## Senior Capstone Projects | Mathematics

The following is a compilation of mathematics senior projects.

### 2018

#### Brenna Calmer

##### Symmetry and Structure: characterizing graphs by automorphisms and fixing numbers

A subset S of the vertices of a graph G is a fixing set for G if and only if the identity automorphism is the only automorphism of G that fixes every vertex of S. That is, each automorphism of G is completely determined by its action on a fixing set S. The fixing number of G, denoted fix(G), is the cardinality of the smallest fixing set of G. Automorphism groups and fixing numbers allow us to describe the symmetry properties and structural complexity of a graph. We begin by examining the automorphism groups and fixing numbers of various classes of finite simple graphs. We then prove two results about graph complements which allow us to assert that all fixing number results for disconnected graphs must also hold for connected graphs. We also look at an interesting example that two graphs with isomorphic automorphism groups may not necessarily have the same fixing number. In addition, we explore bounds for fixing numbers and strategies for identifying fixing sets of a graph based on its automorphism group and vertex set.

#### Aulii Fisher

##### Fixing Numbers of Graph Classes

A relatively new topic in graph theory, fixing sets and numbers of a graph give a new way to determine the symmetries of a graph and thus may be applied to many of different applications. A subset S of vertices of a graph G is called a fixing set if whenever g,h in Aut(g) agree on the vertices of S, they agree on all vertices of G. That is S is a fixing set if whenever g and h are automorphisms with the property g(s) = h(s) for all s in S, then g=h. The fixing number of a graph G is the smallest integer r so that G has a fixing set of size r. First we show the fixing number of different classes of graphs. Then look into the how to apply these classes and other properties to finding the fixing number of graph products, specifically the cartesian product. Looking forward we investigate possible applications of fixing numbers.

#### Abigail Lian

##### The Hungarian Approach to High School Mathematics Curricula Through Combinatorics

Combinatorics is often excluded from traditional secondary mathematics curricula in the U.S., although it meets the goals for various common core state standards and NCTM standards. It was introduced in to the curriculum in Hungary in 1987, using a pedagogy that allows students to collaborate and challenge one another while working through problems. The main goal of this research was to examine how combinatorics could fit into current U.S. math curricula, and to evaluate the efficacy of the Hungarian Approach in mathematics education. We designed problem threads that we presented to high school students, and then analyzed their learning via techniques that include classifying common mistakes. We use our conclusions to pose areas of further research.

#### Kory Melton

##### A 3D Mass-spring Mesh Model of Eye Deformation

Blunt trauma to the eye inflicts long term damage to the eye which can even cause a loss of vision. During sports, like handball, the eye is highly susceptible to injuries resulting from blunt trauma. A recent study at Pacific found the traditional protective eye-wear used by handball athletes does not prevent impact injuries to the eye. Mathematical modeling methods can be used to research the effects sports-related blunt trauma has on the eye. Eyes are elastic solids whose deformations can be modeled using partial differential equations. However, modeling a system in this fashion is very computationally expensive. Thus, an alternative mass-spring mesh system can be used to simplify the calculations to coupled system of ordinary differential equations. A previous student at Pacific developed a two-dimensional mass-spring mesh model to measure deformations of the eye after being subjected to different hypothetical forces. My work extends this model to three-dimensions with a tetrahedral mesh of point masses connected by springs. Our model will be used in future projects to test and recommend design changes to protective eyewear in sports.

#### Emily Barry

##### Voting Rank Methods and the Borda Count

How should a group make a single selection from more than two options? There are many ways to do this and social choice theory is dedicated to the mathematical and logical investigation of these methods. One such method is the Borda Count, a so-called "rank method." The Borda Count is used, for example, in ranking college teams. A recent paper investigated the continuous mean of all rank methods. The authors proved that this mean is equivalent to the Borda Count. However, the use of a continuous mean incorporates unreasonable rank methods. In this talk, we consider the more reasonable discrete mean and show that this mean is also equivalent to the Borda Count.

#### Evan Carlson

##### The Dynamics of Electromagnetically Induced Transparency

At the edge between the fields of physics and computer science lies the field of quantum computing. The aim of quantum computing is to use light, instead of electrons, to transmit, interpret and store information. A central issue in the field is memory. There are currently several possible methods of storing light as a memory for a quantum computer. One potential candidate for an optical memory is a quantum phenomenon known as electromagnetically induced transparency (EIT). During EIT, a transmission window is induced onto an otherwise opaque atomic resonance. Light tuned to this transmission window experiences dramatically reduced group velocities, resulting in a measurable delay of light pulses. The optical properties of the medium are determined by a set of three coupled linear differential equations. We aim to both numerically and analytically solve these equations in order to characterize the optical properties of the medium and investigate the optimal conditions for EIT.

#### Prescott Devinney

##### System Dynamics Model of Schistosomiasis: Comparing Holistic Interventions to Piecemeal

We utilize a system dynamics model to analyze the relative cost and benefits of a holistic versus piecemeal approach to reducing the incidence of schistosomiasis. We develop a thorough understanding of the mathematics behind system dynamics, as well as its applications within mathematical epidemiology. Our use of this method for schistosomiasis is novel, though it has seen limited usage in epidemiology and public health. The implications for epidemiology and mathematics is that system dynamics may be an appropriate technique for quickly and intuitively modeling a system, and it is especially effective at communicating complicated ideas to a non-technical audience.

#### Tayla Kuyl

##### Mathematical Modeling of Parasympathetic Influence in Ventricular Cells

We develop a system of differential equations that models the action potential of ventricular cells when the parasympathetic neurotransmitter acetylcholine is released into the sinoatrial node via the right vagus nerve. This model is based on the model developed by Beeler-Reuter, and is coupled to a model of the sinoatrial node by Demir, Clark, and Giles. The effects of parasympathetic stimulation on the pacing of the heart are investigated and simplified versions of this model are explored.

#### Charles Morse

##### Crystalline Structure

Crystallography is the study of crystals and their structure. This understanding of the internal structure of crystals is important in chemistry and solid state physics. I wish to take the current mathematical understanding of crystals and how they are structured and apply it to crystals not yet discovered. I looked for combinations of two elements not found together in nature and through my study of crystallography I make predictions on the structure of the theoretical crystals and their various properties.

#### Victoria Prawitz

##### Settling for Fairness: A Settlers of Catan Analysis

Settlers of Catan is a popular strategy board game that, due to random board setup and the use of dice, involves probability. Often the game starts unevenly for the players, with the first player receiving a probabilistic advantage. Fair division is an active area of mathematics that addresses the problem of dividing a set of resources between several people. We explore several fair division algorithms, apply the Last Diminisher algorithm to the setup of Settlers of Catan, and investigate the consequences.

#### Jacob Richards

##### Modeling of the Effects of the Cardiac Caveolae on the Action Potential in the Sinoatrial Node Cell.

We developed a mathematical model of the cardiac action potential in the sinoatrial node cells which include the effect of the cardiac caveolae The effects of the caveolae can be modeled through probability density functions. These probability density functions allow us to model the randomness of the opening and closing of the thousands of caveolae formed around the cell. Lastly, these results are compared with those of a previous mathematical modeling study on stochastic caveolae in heart cells.

#### Brendon Walters

##### The Many Masks of Matroids

In comparison to many mathematical subjects, matroids are relatively new, having only been discovered in the mid-1900s. However, matroids are a very flexible subject, as they appear in several seemingly unrelated mathematical fields, including linear algebra, finite geometries, Latin squares, and graph theory. This presentation will look at the properties of matroids, some of the different areas of math to which they apply, some interesting results that have arisen from their discovery, and what these results can tell us about how the different fields of study intersect.

#### Daniel Yates

##### Light Diffraction Patterns for Telescope Application

Modern optical telescopes are giving us unprecedented imaging of the Universe in order to probe into the nature of dark matter and dark energy. Large optical telescopes require complex optical systems-often involving multiple lenses and mirrors. Light diffraction patterns on the focal planes influence imaging resolution, so a good understanding of diffraction is required to ensure high precision imaging. This project examines elements of diffraction-such as the Huygens-Fresnel Principle and the point spread function-to understand how they influence telescope optics. In addition, this project compares different diffraction theories, including Kirchhoff, Fraunhofer, and Fresnel diffraction, in order to determine similarities and differences between the diffraction theories.

### 2016

#### Jeremy Tate Campbell

##### Mathematical Optimization and the Evacuation Problem

Operations research (OR) is a field of mathematics centered about the idea of abstracting essential elements of a problem and analyzing them to yield an optimal solution for implementation. The specific branch of OR we studied was mathematical optimization, the process of maximizing or minimizing a real function subject to a set of constraints. An active area of research in applied mathematics is the application of the theory to model evacuations from buildings. One models them by casting physical locations into a network of nodes and arcs and using linear programming to find the optimal solution for evacuation. We applied this to the second floor of Price Hall, where we used linear programming to find that evacuation time versus building capacity have a somewhat linear relationship.

#### David Carr

##### An Actuarial Model of Student Loan Debt

As average college load debt has increased, default rates have risen quickly. In this presentation, we develop a mathematical model to help us explore the impact that rising default rates have on financial institutions. We consider several factors that are related to the rising default rate.

#### Bridget Daly

##### Use & Misuse of Regression for Chemical Analysis

It is standard practice in analytical chemistry to use linear regression, particularly to calibrate analytical instruments. If a regression line were used to estimate the output of the instrument for a known concentration of analyte, all would be well. However, chemists use this line in reverse, estimating the concentration of an unknown analyte from the output of the instrument. In this talk we will explore how this misuse of the regression line influences the estimation of concentration.

#### Emily Gaub

##### One Rook, Two Rook, Red Rook, Blue Rook: An Examination of Distinctly Colored Rook Polynomials

Rook polynomials are closely connected to placing rooks in a game of chess. First, consider a chessboard of m by n dimensions. Rooks are placed on the board such that they are non-attacking, meaning no two rooks share the same row or column. Then in a real game of chess these rooks could not attack each other. A rook polynomial is a generating function where the coefficients of the degree r term give the number of arrangements for r non-attacking rooks on a given board. This is the standard way to examine rook polynomials. For my original research I consider the case where the rooks are distinctly colored. In particular, there is a set of rooks of two colors, where rooks of the same color can be in the same row or column and not be considered attacking. Rooks are only attacking when sharing a row or column with rooks of a different color. Results include the maximum number of distinctly colored rooks which can be placed on a given board, the number of arrangement of rooks, expressing those values as a polynomial, and expressing rooks polynomials as graphs.

#### Ashley Grogan

##### Gaud

Antoni Gaudi is one of the most famous architects in history. His influence in Barcelona, Spain has become iconic through different pieces such as the Sagrada Familia and Palau Guell. The varying arches within these buildings have left tourists agape, but their significance lies further than just the definition of their visual beauty. From various images of his arches, we are able to plot data sets and construct graphs in order to dissect the underlying mathematical structures. After deriving the equation of a catenary from optimization principles, we utilize the Least-Squares technique to discover the best-fit catenary to the curve of each arch. The catenary is a slice of the catenoid: a surface that minimizes total area in Euclidean space. We find that some of Gaudi's arches are closely related to parabolas and surprisingly, we discover an arch better suited to a hybrid of the parabola and catenary functions. The use of mathematics to create these designs has resulted in structural and visual approval from guests, designers, and many aspiring architects.

#### Hannah Herbert

##### Modeling Deformations of the Eye

Modeling membranes and various organs of the body can be useful for predicting the extents of injuries from forceful impacts. In particular, eyes are especially vulnerable to potential impact injury in sports such as handball. The accurate formulation of a model for the eye would allow for accessible testing and for the design of more effective eyewear protection. Since eyes are examples of elastic solids, their deformation in response to forces can be characterized by systems of partial differential equations. However, finding solutions to such complex problems are most often computationally expensive to solve. Thus, an alternate approach, such as the mass-spring system can be applied. The mass-spring model reduces the problem to a large system of coupled ordinary differential equations of which we use in our model. The basis of our model originated from Sathyanarayanan and his colleagues' model with the exception that we used a different spring mesh configuration to simplify the calculations. We implement a discretization of our model using several numerical methods to analyze the damage done to the eye in several hypothetical impact scenarios.

#### Marisa Huffman

##### The SURF Feature Detector and Descriptor

Computer vision consists of methods for processing and analyzing images in order to produce numerical information. The areas of item detection and recognition have seen recent advancements such as the development of the Speeded Up Robust Features (SURF) algorithm.

#### Ian Kit Nicolas

##### Graph colouring with fewer colours than the chromatic number

We determine a type of proper graph colouring that makes use of a number of colours less than or equal to the chromatic number. Proper graph colouring refers to the assignment of labels onto vertices of a graph such that adjacent vertices have distinct labels. This proper graph colouring, called versicolouring, emerges from a natural way to combine colours via combinations, not unlike combining colours of paint. Versicolouring assigns sets of colour combinations to each vertex such that the sets are completely disjoint for adjacent vertices. In turn, we also introduce a weaker version of versicolouring, called weak-versicolouring, which relaxes the completely disjoint condition. Similar to the chromatic number, graphs have a versicolour number (and weak-versicolour number) which denotes the minimum number of colours needed to versicolour (and weak-versicolour) a graph. We determine the necessary and sufficient conditions under which the versicolour number and weak-versicolour number is at most the chromatic number for any (simple) graph.

#### Esbeida Ramos

##### Colors and products, and graphs, oh my!

Graph coloring is a type of graph labeling where vertices are colored so that no two adjacent vertices are the same color. The smallest number of colors needed to color a given graph is called the chromatic number. There are several ways to combine two graphs by taking their product; we focus on the tensor product or direct product. We explore the properties of the direct product and some ways to determine bounds for the chromatic number. These ideas lead us into a world of unanswered questions. An unproven but widely accepted answer to one of these questions is famously known as Hedetniemi's conjecture and was first posed by Stephen T. Hedetniemi in 1966. His conjecture serves to show that no tensor or direct product can be colored with an unexpectedly small number of colors. Furthermore, we also ask what happens when there are not enough colors available? We use an approach called versicoloring, which uses a graph's color set to create more color options, to answer this question. Our results find a way to examine products of versicoloring problems, as well as methods to calculate the chromatic number of a direct product.

#### Crystal Susbauer

##### Graph Products and Colorings in Relation to the Hedetniemi Conjecture

There are many types of product operations defined in graph theory. These products use the vertices and edges of two (not necessarily distinct) graphs to create a new graph. We focus on four important types of graph products: the Cartesian product, the direct product, the strong product, and the lexicographic product. Additionally, we explore ways of labeling graphs through graph coloring and list coloring, and investigate the chromatic number of a graph. Finally, we relate both topics to one another by examining the Hedetniemi conjecture and researching what work as been done to try to prove or disprove the conjecture.

#### Emma Winkel

##### Pi: Its History, Properties and Introduction in the Middle School Classroom

In this presentation we examine the mathematical constant pi in a variety of contexts. We discuss the history of the use of pi, proofs of some of the most important properties of pi, and the necessary conceptual understanding of pi in an educational setting.

#### Wubin Zhang

##### Combinatorial Game Theory

Losing a game can be much more difficult than winning a game. In combinatorial game theory, trying to win is called "normal play" and trying to lose is called "misere play." The theory of normal play was well understood by the 1950's, while it took another fifty years to make progress in the theory misere play. We investigate new techniques developed to study misere play and use them to solve the misere version of the game Kayles.

### 2015

#### Blake Anthony

##### Quantum State Tomography

Data rates in electronics are reaching their maximums which is causing a shift towards using light to carry information in technology such as computers and televisions. The goal of my project was to develop a technique to determine the frequencies present in the light. The method of storing information in light is to send it through a gas such as rubidium. To understand the effects of the rubidium gas on the information contained in the light, we modeled light going through the gas using Maxwell’s equations. Then we used an array of light detectors to create an intensity graph of two light beams coming onto the detectors at an angle of 12 mrad. The first beam was a high intensity beam with one frequency mode while the other beam was a low intensity beam comprised of multiple frequency modes. From the interference of the two beams, we extracted the frequency modes present with the Discrete Fourier Transform in order to see how the frequencies were affected by the gas.

#### Stephanie Carter

##### Nonlinear Model for Combined Radiopharmaceutical Therapy

This is an extension of a linear model for combined radiopharmaceutical therapy in treating neuroendocrine tumors using 131I-metaiodobenzylguanidine and 90Y-DOTA-D-Phe1-Tyr3-octreotide. Like the original linear model, this model attempts to optimize the amount of radiation delivered to the tumor without exceeding the critical dose absorbed by other organs. However, in this nonlinear model, we are modeling the effects of not just one, but two treatments of combined radiopharmaceutical therapy. The second treatment is based upon the amounts of each radiopharmaceutical used in the first treatment, and the diameter of the tumor before and after the radiation is delivered. The result is an optimization problem in four variables that will provide additional information and treatment options that the linear model was unable to predict.

#### Spencer Chandler

##### Patterns in the (a,b)-Pascal Triangle modulo a prime

The Fibonacci sequence, an important sequence of integers, is generated under the recursion Fn=Fn-1+Fn-2.  It is well known that the Fibonacci sequence can be found as backward diagonal sums in Pascal’s triangle.  In this talk, we consider a generalization of Fibonacci numbers called the (a,b)-Fibonacci numbers.  We construct a modified Pascal’s triangle in which the (a,b)-Fibonacci numbers can be found and investigate patterns in this triangle modulo a prime.

#### Julie Davenport

##### Radionuclide Therapy: Optimization of a Dosing Schedule

In this paper, we propose a mathematical model that minimizes the size of a tumor after two radionuclide treatments. In this investigation, we will show that the way to treat a neuroendocrine tumor with dual-agent therapy is by giving the second dose as a fraction of the maximum dose before the tumor has recovered from the effect of the first dose. The amount of this second dose is determined by patient-specific parameters and the rebound rates of the affected critical organs. Our method use mathematical computation software to optimize the effect of the therapy. Our results give an optimal schedule for two treatment therapies.

#### Brandon Harms

##### Multimode Quantum State Measurement

In our current electronic world, we have found a need to strive for faster speeds. One way of accomplishing this task is by transmitting information via light photonics. In order to advance this technology, it is necessary to be able to characterize light that has been stopped and stored. In my project, we were able to characterize an original weak, signal beam of light by interfering it with a strong reference beam and observing the intensity pattern formed in space on our array detector. We determine the spatial frequencies or "modes" associated with this interference pattern using Fourier Analysis. With our array detection technique, we are able to plot the average photon count vs the plane wave mode angle for every mode that contributes to the pattern.

#### Victor Rielly

##### Differential Equations

We present a few rudimentary and general methods of solving first order linear and nonlinear differential equations which satisfy a specific form. The analysis is designed to present the reader with ideas more so than to present a complete and all-encompassing theory of the differential equations we study. Never the less, we present very general tests which can be run on the differential equations in question. If the equations satisfy the tests, the solutions to the differential equations can be written out directly. The analysis can be extended to a large class of differential equations, and can already tackle the general first order linear differential equation, the Bernoulli equation, separable differential equations, and a few differential equations we may be the first to solve. We end the paper with an extension to higher order differential equations and a discussion of the generality of the methods.

### 2014

#### Alex Caroline

##### Periodic out billiards in the hyperbolic plane

The topic of outer billiards is a relatively new field in geometry with many results waiting to be discovered.  In the broadest sense, outer billiards can be thought of as a crude model on how objects will orbit around a body.  Two common questions about orbits usually arise: which orbits will eventually repeat, or be periodic, and which orbits will eventually become unbounded, getting farther and farther away from the body.  We focus on which orbits in hyperbolic space will be periodic.  We define a particular type of orbit called a special orbit.  We show that if a convex polygon has a special orbit, then it all of its orbits are periodic orbits.

#### Liam Dalton

##### Iterative 3D Scanning with Structured Light

3-dimensional models of real-world objects can be acquired using scanning techniques of varying cost and quality. In this research, we reproduced the cheapest of such techniques: structured-light scanning. We improve on preexisting methods by computing the error in the model as well as the model, and then using this error to determine where to execute further scanning.

#### Eva Forrester

##### The Mathematics of Decision-Making

Linear programming has been widely recognized as one of the most important scientific advances of the mid-20th century. This remarkable field in mathematics has given rise to the tools that have become industry standard in the field of Operations Research. These tools have application to a broad class of decision-making problems pertinent to nearly all industries. The most common type of application involves the general problem of allocating limited resources among competing activities in the best possible (i.e. optimal) way. Our research examines the mathematical theory behind linear programming as well as the many forms these problems can assume. The logic and methods for solving such optimization problems are then discussed in the context of applications to business and manufacturing.

#### Taylor Matyasz

##### On the Number of Minimal Solutions to Diophantine Inequalities

Diophantine equations, those in which only integer solutions are of interest, have been studied since the 3rd century.  Also of great interest are Diophantine inequalities. Indeed, these inequalities arose even in senior capstone projects.  In 2013, Evan Cooper encountered a Diophantine inequality in his work to improve upon a college football ranking system.  Cooper needed to determine the so-called minimal solutions to his inequality.  Inspired by his work, we seek to determine the number of minimal solutions to a general linear Diophantine inequality. We begin by investigating the case of a two dimensional, linear, Diophantine inequality, and find a simple closed expression for the number of minimal solutions.  For higher dimensional problems, we are able to determine a recursive formula for the number of minimal solutions. Finally, we make use of Ehrhart polynomials to find a closed form for both an upper and lower bound on the number of minimal solutions to a general linear Diophantine inequality in n variables.

#### Jeff Mintz

##### Synthetic Wavefront Reconstruction

Traditional 3D displays create the illusion of depth by providing separate images for both eyes, however there is currently no method to produce imagery with the proper focal distance.  Focusing light requires rays come together at the correct angles, creating a wavefront that converges: a wavefront with curvature.  In this project we investigate the use of lenslet arrays to add curvature to the wavefront created by an ordinary display screen.  We show that by selection of the proper pixels, rays can be made to converge at a point in front of the screen.  A vector version of Snell’s law is derived and used to predict the angle at which light from an arbitrary pixel will leave the lens-screen system.   The resulting pixel-to-angle map will facilitate the development of future content for lenslet based display systems.

#### Matthew Morse

##### Modeling the One-Dimensional Propagation of the Caveolae-Inclusive Cardiac Action Potential

Two models (based off of the Luo-Rudy 1 guinea-pig ventricular model) were produced to analyze the effects of caveolar sodium current on a single cardiac action potential and the one-dimensional propagation of a cardiac action potential in a line of cardiomyocytes. Evidence suggests that the opening of caveolae recruits additional sodium channels on the cardiomyocyte membrane that can affect both the peak voltage overshoot and the maximum upstroke velocity of the cardiac action potential—the maximum upstroke velocity in turn can alter the conduction velocity of an electrical signal. We examined two opening mechanisms of caveolae. The first opening mechanism simulated a 1-cm2 patch of membrane perfused with a β-adrenergic agonist that opened a certain number of caveolae on the membrane. The second opening mechanism simulated a 1-cm2 patch of membrane with stochastically opening caveolae that open according to a Poisson process. The effects of these two opening mechanisms of caveolae on a single cardiac action potential using the LR1 model were compared to previous computational results using the Pandit et al rat ventricular model. Our simulations (which incorporated varying capacitance) revealed a 4.1% increase in peak voltage overshoot and a 19.1% increase in the maximum upstroke velocity for a 42% increase in sodium current due to β-adrenergic stimulation. Incorporating stochastically opening caveolae, we observed a delay in ventricular repolarization, a characteristic of a serious heart condition called Long QT Syndrome.

#### Scott Perez

##### Chop it up: An outline of Geometric Dissections

Geometric dissections involve the cutting of one or more figures into pieces that can be rearranged to form other figures.  Dissections can be used to provide visual proofs of mathematical identities, such as the Pythagorean Theorem, and to reveal previously unknown identities.  They can help those not familiar with advanced mathematics to understand certain fundamental results.  In this presentation we introduce the method of dissections and illustrate several interesting examples.  Finally we discuss the Wallace-Bolyai-Gerwien Theorem, which provides the foundation for the method of dissections.

#### Daniel Pitluck

##### A Study of DeBruijn Graphs and Sequences

At the crossroads of mathematics and computer science lies graph theory. Graphs are composed of vertices, or nodes, and edges, which link vertices. Common applications include network models and computer algorithms, and graph theory offers solutions and insights to many practical problems.  A relatively new area of graph theory, deBruijn graphs and sequences originally “solved the problem of finding a minimum-length binary string that contains as a (continuous) substring every binary string of a prescribed length-k.” These graphs have length-k bits as vertices composed of an n-letter alphabet. Character shifts indicate the direction of travel from one vertex to another. Unique characteristics make them strong candidates for modeling peer-to-peer computer networks. Properties, characteristics, and applications of deBruijn and similar graphs are presented.

#### Connor Ryan

Geometric group theory relates algebraic objects to geometric objects and provides a way to use geometry to study algebra.  Given a finitely presented algebraic group, one can define its Cayley graph and study the geometric properties of this graph.  Certain vertices of a Cayley graph are called dead ends and the corresponding elements in the associated group are algebraically important.  We examine dead ends in finite lamplighter groups classify these elements with respect to the natural generating set of the group.  We also investigate the ratio of the number of dead end elements to the order of the group.

#### Joshua Siva

##### Moore Than You Needed To Know About Kautz and De Bruijn Graphs

Numbers and words sometimes fall short when tackling particular problems in mathematics and computer science. For instance, say we want to represent a collection of computers and the connections between them. We could start describing them in words, but instead we put pencil to paper and draw dots (dots, of course, being the galactic standard for representing random objects) to represent the computers and lines connecting the dots to represent the connections. We now have a clear picture of what is going on with what we call a graph! Now let’s say we are faced with the problem of trying to find a path through this graph that hits every dot and returns to the first dot we started at. What if we complicate the graph by forcing the connections to have directions? A logical question is how could these computers be better connected so as to allow for higher connectivity? What if we have a cable shortage and we need to connect the maximum amount of computers without sacrificing connectivity too much? Kautz and De Bruijn graphs model these situations and Moore. These two graphs are used to structure networks such as parallel computing architectures and Peer to Peer networks because they come very close to the Moore Bound. This means the graph has high connectivity in the network in addition to the guarantee on the maximum distance a message must travel before getting to its destination. The Kautz graph is better than the De Bruijn graph in this respect. On the other hand, the initial inspiration for the De Bruijn graph, the De Bruijn sequence, illustrates that there is more going on in these graphs that there appears to be. We construct these graphs, show what questions they can answer, and we tie together the concepts of sequences, necklaces, and Moore graphs.

### 2013

#### Evan Cooper

##### College Football Ranking Systems: Improvements of Two Models

Over the years, the Bowl Championship Series has been widely criticized. Many fans felt that their teams were poorly represented in the rankings and thus left out of more prestigious postseason games. As a result of this dissatisfaction, the NCAA is changing the way the national champion in Division I college football will be determined. Starting in the 2013-14 season, the national champion will be the winner of a four team, two-round playoff. A committee of experts, presumably using a ranking system, will select the teams to participate in the playoff. In this talk we explore two existing mathematical models that can be used to rank college football teams. We discuss methods that improve the accuracies of these models including the use of Diophantine equations in weighting margin of victory and using least squares regression to determine parameters involved in predicting the margin of victory of a game.

#### Briana Flores

##### Tsunami Forecasting: Linear vs. Non-linear Models

Reflecting on tsunami events such as the Tohoku tsunami of March 2011 or the Sumatra tsunami of December 2004 reminds everyone of how destructive and devastating a tsunami can be. Mathematical models provide different types of information regarding tsunamis such as the locations that it will affect, estimated time of arrival to shore, height, and power. By relying on the results of their mathematical models, Tsunami Watch centers are able to determine what types of precautions residents should take. Models can either be linear, which are numerically more stable and faster to compute, or non-linear, which are generally more accurate. Time is of the essence for tsunami forecasting, so a faster model could save lives. So, when do linear models produce good approximations of non-linear models? To study this concept, I use software from the Pacific Tsunami Warning Center and a model based off the Shallow Water Wave Equations to compare simulations.

#### Hanna Landrus

##### Hitches Holdin

A hitch is a tangle about a pole. When climbing a hitch could save a climber's life. When sailing a hitch could prevent a boat from floating away. We will present a model for determining when a hitch will hold and relate the determinant of a certain matrix to the friction required for a hitch to hold. We will consider sequences of hitches and investigate the minimum friction required for the hitches to hold. This involves examining a sequence of polynomials whose coefficients are related to Pascal's Triangle.

#### Kaci Linton

##### Mathematical Illiteracy causes Financial Collapse: Incorrect Implementation of the Black-Scholes Model

The Black-Scholes formula is a mathematical model used to find the value of a European call option in a manner that nearly eliminates risk. Upon publishing the formula, the buying and selling of options increased significantly. The Black-Scholes formula assumes ideal market conditions and derives a partial differential equation, commonly referred to as the Black-Scholes construction; the solution to the PDE is the Black-Scholes model. We investigate what is needed in order for the model to produce accurate results and find that the parameters must follow a normal distribution. It is thought that invalid assumptions of ideal conditions in the market directly contributed to the financial market collapse in 2008. We attempt to determine which parameters do not follow normal distributions and analyze the weaknesses of the Black-Scholes model. In the market situation of 2008, the stock price followed a fat-tailed distribution. Often compared to the normal distribution, fat-tails exhibit a large kurtosis which means observations far from the mean are more likely to occur as compared to normally distributed occurrences. This lead to an incorrect implementation of the Black-Scholes model and ultimately contributed to the financial market collapse in 2008.

#### Jack McCorkel

##### Gambling Partitions: Calculating Electorate and Subelectorate Voter Turnout

Social scientists have long struggled to explain the act of voting. Regardless of the statistical methods used, any model of voting becomes incredibly complex, requiring submodels describing preference, contributing socioeconomic factors, and the importance of a given vote. I begin by looking at the issues of voter turnout through the lens of economics applied to democracy. Describing the choice to vote as a gamble, I look toward econometrics in order to better understand turnout. In an attempt to reconcile data with theory, I also explore the mathematics of incorporating demographic interests into turnout. Finding the current mathematical models lacking in practicality, robustness, and depth, I offer a new formulation and investigate this model's ability to predict voter turnout and preference.

### 2012

#### Jesse Amano

##### Connecting Block Designs and Matroids

Block designs, abstract structures with applications in experiment control and data gathering, have in many cases been shown to be isomorphic to matroids, another broad family of abstract structures. We explore and discuss some of the ways a specific class of designs can also describe a matroid, and some of the ways a specific type of matroid may also constitute a design. We also consider cases where a matroid exists but not a design, and where a design exists but not a matroid. Because certain algorithms are known to produce or manipulate matroids, these may be useful in constructing certain types of designs. Meanwhile, careful study of designs may lead to useful theorems that apply to matroids.

#### Tony Fernandez

##### How Does your Triangle Like To Tile

We investigate which triangles can tile the Euclidean or hyperbolic plane without overlapping and determine exactly which triangles tile without overlapping. Euclidean triangles and hyperbolic triangles have different and this leads to contrasting results about tilings. In particular, while the Euclidean plan can be tiled by arbitrarily small triangles, we show that there is a least area triangle which tiles the hyperbolic plane without overlapping.

#### Kellie Takamor

##### The United States vs Australia: A Comparison of High School Math Curricula

I compare the numeracy levels of public high schools in Australia and the United States by analyzing nationally implemented tests. In particular, I investigate the differences in math curricula and assessment, and correlate it to factors such as high school graduation rates. Because Australia arranges math topics in an integrated fashion as opposed to the United States' organization of math topics by subject content, I first assess the content of Australia's national numeracy test and correlate the content to U.S. math subjects such as Algebra I, Geometry, Algebra II, etc. In particular, I analyze year 9 in public high schools of the state of Oregon to represent the United States and the state of South Australia to represent Australia. I create my own scale of 'proficiency' levels for Australian students by subject and compare the data to the proficiency levels in the U.S. The tests that I use are Australia's National Assessment Program Literacy and Numeracy (NAPLAN) Test and America's Oregon Assessment of Knowledge and Skills (OAKS) Test. I then statistically analyze the benefits of different assessments and curricula, relating the results to high school graduation, first year college enrollment, and proficiency statistics.

#### Qianru Wang

##### Delannoy Numbers: Counting Certain Restricted Paths in an Integer Lattic

Delannoy numbers count the number of paths in an integer lattice from one point to another where legal moves are to move one step up, one step right, or one step diagonally up and right. In this project, we give combinatorial proofs for formulas for the central Dellanoy numbers and general Dellanoy numbers. We proceed to use generating functions, a powerful tool from combinatorics, to find a different formula for the general Delannoy numbers. The latter process generalizes to dimensions higher than two. Finally we consider what will happen if we add an additional legal move. Can we find a formula through the same methods?

#### Travis White

##### Juggling Mathematics: Counting Juggling Sequences

It may come as a surprise that juggling and mathematics are related. But many famous mathematicians are also expert jugglers. In fact, Ronald Graham served as president of both the International Jugglers Association and the American Mathematical Society. The relationship may be because juggling is a repetitive act embedded with patterns. Jugglers describe these patterns using sequences. In this talk we will learn how to categorize juggling patterns by the number of balls being juggled, the length of the pattern and when the balls are scheduled to land. In addition, we will see how to fix a landing schedule and use generating functions to count the number of possible juggling patterns of various periods. These numbers sometimes show up in surprising places and we will investigate a few of these instances.

### 2011

#### Lana Carter

##### A Cops and Robbers Game

Pursuit invasion games are used to model or explain a variety of real life situations where a team of pursers tries to capture an invader. The math that goes along with these games have been studied immensely throughout the years, tracing back to the work of Pierre Bouguer, who in 1732 studied the problem of a pirate ship pursuing a fleeing vessel. Pursuit games have been associated with many different categories and have thus been studied in a variety of ways and techniques. A discrete version of the game was introduced by Nowakowski, which eventually paved the way to the cops and robbers game. This project follows the work that was done by Tina Zhang, of Bard College, who studied the number of cops needed to catch one robber on a finite graph. Taking into consideration the rules of the game, the goal of this project is to calculate the cop number for various classes of graphs and graph minors to generalize theorems or rules that can be used to find the number of cops needed on a certain graph.

#### Rebecca Hoffman

##### Synchronous Fireflies

Fireflies in nature act in a very interesting way. When large groups of fireflies have landed together, the male fireflies begin to blink in synchrony and then continue in this behavior. I have studied the mathematical background of dynamical systems in order to understand a way to model this behavior. That is, fireflies will speed up or slow down their blinking in order to flash together with a stimulus. This behavior can be modeled using an oscillating sinusoidal function. I have manipulated this model further to incorporate a triangular piecewise function to see if it will model natural behavior better. Further, I have used Actionscript 3.0 to visually model this behavior in order to match the natural rhythm.

#### Kelsey Kaku

##### Polynomial Interpolation

Polynomials are used to define various functions in subjects such as chemistry, physics, economics, and social sciences. In the real world, a polynomial is often fit to match finitely many data points, a process known as polynomial interpolation. Polynomial interpolation is mainly practiced within the field of real numbers. It is a well-known fact that for k points and x-distinct coordinates, there exists a unique polynomial of degree k - 1. However, what happens when we replace the field of real numbers, with the ring of integers modulo n? Does the existence and uniqueness we had over the real numbers still hold for the ring of integers modulo n?

#### Duncan McGregor

##### Renormalization Group Flow on a Homogeneous Space

A geometric flow is a system of differential equations which bends a space by changing its metric (the ruler by which distance in a space is measured). The Ricci flow is a geometric flow which "bends" a space according to its curvature, essentially smoothing. This geometric flow was used by Perelman to solve the Poincaré conjecture. The Renormalization Group flow is a generalization of the Ricci flow used in quantum field theory to model changes to the metric caused by quantization a classical action. We investigate existence and uniqueness of solutions of the RGF on constant curvature and homogeneous spaces.

#### Jennifer Novak

##### Modeling Capillary Waves

I look at an equation, called the dispersion relation, which is used to related wave frequency to wave propagation speed with the forces being gravity and capillary action. I first will derive this equation by looking at Rayleigh's The Theory of Sound, and then modify the model to incorporate and arbitrary force in the vertical direction. An application to this process is for agricultural engineering where we study particle size.

#### Brandon Oshiro

##### What's Changing? The Max Flow Problem

Given a single source, single sink flow network, the maximum flow problem's goal is to find the maximum amount of flow from the source to the sink within a network. Since the 1950s, the maximum flow problem can be used for real world situations such as sewage pipes and vehicle traffic within a city. The Ford-Fulkerson algorithm was first created to find the solution to the maximum flow problem. When given a flow network G with a set of vertices V and edges E, we start with f(u,v) = 0 for all u,v contained in V, giving an initial flow of value 0. At each iteration, the flow value is increased by finding an augmenting path. Then the process is repeated until no augmenting path can be found. Throughout the years, new algorithms such as the Dinitz blocking flow algorithm and Push-relabel maximum flow algorithm, have made solving the maximum flow problem much more efficient. This project will look at these algorithms, as well as other algorithms used to solve the maximum flow problem and analyze what differences caused the change in efficiency.

#### James (Alex) Patton

##### Derivations of the Zeta Function

Dirichelet series are infinite series of the form sum(a(n)/n^5, n=1..infinity) when a(n) is an arithmetic function. We note that the Rieman-Zeta function-ζ(s)-is a Dirichelet series with a(n)=1 for all n. Function convolution allows for a useful arithmetic involving Dirichlet series, which Emmons used along with log derivations to create a new representation of ζ(s). By cleverly defining our log derivations, we can create new representations of ζ(s) and show their absolute convergence.

#### Summer Steenberg

##### Blocks and Bases and BIBDs, Oh My!

When does a matroid constitute a block design? Matroids can be found throughout mathematics, constructed of a family of independent sets defined on a ground set. They are essential in our understanding of combinatorial optimization and bridge various fields of discrete mathematics. Block designs allow us to divide a set of varieties into different groups, called blocks. When each block does not contain every element, they are all the same size, and each pair of varieties appears the same number of times we have a BIBD (balanced incomplete block design). Creating the categorization of sets into equal sections, block designs are utilized in practical studies such as shampoo and fertilizer testing. In this talk we explore which types of graphic matroids constitute block designs and what properties are necessary for this to occur. When do the bases of the matroid form the blocks of a design? Is there another way to relate the two? Previous knowledge of matroids or block designs is not required, simply an interest in some graphic material!

#### Catherine Tardif

##### The Real Beyond the 15-Puzzle

The 15-Puzzle has been an object of mathematical study for over 100 years. One can classify which initial positions are solvable, and bound the number of moves needed to solve a puzzle depending on where the pieces lie on the puzzle. We are working to determine how much the puzzle would be simplified, if at all, were another piece to be removed. This removal would make the (n^2-1)-Puzzle (where n = 4, for the 15-Puzzle) into an (n^2-2)-Puzzle. By appropriately modifying the proofs for the (n^2-1)-Puzzle, we work to explicitly quantify how much simpler the (n^2-2)-Puzzle is.

### 2010

#### Michaela Balkus

##### Shuffle Your Way to Order

Throughout time, millions of people have played card games. Since most card games begin with a deck in random order, methods of shuffling cards have been a subject of particular interest. In this talk we will explain and explore the “pinch” shuffle introduced in Number Theory by George Andrews. Our main focus will be the following, how many shuffles does it take to get particular deck of cards back to its original order? We will answer this question for a deck of any size and continue further by discussing upper and lower bounds for the necessary number of “pinch” shuffles to return a deck back to its original order.

#### Stephanie Lowery

##### Tiling Fibonacci Theorems And Generalizing The Theorems To Include M-Ominoes

A common childhood game is dominoes. What if you wanted to construct a row that was eight inches long using dominoes that are two inches long and squares that are one inch long? How many different ways could you combine squares and dominoes to construct such a row? We begin this talk by answering this question in relation to the Fibonacci numbers. We then look at some known theorems related to Fibonacci numbers and discuss how our tilings can be used to understand why they are true. This idea is then expanded using m-ominoes, dominoes of length m, to generalize the previous results.

#### Cody Stein

##### Measuring Symmetry: The Distinguishing Number

What makes a graph symmetric? If a graph can be rotated, reflected, or is in some other way symmetric, what special properties are implied about the graph? Distinguishing graphs is the process of observing these inherent symmetries and assigning a coloring to the vertices such that these symmetries no longer exist. The distinguishing number of a graph, denoted D(G), is the minimum number of colors, r, required to distinguish the graph G. We define an algorithm that will determine if a specific coloring of a graph is distinguished by systematically determining if each similar vertex and set of vertices are distinguished. With these properties, we will formally prove the distinguishing number of the cycle graph, Cn, the wheel graph Wn, the complete graphKn, the complete bipartite graph Kn,m and the star graph Sn. A greedy algorithm for finding D(G)of a tree graph has been found. We seek to elaborate on or create a new algorithm with which to find the distinguishing number of a more general graph.

#### Chris Upshaw

##### Datatypes As Mathematical Objects

Programmers use datatypes to describe what values a particular program uses. This allows the computer to know how to store and work with those values. By investigating this practice mathematically we can discover fascinating and useful new mathematical objects, and find that some familiar ideas have much broader applications then one would first think. Unfortunately there exists very few introductions to this field. I attempt to provide a overview of the basic ideas and show some examples of the types of results this investigation has yielded, in particular showing how to do algebra and calculus on datatypes.

### 2009

#### Marisa Allen

##### Global Warming: the Math Behind Climate Modeling

Global warming is a controversial topic, in part because the complex mathematical concepts that are used to explain this phenomenon are rarely made accessible to the general public. In this project, we explore the mathematics and scientific principles behind climate models, in an attempt to demystify global warming. In particular, we will look at the equations used in complex climate models, as well as a simple radiative forcing equation used in general climate models (GMC’s). We then investigate the effect of rising CO2 levels on the governing equations of climate change.

#### Kristen Almgren

##### Partitions: How many different ways can you get to the center of a tootsie roll pop?

In this presentation we introduce the idea of partitions as well as generating functions. With these ideas in mind we examine a variety of examples in order to give a summary of the techniques that can be applied to partitions and generating functions. With these techniques in mind, we present a problem relating two sets of partitions. After an examination of the known proof of the problem, we extend the problem to a more general setting.

#### Jacob Artz

##### Counting Squares and Dominoes: What's math got to do with it?

While arranging dominoes and squares in different patterns is not necessarily esoteric in nature, its application in mathematics is. We examine a specific way in which squares and dominoes are used, namely through tiling.

We begin with a brief examination of the Fibonacci sequence and its combinatorial representation. We show that the number of ways to arrange dominoes and squares on an n-length board is equal to the nth Fibonacci number. From this basic relationship we determine a number of identities involving Fibonacci numbers and strategies to find these identities. Some methods we use are breaking an n-length board into different portions, considering the position of a specific tile, and finding correspondences between two sets of tilings.

We also explore Zeckendorf's Theorem and its application to combinatorics and specifically tiling. This allows us to produce an explicit definition for some Zeckendorf family identities.

#### Brittany Cuff

##### List Coloring and Rook Polynomials: Using chess to determine how many ways to color a graph

How many ways can you color a map? How many ways can you place r rooks on an mXn chessboard?

In graph theory, we color vertices of a graph in such a way that if two vertices are adjacent, they are colored differently. List coloring is a restriction of this coloring where not every color is available to each vertex. We now assign each vertex a list of allowed colors in which it may be colored.

A rook polynomial is a generating function that represents the number of ways we can place r non-attacking rooks on an mXn chessboard, where non-attacking is a placement such that no rook shares a column or row with any other rook.

We connect list coloring to rook polynomials by transforming the relationship between a graph's vertices and associated list assignment into a rook board. We have proved that a complete graph G that has a valid list assignment will result in a rook polynomial whose leading coefficient gives the number of proper colorings of G. We take this result further by determining the number of proper colorings of G if G is not complete via inclusion.

#### Kerensa Gimre

##### Answering a burning question: Analyzing methods to estimate remaining oil reserves and peak oil production

Peak oil (the time when half of all oil exploitable oil is expended) was introduced in the 1950s by M. King Hubbert who wished to predict the time of maximum oil production for both the United States and the world. Correctly following Hubbert's prediction, US oil production peaked in the 1970s. There currently exist a variety of estimates for the timing of world peak oil production. Due to the vast economic implications of running out of fuel, peak oil is a critical problem.

We investigate methods to estimate remaining amounts of untapped oil supply, specifically the Level Set Method. Developed in the 1980s, the Level Set Method has numerous applications in fluid mechanics, materials science, computer vision, computational geometry, computer-aided design, and image processing. By numerically solving the Hamilton-Jacobi equation and applying an appropriate velocity function dependent on the curvature (curvature in this instance depends on image intensity), an image can be analyzed to eliminate noise. This process assists in "cleaning up" a seismic image of the earth's subsurface. If these images are cleaner, more accurate approximations for subterranean oil can be found. This information is vital to oil companies when deciding if it cost-effective to drill a field, and is also important when predicting total remaining subterranean oil.
We conclude by estimating the timing of peak oil using the results of the Level Set Method and analyzing the popular Hubbert's Method.

#### Bobby Larkins

##### Pondering predictions of the path and pain of powerful pivoting puffs to possibly protect the population (hurricanes)

Hurricanes have been killing people and destroying places for far too long. I, how-
ever willing, can not change this fact. Improving the system in which the people are
warned was the next best thing, limiting damage and saving lives. Currently, the pub-
lic is alerted with the Saffir-Simpson Scale. I will be comparing that to the new idea
of integrated kinetic energy or IKE. We will look at some or the raw data collected,
see how that generates the IKE values, and make a conclusion as to the method that
will be better for warning the public.

#### Stephanie Murayama

##### Is it a really a small world after all? A study of small world networks

Six degrees of separation is a theory that everyone is separated by at most six people. You might not know everyone, but you probably know someone who knows someone who knows that person. What is a small world network? A small world network is this type of social network. One way to display this situation is by modeling the network with a graph.

Graphs of social networks have taken many forms over the years. Sometimes, these graphs are completely random, with the thought that anyone in the world could possibly know any other person in the world. In this project, we consider more structured networks. Graphs such as complete graphs and the 1-lattice graph take into account that it is more likely that you will know your neighbors than know a random person. For example, a complete graph could model a very close group of friends, where everyone in the group knows everyone else in the group, while the 1-lattice could model a neighborhood where a person knows his neighbor and his neighbor’s neighbor, but he doesn’t know anyone else in the neighborhood. Of course, your neighbor might know someone across town, and that person might know someone around the world, making it a small world. Notice there is still a bit of randomness amid the structure. By looking at graphs such as these, we are able to investigate the more traditional, completely random models, and the more modern, structured models.

In this project, we will focus on the more structured, 1-lattice model that Watts and Strogatz created in 1998. We investigate this graph, along with the complete graph and some others models in detail. Which graph models which real-world situation best? We investigate different properties such as clustering and path length. We also attempt to extend generating functions to help us calculate these properties.

#### Meagan Potter

##### Breakdown! Discovering the Relationships Between Designs and Matroids

Introduced in the early 1930s, design theory and matroid theory are two distinct areas of discrete mathematics. Are they related? How? Some matroids are designs; some designs are matroids. A matroidal design occurs when the blocks of a design are the hyperplanes of a matroid; a base design occurs when the blocks of a design are the bases of a matroid. Both designs and matroids have connections to geometry, and we can use geometries to link the two structures. Affine planes give rise to (n^2+n, n^2, n+1, n, 1)-designs, projective planes give rise to (n^2+n+1, n+1, 1)-designs, and all projective geometries are matroids.

The most well-known example that connects designs and matroids is the Fano plane, which is also the Fano matroid and the (7, 3, 1)-design. We use this to investigate the relationship between specific components of the three structures. Then, we look at several different classes of matroids -- uniform, cycle, transversal, etc -- and develop a method of translating among matroids and designs. We find designs given a specific matroid and find matroids given a specific design. Can we generalize this to an entire class of matroids? an entire class of designs? We consider the correlations from these specific examples and seek to make general conclusions about the nature of one structure from the other.

#### Maria Walters

##### Patching the Holes in Quasiderivations

"Number derivatives'' or "quasiderivations'' \$\Delta(x)\$ were first mentioned over 40 years ago in the Putnam Competition as a map from the integers to the integers that would satisfy the product rule. This definition was later expanded to all nonzero rational numbers. In 2007, this allowed Emmons to define the quasiderivation of a function f(x) for any "number quasiderivation'' \$\Delta\$. However, this definition was found to have a large number of ``holes'' whenever \$\Delta(x)\$ = 0. This motivated us to incorporate the limit of an infinitely-dimensional vector in order to patch these holes and define a continuous quasiderivation \$f^{\Delta}(x)\$. We then touch on a few examples where this quasiderivation \$f^{\Delta}(x)\$ resembles our standard derivative f'(x), and many more cases where they differ widely.

### 2008

#### Karsten Gimre

##### A Numerical Analysis of Nonlinear Partial Differential Equations Related to Electrochemistry

In this project, we set up a system of nonlinear partial differential equations to model an electrochemical reaction. Our results answer a long-standing conjecture as to whether current is influenced by a certain reaction parameter. After manipulating the domain of the functions involved, we could solve them numerically with MATLAB. With Laplace Transforms, we were able to convert the system of equations into a single equation. We determined that an analytical solution to the equation seems unknown, and likely not possible. Using existence and uniqueness theorems, together with the Lax Equivalence Theorem, we showed that the numerical solution is indeed the correct solution.

#### Alexis Sakaida-Diaz

##### An Examination of the Effect of Standards Based Curriculum on AP-Calculus Test Scores

The National Council of Teachers of Mathematics has published standards for teaching mathematics to children Pre-K through 12th grade. While the standards appear to be reasonable, there is still controversy as to whether these standards lead to appropriate levels of grade-school learning in mathematics. In this investigation, we examine one measure of the success of the standards by determining if a standards based high school curriculum leads to higher scores on the AP-Calculus exam.

### 2007

#### Dale Blem

##### Logarithmic Differentiation and Higher Order Newton's Method

The Logarithmic Derivative is the derivative of the natural log of a function D(ln(f(x)). This is more commonly expressed as the ratio f'(x)/f(x). It is used throughout mathematics in the fields of Differential Equations and Number Theory. We investigate properties of the Logarithmic Derivative, first strictly as an operator, and then as it is applied to Newton's Method. We illustrate a Modified Newton's Method that uses a higher order logarithmic derivative. We show both algebraically and geometrically that this Modified Newton's Method can converge more quickly than the classical Newton's Method.

#### Tara Fechter

##### Exploring the Derivative of a Natural Number Using the Logarithmic Derivative

We give an expository overview of the concept of the derivative of  a natural number. Several examples of the concept are illustrated. We relate the concept to the logarithmic derivative, n'/n.  We examine this concept in an unfamiliar setting, a function that, at first, appears to have no discernable pattern. We then determine the limit of the average values of this logarithmic derivative by bounding its values between two generating functions.

#### Matthew Rose

##### Wavelets and Filters

Wavelets, which are localized waves, are an exciting new alternative to Fourier series to analyze mathematical signals. Some emerging applications of wavelets include data compression and feature extraction in sound and image processing, including the new JPEG2000 standard. We will specifically explore the Haar Wavelet, which is a pulse of 1 and -1 over a finite range. The Haar Wavelet is just one of many wavelets to have the important property of orthogonality and we will show how this can be used to establish a basis from a collection of wavelets.

### 2006

#### Heather Helmandollar (LuBean)

##### Length Minimizing Paths in the Hyperbolic Plane: Proof Via Paired Subcalibrations

Minimization proofs using paired calibrations have in the past been done with vector fields of divergence zero. In this paper we explore the possibility of using vector fields of nonzero divergence and their applications in paired calibration proofs in the hyperbolic plane.  We will explore Steiner's Problem with three and four points spaced evenly around a circle.

#### Tim Prins

##### Scheduling a Bridge Club: A Case Study in Discrete Optimization

We consider a scheduling problem posed in 1992 to two mathematicians at the University of Michigan by a local bridge club. The club wanted a schedule which would allow each player to play against every other player an equal number of times over their eight yearly meetings. To find such a schedule, we first determine its characteristics and define a function which gives us a numerical representation of how close a given schedule is to optimal. This operations research problem does not submit well to linear programming, so we try other algorithms. Using computer programs we wrote, we attempt to find an optimal schedule using several well known algorithms: exhaustive search, greedy, steepest descent, annealed search, and tabu search. Using the greedy algorithm, we find a schedule which is close to optimal while we find optimal schedules using steepest descent, annealed search, and tabu search. We also compare the run times of these algorithms.

#### Kalei Titcomb

##### Periodicity In Dynamical Systems

Dynamical systems arise in the study of many physical phenomena including the motion of heavenly bodies, variation in weather, and the rise and fall of populations. Often, these phenomena exhibit periodic behavior: planets and solar systems maintain fairly stable orbits; temperature and rainfall display annual patterns; predator populations cycle with prey populations. The mathematics of dynamical systems helps analyze this periodic behavior. In this talk, we investigate theory that guarantees, in such systems, the existence of periodicity and that allows us to numerically estimate periodic points. We apply this theory to the logistic family of functions, a family that arises in population dynamics. We visualize the dynamics through the use of web and bifurcation diagrams..