Advisor: Brian Harbourne

]]>This dissertation focuses on codes based on parity check matrices that are dyadic, *n*-adic, or quasi-dyadic (QD), meaning the parity check matrix representation is block structured with dyadic matrices as blocks. Depending on the number of nonzero positions in the leading row of each block, these codes may be either low density or moderate density. Since each block is reproducible, the resulting QD codes have similar advantages to quasi-cyclic (QC) codes. We examine basic code properties of dyadic, *n*-adic, and QD parity check codes, including bounds on the dimension and minimum distance, cycle structure of the corresponding Tanner graph, and their possible use in quantum code constructions. We also consider the relationship between cycle codes of graphs and cycle codes of their lifts.

Advisor: Christine A. Kelley

]]>Advisors: Richard Rebarber and Brigitte Tenhumberg

]]>This dissertation addresses this gap by presenting a qualitative case study of the LA role in active learning precalculus classrooms at the University of Nebraska Lincoln. Participants included 9 LAs, 18 GSIs, 411 students, and one LA Coordinator. This study aimed to understand the LA role by examining how participants perceived LA-instructor and LA-student interactions. Data included interviews, classroom observations, observations of instructor-LA meetings, and open-ended survey responses.

The findings of this study are presented in three chapters: Chapter 4 describes how instructional rights and duties were distributed through five positions that defined LA-instructor interactions. Chapter 5 describes and compares how instructors and LAs perceived LA-student interactions, focusing on two subcases to highlight common themes from the data. Chapter 6 describes how students viewed LA-student interactions. These findings suggest that the LA role manifests through multiple positions, that those involved in instruction may have conflicting perceptions of the LA, and that although LAs are at times thought of as distinct from the instructor, they are often considered to occupy the same role, particularly by students. The conclusions from this research can help inform professional development for learning assistants, as well as extend our field’s understanding of factors that influence the manifestation of the LA role.

Advisers: Yvonne Lai & Wendy Smith

]]>Adviser: Brian Harbourne

]]>Given $k\in\mathbb{N}$ and $\epsilon\in(0,\infty)$, we define a $(k,\epsilon)$-secluded unit cube partition of $\mathbb{R}^{d}$ to be a unit cube partition of $\mathbb{R}^{d}$ such that for every point $\vec{p}\in\R^d$, the closed $\ell_{\infty}$ $\epsilon$-ball around $\vec{p}$ intersects at most $k$ cubes. The problem is to construct such partitions for each dimension $d$ with the primary goal of minimizing $k$ and the secondary goal of maximizing $\epsilon$.

We prove that for every dimension $d\in\mathbb{N}$, there is an explicit and efficiently computable $(k,\epsilon)$-secluded axis-aligned unit cube partition of $\mathbb{R}^d$ with $k=d+1$ and $\epsilon=\frac{1}{2d}$. We complement this construction by proving that for axis-aligned unit cube partitions, the value of $k=d+1$ is the minimum possible, and when $k$ is minimized at $k=d+1$, the value $\epsilon=\frac{1}{2d}$ is the maximum possible. This demonstrates that our constructions are the best possible.

We also consider the much broader class of partitions in which every member has at most unit volume and show that $k=d+1$ is still the minimum possible. We also show that for any reasonable $k$ (i.e. $k\leq 2^{d}$), it must be that $\epsilon\leq\frac{\log_{4}(k)}{d}$. This demonstrates that when $k$ is minimized at $k=d+1$, our unit cube constructions are optimal to within a logarithmic factor even for this broad class of partitions. In fact, they are even optimal in $\epsilon$ up to a logarithmic factor when $k$ is allowed to be polynomial in $d$.

We extend the techniques used above to introduce and prove a variant of the KKM lemma, the Lebesgue covering theorem, and Sperner's lemma on the cube which says that for every $\epsilon\in(0,\frac12]$, and every proper coloring of $[0,1]^{d}$, there is a translate of the $\ell_{\infty}$ $\epsilon$-ball which contains points of least $(1+\frac23\epsilon)^{d}$ different colors.

Advisers: N. V. Vinodchandran & Jamie Radcliffe

]]>Additionally, we use a condition on determinants of knots one crossing change away from unknotting number one knots to improve KnotInfo’s unknotting number data on 11 and 12 crossing knots. Lickorish introduced an obstruction to unknotting number one, which proves the same result. However, we show that Lickorish’s obstruction does not subsume the obstruction coming from the condition on determinants.

Adviser: Professor Mark Brittenham and Professor Alex Zupan

]]>Groups with the falsification by fellow traveler property are known to have solvable word problem, but they are not known to be automatic or to have finite convergent rewriting systems. We show that groups with this geometric property are geodesically autostackable. As a key part of proving this, we show that a wider class of groups, namely groups with a weight non-increasing synchronously regular convergent prefix-rewriting system, have a bounded regular convergent prefix-rewriting system.

Our second approach to creating prefix-rewriting systems is a more general approach. We design a procedure that, when provided with a finitely presented group G = < A | R > and an ordering < on A*, searches for a bounded convergent prefix-rewriting system. We also create a class of orderings for which each step of this procedure can be practically computed, and which guarantees that any bounded convergent prefix-rewriting system is an autostackable structure.

Adviser: Susan Hermiller

]]>Adviser: Susan Hermiller and Mark Brittenham

]]>The present study is a qualitative phenomenological multiple case study in which the decision making of six mathematics Graduate Teaching Assistants (GTAs) was analyzed through the lens of support. This study consisted of both interviews and classroom observations and aimed to understand the goals and beliefs held by the GTAs in order to explain the decisions they made to offer support in particular ways. The findings of this study are presented in three chapters: Chapter 4 discusses two GTAs whose goals and beliefs provide clear insight into the types of support they offer their students. Chapter 5 discusses two GTAs whose decision making is more nuanced and examines their goals and beliefs through existing frameworks in the literature. Chapter 6 discusses two more experienced GTAs in order to better understand how the constructs explored throughout this dissertation can change over time. The findings of this dissertation suggest that GTAs possess a wide variety of goals and beliefs that impact their decision making in complex ways and that GTAs’ goals, beliefs, and decision-making practices evolve over time. The conclusions from this research can help inform individual instructors’ reflections on their teaching as well as professional development efforts of novice mathematics instructors.

Advisors: Wendy Smith and Nathan Wakefield

]]>Adviser: Jamie Radcliffe

]]>Advisor: Alex Zupan and Mark Brittenham

]]>In this thesis, we investigate other counterexamples of the Generalized Total Rank Conjecture and some of their properties. Under the BGG correspondence, a finite free graded complex over the exterior algebra with small homology corresponds to a free complex over the polynomial ring with a small total Betti number. Therefore, we focus on examples of finite free complexes over the exterior algebra with small homology. The main examples we consider are Koszul complexes of quadrics, and we show the Koszul complex of one general quadric and the Koszul complex of two general quadrics have the smallest possible homology among complexes over the exterior algebra with the same graded Poincaré series. Finally while analyzing these Koszul complexes, we notice the dimension of their total homology has a nice asymptotic behavior and investigate under what conditions other complexes have this same asymptotic behavior.

Adviser: Alexandra Seceleanu and Mark E. Walker

]]>Initially, each vertex of the graph is set active with probability *p* or inactive otherwise. Then, at each time step, every inactive vertex with at least *k* active neighbors becomes active. Active vertices will always remain active. The process ends when it reaches a stationary state. If all the vertices eventually become active, then we say we achieve percolation.

This process has been widely studied on many families of graphs, deterministic and random. We analyze the Bootstrap Percolation process on a Random Geometric Graph.

A Random Geometric Graph is obtained by choosing n vertices uniformly at random from the unit *d*-dimensional cube or torus, and joining any two vertices by an edge if they are within a certain distance, *r*, of each other.

We obtain very precise results that characterize the final state of the Bootstrap Percolation process in terms of the parameters *p* and *r *with high probability as the number *n* of vertices tends to infinity.

Adviser: Xavier Pérez Giménez

]]>Advisers: Kyungyong Lee and Jamie Radclie

]]>Adviser: Tom Marley

]]>Adviser: Mark Brittenham and David Pitts

]]>Adviser: Allan Donsig

]]>Adviser: Richard Rebarber and Brigitte Tenhumberg

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