Mathematics Commons^™

An Introduction To Modern Conversations On Knot Invariants, Stella Shah

Scripps Senior Theses

Knot Theory is a vast and diverse subfield of modern mathematics involving the classification and abstraction of knots and links. In this thesis, we wish to provide the necessary background for and an explanation of two papers in different subfields of knot theory, The Forbidden Quiver of a Link, and Biquandle Fares and Link Invariants.

In Chapter I, we begin with an introduction to knot theory and knot invariants. We continue to present the example of Fox Colorings, and conclude the chapter with an example of the Fox Coloring Number Invariant.

In Chapter II, we explore the derivation and utilization …

Developing Collaboration And Community Through An Online Virtual Modality: Participatory Action Research, Jennifer Reed

Doctor of Education Dissertations

This action research study examined educators’ perceptions of collaboration and community within a virtual professional learning community and investigated how participation influenced collaborative actions over time. The study also compared the needs and goals of secondary and postsecondary educators participating in a shared VPLC model. Grounded in the Community of Inquiry and Online Collaborative Learning theoretical frameworks, this study addressed a growing need for effective, flexible professional learning structures that support collaboration across educational contexts. Data were collected from six educators, including three secondary and three postsecondary instructors, through pre- and post-administration of the Professional Learning Community Assessment–Revised, recorded virtual …

Uncertainty Quantification, Propagation & Conjunction Assessment In Orbital Mechanics Using Generalized Polynomial Chaos Expansion & 2-Dimensional Conjunction Plane Analysis Techniques, Monalisa Karim

Mechanical and Aerospace Engineering Theses

Uncertainties, that are inherent to dynamic models, can be associated with state initial conditions, force modelling errors, navigation and actuation errors. In system modelling stochastic differential equations are used to represent dynamic phenomena with uncertainties, for which the solutions are probability density functions of quantities of interest characterizing the realization of the stochastic processes. In Polynomial Chaos Expansion (PCE) propagation, these solutions are represented as weighted sums of multivariate spectral polynomials that are functions of the input random variables. Generalized polynomial chaos expansion (gPC) is an extension to the original homogenous PCE which projects the random solution onto a basis …

Adaptive Control For A Robotic Bipedal Device Using A Hybrid Discrete-Continuous Reinforcement Learning Strategy, Karla Rincon-Martinez, Wen Yu, Isaac Chairez

Mathematics Faculty Publications

This research develops and implements a novel reinforcement learning (RL) architecture to address the trajectory-tracking problem in bipedal robotic systems under articulated-joint constraints. The proposed RL framework extends previously designed adaptive controllers characterized by state-dependent gain structures. The learning mechanism comprises two hierarchical adaptation layers: the first employs an adaptive dynamic programming (ADP) formulation to approximate the Bellman value function using a class of continuous-time dynamic neural networks. In contrast, the second uses an iterative optimization scheme based on the deep deterministic policy gradient (DDPG) algorithm. The resulting control strategy minimizes a robust performance index defined over the tracking trajectories …

Delaying Cancer Progression By Integrating Toxicity Constraints In A Model Of Adaptive Therapy, Jana L. Gevertz, Harsh Vardhan Jain, Irina Kareva, Kathleen P. Wilkie, Joel Brown, Yitong Pepper Huang, Eduardo Sontag, Vladimir Vinogradov, Mark Davies

Mathematics Sciences: Faculty Publications

Cancer therapies often fail when intolerable toxicity or drug-resistant cancer cells undermine otherwise effective treatment strategies. Over the past decade, adaptive therapy has emerged as a promising approach to postpone emergence of resistance by altering dose timing based on tumor burden thresholds. Despite encouraging results, these protocols often overlook the crucial role of toxicity-induced treatment breaks, which may permit tumor regrowth. Herein, we explore the following question: would incorporating toxicity feedback improve or hinder the efficacy of adaptive therapy? To address this question, we propose a mathematical framework for incorporating toxic feedback into treatment design. We and that the degree …

Balanced Multi-Party Tournament Designs, Parsa Nematollahe

Honors College Theses

This paper introduces Multi-Party Tournament (MPT) designs that generalize established combinatorial structures, including Whist, Pitch, and Generalized Whist tournament designs. This work will formally define MPTs, establish the fundamental properties of resolvability, fullness, and balance, and formulate a mathematical and algorithmic foundation for multi-party tournament scheduling. The primary contributions of this research are the presentation of necessary and sufficient existence conditions for MPTs across various properties and parameters, the identification of connections between MPTs and other fields of mathematics such as combinatorial design theory, graph theory, and probability theory, and the investigation of MPT construction algorithms, including tree-search, finite-field constructions, …

Using Provided Guided Notes In Coordinated Introductory First-Year Mathematics Courses, Jennifer L. Huber

Mathematics Dissertations

The goal of this study is to investigate how standardized guided notes shape instructional practices and student engagement in coordinated introductory first-year college mathematics courses at a large public university. The researcher explored three multi-section introductory mathematics courses with overlapping learning objectives. Each course required students to purchase a student workbook as part of the instructional materials for the class. The instructors taught primarily from the workbook containing guided notes created by a former coordinator of the course. The researcher used a mixed-methods approach. Instructors and students participated in surveys, class observations and provided class meeting notes. Instructors shared additional …

Pursuer Evader Surveillance Game Control Theory And Motion Planning, Zijie Mu

Honors Theses

This thesis studies the pursuer evader surveillance game with a triangular obstacle in the short-term. In the game, the pursuer aims to maintain surveillance of the evader as long as possible while the evader aims to break surveillance in a finite time. We classify the player strategies into ideal ones and best admissible ones. The outcome of the game is determined by line of sight. We reduce the 4D game to a 3D game with an upward motion for a small time interval to terminate the game. When the evader starts outside the threshold, we show that there exists an …

From Shock To Routine: The Evolving Impact Of Shutdown-Related Sentiment On Stock Markets, Yiran Shao

Honors Theses

To address gaps in existing research, this paper selects two U.S. government shutdown periods, 2018-2019 and 2025, as research samples to explore the effect of policy uncertainty on sentiment. This paper primarily analyzes the following two research questions.

First, what is the correlation between government shutdown-related sentiment during the shutdown period and daily market fluctuations? Specifically, can the sentiment index constructed from shutdown-related news effectively predict the next-day stock return during the event period?

Second, does the market have a learning effect? That is, between 2018-2019 and 2025, has the relationship between shutdown-related emotions and market outcomes weakened, shortened the …

Investigating The Connection Between Als Through The Mutation R522s In The Rna Binding Protein, Dennia Estrella-Vargas, Lydia Uptain

Mathematics

Amyotrophic lateral sclerosis (ALS) is a fatal disease that causes the deterioration of motor neurons , death is usually due to respiratory paralysis. The variant R522S was chosen because it is near a hot spot of pathogenic variants. It is an arginine-to-serine swap, this swap is present in pathogenic variants near the 522 position, such as R514S, R521S, R524S. Recent evidence suggests that arginine-deficiency can influence disease progression.

Classifying Mathieu-Zhao Subspaces In Products Of Cyclic Rings, Sarah A. Huber

Theses and Dissertations

Mathieu-Zhao subspaces are a generalization of ideals in an algebra and were introduced by Wenhua Zhao in connection to the Jacobian conjecture and its variants. These subspaces have interesting properties, and often the problem of classification is hard. In this thesis, we investigate the structure of Mathieu-Zhao subspaces of the cartesian product of integers modulo powers of a prime p, Zpr × Zps . We will give a complete classification of the subgroups, maximal subgroups, Mathieu-Zhao subspaces, and maximal Mathieu-Zhao subspaces in these rings.

Fuchs' Problem For Quasi-Cyclic Groups, Dalen F. Elliott

Theses and Dissertations

Fuchs’ problem asks which groups can arise as the group of units of a ring. Although the finite cyclic case has been completely classified, much less is known in the infinite setting. This thesis contributes to this problem by investigating quasi-cyclic. (Pr¨ufer) groups and their finite direct products. We show that for every odd prime p, there is no commutative ring R such that R×∼= Cp∞. This obstruction arises from characteristic restrictions and the algebraic structure of finite fields. More generally, we prove that any group in which every element has order a power of an odd prime p and …

Symmetry In Latin Hypercubes, Levi Neiburger

Theses and Dissertations

Let [n] = {1, ..., n}. A hypercube H of order n and dimension d is a d-dimensional array whose nᵈ cells are indexed by [n]ᵈ. A hyperplane in H is obtained by fixing one coordinate, while allowing the remaining d–1 coordinates to vary. We wish to color each cell of H from a palette of n^d-1 colors such that each hyperplane is polychromatic.

Our main result is the following. Let n be sufficiently large. There exists a symmetric coloring of the d-dimensional hypercubes of order n whose all hyperplanes are polychromatic if and only if: …

On Stripping And Antipodes In Motivic Steenrod Algebras, Joshua A. Peterson

Theses and Dissertations--Mathematics

We generalize the stripping process to the (mod 2) $\mathbb{C}$- and $\mathbb{R}$-motivic settings. Throughout, we include discussion on how the process changes and the difficulties moving to more general settings. We also introduce antipodes and consider what a potential $\mathbb{R}$-motivic analogue may look like. Finally, we elaborate on how the results may be used in future work to generalize a nilpotence result of Walker and Wood.

Extensions Between Modules Defined By Lattice Paths In The Preprojective Algebra, Chloe Napier

Theses and Dissertations--Mathematics

In 2001, Fomin and Zelevinsky introduced cluster algebras which appear as coordinate rings of many varieties. We study cluster algebras coming from Richardson varieties. Leclerc gives a cluster structure on Richardson varieties using the representation theory of preprojective algebras. While this construction is very algebraic, we take a more combinatorial approach. The main goal is to find a combinatorial description for when certain cluster variables are compatible, or equivalently when modules defined by lattice paths in the preprojective algebra have trivial extensions. We extend the known results from Geiss, Leclerc, and Schröer that answer this question in the case of …

A Character Theory For Loop Representations Of Symmetric Monoidal Bicategories, Jordan Sawdy

Theses and Dissertations--Mathematics

Character theory arises in many distinct fields of mathematics, but its many instantiations often share a few key features: they arise in contexts where one object is acted on or parametrized by another, and they are often computed via "trace-like" formulas. Focusing on these properties, we present a categorical formalism for constructing such characters. We first define a notion of "loop representation" for symmetric monoidal bicategories, then build a character for such representations via the canonical symmetric monoidal trace. We then show that this character defines a symmetric monoidal functor which satisfies commutativity properties with respect to both restriction- and …

Combinatorial Models For Nonnegativity In Flag Varieties, Williem L. Rizer

Theses and Dissertations--Mathematics

The nonnegative Grassmannian admits a widely studied cell decomposition due to Alexander Postnikov, whose cells are indexed by positroids and modeled by several equivalent combinatorial objects. Subsequent work by authors including Lauren Williams, Suho Oh, and Carolina Benedetti has further developed the combinatorics and geometry of these structures. In this dissertation, we extend some of Postnikov’s combinatorial framework to the nonnegative flag variety. While cell decompositions in this setting were previously obtained, notably in work of Konstanze Rietsch, our focus is on providing new combinatorial models that make this structure more explicit and computationally tractable. We introduce flag positroid pipe …

The Interaction Between Additive And Multiplicative Structures In Arithmetic Settings, Ali Alsetri

Theses and Dissertations--Mathematics

The first part of this thesis is concerned with Goldbach-type problems. In recent years, there has been an interest in developing density versions of Goldbach-type results. Namely, given a relatively dense subset A of the primes, one may study representations of integers as sums of primes belonging to the subset A. These density Goldbach-type results have been facilitated by the development of new tools from additive combinatorics, in particular the Fourier-analytic transference principle due to Green. We apply the transference principle to obtain a variant of Vinogradov’s theorem involving subsets of primes confined to the residue class 1 (mod 3). …

Local Energy Decay For Non-Stationary Damped Wave Operators, Nicholas Dj Arsenault

Theses and Dissertations--Mathematics

This work establishes integrated local energy decay (ILED) estimates for the damped wave equation on certain non-stationary spacetimes. The main technical result is a high frequency estimate that holds in great generality, provided that null geodesics trapped in a compact region are sufficiently damped. This is combined with low- and medium-frequency estimates to establish full local energy decay. We conclude by providing a counterexample where the damping assumption fails and local energy decay does not hold.

Understanding Möbius Inversion As Composite Of Dual Pairs, Isaac B. Kochmaan

Theses and Dissertations--Mathematics

Möbius inversion is a well-studied computational tool in number theory and combinatorics, but it exhibits a pattern which may be familiar to those who study categorical traces. In 2016, Kate Ponto and Michael Shulman characterized duality and traces in the bicategory of profunctors for a particularly nice class of one-cells. We extend these results. Consequently, we construct, for a given poset, a profunctor whose Euler characteristic is the Möbius function of said poset. In light of this, we propose a categorical notion of Möbius inversion.