2019 Mathematics Capstone Symposium
Join the Mathematics Department to hear the senior capstone presentations. Student presentations will take place Friday. Talks are scheduled in Morken 214 and 216.
2:00pm – Incan Strip Patterns: An Ethnomathematical Exploration
We explore the ethnomathematics of strip patterns found in Incan cultures. We look at and analyze the strip pattern to see if we can expand what see into the mathematical language of symmetry and isometries. In particular, we use this language to classify all possible types of strip patterns.
2:30pm – Polynomial Factorization and How it Relates to the Common Core
In this talk, we investigate complex numbers and polynomials over the complex numbers. We focus on finding roots by stating the Fundamental Theorem of Algebra as well as providing an example. We will then prove a corollary of the Fundamental Theorem of Algebra stating that every (real or complex) polynomial of degree greater than 2 is reducible. We finish by discussing some common core standards and how they are related to the Fundamental Theorem of Algebra.
3:00pm – An Exploration of Egyptian Fractions
Egyptian Fraction Decompositions, named after the culture that used them for division, are a method of writing fractions as a sum of unit fractions. In this presentation, we will explore some useful properties of Egyptian Fractions, including an extensive look into the most commonly used algorithm for obtaining Egyptian Fraction Decompositions, the Greedy Method, discovered by Fibonacci in 1202. In addition, we will explore a brute-force algorithm for generating Egyptian Fraction Decompositions with the smallest possible number of summands, along with a detailed explanation of the mechanics of the algorithm.
3:25 – 4:00pm Coffee and Snacks
4:00pm – A Survey of Generating Functions
In this presentation, we will explore sequences, defined simply as ordered lists of numbers. Sequences are often built recursively: to get the value at index n you need the value(s) of some previous term(s). Such recursive definitions may show us the structure of our sequence, but it is often more useful to have a formula that doesn’t rely on a previous term. Here we’ll define and use generating functions in order to describe methods to find such formulas.
4:30pm – How to Always Win the Game of Nim
We introduce the notion of a combinatorial game by studying the game Nim. We will discuss how Nim can be modeled by binary representations of integers along with an operation known as the Nim-Sum. Finally, we use the mathematical formulation to describe the winning strategy for the game of Nim.
5:00pm – Error Detection and Correction Through Linear Algebra
Error detecting and error correcting codes can be used to ensure reliable transmission of information even when the communication channel experiences noise or other disruptions. Throughout this talk, we will discuss error detecting codes and error correcting codes that can detect and correct a single error, respectively. We will use concepts from Linear Algebra to obtain a single error correcting code and see an example of how it works.
2:00pm – Chutes, Ladders, and Chains
We introduce stochastic processes in order to model the children’s board game Chutes and Ladders as a Markov chain. We then describe our own variant of the game with a transition matrix in order to explore general properties of Markov chains. After providing some general results about transition matrices for Markov chains, we will apply these results to our own board in order to find the average number of turns required to play our version of Chutes and Ladders.
2:30pm – Representing Sound Waves Using Fourier Series
In this presentation, we investigate how to mathematically represent sound waves. We introduce Fourier series as a possible way of representing these sound waves and derive the Fourier series coefficients. After working through an example application of the series, we discuss convergence conditions for the series. We conclude by discussing the wave equation from physics as a way of producing an equation for a sound wave and see how Fourier series can be used to solve the wave equation.
3:00pm – Predicting the Next Debris Flow on Mt. Rainier – A Logistic Regression Method
Debris flows pose a serious threat to Mount Rainier National Park (MORA). While not as destructive as lahars, they occur during long intra-eruptive periods which make them a high concern for MORA staff. While debris flows can be difficult to predict, the park currently uses an effective model to predict wet weather debris flows. In this study we evaluate regression methods and attempt to build a logistic regression model in Minitab to predict warm weather debris flows on MORA.
3:25 – 4:00pm Coffee and Snacks
4:00pm – Predicting March Madness Using a Regression Model
In this talk, we will discuss statistical significance tests with the explanations of R-squared and p-values. These quantities will be used to measure the level of significance of different basketball statistics in comparison to win margin in the previous 15 NCAA division one March Madness basketball tournaments. Finally, we will apply these tests to attempt to predict this most recent 2019 NCAA tournament.
4:30pm – Complex Dynamics: Understanding the Mandelbrot and Julia Sets
The Mandelbrot and Julia Sets are quickly recognized for their beautiful and complex structure, but what are they and how are they generated? The answers to these questions lie within Complex Dynamics, the study of dynamical systems defined by iteration of functions on complex number spaces. In this paper, we seek to understand the Mandelbrot and Julia Sets as well as the Fundamental Dichotomy, which will help us realize the profound connection between the two.
5:00pm – A Tale of Two Topologies
We introduce the sphere and the torus as topological surfaces. Using ideas from topology, we introduce an invariant of a surface known as the Euler characteristic. We will prove that the Euler characteristic of the sphere is 2 and we will describe how known proofs of the Euler characteristic of the sphere may or may not be used to determine the Euler characteristic of the torus.