Other Mathematics Commons^™

An Exposition Of The Curvature Of Warped Product Manifolds, Angelina Bisson

Electronic Theses, Projects, and Dissertations

The field of differential geometry is brimming with compelling objects, among which are warped products. These objects hold a prominent place in differential geometry and have been widely studied, as is evident in the literature. Warped products are topologically the same as the Cartesian product of two manifolds, but with distances in one of the factors in skewed. Our goal is to introduce warped product manifolds and to compute their curvature at any point. We follow recent literature and present a previously known result that classifies all flat warped products to find that there are flat examples of warped products …

Dna Self-Assembly Of Trapezohedral Graphs, Hytham Abdelkarim

Electronic Theses, Projects, and Dissertations

Self-assembly is the process of a collection of components combining to form an organized structure without external direction. DNA self-assembly uses multi-armed DNA molecules as the component building blocks. It is desirable to minimize the material used and to minimize genetic waste in the assembly process. We will be using graph theory as a tool to find optimal solutions to problems in DNA self-assembly. The goal of this research is to develop a method or algorithm that will produce optimal tile sets which will self-assemble into a target DNA complex. We will minimize the number of tile and bond-edge types …

Knot Equivalence, Jacob Trubey

Electronic Theses, Projects, and Dissertations

A knot is a closed curve in R3. Alternatively, we say that a knot is an embedding f : S1 → R3 of a circle into R3. Analogously, one can think of a knot as a segment of string in a three-dimensional space that has been knotted together in some way, with the ends of the string then joined together to form a knotted loop. A link is a collection of knots that have been linked together.

An important question in the mathematical study of knot theory is that of how we can tell when two knots are, or are …

Symmetric Generations And An Algorithm To Prove Relations, Diddier Andrade

Electronic Theses, Projects, and Dissertations

In this thesis we have discovered homomorphic images of several progenitors such as 3^(*56):(23:(3:7), 3^(*14):(23:(3:7)), 5^(∗24) : S5, 2^(∗10) : (10 : 2), 56^(∗24) : (A5 : 2), and 11^(∗12) :m L2(11). We give isomorphism types of each image that we have found.
We then create a monomial representation of L2(11) by lifting 5:11 onto it.
We manually perform Double Coset Enumeration of 3:(2×S5) over D12
to create its Cayley graph. This is achieved by solving many word problems. The
Cayley graph is used to find a permutation representation of 3:(2×S5). We also
perform Double Coset Enumeration S3 × A5 …

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 …

Measure And Integration, Jeonghwan Lee

Electronic Theses, Projects, and Dissertations

Measure and Integral are important when dealing with abstract spaces such as function spaces and probability spaces. This thesis will cover Lebesgue Measure and Lebesgue integral. The Lebesgue integral is a generalized theory of Riemann integral learned in mathematics. The Riemann integral is centered on the domain of the function, but the Lebesgue integral is different in that it is centered on the range of the function, and uses the basic concept of analysis. Measure and integral have widely applied not only to mathematics but also to other fields.

Symmetric Representations Of Finite Groups And Related Topics, Connie Corona

Electronic Theses, Projects, and Dissertations

In this thesis, we have presented our discovery of original symmetric presentations of a number of non-abelian simple groups, including several sporatic groups, linear groups, and classical groups.

We have constructed, using our technique of double coset enumeration, J2, M₁₂, J₁, PSU(3, 3):2, M₁₁, A₁₀, S(4,3), M₂₂:2, PSL(3, 4), S₆, 2:S₅, 2:PSL(3, 4) as homomorphic images of the involutory progenitors 2^*32:(2⁵:A₅), 2^*110: PSL(2, 11), 2^*5:A₅, 3^*4:D₈, 2^*110:PSL(2, 11), …

Sum Of Cubes Of The First N Integers, Obiamaka L. Agu

Electronic Theses, Projects, and Dissertations

In Calculus we learned that 􏰅Sum^{n}_{k=1} k = [n(n+1)]/2 , that Sum^{􏰅n}_{k=1} k^2 = [n(n+1)(2n+1)]/6 , and that Sum^{n}_{k=1} k^{3} = (n(n+1)/2)^{2}. These formulas are useful when solving for the area below quadratic or cubic function over an interval [a, b]. This tedious process, solving for areas under a quadratic or a cubic, served as motivation for the introduction of Riemman integrals. For the overzealous math student, these steps were replaced by a simpler method of evaluating antiderivatives at the endpoints a and b. From my recollection, a former instructor informed us to do the value of memorizing these formulas. …

Tile Based Self-Assembly Of The Rook's Graph, Ernesto Gonzalez

Electronic Theses, Projects, and Dissertations

The properties of DNA make it a useful tool for designing self-assembling nanostructures. Branched junction molecules provide the molecular building blocks for creating target complexes. We model the underlying structure of a DNA complex with a graph and we use tools from linear algebra to optimize the self-assembling process. Some standard classes of graphs have been studied in the context of DNA self-assembly, but there are many open questions about other families of graphs. In this work, we study the rook's graph and its related design strategies.

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:(D₅ X 3) and construction of a monomial representation through the group 2³:3.

We constructed PGL(2,7) over 2³:3 on 6 letters and L₂(11) over 2²:3 on 8 letters. We also give our construction of S₅ X 2 and L₂(25) as homomorphic images of the …

Written Reflections And Discussion Forums-- Math For Elementary School Teachers (Q2s-Ep: Math 301aqbr And Math 301bqbr ), Stephanie Creswell

Q2S Enhancing Pedagogy

Preparing for the transition from quarters to semesters, instructors of the mathematics sequence for future elementary teachers (Math 301ABC, Math 308 and their semester equivalents 3011, 3012 and 3013) applied research about best practices for online learning in mathematics to the quarter bridge courses Math 301AQBR and 301BQBR that each include 0.5 units of online activities. Successful activities piloted in the quarter bridge courses may be implemented in the 3011-3012-3013 semester sequence and their associated lab courses 3011L-3012L-3013L. This paper focuses on written reflections and group discussion forums associated with the class textbook Powerful Problem Solving by Max Ray.

Pascal's Triangle, Pascal's Pyramid, And The Trinomial Triangle, Antonio Saucedo Jr.

Electronic Theses, Projects, and Dissertations

Many properties have been found hidden in Pascal's triangle. In this paper, we will present several known properties in Pascal's triangle as well as the properties that lift to different extensions of the triangle, namely Pascal's pyramid and the trinomial triangle. We will tailor our interest towards Fermat numbers and the hockey stick property. We will also show the importance of the hockey stick properties by using them to prove a property in the trinomial triangle.

Analogues Between Leibniz's Harmonic Triangle And Pascal's Arithmetic Triangle, Lacey Taylor James

Electronic Theses, Projects, and Dissertations

This paper will discuss the analogues between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle by utilizing mathematical proving techniques like partial sums, committees, telescoping, mathematical induction and applying George Polya's perspective. The topics presented in this paper will show that Pascal's triangle and Leibniz's triangle both have hockey stick type patterns, patterns of sums within shapes, and have the natural numbers, triangular numbers, tetrahedral numbers, and pentatope numbers hidden within. In addition, this paper will show how Pascal's Arithmetic Triangle can be used to construct Leibniz's Harmonic Triangle and show how both triangles relate to combinatorics and arithmetic through the …

Calculus Remediation As An Indicator For Success On The Calculus Ap Exam, Ty Stockham

Electronic Theses, Projects, and Dissertations

This study investigates the effects of implementing a remediation program in a high school Advanced Placement Calculus AB course on student class grades and success in passing the AP Calculus AB exam.

A voluntary remediation program was designed to help students understand the key concepts and big ideas in beginning Calculus. Over a period of eight years the program was put into practice and data on student participation and achievement was collected. Students who participated in this program were given individualized recitation activities targeting their specific misunderstandings, and then given an opportunity to retest on chapter exams that they had …

Tribonacci Convolution Triangle, Rosa Davila

Electronic Theses, Projects, and Dissertations

A lot has been said about the Fibonacci Convolution Triangle, but not much has been said about the Tribonacci Convolution Triangle. There are a few ways to generate the Fibonacci Convolution Triangle. Proven through generating functions, Koshy has discovered the Fibonacci Convolution Triangle in Pascal's Triangle, Pell numbers, and even Tribonacci numbers. The goal of this project is to find inspiration in the Fibonacci Convolution Triangle to prove patterns that we observe in the Tribonacci Convolution Triangle. We start this by bringing in all the information that will be useful in constructing and solving these convolution triangles and find a …

Symmetric Presentations And Double Coset Enumeration, Charles Seager

Electronic Theses, Projects, and Dissertations

In this project, we demonstrate our discovery of original symmetric presentations and constructions of important groups, including nonabelian simple groups, and groups that have these as factor groups. The target nonabelian simple groups include alternating, linear, and sporadic groups. We give isomorphism types for each finite homomorphic image that has been found. We present original symmetric presentations of $M_{12}$, $M_{21}:(2 \times 2)$, $L_{3}(4):2^2$, $2:^{\cdot}L_{3}(4):2$, $S(4,3)$, and $S_{7}$ as homomorphism images of the progenitors $2^{*20}$ $:$ $A_{5}$, $2^{*10}$ $:$ $PGL(2,9)$, $2^{*10}$ $:$ $Aut(A_{6})$, $2^{*10}$ $:$ $A_{6}$, $2^{*10}$ $:$ $A_{5}$, and $2^{*24}$ $:$ $S_{5}$, respectively. We also construct $M_{12}$, $M_{21}:(2 \times 2)$, …

Exploring Flag Matroids And Duality, Zachary Garcia

Electronic Theses, Projects, and Dissertations

Matroids capture an abstraction of independence in mathematics, and in doing so, connect discrete mathematical structures that arise in a variety of contexts. A matroid can be defined in several cryptomorphic ways depending on which perspective of a matroid is most applicable to the given context. Among the many important concepts in matroid theory, the concept of matroid duality provides a powerful tool when addressing difficult problems. The usefulness of matroid duality stems from the fact that the dual of a matroid is itself a matroid. In this thesis, we explore a matroid-like object called a flag matroid. In particular, …

Making Models With Bayes, Pilar Olid

Electronic Theses, Projects, and Dissertations

Bayesian statistics is an important approach to modern statistical analyses. It allows us to use our prior knowledge of the unknown parameters to construct a model for our data set. The foundation of Bayesian analysis is Bayes' Rule, which in its proportional form indicates that the posterior is proportional to the prior times the likelihood. We will demonstrate how we can apply Bayesian statistical techniques to fit a linear regression model and a hierarchical linear regression model to a data set. We will show how to apply different distributions to Bayesian analyses and how the use of a prior affects …

Regular Round Matroids, Svetlana Borissova

Electronic Theses, Projects, and Dissertations

A matroid M is a finite set E, called the ground set of M, together with a notion of what it means for subsets of E to be independent. Some matroids, called regular matroids, have the property that all elements in their ground set can be represented by vectors over any field. A matroid is called round if its dual has no two disjoint minimal dependent sets. Roundness is an important property that was very useful in the recent proof of Rota's conjecture, which remained an unsolved problem for 40 years in matroid theory. In this thesis, we …

Bio-Mathematics: Introduction To The Mathematical Model Of The Hepatitis C Virus, Lucille J. Durfee

Electronic Theses, Projects, and Dissertations

In this thesis, we will study bio-mathematics. We will introduce differential equations, biological applications, and simulations with emphasis in molecular events. One of the first courses of action is to introduce and construct a mathematical model of our biological element. The biological element of study is the Hepatitis C virus. The idea in creating a mathematical model is to approach the biological element in small steps. We will first introduce a block (schematic) diagram of the element, create differential equations that define the diagram, convert the dimensional equations to non-dimensional equations, reduce the number of parameters, identify the important parameters, …

The Kauffman Bracket And Genus Of Alternating Links, Bryan M. Nguyen

Electronic Theses, Projects, and Dissertations

Giving a knot, there are three rules to help us finding the Kauffman bracket polynomial. Choosing knot’s orientation, then applying the Seifert algorithm to find the Euler characteristic and genus of its surface. Finally finding the relationship of the Kauffman bracket polynomial and the genus of the alternating links is the main goal of this paper.

A Dual Fano, And Dual Non-Fano Matroidal Network, Stephen Lee Johnson

Electronic Theses, Projects, and Dissertations

Matroidal networks are useful tools in furthering research in network coding. They have been used to show the limitations of linear coding solutions. In this paper we examine the basic information on network coding and matroid theory. We then go over the method of creating matroidal networks. Finally we construct matroidal networks from the dual of the fano matroid and the dual of the non-fano matroid, and breifly discuss some coding solutions.

Realizing Tournaments As Models For K-Majority Voting, Gina Marie Cheney

Electronic Theses, Projects, and Dissertations

A k-majority tournament is a directed graph that models a k-majority voting scenario, which is realized by 2k - 1 rankings, called linear orderings, of the vertices in the tournament. Every k-majority voting scenario can be modeled by a tournament, but not every tournament is a model for a k-majority voting scenario. In this thesis we show that all acyclic tournaments can be realized as 2-majority tournaments. Further, we develop methods to realize certain quadratic residue tournaments as k-majority tournaments. Thus, each tournament within these classes of tournaments is a model for a k …

The Evolution Of Cryptology, Gwendolyn Rae Souza

Electronic Theses, Projects, and Dissertations

We live in an age when our most private information is becoming exceedingly difficult to keep private. Cryptology allows for the creation of encryptive barriers that protect this information. Though the information is protected, it is not entirely inaccessible. A recipient may be able to access the information by decoding the message. This possible threat has encouraged cryptologists to evolve and complicate their encrypting methods so that future information can remain safe and become more difficult to decode. There are various methods of encryption that demonstrate how cryptology continues to evolve through time. These methods revolve around different areas of …

Geometric Constructions From An Algebraic Perspective, Betzabe Bojorquez

Electronic Theses, Projects, and Dissertations

Many topics that mathematicians study at times seem so unrelated such as Geometry and Abstract Algebra. These two branches of math would seem unrelated at first glance. I will try to bridge Geometry and Abstract Algebra just a bit with the following topics. We can be sure that after we construct our basic parallel and perpendicular lines, bisected angles, regular polygons, and other basic geometric figures, we are actually constructing what in geometry is simply stated and accepted, because it will be proven using abstract algebra. Also we will look at many classic problems in Geometry that are not possible …

Unique Prime Factorization Of Ideals In The Ring Of Algebraic Integers Of An Imaginary Quadratic Number Field, Nolberto Rezola

Electronic Theses, Projects, and Dissertations

The ring of integers is a very interesting ring, it has the amazing property that each of its elements may be expressed uniquely, up to order, as a product of prime elements. Unfortunately, not every ring possesses this property for its elements. The work of mathematicians like Kummer and Dedekind lead to the study of a special type of ring, which we now call a Dedekind domain, where even though unique prime factorization of elements may fail, the ideals of a Dedekind domain still enjoy the property of unique prime factorization into a product of prime ideals, up to order …

Radio Number For Fourth Power Paths, Linda V. Alegria

Electronic Theses, Projects, and Dissertations

A path on n vertices, denoted by Pn, is a simple graph whose vertices can be ordered so that two vertices are adjacent if and only if they are consecutive in the order. A fourth power path, Pn4, is obtained from Pn by adding edges between any two vertices, u and v, whose distance in Pn, denoted by dPn(u,v), is less than or equal to four. The diameter of a graph G, denoted diam(G) is the greatest distance between any two distinct vertices of G. A radio labeling of a graph G is a function f that assigns to each …

Monoid Rings And Strongly Two-Generated Ideals, Brittney M. Salt

Electronic Theses, Projects, and Dissertations

This paper determines whether monoid rings with the two-generator property have the strong two-generator property. Dedekind domains have both the two-generator and strong two-generator properties. How common is this? Two cases are considered here: the zero-dimensional case and the one-dimensional case for monoid rings. Each case is looked at to determine if monoid rings that are not PIRs but are two-generated have the strong two-generator property. Full results are given in the zero-dimensional case, however only partial results have been found for the one-dimensional case.