# Algebra Commons™

## All Articles in Algebra

1,093 full-text articles. Page 1 of 41.

Patterns, Symmetries, And Mathematical Structures In The Arts, 2020 Georgia Southern University

#### Patterns, Symmetries, And Mathematical Structures In The Arts, Sarah C. Deloach

##### University Honors Program Theses

Mathematics is a discipline of academia that can be found everywhere in the world around us. Mathematicians and scientists are not the only people who need to be proficient in numbers. Those involved in social sciences and even the arts can benefit from a background in math. In fact, connections between mathematics and various forms of art have been discovered since as early as the fourth century BC. In this thesis we will study such connections and related concepts in mathematics, dances, and music.

The Inverse Eigenvalue Problem For Leslie Matrices, 2019 University of Roma "La Sapienza"

#### The Inverse Eigenvalue Problem For Leslie Matrices, Luca Benvenuti

##### Electronic Journal of Linear Algebra

The Nonnegative Inverse Eigenvalue Problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of $n$ complex numbers to be the spectrum of an entry--wise nonnegative matrix of dimension $n$. This is a very difficult and long standing problem and has been solved only for $n\leq 4$. In this paper, the NIEP for a particular class of nonnegative matrices, namely Leslie matrices, is considered. Leslie matrices are nonnegative matrices, with a special zero--pattern, arising in the Leslie model, one of the best known and widely used models to describe the growth of populations. The lists ...

On The Block Structure And Frobenius Normal Form Of Powers Of Matrices, 2019 Washington State University

#### On The Block Structure And Frobenius Normal Form Of Powers Of Matrices, Mashael M. Al Baidani, Judi J. Mcdonald

##### Electronic Journal of Linear Algebra

The Frobenius normal form of a matrix is an important tool in analyzing its properties. When a matrix is powered up, the Frobenius normal form of the original matrix and that of its powers need not be the same. In this article, conditions on a matrix $A$ and the power $q$ are provided so that for any invertible matrix $S$, if $S^{-1}A^qS$ is block upper triangular, then so is $S^{-1}AS$ when partitioned conformably. The result is established for general matrices over any field. It is also observed that the contributions of the index of cyclicity ...

The Derived Category Of A Locally Complete Intersection Ring, 2019 University of Nebraska - Lincoln

#### The Derived Category Of A Locally Complete Intersection Ring, Joshua Pollitz

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

Let R be a commutative noetherian ring. A well-known theorem in commutative algebra states that R is regular if and only if every complex with finitely generated homology is a perfect complex. This homological and derived category characterization of a regular ring yields important ring theoretic information; for example, this characterization solved the well-known localization problem" for regular local rings. The main result of this thesis is establishing an analogous characterization for when R is locally a complete intersection. Namely, R is locally a complete intersection if and only if each nontrivial complex with finitely generated homology can build a ...

2019 Shanghai University

#### Pure Psvd Approach To Sylvester-Type Quaternion Matrix Equations, Zhuo-Heng He

##### Electronic Journal of Linear Algebra

In this paper, the pure product singular value decomposition (PSVD) for four quaternion matrices is given. The system of coupled Sylvester-type quaternion matrix equations with five unknowns $X_{i}A_{i}-B_{i}X_{i+1}=C_{i}$ is considered by using the PSVD approach, where $A_{i},B_{i},$ and $C_{i}$ are given quaternion matrices of compatible sizes $(i=1,2,3,4)$. Some necessary and sufficient conditions for the existence of a solution to this system are derived. Moreover, the general solution to this system is presented when it is solvable.

Mathematics And Programming Exercises For Educational Robot Navigation, 2019 Loyola University Chicago

#### Mathematics And Programming Exercises For Educational Robot Navigation, Ronald I. Greenberg

##### Computer Science: Faculty Publications and Other Works

This paper points students towards ideas they can use towards developing a convenient library for robot navigation, with examples based on Botball primitives, and points educators towards mathematics and programming exercises they can suggest to students, especially advanced high school students.

Non-Sparse Companion Matrices, 2019 Redeemer University College

#### Non-Sparse Companion Matrices, Louis Deaett, Jonathan Fischer, Colin Garnett, Kevin N. Vander Meulen

##### Electronic Journal of Linear Algebra

Given a polynomial $p(z)$, a companion matrix can be thought of as a simple template for placing the coefficients of $p(z)$ in a matrix such that the characteristic polynomial is $p(z)$. The Frobenius companion and the more recently-discovered Fiedler companion matrices are examples. Both the Frobenius and Fiedler companion matrices have the maximum possible number of zero entries, and in that sense are sparse. In this paper, companion matrices are explored that are not sparse. Some constructions of non-sparse companion matrices are provided, and properties that all companion matrices must exhibit are given. For example, it is ...

2019 John Carroll University

#### Algebraic Topics In The Classroom – Gauss And Beyond, Lisa Krance

##### Masters Essays

No abstract provided.

2019 Augustana College, Rock Island Illinois

#### The Last Digits Of Infinity (On Tetrations Under Modular Rings), William Stowe

##### Celebration of Learning

A tetration is defined as repeated exponentiation. As an example, 2 tetrated 4 times is 2^(2^(2^2)) = 2^16. Tetrated numbers grow rapidly; however, we will see that when tetrating where computations are performed mod n for some positive integer n, there is convergent behavior. We will show that, in general, this convergent behavior will always show up.

Graded Character Rings, Mackey Functors And Tambara Functors, 2019 The University of Western Ontario

#### Graded Character Rings, Mackey Functors And Tambara Functors, Beatrice Isabelle Chetard

##### Electronic Thesis and Dissertation Repository

Let $G$ be a finite group. The ring $R_\KK(G)$ of virtual characters of $G$ over the field $\KK$ is a $\lambda$-ring; as such, it is equipped with the so-called $\Gamma$-filtration, first defined by Grothendieck. In the first half of this thesis, we explore the properties of the associated graded ring $R^*_\KK(G)$, and present a set of tools to compute it through detailed examples. In particular, we use the functoriality of $R^*_\KK(-)$, and the topological properties of the $\Gamma$-filtration, to explicitly determine the graded character ring over the complex numbers of ...

Dehn Functions Of Bestvina-Brady Groups, 2019 Louisiana State University and Agricultural and Mechanical College

#### Dehn Functions Of Bestvina-Brady Groups, Yu-Chan Chang

##### LSU Doctoral Dissertations

In this dissertation, we prove that if the flag complex on a finite simplicial graph is a 2-dimensional triangulated disk, then the Dehn function of the associated Bestvina--Brady group depends on the maximal dimension of the simplices in the interior of the flag complex. We also give some examples where the flag complex on a finite simplicial graph is not 2-dimensional, and we establish a lower bound for the Dehn function of the associated Bestvina--Brady group.

2019 California State University - San Bernardino

#### Analogues Between Leibniz's Harmonic Triangle And Pascal's Arithmetic Triangle, Lacey Taylor James

##### Electronic Theses, Projects, and Dissertations

This paper will discuss the analogues between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle by utilizing mathematical proving techniques like partial sums, committees, telescoping, mathematical induction and applying George Polya's perspective. The topics presented in this paper will show that Pascal's triangle and Leibniz's triangle both have hockey stick type patterns, patterns of sums within shapes, and have the natural numbers, triangular numbers, tetrahedral numbers, and pentatope numbers hidden within. In addition, this paper will show how Pascal's Arithmetic Triangle can be used to construct Leibniz's Harmonic Triangle and show how both triangles ...

Cone-Constrained Rational Eigenvalue Problems, 2019 University of Avignon

#### Cone-Constrained Rational Eigenvalue Problems, Alberto Seeger

##### Electronic Journal of Linear Algebra

This work deals with the eigenvalue analysis of a rational matrix-valued function subject to complementarity constraints induced by a polyhedral cone $K$. The eigenvalue problem under consideration has the general structure $\left(\sum_{k=0}^d \lambda^k A_k + \sum_{k =1}^m \frac{p_k(\lambda)}{q_k(\lambda)} \,B_k\right) x = y , \quad K\ni x \perp y\in K^\ast,$ where $K^\ast$ denotes the dual cone of $K$. The unconstrained version of this problem has been discussed in [Y.F. Su and Z.J. Bai. Solving rational eigenvalue problems via linearization. \emph{SIAM J. Matrix Anal. Appl.}, 32 ...

Minimum Rank, Maximum Nullity, And Zero Forcing Number Of Graphs, 2019 University of Regina

#### Minimum Rank, Maximum Nullity, And Zero Forcing Number Of Graphs, Shaun M. Fallat, Leslie Hogben

##### Leslie Hogben

This chapter represents an overview of research related to a notion of the “rank of a graph" and the dual concept known as the “nullity of a graph," from the point of view of associating a fixed collection of symmetric or Hermitian matrices to a given graph. This topic and related questions have enjoyed a fairly large history within discrete mathematics, and have become very popular among linear algebraists recently, partly based on its connection to certain inverse eigenvalue problems, but also because of the many interesting applications (e.g., to communication complexity in computer science and to control of ...

Pairwise Completely Positive Matrices And Conjugate Local Diagonal Unitary Invariant Quantum States, 2019 Mount Allison University

#### Pairwise Completely Positive Matrices And Conjugate Local Diagonal Unitary Invariant Quantum States, Nathaniel Johnston, Olivia Maclean

##### Electronic Journal of Linear Algebra

A generalization of the set of completely positive matrices called pairwise completely positive (PCP) matrices is introduced. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise non-negative. Basic properties of these matrix pairs are explored and several testable necessary and sufficient conditions are developed to help determine whether or not a pair is PCP. A connection with quantum entanglement is established by showing that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not ...

On The Intersection Number Of Finite Groups, 2019 University of Texas at Tyler

#### On The Intersection Number Of Finite Groups, Humberto Bautista Serrano

##### Math Theses

Let G be a finite, nontrivial group. In a paper in 1994, Cohn defined the covering number of a finite group as the minimum number of nontrivial proper subgroups whose union is equal to the whole group. This concept has received considerable attention lately, mainly due to the importance of recent discoveries. In this thesis we study a dual concept to the covering number. We define the intersection number of a finite group as the minimum number of maximal subgroups whose intersection is equal to the Frattini subgroup. Similarly we define the inconjugate intersection number of a finite group as ...

2019 Kennesaw State University

#### The Effect Of Experiential Learning On Students’ Conceptual Understanding Of Functions In Algebra 1, Jeremiah Veillon

##### Doctor of Education in Secondary Education Dissertations

For years, traditional mathematics instruction has prioritized memory over thought; this often leads to a disconnect between mathematics and real life, which is a contributing factor for the increased rate of college students having to take remedial mathematics. Experiential Learning (EL) seeks to influence students’ decisions to transfer what they are learning to the classroom by engaging students’ emotional processes through concrete experiences. EL has been successful over the years in increasing student engagement and conceptual understanding; however, more research is needed to examine the effect of EL with secondary mathematics students. The purpose of this study was to examine ...

Proof Of A Conjecture Of Graham And Lovasz Concerning Unimodality Of Coefficients Of The Distance Characteristic Polynomial Of A Tree, 2019 Iowa State University

#### Proof Of A Conjecture Of Graham And Lovasz Concerning Unimodality Of Coefficients Of The Distance Characteristic Polynomial Of A Tree, Ghodratollah Aalipour, Aida Abiad, Zhanar Berikkyzy, Leslie Hogben, Franklin H.J. Kenter, Jephian C.-H. Lin, Michael Tait

##### Leslie Hogben

The conjecture of Graham and Lov ́asz that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal is proved; it is also shown that the (normalized) coefficients are log-concave. Upper and lower bounds on the location of the peak are established.

2019 East Tennessee State University

#### Taking Notes: Generating Twelve-Tone Music With Mathematics, Nathan Molder

##### Electronic Theses and Dissertations

There has often been a connection between music and mathematics. The world of musical composition is full of combinations of orderings of diﬀerent musical notes, each of which has diﬀerent sound quality, length, and em phasis. One of the more intricate composition styles is twelve-tone music, where twelve unique notes (up to octave isomorphism) must be used before they can be repeated. In this thesis, we aim to show multiple ways in which mathematics can be used directly to compose twelve-tone musical scores.

Weyl Groups And The Nil-Hecke Algebra, 2019 University at Albany, State University of New York

#### Weyl Groups And The Nil-Hecke Algebra, Arta Holaj

##### Mathematics and Statistics

We begin this paper with a short survey on finite reflection groups. First we establish what a reflection in Euclidean space is. Then we introduce a root system, which is then partitioned into two sets: one of positive roots and one with negative roots. Th is articulates our understanding of groups generated by simple reflections. Furthermore, we develop our insight to Weyl groups and crystallographic groups before exploring crystallographic root systems. The section section of this paper examines the twisted group algebra along with the Demazure element Xi and the Demazure-Lusztig element Ti. Lastly, the third section of this paper ...