Authors:P. D. Beites; A. P. Nicolás Pages: 955 - 964 Abstract: Some identities satisfied by certain standard composition algebras, of types II and III, are studied and become candidates for the characterization of the mentioned types. Composition algebras of arbitrary dimension, over a field F with char \({(F) \neq 2}\) and satisfying the identity \({x^{2}y = n(x)y}\) are shown to be standard composition algebras of type II. As a consequence, the identity \({yx^{2} = n(x)y}\) characterizes the type III. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0668-8 Issue No:Vol. 27, No. 2 (2017)

Authors:Murat Bekar; Yusuf Yayli Pages: 965 - 975 Abstract: In this paper, semi-quaternions are studied with their basic properties. Unit tangent bundle of \({{\mathbb {R}}^2}\) is also obtained by using unit semi-quaternions and it is shown that the set \({{T {\mathbb {R}}^2}}\) of all unit semi-quaternions based on the group operation of semi-quaternion multiplication is a Lie group. Furthermore, the vector space matrix of angular velocity vectors forming the Lie algebra \({{T_{1} {\mathbb {R}}^2}}\) of the group \({{T {\mathbb {R}}^2}}\) is obtained. Finally, it is shown that the rigid body displacements obtained by using semi-quaternions correspond to planar displacements in \({{\mathbb {R}^3}}\) . PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0670-1 Issue No:Vol. 27, No. 2 (2017)

Authors:Fatma Bulut; Salih Celik Pages: 1019 - 1030 Abstract: We present a noncommutative differential calculus on the h-superspace via a contraction of the q-superspace, and find a new quantum matrix superalgebra. We also give a new deformation of the super-Heisenberg algebra. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0672-z Issue No:Vol. 27, No. 2 (2017)

Authors:Tao Cheng; Hua-Lin Huang; Yuping Yang; Yinhuo Zhang Pages: 1055 - 1064 Abstract: By applying the idea of viewing the octonions as an associative algebra in certain tensor categories, or more precisely as a twisted group algebra by a 2-cochain, we show that the octonions form an Azumaya algebra in some suitable braided linear Gr-categories. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0656-z Issue No:Vol. 27, No. 2 (2017)

Authors:Adam Chapman; Casey Machen Pages: 1065 - 1072 Abstract: Given a central division algebra D of degree d over a field F, we associate to any standard polynomial \(\phi (z)=z^n+c_{n-1} z^{n-1}+\cdots +c_0\) over D a “companion polynomial” \(\Phi (z)\) of degree nd with coefficients in F. The roots of \(\Phi (z)\) in D are exactly the set of conjugacy classes of the roots of \(\phi (z)\) . When D is a quaternion algebra, we explain how all the roots of \(\phi (z)\) can be recovered from the roots of \(\Phi (z)\) . Along the way, we are able to generalize a few known facts from \(\mathbb {H}\) to any division algebra. The first is the connection between the right eigenvalues of a matrix and the roots of its characteristic polynomial. The second is the connection between the roots of a standard polynomial and left eigenvalues of the companion matrix. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0740-4 Issue No:Vol. 27, No. 2 (2017)

Authors:Arsenio Moreno García; Tania Moreno García; Ricardo Abreu Blaya; Juan Bory Reyes Pages: 1147 - 1159 Abstract: In this paper we derive a Cauchy integral representation formula for the solutions of the sandwich equation \(\partial _{\underline{x}}f\partial _{\underline{x}}=0\) , where \(\partial _{\underline{x}}\) stands for the first-order vector-valued rotation invariant differential operator in the Euclidean space \({\mathbb R}^m\) , called Dirac operator. Such a solutions are referred in the literature as inframonogenic functions and represent an extension of the monogenic functions, i.e., null solutions of \(\partial _{\underline{x}}\) , which represent higher-dimensional generalizations of the classic Cauchy–Riemann operator. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0745-z Issue No:Vol. 27, No. 2 (2017)

Authors:Abdullah Inalcik Pages: 1329 - 1341 Abstract: In this paper, we establish the concept of similarity and semi-similarity for elements of set of degenerate quaternions, pseudodegenerate quaternions and doubly degenerate quaternions by solving \({ax=xb}\) and \({xay=b, ybx=a}\) , respectively. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0674-x Issue No:Vol. 27, No. 2 (2017)

Authors:Artur Kobus; Jan L. Cieśliński Pages: 1369 - 1386 Abstract: We consider the scator space in 1+2 dimensions—a hypercomplex, non-distributive hyperbolic algebra introduced by Fernández-Guasti and Zaldívar. We find a method for treating scators algebraically by embedding them into a distributive and commutative algebra. A notion of dual scators is introduced and discussed. We also study isometries of the scator space. It turns out that zero divisors cannot be avoided while dealing with these isometries. The scator algebra may be endowed with a nice physical interpretation, although it suffers from lack of some physically demanded important features. Despite that, there arise some open questions, e.g., whether hypothetical tachyons can be considered as usual particles possessing time-like trajectories. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0658-x Issue No:Vol. 27, No. 2 (2017)

Authors:John Leventides; George Petroulakis Pages: 1503 - 1515 Abstract: In this article we define a number of varieties and sets in the projective space \({\mathbb{P} \left(\wedge^{2}\mathbb{R}^{n}\right)}\) , which are obtained from the spectral analysis of a 2-tensor in \({\wedge^2\mathbb{R}^n}\) . We refer to these sets as \({\mathcal{G}_{\mathcal{V}}}\) , special cases of which are the Grassmannian and its extremal variety, as well as other sets. We study the problem of finding the projective variety whose 2-tensors in \({\wedge^{2}\mathbb{R}^{n}}\) have the maximum distance from these varieties, via the use of a special mapping which we call Extr \({(\cdot)}\) . It is shown that successive applications of this mapping lead either to the Grassmannian or its Extremal variety. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0721-7 Issue No:Vol. 27, No. 2 (2017)

Authors:Yuan-Min Li Pages: 1517 - 1530 Abstract: Quadratic representations are very useful in the study of Euclidean Jordan algebras and complementarity problems. In this paper, we provide some characterizations of the complementarity properties for the quadratic representation P a . For example, P a has the E0-property; P a is monotone iff \({\pm a \in {\mathcal K}}\) . In addition, the algebra and cone automorphism invariance of some E-properties are studied. By use of the quadratic representations, the Jordan quad E-property is proved to keep cone automorphism invariant in simple Jordan algebras. The pseudomonotone property is shown to be cone automorphism invariant. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0678-6 Issue No:Vol. 27, No. 2 (2017)

Authors:Vishnu Narayan Mishra; Shikha Pandey Pages: 1633 - 1646 Abstract: In the present paper, we introduce the generalized form of (p, q) Baskakov–Durrmeyer Operators with Stancu type parameters. We derived the local and global approximation properties of these operators and obtained the convergence rate and behaviour for the Lipschitz functions. Moreover, we give comparisons and some illustrative graphics for the convergence of operators to some function. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0738-y Issue No:Vol. 27, No. 2 (2017)

Authors:Craig A. Nolder; Guanghong Wang Pages: 1647 - 1657 Abstract: We use Fourier multipliers of the Dirac operator and Cauchy transform to obtain composition theorems and integral representations. In particular we calculate the multiplier of the \(\Pi \) -operator. This operator is the hypercomplex version of the Beurling-Ahlfors transform in the plane. The hypercomplex Beurling-Ahlfors transform is a direct generalization of the Beurling-Ahlfors transform and reduces to this operator in the plane. We give an integral representation for iterations of the hypercomplex Buerling-Ahlfors transform and we present here a bound for the \(L^p\) -norm. Such \(L^p\) -bounds are essential for applications of the Beurling-Ahlfors transformation in the plane. The upper bound presented here is \(m(p^*-1)\) where m is the dimension of the Euclidean space on which the functions are defined, \(1<p<\infty \) and \(p^*=\max (p,p/(p-1))\) . We use recent estimates on second order Riesz transforms to obtain this result. Using the Fourier multiplier of the \(\Pi \) -operator we express this operator as a hypercomplex linear combination of second order Riesz transforms. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0752-0 Issue No:Vol. 27, No. 2 (2017)

Authors:Guangbin Ren; Zeping Zhu Pages: 1715 - 1740 Abstract: This article focuses on the integral theory of discrete regular quaternionic functions and the connection between discrete and classical quaternionic analysis. We develop a new approach to the discrete integral theory based on the concepts ‘discrete boundary measure’ and ‘discrete normal vector’. By combining the discrete integral theory and our results on discrete approximations to domains in \({\mathbb {R}}^4\) , we reveal a convergence relation between the discrete and continuous regular functions: a quaternionic function is regular if and only if it is the scaling limit of some discrete regular functions. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0743-1 Issue No:Vol. 27, No. 2 (2017)

Authors:Waldyr Alves Rodrigues; Samuel A. Wainer Pages: 1761 - 1767 Abstract: Let \({M = SO(1, 4)/SO(1,3) \simeq S^{3}\times \mathbb{R}}\) (a parallelizable manifold) be a submanifold in the structure \({(\mathring{M}, {\mathring{g}})}\) (hereafter called the bulk) where \({\mathring {M} \simeq \mathbb{R}^{5}}\) and \({{\mathring{g}}}\) is a pseudo Euclidian metric of signature (1,4). Let \({{i}:M\rightarrow\mathbb{R}^{5}}\) be the inclusion map and let \({{g}={i}^{\ast}{\mathring{g}}}\) be the pullback metric on M. It has signature (1,3) Let \({{D}}\) be the Levi-Civita connection of \({{g} }\) . We call the structure \({(M,{g})}\) a de Sitter manifold and \({M^{dSL} = (M = {\mathbb{R}} \times S^{3},g,D,\tau_{g},\uparrow)}\) a de Sitter spacetime structure, which is of course orientable by \({\tau_{g} \in {\rm sec} \bigwedge^{4} T^{\ast}M}\) and time orientable (by \({\uparrow}\) ). Under these conditions (and without using any General Relativity theory concept) we prove that if the motion of a particle restricted to move on M (without the action of any force as detected by observers living in M) happens with constant bulk angular momentum then its motion in the structure M dSL is a timelike geodesic. Also any geodesic motion in the structure M dSL implies that the particle has constant angular momentum in the bulk. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0729-z Issue No:Vol. 27, No. 2 (2017)

Authors:Süleyman Şenyurt; Ceyda Cevahir; Yasin Altun Pages: 1815 - 1824 Abstract: In this study, the normal vector and the unit Darboux vector of spatial involute curve of the spatial quaternionic curve are taken as the position vector, the curvature and torsion of obtained smarandahce curve were calculeted. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0669-7 Issue No:Vol. 27, No. 2 (2017)

Authors:Zafer Ünal; Ümit Tokeşer; Göksal Bilgici Pages: 1907 - 1916 Abstract: Halici (Adv Appl Clifford Algebr 25(4):905–914, 2015) defined dual Fibonacci and dual Lucas octonions by the relations \({\widetilde{Q}_{n}={Q}_n+\varepsilon Q_{n+1}}\) and \({\widetilde{P}_n=P_n+\varepsilon P_{n+1}}\) for every integer n where \({Q_n}\) and \({P_n}\) are the Fibonacci and Lucas octonions respectively, and \({\varepsilon}\) is the dual unit. The aim of this paper is to investigate properties of dual Fibonacci and dual Lucas octonions. After obtaining the Binet formulas for the sequences \({\{\widetilde{Q}_n \}_{n=0}^\infty}\) and \({\{\widetilde{P}_n \}_{n=0}^\infty}\) , we derive some identities for these sequences such as Catalan’s, Cassini’s and d’Ocagne’s identities. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0724-4 Issue No:Vol. 27, No. 2 (2017)

Authors:G. P. Wene Pages: 1917 - 1925 Abstract: Finite dimensional real division algebras are examined under the permutation of their structure constants. It is noted that there is only one finite-dimensional real flexible division algebras with an automorphism group isomorphic to S 3. We show that the 8-dimensional real division algebras with derivation algebra g 2 are isotopic under all permutations of the structure constants. Being a composition algebra is preserved under all permutations. We simultaneously derive information about the quaternion division algebras as subalgebras of the eight dimensional algebras. Finally, a division algebra with derivation algebra \({su(2)\oplus su(2)}\) is looked at under the actions of S 3. The quaternion, octonion and Okubo algebras are special cases of our results. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0712-8 Issue No:Vol. 27, No. 2 (2017)