Patterns, Symmetries, And Mathematical Structures In The Arts, 2020 Georgia Southern University

#### Patterns, Symmetries, And Mathematical Structures In The Arts, Sarah C. Deloach

*University Honors Program Theses*

Mathematics is a discipline of academia that can be found everywhere in the world around us. Mathematicians and scientists are not the only people who need to be proficient in numbers. Those involved in social sciences and even the arts can benefit from a background in math. In fact, connections between mathematics and various forms of art have been discovered since as early as the fourth century BC. In this thesis we will study such connections and related concepts in mathematics, dances, and music.

Group Theoretical Analysis Of Arbitrarily Large, Colored Square Grids, 2019 University of Lynchburg

#### Group Theoretical Analysis Of Arbitrarily Large, Colored Square Grids, Brett Ehrman

*Student Scholar Showcase*

In this research, we examine *n* x *n* grids whose individual squares are each colored with one of *k* distinct colors. We seek a general formula for the number of colored grids that are distinct up to rotations, reflections, and color reversals. We examine the problem using a group theoretical approach. We define a specific group action that allows us to incorporate Burnside’s Lemma, which leads us to the desired general results

Experience Of A Noyce-Student Learning Assistant In An Inquiry-Based Learning Class, 2019 University of Nebraska at Omaha

#### Experience Of A Noyce-Student Learning Assistant In An Inquiry-Based Learning Class, Melissa Riley

*Student Research and Creative Activity Fair*

This presentation refers to an undergraduate course called introduction to abstract mathematics at the University of Nebraska at Omaha. During the academic year 2017-2018, undergraduate, mathematics student Melissa Riley was a Noyce-student learning assistant for the Inquiry Based Learning (IBL) section of the course. She assisted the faculty-in-charge with all aspects of the course. These included: materials preparation, class organization, teamwork, class leading, presentations, and tutoring. This presentation shall address some examples of how the IBL approach can be used in this type of class including: the structure of the course, the activities and tasks performed by the students, learning ...

Some Intuition Behind Large Cardinal Axioms, Their Characterization, And Related Results, 2019 Virginia Commonwealth University

#### Some Intuition Behind Large Cardinal Axioms, Their Characterization, And Related Results, Philip A. White

*Theses and Dissertations*

We aim to explain the intuition behind several large cardinal axioms, give characterization theorems for these axioms, and then discuss a few of their properties. As a capstone, we hope to introduce a new large cardinal notion and give a similar characterization theorem of this new notion. Our new notion of near strong compactness was inspired by the similar notion of near supercompactness, due to Jason Schanker.

Elementary Set Theory, 2018 University of North Dakota

#### Elementary Set Theory, Richard P. Millspaugh

*Open Educational Resources*

This text is appropriate for a course that introduces undergraduates to proofs. The material includes elementary symbolic logic, logical arguments, basic set theory, functions and relations, the real number system, and an introduction to cardinality. The text is intended to be readable for sophomore and better freshmen majoring in mathematics.

The source files for the text can be found at https://github.com/RPMillspaugh/SetTheory

Tutte-Equivalent Matroids, 2018 California State University - San Bernardino

#### Tutte-Equivalent Matroids, Maria Margarita Rocha

*Electronic Theses, Projects, and Dissertations*

We begin by introducing matroids in the context of finite collections of vectors from a vector space over a specified field, where the notion of independence is linear independence. Then we will introduce the concept of a matroid invariant. Specifically, we will look at the Tutte polynomial, which is a well-defined two-variable invariant that can be used to determine differences and similarities between a collection of given matroids. The Tutte polynomial can tell us certain properties of a given matroid (such as the number of bases, independent sets, etc.) without the need to manually solve for them. Although the Tutte ...

Selective Strong Screenability, 2018 Boise State University

#### Selective Strong Screenability, Isaac Joseph Coombs

*Boise State University Theses and Dissertations*

Screenability and strong screenability were both introduced some sixty years ago by R.H. Bing in his paper *Metrization of Topological Spaces*. Since then, much work has been done in exploring selective screenability (the selective version of screenability). However, the corresponding selective version of strong screenability has been virtually ignored. In this paper we seek to remedy this oversight. It is found that a great deal of the proofs about selective screenability readily carry over to proofs for the analogous version for selective strong screenability. We give some examples of selective strongly screenable spaces with the primary example being Pol ...

The Structure Of Models Of Second-Order Set Theories, 2018 The Graduate Center, City University of New York

#### The Structure Of Models Of Second-Order Set Theories, Kameryn J. Williams

*All Dissertations, Theses, and Capstone Projects*

This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of T-realizations of a fixed countable model of ZFC, where T is a reasonable second-order set theory such as GBC or KM, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve ...

Distributive Lattice Models Of The Type C One-Rowed Weyl Group Symmetric Functions, 2018 Murray State University

#### Distributive Lattice Models Of The Type C One-Rowed Weyl Group Symmetric Functions, William Atkins

*Murray State Theses and Dissertations*

We present two families of diamond-colored distributive lattices – one known and one new – that we can show are models of the type C one-rowed Weyl symmetric functions. These lattices are constructed using certain sequences of positive integers that are visualized as ﬁlling the boxes of one-rowed partition diagrams. We show how natural orderings of these one-rowed tableaux produce our distributive lattices as sublattices of a more general object, and how a natural coloring of the edges of the associated order diagrams yields a certain diamond-coloring property. We show that each edge-colored lattice possesses a certain structure that is associated with ...

Introduction To Game Theory: A Discovery Approach, 2018 Linfield College

#### Introduction To Game Theory: A Discovery Approach, Jennifer Firkins Nordstrom

*Linfield Authors Book Gallery*

Game theory is an excellent topic for a non-majors quantitative course as it develops mathematical models to understand human behavior in social, political, and economic settings. The variety of applications can appeal to a broad range of students. Additionally, students can learn mathematics through playing games, something many choose to do in their spare time! This text also includes an exploration of the ideas of game theory through the rich context of popular culture. It contains sections on applications of the concepts to popular culture. It suggests films, television shows, and novels with themes from game theory. The questions in ...

Transfinite Ordinal Arithmetic, 2017 Governors State University

#### Transfinite Ordinal Arithmetic, James Roger Clark

*All Student Theses*

Following the literature from the origin of Set Theory in the late 19th century to more current times, an arithmetic of finite and transfinite ordinal numbers is outlined. The concept of a set is outlined and directed to the understanding that an ordinal, a special kind of number, is a particular kind of well-ordered set. From this, the idea of counting ordinals is introduced. With the fundamental notion of counting addressed: then addition, multiplication, and exponentiation are defined and developed by established fundamentals of Set Theory. Many known theorems are based upon this foundation. Ultimately, as part of the conclusion ...

Joint Laver Diamonds And Grounded Forcing Axioms, 2017 The Graduate Center, City University of New York

#### Joint Laver Diamonds And Grounded Forcing Axioms, Miha Habič

*All Dissertations, Theses, and Capstone Projects*

In chapter 1 a notion of independence for diamonds and Laver diamonds is investigated. A sequence of Laver diamonds for κ is *joint* if for any sequence of targets there is a single elementary embedding *j* with critical point κ such that each Laver diamond guesses its respective target via *j*. In the case of measurable cardinals (with similar results holding for (partially) supercompact cardinals) I show that a single Laver diamond for κ yields a joint sequence of length κ, and I give strict separation results for all larger lengths of joint sequences. Even though the principles get strictly ...

The Classification Problem For Models Of Zfc, 2017 Boise State University

#### The Classification Problem For Models Of Zfc, Samuel Dworetzky

*Boise State University Theses and Dissertations*

Models of ZFC are ubiquitous in modern day set theoretic research. There are many different constructions that produce countable models of ZFC via techniques such as forcing, ultraproducts, and compactness. The models that these techniques produce have many different characteristics; thus it is natural to ask whether or not models of ZFC are classifiable. We will answer this question by showing that models of ZFC are unclassifiable and have maximal complexity. The notions of complexity used in this thesis will be phrased in the language of Borel complexity theory.

In particular, we will show that the class of countable models ...

Classification Of Vertex-Transitive Structures, 2017 Boise State University

#### Classification Of Vertex-Transitive Structures, Stephanie Potter

*Boise State University Theses and Dissertations*

When one thinks of objects with a significant level of symmetry it is natural to expect there to be a simple classification. However, this leads to an interesting problem in that research has revealed the existence of highly symmetric objects which are very complex when considered within the framework of Borel complexity. The tension between these two seemingly contradictory notions leads to a wealth of natural questions which have yet to be answered.

Borel complexity theory is an area of logic where the relative complexities of classification problems are studied. Within this theory, we regard a classification problem as an ...

Combinatorial Polynomial Hirsch Conjecture, 2017 Harvey Mudd College

#### Combinatorial Polynomial Hirsch Conjecture, Sam Miller

*HMC Senior Theses*

The Hirsch Conjecture states that for a *d*-dimensional polytope with *n* facets, the diameter of the graph of the polytope is at most *n-d*. This conjecture was disproven in 2010 by Francisco Santos Leal. However, a polynomial bound in *n* and *d* on the diameter of a polytope may still exist. Finding a polynomial bound would provide a worst-case scenario runtime for the Simplex Method of Linear Programming. However working only with polytopes in higher dimensions can prove challenging, so other approaches are welcome. There are many equivalent formulations of the Hirsch Conjecture, one of which is the Combinatorial ...

The Density Topology On The Reals With Analogues On Other Spaces, 2016 Boise State University

#### The Density Topology On The Reals With Analogues On Other Spaces, Stuart Nygard

*Boise State University Theses and Dissertations*

A point *x* is a *density point* of a set *A* if all of the points except a measure zero set near to *x* are contained in *A*. In the usual topology on ℝ, a set is open if shrinking intervals around each point are eventually contained in the set. The density topology relaxes this requirement. A set is open in the density topology if for each point, the limit of the measure of A contained in shirking intervals to the measure of the shrinking intervals themselves is one. That is, for any point *x* and a small enough interval ...

Development Of Utility Theory And Utility Paradoxes, 2016 Lawrence University

#### Development Of Utility Theory And Utility Paradoxes, Timothy E. Dahlstrom

*Lawrence University Honors Projects*

Since the pioneering work of von Neumann and Morgenstern in 1944 there have been many developments in Expected Utility theory. In order to explain decision making behavior economists have created increasingly broad and complex models of utility theory. This paper seeks to describe various utility models, how they model choices among ambiguous and lottery type situations, and how they respond to the Ellsberg and Allais paradoxes. This paper also attempts to communicate the historical development of utility models and provide a fresh perspective on the development of utility models.

Mathematical Reasoning And The Inductive Process: An Examination Of The Law Of Quadratic Reciprocity, 2016 California State University - San Bernardino

#### Mathematical Reasoning And The Inductive Process: An Examination Of The Law Of Quadratic Reciprocity, Nitish Mittal

*Electronic Theses, Projects, and Dissertations*

This project investigates the development of four different proofs of the law of quadratic reciprocity, in order to study the critical reasoning process that drives discovery in mathematics. We begin with an examination of the first proof of this law given by Gauss. We then describe Gauss’ fourth proof of this law based on Gauss sums, followed by a look at Eisenstein’s geometric simplification of Gauss’ third proof. Finally, we finish with an examination of one of the modern proofs of this theorem published in 1991 by Rousseau. Through this investigation we aim to analyze the different strategies used ...

Exploring Argumentation, Objectivity, And Bias: The Case Of Mathematical Infinity, 2016 University of Ontario Institute of Technology

#### Exploring Argumentation, Objectivity, And Bias: The Case Of Mathematical Infinity, Ami Mamolo

*OSSA Conference Archive*

This paper presents an overview of several years of my research into individuals’ reasoning, argumentation, and bias when addressing problems, scenarios, and symbols related to mathematical infinity. There is a long history of debate around what constitutes “objective truth” in the realm of mathematical infinity, dating back to ancient Greece (e.g., Dubinsky et al., 2005). Modes of argumentation, hindrances, and intuitions have been largely consistent over the years and across levels of expertise (e.g., Brown et al., 2010; Fischbein et al., 1979, Tsamir, 1999). This presentation examines the interrelated complexities of notions of objectivity, bias, and argumentation as ...

The Development Of Notation In Mathematical Analysis, 2016 Loyola Marymount University

#### The Development Of Notation In Mathematical Analysis, Alyssa Venezia

*Honors Thesis*

The field of analysis is a newer subject in mathematics, as it only came into existence in the last 400 years. With a new field comes new notation, and in the era of universalism, analysis becomes key to understanding how centuries of mathematics were unified into a finite set of symbols, precise definitions, and rigorous proofs that would allow for the rapid development of modern mathematics. This paper traces the introduction of subjects and the development of new notations in mathematics from the seventeenth to the nineteenth century that allowed analysis to flourish. In following the development of analysis, we ...