Digital Commons Network^™

Time As A Line: Helping Children Make Abstract Ideas Concrete, Rachel Mae Stenner

WWU Honors College Senior Projects

This is a math education project that included research, a lesson plan, and actual in the classroom work with students. Under the advisement of Dr. Rebecca Borowski, I looked into how time, an abstract idea, is taught to young children who are just starting to learn what measurement is, and examined how teachers can better teach time as a more concrete topic. This focused on the idea of turning the abstract time concepts that are thrown at children into the more abstract ideas of both circular and then linear number lines, using physical materials to help guide the process.

A Change-Point Analysis Of Air Pollution Levels In Silao, Mexico And Fresno, California, Rachael Goodwin

WWU Honors College Senior Projects

We analyzed PM10 levels in the city of Silao, Mexico, as well as PM2.5 and PM10 levels in Fresno, California to determine if there was a shift in air pollution levels in either location. A change point based analysis was used to determine if there was a shift in air pollution levels. In the city of Silao, there was a significant increase in PM10 levels, but there was no significant change in Fresno for either pollutant.

Biological Oscillator Synchronization With The Cellular Potts Model, Rose Una

WWU Honors College Senior Projects

Similar to how neurons synchronize their firing in the brain, individual cells of certain single-celled species can synchronize their internal oscillatory molecular clocks to those of their neighboring cells. This study develops and analyzes an abstract, discrete agent-based computational model to investigate the movement and synchronization of internal oscillators in biological cells. We adapt a Cellular Potts Model to explore this oscillator synchronization process with two-dimensional cells on a square lattice. Model assumptions are motivated by behavior in single-celled species of slime mold (Dictyostelium discoideum) and slime bacteria (myxobacteria). The effects of the spatial attraction parameter and the neighboring clock …

Bootstrapping The Likelihood Ratio Test To Determine Change Points, Lili Donovan

WWU Honors College Senior Projects

Change point analysis is the process of determining changes to the mean of a sequence of independent observations. The goal is to determine the location of the change and how the change impacts the parameter in question. In this project, we applied the likelihood ratio test (LR) which uses a binary segmentation method to split the data at each change point. The data is iteratively split at each change point until every location of change is identified. The p-value is typically computed using the asymptotic distribution of the test statistic, however, it can be unreliable when the number of observations …

Making Upper-Level Math Accessible To A Younger Audience, Allyson Roller

WWU Honors College Senior Projects

Symmetry is all around us. It appears on fabrics and on the buildings that surround us. Believe it or not, there is actually quite a bit of math that goes into generating these patterns, which are known as the seven frieze patterns. In my work, I explain how each unique pattern is generated using different types of symmetries. I also created a PDF of a children’s book about frieze patterns to ensure that people of all ages have the opportunity to learn about seemingly complex patterns.

Quasipositive Braids And Ribbon Surfaces, Rachel Snyder

WWU Honors College Senior Projects

Meant to serve as an accessible exploration of knot theory for undergraduates and those without much experience in topology, this paper will start by exploring the basics of knot theory and will work through investigating the relationships between knots and surfaces, ending with an analysis of the relationship between quasipositive braids and surfaces in 4-space. We will begin by defining a knot and introducing the ways in which we are able to manipulate them. Following that, we will explore the basics of surfaces, building up to a proof that all surfaces are homeomorphic to a series of disks and bands …

Reflections On Setting Up The Cyber Range Intrusion Detection System, William Pearson

WWU Honors College Senior Projects

A short reflection on the project to set up an Intrusion Detection System for the Cyber Range at Western Washington University Poulsbo.

Knitting Math: Geometric Shapes, Cynthia Wright

WWU Honors College Senior Projects

When knitting 3-D objects such as hats or socks, the knitter is using geometry and mathematics to make the 2-dimensional string into 3-dimensional shapes. In this project, I will be creating mathematically accurate, geometric shapes, to directly show the relationship between the mathematical formulas, knitting patterns, and the knitted objects. There is more than one way to understand and perceive math, one of which is knitting. Past mathematical knitters have shown the relationship between algebra and complex shapes (such as a Klein bottle or Möbius strip) and knitting. In an effort to explore how more accessible mathematical shapes and concepts …

Conformal Geometry Of Polygons, Michael Albert

WWU Honors College Senior Projects

Conformal maps are functions from subsets of the complex plane to the complex plane that locally preserve angles. Our goal is to understand conformal maps that pass to and from polygonal domains. In order to do so, we derive some of the basic theory of harmonic functions on simply connected domains. In particular, our goal with the first few sections is to prove the Schwarz Reflection principle. Using this, as well as other tools from complex analysis, we give an in-depth explanation of Tao’s proof of the Schwarz-Christoffel formula. This is a differential equation that allows one to compute a …

Do Men Matter? In Statistics, Probably, Michael Kelly

WWU Honors College Senior Projects

In statistical genetics, there are several parameters of a dataset which a researcher might, but which are difficult to estimate in practice. In this paper, we will be focusing on allele frequencies, null alleles, inbreeding coefficients and, to a certain extent, beta values. A common technique for obtaining these values, developed by Amy Anderson and her co-workers, is to jointly estimate all of them using an EM-algorithm and the method of maximum likelihood. Despite this technique being effective in general, it is currently unable to deal with males at X-linked markers. The purpose of this project is to modify the …

Deblurring Images, Jamie Mcmullen

WWU Honors College Senior Projects

Let the matrix B be a blurred version of a sharp image represented by the matrix X. Given B, we would like to recover X.

To accomplish this, we construct linear models of the blurring process that produced B from X. The idea is that we could then reverse the blurring to reproduce the original image.

For example, if the blurred image satisfies

B = CXR^T

for some invertible matrices C and R, then we could recover X as

X = C-¹B(R^T)^-1.

However, the blurring model …

The World Before Calculus: Historical Approaches To The Tangent Line Problem, Lindsay Skinner

WWU Honors College Senior Projects

Pierre de Fermat and René Descartes were two brilliant 17th century mathematicians who have had lasting impacts on modern mathematics. Descartes laid the groundwork for the Cartesian coordinate system that is frequently employed in modern mathematics and Fermat’s last theorem vexed the mathematics community until Wiles’ proof was published in 1995. Amidst their many ground-breaking accomplishments these two men produced solutions for another mathematical problem - developing a general method to find the tangent line to a curve.

In spite of their apparent genius, neither man’s method had the lasting impact of their other works. Descartes’ and Fermat’s methods were …

A Comparison Of Two Statistical Tests For Interaction In Genetic Data, Clair Smith

WWU Honors College Senior Projects

This paper focuses on statistical methods that test for the effect of a single gene in a way that accounts for interaction with other genes. Such tests of association can be difficult since there may be many genetic and environmental factors that contribute to an effect. A gene is a hereditary DNA sequence that codes for a specific protein. A locus is a gene’s location in the DNA sequence of nucleotides (A, T, G, and C) and an allele is a specific version of a gene that has multiple forms. The existence of interactions between loci makes it difficult to …

The Four Color Theorem, Patrick Turner

WWU Honors College Senior Projects

The history of mathematics is pervaded by problems which can be stated simply, but are difficult and in some cases impossible to prove. The pursuit of solutions to these problems has been an important catalyst in mathematics, aiding the development of many disparate fields. While Fermat’s Last theorem, which states xⁿ + yⁿ = zⁿ has no integer solutions for n > 2 and x, y, z ≠ 0^[12] is perhaps the most famous of these problems, the Four Color Theorem proved a challenge to some of the greatest mathematical minds from its conception 1852 until its …

Topics In Extremal Graph Theory: Ramsey Numbers And The Turan Function, Damon J. (Damon John) Gulczynski

WWU Honors College Senior Projects

"Topics in Extremal Graph Theory: Ramsey Numbers and the Turan" Function by Damon J. Gulczynski