# Algebra Commons™

## All Articles in Algebra

880 full-text articles. Page 6 of 32.

Simple And Semi-Simple Artinian Rings, 2017 California State University - San Bernardino

#### Simple And Semi-Simple Artinian Rings, Ulyses Velasco

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

The main purpose of this paper is to examine the road towards the structure of simple and semi-simple Artinian rings. We refer to these structure theorems as the Wedderburn-Artin theorems. On this journey, we will discuss R-modules, the Jacobson radical, Artinian rings, nilpotency, idempotency, and more. Once we reach our destination, we will examine some implications of these theorems. As a fair warning, no ring will be assumed to be commutative, or to have unity. On that note, the reader should be familiar with the basic findings from Group Theory and Ring Theory.

Elementary Abstract Algebra, 2017 Plymouth State University

#### Elementary Abstract Algebra, Emma Norbrothen Wright

##### Open Educational Resources

No abstract provided.

Projective Partitions Of Vector Spaces, 2017 Siena College

#### Projective Partitions Of Vector Spaces, Mohammad Javaheri

##### Electronic Journal of Linear Algebra

Given infinite-dimensional real vector spaces $V,W$ with $|W| \leq |V|$, it is shown that there exists a collection of subspaces of $V$ that are isomorphic to $W$, mutually intersect only at 0, and altogether cover $V$.

Note On Von Neumann And Rényi Entropies Of A Graph, 2017 Iowa State University

#### Note On Von Neumann And Rényi Entropies Of A Graph, Michael Dairyko, Leslie Hogben, Jephian C.H. Lin, Joshua Lockhart, David Roberson, Simone Severini, Michael Young

##### Mathematics Publications

We conjecture that all connected graphs of order n have von Neumann entropy at least as great as the star K1;n1 and prove this for almost all graphs of order n. We show that connected graphs of order n have Renyi 2-entropy at least as great as K1;n1 and for > 1, Kn maximizes Renyi -entropy over graphs of order n. We show that adding an edge to a graph can lower its von Neumann entropy.

Singular Value And Norm Inequalities Associated With 2 X 2 Positive Semidefinite Block Matrices, 2017 Zarqa University

#### Singular Value And Norm Inequalities Associated With 2 X 2 Positive Semidefinite Block Matrices, Aliaa Burqan, Fuad Kittaneh

##### Electronic Journal of Linear Algebra

This paper aims to give singular value and norm inequalities associated with $2\times 2$ positive semidefinite block matrices.

2017 The College at Brockport

#### A Financial Literacy Curriculum Project On Linear Functions In Algebra I Aligned With New York State Common Core State Standards, Michael T. Hughson Jr

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

There is a lack of financial literacy curriculum available for teachers to support students learning of mathematical skills that are needed in real life. Financial literacy curricula can support students learning real life skills needed before independent living situations in college or career. Direct mathematical modelling and application can support students comprehension of the relationship between financial literacy and algebra. This curriculum highlights the ideology of the need to know drives learning and aligns real world financial literacy problems to the Common Core State Standards (CCSS) algebra standards.

On Rings Of Invariants For Cyclic P-Groups, 2017 University of Arkansas, Fayetteville

#### On Rings Of Invariants For Cyclic P-Groups, Daniel Juda

##### Theses and Dissertations

This thesis studies the ring of invariants R^G of a cyclic p-group G acting on k[x_1,\ldots, x_n] where k is a field of characteristic p >0. We consider when R^G is Cohen-Macaulay and give an explicit computation of the depth of R^G. Using representation theory and a result of Nakajima, we demonstrate that R^G is a unique factorization domain and consequently quasi-Gorenstein. We answer the question of when R^G is F-rational and when R^G is F-regular.

We also study the a-invariant for a graded ring S, that is, the maximal graded degree ...

Application Of Symplectic Integration On A Dynamical System, 2017 East Tennessee State University

#### Application Of Symplectic Integration On A Dynamical System, William Frazier

##### Electronic Theses and Dissertations

Molecular Dynamics (MD) is the numerical simulation of a large system of interacting molecules, and one of the key components of a MD simulation is the numerical estimation of the solutions to a system of nonlinear differential equations. Such systems are very sensitive to discretization and round-off error, and correspondingly, standard techniques such as Runge-Kutta methods can lead to poor results. However, MD systems are conservative, which means that we can use Hamiltonian mechanics and symplectic transformations (also known as canonical transformations) in analyzing and approximating solutions. This is standard in MD applications, leading to numerical techniques known as symplectic ...

2017 Chapman University

#### Relation Algebras, Idempotent Semirings And Generalized Bunched Implication Algebras, Peter Jipsen

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

This paper investigates connections between algebraic structures that are common in theoretical computer science and algebraic logic. Idempotent semirings are the basis of Kleene algebras, relation algebras, residuated lattices and bunched implication algebras. Extending a result of Chajda and Länger, we show that involutive residuated lattices are determined by a pair of dually isomorphic idempotent semirings on the same set, and this result also applies to relation algebras. Generalized bunched implication algebras (GBI-algebras for short) are residuated lattices expanded with a Heyting implication. We construct bounded cyclic involutive GBI-algebras from so-called weakening relations, and prove that the class of weakening ...

Six Septembers: Mathematics For The Humanist, 2017 Duquesne University

#### Six Septembers: Mathematics For The Humanist, Patrick Juola, Stephen Ramsay

##### Zea E-Books

Scholars of all stripes are turning their attention to materials that represent enormous opportunities for the future of humanistic inquiry. The purpose of this book is to impart the concepts that underlie the mathematics they are likely to encounter and to unfold the notation in a way that removes that particular barrier completely. This book is a primer for developing the skills to enable humanist scholars to address complicated technical material with confidence. This book, to put it plainly, is concerned with the things that the author of a technical article knows, but isn’t saying. Like any field, mathematics ...

A Game Of Monovariants On A Checkerboard, 2017 Lynchburg College

#### A Game Of Monovariants On A Checkerboard, Linwood Reynolds

##### Student Scholar Showcase

Abstract: Assume there is a game that takes place on a 20x20 checkerboard in which each of the 400 squares are filled with either a penny, nickel, dime, or quarter. The coins are placed randomly onto the squares, and there are to be 100 of each of the coins on the board. To begin the game, 59 coins are removed at random. The goal of the game is to remove each remaining coin from the board according to the following rules: 1. A penny can only be removed if all 4 adjacent squares are empty. That is, a penny cannot ...

Involutions And Total Orthogonality In Some Finite Classical Groups, 2017 College of William and Mary

#### Involutions And Total Orthogonality In Some Finite Classical Groups, Gregory K. Taylor

A group $G$ is called \emph{real} if every element is conjugate to its inverse, and $G$ is \emph{strongly real} if each of the conjugating elements may be chosen to be an involution, an element in $G$ which squares to the identity. Real groups are called as such because every irreducible character of a real group is real valued. A group $G$ is called \emph{totally orthogonal} if every irreducible complex representation is realizable over the field of real numbers. Total orthogonality is sufficient, but not necessary for reality.

Reality of representations is quantified in the Frobenius-Schur indicator. For ...

Bridging The Gap Between College Algebra And Agronomic Math, 2017 Iowa State University

#### Bridging The Gap Between College Algebra And Agronomic Math, Haleigh Nicole Summers

##### Honors Projects and Posters

Practitioners of agronomy are often faced with scenarios involving math during their daily activities. Students studying agronomy are required to take college algebra but often miss the opportunity to bridge the gap between general algebra and agronomic math. The purpose of this research was to evaluate the effectiveness of two delivery methods for teaching agronomic math. Videos and posters were created to demonstrate: fertilizer application, unit conversions, irrigation, yield estimation, and growing degree day calculation. We predicted using videos to teach agronomic math would be more effective by providing both auditory and visual teaching methods, while posters only provide visual ...

Laurent Series Obtained By Long Division, 2017 Iowa State University

#### Laurent Series Obtained By Long Division, A. Abian, Leslie Hogben, Elgin H. Johnston

##### Leslie Hogben

Let r1,...,rn be the n root-moduli of the polynomial azn+bzm+c, where n>m>0 are integers and a,b,c are nonzero complex numbers. We give a necessary and sufficient condition in order that the long division of .1 by bzm+azn+c (where contrary to traditional long division, the divisor is ordered neither in the ascending nor in the descending powers of z) yield the Laurent series of 1/(azn+bzm+c) valid in the annulus rk< IzI k+1 for some root-modulus rk. Our method gives an effective way of obtaining Laurent series of 1 ...

2017 The University of Western Ontario

#### Metafork: A Compilation Framework For Concurrency Models Targeting Hardware Accelerators, Xiaohui Chen

##### Electronic Thesis and Dissertation Repository

Parallel programming is gaining ground in various domains due to the tremendous computational power that it brings; however, it also requires a substantial code crafting effort to achieve performance improvement. Unfortunately, in most cases, performance tuning has to be accomplished manually by programmers. We argue that automated tuning is necessary due to the combination of the following factors. First, code optimization is machine-dependent. That is, optimization preferred on one machine may be not suitable for another machine. Second, as the possible optimization search space increases, manually finding an optimized configuration is hard. Therefore, developing new compiler techniques for optimizing applications ...

Minimum Rank Of Skew-Symmetric Matrices Described By A Graph, 2017 University of Wyoming

#### Minimum Rank Of Skew-Symmetric Matrices Described By A Graph, Mary Allison, Elizabeth Bodine, Luz Maria Dealba, Joyati Debnath, Laura Deloss, Colin Garnett, Jason Grout, Leslie Hogben, Bokhee Im, Hana Kim, Reshmi Nair, Olga Pryporova, Kendrick Savage, Bryan Shader, Amy Wangsness Wehe

##### Leslie Hogben

The minimum (symmetric) rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. The problem of determining minimum (symmetric) rank has been studied extensively. We define the minimum skew rank of a simple graph G to be the smallest possible rank among all skew-symmetric matrices over F whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. We apply ...

Potentially Eventually Exponentially Positive Sign Patterns, 2017 Columbia College

#### Potentially Eventually Exponentially Positive Sign Patterns, Marie Archer, Minerva Catral, Craig Erickson, Rana Haber, Leslie Hogben, Xavier Martinez-Rivera, Antonio Ochoa

##### Leslie Hogben

We introduce the study of potentially eventually exponentially positive (PEEP) sign patterns and establish several results using the connections between these sign patterns and the potentially eventually positive (PEP) sign patterns. It is shown that the problem of characterizing PEEP sign patterns is not equivalent to that of characterizing PEP sign patterns. A characterization of all 2×2 and 3×3 PEEP sign patterns is given.

2017 Iowa State University

#### The Symmetric M-Matrix And Symmetric Inverse M-Matrix Completion Problems, Leslie Hogben

##### Leslie Hogben

The symmetric M-matrix and symmetric M0-matrix completion problems are solved and results of Johnson and Smith [Linear Algebra Appl. 290 (1999) 193] are extended to solve the symmetric inverse M-matrix completion problem: (1) A pattern (i.e., a list of positions in an n×n matrix) has symmetric M-completion (i.e., every partial symmetric M-matrix specifying the pattern can be completed to a symmetric M-matrix) if and only if the principal subpattern R determined by its diagonal is permutation similar to a pattern that is block diagonal with each diagonal block complete, or, in graph theoretic terms, if and only ...

Spectrally Arbitrary Patterns: Reducibility And The 2n Conjecture For N = 5, 2017 Drake University

#### Spectrally Arbitrary Patterns: Reducibility And The 2n Conjecture For N = 5, Luz M. Dealba, Irvin R. Hentzel, Leslie Hogben, Judith Mcdonald, Rana Mikkelson, Olga Pryporova, Bryan Shader, Kevin N. Vander Meulen

##### Leslie Hogben

A sign pattern Z (a matrix whose entries are elements of {+, −, 0}) is spectrally arbitrary if for any self-conjugate spectrum there is a real matrix with sign pattern Z having the given spectrum. Spectrally arbitrary sign patterns were introduced in [J.H. Drew, C.R. Johnson, D.D. Olesky, P. van den Driessche, Spectrally arbitrary patterns, Linear Algebra Appl. 308 (2000) 121–137], where it was (incorrectly) stated that if a sign pattern Z is reducible and each of its irreducible components is a spectrally arbitrary sign pattern, then Z is a spectrally arbitrary sign pattern, and it was conjectured ...

2017 Iowa State University

#### The Copositive Completion Problem: Unspecified Diagonal Entries, Leslie Hogben

##### Leslie Hogben

In [L. Hogben, C.R. Johnson, R. Reams, The copositive matrix completion problem, Linear Algebra Appl. 408 (2005) 207–211] it was shown that any partial (strictly) copositive matrix all of whose diagonal entries are specified can be completed to a (strictly) copositive matrix. In this note we show that every partial strictly copositive matrix (possibly with unspecified diagonal entries) can be completed to a strictly copositive matrix, but there is an example of a partial copositive matrix with an unspecified diagonal entry that cannot be completed to a copositive matrix.