Mathematics Honors Theses
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Pebbling And Cover Pebbling Numbers Of GraphsCover pebbling is a method in graph theory that was first brought about by Largias and Saks. This topic in graph theory was used to come up with a way to calculate how much of a consumable resource would being needed to begin transportation of said consumable resource. This paper will tackle the basics on cover pebbling, brush upon weight pebbling, and problem 9 of the open problems that can be found in Betsy Crull's paper The cover pebbling number of graphs. Problem 9: What are the cover pebbling number for other graphs G, for example cubes, complete rpartite graphs, etc.

Confidence Levels of SSU Education Students in Writing AssessmentsSalem State University has a newly accredited 4+1 master’s in education program but has been teaching education for decades. Given the newness of the 4+1 program, are the students getting a good idea of what assessments to write, what they look like, and when to do which type of assessment? This study investigates the confidence levels of SSU Education students in writing assessments based on their experiences and classes that they have taken in the School of Education. In order to examine this question, a survey was completed by students in the education program in which they specified their knowledge on each type of assessment (summative and formative), which classes they have taken, what experiences they have had, and how comfortable they are writing both. It is then analyzed by the classes they have taken the program, their comfortability, and knowledge of each assessment.

Exploring The Thirteen Colorful Variations Of Guthrie's FourColor ConjectureColoring is an important part of graph theory. Historically, it was thought that only four colors could be the minimal number of colors. This paper discusses the Four Color Theorem and how the Four Color Theorem is applied to graphs. This paper gives an overview of several different definitions involved with graphs and shows how to create a dual graph. This paper also discusses how a graph of 12 regions has at least one region bounded by less than five edges. The paper includes several figures which include graphs, dual graphs, and different colorings. The paper also provides a proof which shows mathematically why a graph of 12 regions has at least one region bounded by less than five edges.

Ranking College Football Teams Independent of Victory MarginsThis paper addresses David Mease's formula for ranking college football teams. It is just one of the numerous formulas that can be used by the Bowl Championship Series in ranking the top teams in the country. Mease uses his formula to rank teams independent of victory margins, something not all formulas take into consideration, Winning margin may help teams gain higher ranking in some formulas, so this formula ignores that statistic.

NQueens ProblemUsing combinatorics in this paper, we will discuss three different methods in solving the nqueens problem. We will find the maximum and minimum number of queens we can place on an n x n chessboard. Also, we will use latin squares, latin rectangles and circulant matrices as another method of placing the queens on a chessboard.

Mathematical Understandings Of A Rubik's CubeMany people are familiar with the 3x3x3 Rubik’s Cube as a puzzle or a toy. But, what most people do not realize is that the cube is a great physical visual of a group. The goal of this paper is to discuss the Rubik’s Cube as a group and dive into a specific subgroup of the cube. Through this discussion, we will also explore homomorphisms in the slice group. This paper will also give insight on permutations, commutators, and conjugates in terms of the cube, as well as “God’s number”.

Using Matrices And Hungarian Method To Solve The Traveling Salesman ProblemIn this paper, we introduce the Traveling Salesman Problem (TSP) and solve for the most efficient route of the problem using the steps of the Hungarian method. Specifically, this paper discusses the properties of a TSP matrix, provides the steps for the Hungarian method, and presents examples that apply these concepts to a Traveling Salesman Problem. We do not consider any constraints on the order in which the localities are visited, nor do we take into account possible traffic at differing times. We use examples to show how the Hungarian method is used and why it is an efficient way to solve the Traveling Salesman Problem.

Investigation Of Plane Symmetry In Lattice DesignsThe purpose of this research project is to analyze the scholarly article The Plane Symmetry Groups: Their Recognition and Notation by Doris Schattschneider. In this article, Schattschneider discusses an application of abstract algebra which is useful in art as well as crystallography: frieze groups and wallpaper groups. I was interested in pursuing this topic because it combines mathematics with its applications, particularly with my own interest in chemistry. The article provides a compiled resource of terminology and rules of these groups, but not one which was easily accessible to undergraduate students. In my research, I elaborated on the descriptions of certain types of periodic patterns to add to the accessibility, and analyzed a few designs to prove their classification based on the rules from Schattschneider's article. I found that this resource provided a good source of rules for which mathematical proofs could be based, and proved the classification of two different periodic plane designs.

A Combinatorial Method To Producing PortfoliosThere is a lack of research on portfolios and combinatorial methods in finance. In this paper, we outline a new method for producing longterm portfolios of stock using a combinatorial approach. A retrospective data analysis shows that this method produces protable longterm portfolios.

Topspin: OvalTrack Puzzle, Taking Apart The Topspin One Tile At A TimeA Topspin “Oval Track” puzzle consists of 20 numbered tiles in an ovalshaped track and a flipping window that reverses the 4 tiles in the window. The solvability of the puzzle uses permutations which are combinations where the order matters. A puzzle is considered solvable if each permutation in can be mapped to a spot in the original position through the three different moves the puzzle can make; a left shift, a right shift, and the flip which reverses the order of the 4 tiles in the window. I wanted to find out what math was involved in solving this puzzle. I had certain topics that I wanted to find out more information about, but the major question I had was “what is the fewest number of moves it takes to solve a puzzle”. Other topics I had were what made a puzzle unsolvable and what other types of puzzles use the same kind of math to solve them. To construct this research, I had read different scholarly articles that talked about the Topspin as well as physically looked at the puzzle and see how it works. I had found that there was no way to determine the fewest number of moves it takes to solve a puzzle since it’s impossible to decide what a “more scrambled” puzzle is compared to another. One scrambled puzzle might look completely different from another, and still have the same number of moves to solve it. In addition to this, I was also able to find that the Rubik’s Cube is solved like the Topspin.