Scaled Global Operators And Fueter Variables On Non-Zero Scaled Hypercomplex Numbers, 2024 Chapman University

#### 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 H_{1} 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, 2024 Rose-Hulman Institute of Technology

#### 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 …

Q-Rational Functions And Interpolation With Complete Nevanlinna–Pick Kernels, 2024 Chapman University

#### 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, 2024 The Graduate Center, City University of New York

#### 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 …

The Fundamental Groupoid In Discrete Homotopy Theory, 2024 Western University

#### The Fundamental Groupoid In Discrete Homotopy Theory, Udit Ajit Mavinkurve

*Electronic Thesis and Dissertation Repository*

Discrete homotopy theory is a homotopy theory designed for studying graphs and for detecting combinatorial, rather than topological, “holes”. Central to this theory are the discrete homotopy groups, defined using maps out of grids of suitable dimensions. Of these, the discrete fundamental group in particular has found applications in various areas of mathematics, including matroid theory, subspace arrangements, and topological data analysis.

In this thesis, we introduce the discrete fundamental groupoid, a multi-object generalization of the discrete fundamental group, and use it as a starting point to develop some robust computational techniques. A new notion of covering graphs allows us …

Relating Elasticity And Other Multiplicative Properties Among Orders In Number Fields And Related Rings, 2024 Clemson University

#### 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 …

Making Sandwiches: A Novel Invariant In D-Module Theory, 2024 University of Nebraska-Lincoln

#### 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, 2024 Clemson University

#### 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 …

Visualization Of Species Tree Likelihood Under The Multispecies Coalescent Model, 2024 University of New Mexico

#### 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, 2024 Louisiana State University

#### 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, 2024 Chapman University

#### 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 : A^{n} → 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, 2024 University of the Pacific

#### 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, 2024 University of Denver

#### 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, 2024 Chapman University

#### 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, 2024 University of New Mexico

#### 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 …

On Near-Linear Cellular Automata Over Near Spaces, 2024 Western Michigan University

#### 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, 2024 The Graduate Center, City University of New York

#### 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, 2024 California Polytechnic State University, San Luis Obispo

#### 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.

Representation Theory And Its Applications In Physics, 2024 California Polytechnic State University, San Luis Obispo

#### 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 …

Weakly Pseudo Primary 2-Absorbing Submodules, 2024 Department of Mathematics, College of Computer Science and Mathematics, Tikrit University, Tikrit, Iraq

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

*Al-Bahir Journal for Engineering and Pure Sciences*

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.