Mathematics Commons^™

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), …

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.

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 …

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 …

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.

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 …

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 …

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 …

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 …

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.

Syllabus For Semester Bridge Course: Fundamental Concepts Of Math For Educators: Fundamental Concepts Of Algebra And Geometry & Problem Solving Through Theory And Practice (Math 301a Qbr), Lamies Nazzal, Joyce Ahlgren

Q2S Enhancing Pedagogy

The Quarter-to-Semester transition at CSUSB brought a number of challenges for many courses or course series. One of those included the math requirement for Liberal Studies series, Math 30x courses. The challenge here is that the 30x series includes four courses, yet the transition to semesters will yield three courses. In the Fall of 2020, the fourth 2-unit course in the series, Math 308 (Problem Solving Through Theory and Practice), will no longer be offered. Instead, it will be embedded into the first three courses. Students beginning the series after Fall 2019, will not have enough time to complete the …

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 …

Collaboration (Reacting To The Past/Math/History/Writing), James Hayashi

Q2S Enhancing Pedagogy

This is an assignment for a Freshman level course in the College of Natural Science. By the end students will have an understanding of valid research, collaboration and communication skills. Faculty that chooses to use this assignment will be preparing students for an active learning environment, and understanding a “Big Idea”, valid research, technology and communication skills.

Faculty should give an example of what is valid research. As students are completing this assignment mini deadlines (check-ins) shall be set. With the check-ins for this assignment focus on how the group will communicate the check point and the collaboration.

The focus …

Geometry Across The Curriculum, Corey Dunn

Q2S Enhancing Pedagogy

This project is designed for a multicalculus class already familiar with computing the arc length of a parameterized curve in space. The activity asks the student to first recall basic facts about arc length, and then introduces the notion of measuring lengths of vectors differently, depending on where their initial point is. This is a foundational concept in metric differential geometry, and, this activity attempts to motivate this generalization of computing lengths of vectors through this arc length activity. The activity concludes with a short discussion of basic concepts of Lorentzian geometry, including the idea that lightlike vectors have length …

Geogebra Activities: Tracing Points, Jeremy Aikin, Corey Dunn, Jeffrey Meyer, Rolland Trapp

Q2S Enhancing Pedagogy

In this activity, we will learn how to use GeoGebra (www.geogebra.org) to trace the movement of points, which depend on the movement of other objects. The paths of these points determine curves and we will provide algebraic descriptions of these curves.

Math 3100 - Mathematical Thinking: Communication And Proof - Week 1 Outline, Laura J. Wallace, Cory Johnson, Shawn Mcmurran, Min-Lin Lo

Q2S Enhancing Pedagogy

As a group, we hope through this FLC project to develop a course that promotes departmental SLOs, as well as University learning outcomes for a writing-intensive course. We focused on our new foundational course (MATH 3100 Mathematical Thinking: Communication and Proof). This course, designated as a writing-intensive course, introduces students to disciplinary ways of thinking and communicating in mathematics with emphasis on the construction of valid mathematical arguments, critiques of arguments, and structure of professional mathematical writing including typesetting. We would like to develop a library of useful materials that faculty members can adapt in their classrooms in order to …

Math 3100 Mathematical Thinking: Communication And Proof - Sample Syllabus, Cory Johnson, Shawn Mcmurran, Laura Wallace, Min-Lin Lo

Q2S Enhancing Pedagogy

As a group, we hope through this FLC project to develop a course that promotes departmental SLOs, as well as University learning outcomes for a writing-intensive course. We focused on our new foundational course (MATH 3100 Mathematical Thinking: Communication and Proof). This course, designated as a writing-intensive course, introduces students to disciplinary ways of thinking and communicating in mathematics with emphasis on the construction of valid mathematical arguments, critiques of arguments, and structure of professional mathematical writing including typesetting. We would like to develop a library of useful materials that faculty members can adapt in their classrooms in order to …

Math 3100 Mathematical Thinking: Communication And Proof-Sample Writing (Including Typesetting) Assignments, Min-Lin Lo, Cory Johnson, Shawn Mcmurran, Laura Wallace

Q2S Enhancing Pedagogy

As a group, we hope through this FLC project to develop a course that promotes departmental SLOs, as well as University learning outcomes for a writing-intensive course. We focused on our new foundational course (MATH 3100 Mathematical Thinking: Communication and Proof). This course, designated as a writing-intensive course, introduces students to disciplinary ways of thinking and communicating in mathematics with emphasis on the construction of valid mathematical arguments, critiques of arguments, and structure of professional mathematical writing including typesetting. We would like to develop a library of useful materials that faculty members can adapt in their classrooms in order to …

Math 3100: Communication And Proof - Assessment, Shawn Mcmurran, Min-Lin Lo, Corrine Johnson, Laura Wallace

Q2S Enhancing Pedagogy

As a group, we hope through this FLC project to develop a course that promotes departmental SLOs, as well as University learning outcomes for a writing-intensive course. We focused on our new foundational course (MATH 3100 Mathematical Thinking: Communication and Proof). This course, designated as a writing-intensive course, introduces students to disciplinary ways of thinking and communicating in mathematics with emphasis on the construction of valid mathematical arguments, critiques of arguments, and structure of professional mathematical writing including typesetting. We would like to develop a library of useful materials that faculty members can adapt in their classrooms in order to …

Reflection On Use Of The "Reacting To The Past" Pedagogy In A History Of Mathematics Course, Davida Fischman

Q2S Enhancing Pedagogy

This brief report provides a reflection on the use of the "Reacting to the Past" (RTTP) pedagogy in a History of Mathematics classroom. The conclusion is drawn that the RTTP pedagogy is very successful in engaging students in active learning, and appropriate games may be utilized to help students learn about the role of mathematics in historical developments as well as in society today.

Algebraic Methods For Proving Geometric Theorems, Lynn Redman

Electronic Theses, Projects, and Dissertations

Algebraic geometry is the study of systems of polynomial equations in one or more variables. Thinking of polynomials as functions reveals a close connection between affine varieties, which are geometric structures, and ideals, which are algebraic objects. An affine variety is a collection of tuples that represents the solutions to a system of equations. An ideal is a special subset of a ring and is what provides the tools to prove geometric theorems algebraically. In this thesis, we establish that a variety depends on the ideal generated by its defining equations. The ability to change the basis of an ideal …

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 …