Related papers: Product of three octonions

We introduce a nonsymmetric, associative tensor product among representations of Cuntz algebras by using embeddings. We show the decomposition formulae of tensor products for permutative representations explicitly We apply decomposition…

Higher-order tensors appear in various areas of mechanics as well as physics, medicine or earth sciences. As these tensors are highly complex, most are not well understood. Thus, the analysis and the visualization process form a highly…

We define the e\~ne product for the multiplicative group of polynomials and formal power series with coefficients on a commutative ring and unitary constant coefficient. This defines a commutative ring structure where multiplication is the…

The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates,…

As is well-known, the real quaternion division algebra $ {\cal H}$ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra ${\cal O}$ can not be algebraically isomorphic to any matrix algebras…

In this note, we construct and study an algebraic system similar to the natural numbers, but with noncommutative addition. The addition we introduce is a binary operation that commutes with itself in the sense of N. Durov. Neverheless, the…

Using the natural irreducible 8-dimensional representation and the two spin representations of the quantum group $U_q$(D$_4$) of D$_4$, we construct a quantum analogue of the split octonions and study its properties. We prove that the…

Semi-direct products of finite groups have permutation representations that are constructed from the permutation representations of their constituents. One can envision these in a metaphoric sense in which a rope is made from a bundle of…

We give an octonionic formulation of the N = 1 supersymmetry algebra in D = 11, including all brane charges. We write this in terms of a novel outer product, which takes a pair of elements of the division algebra A and returns a real linear…

In this paper we prove that if $R$ is a proper alternative ring whose additive group has no $3$-torsion and whose non-zero commutators are not zero-divisors, then $R$ has no zero-divisors. It follows from a theorem of Bruck and Kleinfeld…

A natural definition of the product of infinite matrices mimics the usual formulation of multiplication of finite matrices with the caveat (in the absence of any sense of convergence) that the intersection of the support of each row of the…

We consider a number of combinatorial problems in which rational generating functions may be obtained, whose denominators have factors with certain singularities. Specifically, there exist points near which one of the factors is asymptotic…

Commutative totally ordered monoids abound, number systems for example. When the monoid is not assumed commutative, one may be hard pressed to find an example. One suggested by Professor Orr Shalit are the countable ordinals with addition.…

We study enumeration problems for multi-operator monomials generated from one indeterminate by an associative multiplication together with finitely many unary operators. We consider four regimes, according to whether multiplication is…

For a composite $n$ and an odd $c$ with $c$ not dividing $n$, the number of solutions to the equation $n+a\equiv b\mod c$ with $a,b$ quadratic residues modulus $c$ is calculated. We establish a direct relation with those modular solutions…

A product difference equation is proved and used for derivation by elementary methods of four combinatorial identities, eight combinatorial identities involving generalized harmonic numbers and eight combinatorial identities involving…

We study the symmetric outer product decomposition which decomposes a fully (partially) symmetric tensor into a sum of rank-one fully (partially) symmetric tensors. We present iterative algorithms for the third-order partially symmetric…

This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a…

Considering commutator monomials of the non-commutative associative variables $X_1,\ldots,X_n$; we determine the maximal possible number of alternating associative monomials in their noncommutative polynomial expansions. This is achieved by…

We show that any two Hadamard subfactors arising from a pair of distinct complex Hadamard matrices of order 3 are either equal or conjugate by a unitary in the relative commutant of their intersection. Moreover, when the Hadamard subfactors…