Physical Sciences and Mathematics Commons^™

Existence And Uniqueness Of Solutions Of Fourth Order Iterative Boundary Value Problems, Zach Whaley

Theses and Dissertations

In this dissertation, we study the fourth-order iterative differential equation \begin{displaymath} x^{(4)}(t) = \f(t, x(t), x^{[2]}(t), \dots, x^{[m]}(t)) \end{displaymath} where $x^{[2]}(t) = x(x(t))$ and $x^{[j]}(t) = x(x^{[j - 1]}(t))$ for $j > 2$. We consider the above equation with multiple sets of boundary conditions, and we state results on the existence and uniqueness of solutions for each set of boundary conditions. In Chapter 2, the boundary conditions are conjugate boundary conditions, \begin{align*} &x(-a) = -a, \ x'(-a) = b, \ x''(-a) = c, \ x(a) = a \\ &x(-a) = -a, \ x(a) = a, \ x'(a) = b, \ x''(a) …

Traveling Waves In Biological Population Models, Ahlam Alzahrani

Theses and Dissertations

In this thesis, we examine the existence of traveling waves in population models. We begin by exploring conditions under which traveling waves exist, even when the migration probability lacks a density function or, if the density function exists, it may be discontinuous. To address this, we impose certain conditions that ensure the monotonicity of the evolution operator, enabling the application of the Monotone Iteration Method. Next, we explore the existence of monotone traveling waves in a general class of integral-difference population models that depend on both the previous state and long-term memory, allowing for the consideration of multiple past states. …

Snap Recipients' Labor Supply In The Presence Of Children In Households, Sanae Tashiro, Jongsung Kim

Mathematics and Economics Faculty Journal Articles

No abstract provided.

An Examination Of Hilbert Spaces, Fnu Sumaiya

Theses and Dissertations

This independent study focuses on the exploration of Hilbert spaces, a cornerstone of modern analysis and a fundamental construct in mathematics. It starts by introducing vector space, metric and metric spaces and then Hilbert spaces and its properties.

This study includes good amount of mathematical proofs, theorems and examples. Through extensive research and analysis, this independent study aims to offer a comprehensive examination of Hilbert spaces, detailing their properties and their pivotal role in modern analysis.

Math Is Good, Elisa Benthem

Student Work

"There are endless ways we can explore creation, each revealing something wonderfully distinct."

Posting about ­­­­­­­­the beauty of mathematics from In All Things, an online hub that offers insight into maintaining and faithful and orthodox Reformed Christian worldview while fearlessly engaging in every aspect of contemporary life – until all is made new.

Math is Good

A Finite-Horizon Approach To Active Level Set Estimation, Phillip Kearns, Bruno Jedynak, John Lipor

Mathematics and Statistics Faculty Publications and Presentations

We consider the problem of active learning for level set estimation (LSE), where the goal is to localize all regions where a function of interest lies above/below a given threshold as quickly as possible. We present a finitehorizon search procedure to perform LSE in one dimension while optimally balancing both the final estimation error and the distance traveled during active learning for a fixed number of samples. A tuning parameter is used to trade off between the estimation accuracy and distance traveled. We show that the resulting optimization problem can be solved in closed form and that the resulting policy …

Exploring The Human Side Of Math, Mike Janssen

Faculty Work Comprehensive List

"Abstract mathematics is an important way to better understand and marvel at the beauty and intricacy of God’s creation."

Posting about ­­­­­­­­experiences in math research from In All Things, an online hub that offers insight into maintaining and faithful and orthodox Reformed Christian worldview while fearlessly engaging in every aspect of contemporary life – until all is made new.

Exploring the Human Side of Math

A Phenomenological Study Of Persistence For Black Women In Mathematics Doctoral Programs, Shushannah Marie Smith

Doctoral Dissertations and Projects

The purpose of this phenomenological study was to describe the doctoral persistence experiences of Black women mathematicians. Guided by McGee’s fragile and robust mathematical identity framework, the research design followed Moustakas’ qualitative transcendental phenomenological approach. The researcher explored the contributing factors to the phenomenon guided by the central research question, “What are the doctoral persistence experiences of Black women mathematicians?” The study involved 11 Black women participants who hold or were within one year of holding a doctoral degree in mathematics and were selected to ensure diversity and representativeness. Data collection encompassed three methods: a mathematical identity development timeline, one-on-one …

Exploring The Roles Of Genetics, Gender, Nurturing, And Learning Methods In Mathematical Skills Development, Laura Alinani, Jana Shyti

Honors Projects

This research explores the multifaceted factors influencing the development of mathematical competence through a combination of cognitive, psychological, and sociocultural frameworks. The study aims to address how genetic predispositions, nurturing environments, teaching methods, and societal influences interact to shape mathematical abilities. Using insights from genetic research, cognitive psychology, and education theory, we analyze the role of effective teaching methods, such as active knowledge construction and differentiated instruction, in fostering mathematical skills. The study also examines how societal biases, particularly gender stereotypes, impact students' confidence and interest in math. The findings highlight the importance of promoting gender-inclusive practices and early exposure …

Mathematics In Contemporary Society, Patrick J. Wallach

Open Educational Resources

Mathematics in Contemporary Society is the textbook that corresponds to MA-321, the course of the same name. The course is designed to provide students with mathematical ideas and methods found in the social sciences, the arts, and in business. Topics will include fundamentals of statistics, scatterplots, graphics in the media, problem solving strategies, dimensional analysis, mathematics in music and art, and mathematical modeling. EXCEL is used to explore real world applications.

Ideals In Semigroups Of Partial Transformations With Invariant Set, Jitsupa Srisawat, Yanisa Chaiya

Turkish Journal of Mathematics

This paper explores the ideals and their structural properties in two generalizations of the partial transformationsemigroup. Furthermore, principal, maximal, and minimal ideals within these semigroups are elucidated.

Oer Textbook Review For Calculus - Openstax Calculus, Jing Hu Ph.D.

Open Educational Resources Publications

This OER textbook review provides a comprehensive evaluation of the "Calculus" textbook series published by OpenStax. The reviewer, Jing Hu, an adjunct lecturer at Bentley University, highlights the textbook's strengths, including its thorough coverage of essential calculus topics, accurate and well-established mathematical principles, practical relevance, and user-friendly design. The open-access nature of the resource is seen as a significant advantage, contributing to its long-term utility and accessibility for both students and educators. Overall, the review concludes that the OpenStax Calculus textbook is a high-quality, comprehensive, and freely available resource that effectively supports the learning and teaching of calculus.

Matrix-Free High-Performance Saddle-Point Solvers For High-Order Problems In H (Div), Will Pazner, Tzanio Kolev, Panayot S. Vassilevski

Mathematics and Statistics Faculty Publications and Presentations

This work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in . The solvers are applicable to grad-div and Darcy problems in saddle-point formulation, and have applications in radiation diffusion and porous media flow problems, among others. Using the interpolation–histopolation basis (cf. [W. Pazner, T. Kolev, and C. R. Dohrmann, SIAM J. Sci. Comput., 45 (2023), pp. A675–A702]), efficient matrix-free preconditioners can be constructed for the -block and Schur complement of the block system. With these approximations, block-preconditioned MINRES converges in a number of iterations that is independent of the mesh size and polynomial degree. The …

Diagrams In Involutive Residuated Lattices, Isis A. Gallardo

Electronic Theses and Dissertations

First, we show that every distributive lattice-ordered pregroup can be embedded into a functional algebra over an integral chain, thereby improving the existing Cayley/Holland style embedding theorem. Using this result, we demonstrate that the variety of all dis tributive lattice-ordered pregroups is generated by the functional algebra on the integers. Additionally, we prove that the equational theory of this variety is decidable.

Next, we establish that DLP is equal to the join of its subvarieties LP_n, where 𝑛 ∈ ℤ⁺, consisting of 𝑛-periodic ℓ-pregroups. We also prove that every algebra in LP_n can be embedded …

B-Chromatic Number Of The Graph Power Of The Star Graph, Erik Dahlen

DePaul Discoveries

In this paper we focus on the newly introduced b-colorings of a graph G. A b-coloring is a proper coloring such that for each color class, there exists at least one vertex which is adjacent to every other color. The b-chromatic number of a graph G is the largest number k such that G admits a b-coloring with k colors. This paper will introduce the b-chromatic number of some interesting graphs. Several operations of graphs are defined, and the b-chromatic number of those operations are found. All graphs in this paper are simple, connected, …

Math 75: Introduction To Linear Algebra, Sarah K. Merz

Pacific Open Texts

This text is intended to use in a first course of Linear Algebra with a prerequisite of Calculus 1. Topics covered include systems of linear equations, matrix operations and inverses, linear transformations, Markov chains, determinants, eigenvalues and eigenvectors, diagonalization, vector geometry, projections and planes, homogeneous coordinates, subspaces, spanning sets, linear independence, orthogonality, fundamental subspaces, and least squares.

Evaluation Of Inner Products Of Implicitly Defined Finite Element Functions On Multiply Connected Planar Mesh Cells, Jeffrey S. Ovall, Samuel E. Reynolds

Mathematics and Statistics Faculty Publications and Presentations

In recent years, there has been significant interest in the development of finite element methods defined on meshes that include rather general polytopes and curvilinear polygons. In the present work, we provide tools necessary to employ multiply connected mesh cells in planar domains, i.e., cells with holes, in finite element computations. Our focus is efficient evaluation of the ��1��1 semi-inner product and ��2��2 inner product of implicitly defined finite element functions of the types arising in boundary element based finite element methods and virtual element methods. Such functions are defined as solutions of Poisson problems having a polynomial source term …

Combinatorial Problems On The Integers: Colorings, Games, And Permutations, Collier Gaiser

Electronic Theses and Dissertations

This dissertation consists of several combinatorial problems on the integers. These problems fit inside the areas of extremal combinatorics and enumerative combinatorics.

We first study monochromatic solutions to equations when integers are colored with finitely many colors in Chapter 2. By looking at subsets of {1, 2, . . . , n} whose least common multiple is small, we improved a result of Brown and Rödl on the smallest integer n such that every 2-coloring of {1, 2, . . . , n} has a monochromatic solution to equations with unit fractions. Using a recent result of Boza, …

Building Blocks For W-Algebras Of Classical Types, Vladimir Kovalchuk

Electronic Theses and Dissertations

The universal 2-parameter vertex algebra W_∞ of type W(2, 3, 4; . . . ) serves as a classifying object for vertex algebras of type W(2, 3, . . . ,N) for some N in the sense that under mild hypothesis, all such vertex algebras arise as quotients of W_∞. There is an ℕ X ℕ family of such 1-parameter vertex algebras known as Y-algebras. They were introduced by Gaiotto and Rapčák are expected to be building blocks for all W-algebras in type A, i.e, every W-(super) algebra in …

Circling The Square: Computing Radical Two, Isaiah Mellace, Joshua Kroeker

NEXUS: The Liberty Journal of Interdisciplinary Studies

Discoveries of equations for irrational numbers are not new. From Newton’s Method to Taylor Series,there are many ways to calculate the square root of two to arbitrary precision. The following method is similar in this way, but it is also a fascinating derivation from geometry that has applications to other irrationals. Additionally, the equation derived has some properties that may lead to fast computation. The first part of this paper is dedicated to deriving the equation, and the second is focused on computer science implementations and optimizations.

Khovanov Homology And Legendrian Simple Knots, Ryan J. Maguire

Dartmouth College Ph.D Dissertations

The Jones polynomial and Khovanov homology are powerful invariants in knot theory. Their computations are known to be NP-Hard and it can be quite a challenge to directly compute either of them for a general knot. We develop explicit algorithms for the Jones polynomial and discuss the implementation of an algorithm for Khovanov homology. Using this we tabulate the invariants for millions of knots, generate statistics on them, and formulate conjectures for Legendrian and transversely simple knots.

A Thesis, Or Digressions On Sculptural Practice: In Which, Concepts & Influences Thereof Are Explained, Set Forth, Catalogued, Or Divulged By Way Of Commentaries To A Poem, First Conceived By The Artist, Fed Through Chatg.P.T., And Re-Edited By The Artist, To Which Are Added, Annotated References, Impressions And Ruminations Thereof, Also Including Private Thoughts & Personal Accounts Of The Artist, Jaimie An

Masters Theses

This thesis is an exercise in, perhaps a futile, attempt to trace just some of the ideas, stories, and musings I might meander through in my process. It’s not quite a map, nor is it a neat catalogue; it is a haphazard collection of tickets and receipts from a travel abroad, carelessly tossed in a carry-on, only to be stashed upon returning home. These ideas are derived from much greater thinkers and authors than myself; I am a mere collector or a translator, if that, and not a very good one, for much is lost. I do not claim comprehensive …

A Brief Introduction To General Topology, Richard P. Millspaugh

Open Educational Resources

The material in this text is intended to be accessible to undergraduates who have had an introduction to elementary set theory and proof techniques. It includes sufficient material from general topology to prove the two main topological results found in a standard first semester calculus course: the Intermediate Value Theorem and the Extreme Value Theorem. This material can be found in Chapters 2 through 6 and makes up the bulk of the text. Rather than approaching these topics through use of the standard euclidean metric, it defines the standard topology on R in terms of the usual order on R. …

Valid Confidence Intervals For Μ,Σ When There Is Only One Observation Available, Anirban Dasgupta, Stephen Portnoy

Mathematics and Statistics Faculty Publications and Presentations

Portnoy (2019) considered the problem of constructing an optimal confidence interval for the mean based on a single observation X ∼ N(µ, σ2). Here we extend this result to obtaining 1-sample confidence intervals for σ and to cases of symmetric unimodal distributions and of distributions with compact support. Finally, we extend the multivariate result in Portnoy (2019) to allow a sample of size m from a multivariate normal distribution where m may be less than the dimension.

Some Hypergeometric Identities And Related Leonard Pairs, Taiyo Summers Terada

Dissertations and Theses

The notion of a Leonard pair was introduced by Terwilliger in 2001 to simplify Leonard's theorem, which classifies the orthogonal polynomials in the terminating branch of the Askey-Wilson scheme. In the same year, Kresch and Tamvakis made a conjecture about a certain ₄F₃ hypergeometric series while studying the arithmetic analogues of the standard conjectures for the Grassmanian G(2,n). The ₄F₃ series appearing in their conjecture is closely related to a family of orthogonal polynomials in the Askey-Wilson scheme. Consequently, the theory of Leonard pairs provides a useful framework for understanding their conjecture.

In …

Struggling To Be Positive Problem Solvers – A Study On The Relationship Of Elementary Mathematical Self-Concept, Gender, Learner Identification, And Productive Struggle Trajectories On Constructed Response Questions When Being Taught With A Standardized Curriculum, Kali D. Rogers

LSU Doctoral Dissertations

Mathematical word problems are some of the most difficult problems for students to solve due to their complexity and relationship to real-world situations (Şahin et al., 2014; Verschaffel et al., 2020). With math being known for right-or-wrong answers, these open-ended questions cause much confusion and frustration, specifically for students of various subgroups. In the last several decades, educational teaching expectations have shifted from engaging, self-created learning to robotic and redundant curriculum expectations that induce a ‘teaching to the test’ environment (Shafiyeva, 2021). Mathematics is the powerhouse of where students can experience and succeed with productive struggle learning that teaches the …

Analysis And Computation Of Constrained Sparse Coding On Emerging Non-Von Neumann Devices, Kyle Henke

Mathematics & Statistics ETDs

This dissertation seeks to understand how different formulations of the neurally inspired Locally Competitive Algorithm (LCA) represent and solve optimization problems. By studying these networks mathematically through the lens of dynamical and gradient systems, the goal is to discern how neural computations converge and link this knowledge to theoretical neuroscience and artificial intelligence (AI). Both classical computers and advanced emerging hardware are employed in this study. The contributions of this work include:

1. Theoretical Work: A comprehensive convergence analysis for networks using both generic Rectified Linear Unit (ReLU) and Rectified Sigmoid activation functions. Exploration of techniques to address the binary …

New Algorithms For The Multiplication Table Problem, Evan Blom

Undergraduate Honors Thesis Collection

In 1955, Paul Erdős initiated the study of a function that counts the number of distinct integers in an (n × n) multiplication table. That is, he studied M(n) = |{i · j, 1 ≤ i, j ≤ n}|. Much research has been done in regards to both asymptotic and exact approximations of M(n) for increasingly large values of n. Recently, Brent et. al. investigated the algorithmic cost in computing this function. Instead of computing M(n) directly, their approach was to compute it incrementally. That is, given M(n−1), they could quickly compute M(n) using another function δ(n) to count the …

Mathematics Majors In Medical School Admissions: A Comparative Evaluation Of Mcat And Gpa Performance, Morgan Baker

Theses/Capstones/Creative Projects

Choosing a major as an incoming undergraduate student can be very stressful. This study investigates the differences in success that come with choice of undergraduate major, particularly focusing on the performance of mathematics majors. A large majority of medical school applicants come from a biological sciences background. Despite this preference, there is evidence that students from nontraditional majors produce higher Medical College Admission Test (MCAT) scores and superior grade point averages (GPAs). Utilizing data visualization and analysis through R programming, this research examines public data from the Association of American Medical Colleges (AAMC) to understand the benefits of pursuing a …

An Analysis Of Instructor Feedback, Olga Hawranick

Boise State University Theses and Dissertations

Carless & Boud (2018) capture the definition of feedback best as “a process through which learners make sense of information… use it to enhance their work or learning strategies (p.1).” This definition goes beyond feedback solely being the role of instructors and highlights the student component in making sense of the feedback which we reference in this study as “actionable feedback.” Are instructors providing meaningful feedback to their students to encourage revision? Past studies offer inconsistent results on most features of feedback and little is known about how some of these factors benefit students’ interactions with feedback.

This study aims …