In this communication, introduce the split Mersenne and Mersenne-Lucas hybrid quaternions, also obtaining generating functions Binet formulas for these quaternions investigating some properties among them.

Polar orderings arose in recent work of Salvetti and the second author on minimal CW-complexes for complexified hyperplane arrangements. We study the combinatorics of these orderings in the classical framework of oriented matroids, and reach thereby a weakening of the conditions required to actually determine such orderings. A class of arrangements for which the construction of the minimal comp...

We explain the orthogonal planes split (OPS) of quaternions based on the arbitrary choice of one or two linearly independent pure unit quaternions f ,g. Next we systematically generalize the quaternionic Fourier transform (QFT) applied to quaternion fields to conform with the OPS determined by f ,g, or by only one pure unit quaternion f , comment on their geometric meaning, and establish invers...

The practical and accurate computation of the singular value decomposition of a quaternion matrix is of importance in vector signal processing using quaternions. We present a Jacobi algorithm for computing such an SVD, and discuss its utility and accuracy. The algorithm is included in an open-source Matlab toolbox for quaternions where it serves as an accurate reference implementation.

In this paper we describe the geometry of the energy momentum mapping of the complexified spherical pendulum. For background on the classical spherical pendulum we refer the reader to [4, chpt. IV]. We show that this complex Hamiltonian system is Mumford-Jacobi completely integrable [9, p.3.51–3.53] and admits an analogue of the period lattice [5] and action angle coordinates [4, Apend. D]. Fol...

Abstract The aim of this article is to investigate two new classes quaternions, namely, balancing and Lucas-balancing quaternions that are based on numbers, respectively. Further, some identities including Binet’s formulas, summation Catalan’s identity, etc. concerning these also established.

Polar orderings arose in recent work of Salvetti and the second author on minimal CW-complexes for complexified hyperplane arrangements. We study the combinatorics of these orderings in the classical framework of oriented matroids, and reach thereby a weakening of the conditions required to actually determine such orderings. A class of arrangements for which the construction of the minimal comp...