Other Mathematics Commons^™

Ultrametrics And Complete Multipartite Graphs, Viktoriia Viktorivna Bilet, Oleksiy Dovgoshey, Yuriy Nikitovich Kononov

Theory & Applications of Graphs

Let (X, d) be a semimetric space and let G be a graph. We say that G is the diametrical graph of (X, d) if X is the vertex set of G and the adjacency of vertices x and y is equivalent to the equality diam X = d(x, y). It is shown that a semimetric space (X, d) with diameter d* is ultrametric if the diametrical graph of (X, d _ε) with d _ε (x, y) = min{d(x, y), ε} is complete multipartite for every ε ∈ (0, d* …

Self-Repelling Elastic Manifolds With Low Dimensional Range, Carl Mueller, Eyal Neumann

Journal of Stochastic Analysis

No abstract provided.

Induced Matrices: Recurrences And Markov Chains, Philip Feinsilver

Journal of Stochastic Analysis

No abstract provided.

Using Graph Theoretical Methods And Traceroute To Visually Represent Hidden Networks, Jordan M. Sahs

UNO Student Research and Creative Activity Fair

Within the scope of a Wide Area Network (WAN), a large geographical communication network in which a collection of networking devices communicate data to each other, an example being the spanning communication network, known as the Internet, around continents. Within WANs exists a collection of Routers that transfer network packets to other devices. An issue pertinent to WANs is their immeasurable size and density, as we are not sure of the amount, or the scope, of all the devices that exists within the network. By tracing the routes and transits of data that traverses within the WAN, we can identify …

Unomaha Problem Of The Week (2021-2022 Edition), Brad Horner, Jordan M. Sahs

UNO Student Research and Creative Activity Fair

The University of Omaha math department's Problem of the Week was taken over in Fall 2019 from faculty by the authors. The structure: each semester (Fall and Spring), three problems are given per week for twelve weeks, with each problem worth ten points - mimicking the structure of arguably the most well-regarded university math competition around, the Putnam Competition, with prizes awarded to top-scorers at semester's end. The weekly competition was halted midway through Spring 2020 due to COVID-19, but relaunched again in Fall 2021, with massive changes.

Now there are three difficulty tiers to POW problems, roughly corresponding to …

Exploring The Numerical Range Of Block Toeplitz Operators, Brooke Randell

Master's Theses

We will explore the numerical range of the block Toeplitz operator with symbol function \(\phi(z)=A_0+zA_1\), where \(A_0, A_1 \in M_2(\mathbb{C})\). A full characterization of the numerical range of this operator proves to be quite difficult and so we will focus on characterizing the boundary of the related set, \(\{W(A_0+zA_1) : z \in \partial \mathbb{D}\}\), in a specific case. We will use the theory of envelopes to explore what the boundary looks like and we will use geometric arguments to explore the number of flat portions on the boundary. We will then make a conjecture as to the number of flat …

On The Numerical Range Of Compact Operators, Montserrat Dabkowski

Master's Theses

One of the many characterizations of compact operators is as linear operators which
can be closely approximated by bounded finite rank operators (theorem 25). It is
well known that the numerical range of a bounded operator on a finite dimensional
Hilbert space is closed (theorem 54). In this thesis we explore how close to being
closed the numerical range of a compact operator is (theorem 56). We also describe
how limited the difference between the closure and the numerical range of a compact
operator can be (theorem 58). To aid in our exploration of the numerical range of
a compact …

Van Kampen Diagrams And Small Cancellation Theory, Kelsey N. Lowrey

Master's Theses

Given a presentation of G, the word problem asks whether there exists an algorithm to determine which words in the free group, F(A), represent the identity in G. In this thesis, we study small cancellation theory, developed by Lyndon, Schupp, and Greendlinger in the mid-1960s, which contributed to the resurgence of geometric group theory. We investigate the connection between Van Kampen diagrams and the small cancellation hypotheses. Groups that have a presentation satisfying the small cancellation hypotheses C'(1/6), or C'(1/4) and T(4) have a nice solution to the word problem known as Dehn’s Algorithm.

Implementation Of A Least Squares Method To A Navier-Stokes Solver, Jada P. Lytch, Taylor Boatwright, Ja'nya Breeden

Rose-Hulman Undergraduate Mathematics Journal

The Navier-Stokes equations are used to model fluid flow. Examples include fluid structure interactions in the heart, climate and weather modeling, and flow simulations in computer gaming and entertainment. The equations date back to the 1800s, but research and development of numerical approximation algorithms continues to be an active area. To numerically solve the Navier-Stokes equations we implement a least squares finite element algorithm based on work by Roland Glowinski and colleagues. We use the deal.II academic library , the C++ language, and the Linux operating system to implement the solver. We investigate convergence rates and apply the least squares …

Estimating Glutamate Transporter Surface Density In Mouse Hippocampal Astrocytes, Anca R. Radulescu, Annalisa Scimemi

Biology and Medicine Through Mathematics Conference

No abstract provided.

Optimal Time-Dependent Classification For Diagnostic Testing, Prajakta P. Bedekar, Paul Patrone, Anthony Kearsley

Biology and Medicine Through Mathematics Conference

No abstract provided.

Sheltered Math Curriculum For Middle School English Learners, Jasmine Ercink

Dissertations, Theses, and Projects

Language barriers have shown a need for differentiation and sheltered instruction in the classroom for English Learners (ELs) to be successful in the United States public school system. This project proposes a mathematics curriculum using SIOP so that both groups of students in the middle school level can increase their proficiency in the mathematics content area as well as experience opportunities for academic and social language development. The purpose of this report is to describe the processes, methods, data, and intent of the mathematics curriculum for these learners. The curriculum acts as an effective intervention to fill gaps in both …

The Mathematical Foundation Of The Musical Scales And Overtones, Michaela Dubose-Schmitt

Theses and Dissertations

This thesis addresses the question of mathematical involvement in music, a topic long discussed going all the way back to Plato. It details the mathematical construction of the three main tuning systems (Pythagorean, just intonation, and equal temperament), the methods by which they were built and the mathematics that drives them through the lens of a historical perspective. It also briefly touches on the philosophical aspects of the tuning systems and whether their differences affect listeners. It further details the invention of the Fourier Series and their relation to the sound wave to explain the concept of overtones within the …

3-Uniform 4-Path Decompositions Of Complete 3-Uniform Hypergraphs, Rachel Mccann

Mathematical Sciences Undergraduate Honors Theses

The complete 3-uniform hypergraph of order v is denoted as K_v and consists of vertex set V with size v and edge set E, containing all 3-element subsets of V. We consider a 3-uniform hypergraph P₇, a path with vertex set {v₁, v₂, v₃, v₄, v₅, v₆, v₇} and edge set {{v₁, v₂, v₃}, {v₂, v₃, v₄}, {v₄, v₅, v₆}, {v₅, v_{6 …}

Analyzing Suicidal Text Using Natural Language Processing, Cassandra Barton

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Using Natural Language Processing (NLP), we are able to analyze text from suicidal individuals. This can be done using a variety of methods. I analyzed a dataset of a girl named Victoria that died by suicide. I used a machine learning method to train a different dataset and tested it on her diary entries to classify her text into two categories: suicidal vs non-suicidal. I used topic modeling to find out unique topics in each subset. I also found a pattern in her diary entries. NLP allows us to help individuals that are suicidal and their family members and close …

The Decomposition Of The Space Of Algebraic Curvature Tensors, Katelyn Sage Risinger

Electronic Theses, Projects, and Dissertations

We decompose the space of algebraic curvature tensors (ACTs) on a finite dimensional, real inner product space under the action of the orthogonal group into three inequivalent and irreducible subspaces: the real numbers, the space of trace-free symmetric bilinear forms, and the space of Weyl tensors. First, we decompose the space of ACTs using two short exact sequences and a key result, Lemma 3.5, which allows us to express one vector space as the direct sum of the others. This gives us a decomposition of the space of ACTs as the direct sum of three subspaces, which at this point …

Improved First-Order Techniques For Certain Classes Of Convex Optimization, Trevor Squires

All Dissertations

The primary concern of this thesis is to explore efficient first-order methods of computing approximate solutions to convex optimization problems. In recent years, these methods have become increasingly desirable as many problems in fields such as machine learning and imaging science have scaled tremendously. Our aim here is to acknowledge the capabilities of such methods and then propose new techniques that extend the reach or accelerate the performance of the existing state-of-the-art literature.

Our novel contributions are as follows. We first show that the popular Conditional Gradient Sliding (CGS) algorithm can be extended in application to objectives with H\"older continuous …

How To Guard An Art Gallery: A Simple Mathematical Problem, Natalie Petruzelli

The Review: A Journal of Undergraduate Student Research

The art gallery problem is a geometry question that seeks to find the minimum number of guards necessary to guard an art gallery based on the qualities of the museum’s shape, specifically the number of walls. Solved by Václav Chvátal in 1975, the resulting Art Gallery Theorem dictates that ⌊n/3⌋ guards are always sufficient and sometimes necessary to guard an art gallery with n walls. This theorem, along with the argument that proves it, are accessible and interesting results even to one with little to no mathematical knowledge, introducing readers to common concepts in both geometry and graph …

Additional Fay Identities Of The Extended Toda Hierarchy, Yu Wan

Rose-Hulman Undergraduate Mathematics Journal

The focus of this paper is the extended Toda Lattice hierarchy, an infinite system of partial differential equations arising from the Toda lattice equation. We begin by giving the definition of the extended Toda hierarchy and its explicit bilinear equation, following Takasaki’s construction. We then derive a series of new Fay identities. Finally, we discover a general formula for one type of Fay identity.

Fock And Hardy Spaces: Clifford Appell Case, Daniel Alpay, Kamal Diki, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we study a specific system of Clifford–Appell polynomials and, in particular, their product. Moreover, we introduce a new family of quaternionic reproducing kernel Hilbert spaces in the framework of Fueter regular functions. The construction is based on a general idea which allows us to obtain various function spaces by specifying a suitable sequence of real numbers. We focus on the Fock and Hardy cases in this setting, and we study the action of the Fueter mapping and its range.

A New Metaphor: How Artificial Intelligence Links Legal Reasoning And Mathematical Thinking, Melissa E. Love Koenig, Colleen Mandell

Marquette Law Review

Artificial intelligence’s (AI’s) impact on the legal community expands exponentially each year. As AI advances, lawyers have more powerful tools to enhance their ability to research and analyze the law, as well as to draft contracts and other legal documents. Lawyers are already using tools powered by AI and are learning to shift their methodologies to take advantage of these enhancements. To continue to grow into their shifting role, lawyers should understand the relationship between AI, mathematics, and legal reasoning.

Finite Subdivision Rules For Matings Of Quadratic Thurston Maps With Few Postcritical Points, Jeremiah Zonio

Undergraduate Theses

A finite subdivision rule is set of instructions for repeatedly subdividing a partitioning of a given space. This turns out to be incredibly useful when attempting to describe a process known as polynomial mating. Polynomial mating is a way of gluing together two spaces which two polynomials may act upon such that the glued borders of each space respects the dynamics described by each polynomial. For matings of Misiurewicz polynomials, the spaces we are gluing together are 1-dimensional and are thus all border. This poses a conceptual difficulty which this paper attempts to resolve by using finite subdivison rules to …

Remotely Close: An Investigation Of The Student Experience In First-Year Mathematics Courses During The Covid-19 Pandemic, Sawyer Smith

Honors Program: Senior Projects (Public)

The realm of education was shaken by the onset of the COVID-19 pandemic in 2020. It had drastic effects on the way that courses were delivered to students, and the way that students were getting their education at the collegiate level. At the University of Nebraska – Lincoln, the pandemic dramatically changed the way that first-year mathematics courses looked for students. By Spring 2021, students had the opportunity to take their first-year math courses either in-person or virtually. This project sought to identify differences between the two methods of course delivery during the Spring 2021 semester, regarding interaction with peers …

Varieties Of Nonassociative Rings Of Bol-Moufang Type, Ronald E. White

All NMU Master's Theses

In this paper we investigate Bol-Moufang identities in a more general and very natural setting, \textit{nonassociative rings}.

We first introduce and define common algebras. We then explore the varieties of nonassociative rings of Bol-Moufang type. We explore two separate cases, the first where we consider binary rings, rings in which we make no assumption of it's structure. The second case we explore are rings in which, $2x=0$ implies $x=0$.

Quadratic Neural Network Architecture As Evaluated Relative To Conventional Neural Network Architecture, Reid Taylor

Senior Theses

Current work in the field of deep learning and neural networks revolves around several variations of the same mathematical model for associative learning. These variations, while significant and exceptionally applicable in the real world, fail to push the limits of modern computational prowess. This research does just that: by leveraging high order tensors in place of 2nd order tensors, quadratic neural networks can be developed and can allow for substantially more complex machine learning models which allow for self-interactions of collected and analyzed data. This research shows the theorization and development of mathematical model necessary for such an idea to …

Superoscillating Sequences And Supershifts For Families Of Generalized Functions, F. Colombo, I. Sabadini, Daniele Carlo Struppa, A. Yger

Mathematics, Physics, and Computer Science Faculty Articles and Research

We construct a large class of superoscillating sequences, more generally of F-supershifts, where F is a family of smooth functions in (t, x) (resp. distributions in (t, x), or hyperfunctions in x depending on the parameter t) indexed by λ ∈ R. The frame in which we introduce such families is that of the evolution through Schrödinger equation (i∂/∂t−H (x))(ψ) = 0 (H (x) = −(∂2/∂x2)/2+V (x)), V being a suitable potential). If F = {(t, x) → ϕλ(t, x) ; λ ∈ R}, where ϕλ is evolved from the initial datum x → eiλx , F-supershifts will be of …

High School Student Perspective: My Njit Stem For Success Internship Experience, Michael Mora

STEM Month

During the 2020-2021 school year, I was a senior at the Academy for Mathematics, Science, and Engineering (AMSE) in Rockaway, NJ. At AMSE, a STEM-focused four-year magnet high school program hosted at Morris Hills High School, participating in an extended internship senior year is a cornerstone of the learning process. Required to complete a STEM-related internship to graduate, Academy students are encouraged to seek out an internship they’re passionate about in a field of their choice. The internship, which must be conducted under the mentorship of an industry professional, must meet the New Jersey-approved standards for a work-based learning experience …

Spectral Theorem Approach To The Characteristic Function Of Quantum Observables, Andreas Boukas

Journal of Stochastic Analysis

No abstract provided.

Construction Of The Canonical Representation From A Noncanonical Representation, Yuji Hibino

Journal of Stochastic Analysis

No abstract provided.

Existence And Uniqueness Of Minimizers For A Nonlocal Variational Problem, Michael Pieper

Honors Program: Senior Projects (Public)

Nonlocal modeling is a rapidly growing field, with a vast array of applications and connections to questions in pure math. One goal of this work is to present an approachable introduction to the field and an invitation to the reader to explore it more deeply. In particular, we explore connections between nonlocal operators and classical problems in the calculus of variations. Using a well-known approach, known simply as The Direct Method, we establish well-posedness for a class of variational problems involving a nonlocal first-order differential operator. Some simple numerical experiments demonstrate the behavior of these problems for specific choices of …