# Algebra Commons™

Articles 1 - 30 of 885

## Full-Text Articles in Algebra

Linear Algebra (Ung), Hashim Saber, Beata Hebda, Piotr Hebda, Benkam Bobga Apr 2018

#### 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:

• Initial Proposal
• Final Report

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

#### 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.

Decoding Book Barcode Images, Yizhou Tao Jan 2018

#### 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, Allen Herman, Mikhael Muzychuk, Bangteng Xu Jan 2018

#### 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 ...

Jan 2018

#### 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), Danielle Cifone, Karan Puri, Debra Masklanko, Ewa Dabkowska Jan 2018

#### 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.

Low Rank Perturbations Of Quaternion Matrices, Christian Mehl, Andre C.M. Ran Dec 2017

#### 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, Zhuo-Heng He, Jianzhen Liu, Tin-Yau Tam Dec 2017

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

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), Irvin Roy Hentzel, David P. Jacobs, Sekhar V. Muddana Dec 2017 #### 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, Irvin R. Hentzel, H. F. Smith Dec 2017 #### 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, Irvin R. Hentzel, Ivan Correa, Luiz Antionio Peresi Dec 2017 #### 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, Irvin R. Hentzel, Luiz Antonio Peresi Dec 2017 #### 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, Kenneth Driessel, Irvin R. Hentzel, Wasin So Dec 2017 #### 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, Giulia Maria Piacentini Cattaneo, Irvin R. Hentzel Dec 2017 #### 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 ... Investigation Of Finite Groups Through Progenitors, Charles Baccari Dec 2017 #### Investigation Of Finite Groups Through Progenitors, Charles Baccari ##### Electronic Theses, Projects, and Dissertations The goal of this presentation is to find original symmetric presentations of finite groups. It is frequently the case, that progenitors factored by appropriate relations produce simple and even sporadic groups as homomorphic images. We have discovered two of the twenty-six sporadic simple groups namely, M12, J1 and the Lie type group Suz(8). In addition several linear and classical groups will also be presented. We will present several progenitors including: 2*12: 22 x (3 : 2), 2*11: PSL2(11), 2*5: (5 : 4) which have produced the homomorphic images: M12 : 2, Suz(8 ... An Introduction To Lie Algebra, Amanda Renee Talley Dec 2017 #### 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 ... Groups Of Matrices That Act Monopotently, Joshua D. Hews, Leo Livshits Nov 2017 #### Groups Of Matrices That Act Monopotently, Joshua D. Hews, Leo Livshits ##### Electronic Journal of Linear Algebra In the present article, the authors continue the line of inquiry started by Cigler and Jerman, who studied the separation of eigenvalues of a matrix under an action of a matrix group. The authors consider groups \Fam{G} of matrices of the form$\left[\small{\begin{smallmatrix} G & 0\\ 0& z \end{smallmatrix}}\right]$, where$z$is a complex number, and the matrices$G$form an irreducible subgroup of$\GL(\C)$. When \Fam{G} is not essentially finite, the authors prove that for each invertible$A$the set$\Fam{G}A$contains a matrix with more than one eigenvalue. The ... On One-Sided (B;C)-Inverses Of Arbitrary Matrices, Julio Benitez, Enrico Boasso, Hongwei Jin Nov 2017 #### On One-Sided (B;C)-Inverses Of Arbitrary Matrices, Julio Benitez, Enrico Boasso, Hongwei Jin ##### Electronic Journal of Linear Algebra In this article, one-sided$(b, c)$-inverses of arbitrary matrices as well as one-sided inverses along a (not necessarily square) matrix, will be studied. In addition, the$(b, c)\$-inverse and the inverse along an element will be also researched in the context of rectangular matrices.

Nilpotent Linear Transformations And The Solvability Of Power-Associative Nilalgebras, Ivan Correa, Irvin R. Hentzel, Pedro Pablo Julca, Luiz Antonio Peresi Nov 2017

#### Nilpotent Linear Transformations And The Solvability Of Power-Associative Nilalgebras, Ivan Correa, Irvin R. Hentzel, Pedro Pablo Julca, Luiz Antonio Peresi

##### Irvin Roy Hentzel

We prove some results about nilpotent linear transformations. As an application we solve some cases of Albert’s problem on the solvability of nilalgebras. More precisely, we prove the following results: commutative power-associative nilalgebras of dimension n and nilindex n − 1 or n − 2 are solvable; commutative power-associative nilalgebras of dimension 7 are solvable.

Nuclear Elements Of Degree 6 In The Free Alternative Algebra, Irvin R. Hentzel, L. A. Peresi Nov 2017

#### Nuclear Elements Of Degree 6 In The Free Alternative Algebra, Irvin R. Hentzel, L. A. Peresi

##### Irvin Roy Hentzel

We construct five new elements of degree 6 in the nucleus of the free alternative algebra. We use the representation theory of the symmetric group to locate the elements. We use the computer algebra system ALBERT and an extension of ALBERT to express the elements in compact form and to show that these new elements are not a consequence of the known degree-5 elements in the nucleus. We prove that these five new elements and four known elements form a basis for the subspace of nuclear elements of degree 6. Our calculations are done using modular arithmetic to save memory ...

Solvability Of Commutative Right-Nilalgebras Satisfying (B (Aa)) A= B ((Aa) A), Ivan Correa, Alicia Labra, Irvin R. Hentzel Nov 2017

#### Solvability Of Commutative Right-Nilalgebras Satisfying (B (Aa)) A= B ((Aa) A), Ivan Correa, Alicia Labra, Irvin R. Hentzel

##### Irvin Roy Hentzel

We study commutative right-nilalgebras of right-nilindex four satisfying the identity (b(aa))a = b((aa)a). Our main result is that these algebras are solvable and not necessarily nilpotent. Our results require characteristic ≠ 2, 3, 5.

Invariant Nonassociative Algebra Structures On Irreducible Representations Of Simple Lie Algebras, Murray Bremner, Irvin R. Hentzel Nov 2017

#### Invariant Nonassociative Algebra Structures On Irreducible Representations Of Simple Lie Algebras, Murray Bremner, Irvin R. Hentzel

##### Irvin Roy Hentzel

An irreducible representation of a simple Lie algebra can be a direct summand of its own tensor square. In this case, the representation admits a nonassociative algebra structure which is invariant in the sense that the Lie algebra acts as derivations. We study this situation for the Lie algebra sl(2).

The Nucleus Of The Free Alternative Algebra, Irvin R. Hentzel, L. A. Peresi Nov 2017

#### The Nucleus Of The Free Alternative Algebra, Irvin R. Hentzel, L. A. Peresi

##### Irvin Roy Hentzel

We use a computer procedure to determine a basis of the elements of degree 5 in the nucleus of the free alternative algebra. In order to save computer memory, we do our calculations over the field Z103. All calculations are made with multilinear identities. Our procedure is also valid for other characteristics and for determining nuclear elements of higher degree.

Rings With (A, B, C) = (A, C, B) And (A, [B, C]D) = 0: A Case Study Using Albert, Irvin R. Hentzel, D. P. Jacobs, Erwin Kleinfeld Nov 2017

#### Rings With (A, B, C) = (A, C, B) And (A, [B, C]D) = 0: A Case Study Using Albert, Irvin R. Hentzel, D. P. Jacobs, Erwin Kleinfeld

##### Irvin Roy Hentzel

Albert is an interactive computer system for building nonassociative algebras [2]. In this paper, we suggest certain techniques for using Albert that allow one to posit and test hypotheses effectively. This process provides a fast way to achieve new results, and interacts nicely with traditional methods. We demonstrate the methodology by proving that any semiprime ring, having characteristic ≠ 2, 3, and satisfying the identities (a, b, c) - (a, c, b) = (a, [b, c], d) = 0, is associative. This generalizes a recent result by Y. Paul [7].

A Variety Containing Jordan And Pseudo-Composition Algebras, Irvin R. Hentzel, Luiz Antonio Peresi Nov 2017

#### A Variety Containing Jordan And Pseudo-Composition Algebras, Irvin R. Hentzel, Luiz Antonio Peresi

##### Irvin Roy Hentzel

We consider 3-Jordan algebras, i.e., the nonassociative commutative algebras satisfying (x^3 y)x=x^3(yx). The variety of 3-Jordan algebras contains all Jordan algebras and all pseudo-composition algebras. We prove that a simple 3-Jordan algebra with idempotent is either a Jordan algebra or a pseudo-composition algebra.

Commutative Finitely Generated Algebras Satisfying ((Yx)X)X=0 Are Solvable, Ivan Correa, Irvin R. Hentzel Nov 2017

#### Commutative Finitely Generated Algebras Satisfying ((Yx)X)X=0 Are Solvable, Ivan Correa, Irvin R. Hentzel

##### Irvin Roy Hentzel

No abstract provided.

Identities Relating The Jordan Product And The Associator In The Free Nonassociative Algebra, Murray R. Bremner, Irvin R. Hentzel Nov 2017

#### Identities Relating The Jordan Product And The Associator In The Free Nonassociative Algebra, Murray R. Bremner, Irvin R. Hentzel

##### Irvin Roy Hentzel

We determine the identities of degree ≤ 6 satisfied by the symmetric (Jordan) product a○b = ab + ba and the associator [a,b,c] = (ab)c - a(bc) in every nonassociative algebra. In addition to the commutative identity a○b = b○a we obtain one new identity in degree 4 and another new identity in degree 5. We demonstrate the existence of further new identities in degree 6. These identities define a variety of binary-ternary algebras which generalizes the variety of Jordan algebras in the same way that Akivis algebras generalize Lie algebras.

Generalized Alternative And Malcev Algebras, Irvin R. Hentzel, H.F. Smith Nov 2017

#### Generalized Alternative And Malcev Algebras, Irvin R. Hentzel, H.F. Smith

##### Irvin Roy Hentzel

No abstract provided.

Generalized Right Alternative Rings, Irvin R. Hentzel Nov 2017

#### Generalized Right Alternative Rings, Irvin R. Hentzel

##### Irvin Roy Hentzel

We show that weakening the hypotheses of right alternative rings to the three identities (1) (ab,c,d) + (a,b,[c,d]) = a(b,c,d) + (a,c,d)b (2) (α,α,α) = 0 (3) ([a,b],b,b) = O for all α, b, c, d in the ring will not lead to any new simple rings. In fact, the ideal generated by each associator of the form (a, b, b) is a nilpotent ideal of index at most three. Our proofs require characteristic ^2 , ^3 .

Fast Change Of Basis In Algebras, Irvin R. Hentzel, David Pokrass Jacobs Nov 2017

#### Fast Change Of Basis In Algebras, Irvin R. Hentzel, David Pokrass Jacobs

##### Irvin Roy Hentzel

Given an n-dimensional algebraA represented by a basisB and structure constants, and given a transformation matrix for a new basisC., we wish to compute the structure constants forA relative to C. There is a straightforward way to solve this problem inO(n5) arithmetic operations. However given an O(nω) matrix multiplication algorithm, we show how to solve the problem in time O(nω+1). Using the method of Coppersmith and Winograd, this yields an algorithm ofO(n3.376).