Abstract This letter will clarify the fundamental properties of a quaternary neuron whose weights, threshold values, input and output signals are all quaternions, which is an extension of a usual real-valued neuron to quaternions. The main results of this letter are summarized as follows. A quaternary neuron has an orthogonal decision boundary. The 4-bit parity problem which cannot be solved wi...

The purpose of this paper is to classify invariant hypercomplex structures on a 4-dimensional real Lie group G. It is shown that the 4dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group H of the quaternions, the multiplicative group H∗ of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyper...

In a quaternion structure composed of four real dimensions, we derive form wherein three complex numbers are combined. Thereafter, examined whether this includes the algebraic properties and transformations were necessary for its application to system. addition, defined regular function in quaternions, expressed as combination numbers. Furthermore, derived Cauchy-Riemann equation investigate qu...

In this paper, we established the connection between generalized quaternion algebra and real (complex) matrix algebras by using Hamilton operators. We obtained complex matrices corresponding to basis of quaternions. Also, investigated features matrices. get Pauli Then, have shown that produced these is isomorphic Clifford Cl(E_αβ^3) space E_αβ^3.
 Finally, studied relations among symplecti...

Quaternions are well suited to describe and combine rotations in the usual 3D euclidian space. Therefore, an increasing interest has been shown especially by engineers for symbolically computing with quaternions. Quaternica is a Mathematica package designed to give the ability to perform manipulations on symbolic expressions involving quaternions. The features of this package include some facil...

If scattering amplitudes are ordinary complex numbers (not quaternions) there is a universal algebraic relationship between the six coherent cross sections of any three scatterers (taken singly and pairwise). A violation of this relationship would indicate either that scattering amplitudes are quaternions, or that the superposition principle fails. Some possible experimental tests involve neutr...

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 ...

The definition of morphological operators for colour images needs a total ordering between the colour points. A colour can be represented according to different algebraic structures, in particular in this paper we focus on real quaternions. The paper presents two main contributions. On the one hand, we have studied different alternatives to introduce the scalar part to obtain full colour quater...

The octonionic X-product gives to the octonions a flexibility not found in the other real division algebras (reals, complexes, quaternions). The pattern of this flexibility is investigated here. supported by a grant from ..., well, no one, really. 0

The main result is that any continuous real-linear operator A on a quaternionic Hubert space has a unique decomposition A=A0+iiAl + izAi+iiA3, where the A„ are continuous linear operators and (fi,f2,'3) is any right-handed orthonormal triad of vector quaternions. Other results concern the place of the colinear and complex-linear operators in this characterisation and the effect on the Av of a r...