Mathematics Commons^™

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 …

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 …

Reverse Mathematics Of Ramsey's Theorem, Nikolay Maslov

Electronic Theses, Projects, and Dissertations

Reverse mathematics aims to determine which set theoretic axioms are necessary to prove the theorems outside of the set theory. Since the 1970’s, there has been an interest in applying reverse mathematics to study combinatorial principles like Ramsey’s theorem to analyze its strength and relation to other theorems. Ramsey’s theorem for pairs states that for any infinite complete graph with a finite coloring on edges, there is an infinite subset of nodes all of whose edges share one color. In this thesis, we introduce the fundamental terminology and techniques for reverse mathematics, and demonstrate their use in proving Kőnig's lemma …

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 …

Symbolic Logic, Tony Roy

Books

Textbook for symbolic logic, beginning at a level appropriate for beginning students, continuing through Godel's completeness and incompleteness theorems. The text naturally divides into two volumes, the first for reasoning in logic, the second for reasoning about it.

The first volume includes parts I and II of the text. Part I introduces the complete classical predicate calculus with equality, including both axiomatic and natural derivation systems. Part II transitions to methods for reasoning about logic, including direct reasoning from definitions and mathematical induction.

The second volume includes parts III and IV of the text. Part III develops basic results in …

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 …

Verifying Sudoku Puzzles, Chelsea Schweer

Electronic Theses, Projects, and Dissertations

Sudoku puzzles, created by Meki Kaji around 1983, consist of a square 9 by 9 grid made up of 9 rows, 9 columns, and nine 3 by 3 square sub-grids called blocks. The goal of the puzzle is to be able to place the numbers 1 through 9 in every row, column, and block where no number is repeated in each row, column, and block. Imagine being given a completed Sudoku puzzle and having to check that it was solved correctly. You could just check all the rows columns and blocks (27 items), but is there a smaller number of …

Error Terms For The Trapezoid, Midpoint, And Simpson's Rules, Jessica E. Coen

Electronic Theses, Projects, and Dissertations

When it is not possible to integrate a function we resort to Numerical Integration. For example the ubiquitous Normal curve tables are obtained using Numerical Integration. The antiderivative of the defining function for the normal curve involves the formula for antiderivative of e^{-x^2} which can't be expressed in the terms of basic functions.

Simpson's rule is studied in most Calculus books, and in all undergraduate Numerical Analysis books, but proofs are not provided. Hence if one is interested in a proof of Simpson's rule, either it can be found in advanced Numerical Analysis books as a special case …

Homomorphic Images And Related Topics, Alejandro Martinez

Electronic Theses, Projects, and Dissertations

In this thesis, we have demonstrated our method of writing symmetric presentations of permutation progenitors, finding monomial representations and symmetric presentations of monomial progenitors. We have also explained how various types of additional relations are found. We have discovered original symmetric presentations and original constructions of numerous groups.

Symmetric Generation, Ana Gonzalez

Electronic Theses, Projects, and Dissertations

We will examine progenitors. We start with progenitors of the form $m^{*n} : N$ where $m^{*n}$ is a free group and $N$ is a permutation group of degree $n$. But, $m^{*n} : N$ is a group of infinite order so we will factor by the necessary relations to get finite homomorphic images. These groups are constructed through the manual double coset enumeration method. We will examine how to construct progenitors for wreath products.

Lattice Reduction Algorithms, Juan Ortega

Electronic Theses, Projects, and Dissertations

The purpose of this thesis is to propose and analyze an algorithm that follows
similar steps of Guassian Lattice Reduction Algorithm in two-dimensions and applying
them to three-dimensions. We start off by discussing the importance of cryptography in
our day to day lives. Then we dive into some linear algebra and discuss specific topics that
will later help us in understanding lattice reduction algorithms. We discuss two lattice
problems: the shortest vector problem and the closest vector problem. Then we introduce
two types of lattice reduction algorithms: Guassian Lattice Reduction in two-dimensions
and the LLL Algortihm. We illustrate how both …

An Exposition Of Elliptic Curve Cryptography, Travis Severns

Electronic Theses, Projects, and Dissertations

Protecting information that is being communicated between two parties over
unsecured channels is of huge importance in today’s world. The use of mathematical concepts to achieve high levels of security when communicating over these unsecured platforms is cryptography. The world of cryptography is always expanding and growing. In this paper, we set out to explore the use of elliptic curves in the cryptography of today, as well as the cryptography of the future.
We also offer our own original cryptosystem, CSDH. This system on its own
offers some moderate level of security. It shares many similarities to the post-quantum, SIDH …

Symmetric Presentations Of Finite Groups And Related Topics, Samar Mikhail Kasouha

Electronic Theses, Projects, and Dissertations

A progenitor is an infinite semi-direct product of the form m∗n : N, where N ≤ Sn and m∗n : N is a free product of n copies of a cyclic group of order m. A progenitor of this type, in particular 2∗n : N, gives finite non-abelian simple groups and groups involving these, including alternating groups, classical groups, and the sporadic group. We have conducted a systematic search of finite homomorphic images of numerous progenitors. In this thesis we have presented original symmetric presentations of the sporadic simple groups, M12, J1 as homomorphic images of the progenitor 2∗12 : …

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 …

De Rham Cohomology, Homotopy Invariance And The Mayer-Vietoris Sequence, Stacey Elizabeth Cox

Electronic Theses, Projects, and Dissertations

This thesis will discuss the de Rham cohomology, homotopy invariance and the Mayer-Vietoris sequence. First the necessary information for this thesis is discussed such as differential p-forms, the exterior derivative as well as pull back of a map. The de Rham cohomology is defined explicitly, some properties of the de Rham cohomology will also be discussed. It will be shown that the de Rham cohomology is in fact a homotopy invariant as well as some examples using homotopy invariance are provided. Finally the Mayer-Vietoris sequence will be established, an example of using the Mayer-Vietoris sequence to compute the de …

The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles

Electronic Theses, Projects, and Dissertations

This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i² = 3, j² = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL₂(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This …

Simple Groups And Related Topics, Simrandeep Kaur

Electronic Theses, Projects, and Dissertations

Since every nonabelian simple group is a homomorphic image of an involutory progenitor 2^(*n):N where N ≤ S_n is transitive, our motivation for the thesis has been to seek finite homomorphic images of such progenitors and construct them using our technique of double coset enumeration. We have constructed U_3 (3):2 over 5^2:S_3, 2x(A_5 x A_5) over D_5 x D_5, S_6 over S_5, 2^5:S_5 over S_5, and 3^3: 2^3 over 3^2:2 . We have discovered original symmetric presentations numerous group as homomorphic images various progenitors. We have also found new monomial representations of groups and given monomial progenitors. We have given …

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), …

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.

A Study In Applications Of Continued Fractions, Karen Lynn Parrish

Electronic Theses, Projects, and Dissertations

This is an expository study of continued fractions collecting ideas from several different sources including textbooks and journal articles. This study focuses on several applications of continued fractions from a variety of levels and fields of mathematics. Studies begin with looking at a number of properties that pertain to continued fractions and then move on to show how applications of continued fractions is relevant to high school level mathematics including approximating irrational numbers and developing new ideas for understanding and solving quadratics equations. Focus then continues to more advanced applications such as those used in the studies of number theory …

Matroids Determinable By Two Partial Representations, Aurora Calderon Dojaquez

Electronic Theses, Projects, and Dissertations

A matroid is a mathematical object that generalizes and connects notions of independence that arise in various branches of mathematics. Some matroids can be represented by a matrix whose entries are from some field; whereas, other matroids cannot be represented in this way. However, every matroid can be partially represented by a matrix over the field GF(2). In fact, for a given matroid, many different partial representations may exist, each providing a different collection of information about the matroid with which they are associated. Such a partial representation of a matroid usually does not uniquely determine the matroid on its …

Partial Representations For Ternary Matroids, Ebony Perez

Electronic Theses, Projects, and Dissertations

In combinatorics, a matroid is a discrete object that generalizes various notions of dependence that arise throughout mathematics. All of the information about some matroids can be encoded (or represented) by a matrix whose entries come from a particular field, while other matroids cannot be represented in this way. However, for any matroid, there exists a matrix, called a partial representation of the matroid, that encodes some of the information about the matroid. In fact, a given matroid usually has many different partial representations, each providing different pieces of information about the matroid. In this thesis, we investigate when a …

Non-Abelian Finite Simple Groups As Homomorphic Images, Sandra Bahena

Electronic Theses, Projects, and Dissertations

The purpose of exploring infinite groups in this thesis was to discover homomorphic images of non-abelian finite simple groups. These infinite groups are semi-direct products known as progenitors. The permutation progenitors studied were: 2^*8 ∶ 2^{2 ∙} A₄, 2^*10 ∶ D₂₀, 2^*4 ∶ C₄, 2^*7 ∶ (7 ∶ 6), 3^*3 ∶ S₃, 2^*15 ∶ ((5 × 3) ∶ 2), and 2^*20 ∶ A₅. When we factored said progenitors by an appropriate number of relations, we produced several original symmetric presentations and constructions …

Symmetric Presentation Of Finite Groups, And Related Topics, Marina Michelle Duchesne

Electronic Theses, Projects, and Dissertations

We have discovered original symmetric presentations for several finite groups, including 2²:^.(2⁴:(2^.S₃)), M₁₁, 3:(PSL(3,3):2), S₈, and 2^.M₁₂. We have found homomorphic images of several progenitors, including 2^*18:((6x2):6), 2^*24:(2^.S₄), 2^*105:A₇, 3^*3:_m(2³:3), 7^*8:_m(PSL(2,7):2), 3^*4:_m(4²:2²), 7^*5:(2xA₅), and 5^*6:_mS₅. We have provided the isomorphism type of …

Optimal Tile-Based Dna Self-Assembly Designs For Lattice Graphs And Platonic Solids, Leyda Almodovar, Joanna Ellis-Monaghan, Amanda Harsy, Cory Johnson, Jessica Sorrells

Mathematics Faculty Publications

A design goal in self-assembly of DNA nanostructures is to find minimal sets of branched junction molecules that will self-assemble into targeted structures. This process can be modeled using techniques from graph theory. This paper is a collection of proofs for a set of DNA complexes which can be represented by specific graphs, namely Platonic solids, square lattice graphs, and triangular lattice graphs. This work supplements the results presented in https://arxiv.org/abs/2108.00035

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.

Assessing Student Understanding While Solving Linear Equations Using Flowcharts And Algebraic Methods, Edima Umanah

Electronic Theses, Projects, and Dissertations

Solving linear equations has often been taught procedurally by performing inverse operations until the variable in question is isolated. Students do not remember which operation to undo first because they often memorize operations with no understanding of the underlying meanings. The study was designed to help assess how well students are able to solve linear equations. Furthermore, the lesson is designed to help students identify solving linear equations in more than one-way. The following research questions were addressed in this study: Does the introduction of multiple ways to think about linear equations lead students to flexibly incorporate appropriate representations/strategies in …

Dna Complexes Of One Bond-Edge Type, Andrew Tyler Lavengood-Ryan

Electronic Theses, Projects, and Dissertations

DNA self-assembly is an important tool used in the building of nanostructures and targeted virotherapies. We use tools from graph theory and number theory to encode the biological process of DNA self-assembly. The principal component of this process is to examine collections of branched junction molecules, called pots, and study the types of structures that such pots can realize. In this thesis, we restrict our attention to pots which contain identical cohesive-ends, or a single bond-edge type, and we demonstrate the types and sizes of structures that can be built based on a single characteristic of the pot that is …