Mathematics Commons^™

Mathematical Melodies: The Exploration Of Music Using Fourier Signal Analysis, Courtney Francois

Honors Theses

Upon initial inspection, mathematics and music appear to be distinct disciplines lacking any connections to one another. However, many complex intersections lie beneath the surface. This paper seeks to explore these connections through the analysis of the Fourier series and the Fourier transformation of musical sound signals. Orthogonal functions of various frequencies are used to approximate and recover sound signals. Following the examination of the basis of Fourier signal analysis, MATLAB is utilized to visualize the connectedness of mathematics and music through sound signals from varied musical instruments and by performing Fourier signal analysis to the sound signals. Then, the …

Logic's Modern British Up-Enders, Defenders, And Extenders: Whately's Revitalization Of Logic, Calvin Jongsma

Faculty Work Comprehensive List

Logic developed dramatically during the last half of the nineteenth century. The baseline for these transformations in Great Britain was the revival of logic by Richard Whately around 1825. Whately successfully defended syllogistic logic as the science of valid reasoning against potent seventeenth and eighteenth-century detractors—Bacon, Locke, Reid, Campbell, Stewart, and others. In so doing, he made logic an intellectually respectable field of investigation for the next generation of logicians to explore and extend. This included John Stuart Mill (inductive logic), Augustus De Morgan (logic of relations), and George Boole (algebraic logic; propositional logic).

Effects Of The Gravity Gradient On The Path Of 1i/‘Oumuamua, Hannah R. Richardson

Poster Presentations

In October 2017, the asteroid 1I/’Oumuamua first passed into viewing range [1]. The asteroid is notable for being the first interstellar object to enter the solar system. 1I/’Oumuamua was also unusual in its geometry; it is thought to have an aspect ratio of 6:1 and a length of approximately 400 m [2] [3]. The asteroid was observed to experience a non-Keplerian acceleration estimated to be on the order of 1×10−6 m s-2 Several theories have been proposed for the cause of this acceleration, all of which are non-gravitational in nature: volatile outgassing, photon pressure, and solar winds [1][4]. However, none …

Inversion Numbers Of Oriented Graphs, Sam Harris

Theses, Dissertations and Culminating Projects

This thesis investigates recent results and conjectures regarding inversion numbers of oriented graphs. We define the inversion number of an oriented graph G as the minimum size of a decycling family of subsets of vertices of G. We focus on three areas: the maximum inversion number of a graph on n vertices, the inversion numbers of dijoins of two oriented graphs, and computational algorithms for deciding the inversion number. The main contribution of this thesis is a proposed construction of a graph with higher inversion number than that of any other currently known construction. We also provide explicit, rather than …

Analysis And Mathematical Recomposition Of The Art Of The Fugue, Joshua Cellar

Theses, Dissertations and Culminating Projects

The Art of the Fugue is a collection of 18 fugues and cannons written by Johann Sebastian Bach in the 1700s. Each piece starts with a single theme which is transformed, developed and mixed in fascinating mathematical combinations resulting in a distinctive musical style. In this paper, we use the method introduced in [7] to analyze 14 pieces from The Art of the Fugue. We focus on the variation of motifs from one piece to another as well as the complexities of mathematical transformations present throughout the entire collection. We also demonstrate how using maps of transformations and the rules …

Improving Research Software Engineering In Mathematics, Abram Miller

Honors Theses

Research Software Engineering is critical to modern mathematical research, enabling the creation, maintenance, and dissemination of computational tools that bridge theory and practice. However, the field faces systemic challenges, including insufficient funding, lack of institutional recognition, and gaps in training and infrastructure. This thesis investigates these challenges through two approaches: (1) a comparative survey study focused on mathematicians and (2) hands-on contributions to an open-source research software project.

The Improving Research Software Engineering in Mathematics survey, conducted from September 2024 to January 2025, adapts the survey framework developed by Carver et al. in A survey of the state of the …

Development Of Saddlepoint Methodologies For Sparse Sample Multiple Parameter Generalized Linear Models And Correlated Data Scenarios, Christopher Johnson

All Graduate Theses and Dissertations, Fall 2023 to Present

Parameter estimation using maximum likelihood techniques may be biased when sample sizes are small, event rates are low, or otherwise sparse counts exist in a parametric model. This in turn may lead researchers to draw invalid statistical conclusions when conventional methods are utilized. The saddlepoint approximation has potential to lessen the degree of bias in sparse data conditions through its use of moments beyond the mean and variance, which allows for more accurate approximations using a smaller number of observations. We propose two novel saddlepoint methods for use in practical analysis scenarios, as an alternative to maximum likelihood estimation. First, …

Computer Vision In Soccer: Yolov11 Analytics Engine For Quantifying Game Strategy, Connor S. Maurer

Data Science Undergraduate Honors Theses

Single-shot object detection capabilities significantly reduce computational overhead for real-time computer vision in sports analytics at 60 FPS. YOLO11’s lightweight CNN gives promising accuracy while meeting the low-latency demand of dynamic soccer matches. As data-driven approaches take over the sport of soccer, efficient player tracking systems become critical for informing coach’s strategies. I prototype the ETL (Extract, Transform, Load) process of data collected from a single- shot detection program and evaluate its viability for estimating player fatigue. YOLO11 detects players, the ball, and other characteristics, with the output transformed by homography to estimate the positions in the real world. These …

Knighted Pawn's Tour, Morgan Wilson

Honors Theses

The Knighted Pawn’s Tour is a variant of the traditional Knight’s Tour problem, whichitself is an instance of the Hamiltonian cycle problem. The deviation from the Knight’s Tour problem is that a pawn begins on any field on the second rank (row), and advances with legal moves that can include captures to the opposite end of the board to become a knight. These fields, used by the pawn on the path to knighthood, are subsequently forbidden for the knight’s return to the starting field. The knight must traverse the rest of the chessboard, visiting each remaining field exactly once …

Two-Player Trick-Taking Games On Bipartite Tournaments., Allan Bagley

College of Arts & Sciences Senior Theses

We consider a two-person game played on a bipartite tournament with equal size parts. The game is modeled after trick-taking card games like bridge or euchre. The two players, South and North, each receive one of the parts as their hand, and the arcs from a vertex in one hand beat the corresponding vertices in the other hand. The game is played over rounds called tricks where the number of tricks is equal to the number of vertices in each player’s hand. At the beginning of the game, one player is designated as the leader, and they play the first …

A Study Of Knots And Quandles, Zhaoqi Wu

Math and Computer Science Honors Theses

We explore the mathematical theory of knots through the lens of algebraic structures known as kei and quandles. We begin by introducing classical knot invariants and then study the fundamental kei of a knot as a tool for distinguishing knot types. We generalize this approach using various kinds of quandles, including Alexander and dihedral quandles, and investigate their associated polynomial invariants. We also examine the connection between quandles and group theory, as well as their algebraic representations in quandle rings. Moreover, we analyze idempotent elements in quandle rings over finite fields, providing both general results and specific examples.

Modules Of Finite Projective Dimension And Singularities, Nawaj Kc

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

In the first part of this thesis, we study liftings of modules of finite projective dimension. We introduce a notion of “Serre liftable” modules and deduce applications to multiplicity conjectures in local algebra. In the second part, we introduce and study a module of mixed K\"ahler differentials for finite algebras over ramified discrete valuation rings of mixed characteristic. We state and prove a Jacobian criterion for computing the singular loci of such algebras.

Advisors: Jack Jeffries and Mark Walker

Using Permutation Groups To Identify Family Of Capacity Achieving Codes, Daniel Joseph Welchons

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

When communicating over a noisy channel, the probability of message interference sets a maximum possible transmission rate known as the channel capacity. Any family of codes which have rates converging to the channel capacity and arbitrarily low probability of decoding failure is called capacity achieving. Such codes have been known to exist since the birth of information theory but are difficult to find explicitly. It has recently been shown that the permutation groups of a family of codes can be used to show that the family is capacity achieving on the q-ary erasure channel.

This this thesis seeks to …

Culture And Context: How Frames Of Teaching And Of Learning Mathematics Form And Change For Graduate Student Instructors, Johan Benedict A. Cristobal

Department of Mathematics: Dissertations, Theses, and Student Research

Why do instructors teach the way that they do? This question is key in understanding how and why asset-based or deficit-based practices are enacted in the classroom. In consequence, this question also gives insight into how the environment where students learn is shaped. Mathematics education research, in particular, has been concerned with how instructors respond to students’ contributions in the classroom and how these ways of responding can (dis)empower students.

One way to answer this question is to understand how instructors frame their own teaching and their students’ learning. Frames are mental constructs which help individuals filter details, interpret information, …

Maximal Independent Set Algorithms Within Procedural Planar Maps: A Large-Scale Evaluation, Chaucer Ihrig

Honors College Theses

Analysis of a childhood game has led us to the problem of maximum independent sets in planar graphs. We wrote a graph creation utility using R to generate a random planar map and its dual graph. This utility then finds a graph’s maximal independent set using a variety of six algorithms. We investigate statistical connections between graph structure, colorability, and the maximal independent sets found using these algorithms over an incredibly large and procedurally generated dataset. We find one can always win the coloring game if the resultant graph is two-colorable. The algorithms perform statistically and practically significantly better on …

Functional Data Analysis Of Weekly Confirmed Covid-19 Cases Per Million In Relation To Unemployment Rates And Emergency Declaration Responses, Elaina Kay Mcgovern

Boise State University Theses and Dissertations

At the onset of the COVID-19 pandemic, U.S. states implemented policies that attempted to mitigate the spread and effects of the disease, such as a state of emergency declaration. This study explores the application of functional data analysis (FDA) as a means to investigate the impact of U.S. state-level emergency declarations on weekly confirmed COVID-19 cases per million during the first year of the pandemic. We focus on the duration and extent to which proactive states showed significantly different patterns of COVID-19 spread than those that implemented policies reactively. Additionally, we examine patterns in monthly state-level unemployment rates over the …

Leading Digits Of Some P-Adic Numbers, Jinha Park

Boise State University Theses and Dissertations

I investigated the leading digits of some sequences in the p-adic numbers Q_p, hoping to find some sequences following Benford's law, or following something completely different. We found that the sequence of partial sums Sum^N_{n=1}\frac{1}{n^s}, where s\in\N is fixed, shows uneven distribution of leading digits, except for s\equiv 0 (mod{p,/em>-1}), where the distribution of leading digits is eventually even. We were able to characterize the distribution of this partial sum for almost all s.