Iztok Fister and Iztok Fister Jr. Abstract This paper introduces a novel idea for representation of individuals using quaternions in swarm intelligence and evolutionary algorithms. Quaternions are a number system, which extends complex numbers. They are successfully applied to problems of theoretical physics and to those areas needing fast rotation calculations. We propose the application of qu...

Quaternions are ordered quadruples of four numbers subject to specified rules of addition and multiplication, which can represent points in four-dimensional (4D) space and which form finite groups under multiplication isomorphic to polyhedral groups. Projection of the 8 quaternions of the dihedral group D2h, with only two-fold symmetry, into 3D space provides a basis for crystal lattices up to ...

In this study, we present higher order Jacobsthal numbers. Then define quaternions by using We give the concept of norm and conjugate for these quaternions. express prove some propositions related to Also, find recurrence relation, Binet formula generating function Finally, calculate Cassini, Catalan, Vajda d’Ocagne identities

The classical matrix groups are of fundamental importance in many parts of geometry and algebra. Some of them, like Sp.n/, are most conceptually defined as groups of quaternionic matrices. But, the quaternions not being commutative, we must reconsider some aspects of linear algebra. In particular, it is not clear how to define the determinant of a quaternionic matrix. Over the years, many peopl...

The four-component rotational operators called quaternions, which represent eye rotations in terms of their axes and angles, have several advantages over other representations of eye position (such as Fick coordinates): they provide easy computations, symmetry, a simple form for Listing's law, and useful three-dimensional plots of eye movements. In this paper we present algorithms for computing...

A heuristic diagram of the evolution of the standard genetic code is presented. It incorporates, in a way that resembles the energy levels of an atom, the physical notion of broken symmetry and it is consistent with original ideas by Crick on the origin and evolution of the code as well as with the chronological order of appearance of the amino acids along the evolution as inferred from work th...