Physical Sciences and Mathematics Commons^™

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.

Exploring Matroid Minors, Jonathan Lara Tejeda

Electronic Theses, Projects, and Dissertations

Matroids are discrete mathematical objects that generalize important concepts of independence arising in other areas of mathematics. There are many different important classes of matroids and a frequent problem in matroid theory is to determine whether or not a given matroid belongs to a certain class of matroids. For special classes of matroids that are minor-closed, this question is commonly answered by determining a complete list of matroids that are not in the class but have the property that each of their proper minors is in the class; that is, minor-minimal matroids that are not in the minor-closed class. These …

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 …

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 …

Minimal Surfaces And The Weierstrass-Enneper Representation, Evan Snyder

Electronic Theses, Projects, and Dissertations

The field of minimal surfaces is an intriguing study, not only because of the exotic structures that these surfaces admit, but also for the deep connections among various mathematical disciplines. Minimal surfaces have zero mean curvature, and their parametrizations are usually quite complicated and nontrivial. It was shown however, that these exotic surfaces can easily be constructed from a careful choice of complex-valued functions, using what is called the Weierstrass-Enneper Representation.

In this paper, we develop the necessary tools to study minimal surfaces. We will prove some classical theorems and solve an interesting problem that involves ruled surfaces. We will …

Excluded Minors For Nearly-Paving Matroids, Vanessa Natalie Vega

Electronic Theses, Projects, and Dissertations

Matroids capture an abstract notion of independence that generalizes linear independence in linear algebra, edge independence in graph theory, as well as algebraic independence. Given a particular property of matroids, all the matroids possessing that property form a matroid class. A common research theme in matroid theory is to characterize matroid classes so that, given a matroid M, it is possible to determine whether or not M belongs to a given class. An excluded minor of a minor-closed class is a matroid N that is, in a sense, minimal with respect to not being in the minor-closed class. An attractive …

Hyperbolic Triangle Groups, Sergey Katykhin

Electronic Theses, Projects, and Dissertations

This paper will be on hyperbolic reflections and triangle groups. We will compare hyperbolic reflection groups to Euclidean reflection groups. The goal of this project is to give a clear exposition of the geometric, algebraic, and number theoretic properties of Euclidean and hyperbolic reflection groups.

Modeling The Spread Of Measles, Alexandria Le Beau

Electronic Theses, Projects, and Dissertations

The measles virus has been around since the 9th century. Throughout the years measles have become less problematic in certain areas of the world due to research and the creation of vaccinations. Sadly, not all countries are fortunate enough to have adequate access to the vaccination, which leads to yearly outbreaks.

The goal of this project is to experiment with different mathematical growth models and examine their suitability for modeling outbreaks of measles. We will compare and contrast the exponential model, the logistic model, the SIR model, and the SEIR model. In addition, we will show how the epidemiological models …

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 …

Symmetric Presentations And Related Topics, Mayra Mcgrath

Electronic Theses, Projects, and Dissertations

In this thesis, we have investigated several permutation and monomialprogenitors for finite images. We have found original symmetric presen-tations for several important non-abelian simple groups, including lineargroups, unitary groups, alternating groups, and sporadic simple groups.We have found a number of finite images, including : L(2,41), PSL(2,11)×2, L(2,8), and L(2,19), as homomorphic images of the permutation progenitors. We have also found PGL(2,16) : 2 =Aut(PSL(2,16)) and PSL(2,16) as homomorphic images of monomial progenitors. We have performed manual double coset enumeration of finte images. In addition, we have given the isomorphism class of each image that we have discovered. Presentation for all …