Algebra Commons^™

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 …

Diagrams In Involutive Residuated Lattices, Isis A. Gallardo

Electronic Theses and Dissertations

First, we show that every distributive lattice-ordered pregroup can be embedded into a functional algebra over an integral chain, thereby improving the existing Cayley/Holland style embedding theorem. Using this result, we demonstrate that the variety of all dis tributive lattice-ordered pregroups is generated by the functional algebra on the integers. Additionally, we prove that the equational theory of this variety is decidable.

Next, we establish that DLP is equal to the join of its subvarieties LP_n, where 𝑛 ∈ ℤ⁺, consisting of 𝑛-periodic ℓ-pregroups. We also prove that every algebra in LP_n can be embedded …

Frames And Spaces For Distributive Quasi Relation Algebras And Distributive Involutive Fl-Algebras, Andrew Craig, Peter Jipsen, Claudette Robinson

Mathematics, Physics, and Computer Science Faculty Articles and Research

Analogous to atom structures for relation algebras, we define partially ordered frames and prove they are duals for complete perfect distributive quasi relation algebras and distributive involutive FL-algebras. We then extend this dual representation to all algebras and their corresponding frames with a Priestley topology.

For relation algebras up to size 16 it has been determined which algebras are representable by binary relations. We compute all finite distributive quasi relation algebras up to 8 elements and provide representations for some of them.

On The Structure Of Balanced Residuated Partially Ordered Monoids, Stefano Bonzio, José Gil-Férez, Peter Jipsen, Adam Přenosil, Melissa Sugimoto

Mathematics, Physics, and Computer Science Faculty Articles and Research

A residuated poset is a structure ⟨A,⩽, ·, \, /, 1⟩ where ⟨A,⩽⟩ is a poset and ⟨A, ·, 1⟩ is a monoid such that the residuation law x · y ⩽ z ⇐⇒ x ⩽ z/y ⇐⇒ y ⩽ x\z holds. A residuated poset is balanced if it satisfies the identity x\x ≈ x/x. By generalizing the well-known construction of Płonka sums, we show that a specific class of balanced residuated posets can be decomposed into such a sum indexed by the set of positive idempotent elements. Conversely, given a semilattice directed system of residuated posets equipped with two …

Making Sandwiches: A Novel Invariant In D-Module Theory, David Lieberman

Department of Mathematics: Dissertations, Theses, and Student Research

Say I hand you a shape, any shape. It could be a line, it could be a crinkled sheet, it could even be a the intersection of a cone with a 6-dimensional hypersurface embedded in a 7-dimensional space. Your job is to tell me about the pointy bits. This task is easier when you can draw the shape; you can you just point at them. When things get more complicated, we need a bigger hammer.

In a sense, that “bigger hammer” is what the ring of differential operators is to an algebraist. Then we will say some things and stuff …

Cohen-Macaulay Type Of Open Neighborhood Ideals Of Unmixed Trees, Jounglag Lim

All Theses

Given a tree T and a field k, we define the open neighborhood ideal N(T) of T in k[V] to be the ideal generated by the open neighborhoods of all vertices in the graph. If T is unmixed with respect to the total domination problem, then it is known that N(T) is Cohen-Macaulay. Our goal is to compute the (Cohen-Macaulay) type of k[V]/N(T) using graph theoretical properties of T. We achieve this by using homological algebra and properties of monomial ideals. Along the way, we also provide a different characterization of unmixed trees and a generalization of the total dominating …

Relating Elasticity And Other Multiplicative Properties Among Orders In Number Fields And Related Rings, Grant Moles

All Dissertations

This dissertation will explore factorization within orders in a number ring. By far the most well-understood of these orders are rings of algebraic integers. We will begin by examining how certain types of subrings may relate to the larger rings in which they are contained. We will then apply this knowledge, along with additional techniques, to determine how the elasticity in an order relates to the elasticity of the full ring of algebraic integers. Using many of the same strategies, we will develop a corresponding result in the rings of formal power series. Finally, we will explore a number of …

Visualization Of Species Tree Likelihood Under The Multispecies Coalescent Model, Jaimasan Sutton

Mathematics & Statistics ETDs

A commonly used tool for evolutionary biologists is a phylogenetic tree that represents the ancestry of a set of species and the evolution of traits. Statistical models can be used to predict the probabilities of gene trees which represent ancestral relationships of genes sampled from species. Because of this, we are able to represent the likelihood of a species tree, which represents the evolutionary history of a set of species, as a function of the counts of gene tree topologies, where each gene tree represents the ancestry of a specific genetic locus for multiple species. Because we can represent these …

Diving Deeper Into Supercuspidal Representations, Prerna Agarwal

LSU Doctoral Dissertations

In 2013, Reeder and Yu introduced certain low positive depth supercuspidal representations of $p$-adic groups called \textit{epipelagic} representations. These representations generalize the simple supercuspidal representations of Gross and Reeder, which have the lowest possible depth. Epipelagic representations also arise in recent work on the Langlands correspondence; for example, simple supercuspidals appear in the automorphic data corresponding to the Kloosterman $l$-adic sheaf. In this thesis, we take a first step towards the construction of ``\textit{mesopelagic} representation (of Iwahori type)'' which are the higher depth analogues of simple supercuspidal representations. We see that these constructions can be done in a similar way …

S-Preclones And The Galois Connection SPol–SInv, Part I, Peter Jipsen, Erkko Lehtonen, Reinhard Pöschel

Mathematics, Physics, and Computer Science Faculty Articles and Research

We consider S-operations f : Aⁿ → A in which each argument is assigned a signum s ∈ S representing a “property” such as being order- preserving or order-reversing with respect to a fixed partial order on A. The set S of such properties is assumed to have a monoid structure reflecting the behaviour of these properties under the composition of S-operations (e.g., order-reversing composed with order-reversing is order- preserving). The collection of all S-operations with prescribed properties for their signed arguments is not a clone (since it is not closed under arbitrary identification of arguments), …

Math 75: Introduction To Linear Algebra, Sarah K. Merz

Pacific Open Texts

This text is intended to use in a first course of Linear Algebra with a prerequisite of Calculus 1. Topics covered include systems of linear equations, matrix operations and inverses, linear transformations, Markov chains, determinants, eigenvalues and eigenvectors, diagonalization, vector geometry, projections and planes, homogeneous coordinates, subspaces, spanning sets, linear independence, orthogonality, fundamental subspaces, and least squares.

Building Blocks For W-Algebras Of Classical Types, Vladimir Kovalchuk

Electronic Theses and Dissertations

The universal 2-parameter vertex algebra W_∞ of type W(2, 3, 4; . . . ) serves as a classifying object for vertex algebras of type W(2, 3, . . . ,N) for some N in the sense that under mild hypothesis, all such vertex algebras arise as quotients of W_∞. There is an ℕ X ℕ family of such 1-parameter vertex algebras known as Y-algebras. They were introduced by Gaiotto and Rapčák are expected to be building blocks for all W-algebras in type A, i.e, every W-(super) algebra in …

Schur Analysis Over The Unit Spectral Ball, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

We begin a study of Schur analysis when the variable is now a matrix rather than a complex number. We define the corresponding Hardy space, Schur multipliers and their realizations, and interpolation. Possible applications of the present work include matrices of quaternions, matrices of split quaternions, and other algebras of hypercomplex numbers.

Nidus Idearum. Scilogs, Xiv: Superhyperalgebra, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this fourteenth book of scilogs – one may find topics on examples where neutrosophics works and others don’t, law of included infinitely-many-middles, decision making in games and real life through neutrosophic lens, sociology by neutrosophic methods, Smarandache multispace, algebraic structures using natural class of intervals, continuous linguistic set, cyclic neutrosophic graph, graph of neutrosophic triplet group , how to convert the crisp data to neutrosophic data, n-refined neutrosophic set ranking, adjoint of a square neutrosophic matrix, neutrosophic optimization, de-neutrosophication, the n-ary soft set relationship, hypersoft set, extending the hypergroupoid to the superhypergroupoid, alternative ranking, Dezert-Smarandache Theory (DSmT), reconciliation between …

(R2087) Modal Operators On Bipolar Intuitionistic Fuzzy Matrices, T. Muthuraji, P. Punitha Elizabeth

Applications and Applied Mathematics: An International Journal (AAM)

Bipolar intuitionistic fuzzy sets are currently a robust area in several industries. Additionally, bipolar intuitionistic fuzzy matrices are a highly regarded topic in many fields including psychology, engineering, qualitative reasoning and multi-criteria decision making. A civilization must deal with negative and positive problems. In this article, we discussed some algebraic properties of modal operators on bipolar intuitionistic fuzzy matrices. Finally, we derive some results of necessity and possibility operators along with Max-Min product.

Representation Theory And Its Applications In Physics, Max Varverakis

Master's Theses

Representation theory, which encodes the elements of a group as linear operators on a vector space, has far-reaching implications in physics. Fundamental results in quantum physics emerge directly from the representations describing physical symmetries. We first examine the connections between specific representations and the principles of quantum mechanics. Then, we shift our focus to the braid group, which describes the algebraic structure of braids. We apply representations of the braid group to physical systems in order to investigate quasiparticles known as anyons. Finally, we obtain governing equations of anyonic systems to highlight the differences between braiding statistics and conventional Bose-Einstein/Fermi-Dirac …

On Near-Linear Cellular Automata Over Near Spaces, Abdul-Rahman M. Nasser

Dissertations

Cellular Automata can be considered as examples of massively parallel machines. They are computational mathematical objects consisting of a grid of cells, each of which can exist in a finite number of states. These cells evolve over discrete time steps according to a set of predefined rules based on the states of neighboring cells. The notion of cellular automata was first introduced by Ulam and von Neumann and then popularized by John H. Conway in the 1970s with one of the most famous examples being The Game of Life.

This research builds on and generalizes the work of Tullio Ceccherini-Silberstein …

Explicit Composition Identities For Higher Composition Laws In The Quadratic Case, Ajith A. Nair

Dissertations, Theses, and Capstone Projects

The theory of Gauss composition of integer binary quadratic forms provides a very useful way to compute the structure of ideal class groups in quadratic number fields. In addition to that, Gauss composition is also important in the problem of representations of integers by binary quadratic forms. In 2001, Bhargava discovered a new approach to Gauss composition which uses 2x2x2 integer cubes, and he proved a composition law for such cubes. Furthermore, from the higher composition law on cubes, he derived four new higher composition laws on the following spaces - 1) binary cubic forms, 2) pairs of binary quadratic …

Hyperbolic Groups And The Word Problem, David Wu

Master's Theses

Mikhail Gromov’s work on hyperbolic groups in the late 1980s contributed to the formation of geometric group theory as a distinct branch of mathematics. The creation of hyperbolic metric spaces showed it was possible to define a large class of hyperbolic groups entirely geometrically yet still be able to derive significant algebraic properties. The objectives of this thesis are to provide an introduction to geometric group theory through the lens of quasi-isometry and show how hyperbolic groups have solvable word problem. Also included is the Stability Theorem as an intermediary result for quasi-isometry invariance of hyperbolicity.

Weakly Pseudo Primary 2-Absorbing Submodules, Omar Hisham Taha, Marrwa Abdulla Salih

Al-Bahir

Let be a commutative ring with identity. In this paper, we introduce the notion of a weakly pseudo primary 2-absorbing sub-module as a generalization of a 2-absorbing sub-module and a pseudo 2-absorbing sub-module. Moreover, we give many basic properties, examples, and characterizations of these notions.