# Algebra Commons™

Articles 1 - 30 of 1266

## Full-Text Articles in Algebra

Jun 2021

#### 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, Muhammad S. Ali Khan, Saleem Abdullah, Kostaq Hila Jun 2021

#### 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, I. Silambarasan Jun 2021

#### 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.

May 2021

#### 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 fle􏰄ible thinking skills. Ho􏰃ever, 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 ...

May 2021

#### 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 ...

May 2021

#### 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.

May 2021

#### 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, Akshata Pisharody May 2021

#### 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, Samantha Phillips May 2021

#### 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, Marina Michelle Duchesne May 2021

#### 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 22:.(24:(2.S3)), M11, 3:(PSL(3,3):2), S8, and 2.M12. We have found homomorphic images of several progenitors, including 2*18:((6x2):6), 2*24:(2.S4), 2*105:A7, 3*3:m(23:3), 7*8:m(PSL(2,7):2), 3*4:m(42:22), 7*5:(2xA5), and 5*6:mS5. We have provided the isomorphism type of all of the finite images that we have discovered. We ...

On The Hamiltonicity Of Subgroup Lattices, Nicholas Charles Fleece May 2021

#### 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, Montana S. Miller May 2021

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

##### MSU Graduate Theses

An elliptic curve over the rational numbers is given by the equation y2 = x3+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, Austin Nicholas Beard May 2021

#### 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, Angsuman Das, Arnab Mandal Apr 2021

#### 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, Grace Weeks, Hallie Kaiser, Katy O'Malley Apr 2021

#### 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, Thomas Lauria Apr 2021

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

##### Open Educational Resources

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

Apr 2021

#### 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, Bernard Lidicky, Connor Mattes, Florian Pfender Feb 2021

#### 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, Christopher James Natoli Feb 2021

#### 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, I. Sabadini, Daniele C. Struppa Jan 2021

#### 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.

A Gentle Introduction To Inequalities: A Casebook From The Fullerton Mathematical Circle, Adam Glesser, Matt Rathbun, Bogdan Suceavă Jan 2021

#### A Gentle Introduction To Inequalities: A Casebook From The Fullerton Mathematical Circle, Adam Glesser, Matt Rathbun, Bogdan Suceavă

##### Journal of Math Circles

Run for nearly a decade, the Fullerton Mathematical Circle at California State University, Fullerton prepares middle and high school students for mathematical research by exposing them to difficult problems whose solutions require only age-appropriate techniques and background. This work highlights one of the avenues of study, namely inequalities. We cover Engel's lemma, the Cauchy--Schwartz inequality, and the AM-GM inequality, as well as providing a wealth of problems where these results can be applied. Full solutions or hints, several written by Math Circle students, are given for all of the problems, as well as some commentary on how or when ...

Mathematical Magic: A Study Of Number Puzzles, Nicasio M. Velez Jan 2021

#### Mathematical Magic: A Study Of Number Puzzles, Nicasio M. Velez

##### Rose-Hulman Undergraduate Mathematics Journal

Within this paper, we will briefly review the history of a collection of number puzzles which take the shape of squares, polygons, and polyhedra in both modular and nonmodular arithmetic. Among other results, we develop construction techniques for solutions of both Modulo and regular Magic Squares. For other polygons in nonmodular arithmetic, specifically of order 3, we present a proof of why there are only four Magic Triangles using linear algebra, disprove the existence of the Magic Tetrahedron in two ways, and utilizing the infamous 3-SUM combinatorics problem we disprove the existence of the Magic Octahedron.

Jan 2021

#### Maximums Of Total Betti Numbers In Hilbert Families, Jay White

##### Theses and Dissertations--Mathematics

Fix a family of ideals in a polynomial ring and consider the problem of finding a single ideal in the family that has Betti numbers that are greater than or equal to the Betti numbers of every ideal in the family. Or decide if this special ideal even exists. Bigatti, Hulett, and Pardue showed that if we take the ideals with a fixed Hilbert function, there is such an ideal: the lexsegment ideal. Caviglia and Murai proved that if we take the saturated ideals with a fixed Hilbert polynomial, there is also such an ideal. We present a generalization of ...

#### Counting Conjugacy Classes Of Elements Of Finite Order In Compact Exceptional Groups, Qidong He

##### Honors Theses

Given a compact exceptional group $G$ and $m,s\in\mathbb{N}$, let $N(G,m)$ be the number of conjugacy classes of elements of order $m$ in $G$, and $N(G,m,s)$ the number of such classes whose elements have $s$ distinct eigenvalues. In string theory, the problem of enumerating certain classes of vacua in the string landscape can be rephrased in terms of the study of these quantities. We develop unified combinatorial algorithms based on Burnside's Lemma that can be used to compute both quantities for each of the five compact exceptional groups. Thus, we provide ...

#### On Properties Of Positive Semigroups In Lattices And Totally Real Number Fields, Siki Wang

##### CMC Senior Theses

In this thesis, we give estimates on the successive minima of positive semigroups in lattices and ideals in totally real number fields. In Chapter 1 we give a brief overview of the thesis, while Chapters 2 – 4 provide expository material on some fundamental theorems about lattices, number fields and height functions, hence setting the necessary background for the original results presented in Chapter 5. The results in Chapter 5 can be summarized as follows. For a full-rank lattice L ⊂ Rd, we are concerned with the semigroup L+ ⊆ L, which denotes the set of all vectors with nonnegative coordinates in L ...

The Complexity Of Symmetry, Matthew Lemay Jan 2021

#### The Complexity Of Symmetry, Matthew Lemay

##### HMC Senior Theses

One of the main goals of theoretical computer science is to prove limits on how efficiently certain Boolean functions can be computed. The study of the algebraic complexity of polynomials provides an indirect approach to exploring these questions, which may prove fruitful since much is known about polynomials already from the field of algebra. This paper explores current research in establishing lower bounds on invariant rings and polynomial families. It explains the construction of an invariant ring for whom a succinct encoding would imply that NP is in P/poly. It then states a theorem about the circuit complexity partial ...

Dec 2020

#### The Beautiful Math Of Everything And You Included, E. Ozie

##### The STEAM Journal

This a reflection on how there is beautiful math to everything. An author's interpretation of matrices and mechanics in its relationship to someone's identity.

Adaptive Analytics: It’S About Time, Charles Dziuban, Colm Howlin, Patsy Moskal, Tammy Muhs, Connie Johnson, Rachel Griffin, Carissa Hamilton Dec 2020

#### Adaptive Analytics: It’S About Time, Charles Dziuban, Colm Howlin, Patsy Moskal, Tammy Muhs, Connie Johnson, Rachel Griffin, Carissa Hamilton

##### Current Issues in Emerging eLearning

This article describes a cooperative research partnership among a large public university, a for-profit private institution and their common adaptive learning platform provider. The focus of this work explored adaptive analytics that uses data the investigators describe as metaphorical “digital learning dust” produced by the platform as a matter of course. The information configured itself into acquired knowledge, growth, baseline status and engagement. Two complimentary models evolved. The first, in the public university, captured end-of-course data for predicting success. The second approach, in the private university, formed the basis of a dynamic real-time data analytic algorithm. In both cases the ...

#### Behavior And Dynamics Of The Set Of Absolute Nilpotent And Idempotent Elements Of Chain Of Evolution Algebras Depending On The Time, Anvar Imomkulov

##### Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

In this paper we construct some families of three-dimensional evolution algebras which satisfies Chapman-Kolmogorov equation. For all of these chains we study the behavior of the baric property, the behavior of the set of absolute nilpotent elements and dynamics of the set of idempotent elements depending on the time.

Dec 2020

#### Using Desmos To Support Conceptual Understanding Of Quadratic Equations And Their Graphs, Shannon Bromley

##### Education and Human Development Master's Theses

When teaching secondary mathematics, there is often a struggle to distinguish true understanding versus the ability to follow a memorized procedure. This commonly occurs in Algebra I content of solving and graphing quadratic equations. Engaging students in Desmos interactive graphing learning activities has been shown to help deepen students’ understanding of such concepts. This curriculum project is designed to deepen students’ understanding and fluency of quadratic equations by including hands-on activities that support students discovering relationships and patterns between quadratic equations and their corresponding graphs. The unit plan aligns to the New York State Common Core State Standards.