Other Mathematics Commons^™

Exploring The Exceptional Extreme Rays Of The Copositive Cone, Trent Holmgren

All NMU Master's Theses

In this paper we will look at the exceptional extreme rays of COP^5 and COP^6 and identify if they are exposed or nonexposed. We start with some necessary background material on copositive matrices and cones. Then we construct two algorithms to see if a matrix is exposed or nonexposed.

Machine Learning: Neural Networking With Relu And Optimization, Aidan Redmond Brownell

Undergraduate Theses, Capstones, and Recitals

At its core, learning is an algorithmic process: it begins with input data, undergoes a series of transformations or computations, and yields an output intended to solve a specific task. This output is then compared against a target or desired result, and the internal mechanisms are updated based on how well the output aligns with expectations. While this feedback-driven process occurs almost effortlessly in humans, it is a far more structured, deliberate, and computationally intensive undertaking for machines.

Basic Theory And Implementations Of Quantum Error Correction, Derek Rodriguez

Undergraduate Theses, Capstones, and Recitals

The introduction of quantum computing has presented algorithmic solutions to computationally difficult challenges that are far more efficient than those of classical computers. These algorithms leverage the properties of quantum mechanics to manipulate the quantum properties of subatomic particles, requiring immense precision and stability. Current quantum hardware, however, is too noisy and introduces too many errors for these algorithms to be useful in practice, necessitating the use of error correction algorithms. This field survey seeks to introduce various principles of quantum mechanics relevant to quantum computing and quantum error correction (QEC), detail the implementation and motivations of a basic QEC …

A Study On Fuzzy Time-Series And Its Applications To Stock Price Forecasting, Takeshi Stormer

University Honors Theses

Fuzzy mathematics looks to incorporate the vagueness that exists within the real world, specifically regarding imprecise classes, or non-numerical information expressed as "linguistic" variables. Since most traditional mathematical theories do not have the ability to be applied with the exactness that is otherwise seen in mathematics. As such, there had been many applications of fuzzy mathematics throughout many different fields of mathematics, including that of forecasting. By exploring the fundamentals of fuzzy mathematics, including fuzzy sets, operations of fuzzy sets, the surface level introduction to fuzzy logic, fuzzy relations, operations of fuzzy relations, and fuzzy time-series, this work looks to …

On Excursions Associated With A Certain Local Time Of Simple Symmetric Random Walks, With Applications, Takahiko Fujita, Naohiro Yoshida

Journal of Stochastic Analysis

In this note, some applications of excursions associated with a certain local time of simple symmetric random walks are presented. Specifically, the excursions are applied to calculate some probability distributions of interest regarding the random walks. Furthermore, a solution of the Skorokhod embedding problem for random walks is obtained through the excursions.

Tilings In The 3 Dimensional Lattice With L-Tetrominoes, Ian N. Bridges

Rose-Hulman Undergraduate Mathematics Journal

We consider three dimensional L-tetrominoes. We show that there exists at least one way to tile every three dimensional rectangle whose side lengths are at least $3$ and area is congruent to $1 \pmod 4$ such that one square goes untiled. In addition, we show that every three dimensional rectangle is tileable provided one side has length at least $2$ and the other is a multiple of $4$.

A Novel Closed Monoidal Structure On The Nucleus Of A Profunctor, Samantha K. Jarvis

Dissertations, Theses, and Capstone Projects

We describe a novel closed monoidal structure on the nucleus of a profunctor enriched over a posetal category, a distinguished subcategory of the category of presheaves given by the invariant part of an adjunction induced by the profunctor. Our structure is motivated by a connection to a notion of type given by orthogonality. For specific examples, we consider the two-element enriching category {0,1} and the reals.

Banach Algebras And The Gelfand Theory Of Group Algebras On Locally Compact Abelian Groups, James Gabriel Bonvanie

Master's Theses

A Banach algebra is a complex algebra that is simultaneously a Banach space in which the norm is submultiplicative. Notably, $L^1(\mathbb{R})$ with the convolutional product is an Abelian, non-unital Banach algebra that admits an approximate identity. We rectify $L^1(\mathbb{R})$ lacking a unit via the unitization $L^1(\mathbb{R})\times\mathbb{C}$ with identity $(0,1)$. Unitization opens the discussion to the spectrum $\sigma(x)$ of a Banach algebra element, in which the spectrum is a nonempty, compact subset of the complex plane. The spectrum of an Abelian Banach algebra is fully characterized with multiplicative linear functionals, and we prove that the Fourier transform is the unique multiplicative …

The Jacod-Yor Theorem For Sigma Martingales And The Second Fundamental, Moritz Sohns

Journal of Stochastic Analysis

In this paper, we prove the Jacod-Yor Theorem for sigma martingales, a class of processes that generalize local martingales and play a pivotal role in financial mathematics. While the Jacod-Yor Theorem has been extensively studied for L2-martingales, martingales, and local martingales, no prior version exists for sigma martingales. Our result establishes the connection between sigma martingales and their martingale representation properties, addressing a critical gap in the literature. As an application, we prove the Second Fundamental Theorem of Asset Pricing for markets where price processes are modeled as sigma martingales.

Similarity Metrics, Metrics, And Conditionally Negative Definite Functions, Daniel Alpay, Liora Mayats-Alpay

Mathematics, Physics, and Computer Science Faculty Articles and Research

We give an example of a similarity metric which is not positive definite, and present a general theorem which provides a large family of similarity metrics which are positive definite.

Analyticity, Superoscillations And Supershifts In Several Variables, Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa, Alain Yger

Mathematics, Physics, and Computer Science Faculty Articles and Research

Superoscillations have roots in various scientific disciplines, including optics, signal processing, radar theory, and quantum mechanics. This intriguing mathematical phenomenon permits specific functions to oscillate at a rate surpassing their highest Fourier component. A different way of thinking about superoscillations consists in realizing that it is possible to reproduce the exponential function far away from the origin by only knowing its value in a countable set of points near the origin. By using this perspective, one can extend the idea of superoscillations to functions that are not a sum of exponential functions, namely to the notion of supershift. The study …

Applications Of The Mathieu Groups And Information Theory In Dna Encoding Functions, Juan C. Nava Jr

Theses and Dissertations

A foundational idea in mathematics lies in breaking down existing components into their bare fundamentals. As evidenced by prime numbers and composites, we learn this idea at an early age. Categorizing these broken-down components into their simplest form allows mathematicians to construct proofs from emergent patterns. John Conway’s Atlas of Finite Groups in the 1990s was particularly concerned with the categorization of structures known as groups. There are certain axioms a group must adhere to, which amount to the retention of symmetry; ultimately a group helps us to better understand symmetric actions performed on a set with a binary operation. …

Statistics - What Does My Data Say About Me?, Taylor Gadsden-Deterville

Student Scholar Symposium Abstracts and Posters

For my Introduction to Statistics Class, I have been tasked with collecting unique, personal data to give insight into my daily routine. I decided to record nine different outcomes (two qualitative and seven quantitative). On February 6, 2025, I began with a blank Excel sheet, and so far, I have 57 full days of data collected. I will continue monitoring my findings for the remainder of the Spring 2025 Semester. Per my project instructions, I must include tables and graphs for my qualitative and quantitative outcomes. So far, I have collected daily quantitative data on my screen time (Instagram and …

A Preliminary Study Of Hilbert–Kunz Functions: Coefficient Behavior In A Normal Affine Semigroup Ring, Jesus A. Mendiola Herrera

Theses and Dissertations

In 1890, David Hilbert published a set of notes on what now constitutes one of the bases of Commutative Algebra; his work would eventually influence the efforts of mathematicians like Ernst Kunz. In 1969, Ernst Kunz introduced a particular mapping regarding modules of regular local rings. His goal was to characterize Noetherian local rings of prime characteristic by computing the length of the composition series under Frobenius power transformations. In this thesis, the focus will be on stating the initial steps on finding the coefficients of the Hilbert-Kunz function of the normal affine semigroup ring of the form R = …

On The Hexgame, Corwin Jones

Mathematical Sciences Technical Reports (MSTR)

The SOMA Cube has been studied by mathematicians for a number of decades, but so far methods for solving three-dimensional space-filling puzzles like the SOMA cube remain numerical; we do not have a means to predict the number of solutions to SOMA-like puzzles. We present a two-dimensional puzzle that shares certain features of the SOMA Cube, with the hope that it will be a more convenient object of study for future research into space-filling/space-covering puzzles.

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 …

Geometric Algebra For Field Theory In Curved Spacetime, Kevin Rhine

All Graduate Reports and Creative Projects, Fall 2023 to Present

Physics seeks to understand the universe by uncovering the fundamental laws that govern matter, energy, space, and time. At its heart lies the challenge of unification: finding a mathematical framework that consistently describes these interactions across all scales, from the subatomic to the cosmological.

This thesis explores geometric algebra, a mathematical language that unifies algebra and geometry, as a tool for advancing this understanding. By extending this framework to curved spacetimes, where gravity influences the structure of space and time, we investigate its ability to describe physical phenomena such as electromagnetism and general relativity. A notable contribution includes the geometric …

The Impact Of Corporate Profits On Gdi Estimates: Forecasting And Data Revisions, Peterson Haas

Honors Theses

The National Income and Product Accounts (NIPAs) produced by the U.S. Bureau of Economic Analysis (BEA) provide key measures of U.S. economic activity, including gross domestic product (GDP) and gross domestic income (GDI). Although conceptually equivalent, GDP and GDI often diverge due to differences in source data and revision timing. This study focuses on corporate profits—a small but volatile component of GDI that is frequently revised, especially during the BEA’s annual (A1) and benchmark (C1) revisions. I examine how revisions to corporate profits influence GDI revisions and whether they can improve real-time estimates of GDI. First, I document the size …

Constructing Binary Encoding Matrices From Joined Graphs, Joshua Avalos

Electronic Theses, Projects, and Dissertations

Codes and technology are part of our daily lives and allow the modern world to function, and for us to have conveniences in our lives such as smartphones that can be used to privately call people on the other side of the planet, and for secure access to the internet. In this thesis we will explore the construction of binary codes created by vertex-edge incidence matrices of planar graphs. The Hamming (7,4) code was an incredible code that allowed the detection and correction of errors after receiving them through a transmission. We will explore the possibility of the creation of …

The Food Truck: A Multi-Product Newsvendor With Trans-Shipment Cost, Samuel Ajibola

Electronic Theses and Dissertations

The Newsvendor Problem is a key model in supply chain management that focuses on determining the optimal order quantity to minimize costs under uncertain demand. This thesis introduces the Food Truck Problem, an extension of the Newsvendor model that incorporates nonlinear transshipment costs for inventory transportation. In this context, a Food Truck must determine the optimal stock levels for multiple products while minimizing costs related to stock shortages, excess inventory, and transportation. Unlike traditional Newsvendor models, our approach explicitly considers a quadratic transshipment cost, which necessitates the use of Lagrangian duality and Karush-Kuhn-Tucker (KKT) conditions for analysis. Moreover, we apply …