Algebra Commons™

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Full-Text Articles in Algebra

Patterns, Symmetries, And Mathematical Structures In The Arts, Sarah C. Deloach Jan 2020

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.

Inequalities For Sector Matrices And Positive Linear Maps, Fuping Tan, Huimin Chen Oct 2019

Inequalities For Sector Matrices And Positive Linear Maps, Fuping Tan, Huimin Chen

Electronic Journal of Linear Algebra

Ando proved that if $A, B$ are positive definite, then for any positive linear map $\Phi$, it holds \begin{eqnarray*} \Phi(A\sharp_\lambda B)\le \Phi(A)\sharp_\lambda \Phi(B), \end{eqnarray*} where $A\sharp_\lambda B$, $0\le\lambda\le 1$, means the weighted geometric mean of $A, B$. Using the recently defined geometric mean for accretive matrices, Ando's result is extended to sector matrices. Some norm inequalities are considered as well.

Oct 2019

A Concise Workbook For College Algebra 2nd Edition, Fei Ye

Open Educational Resources

This is the second edition of the book "A Concise Workbook for College Algebra". In this new edition, some tips and notes, more exercises and examples were added.

Oct 2019

Beauty, Bees, And God: The Fibonacci Sequence As A Theological Springboard In Secondary Mathematics, John D. Brahier

Journal of Catholic Education

Catholic schools primarily should be in the business of making saints. This article identifies and explores a meaningful, engaging point of contact between mathematics and theology for high school math classes, the Fibonacci Sequence. This sequence serves as an engaging introduction to sequences and series; more importantly, the topic can be used as a springboard to theological discussions. The paper will provide a brief historical background to the Fibonacci Sequence, an explanation of how it can be used in a high school math classroom, and an exploration of three different theological touchpoints that the Fibonacci Sequence offers.

Sharp Bounds For Decomposing Graphs Into Edges And Triangles, Adam Blumenthal, Bernard Lidicky, Oleg Pikhurko, Yanitsa Pehova, Florian Pfender, Jan Volec Sep 2019

Sharp Bounds For Decomposing Graphs Into Edges And Triangles, Adam Blumenthal, Bernard Lidicky, Oleg Pikhurko, Yanitsa Pehova, Florian Pfender, Jan Volec

Bernard Lidický

Let pi3(G) be the minimum of twice the number of edges plus three times the number of triangles over all edge-decompositions of G into copies of K2 and K3. We are interested in the value of pi3(n), the maximum of pi3(G) over graphs G with n vertices. This specific extremal function was first studied by Gyori and Tuza [Decompositions of graphs into complete subgraphs of given order, Studia Sci. Math. Hungar. 22 (1987), 315--320], who showed that pi3(n)<9n2/16.
In a recent advance on this problem, Kral, Lidicky, Martins and Pehova [arXiv:1710:08486] proved via flag ...

The Cone Of Z-Transformations On Lorentz Cone, Sandor Zoltan Nemeth, Muddappa S. Gowda Sep 2019

The Cone Of Z-Transformations On Lorentz Cone, Sandor Zoltan Nemeth, Muddappa S. Gowda

Electronic Journal of Linear Algebra

In this paper, the structural properties of the cone of $\calz$-transformations on the Lorentz cone are described in terms of the semidefinite cone and copositive/completely positive cones induced by the Lorentz cone and its boundary. In particular, its dual is described as a slice of the semidefinite cone as well as a slice of the completely positive cone of the Lorentz cone. This provides an example of an instance where a conic linear program on a completely positive cone is reduced to a problem on the semidefinite cone.

Zeta Functions Of Classical Groups And Class Two Nilpotent Groups, Fikreab Solomon Admasu Sep 2019

Zeta Functions Of Classical Groups And Class Two Nilpotent Groups, Fikreab Solomon Admasu

All Dissertations, Theses, and Capstone Projects

This thesis is concerned with zeta functions and generating series associated with two families of groups that are intimately connected with each other: classical groups and class two nilpotent groups. Indeed, the zeta functions of classical groups count some special subgroups in class two nilpotent groups.

In the first chapter, we provide new expressions for the zeta functions of symplectic groups and even orthogonal groups in terms of the cotype zeta function of the integer lattice. In his paper on universal $p$-adic zeta functions, J. Igusa computed explicit formulae for the zeta functions of classical algebraic groups. These zeta ...

Spn Graphs, Leslie Hogben, Naomi Shaked-Monderer Aug 2019

Spn Graphs, Leslie Hogben, Naomi Shaked-Monderer

Electronic Journal of Linear Algebra

A simple graph G is an SPN graph if every copositive matrix having graph G is the sum of a positive semidefinite and nonnegative matrix. SPN graphs were introduced in [N. Shaked-Monderer. SPN graphs: When copositive = SPN. Linear Algebra Appl., 509:82{113, 2016.], where it was conjectured that the complete subdivision graph of K4 is an SPN graph. This conjecture is disproved, which in conjunction with results in the Shaked-Monderer paper show that a subdivision of K_4 is a SPN graph if and only if at most one edge is subdivided. It is conjectured that a graph is an ...

Alpha Adjacency: A Generalization Of Adjacency Matrices, Matt Hudelson, Judi Mcdonald, Enzo Wendler Aug 2019

Alpha Adjacency: A Generalization Of Adjacency Matrices, Matt Hudelson, Judi Mcdonald, Enzo Wendler

Electronic Journal of Linear Algebra

B. Shader and W. So introduced the idea of the skew adjacency matrix. Their idea was to give an orientation to a simple undirected graph G from which a skew adjacency matrix S(G) is created. The -adjacency matrix extends this idea to an arbitrary field F. To study the underlying undirected graph, the average -characteristic polynomial can be created by averaging the characteristic polynomials over all the possible orientations. In particular, a Harary-Sachs theorem for the average-characteristic polynomial is derived and used to determine a few features of the graph from the average-characteristic polynomial.

The Inverse Eigenvalue Problem For Leslie Matrices, Luca Benvenuti Aug 2019

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, Mashael M. Al Baidani, Judi J. Mcdonald Aug 2019

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, Joshua Pollitz Aug 2019

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 ...

Jul 2019

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, Ronald I. Greenberg Jul 2019

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, Louis Deaett, Jonathan Fischer, Colin Garnett, Kevin N. Vander Meulen Jul 2019

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 ...

Algebraic Topics In The Classroom – Gauss And Beyond, Lisa Krance Jul 2019

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

Masters Essays

No abstract provided.

Jun 2019

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, Beatrice Isabelle Chetard Jun 2019

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, Yu-Chan Chang Jun 2019

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.

Analogues Between Leibniz's Harmonic Triangle And Pascal's Arithmetic Triangle, Lacey Taylor James Jun 2019

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, Alberto Seeger May 2019

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 ...

Pairwise Completely Positive Matrices And Conjugate Local Diagonal Unitary Invariant Quantum States, Nathaniel Johnston, Olivia Maclean May 2019

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, Humberto Bautista Serrano May 2019

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 ...

May 2019

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 ...

Taking Notes: Generating Twelve-Tone Music With Mathematics, Nathan Molder May 2019

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, Arta Holaj May 2019

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 ...

One Teacher's Transformation Of Practice Through The Development Of Covariational Thinking And Reasoning In Algebra : A Self-Study, Jacqueline Dauplaise

Theses, Dissertations and Culminating Projects

CCSSM (2010) describes quantitative reasoning as expertise that mathematics educators should seek to develop in their students. Researchers must then understand how to develop covariational reasoning. The problem is that researchers draw from students’ dialogue as the data for understanding quantitative relationships. As a result, the researcher can only conceive the students’ reasoning. The objective of using the self-study research methodology is to examine and improve existing teaching practices. To improve my practice, I reflected upon the implementation of my algebra curriculum through a hermeneutics cycle of my personal history and living educational theory. The critical friend provoked through dialogues ...

Understanding The Impact Of Peer-Led Workshops On Student Learning, Afolabi Ibitoye, Armando Cosme, Nadia Kennedy May 2019

Understanding The Impact Of Peer-Led Workshops On Student Learning, Afolabi Ibitoye, Armando Cosme, Nadia Kennedy

Publications and Research

As students we often wonder why some subjects are easy to understand and requires not much effort in terms of re-reading the material, for us to grasp what it entails. One subject seems to remain elusive and uneasy for a vast majority of learners at all levels of education; that subject is Mathematics, it is one subject that most learners finds difficult even after doubling the amount of time spent on studying the material. My intention is to explore ways to make Mathematics easier for other students using feedback from students enrolled in NSF mathematics peer leading workshops, and use ...

Analysis Of A Group Of Automorphisms Of A Free Group As A Platform For Conjugacy-Based Group Cryptography, Pavel Shostak

All Dissertations, Theses, and Capstone Projects

Let F be a finitely generated free group and Aut(F) its group of automorphisms.

In this monograph we discuss potential uses of Aut(F) in group-based cryptography.

Our main focus is on using Aut(F) as a platform group for the Anshel-Anshel-Goldfeld protocol, Ko-Lee protocol, and other protocols based on different versions of the conjugacy search problem or decomposition problem, such as Shpilrain-Ushakov protocol.

We attack the Anshel-Anshel-Goldfeld and Ko-Lee protocols by adapting the existing types of the length-based attack to the specifics of Aut(F). We also present our own version of the length-based attack that significantly increases ...

May 2019

On The Complexity Of Computing Galois Groups Of Differential Equations, Mengxiao Sun

All Dissertations, Theses, and Capstone Projects

The differential Galois group is an analogue for a linear differential equation of the classical Galois group for a polynomial equation. An important application of the differential Galois group is that a linear differential equation can be solved by integrals, exponentials and algebraic functions if and only if the connected component of its differential Galois group is solvable. Computing the differential Galois groups would help us determine the existence of the solutions expressed in terms of elementary functions (integrals, exponentials and algebraic functions) and understand the algebraic relations among the solutions.

Hrushovski first proposed an algorithm for computing the differential ...