Noncommutative Tensor Triangular Geometry And Its Applications To Representation Theory, 2021 Louisiana State University and Agricultural and Mechanical College

#### Noncommutative Tensor Triangular Geometry And Its Applications To Representation Theory, Kent Barton Vashaw

*LSU Doctoral Dissertations*

One of the cornerstones of the representation theory of Hopf algebras and finite tensor categories is the theory of support varieties. Balmer introduced tensor triangular geometry for symmetric monoidal triangulated categories, which united various support variety theories coming from disparate areas such as homotopy theory, algebraic geometry, and representation theory. In this thesis a noncommutative version will be introduced and developed. We show that this noncommutative analogue of Balmer's theory can be determined in many concrete situations via the theory of abstract support data, and can be used to classify thick tensor ideals. We prove an analogue of prime ...

On The Generalization Of Interval Valued Fuzzy Generalized Bi-Ideals In Ordered Semigroups, 2021 Hazara University

#### On The Generalization Of Interval Valued Fuzzy Generalized Bi-Ideals In Ordered Semigroups, Muhammad S. Ali Khan, Saleem Abdullah, Kostaq Hila

*Applications and Applied Mathematics: An International Journal (AAM)*

In this paper, a new general form than interval valued fuzzy generalized bi-ideals in ordered semigroups is introduced. The concept of interval valued fuzzy generalized bi-ideals is initiated and several properties and characterizations are provided. A condition for an interval valued fuzzy generalized bi-ideal to be an interval valued fuzzy generalized bi-ideal is obtained. Using implication operators and the notion of implication-based an interval valued fuzzy generalized bi-ideal, characterizations of an interval valued fuzzy generalized bi-ideal and an interval valued fuzzy generalized bi-ideal are considered.

Hamacher Operations Of Fermatean Fuzzy Matrices, 2021 Annamalai University

#### Hamacher Operations Of Fermatean Fuzzy Matrices, I. Silambarasan

*Applications and Applied Mathematics: An International Journal (AAM)*

The purpose of this study is to extend the Fermatean fuzzy matrices to the theory of Hamacher operations. In this paper, the concept of Hamacher operations of Fermatean fuzzy matrices are introduced and some desirable properties of these operations, such as commutativity, idempotency, and monotonicity are discussed. Further, we prove DeMorgan’s laws over complement for these operations. Furthermore, the scalar multiplication and exponentiation operations of Fermatean fuzzy matrices are constructed and their algebraic properties are investigated. Finally, some properties of necessity and possibility operators of Fermatean fuzzy matrices are proved.

Developing Mathematical Flexible Thinking In Students With Disabilities, 2021 The College at Brockport: State University of New York

#### Developing Mathematical Flexible Thinking In Students With Disabilities, Bridget Sadler

*Education and Human Development Master's Theses*

This curriculum project was designed to improve mathematical flexible thinking skills, with a focus on students with disabilities. This is achieved through the integration of inquiry-based learning activities and mathematical discourse throughout each lesson. This collection of lessons is not a unit, rather it is a collection of exemplar lessons which detail how to embed these strategies to develop students mathematical fleible thinking skills. Hoever, all lessons are from Alegbra I and Geometry. Teachers can use this curriculum as it is, or as a model of how to support the development of flexible thinking skills for all students ...

Streaming Down The Stern-Brocot Tree: Finding And Expressing Solutions To Pell's Equation In Sl(2,Z), 2021 Jacksonville State University

#### Streaming Down The Stern-Brocot Tree: Finding And Expressing Solutions To Pell's Equation In Sl(2,Z), Marcus L. Shell

*Theses*

This paper explores and elaborates on a method of solving Pell’s equation as introduced by Norman Wildberger. In the first chapters of the paper, foundational topics are introduced in expository style including an explanation of Pell’s equation. An explanation of continued fractions and their ability to express quadratic irrationals is provided as well as a connection to the Stern-Brocot tree and a convenient means of representation for each in terms of 2×2 matrices with integer elements. This representation will provide a useful way of navigating the Stern-Brocot tree computationally and permit us a means of computing continued ...

Frobenius And Homological Dimensions Of Complexes, 2021 University of Nebraska-Lincoln

#### Frobenius And Homological Dimensions Of Complexes, Taran Funk

*Dissertations, Theses, and Student Research Papers in Mathematics*

Much work has been done showing how one can use a commutative Noetherian local ring R of prime characteristic, viewed as algebra over itself via the Frobenius endomorphism, as a test for flatness or projectivity of a finitely generated module M over R. Work on this dates back to the famous results of Peskine and Szpiro and also that of Kunz. Here I discuss what work has been done to push this theory into modules which are not necessarily finitely generated, and display my work done to weaken the assumptions needed to obtain these results.

Adviser: Tom Marley

Algebraic Structures And Variations: From Latin Squares To Lie Quasigroups, 2021 Northern Michigan University

#### Algebraic Structures And Variations: From Latin Squares To Lie Quasigroups, Erik Flinn

*All NMU Master's Theses*

In this Master's Thesis we give an overview of the algebraic structure of sets with a single binary operation. Specifically, we are interested in quasigroups and loops and their historical connection with Latin squares; considering them in both finite and continuous variations. We also consider various mappings between such algebraic objects and utilize matrix representations to give a negative conclusion to a question concerning isotopies in the case of quasigroups.

Determining Quantum Symmetry In Graphs Using Planar Algebras, 2021 William & Mary

#### Determining Quantum Symmetry In Graphs Using Planar Algebras, Akshata Pisharody

*Undergraduate Honors Theses*

A graph has quantum symmetry if the algebra associated with its quantum automorphism group is non-commutative. We study what quantum symmetry means and outline one specific method for determining whether a graph has quantum symmetry, a method that involves studying planar algebras and manipulating planar tangles. Modifying a previously used method, we prove that the 5-cycle has no quantum symmetry by showing it has the generating property.

A Survey Of Methods To Determine Quantum Symmetry Of Graphs, 2021 William & Mary

#### A Survey Of Methods To Determine Quantum Symmetry Of Graphs, Samantha Phillips

*Undergraduate Honors Theses*

We introduce the theory of quantum symmetry of a graph by starting with quantum permutation groups and classical automorphism groups. We study graphs with and without quantum symmetry to provide a comprehensive view of current techniques used to determine whether a graph has quantum symmetry. Methods provided include specific tools to show commutativity of generators of algebras of quantum automorphism groups of distance-transitive graphs; a theorem that describes why nontrivial, disjoint automorphisms in the automorphism group implies quantum symmetry; and a planar algebra approach to studying symmetry.

Symmetric Presentation Of Finite Groups, And Related Topics, 2021 California State University, San Bernardino

#### Symmetric Presentation Of Finite Groups, And Related Topics, Marina Michelle Duchesne

*Electronic Theses, Projects, and Dissertations*

We have discovered original symmetric presentations for several finite groups, including 2^{2}:^{.}(2^{4}:(2^{.}S_{3})), M_{11}, 3:(PSL(3,3):2), S_{8}, and 2^{.}M_{12}. We have found homomorphic images of several progenitors, including 2^{*18}:((6x2):6), 2^{*24}:(2^{.}S_{4}), 2^{*105}:A_{7}, 3^{*3}:_{m}(2^{3}:3), 7^{*8}:_{m}(PSL(2,7):2), 3^{*4}:_{m}(4^{2}:2^{2}), 7^{*5}:(2xA_{5}), and 5^{*6}:_{m}S_{5}. We have provided the isomorphism type of all of the finite images that we have discovered. We ...

On The Hamiltonicity Of Subgroup Lattices, 2021 Missouri State University

#### On The Hamiltonicity Of Subgroup Lattices, Nicholas Charles Fleece

*MSU Graduate Theses*

In this paper we discuss the Hamiltonicity of the subgroup lattices of

different classes of groups. We provide sufficient conditions for the

Hamiltonicity of the subgroup lattices of cube-free abelian groups. We also

prove the non-Hamiltonicity of the subgroup lattices of dihedral and

dicyclic groups. We disprove a conjecture on non-abelian *p*-groups by

producing an infinite family of non-abelian *p*-groups with Hamiltonian

subgroup lattices. Finally, we provide a list of the Hamiltonicity of the

subgroup lattices of every finite group up to order 35 barring two groups.

On Elliptic Curves, 2021 Missouri State University

#### On Elliptic Curves, Montana S. Miller

*MSU Graduate Theses*

An elliptic curve over the rational numbers is given by the equation *y ^{2} = x^{3}+Ax+B*. In our thesis, we study elliptic curves. It is known that the set of rational points on the elliptic curve form a finitely generated abelian group induced by the secant-tangent addition law. We present an elementary proof of associativity using Maple. We also present a relatively concise proof of the Mordell-Weil Theorem.

Proper Sum Graphs, 2021 Missouri State University

#### Proper Sum Graphs, Austin Nicholas Beard

*MSU Graduate Theses*

The Proper Sum Graph of a commutative ring with identity has the prime ideals as vertices, with two ideals adjacent if their sum is a proper ideal. This thesis expands upon the research of Dhorajia. We will cover the groundwork to understanding the basics of these graphs, and gradually narrow our efforts into the minimal prime ideals of the ring.

Classification Of Cayley Rose Window Graphs, 2021 Presidency University, Kolkata

#### Classification Of Cayley Rose Window Graphs, Angsuman Das, Arnab Mandal

*Theory and Applications of Graphs*

Rose window graphs are a family of tetravalent graphs, introduced by Steve Wilson. Following it, Kovacs, Kutnar and Marusic classified the edge-transitive rose window graphs and Dobson, Kovacs and Miklavic characterized the vertex transitive rose window graphs. In this paper, we classify the Cayley rose window graphs.

Normality Properties Of Composition Operators, 2021 Taylor University

#### Normality Properties Of Composition Operators, Grace Weeks, Hallie Kaiser, Katy O'Malley

*Celebration of Scholarship 2021*

We explore two main concepts in relation to truncated composition matrices: the conditions required for the binormal and commutative properties. Both of these topics are important in linear algebra due to their connection with diagonalization.

We begin with the normal solution before moving onto the more complex binormal solutions. Then we cover conditions for the composition matrix to commute with the general matrix. Finally, we end with ongoing questions for future work.

Factoring: Difference Of Squares, 2021 CUNY College of Staten Island

#### Factoring: Difference Of Squares, Thomas Lauria

*Open Educational Resources*

This lesson plan will explain how to factor basic difference of squares problems

Matrix Product Structure Of A Permuted Quasi Cyclic Code And Its Dual, 2021 University of Missouri-St. Louis

#### Matrix Product Structure Of A Permuted Quasi Cyclic Code And Its Dual, Perian Perdhiku

*Dissertations*

In my Dissertation I will work mostly with *Permuted Quasi Cyclic Codes. *They are a generalization of *Cyclic Codes*, one of the most important families of Linear Codes in *Coding Theory*. Linear Codes are very useful in *error detection and correction*. Error Detection and Correction is a technique that first detects the corrupted data sent from some transmitter over unreliable communication channels and then corrects the errors and reconstructs the original data. Unlike linear codes, cyclic codes are used to correct errors where the pattern is not clear and the error occurs in a short segment of the message.

The ...

C5 Is Almost A Fractalizer, 2021 Iowa State University

#### C5 Is Almost A Fractalizer, Bernard Lidicky, Connor Mattes, Florian Pfender

*Mathematics Publications*

We determine the maximum number of induced copies of a 5-cycle in a graph on n vertices for every n. Every extremal construction is a balanced iterated blow-up of the 5-cycle with the possible exception of the smallest level where for n=8, the Möbius ladder achieves the same number of induced 5-cycles as the blow-up of a 5-cycle on 8 vertices.

This result completes work of Balogh, Hu, Lidický, and Pfender [Eur. J. Comb. 52 (2016)] who proved an asymptotic version of the result. Similarly to their result, we also use the flag algebra method but we extend its ...

Some Model Theory Of Free Groups, 2021 The Graduate Center, City University of New York

#### Some Model Theory Of Free Groups, Christopher James Natoli

*Dissertations, Theses, and Capstone Projects*

There are two main sets of results, both pertaining to the model theory of free groups. In the first set of results, we prove that non-abelian free groups of finite rank at least 3 or of countable rank are not A-homogeneous. We then build on the proof of this result to show that two classes of groups, namely finitely generated free groups and finitely generated elementary free groups, fail to form A-Fraisse classes and that the class of non-abelian limit groups fails to form a strong A-Fraisse class.

The second main result is that if a countable group is elementarily ...

Total Differentiability And Monogenicity For Functions In Algebras Of Order 4, 2021 Politecnico di Milano

#### Total Differentiability And Monogenicity For Functions In Algebras Of Order 4, I. Sabadini, Daniele C. Struppa

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

In this paper we discuss some notions of analyticity in associative algebras with unit. We also recall some basic tool in algebraic analysis and we use them to study the properties of analytic functions in two algebras of dimension four that played a relevant role in some work of the Italian school, but that have never been fully investigated.