Mathematics Masters Papers
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Item New Perspectives on Promotion and Rowmotion: Generalizations and Translations(North Dakota State University, 2022) Bernstein, JosephWe define P-strict labelings for a finite poset P as a generalization of semistandard Young tableaux and show that promotion on these objects is in equivariant bijection with a toggle action on B-bounded Q-partitions of an associated poset Q. In many nice cases, this toggle action is conjugate to rowmotion. We apply this result to flagged tableaux, Gelfand-Tsetlin patterns, and symplectic tableaux, obtaining new cyclic sieving and homomesy conjectures. We then study cases in which P is a finite, graded poset other than a chain, yielding new results for products of chains and new perspectives on known conjectures. Additionally, we give resonance results for promotion on P-strict labelings and rowmotion on Q-partitions and demonstrate that P-strict promotion can be equivalently defined using Bender-Knuth and jeu-de-taquin perspectives. Finally, we explore conjectures, related and unrelated to our main theorems, on objects that promise beautiful dynamical properties.Item Enumeration of Reduced Words of Length N for Coxeter Groups via BrinkHowlett Automaton(North Dakota State University, 2022) Allen, Brandon JamesThe overall goal of this paper is to give a method of computing out how many words of length n there are for any Coxeter group via its Brink-Howlett automaton. [6] [7] To build our automaton, we focus on Coxeter systems and root systems honing in on a special set of roots called the small roots. We follow closely [1] [5] for the first two chapters. Finally, we build the Brink-Howlett automaton through literature compiled through the years and present explicit examples of A˜1 and the Coxeter group on three generators which each pair of generators is in a free relation with one another.Item Stochastic Processes, and Development of the Barndorff-Nielsen and Shephard Model for Financial Markets(North Dakota State University, 2022) Uden, AustinIn this paper, we introduce Brownian motion, and some of its drawbacks in connection to the financial modeling. We then introduce geometric Brownian motion as the basis for European call option pricing as we navigate our way through the Black-Scholes-Merton equation. Lévy Processes round out the background information of the paper as we discuss Poisson and compound Poisson processes and the pricing of European call options using the stochastic calculus of jump processes. Ornstein-Uhlenbeck processes are then constructed. Finally we review and analyze the Barndorff-Nielsen and Shepard model. We provide its application to price European call options using the fast Fourier transform and the direct integration method.Item Hook Formula For Skew Shapes(North Dakota State University, 2019) Jensen, Megan LisaThe number of standard Young tableaux is given by the hook-length formula of Frame, Robinson, and Thrall. Recently, Naruse found a hook-length formula for the number of skew shaped standard Young tableaux. In a series of papers, Morales, Pak, and Panova prove the Naruse hook-length formula as well as q-analogues of Naruse's formula. In this paper, we will discuss their work, including connections between excited diagrams and Dyck paths.Item Understanding Students’ Perceptions of Difficulty and the Effect Difficulty Has on Mathematical Anxiety(North Dakota State University, 2018) Larson, Caleb BridgerThere exists a growing atmosphere surrounding mathematics that allows individuals to exclaim their belief they are deficient in math without any societal judgement. Compared to a state like being illiterate, we have reached a stage where it is acceptable to be math illiterate as well as hate math. To discover why so many people have this strong distaste towards math, we look towards the difficulty level of the subject. Students cite difficulty as one of the main reasons that they dislike math, so to fully understand the issue at hand, we must first understand students’ perceptions of difficulty in mathematics. To this end, we use existing research to develop a survey targeting common issues in algebra that asks the students to complete the problems and describe to us what they think may be challenging about that problem. We then compare and contrast students’ reactions to our own hypotheses.Item Colorings of Zero-Divisor Graphs of Commutative Rings(North Dakota State University, 2015) Ramos, Rebecca ElizabethWe will focus on Beck’s conjecture that the chromatic number of a zero-divisor graph of a ring R is equal to the clique number of the ring R. We begin by calculating the chromatic number of the zero-divisor graphs for some finite rings and characterizing rings whose zero-divisor graphs have finite chromatic number, known as colorings. We will discuss some properties of colorings and elements called separating elements, which will allow us to determine that Beck’s conjecture holds for rings that are principal ideal rings and rings that are reduced. Then we will characterize the finite rings whose zero-divisor graphs have chromatic number less than or equal to four. In the general case, we will discuss a local ring that serves as a counterexample to Beck’s conjecture.Item Resolutions and Semidualizing Modules(North Dakota State University, 2014) Feickert, Aaron JamesProjective and injective modules are of key importance in algebra, in part because of their useful homological properties. The notion of C-projective and C-injective modules generalizes these constructions. In particular, these modules may be used to construct resolutions and define related homological dimensions in a natural way. When C is a semidualizing module, the C-projective and C-injective modules have particularly useful homological properties. Further, one may combine projective and C-projective resolutions to construct complete PC-resolutions (and, dually, complete IC-resolutions) that yield other modules with nice homological properties. This paper surveys some of the literature on these constructions.