Algebra Commons^™

Integrating Mathematics And Educational Robotics: Simple Motion Planning, Ronald I. Greenberg, George K. Thiruvathukal, Sara T. Greenberg

George K. Thiruvathukal

This paper shows how students can be guided to integrate elementary mathematical analyses with motion planning for typical educational robots. Rather than using calculus as in comprehensive works on motion planning, we show students can achieve interesting results using just simple linear regression tools and trigonometric analyses. Experiments with one robotics platform show that use of these tools can lead to passable navigation through dead reckoning even if students have limited experience with use of sensors, programming, and mathematics.

Dissertation_Davis.Pdf, Brian Davis

brian davis

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

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, and …

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

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

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

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 algebra …

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

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

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

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

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

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

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

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

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

Irvin Roy Hentzel

No abstract provided.

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

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, 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, 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, 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 …

Recursive Robust Pca Or Recursive Sparse Recovery In Large But Structured Noise, Chenlu Qiu, Namrata Vaswani, Brian Lois, Leslie Hogben

Namrata Vaswani

This paper studies the recursive robust principal components analysis problem. If the outlier is the signal-of-interest, this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors, St, in the presence of large but structured noise, Lt. The structure that we assume on Lt is that Lt is dense and lies in a low-dimensional subspace that is either fixed or changes slowly enough. A key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background (Lt) from moving foreground objects (St) on-the-fly. To solve the above …

Hom Quandles, Alissa S. Crans, Sam Nelson

Alissa Crans

If A is an abelian quandle and Q is a quandle, the hom set Hom(Q,A) of quandle homomorphisms from Q to A has a natural quandle structure. We exploit this fact to enhance the quandle counting invariant, providing an example of links with the same counting invariant values but distinguished by the hom quandle structure. We generalize the result to the case of biquandles, collect observations and results about abelian quandles and the hom quandle, and show that the category of abelian quandles is symmetric monoidal closed.

Higher Dimensional Algebra Vi: Lie 2-Algebra, John C. Baez, Alissa S. Crans

Alissa Crans

The theory of Lie algebras can be categorified starting from a new notion of `2-vector space', which we define as an internal category in Vect. There is a 2-category 2Vect having these 2-vector spaces as objects, `linear functors' as morphisms and `linear natural transformations' as 2-morphisms. We define a `semistrict Lie 2-algebra' to be a 2-vector space L equipped with a skew-symmetric bilinear functor [ . , . ] : L x L -> L satisfying the Jacobi identity up to a completely antisymmetric trilinear natural transformation called the `Jacobiator', which in turn must satisfy a certain law of its …

Musical Actions Of Dihedral Groups, Alissa S. Crans, Thomas M. Fiore, Ramon Satyendra

Alissa Crans

The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords. We illustrate both geometrically and algebraically how these two actions are {\it dual}. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.

From Loop Groups To 2-Groups, John C. Baez, Danny Stevenson, Alissa S. Crans, Urs Schreiber

Alissa Crans

We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie 2-group is a categorified version of a Lie group. If G is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras g_k each having Lie(G) as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on G. There appears to be no Lie 2-group …

Polynomial Knot And Link Invariants From The Virtual Biquandle, Alissa S. Crans, Allison Henrich, Sam Nelson

Alissa Crans

The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek polynomial) for virtual knots and the classical Alexander polynomial for classical knots. For a fixed monomial ordering <, the Gr\"obner bases for these ideals are computable, comparable invariants which fully determine the elementary ideals and which generalize and unify the classical and generalized Alexander polynomials. We provide examples to …

Cohomology Of The Adjoint Of Hopf Algebras, J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi, Masahico Saito

Alissa Crans

A cohomology theory of the adjoint of Hopf algebras, via deformations, is presented by means of diagrammatic techniques. Explicit calculations are provided in the cases of group algebras, function algebras on groups, and the bosonization of the super line. As applications, solutions to the YBE are given and quandle cocycles are constructed from groupoid cocycles.