Mathematics Commons^™

Front Matter

Journal of Humanistic Mathematics

No abstract provided.

A Discrete Analogue For The Poincaré-Hopf Theorem, Savana Ammons

HMC Senior Theses

In this thesis, we develop a discrete analogue to the Poincaré–Hopf Theorem. We define the notion of a vector field on a graph, and establish an index theory for such a field. Specifically, we create well-defined indices for the nodes and “cells" formed by a planar graph. Then, we show that the sum of these indices remains constant for certain types of planar graphs, regardless of the discrete vector fields they have.

On The Mysteries Of Interpolation Jack Polynomials, Havi Ellers

HMC Senior Theses

Interpolation Jack polynomials are certain symmetric polynomials in N variables with coefficients that are rational functions in another parameter k, indexed by partitions of length at most N. Introduced first in 1996 by F. Knop and S. Sahi, and later studied extensively by Sahi, Knop-Sahi, and Okounkov-Olshanski, they have interesting connections to the representation theory of Lie algebras. Given an interpolation Jack polynomial we would like to differentiate it with respect to the variable k and write the result as a linear combination of other interpolation Jack polynomials where the coefficients are again rational functions in k. In this …

Where The Wild Knots Are, Forest Kobayashi

HMC Senior Theses

The new work in this document can be broken down into two main parts. In the first, we introduce a formalism for viewing the signed Gauss code for virtual knots in terms of an action of the symmetric group on a countable set. This is achieved by creating a "standard unknot" whose diagram contains countably-many crossings, and then representing tame knots in terms of the action of permutations with finite support. We present some preliminary computational results regarding the group operation given by this encoding, but do not explore it in detail. To make the encoding above formal, we require …

Use Of Kalman Filtering In State And Parameter Estimation Of Diabetes Models, Cassidy Le

HMC Senior Theses

Diabetes continues to affect many lives every year, putting those affected by it at higher risk of serious health issues. Despite many efforts, there currently is no cure for diabetes. Nevertheless, researchers continue to study diabetes in hopes of understanding the disease and how it affects people, creating mathematical models to simulate the onset and progression of diabetes. Recent research by David J. Albers, Matthew E. Levine, Andrew Stuart, Lena Mamykina, Bruce Gluckman, and George Hripcsak¹ has suggested that these models can be furthered through the use of Data Assimilation, a regression method that synchronizes a model with a …

Pascal's Mystic Hexagon In Tropical Geometry, Hanna Hoffman

HMC Senior Theses

Pascal's mystic hexagon is a theorem from projective geometry. Given six points in the projective plane, we can construct three points by extending opposite sides of the hexagon. These three points are collinear if and only if the six original points lie on a nondegenerate conic. We attempt to prove this theorem in the tropical plane.

An Exploration Of Combinatorial Interpretations For Fibonomial Coefficients, Richard Shapley

HMC Senior Theses

We can define Fibonomial coefficients as an analogue to binomial coefficients as F(n,k) = F_nF_n-1 … F­_n-k+1 / F­_kF_k-­1…F₁, where F_n represents the nth Fibonacci number. Like binomial coefficients, there are many identities for Fibonomial coefficients that have been proven algebraically. However, most of these identities have eluded combinatorial proofs.

Sagan and Savage (2010) first presented a combinatorial interpretation for these Fibonomial coefficients. More recently, Bennett et al. (2018) provided yet another interpretation, that is perhaps more tractable. However, there still has been little progress towards using these interpretations …

A Coherent Proof Of Mac Lane's Coherence Theorem, Luke Trujillo

HMC Senior Theses

Mac Lane’s Coherence Theorem is a subtle, foundational characterization of monoidal categories, a categorical concept which is now an important and popular tool in areas of pure mathematics and theoretical physics. Mac Lane’s original proof, while extremely clever, is written somewhat confusingly. Many years later, there still does not exist a fully complete and clearly written version of Mac Lane’s proof anywhere, which is unfortunate as Mac Lane’s proof provides very deep insight into the nature of monoidal categories. In this thesis, we provide brief introductions to category theory and monoidal categories, and we offer a precise, clear development of …

Preliminary Study Of Highway Pavement And Materials, Omer Eljairi

CGU Theses & Dissertations

This preliminary study covered (a) the effects of in-place air voids and other factors on fatigue cracking using Long-Term Pavement Performance data, (b) fracture properties of asphalt concrete in a semicircular bend (SCB) test using a noncontact camera and crosshead movement, and (c) hot applied modified-binder-chip-seal field performance in California. The objective is to improve pavement performance and life, establish a quality assurance/quality control (QA/QC) tests of fracture properties of asphalt mixtures, and save millions of dollars on maintenance and rehabilitation. Chapter 1 investigated the effect of in-place air voids (AV), asphalt content (AC), bulk-specific gravity (BSG), and maximum specific …

Detection And Localization Of Linear Features Based On Image Processing Methods, Sean Cormick Matz

CGU Theses & Dissertations

In this work, the general problem of the detection of features in images is considered. One of the methods, the orientation detection of lines, utilized the Radon transform (sinogram) of an image to detect lines at different angles in an image. The line thickness algorithm was generated by finding a pattern formed by particular lines in an image. The filtering of reconstructed images dealt with the removal of blur and other artifacts that arose in the course of inverting the Radon transform of an image to attempt to obtain the original image.

Discrete Geometry And Covering Problems, Alexander Hsu

CMC Senior Theses

This thesis explores several problems in discrete geometry, focusing on covering problems. We first go over some well known results, explaining Keith Ball's solution to the symmetric Tarski plank problem, as well as results of Alon and F\"uredi on covering all but vertices of a cube with hyperplanes. The former extensively utilizes techniques from matrix analysis, and the latter applies polynomial method. We state and explore the related problem, asking for the number of parallel hyperplanes required to cover a given discrete set of points in $\mathbb{Z}^{d}$ whose entries are bounded, and prove that there exist sets which are ``difficult'' …

Optimal Execution In Cryptocurrency Markets, Ethan Kurz

CMC Senior Theses

The purpose of this paper is to study the Almgren and Chriss model on the optimal execution of large block orders both on the NYSE and in cryptocurrency exchanges. Their model minimizes execution costs, which include linear temporary and permanent price impacts. We focus on how the stock market microstructure differs from a cryptocurrency exchange microstructure and what that means for how the model functions. Once the model and microstructures are explained, we examine how the Almgren-Chriss model functions with stocks from the NYSE, looking at specifically selling a large number of shares. We then investigate how a large "wholesale" …

Spectral Analysis Of Complex Dynamical Systems, Casey Lynn Johnson

CGU Theses & Dissertations

The spectrum of any differential equation or a system of differential equations is related to several important properties about the problem and its subsequent solution. So much information is held within the spectrum of a problem that there is an entire field devoted to it; spectral analysis. In this thesis, we perform spectral analysis on two separate complex dynamical systems. The vibrations along a continuous string or a string with beads on it are the governed by the continuous or discrete wave equation. We derive a small-vibrations model for multi-connected continuous strings that lie in a plane. We show that …

Stationary Distribution Of Recombination On 4x4 Grid Graph As It Relates To Gerrymandering, Camryn Hollarsmith

Scripps Senior Theses

A gerrymandered political districting plan is used to benefit a group seeking to elect more of their own officials into office. This practice happens at the city, county and state level. A gerrymandered plan can be strategically designed based on partisanship, race, and other factors. Gerrymandering poses a contradiction to the idea of “one person, one vote” ruled by the United States Supreme Court case Reynolds v. Sims (1964) because it values one demographic’s votes more than another’s, thus creating an unfair advantage and compromising American democracy. To prevent the practice of gerrymandering, we must know how to detect a …

About Time: Visualizing Time At Burning Man, Gordon D. Hoople, Austin Choi-Fitzpatrick, Nathaniel Parde, Diane Hoffoss, Max Mellette, Rachel Nishimura, Virginia Gutman

The Transdisciplinary STEAM+ Journal

About Time was a 30 foot long, 3000 pound wooden sundial that went up in flames at Burning Man 2019. The piece reflected on the role time plays in our lives. We organize our lives around time—are enslaved to time—and yet we know so little about it. Physicists and philosophers continue to grapple with deep puzzles of time—Is time a fundamental quantity, independent of human actions or observations or is it an emergent property of our perception? This installation projected time using two sundials: a horizontal dial which swept time out across the desert floor and an …

Third Voices Conference On Teaching Stem With Music, September 22-23, 2019, Lawrence M. Lesser

Journal of Humanistic Mathematics

The third annual VOICES (Virtual Ongoing Interdisciplinary Collaborations on Educating with Song; https://www.causeweb.org/voices/) conference will be held online September 22-23, 2019. Chaired by Tiffany Getty, this conference will explore the use of song to teach STEM (science, technology, engineering, and mathematics) at the postsecondary (or secondary) level.

How To Bake A Theorem, Cache Dexter

Journal of Humanistic Mathematics

We bake bread to satisfy our hunger, and we prove theorems to satisfy our hunger for knowledge and existence. Here I explore this analogy.

Tartaglia Re-Imagined, Aja Juola

Journal of Humanistic Mathematics

Niccolo Fontana wrote that he wore a beard because he feared his appearance was monstrous. No one should ever feel such shame about their physical appearance so I figured after all these years I would tell him to bare his scars proudly as a reminder of what he had endured and the life he led.

Applied Scientific Demiurgy I – Entrance Examination Information Sheet, Mario Daniel Martín

Journal of Humanistic Mathematics

This document provides all the required information needed by aspiring demiurges to sit the entrance examination for the foundation course Applied Scientific Demiurgy I in the scientific stream of the Bachelor of Applied Demiurgy at the Topological Hyper-university of Technological Cosmology.

Our Binary World, Simona Carini

Journal of Humanistic Mathematics

A poem about counting in binary, life, and love.

Tuesday, Ursula Whitcher

Journal of Humanistic Mathematics

"It's Tuesday, and the week's no longer new . . ." This poem's form is taken from the structure of the field with seven elements: the meter, in iambs, follows a pattern based on 5, 4, 6, 2, 3, the nontrivial values taken by powers of 5 (mod 7) as it generates the group of units of the field.

What Is Math?, Christopher Ryan Loga

Journal of Humanistic Mathematics

Here is presented a poem examining what math is and is not and the conclusions (if any) we can draw as such.

A Life Of Equations Shifting To A Life Of Words, Thomas R. Willemain

Journal of Humanistic Mathematics

As my technical life diminishes, my writing life increases. Since 2017 I have been writing, first a memoir of my time in the Intelligence Community, then poetry and flash fiction.

One of the missions I have assigned to my poetry is to expose to `regular' people the inner life of the mathematical person. The poems in this poetry folder develop three themes. "Formulations'' pokes a bit of fun at the bloated (and in this case almost musical) titles that can grow from our research; more seriously, it documents the change in self-definition that flows from recognizing the inevitable drift away …

Book Review: The Seduction Of Curves By Allan Mcrobie, Hans J. Rindisbacher

Journal of Humanistic Mathematics

This review emphasizes, as does the compelling and beautiful book, The Seduction of Curves by Allan McRobie, the “lines of beauty” that link art and mathematics. McRobie and his collaborator on the indispensable visuals of the volume, Helena Weightman, succeed admirably in connecting theoretically and visually the mathematical field of singularity or catastrophe theory and its graphical representations on the one hand and the seemingly intersecting lines around the volumes of the human body in the artistic representation of the nude. This book thus constitutes a creative and illuminating overlap of mathematics and art that lets the practitioners on both …

Book Review: What Is A Mathematical Concept? Edited By Elizabeth De Freitas, Nathalie Sinclair, And Alf Coles, Brendan P. Larvor

Journal of Humanistic Mathematics

This is a review of What is a Mathematical Concept? edited by Elizabeth de Freitas, Nathalie Sinclair, and Alf Coles (Cambridge University Press, 2017). In this collection of sixteen chapters, philosophers, educationalists, historians of mathematics, a cognitive scientist, and a mathematician consider, problematise, historicise, contextualise, and destabilise the terms ‘mathematical’ and ‘concept’. The contributors come from many disciplines, but the editors are all in mathematics education, which gives the whole volume a disciplinary centre of gravity. The editors set out to explore and reclaim the canonical question ‘what is a mathematical concept?’ from the philosophy of mathematics. This review comments …

Ducci’S Four-Number Game: Making Sense Of A Classic Problem Using Mobile Simulation, Lingguo Bu

Journal of Humanistic Mathematics

Ducci’s Four-Number Game is a classic mathematical puzzle appealing to a wide audience for its procedural simplicity, mathematical richness, and aesthetic values. This article first describes a few activities appropriate for school students and mathematics teachers to make sense of the intriguing behavior of the game. Then, using a mobile simulation, we delve into the lengths of the Four-Number Game and the corresponding probability distribution. The Ducci number game is playful, engaging, and full of mathematical surprises.

What Are The Odds At The Russia 2018 Fifa World Cup?, Gunhan Caglayan

Journal of Humanistic Mathematics

In this note we explore the group stage competition format in a standard FIFA World Cup Soccer Championship group phase. The group stage involves thirty-two teams divided into eight groups of four teams each, based on a draw that takes the national teams’ seeding and geographical location into consideration. Each of the four teams in a given group is scheduled to play one match against every other team in the same group. Upon the completion of six games in each of the eight groups (for a total of 48 games), the top two highest scoring teams (the winner and the …

Mathematics Out Of Nothing: Talking About Powerful Mathematical Ideas With Children, Matthew Oldridge

Journal of Humanistic Mathematics

Parents and educators have powerful opportunities to introduce children to big mathematical ideas, when those ideas become necessary. Children are capable and curious. They don’t need to be sheltered from big mathematical ideas. Bring out mathematical ideas when kids are ready, or when they are needed. This article describes one such instance, when I helped my six-year-old son move beyond zero in the negative direction when subtracting.

Everyman's Climb, Charles A. Coppin

Journal of Humanistic Mathematics

Hal and Verity represent two different philosophies of learning, one used by most of us. In today’s world, authentic teaching is indeed a heroic act, but may not be the most popular. This piece draws distinctions between these choices, each time we teach a course, each day we walk into the classroom, and even when working with an individual student; they are ever present.

Differential Equations Of Love And Love Of Differential Equations, Isaac Elishakoff

Journal of Humanistic Mathematics

In this paper, simple ordinary differential equations are discussed against the background of William Shakespeare’s Romeo and Juliet. In addition, a version of this relationship in a somewhat opposite setting is considered. It is proposed that engineering mathematics courses include this topic in order to promote additional interest in differential equations. In the final section it is shown that vibration of a single-degree-of-freedom mechanical system can be cast as a love-hate relationship between its displacement and velocity, and dynamic instability identified as a transition from trigonometric love to hyperbolic.