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The matrices of Pauli quaternions, their De Moivre's and Euler's formulas
(World Scientific Publ Co Pte Ltd, 2022) Akbiyik, Mucahit; Yamac Akbiyik, Seda; Yilmaz, Fatih
In this paper, we obtain Euler's and De Moivre's formulas for the 4 x 4 matrix representation of Pauli quaternions. Moreover, we provide De Moivre's formula for the light-like Pauli quaternions. Additionally, we give the nth roots of the matrix representation of Pauli quaternions. Moreover, we exemplify some of the results with illustrative examples to support the main formulas.
On split-octonionic curves
(Oxford Univ Press, 2024) Alo, Jeta; Akbiyik, Mucahit
In this paper, we first define the vector product in Minkowski space $\mathbb{R}_{4}<^>{7}$ , which is identified with the space of spatial split-octonions. Next, we derive the $G_{2}-$ frame formulae for a seven dimensional Minkowski curve by using the spatial split-octonions and the vector product. We show that Frenet-Serret formulas are satisfied for a spatial split octonionic curve. We obtain the congruence of two spatial split octonionic curves and give relationship between the $G_{2}-$ frame and Frenet-Serret frame. Furthermore, we present the Frenet-Serret frame with split octonions in $\mathbb{R}_{4}<^>{8}$ . Finally, we give illustrative examples with Matlab codes.