Nuclear Elements Of Degree 6 In The Free Alternative Algebra, 2017 Iowa State University

#### 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), 2017 Universidad de Chile

#### 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, 2017 University of Saskatchewan

#### 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, 2017 Iowa State University

#### 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, 2017 Iowa State University

#### 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, 2017 Iowa State University

#### 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, 2017 Universidad Metropolitana de Ciencias de la Educación

#### 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, 2017 University of Saskatchewan

#### 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, 2017 Iowa State University

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

*Irvin Roy Hentzel*

No abstract provided.

Generalized Right Alternative Rings, 2017 Iowa State University

#### 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, 2017 Iowa State University

#### 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).

Complexity And Unsolvability Properties Of Nilpotency, 2017 Iowa State University

#### Complexity And Unsolvability Properties Of Nilpotency, Irvin R. Hentzel, David Pokrass Jacobs

*Irvin Roy Hentzel*

A nonassociative algebra is nilpotent if there is some n such that the product of any n elements, no matter how they are associated, is zero. Several related, but more general, notions are left nilpotency, solvability, local nilpotency, and nillity. First the complexity of several decision problems for these properties is examined. In finite-dimensional algebras over a finite field it is shown that solvability and nilpotency can be decided in polynomial time. Over Q, nilpotency can be decided in polynomial time, while the algorithm for testing solvability uses a polynomial number of arithmetic operations, but is not polynomial time. Also ...

Idempotents In Plenary Train Algebras, 2017 Universidad de Chile

#### Idempotents In Plenary Train Algebras, Antonio Behn, Irvin R. Hentzel

*Irvin Roy Hentzel*

In this paper we study plenary train algebras of arbitrary rank. We show that for most parameter choices of the train identity, the additional identity (x^2 -w(x)x)^2 =0 is satisfied. We also find sufficient conditions for *A* to have idempotents.

Left Centralizers On Rings That Are Not Semiprime, 2017 Iowa State University

#### Left Centralizers On Rings That Are Not Semiprime, Irvin R. Hentzel, M.S. Tammam El-Sayiad

*Irvin Roy Hentzel*

A (left) centralizer for an associative ring *R* is an additive map satisfying *T(xy)* = *T(x)y* for all *x*, *y* in *R*. A (left) Jordan centralizer for an associative ring *R* is an additive map satisfying *T*(*xy*+*yx*) = *T*(*x*)*y* + *T*(*y*)*x* for all *x*, *y* in *R*. We characterize rings with a Jordan centralizer *T*. Such rings have a *T* invariant ideal *I*, *T* is a centralizer on *R/I*, and *I* is the union of an ascending chain of nilpotent ideals. Our work requires 2-torsion free. This result has applications to (right) centralizers ...

Distributed Evolution Of Spiking Neuron Models On Apache Mahout For Time Series Analysis, 2017 Cylance, Inc.

#### Distributed Evolution Of Spiking Neuron Models On Apache Mahout For Time Series Analysis, Andrew Palumbo

*Annual Symposium on Biomathematics and Ecology: Education and Research*

No abstract provided.

On The Location Of Eigenvalues Of Real Matrices, 2017 King Fahd University of Petroleum and Minerals

#### On The Location Of Eigenvalues Of Real Matrices, Rachid Marsli, Frank Hall

*Electronic Journal of Linear Algebra*

The research in this paper is motivated by a recent work of I. Barany and J. Solymosi [I. Barany and J. Solymosi. Gershgorin disks for multiple eigenvalues of non-negative matrices. Preprint arXiv no. 1609.07439, 2016.] about the location of eigenvalues of nonnegative matrices with geometric multiplicity higher than one. In particular, an answer to a question posed by Barany and Solymosi, about how the location of the eigenvalues can be improved in terms of their geometric multiplicities is obtained. New inclusion sets for the eigenvalues of a real square matrix, called Ger\v{s}gorin discs of the second ...

The Roots Of Early Group Theory In The Works Of Lagrange, 2017 Colorado State University-Pueblo

#### The Roots Of Early Group Theory In The Works Of Lagrange, Janet Heine Barnett

*Abstract Algebra*

No abstract provided.

Inquiry Into Saving [Mathematics], 2017 CUNY La Guardia Community College

#### Inquiry Into Saving [Mathematics], Jeanne Funk

*Open Educational Resources*

‘Inquiry Into Saving’ is an assignment originally designed for MAT117, which is a course for students who have been placed in basic skills mathematics and who can apply a college level course in Algebra and Trigonometry to their program. These students should, ideally, be early in their LaGuardia career, though that is frequently not the case. All, however, are novices of mathematics. The assignment was vetted and revised based on feedback from the Inquiry and Problem Solving in STEM CTL seminar and a charrette not affiliated with the seminar. Revisions addressed connections to the Inquiry and Problem Solving/Written competency ...

World Population Dynamics: Modeling Involving Polynomial Functions [Mathematics], 2017 CUNY La Guardia Community College

#### World Population Dynamics: Modeling Involving Polynomial Functions [Mathematics], Mangala Kothari

*Open Educational Resources*

In this Inquiry and Problem Solving Assignment students are expected to reflect on their analysis and compare their results with the actual population by conducting their own elementary level research such as searching databases, gathering information and interpreting. Students are expected to comment on the scope of the mathematical model and connect their learning in context to the real-world problem. The assignment includes open-ended questions such as: Write a paragraph about the dynamics of population for the world. What could be some of the possible parameters that contribute to the change in population size? Reflect on what you learned by ...

On A Frobenius Problem For Polynomials, 2017 Gettysburg College

#### On A Frobenius Problem For Polynomials, Ricardo Conceição, R. Gondim, M. Rodriguez

*Math Faculty Publications*

We extend the famous diophantine Frobenius problem to a ring of polynomials over a field~*k*. Similar to the classical problem we show that the *n* = 2 case of the Frobenius problem for polynomials is easy to solve. In addition, we translate a few results from the Frobenius problem over ℤ to *k*[*t*] and give an algorithm to solve the Frobenius problem for polynomials over a field *k* of sufficiently large size.