Mathematics Commons^™

Tasks For Learning Trigonometry, Sydnee Andreasen

All Graduate Reports and Creative Projects, Fall 2023 to Present

Many studies have been done using task-based learning within different mathematics courses. Within the field of trigonometry, task-based learning is lacking. The following research aimed to create engaging, mathematically rich tasks that meet the standards for the current trigonometry course at Utah State University and align with the State of Utah Core Standards for 7th through 12th grades. Four lessons were selected and developed based on the alignment of standards, the relevance to the remainder of the trigonometry course, and the relevance to courses beyond trigonometry. The four lessons that were chosen and developed were related to trigonometric ratios, graphing …

Analysis And Construction Of Artificial Neural Networks For The Heat Equations, And Their Associated Parameters, Depths, And Accuracies., Shakil Ahmed Rafi

Graduate Theses and Dissertations

This dissertation seeks to explore a certain calculus for artificial neural networks. Specifically we will be looking at versions of the heat equation, and exploring strategies on how to approximate them.

Our strategy towards the beginning will be to take a technique called Multi-Level Picard (MLP), and present a simplified version of it showing that it converges to a solution of the equation (^∂/_∂t u_d ) (t, x) = (∇² _x u_d)(t, x).

We will then take a small detour exploring the viscosity super-solution properties of solutions to such …

How To Explain Allen-Manandhar’S Method To Beginner Mathematicians : A Convergence Analysis Of A Hybrid Method For Variable-Coefficient Boundary Value Problems, Rebecca Scariano

Honors Theses

In this project, analogies are employed to make complex math concepts approachable to beginners who may only have a basic understanding of calculus and linear algebra. Serving as the focal point of this project, Allen-Manandhar’s method solves an equation, known as an ordinary differential equation (ODE). The mentioned equation with its coefficients is comparable to a pie recipe with ingredients. With the outcome to a recipe seen as its solution, the solution to our pie recipe is a perfectly baked pie, as in without error. The chosen method for baking a pie then classifies as its baking approach that when …

An Experimental And Computational Study On The Dynamics Of Vertical Axis Wind Turbines, Stephen Conte

Theses, Dissertations and Culminating Projects

This thesis conducts preliminary analyses on the estimated performance of a novel Vertical-Axis Wind Turbine at a small-scale through experimental and numerical means. This includes the construction of a wind tunnel in the Complex Fluids Lab at Montclair State University for the simulation and analyses of small-scale prototypes of said VAWT; as well as the utilization of Ansys® 2024 Student Fluent R1 for CFD simulation of scaled single-blade pairs, and potentially full-models of said VAWT.

Applications Of Conic Programming Reformulations, Sarah Kelly

All Dissertations

In general, convex programs have nicer properties than nonconvex programs. Notably, in a convex program, every locally optimal solution is also globally optimal. For this reason, there is interest in finding convex reformulations of nonconvex programs. These reformulation often come in the form of a conic program. For example, nonconvex quadratically-constrained quadratic programs (QCQPs) are often relaxed to semidefinite programs (SDPs) and then tightened with valid inequalities. This dissertation gives a few different problems of interest and shows how conic reformulations can be usefully applied.

In one chapter, we consider two variants of the trust-region subproblem. For each of these …

The Nature Of Undergraduate Calculus I Students' Participation In The Parallel Spaces Of Coursework And Peer-Led, Cooperative, Inquiry-Oriented Complementary Instruction, Karmen T. Yu

Theses, Dissertations and Culminating Projects

Calculus has long been known as a “gateway course” to STEM fields in postsecondary education. To moderate this gatekeeping effect, Montclair State University researchers developed a peer-led, inquiry-based instructional support (IBIS) to run parallel to Calculus classes. The design of the IBIS model was informed by an instructional approach called Complex Instruction (CI) and two other peer-led, cooperative learning models, Peer-Led Team Learning (PLTL) and Supplemental Instruction (SI). While the literature on peer-led models of cooperative learning confirms their effectiveness on undergraduate students’ academic outcomes in mathematics, this multiple-case study set out to address the following: What is the nature …

Flipped Classroom For Linear Algebra At Undergraduate Level, M. Thulasidas

Research Collection School Of Computing and Information Systems

In this article, we describe our experience in developing an undergraduate Linear Algebra course tailored to highlight its relevance and applicability in Computer Science. Over the course of three years, the course transitioned from a traditional direct-instruction format to a flipped-classroom design, resulting in positive student learning outcomes. This article covers the course design philosophy, its syllabus, learning objectives, and the incorporation of both quantitative and qualitative student feedback in shaping the course. Furthermore, the article shares the insights gleaned from our experience, which can serve as best practices for instructors aiming to deliver a successful Linear Algebra course for …

(R2082) Two New Operations And Extended Modal Operators On Bipolar Pythagorean Fuzzy Matrices, S. Sriram, K. Sivaranjani

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, two novel binary operations concerning bipolar Pythagorean fuzzy matrices are delineated. Several algebraic properties, such as commutativity and associativity, are explored. Additionally, extended modal operators for Bipolar Pythagorean fuzzy matrices are introduced. Subsequently, these methodologies are applied to a decision-making scenario wherein a scoring matrix is formulated and alternatives are ranked according to their cumulative score values.

Modeling Prices In Limit Order Book Using Univariate Hawkes Point Process, Wenqing Jiang

University of New Orleans Theses and Dissertations

This thesis presents a time-changed geometric Brownian price model with the univariate Hawkes processes to trace the price changes in a limit order book. Limit order books are the core mechanism for trading in modern financial markets, continuously collecting outstanding buy and sell orders from market participants. The arrival of orders causes fluctuations in prices over time. A Hawkes process is a type of point process that exhibits self-exciting behavior, where the occurrence of one event increases the probability of other events happening in the near future. This makes Hawkes processes well-suited for capturing the clustered arrival patterns of orders …

Plumbing The Depths Of The Shallow End: Exploring Persistent Homology Using Small Data, R. Anne Flynn

All NMU Master's Theses

Persistent homology is a prominent tool in topological data analysis. This thesis is designed to be an introduction and guide to a beginner in persistent homology. This comprehensive overview discusses the math used behind it, the code needed to apply it, and its current place in the field. We explain and demonstrate the algebraic topology which fuels persistent homology. Homotopies inspire homology groups, which are able to determine how many holes a shape has. By visualizing data as a shape, persistent homology determines what type of holes are present.

We demonstrate this by using the package TDA in the manipulation …

Hilbert Reciprocity Over Number Fields, Dillon Snyder

Honors Scholar Theses

A Hilbert symbol has the value 1 or −1 depending on the existence of solutions to a certain quadratic equation in a local field, R, or C. Hilbert reciprocity states that for a number field F and two nonzero a and b in F, the product of Hilbert symbols associated to a and b at all the places of F is 1. That is, these Hilbert symbols are −1 for a finite, even number of places of F . Hilbert reciprocity when F = Q is equivalent to the classical quadratic reciprocity law, so Hilbert reciprocity in number fields can …

The Future Of Brain Tumor Diagnosis: Cnn And Transfer Learning Innovations, Shengyuan Wang

Mathematics, Statistics, and Computer Science Honors Projects

For the purpose of improving patient survival rates and facilitating efficient treatment planning, brain tumors need to be identified early and accurately classified. This research investigates the application of transfer learning and Convolutional Neural Networks (CNN) to create an automated, high-precision brain tumor segmentation and classification framework. Utilizing large-scale datasets, which comprise MRI images from open-accessible archives, the model exhibits the effectiveness of the method in various kinds of tumors and imaging scenarios. Our approach utilizes transfer learning techniques along with CNN architectures strengths to tackle the intrinsic difficulties of brain tumor diagnosis, namely significant tumor appearance variability and difficult …

Deterministic Global 3d Fractal Cloud Model For Synthetic Scene Generation, Aaron M. Schinder, Shannon R. Young, Bryan J. Steward, Michael L. Dexter, Andrew Kondrath, Stephen Hinton, Ricardo Davila

Faculty Publications

This paper describes the creation of a fast, deterministic, 3D fractal cloud renderer for the AFIT Sensor and Scene Emulation Tool (ASSET). The renderer generates 3D clouds by ray marching through a volume and sampling the level-set of a fractal function. The fractal function is distorted by a displacement map, which is generated using horizontal wind data from a Global Forecast System (GFS) weather file. The vertical windspeed and relative humidity are used to mask the creation of clouds to match realistic large-scale weather patterns over the Earth. Small-scale detail is provided by the fractal functions which are tuned to …

Conditional Constrained And Unconstrained Quantization For A Uniform Distribution On A Hexagon, Christina Hamilton

Theses and Dissertations

In this thesis, we have considered a uniform distribution on a regular hexagon and the set of all its six vertices as a conditional set. For the uniform distribution under the conditional set first, for all positive integers n ≥ 6, we obtain the conditional optimal sets of n-points and the nth conditional quantization errors, and then we calculate the conditional quantization dimension and the conditional quantization coefficient in the unconstrained scenario. Then, for the uniform distribution on the hexagon taking the same conditional set, we investigate the conditional constrained optimal sets of n-points and the conditional constrained quantization errors …

Conditional Quantization For Uniform Distributions On Line Segments And Regular Polygons, Tsianna Danielle Dominguez

Theses and Dissertations

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the support are preselected, then the quantization is called a conditional quantization. In this thesis, we have investigated the conditional quantization for the uniform distributions defined on the unit line segments and m-sided regular polygons, where m ≥ 3, inscribed in a unit circle.

How Mathematics Instructors Foster The Development Of Black Students' Mathematics Identity In Undergraduate Active Learning Mathematics Courses, Ashly J. Olusanya

Theses and Dissertations

Black students must overcome unique challenges to succeed in mathematics. Educators are tasked with identifying equitable teaching practices to support these students. Active learning (AL) is a teaching pedagogy that engages students in rigorous mathematical activities and encourages student participation. This research study will explore the professors’ beliefs about how students learn mathematics and why they use active learning in their collegiate mathematics courses. The study explores the connections between these beliefs and their reported use of instructional practices. The study also identifies the instructors’ beliefs about developing students’ mathematics identities, particularly their Black …

A Study On A Vector Complex Modified Korteweg-De Vries Equation, Changyan Shi

Theses and Dissertations

In this thesis, we systematically study a vector complex modified Kordeweg-de Vries equation by combining Hirota's bilinear method and the the Kadomtsev–Petviashvili (KP) reduction method. This vector nonlinear equation is a multi-component generalization of the well-known modified Kordeweg-de Vries (mKdV) equation and can be reduced to the known Hirota equation, Sasa-Satsuma (SS) equation, Sasa-Satsuma-mKdV equation as well as coupled Sasa-Satsuma equation. First, we bilinearize the vector complex mKdV equation under both the zero and nonzero boundary conditions by introducing auxiliary tau functions. Then, starting from two sets of bilinear equations of multi-component KP hierarchy and single-component KP-Toda …

The Perspectives Of Using Desmos For Students’ Conceptual Understanding And Procedural Fluency To Solve Linear Equations, Larmel Dimatulac Madrilejos

Theses and Dissertations

The study examines the perspectives of using the Desmos calculator of Algebra I students' conceptual understanding and procedural fluency to write, graph, and solve linear equations in Algebra I STAAR. While the students have continuously used technology for mathematics assessment, emergent bilingual students in South Texas still need help passing high-stakes testing. The framework of the study is grounded in the theory of mathematical education (knowledge of mathematics educators to teach), the theory of mathematical learning (understanding how students learn mathematics), and social constructivism. The study seeks ways to teach all students, mainly the minority, to learn …

Derived Jet Schemes And Arc Spaces, And Arithmetic Arc Space Representability, C. Eric Overton-Walker

Graduate Theses and Dissertations

Associated to a given scheme X one can define geometric and arithmetic notions of jet schemes and arc spaces. We develop a construction of the geometric jets and arcs in the setting of derived schemes and explore consequences thereof. In particular, we prove an analogous theorem to that of one by Tommaso de Fernex and Roi Docampo concerning the cotangent sheaves of geometric jets and arcs. Our version then produces many subsequent results which allow us to prove stronger versions of results concerning geometric jets and arcs by removing unnecessary hypotheses. Separately, we explore evidence as to why, in contrast …

The Mathematics Of Financial Portfolio Optimization Incorporating Environmental, Social, And Governance Score Information, Ian Driskill

Master's Theses

We numerically investigate the effects that Environmental, Social, and Governance (ESG) scores have on portfolio optimization with Modern Portfolio Theory assumptions and how ESG scores correlate with the market returns of a rated company's stock. Additionally, we review and analyze a research paper published in the Journal of Financial Economics regarding ESG investing titled “Responsible investing: The ESG-efficient frontier” by Pedersen, Fitzgibbons, and Lukasz. Our overall goal is provide insight for socially responsible inclined investors, to help them understand what ESG scores tell us and how those scores may effect their overall investment returns."

On The Existence Of Periodic Traveling-Wave Solutions To Certain Systems Of Nonlinear, Dispersive Wave Equations, Jacob Daniels

All Graduate Theses and Dissertations, Fall 2023 to Present

A variety of physical phenomena can be modeled by systems of nonlinear, dispersive wave equations. Such examples include the propagation of a wave through a canal, deep ocean waves with small amplitude and long wavelength, and even the propagation of long-crested waves on the surface of lakes. An important task in the study of water wave equations is to determine whether a solution exists. This thesis aims to determine whether there exists solutions that both travel at a constant speed and are periodic for several systems of water wave equations. The work done in this thesis contributes to the subfields …

Vectors And Vector Borne Disease: Models For The Spread Of Curly Top Disease And Culex Mosquito Abundance, Rachel M. (Frantz) Georges

All Graduate Theses and Dissertations, Fall 2023 to Present

Mathematical models are useful tools in managing infectious disease. When designed appropriately, these models can provide insight into disease incidence patterns and transmission rates. In this work, we present several models that provide information that is useful in monitoring diseases spread by insects.

In the first part of this dissertation, we present two models that predict disease incidence patterns for Curly Top disease (CT) in tomato crops. CT affects a wide variety of plants and is spread through the bite of the Beet Leafhopper. This disease is particularly devastating to tomato crops. When infected, tomato plants present with stunted growth …

A Comprehensive Uncertainty Quantification Methodology For Metrology Calibration And Method Comparison Problems Via Numeric Solutions To Maximum Likelihood Estimation And Parametric Bootstrapping, Aloka B. S. N. Dayarathne

All Graduate Theses and Dissertations, Fall 2023 to Present

In metrology, the science of measurements, straight line calibration models are frequently employed. These models help understand the instrumental response to an analyte, whose chemical constituents are unknown, and predict the analyte’s concentration in a sample. Techniques such as ordinary least squares and generalized least squares are commonly used to fit these calibration curves. However, these methods may yield biased estimates of slope and intercept when the calibrant, substance used to calibrate an analytical procedure with known chemical constituents (x-values), carries uncertainty. To address this, Ripley and Thompson (1987) proposed functional relationship estimation by maximum likelihood (FREML), which considers uncertainties …

Information Based Approach For Detecting Change Points In Inverse Gaussian Model With Applications, Alexis Anne Wallace

Electronic Theses, Projects, and Dissertations

Change point analysis is a method used to estimate the time point at which a change in the mean or variance of data occurs. It is widely used as changes appear in various datasets such as the stock market, temperature, and quality control, allowing statisticians to take appropriate measures to mitigate financial losses, operational disruptions, or other adverse impacts. In this thesis, we develop a change point detection procedure in the Inverse Gaussian (IG) model using the Modified Information Criterion (MIC). The IG distribution, originating as the distribution of the first passage time of Brownian motion with positive drift, offers …

On Cheeger Constants Of Knots, Robert Lattimer

Electronic Theses, Projects, and Dissertations

In this thesis, we will look at finding bounds for the Cheeger constant of links. We will do this by analyzing an infinite family of links call two-bridge fully augmented links. In order to find a bound on the Cheeger constant, we will look for the Cheeger constant of the link’s crushtacean. We will use that Cheeger constant to give us insight on a good cut for the link itself, and use that cut to obtain a bound. This method gives us a constructive way to find an upper bound on the Cheeger constant of a two-bridge fully augmented link. …

A Study Of Quantitative Reasoning Instructors’ Choices And Motivations When Teaching Quantitative Reasoning For The First Time, Trish Ann Harding

Theses and Dissertations

This qualitative study delves into the instructional decision-making processes of post-secondary instructors teaching quantitative reasoning (QR) courses for the first time. The study aims to address the gap in understanding how first-time QR instructors navigate the complexities of curriculum design and pedagogical strategies, and how these experiences contribute to their professional development. The research questions center on identifying the instructional decisions made by these instructors, exploring the factors influencing their decision-making, and understanding the impact of exercising agency has on their professional development. Through in-depth exploration, this study seeks to shed light on the challenges and opportunities faced by first-time …

A Post-Quantum Mercurial Signature Scheme, Madison Mabe

All Theses

This paper introduces the first post-quantum mercurial signature scheme. We also discuss how this can be used to construct a credential scheme, as well as some practical applications for the constructions.

Domination In Graphs And The Removal Of A Matching, Geoffrey Boyer

All Theses

We consider how the domination number of an undirected graph changes on the removal of a maximal matching. It is straightforward that there are graphs where no matching removal increases the domination number, and where some matching removal doubles the domination number. We show that in a nontrivial tree there is always a matching removal that increases the domination number; and if a graph has domination number at least $2$ there is always a maximal matching removal that does not double the domination number. We show that these results are sharp and discuss related questions.

The Brahmagupta Equation And Record Numbers, Christine Patterson

Boise State University Theses and Dissertations

The Diophantine equation x² - Dy² = k has been explored since antiquity and has found significant applications in diverse contexts in mathematics. This thesis focuses on two aspects of solutions of this equation:

1. For fixed k, the sizes of the D values for which there is a solution, and the corresponding sizes of the x and y constituting solutions. This topic is previously uninvestigated and our contribution is data-based.
2. For fixed k and D for which the equation is solvable, statistical properties of the x and y occurring in these solutions. We prove that Benford's Law applies …

A Survey Of The Classification Of 1-Dimensional Shift Systems, Jacob Miller

Boise State University Theses and Dissertations

The one-dimensional full shift over a finite set A is the collection of bi-infinite sequences of symbols in A together with the left-shift map which shifts the indexing of the sequence. A shift space is a subset of a full shift defined by a collection of forbidden blocks, i.e., finite words which are not allowed to appear. Many shift spaces arise as the set of bi-infinite walks on a labeled graph, and many dynamical systems can be encoded as shift spaces where the dynamics are replaced by the left-shift map.

We introduce shift spaces and their basic properties, then …