# MUMS 2012 Abstracts

Here are the abstracts for all student talks and poster presentations. The abstracts are listed in alphabetical order according to the presenter’s last name. If you are a presenter and want to edit your abstract, contact Dr. Bill Schellhorn.

**Ambarlee Berringer** (Simpson College)

*Lights Out*

The game Lights Out can be mathematical if we look closely enough; it is a five by five matrix that is made of lights that either turn on or off. Now this may seem simple, but when pushing a light it not only changes the state of that light but the lights above, below, to the right of, and below it, making the game harder to solve. This poster presentation will give you a detailed example of how to solve a given Lights Out game. (POSTER)

**Amelia Brown** (Simpson College)

*Rotationally Symmetric Rose Links*

Rose links are a special class of links comprised solely of unknots which are arranged systematically to yield a rose projection. To be rotationally symmetric, a rose link must have symmetry in its crossing arrangements. There is a maximum of 2^{ n(n-1)} distinct *n*-component rose links, which is reduced to 2^{ n-1} when rotational symmetry is required. For the purposes of this project, attention was limited to the 3, 4, and 5-component rotationally symmetric rose links. A series of invariants was applied to narrow the bounds on the number of possible distinct links. (TALK)

**Katie Burke** (Loras College)

*Reduction Numbers of Monomial Ideals*

Let *I* be an ideal of the ring *R*. We may then denote the integral closure of *I* in *R* as *Cl(I)*. An ideal*J* is called a reduction of *I* if *J* is a subset of *I* and *J I ^{r} *=

*I*

^{r+1}for some

*r*, and the smallest such

*r*is called the reduction number of

*I*. Monomial ideals possess the unique property that their monomial elements can be represented by an integer point lattice known as the exponent set. We will therefore examine how to determine the exponent set of a monomial ideal and subsequently utilize its exponent set to succinctly define its integral closure. We may then use the property that

*J*is a reduction of

*I*if and only if

*Cl(J)*=

*Cl(I)*to devise correlations between a monomial ideal’s reduction number and the geometry of its exponent set. (TALK)

**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 lithium borate glasses because of their many interesting applications including radiation detectors, safe disposal of nuclear waste, and delivery of medication to specified locations. In lithium borate glasses, the short-range structure of the glass has 5 basic structural units and depends on how much lithium 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. We seek to model the NMR spectrum of lithium 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. (TALK)

**Christopher Cox** (Iowa State University)

*Isospectrality in the Euclidean Spaces*

In 1966, Marc Kac inquired as to whether or not one could hear the shape of the drum. In other words, he was asking if the sets of eigenvalues, found by evaluating the wave equation on a closed domain, are unique with respect to the region. This question was answered negatively by the construction of what are known as isospectral manifolds, i.e. drums of different shapes that sound identical. In this talk, I shall present the methods utilized in the study of isospectrality, present my constructions of explicit examples of isospectral manifolds in 3-space, and discuss my research into the relationships between isospectral manifolds and cospectral graphs. (TALK)

**Jennifer Crumly** (University of Northern Iowa)

*Links in Graphs: Embeddings of K _{6}*

We explore a Theorem of Conway and Gordon from 1983 which says that any embedding of the complete graph on six vertices contains a pair of linked triangles. This is an amazing fact with an elementary proof. (POSTER)

**Differential Analyzer Club** (Simpson College)

*A Mechanical Visualization of Dynamic Equations*

The differential analyzer (DA) demonstrates a mechanical visualization of integration, specifically Riemann sums. The DA has the ability to solve nonlinear differential equations of interest to mathematics researchers in the broad field of differential equations. As the Simpson College DA Club, we are working on perfecting the machine’s accuracy. We are also working together to determine the different constants embedded in the machine because of the gear arrangement. (POSTER)

**Om Gurung** (Loras College)

*Finite Element Analysis*

Finite Element Analysis is a numerical technique for finding approximate solutions of the partial differential equations. The technique has very wide application, and has been used on problems involving stress analysis, fluid mechanics, heat transfer, diffusion, vibrations, electrical and magnetic fields, etc. We will be analyzing one dimensional boundary value problem with springs and two dimensional boundary value problem with heat transfer, using the Finite Element Analysis technique. (TALK)

**Lindsey Harnack** (Simpson College)

*Samuelson’s Multiplier-Accelerator Model*

This is an economic model using discrete dynamical systems to determine multipliers and accelerators that affect the economy. (POSTER)

**Jill Jessee** (Simpson College)

*Physiologically-based pharmacokinetic (PBPK) modeling of metabolic pathways of bromochloromethane in rats*

Bromochloromethane (BCM) is a volatile compound and a by-product of disinfection of water by chlorination. We developed a physiologically-based pharmacokinetic model and explored three hypotheses describing metabolic pathways of BCM in rats. The hypotheses are: 1) Michalis- Menten kinetics with one CYP2E1 binding site, 2) a two-pathway model using both CYP2E1 and glutathione transferase enzymes, and 3) a two-binding site model where metabolism can occur on one enzyme, CYP2E1. One goal of this project is to demonstrate the utility of PBPK modeling for hypothesis testing with BCMs kinetics. Our computer simulations show that all three hypotheses generally describe the experimental data. Of the three kinetic models tested, the two-binding site model provided better fits to the data, producing smaller differences between the data and computer simulations. Finally, we explore the sensitivity of different parameters for each model using our obtained optimized values. (TALK)

**Donna Jones** (Maharishi University of Management)

*The Cauchy-Schwarz Inequality*

The Cauchy-Schwarz Inequality is encountered in a range of mathematical areas from linear algebra to calculus and is even seen in the derivation of the Heisenberg uncertainty principle in physics. Cauchy published the inequality for sums, while the corresponding inequality for integrals was first stated by Bunyakovsky and later rediscovered by Schwarz. This talk covers the definitions both for function spaces and for vectors of real numbers. We prove the inequality and from that we prove the triangle inequality. Then we discuss the geometric implications and prove that, for *n* = 2, the geometric mean never exceeds the arithmetic mean. This inequality has a unifying role in mathematics, applying in analysis, in linear algebra, in probability and statistics, and many other areas. (TALK)

**Ashley Klocke** (Simpson College)

*The Feasibility of Electric Vehicles*

The aim of this project is to determine how much energy is necessary to power and run electric vehicles in comparison to fuel vehicles, and to address the feasibility of switching over. We found how much oil could be saved by having an immediate switch from passenger cars to a model electric vehicle. We estimated how much energy and money it would cost to use an electric vehicle in comparison to an average passenger fuel vehicle and an average new passenger fuel vehicle. We have also created a model to determine if the vehicle has enough energy to take certain trips. (TALK)

**Alicia Kollenkark** (Simpson College)

*The Dating Game*

Assume there are *n* men and *n* women and we need to match them for a date. One goal of such a challenge is to produce an outcome that maximizes everybody’s satisfaction level. We will illustrate a process to accomplish this goal. This process will involve first defining what we will call a stable outcome. We will see that a solution concept we will yield is a gender biased outcome. This process will involve the use of an algorithm that will guarantee the stability of an outcome. (POSTER)

**Nadia Koranteng** (Winona State University)

*Two opposite ends of the spectra*

HIV/AIDS a disease which has been around for decades but is nowhere near its end when it comes to getting rid of it. Several math models have been proposed to suggest how to keep the disease under wraps. This presentation looks at math models which have been used to model the progress of this infection in two countries, South Africa and Cuba. These two countries are opposite ends of the spectrum with one having the highest AIDS prevalence rates and the other having one of the lowest. We look at math models explaining these differences in prevalence rates and make some modifications to these models. (TALK-AM)

**Gaby Liedman** (Simpson College)

*Graceful Labeling*

In this presentation, I will introduce the concept of graceful labeling and different families of graphs that can and cannot be gracefully labeled. In particular, I will discuss the graceful labeling of paths, trees, and wheels. (TALK)

**Brittni Longer** (Simpson College)

*Exploring the Dimension of n×n Magic Squares*

Magic squares have been around since around 2200 BC. It was my goal to find the dimension of the vector space *MS(n)*, where *MS(n)* is the set of all *n*×*n* magic squares. In order to do so, it must be proved that *MS(n)* is a subspace of *R^(n ^{2}+1)*. Before jumping into the proof of

*MS(n)*, a proof of the dimension of the vector space of

*MS(3)*is given, where each entry is represented by a variable. Then, the equations of the rows, columns, and diagonals are written out, put into an augmented matrix and row reduced to get the echelon form. Next the number of free variables can be determined, which corresponds to the dimension of the vector space. So the dimension of the vector space of

*MS(3) = 3*. Then, by applying similar methods, the dimension of

*0MS(n)*, which is the vector space of all

*n*×

*n*magic squares with a magic sum of 0, can be found. The dimension of

*0MS(n)*is

*n*. By using the proof for the dimension of

^{2}–(2n+1)*0MS(n)*, it can be proven that the dimension of the vector space

*MS(n)*is

*n*. (POSTER)

^{2}–2n**Jacqueline Mathenge** (Simpson College)

*Derivation and Modification of Continuous-Time Based Processes to Suit Real-Life Applications*

This project introduces some of the fundamental assumptions used in financial models. It looks into the study of derivatives and stock prices as following continuous-time periods incorporated in stochastic processes. It also generates some vital assumptions that are used in the pricing of derivatives based on the underlying assets modeled by the Black-Scholes formula. (POSTER)

**Madeline McAreavy **(Simpson College)

Spirographs, Geogebra, and Parametric Curves

The Spirograph toy enables children of all ages to draw curves that can be modeled using parametric equations. In my presentation I would like to explain the parametric equations that can be used to create these curves. I would further like to show the two ways Spirograph curves can be modeled using GeoGebra software, using the parametric equations and geometric principles. To end my presentation I would like to compare the GeoGebra models with the actual Spirograph drawings. (TALK)

**Bryan Michelsen** (Simpson College)

Benford’s Law: Unexpected Distributions of Leading Digits

If asked to predict the distribution of leading digits in a set of data many of us would expect to find a uniform distribution where each digit occurs with the same frequency as all the other digits. Our intuition fails us here though and in many sets of data the distribution of the leading digits is not uniform. Benford’s Law shows that the distributions of leading digits in lists of similar data tend to follow a logarithmic curve with the distribution being heavily skewed towards the lower numbers. While this does not hold for all scenarios, Benford’s Law does have valuable applications and can even be used to identify fraudulent tax returns. (TALK)

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

*Numerical Experiments in Chaotic Dynamics*

When studying particular families of discrete dynamical systems it can be difficult to understand which parameter values of these families produce interesting results. Using computers we can generate images which represent the behavior of orbits under many iterations of a desired dynamic. Some images produced this way can be beautiful, explanatory, and shocking all at the same time. (POSTER)

**Bria Morgan and Daniel Schilcher** (University of Wisconsin – Eau Claire)

*Constructing the Subgroup Lattice of D _{8}×D_{8}*

In 1889, Edouard Goursat formulated a theorem in group theory that provides the subgroup structure of a direct product of finite groups *A* and *B*. It does this by relating the isomorphism of a factor group of *A* and a factor group of *B* with a corresponding subgroup of the direct product. This provides the backbone for a powerful method, developed by Dr. Dandrielle Lewis, of determining subgroup containment in a direct product. In this project, we have been using this method to construct the subgroup lattice of the direct product of the dihedral group of order eight,*D _{8} *, with itself. In this talk, we will demonstrate how we have applied this method and show the progress we have made in constructing this subgroup lattice. (TALK)

**Jeanie Mullen** (Simpson College)

*Visualizing & Constructing 3D Images*

Two-dimensional objects are easily visualized in two-dimensional space. However, the projection of three-dimensional objects in two-dimensional space is more difficult to visualize. The use of computer software allows for three-dimensional objects to be constructed with visible faces and edges clearly defined. This presentation will focus on how to construct three-dimensional objects using GeoGebra software, as well as the basics of 3D constructions in projective geometry. (TALK)

**Corie Peterson** (Simpson College)

*Colonel Mustard on the Island with the Fork*

We will investigate a model to be used for fingerprint identification at crime scenes. Parameter values for the model were established from the analysis of a random sample of 49 fingerprints. This procedure will be discussed. We will also investigate the weakness of the model in terms of fingerprint misidentification. (TALK)

**Kyle Riegel** (Loras College)

*Mathematical and experimental investigation of Electromagnetic Field Theory*

James Clerk Maxwell is known as the father of electromagnetic theory, which is the relation between electric fields, magnetic fields and optics. A vector analysis approach will be used to explain Maxwell’s Equations. Concepts will be explained with a vector field mindset. The talk will also include experiments for Faraday’s law, Coulomb’s law, and Biot-Savart’s law, which is used to calculate the strength of Earth’s magnetic field during the presentation. (TALK)

**Kevin Schiroo** (Coe College) **and** **Ryan Lane** (Simpson College)

*Numerical Analysis of Multiple Myeloma Model*

We explored the parameter space of a mathematical model of multiple myeloma. This system of four differential equations models osteoblast, osteoclast, tumor and bone densities. Using MATLAB 7 we identified regions in the parameter space when a treatment regime was effective, that is, when the bone mass did not reach zero. Using Mathematica 8 we extended the stability analysis of healthy osteoclast/osteoblast bone remodeling in to the full system with the tumor growth. In particular, for a given tumor load, the 2×2 Jacobian and the expanded 3×3 Jacobian of the linearized systems yield similar stability conditions. (TALK)

**Jarrod Schott** (Simpson College)

*Solutions to Difficult Crossing Puzzles*

This presentation looks into using graph theory to solve difficult crossing puzzles. In the cases examined in this project there are an equal number of missionaries and cannibals trying to cross a river. We will explore possible scenarios to safely get everyone to the other side without ever allowing the missionaries to be outnumbered by the cannibals. I will also be looking into minimal and unique solutions for various scenarios involving even numbers of missionaries and cannibals, as well as odd numbers of missionaries and cannibals. (POSTER)

**Will Swalwell** (Simpson College)

*Topspin: Determining the Solvability of Different Game Sizes*

The puzzle Topspin is a sliding number game consisting of an oval track containing a random arrangement of numbered discs, and a small turnstile within the track. We will mathematically analyze different versions to determine the solvability of different game types. This analysis will require the use and knowledge of the symmetric group on the set {1, 2, … , *n*}, *S _{n} *. (TALK)

**Cassie Thill** (Loras College)

*Counting Sudoku Puzzles*

In this talk I will be discussing Sudoku puzzles and solutions. I will be looking at how many Sudoku solutions there are and what it means for Sudoku to be “essentially the same.” For simplicity, I will be looking mostly at mini 4×4 solutions. I will then briefly discuss the difference between a puzzle and a solution, followed by a discussion of how many puzzles will lead you to a unique solution. (TALK)

**Kraig Thomas** (Simpson College)

*War Disease: A Spatial-Temporal Analysis of Civil War*

Studies of civil war commonly value socioeconomic conditions. This presentation discusses studies of civil war, spatial processes, and econometric analysis utilizing a case study of the Maoist insurgency in Nepal. We consider civil war much as a disease, spreading by a contagion effect, and we identify the onset and escalation of conflict much as the onset and escalation of an epidemic. Socioeconomic considerations of gross domestic product, education, and growth, indicators of opportunity cost, and other conditions, are included. We find that spatial and temporal effects, however, are more significant in explaining both the onset and escalation of civil war. Special thanks to Shikha Basnet for her time and her support of this project. (TALK-AM)

**Emily Tripp** (Simpson College)

*Parents: Peace, Love and Happiness Is Finally Attainable*

We consider the division of a single homogeneous but divisible object among several individuals who may have different value measures for the object. In this negotiation process, we will allow individuals to receive a larger fraction of the object in exchange for monetary compensation. A procedure will be illustrated that produces possible outcomes as starting points for negotiation where each outcome satisfies the definition of being an envy-free outcome. (TALK)

**Tram Vo** (Winona State University)

*Using mathematics to forecast the changing direction in Foreign Exchange Market
*

Foreign Exchange Market is one of the biggest markets in the world. The main objective of this project is to study the applications of continuity of polynomial function to forecast the changing of ratio in EUR/USD currency pair. Using historical data from last 5 years we were able to not only set up the movement function, but also determine the accuracy of forecasts and the affected spot by inputs such as unemployment rate, value of currency, etc. at specific times. We also define the converging of polynomial function to create long-term forecast (times and positions). Finally, we determine the conditions of changing direction and valuate the affective vectors. (TALK-AM)

**Amber Vogel** (Simpson College)

*Triangular Peg Solitaire*

Triangular peg solitaire is a game that involves a series of jumps performed on a triangular piece of wood with pegs placed in holes on the board. I will be discussing how the board can be labeled and classified according to those labels as well as the number of holes and or pegs on the board. I will also discuss what boards are potentially solvable and what those possible solutions are. (TALK)

**Peter Wiese** (Augustana College)

*Modeling Spiking in Neurons with a Poisson Process*

In the nervous system, nerve cells communicate through changes in ion concentrations called action potentials, or spikes. These spikes have been recorded and studied to understand the change in their distribution due to the presentation of a stimulus. By using a Poisson process, it is possible to model the distribution of spikes in time. Based on physiological properties, changes in the model are made to account for both the absolute refractory period and bursting of spikes. We will present several different models implemented on a spread sheet, both of a single neuron and of small systems of neurons. (TALK)

**Duncan Wright** (University of Northern Iowa)

*Structure in the m-ary Partition Function*

Similar to partitions of the integers, *m*-ary partitions are another way of studying the properties of integers in a very exciting way. Throughout this presentation, you should learn what an *m*-ary partition of an integer is and will be introduced to a clever function that comes out of studying these *m*-ary partitions. The structure of this function is very interesting and there are many amazing properties that have been noticed. I will show you how this function is built and many of the properties that have been discovered, as well as some explanation as to why some of them happen. (TALK)