Search

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. [ABSTRACT FROM AUTHOR]

In this paper the generalization of the rotations on any lightcone in Minkowski 3-space $${\mathbb {R}}_{g}^{1,2}$$ is given. The rotation motion on the lightcone is examined by means of a bilinear form and Lorentzian notions. We use the corresponding Rodrigues and Cayley formulas and benefit from the hyperbolic split quaternion product to obtain the corresponding rotation matrix. [ABSTRACT FROM AUTHOR]

We present a new different polar representation of split and dual split quaternions inspired by the Cayley-Dickson representation. In this new polar form representation, a split quaternion is represented by a pair of complex numbers, and a dual split quaternion is represented by a pair of dual complex numbers as in the Cayley-Dickson form. Here, in a split quaternion these two complex numbers are a complex modulus and a complex argument while in a dual split quaternion two dual complex numbers are a dual complex modulus and a dual complex argument. The modulus and argument are calculated from an arbitrary split quaternion in Cayley-Dickson form. Also, the dual modulus and dual argument are calculated from an arbitrary dual split quaternion in Cayley-Dickson form. By the help of polar representation for a dual split quaternion, we show that a Lorentzian screw operator can be written as product of two Lorentzian screw operators. One of these operators is in the two-dimensional space produced by 1 and i vectors. The other is in the three-dimensional space generated by 1, j and k vectors. Thus, an operator in a four-dimensional space is expressed by means of two operators in two and three-dimensional spaces. Here, vector 1 is in the intersection of these spaces. [ABSTRACT FROM AUTHOR]

Involutions and anti-involutions are self-inverse linear mappings. In three-dimensional Euclidean space $${\mathbb{R}^{3}}$$ , a reflection of a vector in a plane can be represented by an involution or anti-involution mapping obtained by real-quaternions. A reflection of a line about a line in $${\mathbb{R}^{3}}$$ can also be represented by an involution or anti-involution mapping obtained by dual real-quaternions. In this paper, we will represent involution and anti-involution mappings obtaind by dual split-quaternions and a geometric interpretation of each as rigid-body (screw) motion in three-dimensional Lorentzian space $${\mathbb{R}_1^{3} }$$ . [ABSTRACT FROM AUTHOR]

Hyperbolic rotation is hyperbolically the motion of a smooth object on general hyperboloids given by − a 1 x 2 + a 2 y 2 + a 3 z 2 = ± λ , λ ∈ R + . In this paper, we investigate the hyperbolical rotation matrices in order to get the motion of a point about a fixed point or axis on the general hyperboloids by defining the Lorentzian Scalar Product Space R a 1 a 2 a 3 2 , 1 such that the general hyperboloids are the pseudo-spheres of R a 1 a 2 a 3 2 , 1 . We adapt the Rodrigues, Cayley, and Householder methods to R a 1 a 2 a 3 2 , 1 and define hyperbolic split quaternions to obtain an hyperbolical rotation matrix. [ABSTRACT FROM AUTHOR]