Linear Algebra (Ung), 2018 University of North Georgia

#### Linear Algebra (Ung), Hashim Saber, Beata Hebda, Piotr Hebda, Benkam Bobga

*Mathematics Grants Collections*

This Grants Collection for Linear Algebra was created under a Round Seven ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

- Linked Syllabus
- Initial Proposal
- Final Report

Monomial Progenitors And Related Topics, 2018 California State University - San Bernardino

#### Monomial Progenitors And Related Topics, Madai Obaid Alnominy

*Electronic Theses, Projects, and Dissertations*

The main objective of this project is to find the original symmetric presentations of some very important finite groups and to give our constructions of some of these groups. We have found the Mathieu sporadic group M_{11}, HS × D_{5}, where HS is the sporadic group Higman-Sim group, the projective special unitary group U(3; 5) and the projective special linear group L_{2}(149) as homomorphic images of the monomial progenitors 11*^{4} :_{m} (5 :4), 5*^{6 } :_{m} S_{5} and 149*^{2 } :_{m } D_{37}. We have also discovered 2^{4} : S_{3} × C_{2}, 2 ...

Progenitors, Symmetric Presentations, And Related Topics, 2018 California State University-San Bernardino

#### Progenitors, Symmetric Presentations, And Related Topics, Joana Viridiana Luna

*Electronic Theses, Projects, and Dissertations*

Abstract

A progenitor developed by Robert T. Curtis is a type of infinite groups formed by the semi-direct product of a free group m∗n and a transitive permutation group of degree n. To produce finite homomorphic images we had to add relations to the progenitor of the form 2∗n : N. In this thesis we have investigated several permutations progenitors and monomials, 2∗12 : S4, 2∗12 : S4 × 2, 2∗13 : (13 : 4), 2∗30 : ((2• : 3) : 5), 2∗13 :13,2∗13 :(13:2),2∗13 :(13:S3),53∗2 :m (13:4),7∗8 :m (32 :8 ...

Progenitors, Symmetric Presentations And Constructions, 2018 California State University - San Bernardino

#### Progenitors, Symmetric Presentations And Constructions, Diana Aguirre

*Electronic Theses, Projects, and Dissertations*

Abstract

In this project, we searched for new constructions and symmetric presentations of important groups, nonabelian simple groups, their automorphism groups, or groups that have these as their factor groups. My target nonabelian simple groups included sporadic groups, linear groups, and alternating groups. In addition, we discovered finite groups as homomorphic images of progenitors and proved some of their isomorphism type and original symmetric presentations. In this thesis we found original symmeric presentations of M12, J1 and the simplectic groups S(4,4) and S(3,4) on various con- trol groups. Using the technique of double coset enumeration we ...

Algebraic Methods For The Construction Of Algebraic-Difference Equations With Desired Behavior, 2018 Aristotle University of Thessaloniki

#### Algebraic Methods For The Construction Of Algebraic-Difference Equations With Desired Behavior, Lazaros Moysis, Nicholas Karampetakis

*Electronic Journal of Linear Algebra*

For a given system of algebraic and difference equations, written as an Auto-Regressive (AR) representation $A(\sigma)\beta(k)=0$, where $\sigma $ denotes the shift forward operator and $A\left( \sigma \right) $ a regular polynomial matrix, the forward-backward behavior of this system can be constructed by using the finite and infinite elementary divisor structure of $A\left( \sigma \right) $. This work studies the inverse problem: Given a specific forward-backward behavior, find a family of regular or non-regular polynomial matrices $A\left( \sigma \right) $, such that the constructed system $A\left( \sigma \right) \beta \left( k\right) =0$ has exactly the ...

Italian Folk Multiplication Algorithm Is Indeed Better: It Is More Parallelizable, 2018 University of Texas at El Paso

#### Italian Folk Multiplication Algorithm Is Indeed Better: It Is More Parallelizable, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

*Departmental Technical Reports (CS)*

Traditionally, many ethnic groups had their own versions of arithmetic algorithms. Nowadays, most of these algorithms are studied mostly as pedagogical curiosities, as an interesting way to make arithmetic more exciting to the kids: by applying to their patriotic feelings -- if they are studying the algorithms traditionally used by their ethic group -- or simply to their sense of curiosity. Somewhat surprisingly, we show that one of these algorithms -- a traditional Italian multiplication algorithm -- is actually in some reasonable sense better than the algorithm that we all normally use -- namely, it is easier to parallelize.

College Algebra Through Problem Solving (2018 Edition), 2018 CUNY Queensborough Community College

#### College Algebra Through Problem Solving (2018 Edition), Danielle Cifone, Karan Puri, Debra Maslanko, Ewa Dabkowska

*Open Educational Resources*

This is a self-contained, open educational resource (OER) textbook for college algebra. Students can use the book to learn concepts and work in the book themselves. Instructors can adapt the book for use in any college algebra course to facilitate active learning through problem solving. Additional resources such as classroom assessments and online/printable homework is available from the authors.

Abelian Subalgebras Of Maximal Dimension In Euclidean Lie Algebras, 2018 Wilfrid Laurier University

#### Abelian Subalgebras Of Maximal Dimension In Euclidean Lie Algebras, Mark Curro

*Theses and Dissertations (Comprehensive)*

In this paper we define, discuss and prove the uniqueness of the abelian subalgebra of maximal dimension of the Euclidean Lie algebra. We also construct a family of maximal abelian subalgebras and prove that they are maximal.

College Algebra Through Problem Solving (2018 Edition), 2018 CUNY Queensborough Community College

#### College Algebra Through Problem Solving (2018 Edition), Danielle Cifone, Karan Puri, Debra Masklanko, Ewa Dabkowska

*Open Educational Resources*

This is a self-contained, open educational resource (OER) textbook for college algebra. Students can use the book to learn concepts and work in the book themselves. Instructors can adapt the book for use in any college algebra course to facilitate active learning through problem solving. Additional resources such as classroom assessments and online/printable homework is available from the authors.

Decoding Book Barcode Images, 2018 Claremont McKenna College

#### Decoding Book Barcode Images, Yizhou Tao

*CMC Senior Theses*

This thesis investigated a method of barcode reconstruction to address the recovery of a blurred and convoluted one-dimensional barcode. There are a lot of types of barcodes used today, such as Code 39, Code 93, Code 128, etc. Our algorithm applies to the universal barcode, EAN 13. We extend the methodologies proposed by Iwen et al. (2013) in the journal article "A Symbol-Based Algorithm for Decoding barcodes." The algorithm proposed in the paper requires a signal measured by a laser scanner as an input. The observed signal is modeled as a true signal corrupted by a Gaussian convolution, additional noises ...

Noncommutative Reality-Based Algebras Of Rank 6, 2018 University of Regina

#### Noncommutative Reality-Based Algebras Of Rank 6, Allen Herman, Mikhael Muzychuk, Bangteng Xu

*EKU Faculty and Staff Scholarship*

We show that noncommutative standard reality-based algebras (RBAs) of dimension 6 are determined up to exact isomorphism by their character tables. We show that the possible character tables of these RBAs are determined by seven real numbers, the first four of which are positive and the remaining three real numbers can be arbitrarily chosen up to a single exception. We show how to obtain a concrete matrix realization of the elements of the RBA-basis from the character table. Using a computer implementation, we give a list of all noncommutative integral table algebras of rank 6 with orders up to 150 ...

Low Rank Perturbations Of Quaternion Matrices, 2017 TU Berlin

#### Low Rank Perturbations Of Quaternion Matrices, Christian Mehl, Andre C.M. Ran

*Electronic Journal of Linear Algebra*

Low rank perturbations of right eigenvalues of quaternion matrices are considered. For real and complex matrices it is well known that under a generic rank-$k$ perturbation the $k$ largest Jordan blocks of a given eigenvalue will disappear while additional smaller Jordan blocks will remain. In this paper, it is shown that the same is true for real eigenvalues of quaternion matrices, but for complex nonreal eigenvalues the situation is different: not only the largest $k$, but the largest $2k$ Jordan blocks of a given eigenvalue will disappear under generic quaternion perturbations of rank $k$. Special emphasis is also given ...

The General $\Phi$-Hermitian Solution To Mixed Pairs Of Quaternion Matrix Sylvester Equations, 2017 Auburn University

#### The General $\Phi$-Hermitian Solution To Mixed Pairs Of Quaternion Matrix Sylvester Equations, Zhuo-Heng He, Jianzhen Liu, Tin-Yau Tam

*Electronic Journal of Linear Algebra*

Let $\mathbb{H}^{m\times n}$ be the space of $m\times n$ matrices over $\mathbb{H}$, where $\mathbb{H}$ is the real quaternion algebra. Let $A_{\phi}$ be the $n\times m$ matrix obtained by applying $\phi$ entrywise to the transposed matrix $A^{T}$, where $A\in\mathbb{H}^{m\times n}$ and $\phi$ is a nonstandard involution of $\mathbb{H}$. In this paper, some properties of the Moore-Penrose inverse of the quaternion matrix $A_{\phi}$ are given. Two systems of mixed pairs of quaternion matrix Sylvester equations $A_{1}X-YB_{1}=C_{1},~A_{2}Z-YB_{2}=C_ ...

Experimenting With The Identity (Xy)Z = Y(Zx), 2017 Iowa State University

#### Experimenting With The Identity (Xy)Z = Y(Zx), Irvin Roy Hentzel, David P. Jacobs, Sekhar V. Muddana

*Irvin Roy Hentzel*

An experiment with the nonassociative algebra program Albert led to the discovery of the following surprising theorem. *Let G be a groupoid satisfying the identity (xy)z = y(zx). Then for products in G involving at least five elements, all factors commute and associate.* A corollary is that any semiprime ring satisfying this identity must be commutative and associative, generalizing a known result of Chen.

Semiprime Locally(-1, 1) Ring With Minimal Condition, 2017 Iowa State University

#### Semiprime Locally(-1, 1) Ring With Minimal Condition, Irvin R. Hentzel, H. F. Smith

*Irvin Roy Hentzel*

Let L be a left ideal of a right alternative ring A with characteristic ::/=2. If L is maximal and nil, then L is a two-sided ideal. If L is minimal, then it is either a two-sided ideal, or the ideal it generates is contained in the right nucleus of A. In particular, if A is prime, then a minimal left ideal of A must be a two-sided ideal. Let A be a semiprime locally (-1, 1) ring with characteristic ::1=2, 3. Then A is isomorphic to a subdirect sum of an alternative ring, a strong (-1, 1) ring ...

Minimal Identities Of Bernstein Alegebras, 2017 Iowa State University

#### Minimal Identities Of Bernstein Alegebras, Irvin R. Hentzel, Ivan Correa, Luiz Antionio Peresi

*Irvin Roy Hentzel*

We construct the minimal identities for Bernstein algebras, exceptional Bernstein algebras and normal Bernstein algebras. We use the technique of processing identities via the representation of the symmetric groups. The computer algorithms for creating the standard tableaus and the integral representations are summarized.

Counterexamples In Nonassociative Algebra, 2017 Iowa State University

#### Counterexamples In Nonassociative Algebra, Irvin R. Hentzel, Luiz Antonio Peresi

*Irvin Roy Hentzel*

We present a method of constructing counterexamples in nonassociative algebra. The heart of the computation is constructing a matrix of identities and reducing this matrix (usually very sparse) to row canonical form. The example is constructed from the entries in one column of this row canonical form. While this procedure is not polynomial in the degree of the identity, several shortcuts are listed which shorten calculations. Several examples are given.

On Preserving Structured Matrices Using Double Bracket Operators: Tridiagonal And Toeplitz Matrices, 2017 University of Wyoming

#### On Preserving Structured Matrices Using Double Bracket Operators: Tridiagonal And Toeplitz Matrices, Kenneth Driessel, Irvin R. Hentzel, Wasin So

*Irvin Roy Hentzel*

In the algebra of square matrices over the complex numbers, denotes Two problems are solved: (1) Find all Hermitian matrices M which have the following property: For every Hermitian matrix A, if A is tridiagonal, then so is (2) Find all Hermitian matrices M which have the following property: For every Hermitian matrix A, if A is Toeplitz, then so is

On Prime Right Alternative Algebras And Alternators, 2017 Italian National Research Council

#### On Prime Right Alternative Algebras And Alternators, Giulia Maria Piacentini Cattaneo, Irvin R. Hentzel

*Irvin Roy Hentzel*

We study subvarieties of the variety of right alternative algebras over a field of characteristic t2,t3 such that the defining identities of the variety force the span of the alternators to be an ideal and do not force an algebra with identity element to be alternative. We call a member of such a variety a right alternative alternator ideal algebra. We characterize the algebras of this subvariety by finding an identity which holds if and only if an algebra belongs to the subvariety. We use this identity to prove that if R is a prime, right alternative alternator ideal ...

An Introduction To Lie Algebra, 2017 California State University – San Bernardino

#### An Introduction To Lie Algebra, Amanda Renee Talley

*Electronic Theses, Projects, and Dissertations*

An (associative) algebra is a vector space over a field equipped with an associative, bilinear multiplication. By use of a new bilinear operation, any associative algebra morphs into a nonassociative abstract Lie algebra, where the new product in terms of the given associative product, is the commutator. The crux of this paper is to investigate the commutator as it pertains to the general linear group and its subalgebras. This forces us to examine properties of ring theory under the lens of linear algebra, as we determine subalgebras, ideals, and solvability as decomposed into an extension of abelian ideals, and nilpotency ...