Harmony Amid Chaos, 2020 Olivet Nazarene University

#### Harmony Amid Chaos, Drew Schaffner

*Pence-Boyce STEM Student Scholarship*

We provide a brief but intuitive study on the subjects from which Galois Fields have emerged and split our study up into two categories: harmony and chaos. Specifically, we study finite fields with elements where is prime. Such a finite field can be defined through a logarithm table. The Harmony Section is where we provide three proofs about the overall symmetry and structure of the Galois Field as well as several observations about the order within a given table. In the Chaos Section we make two attempts to analyze the tables, the first by methods used by Vladimir Arnold as ...

Superposed Ornstein-Uhlenbeck Processes, 2020 University of Mary Washington

#### Superposed Ornstein-Uhlenbeck Processes, Julius Esunge, Auguste Muhau

*Journal of Stochastic Analysis*

No abstract provided.

Von Neumann's Minimax Theorem For Continuous Quantum Games, 2020 Universitá di Roma Tor Vergata, Via di Torvergata, Roma, Italy

#### Von Neumann's Minimax Theorem For Continuous Quantum Games, Luigi Accardi, Andreas Boukas

*Journal of Stochastic Analysis*

No abstract provided.

On The Extension Of Positive Definite Kernels To Topological Algebras, 2020 Chapman University

#### On The Extension Of Positive Definite Kernels To Topological Algebras, Daniel Alpay, Ismael L. Paiva

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

We define an extension of operator-valued positive definite functions from the real or complex setting to topological algebras and describe their associated reproducing kernel spaces. The case of entire functions is of special interest, and we give a precise meaning to some power series expansions of analytic functions that appears in many algebras.

Restoring The Function Set By Integrals For The Family Of Parabolas On The Plane, 2020 Samarkand State University

#### Restoring The Function Set By Integrals For The Family Of Parabolas On The Plane, Akram Begmatov, Alisher Ismoilov

*Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences*

Integral geometry is one of the most important sections of the theory of ill-posed problems of mathematical physics and analysis. The urgency of the problems of integral geometry is due to the development of tomographic methods, which raise the requirements for the depth of the applied results, the fact that the solution of problems of integral geometry reduces a number of multidimensional inverse problems for partial differential problems, as well as the internal development needs of the theory of ill-posed problems of mathematical physics and analysis. In this work we consider the problem of reconstructing a function from a family ...

Taylor Expansions And Castell Estimates For Solutions Of Stochastic Differential Equations Driven By Rough Paths, 2020 University of Southern California, Los Angeles, CA 90089-2532, USA

#### Taylor Expansions And Castell Estimates For Solutions Of Stochastic Differential Equations Driven By Rough Paths, Qi Feng, Xuejing Zhang

*Journal of Stochastic Analysis*

No abstract provided.

Derivable Single Valued Neutrosophic Graphs Based On Km-Fuzzy Metric, 2020 University of New Mexico

#### Derivable Single Valued Neutrosophic Graphs Based On Km-Fuzzy Metric, Florentin Smarandache, Mohammad Hamidi

*Mathematics and Statistics Faculty and Staff Publications*

In this paper we consider the concept of KM-fuzzy metric spaces and we introduce a novel concept of KM-single valued neutrosophic metric graphs based on KM-fuzzy metric spaces. Then we investigate the finite KM-fuzzy metric spaces with respect to KM-fuzzy metrics and we construct the KMfuzzy metric spaces on any given non-empty sets. We try to extend the concept of KM-fuzzy metric spaces to a larger class of KM-fuzzy metric spaces such as union and product of KM-fuzzy metric spaces and in this regard we investigate the class of products of KM-single valued neutrosophic metric graphs. In the final, we ...

At The Interface Of Algebra And Statistics, 2020 The Graduate Center, City University of New York

#### At The Interface Of Algebra And Statistics, Tai-Danae Bradley

*Dissertations, Theses, and Capstone Projects*

This thesis takes inspiration from quantum physics to investigate mathematical structure that lies at the interface of algebra and statistics. The starting point is a passage from classical probability theory to quantum probability theory. The quantum version of a probability distribution is a density operator, the quantum version of marginalizing is an operation called the partial trace, and the quantum version of a marginal probability distribution is a reduced density operator. Every joint probability distribution on a finite set can be modeled as a rank one density operator. By applying the partial trace, we obtain reduced density operators whose diagonals ...

Permutation And Monomial Progenitors, 2020 California State University, San Bernardino

#### Permutation And Monomial Progenitors, Crystal Diaz

*Electronic Theses, Projects, and Dissertations*

We searched monomial and permutation progenitors for symmetric presentations of important images, nonabelian simple groups, their automorphism groups, or groups that have these as their factor groups. In this thesis, we described our search for the homomorphic images through the permutation progenitor 2^{*15}:(D_{5} X 3) and construction of a monomial representation through the group 2^{3}:3.

We constructed PGL(2,7) over 2^{3}:3 on 6 letters and L_{2}(11) over 2^{2}:3 on 8 letters. We also give our construction of S_{5} X 2 and L_{2}(25) as homomorphic images ...

Acoustic Versus Electromagnetic Field Theory: Scalar, Vector, Spinor Representations And The Emergence Of Acoustic Spin, 2020 Chapman University

#### Acoustic Versus Electromagnetic Field Theory: Scalar, Vector, Spinor Representations And The Emergence Of Acoustic Spin, Lucas Burns, Konstantin Y. Bliokh, Franco Nori, Justin Dressel

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

We construct a novel Lagrangian representation of acoustic field theory that describes the local vector properties of longitudinal (curl-free) acoustic fields. In particular, this approach accounts for the recently-discovered nonzero spin angular momentum density in inhomogeneous sound fields in fluids or gases. The traditional acoustic Lagrangian representation with a scalar potential is unable to describe such vector properties of acoustic fields adequately, which are however observable via local radiation forces and torques on small probe particles. By introducing a displacement vector potential analogous to the electromagnetic vector potential, we derive the appropriate canonical momentum and spin densities as conserved Noether ...

Strategies And Algorithms Of Sudoku, 2020 Louisiana Tech University

#### Strategies And Algorithms Of Sudoku, Callie Weaver

*Mathematics Senior Capstone Papers*

This paper discusses different strategies for the game of Sudoku and how those strategies relate to other problem solving techniques while also attempting to use those other techniques in a way that improves the strategies for Sudoku. This includes a thorough analysis of the general algorithm and an algorithm that is formed by the Occupancy Theorem and Preemptive Sets. This paper also compares these algorithms that directly relate to Sudoku with algorithms to similar combinatorial problems such as the Traveling Salesman problem and more. With the study of game theory becoming more popular, these strategies have also been shown to ...

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 ...

From Branches To Fibers - Investigating F-Actin Networks With Biochemistry And Mathematical Modeling, 2020 James Madison University

#### From Branches To Fibers - Investigating F-Actin Networks With Biochemistry And Mathematical Modeling, Melissa Riddle

*Senior Honors Projects, 2020-current*

F-actin networks have different structures throughout the cell depending on their location or mechanical role. For example, at the leading edge of a migrating cell, F-actin is organized in a region called the lamellipodia as a branched network responsible for pushing the membrane outwards. Behind the lamellipodia is a lamellar actin network where focal adhesions and stress fibers originate, and then within the cell cortex, actin is arranged in a gel-like network. Stress fibers are an important organization of F-actin and how they arise from either the branched lamellipodia network or the gel-like cortex network is poorly understood. Our approach ...

An Analysis And Comparison Of Knot Polynomials, 2020 James Madison University

#### An Analysis And Comparison Of Knot Polynomials, Hannah Steinhauer

*Senior Honors Projects, 2020-current*

Knot polynomials are polynomial equations that are assigned to knot projections based on the mathematical properties of the knots. They are also invariants, or properties of knots that do not change under ambient isotopy. In other words, given an invariant α for a knot K, α is the same for any projection of K. We will define these knot polynomials and explain the processes by which one finds them for a given knot projection. We will also compare the relative usefulness of these polynomials.

Structure Theorems For Idempotent Residuated Lattices, 2020 University of Bern

#### Structure Theorems For Idempotent Residuated Lattices, José Gil-Férez, Peter Jipsen, George Metcalfe

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

In this paper we study structural properties of residuated lattices that are idempotent as monoids. We provide descriptions of the totally ordered members of this class and obtain counting theorems for the number of finite algebras in various subclasses. We also establish the finite embeddability property for certain varieties generated by classes of residuated lattices that are conservative in the sense that monoid multiplication always yields one of its arguments. We then make use of a more symmetric version of Raftery’s characterization theorem for totally ordered commutative idempotent residuated lattices to prove that the variety generated by this class ...

The Fundamental System Of Units For Cubic Number Fields, 2020 University of Wisconsin-Milwaukee

#### The Fundamental System Of Units For Cubic Number Fields, Janik Huth

*Theses and Dissertations*

Let $K$ be a number field of degree $n$. An element $\alpha \in K$ is called integral, if the minimal polynomial of $\alpha$ has integer coefficients. The set of all integral elements of $K$ is denoted by $\mathcal{O}_K$. We will prove several properties of this set, e.g. that $\mathcal{O}_K$ is a ring and that it has an integral basis. By using a fundamental theorem from algebraic number theory, Dirichlet's Unit Theorem, we can study the unit group $\mathcal{O}_K^\times$, defined as the set of all invertible elements of $\mathcal{O}_K$. We ...

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 ...

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 ...

A Note On The Fine Structure Constant, 2020 University of Nebraska-Lincoln

#### A Note On The Fine Structure Constant, Bilal Khan, Irshadullah Khan

*CSE Technical reports*

We derive the numerical value of the fine structure constant in purely number-theoretic terms, under the assumption that in a system of charges between two parallel conducting plates, the Casimir energy and the mutual Coulomb interaction energy agree.

A Note On The Fine Structure Constant, 2020 University of Nebraska-Lincoln

#### A Note On The Fine Structure Constant, Bilal Khan, Irshadullah Khan

*Sociology Department, Faculty Publications*

We derive the numerical value of the fine structure constant $\alpha$ in purely number-theoretic terms, under the assumption that in a system of charges between two parallel conducting plates, the Casimir energy and the mutual Coulomb interaction energy agree.