Authors:Naruhiko Aizawa Abstract: For a given Lie superalgebra, two ways of constructing color superalgebras are presented. One of them is based on the color superalgebraic nature of the Clifford algebras. The method is applicable to any Lie superalgebras and results in color superalgebra of \( {\mathbb Z}_2^{\otimes N} \) grading. The other is discussed with an example, a superalgebra of boson and fermion operators. By treating the boson operators as “second” fermionic sector we obtain a color superalgebra of \({\mathbb Z}_2 \otimes {\mathbb Z}_2\) grading. A vector field representation of the color superalgebra obtained from the boson–fermion system is also presented. PubDate: 2018-02-26 DOI: 10.1007/s00006-018-0847-x Issue No:Vol. 28, No. 1 (2018)

Authors:Caroline Costa; Marcia R. Tenser; Ronni G. G. Amorim; Marco C. B. Fernandes; Ademir E. Santana; J. David M. Vianna Abstract: Using elements of symmetry, as gauge invariance, aspects of field theories represented in symplectic space are introduced and analyzed under physical bases. The states of a system are described by symplectic wave functions, which are associated with the Wigner function. Such wave functions are vectors in a Hilbert space introduced from the cotangent-bundle of the Minkowski space. The symplectic Klein–Gordon and the Dirac equations are derived, and a minimum coupling is considered in order to analyze the Landau problem in phase space. PubDate: 2018-02-26 DOI: 10.1007/s00006-018-0840-4 Issue No:Vol. 28, No. 1 (2018)

Authors:David Eelbode; Tim Janssens; Matthias Roels Abstract: The Cauchy–Kovalevskaya extension for polynomials in several variables is a crucial result in Clifford analysis which describes theoretically how to construct simplicial monogenics in m dimensions starting from certain polynomial spaces in \(m-1\) dimensions, hereby using the so-called branching rules. In [13] it was shown that the Cauchy–Kovalevskaya extension map is an isomorphism of the kernel of the wedge operator onto the space of spherical monogenics in two variables. The aim of this paper is to show that this extension maps simplicial polynomials of the kernel of the wedge operator just onto the simplicial monogenics. PubDate: 2018-02-24 DOI: 10.1007/s00006-018-0846-y Issue No:Vol. 28, No. 1 (2018)

Authors:Mustafa Tarakçioğlu; Tülay Erişir; Mehmet Ali Güngör; Murat Tosun Abstract: In this study, firstly, we give a different approach to the relationship between the split quaternions and rotations in Minkowski space \(\mathbb {R}_1^3\) . In addition, we obtain an automorphism of the split quaternion algebra \(H'\) corresponding to a rotation in \(\mathbb {R}_1^3\) . Then, we give the relationship between the hyperbolic spinors and rotations in \(\mathbb {R}_1^3\) . Finally, we associate to a split quaternion with a hyperbolic spinor by means of a transformation. In this way, we show that the rotation of a rigid body in the Minkowski 3-space \(\mathbb {R}_1^3\) expressed the split quaternions can be written by means of the hyperbolic spinors with two hyperbolic components. So, we obtain a new and short representation (hyperbolic spinor representation) of transformation in the 3-dimensional Minkowski space \(\mathbb {R}_1^3\) expressed by means of split quaternions. PubDate: 2018-02-24 DOI: 10.1007/s00006-018-0844-0 Issue No:Vol. 28, No. 1 (2018)

Authors:Jacques Helmstetter Abstract: In every Clifford algebra \({\mathrm {Cl}}(V,q)\) , there is a Lipschitz monoid (or semi-group) \({\mathrm {Lip}}(V,q)\) , which is in most cases the monoid generated by the vectors of V. This monoid is useful for many reasons, not only because of the natural homomorphism from the group \({\mathrm {GLip}}(V,q)\) of its invertible elements onto the group \({\mathrm {O}}(V,q)\) of orthogonal transformations. From every non-zero \(a\in {\mathrm {Lip}}(V,q)\) , we can derive a bilinear form \(\phi \) on the support S of a in V; it is q-compatible: \(\phi (x,x)=q(x)\) for all \(x\in S\) . Conversely, every q-compatible bilinear form on a subspace S of V can be derived from an element \(a\in {\mathrm {Lip}}(V,q)\) which is unique up to an invertible scalar; and a is invertible if and only if \(\phi \) is non-degenerate. This article studies the relations between a, \(\phi \) and (when a is invertible) the orthogonal transformation g derived from a; it provides both theoretical knowledge and algorithms. It provides an effective tool for the factorization of lipschitzian elements, based on this theorem: if \((v_1,v_2,\ldots ,v_s)\) is a basis of S, then \(a=\kappa \,v_1v_2\ldots v_s\) (for some invertible scalar \(\kappa \) ) if and only if the matrix of \(\phi \) in this basis is lower triangular. This theorem is supported by an algorithm of triangularization of bilinear forms. PubDate: 2018-02-24 DOI: 10.1007/s00006-018-0842-2 Issue No:Vol. 28, No. 1 (2018)

Authors:Ivan Kyrchei Abstract: Within the framework of the theory of quaternion column–row determinants and using determinantal representations of the Moore–Penrose inverse previously obtained by the author, we get explicit determinantal representation formulas of solutions (analogs of Cramer’s rule) to the systems of quaternion matrix equations \( \mathbf{A}_{1}{} \mathbf{X}=\mathbf{C}_{1}\) , \( \mathbf{X}\mathbf{B}_{2}=\mathbf{C}_{2} \) , and \( \mathbf{A}_{1}{} \mathbf{X}=\mathbf{C}_{1}\) , \(\mathbf{A}_{2} \mathbf{X}=\mathbf{C}_{2} \) . PubDate: 2018-02-23 DOI: 10.1007/s00006-018-0843-1 Issue No:Vol. 28, No. 1 (2018)

Authors:Mariusz Klimek; Stefan Kurz; Sebastian Schöps; Thomas Weiland Abstract: We employ classical Maxwell’s equations formulated in space-time algebra to perform discretization of moving geometries directly in space-time. All the derivations are carried out without any non-relativistic assumptions, thus the application area of the scheme is not restricted to low velocities. The 4D mesh construction is based on a 3D mesh stemming from a conventional 3D mesh generator. The movement of the system is encoded in the 4D mesh geometry, enabling an easy extension of well-known 3D approaches to the space-time setting. As a research example, we study a manifestation of Sagnac’s effect in a rotating ring resonator. In case of constant rotation, the space-time approach enhances the efficiency of the scheme, as the material matrices are constant for every time step, without abandoning the relativistic framework. PubDate: 2018-02-14 DOI: 10.1007/s00006-018-0841-3 Issue No:Vol. 28, No. 1 (2018)

Authors:Xinming Huo; Yimin Song Abstract: Finite motion analysis of parallel mechanisms (PMs) denotes formulating the map between finite motion of end-effector and those of its component limbs. By employing conformal geometric algebra (CGA), this paper presents an analytical and accurate method to analyze the finite motions of PMs with parasitic motions. Herein, parasitic motions are defined as the dependent motions in the constraint Degrees-of-Freedom (DoFs) of PMs. Firstly, description of rigid body transformations based on CGA is reviewed. Then, the intersection algorithm of finite motions is introduced by exploiting the algebraic properties of CGA. Based on this, a method to formulate the finite motions of PMs with parasitic motions is proposed. Finally, Z3 mechanism is sketched as example by utilizing the approach. This method facilitates the invention of new mechanisms and can also be applied in the finite motion analysis of other kinds of PMs. PubDate: 2018-02-12 DOI: 10.1007/s00006-018-0832-4 Issue No:Vol. 28, No. 1 (2018)

Authors:Sergio Giardino Abstract: The breakdown of Ehrenfest’s theorem imposes serious limitations on quaternionic quantum mechanics (QQM). In order to determine the conditions in which the theorem is valid, we examined the conservation of the probability density, the expectation value and the classical limit for a non-anti-hermitian formulation of QQM. The results also indicated that the non-anti-hermitian quaternionic theory is related to non-hermitian quantum mechanics, and thus the physical problems described with both of the theories should be related. PubDate: 2018-02-12 DOI: 10.1007/s00006-018-0819-1 Issue No:Vol. 28, No. 1 (2018)

Authors:Xinming Huo; Panfeng Wang; Wanzhen Li Abstract: This paper proposes a geometric algebra (GA) based approach to carry out inverse kinematics and design parameters of a 2-degree-of-freedom parallel mechanism with its topology structure 3-RSR&SS for the first time. Here, R and S denote respectively revolute and spherical joints. The inverse solutions are obtained easily by utilizing special geometric relations of 3-RSR&SS parallel positioning mechanism, which are proven by calculating relations among point, line and plane in virtue of operation rules. Three global indices of kinematic optimization are defined to evaluate kinematic performance of 3-RSR&SS parallel positioning mechanism in the light of shuffle and outer products. Finally, the kinematic optimal design of 3-RSR&SS parallel positioning mechanism is carried out by means of NSGA-II and then a set of optimal dimensional parameters is proposed. Comparing with traditional kinematic analysis and optimal design method, the approach employing GA has following merits, (1) kinematic analysis and optimal design would be carried out in concise and visual way by taking full advantage of the geometric conditions of the mechanism. (2) this approach is beneficial to kinematic analysis and optimal design of parallel mechanisms in automatic and visual manner using computer programming languages. This paper may lay a solid theoretical and technical foundation for prototype design and manufacture of 3-RSR&SS parallel positioning mechanism. PubDate: 2018-02-12 DOI: 10.1007/s00006-018-0829-z Issue No:Vol. 28, No. 1 (2018)

Authors:Miguel Socolovsky Abstract: We review several aspects of anti-De Sitter (AdS) spaces in different dimensions, and of four dimensional Schwarzschild anti-De Sitter (SAdS) black hole. PubDate: 2018-02-08 DOI: 10.1007/s00006-018-0822-6 Issue No:Vol. 28, No. 1 (2018)

Authors:José María Grau; Celino Miguel; Antonio M. Oller-Marcén Abstract: We consider a generalization of the quaternion ring \(\Big (\frac{a,b}{R}\Big )\) over a commutative unital ring R that includes the case when a and b are not units of R. In this paper, we focus on the case \(R=\mathbb {Z}/n\mathbb {Z}\) for and odd n. In particular, for every odd integer n we compute the number of non R-isomorphic generalized quaternion rings \(\Big (\frac{a,b}{\mathbb {Z}/n\mathbb {Z}}\Big )\) . PubDate: 2018-02-07 DOI: 10.1007/s00006-018-0839-x Issue No:Vol. 28, No. 1 (2018)

Authors:Ahmad Hosny Eid Abstract: There is a steadily increasing interest in applying Geometric Algebra (GA) in diverse fields of science and engineering. Consequently, we need better software implementations to accommodate such increasing demands that widely vary in their possible uses and goals. For large-scale complex applications having many integrating parts, such as Big Data and Geographical Information Systems, we should expect the need for integrating several GAs to solve a given problem. Even within the context of a single GA space, we often need several interdependent systems of coordinates to efficiently model and solve the problem at hand. Future GA software implementations must take such important issues into account in order to scale, extend, and integrate with existing software systems, in addition to developing new ones, based on the powerful language of GA. This work attempts to provide GA software developers with a self-contained description of an extended framework for performing linear operations on GA multivectors within systems of interdependent coordinate frames of arbitrary metric. The work explains the mathematics and algorithms behind this extended framework and discusses some of its implementation schemes and use cases. If properly implemented, the extended framework can significantly reduce the memory requirements for implementing Geometric Algebras with larger dimensions, especially for systems based on the symbolic processing of multivector scalar coefficients. PubDate: 2018-02-07 DOI: 10.1007/s00006-018-0827-1 Issue No:Vol. 28, No. 1 (2018)

Authors:Yan-Na Zhang; Bing-Zhao Li Abstract: The uncertainty principle, which offers information about a function and its Fourier transform in the time-frequency plane, is particularly powerful in mathematics, physics and signal processing community. In this paper, based on the fundamental relationship between the quaternion linear canonical transform (QLCT) and quaternion Fourier transform (QFT), we propose two different uncertainty principles for the two-sided QLCT. It is shown that the lower bounds can be obtained on the product of spreads of a quaternion-valued function and its two-sided QLCT from newly derived results. Furthermore, an example is given to verify the consequences. Finally, some possible applications are provided to demonstrate the usefulness of new uncertainty relations in the QLCT domain. PubDate: 2018-02-06 DOI: 10.1007/s00006-018-0828-0 Issue No:Vol. 28, No. 1 (2018)

Authors:Geoffrey Dixon Abstract: The dimensions 2, 8 and 24 play significant roles in lattice theory. In Clifford algebra theory there are well-known periodicities of the first two of these dimensions. Using novel representations of the purely Euclidean Clifford algebras over all four of the division algebras, \({\mathbf{R}}\) , \({\mathbf{C}}\) , \({\mathbf{H}}\) , and \({\mathbf{O}}\) , a door is opened to a Clifford algebra periodicity of order 24 as well. PubDate: 2018-02-05 DOI: 10.1007/s00006-018-0820-8 Issue No:Vol. 28, No. 1 (2018)

Authors:Tongsong Jiang; Zhaozhong Zhang; Ziwu Jiang Abstract: This paper studies the constrained least squares problems of quaternion matrices by means of complex representation of quaternion matrices, and derive a novel algebraic technique for the quaternion equality constrained least squares (LSE) and quadratic inequality constrained least squares (LSQI) problems by generalized singular values decomposition (GSVD) of complex matrices. PubDate: 2018-02-05 DOI: 10.1007/s00006-018-0838-y Issue No:Vol. 28, No. 1 (2018)

Authors:G. Stacey Staples; Alexander Weygandt Abstract: Algebraic properties of zeons are considered, including the existence of elementary factorizations and homogeneous factorizations of invertible zeons. A “zeon division algorithm” is established, showing that every nontrivial invertible zeon can be written as a sum of homogeneously decomposable zeons. Elementary functions (exponential, logarithmic, hyperbolic, and trigonometric) are extended to zeons, and a number of properties and identities are revealed. Finally, fast computation of logarithms is discussed for homogeneously decomposable zeons. PubDate: 2018-02-05 DOI: 10.1007/s00006-018-0836-0 Issue No:Vol. 28, No. 1 (2018)

Authors:Talat Körpinar Abstract: In the present paper, we define new particles by using biharmonic particles in Heisenberg group \(\mathbf{H}\) . We obtain energy and angle of T magnetic biharmonic particles and some vector field. Finally, we draw energy and angle value in terms of Frenet fields in the Heisenberg group \(\mathbf{H}\) . PubDate: 2018-02-02 DOI: 10.1007/s00006-018-0834-2 Issue No:Vol. 28, No. 1 (2018)

Authors:Tevfik Şahin; Fatma Karakuş; Yusuf Yaylı Abstract: In this study, we defined Fermi–Walker derivative in dual space \(\mathbb {D}^3\) . Fermi–Walker transport, non-rotating frame and Fermi–Walker termed Darboux vector by using Fermi–Walker derivative are given in dual space \(\mathbb {D}^3\) . Being conditions of Fermi–Walker transport and non-rotating frame are investigated along any dual curve for dual Frenet frame, dual Darboux frame and dual Bishop frame. PubDate: 2018-02-02 DOI: 10.1007/s00006-018-0837-z Issue No:Vol. 28, No. 1 (2018)

Authors:Mustafa Özdemir Abstract: In this study, we define a new non-commutative number system called hybrid numbers. This number system can be accepted as a generalization of the complex \(\left( {\mathbf {i}}^{2}=-1\right) \) , hyperbolic \(\left( {\mathbf {h}} ^{2}=1\right) \) and dual number \(\left( \varvec{\varepsilon }^{2}=0\right) \) systems. A hybrid number is a number created with any combination of the complex, hyperbolic and dual numbers satisfying the relation \(\mathbf { ih=-hi=i}+\varvec{\varepsilon }.\) Because these numbers are a composition of dual, complex and hyperbolic numbers, we think that it would be better to call them hybrid numbers instead of the generalized complex numbers. In this paper, we give some algebraic and geometric properties of this number set with some classifications. In addition, we examined the roots of a hybrid number according to its type and character. PubDate: 2018-02-02 DOI: 10.1007/s00006-018-0833-3 Issue No:Vol. 28, No. 1 (2018)