Physical Sciences and Mathematics Commons^™

On Loop Commutators, Quaternionic Automorphic Loops, And Related Topics, Mariah Kathleen Barnes

Electronic Theses and Dissertations

This dissertation deals with three topics inside loop and quasigroup theory. First, as a continuation of the project started by David Stanovský and Petr Vojtĕchovský, we study the commutator of congruences defined by Freese and McKenzie in order to create a more pleasing, equivalent definition of the commutator inside of loops. Moreover, we show that the commutator can be characterized by the generators of the inner mapping group of the loop. We then translate these results to characterize the commutator of two normal subloops of any loop.

Second, we study automorphic loops with the desire to find more examples of …

Local-Global Results On Discrete Structures, Alexander Lewis Stevens

Electronic Theses and Dissertations

Local-global arguments, or those which glean global insights from local information, are central ideas in many areas of mathematics and computer science. For instance, in computer science a greedy algorithm makes locally optimal choices that are guaranteed to be consistent with a globally optimal solution. On the mathematical end, global information on Riemannian manifolds is often implied by (local) curvature lower bounds. Discrete notions of graph curvature have recently emerged, allowing ideas pioneered in Riemannian geometry to be extended to the discrete setting. Bakry- Émery curvature has been one such successful notion of curvature. In this thesis we use combinatorial …

Topics In Moufang Loops, Riley Britten

Electronic Theses and Dissertations

We will begin by discussing power graphs of Moufang loops. We are able to show that as in groups the directed power graph of a Moufang loop is uniquely determined by the undirected power graph. In the process of proving this result we define the generalized octonion loops, a variety of Moufang loops which behave analogously to the generalized quaternion groups. We proceed to investigate para-F quasigroups, a variety of quasigroups which we show are antilinear over Moufang loops. We briefly depart from the context of Moufang loops to discuss solvability in general loops. We then prove some results on …

Banach Spaces On Topological Ramsey Structures, Cheng-Chih Ko

Electronic Theses and Dissertations

A Banach space T₁(d, θ) with a Tsirelson-type norm is constructed on the top of the topological Ramsey space T₁ defined by Dobrinen and Todorcevic [6]. Finite approximations of the isomorphic subtrees are utilised in constructing the norm. The subspace on each “branch” of the tree is shown to resemble the structure of an ℓ_∞ⁿ⁺¹ -space where the dimension corresponds to the number of terminal nodes on that branch. The Banach space T₁(d, θ) is isomorphic to (∑_n∊ℕ⊕ℓ_∞ⁿ⁺¹)_p , where d ∈ ℕ with d ≥ 2, …

Measure-Theoretically Mixing Subshifts With Low Complexity, Darren Creutz, Ronnie Pavlov, Shaun Rodock

Mathematics: Faculty Scholarship

We introduce a class of rank-one transformations, which we call extremely elevated staircase transformations. We prove that they are measure-theoretically mixing and, for any f : N → N with f (n)/n increasing and ∑ 1/f (n) < ∞, that there exists an extremely elevated staircase with word complexity p(n) = o(f (n)). This improves the previously lowest known complexity for mixing subshifts, resolving a conjecture of Ferenczi.

Local Finiteness And Automorphism Groups Of Low Complexity Subshifts, Ronnie Pavlov, Scott Schmieding

Mathematics: Faculty Scholarship

We prove that for any transitive subshift X with word complexity function c_n(X), if lim inf(log(c_n(X)/n)/(log log log n)) = 0, then the quotient group Aut(X, σ)/〈 σ〉 of the automorphism group of X by the subgroup generated by the shift σ is locally finite. We prove that significantly weaker upper bounds on c_n(X) imply the same conclusion if the gap conjecture from geometric group theory is true. Our proofs rely on a general upper bound for the number of automorphisms of X of range n in terms of word complexity, which may be …

Subsystems Of Transitive Subshifts With Linear Complexity, Andrew Dykstra, Nicholas Ormes, Ronnie Pavlov

Mathematics: Faculty Scholarship

We bound the number of distinct minimal subsystems of a given transitive subshift of linear complexity, continuing work of Ormes and Pavlov [On the complexity function for sequences which are not uniformly recurrent. Dynamical Systems and Random Processes (Contemporary Mathematics, 736). American Mathematical Society, Providence, RI, 2019, pp. 125--137]. We also bound the number of generic measures such a subshift can support based on its complexity function. Our measure-theoretic bounds generalize those of Boshernitzan [A unique ergodicity of minimal symbolic flows with linear block growth. J. Anal. Math.44(1) (1984), 77–96] and are closely related to those of Cyr and Kra …

On Subshifts With Slow Forbidden Word Growth, Ronnie Pavlov

Mathematics: Faculty Scholarship

In this work, we treat subshifts, defined in terms of an alphabet A and (usually infinite) forbidden list F, where the number of n-letter words in F has ‘slow growth rate’ in n. We show that such subshifts are well behaved in several ways; for instance, they are boundedly supermultiplicative in the sense of Baker and Ghenciu [Dynamical properties of S-gap shifts and other shift spaces. J. Math. Anal. Appl.430(2) (2015), 633–647] and they have unique measures of maximal entropy with the K-property and which satisfy Gibbs bounds on large (measure-theoretically) sets. The main tool in our proofs is a …

Zn Orbifolds Of Vertex Operator Algebras, Daniel Graybill

Electronic Theses and Dissertations

Given a vertex algebra V and a group of automorphisms of V, the invariant subalgebra V^G is called an orbifold of V. This construction appeared first in physics and was also fundamental to the construction of the Moonshine module in the work of Borcherds. It is expected that nice properties of V such as C2-cofiniteness and rationality will be inherited by V^G if G is a finite group. It is also expected that under reasonable hypotheses, if V is strongly finitely generated and G is reductive, V^G will also be strongly finitely generated. This is an analogue …

Exponential Random Graphs And A Generalization Of Parking Functions, Ryan Demuse

Electronic Theses and Dissertations

Random graphs are a powerful tool in the analysis of modern networks. Exponential random graph models provide a framework that allows one to encode desirable subgraph features directly into the probability measure. Using the theory of graph limits pioneered by Borgs et. al. as a foundation, we build upon the work of Chatterjee & Diaconis and Radin & Yin. We add complexity to the previously studied models by considering exponential random graph models with edge-weights coming from a generic distribution satisfying mild assumptions. In particular, we show that a large family of two-parameter, edge-weighted exponential random graphs display a phase …

Discrepancy Inequalities In Graphs And Their Applications, Adam Purcilly

Electronic Theses and Dissertations

Spectral graph theory, which is the use of eigenvalues of matrices associated with graphs, is a modern technique that has expanded our understanding of graphs and their structure. A particularly useful tool in spectral graph theory is the Expander Mixing Lemma, also known as the discrepancy inequality, which bounds the edge distribution between two sets based on the spectral gap. More specifically, it states that a small spectral gap of a graph implies that the edge distribution is close to random. This dissertation uses this tool to study two problems in extremal graph theory, then produces similar discrepancy inequalities based …

Barrier Graphs And Extremal Questions On Line, Ray, Segment, And Hyperplane Sensor Networks, Kirk Anthony Boyer

Electronic Theses and Dissertations

A sensor network is typically modeled as a collection of spatially distributed objects with the same shape, generally for the purpose of surveilling or protecting areas and locations. In this dissertation we address several questions relating to sensors with linear shapes: line, line segment, and rays in the plane, and hyperplanes in higher dimensions.

First we explore ray sensor networks in the plane, whose resilience is the number of sensors that must be crossed by an agent traveling between two known locations. The coverage of such a network is described by a particular tripartite graph, the barrier graph of the …

Applications Of Geometric And Spectral Methods In Graph Theory, Lauren Morey Nelsen

Electronic Theses and Dissertations

Networks, or graphs, are useful for studying many things in today’s world. Graphs can be used to represent connections on social media, transportation networks, or even the internet. Because of this, it’s helpful to study graphs and learn what we can say about the structure of a given graph or what properties it might have. This dissertation focuses on the use of the probabilistic method and spectral graph theory to understand the geometric structure of graphs and find structures in graphs. We will also discuss graph curvature and how curvature lower bounds can be used to give us information about …

Decidability For Residuated Lattices And Substructural Logics, Gavin St. John

Electronic Theses and Dissertations

We present a number of results related to the decidability and undecidability of various varieties of residuated lattices and their corresponding substructural logics. The context of this analysis is the extension of residuated lattices by various simple equations, dually, the extension of substructural logics by simple structural rules, with the aim of classifying simple equations by the decidability properties shared by their extensions. We also prove a number of relationships among simple extensions by showing the equational theory of their idempotent semiring reducts coincides with simple extensions of idempotent semirings. On the decidability front, we develop both semantical and syntactical …

Surface Entropy Of Shifts Of Finite Type, Dennis Pace

Electronic Theses and Dissertations

Let χ be the class of 1-D and 2-D subshifts. This thesis defines a new function, H_S : χ x R → [0,∞] which we call the surface entropy of a shift. This definition is inspired by the topological entropy of a subshift and we compare and contrast several structural properties of surface entropy to entropy. We demonstrate that much like entropy, the finiteness of surface entropy is a conjugacy invariant and is a tool in the classification of subshifts. We develop a tiling algorithm related to continued fractions which allows us to prove a continuity result about surface …

T-De Vries Algebra, Nawal Alznad

Electronic Theses and Dissertations

The main point of this dissertation is to introduce the action on de Vries algebra by a topological monoid and we denoted the resulting category by dVT. In order to reach our goal, we started with introducing new proofs for some well known results in the category of flows. Then, we studied the Generalized Smirnov's Theorem for flows. After we studied the new category (dVT), we were able to provide a new way to construct the Čech-stone flow compactification of a given flow. Finally, we developed the co-free T-de Vries algebra for a special case.

Categories Of Residuated Lattices, Daniel Wesley Fussner

Electronic Theses and Dissertations

We present dual variants of two algebraic constructions of certain classes of residuated lattices: The Galatos-Raftery construction of Sugihara monoids and their bounded expansions, and the Aguzzoli-Flaminio-Ugolini quadruples construction of srDL-algebras. Our dual presentation of these constructions is facilitated by both new algebraic results, and new duality-theoretic tools. On the algebraic front, we provide a complete description of implications among nontrivial distribution properties in the context of lattice-ordered structures equipped with a residuated binary operation. We also offer some new results about forbidden configurations in lattices endowed with an order-reversing involution. On the duality-theoretic front, we present new results on …

Quantum Metrics On Approximately Finite-Dimensional Algebras, Konrad Aguilar

Electronic Theses and Dissertations

Our dissertation focuses on bringing approximately finite-dimensional (AF) algebras into the realm of noncommutative metric geometry. We construct quantum metric structures on unital AF algebras equipped with a faithful tracial state, and prove that for such metrics, AF algebras are limits of their defining inductive sequences of finite dimensional C*-algebras for the quantum Gromov-Hausdorff propinquity. We then study the geometry, for the quantum propinquity, of three natural classes of AF algebras equipped with our quantum metrics: the UHF algebras, the Effros-Shen AF algebras associated with continued fraction expansions of irrationals, and the Cantor space, on which our construction recovers traditional …

Z2-Orbifolds Of Affine Vertex Algebras And W-Algebras, Masoumah Abdullah Al-Ali

Electronic Theses and Dissertations

Vertex algebras arose in conformal field theory and were first defined axiomatically by Borcherds in his famous proof of the Moonshine Conjecture in 1986. The orbifold construction is a standard way to construct new vertex algebras from old ones. Starting with a vertex algebra V and a group G of automorphisms, one considers the invariant subalgebra V^G (called G-orbifold of V), and its extensions. For example, the Moonshine vertex algebra arises as an extension of the Z₂-orbifold of the lattice vertex algebra associated to the Leech lattice.

In this thesis we consider two problems. First, …

Banach Spaces From Barriers In High Dimensional Ellentuck Spaces, Gabriel Girón-Garnica

Electronic Theses and Dissertations

We construct new Banach spaces using barriers in high dimensional Ellentuck spaces following the classical framework under which a Tsirelson type norm is defined from a barrier in Ellentuck space. It is shown that these spaces contain arbitrary large copies of lⁿ_infinity and specific block subspaces isomorphic to l_p. We also prove that they are l_p-saturated and not isomorphic to each other. Finally, a study of alternative norms for our spaces is presented.

Follower And Extender Sets In Symbolic Dynamics, Thomas Kelly French

Electronic Theses and Dissertations

Given a word w in the language of a one-dimensional shift space X, the follower set of w, denoted F_X(w), is the set of all right-infinite sequences which follow w in some point of X. Extender sets are a generalization of follower sets and are defined similarly. To a given shift space X, then, we may associate a follower set sequence {|F_X(n)|} which records the number of distinct follower sets in X corresponding to words of length n. Similarly, we may define an extender set sequence {|E …

The Finite Embeddability Property For Some Noncommutative Knotted Varieties Of Rl And Drl, Riquelmi Salvador Cardona Fuentes

Electronic Theses and Dissertations

Residuated lattices, although originally considered in the realm of algebra providing a general setting for studying ideals in ring theory, were later shown to form algebraic models for substructural logics. The latter are non-classical logics that include intuitionistic, relevance, many-valued, and linear logic, among others. Most of the important examples of substructural logics are obtained by adding structural rules to the basic logical calculus FL. We denote by 𝖱𝖫𝑛 � the varieties of knotted residuated lattices. Examples of these knotted rules include integrality and contraction. The extension of �� by the rules corresponding to these two equations is …

Cayley-Dickson Loops, Jenya Kirshtein

Electronic Theses and Dissertations

In this dissertation we study the Cayley-Dickson loops, multiplicative structures arising from the standard Cayley-Dickson doubling process. More precisely, the Cayley-Dickson loop Q_n is the multiplicative closure of basic elements of the algebra constructed by n applications of the doubling process (the first few examples of such algebras are real numbers, complex numbers, quaternions, octonions, sedenions). Starting at the octonions, Cayley-Dickson algebras and loops become nonassociative, which presents a significant challenge in their study.

We begin by describing basic properties of the Cayley–Dickson loops Q_n. We establish or recall elementary facts about Q_n, e.g., inverses, …

Approximate Transversals Of Latin Squares, Jon Kyle Pula

Electronic Theses and Dissertations

A latin square of order n is an n by n array whose entries are drawn from an n-set of symbols such that each symbol appears precisely once in each row and column. A transversal of a latin square is a subset of cells that meets each row, column, and symbol precisely once.

There are many open and difficult questions about the existence and prevalence of transversals. We undertake a systematic study of collections of cells that exhibit regularity properties similar to those of transversals and prove numerous theorems about their existence and structure. We hope that our results …