Physical Sciences and Mathematics Commons^™

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.

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.

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 …

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 …

Before College Math, Roxane Elena Ronca

A Collection of Open Access Books and Monographs

Most math books for college students start out reviewing “rules” in an introductory chapter. The review usually goes like this: here are the “rules”, here are some examples of using those “rules” and here are 10 to 100 exercises where you will practice using those “rules” and then you’ll be tested on them.

The problem with that approach, even if it seems familiar and comfortable to you, is that people learn, in part, by connecting new ideas and perspectives to what they already understand, and correcting any previous misunderstandings. This process takes time and effort. Memorizing rules to quickly retrieve …

Vibrations On Networks, Zachary Pontrantolfi

Scholars Week

Studying vibrations on networks helps inform our understanding of random processes on other networks with similar geometry. We discuss two physical models to build up intuition about their eigenvectors. We conclude with a hidden connection between the rate of convergence of random walks, and the ground state energies of molecules.

A Quantitative Assessment Of The Diabetes Self-Management Education Program, Grace Mcfarlane

Scholars Week

A Diabetes Self-Management Education (DSME) program offered in an inner-city health center run by the Cincinnati Health Department, which started in 2014, was created to help those in an underserved population learn how to manage their diabetes. Two key measurements, A1C (glycated hemoglobin) and BMI (body mass index), were taken over time to monitor their progress. In this study, we analyzed quantitatively whether or not there was a significant improvement in their BMI and A1C values over the course of two years since they joined DSME program as any improvement would imply a potentially healthier lifestyle in regards to their …

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 …

Modeling And Forecasting Crime Patterns In Bellingham, Washington, Zachary Domingo, Eric Shoner

Scholars Week

Our purpose is to use time series analysis to model and forecast the underlying dynamics behind crime in Bellingham, Washington. Using recent monthly data from the Bellingham Police Department, we considered singular spectrum analysis and autoregressive moving average modelling techniques to estimate significant deterministic patterns in the data. After examining the multitude of data provided, we narrowed down to two categories of crime: alcohol offenses and domestic violence. We created two time series models for each category and compared them to each other. The better performing model was used to forecast the number of crime incidents for ten months and …

Short-Term Volatility Curve Predictions Using Singular Spectrum Analysis, Nick Odell

Scholars Week

This project aims to produce accurate volatility forecasts, using high-frequency financial time series data. The primary mathematical methods used are Functional Data Analysis, time series analysis techniques such as Autoregressive Models and a comparison between Multi-variate and Uni-variate Singular Spectrum Analysis. These results aim to be useful for financial risk quantification.

Modeling Cross-Border Regions, Place-Making, And Resource Management: A Delphi Analysis, Amy D. Anderson, Patrick H. Buckley, John Belec

Mathematics Faculty Publications

Along international borders, spillover of resource management issues is a growing challenge. Development of cross-border regions (CBRs) is seen as an emerging means of addressing these issues. A set of theoretical models, geo-economic mobilization and a resource-focused territorial program of place-making have been proposed as a lens for understanding why such change could occur. From this theory, we identify three C’s as critical initial or necessary conditions to start the process: common territorial identity, convergence of knowledge and values, willingness for cooperation. We then utilize results of a Delphi study in the Fraser Lowland, a sub-district of the American-Canadian Cascadia …

Rainbow Turán Problems For Paths And Forests Of Stars, Daniel Johnston, Cory Palmer, Amites Sarkar

Mathematics Faculty Publications

For a fixed graph F, we would like to determine the maximum number of edges in a properly edge-colored graph on n vertices which does not contain a rainbow copy of F, that is, a copy of F all of whose edges receive a different color. This maximum, denoted by ex^∗ (n, F), is the rainbow Turán number of F, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstraëte [Combinatorics, Probability and Computing 16 (2007)]. We determine ex^∗ (n, F) exactly when F is a forest of stars, and …

Weierstrass Points On X 0+(P) And Supersingular J-Invariants, Stephanie Treneer

Mathematics Faculty Publications

We study the arithmetic properties of Weierstrass points on the modular curves X₀⁺(p) for primes p. In particular, we obtain a relationship between the Weierstrass points on X₀⁺(p) and the j-invariants of supersingular elliptic curves in characteristic p.

Deep Phylogenomics Of A Tandem-Repeat Galectin Regulating Appendicular Skeletal Pattern Formation, Ramray Bhat, Mahul Chakraborty, Tilmann Glimm, Thomas A. Stewart, Stuart (Stuart A.) Newman

Mathematics Faculty Publications

Background: A multiscale network of two galectins Galectin-1 (Gal-1) and Galectin-8 (Gal-8) patterns the avian limb skeleton. Among vertebrates with paired appendages, chondrichthyan fins typically have one or more cartilage plates and many repeating parallel endoskeletal elements, actinopterygian fins have more varied patterns of nodules, bars and plates, while tetrapod limbs exhibit tandem arrays of few, proximodistally increasing numbers of elements. We applied a comparative genomic and protein evolution approach to understand the origin of the galectin patterning network. Having previously observed a phylogenetic constraint on Gal-1 structure across vertebrates, we asked whether evolutionary changes of Gal-8 could have …

Quantum Mock Modular Forms Arising From Eta–Theta Functions, Amanda Folsom, Sharon Garthwaite, Soon-Yi Kang, Holly Swisher, Stephanie Treneer

Mathematics Faculty Publications

In 2013, Lemke Oliver classified all eta-quotients which are theta functions. In this paper, we unify the eta–theta functions by constructing mock modular forms from the eta–theta functions with even characters, such that the shadows of these mock modular forms are given by the eta–theta functions with odd characters. In addition, we prove that our mock modular forms are quantum modular forms. As corollaries, we establish simple finite hypergeometric expressions which may be used to evaluate Eichler integrals of the odd eta–theta functions, as well as some curious algebraic identities.

The Role Of Sister Cities’ Staff Exchanges In Developing “Learning Cities”: Exploring Necessary And Sufficient Conditions In Social Capital Development Utilizing Proportional Odds Modeling, Patrick H. Buckley, Akio Takahashi, Amy D. Anderson

Mathematics Faculty Publications

In the last half century former international adversaries have become cooperators through networking and knowledge sharing for decision making aimed at improving quality of life and sustainability; nowhere has this been more striking then at the urban level where such activity is seen as a key component in building “learning cities” through the development of social capital. Although mega-cities have been leaders in such efforts, mid-sized cities with lesser resource endowments have striven to follow by focusing on more frugal sister city type exchanges. The underlying thesis of our research is that great value can be derived from city-to-city exchanges …

On The Probabilistic Cauchy Theory Of The Cubic Nonlinear Schrödinger Equation On Rd, D≥3, Árpád Bényi, Tadahiro Oh, Oana Pocovnicu

Mathematics Faculty Publications

We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) : i∂_tu + Δu = ±|u|²u on R ^d, d ≥ 3, with random initial data and prove almost sure well-posedness results below the scaling-critical regularity ^scrit = d-2/2. More precisely, given a function on R ^d, we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold. (i) We prove almost sure local well-posedness of the cubic NLS below the scaling-critical regularity …

Directionally Bounded Utility And The Executive Pay Puzzle, Edoh Y. Amiran, Daniel Andreas Hagen

Mathematics Faculty Publications

The pay of CEOs and other top executives has risen disproportionately relative to other earnings. We provide a supply-side explanation based on utility theory using directionally bounded utility functions. As overall income levels have grown, the amount of compensation required to induce top executives to sacrifice a quiet life has risen. We show that directionally bounded utility functions predict a general rise in compensation for stress. More importantly, such utility functions can be used to explain why the CEO pay ratio has risen at an increasing rate, something which other approaches have difficulty explaining.

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 …

Compactness Properties Of Commutators Of Bilinear Fractional Integrals, Árpád Bényi, Wendolin Damián, Kabe Moen, Rodolfo H. (Rodolfo Humberto) Torres

Mathematics Faculty Publications

Commutators of a large class of bilinear operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be jointly compact. Under a similar commutation, fractional integral versions of the bilinear Hilbert transform yield separately compact operators.

Somewhat Stochastic Matrices, Branko Ćurgus, Robert I. Jewett

Mathematics Faculty Publications

The standard theorem for stochastic matrices with positive entries is generalized to matrices with no sign restriction on the entries. The condition that column sums be equal to 1 is kept, but the positivity condition is replaced by a condition on the distances between columns.

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 …

Outer Median Triangles, Árpád Bényi, Branko Ćurgus

Árpád Bényi

We define the notions of outer medians and outer median triangles. We show that outer median triangles enjoy similar properties to that of the median triangle.

Smoothing Of Commutators For A Hörmander Class Of Bilinear Pseudodifferential Operators, Árpád Bényi, Tadahiro Oh

Mathematics Faculty Publications

Commutators of bilinear pseudodifferential operators with symbols in the Hörmander class BS¹_1,0 and multiplication by Lipschitz functions are shown to be bilinear Calderón-Zygmund operators. A connection with a notion of compactness in the bilinear setting for the iteration of the commutators is also made.