In this paper, we introduce Mersenne and Mersenne-Lucas hybrinomial quaternions present some of their properties. Some identities are derived for these polynomials. Furthermore, give the Binet formulas, Catalan, Cassini, d’Ocagne identity generating exponential function quaternions.

A quaternionic representation of the genetic code, previously reported by the authors (BioSystems 141 (10-19), 2016), is updated in order to incorporate chirality of nucleotide bases and amino acids. The original representation associates with each nucleotide base a prime integer quaternion of norm 7 and involves a function that assigns to each codon, represented by three of these quaternions, ...

This note adds three annexes to my previous paper math/9904044 Annex 1. A sufficient condition for self-adjointness Annex 2. Invariant closed operators on locally compact abelian groups Annex 3. The trace of Connes for quaternions This last item is a minor variation on the evaluation of Connes’s trace (math/9811068), which is explained here in the setting of quaternions and can be applied also ...

This paper introduces some construction methods for amicable orthogonal designs over the real and quaternion domain which have not been explored for code design before. The proposed construction methods generate a large number of amicable orthogonal designs of quaternions. It is also shown that amicable orthogonal designs of quaternions can be used to construct restricted quaternion orthogonal ...

In 1853 Hamilton showed that a general linear vector transformation in three dimensions satisfied a third-order equation. In this paper we indicate how Hamilton came to consider this transformation and what he did with it. Hamilton’s work was written in the language of quaternions, which he invented. We describe this work using vectors instead of the quaternions. In addition, we express Hamilto...

We consider a random matrix whose entries are independent Gaussian variables taking values in the field of quaternions with variance 1/n. Using logarithmic potential theory, we prove the almost sure convergence, as the dimension n goes to infinity, of the empirical distribution of the right eigenvalues towards some measure supported on the unit ball of the quaternions field. Some comments on mo...

Following an ‘Ansatz’ of Björner & Ziegler [BZ], we give an axiomatic development of finite sign vector systems that we call complex matroids. This includes, as special cases, the sign vector systems that encode complex arrangements according to [BZ], and the complexified oriented matroids, whose complements were considered by Gel’fand & Rybnikov [GeR]. Our framework makes it possible to study ...

The research on this paper started with an attempt to understand the geometry of Bennett linkages [1–6] from the point of view of dual quaternions. The group of Euclidean displacements can be embedded as an open subset of the Study quadric in the projectivization of the dual quaternions regarded as a real vector space of dimension eight. Rotation subgroups and and their composition then get an ...

Dual quaternions and dual quaternion interpolation are powerful mathematical tools for the spatial analysis of rigid body motions. In this paper, after a review of some basic results and formulas, it will be presented an attempt to use these tools for the the kinematic modeling of human joints. In particular, the kinematic parameters extracted from experimentally acquired data are compared with...