Physical Sciences and Mathematics Commons^™

Counting Conjugates Of Colored Compositions, Jesus Omar Sistos Barron

Honors College Theses

The properties of n-color compositions have been studied parallel to those of regular compositions. The conjugate of a composition as defined by MacMahon, however, does not translate well to n-color compositions, and there is currently no established analogous concept. We propose a conjugation rule for cyclic n-color compositions. We also count the number of self-conjugates under these rules and establish a couple of connections between these and regular compositions.

Jumping Frogs On Cyclic Graphs, Jake Mitchell

Honors College Theses

From the traditional game of Solitaire to modern video games like Candy Crush and Five Nights at Freddy’s, single-player games have captivated audiences for gener- ations. We investigate a lesser-known single-player game, the Jumping Frogs problem, on various classes of simple graphs, a graph with no multiple edges or looped ver- tices. We determine whether frogs can be stacked together on one vertex of a given graph. In a graph with k vertices and one frog on each vertex, the frogs must make legal jumps to form a stack of k frogs. The problem is known to be solvable on …

Long-Time Stability For An Imex Discretization Of The 1d Fujita Equation, Victoria Luongo

Honors College Theses

We study an efficient time-stepping scheme for the 1D Fujita equation that is implicit for the linear terms but explicit for the nonlinear terms. We analyze the long-time stability of the scheme for varying parameter values, which reveal parameter value regimes in which the method is stable. We provide numerical results that illustrate the theory and show the analytically derived stability conditions are sufficient to achieve long-time stability results.

A Graphical User Interface Using Spatiotemporal Interpolation To Determine Fine Particulate Matter Values In The United States, Kelly M. Entrekin

Honors College Theses

Fine particulate matter or PM2.5 can be described as a pollution particle that has a diameter of 2.5 micrometers or smaller. These pollution particle values are measured by monitoring sites installed across the United States throughout the year. While these values are helpful, a lot of areas are not accounted for as scientists are not able to measure all of the United States. Some of these unmeasured regions could be reaching high PM2.5 values over time without being aware of it. These high values can be dangerous by causing or worsening health conditions, such as cardiovascular and lung diseases. Within …

Quantum Dimension Polynomials: A Networked-Numbers Game Approach, Nicholas Gaubatz

Honors College Theses

The Networked-Numbers Game--a mathematical "game'' played on a simple graph--is incredibly accessible and yet surprisingly rich in content. The Game is known to contain deep connections to the finite-dimensional simple Lie algebras over the complex numbers. On the other hand, Quantum Dimension Polynomials (QDPs)--enumerative expressions traditionally understood through root systems--corresponding to the above Lie algebras are complicated to derive and often inaccessible to undergraduates. In this thesis, the Networked-Numbers Game is defined and some known properties are presented. Next, the significance of the QDPs as a method to count combinatorially interesting structures is relayed. Ultimately, a novel closed-form expression of …

Data And Algorithmic Modeling Approaches To Count Data, Andraya Hack

Honors College Theses

Various techniques are used to create predictions based on count data. This type of data takes the form of a non-negative integers such as the number of claims an insurance policy holder may make. These predictions can allow people to prepare for likely outcomes. Thus, it is important to know how accurate the predictions are. Traditional statistical approaches for predicting count data include Poisson regression as well as negative binomial regression. Both methods also have a zero-inflated version that can be used when the data has an overabundance of zeros. Another procedure is to use computer algorithms, also known as …

Sangaku In Multiple Geometries: Examining Japanese Temple Geometry Beyond Euclid, Nathan Hartmann

Honors College Theses

When the country of Japan was closed from the rest of the world from 1603 until
1867 during the Edo period, the field of mathematics developed in a different way
from how it developed in the rest of the world. One way we see this development
is through the sangaku, the thousands of geometric problems hung in various Shinto and Buddhist temples throughout the country. Written on wooden tablets by people from numerous walks of life, all these problems hold true within Euclidean geometry. During the 1800s, while Japan was still closed, non-Euclidean geometries began to develop across the …

The Butterfly Effect Of Fractals, Cody Watkins

Honors College Theses

This thesis applies concepts in fractal geometry to the relatively new field of mathematics known as chaos theory, with emphasis on the underlying foundation of the field: the butterfly effect. We begin by reviewing concepts useful for an introduction to chaos theory by defining terms such as fractals, transformations, affine transformations, and contraction mappings, as well as proving and demonstrating the contraction mapping theorem. We also show that each fractal produced by the contraction mapping theorem is unique in its fractal dimension, another term we define. We then show and demonstrate iterated function systems and take a closer look at …

Designing Efficient Algorithms For Sensor Placement, Gabriel Loos

Honors College Theses

Sensor placement has many applications and uses that can be seen everywhere you go.These include, but not limited to, monitoring the structural health of buildings and bridgesand navigating Unmanned Aerial Vehicles(UAV).We study ways that leads to efficient algorithms that will place as few as possible sen-sors to cover an entire area. We will tackle the problem from both 2-dimensional and3-dimensional points of view. Two famous related problems are discussed: the art galleryproblem and the terrain guarding problem. From the top view an area presents a 2-D im-age which will enable us to partition polygonal shapes and use graph theoretical results …

Grim Under A Compensation Variant, Aaron Davis, Aaron Davis

Honors College Theses

Games on graphs are a well studied subset of combinatorial games. Balance and strategies for winning are often looked at in these games. One such combinatorial graph game is Grim. Many of the winning strategies of Grim are already known. We note that many of these winning strategies are only available to the first player. Hoping to develop a fairer Grim, we look at Grim played under a slighlty different rule set. We develop winning strategies and known outcomes for this altered Grim. Throughout, we discuss whether our altered Grim is a fairer game then the original.

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

Honors College 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 Theory And Particles, Elizabeth V. Hawkins

Honors College Theses

We begin by a brief overview of the notion of groups and Lie groups. We then explain what group representations are and give their main properties. Finally, we show how group representation form a natural framework to understand the Standard Model of physics.

Network Modeling Of Infectious Disease: Transmission, Control And Prevention, Christina M. Chandler

Honors College Theses

Many factors come into play when it comes to the transmission of infectious diseases. In disease control and prevention, it is inevitable to consider the general population and the relationships between individuals as a whole, which calls for advanced mathematical modeling approaches.

We will use the concept of network flow and the modified Ford-Fulkerson algorithm to demonstrate the transmission of infectious diseases over a given period of time. Through our model one can observe what possible measures should be taken or improved upon in the case of an epidemic. We identify key nodes and edges in the resulted network, which …

Drawing Numbers And Listening To Patterns, Loren Zo Haynes

Honors College Theses

The triangular numbers is a series of number that add the natural numbers. Parabolic shapes emerge when this series is placed on a lattice, or imposed with a limited number of columns that causes the sequence to continue on the next row when it has reached the kth column. We examine these patterns and construct proofs that explain their behavior. We build off of this to see what happens to the patterns when there is not a limited number of columns, and we formulate the graphs as musical patterns on a staff, using each column as a line or space …

Gallai-Ramsey And Vertex Proper Connection Numbers, Emily C. Chizmar

Honors College Theses

Given a complete graph G, we consider two separate scenarios. First, we consider the minimum number N such that every coloring of G using exactly k colors contains either a rainbow triangle or a monochromatic star on t vertices. This number is known for small cases and generalized for larger cases for a fixed k. Second, we introduce the vertex proper connection number of a graph and provide a relationship to the chromatic number of minimally connected subgraphs. Also a notion of total proper connection is introduced and a question is asked about a possible relationship between the total proper …

Explorations Of The Collatz Conjecture (Mod M), Glenn Micah Jackson Jr

Honors College Theses

The Collatz Conjecture is a deceptively difficult problem recently developed in mathematics. In full, the conjecture states: Begin with any positive integer and generate a sequence as follows: If a number is even, divide it by two. Else, multiply by three and add one. Repetition of this process will eventually reach the value 1. Proof or disproof of this seemingly simple conjecture have remained elusive. However, it is known that if the generated Collatz Sequence reaches a cycle other than 4, 2, 1, the conjecture is disproven. This fact has motivated our search for occurrences of 4, 2, 1, and …

Teacher Influence On Elementary School Students’ Participation In Science, Technology, Engineering, And Mathematics, Courtney Hartman

Honors College Theses

The purpose of this study is to explore the influence of elementary school teachers on encouraging students’ interest and participation in Science, Technology, Engineering, and Mathematics. The researcher sought to understand what methods teachers use in their classrooms to encourage students to participate in STEM subjects and programs. This mixed methods study consisted of a questionnaire to collect quantitative data, as well as an interview of selected teachers who participated in the questionnaire to collect qualitative data. The data was analyzed to determine the overall perceptions of teachers regarding the importance of encouraging students to participate in STEM. The qualitative …

A Quasi-Classical Logic For Classical Mathematics, Henry Nikogosyan

Honors College Theses

Classical mathematics is a form of mathematics that has a large range of application; however, its application has boundaries. In this paper, I show that Sperber and Wilson’s concept of relevance can demarcate classical mathematics’ range of applicability by demarcating classical logic’s range of applicability. Furthermore, I introduce how to systematize Sperber and Wilson’s concept of relevance into a quasi-classical logic that can explain classical logic’s and classical mathematics’ range of applicability.

Calculator Usage In Secondary Level Classrooms: The Ongoing Debate, Nicole Plummer

Honors College Theses

With technology becoming more prevalent every day, it is imperative that students gain enough experience with different technological tools in order to be successful in the “real-world”. This thesis will discuss the debate and overall support for an increased usage of calculators as tools in the secondary level classroom. When the idea of calculators in the classroom first came to life, many educators were very apprehensive and quite hesitant of this change. Unfortunately, more than 40 years later, there is still hesitation for their usage; and rightfully so. While there are plenty of advantages of calculator use in the classroom, …

Analysis Of Resource Allocation Through Game Theory, Brittney L. Benzio

Honors College Theses

Game theory presents a set of decision-makers in a model in order to simulate how they will interact according to a set of rules. The game is set up with a set of players, actions, and preferences. The model allows each player to, in some way, be affected by the actions of all players. Nash equilibrium illustrates that the best action for any given player depends on the other players’ actions, and so, each player must make some assumption about what the competition will do. The goal of this project is to model situations of different car companies to improve …