# Algebra Commons™

## All Articles in Algebra

1,047 full-text articles. Page 1 of 39.

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.

2019 Georgia State University

#### The Minimal Generating Set Of The Presentation Ideal Of Backelin Semigroup Ring, Arun Suresh

##### Georgia Undergraduate Research Conference (GURC)

Let $K$ be a field, and let n1, ..., nr be positive integers. The Ideal of relations of the semigroup ring K[x1, ..., xr] generated by n1,..., nr is P(n1,..., nr) = {f(x1,...,xr): f(x1,...,xr) \in K[x1,...,xr], f(t^{n1}, ... ,t^{nr}) = 0} with t is transcendental over K.

In 1970, Herzog showed that the least upper bound on the number of generators of $P(n1,..., nr) for r=3 is 3. In 1975, Bresinsky showed that the ... Inequalities For Sector Matrices And Positive Linear Maps, 2019 Shanghai University #### Inequalities For Sector Matrices And Positive Linear Maps, Fuping Tan, Huimin Chen ##### Electronic Journal of Linear Algebra Ando proved that if$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. 2019 Queensborough Community College #### 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. 2019 Divine Child High School #### 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, 2019 Iowa State University #### 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, 2019 University of Birmingham #### 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, 2019 The Graduate Center, City University of New York #### 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, 2019 Iowa State University #### 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, 2019 Washington State University #### 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, 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.