Quadratic formulas for split quaternions

Unlike the Hamilton quaternion algebra, the split-quaternions contain nontrivial zero divisors. In general speaking, it is hard to find the solutions of equations in algebras containing zero divisor. In this paper, we manage to derive explicit formulas for computing the roots of $x^{2}+bx+c=0$ in split quaternion algebra.Comment: 13 page

Algorithms and Polynomiography for Solving Quaternion Quadratic Equations

Solving a quadratic equation $P(x)=ax^2+bx+c=0$ with real coefficients is known to middle school students. Solving the equation over the quaternions is not straightforward. Huang and So \cite{Huang} give a complete set of formulas, breaking it into several cases depending on the coefficients. From a result of the second author in \cite{kalQ}, zeros of $P(x)$ can be expressed in terms of the zeros of a real quartic equation. This drastically simplifies solving a quadratic equation. Here we also consider solving $P(x)=0$ iteratively via Newton and Halley methods developed in \cite{kalQ}. We prove a property of the Jacobian of Newton and Halley methods and describe several 2D polynomiography based on these methods. The images not only encode the outcome of the iterative process, but by measuring the time taken to render them we find the relative speed of convergence for the methods.Comment: 17 pages, 6 figures, 15 images, 1 tabl

Superalgebras of (split-)division algebras and the split octonionic M-theory in (6,5)-signature

The connection of (split-)division algebras with Clifford algebras and supersymmetry is investigated. At first we introduce the class of superalgebras constructed from any given (split-)division algebra. We further specify which real Clifford algebras and real fundamental spinors can be reexpressed in terms of split-quaternions. Finally, we construct generalized supersymmetries admitting bosonic tensorial central charges in terms of (split-)division algebras. In particular we prove that split-octonions allow to introduce a split-octonionic M-algebra which extends to the (6,5) signature the properties of the 11-dimensional octonionic M-algebras (which only exist in the (10,1) Minkowskian and (2,9) signatures).Comment: 16 page

Slice regularity and harmonicity on Clifford algebras

We present some new relations between the Cauchy-Riemann operator on the real Clifford algebra $\mathbb R_n$ of signature $(0,n)$ and slice-regular functions on $\mathbb R_n$. The class of slice-regular functions, which comprises all polynomials with coefficients on one side, is the base of a recent function theory in several hypercomplex settings, including quaternions and Clifford algebras. In this paper we present formulas, relating the Cauchy-Riemann operator, the spherical Dirac operator, the differential operator characterizing slice regularity, and the {spherical derivative} of a slice function. The computation of the Laplacian of the spherical derivative of a slice regular function gives a result which implies, in particular, the Fueter-Sce Theorem. In the two four-dimensional cases represented by the paravectors of $\mathbb R_3$ and by the space of quaternions, these results are related to zonal harmonics on the three-dimensional sphere and to the Poisson kernel of the unit ball of $\mathbb R^4$.Comment: 17 pages. To appear in "Topics in Clifford Analysis - A Special Volume in Honor of Wolfgang Spr\"o{\ss}ig", Springer series Trends in Mathematic

On a Generalization for Tribonacci Quaternions

Let $V_{n}$ denote the third order linear recursive sequence defined by the initial values $V_{0}$, $V_{1}$ and $V_{2}$ and the recursion $V_{n}=rV_{n-1}+sV_{n-2}+tV_{n-3}$ if $n\geq 3$, where $r$, $s$, and $t$ are real constants. The $\{V_{n}\}_{n\geq0}$ are generalized Tribonacci numbers and reduce to the usual Tribonacci numbers when $r=s=t=1$ and to the $3$-bonacci numbers when $r=s=1$ and $t=0$. In this study, we introduced a quaternion sequence which has not been introduced before. We show that the new quaternion sequence that we introduced includes the previously introduced Tribonacci, Padovan, Narayana and Third order Jacobsthal quaternion sequences. We obtained the Binet formula, summation formula and the norm value for this new quaternion sequence

The unifying formula for all Tribonacci-type octonions sequences and their properties

Various families of octonion number sequences (such as Fibonacci octonion, Pell octonion and Jacobsthal octonion) have been established by a number of authors in many different ways. In addition, formulas and identities involving these number sequences have been presented. In this paper, we aim at establishing new classes of octonion numbers associated with the generalized Tribonacci numbers. We introduce the Tribonacci and generalized Tribonacci octonions (such as Narayana octonion, Padovan octonion and third-order Jacobsthal octonion) and give some of their properties. We derive the relations between generalized Tribonacci numbers and Tribonacci octonions.Comment: arXiv admin note: text overlap with arXiv:1710.0060

The Third Order Jacobsthal Octonions: Some Combinatorial Properties

Various families of octonion number sequences (such as Fibonacci octonion, Pell octonion and Jacobsthal octonion) have been established by a number of authors in many different ways. In addition, formulas and identities involving these number sequences have been presented. In this paper, we aim at establishing new classes of octonion numbers associated with the third order Jacobsthal and third order Jacobsthal-Lucas numbers. We introduce the third order Jacobsthal octonions and the third order Jacobsthal-Lucas octonions and give some of their properties. We derive the relations between third order Jacobsthal octonions and third order Jacobsthal-Lucas octonions

On the missing branches of the Bruhat-Tits tree

Let k be a local field and let A be the two-by-two matrix algebra over k. In our previous work we developed a theory that allows the computation of the set of maximal orders in A containing a given suborder. This set is given as a sub-tree of the Bruhat-Tits tree that is called the branch of the order. Branches have been used to study the global selectivity problem and also to compute local embedding numbers. They can usually be described in terms of two invariants. To compute these invariants explicitly, the strategy in our past work has been visualizing branches through the explicit representation of the Bruhat-Tits tree in terms of balls in k. This is easier for orders spanning a split commutative sub-algebra, i.e., an algebra isomorphic to (k x k). In the present work, we develop a theory of branches over field extension that can be used to extend our previous computations to orders spanning a field. We use the same idea to compute branches for orders generated by arbitrary pairs of non-nilpotent pure quaternions. In fact, the hypotheses on the generators are not essential.Comment: This article will be published under the title "Computing embedding numbers and branches of orders via extensions of the Bruhat-Tits tree

Geometrization of the Real Number System

Geometric number systems, obtained by extending the real number system to include new anticommuting square roots of +1 and -1, provide a royal road to higher mathematics by largely sidestepping the tedious languages of tensor analysis and category theory. The well known consistency of real and complex matrix algebras, together with Cartan-Bott periodicity, firmly establishes the consistency of these geometric number systems, often referred to as Clifford algebras. The geometrization of the real number system is the culmination of the thousands of years of human effort at developing ever more sophisticated and encompassing number systems underlying scientific progress and advanced technology in the 21st Century. Complex geometric algebras are also considered.Comment: This new version shows how Cartan periodicity of Clifford algebras is related to Bott periodicity of homotopy groups and Hurwitz-Radon numbers. It also corrects a number of typos and adds a new section. 15 pages, 6 table

Composition algebras

This paper is devoted to survey composition algebras and some of their applications. After overviewing the classical algebras of quaternions and octonions, both unital composition algebras (or Hurwitz algebras) and symmetric composition algebras will be dealt with. Their main properties, as well as their classifications, will be reviewed. Algebraic triality, through the use of symmetric composition algebras, will be considered too.Comment: 21 page