Algebra Commons^™

Automorphism Groups Of N-Graded Vertex Algebras Associated With Cyclic Leibniz Algebras With Small Dimensions, Alexander M. Keene

Theses and Dissertations

A fundamental problem in the study of vertex (operator) algebras V is the determination of the group of (grading-preserving) N-graded vertex algebras associated with cyclic Leibniz algebras of dimensions 2 and 3 that were classified by C. Barnes, E. Martin, J. Service, and G. Yamskulna in [1].

In each case examined, investigation of the automorphism group relies on the key fact that the action of an automorphism σ is determined solely by its value at a single basis element b. Furthermore, we employ a result in [19] by H. Li and G. Yamskulna which states that we can determine the …

The Impact Of Loss Function Topology On Gradient Descent, Robert B. Skudnig Jr.

Theses and Dissertations

Gradient descent is a popular optimization method that utilizes a model’s prediction error to iteratively improve its parameters for a given task. The functions that measure this error can be defined to align with the user’s goals and sometimes satisfy metric or norm properties. It is common for these functions to measure over Rn, but any differentiable space allows for gradient descent to occur. There has been some research investigating the influence of topological spaces on optimization methods, but it is a limited field of study. This thesis further explores this phenomenon by applying a transformation prediction model to multiple …

The Algebra Behind Magic, Lois Carpenter

Undergraduate Research Awards

Card Tricks have long been a staple in the common magician’s repertoire, and while many tricks can be explained through sleight of hand alone, others rely on seemingly random shuffling methods that leave the magician with significant control over the deck. Applied card magic (frequently referred to as ‘cheating’) makes significant use of this ability. Thus, the utility of this topic is clear- anyone with basic mastery of perfect shuffles has complete control over the arrangement of cards in a deck, and with it a fundamental advantage against other players in any game of cards. While most perfect shuffles are …

Algebra Structures For The Koszul Homology Of Minimal Intersections, Kathryn A. Grebel

Mathematics Dissertations - Archive

The Koszul homology of a local ring is a powerful tool in commutative algebra as it provides information on the structure and properties of the ring. In this research, we explore the relationship between quotients of regular local rings and their Koszul homology algebra. One such relationship is detailed by the Tate-Assmus theorem, which asserts, in part, that a ring is a complete intersection if and only if the Koszul homology is generated by its degree 1 homology elements. An objective of this research is to examine and identify the properties of a minimal intersection and its Koszul homology algebra. …

Blow-Ups And Projectivized Toric Vector Bundles, Sara Church

Theses and Dissertations--Mathematics

This dissertation is set in the intersection of toric geometry, tropical geometry, and the theory of vector bundles. We focus on the geometry of projectivized toric vector bundles and their connections to Mori dream spaces, matroid theory, and tropical geometry. We generalize previous results on the quotient construction of Gonzalez, Hering, Payne, and Suss (GHPS) by introducing a new approach to describing the geometry of these bundles via associated blow-ups. Additionally, we examine tautological bundles arising from representable matroids and establish connections between their geometry and the wonderful compactification. Finally, we consider the case where Klyachko filtrations have maximal steps …

Action This Day: The Mathematics And Machinations That Bested The German Enigma, Jonah Weinbaum

Dartmouth College Master’s Theses

This thesis presents a comprehensive and chronological overview of cryptographic techniques designed to break Enigma, beginning in 1932 and culminating in the creation of the Turing-Welchman Bombe. We discuss the mathematical theory and electromechanical implements used to decode one of history's greatest ciphers.

Reexamining the Bombe through the lens of modern group theory, we critique Alan Turing's estimation of the number of "stops" that the Bombe produces for various plaintext-ciphertext pairing structures. To address its limitations, we introduce a new framework for estimating the number of stops by extending John Dixon's theorem concerning the probability that uniformly distributed elements of …

On Linear Invariants Of Hypergraphs, Clara Chaplin

Honors Theses

We introduce linear invariants of hypergraphs as a way to study hypergraphs by their tensor representations. Our primary research goal is to determine what information linear invariants capture about the hypergraphs they arise from. We first investigate the centroid, which is shown to determine the connected components of a hypergraph. Next, we study the derivations of a hypergraph, and use this linear invariant to define a quotient operator $Q_\mathrm{Der}$ on the collection of all hypergraphs. This operator is shown to be a closure operator in that $Q_\mathrm{Der}(Q_\mathrm{Der}(\mathcal{H}))=Q_\mathrm{Der}(\mathcal{H})$ for any hypergraph $\mathcal{H}$. We apply the operator $Q_\mathrm{Der}$ to synthetically generated hypergraphs, …

Multipliers On Weighted Sequence Spaces, Gilbert Acheampong, Raymond Cheng

Mathematics & Statistics Faculty Publications

The space ℓp,α of complex sequences a = (a0, a1,a2,...) for which

[[formula omitted]]

is studied. Each such sequence can be identified with the analytic function with power series

[[formula omitted]]

In this setting, the point evaluation and the difference quotient mappings are shown to be bounded; the cases are identified in which ℓp,α is boundedly contained in ℓr,β. Conditions on the parameters are derived for the analytic functions of ℓp,α to have radial limits almost everywhere on the boundary, and for ℓp,α to be an algebra. Smoothness properties of the boundary function are investigated. Basic properties of multipliers on …

Multipliers Between ℓᴾ Spaces, Raymond Cheng

Mathematics & Statistics Faculty Publications

For 0 < p ⩽ ∞ and 0 < r ⩽ ∞, the space 𝔐_p,r of (coefficient) multipliers from ℓ^p and ℓ^r is completely characterized. This is elementary in most instances. The interesting case 0 < r < p < ∞ requires more effort, and it is shown that a sequence of complex numbers belongs to 𝔐_p,r if and only if the sequence of their absolute values has a non increasing rearrangement (h₀,h₁,h₂,...) satisfying

(∞

Σ (k +1)^(p-r)/p (h^r_k - h^r_k+1)^1/r) < ∞

k = 0

In that case, the expression on the left is the norm of the multiplier, and it is a compact operator. Further upper and lower bounds are given for the multiplier norm.

Moore Graphs, Trevor Saxton

Williams Honors College, Honors Research Projects

A Moore graph is a simple regular graph, with n vertices, degree d, and diameter k, that satisfies the Moore bound: n = 1 + d (d − 1)k − 1 d − 2 . There are graphs for which the bound is met and in which existence and uniqueness are known. For k = 2 it is known that the Moore bound is achieved for d = 2, 3, 7, with the case of d = 57 conjectured to exist. For k = 3 the bound is achieved for only d = 3 [5]. Due to the construction of …

Betti Numbers Of Generic Ideals, Jason R. Howell

Electronic Theses & Dissertations (2024 - present)

We present results related to Betti numbers of so called generic ideals over a polynomial ring $T = \Bbbk[x_1, \ldots, x_n]$. For each $m$ define $T(m) = T/(x_{m+1}, \ldots, x_n)$ and for a homogeneous ideal $J$ we use the notation $J(m) = JT(m)$. Also we set $Q(m) = T(m)/J(m)$, and $L(m) = ann_{Q(m)}(x_m)$. The first main result is Theorem \ref{Long Exact Sequence} where we produce the following long exact sequence \begin{align*} \cdots \rightarrow &Tor_{k+1,j}^{T(m-1)}(Q(m-1),\Bbbk) \rightarrow Tor_{k-1,j+1}^{T(m-1)}(L(m),\Bbbk)_{j-1} \rightarrow Tor_{k,j}^{T(m)}(Q(m),\Bbbk) \rightarrow \\ &Tor_{k,j}^{T(m-1)}(Q(m-1),\Bbbk) \rightarrow Tor_{k-2,j-1}^{T(m-1)}(L(m),\Bbbk) \rightarrow \cdots. \end{align*} The second main result is Theorem \ref{Theorem F(j,m) equiv k(j,m)} where we explicitly describe …

Towards A Baker-Campbell-Hausdorff Theorem In Positive Characteristic, Nikolas A. Koutroulakis

College of Graduate Studies: Theses & Dissertations

We investigate whether there is an analog of the Baker-Campbell-Hausdorff (BCH) theorem for Lie algebras over fields of positive characteristic. We begin by introducing the proof of the BCH formula in characteristic zero. We then introduce the Artin-Hasse exponential and show that it is $p$-integral. Our main result provides sufficient conditions under which a BCH-type formula exists for the Artin-Hasse exponential in positive characteristic. Additionally, we derive a formula for computing an inverse of the Artin-Hasse exponential.

Gorenstein Flat Preenvelopes Over Coherent Rings, Alec Moore

College of Graduate Studies: Theses & Dissertations

The existence of precovers and preenvelopes of Gorenstein flat modules is of great interest in the field of Gorenstein homological algebra. We give a sufficient condition in order for the class of Gorenstein flat modules to be preenveloping. More precisely, we prove that if the ring R is coherent such that every injective module has finite flat dimension, then every R-module has a Gorenstein flat preenvelope.

Local Limit Theorems On Finitely Generated Abelian Groups, Yutong Yan

Honors Theses

In this thesis, we classify the pointwise behavior of finite-range random walks on finitely generated abelian groups in terms of local limit theorems. Random walks are central objects of research in probability theory, and the theory has found applications in statistics, physics, and even card shuffling. One significant topic in this line of study is random walks on finitely generated groups. Starting from the pioneering work of G. Pólya and H. Kesten, random walks on finitely generated groups have been studied extensively. However, many notable results on the subject (local limit theorems, for example) make assumptions about periodicity and irreducibility …

Spectral Theory And The Gelfand Transform, Brenden Schlader

All Graduate Theses, Dissertations, and Other Capstone Projects

The overall goal of this thesis is to study spectral and Gelfand theory as it relates to unital Banach and C∗-algebras. In the first part, we develop the necessary algebraic, analytic, and topological background relevant to the content of this work. We also discuss concrete examples of algebras frequently used in the subsequent sections. In the second part of this thesis, we develop spectral theory by first defining the spectrum of an algebra through the characterization of the invertible and noninvertible elements. In particular, we establish properties of the commutative unital Banach algebra ℓ1(Z). We also establish fundamental results such …

Lipschitz Conditions On Operators And Matrices, Ryan Farrell

UNF Graduate Theses and Dissertations

Lipschitz functions on the real line find various applications across mathematics, including in differential equations, optimization, and machine learning. The goal of this thesis is to investigate functions which satisfy certain Lipschitz conditions when ap- plied to operators and matrices. Our study will review two classes of such functions, the class of Operator Lipschitz functions with respect to a given matrix norm, and the class consisting of functions which do not meet Lipschitz conditions in the traditional sense but satisfy inequalities which are Lipschitz in nature – we call such conditions ”Lipschitz-like”. The thesis concludes with a survey of these …

Rings With Q-Torsionfree Canonical Modules, Naoki William Endo, Laura P. Ghezzi, Shiro S. Goto, Jooyoun Hong Phd, Shin-Ichiro Iai, Toshinori Kobayashi, Naoyuki Matsuoka, Ryo Takahashi

Publications and Research

Let A be a Noetherian local ring with canonical module K A . We characterize A when K A is a torsionless, reflexive, or q-torsionfree module for an integer q ≥ 3 . If A is a Cohen–Macaulay ring, H.-B. Foxby proved in 1974 that the A-module K A is q-torsionfree if and only if the ring A is q-Gorenstein. With mild assumptions, we provide a generalization of Foxby’s result to arbitrary Noetherian local rings admitting the canonical module. In particular, since the reflexivity of the canonical module is closely related to the ring being Gorenstein …

Vanishing Of Local Cohomology With Applications To Hodge Theory, Scott Hiatt

Mathematics Faculty Publications

Let H = ((H, F •), L) be a polarized variation of Hodge structure on a smooth quasi-projective variety U . By M. Saito’s theory of mixed Hodge modules, the variation of Hodge structure H can be viewed as a polarized Hodge module M ∈ HM (U ). Let X be a compactification of U , and j : U ↪→ X is the natural map. In this paper, we use local cohomology with mixed Hodge module theory to study j+M ∈ DbM HM (X). In particular, we study the graded pieces of the de Rham complex GrF p DR(j+M) …

Prime Factorization And Unit Calculations Of Quadratic Integer Rings, Gabriel F. Roca

Honors Undergraduate Theses

The failure of unique factorization in a ring leads to the investigation of the closest algebraic structure, which are prime ideals. Using generalizations that have helped solve questions such as Fermat's Last Theorem, there is interest to study the elements with a multiplicative inverse (units) via the geometry and arithmetic patterns that arise in quadratic integer rings, since they provide tools for other questions in mathematics, ranging from pure algebra to applications in cryptography, and more. Overall, the following thesis provides a small exposition on the theory of integral domains and some specific calculations.

Calculation And Statistical Analysis Of Wins Above Replacement, Joshua Taylor

Departmental Honors & Graduate Capstone Projects

The Wins Above Replacement (WAR) statistic in Major League Baseball is a prominent metric used to estimate player value by quantifying all aspects of play in terms of wins added to a baseball team. We will use R to calculate WAR for all players from 1871 to 2012 and use data from those years to construct multivariate predictive models to attempt to estimate WAR for players from 2013 to 2024. We find strong correlations between predicted and actual WAR values for most models, with the exception of the polynomial predictive model for non-qualified pitchers.

An Analysis Of The Properties Of Polar Codes, Luke Szramowski

All Theses

Polar Codes have risen to the forefront of practical coding theory, due to their incredible efficiency and ease of construction. Originally introduced by Arikan in his 2009 paper, they are the first code defined with an explicit construction that achieved channel capacity. Moreover, polar codes possess some physically practical properties that make their implementation alluring. In the same paper as mentioned above, Arikan elaborated on his construction and noted that the construction given was one specific instance of a polar code and that there is a family of polar codes that can be produced by the same method. Since this …

A Comparative Analysis Of Early Algebraic Thinking Activities From U.S. And Singapore Primary Textbooks, Christian Joel Hernandez

Theses and Dissertations

Many studies highlight the challenges students face when transitioning to algebra at the secondary level. Introducing algebraic concepts and fostering algebraic thinking at the primary level can help mitigate these difficulties. Prior to formal algebra instruction, early algebra can be cultivated as a mode of thinking known as algebraic thinking. Several international curricula, such as Singapore Math, incorporate early algebraic thinking into the early stages of schooling. Singapore Math, renowned for its high performance in international assessments, has been widely adopted by schools seeking to replicate its success.

This study compares two primary-level mathematics curricula—CCSSM-aligned textbooks and Singapore Math—specifically focusing …

Andre-Quillen Homology And Special Classes Of Ring Homomorphisms, Hossein Faridian

All Dissertations

This thesis is comprised of three chapters. The first chapter deals with a purely algebraic proof of a deep result of Quillen stating that the category of simplicial commutative algebras over a commutative ring is a model category. The central focus of our approach is on the study of shuffle product of connective chain complexes that provides a bridge to translate the constructions in the simplicial algebra world to the chain complex world.

The second chapter delves into Quillen's fundamental spectral sequences that relate Andre-Quillen homology and cohomology to Tor and Ext functors. Our comprehensive treatment develops and streamlines the …

Magpy: A Python Package For Magmas, Skylar Korf

All NMU Master's Theses

There exist a multitude of computational tools available to mathematicians, such as interactive and automated theorem provers, finite counter-example generators, and computer algebra systems, most of which have a complicated installation process, unintuitive syntax, a lack of comprehensiveness for mathematical structures, or some combination of these. Computer algebra systems are indispensable tools due to their capacity for creating mathematical objects and performing computations on them. However, there is a lack of computational resources that focus on dealing with explicit examples, and extracting the properties thereof, especially for the most basic algebraic structures. Here, we will dive into several prominent computer …

The Near Normality Of The Commutant Of A Moufang Loop, Evan Phillips

All NMU Master's Theses

We investigate conditions under which the commutant of a Moufang loop is normal and related topics. We begin by giving relevant background on Moufang loops and normality. We then prove a theorem showing how close the commutant is to being normal: cR(x, y)R(x, y)R(x, y) = c. We then prove a couple of theorems emphasizing the importance of cubes in Moufang loops. Finally we prove a decomposition theorem for the multiplication group of a finite commutative Moufang loop.

Scaled Global Operators And Fueter Variables On Non-Zero Scaled Hypercomplex Numbers, Daniel Alpay, Ilwoo Choo, Mihaela Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we describe the rise of global operators in the scaled quaternionic case, an important extension from the quaternionic case to the family of scaled hypercomplex numbers H_t, t ∈ R^∗ , of which the H₋₁ = H is the space of quaternions and H₁ is the space of split quaternions.We also describe the scaled Fueter-type variables associated to these operators, developing a coherent theory in this field. We use these types of variables to build different types of function spaces on H_t. Counterparts of the Hardy space and of the …

Counting The Classes Of Projectively-Equivalent Pentagons On Finite Projective Planes Of Prime Order, Maxwell Hosler

Rose-Hulman Undergraduate Mathematics Journal

In this paper, we examine the number of equivalence classes of pentagons on finite projective planes of prime order under projective transformations. We are interested in those pentagons in general position, meaning that no three vertices are collinear. We consider those planes which can be constructed from finite fields of prime order, and use algebraic techniques to characterize them by their symmetries. We are able to construct a unique representative for each pentagon class with nontrivial symmetries. We can then leverage this fact to count classes of pentagons in general. We discover that there are (1/10)((p+3)(p-3)+4 …

Noncrystallographic Tail-Triangle C-Groups Of Rank 4 And Interlacing Number 2, Mark L. Loyola, Nonie Elvin S. Leyrita, Ma. Louise Antonette N. De Las Peñas

Mathematics Faculty Publications

This work applies the modular reduction technique to the Coxeter group of rank 4 having a star diagram with labels 5, 3, and k = 3,4,5, or 6. As moduli, we use the primes in the quadratic integer ring Z[τ], where τ = (1+√5)/2, the golden ratio. We prove that each reduced group is a C-group, regardless of the prime used in the reduction. We also classify each reduced group as a reflection group over a finite field, whenever applicable.

Q-Rational Functions And Interpolation With Complete Nevanlinna–Pick Kernels, Daniel Alpay, Paula Cerejeiras, Uwe Kaehler, Baruch Schneider

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we introduce the concept of matrix-valued q-rational functions. In comparison to the classical case, we give different characterizations with principal emphasis on realizations and discuss algebraic manipulations. We also study the concept of Schur multipliers and complete Nevanlinna–Pick kernels in the context of q-deformed reproducing kernel Hilbert spaces and provide first applications in terms of an interpolation problem using Schur multipliers and complete Nevanlinna–Pick kernels.

Categorical Chain Conditions For Étale Groupoid Algebras, Sunil Philip

Dissertations, Theses, and Capstone Projects

Let R be a unital commutative ring and G an ample groupoid. Using the topology of the groupoid G, Steinberg defined an étale groupoid algebra RG. These étale groupoid algebras generalize various algebras, including group algebras, commutative algebras over a field generated by idempotents, traditional groupoid algebras, Leavitt path algebras, higher-rank graph algebras, and inverse semigroup algebras. Steinberg later characterized the classical chain conditions for étale groupoid algebras. In this work, we characterize categorically noetherian and artinian, locally noetherian and artinian, and semisimple étale groupoid algebras, thereby generalizing existing results for Leavitt path algebras and introducing new results for inverse …