Physical Sciences and Mathematics Commons^™

Raising Student Awareness Of Environmental Issues Via Writing Assignments With Differential Equations, Michelle L. Ghrist

CODEE Journal

In this paper, I discuss two environmentally-focused writing assignments that I developed and implemented in recent integral calculus and differential equations courses. These models of carbon storage and PCB’s in a river provide interesting applications of one-compartment mixing problems. The assignments were intended to focus student attention on sustainability concerns while also developing other essential skills. I discuss these assignments and their effect on my students’ technical writing and environmental awareness. Detailed introductory instructions and mostly complete solutions to these assignments appear in the appendices, to include sample student work.

Using A Sand Tank Groundwater Model To Investigate A Groundwater Flow Model, Christopher Evrard, Callie Johnson, Michael A. Karls, Nicole Regnier

CODEE Journal

A Sand Tank Groundwater Model is a tabletop physical model constructed of plexiglass and filled with sand that is typically used to illustrate how groundwater water flows through an aquifer, how water wells work, and the effects of contaminants introduced into an aquifer. Mathematically groundwater flow through an aquifer can be modeled with the heat equation. We will show how a Sand Tank Groundwater Model can be used to simulate groundwater flow through an aquifer with a no flow boundary condition.

Fitting A Covid-19 Model Incorporating Senses Of Safety And Caution To Local Data From Spartanburg County, South Carolina, D. Chloe Griffin, Amanda Mangum

CODEE Journal

Common mechanistic models include Susceptible-Infected-Removed (SIR) and Susceptible-Exposed-Infected-Removed (SEIR) models. These models in their basic forms have generally failed to capture the nature of the COVID-19 pandemic's multiple waves and do not take into account public policies such as social distancing, mask mandates, and the ``Stay-at-Home'' orders implemented in early 2020. While the Susceptible-Vaccinated-Infected-Recovered-Deceased (SVIRD) model only adds two more compartments to the SIR model, the inclusion of time-dependent parameters allows for the model to better capture the first two waves of the COVID-19 pandemic when surveillance testing was common practice for a large portion of the population. We find …

Modeling Aircraft Takeoffs, Catherine Cavagnaro

CODEE Journal

Real-world applications can demonstrate how mathematical models describe and provide insight into familiar physical systems. In this paper, we apply techniques from a first-semester differential equations course that shed light on a problem from aviation. In particular, we construct several differential equations that model the distance that an aircraft requires to become airborne. A popular thumb rule that pilots have used for decades appears to emanate from one of these models. We will see that this rule does not follow from a representative model and suggest a better method of ensuring safety during takeoff. Aircraft safety is definitely a matter …

Odes And Mandatory Voting, Christoph Borgers, Natasa Dragovic, Anna Haensch, Arkadz Kirshtein, Lilla Orr

CODEE Journal

This paper presents mathematics relevant to the question whether voting should be mandatory. Assuming a static distribution of voters’ political beliefs, we model how politicians might adjust their positions to raise their share of the vote. Various scenarios can be explored using our app at https: //centrism.streamlit.app/. Abstentions are found to have great impact on the dynamics of candidates, and in particular to introduce the possibility of discontinuous jumps in optimal candidate positions. This is an unusual application of ODEs. We hope that it might help engage some students who may find it harder to connect with the more customary …

Reducing Generalization Error In Multiclass Classification Through Factorized Cross Entropy Loss, Oleksandr Horban

CMC Senior Theses

This paper introduces Factorized Cross Entropy Loss, a novel approach to multiclass classification which modifies the standard cross entropy loss by decomposing its weight matrix W into two smaller matrices, U and V, where UV is a low rank approximation of W. Factorized Cross Entropy Loss reduces generalization error from the conventional O( sqrt(k / n) ) to O( sqrt(r / n) ), where k is the number of classes, n is the sample size, and r is the reduced inner dimension of U and V.

Unveiling The Power Of Shor's Algorithm: Cryptography In A Post Quantum World, Dylan Phares

CMC Senior Theses

Shor's Algorithm is an extremely powerful tool, in utilizing this tool it is important to understand how it works and why it works. As well as the vast implications it could have for cryptography

Analyzing A Smartphone Battle Using Bass Competition Model, Maila Hallare, Alireza Hosseinkhan, Hasala Senpathy K. Gallolu Kankanamalage

CODEE Journal

Many examples of 2x2 nonlinear systems in a first-course in ODE or a mathematical modeling class come from physics or biology. We present an example that comes from the business or management sciences, namely, the Bass diffusion model. We believe that students will appreciate this model because it does not require a lot of background material and it is used to analyze sales data and serve as a guide in pricing decisions for a single product. In this project, we create a 2x2 ODE system that is inspired by the Bass diffusion model; we call the resulting system the Bass …

Fibonacci Differential Equation And Associated Spiral Curves, Mehmet Pakdemirli

CODEE Journal

The Fibonacci differential equation is defined with analogy from the Fibonacci difference equation. The linear second order differential equation is solved for suitable initial conditions. The solutions constitute spirals in the polar coordinates. The properties of the spirals with respect to the Fibonacci numbers and the differences between the new spirals and classical spirals are discussed.

Exploring Parameter Sensitivity Analysis In Mathematical Modeling With Ordinary Differential Equations, Viktoria Savatorova

CODEE Journal

This paper presents an exploration into parameter sensitivity analysis in mathematical modeling using ordinary differential equations (ODEs). Taking the first steps in understanding local sensitivity analysis through the direct differential method and global sensitivity analysis using metrics like Pearson, Spearman, PRCC, and Sobol’, we provide readers with a basic understanding of parameter sensitivity analysis for mathematical modeling using ODEs. As an illustrative application, the system of differential equations modeling population dynamics of several fish species with harvest considerations is utilized. The results of employing local and global sensitivity analysis are compared, shedding light on the strengths and limitations of each …

Modeling Immune System Dynamics During Hiv Infection And Treatment With Differential Equations, Nicole Rychagov

CODEE Journal

An inquiry-based project that discusses immune system dynamics during HIV infection using differential equations is presented. The complex interactions between healthy T-cells, latently infected T-cells, actively infected T-cells, and the HIV virus are modeled using four nonlinear differential equations. The model is adapted to simulate long term HIV dynamics, including the AIDS state, and is used to simulate the long term effects of the traditional antiretroviral therapy (ART). The model is also used to test viral rebound over time of combined application of ART and a new drug that blocks the reactivation of the viral genome in the infected cells …

Beginner's Analysis Of Financial Stochastic Process Models, David Garcia

HMC Senior Theses

This thesis explores the use of geometric Brownian motion (GBM) as a financial model for predicting stock prices. The model is first introduced and its assumptions and limitations are discussed. Then, it is shown how to simulate GBM in order to predict stock price values. The performance of the GBM model is then evaluated in two different periods of time to determine whether it's accuracy has changed before and after March 23, 2020.

Multilayer Network Model Of Gender Bias And Homophily In Hierarchical Structures, Emerson Mcmullen

HMC Senior Theses

Although women have made progress in entering positions in academia and
industry, they are still underrepresented at the highest levels of leadership.
Two factors that may contribute to this leaky pipeline are gender bias,
the tendency to treat individuals differently based on the person’s gender
identity, and homophily, the tendency of people to want to be around those
who are similar to themselves. Here, we present a multilayer network model
of gender representation in professional hierarchies that incorporates these
two factors. This model builds on previous work by Clifton et al. (2019), but
the multilayer network framework allows us to …

Modeling Self-Diffusiophoretic Janus Particles In Fluid, Kausik Das

HMC Senior Theses

We explore spherical Janus particles in which a chemical reaction occurs on one face, depleting a substrate in the suspending fluid, while no reaction occurs on the other face. The steady state concentration field is governed by Laplace’s equation with mixed boundary conditions. We use the collocation method to obtain numerical solutions to the equation in spherical coordinates. The asymmetry of the reaction gives rise to a slip velocity that causes the particle to move spontaneously in the fluid through a process known as self-diffusiophoresis. Using the Lorentz reciprocal theorem, we obtain the swimming velocity of the particle. We extend …

Quantifying The Carbon Stored And Sequestered By The Trees On Pomona College’S Campus, Paola A. Giron-Carson

Scripps Senior Theses

We are experiencing a climate crisis that must be confronted with strategic mitigation. Pomona College contributes to the climate crisis through its emissions for which there is a baseline record. However there is no baseline record of the climate mitigation currently performed by the trees on Pomona’s campus through carbon storage. This study seeks to determine a current baseline quantity of carbon stored and sequestrated by Pomona’s trees as well as possible courses of climate mitigation for Pomona College to take. Initial information gathering was conducted through interviews with several stakeholders. This study was conducted using data collected prior to …

Graph-Based Acoustic Clustering And Classification, Justin Youngho Sunu

CGU Theses & Dissertations

The rapid growth of audio data collection in various domains necessitates advanced techniquesfor efficient analysis and classification. This dissertation proposes new approaches for categorizing acoustic data, using both unsupervised and semi-supervised learning methods. Starting with raw audio, we preprocess the signal to segment it into time windows, each of which we consider as an independent data point. We use the short-time Fourier transform to describe the signal in a given time window as a set of Fourier coefficients. We interpret the resulting frequency signature as a high-dimensional feature description of each data point. We then develop a graph-based approach for …

Measuring Racial Segregation In Los Angeles County Using Random Walks, Zarina Kismet Dhillon

CMC Senior Theses

As of now there is no universal quantitative measure used to evaluate racial segregation in different regions. This paper begins by providing a history of segregation, with an emphasis on the impact of redlining in the early 20th century. We move to its effect on the current population distribution in Los Angeles, California, and then provide an overview of the mathematical concepts that have been used in previous measurements of segregation. We then introduce a method that we believe encompasses the most representative aspects of preceding work, proposed by Sousa and Nicosia in their work on quantifying ethnic segregation in …

Counting Spanning Trees On Triangular Lattices, Angie Wang

CMC Senior Theses

This thesis focuses on finding spanning tree counts for triangular lattices and other planar graphs comprised of triangular faces. This topic has applications in redistricting: many proposed algorithmic methods for detecting gerrymandering involve spanning trees, and graphs representing states/regions are often triangulated. First, we present and prove Kirchhoff’s Matrix Tree Theorem, a well known formula for computing the number of spanning trees of a multigraph. Then, we use combinatorial methods to find spanning tree counts for chains of triangles and 3 × n triangular lattices (some limiting formulas exist, but they rely on higher level mathematics). For a chain of …

Academic Hats And Ice Cream: Two Optimization Problems, Valery F. Ochkov, Yulia V. Chudova

Journal of Humanistic Mathematics

This article describes the use of computer software to optimize the design of an academic hat and an ice cream cone!

Maintaining Ecosystem And Economic Structure In A Three-Species Dynamical System In Chesapeake Bay, Maila Hallare, Iordanka Panayotova

CODEE Journal

We consider a three-species fish dynamical system in Chesapeake Bay consisting of the Atlantic menhaden as the prey and its two competing predators, the striped bass and the catfish. Building on our previous work in this system, we consider the issue of balancing economic harvesting goals (financial gain for fishermen) with ecological harvesting goals (non-extinction of species). In particular, we investigate the bionomic equilibria, maximum sustainable yield, and the maximum economic yield. Analytical computations and numerical simulations are employed to provide some mathematical guidance on fisheries management policies.

Human Impact On Planetary Temperature And Glacial Volume: Extending A Toy Climate Model To A New Millennium, Samantha Secor, Jennifer Switkes

CODEE Journal

Starting with a toy climate model from the literature, we employ a system of two nonlinear differential equations to model the reciprocal effects of the average temperature and the percentage of glacial volume on Earth. In the literature, this model is used to demonstrate the potential for a stable periodic orbit over a long time span in the form of an attracting limit cycle. In the roughly twenty five years since this model appeared in the literature, the effects of global warming and human-impacted climate change have become much more well known and apparent. We demonstrate modification of initial conditions …

The Nature Of Numbers: Real Computing, Bradley J. Lucier

Journal of Humanistic Mathematics

While studying the computable real numbers as a professional mathematician, I came to see the computable reals, and not the real numbers as usually presented in undergraduate real analysis classes, as the natural culmination of my evolving understanding of numbers as a schoolchild. This paper attempts to trace and explain that evolution. The first part recounts the nature of numbers as they were presented to us grade-school children. In particular, the introduction of square roots induced a step change in my understanding of numbers. Another incident gave me insight into the brilliance of Alan Turing in his paper introducing both …

Correlation Does Not Imply Correlation: A Thesis On Causal Influence And Simpson’S Paradox, Emily Naitoh

Scripps Senior Theses

In our data-driven world, it has become commonplace to attempt to find
causal relationships. One of the themes of this thesis is to show methods of
determining causation. The second theme follows a saying in mathematics,
"correlation does not imply causation". We will also discuss situations where
correlation does not even imply correlation itself. These cases are described
by Simpson’s paradox in an exploration of different areas of mathematics
and computer coding.

Smoothed Bounded-Confidence Opinion Dynamics On The Complete Graph, Solomon Valore-Caplan

HMC Senior Theses

We present and analyze a model for how opinions might spread throughout a network of people sharing information. Our model is called the smoothed bounded-confidence model and is inspired by the bounded-confidence model of opinion dynamics proposed by Hegselmann and Krause. In the Hegselmann–Krause model, agents move towards the average opinion of their neighbors. However, an agent only factors a neighbor into the average if their opinions are sufficiently similar. In our model, we replace this binary threshold with a logarithmic weighting function that rewards neighbors with similar opinions and minimizes the effect of dissimilar ones. This weighting function can …

An Adaptive Hegselmann–Krause Model Of Opinion Dynamics, Phousawanh Peaungvongpakdy

HMC Senior Theses

Models of opinion dynamics have been used to understand how the spread
of information in a population evolves, such as the classical Hegselmann–
Krause model (Hegselmann and Krause, 2002). One extension of the model
has been used to study the impact of media ideology on social media
networks (Brooks and Porter, 2020). In this thesis, we explore various
models of opinions and propose our own model, which is an adaptive
version of the Hegselmann–Krause model. The adaptive version implements
the social phenomenon of homophily—the tendency for like-minded agents to
associate together. This is done by having agents dissolve connections …

Check Yourself Before You Wrek Yourself: Unpacking And Generalizing Randomized Extended Kaczmarz, William Gilroy

HMC Senior Theses

Linear systems are fundamental in many areas of science and engineering. With the advent of computers there now exist extremely large linear systems that we are interested in. Such linear systems lend themselves to iterative methods. One such method is the family of algorithms called Randomized Kaczmarz methods.
Among this family, there exists a Randomized Kaczmarz variant called Randomized
Extended Kaczmarz which solves for least squares solutions in inconsistent linear systems.
Among Kaczmarz variants, Randomized Extended Kaczmarz is unique in that it modifies input system in a special way to solve for the least squares solution. In this work we …

An Exploration Of Voting With Partial Orders, Mason Acevedo

HMC Senior Theses

In this thesis, we discuss existing ideas and voting systems in social choice theory. Specifically, we focus on the Kemeny rule and the Borda count. Then, we begin trying to understand generalizations of these voting systems in a setting where voters can submit partial rankings on their ballot, instead of complete rankings.

An Exponential Formula For Random Variables Generated By Multiple Brownian Motions, Maximilian Lawrence Baroi

CGU Theses & Dissertations

The frozen operator has been used to develop Dyson-series like representations for random variables generated by classical Brownian motion, Lévy processes and fractional Brownian with Hurst index greater than 1/2.The relationship between the conditional expectation of a random variable (or fractional conditional expectation in the case of fractional Brownian motion)and that variable's Dyson-series like representation is the exponential formula. These results had not yet been extended to either fractional Brownian motion with Hurst index less than 1/2, or d-dimensional Brownian motion. The former is still out of reach, but we hope our review of stochastic integration for fractional Brownian motion …

Examining Bias Against Women In Professional Settings Through Bifurcation Theory, Lauren Cashdan

CMC Senior Theses

When it comes to women in professional hierarchies, it is important to recognize the lack of representation at the higher levels. By modeling these situations we hope to draw attention to the issues currently plaguing professional atmospheres. In a paper by Clifton et. al. (2019), they model the fraction of women at any level in a professional hierarchy using the parameters of hiring gender bias and internal homophily on behalf of the applicant. This thesis will focus on a key theory in Clifton et. al.’s analysis and explain its role in the model, specifically bifrucation analysis. In order to analyze …

Analyzing Marriage Statistics As Recorded In The Journal Of The American Statistical Association From 1889 To 2012, Annalee Soohoo

CMC Senior Theses

The United States has been tracking American marriage statistics since its founding. According to the United States Census Bureau, “marital status and marital history data help federal agencies understand marriage trends, forecast future needs of programs that have spousal benefits, and measure the effects of policies and programs that focus on the well-being of families, including tax policies and financial assistance programs.”[1] With such a wide scope of applications, it is understandable why marriage statistics are so highly studied and well-documented.

This thesis will analyze American marriage patterns over the past 100 years as documented in the Journal of …