab geometry, the only non-Euclidean geometry mentioned in the NCTM Standards. The goal is to create an answer key to *Taxicab Geometry: An Adventure in Non-Euclidean Geometry* by Eugene F. Krause (1986). This text has been repeatedly cited as the only high school appropriate complete text on the subject. Solutions to the problems in this text, including using Geometer’s Sketchpad, will be discussed. (TALK)

**▶ Jacob Christy** (Simpson College)

*Aquaponic Vertical Farming*

My research is on the profitability of a commercial aquaponic farm, and the implementation of a commercial aquaponic system into a vertical multilevel controlled environment greenhouse structure. (POSTER)

**▶ Laura Collins** (Simpson College)

*Stay Glassy: An Analysis of Borate Glasses*

Glass is an inorganic substance that has passed from a high-temperature liquid state to a solid state without the formation of crystals and is therefore classified as a disorganized solid. Since glass is disorganized, it is difficult to analyze its structure on long-range order and we must analyze the short-order structure of the glass instead; analyzing the structure of glass can help us understand the properties and potential applications of the glass better. We focus on the analysis of borate glasses because of their many interesting applications including radiation detectors, safe disposal of nuclear waste, and delivery of medication to specified locations. In borate glasses, the short-range structure of the glass has five basic structural units and depends on how much glass modifier has been added to the glass mixture. Solid-state NMR spectroscopy can be used to determine which structural units are present and how they are interacting with each other in the glass; specifically, we can use solid-state NMR spectroscopy to determine characteristic parameters of the glass. We seek to model the NMR spectrum of borate glasses and compare the simulated spectrum to experimental results. To model the NMR spectrum, we determine all possible resonance frequencies of the glass and plot them as a histogram to form the simulated NMR spectrum. To determine the accuracy of our model we will compare the simulated NMR spectrum to experimental results. (POSTER)

**▶ Mike Comer** (Simpson College)

*Topology Explains Why Automobile Sunshades Fold Oddly *

If you are interested in topology or knot theory, come explore the math behind how a Magic Sunshade folds. Using ideas from braid theory and permutations we will look at how the Magic Sunshade folds the way it does. Specifically, we will look at how many loops the shade makes when you close it and why that is. (POSTER)

**▶ Clay Daggett** (Simpson College)

*Defining Relation: Observing International Relations with Guttman Scales*

We use Guttman scales to identify the accuracy of different methods of comparison through the ordering of subjects and observation of logical errors. In our research we observe the relationships between countries based on the placement of foreign embassies. Applying Guttman scales we determine the accuracy of examining international relations based on embassies. We found the Guttman scales to be ambiguous and therefore unreliable. We then developed a more specific and accurate version of the Guttman scales utilizing a running average. We found the updated method of Guttman scales to confirm observing international relations through embassies to be accurate. (TALK)

**▶ Michael Frank, Courtney Sherwood, Lauren Tirado** (Simpson College)

*A Model of Invertebrate Richness on Restored Prairies*

We will present a differential equations model of prairie restoration. Here, species richness is considered as an indicator of prairie restoration, with the variables for the equation being invertebrate and plant species richness and time. We will incorporate field work from a prairie in Nebraska as an example of our model. Our main goal is determining if planting fewer seeds will yield similar invertebrate richness as planting more seeds. (TALK)

**▶ Rob Heise** (Simpson College)

*Network Modeling of Earth’s Health *

The condition of the Earth’s health is something that needs to be closely examined. Earth has many global biological and environmental systems that affect each other greatly. The condition of these systems is what determines Earth’s health. We have attempted to model the ways that these many different systems interact with each other using a network model. (POSTER)

**▶ Scott Henry, Luke Kirchner** (Simpson College)

*Magic Squares*

Our goal will be to determine the dimension of 3×3 semi-magic square vector spaces, compare them to the dimension of 3×3 magic square vector spaces, and amend the general formula to include semi-magic squares. Magic squares originated in China and spread around the world from there. Their original use was for “magic” and other events that could not be explained at the time. However, along the way rules were developed until there was a well established theory. While magic squares are not applicable, they do have a number of interesting properties. One of these properties is that the dimension of the vector space for an nxn magic square is *n*^2 – 2*n*. Working with semi-magic squares, our results show that the number of diagonals that do not add to the magic sum impact the dimension of the vector space in a predictable way. It follows that the dimension of an nxn magic square (semi-magic or not) is *n*^2 – 2*n* + *d*, where *d* represents the number of diagonals that do not sum to the magic sum. (POSTER)

**▶ Michelle Hoerber** (Millikin University)

*Security Valuation: Combining the Fundamental and Technical Analysis*

Billions of dollars of securities are traded through exchanges every day. Investors buy and sell securities to earn a rate of return to compensate for the time and uncertainty involved. This project analyzes the two methods of evaluating securities. The fundamental analysis estimates a security’s intrinsic value by examining related economic, financial, and company-specific factors. In contrast, the technical analysis is a method of evaluating securities by analyzing statistics generated by market activity, such as past prices and volume, to identify patterns that can suggest future activity. The two approaches to stock valuation differ significantly; however, each process has support and evidence of predictive power. After analyzing six portfolios of stocks, three created using fundamentals and three created using technicals, this project centers around developing an investment strategy to combine the two methods of valuing securities, in an attempt to find a more predictive model to be used in security valuation. (TALK)

**▶ Zach Huebener** (Simpson College)

*Prime Numbers, Modular Arithmetic, and an Unbreakable Cryptographic Code*

Cryptography is the study of turning information into a code so that it cannot be read and understood by anyone except its intended recipient. Digital signatures are attached to messages to verify the message’s author, the date and time it was sent, and its contents when it was sent. This digital signature prevents hostile parties from forging messages in order to cause problems such as stealing money or acquiring important American intelligence information. Digital signatures also protect virtually all secure electronic information transfers from emails to bank account transactions to online purchases. This project explores the mathematics behind digital signatures that keeps electronic information secure. (POSTER)

**▶ Ryan Lane** (Simpson College)

*The Immune System Response to HIV*

I explore a mathematical model which attempts to describe the interaction of the immune system with the human immunodeficiency virus, HIV. The model specifically monitors CD4+ T cell conditions and concentrations as the disease progresses using a system of differential equations. (POSTER)

**▶ Jon Ludwig** (Simpson College)

*Trinomial Triangles and Tetranomial Tetrahedrons*

I found symmetry when using the multinomial theorem. The coefficients of a trinomial will form a triangle of numbers with simple symmetry and the coefficients of a tetranomial will form a tetrahedron with similar symmetry. The symmetry can be proven with combinatorial formulas. (POSTER)

**▶ Christian Manahl** (Ames High School)

*Distribution of primes in quadratic number fields*

In extensions to the rational integers, such as *Z*(√2) or *Z*(i), there are similarly prime numbers that are only divisible by themselves and one down to units. The question, in this case, is whether or not the primes in *Z*(√2) are uniformly distributed. (TALK)

**▶ Nathan May, Max Nguyen** (Simpson College)

*Study of North American Box Office Dynamics*

There are a number of predictive box office trends that are important in the decision making process and development of a film. Copious amounts of variables and numerous, seemingly unquantifiable parameters need to be identified in order to develop a model that could reveal these trends. We studied a model that was created by David A. Edwards and Ron Buckmire that can predict several box office tendencies and their effect on a film’s total gross. The results are accurate based on the complexity of the model but can still be altered to determine further patterns. We will explain the model and provide additional suggestions for future research. (TALK)

**▶ Jesse Moeller** (University of Northern Iowa)

*The Nature of Fixed Points in a Curve Shortening System*

A dynamical system can be understood as something that makes things move around in a space. There are some points, however, that are fixed. For a dynamical system that moves vertices of polygons in the direction of their local curvature, what do fixed polygons look like? Are they sources, sinks, or something in-between? (TALK)

**▶ Taylor Norland** (Simpson College)

*Lewis Carroll’s Amazing Number-Guessing Game*

Mathematical lecturer and author Lewis Carroll created an unpublished mathematical number-guessing game shortly before he passed away. Without understanding the underlying mathematics, the game truly seems amazing. In order to objectively comprehend the cause of the players’ bewilderment, we will analyze Carroll’s game abstractly, fix his mistakes, and test the limits of the game. (TALK)

**▶ Janak Panthi** (Loras College)

*Linear Algebra in Quantum Mechanics: A Look at the Fourier Transform*

Any wave function in Hilbert space can be treated as a vector. The concepts of linear transformations, linear independence, subspaces, eigenvalues, eigenvectors, etc., make the characteristics of wave functions easily comprehensible. The Fourier Transform is a very useful tool in interpreting complex differential equations like the Schrödinger equation. The aim of this talk is to explore the use of linear algebra in dealing with the Fourier Transform, which is the pillar of quantum mechanics. (TALK)

**▶ Katherine Pearce** (University of Northern Iowa)

*Developing Crochet Patterns for Surfaces*

The purpose of this project is to develop an algorithm to create crochet patterns for a variety of surfaces. I generate patterns for surfaces of revolution by calculating the change in circumference of each row of stitches. My methods suggest an approach to crochet more surfaces such as surfaces whose cross section is not a circle. This research demonstrates how crochet can act as a discrete model of differential geometry. Producing these patterns allows for further research into the surfaces themselves by providing accurate models as well as continue the study of the relationship between crochet and mathematics. (TALK)

**▶ Hailee Peck** (Millikin University)

*A classification of ideals and central sets via ideal-divisor graphs*

The zero-divisor graph of a commutative ring *R*, denoted Γ(*R*), is a graph whose vertices are the nonzero zero-divisors of a ring, and distinct vertices are connected if and only if their product is zero. The study of these graphs has provided algebraists with useful insight into the structure of commutative rings, as is evident in Anderson and Livingston’s paper on zero-divisor graphs. In 2003, Redmond pioneered the notion of the analogously-deﬁned ideal-divisor graph, denoted Γ_{I}(*R*), where the vertices are the nonideal ideal-divisors of *R*, and distinct vertices are connected if and only if their product is an element of the ideal. We look to expand upon Redmond’s results, with particular interest in classifying some ideals of a ﬁnite commutative ring with identity from properties of ideal-divisor graphs. We also provide results classifying the radius and center of ideal-divisor graphs. (TALK)

**▶ Avery Podell-Blume** (Coe College)

*End-View Puzzles*

End-View puzzles are a type of logic puzzle, related to Sudoku, where each row and column contains each letter exactly once for a given set of letters with the rest of the cells left blank, and hints (“views”) are arranged around the edges to indicate the solution of the first non-blank cell it comes across. I explored solving algorithms and methods for determining whether a puzzle has one solution, many solutions, or no solution. In the course of my research, I developed deductive rules and implemented a solving algorithm in C++. I then used these to investigate how the number of hints and blanks affects a puzzle’s solvability. I found that more blanks help to insure that larger puzzles have single solutions while they make the solving more difficult, as well as evidence towards an unexpected limit on single-solution puzzles. (TALK)

**▶ Dan Schilcher** (University of Wisconsin-Eau Claire)

*Subgroup lattice of D _{8}×D_{8}*

Let *A* and *B* be finite groups and consider the direct product *A×B*. What can we say about the subgroups of *A×B*? In 1889, Edouard Goursat proved a theorem stating that there is a bijection between the set of subgroups of a direct product, *A×B*, and the set of all isomorphisms between a factor group of *A* and a factor group of *B*. We used a recently developed containment theorem to create the subgroup lattice of the direct product of the dihedral group of order 8, *D _{8}*, with itself,

*D*×

_{8}*D*. One of our long term goals was to provide the subgroup lattice of one of the extraspecial groups of order 32. Although we already know what the subgroup lattice is going to be from a construction of the subgroup lattice of

_{8}*Q×Q*, where

*Q*is the quaternion group, this construction is interesting because the subgroup structure of

*D*is very different. Specically

_{8}×D_{8}*Q×Q*contains 133 subgroups where as

*D*contains 389 subgroups. We have determined the containment of all of the subgroups of

_{8}×D_{8}*D*, used Geometer’s Sketch Pad to create its subgroup lattice, and provided the subgroup lattice of the extraspecial group of order 32 that lies inside of

_{8}×D_{8}*D*. (TALK)

_{8}×D_{8}**▶ Heidi Scott** (Simpson College)

*Dominating a Flock using the King Chicken Theorems*

In this model, we consider the dominance that occurs among chickens in a flock using the King Chicken Theorems. Our results about pecking orders describe possible patterns of dominance as well as give unexpected results about dominating chickens in a flock. We use several approaches to find the dominating chickens within a flock and analyze the best method and pattern of dominance. (POSTER)

**▶ August Severn** (Simpson College)

*Identifying Critical Locations in a Spatial Network*

Identifying critical locations is important in everything from counter-terrorism to city planning. I will demonstrate what makes a location critical and how to determine its level of importance. These techniques will then be applied to a road network to accurately determine the order of road significance. (POSTER)

**▶ Adam Smith** (Simpson College)

*Correlating Stereo Images to Extract Depth Information*

In computer vision, stereo imaging is the use of two images taken from slightly different vantage points to find the depth of objects in the images. If images are captured by two aligned cameras with only a horizontal displacement, any object visible to both cameras should have the same vertical position in each image. However, horizontal position will differ by an amount dependent on the object’s depth. Using triangulation and appropriate information about the cameras, accurate distances to objects can be extracted from the image pair. This project focused on finding common points between images using normalized cross-correlation in a computationally efficient manner. Additionally, a graphical user interface was developed that allows user-friendly capture and processing of images. (POSTER)

**▶ Taylor Stockdale** (Simpson College)

*Designing a Solar Collector*

Designing a solar collector is an intricate process that, when done properly, can greatly increase the efficiency of the collector. In this project, two different types of solar collectors are observed, flat plate collectors and parabolic trough collectors. First, an equation for a parabolic trough solar collector is derived. Then, geographic information is used to determine the optimal inclination angle for a flat plate solar collector. (POSTER)

**▶ Ayush Subedi** (Loras College)

*Applications of the Monte Carlo Method*

The Monte Carlo Method predicts the aggregate of a system by generating suitable random numbers. To establish some important properties of the Monte Carlo Method, we will begin the presentation by solving the famous *birthday problem*. The *birthday problem* concerns the probability that, in a set of *n* randomly chosen people, some pair will share their birthday. Although the method can be used to solve a wide variety of problems, it is truly useful for obtaining numerical solutions to problems which are too complicated to solve analytically. In the second part of the presentation, we will estimate the area of the Mandelbrot Set using this method. (TALK)

**▶ Jake Sutton** (Simpson College)

*Modeling Unconventional Warfare*

This presentation includes an overview of “Confronting Entrenched Insurgents” by Kaplan, Kress, and Szechtman. Their model for relating forces engaging in battle has been implemented in Maple to demonstrate the outcomes based on given parameters, including the size of the forces, attrition rates, and intelligence levels. (POSTER)

**▶ Whitney Thompson** (Simpson College)

*Cooperation Using Evolutionary Game Theory*

Previous research has shown that humans tend to form social groups consisting of similar individuals. These groups have unique characteristics that give them their own identities, and each group is likely to have its own way of solving problems. When multiple groups are affected by the same problem, such as pollution or international crime, the need for cooperation arises. My research examines what qualities are necessary to facilitate cooperation among these groups. Using evolutionary game theory, previous researchers have created a model which showed that in order to achieve high levels of cooperation, ultimately, the population must become homogeneous. I have modified this model in an attempt to sustain high levels of cooperation without sacrificing the diversity of the population. I use this model to examine what must take place in order to solve worldwide problems where the groups involved are unwilling to change their cultural identity. By changing the parameters of my model, I am able to see how different conditions will affect cooperation levels and the overall diversity of a given population. (TALK)

**▶ Jes Toyne** (Simpson College)

*Introduction to Surreal Numbers*

The theory of surreal numbers, as invented by John Conway, is a largely unexplored area of mathematics. The surreal numbers create a field that contains infinitely large and infinitely small numbers and everything in between. In my presentation I will give an introduction to surreal numbers including their construction, basic properties, field axioms, and dyadic fractions. From this presentation you will gain a basic knowledge of this exciting development in mathematics. (TALK)

**▶ Hannah VanEvera** (Simpson College)

*Crossing Numbers in Mosaic Knot Theory*

This presentation is an introduction to mosaic knot theory as defined by Lomonaco and Kauffman. We will also investigate proposed upper bounds on the crossing numbers for mosaic knots and explain our efforts to prove they are upper bounds. (TALK)

**▶ Rachel Volkert** (University of Northern Iowa)

*Equivalences of Dessins D’Enfants*

*Dessins d’enfants* are bipartite graphs with a cyclic ordering given to the set of edges that meet at each vertex. Merling and Perlis presented a method by which to construct pairs of dessins d’enfants using the permutations induced by the action of a finite group on the cosets of two locally conjugate subgroups of that group. They called these pairs of dessins *Gassmann equivalent* and investigated some of their properties. First, we discuss several properties of pairs of dessins that imply Gassmann equivalence. Then, using elementwise conjugate subgroups, we introduce and investigate a weaker type of equivalence of dessins, which we refer to as *Kronecker equivalence*. (TALK)

**▶ Guanyu Wang** (University of Iowa)

*Tangle Tabulation*

A knot is the image of a circle (i.e, a closed arc) embedded in 3-dimensional space. Tangles are similar to knots, but consist of strings whose endpoints are “nailed down” on the boundary of a 3-dimensional ball. In knot tabulation, knots are tabulated using crossing number (the minimal number of crossings needed to draw the diagram of a knot/tangle). In a similar manner to knot tabulation, we are creating a table of two string tangles ordered/categorized by crossing number. A sequence of numbers is used to represent a tangle which can be visualized by using the software KnotPlot. From there, code is being implemented to generate various invariants, each of which is a quantity that is the same when computed from different descriptions of a knot/tangle. And a webpage is being developed in which users can create a table of tangles, their different invariants, and images. (TALK)

**▶ John Warnke** (Simpson College)

*Modeling of School Attendance with Operant Behavior*

A model examining the Detroit Public Schools count day incentives program and the effects on student attendance utilizing operant behavior models. (POSTER)

**▶ Haining Wei** (Knox College)

*Risk Management in Banking*

It has been said ‘to finance is to create’. From the boom of the railroads to the advent of the Internet, the financial markets have manifested the authenticity of this proverb for centuries. But it seems more reasonable to substitute this saying with ‘to finance is to destroy’ after such events as the financial crisis in 2007 caused by mortgage loan mismanagement. Bankers misunderstood and misused the Gaussian copula function created by David Li to model credit risk so that they allowed themselves to be exposed to the dangers of risky loan portfolios. The talk will study this formula, and illustrate its potential dangers. (TALK)

**▶ “REUs and Study Abroad Opportunities” Panelists**

Lane Bloome (REU at Cornell University)

Rick Gillman (REU at Valparaiso University)

Colin Grove (Budapest Semesters in Mathematics)

Joe Stickles (Millikin University)

Whitney Thompson (REU at University of Michigan School of Information)

**▶ “**Graduate Schools and Careers” Panelists

Maranda Franke (University of Nebraska-Lincoln – Mathematics)

Jennifer Griffiths (Health Care Analyst at Wellmark / University of Iowa – Biostatistics)

Colin Grove (University of Iowa – Mathematics)

Ashley Klocke (Drake University Law School)

Joe Stickles (Millikin University)

Duncan Wright (University of Northern Iowa)