The multi-dimensional nature of quaternions allows for the full characterization of three-phase power systems. This is achieved through the use of quaternions to provide a unified framework for incorporating voltage measurements from all the phases of a three-phase system and then employing the recently introduced HR-calculus to derive a state space estimator based on the quaternion extended Ka...

In the space H of quaternions, we investigate a natural, invariant geometry of the open, unit disc ∆H and of the open half-space H . These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of Möbius transformations of ∆H and H + and identify original ways of representing them in terms of two (isomorphic) groups of matrices wit...

We introduce a model of computation based on quaternions, which is inspired on the quantum computing model. Pure states are vectors of a suitable linear space over the quaternions. Every other aspect of the theory is the same as in quantum computing: superposition and linearity of the state space, unitarity of the transformations, and projective measurements. We then show that this model is no ...

This article is an exhaustive revision of concepts and formulas related to quaternions and rotations in 3D space, and their proper use in estimation engines such as the error-state Kalman filter. The paper includes an in-depth study of the rotation group and its Lie structure, with formulations using both quaternions and rotation matrices. It makes special attention in the definition of rotatio...

Split quaternions are noncommutative and contain nontrivial zero divisors. Generally speaking, it is difficult to solve equations in such an algebra. In this paper, by using the roots of any split two real nonlinear systems, we derive explicit formulas for computing x2+bx+c=0 quaternion

Many quaternion numbers associated with Fibonacci and Lucas or even their generalizations have been defined widely discussed so far. In all the studies, coefficients of these quaternions selected from consecutive terms numbers. this study, we define other for usual quaternions. We also present some properties, including Binet's formulas d'Ocagne's identities, types quaternions.

We give an explicit construction of the simply-connected compact real form of the Lie group of type E7, as a group of 28 × 28 matrices over quaternions, acting on a 28-dimensional left quaternion vector space. This leads to a description of the simply-connected split real form, acting on a 56dimensional real vector space, and thence to the finite quasi-simple groups of type E7. The sign problem...

Quaternions are an important tool to describe the orientation of a molecule. This paper considers the use of quaternions in matching two conformations of a molecule, in interpolating rotations, in performing statistics on orientational data, in the random sampling of rotations, and in establishing grids in orientation space. These examples show that many of the rotational problems that arise in...