Introductory Calculus: Through The Lenses Of Covariation And Approximation, 2020 University of Montana, Missoula

#### Introductory Calculus: Through The Lenses Of Covariation And Approximation, Caleb Huber

*Graduate Student Portfolios, Papers, and Capstone Projects*

Over the course of a year, I investigated reformative approaches to the teaching of calculus. My research revealed the substantial findings of two educators, Michael Oehrtman and Pat Thompson, and inspired me to design a course based upon two key ideas, covariation and approximation metaphors. Over a period of six weeks, I taught a course tailored around these ideas and documented student responses to both classroom activities and quizzes. Responses were organized intonarratives, covariation, rates of change, limits, and delta notation. Covariation with respect to rates of change was found to be incredibly complex, and students would often see it ...

Conjugation By Circulant Matrices In Non-Commutative Cryptography, 2020 University of Mary Washington

#### Conjugation By Circulant Matrices In Non-Commutative Cryptography, Hannah B. Frederick

*Student Research Submissions*

We introduce a procedure in which two trusted individuals, Alice and Bob, may share a secret matrix K from the non-abelian general linear group of matrices. In this procedure, the matrix K is concealed from an eavesdropper, Eve, by a sequence of conjugations by elements from a pre-determined abelian subgroup of the general linear group. We demonstrate that the group of invertible circulant matrices is one abelian subgroup that may be able to withstand a brute force attack. To analyze this we need a technique to determine the order of this group, and to do this we make use of ...

Remote Learning Assignment, 2020 The College at Brockport: State University of New York

Lattice Paths In Diagonals And Dimensions, 2020 Coastal Carolina University

#### Lattice Paths In Diagonals And Dimensions, Freya Bennett

*Honors Theses*

The Lattice Paths of Combinatorics have been used in many applications, normally under the guise of a different name, due to its versatility in surface variety and specificity of answer. The Lattice Path’s of game development, in finding paths around barriers in mazes, is called Path Finder with the A∗ algorithms as its method of solving.

Combinatorial And Asymptotic Statistical Properties Of Partitions And Unimodal Sequences, 2020 Louisiana State University and Agricultural and Mechanical College

#### Combinatorial And Asymptotic Statistical Properties Of Partitions And Unimodal Sequences, Walter Mcfarland Bridges

*LSU Doctoral Dissertations*

Our main results are asymptotic zero-one laws satisfied by the diagrams of unimodal sequences of positive integers. These diagrams consist of columns of squares in the plane; the upper boundary is called the shape. For various types of unimodal sequences, we show that, as the number of squares tends to infinity, 100% of shapes are near a certain curve---that is, there is a single limit shape. Similar phenomena have been well-studied for integer partitions, but several technical difficulties arise in the extension of such asymptotic statistical laws to unimodal sequences. We develop a widely applicable method for obtaining these limit ...

Singular Value Decomposition, 2020 Coastal Carolina University

#### Singular Value Decomposition, Krystal Bonaccorso, Andrew Incognito

*Honors Theses*

A well-known theorem is Diagonalization, where one of the factors is a diagonal matrix. In this paper we will be describing a similar way to factor/decompose a non-square matrix. The key to both of these ways to factor is eigenvalues and eigenvectors.

Exploration Of Solvable Quintic Polynomials, 2020 University of Mary Washington

#### Exploration Of Solvable Quintic Polynomials, Stephen Tivenan

*Student Research Submissions*

A polynomial f(x) with rational coefficients is solvable by radicals if its roots (in the field of complex numbers C) can be expressed in terms of its coefficients using the basic operations and radicals. It is known that for quintic polynomials there is no generic formula for the roots. That is, some quintic polynomials are solvable and some are not. In this paper, we address the mathematical theory that makes the formula for the roots of a polynomial. Primarily we will focus on our methodology of generating and examining quintic polynomials. In one case study, we will examine quintic ...

Anticommutative Associative Algebras And The Binomial Theorem, 2020 University of Mary Washington

#### Anticommutative Associative Algebras And The Binomial Theorem, Ashley Scurlock

*Student Research Submissions*

We examine the binomial theorem and its components in a noncommutative associative algebra. Specifically, we examine the relationship between the 1-binomial and -1-binomial coefficient, as well as exploring alternatives for the exponential identity for non-commutative and anticommutative elements. Through this investigation we found that the 1-binomial can be mapped to the -1-binomial and that the relationship could be used to prove a defined alternative for the exponential identity for anticommutative elements.

The T, T*, Vi, Vni Model For Human Immunodeficiency Virus Type 1 (Hiv-1) Dynamics, 2020 University of Mary Washington

#### The T, T*, Vi, Vni Model For Human Immunodeficiency Virus Type 1 (Hiv-1) Dynamics, Amy Creel

*Student Research Submissions*

In this research project, I investigated deterministic and stochastic versions of a model for Human Immunodeficiency Virus Type 1 (HIV-1) dynamics. First, an analytical solution to a simplified version of the deterministic model is found. Then, numerical techniques are used to obtain an approximate solution to the deterministic model. Finally, a stochastic version of the model is discussed, and numerical methods are used to find an approximate solution to the stochastic system. These results demonstrate the behavior of HIV-1 in an infected patient under the effects of reverse transcriptase and protease inhibitors, and illustrate how the addition of randomness to ...

Nasa L'Space Mission Concept Academy, 2020 Salem State University

#### Nasa L'Space Mission Concept Academy, Anaily Lorenzo

*Research Day*

NASA L'Space Mission Concept Academy is a virtual academy in which we had to complete a 12-week team mission concept. Our mission objective was to design a small mission concept that characterized the polar water ice on Earth's moon. Our mission had to fit within the mission concept constraints (mass, volume, and budget). Our spacecraft design and scientific payload had to reflect what can be afforded within these constraints. To complete this mission concept, our team relied heavily on math, engineering, geology, and chemistry.

A New Asymmetric Encryption Algorithm Involving Both Group And Number Theory: Derivation Of The Lucente Stabile Atkins Cryptosystem Using Gauss’S Generalization Of Wilson’S Theorem, 2020 Salem State University

#### A New Asymmetric Encryption Algorithm Involving Both Group And Number Theory: Derivation Of The Lucente Stabile Atkins Cryptosystem Using Gauss’S Generalization Of Wilson’S Theorem, Francesco Lucente Stabile, Carey Atkins, Arthur James Rosenthal

*Research Day*

Our research led to the discovery of an asymmetric encryption algorithm that follows Kerckhoff's principle and relies on a specific case of Gauss's Generalization of Wilson's Theorem. Unlike prime factorization based algorithms, the eavesdropping cryptanalyst has no indication that he has successfully decrypted the cyphertext. For this reason, we aim to show that this algorithm is not only more secure than existing asymmetric algorithms, but it has the potential to be significantly computationally faster.

Legendrian Dga Representations And The Colored Kauffman Polynomial, 2020 Louisiana State University and Agricultural and Mechanical College

#### Legendrian Dga Representations And The Colored Kauffman Polynomial, Justin Murray, Dan Rutherford

*Faculty Publications*

For any Legendrian knot K in standard contact R-3 we relate counts of ungraded (1-graded) representations of the Legendrian contact homology DG-algebra (A(K), partial derivative) with the n-colored Kauffman polynomial. To do this, we introduce an ungraded n-colored ru-ling polynomial, R-n,K(1)(q), as a linear combination of reduced ruling polynomials of positive permutation braids and show that (i) R-n,K(1)(q) arises as a specialization F-n,F-K(a, q)vertical bar(a-1) = 0 of the n-colored Kauffman polynomial and (ii) when q is a power of two R-n,K(1)(q) agrees with the total ungraded ...

Predictive Modeling Of Iphone 7 Charge Rates Using Least Squares Curve Fitting, 2020 University of Tennessee at Chattanooga

#### Predictive Modeling Of Iphone 7 Charge Rates Using Least Squares Curve Fitting, Grace Cahill

*Honors Theses*

In a time where individuals depend on their cell phones, the need for a long lasting and quick charging battery life is imperative. As information regarding how long a battery can remained charged is highly advertised, there is no information regarding how long it would take for a dead phone battery to completely charge. This study determined the amount of time it will take an iPhone 7 to charge from 0% to 100% using the standard charging cable under four different charging conditions. The charge percentage was recorded every two minutes until it was fully charged with this process being ...

Boundary, Costs And Trade-Offs In Reserve Design Systems, 2020 University of Tennessee at Chattanooga

#### Boundary, Costs And Trade-Offs In Reserve Design Systems, Justus Hurd

*Honors Theses*

Due to limitations in funding and natural resources, it is infeasible to construct perfect reserve systems for large populations of critical species. For this project, our objective is to formulate a reserve design model that minimizes the distance between reserve sites meeting a threshold of biodiversity features subject to a species coverage constraints. Coupled with other spatial characteristics including reserve size and configuration, the boundary of a reserve system is of key importance. While positive area effects are gained when selecting additional sites, negative boundary length effects are also experienced. For example, it is costly to implement and maintain boundary ...

Beginning Algebra Made Useful, 2020 Grand Valley State University

#### Beginning Algebra Made Useful, Charlene E. Beckmann

*Open Textbooks*

*Beginning Algebra Made Useful* addresses the needs of learners to make sense of algebra by quantifying and generalizing everyday occurrences such as commuting to work, buying gas or pizza, and determining the better deal. It requires learners to actively engage with algebraic concepts through physical and thought experiments in ways that help them connect ideas, representations, and contexts, and solve problems that arise in their daily lives. The text* *helps learners grow their brains and develop growth mindsets as they learn algebra conceptually. Problem sets continue the process, extending work begun in each lesson, applying new understandings to new contexts ...

Nonlinear Photonics In Twisted And Nonlocal Structures, 2020 Southern Methodist University

#### Nonlinear Photonics In Twisted And Nonlocal Structures, Austin Copeland

*Mathematics Theses and Dissertations*

We provide a theoretical framework for the observed confinement of light modes within a twisted coreless photonic crystal fiber. Asymptotic methods are applied through ray theory and field theory in both the linear and nonlinear regime. We find the modes have a radially symmetric chirp and the envelope will decay away from the axis of propagation. Secondly, we study the stability and singularity formation of unidirectional beams as described by the Schrodinger equation. We propose a novel extension to the modeling equation to include a fractional Laplacian in one spatial dimension and a standard second derivative in a second dimension ...

Gait Characterization Using Computer Vision Video Analysis, 2020 College of William and Mary

#### Gait Characterization Using Computer Vision Video Analysis, Martha T. Gizaw

*Undergraduate Honors Theses*

The World Health Organization reports that falls are the second-leading cause of accidental death among senior adults around the world. Currently, a research team at William & Mary’s Department of Kinesiology & Health Sciences attempts to recognize and correct aging-related factors that can result in falling. To meet this goal, the members of that team videotape walking tests to examine individual gait parameters of older subjects. However, they undergo a slow, laborious process of analyzing video frame by video frame to obtain such parameters. This project uses computer vision software to reconstruct walking models from residents of an independent living retirement ...

Exploring Pedagogical Empathy Of Mathematics Graduate Student Instructors, 2020 University of Nebraska-Lincoln

#### Exploring Pedagogical Empathy Of Mathematics Graduate Student Instructors, Karina Uhing

*Dissertations, Theses, and Student Research Papers in Mathematics*

Interpersonal relationships are central to the teaching and learning of mathematics. One way that teachers relate to their students is by empathizing with them. In this study, I examined the phenomenon of pedagogical empathy, which is defined as empathy that influences teaching practices. Specifically, I studied how mathematics graduate student instructors conceptualize pedagogical empathy and analyzed how pedagogical empathy might influence their teaching decisions. To address my research questions, I designed a qualitative phenomenological study in which I conducted observations and interviews with 11 mathematics graduate student instructors who were teaching precalculus courses at the University of Nebraska—Lincoln.

In ...

Semisimple Subalgebras Of Semisimple Lie Algebras, 2020 Utah State University

#### Semisimple Subalgebras Of Semisimple Lie Algebras, Mychelle Parker

*All Graduate Theses and Dissertations*

Let g be a Lie algebra. The subalgebra classification problem is to create a list of all subalgebras of g up to equivalence. The purpose of this thesis is to provide a software toolkit within the Differential Geometry package of Maple for classifying subalgebras of In particular the thesis will focus on classifying those subalgebras which are isomorphic to the Lie algebra sl(2) and those subalgebras of which have a basis aligned with the root space decomposition (regular subalgebras).

On Quantum Effects Of Vector Potentials And Generalizations Of Functional Analysis, 2020 Chapman University

#### On Quantum Effects Of Vector Potentials And Generalizations Of Functional Analysis, Ismael L. Paiva

*Computational and Data Sciences (PhD) Dissertations*

This is a dissertation in two parts. In the first one, the Aharonov-Bohm effect is investigated. It is shown that solenoids (or flux lines) can be seen as barriers for quantum charges. In particular, a charge can be trapped in a sector of a long cavity by two flux lines. Also, grids of flux lines can approximate the force associated with continuous two-dimensional distributions of magnetic fields. More, if it is assumed that the lines can be as close to each other as desirable, it is explained how the classical magnetic force can emerge from the Aharonov-Bohm effect. Continuing, the ...