# Algebra Commons™

## All Articles in Algebra

611 full-text articles. Page 1 of 20.

Elimination For Systems Of Algebraic Differential Equations, 2017 The Graduate Center, City University of New York

#### Elimination For Systems Of Algebraic Differential Equations, Richard Gustavson

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

We develop new upper bounds for several effective differential elimination techniques for systems of algebraic ordinary and partial differential equations. Differential elimination, also known as decoupling, is the process of eliminating a fixed subset of unknown functions from a system of differential equations in order to obtain differential algebraic consequences of the original system that do not depend on that fixed subset of unknowns. A special case of differential elimination, which we study extensively, is the question of consistency, that is, if the given system of differential equations has a solution. We first look solely at the algebraic data" of ...

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 ...

The Enhanced Principal Rank Characteristic Sequence For Hermitian Matrices, 2017 Iowa State University

#### The Enhanced Principal Rank Characteristic Sequence For Hermitian Matrices, Steve Butler, M. Catral, H. Tracy Hall, Leslie Hogben, Xavier Martinez-Rivera, Bryan L. Shader, Pauline Van Den Driessche

##### Electronic Journal of Linear Algebra

The enhanced principal rank characteristic sequence (epr-sequence) of an $n\x n$ matrix is a sequence $\ell_1 \ell_2 \cdots \ell_n$, where each $\ell_k$ is ${\tt A}$, ${\tt S}$, or ${\tt N}$ according as all, some, or none of its principal minors of order $k$ are nonzero. There has been substantial work on epr-sequences of symmetric matrices (especially real symmetric matrices) and real skew-symmetric matrices, and incidental remarks have been made about results extending (or not extending) to (complex) Hermitian matrices. A systematic study of epr-sequences of Hermitian matrices is undertaken; the differences with the case of symmetric matrices are quite ...

2017 Shanghai Jiaotong University

#### Self-Interlacing Polynomials Ii: Matrices With Self-Interlacing Spectrum, Mikhail Tyaglov

##### Electronic Journal of Linear Algebra

An $n\times n$ matrix is said to have a self-interlacing spectrum if its eigenvalues $\lambda_k$, $k=1,\ldots,n$, are distributed as follows: $$\lambda_1>-\lambda_2>\lambda_3>\cdots>(-1)^{n-1}\lambda_n>0.$$ A method for constructing sign definite matrices with self-interlacing spectrum from totally nonnegative ones is presented. This method is applied to bidiagonal and tridiagonal matrices. In particular, a result by O. Holtz on the spectrum of real symmetric anti-bidiagonal matrices with positive nonzero entries is generalized.

#### Session A-3: Three-Act Math Tasks, Lindsey Herlehy

##### Professional Learning Day

Participants will engage in a Three-Act Math task highlighting the application of properties of geometrical figures. Developed by Dan Meyer, an innovative and highly regarded mathematics instructor, Three-Act Math tasks utilize pedagogical skills that elicit student curiosity, collaboration and questioning. By posing a mathematical problem through active storytelling, this instructional approach redefines real-world mathematics and clarifies the role that a student plays in the learning process. Participants will be given multiple resources where they can access Three-Act Math tasks appropriate for upper elementary grades through Algebra and Geometry courses.

Generalized Left And Right Weyl Spectra Of Upper Triangular Operator Matrices, 2017 University of Nis

#### Generalized Left And Right Weyl Spectra Of Upper Triangular Operator Matrices, Guojun Hai, Dragana S. Cvetkovic-Ilic

##### Electronic Journal of Linear Algebra

In this paper, for given operators $A\in\B(\H)$ and $B\in\B(\K)$, the sets of all $C\in \B(\K,\H)$ such that $M_C=\bmatrix{cc} A&C\\0&B\endbmatrix$ is generalized Weyl and generalized left (right) Weyl, are completely described. Furthermore, the following intersections and unions of the generalized left Weyl spectra $$\bigcup_{C\in\B(\K,\H)}\sigma^g_{lw}(M_C) \;\;\; \mbox{and} \;\;\; \bigcap_{C\in\B(\K,\H)}\sigma^g_{lw}(M_C)$$ are also described, and necessary and sufficient conditions which two operators $A\in\B(\H)$ and $B\in\B(\K)$ have to satisfy in order for $M_C$ to be a generalized left Weyl operator for each $C\in\B(\K,\H)$, are presented.

Lorentz Transformation From An Elementary Point Of View, 2017 Laboratoire de Physique Th\'{e}orique, Universit\'{e} de Toulouse III \& CNRS

#### Lorentz Transformation From An Elementary Point Of View, Arkadiusz Jadczyk, Jerzy Szulga

##### Electronic Journal of Linear Algebra

Elementary methods are used to examine some nontrivial mathematical issues underpinning the Lorentz transformation. Its eigen-system is characterized through the exponential of a $G$-skew symmetric matrix, underlining its unconnectedness at one of its extremes (the hyper-singular case). A different yet equivalent angle is presented through Pauli coding which reveals the connection between the hyper-singular case and the shear map.

Links With Finite N-Quandles, 2017 Pitzer College

#### Links With Finite N-Quandles, Jim Hoste, Patrick D. Shanahan

##### Patrick Shanahan

We prove a conjecture of Przytycki which asserts that the n-quandle of a link L in the 3-sphere is finite if and only if the fundamental group of the n-fold cyclic branched cover of the 3-sphere, branched over L, is finite.

2017 Dordt College

#### Specifications Grading In A First Course In Abstract Algebra, Mike Janssen

##### Faculty Work: Comprehensive List

Specifications grading offers an alternative to more traditional, points-based grading and assessment structures. In place of partial credit, students are assessed pass/fail on whether or not they have achieved the learning outcomes being assessed on a given piece of work according to certain specifications, with limited opportunities for revision of non-passing work. This talk will describe the learning outcomes and specifications grading system I used in my Fall 2016 abstract algebra course, as well as student responses.

Zero Forcing Propagation Time On Oriented Graphs, 2017 Saint Olaf College

#### Zero Forcing Propagation Time On Oriented Graphs, Adam Berliner, Chassidy Bozeman, Steve Butler, Minerva Catral, Leslie Hogben, Brenda Kroschel, Jephian C.H. Lin, Nathan Warnberg, Michael Young

##### Mathematics Publications

Zero forcing is an iterative coloring procedure on a graph that starts by initially coloring vertices white and blue and then repeatedly applies the following rule: if any blue vertex has a unique (out-)neighbor that is colored white, then that neighbor is forced to change color from white to blue. An initial set of blue vertices that can force the entire graph to blue is called a zero forcing set. In this paper we consider the minimum number of iterations needed for this color change rule to color all of the vertices blue, also known as the propagation time ...

An Introduction To Boolean Algebras, 2016 California State University - San Bernardino

#### An Introduction To Boolean Algebras, Amy Schardijn

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

This thesis discusses the topic of Boolean algebras. In order to build intuitive understanding of the topic, research began with the investigation of Boolean algebras in the area of Abstract Algebra. The content of this initial research used a particular notation. The ideas of partially ordered sets, lattices, least upper bounds, and greatest lower bounds were used to define the structure of a Boolean algebra. From this fundamental understanding, we were able to study atoms, Boolean algebra isomorphisms, and Stone’s Representation Theorem for finite Boolean algebras. We also verified and proved many properties involving Boolean algebras and related structures ...

Mathematics Education From A Mathematicians Point Of View, 2016 University of Tennessee, Knoxville

#### Mathematics Education From A Mathematicians Point Of View, Nan Woodson Simpson

##### Masters Theses

This study has been written to illustrate the development from early mathematical learning (grades 3-8) to secondary education regarding the Fundamental Theorem of Arithmetic and the Fundamental Theorem of Algebra. It investigates the progression of the mathematics presented to the students by the current curriculum adopted by the Rhea County School System and the mathematics academic standards set forth by the State of Tennessee.

2016 Poly-disciplinary Faculty of Safi

#### Controllability And Nonsingular Solutions Of Sylvester Equations, Hamid Maarouf

##### Electronic Journal of Linear Algebra

The singularity problem of the solutions of some particular Sylvester equations is studied. As a consequence of this study, a good choice of a Sylvester equation which is associated to a linear continuous time system can be made such that its solution is nonsingular. This solution is then used to solve an eigenstructure assignment problem for this system. From a practical point view, this study can also be applied to automatic control when the system is subject to input constraints.

2016 Murray State University

#### Teaching Functions: The Good, The Bad, And The Many Ways To Do Better, Melanie Bayens

##### Honors College Theses

The way functions are taught in school needs improvement. Many times when students are introduced to functions in Algebra 2, the definition is glossed over, the instruction is lacking, and deeper understanding of the concept is lost. This causes problems when students are required to use this knowledge of functions in later classes, particularly, in Precalculus and Calculus. First, this paper will give the definition of a function and its role in mathematics. Second, this paper will delve into the problems of teaching functions the standard way. Finally, it will present multiple alternative methods for teaching functions. Specifically, it will ...

Group Inverse Extensions Of Certain $M$-Matrix Properties, 2016 National Institute of Technology Warangal

#### Group Inverse Extensions Of Certain $M$-Matrix Properties, Appi Reddy K., Kurmayya T., K. C. Sivakumar

##### Electronic Journal of Linear Algebra

In this article, generalizations of certain $M$-matrix properties are proved for the group generalized inverse. The proofs use the notion of proper splittings of one type or the other. In deriving certain results, we make use of a recently introduced notion of a $B_{\#}$-splitting. Applications in obtaining comparison results for the spectral radii of matrices are presented.

On The Characterization And Parametrization Of Strong Linearizations Of Polynomial Matrices, 2016 Alexander Technological Education Institute of Thessaloniki

#### On The Characterization And Parametrization Of Strong Linearizations Of Polynomial Matrices, Efstathios Antoniou, Stavros Vologiannidis

##### Electronic Journal of Linear Algebra

In the present note, a new characterization of strong linearizations, corresponding to a given regular polynomial matrix, is presented. A linearization of a regular polynomial matrix is a matrix pencil which captures the finite spectral structure of the original matrix, while a strong linearization is one incorporating its structure at infinity along with the finite one. In this respect, linearizations serve as a tool for the study of spectral problems where polynomial matrices are involved. In view of their applications, many linearization techniques have been developed by several authors in the recent years. In this note, a unifying approach is ...

The Inverse Along An Element In Rings, 2016 Universidad Politecnica de Valencia

#### The Inverse Along An Element In Rings, Julio Benitez, Enrico Boasso

##### Electronic Journal of Linear Algebra

Several properties of the inverse along an element are studied in the context of unitary rings. New characterizations of the existence of this inverse are proved. Moreover, the set of all invertible elements along a fixed element is fully described. Furthermore, commuting inverses along an element are characterized. The special cases of the group inverse, the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with involutions) are also considered.

Exploring Mathematical Strategies For Finding Hidden Features In Multi-Dimensional Big Datasets, 2016 University of Houston

#### Exploring Mathematical Strategies For Finding Hidden Features In Multi-Dimensional Big Datasets, Tri Duong, Fang Ren, Apurva Mehta

##### STAR (STEM Teacher and Researcher) Presentations

With advances in technology in brighter sources and larger and faster detectors, the amount of data generated at national user facilities such as SLAC is increasing exponentially. Humans have a superb ability to recognize patterns in complex and noisy data and therefore, data is still curated and analyzed by humans. However, a human brain is unable to keep up with the accelerated pace of data generation, and as a consequence, the rate of new discoveries hasn't kept pace with the rate of data creation. Therefore, new procedures to quickly assess and analyze the data are needed. Machine learning approaches ...

Algebra Tutorial For Prospective Calculus Students, 2016 Governors State University

#### Algebra Tutorial For Prospective Calculus Students, Matthew Mckain

##### All Capstone Projects

Many undergraduate degrees require students to take one or more courses in calculus. Majors in mathematics, science, and engineering are expected to enroll in several rigorous calculus courses, but those majoring in social and behavioral sciences and business must also have some basic understanding of calculus. The goal of this project is to create a web-based tutorial that can be used by the GSU Mathematics faculty to reinforce the algebra skills needed for introductory or Applied Calculus. The tutorial covers the concepts of the slopes of lines, polynomial arithmetic, factoring polynomials, rational expressions, solving quadratic equations, linear and polynomial inequalities ...

Some 2-Categorical Aspects In Physics, 2016 The Graduate Center, City University of New York

#### Some 2-Categorical Aspects In Physics, Arthur Parzygnat

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

2-categories provide a useful transition point between ordinary category theory and infinity-category theory where one can perform concrete computations for applications in physics and at the same time provide rigorous formalism for mathematical structures appearing in physics. We survey three such broad instances. First, we describe two-dimensional algebra as a means of constructing non-abelian parallel transport along surfaces which can be used to describe strings charged under non-abelian gauge groups in string theory. Second, we formalize the notion of convex and cone categories, provide a preliminary categorical definition of entropy, and exhibit several examples. Thirdly, we provide a universal description ...